[    {        "title": "2402.04225v1.Growth_rate_of_self_sustained_QED_cascades_induced_by_intense_lasers.pdf",        "content": "Growth rate of self-sustained QED cascades induced by intense lasers\nA. Mercuri-Baron,1,∗A. A. Mironov,1,∗C. Riconda,1A. Grassi,1and M. Grech2\n1LULI, Sorbonne Universit´ e, CNRS, CEA, ´Ecole Polytechnique,\nInstitut Polytechnique de Paris, F-75255 Paris, France\n2LULI, CNRS, CEA, Sorbonne Universit´ e, ´Ecole Polytechnique,\nInstitut Polytechnique de Paris, F-91128 Palaiseau, France\nIt was suggested [A. R. Bell & J. G. Kirk, PRL 101, 200403 (2008)] that an avalanche of electron-\npositron pairs can be triggered in the laboratory by a standing wave generated by intense laser\nfields. Here, we present a general solution to the long-standing problem of the avalanche growth\nrate calculation. We provide a simple formula that we apply to the case of the standing wave created\nby two circularly polarized lasers and demonstrate that it allows to predict the particle yield for the\nfull range of intensity able to generate an avalanche. We account for the damping of the growth\nrate due to pair migration from the region of prolific generation and show that above a threshold in\nintensity, this effect is negligible. The growth rate calculation allows us to predict when abundant\npair production will induce a back-reaction on the generating field due to plasma collective effects\nand screening. Our model shows excellent agreement with self-consistent PIC simulations and can be\napplied to study the generation of electron-positron pair avalanches in realistic field configurations\nto plan future experiments at ultra-high-intensity laser facilities.\nI. INTRODUCTION\nRelativistic electron-positron pair (or QED) plasma is\na state of matter that is believed to be responsible for\nmultiple striking and yet not fully explained astrophys-\nical phenomena. It can be generated in the vicinity of\ncompact objects such as black holes [1, 2] or neutron stars\n[3] and pulsars [4–6]. Interactions with such QED plasma\ncan be the source of prominent hard cosmic radiation in\nGamma Ray Bursts and in bright gamma flashes from rel-\nativistic jets [7–10]. Abundant production of e−e+pairs\ncan take place in cascade processes developing in polar\ncaps of a rotating compact star [2, 11]. This mechanism\nopens a path to explain the nature of radio-pulsar emis-\nsion [12, 13] and the source for plasma populating the\nmagnetosphere of a star and the magnetic reconnection\nlayer [14–16].\nA laboratory study of dense relativistic electron-\npositron pair plasma would facilitate a breakthrough in\nour understanding of the impact of QED effects in as-\ntrophysical phenomena. However, the generation of such\nplasma appears to be exceptionally challenging [17, 18].\nThe first observation of a neutral e−e+plasma state was\nreported only a few years ago [19], and the first investiga-\ntion of collective behaviour was done very recently [20].\nStill, in both cases, the density of electrons and positrons,\ngenerated via the Bethe-Heitler process, barely reached\nthe value high enough to form a plasma.\nA prospective path to obtaining e−e+plasma in the\nlaboratory lies in using super-strong electromagnetic\n(EM) fields, for example, generated with ultra-high in-\ntensity lasers [21–25]. When interacting with elementary\n∗These two authors contributed equally.\nanthony.mercuri@protonmail.com\nmironov.hep@gmail.comparticles, such fields can induce a wide variety of non-\nlinear strong-field QED (SFQED) phenomena by shar-\ningN≫1 soft photons γLwith e±[26–30]. The two\nleading order effects are the nonlinear inverse Compton\nscattering (for brevity, Compton emission) e±+NγL→\ne±+γand the nonlinear Breit-Wheeler pair production1\nγ+NγL→e−+e+. The strong-field QED treatment of\nthese processes is essential when the field experienced by\na relativistic particle in its rest frame2is comparable to\nthe critical field of QED ES=m2c3/(eℏ), where mand\n−eare the electron mass and charge, respectively. Se-\nquential Compton emissions and Breit-Wheeler pair pro-\nductions can lead to cascades.\nCascades can be generated as shower-type events by\nincident high-energy particles in a strong field [31–33].\nAccording to the estimates [25, 34, 35], the shower parti-\ncle yield is defined by the energy input from the incident\nbunch. The configuration envisioned in this scheme is\nsimilar to Bethe-Heitler process-based setups [36], with\nthe only difference that in the former the field is provided\nby high-Z nuclei. In the classical limit (namely, in weak\nfields or at low particle energy), the Compton emission\ndescribes classical radiation by a charge, while the Breit-\nWheeler process is suppressed [37]. This naturally limits\nthe shower multiplicity.\nStrong EM fields can induce a different phenomenon\noriginally predicted by Bell & Kirk [38]: electron-seeded\navalanche-type (or self-sustained) QED cascades. They\ncan be triggered by low-energy electrons injected into the\nstrong field region (illustrated in Fig. 1). The initial and\nsecondary charged particles experience ongoing accelera-\ntion by the field, which restores their energy in between\n1For brevity, we omit the word ‘nonlinear’ throughout the paper\nwhen referring to both these processes.\n2For a photon of frequency ωγin the laboratory frame, the anal-\nogous frame can be defined as the one where ℏω′\nγ=mc2.arXiv:2402.04225v1  [physics.plasm-ph]  6 Feb 20242\nFigure 1. Schematic representation of an avalanche-type cas-\ncade. Blue and yellow balls show e±andγ, respectively. The\ninitial electron is injected into the focus of a CP standing wave\nformed by two counterpropagating laser beams. The field at\nelectric antinode accelerates initial and secondary e±in the\ndirection transverse to the optical axis. This acceleration ren-\nders the onset and further development of the cascade with\nprolific production of e+e−pairs.\nhard photon emissions, therefore sustaining the cascade.\nIn an avalanche triggered even by few seed electrons, the\nnumber of produced e−e+pairs can rapidly exponentiate\nwith time and field strength [39, 40]. The process is ac-\ncompanied by a bright gamma-flash [41]. Such cascades\ninduced by lasers can hence mimic the processes develop-\ning in the polar caps of rotating neutron stars [2, 11, 12].\nThe avalanche-type cascades can onset in alternating\nelectric fields that are strong enough [42]. The suit-\nable configurations include electric antinodes of stand-\ning waves formed by two counterpropagating laser beams\nof an optical frequency and circularly [39, 43–45] or lin-\nearly [46, 47] polarized; a multi-beam setup [48, 49] and\na dipole wave as its limiting case [41]; a single ultra-\nintense beam focused in vacuum [42] or reflected from a\nplasma mirror [50]; irradiation of a solid [51–53], a plasma\nslab [54, 55], or a gas target [56, 57], vortex laser pulses\n[58] (for a more detailed review see Refs. [26–28]). Sim-\nulations show that an electron-positron plasma can be\ngenerated in a single laser shot and reach a high den-\nsity demonstrating collective effects [22, 59–61] or even\nscreening the field [45, 51, 62–64].\nSimulations for the mentioned setups show that\nthe intensity required for initiating an electron-seeded\navalanche-type cascade is of the order of 1024W/cm2for\na femtosecond optical laser pulse. Intensities approach-\ning this value are anticipated to be within the reach of\nthe new generation of multi-petawatt laser facilities, such\nas Apollon [65], ELI-Beamlines [66], CoReLS [67], and\nmore worldwide [68–72]. The record intensity of ≈1023\nW/cm2was recently reported [73], and increasing effortsare invested to reach 1024–1025W/cm2in the near future\n[74–77].\nIn this work, we address two general questions:\n(i) What is the scaling of the particle yield with the field\nstrength in an avalanche-type cascade developing in a\nrealistic field configuration? and (ii) What are the re-\nquirements to reach plasma densities in such cascades?\nAssuming that initial electrons are already injected in\nthe strong field, the cascade development splits into the\nonset and exponential phases, with the majority of parti-\ncles produced in the latter. While the general onset con-\nditions seem clear from the theoretical viewpoint [42], a\nuniversal model for the exponential growth rate of parti-\ncles is lacking.\nMost of the progresses in the theory of avalanche-type\ncascades were made for a uniform rotating electric field\n(for brevity, we refer to it as ‘rotating E-field’). This\nfield corresponds to the electric antinode ( E-antinode)\nof a circularly polarized (CP) standing wave formed by\ntwo counterpropagating laser beams. It was initially es-\ntimated in Refs. [40, 43] that cascades can develop in\nfields of the order ∼αES, which corresponds to inten-\nsity∼1025W/cm2for an optical laser field. Here,\nα=e2/(4πε0ℏc)≈1/137 is the fine structure constant.\nYet, simulations show [26, 45] that cascades can be trig-\ngered at fields lower by an order of magnitude, and the\nresulting growth rate of e−e+pairs matches the predic-\ntion of Refs. [40, 43] only at very high fields. Subsequent\nworks [45, 46, 63] provide some phenomenological models\nimproving this estimate in (relatively) low- and high-field\nregimes, and Ref. [78] in the mid-range. However, these\nresults do not provide a general dependence on the field\nstrength valid in the full range and are limited to the\nmodel of a rotating E-field.\nAnother effect that is not treated in the above-\nmentioned models and will be discussed in this paper, is\nparticle migration. In realistic field configurations, par-\nticles can escape from the strong field region, e.g. as\nobserved at the E-antinode in a standing wave [79, 80].\nThis proves important in the low field regime where mi-\ngration can reduce the overall e−e+pair yield [47].\nUnder the semiclassic approximation, the kinetic ap-\nproach provides a general basis for describing the particle\ndistribution evolution in cascades [43, 62, 81, 82]. Indeed,\ncommon numerical approaches solve kinetic equations in\nthe Monte-Carlo scheme or particle-in-cell codes. The\nlatter allows accounting for plasma effects by consistent\ntreatment of the Vlasov and Maxwell equations [83]. No-\ntably, a kinetic description also allows to account for spin\neffects [84–86].\nWe adopt the kinetic approach for an ab-initio deriva-\ntion of the particle growth rate expression in an\navalanche-type cascade and rigorously define the approxi-\nmations in use. Moreover, this approach allows us to the-\noretically account for particle migration and highlight its\neffect on the growth rate. We focus on cascades develop-\ning in the CP standing wave configuration, however, we\nbelieve that our considerations can be generalised to the3\nwide class of fields considered in Ref. [42].\nWe leave aside the nontrivial question of injecting the\ninitial electrons [87, 88] and assume that they are lo-\ncated in the strong field region. Let us mention that\nseeding can be implemented by using a high-energy elec-\ntron bunch [44, 89], ionizing a jet of high-Z atoms [56],\nor solid targets [21, 52]. At extreme fields, seed electrons\ncan be created due to Schwinger e−e+pair production,\nwhich can become feasible due to the large volume of the\nfocal spot [49, 90]. As we aim at finding the field scaling\nof the avalanche-type cascade growth rate, for simplicity,\nwe omit the Schwinger effect, as well as spin contribution\nin the current work.\nThe paper is organized as follows. At the beginning of\nSec. II we introduce our notations and give an overview\nof the approximations in use. In Sec. II A we present the\nprobability rates of the basic processes. In Sec. II B, fol-\nlowing Ref. [42], we postulate the semiclassic short-time\ndynamics of a single electron in a general accelerating\nfield. This is then used in Sec. II C to define the charac-\nteristic time between Compton emissions for the electron\nand to find the key parameters entering in the emission\nprocess. In Sec. II D we discuss the related asymptotic\nlimits. Section III is devoted to our major result, namely,\nthe master equations for the particle numbers and the\ngeneral formula for the avalanche-type cascade growth\nrate (taking into account particle migration) in Sec. III A,\nthe effective model for the growth rate in Sec. III B, and\nits low- and high-field limits in Sec. III C. In Sec. IV,\nwe apply our model to study avalanches developing in a\nCP standing EM wave at relatively low- and mid-range\nfields and test the model against simulations performed\nwith the PIC-QED code SMILEI [91]. Section IV A con-\ntains details about the numerical setup and data analysis,\nwhile in Sec. IV B discuss the physical results and vali-\ndate our model. Section V A continues this discussion\nfor high fields, and in Sec. V B we compare our results\nwith previous works. In Sec. VI we apply our model to\nidentify the field parameters at which the plasma state\ncan be reached in avalanche-type cascades in a rotating\nE-field and test the model against PIC simulations. In\nSec. VII we summarise and discuss our results.\nII. PARTICLE DYNAMICS IN A CASCADE\nThe interaction of a relativistic particle (electron,\npositron, or photon) with a strong EM field can be\ncharacterized by two parameters. First, by the parti-cle Lorentz factor γefore±or by the normalized energy\nγγ=ℏωγ/mc2for a photon with frequency ωγ. Second,\nby the invariant quantum dynamical parameter combin-\ning the electron3or photon momentum pe,γand the elec-\n3Hereinafter, we treat electrons and positrons identically, and for\nbrevity refer to both as just ‘electrons’ unless mentioned explic-\nitly.\ntric and magnetic field components EandB:\nχe,γ=1\nESq\n(γe,γE+cue,γ×B)2−(ue,γ·E)2,(1)\nwhere ue,γ=pe,γ/(mc). For an electron, χeequals the\nrest frame field strength in the units of the QED critical\nfieldES=m2c3/(eℏ).\nParameter (1) allows to assess when the quantum ef-\nfects in the interaction of a particle with an external EM\nfield need to be considered. They become important at\nχe,γ≳1. As mentioned in the introduction, we con-\nsider two leading order field-induced effects: the nonlin-\near Compton emission and nonlinear Breit-Wheeler pro-\ncess. In this work, we adopt the probability rates for\nthese processes within the locally constant field approx-\nimation (LCFA) [37, 92–94]. It implies that the rates\ndepend only on the field strength at the local position\nof the incoming relativistic particle, and the field is close\nto the constant crossed EM field in the particle reference\nframe [26, 37]. Furthermore, in a quantum process, the\nproduced and scattered particles propagate collinearly to\nthe incoming one.\nIt should be noted, that the LCFA breaks if one of the\nparticles in the process is nonrelativistic or the transverse\nfield seen by the particle is relatively low [95] (see more\ndiscussions in Refs. [96–103]. However, the LCFA is rea-\nsonable for modeling avalanche-type cascades, as under\noptimal conditions for their onset, such events are rare\nand also do not contribute to the cascade particle growth\nrate [42, 95].\nA. Photon emission and pair creation probability\nrates\nThe differential probability rate for an electron with\nthe Lorentz factor γeand quantum parameter χeto emit\na photon with energy within the interval mc2×(γγ, γγ+\ndγγ) is given by\ndWCS(γe, χe, γγ)\ndξ=1√\n3πα\nτCγe\u0014\u0012\n1−ξ+1\n1−ξ\u0013\nK2/3(µ)−Z∞\nµdsK 1/3(s)\u0015\n, (2)\nwhere τC=ℏ/(mc2) is the Compton time, and we defined µ= 2ξ/[3χe(1−ξ)],ξ=γγ/γe≡χγ/χe.Knare the n-th\norder modified Bessel functions of the second kind. Note that the χ-parameter is conserved, χe=χγ+χ′\ne, where χ′\ne\ncorresponds to the scattered electron. The differential probability rate for a photon with γγandχγto produce a pair4\nofe−e+, so that either e−ore+gets energy within the interval mc2×(γe, γe+dγe), reads:\ndWBW(χγ, γγ, γe)\ndζ=1√\n3πα\nτCγγ\u0014\u0012ζ\n1−ζ+1−ζ\nζ\u0013\nK2/3(µ′)−Z∞\nµ′dsK 1/3(s)\u0015\n, (3)\nwhere µ′= 2ζ/[3χγ(1−ζ)],ζ=γe/γγ≡χe/χγ, and χγ=χe+χ′\ne. Here, χeandχ′\necorrespond to the electron and\npositron.\nThe total probability rates WCS(γe, χe),WBW(χγ, γγ) are\nobtained by integrating out the final particle energies or,\nequivalently, ξandζ:\nWCS(γe, χe) =Z1\n0dWCS(γe, χe, γγ)\ndξdξ, (4)\nWBW(χγ, γγ) =Z1\n0dWBW(χγ, γγ, γe)\ndζdζ (5)\nIn the asymptotic case, they can be simplified:\nWCS(γe, χe)≈α\nτCγe×\n\n1.44χe, χ e≪1,\n1.46χ2/3\ne, χe≫1,(6)\nWBW(χγ, γγ)≈α\nτCγγ×\n\n0.23χγe−8\n3χγ, χγ≪1,\n0.38χ2/3\nγ, χ γ≫1.(7)\nThe probability rates and the product spectrum width\ngrow with the χ-parameter of the incoming particle (if\nthe Lorentz factor is fixed). On the other hand, the χ-\nparameter is shared among the products in each quantum\nevent, so its value decreases for subsequent particle gen-\neration. As a result, if a high-energy particle in a laser\nfield triggers a cascade, at some point it will stop unless\ntheχ-parameter increases again during the particle prop-\nagation in between the quantum events [33, 44]. For this\nreason, the multiplicity of shower-type cascades is de-\nfined by the energy of incident particles, as χdecreases\nfor each generation of secondary particles.\nB. χetime dependence in accelerating fields\nThe key idea of the avalanche-type cascade mecha-\nnism is that in between emission events electrons are\nre-accelerated by the field, which restores a high value\nofχe. Then the cascade will be sustained as long as par-\nticles experience a strong field. The mean free path time\nof an electron or a photon in an external EM field can\nbe estimated as the inverse total probability rate W−1\nCS,BW.\nLetω−1be the time scale of the field variation. For a\nparticle in a prolific cascade, we expect that ωW−1\nCS,BW<1\n[38, 40], i.e. multiple quantum events take place during\na field cycle. The cascade overall properties, such as the\nparticle number growth rate and spectra, are defined by\nthe single-particle χeevolution at the time scale ωt≪1.\nLet us consider an electron (seed or secondary) in a\nstrong accelerating field. The evolution of γeandχeatωt≪1 can be written explicitly as proposed in Ref. [42]:\nγe(t)≃eϵt\nmc, (8)\nχe(t)≃ϵ2ωeff\nE2\nSτCt2, (9)\nwhere, in simple terms, ϵ∝Ecan be regarded as the\nelectric field strength Eexperienced by the electron at\nits initial position, and the effective frequency ωeff∝ω\nrepresents the time and length scale of the field varia-\ntion. The full expressions for ϵandωeffare presented in\nAppendix A. Let us note here that ϵis Lorentz-invariant\nand defined as the electric field in the frame where the\nelectric and magnetic components are parallel or one of\nthem vanishes [37, 104]. The effective frequency ωeffis\ndefined via the local field derivatives and vanishes for a\nconstant field. The combination ωefft2is also invariant.\nEquations (8) and (9) are general for a wide class of\nEM fields, and are valid under the following assumptions:\n(i)the particle dynamics is semiclassic , namely, pho-\ntons propagate along straight lines and electron\nmotion satisfies the Lorentz equations. This holds\nfor subcritical fields, E≪ES, which is relevant for\nlaser beams and astrophysical applications;\n(ii) the field components at the initial position of the\nelectron satisfy the condition E > cB . We refer to\nsuch configurations as to fields of electric type .\n(iii) by the time moment of emission W−1\nCS≪ω−1, the\nenergy gained by the electron from the field is high\ncompared to its initial energy (i.e. particles rapidly\n‘forget’ their initial conditions).\nThen the electron trajectory can be approximated and\nsubstituted in Eq. (1), which results in Eqs. (9). We\nprovide some details about the derivation in Appendix A,\nwhich follows Ref. [42].\nIt is important to note that fields with E=cBor\nE < cB (so-called null and magnetic-type fields) do not\nfulfil the conditions for sustaining the cascade. In both\ncases, χedoes not increase as the electron propagates,\nand therefore they are not favourable for the avalanche-\ntype cascade development4.\n4A remarkable exception is cascading in pulsar polar caps where\ncurved magnetic lines extend over very large distances. The\nphotons resulting from curvature radiation experience increasing\ntransverse magnetic field as they propagate, and hence χγgrows,\nwhich in turn can supply the cascade development [11–13].5\nOne of the suitable configurations is the electric\nantinode of a standing wave formed by two counter-\npropagating circularly polarized (CP) laser beams [39,\n40, 43, 45, 46, 52, 53, 62], where the magnetic field is\nabsent and the electric component is close to a uniform\nrotating electric field:\nE=E0[cos(ωt)ˆx+ sin( ωt)ˆy], (10)\nwith Ez= 0, B= 0. In this case, one has ϵ=E0,\nωeff=ω/2, and Eqs. (8)-(9) read [40, 43, 62]:\nγe(t)≃eE0t\nmc, χ e(t)≃E2\n0ωt2\n2E2\nSτC. (11)\nC. Characteristic γeand χeat emission\nOnce the evolution of χefor the accelerated electron is\ndetermined, it can be used to calculate the time instant of\nphoton emission and the corresponding values of γeand\nχeat this moment. These quantities identify whether the\nelectron-seeded cascade can onset [42] and the particle\ngrowth rate (as we will show in Sec. III).\nIn a prolific avalanche-type cascade, the number of par-\nticles rapidly exponentiates with time and therefore is\nsensitive to small variations of the parameters that en-\nter in the growth rate. For this reason, in our model,\nwe go beyond the simple estimate for the electron mean\nfree path time as the inverse emission rate, W−1\nCS. Let us\nintroduce the characteristic time of electron propagation\nbetween photon emissions tem:\nZtem\n0WCS(t′)dt′:= 1, (12)\nwhere WCS(t) =WCS(γe(t), χe(t)), and γe(t) and χe(t) are\ngiven by Eqs. (8)-(9). It is convenient to introduce the\ncorresponding characteristic values:\nγem=γe(tem), χ em=χe(tem), (13)\nobtained by substituting teminto Eqs. (8)-(9). As we will\nshow, the cascading regime is characterised by χem. Let\nus stress here, that tem,γem, and χemdepend only on the\nfield parameters via ϵandωeff:\nϵ, ωeff7→ {tem, γem, χem}. (14)\nIn view of Eqs. (8)-(9), χemcan be chosen as an in-\ndependent variable instead of tem(assuming that ϵand\nωeffare nonzero). In particular, by inverting Eq. (9) and\npassing to the corresponding variable in Eq. (12), we can\nrewrite the latter in the Lorentz-invariant form:Zχem\n0dχτCWCS(1, χ)\n2αχ=ϵ\nαES, (15)\nwhere we moved the physical parameters to the RHS,\nand the integrand in the LHS is a dimensionless special\nfunction of χ(note that the dependence on αcancels in\nthe LHS as WCS∝α). This expression defines χem(that\nis Lorentz invariant) in the electron reference frame. As\none may notice, χemis independent of ωeff.\n10−1100ωtem\n102103104γem\n102103104105\neE0/mcω100102χem\nES αESFigure 2. The emission characteristic time temand the corre-\nsponding values γemandχem[see Eqs. (12)-(13)] as functions\nofE0for an electron-seeded cascade in a uniform rotating\nelectric field (solid lines). The dot-dashed lines show the low-\nand high-field asymptotic behavior. The short-time dynam-\nics model given in Eqs. (8)-(9) is valid in the unshaded area,\nwhere ωtem<1. Thin dashed lines in the bottom panel show\nthe field strength at which χem= 1 and 10. The vertical\ngreen line E0=αEScorresponds to eE0/(mcω)≈2400 and\nthe vertical red line E0=ES— to eE0/(mcω)≈3.3×105.\nThe field rotation frequency ωcorresponds to the wavelength\nλ= 0.8µm.\nD. Weak and high field regimes of photon emission\nby accelerated electrons\nFormula (15) establishes the general functional depen-\ndence χem=χem(ϵ/αE S). At χem≪1 or≫1 Eq. (15)\nsimplifies, as we can use the asymptotic expressions (6)\nforWCSto approximate the integral. The result sets the\ndirect correspondence\nχem≪≫1⇐⇒ ϵ≪≫αES, (16)\nso that\nχem≃\n\n1.39ϵ\nαES, ϵ ≪αES,\n0.87\u0012ϵ\nαES\u00133/2\n, ϵ≫αES.(17)\nRelation (16) defines the weak and strong field regime of\nan avalanche-type cascade, with αESbeing the thresh-\nold.\nAnalogously to Eq. (17), we can also calculate the\nasymptotic behavior of temandγem. The resulting ex-\npressions with the full numerical coefficients are pre-\nsented in Appendix B.\nFor illustration, in Fig. 2, we show the dependence of\ntem,γem, and χemon the field strength for an electron\naccelerated by a uniform rotating electric field [Eq. (10)].\nAstemandχemare defined via the full expression (4) for6\nWCS[see Eqs. (12), (15)], we calculate them by numeri-\ncal integration. We confirm that our initial assumption\nωtem<1, required in Eqs. (8)-(9), is met in a wide range\nofE0and improves with a growing field strength. We\nalso plot the asymptotic expressions for χem[given in\nEq. (17)], as well as for temandγem(see Appendix B).\nAs expected, the low and high field asymptotics set in at\nE0≪αESand≫αES, respectively. Finally, let us note\nthat in the transition regime E0∼αES,χemtakes the\nvalue 1 ≲χem<10. This interval will also correspond to\nthe transition between the models for the cascade growth\nrate, which we present in the following Section.\nIII. STEADY-STATE PARTICLE GROWTH\nRATE OF A CASCADE IN A GENERAL FIELD\nLet us now address the evolution of the particle num-\nber in an avalanche-type cascade and its scaling with the\nfield strength, which is the central question of this work.\nWe consider a field that meets the validity conditions for\nEqs. (8), (9) in a localized region. In the context of a\nstanding EM wave, this can be an electric antinode, with\nthe limitation that particles can be expelled (or ’migrate’)\nfrom the region of strong electric field. In this section,\nwe derive the master equations for the particle number\nevolution and take this effect into account.\nWe show explicitly by an ab initio calculation that\nthe master equations, proposed phenomenologically in\nthe absence of migration in Ref. [46] and later used in\nRefs. [45, 61], are exact if the photon emission and pair\ncreation rates entering in the equations are calculated\nby averaging over the particle distributions. However in\ngeneral the analytic expression for the distribution func-\ntions and hence the rates are unknown. Here, we propose\nan approximation for the latter based on the small-time\nelectron dynamics discussed in the previous Section. As\na result, this allows us to solve the master equations and\nfind the explicit scaling of the cascade growth rate with\nthe field strength.\nA. Growth rate in the steady state\nThe onset stage of the avalanche-type cascade is highly\nnonstationary. The seed particle distribution rapidly\nevolves as new e−e+pairs and photons are produced.\nProvided that the particle yield is high at the scale of\nfield duration, the particle energy quickly averages out\nas the cascade develops. The cascade can enter the so-\ncalled (quasi-)steady state, in which the energy distri-\nbution relaxes to a stationary function [43, 45]. We as-\nsume that the system quickly reaches this state (which\nwe further confirm with numerical simulations), in which\nthe particle number grows exponentially at a constant\nrate. However, depending on the field strength, the re-\nlaxation can take up to multiple field periods in time.\nHence, particles can migrate from the strong-field areadue to the field gradient at a rate ν, which in turn can\nreduce the overall particle production rate. This should\nbe accounted for in the cascade models when applied to\nrealistic field configurations such as standing EM waves\nor focused laser pulses.\nThe steady-state master equations for the number of\ngenerated pairs Npand photons Nγcan be derived by\nusing the kinetic approach, as we show in Appendix C:\ndNp\ndt=WcrNγ−νNp, (18)\ndNγ\ndt=−WcrNγ+ 2WradNp, (19)\nwhere Wrad,crare the effective constant rates of photon\nemission and pair creation, respectively, and the migra-\ntion rate νrepresents the average stationary flow of par-\nticles leaving the cascading region. These equations are\nexact , and Wrad,crresult from averaging WCS,BW(γ, χ) over\nthe steady-state particle distributions fp,γ(γ, χ):\nWrad,cr=⟨WCS,BW(γ, χ)⟩. (20)\nIn the derivation of Eqs. (18)-(20), we rely only on the\nsemiclassic approximation and the LCFA.\nThe master Eqs. (18)-(19) implicitly incorporate the\nemission and pair creation spectra without additional ap-\nproximations about their shape. The analytical descrip-\ntion of the particle distribution evolution in a cascade re-\nquires convoluting the differential probability rates from\nEqs. (2), (3) with the distribution functions in the mo-\nmentum space fp,γ(p, t). However, when we consider the\nparticle numbers Np,γ(t), the cascade yield is determined\nonly by the effective total rates (20). We present our\nderivation in Appendix C for a cascade developing in the\nE-antinode of a CP standing wave-like configuration, but\nit can be generalized to other field configurations that al-\nlow a steady state.\nEqs. (18)-(19) can be solved by the substitution eλt, re-\nsulting in two eigenvalues λ. As one of them, denoted by\nΓ, is positive, the particle number increases exponentially\nwith time, Np,γ(t)∼eΓt, and the growth rate reads:\nΓ[Wrad,Wcr] =\nWcr+ν\n2\"s\n1 +4Wcr(2Wrad−ν)\n(Wcr+ν)2−1#\n.(21)\nFor a given EM field configuration, Wrad,crandνdepend\non the field parameters, and hence the growth rate Γ.\nA calculation of the effective rates Wrad,crandνre-\nquires averaging over the particle distribution functions\nfp,γ. As their analytic expressions are unknown even in\nthe simplest case of the rotating electric field configura-\ntion, they can be extracted from numerical simulations.\nAlternatively, they can be estimated under additional ap-\nproximations, as, for example, was done for Wrad,crin\nRefs. [40, 43, 45, 46, 63]. The rate of migration from a\nregion of size ∼2πc/ω can be estimated as ν∼ω/(2π).\nFor higher precision calculations we use the numerical7\ndata. Anticipating further discussion in Section IV, let\nus note here that νweakly scales with the field strength.\nIn contrast to this, Wrad,crstrongly depend on the field\nparameters. In what follows, we propose a refined ana-\nlytical model for Wrad,cr, which is valid in a wide class of\nEM field models.\nB. Effective emission and pair creation rates and\nmodel for Γ\nLet us approximate the effective rates of photon emis-\nsionWradand pair creation Wcr. We adapt the general\nidea of Ref. [40], where it was proposed to estimate Wrad\nat the characteristic γeandχegained by e−in a uniform\nrotating field before the instant of emission. We go be-\nyond this simple estimate as we (i) consider a general ac-\ncelerating field [in the context of Eqs. (8)-(9) validity], (ii)\nuse the refined expressions for γemandχemto describe\nthe radiating e−, which we propose in Section II C, and\n(iii) treat electrons and photons differently depending on\nthe field strength.\nTo evaluate the effective emission rate Wradby an elec-\ntron contributing to an avalanche-type cascade, we plug\nthe characteristic values for γe∼γemandχe∼χem\nat the time of emission [see Eqs. (12)-(13)] into the full\nprobability rate for the Compton emission [Eq. (4)]:\nWrad≃WCS(γem, χem). (22)\nNote that in a strong field such that χem≫1, the field\ndependence of the emission probability rate can be writ-\nten explicitly [see also Eq. (B6) in Appendix B]:\nWCS(χem≫1)≈1.43αrωeff\nτC\u0012ϵ\nαES\u00131/4\n. (23)\nThe estimate for the effective pair creation rate Wcr\nis less straightforward. Unlike for electrons, χγcan vary\nonly at the scale ∼ω−1as the photon traverses through\nthe field. In a cascade, the emission and pair creation\nprocesses happen successively. To select the photons con-\ntributing the most, we should consider the interplay be-\ntween two factors: (i) the pair creation is exponentially\nsuppressed for χγ<1 but contributes at χγ≳1 [see\nEq. (7)], and (ii) in the emission spectrum, softer pho-\ntons with χγ≪χedominate over the hardest ones with\nχγ∼χe. Furthermore, one should note that γγandχγ\nare related, since γeandχeof the emitting electron are\nmutually dependent in view of Eqs. (8)-(9).\nIn Fig. 3, we show examples of the emission spectra\nfor two cases: for electrons with fixed γeandχe, and for\nelectrons accelerated by a uniform rotating electric field\nso that γemandχemare consistent with Eqs. (11) and\n(12). We compare these curves to the probability rate\nofe−e+pair creation by a photon in a rotating electric\nfield. For this, we estimated χγ∼γγϵ/ES. The result\nsuggests that at relatively low and high fields the dom-\ninating contribution to pair production is rendered by\n10−1100101102\nχγ10−210−1100101Rate×105τC\nχe= 2\nχe= 10\nχe= 100Figure 3. The spectrum of photons emitted by an electron\ndW CS(γe, χe, χγ)/dχγ[Eq. (2)] and the total probability rate\nofe−e+pair creation by a photon WBW(γγ, χγ) [Eq. (5)] as a\nfunction of χγ. Solid lines stand for emission by an electron\nwithγe= 103and different values of χe. Dot-dashed lines cor-\nrespond to emission by an electron accelerated by a uniform\nrotating electric field, so that γe=γemandχe=χem[see\nEqs. (11), (12)]. Dashed lines show WBWfor a photon emitted\nin a uniform rotating field, so that γγandχγare consistent,\nγγ∼χγES/E0. The dashed and dot-dashed lines are plot-\nted at the following values of eE0/(mcω): 2550 (purple), 9436\n(black), and 51581 (orange), which gives χem= 2, 10, and 100,\nrespectively, for the field wavelength λ= 2πc/ω = 0.8µm. In\nthe grey-shaded area χγ<1, the pair creation probability is\nsuppressed [see Eq. (7)].\ndifferent parts of the emission spectrum . Let us discuss\nthese cases one by one.\nAt a relatively low field, such that χembarely exceeds\n1, only the rightmost edge of the photon χγdistribution\ncan contribute to the cascade (see the χe= 2 line in\nFig. 3). Then the pair creation rate can be estimated as\nWcr|χem<X∼WBW(γγ=γem, χγ=χem), (24)\nwhere 1 ≲X<10 sets the effective applicability range\nof the formula. We discuss it at the end of this sub-\nsection. Note that this is consistent with the (rough)\ncascade threshold condition proposed in Refs. [40, 42].\nNamely, χem≳1 is the necessary requirement ensuring\nthat the emitted photon can have χγ∼1 and, hence,\ncreate a pair in a successive quantum event.\nIn a high field, electrons emit at χem≫1, as we dis-\ncussed in Section II D. All of the photons with 1 ≲χγ≤\nχemcan contribute to pair production, which represents\na wide part of the distribution (see the orange lines in\nFig. 3 corresponding to χem= 100). Furthermore, the\nfraction of photons with χγ∼1 significantly exceeds\nthat of χγ∼χem, which is not compensated for by the\n(slower) increase of the pair creation rate. Therefore, we\nevaluate the pair creation rate as\nWcr|χem>X∼WBW(γγ=ES\nϵ, χγ= 1)\n=ϵ\nESWBW(1,1).(25)8\nHere, we estimated γγfrom χγ∼γγϵ/ES:= 1. In the\nsecond line, we used that WBWdepends on γγ∼ES/ϵ\nonly in the prefactor [see Eq. (3)].\nSummarising, Γ depends on the (invariant) field\nstrength ϵentering the effective rates parametrically via\nγem(ϵ),χem(ϵ). This dependence can be written for a\ngeneral field by plugging the effective rates into Eq. (21):\nΓ|χem<X(ϵ)≃Γ[WCS(γem, χem), WBW(γem, χem)],(26)\nΓ|χem>X(ϵ)≃Γ[WCS(γem, χem),ϵ\nESWBW(1,1)].(27)\nIn these expressions, the rates WCSandWBWcan be cal-\nculated numerically by using Eqs. (2)-(5), and νcan be\nextracted from simulations. We require that Eqs. (26)\nand (27) match at χem∼ X. This condition can be reex-\npressed in terms of the field strength in view of Eq. (15).\nIt is important to note that the crossover region for\nEqs. (26)-(27) is finite, which is due to the following rea-\nson. The transition between the approximations for Wcr\nin Eqs. (24) and (25) is smeared, as for the electrons ra-\ndiating with 1 ≲χem<10 the relative decrease of the\nnumber of emitted photons with χγ∼1 and χγ∼χem\ncan be comparable [see the black curves in Fig. 3, corre-\nsponding to χem= 10]. The photons with χγ∼1 start to\ndominate in the spectrum at relatively high χem≳10. As\na result, we can estimate the transition point Xby pick-\ning a value from the interval 1 ≲X<10. Alternatively,\nit can be evaluated with higher precision by extracting\nEqs. (26) and (27) from simulations and matching them.\nC. Asymptotic behavior of Γat low and high fields\nand the particle orbit dimensionality\nLet us discuss the behavior of the growth rate in the\nlimiting cases of a low and high field. We define a\nlow/high field by the correspondence given in Eq. (16).\nIn a weak field ϵ≪αES, i.e. at χem≪1, the emitted\nphotons are soft, χγ≪1, and the pair creation rate is\nexponentially small, see Eq. (7). At the same time, the\nprobability for an electron to emit a soft photon stays fi-\nnite. Assuming that ν∼ω/(2π), let us expand Eq. (26)\nat small Wcr≪ν, W radto the leading order:\nΓ|ν>0(ϵ≪αES)≃\nWBW(γem, χem)\nν[2WCS(γem, χem)−ν]∝e−8\n3χem.(28)\nIfν= 0 (e.g. for a cascade developing in a uniform\nrotating electric field), the expansion results in a different\nscaling for the leading term (c.f. Eq. (8) in [45]):\nΓ|ν=0(ϵ≪αES)≃\np\n2WCS(γem, χem)WBW(γem, χem)∝e−4\n3χem.(29)\nHence, the particle migration leads to significant suppres-\nsion of the cascade growth rate in the low field regime. It\ncan be viewed as a topological effect: ν= 0 correspondsto an effective reduction of the particles’ degrees of free-\ndom as if they are confined in the strong field region (e.g.\nthe center of the E-antinode of a standing wave). This\nnaturally amplifies the growth rate. The field depen-\ndence at low ϵcan be found explicitly by using the low- χ\nasymptotic of Eqs. (6), (7) and the low- ϵexpansion for\nγem(ϵ) and χem(ϵ) [see also Eqs. (B4), (B5)]:\nΓ(ϵ≪αES)≈α\nτCrϵωeffτC\nαES\n×\n\n0.44\u0014\n3.40α\nνrϵωeffτC\nαES−1\u0015\ne−1.92αES\nϵ, ν > 0,\n1.23e−0.96αES\nϵ, ν = 0.\n(30)\nThe asymptotic expressions with full numerical coeffi-\ncients are relegated to Appendix B, see Eq. (B7).\nAt high field ϵ > αE S, namely, in the regime when\nEq. (27) for Γ applies, the probability rates increase as\nWrad∝ϵ1/4andWcr∝ϵ[see Eqs. (23) and (25)]. The\nmigration rate is bounded, ν≲ω, hence, as the field\ngrows, we can neglect it compared to WradandWcrin\nEq. (27):\nΓ(ϵ > αE S)≃\nϵWBW(1,1)\n2ES\"s\n1 +8ESWCS(γem, χem)\nϵWBW(1,1)−1#\n.(31)\nIt means that at high ϵ, regardless of the global field\nstructure in space, the cascade will be effectively confined\nto the strong field region, where the particle production\nis the most efficient. Put differently, only the local struc-\nture of the field, where the conditions are optimal, affects\nthe cascade growth rate. This effect should be universal\nfor the fields, in which avalanche cascades can onset and\nreach a steady state.\nBy substituting in Eq. (31) the asymptotic expressions\nfor the rates given by Eqs. (23), (25), we obtain the ex-\nplicit field dependence of the growth rate in the high-field\nregime:\nΓ(ϵ >αE S)≃\nc1ϵ\nτCES\ns\n1 +c2√ωeffτC\u0012αES\nϵ\u00133/4\n−1\n,(32)\nwhere the numerical coefficients c1≈0.52×10−4,c2≈\n1.1×105are defined in Eqs. (B8), (B9) in Appendix B.\nHere, we note that their magnitude is determined by\nτCWBW(1,1)≈1.03×10−4.\nSince Wrad∝ϵ1/4grows slower with ϵas compared\ntoWcr∝ϵ, at asymptotically high fields the latter will\nbecome so large that 8 Wrad/Wcr≪1, and Eqs. (31), (32)\nsimplify even further:\nΓ(ϵ≫αES)≃2WCS(γem, χem)\n∝αrωeff\nτC\u0012ϵ\nαES\u00131/4\n.(33)9\nFor the rotating electric field configuration [see Eq. (10)]\nthis corresponds to the scaling proposed by Fedotov et.\nal. [40] (recall that for this case ϵ=E0,ωeff=ω/2):\nΓrot.-E(E0≫αES)∼αrω\nτC\u0012E0\nαES\u00131/4\n. (34)\nThe condition Wcr>8Wradgives the estimate for the\nfield at which this asymptotic starts to set in. By plug-\nging in the explicit high-field expressions (23) and (25)\nfor the rates, we arrive at the condition E0>3.35×\n106α(ωτC)2/3. For the optical wavelength λ= 0.8µm,\nthis corresponds to the field strength E0>5ES, which\nis beyond the applicability range of the semiclassic ap-\nproach. Notably, the scaling (34) was never confirmed\nby simulations in past works [43, 45, 46], as this range of\nfield strengths was not studied.\nIV. CASCADES IN A CP STANDING WAVE AT\nLOW AND MODERATE FIELDS ( χem<X)\nWe apply our model to study the development of e−-\nseeded cascades in a CP EM standing wave and test the\nmodel against simulations. This field configuration can\nbe formed by two counterstreaming CP EM plane waves\n(with the opposite sign of polarization) so that the total\nelectric and magnetic field components read:\nE=E0[cos(kz) sin(ωt)ˆx+ cos( kz) cos( ωt)ˆy], (35)\nB=−E0\nc[sin(kz) sin(ωt)ˆx+ sin( kz) cos( ωt)ˆy],(36)\nwhere k=ω/c,zis the axis of propagation, and E0is the\nelectric field amplitude. The distribution of the electric\nand magnetic field strength along the longitudinal axis is\nillustrated in Fig. 4. At the electric antinode center, the\nmagnetic field vanishes, and the electric field corresponds\nto the rotating E-field given in Eq. (10). The values of\nϵandωeffdepend on the longitudinal coordinate. At\ntheE-antinode centre ( z= 0 in Fig. 4), ϵ=E0and\nωeff=ω/2 as in a rotating E-field, and the combination\nϵ2ωeffis maximal. In the B-antinodes (e.g. λ/8< z <\n3λ/8),ϵ2ωeffvanishes, hence, electrons do not gain χe\n[see Eq. (9)], and the cascade cannot be sustained in these\nregions.\nA. PIC simulations\nWe study the electron-seeded cascade development nu-\nmerically in both the standing wave and the uniform ro-\ntating E-field configurations. The calculations are per-\nformed with the PIC-QED code SMILEI [91] in 1D3V ge-\nometry. For further comparison, we also run full 3D PIC-\nQED simulations in a realistic setup where the stand-\ning wave is formed by two counterpropagating Gaussian\nbeams.\n−5\n8−1\n2−3\n8−1\n4−1\n801\n81\n43\n81\n25\n8\nz/λ0.00.20.40.60.81.0Field strengthMigration\nRotatingE-fieldcB/E 0\nE/E 0\n/epsilon12ωeff\nE2\n0ωFigure 4. The longitudinal structure of the electric Eand\nmagnetic Bfield amplitude of a CP standing wave [see Eqs.\n(35) and (36)]. At the E-antinode center (shown by the thick\nvertical line), the magnetic field vanishes and the electric\nfield rotates in the transverse plane [described by Eq. (10)].\nThe black line represents the longitudinal coordinate depen-\ndence of the ϵ2ωeffvalue entering the expression for χe(t), see\nEq. (9). In the grey-shaded areas delimited by red dashed\nlines, it is zero, therefore, χe(t) for e±is not restored in be-\ntween the emission events. Electrons and positrons migrate\nfrom the E-antinode to the grey-shaded areas at characteris-\ntic time t∼2π/ω(shown by arrows), as the B-antinodes are\nthe spiraling attractors for charged particles [79, 80, 105–107].\nThe numerical setup is detailed in Appendix E, though,\nlet us mention some important points. The seed elec-\ntrons are injected at time t= 0 into the electric antin-\node centers in the simulations with a standing wave, and\ndistributed homogeneously in the simulation box for the\nrotating E-field configuration. We fix the field frequency\nso that it corresponds to the optical laser wavelength\nλ= 0.8µm, and we study the development of avalanches\nwith time at varying E0. We keep the particle density\nlow during the whole simulation so that collective effects\nare insignificant.\nIn the discussion of our model for Γ [given in Eq. (21)]\nin the standing wave configuration, we use the migration\nrate field dependence ν(E0) calculated as follows. We in-\nject the initial electrons in the vicinity of the E-antinode\ncenter z= 0, where ϵ2ωeffis maximal and the cascade\nonset is optimal (see Fig. 4). As the cascade develops,\nparticles migrate to the B-antinodes, which are the spi-\nraling attractors for charges [79, 80, 105–107]. When\nan electron reaches a spatial region at |z|> λ/ 8 where\nϵ2ωeff= 0, we consider this e−is dropped out of the cas-\ncade. We define νas the average inverse time at which\ninitial particles propagate at distance λ/8 from the origin\nand extract it from simulations (for more details, refer to\nAppendix E).\nIn each simulation, we ensure that the cascade reaches\na steady state so that it exhibits exponential growth of\nthe particle number. We extract the growth rate by us-\ning the two following methods. First, we directly calcu-10\n0.1110Γ×2π/ω\nRotatingE-field (part. number)\nRotatingE-field (average rates)\nCP standing wave (part. number)\nCP standing wave (average rates)\nCP Gaussian (part. number)\nModel [Eq. (26)] ν= 0\nModel [Eq. (26)] ν >0\nAsymptote [Eq. (30)] ν= 0\nAsymptote [Eq. (30)] ν=ω/2π102410251026Intensity of one laser beam, W/cm2\n103104\neE0/mcω012RateWcr/greatermuchν, “rotating E” regimeαES\nν×2π/ω\nWcr×2π/ω\nFigure 5. Top panel: the growth rate of an avalanche-type\ncascade as a function of the field amplitude E0in different\nfield configurations. We plot the results of simulation ob-\ntained with SMILEI PIC for: CP standing wave [see Eqs. (35),\n(36)] (1D3V simulation), a uniform rotating electric field [see\nEq. (10)] (1D3V simulation), CP standing wave formed by\ntwo Gaussian beams (full 3D simulation). Empty circles and\nsquares correspond to Γ extracted directly from the particle\nnumbers [see Appendix E]. Small filled circles and squares cor-\nrespond to Eq. (21) with Wrad,cr=⟨WCS,BW(γ, χ)⟩averaged\nover the particle distributions reached at the end of each simu-\nlation. We compare the numerical data to our model given in\nEq. (26) both with and without accounting for the migration\neffect and to the corresponding asymptotic expressions from\nEq. (30). Bottom panel: migration rate νin a CP standing\nwave extracted from 1D3V PIC simulations, and estimated\npair production rate [see Eq. (24)]. For reference purposes,\nwe put a secondary horizontal axis at the top in the intensity\nunits of a single laser beam of the amplitude E0/2, implying\nthat the standing wave is formed by two of such laser beams\n[c.f. Eqs. (35)-(36)]. For all the curves, the field wavelength\nis set to λ= 0.8µm.\nlate Γ by fitting the e−e+pair number time dependence\nNp(t) with the exponential function eΓt(the full proce-\ndure is described in Appendix E). Second, as discussed\nin Section III A, the exact growth rate given in Eq. (21)\nwith the effective rates Wrad,crevaluated as prescribed by\nEq. (20). We extract the steady-state distribution func-\ntions fe,γ(γe,γ, χe,γ) from the numerical data and cal-\nculate the corresponding values ⟨WCS,BW(γ, χ)⟩[see also\nEqs. (C21)-(C22) in Appendix C]. Then we plug this\nresult and the migration rate ν(E0) (calculated as ex-\nplained above) in Eq. (21) to obtain Γ.\nThe simulation results and the comparison with the\ntheoretical models for the described field configurationsare presented in Fig. 5. We plot the extracted cascade\ngrowth rate for E0varied from a (relatively) low value,\nwhen the cascade multiplicity is exponentially small, to\na strong field with high particle yield. The migration\nrate field dependence ν(E0) is also shown in the bottom\npanel.\nWe confirm the full consistency of the numerical data\nfor Γ obtained directly from Np(t) (shown in Fig. 5 with\nempty circles and squares for the rotating E-field and\nstanding wave configurations, respectively) with formu-\nlas (20)-(21) (depicted with small filled round and square\nmarkers). This result substantiates our argumentation\nfrom Section III A and Appendix C that the master equa-\ntions and their solution, given in Eqs. (18)-(21), are ex-\nact. It also validates our method of accounting for parti-\ncle migration by using the rate ν(E0). Let us note that\na small discrepancy between the two methods for Γ ex-\ntraction is explained by the unideal determination of the\nsteady state due to the limitations of our numerical ap-\nproach in the low field regime.\nB. Validation of the analytical model\nWe plot our model for Γ given in Eq. (26) and\ntest it against simulations (see solid lines in Fig. 5).\nRecall that in this case, we calculate the rates as\nWrad,cr=WCS,BW(γem, χem), where γemandχemare eval-\nuated by combining Eqs. (11) and numerically integrated\nEq. (12).5Apart from the high field region (which we\ndiscuss in the next Section), the model shows excellent\nagreement with the simulation results for both configura-\ntions: the standing wave (compare the purple curve and\nempty squares in Fig. 5) and rotating E-field (compare\nthe orange curve and empty circles ibid). In the former,\nwe take into account the particle migration by using the\nnumerical data for ν(E0). However, we note here that by\nusing a simple estimate ν= 1/T=ω/(2π) we obtain a\nvery close result (refer to the purple line).\nAmong the cases we studied, the uniform rotating E-\nfield provides the highest pair yield. As compared to it, Γ\nfor a cascade in a standing wave is substantially smaller\nat low E0, which is due to e−e+migration. The expo-\nnential decay of the rate for the asymptotically low fields\nis faster for the standing wave configuration, which cor-\nrelates with the prediction by our model, see Eqs. (28)–\n(30) (also plotted in Fig. 5 with thin dot-dashed lines).\nThe effect of migration is even stronger in the field of\ntwo Gaussian beams, as particles can also escape in the\ndirection transverse to the optical axis.\nAt low E0, the growth rate can be small as compared\nto the inverse field period, Γ ×2π/ω≪1, meaning that\n5We assume that the main contribution to the cascade growth\nrate comes from the particles the E-antinode center of a standing\nwave, therefore, when calculating γemandχem, we set ϵ=E0\nandωeff=ω/2 as in a rotating E-field.11\nthe setting time for the steady state can be also much\nlarger than 2 π/ω. In this case, the growth rate can be\nlimited not only by migration but also by the finite pulse\nduration. In this case, our model can be used for an esti-\nmate of the average particle yield during the interaction.\nWith growing E0, the simulation results and the model\npredictions for Γ in the standing wave configuration ap-\nproach the optimal case of a rotating E-field. More-\nover, the growth rate of a cascade in the field of Gaus-\nsian beams also shows the same behavior. At the same\ntime, the migration rate decreases and becomes small\ncompared to the effective pair creation rate by photons,\nas shown in the bottom panel of Fig. 5. In effect, the\ncascade develops at the E-antinode center. The model\nof a uniform rotating E-field becomes universal for all\nCP standing wave-like configurations. Notably, the or-\nbits of individual particles become 2-dimensional. This\nillustrates the discussion in Section III C.\nThe transition to the reduced dynamics takes place at\nE0∼αES, therefore, at higher fields it is enough to\nconsider only the rotating electric field model for calcu-\nlating the particle growth rate. Moreover, as the field\nincreases the growth rate given by Eq. (26), as expected,\ndeviates from the numerical results (see Fig. 5) and has\nto be replaced by Eq. (27). This will be addressed in the\nfollowing Section.\nV. CASCADES IN HIGH CP FIELDS ( χem>X)\nA. Modelling the cascade growth rate in the full\nrange of field strength\nLet us discuss the growth rate of avalanches at very\nhigh fields focusing specifically on the uniform rotating\nelectric field configuration.6We extend our simulations\nwith SMILEI to higher E0using the setup presented in\nSection IV A and Appendix E. In addition, we perform\nindependent 3D simulations with a semiclassic Monte-\nCarlo code [44]. In the latter case, the external field\nis described by Eq. (10), and the results are averaged\nover∼102runs, where each run is initialized with one\nseed electron at rest. The results are presented in Fig. 6.\nThe data obtained with both codes matches with high\nprecision.\nIn Fig. 6, we plot the previously discussed Eq. (26) and\npresent the growth rate model developed for high fields,\nnamely, Eq. (27). The latter expression matches the data\npoints in the high field region, which is not covered by\nEq. (26). The curves overlap in an extended interval of\nE0atE0≳αES. The full model is rendered by linking\nboth expressions (highlighted with green in Fig. 6). Let\nus point out here, that at the overlap χemvaries from\n≈3 to 10 (also shown in Fig. 2). This corresponds to\n6Throughout Section V, we assume that ν= 0.\n103104105\neE0/mcω0.1110102Γ×2π/ω\nES αES\nSMILEI PIC\nMonte-Carlo\nFedotov et al [40]\nBashmakov et al [46]\nGrismayer et al [45]\nGrismayer model at tem\nThis work, Γ( χem<X)\nThis work, Γ( χem>X)\nAsymptotes: Eqs. (30), (31)Figure 6. Cascade growth rate dependence on E0in a uniform\nrotating electric field obtained in simulations (circles and cross\nmarkers). The solid curves represent the model proposed in\nthis work. The full model (highlighted in green) is obtained\nby switching from Eq. (26) [in orange] to Eq. (27) [in black]\nwhen they overlap (marked by the black diamond). Note that\nwe partially duplicate the data from Fig. 5, keeping the same\nnotation. Here, λ= 0.8µm.\nthe matching condition for Eqs. (26) and (27), which we\nconsidered in Section III B.\nWe also plot in Fig. 6 the asymptotic expressions for\nΓ at low and high fields, given in Eqs. (30) and (32),\nrespectively. The latter fits the black curve at E0> αE S\nwith high precision, namely, almost in the whole region\nof Eq. (27) applicability. The two asymptotic expressions\ncross at E0∼αES. As a result, when combined, they\nprovide a simple yet robust explicit formula for the field\ndependence of the avalanche growth rate.\nRecall that our model in (27) for high fields E0>\nαESincorporates the pair creation rate by photons at\nχγ∼1, as we expect them to dominate in the emis-\nsion spectrum. We confirm this assertion with simula-\ntions. We extract the steady-state photon distribution\nfunction fγ(γγ, χγ), which is normalised by the condi-\ntionR\ndγγdχγfγ(γγ, χγ) = 1. We then weight it with\nthe rate WBWto identify the photons that contribute to\nthe process of pair creation. The weighted distribution\ninχγis defined by\n[WBW∗fγ] (χγ) =Z\ndγγWBW(γγ, χγ)fγ(γγ, χγ).(37)\nWe plot this expression for different E0in Fig. 7. As we\nanticipated in our model, the weighted distribution peaks\natχγ≈2 for E0> αE S. At the same time, χemof emit-\nting electrons increases with the field (see also Fig. 2).\nIn the example of a lower field, E0≈0.4αES, the curve\nmaximum is shifted to the left as compared to the oth-\ners, and the corresponding χemis shifted leftwards even\nfurther. Thus only the exponential tail of the photon dis-\ntribution can contribute to pair creation, so the avalanche12\n100101102\nχγ0.00.10.20.30.40.5WBW∗fγ[1015τ−1\nC]\neE0\nmcω= 33000 [E0\nαES= 13.7]\neE0\nmcω= 16000 [E0\nαES= 6.9]\neE0\nmcω= 5000 [E0\nαES= 2.2]\neE0\nmcω= 2000 [E0\nαES= 0.8]\neE0\nmcω= 1000 [E0\nαES= 0.4]\nFigure 7. Distribution of photons fγ(χγ) in a cascade in the\nsteady state weighted with the probability rate of pair cre-\nation WBWfor different values of field strength [for the defini-\ntion, see Eq. (37)]. The data is extracted from Monte-Carlo\nsimulations. The vertical dashed lines represent the charac-\nteristic value of χemfor electrons at the emission event [as\ndefined in Eq. (15)]. For each of the dashed lines, χemis cal-\nculated at E0respective to a Wcrfγgraph of the same color.\ndevelopment is suppressed (see the corresponding growth\nrate in Fig. 6). At E0≈0.8αES, the value of χem≈1.4\nis close to the peak of the corresponding weighted distri-\nbution. Hence, both the emission and pair creation rates\nare far from their asymptotic given in Eqs. (6), (7), and\nwe can associate the region E0∼αESto the transition\nbetween the low and high field regimes.\nB. Comparison to the growth rate models\nproposed in past works\nLet us discuss how our model compares to the results\nof previous works. The available growth rate models for\nthe uniform rotating electric field configuration are sum-\nmarised in Table I. The scaling Γ ∝E1/4\n0atE0≫αES\nwas first obtained by Fedotov et al [40, 43]. The coeffi-\ncient for this scaling can be refined by using the result\nof Bashmakov et al [46], where it was initially proposed\nto use the analog of Eq. (21), which, however, does not\naccount for the migration effect and implies using Wrad,cr\nestimated in the spirit of Refs. [40, 43]. Both results are\nplotted in Fig. 6. Although they give the right order of\nmagnitude estimate at high E0, the E1/4\n0scaling sets in\nonly beyond ES(for a laser field of the optical frequency),\nas we mentioned at the end of Sec. III C. In contrast to\nthat, our formula (32) provides reliable high-field scaling\natE0> αE S.\nIn our notations, the scaling of Fedotov et al [40, 43]\nis based on the assumption that χγ∼χem, therefore, it\ndoes not account for the emission spectrum shape. For\ninstance, the probability rate of two successive events\nof photon emission and pair creation by this photon (in\nthe steady state) would be proportional to Pe−→e−e+∝WradWcr∼W2\nrad. Grismayer et al [45, 63] proposed a\nphenomenological model improving on that front. Put\nconcisely, it is achieved by expressing this probability as\nPe−→e−e+∝Z¯χe\n0dχ′dWCS(¯γe,¯χe, χ′)\ndχ′WBW(ε′, χ′) (38)\nwhere ¯ γe, ¯χeare the characteristic values at the moment\nof emission, which depend on the field and are kept gen-\neral in this step. We re-derive the formula for the growth\nrate proposed by Grismayer et al using the kinetic ap-\nproach, see Appendix D, where we also clarify the ap-\nproximations needed to get the result. It results in tran-\nscendental equation (D7) for Γ, which can be solved nu-\nmerically once the expressions for ¯ γe, ¯χeare known. In\nRef. [45], this calculation was carried out at asymptoti-\ncally low and high fields.\nRecall that in Section III A we argue that our model\nbased on Eqs. (18)-(21) incorporates the emission spectra\nwithout additional approximations. Let us compare our\nresults against the model of Grismayer et al.\nFor a low field, Grismayer et al propose setting ¯ γe∝\nE0, ¯χe∝E2\n0(see Table I) extracted from classical\ndynamics of an electron averaged over the field cycle.\nThe thin dashed line in Fig. 6 illustrates the result-\ning growth rate. Notably, the asymptotic decays as\n∝exp[−8/(3¯χe)], which is steeper than in our model\nEq. (29) [see also Table I]. Though, the numerical data\nfalls in between the two asymptotes.\nIn the high field regime, Grismayer et al use the E0-\nscaling for ¯ γe, ¯χederived by Fedotov et al from the\nshort time dynamics [we write the corresponding expres-\nsions valid for a general field in Eqs. (B4), (B5)] with\none additional modification, such that ¯ χeis defined as\n¯χe:=A[E0/(αES)]3/2, where A:= 1.24 is a free pa-\nrameter used to fit the numerical data. The fit allows\nmatching the data by the asymptote at very high E0, as\nshown in Fig. 6. The results of of Grismayer et al are\nclose to our predictions. The advantage of our model is\nthat it does not require fitting parameters.\nThe study in Ref. [45] did not cover the intermediate\nregime due to the lack of the corresponding expressions\nfor ¯γe, ¯χe. As we propose them in Section II C, namely,\nγemandχem, we tested the model of Grismayer et al\nin the full range. The result is shown in Fig. 6 (the\nsparse-dashed line).7Unfortunately, the result signifi-\ncantly underestimates the particle growth rate, which is,\npresumably, because of the overall pair production rate\nundercount. The radiation and pair creation events are\ncoupled in Eq. (38): χ′of the emitted photon also en-\ntersWBW, namely, the photon supposedly decays into a\npair shortly after the emission event. In our model, these\nprocesses are decoupled, as rates Wrad, Wcrare calculated\nindependently. The possible variation of χe,γin between\n7We assigned ¯ γe:=γem, ¯χe:=χemwithout the fitting parameter\nused in Ref. [45].13\nTable I. Summary of the particle growth rate models in avalanche-type QED cascades developing in a uniform rotating electric\nfield. The table collects the results of Refs. [40, 43, 45, 46, 63], and this paper.\nReference χe γe χγ Wrad Wcr Γ\n[40]Fedotov, [43]Elkinaµ3/2,a µ3/4\n√\nω∗∼χe≫1 ∼αχ2/3\ne\nτCγeWrad ∼αµ1/4\nτC√\nω∗\n[46]Bashmakov0.57µ3/2≈µ3/4\n√\nω∗∼χe≫1 WCS(χe≫1)bWBW(χγ≫1) Eq. (21) at ν= 0\n[45]GrismayerlowE0(αµ)2\nω∗4αµ\nπω∗0< χγ≤χec– WBW(χγ≪1) C1α2µ\nτCe−8ω∗\n3α2µ2,d\n[45]Grismayerhigh E01.24µ3/2 µ3/4\n√\nω∗0< χγ≤χecWCS(χe≫1) WBW(χγ) Root of Eq. (D7)\nThis work E0< αE S C2µ√2C2rµ\nω∗∼χem WCS(χem≪1)WBW(χem≪1) C3α√µω∗\nτCe−5\n3√\n3µ\nThis work E0> αE SC4µ3/2√2C4µ3/4\n√\nω∗∼1 WCS(χem≫1) WBW(1)c1αµ\nτChp\n1 +c2√\nω∗µ3/4−1i\naFor brevity, we define the dimensionless quantities µ=E0/(αES),ω∗=ωτC=ℏω/(mc2).\nbTo keep the table concise, we omit γe,γin the argument of WCS,BW[see Eqs. (4), (5)]; for the asymptotic expressions see Eqs. (6)-(7).\ncTheχγdistribution is given by dW CS/dχγ, see Eq. (2).\ndThe numerical constants: C1=π3/2/(20·61/4)≈0.18,C2= 4√\n3/5≈1.39,C3= 35/6√\n5 Γ2(2/3)/(√\n14π)≈0.87,\nC4= 9 (2 /[7Γ(2 /3)])3/2≈0.87, for c1,2see Eqs. (B8), (B9).\nthe quantum processes is taken into account implicitly.\nAs a result, we were able to reproduce the numerical data\nwith Eqs. (26) and (27) in the full range of E0without\nusing free parameters (apart from the model matching\npoint choice).\nVI. FORMATION OF ELECTRON-POSITRON\nPLASMA IN AVALANCHE-TYPE CASCADES\nWe apply our model to estimate the threshold for\nelectron-positron plasma formation in a cascade in termsof field parameters. Let us assume that a cascade is trig-\ngered in a uniform rotating electric field by electrons with\ninitial density n0. At time tthe particle density will reach\nn(t) =n0exp(Γ t). For a field of an optical frequency, Γ\nis well estimated by the asymptotic expressions (30) and\n(32) matched at their crossing point. By using them, we\ncan calculate the time required to reach density n:\nt≃Tln\u0012n\nn0\u0013\n×\n\n6.74·10−2q\n1000\na0exp\u0014\n2.89\u00121000\na0\u0013\u0012λ\n1µm\u0013\u0015\n, a0<3000\u0012λ\n1µm\u0013\n,\n3.31·10−2\u0010\n1000\na0\u0011\ns\n1 + 278 .54r\u0010\n1000\na0\u00113\u0010\nλ\n1µm\u0011\n−1, a 0>3000\u0012λ\n1µm\u0013 (39)\nwhere we used the notation a0=eE0/(mcω).\nEq. (39) allows to identify two characteristic times.\nFirst, the time tcat which the produced particles reach\nthe critical electron plasma density, n(tc) = nc=\nε0mω2/e2, and second, tscrcorresponding to the time mo-\nment when the plasma starts to screen the external field,\nn(tscr) =a0nc. The field dependence for these quantities\nis plotted in Fig. 8. The time needed to reach an e−e+plasma state is exponentially large at low field, but for\na0≳103it can be reached in several or even one field\ncycle depending on the initial density n0. Backreaction\nfrom the produced plasma should become noticeable af-\nter one field cycle at a0≳5·103(for optical lasers) if\nn0≳10−3nc. Note that the corresponding field strength\nisE0≳αES.\nWe calculated the characteristic screening time tscrin14\nthe rotating E-field configuration through PIC simula-\ntions with SMILEI. The numerical setup is the same as\nfor the previous simulations in Sections IV, V and is de-\ntailed in Appendix E. Here, we extended the time of each\nsimulation so that the produced plasma could generate a\nnoticeable field.\nTo identify the significance of the plasma-generated\nfield, we used the following convenient technical feature\nof SMILEI. We generated two sets of simulations under\nthe same input parameters: with so-called test and real\nparticles. The formers do not generate self-fields, there-\nfore, a corresponding simulation is equivalent to a semi-\nclassic Monte-Carlo computation.8The particle growth\nrate in such simulations remains constant in time once\nthe steady state is reached. In contrast, when the den-\nsity of real particles becomes high, the external field is\nscreened, and the cascade growth rate can be reduced.\nThe magnitude of screening can be guessed from the rel-\native difference in the particle numbers obtained in the\ntwo simulations, δN= (Ntest−Nreal)/Ntest. As δNin-\ncreases with time, tscrcan be associated with the time\nmoment when δNreaches a prescribed threshold value.\nThe results of PIC simulations are presented in Fig. 8.\nEach of the (square) points corresponds to a time mo-\nment tscrsuch that δN(tscr) = 0 .15. The inset illus-\ntrates our method for extracting tscrfor one of the plot-\nted points. Note that variation of the threshold value for\nδNdoes not lead to a significant change in the results.\nThe simulation results show good correspondence to the\nprediction of our model. Let us remark that we did not\naccount for the migration. For tscrat lower fields, that re-\nquire several field revolutions to generate dense plasma,\nmigration will lead to even longer times. However, at\nE0≳αES,tscrbecomes comparable to the field period,\nand at the same time the effect of migration becomes\nnegligible (as we discussed in previous sections).\nVII. SUMMARY AND DISCUSSION\nIn the present work, we considered electron-seeded\navalanche-type (or self-sustained) QED cascades develop-\ning in the field of a strong circularly polarized standing\nEM wave. Cascades in this configuration were investi-\ngated in past works including [38–40, 43, 45–47, 62, 63]\nmainly relying on numerical simulations. Our major goal\nwas identifying the analytical time- and field-dependence\nof the particle numbers produced in such a cascade.\nFor optimal particle yield in such a configuration, it is\nuseful to inject seed particles into the electric antinodes,\nwhere the acceleration of charged particles is maximal.\nHowever, charges can migrate to the magnetic antinodes\nat rate νreducing the cascade efficiency. Our work high-\nlights that this effect is important below a certain field\n8Technical details can be found in the SMILEI code manual\nhttps://smileipic.github.io/Smilei/ .\n103104\neE0/mcω0.1110102t/T\nn(tc) =nc\nn(tscr) =a0nc\nSimulations: δn= 0.151024102510261027Intensity of one laser beam, W/cm2\nαES\n0.0 0.5 1.0 1.5t/T0.00.51.01.5Ne+[a.u.]Test particles\nReal particles\nt:δN(t) = 0.15Figure 8. The field strength dependence of the time required\nto reach the electron critical plasma density nc(blue) and the\nrelativistically opaque regime a0nc(orange) in a cascade in\na uniform rotating electric field. The time depends on the\ninitial electron density n0, which is depicted as color bands:\nthe lower bound corresponds to n0= 10−5nc, and the upper\nton0= 10−1nc. Solid lines correspond to n0= 10−3nc. The\ninset shows two sample simulations in SMILEI: with real and\ntest particles. Squares in the main plot correspond to a time\npoint when δN= (Ntest−Nreal)/Ntestreaches the value of\n0.15. For reference purposes, we put a secondary horizontal\naxis at the top in the intensity units of a single laser beam of\nthe amplitude E0/2. The wavelength is set to λ= 0.8µm.\nthreshold since it raises the field strength necessary to\ntrigger a cascade. Understanding the impact of migra-\ntion is critical to make the optimal choice of field config-\nuration in upcoming experiments at ultra-high-intensity\nlaser facilities, in particular because the first tests will\ntake place near the cascade onset threshold.\nAt the onset stage, the cascade evolution is highly non-\nstationary. Nonetheless, the system can rapidly relax to\nthe so-called steady state, in which the effective rates\nof photon emission Wrad, pair creation Wcr, and migra-\ntionνare constant and depend only on the field param-\neters. The steady-state master equations for the particle\nnumber can be cast in a simple form: Eqs. (18)-(19).\nWe derive these equations by using a rigorous kinetic\napproach (detailed in Appendix C) relying only on the\nLCFA. We demonstrated that the values for Wrad,cr, that\nenter the master Eqs. (18)-(19), are obtained by averag-\ning the total emission and pair creation rates given in\nEqs. (4) and (5) over the particle distributions. The so-\nlution to the steady-state equations shows an exponential\ngrowth Ne±,γ∼eΓt, where the growth rate Γ is given by\nEq. (21).\nAn explicit calculation of Γ would provide the desired\nNe±,γfield dependence, however, the analytic expressions\nfor the distribution functions and, hence, for the average\nrates Wrad,crandνare unknown. Therefore, we devel-\noped a refined model for the rates. Simple relations for\nthe short-time semiclassic electron dynamics in the cas-15\nTable II. Single CP laser beam intensity [W/cm2] required to\nproduce (on average) one e−e+pair per seed electron during\none field period in an avalanche-type cascade triggered in the\nCP field of two such counterpropagating lasers. Here, we\nassume that the laser wavelength is λ= 0.8µm.\nRotating ECP standing wave Gaussian beams\nModel 6 .5×10238.8×1023—\nSimulation 5 ×10238.5×10231.1×1024\ncade allow defining the characteristic values of the key\nparameters χe,γ, at which the photon emission or pair\ncreation events are likely to happen (see Section II C).\nAs a result, we were able to calculate the effective rates\nas given in Eqs. (22), (24), and (25). These expressions\nallowed us to identify the growth rates Eqs. (26), (27) and\ntheir asymptotic forms, (30) and (32). Let us emphasize,\nthat Eqs. (26), (27) for Γ are valid for a wide range of\nfield models and parameters, with the major requirement\nthat a cascade can onset and then reach the steady state.\nTo test the model, we performed extensive numerical\nsimulations with the PIC-QED code SMILEI [91] and a\nMonte Carlo code [42, 44]. We considered three field con-\nfigurations: the electric antinode of an infinite standing\nCP wave, the uniform rotating electric field (migration is\nabsent in this case), and two counterpropagating tightly\nfocused CP Gaussian beams representing a realistic laser\nfield (the setup parameters are detailed in Appendix E).\nFirst, we demonstrated that the steady state equations\n(18)-(19) describe the evolution of the particle numbers\nprecisely when the effective rates are calculated as av-\nerages over the particle distributions. Second, we con-\nfirmed that our model for the growth rate [Eqs. (26)-(27)]\nis in excellent agreement with the numerical data.\nAs we expected, at lower fields the migration of par-\nticles significantly suppresses the cascade growth rate as\ncompared to the best-case scenario of a uniform rotating\nE-field. Notably, this happens in the parameter range\nof the upcoming experiments at the new generation of\nmulti-petawatt laser facilities. We provide sample data\nfor the threshold intensity in Table II for reference. In\na certain sense, the growth rate suppression is related\nto the dimensionality of the particle trajectories in the\ncascade. Thus, the migration of charges in auxiliary di-\nrections from the cascading region reduces the cascade\nmultiplicity. However, if it was possible to confine the\nparticle motion in the plane where the electric field is\nmaximal, the particle yield would become higher or, al-\nternatively, the cascade threshold would lower. This as-\npect can be useful for designing future experiments, e.g.\nto find the optimal setup, geometry, and/or targets that\nallow to trap particles (e.g. by using plasma shutters\n[108]).\nAt fields as high as E0≳αES, migration becomesnegligible for both the infinite standing wave and fo-\ncused Gaussian beam configurations. The correspond-\ning growth rates match the result for a uniform rotating\nE-field (see Fig. 5). First, this is simply because the cas-\ncade reaches the exponential phase at a sub-cycle time\nbefore particles can escape, as WcrT≫1. Second, the\nmigration rate decreases as the field strength increases,\nmeaning that the particles are trapped in the antinode\ncenter, where the electric field is maximal. This resem-\nbles the anomalous radiative trapping effect found for\nelectrons moving in a linearly polarized standing wave\n[109] and in dipole waves [41, 110]. As a consequence, at\na high field, the model of a cascade in a rotating uniform\nelectric field is universal for all the CP standing wave-like\nfields.\nOur model provides the explicit scaling of the particle\nnumber growth rate with the field strength [see Eq. (32)].\nIt significantly improves the previous findings of [40, 43,\n45, 46, 78] (summarized in Table I). The scaling Γ ∝\nE1/4\n0proposed in Ref. [40] reappears in our formula (32)\natE0≫αES, however, we showed that it sets in only\nbeyond the critical field of QED ES, namely, where the\napplicability of the cascade model is debatable.\nWith our model, we also estimate the threshold at\nwhich plasma effects and backreaction become relevant.\nAt the single laser intensity I≈2×1024W/cm2(in a\ntwo-beam setup) the produced electron-positron plasma\ncan reach the critical density in one field cycle, meaning\nthat some collective effects could be observable. At in-\ntensities I≳1025W/cm2, the external field can be fully\nscreened after one cycle. In general, the density of the\nproduced plasma is well-controllable by tuning the laser\nintensity. Once available in the laboratory, avalanche-\ntype cascades will open a path to studying relativistic\ne−e+plasma, relevant to astrophysical phenomena.\nIn this work, we focused on the case of a circularly\npolarized standing wave, as this field is favorable for the\ncascade onset and formation of the steady state. The\ncascades in such a field are also tractable with a theo-\nretical approach. However, linearly polarized fields are\neasier to access experimentally. Also, it was proposed\nthat using the linear polarization can be more beneficial\nat lower field strength near the cascade onset threshold\n[47]. This is due to the properties of particle motion\nat the time scale of the field period. However, in lin-\nearly polarized fields, the cascade steady state cannot\nextend for longer than one field period, and hence cir-\ncular polarization should provide higher particle yield at\nhigher fields. The full comparative study for both po-\nlarizations, as well as the generalization to single and\nmultiple tightly focused laser beams and structured laser\npulses (e.g. Laguerre-Gauss modes), is under progress\nand left for future work.\nFinally, let us make some additional remarks. Our\ngoal was to derive the cascade growth rate scaling with\nthe field strength, therefore, we disregarded the Sauter-\nSchwinger pair creation from vacuum for the sake of\nclarity, which might be important in the high-field limit16\n[26, 40, 90]. For a realistic simulation at ultra-high in-\ntensities, this effect should be accounted for in future\nstudies. Nevertheless, the cascade growth rate should\nnot be affected by the spontaneous pair production as\nlong as the field is not screened. Hence, we believe that\nthe presented results are physically relevant and, more-\nover, can be used to refine the threshold for attainable\nlaser intensities [40].\nACKNOWLEDGEMENTS\nWe express our gratitude to A.M. Fedotov, E.G. Gelfer,\nA. Gonoskov, T. Grismayer and M. Vranic for fruitful\nand elucidating discussions. This work used the open-\nsource PIC code SMILEI, the authors are grateful to all\nSMILEI contributors and to the SMILEI-dev team for\nits support. Simulations were performed on the Irene-\nJoliot-Curie machine hosted at TGCC, France, using\nHigh-Performance Computing resources from GENCI-\nTGCC (Grant No. A0030507678). This work re-\nceived financial support from the French state agency\nAgence Nationale de la Recherche , in the context of the\nInvestissements d’Avenir program (reference ANR-18-\nEURE-0014). A.A.M. was supported by Sorbonne Uni-\nversit´ e in the framework of the Initiative Physique des\nInfinis (IDEX SUPER).\nAppendix A: Small-time dynamics of e±in a cascade\nLet us discuss the motion of electrons in between emis-\nsions of photons in a cascade developing in a general field,\nwhich varies in time and space with the characteristic fre-\nquency ωand the corresponding wavelength λ. We follow\nRef. [42] and provide only the details necessary for the\ncurrent work.\nConsider an electron that is initially located at the ori-\ngin at time t= 0 in an external field. The electric and\nmagnetic fields at the initial position are characterized by\nthe EM field tensor Fµν=Fµν(0) combining the electric\nEand magnetic Bcomponents. As mentioned in Sec-\ntion II, we assume that the electron motion is semiclassic,\nmeaning that it is governed by the Lorentz equations:9\ndpµ\ndτ=−e\nmcFµ\nν(x(τ))pν, p µ(0) = p(0)\nµ, (A1)\nwhere pµandxµare the 4-momentum and time-space\ncoordinate of the electron and τis the proper time.10To\nsolve this equation we impose the following assumptions:\n9The discussion of positron motion will be the same except the\nchange of sign in the RHS of Eq. (A1).\n10In our notations, after every quantum event, an electron is as-\nsigned new initial conditions (this includes scattered electrons in\nthe Compton process).(i) the time between photon emissions is small, hence,\nwe consider time intervals t≪ω−1. The electron\npropagates almost as in a constant field Fµν(0), the\ncorrections are provided by the field derivatives,\nFµν,σ(0)≡∂Fµν(0)/∂xσ;\n(ii) due to acceleration by the field, e−rapidly becomes\nultra-relativistic at time scale mc/eE ≪t;\n(iii) energy gained by e−during acceleration is much\nhigher than the initial energy, eEt/mc ≫γe(0);\n(iv) The field is of electric type, namely, E > B (this\nis a wide class of fields, and the field of the electric\nantinode of a CP standing wave belongs to it).\nWhen solving the equations of motion, it is convenient\nto use the proper reference frame of the field, where the\nelectric and magnetic fields are parallel (or at least one\nof them vanishes), and their magnitude is given by the\ninvariants:\nϵ=qp\nF2+G2+F, (A2)\nη=qp\nF2+G2− F (A3)\nwhere F= (E2−c2B2)/2 and G=cB·Eare the field\ninvariants. Mathematically, αk={ϵ,−ϵ, iη,−iη}are the\neigenvalues of the field tensor Fµ\nν(0) with uµ\nkbeing the\ncorresponding eigenvectors:\nF(0)uk=αk\ncuk, k = 1, . . . , 4. (A4)\nThe solution to Eq. (A1) at the 0th order in ωt≪1 can\nbe expanded in four terms ∝ukeeαkτ/mc. However, in\nthe field of electric type ϵ >0 and ϵ > η . Atτ > mc/eϵ ,\nthe dominating contribution corresponds to ∝u1eeϵτ/mc,\nand the solution simplifies (see Ref. [42] for more details).\nThe eigenvector u1can be expressed in terms of the lab-\noratory frame field components:\nuµ\n1=\u0012\n1,c2(E·B)B+ϵc[E×B] +ϵ2E\nϵ(c2B2+ϵ2)\u0013\n. (A5)\nIt is worth noting that, although ϵrepresents the electric\nfield strength in the proper frame of the field, in the lab-\noratory frame this quantity depends both on the electric\nand magnetic components.\nUnder the above listed conditions, we solve Eq. (A1)\nto the order O((ωt)3) to find the key parameters:\nγe(t)≃eϵt\nmc, (A6)\nχ2\ne(t)≃χ2\ne(0) +\u0012ℏe2ϵ2ωeff\nm3c4t2\u00132\n, (A7)\nwhere χe(0) = eℏq\n−(p(0)\nµFµν(0))2/m3c4, and ωeffis de-\nfined by\nω2\neff=Fµν,σ(0)uµ\n1uσ\n1(J−1)ν\nλFλ\nκ,ρ(0)uρ\n1uκ\n1, (A8)17\nJ=\u0010\n2ϵ\nc1−F(0)\u00112\n, (A9)\nThe initial value of the χ-parameter in Eq. (A7) can be\nomitted too as the second term rapidly becomes domi-\nnating at the time scales of interest for us. As a result, we\nget Eqs. (8), (9). We remark here that the combination\nωefft2is Lorentz invariant, and hence Eq. (A7).\nAppendix B: Electron dynamics and cascade growth\nrate in asymptotically weak and strong fields\nFor reference purposes, we collect all the asymptotic\nexpressions in the low- and high-field regimes with full\nnumerical coefficients for the quantities introduced in the\nmain text.\nThe full probability rates of photon emission and e−e+\npair creation are given by:\nWCS(γe, χe)≃\n\n5\n2√\n3αχe\nτCγe, χ e≪1,\n14Γ(2 /3)\n37/3αχ2/3\ne\nτCγe, χe≫1,(B1)\nWBW(γγ, χγ)≃\n\nr\n3\n23\n16αχγ\nτCγγe−8/3χγ, χγ≪1,\n35/35Γ4(2/3)\n28π2αχ2/3\nγ\nτCγγ, χγ≫1.(B2)\nWe used Eq. (B1) to calculate the asymptotic values for\ntem,γem, and χemas described in Section II C. Thus, the\ntime of emission reads:\ntem≃\n\n2·4√\n3√\n5αrτCαES\nωeffϵ, χ em≪1,\n3\nα\u00122\n7Γ(2/3)\u00133/4rτC\nωeff\u0012αES\nϵ\u00131/4\n, χem≫1.\n(B3)Here, ϵis the invariant field strength as defined in\nEq. (A2), and ωeffis given by Eq. (A8). The values for\nγemandχemare obtained straightforwardly by substitut-\ningtemto Eqs. (8)-(9):\nγem≃\n\n2·4√\n3√\n5rϵ\nωeffτCαES, χ em≪1,\n3√ωeffτC\u00142\n7Γ(2/3)ϵ\nαES\u00153/4\n, χem≫1,\n(B4)\nχem≃\n\n4√\n3\n5ϵ\nαES, ϵ ≪αES,\n9\u00142\n7Γ(2/3)ϵ\nαES\u00153/2\n, ϵ≫αES.(B5)\nNote that in the last equation, we express the asymp-\ntotic conditions in terms of the invariant field ϵ, so that\nthey are consistent with the initial hypothesis χem≪1 or\nχem≫1, respectively. The threshold ϵ=αES≈ES/137\nseparates the regimes of the avalanche-type cascade de-\nvelopment.\nThe effective emission probability rate Wrad =\nWCS(γem, χem) at temat asymptotically low and high\nfields can be expressed as (see the corresponding discus-\nsion in Section III B):\nWCS(γem, χem)≃α\nτC×\n\n√\n5\n4√\n3rϵωeffτC\nαES, ϵ ≪αES,\n√\n2\n9[14 Γ(2 /3)]3/4√ωeffτC\u0012ϵ\nαES\u00131/4\n, ϵ≫αES.(B6)\nFinally, we collect the asymptotic expressions for the particle growth rate derived and discussed in Section III C18\nand the used numerical coefficients:\nΓ(ϵ)≃α\nτC×\n\n9√\n5Γ4(2/3)\n1412√\n3π2rϵωeffτC\nαES\"\n2√\n5\n4√\n3α\nνrϵωeffτC\nαES−1#\nexp\u0012\n−10αES\n3√\n3ϵ\u0013\n, ϵ≪αES, ν > 0,\n35/6√\n5√\n7πΓ2(2/3)rϵωeffτC\nαESexp\u0012\n−5αES\n3√\n3ϵ\u0013\n, ϵ ≪αES, ν= 0,\nc1ϵ\nαES\ns\n1 +c2√ωeffτC\u0012αES\nϵ\u00133/4\n−1\n, αE S< ϵ < c 3α(ωeffτC)2/3,\n2√\n2\n9[14 Γ(2 /3)]3/4√ωeffτC\u0012ϵ\nαES\u00131/4\n, c 3α(ωeffτC)2/3≪ϵ.(B7)\nc1=τCWBW(1,1)\n2≈0.516×10−4, (B8)\nc2=8√\n2 (14Γ(2 /3))3/4\n9τCWBW(1,1)≈1.107×105, (B9)\nc3=\u00122\n3\u00132/3224Γ(2 /3)\n9(τCWBW(1,1))4/3≈5.31×106, (B10)\nτCWBW(1,1)≈1.0318068870 ×10−4. (B11)\nAppendix C: Particle number equations in quasi-steady state\nCascade equations in general form read\n\u0014∂\n∂t+c2pe\nεe·∂\n∂r±∂\n∂pe·FL\u0015\nf±(r,pe, t) =Z\ndpγWrad(pe+pγ→pγ)f±(r,pe+pγ, t)\n−Wrad(pe)f±(r,pe, t) +Z\ndpγWcr(pγ→pe)fγ(r,pγ, t),(C1)\n\u0014∂\n∂t+c2pγ\nεγ·∂\n∂r\u0015\nfγ(r,pγ, t) =Z\ndpeWcr(pe→pγ) [f−(r,pe, t) +f+(r,pe, t)]\n−Wcr(pγ)fγ(r,pγ, t),(C2)\nwhere fa(r,p, t),a={±, γ}, are the electron, positron,\nand photon distribution functions. The LHS of the equa-\ntions describes the continuous dynamics of particles with\nmomenta pe,γand energies εe,γfore±andγ, respec-\ntively; ±FL=±e\u0000\nE+c2pe×B/εe\u0001\nis the Lorentz\nforce acting on a charged particle. In the RHS, we col-\nlect the source terms corresponding to the stochastic\nprocesses of photon emission and pair creation, where\nWrad(pe→pγ) is the differential probability rate for an\nelectron or positron with momentum peto emit a pho-\nton with momentum pγ, and Wcr(pγ→pe) is the dif-\nferential probability rate for a photon with momentum\npγto create a pair such that e−ore+gets momentum\npe. Equations (C1) and (C2) provide a starting point\nto model QED cascades in various scenarios, including\nelectromagnetic air showers [31], shower-type cascades\nin interaction of high-energy beams with laser pulses\n[33], and avalanche-type cascades [43, 62]. Let us pointout here that the probability rates depend on the field\nstrength, which in turn can depend on randt, namely,\nWcr,rad(pa→pb)≡Wcr,rad(pa→pb;E(r, t),B(r, t)),\nWcr,rad(pa)≡Wcr,rad(pa;E(r, t),B(r, t)). Within the\nLCFA, the fields enter through the quantum parameter\nχe,γof the incoming particle [see Eq. (1)]. For brevity,\nwe omit the explicit field dependence in our notation.\nLet us consider an avalanche-type e−-seeded cascade\ndeveloping in an electric antinode of a standing wave,\nformed by two circularly polarized waves propagating\nalong the z-axis. We assume that e−is injected close to\ntheE-antinode centered at the origin r= 0. In the trig-\ngered cascade, the number of produced pairs can grow\nexponentially with time, Np∼eΓt. As the mean free\npath time of e±andγin the cascade is small (see Sec-\ntion II), the majority of particles will be created in the\nsmall vicinity of the E-antinode centre, which we denote\nas domain D. To identify the dependence of Γ on the19\nfield parameters, it is therefore enough to study the par-\nticle distributions within D. Assuming that the field isalmost homogeneous in D, let us integrate out the coor-\ndinate dependence r. For Eq. (C1), we obtain:\n∂\n∂tf±(pe, t) +c2pe\nεe·Z\n∂DdSf±(r,pe, t)±∂\n∂pe·Z\nDdr FLf±(r,pe, t)\n=Z\ndpγWrad(pe+pγ→pγ)f±(pe+pγ, t)\n−Wrad(pe)f±(pe, t) +Z\ndpγWcr(pγ→pe)fγ(pγ, t),(C3)\nwhere we introduced\nf±,γ(pe, t) =Z\nDdrf±,γ(r,pe, t).\nHere, to simplify the RHS, we approximate the field dependence of the probability rates by a uniform field that\ncorresponds to the field at the E-antinode centre, Wcr,rad(pa→pb)≡Wcr,rad(pa→pb;E(0, t),B(0, t)). For the\nsecond term in the LHS, we used the divergence theorem to cast it into the surface integral. It accounts for the\nparticle flux through the surface ∂D. The flux can be caused by the migration of charged particles in the direction\nof the electric field gradient, in our case to the B-antinodes (see Fig. 4). As we consider waves that are infinite\nin the transverse plane, particles migrate only in the longitudinal direction. However, Eq. (C3) can be generalised\nstraightforwardly to account for migration in other directions for other field configurations, e.g. focused laser fields.\nIn the events of emission or pair creation, the incoming and outgoing particles are ultrarelativistic, thus εe≈c|pe|.\nFollowing [43, 62], we assume that in each quantum process the momenta of the initial and secondary particles are\ncollinear:\nWrad(pe→pγ) =Z1\n0dλ δ(pγ−λpe)εedWrad(pe, εγ)\ndεγ\f\f\f\f\nεγ=λεe,\nWcr(pγ→pe) =Z1\n0dλ δ(p−λpγ)εγdWcr(pγ, ε′\ne)\ndε′e\f\f\f\f\nε′e=λεγ.\nNote that within the LCFA, the probability rates explicitly depend only on the particle energy and χparameter:\ndWrad(pe, εγ)\ndεγ=dWrad(εe, χe, εγ)\ndεγ,dWcr(pγ, εe)\ndεe=dWcr(εγ, χγ, εe)\ndεe, (C4)\n[c.f. Eqs. (2) and (3)].11By applying this to the RHS of Eq. (C3), we can integrate out the angular dependence in\npγ. After repeating the same steps for Eq. (C2), we get:\n∂\n∂tf±(pe, t) +c2pe\nεe·Z\n∂DdSf±(r,pe, t)±∂\n∂pe·Z\nDdr FLf±(r,pe, t) =−Wrad(pe)f±(pe, t)\n+Z∞\nεedε′ε′2\nε2e\"\ndWrad(p′, εγ)\ndεγ\f\f\f\f\nεγ=ε′−εef±(p′, t) +dWcr(p′, εe)\ndεefγ(p′, t)#\f\f\f\f\f\np′=ε′cpe\nεe.(C5)\n∂\n∂tfγ(pγ, t) =−Wcr(pγ)fγ(pγ, t)\n+Z∞\nεγdε′ε′2\nε2γdWrad(p′, εγ)\ndεγ[f−(p′, t) +f+(p′, t)]\f\f\f\f\np′=ε′cpγ\nεγ.(C6)\nFor a uniform field, for example, if FL=eE(t) (this includes a uniform rotating electric field), the flux term in\nEq. (C5) vanishes and we recover the equations used in Refs. [43, 62].\n11We use particle energy εe,γ=γe,γmc2as the argument for the\nprobability rates to simplify the notations within Appendix C.The transition to the notation used in Eqs. (2), (3) is straight-\nforward.20\nAs we aim to obtain the equations for particle numbers, we integrate out the particle momenta from Eqs. (C5)-(C6):\n∂\n∂tN±(t) +ν±N±(t) =Z\ndpeZ∞\nεedε′ε′2\nε2edWcr(p′, εe)\ndεefγ(p′, t)\f\f\f\f\np′=ε′cpe\nεe, (C7)\n∂\n∂tNγ(t) =Z\ndpγ\n\n−Wcr(pγ)fγ(pγ, t) +Z∞\nεγdε′ε′2\nε2γdWrad(p′, εγ)\ndεγ[f−(p′, t) +f+(p′, t)]\f\f\f\f\np′=ε′cpγ\nεγ\n\n, (C8)\nwhere\nNa(t) =Z\ndpfa(p, t), (C9)\nj±(r, t) =Z\ndpec2pe\nεef±(r,pe, t), (C10)\nν±(t) =1\nN±(t)Z\n∂DdS j±(r, t) (C11)\nare the particle numbers, flows, and effective migration rates, respectively. As the electric field is symmetric around\ntheE-antinode center, we put ν−(t) =ν+(t)≡ν(t). Note that in Eq. (C7) we used that\nZ\ndpe\n\nZ∞\nεdε′ε′2\nε2e\"\ndWrad(p′, εγ)\ndεγ\f\f\f\f\nεγ=ε′−εef±(p′, t)#\f\f\f\f\f\np′=ε′cpe\nεe−Wrad(pe)f±(pe, t)\n\n= 0,\nwhich can be shown by interchanging the integrals in the first term and noticing that dWrad(p′, εγ−εe)/dεγ=\n−dWrad(p′, εγ−εe)/dεeunder the integral. Physically, it corresponds to the balance of radiation.\nTo simplify the equations further, let us consider the following term in the RHS of Eq. (C8):\nZ\ndpγZ∞\nεγdε′ε′2\nε2γdWrad(p′, εγ)\ndεγf±(p′, t)\f\f\f\f\np′=ε′cpγ\nεγ(C12)\nIt is convenient to pass to spherical coordinates in momentum space, pa→ {εa/c,na}, where na=pa/|pa|is the\nunit vector. The emission collinearity condition p′=ε′cpγ\nεγsimply means that the integration over nγin the outer\nintegral can be replaced by n′:\n1\nc3Z∞\n0dεγdn′ε2\nγZ∞\nεedε′ε′2\nε2γdWrad(ε′, χ′, εγ)\ndεγf±(ε′,n′, t), (C13)\nNote that we wrote the argument of dWrad/dεγso that it corresponds to Eq. (C4), and χ′depends on n′(e.g. in the\nreference frame co-moving with the electric field component [62]). Now we can interchange the dεγanddε′integrals:\n1\nc3Z∞\n0dε′dn′ε′2f±(ε′,n′, t)Zε′\n0dεγdWrad(ε′, χ′, εγ)\ndεγ(C14)\nThe rightmost integral gives Wrad(ε′, χ′), and the whole expression can be written in a compact form:\nZ\ndp′Wrad(p′)f±(p′, t). (C15)\nAfter repeating the same steps for the RHS of Eq. (C7), we can rewrite the particle number equations as follows:\n∂\n∂tN±(t) =−ν(t)N±(t) +Wcr(t)Nγ(t), (C16)\n∂\n∂tNγ(t) =Wrad(t) [N−(t) +N+(t)]−Wcr(t)Nγ(t), (C17)\nwhere we introduced time-dependent effective rates:\nWrad(t) =1\nN−(t) +N+(t)Z\ndp′Wrad(p′) [f−(p′, t) +f+(p′, t)], (C18)21\nWcr(t) =1\nNγ(t)Z\ndp′Wcr(p′)fγ(p′, t). (C19)\nRecall that, here, the probability rates also depend on time, Wrad,cr(p′)≡Wrad,cr(p′;E(0, t),B(0, t)). For a cascade\ndeveloping at the E-antinode center of a CP standing wave, the probability rates are independent of time in the\nframe co-rotating with the electric field. Transition to such the frame can be done for Eqs. (C1), (C2) (see Ref. [62]),\nhowever, does not change the form of Eqs. (C16), (C17).\nTo advance, we assume now that the cascade rapidly reaches the (quasi-)steady state, and that the time dependence\ninfafactorizes in the frame co-rotating with the electric field: fa(pa, t) =fa(pa)ga(t) [anticipating that ga(t)∼eΓt].\nWe use the probabilities in the LCFA, so Wrad(p′) =Wrad(ε′, χ′),Wcr(p′) =Wcr(ε′, χ′), hence, χ′can be used as\nan independent integration variable in Eqs. (C18), (C19). Passing to the new variables p′→ {ε′, χ′, θ′}casts the\ndistribution function to\nf±(p′)dp′=˜f±(ε′, χ′, θ′)dε′dχ′sinθ′dθ′, (C20)\nwhere −π/2< θ′≤π/2 is the azimuthal angle in the system such that the electric field rotates in the plane θ′= 0. As\na result of the distribution function factorisation ga(t) cancels out in Eqs. (C18), (C19) [with Na(t) written explicitly,\nsee Eq. (C9)], and the effective rates become stationary:\nWrad=Z\ndε′dχ′sinθ′dθ′Wrad(ε′, χ′)h\n˜f−(ε′, χ′, θ′) +˜f+(ε′, χ′, θ′)i\nZ\ndε′dχ′sinθ′dθ′h\n˜f−(ε′, χ′, θ′) +˜f+(ε′, χ′, θ′)i , (C21)\nWcr=Z\ndε′dχ′sinθ′dθ′Wcr(ε′, χ′)˜fγ(ε′, χ′, θ′)\nZ\ndε′dχ′sinθ′dθ′˜fγ(ε′, χ′, θ′). (C22)\nWe can apply the same reasoning to the migration rate,\nν(t)→ν, see Eq. (C11). By using these expressions\nin Eqs. (C16), (C17), we finally arrive at the particle\nnumber equations in the steady state:\ndN±(t)\ndt=WcrNγ(t)−νN±(t), (C23)\ndNγ(t)\ndt=Wrad[N−(t) +N+(t)]−WcrNγ(t).(C24)\nBy introducing Np= (N−+N+)/2 we cast these relations\ninto Eqs. (18)-(19). At ν= 0, we get the equations of\nthe form used in Refs. [45, 46, 61].\nThe effective rate of photon emission given in\nEqs. (C21) can be simplified further if we take into ac-\ncount the electron dynamics. Let us consider the integral:\nZ\ndε′dχ′sinθ′dθ′Wrad(ε′, χ′)˜f±(ε′, χ′, θ′)\nThe short-time expressions (8)-(9) relate εeandχeas\nthey are parametrised. We can express χein terms of εe:\nχe(εe) = (ε2\ne/mc2)2ωeffτC, meaning that the distribution\nfunction can be written as\n˜f±(ε′, χ′, θ′) =˜f±(ε′, θ′)δ(χ′−χe(ε′)). (C25)\nNotably, in the uniform rotating electric field con-\nfiguration, we can make an additional simplification:˜f±(ε′, θ′) =˜f±(ε′)δ(θ′)/sinθ′. Then the integral reads\nZ\ndε′sinθ′dθ′Wrad(ε′, χe(ε′))˜f±(ε′, θ′).\nIn Section III B, we propose to approximate the rate\nin this integral by a constant Wrad(ε′, χe(ε′))≈\nWCS(γem, χem), where χem=χe(γemmc2). Then\nthe effective rate in Eq. (C21) simplifies to Wrad≈\nWCS(γem, χem).\nDeveloping a similar line of reasoning applied to\nthe RHS of Eq. (C22), requires coupling εγand\nχγSince photons propagate along straight lines, and\ntheir energy is not altered, let us estimate χγ(εγ)∼\nsin(θγ)ϵεγ/(ESmc2), where θγis some characteristic\nangle between the photon momentum and the elec-\ntric field direction. Then we can rewrite the pho-\nton distribution function as in Eq. (C25), substitute\nit into the effective rate (C22), and obtain Wcr≈\nWBW(εγ/(mc2),sin(θγ)ϵεγ/(ESmc2)). Here, sin( θγ) can\nbe used as a fitting parameter when comparing the re-\nsulting growth rate expressions to numerical data.\nLet us make several remarks. First, our derivation\nshows that the steady-state hypothesis is equivalent to\ntaking the limit Wrad,cr(t→ ∞ )→Wrad,cr. The solution\nof the steady-state equations for the particle numbers\nis exponential in time, as we discuss in Section III. In\nthe derivation, we showed that the steady-state naturally\narises if the distribution function time dependence fac-22\ntorizes at large times, namely, we can replace the steady-\nstate hypothesis with the latter statement.\nSecond, the particular time required for this limit to\nset in, i.e. the steady state formation time, depends on\nthe particular field configuration and its parameters. As\nof now, the formation of the steady distribution has to\nbe confirmed with numerical simulations for each field\nconfiguration, which was done in the past works [42, 43,\n45, 47, 48] and the present study. A rigorous proof of the\nsteady state existence would be very interesting, as well\nas the calculation of the relaxation time, and a systematic\nclassification of the field configurations allowing such a\nstate.\nThird, Eqs. (C21)-(C24) are derived in the context of\nthe CP standing wave configuration, and particularly the\ncascade development at the E-antinode center, where the\nfield simplifies to a rotating E-field. However, these ex-\npressions can be generalized to cascades in other field\nconfigurations that allow a steady state.\nAnd last, to calculate the cascade particle growth rate\nΓ, it is enough to solve Eqs. (C23)-(C24). However, the\neffective rates Wrad,crand the migration coefficient νare\ndefined via the distribution functions, which are unknown\na priori . While solving the full distribution function an-\nalytically is intricate, the rates can be estimated under\nadditional approximations, as we propose in Sec. III B,\nor extracted from numerical simulations, as we do in\nSec. IV.\nAppendix D: The model of Grismayer et al [45]\nThe cascade growth rate model proposed by Grismayer\net al [45] [see Eqs. (9)–(10) therein] for the rotating E-\nfield was advancing the models of Fedotov et al [40] and\nBashmakov et al [46]. The improvement was achieved\nby reconsidering the assumption that almost all of the\nenergy of a radiating electron is transferred to a photon.\nInstead, the electron radiation spectrum dWCS/dχγwith\n0< χγ< χewas fully accounted for in Ref. [45]. At the\nsame time, Grismayer et al follow the preceding works by\nassuming that the electron mean free path time is given\nbyte∼W−1\nrad:\nWrad≡WCS(¯γe,¯χe) =Z¯χe\n0dχγdWCS(¯γe,¯χe, χγ)\ndχγ,where ¯ γeand ¯χeare the characteristic values of γeand\nχeat the moment of emission, which depend only on\nthe field parameters. The corresponding model for the\nnumber of produced pairs was introduced in Refs. [45, 63]\nby using a phenomenological approach. We show, how\nthis model arises from the kinetic approach and discuss\nthe approximations that are required to obtain it.\nLet us take Eq. (C7) coupled to Eq. (C6) as the starting\npoint. We set the migration term to zero in the context\nof the uniform rotating electric field model. As in Ap-\npendix C, we assume that the cascade reaches the steady\nstate. For Eq. (C7), we repeat the steps proposed in\nAppendix C, and arrive at:\n∂\n∂tN±(t) =Z\ndp′Wcr(ε′, χ′)fγ(p′, t). (D1)\nWe aim at obtaining a closed equation on N±(t). For\nthis, we will use Eq. (C6) to express fγ(εe, χe, t): we\napproximate the integral over ε′in the RHS in order to\npass to the particle numbers N±(t), and then integrate\nthe whole equation in time.\nAs we aim at keeping the differential rate in the ex-\npression, dWrad/dχγ=dWCS/dχγ, the second integral in\nthe RHS Eq. (C6) can be simplified only by imposing\nan additional assumption for the distribution functions\nf±(pe, t). In the steady state, the normalized electron\nspectrum does not change with time and has a promi-\nnent peak, as was noted by Grismayer et al [45] and con-\nfirmed in the current work with numerical simulations.\nWe rewrite the steady-state distribution function in the\nform:\nf±(pe, t) =N±(t)1\nNeg(εe), (D2)\nwhere Nnormalizes the electron spectrum, N=R\ndεeeg(εe). Let us assume for the moment that g(εe)\nhas a single maximum located at ¯ εe(see also the discus-\nsion below). Therefore, we can use the Laplace method\nto approximate the integral:\nZ∞\nεγdε′ε′2\nε2γdWrad(p′, εγ)\ndεγf±(p′, t)≈N±(t)\n4πε2γdWCS(¯γe,¯χe, εγ)\ndεγ, (D3)\nwhere we used that N ≈ 4πc−3¯ε2\neeg(¯εe)p\n2π/|g′′(¯εe)|. As a result, we get the equation\n∂\n∂tfγ(pγ, t) =−Wcr(pγ)fγ(pγ, t) +[N+(t) +N−(t)]\n4πε2γdWCS(¯γe,¯χe, εγ)\ndεγ, (D4)\nwhich can be solved explicitly:\nfγ(pγ, t) =c3\n4πε2γdWCS(¯γe,¯χe, εγ)\ndεγZt\n0dt′[N+(t′) +N−(t′)]e−Wcr(εγ,χγ)(t−t′). (D5)23\nHere, we used the initial condition fγ(pγ,0) = 0. Finally, after substituting fγ(pγ, t), putting Wcr=WBWinto\nEq. (D1) and rearranging the expression we get:\n∂\n∂tNp(t) = 2Zt\n0dt′Np(t′)Z¯χe\n0dχ′dWCS(¯γe,¯χe, χ′)\ndχ′WBW(ε′, χ′)e−WBW(ε′,χ′)(t−t′)(D6)\nwhere we passed to the number of pairs Np=N−+N+,\nchanged the integration variable ε′→χ′, and put the\nupper limit of the χ′integral as dWCS(¯γe,¯χe, χ′)/dχ′=\n0 for χ′>¯χe. Following Ref. [45], we formally solve\nEq. (D6) by using the Laplace transform\nNp(s) =Z∞\n0dt N p(t)e−st.\nWhen Eq. (D6) is rewritten in the image space, one can\nshow that the function Np(s) has poles at\ns−2Z¯χe\n0dχ′dWCS(¯γe,¯χe,χ′)\ndχ′ WBW(ε′, χ′)\ns+WBW(ε′, χ′)= 0.(D7)\nA positive solution s+>0 of this equation provides\nthe cascade growth rate in the steady state, Np(t)≃\nNp(0)eΓt, Γ = s+. Eq. (D7) can be solved numerically.\nFor more details, see Refs. [45, 63].\nEquations (D6) and(D7) reproduce the model of Gris-\nmayer et al [see Eq. (9) in Ref. [45]]. It should be noted\nthat in our derivation, we had to assume that the position\nof the electron spectrum coincides with ¯ εe, namely, at the\ncharacteristic energy of the electron at the time of emis-\nsion. This particular supposition (i) allowed decoupling\nf±anddWrad/dεγunder the integral in Eq. (D3), and (ii)\nnaturally cuts off the χ′-integral in Eq. (D7) at ¯ χe. How-\never, our simulations show that for εe≳¯εethe spectrum\ndecays exponentially, f±(εe, t)∝exp[−(εe/¯εe)n]. Such\nbehavior is expected as according to the initial hypoth-\nesis electrons emit photons as they reach such energies.\nTherefore, accounting for this feature may result in a re-\nfined version of Eq. (D7).\nLet us point out here, that we demonstrated in Ap-\npendix C that Eqs. (18)-(19) for N±,γ(t) are exact in\nterms of the particle spectra, as we integrate them out\nwithout additional approximations (except the LCFA).\nThe approximations that were applied in past works\n[40, 43, 46] and the new model that we propose in this\nwork in Section III, also inexplicitly account for the parti-\ncle spectrum precisely. The uncertainty of the mentioned\nmodels is contained in the approximation of ¯ εe,γ, ¯χe,γ,\nthat are used to estimate the emission and pair creation\nrates.\nAppendix E: Simulation setup\nFor the numerical study, we use the PIC code SMILEI\n[91] and perform simulations in the 1D3V geometry (1\n100101102Ne+e∗∗−\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nωt0510/angbracketleftχe/angbracketright\n/angbracketleftχe/angbracketrightstFigure 9. Example of one simulation in a rotating electric\nfield. Top panel: the time dependence of the e−e+pair\nnumber. Bottom panel: the average χevalue of the elec-\ntrons in the cascade. In the steady state, the growth rate\nΓ and ⟨χe⟩become stationary (see the corresponding hori-\nzontal line ⟨χe⟩stin the bottom panel). The growth rate is\nextracted by using the exponential fitting of the dependence\nNe−e+(t)∝eΓtat the data points depicted with the thick\ndashed line. The plot results from a simulation at a0= 5275.\nspatial dimension in the direction transverse to the field\nand 3 momentum projections). The code consistently\nsolves the equations of motion for particles and Maxwell’s\nequations for EM fields, and includes the quantum pro-\ncesses of photon emission by electrons and positrons and\npair creation by photons within the LCFA as given by\nEqs. (2)-(5).\nFor the standing wave configuration, we pick the sim-\nulation box of length 4 λ, and preliminary fill it with the\nfield, so that the box is centered at the electric antinode.\nWe initialize ≈100 seed electrons at rest in the vicinity\nof the center. The spatial and temporal resolution is set\ntoλ/128 and min( T/500, T/a 0), respectively. We per-\nformed the convergence checks to validate the choice of\ntemporal resolution. For the rotating field configuration,\nwe use a 6 λsimulation box. The field is prescribed by\nEq. (10), and seed electrons are distributed in the mid-\ndle 2λ-waist. For both configurations, we pick the initial\nelectron density of 0 .01nc, where nc=ε0mω2/e2is the\ncritical plasma density. Since our aim is testing the cas-\ncade growth model presented in Section III, we stay in the\nregime of relatively low particle densities when plasma ef-\nfects are not significant (we discuss the field screening by\nthe produced plasma in Section VII).\nIn addition, we performed several full 3D PIC simula-\ntions in a realistic configuration, when the standing wave24\nis formed by two focused Gaussian laser beams of circular\npolarization and with the waist w0= 3λ. The seeding\nconditions are the same as for the standing wave config-\nuration, however, in this case, we start the simulation by\ninjecting the beams from two opposite sides of an empty\nbox, and seed electrons later, when the standing wave is\nformed. The EM field is calculated consistently in the\nPIC loop. We keep the total field amplitude at E0to\nmatch simulations in the other field configurations.\nExtraction of the growth rate. In each simula-\ntion run, we observe an avalanche exhibiting exponen-\ntial growth of the particle number, see an example in\nFig. 9. We continue each run for a long enough time\nto ensure that the cascade reaches a steady state. The\ndata collected in the steady state can be used to extract\nthe growth rate Γ. The corresponding time points can\nbe found by studying the time-dependence of the aver-\nage electron Lorentz factor ⟨γe⟩andχ-parameter ⟨χe⟩.\nWhen the cascade reaches the steady state, ⟨χe⟩relaxes\nto the constant value ⟨χe⟩st, as illustrated in Fig. 9. No-\ntably, the characteristic relaxation time depends on the\nfield strength and decreases for higher E0. We select the\nsteady-state time points by the following procedure: (i)\nequate value of ⟨χe⟩reached at the end of the simulation\nto⟨χe⟩st, and (ii) pick time points from the data set tail\nthat satisfy the condition |⟨χe⟩(t)−⟨χe⟩st|/⟨χe⟩st<0.02.\nThe growth rate Γ can be extracted by fitting the data\nforNe−e+(t) with the exponential function Aexp(Γ t) for\nthe selected points, where Aand Γ are the fitting param-\neters.\nEvaluation of the migration rate ν.Let us con-\nsider seed electrons in the field of a standing wave. If\nthey are injected in spatial regions where χe(t) growsfast, they can trigger a cascade. Recall that χe(t) is pro-\nportional to the value of ϵ2ωeff[see Eq. (9)], which we\nplot in Fig. 4. It peaks in the electric node centers and\nrapidly falls off near the magnetic antinodes, where it\nvanishes. It is therefore beneficial to seed particles near\nthe electric field peak, however, they will migrate to the\nmagnetic antinodes, as the latter are spiraling attractors\nfor charges [79, 80, 105–107]. Let us calculate the mi-\ngration rate νofe−e+from the E-antinode center as a\nfunction of E0to evaluate its effect on the cascade growth\nrate.\nThe motion of an electron in a standing wave is known\nto be chaotic already in the classical regime. In our case,\nit is also affected by radiation. As general analytic ex-\npressions for the trajectories for this case are unknown,\nwe perform statistical simulation of particle migration us-\ning SMILEI PIC code [91]. The simulation setup is the\nsame as described above for the standing wave configu-\nration, except that we seed ∼105macro-particles in the\nvicinity of the E-antinode and switch off pair creation\nfor the photon species in the code, as we are interested\nparticularly in the trajectories of the seed particles. Nev-\nertheless, we fully account for the recoil due to photon\nemissions by the electrons. We define νas the inverse\ntime at which half of the particles propagate at distance\nλ/8 from the origin, meaning that they reach the spatial\nregion where ϵ2ωeff= 0 (one of the shaded regions in Fig.\n4). The resulting dependence ν(a0) is presented in the\nbottom panel of Fig. 5. At low intensities, the migration\nrate is close to ν≈T−1=ω/2π, which is consistent with\nthe analytical study in Ref. [106]. However as the am-\nplitude grows, radiation effects become important and ν\nslowly decreases.\n[1] R. D. Blandford and R. L. Znajek, Electromagnetic ex-\ntraction of energy from Kerr black holes, MNRAS 179,\n433 (1977).\n[2] A. L. Ford, B. D. Keenan, and M. V. Medvedev,\nElectron-positron cascade in magnetospheres of spin-\nning black holes, Phys. Rev. D 98, 063016 (2018).\n[3] F. C. Michel and H. Li, Electrodynamics of neutron\nstars, Phys. Rep. 318, 227 (1999).\n[4] P. Goldreich and W. H. Julian, Pulsar electrodynamics,\nAstrophys. J., vol. 157, p. 869 157, 869 (1969).\n[5] J. Arons, Some problems of pulsar physics or I’m madly\nin love with electricity, Space Sci. Rev. 24, 437 (1979).\n[6] R. Gueroult, Y. Shi, J.-M. Rax, and N. J. Fisch, Deter-\nmining the rotation direction in pulsars, Nat. Commun.\n10, 3232 (2019).\n[7] T. Piran, The physics of gamma-ray bursts, Rev. Mod.\nPhys. 76, 1143 (2005).\n[8] P. Chang, A. Spitkovsky, and J. Arons, Long-term evo-\nlution of magnetic turbulence in relativistic collisionless\nshocks: electron-positron plasmas, Astrophys. J. 674,\n378 (2008).\n[9] A. Spitkovsky, Particle acceleration in relativistic col-\nlisionless shocks: Fermi process at last?, Astrophys. J.682, L5 (2008).\n[10] P. Kumar and B. Zhang, The physics of gamma-ray\nbursts & relativistic jets, Phys. Rep. 561, 1 (2015).\n[11] A. N. Timokhin and A. K. Harding, On the maximum\npair multiplicity of pulsar cascades, Astrophys. J. 871,\n12 (2019).\n[12] F. Cruz, T. Grismayer, and L. O. Silva, Kinetic model\nof large-amplitude oscillations in neutron star pair cas-\ncades, Astrophys. J. 908, 149 (2021).\n[13] F. Cruz, T. Grismayer, S. Iteanu, P. Tortone,\nand L. O. Silva, Model of pulsar pair cascades\nin non-uniform electric fields: Growth rate, den-\nsity profile, and screening time, Phys. Plasmas 29,\nhttps://doi.org/10.1063/5.0085847 (2022).\n[14] B. Cerutti, G. R. Werner, D. A. Uzdensky, and M. C.\nBegelman, Simulations of particle acceleration beyond\nthe classical synchrotron burnoff limit in magnetic re-\nconnection: an explanation of the Crab flares, Astro-\nphys. J. 770, 147 (2013).\n[15] A. A. Philippov and A. Spitkovsky, Ab initio pulsar\nmagnetosphere: three-dimensional particle-in-cell simu-\nlations of axisymmetric pulsars, Astrophys. J. Lett. 785,\nL33 (2014).25\n[16] R. Hu and A. M. Beloborodov, Axisymmetric pulsar\nmagnetosphere revisited, Astrophys. J. 939, 42 (2022).\n[17] H. Chen and F. Fiuza, Perspectives on relativistic\nelectron–positron pair plasma experiments of astrophys-\nical relevance using high-power lasers, Phys. Plasmas\n30, https://doi.org/10.1063/5.0134819 (2023).\n[18] K. Qu, S. Meuren, and N. J. Fisch, Creating pair plas-\nmas with observable collective effects, Plasma Phys.\nControl. Fusion 65, 034007 (2023).\n[19] G. Sarri, K. Poder, J. Cole, W. Schumaker, A. Di Pi-\nazza, B. Reville, T. Dzelzainis, D. Doria, L. Gizzi,\nG. Grittani, et al. , Generation of neutral and high-\ndensity electron–positron pair plasmas in the labora-\ntory, Nat. Commun. 6, 6747 (2015a).\n[20] C. D. Arrowsmith, A. F. A. Bott, F. D. Cruz, D. H.\nFroula, et al. , Laboratory realization of relativistic pair-\nplasma beams, arXiv preprint arXiv:2312.05244 (2023).\n[21] C. P. Ridgers, C. S. Brady, R. Duclous, J. G. Kirk,\nK. Bennett, T. D. Arber, and A. R. Bell, Dense\nelectron-positron plasmas and bursts of gamma-rays\nfrom laser-generated quantum electrodynamic plasmas,\nPhys. Plasmas 20, 056701 (2013).\n[22] N. B. Narozhny and A. M. Fedotov, Creation of\nelectron-positron plasma with superstrong laser field,\nEur. Phys. J. Spec. Top. 223, 1083 (2014).\n[23] G. Sarri, M. E. Dieckmann, I. Kourakis, A. Di Piazza,\nB. Reville, C. Keitel, and M. Zepf, Overview of laser-\ndriven generation of electron–positron beams, J. Plasma\nPhys. 81, 455810401 (2015b).\n[24] P. Zhang, S. Bulanov, D. Seipt, A. V. Are-\nfiev, and A. G. R. Thomas, Relativistic plasma\nphysics in supercritical fields, Phys. Plasmas 27,\nhttps://doi.org/10.1063/1.5144449 (2020).\n[25] K. Qu, S. Meuren, and N. J. Fisch, Signature of collec-\ntive plasma effects in beam-driven QED cascades, Phys.\nRev. Lett. 127, 095001 (2021).\n[26] A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt,\nH. Taya, and G. Torgrimsson, Advances in QED with\nintense background fields, Phys. Rep. 1010 , 1 (2023).\n[27] A. Gonoskov, T. G. Blackburn, M. Marklund, and\nS. S. Bulanov, Charged particle motion and radiation\nin strong electromagnetic fields, Rev. Mod. Phys. 94,\n045001 (2022).\n[28] S. V. Popruzhenko and A. M. Fedotov, Dynamics and\nradiation of charged particles in ultra-intense laser\nfields, Phys. Usp. 66, 460 (2023).\n[29] A. Di Piazza, C. M¨ uller, K. Hatsagortsyan, and C. H.\nKeitel, Extremely high-intensity laser interactions with\nfundamental quantum systems, Rev. Mod. Phys. 84,\n1177 (2012).\n[30] R. Ruffini, G. Vereshchagin, and S.-S. Xue, Electron–\npositron pairs in physics and astrophysics: From heavy\nnuclei to black holes, Phys. Rep. 487, 1 (2010).\n[31] T. K. Gaisser, R. Engel, and E. Resconi, Cosmic rays\nand particle physics (Cambridge University Press, Cam-\nbridge, 2016).\n[32] I. V. Sokolov, N. M. Naumova, J. A. Nees, and G. A.\nMourou, Pair creation in QED-strong pulsed laser fields\ninteracting with electron beams, Phys. Rev. Lett. 105,\n195005 (2010).\n[33] S. S. Bulanov, C. B. Schroeder, E. Esarey, and W. P.\nLeemans, Electromagnetic cascade in high-energy elec-\ntron, positron, and photon interactions with intense\nlaser pulses, Phys. Rev. A 87, 062110 (2013).[34] T. Blackburn and C. Murphy, Scaling laws for positron\nproduction in laser–electron-beam collisions, Phys. Rev.\nA96, 022128 (2017).\n[35] A. Mercuri-Baron, M. Grech, F. Niel, A. Grassi, M. Lo-\nbet, A. Di Piazza, and C. Riconda, Impact of the laser\nspatio-temporal shape on Breit–Wheeler pair produc-\ntion, New J. Phys. 23, 085006 (2021).\n[36] C. D. Zerby and H. S. Moran, Studies of the longitudi-\nnal development of electron—photon cascade showers,\nJ. Appl. Phys. 34, 2445 (1963).\n[37] V. I. Ritus, Quantum effects of the interaction of ele-\nmentary particles with an intense electromagnetic field,\nJ. Russ. Laser Res. 6, 497 (1985).\n[38] A. R. Bell and J. G. Kirk, Possibility of prolific pair pro-\nduction with high-power lasers, Phys. Rev. Lett. 101,\n200403 (2008).\n[39] J. G. Kirk, A. R. Bell, and I. Arka, Pair production\nin counter-propagating laser beams, Plasma Phys. Con-\ntrol. Fusion 51, 085008 (2009).\n[40] A. M. Fedotov, N. B. Narozhny, G. Mourou, and\nG. Korn, Limitations on the attainable intensity of high\npower lasers, Phys. Rev. Lett. 105, 080402 (2010).\n[41] A. Gonoskov, A. Bashinov, S. Bastrakov, E. Efimenko,\nA. Ilderton, A. Kim, M. Marklund, I. Meyerov, A. Mu-\nraviev, and A. Sergeev, Ultrabright gev photon source\nvia controlled electromagnetic cascades in laser-dipole\nwaves, Phys. Rev. X 7, 041003 (2017).\n[42] A. A. Mironov, E. G. Gelfer, and A. M. Fedotov, Onset\nof electron-seeded cascades in generic electromagnetic\nfields, Phys. Rev. A 104, 012221 (2021).\n[43] N. V. Elkina, A. M. Fedotov, I. Y. Kostyukov, M. V.\nLegkov, N. B. Narozhny, E. N. Nerush, and H. Ruhl,\nQED cascades induced by circularly polarized laser\nfields, Phys. Rev. STAB 14, 054401 (2011).\n[44] A. A. Mironov, N. B. Narozhny, and A. M. Fedotov, Col-\nlapse and revival of electromagnetic cascades in focused\nintense laser pulses, Phys. Lett. A 378, 3254 (2014).\n[45] T. Grismayer, M. Vranic, J. L. Martins, R. A. Fonseca,\nand L. O. Silva, Seeded QED cascades in counterprop-\nagating laser pulses, Phys. Rev. E 95, 023210 (2017).\n[46] V. F. Bashmakov, E. N. Nerush, I. Y. Kostyukov, A. M.\nFedotov, and N. B. Narozhny, Effect of laser polarization\non quantum electrodynamical cascading, Phys. Plasmas\n21, 013105 (2014).\n[47] M. Jirka, O. Klimo, S. V. Bulanov, T. Z. Esirkepov,\nE. Gelfer, S. S. Bulanov, S. Weber, and G. Korn, Elec-\ntron dynamics and γande−e+production by colliding\nlaser pulses, Phys. Rev. E 93, 023207 (2016).\n[48] E. G. Gelfer, A. A. Mironov, A. M. Fedotov, V. F.\nBashmakov, E. N. Nerush, I. Y. Kostyukov, and N. B.\nNarozhny, Optimized multibeam configuration for ob-\nservation of QED cascades, Phys. Rev. A 92, 022113\n(2015).\n[49] M. Marklund, T. G. Blackburn, A. Gonoskov, J. Mag-\nnusson, S. S. Bulanov, and A. Ilderton, Towards criti-\ncal and supercritical electromagnetic fields, High Power\nLaser Sci. Eng. 11, e19 (2023).\n[50] H. Vincenti, T. Clark, L. Fedeli, P. Martin, A. Sainte-\nMarie, and N. Zaim, Plasma mirrors as a path to the\nSchwinger limit: theoretical and numerical develop-\nments, Eur. Phys. J.: Spec. Top. 232, 2303 (2023).\n[51] C. P. Ridgers, C. S. Brady, R. Duclous, J. G. Kirk,\nK. Bennett, T. D. Arber, A. P. L. Robinson, and A. R.\nBell, Dense electron-positron plasmas and ultraintense26\nγrays from laser-irradiated solids, Phys. Rev. Lett. 108,\n165006 (2012).\n[52] M. Jirka, O. Klimo, M. Vranic, S. Weber, and G. Korn,\nQED cascade with 10 PW-class lasers, Sci. Rep. 7, 15302\n(2017).\n[53] C. Slade-Lowther, D. D. Sorbo, and C. P. Ridgers, Iden-\ntifying the electron–positron cascade regimes in high-\nintensity laser-matter interactions, New J. Phys. 21,\n013028 (2019).\n[54] X.-L. Zhu, T.-P. Yu, Z.-M. Sheng, Y. Yin,\nI. C. E. Turcu, and A. Pukhov, Dense GeV\nelectron–positron pairs generated by lasers in\nnear-critical-density plasmas, Nat. Commun. 7,\nhttps://doi.org/10.1038/ncomms13686 (2016).\n[55] A. S. Samsonov, I. Y. Kostyukov, and E. N. Nerush,\nHydrodynamical model of QED cascade expansion in an\nextremely strong laser pulse, Matter Radiat. Extremes\n6, https://doi.org/10.1063/5.0035347 (2021).\n[56] I. I. Artemenko and I. Y. Kostyukov, Ionization-induced\nlaser-driven QED cascade in noble gases, Phys. Rev. A\n96, 032106 (2017).\n[57] M. Tamburini, A. Di Piazza, and C. H. Keitel, Laser-\npulse-shape control of seeded QED cascades, Sci. Rep.\n7, http://doi.org/10.1038/s41598-017-05891-z (2017).\n[58] M. Jirka and S. Bulanov, Effects of colliding laser pulses\npolarization on e−e+cascade development in extreme\nfocusing, arXiv preprint arXiv:2401.08410 (2024).\n[59] M. Lobet, C. Ruyer, A. Debayle, E. d’Humi` eres,\nM. Grech, M. Lemoine, and L. Gremillet, Ultrafast\nsynchrotron-enhanced thermalization of laser-driven\ncolliding pair plasmas, Phys. Rev. Lett. 115, 215003\n(2015).\n[60] J. Y. Yu, T. Yuan, W. Y. Liu, M. Chen, W. Luo, S. M.\nWeng, and Z. M. Sheng, QED effects induced harmon-\nics generation in extreme intense laser foil interaction,\nPlasma Phys. Control. Fusion 60, 044011 (2018).\n[61] W. Luo, W.-Y. Liu, T. Yuan, M. Chen, J.-Y. Yu, F.-Y.\nLi, D. Del Sorbo, C. P. Ridgers, and Z.-M. Sheng, QED\ncascade saturation in extreme high fields, Sci. Rep. 8,\n8400 (2018).\n[62] E. N. Nerush, V. F. Bashmakov, and I. Y. Kostyukov,\nAnalytical model for electromagnetic cascades in rotat-\ning electric field, Phys. Plasmas 18, 083107 (2011).\n[63] T. Grismayer, M. Vranic, J. L. Martins, R. A. Fonseca,\nand L. O. Silva, Laser absorption via quantum electro-\ndynamics cascades in counter propagating laser pulses,\nPhys. Plasmas 23, 056706 (2016).\n[64] E. S. Efimenko, A. V. Bashinov, S. I. Bastrakov, A. A.\nGonoskov, A. A. Muraviev, I. B. Meyerov, A. V. Kim,\nand A. M. Sergeev, Extreme plasma states in laser-\ngoverned vacuum breakdown, Sci. Rep. 8, 2329 (2018).\n[65] D. Papadopoulos, J.-P. Zou, C. Le Blanc, L. Ranc,\nF. Druon, L. Martin, A. Fr´ eneaux, A. Beluze, N. Lebas,\nM. Chabanis, et al. , First commissioning results of the\napollon laser on the 1 PW beam line, in CLEO: Science\nand Innovations (Optica Publishing Group, 2019) pp.\nSTu3E–4.\n[66] S. Weber, S. Bechet, S. Borneis, L. Brabec, M. Buˇ cka,\nE. Chacon-Golcher, M. Ciappina, M. DeMarco, A. Fajs-\ntavr, K. Falk, et al. , P3: An installation for high-energy\ndensity plasma physics and ultra-high intensity laser–\nmatter interaction at ELI-Beamlines, Matter Radiat. at\nExtremes 2, 149 (2017).\n[67] Center for Relativistic Laser Science, https://corels.ibs.re.kr/html/corels_en/ .\n[68] K. A. Tanaka, K. M. Spohr, D. L. Balabanski, S. Balas-\ncuta, L. Capponi, M. O. Cernaianu, M. Cuciuc, A. Cu-\ncoanes, I. Dancus, A. Dhal, et al. , Current status and\nhighlights of the ELI-NP research program, Matter Ra-\ndiat. at Extremes 5, 024402 (2020).\n[69] Astra Gemini laser, https://www.clf.stfc.ac.uk/\nPages/The-Astra-Gemini-Facility.aspx .\n[70] Z. Gan, L. Yu, S. Li, C. Wang, X. Liang, Y. Liu, W. Li,\nZ. Guo, Z. Fan, X. Yuan, et al. , 200 J high efficiency\nTi:sapphire chirped pulse amplifier pumped by temporal\ndual-pulse, Opt. Express 25, 5169 (2017).\n[71] PEARL laser complex, https://ipfran.ru/\nscience/laser-physics-and-nonlinear-optics/\ngeneration-of-extreme-laser-fields/\nPEARL-laser-complex .\n[72] C. N. Danson, C. Haefner, J. Bromage, T. Butcher, J.-\nC. F. Chanteloup, E. A. Chowdhury, A. Galvanauskas,\nL. A. Gizzi, J. Hein, D. I. Hillier, et al. , Petawatt and\nexawatt class lasers worldwide, High Power Laser Sci.\nEng.7, https://doi.org/10.1017/hpl.2019.36 (2019).\n[73] J. W. Yoon, Y. G. Kim, I. W. Choi, J. H. Sung, H. W.\nLee, S. K. Lee, and C. H. Nam, Realization of laser\nintensity over 1023W/cm2, Optica 8, 630 (2021).\n[74] J. Bromage, S.-W. Bahk, I. A. Begishev, C. Dorrer,\nM. J. Guardalben, B. N. Hoffman, J. B. Oliver, R. G.\nRoides, E. M. Schiesser, M. J. Shoup Iii, et al. , Technol-\nogy development for ultraintense all-OPCPA systems,\nHigh Power Laser Sci. Eng. 7, e4 (2019).\n[75] NSF OPAL, https://www.lle.rochester.edu/\nnsf-opal/ .\n[76] E. Khazanov, A. Shaykin, I. Kostyukov, V. Ginzburg,\nI. Mukhin, I. Yakovlev, A. Soloviev, I. Kuznetsov,\nS. Mironov, A. Korzhimanov, et al. , eXawatt Center\nfor Extreme Light Studies, High Power Laser Sci. Eng.\n11, e78 (2023).\n[77] B. Shao, Y. Li, Y. Peng, P. Wang, J. Qian, Y. Leng, and\nR. Li, Broad-bandwidth high-temporal-contrast carrier-\nenvelope-phase-stabilized laser seed for 100 PW lasers,\nOptics Lett. 45, 2215 (2020).\n[78] I. Y. Kostyukov, I. I. Artemenko, and E. N. Nerush,\nGrowth rate of QED cascades in a rotating electric field,\nProbl. At. Sci. Technol. , 259 (2018).\n[79] T. Z. Esirkepov, S. S. Bulanov, J. K. Koga, M. Kando,\nK. Kondo, N. N. Rosanov, G. Korn, and S. V. Bu-\nlanov, Attractors and chaos of electron dynamics in elec-\ntromagnetic standing waves, Phys. Lett. A 379, 2044\n(2015).\n[80] B. King and H. Hu, Classical and quantum dynamics of\na charged scalar particle in a background of two coun-\nterpropagating plane waves, Phys. Rev. D 94, 125010\n(2016).\n[81] F. Niel, C. Riconda, F. Amiranoff, R. Duclous, and\nM. Grech, From quantum to classical modeling of ra-\ndiation reaction: A focus on stochasticity effects, Phys.\nRev. E 97, 043209 (2018).\n[82] G. Fauth, J. Berges, and A. Di Piazza, Collisional\nstrong-field QED kinetic equations from first principles,\nPhys. Rev. D 104, 036007 (2021).\n[83] A. Gonoskov, S. Bastrakov, E. Efimenko, A. Ilderton,\nM. Marklund, I. Meyerov, A. Muraviev, A. Sergeev,\nI. Surmin, and E. Wallin, Extended particle-in-cell\nschemes for physics in ultrastrong laser fields: Review\nand developments, Phys. Rev. E 92, 023305 (2015).27\n[84] B. King, N. Elkina, and H. Ruhl, Photon polarization\nin electron-seeded pair-creation cascades, Phys. Rev. A\n87, 042117 (2013).\n[85] D. Seipt and A. G. R. Thomas, Kinetic theory for\nspin-polarized relativistic plasmas, Phys. Plasmas 30,\nhttps://doi.org/10.1063/5.0165836 (2023).\n[86] Q. Zhao, T. Sun, K. Xue, F. Wan, and J.-X. Li, Cascade\nof polarized Compton scattering and Breit-Wheeler pair\nproduction, Phys. Rev. D 108, 116012 (2023).\n[87] A. Sampath and M. Tamburini, Towards realis-\ntic simulations of QED cascades: Non-ideal laser\nand electron seeding effects, Phys. Plasmas 25,\nhttps://doi.org/10.1063/1.5022640 (2018).\n[88] M. Jirka, P. Sasorov, S. S. Bulanov, G. Korn, B. Rus,\nand S. V. Bulanov, Reaching high laser intensity by a\nradiating electron, Phys. Rev. A 103, 053114 (2021a).\n[89] A. A. Mironov, A. M. Fedotov, and N. B. Narozhny,\nGeneration of quantum-electrodynamic cascades in\noblique collisions of ultrarelativistic electrons with an\nintense laser field, Quantum Electron. 46, 305 (2016).\n[90] S. S. Bulanov, V. D. Mur, N. B. Narozhny, J. Nees, and\nV. S. Popov, Multiple colliding electromagnetic pulses:\na way to lower the threshold of e+ e- pair production\nfrom vacuum, Phys. Rev. Lett. 104, 220404 (2010).\n[91] J. Derouillat, A. Beck, F. P´ erez, T. Vinci,\nM. Chiaramello, A. Grassi, M. Fl´ e, G. Bouchard,\nI. Plotnikov, N. Aunai, et al. , Smilei: A collaborative,\nopen-source, multi-purpose particle-in-cell code for\nplasma simulation, Comp. Phys. Comm. 222, 351\n(2018).\n[92] A. I. Nikishov and V. I. Ritus, Quantum processes in the\nfield of a plane electromagnetic wave and in a constant\nfield I, Sov. Phys. JETP 19, 529 (1964).\n[93] A. I. Nikishov and V. I. Ritus, Pair production by a\nphoton and photon emission by an electron in the field\nof an intense electromagnetic wave and in a constant\nfield, Sov. Phys. JETP 25, 1135 (1967).\n[94] V. N. Baier and V. M. Katkov, Processes involved in\nthe motion of high energy particles in a magnetic field,\nSov. Phys. JETP 26, 854 (1968).\n[95] E. G. Gelfer, A. M. Fedotov, A. A. Mironov, and S. We-\nber, Nonlinear Compton scattering in time-dependent\nelectric fields beyond the locally constant crossed field\napproximation, Phys. Rev. D 106, 056013 (2022).\n[96] A. Di Piazza, M. Tamburini, S. Meuren, and C. H.\nKeitel, Implementing nonlinear Compton scattering be-yond the local-constant-field approximation, Phys. Rev.\nA98, 012134 (2018).\n[97] A. Di Piazza and S. Meuren, Improved local-constant-\nfield approximation for strong-field QED codes, Phys.\nRev. A 99, 022125 (2019).\n[98] A. Ilderton, B. King, and D. Seipt, Extended locally\nconstant field approximation for nonlinear Compton\nscattering, Phys. Rev. A 99, 042121 (2019).\n[99] E. Raicher, S. Eliezer, C. H. Keitel, and K. Z. Hatsagort-\nsyan, Semiclassical limitations for photon emission in\nstrong external fields, Phys. Rev. A 99, 052513 (2019).\n[100] T. Heinzl, B. King, and A. J. MacLeod, Locally\nmonochromatic approximation to QED in intense laser\nfields, Phys. Rev. A 102, 063110 (2020).\n[101] B. King, Uniform locally constant field approximation\nfor photon-seeded pair production, Phys. Rev. A 101,\n042508 (2020).\n[102] A. Di Piazza, WKB electron wave functions in a tightly\nfocused laser beam, Phys. Rev. D 103, 076011 (2021).\n[103] C. F. Nielsen, R. Holtzapple, and B. King, High-\nresolution modeling of nonlinear Compton scattering in\nfocused laser pulses, Phys. Rev. D 106, 013010 (2022).\n[104] A. H. Taub, Orbits of charged particles in constant\nfields, Phys. Rev. 73, 786 (1948).\n[105] G. Lehmann and K. H. Spatschek, Phase-space contrac-\ntion and attractors for ultrarelativistic electrons, Phys.\nRev. E 85, 056412 (2012).\n[106] Z. Gong, R. Hu, Y. Shou, B. Qiao, C. Chen, F. Xu,\nX. He, and X. Yan, Radiation reaction induced spiral\nattractors in ultra-intense colliding laser beams, Matter\nRadiat. Extremes 1, 308 (2016).\n[107] J. G. Kirk, Radiative trapping in intense laser beams,\nPlasma Phys. Control. Fusion 58, 085005 (2016).\n[108] M. Jirka, O. Klimo, and M. Matys, Relativistic plasma\naperture for laser intensity enhancement, Phys. Rev. R\n3, 033175 (2021b).\n[109] A. Gonoskov, A. Bashinov, I. Gonoskov, C. Harvey,\nA. Ilderton, A. Kim, M. Marklund, G. Mourou, and\nA. Sergeev, Anomalous radiative trapping in laser fields\nof extreme intensity, Phys. Rev. Lett. 113, 014801\n(2014).\n[110] A. V. Bashinov, E. S. Efimenko, A. A. Muraviev, V. D.\nVolokitin, I. B. Meyerov, G. Leuchs, A. M. Sergeev, and\nA. V. Kim, Particle trajectories, gamma-ray emission,\nand anomalous radiative trapping effects in magnetic\ndipole wave, Phys. Rev. E 105, 065202 (2022)."    },    {        "title": "2402.04239v1.CAST__Clustering_Self_Attention_using_Surrogate_Tokens_for_Efficient_Transformers.pdf",        "content": "CAST: C LUSTERING SELF -ATTENTION USING SURRO -\nGATE TOKENS FOR EFFICIENT TRANSFORMERS\nAdjorn van Engelenhoven, Nicola Strisciuglio & Estefanía Talavera\nDepartment of Computer Science\nUniversity of Twente (The Netherlands)\nABSTRACT\nThe Transformer architecture has shown to be a powerful tool for a wide range\nof tasks. It is based on the self-attention mechanism, which is an inherently\ncomputationally expensive operation with quadratic computational complexity:\nmemory usage and compute time increase quadratically with the length of the\ninput sequences, thus limiting the application of Transformers. In this work, we\npropose a novel Clustering self-Attention mechanism using Surrogate Tokens\n(CAST), to optimize the attention computation and achieve efficient transformers.\nCAST utilizes learnable surrogate tokens to construct a cluster affinity matrix,\nused to cluster the input sequence and generate novel cluster summaries. The\nself-attention from within each cluster is then combined with the cluster summaries\nof other clusters, enabling information flow across the entire input sequence. CAST\nimproves efficiency by reducing the complexity from O(N2)toO(αN)where N\nis the sequence length, and αis constant according to the number of clusters and\nsamples per cluster. We show that CAST performs better than or comparable to the\nbaseline Transformers on long-range sequence modeling tasks, while also achieving\nhigher results on time and memory efficiency than other efficient transformers.\n1 I NTRODUCTION\nThe Transformer architecture (Vaswani et al., 2017) has revolutionized many fields within machine\nlearning such as translation (Vaswani et al., 2017), summarization (Miller, 2019), text generation\n(Chen et al., 2019), sentiment classification (Sun et al., 2019), and also tasks like image classification\n(Dosovitskiy et al., 2020), object detection (Liu et al., 2021b), and protein folding (Jumper et al.,\n2021). The self-attention mechanism stands at the core of its strengths. It allows the Transformer to\ndirectly model long-range dependencies within a sequence without the need for a hidden state like in\nrecurrent neural networks (Hochreiter & Schmidhuber, 1997). However, the self-attention mechanism\nhas an inherent large memory cost, since its complexity grows quadratically with the input sequence\nlength. With these memory requirements and the ever-increasing size of large language models, such\nas the GPT series (Brown et al., 2020; OpenAI, 2023) and LLaMA (Touvron et al., 2023), a need\nfor more efficient attention mechanisms has emerged (Dao et al., 2022). Current implementations of\nmore efficient self-attention mechanisms can be roughly grouped into the following categories: (1)\napply self-attention on subsets of the input sequences(sparsification) (Ainslie et al., 2020; Daras et al.,\n2020; Kitaev et al., 2020; Ma et al., 2023; Tay et al., 2020b; Zaheer et al., 2021), (2) approximate the\nself-attention mechanism with a lower complexity (Choromanski et al., 2020; Liu et al., 2021a; Wang\net al., 2020), and (3) remove self-attention in favor of a lower complexity similar operation (Gu et al.,\n2022; Lee-Thorp et al., 2021; Smith et al., 2023; Tolstikhin et al., 2021).\nIn this paper, we introduce a new efficient variant of self-attention, named Clustering Attention using\nSurrogate Tokens (CAST) – see Figure 1. It applies clustering to self-attention and introduces two\nnovel ideas, namely 1) learnable clustering of tokens, and 2) cluster summaries, which allow for\ninformation to flow between tokens from different clusters within the same attention head. CAST\nlearns to cluster tokens that would have a strong connection in the original attention matrix by\nclustering based on a similarity matrix between the surrogate tokens, queries, and keys. Standard\nself-attention is applied within clusters, where its result is combined with cluster summaries based on\na previously created similarity matrix. This allows for each token to retrieve information from the\nrest of the sequence, improving stability and performance. CAST is significantly faster than other\n1arXiv:2402.04239v1  [cs.LG]  6 Feb 2024Figure 1: Sketch of the proposed method. The colors red and blue correspond to two different\nclusters. With the queries ( Q), keys ( K), and values ( V), we create the surrogate token similarities\nAq(similarity between the queries and surrogate tokens) and Ak(similarity between the keys and\nsurrogate tokens). They are combined to create a final similarity Agfor each token to each cluster. We\nthen use this clustering of tokens and create the clustered queries ( Qg), keys ( Kg), and values ( Vg).\nWithin each cluster, self-attention is applied resulting in Rintra . Furthermore, Akis also clustered\nand matrix multiplied with Vgto create a summary per cluster resulting in Rinter . The results Rintra\nandRinter are then combined using Aqas the weights for a weighted sum, resulting in R. Another\nlinear projection Ois then applied on Rand passed on to the feedforward layer of the Transformer.\nefficient Transformer variants, and matches or improves the performance of a standard Transformer\non long-sequence modeling tasks.\nThecontribution of this paper is a novel efficient approach to replace the self-attention in Transform-\ners based on learning token clusters in an unsupervised manner.\n2 R ELATED WORKS\nPrior research on efficient computation of self-attention has focused on chunking attention, clustering\nattention, and the current state-of-the-art, structured-state-space-based models.\nChunking attention. One obvious way to solve the quadratic complexity of self-attention is to chunk\nthe given sequence into smaller pieces and apply self-attention within those pieces. This is known as\nLocal Attention (Luong et al., 2015). However, by chunking the sequence, no information can be\npassed between chunks, causing a decrease in task performance below that of the standard Transformer.\nSeveral works have sparsified the attention matrix by windowing or chunking the sequence (Ainslie\net al., 2020; Beltagy et al., 2020; Child et al., 2019; Luong et al., 2015; Zaheer et al., 2021). Some\nopted for applying attention in a sliding window manner, but the use of global attention to special\ntokens, such as the \"CLS\", is also common among the original efficient Transformers of the LRA\nbenchmark (Ainslie et al., 2020; Beltagy et al., 2020; Zaheer et al., 2021). These models also use the\n\"CLS\" token for the final classification, allowing all parts of the sequence to contribute to the final\nresult. BigBird (Zaheer et al., 2021) combined global attention, window attention, and random blocks\nof attention to achieve state-of-the-art performance on the LRA benchmark. Despite the efficiency\ngains of the chunking of self-attention, it does not necessarily model long-range dependencies\nwell. Multiple rounds of self-attention can be necessary to create a large enough receptive field to\nmodel long-range dependencies. Although chunking has shown its effectiveness, it does not model\nlong-range dependencies well, since no information can flow from distant parts of the input sequence.\nClustering attention. One way to easily model long-range dependencies is the clustering of the\ninput sequence which is only partially dependent on the order of the input. Specifically, the Reformer\n(Kitaev et al., 2020) and its descendant SMYRF (Daras et al., 2020) both use locality-sensitive\nhashing (LSH) to apply clustering to the input sequence and then apply a form of attention. The\nReformer first uses the constraint of the queries and keys being equal such that the attention matrix\nis symmetric. Then to create the clusters they define a random matrix R∈Rdh×Nc/2, which is\nthen matrix multiplied with query-key. The query keys are then clustered based on the result of\nargmax ([XqkR⊕−XqkR]). The symbol ⊕stands for concatenation, while Xqkstands for the shared\nquery-key representation. The resulting clusters are of different sizes, making it difficult to compute\nthem on conventional hardware, harming efficiency, f.i. on the Long Range Arena Benchmark.\n2SMYRF efficiently computes asymmetric clustering of queries and keys, that is a query is not\nnecessarily clustered with its corresponding key. Unlike the Reformer, SMYRF also creates balanced\nclusters of constant size and thus achieving better computational efficiency. Although both these\nclustering Transformers do model long-range dependencies, clustering also creates problems. The\nrandom initialization of the network causes queries and keys to be clustered randomly at first. As a\nresult, the weight update for queries and keys is based only on the information that is inside single\nclusters. Furthermore, gradients could be unstable when a query or key switches from one cluster to\nanother. Ideally, an efficient Transformer retains the original strength of the Transformer, namely the\ninformation flow throughout the entire input sequence via the self-attention mechanism. Our approach\nkeeps this information flow and introduces more stability through the use of cluster summaries, which\nact as an indicator of the type of information that can be obtained in a given cluster.\nStructured State Space Models. More recent work on efficient sequence models includes Structured\nState Space Models (SSSM), such as S4 (Gu et al., 2022), S5 (Smith et al., 2023), and MEGA (Ma\net al., 2023) which are currently the state-of-the-art on long-range sequence modeling benchmarks.\nSSSMs do not use the self-attention mechanism, and rely on a learnable state space to capture relevant\ndependencies in a sequence. However, we do not investigate these types of architectures further, as\nthey are no longer related to the Transformer architecture and thus the self-attention mechanism.\n3 C LUSTERING ATTENTION USING SURROGATE TOKENS\nWe propose CAST, Clustering Attention using Surrogate Tokens, that learns token clusters and\noptimizes the computation of attention in Transformer architectures. We present its versions with\nsingle-headed and multi-headed self-attention, and discuss the computational complexity. A visual-\nization of CAST is in Figure 1, and a module-based description is in Appendix A.1. Moreover, a\nnomenclature with symbol definitions and pseudocodes are in Appendices A.2 and A.3.2, respectively.\n3.1 I NTUITION\nA query ( Q) and key ( K) which are in the same direction with a large enough magnitude will end up\nwith a large score in the self-attention matrix AK. This relationship can be exploited for clustering\nby defining some static clustering directions, determining the similarity of all queries and all keys\nwith these clustering directions, and then clustering based on this similarity. However, this approach\nhas two problems: (1) when clustering, directions are randomly initialized, and their configuration\nmight not be optimal for the task that is trained on, and (2) when training is started, queries and keys\nare clustered randomly. Consequentially, the gradient of the queries and keys is only based on the\nself-attention within their cluster, making it impossible for queries and keys from different clusters to\nalign themselves according to the loss.\nTo alleviate this problem, we design CAST to ensure that the clustering directions are learnable\nand that each token receives information from all clusters. In CAST, surrogate tokens represent the\nlearnable clustering directions and are used as a surrogate for finding similar queries and keys. The\nweight of each cluster is based on the similarity of its query and the clustering direction. Within the\ncluster of a certain token, we apply self-attention. For other clusters, cluster summaries are created,\nbased on the similarity of a token key with the direction of the cluster it belongs to.\n3.2 S INGLE -HEAD CLUSTERING ATTENTION USING SURROGATE TOKENS\nCAST is an extension of the self-attention mechanism in the Transformer architecture (Vaswani et al.,\n2017). We first create query-key-value combinations from the input sequence X∈RN×d, where N\nis the input sequence length, and dthe feature embedding dimension:\nQ=XWq, K=XWk, V=XWv, ∈RN×d, (1)\nwhere Wq, Wk, Wv∈Rd×dare learnable parameters for the queries (Q), keys (K), and values (V)\nrespectively. To create clusters and lower the computational complexity, we define learnable surrogate\ntokens S∈RNc×d, where Ncindicates the number of clusters. The surrogate tokens represent the\nlearnable clustering directions and are used as a surrogate for finding similar queries and keys. Then\nwe compute the similarity matrices with the surrogate tokens for the queries ( Aq) and the keys ( Ak).\nWe combine these similarity matrices using a ratio σ(φ) : 1−σ(φ)based on a linear transformation\n3ofX, where σindicates the sigmoid function.\nAq=QST,Ak=KST∈RN×Nc\nφ=XWφ+bφ ∈RN×1\nAg=σ(φ)⊙f2(Aq) + (1 −σ(φ))⊙f2(Ak) ∈RN×Nc, (2)\nwhere σ(φ)is a sigmoid function applied to a linear transformation of X, which is represented\nasXWφ+bφ. Where Wφ∈Rd×1andbφ∈R1are learnable parameters. The function f(·)\nindicates an attention function, which in this paper includes the classical softmax and the Laplace\nfunction from MEGA (Ma et al., 2023). In the case of softmax, fi(·)indicates that the softmax\nis applied over the dimension iof the matrix. Here the softmax is applied to the dimensions\nholding corresponding to the different clusters. The symbol ⊙represents element-wise multiplication.\nSubsequently, the calculated similarities are used to cluster the input sequence using a clustering\nmechanism G, computed as G:RN×Nc,RN×∗→RNc×κ×∗, where ∗indicates any given shape,\nandκindicates the size of a cluster. Furthermore, let G−1indicate the reverse of the function G, such\nthatG−1:RN×Nc,RNc×κ×∗→RN×∗, where in the event of an input is contained in two clusters\nthe sum is calculated. Then, standard self-attention is applied within each cluster as follows:\nRintra =f \nQgKT\ng\nτ!\nVg ∈RNc×κ×d, (3)\nwhere Qg=G(Ag,Q),Kg=G(Ag,K),Vg=G(Ag,V), and τis a scalar depending on the used\nattention function. Here Rintra indicates the result of attention within the clusters.\nTo create a gradient between tokens from different clusters we apply attention between clusters as\nwell. To do this, we define value summaries Rinter , which is a weighted sum of all values within\neach cluster where the weights Ainter are based on Akandφas follows:\nAinter=G\u0012\nAg,Ak⊙ϕ(−φ)\nτk\u0013\nI′\nNc∈RNc×κ×1,\nRinter=f2(Ainter)VT\ng ∈RNc×1×d, (4)\nwhere I′\nNcindicates the expanded identity matrix INcsuch that I′\nNc∈BNc×Nc×1, the function\nϕ(x) =Softplus (x) + 1 (Zheng et al., 2015), and τkis a scaling factor. We use AQandrto create\nan attention matrix used as a weighted sum for the value summaries and attention within clusters:\nAsum=f3\u0012Aq⊙ϕ(φ)\nτq\u0013\n∈RN×Nc,\nAinter= (Asum⊙ˆM) ∈RN×Nc,\nAintra =G(Ag,Asum⊙M) ∈RN×Nc,\nR=G−1(Ag,Aintra Rintra) +Ainter Rinter ∈RN×d, (5)\nwhere M∈(0,1)N×Ncis a mask where Mi,j= 1 ifXi∈G(Ag,X)jandMi,j= 0 ifXi/∈\nG(Ag,X)j. As a result of this operation, Ris a weighted sum according to Asumof the attention\nwithin clusters (Rintra)and the summaries of the clusters (Rinter). The final output Ois then\ncalculated as O=RWo∈RN×d, where Wo∈Rd×dare learnable parameters. Ois then passed on\nto the rest of the standard Transformer architecture.\nClustering. The clustering mechanism is an integral part of CAST, and serves to group inter-\nimportant tokens of the input sequence. We measure the inter-importance as the similarity scores in\nthe attention matrix Ag. We define two clustering mechanisms to maximize the similarity per cluster:\nA)Top-K Clustering Mechanism. The Top-K clustering mechanism is a naive approach to\nclustering the input sequence, the indices of the largest Kelements in Agare taken per cluster and\nused to index the original sequence. Because Top-K simply maximizes the similarity scores per\ncluster separately, it is possible for any token to be contained in anywhere between 0andNcclusters.\nThis attribute of Top-K can be useful in case padding is used, by setting the similarity scores of\npadding to 0, it can be ensured that padding is never taken into consideration when applying attention\nwithin clusters. However, in static sequence domains, like images, it can also cause certain parts of\nthe input sequence to never be clustered.\n4Figure 2: The practical difference between the Top-K andSA Top-K clustering mechanisms. Here, S\nindicates the clustering direction of two surrogate tokens. The blue and green dashed circles indicate\nthe clusters that the Top-K and SA Top-K clustering mechanisms would create, respectively.\nB)Single Assignment Top-K Clustering Mechanism. This approach has the constraint that every\npart of the sequence can only be assigned to a single cluster, and ensure that every token is part of\nthe result of CAST and thus has a gradient. We implement the constraint by clustering tokens in\ndescending order according to their maximum score in Ag. When a cluster has reached the desired\nsize, we no longer assign tokens to this cluster.\n3.3 M ULTI -HEAD CLUSTERING ATTENTION USING SURROGATE TOKENS\nTo apply CAST in a multi-headed scenario, the surrogate tokens Sare also split into multiple heads\nsuch that S∈RNc×h×dh, where his the number of heads, and dh=d\nh. The score Agis then\ncomputed as follows:\nAq=QST,Ak=KST∈RN×h×Nc,\nφ=XWφ+bφ ∈RN×1,\nAs\nq=σ(φ)⊙f2(X\nhAq:,h, :) ∈RN×Nc,\nAs\nk= (1−σ(φ))⊙f2(X\nhAk:,h, :)∈RN×Nc,\nAg=As\nq+As\nk ∈RN×Nc. (6)\nIn short, we sum the similarity scores AqandAkover the head dimension to get the similarity of\neach token to each cluster. After this step CAST works as described in Section 3.2, but with an\nadded constant dimension h. Before the result Ois calculated, the result of the different heads Ris\nconcatenated such that R∈RN×d.\n3.4 C OMPLEXITY\nWith the use of CAST, the original quadratic complexity of the self-attention mechanism is signifi-\ncantly reduced. The complexity of CAST without added constants regarding the number of layers,\nbatch size, and hidden dimensions is O(αN). Here α=max(κ, N2\nc), where κis the number of\nelements in a cluster, and Ncthe number of clusters. Here, the complexity O(Nκ)is derived from\nthe computation of Rintra being Ncκ2, which can be rewritten as Nκ. The complexity O(NN2\nc)is\nderived from the computation of Rinter . Here, we have set the relation of κto be κ=N\nNc, although\ntechnically not necessary this would allow for each token to be clustered. Theoretically, the memory\nusage is lowest with a configuration where N2\nc=κ.\n4 E XPERIMENTAL SETUP\nWe use the Long Range Arena Benchmark (Tay et al., 2020c) (LRA) to evaluate the performance\nof CAST, and compare the results with those of other methods for efficient Transformers. The\nLRA benchmark is composed of six complex tasks on different data modalities and sequence length\n(1K-16K tokens), considered to theoretically represent various challenging tasks and complexity\nlevels. The LRA dataset and training rules were proposed with the aim of allowing researchers to\ncompare efficient Transformers without the need for large computational power. We further perform\nan ablation study on the surrogate tokens to determine how the number of clusters influences the\n5performance, peak memory usage, and the training steps efficiency. Lastly, we analyze the learned\nclusters to gain insights into why CAST works.\nThe tasks we consider serve to investigate the capability of models to deal with a diverse range of data\nmodalities and structures, such as natural language, images, and mathematics. The LRA benchmark\nis currently being used as the main benchmark for efficient Transformers and long-range sequence\nmodeling. The evaluation metric for all the tasks in LRA is classification accuracy. More details on\nthe pertinence of the LRA and its six tasks can be found in Appendix A.4.\n4.1 E XPERIMENTS\nWe carried out experiments on a variety of hardware, and expanded upon in more detail per experiment\nwhere significant. For reproducibility and fair comparison, we keep the number of layers and features\ncomparable to those used in efficient Transformers in the original LRA paper (Tay et al., 2020c).\nCAST efficiency. We evaluate the efficiency of CAST by running it on the Texttask of LRA with a\nvarying sequence length of 1K, 2K, 3K, and 4K. For each of these sequence tasks, we determine the\npeak memory usage and the number of training steps per second relative to the original Transformer\narchitecture. For comparison with other efficient Transformers we take their performance reported in\nthe original LRA paper (Tay et al., 2020c). We ensure that CAST and the Transformer use the exact\nsame hyperparameters, such as the number of layers, the number of heads, and the size of the feature\nspace. CAST uses a cluster size of 200 throughout all sequence lengths. All experiments regarding\nmemory and time efficiency were run on a single A40 GPU.\nLong Range Arena performance. We evaluate the performance of CAST on the LRA dataset\nby performing a small hyperparameter sweep. In total, we ran ten full-length training sessions per\ntask, where the checkpoint with the lowest validation loss was used to evaluate the performance of\nCAST. In Appendix A.5, a more detailed description regarding the hyperparameters can be viewed.\nFurthermore, a Weights & Biases (Biewald, 2020) report with all hyperparameters and loss curves\ncan be found here1.\nClustering ablation. We further perform an ablation study on how the number of surrogate tokens,\ni.e. the number of clusters, affects the performance, peak memory usage, and number of training\nsteps per second. We also investigate whether there is a difference in using the Top-K or Single\nAssignment Top-K clustering mechanisms in the Image task. For this ablation, we use the Text and\nImage tasks to determine whether there is a difference between modalities. For each task, we take\nthe best-performing models from the hyperparameter sweep but vary the cluster size κsuch that\nκ∈ {32,64,128,256,512}.\nVisual analysis on clusters. Lastly, we perform a visual analysis on the learned clusters in the\nImage task of the LRA dataset. From the ablations, we take a single model with two CAST layers\nand eight surrogate tokens. We then visualize which tokens are clustered together and have a more\nin-depth look at the obtained similarity scores Ag.\n5 R ESULTS AND DISCUSSION\n5.1 L ONG RANGE ARENA EFFICIENCY\nIn Table 1, we compare the speed and memory efficiency of several notable architectures. We\nobserve that CAST with Top-K is significantly faster compared to both the original Transformer and\nother efficient Transformers for all sequence lengths, with it being 6.18 times faster during training\nthan the original Transformer on a sequence length of 4K. Furthermore, CAST needs slightly less\nmemory than other efficient Transformers, only needing 10% of the memory compared to the original\nTransformer architecture at a sequence length of 4K. The use of SA Top-K lowers the speed of CAST\nsignificantly but does not affect the memory efficiency. Further results regarding the efficiency during\ninference can be found in Appendix A.6.1.\n5.2 L ONG RANGE ARENA PERFORMANCE\nTable 2 reports the performance results of CAST compared to those of the baseline Transformer and\nits efficient variations, and the current state-of-the-art models. CAST achieves performance between\nthat of the state space models and the other efficient Transformers. Although structured state space\n1Link to be publicly available upon acceptance\n6Table 1: Speed and Memory efficiency of the LRA Benchmark with the average performance (Avg.).\nThe Transformer and CAST were created using the same hyperparameters. A batch size of 25 was\nused and CAST uses a constant cluster size of 200. Speed and Memory increase/decrease are reported\nrelative to the results of the original Transformer architecture. Models annotated with the †symbol\nhad their relative speed and memory taken from the LRA benchmark (Tay et al., 2020c).\nModelSteps Per Second ↑ Peak Memory Usage ↓ Avg.\n1K 2K 3K 4K 1K 2K 3K 4K Performance\nTransformer (Vaswani et al., 2017) 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 57.71\nReformer†(Kitaev et al., 2020) 0.5 0.4 0.7 0.8 0.56 0.37 0.28 0.24 50.56\nSinkhorn Trans.†(Tay et al., 2020b) 1.1 1.6 2.9 3.8 0.55 0.31 0.21 0.16 51.23\nPerformer†(Choromanski et al., 2020) 1.2 1.9 3.8 5.7 0.44 0.22 0.15 0.11 51.18\nLuna-16 (Ma et al., 2021) 1.2 1.8 3.7 5.5 0.44 0.23 0.17 0.10 59.55\nS4 (Gu et al., 2022) - - - 4.8 - - - 0.14 86.09\nMEGA (Ma et al., 2023) - - - 2.9 - - - 0.31 88.21\nMEGA-Chunk (Ma et al., 2023) - - - 5.5 - - - 0.13 85.66\nCAST (Top-K) 1.76 3.25 4.48 6.18 0.33 0.18 0.13 0.10 59.32\nCAST (SA Top-K) 1.47 2.24 2.33 2.62 0.33 0.18 0.13 0.10 57.57\n.\nTable 2: The performance of different architectures on the Long Range Arena benchmark in classifi-\ncation accuracy. We divide these works in (A) Transformer architectures that do not use Structured\nState Spaces or any derivation of this, and (B) Architectures using Structured State Spaces. (A-Top)\nThe original Transformer architecture. (A-Middle) Efficient Transformer architectures that came out\nwith the LRA benchmark. (A-Bottom) Notable models that came out after the release of the LRA\nbenchmark. (B) Architectures using Structured State Spaces. here the symbol †indicates that the\nresults came from the original paper from the LRA dataset (Tay et al., 2020c). Furthermore, the\nsymbol ×indicates that the Transformer variant either ran out of memory and −indicates that results\nwere not reported.\nModel Year ListOps Text Retrieval Image Pathfinder Path-X Avg.\nRandom 10.00 50.00 50.00 10.00 50.00 50.00 36.67\n(A) Transformer Based Architectures\nTransformer†(Vaswani et al., 2017) 2017 36.37 64.27 57.46 42.44 71.40 × 53,66\nTransformer (re-impl (Ma et al., 2021)) 2017 37.11 65.21 79.14 42.94 71.83 × 57.71\nLocal Att.†(Tay et al., 2020c) 2017 15.82 52.98 53.39 41.46 66.63 × 46.71\nSparse Trans.†(Child et al., 2019) 2019 17.07 63.58 59.59 44.24 71.71 × 51.03\nPerformer†(Choromanski et al., 2020) 2020 18.01 65.40 53.82 42.77 77.05 × 51.18\nReformer†(Kitaev et al., 2020) 2020 37.27 56.10 53.40 38.07 68.50 × 50.56\nSinkhorn Trans.†(Tay et al., 2020b) 2020 33.67 61.20 53.83 41.23 67.45 × 51.23\nBigBird†(Zaheer et al., 2021) 2021 36.05 64.02 59.29 40.83 74.87 × 54.18\nFNet (Lee-Thorp et al., 2021) 2021 35.33 65.11 59.61 38.67 77.80 × 54,42\nLuna-16 (Ma et al., 2021) 2021 37.43 65.74 79.38 46.39 78.36 - 59.55\nCAST Top-K ( Ours ) 2023 39.90 65.45 78.01 52.37 70.18 × 59.32\nCAST SA Top-K ( Ours ) 2023 40.70 65.13 74.64 52.78 62.22 × 57.57\n(B) Structured State Space Architectures\nS4 (Gu et al., 2022) 2021 59.60 86.82 90.90 88.65 94.20 96.35 86.09\nS5 (Smith et al., 2023) 2022 62.15 89.31 91.40 88.00 95.33 98.58 87.46\nMEGA-Chunk (Ma et al., 2023) 2022 58.76 90.19 90.97 85.80 94.41 93.81 85.66\nMEGA (Ma et al., 2023) 2022 63.14 90.43 91.25 90.44 96.01 97.98 88.21\nmodels are state-of-the-art, they cannot be directly compared to other efficient Transformers since\nthey apply global convolutions and are not solely relying on attention. CAST has a relatively high\nscore for the Image task and a relatively low score for the Pathfinder task compared to that of the\nother efficient Transformers. The low score of the Pathfinder task could be explained by the fact that\nmany of the pixels in the Pathfinder image are black, which makes their query-key pairs similar and\nput in the same cluster. An extended version of Table 2 can be found in Appendix A.6.2.\n5.3 C LUSTERING ABLATION\nClustering mechanisms. Figure 3 shows the difference in performance, memory footprint, and\ntime efficiency between the clustering mechanisms Top-K and SA Top-K on the TextandImage tasks\n7(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 3: Ablations on the cluster size using CAST with Top-K Clustering Mechanism (blue) and\nSingle Assignment Top-K Clustering Mechanism (orange) on the Text andImage tasks of the LRA\nbenchmark against (a & d) the performance, (b & e) the peak memory allocated, and (c & f) the time\nefficiency, respectively.\nof LRA. In Figure 3d, it can be seen that the choice in the clustering mechanism slightly affects\nthe resulting performance on the Image task at a cluster size of 128 and 256. It can be observed\nfrom Figure 3b and Figure 3e, that the cluster mechanism does not affect the Peak Memory Usage.\nFurthermore, it can be seen from Figure 3c and Figure 3f that the Top-K clustering mechanism\nis overall significantly faster than the SA Top-K clustering mechanism. The SA Top-K clustering\nmechanism in particular is much slower when using small cluster sizes on large input sequences, like\nfor the Text task.\nPerformance. In Figure 3a and Figure 3d, we show a comparison of the effect of cluster sizes and\ncluster mechanism on the performance of CAST on Text andImage task of the LRA dataset. For\ntheText task, the cluster size does not significantly impact the resulting accuracy, although a slight\nincrease in accuracy can be observed at a larger cluster size. However, cluster size does impact the\nperformance of the Image task significantly for both Top-K and SA Top-K . It can be observed that the\nperformance on the Image task dips around a cluster size of 64 to 128, but peaks at a cluster size of\n32 and 256.\nPeak memory usage. In Figure 3b and Figure 3e, we show measurements of the influence of the\ncluster sizes on the peak memory usage of the Image andText task. At its lowest, CAST only uses\naround 1.35 gigabytes of memory for the Image and 6.3 gigabytes of memory for the Text task. The\nmemory curves represent a quadratic relationship, with an increase in memory when the number of\nclusters becomes too large, which was expected from Section 3.4. For both tasks, it can be seen that\nthe least amount of memory is used when the number of clusters and the cluster size is close to the\nrelation N2\nc=κ. However, knowing that CAST achieves similar performance across different cluster\nsizes, we can use the cluster size that minimizes the memory footprint without a large decrease in\nperformance.\nTime efficiency. In Figure 3c and Figure 3f, we report measurements of the influence of the cluster\nsize and clustering mechanism on the training steps per second of the Text andImage task. It can be\nobserved that the number of training steps per second for the SA Top-K clustering mechanism (orange)\nis significantly lower than that of the standard Top-K clustering mechanism, especially at smaller\ncluster sizes. This is due to the constraint of SA Top-K must ensure every token is contained only in\none cluster. However, knowing that the change of performance between clustering mechanisms and\n8(a) Image of the LRA Image task.\n (b) Visualizations of learned clusters of CAST per layer.\nFigure 4: Visualizations of the learned clusters of a CAST model with SA Top-K on the LRA Image\ntask. The number of clusters is 8. (a) An example image. (b- Left) Clustered pixels, where each color\nrepresents a different cluster. Example scores for clusters of Ag(b- Middle & Right), each image\ncorresponds to a different cluster for the first (b-Top) and last layer (b-Bottom), respectively.\ncluster sizes is small, the Top-K clustering mechanism can be chosen at a cluster size that maximizes\nthe number of steps per second.\n5.4 V ISUAL ANALYSIS ON CLUSTERS\nThe clusters created by CAST do seem to hold visuospatial information on image tasks. More\nspecifically, CAST seems to separate background from foreground in images. In Figure 6a, we show\nan example of an input image from the Image task which depicts a horse and its rider. In Figure\n4b, the clustered pixels of the two layers of CAST are depicted, where each color corresponds to\none of the clusters. In the first layer, we can observe that the clusters are approximately slices of the\noriginal image. In the last layer, it can be observed that the background and foreground of the image\nare roughly separated in different clusters. This behavior is observed for most of the images in the\nImage task – see Appendix A.6.3 for more examples. We further analyze the clusters by visualizing\nthe scores of Agin Figure 4b, where the separation of foreground and background is more evident,\ntogether with the separation per slice of the image.\n5.5 L IMITATIONS\nCAST is significantly faster than other methods based on efficient transformers. In terms of benchmark\nresults, however, efficient transformers including CAST perform lower than Structured State Space\nmodels. While a direct comparison is unfair, as the architecture and working principle of these\nmethods are different from transformers, with higher complexity, we report their results for the sake\nof completeness. A current limitation of CAST is the absence of a decoding version for generative\nnatural language. While the focus of the paper is however on the optimization of the attention\ncomputation via a novel clustering of surrogate tokens approaches, we foresee that CAST could be\nadapted using asymmetric clustering and casual masking to create a decoder and be deployed in\ngenerative models as well.\n6 C ONCLUSIONS\nWe present CAST, a more efficient drop-in replacement for self-attention, which lowers the complexity\nof computing the self-attention in Transformers. The solution that we propose is based on clustering\nsurrogate tokens, a novel approach in which the cluster directions are learnable, in contrast with static,\nalgorithmically defined cluster directions of previous works. While our contribution is potentially\ngeneral, and applicable to many tasks, we focus on analyzing its impact towards improving efficient\ncomputation of self-attention in transformers, especially at inference time, while maintaining the\nhigh results of standard transformer architectures. Our experiments demonstrate that the memory\nand computational efficiency of CAST is significantly better than other efficient Transformers (e.g.\nReformer (Kitaev et al., 2020), Performer (Choromanski et al., 2020)) in the Long Range Arena\nefficiency benchmark. CAST uses less memory than existing methods for all tested sequence lengths,\nbeing up to about 6 ×faster than and using 10% of the memory of the original Transformer. Our\nfuture work will explore increasing the efficiency of CAST by parallelizing attention within clusters,\nand finding more efficient clustering mechanisms.\n9REFERENCES\nJoshua Ainslie, Santiago Ontanon, Chris Alberti, Vaclav Cvicek, Zachary Fisher, Philip Pham,\nAnirudh Ravula, Sumit Sanghai, Qifan Wang, and Li Yang. Etc: Encoding long and structured\ninputs in transformers, 2020.\nIz Beltagy, Matthew E. Peters, and Arman Cohan. Longformer: The long-document transformer,\n2020.\nLukas Biewald. Experiment tracking with weights and biases, 2020. URL https://www.wandb.\ncom/ . Software available from wandb.com.\nTom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal,\nArvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel\nHerbert-V oss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler,\nJeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott\nGray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya\nSutskever, and Dario Amodei. Language models are few-shot learners, 2020.\nYen-Chun Chen, Zhe Gan, Yu Cheng, Jingzhou Liu, and Jingjing Liu. Distilling knowledge learned\nin bert for text generation, 2019. URL https://arxiv.org/abs/1911.03829 .\nLei Cheng, Ruslan Khalitov, Tong Yu, and Zhirong Yang. Classification of long sequential data using\ncircular dilated convolutional neural networks, 2022.\nRewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse\ntransformers, 2019. URL https://arxiv.org/abs/1904.10509 .\nKrzysztof Choromanski, Valerii Likhosherstov, David Dohan, Xingyou Song, Andreea Gane, Tamas\nSarlos, Peter Hawkins, Jared Davis, Afroz Mohiuddin, Lukasz Kaiser, David Belanger, Lucy\nColwell, and Adrian Weller. Rethinking attention with performers, 2020. URL https://\narxiv.org/abs/2009.14794 .\nTri Dao, Daniel Y . Fu, Stefano Ermon, Atri Rudra, and Christopher Ré. Flashattention: Fast and\nmemory-efficient exact attention with io-awareness, 2022.\nTri Dao, Daniel Y . Fu, Khaled K. Saab, Armin W. Thomas, Atri Rudra, and Christopher Ré. Hungry\nhungry hippos: Towards language modeling with state space models, 2023.\nGiannis Daras, Nikita Kitaev, Augustus Odena, and Alexandros G. Dimakis. Smyrf: Efficient\nattention using asymmetric clustering, 2020.\nAlexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas\nUnterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit,\nand Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale,\n2020. URL https://arxiv.org/abs/2010.11929 .\nAlbert Gu, Karan Goel, and Christopher Ré. Efficiently modeling long sequences with structured\nstate spaces, 2022.\nRamin Hasani, Mathias Lechner, Tsun-Hsuan Wang, Makram Chahine, Alexander Amini, and\nDaniela Rus. Liquid structural state-space models, 2022.\nSepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation , 9(8):\n1735–1780, 1997.\nJohn M. Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ron-\nneberger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Zídek, Anna Potapenko, Alex Bridg-\nland, Clemens Meyer, Simon A A Kohl, Andy Ballard, Andrew Cowie, Bernardino Romera-\nParedes, Stanislav Nikolov, Rishub Jain, Jonas Adler, Trevor Back, Stig Petersen, David A.\nReiman, Ellen Clancy, Michal Zielinski, Martin Steinegger, Michalina Pacholska, Tamas Bergham-\nmer, Sebastian Bodenstein, David Silver, Oriol Vinyals, Andrew W. Senior, Koray Kavukcuoglu,\nPushmeet Kohli, and Demis Hassabis. Highly accurate protein structure prediction with alphafold.\nNature , 596:583 – 589, 2021.\n10Ruslan Khalitov, Tong Yu, Lei Cheng, and Zhirong Yang. Sparse factorization of large square\nmatrices, 2021.\nRuslan Khalitov, Tong Yu, Lei Cheng, and Zhirong Yang. Chordmixer: A scalable neural attention\nmodel for sequences with different lengths, 2023.\nNikita Kitaev, Łukasz Kaiser, and Anselm Levskaya. Reformer: The efficient transformer, 2020.\nURLhttps://arxiv.org/abs/2001.04451 .\nAlex Krizhevsky. Learning multiple layers of features from tiny images. pp. 32–33, 2009. URL\nhttps://www.cs.toronto.edu/~kriz/learning-features-2009-TR.pdf .\nJames Lee-Thorp, Joshua Ainslie, Ilya Eckstein, and Santiago Ontanon. Fnet: Mixing tokens with\nfourier transforms. arXiv preprint arXiv:2105.03824 , 2021.\nYuhong Li, Tianle Cai, Yi Zhang, Deming Chen, and Debadeepta Dey. What makes convolutional\nmodels great on long sequence modeling?, 2022.\nDrew Linsley, Junkyung Kim, Vijay Veerabadran, and Thomas Serre. Learning long-range spatial\ndependencies with horizontal gated-recurrent units. CoRR , abs/1805.08315, 2018. URL http:\n//arxiv.org/abs/1805.08315 .\nLiu Liu, Zheng Qu, Zhaodong Chen, Yufei Ding, and Yuan Xie. Transformer acceleration with\ndynamic sparse attention. CoRR , abs/2110.11299, 2021a. URL https://arxiv.org/abs/\n2110.11299 .\nZe Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining\nGuo. Swin transformer: Hierarchical vision transformer using shifted windows, 2021b. URL\nhttps://arxiv.org/abs/2103.14030 .\nMinh-Thang Luong, Hieu Pham, and Christopher D. Manning. Effective approaches to attention-\nbased neural machine translation, 2015. URL https://arxiv.org/abs/1508.04025 .\nXuezhe Ma, Xiang Kong, Sinong Wang, Chunting Zhou, Jonathan May, Hao Ma, and Luke Zettle-\nmoyer. Luna: Linear unified nested attention, 2021. URL https://arxiv.org/abs/2106.\n01540 .\nXuezhe Ma, Chunting Zhou, Xiang Kong, Junxian He, Liangke Gui, Graham Neubig, Jonathan May,\nand Luke Zettlemoyer. Mega: Moving average equipped gated attention, 2023.\nAndrew L. Maas, Raymond E. Daly, Peter T. Pham, Dan Huang, Andrew Y . Ng, and Christopher Potts.\nLearning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the\nAssociation for Computational Linguistics: Human Language Technologies - Volume 1 , HLT ’11,\npp. 142–150, Stroudsburg, PA, USA, 2011. Association for Computational Linguistics. ISBN 978-\n1-932432-87-9. URL http://dl.acm.org/citation.cfm?id=2002472.2002491 .\nDerek Miller. Leveraging bert for extractive text summarization on lectures, 2019. URL https:\n//arxiv.org/abs/1906.04165 .\nOpenAI. Gpt-4 technical report, 2023.\nDragomir R. Radev, Pradeep Muthukrishnan, Vahed Qazvinian, and Amjad Abu-Jbara. The acl an-\nthology network corpus. Lang. Resour. Eval. , 47(4):919–944, dec 2013. ISSN 1574-020X. doi: 10.\n1007/s10579-012-9211-2. URL https://doi.org/10.1007/s10579-012-9211-2 .\nJimmy T. H. Smith, Andrew Warrington, and Scott W. Linderman. Simplified state space layers for\nsequence modeling, 2023.\nChi Sun, Luyao Huang, and Xipeng Qiu. Utilizing bert for aspect-based sentiment analysis via\nconstructing auxiliary sentence, 2019. URL https://arxiv.org/abs/1903.09588 .\nYi Tay, Dara Bahri, Donald Metzler, Da-Cheng Juan, Zhe Zhao, and Che Zheng. Synthesizer:\nRethinking self-attention in transformer models, 2020a. URL https://arxiv.org/abs/\n2005.00743 .\n11Yi Tay, Dara Bahri, Liu Yang, Donald Metzler, and Da-Cheng Juan. Sparse sinkhorn attention, 2020b.\nURLhttps://arxiv.org/abs/2002.11296 .\nYi Tay, Mostafa Dehghani, Samira Abnar, Yikang Shen, Dara Bahri, Philip Pham, Jinfeng Rao,\nLiu Yang, Sebastian Ruder, and Donald Metzler. Long range arena: A benchmark for efficient\ntransformers, 2020c.\nIlya Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Un-\nterthiner, Jessica Yung, Andreas Steiner, Daniel Keysers, Jakob Uszkoreit, Mario Lucic, and\nAlexey Dosovitskiy. Mlp-mixer: An all-mlp architecture for vision, 2021.\nHugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet, Marie-Anne Lachaux, Timothée\nLacroix, Baptiste Rozière, Naman Goyal, Eric Hambro, Faisal Azhar, Aurelien Rodriguez, Armand\nJoulin, Edouard Grave, and Guillaume Lample. Llama: Open and efficient foundation language\nmodels, 2023.\nAshish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz\nKaiser, and Illia Polosukhin. Attention is all you need, 2017.\nSinong Wang, Belinda Z. Li, Madian Khabsa, Han Fang, and Hao Ma. Linformer: Self-attention\nwith linear complexity, 2020.\nTong Yu, Ruslan Khalitov, Lei Cheng, and Zhirong Yang. Paramixer: Parameterizing mixing links in\nsparse factors works better than dot-product self-attention, 2022.\nManzil Zaheer, Guru Guruganesh, Avinava Dubey, Joshua Ainslie, Chris Alberti, Santiago Ontanon,\nPhilip Pham, Anirudh Ravula, Qifan Wang, Li Yang, and Amr Ahmed. Big bird: Transformers for\nlonger sequences, 2021.\nHao Zheng, Zhanlei Yang, Wenju Liu, Jizhong Liang, and Yanpeng Li. Improving deep neural\nnetworks using softplus units. In 2015 International Joint Conference on Neural Networks\n(IJCNN) , pp. 1–4, 2015. doi: 10.1109/IJCNN.2015.7280459.\nA A PPENDIX\nA.1 M ODULARIZED VISUALISATION OF CAST\nIn Figure 5, a modularized verison of the visualization of CAST can be seen. In Figure 5a, we\napply Intra Cluster Attention by clustering the input Xbased on the values in Ag, and then apply\nself-attention within the created clusters. In Figure 5b, we create the cluster summaries Rinter ,\nwhich is based on a sum values weighted based on the the key-surrogate token similarity matrix Ak.\nIn Figure 5c, we lastly combine the results Rintra andRinter based on the query-surrogate token\nsimilarity matrix Aq, resulting in the output of CAST.\nA.2 N OMENCLATURE\nWe include a nomenclature in Table 3 to aid in the reading of Section 3.2. Here the symbols are given\ntogether with what they are representing.\nTable 3: A nomenclature of symbols used in the equations defining CAST.\nSymbol Meaning/Representation\nAq The dot product similarity between the queries ( Q) and the surrogate tokens ( S).\nAk The dot product similarity between the keys(K) and the surrogate tokens ( S).\nϕ A learned value similar to the queries/keys, represents whether it is more important to share or receive information.\nAgThe combined similarity of AqandAk, where Aqhas more weight if phi is high,\nandAkhas more weight when phi is low. Used as the basis for clustering.\nXg,Qg,Kg,VgThe clustered tokens, queries, keys, and values.\nRintra The result of self-attention within each cluster.\nAvalue The weights for the weighted sum of the cluster summaries.\nRinter The cluster summaries.\n12(a) Intra Cluster Attention\n (b) Creation of the cluster summaries Rinter , which is a\nweighted sum of each cluster’s values based on the weights in\nAk.\n(c) The combining of Rinter andRintra us-\ningAqas the weights for the weighted sum.\nFigure 5: A modularized sketch of the proposed method. Here, some details are omitted to make it\neasier to read. (a) shows intra-cluster self-attention, (b) shows the creation of the cluster summaries\nRinter , and (c) shows how Rinter andRintra are combined.\nA.3 D ETAILS OF THE CLUSTERING MECHANISMS\nIn this section, we describe our proposed Top-K clustering mechanism and the SA Top-K clustering\nmechanism.\nA.3.1 T OP-KCLUSTERING\nTheTop-K clustering mechanism groups the indices with the largest similarity scores in Ag, it allows\nfor a token to be clustered into two clusters, but also for a token not to be clustered at all. A formal\ndefinition of the Top-K clustering mechanism is in Algorithm 1, where A∈RN×Ncis the similarity\nscores for each token to each cluster, and X∈RN×∗is a matrix of feature vectors that we wish to\ncluster, where ∗indicates any shape.\nAlgorithm 1 Implementation of the proposed Top-K clustering mechanism.\nInput X, A\nOutput C\n1:function SATop-K (A, X )\n2: C={C1...CNc} ▷Initialize result\n3: I, Atop=Top-K (A) ▷Get the indices of the largest values per cluster\n4: fori←1toNcdo\n5: forj←1toN\nNcdo\n6: itoken =Ii,j\n7: Ci.insert (Xitoken)\n8: end for\n9: end for\n10:end function\nA.3.2 S INGLE ASSIGNMENT TOP-KCLUSTERING\nThe single assignment Top-K clustering mechanism has the constraint that each token is assigned to\nonly a single cluster, while also maximizing the total similarity for all clusters combined. The SA\nTop-K clustering mechanism is formally defined in Algorithm 2, where A∈RN×Ncrepresents the\n13similarity scores for each token to each cluster, and X∈RN×∗represents a matrix of feature vectors\nthat we wish to cluster.\nAlgorithm 2 Implementation of the proposed Single Assignment Top-K clustering mechanism.\nInput X, A\nOutput C\n1:function SATop-K (A, X )\n2: Ac, Ic=sort 2(A) ▷Sort from highest to lowest cluster\n3: Ar, Ir=sort 1(Ac) ▷Sort from highest to lowest token\n4: C={C1...CNc} ▷Initialize result\n5: M=0N▷Initialize Assignment Mask\n6: fori←1toNcdo\n7: forj←1toNdo\n8: jtoken =Ir\nj\n9: icluster =Ic\njtoken\n10: ifMj= 1orlength (Cicluster ) =N\nNcthen\n11: continue for loop\n12: end if\n13: Cicluster .insert (Xjtoken)\n14: Mjtoken= 1\n15: end for\n16: end for\n17: return C\n18:end function\nA.4 L ONG RANGE ARENA BENCHMARK\nThe Long Range Arena (LRA) benchmark contains six tasks [ListOps, Text, Retrieval, Image, Path,\nand Path-X] that represent a diverse and intricate spectrum of challenges. Each task demands a distinct\nset of skills, ranging from semantic understanding and reasoning to image comprehension and logical\nmanipulation. This diverse selection of tasks aims to assess the capabilities of architectures, ensuring\na comprehensive evaluation that goes beyond singular skill acquisition. Furthermore, the LRA\nbenchmark holds a unique position within the research community as it is the common benchmark\nfor comparing efficiency and performance, such as for the architectures in Luna (Ma et al., 2021),\nMEGA (Ma et al., 2023), and S4 (Gu et al., 2022). This widespread adoption signifies a consensus\namong researchers regarding its suitability for assessing the performance of experimental frameworks,\nincluding our proposed CAST. Next, we describe how these six tasks are treated.\nListOps. The ListOps dataset was created for testing the parsing ability of latent tree models, but a\nlarger version is now used in the LRA to test the capability of Transformers to learn the hierarchical\nstructures. The data is a sequence of tokens representing a large mathematical operation on lists of\nnumbers. The numbers 0to9are available as both the input of the operations and the final result.\nThere are four base mathematical operations :\n• MAX: The largest value in a given list.\n• MIN: The smallest value in a given list.\n• MED: The median value in a given list.\n• SUM MOD: The sum of the list module 10.\nIn the LRA the maximal length of the input sequence is set to 2K tokens. This is a ten-way\nclassification task where accuracy is used as the evaluation metric.\nText. TheText task takes the IMDb reviews sentiment classification task (Maas et al., 2011) and the\ncharacters as tokens in the input sequence. The maximum length of the input sequences is truncated\nor padded to 4K tokens. This task is a binary classification task with accuracy as its metric.\nRetrieval. For the Retrieval task the ACL Anthology Network dataset (Radev et al., 2013) is\nused. For this dataset, the task is to determine whether two papers are linked by a citation. Both\npapers are passed to the Transformer variant, creating compressed representations, which are then\n14combined and passed into a classification head. With this setup the Retrieval task can be considered\na binary classification task. To make the task more challenging, character-level tokens like in the text\nclassification task are used in the setup. A sequence length of 4K tokens is used per document the use\nof 8K tokens per example.\nImage. TheImage task takes the CIFAR-10 dataset (Krizhevsky, 2009) as its base. The images\nare first greyscaled into a single channel with each pixel having an 8-bit pixel intensity as its\nrepresentation. This results in a 32×32image which is unrolled into a 1-D sequence, this sequence\nis then used as input for a ten-way classification task.\nPathfinder. ThePathfinder task (Linsley et al., 2018) consists of images of 32×32where two\ndots, represented by circles, are connected by dashed lines. A model is required to make a binary\ndecision of whether the two dots are connected by the dashed lines, however, there are also distraction\nlines that are not connected to any of the dots. Just like in the image classification task the image is\nunrolled into a sequence length of 1024 and used as input for this task.\nPath-X. ThePath-X task is a more extreme case of the original Pathfinder, instead of the image\nbeing 32×32it is128×128making the sequence length 16 times larger than the original. Apart\nfrom the size this task is exactly the same as the original Pathfinder. It should be noted that this task\nhas not yet been achieved with a higher-than-random accuracy with the constraints of the LRA.\nA.5 E XPERIMENT DETAILS\nFor all tasks, we follow the standards given in the original Long Range Arena paper (Tay et al., 2020c)\nregarding the data processing and task setup. For our choices in most hyperparameters, we used the\ncurrent state-of-the-art, MEGA (Ma et al., 2023), as our baseline regarding the number of weight\nupdates. Furthermore, we use their data splits regarding all tasks. The final hyperparameters used\nfor our reported accuracy are in Table 4. For both the reported performance of Top-K and SA Top-K\nthe same hyperparameters are used. General hyperparameters include the averaging of the output\nfeatures over the sequence for the classification features, the use of linear feature embeddings for\npixel tasks, the use of sinusoidal positional embeddings for all tasks, and an extra normalization layer\non the output features when pre-normalization is used.\nTable 4: Final hyperparameters for the best performing CAST models in our hyperparameter sweep.\nHere, Depth indicates the number of Transformer blocks, hthe number of heads, dthe number of\nfeatures in the self-attention block, dff, the number of features in the feedforward block, dembthe\nnumber of features in the embedding, Ncthe number of clusters, Norm the type of normalization\nbeing used, BS the batch-size, LR the learning rate, WD the weight decay, and Epochs the number of\nepochs that were trained for.\nTask Depth h d d ffdemb NcNorm Pre-norm BS LR WD Epochs\nListOps 4 8 64 128 256 10 Layer False 64 1e−31e−260\nText 4 4 64 128 256 20 Scale False 25 1e−31e−225\nRetrieval 2 8 256 256 256 20 Layer False 8 1e−21e−25\nImage 2 2 128 128 256 16 Batch True 50 5e−31e−2200\nPathfinder 2 2 32 32 64 16 Batch True 128 1e−31e−2200\nA.6 D ETAILED RESULTS\nIn this section, we go into more depth regarding the results of CAST. We first give an extensive\noverview of other long-range sequence modeling architectures, and then show more examples of the\nvisual analysis that was done on the Image task.\nA.6.1 L ONG RANGE ARENA EFFICIENCY\nWe further compare the relative efficiency of CAST Top-K with the vanilla Transformer during\ninference time in Table 5. Here, we observe that during inference CAST is still significantly faster\nand more memory efficienct than the vanilla Transformer.\nA.6.2 L ONG RANGE ARENA PERFORMANCE\nIn Table 6, we report a detailed list of results of different efficient Transformer variants, and other\nlong-range sequence models on the Long Range Arena benchmark. We divide these models into\n15Table 5: Speed and Memory efficiency of the LRA Benchmark during inference.\nModelSteps Per Second ↑ Peak Memory Usage ↓\n1K 2K 3K 4K 1K 2K 3K 4K\nTransformer 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0\nCAST (Top-K) 1.87 3.95 5.27 6.91 0.280 0.150 0.102 0.081\nthe following categories: Transformer Based Architectures, Structured State Space Architectures,\nand Other Architectures. We can see that among the Transformer Based Architectures (A) Luna\n(Ma et al., 2021) and CAST similarly strong performance. When it comes to the Structured State\nSpace Architectures (B), it can be observed that all models perform similarly, with MEGA (Ma et al.,\n2023) being slightly better than the rest. As for the other types of architectures, they neither use\nself-attention nor structured state spaces to \"mix\" their input sequence. Among them, the recent\nChordMixer (Khalitov et al., 2023) stands out, ChordMixer was created for handling data with\nextremely long sequence lengths (in the order of 100K tokens), but has shown impressive results on\nthe LRA benchmark too.\nTable 6: The performance of different architectures on the Long Range Arena benchmark in classifi-\ncation accuracy. We divide these works into (A) Transformer architectures that do not use Structured\nState Spaces or any derivation of this, (B) Architectures using Structured State Spaces, and (C) Other\ntypes of architectures. The (A)-related models are grouped as; (A-Top) The original Transformer\narchitecture, (A-Middle) efficient Transformer architectures that came out with the LRA benchmark,\nand (A-Bottom) notable models that came out after the release of the LRA benchmark. Here the\nsymbol †indicates that the results came from the original paper from the LRA dataset (Tay et al.,\n2020c). Furthermore, the symbol ×indicates that the Transformer variant either ran out of memory\nand−indicates that results were not reported.\nModel Year ListOps Text Retrieval Image Pathfinder Path-X Avg.\nRandom 10.00 50.00 50.00 10.00 50.00 50.00 36.67\n(A) Transformer Based Architectures\nTransformer†(Vaswani et al., 2017) 2017 36.37 64.27 57.46 42.44 71.40 × 53.66\nTransformer (re-impl (Ma et al., 2021)) 2017 37.11 65.21 79.14 42.94 71.83 × 57.71\nSparse Trans.†(Child et al., 2019) 2019 17.07 63.58 59.59 44.24 71.71 × 51.03\nLocal Att.†(Tay et al., 2020c) 2020 15.82 52.98 53.39 41.46 66.63 × 46,71\nReformer†(Kitaev et al., 2020) 2020 37.27 56.10 53.40 38.07 68.50 × 50.56\nSinkhorn Trans.†(Tay et al., 2020b) 2020 33.67 61.20 53.83 41.23 67.45 × 51.23\nPerformer†(Choromanski et al., 2020) 2020 18.01 65.40 53.82 42.77 77.05 × 51.18\nLinformer†(Wang et al., 2020) 2020 35.70 53.94 52.27 38.56 76.34 × 51.36\nLongformer†(Beltagy et al., 2020) 2020 35.63 62.85 56.89 42.22 69.71 × 53.46\nSynthesizer†(Tay et al., 2020a) 2021 36.99 61.68 54.67 41.61 69.45 × 52.40\nBigBird†(Zaheer et al., 2021) 2021 36.05 64.02 59.29 40.83 74.87 × 54.18\nLuna-16 (Ma et al., 2021) 2021 37.43 65.74 79.38 46.39 78.36 - 59.55\nLuna-128 (Ma et al., 2021) 2021 38.01 65.74 79.55 47.47 78.89 - 59,94\nLuna-256 (Ma et al., 2021) 2021 37.98 65.78 79.56 47.86 78.55 - 59,96\nPSF (Khalitov et al., 2021) 2021 38.85 77.32 - 45.01 80.49 - 56.95\nFNet (Lee-Thorp et al., 2021) 2021 35.33 65.11 59.61 38.67 77.80 × 54.42\nCAST Top-K ( Ours ) 2023 39.90 65.45 78.01 52.37 70.18 × 59.32\nCAST SA Top-K ( Ours ) 2023 40.70 65.13 74.64 52.78 62.22 × 57.57\n(B) Structured State Space Architectures\nS4 (Gu et al., 2022) 2021 59.60 86.82 90.90 88.65 94.20 96.35 86.09\nH3 (Dao et al., 2023) 2022 57.50 88.20 91.00 87.30 93.00 91.80 84.80\nLiquid-S4 (Hasani et al., 2022) 2022 62.75 89.02 91.20 89.50 94.8 96.66 87.32\nSGConv (Li et al., 2022) 2022 61.45 89.20 91.11 87.97 95.46 97.83 87.17\nS5 (Smith et al., 2023) 2022 62.15 89.31 91.40 88.00 95.33 98.58 87.46\nMEGA-Chunk (Ma et al., 2023) 2022 58.76 90.19 90.97 85.80 94.41 93.81 85.66\nMEGA (Ma et al., 2023) 2022 63.14 90.43 91.25 90.44 96.01 97.98 88.21\n(C) Other Architectures\nParamixer (Yu et al., 2022) 2022 39.57 83.32 - 46.58 80.49 - 58.33\nCDIL (Cheng et al., 2022) 2022 - 87.61 84.27 64.49 91.00 - 64.56\nChordMixer (Khalitov et al., 2023) 2023 60.12 88.82 89.98 90.17 96.69 98.63 87.40\n16(a) Example Image of\nthe LRA Image task.\n(b) Clustered Image,\n8 hashes, first layer.\n(c) Clustered Image,\n8 hashes, last layer.\nFigure 6: A visualization of learned clusters when using LSHAttention. Here 6a shows the original\ninput image, 6b shows the way pixels are clustered in the first layer of LSHAttention, and 6c shows\nthe way pixels are clustered in the last layer of LSHAttention\nA.6.3 F URTHER VISUAL ANALYSIS\nAdditional visualizations of the clusters created for different samples from the Image task of LRA,\ncan be seen in Figure 7 (a horse), Figure 8 (a deer), and Figure 9 (an automobile). For each of\nthese figures, subfigure (a) shows the original input image, (b) shows the assignment of clusters for\neach pixel in the first layer, and (c) shows the assignment of clusters for each pixel in the last layer.\nSubfigure (d) shows for each cluster the score in Agthat each token had for the first layer. Subfigure\n(e) shows for each cluster the score in Agthat each pixel had for the last layer.\nFor the mentioned sample images, it can be seen that the in the first layer, i.e. in Figures 7d, 8d,\nand Figure 9d, each cluster roughly clusters the same pixels. This behavior could occur, because the\npositional embeddings are most prominent in the first layer, causing the surrogate tokens to cluster\nbased on this positional embedding. Furthermore, it also shows that CAST learns to cluster slices\nof the image first, similar to convolution. In Figures 7e, 8e and, Figure 9e, the scores in Agfor the\nlast layer can be seen. This layer (e) shows more image-specific clustering. For instance, from these\nscores, we can observe the outline and inverse outline of a horse, a deer in a forest, and an automobile,\nrespectively. We interpret this as the separation of background and foreground. In the case of the\ndeer, Figure 8e, we observe a more rough outline, which can be due to the fact that the background\nand foreground of this image are much more similar.\nA.6.4 R EFORMER VISUAL ANALYSIS\nTo determine whether the visuospatial information contained in the clustering of CAST is general\nfeature of clustering Transformers like the Reformer and SMYRF, we trained a Reformer model\non the CIFAR-10 dataset and inspected the clustered images. In Figure 6, an example image for\nclustering created by the Reformer’s LSHAttention can be seen.\n17(a) Example Image of the LRA\nImage task.\n(b) Clustered Image, Nc= 8 ,\nFirst Layer.\n(c) Clustered Image, Nc= 8 ,\nLast Layer.\n(d) Scores of Agper cluster for the first layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\n(e) Scores of Agper cluster for the last layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\nFigure 7: A visualization learned clusters in different layers of CAST. Here, (a) is a sample from the\nImage of LRA, depicting a horse with a rider, (b) is an image representing the clustered pixels in the\nfirst layer of CAST, (c) is an image representing the clustered pixels in the last layer of CAST, (d) the\nscores in Agfor every pixel in each of the eight clusters in the first layer of CAST, (e) the scores in\nAgfor every pixel in each of the eight clusters in the last layer of CAST.\n18(a) Example Image of the LRA\nImage task.\n(b) Clustered Image, Nc= 8 ,\nFirst Layer.\n(c) Clustered Image, Nc= 8 ,\nLast Layer.\n(d) Scores of Agper cluster for the first layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\n(e) Scores of Agper cluster for the last layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\nFigure 8: A visualization of the learned clusters in different layers of CAST. Here, (a) is a sample\nfrom the Image of LRA, depicting a deer in a forest, (b) is an image representing the clustered pixels\nin the first layer of CAST, (c) is an image representing the clustered pixels in the last layer of CAST,\n(d) the scores in Agfor every pixel in each of the eight clusters in the first layer of CAST, (e) the\nscores in Agfor every pixel in each of the eight clusters in the last layer of CAST.\n19(a) Example Image of the LRA\nImage task.\n(b) Clustered Image, Nc= 8 ,\nFirst Layer.\n(c) Clustered Image, Nc= 8 ,\nLast Layer.\n(d) Scores of Agper cluster for the first layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\n(e) Scores of Agper cluster for the last layer of a CAST model trained on the Image task of LRA. Here, each\nimage corresponds to a cluster, i.e. a single column of Ag.\nFigure 9: A visualization learned clusters in different layers of CAST. Here, (a) is a sample from the\nImage of LRA, depicting an automobile, (b) is an image representing the clustered pixels in the first\nlayer of CAST, (c) is an image representing the clustered pixels in the last layer of CAST, (d) the\nscores in Agfor every pixel in each of the eight clusters in the first layer of CAST, (e) the scores in\nAgfor every pixel in each of the eight clusters in the last layer of CAST.\n20"    },    {        "title": "2402.04240v1.Novel_IMU_based_Adaptive_Estimator_of_the_Center_of_Rotation_of_Joints_for_Movement_Analysis.pdf",        "content": "arXiv:2402.04240v1  [eess.SP]  6 Feb 2024JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 1\nNovel IMU-based Adaptive Estimator of the Center\nof Rotation of Joints for Movement Analysis\nSara Garc´ ıa-de-Villa, Ana Jim´ enez-Mart´ ın and J. Jes´ us Garc´ ıa-Dom´ ınguez\nAbstract —The location of the center of rotation (COR) of joints\nis a key parameter in multiple applications of human motion\nanalysis. The aim of this work was to propose a novel real-\ntime estimator of the center of fixed joints using an inertial\nmeasurement unit (IMU). Since the distance to this center\ncommonly varies during the joint motion due to soft tissue\nartifacts (STA), our approach is aimed at adapting to these s mall\nvariations when the COR is fixed. Our proposal, called ArVE d,\nto the best of our knowledge, is the first real-time estimator of\nthe IMU-joint center vector based on one IMU. Previous works\nare off-line and require a complete measurement batch to be\nsolved and most of them are not tested on the real scenario.\nThe algorithm is based on an Extended Kalman Filter (EKF)\nthat provides an adaptive vector to STA motion variations at\neach time instant, without requiring a pre-processing stag e to\nreduce the level of noise. ArVE dhas been tested through different\nexperiments, including synthetic and real data. The synthe tic data\nare obtained from a simulated spherical pendulum whose COR\nis fixed, considering both a constant and a variable IMU-join t\nvector, that simulates translational IMU motions due to STA .\nThe results prove that ArVE dis adapted to obtain a vector per\nsample with an accuracy of 6.8±3.9mm on the synthetic data,\nthat means an error lower than 3.5% of the simulated IMU-joint\nvector. Its accuracy is also tested on the real scenario esti mating\nthe COR of the hip of 5volunteers using as reference the results\nfrom an optical system. In this case, ArVE dgets an average error\nof9.5% of the real vector value. In all the experiments, ArVE d\noutperforms the published results of the reference algorit hms.\nIndex Terms —motion analysis, rehabilitation, inertial sensor,\nIMU, biomechanical model, center of rotation, sensor calib ration.\nI. I NTRODUCTION\nHuman motion analysis is a fundamental support tool to\nobtain objective information about the movement parameter s,\nwhich are especially important in sports or clinical reha-\nbilitation routines and preventive treatments [1]. With th e\ncurrent ageing in developed countries, the demand for home\nrehabilitation has increased and, with it, the need to obtai n\nquantitative exercise data remotely. Optical motion captu re is\nconsidered the gold standard technique in the motion analysis\nfield, but the systems that use it are very expensive and requi re\nhigh computational cost. Recently, low-cost optical metho ds\nhave been developed, based on the analysis through differen t\nsoftware applications of recordings captured by video came ras\nor smart mobile devices [2]. However, optical systems are\nlimited to controlled environments and suffer from occlusi ons,\nso wearable systems, such as inertial systems, have emerged as\nalternatives for motion analysis [3], [4], [5]. Focusing on the\ninertial capture systems, we can find two main alternatives:\nusing an inertial measurement unit (IMU) attached to each\nsegment of the human body to measure its orientation [6], [7] ,[8], [9], [10], [11], [12]; or using one IMU to measure the\norientation of the attached and the adjacent segments [8].\nSome inertial systems require the characterization of per-\nsonalized multi-body kinematic models in order to describe\nmovements. In this case, as IMU measurements are given in\nthe sensor frame, the relationships between the sensor and\nanatomical frames are needed for the calculation of the moti on\nof the joints and segments. Thus, most of algorithms aimed at\ncalibrating IMUs with respect to the human body focus only\non identifying the orientation of the axes of rotation of joi nts\nwith respect to the IMU axes, as in [13], [14], [15]. However,\nrecent studies have shown improvements in the accuracy of\nsegments orientation estimation by exploiting the equatio ns\nof motion for the translational acceleration of rotating bo dies\nfusing information from several IMUs [7], [8], [9], [10], [1 1],\n[12]. In order to use this approach, the location of the cente r\nof rotation (COR) of joints with respect to IMUs is needed. In\ninertial systems, the COR is determined through the oriente d\nvectorr, defined from the IMU accelerometers to the COR of\njoints, see Fig. 1. The estimation of the orientation of join ts\nis highly sensitive to the accuracy in obtaining this vector r.\nIMU-based algorithms have been proposed to determine this\nvector [16], [17], [18], [19], [20], [21], [22]. These algor ithms\ncan be separated between those that get an average rand\nthose that obtain an adaptive r. In the former, it is assumed\nthat changes in rcaused by soft tissue artifacts (STA) are\neliminated by using signals of several seconds duration [21 ],\n[17], [23]. However, these approaches lead to errors on\nscenarios with STA [22], [18]. As an alternative, Frick and\nRahmatalla [18] propose an adaptive gradient descent metho d\nto obtain an rat each time instant in fixed CORs. This\nproposal is tested with synthetic data but it has not been tes ted\non the real scenario of human joints.\nIn this work, we propose a novel estimator of rin fixed\njoints, which is adapted to variations in the relative posit ions\nof IMUs caused by STA. Our proposal is called ArVEd, that\nstands for Adaptive rVector Estimator. This algorithm is\nbased on the method introduced in [24]. To evaluate the\nperformance of our proposal, we compare ArVE dwith the\napproach described in [17], used as a reference to estimate a n\naverager, hereinafter MrVS , that stands for Mean rVector\nleast-Squares-based estimator.\nTherefore, the main goal of this work is to demonstrate\nthat ArVE dis a competitive option to estimate the location\nof CORs of fixed human joints with one IMU, assuming that\nthe IMU undergoes STA during motions. The objectives are\nas follows:\n• Validate the proposed algorithm, ArVE d, with syntheticJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 2\ndata.\n• Evaluate the effect of the adaptation to variations in r\ncomparing ArVE dwith MrVS, which does not consider\nthe effect of STA.\n• Study the accuracy of ArVE dwith real data of the COR\nof hips.\nThis document has five sections besides the introduction.\nSection II includes a revision of the state of the art of inert ial\nmethods to obtain the location of CORs. Section III details\nthe proposed ArVE dmethod together with our implementation\nof MrVS. Section IV explains the experiments with synthetic\ndata and the achieved results of ArVE d, compared with the\nresults of our adaptation of MrVS. Section V describes the\nexperiments on the real scenario of calibration of the hip\ncenter in five volunteers, together with the discussion of th e\nresults from ArVE dand MrVS, versus the results obtained\nwith an optical system. Finally, section VI summarizes the\nmain conclusions of this work.\nII. R ELATED WORKS\nThere are different works focused on the location of the\nCOR of human joints, since the determination of an internal\npoint of the body, as this center, is not trivial. The most\naccurate approaches to determine the position and location\nof IMUs with respect to anatomical CORs or joint axes\norientation are based on X-ray or magnetic resonance image,\nbut both approaches are high priced, invasive and, ultimate ly,\nimpractical [23]. Therefore, CORs in the motion analysis\nfield are commonly determined through palpation of external\nanatomic landmarks by expert therapists or by the use of\noptical systems [8], [9], [10]. Optical systems use sphere-\nfitting approaches to find the radius that best fits a trajec-\ntory described by optical markers [25], [26], [27]. Both the\npalpation and the optical systems require expert hands to\nplace the markers and are limited to controlled environment s.\nThese methods to obtain ruse external information other than\nIMU-derived data, such as the location of the optical system\nmarkers. Thus, it limits the use of the biomechanical model-\nbased inertial motion analysis to environments where optic al\nsystems are available.\nAs aforementioned, there are different proposals of IMU-\nbased algorithms to determine this vector in the literature .\nIn [20], a method for estimating the location of knees, mod-\nelled as hinge joints, is introduced and tested by adding\ndifferent levels of signal-to-noise ratio. This algorithm is also\ntested on the human gait scenario [16], but no conclusions\non the absolute error in the 3D joint location estimates are\ngiven. For locating fixed CORs instead of axes, McGinnis\nand Perkins propose in [21] an algorithm based on exploiting\nthe relationship between linear acceleration and turn rate in\nrigid solids. The algorithm is based on solving the equation\nof accelerations of a rigid-solid body moving in the 3D space\naround a fixed COR whose linear acceleration is equal to zero.\nThey reported an error of 3.1mm in tests with a mechanical\nanalogue of the hip joint performing a determined joint moti on\n(with a specific trajectory, range and velocity). The depend ence\nof this method to different types of motions, ranges of motio nposition of IMUs and joint velocity is assessed in [17]. This\nstudy reports a considerable impact of the angular velocity on\nthe COR identification and non-critical relations with the t ype\nand ranges of motion. This algorithm is tested on the glen-\nhumeral joint estimation scenario [23], reporting an accur acy\nof21mm compared with magnetic resonance images. It was\nconcluded that the location of fixed CORs is more accurate\nusing the information of one IMU in the algorithm of [17]\nthan using data from two devices due to the small amplitude\nof the signals recorded by the IMU placed on the fixed segment\nand the difficulty related to its tracking. Olsson and Halvor sen\ntested the same proposal in the case of moving CORs in\nmechanical simulations, studying different methodologie s of\nsolving the acceleration equation [22]. However, the accur acy\nof this approach on the real scenario of human joints is not\nreported.\nThe previous algorithms obtain a mean value of rfor each\ntest, averaging the STA. However, the effect of STA is studie d\nin [22] the least squares method used in previous studies (su ch\nas [21], [17], [23]) shows no robustness to outliers that may\noccur as a consequence of STA. And as stated in [18], the STA\ncan introduce significant errors in the location of the cente r of\njoints when assuming an average value of r.\nTo overcome this limitation associated with the STA, Frick\nand Rahmatalla propose a gradient descent method to obtain\nanrat each time instant with an IMU attached to a hinge\njoint [18]. However, this adaptive method requires its init ial-\nization using the complete test data with a duration around\n25s, otherwise it may reach a local minimum. This proposal is\ntested with synthetic data from a 2D-pendulum simulating STA\nwith an attached spring and reports errors of 7.53mm. In [28],\nthe algorithm is evaluated with a mechanical hinge joint in\nwhich the effect of STA is replicated with the IMU placed on\na piece of raw meat. The authors provide results on synthetic\ndata, where the errors range from 10.8mm to21.4mm on the\nhighest STA scenarios. However, this algorithm has not been\ntested on the real scenario of human joints.\nIt is remarkable that all aforementioned inertial approach es\nentail different signal pre-processing to reduce noise in t he\nIMU data and to estimate the angular acceleration, ˙ω, a\ncommon parameter required in these methods. In contrast,\nour initial proposed method, called ArVE, estimates an rat\neach time instant without signal pre-processing [24]. In th e\ninitial evaluation, ArVE provides average errors of 1.5mm\nand6.0mm in the fixed and changing rcases, respectively.\nIII. P ROPOSED ALGORITHM\nThe main goal of our proposal is to obtain the location of\nthe COR as an adaptive IMU-joint vector, r= [rx,ry,rz]⊤,\ndefined from the accelerometer to this COR in the sensor\nframe. We aim at estimating rwith one IMU by using the\nmeasures of turn rate ωIand specific force fA,I, that is the\nlinear acceleration aAinfluenced by the gravity acceleration\ng. Subindex Iindicates the measurements obtained directly\nfrom the IMU in its reference system. We obtain the IMU-JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 3\njoint vector ron the basis of the equation of accelerations of\na rigid-solid body moving in the 3D space (1).\nak\n0=ak\nA+˙ωk×rk+ωk\nI×/parenleftbig\nωk\nI×rk/parenrightbig\n, (1)\nWhereak\n0andak\nAare the linear accelerations in the COR\nand the IMU, respectively, ωk\nIis the turn rate of the rigid-\nsolid body and ˙ωkis its first-order derivative. As the aim is to\nestimate the location of fixed CORs, we assume ak\n0negligible.\nAll parameters are expressed in the sensor frame. Superscri pt\nkdenotes the time instant of parameters. Rigid-solid bodies\npresent a constant rvector, but in this study we focus on\nhuman bodies in which STA modify rkat each time k. Besides,\nusing (1) to estimate an adaptive rk, we assume negligible the\nlinear acceleration caused by STA.\nFig. 1 depicts the relation of these magnitudes measured\nwith one IMU and the estimation of the COR. This figure\nshows the global frame with the subscript gand the sensor\nframe, which is attached to the IMU.\nIMUCOR\nFig. 1. Scheme of the relationship between the magnitudes in (1). The rigid-\nsolid body moves with turn rate ωIand angular acceleration ˙ω, whereas the\nIMU suffers a linear acceleration aA, but it measures the specific force fA,I.\nThe specific force fA,Iis the result of aA−gboth expressed in the sensor\nframe. Conversely, the linear velocity of CORs is v0=0by definition and\nas it is fixed, its linear acceleration a0is also equal to zero.\nTo obtain rkwith (1), the linear acceleration ak\nAis required.\nAs IMUs provide the specific force fk\nA,Iundergone by the\naccelerometer, we obtain ak\nAcorrecting the effect of the gravity\nthrough the projection of the gravity vector ginto the frame\nof IMUs as follows:\nak\nA=fk\nA,I+(Ck)⊤g, (2)\nwhereCkis the Direction Cosine Matrix that relates the global\nframe with the sensor frame and gis the gravity vector defined\ndownwards in the global frame with a value of 9.8m/s2. We\ndo not use the direct measures of orientation with respect\nto the global frame in order to provide an algorithm usable\nwith any generic IMU. We calculate the transformation matri x\nCfusing the measures of turn rate ωk\nIand specific force\nfk\nA,Iof the IMU using the algorithm introduced in [29]. This\nalgorithm estimates through an unscented Kalman filter (UKF )\nthe Euler angles of the IMU from the measures of turn rate\nωk\nIand updates these estimations with the specific force fk\nA,I\nmeasured in those moments when its norm is close to the\ngravity vector norm.In this work, we evaluate three different ways to estimate\nr: ArVE destimates a dynamic rkat each time kand MrVS\nobtains, on the one hand, an averaged rfor complete tests and,\non the other hand, a dynamic rk\nnfor a determined number\nof samples nwith an overlap of n−1samples between\nconsecutive estimations of rk\nn. These methods are explained\nin the following two sub-sections: ArVE din section III-A and\nMrVS in section III-B.\nA. Proposed algorithm: ArVE d\nWe propose ArVE dto estimate rkat each time instant based\non the assumption of fixed CORs using an EKF. Fig. 2 depicts\nthe two stages of ArVE dat each time k: an initial stage to\nobtain the linear acceleration ak\nAfollowed by the second stage\nthat consists in an EKF to determine rk. The EKF fuses the\nmeasured turn rate ωk\nIand the calculated linear acceleration\nak\nA.\nFig. 2 shows also the two steps of this EKF. The proposed\nEKF, from the second stage of ArVE d, minimizes the predic-\ntion error of the state vector xk, composed of the searched\noriented vector rk, its first-order derivative ˙rk, the turn rate\nωkand the angular acceleration ˙ωk, given the measurements\nfrom the IMU.\nIn the estimation step of the EKF, we assume ˆ˙rkandˆ˙ωk\nconstant, whereas ˆrkandˆωkare the integral at each time of\nthese terms. Thus, the estate vector ˆxkis estimated at each\ntimekas follows:\n\n\nˆrk=rk−1+ˆ˙rk∆t\nˆ˙rk=˙rk−1\nˆωk=ωk−1+ˆ˙ωk∆t\nˆ˙ωk=˙ωk−1(3)\nThe observations consist of the measured turn rate ωk\nIand\nthe linear acceleration ak\nAobtained in the initial stage of\nArVEd. ArVE dthen updates the estimations exploiting the\nrelationship between the estimations and the estimated lin ear\nacceleration ˆak\nA, using (4), and the direct relation between the\nestimated turn rate ˆωkand the measured one ωk\nI.\nˆak\nA=−ˆ˙ωk׈rk−ˆωk×(ˆωk׈rk) (4)\nThe linear acceleration ak\nAand the turn rate ωk\nIare then used\nto obtain the innovation of the EKF to update the estimations\nat each time k.\nConsidering ωkand˙ωkin the state vector, we obtain\nan estimation of ˙ωkusing the raw data from gyroscopes,\nfacilitating the generalized use of the algorithm. Since EK Fs\nminimize the variance of the estimation error, noisy data fr om\nIMUs do not require an initial signal filtering and avoids\nthe post-processing suggested in [18]. Notice also that ˙rkis\nnot related to any measured magnitude, so it is not directly\nupdated, but used as a parameter of adjustment of the EKF.\nThe use of the derivative of rin the state vector of the EKF\nis one of the main differences between ArVE dand ArVE,\nour initial approach proposed in [24], differentiated with the\nsubindex d.\nThe covariance parameters of the Qmatrix in the EKF are\nset according to [30]. We select a constant value of covarian ceJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 4\nFig. 2. Flowchart to obtain the adaptive rkat each time instant. In the initial stage, we fuse the IMU mea surements of turn rate ωk\nIand specific force fk\nA,I\nusing the UKF introduced in [29] to obtain the linear acceler ationak\nA. In the second stage, we obtain rkwith this signal of linear acceleration combined\nwith the turn rate through the EKF.\nfor each kind of tests, with synthetic and real data, indicat ed\nin section IV-C and section V-A, respectively, together wit h\nthe explanation of the estimation of the covariance matrice s\nPandR.\nB. MrVS\nThe original approach of MrVS proposed in [23] uses the\ncomplete several-second long signals of tests of the measur ed\nturn rate ωIand the linear acceleration aAobtained from the\nmeasured specific force fA,I, and it also requires computing\n˙ω. This parameter is obtained by discrete derivative of the\nturn rate ωImeasured with the IMU. Since a0is negligible\nin fixed CORs, the only unknown term in (1) is r, so it can\nbe rearranged as follows:\naA=Mr, (5)\nwhere\nM=\n−ω2\ny−ω2\nz−˙ωz+ωxωy˙ωy+ωxωz\n˙ωz+ωxωy−ω2\nx−ω2\nz−˙ωx+ωyωz\n−˙ωy+ωxωz˙ωx+ωyωz−ω2\nx−ω2\ny\n(6)\nis the matrix introduced in [17]. Variables ωx,ωyandωz\nare the components of the measured turn rate ωI. In both (5)\nand (6), the vector aAand the matrix Msymbolize a set of\ntemporal measurements of the corresponding parameters, so no\nsuperscript kis used. An averaged ris obtained solving (6)\nwith least squares for complete tests.\nWhen we work with several-second long IMU signals to\nobtain an average r, theMmatrix from MrVS has full rank\non the scenario of CORs of ball joints, as hips. However, when\nwe look for an adaptive calculation of r, uncertainties appear\nin (5) when ωis negligible. In these points, Mbecomes\nantisymmetric, so its determinant is zero and the system is\nundetermined. Therefore, MrVS cannot be implemented in\nreal-time applications in a straightforward way to obtain o ne\nvector per sample. Thus, we test two approaches of MrVS:\nobtaining an averaged rfor the complete test as proposedin [23] and estimating an adaptive rk\nnin a sliding window\nwith annnumber of samples.\nIV. E XPERIMENTS ON SYNTHETIC DATA\nWe carry out two experiments with synthetic data to test the\nperformance of ArVE dand MrVS. The experiments simulate\nthe motion of a pendulum moving in circles from a fixed\nball joint. This pendulum imitates a limb carrying out circl es\nfrom a fixed COR, as a leg moving from the hip. The first\nexperiment consists in an IMU moving around a fixed COR\nwith a constant rvector to assess the accuracy of the evaluated\nsystems in the ideal case. The second experiment imitates th e\nmotion of an IMU around a fixed COR with variations of\nrover the test caused by simulated STA that involve small\ntranslations of the IMU. In this experiment, we study the err or\ncaused by assuming a constant rwhereas it varies over time.\nThe experiments with synthetic data are presented in four\nsub-sections. We describe the spherical pendulum simulate d to\nobtain the synthetic data in section IV-A and detail the metr ics\nused to evaluate the inertial-based methods in section IV-B .\nThen, section IV-C and section IV-D introduce the results fo r\nthese experiments carried out on synthetic data.\nA. Simulation of a spherical pendulum\nWe simulate the movement of a spherical pendulum rotating\nin the3D space during 10s, around a fixed COR and around\nthe main axis of the pendulum. The pendulum describes an\nellipse with two main rotations around the xandyaxes of\nthe simulated IMU, and a partial rotation around its zaxis,\ncombining the three motions around the three IMU axes. The\namplitudes of the movements around the x,yandzaxes are\n17◦,9◦and3◦, respectively, and the motion of the pendulum\naround the xandyaxes lasts 1s; and1.5s around the zaxis.\nThe parameters of the simulated motions are set according\nto the motions observed during the lower-limb calibration\nobserved in [31], as we do in the simulations reported in [24] .JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 5\nFig. 3 depicts these axes of the IMU together with a scheme\nof its motion.\nIMU\nFig. 3. Scheme of the pendulum designed for simulations. The IMU (orange\nbox) moves around the COR and its coordinate system moves wit h the device\nfrom positions of the initial x,yandzto each corresponding x′,y′andz′. In\nthe first experiment, rremains constant and, in the second one, its coordinates\nchange over time.\nWe establish rconsidering the most likely configuration on\nthe real scenario, where the IMU is placed over the thigh and\nnot in contact with the femur. In the simulation, a displacem ent\nbetween the main axis of the pendulum and the origin of\ncoordinates of IMUs is taken into account and the IMU axes\nare misaligned with r, as shown in Fig. 3. The rx,ryand\nrzcomponents are −60,20, and200mm, respectively, so the\nnorm of the vector is 209.8mm.\nThe inertial data are simulated at a sampling rate of 100Hz.\nIn both experiments, we add a Gaussian noise in the simu-\nlated turn rate ωIand specific force fA,Iaccording to the\nspecifications of the MTw Awinda sensors from Xsens [32],\nsince we use these sensors in the real data experiments. The\nstandard deviation of noise in the measurements of gyroscop e\nof turn rate ωis0.0017◦/s, and in the specific force fAfrom\nthe accelerometer is 0.02m/s2. Bias is not considered since\nsimulations and tests are short enough in time to be affected\nby it, as done in [18] and because the estimation of rdoes not\ninclude integration, so its estimations are insensitive to bias,\naccording to [20]. In order to provide more significant resul ts\nthan in our previous work [24], we carry out 100tests for each\nexperiment.\nOn both scenarios we set the observation noise, R, equally\nsince it depends on the noise of the simulated sensors, but we\nadjust the estimate covariance, P, and the process covariance,\nQ, for each scenario.\nB. Metrics and errors\nWe quantify the accuracy of the proposals using three\ndifferent metrics:\n1) The Euclidean norm of the vector difference between the\nreference rrvector, the ground truth , and the estimated\nrusing the measurements from IMUs, noted with |∆r|.\nIn order to consider one method competitive for its use\nin orientation tracking, we define the upper limit of |∆r|\nin the10% of the Euclidean norm of rr, because in [8]\nit is reported that errors over this 10% double errors in\nestimations of the orientation of limbs.2) The difference between the norms of rrandr, defined\nas∆|r|.\n3) The deviation angle, γ, between rrandr.\nWe consider these three metrics because each considered err or\nhas a source related with the different parameters in (1). Th e\ndifference of norms ∆|r|is mainly caused by errors in the\ndetermination of ˙ω. The deviation angle γis mostly affected\nby the accuracy of the measured linear acceleration aA,Iand\nturn rate ωI. Finally, |∆r|is affected by both the difference\nbetween norms and the deviation angle.\nC. Results on a constant IMU-joint vector\nUsing the experiments of a simulated 3D pendulum with\na constant IMU-joint vector detailed in section IV-A, we\nevaluate the accuracy of MrVS and ArVE dto obtain an r\nper window and per sample, respectively.\nWe assess the proposal of MrVS in a sliding window as\nan alternative to estimate a variable rvector. We test different\nwindow sizes in order to study the accuracy obtained with eac h\nconsidered number of samples n. The evaluated window sizes\nare from n= 5 untiln= 100 samples, increasing 5samples\nbetween tests. We stop at 100samples since it would average\nthe STA of a complete cycle in the simulations. Windows slide\n1sample to obtain each r, so they overlap n−1samples. Since\nthe norm of the reference vector is 209.8mm, we define the\nupper limit in 20mm, which is the 10% of the vector norm.\nThe resulting average |∆r|of each test is depicted in Fig. 4.\n0 10 20 30 40 50 60 70 80 90 100\nWindow size (number of samples =    )050100150 (mm)\n20\nFig. 4. Average and maximum |∆r|of the100tests carried out with our\nproposal of MrVS in a sliding window with the corresponding w indow size\nused to estimate r= [200,20,−60]⊤mm. The horizontal red line depicts\nthe upper limit.\nResults in Fig. 4 show that the errors reduce as the number\nof samples increases, reaching an error smaller than 10mm\nfrom the window size of n= 80 samples. The maximum errors\nalso decrease when increasing of n, obtaining bearable error\nvalues under the upper limit with n= 90 samples. However,\nusing95samples, the information of almost 1s is averaged,\nwhich reduces the sensitivity to changes in r.\nThe use of n= 45 samples in each window is a trade-off\nsolution between the nnumber of samples and the averaged\n|∆r|error. In this case, the average error is 17.6mm, which is\nlower than the upper limit of 20mm. Nevertheless, maximum\nerrors are larger than 100mm.\nFig. 5 a) shows the results of MrVS with a sliding window\nsize of45samples over the initial 2.5s of the constant rtest.\nThe purple circles points out the intervals where errors of\nMrVS increase when the norm of ωis negligible. The required\nnumber of samples to obtain an accurate estimation of theJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 6\nIMU-joint vector is too long to estimate a variable vector, s o\nMrVS is not able to adapt to variations in the IMU-joint vecto r.\nFig. 5. Results on the fixed rscenario, in which the ground truth is\nblack depicted. a) Vector rk\nnobtained using MrVS in a sliding window\nofn= 45 samples. b) Resulting rkusing ArVE dand setting the initial\nr0= [0,0,0]⊤in the EKF. During the first second the estimations are\ninaccurate, until the filter convergence. After this transi tory time, estimations,\ndepicted in blue, red and yellow, are similar to the ground tr uth.\nConversely, we use ArVE dto combine the information of\nthe IMU signals at each time instant, avoiding the inversion\nof the system matrix and the calculation of ˙ω. Fig. 5 b) shows\nthe resulting rvector when using ArVE dwith an initial r0\ncomposed of zeros. In this particular case, the EKF takes one\nsecond to converge. After this transitory time, the estimat ions\ndo not suffer from miscalculations even when ωis close to\nzero. We can conclude that ArVE dprovides stable estimations\neven in the intervals where MrVS was not able to provide an\naccurate result, highlighted with pink circles.\nApart from the parameters of covariance in the EKF, the\nperformance of ArVE ddepends on the initial state vector, so\nwe test the proposed method with different r0vectors. We\ncalculate an r0as an average vector similarly than in MrVS,\nby using (5) with the initial samples of tests. We use from 20\nuntil140samples to estimate this r0, increasing 20samples\nbetween tests. We repeat 100times every test to evaluate the\naccuracy of ArVE dwith each initial vector through the metrics\nintroduced in section IV-B. Fig. 6 depicts the average of err ors,\ntogether with their maximum and minimum errors.\nFig. 6 shows the evaluated errors and their range of values\ndecrease as the number of samples considered to estimate\nr0increases. These errors become stable when we obtain r0\nwith60samples. From this number of samples, |∆r|is around\n2mm, so using more than these 60samples (that means 0.6s of\nsignals since fs= 100 Hz) does not improve the accuracy of\nArVEdsince the EKF converges from the initial samples. For\nthat reason, on the following we use 60samples to calculate\nr0for the initialization of the EKF of ArVE d.\nAccording to the errors shown in Fig. 6, ArVE doutperforms\nin the evaluated cases our proposal of MrVS in a sliding0510\n024\n20 40 60 80 100 120 1400123(mm) (mm) (º)\nSize of (number of samples)\nFig. 6. Errors with their corresponding ranges of values in t he estimation of r\nwith ArVE dover the10second-experiment with the different r0considered.\nThe Euclidean norm of the vector difference, |∆r|, is depicted with blue bars,\nthe difference between norms, ∆|r|, with green bars and the deviation angle\nγin orange bars. Each bar corresponds to the average error of t he100tests\ncarried out with this number of initial samples and the verti cal lines depict\nthe range of these errors.\nwindow. This improvement in accuracy is due to the fact that\nArVEdhas no problems with the singular points of the signal\nas happens with MrVS. Despite the inaccurate estimations of\nMrVS near the instants when ωis negligible, its estimations\nare accurate in the remaining time intervals.\nD. Results on variable IMU-joint vectors\nWe simulate the STA of the real scenario as variations in r\nthat imitate the translation of the IMU with respect to the fix ed\nCOR. The variation of rover time is presented as a sinusoidal\nsignal of frequency 1Hz and amplitude 20mm inrxandrz,\nand5mm inry, components previously shown in Fig. 3. We\nset this frequency of the translational STA to make it simila r\nto the frequency of motion of the pendulum, as suggested\nin [33], and the amplitude values are also set according to\nthe results in the same work. Since IMUs are taped to the\nbody, lateral motions over the y-axis are restricted, whereas\nthe muscle contractions entail translations in the x-andz-axis.\nIn this case, we compare ArVE dwith ArVE to evaluate the\ninfluence of the new parameter of adjustment introduced in\nArVEdon the accuracy of this proposal. Fig. 7 depicts in black\nthese components of the variable rvector used as ground truth\nover the test, together with the components of the variable r\nusing ArVE and ArVE d.\nFig. 7 shows that ArVE dand ArVE adapt to most of\nchanges of rover time. Thus, both method provide adapt-\nability to a variable r. Besides, according to these results,\nestimations using ArVE dare closer to the ground truth, with\nan improvement of 7% compared to the results obtained by\nusing ArVE. In this way, ArVE doutperforms ArVE through\nthe adjustment of the noise parameters of ˙rin the EKF. Both\nmethods are also evaluated by means of a Bland-Altman plot\ncompared with the ground truth in Fig. 8.\nAs shown in Fig. 8, errors in the estimation of rxandrzare\nsimilar using ArVE dand ArVE. They are in the approximateJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 7\nFig. 7. Coordinates of rover the test. The ground truth is depicted in black and the es timatedrx,ryandrzin blue, red and yellow, respectively. We\nestimaterusing ArVE and ArVE d, and their results correspond to the images presented on the left and on the right, respectively.\n10 15 20 25ArVE ArVEd\n-80 -70 -60 -50-15-10-505101520Error of ArVE & ArVE  (mm)d+1.96 SD =\n+4.1 mm\n-1.96 SD =\n- 4.0 mm+1.96 SD =\n+3.4 mm\n-1.96 SD =\n- 3.1 mm+1.96 SD =+12.5 mm\n-1.96 SD =\n-10.6 mm+1.96 SD =\n+14.7 mm\n- 1.96 SD =\n-12.3 mm\n180 190 200 210+1.96 SD =\n+7.9 mm+1.96 SD =\n+8.5 mm\n-1.96 SD =\n-7.4 mm\n-1.96 SD =\n-7.9 mm\n(mm) (mm) (mm)\nFig. 8. Bland-Altman plot with the comparison of ArVE dand ArVE. The\ndotted lines point out the confidence interval in which the 95% of the errors\nobtained with each methods are contained, so these lines cor respond with the\nvalue of1.96times the standard deviation (SD).\nrange of ±4mm forrxand±8mm forrz. Conversely, there\nare differences in accuracy estimating ry. The improvement in\nthis coordinate is specially remarkable in the error disper sion,\nlowered2mm in each upper-and lower-bounds. The results\nmay not be as good with rybecause its range of variation is\nat the noise level in the filter, so none of these methods adapt s\nto its variations.\nThese simulations are closer to the real scenario, where r\nchanges due to STA, so we use the simulated data to evaluate\nthe three methods, whose results are shown in Table I. In this\ncase, MrVS estimates a unique r, constant for the complete\ntest, whereas ArVE dand ArVE adapt to the variable vector,\nobtaining an instantaneous vector per sample.\nTABLE I\nERRORS IN THE DETERMINATION OF rWITH INERTIAL METHODS WITH AN\nAVERAGE r= [200,20,−60]⊤\n|∆r|mm∆|r|mm γ◦\nArVEd6.8±3.93.4±2.41.6±0.9\nArVE 7.3±4.63.8±2.81.7±1.1\nMrVS 14.4±3.85.7±4.93.5±0.7\nThe|∆r|error is the most remarkable since it shows\nthe highest improvement, from 14.4mm using MrVS, until\n6.8mm using ArVE d, lowering errors more than a 50%. This\nerror decreases more than the difference of norms ∆|r|andthe angle γbecause the distance vector is affected by ∆|r|\nandγ, and both errors are smaller using ArVE d. According\nto these results, ArVE dis the method that best adapts to a\nvariabler, which justifies our proposal for improvement by\nintroducing the derivative of rin the estate vector.\nIt is noticeable that the errors of using ArVE din simulations\nthat include the simulated effect of the translational STA a re\nsimilar to those presented in [18], but they do not consider\nthe three components in the reported errors. So, even if we\ncannot compare directly our results, we can conclude that ou r\nalgorithm is at least as accurate as the methods in the litera ture.\nFurthermore, ArVE donly needs the initial data during 0.6s to\ninitialize the algorithm and we get rid of the low-pass filter ing\nof the IMU signals and the analytic derivation of the measure d\nturn rate ωI, used in other works as [17], [23], [18], [28].\nFinally, Fig. 9 depicts the estimated COR in the simulations\nof the variable rusing ArVE dand, drawn in red, its actual\nposition in two different planes. According to these result s, the\nrelative errors depend on the component of r, being larger for\nthey-axis and the smallest variation on the x-axis. But it is\nremarkable that the 93% of the COR estimated by ArVE dare\nin a sphere with a radius of 6mm. This estimated radius only\nis around a fifth of the hip joint radius, commonly included\nin the range of 25mm to30mm according to [34].\nV. E XPERIMENTS ON THE REAL SCENARIO\nWe study now the performance of ArVE don a real scenario.\nWe obtain the COR of the hip of five volunteers with respect\nto one IMU, using the inertial-based systems and an optical\nsystem at the same time. We compare ArVE dand our imple-\nmentation of [17], MrVS, with the optical method introduced\nin [27]. Although other methods exist, as explained in secti on I,\nwe use this one because it aims at obtaining a variable r.\nA. Experimental setup\nFive volunteers with a height of 165±8cm participate\nin this study. During the experiments, they repeat 10times\nthe hip circles motion depicted in Fig. 10 a), being equipped\nwith one IMU placed on the thigh and four optical markers\nlocated around the IMU, as shown in Fig. 10 b). We choose\nthis motion to ensure the presence of a unique COR at each\ntime instant instead of an axis of rotation, so our system has\na unique solution, that is the searched COR. The concernsJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 81 \u0000 1\n\u00011\n\u0002 202 \u0000 222\n\u0001 - \u0001 \u0001 - \u0001\u0000\n- \u00012\n- \u0001 \u0003 - \u0004 \u0001 - \u0004 \u0002 - \u0004\u0000\n2\n\u0003 \u00012\n\u0003\u0000\n202\n2001 \u0005\n\u0001\n1 \u0005\n\u00021 \u0005 \u0000\n(mm)(mm)\n(mm)\nFig. 9. Projection of the points estimated by ArVE din planes XZ, depicted in the image on the left, and YZ, in the i mage on the right, over one second of\nthe experiment. The estimated points are depicted in blue an d the ground truth in red.\nabout estimations of axes using (1) are more detailed in [35] .\nThe motion of hip circles is performed maintaining both legs\nstraightened, one foot is placed on the floor while the other l eg\nperforms circles from the hip. To do this exercise, the stabi lity\nof the volunteers is important to keep the hip still, so their\nbacks rest on a stable surface and we ensure that their motion s\nare according to the requirements for these experiments. We\neliminate the first and last signal segments of 1.5s long of\ntests to remove movements other than hip circles.\nFig. 10. Experimental setup. a) Illustration of the movemen t performed by the\nvolunteer in order to calibrate her/his hip. The COR, the IMU , its reference\nsystem and the global frame are shown. b) Picture of the mount ing board with\nthe IMU together with the four optical markers on the thigh of the volunteer.\nThe inertial sensor is the MTw Awinda from Xsens [32]\nand the optical system consists in the Vicon equipment [36]\ntogether with the method proposed in [27] to estimate CORs.\nBoth inertial and optical measurements are recorded at a\nsampling rate of 100Hz. We synchronize both systems with\nan initial motion of flex-extension of the hip and the followi ng\ndetection of the maximum position and null turn rate measure d\nwith the optical system and the IMU, respectively, in the sig -\nnals measured during calibration movement. In the definitio n\nof the mounting board, we ensure that its axes are aligned wit h\nthe axes of the IMU and its location center is placed at theIMU accelerometer. Using the spatial location and position of\nthe mounting board in the reference system of the Vicon, we\ntranslate the estimations of our reference IMU-joint vecto rv\ninto the IMU system. We use the method proposed in [27]\nbecause it is aimed at estimating an adaptive v, although\nwe finally obtain an average vector for the complete test to\nimprove its accuracy and eliminate the dependence with the\nnumber of samples considered for each estimation.\nWe obtain the location of the mounting board in which the\nIMU is placed, set at the location of the accelerometer in the\ndevice, its orientation and the position of each marker from the\noptical system. Since the IMU is aligned with the mounting\nboard, we consider the data from the mounting board as the\norientation and location of the IMU.\nBesides, the covariance parameters are estimated as follow s:\nwe use the measurements of a static IMU to calculate the\nstandard deviation needed to obtain R; we estimate PandQ\nusing the reference data of one subject and adapting them to\nthe best performance of ArVE d, and we use these parameters\nfor all the other subjects.\nB. Metrics and errors\nAs optical methods are commonly used as baseline because\nthey provide a high accuracy, we compare the outputs from\nboth, ArVE dand our implementation of MrVS, with the\nresults obtained trough the measurements from the optical\nsystem. We evaluate those methods with the same metrics\nthat we used in the experiments on simulations to study the\ndifferent sources of errors, described in section IV-B. In t his\ncase, our reference to determine the errors of ArVE dand\nMrVS is the vvector, obtained with the optical system and\ntranslated into the IMU system.\nC. Evaluation of the adaptive r on human joints\nOn the real scenario of human hips, we have a reference\nvfrom the optical system, depicted in gray in Fig. 11. We\nuse this reference to evaluate the estimations of the variab le\nradapted to these changes caused by the STA using ArVE dJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 9\nand the average rfor the complete test with MrVS. Fig. 11\nshows that results from both adaptive algorithms, ArVE dand\nthe optical system, experience periodic changes caused by t he\nSTA in legs during the experiment. This is coherent with the\nexercise since it consists in repetitions of the circles per formed\nfrom the hip.\nFig. 11. IMU-joint vector obtained with ArVE d, depicted in blue, red and\nyellow dotted lines; with MrVS, presented in cyan, red and mu stard stripes;\nand with the visual-based method, black and gray depicted in continuous lines.\nFig. 12 depicts the norm of the difference vector |∆r|, the\ndifference of norms ∆|r|and the deviation angle γof the\nestimations of ArVE dand MrVS with respect to vin the case\nof each evaluated volunteer. According to this figure, the er rors\nobtained with ArVE dare similar for all volunteers and MrVS\nprovides errors with great variability, so the accuracy of M rVS\ndepends more on the volunteer. Also, these results show the\ndecrement of |∆r|and∆|r|errors using ArVE dversus using\nMrVS in all cases and the deviation angle is similar in most\ncases around 5◦with both methods, so the adaptive method\nis the most accurate. The differences in the average values o f\nthose errors are also consistent with this affirmation.\n0100200(mm)\nM \u0006 \u0007 \b ArVEd(mm)\n1 2 3 4 5V \t \n \u000b \f \r \u000e \u000e \u000f(º)\nAvg.0100200\n010\n5\nFig. 12. Errors obtained for each evaluated volunteer with M rVS and ArVE d\ncompared with v. The average values are also included with the label Avg.\nTable II shows the values of the errors depicted in Fig. 12\ntogether with their corresponding reference vector vof both\nIMU-based approaches compared with vfrom the optical\nsystem. Differences exist between the metrics of both meth-\nods, being specially noticeable with respect to the differe ncebetween modulus ∆|r|that decreases from 62mm (28.0%)\nuntil10mm (4.5%).\nTABLE II\nAVERAGE DIFFERENCES BETWEEN ARVEdAND MRVS COMPARED WITH\nTHE OPTICAL METHOD FOR THE FIVE VOLUNTEERS ,TOGETHER WITH THE\nAVERAGE (AVG.)VALUES\nV olunteer v Method |∆r|mm∆|r|mm γ◦\n1166±1ArVEd17±511±64±2\nMrVS 38±137±14±1\n2249±4ArVEd12±28±42±1\nMrVS 126±3124±49±1\n3228±1ArVEd23±610±75±1\nMrVS 40±140±11±1\n4235±2ArVEd31±39±67±1\nMrVS 68±358±210±2\n5226±2ArVEd22±513±74±1\nMrVS 55±250±26±1\nAvg. 221±2ArVEd21±210±34±1\nMrVS 65±162±16±1\nThe variation between the different errors used to evaluate\nMrVS is remarkable. The average deviation angle γversusv\nis only around 6◦, but the error in the estimated norm ∆|r|\nis62mm, that is a 30% of the norm of v. This variation\nis a consequence of errors in the estimation of the angular\nacceleration ˙ω, since these estimations contain the propagation\nof errors in the measurements of turn rate ω. It only affects the\nnorm since during the derivation the direction of this vecto r\ndoes not change.\nAccording to Table II, we achieve a decrease in |∆r|larger\nthan a60% with ArVE dcompared to using MrVS. ArVE d\noutperforms MrVS also in experiments on the real scenario.\nIn addition, the |∆r|difference is in most of volunteers under\nthe upper limit, achieving accurate results, and the averag ed\naccuracy is 9.5% of the average norm of v, which is lower\nthan the upper limit.\nOur adaptation of MrVS obtaining an average rfor the\ncomplete test entails |∆r|differences of 65mm, meaning a\n29% of relative error. Therefore, this method is not accurate\nenough to be used in the estimation of CORs with a low speed\nin limbs with STA. Differences are larger than the reported i n\nthe previous studies that use MrVS because the conditions\nof the evaluated tests are different, as in [23]. In particul ar,\nthe authors evaluate the arm and the experiments are based on\ntwo perpendicular linear motions, as crosses, with a maximu m\nturn rate of 1.6rad/s. However, our experiments are based on\ncircular trajectories of the evaluated leg with an average t urn\nrate of0.8±0.1rad/s. This is remarkable since some of the\nleg exercises that may be prescribed to improve hip mobility\ninclude circular components, while not as many consider two\nperpendicular linear movements in a row. The speed differen ce\nis important, as it is stated in [17], because errors are larg er\nin the experiments carried out at a lower speed.\nFurthermore, ArVE dis able to be implemented in real-time\nsince it obtains one vector rper sample and only requires of\nthe first0.6s of tests for the initialization. The algorithms are\nprogrammed in MATLAB R2019b, running in a personal com-\nputer (processor i 7-8700 at3.2GHz, RAM memory 16GB). In\nthis platform, the average time for the execution of a sample\nwith ArVE dis0.03ms. As the sampling rate is 100Hz theJOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 10\navailable time for processing a sample is 10ms. Since the\nexecution time of a sample, i.e. the time to obtain one vector r,\nis far lower than the sampling period, we consider that ArVE d\nis suitable for real-time applications, such as monitoring of\nrehabilitation exercises where null acceleration points e xist (as\nin the orientation estimation of lower limbs presented in [7 ]\nor [8]). These results of accuracy prove the usability of ArV Ed\nas an alternative to the optical methods, being adapted to th e\nhuman lower-limb scenario when the joint is constraint to be\nfixed.\nVI. C ONCLUSIONS\nA novel adaptive method for IMU-joint vector determination\nis proposed and validated in this work using synthetic and\nreal data from a hip. The method, called ArVE d, uses raw\ndata from an IMU to determine in real-time the COR of fixed\njoints with respect to the IMU location at each time instant.\nWith the synthetic data, ArVE dachieves an accuracy higher\nthan1% of length and 1◦of deviation when the IMU-joint\nvector is constant and shows adaptability to variable vecto rs\nwith an error around 3% of length and 1◦of deviation. This\nscenario of variable vector is in which the IMU undergoes\nfrom translational movements caused by STA apart from the\nmain rotations. Besides, ArVE dhas also been compared with\nour implementation of one state-of-the-art algorithm that we\nhave called MrVS [17] in this work. In all cases, the proposed\nArVEdoutperforms MrVS decreasing its errors around a\n50%. The accuracy of ArVE dis10% when is tested on real\nvolunteers performing standardized and repetitive exerci ses. In\nthis case, the reference has been obtained with an optical\nsystem. Thus, ArVE dcan be considered as an alternative\nto estimate the IMU-joint vector, obtaining a precise COR,\nsuitable for monitoring rehabilitation therapies that imp ly the\nmotion of ball joints, as shoulders or hips.\nOne of the limitations of the proposal is that it is adapted to\nfixed joints. As future work, we are working on the adaptation\nof ArVE dto be applied to motions in which CORs do not\nshow a negligible linear acceleration, such as gait or runni ng\nanalysis. Nevertheless, ArVE dcan be used in a previous\ncalibration step to obtain an average IMU-joint vector for o ff-\nline applications. In addition, we will adapt the algorithm s\nto be used on portable systems, e.g. smartphones, wirelessl y\nconnected to our IMU. It will allow to provide real-time\nfeedback in rehabilitation therapies.\nACKNOWLEDGMENT\nThis work was supported by Junta de Comunidades de\nCastilla La Mancha (FrailCheck SBPLY/17/180501/000392),\nthe Spanish Ministry of Science, Innovation and Universi-\nties (MICROCEBUS RTI2018-095168-B-C51) and the Youth\nEmployment Program (PEJ-2017-AI/TIC-7372). The authors\nwould like to thank the Deutschen Zentrums fur Luft-und\nRaumfahrt (DLR) for its collaboration in the measurement\ncampaign and the borrowing of the necessary infrastructure\nand equipment.ABBREVIATIONS\nThe following abbreviations are used in this manuscript:\nArVE adaptive r vector estimator\nArVEdadaptive r vector estimator, considering ˙r\nMrVS mean r vector least-squares-based estimator\nCOR center of rotation\nSTA soft tissue artifacts\nIMU inertial measurement unit\nEKF extended Kalman filter\nUKF unscented Kalman filter\nSD standard deviation\nREFERENCES\n[1] D. Fitzgerald, J. Foody, D. Kelly, T. Ward, C. Markham, J. McDonald,\nand B. Caulfield, “Development of a wearable motion capture s uit and\nvirtual reality biofeedback system for the instruction and analysis of\nsports rehabilitation exercises,” Annual International Conference of the\nIEEE Engineering in Medicine and Biology - Proceedings , pp. 4870–\n4874, 2007.\n[2] K. Mills, “Motion analysis in the clinic: There’s an\napp for that,” pp. 49–50, jan 2015. [Online]. Available:\nhttp://dx.doi.org/10.1016/j.jphys.2014.11.014\n[3] J. J. Kavanagh and H. B. Menz, “Accelerometry: a techniqu e for\nquantifying movement patterns during walking,” Gait & posture , vol. 28,\nno. 1, pp. 1–15, 2008.\n[4] I. Weygers, M. Kok, M. Konings, H. Hallez, H. De Vroey, and K. Claeys,\n“Inertial sensor-based lower limb joint kinematics: A meth odological\nsystematic review,” Sensors (Switzerland) , vol. 20, no. 3, pp. 1–23, 2020.\n[5] L. De Baets, S. Vanbrabant, C. Dierickx, R. van der Straat en, and\nA. Timmermans, “Assessment of Scapulothoracic, Glenohume ral, and\nElbow Motion in Adhesive Capsulitis by Means of Inertial Sen sor Tech-\nnology: A Within-Session, Intra-Operator and Inter-Opera tor Reliability\nand Agreement Study,” Sensors (Switzerland) , vol. 20, no. 1, 2020.\n[6] E. Allseits, J. Luˇ carevi´ c, R. Gailey, V . Agrawal, I. Ga unaurd, and C. Ben-\nnett, “The development and concurrent validity of a real-ti me algorithm\nfor temporal gait analysis using inertial measurement unit s,”Journal\nof Biomechanics , vol. 55, pp. 27–33, apr 2017. [Online]. Available:\nhttps://linkinghub.elsevier.com/retrieve/pii/S00219 29017301124\n[7] J. F. Lin and D. Kuli´ c, “Human pose recovery using wirele ss inertial\nmeasurement units,” Physiological Measurement , vol. 33, no. 12, pp.\n2099–2115, 2012.\n[8] V . Bonnet, V . Joukov, D. Kuli´ c, P. Fraisse, N. Ramdani, a nd G. Venture,\n“Monitoring of hip and knee joint angles using a single inert ial mea-\nsurement unit during lower limb rehabilitation,” IEEE Sensors journal ,\nvol. 16, no. 6, pp. 1557–1564, 2015.\n[9] V . Joukov, V . Bonnet, M. Karg, G. Venture, and D. Kuli´ c, “ Rhythmic\nextended kalman filter for gait rehabilitation motion estim ation and\nsegmentation,” IEEE Transactions on Neural Systems and Rehabilitation\nEngineering , vol. 26, no. 2, pp. 407–418, 2017.\n[10] M. El-Gohary and J. McNames, “Human joint angle estimat ion with\ninertial sensors and validation with a robot arm,” IEEE Transactions on\nBiomedical Engineering , vol. 62, no. 7, pp. 1759–1767, 2015.\n[11] I. Weygers, M. Kok, H. De Vroey, T. Verbeerst, M. Verstey he, H. Hallez,\nand K. Claeys, “Drift-free inertial sensor-based joint kin ematics for long-\nterm arbitrary movements,” IEEE Sensors Journal , vol. 20, no. 14, pp.\n7969–7979, 2020.\n[12] W. Teufl, M. Miezal, B. Taetz, M. Fr¨ ohlich, and G. Bleser , “Validity,\ntest-retest reliability and long-term stability of magnet ometer free inertial\nsensor based 3d joint kinematics,” Sensors , vol. 18, no. 7, p. 1980, 2018.\n[13] P. M¨ uller, M.-A. S. T. B´ egin, and T. Seel, “Alignment- Free , Self-\nCalibrating Elbow Angles Measurement Using Inertial Senso rs,”IEEE\nJournal of Biomedical and Health Informatics , vol. 21, no. 2, pp. 312–\n319, 2017.\n[14] A. G. Cutti, A. Giovanardi, L. Rocchi, A. Davalli, and R. Sacchetti,\n“Ambulatory measurement of shoulder and elbow kinematics t hrough\ninertial and magnetic sensors,” Medical & biological engineering &\ncomputing , vol. 46, no. 2, pp. 169–178, 2008.\n[15] J. Favre, R. Aissaoui, B. M. Jolles, J. A. de Guise, and K. Aminian,\n“Functional calibration procedure for 3d knee joint angle d escription\nusing inertial sensors,” Journal of biomechanics , vol. 42, no. 14, pp.\n2330–2335, 2009.JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, SEPTEMBER 2020 11\n[16] T. Seel, J. Raisch, and T. Schauer, “IMU-Based joint ang le measurement\nfor gait analysis,” Sensors , pp. 6891–6909, 2014.\n[17] M. Crabolu, D. Pani, L. Raffo, and A. Cereatti, “Estimat ion of the\ncenter of rotation using wearable magneto-inertial sensor s,”Journal\nof Biomechanics , vol. 49, no. 16, pp. 3928–3933, 2016. [Online].\nAvailable: http://dx.doi.org/10.1016/j.jbiomech.2016 .11.046\n[18] E. Frick and S. Rahmatalla, “Joint center estimation us ing single-frame\noptimization: Part 1: Numerical simulation,” Sensors (Switzerland) ,\nvol. 18, no. 4, pp. 1–17, 2018.\n[19] P. M¨ uller, M.-A. B´ egin, T. Schauer, and T. Seel, “Alig nment-Free , Self-\nCalibrating Elbow Angles Measurement Using Inertial Senso rs,”IEEE\nJournal of Biomedical and Health Informatics , vol. 21, no. 2, pp. 312–\n319, 2017.\n[20] T. Seel and T. Schauer, “Joint Axis and Position Estimat ion from Inertial\nMeasurement Data by Exploiting Kinematic Constraints,” 2012 IEEE\nInternational Conference on Control Applications (CCA) , pp. 0–4, 2012.\n[21] R. S. McGinnis and N. C. Perkins, “Inertial sensor based method for\nidentifying spherical joint center of rotation,” Journal of Biomechanics ,\nvol. 46, no. 14, pp. 2546–2549, 2013.\n[22] F. Olsson and K. Halvorsen, “Experimental evaluation o f joint position\nestimation using inertial sensors,” 20th International Conference on\nInformation Fusion, Fusion 2017 - Proceedings , 2017.\n[23] M. Crabolu, D. Pani, L. Raffo, M. Conti, P. Crivelli, and A. Cereatti, “In\nvivo estimation of the shoulder joint center of rotation usi ng magneto-\ninertial sensors: MRI-based accuracy and repeatability as sessment,”\nBioMedical Engineering Online , vol. 16, no. 1, pp. 1–18, 2017.\n[24] S. Garc´ ıa de Villa, A. Jim´ enez Mart´ ın, and J. J. Garc´ ıa Dom´ ınguez,\n“Adaptive IMU-based Calibration of the Center of Joints for Movement\nAnalysis: One Case Study,” in IEEE International Symposium on\nMedical Measurements and Applications , Bari - online, 2020, pp. 1–10.\n[25] C. Huang, “On definitions of pitches and the finite screw s ystem for\ndisplacing a line,” in Proceedings of a symposium commemorating the\nlegacy, works and life of Sir Robert Stawell Ball upon the 100 th anniver-\nsary of A Treatise on the Theory of Screws. Cambridge: Univer sity of\nCambridge . Citeseer, 2000.\n[26] R. M. Ehrig, W. R. Taylor, G. N. Duda, and M. O. Heller, “A s urvey\nof formal methods for determining the centre of rotation of b all joints,”\nJournal of biomechanics , vol. 39, no. 15, pp. 2798–2809, 2006.\n[27] H. De Rosario, ´A. Page, and V . Mata, “Point of optimal\nkinematic error: Improvement of the instantaneous helical pivot\nmethod for locating centers of rotation,” Journal of Biomechanics ,\nvol. 47, no. 7, pp. 1742–1747, 2014. [Online]. Available:\nhttp://dx.doi.org/10.1016/j.jbiomech.2014.02.003\n[28] E. Frick and S. Rahmatalla, “Joint center estimation us ing single-frame\noptimization: Part 2: Experimentation,” Sensors (Switzerland) , vol. 18,\nno. 8, pp. 1–22, 2018.\n[29] E. M. Diaz, D. B. Ahmed, and S. Kaiser, “A review of indoor localization\nmethods based on inertial sensors,” in Geographical and Fingerprinting\nData to Create Systems for Indoor Positioning and Indoor/Ou tdoor\nNavigation . Elsevier, 2019, pp. 311–333.\n[30] P. Cerveri, A. Pedotti, and G. Ferrigno, “Robust recove ry of human\nmotion from video using kalman filters and virtual humans,” Human\nmovement science , vol. 22, no. 3, pp. 377–404, 2003.\n[31] S. Garcia de Villa, E. Munoz Diaz, D. Bousdar Ahmed, A. Ji menez Mar-\ntin, and J. J. Garcia Dominguez, “IMU-based Characterizati on of the\nLeg for the Implementation of Biomechanical Models,” in International\nConference on Indoor Positioning and Indoor Navigation - Pr oceedings ,\n2019.\n[32] “Xsens Motion Tracking,” https://www.xsens.com/ Last visit: 2020-01-\n15. [Online]. Available: https://www.xsens.com/\n[33] V . Camomilla, A. Cereatti, L. Ch` eze, and A. Cappozzo, “ A hip joint\nkinematics driven model for the generation of realistic thi gh soft tissue\nartefacts,” Journal of biomechanics , vol. 46, no. 3, pp. 625–630, 2013.\n[34] D. Calin, D. Tarnit ¸˘ a, D. Popa, D. Calafeteanu, and D. T arnita, “Virtual\nModel and Simulation of the Normal and Affected Human Hip Joi nt,”\nApplied Mechanics and Materials , vol. 823, no. January, pp. 167–172,\n2016.\n[35] M. Crabolu, D. Pani, L. Raffo, M. Conti, and A. Cereatti, “Functional\nestimation of bony segment lengths using magneto-inertial sensing:\nApplication to the humerus,” PLoS ONE , vol. 13, no. 9, pp. 1–11, 2018.\n[36] “Vicon Motion Capture,” https://www.vicon.com/ Last visit: 2020-01-28.\n[Online]. Available: https://www.vicon.com/"    },    {        "title": "2402.04252v1.EVA_CLIP_18B__Scaling_CLIP_to_18_Billion_Parameters.pdf",        "content": "EV A-CLIP-18B: Scaling CLIP to 18 Billion Parameters\nQuan Sun1*Jinsheng Wang1∗Qiying Yu1,2∗Yufeng Cui1\nFan Zhang1Xiaosong Zhang1Xinlong Wang1\n1Beijing Academy of Artificial Intelligence2Tsinghua University\n∗equal contribution\ncode & models: baaivision/EVA/EVA-CLIP-18B\nAbstract\nScaling up contrastive language-image pretraining\n(CLIP) is critical for empowering both vision and multi-\nmodal models. We present EVA-CLIP-18B , the largest\nand most powerful open-source CLIP model to date, with\n18-billion parameters. With only 6-billion training sam-\nples seen, EVA-CLIP-18B achieves an exceptional 80.7%\nzero-shot top-1 accuracy averaged across 27 widely rec-\nognized image classification benchmarks, outperforming\nits forerunner EVA-CLIP (5-billion parameters) and other\nopen-source CLIP models by a large margin. Remarkably,\nwe observe a consistent performance improvement with the\nmodel size scaling of EVA-CLIP , despite maintaining a\nconstant training dataset of 2-billion image-text pairs from\nLAION-2B and COYO-700M. This dataset is openly avail-\nable and much smaller than the in-house datasets (e.g., DFN-\n5B, WebLI-10B) employed in other state-of-the-art CLIP\nmodels. EVA-CLIP-18B demonstrates the potential of EVA-\nstyle [ 30,29,63] weak-to-strong visual model scaling. With\nour model weights made publicly available, we hope to fa-\ncilitate future research in vision and multimodal foundation\nmodels.\n1. Introduction\nRecent years witnessed the rapid growth of Large Mul-\ntimodal Models (LMMs) [ 3,64,62,69,5,46], with CLIP\nmodels [ 53,19,63,43,75,28,17] serving as a foundational\nvision encoder to deliver robust and transferable visual rep-\nresentations, and Large Language Models (LLMs) [ 65,54]\nserving as a general interface for reasoning across different\nmodalities. However, as LLMs have scaled up to around\n100B parameters or higher [ 11,20,65], the adopted vision\nfoundation models continue to operate at a much smaller\nscale, lagging far behind the LLMs.\n*Correspondence to wangxinlong@baai.ac.cn\nFigure 1: Scaling behavior of EV A-CLIP with zero-shot classi-\nfication performance averaged across 27 image classification\nbenchmarks , compared with the current state-of-the-art and largest\nCLIP models (224px). The diameter of each circle demonstrates\nthe forward GFLOPs ×the number of training samples seen. The\nperformance of EV A-CLIP consistently improves as scaling up.\nThis paper introduces EV A-CLIP-18B , the largest open-\nsource CLIP model with 18-billion parameters to narrow\nthis gap. EV A-CLIP [63] open-sources a series of effective\nand efficient CLIP models, which have been leveraged as the\nvision foundation by numerous impactful works across 2D /\n3D vision and multimodal modeling [ 42,78,77,50,69,64].\nWe further scale up EV A-CLIP to this significant parameter\nsize building upon the scaling philosophy of EV A [30,29]\nandEV A-CLIP [63]. With merely 6-billion training samples\nseen and trained on publicly available datasets, EV A-CLIP-\n18B achieves the exceptional 80.7% average zero-shot top-\n1 accuracy on 27 widely recognized image classification\nbenchmarks, significantly surpassing its forerunner EV A-02-\nCLIP -E/14+ (5-billion parameters) and other open-source\nCLIP models. Besides, the models have not exhibited any\n1arXiv:2402.04252v1  [cs.CV]  6 Feb 2024total image text samples image batch image cls. video cls. retrieval\nmodel + #param. #param. #param. data seen size size gpus for training avg. acc. avg. acc. MR\nEV A-01-CLIP -g/14+ [63] 1.1B 1.0B 124M LAION-400M [59] 11B 224241k 256×A100 (40GB) 72.2 66.2 80.9\nEV A-01-CLIP -g/14+ [63] 1.3B 1.0B 354M Merged-2B [63] 11B 2242114k 112×A100 (40GB) 75.1 68.8 85.3\nOpenCLIP-G/14+ [2] 2.5B 1.8B 695M LAION-2B [58] 39B 2242160k 512 ×A100 (80GB) 76.2 68.7 85.7\nInternVL-C+ [17] 14.0B 6.0B 8.0B custom [17] 29B 2242164k 640 ×A100 (80GB) 78.0 73.7 86.6\nDFN5B-CLIP-H/14+ [28] 1.0B 632M 354M DFN-5B [28] 39B 224279k TPUv4 78.3 67.0 86.6\nEV A-02-CLIP -E/14+ [63] 5.0B 4.4B 695M LAION-2B [58] 9B 2242144k 144×A100 (80GB) 78.7 72.1 85.7\nDFN5B-CLIP-H/14+ [28] 1.0B 632M 354M DFN-5B [28] 5B 378279k TPUv4 79.2 68.4 87.2\nEV A-CLIP-8B + 8.1B 7.5B 695M Merged-2B [63] 9B 2242178k 384×A100 (40GB) 79.4 73.6 86.2\nEV A-CLIP-18B + 18.1B 17.5B 695M Merged-2B+ 6B 2242108k 360×A100 (40GB) 80.7 75.0 87.8\nTable 1: CLIP model configurations and zero-shot performance on 33 benchmarks including 27 image classification, 4 video\nclassification and 2 image-text retrieval datasets. DFN-5B [ 28] are 5B images filtered from a pool of 43B uncurated image-text pairs\nconsisting of 12.8B image-text pairs from CommonPool-12.8B [ 32] and 30B additional public image-text pairs. The dataset used for training\nInternVL-C [17] is custom mixtures, see detail in [17].\nmethod\nImageNet-1K [26]\nImageNet-V2 [57]\nImageNet-Adv. [36]\nImageNet-Ren. [35]\nImageNet-Ske. [68]\nObjectNet [8]\nCIFAR-10 [40]\nCIFAR-100 [40]\nMNIST [41]\nCaltech101 [31]\nSUN397 [72]\nFGVC Aircraft [47]\nCountry-211 [53]\nStanford Cars [39]\nBirdsnap [9]\nDTD [21]\nEuroSAT [34]\nFER2013 [33]\nFlowers-102 [49]\nFood-101 [10]\nGTSRB [61]\nPCam [67]\nPets [51]\nRendered SST2 [53]\nRESISC45 [18]\nSTL-10 [23]\nVOC2007 [27]\n.avg. top-1 acc.\nEV A-01-CLIP -g/14+ 78.5 71.5 73.6 92.5 67.6 72.3 98.3 88.7 62.6 87.7 74.2 32.4 28.9 91.7 65.8 61.7 73.8 52.2 74.5 93.5 49.3 49.9 94.2 58.4 70.3 98.9 85.7 72.2\nEV A-01-CLIP -g/14+ 79.3 72.5 74.1 92.7 68.4 75.3 99.1 90.1 72.0 89.5 74.7 39.9 31.8 90.7 70.2 67.8 73.2 56.0 79.7 93.7 66.5 62.4 94.9 58.6 71.4 99.5 84.7 75.1\nOpenCLIP-G/14+ 80.4 73.6 69.3 92.8 69.9 73.0 98.3 87.5 71.6 89.4 75.0 53.6 34.9 94.9 73.0 69.1 71.1 59.6 81.5 93.1 62.7 63.6 95.3 65.3 72.6 98.5 87.4 76.2\nInternVL-C + 83.2 77.3 83.8 95.7 74.3 80.6 99.4 93.1 80.6 89.5 76.3 53.3 35.1 94.4 69.2 70.8 79.4 56.2 85.8 95.3 65.5 48.7 96.3 68.4 74.4 99.4 80.0 78.0\nDFN5B-CLIP-H/14 + 83.5 77.4 71.7 92.9 72.8 76.7 98.8 90.5 85.8 89.5 77.0 71.4 34.4 95.8 77.4 70.7 65.2 54.7 92.5 95.8 67.7 65.2 96.5 54.8 76.1 98.9 81.5 78.3\nEV A-02-CLIP -E/14+ 82.1 75.7 82.1 94.7 72.2 79.6 99.3 93.2 74.7 90.5 75.3 58.7 37.0 94.7 77.6 68.2 75.9 59.0 84.5 94.9 67.7 64.4 96.0 62.6 75.7 99.3 87.9 78.7\nDFN5B-CLIP-H/14+ 84.3 78.3 79.6 93.6 73.3 79.6 98.8 90.5 83.6 88.9 77.4 72.5 37.9 96.0 80.5 70.9 61.1 56.1 91.6 96.2 67.9 69.6 96.8 55.5 75.9 99.1 81.9 79.2\nEV A-CLIP-8B + 83.5 77.7 85.2 95.3 74.3 81.2 99.3 92.3 84.8 89.6 76.2 60.5 41.7 94.8 79.0 71.0 68.9 56.1 86.4 95.5 70.9 58.1 96.4 66.2 75.3 99.3 85.1 79.4\nEV A-CLIP-18B + 83.8 77.9 87.3 95.7 74.7 82.2 99.4 93.8 83.0 89.8 77.7 59.7 43.1 94.9 79.9 72.1 79.8 59.3 86.0 95.8 72.4 65.2 96.1 67.5 76.9 99.6 85.8 80.7\nTable 2: EV A-CLIP zero-shot image classification performance on 27 datasets. We report top-1 accuracy on all datasets. The best results\nare in bold and the second best are underlined .\nsignal of performance saturation , shedding light on further\nscaling of vision models. An intuitive demonstration is\nshown in Figure 1.\nThe successful training of EV A-CLIP-18B exemplifies\nthe potential of the EV A-style visual model scaling philoso-\nphy. We keep open-sourcing the training code and weights of\nour models to encourage further research and empower the\ndevelopment of vision and multimodal foundation models.\n2. Weak-to-Strong Vision Scaling\nOur scaling-up procedure is guided by the principles of\nEV A [30] and EV A-CLIP [63]. The EV A philosophy for\nscaling visual models follows a weak-to-strong paradigm,\ndesigned to improve visual models through a strategic pro-\ngression. This process begins with a large EV A vision model\ndistilling knowledge from a small EV A-CLIP model, which\nin turn serves as the vision encoder initialization to stabilize\nand accelerate the training of a larger EV A-CLIP . After\nthat, the closed-loop scaling-up cycle continues and a larger\nEV A is distilled out. Throughout our model scaling cycle,\nthe training dataset remains largely fixed to demonstrate theeffectiveness of our model-scale specific scaling philosophy,\nalthough scaling up datasets can further unleash the scaling\npower of our method.\nSpecifically, in this work, we pre-train a large EV A model\nnamed EV A -18B using a small EV A-CLIP (EV A-02-CLIP -\nE/14+) [ 63] as the teacher, which is trained to reconstruct\nmasked image-text aligned vision features from EV A-02-\nCLIP -E/14+. EV A -18B omits bias terms of QKV projec-\ntions and uses RMSNorm [ 76] instead of LayerNorm [ 4]\nfollowing LLaMA [ 65]. Subsequently, we leverage the EV A\nmodel as the vision encoder initialization for EV A-CLIP\npre-training with the image-text contrastive learning objec-\ntive. Besides, we also introduce a smaller counterpart, EV A-\nCLIP-8B , which undergoes similar pre-training methodolo-\ngies. Notably, our experiments demonstrate sustained perfor-\nmance improvement with the progressive weak-teach-strong\nscaling up of EV A-CLIP .\n3. Experiments\nSettings. Following EV A-CLIP [63], we initialized the\nmodel with pre-trained vision and text encoders. Specifi-\n2zero-shot textretrieval zero-shot image retrieval\nFlickr30K COCO Flickr30K COCO\nmethod + R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 MR\nEV A-01-CLIP -g/14 + 87.9 98.0 99.5 61.7 83.2 89.9 72.5 91.5 95.4 44.5 69.1 77.7 80.9\nEV A-01-CLIP -g/14+ 93.3 99.5 99.9 69.4 88.3 93.2 79.2 95.2 97.3 51.1 74.7 82.5 85.3\nOpenCLIP-G/14+ 93.5 99.3 99.7 69.0 87.8 93.1 80.9 95.1 97.2 52.6 76.1 83.6 85.7\nEV A-02-CLIP -E/14+ 94.3 99.6 99.8 69.4 88.6 93.3 79.7 94.9 97.3 52.5 75.9 83.4 85.7\nEV A-CLIP-8B + 95.6 99.6 99.9 70.3 89.3 93.9 80.8 95.5 97.6 53.0 76.0 83.4 86.2\nDFN5B-CLIP-H/14 + 92.9 99.3 99.9 72.3 90.2 94.6 80.1 95.2 97.3 53.9 78.0 85.6 86.6\nInternVL-C + 93.8 99.7 100.0 70.3 89.2 93.8 82.1 96.0 98.1 54.1 77.1 84.8 86.6\nDFN5B-CLIP-H/14 + 93.6 99.3 99.6 71.8 90.4 94.9 82.1 96.0 97.9 55.6 79.2 86.3 87.2\nEV A-CLIP-18B + 96.7 99.7 100.0 73.6 90.9 95.0 83.3 96.3 98.3 56.2 78.5 85.6 87.8\nTable 3: Zero-shot retrieval performance on Flickr30K [74] and COCO [45].\ncally, we employ a pre-trained EV A -18B [ 30,29] as the\nvision encoder and EV A-02-CLIP -E/14+ [ 63] for the text\nencoder. We adopt the LAMB optimizer [ 73] with β1= 0.9,\nβ2=0.95, and a weight decay of 0. We apply different learn-\ning rates and layer decay rates to the vision encoder and text\nencoder to ensure optimal training. We set the peak learning\nrate as 4e-4 and 4e-5 for the vision encoder and the text\nencoder respectively, with 2000 warm-up steps. Afterwards,\nthe learning rates decay to 0 with a cosine schedule. The\nlearning rate layer decay rates are configured as 0.9 and 0.75\nfor the vision and text encoders. The temperature parameter\nremains constant at 0.01. Further, we use the DeepSpeed\noptimization library [ 56] with ZeRO stage-3 partition [ 55],\ngradient checkpointing [ 16] and flash attention [ 24] to opti-\nmize the training cost.\nDataset. Our Merged-2B dataset consists of 1.6 billion sam-\nples from LAION-2B [ 58] and 0.4 billion samples from\nCOYO-700M [ 12]. Note that the use of a subset from\nLAION-2B is not the result of deliberate filtering, but rather\ndue to image downloading failures. The use of 0.4 billion\nCOYO-700M samples aims to complement the number of\ntraining samples to nearly the same as LAION-2B. Merged-\n2B+ consists of all samples from Merged-2B, along with ad-\nditional 20 million samples from LAION-COCO [ 1] and 23\nmillion samples from Merged-video including VideoCC [ 48],\nInternVid [ 70] and WebVid-10M [ 6]. Merged-video is in-\ncluded at the end of the training process.\nEV A-CLIP-18B pre-trains with 5.4 billion samples from\nMerged-2B seen with 50% of patch dropout ratio [ 44], 0.6\nbillion samples from Merged-2B and 20 million samples\nfrom LAION-COCO without patch dropout, and 24 million\nsamples from Merged-video with 50% of patch dropout ratio.\nEvaluation. We evaluate on 33 widely used datasets across\nimage, video classification and image-text retrieval. All\ndatasets used to evaluate EV A-CLIP-18B are reported in\nTable 11. We utilize the specified prompt templates follow-\ning [53, 38].image encoder text encoder # params\nmethod layers width heads layers width heads image text total\nEV A-CLIP-8B + 32 4096 32 32 1280 20 7.5B 695M 8.1B\nEV A-CLIP-18B +48 5120 40 32 1280 20 17.5B 695M 18.1B\nTable 4: Architecture configurations.\nZero-Shot Image Classification. We show the exceptional\nperformance of EV A-CLIP on all 27 zero-shot image classi-\nfication benchmarks in Table 2. EV A-CLIP-18B achieves\n80.7% top-1 accuracy averaged across all 27 benchmarks.\nThese results significantly outperform the previous best open-\nsourced DFN5B-CLIP-H/14+ [ 28] by+1.5% , and the largest\nexisting CLIP model, InternVL-C [ 17], by +2.7% . For Bird-\nsnap dataset, the download was limited to 2195 test images\ndue to broken links.\nmethod+ #Frames UCF-101 K-400 K-600 K-700 avg.\nEV A-01-CLIP -g/14+ 1 76.0 65.4 64.5 58.8 66.2\nDFN5B-CLIP-H/14+ 1 78.2 65.2 65.5 59.2 67.0\nDFN5B-CLIP-H/14+ 1 79.2 66.7 67.0 60.7 68.4\nOpenCLIP-G/14+ 1 80.5 67.1 66.9 60.3 68.7\nEV A-01-CLIP -g/14+ 1 78.9 67.3 67.3 61.5 68.8\nEV A-02-CLIP -E/14+ 1 83.1 70.7 70.0 64.4 72.1\nEV A-CLIP-8B + 1 85.7 71.3 71.2 66.1 73.6\nInternVL-C + 1 85.2 71.8 71.7 66.4 73.7\nEV A-CLIP-18B + 1 86.0 72.9 72.9 68.2 75.0\nEV A-CLIP-18B + 8 88.2 79.3 79.2 72.1 79.7\nEV A-CLIP-18B + 16 88.4 79.4 79.4 72.2 79.8\nTable 5: EV A-CLIP zero-shot video classification performance.\nWe report top1 accuracy for UCF-101 [ 60], average of top1\nand top5 accuracy for Kinetics-400 [ 15], Kinetics-600 [ 13] and\nKinetics-700 [14].\nZero-Shot Video Classification. We report the top-1 accu-\nracy for UCF-101 [ 60] and the mean of top-1 and top-5 accu-\nracy for Kinetics-400 [ 15], Kinetics-600 [ 13] and Kinetics-\n700 [ 14]. In Table 5 we demonstrate that EV A-CLIP-18B\nalso outperforms other CLIP models on zero-shot video clas-\n3method+ IN-1K IN-A IN-R IN-V2 IN-Sketch ObjectNet ∆↓avg. acc.\nDFN5B-CLIP-H/14+ 83.5 71.7 92.9 77.4 72.8 76.7 4.4 79.2\nOpenCLIP-G/14+ 80.4 69.3 92.8 73.6 69.9 73.0 3.9 76.5\nSigLIP-SO [75] (reported) 82.0 71.9 95.1 76.1 74.0 70.6 3.7 78.3\nDFN5B-CLIP-H/14+ 84.3 79.6 93.6 78.3 73.3 79.6 2.8 81.5\nEV A-01-CLIP -g/14 + 78.5 73.6 92.5 71.5 67.6 72.3 2.5 76.0\nEV A-01-CLIP -g/14+ 79.3 74.1 92.7 72.5 68.4 75.3 2.2 77.1\nBASIC-L [52] (reported) 85.7 85.6 95.7 80.6 76.1 82.3 1.4 84.3\nSigLIP-SO+ [75] (reported) 83.0 82.5 95.8 77.2 74.5 77.0 1.3 81.7\nEV A-02-CLIP -E/14+ 82.1 82.1 94.7 75.7 72.2 79.6 1.0 81.1\nInternVL-C + 83.2 83.8 95.7 77.3 74.3 80.6 0.7 82.5\nEV A-CLIP-8B + 83.5 85.2 95.3 77.7 74.3 81.2 0.6 82.9\nEV A-CLIP-18B + 83.8 87.3 95.7 77.9 74.7 82.2 0.2 83.6\n(a)Zero-shot performance on ImageNet variants and ObjectNet. “avg. acc.”: the averaged top-1 accuracy on different ImageNet variants ( i.e., IN-{1K, V2,\nReaL, Adv., Ren., Ske. }), and ObjectNet. “ ∆↓”: The gap between the averaged top-1 accuracy and the ImageNet-1K top-1 accuracy (the lower the better).\nEV A-CLIP suffers from the smallest performance drop (only 0.2% top-1 accuracy gap for EV A-CLIP-18B ) while EV A-CLIP-18B achieves 83.6% top-1\naccuracy averaged on all 6 benchmarks.\nmethod\nImageNet-1K [26]\nImageNet-V2 [57]\nImageNet-Adv. [36]\nImageNet-Ren. [35]\nImageNet-Ske. [68]\nObjectNet [8]\nCIFAR-10 [40]\nCIFAR-100 [40]\nMNIST [41]\nSUN397 [72]\nBirdsnap [9]\nDTD [21]\nEuroSAT [34]\nFood-101 [10]\nPCam [67]\nRESISC45 [18]\nSTL-10 [23]\n.avg. top-1 acc.\nBASIC-L [52] (reported) 85.7 80.6 85.6 95.7 76.1 82.3 97.5 82.3 40.3 76.2 59.2 64.6 51.0 95.1 59.6 72.7 99.6 76.7 (77.8)\nEV A-CLIP-18B + 83.8 77.9 87.3 95.7 74.7 82.2 99.4 93.8 83.0 77.7 79.9 72.1 79.8 95.8 65.2 76.9 99.6 84.9 (84.1)\n-1.9 -2.7 +1.7 +0.0 -1.4 -0.1 +1.9 +11.5 +42.7 +1.5 +20.7 +7.5 +28.8 +0.7 +5.6 +4.2 +0.0 +8.2 (+6.3)\n(b)Comparison EV A-CLIP-18B’s zero-shot image classification performance with BASIC-L [ 52] on 17 datasets. Our report includes the top-1 accuracy\nfor all datasets, considering that BASIC-L only provided top-1 accuracy for these specific 17 datasets. ( ) is the average top-1 accuracy removing Birdsnap due\nto the different test size between EV A-CLIP-18B and BASIC-L. EV A-CLIP-18B outperforms BASIC-L with a notable margin of +8.2 (+6.3) in average top-1\naccuracy, despite exhibiting lower performance on ImageNet variants.\nTable 6: Robustness evaluation of CLIP models and comparison with BASIC-L [52] on 17 Benchmarks.\nsification benchmarks by a large margin. When sampling\na single center frame per video, EV A-CLIP-18B achieves\naccuracies of 86.0%, 72.9%, 72.9%, and 68.2% across the\nfour evaluated benchmarks. Further, when uniformly sample\n8 or 16 frames per video, we observe an improvement of\n+4.7% /+4.8% averaged across four benchmarks compared\nto the single-frame setting.\nZero-Shot Image-Text Retrieval. In Table 3, we report the\nzero-shot image and text retrieval results on Flickr30K [ 74]\nand COCO [ 45].EV A-CLIP-18B achieves an average re-\ncall of 87.8% across all retrieval benchmarks, significantly\noutperforming competitors.\nRobustness. In Table 6, we demonstrate that scaling up\nEV A-CLIP significantly enhances the robustness of visual\nrepresentations. EV A-CLIP suffers from the smallest per-\nformance drop ( ∆↓) between ImageNet-1K and ImageNet\nvariants including adversarial ones, with merely 0.2% top-1\naccuracy gap for EV A-CLIP-18B .\nFor a more robust and comprehensive evaluation of ro-\nbustness and zero-shot performance, it is advisable to include\nmore benchmarks covering more image distributions. How-\never, we want to note that higher ImageNet top-1 accuracydoes not necessarily lead to better overall performance, as\nevidenced in Table 6b, where BASIC-L [ 52] exhibits higher\nImageNet-related top-1 accuracy but considerably lower\noverall average top-1 accuracy compared to EV A-CLIP-\n18B across a broader range of datasets and distributions,\nshowing a difference of -8.2%.\nLinear Probing on ImageNet-1K. In Table 7, we present\nthe results of linear probing on ImageNet-1K [ 26].EV A-\nCLIP-18B achieves an average top-1 accuracy of 88.9%,\nsurpassing InternVL-C [17] by 0.7%.\nmethod #param top1 acc.\nOpenCLIP-G/14 (reported) 1.8B 86.2\nEV A-01-CLIP -g/14 + 1.0B 86.5\nEV A-02-CLIP -E/14+ 4.4B 88.1\nInternVL-C (reported) 5.9B 88.2\nEV A-CLIP-8B + 7.5B 88.5\nEV A-CLIP-18B + 17.5B 88.9\nTable 7: Linear Probing on ImageNet-1K [ 26].The top-1 ac-\ncuracy shows a continuous improvement with the scaling up of\nEV A-CLIP .\n43D Representation. We adopt the Uni3D [ 77] setting to\nexplore the effectiveness of scaling up teachers. With the\nscaling up of EV A-CLIP in Table 8, we observe consis-\ntent improvements in 3D representation learning capabili-\nties. Further, Uni3D-base equipped with EV A-CLIP-18B\nsets new records on ModelNet [ 71] and ScanObjectNN [ 66]\nbenchmarks.\nteacher data O-LVIS MNet40 ScanObjNN\nOpenCLIP-G/14 + w/o LVIS 44.5 85.8 58.9\nEV A-02-CLIP -E/14+ w/o LVIS 45.8 86.1 61.7\nEV A-CLIP-8B + w/o LVIS 46.2 87.3 62.7\nEV A-CLIP-18B + w/o LVIS 47.0 87.6 65.3\nEV A-02-CLIP -E/14+ Ensembled 51.7 86.3 63.8\nEV A-CLIP-18B + Ensembled 53.2(+1.5) 88.6(+2.3) 67.8(+4.0)\nTable 8: EV A-CLIP-18B enhances zero-shot 3d classification\nperformance. We use Uni3D-base [ 77] as the baseline and scale\nthe teacher from 5B to 18B. We report top-1 accuracy on Objaverse-\nLVIS [25], ModelNet40 [71] and ScanObjectNN [66].\n4. Ablation Studies\nVideo Data. In Table 9, we conduct ablations on EV A-\nCLIP-18B ’s zero-shot performance, comparing results when\ntrained with and without Merged-Video. The training ob-\njective for the video data aligns with that of images, encom-\npassing the extraction of features from video where 8 frames\nare uniformly sampled. The mean of all [CLS] embeddings\nserves as a representation for the video. The outcomes reveal\nsubstantial performance improvements associated with train-\ning using Merged-Video. The zero-shot performance, aver-\naged across UCF-101 [ 60] and Kinetics-400 [ 15] / 600 [ 13]\n/ 700 [ 14], indicates a gain of +0.7 for evaluation with one\nmiddle frame and +0.8 for evaluation with 8 frames.\nclassification retrieval\nimage video (#F 1) video (#F 8) avg. recall\nw/o video data 80.7 74.3 78.9 87.9\nw/ video data 80.7 75.0 ( +0.7) 79.7 ( +0.8)87.8 ( -0.1)\nTable 9: Video data enhances zero-shot video classification per-\nformance . We respectively report performances averaged on 27\nimage classification benchmarks, 4 video benchmarks and 2 image-\ntext retrieval benchmarks.\nImage Resolution. In Table 10, we investigate the impact of\nlarger image resolutions on zero-shot performance. Notably,\nthere is an average top-1 accuracy gain of +0.9 when the\nresolution increases from 2242to 4482forEV A-CLIP-8B .\nSimilarly, an increase from 2242to 3362results in a gain of\n+0.5, even when trained with low global batch sizes of 24k\nforEV A-CLIP-8B + and 23k for EV A-CLIP-18B +.method+ resolution IN-1K IN-A IN-R IN-V2 IN-Ske. ObjectNet avg.\nEV A-CLIP-8B +224×224 83.5 85.2 95.3 77.7 74.3 81.2 82.9\nEV A-CLIP-8B +448×448 83.8 88.7 95.4 77.7 74.1 82.9 83.8\n+0.3 +3.5 +0.1 +0.0 -0.2 +1.7 +0.9\nEV A-CLIP-18B +224×224 83.8 87.3 95.7 77.9 74.7 82.2 83.6\nEV A-CLIP-18B +336×336 83.9 88.9 95.6 78.2 74.3 83.6 84.1\n+0.1 +1.6 -0.1 +0.3 -0.4 +1.4 +0.5\nTable 10: Increasing resolution. We report zero-shot performance\non ImageNet variants and ObjectNet.\n5. Conclusion\nWe present EV A-CLIP-18B , the currently largest and\nmost performant open-sourced CLIP model with 18-billion\nparameters. We show that following EV A ’s weak-to-strong\nvision scaling principle, we can further scale up CLIP mod-\nels to a new record and advance SOTA on multiple prevalent\nbenchmarks across image, video and 3D domains. Impor-\ntantly, we demonstrate that scaling up the size of EV A-CLIP\nmodels consistently boosts performance with no sign of sat-\nuration, shedding light on future vision model scaling.\nReferences\n[1]Laion coco: 600m synthetic captions from laion2b-en. https:\n//laion.ai/blog/laion-coco/ . 3\n[2]Reaching 80 zero-shot accuracy with openclip: Vit-g/14 trained on\nlaion-2b. https://laion.ai/blog/giant-openclip/ .\n2\n[3]Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech,\nIain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katie Millican,\nMalcolm Reynolds, et al. Flamingo: a visual language model for\nfew-shot learning. arXiv preprint arXiv:2204.14198 , 2022. 1\n[4]Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer\nnormalization. arXiv preprint arXiv:1607.06450 , 2016. 2\n[5]Jinze Bai, Shuai Bai, Shusheng Yang, Shijie Wang, Sinan Tan, Peng\nWang, Junyang Lin, Chang Zhou, and Jingren Zhou. Qwen-vl: A\nversatile vision-language model for understanding, localization, text\nreading, and beyond. arXiv preprint arXiv:2308.12966 , 2023. 1\n[6]Max Bain, Arsha Nagrani, G ¨ul Varol, and Andrew Zisserman.\nFrozen in time: A joint video and image encoder for end-to-end\nretrieval. In Proceedings of the IEEE/CVF International Conference\non Computer Vision , pages 1728–1738, 2021. 3\n[7]Hangbo Bao, Li Dong, and Furu Wei. Beit: Bert pre-training of\nimage transformers. arXiv preprint arXiv:2106.08254 , 2021. 9\n[8]Andrei Barbu, David Mayo, Julian Alverio, William Luo, Christo-\npher Wang, Dan Gutfreund, Josh Tenenbaum, and Boris Katz. Ob-\njectnet: A large-scale bias-controlled dataset for pushing the limits\nof object recognition models. In NeurIPS , 2019. 2, 4, 9, 10\n[9]Thomas Berg, Jiongxin Liu, Seung Woo Lee, Michelle L Alexander,\nDavid W Jacobs, and Peter N Belhumeur. Birdsnap: Large-scale\nfine-grained visual categorization of birds. In CVPR , 2014. 2, 4, 9,\n10\n[10] Lukas Bossard, Matthieu Guillaumin, and Luc Van Gool. Food-101–\nmining discriminative components with random forests. In ECCV ,\n2014. 2, 4, 9, 10\n5[11] Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah,\nJared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav\nShyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel\nHerbert-V oss, Gretchen Krueger, Tom Henighan, Rewon Child,\nAditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter,\nChristopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott\nGray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCan-\ndlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language\nmodels are few-shot learners. arXiv preprint arXiv:2005.14165 ,\n2020. 1\n[12] Minwoo Byeon, Beomhee Park, Haecheon Kim, Sungjun Lee,\nWoonhyuk Baek, and Saehoon Kim. Coyo-700m: Image-\ntext pair dataset. https://github.com/kakaobrain/\ncoyo-dataset , 2022. 3\n[13] Joao Carreira, Eric Noland, Andras Banki-Horvath, Chloe Hillier,\nand Andrew Zisserman. A short note about kinetics-600. arXiv\npreprint arXiv:1808.01340 , 2018. 3, 5, 9\n[14] Joao Carreira, Eric Noland, Chloe Hillier, and Andrew Zisserman. A\nshort note on the kinetics-700 human action dataset. arXiv preprint\narXiv:1907.06987 , 2019. 3, 5, 9\n[15] Joao Carreira and Andrew Zisserman. Quo vadis, action recogni-\ntion? a new model and the kinetics dataset. In CVPR , 2017. 3, 5,\n9\n[16] Tianqi Chen, Bing Xu, Chiyuan Zhang, and Carlos Guestrin. Train-\ning deep nets with sublinear memory cost, 2016. 3\n[17] Zhe Chen, Jiannan Wu, Wenhai Wang, Weijie Su, Guo Chen, Sen\nXing, Muyan Zhong, Qinglong Zhang, Xizhou Zhu, Lewei Lu, Bin\nLi, Ping Luo, Tong Lu, Yu Qiao, and Jifeng Dai. Internvl: Scaling up\nvision foundation models and aligning for generic visual-linguistic\ntasks. arXiv preprint arXiv:2312.14238 , 2023. 1, 2, 3, 4\n[18] Gong Cheng, Junwei Han, and Xiaoqiang Lu. Remote sensing image\nscene classification: Benchmark and state of the art. Proceedings of\nthe IEEE , 2017. 2, 4, 9, 10\n[19] Mehdi Cherti, Romain Beaumont, Ross Wightman, Mitchell Worts-\nman, Gabriel Ilharco, Cade Gordon, Christoph Schuhmann, Ludwig\nSchmidt, and Jenia Jitsev. Reproducible scaling laws for contrastive\nlanguage-image learning, 2022. 1\n[20] Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten\nBosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won\nChung, Charles Sutton, Sebastian Gehrmann, Parker Schuh, Kensen\nShi, Sasha Tsvyashchenko, Joshua Maynez, Abhishek Rao, Parker\nBarnes, Yi Tay, Noam Shazeer, Vinodkumar Prabhakaran, Emily\nReif, Nan Du, Ben Hutchinson, Reiner Pope, James Bradbury, Ja-\ncob Austin, Michael Isard, Guy Gur-Ari, Pengcheng Yin, Toju\nDuke, Anselm Levskaya, Sanjay Ghemawat, Sunipa Dev, Hen-\nryk Michalewski, Xavier Garcia, Vedant Misra, Kevin Robinson,\nLiam Fedus, Denny Zhou, Daphne Ippolito, David Luan, Hyeon-\ntaek Lim, Barret Zoph, Alexander Spiridonov, Ryan Sepassi, David\nDohan, Shivani Agrawal, Mark Omernick, Andrew M. Dai, Thanu-\nmalayan Sankaranarayana Pillai, Marie Pellat, Aitor Lewkowycz,\nErica Moreira, Rewon Child, Oleksandr Polozov, Katherine Lee,\nZongwei Zhou, Xuezhi Wang, Brennan Saeta, Mark Diaz, Orhan\nFirat, Michele Catasta, Jason Wei, Kathy Meier-Hellstern, Douglas\nEck, Jeff Dean, Slav Petrov, and Noah Fiedel. Palm: Scaling lan-\nguage modeling with pathways. arXiv preprint arXiv:2204.02311 ,\n2022. 1\n[21] M. Cimpoi, S. Maji, I. Kokkinos, S. Mohamed, , and A. Vedaldi.\nDescribing textures in the wild. In CVPR , 2014. 2, 4, 9, 10\n[22] Kevin Clark, Minh-Thang Luong, Quoc V Le, and Christopher D\nManning. ELECTRA: Pre-training text encoders as discriminators\nrather than generators. arXiv preprint arXiv:2003.10555 , 2020. 9[23] Adam Coates, Andrew Ng, and Honglak Lee. An analysis of single-\nlayer networks in unsupervised feature learning. In AISTAT , 2011.\n2, 4, 9, 10\n[24] Tri Dao, Daniel Y . Fu, Stefano Ermon, Atri Rudra, and Christopher\nR´e. Flashattention: Fast and memory-efficient exact attention with\nio-awareness, 2022. 3\n[25] Matt Deitke, Dustin Schwenk, Jordi Salvador, Luca Weihs, Oscar\nMichel, Eli VanderBilt, Ludwig Schmidt, Kiana Ehsani, Aniruddha\nKembhavi, and Ali Farhadi. Objaverse: A universe of annotated 3d\nobjects. In Proceedings of the IEEE/CVF Conference on Computer\nVision and Pattern Recognition , pages 13142–13153, 2023. 5\n[26] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li\nFei-Fei. Imagenet: A large-scale hierarchical image database. In\nCVPR , 2009. 2, 4, 9, 10\n[27] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn,\nand A. Zisserman. ”the PASCAL Visual Object Classes\nChallenge 2007 (VOC2007) Results. ”http://www.pascal-\nnetwork.org/challenges/VOC/voc2007/workshop/index.html”, 2007.\n2, 9, 10\n[28] Alex Fang, Albin Madappally Jose, Amit Jain, Ludwig Schmidt,\nAlexander Toshev, and Vaishaal Shankar. Data filtering networks.\narXiv preprint arXiv:2309.17425 , 2023. 1, 2, 3\n[29] Yuxin Fang, Quan Sun, Xinggang Wang, Tiejun Huang, Xinlong\nWang, and Yue Cao. Eva-02: A visual representation for neon\ngenesis. arXiv preprint arXiv:2303.11331 , 2023. 1, 3\n[30] Yuxin Fang, Wen Wang, Binhui Xie, Quan Sun, Ledell Wu, Xing-\ngang Wang, Tiejun Huang, Xinlong Wang, and Yue Cao. Eva:\nExploring the limits of masked visual representation learning at\nscale. arXiv preprint arXiv:2211.07636 , 2022. 1, 2, 3\n[31] Li Fei-Fei, Rob Fergus, and Pietro Perona. Learning generative\nvisual models from few training examples: An incremental bayesian\napproach tested on 101 object categories. In CVPRW , 2004. 2, 9, 10\n[32] Samir Yitzhak Gadre, Gabriel Ilharco, Alex Fang, Jonathan Hayase,\nGeorgios Smyrnis, Thao Nguyen, Ryan Marten, Mitchell Wortsman,\nDhruba Ghosh, Jieyu Zhang, Eyal Orgad, Rahim Entezari, Gian-\nnis Daras, Sarah Pratt, Vivek Ramanujan, Yonatan Bitton, Kalyani\nMarathe, Stephen Mussmann, Richard Vencu, Mehdi Cherti, Ran-\njay Krishna, Pang Wei Koh, Olga Saukh, Alexander Ratner, Shu-\nran Song, Hannaneh Hajishirzi, Ali Farhadi, Romain Beaumont,\nSewoong Oh, Alex Dimakis, Jenia Jitsev, Yair Carmon, Vaishaal\nShankar, and Ludwig Schmidt. Datacomp: In search of the next gen-\neration of multimodal datasets. arXiv preprint arXiv:2304.14108 ,\n2023. 2\n[33] Ian J Goodfellow, Dumitru Erhan, Pierre Luc Carrier, Aaron\nCourville, Mehdi Mirza, Ben Hamner, Will Cukierski, Yichuan\nTang, David Thaler, Dong-Hyun Lee, et al. Challenges in repre-\nsentation learning: A report on three machine learning contests. In\nICONIP , 2013. 2, 9, 10\n[34] Patrick Helber, Benjamin Bischke, Andreas Dengel, and Damian\nBorth. Eurosat: A novel dataset and deep learning benchmark for\nland use and land cover classification. IEEE J. Sel. Top. Appl. Earth\nObs. Remote Sens. , 2019. 2, 4, 9, 10\n[35] Dan Hendrycks, Steven Basart, Norman Mu, Saurav Kadavath,\nFrank Wang, Evan Dorundo, Rahul Desai, Tyler Zhu, Samyak Para-\njuli, Mike Guo, et al. The many faces of robustness: A critical\nanalysis of out-of-distribution generalization. In CVPR , 2021. 2, 4,\n9, 10\n[36] Dan Hendrycks, Kevin Zhao, Steven Basart, Jacob Steinhardt, and\nDawn Song. Natural adversarial examples. In CVPR , 2021. 2, 4, 9,\n10\n6[37] Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q\nWeinberger. Deep networks with stochastic depth. In ECCV , 2016.\n9\n[38] Gabriel Ilharco, Mitchell Wortsman, Ross Wightman, Cade Gor-\ndon, Nicholas Carlini, Rohan Taori, Achal Dave, Vaishaal Shankar,\nHongseok Namkoong, John Miller, Hannaneh Hajishirzi, Ali\nFarhadi, and Ludwig Schmidt. Openclip. https://github.\ncom/mlfoundations/open_clip , 2021. 3\n[39] Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei. 3d object\nrepresentations for fine-grained categorization. In ICCVW , 2013. 2,\n9, 10\n[40] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers\nof features from tiny images. 2009. 2, 4, 9, 10\n[41] Yann LeCun, L ´eon Bottou, Yoshua Bengio, and Patrick Haffner.\nGradient-based learning applied to document recognition. Proceed-\nings of the IEEE , 1998. 2, 4, 9, 10\n[42] Junnan Li, Dongxu Li, Silvio Savarese, and Steven Hoi. Blip-2:\nBootstrapping language-image pre-training with frozen image en-\ncoders and large language models. arXiv preprint arXiv:2301.12597 ,\n2023. 1\n[43] Xianhang Li, Zeyu Wang, and Cihang Xie. Clipa-v2: Scaling clip\ntraining with 81.1 arXiv preprint arXiv:2306.15658 , 2023. 1\n[44] Yanghao Li, Haoqi Fan, Ronghang Hu, Christoph Feichtenhofer,\nand Kaiming He. Scaling language-image pre-training via masking,\n2022. 3, 9\n[45] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro\nPerona, Deva Ramanan, Piotr Doll ´ar, and C Lawrence Zitnick. Mi-\ncrosoft coco: Common objects in context. In European conference\non computer vision , pages 740–755. Springer, 2014. 3, 4, 9\n[46] Haotian Liu, Chunyuan Li, Qingyang Wu, and Yong Jae Lee. Visual\ninstruction tuning. arXiv preprint arXiv:2304.08485 , 2023. 1\n[47] Subhransu Maji, Esa Rahtu, Juho Kannala, Matthew Blaschko, and\nAndrea Vedaldi. Fine-grained visual classification of aircraft. arXiv\npreprint arXiv:1306.5151 , 2013. 2, 9, 10\n[48] Arsha Nagrani, Paul Hongsuck Seo, Bryan Seybold, Anja Hauth,\nSantiago Manen, Chen Sun, and Cordelia Schmid. Learning audio-\nvideo modalities from image captions, 2022. 3\n[49] Maria-Elena Nilsback and Andrew Zisserman. Automated flower\nclassification over a large number of classes. In ICVGIP , 2008. 2, 9,\n10\n[50] Ting Pan, Lulu Tang, Xinlong Wang, and Shiguang Shan. Tokenize\nanything via prompting. arXiv preprint arXiv:2312.09128 , 2023. 1\n[51] Omkar M. Parkhi, Andrea Vedaldi, Andrew Zisserman, and C. V .\nJawahar. Cats and dogs. In CVPR , 2012. 2, 9, 10\n[52] Hieu Pham, Zihang Dai, Golnaz Ghiasi, Hanxiao Liu, Adams Wei\nYu, Minh-Thang Luong, Mingxing Tan, and Quoc V Le. Com-\nbined scaling for zero-shot transfer learning. arXiv preprint\narXiv:2111.10050 , 2021. 4\n[53] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh,\nGabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell,\nPamela Mishkin, Jack Clark, et al. Learning transferable visual\nmodels from natural language supervision. In ICML , 2021. 1, 2, 3,\n9, 10\n[54] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan\nNarang, Michael Matena, Yanqi Zhou, Wei Li, Peter J Liu, et al.\nExploring the limits of transfer learning with a unified text-to-text\ntransformer. JMLR , 2020. 1\n[55] Samyam Rajbhandari, Jeff Rasley, Olatunji Ruwase, and Yuxiong\nHe. Zero: Memory optimizations toward training trillion parameter\nmodels. In SC20 , 2020. 3, 9[56] Jeff Rasley, Samyam Rajbhandari, Olatunji Ruwase, and Yuxiong\nHe. Deepspeed: System optimizations enable training deep learning\nmodels with over 100 billion parameters. In KDD , 2020. 3, 9\n[57] Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal\nShankar. Do imagenet classifiers generalize to imagenet?, 2019. 2,\n4, 9, 10\n[58] Christoph Schuhmann, Romain Beaumont, Richard Vencu, Cade\nGordon, Ross Wightman, Mehdi Cherti, Theo Coombes, Aarush\nKatta, Clayton Mullis, Mitchell Wortsman, et al. Laion-5b: An open\nlarge-scale dataset for training next generation image-text models.\narXiv preprint arXiv:2210.08402 , 2022. 2, 3\n[59] Christoph Schuhmann, Richard Vencu, Romain Beaumont, Robert\nKaczmarczyk, Clayton Mullis, Aarush Katta, Theo Coombes, Je-\nnia Jitsev, and Aran Komatsuzaki. Laion-400m: Open dataset\nof clip-filtered 400 million image-text pairs. arXiv preprint\narXiv:2111.02114 , 2021. 2\n[60] Khurram Soomro, Amir Roshan Zamir, and Mubarak Shah. Ucf101:\nA dataset of 101 human actions classes from videos in the wild.\narXiv preprint arXiv:1212.0402 , 2012. 3, 5, 9\n[61] Johannes Stallkamp, Marc Schlipsing, Jan Salmen, and Christian\nIgel. Man vs. computer: Benchmarking machine learning algorithms\nfor traffic sign recognition. Neural networks , 2012. 2, 9, 10\n[62] Quan Sun, Yufeng Cui, Xiaosong Zhang, Fan Zhang, Qiying Yu,\nZhengxiong Luo, Yueze Wang, Yongming Rao, Jingjing Liu, Tiejun\nHuang, and Xinlong Wang. Generative multimodal models are\nin-context learners. arXiv preprint arXiv:2312.13286 , 2023. 1\n[63] Quan Sun, Yuxin Fang, Ledell Wu, Xinlong Wang, and Yue Cao.\nEva-clip: Improved training techniques for clip at scale. arXiv\npreprint arXiv:2303.15389 , 2023. 1, 2, 3\n[64] Quan Sun, Qiying Yu, Yufeng Cui, Fan Zhang, Xiaosong Zhang,\nYueze Wang, Hongcheng Gao, Jingjing Liu, Tiejun Huang, and\nXinlong Wang. Generative pretraining in multimodality. arXiv\npreprint arXiv:2307.05222 , 2023. 1\n[65] Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier Martinet,\nMarie-Anne Lachaux, Timoth ´ee Lacroix, Baptiste Rozi `ere, Na-\nman Goyal, Eric Hambro, Faisal Azhar, Aurelien Rodriguez, Ar-\nmand Joulin, Edouard Grave, and Guillaume Lample. Llama:\nOpen and efficient foundation language models. arXiv preprint\narXiv:2302.13971 , 2023. 1, 2\n[66] Mikaela Angelina Uy, Quang-Hieu Pham, Binh-Son Hua, Thanh\nNguyen, and Sai-Kit Yeung. Revisiting point cloud classification:\nA new benchmark dataset and classification model on real-world\ndata. In Proceedings of the IEEE/CVF international conference on\ncomputer vision , pages 1588–1597, 2019. 5\n[67] Bastiaan S Veeling, Jasper Linmans, Jim Winkens, Taco Cohen, and\nMax Welling. Rotation equivariant cnns for digital pathology. In\nMICCAI , 2018. 2, 4, 9, 10\n[68] Haohan Wang, Songwei Ge, Zachary Lipton, and Eric P Xing.\nLearning robust global representations by penalizing local predictive\npower. NeurIPS , 2019. 2, 4, 9, 10\n[69] Weihan Wang, Qingsong Lv, Wenmeng Yu, Wenyi Hong, Ji Qi,\nYan Wang, Junhui Ji, Zhuoyi Yang, Lei Zhao, Xixuan Song, et al.\nCogvlm: Visual expert for pretrained language models. arXiv\npreprint arXiv:2311.03079 , 2023. 1\n[70] Yi Wang, Yinan He, Yizhuo Li, Kunchang Li, Jiashuo Yu, Xin Ma,\nXinhao Li, Guo Chen, Xinyuan Chen, Yaohui Wang, Conghui He,\nPing Luo, Ziwei Liu, Yali Wang, Limin Wang, and Yu Qiao. Intern-\nvid: A large-scale video-text dataset for multimodal understanding\nand generation, 2024. 3\n7[71] Zhirong Wu, Shuran Song, Aditya Khosla, Fisher Yu, Linguang\nZhang, Xiaoou Tang, and Jianxiong Xiao. 3d shapenets: A deep\nrepresentation for volumetric shapes. In Proceedings of the IEEE\nconference on computer vision and pattern recognition , pages 1912–\n1920, 2015. 5\n[72] Jianxiong Xiao, James Hays, Krista A Ehinger, Aude Oliva, and\nAntonio Torralba. Sun database: Large-scale scene recognition from\nabbey to zoo. In CVPR , 2010. 2, 4, 9, 10\n[73] Yang You, Jing Li, Sashank Reddi, Jonathan Hseu, Sanjiv Kumar,\nSrinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer,\nand Cho-Jui Hsieh. Large batch optimization for deep learning:\nTraining bert in 76 minutes, 2019. 3, 9\n[74] Peter Young, Alice Lai, Micah Hodosh, and Julia Hockenmaier.\nFrom image descriptions to visual denotations: New similarity met-\nrics for semantic inference over event descriptions. TACL , 2014. 3,\n4, 9\n[75] Xiaohua Zhai, Basil Mustafa, Alexander Kolesnikov, and Lucas\nBeyer. Sigmoid loss for language image pre-training. arXiv preprint\narXiv:2303.15343 , 2023. 1, 4\n[76] Biao Zhang and Rico Sennrich. Root mean square layer normaliza-\ntion. arXiv preprint arXiv:1910.07467 , 2019. 2\n[77] Junsheng Zhou, Jinsheng Wang, Baorui Ma, Yu-Shen Liu, Tiejun\nHuang, and Xinlong Wang. Uni3d: Exploring unified 3d representa-\ntion at scale. arXiv preprint arXiv:2310.06773 , 2023. 1, 5\n[78] Deyao Zhu, Jun Chen, Xiaoqian Shen, Xiang Li, and Mohamed El-\nhoseiny. Minigpt-4: Enhancing vision-language understanding with\nadvanced large language models. arXiv preprint arXiv:2304.10592 ,\n2023. 1\n8EV A-CLIP-18B: Scaling CLIP to 18 Billion Parameters\nSupplementary Material\nDataset Classes Test size Evaluation Metric\nImageNet-1K [26] 1000 50,000 accuracy\nImageNet-V2 [57] 1000 10,000 accuracy\nImageNet-Adversarial [36] 1000 7,500 accuracy\nImageNet-R(endition) [35] 1000 30,000 accuracy\nImageNet-Sketch [68] 1000 50,899 accuracy\nObjectNet [8] 1000 50,273 accuracy\nCIFAR-10 [40] 10 10,000 accuracy\nCIFAR-100 [40] 100 10,000 accuracy\nMNIST [41] 10 10,000 accuracy\nCaltech101 [31] 101 9144 accuracy\nSUN397 [72] 397 108,754 accuracy\nFGVC Aircraft [47] 100 3,333 accuracy\nCountry-211 [53] 211 21,100 accuracy\nStanford Cars [39] 196 8,041 accuracy\nBirdsnap [9] 500 2,195 accuracy\nDescribable Textures [21] 47 1,880 accuracy\nEuroSAT[34] 10 27,000 accuracy\nFacial Emotion Recognition 2013 [33] 8 3,574 accuracy\nOxford Flowers 102 [49] 102 6,149 accuracy\nFood-101 [10] 102 25,250 accuracy\nGTSRB [61] 43 12,630 accuracy\nPatchCamelyon [67] 2 32,768 accuracy\nOxford-IIIT Pets [51] 37 3,669 accuracy\nRendered SST2 [53] 2 1,821 accuracy\nRESISC45 [18] 45 31,500 accuracy\nSTL-10 [23] 10 8000 accuracy\nPascal VOC 2007 Classification [27] 20 4,952 accuracy\nUC-F101 [60] 101 11,213 accuracy\nKinetics-400 [15] 400 19,240 mean(top1, top5)\nKinetics-600 [13] 600 29,788 mean(top1, top5)\nKinetics-700 [14] 700 33,966 mean(top1, top5)\nFlickr30K [74] - 1000 recall\nCOCO [45] - 5000 recall\nTable 11: Datasets used to evaluate EV A-CLIP models.\n6. Training Settings\nWe present detailed training settings of EV A-CLIP-8B\nandEV A-CLIP-18B in Tabs. 12 and 13.\n7. Image Transformations for Evaluation\nTwo prevalent image transformations utilized in zero-shot\nevaluation are: 1) direct resizing of images to a fixed size,\nsuch as 224 ×224, and 2) resizing images based on the short-\nest side, followed by center cropping to achieve a fixed size.\nIn Table 14, our study systematically investigates the impactconfig EV A-CLIP -{8B, 8B+ }\nimage enc. weight init. EV A -8B / EV A-CLIP-8B\ntext enc. weight init. EV A-02-CLIP -E/14+ / EV A-CLIP-8B\nimage-text data Merged-2B\nimage enc. peak learning rate 4e-4 / 2e-4\nimage enc. layer-wise lr decay [22, 7] 0.9 / 0.85\ntext enc. peak learning rate 4e-5 / 2e-5\ntext enc. layer-wise lr decay [22, 7] 0.75\nlearning rate schedule cosine decay\noptimizer LAMB [73]\noptimizer hyper-parameters β1,β2,ϵ= 0.9, 0.95, 1e-6\nweight decay 0\ninput resolution 2242/ 4482\npatch size 142\nbatch size 178k / 24k\nsamples seen 9B / 800M\ndrop image patch [44] 0.5 / 0.0\ndrop path [37] 0.0\nrandom resized crop (0.9, 1)\nnumerical precision DeepSpeed bf16 [56]\nZeRO optimizer [55] stage 3\nTable 12: EV A-CLIP-8B andEV A-CLIP-8B + training settings.\nconfig EV A-CLIP -{18B,18B+ }\nimage enc. weight init. EV A -18B / EV A-CLIP-18B\ntext enc. weight init. EV A-02-CLIP -E/14+ / EV A-CLIP-18B\nimage-text data Merged-2B+\nimage enc. peak learning rate 4e-4 / 2e-4\nimage enc. layer-wise lr decay [22, 7] 0.9 / 0.85\ntext enc. peak learning rate 4e-5 / 2e-5\ntext enc. layer-wise lr decay [22, 7] 0.75\nlearning rate schedule cosine decay\noptimizer LAMB [73]\noptimizer hyper-parameters β1,β2,ϵ= 0.9, 0.95, 1e-6\nweight decay 0\ninput resolution 2242/ 3362\npatch size 142\nbatch size 108k / 23k\nsamples seen 6B / 400M\ndrop image patch [44] 0.5 / 0.0\ndrop path [37] 0.0\nrandom resized crop (0.9, 1)\nnumerical precision DeepSpeed bf16 [56]\nZeRO optimizer [55] stage 3\nTable 13: EV A-CLIP-18B andEV A-CLIP-18B + training settings.\nof these two image transformations in zero-shot evaluations.\nNotably, there exists a significant performance gap between\nthe two transformations, observed particularly in zero-shot\nimage classification on ObjectNet [ 8] and VOC2007 [ 27],\nand zero-shot retrieval on Flickr30K [ 74] and COCO [ 45].\nEV A-CLIP-18B shows robustness with almost the same\naverage accuracy across different image transformations in\nzero-shot image/video classification.\nFor zero-shot image classification and video classification,\n9method\nImageNet-1K [26]\nImageNet-V2 [57]\nImageNet-Adv. [36]\nImageNet-Ren. [35]\nImageNet-Ske. [68]\nObjectNet [8]\nCIFAR-10 [40]\nCIFAR-100 [40]\nMNIST [41]\nCaltech101 [31]\nSUN397 [72]\nFGVC Aircraft [47]\nCountry-211 [53]\nStanford Cars [39]\nBirdsnap [9]\nDTD [21]\nEuroSAT [34]\nFER2013 [33]\nFlowers-102 [49]\nFood-101 [10]\nGTSRB [61]\nPCam [67]\nPets [51]\nRendered SST2 [53]\nRESISC45 [18]\nSTL-10 [23]\nVOC2007 [27]\n.avg. top-1 acc.\nDFN5B-CLIP-H/14+* 84.0 77.8 79.6 92.9 72.4 79.6 98.8 90.5 83.6 88.7 77.0 64.9 36.1 95.7 80.5 70.9 61.1 56.1 91.6 96.1 67.8 69.6 96.7 55.5 75.9 99.1 78.2 78.5\nDFN5B-CLIP-H/14+ †84.3 78.3 79.3 93.6 73.3 73.5 98.8 90.5 83.6 88.9 77.4 72.5 37.9 96.0 80.3 70.9 61.1 56.1 91.4 96.2 67.9 69.6 96.8 55.5 75.9 99.1 81.9 78.9\nEV A-CLIP-18B * 83.7 77.9 87.3 95.6 74.4 82.2 99.4 93.8 83.0 89.4 77.5 58.4 41.8 94.9 79.9 71.9 79.8 59.3 85.9 95.8 72.4 65.2 96.0 67.5 76.8 99.6 82.4 80.4\nEV A-CLIP-18B † 83.8 77.7 86.2 95.7 74.7 76.2 99.4 93.8 83.0 89.8 77.7 59.7 43.1 94.9 78.4 72.1 79.8 59.3 86.0 95.7 72.3 65.2 96.1 67.5 76.9 99.6 85.8 80.4\n(a)Impact of image transformations on zero-shot image classification performance. Different transformations can significantly influence zero-shot image\nclassification performance, particularly for ObjectNet [ 8].EV A-CLIP-18B shows robustness with the same average top-1 accuracy across different image\ntransformations.\nmethod+ #Frames UCF-101 K-400 K-600 K-700 avg. acc.\nDFN5B-CLIP-H/14+* 1 78.5 65.2 66.0 59.2 67.2\nDFN5B-CLIP-H/14+ † 1 79.2 66.7 67.0 60.7 68.4\nEV A-CLIP-18B * 1 86.0 72.2 72.6 67.4 74.6\nEV A-CLIP-18B † 1 85.6 72.9 72.9 68.2 74.9\nEV A-CLIP-18B * 8 88.2 79.3 79.2 72.0 79.7\nEV A-CLIP-18B † 8 87.9 79.2 79.1 72.1 79.6\n(b)Impact of image transforms on zero-shot video classification performance.\nzero-shot textretrieval zero-shot image retrieval\nFlickr30K COCO Flickr30K COCO\nmethod + R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 R@1 R@5 R@10 MR\nDFN5B-CLIP-H/14 +* 92.3 99.1 99.7 70.6 89.6 94.4 80.7 95.5 97.7 54.1 78.0 85.4 86.4\nDFN5B-CLIP-H/14 + †93.6 99.3 99.6 71.8 90.4 94.9 82.1 96.0 97.9 55.6 79.2 86.3 87.2\nEV A-CLIP-18B * 95.4 99.5 99.8 72.8 89.7 94.3 83.2 95.9 97.8 55.6 77.9 85.3 86.7\nEV A-CLIP-18B † 96.7 99.7 100.0 73.6 90.9 95.0 83.3 96.3 98.3 56.2 78.5 85.6 87.8\n(c)Impact of image transforms on zero-shot retrieval performance.\nTable 14: Impact of image transformations on zero-shot evaluation. †denotes the direct resizing of images to a fixed size, while *\nindicates resizing images based on the shortest side and subsequently center cropping to achieve a fixed size.\nwe present results obtained by selecting the best-performing\ntransformation between the two. In the case of zero-shot\nretrieval tasks, we specifically choose the transformation that\ninvolves direct resizing of images to a fixed size.\n10"    },    {        "title": "2402.04283v1.On_the_Continuity_Equation_in_Space_Time_Algebra__Multivector_Waves__Poynting__Diffusion__and_a_Derivation_of_Maxwell_s_Equations_by_Symmetries.pdf",        "content": "arXiv:2402.04283v1  [physics.class-ph]  6 Feb 2024On the Continuity Equation in Space-Time Algebra:\nMultivector Waves, Poynting, Diffusion, and a Derivation of\nMaxwell’s Equations by Symmetries\nManuel Beato V´ asquez1and Melvin Arias Polanco∗1, 2\n1Escuela de F´ ısica, Facultad de Ciencias, Universidad Aut´ onoma de Santo Domingo, Av. Alma Mater,\nSanto Domingo 10105, Dominican Republic\n2Laboratorio de Nanotecnolog´ ıa, ´Area de Ciencias B´ asicas y Ambientales, Instituto Tecnol´ ogico de\nSanto Domingo, Av. Los Pr´ oceres, Santo Domingo 10602, Domi nican Republic\nFebruary 2024\nAbstract\nHistorically and to date, the continuity equation has serve d as a consistency criterion for\nthe development of physical theories. Employing Clifford’s geometric algebras, a system of\ncontinuity equations for a generalised multivector of the s pace-time algebra (STA) is con-\nstructed. Associated with this continuity system, a system of wave equations is constructed,\nthe Poynting multivector is defined, and decoupling conditi ons are determined. The diffu-\nsion equation is explored from the continuity system, where it is found that for decoupled\nsystems with constant or explicitly-dependent diffusion co efficients the absence of exter-\nnal vector sources implies a loss in the diffusion equation st ructure being transformed to\nHelmholtz-like or wave systems. From the symmetry transfor mations that make the conti-\nnuity equations system’s structure invariant, a system wit h the structure of Maxwell’s field\nequations is derived. The Maxwellian system allows for the c onstruction of potentials and\nfields directly linked with the continuity of a generalised m ultivector in STA. The results\nfound are consistent with the classical electromagnetic th eory and hydrodynamics.\n1 Introduction\nThe continuity equation (C.E.) is a first-order partial differential eq uation that describes the\ntransport in time of a physical quantity through a region of space in terms of its density, flux,\nand an external source (or sink) which increases (or decreases) the quantity from the system con-\ntinuously. If the source is null, then the transported physical qua ntity is said to be conservative,\nand the C.E. represents the principle of local conservation of said q uantity [1]. In this form, the\nC.E. permeates virtually all branches of physics (table 1).\nAn iconic case where the continuity equation played a fundamental r ole in the development of\na physical theory occurred when J. C. Maxwell noticed the inconsis tency between the primitive\nfield equations of the electromagnetic theory and the C.E. for the e lectric charge (a principle of\nconservation well established at the time). Maxwell is capable of solv ing said inconsistency by\nintroducing the now renowned displacement ‘current’. Thanks to th is addition, he was able to\n∗melvin.arias@intec.edu.do\n1derive wave equations for the electric and magnetic fields, subsequ ently discovering that their\nspeed of propagation is precisely the speed of light [ 2]. Another instance was in the development\nof the quantum-relativistic equation by P. A. M. Dirac following previo us inconsistent attempts\nto reconcile both theories under a unified model. One such attempt is the Klein-Fock-Gordon\nequation, whichuponitsreductiontoaC.E.yieldsnegativeenergyeig envaluesandanon-positive\ndefinite probability density. Dirac, using a matrix analysis and ensurin g consistency with the\nC.E., constructed what we today know as Dirac’s gamma matrices, alg ebra, and equation. Thus,\nallowing for a description of spin-1 /2 massive particles in the presence of external fields while\ntaking into account relativistic effects [ 3].\nContinuum MechanicsMass\nEnergy\nLinear momentum\nAngular momentum\nElectrodynamicsElectric charge\nElectromagnetic energy\nElectromagnetic momentum\nThermal PhysicsParticle number\nThermal energy\nEntropy\nProbability (Liouville, Fokker-Planck)\nQuantum MechanicsProbability (Schr¨ odinger, Dirac)\nSpin angular momentum\nTable 1: Conservative physical quantities.\nUndoubtedly, the C.E. has served as a consistency criterion for th e development of physical\ntheories. A natural language to express and study the C.E. are W. K. Clifford’s geometric\nalgebras (GA). In physics, GA have enjoyed a vast popularisation s ince their ‘rediscovery’ by W.\nE. Pauli and Dirac in the 1920’s, and D. Hestenes in the 1960’s. By des ign, GA establishes itself\nas a holistic mathematical framework capable of algebraically manipula ting multi-dimensional\nobjects, operatingon them with a clear geometricinterpretation, and nesting within itself several\nsub-algebrassuch as complex numbers, quaternions, four-vect ors,and spinors[ 4]–[8]. As a result,\nGA finds numerous applications not only in physics, but also in mathema tics, computer science,\nand engineering [ 9], [10].\nAxiomatically, a geometric algebra G(p,q) is a graded and associative algebra of dimension 2p+q\ndefined by a vector space Vp+qover the field of real numbers R[11]–[14]. Elements of a GA\nare called multivectors and operate through the so-called geometr ic product. The geometric\nproduct—denoted by juxtaposition—is closed, distributive, ‘well be haved’ with scalars, invert-\nible, and associative. For vectors, the geometric product is linked t o the inner product of the\nvector space and H. G. Grassmann’s outer product via the so-calle d fundamental identity\nγµγν=γµ·γν+γµ∧γν (1.1)\nIn general, for two homogeneous multivectors Ψ α=/an}b∇acketle{tΨα/an}b∇acket∇i}htαand Φ β=/an}b∇acketle{tΦβ/an}b∇acket∇i}htβof gradesαand\nβrespectively, the inner and outer products are defined in terms of the geometric product as\n[15]–[17]:\nΨα·Φβ=/an}b∇acketle{tΨαΦβ/an}b∇acket∇i}ht|α−β| (1.2)\nΨα∧Φβ=/an}b∇acketle{tΨαΦβ/an}b∇acket∇i}htα+β (1.3)\n2Consider the geometric algebra G(1,3), known as the space-time algebra (STA), generated by\nthe orthonormal basis vectors {γ0,γ1,γ2,γ3}which satisfy [ 18]–[20]:\nγµ·γν=1\n2(γµγν+γνγµ) =ηµν (1.4)\nwhereηµν= diag(+1,−1,−1,−1) is Minkowski’s metric tensor. The relations with the dual\nbasis are\nγµ=ηµνγν, γ µ·γν=δν\nµ (1.5)\nwhereδν\nµ=ηναηαµis the symmetric Kronecker delta tensor. A key corollary of eq. ( 1.4) is\nthat orthogonal vectors anticommute under the geometric prod uct:γµγν=−γνγµ,∀µ/ne}ationslash=ν.\nHenceforth we adopt Einstein’s summation convention; Greek indice s run from 0 to 3, and Latin\nindices from 1 to 3 (except when explicitly stated otherwise); and we make use of the so-called\n‘natural units’ where c= 1. LetI=γ0γ1γ2γ3be the unit pseudo-scalarof G(1,3) with properties\nII=−1, Iγ µ=−γµI (1.6)\nLetusintroducethespace-timesplitgivenbythesub-algebraofph ysicalspace(APS), G+(1,3)≃\nG(3,0), generated by the bivectors\nσi=γiγ0,σiσj=δij+εijkIσk (1.7)\nwhereεijkis the antisymmetric Levi-Civitatensorand such that the relativeve ctorsaredefined1:\nψα=ψi\nασi,∇=σi∂i (1.8)\nIn STA, the vector derivative operator with respect to space-tim e positionx=xµγµ= (t+x)γ0\nis defined as\n∇=γµ∂µ= (∂t−∇)γ0 (1.9)\nFor brevity, we will refer to eq. ( 1.9) as the nabla operator. Let us also define the structure of a\ngeneralised multivector of STA as:\nΨ =/an}b∇acketle{tΨ/an}b∇acket∇i}ht0+/an}b∇acketle{tΨ/an}b∇acket∇i}ht1+/an}b∇acketle{tΨ/an}b∇acket∇i}ht2+/an}b∇acketle{tΨ/an}b∇acket∇i}ht3+/an}b∇acketle{tΨ/an}b∇acket∇i}ht4\n=ψ0\n2+ψµ\n1γµ+ψi\n2γiγ0+ψi\n3Iγiγ0+ψµ\n4Iγµ+ψ0\n3I\n=ψ0\n2+/parenleftbig\nψ0\n1+ψ1/parenrightbig\nγ0+ψ2+ψ3I+/parenleftbig\nψ0\n4+ψ4/parenrightbig\nIγ0+ψ0\n3I(1.10)\nwhere2ψµ\nα=ψµ\nα(xµ) are scalar functions of the coordinates of x. Respectively, they constitute\nthe scalar/an}b∇acketle{tΨ/an}b∇acket∇i}ht0=ψ0\n2, vector/an}b∇acketle{tΨ/an}b∇acket∇i}ht1=/parenleftbig\nψ0\n1+ψ1/parenrightbig\nγ0, timelike bivector /an}b∇acketle{tΨ/an}b∇acket∇i}ht2t=ψ2, spacelike\nbivector3/an}b∇acketle{tΨ/an}b∇acket∇i}ht2s=ψ3I, pseudo-vector/an}b∇acketle{tΨ/an}b∇acket∇i}ht3=/parenleftbig\nψ0\n4+ψ4/parenrightbig\nIγ0, and pseudo-scalar part /an}b∇acketle{tΨ/an}b∇acket∇i}ht4=\n1Another property we shall use numerously but implicitly is t he commutation of the pseudoscalar with the\nrelative vectors: Iσi=σiI.\n2Throughout the document: α,β= 1,2,3,4.\n3The integral bivector part of Ψ is /angbracketleftΨ/angbracketright2=/angbracketleftΨ/angbracketright2t+/angbracketleftΨ/angbracketright2s=ψ2+ψ3I.\n3ψ0\n3Iof the multivector Ψ. For the generalised multivector defined in eq. ( 1.10) the following\nconjugation operations are defined:\n•Inverse conjugation or reversal ( γα∧γβ∧···∧γµ∧γν)∼=γν∧γµ∧···∧γβ∧γα,\n/tildewideΨ =/parenleftbig\nψ0\n1+ψ1/parenrightbig\nγ0+/parenleftbig\nψ0\n2−ψ2/parenrightbig\n+/parenleftbig\nψ0\n3−ψ3/parenrightbig\nI−/parenleftbig\nψ0\n4+ψ4/parenrightbig\nIγ0 (1.11)\n•Space conjugation Ψ∗=γ0Ψγ0,\nΨ∗=/parenleftbig\nψ0\n1−ψ1/parenrightbig\nγ0+/parenleftbig\nψ0\n2−ψ2/parenrightbig\n−/parenleftbig\nψ0\n3−ψ3/parenrightbig\nI−/parenleftbig\nψ0\n4−ψ4/parenrightbig\nIγ0(1.12)\n•Hermitian conjugation Ψ†=/tildewideΨ∗,\nΨ†=/parenleftbig\nψ0\n1−ψ1/parenrightbig\nγ0+/parenleftbig\nψ0\n2+ψ2/parenrightbig\n−/parenleftbig\nψ0\n3+ψ3/parenrightbig\nI+/parenleftbig\nψ0\n4−ψ4/parenrightbig\nIγ0 (1.13)\nIt is assumed the invariance of the scalar functions ψµ\nαunder all conjugations, that they satisfy\nClairaut-Schwarz’s theorem, and commute with the basis vectors a nd the pseudoscalar. Attend-\ning to these definitions, the reversal of the nabla eq. ( 1.9) indicates that the partial derivatives\nact to the left, /tildewide∇=← −∇, while hermitian conjugation yields\n∇†=/parenleftig← −∂t+← −∇/parenrightig\nγ0 (1.14)\nWith this preamble, in the present paper, we propose to employ the m athematical framework of\nClifford’sgeometricalgebrastostudysystemsofcontinuityequatio nsforgeneralisedmultivectors\nin STA. Our starting point in section 2will consist of establishing an equation of structure\n∇Ψ = Φ and identifying the systems of continuity equations that emerg e from each component\nof the multivector defined by eq. ( 1.10). Then, we will construct a system of wave equations\nand define the generalised Poynting multivector (along with its contin uity) associated with the\ncontinuity system of the generalised multivector. STA contains APS which further contains\nthe quaternion algebra, G+(3,0)≃H. So, our approach with generalised multivectors and\nnotably the formalism of STA distinguishes this work from similar ones u sing a quaternionic\nformalism[ 21], [22], and with explorationsofthe Poyntingvectorrestrictedtoan elec tromagnetic\ntreatment in GA [ 23], [24] and hydrodinamical without GA [ 25]. In section 3we will explore\nsome decoupling cases for the different parts of Ψ which are coupled with each other through\nthe continuity system. Subsequently, in section 4we shall determine transformation symmetries\nfor the functions involved in the continuity system such that the sy stem of continuity equations\nremains invariant in structure. From these symmetries we plan to de rive a system of equations\nwith the same structure as Maxwell’s field equations. Also, together with it, see an application\nof the wave equations system, the Poynting multivector, and the d ecoupling conditions for the\nfields and potentials to be determined. Alternative derivations of Ma xwell’s equations from the\nC.E. can be found in [ 26], [27] by means of retarded potentials and the Poincar´ e lemma in GA;\nalso in [28], [29] using Jefimenko’s equations and retarded tensor fields without GA . Finally, in\nsection5we will employ the continuity system to study the diffusion equation an d the equations\ncoupled to it.\n42 Continuity Equations\nConsider the equation\n∇Ψ = Φ (2.1)\nwhere Φ = Φ( xµ) is another multivector with the structure of eq. ( 1.10). The geometric product\nbetween the nabla and the multivector Ψ yields\n∇Ψ =/parenleftbig\n∂tψ0\n1+∇·ψ1/parenrightbig\n+/parenleftbig\n∂tψ0\n2+∇·ψ2−∂tψ2−∇ψ0\n2−I∇∧ψ3/parenrightbig\nγ0\n−/parenleftbig\n∂tψ1+∇ψ0\n1+I∇∧ψ4/parenrightbig\n+/parenleftbig\n∂tψ4+∇ψ0\n4−I∇∧ψ1/parenrightbig\nI\n+/parenleftbig\n−∂tψ0\n3−∇·ψ3+∂tψ3+∇ψ0\n3−I∇∧ψ2/parenrightbig\nIγ0\n−/parenleftbig\n∂tψ0\n4+∇·ψ4/parenrightbig\nI(2.2)\nThe terms in eq. ( 2.2) have been organised by parenthesis into scalar, vector, timelike b ivector,\nspacelike bivector, pseudovector, and pseudoscalar part, resp ectively. By hypothesis, Φ has\nstructure eq. ( 1.10), so matrix notation allows us to identify the following pair of systems of\nequations which follow from the components of eq. ( 2.1). On one hand, the scalar structure:\n∂t\nψ0\n1\nψ0\n2\nψ0\n3\nψ0\n4\n+∇·\nψ1\nψ2\nψ3\nψ4\n=\nφ0\n2\nφ0\n1\n−φ0\n4\n−φ0\n3\n(2.3)\nAnd on the other hand the vector structure:\n∂t\nψ1\nψ2\nψ3\nψ4\n+∇\nψ0\n1\nψ0\n2\nψ0\n3\nψ0\n4\n−I∇∧\n−ψ4\n−ψ3\nψ2\nψ1\n=\n−φ2\n−φ1\nφ4\nφ3\n(2.4)\nThesystemofeq.( 2.3)clearlypossessesthecanonicalstructureofthe(scalar)cont inuityequation\nwith existence of non-zero sources given by Φ. To observe that th e system of eq. ( 2.4) has a\ncontinuity structure as well, but one of vector character, we use the dual identity\nI∇∧ψα=∇·(Iψα) (2.5)\nand define the second-rank tensors ( Tα)ijsuch that its components are connected with the ψ0\nα\nthrough the restriction\n∂iTij\nα=∂jψ0\nα,∇·Tα=∇ψ0\nα (2.6)\n5With these results, eqs. ( 2.3) and (2.4) compact into a visibly single continuity structure:\n∂t\nψ0\n1\nψ0\n2\nψ0\n3\nψ0\n4\nψ1\nψ2\nψ3\nψ4\n+∇·\nψ1\nψ2\nψ3\nψ4\nT1+Iψ4\nT2+Iψ3\nT3−Iψ2\nT4−Iψ1\n=\nφ0\n2\nφ0\n1\n−φ0\n4\n−φ0\n3\n−φ2\n−φ1\nφ4\nφ3\n(2.7)\n2.1 Wave Equations\nBy applying from the left the nabla operator to eq. ( 2.1) we obtain the second-order differential\nequation\n/squareΨ =∇Φ (2.8)\nwhere/square=∇∇=∇·∇=ηµν∂2\nµνis the d’Alembertian operator. The multivector ∇Φ will have\nby structure eq. ( 2.2), so by separating the components of eq. ( 2.8) we will get two systems of\nequations with the form of eqs. ( 2.3) and (2.4). Defining the tensors ( Nα)ijrestricted to the\nanalogous condition of eq. ( 2.6) but connected to the φ0\nα, the following system of wave equations\nis then obtained:\n∂t\nφ0\n1\nφ0\n2\nφ0\n3\nφ0\n4\nφ1\nφ2\nφ3\nφ4\n+∇·\nφ1\nφ2\nφ3\nφ4\nN1+Iφ4\nN2+Iφ3\nN3−Iφ2\nN4−Iφ1\n=/square\nψ0\n2\nψ0\n1\n−ψ0\n4\n−ψ0\n3\n−ψ2\n−ψ1\nψ4\nψ3\n(2.9)\nThe scalar continuity system of eq. ( 2.3) shows the direct coupling between the vector compo-\nnents (ψ0\n1,ψ1), the pseudovector components ( ψ0\n4,ψ4), between the scalar and timelike bivector\n(ψ0\n2,ψ2), and between the pseudoscalar and spacelike bivector ( ψ0\n3,ψ3). We shall call these\nrelations the ‘first couplings’. The first couplings remain in the vector continuity system of\neq. (2.4), and additionally the relative pseudovector with the vector ( ψ0\n1,ψ1,ψ4), the relative\nvector with the pseudovector ( ψ0\n4,ψ4,ψ1), and the timelike and spacelike bivectors with each\nscalar(ψ0\n2,ψ2,ψ3), (ψ0\n3,ψ3,ψ2)arecoupled. Forconsistency,wecallthesethe‘secondcouplings ’.\nAn analogous argument applies to the components of the source Φ. The cost for decoupling—\nfirst or second— is a transition to a higher-order system of equatio ns, as can be seen in eq. ( 2.9)\nwhere each component ψµ\nαsatisfies a non-homogeneouswaveequation independently ofany o ther\ncomponent of Ψ. In section 3we will explore some decoupling cases.\n2.2 Poynting Multivector\nAttending to the conjugations eqs. ( 1.13) and (1.14), the hermitian conjugation of eq. ( 2.1)\nis (∇Ψ)†= Ψ†∇†= Φ†. The well-known method to derive the continuity equation from\n6Schr¨ odinger, Pauli, Klein-Fock-Gordon, and Dirac’s equation\n(∇Ψ)†(γ0Ψ)+(γ0Ψ)†(∇Ψ) = Φ†(γ0Ψ)+(γ0Ψ)†Φ\n∂µ/parenleftbig\nΨ†γ0γµΨ/parenrightbig\n= Φ†γ0Ψ+Ψ†γ0Φ\n∇·J=γ01\n2/parenleftig\n/tildewideΦΨ+/tildewideΨΦ/parenrightig(2.10)\nallows us to define\nJµ=1\n2/parenleftbig\nΨ†γ0γµΨ/parenrightbig\n(2.11)\nEquation ( 2.11) defines four multivectors which may be associated with a second-r ank tensor,\nEµν=Jµ·γν. Let us define the Poynting multivector as\nS=γ0J0=1\n2γ0Ψ†Ψ (2.12)\nIn terms of the components of Ψ, it has the explicit form\nS=/parenleftbig\nψ0\n1ψ0\n2+ψ1·ψ2+ψ0\n3ψ0\n4+ψ3·ψ4/parenrightbig\n+1\n2/bracketleftbig\n(ψ0\n1)2+ψ1·ψ1+(ψ0\n2)2+ψ2·ψ2+(ψ0\n3)2+ψ3·ψ3+(ψ0\n4)2+ψ4·ψ4/bracketrightbig\nγ0\n+/parenleftbig\nψ0\n1ψ1−ψ0\n2ψ2−ψ0\n3ψ3+ψ0\n4ψ4+Iψ1∧ψ4−Iψ2∧ψ3/parenrightbig\nγ0\n+/parenleftbig\nψ0\n1ψ0\n3+ψ1·ψ3−ψ0\n2ψ0\n4−ψ2·ψ4/parenrightbig\nI(2.13)\nThe terms in eq. ( 2.13) have been organised by parenthesis with respect to the ‘canonica l’ struc-\nture of the multivector eq. ( 1.10),S=S0\n2+ (S0\n1+S1)γ0+S0\n3I. Interestingly, the Poynting\nmultivector lacks bivector and pseudovector parts, /an}b∇acketle{tS/an}b∇acket∇i}ht2=/an}b∇acketle{tS/an}b∇acket∇i}ht3= 0. Let us establish the equa-\ntion,\n∇·S=W (2.14)\nThen, according to the general structure of eq. ( 2.7), the continuity system which follows from\neq. (2.14) is:\n∂tS0\n1+∇·S1=/an}b∇acketle{tW/an}b∇acket∇i}ht0\n−∇S0\n3=/an}b∇acketle{tW/an}b∇acket∇i}ht3(2.15)\nNo second couplings are present; only the first coupling of Sµ\n1.\n72.2.1 Bivector & Vector Cases\nAs a special case, consider a multivector for which ψµ\n1=ψµ\n4= 0 such that\nΨ = (ψ0\n2+ψ2)+(ψ0\n3+ψ3)I (2.16)\nThe associated Poynting multivector has a purely vector structur e\nS=1\n2/bracketleftbig\n(ψ0\n2)2+ψ2·ψ2+(ψ0\n3)2+ψ3·ψ3/bracketrightbig\nγ0−/parenleftbig\nψ0\n2ψ2+ψ0\n3ψ3+Iψ2∧ψ3/parenrightbig\nγ0(2.17)\nI.e.,S=/parenleftbig\nS0\n1+S1/parenrightbig\nγ0. Correspondingly, the continuity system eq. ( 2.15) reduces to the single\nequation∂µSµ\n1=/an}b∇acketle{tW/an}b∇acket∇i}ht0. Now, consider a multivector for which ψµ\n2=ψµ\n3= 0,\nΨ =/parenleftbig\nψ0\n1+ψ1/parenrightbig\nγ0+/parenleftbig\nψ0\n4+ψ4/parenrightbig\nIγ0 (2.18)\nPoynting multivector:\nS=1\n2/bracketleftbig\n(ψ0\n1)2+ψ1·ψ1+(ψ0\n4)2+ψ4·ψ4/bracketrightbig\nγ0+/parenleftbig\nψ0\n1ψ1+ψ0\n4ψ4+Iψ1∧ψ4/parenrightbig\nγ0(2.19)\nThat is,S(ψµ\n2=ψµ\n3= 0) has a purely vector structure, just like S(ψµ\n1=ψµ\n4= 0). Thus, their\nreduced continuity system will share the same structure.\n3 ‘Linear’ Decoupling\nBecause of the independence and symmetry between the continuit y equations regarding the\nsecond couplings, ( ψµ\n1,ψµ\n4) & (ψµ\n2,ψµ\n3), consider the simplified system of eq. ( 2.7):\n∂tψ0\nα+∇·ψα=φα (3.1)\n∂tψ0\nβ+∇·ψβ=−φβ (3.2)\n∂tψα+∇ψ0\nα+I∇∧ψβ=−φα (3.3)\n∂tψβ+∇ψ0\nβ−I∇∧ψα=φβ (3.4)\nWhere it is understood that if α= 1 thenβ= 4, and correspondingly if α= 2 thenβ= 3. The\nsource terms φα,φα, etc. are given by eq. ( 2.7) although for now we are not interested in their\nform. Let us proceed to determine some cases for ‘linear’ decouplin g—for which we simply mean\nthat we wish to decouple any one of the functions ψµ\nαandψµ\nβfrom one another in the system\neqs. (3.1) to (3.4) without necessarily recurring to the wave system eq. ( 2.9) or the trivial case.\n3.1 Mutual Decoupling\nThe simplest case of linear decoupling consists of imposing on both rela tive vectors the mutual\nrestriction ∇∧ψα,β= 0. Whence,\n8ψα,β=∇Γα,β (3.5)\nwhere the Γ α,β= Γα,β(xµ) are scalar functions, and the index ( α,β) denotes that the above\nequations apply for both functions simultaneously. With the mutual simultaneous decoupling of\neq. (3.5), the system of eqs. ( 3.1) to (3.4) reduces to:\n∂tψ0\nα,β+∇2Γα,β=±φα,β\n∇/parenleftbig\nψ0\nα,β+∂tΓα,β/parenrightbig\n=∓φα,β(3.6)\nwith∇2=∇·∇=δij∂2\nijthe Laplacian operator of APS. Thus, imposing the restriction of\neq. (3.5) simultaneously to both relative vectors ψαandψβdecouples them in two pairs of\nequations with the same structure but independent of each other .\n3.2 Unilateral Decoupling\nSuppose we now impose a restriction to just one of the terms, say ψµ\nβwith respect to ψµ\nα.\nDecoupling is achieved with the condition,\nψµ\nβ=∂µΓβ (3.7)\nOr explicitly, ψ0\nβ=∂tΓβ,ψβ=−∇Γβ, which ensures\n∇∧ψβ=−∇∧∇Γβ= 0\n∂tψβ+∇ψ0\nβ=−∂t(∇Γβ)+∇(∂tΓβ) = 0(3.8)\nThen, the system of eqs. ( 3.1) to (3.4) takes the form:\n∂t\nψ0\nα\n∂tΓβ\nψα\n0\n+∇·\nψα\n−∇Γβ\nTα\n−Iψα\n=\nφα\n−φβ\n−φα\nφβ\n(3.9)\nAt the cost of restricting the structure of ψµ\nβwe get a wave equation for Γ β, however, we free ψµ\nα\nin three equations independent of any other component of Ψ.\n3.3 Mixed Coupling\nAs a last notable case, consider instead a multivector for which its co mponents are coupled in\nthe form:\nΨ = (∂tf−∇f−I∇∧F)γ0+(−∇·F+∂tF)Iγ0 (3.10)\nwheref=f(xµ) is a scalar function and F=F(xµ) is a vector function. With this structure,\nthe system eqs. ( 3.1) to (3.4) directly yields the two wave equations\n9/squaref=φα\n/squareF=φβ(3.11)\nwhere we have used the identities\n∇·(I∇∧F) =I∇∧(∇∧F) = 0 (3.12)\n∇·(∇∧F) =∇2F−∇(∇·F) (3.13)\nUpon comparing with the general case, we see that eqs. ( 3.10) and (3.11) can be put in terms of\na potential multivector defined by fandF:\nΨ =∇(f+FI)\nΦ =∇Ψ =/square(f+FI)(3.14)\n4 Symmetries of the Continuity Equation\nFrom the system of eqs. ( 3.1) to (3.4) consider the proposal:\n∂tψ0\nα+∇·ψα=∂t(ψ0\nα)′+∇·(ψα)′(4.1)\nIn other words, how can the functions ψµ\nαtransform in such a way that the structure of its scalar\ncontinuity equation remains invariant? Making use of the identity of e q. (3.12), it is found:\nψ0\nα= (ψ0\nα)′+∇·Λ (4.2)\nψα= (ψα)′−∂tΛ−I∇∧Θ (4.3)\nIf these symmetry transformations for the scalar continuity are employed in eq. ( 3.3), we obtain\n∂t(ψα)′+∇(ψ0\nα)′+I∇∧ψβ−[/squareΛ+∇·(I∂tΘ+∇∧Λ)] =−φα (4.4)\nwhere we have used identity eq. ( 3.13). Independently of decoupling or not, let us impose an\ninvariance in the vector continuity eq. ( 3.3) upon the transformations eqs. ( 4.2) to (4.3):\n∂tψα+∇ψ0\nα+I∇∧ψβ=∂t(ψα)′+∇(ψ0\nα)′+I∇∧ψβ (4.5)\nWhich then delivers the condition\n/squareΛ=−I∇∧(∂tΘ−I∇∧Λ) (4.6)\nThis allows us to define\n10∂tΘ−I∇∧Λ=Ω (4.7)\nBy applying the ∂toperator to eq. ( 4.7), theI∇∧operator to eq. ( 4.3), and combining the two\nexpressions we obtain\n/squareΘ=∂tΩ−∇(∇·Θ)−I∇∧[ψα−(ψα)′] (4.8)\nFinally, it is convenient to define\n∇·Θ=−ω (4.9)\nThus, grouping all the results into matrix-notation continuity syst ems:\n∂t\n0\n0\nΛ\nΘ\n+∇·\nΛ\nΘ\nIΘ\n−IΛ\n=\nψ0\nα−(ψ0\nα)′\n−ω\n−/bracketleftbig\nψα−(ψα)′/bracketrightbig\nΩ\n(4.10)\n∂t/parenleftbiggψ0\nα−(ψ0\nα)′\nω/parenrightbigg\n+∇·/parenleftbiggψα−(ψα)′\nΩ/parenrightbigg\n=/square/parenleftbigg0\n0/parenrightbigg\n(4.11)\n∂t/parenleftbiggψα−(ψα)′\nΩ/parenrightbigg\n+∇/parenleftbiggψ0\nα−(ψ0\nα)′\nω/parenrightbigg\n−I∇∧/parenleftbigg−Ω\nψα−(ψα)′/parenrightbigg\n=/square/parenleftbigg−Λ\nΘ/parenrightbigg\n(4.12)\nErgo, by comparing with the general structures eqs. ( 2.7) and (2.9), it follows that if\nQ=Λ+ΘI (4.13)\nM=/bracketleftbig\nψ0\nα−(ψ0\nα)′+ψα−(ψα)′/bracketrightbig\nγ0+(ω+Ω)Iγ0 (4.14)\nthen eqs. ( 4.10) to (4.12) are nothing but\n∇Q=M (4.15)\n/squareQ=∇M (4.16)\n4.1 Free Waves\nThe condition for homogeneous wave equations for the fields ΛandΘ,\n/squareΛ=/squareΘ= 0 (4.17)\non one hand imposes the invariance of eq. ( 3.4),\n11−I∇∧ψα=−I∇∧(ψα)′(4.18)\nand on the other hand the restrictions on /an}b∇acketle{tM/an}b∇acket∇i}ht3:\n∇∧Ω= 0\n∂tΩ+∇ω= 0(4.19)\nBut from section 3.2these are just the conditions for the unilateral decoupling of /an}b∇acketle{tM/an}b∇acket∇i}ht3with\nrespect to/an}b∇acketle{tM/an}b∇acket∇i}ht1. Thus, the pseudovector of Mtakes the form\n/an}b∇acketle{tM/an}b∇acket∇i}ht3= (∂tζ−∇ζ)Iγ0 (4.20)\n4.2 Potentials\nSection3.3and eqs. ( 4.15) and (4.16) allow us to consider the equations:\n∇P=Q (4.21)\n/squareP=∇Q=M (4.22)\nFrom structure eq. ( 2.7) we see that the bivector Qis completely characterised by a potential\nmultivector of the form\nP= (p0\n1+p1)γ0+(p0\n4+p4)Iγ0 (4.23)\nTherefore, eqs. ( 4.21) and (4.22) in explicit form are\n∂t/parenleftbigg\np0\n1\np0\n4/parenrightbigg\n+∇·/parenleftbigg\np1\np4/parenrightbigg\n=/parenleftbigg\n0\n0/parenrightbigg\n(4.24)\n∂t/parenleftbiggp1\np4/parenrightbigg\n+∇/parenleftbiggp0\n1\np0\n4/parenrightbigg\n−I∇∧/parenleftbigg−p4\np1/parenrightbigg\n=/parenleftbigg−Λ\nΘ/parenrightbigg\n(4.25)\n/square\np0\n1\n−p0\n4\n−p1\np4\n=∂t\n0\n0\nΛ\nΘ\n+∇·\nΛ\nΘ\nIΘ\n−IΛ\n=\nψ0\n1−(ψ0\n1)′\n−ω\n−/bracketleftbig\nψ1−(ψ1)′/bracketrightbig\nΩ\n(4.26)\nImposing the decoupling of /an}b∇acketle{tP/an}b∇acket∇i}ht3with respect to/an}b∇acketle{tP/an}b∇acket∇i}ht1through eq. ( 3.7) permits to express the\nfields in terms of the vector potential exclusively,\nΛ=−∇p0\n1−∂tp1\nΘ=−I∇∧p1(4.27)\n124.3 Poynting\nBy the structure of eq. ( 4.13), the Poynting multivector associated with the field Qhas the\nstructure described in section 2.2.1. That is,\nSQ=1\n2/parenleftig\nΛ·Λ+Θ·Θ/parenrightig\nγ0−IΛ∧Θγ0 (4.28)\nFurther, by the structure of eq. ( 4.23) we may also construct a Poynting multivector for the\npotentialP. Its general form corresponds to that described in eq. ( 2.19). We show the special\ncase where pµ\n4is null:\nSP=1\n2/bracketleftbig\n(p0\n1)2+p1·p1/bracketrightbig\nγ0+p0\n1p1γ0 (4.29)\nThe respective continuity equations are:\n∂t/bracketleftig1\n2/parenleftig\nΛ2+Θ2/parenrightig/bracketrightig\n+∇·/parenleftig\n−IΛ∧Θ/parenrightig\n=/an}b∇acketle{tW/an}b∇acket∇i}ht0 (4.30)\n∂t/bracketleftig1\n2/parenleftbig\np0\n1p0\n1+p1·p1/parenrightbig/bracketrightig\n+∇·/parenleftig\np0\n1p1/parenrightig\n=/an}b∇acketle{tU/an}b∇acket∇i}ht0 (4.31)\n4.4 Discussion\nWehavefoundthattheinvarianceofthesystemofcontinuityequa tionsforamultivector /an}b∇acketle{tΨ/an}b∇acket∇i}ht1due\nto symmetry transformations naturally allows for the constructio n of a system of eqs. with the\nstructure of Maxwell’s field equations4, eq. (4.10), for the bivector field Qdefined by eq. ( 4.13).\nBecause of the symmetric nature in the system of eqs. ( 3.1) to (3.4) for/an}b∇acketle{tΨ/an}b∇acket∇i}ht1+/an}b∇acketle{tΨ/an}b∇acket∇i}ht3and/an}b∇acketle{tΨ/an}b∇acket∇i}ht0+\n/an}b∇acketle{tΨ/an}b∇acket∇i}ht2+/an}b∇acketle{tΨ/an}b∇acket∇i}ht4, all of these relations hold if Mhas a bivector character and Qa vector character.\nThe emergence of the new pseudovector term /an}b∇acketle{tM/an}b∇acket∇i}ht3= (ω+Ω)Iγ0exists in independence of the\npseudovector part /an}b∇acketle{tΨ/an}b∇acket∇i}ht3=/parenleftig\nψ0\nβ+ψβ/parenrightig\nIγ0of Ψ,\nM=/an}b∇acketle{tΨ/an}b∇acket∇i}ht1−/an}b∇acketle{tΨ′/an}b∇acket∇i}ht1+/an}b∇acketle{tM/an}b∇acket∇i}ht3 (4.32)\nThe symmetries do admit /an}b∇acketle{tΨ/an}b∇acket∇i}ht3=/an}b∇acketle{tM/an}b∇acket∇i}ht3, although this does not seem to be a requirement but a\nspecial case. The whole system is obtained by only the symmetries of eqs. (3.1) and (3.3) insofar\nas the invariance of eq. ( 3.4) emerges as a condition for the existence of free waves for the fie lds.\nThe vector part/an}b∇acketle{tM/an}b∇acket∇i}ht1=/an}b∇acketle{tΨ/an}b∇acket∇i}ht1−/an}b∇acketle{tΨ′/an}b∇acket∇i}ht1may be regarded as a displacement in the transported\nphysical quantity with a fixed coordinate system.\nIf, by hypothesis, /an}b∇acketle{tΨ/an}b∇acket∇i}ht1→Jγ0=ρ+jin eq. (4.14) is the space-time electric current density, then\nit can be immediately identified Q→F=E+IBfrom eq. ( 4.13) as Faraday’s electromagnetic\nfield,/an}b∇acketle{tP/an}b∇acket∇i}ht1→Aγ0=φ+Afrom eq. ( 4.23) as the space-time electromagnetic potential, and\nS0\n1→u=1\n2/parenleftig\nE2+B2/parenrightig\n,S1→s=E×Bof eq. (4.28) as the electromagnetic energy density\nand Poynting vector, respectively. /an}b∇acketle{tM/an}b∇acket∇i}ht3may be identified as the space-time magnetic current\ndensity, and/an}b∇acketle{tP/an}b∇acket∇i}ht3as a new space-time magnetic potential. Correspondingly, eq. ( 4.11) represents\nthe electric and magnetic charge conservation, eq. ( 4.24) the Lorenz gauge, and eq. ( 4.30) the\n4To express in Gibbs-Heaviside’s vector calculus notation: ∇·(IF) =I∇∧F=−∇×F.\n13conservation of electromagnetic energy5. The fact that the condition of free wavesfor the electric\nand magnetic fields implies the decoupling of the magnetic current den sity/an}b∇acketle{tM/an}b∇acket∇i}ht3with respect to\nthe electric current density J(section4.1) and also that the magnetic charge density is negative,\neq. (4.9), may shed light to the nature of magnetic monopoles and their expe rimental elusiveness\nso far.\nIf/an}b∇acketle{tΨ/an}b∇acket∇i}ht1→Jγ0=n+jis the turbulent or fluid-mechanical ‘charge’ density, then Λ→Lis\nLamb’s vector, Ω→wis the vorticity, p1→vis the fluid’s velocity, p0\n1→his the enthalpy\nper unit mass or equivalently p0\n1→ϕBernoulli’s energy function, and eq. ( 4.10) are Maxwell’s\nfield equations for a compressible fluid [ 31], [32]. But, of course, the results found that Maxwell’s\nequations are a consequence of the continuity equation’s symmetr ies are not limited to either\nelectrodynamics or hydrodynamics.\n5 Diffusion Equation\nTo conclude, consider the case where one of the relative vectors s atisfies Fick’s laws of diffusion.\nE.g.,\nψα=−D∇ψ0\nα (5.1)\nwhereD=D/parenleftbig\nψ0\nα;xµ/parenrightbig\nis called the diffusion coefficient. With eq. ( 5.1), the continuity system of\neqs. (3.1) to (3.4) takes the form:\n∂tψ0\nα−∇·(D∇ψ0\nα) =φα (5.2)\n∂tψ0\nβ+∇·ψβ=−φβ (5.3)\n(1−∂tD)∇ψ0\nα−D∇/parenleftbig\n∂tψ0\nα/parenrightbig\n+I∇∧ψβ=−φα (5.4)\n∂tψβ+∇ψ0\nβ+I∇D∧∇ψ0\nα=φβ (5.5)\nEquation ( 5.1) establishes a diffusion equation for ψ0\nαin eq. (5.2). However, the solution and\nstructure of this equation are conditioned by the rest of the equa tions in the system.\n5.1 Constant Diffusion Coefficient\nConsider the case D= constant. For this condition, the system of eqs. ( 5.2) to (5.5) reduces to:\n∂tψ0\nα−D∇2ψ0\nα=φα\n∂tψ0\nβ+∇·ψβ=−φβ\n∇/parenleftbig\nψ0\nα−D∂tψ0\nα/parenrightbig\n+I∇∧ψβ=−φα\n∂tψβ+∇ψ0\nβ=φβ(5.6)\n5In STA’s treatment of electromagnetic theory it is well-kno wn that the expansion of eq. ( 2.10) and1\n2/parenleftBig\n/tildewideΦΨ+\n/tildewideΨΦ/parenrightBig\nis connected to Poynting’s theorem and Lorentz force [ 30].\n14As expected, a constant diffusion coefficient yields a heat equation f orψ0\nα. The remaining\nequations are still coupled with it, so let us inspect the case for which ψβis subjected to the\nimposition of eq. ( 3.5) such that eq. ( 5.6) takes the form:\n∂tψ0\nα−D∇2ψ0\nα=φα\n∂tψ0\nβ+∇2Γβ=−φβ\n∇/parenleftbig\nψ0\nα−D∂tψ0\nα/parenrightbig\n=−φα\n∇/parenleftbig\nψ0\nβ+∂tΓβ/parenrightbig\n=φβ(5.7)\nDecoupling of ψµ\nαandψµ\nβis indeed achieved. However, it can be seen that the absence of ext ernal\nsources,∇Ψ = 0, represents a separation of space and time variables for ψ0\nα. That is, Φ = 0\nimplies\nψ0\nα−D∂tψ0\nα= constant\nψ0\nβ+∂tΓβ= constant(5.8)\nand thus\n/parenleftbig\n∇2−D−2/parenrightbig\nψ0\nα= const. (5.9)\n/squareΓβ= 0 (5.10)\nFor a pseudoscalardiffusion coefficient, D= (D′)−1I, eq. (5.9) takes the structure ofa Helmholtz\nequation:/parenleftbig\n∇2+D′2/parenrightbig\nψ0\nα= const. The separation of variables for ψ0\nαconstitutes by itself a\ntransformation that decouples the system, as can be easily verifie d by imposing a priorieq. (5.8)\nto the system of eq. ( 5.6). Furthermore, the decoupling-by-gradient hypothesis for ψβmay also\nadopt a diffusion structure by considering a multivector Ψ = ( ψα−Dα∇ψα)+(ψβ−Dβ∇ψβ)I,\nwhereDα,βare both constants. Equation ( 2.1) yields the ‘secondly decoupled’ system:\n∂tψα,β−Dα,β∇2ψα,β=±φα,β\n∇(ψα,β−Dα,β∂tψα,β) =∓φα,β(5.11)\nOnce again, the absence of external sources Φ = 0—or specifically t he absence of vector field\nsourcesφα,β= 0—reduces the system into two Helmholtz-like equations, eq. ( 5.9).\n5.2 Non-Constant Diffusion Coefficient\nLet us decouple ψµ\nβfrom the general diffusion system of eqs. ( 5.2) to (5.5):\n∂tψ0\nα−∇·(D∇ψ0\nα) =φα (5.12)\n(1−∂tD)∇ψ0\nα−D∇/parenleftbig\n∂tψ0\nα/parenrightbig\n=−φα (5.13)\n15I∇D∧∇ψ0\nα=φβ (5.14)\nConsider the case where the diffusion coefficient is an explicit function ofψ0\nαbut not on the\nspace-time coordinates, D=D/parenleftbig\nψ0\nα/parenrightbig\n:\n∂tD=∂D\n∂ψ0α∂tψ0\nα\n∇D=∂D\n∂ψ0α∇ψ0\nα(5.15)\nThen, from eqs. ( 5.12) to (5.14) the following non-linear system is obtained\n∂tψ0\nα−∇·(D∇ψ0\nα) =φα\n∇ψ0\nα−∂D\n∂ψ0α∂tψ0\nα∇ψ0\nα−D∇/parenleftbig\n∂tψ0\nα/parenrightbig\n=−φα(5.16)\nStructureD=D/parenleftbig\nψ0\nα/parenrightbig\nby virtue of eq. ( 5.15) implies that the diffusion flux is a conservative\nvector field, which further enables the introduction of a potential gradient:\n∇ϑ=D∇ψ0\nα (5.17)\nErgo, system eq. ( 5.16) can also be expressed as\n∂tψ0\nα−∇2ϑ=φα\n∇/parenleftbig\nψ0\nα−∂tϑ/parenrightbig\n=−φα(5.18)\nOnce again, in the absence of external sources ψ0\nα−∂tϑ= const, and therefore\n/squareϑ= 0 (5.19)\n5.2.1 Interpretation\nConsider the system of eqs. ( 5.12) to (5.14) product of∇/parenleftbig\nψ0\nα−D∇ψ0\nα/parenrightbig\n=/parenleftbig\nφ0\nα+φα+Iφβ/parenrightbig\nγ0.\nEquations( 5.13)and(5.14)conditionthefunction ψ0\nαinsuchawaythatforconstantorexplicitly-\ndependent coefficients, the absence of external sources implies t hat the diffusion structure of\neq. (5.12) is lost. This situation can be understood from a perspective of bi-d ependence between\nthe components of Ψ and Φ. From eq. ( 5.12), the density/concentration ψ0\nαcan be determined\nin terms of the scalar source φ0\nαand the diffusive coefficient D. Then, (φα+Iφβ) may be\nconceived as an external vector-field source definedby eqs. (5.13) and (5.14) in terms of ψ0\nαand\nD. From the constant and explicitly-dependent coefficient cases, we saw that if this field is null\nthen the diffusive structure is transformed into Helmholtz-like and w ave structures. Therefore,\nthe external field ( φα+Iφβ) admits the interpretation of being the source that produces itse lf\nthe diffusion phenomena.\n166 Conclusion\nIn synthesis, we have shown that for a generalised STA multivector , equation∇Ψ = Φ yields\na system of eight continuity equations—four of scalar character a nd four of vector character.\nFrom this, we constructed the system /squareΨ =∇Φ where each wave equation for the components\nof Ψ plays the role of the source for the continuity system of Φ. With the same method for\nwhich the continuity equation is derived from Schr¨ odinger and Dirac equations, we have defined\nthe Poynting multivector associated with the density of the genera lised multivector Ψ, together\nwith its continuity system. These systems allow the identification of t he coupled structures\n/an}b∇acketle{tΨ/an}b∇acket∇i}ht1+/an}b∇acketle{tΨ/an}b∇acket∇i}ht3and/an}b∇acketle{tΨ/an}b∇acket∇i}ht0+/an}b∇acketle{tΨ/an}b∇acket∇i}ht2+/an}b∇acketle{tΨ/an}b∇acket∇i}ht4. The cost of complete decoupling for either structure is a\ntransition to a system of higher-order equations, such as the wav e system. Faced with this, we\nhave determined three cases of decoupling by imposing restrictions on the structure of at least\none of the functions involved in the continuity equations: if both fun ctions are APS gradients;\nif one of them is a space-time gradient; and, considering instead a mix ed coupling, a potential\nmultivector structure was constructed for which the continuity s ystem yields directly a wave\nsystem. Symmetry transformations that make the (scalar) cont inuity equation invariant are\nfound in terms of vector fields. By imposing that the continuity equa tions system be invariant\nunder these transformations,asystem offield and waveequation swith the structure ofMaxwell’s\nequations was derived. This Maxwellian system along with an application of the continuity,\nwave, and Poynting multivector systems permitted the construct ion of fields and potentials\ndirectly connected to the generalised STA multivector: /squareP=∇Q=/an}b∇acketle{tΨ/an}b∇acket∇i}ht1−/an}b∇acketle{tΨ′/an}b∇acket∇i}ht1. The results\nfound areconsistentwith the classicalelectromagnetictheoryan d with hydrodynamicsfrom fluid\nmechanics. When one of the functions in the continuity system satis fies Fick’s laws, a diffusion\nequation arises coupled with the remaining continuity equations of th e system. It was found that\nfor a decoupled system with a constant or explicitly-dependent diffu sion coefficient, the absence\nof external sources implies a loss in the diffusion equation structure being transformed to a\nHelmholtz-like structure and a wave system. For both cases consid ered, the external vector-field\nsource can be defined by the complementary equations in the syste m in terms of the density and\ndiffusion coefficient, and admit the interpretation of the diffusion phe nomena producer.\nReferences\n[1] K.S.ThorneandR.D.Blandford, Modern Classical Physics: Optics, Fluids, Plasmas, Elasti city, Relativity,\nand Statistical Physics . Princeton University Press, 2017, isbn: 978-0691159027.\n[2] J. D. Jackson, Classical Electrodynamics , 3rd. Wiley, 1999, ch. 6, isbn: 978-0-471-30932-1.\n[3] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics , 3rd ed. Cambridge University Press, 2020,\nch. 8,isbn: 9781108587280.\n[4] J.Lasenby, A.N.Lasenby, and C.J.L.Doran,“Aunifiedmat hematical language forphysicsand engineering\nin the 21st century,” Philosophical Transactions of the Royal Society of London. Series A: Mathematical,\nPhysical and Engineering Sciences , vol. 358, no. 1765, pp. 21–39, 2000. doi:10.1098/rsta.2000.0517 .\n[5] S. Gull, A. Lasenby, and C. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,”\nFoundations of Physics , vol. 23, no. 9, pp. 1175–1201, Sep. 1993. doi:10.1007/BF01883676 .\n[6] A. N. Lasenby, “Geometric algebra as a unifying language for physics and engineering and its use in the\nstudy of gravity,” Advances in Applied Clifford Algebras , vol. 27, no. 1, pp. 733–759, Mar. 2017. doi:\n10.1007/s00006-016-0700-z .\n[7] D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” American\nJournal of Physics , vol. 71, no. 2, pp. 104–121, Feb. 2003, issn: 0002-9505. doi:10.1119/1.1522700 .\n[8] W. E. Baylis, J. Huschilt, and J. Wei, “Why i?” American Journal of Physics , vol. 60, no. 9, pp. 788–797,\nSep. 1992, issn: 0002-9505. doi:10.1119/1.17060 .\n[9] E. Hitzer, C. Lavor, and D. Hildenbrand, “Current survey of clifford geometric algebra applications,”\nMathematical Methods in the Applied Sciences , vol. 47, no. 3, pp. 1331–1361, 2024. doi:10.1002/mma.8316 .\n[10] S. Breuils, K. Tachibana, and E. Hitzer, “New applicati ons of clifford’s geometric algebra,” Advances in Ap-\nplied Clifford Algebras , vol. 32, no. 2, p. 17, Feb. 2022, issn: 1661-4909. doi:10.1007/s00006-021-01196-7 .\n17[11] A. Macdonald, “An elementary construction of the geome tric algebra,” Advances in Applied Clifford Alge-\nbras, vol. 12, no. 1, pp. 1–6, Jun. 2002, issn: 1661-4909. doi:10.1007/BF03161249 .\n[12] R. D. Arthan, A minimalist construction of the geometric algebra , 2006. arXiv: math/0607190 [math.RA] .\n[13] A. Macdonald, “A survey of geometric algebra and geomet ric calculus,” Advances in Applied Clifford\nAlgebras , vol. 27, no. 1, pp. 853–891, Mar. 2017, issn: 1661-4909. doi:10.1007/s00006-016-0665-y .\n[14] E. Hitzer, “Introduction to clifford’s geometric algeb ra,”Journal of The Society of Instrument and Control\nEngineers , vol. 51, no. 4, pp. 338–350, 2012. doi:10.11499/sicejl.51.338 .\n[15] D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language fo r Mathematics\nand Physics . Springer Dordrecht, 1984, isbn: 978-94-009-6292-7. doi:10.1007/978-94-009-6292-7 .\n[16] C. Doran and A. Lasenby, Geometric Algebra for Physicists . Cambridge University Press, 2003, isbn:\n9780511807497. doi:10.1017/CBO9780511807497 .\n[17] C. J. L. Doran, “Geometric algebra and its application t o mathematical physics,” Ph.D. dissertation, Apollo\n- University of Cambridge Repository, 1994. doi:10.17863/CAM.16148 .\n[18] D.Hestenes, Space-Time Algebra .Birkh¨ auserCham,2015, isbn:978-3-319-18413-5. doi:10.1007/978-3-319-18413-5 .\n[19] D. Hestenes, “Spacetime physics with geometric algebr a,”American Journal of Physics , vol. 71, no. 7,\npp. 691–714, Jul. 2003, issn: 0002-9505. doi:10.1119/1.1571836 .\n[20] W. E. Baylis and G. Sobczyk, “Relativity in clifford’s ge ometric algebras of space and spacetime,” Inter-\nnational Journal of Theoretical Physics , vol. 43, no. 10, pp. 2061–2079, Oct. 2004, issn: 1572-9575. doi:\n10.1023/B:IJTP.0000049010.53558.b7 .\n[21] A. I. Arbab and H. M. Widatallah, “On the generalized con tinuity equation,” Chinese Physics Letters ,\nvol. 27, no. 8, p. 084703, Aug. 2010. doi:10.1088/0256-307X/27/8/084703 .\n[22] A. Arbab and N. N. Alsaawi, “The dual continuity equatio ns,”Optik, vol. 248, p. 168095, 2021, issn:\n0030-4026. doi:10.1016/j.ijleo.2021.168095 .\n[23] Q. M. S. Jr. and D. J. McNamara, “Electromagnetic energy -momentum equation without tensors: A geo-\nmetric algebra approach,” 2008. arXiv: 0807.1382 [physics.class-ph] .\n[24] M. Castilla, J. C. Bravo, M. Ordonez, and J. C. Montano, “ An approach to the multivectorial apparent\npower in terms of a generalized poynting multivector,” Progress In Electromagnetics Research B , vol. 15,\npp. 401–422, 2009. doi:10.2528/PIERB09042402 .\n[25] D. F. Scofield and P. Huq, “Fluid dynamical lorentz force law and poynting theorem—introduction,” Fluid\nDynamics Research , vol. 46, no. 5, p. 055513, Aug. 2014. doi:10.1088/0169-5983/46/5/055513 .\n[26] L. Burns, “Maxwell’s equations are universal for local ly conserved quantities,” Advances in Applied Clifford\nAlgebras , vol. 29, no. 4, p. 62, Jul. 2019, issn: 1661-4909. doi:10.1007/s00006-019-0979-7 .\n[27] A.Macdonald, Potentials, fields, and sources with geometric calculus , 2019. [Online].Available: https://api.semanticscholar.org/Corp \n[28] J. A. Heras, “Can Maxwell’s equations be obtained from t he continuity equation?” American Journal of\nPhysics, vol. 75, no. 7, pp. 652–657, Jul. 2007, issn: 0002-9505. doi:10.1119/1.2739570 .\n[29] J. A. Heras, “How to obtain the covariant form of maxwell ’s equations from the continuity equation,”\nEuropean Journal of Physics , vol. 30, no. 4, p. 845, May 2009. doi:10.1088/0143-0807/30/4/017 .\n[30] T. G. Vold, “An introduction to geometric calculus and i ts application to electrodynamics,” American\nJournal of Physics , vol. 61, no. 6, pp. 505–513, Jun. 1993, issn: 0002-9505. doi:10.1119/1.17202 .\n[31] S. Demir and M. Tanı¸ slı, “Spacetime algebra for the ref ormulation of fluid field equations,” International\nJournal of Geometric Methods in Modern Physics ,vol.14,no.05,p.1750075,2017. doi:10.1142/S021988781750075X .\n[32] D. Sen, “Field theoretic formulation of fluid mechanics according to the geometric algebra,” Pramana ,\nvol. 97, no. 3, p. 132, Aug. 2023, issn: 0973-7111. doi:10.1007/s12043-023-02617-x .\n18"    },    {        "title": "2402.04305v2.Cosmological_Observatories.pdf",        "content": "Cosmological Observatories\nDionysios Anninos1,2, Dami´ an A. Galante1, and Chawakorn Maneerat1\n1Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK\n2Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium\ndionysios.anninos@kcl.ac.uk, damian.galante@kcl.ac.uk, chawakorn.maneerat@kcl.ac.uk\nAbstract\nWe study the static patch of de Sitter space in the presence of a timelike boundary. We impose\nthat the conformal class of the induced metric and the trace of the extrinsic curvature, K, are\nfixed at the boundary. We present the thermodynamic structure of de Sitter space subject to\nthese boundary conditions, for static and spherically symmetric configurations to leading order\nin the semiclassical approximation. In three spacetime dimensions, and taking Kconstant on a\ntoroidal Euclidean boundary, we find that the spacetime is thermally stable for all K. In four\nspacetime dimensions, the thermal stability depends on the value of K. It is established that\nfor sufficiently large K, the de Sitter static patch subject to conformal boundary conditions\nis thermally stable. This contrasts the Dirichlet problem for which the region encompassing\nthe cosmological horizon has negative specific heat. We present an analysis of the linearised\nEinstein equations subject to conformal boundary conditions. In the worldline limit of the\ntimelike boundary, the underlying modes are linked to the quasinormal modes of the static\npatch. In the limit where the timelike boundary approaches the cosmological event horizon, the\nlinearised modes are interpreted in terms of the shear and sound modes of a fluid dynamical\nsystem. Additionally, we find modes with a frequency of positive imaginary part. Measured in a\nlocal inertial reference frame, and taking the stretched cosmological horizon limit, these modes\ngrow at most polynomially.arXiv:2402.04305v2  [hep-th]  13 Feb 2024Contents\n1 Introduction 3\n2 General framework 6\n2.1 Conformal thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3 dS 3conformal thermodynamics 9\n3.1 Pole patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.2 Cosmic patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3.3 Pure dS 3patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n3.4 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n3.5 A two-sphere perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n4 dS 4conformal thermodynamics 17\n4.1 Pole patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n4.2 Cosmic patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n4.3 Black hole patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.4 Pure dS 4patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n4.5 Nariai patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n4.6 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n5 Linearised dynamics 28\n5.1 Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n5.2 Vector perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n5.3 Scalar perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n5.4 Dynamics of the pole patch, briefly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\nA dS 3Dirichlet thermodynamics 34\nB Useful formulae for D= 4 36\nC Details of black hole patch computations in D= 4 36\nD Plots of the regulated action, Econf, and CKinD= 4 38\nEl= 0and l= 1modes 38\nF Near-horizon diffeomorphisms 40\n21 Introduction\nIn the absence of an asymptotic spatial or null boundary, the construction of gauge invariant\nobservables subject to the constraints of diffeomorphism redundancies in a theory of gravity becomes\na challenging task. One is often led to relational notions [1–3] (for a review on recent work see [4])\nwhereby a given physical phenomenon is measured in relation to some other semiclassical feature.\nFor instance, in inflationary models of the early Universe, we can measure the time-dependence of\nphysical phenomena with respect to the slow classical roll of the background inflaton field. From a\nmore quasilocal perspective, one might imagine decorating spacetime with a worldline [5–8], perhaps\nslightly thickened into a worldtube, and use this as a reference frame for ambient phenomena. This\nperspective appears to be of particular value for an asymptotically de Sitter spacetime [9–16], where\nnot only are the Cauchy spatial slices potentially compact, but quasilocal entities are moreover\nsurrounded by a cosmological event horizon rendering most of the expanding portion of spacetime\nphysically obscure. A drawback of the relational approach is that it often necessitates the presence\nof a semiclassical feature in spacetime, making the general picture away from the semiclassical or\nperturbative regime difficult to control.\nAn alternative, complementary, route may be to study the gravitational theory on a manifold\nendowed with a quasi-auxiliary timelike boundary Γ, much like we do when considering gravitational\nphysics in anti-de Sitter space, and try to make sense of general relativity in such a setting. This\nsetup has been the focus of recent work in mathematical relativity [17–22], accompanied by [23–25]\n(see also [26,27] for related work). Of particular interest to our work is the proposal of [18,21] that\nin four spacetime dimensions certain conformal data along Γ lead to a well-posed initial boundary\nvalue problem. In [18,21] it is further established that generically, the Dirichlet problem in general\nrelativity—whereby one fixes the induced metric along Γ—suffers from potential existence and non-\nuniqueness issues for both Euclidean and Lorentzian signature. Concretely, the conformal boundary\nconditions of interest fix the conformal class of the induced metric, [ gmn|Γ](conf), and the trace of\nthe extrinsic curvature, K, along Γ whilst also specifying standard Cauchy data along a spalike\nsurface Σ intersecting Γ at its boundary.\nIn this paper, we explore the conformal boundary conditions of [18, 21] for general relativity with\na positive cosmological constant Λ [28–30]. We consider the problem in both Euclidean and\nLorentzian signatures. In Euclidean signature, our main goal is to compute the semiclassical ap-\nproximation of the gravitational path integral and consider its interpretation from the point of\nview of Euclidean gravitational thermodynamics [31,32]. In the absence of a boundary, the natural\nEuclidean geometry for general relativity with Λ >0 is the sphere [31], void of any external data\nsuch as the size of a thermal circle, and one is led to path integrate fields on top of it. In adding a\nboundary to our Euclidean manifold, as pointed out in [12,33,34] among other places, one has the\npossibility of providing a novel perspective to the sphere path integral and its rich though elusive\nphysical content [35].1In Lorentzian signature, one is led to the question of dynamical features of de\nSitter space, known to be dynamically stable at the classical level [38], in the presence of a timelike\n1More speculatively, as suggested in [36], the presence of boundaries in the Λ >0 Euclidean path integral might be\nnecessitated in parallel to the non-perturbative necessity of boundaries on the two-dimensional worldsheet of closed\nstrings [37] upon trying to make sense of the sum over topologies. From a Lorentzian perspective, we may imagine\nthe creation of a long-lived heavy particle from the vacuum. These events are Boltzmann suppressed but can occur.\nIn the semiclassical limit, such timelike features become parametrically long-lived and may warrant a treatment\ninvolving timeilke boundaries.\n3boundary. There are two timelike surfaces of particular interest. One of these is the worldline limit,\nwhereby the spatial size of Γ becomes small in units of Λ. The other is the cosmological horizon\nlimit, whereby Γ approaches the cosmological de Sitter horizon. The former limit is of interest in\ndescribing the theory of a quasilocal entity, whilst the latter is of interest if one wishes to describe\nphysics from the perspective of a stretched horizon [39–43].\nOrganisation and summary of main results\nIn section 2, we present the general framework, and provide an explicit definition of the conformal\nboundary conditions. As noted, our gravitational theory is endowed with a positive cosmological\nconstant Λ = +( D−1)(D−2)/2ℓ2inDspacetime dimensions.\nIn sections 3 and 4, we consider the problem in Euclidean signature for D= 3 and D= 4 spacetime\ndimensions respectively. The boundary of our manifold is taken to have an S1×SD−2topology. In\nthe standard treatment of semiclassical black hole thermodynamics [31] with Dirichlet boundary\nconditions, one defines the canonical ensemble by fixing the size of the boundary S1to be the inverse\ntemperature βand the radius of the spatial sphere to be fixed to some size r. In our treatment,\nwe will instead fix the conformal class of the boundary metric. As such, we fix a conformal version\nof the inverse temperature, ˜β≡β/r. The other boundary data we fix is the trace of the extrinsic\ncurvature, K. We refer to this ensemble as the conformal canonical ensemble. Gravitational\nsolutions in the conformal canonical ensemble include patches with no horizons, referred to as pole\npatches, patches with cosmological horizons, referred to as cosmic patches, and patches with black\nhole horizons, referred to as black hole patches. Upon tuning ˜βandK, one finds that the space is\nfilled with a pure de Sitter spacetime, and we refer to this as a pure de Sitter patch.\nThe complete thermal phase space of static and spherically symmetric solutions at a given ˜β >0\nandKℓ∈Ris provided in both D= 3 and D= 4 spacetime dimensions. Below we summarise the\nmain results, with an emphasis on the conformal thermodynamics of the pure de Sitter patch.\nThree-spacetime dimensions. Our analysis in D= 3 spacetime dimensions naturally builds on\nrecent developments on dS 3[12,44–46]. We find that upon imposing conformal boundary conditions:\n•Both a pole and a cosmic patch exist at any value of ˜βandKℓ.\n•The entropy of the cosmic patch is given by the Gibbons-Hawking entropy of the cosmological\nhorizon, and the specific heat is positive for all ˜βandKℓ.\n•The thermodynamic quantities take the form of a two-dimensional conformal field theory.\nViewed as such, we identify a c-function\ncconf=3ℓ\n4GN\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n, (1.1)\nwhich decreases monotonically as one goes from the worldline limit, where Kℓ→ −∞ , to the\nstretched horizon, where Kℓ→+∞.\n•There is a phase transition at ˜βc= 2π(for all values of Kℓ). At ˜β >˜βc, the cosmic patch is\nthe thermally preferred solution. In the stretched horizon limit, the cosmic patch is thermally\nstable, while in the worldline limit it is metastable.\nThe thermodynamic picture in D= 3 contrasts that of the canonical ensemble, obtained by im-\nposing Dirichlet boundary conditions. In the canonical ensemble, the specific heat of the cosmic\n4patch is always negative.\nFour-spacetime dimensions. We now summarise the situation in D= 4 spacetime dimensions,\nwhich naturally builds on previous work in [33,34,47–49]. We find that upon imposing conformal\nboundary conditions:\n•A pole patch solution exists for any value of ˜βandKℓ. On the other hand, cosmic/black\nhole horizon patch solutions only exist for certain values of ˜βandKℓ. When they do exist,\nwe identify three distinct solutions for a given ˜βandKℓ—one pole patch and two horizon\npatches, which can be of the cosmic or black hole type.\n•The entropy of the solutions with horizons is always given by their respective horizon area\nformulas. The solution with the larger horizon area always has positive specific heat, while\nthe one with a smaller horizon area always has negative specific heat.\n•When the cosmic patches have positive specific heat, one can take a large temperature limit.\nIn this limit, the entropy goes as Nd.o.f./˜β2, and thus resembles that of a conformal field\ntheory in three dimensions. We identify\nNd.o.f.=32π3ℓ2\n81GN\u0010p\nK2ℓ2+ 9−Kℓ\u00112\n. (1.2)\nFurther taking the large Kℓlimit of the above expression yields Nd.o.f.≈8π3\nGNK2, a behaviour\nidentified in [25] for black holes in Minkowski space subject to conformal boundary conditions.\n•We identify pure dS 4patches with positive specific heat. These solutions exist when the\ntube is positioned sufficiently near the cosmological horizon, starting from rtube≈0.259ℓ.\nDepending on Kℓ, these solutions are either metastable or globally stable. The pure dS 4patch\nis thermally stable in the stretched horizon limit, at least among the spherically symmetric\nsector. Near the worldline regime, pure dS 4patches have negative specific heat. The full\nphase diagram is presented in figure 10.\nWe can contrast the thermodynamic behaviour above to that of the canonical thermal ensemble\nstemming from Dirichlet boundary conditions [33,34,47–49]. In the latter, the specific heat of the\npure dS 4patch is always negative.\nIn section 5, we consider the four-dimensional Lorentzian picture. The theory is placed on a\nmanifold with timelike boundary of R×S2topology. In the Lorentzian case, one must further\nsupply standard Cauchy data along the initial time spatial slice Σ. Employing the Kodama-\nIshibashi method [50], we present the linearised gravitational dynamics about the pure de Sitter\nsolution. The linearised solutions split into vector and scalar modes concerning their transformation\nproperties under SO(3). We are mainly interested in two limits: one in which the boundary is close\nto the worldline observer, which we call the worldline limit; and a second one, in which the boundary\nbecomes close to the cosmological horizon, which we call the stretched horizon limit. In each case,\nthe main results of our analysis are:\n•In the worldline limit of the cosmic patch, we retrieve a set of modes that approximate the\nquasinormal modes of the static patch [51], whilst also uncovering a family of modes in the\nscalar sector with a negative imaginary part. The latter modes have a Minkowskian analogue\nuncovered in [25].\n•In the stretched horizon limit, our modes degenerate into a variety of modes. The low-lying\n5vector modes match a set of modes identified in [52] as a type of linearised shear mode for\nan incompressible non-relativistic Navier-Stokes equation. The scalar modes take either the\nform of a sound mode with diverging speed of sound as we approach the horizon limit, or a\npair of modes with ωℓ=±i. We provide an understanding of these two modes from a purely\nRindler perspective and note that in a local inertial frame the exponential behavior becomes\npolynomial.\nAdditional technical details are provided in the various appendices.\n2 General framework\nWe consider vacuum solutions to general relativity with positive cosmological constant Λ = +( D−\n1)(D−2)/2ℓ2inD= 3, and D= 4 spacetime dimensions. In Euclidean signature, the action IE\nis given by\nIE=−1\n16πGNZ\nMdDxp\ndetgµν(R−2Λ)−αb.c.\n(D−1)8πGNZ\nΓdD−1xp\ndetgmnK , (2.1)\nwhere GNis the Newton’s constant, Γ = ∂M,gmndenotes the induced metric at Γ, and the trace\nof the extrinsic curvature Kis given by\nK=gmnKmn, K mn=1\n2Lˆngmn. (2.2)\nHere, ˆ n= ˆnµ∂µis an outward pointing unit normal vector associated with the boundary, and\nLˆndenotes a Lie derivative with respect to ˆ nµ. We adopt the notation in which Greek indices\nµ= 0, ..., D −1 are used for spacetime indices and m= 0, ..., D −2 are used for spacetime indices\ntangent to the boundary.\nThe constant αb.c.in (2.1) depends on the choice of boundary conditions. In most of the paper we\nwill consider conformal boundary conditions, in which we fix the conformal class of the induced\nmetric and the trace of the extrinsic curvature at the boundary,\nConformal boundary conditions : {[gmn|Γ]conf, K|Γ}= fixed . (2.3)\nWith this set of boundary conditions, the initial boundary value problem in general relativity\nis proven to be well-posed in Euclidean signature [18, 23] and conjectured to be well-posed in\nLorentzian signature [21, 24]. This is in contrast to Dirichlet or Neumann boundary conditions,\nwhere general relativity does not permit a well-posed initial boundary value problem for generic\nboundary data. On occasion, it will be useful to contrast results between different boundary\nconditions. One has\nαb.c.=\n\n1 for conformal boundary conditions ,\n(D−1) for Dirichlet boundary conditions ,\n(4−D)(D−1)\n2for Neumann boundary conditions ,(2.4)\nto ensure the variational principle is well-defined. Note that for Dirichlet boundary conditions,\nthis gives the standard Gibbons-Hawking-York term [31, 53] and that conformal and Neumann\nboundary conditions have the same action in D= 3 [54].\n6Regardless of the choice of the boundary term, the equations of motion satisfied in the interior\nmanifold are the Einstein field equations\nRµν−1\n2gµνR+ Λgµν= 0. (2.5)\n2.1 Conformal thermodynamics\nFollowing [25], we would like to study the thermodynamic behaviour of solutions subject to con-\nformal boundary conditions, but now in the presence of Λ >0.\nFor this, we take the topology of the boundary to be S1×SD−2, and consider the following boundary\ndata,\nds2\f\f\nΓ=e2ω\u0000\ndτ2+r2dΩ2\nD−2\u0001\n, K = constant , (2.6)\nwhere ωis an unspecified function that in principle could depend on boundary coordinates,2and\ndΩ2\nD−2is the round metric of the unit ( D−2)-sphere. The Euclidean time coordinate τ∼τ+β\nparameterises the S1factor. The parameter rcharacterises the size of the SD−2. Given that\nonly the conformal class of the metric is specified, only the dimensionless parameter ˜β≡β/ris\ngeometrically meaningful.\nTo define the conformal canonical ensemble, we consider a partition function Z(˜β, K) as\nZ(˜β, K)≡X\ng∗µνe−IE[g∗\nµν], (2.9)\nwhere g∗\nµνare Euclidean metrics satisfying the Einstein field equation (2.5) and obeying the bound-\nary conditions (2.6). Note that if there is more than one solution with the same boundary data,\nwe sum all of them.\n2In most of the paper, we will consider solutions that have constant ω=ω, but it is also possible to find non-static\nsolutions where ωdepends on the boundary coordinates. For instance, consider the case where the metric in the bulk\nis purely de Sitter in D= 3, with radius ℓ. Fixing the trace of the extrinsic curvature at r=rto be a constant, K,\nimposes the following differential equation on ω(assuming it is only a function of boundary time τ),\nr2∂2\nτω= 1−(r∂τω)2−2r2e2ω\nℓ2−Kreωr\n1−(r∂τω)2−r2e2ω\nℓ2. (2.7)\nThis equation may have solutions apart from ω=ωconstant (a preliminary numerical analysis indeed suggests\nsolutions periodic in τ). A similar phenomenon occurs in Lorentzian signature, for higher dimensional cases, and\nalso for Λ = 0. We leave a full analysis of these solutions for future work. Just as a simple concrete example, one\ncan consider Euclidean de Sitter solutions in D= 3. One solution is simply given by choosing ωconstant, which\ngives Kℓas in (3.4). One could also consider the (Euclidean) de Sitter slicing. In this case, the bulk metric can be\nconveniently written as\nds2\nℓ2=dρ2+sin2ρ\nr2cosh2τ/r\u0000\ndτ2+r2dϕ2\u0001\n, (2.8)\nwhich at a constant ρ=ρ0, has the same boundary conditions as in (2.6), but now ωdepends on τ. The trace of\nthe extrinsic curvature at the boundary is given by Kℓ= 2 cot ρ0, so one can choose ρ0so that both solutions have\nthe same boundary data. Nonetheless, the time-symmetric spatial slice with τ= 0, which has vanishing extrinsic\ncurvature, has a different proper area than the constant ωsolution. Moreover, the above metric is not periodic in τ.\nA Lorentzian version of these configurations is obtained by taking τ→it. A subset of solutions to (2.7) will appear\nat the linearised level, and we analyse them in appendix E. These additional solutions need not spoil the uniqueness\nproperties of the Lorentzian conformal boundary conditions, as they have distinguishable Cauchy data.\n7According to the Gibbons-Hawking prescription [31], we interpret Z(˜β, K) as a leading contribution\nto the thermodynamics partition function in the GN→0 limit. Since we do not fix the Euclidean\ntime periodicity but rather the dimensionless ratio ˜β, we interpret this as a thermal system in a\nconformal canonical ensemble at a fixed conformal temperature ˜β−1.\nGiven the partition function Z(˜β, K), one can compute different thermodynamic quantities. For\ninstance, the conformal energy, conformal entropy, and specific heat at fixed Kare given by\nEconf≡ − ∂˜β\f\f\f\nKlogZ,Sconf≡\u0010\n1−˜β∂˜β\u0011\f\f\f\nKlogZ, C K≡˜β2∂2\n˜β\f\f\f\nKlogZ. (2.10)\nRegular Euclidean solutions. For certain ranges of ˜βandK, the Einstein field equation may\ngive rise to a solution g∗\nµνwhich contains a Euclidean horizon. In analogy to the Dirichlet case,\nrequiring the solution to be regular at the horizon fixes its size in terms of ˜βandK. We will\nconsider g∗\nµνthat are both static and spherically symmetric, taking the explicit form\nds2=e2ω\u0012f(r)\nf(r)dτ2+dr2\nf(r)+r2dΩ2\nD−2\u0013\n, (2.11)\nwhere ωis a constant and f(r), a function of ronly. (Although it would be interesting to explore\nthe existence of saddles subject to conformal boundary conditions with less restrictive symmetry\nproperties than (2.11), we will postpone such an analysis to future work.) Note that at the boundary\nr=rwith an outward3normal vector ˆ n=p\nf(r)∂r,\nds2\f\f\nΓ=e2ω\u0000\ndτ2+r2dΩ2\nD−2\u0001\n, K |Γ=f′(r)\n2eωp\nf(r)+D−2\neωrp\nf(r), (2.12)\nso this metric satisfies conformal boundary conditions (2.6). We further assume that f(r) has a\nsimple root at r=r+, so that\nf(r) = ( r−r+)f′(r+) +O(r−r+)2. (2.13)\nThen, close to r+, its near horizon geometry (to leading order) is given by,\nds2=ρ2 \nf′(r+)\n2p\nf(r)!2\ndτ2+dρ2+e2ωr2\n+dΩ2\nD−2, (2.14)\nwhere ρ≡2eωq\nr−r+\nf′(r+). This geometry has a conical singularity near r=r+unless one identifies\nτ∼τ+βwith\nβ=4πp\nf(r)\n|f′(r+)|. (2.15)\nThus, regularity near the horizon fixes the horizon radius r+in terms of boundary data ˜βandK.\n3It is also possible to consider solutions with the inward normal vector. In such cases, the trace of the extrinsic\ncurvature at the boundary will have an additional minus sign. For our cases of interest, such choice will give rise to\nthe pole and black hole patches in the next two sections.\n83 dS 3conformal thermodynamics\nWe first study conformal thermodynamics of three-dimensional gravity with Λ = +1 /ℓ2>0. A\nfamily of static Euclidean solutions to (2.5) is given by\nds2=e2ω\u0012f(r)\nf(r)dτ2+dr2\nf(r)+r2dϕ2\u0013\n, f (r) =r2\nc−e2ωr2\nℓ2, (3.1)\nwhere τ∼τ+βandϕ∼ϕ+ 2π. The choice for this particular parameterisation of the solution\nwill soon become evident. The parameter ωis an unspecified constant and directly controls the\nphysical size of the boundary, namely rtube=eωr. The cosmological horizon is located at r=e−ωrc\nand has a physical radius rc>0. There is no black hole horizon in the present setup.4However,\nforrc̸=ℓ, there is a conical defect located at the origin r= 0.\nNote that one can recover the standard dS static patch coordinates ( τstatic, rstatic, ϕstatic) via the\nidentification\nτstatic =eωτp\nf(r), r static =eωr , ϕ static =ϕ . (3.2)\nIt is straightforward to verify that due to the choice of parameterisation (3.1), the metric auto-\nmatically satisfies the first boundary condition in (2.6) at r=r. Requiring that the trace of\nthe extrinsic curvature at the boundary is constant, further fixes the parameter ωin terms of the\nboundary data,\ne2ω=r2\nc\n2r2√\nK2ℓ2+ 4±Kℓ√\nK2ℓ2+ 4, (3.3)\nwhere the ±corresponds to two different spacetime regions of interest, which we discuss below.\nWe also note that, assuming that ωdoes not depend on τ, we find that (3.1) is the most general\nsolution to the Einstein field equation in three dimensions.\nWe consider two classes of solutions, which we call the pole and the cosmic patch [12]. The first\none is a patch of spacetime which does not contain the cosmological horizon, while the second one\ndoes. In Lorentzian signature they would correspond to the regions shown in figure 1. Below we\nstudy the two solutions and their corresponding thermodynamic quantities, separately.\n3.1 Pole patch\nIn the first class of solutions, the cosmological horizon is fixed to be at rc=ℓ, leading to the\nabsence of the conical defect. The spacetime region of interest is r∈[0,r], with the boundary at\nr=r≤ℓ. We call this spacetime the pole patch of dS. This solution can be obtained by choosing\nthe minus sign in (3.3).\nImposing that the boundary has a constant trace of the extrinsic curvature Kleads to\nKℓ=ℓ2−2r2\ntube\nrtubeq\nℓ2−r2\ntube, (3.4)\n4Note, however, that related theories of three-dimensional gravity with a gravitational Chern-Simons term admit\nblack hole solutions [55]. Moreover, quantum black holes have recently been constructed in dS 3[56,57].\n9<latexit sha1_base64=\"Sj0BNEyFb7LAm7obLfc5Rb31yh8=\">AAAB9XicbVDLSgMxFL2pr1pfVZdugkVwVWbE10YounFZwT6gHUsmzbShmcyQZJQy9D/cuFDErf/izr8x085CWw8EDufcyz05fiy4No7zjQpLyyura8X10sbm1vZOeXevqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dJP5rUemNI/kvRnHzAvJQPKAU2Ks9KCuuiExw0CRUaomvXLFqTpT4EXi5qQCOeq98le3H9EkZNJQQbTuuE5svJQow6lgk1I30SwmdEQGrGOpJCHTXjpNPcFHVunjIFL2SYOn6u+NlIRaj0PfTmYZ9byXif95ncQEl17KZZwYJunsUJAIbCKcVYD7XDFqxNgSQhW3WTEdEkWosUWVbAnu/JcXSfOk6p5Xz+5OK7XrvI4iHMAhHIMLF1CDW6hDAygoeIZXeENP6AW9o4/ZaAHlO/vwB+jzB/NzktM=</latexit>r=r<latexit sha1_base64=\"H734ctACTCKkxuqiwKBxyLGiQc8=\">AAAB/HicbVDLSgNBEJz1GeNrNUcvg0HwFHbF1zGYi8cI5gHJEmYnk2TIzM4y0ysuS/wVLx4U8eqHePNvnCR70MSChqKqm+6uMBbcgOd9Oyura+sbm4Wt4vbO7t6+e3DYNCrRlDWoEkq3Q2KY4BFrAAfB2rFmRIaCtcJxbeq3Hpg2XEX3kMYskGQY8QGnBKzUc0tdYI+Q1ZSRnOKYAB1Nem7Zq3gz4GXi56SMctR77le3r2giWQRUEGM6vhdDkBENnAo2KXYTw2JCx2TIOpZGRDITZLPjJ/jEKn08UNpWBHim/p7IiDQmlaHtlARGZtGbiv95nQQG10HGozgBFtH5okEiMCg8TQL3uWYURGoJoZrbWzEdEU0o2LyKNgR/8eVl0jyr+JeVi7vzcvUmj6OAjtAxOkU+ukJVdIvqqIEoStEzekVvzpPz4rw7H/PWFSefKaE/cD5/ACRTlRo=</latexit>Cosmic patch<latexit sha1_base64=\"GH+7XeguUc4RS4LQA5CcXRcnMrI=\">AAAB+nicbVDJSgNBEO1xjXGb6NFLYxA8hRlxOwa9eIxgFkiG0NOpSZr0LHTXqGHMp3jxoIhXv8Sbf2MnmYMmPih4vFdFVT0/kUKj43xbS8srq2vrhY3i5tb2zq5d2mvoOFUc6jyWsWr5TIMUEdRRoIRWooCFvoSmP7ye+M17UFrE0R2OEvBC1o9EIDhDI3XtUgfhEbNaLIEmDPlg3LXLTsWZgi4SNydlkqPWtb86vZinIUTIJdO67ToJehlTKLiEcbGTakgYH7I+tA2NWAjay6anj+mRUXo0iJWpCOlU/T2RsVDrUeibzpDhQM97E/E/r51icOllIkpShIjPFgWppBjTSQ60JxRwlCNDGFfC3Er5gCnG0aRVNCG48y8vksZJxT2vnN2elqtXeRwFckAOyTFxyQWpkhtSI3XCyQN5Jq/kzXqyXqx362PWumTlM/vkD6zPH5NulDg=</latexit>Pole patch\n<latexit sha1_base64=\"Sj0BNEyFb7LAm7obLfc5Rb31yh8=\">AAAB9XicbVDLSgMxFL2pr1pfVZdugkVwVWbE10YounFZwT6gHUsmzbShmcyQZJQy9D/cuFDErf/izr8x085CWw8EDufcyz05fiy4No7zjQpLyyura8X10sbm1vZOeXevqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dJP5rUemNI/kvRnHzAvJQPKAU2Ks9KCuuiExw0CRUaomvXLFqTpT4EXi5qQCOeq98le3H9EkZNJQQbTuuE5svJQow6lgk1I30SwmdEQGrGOpJCHTXjpNPcFHVunjIFL2SYOn6u+NlIRaj0PfTmYZ9byXif95ncQEl17KZZwYJunsUJAIbCKcVYD7XDFqxNgSQhW3WTEdEkWosUWVbAnu/JcXSfOk6p5Xz+5OK7XrvI4iHMAhHIMLF1CDW6hDAygoeIZXeENP6AW9o4/ZaAHlO/vwB+jzB/NzktM=</latexit>r=rFig. 1: Penrose diagram of dS space, with a timelike boundary at r=r. On the left static patch, the shaded\nregion corresponds to the pole patch, while on the right, it corresponds to the cosmic patch.\nwhich can be inverted to obtain\nr2\ntube=ℓ2\n2√\nK2ℓ2+ 4−Kℓ√\nK2ℓ2+ 4. (3.5)\nThe dimensionless parameter Kℓ∈Rcontrols the size of the boundary. For Kℓ→+∞the\nboundary locates near the origin, whilst for Kℓ→ −∞ it is located near the cosmological horizon.\nWhen Kℓ= 0, the boundary is located exactly at rtube=ℓ/√\n2.\nSince the cosmological horizon is not part of the pole patch, the parameter ˜βis free, and there is\na pole patch solution for all values of ˜βandK.\nPole patch thermodynamics. By evaluating (2.1) with αb.c.= 1 and D= 3 on the pole patch\nsolution, the on-shell Euclidean action in terms of the boundary data becomes,\nI(pole)\nE=−˜βℓ\n16GN\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n. (3.6)\nSince the action depends linearly on ˜β, one immediately finds that Sconf=CK= 0 and that\nEconf=I(pole)\nE\n˜β=−ℓ\n16GN\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n, (3.7)\nwhich is independent of ˜β. Note that small fluctuations of the energy can be written as,\nδEconf=r2\ntube\n8GNδK . (3.8)\nFollowing [58], we treat the pole patch of dS as a reference configuration and the on-shell action\n(3.6) as a subtraction term. Therefore, the conformal energy (3.7) plays the role of a vacuum\nenergy. From now onwards, we will compute subtracted quantities such that the energy of the pole\npatch solution with trace of extrinsic curvature Kvanishes.\n103.2 Cosmic patch\nWe now consider the second class of static geometries (3.1), which contain the cosmological horizon\nand hence are dubbed as cosmic patches of dS. This is achieved by choosing the plus sign on (3.3)\nand considering the region r∈[r, e−ωrc]. As a consequence, the conical defect at r= 0 is not part\nof the cosmic patch.\nRegularity of the geometry near the cosmological horizon imposes that the inverse conformal tem-\nperature of the cosmic patch is given by\n˜β=2πℓq\nr2c−r2\ntube\nrcrtube, (3.9)\nwhich is always greater than zero. The conformal temperature ˜β−1becomes zero as the boundary\napproaches the origin. On the other hand, the conformal temperature diverges to infinity as the\nboundary approaches the cosmological horizon.\nRequiring that the boundary has a constant trace of the extrinsic curvature K, fixes\nKℓ=−r2\nc−2r2\ntube\nrtubeq\nr2c−r2\ntube, (3.10)\nwhich can take any real value. Contrary to the pole patch, the limit of Kℓgoing to positive\nand negative infinity now corresponds to the limit of the boundary approaching the cosmological\nhorizon and the origin, respectively. This is expected as the normal vector now points in the\nopposite direction.\nUsing (3.9) and (3.10), we can express rcand rtubein terms of the boundary data ˜βandKℓ,\nrc=πℓ\n˜β\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n, rtube=√\n2πℓ\n˜βs√\nK2ℓ2+ 4−Kℓ√\nK2ℓ2+ 4. (3.11)\nInterestingly, both rcand rtubedepend linearly on the conformal temperature ˜β−1. This fact\nimplies that the cosmological horizon rcis a monotonically increasing function of the conformal\ntemperature, which contrasts with the Dirichlet problem, where one finds an opposite behaviour,\nsee appendix A.\nAdditionally, for any positive ˜βand real K, one always finds that 0 <rtube<rc. We showrc˜β\nℓ\nandrtube˜β\nℓas functions of Kℓin figure 2.\nCosmic patch thermodynamics. To compute thermodynamic quantities, we evaluate the cosmic\npatch solution on-shell in the Euclidean action (2.1). It is useful to define a regulated quantity\nby subtracting the pole patch action (3.6) with the same boundary data. Then, the associated\nregulated action corresponding to the cosmic patch solution is given by\nI(cosmic)\nE, reg(K)≡I(cosmic)\nE−I(pole)\nE=−πℓ\n8GN \n2π\n˜β−˜β\n2π!\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n. (3.12)\n11-6 -4 -2 0 2 4 6010203040Fig. 2: Plot ofrc˜β\nℓ(green) andrtube˜β\nℓ(yellow) as functions of Kℓ. For Kℓ→ ∞ , the tube becomes very close\nto the cosmological horizon; for Kℓ→ −∞ , the tube approachesrtube˜β\nℓ= 2π.\nWe emphasise that the pole action that we subtract has the same trace of extrinsic curvature K. We\nfurther note that I(cosmic)\nE, reg(K) is invariant under˜β\n2π→ −2π\n˜β. The conformal energy and conformal\nentropy are given by\n\n\nEconf =πℓ\n8GN˜β\u0010\n2π\n˜β+˜β\n2π\u0011\u0010√\nK2ℓ2+ 4−Kℓ\u0011\n,\nSconf =π2ℓ\n2GN˜β\u0010√\nK2ℓ2+ 4−Kℓ\u0011\n.(3.13)\nThe entropy Sconfagrees with the Gibbons-Hawking entropy Ahorizon /4GNof the cosmological\nhorizon. We can also compute the specific heat at constant K, which is given by\nCK=π2ℓ\n2GN˜β\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n. (3.14)\nThe specific hear is positive for all allowed values of ˜βandK, which means that this configuration\nis thermally stable under small thermal fluctuations. This is in contrast to the specific heat of\nthe cosmic patch with Dirichlet boundary conditions, which is always negative, as reviewed in ap-\npendix A. Moreover, note that CKgrows linearly with the conformal temperature. This behaviour\nresembles that of a two-dimensional conformal field theory at finite temperature. This observation\ncan be sharpened upon expressing ˜βin terms of Econf, whereby the conformal entropy and specific\nheat can be written as\nSconf= 2πr\ncconf\n3\u0010\nEconf−cconf\n12\u0011\n, C K=2π2cconf\n3˜β, (3.15)\nwhere\ncconf≡3ℓ\n4GN\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n, (3.16)\n12which is a monotonically-decreasing function of Kℓdisplayed in figure 3.5Note that the en-\ntropy in (3.15) resembles the Cardy formula, describing the growth of states of energy Econfin\na two-dimensional conformal field theory of central charge cconf[61]. By considering large posi-\ntive/negative Kℓlimits,\ncconf→(3\n2GNK,asKℓ→ ∞ ,\n3|K|ℓ2\n2GN,asKℓ→ −∞ .(3.17)\n-6 -4 -2 0 2 4 60246810\nFig. 3: The quantity cconfas a function of Kℓ. This central charge decreases monotonically as a function\nofKℓ. As comparison, we included the plot of the analogous cBTZ\nconf, for the BTZ black hole with conformal\nboundary conditions. The dS and AdS radii are chose to be equal. In the limit where Kℓ→ ∞ both central\ncharges coincide, as the tube in both cases gets very close to the horizon and they both exhibit Rindler\nbehaviour. In dashed lines, we show the position of the conformal boundary in the AdS case, where we recover\nthe Brown-Henneaux central charge.\nFinally, using (3.15), we find a first-law type of relation in which\nδ\u0010\nEconf−cconf\n12\u0011\n=˜β−1δSconf−µKδK , (3.18)\nwhere µKcan be understood as the chemical potential associated to K,\nµK≡ −r2\ntube\n8GN. (3.19)\n3.3 Pure dS 3patch\nWe now discuss a particular solution, denoted as the pure dS 3solution. The solution arises from\ntuning the conformal temperature of the system such that rc=ℓ, leading to a pure dS 3with a\nboundary. For the cosmic patch, this happens at the conformal temperature ˜β=˜βdS, which is a\nfunction of Kℓgiven by\n˜βdS(Kℓ)≡π\u0010p\nK2ℓ2+ 4−Kℓ\u0011\n= 2π\u00122GNcconf\n3ℓ\u0013\n. (3.20)\n5A similar computation can be done for the BTZ black hole in AdS 3with conformal boundary conditions [45,\n59]. The resulting value for this function is now given by cBTZ\nconf=3ℓAdS\n4GN(KℓAdS−p\nK2ℓ2\nAdS−4). This is also a\nmonotonically decreasing function of KℓAdS. It approaches its maximal real value as KℓAdS→2, which corresponds\nto the AdS conformal boundary, and for which we recover the standard Brown-Henneaux central charge [60].\n13By using (3.10) with rc=ℓ, we can re-express this in terms of the physical size of the boundary\nrtubeas\n˜βdS(rtube) = 2 πs\nℓ2\nr2\ntube−1. (3.21)\nWe can recover the standard dS temperature when the tube becomes small,\n˜βdSrtube→2πℓ as rtube→0. (3.22)\nThis is the worldline limit of the solution. It also corresponds to a small conformal temperature. On\nthe other hand, we can consider the stretched horizon limit, which is defined by taking rtube→ℓ.\nThis is equivalent to a high conformal temperature limit,\n˜βdS→0 as rtube→ℓ . (3.23)\nNow we can use (3.13) and (3.14) to calculate the thermodynamic properties of pure dS 3. The\nconformal entropy and specific heat at constant Kare equal and independent of rtube,\nSconf=CK=πℓ\n2GN. (3.24)\nWe stress again that the specific heat CKis positive for all values of rtube. The conformal energy\nreads\nEconf=cconf\n12\u0012\n1−r2\ntube\nℓ2\u0013−1\n. (3.25)\nNote that in the worldline limit, the energy reduces to the energy cconf/12. One may wonder why\nthe energy does not go to zero as we take worldline limit. The reason is that we are considering\nsubtracted energies.\nTo reproduce the full static patch thermodynamics from a cosmic patch solution, we must include\nthe pole patch to complete a full static patch. Note that this pole patch is not the same that we\nare using to define the regulated action, as this one has the same ˜βbut the opposite trace of the\nextrinsic curvature, −K. Now it is straightforward to check that, for ˜β=˜βdS, a combination of\na pure de Sitter patch with Kand a pole patch with −K(without the subtraction terms) indeed\nreproduces the Gibbons-Hawking result,\nZ(cosmic)(˜βdS, K)Z(pole)(˜βdS,−K) = expπℓ\n2GN, (3.26)\nas the right-hand side is the exponential of the Gibbons-Hawking entropy for the de Sitter horizon.\nFor the energy, we can define a regulated action for this pole patch with opposite K,\nI(pole)\nE, reg(−K)≡I(pole)\nE(−K)−I(pole)\nE(K) =−˜βKℓ2\n8GN. (3.27)\nIn the pure dS 3solution, the conformal energy of the pole patch with −Kis negative and it is\nexactly opposite to the conformal energy of the cosmic patch with Ksuch that\n\u0010\nE(pole)\nconf(−K)−cconf\n12\u0011\n+\u0010\nE(cosmic)\nconf(K)−cconf\n12\u0011\f\f\f˜β=˜βdS= 0, (3.28)\nas we expect for the full static patch of de Sitter space.\n143.4 Phase diagram\nNow we can combine the results from pole patch and cosmic patch thermodynamics. Recall that in\nD= 3, both solutions exist for all values of ˜βandK. This means that one may write the partition\nfunction of the total system in the semiclassical limit as\nZ(˜β, K) = max\u0012\ne−I(cosmic)\nE, reg(˜β,K),1\u0013\ne−I(pole)\nE(˜β,K), (3.29)\nwhere I(pole)\nE(˜β, K) and I(cosmic)\nE, reg(˜β, K) are given by (3.6) and (3.12), respectively. The sign of\nI(cosmic)\nE, reg(˜β, K) therefore determines which configuration is stable/meta-stable. It can be shown that,\nindependently of Kℓ, there is a critical inverse temperature ˜β= 2π≡˜βc, for which I(cosmic)\nE(˜β, K)\nchanges sign, see figure 4. There is a first-order phase transition at ˜β=˜βc.\n•In the low-temperature regime with ˜β > ˜βc,I(cosmic)\nE, regis positive for all values of Kℓwhich\nimplies that the pole patch is thermodynamically favoured. Given it has positive specific\nheat, the cosmic patch is then metastable.\n•In the high-temperature regime with ˜β < ˜βc,I(cosmic)\nE, regbecomes negative, so it becomes\nthermodynamically favored and a stable configuration.\n0 2 4 6 8 10 12-4-3-2-101\nFig. 4: The regulated on-shell action for the cosmic patch solution, as a function of boundary data ˜β. The\ndashed vertical line indicates the critical inverse temperature ˜βc= 2π.\nIt is interesting to analyse the phase structure of conformal thermodynamics at fixed Kℓ. Consider\nthe conformal energy along the path of lowest free energy at fixed Kℓ. Using (3.7) and (3.13) and\nexpressing them in terms of the central charge cconf, we find that\nEconf=(cconf\n12\u0010\n1 +˜β2\nc\n˜β2\u0011\n, ˜β < ˜βc,\n0, ˜β≥˜βc.(3.30)\nThe discontinuity of Econfat˜β=˜βcreflects a first-order phase transition.\nFor the conformal entropy, we find a behaviour similar to the Hawking-Page transition of the AdS\nblack hole [62, 63]. For temperatures lower than ˜β−1\nc, the conformal entropy is zero. There is a\n15discontinuity in the entropy at the critical temperature ˜β−1\nc, after which, in the high temperature\nregime, the entropy is precisely given by the Gibbons-Hawking entropy.\nPure dS 3phase structure. For the pure dS 3solution, we must constrain the inverse temperature\nto the dS inverse temperature (3.20). In this case, Kℓ= 0 corresponds to rtube=ℓ/√\n2.\n•Kℓ > 0 implies that cconf<3ℓ\n2GN. In this regime, the dS temperature is higher than the\ncritical temperature, ˜βdS<˜βc. As a consequence, in this regime, the pure dS 3has free\nenergy lower than the pole patch and hence is thermodynamically favoured.\n•ForKℓ < 0 or cconf>3ℓ\n2GN, we find that ˜βdS>˜βc, so the pure dS solution is only metastable.\nLastly, at cconf=3ℓ\n2GN, the phase transition and dS 3temperature coincide.\nThe full phase diagram, including the curve of pure dS 3solutions, is depicted in figure 5.\n0 -∞ -10 -2 2 10 ∞021050∞\nFig. 5: Phase diagram of conformal dS 3thermodynamics. In each point of the diagram, there co-exist a pole\nand a cosmic patch solution. There is a critical conformal inverse temperature at ˜βc= 2π, marked in black.\nFor˜β < ˜βc(shaded in green), the cosmic patch is the most favourable configuration. For ˜β > ˜βc(in white),\nthe cosmic patch is metastable. The darker green curve shows pure stable (solid) and metastable (dashed) dS 3\nsolutions, that follow relation (3.20). Worldline and stretched horizon limits of pure dS 3are further indicated.\n3.5 A two-sphere perspective\nAs a final remark before moving on to the four-dimensional case, we consider a two-sphere rather\nthan toroidal boundary topology. As our coordinate system, we take\nds2=ℓ2dρ2\nℓ2−ρ2+ℓ2−ρ2\nℓ2\u0000\ndθ2+ sin2θdϕ2\u0001\n, (3.31)\nwhere for the full three-sphere we have ρ∈(−ℓ, ℓ),θ∈(0, π), and ϕ∼ϕ+ 2π. We take the\nconformal boundary to be located at constant ρ=ρ0. The induced metric has the conformal\n16structure of the unit S2metric. Requiring further that the boundary has a constant Kfixes\nKℓ=−2ρ0p\nℓ2−ρ2\n0, (3.32)\nwhere we have used a unit normal vector ˆ n=ℓ√\nℓ2−ρ2∂ρ. By computing the on-shell action (2.1)\nfor this solution, we can approximate the path integral as\nZ[S2]≈\u0012Kℓ−2i√\nK2ℓ2+ 4\u0013i\n3×3ℓ\n2GN. (3.33)\nThe above expression is real valued. Unlike the thermodynamic expression (3.13) and (3.15), the\nabove expression does not immediately take the form of the two-sphere path integral of a two-\ndimensional conformal field theory of central charge cconf. A similar observation will hold for\nEuclidean AdS 3with a two-sphere boundary subject to conformal boundary conditions.\n4 dS 4conformal thermodynamics\nIn this section, we study conformal thermodynamics of four-dimensional gravity with Λ = +3 /ℓ2\nfor the following family of static and spherically symmetric Euclidean solutions\nds2=e2ω\u0012f(r)\nf(r)dτ2+dr2\nf(r)+r2\u0000\ndθ2+ sin2θdϕ2\u0001\u0013\n, f (r) = 1 −2µ\nr−e2ωr2\nℓ2, (4.1)\nwhere τ∼τ+β,θ∈(0, π), and ϕ∼ϕ+ 2π. Similarly to the three-dimensional case, the\nparameter ωcontrols the size of the boundary, namely rtube=eωr.\nInD= 4, one further has the Euclidean Schwarzschild-de Sitter solution corresponding to a black\nhole placed inside the cosmological horizon. The parameter µis related to the size of the cosmo-\nlogical horizon rcthrough\nµ=e−ωrc\n2\u0012\n1−r2\nc\nℓ2\u0013\n. (4.2)\nForµ= 0, we obtain an empty de Sitter solution with cosmological horizon at rc=ℓ. For µ > 0,\nthere is a black hole horizon located at r=e−ωrbhwith horizon radius rbh. The cosmological\nhorizon in this case is smaller than the one without the black hole, rc< ℓ. Note that the size of\nthe two horizons are related by\nrbh=1\n2\u0010p\n4ℓ2−3r2c−rc\u0011\n. (4.3)\nBoth horizon sizes coincide when rc=rbh=ℓ√\n3. This is known as the Nariai radius, which also\nserves as a lower bound of rc. For µ < 0, there is a naked singularity located at r= 0, and the\ncosmological horizon is greater than the de Sitter length, rc> ℓ.\nAs in dS 3, we note that the de Sitter static patch coordinates ( τstatic, rstatic, θstatic, ϕstatic) can be\nrecovered via the rescaling\nτstatic =eωτp\nf(r), r static =eωr , θ static =θ , ϕ static =ϕ . (4.4)\n17We now study these solutions on a four-manifold with an S1×S2boundary subject to the conformal\nboundary data (2.6). Solving the Einstein equation in terms of boundary data, we obtain multiple\nexpressions for e2ω. The exact expressions in the different ranges of parameters are provided in\nappendix B.\nThere are three different classes of solutions denoted by the pole patch, the black hole patch, and\nthe cosmic patch. Examples of them are displayed in figure 6.\n<latexit sha1_base64=\"Sj0BNEyFb7LAm7obLfc5Rb31yh8=\">AAAB9XicbVDLSgMxFL2pr1pfVZdugkVwVWbE10YounFZwT6gHUsmzbShmcyQZJQy9D/cuFDErf/izr8x085CWw8EDufcyz05fiy4No7zjQpLyyura8X10sbm1vZOeXevqaNEUdagkYhU2yeaCS5Zw3AjWDtWjIS+YC1/dJP5rUemNI/kvRnHzAvJQPKAU2Ks9KCuuiExw0CRUaomvXLFqTpT4EXi5qQCOeq98le3H9EkZNJQQbTuuE5svJQow6lgk1I30SwmdEQGrGOpJCHTXjpNPcFHVunjIFL2SYOn6u+NlIRaj0PfTmYZ9byXif95ncQEl17KZZwYJunsUJAIbCKcVYD7XDFqxNgSQhW3WTEdEkWosUWVbAnu/JcXSfOk6p5Xz+5OK7XrvI4iHMAhHIMLF1CDW6hDAygoeIZXeENP6AW9o4/ZaAHlO/vwB+jzB/NzktM=</latexit>r=r<latexit sha1_base64=\"b+h58oN1xbRcT/uEqHOPeKn694g=\">AAACB3icbVDLSsNAFJ34rPUVdSlIsAhuLIn42ghFNy4r2Ac0sUymN+3QySTMTIQSsnPjr7hxoYhbf8Gdf+OkzUJbD1w4nHMv997jx4xKZdvfxtz8wuLScmmlvLq2vrFpbm03ZZQIAg0SsUi0fSyBUQ4NRRWDdiwAhz6Dlj+8zv3WAwhJI36nRjF4Ie5zGlCClZa65p64hPv0yI1C6OPMDbEaBAIPU5F1U3+Qdc2KXbXHsGaJU5AKKlDvml9uLyJJCFwRhqXsOHasvBQLRQmDrOwmEmJMhrgPHU05DkF66fiPzDrQSs8KIqGLK2us/p5IcSjlKPR1Z36onPZy8T+vk6jgwkspjxMFnEwWBQmzVGTloVg9KoAoNtIEE0H1rRYZYIGJ0tGVdQjO9MuzpHlcdc6qp7cnldpVEUcJ7aJ9dIgcdI5q6AbVUQMR9Iie0St6M56MF+Pd+Ji0zhnFzA76A+PzBwOVmhI=</latexit>r=e\u0000!rbh<latexit sha1_base64=\"dY4/smsXs0WV9hPY7uTF5qvBVIw=\">AAACBnicbVDLSsNAFJ34rPUVdSlCsAhuLIn42ghFNy4r2Ac0sUymN+3QySTMTIQSsnLjr7hxoYhbv8Gdf+OkzUJbD1w4nHMv997jx4xKZdvfxtz8wuLScmmlvLq2vrFpbm03ZZQIAg0SsUi0fSyBUQ4NRRWDdiwAhz6Dlj+8zv3WAwhJI36nRjF4Ie5zGlCClZa65p64hPv0yI1C6OPMDbEaBAIPU5F1U5J1zYpdtcewZolTkAoqUO+aX24vIkkIXBGGpew4dqy8FAtFCYOs7CYSYkyGuA8dTTkOQXrp+I3MOtBKzwoioYsra6z+nkhxKOUo9HVnfqec9nLxP6+TqODCSymPEwWcTBYFCbNUZOWZWD0qgCg20gQTQfWtFhlggYnSyZV1CM70y7OkeVx1zqqntyeV2lURRwnton10iBx0jmroBtVRAxH0iJ7RK3oznowX4934mLTOGcXMDvoD4/MHNnyZoQ==</latexit>r=e\u0000!rc\nFig. 6: Penrose diagram of the Schwarzschild de Sitter spacetime. The boundary is given by r=r. The\nshaded blue area corresponds to a cosmic patch, while the yellow one, to a black hole patch. The pole and the\npure dS 4patches can be obtained when rc=ℓ.\n4.1 Pole patch\nWe begin with the solutions with rc=ℓand take the spacetime region of interest to be r∈[0,r].\nAs a consequence, the pole patches contain the worldline at r= 0 without any horizon.\nImposing that the boundary has a constant trace of the extrinsic curvature Kleads to\nKℓ=2ℓ2−3r2\ntube\nrtubeq\nℓ2−r2\ntube. (4.5)\nBy inverting this equation, we find that the physical size of the boundary can be written as a\nfunction of Kℓas\nr2\ntube=ℓ2 \nK2ℓ2+ 12−Kℓ√\nK2ℓ2+ 8\n2K2ℓ2+ 18!\n. (4.6)\nThe parameter Kℓcan take any real value. The limit of large positive and large negative Kℓ\ncorresponds to pushing the boundary to the origin and the cosmological horizon, respectively.\nSince the pole patches do not contain any horizon, the parameter ˜βis free and can take any positive\nvalue. Hence, the pole patch exists for all values of ˜βandK.\nPole patch thermodynamics. We now evaluate (2.1) with αb.c.= 1 and D= 4 on the pole\npatch solution. The on-shell Euclidean action in terms of the boundary data reads,\nI(pole)\nE=−˜βℓ2\n6GNvuut\u0010\nK2ℓ2+ 4−Kℓ√\nK2ℓ2+ 8\u0011\u0010\nK2ℓ2+ 12−Kℓ√\nK2ℓ2+ 8\u0011\nK2ℓ2+ 9. (4.7)\n18By taking the limit Kℓ→ ∞ , we find that I(pole)\nE→ −4˜β\n3GNK2, retrieving the result in flat space\nobtained in [25]. This is expected as this limit corresponds to a boundary size that is parameterically\nsmall compared to the cosmological horizon.\nAs the action depends linearly on ˜β, one immediately finds that Sconf=CK= 0 and that\nEconf=I(pole)\nE\n˜β=−ℓ2\n6GNvuut\u0010\nK2ℓ2+ 4−Kℓ√\nK2ℓ2+ 8\u0011\u0010\nK2ℓ2+ 12−Kℓ√\nK2ℓ2+ 8\u0011\nK2ℓ2+ 9,(4.8)\nwhich is independent of ˜β. Note that small fluctuations of the energy can be written as\nδEconf=r3\ntube\n3GNδK . (4.9)\nCuriously, the coefficient in front of δKis independent of ℓ. As in D= 3, we treat the pole patch\nof de Sitter as a reference configuration, and the on-shell action (4.7) as a subtraction term.\n4.2 Cosmic patch\nIn this section, we consider a class of geometries (4.1) which contain the cosmological horizon. We\ncall these cosmic patches of dS. The spacetime region of interest is taken to be r∈[r, e−ωrc]. For\nℓ/√\n3<rc< ℓ, the full solution has a black hole horizon that lies outside the boundary, so it\nis not present in the cosmic patch. Similarly, for rc> ℓ, there is a naked timelike singularity at\nthe origin in Lorentzian signature, which would be associated with the presence of negative energy.\nAgain, since this region is not part of the cosmic patch, we also allow for rc> ℓ.\nRegularity of the geometry near the cosmological horizon fixes the conformal temperature of the\ncosmic patch to be\n˜β=4πrcℓq\nrtubeℓ2−r3\ntube−rcℓ2+r3c\nr3/2\ntube(3r2c−ℓ2), (4.10)\nwhich is always greater than zero. In this case, the conformal temperature ˜β−1does not have\na lower bound. Specifically, for ℓ/√\n3<rc≤ℓ, the zero temperature limit can be reached by\nsetting rc=ℓand taking rtube/ℓ→0. For rc> ℓ, there are also cosmic patches with zero\nconformal temperature. They have the boundary located closed to the naked singularity, that is to\nsay rtube/ℓ→0.\nThe high conformal temperature limit ˜β→0 can be achieved in different ways, for instance, by\ntaking the near horizon limits, rtube→rbhorrtube→rc.\nSetting the trace of the extrinsic curvature at the boundary to be constant leads to\nKℓ=−4rtubeℓ2−6r3\ntube−3rcℓ2+ 3r3\nc\n2r3/2\ntubeq\nrtubeℓ2−r3\ntube−rcℓ2+r3c. (4.11)\nThere is no upper or lower bound on Kℓ. The limit of large positive and large negative Kℓ\ncorrespond to taking the boundary to be near the cosmological horizon and the black hole horizon,\n19respectively. In the case of rc≥ℓ, there is no black hole and so the large negative limit of Kℓ\ncorresponds to taking the boundary to be near the origin.\nUnlike the D= 3 case, we could not find analytic expressions for rtubeand rcin terms of the\nboundary data ˜βandK, but we relegate some useful analytical expressions to appendix B. We\nlater present examples of rtubeand rcas functions of ˜βat fixed Kℓin figure 9, together with the\nblack hole patch solutions.\nCosmic patch thermodynamics. We start by computing the on-shell action (2.1) of the cosmic\npatch. We define a regulated action as the on-shell action of the cosmic patch subtracted by the\npole patch action with the same ˜βandK. By expressing it in terms of rbhand rtube, we find that\nI(cosmic)\nE, reg=−πrc\u0000\n4rtubeℓ2−3rcℓ2−3r3\nc\u0001\n3GN(ℓ2−3r2c)−I(pole)\nE, (4.12)\nwhere I(pole)\nEis given by (4.7).\nThe conformal energy and the conformal entropy of the cosmic patch are\nEconf=r3/2\ntube\u0000\n2rtubeℓ2−3rcℓ2+ 3r3\nc\u0001\n6GNℓq\nrtubeℓ2−r3\ntube−rcℓ2+r3c−I(pole)\nE\n˜β, Sconf=πr2\nc\nGN. (4.13)\nThe conformal entropy Sconfagrees with the Gibbons-Hawking entropy of the cosmological horizon,\nAhorizon /4GN. The specific heat at constant Kof the cosmic patch is given by\nCK=2πr2\nc\u0000\n−ℓ2+ 3r2\nc\u0001\u0010\n9r2\nc\u0000\nr2\nc−ℓ2\u00012+ 16 rc\u0000\nr2\nc−ℓ2\u0001\nrtubeℓ2+ 8r2\ntubeℓ4−4r4\ntubeℓ2\u0011\nGN(ℓ2+ 3r2c)\u0010\n9r2c(r2c−ℓ2)2+ 2rc(−9ℓ4−10r2cℓ2+15 r4c)\n(ℓ2+3r2c)rtubeℓ2+ 8r2\ntubeℓ4−4r4\ntubeℓ2\u0011.(4.14)\nIt is interesting to remark that in the limit where the boundary approaches the cosmological horizon,\nthe specific heat becomes\nCK→2πr2\nc\nGN, asrtube\nrc→1. (4.15)\nThis positive specific heat is to be contrasted with the negative specific heat that is obtained when\nDirichlet boundary conditions are imposed on the cosmic patch [34, 48]. We will further discuss\nthis fact when we consider the pure dS 4solutions.\nInterestingly, the high conformal temperature limit of the specific heat (4.15) at finite Kℓis given\nby\nCK→32π3ℓ2\n81˜β2GN\u0010p\nK2ℓ2+ 9−Kℓ\u00112\n, as˜β→0. (4.16)\nThis takes the form of the specific heat of a three-dimensional conformal field theory. Under this\ninterpretation, the putative number of degrees of freedom goes as\nNd.o.f. =32π3ℓ2\n81GN\u0010p\nK2ℓ2+ 9−Kℓ\u00112\n, (4.17)\nwhich is a monotonically decreasing function of Kℓ, displayed in figure 7. Further taking Kℓ→+∞\nyields Nd.o.f.→8π3/GNK2matching the Λ = 0 result in [25].\n20Finally, we find that the thermodynamic quantities satisfy a first-law type of relation\nδEconf=˜β−1δSconf−µKδK , (4.18)\nwhere µK, similarly to the three-dimensional case, is interpreted as the chemical potential associated\ntoK,\nµK≡ −r3\ntube\n3GN−1\n˜β∂I(pole)\nE\n∂K\f\f\f\f\f˜β. (4.19)\nNote that the first term in µKlooks identical to the one that appears for the pole patch in D= 4,\nsee (4.9).\n-4 -2 0 2 40200400600800100012001400\nFig. 7: The number of degrees of freedom Nd.o.f. as a function of Kℓ. The number of degrees of freedom\ndecreases monotonically as a function of Kℓ.\n4.3 Black hole patch\nWe now consider a class of geometries (4.1) which contain the black hole horizon. We refer to these\nas black hole patches of dS 4. These solutions exist as long as the cosmological horizon radius takes\nvalues between the dS length and the Nariai radius,ℓ√\n3<rc< ℓ. The spacetime region of interest\nis then given by r∈[e−ωrbh,r] where rbhis related to rcthrough (4.3). For convenience, in this\nsection, we will always express rcin terms of rbh.\nGiven that these regions are complementary to the cosmic patches (see figure 6) many results for\nthese solutions are closely related to those in the cosmic patch, upon replacing rc→rbh. Here\nwe point out the main differences between the two patches and relegate the explicit formulae to\nappendix C.\nIt is easy to obtain the boundary data from the results of the cosmic patch. The inverse conformal\ntemperature is the same as in (4.10), but with rc→rbhand an extra minus sign. The trace of the\nextrinsic curvature is minus the expression that appears in (4.11), again with rc→rbh. As opposed\nto the cosmic patch, in this case, the conformal temperature ˜β−1has a lower bound ˜β−1\nmin= 2π,\nwhich occurs in the Nariai limit, by setting rtube=ℓ/√\n3 and taking rbh→ℓ/√\n3 from below.\nBelow this conformal temperature, the black hole patch solution does not exist. For larger conformal\ntemperatures, ˜β−1>˜β−1\nmin, there is a one-parameter family of black hole patches.\n21Regarding the trace of the extrinsic curvature, the limit of Kℓapproaching negative infinity cor-\nresponds to pushing the boundary to be near the cosmological horizon. For Kℓgoing to positive\ninfinity, the boundary is pushed near the black hole horizon. The behaviour of Kℓas a function of\nrcand rtubeis exactly opposite to the cosmic patch since the normal vector points in the opposite\ndirection.\nBlack hole patch thermodynamics. Similarly, thermodynamic quantities can also be obtained\nfrom those of the cosmic patch. In particular, the regulated action I(bh)\nE, regis the same as in the\ncosmic patch, but with rc→rbh. The conformal energy is also the same as in (4.13), but with a\nminus sign in front of the first term and the replacement of rc→rbh. The entropy is now given by\nSconf=πr2\nbh\nGN, (4.20)\nwhich agrees with the Bekenstein-Hawking entropy Ahorizon /4GNwhere the horizon in this formula\nnow corresponds to the black hole horizon. The specific heat can also be obtained from (4.14), with\nthe replacement rc→rbh. In particular, it is also positive as the tube approaches the black hole\nhorizon. Explicit expressions and further interesting limits are shown in appendix C.\n4.4 Pure dS 4patch\nAs in the dS 3case, we can recover the pure dS 4solution for a particular family of conformal\ntemperatures. Using the boundary data of the cosmic patch (4.10) and (4.11), the conditions for\nhaving pure dS 4are\n˜β=˜βdS≡2πq\nℓ2−r2\ntube\nrtube, Kℓ =−2ℓ2−3r2\ntube\nrtubeq\nℓ2−r2\ntube. (4.21)\nFrom these, it follows that rc=ℓ. One can further solve for ˜βdSin terms of Kℓ,\n˜βdS=π\n2\u0010p\nK2ℓ2+ 8−Kℓ\u0011\n. (4.22)\nConsider now the worldline limit rtube→0. The standard dS temperature is recovered when\n˜βdSrtube→2πℓ as rtube→0. (4.23)\nThe stretched horizon limit, corresponds to the high conformal temperature limit in which\n˜βdS→0 as rtube→ℓ . (4.24)\nNow we can use (4.13) and (4.14) to calculate the thermodynamic properties of pure dS 4. The\nconformal entropy and the specific heat at constant Kof the pure dS 4are given by\nSconf=πℓ2\nGN, C K=−2πrtubeℓ2\u0000\n2ℓ2−r2\ntube\u0001\nGN\u0000\nℓ3−4rtubeℓ2+ 2r3\ntube\u0001. (4.25)\nThe conformal entropy is constant regardless of rtubeand is given by the Gibbons-Hawking entropy\nof the cosmological horizon, as expected.\n22It is interesting to note the behaviour of the specific heat at constant K. Close to the worldline,\nCKin (4.25) is negative. In fact, in the worldline limit, the specific heat converges to zero from\nbelow, CK→ − 4πrtubeℓ/G N. This is similar to what happens for the Dirichlet case [34,48] where\nC(Dirichlet) =−2πrtubeℓ2\u0000\nℓ2−r2\ntube\u0001\nGN\u0000\nℓ3−2rtubeℓ2+ 2r3\ntube\u0001. (4.26)\nWe plot both specific heats in figure 8. A notable feature is that for the case of conformal boundary\nconditions, the specific heat diverges as rtube→r0, with r0given by6\nr0≈0.259ℓ ⇔ Kℓ≈ −7.202. (4.28)\nFor rtube>r0, we find that the specific heat is positive and approaches a constant CK→2πℓ2/GN\nas we take the stretched horizon limit. The pure dS 4patch with a conformal boundary sufficiently\nclose to the de Sitter horizon is thus thermally stable.\n0.0 0.2 0.4 0.6 0.8 1.0-30-20-1001020\nFig. 8: A plot of the specific heat of the pure de Sitter patch for conformal (green) and Dirichlet (yellow)\nboundary conditions. For the Dirichlet case, the specific heat is never positive.\nThe conformal energy of the pure dS 4is given by\nEconf=4ℓ2\u0000\nℓ2−r2\ntube\u0001\n3rtubeGNq\n4ℓ2−3r2\ntube+r2\ntubeℓ\n3GNq\nℓ2−r2\ntube, (4.29)\nwhich is positive for all 0 <rtube< ℓwith a lowest value given by\nEconf|rtube=q\n2\n3ℓ=4ℓ2\n3√\n3GN. (4.30)\n6This is the only root to the cubic equation ℓ3−4r0ℓ2+ 2r3\n0= 0, satisfying 0 <r0< ℓ,\nr0\nℓ= 2r\n2\n3cos\u00121\n3cos−1\u0012\n−3√\n3\n8√\n2\u0013\n−2π\n3\u0013\n. (4.27)\n23In terms of the boundary data, the minimum energy (4.30) is obtained precisely when Kℓ= 0.\nBoth in the worldline and in the stretched horizon limit, the energy is divergent. In particular,\nEconf→2ℓ3\n3rtubeGN, in the worldline limit and Econf→ℓ2\n3√\n2GN√\n1−rtube/ℓ, in the stretched horizon\nlimit.\nAs in the pure dS 3discussion, one could consider a pole patch which, together with the cosmic\npatch of pure dS 4, completes the full Euclidean static patch geometry, which is the four-sphere.\nThis is achieved by considering a pole patch which has the same ˜βbut an opposite trace of the\nextrinsic curvature −K. As in D= 3, it is straightforward to check that at the saddle point level\nZ(cosmic)(˜βdS, K)Z(pole)(˜βdS,−K) = expπℓ2\nGN. (4.31)\nIn the above, the Zare computed with the bare action, without subtracting the pole on-shell action.\nThe expression parallels a similar observation for two-dimensional near Nariai geometries [64], and\nsuggests that the thermodynamic content of the empty static patch is purely entropic. It would be\ninteresting to understand the above expression at the one-loop level or higher.\nFor the black hole patch of pure dS 4, we find that, by setting rbh= 0 in (C.4) and (C.6), the\nthermodynamic quantities are trivial, Econf=Sconf=CK= 0, as can be expected.\n4.5 Nariai patch\nIn addition to the pure dS 4solution, there is another interesting geometry that can be reached\nby tuning the conformal temperature ˜β−1to a particular value set by the trace of the extrinsic\ncurvature Kℓ. We call these Nariai patches, which exist as the size of the cosmological and black\nhorizon become close to each other.\nTo find the Nariai temperature ˜β−1\nN, we consider the following limit. Let ρ≡1\nϵ\u0010\nrtube−ℓ√\n3\u0011\nbe a\ndimensionless parameter describing deviation of rtubefrom the Nariai radius. The Nariai geometry\nis obtained by setting rc=ℓ√\n3+ϵand taking ϵ/ℓ→0 while keeping ρfixed. Let us first consider\nthe Nariai solution from the cosmic patch perspective. The cosmological horizon and black hole\nhorizons are, respectively, located at ρ= 1 and −1. Using (4.10) and (4.11), the Nariai limit fixes\n˜βN≡2πp\n1−ρ2, Kℓ =√\n3ρp\n1−ρ2. (4.32)\nThese equations can be inverted analytically to obtain ˜βNin terms of Kℓ,\n˜βN=2√\n3π√\nK2ℓ2+ 3. (4.33)\nWe note that ˜βNexists for all values of Kℓ∈R. This is yet another difference with the Dirichlet\nproblem, where to obtain the Nariai solution one needs to fix the value of rtubeto be very close to\nthe Nariai radius.\nNariai patch thermodynamics. Using (4.13), the conformal energy of the Nariai solution is\ngiven by\nEconf=ρℓ2\n9GNp\n1−ρ2+2ℓ2\n3GNs\nρ2−4 +√\n3ρp\n8−5ρ2\n9ρ2−12−√\n3ρp\n8−5ρ2. (4.34)\n24The conformal entropy and specific heat, evaluated at constant Kand˜β=˜βN, are respectively\ngiven by\nSconf=πℓ2\n3GN, C K=−2πℓ2\nGN(2−9ρ+ 4ρ3). (4.35)\nThe specific heat CKbecomes positive (negative) as one pushes the boundary to the cosmological\n(black hole) horizon. In particular, CKchanges sign at ρ=ρ0where7\nρ0≈0.227 ⇔ Kℓ≈0.405. (4.37)\nWe can also have black hole solutions in the Nariai patch. These are simply obtained by changing\nρ→ −ρin the expressions above.\nThe Dirichlet thermodynamics of configurations near the Nariai geometry, from the perspective of\na dimensionally reduced theory, was studied in [64,65] where it was shown that the cosmic Nariai\npatch always has negative specific heat.\n4.6 Phase diagram\n0 5 10 150.00.20.40.60.81.01.2\n(a)\n0 5 10 150.00.20.40.60.81.01.2 (b)\nFig. 9: Plots of rhorizon /ℓand rtube/ℓas a function of ˜βat fixed Kℓ. The solid curves correspond to cosmic\npatches, while the dashed ones, to black hole patches. rhorizon denotes the radius of the black hole (cosmological)\nhorizon when considering the black hole (cosmic) patch. The two horizontal black dashed lines correspond to\nthe dS 4patch (upper line) and Nariai patch (lower line).\n7This is given by the only solution to the cubic equation 4 ρ3\n0−9ρ0+ 2 = 0 with 1 > ρ0>−1,\nρ0=√\n3 cos\u00121\n3cos−1\u0012\n−2\n3√\n3\u0013\n−2π\n3\u0013\n. (4.36)\n25In this section, we discuss the thermodynamics of four-dimensional spacetime with Λ >0 by\ncombining the results from the pole patch, cosmic patch, and black hole patch solutions. In the\nGN→0 limit, the partition function of the total system is generally given by a sum of all possible\npatches which have the same boundary data ˜βandK,\nZ(˜β, K) =e−I(pole)\nE(1 +. . .). (4.38)\nA pole patch solution exists for all ˜β∈R+andKℓ∈R. The omitted terms are additional\ncontributions stemming from the co-existing cosmic/black hole patch solutions, i.e. e−I(cosmic)\nE, reg or\ne−I(black hole)\nE, reg . The number of these terms and the details of the solutions depend on the value of the\nboundary data, as we will discuss below. For patches with positive specific heat, the one with lowest\nregulated action is thermodynamically stable; otherwise, they are thermodynamically metastable.\nPatches with negative specific heat are thermodynamically unstable.\nAs opposed to D= 3, solutions with horizons do not exist for all values of ˜βandKℓ. At low\ntemperatures, only one pole patch solution exists. To separate the phase space regions with no\nhorizon patches, we define the inverse conformal temperature ˜β0(Kℓ). Note that it depends on the\nvalue of Kℓ, so that at a given Kℓ, horizon patches only exist for conformal temperatures such\nthat ˜β≤˜β0. The curve ˜β0(Kℓ) can be found numerically and is shown in figure 10. The rest of\nthe phase diagram can be decribed as follows:\n•Exactly at ˜β0(Kℓ), there are two solutions: one pole patch and one horizon patch. The latter\nis a cosmic patch if Kℓ≲0.405. Otherwise, it is a black hole patch. Note the transition\nhappens at the Kℓin which the Nariai patch specific heat changes sign. In both cases, the\nhorizon patch has positive regulated action and, therefore, it is always sub-dominant.\n•For lower inverse temperatures, ˜β < ˜β0(Kℓ), we always find three solutions for any given ˜β\nandKℓ. There is always one pole patch and two horizon patches. The horizon patches can\nbe either cosmic or black hole patches, but always the horizon patch with larger horizon size\nhas positive specific heat, while the one with smaller size, has negative specific heat. This can\nbe confirmed by observing that the large (small) horizon patch has a horizon radius which\nis an increasing (decreasing) function of the conformal temperature, as we show in figure 9.\nMoreover, if the horizon radius is larger (smaller) than the Nariai radius rN=ℓ/√\n3, the\ncorresponding horizon patch is a cosmic (black hole) patch. There exists a smooth transition\nbetween black hole patch and cosmic patch as one varies the conformal temperature.\n•At a given critical conformal temperature that depends on the value of Kℓ, there is a first-\norder phase transition, similar to the Hawking-Page transition. We call this temperature\n˜βc(Kℓ) and show it numerically in figure 10. For ˜β < ˜βc, the large horizon patch solution\ndominates over the pole patch, while the opposite happens for ˜β >˜βc.\nConsequently, for ˜β0>˜β > ˜βc, the large horizon patch is metastable, while for ˜β < ˜βc, the large\nhorizon patch becomes stable and the dominant configuration. If a small horizon patch exists, then\nit is always subdominant. We display various plots of the regulated action, conformal energy, and\nspecific heat as a function of the inverse conformal temperature at fixed Kℓin appendix D, see\nfigure 12.\nPure dS 4phase structure. For the pure dS 4solution, we constrain the inverse conformal\ntemperature to be given by the dS inverse temperature (4.22). We note that Kℓ≈0.256 is\n26Fig. 10: Phase diagram of conformal dS 4thermodynamics for static and spherically symmetric configurations.\nThe number of different solutions co-existing at a given point in the phase diagram depends on whether the\npoint lies above or below the ˜β0curve (dot-dashed black curve). Above that curve, only one pole patch solution\nexists. Below the ˜β0curve, apart from a pole patch solution, there co-exist two additional cosmic/black hole\npatches, one with negative and one with positive CK. The curve of critical inverse conformal temperature ˜βcis\nshown in thick black, above which the pole patch is thermodynamically preferred. In the region bounded by ˜β0\nand˜βccurves, shaded in green (yellow) halftone, the cosmic (black hole) patch is metastable. For ˜βc>˜β, the\ncosmic (black hole) patch is stable with the associated region shaded in solid green (yellow). The dark green\nand purple curves represent pure dS 4and Nariai patches. Both curves are divided into three segments: stable,\nmetastable, and unstable, which are shown as thick, dashed and dotted curves, respectively. The (meta)stable\nNariai curve marks the separation of the (meta)stable cosmic patch and black hole patch regions.\nequivalent to rtube/ℓ≈0.840.\n•ForKℓ < −7.202, the dS temperature lies in the intermediate temperature regime implying\nthat the corresponding pure dS 4is sub-dominant. We find that these pure dS 4have negative\nspecific heat and are thus unstable. The worldline limit is included in this regime. At\nKℓ≈ −7.202, the dS temperature coincides with ˜β−1\n0.\n•For−7.202< Kℓ < 0.256, the dS temperature remains in the intermediate regime, but\nnow the associated specific heat becomes positive. Therefore, these pure dS 4patches are\nmetastable. At Kℓ≈0.256, the dS temperature coincides with the critical temperature,\n˜β−1\nc.\n•For 0 .256< Kℓ , the dS temperature is higher than the critical temperature. As a conse-\nquence, the pure dS 4has regulated action lower than the pole patch. It has also positive\nspecific heat. We therefore find that the pure dS 4, in this regime, is thermodynamically\nstable. We note that the strechted horizon is included in this case.\n27Nariai phase structure. To obtain the Nariai solution, we constrain the inverse conformal\ntemperature to the Nariai inverse temperature (4.33).\n•ForKℓ < 0.405, the Nariai temperature is higher than ˜β−1\n0temperature. In this regime, the\nNariai solution has negative specific heat, so it is thermally unstable.\n•For 0 .405 < Kℓ < 2.239, the Nariai temperature lies in the intermediate temperature\nregime. The corresponding Nariai solution has positive specific heat and positive regulated\naction. This means that the Nariai soution, in this regime, is metastable.\n•For 2 .239< Kℓ , the Nariai temperature is higher than the critical temperature. We find\nthat the Nariai solution now becomes thermodynamically stable.\n5 Linearised dynamics\nSo far our treatment has been largely based on a quasi-equilibrium Euclidean picture. The aim of\nour final section is to complement the Euclidean analysis with a Lorentzian analysis. Concretely, we\nwill consider solutions to the four-dimensional linearised Einstein equations equipped with a positive\ncosmological constant Λ >0. As boundary conditions, we will once again consider conformal\nboundary conditions for the induced metric gmnand mean curvature Kon a topologically R×S2\ntimelike boundary Γ. Our treatment parallels that for Minkowski space [25], here extended to the\ncase of Λ >0. A portion of our linearised analysis was already treated in [52], in the context of\nthe fluid-gravity correspondence applied to de Sitter horizons.8\n5.1 Basic setup\nWe will consider the linearised Einstein equations about the static patch metric,\nds2=−f(r)dt2+dr2\nf(r)+r2dΩ2, f (r) = 1 −r2\nℓ2, d Ω2=dθ2+ sin2θdϕ2. (5.1)\nThe timelike boundary Γ is located at r=r∈(0, ℓ). As in [52], we are primarily interested in\ndynamical features of the cosmological horizon, but we also report on the dynamical features of\nthe pole patch below. As such, the spacetime region of interest is taken to be the Lorentzian\ncosmological patch r∈(r, ℓ),t∈R, and θ∈(0, π),ϕ∼ϕ+ 2π. The induced metric on Γ is given\nby\nds2\f\f\nr=r=−f(r)dt2+r2dΩ2. (5.2)\nUsing an inward-pointing normal vector ˆ n=−p\nf(r)∂r, the extrinsic curvature and its trace are\ngiven by,\nKmndxmdxn|r=r=rp\nf(r)\nℓ2\u0000\ndt2+ℓ2dΩ2\u0001\n, Kℓ |r=r=3r2−2ℓ2\nr√\nℓ2−r2. (5.3)\nWe denote linearised perturbations about the background (5.1) as\ngµν= ¯gµν+ε hµν, |ε| ≪1, (5.4)\n8The conformal boundary conditions were necessitated in [52], as well as [66], due to an obstruction in solving the\nnon-linear Einstein equations with Dirichlet data on a timelike surface in the near-horizon expansion. The Lorentzian\nproblem with Dirichlet conditions was considered in [67], where exponentially growing modes were found for the\ncosmic patch, as well as the pole patch at sufficiently large worldtube size.\n28where the background metric ¯ gµνis given in (5.1). The equation of motion for hµνis obtained by\nexpanding (2.5) to first order in ε. Further demanding that the conformal boundary data remains\ninvariant under arbitrary perturbation hµνimplies that\n(\nhmn|r=r =γ(x)¯gmn|r=r,\nεδK(hµν)|r=r≡K(¯gµν+ε hµν)−K(¯gµν)|r=r= 0,(5.5)\nwhere γ(x) is an arbitrary function, which will depend on the initial data of the linearised metric\nhµν, and ¯ gmn|r=ris the induced metric (5.2). By contracting the first expression in (5.5) with ¯ gmn,\none may write the first boundary condition in a form that does not contain γ(x) as\nhmn−1\n3¯gmnhpp\f\f\f\f\nr=r= 0. (5.6)\nUsing (2.2), the variation of the trace of the extrinsic curvature to first order in ϵis given by\nδK(hµν)p\nf(r)\f\f\f\f\f\nr=r=1\n2∂rhmm− Dmhrm−p\nf(r)\n2Khrr\f\f\f\f\f\nr=r= 0, (5.7)\nwhere Dndenotes the covariant derivative with respect to the boundary metric ¯ gmn. In the fol-\nlowing analysis, we will take (5.6) and (5.7) as the conformal boundary conditions for linearised\ngravity with Λ >0. We must also impose conformal boundary conditions on the space of allowed\ndiffeomorphisms ξµ. Finally, we require that ξr|r=r= 0, such that the allowed diffeomorphisms do\nnot move the location of the boundary.\nKodama-Ishibashi method. Following the treatment of Kodama and Ishibashi [50, 68], due\nto the spherical symmetry and time-translation invariance of the background, we can split our\nlinearised solutions into vector and scalar perturbations, denoted by h(V)\nµνandh(S)\nµνrespectively.\nOur details and conventions follow directly those in appendix C of [25]. As such, our treatment\nwill be brief in what follows and mostly focused on presenting the main results for Λ >0.\nTheSO(3) content of h(V)\nµνis captured by the vectorial spherical harmonics, Vi, which are transverse\neigenfunctions of the unit two-sphere Laplacian acting on vectors, with eigenvalues kV=l(l+1)−1\nforl= 1,2, . . .TheSO(3) content of h(S)\nµνis captured by the scalar spherical harmonics, S, which\nare transverse eigenfunctions of the unit two-sphere Laplacian with eigenvalues kS=l(l+ 1) for\nl= 0,1,2, . . .We note that the l= 0 and l= 1 modes require a separate treatment and we discuss\nthem in appendix E. Together, h(V)\nµνandh(S)\nµνencode the two propagating degrees of freedom of the\nfour-dimensional metric at the linearised level.\nIn the absence of timelike boundaries, the Kodama-Ishibashi formalism is gauge invariant and\nreduces the linearised Einstein equations to a set of ‘master equations’ governing the vectorial and\nscalar master fields Φ(V)and Φ(S)which are directly linked to h(V)\nµνandh(S)\nµν. It proves convenient\nfor our analysis, as it did in [25], to select a gauge where the linearised boundary conditions (5.6)\nand (5.7) act only on h(V)\nµνandh(S)\nµνrespectively. This gauge choice is indeed possible, and in this\n29gauge the components of our metric perturbation read\n\n\nhmn =−¯gmn1\n2r\u0014\nl(l+ 1)\u0010\n1−2r2\nℓ2\u0011\n+ 2r2∂2\nt+ 2\u0010\n1−r2\nℓ2\u00112\nr∂r\u0015\nΦ(S)S\n+\u0000\nδi\nmδt\nn+δi\nnδt\nm\u0001\u0010\n1−r2\nℓ2\u0011\n∂r\u0000\nrΦ(V)\u0001\nVi,\nhrr =−1\nr\u0010\n1−r2\nℓ2\u00112\"\nl(l+1)\n2\u0010\n3−7r2\nℓ2+4r4\nℓ4\u0011\n+\u0010\n3−2r2\nℓ2\u0011\nr2∂2\nt\n+\u0010\n1−r2\nℓ2\u0011\u0010\u0010\n1−r2\nℓ2\u0011\u0010\nl(l+ 1) + 1 −2r2\nℓ2\u0011\n+r2∂2\nt\u0011\nr∂r#\nΦ(S)S,\nhtr =−1\n2\u0010\n1−r2\nℓ2\u0011∂th\nl(l+ 1)\u0010\n1−r2\nℓ2\u0011\n−2 +r2∂2\nt−\u0010\n1−r2\nℓ2\u0011\nr2\nℓ2r∂ri\nΦ(S)S,\nhri =√\nl(l+1)\n2\u0010\n1−r2\nℓ2\u0011h\nl(l+ 1)\u0010\n1−r2\nℓ2\u0011\n+r2∂2\nt+\u0010\n2−\u0010\n3−r2\nℓ2\u0011\nr2\nℓ2\u0011\nr∂ri\nΦ(S)Si+r\n1−r2\nℓ2∂tΦ(V)Vi.(5.8)\nIn the above the indices mandndenote indices with respect to ( t, r), while the index idenotes\nindices on the two-sphere.\n5.2 Vector perturbation\nThe master equation for Φ(V), for given angular momentum l∈Z+, is given by\n\u0012\n−∇2+l(l+ 1)\nr2\u0013\nΦ(V)(t, r) = 0 , (5.9)\nwhere∇2denotes the Laplacian on a two-dimensional de Sitter space with curvature +2 /ℓ2. The\nsolutions can be expressed as hypergeometric functions (see for instance [66]). For a given frequency,\nthey take the form\nΦ(V)=ℜe−iωt\u0012\n1−r2\nℓ2\u0013−iωℓ/2\u0012r2\nℓ2\u0013iωℓ/2\n2F1\u0012\n−l−iωℓ,1 +l−iωℓ; 1−iωℓ;1\n2−ℓ\n2r\u0013\n.(5.10)\nThe boundary condition (5.7) is automatically satisfied while the boundary condition (5.6) imposes\nΦ(V)\nr+∂rΦ(V)\f\f\f\f\f\nr=r= 0. (5.11)\nUpon scanning numerically for solutions in the complex frequency plane, we find that all vec-\ntor modes satisfying the conformal boundary conditions have a negative imaginary part, and are\ntherefore dissipative, decaying at late times.\nWorldline limit. In the worldline limit, where Kℓ→ −∞ , we find two sets of modes. One set\nis found to be a small deformation of the quasinormal modes of the de Sitter static patch, whose\nanalytic form (in the worldline limit) is given by [51] ωqnmℓ=−i(l+n+ 1) where n∈N. As for\nthe exact quasinormal modes, the modes we find are also purely negative imaginary and their size\nis of the order of the de Sitter length ℓ. The other set of modes have a real part also and are the\ncounterpart of the vectorial Minkowski modes uncovered in [25]. For each l≥2, the second set is\na discrete tower of modes with increasing negative imaginary parts, and their size scales with the\nsize of the worldtube r, rather than the de Sitter length.\n30Cosmological horizon limit. In the cosmological horizon limit, where Kℓ→+∞, the structure\nof the modes is altered. The purely imaginary modes degenerate into a set of modes that approach\nωshearℓ=−i(l(l+ 1)−2)1\n2K2ℓ2+O(K−3ℓ−3). (5.12)\nThese modes were identified in [52] where they were interpreted as shear modes of a linearised\nincompressible non-relativistic fluid dynamical behavior near the horizon, paralleling other consid-\nerations of the fluid/gravity relation [66,69–72]. For each l≥2, the second set is a discrete tower\nof modes with increasing negative imaginary parts, and their size scales with the de Sitter length.\n5.3 Scalar perturbation\nThe master equation for Φ(S)is given by\n\u0012\n−∇2+l(l+ 1)\nr2\u0013\nΦ(S)= 0. (5.13)\nThe solutions can be expressed as hypergeometric functions, and take the form\nΦ(S)=ℜe−iωt\u0012\n1−r2\nℓ2\u0013−iωℓ/2\u0012r2\nℓ2\u0013iωℓ/2\n2F1\u0012\n−l−iωℓ,1 +l−iωℓ; 1−iωℓ;1\n2−ℓ\n2r\u0013\n.(5.14)\nThe boundary condition (5.6) is automatically satisfied while the boundary condition (5.7) imposes\nFl(Kℓ , ωℓ )≡\u0010a1\nr4+a2\nr2∂2\nt−2∂4\nt\u0011\nΦS+\u0010a3\nr2−2∂2\nt\u0011\u0012\n1−r2\nℓ2\u00132∂rΦ(S)\nr\f\f\f\f\f\nr=r= 0, (5.15)\nwhere\n\n\na1=l(l+ 1)\u0010\n1−r2\nℓ2\u0011\u0010\n3−2r2\nℓ2−2l(l+ 1)\u0010\n1−r2\nℓ2\u0011\u0011\n,\na2= 4−2r2\nℓ2−4l(l+ 1)\u0010\n1−r2\nℓ2\u0011\n,\na3= 4−2r2\nℓ2−l(l+ 1)\u0010\n3−2r2\nℓ2\u0011\n.(5.16)\nUpon scanning numerically for solutions in the complex frequency plane, we find that some scalar\nmodes satisfying the conformal boundary conditions have a positive imaginary part.\nWorldline limit. In the worldline limit, where Kℓ→ −∞ , we find two sets of allowed frequencies.\nOne set corresponds to a small deformation of the scalar quasinormal modes in the pure static\npatch, which are of the order of the de Sitter length scale ℓ. The deformed quasinormal mode\nfrequencies have a negative imaginary part, and are hence dissipative. The other set of allowed\nfrequencies is borrowed from the analogous modes in Minwkoski space uncovered in [25] and are of\nthe order of the worldtube size r≪ℓ. For this set of modes, we find a pair of allowed frequencies\nwith positive imaginary part for each l.\nCosmological horizon limit. In the cosmological horizon limit, where Kℓ→+∞, the structure\nof the modes is altered. There is still a collection of fluid-type modes, but they take a relativistic\n31dispersion relation. Upon expanding rnear ℓin (5.14), implementing the conformal boundary\nconditions, and scaling ωwith 1 /K, we find the analytic expansion\nωsoundℓ=±1√\n2Kℓp\nl(l+ 1)−il(l+ 1)−2\n21\n2K2ℓ2+O(K−3ℓ−3). (5.17)\nIt is natural to interpret the above modes as the sound mode counterpart to the fluid dynamical\nshear modes in (5.12). It is worth noting, however, that they scale differently with Kℓ, such that in\nthe strict horizon limit only the shear modes survive. This is one of the reasons the sound modes,\nwhose speed of sound becomes infinite in this limit, does not appear in the previous analyses\nof [52,66].\n(a)l= 2\n (b)l= 10\nFig. 11: Density plot of absolute value of log\u0000\ne−4ωℓiFl(Kℓ , ωℓ )\u0001\nin the complex ωℓplane for l= 2 and l= 10,\nwhere Fl(Kℓ , ωℓ ) is defined in (5.15). In both plots, Kℓis fixed to be 40. Both the ωℓ≈ ±iandωsoundℓare\ndisplayed in both plots. For l= 10, the sound modes develop a small negative imaginary part.\nIn addition to the sound modes, upon taking the strict horizon limit, Kℓ→+∞, of the scalar\nsolutions (5.14) for each l, and implementing the conformal boundary conditions, any solutions\nwith modes with positive imaginary frequency coalesce either to ωℓ= + iorωℓ=−i. We\nshow these in figure 11. One can identify these modes in a Rindler analysis, subject to conformal\nboundary conditions. Concretely, we take the Rindler metric to be\nds2=−z2\nz2\n0dt2+dz2+dx2+dy2, (5.18)\nwith ( t, x, y )∈R3andz∈(0, z0]. We then perform a straightforward analysis of the linearised\nEinstein equations, with vanishing Λ, subject to conformal boundary conditions at z=z0. One\nobserves that the following configuration (chosen for simplicity to have spatial momentum entirely\n32along the x-direction)\n\n\nhtt=−z2\nz2\n0hxx=−z2\nz2\n0hyy=e−i(ωt−kxx)J−iωz0(−ikz),\nhzz=e−i(ωt−kxx)\nk2z2\u0000\u0000\nk2z2+ 2ω2z2\n0\u0001\n+\u0000\n2k2z2−2ω2z2\n0\u0001\nz∂z\u0001\nJ−iωz0(−ikz),\nhzt=iωe−i(ωt−kxx)\nk2z\u0000\u0000\n2−k2z2+ω2z2\n0\u0001\n−z∂z\u0001\nJ−iωz0(−ikz),\nhzx=−ie−i(ωt−kxx)\nkxz\u0000\u0000\n−k2z2+ω2z2\n0\u0001\n+z∂z\u0001\nJ−iωz0(−ikz),(5.19)\nsolves the linearised Einstein equations with Λ = 0, subject to conformal boundary conditions\natz=z0, for a selection of complex frequencies. In the limit kz0→0 the allowed frequencies\ncoalesce to ωz0=±i. This mode coincides with the linearised de Sitter mode with ωℓ=±i. Upon\nexpressing the Rindler solution (5.19) in terms of a local inertial time coordinate, one notes that it\ngrows at most polynomially. As such, although growing in time, the exponential growth of (5.19)\nis more tame than an ordinary unstable mode. In fact, to leading order at small kz0, the Rindler\nmode is locally a pure diffeomorphism.\nAs shown in appendix F, the leading contribution to the de Sitter scalar modes (5.14) with ω=\n±iℓ, in the stretched cosmological horizon limit, are locally pure gauge. This leads to a double\nsuppression effect whereby the physical contribution of the modes is not only small due to the\nlinearised nature of hµν, but also due to a suppression factor that goes as 1 /K2ℓ2.\n5.4 Dynamics of the pole patch, briefly\nOne can also consider linearised gravity subject to conformal boundary conditions in the pole\npatch. Here we further impose that the gravitational solutions are smooth throughout the whole\ninterior. The pole patch has been explored as a potential candidate for static patch holography\nin [40,41,44,73,74] among other places.\nWorldline limit. ForKℓ→ −∞ the size of the timelike boundary is small in units of the de Sitter\nlength ℓ. Here our analysis matches the Minkowski analysis presented in [25], where it was observed\nthat the allowed vector modes frequencies are all real-valued, whilst the scalar mode frequencies\npermit a subset of modes ω(S)with positive real imaginary frequency for each l. At large l, these\nmodes were numerically found to scale as ω(S)r≈ ±l+icl1/3, where cis an order one number.\nThus, the pole patch in the thin world tube limit mimics the Minkowskian picture.\nCosmological horizon limit. ForKℓ→+∞the timelike boundary of the pole patch approaches\nthe cosmological horizon. In this regime, the analysis differs from the thin worldline limit. Nonethe-\nless, we observe the presence of scalar modes with complex frequencies of positive imaginary part\nfor each l. These modes, similarly to the cosmic patch, coalesce onto ωpoleℓ=±i. Moreover, we\nfind two additional sets of modes. The first set is a pair of real valued scalar modes for each l. Nu-\nmerically, they are found to linearly depend on l, with the following large lbehaviour ωℓ≈ ±0.88l√\n2Kℓ.\nThe second set is an infinite tower of real valued modes, for each l, which are numerically found to\nbe evenly spaced. This set of modes appears in both the vector and scalar sector, and their large l\nbehaviour is found to be ωℓ≈3n\nlog|√\n2Kℓ|, with n= 0,1,2. . .\nWe can synthesise, in short. We have provided evidence that, subjected to conformal boundary\nconditions, the stretched horizon limit of the pure de Sitter patch is a thermodynamically stable\n33portion of spacetime containing a cosmological horizon. Dynamically, the majority of linearised\ngravitational perturbations about this portion of spacetime decay at late times, save one mode for\neach lof total angular momentum. These modes have a purely imaginary frequency ωℓ= +i, and\nare moreover retrieved from a Rindler analysis. We take this latter property as an indication that\nthey are not endemic to de Sitter space, but rather a universal property of near horizon physics\nsubjected to conformal boundary conditions. Their fate, whose behavior as measured by a local\ninertial clock is at most polynomial in time, is a remaining obstacle in obtaining a portion of\nspacetime with Λ >0 that is both thermodynamically stable, as well as dynamically stable at the\nlinearised level. Perhaps to tame this mode we must impose an additional boundary condition,\nalways ensuring that in doing so we do not overly restrict any interesting dynamics. Or perhaps\nwe must relax the condition of a constant K. A careful examination is left to the future.\nAcknowledgements\nIt is a pleasure to acknowledge Tarek Anous, Eleanor Harris, Diego Hofman, Juan Maldacena,\nBeatrix M¨ uhlmann, Edgar Shaghoulian, Eva Silverstein and Manus Visser for interesting discus-\nsions. We are particularly grateful for interesting discussions with Andrew Svesko. D.A. is funded\nby the Royal Society under the grant “Concrete Calculables in Quantum de Sitter”. The work of\nD.A.G. is funded by UKRI Stephen Hawking Fellowship “Quantum Emergence of an Expanding\nUniverse”. D.A. and D.A.G. are further funded by STFC Consolidated grant ST/X000753/1. C.M.\nis funded by STFC under grant number ST/X508470/1.\nA dS 3Dirichlet thermodynamics\nIn this appendix, we review the thermodynamics of the three-dimensional dS using Dirichlet bound-\nary condition [12]. Here, we use the standard coordinate for Euclidean Schwarzschild-de-Sitter\nspace\nds2=f(r)\nf(rtube)dτ2+dr2\nf(r)+r2dϕ2, f (r) =r2\nc−r2\nℓ2, (A.1)\nwhere τ∼τ+βDandϕ∼ϕ+ 2π. The Euclidean time τis measured with respect to the clock\ndefined on the boundary r=rtube. Similar to the analysis in the main text, rcrepresents the\nphysical size of the cosmological horizon. There is no black hole in this geometry. When rc̸= 1,\nthere is a conical defect at the origin r= 0.\nThe boundary r=rtubehas topology of S1×S1and obeys Dirichlet boundary data, namely the\ninduced metric at r=rtubeis fixed to be\nds2\f\f\nr=rtube=dτ2+r2\ntubedϕ2. (A.2)\nIn this case, the boundary data consists of the Euclidean time period βDand the physical size of\nthe boundary rtube. The Euclidean action IEis given by (2.1) with αb.c.= (D−1) and D= 3.\nSince the Dirichlet problem fixes βDand rtube, the partition function is a function of these, Z=\nZ(βD,rtube). The energy Eand entropy Sfollow from (2.10) with βDreplacing ˜β. The specific\nheat now becomes the specific heat at constant r,Cr, and is defined as in (2.10), but with rtube\nreplacing K.\n34The solutions are divided into two classes as in the conformal problem, the pole patch and cosmic\npatch.\nPole patch. The first class of solutions is the pole patch solutions which restrict the spacetime to\nr∈[0,rtube]. In order to have a regular geometry, we must set rc=ℓ. Consequently, βDis a free\nparameter, and the Euclidean action is given by\nI(pole)\nE=−βDrtube\n4GNℓs\nℓ2\nr2\ntube−1. (A.3)\nIt follows that the energy is given by\nβDE=I(pole)\nE, (A.4)\nand the entropy and specific heat at constant rtubeare zero.\nCosmic patch. We now consider solutions with r∈[rtube,rc], so they contain the cosmological\nhorizon. Regularity near the horizon rcfixes\nβD= 2πℓs\n1−r2\ntube\nr2c. (A.5)\nThe Euclidean action for this case is given by\nI(cosmic)\nE=−βDrtube\n4GNℓs\u0012\n1−r2\ntube\nr2c\u0013−1\n−1. (A.6)\nWe note that, by setting rc=ℓ, (A.3) and (A.6) reproduce (B.8) and (B.16), respectively, in [12],\nbut without adding the additional boundary subtraction term.\nThe energy of the cosmic patch is given by\nE=1\n4GNℓq\nr2c−r2\ntube. (A.7)\nDefining a dimensionless energy E ≡ 2πErtube, we find that\nE+πr2\ntube\n2GNℓ=πr2\ntube\n2GNℓ \n1 +s\n−1 +r2c\nr2\ntube!\n, (A.8)\nwhich agrees with the energy in [12], upon identifying\n3ℓ\n2GN→c ,r2\nc\nℓ2→12∆\nc−1,2GNℓ\nπ2r2\ntube→y . (A.9)\nWe can also compute the entropy and the specific heat with Dirichlet boundary conditions. For\nthe entropy we obtain,\nS=πrc\n2GN, (A.10)\n35which agrees with the Gibbons-Hawking entropy. In fact, this entropy (including a sub-leading\nlogarithmic correction) can be matched to the entropy in a T¯T+ Λ 2-deformed CFT 2[12]. Finally,\nfor the specific heat we obtain,\nC(Dirichlet) =−πrc\n2GN\u0012r2\nc\nr2\ntube−1\u0013\n, (A.11)\nwhich is negative for all values of rtube. Moreover, in the worldline limit, the specific heat diverges.\nB Useful formulae for D= 4\nIn this appendix, we provide useful formulae to study conformal thermodynamics of four-dimensional\nspacetime with Λ >0.\nFirst, we recall the metric (4.1),\nds2=e2ω\u0012f(r)\nf(r)dτ2+dr2\nf(r)+r2dθ2+r2sin2θdϕ2\u0013\n, f (r) = 1 −2µ\nr−e2ωr2\nℓ2. (B.1)\nUsing a normal vector ˆ n=p\nf(r)∂r, one can express e2ωin terms of Kℓandµ/r. For 0 < µ/ r<\n1/3, there exists a unique eωfor any real Kℓgiven by\ne2ω=3 +\u0000\nK2ℓ2+ 9\u0001\u0010\n1−2µ\nr\u0011\n−Kℓr\n(K2ℓ2+ 9)\u0010\n1−2µ\nr\u00112\n−1\n2 (K2ℓ2+ 9) r2/ℓ2, Kℓ ∈R. (B.2)\nFor 1 /3< µ/ r<1/2, there are two branches of eωwhich gives rise to the same Kℓwhen Kℓis\npositive. They are given by\ne2ω±=3 +\u0000\nK2ℓ2+ 9\u0001\u0010\n1−2µ\nr\u0011\n±Kℓr\n(K2ℓ2+ 9)\u0010\n1−2µ\nr\u00112\n−1\n2 (K2ℓ2+ 9) r2/ℓ2, Kℓ ≥0. (B.3)\nThis also means that, for 1 /3< µ/ r<1/2, there is no eωthat leads to negative Kℓ.\nWe can write the radius of the boundary as rtube=eωr. The cosmological horizon rcand black\nhole horizon rbhcan also be written in terms of eωandµby\nrc=2√\n3cos\u00121\n3cos−1\u0010\n−3√\n3eωµ\u0011\u0013\n, rbh=2√\n3sin\u00121\n3sin−1\u0010\n3√\n3eωµ\u0011\u0013\n. (B.4)\nGiven (B.2) and (B.3), one can use these formulae to investigate the behaviour of rtube,rc, and rbh\nwhile keeping Kℓfixed and varying µ/r.\nC Details of black hole patch computations in D= 4\nIn this appendix we give some more details and explicit expressions of the computations done in\nsection 4.3, for the black hole patch solutions in D= 4. For convenience, in this section, we will\nalways express rcin terms of rbh.\n36Regularity of the geometry near the black hole horizon determines the conformal temperature of\nthe black hole patch to be\n˜β=4πrbhℓq\nrtubeℓ2−r3\ntube−rbhℓ2+r3\nbh\nr3/2\ntube\u0000\nℓ2−3r2\nbh\u0001 , (C.1)\nwhich is greater than zero. The conformal temperature ˜β−1has a lower bound ˜β−1\nmin= 2π, which\noccurs in the Nariai limit, by setting rtube=ℓ/√\n3 and taking rbh→ℓ/√\n3 from below.\nBelow this conformal temperature, the black hole patch solution does not exist. For larger conformal\ntemperatures, ˜β−1>˜β−1\nmin, there is a one-parameter family of black hole patches. To reach the high\nconformal temperature regime, ˜β→0, one can take the near horizons limit, i.e. either rtube→rbh\norrtube→rc, or the small black hole limit rbh/ℓ→0.\nRequiring that the boundary has a constant trace of the extrinsic curvature Kfixes\nKℓ=4rtubeℓ2−6r3\ntube−3rbhℓ2+ 3r3\nbh\n2r3/2\ntubeq\nrtubeℓ2−r3\ntube−rbhℓ2+r3\nbh. (C.2)\nThere is no upper or lower bound on Kℓ. Similarly to the pole patch, the limit of Kℓapproaching\nnegative infinity corresponds to pushing the boundary to be near the cosmological horizon. For Kℓ\ngoing to positive infinity, the boundary is pushed near the black hole horizon.\nBlack hole patch thermodynamics. The regulated action is given by\nI(bh)\nE, reg=−πrbh\u0000\n4rtubeℓ2−3rbhℓ2−3r3\nbh\u0001\n3GN\u0000\nℓ2−3r2\nbh\u0001 −I(pole)\nE, (C.3)\nwhere we used (C.1) and (C.2) to simplify the expression. The corresponding conformal energy\nand conformal entropy for the black hole patch are given by\nEconf=−r3/2\ntube\u0000\n2rtubeℓ2−3rbhℓ2+ 3r3\nbh\u0001\n6GNℓq\nrtubeℓ2−r3\ntube−rbhℓ2+r3\nbh−I(pole)\nE\n˜β, Sconf=πr2\nbh\nGN. (C.4)\nThe conformal entropy agrees with the Bekenstein-Hawking entropy Ahorizon /4GNwhere the hori-\nzon in the formula corresponds to the black hole horizon.\nTaking a small black hole limit, we find that\nEconf→rbh\n2GNrtubeq\n1−r2\ntube\nℓ2, as rbh/ℓ→0. (C.5)\nProvided that rbh/2GNis the physical mass of the small black hole, we can see that Econfis indeed\nthe energy as measured by the conformal clock defined on the boundary.\nThe specific heat at constant Kof the black hole patch is given by\nCK=2πr2\nbh\u0000\n−ℓ2+ 3r2\nbh\u0001\u0010\n9r2\nbh\u0000\nr2\nbh−ℓ2\u00012+ 16 rbh\u0000\nr2\nbh−ℓ2\u0001\nrtubeℓ2+ 8r2\ntubeℓ4−4r4\ntubeℓ2\u0011\nGN\u0000\nℓ2+ 3r2\nbh\u0001\u0012\n9r2\nbh\u0000\nr2\nbh−ℓ2\u00012+ 2rbh(−9ℓ4−10r2\nbhℓ2+15 r4\nbh)\n(ℓ2+3r2\nbh)rtubeℓ2+ 8r2\ntubeℓ4−4r4\ntubeℓ2\u0013.\n(C.6)\n37Note that\nCK→(\n−2πr2\nbh\nGN,as rbh/ℓ→0,\n+2πr2\nbh\nGN,as rbh/rtube→1.(C.7)\nThe first limit corresponds to a small black hole and, in that case, the specific heat is negative. On\nthe opposite limit, when the black hole size is comparable to the boundary size, then the specific\nheat is positive.\nD Plots of the regulated action, Econf, and CKinD= 4\nIn this appendix, we give numerical examples of the regulated action, conformal energy, and specific\nheat for dS 4conformal thermodynamics as functions of ˜βfor various values of Kℓ. These are\ndisplayed in figure 12 for Kℓ=−7.5,−1.5, and 6 .0. These values of Kℓare chosen to show pure\ndS4patches which are unstable, metastable, and stable, respectively.\nEl= 0andl= 1modes\nIn this appendix, we consider linearised dynamics of gravitational l= 0 and l= 1 modes. Fol-\nlowing the analysis in [25], these modes are locally pure diffeomorphisms that become physical by\nthe presence of the timelike boundary with fixed boundary data. In particular, we are interested\nin linearised perturbation hµνof the form,\nhµν=∇µξν+∇νξµ, (E.1)\nfor an arbitrary vector field ξµ. The perturbation (E.1) automatically satisfies the linearised Ein-\nstein field equation. The conditions that this perturbation preserves the conformal boundary data\natr=rlead to\n\n\n2\u0000\nKmn−K\n3¯gmn\u0001\u0012q\n1−r2\nℓ2ξr\u0013\n+\u0010\nDmξn+Dnξm−2¯gmn\n3Dpξp\u0011\f\f\f\f\nr=r= 0,\n\u0012q\n1−r2\nℓ2∂rK− DmDm\u0013\u0012q\n1−r2\nℓ2ξr\u0013\n+ξmDmK\f\f\f\f\nr=r= 0,(E.2)\nwhere Kmnand ¯gmnare the extrinsic curvature (5.3) and induced metric (5.2) of Γ, respectively.\nThe covariant derivative Dmis that associated to the induced metric ¯ gmn.\nAt the linearised level, the perturbation (E.1) is subject also to gauge redundancy in the form of\nthe diffeomorphism\nxµ→xµ+ϵ ξ′µ, h µν→hµν− ∇ µξ′\nν− ∇ νξ′\nµ, (E.3)\nfor an arbitrary vector field ξ′µ. Due to the presence of the boundary, the vector field ξ′µmust\npreserve the boundary data and location of the boundary leading to (E.2) and ξ′r|r=r= 0, respec-\ntively. This means that a large number of the perturbations (E.1) obeying (E.2) can be gauged\naway by some suitable diffeomorphism (E.3). The exception is when the perturbation (E.1) is\nconstructed from the vector field ξµwhich disturbs the location of the boundary, i.e.\nξr|r=r̸= 0. (E.4)\n380 5 10 15 20 25 30-60-40-200204060(a) reg. action, Kℓ=−7.5\n0 2 4 6 8 10-10-505 (b) reg. action, Kℓ=−1.5\n0 1 2 3 4 5 6-1.2-1.0-0.8-0.6-0.4-0.20.0 (c) reg. action, Kℓ= 6\n0 5 10 15 20 25 300123456\n(d)Econf,Kℓ=−7.5\n0 2 4 6 8 10012345 (e)Econf,Kℓ=−1.5\n0 1 2 3 4 5 60.00.51.01.52.02.53.0 (f)Econf,Kℓ= 6\n0 5 10 15 20 25 30-100-50050100\n(g)CK,Kℓ=−7.5\n0 2 4 6 8 10-20-1001020304050 (h)CK,Kℓ=−1.5\n0 1 2 3 4 5 6-20246 (i)CK,Kℓ= 6\nFig. 12: Plots of regulated action, Econf, and CKas a function of ˜βat fixed Kℓ. When evaluated on the\ncosmic (black hole) patch, the curve colour is green (yellow). The pure dS 4and Nariai solutions are marked in\ndark green and purple dots, respectively. For the plots of CK, we also display the conformal answer Nd.o.f./˜β2\nas a black dashed curve.\n39We therefore take (E.2) and (E.4) as boundary conditions for a physical metric perturbation (E.1).\nl= 0modes. Choosing the htr= 0 gauge, the general spherically symmetric ( l= 0) vector field\nξµsatisfying (E.2) and (E.4) is given by\nξµdxµ∼r\nℓ\u0012\n1−r2\nℓ2\u0013ω(l=0)2\n±ℓ2/2\ne−iω(l=0)\n±t \ndr−i1−r2/ℓ2\nω(l=0)\n±rdt!\n, (E.5)\nwhere the frequency ω(l=0)\n± r=±iq\n2−r2\nℓ2is purely imaginary. In the worldline limit, where\nr/ℓ→0, we match the result of l= 0 modes found in [25]. In the strechted horizon limit, where\nr/ℓ→1, we find that these pair of modes coalesce to ω(l=0)\n±ℓ=±i.\nl= 1modes. Taking again the htr= 0 gauge, the general l= 1 vector field ξµsatisfying (E.2)\nand (E.4) is given by\nξµdxµ∼e−iω(l=1)\n±t\np\n1−r2/ℓ2\u0010\nSdr+iω(l=1)\n±r\u0000\n1−r2/ℓ2\u0001\nSdt+√\n2\u0000\n1−r2/ℓ2\u0001\nSir dΩi\u0011\n(E.6)\nwhere the frequency ω(l=1)\n±ℓ=±iis pure imaginary and r-independent. Unlike the l= 0 modes,\nthe metric perturbation constructed from (E.6) is vanishing everywhere implying that (E.6) is a\nKilling vector of the background dS 4. Near the worldline, where r/ℓ→0, these modes reproduce\nthree translations and three Lorentz boosts of the flat spacetime. In the strectched horizon limit,\nwhere r/ℓ→1, we find that these modes become combinations of translations and Lorentz boosts\nin a local inertial frame near the boundary.\nF Near-horizon diffeomorphisms\nLet us first define a near-horizon parameter ϵ≡1−r\nℓand a near-horizon radial coordinate\nρ≡1\nϵ\u0000\n1−r\nℓ\u0001\n. It follows that in terms of ρ, the boundary is located at ρ= 1.\nConsider the following linearised diffeomorphism with complex frequency ωℓ=±i,\nξµdxµ=e±tSdr−e±t\u0012\n1−r2\nℓ2\u0013\u0000\n±Sdt−∂iSdΩi\u0001\n, (F.1)\nwhere Shere is an arbitrary angle-dependent function obeying\f\f\f∂iS\nS\f\f\f≪1\nϵ. By considering the\nassociated linearised metric perturbation hµν=∇µξν+∇νξµ, we find that\nδK(hµν)|r=r=O(√ϵ), δhmn|r=r=O(ϵ)δm\nn. (F.2)\nIn terms of the near-horizon coordinate, we find that\nξθ\f\f\f\nρ=1=ξϕ\f\f\f\nρ=1=O(ϵ),−ℓ\n2ξρ|ρ=1=±ξt\f\f\nρ=1=e±t/ℓS+O(ϵ). (F.3)\nThis means that, in the ϵ→0 limit, the diffeomorphism (F.1) preserves the conformal boundary\ndata but not the location of the boundary. In particular, in the local inertial frame, these modes\nbecome angle-dependent radial/time translations.\n40References\n[1] J. G´ eh´ eniau and R. Debever, Les invariants de courbure de l’espace de riemann ` a quatre\ndimensions ,Bulletins de l’Acad´ emie Royale de Belgique 42(1956) 114.\n[2] A. Komar, Construction of a Complete Set of Independent Observables in the General\nTheory of Relativity ,Phys. Rev. 111(1958) 1182.\n[3] P. G. Bergmann, Observables in General Relativity ,Rev. Mod. Phys. 33(1961) 510.\n[4] J. Tambornino, Relational Observables in Gravity: a Review ,SIGMA 8(2012) 017\n[1109.0740 ].\n[5] E. Fermi , Atti Acad. Naz. Lincei Rend. Cl. Sci. Fiz. Mat. Nat. 31(1922) 21.\n[6] F. A. E. Pirani, On the Physical significance of the Riemann tensor ,Acta Phys. Polon. 15\n(1956) 389.\n[7] F. K. Manasse and C. W. Misner, Fermi Normal Coordinates and Some Basic Concepts in\nDifferential Geometry ,J. Math. Phys. 4(1963) 735.\n[8] W. G. Unruh, Notes on black hole evaporation ,Phys. Rev. D 14(1976) 870.\n[9] D. Anninos, S. A. Hartnoll and D. M. Hofman, Static Patch Solipsism: Conformal Symmetry\nof the de Sitter Worldline ,Class. Quant. Grav. 29(2012) 075002 [ 1109.4942 ].\n[10] D. Anninos and D. M. Hofman, Infrared Realization of dS 2in AdS 2,Class. Quant. Grav. 35\n(2018) 085003 [ 1703.04622 ].\n[11] D. Anninos, D. A. Galante and D. M. Hofman, De Sitter horizons & holographic liquids ,\nJHEP 07(2019) 038 [ 1811.08153 ].\n[12] E. Coleman, E. A. Mazenc, V. Shyam, E. Silverstein, R. M. Soni, G. Torroba et al., De Sitter\nmicrostates from T T+Λ2and the Hawking-Page transition ,JHEP 07(2022) 140\n[2110.14670 ].\n[13] E. Witten, Algebras, Regions, and Observers ,2303.02837 .\n[14] M. J. Blacker and S. A. Hartnoll, Cosmological quantum states of de Sitter-Schwarzschild are\nstatic patch partition functions ,2304.06865 .\n[15] R. Loganayagam and O. Shetye, Influence Phase of a dS Observer I : Scalar Exchange ,\n2309.07290 .\n[16] J. Kudler-Flam, S. Leutheusser and G. Satishchandran, Generalized Black Hole Entropy is\nvon Neumann Entropy ,2309.15897 .\n[17] H. Friedrich and G. Nagy, The Initial boundary value problem for Einstein’s vacuum field\nequations ,Commun. Math. Phys. 201(1999) 619.\n[18] M. T. Anderson, On boundary value problems for einstein metrics ,Geometry and Topology\n12(2008) 2009 [ math/0612647 ].\n[19] O. Sarbach and M. Tiglio, Continuum and Discrete Initial-Boundary-Value Problems and\nEinstein’s Field Equations ,Living Rev. Rel. 15(2012) 9 [ 1203.6443 ].\n[20] G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem for the Einstein\n41Equations with Totally Geodesic Timelike Boundary ,Commun. Math. Phys. 385(2021) 1615\n[2006.01498 ].\n[21] Z. An and M. T. Anderson, The initial boundary value problem and quasi-local Hamiltonians\nin General Relativity ,2103.15673 .\n[22] G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem in General\nRelativity: The Umbilic Case ,Int. Math. Res. Not. 2023 (2023) 3790 [ 2104.08851 ].\n[23] E. Witten, A note on boundary conditions in Euclidean gravity ,Rev. Math. Phys. 33(2021)\n2140004 [ 1805.11559 ].\n[24] D. Anninos, D. A. Galante and B. M¨ uhlmann, Finite features of quantum de Sitter space ,\nClass. Quant. Grav. 40(2023) 025009 [ 2206.14146 ].\n[25] D. Anninos, D. A. Galante and C. Maneerat, Gravitational observatories ,JHEP 12(2023)\n024 [ 2310.08648 ].\n[26] P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow, and strongly coupled CFT\nin the Schwarzschild Unruh or Boulware vacua ,Class. Quant. Grav. 28(2011) 215018\n[1104.4489 ].\n[27] A. Adam, S. Kitchen and T. Wiseman, A numerical approach to finding general stationary\nvacuum black holes ,Class. Quant. Grav. 29(2012) 165002 [ 1105.6347 ].\n[28] M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space , inLes\nHouches Summer School: Session 76: Euro Summer School on Unity of Fundamental\nPhysics: Gravity, Gauge Theory and Strings , pp. 423–453, 10, 2001, hep-th/0110007 .\n[29] D. Anninos, De Sitter Musings ,Int. J. Mod. Phys. A 27(2012) 1230013 [ 1205.3855 ].\n[30] D. A. Galante, Modave lectures on de Sitter space & holography ,PoSModave2022 (2023)\n003 [ 2306.10141 ].\n[31] G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum\nGravity ,Phys. Rev. D 15(1977) 2752.\n[32] G. W. Gibbons and S. W. Hawking, Cosmological Event Horizons, Thermodynamics, and\nParticle Creation ,Phys. Rev. D 15(1977) 2738.\n[33] G. Hayward, Euclidean action and the thermodynamics of manifolds without boundary ,Phys.\nRev. D 41(1990) 3248.\n[34] B. Banihashemi and T. Jacobson, Thermodynamic ensembles with cosmological horizons ,\nJHEP 07(2022) 042 [ 2204.05324 ].\n[35] D. Anninos, F. Denef, Y. T. A. Law and Z. Sun, Quantum de Sitter horizon entropy from\nquasicanonical bulk, edge, sphere and topological string partition functions ,JHEP 01(2022)\n088 [ 2009.12464 ].\n[36] D. Anninos and B. M¨ uhlmann, The semiclassical gravitational path integral and random\nmatrices (toward a microscopic picture of a dS 2universe) ,JHEP 12(2021) 206\n[2111.05344 ].\n[37] J. Polchinski, Combinatorics of boundaries in string theory ,Phys. Rev. D 50(1994) R6041\n42[hep-th/9407031 ].\n[38] H. Friedrich, On the existence of n-geodesically complete or future complete solutions of\nEinstein’s field equations with smooth asymptotic structure ,Commun. Math. Phys. 107\n(1986) 587.\n[39] L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole\ncomplementarity ,Phys. Rev. D 48(1993) 3743 [ hep-th/9306069 ].\n[40] T. Banks, Holographic spacetime ,International Journal of Modern Physics D 21(2012)\n1241004.\n[41] E. Shaghoulian, The central dogma and cosmological horizons ,JHEP 01(2022) 132\n[2110.13210 ].\n[42] E. Shaghoulian and L. Susskind, Entanglement in De Sitter space ,JHEP 08(2022) 198\n[2201.03603 ].\n[43] V. Narovlansky and H. Verlinde, Double-scaled SYK and de Sitter Holography ,2310.16994 .\n[44] E. Coleman and V. Shyam, Conformal boundary conditions from cutoff AdS 3,JHEP 09\n(2021) 079 [ 2010.08504 ].\n[45] V. Shyam, T T+Λ2deformed CFT on the stretched dS 3horizon ,JHEP 04(2022) 052\n[2106.10227 ].\n[46] D. Anninos and E. Harris, Three-dimensional de Sitter horizon thermodynamics ,JHEP 10\n(2021) 091 [ 2106.13832 ].\n[47] B. B. Wang and C. G. Huang, Thermodynamics of de Sitter space-time in York’s formalism ,\nMod. Phys. Lett. A 16(2001) 1487.\n[48] P. Draper and S. Farkas, Euclidean de Sitter black holes and microcanonical equilibrium ,\nPhys. Rev. D 105(2022) 126021 [ 2203.01871 ].\n[49] B. Banihashemi, T. Jacobson, A. Svesko and M. Visser, The minus sign in the first law of de\nSitter horizons ,JHEP 01(2023) 054 [ 2208.11706 ].\n[50] H. Kodama, A. Ishibashi and O. Seto, Brane world cosmology: Gauge invariant formalism\nfor perturbation ,Phys. Rev. D 62(2000) 064022 [ hep-th/0004160 ].\n[51] A. Lopez-Ortega, Quasinormal modes of D-dimensional de Sitter spacetime ,Gen. Rel. Grav.\n38(2006) 1565 [ gr-qc/0605027 ].\n[52] D. Anninos, T. Anous, I. Bredberg and G. S. Ng, Incompressible Fluids of the de Sitter\nHorizon and Beyond ,JHEP 05(2012) 107 [ 1110.3792 ].\n[53] J. W. York, Role of conformal three-geometry in the dynamics of gravitation ,Phys. Rev. Lett.\n28(1972) 1082.\n[54] G. Odak and S. Speziale, Brown-York charges with mixed boundary conditions ,JHEP 11\n(2021) 224 [ 2109.02883 ].\n[55] D. Anninos, Sailing from Warped AdS(3) to Warped dS(3) in Topologically Massive Gravity ,\nJHEP 02(2010) 046 [ 0906.1819 ].\n43[56] R. Emparan, J. F. Pedraza, A. Svesko, M. Tomaˇ sevi´ c and M. R. Visser, Black holes in dS 3,\nJHEP 11(2022) 073 [ 2207.03302 ].\n[57] E. Panella and A. Svesko, Quantum Kerr-de Sitter black holes in three dimensions ,JHEP 06\n(2023) 127 [ 2303.08845 ].\n[58] J. W. York, Jr., Black hole thermodynamics and the Euclidean Einstein action ,Phys. Rev. D\n33(1986) 2092.\n[59] E. Shaghoulian, Unpublished , .\n[60] J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic\nSymmetries: An Example from Three-Dimensional Gravity ,Commun. Math. Phys. 104\n(1986) 207.\n[61] J. L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories ,Nucl.\nPhys. B 270(1986) 186.\n[62] S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space ,\nCommun. Math. Phys. 87(1983) 577.\n[63] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories ,\nAdv. Theor. Math. Phys. 2(1998) 505 [ hep-th/9803131 ].\n[64] D. Anninos and E. Harris, Interpolating geometries and the stretched dS 2horizon ,JHEP 11\n(2022) 166 [ 2209.06144 ].\n[65] A. Svesko, E. Verheijden, E. P. Verlinde and M. R. Visser, Quasi-local energy and\nmicrocanonical entropy in two-dimensional nearly de Sitter gravity ,JHEP 08(2022) 075\n[2203.00700 ].\n[66] I. Bredberg and A. Strominger, Black Holes as Incompressible Fluids on the Sphere ,JHEP\n05(2012) 043 [ 1106.3084 ].\n[67] T. Andrade, W. R. Kelly, D. Marolf and J. E. Santos, On the stability of gravity with\nDirichlet walls ,Class. Quant. Grav. 32(2015) 235006 [ 1504.07580 ].\n[68] H. Kodama and A. Ishibashi, A Master equation for gravitational perturbations of maximally\nsymmetric black holes in higher dimensions ,Prog. Theor. Phys. 110(2003) 701\n[hep-th/0305147 ].\n[69] T. Damour, Black Hole Eddy Currents ,Phys. Rev. D 18(1978) 3598.\n[70] R. Znajek, The electric and magnetic conductivity of a kerr hole ,Monthly Notices of the\nRoyal Astronomical Society 185(1978) 833.\n[71] R. H. Price and K. S. Thorne, Membrane Viewpoint on Black Holes: Properties and\nEvolution of the Stretched Horizon ,Phys. Rev. D 33(1986) 915.\n[72] S. Bhattacharyya, S. Minwalla and S. R. Wadia, The Incompressible Non-Relativistic\nNavier-Stokes Equation from Gravity ,JHEP 08(2009) 059 [ 0810.1545 ].\n[73] L. Susskind, Entanglement and Chaos in De Sitter Space Holography: An SYK Example ,\nJHAP 1(2021) 1 [ 2109.14104 ].\n[74] E. Jørstad, R. C. Myers and S.-M. Ruan, Holographic complexity in dS d+1,JHEP 05(2022)\n44119 [ 2202.10684 ].\n45"    },    {        "title": "2402.04307v2.The_fermion_self_energy_and_damping_rate_in_a_hot_magnetized_plasma.pdf",        "content": "arXiv:2402.04307v2  [hep-ph]  12 Feb 2024The fermion self-energy and damping rate in a hot magnetized plasma\nRitesh Ghosh1,∗and Igor A. Shovkovy1,2,†\n1College of Integrative Sciences and Arts, Arizona State Uni versity, Mesa, Arizona 85212, USA\n2Department of Physics, Arizona State University, Tempe, Ar izona 85287, USA\n(Dated: February 13, 2024)\nWe derive a general expression for the fermion self-energy i n a hot magnetized plasma by using\nthe Landau-level representation. In the one-loop approxim ation, the Dirac structure of the self-\nenergy is characterized by five different functions that depe nd on the Landau-level index nand\nthe longitudinal momentum pz. We derive general expressions for all five functions and obt ain\nclosed-form expressions for their imaginary parts. The lat ter receive contributions from three types\nof on-shell processes, which are interpreted in terms of Lan dau-level transitions, accompanied by\na single photon (gluon) emission or absorption. By making us e of the imaginary parts of the self-\nenergy functions, we also derive the Landau-level dependen t fermion damping rates Γ n(pz)and\nstudy them numerically in a wide range of model parameters. W e also demonstrate that the two-\nspin degeneracy of the Landau levels is lifted by the one-loo p self-energy corrections. While the\nspin splitting of the damping rates is small, it may be import ant for some spin and chiral effects.\nWe argue that the general method and the numerical results fo r the rates can have interesting\napplications in heavy-ion physics, astrophysics, and cosm ology, where strongly magnetized QED or\nQCD plasmas are ubiquitous.\nI. INTRODUCTION\nThe influence of magnetic fields on relativistic matter has been a topic of continued investigations and interest\nfor decades. Strong magnetic fields appear and play an important r ole in cosmology [ 1,2], and astrophysics [ 3,4],\nand heavy-ion collisions [ 5–8]. They can affect physics of magnetars [ 9], supernovae [ 10], and gamma ray bursts [ 11].\nTheoretical estimates show that extremely strong magnetic fields up to/divides.alt0eB/divides.alt0≃m2\nπare produced in high-energy\nnoncentral heavy-ion collisions [ 12–16]. Of course, the strength and temporal evolution of these fields c an be affected\nby many factors, including the collision energy, the impact paramete r, and the electrical conductivity of the plasma\n[17–22]. Even in condensed matter physics, strong magnetic fields can trig ger some relativisticlike phenomena when\ntopological features of the band structure give rise to low-energ yquasiparticlesdescribed by Dirac and Weyl equations\n[23].\nThe groundwork for understanding relativistic systems in the pres ence of a magnetic field was laid by Heisenberg\nand Euler [ 24] and later by Schwinger [ 25]. Many field-theoretical studies have been done over the years sin ce. The\nkey developments and foundations can be found in many books and r eviews, e.g., see Refs. [ 26–28]. Despite broad\ntheoretical knowledge gained, surprisingly few quantitive results a re known about the Green functions and radiative\ncorrections for relativistic plasmas in background magnetic fields be yond the two extremes of the lowest Landau level\napproximation and the weak-field limit [ 29–31]. For some of the recent developments, see Refs. [ 32–47].\nIn a uniform magnetic field, the translation symmetry in the plane per pendicular to the field is broken. As a result,\nthe usual transverse momenta are not good quantum numbers fo r charged particles. Instead, their eigenstates are\ngiven by the Landau-level orbitals. This fact has profound implicatio ns on the field theory formalism. The most\nnatural form of the fermion propagator is given in the Landau-leve l representation [ 28]. The inherent complexity of\nsuch a representation makes the evaluation of Feynman diagrams d ifficult even at the lowest one-loop order.\nThe main objective of this study is a rigorous derivation of the fermio n self-energy in a strongly magnetized hot\nrelativistic plasma. In particular, the emphasis will be made on the pro per treatment of the self-energy in the Landau-\nlevel representation. We will follow the approach developed previou sly in the context of the quantum Hall effect in\ngraphene [ 48,49]. Similar methodology was also utilized in the studies of chiral asymmetr y in magnetized QED\nat nonzero density [ 50,51]. Here we will focus on the fermion self-energy in the Landau-level r epresentation and\ninvestigate in detail its imaginary part. Such an imaginary part define s the fermion damping rate in the plasma. It is\nalso a critical input in determining the particle mean free path and som e transport properties. We will derive explicit\nexpressions for different components of the self-energy and disc uss their interpretation in terms of underlying physical\nprocesses. We will also study the quantitative dependence of the f ermion damping rate on the Landau level index\nand the longitudinal momentum.\n∗Ritesh.Ghosh@asu.edu\n†igor.shovkovy@asu.edu2\nSeveral attempts at studying the fermion self-energy in strongly magnetized vacuum can be found in the literature\n[52–56]. Most notably, the authors of Refs. [ 57–59] had the most of progress in recent years, where they calculated\nthe Fourier transform of the transitionary invariant part of the s elf-energy but stopped short of projecting the results\nonto the Landau levels. As we argue here, the latter procedure is n ecessary in order to extract observable features of\nthe self-energy.\nThe paper is organized as follows. We start from the definition of the fermion self-energy in coordinate space in\nSec.II. After removing the Schwinger phase and performing a Fourier tra nsform on the translation invariant part of\nthe self-energy, we derive a relation that resembles but is not the u sual momentum space representation. To extract\nphysics information, the corresponding result is mapped onto the L andau levels in Sec. IIIA. The numerical results\nfor the imaginary parts of the functions, defining the Dirac struct ure of the self-energy, are presented in Sec. IIIB.\nBy utilizing the imaginary part of the self-energy, we derive the ferm ion damping rate and study its dependence on\nthe Landau-level index nand the longitudinal momentum pzin Sec.IV. Note that we use two different methods in\nSubsecs. IVAandIVB, but they give the same spin-averaged expression for the damping rate. However, the use of\nthe poles of the full propagator in Subsec. IVBreveals that the rates for the two spin states of each Landau leve l are\nslightly different. Finally, we summarize our main results and conclusion s in Sec. V. Several technical derivations and\nauxiliary results are given in the appendices at the end of the paper.\nII. FERMION SELF-ENERGY IN MAGNETIZED PLASMA\nTo keep our analysis as simple as possible, we consider a hot magnetize d QED-like plasma with a single fermion\nflavor of mass ¯ m0and charge q. With minor adjustments, accounting for a different coupling const ant and the number\nof gauge bosons, the one-loop expression for the self-energy will be also valid for the QCD plasma. Without loss of\ngenerality, we will assume that the background magnetic field Bpoints in the+zdirection.\nAt the leading order in coupling, the coordinate space representat ion of the fermion self-energy is given by\nΣ(u,u′)=−4iπαγµS(u,u′)γνDµν(u−u′), (1)\nwhereα=q2/slash.left(4π)in the coupling constant, S(u,u′)is the free fermion propagator, and Dµν(u−u′)is the photon\n(gauge-field)propagator. Notethat, bydefinition, Σ (u,u′)=i/bracketleft.alt1S−1(u,u′)−G−1(u,u′)/bracketright.alt, whereG−1(u,u′)is the inverse\nof the full fermion propagator (at the leading one-loop order). In the case of the QCD plasma, one would need to\nreplace the coupling constant αwithαsCF, whereαs=g2\ns/slash.left(4π)andCF=(N2\nc−1)/slash.left(2Nc).\nBecause of the broken translation symmetry, the free fermion pr opagatorS(u,u′)and, in turn, the self-energy\nΣ(u,u′)depend on spacetime coordinates u=(t,x,y,z)andu′=(t′,x′,y′,z′)as follows [ 25]:\nS(u,u′) =eiΦ(u⊥,u′\n⊥)¯S(u−u′), (2)\nΣ(u,u′) =eiΦ(u⊥,u′\n⊥)¯Σ(u−u′), (3)\nwhere Φ(u⊥,u′\n⊥)is the famous Schwinger phase. Note that the translation-invarian t parts¯S(u−u′)and¯Σ(u−u′)\ndepend on the difference u−u′only. Assuming the Landau gauge for the background field, i.e., A=(0,Bx,0), the\nexplicit form of the Schwinger phase is given by Φ (u⊥,u′\n⊥)=qB\n2(x+x′)(y−y′), whereqis the fermion charge.\nFor reference, we derive an explicit form of the fermion propagato r in a background magnetic field in Appendix A.\nWe makesuretoemphasizeitscoordinatespacedependence andth e Landau-levelstructure. Usingthesameapproach,\nwe also obtain the inverse fermion propagator in Appendix B. Note that both propagator and its inverse (and, by\nextension, the self-energy) have exactly the same Schwinger pha se. It is consistent with the structure of Eq. ( 1) and\nthe spacetime dependence in Eqs. ( 2) and (3).\nAfter removing the Schwinger phase and performing the Fourier tr ansform on both sides of Eq. ( 1), we arrive at\nthe following expression for the self-energy function:\n¯Σ(p∥,p⊥)=−4iπα/integral.dispd2k∥d2k⊥\n(2π)4γµ¯S(k∥,k⊥)γνDµν(p−k), (4)\nwherepµ\n∥=(p0,pz)andpµ\n⊥=(px,py). Interestingly, thisexpressioncoincideswith theusualdefinition o fthe self-energy\nin a theory with unbroken translation symmetry. Clearly, it is not the case, and the vector-like variable p⊥cannot\nbe interpreted as a conserved transverse momentum in a magnetiz ed plasma. (In contrast, the two components of p∥,\ni.e., the energy p0and the longitudinal momentum pz, are conserved quantities in a uniform magnetic field.) Despite\nthe appearance, the functions ¯S(k∥,k⊥)and¯Σ(p∥,p⊥)are not the momentum-space representations of the fermion\npropagator and the self-energy, respectively. Yet, they encod e all information about the propagator and self-energy.3\nq\n(n,p  ) z\n(n ,k  ) z(n,p  ) z\nFIG. 1. The leading order Feynman diagram for the fermion sel f-energy in the Landau-level representation.\nThe main advantage of the representation in Eq. ( 4) is its simplicity. To extract its observables effects, however,\nwe will need to render it in the Landau-level basis. The correspondin g projection will be discussed and implemented\nin Sec.III. While technically nontrivial, its outcome is obvious in the diagrammatic f orm shown in Fig. 1.\nAt this point, we will proceed with the calculation of the self-energy in Eq. (4). In the derivation, we will use the\nfollowing Feynman gauge for the free gauge-field propagator:\nDµν(p−k)=−igµν\n(p−k)2, (5)\nwheregµν=diag(1,−1,−1,−1)is the Minkowski metric. We note that a more refined analysis of a hot magnetized\nplasma may require using the hard-thermal [ 60] and hard-magnetic loop [ 28] resummations. The corresponding\nrefinements are beyond the scope of the present exploratory st udy but should be undertaken in the future.\nIt is instructive to emphasize that the Feynman gauge for the gaug e-field propagatoris convenient but not the most\ngeneral. In fact, it is well known that the fermion self-energy depe nds on a gauge choice. In this study, however, we\nwill be concerned primarily with the imaginary (dissipative) part of the self-energy and the fermion damping rate.\nFor these purposes, the simplest Feynman gauge should be sufficien t [61,62].\nBy substituting the free fermion propagator,whose explicit form is given in Appendix A, and the photon propagator\nin Eq. (5) into the expression for the self-energy in Eq. ( 4), we obtain\n¯Σ(p∥,p⊥)=−4iπα∞\n/summation.disp\nn′=0/integral.dispd2k∥d2k⊥\n(2π)4e−k2\n⊥l2γµ(−1)n′D(0)\nn′(k∥,k⊥)\nk2\n∥−¯m2\n0−2n′/divides.alt0qB/divides.alt0γµ1\nq2\n∥−q2⊥. (6)\nHereq∥=p∥−k∥,q⊥=p⊥−k⊥, and\nD(0)\nn′(k∥,k⊥)=2/bracketleft.alt1(k∥⋅γ∥)+¯m0/bracketright.alt/bracketleft.alt1P+Ln′/parenleft.alt12k2\n⊥ℓ2/parenright.alt1−P−Ln′−1/parenleft.alt12k2\n⊥ℓ2/parenright.alt1/bracketright.alt+4(k⊥⋅γ⊥)L1\nn′−1/parenleft.alt12k2\n⊥ℓ2/parenright.alt1, (7)\nwhereP±=/parenleft.alt11±s⊥iγ1γ2/parenright.alt1/slash.left2 are spin projectors, ℓ=1/slash.left/radical.alt1\n/divides.alt0qB/divides.alt0is the magnetic length, s⊥=sign(qB), andLα\nn(z)are the\ngeneralized Laguerre polynomials [ 63]. We assume that, by definition, Lα\n−1(z)=0.\nTo account for a nonzero temperature T, we use Matsubara’s formalism. In particular, we replace the fermio n\nenergiesp0andk0withiωnp≡iπT(2np+1)andiωnk≡iπT(2nk+1), respectively, and replace the integral over k0\nwith the Matsubara sum, i.e.,\n/integral.dispdk0\n2πF(p0,k0)→iT∞\n/summation.disp\nnk=−∞F/parenleft.alt1iωnp,iωnk/parenright.alt1. (8)\nThen, the self-energy ( 6) becomes\n¯Σ(iωnp,pz,p⊥)=4παT∞\n/summation.disp\nn′=0∞\n/summation.disp\nnk=−∞/integral.dispdkzd2k⊥\n(2π)3(−1)n′e−k2\n⊥l2˜D(0)\nn′(iωnk,kz,k⊥)\n/parenleft.alt2ω2nk+E2\nn′,kz/parenright.alt2/bracketleft.alt1(ωnp−ωnk)2+E2q/bracketright.alt, (9)\nwhere we used the shorthand notations for the Landau-level ene rgiesEn′,kz≡/radical.alt1\n2n′/divides.alt0qB/divides.alt0+¯m2\n0+k2zand the gauge boson\nenergyEq≡/radical.alt1\nq2⊥+q2z, and introduced the following new function:\n˜D(0)\nn′(iωnk,kz,k⊥)≡γµD(0)\nn′(iωnk,kz,k⊥)γµ\n=−4/parenleft.alt1iωnkγ0−kzγz−2¯m0/parenright.alt1/bracketleft.alt1P+Ln′/parenleft.alt12k2\n⊥ℓ2/parenright.alt1−P−Ln′−1/parenleft.alt12k2\n⊥ℓ2/parenright.alt1/bracketright.alt−8(k⊥⋅γ⊥)L1\nn′−1/parenleft.alt12k2\n⊥ℓ2/parenright.alt1.(10)\nAfter performing the Matsubara sum, we obtain\n¯Σ(p∥,p⊥) =4πα∞\n/summation.disp\nn′=0/summation.disp\ns1=±/summation.disp\ns2=±(−1)n′\n/integral.dispdkzd2k⊥\n(2π)3e−k2\n⊥l2s1s2[1−nF(s1En′,kz)+nB(s2Eq)]\nEn′,kzEq(p0−s1En′,kz−s2Eq+iǫ)\n×/brac⟩l⟩ft.alt4/par⟩nl⟩ft.alt1s1En′,kzγ0−kzγz−2¯m0/par⟩nright.alt1/brack⟩tl⟩ft.alt1P+Ln′/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1−P−Ln′−1/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1/brack⟩tright.alt+2(k⊥⋅γ⊥)L1\nn′−1/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1/brac⟩right.alt4,(11)4\nwhere we used the standard Fermi-Dirac and Bose-Einstein distribu tion functions, nF(E)=/par⟩nl⟩ft.alt1eE/slash.leftT+1/par⟩nright.alt1−1andnB(E)=\n/par⟩nl⟩ft.alt1eE/slash.leftT−1/par⟩nright.alt1−1, respectively. In the derivation, we used the following result for th e Matsubara sum:\nT∞\n/summation.disp\nnk=−∞iωnkA+B\n/par⟩nl⟩ft.alt1ω2nk+E2a/par⟩nright.alt1/brack⟩tl⟩ft.alt1(ωnp−ωnk)2+E2\nb/brack⟩tright.alt=−1\n4/summation.disp\ns1,s2=±(s1EaA+B)[1−nF(s1Ea)+nB(s2Eb)]\ns1s2EaEb/par⟩nl⟩ft.alt1iωnp−s1Ea−s2Eb/par⟩nright.alt1.(12)\nTo separate the real and imaginary parts of the self-energy, we p erform the analytical continuation iωnp→p0+iǫand\nuse the Sokhotski formula,\n1\np0−s1En′,kz−s2Eq+iǫ=P1\np0−s1En′,kz−s2Eq+iǫ−iπδ(p0−s1En′,kz−s2Eq). (13)\nIn the rest, we will concentrate on the imaginary (absorptive) par t. The corresponding expression can be simplifies\nby taking into account that\nδ(p0−s1En′,kz−s2Eq)=/summation.disp\ns′=±En′,kzEqδ(kz−ks′\nz)\n/divid⟩s.alt0(ks′\nz−pz)s1En′,kz+ks′\nzs2Eq/divid⟩s.alt0=/summation.disp\ns′=±2En′,kzEqδ(kz−ks′\nz)/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2], (14)\nwhereq±\n⊥=/divid⟩s.alt0/radical.alt1\n2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0±/radical.alt1\np2\n0−p2z/divid⟩s.alt0and the explicit expressions for the two solutions k±\nzto the energy-conservation\ncondition read\nk±\nz=pz\n2/par⟩nl⟩ft.alt41+2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0−q2\n⊥\np2\n0−p2z±p0\npz(p2\n0−p2z)/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]/par⟩nright.alt4. (15)\nNote that, for the fermions on the mass shell, we should set p2\n0−p2\nz=2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0, and the two thresholds will become\nq±\n⊥=/divid⟩s.alt2/radical.alt1\n2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0±/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0/divid⟩s.alt2.\nBy substituting the solutions of the energy-conservationconditio n (kz=k±\nz), we derive the following two expressions\nfor the particle energies:\nEn′,kz/divid⟩s.alt2\nkz→k±\nz=s1p0\n2/par⟩nl⟩ft.alt41+2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0−q2\n⊥\np2\n0−p2z±pz\np0(p2\n0−p2z)/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]/par⟩nright.alt4, (16)\nEq/divid⟩s.alt2\nkz→k±z=s2p0\n2/par⟩nl⟩ft.alt41−2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0−q2\n⊥\np2\n0−p2z∓pz\np0(p2\n0−p2z)/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]/par⟩nright.alt4. (17)\nWithout loss of generality, below we will concentrate on the case of L andau-level states with positive-energies, p0>0.\nOn the mass shell, they will be given by the Landau-level energies, p0=/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0+p2z. If needed, the self-energy\nresults forthe Landau-levelstateswith negativeenergiescould b e obtained by using the charge-conjugationsymmetry.\nBy analyzing the solutions for energy-conservation relation p0=s1En′,kz+s2Eqwith the assumption p0>0, we\nidentify the following three kinematic cases:\ns1>0, s2>0∶0<q⊥<q−\n⊥, (18)\ns1>0, s2<0∶0<q⊥<q−\n⊥, (19)\ns1<0, s2>0∶q+\n⊥<q⊥<∞. (20)\nThe first one describes a transition to a lower energy particle state with emission of a photon ( ψn→ψn′+γwith\nn>n′), see Fig. 2a. The second one describes a transition to a higher energy particle state with absorption of a\nphoton (ψn+γ→ψn′withn<n′), see Fig. 2b. Finally, the third one describes a transition to an antiparticle stat e\nwith emission of a photon (i.e., annihilation process ψn+¯ψn′→γfor anynandn′), see Fig. 2c.\nIt is instructive to emphasize that the three processes in Fig. 2contribute to the fermion damping rate only when\nthe background magnetic field is nonzero. Without magnetic field, th ese processes of order αare forbidden by the\nenergy-momentum conservation. Instead, the fermion damping is dominated by diagrams of order α2such as two-to-\ntwo scattering and annihilation processes (i.e., ψk+γ→ψk′+γandψk+¯ψk′→γ+γ). Turning the argument around,\nthis also implies that contributions from higher-order processes will compete with those in Fig. 2when the magnetic\nfield is sufficiently weak.\nThe final expression for the imaginary part reads\nIm/brack⟩tl⟩ft.alt1¯Σ(p∥,p⊥)/brack⟩tright.alt=−4πα∞\n/summation.disp\nn′=0/summation.disp\n{s}(−1)n′\n/integral.dispd2k⊥\n(2π)2e−k2\n⊥l21−nF(s1En′,ks′\nz)+nB(s2Eq)\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]\n×/brac⟩l⟩ft.alt4/par⟩nl⟩ft.alt2s1En′,ks′\nzγ0−ks′\nzγz−2¯m0/par⟩nright.alt2/brack⟩tl⟩ft.alt1P+Ln′/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1−P−Ln′−1/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1/brack⟩tright.alt+2(k⊥⋅γ⊥)L1\nn′−1/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1/brac⟩right.alt4,(21)5\nq\n(n,p  ) z\n(n ,k  ) z\n(a)q(n ,k  )z(n,p  ) z\n(b)q(n,p  ) z\n(n ,k  )z\n(c)\nFIG. 2. Feynman diagrams for the three processes contributi ng to the fermion damping in the nth Landau-level state: (a)\nquantum transition to a lower Landau level with emission of a photonψn→ψn′+γ(n>n′), (b) quantum transition to a higher\nLandau level with absorption of a photon ψn+γ→ψn′(n<n′), (c) particle-antiparticle annihilation ψn+¯ψn′→γ.\nwhere the shorthand notation ∑{s}represents the sum over three signs, i.e., s1,s2,s′=±1. This expression for the\nFourier transform of the translation invariant part of the self-en ergy, as defined in Eq. ( 3), does not reveal explicitly\nthe Landau-level structure. Indeed, as we show in Appendix B1, its Landau-level representation ( B7) should be given\nas an expansion in Laguerre polynomials Lα\nn/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1, wherep⊥is the Fourier variable for the external line. In the\nnext section, we will use the properties of the Laguerre polynomials to render the self-energy in such a Landau-level\nform.\nIII. FERMION SELF-ENERGY IN LANDAU-LEVEL REPRESENTATION\nA. Analytical expressions for the self-energy functions\nBy making use of explicit wave functions for the Landau-level orbita ls in Appendix B1, we find that the self-energy\nmust take the following general form:\n¯Σ(p∥,p⊥)=−2e−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)n/brack⟩tl⟩ft.alt1δv∥,n(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)˜vn−δmn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt1P+Ln(2p2\n⊥ℓ2)−P−Ln−1(2p2\n⊥ℓ2)/brack⟩tright.alt\n−4e−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)nδv⊥,n(γ⊥⋅p⊥)L1\nn−1(2p2\n⊥ℓ2). (22)\nAs is easy to verify, it contains all the same Dirac matrices as the main expression for the one-loop self-energy\n¯Σ(p∥,p⊥)in Eq. (11), or its imaginary part in Eq. ( 21). Of course, it is not accidental since we made an educated\nchoice for a general form of the full propagator in Appendices AandB. In this connection, we should mention that,\nif higher-order calculations would reveal the need for additional Dir ac structures (allowed by symmetries), they could\nbe easily incorporated into a general ansatz for the full propagat or.\nThe physical meaning of δv∥,n,δv⊥,n, andδmnis clear. They measure the one-loop corrections to the maximum\nspin-averaged particle speed (in the directions parallel and perpen dicular to the magnetic field) and the corrections\nto the particle mass in the nth Landau level. As for the other two functions, i.e., ˜ vnand ˜mn, they determine the\nsplitting of the parallel velocities and masses of the two spin states.\nThe functional form of the self-energy dependence on p⊥, obtained in the previous section, does not seem to match\nthe Landau-level representation in Eq. ( 22), which is an expansion in the Laguerre polynomials Lα\nn/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1. However,\nby making use of the orthogonality property of the Laguerre polyn omials, i.e.,\n/integral.disp∞\n0e−xxαLα\nn(x)Lα\nn′(x)dx=δnn′Γ(n+α+1)\nn!, (23)\nit is straightforward to render the result in the form of such an exp ansion.\nIn particular, after separating different Dirac structures, we ca n match the self-energy functions δv∥,n,δv⊥,n,δmn,\n˜vn, and ˜mnin the Landau-level representation to the projection of function ¯Σ(p∥,p⊥)onto the Landau-level orbitals.6\nThe corresponding relations read:\nδv+\n∥,n≡δv∥,n+s⊥˜vn=−(−1)nℓ2\n2πp2\n∥/integral.dispd2p⊥e−p2\n⊥ℓ2tr/brack⟩tl⟩ft.alt1(p∥⋅γ∥)P+Σ(p∥,p⊥)/brack⟩tright.altLn(2p2\n⊥ℓ2), (24)\nδv−\n∥,n≡δv∥,n−s⊥˜vn=(−1)nℓ2\n2πp2\n∥/integral.dispd2p⊥e−p2\n⊥ℓ2tr/brack⟩tl⟩ft.alt1(p∥⋅γ∥)P−Σ(p∥,p⊥)/brack⟩tright.altLn−1(2p2\n⊥ℓ2), (25)\nδm+\nn≡δmn+s⊥˜mn=(−1)nℓ2\n2π/integral.dispd2p⊥e−p2\n⊥ℓ2tr/brack⟩tl⟩ft.alt1P+Σ(p∥,p⊥)/brack⟩tright.altLn(2p2\n⊥ℓ2), (26)\nδm−\nn≡δmn−s⊥˜mn=−(−1)nℓ2\n2π/integral.dispd2p⊥e−p2\n⊥ℓ2tr/brack⟩tl⟩ft.alt1P−Σ(p∥,p⊥)/brack⟩tright.altLn−1(2p2\n⊥ℓ2), (27)\nδv⊥,n=(−1)nℓ4\n4πn/integral.dispd2p⊥e−p2\n⊥ℓ2tr/brack⟩tl⟩ft.alt1(γ⊥⋅p⊥)Σ(p∥,p⊥)/brack⟩tright.altL1\nn−1(2p2\n⊥ℓ2). (28)\nNote that the self-energy component functions δv±\n∥,nandδm±\nnhave a simple meaning. They describe corrections to\nthe velocity and mass parameters for the two spin states in the nth Landau level. Only two of such functions, namely\nEqs. (24) and (26), are defined for all the Landau levels, n≥0. The other three are defined only for the higher\nLandau levels with n≥1. This is related to the unique property of the lowest Landau level, w hich has only one spin\npolarization (i.e., pointing along the field direction if the fermions carry a positive charge, or opposite to the field if\nthey carry a negative charge). As a result, the self-energy in the lowest Landau level ( n=0) is fully characterized\nby the longitudinal velocity (or the wave-function renormalization) v+\n∥,n=δv∥,n+s⊥˜vnand the mass renormalization\nδm+\nn=δmn+s⊥˜mn.\nBy substituting the expression for the one-loop result ( 11) into the above definitions ( 24) through ( 28), we will have\nall Dirac components of the self-energy in the Landau-level repre sentation. Here we will concentrate on the imaginary\nparts of the self-energy functions by using the result in Eq. ( 21). The corresponding results read\nIm/brack⟩tl⟩ft.alt1δv+\n∥,n/brack⟩tright.alt=α∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In,n′\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4/par⟩nl⟩ft.alt2s1En′,ks′\nzp0−ks′\nzpz/par⟩nright.alt2/brack⟩tl⟩ft.alt21−nF(s1En′,ks′\nz)+nB(s2Eq)/brack⟩tright.alt2\ns1s2p2\n∥/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2], (29)\nIm/brack⟩tl⟩ft.alt1δv−\n∥,n/brack⟩tright.alt=α∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In−1,n′−1\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4/par⟩nl⟩ft.alt2s1En′,ks′\nzp0−ks′\nzpz/par⟩nright.alt2/brack⟩tl⟩ft.alt21−nF(s1En′,ks′\nz)+nB(s2Eq)/brack⟩tright.alt2\ns1s2p2\n∥/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2],(30)\nIm[δm+\nn]=2α¯m0∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In,n′\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt41−nF(s1En′,ks′\nz)+nB(s2Eq)\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2], (31)\nIm[δm−\nn]=2α¯m0∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In−1,n′−1\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt41−nF(s1En′,ks′\nz)+nB(s2Eq)\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2], (32)\nIm[δv⊥,n]=α\n2n∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In−1,n′−1\n2/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt41−nF(s1En′,ks′\nz)+nB(s2Eq)\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]. (33)\nHere we introduced two unitless kernel functions that depend on q2\n⊥ℓ2/slash.l⟩ft2. They are defined in Appendix C. There we\nalso prove that the kernels reduce to functions In,n′\n0(ξ)andIn,n′\n2(ξ)introduced previously in Ref. [ 44]. The explicit\nexpressions for these functions are given in Eqs. ( C7) – (C8) of our Appendix C.\nWe can further simplify the integrand in Eqs. ( 29) and (30) by taking into account the following relation:\ns1En′,ks′\nzp0−ks′\nzpz\np2\n∥/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rtkz→k±z=1\n2/par⟩nl⟩ft.alt41+2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0−q2\n⊥\np2\n0−p2z/par⟩nright.alt4/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rt/arrowv⟩rtm.s.=1\n2/par⟩nl⟩ft.alt41+2n′/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0−q2\n⊥\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0/par⟩nright.alt4. (34)\nInthe lastexpression,weusedthe mass-shellconditiontoexpres sedtheparallelcomponentsofthe fermionmomentum\nin terms of the Landau-level index: p2\n0−p2\nz=2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0. For numerical calculations later, it will help that the result\nis independent of the signs s1ands′.\nWe should note that, despite the appearance, the combination of t he Fermi-Dirac and Bose-Einstein distribution\nfunctions, 1−nF(s1En′,ks′\nz)+nB(s2Eq), is also independent of the signs s1ands2. Indeed, it is obvious after taking\ninto account the energy expressions in Eqs. ( 16) and (17), which contain the overall factors of s1ands2, respectively.\nThe only dependence of the integrands in Eqs. ( 29) through ( 33) on the signs s1ands2comes from the overall\nfactors1s2. It is instructive to recall that different sign choices determine the process types contributing to the7\nimaginary part, see Eqs. ( 18) – (20). Therefore, up to overall sign s1s2, the integrands are formally the same for all\nprocesses. The contributions of quantum transitions of fermions to lower Landau-level states (accompanied by photon\nemission) come with a plus sign. The contributions of transitions to hig her Landau-level states (accompanied by\nphoton absorption) and the annihilation processes (accompanied b y photon emission) come with a minus sign. While\nthe integrands are formally the same for all three processes (up t o a sign), the range of integration over q⊥differs.\nNamely, it is 0<q⊥<q−\n⊥for transitions to lower/higher Landau-level states and q+\n⊥<q⊥<∞for the annihilation\nprocesses.\nBy using the five functions in Eqs. ( 29) through ( 32), we can obtain the spin-averageLandau-level dependent values\nof the parallel velocity and mass, i.e.,\nIm/brack⟩tl⟩ft.alt1δv∥,n/brack⟩tright.alt=1\n2Im/brack⟩tl⟩ft.alt1δv+\n∥,n+δv−\n∥,n/brack⟩tright.alt, (35)\nIm[δmn]=1\n2Im[δm+\nn+δm−\nn]. (36)\nas well as the corresponding spin-splitting functions, i.e.,\nIm[˜vn]=s⊥\n2Im/brack⟩tl⟩ft.alt1δv+\n∥,n−δv−\n∥,n/brack⟩tright.alt, (37)\nIm[˜mn]=s⊥\n2Im[δm+\nn−δm−\nn]. (38)\nAs expected, all of these parameters, as well as Im [δv⊥,n], are Landau-level dependent functions of the longitudinal\nmomentum pz.\nB. Self-energy in QCD plasma\nTo demonstrate the proof of concept, here we study numerically t he imaginary part of the self-energy functions in a\nhot magnetized QCD plasma. Keeping in mind their potential application s to heavy-ion physics, we will assume that\nthe plasma temperature Tis of the order of 200 MeV to 400 MeV and the magnetic field is of the or der of/divid⟩s.alt0qB/divid⟩s.alt0∼m2\nπ,\nwheremπ=135 MeV.\nBecause of different electric charges of the up- and down-quarks (qu=+2e/slash.l⟩ft3 andqd=−e/slash.l⟩ft3), the effect of a\nbackground magnetic field on their self-energies differs. Neverthe less, their dependence on /divid⟩s.alt0qB/divid⟩s.alt0will remain essentially\nthe same. (Strictly speaking, the roles of spin-up and spin-down st ates in the lowest Landau level will be interchanged\nbecause their charges have opposite signs.) Instead of considerin g the cases of up- and down-quarks separately, we\nwill consider several fixed values of /divid⟩s.alt0qB/divid⟩s.alt0. This will suffice to demonstrate the qualitative effects of the magne tic field\non the quark self-energy in the QCD plasma. We will also assume that t he quark mass is the same for both flavors,\ni.e., ¯m0=5 MeV.\nIn the case of QCD plasma, the expressions for the self-energy fu nctions have the same form as in Eqs. ( 29) – (33),\nbut the coupling constant αshould be replaced with αsCF, whereαs=g2/slash.l⟩ft(4π)andCF=(N2\nc−1)/slash.l⟩ft(2Nc)=4/slash.l⟩ft3. To get\nan order of magnitude estimate, we will assume that the strong cou pling isαs≃1/slash.l⟩ft2. In this case, αsCF=2/slash.l⟩ft3. This\nchoice is sufficient to get order of magnitude estimates. One could tr y to improve the approximation, for example, by\nincorporating the running of the coupling constant at the scale of t emperature or the momentum transfer. For the\npurposes of the current proof-of-concept study, however, it is unnecessary. In any case, the overall benefit from this\nand other improvements is likely very limited. Because of the strong c oupling in QCD, the quantitative validity of\nthe one-loop correction will remain questionable. Thus, our numeric al result should be interpreted with great caution\nand, at best, viewed as reasonable estimates rather than true qu antitative results.\nWhen calculatingthe self-energyfunctions defined in Eqs.( 29) through ( 33), oneneeds to addup contributionsfrom\nall three processes, sum overLandau level index n′, and integrate over the transverse momentum q⊥in the appropriate\nkinematic range, see Eqs. ( 18) – (20). We will limit the analysis to the first 50 Landau levels (i.e., n≤nmax=50). In\nthis case, to achieve a good numerical precision in calculations, we inc lude all transitions to Landau levels with the\nindices up to n′\nmax=100.\nThe representativeresultsforthe imaginarypartsofthe velocity and the massasfunctions ofthe Landau-levelindex\nnare shown in Fig. 3. Each panel displays numerical data for three different temperat ures, i.e.,T=200 MeV (blue\nlines),T=300 MeV (green lines), T=400 MeV (red lines), and two different magnetic fields, i.e., /divid⟩s.alt0qB/divid⟩s.alt0=(75 MeV)2\n(open circles),/divid⟩s.alt0qB/divid⟩s.alt0=(200 MeV)2(filled squares). The top three panels show the results for pz=0, while the bottom\nthree panels show the results for pz=1000 MeV.\nThe multi-panel Fig. 3provides only a limited view of the numerical data for two fixed values o f the longitudinal\nmomentum. A large set of additional data for a wide range of pzvalues is included as Supplementary Data files [ 64].8\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)\n2\n■qB=(200MeV )2\nT=200MeVT=3 \u0000 \u0000MeVT=4\n\u0000 \u0000MeV\n01\u0001 20\u0002 \u0001 \u0003\u0001 50\n\u0001\n0\u0001\n\u0001\n0\u0004\n1\n0\u0001\n1\n0\u0004\n\u0005\n0\u0001\n\u0005\n0\u0004\npz=0\n○○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10\u0006\u0007 30 405\u0007\n\u0007\n\b\u0007\n\u0007\n\b5\n\t\n\b\u0007\n\t\n\b5\n\u0006\n\b\u0007\n\u0006\n\b5\npz=0\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.00.51.01.52.02.5\npz=0\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.00.10.20.30.40.5\npz=1000MeV\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.00.10.20.30.40.5\npz=1000MeV\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.00.20.40.60.81.0\npz=1000MeV\nFIG. 3. The dependence of the self-energy functions Im [δv∥,n], Im[δv⊥,n], and Im[δmn]/slash.left¯m0on the Landau-level index nfor\ntwo fixed values of the longitudinal momentum: pz=0 (top panels) and pz=1000 MeV (bottom panels). Each panel contains\nresults for three different temperatures: T=200 MeV (blue), T=300 MeV (green), and T=400 MeV (red); and two magnetic\nfields:/divides.alt0qB/divides.alt0=(75 MeV)2(open circles) and /divides.alt0qB/divides.alt0=(200 MeV)2(filled squares).\nOverall, we find that the imaginary parts of the velocity and mass fun ctions tend to increase with the temperature and\ndecreasewith themagneticfield. Beyondthesegeneraltendencie s, onefinds thattheirdependence onthe Landau-level\nindex is nonmonotonous in general and differs at small and large value s ofpz.\nSince the imaginaryparts ofthe Landau-leveldependent velocity a nd mass functions have no clearphysical meaning\nby themselves, we will not be discussing them in more detail. We note, h owever, that they are needed as an input to\ncalculate the fermion damping rate. The latter will be discussed in the next section.\nIV. DAMPING RATE\nIn quantum field theory without a background magnetic field, the fe rmion damping rate is related to the imaginary\npart of self-energy [ 65]. In some recent studies, e.g., see Refs. [ 66,67], a similar formula was used rather heuristically\nin the case of magnetized plasmas. It should be noted, however, th at no formal justification was given to utilize the\nFouriertransformofthetranslationinvariantpartoftheself-en ergyinsuchcalculations. Sincethetransversemomenta\narenotgoodquantumnumbersinthefieldtheoryinamagneticfield, t heunderlyingfoundationofWeldon’sarguments\nin Ref. [65] cannot be transferred to an unphysical representation. Below we provide a more rigorous derivation of\nthe damping rate in terms of the self-energy in Landau-level repre sentation.\nA. Damping rate from the imaginary part of self-energy\nFollowing the general approach of Ref. [ 65], we define the damping rate using the wave functions in coordinate\nspace as follows:\nΓn(pz)=1\n2p0/integral.dispd4u′/integral.dispd4uTr/brack⟩tl⟩ft.alt42πℓ2\nV⊥/integral.dispdp/summation.disp\ns¯Ψn,p,s(u′)ImΣ(u′,u)Ψn,p,s(u)/brack⟩tright.alt4. (39)\nNote that 1/slash.l⟩ft(2πℓ2)is the number of degenerate states per unit transverse area (ex cluding the spin degeneracy) and\nV⊥is the volume (area) of the transverse plane. Thus, V⊥/slash.l⟩ft(2πℓ2)is the total number of such degenerate states.9\nBy making use of the fermion wave functions in a constant magnetic fi eld, discussed in Appendix D, we then derive\nΓn(pz)=1\np0⎧⎪⎪⎨⎪⎪⎩δn,0\n2/brack⟩tl⟩ft.alt1p2\n∥Im(δv∥,n+s⊥˜vn)−¯m0Im(δmn+s⊥˜mn)/brack⟩tright.alt\n+(1−δn,0)/brack⟩tl⟩ft.alt1p2\n∥Im(δv∥,n)−¯m0Im(δmn)−2n/divid⟩s.alt0qB/divid⟩s.alt0Im(δv⊥,n)/brack⟩tright.alt⎫⎪⎪⎬⎪⎪⎭, (40)\nwhere we used the result in Eq. ( D10). In the final expression, one should assume that the fermion is on the mass\nshell, i.e.,p0=/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0+p2z.\nBy definition, the Landau-level dependent fermion damping rate in E q. (40) is a spin-averaged quantity. Indeed,\nin the derivation, we summed up contributions of the spin states indis criminately. In the presence of a nonzero\nmagnetic field, however, the spin-degeneracy of each Landau leve l is likely to be lifted. Thus, the damping rates of the\ncorresponding states are expected to be different. As we show in t he next subsection, it is indeed the case. Moreover,\nwe will be able to calculate the spin-dependent damping rates from th e imaginary part of the one-loop self-energy.\nBy substituting the results in Eqs. ( 29) through ( 33) into the general expression for the rate ( 40), we derive the\nfollowing damping rate in the zeroth Landau level:\nΓ0(pz)=α\n2p0∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥I0,n′\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4/par⟩nl⟩ft.alt2s1En′,ks′\nzp0−ks′\nzpz−2¯m2\n0/par⟩nright.alt2/brack⟩tl⟩ft.alt21−nF(s1En′,ks′\nz)+nB(s2Eq)/brack⟩tright.alt2\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2],(41)\nand following damping rate in the higher Landau levels ( n≥1):\nΓn(pz)=α\n2p0∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥/brack⟩tl⟩ft.alt4In,n′\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4+In−1,n′−1\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4/brack⟩tright.alt4\n×/par⟩nl⟩ft.alt2s1En′,ks′\nzp0−ks′\nzpz−2¯m2\n0/par⟩nright.alt2/brack⟩tl⟩ft.alt21−nF(s1En′,ks′\nz)+nB(s2Eq)/brack⟩tright.alt2\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]\n−α\np0ℓ2∞\n/summation.disp\nn′=0/summation.disp\n{s}/integral.dispq⊥dq⊥In−1,n′−1\n2/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt41−nF(s1En′,ks′\nz)+nB(s2Eq)\ns1s2/radical.alt1\n[q2⊥−(q−⊥)2][q2⊥−(q+⊥)2]. (42)\nThe corresponding numerical results for the fermion damping rate as a function of the Landau-level index nand the\nlongitudinal momentum are shown in Fig. 4. Note that the values of the rate and the longitudinal momentu pzare\ngiven in units of the pion mass mπ=135 MeV. We use the same value of the QCD coupling as in Subsec. IIIB. Four\ndifferent panels display results for two different temperatures, i.e., T=200 MeV (left panels) and T=400 MeV (right\npanels), and two different magnetic fields, i.e., /divid⟩s.alt0qB/divid⟩s.alt0=(75 MeV)2(top panels) and/divid⟩s.alt0qB/divid⟩s.alt0=(200 MeV)2(bottom panels).\nBy comparingthe compilationof numericaldata in the four panelsof F ig.4, representingdifferent temperaturesand\nmagnetic fields, we see that both temperature and magnetic field ha ve a tendency to increase the damping rates. Such\nan enhancement is not surprising since both have the tendency to in crease the phase space for transitions to other\nLandau levels. In connection to the magnetic field, in particular, its p resence is critical to trigger the three processes\nresponsible for the damping rate at the leading order in coupling. In t he absence of the field, the only processes\ncontributing to the fermion damping rate are of the subleading orde r in coupling. The findings are further reinforced\nby the numerical data for intermediate values of temperature, T=300 MeV, and magnetic field, /divid⟩s.alt0qB/divid⟩s.alt0=(125 MeV)2,\nwhich are not shown in the figures but included in the Supplementary D ata files [ 64].\nA careful analysis shows that the enhancement factors, resultin g from increasing the temperatures and magnetic\nfield, are nonuniform functions of the Landau-level index nand longitudinal momentum pz. For example, the increase\nof temperature from T=200 MeV to T=400 MeV leads to enhancement factors of the order of 3 to 4 in the r egion of\nsmall values of nandpz. At largenorpz, on the other hand, the damping rates are nearly insensitive to the thermal\neffects.\nThe effect of the magnetic field is also nonuniform across the whole ra nge ofnandpzvalues. Quantitatively, the\nincrease of the magnetic field from /divid⟩s.alt0qB/divid⟩s.alt0=(75 MeV)2to/divid⟩s.alt0qB/divid⟩s.alt0=(200 MeV)2gives the largest enhancement factors of\nthe order of 6 to 7, which occurs at large values of nandpz. While, in absolute terms, the damping rates are the\nhighest at small values of pz, the increase due to the magnetic field is moderate (of the order of 2 or less). In fact,\nwhen both nandpzare small, we find that the rate can even decrease by a factor of ab out 2 or less. We should note,\nhowever, that this part of the parameter space must be treated with great caution because of a limited validity of the\none-loop approximation.10\nFIG. 4. The fermion damping rate as a function of the longitud inal momentum pzand the Landau-level index n. The damping\nrate is measured in units of the temperature. The results in t hree columns corresponding to two different temperatures:\nT=200 MeV (left panels) and T=400 MeV (right panels). The magnetic field is /divides.alt0qB/divides.alt0=(75 MeV)2(top panels) and\n/divides.alt0qB/divides.alt0=(200 MeV)2(bottom panels).\nBefore proceeding further, it is instructive to investigate the rat io of the damping rate and the real part of the\nfermion energy, Γ n(pz)/slash.l⟩ftEn,pz, presented in Fig. 5. The four panels correspondto the samechoicesoftwo temperat ures\nand two magnetic fields. In essence, this is the ratio of the imaginary and real parts of the fermion energy that shows\nwhether the quantum state(with given nandpz) is awell-defined quasiparticle. When the ratiovalueis comparableto\n1 or larger, the quasiparticle description is inapplicable. Indeed, this is the case when the particle’s lifetime τn=1/slash.l⟩ftΓn\nis comparable to or shorter than the time needed to measure its ene rgy ∆t≲1/slash.l⟩ftEn,pzaccording to the uncertainty\nprinciple. Alternatively, the uncertainty in particle’s energy Γ nis larger than the energy En,pzitself.\nAs we see from Fig. 5, the ratio Γ n(pz)/slash.l⟩ftEn,pzremains small almost in the whole range of nandpzvalues. However,\nthe damping rate becomes very large in the lowest few Landau levels ( n≲1) when the longitudinal momentum pz\nis sufficiently small ( pz≲mπ). Formally, these results indicate that the concept of well-defined quasiparticles breaks\ndown for the corresponding lowest Landau level states. We believe this might be a premature conclusion, however.\nIt seems more likely that the validity of the perturbative one-loop ca lculation breaks down in this case. Because of\nthe high degeneracy of the Landau levels, it is plausible that the one- loop calculation breaks down, especially in the\nregion of small fermion energies.\nAs in the absence of a magnetic field, hard thermal loop resummation s might be very important in the strongly\nmagnetized QCD plasma [ 60]. Additionally, somewhat similar hard magnetic loop resummation [ 28] may be needed\nwhen there is a strong magnetic field. Both are very likely to affect th e self-energy at small energies. Therefore, we\nreiterate that the large damping rates at small nandpzshould be accepted with great caution. Most likely, the\ncorresponding results are outside of the range of validity of the ap proximations used. Qualitatively, however, it is\nintriguing to think that the damping ratescan be indeed largein the low -lyingLandau levels. They could dramatically\naffect some observables in heavy-ion collisions, e.g., the electrical co nductivity of plasma [ 66] and the heavy-quark\nenergy loss and dissipation rate [ 67].\nB. Damping rates from the poles of the propagator\nIn the previous subsection, we used the definition of the damping ra te in terms of the imaginary part of the self-\nenergy by generalizing the general approach of Ref. [ 65] to the case of quantum field theory in a quantizing magnetic\nfield. Here we consider an alternative definition that follows from the structure of the full propagator, calculated in11\nFIG. 5. The fermion damping rate as a function of the longitud inal momentum pzand the Landau-level index n. The damping\nrate is measured in units of the fermion energy. The results i n three columns corresponding to three different temperatur es:\nT=200 MeV (left panels) and T=400 MeV (right panels). The magnetic field is /divides.alt0qB/divides.alt0=(75 MeV)2(top panels) and\n/divides.alt0qB/divides.alt0=(200 MeV)2(bottom panels).\nthe one-loop approximation.\nWhen the full propagatorisknown, the fermiondampingratecanbe alsodetermined fromthe locationofits polesin\nthe complex energy plane. At the leading order in coupling, the explicit structure of the fermion propagator is derived\nin Appendix A. As expected, the self-energy functions v∥,n,mn,v⊥,n, ˜vn, and ˜mnmodify the fermion propagator,\nsee Eqs. ( A9), (A15) and (A16). Most importantly for our purposes here, one can extract the q uasiparticles energies\nfrom the location of the poles in the propagator, see Eq. ( A17). Assuming that the self-energy corrections are small,\nthe approximate expressions for the (positive) energies can be wr itten as follows:\np(±)\n0≃/radical.alt2\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0+p2z⎛\n⎝1+¯m0δmn−(2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0)δv∥,n+2n/divid⟩s.alt0qB/divid⟩s.alt0δv⊥,n±/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0(¯m0˜vn−˜mn)\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0+p2z⎞\n⎠.(43)\nNotethattherearetwodifferentbranchesofsolutionsthatcorr espondtotwospinstates. Recallthatthecorresponding\ntwo states were degenerate in the free propagator. However, a lready at the leading order in coupling, the degeneracy\nis lifted by the self-energy corrections ˜ vnand ˜mn. Since we did not calculate explicitly the real parts of the self-energ y\nfunctionsv∥,n,mn,v⊥,n, ˜vn, and ˜mn, we cannot quantify the corresponding corrections to the real p arts of particle\nenergies.\nNevertheless, using the imaginary parts of self-energy functions , see Eqs. ( 29) through ( 33), we can determine\nleading-order corrections to the imaginary parts of particle energ ies, i.e., Im[δp(±)\n0,n]. Since the latter should coincide\nup an overall sign with the damping rate, we derive\nΓ(±)\nn≃(2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0)Im[δv∥,n]−¯m0Im[δmn]−2n/divid⟩s.alt0qB/divid⟩s.alt0Im[δv⊥,n]∓/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0(¯m0Im[˜vn]−Im[˜mn])/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0+¯m2\n0+p2z.(44)\nAs expected, this result demonstrates that the two spin-split Lan dau-level states have different damping rates. At\nthe same time, it is rewarding to see that the spin-averaged damping rate, Γ(ave)\nn≡(Γ(+)\nn+Γ(−)\nn)/slash.l⟩ft2, agrees perfectly\nwith the result obtained by a very different method in the previous su bsection, see Eq. ( 40).\nIt is natural to ask how large the spin splitting effects on the quasipa rticle damping rate are. As we see from\nEq. (44), they are determined by the self-energy functions Im [˜vn]and Im[˜mn]. The representative results for both,\nas functions of the Landau-level index n, are shown in Fig. 6. Each panel displays numerical data for three different\ntemperatures, i.e., T=200 MeV (blue lines), T=300 MeV (green lines), T=400 MeV (red lines), and two different12\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.00.20.40.60.81.0\npz=0\n○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.000.020.040.060.080.100.120.14\npz=0\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.000.050.100.15\npz=1000MeV\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■○qB=(75MeV)2\n■qB=(200MeV )2T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.000.020.040.060.080.100.120.14\npz=1000MeV\nFIG. 6. The dependence of the self-energy functions Im [˜vn]and Im[˜mn]/slash.left¯m0on the Landau-level index nfor two fixed values\nof the longitudinal momentum: pz=0 (top panels) and pz=1000 MeV (bottom panels). Each panel contains results for\nthree different temperatures: T=200 MeV (blue), T=300 MeV (green), and T=400 MeV (red); and two magnetic fields:\n/divides.alt0qB/divides.alt0=(75 MeV)2(open circles) and /divides.alt0qB/divides.alt0=(200 MeV)2(filled squares).\nmagnetic fields, i.e., /divid⟩s.alt0qB/divid⟩s.alt0=(75 MeV)2(open circles),/divid⟩s.alt0qB/divid⟩s.alt0=(200 MeV)2(filled squares). The top panels show the\nresults forpz=0, while the bottom panels show the results for pz=1000 MeV. Since the imaginary parts of ˜ vnand\n˜mnthemselves have no direct physical meaning, there is no need to disp lay more data here. However, an interested\nreader could find a large set of additional data for a wide range of pzvalues in the Supplementary Data files [ 64].\nLet us now turn to the spin-splitting effects on the damping rates. T wo sets of representative results are shown\nin Fig.7. We display the rates for three different temperatures but only on e (largest) value of the magnetic field,\n/divid⟩s.alt0qB/divid⟩s.alt0=(200 MeV)2. Even for this strong field, the effect of spin splitting is relatively sma ll. Quantitatively, the\ndifference between the spin-split rates does not exceed about 8% o f the average rate in the whole range of model\nparametersthat we explored. We verified that the same remainstr ue for the weakermagnetic fields as well. Therefore,\none can argue that, for most purposes, it is sufficient to use the sp in-averaged damping rate, Γ(ave)\nn≡(Γ(+)\nn+Γ(−)\nn)/slash.l⟩ft2,\nwhichwasinvestigatedin detailin theprevioussubsection. Thisargu mentcanbe furtherreinforcedbythe observation\nthat systematic uncertainties of the one-loop approximation used in the study are probably larger than the effects of\nspin splitting.\nIn conclusion of this section, let us emphasize that the spin splitting is a qualitatively new feature that can play an\nimportant role in strongly magnetized plasmas. While the differences b etween the damping rates for spin-split states\nin each Landau level remain quantitatively small, they may affect some spin physics phenomena, chiral magnetic or\nchiral separation effects. In this connection, it should be emphasiz ed that not only the imaginary parts of the Landau-\nlevel energies but also their real parts will be spin split. While we did not calculate the latter, such a conclusion is\nsupported by the general expression for the self-energy derive d.13\n✶\n✶\n✶\n✶\n✶\n✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶\n✶\n✶\n✶\n✶\n✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶\n✶\n✶\n✶\n✶\n✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶◆\n◆\n◆\n◆\n◆\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶\n◆T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.000.020.040.060.080.100.120.14\nqB=(200MeV )2, pz=0\n✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶✶◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆✶\n◆T=200MeV\nT=300MeV\nT=400MeV\n0 10 20 30 40 500.000.020.040.060.080.100.120.14\nqB=(200MeV )2\n \u000bz=1000MeV\nFIG. 7. The damping rates for the two spin states Γ(±)\nnas functions of Landau-level index nfor two fixed values of the\nlongitudinal momentum: pz=0 (left panel) and pz=1000 MeV (right panel). The magnetic field is /divides.alt0qB/divides.alt0=(200 MeV)2. Each\npanel contains results for three different temperatures: T=200 MeV (blue), T=300 MeV (green), and T=400 MeV (red).\nV. DISCUSSION AND SUMMARY\nIn this paper we derived a general expression for the fermion self- energy in a hot and strongly magnetized plasma\nby using the Landau-level representation. As we show, the leading -order one-loop expression for the self-energy is\ncharacterized by three velocity and two mass functions. The veloc ity functions include a pair of spin-split parallel\ncomponents and a perpendicular component of the velocity. The ot her two functions are the masses of the spin-split\npair of states in each Landau level. As we demonstrated, all of thes e five functions have a nontrivial dependence on\nthe Landau-level index nand the longitudinal momentum pz.\nHere we focused primarily on the imaginary (dissipative) part of the f ermion self-energy. We derived closed-form\nexpressionsforthe imaginarypartsofallfivefunctions thatdefin e the Diracstructureofthe self-energy. At the leading\norder in coupling, the contributions to the imaginary parts of the ve locity and mass functions in the nth Landau level\ncome from the following three types of on-shell processes: (i) tra nsitions to other Landau levels with lower indices n′\n(ψn→ψn′+γwithn>n′), (ii) transitions to other Landau levels with higher indices n′(ψn+γ→ψn′withn<n′),\nand (iii) transitions to Landau-level states with negative energies ( i.e., the annihilation process ψn+¯ψn′→γfor any\nnandn′).\nWe used the imaginary parts of the self-energy functions to derive the Landau-level dependent fermion damping\nrates Γ n(pz). We employed two different methods to get the corresponding resu lts. On one hand, we obtain the\ndamping rate by utilizing the general approach of Weldon [ 65]. To apply it to the case of hot plasma in a quantizing\nmagnetic field, first we had to modify the method to account for the correct set of quantum numbers characterizing\nthe Landau-level states. As expected, the final result is expres sed in terms of the imaginary parts of the spin-averaged\nvelocity and mass functions, see Eq. ( 40).\nThe second method for extracting the damping rates used the loca tion of the poles in the full propagator. This\napproachrevealed that the two-spin degeneracy of the Landau le vel states was lifted by radiative corrections. Further-\nmore, by using the imaginary parts of particle energies, we were able to extract the damping rates for the spin-split\nstates Γ(±)\nn(pz). It is important to note that the spin-averaged rate, Γ(ave)\nn≡(Γ(+)\nn+Γ(−)\nn)/slash.l⟩ft2, agrees perfectly with the\nresult obtained by Weldon’s method. Since the effect of spin splitting o n the rate is not large, one may argue that\nthe use of Weldon’s method might be sufficient in most applications.\nTodemonstratetheLandau-leveldependentdescriptionofthes elf-energyeffects, westudied numericallythefermion\ndamping rates in a wide range of model parameters, considering thr ee different temperatures and three different\nmagnetic fields. The choice of model parameters, with temperatur es between 200 MeV and 400 MeV and magnetic\nfields of the order of m2\nπ, were motivated by potential applications in heavy-ion physics. The main results are\nsummarized in Figs. 4and5. In absolute terms, the largest values of the rates are found for the low-lying Landau\nlevels and small values of the longitudinal momentum. In fact, in some cases (at small nandpz), the damping rates\nappear to be formally much larger than the real parts of the partic le energies. This suggests that the quasiparticle\npicture may fail for such quantum states. These extreme cases s hould be treated with great caution, however, since\nthe one-loop approximation may become particularly bad in those reg ions of the parameter space.\nGenerally, we find that the rates have an overall tendency to grow with increasing both temperature and magnetic\nfield. However, the enhancement is nonuniform in the range of Land au-level indices nand longitudinal momenta pz14\nexplored. The thermal effects are pronounced the most in the reg ion of small values of nandpz. The magnetic field\nenhancement, in contrast, also extends to large values of pz. The latter may not be as surprising after one recalls that\nthe magnetic field is essential for allowing the leading-order, one-ph oton processes (i.e., ψn→ψn′+γ,ψn+γ→ψn′,\nandψn+¯ψn′→γ) to occur in the first place.\nWe hope that the results for the fermion damping rates, as well as t he general method for calculating the self-energy\nin the Landau-level representation, can be useful in a wide range o f studies of strongly magnetized relativistic plasma.\nThey can be useful in the calculation of transport properties such as the electrical conductivity [ 66] and the particle\nloss or dissipation rate [ 67]. In addition to heavy-ion physics, our self-energy results can be u seful in studies of QED\nplasmas in astrophysics and cosmology.\nWhilethisstudyprovidesaclearproofofconceptforutilizingtheLan dau-levelrepresentationtodescribeself-energy\neffectsinstronglymagnetizedrelativisticplasmas,therearemanyt heoreticalissuesleftoutstanding. Themostobvious\nof them is the calculation of the real part of the self-energy. Unlike the imaginary part, the expression for the real\npart contains ultraviolet divergences. Therefore, its evaluation r equires a careful renormalization procedure, which is\ncomplicated by the Landau-level structure of the self-energy. D espite these difficulties, we believe, the problem can\nbe solved by using the general expression for the self-energy der ived here as the starting point. We plan to consider\nthis problem in the follow-up studies.\nThe analytical expression for the damping rate in Eq. ( 42) is remarkable in many ways. Surprisingly, however,\nit is not a positive definite quantity. This is puzzling since such a rate sh ould be positive for any choice of model\nparameters. If taken at face value, a negative value of the rate w ould indicate an unknown type of plasma instability\nthat could develop for some choices of model parameters. We have no resolution to this puzzle but plan to investigate\nit in detail in the future.\nACKNOWLEDGMENTS\nThis research was funded in part by the U.S. National Science Found ation under Grant No. PHY-2209470.\nAppendix A: Fermion propagator in the Landau-level represe ntation\nIn this Appendix, we derive an explicit form of the fermion propagato r in a magnetic field in the Landau-level\nrepresentation by using the method developed in Ref. [ 28]. By definition, the corresponding propagator in coordinate\nspace is given by the following matrix element:\nG(u,u′)=i⟨u/divid⟩s.alt0/brack⟩tl⟩ft.alt1(i∂tγ0−π3γ3)−(π⊥⋅γ⊥)−¯m0−Σ/brack⟩tright.alt−1/divid⟩s.alt0u′⟩\n=i⟨u/divid⟩s.alt0/brack⟩tl⟩ft.alt1v∥(i∂tγ0−π3γ3)−v⊥(π⊥⋅γ⊥)+iγ1γ2˜v(i∂tγ0−π3γ3)−m−iγ1γ2˜m/brack⟩tright.alt−1/divid⟩s.alt0u′⟩, (A1)\nwhereπ⊥=−i(∇−iqA)and the vector potential in the Landau gauge is used, i.e., A=(0,Bx,0). Here we took into\naccount all possible Dirac structures of the full propagator at th e leading order in coupling. In particular, functions\nm,v∥andv⊥include radiative corrections to the mass, the parallel and perpend icular components of the velocity,\nrespectively. The two additional functions ˜ vand ˜mcapture the effects of spin splitting corrections to the parallel\nvelocity and the mass. A self-consistency check shows that there is no spin splitting correction to v⊥. Note that,\nstrictly speaking, all five are operator-valued functions. When ac ting on the Landau-level orbitals (see below), they\nwill become functions of the Landau-level index nand the longitudinal momentum p∥. For example, mnwill be the\nmass function in the nth Landau level. (Note that we use notation ¯ m0for the tree-level mass to distinguish it from\nthe mass in the lowest Landau level m0.)\nConsidering that translation symmetry remains intact in the time and thezdirection, it is convenient to switch\nto the corresponding momentum subspace represented by the lon gitudinal momentum p∥=(p0,pz). The resulting\npropagator in a mixed representation reads\nG(p∥,u⊥,u′\n⊥)=⟨u⊥/divid⟩s.alt0/brack⟩tl⟩ft.alt1v∥(p∥⋅γ∥)−v⊥(π⊥⋅γ⊥)−iγ1γ2˜v(p∥⋅γ∥)+m−iγ1γ2˜m/brack⟩tright.alt\n×/brack⟩tl⟩ft.alt2/par⟩nl⟩ft.alt1v2\n∥−˜v2/par⟩nright.alt1p2\n∥−v2\n⊥π2−m2+˜m2+2iγ1γ2(m˜v−˜mv∥)(p∥⋅γ∥)−iγ1γ2v2\n⊥qB/brack⟩tright.alt2−1\n/divid⟩s.alt0u′\n⊥⟩.(A2)\nHere we took into account that −(π⊥⋅γ⊥)2=π2\n⊥−qBiγ1γ2. The eigenvalues of the operator π2\n⊥are(2n+1)/divid⟩s.alt0qB/divid⟩s.alt0, where\nn=0,1,2,..., and the corresponding normalized eigenfunctions are given by the Landau orbitals, i.e.,\nψnp(u⊥)≡⟨u⊥/divid⟩s.alt0np⟩=1√\n2πℓ1/radical.alt1\n2nn!√πHn/par⟩nl⟩ft.alt3x\nℓ+pℓ/par⟩nright.alt3e−1\n2ℓ2(x+pℓ2)2e−is⊥py, (A3)15\nwheres⊥=sign(qB),ℓ=1/slash.l⟩ft/radical.alt1\n/divid⟩s.alt0qB/divid⟩s.alt0is the magnetic length, and Hn(x)are the Hermite polynomials. It is useful to note\nthat\nπxψn,p(u⊥)=−i∂xψn,p(u⊥)=i\n2ℓ/par⟩nl⟩ft.alt2/radical.alt1\n2(n+1)ψn+1,p(u⊥)−√\n2nψn−1,p(u⊥)/par⟩nright.alt2, (A4)\nπyψn,p(u⊥)=(−i∂y−qBx)ψn,p(u⊥)=−s⊥\n2ℓ/par⟩nl⟩ft.alt2/radical.alt1\n2(n+1)ψn+1,p(u⊥)+√\n2nψn−1,p(u⊥)/par⟩nright.alt2, (A5)\n(π⊥⋅γ⊥)ψn,p(u⊥)=i\n2ℓ/radical.alt1\n2(n+1)ψn+1,p(u⊥)/par⟩nl⟩ft.alt1γ1+is⊥γ2/par⟩nright.alt1−i\n2ℓ√\n2nψn−1,p(u⊥)/par⟩nl⟩ft.alt1γ1−is⊥γ2/par⟩nright.alt1, (A6)\nwhere we took into account that H′\nn(x)=2nHn−1(x)andHn+1(x)=2xHn(x)−2nHn−1(x).\nThese wave functions satisfy the condition of completeness\n∞\n/summation.disp\nn=0∞\n/integral.disp\n−∞dpψnp(u⊥)ψ∗\nnp(u′\n⊥)=δ2(u⊥−u′\n⊥), (A7)\nwhich can be written in a compact form as ∑n,p⟨u⊥/divid⟩s.alt0np⟩⟨pn/divid⟩s.alt0u′⟩=⟨u⊥/divid⟩s.alt0u′⟩.\nBy inserting the unit operator ∑n,p/divid⟩s.alt0np⟩⟨pn/divid⟩s.alt0in front of/divid⟩s.alt0u′\n⊥⟩on the right hand of Eq. ( A2) and making use of the\nproperties in Eqs. ( A4) – (A6), we derive the propagator in the following form:\nG(p∥,u⊥,u′\n⊥)=eiΦ(u⊥,u′\n⊥)¯G(p∥,u⊥−u′\n⊥), (A8)\nwhere Φ(u⊥,u′\n⊥)=qB\n2(x+x′)(y−y′)is the Schwinger phase, and\n¯G(p∥,u⊥)=ie−u2\n⊥/slash.left(4ℓ2)\n2πℓ2∞\n/summation.disp\nn=0⎧⎪⎪⎨⎪⎪⎩/brack⟩tl⟩ft.alt1v∥,n(p∥⋅γ∥)−iγ1γ2˜vn(p∥⋅γ∥)+mn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt4Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4P++Ln−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4P−/brack⟩tright.alt4\n−iv⊥,n\nℓ2(u⊥⋅γ⊥)L1\nn−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4⎫⎪⎪⎬⎪⎪⎭1\nMn−2nv2⊥,n/divid⟩s.alt0qB/divid⟩s.alt0, (A9)\nwhereLα\nn(z)are the Laguerre polynomials (by definition, Lα\n−1(z)≡0),P±=/par⟩nl⟩ft.alt11±s⊥iγ1γ2/par⟩nright.alt1/slash.l⟩ft2 are spin projectors, and\nMn=/par⟩nl⟩ft.alt1v2\n∥,n−˜v2\nn/par⟩nright.alt1p2\n∥−m2\nn+˜m2\nn+2iγ1γ2(mn˜vn−˜mnv∥,n)(p∥⋅γ∥). (A10)\nIn derivation, we used the following relations:\n/integral.disp∞\n−∞dpψnp(u⊥)ψ∗\nnp(u′\n⊥)=e−ζ/slash.left2+iΦ(u⊥,u′\n⊥)\n2πℓ2Ln/par⟩nl⟩ft.alt4(u⊥−u′\n⊥)2\n2ℓ2/par⟩nright.alt4, (A11)\n/integral.disp∞\n−∞dpψn+1,p(u⊥)ψ∗\nnp(u′\n⊥)=e−ζ/slash.left2+iΦ(u⊥,u′\n⊥)\n2πℓ2/radical.alt1\n2(n+1)x−x′−is⊥(y−y′)\nℓL1\nn/par⟩nl⟩ft.alt4(u⊥−u′\n⊥)2\n2ℓ2/par⟩nright.alt4, (A12)\n/integral.disp∞\n−∞dpψn−1,p(u⊥)ψ∗\nnp(u′\n⊥)=e−ζ/slash.left2+iΦ(u⊥,u′\n⊥)\n2πℓ2√\n2nx′−x−is⊥(y−y′)\nℓL1\nn−1/par⟩nl⟩ft.alt4(u⊥−u′\n⊥)2\n2ℓ2/par⟩nright.alt4, (A13)\nwhereζ=(u⊥−u′\n⊥)2/slash.l⟩ft(2ℓ2). To obtain Eq. ( A13), we used the following table integral 7.377 [ 63]:\n/integral.disp∞\n−∞e−x2Hm(x+y)Hn(x+z)dx=2n√πm!zn−mLn−m\nm(−2yz), (A14)\nwhich assumes m≤n.\nNote that the last factor in Eq. ( A9) is a matrix. It can be rendered in the following more convenient form :\n1\nMn−2nv2⊥,n/divid⟩s.alt0qB/divid⟩s.alt0=1\nUn/brack⟩tl⟩ft.alt1/par⟩nl⟩ft.alt1v2\n∥,n−˜v2\nn/par⟩nright.alt1p2\n∥−2nv2\n⊥,n/divid⟩s.alt0qB/divid⟩s.alt0−m2\nn+˜m2\nn−2iγ1γ2(mn˜vn−˜mnv∥,n)(p∥⋅γ∥)/brack⟩tright.alt,(A15)\nwhere\nUn=/brack⟩tl⟩ft.alt1/par⟩nl⟩ft.alt1v2\n∥,n−˜v2\nn/par⟩nright.alt1p2\n∥−2nv2\n⊥,n/divid⟩s.alt0qB/divid⟩s.alt0−m2\nn+˜m2\nn/brack⟩tright.alt2−4p2\n∥/par⟩nl⟩ft.alt1mn˜vn−˜mnv∥,n/par⟩nright.alt12. (A16)16\nThe poles of the full propagator are determined by setting Un=0. Its solutions determine the modified energies of\nthe Landau-level states, i.e.,\np2\n0=p2\nz+/par⟩nl⟩ft.alt2v2\n∥,n−˜v2\nn/par⟩nright.alt2/par⟩nl⟩ft.alt12nv2\n⊥,n/divid⟩s.alt0qB/divid⟩s.alt0+m2\nn−˜m2\nn/par⟩nright.alt1+2/par⟩nl⟩ft.alt1mn˜vn−˜mnv∥,n/par⟩nright.alt12±2/par⟩nl⟩ft.alt1mn˜vn−˜mnv∥,n/par⟩nright.alt1√Vn\n(v2\n∥,n−˜v2n)2,(A17)\nwhere\nVn=/par⟩nl⟩ft.alt1v2\n∥,n−˜v2\nn/par⟩nright.alt1/par⟩nl⟩ft.alt12nv2\n⊥,n/divid⟩s.alt0qB/divid⟩s.alt0+m2\nn−˜m2\nn/par⟩nright.alt1+/par⟩nl⟩ft.alt1mn˜vn−˜mnv∥,n/par⟩nright.alt12. (A18)\n1. Fourier transform of the translation invariant part of th e propagator\nBy performing the Fourier transform of the translation invariant p art of the propagator in Eq. ( A9), we derive\n¯G(p∥,p⊥)=ie−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)nDn(p∥,p⊥)1\nMn−2nv2⊥,n/divid⟩s.alt0qB/divid⟩s.alt0, (A19)\nwhere thenth Landau level contribution is determined by\nDn(p∥,p⊥)=2/brack⟩tl⟩ft.alt1v∥,n(p∥⋅γ∥)−iγ1γ2˜vn(p∥⋅γ∥)+mn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt1P+Ln/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1−P−Ln−1/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1/brack⟩tright.alt\n+4v⊥,n(p⊥⋅γ⊥)L1\nn−1/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1. (A20)\nIn derivation, we used the following table integrals:\n/integral.disp2π\n0e−i(k⊥⋅u⊥)dφ=2πJ0(k⊥u⊥), (A21)\n/integral.disp2π\n0(γ⊥⋅ˆu⊥)e−i(k⋅u⊥)dφ=2iπ(γ⊥⋅ˆk⊥)J1(k⊥u⊥), (A22)\n/integral.disp∞\n0rν+1e−βr2Lν\nn/par⟩nl⟩ft.alt1αr2/par⟩nright.alt1Jν(rk)dr=kν\n(2β)1+ν/par⟩nl⟩ft.alt3β−α\nβ/par⟩nright.alt3n\ne−k2\n4βLν\nn/par⟩nl⟩ft.alt4αk2\n4β(α−β)/par⟩nright.alt4. (A23)\n2. Free fermion propagator\nThe free fermion propagator is obtained from the full propagator by replacing v∥,n,v⊥,n→1,mn→¯m0, and setting\nzero values to spin-splitting functions ˜ vnand ˜mn. Then, the Fourier transform of the translation invariant part of the\nfree propagator takes the form:\n¯S(p∥,p⊥)=ie−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)nD(0)\nn(p∥,p⊥)\np2\n∥−¯m2\n0−2n/divid⟩s.alt0qB/divid⟩s.alt0, (A24)\nwhere\nD(0)\nn(p∥,p⊥)=2/brack⟩tl⟩ft.alt1(p∥⋅γ∥)+¯m0/brack⟩tright.alt/brack⟩tl⟩ft.alt1P+Ln/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1−P−Ln−1/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1/brack⟩tright.alt+4(p⊥⋅γ⊥)L1\nn−1/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1.(A25)\nAppendix B: Inverse fermion propagator in the Landau-level representation\nBy definition, the inverse of the full propagator is given by the follow ing matrix element:\nG−1(p∥,u⊥,u′\n⊥)=−i⟨u⊥/divid⟩s.alt0/brack⟩tl⟩ft.alt1v∥(p∥⋅γ∥)−v⊥(π⊥⋅γ⊥)+iγ1γ2(p∥⋅γ∥)˜v−m−iγ1γ2˜m/brack⟩tright.alt/divid⟩s.alt0u′\n⊥⟩. (B1)\nAs in the derivation of the propagator in Appendix A, we insert the unit operator ∑n,p/divid⟩s.alt0np⟩⟨pn/divid⟩s.alt0in front of/divid⟩s.alt0u′\n⊥⟩to\nderive the following representation for the inverse propagator:\nG−1(p∥,u⊥,u′\n⊥)=eiΦ(u⊥,u′\n⊥)¯G−1(p∥,u⊥−u′\n⊥), (B2)17\nwhere the translation invariant part of the propagator is given by a sum over Landau levels\n¯G−1(p∥,u⊥)=−ie−u2\n⊥/slash.left(4ℓ2)\n2πℓ2∞\n/summation.disp\nn=0⎧⎪⎪⎨⎪⎪⎩/brack⟩tl⟩ft.alt1v∥,n(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)˜vn−mn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt4P+Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4+P−Ln−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4/brack⟩tright.alt4\n+1\nℓ2v⊥,n(u⊥⋅γ⊥)L1\nn−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4⎫⎪⎪⎬⎪⎪⎭. (B3)\nRecall that, by definition, Lα\n−1≡0.\nThe corresponding Fourier transform reads\n¯G−1(p∥,p⊥)=−2ie−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)n/brack⟩tl⟩ft.alt1v∥,n(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)˜vn−mn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt1P+Ln(2p2\n⊥ℓ2)−P−Ln−1(2p2\n⊥ℓ2)/brack⟩tright.alt\n−4ie−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)nv⊥,n(γ⊥⋅p⊥)L1\nn−1(2p2\n⊥ℓ2). (B4)\n1. Self-energy in the Landau-level representation\nBy making use of the inverse full and free propagators, we derive\n¯Σ(p∥,u⊥)=i¯S−1(p∥,u⊥)−i¯G−1(p∥,u⊥). (B5)\nBy using the Landau-level representation for the inverse propag ator, we obtain\n¯Σ(p∥,u⊥)=−e−u2\n⊥/slash.left(4ℓ2)\n2πℓ2∞\n/summation.disp\nn=0⎧⎪⎪⎨⎪⎪⎩/brack⟩tl⟩ft.alt1δv∥,n(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)˜vn−δmn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt4P+Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4+P−Ln−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4/brack⟩tright.alt4\n+δv⊥,n\nℓ2(u⊥⋅γ⊥)L1\nn−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4⎫⎪⎪⎬⎪⎪⎭. (B6)\nwhereδv∥,n=v∥,n−1,δv⊥,n=v⊥,n−1, andδmn=mn−¯m0. The corresponding Fourier transform reads\n¯Σ(p∥,p⊥)=−2e−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)n/brack⟩tl⟩ft.alt1δv∥,n(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)˜vn−δmn−iγ1γ2˜mn/brack⟩tright.alt/brack⟩tl⟩ft.alt1P+Ln(2p2\n⊥ℓ2)−P−Ln−1(2p2\n⊥ℓ2)/brack⟩tright.alt\n−4e−p2\n⊥ℓ2∞\n/summation.disp\nn=0(−1)nδv⊥,n(γ⊥⋅p⊥)L1\nn−1(2p2\n⊥ℓ2). (B7)\nAppendix C: Calculation of the kernels\nIn the derivation of the Landau-level representationfor the five component functions of the self-energy, see Eqs. ( 24)\nthrough ( 28), one encounters the two different types of kernel functions de fined by the following expressions:\nKn,n′=(−1)n+n′2ℓ2\nπ/integral.dispd2k⊥e−k2\n⊥ℓ2e−(k⊥−q⊥)2ℓ2Ln′/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1Ln/par⟩nl⟩ft.alt12(k⊥−q⊥)2ℓ2/par⟩nright.alt1, (C1)\n¯Kn,n′=(−1)n+n′8ℓ4\nπ/integral.dispd2k⊥e−k2\n⊥ℓ2e−(k⊥−q⊥)2ℓ2(k⊥⋅(k⊥−q⊥))L1\nn′−1/par⟩nl⟩ft.alt12k2\n⊥ℓ2/par⟩nright.alt1L1\nn−1/par⟩nl⟩ft.alt12(k⊥−q⊥)2ℓ2/par⟩nright.alt1.(C2)\nTo calculate the first kernel, it is convenient to start by noting the f ollowing Fourier transform:\n1\n4πℓ2/integral.dispd2u⊥e−u2\n⊥/slash.left(4ℓ2)Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−ip⊥⋅u⊥=1\n4πℓ2/integral.disp∞\n0u⊥du⊥e−u2\n⊥/slash.left(4ℓ2)Ln/par⟩nl⟩ft.alt4u2\n⊥\n4πℓ2/par⟩nright.alt4/integral.disp2π\n0dφe−ip⊥u⊥cosφ\n=1\n2/integral.disp∞\n0¯rd¯re−¯r2/slash.left4Ln/par⟩nl⟩ft.alt4¯r2\n2/par⟩nright.alt4J0(p⊥ℓ¯r)=(−1)ne−p2\n⊥ℓ2Ln/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1,(C3)18\nwhereweintroducedthefollowingdimensionlessvariable ¯ r=u⊥/slash.l⟩ftℓandusedtableintegral7.4191in Ref.[ 63]. Similarly,\nin the calculation of the second kernel, it is useful to utilize another F ourier transform\ni\n8πℓ4/integral.dispd2u⊥(u⊥⋅a)e−u2\n⊥/slash.left(4ℓ2)L1\nn/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−ip⊥⋅u⊥=(ˆp⊥⋅a)\n4ℓ/integral.disp∞\n0¯r2d¯re−¯r2/slash.left4L1\nn/par⟩nl⟩ft.alt4¯r2\n2/par⟩nright.alt4J1(p⊥ℓ¯r)\n=(−1)n(p⊥⋅a)e−p2\n⊥ℓ2L1\nn/par⟩nl⟩ft.alt12p2\n⊥ℓ2/par⟩nright.alt1, (C4)\nwhereais an arbitrary transverse 2D vector. In the derivation, we used t able integral 7.419 4 in Ref. [ 63].\nBy making use of the first result, we derive\nKn,n′=/integral.dispd2k⊥\n8π3ℓ2/integral.dispd2u⊥e−u2\n⊥/slash.left(4ℓ2)Ln′/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−ik⊥⋅u⊥/integral.dispd2u′\n⊥e−(u′\n⊥)2/slash.left(4ℓ2)Ln/par⟩nl⟩ft.alt4(u′\n⊥)2\n2ℓ2/par⟩nright.alt4e−i(k⊥−q⊥)⋅u′\n⊥\n=/integral.dispd2u⊥\n2πℓ2e−u2\n⊥/slash.left(2ℓ2)Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4Ln′/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−iq⊥⋅u⊥\n=1\nℓ2/integral.disp∞\n0u⊥du⊥e−u2\n⊥/slash.left(2ℓ2)Ln/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4Ln′/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4J0(q⊥u⊥)=In,n′\n0/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4. (C5)\nBy making use of the second result, we derive\n¯Kn,n′=−/integral.dispd2k⊥\n8π3ℓ4/integral.dispd2u⊥e−u2\n⊥/slash.left(4ℓ2)L1\nn′−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−ik⊥⋅u⊥/integral.dispd2u′\n⊥(u⊥⋅u′\n⊥)e−(u′\n⊥)2/slash.left(4ℓ2)L1\nn−1/par⟩nl⟩ft.alt4(u′\n⊥)2\n2ℓ2/par⟩nright.alt4e−i(k⊥−q⊥)⋅u′\n⊥\n=/integral.dispd2u⊥\n2πℓ4u2\n⊥e−u2\n⊥/slash.left(2ℓ2)L1\nn−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4L1\nn′−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4e−iq⊥⋅u⊥\n=1\nℓ4/integral.disp∞\n0u3\n⊥du⊥e−u2\n⊥/slash.left(2ℓ2)L1\nn−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4L1\nn′−1/par⟩nl⟩ft.alt4u2\n⊥\n2ℓ2/par⟩nright.alt4J0(q⊥u⊥)=In−1,n′−1\n2/par⟩nl⟩ft.alt4q2\n⊥ℓ2\n2/par⟩nright.alt4. (C6)\nwhereIn,n′\n0(ξ)andIn,n′\n2(ξ)are the same function that were introduced in Ref. [ 44], i.e.,\nIn,n′\n0(ξ)=(n′)!\nn!e−ξξn−n′/par⟩nl⟩ft.alt2Ln−n′\nn′(ξ)/par⟩nright.alt22\n=n!\n(n′)!e−ξξn′−n/par⟩nl⟩ft.alt2Ln′−n\nn(ξ)/par⟩nright.alt22\n, (C7)\nIn,n′\n2(ξ)=2(n′+1)!\nn!e−ξξn−n′Ln−n′\nn′(ξ)Ln−n′\nn′+1(ξ)=2(n+1)!\n(n′)!e−ξξn′−nLn′−n\nn(ξ)Ln′−n\nn+1(ξ). (C8)\nNote thatIn,n′\n2(ξ)can be also expressed in terms of In,n′\n0(ξ)[44], i.e.,\nIn,n′\n2(ξ)=n+n′+2\n2/brack⟩tl⟩ft.alt2In,n′\n0(ξ)+In+1,n′+1\n0(ξ)/brack⟩tright.alt2−ξ\n2/brack⟩tl⟩ft.alt2In+1,n′\n0(ξ)+In,n′+1\n0(ξ)/brack⟩tright.alt2. (C9)\nAppendix D: Wave functions for fermions in a magnetic field\nLet us consider the spinor wave function in a given Landau (labeled by indexn) level with a positive energy:\nΨn,p(u)=e−ip∥⋅u∥[ψn,p(u⊥)P++iψn−1,p(u⊥)P−]v. (D1)\nSubstituting it into the Dirac equation gives\n(iγµDµ−¯m0)Ψn,p(u)=/brack⟩tl⟩ft.alt1/par⟩nl⟩ft.alt1p∥⋅γ∥/par⟩nright.alt1−(π⊥⋅γ⊥)−¯m0/brack⟩tright.altΨn,p(u)\n=e−ip∥⋅u∥[ψn,p(u⊥)P++iψn−1,p(u⊥)P−]/brack⟩tl⟩ft.alt2/par⟩nl⟩ft.alt1p∥⋅γ∥/par⟩nright.alt1+γ1/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0−¯m0/brack⟩tright.alt2v, (D2)\nwhere we used the property in Eq. ( A6). The spinor vmust satisfy the equation:\n/brack⟩tl⟩ft.alt2/par⟩nl⟩ft.alt1p∥⋅γ∥/par⟩nright.alt1+γ1/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0−¯m0/brack⟩tright.alt2v=0. (D3)\nIt has nonzero solution when\nDet/brack⟩tl⟩ft.alt2/par⟩nl⟩ft.alt1p∥⋅γ∥/par⟩nright.alt1+γ1/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0−¯m0/brack⟩tright.alt2=/brack⟩tl⟩ft.alt1p2\n∥−2n/divid⟩s.alt0qB/divid⟩s.alt0−¯m2\n0/brack⟩tright.alt2=0. (D4)19\nUsing the following representation of Dirac matrices:\nγ0=/par⟩nl⟩ft.alt3I20\n0−I2/par⟩nright.alt3, γi=/par⟩nl⟩ft.alt30σi\n−σi0/par⟩nright.alt3, (D5)\nwhereσiare the Pauli matrices, we derive explicit solutions for spinor v, i.e.,\nv=/radical.alt1\n¯m0+En,pz⎛\n⎜⎜⎜⎜⎜\n⎝a1\na2\npza1−/radical.alt1\n2n/divides.alt0qB/divides.alt0a2\n¯m0+En,pz\n−/radical.alt1\n2n/divides.alt0qB/divides.alt0a1+pza2\n¯m0+En,pz⎞\n⎟⎟⎟⎟⎟\n⎠. (D6)\nThese spinor are similar but different from those in Ref. [ 68]. However, here we use a slightly different ansatz for the\nwave function ( D1) and a different normalization convension for the spinors, i.e.,\n¯vv=2¯m0(a2\n1+a2\n2). (D7)\nTherefore, the final spinor wave function reads\nΨn,p(u)=e−ip∥⋅u∥⎛\n⎜⎜⎜⎜⎜⎜\n⎝/radical.alt1\n¯m0+En,pzψn,p(u⊥)a1\ni/radical.alt1\n¯m0+En,pzψn−1,p(u⊥)a2\npza1−/radical.alt1\n2n/divides.alt0qB/divides.alt0a2/radical.alt1\n¯m0+En,pzψn,p(u⊥)\n−i/radical.alt1\n2n/divides.alt0qB/divides.alt0a1+pza2/radical.alt1\n¯m0+En,pzψn−1,p(u⊥)⎞\n⎟⎟⎟⎟⎟⎟\n⎠, (D8)\nwhere we set s⊥=1 for simplicity. (Note that ψn,p(u⊥)andiψn−1,p(u⊥)switch places when s⊥=−1.) Two independent\nstates Ψ n,p,s(u)are obtained by setting either (i) a1=1,a2=0 or (ii)a1=0,a2=1. Then, we check that the sum\nover both spin states gives\n/summation.disp\nsΨn,p,s(u)¯Ψn,p,s(u′)=e−ip∥⋅(u∥−u′\n∥)\n2/brack⟩tl⟩ft.alt2/par⟩nl⟩ft.alt1ψn,p(u⊥)ψ∗\nn,p(u′\n⊥)+ψn−1,p(u⊥)ψ∗\nn−1,p(u′\n⊥)/par⟩nright.alt1/par⟩nl⟩ft.alt1En,pzγ0−pzγ3+¯m0/par⟩nright.alt1\n+iγ1γ2/par⟩nl⟩ft.alt1ψn,p(u⊥)ψ∗\nn,p(u′\n⊥)−ψn−1,p(u⊥)ψ∗\nn−1,p(u′\n⊥)/par⟩nright.alt1/par⟩nl⟩ft.alt1En,pzγ0−pzγ3+¯m0/par⟩nright.alt1\n+/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0(iγ1+γ2)ψn−1,p(u⊥)ψ∗\nn,p(u′\n⊥)+/radical.alt1\n2n/divid⟩s.alt0qB/divid⟩s.alt0(−iγ1+γ2)ψn,p(u⊥)ψ∗\nn−1,p(u′\n⊥)/brack⟩tright.alt2.(D9)\nFinally, when also integrated over the quantum number p, we obtain\n/integral.dispdp/summation.disp\nsΨn,p,s(u)¯Ψn,p,s(u′)=e−ip∥⋅(u∥−u′\n∥)e−(u⊥−u′\n⊥)2/slash.left(2ℓ2)+iΦ(u⊥,u′\n⊥)\n2πℓ2\n×/brack⟩tl⟩ft.alt4/par⟩nl⟩ft.alt1En,pzγ0−pzγ3+¯m0/par⟩nright.alt1/brack⟩tl⟩ft.alt1P+Ln(ζ)+P−Ln−1(ζ)/brack⟩tright.alt+(u⊥⋅γ⊥)\nℓ2L1\nn−1(ζ)/brack⟩tright.alt4,(D10)\nwhere we used the shorthand notation ζ=(u⊥−u′\n⊥)2/slash.l⟩ft(2ℓ2). By making use of this result, the expression for the\nfermion damping rate ( 39) becomes\nΓn(pz)=1\n2p0/integral.dispd2u⊥e−ζ\n2πℓ2Tr⎧⎪⎪⎨⎪⎪⎩/brack⟩tl⟩ft.alt4/brack⟩tl⟩ft.alt1(p∥⋅γ∥)+¯m0/brack⟩tright.alt[P+Ln(ζ)+P−Ln−1(ζ)]+(u⊥⋅γ⊥)\nℓ2L1\nn−1(ζ)/brack⟩tright.alt4\n×∞\n/summation.disp\nn′=0/brack⟩tl⟩ft.alt2/brack⟩tl⟩ft.alt1Imδv∥,n′(p∥⋅γ∥)+iγ1γ2(p∥⋅γ∥)Im˜vn′−Imδmn′−iγ1γ2Im˜mn′/brack⟩tright.alt[P+Ln′(ζ)+P−Ln′−1(ζ)]\n+Imδv⊥,n′\nℓ2(u⊥⋅γ⊥)L1\nn′−1(ζ)/brack⟩tright.alt2⎫⎪⎪⎬⎪⎪⎭. (D11)\nAfter integrating over ζ, this reduces to the final expression for the spin-averaged damp ing rate, which is given in\nEq. (40) in the main text.\n[1] T. Vachaspati, Magnetic fields from cosmological phase t ransitions, Phys. Lett. B 265, 258 (1991) .20\n[2] L. M. Widrow, Origin of galactic and extragalactic magne tic fields, Rev. Mod. Phys. 74, 775 (2002) ,\narXiv:astro-ph/0207240 .\n[3] G. G. Raffelt, Stars as Laboratories for Fundamental Physics: The Astroph ysics of Neutrinos, Axions, and Other Weakly\nInteracting Particles (University of Chicago Press, Chicago, IL, 1996).\n[4] A. K. Harding and D. Lai, Physics of Strongly Magnetized N eutron Stars, Rept. Prog. Phys. 69, 2631 (2006) ,\narXiv:astro-ph/0606674 .\n[5] J. Liao, Anomalous transport effects and possible enviro nmental symmetry ‘violation’ in heavy-ion collisions,\nPramana 84, 901 (2015) ,arXiv:1401.2500 [hep-ph] .\n[6] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chira l magnetic and vortical effects in high-energy nuclear colli -\nsions—A status report, Prog. Part. Nucl. Phys. 88, 1 (2016) ,arXiv:1511.04050 [hep-ph] .\n[7] X.-G. Huang, Vorticity and Spin Polarization — A Theoret ical Perspective, Nucl. Phys. A 1005, 121752 (2021) ,\narXiv:2002.07549 [nucl-th] .\n[8] I. A. Shovkovy, Anomalous plasma: chiral magnetic effect and all that, in Peter Suranyi 87th Birthday Festschrift , edited\nby P. Argyres, G. Dunne, G. Semenoff, and R. Wijewardhana (Wor ld Scientific, Singapore, 2022) Chap. 18, pp. 291–316,\narXiv:2111.11416 [nucl-th] .\n[9] V. M. Kaspi and A. Beloborodov, Magnetars, Ann. Rev. Astron. Astrophys. 55, 261 (2017) ,\narXiv:1703.00068 [astro-ph.HE] .\n[10] S. J. Hardy and M. H. Thoma, Neutrino electron processes in a strongly magnetized thermal plasma,\nPhys. Rev. D 63, 025014 (2001) ,arXiv:astro-ph/0008473 .\n[11] J. Granot, T. Piran, O. Bromberg, J. L. Racusin, and F. Da igne, Gamma-Ray Bursts as Sources of Strong Magnetic Fields ,\nSpace Sci. Rev. 191, 471 (2015) ,arXiv:1507.08671 [astro-ph.HE] .\n[12] V. Skokov, A. Y. Illarionov, and V. Toneev, Estimate of t he magnetic field strength in heavy-ion collisions,\nInt. J. Mod. Phys. A24, 5925 (2009) ,arXiv:0907.1396 .\n[13] V. Voronyuk, V. Toneev, W. Cassing, E. Bratkovskaya, V. Konchakovski, et al., (Electro-)magnetic field evolution in\nrelativistic heavy-ion collisions, Phys. Rev. C83, 054911 (2011) ,arXiv:1103.4239 .\n[14] W.-T. Deng and X.-G. Huang, Event-by-event generation of electromagnetic fields in heavy-ion collisions,\nPhys. Rev. C85, 044907 (2012) ,arXiv:1201.5108 .\n[15] J. Bloczynski, X.-G. Huang, X. Zhang, and J. Liao, Azimu thally fluctuating magnetic field and its impacts on observab les\nin heavy-ion collisions, Phys. Lett. B718, 1529 (2013) ,arXiv:1209.6594 .\n[16] X. Guo, J. Liao, and E. Wang, Spin Hydrodynamic Generati on in the Charged Subatomic Swirl, Sci. Rep. 10, 2196 (2020) ,\narXiv:1904.04704 [hep-ph] .\n[17] L. McLerran and V. Skokov, Comments About the Electroma gnetic Field in Heavy-Ion Collisions,\nNucl. Phys. A 929, 184 (2014) ,arXiv:1305.0774 [hep-ph] .\n[18] K. Tuchin, Time and space dependence of the electromagn etic field in relativistic heavy-ion collisions,\nPhys. Rev. C 88, 024911 (2013) ,arXiv:1305.5806 [hep-ph] .\n[19] U. Gursoy, D. Kharzeev, and K. Rajagopal, Magnetohydro dynamics, charged currents and directed flow in heavy ion\ncollisions, Phys. Rev. C 89, 054905 (2014) ,arXiv:1401.3805 [hep-ph] .\n[20] Y. Zhong, C.-B. Yang, X. Cai, and S.-Q. Feng, A systemati c study of magnetic field in Relativistic Heavy-ion Collisio ns\nin the RHIC and LHC energy regions, Adv. High Energy Phys. 2014, 193039 (2014) ,arXiv:1408.5694 [hep-ph] .\n[21] K. Tuchin, Initial value problem for magnetic fields in h eavy ion collisions, Phys. Rev. C 93, 014905 (2016) ,\narXiv:1508.06925 [hep-ph] .\n[22] H. Li, X.-l. Sheng, and Q. Wang, Electromagnetic fields w ith electric and chiral magnetic conductivities in heavy io n\ncollisions, Phys. Rev. C 94, 044903 (2016) ,arXiv:1602.02223 [nucl-th] .\n[23] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O. Suk hachov,Electronic Properties of Dirac and Weyl Semimetals\n(World Scientific, Singapore, 2021).\n[24] W. Heisenberg and H. Euler, Consequences of Dirac’s the ory of positrons, Z. Phys. 98, 714 (1936) .\n[25] J. S. Schwinger, On gauge invariance and vacuum polariz ation,Phys. Rev. 82, 664 (1951) .\n[26] W. Dittrich and M. Reuter, Effective Lagrangians in Quantum Electrodynamics, Lecture N otes in Physics , Vol. 220\n(Springer, Berlin, 1985).\n[27] J. O. Andersen, W. R. Naylor, and A. Tranberg, Phase diag ram of QCD in a magnetic field: A review,\nRev. Mod. Phys. 88, 025001 (2016) ,arXiv:1411.7176 [hep-ph] .\n[28] V. A. Miransky and I. A. Shovkovy,Quantum field theory in a magnetic field: From quantum chromodynamics to graphene\nand Dirac semimetals, Phys. Rept. 576, 1 (2015) ,arXiv:1503.00732 [hep-ph] .\n[29] A. Bandyopadhyay, C. A. Islam, and M. G. Mustafa, Electr omagnetic spectral properties and Debye screening of a stro ngly\nmagnetized hot medium, Phys. Rev. D 94, 114034 (2016) ,arXiv:1602.06769 [hep-ph] .\n[30] A. Das, N. Haque, M. G. Mustafa, and P. K. Roy, Hard dilept on production from a weakly magnetized hot QCD medium,\nPhys. Rev. D99, 094022 (2019) ,arXiv:1903.03528 .\n[31] R. Ghosh, B. Karmakar, and M. G. Mustafa, Soft contribut ion tothe dampingrate of ahard photon in aweakly magnetized\nhot medium, Phys. Rev. D 101, 056007 (2020) ,arXiv:1911.00744 [hep-ph] .\n[32] K. Hattori and K. Itakura, Vacuum birefringence in stro ng magnetic fields: (I) Photon polarization tensor with all t he\nLandau levels, Annals Phys. 330, 23 (2013) ,arXiv:1209.2663 .\n[33] K. Hattori and K. Itakura, Vacuum birefringence in stro ng magnetic fields: (II) Complex refractive index from the lo west\nLandau level, Annals Phys. 334, 58 (2013) ,arXiv:1212.1897 .21\n[34] K. Tuchin, Magnetic contribution to dilepton producti on in heavy-ion collisions, Phys. Rev. C88, 024910 (2013) ,\narXiv:1305.0545 .\n[35] F. Karbstein, Photon polarization tensor in a homogene ous magnetic or electric field, Phys. Rev. D88, 085033 (2013) ,\narXiv:1308.6184 .\n[36] K.-I. Ishikawa, D. Kimura, K. Shigaki, and A. Tsuji, A nu merical evaluation of vacuum polarization tensor in consta nt\nexternal magnetic fields, Int. J. Mod. Phys. A28, 1350100 (2013) ,arXiv:1304.3655 .\n[37] N. Sadooghi and F. Taghinavaz, Dilepton production rat e in a hot and magnetized quark-gluon plasma,\nAnnals Phys. 376, 218 (2017) ,arXiv:1601.04887 .\n[38] A. Ayala, J. D. Casta˜ no Yepes, M. Loewe, and E. Mu˜ noz, G luon polarization tensor in a magnetized medium: Analytic\napproach starting from the sum over Landau levels, Phys. Rev. D 101, 036016 (2020) ,arXiv:1912.07136 [hep-th] .\n[39] A. Ayala, J. D. Casta˜ no Yepes, L. A. Hern´ andez, J. Sali nas San Mart´ ın, and R. Zamora, Gluon polarization tensor an d\ndispersion relation in a weakly magnetized medium, Eur. Phys. J. A 57, 140 (2021) ,arXiv:2009.00830 [hep-ph] .\n[40] K.Hattori, H.Taya,andS.Yoshida,Di-leptonproducti onfromasingle photoninstrongmagnetic fields: vacuumdich roism,\nJHEP01, 093,arXiv:2010.13492 [hep-ph] .\n[41] S. Ghosh and V. Chandra, Electromagnetic spectral func tion and dilepton rate in a hot magnetized QCD medium,\nPhys. Rev. D98, 076006 (2018) ,arXiv:1808.05176 .\n[42] S. Ghosh, N. Chaudhuri, S. Sarkar, and P. Roy, Effects of t he anomalous magnetic moment of quarks on the dilep-\nton production from hot and dense magnetized quark matter us ing the NJL model, Phys. Rev. D 101, 096002 (2020) ,\narXiv:2004.09203 [nucl-th] .\n[43] X. Wang, I. A. Shovkovy, L. Yu, and M. Huang, Ellipticity of photon emission from strongly magnetized hot QCD plasma,\nPhys. Rev. D 102, 076010 (2020) ,arXiv:2006.16254 .\n[44] X. Wang and I. Shovkovy, Photon polarization tensor in a magnetized plasma: Absorptive part,\nPhys. Rev. D 104, 056017 (2021) ,arXiv:2103.01967 [nucl-th] .\n[45] N.Chaudhuri, S.Ghosh, S.Sarkar,andP.Roy,Dileptonp roductionfrom magnetized quarkmatterwithananomalous ma g-\nneticmomentofthequarksusingathree-flavorPNJLmodel, Phys. Rev. D 103, 096021 (2021) ,arXiv:2104.11425 [hep-ph] .\n[46] A. Das, A. Bandyopadhyay, and C. A. Islam, Lepton pair pr oduction from a hot and dense QCD medium in the presence\nof an arbitrary magnetic field, Phys. Rev. D 106, 056021 (2022) ,arXiv:2109.00019 [hep-ph] .\n[47] X. Wang and I. A. Shovkovy, Rate and ellipticity of dilep ton production in a magnetized quark-gluon plasma,\nPhys. Rev. D 106, 036014 (2022) ,arXiv:2205.00276 [nucl-th] .\n[48] E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shov kovy, Coulomb interaction and magnetic catalysis in the\nquantum Hall effect in graphene, Phys. Scripta T 146, 014018 (2012) ,arXiv:1105.1360 [cond-mat.mes-hall] .\n[49] I. A. Shovkovy and L. Xia, Generalized Landau level repr esentation: Effect of static screening in the quantum Hall eff ect\nin graphene, Phys. Rev. B 93, 035454 (2016) ,arXiv:1508.04471 [cond-mat.mes-hall] .\n[50] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and X. Wang, Chiral asymmetry in QED matter in a magnetic field,\nPhys. Rev. D 88, 025043 (2013) ,arXiv:1306.3245 [hep-ph] .\n[51] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and X. Wang, Radiative corrections to chiral separation effect in QED,\nPhys. Rev. D 88, 025025 (2013) ,arXiv:1304.4606 [hep-ph] .\n[52] W.-y. Tsai, Modified electron Propagation Function in S trong Magnetic Fields, Phys. Rev. D 10, 1342 (1974) .\n[53] B. Jancovici, Radiative correction to the ground-stat e energy of an electron in an intense magnetic field,\nPhys. Rev. 187, 2275 (1969) .\n[54] R. Gepraegs, H. Riffert, H. Herold, H. Ruder, and G. Wunne r, Electron selfenergy in a homogeneous magnetic field,\nPhys. Rev. D 49, 5582 (1994) .\n[55] V. P. Gusynin and A. V. Smilga, Electron selfenergy in st rong magnetic field: Summation of double logarithmic terms,\nPhys. Lett. B 450, 267 (1999) ,arXiv:hep-ph/9807486 .\n[56] B. Machet, The 1-loop self-energy of an electron in a str ong external magnetic field revisited,\nInt. J. Mod. Phys. 31, 1650071 (2016) ,arXiv:1510.03244 [hep-ph] .\n[57] A. Das, A. Bandyopadhyay, P. K. Roy, and M. G. Mustafa, Ge neral structure of fermion two-point function and its spect ral\nrepresentation in a hot magnetized medium, Phys. Rev. D 97, 034024 (2018) ,arXiv:1709.08365 [hep-ph] .\n[58] A. Ayala, J. D. Casta˜ no Yepes, M. Loewe, and E. Mu˜ noz, F ermion mass and width in QED in a magnetic field,\nPhys. Rev. D 104, 016006 (2021) ,arXiv:2104.04019 [hep-ph] .\n[59] N. Chaudhuri, S. Ghosh, P. Roy, and S. Sarkar, Collectiv e modes of a massive fermion in a magnetized medium with finite\nanomalous magnetic moment, Phys. Rev. D 108, 116006 (2023) ,arXiv:2310.05769 [hep-ph] .\n[60] E.BraatenandR.D.Pisarski, SoftAmplitudesinHotGau geTheories: AGeneralAnalysis, Nucl. Phys. B 337, 569 (1990) .\n[61] A. Rebhan, Comment on ‘high temperature fermion propag ator: Resummation and gauge dependence of the damping\nrate’,Phys. Rev. D 46, 4779 (1992) ,arXiv:hep-ph/9204210 .\n[62] H. Nakkagawa, A. Niegawa, and B. Pire, Resolution of the gauge dependence problem of the fermion damping rate in hot\ngauge theories, Phys. Lett. B 294, 396 (1992) .\n[63] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , 5th ed. (Academic Press, New York, 1980).\n[64] R. Ghosh and I. A. Shovkovy,Data Files for the Fermion Se lf-Energy Function and DampingRate in a Strongly Magnetize d\nPlasma,https://www.dropbox.com/scl/fo/r95a1va222rph16k2ktf 2/h?rlkey=dl4c5ty7vbmy7ltgj3vhhnqhl (2024).\n[65] H. A. Weldon, Simple Rules for Discontinuities in Finit e Temperature Field Theory, Phys. Rev. D 28, 2007 (1983) .\n[66] K. Hattori and D. Satow, Electrical Conductivity of Qua rk-Gluon Plasma in Strong Magnetic Fields,\nPhys. Rev. D 94, 114032 (2016) ,arXiv:1610.06818 [hep-ph] .22\n[67] A. Bandyopadhyay, J. Liao, and H. Xing, Heavy quark dyna mics in a strongly magnetized quark-gluon plasma,\nPhys. Rev. D 105, 114049 (2022) ,arXiv:2105.02167 [hep-ph] .\n[68] K. Bhattacharya and P. B. Pal, Neutrinos and magnetic fie lds: A Short review, Proc. Indian Natl. Sci. Acad. A 70, 145\n(2004),arXiv:hep-ph/0212118 ."    },    {        "title": "2402.04308v2.New_Well_Posed_Boundary_Conditions_for_Semi_Classical_Euclidean_Gravity.pdf",        "content": "New Well-Posed Boundary Conditions for Semi-Classical\nEuclidean Gravity\nXiaoyi Liu,aJorge E. Santos,bToby Wisemanc\naDepartment of Physics, University of California, Santa Barbara, CA 93106, USA\nbDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge,\nCB3 0WA, UK\ncTheoretical Physics Group, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom\nE-mail: xiaoyiliu@ucsb.edu ,jss55@cam.ac.uk ,t.wiseman@imperial.ac.uk\nAbstract:\nWe consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a\nwell-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that\nthere exists a one-parameter family of boundary conditions, parameterized by a constant p, where a suitably\nWeyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet\nboundary conditions can be seen as the limits p→0 and ∞of these. Focussing on static Euclidean solutions,\nwe derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space\nor the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth\nEuclidean fluctuations about the flat space saddle; for p >1/6 the spectrum of the Lichnerowicz operator\nis stable – its eigenvalues have positive real part. Thus we may regard large pas a regularization of the ill-\nposed Dirichlet boundary conditions. However for p <1/6 there are unstable modes, even in the spherically\nsymmetric and static sector. We then turn to Lorentzian signature. For p <1/6 we may understand this\nspherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics\nof the boundary itself. However, a mystery emerges when we consider perturbations that break spherical\nsymmetry. Here we find a plethora of dynamically unstable modes even for p > 1/6, contrasting starkly\nwith the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but\nunstable dynamics, calling into question the standard assumption of smoothness that we have implemented\nwhen discussing the Euclidean theory.arXiv:2402.04308v2  [hep-th]  4 Apr 2024Contents\n1 Introduction 1\n2 A New One-Parameter Family of Boundary Conditions 3\n2.1 The Boundary Conditions 4\n2.2 Ellipticity of the Conformal Boundary Conditions 7\n3 Euclidean Gravitational Path Integral 9\n3.1 First law of black hole mechanics 10\n3.2 Saddle points for spherical cavities: hot space and black holes 13\n3.3 Fluctuations about the empty spherical cavity 16\n3.3.1 Static spherical modes 19\n3.3.2 Instabilites for p <1/6 for the static non-spherical modes 20\n3.3.3 Stability for p >1/6 for all modes 21\n3.4 Black hole fluctuations: spherical static negative modes 24\n4 Dynamical Lorentzian Stability of Generalized Conformal Boundary Conditions 27\n4.1 Spherical stability 28\n4.1.1 Linear instabilities for spherical cavities and black holes 30\n4.1.2 Revisiting the spherical stability calculation 31\n4.1.3 End-point of the instabilities 33\n4.2 Stability beyond spherical symmetry 36\n5 Conclusion and Discussion 42\nA Euclidean Fluctuations of the Empty Spherical Cavity 46\nA.1 Modes with n=ℓS= 0 46\nA.2 Modes with n̸= 0 and ℓS= 0 47\nA.3 Non-spherical scalar-derived gravitational Euclidean modes with ℓS≥2 48\nA.3.1 Static modes 48\nA.3.2 Generic modes 51\nA.4 Modes with n= 0 and ℓS= 1 54\nA.5 Modes with n̸= 0 and ℓS= 1 56\nA.6 Non-spherical Euclidean modes with ℓV≥2 59\nA.6.1 Static modes 60\nA.6.2 Generic modes 61\nA.7 Modes with n= 0 and ℓV= 1 62\nA.8 Modes with n̸= 0 and ℓV= 1 62\nB Analyticity of pin the Unstable Complex ˜λPlane 63\nB.1 The static case ϖ= 0 63\nB.2 The general case ϖ̸= 0 65\n– i –C Static and Spherical Euclidean Modes for Vacuum Filled Cavity with Λ̸= 0 68\nD Euclidean Spherical Static Fluctuations of (A)dS Black Holes 71\nReferences 72\n1 Introduction\nYork originally considered the Einstein equations in a cavity whose boundary is a product of time with\na sphere in order to obtain a sensible canonical thermodynamic ensemble for gravity [1]. Gross, Perry\nand Yaffe had previously discovered that the Schwarzschild solution in asymptotically flat spacetime\nsuffers from a Euclidean negative mode [2], later shown to be directly related to its negative specific\nheat capacity [3–5], and thus it can never be a dominant saddle point in the Euclidean path integral\nwhich computes the canonical partition function. Introducing a finite boundary, York then found that\nthe Schwarzschild solution continuously connected to the asymptotically flat one when the the cavity\nsize is increased, the ‘small black hole’, remains unstable with a negative mode. However a second\n‘large’ black hole is now found to exist, always with a size comparable to the sphere of the cavity\nboundary, but now stable, and for sufficient temperature it dominates the partition function. Shortly\nafterwards, Hawking and Page discovered precisely the same phenomenon occurs in Anti de Sitter\n(AdS) spacetime, with the AdS length scale playing an analog role to the size of the cavity in York’s\nconstruction. For many years this was a curiosity, but with the advent of AdS-CFT it took on a crucial\nrole [6–8]. A CFT on a sphere should have a sensible partition function, and at high temperature it\nshould have a free energy which scales with the number of local degrees of freedom in the theory. It is\nprecisely the large black hole which provides the gravitational dual to this behaviour [9].\nHere we restrict our focus to four-dimensional pure gravity, possibly including a cosmological\nconstant. Thus solutions to the Einstein equations are Einstein metrics, and we are primarily interested\nin Euclidean geometries with a boundary that takes the form of S1×Σ, where S1is interpreted as\nbeing Euclidean time, τ, with length βsoτ∼τ+β, and Σ is a two-dimensional compact Riemannian\nmetric. Then βhas the thermodynamic interpretation of the inverse temperature. The canonical\ncavity to consider is a spherical one, so that Σ = S2, a round two sphere with a radius R, but we\nwish to consider here the more general setting where the gravitational path integral is thought of as a\nfunctional of βand Σ.\nAn important issue was pointed out by Avramiki and Esposito [10, 11] who considered Euclidean\nquantum gravity on manifolds with boundary, and then discussed by Anderson [12] in the context of\nEinstein metric infillings in Riemannian geometry. The natural boundary condition one would wish to\nimpose, namely fixing the induced metric on the cavity boundary, so here fixing βand the two-geometry\nΣ, does not yield a well-posed infilling problem. In Euclidean signature the Einstein equation locally\nhas an elliptic character (after dealing with the issue of coordinate freedom), but this analog of Dirichlet\nboundary conditions, so fixing the metric of the boundary, does not preserve elliptic regularity of the\nproblem. In [12] it was argued that this may be understood in terms of the Gauss constraint of the\nboundary. Anderson proposed a well-posed set of boundary conditions, which we refer to as ‘Anderson\n– 1 –boundary condition’, which entails fixing the conformal class of the boundary (so the induced metric\nup to a Weyl rescaling) together with the trace of the extrinsic curvature. The lack of well-posedness\npresents an obstruction to York’s original formulation of a stable canonical ensemble for gravity, where\nthe induced metric of the spatial cavity wall, βand Σ, is given as data. In the semi-classical limit we\nexpect a continuous map from the boundary data at the surface of the cavity to infilling saddle point\nsolutions, and lack of well-posedness for Dirichlet boundary conditions exactly implies the lack of such\na map. As emphasized in [10, 11] this leads to pathologies when computing the Euclidean partition\nfunction at 1-loop as the lack of well-posedness may lead to infinitely many zero modes. Hence one\nis naively motivated to consider instead the thermodynamic ensemble given by Anderson’s well-posed\nboundary condition.\nThere has been limited study of Anderson boundary condition in the physics context. The impli-\ncations for gravity were discussed in [13] and reviewed in detail in [14]. Recently [15] has shown that\nin the Lorentzian setting there are linear dynamical instabilities for static spherical cavities, without\ncosmological constant, that are filled with flat spacetime. They also considered a first law of thermo-\ndynamics for static cavities with spherical boundary. In fact stationary rotating black holes in such\ncavities were actually constructed some time ago in [16] – these are deformed from Kerr due to the\nrequirement the boundary is a round sphere. Perhaps unsurprisingly given the dynamical instabilities\nfound in [15], in this work we will later see that Anderson boundary condition already yields a patho-\nlogical partition function for a static spherical cavity filled with flat spacetime due to the existence of\nEuclidean negative modes.\nThe main purpose of our paper is to point out a natural generalization of Anderson boundary\ncondition. The gravitational actions relevant for Dirichlet and Anderson boundary conditions differ by\nthe coefficient of the Gibbons-Hawking-York term [17]. Our generalized boundary conditions derive\nfrom considering the coefficient of this term to parameterize a family of choices. For these the conformal\nclass is again fixed, with the trace of the extrinsic curvature weighted by a power of the volume element\nof the induced metric. Specifically letting the boundary induced metric be γ, and the trace of the\nextrinsic curvature be K, these fix the conformal class [ γ] and,\nγpK= fixed (1.1)\nwhere the constant pis related to the coefficient of the Gibbons-Hawking-York term. The case of p→0\ngives the Anderson boundary condition, and p→ ∞ corresponds to Dirichlet conditions. Alternatively\n(provided we wish Kto be non-vanishing) we may define the Weyl rescaled boundary metric,\nΓµν=K1\n(d−1)pγµν (1.2)\nand then the boundary condition neatly corresponds to fixing this rescaled metric.\nWe will thus call this family of boundary conditions the ‘generalized conformal boundary condi-\ntions’. We show that these are all well-posed in the same sense as Anderson boundary condition is, and\nfurther one can regard the (ill-posed) Dirichlet case as a limit of this family of boundary conditions.\nWe then explore the physical consequences of these boundary conditions. We begin with the\nEuclidean theory. Firstly we show that the first law of thermodynamics can be naturally formulated\nfor static Euclidean cavities, where the rescaled boundary metric Γ above is a product of a Euclidean\ntime circle, whose size is the inverse to temperature, and a spatial 2-geometry. Secondly we study the\n– 2 –Euclidean stability of a static spherical cavity, without cosmological constant, filled with flat spacetime.\nAfter an extensive mode analysis we find that for p <1/6 (which includes the case of Anderson) there\nare Euclidean negative modes, but for p >1/6 there are none. The Euclidean mode that becomes\nunstable at p= 1/6 is in fact spherically symmetric. We also look at the Euclidean stability of black\nholes, restricted to static spherical fluctuations, and find a similar pattern of instability to the Dirichlet\ncase – small black holes have an additional unstable mode relative to that of the flat space cavity – and\nthe thermodynamics reflect this with a York-Hawking-Page first order phase transition between the\nflat cavity and the large black holes. Thus we are lead to conclude that a sensible Euclidean partition\nfunction exists for our well-posed boundary conditions provided we take p >1/6. However, there is\none mystery. When we consider the specific heat capacity, for p >1/6 and sufficiently large black holes\nthis becomes negative even though they dominate the thermodynamic ensemble and seemingly have\nno Euclidean negative modes.\nWe then move on to consider the Lorentzian dynamical stability, as done for Anderson boundary\ncondition in [15]. We begin with a spherical cavity filled with flat spacetime and consider the spherically\nsymmetric sector. Here Birkhoff’s theorem enforces that the interior geometry is static. However, the\nboundary may dynamically move in this geometry in a manner similar to the Z2-symmetric branes\nof the Randall-Sundrum models [18]. In fact we may simply solve this dynamics of the boundary\nnon-linearly. In the case p <1/6 we find an instability which acts to shrink the boundary to zero\nsize in finite time, or expand it forever. For p >1/6 the cavity is stable, and at the marginal point\np= 1/6 this Lorentzian mode is static and precisely corresponds the marginal mode in the Euclidean\nanalysis. For cavities filled with black holes, we find an additional instability for p >1/6 and sufficiently\nlarge (depending on the value of p) black holes, which precisely correlates with when the specific heat\ncapacity becomes negative. Thus we see that such black holes appear well behaved from a Euclidean\nperspective, but unstable from the Lorentzian one, which is unexpected. The situation becomes yet\nmore confusing when we consider Lorentzian perturbations that break spherical symmetry. Restricting\nto a spherical cavity filled with flat spacetime we now find a plethora of unstable ( i.e.exponentially\ngrowing) dynamical modes even for p > 1/6.Thus while these generalized boundary conditions\napparently give a sensible Euclidean partition function for p > 1/6, they are apparently unstable\nin the Lorentzian setting. This is a physically unusual situation because one would not normally\nexpect sensible thermodynamics from an unstable dynamical system. It suggests to us that one of\nthe assumptions we have made in considering the Euclidean fluctuations, such as the assumption of\nsmoothness of the geometry, may be invalid.\nThe plan of this paper is as follows. We begin in section 2 by introducing our new boundary\nconditions, which include the case of Anderson, and confirm that they are well-posed in the Euclidean\nsetting. In section 3 we derive a first law of the thermodynamics and then compute the Euclidean\nstability of vacuum filled spherical cavities. We then consider the Lorentzian stability of spherical\ncavities in section 4, before concluding with a discussion.\n2 A New One-Parameter Family of Boundary Conditions\nIn this section, we first introduce our new one-parameter family of boundary conditions in section 2.1,\nwhich is adapted to general coefficients of the Gibbons-Hawking-York term. Then we prove they yield\na well-posed elliptic infilling problem in section 2.2.\n– 3 –2.1 The Boundary Conditions\nBefore we detail our novel boundary conditions, let us introduce some terminology. We will be inter-\nested in spacetimes ( M, g), where Mis ad-dimensional manifold-with-boundary. We are interested\nin determining which choice of boundary conditions we can impose on the boundary ∂Mso that we\nhave a well-defined variational problem. For the moment, we will work in Euclidean signature, so that\ngis Riemannian, since it is for this class of spacetimes that we can prove the well-posedness of the new\nboundary conditions.\nLet us first review the action with respect to which variations of the metric satisfying Dirichlet\nboundary conditions on the boundary ∂Mare stationary. It is well known that for the Dirichlet\nproblem the action reads\nS=−1\n16πGZ\nMddx√g(R−2Λ)−1\n8πGZ\n∂Mdd−1x√γ K , (2.1)\nwith Kthe trace of the extrinsic curvature Kµνcomputed using an outward pointing unit normal n,\nγthe induced metric on ∂M,Rthe Ricci scalar associated with g,Gthe Newton’s constant and Λ a\ncosmological constant. The first term is the standard Einstein-Hilbert action, whereas the second is\nthe Gibbons-Hawking-York term [17]. Varying the action with respect to gyields\nδS=1\n16πGZ\nMddx√gEabδgab+1\n16πGZ\n∂Mdd−1x√γ Tµνδγµν (2.2)\nwhere lower case Latin indices are bulk indices, lower case Greek indices run over the boundary ∂M\nand\nTµν≡Kµν−γµνKand Eab≡Rab−1\n2gabR+ Λgab, (2.3)\nwith Rabthe components of the Ricci tensor associated with g. One can see from (2.2) that imposing\nDirichlet conditions on the boundary ∂Mand the bulk equations of motion Eab= 0, do give rise to a\nwell-defined variational problem.\nWe will now ask what boundary conditions we should choose to render the first variation of the\naction\nSΘ=−1\n16πGZ\nMddx√g(R−2Λ)−Θ\n8πGZ\n∂Mdd−1x√γ K , (2.4)\nstationary with respect to metrics satisfying Eab= 0 for fixed values of Θ. To our knowledge such\ngeneral class of boundary term has never been considered in the literature and is the main objective\nof this section of our paper.\nWe now vary the action (2.4) with respect to gand keep track of all boundary terms. This yields\nδSΘ=1\n16πGZ\nMddx√g Eabδgab+1\n16πGZ\n∂Mdd−1x√γ Tµνδγµν\n+1\n16πGZ\n∂Mdd−1x δ(2√γK)−Θ\n8πGZ\n∂Mdd−1x δ(√γ K). (2.5a)\nwith\nEab≡Rab−1\n2gabR+ Λgaband Tµν≡Kµν−K γµν, (2.5b)\n– 4 –with the latter being the so called Brown-York tensor [19]. Since we are interested in spacetimes that\nsatisfy the bulk Einstein equation Eab= 0, the first term in (2.5a) vanishes.\nTo proceed, we decompose the variation of the boundary metric δγµνas\nδγµν=δ˜γµν+1\nd−1δγ\nγγµν, (2.6)\nwith δ˜γµνbeing traceless with respect to γµν,i.e.\nγµνδ˜γµν= 0. (2.7)\nOne might wonder why we chose to introduce a factor of 1 /γin the last term of the decomposition\n(2.6). This was done to align variations of the determinant of γµνwith δγ, thereby justifying the\nnomenclature.\nUsing our decomposition (2.6) we can write the first order variation as\nδSΘ=1\n16πGZ\n∂Mdd−1x√γ˜Kµνδ˜γµν+1−Θ\n8πGZ\n∂Mdd−1x√γ γ−pδ(γpK). (2.8a)\nwhere we identify\np=1\n21\n1−Θ\u0012\n1−Θ−d−2\nd−1\u0013\n⇔ Θ =2p(d−1)−1\n(d−1)(2p−1)(2.8b)\nand where we have defined ˜Kµνas the traceless part of the extrinsic curvature, which equals the\ntraceless part of the Brown-York stress tensor,\n˜Kµν=Kµν−γαβKαβ\nd−1γµν. (2.9)\nNote that Θ <0 when1\n2(d−1)< p <1\n2.\nNow we see that we have a good variational principle if we require boundary conditions on ∂M\nsuch that,\nδ(γpK) = 0 (2.10)\ntogether with either,\nδ˜γµν= 0 (2.11)\nor,\n˜Kµν= 0. (2.12)\nIn this paper we will study the first class of boundary conditions. The second class of boundary\nconditions is also of interest, providing a generalization of the Neumann condition Kµν= 0, but will\nbe studied elsewhere.\nThe condition δ˜γµν= 0 implies that we are making a statement about the conformal structure of\nthe boundary metric, since δ˜γµνhas a well defined conformal weight under Weyl transformations. We\ncan formulate our boundary condition using this language.\n1. Fix the conformal class of the boundary metric γµν– that is to say we fix the boundary metric\nγµνto be some given metric up to a Weyl transformation. This precisely yields δ˜γµν= 0.\n– 5 –2. Fix,\nγpK=f (2.13)\nwhere fis a scalar density on ∂Mwith the appropriate weight (given by p) such that this\nexpression transforms correctly under coordinate changes.\nWe see that for p= 0, so Θ = 1 /(d−1), then fis simply a function and these reduce to pure\nAnderson boundary conditions as described in [12]. Furthermore, it is evident from the expression for\nΘ (see Eq. (2.8b)) that Dirichlet boundary conditions can be attained as a somewhat singular limits\nwhen p→ ±∞ when Θ →1. In this scenario, we may more conveniently write the second condition\nas fixing γK1\np, and then taking p→ ∞ we see that this is essentially equivalent to fixing γ, the\ndeterminant of the metric. Then in combination with fixing the conformal class, this yields standard\nDirichlet boundary conditions. As we will soon show, these boundary conditions are well-posed for\nany finite p, and hence we may regard large positive pas a regulator for Dirichlet boundary conditions\nthat makes them well-posed. For large |p|and fixed length scales, the boundary conditions tend to the\nbehaviour of Dirichlet. For example, the spectrum of the Lichnerowicz operator will agree for modes\nof eigenvalue with magnitude below some fixed large value as p→ ±∞ . However, a subtle point is\nthat there is no guarantee that there are no instabilities that are introduced at short distances, or\nhigh eigenvalues. Indeed as p→ −∞ we will see exactly this, that in the Euclidean setting, there\nare negative modes of the Lichnerowicz operator that have very large magnitude as p→ −∞ . In the\npositive p→+∞the high-lying spectrum appears to be stable, and thus we may regard p→+∞as\na good regulator of Dirichlet boundary conditions, whereas p→ −∞ is not.\nWe noted above that two special values of parep=1\n2(d−1)andp=1\n2, where Θ vanishes and\ndiverges, respectively. For p=1\n2(d−1), which corresponds to p=1\n6in four spacetime dimensions, we\nobserve that Θ vanishes. Therefore, the action in (2.4) consists solely of the bulk Einstein-Hilbert and\ncosmological terms. In the case where there is no cosmological constant, this implies that the solutions\nexhibit scale invariance, given that the Einstein-Hilbert term possesses this property. Taking any\nsolution, d s2=gab(x)dxadxb, then for any constant λ >0, ds2=λ2gab(x)dxadxbis also a solution to\nthe bulk Einstein equation and boundary conditions. This can be explicitly observed in the boundary\nconditions mentioned above. Fixing the conformal class is a scale-invariant condition. Given the scale\ntransformation γ→λd−1γandK→λ2K, fixing γpKis scale-invariant precisely for p=1\n2(d−1).\nNow, consider the special case p→1\n2where instead Θ → ∞ . Fixing γpKbecomes fixing√γK,\nwhich is precisely the integrand of the boundary term. As a consequence, according to equation (2.5a),\nit is evident that Θ must diverge to counterbalance the boundary term arising from the integration by\nparts of the variation of the Einstein-Hilbert term in the action.\nIn the case that we impose the trace of the extrinsic curvature is non-vanishing, so the scalar\ndensity fis chosen not to vanish, then we may reformulate the boundary condition by defining,\nΓµν=K1\n(d−1)pγµν (2.14)\nand noting that since K̸= 0 then Γ µνis also a metric on the boundary, in the same conformal class as\nγµν. We will refer to this as the rescaled boundary metric .1One can then show that the variation of\n1Strictly we see that Γ µνas defined in (2.14) doesn’t have the correct units for a metric. We can correct this simply by instead\ndefining,\nΓµν=\u0012K\nK0\u0013 1\n(d−1)p\nγµν\n– 6 –the action can neatly be written as,\nδSΘ=1\n2Z\n∂Mdd−1x√\nΓTµνδΓµν. (2.15a)\nwith,\nTµν≡1\n8πGK−d+1\n2(d−1)p(Kµν−ΘγµνK). (2.15b)\nbeing a generalization of the Brown-York stress tensor. Now we may simply state the boundary\ncondition as being a Dirichlet condition on this rescaled boundary metric Γ µν. We will find this\nperspective very useful when considering thermodynamics in a cavity with these boundary conditions\nlater.\nIt is important to emphasize that this formulation of the boundary condition, while elegant, cannot\ncapture the case where we wish to impose that Kvanishes or changes sign. However, we note that\nparticularly for spherical cavities, it is natural to wish to impose a strictly positive extrinsic curvature.\n2.2 Ellipticity of the Conformal Boundary Conditions\nThe main advantage of conformal boundary conditions is that these give rise to well-posed elliptic\ninfilling problem. In what follows we will show that this is the case for any finite value of p.\nTo demonstrate the well-posedness of our novel boundary conditions, we will work at the linear\nlevel by setting gab= ˆgab+hab, where the hatted quantities denote a background metric, and habis\nassumed to be infinitesimally small compared to ˆ gab. Next, we will choose a suitable gauge for our\npurposes, which, as it turns out, is the de Donder gauge:\nCb≡ˆ∇ahab−1\n2ˆ∇bh= 0, (2.16)\nwhere h= ˆgabhabrepresents the trace of the metric perturbation.\nWe will follow the approach presented in [14] to review why Dirichlet boundary conditions are not\nwell-posed in this gauge. Since the issue of well-posedness pertains to high-frequency behavior, we will\nconsider our manifold with boundary to be the half-space Rd\n+within a flat Euclidean space Rd, with\nx⊥≥0. The boundary is simply defined by x⊥= 0, and has vanishing extrinsic curvature.\nA general plane wave solution takes the form:\nhab=αabeiy·K, (2.17)\nwhere y∈Rd. We aim to explore solutions that may have non-trivial behavior along the boundary\nwhile decaying away from it. For this reason, we consider y={⃗ x, x⊥}andK={⃗k, ik⊥}, where ⃗ xand\n⃗kbelong to Rd−1andk⊥>0. This choice yields\nhab=αabei⃗ x·⃗k−|⃗k|x⊥, (2.18)\nwith the perturbed Einstein equation locking k⊥=|⃗k|. Our goal is to investigate whether nontrivial\nsolutions of the above form can persist when we impose Dirichlet boundary conditions and the de\nfor an appropriate constant K0. However since this constant plays no role except to rescale Γ µν, for convenience we will always\nset it to one in our units.\n– 7 –Donder gauge for arbitrary wavevectors ⃗k. If this turns out to be the case, it indicates that the\nboundary conditions are not well-posed in the chosen gauge. Physically we may view this as saying\nthat we would then have a class of modes that have arbitrarily short wavelength along the boundary\ndirection, while decaying immediately away from the boundary into the bulk. Hence the bulk solution\nwould not correspond to a unique boundary geometry.\nWe can now analyze this component by component. When x⊥= 0, the Dirichlet boundary\nconditions imply αij= 0, where iandjdenote the directions parallel to the boundary. Let ⃗ αi=αi⊥\nandβ=α⊥⊥, with i= 1,2, . . . , d −1. It then follows that\nC⊥= 0⇒i⃗k·⃗ α−1\n2β|⃗k|= 0 (2.19a)\nand\nCi= 0⇒ −⃗ α|⃗k| −1\n2β i⃗k= 0. (2.19b)\nBoth of the above can be solved if we take\n⃗ α=−i\n2β⃗k\n|⃗k|, (2.20)\nthus showing that Dirichlet conditions are not well-posed in the de Donder gauge.\nWe would like to repeat the above for our general class of boundary conditions and show that\nfor any non-trivial ⃗kwe necessarily have αµν= 0. While the conformal class is fixed, our boundary\nconditions do allow the perturbation in the induced metric, hij, to be non-zero at the boundary x⊥= 0,\nso long as hij=δγ\nd−2ˆgijatx⊥= 0, with δγnow being infinitesimally small. This is a fluctuation resulting\nfrom an infinitesimal Weyl rescaling.\nThe linearisation of the condition δ(γpK) = 0, in coordinates where ˆ g⊥⊥= 1 and ˆ g⊥i= 0, about\nour simple wall yields\n∂x⊥hi\ni−2∂ih⊥i= 0. (2.21)\nNext we consider solutions of the form (2.18), which brings the above to\n2i⃗ α·⃗k+|⃗k|δγ= 0, (2.22a)\nwhile from the de Donder gauge condition we get\ni⃗k·⃗ α−1\n2β|⃗k|+1\n2δγ|⃗k|= 0 (2.22b)\nand\n−⃗ α|⃗k| −1\n2β i⃗k+1\n2δγ i⃗k= 0. (2.22c)\nIt is a simple exercise to check that all the above conditions can only be satisfied if ⃗ α= 0, β= 0,\nandδγ= 0, thus enforcing αab= 0 and showing that our new boundary conditions give rise to a\nwell-posed elliptic problem. The reader might note that the exponent pdoes not make an appearance\nin the above calculations. This is due to the fact that when discussing ellipticity we focus on short\nwavelengths, so that the boundary effectively has zero extrinsic curvature, so then δ(γpK) =γpδ(K),\nwhich simply implies δK= 0 – this is the same condition as for the Anderson case ( p= 0), and hence\n– 8 –our generalized conformal boundary conditions are well-posed for the same reason as for Anderson\nboundary condition, with no dependence on the (finite) value of p.\nWe note that the ill-posed Dirichlet boundary conditions correspond to the limit p→ ∞ , and then\none must take care considering the short wavelength limit as we do here – in fact the order of limits\nis important, and taking p→ ∞ first leads to the last term in each of the equations (2.22) vanishing,\nand yields the usual ill-posedness of Dirichlet conditions.\n3 Euclidean Gravitational Path Integral\nNext we delve into the realm of Euclidean gravitational path integrals, drawing inspiration from the\nperspective introduced by Gibbons and Hawking [17]. Again we consider a d-dimensional manifold\nMwith boundary ∂Mequipped with a metric gaband boundary metric γµν. On ∂Mwe impose the\nboundary conditions discussed in section 2 where (provided Kis non-vanishing) we fix the geometry\n(Γ, ∂M) defined by the rescaled boundary metric, Γ µν. In the event Kdoes vanish somewhere on the\nboundary, we should use the alternate formulation where we fix the conformal class together with γpK.\nThe discussion below can be simply rewritten using that formulation, but is less elegant, and thus we\nprefer to phrase it in terms fixing (Γ , ∂M).\nThe Euclidean gravitational path integral\nZ[Γµν] =Z\nDg e−S[g], (3.1)\nis then computed by integrating over all metrics g, satisfying the prescribed boundary conditions,\nwhere S[g] is the action given in equation (2.4).\nRegrettably, it is well-known that the Euclidean gravitational action is unbounded from below,\nowing to the so-called conformal factor problem [17]. Consequently, the integral cannot be taken over\nRiemannian metrics g. Instead, one is compelled to choose a contour of integration that ensures the\nconvergence of the path integral. Alternatively, one could initiate the analysis with Lorentz signature\n(where the path integral is always well-defined, at least in a distributional sense), leading to implications\nfor Euclidean path integrals determined through a functional Cauchy-type theorem and a careful study\nof the analytic properties of the gravitational path integral [20]. Unfortunately, we do not understand\nthe Lorentzian path integral well enough to derive consequences for the Euclidean path integral.\nNevertheless, progress can be made by focusing on saddle points. In semiclassical quantum gravity,\nthe Euclidean gravitational path integral is calculated through saddle point approximation\nZ[Γµν]∼e−S[ˆg](3.2)\nwith ˆ gsatisfying the classical equations of motion derived from S[g], subject to the appropriate bound-\nary conditions. The Euclidean stability of such saddles is then investigated by expanding the metric\ngab= ˆgab+haband computing the action to quadratic order in hab. For the case of pure gravity,\npossibly in the presence of a cosmological constant, the resulting quadratic action can be written as\nan expectation value[21, 22]\nS(2)[h] =1\n32πGZ\nMddxp\nˆg habˆGab cd(Lh)cd (3.3)\n– 9 –where ˆGab cdis a particular DeWitt ultralocal metric\nˆGab cd=1\n2\u0000\nˆgacˆgbd+ ˆgadˆgbc−ˆgabˆgcd\u0001\n, (3.4)\nand\n(Lh)ab= (ˆ∆Lh)ab+ 2ˆ∇(aˆ∇p¯hb)p (3.5)\nwhere\n¯hab=hab−ˆgab\n2h , (3.6)\nand\n(ˆ∆Lh)ab=−ˆ∇pˆ∇phab−2ˆRacbdhcd(3.7)\nis the Lichnerowicz operator. In particular, Lreduces to the usual Lichnerowicz operator ˆ∆Lon\nperturbations that satisfy the de Donder gauge (2.16), i.e., ˆ∇a¯hab= 0.\nUnlike the leading saddle point approximation, the quadratic contribution to the Euclidean path\nintegral does exhibit a linearized version of the conformal factor problem [23]. Fortunately, this issue\nhas been studied in detail in [22, 24, 25], where a rule of thumb to investigate the stability of a given\nsaddle has been presented and thoroughly examined. For the type of boundary conditions under\nconsideration, the rule of thumb in [22, 24, 25] is equivalent to studying the following eigenvalue\nproblem\n(∆Lh)ab=λhab, (3.8)\nwhere habis assumed to be in the de Donder gauge. A given mode is then stable when it satisfies\nReλ > 0, with marginal stability for Re λ= 0 and strict instability for Re λ < 0. We note that\nstability of saddle points can also be determined by their stability under Ricci flow since they are fixed\npoint of the flow, as discussed in [21], and the same condition for stability is found.\n3.1 First law of black hole mechanics\nThe canonical ensemble for gravity is defined by the semiclassical Euclidean path integral via an analysis\nof its saddle points. Before delving into the on-shell action of well-known saddles in a spherical cavity,\nthe black hole saddles and the hot flat space infilling, let us take a moment to discuss a version\nof the first law of black hole mechanics that will prove useful in the upcoming discussion. We will\nobserve that our boundary conditions give rise to a natural definition of an ensemble, accompanied\nby a corresponding thermodynamic potential, temperature, and energy. Interestingly, these quantities\nare naturally defined with respect to the rescaled boundary metric, Γ µνrather than the usual induced\nmetric, γµν.\nWe first note that general infinitesimal diffeomorphisms xa→xa+vainduce boundary diffeomor-\nphisms xµ→xµ+vµ. Naturally, these should leave the action invariant. Since Γ µνis a symmetric\ntwo-tensor, we also know that\nδΓµν= (£vΓ)µν= 2∇Γ\n(µvν), (3.9)\nwhere ∇Γis a connection that preserves the metric Γ µν. This implies that variations of the action\ngenerated by infinitesimal coordinate transformations vashould satisfy\n0 =δSΘ=Z\n∂Mdd−1√\nΓTµν∇Γ\n(µvν)=−Z\n∂Mdd−1√\nΓvν∇Γ\nµTµν(3.10)\n– 10 –where in the last step we assumed that the boundary is compact so that boundary terms can be\ndiscarded. Since the above holds for arbitrary choices of vµit must be that Tµνis covariantly conserved\nwith respect to ∇Γ,i.e.,\n∇Γ\nµTµν= 0. (3.11)\nThus we interpret Tµνas a boundary stress tensor with respect to the rescaled boundary geometry\nΓµν, and define all thermodynamic quantities with respect to the pair (Γ µν,Tµν) on∂M.\nTo derive the first law, we therefore assume that Γ µνcan be written as a thermal ultrastatic metric,\ni.e.\nΓµνdxµdxν=b2\n(2π)2dτ2+nijdxidxj(3.12)\nwhere bis a constant, and nijis a (d−2)-dimensional Riemannian metric on the space g∂Mspanned by\nthe non-thermal boundary spatial coordinates xi, and τvaries over the range τ∼τ+ 2π, denoting an\nangle along the thermal circle. The thermal circle in this rescaled geometry (Γ , ∂M) then has length\nb, and we define its inverse, t= 1/b, as the natural measure of temperature in this ensemble. We will\nassume that the infilling metric has a Killing field ∂/∂τ that remains hypersurface orthogonal, so that\nin particular Tτi= 0. It is worth noting that while the time-time component of the rescaled metric,\nΓττ, is constant, this will not generally be true for the induced metric, where γττ∼K1\n(d−1)pmight\nspatially vary over the boundary.\nLet us define an energy density ρ= Γ ττTττ, so that one can expand the variations of the action as\nδSΘ=1\n2Z\n∂Mdd−1x√\nΓTµνδΓµν\n=δbZ\ng∂Mdd−2x√nρ+b\n2Z\ng∂Mdd−2x√nTijδnij. (3.13)\nAt this stage we introduce a potential F, and an energy E,\nSΘ=b F and E=Z\ng∂Mdd−2x√nρ (3.14)\nWe note that the energy we define here is defined with respect to the Killing vector ∂/∂τ in the rescaled\ngeometry (Γ , ∂M), and hence differs from the energy defined in [26] that applies to general dynamical\ncavity geometries and hence is not defined with respect to a bulk or boundary Killing vector. Then\nusing b= 1/tyields\nδF=−E−F\ntδt+1\n2Z\ng∂Mdd−2x√nTijδnij. (3.15)\nThis formula for the variation δFhas a striking resemblance with the first law of black hole thermo-\ndynamics if one could identify b(E−F) as the Bekenstein-Hawking entropy [27, 28]. We shall shortly\nsee that this will be the case.\nTo show this we start by looking at the on-shell action which determines our potential Fas,\nb F=SΘ=−Θ\n8πGZ\n∂Mdd−1x√γK−1\n16πGZ\nMddx√g(R−2Λ)\n=−Θ\n8πGbZ\ng∂Mdd−2x√nK1−1\n2p−1\n16πGZ\nMddx√g(R−2Λ). (3.16)\n– 11 –On the other hand, for the energy we find\nb E=Z\ng∂Mdd−2x√nρ\n=b1\n8πGZ\ng∂Mdd−2x√nK−1\n2pKττγττ−bΘ\n8πGZ\ng∂Mdd−2x√nK1−1\n2p. (3.17)\nCombining these expressions gives,\nE−F\nt=1\n8πGZ\n∂Mdd−1x√γKττγττ+1\n8πG2Λ\nd−2Z\nMddx√g , (3.18)\nwhere we have used that on-shell R= 2dΛ/(d−2). Due to our symmetry assumptions, the infilling\nmetric can be written as\nds2=N(x)2dτ2+HIJ(x)dxIdxJ, (3.19)\nwhere HIJis the metric on the ( d−1)-dimensional space Hspanned by all the bulk non-thermal\ndirections, and we recall that τ∼τ+ 2π. Using the above form of the metric, we can write the ττ\ncomponent of the Einstein equation as\nRττ−2Λ\nd−2gττ= 0⇒ −N\u0012\n∇2\n(H)N+2\nd−2ΛN\u0013\n= 0⇒ ∇I\n(H)JI+2Λ\nd−2ΛN= 0, (3.20)\nwhere we defined JI≡HIJ∇(H)INand∇(H)is the metric preserving connection on H. Let us look\nat the last term appearing in Eq. (3.18) and write it as a function of JI\n2Λ\nd−2Z\nMddx√g= 2π2Λ\nd−2Z\nHdd−1x√\nHN=−2πZ\nHdd−1x√\nH∇I\n(H)JI. (3.21)\nWe see that this is a total divergence, and so can be readily evaluated on ∂H. Let us now assume that\nthe cavity is filled with a black hole with a single horizon component. The boundary of ∂Hthen has\ntwo parts: the cavity boundary, g∂M, and the bifurcating Killing horizon Hof the black hole. Then\neach component yields a contribution to the boundary terms as,\n2Λ\nd−2Z\nMddx√g=−2πZ\n∂Hdd−2x√¯γ niJi=−2πZ\nHdd−2x√¯γ niJi−2πZ\ng∂Mdd−2x√¯γ niJi,(3.22)\nwhere niis the unit outer normal to a given boundary and ¯ γabis the spatial ( d−2)-dimensional\nmetric on that boundary. In order to evaluate the contributions at the boundaries it is convenient\nto use Gaussian normal coordinates, that is to say in the neighbourhood of a boundary we choose\ncoordinates,\nds2= dr2+ ¯γijdxidxj. (3.23)\nso that the boundary is at r= 0 and the interior of the geometry is r >0.\nIt then follows that√¯γniJi=√¯γnrJr=−√¯γ∂rN . (3.24)\nFor a bifurcating Killing horizon, N=r+o(r), since we have fixed the period of τto be 2 πand so,\n−2πZ\nHdd−2x√¯γ niJi= 2πZ\nHdd−2x√¯γ= 2πA , (3.25)\n– 12 –where Ais the horizon area. For the cavity boundary, in these normal coordinates,\nKττ=−N∂rN|r=0 (3.26)\nand thus,\n−2πZ\ng∂Mdd−2x√¯γ niJi=−2πZ\ng∂Mdd−2x√γKττγττ=−Z\n∂Mdd−1x√γKττγττ, (3.27)\nwhere we used that γττ= 1/N2|r=0. We have thus found that\n2Λ\nd−2Z\nMddx√g= 2πA−Z\n∂Mdd−1x√γKττγττ, (3.28)\nwhich we can substitute into Eq. (3.18) to confirm our claim above, concluding that for a black hole\nin a cavity,\nE−F\nt=SBH (3.29)\nwhere SBH=A/(4G) is the Bekenstein-Hawking entropy. Then from (3.15) we obtain the first law,\nδF=−SBHδt+1\n2Z\ndd−2x√nTijδnij. (3.30)\nWe emphasize that the temperature tand metric nijare those associated to the rescaled metric Γ µν,\nand not those associated with induced boundary metric γµν. The second term above is a work term\nassociated to deforming the boundary spatial geometry. It is a generalization of the familiar ∼PdV\nterm, and reduces to that if we restrict to a round sphere cavity so that the change in volume solely\ncomes from a change of radius. We also note that the first law above also applies to empty cavities\nwhich contain no black hole horizons, where one would simply drop the horizon entropy term. Like-\nwise if interior solutions with multiple horizon components were to exist – for example with a positive\ncosmological term [29] – then the horizon entropy term would be the sum of that from each horizon\ncomponent. A reduced first law was considered for Anderson boundary condition in [15]. This re-\nstricted to cavities with a round sphere spatial geometry, and only considered changes in dimensionless\nquantities, so in that context a dimensionless free energy as a function of the conformal class. We\nbelieve our first law is consistent with their results in that special situation.\n3.2 Saddle points for spherical cavities: hot space and black holes\nTo evaluate the gravitational path integral (3.1), we will use the saddle point approximation (3.2). We\nwant to consider spherical cavities, and we will assume that spherical symmetry is preserved in the\ninterior of the cavity. The metric γµνon the boundary ∂Mwill be S1\nβ×S2\nR0, with S1\nβhaving length β\nand parametrizing a periodic Euclidean time coordinate τ∼τ+2π, while S2\nR0represents the metric on\na round two-sphere with a radius of R0. As in the Dirichlet case, the infilling saddle points are either\nblack holes, or the vacuum spacetime with compact Euclidean time circle which we term ‘hot space’.\nWe will compare actions for the various saddles while keeping\nB ≡β\nR0andK ≡p\nγ(R0)K(R0)1\n2p (3.31)\n– 13 –constant, noting that Bdetermines the conformal class of the static spherical boundary. It is important\nto note that the different saddles we may consider might have distinct values for R0andβ, but equal\nvalues for BandK. Note that from our discussion in section 2 this is equivalent to fixing Γ µνunder\nspherical symmetry. An important point is that while βgives the size of the Euclidean time circle\nas measured in the metric γµν, we will find that it is the size of the circle in the rescaled metric Γ µν,\nwhich we denote as b, that will determine the notion of inverse temperature in the thermodynamics\nwith our conformal boundary conditions. We may compute this rescaled boundary metric to find,\nΓµνdxµdxν=b2\n(2π)2dτ2+r2dΩ2\n2with b= (2πK)1/3B2/3and r=\u00122πK\nB\u00131/3\n. (3.32)\nWe will ignore the cosmological constant here, although we do give a discussion of the case with\ncosmological constant in the Appendix D. Thus our infilling solutions must be Ricci flat and, in\nparticular, have a vanishing Ricci scalar R. As a result, the bulk action SEHvanishes and the action\nis given by the boundary term only:\nS[B,K] =−Θ\n8πGZ\n∂Md3x√γ K . (3.33)\nBy virtue of Birkhoff’s theorem [30], there are only two different line elements to investigate,\ncorresponding to Euclidean space and Euclidean Schwarzschild black hole, which we will analyse in\nturn.\nLet us first consider Euclidean space with a cavity of radius r=rE\nds2\nE=β2\nE\n(2π)2dτ2+ dr2+r2dΩ2\n2, (3.34)\nfor which the Euclidean action reads\nSE[B,K] =−rEβEΘ\nG. (3.35)\nFurthermore, it a simple exercise to compute\nBE=βE\nrEandKE= 21\n2pr2−1\n2p\nEβE\n2π, (3.36)\nwhich we could use to rewrite SE[B,K] as a function of BandKexplicitly.\nLet us turn our attention now to the line element of Euclidean Schwarzschild inside a cavity of\nradius r=rB\nds2\nB=β2\nBF(r)2\n(2π)2F(rB)2dτ2+dr2\nF(r)2+r2dΩ2\n2 (3.37a)\nwith\nF(r)2= 1−r+\nr. (3.37b)\nThe horizon, located at r=r+, is an S2bolt [31] of area 4 πr2\n+and regularity there requires the\nstandard choice [32–34] for the length of the thermal circle\nβB=2π\nF′(r+). (3.38)\n– 14 –The on-shell action now reads\nSB[B,K] =−rBβBΘ\n4G4−3x√1−x(3.39)\nand furthermore\nBB=βB\nrB= 4πx√\n1−xandKB= 2−1\n2pr2−1\n2p\nB(4−3x)1\n2p(1−x)−1\n4pβB\n2π, (3.40)\nwhere we defined x≡r+/rB∈(0,1). We see from above that, at a fixed and sufficiently small BB\nthere are exactly twoblack hole saddles with distinct values of x, one with x >2/3 and the other with\nx <2/3, as first observed by York [1]. The saddle with larger (smaller) values of xis denoted by large\n(small) Euclidean Schwarzschild black hole. We note that since xis determined by the conformal class\nof the boundary it is not sensitive to the value of p, and thus the boundary between small and large\nblack holes, x= 2/3, is the same for all the boundary conditions we consider.\nWe now impose our boundary conditions ΓE\nµν= ΓB\nµν, which in turn demands\nβB\nrB=βE\nrE= 4πx√\n1−xand rB= 4−1\n1−6p(4−3x)1\n1−6p(1−x)−1\n2(1−6p)rE, (3.41)\nand automatically ensures BE=BH≡ B andKE=KH≡ K. The difference between the on-shell\nactions can then be readily computed and we find\n∆S[B,K]≡SB[B,K]−SE[B,K] =−22\n1−6pB1+4p\n1−6p(2πK)−4p\n1−6pΘ\nGH(p, x) (3.42a)\nwith\nH(p, x)≡\u0012\n1−3x\n4\u00131+2\n1−6p\n(1−x)−1\n2−1\n1−6p−1. (3.42b)\nFor any pwe see that H(p,8/9) = 0 indicating a first order transition between the large black hole\nand hot flat space. For small xwe have,\n∆S[B,K] = 22\n1−6pB1+4p\n1−6pK−4p\n1−6p1\n4G\u0002\nx+O(x2)\u0003\n(3.43)\nshowing that the small Euclidean Schwarzschild saddle is never dominant. Furthermore, since\nΘH∼ −x\n4(3.44)\nnear x= 0 and has a single vanishing derivative at x= 2/3 in the interval x∈(0,1), we conclude that\nΘHhas a minimum at x= 2/3. This shows that the large Euclidean Schwarzschild saddle becomes\ndominant for B<8π/(3√\n3), whereas hot Euclidean space dominates in the complementary regime.\nIndeed, this result is independent of p! This is the same result that York found for Dirichlet boundary\nconditions [1] and analogous to the Hawking-Page transition in AdS spacetimes [35]. One might wonder\nwhy the point of transition is independent of p– this can be understood as being due to the on-shell\naction reducing simply to a boundary term, and changing ponly changes the coefficient of this term.\nThus the action of the competing saddles are all scaled together as pis varied, and so the value of B\nwhere the large black hole and hot space have equal action is not changed.\n– 15 –Now let us have a look at the first law of black hole thermodynamics derived in section 3.1 again.\nIn the black hole background (3.37a), we have\nb= 22−1\n6pπx(4−3x)\u00124−3x√1−x\u00131\n6p−1\nr1−1\n6p\nB,r= 2−1\n6p\u00124−3x√1−x\u00131\n6p\nr1−1\n6p\nB, (3.45)\nwhile for the potential Fand the energy Ewe find\nF=SB\nb=−21\n6pΘ\n4G·r1+1\n6p\nB·\u00124−3x√1−x\u00131−1\n6p\n, (3.46a)\nE=Z\ng∂Mdd−2x√n ρ=21\n6pΘ\n4G·r1+1\n6p\nB·(x−4Θ + 3 xΘ)·(1−x)1\n12(−6+1\np)\n(4−3x)1\n6p. (3.46b)\nCombining these together gives an explicit check of our earlier equation (3.29),\nb(E−F) =πr2\nBx2\nG=πr2\n+\nG=A\n4G, (3.47)\nwhere A= 4πr2\n+is the area of the black hole horizon, recovering the standard Bekenstein-Hawking\nentropy of a black hole.\nGiven the inverse temperature band entropy SBH, we can calculate the black hole specific heat via\nC=−b∂SBH\n∂b. (3.48)\nThis is simple to compute, but we are interested only in its sign, which is governed by the conformal\nclass as,\nC= (some positive expression) ·16x−8−9x2+ 12p(1−x)(4−3x)\n(6p−1)(3x−2). (3.49)\nWe note that for p= 0 this agrees with the specific heat computed in [15]. As for Dirichlet boundary\nconditions, for the small black hole, x <2\n3, the specific heat is always negative, indicating thermody-\nnamic instability. For the large black hole, x >2\n3, ifp <1\n6then the specific heat is positive as in the\nDirichlet case. However for p >1\n6then the specific changes from positive to negative when the black hole\nis large enough, as can be seen because C∼(positive expression)1\n1−6p(1 + (1 −12p)(1−x))+O(1−x)2\nnear x= 1. This transition happens at x→1 ifp→+∞so we still recover the Dirichlet result that\nthe large black hole is thermodynamically stable.\nThus in the case p >1/6 we obtain the novel behaviour that the specific heat capacity of sufficiently\nlarge large black holes is negative, naively indicating thermodynamic instability. We will later explore\nwhether this potential instability shows up in either the fluctuations about this Euclidean saddle point,\nor its dynamical stability.\n3.3 Fluctuations about the empty spherical cavity\nThe aim of this subsection is to demonstrate that an empty spherical cavity filled with hot Euclidean\nspace is devoid of fluctuation negative modes if pis sufficiently large, specifically if p > 1/6. Our\nfindings can be extended to Λ ̸= 0, and we defer discuss this explicitly in appendix C. While we will\nprimarily concentrate on four spacetime dimensions for the sake of presentation, we have verified that\n– 16 –our results generalize to d≥4mutatis mutandis . Note however that in d≥5 one should also consider\ntensor modes, but these are the same as in the Dirichlet case. The eigenvalue problem to study is\nsimply that given in Eq. (3.8)\n(∆Lh)ab=λ habwith ∇aha\nb=1\n2∇bh (3.50)\naround hot Euclidean space with a cavity of radius r=rE(see Eq. (3.34)), so,\nds2\nE=β2\nE\n(2π)2dτ2+ dr2+r2dΩ2\n2, (3.51)\nwhere recall that τ∼τ+ 2πso that the length of the thermal circle is βE.\nAs emphasized in [22], when gravity is placed in a cavity, due to the mixing of the transverse\ntraceless and conformal modes the Lichnerowicz operator is self-adjoint with respect to the non-positive\nDeWitt metric, and thus its eigenvalues need not be real. It was argued that stability is determined by\nthe real part of the eigenvalues being positive. This precisely correlates with stability of a saddle point\nunder Ricci flow – a saddle is a fixed point of Ricci flow, and a negative real eigenvalue corresponds\nto an unstable mode under the flow that takes one away from this fixed point in flow time. Thus for\nstability we require,\nReλ≥0 (3.52)\nfor all eigenvalues λ.\nSince ∂/∂τ is a Killing isometry, we can decompose all the perturbations into Fourier modes\neinτwith n∈Z. Furthermore, we can exploit the background SO(3) symmetry and decompose all\nmodes based on their transformations under diffeomorphisms on the S2. In four spacetime dimensions,\nthese fall into two classes: scalar-derived gravitational perturbations and vector-derived gravitational\nperturbations. Scalar-derived modes are built from spherical harmonics YℓSmSonS2, while vector-\nderived modes are built from vector harmonics on S2YℓVmV\nI , where upper case Latin indices will run\nover the sphere directions. In dimensions d≥5, there are also tensor harmonics to consider. There is\none further simplification that occurs for four-spacetime dimensions. Let GIJbe the metric on a round\ntwo sphere of unit radius. Let also DIbe the metric-preserving covariant derivative on the unit radius\nS2. It is easy to show that YℓVmV\nI are simply two-dimensional Hodge duals of gradients of YℓSmS. For\nspherical harmonics, we have ℓS≥0, while for vector harmonics ℓV≥1. For later convenience, we\ndefine\n˜τ=βE\n2πτand ei n τ=ei˜n˜τ, (3.53)\nwhere we note that n∈Z, but ˜ nis generally notan integer.\nThe analysis of the scalar fluctuations is rather technical, with various special cases to consider. The\nfull details are reserved for the Appendix A. The upshot is that the vector modes of the Lichnerowicz\noperator for our new conformal boundary condition have precisely the same spectrum as in the Dirichlet\ncase – there is no dependence on the boundary condition, and further they are all real and stable. Thus\nwe will concentrate on the scalar modes which need not be real, and depend in detail on the boundary\nconditions.\n– 17 –Non-spherical scalar-derived gravitational modes are constructed from scalar harmonics YℓSmS,\nwhich satisfy the usual eigenvalue equation\nDIDIYℓSmS+ℓS(ℓS+ 1)YℓSmS= 0, (3.54)\nwhere we recall that Iis an index on the two sphere and Dis the metric preserving covariant derivative\non the unit round two-sphere. Let lower case hatted Latin indices run over ˜ τandr. Scalar derived\nperturbations are then written as\nδds2≡+∞X\nℓS=0ℓSX\nmS=−ℓShℓSmS\nab(˜τ, r, θ, ϕ )dxadxb(3.55a)\nwith θandϕthe usual polar and azimuthal angles on the two sphere and\nhℓSmS\nˆaˆb=ˆfℓSmS\nˆaˆb(˜τ, r)YℓSmS(3.55b)\nhℓSmS\nˆaI =ˆfℓSmS\nˆa(˜τ, r)DIYℓSmS(3.55c)\nhℓSmS\nIJ =ˆhℓSmS\nL(˜τ, r)GIJYℓSmS+ˆhℓSmS\nT(˜τ, r)SℓSmS\nIJ (3.55d)\nwhere SℓSmS\nIJ is a traceless symmetric two tensor defined as\nSℓSmS\nIJ =DIDJYℓSmS+ℓS(ℓS+ 1)\n2GIJYℓSmS(3.55e)\nand\nGIJdxIdxJ= dΩ2\n2. (3.55f)\nSince all the angular dependence is fixed, we are left with determining ˆfˆaˆb,ˆfˆa,ˆhLandˆhT. At this\npoint we use the fact that ∂/∂˜τis a Killing vector field of the background and further expand ˆfˆaˆb,ˆfˆa,\nˆhLandˆhTas\nˆfℓSmS\nˆaˆb(˜τ, r) =ei˜n˜τfℓSmS˜n\nˆaˆb(r),ˆfℓSmS\nˆa(˜τ, r) =ei˜n˜τfℓSmS˜n\nˆa (r),\nˆhℓSmS\nL(˜τ, r) =ei˜n˜τhℓSmS˜n\nL (r) and ˆhℓSmS\nT(˜τ, r) =ei˜n˜τhℓSmS˜n\nT (r) (3.56)\nModes with different values of ℓS,mSand/or ˜ ndecouple from each other at the quadratic level in the\naction. For this reason, we shall drop the subscript ℓSmS˜nin what follows.\nIn the analysis of these fluctuations, given in detail in the Appendix A, the spherically symmetric\nmodes with ℓS= 0, and those with ℓS= 1 are special, since for those S1mS\nIJ= 0, and must be treated\nas separate cases. Furthermore the modes that are static with n= 0 must be treated separately.\nIn that Appendix we give analytic expressions for the various fluctuation component functions in\nequation (3.56). In all cases, included those that must be treated separately, the result of the scalar\nperturbation mode analysis may be summarized in the following condition on the eigenvalue:\np(ℓS, n, λ) =AℓS(˜Λ) + ϖ2CℓS(˜Λ)\nBℓS(˜Λ) + ϖ2DℓS(˜Λ)(3.57)\nwhere the lefthand side encodes the choice of boundary condition. Furthermore, we defined\n˜Λ≡˜λ−ϖ2, ˜λ≡λ r2\nE, ϖ ≡˜n rE (3.58)\n– 18 –and we have somewhat complicated expressions for AℓS,BℓS,CℓSandDℓS. These take the form,\nAℓS(˜Λ) =4X\nq=0aq(p\n˜Λ)J 1\n2+ℓS(p\n˜Λ)qJ3\n2+ℓS(p\n˜Λ)4−q(3.59)\nwhere the five coefficients aq(p\n˜Λ, ℓS) are polynomials in the argumentsp\n˜Λ and ℓS, up to quadratic\norder in ℓS, but have no explicit pdependence. These can be explicitly read off from the expressions\nin equations (A.26). The expressions BℓS,CℓSandDℓStake the same form. Ideally we would solve\nthe above expression (3.57) to obtain the eigenvalue λas a function of ℓS,nand the cavity radius rE.\nUnfortunately this appears not to be analytically tractable. However the above form is convenient to\nstudy the Euclidean stability of this background, hot Euclidean space, and we now discuss this.\nWe note that for Dirichlet boundary conditions, a similar analysis yields the condition on the\nmodes,\nBℓS(˜Λ) + ϖ2DℓS(˜Λ) = 0 (3.60)\nwhich corresponds to the vanishing of the denominator in (3.57), and hence to the infinite |p|limit.\nThis demonstrates that in the infinite |p|limit, modes with finite eigenvalue ˜λwill limit to those of\nthe Dirichlet problem. Conversely there will be modes whose magnitude of their eigenvalue diverges\nin the p→ ±∞ limit, and are not shared by the Dirichlet boundary conditions.\n3.3.1 Static spherical modes\nIn the case that p <1/6, we indeed find eigenmodes with Re λ <0. The simplest sector to see these\nis the static spherically symmetric scalar sector, n=ℓS= 0, where the condition above yields,\np\u0010\n˜λ\u0011\n=1\n12\n1 + 5 ˜λ−3˜λ3/2cot\u0010p\n˜λ\u0011\n−˜λcot\u0010p\n˜λ\u00112\n1 +˜λ−p\n˜λcot\u0010p\n˜λ\u0011\n. (3.61)\nNow expanding about ˜λ= 0 we find,\np=1\n6+1\n18˜λ+O\u0010\n˜λ2\u0011\n. (3.62)\nThus for p= 1/6 the spectrum has a marginal mode, with λ= 0, and near to this value, for p <1/6\nwe see that ˜λbecomes real and negative, and hence we see that an unstable mode with Re λ < 0\nexists. For real ˜λthen pis real and is given in figure 1. We may then follow the unstable mode above.\nAsymptotically one finds,\np=∓p\n−˜λ\n4+3\n4+O\u0010\n˜λ−1/2\u0011\n(3.63)\nwith the upper minus sign for Im ˜λ≥0 and the lower positive sign for Im ˜λ <0. This is shown in\nred on the figure, and so p→ −∞ as˜λ→ −∞ . From the figure we see the relation is monotonic\nfor˜λ <0, and hence the negative mode exists for all p <1/6. This asymptotics precisely illustrates\nthe behaviour mentioned above, namely that there are modes in the eigenspectrum with diverging\n– 19 –eigenvalue as p→ ±∞ . Here we see that for p→ −∞ this is a negative mode, whereas for p→+∞\nit is a stable mode.\nForp >1/6 there are no unstable modes in this static spherical sector. We see from the figure that\nmodes with positive real ˜λexist for p >1/6. Indeed for all values of pwe may also find modes with\ncomplex ˜λ, but as we will soon argue, these are stable having Re ˜λ >0. In figure 11 in Appendix A.1,\na variety of modes are shown for different p.\n-100 -50 0 50-2-10123\nFigure 1 :pas a function of ˜λfor the static spherical modes n=ℓS= 0. The green dot marks p=1\n6, where\n˜λis precisely 0. The red dashed curve gives the asymptotic behavior of p(˜λ) as˜λ→ −∞ .\n3.3.2 Instabilites for p <1/6for the static non-spherical modes\nForp <1/6 we have seen that there is an unstable spherically symmetric static mode. However for\nsufficiently small pthere are other unstable modes, both with real but also complex ˜λ. This can already\nbe seen in the static sector with ϖ= 0 (and so n= 0).\nIn figure 2 we plot pagainst negative ˜λfor both ℓS= 2 and 4. For ℓS= 2 then p=−2/3 at˜λ= 0,\nand we see that for the range −2\n3< p < −0.64422 there are a pair of unstable modes with real ˜λ. For\np <−2\n3there persists a single unstable mode. Likewise for ℓS= 4 then p=−17\n6at˜λ= 0 and for the\nrange −17\n6< p < −1.98491 there is a pair of unstable modes with real ˜λ, and for p <−17\n6there is a\nsingle real unstable mode. This phenomena holds for all ℓS, each giving rise to a pair of real unstable\nmodes for sufficiently small p, and a single mode for smaller p≤1\n12(2−ℓS−2ℓ2\nS), where the righthand\nside of this inequality derives from the value of pat˜λ= 0.\nHowever the situation is more complicated as for certain ranges of pthere may also exist complex\nunstable modes, so modes with complex ˜λwhere Re ˜λ <0. We may see these in the example of ℓS= 2\nand 4 depicted in figure 3. Here a contour plot of the real part of p, Rep, is shown in the complex\n˜λplane in the potentially unstable region, Re ˜λ < 0. Now pmust be a real value, and so only if\nIm(p) = 0 can this value of pcorrespond to a physical unstable mode. On the figures we have plotted a\nred curve showing where Im( p) vanishes. While it vanishes along the negative real axis, for ℓS= 2,4 it\n– 20 –-8 -6 -4 -2 0-0.71-0.70-0.69-0.68-0.67-0.66-0.65\n-140 -120 -100 -80 -60 -40 -20 0-3.0-2.8-2.6-2.4-2.2-2.0Figure 2 :pas a function of ˜λfor the static non-spherical modes with ℓS= 2 (left panel) and ℓS= 4 (right\npanel). The green dots mark\u0000\n0,1\n12(2−ℓS−2ℓ2\nS)\u0001\n.\nvanishes also on a curve that then connects to the imaginary ˜λaxis. This leaves the negative real axis\nprecisely at the point which maximizes p, so at p=−0.644 and −1.98 for ℓS= 2 and 4 respectively.\nWe find that pincreases along the curve towards the imaginary axis where p=−0.549 and −1.39 for\nℓS= 2 and 4 respectively. Therefore we conclude that for ℓS= 2 and p∈(−0.644,−0.549) we have\nunstable modes with complex eigenvalue, and likewise for ℓS= 4 we have p∈(−1.98,−1.39). In fact,\nwe observe this same feature for all ℓS≥2. The lowest few modes for ℓS= 2 are given in figure 13 in\nappendix A.3.1. The green dashed curve below the line Re ˜λ= 0 is consistent with our analysis here.\n3.3.3 Stability for p >1/6for all modes\nHaving seen that for p <1/6 there is at least one unstable mode, and that these modes may have\ncomplex eigenvalue, we now argue that for p >1/6 there are no unstable modes for any ℓSandϖ.\nBy inspecting the function for p(˜Λ) for different ℓSandϖin the complex ˜Λ plane one can verify this.\nHowever, the presence of unstable modes off the real ˜λaxis means that it is challenging to rule out\nunstable modes for all ℓSandϖ. Thus rather than relying on inspection of the function, we give a\nneater argument based on analyticity properties of the function p. The details of the argument are\npresented in Appendix B. The main claims are the following:\nClaim 1. : Repis maximized in the complex half plane Re ˜λ≤0 by its value on the imaginary axis\nRe˜λ= 0.\nWe are not able to give a rigorous proof of this, but are able to give a convincing demonstrate\nof this by a combination of analytic and numerical work presented in that Appendix. It follows from\nshowing that the function pis analytic in the complex half plane Re ˜Λ<0 and further observing that\n– 21 –-4 -3 -2 -1 002468\n-0.66-0.62-0.58-0.54-0.50\n-30 -25 -20 -15 -10 -5 0051015202530\n-2.8-2.6-2.4-2.2-2.0-1.8-1.6-1.4Figure 3 : Contour plot of Re pin the complex ˜λplane for ℓS= 2 (left panel) and ℓS= 4 (right panel). The\nred dashed curves are where Im p= 0 that corresponds to a physical unstable mode.\nRepgoes to −∞ asymptotically in all directions in that half plane. Recalling that ˜Λ = ˜λ−ϖ2with\nϖ > 0 and real, then this implies the same results hold in the complex half plane Re ˜λ < 0 of the\neigenvalue ˜λ, with this half plane being the one of interest for stability. Being analytic then writing\n˜λ=x+iy, it obeys the Laplace equation in the variables x, y, and by the maximum principle must\ntake its greatest value on the boundary of the region.\nA corollary of this result is that if Re pis bounded from above on the imaginary ˜λaxis, then that\nbounds Re pfor all ˜λin the complex half plane where Re ˜λ <0 and thus putative instabilities might lie.\nClaim 2. : Rep <1/6 for all ℓSandϖon the imaginary axis ˜λaxis.\nThe combination of these two claims then imply that Re p <1/6 for all ˜λwith Re ˜λ <0. While of\ncourse pmust be real for a physical mode to exist, this bound on the real part of pimplies that there\ncan be no instabilities for p >1/6.\nWe relegate showing analyticity to the Appendix, but here simply note that both the numerator\nand denominator of the expression for pin (3.57), given by AℓS(˜Λ) + ω2CℓS(˜Λ) and BℓS(˜Λ) + ω2DℓS(˜Λ)\nrespectively, are clearly analytic for Re ˜Λ<0 except for branch cuts along the negative real axis\nfrom the origin, from the properties of Bessel functions. Since ˜Λ = ˜λ−ϖ2, then the same holds\nfor Re ˜λ < 0. In fact pis analytic at the origin, the non-analytic behaviour of the numerator and\ndenominator cancelling there. Hence showing that pis analytic for Re ˜λ <0 comes down to showing\nthat the denominator has no zeros in that region, and this is the work detailed in Appendix B.\nOnce analyticity is shown then Re pis maximized on a boundary or asymptotically. Due to the\nreality of the components of the expression for p, we have,\nRep(x, y) = Re p(x,−y),Imp(x, y) =−Imp(x,−y) (3.64)\nwhere again ˜λ=x+iy, so that Re pis an even function of y= Im ˜λ. Further the leading asymptotics\n– 22 –for the static spherical fluctuation case in equation (3.63) are similar to those when ℓS, ϖ̸= 0. We\nrecall that ℓSis a positive number, and ϖis a positive real. Using the asymptotic properties of Bessel\nfunctions we find in the upper unstable (ie. Re ˜λ <0) plane, so x∈(−∞,0],y∈[0,∞), that pgoes\nas,\np=−p\n−˜Λ\n4+3\n4+O\u0010\n˜Λ−1/2\u0011\n(3.65)\nwith ℓSandϖonly affecting the subleading terms, and since ˜Λ = ˜λ−ϖ2we recover the same\nasymptotics in ˜λas above in (3.63). Then for any finite ϖ, writing ˜λ=reiθthen the leading asymptotic\nbehaviour as r→ ∞ is, Re p→ −√r\n4sin\u0000θ\n2\u0001\nfor the unstable region of the upper half complex plane\nπ\n2≤θ≤π. Since sin\u0000θ\n2\u0001\n>1√\n2then as r→ ∞ then Re p→ −∞ . As a result of this, the only\nasymptotic region or boundary remaining for x≤0 where Re pcould be maximized is the imaginary\n˜λaxisx= 0 where Re ˜λ= 0, or equivalently, Re ˜Λ =−ϖ2.\nWe now consider the behaviour of Re pon the imaginary ˜λaxis. To address the second claim we\nwill argue that at fixed ˜λon the imaginary axis, Re pfor all positive numbers ℓSand real ϖ≥0 takes\nits maximum value for the static spherical case ℓS=ϖ= 0. This follows from the fact that we find\nthe behaviour of Re pat fixed ˜λdecreases with increasing ℓSand with increasing ϖ.\nFirstly in figure 4 we show the value of Re pat the origin, ˜λ= 0, so at Re ˜Λ =−ϖ2, as a function\nofϖfor various ℓS= 0, . . . , 6. We see that indeed Re pdecreases for increasing ℓSand increasing ϖ.\nThe maximum value of 1 /6 is attained for the spherical static case. The asymptotic behaviour of these\ncurves is given by,\nRep=−ϖ2\n12+4−3ℓS−3ℓ2\nS\n24+O\u0000\nϖ−2\u0001\n(3.66)\nand is shown in figure 4 by the dashed curves. The value of Re pat the origin in the figure, ϖ= 0, is,\np=2−ℓS−2ℓ2\nS\n12. (3.67)\nWe see both the value at the origin and the asymptotic behaviour decrease for increasing ℓS.\nSecondly in figure 5 we show Re pwith ϖ= 0 on the imaginary ˜λaxis again for ℓS= 0, . . . , 6. We\nshow the asymptotic behaviour, at fixed ℓS, which goes as,\nRep=−√y\n4√\n2+3\n4−24 + 11 ℓS+ 11ℓ2\nS\n24√2y+O\u0000\ny−3/2\u0001\n(3.68)\nas dashed lines, which we see decreases in value as ℓSincreases. The plot confirms that increasing ℓS\nindeed decreases the maximum value of Re pon the imaginary axis. For ℓS= 0 the maximum is the\nvalue at the origin, where p= 1/6. For ℓS>0, where the value at the origin is given above in (3.67),\nin fact the maximum of Re poccurs away from the origin on the imaginary ˜λaxis.\nThus from the previous plots we have seen that Re pis maximized for ϖ= 0 on the imaginary\naxis by the spherically symmetric modes ℓS= 0. At fixed location on the imaginary axis, increasing\nℓSdecreases Re p. Further we have seen that for ˜λ= 0 increasing ϖalso decreases Re p. Generally\nwe find that increasing ϖalways decreases Re pon the imaginary axis. In figure 6 we show surfaces of\nRepplotted on the imaginary axis and also against ϖ, with various ℓS= 0, . . . , 6. For fixed Im ˜λand\n– 23 –0 2 4 6 8 10-14-12-10-8-6-4-20Figure 4 : Repas a function of ϖat˜λ= 0 for different ℓS’s. The dashed lines give the asymptotic behaviors.\nThe black solid line is Re p= 0.\n0.01 0.10 1 10 100 1000-6-5-4-3-2-10\nFigure 5 : Re pas a function of Im ˜λatϖ= 0 for different ℓS’s. The dashed lines give the asymptotic\nbehaviors. The black solid line is Re p= 0.\nφwe again see a decrease with increasing ℓS. For fixed Im ˜λandℓSwe see a monotonic decrease with\nincreasing φ. Hence the ℓS= 0 curve in figure 5 bounds all other values of ℓS= 0 and ϖ= 0, and its\nmaximum value is at the origin ˜λ= 0, where p= 1/6. Thus we have that Re p <1/6 for all ℓSand\nϖon the imaginary axis, and hence the same is true in the unstable part of the complex ˜λplane (i.e.\nwhere Re ˜λ <0). Finally then, we arrive at the conclusion that there can be no Euclidean instability\nforp >1/6.\n3.4 Black hole fluctuations: spherical static negative modes\nThe existence of negative modes for black holes is a sign of local thermodynamic instability [2]. In\n[22], the authors showed that if we compute the spectrum of the Lichnerowicz operator acting on static\n– 24 –Figure 6 : Plot of Re pagainst ˜λon the imaginary axis and also against ϖ. This is plotted for various values\nofℓs= 0, . . . , 6. The orange surface is that with ℓS= 0 and bounds all the others from above. At fixed Im ˜λ\nandϖincreasing ℓSdecreases Re p. At fixed Im ˜λ, increasing ϖdecreases Re pfor all ℓS.\nspherically symmetric perturbations, then the path integral stability matches with thermodynamic\nstability for Euclidean Schwarzschild (AdS) black holes under Dirichlet boundary conditions. We\nwill now consider the same problem, namely that of computing the eigenvalues of the Lichnerowicz\noperator for our generalized conformal boundary condition on a black hole background, and comparing\nthe Euclidean stability determined by the existence of unstable modes to the thermodynamic stability.\nUnlike the previous section, where we considered the hot flat space saddle point, here we are unable\nto solve the fluctuations analytically for a black hole background. We therefore proceed numerically,\nand restrict to considering only the static spherically symmetric sector. For simplicity here we present\nonly the case without cosmological constant. However for completeness in the Appendix D we give\nresults where the cosmological constant is included. To proceed we take the background Schwarzschild\nsolution in equation (3.37a), and consider the fluctuation,\nδds2≡habdxadxb=F(r)2a(r) dτ2+b(r)dr2\nF(r)2+r2c(r) dΩ2\n2, (3.69)\nwhere a,bandcare unknown functions of r. We impose our boundary conditions given in (C.3)\nand the gauge condition (C.5), and then examine the existence of negative modes. The numerical\nmethod we use here is the same as the one proposed in [22] and reviewed in [25]. One can also use the\ntraditional operator approach, e.g., in [21].\n– 25 –Figure 7 gives the lowest four modes of the Lichnerowicz operator in the background of a Schwarzschild\nblack hole inside a cavity with our boundary conditions for ptaking three values p= 0,1/6 and 1 /3.\nIt is worth recapping what happens for Dirichlet boundary conditions [22]. There a small black hole,\nx <2/3, has a negative mode and large black holes where x >2/3 have none. This negative mode\nis the generalization of the Gross-Perry-Yaffe negative mode of asymptotically flat Schwarzschild [2],\nand indeed becomes precisely that mode in the very small black hole limit x→0. As one moves from\nthe small black hole branch to x= 2/3 the negative mode becomes a marginal zero mode, and is\nsimply given by the tangent to the space of Schwarzschild solutions, and so is generated by perturbing\nthe Schwarzschild radius r+. This occurs as at this special point the conformal class, and hence the\ntemperature measured in units of the 2-sphere radius, is stationary as r+is deformed. This Euclidean\nstability exactly coincides with thermodynamic stability, the small black holes having negative specific\nheat and the large ones having positive specific heat. Now with our generalized boundary conditions\nwe recover what we might have naively expected. The number of static spherical negative modes for\na black hole in a cavity can be understood by taking the number of static spherical negative modes in\nan empty cavity, and adding a negative mode in the case the black hole is small. Firstly consider the\ncasep <1/6, illustrated in the figure in the left panel for p= 0. Here for p <1/6 we find two static\nspherical negative modes for small black holes, so x <2/3, one associated to the cavity boundary, and\none to the ‘small’ horizon. For large black holes with p <1/6 we find only one negative mode, which\nwe take to be associated to the cavity. At x= 2/3 there is an exactly marginal zero mode that again is\nsimply tangent to the Schwarzschild family and is generated by perturbing r+, and exists since as we\nhave seen the dimensionless temperature B=β/R 0has a minimum and so is stationary as r+varies\nprecisely at x= 2/3. Looking back to our thermodynamic calculation of specific heat, the change in\nthe number of negative modes at x= 2/3 agrees with the change in sign of the specific heat there.\nWe note that while the p < 1/6 large black holes have a positive specific heat, they suffer from a\nEuclidean negative mode associated to the cavity, and thus presumably are not good saddle points of\nour ensemble.\nNow we consider the special case p= 1/6, illustrated in the middle panel of the figure. Now we\nhave a negative mode associated to the horizon for small black holes and none for large, and for all\nblack holes we have an exact zero mode. The cavity instability for p <1/6 becomes precisely marginal\nforp= 1/6. In fact one can quickly see that this marginal mode simply corresponds to a global scaling\nof the cavity, where both the black hole radius r+and the cavity radius R0are perturbed together,\nand arises as for the special value p= 1/6 the boundary condition, γpK=constant is scale invariant,\nwhile the conformal class is always scale invariant.\nThe most interesting case is p >1/6. Here we have seen an empty cavity has no unstable modes\nand thus naively we expect that a small black hole will have one negative mode, a generalization of that\nof Gross-Perry-Yaffe, and a large black hole will have no unstable modes. This is precisely what we\nobserve in the numerics, and can be seen in the right panel of the figure where the low-lying spectrum\nfor the representative value p= 1/3 is shown. But this gives rise to a strange situation – we have seen\nthat the thermodynamics of black holes for p >1/6 has the novelty that very large black holes have\nnegative specific heat capacity, and thus we would expect them to be thermodynamically unstable.\nHowever we see no hint of this in the Euclidean fluctuation analysis where the saddle point appears to\nbe stable, at least to static spherically symmetry modes. In particular in that figure where p= 1/3 we\ndenote the range x≥0.845 for which the large black hole negative specific heat is negative, and we see\n– 26 –0.0 0.2 0.4 0.6 0.8 1.0-10123\n0.0 0.2 0.4 0.6 0.8 1.0-10123\n0.0 0.2 0.4 0.6 0.8 1.0-10123Figure 7 : The real part of the four lowest lying eigenvalues of ˆ∆Las a function of xfor the case with Λ=0.\nThe left, middle, and right panels correspond to p= 0,p=1\n6, and p=1\n3, respectively. The green dots marks\nx=2\n3. The gray solid line and the green dashed line are Re ˜λ= 0 and Re ˜λ=˜λGPY=−0.7677, respectively.\nThe dashed red line in the red panel denotes x=2\n3\u0000\n3−√\n3\u0001\n≈0.845, where the specific heat changes sign,\naccording to equation (3.49). Since only the sign of the real part of the eigenvalues determines the path\nintegral stability, we do not distinguish between real modes and complex modes.\nno hint of negative modes. We have carefully checked for other values of pfor the existence of static\nspherically symmetric negative modes in the range of xwhere large black holes are thermodynamically\nunstable, finding none.2While it is reasonable that the specific heat might be too crude a measure to\ncapture a Euclidean instability – for example the large black holes with p <1/6 have an unstable mode\ndue to the cavity, but positive specific heat – it seems far less reasonable that on can have unstable\nthermodynamics but no sign of this when considering Euclidean fluctuations. It is worth emphasizing\nthat we are only considering static and spherical fluctuations, but naively we might have expected\nthese to capture the negative specific heat of large black holes for p >1/6. We will discuss this further\nwhen we consider the Lorentzian stability of these solutions.\n4 Dynamical Lorentzian Stability of Generalized Conformal Boundary Conditions\nHaving discussed a new well-posed set of Euclidean boundary conditions, which apparently give a\nsensible thermodynamics in the case of p >1/6 for Euclidean spherical cavities, we now consider the\nLorentzian stability of these systems. We will lead to two related puzzles.\nFirstly in section 4.1, we consider the Lorentzian dynamics in the spherical subsector of the theory.\nDue to the Birkhoff theorem the bulk geometry is fixed, and we may analyse the behaviour by consid-\nering purely the dynamics of the cavity boundary, which is analogous to the motion of a ‘brane’, and\ncan be solved non-linearly. We find that the Lorentzian stability of flat spacetime filled cavities is the\nsame as that discussed above in the Euclidean theory, namely instability for p <1/6 and stability for\np >1/6. The Lorentzian instability we see is identified to the dynamics of the boundary, and gives a\n2As a further check for p >1/6 we have applied Ricci flow to static spherical perturbations of large thermodynamically unstable\nSchwarzschild black holes with our boundary conditions, following the methods in [21]. We will not give the full details here,\nbut report that generic static spherical perturbations quickly decay back to the large black hole fixed point, again confirming our\nconclusion that there are no Euclidean negative modes in the static spherical sector.\n– 27 –physical interpretation for the Euclidean instability observed above. The unstable cavity walls wants\nto shrink to a point, or to expand forever.\nThen in section 4.1.2, we consider the spherical dynamics of black holes, which produces a mystery.\nForp <1/6 the picture is as expected – there is a dynamical instability associated to the boundary\nwhich is paired to a Euclidean instability. However for sufficiently large black holes, for p >1/6 we see\na new dynamical instability, again associated to the boundary dynamics. This time this is not seen\nin the preceding Euclidean analysis, although interestingly it is hinted at in the earlier calculation of\nthe specific heat. Thus we have our first puzzle; the strange situation where a cavity with a black\nhole is stable in the Euclidean setting, but is unstable in Lorentzian signature. We should note that\nour Euclidean analysis of the Euclidean black hole was restricted to static and spherically symmetric\nmodes, and this might be a potential resolution, although we suspect that this is not the case.\nOur second puzzle then arises, in section 4.2, when we go beyond the spherical sector. We may\nanalyse this only using linear perturbation theory. For flat spacetime filled cavities we find Lorentzian\ninstabilities even forp >1/6, when they are stable in Euclidean signature. Thus going beyond the\nspherical sector we find a similar situation to that for the black holes, namely a stable Euclidean\nbehaviour but unstable Lorentzian dynamics, even for an empty cavity.\n4.1 Spherical stability\nWe begin with a spherically symmetric infilling geometry of a spherical cavity,\nds2=−V(t)F(r)2dt2+dr2\nF(r)2+r2dΩ2. (4.1)\nOf course in this situation for gravity with a cosmological constant Birkhoff’s theorem dictates the\ninfilling solution must be static (as in our ansatz above), and must in fact be Schwarzschild or its\ngeneralizations to (A)dS-Schwarzschild. Explicitly then it dictates the general infilling solution is:\nF(r)2=−r2Λ\n3+ 1−r0\nr. (4.2)\nHere we have also included a reparameterization of time for convenience through the introduction\nof the (strictly positive) function V(t). One might then naively imagine that with no gravitational\ndynamics for such spherical symmetry, there is no possibility to have an instability3.\nThe main point we wish to make here is that while the interior geometry is fixed, the boundary\nconditions for the cavity wall can induce an instability entirely localized there. In fact this instability\nis quite unrelated to the interior Einstein equation, and hence to emphasize this we will leave the\nfunction F(r) entirely free, assuming only the static, spherical interior whose line element is above.\nLet us consider the boundary which shares the spherical symmetry. Hence in our coordinate chart this\nwill follow a trajectory in the t-rplane – we may think of this as the dynamics of a ‘brane’ – which\n3We note that in fact one may take any 2-dimensional homogeneous space, so that,\nds2=−V(t)F(r)2dt2+dr2\nF(r)2+r2dΩ2\nk (4.3)\nwhere dΩ2\nkis the metric on the 2-dimensional homogeneous space, which explicitly we can take as dΩ2\nk=1\n1−kx2dx2+x2dy2,\nso that its 2d Ricci scalar curvature is equal to 2 k. The calculation leads to precisely the same equation (4.8) below, and is\nindependent of k(except through the function F(r)).\n– 28 –we parameterize as r=R(t), and we assume the interior of the spacetime near the cavity is r < R (t),\nso that the two dimensional homogeneous space shrinks inside the cavity.\nWe will impose our generalized conformal boundary conditions, so fixing the conformal class and\nγpK, or equivalently for non-vanishing Kfixing the rescaled boundary metric Γ µν, and ask whether\nthis is stable. The outward directed unit normal form, nµ, to the boundary is then,\nnµ=1q\nF(r)2−R′(t)2\nF(r)2V(t)(−R′(t),1,0,0) (4.4)\nwriting the coordinates as xµ= (t, r, x, y ). Using this we compute the projection tensor ⊥µν=gµν−\nnµnν, which can be used to pull the metric back to the cavity hypersurface to get the induced metric γµν,\nand the extrinsic curvature Kµν=⊥α\nµ∇αnν, from which we can extract its trace, K. We assume that\nfor the bulk infilling there is a static boundary location at radius r=R0. Taking V(t) = 1 /F(R0)2and\nR(t) =R0, we term the induced metric, γ0, and trace of the extrinsic curvature of this static boundary\nasK0, where these are given as,\nds2\nγ0=−dt2+R2\n0dΩ2, K 0=F′(R0) +2\nR0F(R0). (4.5)\nWe wish to consider the dynamics of the cavity, where the conformal class of the boundary γis fixed\nto that of the induced metric above, γ0, and γpKis fixed to its value for this static case, so equal to\nγp\n0K0.\nFor a dynamical cavity wall, so R=R(t) we make the choice that,\nV=˙R2\nF(R)4+\u0012R\nF(R)R0\u00132\n, (4.6)\nand then we see that indeed the induced boundary metric is conformal to the static one above,\nds2\nγ=R2\nR2\n0ds2\nγ0. (4.7)\nNow the condition on γpKcan be written explicitly as the ‘brane’ equation of motion,\n¨R\nR+˙R2\nR2−\u0012R\nR0\u00131−6p\u0012\nF′(R0) +2\nR0F(R0)\u0013s\n˙R2\nR2+F(R)2\nR2\n0=−R\nR2\n0F(R)\u0012\nF′(R) +2\nRF\u0013\n(4.8)\nand we may then regard the position of the cavity wall as a ‘brane’ undergoing this dynamics.\nIf we perturb about the static point, so R=R0+δR, then linearizing we find,\nδ¨R=−αδR , α =F(R0)\u0012\nF′′(R0) +2(1 + 3 p)\nR0F′(R0)−2(1−6p)\nR2\n0F(R0)\u0013\n. (4.9)\nThus the stability is simply determined by the sign of the coefficient α, positive implying stability and\nnegative implying instability. Again we emphasize this depends on the Einstein equation only through\nthe geometry of the static spacetime, and specifically the function F. In the case that the fixed point\nR=R0is unstable, we may use the dynamics above to solve the full non-linear evolution to see where\nthe instability leads.\n– 29 –We might wonder what happens if we instead take the spacetime interior to be r > R (t) so that the\nspherical spatial sections increase in size away from the cavity wall. This flips the sign of the extrinsic\ncurvature, but the last two equations above governing the dynamics of cavity boundary, so R(t), are\nunchanged.\n4.1.1 Linear instabilities for spherical cavities and black holes\nNow let us specialize to the case of a spherical cavity with no cosmological constant filled by flat\nspacetime, so F(r) = 1. Then,\nα=−2(1−6p)\nR2\n0, (4.10)\nso that for cavity boundary conditions with p < 1/6 (including Anderson boundary condition) the\ncavity is unstable, but for p > 1/6 we have stability. We note that for this spherically symmetric\nsector the stability precisely agrees with that of the Euclidean fluctuation stability. We might expect\nthis since for the marginal case of dynamical stability, p= 1/6, the Lorentzian static perturbation\ncontinues to a Euclidean zero mode, and hence marks the boundary of Euclidean stability as pvaries.\nWe will show this is indeed the case shortly. We note that just as in the Euclidean case, we expect the\nfrequency spectrum of perturbations to agree with the Dirichlet case in the |p| → ∞ limit. However\nthere may be modes with diverging frequency that do not coincide with those of the Dirichlet problem,\nand we may view this static spherical dynamics exactly as such a perturbation – we see the frequency,\ngoing as√αdiverges in magnitude with |p| → ∞ . For p→ −∞ this leads to an increasingly rapid\ninstability, whereas for p→+∞it is a stable mode.\nWe may consider the case with cosmological constant too. Then writing ˜ x= ΛR2\n0, we have that\n˜x <0 is the Anti de Sitter case, and 0 <˜x <3 is the de Sitter case where the cavity lies between\nthe origin at the de Sitter horizon. Finally ˜ x= 0 corresponds to the flat case with zero cosmological\nconstant. Then we find,\nαR2\n0=−1 + 6 p(2−˜x)−3\n3−˜x. (4.11)\nAs ˜x→3 then αis negative, and hence a cavity sufficiently close to the de Sitter horizon size is always\nunstable for any p. Conversely for a large AdS cavity, so x→ −∞ , then for any p >0 the cavity is\nstable, while for p >1/6 any AdS cavity is stable.\nNow let us consider the linear stability of a Schwarzschild black hole background with no cosmo-\nlogical constant, so F(r)2= 1−r+\nr. Then from equation (4.9), we obtain,\nα=F(R0)2\n4R2\n0(1−x)2·[12p(x−1)(3x−4) +x(16−9x)−8] (4.12)\nwhere x≡r+/R0∈(0,1) and so a marginal static perturbation implies\np∗=8−16x+ 9x2\n12(1−x)(4−3x). (4.13)\nThen p > p ∗is the criteria for the black hole to be stable under our one-parameter family of boundary\nconditions. In the small black hole limit x→0 this reproduces the flat space criteria. When 0 < x <\n2/3 then p∗is in the narrow interval p∗∈(pmin,1/6) with pmin= (√\n6−2)/3≃0.150. Instead when\n– 30 –2/3< x < 1, then we have p∗>1/6, and further we see from the asymptotic behaviour, p∗≃1\n12(1−x)\nasx→1 that p∗diverges to positive infinity as xapproaches to 1, indicating that large enough black\nholes are never stable unless we have strict Dirichlet boundary conditions. It appears that the close\nproximity of the cavity to the horizon leads to instability independent of psimilar to the case of the\nde Sitter cavity discussed above, where the cavity boundary becomes unstable if it gets to close to the\nde Sitter horizon.\nNow if we compare the above equation with the black hole specific heat given in equation (3.49), we\nsee that the minimal value of xfor stability for p >1/6 is precisely the same as the xwhere the specific\nheat changes sign. As a result, there is a dynamical instability associated with the thermodynamically\nunstable large black holes.\nThus our first mystery is revealed. We have seen a correspondence between the Lorentzian and\nEuclidean stability of flat space cavities. However, for black hole interiors and taking p > 1/6 the\nthermodynamics, specifically the specific heat, suggests an instability for sufficiently large black holes.\nWe have now seen this is reflected in a Lorentzian dynamical instability. However, at least in the\nspherically symmetric static Euclidean sector, the corresponding Euclidean saddle point appears to\nbe stable. Therefore this looks to be a system that is dynamically unstable, but nonetheless has a\ngood ( i.e.,stable) thermodynamic description via the partition function. It is worth emphasizing that\nthese large black holes still appear to be the dominant saddle point, as compared to hot flat space\nor the small black hole. We might wonder what happened to the na¨ ıve intuition that the Euclidean\nand Lorentzian stability should agree if the stability for one changes at a marginal perturbation, since\nthis marginal perturbation, being static, should be shared between both the Euclidean and Lorentzian\npictures. It is instructive to see why this is not the case here, so we briefly revisit this Lorentzian linear\nstability calculation by considering a spherical perturbation of the interior of the cavity, rather than\nthe ‘brane’ style calculation we have done above, in order that we can make contact with our previous\nEuclidean analysis.\n4.1.2 Revisiting the spherical stability calculation\nAn equivalent way to study the spherically symmetric dynamics is by directly perturbing the interior\nof the cavity. Consider the background metric\nds2= ¯gabdxadxb=−F(r)2dt2+1\nF(r)2dr2+r2dΩ2(4.14)\nwhich we take either to be Schwarzschild, with F(r)2= 1−r+\nr, or flat spacetime, F(r) = 1, and then\nconsider a perturbation induced by the diffeomorphism,\nr→r(1 +ϵf(r)), t→t(1 +ϵq), (4.15)\nwhich induces a new metric, related to the original one by the perturbation,\nds2=gabdxadxb= (¯gab+ϵhab) dxadxb, (4.16)\nsuch that,\nhabdxadxb=−F(r)2[1 +ϵ δT(r)]dt2+1\nF(r)2[1 +ϵ δA(r)]dr2+r2[1 +ϵ δS(r)]dΩ2, (4.17)\n– 31 –with\nδT(r) =−2q−2rf(r)F′(r)\nF(r),\nδA(r) =−2rf′(r) +f(r)\u0012\n2rF′(r)\nF(r)−2\u0013\n,\nδS(r) =−2f(r). (4.18)\nSince this new metric is related to the original metric by a diffeomorphism, the new metric also solves\nthe bulk equations of motion. Furthermore, this metric is static for any choice of f(r).\nHowever, now we do notperturb the boundary location to r=R0(1 +ϵf(R0)), because otherwise\nwe have not done anything! Instead, we keep the boundary fixed at r=R0, and focus on f(r) such\nthat f(R0)̸= 0. Then if we obey the boundary conditions at r=R0this is a genuine physical static\nperturbation of the system.\nWithout loss of generality, we choose units such that R0= 1. The boundary induced metric is\ngiven by\nds2\nγ=γµνdxµdxν=−F(R0)2[1 +ϵδT(R0)]dt2+R2\n0[1 +ϵδS(R0)]dΩ2. (4.19)\nNow we wish to fix,\nΓµν=K1\n3pγµν\f\f\f\nr=1, (4.20)\nwhere at the boundary r= 1,\nKr=1=K0+ϵδK , (4.21)\nwith\nK0= 2F(1) + F′(1), δK =F(1)\u0012\nδS′(1) +1\n2δT′(1)−δA(1)\u0013\n−1\n2F′(1)δA(1). (4.22)\nNow,\nδΓtt\nΓtt=−2q−2fF′\nF+f(2F−2F′−F′′)\n3p(2F+F′),\nδΓθθ\nΓθθ=−f(−2(1−6p)F+ 2(1 + 3 p)F′+F′′)\n3p(2F+F′). (4.23)\nwhere the r.h.s are evaluated at r= 1. Given that we must have f(1)̸= 0, or we have not done\nanything, the solution to these is given by,\nq=f(1)\u0012\n1−F′(1)\nF(1)\u0013\n,0 =F′′(1) + 2(1 + 3 p)F′(1)−2(1−6p)F(1). (4.24)\nNote that compared to the ‘brane’ calculation, the second condition is precisely the ‘brane’ linearized\ncondition in equation (4.9) (for R0= 1) that α= 0. We emphasize that the function f(r) away\nfrom the boundary is arbitrary, except that it should preserve the position of the origin of spherical\ncoordinates, or the position of the horizon. However, we could simply choose that it is compactly\nsupported close to the boundary and then these would follow. It is only the fact that it is non-zero at\nthe boundary that is important.\n– 32 –Consider the case of a perturbed flat cavity, where we see the Lorentzian and Euclidean stability\nprecisely agree, with the transition being at p= 1/6. The agreement may be understood as the marginal\nLorentzian mode at p= 1/6, being static, can be analytically continued to give a marginal Euclidean\nmode (a zero mode, so ˜λ= 0, of section 3.3.1). Why then is there disagreement for a black hole interior\nandp >1/6? The Euclidean fluctuations are stable in the spherical sector, but the Lorentzian ones\nare unstable for sufficiently large black holes. Consider now the Lorentzian marginal perturbation, so\np=p∗, for a suitably large black hole. This is again a static mode, and so the question is, why can\nwe not continue this to give a Euclidean zero mode, which would separate Euclidean stability from\ninstability?\nIndeed, we can continue the static mode to the Euclidean signature. However, if we consider\nthe perturbation above in equation (4.18) since necessarily q̸= 0 it induces a conical deficit at the\nEuclidean horizon, even if f(r) is compactly supported near the cavity wall, and has no support near\nthis horizon. In the flat space infilling there is no horizon, and so the corresponding marginal mode\ncontinues without a problem to the Euclidean setting4.\nThis suggests that enforcing smoothness of the Euclidean horizon is then the underlying reason\nthat for p >1/6 we appear to have dynamically unstable large black holes that nonetheless are stable\nEuclidean saddle points. It points to the possibility that perhaps the Euclidean path integral should\nbe taken over metrics that are allowed to have conical deficits.\n4.1.3 End-point of the instabilities\nLet us return to the ‘brane’ calculation of spherical stability, as it allows us to consider the full non-\nlinear evolution in the spherical sector. An obvious question is, where do the dynamical instabilities\nfind above evolve the system to? Let us start by considering the non-linear evolution of an empty\nflat spherical cavity, so Λ = 0 and F= 1, where we have seen that for p <1/6 the dynamics of the\nboundary are unstable. In this case where the only scale is set by the boundary, we may choose units\nwhere the static boundary position is R0= 2 without loss of generality to simplify the equations. Then\nwriting R(t) = 2p\nW(t) the dynamical equation for the boundary, equation (4.8), becomes5\n¨W\nW=W1−6p\n2s\n1 +˙W2\nW2−1. (4.26)\nThe static point then corresponds to the solution W(t) = 1. In the special case of marginal stability,\np= 1/6, we see that in fact the cavity wall can be placed anywhere and will remain exactly static.\nThus W=W0>0 is a solution. As noted earlier, this is a result of the boundary condition, which\nfixes γ1\n6Kand the conformal class of γµν, having an exact global scale symmetry.\n4One might wonder whether one could consider a perturbation generated by the diffeomorphism, r→r(1 + ϵf(r)) and\nt→t(1 +ϵq(r)), where q(r) is chosen to be compactly supported near the cavity boundary as for f(r). However now the metric\nperturbation will have explicit time dependence in the off-diagonal terms going linearly in t, so as ∼F(r)q′(r)tdtdr. After\nEuclidean continuation the τdependence, ∼τdτdr, will not be compatible with having a compact Euclidean time identified as\nτ∼τ+β.\n5Equation (4.26) admits an analytic first integral,\n˙W2=C2\n0−W2+ 2C0Wn+1\nn+ 1+W2(n+1)\n(n+ 1)2(4.25)\nwhere C0is a constant of integration and we defined n=1\n2(1−6p). We note that for some values of pit can actually be solved\nanalytically. For instance, for p=−1/6 it can be solved in terms of some Jacobi (sn) function.\n– 33 –0.0 0.5 1.0 1.5 2.00.00.51.01.52.02.53.0\n0.0 0.5 1.0 1.5 2.0012345Figure 8 : Behavior of the function W(t) under different p’s and initial conditions. The left panel has p=1\n6\nand the right panel has p=1\n12<1\n6.\nHowever as we see from the general p= 1/6 solution,\nW=W0\u0012\n1 +2vt+v2t2\n1−v2\u0013\n(4.27)\nfor constants W0andv, at the non-linear level these marginally stable fixed points are in fact unstable.\nIfWis perturbed at t= 0 so that ˙W=2vW0\n1−v2̸= 0, then if v <0 orv >1 the cavity boundary will\nreach zero size in a finite time,\ntshrink =(\n1−1\nv,|v|>1\n1\n|v|−1, v∈(−1,0). (4.28)\nFor−1< v < 0 and v >1 the boundary initially shrinks, but not with the exponential time dependence\nof the instabilities we have seen in the case p <1/6. Thus this is a non-linear instability. Further,\nforv <−1 initially the cavity expands, before then turning around and shrinking to zero size. For\nthe remaining range 0 < v < 1, the cavity wall initially expands, and continues to expand forever,\napproaching asymptotically the accelerated expansion W∼v2W0\n1−v2t2. The behavior of W(t) for different\nv’s are plotted in the left panel of figure 8.\nNow let us consider the unstable case p < 1/6 where the fixed point is R=R0, soW= 1.\nConsider that the fixed point is perturbed and then the dynamics takes Waway from one. To explore\nthe behaviour let us consider whether we may find ˙W= 0 at a later time, so that Whas an extremum.\nThis condition implies that where ˙W= 0 then,\n¨W\nW=W1−6p\n2−1. (4.29)\nNow given that p <1/6, this implies that at an extremum, ¨W > 0 ifW > 1, and ¨W < 0 if 0 < W < 1.\nThus Wcan’t have a maximum if W > 1, and it can’t have a minimum if 0 < W < 1. As a result,\n– 34 –we conclude that if Winitially is perturbed in the positive sense from W= 1 (so initially W≥1\nand ˙W≥0), it will continue to grow indefinitely. Likewise if it is perturbed in the negative sense (so\ninitially W≤1 and ˙W≤0), then Wwill decrease indefinitely towards W= 0. An example of this\nbehavior of W(t) taking p=1\n12<1\n6is plotted in the right panel of figure 8.\nWe may consider the cavity motion in the general background as for the dynamics (4.8). Then\nR=R0is a fixed point for the cavity radius. Let’s consider a perturbation of this, and ask again\nwhether we may have ˙R= 0. This implies the condition,\n¨R\nR=R1−6pF(R)\nR2\n0(Q(R)−Q(R0)), (4.30)\nwhere we have defined,\nQ(R) =−RF′(R) + 2F\nR1−6p. (4.31)\nNow if R > R 0implies Q(R)> Q(R0), then again ¨R > 0 and there can be no maximum in R, and\nhence the cavity will indefinitely grow if initially it is perturbed with R≥R0and ˙R≥0. Suppose the\nminimum size the cavity could be is Rmin– this might be zero size, or it might be the horizon size if\nthe cavity contains a black hole. Then if Rmin< R < R 0implies Q(R)< Q(R0) then ¨R <0 and there\ncan be no minimum in R, and Rwill decrease to Rminif initially it is perturbed with R≤R0and\n˙R≤0. A particular case of such behaviours is found if Q(R) is a monotonically increasing function of\nRforR > R min.\nOne example of this is the case of Anderson boundary conditions, so that p= 0, and an infilling\nthat is Schwarzschild, possibly including a cosmological constant, so,\nF(r) =r\n−r2Λ\n3+ 1−r0\nr. (4.32)\nThen Q(R) is indeed monotonically increasing for all values of r0and Λ, and for cavity radii greater\nthan the black hole horizon size. We have previously seen that Anderson boundary conditions are\nlinearly unstable for spherical cavities infilled with black holes. Now we see that this instability leads\nto unbounded expansion of the cavity, or collapse of the boundary to the horizon.\nLet us now consider Schwarzschild with no cosmological constant, so Λ = 0, and our generalized\nboundary conditions. We have seen above that for a given xthen for p < p ∗(x), with p∗given in\nequation (4.13), the Schwarzschild infilling is linearly unstable. For small black holes with 0 < x < 2/3,\nthen p∗∈(√\n6−2\n3,1/6), and so these mirror the stability of the empty cavity in the sense that for p >1/6\nthey are stable. However for large black holes with x >2/3 we saw that p∗>1/6, and further that\np∗∼1\n12(1−x)asx→1, and so while an empty cavity is stable for p > 1/6, if it is filled with a\nsufficiently large black hole it then becomes unstable in this spherical dynamical sector. What can we\nsay non-linearly about these dynamics?\nFirstly it is easy to show that for −√\n6+2\n3< p <√\n6−2\n3then Q(R) is an increasing monotonic\nfunction for any Rgreater than the horizon size. Small black holes in cavities with such pand are\nlinearly unstable, and this monotonicity of Q(R) shows this instability collapses the cavity to the\nhorizon or expands it forever.\n– 35 –In order to consider the other ranges of pwe note that the behaviour of Q(R) near the horizon\nand asymptotically are,\nQ(R) =−1\n2r6p−1\n2\n0√R−r0+O(p\nR−r0), Q (R) =−2R6p−1\u0000\n1 +O(R−1)\u0001\n(4.33)\nrespectively. Hence for p <1/6 we have Q(R)< Q(R0) for Rnear the horizon and Q(R0)< Q(R)\nfor large radius too. Consider the case of small black holes which are unstable with pin the range\np∈(√\n6−2\n3,1/6). Then the function Q(R) is not monontonic, but has both a maximum at the radius\nR=R−and a minimum at the radius R=R+, where these are given by,\nR±\nr0=1−21\n4p\n1−6p±p\n9p2+ 12p−2\n4(1−6p), (4.34)\nso we see that r0< R−< R +. Given that the cavity is unstable then either the unperturbed cavity\nradius R=R0is at a smaller radius than the maximum, or a larger radius than the minimum; so\neither r0< R 0< R−orR+< R 0. In the first case then Q(R) is monotonic for r0< R < R 0and so\nif the cavity wall is perturbed so that R≤R0and ˙R≤0 then it will continue to fall to the horizon.\nIn the second case it is monotonic for R0< R and so if it is perturbed in an increasing sense it will\ncontinue to expand forever.\nIn the interesting case of p >1/6 the function Q(R) has a single maximum at the value given\nbyR−above for an unstable cavity, and in particular R0< R −. Thus of the cavity is perturbed in\nthe shrinking sense, so initially R≤R0and ˙R≤0 then it will continue to fall to the horizon. If\nperturbed in the increasing sense is initially grows, but since for large Rthen Q(R)≃ −2R6p−1, so\nQ(R)−Q(R0)<0, then generically it reaches a maximum size and recontracts.\nThus in summary, we appear to find that where a linear dynamical instability exists for a cavity\nboundary, it tends to act to either shrink the cavity boundary down to a minimal size (either zero radius\nin an empty cavity or to the black hole horizon) or it acts to expand the cavity radius indefinitely.\n4.2 Stability beyond spherical symmetry\nHaving explored dynamical stability for spherical symmetry, we now explore the linear stability of our\nnovel boundary conditions concerning perturbations that break spherical symmetry , typically examined\nthrough the Kodama-Ishibashi formalism [36]. We focus on the case of an empty spherical cavity with\na vanishing cosmological constant. The background metric thus reads\nds2=−dt2+ dr2+r2dΩ2\n2, (4.35)\nand we take the cavity to be located at r=R0.\nOur initial objective is to understand how our cavity boundary conditions influence the boundary\nconditions for the Kodama-Ishibashi master variables. These variables govern gravitational perturba-\ntions derived from both scalar and vector harmonics on the two-sphere. The vector-derived pertur-\nbations can be shown to have the same boundary conditions as in the Dirichlet problem, and were\nthus studied in [37] and no instabilities were found. For this reason, we will focus on scalar-derived\nperturbations.\n– 36 –The generic form of scalar-derived gravitational perturbations has been explained in section 3.3 in\nthe Euclidean context. The same reasoning applies to the Lorentzian setup so that,\nhℓSmS\nˆaˆb=ˆfℓSmS\nˆaˆb(t, r)YℓSmS(4.36a)\nhℓSmS\nˆaI =−rˆfℓSmS\nˆa(t, r)DIYℓSmS\np\nℓS(ℓS+ 1)(4.36b)\nhℓSmS\nIJ = 2r2\"\nˆhℓSmS\nL(t, r)GIJYℓSmS+ˆhℓSmS\nT(t, r)SℓSmS\nIJ\nℓS(ℓS+ 1)#\n(4.36c)\nwhere SℓSmS\nIJ is the traceless symmetric two tensor defined in Eq. (3.55e) and recall that YℓSmSis a\nspherical harmonic on the two-sphere obeying the earlier equation (3.54). Note that there are a few\nfactors of 2, ℓS(ℓS+ 1), randr2appearing in (4.36) that differ from those of the Euclidean analysis\ndetailed in Appendix A.3. These have been added to make contact with [36, 37]. Since the background\nis static, i.e.has a timelike Killing vector field ∂/∂t, we can decompose all perturbations in Fourier\nmodes as\nˆfℓSmS\nˆaˆb(t, r) =e−i ω tfℓSmSω\nˆaˆb(r),ˆfℓSmS\nˆa(t, r) =e−i ω tfℓSmSω\nˆa (r),\nˆhℓSmS\nL(t, r) =e−i ω thℓSmSω\nL (r),and ˆhℓSmS\nT(t, r) =e−i ω thℓSmSω\nT (r).(4.37)\nFor given boundary conditions, we would like to determine\nn\nfℓSmSω\nˆaˆb(r), fℓSmSω\nˆa (r), hℓSmSω\nL (r), hℓSmSω\nT (r), ωo\n. (4.38)\nAn instability is then identified by a mode with Im( ωR0)>0.\nAt this point, we could introduce a particular gauge, and make progress. However, using the\nprocedure outlined by Kodama and Ishibashi in [36] we will bypass this altogether and instead work\nwith a master variable that is manifestly gauge invariant. To decrease the clutter, we will drop all the\nsuperscripts ℓSmSω.\nWe start by presenting what our boundary conditions imply for the metric components. These are\nsimply given by\nft(R0) = 0 , h T(R0) = 0 , f tt(R0) + 2hL(R0) = 0 ,\nf′\ntt(R0)−4h′\nL(R0)−24p\nR0hL(R0) +2\nR0frr(R0) +2p\nℓS(ℓS+ 1)\nR0fr(R0) + 2 i ω f tr(R0) = 0 .(4.39)\nThese are the boundary conditions that we would like to rewrite in terms of the corresponding Kodama-\nIshibashi variable.\nRecall that any gauge transformation can also be decomposed in terms of the harmonic decom-\nposition that we used to write the metric perturbations, and in particular, if ξaparametrizes such an\ninfinitesimal diffeomorphism we have\nξˆa(t, r, θ, ϕ ) =ξℓSmSω\nˆa (r)e−iωtYℓSmS(θ, ϕ), (4.40a)\nξI(t, r, θ, ϕ ) =−ξℓSmSω(r)e−iωtDIYℓSmS(θ, ϕ)p\nℓS(ℓS+ 1). (4.40b)\n– 37 –One can check that the following metric combinations are invariant under infinitesimal transformations\ngenerated by ξ\nF=hL+1\n2hT+1\nr(ˆ∇ˆar)ˆgˆaˆbXˆa (4.41a)\nFˆaˆb=fˆaˆb+ˆ∇ˆaXˆb+ˆ∇ˆbXˆa (4.41b)\nwhere ˆ∇is the metric preserving connection associated with the two-dimensional Minkowski spacetime\nˆgˆaˆbdxˆadxˆb=−dt2+ dr2, (4.41c)\nand\nXˆa≡1p\nℓS(ℓS+ 1)\"\nfˆa+rp\nℓS(ℓS+ 1)ˆ∇ˆahT#\n. (4.41d)\nOne can rewrite the boundary conditions (4.39) in terms of FandFˆaˆb. Three of these boundary\nconditions are manifestly gauge dependent (for instance, one of the boundary conditions demands\nhT(R0) = 0 and hTitself is gauge dependent), but can be solved to determine linear combinations of\nmetric functions. However, one can show that a single gauge invariant boundary condition emerges\nthat can be expressed in terms of FandFˆaˆb\nF′\ntt(R0)−4F′(R0) +1\nR0\u0002\nℓS(ℓS+ 1)−2−ω2R2\n0+ 12p\u0003\nFtt(R0)\n+2\nR0\u0002\nℓS(ℓS+ 1)−2−ω2R2\n0\u0003\nF(R0) +2\nR0Frr(R0) + 2iωF tr(R0) = 0 .(4.42)\nTo proceed, we introduce three auxiliary quantities X,YandZso that\nFtt(r) =−X(r)−Y(r)\n2, (4.43a)\nFtr(r) =i ω Z(r), (4.43b)\nFrr(r) =−X(r)−Y(r)\n2, (4.43c)\nF(r) =−X(r) +Y(r)\n4. (4.43d)\nOne might wonder about the motivation behind this change of variables. Indeed, using the Einstein\nequation, it can be demonstrated that both FˆaˆbandFadhere to two algebraic constraints. The\nvariables X,Y, and Zare selected to retain a single algebraic constraint. Consequently, the Einstein\nequation is simplified to three first-order differential equations for X,Y, and Z, along with a lone\nalgebraic constraint linking them. These equations can be found in [36] and turn out to be given by\nX′(r)−Z(r)ℓS(ℓS+ 1) + rY(r)\nr2+3Y(r)\nr+ω2Z(r) = 0 , (4.44a)\nY′(r)−ω2Z(r) = 0 , (4.44b)\nZ′(r)−X(r) = 0 , (4.44c)\nrω2[rX(r) +rY(r)−2Z(r)]−Y(r) [ℓS(ℓS+ 1)−2] = 0 . (4.44d)\n– 38 –Finally, one can introduce a variable Φ that is constructed from X,YandZin terms of which\nthe gravitational perturbations simplify immensely (particularly if we were to study black holes in the\ninterior of the cavity, or higher-dimensional cavities). Defining,\nΦ(r) =2Z(r)−r X(r)−r Y(r)\nℓS(ℓS+ 1)−2, (4.45a)\nthen we find that it obeys the second-order differential condition,\nΦ′′+\u0014\nω2−ℓS(ℓS+ 1)\nr2\u0015\nΦ = 0 . (4.45b)\nFrom the definition of Φ and from Eqs. (4.44), we reconstruct X,YandZfrom Φ as,\nX=−ℓS(ℓS+ 1)−r2ω2\nrΦ−2Φ′, (4.46a)\nY=−rω2Φ, (4.46b)\nZ=−Φ−rΦ′. (4.46c)\nFinally, using the above relations, and the definition of Ftt,Ftr,FrrandFin terms of X,Y,Zone\ncan rewrite the boundary condition (4.42) solely in terms of Φ and Φ′finding,\n\u0002\n4(1−6p)−3ℓS(ℓS+ 1) + 2 R2\n0ω2\u0003\nR0Φ′(R0)\n−\b\n2R4\n0ω4+ 4[1−6p−ℓS(ℓS+ 1)] R2\n0ω2−ℓS(ℓS+ 1)[3(1 −4p)−2ℓS(ℓS+ 1)]\t\nΦ(R0) = 0 .(4.47)\nForp= 0, i.e.Anderson boundary conditions, the above boundary condition agrees with the boundary\nconditions presented in [15], while as p→+∞we recover those reported in [37].\nThe generic solution to Eq. (4.45b) can be readily found in terms of Bessel functions\nΦ(r) =√rωh\nC1JℓS+1\n2(rω) +C2YℓS+1\n2(rω)i\n. (4.48)\nRegularity at the origin demands C2= 0, and then our boundary condition yields,\n\u0002\n4(1−6p)−3ℓS(ℓS+ 1) + 2 R2\n0ω2\u0003\nR0ωJℓS+3\n2(R0ω)\n+n\n2R4\n0ω4+ 2[2(1 −6p)−(2ℓS+ 1)( ℓS+ 1)] R2\n0ω2\n−(ℓS+ 1) [3(1 −4p)ℓS−(2ℓS+ 3)( ℓS+ 1)ℓS+ 4(1−6p)]o\nJℓS+1\n2(R0ω) = 0 .(4.49)\nThe above transcendental equation governs how empty spherical cavities in flat spacetime react to non-\nspherical gravitational perturbations induced by scalar-derived gravitational deformations. This is the\nanalog expression to that in equation (3.57) that we analysed earlier for the Euclidean fluctuations.\nBefore commenting on numerical results obtained from solving Eq. (4.49), let us analyse the large\nℓSlimit of ωR0with pbeing held fixed. Using uniform asymptotic expansions for Bessel functions at\nlarge order in terms of Airy functions (see for instance [38]) one can solve for the large ℓSlimit of ωR0\nappearing in (4.49), which turns out to be given by\nR0ω=ℓS−Γ\u0012ℓS\n2\u00131/3\n+1\n2+1\n16Γ11−96p−128Γ3\n7 + 16Γ3\u00122\nℓS\u00131/3\n+O(ℓ−2/3\nS) (4.50a)\n– 39 –where Γ can be any solution to the following transcendental equation:\nAi′(Γ) = 4Γ2Ai(Γ)\nwith Ai the Airy function and Ai′its first derivative. In particular, we find (at least) two complex\nroots :\nΓ≈ −0.0674243 + 0 .427905 ior Γ ≈ −0.0674243 −0.427905 i (4.50b)\nshowing that an instability exists in the spectrum regardless of p, so long as ℓSis large enough while p\nis held fixed. We note that this is in broad agreement with the particular case of Anderson boundary\nconditions studied in [15] which derived similar asymptotics for instabilities for large ℓS, but which\ndiffer in detail6. The point to emphasize here is that for large ℓSall our family of boundary conditions\nshare the same large ℓSinstabilities originating from the imaginary part of Γ above. This is one of the\nmain results of this paper, since for the case p >1/6 it shows that a spherical cavity endowed with\nour boundary conditions appears to be free of Euclidean unstable modes (as argued in the previous\nsections), and yet is dynamically unstable in the Lorentzian section with infinitely many unstable\nnon-spherical modes!\nUnlike the case of ℓS= 0, which can be followed nonlinearly, here we can only speculate as to what\nthe endpoint of this instability might look like. We note also that the instability growth rate grows at\nlarge ℓS, indicating that it is going to be hard to shut down the instability in a simple manner.\nThis is suggestive that the instability leads to a ‘turbulent’ behaviour with power being transferred\nto shorter and shorter scales as these large ℓSinstabilities are triggered by non-linear interactions.\nThe phenomenon of the superradiant instability of Kerr-AdS is a gravitational example where it is\nconjectured that a linear instability leads to such a non-smooth final state, again with power transferred\nto arbitrarily short scales [39–42]. It is worth noting that in the case of superradiance the instability\ntimescale becomes slower for modes with increasingly short scale azimuthal dependence, whereas in our\ncase the instability actually grows with ℓSas the power ℓ1/3\nS, so it may be that the turbulent cascade\nhappens even more rapidly.\nWe validate the expansion (4.50a) by comparing it with numerical results obtained through the\ndirect solution of (4.49) using a Newton-Raphson solver. In Fig.9 we compute the real (left panel)\nand imaginary (right panel) parts of ωas functions of ℓSfor the case p= 0. We successfully tracked\nthe mode up to ℓS= 2000. Achieving such large values of ℓSrequires resorting to extended precision,\nkeeping at least the first five hundred digits. This is necessary because evaluating the Bessel functions\nin (4.49) at such high orders is notoriously challenging. On both panels, the solid black line is the\nasymptotic prediction (4.50a) while the blue disks represent the numerical data, and we see good\nagreement at large ℓS.\nNext, we investigate the spectrum, that is to say R0ω, for a fixed ℓS= 2 while varying p. The\nresults are shown in the top row of Fig. 10, where the real (imaginary) part of ωR0is plotted on the left\n(right) panel. When the mode becomes complex, we represent it with a dashed green curve; for purely\nrealω, a solid blue line is used. The structure of the spectrum in the complex plane is intricate. For\nsufficiently small p, here p <−2/3, there coexist stable modes with purely real frequency and unstable\nmodes with purely imaginary frequency. For larger pthere are intervals of pwhere the ℓS= 2 modes\n6In that work they found instabilities going roughly as R0ω≃ ±ℓS±iℓ1/3\nSfor large ℓSfor the Anderson boundary conditions.\nOur result above gives the precise asymptotic behaviour which is slightly different in the crucial imaginary subleading term.\n– 40 –510 50100 50010000.340.360.380.400.420.44\n510 50100 50010000.10.20.5\n200 500 1000 20000.180.200.220.250.280.30\n200 500 1000 20000.100.110.120.130.140.15Figure 9 : The real part (left panel) and imaginary part (right panel) of R0ωas a function of ℓSfor fixed\np= 0 (top row) and p= 2 (bottom row). In both panels, the solid black line corresponds to the prediction\n(4.50a), while the blue disks depict the exact numerical data.\n– 41 –appear to be stable, with purely real frequency. However these regions are interrupted by ‘bubbles’\nwhere the mode frequencies become complex, and in particular modes with Im( ωR0)>0 exist that\ncorrespond to instabilities, as seen in the righthand panel of the figure. The mode tracked in Fig. 9\nconnects to the middle ‘bubble’ in Fig. 10 with p= 0. In fact, for this mode, we recover the same\nvalue as in [15], which studied the case of Anderson boundary conditions, so p= 0. The unstable\nmodes observed in the righthand panel of the figure appear to occur also at larger values of pthan\nthose shown, with the existence of further complex ‘bubbles’ at ever greater values of p. Aspincreases,\nthese ’bubbles’ become progressively smaller. For instance, there is a complex ’bubble’ in the interval\np∈(184.12619 ,184.15074) with Re( ωR0)∈(47.08062 ,47.08219) for ℓS= 2. We have also explored\nother values of ℓSand observed qualitatively similar behavior, as shown in the bottom row of Fig. 10,\nwhere we present the spectrum for ℓS= 4.\n5 Conclusion and Discussion\nWe have introduced a one-parameter set of elliptic boundary conditions within the framework of Ein-\nstein metric infillings in Riemannian geometry. The conventional Dirichlet and Anderson boundary\nconditions are derived as limits in this context. Additionally, we have demonstrated that these bound-\nary conditions lead to a well-defined infilling problem, except in the strict Dirichlet limit. For these\nboundary conditions, one fixes the conformal structure of the boundary metric [ γµν] and γpK, with\np∈RandKbeing the extrinsic curvature of the boundary metric in the infilling space ( M, g). We\nhave referred to these as ‘generalized conformal boundary conditions’. In the case that we require the\nfunction Kto be non-vanishing we may elegantly phrase these boundary conditions in terms of fixing\nthe rescaled boundary metric Γ µν=K1\n(d−1)pγµνon∂M.\nThese boundary conditions naturally emerge when considering the Gibbons-Hawking-York bound-\nary term to have an arbitrary real coefficient Θ which then determines pvia the relation (2.8b). To the\nbest of our knowledge, this is the first exploration of these boundary conditions. Perhaps amusingly,\nthese include the case Θ = 0 for which no Gibbons-Hawking-York term is needed, corresponding to\nthe particular case of p=1\n2(d−1). We have observed that rather than fixing γpKand [ γµν], instead one\nmay fix γpKand the traceless part of the Brown-York stress tensor which also results in a well-defined\nvariational problem. We have not explored this second option here, but believe it is interesting to do\nso.\nGiven the well-posed nature of these new boundary conditions, it is logical to define an ensemble\nin reference to them, akin to the traditional concepts introduced by York. We then think of the\nEuclidean path integral as being a functional of the rescaled boundary metric Γ µν. We have derived a\nfirst law of thermodynamics applicable to a broad range of geometries, including those with bolts, such\nas the Euclidean Schwarzschild black hole. We derived this first law assuming 1) that a hypersurface-\northogonal Killing vector field exists, and extends into the bulk; 2) that Γ µνis ultrastatic. Exploring\nthe implications and consequences of relaxing these assumptions would be an intriguing avenue for\nfurther study. For instance, the Euclidean version of Kerr black holes fails to satisfy both. This first\nlaw involves the entropy and temperature (measured with respect to Γ µν) associated to possible bolts,\na work term related to deforming the boundary geometry and a free energy Fthat characterises the\nensemble. The free energy can be computed from the Euclidean path integral using the leading saddle\npoint approximation for fixed Γ µν.\n– 42 –-4 -2 0 2 4-505\n-5-4-3-2 -1 0 1 2-505\n-15 -10 -5 0 5 10 15-10-50510\n-15 -10 -5 0 5 10 15-15-10-5051015Figure 10 : The real (left panel) and imaginary (right panel) parts of R0ωas a function of pfor fixed ℓS= 2\n(top row) and ℓS= 4 (bottom row). If the mode turns complex, we represent it with a dashed green curve;\nwhen ωis purely real, we depict it with a solid blue line.\nFrom this point onward, we specialised to d= 4 and it would be interesting to explore any\ndimensionally dependent phenomena. We focused on spherically symmetric infilling solutions with the\nboundary metric γµνchosen as S1\nβ×S2\nR0. Identifying up to three three saddle points for a given Γ µν, we\nscrutinized both their global and local thermodynamics. Our findings, while differing in some details\nfrom those presented in [15] for the case of Anderson boundary conditions, so p= 0, align on the\naspects of global and local thermodynamic stability for that case.\nWe then investigated the Euclidean gravitational path integral beyond the leading saddle point\napproximation. In particular, we investigated the possible existence of fluctuation instabilities, so\n– 43 –Euclidean negative modes of the Lichnerowicz operator, using the methods of [22, 25] for the case of\nan empty cavity, i.e.,no bolt, with boundary metric S1\nβ×S2\nR0. Using a mixture of numerical and\nanalytic work, we believe we have provided convincing evidence that for p≥1/6, there are no such\nEuclidean negative mode instabilities. However, for p < 1/6 we explicitly constructed fluctuations\nwhose Lichnerowicz operator eigenvalues have negative real part, yielding instabilities. Indeed, for\ncertain ranges of pwe find Euclidean negative modes even for deformations that break spherical\nsymmetry. The limit |p| → ∞ corresponds to Dirichlet boundary conditions. Thus for Euclidean\nsignature we may regard large positive pas a regulator that deforms the Dirichlet boundary conditions\nto be well-posed. Then removing the regulator recovers the Dirichlet behaviour. While p→ −∞ also\ncorresponds to the Dirichlet limit, this is not a good regulator, as for large magnitude negative p, very\nnegative modes exist that would dominate the behaviour of the system.\nWe have also explored the existence of negative mode instabilities around the Euclidean-Schwarzschild\nblack hole and found an interesting puzzle that we have not fully solved. In particular, we find that\nfor small black holes an additional negative mode exists beyond any present for the flat spacetime\nfilled cavity, in agreement with the analysis based on local thermodynamic stability. However, we find\nno evidence for Euclidean negative modes for large black holes when p >1/6, even though we find a\nnegative specific heat for these. Our analysis imposed smoothness at the bolt, i.e.at the black hole\nevent horizon, and we suspect that if one relaxes this assumption, a negative mode will exist.\nWhile in this paper we focused mainly on boundary metrics with S1×S2topology, it would be\ninteresting to also investigate boundary metrics with S3topology. Furthermore, we have only studied\nthe static and spherically symmetric limit in the presence of a cosmological constant (see appendix C),\nbut it would be also interesting to perform a more systematic analysis for this more general class of\ngeometries. It would also be very interesting to understand how RG flow of our boundary term works,\nfollowing previous work looking at the Gibbons-Hawking-York term [43, 44].\nHaving studied in some detail the Euclidean problem, we turned our attention to the Lorentzian\nproblem with our generalized conformal boundary conditions. While some boundary conditions have\nbeen shown to be well-posed and geometrically unique, much less is known about initial-boundary-value\nproblems. In particular, the class of boundary conditions for which theorems exist (see for instance\n[45–47]) do not include Anderson boundary conditions or Dirichlet boundary conditions. In fact, it is\nstrongly believed that Dirichlet boundary conditions will not be consistent with geometric uniqueness\n(see for instance [48]). In this paper we did not address the issue of geometric uniqueness, and indeed\nwe only investigated the linear stability problem for our new boundary conditions. Interestingly for\nthe special case of Anderson boundary conditions the well-posedness of the linear Lorentzian evolution\nwas shown in [49] and it seems likely that this will generalize to our more general class of boundary\nconditions.\nWe examined four-dimensional spacetimes with a spherically symmetric infilling and boundary\ngeometry, without cosmological constant. Then Birkhoff’s theorem constrains the interior geometry\nto be flat spacetime, or the Schwarzschild black hole. We then considered dynamics which preserves\nthe spatial spherical symmetry. In the case of a cavity filled with Minkowski spacetime we identified\na linear instability for p <1/6. For the black hole case, we have shown that large enough black holes\nare linearly unstable so long as p >1/6, but with the Dirichlet case limit p→ ∞ being stable (in\naccordance to the results reported in [37]). Furthermore, the onset of the instability agrees precisely\nwith a change in sign of the specific heat capacity, and hence with the local thermodynamic stability for\n– 44 –the large black holes. Because of the spherical symmetry of the dynamics, we were able to determine\nthe fate of the nonlinear evolution. Indeed, we found that, depending on the initial data, either the\ncavity expands indefinitely or contracts to zero size in the case of flat spacetime, or to the horizon for\na black hole interior, in finite boundary time.\nMotivated by the analysis of the spherically symmetric sector, we investigated the linear stabil-\nity of perturbations that disrupt spherical symmetry. We took a static spherical cavity filled with\nflat Minkowski spacetime, and then considered dynamical perturbations to this that break the spher-\nical symmetry. Our analysis focused on scalar-derived perturbations which are characterized by the\nspherical harmonic index ℓS.\nOur findings indicate that, for any value of p, there exists a critical value of ℓSabove which an\ninstability occurs. Notably, the instability persists unless p→+∞, aligning with the Dirichlet case\nas analyzed in [37]. Additionally, for the particular case of Anderson boundary conditions, p= 0, our\nresults agree with those for the values of ℓSreported in [15]. The fate of the linear instability under\nperturbations that disrupt spherical symmetry remains uncertain but since an instability exists for all\nℓSlarger than a critical value, and the instability time scale grows as ℓ1/3\nSasymptotically, we might\nexpect a ‘turbulent’ behaviour where energy cascades to shorter and shorter scales.\nTo the best of our knowledge, our results reveal the first example of a rather surprising physical\nsystem. Euclidean gravity with our cavity boundary conditions with p >1/6 admits a flat spacetime\ninfilling that appears to be a stable saddle point. It apparently possesses no Euclidean negative modes,\nand for small temperatures gives the dominant saddle point contribution to the partition function.\nHowever, the corresponding Lorentzian cavity with the same cavity boundary conditions appears to\nbe dynamically unstable, with infinitely many unstable perturbations associated to ℓS→ ∞ . Thus we\nhave a system that appears to be thermodynamically stable and well behaved, and yet is dynamically\nunstable!\nIt is worth noting that numerous examples exist in the opposite direction, the classic example for\ngravity being the Schwarzschild black hole which is known to be dynamically stable [50–57], but in the\nEuclidean context has been shown to contain negative modes [2]. However, we know of no examples,\nin gravity or otherwise, of a system that is thermodynamically stable and yet dynamically unstable.\nAcknowledgments\nWe would like to express gratitude to Don Marolf for many insightful discussions of this work. Ad-\nditionally, J. E. S. extends thanks to Aron Wall for helpful discussions. X. L. thanks Gary Horowitz\nand Zhencheng Wang for illuminating discussions. J. E. S. is partially supported by STFC consol-\nidated grants ST/T000694/1 and ST/X000664/1. X. L. is supported by NSF grant PHY-2107939,\nand by funds from the University of California. T. W. is supported by the STFC consolidated grant\nST/T000791/1.\n– 45 –A Euclidean Fluctuations of the Empty Spherical Cavity\nIn this Appendix we give details of the analysis of the Euclidean fluctuations of an empty spherical\ncavity filled with flat space. We have given a general argument in the main text that for p >1/6 we\nsee no Euclidean instabilities based on the form of the condition in equation (3.57) that governs the\nmodes. In this Appendix we will derive this condition in detail, taking into account various special\ncases that require separate treatment. In addition we shall explicitly give examples of the behaviour\nof modes and their instabilities in the case p <1/6.\nA.1 Modes with n=ℓS= 0\nThese are the simplest modes, and yet, the ones that will impose the most stringent constraints on p.\nA static, i.e.n= 0, and spherically symmetric mode can be written as\nδds2=a(r) d˜τ2+ 2χ(r)d˜τdr+b(r)dr2+c(r)r2dΩ2\n2, (A.1)\nwhere a,bandcparametrize the mode in question and are functions of ronly. We introduce an\nauxiliary variable\nc(r) =f(r)−a(r)−b(r)\n2, (A.2)\nso that h= f(r). Imposing the de Donder gauge and regularity at the origin further restricts χto\nvanish, and ato take the simple form,\na(r) =−3b(r) + f( r)−r b′(r) +1\n2rf′(r). (A.3)\nThe resulting equations for band f can be determined analytically\nf(r) =C1sin\u0010√\nλr\u0011\nr√\nλ+C2cos\u0010√\nλr\u0011\nr√\nλ, (A.4a)\nb(r) =1\n2f(r) +C3\nr2λ\ncos\u0010√\nλr\u0011\n−sin\u0010√\nλr\u0011\nr√\nλ\n+C4\nr2λ\ncos\u0010√\nλr\u0011\nr√\nλ+ sin\u0010√\nλr\u0011\n. (A.4b)\nRegularity at the origin demands C2=C4= 0, so that we are left with imposing the boundary\nconditions at the cavity wall. Linearising our boundary conditions around the spherical cavity at\nradius r=rEin Euclidean space yields\na(rE) =c(rE) and1\n2a′(rE) +c′(rE)−b(rE)\nrE+6p\nrEc(rE) = 0 . (A.5)\nSubstituting the explicit expressions for a,bandcin terms of the mode (A.4) yields\n\n−1√\n˜λsin\u0010p\n˜λ\u0011\n1\n2˜λ3/2h\n3˜λsin\u0010p\n˜λ\u0011\n−sin\u0010p\n˜λ\u0011\n+p\n˜λcos\u0010p\n˜λ\u0011i\n1\n4cos\u0010p\n˜λ\u0011\n+3(4p−1)\n4√\n˜λsin\u0010p\n˜λ\u0011\n1−6p\n2˜λ3/2h\n˜λsin\u0010p\n˜λ\u0011\n−sin\u0010p\n˜λ\u0011\n+p\n˜λcos\u0010p\n˜λ\u0011i\n\nC1\nC3\n= 0,\n(A.6)\n– 46 –-10 0 10 20 30 40-10010203040\n0.0 0.1 0.2 0.3 0.4 0.5-4-2024\n0 5 10 15 20 25 30-1.0-0.50.00.51.0Figure 11 : The real part (left and middle panels) and imaginary part (right panel) of ˜λas a function of p.\nThe point ( p,˜λ) = (1 /6,0) is marked as a red disk. For p <1/6, there exists a single mode with Re ˜λ <0,\nindicating the presence of a field-theoretic negative mode. The green dashed curves indicate that the mode is\ncomplex.\nwhere we defined ˜λ≡r2\nEλ. Any nontrivial solution can only exist if the determinant of the above\nmatrix vanishes. This imposes a condition on the possible values of ˜λ, namely the following must be\ntrue\n1\n16˜λ2(\n(1 + 6 ˜λ) cos\u0010\n2p\n˜λ\u0011\n+ 3˜λ3/2sin\u0010\n2p\n˜λ\u0011\n−1−4˜λ\n+ 12ph\n1 +˜λ−(1 + ˜λ) cos\u0010\n2p\n˜λ\u0011\n−p\n˜λsin\u0010\n2p\n˜λ\u0011i)\n= 0.(A.7)\nWe emphasize that this is the same expression as follows from our one in the main text in section 3.3,\nequation (3.57), in the special case n=ℓS= 0. As discussed in that section there will be no instabilities\nforp >1/6. In Fig. 11 we plot the real part of ˜λ(left and middle panels) as well as its imaginary part\n(right panel) as a function of p. The middle plot shows a zoom in the red region marked on the left\npanel. It is clear that a single negative mode exists if p <1/6. This critical value of pcan also be\ninferred analytically from Eq.(A.7) evaluated at ˜λ= 0.\nA.2 Modes with n̸= 0and ℓS= 0\nThe modes in this section are allowed to depend on the thermal circle, so that n̸= 0, but that remain\nspherically symmetric with ℓS= 0. We write these as\nδds2=\u0002\na(r) d˜τ2+ 2χ(r)d˜τdr+b(r)dr2+c(r)r2dΩ2\n2\u0003\nei˜n˜τ. (A.8)\nOnce again, we can solve for all metric functions\nc(r) =f(r)−a(r)−b(r)\n2, (A.9a)\nχ(r) =i˜n\nλ−˜n2a′(r)−i˜n\n2(λ−˜n2)f′(r), (A.9b)\n– 47 –b(r) =˜n2\nλ−˜n2a(r) +λ−2˜n2\n2(λ−˜n2)f(r) +λ+ 2˜n2\nr(λ−˜n2)2a′(r) +λ−4˜n2\n2r(λ−˜n2)2f′(r), (A.9c)\na(r) =C1sin\u0000√\nλ−˜n2r\u0001\n√\nλ−˜n2r+C2cos\u0000√\nλ−˜n2r\u0001\n√\nλ−˜n2r(A.9d)\nf(r) =C3sin\u0000√\nλ−˜n2r\u0001\n√\nλ−˜n2r+C4cos\u0000√\nλ−˜n2r\u0001\n√\nλ−˜n2r. (A.9e)\nRegularity at the origin demands C2=C4= 0, while our boundary conditions now impose\na(rE) =c(rE) and1\n2a′(rE) +c′(rE)−b(rE)\nrE+6p\nrEc(rE)−i˜nχ(rE) = 0 . (A.10)\nThese can be simultaneously solved so long as\np=˜Λ + 6 ˜Λ2−csc2\u0010p\n˜Λ\u0011\n˜Λ\u0010\nϖ2+˜Λ\u0011\n+ϖ2\u0010\n2˜Λ−7\u0011\n+ cot\u0010p\n˜Λ\u0011p\n˜Λh\nϖ2\u0010\n8 +˜Λ\u0011\n−3˜Λ2i\n12h\nϖ2\u0010\n˜Λ−3\u0011\n+˜Λ +˜Λ2−cot\u0010p\n˜Λ\u0011p\n˜Λ\u0010\n˜Λ−3ϖ2\u0011i\n(A.11a)\nwhere\nϖ= ˜n rEand ˜Λ = ( λ−˜n2)r2\nE. (A.11b)\nAgain this is consistent with the expression (3.57) in section 3.3 and hence will give rise to no unstable\nmodes for p >1/6. We now look at the explicit behaviour of modes, and in particular find negative\nmodes for sufficiently small p. We start by noting that λ= 0 now requires p= (2−ϖ2)/12<1/6, so\na zero mode (marking a transition to a negative mode) can only exist in a regime where pis smaller\nthan the case for which ϖ= 0. Of course, it could still happen that Re ˜λ= Re( r2\nEλ) becomes negative\nsomewhere on the complex plane. However, we have explicitly checked that this is not the case for\nϖ∈R. See, for instance, Fig. 12, where we plot the real part of ˜λ(left and middle panels) as well as\nits imaginary part (right panel) as a function of pforϖ= 1. The middle panel provides a zoom near\nthe transition point, and shows a negative mode.\nA.3 Non-spherical scalar-derived gravitational Euclidean modes with ℓS≥2\nWe now consider non-spherical Euclidean fluctuations, so ℓS̸= 0. Our treatment requires ℓS≥2 and\nwe later consider the special case ℓS= 1 separately.\nA.3.1 Static modes\nModes with ˜ n= 0 have to be dealt with separately. Due to the staticity of the modes in question, the\ncomponents f˜τrandf˜τdecouple from the remaining ones, and we shall return to these at the end of\nthis section. There are then a total of five functions to solve for: {f˜τ˜τ(r), frr(r), fr(r), hL(r), hT(r)},\nwhich we turn our attention to next.\nTo simplify our expressions, we define\nhL=r2\n2(f−f˜τ˜τ−frr), (A.12)\n– 48 –-10 0 10 20 30 40-10010203040\n0.0 0.1 0.2 0.3 0.4 0.5-4-2024\n0 5 10 15 20 25 3035-1.0-0.50.00.51.0Figure 12 : The real part (left and middle panels) and imaginary part (right panel) of ˜λas a function of p.\nThe point ( p,˜λ) =\u0010\n2−ϖ2\n12,0\u0011\nis marked as a red disk. For p <2−ϖ2\n12, there exists a single mode with Re ˜λ <0,\nindicating the presence of a Euclidean negative mode. This plot was generated with ϖ= 1 and uses the same\ncolour coding as Fig. 11.\nso that the trace of the mode is simply h= fYℓSmS. Using the de Donder gauge condition, as well as\nthe Lichnerowicz eigenvalue equation (3.50) we find\nhT(r) =−r2\nℓS(ℓS+ 1)−2\u0014\nf˜τ˜τ(r) +frr(r)−4fr(r)\nr−2f′\nr(r)\u0015\n, (A.13a)\nfrr(r) =q1(r) +f(r)\n2+f′\n˜τ˜τ(r)\nr λ+f′(r)\n2rλ, (A.13b)\nfr(r) =q2(r) +f˜τ˜τ(r)\nr λ+f(r)\n2r λ, (A.13c)\nq2(r) =3r\nℓS(ℓS+ 1)q1(r) +r2\nℓS(ℓS+ 1)q′\n1(r), (A.13d)\nwith\n(r2f′)′\nr2+\u0014\nλ−ℓS(ℓS+ 1)\nr2\u0015\nf = 0 , (A.13e)\n(r2f′\n˜τ˜τ)′\nr2+\u0014\nλ−ℓS(ℓS+ 1)\nr2\u0015\nf˜τ˜τ= 0, (A.13f)\n(r6q′\n1)′\nr6+\u0014\nλ−ℓS(ℓS+ 1)−6\nr2\u0015\nq1= 0. (A.13g)\nNote that once the solutions for f, f˜τ˜τandq1are known via Eqs. (A.13e)-(A.13g), all the remaining\nfunctions can be found via Eqs. (A.13a)-(A.13d). Indeed, one can readily solve in full generality for f,\nf˜τ˜τandq1\nf(r) =C1\n(r√\nλ)1/2JℓS+1\n2\u0010\nr√\nλ\u0011\n+C2\n(r√\nλ)1/2YℓS+1\n2\u0010\nr√\nλ\u0011\n, (A.14a)\nf˜τ˜τ(r) =C3\n(r√\nλ)1/2JℓS+1\n2\u0010\nr√\nλ\u0011\n+C4\n(r√\nλ)1/2YℓS+1\n2\u0010\nr√\nλ\u0011\n, (A.14b)\n– 49 –q1(r) =C5\n(r√\nλ)5/2JℓS+1\n2\u0010\nr√\nλ\u0011\n+C6\n(r√\nλ)5/2YℓS+1\n2\u0010\nr√\nλ\u0011\n, (A.14c)\nwhere J pand Y pare Bessel functions of the first and second kind of order p, respectively. Regularity\nat the origin requires C2=C4=C5= 0, and we are left with three unknown constants {C1, C2, C3}\ntogether with λto determine in what follows.\nImposing our boundary conditions at the cavity wall yields three conditions\nhL(rE) =r2\nEf˜τ˜τ(rE), h T(rE) = 0 ,\nand f′\n˜τ˜τ(rE) +2\nr2\nEh′\nL(rE)−4(1−3p)\nrEf˜τ˜τ(rE)−2\nrEfrr(rE) +2ℓS(ℓS+ 1)\nr2\nEfr(rE) = 0 .(A.15)\nAll the above conditions can be simultaneously solved so long as\n˜λJ3\nℓS+3\n2\u0010p\n˜λ\u0011\n+p\n˜λΘ1(p, ℓS,˜λ) J3\nℓS+1\n2\u0010p\n˜λ\u0011\n−Θ2(p, ℓS,˜λ) J2\nℓS+1\n2\u0010p\n˜λ\u0011\nJℓS+3\n2\u0010p\n˜λ\u0011\n−p\n˜λΘ3(p, ℓS,˜λ) JℓS+1\n2\u0010p\n˜λ\u0011\nJ2\nℓS+3\n2\u0010p\n˜λ\u0011\n= 0,(A.16a)\nwith ˜λ≡λr2\nEand\nΘ1(p, ℓS,˜λ) = (12 p−2 + 3 ℓS)˜λ−(1 + 2 ℓS)(12p−2 +ℓS+ 2ℓ2\nS), (A.16b)\nΘ2(p, ℓS,˜λ) = (1 + 2 ℓS)(12p−2 +ℓS+ 2ℓ2\nS)−[24p−1 + 4 ℓS(2 +ℓS)]˜λ+ 3˜λ2, (A.16c)\nΘ3(p, ℓS,˜λ) = 3−12p+ℓS−2ℓ2\nS+ 2˜λ . (A.16d)\nAgain this is consistent with equation (3.57), and hence will give no negative modes for p >1/6. We\nnow give an example of the behaviour, determining allowed values of ˜λfor a given value of p. The\nresults for ℓS= 2 are shown in Fig. (13) and similar qualitative results hold for all other values of\nℓSwe have investigated. In the region p >1/6 we find no modes with Re ˜λ <0 consistent with our\ngeneral argument in Section 3.3. Note that negative modes do exist for p≲−0.64422, and indeed it is\nrelatively simple to show that as pbecomes more and more negative, a single negative mode persists\nwith\n˜λ=−16p2+ 24p+1\n3[11ℓS(ℓS+ 1)−3] +29ℓS(ℓS+ 1) + 40\n12p+O(p−2). (A.17)\nWe now return to the f˜τrandf˜τcomponents. As anticipated earlier, these metric components\ndecouple from the remaining ones. The de Donder gauge condition demands\nf˜τ(r) =r\nℓS(ℓS+ 1)[r f′\n˜τr(r) + 2f˜τr(r)] (A.18a)\nwith\n(r4f′\n˜τr)\nr4+\u0014\nλ−ℓS(ℓS+ 1)−2\nr2\u0015\nf˜τr= 0. (A.18b)\nThe equation for f˜τrcan be readily integrated to find\nf˜τr=1\n(r√\nλ)3/2h\nC1JℓS+1\n2\u0010\nr√\nλ\u0011\n+C2YℓS+1\n2\u0010\nr√\nλ\u0011i\n. (A.19)\n– 50 –-10-5 0 5 10 15 20 25-100102030\n0 5 10 15 20 25-6-4-20246Figure 13 : The real part (left panel) and imaginary part (right panel) of ˜λas a function of pforℓS= 2. The\npoint ( p,˜λ) = (1 /6,0) is marked as a red disk. This plot uses the same colour coding as Fig. 11.\nRegularity at the origin once again demands C2= 0, while our boundary conditions at the cavity wall\ndemand\nf˜τ(rE) = 0⇒JℓS+3\n2\u0010p\n˜λ\u0011p\n˜λ−(ℓS+ 1) JℓS+1\n2\u0010p\n˜λ\u0011\n= 0. (A.20)\nThis condition is independent of pand thus will yield the same results as for the Dirichlet case where\nstability was seen in [22]. It is easy to show that the above equation only has real roots, and that for\nℓS= 2 the first four modes are approximately given by\n˜λ={14.97874(7) ,55.39954(5) ,114.76860(4) ,193.78091(7) , . . .}, (A.21)\nthus showing that no negative modes exist in this sector as well.\nA.3.2 Generic modes\nThis is the most complicated sector, since both ˜ n̸= 0 and ℓS≥2. Just as in previous sections, we\nbegin by introducing an auxiliary quantity f\nhL=r2\n2(f−f˜τ˜τ−frr). (A.22)\nThe de Donder gauge condition, together with the Lichnerowicz eigenvalue equation (3.50), now imply\nthe following relations\nf˜τ(r) =i˜n\nλ−˜n2f˜τ˜τ(r)−i\n2˜n\nλ−˜n2f(r) +iℓS(ℓS+ 1) + 9 −r2(λ−˜n2)\nℓS(ℓS+ 1) ˜nq1(r)\n−4\nr˜ni q2(r) +3r\nℓS(ℓS+ 1) ˜ni q′\n1(r)−i\n˜nq′\n2(r),(A.23a)\n– 51 –f˜τr=i3\nr˜nq1(r)−iℓS(ℓS+ 1)\nr2˜nq2(r) +i˜n\nλ−˜n2f′\n˜τ˜τ(r)−i\n2˜n\nλ−˜n2f′(r) +i\n˜nq′\n1(r) (A.23b)\nhT(r) =1\nℓS(ℓS+ 1)−2(\n(λ+ 2˜n2) [2−r2(λ−˜n2)]\n(λ−˜n2)2f˜τ˜τ(r) +(λ−4˜n2) [2−r2(λ−˜n2)]\n2 (λ−˜n2)2f(r)\n−r2[3ℓS(ℓS+ 1) + 18 −2r2(λ−˜n2)]\nℓS(ℓS+ 1)q1(r) + 12 r q2(r) +λ+ 2˜n2\n(λ−˜n2)2r f′\n˜τ˜τ(r)\n+λ−4˜n2\n2 (λ−˜n2)2rf′(r)−6r3\nℓS(ℓS+ 1)q′\n1(r) + 4r2q′\n2(r))\n,(A.23c)\nfrr(r) =q1(r) +1\n2λ−2˜n2\nλ−˜n2f(r) +˜n2\nλ−˜n2f˜τ˜τ(r) +λ+ 2˜n2\n(λ−˜n2)2rf′\n˜τ˜τ(r) +λ−4˜n2\n2 (λ−˜n2)2rf′(r),(A.23d)\nfr(r) =q2(r) +λ+ 2˜n2\n(λ−˜n2)2rf˜τ˜τ(r) +λ−4˜n2\n2 (λ−˜n2)2rf(r) (A.23e)\nwhere\n(r2f′)′\nr2+\u0014\nλ−ℓS(ℓS+ 1)\nr2−˜n2\u0015\nf = 0 , (A.23f)\n(r2f′\n˜τ˜τ)′\nr2+\u0014\nλ−ℓS(ℓS+ 1)\nr2−˜n2\u0015\nf˜τ˜τ= 0, (A.23g)\n(r2q′\n1)′\nr2+\u0014\nλ−ℓS(ℓS+ 1) + 6\nr2−˜n2\u0015\nq1+4ℓS(ℓS+ 1)\nr3q2(r) = 0 , (A.23h)\n(r4q′\n2)′\nr4+\u0014\nλ−ℓS(ℓS+ 1)−8\nr2−˜n2\u0015\nq2−6\nℓS(ℓS+ 1)(r3q1)′\nr3+2r\nℓS(ℓS+ 1)(λ−˜n2)q1= 0.(A.23i)\nNote that once the solutions for f, f˜τ˜τ,q1andq2are known via Eqs. (A.23f)-(A.23i), all the remainder\nfunctions can be found via Eqs. (A.23a)-(A.23e). Indeed, all the above four equations can be integrated\nin full generality\nf(r) =C1\n(r√\nλ−˜n2)1/2JℓS+1\n2\u0010\nr√\nλ−˜n2\u0011\n+C2\n(r√\nλ−˜n2)1/2YℓS+1\n2\u0010\nr√\nλ−˜n2\u0011\n(A.24a)\nf˜τ˜τ(r) =C3\n(r√\nλ−˜n2)1/2JℓS+1\n2\u0010\nr√\nλ−˜n2\u0011\n+C4\n(r√\nλ−˜n2)1/2YℓS+1\n2\u0010\nr√\nλ−˜n2\u0011\n(A.24b)\nq1(r) =C5\n(r√\nλ−˜n2)3/2JℓS+3\n2\u0010\nr√\nλ−˜n2\u0011\n+C6\n(r√\nλ−˜n2)3/2YℓS+3\n2\u0010\nr√\nλ−˜n2\u0011\n+C7\n(r√\nλ−˜n2)3/2JℓS−1\n2\u0010\nr√\nλ−˜n2\u0011\n+C8\n(r√\nλ−˜n2)3/2YℓS−1\n2\u0010\nr√\nλ−˜n2\u0011\n,(A.24c)\nand\nq2(r) =1\n2ℓS(ℓS+ 1)(\n1\n2r\u0002\nℓS(ℓS+ 1) + 6 −r2\u0000\nλ−˜n2\u0001\u0003\nq1(r)−r2q′\n1(r)−1\n2r3q′′\n1(r))\n. (A.24d)\n– 52 –Imposing regularity at the origin in the above expressions demands C2=C4=C6=C8= 0. At this\nstage we impose our boundary conditions, which yield\nhL(rE) =r2\nEf˜τ˜τ(rE), f ˜τ(rE) = 0 , h T(rE) = 0 ,\nand f′\n˜τ˜τ(rE) +2\nr2\nEh′\nL(rE)−4(1−3p)\nrEf˜τ˜τ(rE)−2\nrEfrr(rE) +2ℓS(ℓS+ 1)\nr2\nEfr(rE)−2i˜nf˜τr(rE) = 0 .\n(A.25)\nImposing all of these conditions simultaneously constrains the possible values of λto obey the following\ncomplicated relation summarized in equation (3.57) in the main text in Section 3.3,\n4X\ni=0h\nηi(ϖ,˜Λ, ℓS) +ιi(ϖ,˜Λ, ℓS)pi\n·h\nJℓS+1\n2\u0010p\n˜Λ\u0011i4−ih\nJℓS+3\n2\u0010p\n˜Λ\u0011ii\n= 0 (A.26a)\nwhere ˜Λ≡(λ−˜n2)r2\nEandϖ≡˜n rEand\nη0(ϖ,˜Λ, ℓS) =p\n˜Λ(ℓS+ 1)( ˜Λ +ϖ2)h\n2−2˜Λ + 3(1 + ˜Λ)ℓS−4ℓ3\nS−4ℓ2\nSi\n, (A.26b)\nη1(ϖ,˜Λ, ℓS) = 3 ϖ2˜Λ2+ϖ2˜Λ−˜Λ3−3˜Λ2+ 2˜Λ−8ϖ2˜Λℓ3\nS−16ϖ2˜Λℓ2\nS+ 2ϖ2˜Λ2ℓS\n−4˜Λℓ4\nS+ 8˜Λ2ℓ3\nS−8˜Λℓ3\nS+ 16˜Λ2ℓ2\nS−˜Λℓ2\nS−6˜Λ3ℓS+ 4˜Λ2ℓS+ 5˜ΛℓS+ 8ϖ2ℓ5\nS+ 28ϖ2ℓ4\nS\n+ 26ϖ2ℓ3\nS−7ϖ2ℓ2\nS−19ϖ2ℓS−6ϖ2,(A.26c)\nη2(ϖ,˜Λ, ℓS) =˜Λ1/2\"\n3˜Λ3−ϖ2˜Λ2−ϖ2˜Λ−˜Λ2−5˜Λ + 4 ϖ2˜Λℓ2\nS+ 10ϖ2˜ΛℓS+ 6˜Λℓ3\nS\n−4˜Λ2ℓ2\nS+ 5˜Λℓ2\nS−10˜Λ2ℓS−7˜ΛℓS−4ϖ2ℓ4\nS−20ϖ2ℓ3\nS−23ϖ2ℓ2\nS+ 11ϖ2ℓS+ 11ϖ2#\n,(A.26d)\nη3(ϖ,˜Λ, ℓS) = 2 ˜Λ3−2ϖ2˜Λ2−4ϖ2˜Λ + 4 ˜Λ2+ 4ϖ2˜Λℓ2\nS+ 8ϖ2˜ΛℓS−2˜Λ2ℓ2\nS+ 2˜Λ2ℓS,(A.26e)\nη4(ϖ,˜Λ, ℓS) =−˜Λ3/2(˜Λ +ϖ2), (A.26f)\nand\nι0(ϖ,˜Λ, ℓS) =−12p\n˜Λ(ℓS+ 1)( ˜Λ +ϖ2)(1−˜Λ + 2 ℓS), (A.26g)\nι1(ϖ,˜Λ, ℓS) = 36 ˜Λ2−12ϖ2˜Λ2−12ϖ2˜Λ−12˜Λ3−12˜Λ\n−48ϖ2˜ΛℓS+ 48˜Λ2ℓS−24˜Λℓ2\nS−36˜ΛℓS+ 48ϖ2ℓ3\nS+ 144 ϖ2ℓ2\nS+ 132 ϖ2ℓS+ 36ϖ2,(A.26h)\nι2(ϖ,˜Λ, ℓS) =˜Λ1/2\u0010\n24ϖ2˜Λ−24˜Λ2+ 24˜Λ + 36 ˜ΛℓS−24ϖ2ℓ2\nS−132ϖ2ℓS−72ϖ2\u0011\n, (A.26i)\nι3(ϖ,˜Λ, ℓS) = 36 ϖ2˜Λ−12˜Λ2, (A.26j)\nι4(ϖ,˜Λ, ℓS) = 0 . (A.26k)\n– 53 –-4 -2 0 2 4-100102030\n-0.6 -0.4 -0.2 0.0 0.2-6-4-20246Figure 14 : The real part (left panel) and imaginary part (right panel) of ˜λ=λr2\nEas a function of pfor\nℓS= 2 and ϖ= 1. The point ( p,˜λ) = (1 /6,0) is marked as a red disk. This plot uses the same colour coding\nas Fig. 11.\nNaturally, this transcendental equation has no analytic solutions, but we can proceed numerically, just\nas in previous sections. Our general argument implies no negative modes can exist for p >1/6. We\ndisplay explicit results for the example mode with ℓS= 2 and ϖ= 1 in Fig. (14) and similar qualitative\nresults hold for all other values of ℓSand/ or ϖwe have investigated. Consistent with our general\nargument in Section 3.3, in the region p >1/6, there are no modes with Re ,˜λ <0. Finally, we note\nthat at large negative values of pwe obtain a single negative mode which obeys\n˜Λ =−16p2+ 24p+1\n3\u0002\n11ℓS(ℓS+ 1)−3 + 8 ϖ2\u0003\n+29ℓS(ℓS+ 1) + 40 + 40 ϖ2\n12p+O(p−2).(A.27)\nA.4 Modes with n= 0and ℓS= 1\nNext we consider scalar-derived gravitational modes with ℓS= 1. On-shell, these modes represent\ninfinitesimal translations. For these modes, SℓSmS\nIJ = 0 and the equivalent of hℓSmS\nT in Eq. (3.55e) is\naltogether absent. This is the reason why these modes have to be analysed separately. Due to staticity,\nthe metric components f˜τandf˜τrdecouple from the remaining and shall be analysed at the end of\nthis section. The analysis is very similar to the previous sections. We first introduce the trace variable\nf via\nhL=r2\n2(f−f˜τ˜τ−frr). (A.28)\nSince we are interested in modes with n= 0, the metric components f˜τandf˜τrdecouple and shall\nbe discussed at the end. After some algebra, the de Donder condition together with the Lichnerowicz\neigenvalue equation (3.50) yield\nfrr(r) =−1\nr2λf˜τ˜τ(r) +\u00121\n2−1\n2r2λ\u0013\nf(r) +1\nrλf′\n˜τ˜τ(r) +1\n2rλf′(r) (A.29a)\n– 54 –fr(r) =1\n2rλf˜τ˜τ(r) +1\n4rλf(r)−1\n2λf′\n˜τ˜τ(r)−1\n4λf′(r), (A.29b)\ntogether with\n(r2f′\n˜τ˜τ)′\nr2+\u0012\nλ−2\nr2\u0013\nf˜τ˜τ= 0, (A.29c)\nand\n(r2f′)′\nr2+\u0012\nλ−2\nr2\u0013\nf = 0 . (A.29d)\nOnce f˜τ˜τand f are known, one can determine frrandfrvia Eq. (A.29a) and Eq. (A.29b), respectively.\nEq. (A.29c) and Eq. (A.29d) can be readily solved in full generality to yield\nf(r) =C1\nr√\nλ\ncos\u0010√\nλr\u0011\n−sin\u0010√\nλr\u0011\n√\nλr\n+C2\nr√\nλ\ncos\u0010√\nλr\u0011\n√\nλr+ sin\u0010√\nλr\u0011\n, (A.30)\nf˜τ˜τ(r) =C3\nr√\nλ\ncos\u0010√\nλr\u0011\n−sin\u0010√\nλr\u0011\n√\nλr\n+C4\nr√\nλ\ncos\u0010√\nλr\u0011\n√\nλr+ sin\u0010√\nλr\u0011\n. (A.31)\nOnce again, regularity demands C2=C4= 0. On the other hand, our boundary conditions at the\ncavity wall require\nhL(rE) =r2\nEf˜τ˜τ(rE) (A.32a)\nand\nf′\n˜τ˜τ(rE) +2\nr2\nEh′\nL(rE)−4(1−3p)\nrEf˜τ˜τ(rE)−2\nrEfrr(rE) +4\nr2\nEfr(rE) = 0 . (A.32b)\nBoth conditions can be satisfied so long as ˜λ=λr2\nEobeys\n4˜λ2+ 3˜λ−12 + 3 cos\u0010\n2p\n˜λ\u0011\u0010\n4−9˜λ+ 4˜λ2\u0011\n+p\n˜λ\u0010\n24−23˜λ+ 3˜λ2\u0011\nsin\u0010\n2p\n˜λ\u0011\n−12ph\n3 + 3 ˜λ+˜λ2+ cos\u0010\n2p\n˜λ\u0011\u0010\n˜λ2+ 3˜λ−3\u0011\n−p\n˜λ\u0010\n6 +˜λ\u0011\nsin\u0010\n2p\n˜λ\u0011i\n= 0,(A.33)\nagain consistent with equation (3.57) showing there are no negative modes for p >1/6. The point\np=−1/12 marks the location with ˜λ= 0, so that the modes develop a positive real part for p >−1/12,\nand there is a single negative mode for p <−1/12. Away from this special point and at large values\nof−pwe find that the negative mode approaches\nλ=−16p2+ 24p+19\n3+49\n6p+O(p−2). (A.34)\nOnce again, consistent with our general argument in the range p >1/6, we find no negative modes. In\nFig. 15 we plot Re ˜λ(left and middle panels) and Im ˜λ(right panel) as a function of p. The solid black\nlines indicate regions of moduli space where the mode is purely real, while the dashed green line shows\nregions where the modes become complex. The point ( p,˜λ) = (−1/12,0) is marked as a red disk, and\nthe middle panel provides a zoom of the pink-shaded region of the left panel.\n– 55 –-10 0 10 20 30 40 50-100102030405060\n-0.3 -0.2 -0.1 0.0 0.1-4-2024\n10 20 30 40-1.0-0.50.00.51.0Figure 15 : The real part (left and middle panels) and imaginary part (right panel) of ˜λas a function of\np. The point ( p,˜λ) = (−1/12,0) is marked as a red disk. For p <−1/12, there exists a single mode with\nRe˜λ <0, indicating the presence of a field-theoretic negative mode. The green dashed curves indicate that\nthe mode is complex.\nWe now return to the metric components f˜τandf˜τr. The de Donder gauge condition imposes\nf˜τ(r) =r\n2[rf′\n˜τr(r) + 2f˜τr(r)], (A.35)\nwhile from the Lichnerowicz eigenvalue equation (3.50) we find\n(r4f′\n˜τr)′\nr4+λf˜τr= 0. (A.36)\nThe equation for f˜τrcan be easily integrated in full generality and one finds\nf˜τr(r) =C1\n(r√\nλ)3h\nr√\nλcos\u0010\nr√\nλ\u0011\n−sin\u0010\nr√\nλ\u0011i\n+C2\n(r√\nλ)3h\ncos\u0010\nr√\nλ\u0011\n+r√\nλsin\u0010\nr√\nλ\u0011i\n.(A.37)\nRegularity at the origin demands C2= 0, whereas our boundary conditions require\nf˜τ(rE) = 0⇒1\n˜λ3/2hp\n˜λcos\u0010p\n˜λ\u0011\n−\u0010\n1−˜λ\u0011\nsin\u0010p\n˜λ\u0011i\n= 0, (A.38)\nwith ˜λ=λr2\nE. We see this is independent of pand hence yields the same results as in the Dirichlet case\nstudied in [22] where stability was found. It is a relatively simple exercise to show that no complex\nzeroes of the above transcendental equation exist, and that the real zeroes all like on the positive axis.\nThe first few are approximately given by\n˜λ={7.52793(0) ,37.41480(5) ,86.79932(7) ,155.89863(2) , . . .}. (A.39)\nA.5 Modes with n̸= 0and ℓS= 1\nWe finally reach the last scalar mode to consider. Again SℓSmS\nIJ vanishes identically and the equivalent\nofhℓSmS\nT in Eq. (3.55e) is altogether absent. However, in general, f˜τandf˜τrwill not decouple from\nthe remaining metric components. We follow a familiar procedure, mirroring the preceding sections.\nWe commence by introducing the trace variable f through the equation:\nhL=r2\n2(f−f˜τ˜τ−frr). (A.40)\n– 56 –Following some algebraic manipulations, the conjunction of the de Donder condition and the Lich-\nnerowicz eigenvalue equation (3.50) produces\nf˜τ(r) =i\n4˜n(\u0012\n2 +r2˜n2−λ\nλ−˜n2\u0013\nf˜τ˜τ(r) +\u0002\n8−r2\u0000\nλ−˜n2\u0001\u0003\nfrr(r)\n−1\n2\u0014\n8−r2\u0000\nλ−2˜n2\u0001\n−λ\nλ−˜n2\u0015\nf(r) +rλ\nλ−˜n2f′\n˜τ˜τ(r) + 2rf′\nrr(r)−rλ\n2 (λ−˜n2)f′(r))\n,(A.41a)\nf˜τr(r) =i\n2˜n\"\n4\nrfrr(r)−1\n2r\u0012\n4−λ\nλ−˜n2\u0013\nf(r) +1\nr\u0012\n2−λ\nλ−˜n2\u0013\nf˜τ˜τ(r)\n+˜n2\nλ−˜n2f′\n˜τ˜τ(r) +f′\nrr(r)−λ\n2 (λ−˜n2)f′(r)#\n,(A.41b)\nfr(r) =r\n2\"\nfrr(r) +1\n2λ\nλ−˜n2f˜τ˜τ(r)−1\n4λ\nλ−˜n2f(r)−r\n2˜n2\nλ−˜n2f′\n˜τ˜τ(r) +r\n2f′\nrr(r)−r\n4λ−2˜n2\nλ−˜n2f′(r)#\n(A.41c)\nhrr(r) =q1(r) +˜n2\nλ−˜n2f˜τ˜τ(r) +1\n2λ−2˜n2\nλ−˜n2f(r), (A.41d)\ntogether with\n(r2f′\n˜τ˜τ)′\nr2+\u0012\nλ−2\nr2−˜n2\u0013\nf˜τ˜τ= 0, (A.41e)\n(r2f′)′\nr2+\u0012\nλ−2\nr2−˜n2\u0013\nf = 0 . (A.41f)\nand\n(r4q′\n1)′\nr4+\u0012\nλ−4\nr2−˜n2\u0013\nq1= 0. (A.41g)\nSolutions to f˜τ˜τ, f and q1are found via Eqs. (A.41e)-(A.41g) and one can then determine all remaining\nvariables using Eqs. (A.41a)-(A.41d). For f˜τ˜τ, f and q1we find\nf(r) =C1\nr√\nΛ\ncos\u0010√\nΛr\u0011\n−sin\u0010√\nΛr\u0011\n√\nΛr\n+C2\nr√\nΛ\ncos\u0010√\nΛr\u0011\n√\nΛr+ sin\u0010√\nΛr\u0011\n, (A.42)\nf˜τ˜τ(r) =C3\nr√\nΛ\ncos\u0010√\nΛr\u0011\n−sin\u0010√\nΛr\u0011\n√\nΛr\n+C4\nr√\nΛ\ncos\u0010√\nΛr\u0011\n√\nΛr+ sin\u0010√\nΛr\u0011\n, (A.43)\nq1(r) =C5\nr2Λ\u00143\nr√\nλcos\u0010\nr√\nλ\u0011\n−3\nr2Λsin\u0010\nr√\nλ\u0011\n+ sin\u0010\nr√\nλ\u0011\u0015\n– 57 –+C6\nr2Λ\u00143\nr√\nλsin\u0010\nr√\nλ\u0011\n+3\nr2Λcos\u0010\nr√\nλ\u0011\n−cos\u0010\nr√\nλ\u0011\u0015\n(A.44)\nwhere we defined Λ ≡λ−˜n2. Regularity at the origin demands C2=C4=C6= 0, while our boundary\nconditions impose\nhL(rE) =r2\nEf˜τ˜τ(rE), f ˜τ(rE) = 0 , (A.45a)\nand\nf′\n˜τ˜τ(rE) +2\nr2\nEh′\nL(rE)−4(1−3p)\nrEf˜τ˜τ(rE)−2\nrEfrr(rE) +4\nr2\nEfr(rE)−2i˜nf˜τr(rE) = 0 . (A.45b)\nAll the last three conditions are satisfied if the eigenvalue λ(or equivalently Λ) satisfies\np\n˜Λ sin\u0010p\n˜Λ\u0011\"\n324ϖ2˜Λ2−9ϖ2˜Λ5−198ϖ2˜Λ4−81ϖ2˜Λ3−9˜Λ6+ 126 ˜Λ5−405˜Λ4+ 324 ˜Λ3\n+p\u0010\n3888ϖ2˜Λ3−108ϖ2˜Λ5−108ϖ2˜Λ4+ 972 ϖ2˜Λ2−108˜Λ6−108˜Λ5+ 972 ˜Λ3\u0011#\n+p\n˜Λ sin\u0010\n3p\n˜Λ\u0011\"\n459ϖ2˜Λ3−9ϖ2˜Λ5−306ϖ2˜Λ4−108ϖ2˜Λ2+ 135 ˜Λ6−558˜Λ5+ 567 ˜Λ4−108˜Λ3\n+p\u0010\n1620ϖ2−108ϖ2˜Λ5˜Λ4−324ϖ2˜Λ2−108˜Λ6−108˜Λ5+ 1296 ˜Λ4−324˜Λ3\u0011#\n+ cos\u0010\n3p\n˜Λ\u0011\"\n9ϖ2˜Λ6+ 90ϖ2˜Λ5−450ϖ2˜Λ4+ 324 ϖ2˜Λ3−27˜Λ7+ 342 ˜Λ6−666˜Λ5+ 324 ˜Λ4\n+p\u0010\n1728ϖ2˜Λ4−648ϖ2˜Λ5+ 972 ϖ2˜Λ3−216˜Λ6−864˜Λ5+ 972 ˜Λ4\u0011#\n+ cos\u0010p\n˜Λ\u0011\"\n198ϖ2˜Λ5−9ϖ2˜Λ6+ 18ϖ2˜Λ4−324ϖ2˜Λ3+ 27˜Λ7−54˜Λ6+ 234 ˜Λ5−324˜Λ4\n+p\u0010\n216ϖ2˜Λ5−3024ϖ2˜Λ4−972ϖ2˜Λ3−216˜Λ6−432˜Λ5−972˜Λ4\u0011#\n= 0,(A.46)\nwhere we defined ˜Λ = Λ r2\nE. Again this expression is consistent with (3.57) in the main text, showing\nthere are no negative modes for p >1/6.\nUsing the above expression we can verify that a zero mode exists if p=−(1 + ˜n2)/12, becoming a\nnegative mode in the region p <−(1 + ˜n2)/12 and positive otherwise. Away from this special point,\nwe have to proceed numerically and we find that for p <−(1 + ˜n2)/12 a single negative mode exists,\nand that at large values of −pit approaches\nλ=−16p2+ 24p+1\n3\u0000\n19 + 8˜ n2\u0001\n+49 + 20˜ n2\n6p+O(p−2). (A.47)\nConsistent with our general argument, within the range p >1/6, we observe the absence of negative\nmodes. To conclusively affirm the non-problematic nature of this mode, numerical exploration is once\n– 58 –0 5 10 15 20 25-10-505101520\n-0.25 -0.20 -0.15 -0.10-2-1012\n0 5 10 15 20-1.0-0.50.00.51.0Figure 16 : The real part (left and middle panels) and imaginary part (right panel) of ˜λas a function of p.\nThe point ( p,˜λ) = (−(1 + ˜n2)/12,0) is marked as a red disk. For p <−(1 + ˜n2)/12, there exists a single mode\nwith Re ˜λ <0, indicating the presence of a field-theoretic negative mode. The green dashed curves indicate\nthat the mode is complex. In generating this plot we took ˜ n= 1.\nagain undertaken. In Fig. 16 we depict Re ˜λ(left and middle panels) and Im ˜λ(right panel) against\np. Solid black lines delineate regions in moduli space where the mode is exclusively real, while dashed\ngreen lines demarcate areas where the modes become complex. A red disk at ( p,˜λ) = (−(1+ ˜n2)/12,0)\nserves as a reference point, and the middle panel offers an amplified view of the pink-shaded region in\nthe left panel. In generating this figure we took ˜ n= 1, but we found similar results for different values\nof ˜n.\nA.6 Non-spherical Euclidean modes with ℓV≥2\nWe now turn our attention to vector-derived gravitational perturbations. These are easier to present\nthan the scalars, owing to the fact that less components of the metric are non-zero. Importantly we find\nthat the behaviour of the vector modes is precisely the same as that in the Dirichlet case. Following\nthe stability of the Dirichlet case studied in [22] then all vector modes will be stable for our general\nconformal boundary condition for any p.\nThese modes are built from harmonic vectors on the two-sphere, i.e.\nDIYℓVmV\nI = 0 and DIDIYℓVmV\nJ + [ℓV(ℓV+ 1)−1]YℓVmV\nJ = 0. (A.48)\nwith ℓV≥1. For the case of a two-sphere, it turns out to be relatively simple to write these modes in\nterms of scalar spherical harmonics. Namely,\nYℓVmV≡YℓVmV\nI dxI=⋆S2\u0000\ndYℓSmS\u0001\n(A.49)\nwhere ⋆S2is the Hodge dual on the two-sphere with ℓV=ℓS, and non-zero modes start with ℓV≥1.\nA general vector-derived mode can now be written as\nhℓVmV\nˆaˆb= 0 (A.50a)\nhℓVmV\nˆaI =ˆfℓVmV\nˆa (˜τ, r)YℓVmV\nI (A.50b)\nhℓVmV\nIJ =ˆhℓVmV\nT (˜τ, r)SℓVmV\nIJ (A.50c)\n– 59 –where SℓSmS\nIJ is a traceless symmetric two tensor defined as\nSℓVmV\nIJ =DIYℓVmV\nJ +DJYℓVmV\nI . (A.50d)\nFor each mode, there are a total of (at most) three functions to be determined. Owing to the background\n∂/∂τ Killing vector field, we can further decompose the scalars ˆfℓVmV\nˆa (˜τ, r) and ˆhℓVmV\nT (˜τ, r) as\nˆfℓVmV\nˆa (˜τ, r) =ei˜n˜τfℓVmV˜n\nˆa (r) and ˆhℓVmV\nT (˜τ, r) =ei˜n˜τhℓVmV˜n\nT (r). (A.51)\nJust like with scalars, modes featuring distinct values of ℓV,mV, and/or ˜ ndecouple from one another at\nthe quadratic level in the action. Consequently, we will omit the subscripts ℓV, mV,˜nin the subsequent\ndiscussion. It is worth noting that modes with ℓV= 1 hold a unique status, as for these modes,\nS1,mV\nIJ = 0. The treatment of these specific modes will be addressed in sections A.7 and A.8. Finally,\nwe also note that for the vectors, our boundary conditions reduce to those of the standard Dirichlet\nproblem, being independent of p, and thus there are no negative modes.\nA.6.1 Static modes\nIn section we analyse vector-derived perturbations with ˜ n= 0. Due to staticity, the metric component\nf˜τdecouples from the remaining ones, and we shall discuss it at the end of this section. We are left\nwith determining frandhT. Imposing the de Donder gauge condition and the Lichnerowicz eigenvalue\nequation (3.50) yields\nhT=r\nℓV(ℓV+ 1)−2[r f′\nr(r) + 2 fr(r)] (A.52a)\nand\n(r2f′\nr)′\nr2+\u0014\nλ−ℓV(ℓV+ 1)\nr2\u0015\nfr= 0. (A.52b)\nThe last equation for frcan be readily solved as\nfr(r) =C1\n(r√\nλ)1/2JℓV+1\n2\u0010\nr√\nλ\u0011\n+C2\n(r√\nλ)1/2YℓV+1\n2\u0010\nr√\nλ\u0011\n. (A.53)\nRegularity at the origin demands C2= 0 while our boundary conditions require\nhT(rE) = 0⇒JℓV+3\n2\u0010p\n˜λ\u0011p\n˜λ−(ℓV+ 2) JℓV+1\n2\u0010p\n˜λ\u0011\n= 0, (A.54)\nagain showing the mode behaviour is independent of p, and thus the same as for the Dirichlet case. It\nis easy to show that the above equation only has real roots, and that for ℓV= 2 the first four modes\nare approximately given by\n˜λ={17.91715(3) ,57.60145(7) ,116.86211(3) ,195.83544(5) , . . .}. (A.55)\nTo sum up, there are no negative modes for this sector of modes. We now return to analysing f˜τ. This\ncomponent turns out to be gauge invariant. The Lichnerowicz eigenvalue equation (3.50) yields\nf′′\n˜τ+\u0014\nλ−ℓV(ℓV+ 1)\nr2\u0015\nf˜τ= 0. (A.56)\n– 60 –The generic solution to f˜τreads\nf˜τ= (r√\nλ)1/2h\nC1JℓV\n2+1\u0010\nr√\nλ\u0011\n+C2YℓV\n2+1\u0010\nr√\nλ\u0011i\n. (A.57)\nOnce again, regularity at the origin demands C2= 0, while our boundary conditions give\nf˜τ(rE) = 0⇒JℓV\n2+1\u0010p\n˜λ\u0011\n, (A.58)\nwith ˜λ=λr2\nE, again showing no dependence on p. There are no Euclidean modes lying on the complex\nplane, and in fact λis positive definite. For the first four modes we find\n˜λ={26.37461(7) ,70.84999(9) ,135.02070(9) ,218.92018(9) , . . .}. (A.59)\nA.6.2 Generic modes\nHaving discussed the case with ˜ n= 0, we now turn to the more complicated case with ˜ n̸= 0. In\nthis case all three functions f˜τ(r),fr(r) and hT(r) can in principle be non-vanishing. Just as for the\nscalar-derived modes, these can also be solved in terms of Bessel functions as follows\nhr(r) =q1(r) +ℓV−1\nrq2(r), (A.60a)\nhT(r) =−r\nℓV+ 2q1(r) +q2(r), (A.60b)\nand\nh˜τ(r) =i\n˜nr\u0014\n(1 +ℓV)q1(r) +1−ℓ2\nV\nrq2(r) +rq′\n1(r)−(1−ℓV)q′\n2(r)\u0015\n, (A.60c)\nwith\nq′′\n1+\u0014\nλ−(ℓV+ 1)( ℓV+ 2)\nr2−˜n2\u0015\nq1= 0, (A.60d)\nand\nr2\u0012q′\n2\nr2\u0013′\n+\u0014\nλ−ℓV(ℓV−1)−2\nr2−˜n2\u0015\nq2= 0. (A.60e)\nThe equations for q1andq2can be readily solved as,\nq1(r) = (rΛ)1/2h\nC1JℓV+3\n2\u0010\nr√\nΛ\u0011\n+C2YℓV+3\n2\u0010\nr√\nΛ\u0011i\n(A.61)\nq2(r) = (rΛ)3/2h\nC3JℓV−1\n2\u0010\nr√\nΛ\u0011\n+C4YℓV−1\n2\u0010\nr√\nΛ\u0011i\n, (A.62)\nwhere we defined Λ ≡λ−˜n2. Regularity at the origin demands C2=C4= 0, while our boundary\nconditions demand\nf˜τ(rE) =hT(rE) = 0 , (A.63)\nwhich translates into\nJℓV+1\n2\u0010p\n˜Λ\u0011hp\n˜ΛJℓV+3\n2\u0010p\n˜Λ\u0011\n−(ℓV+ 2) JℓV+1\n2\u0010p\n˜Λ\u0011i\n= 0 (A.64)\nwhere we have defined ˜Λ = Λ r2\nE. Since each factor can be separately vanishing we see that we recover\nthe same values eigenvalues as in the previous section, except that we should make the replacement ˜λ\nby˜Λ.\n– 61 –A.7 Modes with n= 0and ℓV= 1\nThis mode is perhaps the easiest to analyse. Here hT= 0 in Eq. (A.50d) and additionally, since the\nmetric is static, frandf˜τdecouple from each other. Indeed, the de Donder gauge fixes frto be\nfr(r) =C1\nr2, (A.65)\nand then regularity at the origin demands C1= 0, establishing that no modes exist with fr̸= 0 in this\nsector. For f˜τwe find\nf˜τ(r) =C1\ncos\u0010\nr√\nλ\u0011\n−sin\u0010\nr√\nλ\u0011\nr√\nλ\n+C2\ncos\u0010\nr√\nλ\u0011\nr√\nλ+ sin\u0010\nr√\nλ\u0011\n (A.66)\nand regularity at the origin demands C2= 0, while our boundary conditions at the cavity wall impose\nf˜τ(rE) = 0⇒cosp\n˜λ−sinp\n˜λp\n˜λ(A.67)\nwith ˜λ=λr2\nE. Again there is no dependence on pso the behaviour is as for Dirichlet boundary\nconditions. There are no zeroes on the complex plane, and the first four real zeroes are positive (note\nthat ˜λ= 0 is not a valid mode, since then the perturbation vanishes identically) and read\n˜λ={4.49340(9) ,7.72525(2) ,10.90412(2) ,14.06619(4) }. (A.68)\nA.8 Modes with n̸= 0and ℓV= 1\nThis is the last case to analyse. The de Donder gauge condition imposes\nf˜τ(r) =i\nr˜n[rf′\nr(r) +fr(r)], (A.69)\nwhile the Lichnerowicz eigenvalue equation (3.50) yields\nf′′\nr+\u0012\nλ−˜n2−6\nr2\u0013\nfr= 0. (A.70)\nThe equation for hrcan be found in full generality\nfr(r) =C1\n3 cos\u0010\nr√\nΛ\u0011\nr√\nΛ−3 sin\u0010\nr√\nΛ\u0011\nr2Λ+ sin\u0010\nr√\nΛ\u0011\n\n+C2\ncos\u0010\nr√\nΛ\u0011\n−3 cos\u0010\nr√\nΛ\u0011\nr2Λ−3 sin\u0010\nr√\nΛ\u0011\nr√\nΛ\n,(A.71)\nwith Λ ≡λ−˜n2. Regularity at the origin demands C2= 0, while our boundary condition imposes\nf˜τ(rE) = 0⇒p\n˜Λ cosp\n˜Λ−sinp\n˜Λ = 0 , (A.72)\nwith ˜Λ = Λ r2\nE. Once again, the behaviour is independent of p, and so is the same as in the Dirichlet\ncase. Furthermore, there are no zeroes of the above equation on the real axis. Furthermore, the mode\nwith ˜Λ = 0 corresponds to a vanishing mode function f˜τand on the positive real axis we find\n˜Λ ={20.19072(9) ,59.67951(6) ,118.89986(9) ,197.85781(1) , . . .}. (A.73)\n– 62 –B Analyticity of pin the Unstable Complex ˜λPlane\nWe now argue that Re pis analytic in the complex half plane Re ˜λ≤0. While the expression for p\ngiven in (3.57) is given in terms of Bessel functions ofp\n˜Λ, the form of the expressions ensures that\nin fact pis analytic at ˜Λ = 0. As noted in the main text, the numerator in this expression has no\npoles for Re ˜Λ≤0, and other than a branch cut at ˜Λ = 0 is analytic, and so any non-analyticity in\nthe expression for pwould have to derive from the denominator BℓS(˜Λ) +ϖ2DℓS(˜Λ). Since the explicit\nform of BℓS(˜Λ) and DℓS(˜Λ) in terms of Bessel functions again guarantee analyticity for Re ˜Λ≤0,\nexcept for branch cuts at ˜Λ (which cancel with the denominator) then non-analyticity of pmay only\nderive from zeros of this denominator. Now an important point is that ˜Λ =˜λ−ϖ2, and ϖ2>0 and\nshould be real. Thus to show that Re pis analytic for Re ˜λ≤0 we should show that the denominator,\nBℓS(˜Λ) + ϖ2DℓS(˜Λ), has no zeros for Re ˜Λ≤ −ϖ2.\nFirstly we will consider the special case ϖ= 0 where we can make an analytic argument. Then\nwe will consider the general case ϖ̸= 0 where we will use a combination of analytic and numerical\nmethods to make the argument.\nB.1 The static case ϖ= 0\nIn the static case the denominator is simply the function BℓS(˜Λ). We note that in this case ˜Λ = ˜λ.\nLooking at BℓS(˜Λ) we find it factorizes as,\nBℓS(˜Λ) = 12 ˜Λ J 1\n2+ℓS(p\n˜Λ)F1F2F3 (B.1)\nwith factors,\nF1(˜Λ) =p\n˜ΛJ 1\n2+ℓS(p\n˜Λ) + J 3\n2+ℓS(p\n˜Λ)\nF2(˜Λ) = (1 + ℓS)J1\n2+ℓS(p\n˜Λ)−p\n˜ΛJ 3\n2+ℓS(p\n˜Λ)\nF3(˜Λ) = (1 + 2 ℓS−˜Λ)J 1\n2+ℓS(p\n˜Λ)−p\n˜ΛJ 3\n2+ℓS(p\n˜Λ). (B.2)\nWe now provide a simple proof, adapted from [58], that all the functions Fi(˜Λ) only admit zeroes on\nthepositive real axis, and hence the denominator in this static case, BℓS(˜Λ), has no zeros for Re ˜λ≤0.\nFor the first two conditions, we note that the statement we are after is equivalent to proving that\nfi(x)≡Fi(x2) only admits zeroes xifor real values of x. Indeed, the first two conditions can be written\nin the following form\nAiJνi(x) +xJ′\nνi(x) = 0 (B.3a)\nwith\nA1=ℓS+5\n2, ν 1=ℓS+3\n2. (B.3b)\nA2=1\n2, ν 2=ℓS+1\n2. (B.3c)\nNext, we note that a Bessel function Jνsatisfies the following differential equation\n∂x(x∂xJν)−ν2\nxJν=−xJν. (B.4)\n– 63 –We now perform a change of variable, and set x=yα, and take y∈(0,1). Later on, αwill be related\nto the zeroes of the fi, but for now it is an arbitrary complex number. Under this change of variable,\nEq. (B.4) is transformed into\n∂y(y∂yjα\nν)−ν2\nyjα\nν=−α2yjα\nν, (B.5)\nwith jα\nν(y)≡Jν(αy). We recognise the above as a St¨ urm-Liouville problem (if appropriate boundary\nconditions are chosen) with α2playing the role of eigenvalue and jα\nνthe corresponding eigenfunction.\nConsider now two distinct eigenfunctions jβ\nνandjα\nν. Consider the following combination\n−(α2−β2)Zy\n0˜yjβ\nνjα\nνd˜y=Zy\n0\u001a\njβ\nν\u0014\n∂˜y(˜y∂˜yjα\nν)−ν2\n˜yjα\nν\u0015\n−jα\nν\u0014\n∂˜y(˜y∂˜yjβ\nν)−ν2\n˜yjβ\nν\u0015\u001b\nd˜y\n=y\u0000\njβ\nν∂yjα\nν−jα\nν∂yjβ\nν\u0001\n, (B.6)\nwhere we assumed that ν >−1, so that the boundary term evaluated at y= 0 vanishes. We have thus\nconcluded that Zy\n0˜yjα\nνjβ\nνd˜y=y\nα2−β2\u0000\njα\nν∂yjβ\nν−jβ\nν∂yjα\nν\u0001\n. (B.7)\nLet us take αto be a complex zero of fiandβ= ¯αto be its complex conjugate. We will assume that\nαisnotpurely imaginary for the moment, and return to this case later. Note that if αwere purely\nimaginary, we would have α2= ¯α2, and the argument below would not work. Note that if αis a zero\noffi, so will ¯ αbecause fiis a real function of its arguments. In terms of the jα\nν, the condition (B.3a)\ncan be written as\nAijα\nνi(1) + jα\nνi′(1) = 0 . (B.8)\nEvaluating (B.7) with y= 1 (and noting that α2̸= ¯α2) yields\nZy\n0˜yjα\nνij¯α\nνid˜y= 0 (B.9)\nwith Eq. (B.8) regarded as the relevant boundary condition. But the left hand side is positive definite,\nsince j¯α\nνi=¯jα\nνi. This is a contradiction, and as such complex zeroes of (B.3a) cannot exist.\nWe are left with analysing the case where αcan potentially be purely imaginary. For this, we\nrecall the series definition of the Bessel function\nJν(α) =+∞X\nm=0(−1)m\nm!Γ(ν+m+ 1)\u0010α\n2\u00112m+ν\n, (B.10)\nwhere Γ is a Gamma function. Our condition (B.3a) can be written as\nd\ndα\u0014\u0010α\n2\u0011AiJνi(α)\u0015\n= 0⇒\u0010α\n2\u0011νi+Ai+∞X\nm=0(2m+νi+Ai)(−1)m\nm!Γ(ν+m+ 1)\u0010α\n2\u00112m\n= 0. (B.11)\nSo long as νi+Ai>0, all the terms on the right hand side of the above expression that are being\nsummed over are positive definite for αbeing purely imaginary, thus excluding the possibility of purely\nimaginary zeroes. For our choice of fithen νi>0 and Ai>0, thus establishing our proof. In fact,\none can show with some effort that for νi+Ai<0 two purely imaginary zeroes do exist.\n– 64 –The last condition in (B.2) is a little more subtle, because it cannot be written as in (B.3a).\nHowever, it can be written as\n(ν−x2)Jν(x) +xJ′\nν(x) = 0 with ν=ℓS+1\n2. (B.12)\nOne could repeat the above argument replacing (B.8) with\n(ν−α2)jα\nν(1) + jα\nν′(1) = 0 (B.13)\nas the boundary condition to use on the St¨ urm-Liouville problem. The right hand side in Eq. (B.9)\nno longer vanishes, but it is purely imaginary, and thus we still reach a contradiction for complex\nmodes that are not purely imaginary or purely real. Finally, the case with purely imaginary zeroes\ncan be dealt with in a slightly different manner. We note that this final boundary condition can also\nbe written as\nν(1−ν) Jν(α) + 2αJ′\nν(α) +α2J′′\nν(α) = 0 with ν=ℓS+1\n2. (B.14)\nUsing the series definition for J νin (B.10) yields\n\u0010α\n2\u0011ν+∞X\nm=02(2m+ 1)( m+ν)(−1)m\nm!Γ(ν+m+ 1)\u0010α\n2\u00112m\n= 0 (B.15)\nwhich again is positive definite so long as ν >0 and αis purely imaginary, establishing that F3only\nhas zeroes on the positive real axis, just like F1andF2.\nThus we conclude that BℓS(˜Λ) has no zeros for Re ˜Λ≤0, which implies that in the static case\nϖ= 0 then the denominator in the expression for phas no zeros for Re ˜λ≤0, and hence pis analytic\nin that region.\nB.2 The general case ϖ̸= 0\nNow let us consider the general non-static case where the denominator for pisBℓS(˜Λ) + ϖ2DℓS(˜Λ).\nNow DℓStakes a more complicated form than BℓSand doesn’t have a simple factorization. We proceed\nby noting that at a zero of the denominator,\nϖ2=−BℓS(˜Λ)\nDℓS(˜Λ)(B.16)\nandϖshould be positive and real. We now aim to show that such a zero cannot occur for Re ˜λ≤0\nby showing that ϖcannot be positive real in the above expression there.\nFrom their explicit forms both BℓSandDℓSare analytic for Re ˜Λ<0, except for the branch cut at\n˜Λ = 0 along the negative real axis. Hence, away from this negative real axis, ϖ2is an analytic function\nfor Re ˜Λ<0, and thus for Re ˜λ <0, provided that DℓShas no zeros. To analyze this it is convenient\nto normalize DℓSby the function\nFℓS(˜Λ) =−˜Λ3J1\n2+ℓS\u0010p\n˜Λ\u00113\nJ3\n2+ℓS\u0010p\n˜Λ\u0011\n. (B.17)\nand consider the ratio,\nIℓS(˜Λ)≡DℓS(˜Λ)\nFℓS(˜Λ). (B.18)\n– 65 –Writing ˜Λ = x+iy, then due to the properties of Bessel functions and our choice of normalizing\nfactor, then IℓSis analytic for x <0 and y >0, and zeros of DℓScorrespond to zeros of IℓSin that\nregion. Now asymptotically, IℓS(˜Λ) =−12/˜Λ +O\u0010\n˜Λ−3/2\u0011\nforx≤0 and y≥0. Hence Im IℓS(˜Λ)→0\nasymptotically. We also note that Im IℓSvanishes on the real negative ˜λaxis. Thus on all boundaries\nor asymptotic regions of the quadrant x≤0 and y≥0 except the imaginary axis then IℓS(˜Λ)→0. So\nnow we may plot this on the imaginary axis. This is shown in figure 17 for a range of ℓS= 0, . . . , 6.\nThe lowest curve is the one for ℓS= 0, and increasing ℓSgives the curves with greater value. This is\ntrue for the values of ℓSshown, but is true for all other values of ℓSwe have tested. Thus it is positive\non the imaginary axis. We note that it has a singularity at ˜Λ = 0 and hence is discontinuous between\nthe real and imaginary axes at the origin. Since IℓSis analytic for x < 0 and y >0 then Im IℓSis\nharmonic in the complex plane, and hence its minimum is found on a boundary or asymptotically.\nHence we conclude that Im IℓSis strictly positive for x <0 and y >0, vanishing only on the negative\nreal axis, y= 0, or asymptotically, and hence Im IℓScannot have a zero except on the negative real axis.\nThen IℓScan have a zero only on the negative real axis. We may then check whether this occurs by\nconsidering Re IℓSon the negative real axis. This is also plotted in the same figure 17 for ℓS= 0, . . . , 6\nand again the lower curve is ℓS= 0, bounding the others from below, and is everywhere greater than\nzero. Again this is true for the values of ℓSshown and all other values of ℓSwe have tested. Thus we\nconclude that IℓSand hence DℓShas no zeros for Re ˜Λ<0, and hence no zeros in the unstable complex\nplane Re ˜λ <0.\n0 20 40 60 80 1000.00.51.01.5\n-100 -80 -60 -40 -20 00.00.51.01.52.02.53.03.5\nFigure 17 : The lefthand figure shows a plot of Im IℓSas a function of ˜Λ on the positive imaginary axis for\nvarious ℓS= 0,1, . . . , 6. We see that the curve ℓS= 0 which is positive bounds those for the higher ℓSshown\nfrom below, so that the Im IℓSare strictly positive on the positive imaginary axis. This holds for all other ℓS\ntested. The righthand figure shows Re IℓSon the negative real ˜Λ axis for the same ℓS. Again we see the curve\nforℓS= 0 bounds the others from below, and being positive implies Re IℓS>0 on this negative real ˜Λ axis.\nAgain this holds for all other ℓStested.\n– 66 –As a consequence we have that at a zero of the denominator ϖ2is analytic as a function of ˜Λ in\nthe half plane Re ˜Λ<0. While the numerator and denominator are singular at ˜Λ = 0, their ratio\nwhich gives ϖ2is analytic there. Asymptotically it goes as,\nϖ2=−˜Λ−4ip\n˜Λ +O(1) (B.19)\nforx <0 and y >0 with the ℓSdependence being subleading. Now Im ϖ2= 0 on the negative real\naxis and its value on the positive imaginary axis is shown in figure 18 for ℓS= 0, . . . , 6, together with\nthe asymptotic behaviour shown as a black dashed curve. We see it is everywhere negative on the\npositive imaginary axis, except at the origin. Its value asymptotically in the x <0 and y >0 quadrant\naway from the axes is Im ϖ2→ −Im˜Λ for large |˜Λ|, and so is negative for x <0 and y >0. This holds\nfor all other values of ℓStested. Thus we conclude that since ϖ2is analytic for x <0 and y >0, then\nImϖ2is harmonic in xandy, so its minimum value lies on a boundaries or asymptotically, and hence\nit vanishes only on the negative real axis, and otherwise has non-zero imaginary part.\n0 5 10 15 20 25 30-40-30-20-100\n-40 -30 -20 -10 00204060\nFigure 18 : The lefthand figure shows a plot of Im ϖ2on the positive imaginary ˜Λ axis for various ℓS=\n0,1, . . . , 6. The black dotted curve is the asymptotic behaviour given in the text. We find that Im ϖ2is\nnegative away from the origin on this positive imaginary axis for these ℓSand all other values examined. The\nrighthand plot shows Re ϖ2on the negative real ˜Λ axis again for the same ℓS. Also plotted as a black dotted\nline is the condition ϖ2=−˜Λ, which lies below all the curves. We see that for these ℓSshown that while zeros\nof the denominator of pcan occur for Re ˜Λ<0, they do not occur for Re ˜Λ =−ϖ2, and hence for Re ˜λ <0\nwhere the corresponding eigenvalue would be unstable. This holds for all other values of ℓSwe have tested.\nSince ϖ2must be real and positive, then for Re ˜Λ<0 a zero of the denominator can only occur\non the negative real ˜Λ axis, as ϖ2is otherwise complex. The question is then the behaviour of ϖ2\non this negative real axis. Thus finally in figure 18 we also plot ϖ2on the negative real ˜Λ axis for\nℓS= 0, . . . , 6. For these values shown, and all others tested, we see ϖ2is positive there, and hence zeros\nare potentially allowed for Re ˜Λ<0. However as we now argue, they are not allowed for Re ˜λ <0,\nwhich implies the tighter condition Re ˜Λ<−ϖ2. In order to see this on the plot we show a black\n– 67 –dashed curve the condition ˜Λ =−ϖ2. Thus for all these ℓSshown, and all others checked, we find that\nfor any given ˜Λ<0 we see that for a zero of the pdenominator, ϖ2is greater than −˜Λ, and hence\ncorresponds to real strictly positive ˜λ=ϖ2+˜Λ, rather than negative.\nThus finally we conclude that for Re ˜λ <0 there are no zeros of the denominator of p. Hence pis\nanalytic in the half plane Re ˜λ <0 corresponding to unstable eigenmodes.\nC Static and Spherical Euclidean Modes for Vacuum Filled Cavity with Λ̸= 0\nIn this section we will restrict to four spacetime dimensions and study the eigenspectrum of (3.8) in\nthe static spherically symmetric sector for a cavity filled with the Euclidean vacuum (A)dS geometry.\nWe write this background as,\ncds2≡ˆgabdxadxb=F(r)2dτ2+dr2\nF(r)2+r2dΩ2\n2, (C.1a)\nwith τ∼τ+βand\nF(r)2= 1−3Λr2. (C.1b)\nWe will take the boundary to be the surface r=R0endowed with the metric on S1\nβ×S2\nR0. For positive\nΛ, demanding F(r)2>0 yields the bound R0≤1/√\n3Λ.\nWe restrict our attention to modes that preserve spherical symmetry and are τindependent. Such\nmodes can always be written as\nδds2≡habdxadxb=F(r)2a(r) dτ2+b(r)dr2\nF(r)2+r2c(r) dΩ2\n2, (C.2)\nwhere a,bandcare to be determined in what follows. The boundary condition δΓµν= 0 discussed in\nsection 2 implies that the following relations must be obeyed at the cavity wall r=R0:\na(R0) =c(R0) (C.3a)\nand\n2F(r0)R0[a′(R0) + 2c′(R0)]−b(R0) [4F(r0) + 2R0F′(R0)] + 6 p a(R0) [4F(r0) + 2R0F′(R0)] = 0 .\n(C.3b)\nwith′denoting differentiation with respect to r. To proceed, it is convenient to introduce the scalar\nf≡ˆgabhab=a(r) +b(r) + 2c(r). (C.4)\nExchanging cby f and solving the de Donder gauge condition yields\na(r) =F(r)2f(r) +1\n2rF(r)2f′(r)−rF(r)2b′(r) + [1−4F(r)2]b(r), (C.5a)\nc(r) =1\n2[f(r)−a(r)−b(r)]. (C.5b)\nThe eigenvalue equation (3.8) is then solved with respect to f and b, yielding\n1\nr2\u0002\nr2F(r)2f′(r)\u0003′+ 2 Λ f( r) =−λf(r) (C.6a)\n– 68 –1\nr4F(r)2\u0002\nr4F(r)4b′(r)\u0003′+4\n3Λ f(r)−2F(r)2−1\nrf′(r)−10\n3Λb(r) =−λ b(r). (C.6b)\nFor Λ ̸= 0, these equations can be solved in terms of hypergeometric functions 2F1(a, b;c;z)\nb(r) =2−Λr2\n3λ+ 2Λf′(r)\nr+3λ+ 4Λ\n6λ+ 4Λf(r) +C1\nr32F1\u00121\n4−ηb,1\n4+ηb;−1\n2;r2Λ\n3\u0013\n+C2 2F1\u00127\n4−ηb,7\n4+ηb;5\n2;r2Λ\n3\u0013\n(C.7a)\nf(r) =C3\nr2F1\u00121\n4−ηf,1\n4+ηf;1\n2;r2Λ\n3\u0013\n+C4 2F1\u00123\n4−ηf,3\n4+ηf;3\n2;r2Λ\n3\u0013\n(C.7b)\nwhere we have defined\nηb=1\n4r\n9 +12λ\nΛand ηf=1\n4r\n33 +12λ\nΛ. (C.7c)\nRegularity at the origin demands C1=C3= 0, while our boundary conditions (C.3b) can be schemat-\nically written as\nA.\u0014C2\nC4\u0015\n= 0 (C.8)\nwith A a 2 ×2 matrix that is a complicated function of the hypergeometric functions appearing in\nEqs. (C.7), and their derivatives, evaluated at r=R0. We provide the explicit form at the end of this\nAppendix. To determine the eingenvalue λwe simply demand det A = 0, and solve numerically for λ\nusing a simple Newton-Raphson algorithm.\nBefore proceeding, let us note that for λ= 0 the solutions we have determined coincide with those\nof the dynamical linear problem that we have investigated in section 4. Indeed, it is a tedious but\nsimple exercise to show that det A = 0 when λ= 0 and\np≡p⋆(R2\n0Λ) =6−R2\n0Λ\n6(3−R2\n0Λ)(2−R2\n0Λ), (C.9)\nin agreement with Eq. (4.11) with α= 0.\nIn the case Λ = 0 this reduces to the case analysed in Section 3.3.1. Here we are interested in\nthe case where Λ ̸= 0 where we can repeat the analysis but the calculations quickly become rather\ncomplicated. However, with some effort, one can find the equivalent of Eq. (3.62) which we write as\np=p⋆(R2\n0Λ)\u0002\n1 + Ξ( R2\n0Λ)R2\n0λ+O(λ2)\u0003\n⇔R2\n0λ=p−p∗(R2\n0Λ)\np∗(R2\n0Λ)Ξ (Λ R2\n0)(C.10a)\nwhere\nΞ(˜Λ) =1\n(˜Λ−6)(˜Λ−3)(\n9\n4(˜Λ−4)2\n(˜Λ−3)χ(˜Λ)−9√\n3\n2˜Λ3/2arctanh p\n˜Λ√\n3!\n−1\n(˜Λ−3)˜Λ\"\n27\n2+ 51˜Λ−44˜Λ2+ 12˜Λ3−9˜Λ4\n8+1\n8√\n33(˜Λ−4)2˜Λ2#)\n,(C.10b)\n– 69 –-4 -3 -2 -1 0 1 2 30.00.51.01.52.0Figure 19 : The function Ξ( ˜Λ) defined in Eq. (C.10b) as a function of its argument.\nand\nχ(˜Λ) =2F1\u0010\n1\n4\u0000\n7−√\n33\u0001\n,1\n4\u0000\n3 +√\n33\u0001\n;3\n2;˜Λ\n3\u0011\n2F1\u0010\n1\n4\u0000\n7−√\n33\u0001\n,1\n4\u0000\n7 +√\n33\u0001\n;5\n2;˜Λ\n3\u0011. (C.10c)\nIn Fig. 19 we plot Ξ( ˜Λ) as a function of ˜Λ and find that it is positive definite, diverging as Ξ( ˜Λ)∝\n(˜Λ−3)−1near ˜Λ = 3.\nSince we have established positivity of Ξ( ˜Λ) for ˜Λ≤3, we can go back to Eq. (C.10a) and read\noff the consequences for λ. We see that if p⋆(R2\n0Λ)>0, then we must require p > p⋆(R2\n0Λ) in order to\nhave stability. This will certainly happen for Λ R2\n0<2. However, if p⋆(R2\n0Λ)<0 stability now requires\np < p⋆(R2\n0Λ). Indeed, in the range 2 < R2\n0Λ<3 this is precisely what happens. Note that for R2\n0Λ = 2\none can argue that only Dirichlet boundary conditions yield stability, since limR2\n0Λ→2−p⋆(R2\n0Λ) = + ∞.\nFinally, as promised above, we give the explicit form for the 2 ×2 matrix A appearing in Eq. (C.8)\n4A11=\u0002\nR2\n0(1 + 4 ηb) Λ + 5 −12ηb\u0003\n2F1\u00127\n4−ηb,7\n4+ηb;5\n2;R2\n0Λ\n3\u0013\n−(4ηb−7)\u0000\nR2\n0Λ−3\u0001\n2F1\u001211\n4−ηb,7\n4+ηb;5\n2;R2\n0Λ\n3\u0013\n,(C.11)\n8(3λ+ 2Λ) R2\n0A12= 4 (3 −4ηf)2F1\u00127\n4−ηf,3\n4+ηf;3\n2;ΛR2\n0\n3\u0013\n+\u0002\n16ηf−12−16η2\nfΛR2\n0\u0000\nΛR2\n0−2\u0001\n+ 3(4 λ+ 11Λ) R2\n0\u0000\nΛR2\n0−2\u0001\u0003\n2F1\u00123\n4−ηf,3\n4+ηf;3\n2;ΛR2\n0\n3\u0013\n,\n(C.12)\n– 70 –A21= 2(1 + 2 p)2F1\u00127\n4−ηb,7\n4+ηb;5\n2;ΛR2\n0\n3\u0013\u0000\nΛR2\n0−2\u0001\n−1\n90\u0000\n16η2\nb−49\u0001\nΛR2\n0\u0000\nΛR2\n0−3\u0001\n2F1\u001211\n4−ηb,11\n4+ηb;7\n2;ΛR2\n0\n3\u0013\n,(C.13)\nand\n3240(3 λ+ 2Λ) A 22=−3240[(6 p−3)λ−4Λ]\u0000\nΛR2\n0−2\u0001\n2F1\u00123\n4−ηf,3\n4+ηf;3\n2;ΛR2\n0\n3\u0013\n−Λ\u0000\n16η2\nf−9\u0001(\nΛR2\n0\u0002\n6 + Λ R2\n0\u0000\nΛR2\n0−5\u0001\u0003\u0000\n16η2\nf−49\u0001\n2F1\u001211\n4−ηh,11\n4+ηf;7\n2;ΛR2\n0\n3\u0013\n−\n30\u0002\n24−9(λ+ 4Λ) R2\n0+ Λ(3 λ+ 10Λ) R4\n0+ 12p\u0000\nΛR2\n0−2\u00012\u0003\n2F1\u00127\n4−ηh,7\n4+ηf;5\n2;ΛR2\n0\n3\u0013)\n.(C.14)\nD Euclidean Spherical Static Fluctuations of (A)dS Black Holes\nIn this section, we numerically compute the spectrum of the Lichnerowicz operator of the Euclidean\nSchwarzschild black hole background with a non-vanishing cosmological constant.\nThe black hole solution to the bulk equations of motion inside a cavity at radius r=rBhas the\nsame form as that in equation (3.37a), with the blackening factor now given by\nF(r)2=−κr2\nℓ2+ 1−r+\nr\u0012\n−κr2\n+\nℓ2+ 1\u0013\n, (D.1)\nwhere ℓis the (A)dS length and is related to the cosmological constant via\nℓ=r\n3\nκΛ. (D.2)\nHere κ= 1 and κ=−1 correspond to dS and AdS spacetimes, respectively. In addition to x≡r+/rB,\nwe have another length scale y≡r+/ℓ > 07.\nThe action is now given by\nS=SEH+Sbdy=2πℓ2\nG·(1−x3)y4\nx3(−κ+ 3y2)−ΘβBℓy\n4Gx2·[(4−3x)x2+ 3(−2 +x3)y2κ]p\nx2(1−x) +κy2(x3−1), (D.3)\nand we have\nB=4π\n1−3κy2p\nx2(1−x) + (x3−1)κy2. (D.4)\nThe expression for Kis rather tedious and not very illuminating, so we do not write it down explicitly\nhere. Similar to the asymptotically flat case, there are two families of saddles with the same BandK,\nparametrized by their own xandy. We refer to the family with larger/smaller xat a fixed BandK\nas the large/small black hole. One can verify that the small black hole always has a larger action than\nthe large black hole, thus never dominates the ensemble.\n7For dS black holes, we further require that the cavity is inside the cosmological horizon, which puts another constraint on the\nrange of xandy.\n– 71 –0.0 0.1 0.2 0.3 0.4 0.5 0.60.10.20.30.40.50.6(a)\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.00000.00050.00100.00150.00200.00250.00300.0035 (b)\n0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.6 (c)\nFigure 20 : The left/right panel gives the relationship between yandxat the place where Bis extremized\nfor Schwarzschild AdS/dS black holes. The middle panel gives the difference between the critical yfor the\nAnderson boundary condition and that for the Dirichlet boundary condition for Schwarzschild AdS black\nholes. At the place where Bis extremized, we expect a marginal stability of the black hole.\nUnlike the Dirichlet case8, there is no analytical relationship between yandxat the place where\nBand the action are both extremized. This is also the place where we expect the black hole to be\nmarginally stable. In figure 20 we plot the numerical relationship between yandxat the critical place\nforp= 0 (Anderson boundary condition). We also include the difference between critical yforp= 0\nand that for the Dirichlet case p→+∞in figure 20b to show explicitly that the critical points are\np-dependent. The calculations become numerically challenging for general pbut in principle, one can\nget similar figures using a high enough precision.\nFigures 21 and 22 give the lowest four modes of the Lichnerowicz operator in the background of a\nSchwarzschild black hole in AdS and dS spacetimes inside a cavity, respectively. Our numerical data\nmatches precisely (to numerical precision) with the prediction of marginal stability in section D. Again,\nthe existence of another negative mode reflects the instability of the vacuum solution under Anderson\nboundary conditions.\nReferences\n[1] J. W. York, Jr., Black hole thermodynamics and the Euclidean Einstein action ,Phys. Rev. D 33(1986) 2092.\n[2] D. J. Gross, M. J. Perry and L. G. Yaffe, Instability of Flat Space at Finite Temperature ,Phys. Rev. D 25(1982)\n330.\n[3] B. F. Whiting and J. W. York, Jr., Action Principle and Partition Function for the Gravitational Field in Black\nHole Topologies ,Phys. Rev. Lett. 61(1988) 1336.\n[4] T. Prestidge, Dynamic and thermodynamic stability and negative modes in Schwarzschild-anti-de Sitter ,Phys.\nRev. D 61(2000) 084002 [ hep-th/9907163 ].\n8See equation (B.11) in [22] for the expression for Euclidean Schwarzschild AdS black holes. Our xandyare called y−1\n0and\ny+in [22], respectively.\n– 72 –1.0 1.5 2.0 2.5 3.0-20246810\n1.840 1.845 1.850 1.855 1.860-0.010-0.0050.0000.005Figure 21 : The left panel shows the real part of the four lowest lying eigenvalues of ˆ∆Las a function of x−1\nfor the case with y≡r+/ℓ∼0.3551 and Λ <0. The right panel is a zoom-in version of the left panel at the\nplace where there is a marginal stable mode. The green dots indicate the place of critical x−1predicted by\np= 0, as in figure 20a. The red dots indicates the place of critical x−1predicted by the Dirichlet formula,\nequation (B.11) in [22]. The gray line is Re ˜λ= 0.\n1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35-202468\nFigure 22 : The left panel shows the real part of the four lowest lying eigenvalues of ˆ∆Las a function of x−1\nfor the case with y≡r+/ℓ= 0.4 and Λ >0. The green dots indicate the place of critical x−1predicted by\np= 0, as in figure 20c. The gray line is Re ˜λ= 0.\n[5] H. S. Reall, Classical and thermodynamic stability of black branes ,Phys. Rev. D 64(2001) 044005\n[hep-th/0104071 ].\n[6] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity ,Adv. Theor. Math. Phys. 2\n(1998) 231 [ hep-th/9711200 ].\n[7] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncritical string theory ,Phys.\nLett. B 428(1998) 105 [ hep-th/9802109 ].\n[8] E. Witten, Anti-de Sitter space and holography ,Adv. Theor. Math. Phys. 2(1998) 253 [ hep-th/9802150 ].\n[9] E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories ,Adv. Theor. Math.\n– 73 –Phys. 2(1998) 505 [ hep-th/9803131 ].\n[10] I. G. Avramidi and G. Esposito, Lack of strong ellipticity in Euclidean quantum gravity ,Class. Quant. Grav. 15\n(1998) 1141 [ hep-th/9708163 ].\n[11] I. G. Avramidi and G. Esposito, Gauge theories on manifolds with boundary ,Commun. Math. Phys. 200(1999)\n495 [ hep-th/9710048 ].\n[12] M. T. Anderson, On boundary value problems for Einstein metrics ,Geom. Topol. 12(2008) 2009 [ math/0612647 ].\n[13] P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow, and strongly coupled CFT in the\nSchwarzschild Unruh or Boulware vacua ,Class. Quant. Grav. 28(2011) 215018 [ 1104.4489 ].\n[14] E. Witten, A note on boundary conditions in Euclidean gravity ,Rev. Math. Phys. 33(2021) 2140004\n[1805.11559 ].\n[15] D. Anninos, D. A. Galante and C. Maneerat, Gravitational Observatories ,2310.08648 .\n[16] A. Adam, S. Kitchen and T. Wiseman, A numerical approach to finding general stationary vacuum black holes ,\nClass. Quant. Grav. 29(2012) 165002 [ 1105.6347 ].\n[17] G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity ,Phys. Rev. D\n15(1977) 2752.\n[18] L. Randall and R. Sundrum, An Alternative to compactification ,Phys. Rev. Lett. 83(1999) 4690\n[hep-th/9906064 ].\n[19] J. D. Brown and J. W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action ,\nPhys. Rev. D 47(1993) 1407 [ gr-qc/9209012 ].\n[20] D. Marolf, Gravitational thermodynamics without the conformal factor problem: partition functions and Euclidean\nsaddles from Lorentzian path integrals ,JHEP 07(2022) 108 [ 2203.07421 ].\n[21] M. Headrick and T. Wiseman, Ricci flow and black holes ,Class. Quant. Grav. 23(2006) 6683 [ hep-th/0606086 ].\n[22] D. Marolf and J. E. Santos, The canonical ensemble reloaded: the complex-stability of Euclidean quantum gravity\nfor black holes in a box ,JHEP 08(2022) 215 [ 2202.11786 ].\n[23] G. W. Gibbons, S. W. Hawking and M. J. Perry, Path Integrals and the Indefiniteness of the Gravitational\nAction ,Nucl. Phys. B 138(1978) 141.\n[24] D. Marolf and J. E. Santos, Stability of the microcanonical ensemble in Euclidean Quantum Gravity ,JHEP 11\n(2022) 046 [ 2202.12360 ].\n[25] X. Liu, D. Marolf and J. E. Santos, Stability of saddles and choices of contour in the Euclidean path integral for\nlinearized gravity: Dependence on the DeWitt Parameter ,2310.08555 .\n[26] G. Odak and S. Speziale, Brown-York charges with mixed boundary conditions ,JHEP 11(2021) 224\n[2109.02883 ].\n[27] J. D. Bekenstein, Black holes and entropy ,Phys. Rev. D 7(1973) 2333.\n[28] S. W. Hawking, Particle Creation by Black Holes ,Commun. Math. Phys. 43(1975) 199.\n[29] O. J. C. Dias, G. W. Gibbons, J. E. Santos and B. Way, Static Black Binaries in de Sitter Space ,Phys. Rev. Lett.\n131(2023) 131401 [ 2303.07361 ].\n[30] G. D. Birkhoff and R. E. Langer, Relativity and modern physics . Harvard University Press, 1923.\n[31] G. W. Gibbons and S. W. Hawking, Classification of Gravitational Instanton Symmetries ,Commun. Math. Phys.\n66(1979) 291.\n[32] B. Zumino, Supergravity ,Annals of the New York Academy of Sciences 302(1977) 545\n[https://nyaspubs.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1749-6632.1977.tb37073.x ].\n– 74 –[33] G. W. Gibbons and M. J. Perry, Black Holes and Thermal Green’s Functions ,Proc. Roy. Soc. Lond. A 358\n(1978) 467.\n[34] S. W. Hawking, Gravitational Instantons ,Phys. Lett. A 60(1977) 81.\n[35] S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space ,Commun. Math. Phys.\n87(1983) 577.\n[36] H. Kodama and A. Ishibashi, A Master equation for gravitational perturbations of maximally symmetric black\nholes in higher dimensions ,Prog. Theor. Phys. 110(2003) 701 [ hep-th/0305147 ].\n[37] T. Andrade, W. R. Kelly, D. Marolf and J. E. Santos, On the stability of gravity with Dirichlet walls ,Class.\nQuant. Grav. 32(2015) 235006 [ 1504.07580 ].\n[38] F. W. J. Olver and E. C. Bullard, The asymptotic expansion of bessel functions of large order ,Philosophical\nTransactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 247(1954) 328\n[https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1954.0021 ].\n[39] V. Cardoso, O. J. C. Dias, G. S. Hartnett, L. Lehner and J. E. Santos, Holographic thermalization, quasinormal\nmodes and superradiance in Kerr-AdS ,JHEP 04(2014) 183 [ 1312.5323 ].\n[40] B. E. Niehoff, J. E. Santos and B. Way, Towards a violation of cosmic censorship ,Class. Quant. Grav. 33(2016)\n185012 [ 1510.00709 ].\n[41] P. M. Chesler and D. A. Lowe, Nonlinear Evolution of the AdS 4Superradiant Instability ,Phys. Rev. Lett. 122\n(2019) 181101 [ 1801.09711 ].\n[42] P. M. Chesler, Hairy black resonators and the AdS4 superradiant instability ,Phys. Rev. D 105(2022) 024026\n[2109.06901 ].\n[43] T. Jacobson and A. Satz, On the renormalization of the Gibbons-Hawking boundary term ,Phys. Rev. D 89(2014)\n064034 [ 1308.2746 ].\n[44] G. Neri and S. Liberati, On the resilience of the gravitational variational principle under renormalization ,JHEP\n10(2023) 054 [ 2306.17505 ].\n[45] G. Fournodavlos and J. Smulevici, On the initial boundary value problem for the Einstein vacuum equations in\nthe maximal gauge ,1912.07338 .\n[46] G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem for the Einstein Equations with Totally\nGeodesic Timelike Boundary ,Commun. Math. Phys. 385(2021) 1615 [ 2006.01498 ].\n[47] G. Fournodavlos and J. Smulevici, The Initial Boundary Value Problem in General Relativity: The Umbilic Case ,\nInt. Math. Res. Not. 2023 (2023) 3790 [ 2104.08851 ].\n[48] Z. An and M. T. Anderson, On the initial boundary value problem for the vacuum Einstein equations and\ngeometric uniqueness ,2005.01623 .\n[49] D. Anninos, D. A. Galante and B. M¨ uhlmann, Finite features of quantum de Sitter space ,Class. Quant. Grav. 40\n(2023) 025009 [ 2206.14146 ].\n[50] T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity ,Phys. Rev. 108(1957) 1063.\n[51] C. V. Vishveshwara, Stability of the schwarzschild metric ,Phys. Rev. D 1(1970) 2870.\n[52] F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations ,Phys. Rev.\nLett.24(1970) 737.\n[53] V. Moncrief, Gravitational perturbations of spherically symmetric systems. I. The exterior problem. ,Annals Phys.\n88(1974) 323.\n[54] J. M. Bardeen and W. H. Press, Radiation fields in the schwarzschild background ,J. Math. Phys. 14(1973) 7.\n– 75 –[55] M. Dafermos, G. Holzegel and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational\nperturbations ,Acta Math. 222(2019) 1 [ 1601.06467 ].\n[56] S. Klainerman and J. Szeftel, Global Nonlinear Stability of Schwarzschild Spacetime under Polarized\nPerturbations ,1711.07597 .\n[57] M. Dafermos, G. Holzegel, I. Rodnianski and M. Taylor, The non-linear stability of the Schwarzschild family of\nblack holes ,2104.08222 .\n[58] G. N. Watson, A Treatise on the Theory of Bessel Functions , ch. XV, p. 482. Cambridge University Press,\nCambridge, 1922.\n– 76 –"    },    {        "title": "2402.04385v2.Locating_the_roots_of_a_quadratic_equation_in_one_variable_through_a_Line_Circumference__LC__geometric_construction_in_the_plane_of_complex_numbers.pdf",        "content": "1 \n Locating the roots of a quadratic equation in one variable through a Line -Circumference  (LC)     \ngeometric  construction in the plane of complex  numbers  \n \nDaniel Alba Cuell ar \nalbacd@cimat.mx  \n \nAbstract  \nThis paper  describe s a geometrical  method for finding the roots  𝑟1, 𝑟2 of a quadratic equation in one complex \nvariable  of the form 𝑥2+𝑐1𝑥+𝑐2=0, by means of a Line 𝐿 and a Circumference  𝐶 in the complex plane, \nconstructed from known coefficients  𝑐1, 𝑐2. This Line -Circumference (LC) geometric structure contains the  sought  \nroots  𝑟1, 𝑟2 at the intersections of its component elements  𝐿 and 𝐶. Line 𝐿 is mapped onto Circumference  𝐶 by a \nMöbius transformation . The location and inclination angle of 𝐿 can be computed directly from coefficients  𝑐1, 𝑐2, \nwhile  𝐶 is constructed by dividing the constant term  𝑐2 by each point from  𝐿. This paper describes the t echnical  \ndetails for the quadratic LC method, and then shows how the quadratic LC method works through a numerical \nexample. The quadratic LC method described here , although more elaborate than the traditional  quadratic formula,  \ncan be extended to find initial approximations to the roots of polynomials  in one variable of degree 𝑛≥3. As an \nadditional feature , this paper also studies an interesting property of the rectilinear segments connecting key points \nin a quadratic  LC structure.  \nKeywords : Quadratic Equation; Roots of an Equation; Complex Number s; Line; Circumference ; Numerical Method; \nGeometrical  Method; Möbius Transformation; Polynomial  Root-finding  Algorithm .   \n \nDescription of the LC method for finding the roots of a quadratic equation  \nLet’s c onsider the quadratic equation  \n𝑥2+𝑐1𝑥+𝑐2=(𝑥−𝑟1)(𝑥−𝑟2)=0.        (1) \nIn equation (1), we will assume that both the coefficients 𝑐1, 𝑐2 of the polynomial on the left and the \nroots 𝑟1, 𝑟2 in the central factorization are elements of the  set of complex numbers, denoted as ℂ. \nMoreover, we will suppose that roots  𝑟1, 𝑟2 are both different from  zero, and different from each other.    \nThe geometrical  method described below for finding the unknown roots  𝑟1, 𝑟2 from known coefficients  \n𝑐1, 𝑐2, can be used  when the roots 𝑟1, 𝑟2 of equation (1) are on a line  𝐿1⊂ℂ that does not pass through \nthe origin (i.e.,  0∉𝐿1, 𝐿1 being a continuous line of infinite length, with no gaps). This line  𝐿1 can be \nexpressed as a parametric trajectory of the form  \n𝐿1:𝑝1+𝑡𝑣𝜃∗,           (2) \nwhere 𝑝1,𝑣𝜃∗∈ℂ are fixed values, and  𝑡∈ℝ is a parameter that determines each point contained in 𝐿1. \nPoint 𝑝1, define d as 𝑝1:=−𝑐1/2=(𝑟1+𝑟2)/2, is called the fixed point of 𝐿1, while point 𝑣𝜃∗, define d \nas 𝑣𝜃∗:=𝑒𝑖𝜃∗=cos𝜃∗+𝑖sin𝜃∗ (𝑖=√−1), is called the directi on vecto r of 𝐿1. Notice that 𝜃∗ is the \nargument of 𝑣𝜃∗, and  also,  |𝑣𝜃∗|=1; that is  to say , 𝑣𝜃∗ is a unit directi on vector of 𝐿1, and  𝜃∗ is the \ninclination  angle  of line 𝐿1 that causes 𝐿1 to contain 𝑟1, 𝑟2. Notice that 𝑝1 is the midpoint, on 𝐿1, \nbetween roots  𝑟1, 𝑟2, so |𝑟1−𝑝1|=|𝑟2−𝑝1|>0. LC method for locating the roots of a quadratic equation  \n2 \n How do we compute  the angle  𝜃∗ from coefficients 𝑐1, 𝑐2 in equation (1)? From line  𝐿1, it is possible to \nconstruct a semi -line of the form  \n𝐿𝑑:(𝑝1+𝑡𝑣𝜃∗)(𝑝1−𝑡𝑣𝜃∗)=𝑝12+𝑡2(−𝑣𝜃∗2),       (3) \nwith fixed point  𝑝12=𝑐12/4 and direction vector −𝑣𝜃∗2. Noti ce that the inclination  angle  of 𝐿𝑑 in (3) is  \narg(−𝑣𝜃∗2)=2𝜃∗+𝜋 radians, and that 𝑟1𝑟2=𝑐2∈𝐿𝑑. This means that we can construct  𝐿𝑑 directly \nfrom the polynomial coefficients 𝑐1, 𝑐2, and therefore we can derive angle  𝜃∗ from the inclination  angle \nof semi -line 𝐿𝑑. \nOperationally, we obtain  𝜃∗ in the following  way: \n1. We construct a direction vector  for 𝐿𝑑 from its two known points 𝑝12, 𝑐2; let's call this direction \nvector 𝑣𝑑. In this way, 𝑣𝑑=𝑐2−𝑝12. \n2. Since  arg(𝑣𝑑)=2𝜃∗+𝜋, we have that 𝜃∗=arg(−𝑣𝑑)/2. \nIn conclusi on, \n𝜃∗=arg(𝑐12/4−𝑐2)/2.         (4) \nNow, we know the fixed point 𝑝1 and the inclination  angle  𝜃∗ of the line 𝐿1 that contain s the roots 𝑟1, 𝑟2 \nof equation (1), so we can trace  it on the plane ℂ. To determine the location of the roots  𝑟1, 𝑟2 within 𝐿1, \nlet's consider the circumference  \n𝐶:𝑐2/𝐿1=𝑐2/(𝑝1+𝑡𝑣𝜃∗).         (5) \nNoti ce that 𝐶→0 when 𝑡→±∞. Noti ce also that roots  𝑟1, 𝑟2 are contained in circumference  𝐶. \nExpression (5) actually comes from a bijective mapping  𝐿1→𝑐2/𝐿1, called Möbius transformation. A \ndemonstration that expression (5) is indeed a circ umference  in the plane  ℂ can be found in Appendix 1.  \nFrom the above, we see that the unknown roots  𝑟1, 𝑟2 are the intersections between  𝐿1 and 𝐶. \nTo determine the center and radius of circumference 𝐶 defined by expression (5), it is sufficient to know \ntwo distinct points on it; one of them could  be 𝑤1=𝑐2/𝑝1, and another could  be 𝑤2=𝑐2/(𝑝1+𝑣𝜃∗). It \nis then possible to prove that the center 𝑐 of circumference  𝐶 is the point  \n𝑐=−𝑦1(𝑥22+𝑦22)+𝑦2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1+𝑖𝑥1(𝑥22+𝑦22)−𝑥2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1,      (6) \nwhere 𝑥1=Re(𝑤1), 𝑦1=Im(𝑤1), 𝑥2=Re(𝑤2), and 𝑦2=Im(𝑤2). The radius of circumference  𝐶 is \nsimply  |𝑐| (see Appendix 2 for details).  \nIn this way, we have constructed a computable geometrical  method to determine the location of the \nroots of quadratic equations of the form (1); as previously stated, this method will work as long as the \ndistinct roots  𝑟1, 𝑟2 are contained in a continuous line  𝐿1⊂ℂ\\{0} with no gaps .  \nIn summary, the steps to computationally implement this geometrical  method are as follows:  \n \n LC method for locating the roots of a quadratic equation  \n3 \n  \nLC method for finding the roots  𝒓𝟏,𝒓𝟐∈ℂ of quadratic equation 𝒙𝟐+𝒄𝟏𝒙+𝒄𝟐=𝟎 \n \na) Compute  the inclination angle 𝜃∗ for line 𝐿1 by using expression (4).  \nb) Determine line  𝐿1 defined by expression (2), using fixed point  𝑝1=−𝑐1/2 and direction  \nvector  𝑣𝜃∗=cos𝜃∗+𝑖sin𝜃∗=𝑒𝑖𝜃∗. \nc) Determine circumference  𝐶:𝑐2/𝐿1 with center 𝑐 given by (6), and with radius |𝑐|. \nd) Locate the intersections between  𝐿1 and 𝐶. These are the roots 𝑟1, 𝑟2 of equation (1).  \n \n \nExample  \nFind the roots of equation  \n𝑥2+(−1−7𝑖)𝑥+(−18+𝑖)=0        (7) \nusing the LC method described above.  \n \nSolution  \nAccording  to the form of equation (1), we designate coefficients in equation (7) as 𝑐1=−1−7𝑖, and \n𝑐2=−18+𝑖. Now let's carry out the steps of the LC method described above:  \na) First, we c ompute  the inclination angle 𝜃∗ for line 𝐿1 via the coefficients 𝑐1, 𝑐2:    \n𝜃∗=arg(𝑐12/4−𝑐2)/2=arg(6+2.5𝑖)/2=0.1973956  radians =11°18′35.757\". \n \nb) Now, we obtain the parametric expression for line 𝐿1:      \n𝐿1:(1+7𝑖)/2+𝑡𝑒0.1973956 𝑖=(0.5+3.5𝑖)+𝑡(0.9805807 +0.1961161 𝑖). \n \nc) From b), we immediately obtain  the parametric expression for 𝐶:             \n𝐶:𝑐2/𝐿1=(−18+𝑖)/[(0.5+3.5𝑖)+𝑡(0.9805807 +0.1961161 𝑖)].                    \nAccording to (6), this parametric expression  𝐶 is a circ umference  with center           \n𝑐=0.676471+2.617647𝑖 and radius |𝑐|=2.703644 . Figure 1 shows the parametric \ntrajectories  𝐿1 and 𝐶 obtained in this example.  \n \nd) From Figure 1 we can see that the intersections between line  𝐿1 and circumference  𝐶 occur at \npoints 𝑟1=−2+3𝑖 and 𝑟2=3+4𝑖. We can see that these values obtained are in fact the ones \nthat satisfy equation (7). It is perfectly feasible to construct a computational algorithm that \nnumerically approximates the intersections between a line and a circ umference  in the plane  of \ncomplex numbers ℂ, given a fixed point and a directi on vector for the line, as well as the center \nand radius of the circ umference . LC method for locating the roots of a quadratic equation  \n4 \n  \nFigure 1 . Line 𝐿1 and circumference 𝐶 associated with equation (7), plotted on the plane ℂ. The intersection s \nbetween these two trajectories coincide with the roots of (7).  \n \nConclusion  \nIn this paper  we have described a geometrical method that helps us locate the roots of a quadratic \nequation in a single complex variable of the form (1), under the assumption that the roots are di fferent  \nfrom each other and are on a line in the complex plane that does not pass through the origin. Clearly, \nthe quadratic LC method described here is more elaborate than the general  formula  for solving  quadratic \nequations ; however, the quadratic LC method serves as the basis for a general algorithm  that can be \nused  to find initial approximati ons to  the roots of  polynomial  equations  in one variable of degree  𝑛≥3. \nFor more details, see [1]. \nAs an additional note, referring to Figure 1, it is possible to verify numerically that angle ∠0𝑝1𝑟1 is equal \nto angle  ∠𝑟1𝑝1(𝑐2/𝑝1), or equivalently, that angle  ∠0𝑝1𝑟2 is equal to the angle ∠𝑟2𝑝1(𝑐2/𝑝1); this does \nnot happen by chance. In Appendix 3 it is show n that this is in fact a property that holds  for geometric \nconstructions with characteristics similar to the one in Figure 1. From this, we can conclude that an \nalternative method for determining the inclination  angle 𝜃∗ for line 𝐿1 can be based on the bisection of \nangle  ∠0𝑝1(𝑐2/𝑝1), which can be constructed directly from the coefficients 𝑐1, 𝑐2 of equation (1).  \n \n \n \nLC method for locating the roots of a quadratic equation  \n5 \n Appendix 1  \nProposition . If 𝑧∈ℂ is on a line 𝐿 that does not pass through the origin, then 𝑏/𝑧 (𝑏∈ℂ,𝑏≠0) is on a \ncircumference 𝐶 that passes through the origin without containing it.  \nDemonstration . In notes by Professor Carl Eberhart (University of Kentucky), available at [ 2], it is shown \nthat if 𝑧 is on a line  𝐿 that does not pass through the origin, then 1/𝑧 is on a circ umference 𝐶 that passes \nthrough the origin without containing it. We shall see the demonstration of this lemma  below, based on \nthe notes [ 2].  \nWe start from the assumption that 𝑧 is any point on an arbitrary line  𝐿⊂ℂ\\{0} that does  not pass \nthrough the origin, so we could say that 𝑧 is a line that does not pass through the origin. Then, there is a \nvector 𝑢∈ℂ, |𝑢|=1 (𝑢 is a unit vector), such that 𝑢∙𝑧 is a vertical line to the right of the origin;  \nalgebraically, this vertical line can be expressed as  \n𝑢∙𝑧=1\n𝑐+𝑐𝑒𝑖𝜃=1\n𝑐+𝑐(cos𝜃+𝑖sin𝜃), 𝑐>0, 𝜃∈(−𝜋,𝜋).      (A1.1)  \nExpression (A1.1) refers to  a vertical line, because Re(𝑢∙𝑧)=1\n2𝑐 is a constant value, while       \nIm(𝑢∙𝑧)=−𝑐sin𝜃\n(𝑐+𝑐cos𝜃)2+(𝑐sin𝜃)2 is a real-valued  continuous function of a real variable  𝜃 that decreases \nmonotonically, with dom ain (−𝜋,𝜋) and range  (−∞,+∞)=ℝ; it can be seen that  Im(𝑢∙𝑧)→+∞ \nwhen 𝜃→−𝜋+, and  Im(𝑢∙𝑧)→−∞ when 𝜃→𝜋−. Im(𝑢∙𝑧) changes most rapidly near 𝜃=±𝜋, and \nit is symmetrical with respect to the origin, as can be seen in Figure A1.  \n \nFigure A1 . Behavior of the imaginary part of 𝑢∙𝑧, within interval 𝜃∈(−𝜋,𝜋), for the parametric value 𝑐=10. \nNote that Im(𝑢∙𝑧)=0 when 𝜃=0. \nLC method for locating the roots of a quadratic equation  \n6 \n Now, it is evident that 1\n𝑢∙𝑧=𝑐+𝑐𝑒𝑖𝜃 is a circ umference  with center  𝑐 on the real axis of the plane ℂ, and \nradius 𝑐. Then,  \n𝑢1\n𝑢∙𝑧=1\n𝑧=𝑢𝑐+𝑢𝑐𝑒𝑖𝜃=𝑐𝑒𝑖arg(𝑢)+𝑐𝑒𝑖[𝜃+arg(𝑢)].      (A1.2)  \nExpression (A1.2) tells us that 1\n𝑧 is a circ umference  with center  𝑐𝑒𝑖arg(𝑢) and radius  𝑐. Moreover, \ncircumference 1\n𝑧 passes through the origin (without containing it), since  \n|𝑐𝑒𝑖arg(𝑢)−0|=|𝑐𝑒𝑖arg(𝑢)|=|𝑐||𝑒𝑖arg(𝑢)|=|𝑐|=𝑐;      (A1.3)  \nthat is  to say , the distance between the origin and the center of circumference  1\n𝑧 is equal to its radius. \nWith this, we have proved the lemma  stated at the beginning of this demonstration.  \nTo conclude, it is now evident , from expression ( A1.2 ), that if we multiply each point of circumference \n1/𝑧 by a constant value  𝑏∈ℂ,𝑏≠0, 𝑏/𝑧 is still a circ umference  passing through the origin (without \ncontaining it), a lbeit  now with center  𝑏𝑐𝑒𝑖arg(𝑢) and radius  |𝑏|𝑐. ∎ \n \nAppendix 2  \nProposition . The center  𝑐 of circumference  𝐶, which passes through three di fferent  points 0,𝑤1,𝑤2∈ℂ, \nis given by  \n𝑐=−𝑦1(𝑥22+𝑦22)+𝑦2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1+𝑖𝑥1(𝑥22+𝑦22)−𝑥2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1      (A2.1)  \nwhere  𝑥1=Re(𝑤1), 𝑦1=Im(𝑤1), 𝑥2=Re(𝑤2), and 𝑦2=Im(𝑤2). Also , the radius of  𝐶 is |𝑐|. \nDemonstration : The center  𝑐=ℎ+𝑖𝑘 and radius  𝑟 of the circ umference  with known points 0,         \n𝑤1=𝑥1+𝑖𝑦1, 𝑤2=𝑥2+𝑖𝑦2, are obtained by solving the following system of three equations in three  \nunknowns  ℎ, 𝑘, 𝑟:  \n(𝑥1−ℎ)2+(𝑦1−𝑘)2=𝑟2  \n(𝑥2−ℎ)2+(𝑦2−𝑘)2=𝑟2          (A2.2)  \n  (0−ℎ)2+(0−𝑘)2=𝑟2    \nEach  of the three equations in system (A2.2) comes from the formula for calculating the square d \ndistance between a known point on a circumference  and its center; the third equation in system (A2.2) \ntells us immediately that 𝑟=|𝑐|. If we expand  the square d binomials in the first two equations of (A2.2) \nand subtract the third equation of (A2.2) from each of these, we obtain a  reduced  system of two linear \nequations in two unknowns  ℎ, 𝑘: \n2𝑥1ℎ+2𝑦1𝑘=𝑥12+𝑦12          (A2.3)  \n2𝑥2ℎ+2𝑦2𝑘=𝑥22+𝑦22  \n LC method for locating the roots of a quadratic equation  \n7 \n The solution  of system  (A2.3) is  \nℎ=−𝑦1(𝑥22+𝑦22)+𝑦2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1          (A2.4)  \n𝑘=𝑥1(𝑥22+𝑦22)−𝑥2(𝑥12+𝑦12)\n2𝑥1𝑦2−2𝑥2𝑦1  \nThe expressions in (A2.4) correspond to  the real and imaginary parts on the right side of expression \n(A2.1). Solution  (A2.4)  holds  as long as 𝑥1𝑦2≠𝑥2𝑦1 (if 𝑥1𝑦2=𝑥2𝑦1, this means that the points  0, 𝑤1, \n𝑤2 are collinear). ∎ \n \nAppendix 3  \nProposition . If line  𝐿1⊂ℂ\\{0}, defined in expression (2) , contains the roots  𝑟1, 𝑟2 of equation        \n𝑥2+𝑐1𝑥+𝑐2=0, and 𝑟1≠𝑟2, then ∠0𝑝1𝑟1=∠𝑟1𝑝1(𝑐2/𝑝1), with  𝑝1=−𝑐1/2=(𝑟1+𝑟2)/2. \nDemonstration : Let's consider the following rectilinear segments:  \n𝐴≡0𝑝1̅̅̅̅̅,  𝐵≡𝑝1𝑟1̅̅̅̅̅̅,  𝐶≡0𝑟1̅̅̅̅; \n𝐴′≡𝑟1𝑝1̅̅̅̅̅̅,  𝐵′≡𝑝1(𝑐2/𝑝1)̅̅̅̅̅̅̅̅̅̅̅̅̅, 𝐶′≡𝑟1(𝑐2/𝑝1)̅̅̅̅̅̅̅̅̅̅̅̅. \nThus,  ∠0𝑝1𝑟1 is the angle formed by segments  𝐴 and 𝐵, and  △0𝑝1𝑟1 is the triangle formed by segments  \n𝐴, 𝐵 and 𝐶; analogously, ∠𝑟1𝑝1(𝑐2/𝑝1) is formed by segments  𝐴′ and 𝐵′, and  △𝑟1𝑝1(𝑐2/𝑝1) is formed \nby segments  𝐴′, 𝐵′ and 𝐶′. If we denote (for example) the length of the segment 𝐵 as |𝐵|, we can see \nthat |𝐵|=|𝑝1𝑟1̅̅̅̅̅̅|=|𝑟1−(𝑟1+𝑟2)/2|=|(𝑟1−𝑟2)/2|; using this basic approach, it is feasible to verify \nthat \n|𝐴|\n|𝐴′|=|𝐵|\n|𝐵′|=|𝐶|\n|𝐶′|=|𝑟1+𝑟2|\n|𝑟1−𝑟2|. \nThis last expression tells us that triangles △0𝑝1𝑟1 and △𝑟1𝑝1(𝑐2/𝑝1) are similar, because  their  \ncorresponding sides are proportional; therefore, the angle formed by segments 𝐴 and 𝐵 is equal to the \nangle formed by corresponding segments 𝐴′ and 𝐵′; i.e., ∠0𝑝1𝑟1=∠𝑟1𝑝1(𝑐2/𝑝1). ∎ \n \nReferences  \n[1] Alba -Cuellar, D aniel  (2024). The LC Method: A parallelizable numerical method for approximating the \nroots of single -variable polynomials.  arXiv preprint  arXiv:2402.15554  \n[2] Eberhart, Carl (1999). Moebius transformations of the plan e.  \nhttp://www.ms.uky.edu/~carl/ma502/html/moebsln1.html  \n "    },    {        "title": "2402.04387v1.Giant_piezoelectricity_in_group_IV_monochalcogenides_with_ferroelectric_AA_layer_stacking.pdf",        "content": "Giant piezoelectricity in group IV monochalcogenides with\nferroelectric AA layer stacking\nSeungjun Lee,1Hyeong-Ryul Kim,2Wei Jiang,1Young-Kyun Kwon,2, 3,∗and Tony Low1,†\n1Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Physics and Research Institute for Basic Sciences,\nKyung Hee University, Seoul 02447, Korea\n3Department of Information Display,\nKyung Hee University, Seoul 02447, Korea\n(Dated: February 8, 2024)\nAbstract\nThe piezoelectricity of group IV monochalcogenides (MXs, with M = Ge, Sn and X = S, Se) has\nattracted much attention due to their substantially higher piezoelectric coefficients compared to\nother 2D materials. However, with increasing layer number, their piezoelectricity rapidly disappears\ndue to the antiferroelectric stacking order, severely limiting their practical applications. Using\nfirst-principles calculations, we investigated the piezoelectricity of MXs with the ferroelectric AA\nstacking configuration, which has recently been stabilized in experiments. We found that AA-\nstacked MXs have a ferroelectric ground state with the smallest lattice constant among other\nstacking configurations, resulting in a giant piezoelectric coefficient, which is the first demonstration\nof a strategy where the piezoelectric coefficients can increase with the number of layers. This can be\nattributed to a strong negative correlation between the lattice constant along the armchair direction\nand the piezoelectric coefficient, and spontaneous compressive strain stabilized in ferroelectric AA\nstacking configuration.\n1arXiv:2402.04387v1  [cond-mat.mtrl-sci]  6 Feb 2024I. INTRODUCTION\nAn interesting connection between mechanical energy and electricity in piezoelectric ma-\nterials makes them highly valuable in various applications, such as energy harvesting, sensors,\nand optoelectronic devices.1–3Moreover, two-dimensional piezoelectric materials (2DPMs)\nare of practical interest compared to their bulk counterparts because they are robust against\ndepolarization fields, enabling a reduction of device size.4–6Therefore, many efforts con-\ntributed in the last decade have led to the discovery of various types of monolayer or one-\nlayer (1L) piezoelectric materials such as hexagonal (h-) group III-IV materials,7h-group\nII oxides,8H-phase transition metal dichalcogenides (TMDs),9,10In2Se3,11,12CuInP 2S6,13,14\nand group IV monochalcogenides (MXs, with M = Ge, Sn and X = S, Se).15–19Among the\nvarious candidates, MXs, in particular, have received much attention due to their piezo-\nelectric coefficients, which surpass those of other two-dimensional materials by orders of\nmagnitude.\nDespite these promising advancements, the practical utilization of 2DPMs has been lim-\nited because ground-state multilayer stacking configurations often lose their piezoelectric\nresponse.9,20The challenge arises from the interlayer dipole-dipole interactions, which nat-\nurally favor an antiparallel order, resulting in the restoration of spatial inversion symmetry.\nNevertheless, recent advances in materials handling and various growth techniques have\npaved the way for the experimental stabilization of metastable stacking configurations,18,21\nresulting in novel ferroelectric (FE) order in 2DPMs. For example, interlayer twisting22,23\nor sliding19,24,25give rise to additional out-of-plane FE order, which is not possible in a 1L\nor minimum energy stacking configuration. However, despite the fact that in-plane ferro-\nelectricity is also crucial for piezoelectric response, there is limited understanding of the\neffect of interlayer interactions on the in-plane FE order and the corresponding piezoelectric\nresponse.\nThrough first-principles calculation based on density functional theory (DFT), we inves-\ntigated the stacking-dependent piezoelectric responses of MXs. We first carefully evaluated\nthe piezoelectric coefficients of monolayer MX to understand the large variations in their\npiezoelectric coefficients reported previously. We found that an in-plane lattice constant\n(LC) along the armchair direction primarily controls the piezoelectric coefficients of MX due\nto the unique puckered structures. We further revealed that, the choice of van der Waals\n2(vdW) correction is particularly crucial for MXs, as it significantly affects their in-plane lat-\ntice constants not only in multilayer cases but also in the 1L limit, which is not expected for\nother 2D materials. Therefore, we investigated the piezoelectric coefficients of AA and AC\nstacked MX with carefully chosen vdW corrections. We found a significant enhancement of\nthe piezoelectric coefficients in AA stacked MX regardless of the choice of vdW corrections.\nThe physical origin of this enhancement is a spontaneous compressive strain emerging in\nthe AA stacking. Our results provide a deeper understanding of the piezoelectric responses\nof MXs and pave the way for a novel approach to optimize the piezoelectricity in 2DPMs\nthrough stacking configuration.\nII. COMPUTATIONAL DETAILS\nThe first-principles DFT calculation26was carried out using Vienna ab initio simulation\npackage (VASP).27We employed the plane wave basis to expand the electronic wavefunctions\nwith a kinetic energy cutoff of 500 eV. The projector-augmented wave pseudopotentials28,29\nwere used for the valence electrons, and the exchange-correlation (XC) functional was treated\nwithin the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE).30A suf-\nficiently large vacuum region ( >20˚A) was included in the unitcell to avoid any spurious\ninterlayer interactions. The atomic basis was carefully relaxed until the Helmann-Feynman\nforce acting on every atom was smaller than 0.001 eV/ ˚A, which is very crucial to get a con-\nverged piezoelectric coefficient. The Monkhorst-Pack 21 ×21×1k-mesh was used to sample\nthe Brillouin zone. To comprehensively investigate the effects of interlayer interactions, we\nused various types of vdW interactions including, simple Grimme methods,31,32as well as\nvarious non-local vdW-DF functionals.33–36\nThe spontaneous polarization, P, of various MXs were calculated in the framework of the\nmodern theory of polarization based on the Berry phase.37Then, the planar elastic stiffness\ncoefficients Cijand piezoelectric strain coefficients eijkwere evaluated as15\nCij=1\nA0∂2U\n∂εii∂εjj, (1)\neijk=∂Pi\n∂εjk, (2)\nwhere ε,U, and A0are the strain, the total energy and the unitcell area. The effective\nthickness of the 1L MXs was treated as half of the out-of-plane LC of their bulk counterparts,\n3TABLE I. Lattice constants, aandb, spontaneous polarization, P, piezoelectric strain coefficients,\neijk, planar elastic stiffness coefficients, Cij, and piezoelectric stress coefficients, dijof 1L SnSe\ncalculated by various functionals.\nvdWa\n(˚A)b\n(˚A)P\n(pC/m)e111\n(nC/m)e122\n(nC/m)C11\n(N/m)C22\n(N/m)C12\n(N/m)d11\n(pm/V)d12\n(pm/V)\nnone (PBE) 4.383 4.292 195.7 3.17 0.48 21.47 44.73 17.6 205.01 -69.98\nnone (PBE)154.35 4.24 3.49 1.08 19.88 44.49 185.7 250.57 -80.31\nnone (PBE)384.41 4.29 2.81 0.52 23.06 42.82 18.89 175.32 -65.11\nnone (PBE)17181\nD3 4.509 4.265 281.7 1.50 -0.36 25.47 42.06 21.02 112.29 -64.61\nD2 4.305 4.248 150.6 5.31 1.46 17.85 41.64 17.24 439.27 -146.81\noptB88 4.384 4.284 192.5 2.74 0.27 25.02 45.00 20.26 164.7 -68.15\noptB88394.41 4.27 2.35 0.76 20 40.4 17 158.2 -47.7\nrev-vdW-DF2 4.331 4.271 153.8 3.91 0.59 26.61 47.6 20.19 202.8 -73.62\nwhich was used to evaluate an effective bulk polarization. Then, the piezoelectric stress\ncoefficients dijcan be calculated as\nd11=e111C22−e122C12\nC11C22−C2\n12, (3)\nd12=e122C11−e111C12\nC11C22−C2\n12. (4)\nIII. RESULTS AND DISCUSSION\nFigure 1(a) shows the crystal structure of a typical 1L MX. We set x- and y-axes as\nin-plane directions with corresponding LCs of aandb, and z-axis as out-of-plane direction.\nIt has puckered atomic structures with a mirror plane of Myand belongs to Pmn 21space\ngroup.16More importantly, its broken Mxsymmetry leads to a non-zero spontaneous polar-\nization Palong xor armchair direction. Indeed, the spontaneous polarization of 1L MXs\ncan be understood as a rearrangement of the atomic basis along the armchair direction. It\nhas also been reported that the LC along the armchair direction plays a significant role in\ndetermining its P.17Therefore, we begin with our discussion on the LCs of 1L MXs using\n4(a)\nGe,Sn\nS,Se\n𝑎𝑎𝑏𝑏\nxy\nxz𝑴𝑴𝒚𝒚\n𝑎𝑎 (Å)\n4.4\n4.34.5𝑏𝑏 (Å)\n4.3\n4.24.4(b)𝑏𝑏 (Å)\n𝒂𝒂 (Å)𝑃𝑃 (pC/m)\n4.24.4\n4.34.5\n4.2 4.4 4.3 4.5050100150200250(c)\n𝑃𝑃 (pC/m)\n150200250300𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (nC/m )\n-2246\n0𝑒𝑒122𝑒𝑒111(e)\n𝑑𝑑𝑖𝑖𝑖𝑖 (pm /V)\n𝑑𝑑12𝑑𝑑11\n-200200400600\n0300\nPBE\nPBE+D2PBE+D3\noptB88- vdWrev-vdW -DF2\n𝑏𝑏 (Å)\n𝒂𝒂 (Å)𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖 (nC/m )\n4.24.4\n4.34.5\n4.2 4.4 4.3 4.50468101214(d)\n(f)\n2\n𝑒𝑒\n111\n𝑒𝑒\n122FIG. 1. (a) Top and side views of a typical monolayer (1L) group IV monochalcogenides (MXs\nwith M = Ge or Sn and X = S or Se) crystal structure. (b) Lattice constants, aandb, of 1L SnSe\nobtained from different van der Waals (vdW). (c) Spontaneous polarization Pcontour map of 1L\nSnSe, calculated by the PBE functional without vdW correction. Equilibrium lattice constants\nshown in (b) were overlaid for comparison. (d) 2D piezoelectric strain coefficient ( eijk) contour\nmap of 1L SnSe calculated by the derivative of (c) with strain along a. As a result, the upper left\nand lower right triangles represent e111ande122, respectively. (e) Pandeijk, and (f) piezoelectric\nstress coefficients ( dij) of 1L SnSe obtained from different vdW functionals.\n5SnSe as an exemplary material and will carefully re-examine its piezoelectric coefficients.\nAlthough the LCs of bulk MXs have been experimentally reported,40–42those of 1L MXs\nhave not yet been done but are only available through DFT calculations. In most previous\nstudies,15,17,19the LCs of 1L MXs have been obtained only with the PBE XC functional\nwithout vdW corrections, which were considered only in the multilayer case. This is the\nmost common approach to determine the LCs of various 2D materials, since the effect of the\nvdW correction may be negligible in the 1L case. (See Fig. S1 in Supplementary Information\n(SI) for more details.) However, we found that the LC of 1L SnSe strongly depends on the\nchoice of vdW correction, as shown in Fig. 1(b). For example, the LC , a, obtained along the\narmchair direction ranges from 4.509 ˚A (PBE+D3) to 4.305 ˚A (PBE+D2), representing a\ndeviation of almost 5%. Such deviation is attributed to its puckered structure with a weak\nstiffness along the armchair direction.\nWe found that a small change in the LCs of 1L MXs would cause a large variation in their\npiezoelectric coefficients. To understand this result, we calculated Pof 1L SnSe as a function\nof the two in-plane lattice constants aandbusing PBE XC, which is shown as a contour\nmap in Fig. 1(c), where other calculated LCs using other functions are superimposed. The\ncontour map is symmetric with respect to the b=aline with C4zrotational symmetry due\nto the degenerate ground states for a > b anda < b . As clearly visualized in Fig. 1(c), Pis\npositively correlated with a, and the aspect ratio between aandbalso has some secondary\neffects. As the LC decreases, 1L SnSe loses its Pand becomes paraelectric. Therefore,\nthe vdW correction, which yields a smaller (larger) LC along the armchair direction, may\nunderestimate (overestimate) P. For example, the Pvalue of 281.7 pC/m estimated with the\nPBE+D3 correction is almost twice the value of 150.6 pC/m estimated with the PBE+D2\ncorrection. Figure 1(d) shows the contour map of the 2D piezoelectric strain coefficients,\neijk, computed by taking the first derivative of Pshown in Fig. 1(c) with respect to strain\nεjk, as explained in Computational details and Fig S2 in SI. We found significantly stronger\ne111near the boundary between the paraelectric and FE phases due to the rapid emergence\nof non-zero P, and a small enhancement in e122, although much smaller than e111. This\nresult clearly indicates that a compressive strain along the armchair direction may play a\nmajor role in the enhancement of e111of SnSe. Simultaneously, it also implies that the\nfluctuation of the LC according to the choice of vdW functionals makes a large uncertainty\nin predicting its piezoelectric coefficients.\n6FIG. 2. Calculated lattice constants of bulk group IV monochalcogenides, obtained using various\nvan der Waals corrections (red dots). Experimental values obtained from Ref. [40–42] are also\nshown as red solid lines for comparison.\nFor a more systematic study, we evaluated the piezoelectric coefficients of 1L SnSe using\nvarious vdW functionals, as summarized in Fig. 1(e, f), and Table I. Due to the strong\ncorrelation between Panda, both the strain and stress piezoelectric coefficients ( eijkanddij)\nalso strongly depend on the choice of functional. For example, e111andd11range from from\n5.31 nC/m to 1.50 nC/m, and from 439.27 pm/V to 112.29 pm/V, respectively. As predicted\nabove, the smaller aleads to the higher eijkanddij. We further confirmed that our results\nare consistent with previously reported studies, and the observed variation in both eijkand\ndijis simply understood as the result of variation in the relaxed a. Therefore, we emphasize\nthat the choice of computational options is extremely important in the quantitative analysis\nand the vdW functional should be used consistently in both 1L and multilayer cases.\nA careful investigation in the previous section indicates that there is a significant un-\ncertainty in determining the exact value of the piezoelectric coefficients of 1L MXs without\nexperimental input of LCs. Therefore, before investigating the piezoelectric coefficients of\nmultilayer MXs, we calculated the LCs of bulk MXs, and then compared them with ex-\nperimentally available values to validate which vdW correction is most appropriate among\nvarious vdW functionals. Figure 2 shows the LCs of four different bulk MXs calculated\nwith seven different functionals (red dots) together with their experimental values (red solid\nlines) for comparison. Among various functionals, the rev-vdW-DF2 method shows better\n7agreement with experimental values than the other functionals. (See also Figure S3 and\nTable S1 in SI) Therefore, here, we primarily use the rev-vdW-DF2 vdW correction in in-\nvestigating piezoelectric coefficients of 1L and multilayer MXs. For better understanding,\nwe also repeated all calculations using the Grimme-D3 vdW corrections, with all results\npresented in SI. Note that, all physical results are qualitatively consistent when using the\nsame vdW functionals for both 1L and multilayer cases, regardless of which vdW functionals\nwere used. However, the conventional approach (i.e., vdW functional is only considered in\nmultilayers) may lead to misinterpretation of the results, which will be discussed later.\nFigure 3(a) and (b) show antiferroelectric (AFE) and FE stacking orders and four possible\nsliding configurations. Thus, there are a total of eight possible stacking configurations.\nAlthough it is known that the ground state stacking configuration of MXs is the AFE AB\nstacking,18,19it has also been predicted that the FE order can be stabilized in both the AA\nand AC stacking configurations.19In addition, it has been experimentally confirmed that\nthe AA-stacked FE SnS can exist on a mica substrate below a certain critical thickness.18\nTherefore, here, we first focus on 2L AA and AC stacked SnSe to understand the FE order\nin multilayer MXs.\nFigure 3(c) and (d) show the Pcontour maps of FE 2L SnSe in AA and AC stacking con-\nfigurations, respectively. Both contour maps are nothing but those of their 1L counterparts\nshown in Fig. 1(c) with twice larger Pvalues, implying that the Pof 2L SnSe is mainly due\nto the intralayer contribution, regardless of the stacking configuration. Therefore, similar\nto the 1L case, the equilibrium LCs primarily determine their P. The obtained LCs of\nmultilayer SnSe were overlaid in Fig. 3(c) and (d). The LCs of 2L AA SnSe were evalu-\nated to be a= 4.302˚A and b= 4.263˚A, whereas those of 2L AC phase are a= 4.340˚A\nandb= 4.251˚A. Especially we focus on change in aalong the armchair direction when\nstacked from 1L to 2L. As shown in Fig. 3(c) and (d), where the LCs of SnSe with different\nnumber of layers are overlaid, AA stacking significantly reduces a, while AC stacking does\nnot change aconsiderably, compared to the 1L case. Thus, 2L AA SnSe has a smaller P\nand may have a larger eijkthan 2L AC SnSe. It is worth noting that the reduction of LCs\nof 2L AA SnSe can be understood as a result of dipole-dipole interactions. Since parallel\ndipole-dipole interaction is inherently unfavorable, relaxed structures can form in ways that\nfavor either a decrease in Por an increase in the P-Pdistance. Notably, this result remains\nrobust across different vdW functionals and is thus applicable to other MXs, emphasizing\n8the generalizability of the observed phenomenon. (See Table S2 in SI)\nFigure 4 shows the variation of four available piezoelectric coefficients ( e111,e122,d11, and\nd12) of SnSe with the number of layers up to 3L in AA, AB, and AC stacking configurations.\n(See Table S3 in SI for more details) For the AB stacking configuration, due to its AFE\norder, the piezoelectric coefficients of the even-numbered layers are calculated to be zero,\n(a)\n𝐀𝐀𝐀𝐀𝐀𝐀\n(c)\n𝑏𝑏 (Å)\n𝑎𝑎 (Å)𝑃𝑃 (pC/m)\n4.24.4\n4.34.5\n4.2 4.4 4.3 4.50100200300400500 (d)xz𝑏𝑏 (Å)\n𝑎𝑎 (Å)𝑃𝑃 (pC/m)\n4.24.4\n4.34.5\n4.2 4.4 4.3 4.50100200300400500\n𝐀𝐀𝐀𝐀 𝐀𝐀𝐀𝐀\n𝐀𝐀𝐀𝐀\nxy\nGe,Sn\n S,Se𝐀𝐀𝐀𝐀 𝐀𝐀𝐀𝐀\n1st unitcell 2nd unitcell\n𝐀𝐀𝐀𝐀−𝐀𝐀𝐀𝐀 𝟐𝟐𝟐𝟐 𝐀𝐀𝐀𝐀−𝐀𝐀𝐀𝐀 𝟐𝟐𝟐𝟐\n1L2L3L bulk\n1L\n2L\n3L\nbulk(b)\nFIG. 3. (a) Side views of antiferroelectric (AFE) and ferroelectric (FE) stacking configurations of\nbilayer MX and (b) top views of four possible in-plane sliding configurations. In (a), blue arrows\nindicate the direction of the spontaneous polarization of each layer. In (b), grey and blue dashed\nrectangles visualize unitcells of each layer. (c) Contour maps of the spontaneous polarization P\nof the FE 2L SnSe in (c) AA and (d) AC stacking configurations calculated by PBE with the\nrev-vdW-DF2 vdW correction. Equilibrium lattice constants of 1L, 2L, 3L, and bulk AA and AC\nSnSe are overlaid on (c) and (d), respectively.\n9(b) (a)\n1216\n8\n4\n01L 2L 3LPBE\nD3AA\nAB\nAC𝑒𝑒111 (nC/m )46\n3\n1\n−11L 2L 3LAA\nAB\nAC𝑒𝑒122 (nC/m )5\n2\n0\n(d) (c)350\n01L 2L 3LAA\nAB\nAC𝑑𝑑11(pm /V)300\n250200\n150\n100\n500\n−801L 2L 3LAA\nAB\nAC𝑑𝑑12(pm /V)\n−60−40−20FIG. 4. 2D piezoelectric strain coefficients (a) e111and (b) e122, and piezoelectric stress coefficients\n(c)d11and (d) d12) of 1L, 2L, and 3L SnSe with AA, AB, and AC stacking configurations, calculated\nwith the rev-vdW-DF2 vdW correction (black). For comparison, the results of the Grimme-D3\ncorrections (grey) and PBE (blue) are also shown for the AA SnSe and 1L SnSe, respectively.\nand those of the odd-numbered layers are also smaller than those of the monolayer. On\nthe other hand, the 2D piezoelectric strain coefficients ( e111ande122, in the unit of nC/m)\nincrease with the number of layers in both AA and AC stacking configurations. Moreover,\nAA stacking exhibits higher piezoelectric coefficients than AC stacking, which is consistent\nwith our prediction based on the LCs. For example, the e111of 3L AA SnSe was calculated\nto be 13.5 nC/m which is larger than that of AC SnSe (8.53 nC/m), and also six times larger\nthan that of 3L AB SnSe (2.39 nC/m). Note that the effective bulk piezoelectric coefficient\n(C/m2) of 3L AA SnSe is also 15% larger than that of 1L SnSe thanks to the smaller LC.\nThe other important coefficients, the piezoelectric stress coefficients ( d11andd12) of\nmultilayer AA SnSe are comparable or slightly larger than those of 1L SnSe, depending on\n10the choice of vdW correction. For all other AA MXs, we calculated e111,e122,d11, and d12up\nto 2L using both rev-vdW-DF2 and Grimme-D3 corrections, as summarized in Tables S4-S6\nin SI. We consistently found that as the LC decreases, e111ande122continue to enhance,\nwhile d11andd12remain comparable to those of 1L MXs. It is worth noting that although\nthere is no meaningful improvement in d11andd12in multilayer AA MXs compared to 1L,\nthese values are much larger than those of AB MXs. For example, the d11of 3L AA SnSe\n(155.96 pm/V) is almost three times larger than that of 3L AB SnSe (55.2 pm/V), suggesting\nthat AA stacking MX is a promising candidate in piezoelectric applications.\nAs shown in Fig. 3(c), the LC aof AA SnSe continues to shrink with stacking, but\nthe bulk AA SnSe loses Pand becomes paraelectric, implying that there exists a critical\nthickness below which non-zero Pis accommodated. In fact, a previous experiment has\nsuccessfully grown AA SnS on mica substrates, exhibiting room temperature ferroelectricity\nup to 15 layers,18indicating that the critical thickness is not too thin and could be further\noptimized by adjusting the experimental conditions. This observation supports our predic-\ntion of optimizing the piezoelectric coefficients of multilayer MXs by utilizing AA stacking\nconfiguration.\nFinally, we summarize the piezoelectric coefficients of multilayer MXs, as shown in\nFig. 5(a). SnSe shows the highest e111andd11, followed by SnS, GeSe, and GeS. In all\nMXs, the AA stacking boasts higher e111than any other stacking, while d11is more or less\nthan that of the corresponding 1L case regardless of the stacking configuration. In addition\ntoe111andd11, other physical properties such as stability and switching barrier between the\nFE and AFE phases are also important in piezoelectric applications. Therefore, all MXs can\nbe practically useful, and utilizing AA stacking configuration is an efficient way to optimize\ntheir piezoelectric response.\nFor comparison, we also summarized the piezoelectric coefficients of several other 2D\nmaterials, as shown in Fig. 5(b). Consistent with the previous reports, we observed that 1L\nMXs have exceptionally large piezoelectric coefficients compared to the other 2D materials,\neven larger coefficients in AA stacked MXs. It is worthy of note that the giant piezoelectric\ncoefficients observed in AA MXs not only stand out among 2D materials, but are also com-\nparable to bulk piezoelectric materials. Recent high-throughput DFT calculations revealed\nthat among 941 bulk piezoelectric materials, only 5% of materials satisfy |eij|>3 C/m2, in-\ncluding experimentally-confirmed giant piezoelectric materials such as BaTiO 3(3.49 C/m2),\n11(b)\n𝑒𝑒111 (nC/m )𝑑𝑑11(pm /V)\n0.001 100 10 1 0.01 0.11000\n100\n10\n1\n0.1\n0.01h III−V \nh II Oxides  \n2H−TMD  \n1L MXs \nmultilayer  MXs Our works(a)\n𝑑𝑑11(pm /V)250\n200\n150\n100\n50\n0\n𝑒𝑒111 (nC/m )0 15 12 9 3 6AA SnSe  \nAA SnS \nAA GeSe \nAA GeS AB SnSe  \nAC SnSe  1L\n3L 2L 3L\n2L1L2L\n2L3L2L\n1L 2L1LFIG. 5. Summarized piezoelectric coefficients of (a) group IV monochalcogenides (MXs) and (b)\nvarious 2D materials. In (a), purple, dark-blue, yellow, green, skyblue, and orange circles represent\nAA (stacking) SnSe, AB SnSe, AC SnSe, AA SnS, AA GeSe, and AA GeS, respectively. In (b),\nyellow, orange, and purple circles denote materials of hexagonal Group III-V (h III-V), hexagonal\nGroup II oxides (h II Oxides), and H phase transition metal dichalcogenides (2H-TMD), respec-\ntively, obtained from Ref. [8]. Skyblue and red circles surrounded by a blue dashed rectangle\ndenote our results of 1L MXs and multilayer MXs.\nSrHfO 3(8.73 C/m2), RbTaO 3(8.93 C/m2), and BaNiO 3(27.46 C/m2).43Using the effec-\ntive thickness of 1.74 nm (1.74 nm=1.5 c,cis a out-of-plane lattice constant of bulk SnSe),\nthe effective bulk e111value of 3L AA SnSe was calculated to be 7.76 C/m2, which is even\nlarger than that of BaTiO 3. Within the Grimme-D2 vdW correction, which provides the\nsmallest LCs among all vdW functionals, the e111of 2L AA SnSe was estimated to be\n12.66 C/m2(14.68 nC/m). It is even larger than not only that of 3L AA SnSe estimated\nby rev-vdW-DF2, but also that of SrHfO 3and RbTaO 3. We now conclude that our the-\noretical calculations have undoubtedly revealed that there is a strong correlation between\nthe reduction of LCs and giant piezoelectric coefficients in AA-stacked MXs, regardless of\nthe vdW functional used, making it a promising candidate for low-dimensional piezoelectric\napplications.\n12IV. CONCLUSION\nIn summary, we have used first-principles calculations to study the piezoelectric coef-\nficients of multilayer MXs with AA and AC stacking configurations, which are known to\nstabilize ferroelectric order. Our re-examination of 1L MXs revealed the pivotal role of\nthe van der Waals interaction in ensuring accurate and reliable predictions of piezoelectric\ncoefficients. Through systematic DFT calculations, we found that the AA-stacked configu-\nrations exhibit remarkably larger piezoelectric coefficients compared to all reported layered\nmaterials, including their monolayer counterparts. The origin of this enhancement lies in\nthe compressive strain along the armchair direction, which is spontaneously introduced in\nthe ferroelectric AA stacking. These findings not only provide a comprehensive understand-\ning of the piezoelectric behavior of MXs, but also suggest a novel strategy for optimizing\npiezoelectricity in low-dimensional materials through stacking configuration.\nACKNOWLEDGMENTS\nS.L., W.J., and T.L. were supported by the National Science Foundation (NSF) through\nthe DMREF program under Award No. DMR-1921629. S.L. is also supported by Ba-\nsic Science Research Program through the National Research Foundation of Korea (NRF)\nfunded by the Ministry of Education (NRF-2021R1A6A3A14038837). H.-R.K. and Y.-\nK.K. acknowledge financial support from the Korean government through the NRF (NRF-\n2022R1A2C1005505, NRF2022M3F3A2A01073562). The computational work was partially\ndone using the resources of the KISTI Supercomputing Center (KSC-2023-CRE-0053).\n∗ykkwon@khu.ac.kr\n†tlow@umn.edu\n1Wang, Z. L. & Song, J. Piezoelectric nanogenerators based on zinc oxide nanowire arrays.\nScience 312, 242–246 (2006).\n2Wang, Z. L. Nanopiezotronics. Adv. Mater. 19, 889–892 (2007).\n3Bao, R., Tao, J., Pan, C. & Wang, Z. L. Piezophototronic effect in nanosensors. Small Sci. 1,\n2000060 (2021).\n134Hinchet, R., Khan, U., Falconi, C. & Kim, S.-W. Piezoelectric properties in two-dimensional\nmaterials: Simulations and experiments. Mater. Today 21, 611–630 (2018).\n5Zhang, Q., Zuo, S., Chen, P. & Pan, C. Piezotronics in two-dimensional materials. InfoMat 3,\n987–1007 (2021).\n6Jin, C.-C., Liu, D.-M. & Zhang, L.-X. An emerging family of piezocatalysts: 2D piezoelectric\nmaterials. Small 19, 2303586 (2023).\n7Ares, P. et al. Piezoelectricity in monolayer hexagonal boron nitride. Adv. Mater. 32, 1905504\n(2020).\n8Blonsky, M. N., Zhuang, H. L., Singh, A. K. & Hennig, R. G. Ab initio prediction of piezoelec-\ntricity in two-dimensional materials. ACS nano 9, 9885–9891 (2015).\n9Zhu, H. et al. Observation of piezoelectricity in free-standing monolayer MoS 2.Nat. Nanotech-\nnol.10, 151–155 (2015).\n10Alyoruk, M. M., Aierken, Y., C ¸akır, D., Peeters, F. M. & Sevik, C. Promising piezoelectric\nperformance of single layer transition-metal dichalcogenides and dioxides. J. Phys. Chem. C\n119, 23231–23237 (2015).\n11Zhou, Y. et al. Out-of-plane piezoelectricity and ferroelectricity in layered α-In2Se3nanoflakes.\nNano Lett. 17, 5508–5513 (2017).\n12Xue, F. et al. Multidirection piezoelectricity in mono-and multilayered hexagonal α-In2Se3.\nACS nano 12, 4976–4983 (2018).\n13Liu, F. et al. Room-temperature ferroelectricity in cuinp2s6 ultrathin flakes. Nat. Commun. 7,\n12357 (2016).\n14Brehm, J. A. et al. Tunable quadruple-well ferroelectric van der waals crystals. Nat. Mater.\n19, 43–48 (2020).\n15Fei, R., Li, W., Li, J. & Yang, L. Giant piezoelectricity of monolayer group IV monochalco-\ngenides: SnSe, SnS, GeSe, and GeS. Appl. Phys. Lett. 107, 173104 (2015).\n16Gomes, L. C., Carvalho, A. & Neto, A. C. Enhanced piezoelectricity and modified dielectric\nscreening of two-dimensional group-IV monochalcogenides. Phys. Rev. B 92, 214103 (2015).\n17Wang, H. & Qian, X. Two-dimensional multiferroics in monolayer group IV monochalcogenides.\n2D Mater. 4, 015042 (2017).\n18Higashitarumizu, N. et al. Purely in-plane ferroelectricity in monolayer SnS at room tempera-\nture. Nat. Commun. 11, 2428 (2020).\n1419Xu, B., Deng, J., Ding, X., Sun, J. & Liu, J. Z. Van der Waals force-induced intralayer\nferroelectric-to-antiferroelectric transition via interlayer sliding in bilayer group-IV monochalco-\ngenides. Npj Comput. Mater. 8, 47 (2022).\n20Li, Y. et al. Probing symmetry properties of few-layer MoS 2and h-BN by optical second-\nharmonic generation. Nano Lett. 13, 3329–3333 (2013).\n21Yoo, H. et al. Atomic and electronic reconstruction at the van der waals interface in twisted\nbilayer graphene. Nat. Mater. 18, 448–453 (2019).\n22Woods, C. et al. Charge-polarized interfacial superlattices in marginally twisted hexagonal\nboron nitride. Nat. Commun. 12, 347 (2021).\n23Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered\nferroelectricity in bilayer boron nitride. Science 372, 1458–1462 (2021).\n24Meng, P. et al. Sliding induced multiple polarization states in two-dimensional ferroelectrics.\nNat. Commun. 13, 7696 (2022).\n25Deb, S. et al. Cumulative polarization in conductive interfacial ferroelectrics. Nature 612,\n465–469 (2022).\n26Kohn, W. & Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.\nPhys. Rev. 140, A1133–A1138 (1965).\n27Kresse, G. & Furthm¨ uller, J. Efficient Iterative Schemes for ab initio Total-Energy Calculations\nUsing a Plane-Wave Basis Set. Phys. Rev. B 54, 11169–11186 (1996).\n28Bl¨ ochl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 50, 17953–17979 (1994).\n29Kresse, G. & Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave\nMethod. Phys. Rev. B 59, 1758–1775 (1999).\n30Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple.\nPhys. Rev. Lett. 77, 3865–3868 (1996).\n31Grimme, S. Semiempirical GGA-Type Density Functional Constructed with a Long-Range\nDispersion Correction. J. Comput. Chem. 27, 1787–1799 (2006).\n32Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametriza-\ntion of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. chem.\nphys.132(2010).\n33Dion, M., Rydberg, H., Schr¨ oder, E., Langreth, D. C. & Lundqvist, B. I. Van der Waals density\nfunctional for general geometries. Phys. Rev. Lett. 92, 246401 (2004).\n1534Lee, K., Murray, ´E. D., Kong, L., Lundqvist, B. I. & Langreth, D. C. Higher-accuracy van der\nWaals density functional. Phys. Rev. B 82, 081101 (2010).\n35Klimeˇ s, J., Bowler, D. R. & Michaelides, A. Chemical accuracy for the van der Waals density\nfunctional. J. Condens. Matter Phys. 22, 022201 (2009).\n36Hamada, I. van der waals density functional made accurate. Phys. Rev. B 89, 121103 (2014).\n37Spaldin, N. A. A beginner’s guide to the modern theory of polarization. J. Solid State Chem.\n195, 2–10 (2012).\n38Guo, S.-D., Guo, X.-S., Zhang, Y.-Y. & Luo, K. Small strain induced large piezoelectric\ncoefficient in α-AsP monolayer. J. Alloys Compd. 822, 153577 (2020).\n39Fang, W. et al. Layer dependence of geometric, electronic and piezoelectric properties of snse.\narXiv preprint arXiv:1603.01791 (2016).\n40Chattopadhyay, T., Pannetier, J. & Von Schnering, H. G. Neutron diffraction study of the\nstructural phase transition in SnS and SnSe. J. Phys. Chem. Solids 47, 879–885 (1986).\n41Grandke, T. & Ley, L. Angular-resolved uv photoemission and the band structure of GeS. Phys.\nRev. B 16, 832–842 (1977).\n42Murgatroyd, P. A. E. et al. GeSe: Optical Spectroscopy and Theoretical Study of a van der\nWaals Solar Absorber. Chem. Mater. 32, 3245–3253 (2020).\n43De Jong, M., Chen, W., Geerlings, H., Asta, M. & Persson, K. A. A database to enable discovery\nand design of piezoelectric materials. Sci. Data 2, 150053 (2015).\n16Supplementary Information\nGiant piezoelectricity in group IV monochalcogenides with\nferroelectric AA layer stacking\nSeungjun Lee,1Hyeong-Ryul Kim,2Wei Jiang,1Young-Kyun Kwon,2, 3,∗and Tony Low1,†\n1Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455, USA\n2Department of Physics and Research Institute for Basic Sciences,\nKyung Hee University, Seoul 02447, Korea\n3Department of Information Display,\nKyung Hee University, Seoul 02447, Korea\n(Dated: February 8, 2024)\n1arXiv:2402.04387v1  [cond-mat.mtrl-sci]  6 Feb 20243.2\n3.13.3𝑎𝑎(Å)𝐌𝐌𝐌𝐌𝐌𝐌𝟐𝟐\n2.5\n2.42.6𝑎𝑎(Å)𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆𝐆\nFIG. S1. Calculated lattice constants of graphene and monoalyer H-MoS 2, obtained using different\nexchange-correlation functionals.\n2Energy  (meV /cell )30\n20(a) (b)\n𝑃𝑃 (pC/m)\n10\n0\n-10\n-20\n-30-300 300 -200-100 0100 200𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒\n𝜀𝜀𝑥𝑥𝑥𝑥=−1% \n 0%\n+1% \n𝑎𝑎−𝑃𝑃 𝑃𝑃=0\n𝑏𝑏𝑃𝑃\n(c) (d) (d)\n𝑏𝑏 (Å)\n𝑎𝑎 (Å)4.31Energy  (meV /cell )\n4.264.30\n4.29\n4.28\n4.27\n4.36 4.41 4.374.384.394.4000.51.01.52.02.53.0\n𝑃𝑃 (pC/m)280\n80240\n200160120\nStrain  (%)−2 2 −1 0 1𝜀𝜀𝑥𝑥𝑥𝑥\n𝜀𝜀𝑦𝑦𝑦𝑦xy\nxzFIG. S2. Computational procedure in determining piezoelectric coefficient of 1L SnSe. (a) Two\nmirror symmetric ground states of 1L SnSe ( Pand−P) and corresponding inversion symmetric\nstructures with P= 0 between their reaction path. (b) Calculated spontaneous polarization P\nalong with the reaction path between Pand−Pstates. The energy barrier was evaluated using\nthe nudged elastic band (NEB) calculation method.S1As discussed in the main manuscript, the\ncompressive (tensile) strain along the armchair direction decreases (increases) its P. (c) Total\nenergy contour map of 1L SnSe as a function of lattice constants. We calculated elastic coefficients\nofCijusing Eq. (1) in the main manuscripts. (d) Calculated Pof 1L SnSe as function of uniaxial\nstrain. Dots indicate DFT results, and the dashed lines show the polynomial fitting function.\nPiezoelectric strain coefficient eijkwas evaluated through Eq. (2) in the main manuscripts, and\npiezoelectric stress coefficient dijwas evaluated through Eq. (3, 4) using obtained Cijandeijk. In\nthis example, all calculation results were obtained within PBE XC functional. Similar calculations\nwere repeated to obtain piezoelectric coefficients of other materials, multilayer cases, as well as the\nsame material with different van der Waals functionals.\n3Error0.1\n0.01\n0.001FIG. S3. Performance of van der Waals functionals for bulk group IV monochalcogenides. Error\nis evaluated by/summationtext\nij((ai,j\nthy−ai,j\nexp)/ai,j\nexp)2, where athyandaexpare the lattice constants obtained\nfrom DFT calculation and experiment, iis a index for the three lattice constants ( a, b,andcin\nTable S1) and jis a index for the four materials studied in this work (SnSe, SnS, GeSe, and GeS).\n4TABLE S1. Calculated lattice constants of bulk group IV monochalcogenides, obtained using\nvarious van der Waals corrections. Experimental values are also shown for comparison.\nbulk-SnSe a(˚A) b(˚A) c(˚A) bulk-SnS a(˚A) b(˚A) c(˚A)\nPBE 4.546 4.207 11.762 PBE 4.426 4.023 11.415\nD2 4.349 4.197 11.631 D2 4.261 4.012 11.384\nD3 4.521 4.166 11.553 D3 4.409 3.990 11.184\nvdW-DF 4.791 4.263 12.295 vdW-DF 4.647 4.073 11.920\nvdW-DF2 4.857 4.334 12.512 vdW-DF2 4.703 4.133 12.076\noptB88 4.501 4.22 11.761 optB88 4.365 4.039 11.400\nrev-vdW-DF2 4.429 4.201 11.627 rev-vdW-DF2 4.291 4.025 11.282\nExperimentS24.445 4.153 11.501 ExperimentS24.34 3.97 11.14\nbulk-GeSe a(˚A) b(˚A) c(˚A) bulk-GeS a(˚A) b(˚A) c(˚A)\nPBE 4.514 3.892 11.179 PBE 4.428 3.679 10.757\nD2 4.469 3.849 11.011 D2 4.339 3.670 10.660\nD3 4.513 3.877 10.978 D3 4.452 3.671 10.634\nvdW-DF 4.779 3.942 11.802 vdW-DF 4.664 3.727 11.337\nvdW-DF2 4.782 4.027 11.924 vdW-DF2 4.669 3.795 11.388\noptB88 4.443 3.927 11.158 optB88 4.338 3.720 10.712\nrev-vdW-DF2 4.339 3.915 10.977 rev-vdW-DF2 4.229 3.713 10.544\nExperimentS34.395 3.836 10.833 ExperimentS44.30 3.64 10.47\n5TABLE S2. Calculated lattice constants of multilayer group IV monochalcogenides, obtained using\nvarious van der Waals corrections.\nMaterials vdW Stacking Thickness a(˚A) b(˚A)\nSnSe rev-vdW-DF2 2L AA 4.302 4.263\nAB 4.392 4.230\nAC 4.340 4.251\n3L AA 4.290 4.257\nAB 4.394 4.224\nAC 4.339 4.248\nbulk AA 4.257 4.257\nAB 4.429 4.201\nAC 4.339 4.238\nD3 2L AA 4.351 4.265\nAC 4.464 4.251\nD2 2L AA 4.251 4.229\nAC 4.334 4.215\noptB88 2L AA 4.356 4.281\nAC 4.398 4.27\nSnS rev-vdW-DF2 2L AA 4.170 4.071\nAC 4.195 4.052\nGeSe rev-vdW-DF2 2L AA 4.150 3.972\nAC 4.254 3.961\nGeS rev-vdW-DF2 2L AA 4.089 3.780\nAC 4.241 3.721\n6TABLE S3. Lattice constants, aandb, spontaneous polarization, P, piezoelectric strain coefficients,\neijk, planar elastic stiffness coefficients, Cij, and piezoelectric stress coefficients, dijof multilayer\nSnSe calculated by various functionals.\nLayer vdWa\n(˚A)b\n(˚A)P\n(pC/m)e111\n(nC/m)e122\n(nC/m)C11\n(N/m)C22\n(N/m)C12\n(N/m)d11\n(pm/V)d12\n(pm/V)\n2L AA ref-vdW-DF2 4.302 4.263 249.1 8.32 1.65 65.69 97.67 43.41 163.52 -55.78\nD3 4.351 4.265 398.0 6.31 1.55 22.45 70.56 16.09 317.24 -50.38\nD2 4.251 4.229 146.0 14.68 5.41 46.97 87.19 37.01 396.15 -155.67\noptB88 4.356 4.281 336.1 6.02 0.9 56.51 96.68 40.96 144 -51.7\n3L AA ref-vdW-DF2 4.290 4.257 347.5 13.5 3.05 107.82 152.92 67.63 155.96 -49.03\nD3 4.308 4.256 498.7 15.76 5.47 59.18 121.46 40.74 305.95 -57.59\n2L AC ref-vdW-DF2 4.340 4.251 381.9 5.7 0.98 51.89 92.05 37.29 144.17 -47.76\nD3 4.464 4.251 556.2 2.54 -0.38 31.89 80.25 26.75 116.08 -43.43\n3L AC ref-vdW-DF2 4.339 4.248 605.1 8.53 1.17 72.58 138.38 57.12 164.22 -59.33\n2L AB ref-vdW-DF2 4.392 4.230\n3L AB ref-vdW-DF2 4.394 4.224 230.9 2.39 0.32 65.26 129.79 56.37 55.2 -21.51\n7TABLE S4. Lattice constants, aandb, spontaneous polarization, P, piezoelectric strain coefficients,\neijk, planar elastic stiffness coefficients, Cij, and piezoelectric stress coefficients, dijof 1L SnS, GeSe,\nand GeS calculated by PBE functional.\nMaterial layera\n(˚A)b\n(˚A)P\n(pC/m)e111\n(nC/m)e122\n(nC/m)C11\n(N/m)C22\n(N/m)C12\n(N/m)d11\n(pm/V)d12\n(pm/V)\nSnS 1L 4.284 4.082 256.2 1.8 -0.13 22.33 38.97 15.98 117.47 -51.5\nGeSe 1L 4.275 3.981 330.6 1.01 -0.62 19.43 52.92 17.47 88.9 -41.06\nGeS 1L 4.487 3.660 448.5 0.41 -0.81 11.62 52.52 18.5 136.25 -63.41\nTABLE S5. Lattice constants, aandb, spontaneous polarization, P, piezoelectric strain coefficients,\neijk, planar elastic stiffness coefficients, Cij, and piezoelectric stress coefficients, dijof 1L and AA-\nstacked 2L SnS, GeSe, and GeS calculated by PBE with ref-vdW-DF2 functional.\nMaterial layera\n(˚A)b\n(˚A)P\n(pC/m)e111\n(nC/m)e122\n(nC/m)C11\n(N/m)C22\n(N/m)C12\n(N/m)d11\n(pm/V)d12\n(pm/V)\nSnS 1L 4.175 4.056 210.2 2.27 -0.14 18.68 39.21 14.27 172.09 -66.20\n2L AA 4.170 4.071 384.2 5.15 -0.58 42.88 77.82 32.07 181.67 -82.32\nGeSe 1L 4.201 3.985 274.1 1.51 -0.46 23.03 53.78 20.70 112.00 -51.66\n2L AA 4.150 3.972 597.0 2.71 -1.09 56.15 114.36 44.07 79.92 -40.33\nGeS 1L 4.253 3.722 395.5 0.68 -0.81 21.58 54.50 23.85 92.84 -55.49\n2L AA 4.089 3.780 685.1 1.64 -1.55 44.04 110.96 48.38 100.93 -57.97\n8TABLE S6. Lattice constants, aandb, spontaneous polarization, P, piezoelectric strain coefficients,\neijk, planar elastic stiffness coefficients, Cij, and piezoelectric stress coefficients, dijof 1L and AA-\nstacked 2L SnS, GeSe, and GeS calculated by PBE with Grimme-D3 functional.\nMaterial layera\n(˚A)b\n(˚A)P\n(pC/m)e111\n(nC/m)e122\n(nC/m)C11\n(N/m)C22\n(N/m)C12\n(N/m)d11\n(pm/V)d12\n(pm/V)\nSnS 1L 4.441 4.032 327.4 0.99 -0.35 14.8 41.62 16.53 137.03 -62.82\n2L AA 4.455 4.03 653.3 2.01 -0.87 24.18 74.04 27.61 168.14 -74.45\nGeSe 1L 4.266 3.978 334.7 0.95 -0.66 22.8 57.03 23.14 90.81 -48.41\n2L AA 4.255 3.967 724.5 1.49 -1.52 62.95 122.98 55.02 56.61 -37.69\nGeS 1L 4.530 3.631 461.3 0.34 -0.81 10.2 55 18.18 145.02 -62.66\n2L AA 4.428 3.656 856.3 0.69 -1.65 35.9 119.75 39.53 54.03 -31.61\n∗ykkwon@khu.ac.kr\n†tlow@umn.edu\n[S1] G. Henkelman, B. P. Uberuaga, and H. J´ onsson, J. Chem. Phys. 113, 9901 (2000).\n[S2] T. Chattopadhyay, J. Pannetier, and H. G. Von Schnering, J. Phys. Chem. Solids 47, 879\n(1986).\n[S3] P. A. E. Murgatroyd, M. J. Smiles, C. N. Savory, T. P. Shalvey, J. E. N. Swallow, N. Fleck,\nC. M. Robertson, F. J¨ ackel, J. Alaria, J. D. Major, et al., Chem. Mater. 32, 3245 (2020),\nISSN 0897-4756.\n[S4] T. Grandke and L. Ley, Phys. Rev. B 16, 832 (1977).\n9"    },    {        "title": "2402.04441v1.Galois_invariants_of_finite_abelian_descent_and_Brauer_sets.pdf",        "content": "arXiv:2402.04441v1  [math.NT]  6 Feb 2024GALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER\nSETS\nBRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nAbstract. For a variety over a global field, one can consider subsets of t he set of adelic points\nof the variety cut out by finite abelian descent or Brauer-Man in obstructions. Given a Galois\nextension of the ground field one can consider similar sets ov er the extension and take Galois\ninvariants. In this paper, we study under which circumstanc es the Galois invariants recover the\nobstruction sets over the ground field. As an application of o ur results, we study finite abelian\ndescentand Brauer-Maninobstructions forisotrivialcurv es overfunction fields andextend results\nobtained by the first and last authors for constant curves to t he isotrivial case.\n1.Introduction\nLetX/kbe a smooth projective and geometrically connected variety over a global field kwith\nseparableclosure ks. Itiswellknownthattheset X(k)ofk-rationalpointsof Xmayfailtobedense\nin the set X(Ak) =/producttext\nvX(kv) of adelic points of X, as there can be cohomological obstructions\nmediated by the Brauer group Br( X) and/or by the finite abelian descent obstruction [14, 19].\nThis remains true even if one replaces X(Ak) with its space X(Ak)•of connected components,\nwhich only differs from X(Ak) at the archimedean places. By abuse of language we will refer to\nelements of X(Ak)•also as adelic points. It is clear that the assignment L/mapsto→X(L) defines a\nsheaf of sets on Spec( k)´ et(i.e., an ´ etale sheaf) in the sense that, for any finite Galois extensio n\nL/k, we have X(L)Gal(L/k)=X(k). In this note we investigate whether the sets cut out by these\ncohomological obstructions also define ´ etale sheaves in this sense .\nFora torsor f:Y→Xunder afinite abelian algebraicgroup G/k, letX(Ak)f\n•⊂X(Ak)•denote\nthe set adelic points which survive f[21, Definition 5.2]. Then X(Ak)f\n•contains the topological\nclosureX(k) ofX(k) inX(Ak)•. For a finite separable extension L/kwe useX(AL)f\n•to denote\nthe set of L-adelic points on Xwhich lift to a twist of the base changed torsor fL:YL→XL.\nWhenL/kis Galois, there is a natural inclusion X(Ak)f\n•⊂(X(AL)f\n•)Gal(L/k). Our first main\nresult shows that this will be an equality for all Galois extensions L/kif and only if Gdoes not\ncontain any nontrivial ´ etale subgroup.\nTheorem 1.1 (Theorems 2.1 and 2.3) .LetXbe a smooth projective and geometrically connected\nvariety over kadmitting a geometrically connected torsor f:Y→Xunder a finite abelian group\nschemeGover a global field k. Then the following are equivalent.\n(1)There exists a finite separable extension K/kand a Galois extension L/Ksuch that\nX(AK)f\n•/\\e}atio\\slash= (X(AL)f)Gal(L/K)\n•;\n(2)G(ks)/\\e}atio\\slash= 0.\nFor subvarieties of abelian varieties with torsion free N´ eron-Seve ri group (e.g., if Xis a curve\nor an abelian variety) we will show that the failure of finite abelian desc ent sets to determine ´ etale\nsheaves goes away in the limit. Define X(Ak)f-ab\n•:=/intersectiontext\nf:Y→XX(Ak)f\n•, where the intersection is\ntaken over all geometrically connected torsors under finite abelian group schemes.\nTheorem 1.2. Suppose X⊂Ais a subvariety of an abelian variety over a global field with t orsion\nfree N´ eron-Severi group. Then for any finite Galois extensi onL/kwe have\nX(Ak)f-ab\n•=/parenleftbig\nX(AL)f-ab\n•/parenrightbigGal(L/k).\n12 BRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nWhenXis a subvariety ofan abelian variety A/kthe finite abelian descent obstruction is closely\nrelated to the Brauer-Manin obstruction. There are containment s\n(†) X(k)⊂\n⊂X(Ak)Br\n•⊂\n⊂X(Ak)f-ab\n•\n⊂\nX(Ak)•∩A(k)⊂X(Ak)•∩A(Ak)Br\n•⊂X(Ak)•∩A(Ak)f-ab\n•.\nFor the following conjecture we refer to [19, p. 133], [21, Section 8], [13, Question 7.4] and [15,\nConjecture C].\nConjecture 1.3. ForXa smooth closed subvariety of an abelian variety Aover a global field k\nall of the containments in ( †) are equalities.\nThisconjecturehasbeenprovenwhen A(k)andX(A/k)arefinite[17]andformostnonisotrivial\ncoset-free subvarieties of abelian varieties over global function fi elds [15], including all nonisotrivial\ncurves of genus at least 2 over global function fields [6].\nIt is not difficult to show that the bottom left set of ( †) defines an ´ etale sheaf (see Lemma 3.5).\nThus, Theorem 1.2 was predicted by Conjecture 1.3 and can be seen as some modest evidence for\nit. In Section 3 we verify that the other terms in ( †) define ´ etale sheaves in a number of cases,\nincluding the case when Xis a curve (See Theorem 3.9). We also show that, in general, none of\nthe sets in the top row of ( †) define ´ etale sheaves if we consider varieties that do not embed int o\nan abelian variety (See Propositions 3.11 and 3.13). These examples c omplement various recent\nresults studying the behaviour of obstructions under base exten sion [1, 10, 16, 23].\nPart of our motivation for studying these questions comes from th e desire to extend the results\nof [5] concerning the Brauer-Manin obstruction for constant cur ves over global function fields to\nthe case of isotrivial curves. Recall that a variety Xover the function field k=F(D) of a smooth\nprojective curve Dover a finite field Fis called constant if there is a variety X0/Fsuch that\nX≃X0×Fk. A variety X/kis called isotrivial if there is a finite extension L/ksuch that X×kL\nis constant. We note that, unlike the nonisotrivial case, it is not imme diate that X(k) is equal to\nthe subset of X(L) fixed by Galois, as there are isotrivial curves of every genus with X(k)/\\e}atio\\slash=X(k).\nAs an application of the results above we generalize [5, Theorems 1.1 and 1.2] to the case of\nisotrivial curves. See Theorems 4.13 and 4.11 below. We also deduce t he following.\nTheorem 1.4. LetXbe a smooth projective and isotrivial curve over a global fun ction field k.\nSuppose Xbecomes constant after base change to L/k. Then X(k) =X(Ak)Brif and only if\nX(L) =X(AL)Br. Moreover, Conjecture 1.3 holds for all isotrivial curves o ver global function\nfields if it holds for all constant curves over global functio n fields.\nNote that we have X(Ak) =X(Ak)•in the function field case. Some instances where the\nequality X(L) =X(AL)Bris known to hold for a constant curve over Lare given in [7, Theorem\n2.14] and [5, Theorem 1.5].\n2.Descent sets for torsors under finite abelian group schemes\nIn this section we will prove Theorem 1.1.\n2.1.The nontrivial ´ etale subgroup scheme case. For a finite abelian group scheme Gover\nk, there is a unique maximal ´ etale subgroup scheme Ge. It is determined by the property that\nGe(ks) =G(ks). We begin by proving Theorem 1.1 in the case that Gcontains a nontrivial ´ etale\nsubgroup scheme.\nTheorem 2.1. LetGbe a finite abelian group scheme over a global field ksuch that G(ks)/\\e}atio\\slash= 0.\nLetXbe a smooth projective over kwhich admits a geometrically connected torsor f:Y→XGALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER SETS 3\nunderG. Then there exists a finite separable extension K/kand a finite Galois extension L/K\nsuch that\nX(AK)f\n•/subsetnoteql/parenleftbig\nX(AL)f\n•/parenrightbigGal(L/K).\nIn particular, the assignment L/mapsto→X(AL)f\n•does not define a sheaf of sets on Spec(k)´ et.\nLemma 2.2. LetX/kbe a smooth projective variety over a global field kand letf:Y→Xbe a\ntorsor under a finite ´ etale abelian group scheme G/k. IfY(Ak)•/\\e}atio\\slash=∅, then there exists an adelic\npointx∈X(Ak)•and a finite separable extension L/ksuch that x∈X(AL)f\n•\\X(Ak)f\n•.\nProof.Call a class ξ∈H1(k,G) finitely supported if the image of ξunder the restriction map\nH1(k,G)→H1(kv,G) is trivial for all but finitely many places vofk. LetK/kbe the splitting\nfield ofG, i.e., the minimal Galois extension K/ksuch that G(K) =G(k). By [12, I.9.3] any\nfinitely supported class lies in the image of the inflation map H1(K/k,G)→H1(k,G). It follows\nthatH1(k,G) contains only finitely many finitely supported classes. Since H1(k,G) is infinite,\nthere are infinitely many classes that are not finitely supported.\nWe claim that there are infinitely many places vofksuch that the map f:Y(kv)→X(kv)\nis not surjective. To see this, let ξ∈H1(k,G) be a class which is not finitely supported and let\nfξ:Yξ→Xbe the corresponding twist. Then Yξis smooth and geometrically connected so\nhas points over kvfor all but finitely many vby the Lang-Weil estimates and Hensel’s lemma. A\npointx∈fξ(Yξ(kv))⊂X(kv) lies in f(Y(kv)) if and only if res v(ξ) = 0. Since ξis not finitely\nsupported,thereareinfinitelymanyplaceswhere fξ(Yξ(kv))isanonemptysubsetof X(kv)disjoint\nfromf(Y(kv)).\nLetMbe the maximum number of places at which a finitely supported class in H1(k,G) is\nnonzero and let w0,...,w MbeM+1 places of ksuch that f:Y(kwi)→X(kwi) is not surjective.\nLet (xv)∈Y(Ak)•and choose ywi∈X(kwi)\\f(Y(kwi)) fori= 0,...,M. Define ( zv)∈X(Ak)•\nby setting zv=f(xv)∈X(kv) forv /∈{w0,...,w M}and setting zwi=ywifori= 0,...,M.\nFor all places v/\\e}atio\\slash∈{w0,...,w M}we have that z∗\nvf=f(xv)∗f= 0∈H1(kv,G). It follows that\n(zv)∗fis nonzero precisely at the M+ 1 places w0,...,w M. From the definition of the integer\nMit follows that ( zv)∗fdoes not lie in the diagonal image H1(k,G)→/producttext\nvH1(kv,G), and so\n(zv)∈X(Ak)•\\X(Ak)f\n•.\nSinceGis ´ etale we have that for each i= 0,...,M, there is a finite separable extension Lwi/kwi\nthat kills z∗\nwif. Moreover, we can find a global Galois extension L/ksuch that, for all i= 0,...,M\nand all primes u|wiwe have that the extension Lu/kwikillsz∗\nwif. Then (zv)∈X(AL)f\n•∩X(Ak)•\nas required. /square\nProof of Theorem 2.1. LetK/kbe a finite separable extension such that X(K)/\\e}atio\\slash=∅. Then there\nexists a twist f′:Y′→XKofYK→XKwithY′(K)/\\e}atio\\slash=∅. By [21, Lemma 5.3(5)] we have that\nX(AK)f′\n•=X(AK)f\n•and similarly over L, so replacing Yby this twist if necessary we may assume\nY(K)/\\e}atio\\slash=∅. LetGe⊂Gbe the maximal ´ etale subgroup scheme (i.e., defined by Ge(ks) =G(ks)).\nThenffactors as a composition of torsors fe:Y→Zandg:Z→XunderGeandG/Ge,\nrespectively.\nThenfesatisfies the conditions of Lemma 2.2 over K, so there is a finite Galois extension L/K\nand some z∈Z(AL)fe•∩Z(AK)•which does not lie in Z(AK)fe•. Define x:=g(z), which by\nconstruction lies in g/parenleftBig\nZ(AL)fe•∩Z(AK)•/parenrightBig\n⊂X(AL)f\n•∩X(AK)•. To complete the proof it is\nenough to show that x /∈X(AK)f\n•.\nBy way of contradiction, suppose x∈X(AK)f\n•⊂X(AK)g\n•. Thenxlifts to a twist f′=g′◦f′\ne\noff=g◦fby a class in H1(K,G) andxlifts to the twist g′ofgby the image ξof this class\ninH1(K,G/G e). Since xalso lifts under gwe must have that ξ∈X1(K,G/G e). The map\nG→G/Geis ´ etale, so any separable point of the quotient will lift to a separable point of G. It\nfollows that ( G/Ge)(ks) = 0. Then X1(K,G/G e) = 0 by [8, Main Theorem]. So g=g′and4 BRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nwe see that x∈g(Z(AK)f′\ne•). Again using [21, Lemma 5.3(5)] we have Z(AK)fe•=Z(AK)f′\ne•, so\nx∈g(Z(AK)fe•)⊂X(AK)f\n•which is a contradiction. /square\n2.2.The case G(ks) = 0.It remains to prove Theorem 1.1 in the case that Gdoes not contain a\nnontrivial ´ etale subscheme. This follows immediately from the next r esult.\nTheorem 2.3. LetGbe a finite abelian group scheme over a global field ksuch that G(ks) = 0.\nLetXbe a smooth projective variety admitting a torsor f:Y→XunderG. Then for any finite\nGalois extension L/k,/parenleftBig\nX(AL)f\n•/parenrightBigGal(L/k)\n=X(Ak)f\n•.\nLemma 2.4. LetGbe a finite abelian group scheme over field Kand letL/Kbe a Galois extension\nsuch that G(L) = 0. Then the restriction map H1(K,G)→H1(L,G)Gal(L/K)is an isomorphism.\nProof.The inflation-restriction sequence in flat cohomology (see [18, p. 42 2]) gives an exact\nsequence\nH1(L/K,G(L))inf→H1(K,G)res→H1(L,G)Gal(L/K)→H2(L/K,G(L)).\nThe outer two terms are trivial because G(L) = 0 as L/Kis separable. Thus, the restriction map\nis an isomorphism. /square\nProof of Theorem 2.3. Noting that X(AL)Gal(L/k)\n•=X(Ak)•, it will be enough to show that we\nhaveX(Ak)f\n•=X(AL)f\n•∩X(Ak)•. The inclusion⊆is clear, so we show the reverse inclusion. We\nhave a commutative diagram\n(1) H1(k,G)/d31 /d127/d47/d47/d127 /d95\n/d15/d15/producttext\nvH1(kv,G)/d127 /d95\n/d15/d15\nH1(L,G)/d31 /d127/d47/d47/producttext\nv/producttext\nw|vH1(Lw,G)\nwherethe injectivity ofthe verticalmapscomefromLemma2.4andt he injectivity ofthe horizontal\nmaps is [8, MainTheorem].\nLetx∈X(AL)f\n•∩X(Ak)•and consider the image x∗fofxunder the map X(Ak)•→/producttext\nvH1(kv,G) induced by f. The image of x∗fin/producttext\nv/producttext\nw|vH1(Lw,G) is the image of a unique\nξ∈H1(L,G) under the bottom horizontal map of (1) because x∈X(AL)f\n•. For any vthere is a\nnatural action of Gal( L/k) on/producttext\nw|vH1(Lw,G) which is compatible with the action of Gal( L/k)\nonH1(L,G). The image of x∗fin/producttext\nw|vH1(Lw,G) is fixed by this action, so we conclude that\nξ∈H1(L,G)Gal(L/k). By Lemma 2.4 we have the ξis the image of some ξ′∈H1(k,G). Since the\nmaps are all injective, x∗fmust be the image of ξ′. This means x∈X(Ak)f\n•. /square\nRemark 2.5. In the proof above, the condition G(L) = 0 is only used to ensure the existence of\na lift ofx∗(f) from/producttext\nvH1(kv,G) toH1(k,G), using that its image in/producttext\nv/producttext\nw|vH1(Lw,G) comes\nfrom an element in H1(L,G). The conclusion of Theorem 2 .3 therefore holds for any (potentially\ninfinite) group scheme Gand any Galois extension L/ksuch that Diagram (1) is Cartesian.\n3.Subvarieties of abelian varieties\nThroughout this section X⊂Adenotes a smooth closed subvariety of an abelian variety over a\nglobal field k.\nDefinition 3.1. We say that X(Ak)Br\n•defines an ´ etale sheaf if, for any finite Galois extension\nL/k, we have/parenleftbig\nX(AL)Br\n•/parenrightbigGal(L/k)=X(Ak)Br\n•, and similarly for the other sets appearing in ( †).\nThis condition is equivalent to the assignment L/mapsto→X(Ak)Br\n•defining a sheaf of sets on Spec( k)´et.GALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER SETS 5\n3.1.The finiteabelian descent set. Following[21, p. 352]wedefine /hatwiderSel(A/k) = lim←−nSeln(A/k).\nThis fits into the Cassels-Tate dual exact sequence [15, Propositio n 4.3], which reads\n0→/hatwiderSel(A/k)→A(Ak)•φ→H1(k,A∨)∗.\nThe map φis induced by the sum of the local Tate pairings /a\\}bracketle{t,/a\\}bracketri}htkv:A(kv)×H1(kv,A∨)→Q/Z.\nThis sequence identifies /hatwiderSel(A/k) with a subset of A(Ak)•.\nTheorem 3.2. The sets /hatwiderSel(A/k)andX(Ak)•∩/hatwiderSel(A/k)define ´ etale sheaves.\nProof.LetL/kbe any finite Galois extension. It suffices to prove the result for /hatwiderSel(A/k), since\n/parenleftBig\nX(AL)•∩/hatwiderSel(A/L)/parenrightBigGal(L/k)\n=X(Ak)•∩/hatwiderSel(A/L)Gal(L/k).\nSuppose x∈/hatwiderSel(A/L)Gal(L/k)=/hatwiderSel(A/L)∩A(Ak)•and letd= [L:k]. We first claim that\ndx∈/hatwiderSel(A/k)⊂A(Ak)•. For this we use the Cassels-Tate dual exact sequence. For any p lacev,\npassing to an extension Lv/kvof degree dvmultiplies the local Tate pairing by dv, i.e., we have\n/a\\}bracketle{tx,resLv/kv(α)/a\\}bracketri}htLv=dv/a\\}bracketle{tx,α/a\\}bracketri}htkv. From this we deduce that for any finite extension K/kwe have\n[K:k]φ(x) =φK(x)◦resK/k∈H1(k,A∨)∗. The assumption that x∈/hatwiderSel(A/L)Gal(L/k)together\nwith exactness of the sequence gives φL(x) = 0, so φ(dx) =dφ(x) =φL(x)◦resL/k= 0, showing\nthatdx∈/hatwiderSel(A/k).\nSincedx∈/hatwiderSel(A/k) = lim←−nSeln(A/k), there is a compatible system of geometrically connected\ntorsorsfn:Yn→AunderA[n] containing lifts yn∈Yn(Ak) ofdx. Here compatible means that\nfor each m,n, we have a torsor structure Ymn→YnunderA[m] sending ymntoyn. The trivial\ntorsor [d] :A→Acontains a lift of dxby hypothesis, so fdmust be a twist of this trivial covering\nby an element ξ∈X1(k,A[d]). For each n≥1, letgnd:Znd→Abe the twist of fndby the image\nof−ξunder the map X1(k,A[d])→X1(k,A[nd]) induced by the inclusion A[d]֒→A[nd]. Then\ngd= [d] :A→Aandgdnfactors as gdn= [d]◦hnfor some torsor hn:Znd→AunderA[n]. Since\nξis locally trivial and the fndcontain lifts of dx, we have we have dx∗gnd=dx∗fnd= 0, for all n.\nIt follows that the family hndetermines an element in /hatwiderSel(A/k), and consequently an adelic point\nx′∈A(Ak)•. By construction dx′=dx, sox−x′∈A[d](Ak)•is an adelic point contained in a\nfinite subscheme of A.\nBy assumption x′−xlies in/hatwiderSel(A/L). Using [21, Proposition 3.6] in the number field case and\n[15, Proposition 5.3] in the function field case (Note that the addition al hypothesis on Athere can\nbe dropped thanks to work of R¨ ossler; see [6, Proposition 3.1]) we conclude that x′−x∈A(L).\nThenx′−x∈A(L)∩A(Ak)•=A(k). Sox∈x′+A(k)⊂/hatwiderSel(A/k) as required. /square\nProof of Theorem 1.2. As noted on [21, p. 373] the image of /hatwiderSel(A/k) inA(Ak)•is equal to\nA(Ak)f-ab\n•. This follows from the fact that the N´ eron-Severi group of Ais torsion free, and so\nthe geometrically abelian fundamental group of Ais isomorphic to its Tate module. The same\nargument works to show X(Ak)f-ab\n•=X(Ak)•∩/hatwiderSel(A/k) for any X⊂Awith torsion free N´ eron-\nSeveri group. So Theorem 1.2 follows from Theorem 3.2. /square\nAs an amusing corollary of Theorem 1.2 we have the following.\nCorollary 3.3. Suppose X⊂Ais defined over a number field and satisfies the following:\n(1)X(k)defines an ´ etale sheaf,\n(2)the N´ eron-Severi group of Xis torsion free, and\n(3) Br(Xks) = 0.\nIfX(L) =X(AL)Br\n•for all finite separable extensions L/k, thenX(L) =X(AL)f-ab\n•for all finite\nseparable extensions L/k. In other words, if the Brauer-Manin obstruction is the only obstruction\nto weak approximation for such X, then the a priori weaker obstruction coming from finite abel ian\ndescent is already sufficient to capture this obstruction.6 BRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nProof.Assumption (1) is that F(L) =X(L) =X(AL)Br\n•is an ´ etale sheaf, and G(L) =X(AL)f-ab\n•\nis an ´ etale sheaf by Theorem 1.2. The inclusion X(AL)Br\n•⊂X(AL)f-ab\n•shows thatFis a subsheaf\nofG. It is enough to show that this inclusion is an isomorphism´ etale locally, for thenF=Gby [9,\n§II Proposition 1.1], in which case we have X(L) =F(L) =G(L) =X(AL)f-ab\n•for any separable\nextension L/k.\nBy condition (3) we have Br( X) = Br 1(X) := ker(Br( X)→Br(Xks)). Condition (2) gives that\nNS(Xks) is a finitely generated free Z-module, so H1(K,NS(Xks)) = 0 for all sufficiently small\nneighbourhoods K/k. For such K/kwe have X(AK)Br\n•=X(AK)f-ab\n•as in [21, Corollary 7.8],\nshowing thatFandGare isomorphic ´ etale locally. /square\nRemark 3.4. Condition (1) of Corollary 3.3 is satisfied if X⊂Ais coset free, since in this case\nX(L) is finite for all L/k. Using [4, Lemma 3.1], Condition (3) can be replaced by the weaker\ncondition that Br( Xks) is finite, or the even weaker condition\n(3’) For every sufficiently small neighbourhood K/kof Spec(k)´ et, there is a separable extension\nL/Ksuch that\n(a) [L:K]Br(XK)⊂Br1(XK), and\n(b) res L/K: Br(XK)/Br1(XK)→Br(XL)/Br1(XL) is surjective.\n3.2.Sheafiness of the other terms in (†).\nLemma 3.5. The sets A(k)andX(Ak)•∩A(k)define ´ etale sheaves.\nProof.The topological closure of A(L) and the profinite completion /hatwideA(L) are isomorphic as\nGal(L/k)-modulesby[15,TheoremE].Since A(L)isfinitelygeneratedwealsohaveanisomorphism\nof Gal(L/k)-modules /hatwideA(L)≃A(L)⊗/hatwideZ. Then (A(L)⊗/hatwideZ)Gal(L/k)=A(L)Gal(L/k)⊗/hatwideZ=A(k)⊗/hatwideZ,\nthe latter being identified with A(k) by [15, Theorem E]. Thus A(k) defines an ´ etale sheaf. The\nstatement about X(Ak)•∩A(k) follows instantly since\n/parenleftBig\nX(AL)•∩A(L)/parenrightBigGal(L/k)\n=X(AL)Gal(L/k)\n•∩A(L)Gal(L/k)=X(Ak)•∩A(k).\n/square\nLemma 3.6. Ifkis a number field or the maximal divisible subgroup of X(A/k)is trivial, then\nA(Ak)Br\n•defines an ´ etale sheaf.\nProof.In the number field case, [3, Theorem 1] implies A(Ak)Br\n•=A(Ak)f-ab\n•, so this follows from\nTheorem 3.2. If the divisible subgroup of X(A/k) is trivial then A(k) =A(Ak)Br\n•=A(Ak)f-ab\n•\n(See [15, Remark 4.5]), so this follows from either Lemma 3.5 or Theore m 3.2. /square\nRemark 3.7. WhenA/kis an isotrivial abelian variety, we have that X(A/k) is finite. For A/k\nconstant, this is shown in [11], and as mentioned in Remark 6.27 of [12] it is an easy extension to\nthe isotrivial case.\nLemma 3.8. IfX⊂Ais a curve embedded in its Jacobian, then X(k)defines an ´ etale sheaf.\nProof.IfXhas genus 1, then X=Aand this follows from Lemma 3.5. So suppose Xhas genus\n≥2. Ifkis a number field or Xis a nonisotrivial, then X(k) is finite. Then X(k) =X(k) which\nclearly defines an ´ etale sheaf. We may therefore suppose Xis an isotrivial curve of genus ≥2.\nThis case is proved in Lemma 4.9 of the following section. /square\nTheorem 3.9. Suppose X⊂Ais a curve embedded in its Jacobian by an Albanese map (i.e., a\nmap sending a point Pto the class of P−Dfor a fixed k-rational divisor D∈Div(X)of degree\n1). Then all of the sets in (†)define ´ etale sheaves.\nProof.First note that X(Ak)•∩A(Ak)Br\n•=X(Ak)Br\n•=X(Ak)f-ab\n•=X(Ak)•∩A(Ak)f-ab\n•(See [21,\nCorollary 7.3]). So all of these define ´ etale sheaves by Theorem 3.2. Moreover, X(Ak)•∩A(k) and\nX(k) define ´ etale sheaves by Lemma 3.5 and Lemma 3.8. /squareGALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER SETS 7\nRemark 3.10. Suppose Cis a curve of genus ≥1 that does not have a k-rational divisor of\ndegree 1, so that it is not embedded in its Jacobian. Then Cembeds canonically in Pic1\nCwhich is\na nontrivial torsor under A= Jac(C). IfC(Ak)Br\n•/\\e}atio\\slash=∅, then Pic1\nCis a nontrivial divisible element\ninX(A/k) by [19, Proposition 3.3.5 and Theorem 6.1.2]. By Remark 3.7 this cannot occur ifCis\nisotrivial, so for isotrivial curves not embedded in their Jacobian, all of the sets in (†) are trivial.\n3.3.Counterexamples among general varieties. We now show that, in general, none of the\nsets in the top row of ( †) define ´ etale sheaves, if we consider varieties that do not admit an\nembedding into an abelian variety.\nProposition 3.11. LetY/kbe a smooth projective variety over a number field such that Pic(Y)\nis torsion free, Br(Y)is finite, and Br(Y)→Br(Y)Gal(k)is surjective.\n(1)IfY(Ak)•/\\e}atio\\slash=Y(Ak)Br\n•, then there exists a finite Galois extension L/ksuch that\n/parenleftbig\nY(AL)Br\n•/parenrightbigGal(L/k)/\\e}atio\\slash=Y(Ak)Br\n•.\n(2)IfY(k) =∅,Y(Ak)•/\\e}atio\\slash=∅andY(K) =Y(AK)Br\n•for all finite Galois extensions K/k, then\nthere exists a finite Galois extension L/ksuch that Y(L)Gal(L/k)/\\e}atio\\slash=∅=Y(k).\nProof.The assumptions in the first sentence imply that Br( Y)/Br0(Y) is finite, say of order d.\nBy [4, Lemma 3.3] there are infinitely many Galois extensions of degree divisible by dsuch that the\nrestriction map Res L/k: Br(Y)/Br0(Y)→Br(YL)/Br0(YL) is surjective. For any such extension,\n[4, Lemma 3.1(2)] gives that Y(Ak)•⊂Y(AL)Br\n•, so every element Y(Ak)•lies in the Gal( L/k)-\ninvariant subset of Y(AL)Br\n•. The claims in (1) and (2) follow immediately. /square\nRemark 3.12. Examples satisfying the conditions of Proposition 3.11(1) can be fou nd among del\nPezzo surfaces and Chˆ atelet surfaces. Chˆ atelet surfaces th at are counterexamples to the Hasse\nprinciple satisfy the conditions in Proposition 3.11(2) by [2, Theorem B ].\nProposition 3.13. LetY/kbe a smooth projective Enriques surface over a global field kof char-\nacteristic not equal to 2. Then the assignment L/mapsto→Y(AL)f-ab\n•does not define an ´ etale sheaf.\nProof.Letf:Z→Ybe aK3 cover of the Enriques surface. Then Zis ´ etale simply connected,\nand soY(AL)f\n•=Y(AL)´ et\n•=Y(AL)f-ab\n•. The result follows by Theorem 2 .1. /square\n4.Isotrivial curves\nIn this section we use results of the previous sections to generalize the main results of [5] to the\ncase of isotrivial curves.\nFix a finite field Fof characteristic p. LetDbe a smooth projective geometrically connected\ncurve over F, and let k=F(D) denote the function field of D. Throughout this section Cwill\ndenote a smooth projective and geometrically connected curve ov erkof genus g=g(C)≥2 which\nwe assume to be isotrivial. We also assume that Chas ak-rational divisor zof degree 1 and use\nthis to define an embedding of Cinto its Jacobian J= Jac(C) by the rule x/mapsto→[x−z]. This\nassumption is justified by Remark 3.10.\nThere exists a finite extension L/kwith corresponding extension of constant fields FL/Fand\na curveC0/FLsuch that C×kL≃C0×FLL. We note that for any such L/kwe also have that\nJ×kL≃J0×FLL, whereJ0= Jac(C0) is an abelian variety defined over FL. One can take\nL/kto be separable because the moduli space of curves with sufficiently large level structure is a\nfine moduli space. Moreover, replacing Lby its Galois closure we obtain a Galois extension of k\ntrivialising C.\nRemark 4.1. There are isotrivial varieties that are not separably isotrivial, mean ing that they\nonly become constant after a non-separable field extension, e.g. s ingular genus-changing curves in\nthe sense of [22]. It is possible that there exist smooth examples in hig her dimension but we do\nnot know of any.8 BRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nThe set of places of kis in bijection with the set D1of closed points of D. Forv∈D1\nwe denote the residue field of the completion kvbyFv. Note that since the valued field kvhas\nequicharacteristic p,Fv⊆kv. We define Ak,F:=/producttext\nv∈D1Fv, which is an F-subalgebra of the usual\nadele ring over Ak. IfL/kis a finite extension with constant field extension FL/F, then there exists\na smooth projective curve D′/FLwithL=FL(D′), so we may define AL,FLsimilarly.\n4.1.Locally constant adelic points. We recall the definition of locally constant adelic points\nas in [5, Section 2.2] (where they were called reduced adelic points) be fore extending this definition\nto isotrivial varieties. Suppose that X/kis a constant variety so that X=X0×Fk. SinceD/Fis\ngeometrically connected, we have natural equalities of sets\nHom/k(Spec(Ak),X) = Hom /F(Spec(Ak),X) = Hom /F(Spec(Ak),X0).\nDefinition 4.2. For a constant variety X=X0×Fkoverkwe define the locally constant adelic\npointsto be the set\nX(Ak,F) := Hom /F(Spec(Ak,F),X0).\nNote that Ak,Fis anF-subalgebra of Ak, so we have an inclusion\nX(Ak,F)⊆Hom/F(Spec(Ak),X0) = Hom /k(Spec(Ak),X) =X(Ak).\nConcretely, X(Ak,F) =/producttext\nv∈D1X0(Fv).\nDefinition 4.3. Suppose X/kis an isotrivial variety and L/kis a Galois extension with cor-\nresponding residue extension FL/Fsuch that XL/Lis a constant variety. Let X0/FLbe the\ncorresponding constant variety and let φ:XL→X0×FLLbe an isomorphism. Define the locally\nconstant adelic points ofX/kto be the set\nX(Ak,F) :=X(AL)Gal(L/k)∩φ−1((X0×FLL)(AL,FL))⊂X(AL)Gal(L/k)=X(Ak).\nThe following two lemmas show that this definition does not depend on o ur choice of trivialising\nextension Land isomorphism φ.\nLemma 4.4. LetX/kbe an isotrivial variety, and let φ,φ′:XL→X0×FLbe two isomorphisms\nofLvarieties. Then φ−1((X0×FL)(AL,F)) =φ′−1((X0×FL)(AL,F)). In particular, the set X(Ak,F)\ndoes not depend on our choice of trivialising isomorphism φ.\nProof.Note that φ′◦φ−1gives us an automorphism of X0×FL, so the result is equivalent to saying\nthat the set X0×FL(AL,F) is stable under automorphisms of X0×F(AL,F). This is immediate by\nthe definition of locally constant adelic points for constant varieties . /square\nLemma 4.5. LetX/kbe an isotrivial variety. Then the definition of X(Ak,F)does not depend on\nour choice of trivialising extension.\nProof.LetL,L′be two trivialisingextensions for X/k, and considerthe extension K:=LL′. Since\nXLis constant, we have that XL(AL,FL) =XK(AK,FK)∩XL(AL), and similarly for XL′(AL′,FL′).\nTherefore\nXL(AL,FL)∩X(Ak) =XK(AK,FK)∩X(Ak) =XL′(AL′,FL′)∩X(Ak)\nas required. /square\nRemark 4.6. Locally constant adelic points are defined precisely so that they defi ne an ´ etale\nsheaf in the sense of Definition 3 .1, and their use is justified by the property they are shown to\nsatisfy in Theorem 4 .11.GALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER SETS 9\n4.2.The Frobenius map on isotrivial varieties.\nLemma 4.7. Suppose X/kis an isotrivial variety and L/kis a Galois extension such that there is\nan isomorphism XL≃X0×FLLfor some X0/FLwhereFL/Fis the residue extension corresponding\ntoL/k. Letm:= [FL:F]. The relative Frobenius morphism FX0/FL:X0→X0induces a\nmorphism induces a morphism FXL/L:XL→XLthat is compatible with descent data, and so\nyields a morphism X→X, which we call Fm\nX/k.\nExample 4.8. Consider an isotrivial curve C:ty2=f(x) overFp(t) withf(x)∈Fp[x] for an\nodd prime p. The relative Frobenius for X/kis the morphism F:C→C(p)given on coordinates\nby raising to the p-th power, where C(p)/Fis the curve given by tpy2=f(x). Since pis odd, we\nhaveC(p)≃Cby the map ( x,y)/mapsto→(x,tp−1\n2y). The Galois extension L=Fp(t1/2)/Fp(t) trivializes\nC. The morphism F1\nX/kconstructed in the lemma is the composition of Fand the isomorphism\nC(p)≃C.\nProof of Lemma 4.7. Letqbe the cardinality of FL. On any open subset UofXLthe morphism\nFX/Lis given by\nFX/L:OXL(U)→OXL(U)\nx/mapsto→xq.\nFrom this it is clear that FXL/Lcommutes with the descent data on XL/L. /square\n4.3.Proof of Lemma 3.8 in the isotrivial case. The following lemma completes the proof of\nLemma 3.8 which stated that C(k) defines an ´ etale sheaf.\nLemma 4.9. LetL/kbe a Galois extension. Then C(L)Gal(L/k)=C(k).\nProof.We claim that we can assume without loss of generality that L/ktrivialises C/k, i.e.,L/k\nis such that C×kL≃C0×FLLfor some curve C0/FL. LetK/kbe a Galois extension such that\nL⊆KandK/ktrivialises C/k. ThenC(K)Gal(K/k)= (C(K)(Gal(K/L)))Gal(L/k), so showing the\nresult for extensions that trivialise C/kimplies the result for all Galois extensions.\nLetDLbe the smooth projective curve with L=FL(DL). LetF=FC0/FLbe the relative\nFrobenius morphism. By a theorem of de Franchis the set C(L)/Fis finite. Define a set of distinct\nelements φ1,...,φ m∈C(L) = Mor FL(DL,C0) representing all morphisms from DLtoC0up to\nFrobenius twisting. Concretely this means that for any φ∈C(L), there exists integers a,b≥0 and\nani∈{1,...,m}such that Faφ=Fbφi. We can assume that the φiare distinct, separable and\nthe set{φ1,...,φ m}is closed under the action of Gal( L/k). The map C(L)/F→Map(D1\nL,C1\n0)\nis injective by [20, Proposition 2.3] and there are only finitely many φi, so we can find a finite set\nSof places v∈D1\nLsuch that the elements rS(φi) := (rv(φi))v∈S∈/producttext\nv∈SC0(Fv) are distinct for\ni/\\e}atio\\slash=j, whererv:C0(Lv)→C0(Fv) is the reduction map. Enlarging Sif needed, we can assume\nthatrS(Faφi)/\\e}atio\\slash=rS(Fbφj) for any a,b≥0 andi/\\e}atio\\slash=j. Note that rv(Faφi) =Fa(rv(φi)), where on\nthe right Facts via Gal( FL) onC0(Fv).\nLetρvbe metrics inducing the v-adic topology on C(Lv) and let ρbe a product metric\nof these inducing the adelic topology on C(AL). The reduction maps give rise to a continu-\nous retraction r= (rv) :C(AL)→C(AL,FL). By the discussion above, the set of distances\n{ρ(r(Faφi),r(Fbφj))|a,b≥0 andi/\\e}atio\\slash=j}is bounded away from 0.\nSuppose Pn∈C(L) converge to P∈C(L)∩C(Ak). For any σ∈Gal(L/k) the sequence σ(Pn)\nconverges to σ(P) =PinC(AL), since the induced map σ:C(AL)→C(AL) is continuous. Thus\nthe sequence of real numbers ρ(Pn,σ(Pn)) convergesto 0. For all n≥1 we have integers an,bn≥0\nandin∈{1,...,m}such that FanPn=Fbnφin. SinceF:C(AL)→C(AL) is continuous and\ncommutes with the action of σ, we find that the sequence ρ(Fbnφin,Fbnσ(φin)) converges to 0.\nSince the reduction map is continuous this implies that ρ(r(Fbnφin),r(Fbnσ(φin))) converge to\n0. These distances are bounded away from zero when φin/\\e}atio\\slash=σ(φin). So for all large enough n10 BRENDAN CREUTZ, JESSE PAJWANI AND JOS ´E FELIPE VOLOCH\nwe must have σ(φin) =φin. Since the action of σcommutes with Fwe find that, for all large\nenoughn, we have Fanσ(Pn) =FanPnand soPn=σ(Pn) as well. Thus the Pnare eventually in\nC(L)Gal(L/k)=C(k). Hence P∈C(k). /square\n4.4.Frobenius descent. LetL/kbe the minimal Galois extension trivializing the isotrivial curve\nC/k. ThenL/kalso trivializes J= Jac(C) and Lemma 4.7 gives an isogeny Fm\nJ/k:J→Jwhere\nm= [FL:F].\nLemma 4.10. The locally constant adelic points of the Jacobian satisfy J(Ak,F)⊂Fm\nJ/k(J(Ak)).\nProof.This is clear for constant varieties and Frobenius commutes with the Galois action. /square\nFor anyn≥1, then-fold composition of Fm\nJ/kis an isogeny φn= (Fm\nJ/k)◦n:J→Jwhose kernel\nis a finite connected abelian group scheme. The pullback of φnalong the embedding C→Jis a\ntorsorFn:Y→Cunder ker( φn). We define C(Ak)F∞=/intersectiontext\nn≥1C(Ak)Fn.\nThe following generalizes [5, Theorem 1.2] to the case of isotrivial cur ves.\nTheorem 4.11. Only global and locally constant adelic points survive infin ite Frobenius descent,\ni.e.,C(Ak)F∞=C(k)∪C(Ak,F).\nProof.For a finite separable extension L/kwith corresponding residue extension FL/Fdefine\nF(L) :=C(L)∪C(AL,FL) andG(L) :=C(AL)ker(F∞). ThenFandGare sheaves of sets on\nSpec(k)´ et, the latter by Theorem 2.3. By Lemma 4.10, all reduced adelic points s urvive Frobenius\ndescent, soFis a subsheaf ofG. For any L/ktrivializing C, we may apply [5, Theorem 1.2] in\norder to obtainF(L) =G(L). We conclude that F(k) =G(k) as in the proof of Theorem 4.13. /square\n4.5.The Mordell-Weil Sieve.\nDefinition 4.12. We define the Mordell–Weil Sieve set for an isotrivial curve embedded in its\nJacobian to be the set\nCMW-sieve:=C(Ak,F)∩J(k),\nwith this intersection taking place in J(Ak).\nWhenC/kis a constant curve this agrees with the definition of CMW-sievein [5] so the following\nresult generalizes [5, Theorem 1.1] to isotrivial curves.\nTheorem 4.13. ForC/ka smooth, projective, isotrivial curve embedded in its Jaco bian, we have\nan equality C(Ak)Br=C(k)∪CMW-sieve.\nProof.The assignments L/mapsto→G(L) :=C(AL)BrandL/mapsto→F(L) :=C(AL,F′)∩J(L) define ´ etale\nsheaves by Theorem 3.9 and Lemma 3.5. Moreover, F(L)⊂G(L) for all finite extensions L/k\nsinceC(AL)Br⊂J(AL)Br=J(L), where the final equality is given in the proof of Lemma 3.6,\nnoting that X(A/L) is finite by Remark 3.7. For any L/ktrivializing Cwe haveF(L) =G(L)\nby [5, Theorem 1.1], so we proceed as in the proof of Theorem 4 .13 to obtainG(k) =F(k) as\nrequired. /square\nRemark 4.14. IfCis an isotrivial curve that does not admit an embedding in its Jacobian,\nRemark 3 .10 shows that C(Ak)Br=∅. ForC/knot isotrivial and of genus ≥2, [6, Theorem 1.1]\ngives that C(Ak)Br=C(k). The above result is a step towards a characterisation of C(Ak)Brfor\ncurves in the remaining cases.\nReferences\n[1] Yang Cao and Yongqi Liang, Etale Brauer-Manin obstruction for Weil restrictions , Advances in Mathematics,\n410(2022).\n[2] J. L. Colliot-Th´ el` ene and J. J. Sansuc, Intersections of two quadrics and Chˆ atelet surfaces I. , Journal f¨ ur die\nreine und angewandte Mathematik, 373, (1987), 37–107.\n[3] B. Creutz, There are no transcendental Brauer–Manin obstructions on a belian varieties , International Math-\nematics Research Notices 2020, (2020), 2684–2697.GALOIS INVARIANTS OF FINITE ABELIAN DESCENT AND BRAUER SETS 11\n[4] B. Creutz and B. Viray, Quadratic points on intersections of two quadrics , Algebra and Number Theory 17,\n2023, 1411-1452.\n[5] B. Creutz and J. F. Voloch, The Brauer-Manin obstruction for constant curves over glob al function fields ,\nAnnales de l’Institute Fourier 72, (2022) 43–58.\n[6] B. Creutz and J. F. Voloch, The Brauer-Manin obstruction for nonisotrivial curves ove r global function fields ,\npreprint, (2023).\n[7] B. Creutz, B. Viray and J. F. Voloch, Thed-primary Brauer-Manin obstruction for curves , Res. Number\nTheory4, 26, (2018).\n[8] C. Gonz´ alez-Avil´ es and Ki-Seng Tan, On the Hasse principle for finite group schemes over global fu nction\nfields, Math. Res. Lett. 19(2012), no.2, 453–460.\n[9] R. Hartshorne, Algebraic Geometry , Grad. Texts in Math., 52, Springer-Verlag, (1977).\n[10] Guang Hu and Yongqi Liang, Arithmetic of Chˆ atelet surface bundles revisited , Forum Mathematicum, 35no.\n4, (2023), 883–900.\n[11] J. S. Milne, The Tate–Shaferevich group of a constant abelian variety , Invent. Math. 6, (1968), 91–105.\n[12] J. S. Milne Arithmetic Duality Theorems , Second Edition, BookSurge LLC, (2006).\n[13] B. Poonen, Heuristics for the Brauer-Manin obstruction for curves , Exp.Math. 15(4), (2006),415–420.\n[14] B. Poonen, Rational points on varieties , Graduate Studies in Mathematics 186, American Mathematical Soci-\nety, Providence, RI, (2017).\n[15] B. Poonen and J. F. Voloch, The Brauer-Manin obstruction for subvarieties of abelian v arieties over function\nfields, Annals of Mathematics 171, 1, (2010) 511–532.\n[16] C. Rivera and B. Viray, Persistence of the Brauer-Manin obstruction for cubic surf aces, Math. Research Letters\n29, (2022), 1881–1889.\n[17] V. Scharaschkin Local-global problems and the Brauer-Manin obstruction Thesis (Ph.D.) - University of Michi-\ngan, ProQuest LLC, Ann Arbor, MI (1999).\n[18] Shatz, Stephen S Cohomology of artinian group schemes over local fields , Ann. of Math. (2) 79(1964), 411–449.\n[19] A. Skorobogatov, Torsors and Rational Points , Cambridge Tracts in Mathematics 144, (2001).\n[20] J. Stix Affine anabelian curves in positive characteristic , Compositio Math. 134(2002), 75–85.\n[21] M.Stoll Finite descent obstructions and rational points on curves , Algebra Number Theory. 1:4(2007) 349-391.\n[22] J. Tate, Genus change in inseparable extensions of function fields , Proc. Amer. Math. Soc. 3, (1952) 400–406.\n[23] H. Wu, Arithmetic of Chˆ atelet surfaces under extensions of base fi elds, The Ramanujan Journal, 62, (2023) ,\n997–1010."    },    {        "title": "2402.04475v1.Quantum_vortex_phases_of_charged_pion_condensates_induced_by_rotation_in_a_magnetic_field.pdf",        "content": "Quantum vortex phases of charged pion condensates induced by rotation in a\nmagnetic field\nTao Guo1and Yanmei Xiao1\n1School of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China\n(Dated: February 8, 2024)\nUsing the relativistic complex scalar field model with a repulsive self-interaction, we discuss the\nground state structure of charged pion condensation under the coexistence of parallel rotation and\nmagnetic field. Our previous study found that the density distribution profile of the condensates\nis a supergiant quantum vortex phase and change with rotational speed and coupling constant.\nIn this work, we further discover vortex lattice structures in the condensates under conditions\nof small rotation and strong coupling constant. This mechanism can be thought of as electrical\nsuperconductivity: Vortex lattices are created to better adapt to changes in rotation and interaction.\nFurthermore, large rotation and weak coupling constant are more likely to cause the vortex lattices\nto be destroyed and form a giant quantum vortex similar to a doughnut. We expect this phenomenon\ncan be observed in the relativistic non-central heavy ion collisions with large rotation and strong\nmagnetic field.\nI. INTRODUCTION\nThe realization of the Bose-Einstein condensate (BEC)\nwith alkali metal elements provides physicists with a huge\nopportunity to study this new state of matter and marks\nthe breakthrough development of modern physics. The\napplications of BEC involve theoretical and experimental\nresearch in many fields, such as ultracold atoms [1–3], su-\nperfluidity and quantum vortices [4–6]. With in-depth re-\nsearch on BEC, physicists have extended traditional con-\ndensed matter physics to relativistic Bose-Einstein con-\ndensates (RBECs), i.e., the condensates are composed of\nrelativistic microscopic constituents [7–9]. As a conse-\nquence, in this era of rapid development of experimental\nfacilities such as relativistic heavy ion collisions, it is of\ngreat physical significance to study the properties and\nforming mechanism of RBECs.\nRotation and magnetic field in general play very im-\nportant roles in large variety of physical environments,\nsuch as rapidly rotating neutron stars [10–12], binary\nblack hole mergers [13, 14] and non-central heavy ion\ncollision experiments [15–17]. Especially, in non-central\nheavy ion collisions, the extreme conditions of strong\nmagnetic fields and high rotational speeds have been real-\nized [18–20]. By analyzing experimental results, it can be\nconcluded that the strong magnetic field generated dur-\ning the collisions can be e B∼m2[21, 22], where e is the\nvalue of an electron charge and mis the mass of a charged\npion. In addition, the numerical simulations indicate that\nthe angular momentum generated in the collisions can in-\nvolve in the range [103,105]ℏ[23, 24]; and the rotational\nangular velocity has reached Ω ≈(9±1)×1021Hz∼\n6.2 MeV [25, 26]. Generally, a strong external magnetic\nfield can enhance a fermion-antifermion condensation for\nleading to generate a fermion dynamical mass [27–29].\nResearch shows that the rotation has similar effect to a\nmagnetic field and can cause certain anomalous transport\nprocesses, for example, chiral vortical effect [30, 31] and\nchiral vortical wave [32, 33]. While, opposite to the mag-netic catalysis effect, the rotation generally suppresses\nthe chiral condensation at finite temperature according\nto effective models [34–36]. However, the lattice QCD\nsimulations at imaginary rotation seem to deny the latter\nfeature, which then cause a lot of debate and discussion\non rotation effect [37–39].\nRecently, many studies have shown that the combi-\nnation of parallel rotation and magnetic field (PRM)\ncan also induce a variety of condensed distributions.\nFor instance, based on the solutions of the Dirac equa-\ntion, the condensed characteristics of free fermionic sys-\ntems in PRM have been discussed [40–43]. Also, for in-\nteracting rotating fermion systems in a magnetic field,\nmore research focuses on the possible effects of the edge\nstates in phase structure [44–47]. Within the three-flavor\nNambu–Jona-Lasinio (NJL) model, the PRM can induce\ncharged rho ( ρ±) superconductor even at a small ro-\ntational angular velocity [48]. The combined effects of\nPRM can make non-interacting charged pions condense\nboth in the vacuum and at finite temperature [49]. How-\never, other calculations show that the charged pion con-\ndensates can only occur under the conditions of a strong\ncoupling constant and negatively large baryon chemical\npotential [50]. In our previous study [51], based on the\nviewpoint of spontaneous symmetry breaking, the results\nshow that the profile of ground state formed by inter-\nacting charged pions in PRM is a supergiant quantum\nvortex. Such possibility was verified by applying the\nGinzburg-Landau analysis in the NJL model [52].\nThis work extends the previous one [51] by looking\ninto a wider range of interaction strength, espacially,\nthe strong case. This paper is organized as follows.\nIn Sec. II, we introduce the relativistic complex scalar\nfield model with a repulsive self-interaction. In Sec. III,\nwe obtain the energy dispersion relation and wave func-\ntion of free charged pion fields by solving the Klein-\nGordon equation, and calculate the ground state density\ndistribution of interacting charged pion fields induced\nby PRM. We summarize in Sec. IV. The nature unitsarXiv:2402.04475v1  [hep-ph]  6 Feb 20242\nc=ℏ=kB= 1 are used throughout.\nII. RELATIVISTIC COMPLEX SCALAR FIELD\nMODEL\nIn order to explore the ground state formation of the\ncharged pion condensates induced by PRM, we adopt the\nrelativistic complex scalar field model with a repulsive\nself-interaction. In the cylindrical coordinates ( r, θ, z ),\nthe action of the relativistic complex scalar field can be\ngiven in a curved spacetime by\nS=Z\ndtZ\nd3xq\n−det(gµν)L(Φ∗,Φ), (1)\nwhere d3x=rdrdθdz , and the Lagrangian density is de-\nfined by\nL= (DµΦ)∗(DµΦ)−m2|Φ|2−g|Φ|4. (2)\nHere the covariant derivative Dµ=∂µ+ieAµ, where Aµ\nis the vector potential. Φ represents a charged complex\nscalar field. gis a coupling constant that reflects the\nstrength of the self-interaction. The spacetime metric\ngµνof the rotating frame reads\ngµν=\n1−r2Ω2yΩ−xΩ 0\nyΩ−1 0 0\n−xΩ 0 −1 0\n0 0 0 −1\n, (3)\nwhere r=p\nx2+y2is the radius of the cylinder sys-\ntem, the notation Ω is the rotational angular velocity,\nandp\n−det(gµν) = 1. It is convenient to rewrite the\nLagrangian density of the system as\nL=|(Dt+ ΩyDx−ΩxDy) Φ|2− |DiΦ|2\n−m2|Φ|2−g|Φ|4. (4)\nHere the Lagrangian density is obviously invariant un-\nder the local U(1) symmetry. Moreover, physical prop-\nerties are generally not influenced by the specific choice\nof gauge. For convenience, we choose the symmetrical\ngauge in the following calculations.\nIn the rotating frame, the Lagrangian density for the\ninteracting charged pion fields in the symmetric gauge\ncan be expressed as\nL=|(∂t−iΩLz)Φ|2− |DiΦ|2−m2|Φ|2−g|Φ|4,(5)\nwhere Lz=−i∂θ=−i(x∂y−y∂x) is the angular momen-\ntum along the z-axis. In the imaginary time formalism,\nthe partition function of the system is given by\nZ=Z\n[dΦ] [dΦ∗] exp Zβ\n0dτZ\nd3xL!\n. (6)\nHere β(= 1/T) is defined as the inverse temperature,\nandτis the imaginary time. By integrating the abovepartition function (6) by parts, we can rewrite the action\nof the system as\nS=−Zβ\n0dτZ\nd3xL=Zβ\n0dτZ\nd3xH, (7)\nwhere the notation Hreads\nH= Φ∗h\n−(∂τ−ΩLz)2i\nΦ\n+ Φ∗\u0012\n−∆ +1\n4e2B2r2−eBLz+m2\u0013\nΦ\n+g|Φ|4. (8)\nThe operator ∆ represents the Laplace operator in the\ncylindrical coordinate frame\n∆≡ ∇2=∂2\nr+1\nr∂r−L2\nz\nr2+∂2\nz. (9)\nIII. GROUND STATE FORMATION OF\nCHARGED PION CONDENSATES\nIf the ground state density distribution of the system\nis a Bose-Einstein condensate, the charged pion field Φ\nacquires a nonzero expectation value. Thus we can de-\ncompose the charged pion field Φ into a classical part\nϕ0(x) and a quantum fluctuation part ϕ(τ,x), i.e.,\nΦ =ϕ0(x) +ϕ(τ,x). (10)\nWe note that in a finite-size system, the condensate\nϕ0(x) is generally inhomogeneous. At zero temperature,\nthe density distribution profile of the charged pion con-\ndensates can be determined by minimizing the Gross-\nPitaevskii-like free energy\nE=Z\nd3x[ϕ∗\n0(x) (−∆ +H)ϕ0(x)]\n+gZ\nd3x|ϕ0(x)|4, (11)\nwhere the operator His defined as\nH=1\n4e2B2r2−eBL z−Ω2L2\nz+m2. (12)\nIn principle, considering that the condensate in the z-\naxis direction is homogeneous, we can simplify the Gross-\nPitaevskii-like free energy as\nK=ER\ndz\n=Z\ndr\u0014\nϕ∗\n0(r)\u0012\n−∂2\nr−1\nr∂r+L2\nz\nr2+H\u0013\nϕ0(r)\u0015\n+gZ\ndr|ϕ0(r)|4, (13)\nwhereR\ndr=R2π\n0dθRR\n0rdrwithr≡(r, θ). And Ris the\nradius of the cylinder cross section.\nIn the following, we define a certain number of con-\nserved charges. The ground state features of free and\ninteracting charged pion fields are studied separately.3\nA. Klein-Gordon equation of free charged pion\nfields\nLet us first solve the more general Klein-Gordon equa-\ntion of free charged pion fields. We consider the charged\npion fields in the cylindrical coordinate with a uniform\nmagnetic field B=B⃗ zand a constant rotation Ω= Ω⃗ z.\nIn this paper, we always take e B > 0 and Ω >0, un-\nless otherwise stated. The Klein-Gordon equation of free\ncharged pion fields in a rotating frame can be given by\n−(∂t−iΩLz)2Φ +D†\niDiΦ−m2Φ = 0 , (14)\nwith the derivative Dt=∂t−ieBΩr2/2,Dx=∂x+\nieBy/2,Dy=∂y−ieBx/2,Dz=∂z. Therefore, the\nsolution for the above equation (14) can be written as\nΦ =e−iεt+ipzzϕnl(r), (15)\nwith\nϕnl(r) =Ceilθϕnl(r). (16)\nThe notation pzis the momentum along the z-direction,\nCis the normalization factor, and lis the azimuthal an-\ngular quantum number.\nIn the infinite-size volume case, substituting (15) to\n(14), we can rewrite the Klein-Gordon equation\nh\n(ε+ ΩLz)2+ ∆−Hi\nϕnl(r) = 0 . (17)\nDue to the need to guarantee the causal conditions of\nrelativity, we must consider the rotation with a finite ve-\nlocity v= Ωr≤1 and the quantization condition justify\nwith r≫1/√\neB. When the first inequality is imposed,\nthe frame does not move faster than light to avoid patho-\nlogical effects of particle spectrum. The second inequal-\nity is satisfied for keeping the wave function localization\nat the boundary. In this way, the interference of some\nnon-physical factors can be eliminated, so that the real\nphysical results can be better obtained. In general, we\nassume that the rotation is rigid, so the rotational an-\ngular velocity Ω does not depend on the distance to the\naxis-of-rotation.\nIn the above case, it is convenient for obtaining the\nradial solution of the equation (17)\nϕnl(r) =r|l|e−1\n4eBr2\n1F1\u0012\n−anl,|l|+ 1,eBr2\n2\u0013\n,(18)\nwhere 1F1is a confluent hypergeometrical function with\nthe parameter\nanl=1\n2eBh\n(ε+ Ωl)2−p2\nz−m2i\n−1\n2(|l| −l+ 1).(19)\nTherefore, we can derive the expression of the energy\ndispersion relationship from (19) as\n(ε+ Ωl)2=p2\nz+m2+ eB(2anl+|l| −l+ 1).(20)Here the parameter anlis the n-th zero point value of the\nconfluent hypergeometric function 1F1when the angular\nquantum number lis given, i.e., anlcan be obtained by\nsolving the equation\n1F1\u0012\n−anl,|l|+ 1,eBr2\n2\u0013\n= 0. (21)\nEspecially, if we consider an infinite system with a ra-\ndius r→ ∞ ,anlis a set of non-negative integers n.\nThe function 1F1can safely be simplified to an associ-\nated Laguerre polynomial. Considering the non-rotation\ncondition again, the energy dispersion relation of the sys-\ntem returns to the Landau levels (LL): ε2=p2\nz+m2+\neB(2n+ 1). It is obvious that the LL with different lis\ndegenerate. However, when a rotational angular veloc-\nity Ω is imposed on the system, the LL produces a shift\n∓Ωl. Here, the angular quantum number of the positive\ncharged particles is land the angular quantum number\nof the negative charged particles is −l. As a result, this\nmeans that the positive charged particles split downward\nand the negative charged particles split upward.\nIn a real physical system, we choose the cylinder with\ncross-sectional radius r=R. And we impose the Dirich-\nlet boundary condition that ensures that the wave func-\ntion of the charged pion fields must vanish at the edge\nof the cylinder. In the disc plane transverse to the rotat-\ning axis ⃗ z, each momentum pzcorresponds to a Landau\ndegeneracy factor\nNf=\u0014eBS\n2π\u0015\n=\u0014eBR2\n2\u0015\n, (22)\nwhere the square bracket [ ···] means a rounding func-\ntion. For the LL to fit into the cross section disc with\nthe area S=πR2, the degeneracies of the LL are identi-\nfied with the z-component of the angular momentum in\nposition space. As a consequence, the possible range of\nthelshould be −n≤l≤Nf−nwhere nlabels the LL.\nAnd when n= 0, it means the lowest Landau level (LLL)\nwith 0 ≤l≤Nf.\nNow we turn the problem to the free charged pion fields\n(i.e., g= 0) in the coexistence of PRM. Compared with\nthe solution of the corresponding Klein-Gordon equation,\nthe global minimum of the Gross-Pitaevskii-like free en-\nergy of the system can be obtained by solving for the\nglobal minimum of Knl, where the Knlreads\nKnl=−Ω2l2+ eB(2anl+|l| −l+ 1) + m2.(23)\nIn order to conveniently consider the relativistic causal-\nity, the above (23) is rewritten as\nEnl=−(ΩR)2l2+ 2Nf(2anl+|l| −l+ 1) + m2\n0,(24)\nwhere Enl=R2Knlandm0=R2m. In this paper, we\ntake the uniform magnetic field e B=m2≈0.5 fm−2.\nThrough the above results, we plot the two lower energy\nspectra of the free charged pion fields in (24) as FIG. 1.\nThe behavior of other higher energy levels is similar.4\n150200250300350400450500550600650700-\n30- 20- 100 05001000150020002500300035000\n1 02 03 0-5005010015020025030035040005001000150020002500300035004000(a)  n = 1(\nb)  n = 0/s937R = 0/s937\nR = 0.1/s937\nR = 0.2/s937\nR = 0.4/s937\nR = 0.6/s8455nl l\nFIG. 1. Energy spectra of the lowest two energy levels ( n= 0 and n= 1 ) as a function of the angular quantum number l\nand different rotational speed Ω R. In this figure we take Nf= 25 and R= 10 fm.\nIt can be clearly concluded from FIG. 1 that the low-\nest energy of the l <0 is much higher than the lowest\nenergy of the l >0. And with the increase of rotational\nangular velocity Ω, the ground state of the free charged\npion system is more inclined to the position with positive\nlarger l. Obviously, this shows that the charged pion con-\ndensates of the positive lmodes are more favorable than\nthe negative lmodes, i.e., the l≥0 modes always cor-\nrespond to the ground state of the system. Specifically,\nthelcorresponding to the ground state is the angular\nquantum number at the global minimum of Knl(orEnl).\nFor example, when the rotational speeds Ω Rare 0, 0 .1,\n0.2, 0.4 and 0 .6, the locations of the lcorresponding to\nthe ground state are 0, 9, 12, 16 and 24, respectively. By\nconsidering (23), we can get that the global minimum of\nthe system depends on the competition between −Ω2l2\nand 2e B(2anl+ 1) + m2. This indicates that Ω lin the\nrotating system plays the role of an effective l-dependent\nchemical potential.B. Quantum vortex phases of interacting charged\npion fields\nAt zero temperature, the minimization of the Gross-\nPitaevskii-like free energy (13) leads to the ground state\nenergy spectra of interacting charged pion fields. The\nequation of motion of interacting charged pion system,\nhowever, is nonlinear and cannot be solved analytically.\nTherefore, we choose the variational method to numeri-\ncally calculate the lowest energy eigenstate of the system.\nConsidering the basic features of interacting charged pion\nfields, the trial wavefunction ϕ0(r) can be expressed by\nthe complete basis vectors composed of the wave func-\ntions of free charged pion fields as\nϕ0(r) =∞X\nn=0∞X\nl=−∞cnlϕnl(r). (25)\nThe variational parameters cnlare determined by mini-\nmizing the Gross-Pitaevskii-like free energy. For a given5\nFIG. 2. In the coexistence of PRM, the ground state density distribution |ϕ0(r)|2of interacting charged pion fields changes\nwith different rotational speeds Ω Rand coupling constants g. Each column has the same Ω R, and each row has the same g.\nThe rotational speeds of different columns increase from left to right and are Ω R= 0.1,0.2,0.4,0.6, respectively. The coupling\nconstants of different rows increase from top to bottom and are g= 0,0.01,0.1,0.3, respectively. In this figure we take Nf= 25\nandR= 10 fm.\nnumber of charged pions, the conserved charges δNis\ndefined as\nδN≡X\nnl\u0000\nNnl−Nnl\u0001\n=X\nnl|cnl|2, (26)\nwhere Nnl\u0000\nNnl\u0001\nis the number of positive pions (nega-\ntive pions). In realistic numerical calculations, we can\nequally choose the probability densityP\nnl|cnl|2= 1.\nSubstituting the trial wavefunction (25) into (13), this\nGross-Pitaevskii-like free energy can be written as the\nsum of two parts: K=K0+Kint. Here we take\nthe non-interacting part K0=P∞\nn=0P∞\nl=−∞|cnl|2εnl\nand the interacting part Kint=gR\ndr|ϕ0(r)|4. Accord-\ning to the dispersion relationship (23), we can obtainεnl=−Ω2l2+ eB(2anl+ 1) + m2. Obviously, Kis the\nsuperposition of the quadratic and quartic term. The\nminimum of the system still depends mainly on the com-\npetition between −Ω2l2and e B(2anl+ 1) + m2.\nBy doing complicated numerical solution, we get the\nglobal minimum (i.e., the ground state energy of inter-\nacting charged pions) of the Gross-Pitaevskii-like free\nenergy (13) and the corresponding ground state ϕ0(r).\nWithout loss of generality, we take four different rota-\ntional speeds Ω Rand four different coupling constants\ng. The ground state phase diagrams of the interacting\ncharged pion condensates are shown in FIG. 2. It should\nbe noted that we did not plot ground state density dis-\ntribution for the Ω = 0 here. Because when Ω = 0 and6\ng= 0, the interacting charged pion fields are restored\nto the free non-rotational charged pion fields. Profile of\nthe condensate shows that almost all charged pions con-\ndense in the center (i.e., the state with l= 0) of the disc.\nThis phenomenon is similar to the traditional BEC, in\nwhich all bosons occupy the same quantum state to form\na macroscopic observable. When Ω = 0 and g̸= 0, the\nground state density distribution is diffuse in the two-\ndimensional plane. In FIG. 2, it can be found that the\nprofiles of these charged pion ground states are various\nquantum vortex phases. And some of these condensates\nare characterized by the creation of a certain number of\nvortex lattices. These vortex lattices are similar to small\ntwisters inside the flowing liquids.\nIn the absence of interaction ( g= 0, see the first row in\nFIG. 2), the profile of the condensates will gradually ex-\npand from the center to the edge position with increasing\nof the Ω R. However, it is worth noting that this expan-\nsion change is not continuous, but quantized. Besides,\npure rotation cannot effectively induce multiple vortex\nlattices, but makes the charged pions condense into a\nspecific single particle state with a determined l. As pre-\nviously analyzed, the rotation causes a shift in the ground\nstate energy level. Specifically, the ground state changes\nfrom εtoε−Ωlcorresponding to the angular quantum\nnumber from l= 0 to l̸= 0, respectively. Therefore,\nthe profile of the charged pion condensates is actually a\nsingle quantum vortex state when Ω R̸= 0 and g= 0.\nThe radius of the single quantum vortex depends on the\nrotational angular velocity. Quantum vortices generally\nhave unique intrinsic angular momentum that is differ-\nent from spin. In a cylinder, the wave function of the\nquantum vortex states can be written as\nϕ0(r) =η(r)eilcθ, (27)\nwhere η(r) is just a r-dependent function, and η(r)∝\nr|lc|.lcis a non-negative integer and is also defined as\nthe winding number (or topological charge) in the vortex\nstate. Theoretically, lcis proportional to the angular mo-\nmentum quantum number. From the first row in FIG. 2,\nwe can find that the larger Ω R, the larger lc. Numerical\ncalculations show that lc=l, when the rotating charged\npions are in the ground state. In particular, the results\nshow that rotation has a certain catalytic effect in a mag-\nnetic field. This allows the non-interacting charged pions\nto form a single giant quantum vortex with a winding\nnumber lc≫1 induced by PRM.\nIn the following, we turn our attention to the system\nof interacting charged pion fields ( g̸= 0, see the rows 2-4\nin FIG. 2). We find that the formation of vortex lattices\nresults from conditions of small rotation and strong cou-\npling constant. In a uniform magnetic field, the possible\nreason for the formation of vortex lattices is to trigger\nelectrical superconductivity to adapt to the combined ef-\nfects of interaction and rotation. The number of vortex\nlattices reflects the weight of free charged pion states with\ndifferent l. It can generally be represented by the values\nof the non-vanishing parameters cnlin (25). For exam-ple, when g= 0.1 and Ω R= 0.1 (i.e., the subfigure in\nthe third row and first column in FIG. 2), there are two\nrings of vortex lattices in the ground state. The non-\nvanishing parameters cnlarec0l= (0.144,0.516,0.835)\nforl= (0,2,11), respectively. In this quantum vortex\nstructure, it is shown that there are two vortex lattices\nin the inner ring, nine vortex lattices in the outer ring\nand the total number is eleven. This result accurately\nshows that the maximum lcontained in the ground state\nis the total number of vortex lattices. In particular, the\nfeatures of other subfigures in FIG. 2 are similar.\nLarge rotation and weak coupling constant are more\nlikely to cause the vortex lattices to be destroyed and\nform a new single giant quantum vortex. As the Ω R\ncontinues to increase, more charged pions are condensed\non the edge of the disc to form a doughnut-like struc-\nture. This change demonstrates that the larger rota-\ntion destroys the equilibrium state initially formed in the\ncondensates, prompting the transformation of multiple\nvortex lattices into a single giant quantum vortex state.\nThe reason for this transformation process is probably\nthe effect of centrifugal force caused by large rotation.\nAnd the single giant quantum vortex state has a certain\nwinding number lc. For example, when Ω R= 0.6 (see\nthe fourth column in FIG. 2), the winding numbers are\nlc= 24 ,24,24,23 for g= 0,0.01,0.1,0.3, respectively.\nHere, the small difference in the lcresults from the cou-\npling of adjacent energy levels due to the existence of an\ninteraction. At this point, the interaction effect is sig-\nnificantly smaller compared to rotation. Obviously, this\nsuggests that rotation and interaction form an entangle-\nment effect in the charged pion condensates.\nThe profile of the condensates may be even richer if we\nconsider dynamic electromagnetic fields. As we all know,\nthe rotation and magnetic field have similar effects. And\nthe effect of rotation applies not only to charged parti-\ncles, but also to neutral particles. For free charged pion\nfields, the rotation causes a change in the ground state of\nthe system from l= 0 to l=lc. This change is generally\nexpressed as an effective l-dependent chemical potential.\nFor interacting charged pion fields, the combined effect\nof rotation and magnetic field may produce more mean-\ningful ground state density distribution with the various\nvortex lattice structures. These provide us with a deeper\nunderstanding of the evolution of condensates induced by\nPRM. Of course, some signals of QGP and CME can also\nbe studied through the properties of vortex phase formed\nin the condensates. Therefore, it is of principal interest\nthat we look forward to more quantitative studies in the\nnext work.\nIV. SUMMARY\nIn PRM, we calculate the ground state density distri-\nbution of charged pions formed under a wider range of\ninteractions and rotational speeds. The method is based\non the relativistic complex scalar field model with a re-7\npulsive self-interaction and then solved by variational cal-\nculations. We conclude that a certain number of vortex\nlattices are found in the condensates under conditions of\nsmall rotation and strong coupling constant.\nOur calculation results show that for free charged pion\nfields, the condensate always tends to angular quantum\nnumber l≥0 modes. And as the rotational angular\nvelocity Ω increases, the corresponding lof the ground\nstate increases. This indicates that Ω lin the rotating\nsystem plays the role of an effective l-dependent chemi-\ncal potential. For interacting charged pion fields, we find\nthat as coupling constant increases, the vortex lattices\nare generated in the ground state of charged pions. In\nthis case, this mechanism can be thought of as electrical\nsuperconductivity. These vortex lattices are created to\nbetter adapt to changes in rotation and interaction. Un-\nder conditions of large rotation and weak coupling con-\nstant, the vortex lattices inside disc tends to be destroyed\nand more pions condense on the edge of the disc to form\na doughnut-like structure. This phenomenon is similar to\nthe condensate of free charged pions in a finite-size cylin-\nder system with a large rotation, and the reason can be\nconsidered to be the result of centrifugal force. Moreover,these vortex structures are theoretically observable.\nFinally, a natural extension of the present paper will be\na more self-consistent and realistic research of the profile\nof charged pion condensates (i.e., the condensed distribu-\ntion under dynamic electromagnetic fields, the pion su-\nperfluid in non-central heavy-ion collisions), which may\nlead to more discussions about the features of the RBECs\nin a finite-size rotating system. We hope the measure-\nment of multipion correlations can help us to check the\nformation and evolution of quantum vortex phases in the\ncharged pion condensates under parallel magnetic field\nand rotation.\nACKNOWLEDGMENTS\nThe authors would like to thank Fan Wu for helpful\ncomments and discussions. This work was supported\nby the Scientific Research Foundation of Chengdu Uni-\nversity of Technology under grant No.10912-KYQD2022-\n09557.\n[1] M. Anderson, J. Ensher, M. Matthews, C. Wieman, and\nE. Cornell, Observation of Bose-Einstein condensation in\na dilute atomic vapor, Science 269, 198 (1995).\n[2] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.\nHulet, Evidence of bose-einstein condensation in an\natomic gas with attractive interactions, Phys. Rev. Lett.\n75, 1687 (1995).\n[3] A. Tononi and L. Salasnich, Bose-einstein condensation\non the surface of a sphere, Phys. Rev. Lett. 123, 160403\n(2019).\n[4] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S.\nHall, C. E. Wieman, and E. A. Cornell, Vortices in a bose-\neinstein condensate, Phys. Rev. Lett. 83, 2498 (1999).\n[5] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dal-\nibard, Vortex formation in a stirred bose-einstein con-\ndensate, Phys. Rev. Lett. 84, 806 (2000).\n[6] S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur,\nA. G¨ orlitz, T. L. Gustavson, A. E. Leanhardt, D. E.\nPritchard, and W. Ketterle, Observation of vortex phase\nsingularities in bose-einstein condensates, Phys. Rev.\nLett. 87, 080402 (2001).\n[7] R. Beckmann, F. Karsch, and D. E. Miller, Bose-einstein\ncondensation of a relativistic gas in ddimensions, Phys.\nRev. Lett. 43, 1277 (1979).\n[8] M. Grether, M. de Llano, and G. A. Baker, Bose-einstein\ncondensation in the relativistic ideal bose gas, Phys. Rev.\nLett. 99, 200406 (2007).\n[9] S. Fagnocchi, S. Finazzi, S. Liberati, M. Kormos, and\nA. Trombettoni, Relativistic bose–einstein condensates:\na new system for analogue models of gravity, New Journal\nof Physics 12, 095012 (2010).\n[10] G. B. Cook, S. L. Shapiro, and S. A. Teukolsky, Rapidly\nrotating neutron stars in general relativity: Realistic\nequations of state, Astrophys. J. 424, 823 (1994).[11] M. Bocquet, S. Bonazzola, E. Gourgoulhon, and J. No-\nvak, Rotating neutron star models with magnetic field,\nAstron. Astrophys. 301, 757 (1995).\n[12] N. Andersson, A new class of unstable modes of rotat-\ning relativistic stars, The Astrophysical Journal 502, 708\n(1998).\n[13] P. Marronetti, W. Tichy, B. Bruegmann, J. Gonzalez,\nand U. Sperhake, High-spin binary black hole mergers,\nPhys. Rev. D 77, 064010 (2008).\n[14] D. Pook-Kolb, O. Birnholtz, B. Krishnan, and E. Schnet-\nter, Interior of a Binary Black Hole Merger, Phys. Rev.\nLett. 123, 171102 (2019).\n[15] F. Becattini, F. Piccinini, and J. Rizzo, Angular momen-\ntum conservation in heavy ion collisions at very high en-\nergy, Phys. Rev. C 77, 024906 (2008).\n[16] F. Becattini, G. Inghirami, V. Rolando, A. Beraudo,\nL. Del Zanna, A. De Pace, M. Nardi, G. Pagliara, and\nV. Chandra, A study of vorticity formation in high en-\nergy nuclear collisions, Eur. Phys. J. C 75, 406 (2015),\n[Erratum: Eur.Phys.J.C 78, 354 (2018)].\n[17] Y. Jiang, Z. Lin, and J. Liao, Rotating quark-gluon\nplasma in relativistic heavy ion collisions, Phys. Rev.\nC94, 044910 (2016), [Erratum: Phys.Rev.C 95, 049904\n(2017)].\n[18] V. Voronyuk, V. D. Toneev, W. Cassing, E. L.\nBratkovskaya, V. P. Konchakovski, and S. A. Voloshin,\n(Electro-)Magnetic field evolution in relativistic heavy-\nion collisions, Phys. Rev. C 83, 054911 (2011).\n[19] A. Bzdak and V. Skokov, Event-by-event fluctuations of\nmagnetic and electric fields in heavy ion collisions, Phys.\nLett. B 710, 171 (2012).\n[20] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang,\nChiral magnetic and vortical effects in high-energy nu-\nclear collisions—A status report, Prog. Part. Nucl. Phys.8\n88, 1 (2016).\n[21] K. Tuchin, Particle production in strong electromagnetic\nfields in relativistic heavy-ion collisions, Adv. High En-\nergy Phys. 2013 , 490495 (2013).\n[22] M. Abdallah et al. (STAR), Search for the chiral mag-\nnetic effect with isobar collisions at√sNN=200 GeV by\nthe STAR Collaboration at the BNL Relativistic Heavy\nIon Collider, Phys. Rev. C 105, 014901 (2022).\n[23] W. Deng and X. Huang, Vorticity in Heavy-Ion Colli-\nsions, Phys. Rev. C 93, 064907 (2016).\n[24] F. Becattini and M. A. Lisa, Polarization and Vorticity\nin the Quark–Gluon Plasma, Ann. Rev. Nucl. Part. Sci.\n70, 395 (2020).\n[25] L. Adamczyk et al. (STAR), Global Λ hyperon polariza-\ntion in nuclear collisions: evidence for the most vortical\nfluid, Nature 548, 62 (2017).\n[26] J. Adam et al. (STAR), Polarization of Λ ( ¯Λ) hyperons\nalong the beam direction in Au+Au collisions at√sNN\n= 200 GeV, Phys. Rev. Lett. 123, 132301 (2019).\n[27] K. G. Klimenko, Three-dimensional Gross-Neveu model\nat nonzero temperature and in an external magnetic field,\nZ. Phys. C 54, 323 (1992).\n[28] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy,\nCatalysis of dynamical flavor symmetry breaking by a\nmagnetic field in 2 + 1 dimensions, Phys. Rev. Lett. 73,\n3499 (1994).\n[29] V. A. Miransky and I. A. Shovkovy, Quantum field the-\nory in a magnetic field: From quantum chromodynamics\nto graphene and Dirac semimetals, Phys. Rept. 576, 1\n(2015).\n[30] D. Kharzeev and A. Zhitnitsky, Charge separation in-\nduced by P-odd bubbles in QCD matter, Nucl. Phys. A\n797, 67 (2007).\n[31] D. E. Kharzeev and D. T. Son, Testing the chiral mag-\nnetic and chiral vortical effects in heavy ion collisions,\nPhys. Rev. Lett. 106, 062301 (2011).\n[32] Y. Jiang, X.-G. Huang, and J. Liao, Chiral vortical wave\nand induced flavor charge transport in a rotating quark-\ngluon plasma, Phys. Rev. D 92, 071501 (2015).\n[33] K. Kamada, N. Yamamoto, and D.-L. Yang, Chiral ef-\nfects in astrophysics and cosmology, Prog. Part. Nucl.\nPhys. 129, 104016 (2023).\n[34] Y. Jiang and J. Liao, Pairing Phase Transitions of Matter\nunder Rotation, Phys. Rev. Lett. 117, 192302 (2016).\n[35] S. Ebihara, K. Fukushima, and K. Mameda, Boundary\neffects and gapped dispersion in rotating fermionic mat-\nter, Phys. Lett. B 764, 94 (2017).[36] Y. Chen, D. Li, and M. Huang, Inhomogeneous chiral\ncondensation under rotation in the holographic QCD,\nPhys. Rev. D 106, 106002 (2022).\n[37] V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev, and\nA. A. Roenko, Influence of relativistic rotation on the\nconfinement-deconfinement transition in gluodynamics,\nPhys. Rev. D 103, 094515 (2021).\n[38] J. Yang and X. Huang, QCD on Rotating Lattice with\nStaggered Fermions, (2023), arXiv:2307.05755 [hep-lat].\n[39] G. Cao, Effects of imaginary and real rotations on QCD\nmatters, Phys. Rev. D 109, 014001 (2024).\n[40] B. R. Iyer, Dirac field theory in rotating coordinates,\nPhys. Rev. D 26, 1900 (1982).\n[41] H. Chen, K. Fukushima, X. Huang, and K. Mameda,\nAnalogy between rotation and density for Dirac fermions\nin a magnetic field, Phys. Rev. D 93, 104052 (2016).\n[42] Y. Liu and I. Zahed, Rotating Dirac fermions in a mag-\nnetic field in 1+2 and 1+3 dimensions, Phys. Rev. D 98,\n014017 (2018).\n[43] K. Fukushima, T. Shimazaki, and L. Wang, Mode de-\ncomposed chiral magnetic effect and rotating fermions,\nPhys. Rev. D 102, 014045 (2020).\n[44] B. McInnes, A rotation/magnetism analogy for the\nquark–gluon plasma, Nucl. Phys. B 911, 173 (2016).\n[45] K. Hattori and Y. Yin, Charge redistribution from\nanomalous magnetovorticity coupling, Phys. Rev. Lett.\n117, 152002 (2016).\n[46] M. N. Chernodub and S. Gongyo, Interacting fermions in\nrotation: chiral symmetry restoration, moment of inertia\nand thermodynamics, JHEP 01, 136, arXiv:1611.02598\n[hep-th].\n[47] K. Fukushima, Extreme matter in electromagnetic fields\nand rotation, Prog. Part. Nucl. Phys. 107, 167 (2019).\n[48] G. Cao, Charged rho superconductor in the presence of\nmagnetic field and rotation, Eur. Phys. J. C 81, 148\n(2021).\n[49] Y. Liu and I. Zahed, Pion condensation by rotation in a\nmagnetic field, Phys. Rev. Lett. 120, 032001 (2018).\n[50] H. Chen, X. Huang, and K. Mameda, Do charged pi-\nons condense in a magnetic field with rotation? (2019),\narXiv:1910.02700 [nucl-th].\n[51] T. Guo, J. Li, C. Mu, and L. He, Formation of a su-\npergiant quantum vortex in a relativistic Bose-Einstein\ncondensate driven by rotation and a parallel magnetic\nfield, Phys. Rev. D 106, 094010 (2022).\n[52] G. Cao and L. He, Rotation induced charged pion con-\ndensation in a strong magnetic field: A Nambu–Jona-\nLasino model study, Phys. Rev. D 100, 094015 (2019)."    },    {        "title": "2402.04487v1.Item_Level_Heterogeneous_Treatment_Effects_of_Selective_Serotonin_Reuptake_Inhibitors__SSRIs__on_Depression__Implications_for_Inference__Generalizability__and_Identification.pdf",        "content": "Item-Level Heterogeneous Treatment Effects of Selective\nSerotonin Reuptake Inhibitors (SSRIs) on Depression:\nImplications for Inference, Generalizability, and\nIdentification\nJoshua B. Gilbert1, Fredrik Hieronymus2, Elias Eriksson2, and Benjamin W.\nDomingue3\n1Harvard Graduate School of Education\n2University of Gothenburg\n3Stanford Graduate School of Education\nAbstract\nObjectives: In analysis of randomized controlled trials (RCTs) with patient-reported\noutcome measures (PROMs), Item Response Theory (IRT) models that allow for het-\nerogeneity in the treatment effect at the item level merit consideration. These models\nfor “item-level heterogeneous treatment effects” (IL-HTE) can provide more accurate\nstatistical inference, allow researchers to better generalize their results, and resolve\ncritical identification problems in the estimation of interaction effects. In this study, we\nextend the IL-HTE model to polytomous data and apply the model to determine how\nthe effect of selective serotonin reuptake inhibitors (SSRIs) on depression varies across\nthe items on a depression rating scale.\nMethods: We first conduct a Monte Carlo simulation study to assess the performance\nof the polytomous IL-HTE model under a range of conditions. We then apply the\nIL-HTE model to item-level data from 28 RCTs measuring the effect of SSRIs on de-\npression using the 17-item Hamilton Depression Rating Scale (HDRS-17) and estimate\npotential heterogeneity by subscale (HDRS-6).\nResults: Our results show that the IL-HTE model provides more accurate statistical\ninference, allows for generalizability of results to out-of-sample items, and resolves iden-\ntification problems in the estimation of interaction effects. Our empirical application\nshows that while the average effect of SSRIs on depression is beneficial (i.e., negative)\nand statistically significant, there is substantial IL-HTE, with estimates of the standard\ndeviation of item-level effects nearly as large as the average effect. We show that this\nsubstantial IL-HTE is driven primarily by systematically larger effects on the HDRS-6\nsubscale items.\n1arXiv:2402.04487v1  [stat.ME]  7 Feb 2024Conclusions: The IL-HTE model has the potential to provide new insights for the\ninference, generalizability, and identification of treatment effects in clinical trials using\npatient reported outcome measures.\nKeywords : causal inference, heterogeneous treatment effects, item response theory,\ndepression, SSRIs\n21 Introduction\nHeterogeneous treatment effects (HTE) are crucial for epidemiological research and public\nhealth policy because understanding for whom and under what conditions a medical treatment\nworks allows policy makers to best target treatments to populations or subgroups that would\nmost benefit (Beghi et al., 2011; Cordero & Dans, 2021; Kent et al., 2016; Lesko et al., 2018;\nRobertson et al., 2021; Varadhan et al., 2013). One limitation of standard statistical methods\nfor HTE analysis is that they focus on person characteristics (e.g., age, gender, etc.) and\nmay ignore the potential HTE that exists among the items used to measure a latent variable\nof interest. That is, many outcomes relevant to clinical trials or epidemiological research\ncan only be assessed indirectly through multi-item surveys or psychometric instruments,\nsuch as patient-reported outcome measures (PROMs) for well-being (McEvoy et al., 2011),\ndepression (Hieronymus et al., 2019; Sajobi et al., 2023), perceptions of hearing loss (Jessen\net al., 2018), pain (Dworkin et al., 2009), or recovery after childbirth (Sultan et al., 2020,\n2021), in contrast to simple biometric measures that can be measured to arbitrary levels of\nprecision and captured in a single number (e.g., height, weight, blood pressure, mortality,\netc.). Therefore, when we use a PROM to construct a sum or factor score to serve as an\noutcome measure in clinical trials or epidemiological studies, we may be ignoring potential\nHTE that exists among the individual items of the PROM. As a result, our understanding of\nthe consistency or generalizability of treatment effects may be limited (Ahmed et al., 2023;\nGilbert, Kim, & Miratrix, 2023; Sales et al., 2021), and—when there is interest in treatment\nby baseline covariate interactions—create causal identification challenges that can only be\nresolved by leveraging item-level data (Gilbert, Miratrix, et al., 2023).\nTo address these limitations, recent work applies techniques from Item Response Theory\n(IRT; Van der Linden, 2017) to allow for the assessment of treatment effects that vary at\nthe outcome item level (Ahmed et al., 2023; Gilbert, Kim, & Miratrix, 2023; Sales et al.,\n2021). These techniques allow us to determine whether treatment effects are consistent and\n1impact all PROM items equally or, alternatively, vary across the items within the PROM. To\nour knowledge, however, the “item-level heterogeneous treatment effects” (IL-HTE) model\nhas not yet been applied beyond standardized tests in education research; clinical trial or\nepidemiological studies that similarly rely on latent variable outcomes such as PROMs may\nalso benefit from such approaches. Our work therefore builds on the previous interest in the\naffordances of item-level analysis in clinical trials and epidemiological research (Andresen\net al., 2013; Barger, 2023; Chan et al., 2004; Grayson et al., 2000; Hieronymus et al., 2019;\nJones, 2019).\nOur study pursues two aims. First, we apply the IL-HTE model to polytomous item\nresponse data, thereby extending past analyses limited to dichotomous (i.e., correct vs.\nincorrect) item responses. Polytomous data is of interest in epidemiological settings given\nthe widespread use of, for example, Likert scales (Capuano et al., 2016; Esterman, 2003).\nSecond, we apply the IL-HTE model to a clinical trial context using item-level data from a\nset of 28 randomized controlled trials (RCTs) evaluating the effect of SSRIs on the 17-item\nHamilton Depression Rating Scale (HDRS-17). The study is organized as follows. In Section\n1.1, we contextualize the IL-HTE model within the broader context of HTE analysis, for both\ndichotomous and polytomous responses. In Section 2, we describe our Monte Carlo simulation\ndesign and our empirical data. In Section 3, we present the results of the simulation and the\nresults of our models fit to the empirical data. We conclude in Section 4 with a discussion of\nthe implications of our findings for HTE analysis in clinical trials and epidemiology.\n1.1 Estimating Heterogeneous Treatment Effects\nConsider the following standard regression model for HTE:\nYj=β0+β1Tj+β2Xj+β3Tj×Xj+ej (1)\nej∼N(0, σθ), (2)\n2in which Yjis the outcome variable for individual j,Tjis a dichotomous indicator for\nrandomized treatment assignment, and Xjis a person characteristic (e.g., age, gender, etc.).\nβ1is the conditional average treatment effect (CATE) when X= 0,β2is the main effect of\nXwhen T= 0, and β3is our HTE parameter, capturing how the CATE depends on the\nlevel of X. When β3= 0, the ATE is constant across all level of X; when β3>0, the ATE\nis larger at higher levels of X. A concrete example of such a model in epidemiology would\nbe how the effect of COVID vaccinations ( T) on mortality ( Y) varies by patient age ( X)\n(Collier et al., 2021; Faro-Viana et al., 2022).\nWhen Yjis a latent outcome such as a PROM, such as a depression rating scale with\nmultiple items reflecting various symptoms of depression, the treatment may differentially\naffects individual symptoms or symptom clusters represented by the individual items of\nthe PROM. The standard HTE model above represents one extreme based on analysis of\na single number such as the sum score as the outcome (a common practice; Flake et al.,\n2017; McNeish and Wolf, 2020) that ignores all such differentiation. On the other extreme,\nresearchers could analyze treatment effects on each item separately, but this approach is\ndifficult to interpret and suffers from multiple comparisons problems, particularly when the\nnumber of items is large. As a compromise, researchers examine effects on subscales or item\nclusters, but this approach requires an a priori specification of which subscales to evaluate\nand the assumption that within each subscale, the item effects are constant (Gilbert, 2023b),\nand as such is somewhat ad hoc.\nAn elegant solution that leverages the PROM item responses directly and thus makes use\nof all available information without the need to compute total or subscale scores in a separate\nstep of the analysis is the Explanatory Item Response Model (EIRM) (De Boeck et al., 2016;\nPetscher et al., 2020; Wilson & De Boeck, 2004; Wilson et al., 2008). For example, when\nitems are dichotomous (e.g., 0 = symptom absence, 1 = symptom presence), we can use a\n3cross-classified logistic regression model with a main effect for treatment, such as,\nlogit( Yij= 1) = ηij=θj+bi (3)\nθj=β0+β1Tj+ej (4)\nbi=bi (5)\nbi∼N(0, σb) (6)\nej∼N(0, σθ), (7)\nwhere Yijis the response of person jto item i,θjis the unobserved or latent person trait (e.g.,\ndepression), and biis item location (i.e., “easiness” in educational measurement). The EIRM\nis equivalent to a one-parameter logistic (1PL) or Rasch IRT model when the item location\nparameters are considered fixed (De Boeck, 2008). β1represents the ATE, but estimated\ndirectly on the latent trait θjwithout the need to compute a sum or factor score to be used\nas an outcome in a two-step analysis (Christensen, 2006; Gilbert, 2023a, 2023b, 2024; Gilbert,\nKim, & Miratrix, 2023; Zwinderman, 1991). Even without considering the possibility of\nIL-HTE, the EIRM still provides some benefits over a sum score analysis as it can be more\nrobust to violations of model assumptions such as heteroskedasticity or missing data (Gilbert,\n2024) and can provide unbiased estimates of standardized effect sizes that are attenuated by\nmeasurement error (Gilbert, 2023a; Hedges, 1981).\nWe can allow for IL-HTE by introducing a random interaction between treatment and\nitem in a random slope term in the equation for bij, where the added subscript jallows for\nseparate bparameters for each treatment group (Ahmed et al., 2023; Gilbert, 2023b; Gilbert,\nKim, & Miratrix, 2023; Gilbert, Miratrix, et al., 2023; Sales et al., 2021):\n4logit( Yij= 1) = ηij=θj+bij (8)\nθj=β0+β1Tj+ej (9)\nbij=bi+ζiTj (10)\n\nbi\nζi\n∼N(0,\nσbρ\nρ σ ζ\n) (11)\nej∼N(0, σθ). (12)\nHere, β1still represents the ATE on the latent trait θj, but now for the average item on\nthe scale. Item-specific residual treatment effects are represented by ζi, the item-specific\ndeviation from β1. Ifζi>0, then item iis more affected by treatment than the average item.\nTheζiare equivalent to uniform differential item function (DIF) caused by the treatment\n(Gilbert, Kim, & Miratrix, 2023; Montoya & Jeon, 2020). σζprovides a direct parameter\nestimate for the degree of IL-HTE in the data by providing the standard deviation (SD) of\nthe item-specific treatment effects around the ATE β1andρindexes the correlation between\nitem location and item-specific treatment effect size. That is, if ρ >0, then items representing\nmore commonly endorsed symptoms show systematically larger or smaller treatment effects.\nρmay be of interest in itself (Gilbert, Kim, & Miratrix, 2023, pp. 893-894), but is perhaps\nmost critical in that ρ̸= 0 can create a causal identification problem. That is, ρ̸= 0 can\ninduce spurious treatment by baseline covariate interaction effects that can only be resolved\nwith item-level analysis (Gilbert, Miratrix, et al., 2023), a result with critical implications for\nthe analysis of HTE that we will return to in our simulations, particularly given the great\ninterest in HTE by baseline severity in depression treatments (Fournier et al., 2010; Kirsch\net al., 2008). A directed acyclic graph (DAG) (Glymour, 2006; Greenland et al., 1999; Joffe\net al., 2012; Tennant et al., 2021) representation of the IL-HTE model is presented in Figure\n1.\n5Figure 1: Directed Acyclic Graph of the IL-HTE Model\nNotes: Squares indicate observed variables, hollow circles indicate latent variables, and solid circles represent\ncross product interaction terms. Inare item responses and Tjis the treatment indicator. β1represents the\naverage treatment effect. ρrepresents the correlation between item location and item-specific treatment effect\nsize. Path coefficients are fixed at 1 unless otherwise indicated.\n6A large value of σζsuggests that there is substantial IL-HTE in the data. We can test\nwhether item features can explain the IL-HTE by interacting item properties (e.g., item\nsubscale, item type, item modality, etc.) with the treatment indicator to explain some of the\nIL-HTE. For example, we can extend Eqn. 10 as follows:\nθj=β0+β1Tj+ej (13)\nbij=bi+γ1Si+γ2Si×Tj+ζiTj. (14)\nHere, Siis an indicator for whether item iis part of subscale S. Accordingly, β1represents\nthe CATE for items in the reference set (assumed here to be impacting θjdirectly), γ1is\na main effect for item location, and γ2provides the difference in the CATE for items on\nsubscale S. For example, education researchers have used the subscale effects IL-HTE model\nto look at treatment effects on items related to different reading comprehension passages\n(Gilbert, 2023b; Gilbert, Kim, & Miratrix, 2023; J. S. Kim et al., 2023).\nApplication of the EIRM to polytomous responses is less common, but it is straightforward\nto extend the IL-HTE model to polytomous data. In short, we can leverage the computational\nmachinery of binary logistic regression to fit polytomous models by reshaping the data to\nrepresent pairwise contrasts between the ordered response categories. We then fit a logistic\nregression model that includes fixed effects for item threshold parameters representing the\nboundaries between each cut point on the scale (Bulut et al., 2021). If we assume that\nthe distances between thresholds are equal across items, the result is a Rating Scale Model\n(RSM). In contrast, if we assume that each item has a unique distance between each threshold,\nthe result is a Partial Credit Model (PCM). While such models are typically implemented\nwith fixed item and threshold parameters, they can be extended to the random item case\nthat better allows for IL-HTE modeling (J. Kim & Wilson, 2020). For readers interested\nin applying these models to their own data sets, our references provide various tutorials in\n7the R programming language: the general EIRM for dichotomous items (De Boeck et al.,\n2011), extending the EIRM to polytomous items (Bulut et al., 2021), the IL-HTE model for\ndichotomous items (Gilbert, 2023b), and a Bayesian approach that allows for extensions such\nas 2PL models or the Graded Response Model (GRM) (B¨ urkner, 2021; Gilbert, 2023b).\n2 Methods\n2.1 Monte Carlo Simulation\nWe use Monte Carlo simulation to examine the performance of the IL-HTE model applied\nto polytomous item responses. Previous simulation studies of the IL-HTE model using\ndichotomous item responses have demonstrated two key results. First, in terms of statistical\ninference, they show that substantial IL-HTE inflates the standard error of the ATE ( SE(ˆβ1))\nbut does not cause bias (Gilbert, Kim, & Miratrix, 2023). Second, in terms of causal\nidentification, they show that the correlation ρbetween item location biand item-specific\ntreatment effect size ζiinduces spurious treatment by covariate interaction effects, which is\ncritical because it suggests that standard HTE analysis of composite outcomes (e.g., Eqn.\n1) can be misleading (Gilbert, Miratrix, et al., 2023). We hypothesize that same pattern\nof results will apply to polytomous data, because ordered logit models are approximately\ninvariant to the collapsing of response categories (Steele, 2011, p. 13).\nWe simulate polytomous item responses from the RSM and fit each model with the glmer\nfunction from the lme4 package in R with fixed item thresholds and random item location\nparameters (Bulut et al., 2021; De Boeck et al., 2011; Gilbert, 2023b; J. Kim & Wilson,\n2020). We fit two sets of simulations, one to examine the effect of IL-HTE on ATEs and\nassociated SEs (i.e., replicating and extending Gilbert, Kim, and Miratrix, 2023), and the\nother to examine the effect of ρon treatment by covariate interaction effects (i.e., replicating\nand extending Gilbert, Miratrix, et al., 2023). We fix the following parameters across our\n8simulations: 500 patients, 20 items, an ATE of .20 on the logit scale, σb= 1,σθ=.5, and a\nbaseline covariate with a coefficient of 1.\nIn our first set of simulations, we vary the following factors in a fully crossed 3x3 design:\nthe number of categories kat 3, 5, 7 and σζat 0, .2, .4 SDs representing no, moderate,\nand large IL-HTE. We fit two models to each simulated data set, one assuming constant\ntreatment effects, the other allowing for IL-HTE. In our second set of simulations, we fix\nk= 3 and σζ=.4 and vary ρfrom -1 to 1 in increments of .25 to determine how an estimated\ntreatment by baseline covariate interaction becomes biased when we assume a constant effects\nmodel. We fit two models to each simulated data set, one allowing for a treatment by baseline\ncovariate interaction but constant item effects, the other allowing for both the interaction\nand IL-HTE. We repeat the process 200 times per condition.\n2.2 Empirical Application\n2.2.1 The Hamilton Depression Rating Scale (HDRS)\nThe Hamilton Depression Rating Scale (HDRS) has been the de facto“gold standard” in\nantidepressant research for over half a century (Ruh´ e et al., 2005). The HDRS has its origins\nin the 1950s in England and has since been applied worldwide (Hamilton, 1960; Obeid et al.,\n2018). The most commonly used version of the HDRS (which we use here) consists of 17\npolytomous (3- and 5-category) items (Williams, 2001) (HDRS-17).\nThere is a long history of psychometric analysis of the HDRS-17 and derivative measures,\nincluding questioning its psychometric properties (Bagby et al., 2004) or focusing on the\ndimensionality and applying various techniques aimed at distinguishing an HDRS subscale\nwhich can function as a unidimensional measure of depressive severity across different\npopulations and treatments (Bech et al., 1975; Bech et al., 1981; Gibbons et al., 1993). Prior\nanalyses of item-level HDRS data include the reliability of subscales or items (Luckenbaugh\net al., 2015), treatment heterogeneity by baseline depression severity (Hieronymus et al.,\n9Table 1: HDRS-17 Items\nItem Number Subject HDRS-6 Item Range\n1 Depressed Mood Yes 0-4\n2 Feelings of Guilt Yes 0-4\n3 Suicide 0-4\n4 Insomnia: Early Night 0-2\n5 Insomnia: Middle Night 0-2\n6 Insomnia: Early Morning 0-2\n7 Work and Activities Yes 0-4\n8 Retardation Yes 0-4\n9 Agitation 0-4\n10 Anxiety: Psychic Yes 0-4\n11 Anxiety: Somatic 0-4\n12 Gastro-Intestinal Symptoms 0-2\n13 General Somatic Symptoms Yes 0-2\n14 Genital Symptoms 0-2\n15 Hypochondriasis 0-4\n16 Loss of Weight 0-2\n17 Insight 0-2\n2019), the properties of the unidimensional HDRS-6 subscale (also known as the Bech or\nmelancholia subscale; items 1, 2, 7, 8, 10, 13) (Bech et al., 1981; Park et al., 2017; Rush et al.,\n2021), and patterns of treatment effects when individual items or subscales are analyzed\nseparately (Hieronymus et al., 2015). We emphasize that our analysis of the HDRS-17 data\nis intended to be illustrative of the affordances of the IL-HTE model rather than a definitive\nanalysis of the measurement properties of the HDRS-17 scale. The HDRS-17 items are\nsummarized in Table 1.\n2.2.2 Data\nFor our empirical application, we use a subset of data from a prior study that evaluated the\neffects of SSRIs on depression measured with the HDRS-17 (Hieronymus et al., 2019). Our\nsample is comprised of data from 28 RCTs comparing acute-phase SSRI treatment to placebo\nin patients diagnosed with depression. 8262 participants were included in the studies and\nin this analysis we focus on those 5313 patients who had HDRS-17 data available after six\n10Table 2: Taxonomy of EIRMs fit to the SSRI Data\nLabel Model Treatment Effect\n1A 1PL Constant TE\n1B 1PL Randomly Varying IL-HTE\n1C 1PL Subscale Effects IL-HTE\n2A RSM Constant TE\n2B RSM Randomly Varying IL-HTE\n2C RSM Subscale Effects IL-HTE\nNotes: 1PL = One-parameter logistic. RSM = Rating Scale Model. EIRM = Explanatory Item Response\nModel.\nweeks of treatment (i.e., excluding patients who dropped out of treatment due to e.g., adverse\nevents or lack of efficacy; for further details see Hieronymus et al., 2015; Hieronymus et al.,\n2019). In total 90313 person-item combinations were included. There were 8 missing item\nresponses; one affordance of the EIRM approach is that it employs Maximum Likelihood for\nmissing item response data (Gilbert, 2024). We included baseline HDRS-17 sum scores as a\ncovariate to improve the precision of our estimates.\n2.2.3 Model Building Strategy\nWe fit a taxonomy of six primary models, summarized in Table 2. Models 1A, 1B, and 1C use\ndichotomized item responses in a 1PL or Rasch model, where 0 indicates symptom absence\nand 1 indicates symptom presence. We begin with this simpler dichotomous model to provide\na baseline for interpretation and build on past work of the IL-HTE model of dichotomous\nitems, and to determine how sensitive the results are to dichotomization. Models 2A, 2B,\nand 2C are RSMs that account for the polytomous data, using fixed effects for average item\nthresholds with random uniform item location shifts. Models 1A and 2A allow for a constant\ntreatment effect and provide a baseline for comparison (Eqn. 3). Models 1B and 2B allow for\nrandomly varying IL-HTE by including a random slope for treatment at the item level (Eqn.\n8). Models 1C and 2C add a main effect and interaction for the HDRS-6 subscale items (Eqn.\n13) to determine whether treatment effects systematically vary by subscale.\n11We also examined two additional modeling strategies to probe the sensitivity of our results.\nFirst, to examine how ρcan induce spurious interaction effects, we fit an additional set of\nmodels interacting treatment status with baseline depression to determine how sensitive\nthe model is to the inclusion or exclusion of ρ. Second, while the RSM is not typically\napplied when items have different numbers of response categories, we determined it would\nbe appropriate in our case because item responses with ratings of 4 or 5 were (a) quite rare\nin our sample (see online supplemental materials) and (b) represented a qualitatively more\nextreme range of symptoms than the 3-category items. In our online supplement, we fit the\nmore flexible PCM that allows for separate thresholds for each item and show that the results\nof our analysis are unchanged.\n3 Results\n3.1 Monte Carlo Simulation\nOur simulation results replicate past findings from the dichotomous IL-HTE model in the\npolytomous setting (Gilbert, Kim, & Miratrix, 2023). That is, in terms of statistical inference,\nIL-HTE does not cause bias in treatment effect point estimates but IL-HTE vastly inflates\nthe SE of the ATE β1. Figure 2 shows the distribution of bias by condition and we see\nthat the degree of bias is small across all conditions and does not vary by constant effects\nor IL-HTE model. While there is a slight negative bias in both models and at all levels of\nIL-HTE as the number of response categories increases, the absolute magnitude of the bias\nis small, and additional simulations in our online supplement show that the magnitude of\nthe bias approaches 0 as the number of items increases. Figure 3 shows the calibration of\nthe SEs of the model by plotting the mean model-based SE against the observed SD of the\ntreatment effect point estimates (i.e., the empirical SE). A ratio of 100% indicates the mean\nmodel-based SE is equivalent to the empirical SE. We clearly see that as IL-HTE increases,\nthe constant TE model SEs are systematically far too low, at about 50% of their true values.\n12In contrast, the SEs of the IL-HTE model are much better calibrated regardless of the level\nof IL-HTE in the data, an important result for accurate statistical inference.\nThe inflation of SEs occurs because IL-HTE accounts for an additional source of uncertainty\nin the estimation of the ATE. That is, if there is IL-HTE in the population of items from\nwhich a PROM is constructed, any finite draw of items for a PROM administration will have\na sample ATE that varies from the population ATE due to sampling error. The random\nslope variance of the IL-HTE model ( σζ) accounts for this source of uncertainty and provides\nadjusted estimates of the SE of the treatment effect, whereas the constant effects model\nprovides uncertainty consistent with the set of realized items treated as fixed (Gilbert, Kim,\n& Miratrix, 2023; Miratrix et al., 2021). The implication of this result is that researchers\nmay be vastly overestimating their precision when they assume a constant effect model and\nintend to generalize their results to a latent trait that could have been measured with an\nalternative set of items.\nOur second set of simulations examines the impact of ρon treatment by baseline covariate\ninteractions also confirms previous simulation findings from dichotomous items (Gilbert,\nMiratrix, et al., 2023). That is, ρinduces a spurious interaction between treatment and\nbaseline variables. Figure 4 shows the bias in an estimated baseline by treatment interaction\nterm (the true data-generating value is zero), and we see that high magnitudes of ρ, both\npositive and negative, induce a substantial bias in the estimated treatment by baseline\ninteraction term in the constant effects model. In contrast, the IL-HTE model that allows for\nρsuccessfully eliminates this bias. This result suggests that treatment by baseline covariate\ninteraction effects estimated on PROMs should be interpreted cautiously when item-level\ndata is not available (B. W. Domingue et al., 2022; Gilbert, Miratrix, et al., 2023).\n3.2 Empirical Application\nModel results are presented in Table 3. In Models 1A, 1B, 2A, and 2B, we see a negative ATE\nof SSRIs on depression. That is, patients randomly assigned to the SSRI condition reported\n13Figure 2: Estimated Bias of Treatment Main Effect by Simulation Condition\n3 5 7\n0 0.2 0.4 0 0.2 0.4 0 0.2 0.4−0.20.00.2\nSD of Treatment EffectsBias\nModel Constant TE IL−HTE\nNotes: Each panel presents a different number of response categories k.\nsubstantial reductions in depression symptoms compared to patients assigned to the placebo\ncondition, at about -.25 units on the logit scale ( p < . 001). The 1PL and RSM provide similar\nresults in terms of magnitude and statistical significance, though the z-statistics are slightly\nlarger in the RSM, consistent with the polytomous items providing more information than\nthe dichotomized items (Steele, 2011).\n14Figure 3: Estimated Standard Error Calibration of Treatment Main Effect by Simulation\nCondition\n3 5 7\n0 0.2 0.4 0 0.2 0.4 0 0.2 0.46080100\nSD of Treatment EffectsRelative SE (%)\nmodel Constant TE IL−HTE\nNotes: Each panel presents a different number of response categories k. A value of 100% indicates that the\nmodel-based SE was equivalent to the empirical SE, on average.\n15Figure 4: Estimated Bias in Treatment by Baseline Interaction Term\n−0.10−0.050.000.05\n−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1\nItem Location−Treatment Effect CorrelationMean Bias\nModel Constant TE IL−HTE\n16Table 3: EIRMs fit to the SSRI Data\nM1A M1B M1C M2A M2B M2C\n1 = SSRI −.272∗∗∗−.274∗∗∗−.136∗−.224∗∗∗−.204∗∗∗−.111∗\n(.036) ( .071) ( .062) ( .027) ( .052) ( .050)\nBaseline Depression (Std.) .279∗∗∗.280∗∗∗.280∗∗∗.200∗∗∗.201∗∗∗.201∗∗∗\n(.017) ( .017) ( .017) ( .012) ( .012) ( .012)\n1 = HDRS6 Item 1 .840∗∗∗1.328∗∗∗\n(.489) ( .376)\nSSRI x HDRS6 −.398∗∗∗−.265∗∗∗\n(.091) ( .072)\nThreshold 2 −.764∗∗∗−.772∗∗∗−.772∗∗∗\n(.016) ( .016) ( .016)\nThreshold 3 −2.618∗∗∗−2.642∗∗∗−2.642∗∗∗\n(.029) ( .029) ( .029)\nThreshold 4 −3.818∗∗∗−3.870∗∗∗−3.872∗∗∗\n(.075) ( .076) ( .076)\nAIC 99389 .193 99251 .523 99237 .694 146967 .819 146838 .173 146827 .259\nBIC 99436 .249 99317 .400 99322 .393 147046 .163 146936 .103 146944 .775\nLog Likelihood −49689 .597−49618 .762−49609 .847−73475 .910−73409 .086−73401 .629\nNum. obs. 90313 90313 90313 132326 132326 132326\nNum. groups: PID 5313 5313 5313 5313 5313 5313\nNum. groups: itemID 17 17 17 17 17 17\nVar: PID (Intercept) 1 .119 1 .127 1 .128 .562 .571 .570\nVar: itemID (Intercept) 1 .445 1 .678 .898 .867 .985 .549\nVar: itemID SSRI .064 .026 .034 .017\nCov: itemID (Intercept) SSRI −.183 −.010 −.090 −.004\n∗∗∗p <0.001;∗∗p <0.01;∗p <0.05\nNotes: PID = Person identifier. itemID = item identifier.\n17Comparing the constant treatment effect models (1A, 2A) to the IL-HTE models (1B,\n2B), we see substantial IL-HTE in the data, with large increases to the log likelihood for the\nIL-HTE models and σζ=.25 for the 1PL model and .18 for the RSM, SDs nearly as large as\nthe point estimates for the ATEs themselves. Accordingly, we can see that SE(ˆβ1) nearly\ndoubles in both cases, capturing the added uncertainty of which items were selected for the\nHDRS-17 from the population of potential items that could have been selected. In other\nwords, the estimated treatment effect of SSRIs on depression measured with a different set\nof items from those that could have been selected might be quite different than what was\nobserved on these items. The approximate doubling of SE(ˆβ1) is equivalent to reducing the\neffective sample size by a factor of four.\nAn intuitive way to interpret the IL-HTE parameter σζis as a measure of generalizability.\nThat is, we can calculate a 95% prediction interval (PI) for a range of item-specific treatment\neffects on out-of-sample depression items (i.e., items that are similar to those on the HDRS-17),\nusing the formula PI=ˆβ1±1.96q\nσ2\nζ+V ar(ˆβ1)(Borenstein et al., 2009, p. 130). Applied\nto Models 1B and 2B, we see that,\nPIM1B=−.274±1.96√\n.064 + .005 (15)\n=−.274±.515 (16)\n= [−.789, .241] (17)\nPIM2B=−.204±1.96√\n.034 + .003 (18)\n=−.204±.377 (19)\n= [−.581, .173], (20)\nwhich suggests that treatment effects on items measuring other symptoms that could reason-\nably be included in a depression rating scale could show anywhere from large negative effects\n18of SSRIs to moderate positive effects (i.e., be made worse by SSRIs), a key finding for the\ngeneralizability of SSRI effects on PROMs.\nThe correlation between item location and treatment effect size of about ρ=−.5 in\nModels 1B and 2B suggests that the most commonly endorsed symptoms saw the largest\n(negative) effects, as displayed in Figure 5, which plots empirical Bayes estimates of item-\nspecific treatment effects ( β1+ζi, y-axis) against item location in the control group ( bi,\nx-axis). As discussed earlier, ρcan create causal identification challenges by inducing spurious\ninteraction effects when not appropriately modeled, as shown in our simulation results in\nFigure 4. We examine this phenomenon in our online supplement by fitting additional models\nthat allow for a baseline depression by SSRI interaction, and see that the inclusion of ρin the\nmodel shifts the point estimate of the interaction effect, in line with our simulation results\nand prior research. However, the shift in the estimated interaction term in our data set is\nnot large in magnitude, changing from -.015 in the constant effects model to -.024 in the\nIL-HTE model, and neither is statistically significant. Figure 5 also highlights the HDRS-6\nitems, and it appears that the treatment effects on the HDRS-6 subscale are systematically\nlarger than treatment effects on the 11 remaining items, and that the negative correlation\nmay be partially or entirely driven by the larger effects on the HDRS-6 subscale.\nModels 1B and 2B show substantial IL-HTE and make appropriate adjustments to our\nSEs, and as such are preferable to Models 1A and 2A. However, with such a large value of\nσζ, a natural question is what item characteristics might explain this IL-HTE. Accordingly,\nModels 1C and 2C include HDRS-6 subscale by SSRI interaction effects to allow the treatment\neffect to systematically differ between the HDRS-6 items and the remaining 11 items (Gilbert,\n2023b; Gilbert, Kim, & Miratrix, 2023; J. S. Kim et al., 2023). Both models confirm\nthat treatment effects on HDRS-6 items are systematically larger than the other 11 items\n(βRSM=−.265, p < . 001). The main effect of treatment on the remaining 11 items is negative\nand statistically significant, though much reduced in magnitude ( βRSM=−.111, p < . 05).\nThese results are in line with prior analysis of differential effects when each subscale is\n19Figure 5: Correlation between Item Location and Item-Specific Treatment Effect Size\n1716\n12\n315\n89\n4\n65\n214\n11\n13\n107\n1−0.6−0.4−0.20.0\n−2 −1 0 1\nItem LocationItem−Specific Treatment Effect (Logits)\nHDRS−6 Subscale Item aa01\nNotes: Empirical Bayes estimates for item-specific treatment effect ( β1+ζi) on item location ( bi) derived\nfrom Model 2B.\n20Figure 6: Conditional Average Treatment Effects of SSRIs on HDRS-6 Items and the\nRemaining 11 Items\nHDRS6 Non−HDRS6\n−2.5 0.0 2.5 5.0 −2.5 0.0 2.5 5.0−2−1012\nBaseline Depression (Std.)Log Odds\ngroup 01\nNotes: Estimates derived from Model 2C. Group 1 is the SSRI treatment, Group 0 is placebo.\nconsidered separately (Hieronymus et al., 2019); one advantage of the IL-HTE model is\nthat we obtain a direct hypothesis test of differences in effect size by subscale in a single\nmodel. The results of Model 2C are displayed in Figure 6, which shows the fitted log-odds\nof exceeding the average category on an average item in each subscale (y-axis) by baseline\ndepression (x-axis) and treatment status (color). The CATE is represented by the vertical\ndistance between the fitted lines, and we can see that it is much larger for the HDRS-6 items.\nThe larger effects on the HDRS-6 subscale have important policy implications for clinical\ntrial evaluation and interpretation, because, “if researchers could somehow a priori select\nthose items known to be more sensitive to the treatment, they would obtain a larger measured\ntreatment impact as an artifact of the selected items, rather than a truly more effective\n21treatment” (Gilbert, Kim, & Miratrix, 2023, p. 895). In the context considered here,\nevaluations of SSRIs using HDRS-6 might appear to be more effective than those using the\nfull HDRS-17, not because the treatment is more effective on depression as a whole, but\nbecause the specific symptoms of depression assessed on the HDRS-6 subscale are more\nsensitive to SSRIs. Conversely, it is also the case that some symptoms that are common in\ndepression covary also with other psychiatric and somatic conditions, as well as with age and\nsex; insomnia being more common in old age, for example. In such cases one might expect\nsome “depressive symptoms” to persist after remission of the depressive episode because in\nsuch a case they are not causally related to the depression, but to an alternative explanation\n(e.g., added exogeneous causes of the item responses over and above ηijin Figure 1). That\nis, such symptoms may reflect some construct-irrelevant variance (Downing, 2002). Indeed,\nit is well known that symptomatic remission as measured by the HDRS-17 does not map\nparticularly well to patient-defined remission (in either direction) (Zimmerman, Martinez,\net al., 2012; Zimmerman, Martinez, et al., 2012); such an inference is similarly supported by\nour IL-HTE analysis.\nFinally, we can calculate an approximate R2of the proportion of IL-HTE explained by\nthe SSRI by HDRS-6 interaction effect comparing σ2\nζbetween models B and C. We see that\nthe SSRI by HDRS-6 interaction explains 59% percent of the IL-HTE in the 1PL model\nand 50% in the RSM, and that after accounting for the differential effects by subscale, our\nestimate of ρgoes nearly to 0, in line with Figure 5, suggesting that the majority of the\nIL-HTE in the data is explained by the differential subscale effects.\n4 Discussion\nModels for heterogeneous treatment effects (HTE) have typically focused on person charac-\nteristics as moderators of treatment efficacy. While valuable, such approaches may generate\nfalse positives given the inference, generalizability, and identification challenges that arise\n22when the outcome of interest is a latent variable such as a PROM constructed from a set of\nitems. In this study, we extend novel Item Response Theory methods to the estimation of\nitem-level HTE (IL-HTE) using an analysis of clinical trial data of the effects of SSRIs on\ndepression as measured by the 17-item Hamilton Depression Rating Scale as an illustration.\nAs such, our study represents an important contribution to epidemiological methodology,\nfrom both statistical and substantive perspectives.\nWhen IL-HTE is present in the data but ignored in the model, the standard errors of\ntreatment effects are underestimated, leading to spurious estimates of precision and potential\nfalse positives. For example, the near doubling of the SE of the treatment effect we observed\nin our empirical analysis is equivalent to a reduction in the sample size by a factor of four,\na finding with implications for both statistical inference generally and prospective power\nanalysis. The IL-HTE model better accounts for the uncertainty of which items were chosen\nfor the PROM and provides a metric for how generalizable our results might be if we were\nto use different items to assess treatment impact. Similarly, the ability to model subscale\neffects can provide insight as to which symptom clusters are most sensitive to treatment,\nallowing researchers to test explicit hypotheses about differential subscale effects. Furthermore,\ncorrelation between item location and item-specific treatment effect sizes can induce spurious\ninteractions between treatment and baseline characteristics that only item-level analysis\ncan appropriately identify. While spurious interactions did not emerge in our empirical\nillustration, it would be prudent to complement analyses that examine HTE by baseline\ncovariates by the kind of item-level analysis advocated here to ensure accurate identification\nof interaction effects. Even when HTE by baseline covariates is not of primary interest,\nnon-zero correlations will also create bias in the main effects of covariates, because the main\neffect in a model without interaction is an average of the effects in each subgroup, weighted\nby subgroup sample size.\nSubstantively, the application of the IL-HTE model to the placebo-controlled SSRI trials\nshowed substantial IL-HTE in the HDRS-17. Estimates of the standard deviation of item-\n23specific treatment effects were nearly as large as the point estimates for the average treatment\neffect themselves and standard errors nearly doubled when we accounted for IL-HTE in the\nmodel. Prediction intervals for out-of-sample items suggested that the impact of SSRIs could\nbe anywhere from slightly harmful to strongly beneficial. Such a fine-grained level of insight\non the extent of SSRI impact on depression would be masked by a traditional analysis of a\nsingle number summary outcome, such as a sum score.\nFurthermore, our analysis of subscale effects on the HDRS-6 items showed systematically\nhigher treatment effects on HDRS-6 than the other eleven items, a finding that has potentially\ncritical policy implications for how we judge the effectiveness of treatments using different\noutcome measures for the same underlying construct such as depression. The HDRS-6, which\nwas designed to be a unidimensional measure of depressive severity, was developed well before\nmodern SSRIs came to market and it is therefore unlikely that researchers developing the\nHDRS-6 would have chanced upon a collection of items that would in the future make SSRIs\nappear particularly effective (Bech et al., 1975; Bech et al., 1981), especially considering that\nall efforts aimed at developing unidimensional subscales have arrived at subscales that closely\nresemble the HDRS-6 (i.e., always including the items depressed mood, feelings of guilt, work\nand activities, and psychic anxiety) (Bagby et al., 2004). Nevertheless, the HTE by subscale\nhas important implications for the interpretation of HDRS-6 or other subscales as outcome\nmeasures given that it demonstrates the extent to which efficacy estimates are conditional on\nwhich rating instrument is applied (Luckenbaugh et al., 2015; Ruh´ e et al., 2005).\nIn the same vein, SSRIs—the antidepressant class examined in this study—have well-known\nside effects related to sexual dysfunction, decreased appetite, gastrointestinal complaints and\ninsomnia (Ferguson, 2001). These side effects, which are present also in healthy volunteers\n(Knorr et al., 2019), have been shown to correlate with ratings on corresponding HDRS items\nthereby introducing another way in which ratings of individual depressive symptoms may\ndiverge from the latent variable that they are intended to measure (Hieronymus et al., 2021).\nNotably, other antidepressants such as amitriptyline and mirtazapine have the opposite effects\n24in that they are hypnotics and increase appetite and could therefore potentially introduce\nthe opposite biases, that is, making them appear as more effective antidepressants when\nmeasured on a scale including many such items.\n4.1 Limitations\nClearly, IL-HTE has both statistical and substantive implications for the analysis of treatment\neffects in clinical trials and epidemiology, but we acknowledge the following four limitations of\nthe IL-HTE, building on the limitations described in prior research (Gilbert, Kim, & Miratrix,\n2023; Gilbert, Miratrix, et al., 2023). First, it is unknown how common IL-HTE is in clinical\ntrial or epidemiological PROMs data, and we leave this largely as an open question. Some\nsystematic evidence comes from education research, where an analysis of 15 RCTs showed\nsubstantial IL-HTE, including a case where 40% of the overall treatment effect was driven\nby a single test item (Ahmed et al., 2023). Therefore, future research using the IL-HTE\nmodel in other clinical trial or epidemiological contexts has the potential to shed light on\nhow widespread the IL-HTE phenomenon is in this field.\nSecond, estimating the IL-HTE model necessitates the availability of item-level data,\nwhich may not be available, particularly in secondary analyses. As such, one practical\nimplication of our results is that researchers should heed calls to share item-level outcome\ndata (B. Domingue & Kanopka, 2023), so that researchers and secondary analysts can evaluate\nthe extent of IL-HTE in a data set given the statistical and substantive insights allowed by\nIL-HTE analysis.\nThird, the use of latent variable models such as the EIRM may be more difficult to\ninterpret and justify to practitioners, particularly when coefficients are on the logit scale\n(Breen et al., 2018; Mood, 2010). One approach to improve communicability is to convert\nthe logit coefficients to standardized effect sizes by dividing the regression coefficient by the\nestimated standard deviation of the latent variable θj, a value that can be derived from model\nresults (Gilbert, Kim, & Miratrix, 2023, pp. 907-908).\n25Finally, the EIRM can be substantially more computationally demanding than alternative\nmodels such as OLS regression because the cross-classified multilevel structure of the item\nresponse data requires numerical integration approaches (Rabe-Hesketh & Skrondal, 2022)\nor Markov chain Monte Carlo (MCMC) methods in Bayesian applications (B¨ urkner, 2021;\nGilbert, 2023b). Thus, when the number of items and persons is large, IL-HTE methods may\nbecome computationally prohibitive.\n4.2 Conclusion\nIn sum, applying measurement and psychometric principles to causal inference in epidemi-\nological and public health research provides a powerful opportunity anywhere multi-item\npatient-reported outcome measures are used to assess treatment impact. By using the IL-\nHTE model, both clinical and epidemiological researchers can gain more accurate statistical\ninference, better estimates of generalizability to new symptoms, and unbiased estimates of\ninteraction effects, all essential qualities for clinical trial and epidemiological research that\naims to inform public health policy.\n265 Declarations\nResearch Ethics\nNot applicable.\nInformed Consent\nNot applicable.\nAuthor Contributions\nConceptualization: Author 1, Author 4\nMethodology: Author 1, Author 4\nSoftware: Author 1\nFormal Analysis: Author 1, Author 2\nWriting—original draft preparation: Author 1\nWriting—review and editing: Author 1, Author 2, Author 3, Author 4\nCompeting Interests\nAuthors 1 and 4 report no conflicts of interest.\nAuthor 2 has received speaker’s fees from Janssen Pharmaceuticals in the last five years\nand is a board member of the Swedish Serotonin Society.\nAuthor 3 has received speaker’s fees from Janssen Pharmaceuticals in the last five years.\nResearch Funding\nThis work was funded in part by the Jacobs Foundation.\nData Availability\nThe data analyzed in this study are proprietary. However, we share a simulated version\nof the data set based on the results of our models in our online supplemental file. The code\nfor this article is available as an online supplemental file (https://researchbox.org/2494&\nPEER REVIEW passcode=HEOZYC).\nThe full HDRS-17 is available at the following URL: https://dcf.psychiatry.ufl.edu/files/\n2011/05/HAMILTON-DEPRESSION.pdf\n27References\nAhmed, I., Bertling, M., Zhang, L., Ho, A. D., Loyalka, P., Xue, H., Rozelle, S., & Ben-\njamin, W. (2023). Heterogeneity of item-treatment interactions masks complexity and\ngeneralizability in randomized controlled trials. Edworkingpapers. com .\nAndresen, E. M., Byers, K., Friary, J., Kosloski, K., & Montgomery, R. (2013). Performance\nof the 10-item center for epidemiologic studies depression scale for caregiving research.\nSAGE Open Medicine ,1, 2050312113514576.\nBagby, R. M., Ryder, A. G., Schuller, D. R., & Marshall, M. B. (2004). The hamilton\ndepression rating scale: Has the gold standard become a lead weight? American Journal\nof Psychiatry ,161(12), 2163–2177. https://doi.org/10.1176/appi.ajp.161.12.2163\nBarger, B. (2023). Epidemiology with psychometric spirit: Moba leads autism’s interdisci-\nplinary future—a commentary on havdahl et al.(2023). Journal of Child Psychology\nand Psychiatry .\nBech, P., Gram, L. F., Dein, E., Jacobsen, O., Vitger, J., & Bolwig, T. G. (1975). Quantitative\nrating of depressive states: Correlation between clinical assessment, beck’s self-rating\nscale and hamilton’s objective rating scale. Acta Psychiatrica Scandinavica ,51(3),\n161–170. https://doi.org/10.1111/j.1600-0447.1975.tb00002.x\nBech, P., Allerup, P., Gram, L., Reisby, N., Rosenberg, R., Jacobsen, O., & Nagy, A. (1981).\nThe hamilton depression scale: Evaluation of objectivity using logistic models. Acta\nPsychiatrica Scandinavica ,63(3), 290–299.\nBeghi, E., Chi` o, A., Couratier, P., Esteban, J., Hardiman, O., Logroscino, G., Millul, A.,\nMitchell, D., Preux, P. -M., Pupillo, E., et al. (2011). The epidemiology and treatment\nof als: Focus on the heterogeneity of the disease and critical appraisal of therapeutic\ntrials. Amyotrophic Lateral Sclerosis ,12(1), 1–10.\nBorenstein, M., Hedges, L. V., Higgins, J. P., & Rothstein, H. R. (2009). Introduction to\nmeta-analysis . John Wiley & Sons.\n28Breen, R., Karlson, K. B., & Holm, A. (2018). Interpreting and understanding logits, probits,\nand other nonlinear probability models. annual review of sociology ,44, 39–54.\nBulut, O., Gorgun, G., & Yildirim-Erbasli, S. N. (2021). Estimating explanatory extensions\nof dichotomous and polytomous rasch models: The eirm package in r. Psych ,3(3),\n308–321.\nB¨ urkner, P. -C. (2021). Bayesian item response modeling in r with brms and stan. Journal of\nStatistical Software ,100(5), 1–54. https://doi.org/10.18637/jss.v100.i05\nCapuano, A. W., Dawson, J. D., Ramirez, M. R., Wilson, R. S., Barnes, L. L., & Field, R. W.\n(2016). Modeling likert scale outcomes with trend-proportional odds with and without\ncluster data. Methodology .\nChan, K. S., Orlando, M., Ghosh-Dastidar, B., Duan, N., & Sherbourne, C. D. (2004). The\ninterview mode effect on the center for epidemiological studies depression (ces-d) scale:\nAn item response theory analysis. Medical care , 281–289.\nChristensen, K. B. (2006). From rasch scores to regression. Journal of Applied Measurement ,\n7(2), 184.\nCollier, D. A., Ferreira, I. A., Kotagiri, P., Datir, R. P., Lim, E. Y., Touizer, E., Meng, B.,\nAbdullahi, A., et al. (2021). Age-related immune response heterogeneity to sars-cov-2\nvaccine bnt162b2. Nature ,596(7872), 417–422.\nCordero, C. P., & Dans, A. L. (2021). Key concepts in clinical epidemiology: Detecting and\ndealing with heterogeneity in meta-analyses. Journal of Clinical Epidemiology ,130,\n149–151.\nDe Boeck, P. (2008). Random item irt models. Psychometrika ,73, 533–559.\nDe Boeck, P., Bakker, M., Zwitser, R., Nivard, M., Hofman, A., Tuerlinckx, F., & Partchev, I.\n(2011). The estimation of item response models with the lmer function from the lme4\npackage in r. Journal of Statistical Software ,39, 1–28.\n29De Boeck, P., Cho, S. -J., & Wilson, M. (2016). Explanatory item response models. The Wiley\nhandbook of cognition and assessment: Frameworks, methodologies, and applications ,\n247–266.\nDomingue, B., & Kanopka, K. (2023). The item response warehouse (irw).\nDomingue, B. W., Kanopka, K., Trejo, S., Rhemtulla, M., & Tucker-Drob, E. M. (2022).\nUbiquitous bias and false discovery due to model misspecification in analysis of\nstatistical interactions: The role of the outcome’s distribution and metric properties.\nPsychological Methods .\nDowning, S. M. (2002). Threats to the validity of locally developed multiple-choice tests in\nmedical education: Construct-irrelevant variance and construct underrepresentation.\nAdvances in Health Sciences Education ,7, 235–241.\nDworkin, R. H., Turk, D. C., Revicki, D. A., Harding, G., Coyne, K. S., Peirce-Sandner, S.,\nBhagwat, D., Everton, D., Burke, L. B., Cowan, P., et al. (2009). Development and\ninitial validation of an expanded and revised version of the short-form mcgill pain\nquestionnaire (sf-mpq-2). Pain®,144(1-2), 35–42.\nEsterman, A. (2003). The likert scale. Australasian epidemiologist ,10(2).\nFaro-Viana, J., Bergman, M. -L., Gon¸ calves, L. A., Duarte, N., Coutinho, T. P., Borges, P. C.,\nDiwo, C., Castro, R., Matoso, P., Malheiro, V., et al. (2022). Population homogeneity\nfor the antibody response to covid-19 bnt162b2/comirnaty vaccine is only reached\nafter the second dose across all adult age ranges. Nature Communications ,13(1), 140.\nFerguson, J. M. (2001). Ssri antidepressant medications: Adverse effects and tolerability. The\nPrimary Care Companion For CNS Disorders ,3(1). https://doi.org/10.4088/pcc.\nv03n0105\nFlake, J. K., Pek, J., & Hehman, E. (2017). Construct validation in social and personality\nresearch: Current practice and recommendations. Social Psychological and Personality\nScience ,8(4), 370–378.\n30Fournier, J. C., DeRubeis, R. J., Hollon, S. D., Dimidjian, S., Amsterdam, J. D., Shelton,\nR. C., & Fawcett, J. (2010). Antidepressant drug effects and depression severity: A\npatient-level meta-analysis. Jama ,303(1), 47–53.\nGibbons, R. D., Clark, D. C., & Kupfer, D. J. (1993). Exactly what does the hamilton\ndepression rating scale measure? Journal of psychiatric research ,27(3), 259–273.\nGilbert, J. B. (2023a). How measurement affects causal inference: Attenuation bias is (usually)\nmore important than scoring weights. Edworkingpapers. com .\nGilbert, J. B. (2023b). Modeling item-level heterogeneous treatment effects: A tutorial with\nthe glmer function from the lme4 package in r. Behavior Research Methods , 1–13.\nGilbert, J. B. (2024). Estimating treatment effects with the explanatory item response model.\nJournal of Research on Educational Effectiveness , 1–19.\nGilbert, J. B., Kim, J. S., & Miratrix, L. W. (2023). Modeling item-level heterogeneous\ntreatment effects with the explanatory item response model: Leveraging large-scale\nonline assessments to pinpoint the impact of educational interventions. Journal of\nEducational and Behavioral Statistics , 10769986231171710.\nGilbert, J. B., Miratrix, L. W., Joshi, M., & Domingue, B. W. (2023). Disentangling person-\ndependent and item-dependent causal effects: Applications of item response theory to\nthe estimation of treatment effect heterogeneity. (881). https://doi.org/10.26300/6b7w-\nvp07\nGlymour, M. M. (2006). Using causal diagrams to understand common problems in social\nepidemiology. Methods in social epidemiology , 393–428.\nGrayson, D. A., Mackinnon, A., Jorm, A. F., Creasey, H., & Broe, G. A. (2000). Item bias\nin the center for epidemiologic studies depression scale: Effects of physical disorders\nand disability in an elderly community sample. The Journals of Gerontology Series B:\nPsychological Sciences and Social Sciences ,55(5), P273–P282.\nGreenland, S., Pearl, J., & Robins, J. M. (1999). Causal diagrams for epidemiologic research.\nEpidemiology , 37–48.\n31Hamilton, M. (1960). A rating scale for depression. Journal of Neurology, Neurosurgery &\nPsychiatry ,23(1), 56–62. https://doi.org/10.1136/jnnp.23.1.56\nHedges, L. V. (1981). Distribution theory for glass’s estimator of effect size and related\nestimators. journal of Educational Statistics ,6(2), 107–128.\nHieronymus, F., Emilsson, J. F., Nilsson, S., & Eriksson, E. (2015). Consistent superiority\nof selective serotonin reuptake inhibitors over placebo in reducing depressed mood\nin patients with major depression. Molecular Psychiatry ,21(4), 523–530. https :\n//doi.org/10.1038/mp.2015.53\nHieronymus, F., Lisinski, A., Eriksson, E., & Østergaard, S. D. (2021). Correction: Do side\neffects of antidepressants impact efficacy estimates based on the hamilton depression\nrating scale? a pooled patient-level analysis. Translational Psychiatry ,11(1). https:\n//doi.org/10.1038/s41398-021-01403-w\nHieronymus, F., Lisinski, A., Nilsson, S., & Eriksson, E. (2019). Influence of baseline severity\non the effects of ssris in depression: An item-based, patient-level post-hoc analysis.\nThe Lancet Psychiatry ,6(9), 745–752.\nJessen, A., Ho, A. D., Corrales, C. E., Yueh, B., & Shin, J. J. (2018). Improving measurement\nefficiency of the inner ear scale with item response theory. Otolaryngology–Head and\nNeck Surgery ,158(6), 1093–1100.\nJoffe, M., Gambhir, M., Chadeau-Hyam, M., & Vineis, P. (2012). Causal diagrams in systems\nepidemiology. Emerging themes in epidemiology ,9, 1–18.\nJones, R. N. (2019). Differential item functioning and its relevance to epidemiology. Current\nepidemiology reports ,6, 174–183.\nKent, D. M., Nelson, J., Dahabreh, I. J., Rothwell, P. M., Altman, D. G., & Hayward, R. A.\n(2016). Risk and treatment effect heterogeneity: Re-analysis of individual participant\ndata from 32 large clinical trials. International journal of epidemiology ,45(6), 2075–\n2088.\n32Kim, J. S., Burkhauser, M. A., Relyea, J. E., Gilbert, J. B., Scherer, E., Fitzgerald, J.,\nMosher, D., & McIntyre, J. (2023). A longitudinal randomized trial of a sustained\ncontent literacy intervention from first to second grade: Transfer effects on students’\nreading comprehension. Journal of Educational Psychology ,115(1), 73.\nKim, J., & Wilson, M. (2020). Polytomous item explanatory item response theory models.\nEducational and psychological measurement ,80(4), 726–755.\nKirsch, I., Deacon, B. J., Huedo-Medina, T. B., Scoboria, A., Moore, T. J., & Johnson, B. T.\n(2008). Initial severity and antidepressant benefits: A meta-analysis of data submitted\nto the food and drug administration. PLoS medicine ,5(2), e45.\nKnorr, U., Madsen, J. M., & Kessing, L. V. (2019). The effect of selective serotonin reuptake\ninhibitors in healthy subjects revisited: A systematic review of the literature. Experi-\nmental and Clinical Psychopharmacology ,27(5), 413–432. https://doi.org/10.1037/\npha0000264\nLesko, C. R., Henderson, N. C., & Varadhan, R. (2018). Considerations when assessing\nheterogeneity of treatment effect in patient-centered outcomes research. Journal of\nclinical epidemiology ,100, 22–31.\nLuckenbaugh, D. A., Ameli, R., Brutsche, N. E., & Zarate Jr, C. A. (2015). Rating depression\nover brief time intervals with the hamilton depression rating scale: Standard vs.\nabbreviated scales. Journal of psychiatric research ,61, 40–45.\nMcEvoy, P. M., Grove, R., & Slade, T. (2011). Epidemiology of anxiety disorders in the\naustralian general population: Findings of the 2007 australian national survey of\nmental health and wellbeing. Australian & New Zealand Journal of Psychiatry ,45(11),\n957–967.\nMcNeish, D., & Wolf, M. G. (2020). Thinking twice about sum scores. Behavior research\nmethods ,52, 2287–2305.\n33Miratrix, L. W., Weiss, M. J., & Henderson, B. (2021). An applied researcher’s guide to\nestimating effects from multisite individually randomized trials: Estimands, estimators,\nand estimates. Journal of Research on Educational Effectiveness ,14(1), 270–308.\nMontoya, A. K., & Jeon, M. (2020). Mimic models for uniform and nonuniform dif as\nmoderated mediation models. Applied psychological measurement ,44(2), 118–136.\nMood, C. (2010). Logistic regression: Why we cannot do what we think we can do, and what\nwe can do about it. European sociological review ,26(1), 67–82.\nObeid, S., Hallit, C. A. E., Haddad, C., Hany, Z., & Hallit, S. (2018). Validation of the\nhamilton depression rating scale (hdrs) and sociodemographic factors associated with\nlebanese depressed patients. L’encephale ,44(5), 397–402.\nPark, S. -C., Jang, E. Y., Kim, J. -M., Jun, T. -Y., Lee, M. -S., Kim, J. -B., Yim, H. -W., &\nPark, Y. C. (2017). Clinical validation of the psychotic depression assessment scale,\nhamilton depression rating scale-6, and brief psychiatric rating scale-5: Results from\nthe clinical research center for depression study. Psychiatry Investigation ,14(5), 568.\nPetscher, Y., Compton, D. L., Steacy, L., & Kinnon, H. (2020). Past perspectives and new\nopportunities for the explanatory item response model. Annals of dyslexia ,70, 160–179.\nRabe-Hesketh, S., & Skrondal, A. (2022). Multilevel and longitudinal modeling using stata .\nSTATA press.\nRobertson, S. E., Leith, A., Schmid, C. H., & Dahabreh, I. J. (2021). Assessing heterogeneity of\ntreatment effects in observational studies. American Journal of Epidemiology ,190(6),\n1088–1100.\nRuh´ e, H. G., Dekker, J. J., Peen, J., Holman, R., & De Jonghe, F. (2005). Clinical use\nof the hamilton depression rating scale: Is increased efficiency possible? a post hoc\ncomparison of hamilton depression rating scale, maier and bech subscales, clinical\nglobal impression, and symptom checklist-90 scores. Comprehensive Psychiatry ,46(6),\n417–427.\n34Rush, A. J., South, C., Jain, S., Agha, R., Zhang, M., Shrestha, S., Khan, Z., Hassan, M., &\nTrivedi, M. H. (2021). Clinically significant changes in the 17-and 6-item hamilton\nrating scales for depression: A star* d report. Neuropsychiatric Disease and Treatment ,\n2333–2345.\nSajobi, T. T., Sanusi, R. A., Mayo, N. E., Sawatzky, R., Kongsgaard Nielsen, L., Sebille,\nV., Liu, J., Bohm, E., Awosoga, O., Norris, C. M., et al. (2023). Unsupervised item\nresponse theory models for assessing sample heterogeneity in patient-reported outcomes\nmeasures. Quality of Life Research , 1–12.\nSales, A., Prihar, E., Heffernan, N., & Pane, J. F. (2021). The effect of an intelligent tutor\non performance on specific posttest problems. International Educational Data Mining\nSociety .\nSteele, F. (2011). Module 9: Single-level and multilevel models for ordinal responses concepts.\nCentre for Multilevel Modelling, University of Bristol .\nSultan, P., Sadana, N., Sharawi, N., Blake, L., El-Boghdadly, K., Falvo, A., Ciechanowicz, S.,\nAthar, W., Shah, R., Guo, N., et al. (2020). Evaluation of domains of patient-reported\noutcome measures for recovery after childbirth: A scoping and systematic review.\nJAMA network open ,3(5), e205540–e205540.\nSultan, P., Sharawi, N., Blake, L., Ando, K., Sultan, E., Aghaeepour, N., Carvalho, B., &\nSadana, N. (2021). Use of patient-reported outcome measures to assess outpatient\npostpartum recovery: A systematic review. JAMA Network Open ,4(5), e2111600–\ne2111600.\nTennant, P. W., Murray, E. J., Arnold, K. F., Berrie, L., Fox, M. P., Gadd, S. C., Harrison,\nW. J., Keeble, C., Ranker, L. R., Textor, J., et al. (2021). Use of directed acyclic graphs\n(dags) to identify confounders in applied health research: Review and recommendations.\nInternational journal of epidemiology ,50(2), 620–632.\nVan der Linden, W. J. (2017). Handbook of item response theory: Volume 1: Models . CRC\npress.\n35Varadhan, R., Segal, J. B., Boyd, C. M., Wu, A. W., & Weiss, C. O. (2013). A framework for\nthe analysis of heterogeneity of treatment effect in patient-centered outcomes research.\nJournal of clinical epidemiology ,66(8), 818–825.\nWilliams, J. B. W. (2001). Standardizing the hamilton depression rating scale: Past, present,\nand future. European Archives of Psychiatry and Clinical Neuroscience ,251(S2), 6–12.\nhttps://doi.org/10.1007/bf03035120\nWilson, M., & De Boeck, P. (2004). Descriptive and explanatory item response models.\nInExplanatory item response models: A generalized linear and nonlinear approach\n(pp. 43–74). Springer.\nWilson, M., De Boeck, P., & Carstensen, C. H. (2008). Explanatory item response models: A\nbrief introduction. Assessment of competencies in educational contexts , 91–120.\nZimmerman, M., Martinez, J., Attiullah, N., Friedman, M., Toba, C., & Boerescu, D. A.\n(2012). Why do some depressed outpatients who are not in remission according to the\nhamilton depression rating scale nonetheless consider themselves to be in remission?:\nResearch article: Why do some depressed outpatients consider themselves to be in\nremission? Depression and Anxiety ,29(10), 891–895. https://doi.org/10.1002/da.21987\nZimmerman, M., Martinez, J. A., Attiullah, N., Friedman, M., Toba, C., Boerescu, D. A., &\nRahgeb, M. (2012). Why do some depressed outpatients who are in remission according\nto the hamilton depression rating scale not consider themselves to be in remission? The\nJournal of Clinical Psychiatry ,73(06), 790–795. https://doi.org/10.4088/jcp.11m07203\nZwinderman, A. H. (1991). A generalized rasch model for manifest predictors. Psychometrika ,\n56, 589–600.\n36"    },    {        "title": "2402.04552v1.Convexity_restoration_from_hairy_black_hole_in_Einstein_Maxwell_charged_scalar_system_in_AdS.pdf",        "content": "RUP-24-2\nYITP-24-11\nConvexity restoration from hairy black hole in\nEinstein-Maxwell-charged scalar system in AdS\nTakaaki Ishii and Yu Nakayama\nDepartment of Physics, Rikkyo University, Nishi-Ikebukuro, Toshima-ku,\nTokyo 171-8501, Japan\nYukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho,\nSakyo-ku, Kyoto 606-8502, Japan\nAbstract\nIn the Einstein-Maxwell-charged scalar system with a negative cosmological con-\nstant in arbitrary dimensions higher than three, there exists a horizonless charged\nsoliton solution, which we construct explicitly for an arbitrary mass of the scalar\nin perturbative series in small charge. We find that the stability of the soliton is\ndetermined by the validity of the AdS weak gravity conjecture. The existence of\na stable soliton might endanger the convexity of the (free) energy as a function\nof the charge because the phase transition between the soliton and the extremal\nReissner-Nordstrom black hole would be discontinuous. We, however, argue that\nthe existence of the hairy black hole solution circumvents the violation of convexity.\nThe thermodynamic properties of the hairy black hole show that the phase transi-\ntion becomes continuous irrespective of whether the AdS weak gravity conjecture\nholds. When it holds, the phase transition occurs between the soliton and the hairy\nblack hole, and when it is violated, the phase transition occurs between the extremal\nReissner-Nordstrom black hole and the hairy black hole.arXiv:2402.04552v1  [hep-th]  7 Feb 20241 Introduction\nConvexity of the (free) energy is at the heart of thermodynamic stability [1]. Still, it\nis a non-trivial problem whether this is realized in statistical mechanics with a given\nmicroscopic Hamiltonian. While it is generically very difficult to verify the convexity, the\ngravitational system may admit explicit analysis because we can read the thermodynamic\nproperties from the classical solutions, circumventing microscopic statistical mechanical\ncomputations. For example, a Schwarzschild black hole in asymptotically flat spacetime or\na small Schwarzschild-AdS black hole shows negative specific heat, which means that the\nconvexity (as a function of temperature) is violated while the microscopic understanding\nof the violation as well as its implication would be quite challenging.\nIn this paper, we would like to focus on the convexity of the (lowest) energy as a\nfunction of the conserved charge. Again, in generic statistical mechanics, while we ex-\npect the convexity of the (free) energy as a function of charge from the thermodynamic\nstability, it is a highly non-trivial task to verify it in interacting systems. The gravita-\ntional benchmark would be an extremal Reissner-Nordstrom black hole, which does show\nconvexity.\nThe recent developments in theoretical physics, however, seem to suggest that the\n(extremal) Reissner-Nordstrom black hole is not the lowest energy state with a given\ncharge. The weak gravity conjecture [2], for example, states that it should be unstable\nin any consistent theory of quantum gravity (see e.g. [3] for review). Indeed, with (light)\ncharged matter, we observe its instability caused by various mechanisms such as classical\ninstability from superradiance or quantum tunneling (e.g. Schwinger effect). Then the\ntask of verifying the convexity requires a study of the onset of such instability and becomes\nmore involved, which we will undertake in this paper.\nIn terms of the AdS/CFT correspondence, the convexity of the global energy in the\nAdS space-time as a function of the charge is related to the charge convexity conjecture\nof the conformal dimensions in conformal field theories. This was first proposed in [4] and\nthen checked in many examples [5][6][7][8][9].1A counter-example in three-dimensional\nconformal field theories was found in [10], whose significance should be understood better.\n1The convexity and the superadditivity are closely related but they can differ. Strictly speaking, in\nthe original paper [4], the charge superadditivity is conjectured. See Appendix A for the connection. See\n[5] as well.\n1A closely related subject is the interpretation of the weak gravity conjecture in terms of\nconformal field theories [11]. We do have a general picture in many examples, but the\nprecise bound has not been established.\nIn this paper, we investigate solutions of the Einstein-Maxwell-charged scalar system\nwith an arbitrary mass for the scalar in asymptotically AdS spacetime. We find there ex-\nists a horizonless charged soliton solution, which may have a lower global energy than the\nextremal Reissner-Nordstrom black hole. We constructed them in perturbative series in\nsmall charge, and they are generalizations of massless cases in five dimensions [12][13][14]\nto arbitrary dimensions with arbitrary mass.2See also charged solitons in four dimen-\nsions with and without scalar masses [15][16][17]. We first point out that the existence of\nsuch a charged soliton might endanger the convexity of the lowest energy as a function of\nthe charge because the phase transition between the soliton and the extremal Reissner-\nNordstrom black hole would be discontinuous.3We, however, argue that the convexity is\nrestored thanks to the existence of the hairy black hole solution. In particular, we show\nthat the existence of the hairy black hole makes the phase transition continuous.\nWhen the AdS weak gravity conjecture is satisfied by the scalar field, the soliton is\nlighter than the extremal Reissner-Nordstrom black hole of the same charge in the small\ncharge limit. If we increase the charge, the charged soliton becomes a hairy black hole,\nand the latter will be the lowest energy state. When the AdS weak gravity conjecture is\nviolated, the soliton is heavier than the extremal Reissner-Nordstrom black hole in the\nsmall charge limit. If we increase the charge, the extremal Reissner-Nordstrom black hole\nbecomes unstable to form the hairy black hole, which will be the lowest energy state. In\nboth cases, the phase transition will be continuous and the convexity will be restored. A\nschematic plot for the case when the AdS weak gravity conjecture is satisfied is given in\nFig. 1.\n2A direct motivation of our paper comes from Figures 2 and 3 in [14], which show the apparent\nviolation of the charge convexity. Their figures were “schematic” as they say, and if we closely follow\ntheir computations, their results do not violate the convexity. Compare them with Fig. 1.\n3In this paper, we consider the canonical ensemble at zero temperature. Meanwhile, the phase tran-\nsition between the (finite temperature) charged AdS black hole and soliton is often discussed in the\ngrand canonical ensemble [18] through the grand potential. The convexity of the (canonical) free energy\ndiscussed in this paper is equivalent to the concavity of the grand potential due to the thermodynamic\nrelation [1].\n2qcq*\n0.1 0.2 0.3 0.4 0.5 0.6 0.7q0.51.01.52.0mFigure 1: A schematic plot for the mass (or global energy) of the (extremal) Reissner-\nNordstrom black hole (blue), charged soliton (orange), and the hairy black hole (red) in\nd= 3. Here, the AdS weak gravity conjecture is satisfied. See section 4 for details.\nThe organization of the paper is as follows. In section 2, we construct the charged\nsoliton solution in perturbative series in small charge and compare the thermodynamic\nproperties with those of the extremal Reissner-Nordstrom black hole. In section 3, we\nstudy the thermodynamic properties of a hairy black hole. In section 4, we study the\nphase transition between these solutions to see if the convexity is violated or not. Section 5\nis devoted to discussion. Appendices contain details of calculations.\n2 Reissner-Nordstrom black hole and charged soliton\nWe study solutions of the Einstein-Maxwell-charged scalar theory described by the action\nS=Z\ndd+1x√−g\u00121\n16πGN(R−2Λ)−1\n4FµνFµν− |Dµϕ|2−m2\nϕ|ϕ|2\u0013\n, (1)\nwhere Λ = −d(d−1)/2 is the negative cosmological constant in units in which the AdS\nradius is unity, and the gauge covariant derivative is defined as Dµϕ=∂µϕ−ieAµϕwith\nebeing the U(1) coupling constant. For convenience, we will parameterize the mass of\nthe charged scalar by the conformal dimension ∆ as m2\nϕ= ∆(∆ −d). In this paper, we\nassume d≥3.\n3Without the scalar hair ( ϕ= 0), the supposedly lowest energy solution with charge Q\nis the Reissner-Nordstrom-AdS black hole solution. In the conventional radial coordinate,\nthe Reissner-Nordstrom-AdS black hole has the metric and gauge field:\nds2=−f(r)dt2+dr2\nf(r)+r2dΩd−1,\nAµdxµ=\u0010\nµ−q\nrd−2\u0011\ndt ,\nf(r) =r2+ 1−m\nrd−2+d−2\nd−1κ2q2\nr2(d−2), (2)\nwhere we use the notation κ2= 8πGN. The event horizon of the black hole is located at\nr=rhgiven by the largest root of f(rh) = 0. We will fix the U(1) gauge by demanding\nthe gauge field satisfies At(rh) = 0, which specifies that µis the chemical potential of the\nU(1). From these conditions, we obtain\nm=rd−2\nh \nr2\nh+ 1 +d−2\nd−1κ2q2\nr2(d−2)\nh!\n,\nµ=q\nrd−2\nh. (3)\nThe thermodynamic properties of the Reissner-Nordstrom-AdS black hole can be ob-\ntained from the above solution in a standard manner. The global energy and charge are\nrelated to the parameters in the solution as\nE=d−1\n16πGNωd−1m ,\nQ= (d−2)ωd−1q , (4)\nwhere ωd−1denotes the volume of a unit ( d−1)-sphere, ωd−1= 2πd\n2/Γ(d\n2). The chemical\npotential can then be written as a function of the charge as\nµ=Q\n(d−2)rd−2\nhωd−1. (5)\nThe temperature and the entropy are given by\nT=1\n4πf′(rh),\nS=rd−1\nhωd−1\n4GN. (6)\nThese thermodynamic quantities satisfy the first law of thermodynamics dE=TdS+µdQ.\n4In the following, we focus on the zero temperature limit, where the Reissner-Nordstrom\nblack hole becomes extremal. Note that in the extremal limit, we have µ=∂E\n∂Qbecause\nthe extremal black hole has T= 0 so F=E−TS=E, and µ=∂F\n∂Q=∂E\n∂Q. Then, we\nwant to express the energy Eas a function of charge Qexplicitly. We can obtain the\nclosed formulae valid for any charge in some particular dimensions, say d= 3,4 and 5.4\nFor example, in d= 3, we find (see e.g. [19][20])\nEd=3=4π√\n6\n9κ2r\n−1 +p\n1 + 6 κ2q2+ 6κ2q2\u0010\n3 +p\n1 + 6 κ2q2\u0011\n, (7)\nwhere qis related to Qas in (4). In this paper, instead, we will only use the small charge\nexpansion. The explicit form of the energy and chemical potential in the small charge\nexpansion is given in general dimensions by (see Appendix B for calculations)\nE=1\nκr\nd−1\nd−2Q\u0012\n1 +1\n2eQ2\nd−2−d2\n8(d−2)2eQ4\nd−2+d4\n16(d−2)4eQ6\nd−2+···\u0013\n,\nµ=1\nκr\nd−1\nd−2\u0012\n1 +d\n2(d−2)eQ2\nd−2−d2(d+ 2)\n8(d−2)3eQ4\nd−2+d4(d+ 4)\n16(d−2)5eQ6\nd−2+···\u0013\n,(8)\nwhere\neQ≡κQ\nωd−1p\n(d−1)(d−2)=κqr\nd−2\nd−1. (9)\nAt this point, we can verify that the global energy of the extremal Reissner-Nordstrom\nblack hole is a convex function of the charge. Since E(0) = 0, this implies that it is\nsuperadditive (see Appendix A). It is a non-trivial prediction of the black hole solution\nthat the expansion parameter is Q2\nd−2, which is dimension specific (rather than Q), and\nit should be contrasted with the charged soliton solution whose expansion parameter is\nQin any dimensions as we will see later.\nThe slope of the energy as a function of the charge sets the weak gravity conjecture\nbound. In this paper, we say that the weak gravity conjecture is satisfied when there is a\nstate whose energy-charge ratio is smaller than that of the extremal Reissner-Nordstrom\nblack hole in the small charge limit:\nE(Q)\nQ≤1\nκr\nd−1\nd−2. (10)\n4Eq. (52) is a quadratic, cubic, or quartic equation of r2\nhind= 3,4, or 5, respectively, and can be\nsolved for r2\nh, but it cannot be algebraically solved in general in d≥6, where the equation is quintic or\nof higher degrees.\n5Alternatively, we say that the weak gravity conjecture is violated when there is no state\nwhose energy-charge ratio is smaller than that of the extremal Reissner-Nordstrom black\nhole in the small charge limit: all the states satisfy\nE(Q)\nQ≥1\nκr\nd−1\nd−2. (11)\nIn the literature, there are some different notions of AdS weak gravity conjecture than\nthat used here (see e.g. [11][21][22][23][24][25] in relation to CFTs). At least for the (non-\ninteracting) charged scalar matter, we will see that this bound is relevant in our analysis\nof the hairy black hole. We also note that the bound we consdier is the same as the bound\nused in [26] to avoid the formulation of the naked singularity in the Einstein-Maxwell-\ncharged scalar system in AdS (in d= 3 studied there). We will discuss this point further\nbelow when we introduce the critical coupling ec.\nIn addition to the extremal Reissner-Nordstrom black hole, in this theory, there exists\na charged soliton solution, that has non-zero scalar condensate. When the weak gravity\nconjecture holds, we expect that the soliton solution, at least in the small charge limit,\nhas a lower energy than the extremal Reissner-Nordstrom black hole.\nWe have explicitly constructed the horizonless soliton solution within a perturbative\nexpansion with respect to the scalar condensate (or equivalently charge). See Appendix C\nfor details. They are reliable in the small charge limit, which we will focus on. Some\nexamples of the solutions in various dimensions with various scalar masses can be found\nin Appendix C. Here we present the final result of the energy and the chemical potential\nof the soliton as functions of the charge up to the second order in small charge expansion:\nE= (d−2)ωd−1\u0014∆\neq+Γ(∆)2Γ(2∆ + 1 −d/2)\n2 Γ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\u0012\n1−(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2\ne2\u0013\nq2\u0015\n=Q\u0014∆\ne+Γ(∆)2Γ(2∆ + 1 −d/2)\n2 Γ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\u0012\n1−(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2\ne2\u0013\nq\u0015\n,\nµ=∆\ne+Γ(∆)2Γ(2∆ + 1 −d/2)\nΓ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\u0012\n1−(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2\ne2\u0013\nq ,\n(12)\nwhere qis related to Qas in (4). We see that µ=∂E\n∂Q, which implies TS= 0 because\nE=F. There is no horizon, so the gravitational entropy is zero. In the Euclidean\ncontinuation of the horizonless soliton, the periodicity of the compactified time direction\nis arbitrary, so is the temperature. In this paper, we consider zero temperature.\n6Let us introduce the notion of the critical coupling ec. When the charge Qis small,\nthe energy of the extremal Reissner-Nordstrom black hole (8) and that of the soliton (12)\nbehave as\nERN≃1\nκr\nd−1\nd−2Q , E sol≃∆\neQ . (13)\nWe identify the critical coupling ecas the value of the coupling that equates these energies,\nec= ∆κr\nd−2\nd−1. (14)\nThis critical coupling is exactly the lower bound of the gauge coupling constant to preserve\ncosmic censorship discussed in [26] for d= 3.5We expect (14) gives the generalization of\nthe bound of [26] to general dimensions d≥3.\nThe critical coupling connects the soliton with the weak gravity conjecture mentioned\nabove. Given ∆ and small Q, the small charged soliton has a smaller energy than the\nextremal Reissner-Nordstrom black hole when e > e c. This is equivalent to saying that\nthe weak gravity conjecture is satisfied by the charged soliton. We can also say that\ngravity is weaker than the electric force and a horizonless charged soliton can be formed\nwithout gravitationally collapsing to a black hole. Alternatively, the small charged soliton\nhas a larger energy than the extremal Reissner-Nordstrom black hole when e < e c. This\nis equivalent to saying that the weak gravity conjecture does not hold because the electric\nforce is weaker than gravity.\nAnother point we would like to discuss about the properties of the charged soliton\nis the convexity of the energy. The energy of the soliton is convexd2E\ndQ2>0 when the\ncoupling constant satisfies\ne2>(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2=∆−d/2\n∆−d/4e2\nc. (15)\nThe right-hand side is not positive for ∆ ≤d/2 ind≥4 and 3 /4<∆≤3/2 ind= 3.6\nThis means that, when ∆ is in these ranges, the energy of the soliton is convex for\n5In [26], the lower bound of the scalar field charge (i.e. U(1) coupling constant) qWis given in their\nnormalization as q(theirs)\nW = ∆, where the AdS radius is set unity, but our normalization of the fields and\ncoupling constant is different from theirs. Rescaling the gauge field and gauge coupling constant in [26]\nso that they match ours, the lower bound reads q(ours)\nW = ∆κ/√\n2, which is nothing but (14) for d= 3:\nq(ours)\nW =ec|d=3.\n6Recall that the range that ∆ can take is ∆ ≥d/2−1 (unitarity bound).\n7any coupling constant e. Otherwise, the right-hand side is positive. Accordingly, if the\ncoupling constant is too small, the energy of the soliton is not convex. However, such a\ncoupling is smaller than the critical coupling (14) (for ∆ > d/2) below which the soliton\nis not the configuration with the smallest energy, and the convexity is irrelevant. If we\nfurther believe in the weak gravity conjecture, this parameter region does not arise.\nWe here observe that in d= 3,4 the inequality (15) obtained from solving Einstein’s\nequation is equivalent to demanding positivity of the binding energy γgrav+γphoton >0\nfrom the perturbative formula in [27] ( d= 4) and [28] ( d= 3). See also eq. (2.5) of [4]\nand discussions there. We expect a similar comparison can be done in other dimensions.\nThe only subtle case is 1 /2<∆<3/4 for d= 3, where the right-hand side of\nthe bound (15) is larger than the critical coupling, i.e. e2>4e2\nc. This means that the\nconvexity can be violated near e∼ec. Our formula implies that the attractive force from\nthe graviton exchange (i.e. γgravin [28]) becomes infinite at ∆ = 3 /4 and continues to be\nlarge down to the unitarity bound ∆ = 1 /2. We might want to say that some extended\nnotion of weak gravity conjecture is violated in this case even though e > e c. If the soliton\nis stable, which we do not know, it may violate the charge convexity. We, however, note\nthat our formula for 1 /2<∆<3/4 is obtained from the analytic continuation of ∆ >3/4\nand we have not constructed an explicit solution in this range, which may be singular.\nTo avoid the subtlety, we assume ∆ >3/4 for d= 3 hereafter.\n3 Hairy black hole\nAs we have seen, if the weak gravity conjecture is violated, even with the existence of the\nsoliton solution, the extremal Reissner-Nordstrom black hole solution has a lower energy\nin the small charge limit. What happens if we increase the charge? Does it show a phase\ntransition to the charged soliton? Or, does it continue to be the lowest energy state? In the\nfollowing, we will argue that at a certain critical charge, the extremal Reissner-Nordstrom\nblack hole forms a scalar hair. The formation of the hair is due to superradiant instability\nof the Reissner-Nordstrom-AdS black hole [17][29][30][31]. Similarly, if the weak gravity\nconjecture holds, the lowest energy solution with a given charge is the soliton (in the small\ncharge limit), but if we increase the charge further (as well as the energy), it collapses into\na hairy black hole. In this section, we will predict the thermodynamic properties of the\n8hairy black hole in the Einstein-Maxwell-charged scalar system, and in the next section,\nwe will study the nature of the phase transition.\nWe have already seen that both the horizonless charged soliton solution and the\nReissner-Nordstrom black hole solution are compatible with the thermodynamic relation\nµ=∂E\n∂Qat zero temperature. It is natural to expect that the hairy black hole solution\nalso obeys the thermodynamic relations. This observation enables us to use the thermo-\ndynamic equilibrium argument to predict the property of the hairy black hole as well.\nAs discussed in [14], when the charge is small, we will regard the extremal hairy black\nhole as a thermodynamic mixture of the extremal Reissner-Nordstrom black hole and the\nhorizonless charged soliton. We express the charge and the energy of the hairy black hole\nas a non-interacting sum as\nQhairy=aQRN+bQsol,\nEhairy=aERN(QRN) +bEsol(Qsol), (16)\nwhere the mixing parameters a, b≥0 are determined by the thermodynamic equilibrium\ncondition\nµRN(QRN) =µsol(Qsol). (17)\nThe thermodynamic equilibrium condition also demands that the temperature must be\nequal, and our assumption that the charged soliton has zero temperature makes it equi-\nlibrium with the extremal Reissner-Nordstrom black hole.\nThis simple picture gives a determination of the critical charge Qcabove which we\nwill see that the hairy black hole becomes the lowest energy state. When e < e c(i.e.\nweak gravity conjecture is violated), it is determined by equating the chemical potential\nof the Reissner-Nordstrom black hole at QRN=Qcand that of the soliton at Qsol= 0:\nµRN(Qc) =µsol(0). In the same way, when e > e c(i.e. weak gravity conjecture holds), it\nis determined by equating the chemical potential of the soliton at Qsol=Qcand that of\nthe Reissner-Nordstrom black hole at QRN= 0: µRN(0) = µsol(Qc).\nThe critical charge Qccan be analytically expressed in arbitrary dimensions.7For\ne < e c, let us take e2=e2\nc(1−θ)< e2\nc, where θ >0 is a small parameter. With this\n7Ford= 3, we assume ∆ >3\n4. See discussions after (15).\n9coupling, the chemical potential of the soliton at Qsol= 0 is given by\nµsol(0) =∆\ne=1\nκr\nd−1\nd−2\u0012\n1 +θ\n2+···\u0013\n. (18)\nComparing this with µRN(Qc) in (8) to the subleading terms, we obtain\nQc=ωd−1p\n(d−1)(d−2)d−1\nκ\u0012θ\nd\u0013d−2\n2\n. (19)\nFore > e c, we take e2=e2\nc(1 + θ)> e2\nc, where θ >0. The charge of the soliton that\nbalances with this shift of the coupling can be parametrized as Qc=qsol=bcθ. Then,\nthe series expansion of (12) in θbecomes\nµsol(Qc) =1\nκr\nd−1\nd−2+ \nbcdΓ(∆)2Γ(2∆ + 1 −d/2)\n(4∆−d)Γ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2−1\n2κr\nd−1\nd−2!\nθ+···.\n(20)\nThe leading order term is the same as µRN(0) from eq. (8). Then, from µRN(0) = µsol(Qc),\nthe critical charge is given by the vanishing of the subleading term in (20) as\nQc=4∆−d\n2κdωd−1p\n(d−1)(d−2)Γ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\nΓ(∆)2Γ(2∆ + 1 −d/2)θ . (21)\nNow we have determined the critical charge, let us study the energy of the hairy black\nhole above Qc. We solve the equilibrium condition (16) with suitable ansatz in the small\ncharge limit. The results are summarized below for d= 3,4,5 and Q > Q c. We will also\ngive comments on the hairy black holes in d≥6. For Q < Q c, we find that the energy\nof the hairy black hole is larger than that of the soliton or extremal Reissner-Nordstrom\nblack hole (and one of the mixing parameters in (16) becomes negative), so below we will\nnot discuss the hairy black holes with Q < Q c. For derivation, see Appendix D.\n3.1 Hairy black hole in e < e c\nWhen the coupling constant is smaller than the critical value as e2=e2\nc(1−θ), we can\nconstruct a hairy black hole with Q∼θ(d−2)/2.\nAdS 4(d= 3) The critical coupling is\nQc=ω2qc, q c=√\n6\n3κθ1/2. (22)\n10The energy of the hairy black hole in Q > Q cis\nE=ω2\nκ2mhairy, m hairy=mRN−3×22∆−5Γ(∆−1/2)2Γ(∆ + 1 /2)\nΓ(∆) Γ(2∆ −3/2)κ4(q2−q2\nc)2,\n(23)\nwhere\nmRN=√\n2κq+κ3q3\n2√\n2+O(q5). (24)\nThe energy satisfies mhairy < m RNif ∆ >3/4. The energy of the hairy black hole is\na convex function of the charge as we can see at O(q3) of the Reissner-Nordstrom black\nhole (24).\nAdS 5(d= 4) The critical coupling is\nQc= 2ω3qc, q c=√\n6\n4κθ . (25)\nThe energy of the hairy black hole in Q > Q cis\nE=3ω3\n2κ2mhairy, m hairy=mRN−2(2∆−1)\n3(3∆−2)κ2(q−qc)2, (26)\nwhere\nmRN=2√\n6\n3κq+2\n3κ2q2. (27)\nFor ∆ >1 (=d\n2−1 with d= 4), we have2\n3<2∆−1\n3∆−2<1. Hence, we find that the energy\nof the hairy black hole satisfies mhairy< m RNand also is a convex function of the charge.\nAdS 6(d= 5) The critical coupling is\nQc= 3ω4qc, q c=6\n5√\n5κθ3/2. (28)\nThe energy of the hairy black hole in Q > Q cis\nE=2ω4\nκ2mhairy, m hairy=√\n3κq+ √\n3\n2κqθ−6√\n15\n125θ5/2!\n=mRN− \n35/6κ5/3q5/3\n25/3−√\n3\n2κqθ+6√\n15\n125θ5/2!\n,(29)\n11where\nmRN=√\n3κq+35/6κ5/3q5/3\n25/3. (30)\nOne can check that mhairy< m RNanddmhairy\ndq|q=qc=dmRN\ndq|q=qcby expanding (29) around\nq=qc, where |q−qc| ≪θ3/2, as\nmhairy=mRN−5√\n15\n36√\nθκ2(q−qc)2+O\u0000\n(q−qc)3\u0001\n. (31)\nThe result (29) given up to O(q) is not sufficient to discuss the convexity of mhairy, so we\nalso calculate the next order O(q2) term and find the convexity as\nd2mhairy\ndq2=5κ2Γ(∆) Γ(2∆ −5/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −3/2)2>0. (32)\n3.2 Hairy black hole in e > e c\nWhen the coupling constant is larger than the critical value as e2=e2\nc(1 + θ), we can\nconstruct a hairy black hole with Q∼θ.\nAdS 4(d= 3) The critical coupling is\nQc=ω2qc, q c=22∆−3/2Γ(∆−1/2)2Γ(∆ + 1 /2)\n3κΓ(∆) Γ(2∆ −3/2)θ . (33)\nThe energy of the hairy black hole in Q > Q cis\nE=ω2\nκ2mhairy, m hairy=√\n2κq−22∆−3Γ(∆−1/2)2Γ(∆ + 1 /2)\n3 Γ(∆) Γ(2∆ −3/2)θ2\n=msol−3 Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆−1/2)2Γ(∆ + 1 /2)κ2(q−qc)2,(34)\nwhere\nmsol=√\n2κq+\u00123 Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆−1/2)2Γ(∆ + 1 /2)κ2q2−κqθ√\n2\u0013\n. (35)\nThe energy satisfies mhairy < m solif ∆ >3/4. Because the coefficient of q2inmhairy\nvanishes, we need to go to the next order to check the convexity. From the calculations\nincluding O(θ3) terms, we find\nd2mhairy\ndq2=3√\n2\n2κ3(q−qc), (36)\nso the convexity is satisfied in Q > Q c.\n12AdS 5(d= 4) The critical coupling is\nQc= 2ω3qc, q c=√\n6(2∆−1)\n4(∆−1)κθ . (37)\nThe energy of the hairy black hole in Q > Q cis\nE=3ω3\n2κ2mhairy, m hairy=msol−2(∆−1)2\n3(2∆−1)(3∆ −2)κ2(q−qc)2, (38)\nwhere\nmsol=2√\n6\n3κq+ \n2(∆−1)\n3(2∆−1)κ2q2−√\n6\n3κqθ!\n. (39)\nThe energy satisfies mhairy< m sol. Because 1 −∆−1\n3∆−2is positive in ∆ >1, the convexity\nis satisfied.\nAdS 6(d= 5) The critical coupling is\nQc= 3ω4qc, q c=22∆−2√\n3 Γ(∆ −3/2)2Γ(∆ + 1 /2)\n5κΓ(∆) Γ(2∆ −5/2)θ . (40)\nThe energy of the hairy black hole in Q > Q cis\nE=2ω4\nκ2mhairy, m hairy=msol−31/4Γ(∆)5/2Γ(2∆−5/2)5/2\n25∆−6Γ(∆−3/2)5Γ(∆ + 1 /2)5/2κ5/2(q−qc)5/2,\n(41)\nwhere\nmsol=√\n3κq+ \n5 Γ(∆) Γ(2∆ −5/2)\n22∆Γ(∆−3/2)2Γ(∆ + 1 /2)κ2q2−√\n3\n2κqθ!\n. (42)\nWe can see that mhairy< m sol. The convexity of mhairyis satisfied because the energy of\nthe soliton (42) is already convex and it dominates over the added term in (41).\nLet us conclude this section with comments on hairy black holes in d≥6. When\ne < e c(i.e. the weak gravity conjecture is violated), the similar construction outlined in\nAppendix D leads to the expression mhairy(q) =mRN(q) +CRN(q−qc)2+O((q−qc)3)\naround q=qc. The coefficient CRNis such that qd/d−2term in the small qexpansion\nofmhairy(q) vanishes and mhairy(q) is an integer power series in q. The above form of\nmhairy(q) implies that the phase transition is continuous between the Reissner-Nordstrom\nblack hole and the hairy black hole (i.e. the first derivatives of mwith respect to qare the\n13same at q=qc). The convexity of mhairy(q) can be checked from the explicit computation\nof the O(q2) term.\nWhen e > e c(i.e. the weak gravity conjecture holds), the similar ansatz in Appendix D\ninstead leads to the expression mhairy(q) =msol(q) +Csol(q−qc)d−2\n2around q=qc. To\ndetermine the precise numerical constant Csol(independent of θ), we need the explicit\nform of the soliton to higher orders than we calculated. This is beyond the scope of this\npaper and we do not attempt. Nevertheless, we can still argue that the energy of the\nhairy black hole must be convex for d≥6 because q2term in msol(q) is dominant over\nC(q−qc)d−2\n2in the small charge expansions of mhairy(q) and we know its convexity (as\ndiscussed at the end of the previous section).\n4 Phase transition and restoration of convexity\nLet us now study the would-be phase transition between the Reissner-Nordstrom solution\nand the soliton solution that we constructed in section 2. Later in this section, within the\nparameter space where our solution is valid, we will show that this type of phase transition\nnever happens due to the existence of the hairy black hole solution. We, nevertheless,\nfirst ask ourselves what would happen if the hairy black hole solution did not exist.\nFirst, consider the case when the weak gravity conjecture is violated (i.e. e < e c). In\nthe small charge limit, the extremal Reissner-Nordstrom black hole has the lowest energy.\nDepending on the parameter, it may or may not happen that if we increase the charge,\nthe charged soliton solution starts to possess a lower energy than the extremal Reissner-\nNordstrom black hole. When it does, generically there is a (would-be) discontinuous phase\ntransition from the Reissner-Nordstrom black hole to the charged soliton solution. It is\n(generically) discontinuous because the first derivative of the energy with respect to the\ncharge (i.e. chemical potential) at the phase transition point is discontinuous.\nSimilarly, when the weak gravity conjecture holds (i.e. e > e c), in the small charge\nlimit the charged soliton has the lowest energy. Then there may exist a phase transition\nto the extremal Reissner-Nordstrom black hole if we increase the charge. If it does, again\nthe phase transition would be discontinuous (see Fig. 1).\nOne can compute the would-be phase transition point more explicitly in the small\ncharge limit where our solutions are reliable. In AdS 4, solving mRN(q∗) =msol(q∗) in the\n14small charge limit (i.e. |θ| ≪1), we obtain for e2=e2\nc(1 +θ)> e2\nc\nq∗=22∆−1/2Γ(∆−1/2)2Γ(∆ + 1 /2)\n3κΓ(∆) Γ(2∆ −3/2)θ , (43)\nwhile there is no q∗>0 for e < e c(we assume ∆ >3/4), so the (would-be) phase\ntransition occurs when e > e c(i.e. the AdS weak gravity conjecture holds).\nIn AdS 5, a similar calculation gives for e2=e2\nc(1−θ)< e2\nc\nq∗=√\n6(2∆−1)\n2κ∆θ , (44)\nwhile there is no q∗>0 for e > e c, so the (would-be) phase transition occurs when e < e c\n(i.e. the AdS weak gravity conjecture does not hold).\nIn AdS d+1with d≥5, we observe that the (would-be) phase transition occurs when\ne < e c(i.e. the AdS weak gravity conjecture does not hold). For e2=e2\nc(1−θ)< e2\nc, we\nobtain\nq∗=1\nκr\nd−1\nd−2θd−2\n2. (45)\nwhile there is no q∗>0 for e > e c. Note that in all these cases, the scaling of q∗with\nrespect to θis the same as that for qcstudied in the previous section.\nWhen the phase transition is discontinuous, the convexity of the energy is violated.\nThe second derivative of the energy as a function of the charge becomes negative (in the\ndelta function sense) at the phase transition, and we can easily find three charges around\nthe phase transition point where the convexity inequality λE(Q1) + (1 −λ)E(Q2)≥\nE(λQ1+ (1−λ)Q2) is violated.\nNote, however, that this does notmean that the discontinuous phase transition causes\nthe violation of the superadditivity. The convex function with E(0) = 0 is always super-\nadditive (see Appendix A), but the converse is not necessarily true. Indeed, we can show\nE(Q) = min( ERN(Q), Esol(Q)) is superadditive. The elementary proof can be found in\nAppendix A.\nThe existence of the hairy black hole solution that we found in the previous section\ncompletely changes the story above. This is because it is always the case that the hairy\nblack hole enters the phase diagram before the would-be phase transition discussed above\n(i.e.qc< q∗). See Fig. 1 for a schematic plot of this situation. In particular, the would-be\nviolation of the convexity of the energy is circumvented as we will see.\n15The study in section 3 shows that above Qc, the hairy black hole is the lowest energy\nsolution of the system with a given charge Q.8The salient feature of the thermodynamic\nproperty of the hairy black hole solution is that not only the energy but also the chemical\npotential is a continuous function with respect to Q(while the higher derivative can be\ndiscontinuous). This can be explicitly seen from our formula in the previous section where\nthe energy difference is O((q−qc)2) or higher. It is continuous but is not analytic because\nthe higher derivative shows the discontinuity at Q=Qc. We further realize that it is a\nconvex function (at least within the small charge limit we have studied). We therefore\nconclude that the convexity of the energy is restored by the existence of the hairy black\nhole solution. Since the convex function with E(0) = 0 is superadditive (see Appendix A),\nthe energy is superadditive with respect to charge.\nLet us briefly discuss the nature of the phase transition from the AdS/CFT viewpoint.\nThe big difference between the (extremal) Reissner-Nordstrom black hole and the hairy\nblack hole is that the former does not have scalar hair. The scalar hair or the scalar\ncondensation induced in the hairy black hole suggests that in the dual CFT, the scalar\noperator which is dual to ϕ(with conformal dimension ∆) has a non-zero expectation\nvalue on the cylinder in this heavy state. It is distinct from the one-particle state that is\ndual to low-dimensional single trace operators with zero scalar expectation value. In the\nplaner black hole (or black membrane), the analogous phase transition discussed here is\nreferred to as a “holographic superconductor” [32][33] (or holographic superfluid), but we\nshould point out that the phase transition here is about the properties of operators on\nthe plane (or states in the cylinder).9\n5 Discussions\nIn this paper, we have studied the convexity of the lowest energy of the Einstein-Maxwell-\ncharged scalar system in AdS under the presence of a hairy black hole. In the small\n8Logically speaking, we do not exclude the possibility that a non-spherical solution, which we have\nnot studied, has lower energy than the spherical hairy black hole.\n9Schematically, one may say that while the Reissner-Nordstrom black hole is dual to Tr(Φm) with\nm∼O(N2), the hairy black hole obtained as adding soliton should look like Tr(Φm)(Tr(Φ2))l. It may\nbe interesting to study the implication of the phase transition in terms of the recent studies of black hole\noperators in CFTs [34][35].\n16charge limit, the existence of the hairy black hole solution restores the would-be violated\nconvexity due to the phase transition between a charged soliton and a Reissner-Nordstrom\nblack hole. It should be interesting to perform (numerical) analysis in the larger charge\nregime and examine if the convexity remains true.\nIndeed, there has been some interest in studying the lowest conformal dimensions\nof conformal field theories as a function of the charge, in particular in the large charge\nexpansion. Recent studies show that the large charge behavior of the generic conformal\nfield theories should be E(Q)∼Qd\nd−2[36][37]. We can easily check that the extremal\nReissner-Nordstrom black hole satisfies this scaling. It was also numerically verified in\ncharged soliton (with ∆ = 2 in d= 3) in [38]. It will be interesting to verify the behavior\nin the large charge hairy black holes with arbitrary ∆. Note that in some supergravity,\nthey found BPS hairy black hole solutions where E(Q)∝Qrather than Qd\nd−2scaling.\nApparently, the presence of the moduli allows the linear behavior. It is generally believed\nthat it should not exhibit such a behavior without supersymmetry, but there is no proof.10\nIn relation, it is highly doubtful but is not proved if there is any solution whose scaling\nbehavior is in between Qαwhere 1 < α <d\nd−2. It is an interesting study if we can or\ncannot engineer such solutions in gravity.\nThe study of the large charge behavior should shed light on the analytic property of\nlarge charge expansions. From the effective field theory viewpoint, it was not obvious if the\nlarge charge expansion is converging (down to Q∼0), has a finite radius of convergence,\nor is just an asymptotic expansion. The extreme Reissner-Nordstrom solution tells that\nthe function is smooth for the entire range of charge, but the large charge expansion\nhas a finite radius of convergence.11In this paper, we have shown that the hairy black\nhole solution shows a continuous phase transition, at which the function is not analytic.\nWhether the large charge expansion breaks down much above this phase transition point\nshould be studied further in detail.\nIn this paper, we have studied charged solitons and hairy black holes under the Dirich-\nlet boundary condition at the AdS boundary. It is an interesting question to allow more\ngeneral boundary conditions such as the Robin boundary condition to study the charged\n10By introducing |ϕ|4interaction with a suitable coefficient, we expect that we can adjust the Q2term\ninE(Q) in the charged soliton solutions so that it has a linear behavior in the small charge limit.\n11One can see that there is a branch cut in the negative QinERN(Q), so one can read the radius of\nconvergence in the positive Q∼1\nκ, which can be verified against d’Alembert’s ratio test.\n17solitons and hairy black holes [16][39]. In the dual field theory perspective, it would cor-\nrespond to double trace deformations [40][41]. The stability condition may change and\nwhether the convexity remains to hold is a non-trivial question to be addressed.\nOur analysis of the charged solitons and hairy black holes was limited to d≥3 in this\npaper. For d= 2, it is necessary to consider scalar hair around the charged BTZ black\nhole, and (extremal) hairy black holes can be drastically different from higher dimensions.\nThe charge convexity conjecture in [4] is also restricted to d >2, and it remains an open\nquestion if similar arguments can be established for d= 2 CFT. It would be interesting\nif this situation could be approached from three-dimensional gravity duals.\nFinally, it may be an interesting question to address the higher derivative gravitational\ncorrections to the small charge (hairy) black holes and solitons. To satisfy the weak\ngravity conjecture in the Minkowski space-time, it is often suggested [42][43] that the\nhigher derivative corrections make the extremal Reissner-Nordstrom black hole lighter. If\nthe same mechanism is at work in the AdS, it may conflict with the convexity conjecture.\nThe competition then may lead to new constraints on effective gravitational field theory\nin AdS.\nAcknowledgments\nThe authors would like to thank Yoshihiko Abe, Nick Dorey, and Toshifumi Noumi\nfor fruitful discussions. T.I. is supported in part by JSPS KAKENHI Grant Number\n19K03871. Y.N. is in part supported by JSPS KAKENHI Grant Number 21K03581.\nThe authors thank the Yukawa Institute for Theoretical Physics at Kyoto University,\nwhere this work was initiated during the YITP workshop “Strings and Fields 2023”.\nA Convexity vs Superadditivity\nIn this appendix, we collect some mathematical facts about convex functions in relation\nto superadditivity.\nWhen a function f(x) is convex between x1≤x≤x2, for all 0 ≤λ≤1,f(x) satisfies\nthe inequality\nλf(x1) + (1 −λ)f(x2)≥f(λx1+ (1−λ)x2). (46)\n18When f(x) is twice differentiable, f′′(x)≥0 in a segment implies the convexity in the\nsame segment. The proof is based on Taylor’s theorem\nf(x) =f(x0) +f′(x0)(x−x0) +f′′(x∗)\n2(x−x0)2, (47)\nwhere x0≤x∗≤x. By noting f′′(x∗)≥0 and setting x0=λx1+ (1−λ)x2with x=x1\nandx=x2, we have\nf(x1)≥f(x0) +f′(x0)(1−λ)(x1−x2),\nf(x2)≥f(x0) +f′(x0)λ(x2−x1). (48)\nAdding λtimes the first line and (1 −λ) times the second line, we obtain (46).\nWhen f(x) is convex in x≥0 and f(0) = 0, f(x) is superadditive (i.e. f(a) +f(b)≤\nf(a+b)). To show this, we first set x2= 0 in (46) to obtain f(λx1)≤λf(x1). Then,\nnoting f(a) =f(a\na+b(a+b))≤a\na+bf(a+b), we obtain\nf(a) +f(b)≤a\na+bf(a+b) +b\na+bf(a+b) =f(a+b). (49)\nLetg1(x) and g2(x) be two twice differentiable convex functions with g1(0) = g2(0) =\n0,g1(x)≤g2(x) when 0 ≤x≤x∗, and g2(x)≤g1(x) when x∗≤x. Let f(x) =\nmin[g1(x), g2(x)] for x≥0. Generically, f(x) is not convex, but we will show that it is\nsuperadditive.\nWhen a, b≥x∗ora+b≤x∗, the convexity of g1(x) org2(x) immediately implies the\nclaim. The nontrivial case is when a, b≤x∗buta+b≥x∗, or when a≥x∗, b≤x∗. In\nthe former case,\nf(a+b)−f(a)−f(b) =g2(a+b)−g1(a)−g1(b)≥g2(a+b)−g2(a)−g2(b)≥0,\n(50)\nand in the latter case,\nf(a+b)−f(a)−f(b) =g2(a+b)−g2(a)−g1(b)≥g2(a+b)−g2(a)−g2(b)≥0,\n(51)\nso the claim holds in all cases.\n19B Small charge extremal Reissner-Nordstrom-AdS\nblack hole\nWe want to express the energy and chemical potential (3) of the extremal Reissner-\nNordstrom black hole as functions of the charge. At T= 0, from f′(rh) = 0, the horizon\nradius and charge are related as\nq=rd−2\nh\n(d−2)κq\n(d−1)(d−2 +d r2\nh). (52)\nWhen the charge qis small, this equation can be inverted by a series expansion as\nr2\nh=eQ2\nd−2−d\n(d−2)2eQ4\nd−2+d2(d+ 1)\n2(d−2)4eQ6\nd−2−d4(d+ 2)\n3(d−2)6eQ8\nd−2+···, (53)\nwhere eQis defined in (9). Substituting the above expansion into Eandµin (3) and (4),\nwe obtain (8).\nC Explicit solution for the charged soliton\nIn this appendix, we present the explicit form of the solution of the equations of motion\ndescribing a horizonless charged soliton in the small charge expansion. We begin with\nthe spherically symmetric ansatz for the static solution of the Einstein-Maxwell-charged\nscalar system,\nds2=−f(r)dt2+g(r)dr2+r2dΩ2\nd−1,\nAµdxµ=At(r)dt ,\nϕ=ϕ(r), (54)\nwhere dΩ2\nd−1is the line element of a unit Sd−1. We have chosen the gauge so that the\ncomplex scalar field ϕis a real function of r. Substituting the ansatz into the equations\nof motion of (1), we obtain a set of equations to be solved:\nd−1\n2rf′+\u0012(d−1)(d−2)\n2(1−g) + Λ r2g−κ2r2ϕ′2+κ2m2\nϕr2gϕ2\u0013\nf\n+κ2\n2r2A′2\nt−κ2e2r2gA2\ntϕ2= 0,\nd−1\n2rg′−\u0012(d−1)(d−2)\n2(1−g) + Λ r2g+κ2r2ϕ′2+κ2m2\nϕr2gϕ2\u0013\ng\n20−κ2g\n2fr2A′2\nt−κ2e2r2g2\nfA2\ntϕ2= 0,\nA′′\nt+\u0012d−1\nr−f′\n2f−g′\n2g\u0013\nA′\nt−2e2gϕ2At= 0,\nϕ′′+\u0012d−1\nr+f′\n2f−g′\n2g\u0013\nϕ′+\u0012e2A2\nt\nf−m2\nϕ\u0013\ngϕ= 0.(55)\nHere,′denotes the derivative with respect to r.\nAround the background with ϕ= 0, we will solve these equations in perturbative\nseries with respect to a small parameter ϵas\nf(r) =f(0)(r) +f(2)(r)ϵ2+f(4)(r)ϵ4+···,\ng(r) =g(0)(r) +g(2)(r)ϵ2+g(4)(r)ϵ4+···,\nAt(r) =At(0)(r) +At(2)(r)ϵ2+At(4)(r)ϵ4+···,\nϕ(r) =ϕ(1)(r)ϵ+ϕ(3)(r)ϵ3+ϕ(5)(r)ϵ5+···. (56)\nThe strategy is to first solve the scalar equation at O(ϵ), where the integration constants\nare fixed by the asymptotic behavior. Then, at O(ϵ2), we first solve gandf. Finally,\nwe solve At. One integration constant appearing in Atis not fixed at this point but will\nbe fixed by studying the regularity of the scalar function in O(ϵ3). In this sense, the\nequations at O(ϵ2) and O(ϵ3) are bundled. Then, we continue to O(ϵ4) for f,g, and At,\nand so on.\nIn the zeroth order in ϵ, the background is the empty AdS spacetime with a constant\ngauge field specified by the chemical potential µ(0),\nf(0)(r) = 1 + r2, g (0)(r) =1\n1 +r2, A t(0)(r) =µ(0), (57)\nwhere we assume µ(0)>0. The empty AdS spacetime has zero charge.\nAtO(ϵ), we turn on the scalar field. The equation for ϕ(1)reads\nϕ′′\n(1)+\u0012d−1\nr+2r\n1 +r2\u0013\nϕ′\n(1)+1\n1 +r2 \ne2µ2\n(0)\n1 +r2−∆(∆−d)!\nϕ(1)= 0, (58)\nwhere we used m2\nϕ= ∆(∆ −d). At the center of the AdS r= 0, this second-order\ndifferential equation has regular and diverging solutions. To solve this equation (without\nhorizon), we impose regularity at r= 0. In the AdS boundary r→ ∞ , we impose the\nabsence of the source discussed as follows.\n21Inr→ ∞ , the behavior of the scalar field (not specific to O(ϵ) but to any orders) is\nϕ=c0(r∆−d+···) +c1(r−∆+···), where c0andc1are constants. For the absence of the\nsource in the boundary field theory, we require c0= 0. When ∆ > d/2,r−∆decays faster\nthan r∆−d. Hence, setting c0= 0 corresponds to keeping the subleading behavior of ϕ.\nThis case is called the standard quantization. When ∆ < d/ 2,r−∆decays slower than\nr∆−d, but the leading behavior is normalizable for ∆ in the range d/2−1<∆< d/2 [44].\nThis case is called the alternative quantization. The border ∆ = d/2 is the case that the\nscalar mass saturates the Breitenlohner-Freedman (BF) bound [45][46], where we have\nϕ=c0(r−d/2logr+···) +c1(r−d/2+···).\nIn the absence of the scalar field source c0= 0, the equation (58) is solved at discrete\nvalues of µ(0)satisfying\nµ(0)=∆ + 2 n\ne, (59)\nwhere nis a non-negative integer. This is nothing but the normal mode frequency ωof\nthe massive scalar field in the global AdS as ω=eµ(0)= ∆ + 2 n. At this µ(0), the regular\nsolution to (58) is given by the hypergeometric function as\nϕ(1)= (1 + r2)∆\n2+n\n2F1\u0012d\n2+n,∆ +n;d\n2;−r2\u0013\n, (60)\nwhose normalization is absorbed by ϵ. Using this as the seed, we proceed to the higher\norders in the perturbative series. In the following, we consider the solution with the lowest\nenergy for a given ∆, and hence we set n= 0. For n= 0, the above chemical potential\nand scalar field solution become\nµ(0)=∆\ne, ϕ (1)=1\n(1 +r2)∆/2. (61)\nForO(ϵ2) and higher, we have solved these equations for integer values of ∆ in various\ndimensions d. Some sample solutions are presented here.\nFord= 4,∆ = 4 ( m2\nϕ= 0), we reproduce the results in [12]. We obtain\nf(r) = 1 + r2−8(3 + 3 r2+r4)\n9(1 + r2)3κ2ϵ2+···,\ng(r) =1\n1 +r2+8r2(3 +r2)\n9(1 + r2)5κ2ϵ2+···,\nAt(r) =4\ne+\u00123e\n14−(3 + 3 r2+r4)e\n6(1 + r2)3−32κ2\n21e\u0013\nϵ2+···,\n22ϕ(r) =ϵ\n(1 +r2)2+···. (62)\nHere, we present the results up to O(ϵ2), but the perturbative solutions can be obtained\nalso in higher orders in ϵ. Once the equations of motion are solved, the mass m, chemical\npotential µ, and charge qcan be read off from the asymptotic behavior of the fields in\nr→ ∞ as\nf(r) =r2+ 1 + ··· −m\nrd−2+···,\nAt(r) =µ+··· −q\nrd−2+···. (63)\nThese are chosen to agree with the notation of m,µ, and qused in the Reissner-Nordstrom\nblack hole solution (2). For d= 4,∆ = 4, we find\nm=8κ2\n9ϵ2+(78336 κ2−6767e2)κ2\n39690ϵ4+···,\nµ=4\ne+\u00123e\n14−32κ2\n21e\u0013\nϵ2+122400480 κ2e2−574944256 κ4−6383817 e4\n97796160 eϵ4+···,\nq=e\n6ϵ2+(2658 κ2−241e2)e\n6615ϵ4+···, (64)\nwhere we explicitly included the O(ϵ4) contributions that are necessary for obtaining\nmtoO(q2). In the above expression, we have used the degrees of freedom to rescale\nϵto set the coefficient of r−∆inϕ(r) inr→ ∞ to be exactly ϵ(no higher orders in\nϵ):ϕ(r) =ϵ/r4+···. From this we can also read off the scalar condensate (vacuum\nexpectation value), but we will not use it, so we omit it. Because we wish to express m\nandµperturbatively as functions of small q, we invert qas\nϵ=r\n6\neq1/2+2√\n6(241 e2−2658κ2)\n735e3/2q3/2+···. (65)\nSubstituting this into mandµ, we obtain\nm=16κ2\n3eq+2κ2\n21\u0012\n9−64κ2\ne2\u0013\nq2+···,\nµ=4\ne+\u00129\n7−64κ2\n7e2\u0013\nq+···. (66)\nThus, we reproduce the results for the massless scalar in AdS 5[12].\nWe also provide some details for massive scalar in asymptotically AdS 4(d= 3) with\n∆ = 2 ( m2\nϕ=−2) [16]. We find\nf(r) = 1 + r2−r+ (1 + r2) tan−1r\nr(1 +r2)κ2ϵ2+···,\n23g(r) =1\n1 +r2−r−(1 +r2) tan−1r\nr(1 +r2)3κ2ϵ2+···,\nAt(r) =2\ne+\u0012e(1 + 5 r2)\n8(1 + r2)−κ2\n2e−etan−1r\n2r\u0013\nϵ2+···,\nϕ(r) =ϵ\n1 +r2+···. (67)\nFrom these, up to O(ϵ4), we obtain12\nm=πκ2\n2ϵ2+πκ2\n192\u0000\n(65 + 4 π2)e2−4(61 + 4 π2)κ2\u0001\nϵ4+···,\nµ=2\ne+\u00125e\n8−κ2\n2e\u0013\nϵ2+41e4(2π2−15) + 4 κ2e2(661−93π2) + 4κ4(98π2−821)\n384eϵ4+···,\nq=πe\n4ϵ2+πe\n192\u0000\ne2(25 + 2 π2)−4κ2(29 + 2 π2)\u0001\nϵ4+···. (68)\nRewriting mandµas functions of q, we find\nm=2κ2\neq+κ2\n4π\u0012\n5−4κ2\ne2\u0013\nq2+···,\nµ=2\ne+1\n2π\u0012\n5−4κ2\ne2\u0013\nq+···. (69)\nWe repeat the above calculations for various integer values of ∆ in general dimensions\nd. Some results of mandµin small qexpansion are listed below.\n•d= 3,∆ = 1 (alternative quantization):\nm=κ2\neq+κ2\n2π\u0012\n1 +κ2\ne2\u0013\nq2+···,\nµ=1\ne+1\nπ\u0012\n1 +κ2\ne2\u0013\nq+···. (70)\n•d= 3,∆ = 2:\nm=2κ2\neq+κ2\n4π\u0012\n5−4κ2\ne2\u0013\nq2+···,\nµ=2\ne+1\n2π\u0012\n5−4κ2\ne2\u0013\nq+···. (71)\n•d= 3,∆ = 3:\nm=3κ2\neq+7κ2\n4π\u0012\n1−3κ2\ne2\u0013\nq2+···,\nµ=3\ne+7\n2π\u0012\n1−3κ2\ne2\u0013\nq+···. (72)\n12Note that our expansion parameter ϵis different from [16], and therefore the coefficients in (68) are\ndifferent in higher orders in ϵ. However, the ambiguities in the parametrization of ϵcancel out when m\nandµare written as functions of qas in (69).\n24•d= 4,∆ = 2 (BF bound):\nm=8κ2\n3eq+2κ2\n9q2+···,\nµ=2\ne+1\n3q+···. (73)\n•d= 4,∆ = 3:\nm=4κ2\neq+8κ2\n15\u0012\n1−3κ2\ne2\u0013\nq2+···,\nµ=3\ne+4\n5\u0012\n1−3κ2\ne2\u0013\nq+···. (74)\n•d= 4,∆ = 4:\nm=16κ2\n3eq+2κ2\n21\u0012\n9−64κ2\ne2\u0013\nq2+···,\nµ=4\ne+\u00129\n7−64κ2\n7e2\u0013\nq+···. (75)\n•d= 5,∆ = 2 (alternative quantization):\nm=3κ2\neq+κ2\n8π\u0012\n1 +2κ2\ne2\u0013\nq2+···,\nµ=2\ne+1\n6π\u0012\n1 +2κ2\ne2\u0013\nq+···. (76)\n•d= 5,∆ = 3:\nm=9κ2\n2eq+κ2\n16π\u0012\n14−27κ2\ne2\u0013\nq2+···,\nµ=3\ne+1\n12π\u0012\n14−27κ2\ne2\u0013\nq+···. (77)\n•d= 5,∆ = 4:\nm=6κ2\neq+3κ2\n16π\u0012\n11−72κ2\ne2\u0013\nq2+···,\nµ=4\ne+1\n4π\u0012\n11−72κ2\ne2\u0013\nq+···. (78)\nBy comparing such results for integer values of ∆ in general dimensions d, we find\nthat the general expression for mandµof the soliton up to O(q2) is given by\nm=2(d−2)∆κ2\n(d−1)eq+(d−2)κ2Γ(∆)2Γ(2∆ + 1 −d/2)\n(d−1)Γ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\u0012\n1−(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2\ne2\u0013\nq2,\n25µ=∆\ne+Γ(∆)2Γ(2∆ + 1 −d/2)\nΓ(d/2) Γ(2∆) Γ(∆ + 1 −d/2)2\u0012\n1−(d−2)∆2(∆−d/2)\n(d−1)(∆−d/4)κ2\ne2\u0013\nq .\n(79)\nWe note that coefficients diverge when ∆ = d/4, which can be realized only in d= 3\nat ∆ = 3 /4 (in d≥4, we always have ∆ > d/ 2−1≥d/4). This expansion formula,\ndeduced from the results of integer ∆, hence might not be suitable for ∆ ≤3/4 ind= 3.\nAt the end of section 2, we have discussed the same issue from the viewpoint of charge\nconvexity. To avoid subtleties, in this paper we assume ∆ >3/4 when d= 3.\nD Construction of small charge hairy black holes\nThe critical coupling e=ecis where the extremal Reissner-Nordstrom black hole and\nsoliton have the same energy for the same charge. Here, near e=ec, we evaluate the\nenergy of the hairy black hole as a non-interacting mix of the extremal Reissner-Nordstrom\nblack hole and soliton. This method was successfully employed for massless scalar in d= 4\nin [14], and here we generalize the calculations to other ∆ and d.\nD.1 Construction in e < e c\nIne < e c, the Reissner-Nordstrom black hole has a lower energy than the soliton around\nQ=Qc. At Q=Qc, we start to mix small charge soliton contributions to the Reissner-\nNordstrom black hole to evaluate the energy of a hairy black hole. We will see that the\nReissner-Nordstrom black hole is the lowest energy state in Q < Q c, while the hairy black\nhole is the lowest energy state in Q > Q c.\nIne < e c, we express the small difference of the coupling efrom ecby introducing a\nsmall parameter θas\ne2=e2\nc(1−θ). (80)\nFrom (52), the chemical potential and charge of the extremal Reissner-Nordstrom\nblack hole depends on small rh,\nµRN=1\nκr\nd−1\nd−2\u0012\n1 +d\n2(d−2)r2\nh+O(r4\nh)\u0013\n,\nqRN=rd−2\nh\nκr\nd−1\nd−2\u0012\n1 +d\n2(d−2)r2\nh+O(r4\nh)\u0013\n. (81)\n26To mix the Reissner-Nordstrom black hole and soliton, we parametrize rhand the charge\nof the soliton as perturbative series of θso that they can be compared at each order in the\nperturbative series. In e < e c, we introduce the ansatz that small soliton contributions\nare put on top of the Reissner-Nordstrom black hole. Hence, we devise the perturbative\nseries so that µRNandqRNare larger than µsolandqsol.\nD.1.1 d= 3\nTo mix the Reissner-Nordstrom black hole and soliton for d= 3 in e < e c, we take\nqRN∼rh∼θ1/2. We use the following ansatz for rh,\nrh=a1θ1/2+a2θ+a3θ3/2+a4θ2+···, (82)\nwhere we have explicitly included the terms necessary for calculating the hairy black hole\nenergy up to O(q2). With this ansatz, µRN,qRN, and mRNof the Reissner-Nordstrom\nblack hole can be expanded in powers of θas\nµRN=√\n2\nκ+3a2\n1√\n2κθ+3√\n2a1a2\nκθ3/2+3(3a4\n1−4a2\n2−8a1a3)\n4√\n2κθ2+−9a3\n1a2+ 6a2a3+ 6a1a4√\n2κθ5/2+···,\nqRN=√\n2a1\nκθ1/2+√\n2a2\nκθ+√\n2(3a3\n1+ 2a3)\n2κθ3/2+√\n2(9a2\n1a2+ 2a4)\n2κθ2+···,\nmRN= 2a1θ1/2+ 2a2θ+ 2(2 a3\n1+a3)θ3/2+ (6a2\n1a2+a4)θ2+···.\n(83)\nTo balance the chemical potential, we take the ansatz for the charge of the soliton as\nqsol=b1θ+b2θ3/2+b3θ2+···. (84)\nHere, the leading behavior of the soliton is chosen as qsol∼θin order to balance the\nchemical potential. Then, the chemical potential and mass of the soliton can be expressed\nas\nµsol=√\n2\nκ+\u00121√\n2κ+3b1Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ+3b2Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2θ3/2\n+\u00123\n4√\n2κ+(3b3+ 2b1(3−2∆))Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ2+···,\nmsol=√\n2b1κθ+√\n2b2κθ3/2+\u0012(b1+ 2b3)κ√\n2+3b2\n1κ2Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ2+···,\n(85)\n27where b1>0 is assumed so that msol>0. Note that, with the above ansatz, the charge\nof the hairy black hole has the series expansion of the form\nq=qRN+qsol=√\n2a1\nκθ1/2+ √\n2a2\nκ+b1!\nθ+···. (86)\nAt this point, we fix the redefinition ambiguities in the θexpansion by demanding\nq∼θ1/2with no higher θcorrections.13Then, each term in the above expansion should\nsatisfy\nq=√\n2a1\nκθ1/2,2a2\nκ+b1= 0, (87)\nand so on for the higher orders involving a3,4andb2,3. The balancing of the chemical\npotential µRN=µsolalso gives the equations\n3a2\n1√\n2κ=1√\n2κ+3b1Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2,\n3√\n2a1a2\nκ=3b2Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2, (88)\nand so on for a3andb3. From these equations, we can determine the coefficients a1,2,3,4and\nb1,2,3. The upshot is that, in q > q c, the mass of the hairy black hole mhairy=mRN+msol\nis given by\nmhairy=mRN−3×22∆−5Γ(∆−1/2)2Γ(∆ + 1 /2)\nΓ(∆) Γ(2∆ −3/2)κ4(q2−q2\nc)2, (89)\nwhere\nqc=√\n6\n3κθ1/2,\nmRN=√\n2κq+κ3q3\n2√\n2+O(q5). (90)\nRecall we assume ∆ >3/4. Then, from (89), we see that mhairy< m RN. Note that we\nalso find b1→0 inq→qcfrom above.\nIn fact, we find qsol<0 in q < q c, and therefore the above expression cannot be\napplied in q < q c, where msolwould be msol<0 ifmsol=qsol+···is assumed. This\n13In [14], a different choice to fix the ambiguities was made (i.e. qsolhas no higher order θcorrections),\nbut eventually, if we express mhairy as a function of q, the ambiguities will be gone.\n28means that we need to parametrize the soliton mass as msol=|qsol|+···. Then, repeating\nthe calculations, we find that the mass of the hairy black hole in q < q cis given by\nmhairy=mRN+22∆−1Γ(∆−1/2)2Γ(∆ + 1 /2)\nΓ(∆) Γ(2∆ −3/2)κ2(q2\nc−q2). (91)\nThus, mhairy > m RNinq < q cif ∆ >3/4. This is reasonable because in q < q cthis\nconstruction attempts to approximate the hairy black hole as a non-interacting mix of\nthe Reissner-Nordstrom black hole and soliton with the opposite charges.\nD.1.2 d= 4\nInd= 4, we use qRN∼r2\nh∼θ. The ansatz for rhis\nr2\nh=a1θ+a2θ2+···. (92)\nWith this ansatz, µRN,qRN, and mRNtakes the form\nµRN=√\n6\n2κ+√\n6a1\n2κθ+√\n6(2a2−a2\n1)\n4κθ2+···,\nqRN=√\n6a1\n2κθ+√\n6(a2\n1+a2)\n2κθ2+···,\nmRN= 2a1θ+ (3a2\n1+ 2a2)θ2+···. (93)\nTo match this, we use the ansatz for the charge of the soliton as\nqsol=b1θ+b2θ2+···. (94)\nThe chemical potential and mass of the soliton are\nµsol=√\n6\n2κ+ \nb1(∆−1)\n2∆−1+√\n6\n4κ!\nθ+ \n(∆−1)(b2−b1(∆−2))\n2∆−1+3√\n6\n16κ!\nθ2+···,\nmsol=2√\n6κb1\n3θ+1\n3\u0012√\n6κ(b1+ 2b2) +2κ2b2\n1(∆−1)\n2∆−1\u0013\nθ2+···,\n(95)\nwhere b1>0. The charge of the hairy black hole has the series expansion of the form\nq=qRN+qsol= √\n6a1\n2κ+b1!\nθ+ √\n6(a2\n1+a2)\n2κ+b2!\nθ2+···. (96)\n29Solving µRN=µsoland fixing the redefinition ambiguities by demanding q∼θwith no\nhigher θcorrections, we can determine a1,2andb1,2. The mass mhairy =mRN+msolis\nobtained in q > q cas\nmhairy=mRN−2(2∆−1)\n3(3∆−2)κ2(q−qc)2, (97)\nwhere\nqc=√\n6\n4κθ ,\nmRN=2√\n6\n3κq+2\n3κ2q2. (98)\nInq < q c, where qsol<0, we find\nmhairy=mRN+4√\n6(2∆−1)κ\n3(3∆−2)(qc−q)> m RN. (99)\nD.1.3 d= 5\nInd= 5, we use qRN∼r3\nh∼θ3/2. The ansatz for rhwe choose is\nrh=a1θ1/2+a2θ+a3θ3/2+a4θ2+···. (100)\nWith this ansatz, µRN,qRN, and mRNcan be parametrized as\nµRN=2√\n3κ+5a2\n1\n3√\n3κθ+10a1a2\n3√\n3κθ3/2+5(−5a4\n1+ 12a2\n2+ 24a1a3)\n36√\n3κθ2\n+5(−5a3\n1a2+ 6a2a3+ 6a1a4)\n9√\n3κθ5/2+···,\nqRN=2a3\n1√\n3κθ3/2+2√\n3a2\n1a2\nκθ2+5a5\n1+ 18a1a2\n2+ 18a2\n1a3\n3√\n3κθ5/2\n+25a4\n1a2+ 6a3\n2+ 36a1a2a3+ 18a2\n1a4\n3√\n3κθ3+···,\nmRN= 2a3\n1θ3/2+ 6a2\n1a2θ2+\u00128a5\n1\n3+ 6a1a2\n2+ 6a2\n1a3\u0013\nθ5/2\n+\u001240a4\n1a2\n3+ 2a3\n2+ 12a1a2a3+ 6a4\n1a4\u0013\nθ3+···. (101)\nFor the matching of the scaling of the above charge, we take the ansatz for the charge of\nthe soliton as\nqsol=b1θ3/2+b2θ2+b3θ5/2+···. (102)\n30The chemical potential and mass of the soliton are\nµsol=2√\n3κ+1√\n3κθ+5×22−2∆b1Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2θ3/2\n+ √\n3\n4κ+5×22−2∆b2Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2!\nθ2\n+22−2∆(5b3−2b1(2∆−5))Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2θ5/2+···,\nmsol=√\n3κb1θ3/2+√\n3κb2θ2+√\n3\n2κ(2b3+b1)θ5/2+···, (103)\nwhere b1>0. The charge of the hairy black hole has the series expansion of the form\nq=qRN+qsol=\u00122a3\n1√\n3κ+b1\u0013\nθ3/2+ \n2√\n3a2\n1a2\nκ+b2!\nθ2+···. (104)\nWe solve µRN=µsolandq∼θ3/2fora1,2,3,4andb1,2,3. The mass mhairy=mRN+msolis\nthen given in q > q cby\nmhairy=√\n3κq+ √\n3\n2κqθ−6√\n15\n125θ5/2!\n=mRN− \n35/6κ5/3q5/3\n25/3−√\n3\n2κqθ+6√\n15\n125θ5/2!\n, (105)\nwhere\nqc=6\n5√\n5κθ3/2,\nmRN=√\n3κq+35/6κ5/3q5/3\n25/3. (106)\nInq < q c, where qsol<0, we find\nmhairy=mRN+ 2√\n3κ(qc−q)> m RN. (107)\nD.1.4 d≥6\nIne < e c, hairy black holes for d≥6 can be constructed in a similar way as d= 5 and\nare of no qualitative difference. We first notice qRN∼rd−2\nh∼θ(d−2)/2(see (19) and (52)).\nThen, we parametrize rhandqsolby series expansion in θso that qRNandqsolhave the\nsame powers of θ. To calculate the energy up to O(q2), for even d, we use the ansatz\nr2\nh=a1θ+a2θ2+···+ad/2θd/2+···,\n31qsol=b1θ(d−2)/2+b2θ(d−2)/2+1+···+bd/2θd−2+···. (108)\nand for odd d, we alternatively use the ansatz\nrh=a1θ1/2+a2θ+···+ad−1θ(d−1)/2+···,\nqsol=b1θ(d−2)/2+b2θ(d−2)/2+1/2+···+bd−2θd−2+···. (109)\nFor example, for d= 6, we have\nr2\nh=a1θ+a2θ2+a3θ3+···,\nqsol=b1θ2+b2θ3+b3θ4+···. (110)\nAs we have done above, we can determine parameters aiandbiby balancing the\nchemical potential µRN=µsoland demanding the charge to be q∼θ(d−2)/2(with no\nhigher order corrections). We can systematically calculate the energy of the hairy black\nhole for any d≥6. Here, we list some results.\n•d= 6: We have q∼θ2. The phase transition at q=qcis continuous as\nmhairy=mRN−\u001223/2κ3/2q3/2\n53/4−2√\n5κqθ+4\n27θ3\u0013\n=mRN−9κ2\n20θ(q−qc)2+O\u0000\n(q−qc)3\u0001\n, (111)\nwhere the second line is the Taylor expansion around q=qc, where |q−qc| ≪θ2,\nand\nmRN=4√\n5κq+23/2κ3/2q3/2\n53/4+···,\nqc=2√\n5\n9κθ2. (112)\nBy considering the O(q2) terms as well, we can check the convexity (because ∆ >2)\nas\nd2mhairy\ndq2=6(∆−1)(∆−2)2κ2\n5(2∆−1)(2∆ −3)>0. (113)\n•d= 7: We have q∼θ5/2. The phase transition at q=qcis continuous as\nmhairy=mRN− \n57/10κ7/5q7/5\n67/10−r\n5\n6κqθ+2×55/2\n77/2θ7/2!\n32=mRN−75/2κ2\n6×55/2θ3/2(q−qc)2+O\u0000\n(q−qc)3\u0001\n, (114)\nwhere the second line is the Taylor expansion around q=qc, where |q−qc| ≪θ5/2,\nand\nmRN=r\n10\n3κq+57/10κ7/5q7/5\n67/10+···,\nqc=25√\n6\n75/2κθ5/2. (115)\nBy considering the O(q2) terms as well, we can check the convexity (because ∆ >\n5/2) as\nd2mhairy\ndq2=7κ2Γ(∆) Γ(2∆ −7/2)\n9×22∆−3Γ(∆ + 1 /2) Γ(∆ −5/2)2>0. (116)\n•d= 8: We have q∼θ3. The phase transition at q=qcis continuous as\nmhairy=mRN− \n62/3κ4/3q4/3\n72/3−r\n6\n7κqθ+27\n256θ4!\n=mRN−64κ2\n189θ2(q−qc)2+O\u0000\n(q−qc)3\u0001\n, (117)\nwhere the second line is the Taylor expansion around q=qc, where |q−qc| ≪θ3,\nand\nmRN= 2r\n6\n7κq+62/3κ4/3q4/3\n72/3+···,\nqc=9√\n42\n128κθ3. (118)\nBy considering the O(q2) terms as well, we can check the convexity (because ∆ >3)\nas\nd2mhairy\ndq2=2(∆−1)(∆−2)(∆−3)2κ2\n7(2∆−1)(2∆ −3)>0. (119)\nD.2 Construction in e > e c\nIne > e c, the soliton has a lower energy than the Reissner-Nordstrom black hole around\nQ=Qc. At Q=Qc, we start to mix small charge Reissner-Nordstrom black hole\ncontributions to the soliton to obtain a hairy black hole. We will see that the soliton is\n33the lowest energy state in Q < Q c, while the hairy black hole is the lowest energy state\ninQ > Q c.\nIne > e c, we write the small difference of the coupling efrom ecas\ne2=e2\nc(1 +θ). (120)\nD.2.1 d= 3\nInd= 3 and e > e c, we take qRN∼rh∼θso that qRNis not more dominant than qsol.\nThe ansatz for rhcan be taken as\nrh=a1θ+a2θ2+···. (121)\nWith this ansatz, µRN,qRN, and mRNcan be parametrized as\nµRN=√\n2\nκ+3a2\n1√\n2κθ2+···,\nqRN=√\n2a1\nκθ+√\n2a2\nκθ2+···,\nmRN= 2a1θ+ 2a2θ2+···. (122)\nTo match this, we use the ansatz for the charge of the soliton as\nqsol=b1θ+b2θ2+···. (123)\nThe chemical potential and mass of the soliton are then\nµsol=√\n2\nκ+\u0012\n−1√\n2κ+3b1Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ+\n+\u00123\n4√\n2κ+(3b2+ 2b1(2∆−3))Γ(∆) Γ(2∆ −3/2)\n22∆−1Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ2+···,\nmsol=√\n2b1κθ+\u0012(2b2−b1)κ√\n2+3b2\n1κ2Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆ + 1 /2) Γ(∆ −1/2)2\u0013\nθ2+···,(124)\nwhere b1>0. The charge of the hairy black hole takes the form\nq=qRN+qsol= √\n2a1\nκ+b1!\nθ+ √\n2a2\nκ+b2!\nθ2+···. (125)\nWe solve µRN=µsolandq∼θfora1,2andb1,2. The mass mhairy=mRN+msolis then\ngiven in q > q cas\nmhairy=√\n2κq−22∆−3Γ(∆−1/2)2Γ(∆ + 1 /2)\n3 Γ(∆) Γ(2∆ −3/2)θ2\n34=msol−3 Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆−1/2)2Γ(∆ + 1 /2)κ2(q−qc)2, (126)\nwhere\nqc=22∆−3/2Γ(∆−1/2)2Γ(∆ + 1 /2)\n3κΓ(∆) Γ(2∆ −3/2)θ ,\nmsol=√\n2κq+\u00123 Γ(∆) Γ(2∆ −3/2)\n22∆Γ(∆−1/2)2Γ(∆ + 1 /2)κ2q2−κqθ√\n2\u0013\n. (127)\nInq < q c, we find qRN<0, and hence we repeat the calculations by using the expression\nof the mass to be mRN= 2|a1|θ+···. Then, we obtain\nmhairy=msol+ 2√\n2κ(qc−q)> m sol. (128)\nD.2.2 d= 4\nInd= 4, we use qRN∼r2\nh∼θ. The ansatz for rhis\nr2\nh=a1θ+a2θ2+···. (129)\nWith this ansatz, µRN,qRN, and mRNare given by\nµRN=√\n6\n2κ+√\n6a1\n2κθ+√\n6(2a2−a2\n1)\n4κθ2+···,\nqRN=√\n6a1\n2κθ+√\n6(a2\n1+a2)\n2κθ2+···,\nmRN= 2a1θ+ (3a2\n1+ 2a2)θ2+···. (130)\nTo match this, we use the ansatz for the charge of the soliton as\nqsol=b1θ+b2θ2+···. (131)\nThe chemical potential and mass of the soliton are\nµsol=√\n6\n2κ+ \nb1(∆−1)\n2∆−1−√\n6\n4κ!\nθ+ \n(∆−1)(b1(∆−2) +b2)\n2∆−1+3√\n6\n16κ!\nθ2+···,\nmsol=2√\n6κb1\n3θ+1\n3\u0012√\n6κ(2b2−b1) +2κ2b2\n1(∆−1)\n2∆−1\u0013\nθ2+···.\n(132)\nThe charge of the hairy black hole is\nq=qRN+qsol= √\n6a1\n2κ+b1!\nθ+ √\n6(a2\n1+a2)\n2κ+b2!\nθ2+···. (133)\n35We solve µRN=µsolandq∼θfora1,2andb1,2. Then, we obtain the mass mhairy =\nmRN+msolas a function of q. The result is, in q > q c,\nmhairy=msol−2(∆−1)2\n3(2∆−1)(3∆ −2)κ2(q−qc)2, (134)\nwhere\nqc=√\n6(2∆−1)\n4(∆−1)κθ ,\nmsol=2√\n6\n3κq+ \n2(∆−1)\n3(2∆−1)κ2q2−√\n6\n3qθ!\n. (135)\nInq < q c, where qRN<0, we find\nmhairy=msol+4√\n6(∆−1)κ\n3(3∆−2)(qc−q)> m sol. (136)\nD.2.3 d= 5\nInd= 5, we use qRN∼r3\nh∼θ3/2. The ansatz for rhwe choose is\nrh=a1θ1/2+a2θ+a3θ3/2+a4θ2+···. (137)\nWith this ansatz, µRN,qRN, and mRNcan be parametrized as\nµRN=2√\n3κ+5a2\n1\n3√\n3κθ+10a1a2\n3√\n3κθ3/2+5(−5a4\n1+ 12a2\n2+ 24a1a3)\n36√\n3κθ2\n+5(−5a3\n1a2+ 6a2a3+ 6a1a4)\n9√\n3κθ5/2+···,\nqRN=2a3\n1√\n3κθ3/2+2√\n3a2\n1a2\nκθ2+5a5\n1+ 18a1a2\n2+ 18a2\n1a3\n3√\n3κθ5/2\n+25a4\n1a2+ 6a3\n2+ 36a1a2a3+ 18a2\n1a4\n3√\n3κθ3+···,\nmRN= 2a3\n1θ3/2+ 6a2\n1a2θ2+\u00128a5\n1\n3+ 6a1a2\n2+ 6a2\n1a3\u0013\nθ5/2\n+\u001240a4\n1a2\n3+ 2a3\n2+ 12a1a2a3+ 6a4\n1a4\u0013\nθ3+···. (138)\nFor the balancing of the chemical potential, the ansatz for the charge of the soliton is\nchosen as\nqsol=b1θ+b2θ3/2+b3θ2+b4θ5/2+···. (139)\n36The chemical potential and mass of the soliton are\nµsol=2√\n3κ+\u0012\n−1√\n3κ+5×22−2∆b1Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2\u0013\nθ\n+5×22−2∆b2Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2θ3/2\n+ √\n3\n4κ+22−2∆(5b3+ 2b1(2∆−5))Γ(∆) Γ(2∆ −5/2)\n3 Γ(∆ + 1 /2) Γ(∆ −3/2)2!\nθ2+···,\nmsol=√\n3κb1θ+√\n3κb2θ3/2+ √\n3\n2κ(2b3−b1) +5b2\n1κ2Γ(∆) Γ(2∆ −5/2)\n22∆Γ(∆ + 1 /2) Γ(∆ −3/2)2!\nθ2\n+1\n2\u0012√\n3κ(2b4−b2) +5b1b2κ2Γ(∆) Γ(2∆ −5/2)\n22∆−2Γ(∆ + 1 /2) Γ(∆ −3/2)2\u0013\nθ5/2+···, (140)\nwhere b1>0. The charge of the hairy black hole is\nq=qRN+qsol=b1θ+\u00122a3\n1√\n3κ+b2\u0013\nθ3/2+···. (141)\nSolving µRN=µsolandq∼θ, we obtain a1,2,3,4andb1,2,3,4. The mass mhairy=mRN+msol\nis then expressed as the following function of q, inq > q c,\nmhairy=msol−31/4Γ(∆)5/2Γ(2∆−5/2)5/2\n25∆−6Γ(∆−3/2)5Γ(∆ + 1 /2)5/2κ5/2(q−qc)5/2, (142)\nwhere\nqc=22∆−2√\n3 Γ(∆ −3/2)2Γ(∆ + 1 /2)\n5κΓ(∆) Γ(2∆ −5/2)θ ,\nmsol=√\n3κq+ \n5 Γ(∆) Γ(2∆ −5/2)\n22∆Γ(∆−3/2)2Γ(∆ + 1 /2)κ2q2−√\n3\n2κqθ!\n. (143)\nInq < q c, no real solution satisfying µRN=µsolwas obtained.\nReferences\n[1] L. D. Landau and E. M. Lifshitz, Butterworth-Heinemann, 1980, ISBN 978-0-7506-\n3372-7\n[2] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, JHEP 06, 060 (2007)\ndoi:10.1088/1126-6708/2007/06/060 [arXiv:hep-th/0601001 [hep-th]].\n[3] D. Harlow, B. Heidenreich, M. Reece and T. Rudelius, Rev. Mod. Phys. 95, no.3, 3\n(2023) doi:10.1103/RevModPhys.95.035003 [arXiv:2201.08380 [hep-th]].\n37[4] O. Aharony and E. Palti, Phys. Rev. D 104, no.12, 126005 (2021)\ndoi:10.1103/PhysRevD.104.126005 [arXiv:2108.04594 [hep-th]].\n[5] O. Antipin, J. Bersini, F. Sannino, Z. W. Wang and C. Zhang, JHEP 12, 204 (2021)\ndoi:10.1007/JHEP12(2021)204 [arXiv:2109.04946 [hep-th]].\n[6] R. Moser, D. Orlando and S. Reffert, JHEP 02, 152 (2022)\ndoi:10.1007/JHEP02(2022)152 [arXiv:2110.07617 [hep-th]].\n[7] E. Palti and A. Sharon, JHEP 09, 078 (2022) doi:10.1007/JHEP09(2022)078\n[arXiv:2206.06703 [hep-th]].\n[8] D. Orlando and E. Palti, Phys. Rev. D 108, no.8, 085002 (2023)\ndoi:10.1103/PhysRevD.108.085002 [arXiv:2303.02178 [hep-th]].\n[9] O. Aharony and Y. N. Breitstein, JHEP 08, 044 (2023)\ndoi:10.1007/JHEP08(2023)044 [arXiv:2305.08947 [hep-th]].\n[10] A. Sharon and M. Watanabe, JHEP 05, 202 (2023) doi:10.1007/JHEP05(2023)202\n[arXiv:2301.08262 [hep-th]].\n[11] Y. Nakayama and Y. Nomura, Phys. Rev. D 92, no.12, 126006 (2015)\ndoi:10.1103/PhysRevD.92.126006 [arXiv:1509.01647 [hep-th]].\n[12] P. Basu, J. Bhattacharya, S. Bhattacharyya, R. Loganayagam, S. Minwalla and\nV. Umesh, JHEP 10, 045 (2010) doi:10.1007/JHEP10(2010)045 [arXiv:1003.3232\n[hep-th]].\n[13] S. Bhattacharyya, S. Minwalla and K. Papadodimas, JHEP 11, 035 (2011)\ndoi:10.1007/JHEP11(2011)035 [arXiv:1005.1287 [hep-th]].\n[14] O. J. C. Dias, P. Figueras, S. Minwalla, P. Mitra, R. Monteiro and J. E. Santos,\nJHEP 08, 117 (2012) doi:10.1007/JHEP08(2012)117 [arXiv:1112.4447 [hep-th]].\n[15] K. Maeda, S. Fujii and J. i. Koga, Phys. Rev. D 81, 124020 (2010)\ndoi:10.1103/PhysRevD.81.124020 [arXiv:1003.2689 [gr-qc]].\n[16] S. A. Gentle, M. Rangamani and B. Withers, JHEP 05, 106 (2012)\ndoi:10.1007/JHEP05(2012)106 [arXiv:1112.3979 [hep-th]].\n38[17] ´O. J. C. Dias and R. Masachs, JHEP 02, 128 (2017) doi:10.1007/JHEP02(2017)128\n[arXiv:1610.03496 [hep-th]].\n[18] P. Basu, C. Krishnan and P. N. Bala Subramanian, JHEP 06, 139 (2016)\ndoi:10.1007/JHEP06(2016)139 [arXiv:1602.07211 [hep-th]].\n[19] O. Loukas, D. Orlando, S. Reffert and D. Sarkar, Nucl. Phys. B 934, 437-458 (2018)\ndoi:10.1016/j.nuclphysb.2018.07.020 [arXiv:1804.04151 [hep-th]].\n[20] Y. Nakayama, Phys. Lett. B 808, 135677 (2020) doi:10.1016/j.physletb.2020.135677\n[arXiv:2004.08069 [hep-th]].\n[21] S. Giombi and E. Perlmutter, JHEP 03, 026 (2018) doi:10.1007/JHEP03(2018)026\n[arXiv:1709.09159 [hep-th]].\n[22] A. Urbano, [arXiv:1810.05621 [hep-th]].\n[23] M. Montero, JHEP 03, 157 (2019) doi:10.1007/JHEP03(2019)157 [arXiv:1812.03978\n[hep-th]].\n[24] L. F. Alday and E. Perlmutter, JHEP 08, 084 (2019) doi:10.1007/JHEP08(2019)084\n[arXiv:1906.01477 [hep-th]].\n[25] M. Cho, S. Choi, K. H. Lee and J. Song, JHEP 11, 118 (2023)\ndoi:10.1007/JHEP11(2023)118 [arXiv:2308.01717 [hep-th]].\n[26] T. Crisford, G. T. Horowitz and J. E. Santos, Phys. Rev. D 97, no.6, 066005 (2018)\ndoi:10.1103/PhysRevD.97.066005 [arXiv:1709.07880 [hep-th]].\n[27] A. L. Fitzpatrick and D. Shih, JHEP 10, 113 (2011) doi:10.1007/JHEP10(2011)113\n[arXiv:1104.5013 [hep-th]].\n[28] S. Andriolo, M. Michel and E. Palti, JHEP 02, 078 (2023)\ndoi:10.1007/JHEP02(2023)078 [arXiv:2211.04477 [hep-th]].\n[29] S. W. Hawking and H. S. Reall, Phys. Rev. D 61, 024014 (2000)\ndoi:10.1103/PhysRevD.61.024014 [arXiv:hep-th/9908109 [hep-th]].\n[30] N. Uchikata and S. Yoshida, Phys. Rev. D 83, 064020 (2011)\ndoi:10.1103/PhysRevD.83.064020 [arXiv:1109.6737 [gr-qc]].\n39[31] S. R. Green, S. Hollands, A. Ishibashi and R. M. Wald, Class. Quant. Grav. 33, no.12,\n125022 (2016) doi:10.1088/0264-9381/33/12/125022 [arXiv:1512.02644 [gr-qc]].\n[32] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev. Lett. 101, 031601 (2008)\ndoi:10.1103/PhysRevLett.101.031601 [arXiv:0803.3295 [hep-th]].\n[33] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP 12, 015 (2008)\ndoi:10.1088/1126-6708/2008/12/015 [arXiv:0810.1563 [hep-th]].\n[34] P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, Phys. Rev. D 103, no.12,\n126006 (2021) doi:10.1103/PhysRevD.103.126006 [arXiv:2005.11240 [hep-th]].\n[35] S. Choi, S. Jeong, S. Kim and E. Lee, JHEP 09, 138 (2023)\ndoi:10.1007/JHEP09(2023)138 [arXiv:2111.10720 [hep-th]].\n[36] S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, JHEP 12, 071 (2015)\ndoi:10.1007/JHEP12(2015)071 [arXiv:1505.01537 [hep-th]].\n[37] D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, JHEP 05, 043 (2018)\ndoi:10.1007/JHEP05(2018)043 [arXiv:1710.11161 [hep-th]].\n[38] A. de la Fuente and J. Zosso, JHEP 06, 178 (2020) doi:10.1007/JHEP06(2020)178\n[arXiv:2005.06169 [hep-th]].\n[39] T. Harada, T. Ishii, T. Katagiri and N. Tanahashi, JHEP 06, 106 (2023)\ndoi:10.1007/JHEP06(2023)106 [arXiv:2304.02267 [hep-th]].\n[40] E. Witten, [arXiv:hep-th/0112258 [hep-th]].\n[41] M. Berkooz, A. Sever and A. Shomer, JHEP 05, 034 (2002) doi:10.1088/1126-\n6708/2002/05/034 [arXiv:hep-th/0112264 [hep-th]].\n[42] C. Cheung, J. Liu and G. N. Remmen, JHEP 10, 004 (2018)\ndoi:10.1007/JHEP10(2018)004 [arXiv:1801.08546 [hep-th]].\n[43] Y. Hamada, T. Noumi and G. Shiu, Phys. Rev. Lett. 123, no.5, 051601 (2019)\ndoi:10.1103/PhysRevLett.123.051601 [arXiv:1810.03637 [hep-th]].\n[44] V. Balasubramanian, P. Kraus and A. E. Lawrence, Phys. Rev. D 59, 046003 (1999)\ndoi:10.1103/PhysRevD.59.046003 [arXiv:hep-th/9805171 [hep-th]].\n40[45] P. Breitenlohner and D. Z. Freedman, Phys. Lett. B 115, 197-201 (1982)\ndoi:10.1016/0370-2693(82)90643-8\n[46] P. Breitenlohner and D. Z. Freedman, Annals Phys. 144, 249 (1982)\ndoi:10.1016/0003-4916(82)90116-6\n41"    },    {        "title": "2402.04553v1.Curvature_Informed_SGD_via_General_Purpose_Lie_Group_Preconditioners.pdf",        "content": "Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nOmead Pooladzandi* 1Xi-Lin Li* 2\nAbstract\nWe present a novel approach to accelerate\nstochastic gradient descent (SGD) by utilizing\ncurvature information obtained from Hessian-\nvector products or finite differences of param-\neters and gradients, similar to the BFGS algo-\nrithm. Our approach involves two precondition-\ners: a matrix-free preconditioner and a low-rank\napproximation preconditioner. We update both\npreconditioners online using a criterion that is\nrobust to stochastic gradient noise and does not\nrequire line search or damping. To preserve the\ncorresponding symmetry or invariance, our pre-\nconditioners are constrained to certain connected\nLie groups. The Lie group’s equivariance prop-\nerty simplifies the preconditioner fitting process,\nwhile its invariance property eliminates the need\nfor damping, which is commonly required in\nsecond-order optimizers. As a result, the learning\nrate for parameter updating and the step size for\npreconditioner fitting are naturally normalized,\nand their default values work well in most scenar-\nios. Our proposed approach offers a promising\ndirection for improving the convergence of SGD\nwith low computational overhead. We demon-\nstrate that Preconditioned SGD (PSGD) outper-\nforms SoTA on Vision, NLP, and RL tasks across\nmultiple modern deep-learning architectures. We\nhave provided code for reproducing toy and large\nscale experiments in this paper.\n1 Introduction\nOptimizing machine learning models with millions of free\nparameters presents a significant challenge. While conven-\ntional convex optimization algorithms such as Broyden-\nFletcher-Goldfarb-Shanno (BFGS), conjugate gradient\n(CG), & nonlinear versions like Hessian-free (HF) opti-\nmization (Martens and Sutskever, 2012) have succeeded in\n*Equal contribution1Department of Electical Engineer-\ning, University of California, Los Angeles, USA2Independent\nresearcher, San Jose, CA 95132. Correspondence to:\nOmead Pooladzandi <opooladz@g.ucla.edu>, Xi-Lin Li <lixil-\ninx@gmail.com>.\nPreprint , Copyright 2024 by the author(s).small-scale, convex mathematical optimization problems,\nthey are rarely used for large-scale, stochastic optimiza-\ntion problems that arise in machine learning (ML). One of\nthe main reasons is their reliance on the line search step.\nIn many ML models, such as variational & reinforcement\nlearning models, cost functions are defined as expectations\n& can only be evaluated through Monte Carlo (MC) sam-\npling averages. This can result in large variances, making\noptimizers that rely on line search to ensure convergence\nproblematic. Several recent extensions of these methods to\ndeep learning, such as K-BFGS & KFAC, have foregone\nthe line search step in favor of damping (Goldfarb et al.,\n2020; Martens and Grosse, 2015). However, this adds com-\nplexity by introducing extra hyperparameters.\nEmpirical results indicate that plain SGD is a highly effi-\ncient optimizer for most ML problems. However, for prob-\nlems with a large eigenvalue spread, SGD may converge\nslowly once the solution is located in a basin of attraction.\nAdaptive optimizers such as RMSProp & Adam (Kingma\nand Ba, 2015) converge faster but have been shown to gen-\neralize worse on many problems (Wilson et al., 2017; Zhou\net al., 2020a). Reducing the generalization gap between\nSGD & Adam remains an active topic of research (Zhuang\net al., 2020). This work focuses on providing SGD with\na good preconditioner to accelerate its convergence around\nthe basin of attraction without undermining its generaliza-\ntion capacity. The curvature information for preconditioner\nfitting can be sampled from Hessian-vector products or fi-\nnite differences of parameters & gradients, similar to the\nBFGS algorithm. However, constructing a preconditioner\nin a deterministic way, as in BFGS (Boyd and Vanden-\nberghe, 2004; Goldfarb et al., 2020), may not be possi-\nble due to potential issues with line search & damping.\nTherefore, we adopt the more general & gradient-noise-\nrobust preconditioner fitting criterion proposed in Li (2015)\n& fit the preconditioner online with another “gradient de-\nscent”(GD) algorithm. The key is to avoid making the pre-\nconditioner fitting more difficult & computationally expen-\nsive than the original parameter-learning problem.\nIn this paper, we propose using Lie groups as a tool for\npreconditioner fitting. The “GD” on a Lie group is simi-\nlar to common gradient descent in Euclidean space. It in-\nvolves applying a series of small transforms via multiplica-\ntion with I+µG, where µis a small scalar & Gis the group\n1arXiv:2402.04553v1  [cs.LG]  7 Feb 2024Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\ngenerator (see D.1). The Lie group is a rich & convenient\nspace to work in as moving a preconditioner around any\npoint on the group behaves similarly to moving it around\nthe identity element of the group, i.e., the identity matrix I.\nThis is known as the Lie group equivariance property.\nRecent curvature-aware optimization methods such as Hes-\nsianFree, KFAC, AdaHessian (Yao et al., 2021), K-BFGS,\nShampoo (Gupta et al., 2018) & GGT (Agarwal et al.,\n2018) have shown moderate empirical results in Deep\nLearning (Osawa et al., 2022). Yet, they require damping,\nline search, or regret & thus require many hyper-paramers\n& are prone to pitfalls that do not affect PSGD where gradi-\nent noises regularize parameter & preconditioner updates.\nContributions. In our paper, we are interested in the per-\nformance of two novel families of preconditioners (Sparse\nMatrix-Free & Low Rank Approximation) in the PSGD\nframework & the structures of the solutions learned by the\nnetwork. In particular, our main observation is that:\nPrecontioned Stochastic Gradient Descent on over-\nparametrized neural networks leads to flat generalized so-\nlutions. While standard optimizers can find solutions that\nare over confident in their predictions, PSGD finds solu-\ntions that are significantly flatter, generalize better while\nbeing drastically less confident.\nWe make the following contributions:\nNovel Preconditioners: In Section 3 we propose two novel\nprecondtioners for PSGD: an extreamly light weight diago-\nnal & coross-diagonal approximation to the Hessian coined\nXM AT, & a powerful light weight Low Rank Approxima-\ntion to the Hessian LRA. These preconditioners are de-\nsigned to leverage curvature information, enhancing the\nperformance of SGD. Both of these preconditioners can be\nused without any adjustment to NN architecture while set-\nting a new SOTA across many deep learning workloads.\nGlobal Curvature Information: The Affine version of\nPSGD as well as KFAC & Shampoo type precondition-\ners all only take each layer’s curvature information into ac-\ncount independently without consideration of interaction of\nlayers. This is clearly an oversimplification. PSGD XM AT\n& LRA are two proposed perconditoners that take into ac-\ncount interaction between all NN layers.\nLie Group Framework: Our approach utilizes a unique\nLie group framework for preconditioner fitting. This\nmethodology simplifies the preconditioner fitting process\nwhile preserving the symmetry & invariance properties,\ncontributing to more stable & efficient optimization.\nPSGD finds Flat Solutions Fast: In Section 5.1 we\npresent toy experiments showing PSGD is the only opti-\nmizer achieving quadratic convergence on the Rosenbrock\nobjective minimization as well as showing clear evidence\nthat PSGD finds flatter solutions while outperforming both\nstandard & sharpness aware optimization methods. Thereexists empirical evidence that width of a local optimum\nis related to generalization (Keskar et al., 2016; Izmailov\net al., 2018; Chaudhari et al., 2019).\nEmpirical Validation: In Section 5.2-5.4 we present\nextensive empirical evidence of PSGD setting a new\nSoTA across vision, natural language processing (NLP),\n& reinforcement learning (RL) tasks, & establishes new\nSoTA for many networks & settings, e.g. ResNet,\nLSTMs (Hochreiter and Schmidhuber, 1997) & GPT-\n2 (Radford et al., 2019). We consider optimization prob-\nlems involving MNIST (LeCun and Cortes, 2010), CIFAR-\n10 (CF10) (Krizhevsky, 2009), Penn Treebank (Marcus\net al., 1993), the works of Shakespeare, the Open Web Text\n(OWT) dataset (Gokaslan and Cohen, 2019), HalfCheetah\n& RoboWalker (Brockman et al., 2016). PSGD outper-\nforms SoTA methods with negligible overhead compared\nto SGD on a wide range of optimization tasks.\nIntuitive Experiments: In Section 6 we provide intuition\nfor understanding characteristics for solutions founds by\nPSGD. In addition to flat solutions; we observe four other\nproperties of solutions found by PSGD. textbfUncertanty\nAnalysis: PSGD does not overfit, converging to solutions\nthat are less certain about predictions while outperform-\ning other optimizers. Forgettability Statistics: PSGD fo-\ncuses on points that are important to generalization & not\nmemorization. Neuro-Plasticity: NNs trained with PSGD\nare flexible; usually NNs trained with corrupt data early in\ntraining concede significant performance even if the data\nis corrected later in training. NNs trained with PSGD can\nrecover this performance loss. Learning Long Term De-\npendence: With this simple scenario we show PSGD is the\nonly optimizer that solves the XOR problem, learning long\nterm dependence without memorizing partial solutions.\nThus, we conclude that PSGD is unique in its ability to\nboth improve generalization, decrease solution curvature\nwithout memorization in a data-adaptive fashion. PSGD\nprovides practitioners with a powerful, stable, & efficient\noptimization tool that can significantly enhance the perfor-\nmance of deep learning models in various domains.\n2 Related work & background on PSGD\n2.1 Notations\nThe objective is to minimize a loss function defined as an\nexpectation, L(θ) =Ez[ℓ(θ, z)], where θ∈Rnis the pa-\nrameter vector to be optimized & zis a random vector that\ncan be sampled to evaluate the loss ℓ(θ, z). We assume\nthat the considered problem is second-order differentiable.\nTo simplify the notation, we use ˆL(θ)to denote a sampled\nnoisy evaluation of L(θ). A step of SGD with learning rate\nµ& an optional positive definite preconditioner Pis:\nθi+1=θi−µP ∂ ˆL(θ)/∂θ|θ=θi(1)\n2Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nwhere iis the iteration index, µ >0is the learning rate, &\nPtypically is a variable or adaptive preconditioner. Once\nthe solution enters a basin of attraction centered at a local\nminimum θ∗, we approximate the iteration step in Eq 1 as:\nθi+1−θ∗≈(I−µPˆH)(θi−θ∗) (2)\nwhere ˆH=∂2ˆL(θ)\n∂θT∂θ|θ=θ∗is the sampled Hessian at the\nlocal minimum. Conceivably, the eigenvalue spread of PˆH\nlargely determines the speed of convergence of the quasi-\nlinear system in equation 2. Nearly quadratic convergence\nis possible if we can find a good approximation for H−1.\nHowever, ˆHis a noisy Hessian & is not necessarily positive\ndefinite, even if the exact one at θ∗, i.e., H, is.\n2.2 The Preconditioner Fitting Criterion\nWe adopt the preconditioner fitting criterion proposed in\nLi (2015). Let (δg, δθ )be associated gradient parameter\nperturbations. Then, the fitting criterion is:\nc(P) =Eδθ[δgTPδg+δθTP−1δθ] (3)\nWith autograd tools, we can replace the pair (δθ, δg )with\n(v,ˆHv), where vis a random vector, & ˆHvis the noisy\nHessian-vector product, which can be evaluated as cheaply\nas gradients. Criterion equation 3 only has one positive def-\ninite solution, P= (H2+Ev[ϵ2])−1\n2, even for indefinite H,\nwhere ϵ=ˆH−His a stochastic noise term. This precon-\nditioner automatically dampens gradient noise. It is worth\nnoting that criterion equation 3 gives the same precondi-\ntioner used in equilibrated SGD (ESGD) (Dauphin et al.,\n2015) & AdaHessian (Yao et al., 2021) when Pis diago-\nnal, i.e., E[v⊙v]⊘E[(ˆHv)⊙(ˆHv)], where ⊙&⊘denote\nelement-wise product & division, respectively.\n2.3 Preconditioners on Lie Groups Naturally Arise\nIt is natural to fit the preconditioner on a Lie group for sev-\neral reasons. First, by rewriting equation equation 1 as\nP−1\n2θi+1=P−1\n2θi−µ∂ˆL(θ)/∂(P−1\n2θ)|θ=θi,it is clear\nthat a preconditioned SGD is equivalent to SGD in a new\nset of coordinates defined by ϑ=P−1\n2θ. This coordinate\nchange consists of rotations & scalings, i.e., operations on\nthe orthogonal group O(n)& the group of nonsingular di-\nagonal matrices. We can represent this coordinate trans-\nform with matrix Q−1&, accordingly, P=QTQ. Thus,\nwe pursue a variable Qon the Lie group to fit it.\nSecond, PSGD can also be viewed as SGD with trans-\nformed features when the parameters to be learned are a\nlist of affine transform matrices (Li, 2019). Specifically, the\nmost commonly used feature transformations (e.g., whiten-\ning, normalization, & scaling) can be represented as matri-\nces on the affine groups. For example, the popular batch\nnormalization (Ioffe and Szegedy, 2015), layer normaliza-\ntion (Ba et al., 2016), & group normalization (Wu and He,\n2018) can be represented as a sparse affine Lie group matrix\nwhere only the diagonal & last column can have nonzerovalues (Li, 2019) (See Appendix ??). The decorrelated\nbatch normalization (Huang et al., 2018) is related to the\nwhitening affine preconditioner in Li (2019). Thus, the Lie\ngroup arises as a natural object to work with.\n3 General Lie Group Preconditioners\nHere we discuss relevant Lie groups properties, PSGD con-\nvergence & present two novel Lie group preconditioners.\n3.1 Properties and Convergence of PSGD\nLie groups have two properties that are particularly suit-\nable for our task. Like any group, a specific Lie group\npreserves certain symmetries or invariances. For exam-\nple, with Q∈GL+(n,R), the general linear group with\npositive determinant, ϑ&θwill always have the same ori-\nentation. This eliminates the need for damping to avoid\ndegenerate solutions, since P=QTQis guaranteed to be\ninvertible. The equivariance property of Lie groups further\nfacilitates the preconditioner fitting. The same group gen-\nerator, i.e., the one at the identity matrix, is used to move a\npreconditioner on any point of the Lie group.\nIn fact, the preconditioner Pestimated by PSGD converges\nto the inverse of “absolute” Hessian regardless of the defi-\nniteness of Hessian. From this, one can show that the pa-\nrameters converge following the established results in open\nliterature. For more details & proof see A. Note that the\nfollowing statements are general to PSGD.\nProposition 3.1. Assume that His invertible, & dQ=\n−µ∂c\n∂QorE=−µQT∂c\n∂Q. Then, Qconverges to ±|H|−0.5\nby update equation 6, Qnew= [(Q′)TQ′]0.5withQ′=\nQold+QE, & a small enough positive step size µ.\nCorollary 3.1.1. Assume L(θ)is second order differ-\nentiable with absolute eigenvalues of the Hessian well\nbounded, i.e., 0< l≤ |λ(H)| ≤u <∞. Then with\nPSGD, the loss drops at least with a linear rate, & param-\neters converge at least linearly to the optimal solution θ∗.\nIt is worth mentioning no convergence rate beyond lin-\near are observed for first or second order line-search-free\nstochastic optimization. Proposition 3.1 & Corollary 3.1.1\n(proved in A.1, A.2) are not aimed to push the theoreti-\ncal convergence limits. Instead they investigate how PSGD\nconverges asymptoticlly with small enough step size.\nThe preconditioners proposed in Li (2019) can only be ap-\nplied to a list of affine transform matrix parameters. Al-\nthough many machine learning models exclusively con-\nsist of affine transforms & activation functions, this is not\nalways the case. Additionally, it can be impractical to\nreparameterize many existing modules, such as a convo-\nlutional layer, into their equivalent affine transformation\nform. Hence, in this paper, we propose two types of novel\ngeneral purpose preconditioners.\n3Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n3.2 Sparse Matrix-Free Preconditioners\nLet us consider bijective mappings that take vectors in Rn\n& map them to other vectors in the same space, i.e., T:\nRn7→Rn. The following claim gives one way to construct\nsuch sparse matrix-free Lie group preconditioners.\nClaim 3.1. LetK={σ1, . . . , σ m}be a subgroup of the\npermutation group Sn. Then, linear transform T:Rn7→\nRn, T (x|a1, . . . , a m) =Pm\ni=1ai⊙σi(x), forms a sub-\ngroup of GL(n,R)parameterized with {a1, . . . , a m}if\nT(·|a1, . . . , a m)is bijective, where both ai&xare∈Rn.\nSee proof of Claim 3.1 in Appendix B.\nExample 1 : the group of invertible diagonal matrices. We\nmust have K={e}if|K|= 1, where eis the identity ele-\nment of Sn, i.e., e(x) =x. Then, Tsimply has a diagonal\nmatrix representation, i.e., T(x|a1) = diag( a1)x. Crite-\nrion equation 3 gives the preconditioner in ESGD (Dauphin\net al., 2015) & AdaHessian (Yao et al., 2021) as a special\ncase when Pis on this group.\nExample 2 : The group of “X-shape matrices.” Let K=\n{e, σf}, where σfdenotes the flipping permutation. Then:\nT(·|a, b)T(·|u, v) =T(·|a⊙u+b⊙σf(v),\na⊙v+b⊙σf(u))\nT−1(·|a, b) =T(·|σf(a)⊘c,−b⊘c)\nwhere c=a⊙σf(a)−b⊙σf(b). Clearly, such trans-\nforms form a Lie group if they are invertible, i.e., no ele-\nment of cis zero. The matrix representation of this Tonly\nhas nonzero diagonal & anti-diagonal elements, thus the\nname X-shape matrix (XM AT). This becomes our minimal\noverhead general purpose preconditioner for PSGD.\nExample 3 : The butterfly matrix. For an even n, subgroup\nK={e, sn\n2}induces a Lie group whose representations\nare invertible butterfly matrices, where sn\n2denotes circu-\nlar shifting byn\n2positions. This group of matrices are the\nbuilding blocks of Kaleidoscope matrices.\nAdditionally, the group GL(n,R)can be recovered by let-\ntingK={e, s1, . . . , s n−1}, where sidenotes circular\nshifting by ipositions. But, GL(n,R)is too expensive for\nlarge scale problems. The group of diagonal matrices, i.e.,\nthe Jacobi preconditioner, is sparse but empirically shown\nto be less effective without the help of momentum for cer-\ntain machine learning problems (Dauphin et al., 2015). We\nare mostly interested in the cases with 2≤ |K| ≤4. These\nLie groups are sparse enough, yet simple enough to derive\ntheir inverse manually, & at the same time can significantly\naccelerate the convergence of SGD by shortcutting gradi-\nents separated far away in positions.\n3.3 Low-Rank Approximation Preconditioner\nLow-rank approximation (LRA) is a standard technique for\nprocessing large-scale matrices. Commonly adopted formsof positive definite low-rank approximation, such as P=\nρI+UUT, cannot always be factorized as P=QTQfor\ncertain Lie groups, where ρ >0is a small positive number.\nAdditionally, this form of approximation is not effective for\nreducing eigenvalue spread. In many real-world problems,\nthe Hessian has a few very large & very small eigenvalues,\ni.e., tails on both ends of the spectra (Sagun et al., 2016;\n2017). However, all the eigenvalues of Pin this form are\nlower bounded by ρ, meaning that it can only fit one tail of\nthe spectra when rank (U)≪n.\nThus, we propose a new LRA with form Q=ρ(I+UVT),\nwhere ρis not necessarily small nor positive, & U&V\nhave rcolumns with r≪n. To justify this form of ap-\nproximation, we need to establish two facts. First, it forms\na Lie group. Second, P=QTQwith this form can fit\nboth tails of the spectra of Hessian, providing an accurate\ncharacterization of the curvature of a function, improving\noptimization algorithms, & assessing their robustness.\nClaim 3.2. Preconditioner P=QTQwithQ=ρ(I+\nUVT)can have positive eigenvalues arbitrarily larger\nthanρ2& arbitrarily less than ρ2with proper U&V.\nClaim 3.3. Ifρ̸= 0 &(I+VTU)−1or(I+UTV)−1\nexists, AV(ρ, U) = ρ(I+UVT)defines a subgroup\nofGL(n,R)parameterized with ρ&U. Similarly,\nAU(ρ, V) = ρ(I+UVT)defines another subgroup of\nGL(n,R)parameterized with ρ&V.\nSee proofs of Claim 3.2 &3.3 in Appendix C.1 & C.1.\nThe form of Qin Claim 3.2 is rather constrained as ρis\na scalar. In practice, we replace ρwith another Lie group\nmatrix & define QasQ=B(I+UVT). In our imple-\nmentations, we choose Bto be the group of diagonal ma-\ntrix with positive diagonals & update Balong with U&V\non two separate Lie groups. Note that now B(I+UVT)\ngenerally no longer forms a single Lie group.\n4 Practical Considerations\nAbove-proposed preconditioners can be fit online by min-\nimizing criterion equation 3 using GD on Lie groups. Un-\nlike traditional GD, moving an object on a Lie group is\nachieved by multiplying it with I+µG, where Gis the\ngroup generator & µis small enough such that ∥µG∥<1.\nThis series of small movements trace a curve on the Lie\ngroup manifold, & Gis in the tangent space of the group as\nthe Lie algebra is closed. See App D.\nNote that optimizer damping is neither necessary nor gen-\nerally feasible on a Lie group, although it is widely used\nin other second-order optimizers to avoid degenerate solu-\ntions. On one hand, by fitting Qon a connected Lie group,\nP=QTQcannot be singular. On the other hand, damping\ncould be incompatible with certain forms of Lie groups, as\nwe may not always be able to find another Q′on the same\ngroup such that Q′TQ′=QTQ+λI, where λ >0. This\n4Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\neliminates the need for setting up a proper damping sched-\nule. However, gradient clipping can be helpful in promot-\ning stability. The quadratic approximation leading to the\nquasi-linear system equation 2 is only valid within a certain\nregion around θ. Thus, ∥δθ∥=µ∥P∂ˆL(θ)/∂θ∥should be\nsmall enough such that θ+δθstill locates in this trust re-\ngion. We can adjust µor clip ∥P∂ˆL(θ)/∂θ∥to ensure that\n∥δθ∥is small enough.\nTheoretically, one Hessian-vector product evaluation dou-\nbles the complexity of one gradient evaluation. In prac-\ntice, we could update the curvature estimation with proba-\nbility 0.1. The cost of updating Pis negligible for r≤100\ncompared with SGD. Then the per iteration complexity of\nPSGD becomes 1 + 2×0.1 = 1 .2times of that of SGD, as\nshown empirically in 15 & 9. For time & space complexity\nanalysis see App. Tables 5 & 6. Lastly, we use a learning\nrate of 0.01 but find PSGD is quite robust to learning rate\nand weight decay, see App K.5.\nFor the LRA preconditioner, the gradients on the Lie\ngroups are given by 0.5∇B= diag[( Ph)hT−v(P−1v)T],\n0.5∇U= [( Qh)(Qh)T−(Q−Tv)(Q−Tv)T]V&\n0.5∇V= [(Qh)(Qh)T−(Q−Tv)(Q−Tv)T]U, respec-\ntively. Interested readers can refer to the App for details.\nTo put these together, we summarize the proposed PSGD\nmethods into Algorithms 1 2,4 and App E.\nAlgorithm 1 PSGD Optimizer\nInitialize: θ0,t←0,Q0∝I\nWhile θtnot converged do\nt←t+ 1\ngt← ∇ θft(θt−1)\nIfu < p withu∼ U(0,1)then\nht← ∇ θ(vT\ntgt), where vt∼ N(0, I)\nUpdate Qtvia(vt, ht):\nQ Update Step\nElseQt←Qt−1\ngt←QT\ntQtgt\nθt←θt−1−µ1gt\nend while\nAlgorithm 2 LRA Q Update Step\nPh=QT(Qh)\nP−1v=Q−1(Q−Tv) via Woodbury identity 2x\n∇d= (Ph)⊙h−v⊙(P−1v)\nd←d−µ2d⊙ ∇d/max(|∇d|)\nIfu <0.5withu∼ U(0,1)\n∇U= (Qh)(Qh)TV−(Q−Tv)(Q−Tv)TV\nU←U−µ2∥∇UVT∥−1∇U(I+VTU)\nElse\n∇V= (Qh)(Qh)TU−(Q−Tv)(Q−Tv)TU\nV←V−µ2∥U∇T\nV∥−1(I+V UT)∇V\nReturn Q= (I+UVT)diag( d)5 Empirical Results\nIn this work, we evaluate the performance of the PSGD\nalgorithm on a diverse set of tasks. First we consider\ntwo toy problems, Rosenbrock objective minimization to\nshow quadratic convergence of PSGD, as well as using a\nLeNet5 (Lecun et al., 1998) (see Figure 1b & Table 13)\nfor MNIST (LeCun and Cortes, 2010) digit recognition for\nstudying the generalization property of PSGD.\nNext, we benchmark more large-scale vision, natural\nlanguage processing (NLP), & reinforcement learning\n(RL) tasks. For each task we benchmark PSGD vs\nthe leading SoTA optimizers. In the domain of com-\nputer vision, we evaluate algorithm performance on the\nMNIST dataset via convex large-scale logistic regres-\nsion (see Appendix K.6). Additionally we consider the\nCIFAR10 (CF10) (Krizhevsky, 2009) & CF10 derived\ndatasets, namely noisy labels, blur-deficit, adversarial at-\ntacks, & class imbalances, (see Figure 2a & Table 15) with\nResNet18 (RN18) to explore generalization performance.\nFor NLP tasks, we study the use of LSTMs (Hochreiter\nand Schmidhuber, 1997) & GPT-2 style transformers (Rad-\nford et al., 2019), on various text corpora, including the\nPenn Treebank (Marcus et al., 1993), the complete works\nof Shakespeare, & the Open Web Text (OWT) dataset\n(Gokaslan and Cohen, 2019) (see Table 17 & 1). In the\nRL setting, we consider a Proximal Policy Optimization\n(PPO) (Schulman et al., 2017) applied to the HalfCheetah\n& RoboWalker environments using OpenAI’s gym envi-\nronments (Brockman et al., 2016) (see Fig. 5).\nTo provide insight into the fundamental differences be-\ntween SGD & PSGD, we perform uncertainty & forget-\ntability analysis (Toneva et al., 2018). Finally, we exam-\nine the pathological delayed XOR problem, first introduced\nin Hochreiter and Schmidhuber (1997), in order to further\nour understanding of the strengths & limitations of differ-\nent optimization methods.\n5.1 Performance Study with Toy Examples\nQuadratic Convergence: The first toy example is the\nminimization of the Rosenbrock function demonstrating\nthe recovery of curvature information by PSGD. As shown\nby Proposition 3.1, preconditioner Pfollows (HTH)−1/2.\nThis property helps PSGD to escape saddle points as P→\n∞when H→0, & accelerate convergence around the\nbasin of attraction. This quadratic convergence behavior of\nPSGD is clearly shown by the convergence curve solving\nthe Rosenbrock benchmark problem in Fig. 1a. For more\ncomparison see App Fig 18.\nPSGD Finds Flatter NNs: Next, we demonstrates that\nPSGD preserves the generalization property of its kin,\nSGD, as the gradient noises regularize both parameter &\npreconditioner updates. The task is the MNIST digit recog-\nnition with the LeNet5. Adam is selected as the counter-\n5Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n(a) Rosenbrock objective minimization\n (b) LeNet5 minima pair\n (c) PSGD excels in bfloat16\nFigure 1: (a) Rosenbrock objective minimization comparison among PSGD & its competitors. Only PSGD shows a clear quadratic\nconvergence curve. (b) MNIST hand written digit recognition with LeNet5. Hessians at the minima of Adam are estimated with a\ndummy LRA PSGD optimizer that only updates the preconditioner. (c) PSGD is able to significantly outperform closed form solutions\nfor curvature at bfloat16. This exemplifies the stability of our method.\nexample as it is known to be more easily trapped in sharp\nminima than SGD (Zhou et al., 2020b). Fig. 1b shows ten\npairs of minima, each starting from the same random initial\ninitialization. We see that PSGD converges to minima with\nflatter or smaller Hessian, i.e., larger preconditioner. From\nthe view of information theory, the total cross entropy &\n0.5 log det( H)≈ −0.5 log det( P)are good proxies of the\ndescription lengths of the train image-label pairs & model\nparameters, respectively. Fig. 1b shows that minima with\nsmaller description lengths tend to perform better on the\ntest sets as well, as suggested by an arrow pointing to the\ndown-left-front corner of the cube.\nCurvature Representations at Low Precision: Here we\nexemplify PSGD’s robust performance & resilience to\nnoise in low-precision settings. All PSGD implementations\nmaintain stability across both single & half (bfloat16) pre-\ncision computations. The inherent equivariance property\nassociated with the Lie group is instrumental in PSGD’s\napproach, leading to a scenario where PSGD effectively\nfits the product PH, which asymptotically approximates\nthe identity matrix I, irrespective of the condition number\nofH. In figure 1c we compare PSGD’s adaptive solution\nvs prevalent closed-form methods. Specifically we com-\npare fitting PSGD via P= tr( PH2+P−1−2H)vs the\nwell known closed form solution P=E[hhT]−0.5when\nv∼N(0, I). This showcases PSGD’s efficiency in solving\nthe secant equation. Moreover, PSGD circumvents numer-\nical challenges commonly encountered in operations such\nas matrix inversion, Cholesky decomposition, & eigenvalue\ndecompositions. These attributes contribute to PSGD’s ro-\nbustness & efficiency, especially in low-precision compu-\ntational environments. For more see App Fig 19.\n5.2 CIFAR-10 and Friends on ResNet18\nWe train an RN18 on the CF10 image classification task us-\ning PSGD, SGD, Adam, & Apollo (Adaptive Quasi New-\nton Diagonal). We adopt the cosine learning rate scheduler(Loshchilov and Hutter, 2016) & train for 200 epochs (see\nK.8). We find that as other first & second-order optimizers\nperformance reduces & increases in variance as task com-\nplexity increases, PSGD is able to sustain the classification\naccuracy of the RN18, outperforming other optimizers by\nas much as much as 64%, see Figure 2a Table 15 for details.\nThese tasks provide a rigorous test for the generalization\nability of optimizers. We maintain the tuned hyperparame-\nters from the standard RN18 CF10 classification task for a\nfair comparison between the optimizers. For robustness &\nsensitivity analysis on PSGD’s hyper-parameters as well as\nmore experimental details see K.5. We update the precon-\nditioner of PSGD every ten iterations, resulting in a 20%\noverhead over SGD, see Table 15 & K.4 for empirical &\ntheoretical timing analysis.\nCIFAR10 with Asymmetric Noisy Labels: We asym-\nmetrically perturb 60% of the CF10 training labels ran-\ndomly, resulting in one of the classes having approximately\n55% incorrect labels & the other 9 classes having 5%in-\ncorrect labels, yielding asymmetric label noise . We use\n(P)SGD, Adam & Apollo to train an RN18 on this task for\n100 epochs with 8 different seeds & compare train & test\nclassification accuracy Fig. 6a & 6b.\nFigure 6a & 6b, show that PSGD achieves the lowest av-\nerage train accuracy (assuming incorrect noisy labels) with\nthe highest ground truth test accuracy. SGD gets an average\ntraining accuracy between Adam & Apollo. SGD has seven\nruns reaching 55% memorizing the train set, yielding 10%\naccuracy on the test set. & one run (due to lucky initializa-\ntion) reaching 34% on the train set, learning an underfit yet\ngeneralizing solution, yielding 77% on the test set.\nFor SGD, this is not a standard case of over-fitting nor a\ncase of catastrophic forgetting, since the training accuracy\ndoes not change. Instead, our experiments show there ex-\nists a tipping point in the test accuracy at around epoch 35,\nwhere within a single epoch the test accuracy drops from\n71% to10% while the training accuracy shows no intelli-\n6Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nStandardResidual FreeBlur Deficit\nClass ImbalanceAdvaserial Noisy Best Noisy Final65707580859095Accuracy\nApollo\nAdam\nSGD\nPSGD\n(a) CF10 RN18 Experiments\n (b) Train Acc: Noisy Label CF10\n (c) Test Acc: Noisy Label CF10\nFigure 2: CIFAR-10 ResNet-18: (a) Robustness of PSGD: We clearly see as classification task increases in complexity( →), PSGD\nis able to consistently outperform other first and second order optimizers. (b) Assym Label Noise Train Acc: Accuracy plots based\non incorrect noisy labels. PSGD effectively mitigates label noise, learning the true underlying solution with low variance, while other\noptimizers tend to overfit/memorize the miss-leading trainset. (c) Assym Label Noise Test Acc: Under ground truth test labels, we see\nthat PSGD reaches a significantly better test accuracy with a very low variance compared to Apollo, Adam, and SGD.\ngible indication of this change. Furthermore, the test ac-\ncuracy does not increase for the rest of the 65 epochs. At\nthe 35 epoch mark Adam & Apollo both go through a pe-\nriod of high variance over-fitting that eventually converges.\nNote that the average margins or predicted probabilities in\nTable 3 indicate that PSGD finds a generalizable solution\nwhereas other first & second-order optimizers fall into a\nregime of overfitting/memorization further discussed in 6.\nIn conclusion, we show PSGD finds a generalizable so-\nlution that can mitigate both over & underfitting. Other\noptimizers easily over/under-fit or memorize incorrect im-\nage label pairs, have large optimization instabilities during\ntraining, & reach degenerate solutions where the NNs have\nreasonable train accuracy but random test accuracy. For\nmore details & experiments on learning under asymmetric\n&symmetric label noise see K.2 & K.2.\n5.3 Language Modeling: nanoGPT\nTransformers have become the de facto language modeling\narchitecture, largely replacing RNN-based models. Trans-\nformers employ sparse self-attention mechanisms that al-\nlow for efficient computation of gradients. Thus, they are\ntypically trained using first-order methods. In this section,\nwe study the effect of curvature information in training\ntransformer models. And provide results for LSTM-based\nexperiments for predicting the Penn TreeBank Dataset us-\ning Zhuang et al. (2020)’s framework in Table 17.\nIn a recent study, Karpathy (2022) proposed a frame-\nwork for reproducing GPT-2 results using the OpenWeb-\nText dataset. Here, we expand upon this by investigating\nthe training of GPT-2 models of different sizes on both the\nOpenWebText & Shakespeare character datasets. Our pri-\nmary objective is to benchmark the performance of PSGD\nagainst other SoTA optimization algorithms.\nAs shown in Table 1, our results indicate that PSGD con-\nsistently outperforms other optimization algorithms acrossTable 1: Comparing different test loss of optimizers over GPT-2\nstyle transformers on the Shakespeare-char (SC) & OpenWebText\n(OWT) datasets. PSGD outperforms other optimizers including\nthe SoTA optimizer AdamW over various model sizes. Trainined\non a single NVIDIA 3080 GPU.\nnanoGPT PSGD SGD AdamW AdanW AdaBelief AdanBelief\nSC: 0.82M 4.52 4.68 4.68 5.52 5.94 6.66\nSC: 1.61M 4.47 4.75 5.03 5.05 5.06 6.47\nSC: 6.37M 4.53 5.31 19.53 4.92 21.04 5.34\nOWT: 50M 197.07 257.86\nvarious model sizes. Notably, a moderate gap in perplex-\nity is observed for the smaller GPT-2 model trained for\n5k iterations on the Shakespeare-char dataset, with a sig-\nnificant gap observed on the 50M parameter GPT-2 LLM\ntrained on the OpenWebText dataset, for 600k iterations\nusing AdamW & 100k iterations using PSGD. We found\nthat decreasing the precond lr to 0.001 greatly improved the\nperformance of transformer models. Lowering the precond\nlrsmoothens the curvature in the sparse embedding layer\nof GPT-2 over time & enables the optimizer to consider a\nlarger window of curvature information. This “curvature\nmemory” improves performance & prevents divergence re-\nsulting from the otherwise sparse embedding space. For\nLSTM experiments see Table 17.\n5.4 Reinforcement Learning\nHere we consider two standard Proximal Policy Optimiza-\ntion (PPO) problems in Reinforcement Learning (RL):\nWalker2d and HalfCheetah. We optimize the actor and\ncritic independently. We compare the performance of the\ntwo SOTA optimizers AdaBelief and Adan to PSGD. We\noptimize the actor and critic independently.We find that\nPSGD can find a higher reward for both Walker2d &\nHalfCheetah as shown in Figure 5.\n7Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n(a)Walker2d\n (b)HalfCheetah\nFigure 3: PSGD outperforms SOTA optimizers on PPO RL on Walker2d & HalfCheetah.\n6 Understanding Second Order Optimizers\nWith the performance distinctions between PSGD & the\nSoTA optimiziers now apparent, we aim to conduct a series\nof simple experiments to better understand the nature of\nsolutions PSGD finds.\nUncertainty Analysis: We train a standard RN18 on CF10\nwith PSGD, SGD, Adam, & Apollo, & after 200 epochs\nwe check the entropy over the softmax-logits & miss-\nclassification margin Toneva et al. (2018) of both NNs. We\nsee in Table 3 that PSGD has higher entropy & a lower\nmargin of classification compared to other optimizers. This\nvery low minimum entropy of other optimizers may be in-\nterpreted as a form of overfitting or memorization. Intu-\nitively, some data points that are low entropy have infor-\nmation that is easily memorized by NNs, giving up network\ncapacity learning points that do not improve generalization.\nConversely, we observe that PSGD never becomes overly\ncertain about any data point, with the minimum entropy\nbeing six orders of magnitude larger than other optimizers.\nSimilar trends can be observed in the mean & maximum\nentropy values. From these statistics, we believe standard\nfirst & second-order optimizers can push NNs into a dan-\ngerous regime of overconfidence which given clean labels\ncan reduce their generalization capacity with the potential\nof catastrophic forgetting. In the scenario where a network\nis given noisy labels (or imaginably poisoned points), train-\ning other SoTA methods may lead to memorization of the\ntraining set with little to no generalization capacity as seen\nin the noisy labeled experiments 2a.\nPSGD’s higher entropy & lower margins are indications\nof a larger exploration of the parameter space, more di-\nverse solutions, & a lack of memorization. Suggesting that\nPSGD is able to better balance the trade-off between over-\nfitting & underfitting, without memorizing points.\nFor more on the nature of low & high entropy points and\nits connection to Forgettability see K.7 &K.8.\nForgettability Statistics & Learning Toneva et al. (2018)\nfound that one can prune 30% of the CF10 train samples\nwithout loss of generalization via Forgettability. Toneva\nfound point’s generalization utility is directly correlatedwith the number of times it is learned and forgotten during\ntraining. We investigate whether the performance differ-\nence between PSGD & SGD’s generalization performance\ncan be attributed to this forgettability ordering.\nWe train the RN18 and keep the top Nimportant points\nbased on each optimizer’s expected forgettability score. Ta-\nble 16 shows that PSGD focuses on points that are central\nto generalization. When we limit the dataset to only the\n5k most forgettable data points deemed by each optimizer,\nPSGD is able to outperform SGD by nearly 14%.\nPSGD Preserves Neuro-Plasticity: Recently, Achille\net al. (2017) studied the phenomenon of neuroplasticity\nin NNs. They found that NNs exhibit a critical learning\nperiod, during which if a learning deficit is not removed,\nthe NN becomes unable to learn effectively. To simulate\nthis deficit, CIFAR10 images are blurred for the first half\nof training, after which the deficit is removed. Following\nAchille et al. (2017), we used an exponential learning rate\ndecay. To investigate the impact of optimization on neuro-\nplasticity we compare SGD & PSGD. We find that PSGD\nretains neuro-plasticity outperforming SGD by 6% & 3%\non test & train sets seen in Fig 2a, 16b & Table 15.\nFor more on critical learning periods & the importance of\ncurvature information early in training for improving dis-\ntributional shift for transfer learning, see & K.8 & K.8.\nUnderstanding long term dependence and memoriza-\ntion via the Delayed XOR Problem: The task is to\npredict the XOR relation of aandbrandomly scattered\nfar away in a long sequence. This problem is challeng-\ning for many architectures and optimizers because it can-\nnot be “partially solved” as memorizing either aorbalone\ndoes not help to predict XOR (a, b). We consider solv-\ning it with the vanilla RNNs and LSTMs optimized us-\ning SGD, AdaBelief, Adan, AdaHessian, Apollo, Hessian\nFree, Shampoo, KFAC, and PSGD at different sequence\nlengths. Apollo was not able to solve the XOR problem\nin any scenario. The success rates for each optimizer are\nshown in Table 2. Clearly, PSGD LRA is the only method\nthat can solve the XOR problem passing sequence length\n32 with an RNN. Furthermore, LSTMs show no benefit to\n8Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nTable 2: Success rate on the delayed XOR problem with variant sequence lengths, optimizers, & networks.\nXOR PSGD LRA AdaHessian KFAC HessianFree Shampoo SGD AdaBelief Adan\nLength RNN LSTM RNN LSTM RNN LSTM RNN LSTM RNN LSTM RNN LSTM RNN LSTM RNN LSTM\n32 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1\n55 1 1 0 1 0 0 0 0.6 0 0.8 0 0 0 0 0 0\n64 1 1 0 0.8 0 0 0 0 0 0.6 0 0 0 0 0 0\n112 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0\nTable 3: Uncertainty Statistics: PSGD’s higher uncertainty leads\nto better generalization & less over-fitting.\nNTK Entropy Margin\nMin Mean Max Min Mean Max\nPSGD 0.139 0.260 1.545 0.144 0.956 0.994\nSGD 7x 10−70.01 0.8975 0.3925 0.999 1\nAdam 1.5x 10−70.009 0.8645 0.3625 0.999 1\nApollo 1x 10−60.05 0.8851 0.4326 0.999 1\nTable 4: Forgetting statistics for CF10 on RN18. PSGD finds\nbetter forgettability statistics outperforming SGD.\nForgetting 50k 25k 15k 5k\nPSGD 96.65 95.83 94.95 56.46\nSGD 96.21 95.56 93.7 42.48\nRNNs without using curvature information. Also, RNN op-\ntimized with PSGD outperforms LSTM optimized by any\nother methods. These results hint at two points. First, the\nchoice of optimizer could be equally important as model\narchitecture. Second, similar to the CF10 tasks, PSGD re-\nlies less on the memorization of train samples in problem-\nsolving. See K.8 for rank analysis.\n7 Conclusion\nThis work presents a comprehensive study of the proposed\ngeneral-purpose Lie group preconditioners for optimiza-\ntion of deep learning problems. We’ve provided theoretical\nguarantees for the convergence of Lie group preconditioned\noptimization methods, & empirical results demonstrating\nPSGD outperforms SoTA optimizers in generalization, ro-\nbustness, & stability across various tasks in Vision, NLP,\n& RL. Our analysis of forgettability & entropy shows that\nPSGD effectively focuses on forgettable points, leading to\nimproved generalization. These findings provide valuable\ninsights for further developing efficient & effective opti-\nmization techniques for deep learning models.\n8 Acknowledgments\nThis work was partially supported with Cloud TPUs from\nGoogle’s Tensorflow Research Cloud (TFRC). We would\nlike to acknowledge Yang Gao and Lalin Theverapperuma\nfor their discussion on PSGD as well as Yaroslav Bulatov\nfor his continued encouragements of the work on PSGD.REFERENCES\nAlessandro Achille, Matteo Rovere, and Stefano Soatto.\nCritical learning periods in deep neural networks. CoRR ,\nabs/1711.08856, 2017. URL http://arxiv.org/\nabs/1711.08856 .\nNaman Agarwal, Brian Bullins, Xinyi Chen, Elad Hazan,\nKaran Singh, Cyril Zhang, and Yi Zhang. Effi-\ncient full-matrix adaptive regularization. arXiv preprint\narXiv:1806.02958 , 2018.\nJimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton.\nLayer normalization, 2016. URL https://arxiv.\norg/abs/1607.06450 .\nS. Boyd and L. Vandenberghe. Convex Optimization . Cam-\nbridge University Press, 2004.\nGreg Brockman, Vicki Cheung, Ludwig Pettersson,\nJonas Schneider, John Schulman, Jie Tang, and Wo-\njciech Zaremba. Openai gym. arXiv preprint\narXiv:1606.01540 , 2016.\nYuan Cao and Quanquan Gu. Generalization error bounds\nof gradient descent for learning over-parameterized deep\nrelu networks, 2019. URL https://arxiv.org/\nabs/1902.01384 .\nPratik Chaudhari, Anna Choromanska, Stefano Soatto,\nYann LeCun, Carlo Baldassi, Christian Borgs, Jennifer\nChayes, Levent Sagun, and Riccardo Zecchina. Entropy-\nsgd: Biasing gradient descent into wide valleys. Journal\nof Statistical Mechanics: Theory and Experiment , 2019\n(12):124018, 2019.\nY . N. Dauphin, H. Vries, and Y . Bengio. Equilibrated adap-\ntive learning rates for non-convex optimization. In NIPS ,\npages 1504–1512. MIT Press, 2015.\nJia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and\nLi Fei-Fei. Imagenet: A large-scale hierarchical image\ndatabase. In 2009 IEEE Conference on Computer Vision\nand Pattern Recognition , pages 248–255, 2009. doi: 10.\n1109/CVPR.2009.5206848.\nAlexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov,\nDirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner,\nMostafa Dehghani, Matthias Minderer, Georg Heigold,\nSylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An\n9Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nimage is worth 16x16 words: Transformers for image\nrecognition at scale. CoRR , abs/2010.11929, 2020. URL\nhttps://arxiv.org/abs/2010.11929 .\nAaron Gokaslan and Vanya Cohen. Openwebtext corpus,\n2019.\nDonald Goldfarb, Yi Ren, and Achraf Bahamou. Practi-\ncal quasi-newton methods for training deep neural net-\nworks. CoRR , abs/2006.08877, 2020. URL https:\n//arxiv.org/abs/2006.08877 .\nVineet Gupta, Tomer Koren, and Yoram Singer. Shampoo:\nPreconditioned stochastic tensor optimization. In Inter-\nnational Conference on Machine Learning , pages 1842–\n1850. PMLR, 2018.\nS. Hochreiter and J. Schmidhuber. Long short-term mem-\nory. Neural Computation , 9(8):1735–1780, 1997.\nLei Huang, Dawei Yang, Bo Lang, and Jia Deng. Decorre-\nlated batch normalization. CoRR , abs/1804.08450, 2018.\nURL http://arxiv.org/abs/1804.08450 .\nSergey Ioffe and Christian Szegedy. Batch normalization:\nAccelerating deep network training by reducing inter-\nnal covariate shift. CoRR , abs/1502.03167, 2015. URL\nhttp://arxiv.org/abs/1502.03167 .\nPavel Izmailov, Dmitrii Podoprikhin, Timur Garipov,\nDmitry Vetrov, and Andrew Gordon Wilson. Averaging\nweights leads to wider optima and better generalization.\narXiv preprint arXiv:1803.05407 , 2018.\nAndrej Karpathy. Nanogpt: Small gpt implementations.\nhttps://github.com/karpathy/nanoGPT ,\n2022. Accessed on 22 February 2023.\nNitish Shirish Keskar, Dheevatsa Mudigere, Jorge No-\ncedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang.\nOn large-batch training for deep learning: Gener-\nalization gap and sharp minima. arXiv preprint\narXiv:1609.04836 , 2016.\nD. P. Kingma and J. L. Ba. Adam: a method for stochastic\noptimization. In ICLR . Ithaca, NY: arXiv.org, 2015.\nAlex Krizhevsky. Learning multiple layers of fea-\ntures from tiny images. pages 32–33, 2009.\nURL https://www.cs.toronto.edu/~kriz/\nlearning-features-2009-TR.pdf .\nJungmin Kwon, Jeongseop Kim, Hyunseo Park, and\nIn Kwon Choi. Asam: Adaptive sharpness-aware mini-\nmization for scale-invariant learning of deep neural net-\nworks. In International Conference on Machine Learn-\ning, pages 5905–5914. PMLR, 2021.Y . Lecun, L. Bottou, Y . Bengio, and P. Haffner. Gradient-\nbased learning applied to document recognition. Pro-\nceedings of the IEEE , 86(11):2278–2324, 1998. doi:\n10.1109/5.726791.\nYann LeCun and Corinna Cortes. MNIST handwritten digit\ndatabase. 2010. URL http://yann.lecun.com/\nexdb/mnist/ .\nX. L. Li. Preconditioner on matrix lie group for SGD.\nIn7th International Conference on Learning Represen-\ntations, ICLR 2019, New Orleans, LA, USA, May 6-9,\n2019 . OpenReview.net, 2019.\nXi-Lin Li. Preconditioned stochastic gradient descent,\n2015. URL https://arxiv.org/abs/1512.\n04202 .\nIlya Loshchilov and Frank Hutter. Sgdr: Stochastic\ngradient descent with warm restarts. arXiv preprint\narXiv:1608.03983 , 2016.\nMitchell P. Marcus, Beatrice Santorini, and Mary Ann\nMarcinkiewicz. Building a large annotated corpus\nof English: The Penn Treebank. Computational\nLinguistics , 19(2):313–330, 1993. URL https://\naclanthology.org/J93-2004 .\nJ. Martens and R. B. Grosse. Optimizing neural net-\nworks with Kronecker-factored approximate curvature.\nInICML , pages 2408–2417, 2015.\nJ. Martens and I. Sutskever. Training deep and recur-\nrent neural networks with hessian-free optimization. In\nG. Montavon, G. B. Orr, and K. R. Muller, editors, Neu-\nral Networks: Tricks of the Trade . Springer, Berlin Hei-\ndelberg, 2012.\nJames Martens. Second-order Optimization for Neural Net-\nworks . PhD thesis, University of Toronto, Canada, 2016.\nURL http://hdl.handle.net/1807/71732 .\nMykola Novik. torch-optimizer – collection of optimiza-\ntion algorithms for PyTorch., January 2020.\nKazuki Osawa, Satoki Ishikawa, Rio Yokota, Shigang Li,\nand Torsten Hoefler. Asdl: A unified interface for gradi-\nent preconditioning in pytorch. In Order Up! The Ben-\nefits of Higher-Order Optimization in Machine Learn-\ning, NeurIPS 2022 Workshop , 2022. URL https://\norder-up-ml.github.io/papers/19.pdf .\nAdam Paszke, Sam Gross, Soumith Chintala, Gregory\nChanan, Edward Yang, Zachary DeVito, Zeming Lin,\nAlban Desmaison, Luca Antiga, and Adam Lerer. Auto-\nmatic differentiation in pytorch. In NIPS-W , 2017.\nBarak A Pearlmutter. Fast exact multiplication by the hes-\nsian. Neural computation , 6(1):147–160, 1994.\n10Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nOmead Pooladzandi, David Davini, and Baharan Mirza-\nsoleiman. Adaptive second order coresets for data-\nefficient machine learning. In Kamalika Chaud-\nhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari,\nGang Niu, and Sivan Sabato, editors, Proceedings of\nthe 39th International Conference on Machine Learn-\ning, volume 162 of Proceedings of Machine Learn-\ning Research , pages 17848–17869. PMLR, 17–23\nJul 2022. URL https://proceedings.mlr.\npress/v162/pooladzandi22a.html .\nOmead Pooladzandi, Pasha Khosravi, Erik Nijkamp, and\nBaharan Mirzasoleiman. Generating high fidelity syn-\nthetic data via coreset selection and entropic regular-\nization, 2023. URL https://arxiv.org/abs/\n2302.00138 .\nAlec Radford, Jeff Wu, Rewon Child, David Luan, Dario\nAmodei, and Ilya Sutskever. Language models are unsu-\npervised multitask learners. 2019.\nLevent Sagun, Léon Bottou, and Yann LeCun. Sin-\ngularity of the hessian in deep learning. CoRR ,\nabs/1611.07476, 2016. URL http://arxiv.org/\nabs/1611.07476 .\nLevent Sagun, Utku Evci, V . Ugur Güney, Yann N.\nDauphin, and Léon Bottou. Empirical analysis of the\nhessian of over-parametrized neural networks. CoRR ,\nabs/1706.04454, 2017. URL http://arxiv.org/\nabs/1706.04454 .\nJohn Schulman, Filip Wolski, Prafulla Dhariwal, Alec Rad-\nford, and Oleg Klimov. Proximal policy optimization al-\ngorithms, 2017. URL https://arxiv.org/abs/\n1707.06347 .\nMariya Toneva, Alessandro Sordoni, Remi Tachet des\nCombes, Adam Trischler, Yoshua Bengio, and Geof-\nfrey J Gordon. An empirical study of example forget-\nting during deep neural network learning. arXiv preprint\narXiv:1812.05159 , 2018.\nAndreas Veit, Michael Wilber, and Serge Belongie. Resid-\nual networks behave like ensembles of relatively shallow\nnetworks, 2016. URL https://arxiv.org/abs/\n1605.06431 .\nAshia C. Wilson, Rebecca Roelofs, Mitchell Stern, Nathan\nSrebro, and Benjamin Recht. The marginal value of\nadaptive gradient methods in machine learning, 2017.\nURL https://arxiv.org/abs/1705.08292 .\nYuxin Wu and Kaiming He. Group normalization. CoRR ,\nabs/1803.08494, 2018. URL http://arxiv.org/\nabs/1803.08494 .Zhewei Yao, Amir Gholami, Sheng Shen, Mustafa\nMustafa, Kurt Keutzer, and Michael W. Mahoney. Ada-\nhessian: an adaptive second order optimizer for machine\nlearning. In AAAI , 2021.\nChia-Hung Yuan and Shan-Hung Wu. Neural tangent\ngeneralization attacks. In Marina Meila and Tong\nZhang, editors, Proceedings of the 38th Interna-\ntional Conference on Machine Learning , volume\n139 of Proceedings of Machine Learning Research ,\npages 12230–12240. PMLR, 18–24 Jul 2021. URL\nhttps://proceedings.mlr.press/v139/\nyuan21b.html .\nGuodong Zhang, Aleksandar Botev, and James\nMartens. Deep learning without shortcuts: Shap-\ning the kernel with tailored rectifiers. arXiv preprint\narXiv:2203.08120 , 2022.\nPan Zhou, Jiashi Feng, Chao Ma, Caiming Xiong, Steven\nHoi, and Weinan E. Towards theoretically understanding\nwhy sgd generalizes better than adam in deep learning,\n2020a. URL https://arxiv.org/abs/2010.\n05627 .\nPan Zhou, Jiashi Feng, Chao Ma, Caiming Xiong, Steven\nChu Hong Hoi, et al. Towards theoretically understand-\ning why sgd generalizes better than adam in deep learn-\ning. Advances in Neural Information Processing Sys-\ntems, 33:21285–21296, 2020b.\nJuntang Zhuang, Tommy Tang, Yifan Ding, Sekhar\nTatikonda, Nicha Dvornek, Xenophon Papademetris,\nand James S. Duncan. Adabelief optimizer: adapting\nstepsizes by the belief in observed gradients. In NeurIPS\n2020 , 2020.\n11Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nA On the Convergence of PSGD\nIn this section, we first show that the preconditioner Pestimated by PSGD converges to the inverse of Hessian under mild\nconditions. We then show the quadratic convergence of PSGD under the convex setting.\nA.1 PSGD’s preconditioner Precovers H−1\nTo begin, let us formulate the problem clearly. Let (v, h)be a pair of vector and the associated noisy Hessian-vector\nproduct. We assume that vis drawn from distribution N(0, I). The noisy Hessian-vector product his modeled by\nh=H0v+ε\nwhere H0is the true Hessian, and εis a noise term independent of vdue to use of sample averaged loss instead of the true\nloss. Note, the full Hessian expansion is circumvented, and instead only requires us to calculate the much cheaper Hessian\nvector product H0v. Then, we can write the criterion for the preconditioner estimation in PSGD as\nc(P) =E[hTPh+vTP−1v]\n=tr(PE[hhT] +P−1E[vvT])\n=tr(PH2+P−1) (4)\nwhere\nH2=H2\n0+E[εεT]\nClearly, H2is positive semi-definite regardless of the definiteness of H0.\nNote, we do not assume any definiteness or sparsity properties for H0. Hence, in general, we assume that Qis fitted on a\nconnected branch of the general linear group with learning rule\nQnew=Qold+dQ=Qold+QE (5)\nwhere Qrelates to Pas\nP=QTQ,\nand both dQandEare sufficiently small matrices related by dQ=QE. One technical difficulty in proving the convergence\nofQwith the learning rule equation 5 is that it cannot exclude any rotation ambiguity of Qas suggested by\n(UQ)T(UQ) =QT(UTU)Q=QTQ\nwhere Ucan be any matrix on the orthogonal group. To ameliorate this technical issue, we constrain dQto take on certain\nstructure. In our proof, we assume that both QoldanddQare symmetric such that Qnewis always symmetric as well. To\nachieve such a constraint, we should update Qoldon the Lie group as\nQ′=Qold+QE\nQnew=[(Q′)TQ′]0.5(6)\nIn this way, dQ=Qnew−Qoldis guaranteed to be symmetric as long as the starting guess Qoldis symmetric as well.\nStill, in practice we have used the learning rule equation 5, and equation 6 only serves for the purpose of proof.\nProposition 3.1. Assume that His invertible, & dQ=−µ∂c\n∂QorE=−µQT∂c\n∂Q. Then, Qconverges to ±|H|−0.5by\nupdate equation 6, Qnew= [(Q′)TQ′]0.5withQ′=Qold+QE, & a small enough positive step size µ.\nProof. Given a small perturbation dQofQ, the change of Pis given by\ndP=(Q+dQ)T(Q+dQ)−QTQ\n=QTdQ+dQTQ+dQTdQ\nThe change of P−1is a little more complicated. We omit any terms smaller than O[(dQ)2], and can expand dP−1as below\n12Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\ndP−1=(P+dP)−1−P−1\n=−P−1dPP−1+P−1dPP−1dPP−1\n=\u0000\n−Q−1dQP−1−P−1dQTQ−T−P−1dQTdQP−1\u0001\n+ (Q−1dQQ−1dQP−1+Q−1dQP−1dQTQ−T\n+P−1dQTdQP−1+P−1dQTQ−TdQTQ−T)\n=−Q−1dQP−1−P−1dQTQ−T+Q−1dQQ−1dQP−1\n+Q−1dQP−1dQTQ−T+P−1dQTQ−TdQTQ−T\nNow, the change of the PSGD criterion is given by\ndc=tr(dPH2+dP−1)\n=tr(H2QTdQ+H2dQTQ+H2dQTdQ)\n+ tr(−Q−1dQP−1−P−1dQTQ−T+Q−1dQQ−1dQP−1\n+Q−1dQP−1dQTQ−T+P−1dQTQ−TdQTQ−T)\n=2tr( H2QTdQ−P−1Q−1dQ) (7)\n+ tr(dQTdQH2+ 2dQQ−1dQP−1Q−1+dQP−1dQTQ−TQ−1)\nFrom equation 7, the first order derivatives of cwith respect to Qis given by\n∂c\n∂Q=QH2−Q−TP−1\nA stationary point is obtained by letting∂c\n∂Q= 0, which leads to H2=P−2. Hence, such a Palways exists as long as H2\nis invertible. Via relationship dQ=QE, the gradient on the Lie group is\n∇E=QT∂c\n∂Q\nThus, fitting Qon the Lie group yields the same stationary point since Qis invertible. Note that the stationary points only\ntell us that QTQ=H−1without excluding the rotation ambiguity.\nWithout the constraint dQ=dQT, the second order derivative of cwith respect to Qcan be shown to be\n∂2c\n∂(vec(Q))2= 2I⊗H2+ 2JT[Q−1⊗(QP)−T] + 2[ Q−T⊗(QP)−1]J+ 2(QQT)−1⊗P−1\nwhere Jis the permutation matrix satisfying\nvec(dQT) =Jvec(dQ)\nVia relationship dQ=QE, the second order derivative on the Lie group is\n∇2\nE= (QT⊗I)∂2c\n∂(vec(Q))2(Q⊗I) (8)\nWe see that neither∂2c\n∂(vec(Q))2nor∇2\nEis guaranteed to be positive definite. Hence, the learning rule equation 5 does not\nconvergence to a point as expected due to the rotation ambiguity.\nOn the other hand, both QanddQare symmetric with the learning rule equation 6. Thus, the second order derivative of c\nwith respect to Qsimplifies to\n∂2c\n∂(vec(Q))2= 2I⊗H2+ 2Q−1⊗(QP)−T+ 2Q−T⊗(QP)−1+ 2(QQT)−1⊗P−1\n13Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nAt a stationary point, we have Q=±P0.5=±|H|−0.5. Thus, this second order derivative reduces to\n∂2c\n∂(vec(Q))2|Q=±|H|−0.5= 2I⊗H2+ 4H0.5⊗H1.5+ 2H⊗H (9)\nBy equation 8, the second order derivative on the Lie group is\n∇2\nE|Q=±|H|−0.5= 2H−1⊗H2+ 4H−0.5⊗H1.5+ 2I⊗H (10)\nNow, both∂2c\n∂(vec(Q))2and∇2\nEare positive definite for an invertible Hat the stationary point. Thus, Qconverges to either\nstationary point, i.e., |H|−0.5or−|H|−0.5.\nFrom Proposition 3.1, we see that Pconverges to |H0|−1when ε= 0, i.e., inverse of the “absolute” Hessian. With\nstochastic gradient noises, ε̸= 0 and we always have P≺ |H0|−1. This is not a bug but rather a feature of PSGD that\ndamps the gradient noises perfect such that PE[δgδgT]P=E[δθδθT](Li, 2015), a relationship generalizing the Newton\nmethod to non-convex and stochastic optimizations. Unlike the damping strategies in other second order methods, this\nbuilt-in gradient noise damping mechanics of PSGD does not requires any tuning effort.\nA.2 Linear Convergence of PSGD under General Setting\nCorollary 3.1.1. Assume L(θ)is second order differentiable with absolute eigenvalues of the Hessian well bounded, i.e.,\n0< l≤ |λ(H)| ≤u <∞. Then with PSGD, the loss drops at least with a linear rate, & parameters converge at least\nlinearly to the optimal solution θ∗.\nProof. For a general nonconvex problems, we assume that the eigenvalues of Hessian is well bounded as\n0< l≤ |λ(H)| ≤u <∞\nThen by Proposition 2.1, Pconverges to |H|−1. Thus\n0<1/u≤λ(P)≤1/l <∞\nThe learning rule for parameters with positive step µand preconditioner Pfollows as\ndL(θ) =(dθ)T∂L(θ)\n∂θ\n=−µP\u0012∂L(θ)\n∂θ\u0013T∂L(θ)\n∂θ\n=−µtr\"\n∂L(θ)\n∂θP\u0012∂L(θ)\n∂θ\u0013T#\n≤ −µ\nutr\"\n∂L(θ)\n∂θ\u0012∂L(θ)\n∂θ\u0013T#\n=−µ\nu\r\r\r\r∂L(θ)\n∂θ\r\r\r\r2\nThus the loss decreases at least with a linear rate. Convergence to a local minimum, if exists, with at least a linear rate\nimmediately follows from the convergence of loss as the Hessian is assumed to be nonsingular everywhere.\nA.3 Quadratic Convergence of PSGD under Convex Setting\nIn the previous subsection we proved that the estimate of the preconditioner Precovers the true inverse Hessian. As such,\nunder the assumptions detailed in A.0.1 bellow, PSGD recovers Newton’s method.\nCorollary A.0.1. Assume that L(θ)isα-strongly convex & β-smooth function. Then with learning rate µ=α/β, PSGD\nrecovers Newton’s method with update rule of Eq. equation 1, & convergences to the optimal solution θ∗at least with\nlinear rate L(θt+1)− L(θt)≤ −α\n2β2∥∂L(θt)\n∂θt∥2.\n14Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nProof. We prove Corollary A.0.1 (similarly to the proof of Newton’s method in (Boyd and Vandenberghe, 2004)) for the\nfollowing general update rule: Note from Proposition 3.1, we have that Pconverges to |H0|−1when ε= 0, i.e., inverse of\nthe “absolute” Hessian. With stochastic gradient noises, ε̸= 0and we always have P≺ |H0|−1.\n∆θt=H−1\ntgt (11)\nθt+1=θt−µ∆θt (12)\nDefine λ(θt) = (gT\ntH−1\ntgt)1/2. Since L(w)isβ-smooth, we have\nL(θt+1)≤ L(θt)−µgT\nt∆θt+µ2β∥∆θt∥2\n2(13)\n≤ L(θt)−µλ(θt)2+β\n2αµ2λ(θt)2, (14)\nwhere in the last equality, we used\nλ(θt)2= ∆θtHt∆θT\nt≥α||∆θt||2. (15)\nTherefore, using step size ˆµ=α\nβwe have θt+1=θt−ˆµ∆θt\nL(θt+1)≤ L(θt)−1\n2ˆµλ(θt)2(16)\nSince αI⪯Ht⪯βI, we have\nλ(θt)2=gT\ntH−1\ntgt≥1\nβ∥gt∥2, (17)\nand therefore Ldecreases as follows,\nL(θt+1)− L(θt)≤ −1\n2βˆµ∥gt∥2=−α\n2β2∥gt∥2. (18)\nB Construction of matrix-free preconditioner\nThe following statement gives a systematic way to construct a family of black-box matrix-free preconditioners.\nClaim 3.1. LetK={σ1, . . . , σ m}be a subgroup of the permutation group Sn. Then, linear transform T:Rn7→\nRn, T (x|a1, . . . , a m) =Pm\ni=1ai⊙σi(x), forms a subgroup of GL(n,R)parameterized with {a1, . . . , a m}if\nT(·|a1, . . . , a m)is bijective, where both ai&xare∈Rn.\nProof. The proof follows by showing that such matrices have the following four properties required to form a Lie group.\nFirst, we show that Iis the identity element. Note that Khas element esince it is a subgroup of Sn. Then without the loss\nof generality, we can set σ1=eanda1to be a vector of ones, and all the other ai,i >1, to be vectors of zeros. This leads\ntoT(x|a1, . . . , a m) =x, and thus T(·|a1, . . . , a m) =Iwhen represented as a matrix.\nSecond, we show that such transforms are closed with binary operation T(1)◦T(2)defined as [T(1)◦T(2)](x) =\nT(1)[T(2)(x)]. Specifically, we have\n[T(1)◦T(2)](x) =mX\ni=1a(1)\ni⊙σ(1)\ni\nmX\nj=1a(2)\nj⊙σ(2)\nj(x)\n\n=mX\ni=1mX\nj=1[a(1)\ni⊙σ(1)\ni(a(2)\nj)]⊙σ(1)\ni(σ(2)\nj(x))\nSince Kis a subgroup of Sn,σ(1)\ni(σ2\nj(·))still belongs to K. Hence, T(1)◦T(2)will have the same form as T(·|a1, . . . , a m)\nafter merging like terms.\nBy representing T(·|a1, . . . , a m)as a matrix, it is clear that the associativity property, i.e., (T(1)◦T(2))◦T(3)=T(1)◦\n(T(2)◦T(3)), holds since matrix multiplication is associative. Lastly, the inverse of T(·|a1, . . . , a m)exists since we assume\nthatT(·|a1, . . . , a m)is bijective, and thus its representation is an invertible matrix.\n15Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nWe want to point out that not all simple matrix-free preconditions can be constructed by Theorem 3.1. Let us take the\nwidely used feature normalization transform, e.g., batch normalization (Ioffe and Szegedy, 2015), layer normalization (Ba\net al., 2016) and group normalization (Wu and He, 2018) as an example. We have\nT(x|µ, σ) = (x−µ)/σ\nwhere µandσare either scalar or vector mean and standard deviation of x, respectively. This T(·|µ, σ)forms a sparse\naffine group for σ̸= 0(Li, 2019). However, we cannot use such preconditioners as black-box ones.\nC Construction of low-rank approximation preconditioners\nThe following property states that the proposed low-rank approximation preconditioners can fit both ends of the spectra\nof Hessian. It is not difficult to prove this statement. But, it reveals an important advantage over traditional low-rank\napproximation preconditions with form P=ρI+UUT, whose eigenvalues are lower bounded by ρ.\nC.1 Notations\nLetQ= (I+UVT)B, where UandVare two tall thin matrices, and Bis a matrix on certain group, e.g., the\ngroup of diagonal matrix, or simply a scalar. It is an important preconditioner as after reparameterization, we can have\nQ= diag( d) +UVTfor diagonal matrix B, which relates to the LM-BFGS, HF, and conjugate gradient (CG) meth-\nods. This preconditioner is efficient if low-rank modication can significantly further reduce the condition number1after\npreconditioning the Hessian with a diagonal matrix, i.e., a Jacobi preconditioner. One also can think Qas a low-rank\napproximation of the inverse square root of a positive definite Hessian. Note that this form of preconditioner can fit both\ntails of the spectra of Hessian.\nClaim 3.2. Preconditioner P=QTQwithQ=ρ(I+UVT)can have positive eigenvalues arbitrarily larger than ρ2&\narbitrarily less than ρ2with proper U&V.\nProof. Let us check the simplest case, i.e., ρ= 1,U=uandV=v. Then, Pis shown to be\nP=(I+vuT)(1 + uvT)\n=I+vuT+uvT+ (uTu)vvT\nThisPhas two eigenvalues determined by uandv, sayλ1andλ2. They satisfy\nλ1λ2=(1 + uTv)2\nλ1+λ2=2 + 2 uTv+∥u∥2∥v∥2\nBy choosing uandvproperly, these two eigenvalues can be arbitrarily smaller or larger than 1. For example, by letting\nuTv= 0 and∥u∥=∥v∥ → ∞ , we have λ1λ2= 1 andλ1+λ2→ ∞ . Hence, we must have one eigenvalue arbitrarily\nlarge, and the other one arbitrarily small. In general, the order of rank can be larger than 1, and thus more degree of\nfreedoms for fitting the spectra of Hessian.\nClaim 3.3. Ifρ̸= 0&(I+VTU)−1or(I+UTV)−1exists, AV(ρ, U) =ρ(I+UVT)defines a subgroup of GL(n,R)\nparameterized with ρ&U. Similarly, AU(ρ, V) =ρ(I+UVT)defines another subgroup of GL(n,R)parameterized\nwithρ&V.\nProof. Without the loss of generality, we assume ρ= 1, and simplify rewrite AV(1, U)asAV(U) =I+UVT. We can\nshow that AV(U)forms a Lie group by revealing the following facts\nAV(0) = I\nAV(U1)AV(U2) =AV(U1+U2+U1VTU2)\nA−1\nV(U) =AV[−U(I+VTU)−1]\n1Since PH is not symmetric, smaller eigenvalue spread does not always suggest smaller condition number, e.g., [1, a; 0,1]has\narbitrarily large conditioner number for |a| → ∞ . In PSGD, Pdoes not amplify the gradient noise (ref (Li, 2015), page 5-6, section\nIV .B), and thus avoids such degraded solution.\n16Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\ni.e., the existence of identity element, closed with respect to matrix multiplication, and the existence of inverse, respectively.\nThe last equation is simply the rewriting of the Woodbury matrix identity, i.e.,\n[I+UVT]−1=I−U(I+VTU)−1VT\nThe condition that (I+VTU)−1exists is necessary as otherwise AV(U)is singular. Lastly, the associativity property\nclearly holds for matrix multiplications. Hence, all such matrices form a Lie group. Similarly, we can show that AU(V) =\nI+UVTdefines another Lie group.\nTHE ROTATION AMBIGUITY AND SCHUR DECOMPOSITION\nNote that UVT=UQ(V Q)Tfor any orthogonal matrix Q. Thus, we can remove this rotation ambiguity by selecting a\nQsuch that QT(VTU)Qis an upper triangular block matrix with block size 1or2, i.e., the Schur decomposition. AV(U)\nandAU(V)still form groups by constraining VTUto be quasi-triangular.\nD Low-rank approximation preconditioner fitting\nIn practice, we seldom use Q=ρ(I+UVT)as a preconditioner directly. Its limitation is clear, i.e., most of its eigenvalues\nareρ2withr≪n. In our method, we start from a rough preconditioner guess, say B, and modify it with I+UVTto\nhave the refined preconditioner as\nQ= (I+UVT)B\nIf matrix Bis from another Lie group, we still can update Qefficiently. For example, Bcan be a diagonal matrix with\nnonzero diagonals. Then this composited preconditioner reduce to a diagonal one when r= 0.\nComputation Note that neither of or methods require direct formation of a curvature matrix. Instead we use Hessian\nvector products. One can utilize auto-differentiation packages to calculate the exact Hessian vector product or approximate\nthem with finite differences both detailed by (Pearlmutter, 1994). Given a neural network with N parameters, the Hessian\nvector calculation can be done in O(N) time and space, and does not make any approximation. Additionally, we often only\ncalculate the preconditioner with probability p = 0.1, making the PSGD as practical as SGD.\nD.1 Fundamental operations on Lie Group\nTHE CONCEPT OF GROUP GENERATOR\nUnlike the additive update to move a point in a Euclidean space, we use multiplicative update to move a point on the Lie\ngroup. For example, we can move AV(U)to any its neighbor, say AV(U+E), as\nAV\u0000\nE(I+VTU)−1\u0001\nAV(U) =AV(U+E)\nSince AV(µU) =I+µUVT=eµUVTforµ→0,UVTis a group generator for any U̸= 0. Indeed, the Lie algebra is\nclosed as shown by\n< U 1VT, U2VT>=U1VTU2VT−U2VTU1VT= (U1VTU2−U2VTU1)VT\nwhere <·,·>is the Lie bracket.\nTHE GRADIENTS FOR PRECONDITIONER UPDATING\nHere, the gradient always refers to the one on the Lie group. Thus dA, say on group AV(U), is either AV(E)AV(U)or\nAV(U)AV(E), where E → 0. Since we will update both groups, we simple put it as dA=E1AordA=AE2. Let’s drop\ntheEin the PSGD preconditioner fitting criterion and derive the stochastic gradient as below,\nd(hTPh+vTP−1v)\n=hTdPh−vTP−1dPP−1v\n=2hTdQTQh−2vTP−1dQTQP−1v\n17Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nSince we are to fit AandBon the Lie groups, thus let\ndQ=dAB +AdB\n=E1AB+ABE2\n=E1Q+QE2\nThen, we have\nd(hTPh+vTP−1v)\n=2hTdQTQh−2vTP−1dQTQP−1v\n=2hTQTET\n1Qh+ 2hTET\n2QTQh−2vTP−1QTET\n1QP−1v−2vTP−1ET\n2QTQP−1v\n=2hTQTET\n1Qh+ 2hTET\n2Ph−2vTQ−1ET\n1Q−Tv−2vTP−1ET\n2v\n=2tr\b\nET\n1\u0002\n(Qh)(Qh)T−(Q−Tv)(Q−Tv)T\u0003\t\n+ 2tr\b\nET\n2\u0002\n(Ph)hT−v(P−1v)T\u0003\t\n(19)\n(20)\nD.1.0.1 Gradient with respect to B\nFor diagonal matrix, we simply have\n0.5∇B= diag[( Ph)hT−v(P−1v)T]\nfrom (19), and thus update Bas\nB←B(I−µ∇B)\nwhere the step size µis small enough such that µ∥∇B∥<1, and∥∇B∥is just the max absolute diagonal element for a\ndiagonal matrix. Here, ∥ · ∥ denotes spectral norm.\nFor a diagonal matrix, we also can update its diagonals with element-wise step sizes as the Lie group reduces to the direct\nsum of a series smaller ones with dimension one. This could lead to faster convergence when the true Hessian is diagonal.\nOtherwise, we do not observe any significant difference between these two step size selection strategies in our numerical\nresults.\nD.1.0.2 Gradient with respect to Uon group AV(U)\nSince\ndA= (I+EVT)(I+UVT)−(I+UVT) =EVTA\nwe replace the E1in (19) with EVTto obtain gradient\n0.5∇U= [(Qh)(Qh)T−(Q−Tv)(Q−Tv)T]V\nThen, we update Aas\nA←A−µ∇UVTA\nwhich suggests its parameter can be updated as\nU←U−µ∇U(I+VTU)\nsince we are operating on the group AV(U), where the step size is small enough such that µ∥∇UVT∥<1. Note that\n∇UVThas at most rank two. Hence, its Frobenius norm can be used to bound its spectral norm tightly, as shown by\n∥∇UVT∥F/√\n2≤ ∥∇ UVT∥ ≤ ∥∇ UVT∥F\n18Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nD.1.0.3 Gradient with respect to Von group AU(V)\nAs\ndA= (I+UET)(I+UVT)−(I+UVT) =UETA\nwe replace the E1in (19) with UETto give gradient\n0.5∇V= [(Qh)(Qh)T−(Q−Tv)(Q−Tv)T]U\nThus, we update Aas\nA←A−µU∇T\nVA\nwhich implies that its parameter Vshould be updated as\nV←V−µ(I+V UT)∇V\nwhere the step size is small enough such that µ∥U∇T\nV∥<1. Again, U∇T\nVhas at most rank two, and its Frobenius norm\ngives a tight enough estimation of its spectral norm.\nE Algorithm\nAlgorithm 3 PSGD\n1:Initialize parameter θand its learning rate 0< µ 1<1\n2:Initialize preconditioner as Q∝I, and its update rate and frequency as 0< µ 2<1,0< p≤1, respectively\n3:foriteration = 1,2, . . .do\n4: Sample a stochastic loss ℓ(θ, z)\n5: Compute stochastic gradient g=∂ℓ(θ,z)\n∂θ\n6: ifu < p withu∼ U(0,1)then\n7: Sample vector v∼ N(0,I)\n8: Compute Hessian-vector product h=∂(v⊤g)\n∂θ\n9: Update preconditioner Qwith pair (v, h)and rate µ2\n10: end if\n11: Compute preconditioned gradient g=Q⊤Qg\n12: Update parameter as θ←θ−µ1g\n13: Adjust (µ1, µ2, p)if needed; stop iteration if certain criteria are met\n14:end for\nA few notes for Algorithm 1. Both learning rates, µ1andµ2, are normalized by design. We should not set either of them\nlarger than 1. When the second order derivative is not supported by a certain automatic differentiation tool, we approximate\nthe Hessian-vector product via finite difference as\nv∼ N(0, εI), h≈∂ℓ(θ+v, z)\n∂θ−∂ℓ(θ, z)\n∂θ\nwhere εis a small positive number, e.g., the machine precision. We should avoid forming the preconditioner Pexplicitly as\nP=QTQ. Instead, the preconditioned gradient is always calculated as g=QT(Qg). Algorithm 1 is for PSGD with any\npreconditioner design. Here, we elaborate its preconditioner update algorithms on the two specific Lie groups proposed in\nthis paper. We are not going to detail the algorithm on the calculation of g=QT(Qg)as its math is pretty straightforward.\n19Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nAlgorithm 4 XMat Preconditioner Update\n1:Prepare inputs Q= diag( a) + adiag( b)and pair (v, h)\n2:Calculate Qh=a⊙h+b⊙flip(h)\n3:Calculate Q−Tv= (flip( a)⊙v−flip(b)⊙flip(v))⊘(a⊙flip(a)−b⊙flip(b))\n4:Calculate gradient ∇a= (Qh)⊙(Qh)−(Q−Tv)⊙(Q−Tv)\n5:Calculate gradient ∇b= (Qh)⊙flip(Qh)−(Q−Tv)⊙flip(Q−Tv)\n6:ifbhas odd length then\n7: Set central element of ∇bto zero\n8:end if\n9:Calculate step size µ=µ2\nmax[max( |∇a|),max(|∇b|)]\n10:Update aasa←a−µ(∇a⊙a+∇b⊙flip(b))\n11:Update basb←b−µ(∇a⊙b+∇b⊙flip(a))\n12:Return updated preconditioner as Q= diag( a) + adiag( b)\nHere are a few notes for Algorithm 2. Notation adiag( b)denotes the anti-diagonal matrix with elements in vector bas its\nentries. Notation flip(a)denotes the operation of flipping the order of elements in vector a. For Qwith an odd size, we\nassume that the central element of bis zero to obtain a unique form of representation of the Lie group.\nAlgorithm 5 Low Rank Approximation Preconditioner Update\n1:Prepare inputs Q= (I+UVT)diag( d)and pair (v, h)\n2:Calculate Qh\n3:Calculate Ph=QT(Qh)\n4:Calculate Q−Tvusing the Woodbury matrix identity\n5:Calculate P−1v=Q−1(Q−Tv)using the Woodbury matrix identity\n6:Calculate gradient ∇d= (Ph)⊙h−v⊙(P−1v)\n7:Update dasd←d−µ2\nmax(|∇d|)d⊙ ∇ d\n8:ifu <0.5withu∼ U(0,1)then\n9: Calculate gradient ∇U= (Qh)(Qh)TV−(Q−Tv)(Q−Tv)TV\n10: Update UasU←U−µ2\n∥∇UVT∥∇U(I+VTU)\n11:else\n12: Calculate gradient ∇V= (Qh)(Qh)TU−(Q−Tv)(Q−Tv)TU\n13: Update VasV←V−µ2\n∥U∇T\nV∥(I+V UT)∇V\n14:end if\n15:Return the updated preconditioner as Q= (I+UVT)diag( d)\nHere are a few notes on Algorithm 3. With the Woodbury matrix identity, linear system Qx=bcan be solved as\nx=Q−1b= diag(1 ⊘d)[b−U(I+VTU)−1(VTb)]\nwhich can be separate into the following two steps\nsolve ( I+VTU)y=VTb\nx= diag(1 ⊘d)(b−Uy)\nwhere I+VTUis a square matrix with size r, i.e., the rank or order of low rank approximation. Solving for such a linear\nsystem should not be considered as a burden for a moderate r, say up to thousands, on today’s hardware. Note that ∇Uis\na matrix with rank 2at most. Thus, we have no need to form ∇Uexplicitly in the actual implementation by writing it as a\ndifference of two outer products,\n∇U= (Qh)[(Qh)TV]−(Q−Tv)[(Q−Tv)TV]\nThis saves considerable memory and computes for a large r. The spectral norm of ∇UVTis approximated with its\nFrobenius norm. Again, since the rank of ∇UVTis at most 2, relative error of this approximation is bounded by 3dB. The\n20Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nsame processing techniques apply to the update of Vas well. Note that we cannot update both UandVin a single step due\nto the special way we have constructed these two “twin” Lie groups for low rank approximation. In our implementation,\nwe choose to update either one with equal probability.\nIn the style of the main here are the algorithms with explicit notation.\nNotations we use the following notations:\n•f(θ)∈R, θ∈Rd:fis the loss function to minimize, θis the parameter in Rd\n•gt: the gradient at step t\n•p: preconditioner update probability default is 0.1\n•ht, vt:htthe Hessian vector product; vtdrawn from ∼ N(0, I)\n•P, Q :Pis the gradient preconditioner (never explicitly formed) defined as P=QTQ\n–UV d : For UV d ,Qtakes form of Q= (I+UVT)diag( d)\n–XMat: For XMat,Qtakes form of Q= diag( a) + adiag( b)\n*adiag( b): the anti-diagonal matrix with elements in vector bas its entries.\n*a: the operation of flipping the order of elements in vector a.\n•µ1, µ2:µ1is the optimizer step size; µ2is the preconditioner step size, both default to 10−2\nAlgorithm 6 PSGD Optimizer\nInitialize θ0,t←0,Q0∝I\nWhile θtnot converged\nt←t+ 1\ngt← ∇ θft(θt−1)\nIfu < p withu∼ U(0,1)\nht← ∇ θ(vT\ntgt), s.t.vt∼ N(0, I)\nUpdate Qtvia(vt, ht)\nQ Update Step\nElse\nQt←Qt−1\ngt←QT\ntQtgt\nθt←θt−1−µ1gtAlgorithm 7 UVd Q Update Step\nPh=QT(Qh)\nP−1v=Q−1(Q−Tv) via Woodbury identity 2x\n∇d= (Ph)⊙h−v⊙(P−1v)\nd←d−µ2d⊙ ∇d/max(|∇d|)\nIfu <0.5withu∼ U(0,1)\n∇U= (Qh)(Qh)TV−(Q−Tv)(Q−Tv)TV\nU←U−µ2∥∇UVT∥−1∇U(I+VTU)\nElse\n∇V= (Qh)(Qh)TU−(Q−Tv)(Q−Tv)TU\nV←V−µ2∥U∇T\nV∥−1(I+V UT)∇V\nReturn Q= (I+UVT)diag( d)\nAlgorithm 8 XMat Q Update Step\nQ−Tv= (a⊙v−b⊙v)⊘(a⊙a−b⊙b))\n∇a= (Qh)⊙(Qh)−(Q−Tv)⊙(Q−Tv)\n∇b= (Qh)⊙(Qh)−(Q−Tv)⊙(Q−Tv)\nIfbhas odd length\nSet central element of ∇bto zero µ =\nµ2\nmax[max( |∇a|),max(|∇b|)]\na←a−µ(∇a⊙a+∇b⊙b))\nb←b−µ(∇a⊙b+∇b⊙a)\nReturn Qt←diag( a) + adiag( b)\n21Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nF Space and Time Complexity\nPSGD has the benefit of the equivariance of the Lie group and as such does not require dampening parameters such as K\norC.This clearly reduces both space and time complexity. We see that calculation of the descent direction for PSGD is\nsignificantly cheaper than other second order methods. In terms of memory usage PSGD XM AThas the same memory usage\nas Adam while PSGD LRA has rank multiplier on complexity. On modern GPUs r <100makes negligible difference.\nIteration Cost Method △µ(descent direction) Update SKorK Update SCorC ∇µℓ(BackProp)\nKFAC O(d2\nido+d2\nodi) O(1\nT(md2\ni+d3\ni)) O(1\nT(md2\no+d3\no)) O(mdido)\nShampoo O(d2\nido+d2\nodi) O(1\nT(md2\ni+d3\ni)) O(1\nT(md2\no+d3\no)) O(mdido)\nAdamW O(dido) NA NA O(mdido)\nPSGD LRA (with rank parameter r) O(rdido) NA NA O(mdido)\nPSGD XM AT O(dido) NA NA O(mdido)\nTable 5: Iteration cost for a non-weight-sharing layer, where mis the size of a mini-batch and µ∈di×dois a learnable weight matrix.\nTable 6: Additional Storage\nMemory Usage Method P ∇µℓ⊙ ∇ µℓ S KorK S CorC\nKFAC NA NA O(d2\ni) O(d2\no)\nShampoo NA NA O(d2\ni) O(d2\no)\nAdamW NA O(dido) NA NA\nPSGD LRA O(rdido) NA NA NA\nPSGD XM AT O(dido) NA NA NA\nG More on: Lie Groups, Preconditioners and Practical Considerations\nAs Lie Groups are not a ubiquitous framework for optimization and even less for machine learning, we provide an overview\nof why we need a general-purpose preconditioner, and practical considerations/timings under different frameworks. Fur-\nthermore, we consider different Hessian structures not included in the main paper. We consider fine and coarse grids for\nfuture ways to update preconditioners, theoretical connections to PCA, FFT, and DCT preconditioners, and more.\nNote we have fundamentals on Lie Groups for low-rank approximations and XMat preconditioners in Appx B & D\nG.1 The Need for a General Purpose Preconditioner\nAmong the Lie groups listed in Table 7, the Kronecker-product one has been proven successful for many tasks (Martens and\nGrosse, 2015; Li, 2019; Goldfarb et al., 2020). However, it is inconvenient to use as we need to sort out the parameters to\nbe optimized as a list of tensors in a certain way such that those preconditioners are doing meaningful operations. Here, we\nare looking for some general-purpose black box preconditioners to avoid the need to rearrange the tensors to be optimized\nin certain specific ways.\nG.2 Practical considerations\nClearly, we cannot initialize either UorVor any diagonal element of Bto zero. We can only update UandVsequentially.\nIn my implementations, I update either UorVin a single step, determined in a random way, to save computations. I call\nit the UVd preconditioner. Another form, dUV has the same capacity as shown by\n(I+UVT)diag( d) = diag( d) +U[diag( d)V]T= diag( d)\b\nI+ [diag( d−1)U][diag( d)V]T\t\nThe Woodbury matrix identity turns Q−Tvinto solving for a system of rlinear equations, where ris the rank of Uand\nV. Table 8 summarizes the wall time comparison results on a few typical solvers. We see that their efficiency varies a lot.\nThe fastest one is about two orders of magnitude faster than the slowest one for r= 10 . A poor combination of hardware\nand solver could slow down the updating of this preconditioner. Note that theoretically, we could use the Woodbury\nmatrix identity to update the inverse of I+VTUrecursively as well. However, similar to a Kalman filter, this process\ncould be numerically unstable. Directly solving the system of linear equations should be cheap enough for a typical r, say\n1≤r≤20. Again, the low efficiency of certain linear solvers for a small ris another issue, but solvable.\nH Hessians with certain structures\nOne import family is the preconditioners for a list of affine transform matrices studied in (Li, 2019). Here we discuss some\nother ideas.\n22Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nTable 7: Useful Lie groups (one form may have several disconnected groups)\nexample forms parameters notes\n\n·\n·\n·\n non-zero diagonals diagonal matrices, scaling\n\n· · ·\n· ·\n·\n non-zero diagonals Triangular matrices (lower or upper), feature whitening\n\n· ·\n· ·\n· ·\n·\nnon-zero diagonals feature normalization as in batch normalization\n\n· · · ·\n· · ·\n·\n·\nnon-zero diagonals incomplete triangular matrices\n\n· ·\n· ·\n· ·\n· ·\ninvertible butterfly matrices, building blocks of Kaleidoscope/FFT/DCT matrices\n\n· ·\n· ·\n· ·\n· ·\ninvertible similar to the butterfly matrices, but defined for both odd and even dims\n\n· · ·\n· · ·\n· · ·\n invertible plain dense invertible matrices, i.e., the general linear (GL) group\nC Cis invertible and circulant cyclic or anti-cyclic group, fast matrix-vector product via FFT\nU Uis orthogonal/unitary the groups of rotations (reflections are not continuous)\nA |det(A)|= 1 traceless, tr log( A) = 0\nC−1AC Ais on a Lie group, and Cis invertible useful for blending when C−1is cheap to calculate\nUTAU Ais on a Lie group, and Uis orthogonal useful when Uis DFT/DCT/Hadamard like transformations\nA⊕B⊕. . . AandBare on the same or diff groups direct sum as in block diagonal matrices\nA⊗B⊗. . . AandBare on the same or diff groups good for matrix/tensor gradient preconditioning\nI+UVTinvertible, either fixed UorV useful for preconditioning via low-rank approximation\nA B . . .\nC D\n......\n invertible and all blocks on the same group large sparse preconditioner construction; special case: butterfly matrix\nH.1 Band Hessian\nThe most common assumption is that the Hessian is roughly a band matrix if parameters far away are not strongly coupled.\nStill, band matrices generally do not form groups, and their inverses are not necessarily band-limited. To obtain a tractable\nsolution, we can approximate Qwith\nQ= (C−1AC)B\nwhere both AandBare block-diagonal matrices with block size K×K, and Cis a left or right circular shifting matrix\nthat shifts K/2positions. The following equation shows this clearly when K= 2and the first block of Ais diagonal.\n\nC−1AC:\n·\n· ·\n· ·\n· ·\n· ·\n· ·\n· ·\n·\n\n×\nB:\n· ·\n· ·\n· ·\n· ·\n· ·\n· ·\n· ·\n· ·\n\n⇒\n· ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· ·\n\nFor preconditioner estimation, we do not need to constrain the two 0.5K×0.5Kanti-diagonal blocks of A’s first K×K\nblock to be zeros (the resultant Qis ‘circular’ band). Note that the circular shifting matrix is unitary, i.e., C−1=CT.\nThus, P=QTQ= (ACB )T(ACB ). Hence, we can simply redefine Qas\nQ=ACB\n23Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nTable 8: Mean wall time (ms) over 3000 runs on solving the system of rlinear equations, Ax=b. Hardware: Xeon (R) W-2135 CPU,\nand GeForce RTX 2080 GPU.\nr= 10 r= 100 r= 1000\nMatlab ( CPU, double, x=A \\b) 0.0078 0.10 14.2\nNumpy ( CPU, double, x=np.linalg.solve(A,b ) 0.17 6.7 57.2\nScipy ( CPU, double, x=scipy.linalg.solve(A,b ) 0.016 0.14 20.5\nPytorch ( GPU, single, x=torch.linalg.solve(A,b ) 0.17 0.53 6.2\nTensorflow ( GPU, single, x=tf.linalg.solve(A,b ) 0.67 0.92 6.6\nGRADIENTS\nLet us have\ndQ=dACB +ACdB\n=E1ACB +ACBE2\n=E1Q+QE2\nNow, it is clear that Eq. equation 19 still can be used to calculate the gradients w.r.t. AandB.\nLetdB=BE. Then from Eq. equation 19, the gradient w.r.t. Bis\n0.5∇B= blkdiag[( Ph)hT−v(P−1v)T]\nand thus we update Bas\nB←B−µB∇B\nSimilarly, let dA=EA, we can show the gradient w.r.t. Ato be\n0.5∇A= blkdiag {[(Qh)(Qh)T−(Q−Tv)(Q−Tv)T]}\nThen, we update Aby\nA←A−µ∇AA\nAs usual, the step size should be small enough such that µ∥∇B∥<1andµ∥∇A∥<1. It is also possible to use block-wise\nstep sizes.\nON THE ESTIMATION OF ∥∇B∥AND∥∇A∥\nFirst, the norm of a block diagonal matrix is the maximum norm of its blocks. Second, note that each block has form\nabT−uvT, which has at most rank 2. Thus, ∥abT−uvT∥F/√\n2≤ ∥abT−uvT∥ ≤ ∥ abT−uvT∥F. Hence, the maximum\nFrobenius norm of blocks gives a tight enough spectral norm estimation for the two block diagonal matrices ∥∇B∥and\n∥∇A∥.\nH.2 The two-grid preconditioner\nInspired by the algebraic multigrid method (AMG), we may precondition on two half-overlapped coarse grids, i.e.,\nQ=C−1(A⊗I)C(B⊗I)\nwhere AandBare two small matrices, and Cis a circular-shifting matrix. The ‘coarseness’ is determined by the size of I.\nClearly, it reduces to a dense preconditioner for I= 1. When the size of Iis large, we only precondition the ‘low-frequency\ncomponents’ of the Hessian.\nThe popular Kronecker product preconditioner also can be viewed as preconditioning on a coarse grid. Unlike AMG, we\ndo not have a prolongation/interpolation step to refine the coarse error since the target is the unknown Hessian.\n24Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nH.3 Preconditioning as PCA or Karhunen–Loeve transform\nTHE CONNECTION TO PCA\nIfvis drawn from N(0, I), then we can simplify Eq. equation 3 as below\nE[hTPh+vTP−1v] =E[hTPh] +E{trace[ P−1vvT]}=E[hTPh] + trace( P−1)\nThen, the optimal Psimply whitens has shown by E[(Ph)(Ph)T] =I. Thus, the optimal preconditioner is performing\nPCA (principal component analysis). We can even reformulate the preconditioner estimation problem as an online PCA\none with a cost\nE[∥QTQh∥2]−4 log|detQ| (21)\nHowever, fitting Qin this way converges slowly as this criterion does not exploit all the information encoded in pair (v, h).\nStill, this connection suggests that we could use certain PCA tools for preconditioning.\nKALEIDOSCOPE , FFT AND DCT MATRICES\nIf the Hessian has certain sparsities, a Kaleidoscope matrix like Qcould do a good job, e.g.\nQ=\n· ·\n· ·\n· ·\n·\n· ·\n· ·\n· ·\n×\n· ·\n·\n· ·\n· ·\n· ·\n· ·\n· ·\n×\n· ·\n· ·\n· ·\n· ·\n· · ·\n· · ·\n· · ·\n\nfor a 7×7Hessian. Theoretically, such a preconditioner has complexity O(Nlog2N)for an N×NHessian. But,\npractically, this will take a lot of effort to achieve such efficiency. For now, I rely on the FFT libs to approximate the KLT.\nDCT PRECONDITIONER\nPractically, DCT is asymptotically equivalent to KLT for a first-order Markov process with strong adjacent sample correla-\ntion. If this is the case, a diagonal or very narrow band preconditioner is sufficient. If the Hessian, H, has certain sparsity\nor regularity like nature signals, e.g., images, then UHUTwill be a highly sparse matrix with most energies concentrated\non the upper left corner, where Uis the DCT matrix. Hence, we could try a preconditioner like\nQ=AUB\nwhere Ais a proper sparse matrix, and Bis a diagonal matrix for preconditioning the Hessian-vector products. We call it a\nDCT preconditioner as UHUTperforms a 2D discrete cosine transform of H. Since we are to fit AandBon Lie groups,\nthus let\ndQ=dAUB +AUdB\n=E1AUB +AUBE2\n=E1Q+QE2\nNow, it is clear that we can use the same equation in equation 19 for gradient derivation.\nH.4 Practical considerations and limitations\nTo facilitate the implementations, we may require Nto have special values, e.g., mod( N, K ) = 0 with block size K, or\nN= 2a3b5c7dfor most FFT libs. If Nis not such a number, we could augment the cost as\nL(θ) + 0.5ϑTϑ\nand optimize vector [θ, ϑ], which has a conformable length. This trick works well in practice.\nAll the preconditioners here have certain limitations. Without knowing the block structure, a band preconditioner scales\npoorly. The DCT preconditioner indeed can de-correlate the input features very well, and thus is good for layer-wise\npreconditioning, but not necessarily good for preconditioning the whole flattened gradient. Similar to the AMG method,\nit is very difficult to define a reasonable coarse grid for preconditioning without knowing the connection of weights in the\nnetwork.\n25Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n100101102103104\nnumber of iterations118120122124126128130132preconditioner fitting lossclosed-form solution\nelement-wise step size\none single step size\nFigure 4: Comparison of three diagonal preconditioner fitting methods on a random dense 100×100Hessian with eigenvalues drawn\nfrom the standard uniform distribution.\nI Preconditioner on a single Lie group\nI.1 Diagonal/Jacobi preconditioner\nPossibly the simplest preconditioner with closed-form solution p=p\nE[v⊙v]/E[h⊙h], where P= diag( p). A Jacobi\npreconditioner is not very competitive. Still, it is simple enough for performance study. We have compared three ways\nfor the estimation of P: closed-form solution where expectations are replaced with sample averages; updating with a\nsingle-step size as\np←p−µ(h2⊙p−v2⊘p)⊙p\nwhere µis small enough such that µmax|h2⊙p−v2⊘p|<1; and updating with element-wise step size as (reduce to\nsign SGD)\np←p−µ⊙sign(h2⊙p−v2⊘p)⊙p\nwhere 0< µ < 1. The closed-form solution performs the best, and the single-step size updating may perform slightly\nbetter than the element-wise one.\nI.2 X-matrix preconditioner\nThis simple preconditioner brings a short-cut connection among gradients far away in positions and performs better than\nthe diagonal one. The X-shape can be nested to knit a fishnet-like preconditioner. The butterfly matrices also work well.\nThese simple preconditioners are very light and can be readily adopted to boost the performance of SGD with marginal\noverhead. Note that these preconditioners can be reduced to the direct sum of smaller groups, i.e., they are reducible.\nDiagonal /Jacobi :\n·\n·\n·\n\nX shape matrix :\n· ·\n·\n· ·\n,\n· ·\n· ·\n· ·\n· ·\n\nFishnet like matrix :\n· · ·\n· ·\n· · ·\n· ·\n· · ·\n,\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n· · · ·\n\nI.3 Triangular matrix preconditioner\nPreviously proposed in (Li, 2015) called a dense preconditioner. This calculated the full rank preconditioner and is only\napplicable to small-scaled problems due to memory and complexity constraints.\n26Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nJ The group of X-matrix\nAll invertible matrices with form\nA= diag( a) + adiag( b)\nform a Lie group, where adiag means skew- or anti-diagonal. Clearly, Acan be reduced to the direct sum of ⌈N/2⌉smaller\ngroups. We assume that the central element of bis zero for Awith odd dimensions.\nShort-hands:\nabdenotes the element-wise product of two vectors aandb.\n← −adenotes the flipped version of a.\nProjX(A)denotes projecting a dense matrix Aonto an X-matrix.\nThen, we have properties:\n[diag( a) + adiag( b)]x=ax+b← −x\ndiag( a) diag( b) =diag( ab)\ndiag( a) adiag( b) =adiag( ab)\nadiag( a) diag( b) =adiag( a← −b)\nadiag( a) adiag( b) =diag( a← −b)\n[diag( a) + adiag( b)] [diag( u) + adiag( v)] =diag( au+b← −v) + adiag( av+b← −u)\n[diag( a) + adiag( b)]−1=diag\u0012← −a\na← −a−b← −b\u0013\n−adiag\u0012b\na← −a−b← −b\u0013\n[diag( a) + adiag( b)]T=diag( a) + adiag(← −b)\n∥diag( a) + adiag( b)∥ ≤∥diag( a)∥+∥adiag( b)∥= max |a|+ max |b|\n∥diag( a) + adiag( b)∥ ≥max(max |a|,max|b|) (for even dim)\nProjX(abT) =diag( ab) + adiag( a← −b) (for even dim)\nwhere for odd dimensionalities, the central element of adiag( ·)must be set to zero so that the last two equations hold as\nwell.\n27Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nK More Experimental Results\nK.1 RL Problems\nHere we consider two standard Proximal Policy Optimization (PPO) problems in Reinforcement Learning (RL): Walker2d\nand HalfCheetah. We optimize the actor and critic independently. We compare the performance of the two SOTA optimiz-\ners AdaBelief and Adan to PSGD. We optimize the actor and critic independently. We find that PSGD can find a higher\nreward for both Walker2d & HalfCheetah as shown in Figure 5.\n(a)Walker2d\n (b)HalfCheetah\nFigure 5: PSGD outperforms SOTA optimizers on PPO RL on Walker2d & HalfCheetah.\nK.2 Noisy Label CIFAR10\nSymmetric Noisy Label\nIn this section, we consider training a ResNet18 under 60% symmetric label noise. We randomly select 60% of the\nCIFAR10 training labels, resulting in each class having approximately 46% correct labels and the 54% consisting of 6%\ndrawn from the other nine classes. Recently, (Kwon et al., 2021) proposed a novel flat basin-seeking algorithm named\nAdaptive Sharpness-Aware Minimization (ASAM) that set a new SoTA for symmetric noisy label for CIFAR10.\nSymmetric Noisy Label Cross Entropy Smoothened Cross Entropy (0.1)\nPSGD (Ours) 77.0 77.0\nASAM (SoTA) 70.55 70.19\nWe benchmark PSGD against ASAM and see that PSGD outperforms ASAM by 7%.\nASYMMETRIC NOISY LABEL\nNext, we consider asymmetric label noise. We asymmetrically perturb 60% of the CIFAR10 training labels randomly,\nresulting in one of the classes having approximately 55% incorrect labels and the other 9 classes having 5%incorrect\nlabels. We use SGD, PSGD, Adam, and Apollo to train a ResNet18 on this task for 100 epochs with 10 different seeds\nand compare train and test classification accuracy 6. Additionally, we consider the recently proposed flat basin-seeking\nalgorithm named Adaptive Sharpness-Aware Minimization (ASAM) (Kwon et al., 2021) that set a new SoTA for symmetric\nnoisy label for CIFAR10 under the asymmetric label noise setting. We compare ASAM to SGD and PSGD (see Table 9)\nseparately from the other experiments since ASAM is orthogonal yet complementary to SGD and PSGD. To the best of\nour knowledge, no optimizer has been designed specifically for asymmetric label noise.\nStandard Optimization Techniques\nFirst looking at Figure 6, we see that PSGD achieved the lowest average training accuracy of around 40%, with Apollo\nreaching the highest average train accuracy of 57%. While SGD gets an average training accuracy between Adam and\nApollo (with 7/8 runs getting 55%, and 1/8 getting 34%), and well above PSGD. During testing, SGD exhibits clear\ninstabilities, falling into a regime of pure memorization of the training set. This can be seen as among the 8 runs of SGD,\nonly one lucky initialization achieved a test accuracy of 77% (learning regime), while the other initializations had a test\naccuracy of 10% (memorization regime) with an average predicted probability of 0.999. This is not a standard case of\nover-fitting nor a case of catastrophic forgetting, since the training accuracy does not change. Instead, our experiments\nshow there exists a tipping point in the test accuracy at around epoch 35, where within a single epoch the test accuracy\ndrops from 71% to10% while the training accuracy shows no intelligible indication of this change. Furthermore, the test\n28Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\naccuracy does not increase for the rest of the 65 epochs. At the 35 epoch mark, Adam and Apollo both go through a period\nof high variance but eventually converge (due to the cosine learning rate decay). PSGD achieved an average accuracy of\n82.63%±0.11after 100 epochs, outperforming SGD by 63.93% and exhibiting no optimization instabilities (See Figure\n6).\n(a) Train Accuracy: Noisy Label CIFAR-10\n (b) Test Accuracy: Noisy Label CIFAR-10\nFigure 6: CIFAR-10 Noisy Label ResNet-18: (a) Train Acc: We clearly see that Apollo (Quazi-Newton diagonal) over-fits the training\nset which includes 60% noisy labels, whereas PSGD finds a solution that results in a significantly worse train accuracy. Furthermore, we\nsee that the variance for Apollo, Adam, and SGD is very high during training starting around epoch 40. Note SGD had 7 runs with train\naccuracy near 55%, and one with train accuracy around 34%. (b) Test Acc: We see that PSGD reaches a significantly better test accuracy\nwith a very low training variance compared to Apollo, Adam, and SGD. In this run, SGD had 7 NNs that got 10% test accuracy and\none network that got 72% test accuracy. For more on SGD’s performance see main paper CIFAR with Noisy Lables . The combination\nof these two plots shows other optimizers can easily over-fit/memorize incorrect image label pairs, can often have large optimization\ninstabilities during training, and even reach degenerate solutions where the NNs have reasonable train accuracy but random test accuracy.\nSharpness Aware Optimizers\nIn this section we compare Adaptive Sharpness-Aware Minimization (ASAM) (Kwon et al., 2021) that set a new SoTA for\nsymmetric noisy label for CIFAR10 under the asymmetric label noise setting to PSGD and SGD. Note to the best of our\nknowledge there has not been an optimizer specifically designed for asymmetric label noise and no specific optimizer has\nbeen deemed SoTA.\nTable 9: Comparing (P)SGD to Sharpness Aware Optimizers. Here we see that PSGD can outperform SGD and ASAM by 12.38% and\n76.31% respectively.\nAsymmetric Noisy Label Final Best\nCIFAR10\nTrain Test Train Test\nPSGD (Ours) 40.44% ± 0.12 82.63% ± 0.11 41.03% ± 0.11 82.63% ± 0.11\nSGD 52.5% ± 19.87 18.7% ± 33.75 56.3% ± 0.1 73.98% ± 5.65\nASAM 64.03% ± 1.12 6.32% ± 2.6 64.03% ± 0.1 40.48% ± 0.8\nWe see in Table 9 that while ASAM is able to achieve the best training accuracy we see that greatly overfits the dataset\nresulting 6%test accuracy at the end of 100 epochs, 12.38% below SGD and 76.31% below PSGD. While ASAM may\nhave been the SoTA for Symmetric label noise, clearly another solution is needed under the Asymmetric setting.\nAs such we see that PSGD, which is a general-purpose optimizer, is able to outperform general first and second-order\noptimizers as well as flat minima-seeking optimizers under the noisy label regime.\n29Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nK.3 MNIST Handwriting Digit Recognition\nTo have a fair comparison between the diagonal and low-rank approximation (LRA) preconditioners, we slightly upgrade\nQin the LRA preconditioner to form\nQ= diag( d)(I+UVT)\nwhere dis a vector. This form of Qcannot form a Lie group. Still, its two factors, diag( d)andI+UVT, can be fitted on\ntheir own Lie groups. Now, the diagonal preconditioner is a special case of this LRA one with order r= 0.\nWe have tested orders r= 0,1,2,5,and10. The batch size is 64. Totally ten epochs of training are performed. The\nlearning rate for parameter updating is annealed exponentially from 0.1for the first epoch to 0.001for the tenth epoch.\nThe step size for preconditioner fitting is annealed exponentially from 0.1for the first epoch to 0.01for the tenth epoch.\nThe preconditioned gradient norm is clipped to 10if too large. No momentum is used. Figure 1 summarizes the test\nclassification error rates for preconditioners with different orders of approximations over 50runs. From Fig. 1, we see that\nthe simple diagonal preconditioner performs well. Still, LRA brings marginal gains up to order r= 5. This cost function\nis fairly ‘flat’ since the only nonlinearities in LeNet5 are two piece-wise linear functions, i.e., the activation function ReLU\nand max pooling one. Only the cross entropy loss introduces the ‘curvature’.\n0 1 2 5 10\nr6.577.588.599.5test error rate10-3\nFigure 7: MNIST test classification error rates over 50runs using preconditioners with different orders of LRA. The one with order 0\nreduces to the diagonal preconditioner. Table 1 reports results of classification accuracy for r= 5. Higher is better.\nK.4 Toy Example: Investigating the Flatness of SAM based solution\nVery recently sharpness-aware minimization (SAM) based optimizers have become a popular area of research. We were in-\nterested to see if PSGD compares to SAM in a toy MNIST LeNet5 classification task. We discussed in 5.1, how the smaller\nthe−logdet (P)of a NN is, the flatter the solution found by the optimizers. Since SAM has been specifically designed to\nfind flat optima we would like to compare. Fig. 8 shows ten pairs of minima, each starting from the same random initial\ninitialization. We see that PSGD converges to minima with flatter or smaller Hessian, i.e., larger preconditioners. From\nthe view of information theory, the total cross entropy and 0.5 log det( H)≈ −0.5 log det( P)are good proxies of the de-\nscription lengths of the train image-label pairs and model parameters, respectively. Fig. 8 shows that minima with smaller\ndescription lengths tend to perform better on the test sets as well, suggested by an arrow pointing to the down-left-front\ncorner of the cube.\n30Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nCross entropy0\n200\n400\n600\n800\n1000\n1200\n1400logdet(P)\n240000\n220000\n200000\n180000\n160000\n140000\n120000\n100000\n80000\nT est error rate\n0.00700.00750.00800.00850.00900.00950.01000.01050.0110\nFigure 8: (Adam,ASAM) →PSGD: MNIST hand written digit recognition with LeNet5. Hessians at the minima of Adam are estimated\nwith a dummy LRA PSGD optimizer that only updates the preconditioner.\nThis shows that we can actually find flatter minima compared to SAM-based optimizers without explicitly optimizing for\nthem.\nWALL-CLOCK TIMINGS\nTheoretically, one Hessian-vector evaluation doubles the complexity of one gradient evaluation, since we do this with a\nprobability of 0.1, the per iteration complexity of PSGD becomes (1+0.1*2) = 1.2 times that of SGD. Empirical timings for\nLeNet5 added in the general response (Figure 9) align with the theoretical calculation. We see that PSGD LRA (Low-Rank\nHessian Approximation) with a rank of 10’s overhead is minimal compared to SGD and is 2x faster than AdaHessian which\nuses a diagonal Hessian estimate.\nDynamics of Preconditioner\nThe dynamics of the preconditioner depend on the specific network structure and task to study. Suppose we start from\na moderate initial guess for the preconditioner (by default it is set to 1). For many tasks, we have observed that the\nmax eigenvalue of the preconditioner keeps increasing until saturation to a point during the whole learning process. As\nthe maximum eigenvalue Pof corresponds to the inverse of the minimum eigenvalue of H, this is expected for over-\nparameterized NNs as their minimum eigenvalue approaches 0. The minimum eigenvalue of Pfirst increases during the\nearly stages of learning, and then eventually drops and settles down on a small value, suggesting parameters are locked\nalong those eigenvector directions associated with small eigenvalues. These empirical results coincide with independent\nfindings from many authors showing that network learning has two stages: the early growing one and the latter refining one.\nWe showed the dynamics of the preconditioner on the LeNet5 (see Figure 10). For this specific task, the min eigenvalue\noccasionally drops but keeps increasing during the last stage of learning due to the vanishing Hessian phenomena when the\ntrain cross-entropy loss approaches zero (meaning the absolute logistic outputs can be scaled to be arbitrarily large, thus\ngradient and Hessian all approaching 0 on the train set).\n31Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n0 20 40 60 80 100 1200.00.51.01.52.0(a) Train CE lossSGD\nAdaHessian\nPSGD\n0 20 40 60 80 100 120\nWalltime (s)0.000.010.020.030.040.05(b) T est error rateSGD\nAdaHessian\nPSGD\nFigure 9: Wall-Time: LeNet5 Comparing SGD, PSGD, and AdaHessian. We see that PSGD UVd (rank 10 Hessian estimate) takes 1.2x\nthe time of SGD but finds a better solution. AdaHessian (Hessian diagonal estimate) takes 2x to complete the run and reaches an error\nrate compared to both SGD and PSGD.\nFigure 10: Dynamics of Preconditioner for LeNet5: The max eigenvalue of preconditioner keeps increasing until saturation to a point\nduring the whole learning process. For this specific task, the min eigenvalue also keeps increasing during the last stage of learning due\nto the vanishing Hessian phenomena when the train cross-entropy loss approaches zero.\n32Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nK.5 CIFAR10 Image Classification with ResNet18\nWe follow the implementations of Adabelief2algorithm (Zhuang et al., 2020) to test preconditioned SGD (PSGD) on the\nCIFAR10 image classification task with the ResNet18 model. One main difference from the implementations in (Zhuang\net al., 2020) is that we reduce the learning rate by tenfold twice for all the optimizers, while the original stage learning rate\nscheduler only anneals the step size once. We also consider the cosine learning rate scheduler, which helps SGD to achieve\nthe state-of-the-art (SOTA) test accuracy about 95.5%. Training and testing accuracy convergence curves over 16runs are\nplotted in Fig. 11. We only show the results of PSGD and SGD here as SGD is known to achieve the SOTA results for this\nproblem.\nFor PSGD, we use step size 0.02for parameter updating and 0.01for preconditioner fitting. The preconditioner is only\nupdated once per ten iterations, and thus its overhead over SGD is marginal. The same momentum factor, 0.9, is used for\nboth SGD and PSGD. Since the step size in PSGD is normalized, we update the momentum as m←0.9m+ 0.1g, instead\nofm←0.9m+gas in the SGD. No gradient clipping is used. Weight decay is realized by adding the L2 regularization\nterm0.5λθTθto the cross entropy loss. We have found that λbetween 0.01and0.02performs the best.\nFrom Fig. 11, we observe that SGD performs very well with the cosine learning rate scheduler. This is expected as these\nresidual networks are highly evolved to be first-order optimizer-friendly. The extensive use of piece-wise linear functions,\nresidual connections, and batch normalizations make these models fairly ‘flat’ and resemble shallow models, instead of\ndeep ones (Veit et al., 2016). Still, PSGD slightly outperforms SGD when we remove the shortcut connections or use a\nless-tuned learning rate scheduler, e.g., stage one here.\nPSGD IS ROBUST TO HYPER -PARAMETERS\nTo showcase the robustness of PSGD to hyper-parameters we show in table 10 the test classification accuracy of a ResNet18\ntrained on CIFAR10 using different learning rates and weight decay.\nPSGD Learning Rate Weight Decay Test Accuracy\nXMat 2e-2 2e-2 95.32\nLRA 2e-2 2e-2 95.69\nLRA 5e-2 2e-2 95.48\nLRA 5e-2 2e-2 95.46\nLRA 4e-2 2e-2 95.55\nLRA 3e-2 2e-2 95.57\nLRA 2e-2 1e-2 95.45\nLRA 2e-2 3e-2 95.51\nTable 10: We see that PSGD is robust to hyper-parameter selection making tuning significantly easier compared to other optimization\nmethods.\nWe clearly see the test accuracy of PSGD in a wide range of learning rates and weight decay converges within the expected\nrange of 95.49±0.08%.\nK.6 A Large-Scale Logistic Regression Problem\nWe consider a simple logistic regression model to solve the MNIST classification problem, with and without Bernoulli\nnoise. We considered AdaHessian, AdaBelief, SGD, LBFGS, PSGD LRA/XMat, KFAC, and HessianFree optimizers. Let\nxbe the vector of the image with length 282. Instead of regression on vector x, we do the regression on the outer product\nvector of x, which has length 284. This significantly increases the test classification accuracy but leads to a large regression\nmatrix with over six million coefficients. The KFAC preconditioner did not fit on our GPU and the HessianFree optimizer\nwould diverge far from all other optimizers in more than 50% of our runs. Both are omitted from further discussion in this\nsection.\nNo momentum is considered since this is the case for LM-BFGS. The train batch size is 500. It is tricky to select the\ninitial learning rate for LM-BFGS even though we exponentially anneal it. We have found that LM-BFGS diverges on\nroughly one-third of the trials with an initial learning rate of 0.1, but0.05is too small and may lead to worse performance\nthan SGD. For PSGD, we consider the LRA preconditioner with order 10and set the learning rates for parameters and\n2https://github.com/juntang-zhuang/Adabelief-Optimizer\n33Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n0 50 100 150 200\nEpoch859095100Train accuracy (%)stage lr\n0 50 100 150 200\nEpoch8284868890929496Test accuracy (%)stage lr\n0 50 100 150 200\nEpoch859095100Train accuracy (%)cos lr\n0 50 100 150 200\nEpoch8284868890929496Test accuracy (%)cos lr\n0 50 100 150 200\nEpoch859095100Train accuracy (%)cos lr, no shortcut\n0 50 100 150 200\nEpoch8284868890929496Test accuracy (%)cos lr, no shortcut\nFigure 11: CIFAR10 image classification with ResNet18. The order of low-rank Hessian approximation is 10. Mean and variance are\nestimated over 16runs. Higher is better.\n34Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\n2 4 6 8 10 12 14 16 18 20\nEpoch10-310-210-1100Train cross entropySGD\nLM-BFGS\nPSGD\nFigure 12: Standard MNIST dataset: Typical convergence curves on the logistic regression problem. Lower is better. When comparing\nthe convergence speed, one should be aware that one step of LM-BFGS may take up to ten iterations, while SGD and PSGD always have\none iteration per step.\npreconditioner to 0.05and0.1, respectively. Since LM-BFGS might diverge with a learning rate of 0.1, we only show a\nfew typical convergence curves of SGD, LM-BFGS, and PSGD in Fig. 12. LM-BFGS converges to regression losses a few\ntimes smaller than SGD. PSGD could converge to losses about one order of magnitude lower than that of SGD and LM-\nBFGS. Regarding test classification error rate, we have 2.37%±0.08,2.09%±0.18,1.98%±0.08for SGD, LM-BFGS,\nand PSGD, respectively, averaged over ten runs. Again, LM-BFGS outperforms SGD, and PSGD performs the best on the\ntest classification error rate as well.\nNext, we simply randomly add Bernoulli noise to the MNIST dataset to add diversity to the dataset. Even though LM-\nBFGS is the standard optimizer for large-scale logistic regression we wanted to consider some other SOTA optimizers for\ndeep learning. PSGD UVd/XMat performed the best in this scenario but we found that AdaBelief did surprisingly well\ngiven its lack of second-order information. Losses and Accuracies plotted in Fig 13\n(a)Train Loss\n(b)Test Accuracy\nFigure 13: We consider logistic regression on the MNIST dataset with Bernoulli noise. We see PSGD finds the lowest loss on the train\nset, with the Belief mechanism also working well. PSGD and AdaBelief generalize well to the Accuracy of the Test Dataset.\nWe see that PSGD significantly outperforms the SOTA optimizers in the convex logistic regression setting under noise-free\nor noisy domains while being memory efficient.\n35Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nK.7 Forgettability & Uncertainty: Rank Analysis\nVery recently, (Toneva et al., 2018) shows that Forgettable points are crucial for generalization in the supervised setting.\nIn the unsupervised, semi-supervised, self-supervised, and generative setting (Pooladzandi et al., 2023) shows that one\ncan use the low entropy samples generated by the generative model to significantly boost the classification performance\nof the Latent Energy Based Model which acts as generative-classifier. Here we acknowledge the previous findings and\nfocus on the supervised setting. We train a ResNet18 on CIFAR-10 and record entropy and forgettability statistics. We\nplot the strong correlation between low forgettability score and low entropy found in our statistics in Fig. 14 and provide\ncorrelation coefficients in Table 11.\nFigure 14: There is a strong correlation between the forgettability ordering and the entropy ordering. i.e. unforgettable points have a\nvery low entropy.\nEntropy v Forgettability Score Correlation Coefficient p-value\nSGD PSGD SGD PSGD\nSpearman 0.72 0.73 0 0\nPearson 0.57 0.65 0 0\nKedal Tau 0.54 0.56 0 0\nWeighted Tau 0.60 0.75 0 0\nTable 11: Comparison of correlation metrics comparing Entropy Score vs Forgettability score between SGD and PSGD. We see that\nPSGD’s average correlation between entropy and forgettability is stronger than that of SGD.\nGENERALIZATION GAPBETWEEN HIGH AND LOWENTROPY SUBSETS\nWe train an Oracle LeNet5 on the full MNIST dataset for 20 epochs, we categorize the train set into two subsets; a high\nentropy subset and a low entropy subset, each consisting of 10k points. The entropy is defined over the softmax of the\nlogits. We then train two different LeNet5s on each dataset. We find that the low entropy dataset does not generalize to\nthe test set well achieving a test accuracy of 74%, whereas the high entropy dataset achieves a 99.3% which is on par\nwith Oracle LeNet5 (see 12). This clearly shows for supervised learning the high entropy points are most important for\ngeneralization.\nFurthermore, we find that training on low entropy points results in low entropy and a lower mean entropy network. This\nsupports the hypothesis that there are certain points that the net can lower the entropy over which do not lead to general-\nization. We see in Table 12 that training over the full dataset resulted in a higher mean low entropy compared to that of the\nfull dataset. This supports the idea that high entropy points are important for supervised learning.\nIn terms of distance from initialization, we see that the network is pushed farther from initialization when using the\nfull dataset than while using the high entropy points and the least when using the low entropy points. This supports\nthat training with low-entropy or unforgettable points unnecessarily pushes a network’s parameters far from initialization\nreducing generalization capacity. Furthermore, we see that when training on low entropy points, PSGD does not push the\n36Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nneural network far from initialization preserving generalization capacity (Cao and Gu, 2019).\nMNIST\nStatisticsAccuracyExpected\nLow EntropyExpected\nHigh EntropyDistance from\nInitilization\nSGDDistance from\nInitialization\nPSGD\nFull Dataset 99.3% 5e-16 6e-3 23.1 26\nHigh Entropy Subset 99.3% 1e-10 2e-2 21.7 21.2\nLow Entropy Subset 74.2% 6e-18 4e-4 9.2 8.7\nTable 12: Comparing generalization accuracy of a LeNet5 trained on either 10k high or 10k low entropy datapoints. We see high entropy\ndatasets can match generalization of the full dataset whith a higher test entropy compared to the other datasets. We see that the distance\nfrom initialization is less for the low entropy points.\nFinally, we see that an NN trained on low entropy points has a mean low entropy 2 orders of magnitude smaller than\ntraining on the full dataset, and 8 orders of magnitude smaller than training on the high entropy subset. This shows that low\nentropy points cause a level of confidence in a NN which is not needed and in some cases can be hurtful to generalization\n5.2.\nK.8 Effect of Rank on XOR\nThe full rank version of PSGD (Li, 2019) can solve the XOR problem with an RNN past delay of 128. Since we can\narbitrarily increase the rank of Hessian approximation in our LRA version of PSGD, we consider the effect of rank on\nconvergence on the XOR problem using an RNN. We find rank approximations of r= 0,1,2,5,and10converge with\nprobability p= 0.1,0.4,0.8,1and1respectively.\n0 1 2 3 4 5 6 7 8 9 10\nr0.10.20.30.40.50.60.70.80.91success rate\nFigure 15: Success rate over ten runs in solving the XOR problem with a simple RNN and LRA preconditioners of orders 0, 1, 2, 5, and\n10. Higher is better.\nThis example shows a typical problem where the diagonal preconditioner struggles, while a low-order Hessian approxima-\ntion works perfectly for preconditioning.\nFor all experiments, both step sizes for parameter and preconditioner updating are fixed at 0.01. The gradient clipping\nthreshold is set to 1. No momentum is used. We run 10 runs per optimizer for 100,000iterations or until convergence.\nThe success rates over ten runs for each r are plotted in Fig. 15. This example shows a typical problem where the diagonal\npreconditioner struggles, while a low-order Hessian approximation works perfectly for preconditioning.\nSETTING A BASELINE USING KFAC, PSGD APSGD LRA/XMat , SGD AND ADAM\nIn this section, we compare general-purpose black box versions of PSGD, namely PSGD LRA andPSGD XMat to the\nAffine or Kronecker Factorized version of PSGD A,to KFAC, SGD, and Adam. While PSGD Ais able to outperform the\nother optimizers, the Affine version of PSGD requires careful adjustment of neural network architecture to be used. This\n37Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nTable 13: Test accuracy of LeNet5 on MNIST over 10runs.\nSGD Adam KFAC PSGD APSGD XMat PSGD LRA\n99.04 99.12 99.16 99.26 99.22 99.22Table 14: Stage & cosine learning rate and residual-free\nResNet18-RF on CIFAR10.\nResNet18 ResNet18-RF\nlr cos stage cos stage\nSGD 95.51 95.07 94.97 94.35\nPSGD 95.54 95.43 95.36 95.17\nTable 15: Test Accuracy of RN18 on diverse CIFAR10 derived tasks with (P)SGD, Adam & Apollo (2nd order). The current SoTA\noptimizer is italicized , and the best accuracies are bolded. ESAM and AESAM took far too long to run on our TPU based system and\nhad sub-par performance, as such we did not continue conducting sharpness aware minimization examples.\nCIFAR10 Standard No Shortcut Class Imb Noisy Final Noisy Best Adversarial Blur Deficit\nPSGD (Ours) 95.490.08(5.5hrs) 95.270.0987.160.8582.630.11 82.630.11 85.170.04 89.510.12\nSGD 95.47 0.14(4.5hrs) 94.66 0.16 86.32 0.84 18.733.75 73.9755.65 83.82 0.18 83.51 0.15\nAdam 93.150.02(4.5hrs) 92.700.02 83.181.2 72.47 2.17 74.47 1.06 82.260.49 82.620.015\nApollo 90.590.252(5hrs) 92.000.20 78.231.3 67.711.49 72.580.92 73.90.57 79.250.46\nESAM 93.70(days) - - - - - -\nAESAM 93.71(days) - - - - - -\nbecomes infeasible for large and intricate modern NN architectures. We see in Table 13 that the black box variants nearly\nmatch the test accuracy of PSGD Aoutperforming the memory-hungry KFAC (second order) optimizer, as well as first\norder SGD and Adam.\nRemoving Skip Connections To increase curvature in the relatively flat modern ResNet architecture, we remove the\nresidual skip connections that gave ResNets their namesake. A recent study (Zhang et al., 2022) demonstrated the use\nof Tailored Activation Transformation (TAT) in conjunction with K-FAC (Martens and Grosse, 2015) (another second-\norder optimizer) to close the gap between residual networks with and without residual skip connections. However, in our\nexperiment, we do not utilize TAT and instead compare the performance of SOTA optimizers on both residual-free and\nstandard ResNet18 models. The results, summarized in Table 14, indicate that PSGD outperforms SGD by 0.61% and\n0.20% on residual-free and standard ResNet18, respectively. Our findings are consistent with the results from (Zhang\net al., 2022), where a difference of 0.6%was observed between the optimization of residual-free using TAT and standard\nResNet18.\nClass Imbalanced CIFAR10 We evaluate the optimization performance on a class-imbalanced version of the CIFAR10\ndataset, where 50% of the classes are randomly reduced by an order of magnitude. We compare SGD and PSGD on\noptimizing ResNet18 and report the results in Table 15. Our results show that PSGD outperforms the SOTA by 1.03% on\nthis dataset.\nAdversarial Attacked CIFAR10 Finally, we trained a ResNet18 on 10k unmodified CIFAR10 images and evaluated it\non a test set consisting of 40k samples perturbed using Neural Tangent Generalization Attacks (Yuan and Wu, 2021). As\nshown in Table 15, PSGD outperformed SGD by 1.35%.\nForgettability Statistics & Learning We revisit Toneva et al. (2018)’s forgettability experiments, which found that one can\nprune 30% of the CF10 train samples without loss of generalization. The study found that a point’s utility for generalization\nincreases as it is learned & then subsequently forgotten during training, regardless of the optimizer or architecture used.\nEssentially, forgettability ordering shows which data points an NN uses to define high-dimensional boundaries akin to\nsupport vectors. We investigate whether the performance difference between PSGD & SGD’s generalization performance\ncan be attributed to this forgettability ordering.\nWe train the RN18 four times, keeping the top Nimportant points based on each optimizer’s expected forgettability score.\nTable 16 shows that PSGD focuses on points that are central to generalization. We see this since, when we limit the dataset\nto only the 5k most forgettable data points deemed by each optimizer, we see PSGD is able to outperform SGD by nearly\n14%.\nTable 16: Forgetting statistics for CIFAR10 on ResNet18. PSGD finds better forgettability statistics outperforming SGD.\nForgetting 50k 25k 15k 5k\nPSGD 96.65 95.83 94.95 56.46\nSGD 96.21 95.56 93.7 42.48\n38Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nSGD and PSGD exhibit fundamentally different importance orderings, which is evident from the significant generalization\ngap observed when training on pruned datasets using these orderings at various degrees. We find that PSGD and SGD\nhave a statistically significant correlation between their forgettability orderings, but the strength of the correlation is weak\n(Spearman coefficient of 0.02 and a p-value of p= 1x10−12). This indicates that the nature of training observed through\nforgettability is different for PSGD compared to first-order optimizers.\nFurthermore, we see a strong correlation coefficient of 0.75 and 0.60 between a low forgetability score and low entropy\nscore with a p-value of p= 0 for PSGD and SGD respectively (see Fig 14). Hence, PSGD does a better job of shaping\nthe NN to focus on the highly forgettable points that are important for generalization while not becoming overconfident on\nthe easy unforgettable points giving up too much capacity, unnecessarily pushing the parameters away from initialization\nreducing generalization ability (Cao and Gu, 2019) (see 12). Note while (Toneva et al., 2018) shows that forgettable points\nare more important for generalization for supervised learning, very recently (Pooladzandi et al., 2023) showed low entropy\npoints are the more important points for learning in the unsupervised, semi-supervised, and generative model setting. See\nTable 12 showing generalization gap training low and high entropy points and how it affects the distance from initialization.\nStubborn Solutions of First Order Optimizers From the previous examples, we learned that first-order optimizers tend\nto heavily overfit certain data points, reducing entropy. In the case of noisy labels, this often results in pure memorization\nwhere the network achieves the training accuracy of PSGD on the train set, but only 10% accuracy on the test dataset\nwith an average confidence of 99.9%. To better understand the stubbornness of SGD solutions, we consider the lucky\ninitialization, which resulted in a test accuracy of 77% under noisy label conditions. Here, we examine a transfer learning\nproblem, where a ResNet18 was trained on noisy data for 100 epochs and then on clean labels for another 100 epochs. This\nsimulates a real-world distributional shift where noisy labels may be refined throughout training, leading to a distributional\nshift. We compare neural networks trained with each optimizer, as well as those that changed optimizers after 100 epochs\nof training.\nAs seen in Fig 16a, PSGD finds a flexible solution when given noisy labels. This is seen since when we correct the labels\nboth PSGD and SGD can reach an accuracy of 92.42%. In contrast, the solution found by SGD seems to be stubborn since\nwhen we correct the noisy labels, PSGD then SGD reaches an accuracy of 91.7% and SGD then PSGD has an accuracy of\n88%. This shows the importance of curvature information in the early periods of training.\n(a)Noisy Label CIFAR-10\n (b)Nero-Plasticity\nFigure 16: a) Solutions found by SGD are harder to optimize compared to PSGD. Note for SGD we used a lucky initialization (see 5.2).\nThe blue and yellow dotted line are the average accuracy of PSGD and SGD after 100 epochs respectivly. b) Removing the blur-deficit\nat epoch 100, PSGD is be more neuro-plastic compared to SGD, achieving better train and test performance.\nIntuitively, first-order methods cause NNs to dedicate parameters to memorizing some data points, unnecessarily reducing\nentropy. When memorization occurs given incorrect labels, it may be difficult to reshape the NN when the labels are\ncorrected, leading to the loss in generalization accuracy seen in Fig 16a. In contrast, as PSGD finds a less certain or suborn\nsolution in the first stage of training, either optimizer can then reshape the NN to their liking in the second stage.\nWe believe the noisy-label and neuro-plasticity results have strong applicability to transfer learning and coreset selection\n(Pooladzandi et al., 2022), particularly in scenarios where the training distribution changes during training. Recently,\n(Osawa et al., 2022) demonstrated that the affine Lie group variant of PSGD outperforms other optimizers when fine-\ntuning a ResNet18 and ViT (Dosovitskiy et al., 2020) on CIFAR10 that were pre-trained on ImageNet (Deng et al., 2009).\n39Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nMore Experiments on Neuro Plasticity\nNeuroplasticity experiments done by (Achille et al., 2017) provided an interesting insight into the critical learning periods\nof NNs. Here we consider whether PSGD can extend the critical learning period of an NN. The summary of (Achille et al.,\n2017) , is that if there is a deficit in learning that is not removed by the first 80 epochs, the final test accuracy will be\nsignificantly hindered.\nFigure 17: Defecit plot: We see that PSGD is able to close the gap\nIn Figure 17, we see that if we remove the blur at epoch 100, well after the end of the critical learning period of an NN\ntrained with SGD/Adam, PSGD is able to retain classification accuracy as if we removed the deficit around epoch 50. If we\nswitch to a cosine learning rate schedule, PSGD is able to recover the accuracy as if one removed the deficit at 20 epochs.\nWe believe that this nature of PSGD is due to us finding a flat generalizable solution that is not memorizing points. Since\nwe are not memorizing, and keeping our entropy relatively high compared to other first and second-order optimization\nmethods, we are able to recover accuracy and reshape our NN when the deficit is removed.\n40Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nTable 17: Test perplexity (lower is better) on Penn Treebank for one-, two- and three-layered LSTMs.\nLSTM PSGD Adan AdaBelief SGD AdaBound Adam AdamW Padam RAdam Yogi\n1 layer 83.5 83.6 84.2 85.0 84.3 85.9 84.7 84.2 86.5 86.5\n2 layers 64.9 65.2 66.3 67.4 67.5 67.3 72.8 67.2 72.3 71.3\n3 layers 59.7 59.8 61.2 63.7 63.6 64.3 69.9 63.2 70.0 67.5\nLSTM based models We conduct a performance comparison between second-order optimization methods and first-order\nmethods for training recurrent neural networks (RNNs) and transformer models. RNNs have a recurrent structure that\nexhibits strong curvature properties, making them particularly suitable for second-order optimization methods. Such meth-\nods can efficiently leverage this structure to converge quickly to good solutions. Consequently, RNNs usually perform\nbetter when trained with second-order optimization methods (Martens, 2016), particularly when the objective function has\nextreme local curvature properties.\nWe benchmark PSGD by predicting the Penn TreeBank Dataset using LSTMs using (Zhuang et al., 2020)’s framework.\nOur results (see Table 17 ) indicate that PSGD consistently outperforms (lower is better) other first-order optimization\nalgorithms, including the SoTA AdamW, on 1, 2, and 3 layer LSTMs.\nRosenbrock: A degenerate problem with known solution\nConsider applying PSGD to the limited complexity two-dimensional stochastic Rosenbrock minimization problem, with\nthe goal of exploring the strengths of our approach in an interpretable domain. We perform a comparison with popular first\nand second order solvers, as well as with more traditional methods such as L-BFGS (results can be seen in Table 18). The\nfirst problem we consider is the search for the minimum of the two-dimensional Rosenbrock test function, which has the\nuseful benefit of enabling us to visualise the trajectories found by each optimiser. Specifically, we use the stochastic variant\nof this function, R:R2→R:R(u, v) = (1 −u)2+ 100 ϵi(v−u2)2,where at each evaluation of the function, a noise\nsample ϵiis drawn from a uniform distribution U[λ1, λ2]withλ1, λ2∈R(we can recover the deterministic Rosenbrock\nfunction with λ1=λ2= 1). To assess robustness to noise, we compare each optimizer on the deterministic formulation\nand two stochastic variants (with differing noise regimes).\nWe clearly see that PSGD is able to outperform all the standard deep learning optimizers as well as LBFGS which is\noptimized for mathematical optimization. We see that PSGD quadratically converges exactly to the minimum of the\nRosenbrock problem which is at (1,1) seen in Figure 18 and Table 18. We utilized (Novik, 2020) to optimize all other\noptimizer’s hyper-parameters while leaving PSGD on its default params.\nFigure 18: PSGD optimization trajectory on the Rosenbrock problem. We see PSGD converges quadradically to the optimal solution.\n41Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nOptimizer Best Loss Achieved\nAdam 0.00077\nSGD 1.59\nAdaBound 1.45e-04\nAdahessian 0.0011\nAdaMod 2.06\nAdamP 6.74e-04\nDiffGrad 0.0028\nLamb 2.06\nMADGRAD 1.04e-04\nNovoGrad 6.38e-05\nRAdam 1.95e-06\nYogi 0.0013\nAccSGD 0.55\nSGDw 3.45\nSGDP 3.45\nPID 1.59\nQHM 0.78\nQHAdam 9.11e-04\nRanger 1.35\nRangerQH 0.67\nRangerV A 7.80e-05\nShampoo 0.074\nLookaheadYogi 0.025\nAggMo 7.13e-04\nSWATS 5.48e-04\nAdafactor 7.48e-03\nA2GradUni 6.77e-03\nA2GradInc 5.53e-03\nA2GradExp 7.15e-03\nAdaBelief 0.0016\nLBFGS 4.92-6\nPSGD 0.0\nTable 18: Considering different optimizers on the Rosenbrock problem. We see PSGD outperforms all other first and second order\noptimizers converging to the exact numerical solution.\n42Curvature-Informed SGD via General Purpose Lie-Group Preconditioners\nK.9 Effectiveness of Lie Group Preconditioner under limited precision\nIn this section we continue the comparison of the proposed adaptive Lie group preconditioner namely PSGD to the closed\nform solution. We clearly see that there is only a minmal gap between the solution found by PSGD in float32 vs bfloat16.\nFurthermore both of these solutions have significantly lower loss compared to the closed form solution seen in 19.\n(a)λ(H)∼from Uniform distribution\n (b)λ(H)∼from Exponential distribution\n (c)λ(H)∼from LogNormal distribution\nFigure 19: Comparing Preconditioner fitting loss for PSGD style vs closed-form solution like used in AdaHessian, under under float32\nand bfloat16. We see adaptive learning on Lie group under either float32 or bloaf16 significanly outperforms closed-form solutions\nunder all eigenvalue distributions of H.\nK.10 Potential Social Impacts and Limitations\nRegarding social impact, our optimization method can outperform other optimization methods in generalization accuracy\nas well as minimizing loss with negligible computational overhead and less tuning efforts due to normalized learning rates.\nThis will enable better training of machine learning systems. Furthermore, since PSGD is better at generalization based on\nimbalanced datasets, it has the potential to reduce bias and provide better representation for under-represented classes and\nsub-classes.\nThe main potential limitation for PSGD we see, is that its certain forms. e.g., low-rank approximation one, require more\nmemory to store the curvature information. Yet, it is still negligible compared to other popular second-order methods like\nKFAC that require per-sample derivatives. XM ATwhich is another variant of PSGD takes the same memory complexity\nas Adam.\nK.11 Hardware & Software\nExperiments were run on a single NVIDIA 3080 10GB GPU with an 11th gen Intel i7 processor or TPUs provided by\nTFRC in the later stages of the experimentation. We utilized PyTorch (Paszke et al., 2017) version 1.13. Base PSGD code\nfrom Xi-Lin Li can be found at Xilin’s github repo while more extensive experiments can be found on Omead’s giuthub\nrepo.\nL Reproducibility\nWe ran all our code with reproducibility in mind. We have kept seeds as well as hyper-parameters for each experiment\nand will release them to public once published. We have included a codebase as a zip for now to retain anonymity. Note\nall results have been averaged over 8-16 experiments with variances provided. Also for PSGD we have hyper-parameter\nsweeps to show that we are robust to changes in these values (see Table K.5).For all Propositions, Corollarys and Claims\nin the main we provide proofs with all assumptions states in the appendix (see Appendix A, B, and C). Additionally, since\nLie Groups are not a ubiquitous framework for machine learning, we provide both crucial as well as extra math for those\ninterested (see D.1 and G). Furthermore, we provide many practical considerations and limitations that come-about in\nimplementation G.2.\n43"    },    {        "title": "2402.04556v1.Large_Eddy_Simulation_of_the_evolution_of_the_soot_size_distribution_in_turbulent_nonpremixed_flames_using_the_Bivariate_Multi_Moment_Sectional_Method.pdf",        "content": "Large Eddy Simulation of the evolution of the soot size\ndistribution in turbulent nonpremixed flames using the\nBivariate Multi-Moment Sectional Method\nHernando Maldonado Colm´ ana,∗, Michael E. Muellera\naDepartment of Mechanical and Aerospace Engineering, Princeton University, Princeton,\nNJ 08544, USA\nAbstract\nA joint volume-surface formalism of the Multi-Moment Sectional Method\n(MMSM) is developed to describe the evolution of soot size distribution in\nturbulent reacting flows. The bivariate MMSM (or BMMSM) considers three\nstatistical moments per section, including the total soot number density, total\nsoot volume, and total soot surface area per section. A linear profile along the\nvolume coordinate is considered to reconstruct the size distribution within\neach section, which weights a delta function along the surface coordinate.\nThe closure for the surface considers that the primary particle diameter is\nconstant so the surface/volume ratio constant within each section. The in-\nclusion of the new variable in BMMSM allows for the description of soot’s\nfractal aggregate morphology compared to the strictly spherical assumption\nof its univariate predecessor. BMMSM is shown to reproduce bimodal soot\nsize distributions in simulations of one-dimensional laminar sooting flames\nas in experimental measurements. To demonstrate its performance in turbu-\n∗Corresponding author.\nEmail address: hm2524@princeton.edu (Hernando Maldonado Colm´ an)arXiv:2402.04556v1  [physics.flu-dyn]  7 Feb 2024lent reacting flows, BMMSM is coupled to a Large Eddy Simulation (LES)\nframework to simulate a laboratory-scale turbulent nonpremixed jet flame.\nComputational results are validated against available experimental measure-\nments of soot size distribution, showing the ability of BMMSM to reproduce\nthe evolution of the size distribution from unimodal to bimodal moving down-\nstream in the flame. In general, varying the number of sections has limited\ninfluence on results, and accurate results are obtained with as few as eight\nsections so 24 total degrees of freedom. The impact of using a different\nstatistical model for soot, the Hybrid Method of Moment (HMOM), is also\ninvestigated. Aside from the fact that HMOM cannot provide information\nabout the soot size distribution, the most significant qualitative difference\nbetween HMOM and BMMSM is in number density in the oxidation region\nof the flame, suggesting that BMMSM outperforms HMOM in reproducing\nkey aspects of the soot oxidation process. Finally, the total computational\ncost of using BMMSM as low as approximately 44% more than the cost of\nHMOM. Therefore, the new formulation results in a computationally efficient\napproach for the soot size distribution in turbulent reacting flows, enabling\nsimulations of the soot size distribution in complex industrial configurations\nthat are unattainable using traditional sectional models.\nKeywords:\nSoot; Size distribution; Multi-Moment Sectional Method; Large Eddy\nSimulation (LES); Turbulent nonpremixed combustion\nNovelty and Significance Statement\nThis paper introduces a joint volume-surface formulation of the Multi-\nMoment Sectional Method (MMSM), called the bivariate MMSM (BMMSM),\n2which allows for tracking the soot size distribution in turbulent reacting\nflows including soot’s fractal aggregate morphology. Also, this model has\nbeen implemented in a Large Eddy Simulation framework, which allows for\nthe description of the evolution of the soot size distribution in turbulent\ncombustion. BMMSM is shown to qualitatively and quantitatively predict\nthe evolution of the soot size distribution in laminar and turbulent flames.\nOverall, the total computational cost for turbulent reacting flows is only\nmarginally more (44%) than the cost of traditional moment methods, which\ndo not provide the soot size distribution.\nAuthors contribution\nHMC : performed research, software development, writing – original draft\n& editing\nMEM : designed research, writing – review & editing, project administra-\ntion\n1. Introduction\nThe need for clean, efficient combustion devices is imperative to prevent\nemissions from propulsion and power systems. In this context, mitigating\npollutants is crucial due to their negative effects on the environment and\nliving organisms, which includes addressing the formation of soot particles.\nSoot formation is a significant concern for engineers and researchers, for fine\nsoot particles directly affect air quality (hazardous for the respiratory system)\nand can contribute to global warming [1]. However, tracking the evolution of\na tremendous number of soot particles in turbulent reacting flows is a grand\nexperimental and computational challenge. In computational modeling, this\n3would entail of a mathematical framework capable of handling a transport\nequation for individual soot particles. For that reason, statistical soot mod-\nels are required in simulations, which amounts to tracking the evolution of\nthe soot Number Density Function (NDF). Its evolution is governed by a\nPopulation Balance Equation (PBE). For instance, Monte Carlo solves for\nthe evolution of a downsampled representative set of particles [2]. However,\nMonte Carlo is prohibitively expensive beyond 0D or 1D systems so would\nnot be suitable for large-scale industrial systems. A computationally efficient\napproach for solving the PBE is through the transport of a few statistical\nmoments of the NDF, known as the method of moments [3]. These statistical\nmodels can accurately represent global soot attributes from the NDF, such\nas soot volume fraction or number density, but they are unable to track the\ndetailed evolution of the size distribution.\nComputationally, the traditional approach for modeling the evolution of\nthe soot size distribution is the sectional method. In the sectional method,\nthe size distribution is subdivided into a set of “sections” that are governed\nby statistical physics of particle dynamics and surface reactivity with the sur-\nrounding gas [4]. In traditional soot sectional methods, the NDF evolution\nis described by transporting the number density per soot section, which is\noften defined with respect to discrete volume intervals. The primary compu-\ntational challenge with the sectional method is cost since it typically requires\na range of 30 to 100 sections to accurately track the soot size distribution.\nAdditionally, these volume-only models can only consider spherical soot and\ncannot characterize fractal aggregate morphology. While some approaches\nattempt to model a fixed size at which aggregation begins to occur [5–9],\n4the use of multivariate sections directly addresses this issue of morphology\nwithout ad hoc size limits. For instance, Zhang et al. [10, 11] developed\na sectional method considering two internal coordinates within soot mass-\nbased sections, namely number densities of soot primary particles and soot\naggregates. However, given the added model fidelity and improved accuracy,\nsuch an investment in higher computational resources is not without merit.\nRather than adding unknowns per section for morphology considerations,\nother approaches instead add unknowns per section to account for different\nchemical reactivity (H /C ratio) within each section [12].\nWhile the multivariate sectional methods aforementioned are simply too\nexpensive for simulating turbulent sooting flames, simplified sectional models\nhave been successful in reproducing the evolution of the soot size distribution\nin simulations of turbulent sooting flames [8, 9, 13, 14], which may consider\ndirectly spherical soot or including the aggregation limit occurring at a crit-\nical size.\nOther (non-sectional) methodologies have also been extended to include\nthe soot size distribution via mathematical reconstruction of the NDF. For\ninstance, the Extended Quadrature Method of Moments for soot [15, 16] was\nimplemented in an LES framework [17, 18] and allowed the reconstruction\nof the NDF by approximating them using weighted kernel distribution func-\ntions (KDFs). Despite this, the model was not validated against experimental\ndata of turbulent flames [18]. A three-equation model for soot formation was\nintroduced by Franzelli et al. [19], in which global soot quantities are trans-\nported. Similar to EQMOM, the NDF was retrieved by approximating it\nas weighted Pareto distributions, which was numerically validated against a\n5sectional method in laminar and turbulent flames. The open question with\nthese mathematical reconstruction is accuracy of the resulting size distribu-\ntion since it does not rely on the underlying PBE.\nTo address the computational cost shortcomings of traditional sectional\nmethods, Yang and Mueller [20] introduced a Multi-Moment Sectional Method\n(MMSM) that essentially combines a sectional method with the method of\nmoments. Within each section, the approach reconstructs the NDF using a\nlinear distribution whose parameters are defined using two section-local mo-\nments. MMSM has a lower computational cost than a traditional sectional\nmethod since fewer sections and overall degrees of freedom are required,\ndue to an improved rate of convergence compared to traditional sectional\nmethods [20]. This implies that MMSM can be as accurate but faster than\ntraditional sectional methods. However, fractal aggregate morphology was\nnot fully described since previous work only considered a univariate (strictly\nspherical) formulation, failing to capture bimodal soot size distributions in\nlaminar flames. Additionally, MMSM was not applied to turbulent flames\nIn this work, a bivariate formulation of MMSM is developed, called BMMSM,\nwhich introduces a joint volume-surface description of soot. Following the\nprevious work of the senior author’s group, BMMSM includes a bivariate\nformulation based on the formulation of the Hybrid Method of Moments\n(HMOM) [21]. BMMSM is first tested in 1D laminar sooting flames and\ncompared to experimental measurements. BMMSM is then integrated in\nan LES framework in order to simulate a laboratory-scale turbulent non-\npremixed jet flame. Combustion and subfilter closures are adapted from\nRef. [22], which were initially developed for HMOM. Computational results\n6are analyzed and compared against available experimental data of soot size\ndistributions. The LES results with BMMSM are then analyzed to provide\nbetter understanding of key aspects of the evolution of the soot size distri-\nbution. Finally, the computational cost of turbulent combustion simulations\nwith BMMSM is then compared to simulations using HMOM.\n2. The Bivariate Multi-Moment Sectional Method\nThe univariate Multi-Moment Sectional Method developed by Yang and\nMueller [20] considered strictly spherical soot particles described by the soot\nparticle volume V. A statistical representation of the particle size distribu-\ntion was considered through a NDF n(V), which was discretized in volume\nsections Vi. In each section, two local (up to first-order) moments were solved,\nthe number density Mi\n0and the volume moments Mi\n1, defined as\nMi\nx=Z\nΩV\nini(V)VxdV, (1)\nwhere ΩV\niindicates the support of the ithsection (out of Ns). The local\nmoments were used to reconstruct the local size distribution with linear ap-\nproximation within each section. The section size ∆ Vi, except for the last\nsection, was centered at Vi. The total number of variables Ns\nvconsidered\nin this approach was Nv= 2Ns. In the following subsections, the MMSM\nformalism is extended to a bivariate soot model.\n2.1. Soot governing equations\nThe bivariate MMSM, hereafter referred as BMMSM, accounts for a joint\nvolume-surface area formalism and can describe the fractal aggregate mor-\nphology of soot particles [23]. The NDF n(V, S) is then a function of two\n7internal coordinates: the soot particle volume Vand surface area S. Mueller\net al. [21] have shown computationally, through Monte Carlo simulations,\nthat the joint surface-volume NDF is divided in two regions with fundamen-\ntally different S−Vdistributions: spherical particles ( S∝V2/3) and fractal\naggregates. Therefore, while a global relationship between SandVis not\nexpected to be accurate or general, section-local relationships between Sand\nV, allowed to vary based on the dynamics of the PBE, are expected to be\naccurate, and a fully multidimensional MMSM is not required. Given this as-\nsumption, the bivariate NDF n(V, S) in section iis modeled with a presumed\ndependence of SandVwithin the section:\nn(V, S) =n(V)δ(S−ˆSi(V)), (2)\nwhere ˆS(V) is the soot surface area for a soot particle of volume Vin the ith\nsection, which needs closure. Then, the bivariate moment in the ithsection\nis defined as\nMi\nx,y=x\nΩV,S\nin(V, S)VxSydVdS\n=Z\nΩV\nin(V)VxˆSy\ni(V)dV.(3)\nAs a first approximation, the primary particle diameter dp= 6V/Sis assumed\nconstant within each section, that is, S/V remains constant within each\nsection and equal to αi. The model variable αiestimates the proportionality\nS/V and is an index of the sphericity or degree of aggregation of the soot\nmorphology in a given section. Low αivalues indicate that soot approaches\na spherical shape, with lower limit (36 π/V i)1/3. To ensure nucleated particles\nare spherical [24], the first section considers α1= (36 π/V 1)1/3and is centered\n8atV1=V0, which corresponds to the nucleated soot particle volume V0.\nThen, the soot surface area ˆSi(V) of a soot particle of volume Vin the ith\nsection can be approximated as\nˆSi(V) =Mi\n0,1\nMi\n1,0V=αiV, (4)\nwhich, inserted to Eq. 3, leads to\nMi\nx,y=αy\niZ\nΩV\nin(V)Vx+ydV=αy\niMi\nx+y,0. (5)\nThis new closure requires an additional degree of freedom per section: the\ntotal surface moment Mi\n0,1. This increases the total number of unknowns Ns\nv\ntoNv= 3Ns.\nThe spatiotemporal and size ( V-S) evolution of the NDF is governed\nby the Population Balance Equation. In the context of the Multi-Moment\nSectional Method, the PBE is integrated to obtain the moment transport\nequations of sections i= 1, . . . , N s, which leads to\n∂Mi\nx,y\n∂t+∂u∗\njMi\nx,y\n∂xj=˙Mi\nx,y, (6)\nwhere u∗\njis the total velocity, including flow and thermophoretic effects [25],\nof diffusionless soot particles [26], and ˙Mi\nx,yare the soot source terms, whose\nclosures are developed in subsection 2.3.\n2.2. Local volume-dependent NDF reconstruction\nThe bivariate NDF defined above needs closure for the volume-dependent\nNDF n(V) within each section. The closure chosen by Yang and Mueller [20]\nis retained. Here, n(V) is modeled with a linear distribution in the first\nNs−1 sections as follows:\n∆Vin(V)|Ωi=a(V−Vi) +b, (7)\n9where Viis the ithsection with section size ∆ Vialong the volume coordinate\nandaandbare local moment-dependent model parameters, defined as\na=12\n(∆Vi)2\u0000\nMi\n1,0−Mi\n0,0Vi\u0001\n(8)\nand\nb=Mi\n0,0, (9)\nrespectively. In the last section ( i=Ns), an exponential distribution is\nconsidered:\n∆VNsn(V)|ΩNs= ∆VNsbaexp\u001a\n−a\u0014\nV−\u0012\nVNs−∆VNs\n2\u0013\u0015\u001b\n, (10)\nwith aas\na=MNs\n0,0\nMNs\n1,0−2VNsMNs\n0,0/(fs+ 1)(11)\nandbthe same as in Eq. 9. The model considers a geometrically increasing\nsection size ∆ Vi, to reproduce rapid coagulation processes, with a spacing\nfactor fsdefined as\nfs=∆Vi+1\n∆Vi, (12)\nleading to\n∆Vi= 2Vi\u0012fs−1\nfs+ 1\u0013\n. (13)\nThe section support in volume space is Ω i= [Vi−∆Vi/2, Vi+ ∆Vi/2) in\nthe first Ns−1 sections (finite), while in the last section is Ω Ns= [VNs−\n∆VNs/2,∞) (semi-infinite). In the last section, the true barycenter is\nV∗\nNs=2VNs\nfs+ 1+1\na. (14)\n10The moment source terms of Eq. 6, defined in the following subsection, are\ncomputed using the bivariate NDF and integrated as in Eq. 3. To limit the\ncomputational costs of those operations, integration using two-point Gauss-\nLegendre quadrature for i < N sand, similarly, two-point Gauss-Laguerre\nquadrature for i=Nsis performed [20].\n2.3. Source terms\nThe sole term on the right-hand-side of Eq. 6 represents the physiochem-\nical phenomena affecting soot evolution: particle dynamics and soot surface\nreactivity with the surrounding gas. This work utilizes the formulations of\nMueller and co-workers [20, 21, 23] with contributions from nucleation, coagu-\nlation, condensation, surface growth, and oxidation. Oxidation-induced frag-\nmentation [27] is not considered in this work and is negligible in the valida-\ntion and application configurations presented in subsequent sections. These\nsource terms, previously formulated within the univariate MMSM model [20],\nare extended to the bivariate formalism below.\n2.3.1. Nucleation\nIn this model, a collision of a pair of PAH dimers leads to a spherical soot\nparticle [28] of fixed volume V0, which also corresponds to the first section\nsizeV1=V0. The nucleation source term is non-zero in the first section and\nzero otherwise and is modeled as follows:\n˙M1\nx,y|nuc=1\n2βnuc[DIMER]2Vx\n0ˆSy(V0), (15)\nwhere βnucis the dimer collision rate and [DIMER] is the PAH dimer con-\ncentration.\n112.3.2. Coagulation\nAfter a reasonably straightforward integration along the soot surface area\ncoordinate, in similar fashion as Eq. 3, the coagulation source term is com-\nputed using a two-point Gauss-Legendre or Gauss-Laguerre quadrature in\neach section along the volume coordinate as in Ref. [20], resulting in\n˙Mi\nx,y|coag=k≤j≤iX\nV(j+k)±∈Ωi\u0012\n1−δj,k\n2\u0013\nβj±,k±Vx\n(j+k)±Sy\n(j+k)±wj±Nj±wk±Nk±\n−wi±Ni±NsX\nk=1βi±,k±Vx\nk±Sy\nk±wk±Nk±,(16)\nwhere the wi,±andVi,±represent the quadrature weights and nodes, with ±\nas a root pair combination. Here, the surface area Si±and number density\nNi±are evaluated from Eqs. 4 and 7 at Vi±, respectively. Then, each pair\nof products (evaluated with opposite root signs) is summed. Within each\nsection, n(V) is equal to the derivative of the local number density Nas\ndN/dV =n(V) (see Eq. 7). δi,jis the Kronecker delta function. βi,jis\nthe soot particle collision frequency, obtained by the harmonic mean of the\ncollision frequencies in the free-molecular and continuum regimes [29].\nThe collision frequency in the free-molecular regime is calculated from\nkinetic theory:\nβfm\nj,k=K\u00121\nVj+1\nVk\u00131/2\n(dc,j+dc,k)2, (17)\nwhere dcare the collision diameters and Kis a constant that considers the\nvan der Waals enhancement factor equal to 2.2, the Boltzmann constant kB,\nand (constant) soot density ( ρsoot= 1800 kg m−3) [30]:\nK= 2.2\u0012πkBT\n2ρsoot\u00131/2\n. (18)\n12The collision diameters dcfor fractal aggregates are calculated as Kruis et\nal. [31] following the expression\ndc,i=dp,in1/Df\np,i=6\n(36π)1/DfV1−2/Df\ni\nS1−3/Df\ni, (19)\nwhere dp,i= 6Vi/Siandnp,i=V−2\niS3\ni/(36π) are the primary particle diam-\neter and number density in the ithsection, respectively, and Df= 1.8 is the\nfractal dimension of soot particles. The collision diameter is then function of\nViandSi=ˆS(Vi), which is closed using Eq. 4.\nThe collision frequency in the continuum regime is given by the Stokes-\nEinstein equation:\nβcont\nj,k=2kBT\n3µ\u0012Cj\ndm,j+Ck\ndm,k\u0013\n(dc,j+dc,k), (20)\nwhere µis the dynamic viscosity, dm,iis the mobility diameter of the ith\nsection, which is assumed equal to the collision diameter dc,i(Eq. 19), and\nCi= 1 + 1 .257Kn iis the Cunningham slip correction factor with Knudsen\nnumber Kn based on the collision diameter dc,i.\nFinally, different collisions types are considered for ˙M0,1|coagand are mod-\neled as in Mueller et al. [21], with both pure aggregation and coalescence as\nlimits. The particle volume resulting from the collision V(j+k)±(in the pro-\nduction part of Eq. 16) is equal to the sum of each particle volume as it is\nconserved: V(j+k)±=Vj±+Vk±. Regarding the surface sizes S(j+k)±, if the\ntwo particles are from the first section ( j=k= 1), the pure coalescence\nlimit is considered:\nS(1+1)±= (36 π)2/3(V1+V1)2/3= (72 πV0)2/3. (21)\n13Then, if one of the particles is from the first section and the other one from\na larger section ( k= 1 and j=iork=iandj= 1, for i >1), then the\nresulting larger particle becomes slightly more spherical:\nS(i+1)±=Si+δS1. (22)\nFor all other collisions ( j >1 and k >1), pure aggregation occurs:\nS(j+k)±=Sj+Sk. (23)\nIn Eq. 22, the change of surface area δS1, due to the collision with a spherical\nparticle from section 1 with volume δV1=V1=V0, is given by the power law\nderived in the work of Mueller et al. [21]:\nδS1\nSi=δV1\nVi\u00122\n3n−0.2043\np,i\u0013\n, (24)\nwhere np,iis the primary particle number at the ithsection.\n2.3.3. Condensation, surface growth, and oxidation\nSource terms for soot growth, such as condensation (particle and PAH\ndimer collision) and surface growth (carbon addition to soot surface via the\nHACA mechanism [24]), and soot destruction, such as oxidation (by oxygen\nmolecule [32] and hydroxyl radical [33]), are modeled using two-point quadra-\nture in three consecutive sections, based on a traditional sectional method\nof Park and Rogak [4] and extended for the univariate MMSM by Yang and\nMueller [20]. The expression for Mi\n0,0source terms is\n˙Mi\n0,0(Ii) =Ai−1\u0002\nw(i−1)±I(i−1)±N(i−1)±\u0003\nV∗\ni−1+Bi[wi±Ii±Ni±]\nV∗\ni\n+Ci+1\u0002\nw(i+1)±I(i+1)±N(i+1)±\u0003\nV∗\ni+1,(25)\n14where the root pair of a quadrature is symbolized by ±,V∗\ni=Vifori < N s,\nand, due to the barycenter variation, V∗\ni=V∗\nNsfori=Ns.Iirepresents the\ncondensation, surface growth, and oxidation rates. The model coefficients\nAi,Bi, and Ciobey\nAi+Bi+Ci= 0, (26)\nsuch that the change of Mi\n0,0is conserved [4]. The expression for Aiis\nAi=fs−Bi(fs−1)\nf2\ns−1, (27)\nand for Bi:\nBi=\n\n−1\nfs+1erf\u0010\n1\n4d lnNi\nd lnVi\u0011\nifd lnNi\nd lnVi>0,\n−fs\nfs+1erf\u0010\n1\n4d lnNi\nd lnVi\u0011\notherwise ,(28)\nwhich is selected to reduce numerical instability and numerical diffusion is-\nsues [4]. The derivative d ln Ni/d lnViis computed using a second-order\ncentral difference. Finally, Ciis obtained from Eq. 26:\nCi=−(Ai+Bi). (29)\nEquations 27-29 are valid for i= 2, . . . , N s−1. The expressions for i= 1\nandi=Nsare derived with the intention of conserving both soot number\nand mass [4]. C1is set to zero since is not needed in Eq. 25. Then, following\nEq. 29, the coefficients in the first section are given by\nA1=−B1=1\nfs−1, C1= 0. (30)\nThe same reasoning is made for ANs, and the model coefficients in the last\nsection are given by\nANs= 0, BNs=−CNs=f∗\ns\nf∗\ns−1. (31)\n15Finally, for i= 1 or i=Nsin Eq. 25, A0=CNs+1= 0 since those sections\ndo not exist. For the last section, to accommodate the barycenter variation,\na modified section spacing factor f∗\ns[20] is considered:\nf∗\ns=1\na\u0012∆VNs−1\n2\u0013\n. (32)\nThe source terms for Mi\n1,0are derived from the number density source\nterms and given by\n˙Mi\n1,0=˙Mi\n0,0(Ii)V∗\ni, (33)\nwhich is obtained by multiplying Eq. 25 by V∗\ni. Finally, the source terms for\nMi\n0,1are\n˙Mi\n0,1=˙Mi\n0,0\u0012\nIiδSi\nδV\u0013\nV∗\ni. (34)\nNote that the ˙Mi\n0,1in Eq. 34 is recast from Eq. 33 by introducing the argu-\nment Ii(δSi/δV), similar to Mueller et al. [21]. δVis the change of volume\nassociated with growth or destruction processes. Then, δSi/δVis calculated\nfollowing Eq. 24 for condensation and surface growth, whereas for oxidation\n[28] the surface area change is equal to\nδSi\nδV=2\n3Si\nVi. (35)\nThe condensation, surface growth, and oxidation rates Iiare based on\ndetailed models from Mueller et al. [21, 23, 27]. The condensation rate is\nwritten as\nIi|cd=βCiδV[DIMER] , (36)\n16where βCiis the collision rate of PAH dimers and soot particles and δVis\nequivalent to the volume of a PAH dimer [21]. Then, the surface growth rate\nis given by\nIi|sg=ksgχSiδV, (37)\nwhere ksgis surface growth rate coefficient [24], χ= 1.7×1019m−2is the\nhydrogenated sites surface density, and δVis equal to the volume two carbon\natoms [21]. Finally, the oxidation rate is\nIi|ox=−koxχSiδV, (38)\nwhere koxis the oxidation rate coefficient including oxidation by both OH\n[33] and O 2[32] and δVis similar as in surface growth [21]. Further details\non the formulation of these terms can be found in Mueller et al. [21].\n3. Validation in laminar flames\n3.1. Experimental and computational setups\nThe new BMMSM model is validated in Flame C4 from Abid et al. [34],\na burner-stabilized laminar premixed ethylene-oxygen-argon flame at atmo-\nspheric conditions. The inlet composition corresponds to ϕ= 2.07 (C 2H4:\n16.3%, O 2: 23.7%, Ar: 60%, by volume). The inlet velocity is 6 .53 cm /s.\nComputational results using the univariate MMSM are also shown to evaluate\nthe performance of the new bivariate BMMSM. Computations where carried\nout in FlameMaster [35], with an imposed temperature profile from experi-\nmental measurements and a gas-phase kinetic mechanism consisting of 158\nspecies and 1804 reactions, including PAHs up to four aromatic rings from\n17Blanquart and coworkers [36, 37]. Simulations were performed with multiple\nnumbers of sections, Ns= 6, 8, 12, 16, and 32, in order to compare the\naccuracy of the models with respect to the number of sections. The section\nspacing factor fsis considered such that the ratio VNs/V1remains the same\nin all cases, which gives fs= 333 .59, 63.43, 14.025, 6.94, and 2.56, respec-\ntively. Results presented in the following subsections are labeled according\nto the total number of (soot) variables solved in each case, NMMSM\nv = 2Ns\nandNBMMSM\nv = 3Ns.\n3.2. Results\nSoot volume fraction and number density are calculated considering soot\nparticles with particle diameters larger than 2 .5 nm, as specified in Ref. [34]\ndue to instrument limitations. Yang and Mueller [20] previously showed ac-\nceptable accuracy in soot volume fraction fvwith the univariate model with\nas few as eight sections, which is confirmed in Fig. 1 (top). With as few\nas six sections, BMMSM is able to predict the soot volume fraction accu-\nrately compared to both a larger number of sections and the experimental\nmeasurements. Fig. 1 (bottom) shows the number density Nprofiles. Com-\nputational results using the univariate MMSM overpredict Nall along the\nflame profile. Conversely, BMMSM improves these results by capturing the\ndecrease of Ndue to the bivariate characterization: for the same volume,\naggregates will coagulate faster, leading to an ultimately lower number den-\nsity. The variation of the soot number density with the number of sections is\nmore significant than for the soot volume fraction, but the number density is\npredicted within experimental uncertainty again with as few as six sections\nat and beyond the peak at 4 mm.\n18Figure 1: Soot volume fraction (top) and number density (bottom) profiles calculated\nconsidering soot particles with particle diameters larger than 2 .5 nm. Computational\nresults using the univariate MMSM (left, dash-dotted) and BMMSM (right, solid) are\ncompared against experimental measurements [34]. The color code indicates those results\nusing the same number of sections.\nResults for the particle size distribution function (PSDF) ˆ n(dp) are shown\nin Fig. 2. The PSDF is defined as in the work of Abid et al. [34] as\nˆn(dp) =1\nNdN(dp)\nd log dp\f\f\f\f\ndp, (39)\nwhere dpis the particle diameter, N(dp) is the (reconstructed) soot particle\n19cumulative distribution function (CDF), and N=N(+∞) is the total par-\nticle number density. Similarly to soot volume fraction and number density,\nthe PSDF is computed by only considering soot particles with particle di-\nameters larger than 2 .5 nm (in the normalizing N) [34]. Figure 2 shows the\nPSDF at two different heights above the burner: H= 3.5 mm (left column)\nand 5 .5 mm (right column). Abid et al. [34] observed spherical particles us-\ning a scanning mobility particle sizer (SMPS) and have defined a corrected\nspherical particle diameter. Therefore, BMMSM results are plotted using\nboth the spherical diameter ( dp∝V1/3), as the univariate model, and mobil-\nity diameter ( dp=dm), which is assumed equal to the collision diameter dc\nas stated in Section 2. A unimodal PSDF is obtained at H= 3.5 mm. Both\nunivariate MMSM and BMMSM are capable of reproducing the experimen-\ntal observations, with no major difference between results. At this location,\nthe soot population is nucleation dominated, so both models predict essen-\ntially all spherical soot. A slight overprediction is observed in the bottom\nrow using BMMSM with small Ns. Increasing Nsimproves the accuracy of\nresults in all cases, due to improvements in predicting the initial coagulation\nprocess. A bimodal PSDF is observed at H= 5.5 mm in the experimental\nmeasurements. For the most part, MMSM still predicts a unimodal distribu-\ntion and a maximum size far smaller than the experimental measurements.\nConversely, BMMSM correctly predicts both a bimodal size distribution with\nthe presence of much larger particles, with a small deviation between results\nusing different dpdefinitions. Unsurprisingly, the bivariate description of soot\nis required to accurately reproduce the evolution of the size distribution. In\nall cases, the PSDF within each section present a parabolic shape, which is\n20Figure 2: PSDF at two locations above the burner: H= 3.5mm (left column) and 5 .5mm\n(right column). Computational results using the univariate MMSM (dash-dotted, top\nrow) and BMMSM (solid), considering dpequal to the spherical (middle) and mobility\n(bottom) diameters, are compared against experimental measurements (symbols) [34].\nThe color code indicates those results using the same number of sections.\n21due to the linear approximation of the NDF in volume as indicated in Eq. 7\n(fori < N s). Indeed, the soot volume Vis proportional to the cube of the\nparticle diameter V∝d3\np, so the derivative in Eq. 39 in terms of volume is\nproportional to V2.\n4. Application to turbulent sooting flames\n4.1. LES modeling framework\nIn this work, the Large Eddy Simulation (LES) modeling framework in-\ncludes the aforementioned soot model, a combustion model, and a turbulence-\nchemistry-soot interactions model. The latter two models are succinctly de-\nscribed below with complete details provided in the cited works.\nThe Radiation Flamelet/Progress varable (RFPV) with soot considera-\ntions [38, 39] is utilized to model the combustion. This approach character-\nizes the thermochemical state by parameterizing it in terms of the mixture\nfraction ( Z), progress variable ( C), and heat loss parameter ( H), which are\nobtained from solving the nonpremixed manifold equations in mixture frac-\ntion (steady flamelet equations). To account for soot formation, the mixture\nfraction has a compensatory source term ˙ mZfor the local mixture leaning\ncaused by the removal of PAHs from the gas phase. Similarly, the source\nterm for the progress variable is adjusted to consider the local variation in\neffective fuel resulting from the removal of PAHs [38]. The heat loss param-\neter source term captures radiative heat losses with H= 0 corresponding\nto the adiabatic state. The radiation model employs an optically thin gray\napproach, including gas effects, as in Barlow et al. [40], and soot effects, as\nin Hubbard and Tien [41]. A Strain-Sensitive Transport Approach (SSTA)\n22is utilized to manage different effective Lewis numbers of species, based on\ntheir characteristic length scales [42]. In addition, a lumped PAH transport\nequation is solved to address their slower chemistry compared to other com-\nbustion products in the thermochemical database, following the approach\ndeveloped by Mueller and Pitsch [38]. The lumped PAH source term is sep-\narated into contributions from chemical production, chemical consumption,\nand dimerization.\nTurbulence-chemistry-soot interactions require closure at the subfilter\nscales. To close turbulence-chemistry subfilter interactions, convolution is\nperformed for each manifold solution in the database with a presumed beta\nsubfilter Probability Density Function (PDF) for the mixture fraction. The\nresulting thermochemical database is then stored in a lookup table, in terms\nof the filtered mixture fraction eZ, subfilter mixture fraction variance Zv,\nfiltered progress variable eC, and filtered heat loss parameter eH. To close\nturbulence-soot subfilter interactions, a presumed soot subfilter PDF model\nthat captures finite-rate oxidation of soot is employed, following the work\nof Maldonado Colm´ an et al. [22], which was validated for turbulent non-\npremixed jet flames [22, 43] and bluff body flames [44]. This model is based\non the presumed bimodal PDF of Mueller and Pitsch [27], which considered\nsooting and non-sooting modes. The sooting mode profile accounts for a\ntransition from rich to lean mixtures to account for the oxidation of soot\nas it encounters rich mixtures. Previous work of Yang et al. [39] considered\nan abrupt transition as soon as the soot oxidation rate surpassed the soot\nsurface growth rate, which assumes that soot oxidation is strictly mixing\ncontrolled so oxidation infinitely fast. In the work of Maldonado Colm´ an\n23et al. [22], which is adapted to BMMSM here, this abrupt transition was\nreplaced with a smooth transition to account for finite-rate soot oxidation\ncompared to the local soot transport rate, which depends on the local mix-\nture fraction dissipation rate. The subfilter intermittency, that is, the weight\nbetween the sooting and non-sooting modes, is obtained by solving an ad-\nditional transport equation for N2, where N=P\niMi\n0,0, which is similar to\nHMOM [38, 45].\n4.2. Experimental and computational details\nBMMSM is evaluated with the KAUST turbulent nonpremixed jet flame,\ninvestigated experimentally by Boyette and coworkers [46, 47]. The experi-\nmental configuration is similar to the Sandia sooting flame [48] but with a\nnitrogen-diluted central jet (C 2H4: 35%, N 2: 65%, by volume). The inner\njet diameter is D= 3.2 mm and bulk velocity 54 .7 m/s, with a Re = 20 ,000.\nBoth pilot flame (ethylene-air mixture with ϕ= 0.9) and surrounding air\ncoflow are kept the same as the original configuration. Further details about\nthis burner can be found in Refs. [46–48]. Experimental measurements of the\nparticle size distribution function were obtained with a SMPS in Refs. [46, 47]\nat several positions along the flame centerline.\nLES computations were carried out using the NGA structured finite dif-\nference solver for low Mach number turbulent reacting flows [49, 50]. The\ngrid-filtered LES equations are computed in a similar domain as in Mal-\ndonado Colm´ an et al. [22] for the Sandia sooting flame, with dimensions of\n300D×75Din the axial and radial directions, respectively, and discretized\nwith 192 ×96×32 grid points in the axial, radial, and circumferential direc-\ntions, respectively. Inlet boundary conditions are prescribed as in Ref. [22],\n24precomputing the unsteady velocity inflow of the central jet following the\nexperimental conditions and the coflow velocity set to 0 .6 m/s. Only the\ncentral jet mixture composition is modified to match the current case. La-\ngrangian dynamic Smagorinsky(-like) models [51, 52] are considered to close\nthe subfilter stress and scalar flux terms. The kinetic mechanism for the\ngas-phase is the same as the one from Section 3 [36, 37].\nTwo simulations are performed with BMMSM with 8 and 12 sections,\nresulting in 24 and 36 unknowns, with spacing factors fs= 8.8327 and 4.0,\nrespectively, in order to focus in the experimental soot size region (2 to 70\nnm [46]). A simulation using the Hybrid Method of Moments (HMOM)\n[21] is also performed in order to compare BMMSM’s performance. In all\ncases, the presumed subfilter PDF model for finite-rate oxidation of soot from\nRef. [22] is considered for subfilter turbulence-chemistry-soot interactions as\ndescribed in the previous section. The total duration of simulation is 150 ms,\nwhich is equivalent to approximately 5 flow-through times through along the\nsooting region of the flame ( x/D≈140), which is sufficient for statistical\nconvergence.\n4.3. Temperature, soot volume fraction, and total number density\nFigure 3 shows centerline profiles of the mean (continuous) and rms re-\nsolved fluctuations (dashed) of the temperature (top), soot volume fraction\nfv(center), and total number density N(bottom). Although no experimen-\ntal measurements exist for these three quantities, computational results are\nshown using HMOM and using BMMSM with 8 and 12 sections, with the aim\nto assess the influence of soot statistical approach and number of sections.\nThe mean and rms temperature profiles show a good agreement between\n250 5 0 1 0 0 1 5 05 0 01 0 0 01 5 0 02 0 0 0h T i ( K )\n01 0 02 0 03 0 0\nTr m s( K )H M O M 2 4 B M M S M 3 6 B M M S M\n0 5 0 1 0 0 1 5 002468h f v i ( ! )# 1 0! 9\n0246\nfr m s\nv ( ! )# 1 0! 9\n0 5 0 1 0 0 1 5 0\nx = D ( ! )0246h N i ( c m! 3)# 1 01 1\n0246\nNr m s( c m! 3)# 1 01 1Figure 3: Centerline profiles of mean (left axes, solid lines) and rms (right axes, dashed\nlines) temperature (top), soot volume fraction (center), and total number density (bot-\ntom). Computational results using the BMMSM with 8 and 12 sections and HMOM are\ncompared.\n26models with a slight difference after the temperature peak. Note the signifi-\ncant increase in the relative magnitude of the rms temperature fluctuations\nin the downstream portion of the flame, which corresponds to the maximum\nin the rms soot volume fraction fluctuations so likely due to soot radiation\nfluctuations.\nThefvprofiles show that BMMSM predicts the same or lower soot vol-\nume fraction compared to HMOM, which is favorable because this modeling\nframework with HMOM has been shown to overpredict the volume fraction\nin such configurations [22, 43]. For both the mean soot volume fraction and\nthe mean soot number density, the BMMSM results with fewer sections are\ncomparable to HMOM, which is consistent with prior work in laminar flames\n[20]. While the maximum mean soot volume fraction in BMMSM is some-\nwhat sensitive to the number of sections (about 1.4 ppb difference between\n8 and 12 sections), the results are not strongly sensitive. The soot volume\nfraction fluctuation profiles indicate that the number of sections does not\nhave as strong of an influence as the soot model, with differences between\nHMOM and BMMSM of about 2 ppb in the peak. The mean number density\nprofiles using BMMSM show little variation with respect to the number of\nsections and are lower compared to using HMOM, whose peak value is about\n20% higher. The most significant qualitative difference between HMOM and\nBMMSM is observed in the oxidation region of the flame, where HMOM\npredicts significant number density for regions up to x/D≈115. Here,\ntheNrms fluctuations are much greater using HMOM compared to using\nBMMSM. Clearly, HMOM and BMMSM are predicting fundamentally differ-\nent soot oxidation processes. The explanation for these trends in the number\n27density findings are further analyzed later.\n4.4. Soot intermittency and soot temperature\nTo understand better the nature of soot fluctuations in the previous sub-\nsection, computational results of resolved soot intermittency and soot tem-\nperature ( Tsoot) are analyzed. The resolved soot intermittency denotes the\nprobability of not finding soot at a spatiotemporal location conditioned on\na threshold criterion of soot volume fraction, which is usually established by\nthe experimentalist. Essentially, the soot intermittency is obtained by time-\naveraging binary detection of soot, where unity indicates that no soot was\npresent over the threshold. Since no experimental measurements of soot inter-\nmittency or volume fraction are available for this configuration, the threshold\nis set at 3% of the maximum value of soot volume fraction observed in sim-\nulations, i.e., about 0.5 ppb, which was utilized in recent experiments by\nBoyette and coworkers [53]. Figure 4 (top) shows computational results of\nsoot intermittency using BMMSM and HMOM. The sooting region is well\ndefined between the abrupt intermittency drop at about x/D = 40 in the\nsoot growth region, consistent with low rms soot volume fraction in Fig. 3,\nand by a more moderate increase downstream starting at x/D = 75 in the\noxidation region, where larger fluctuations in the soot volume fraction are\nobserved in Fig. 3. Interestingly, the differences in the intermittency more\nor less mirror the differences in the soot volume fraction profiles, with only\nsmall differences in the oxidation region. Therefore, the differences in the\noxidation region in the number density and its fluctuation are due to inher-\nent differences in the soot statistical models and how particles of different\nsizes interact with turbulence rather than some fundamental difference in the\n28overall sooting flame dynamics.\nFigure 4: Centerline profiles of resolved intermittency (top) and mean soot temperature\n(bottom). Computational results using the BMMSM with 8 and 12 sections and HMOM\nare compared.\nMean soot temperature profiles are plotted in Fig. 4 (bottom), which is\ncomputed by locally averaging temperature values when the instantaneous\nresolved soot intermittency is zero. BMMSM profiles are similar to that of\nHMOM with differences only appearing once soot has nearly disappeared.\nThe influence of number of sections on soot temperature is negligible. This\nmeans that the location of soot with respect to mixture fraction is essentially\nthe same between the statistical models and is much more sensitive to the\n29subfilter soot-turbulence interactions model [22], which is common between\nBMMSM and HMOM.\n4.5. Size distribution\nThe computational results of the PSDF using BMMSM were obtained at\nseveral locations downstream of the burner along the centerline from x/D =\n60 to 90, with ∆ x/D = 5, and an eighth one further downstream at x/D =\n110 and are plotted in Fig. 5. Experimental data are available only for the\nfirst seven locations [46], including only soot particles with particle diameter\nlarger than 2nm and smaller than 225nm. The PSDF is calculated following\nEq. 39, which is normalized using the total particle number density within the\nexperimental detection limits. The PSDF evolution is well captured: a small\namount of large soot particles is found closer to the burner, which effectively\ngrows as they travel further downstream from the burner. The different plots\nsuggest that BMMSM is capable of reproducing the transition from unimodal\nPSDF in the first seven locations, which is supported by the experimental\ndata, to bimodal PSDF in the last location. It is also noticeable that the\npredicted PSDF is not a strong function of the number of sections. Overall,\nBMMSM is very accurate along the entire flame compared to measurements.\nNevertheless, the model exhibits a slight overprediction of the PSDF in the\nmid-size region ( dp≃20 nm) near the burner ( x/D≤70), which diminishes\nas it progresses downstream. Moreover, BMMSM results are in line with\nother results in the literature [9] but requiring fewer degrees of freedom.\nIn the present study, the use of the BMMSM method reduces the required\nnumber of soot scalars to 24 or 36 for fractal aggregate soot, whereas the\nsectional model from Ref. [9] used 62 soot scalars for only spherical soot.\n30Figure 5: PSDF at several locations along the centerline depicting the progress from a\nunimodal to a bimodal distribution. Computational results using BMMSM with 8 and 12\nsections (lines) are compared against experimental measurements (symbols) [46].\n31Since experimental measurements revealed the evolution of the PSDF in\nthe streamwise direction only, additional computational analysis is conducted\nto assess the radial variation of the PSDF at different locations downstream\nof the burner. Figure 6 shows the radial evolution of the PSDF at four\nlocations downstream of the burner: x/D = 50, 70, 90, and 110. Each\nlocation is representative of a portion of the sooting region (see Fig. 4).\nThe colorbar represents the normalized radial distance ( r/D) of the PSDF,\nwith darker colors being nearer the centerline. For particles with small sizes\n(dp≤30 : nm), the PSDF decreases radially outward in the first three loca-\ntions, which shifts from nucleation-dominated to growth-dominated moving\naway from the centerline. In the last location, where oxidation dominates,\nthe PSDF remains nearly constant. Furthermore, in the first three locations,\nmoving away from the centerline, a “trough” emerges in the PSDF for par-\nticle diameters ranging from 20 to 30 nm, indicating the formation of the\nsecond mode. At x/D = 110, the already existent trough is slightly shifted\nto smaller diameters, which is attributed to the oxidation process, whereas\nthe magnitude of PSDF is practically unchanged or increases (as the total\nnumber density decreases). For particles with larger sizes ( dp>30 nm), the\npresence of a second mode is evident already near the burner at large radial\ndistances and becomes more apparent closer to the centerline with increasing\ndownstream distance. Basically, the core of the flame near the centerline is\nmore unimodal while moving toward the periphery of the sooting region of\nthe flame results in a bimodal distribution. However, in the last location,\nthe second mode remains practically unchanged, with a slight shift towards\nsmaller particle diameters.\n32Figure 6: Variation of the PSDF in the radial direction (radial distance indicated by the\ncolorbar) at four different locations downstream of the burner. Computational results\nusing BMMSM with 12 sections are shown.\n4.6. Source terms\nA comprehensive examination of the soot source terms within the flame\nenables a more refined understanding of the mechanism of evolution of the\nsoot particle size distribution. Mean soot source term fields are evaluated\nfrom simulations using BMMSM with 12 sections and are compared to those\nusing HMOM. The findings obtained with 8 sections closely align with those\nobtained with 12 sections, leading to similar conclusions. Figure 7 shows\nresults using HMOM (top row) and BMMSM (bottom row), and the columns\ncorrespond to fields of mean soot-related quantities: soot volume fraction\n33Figure 7: Fields of mean soot-related quantities by columns: ( a) soot volume fraction\nfv; (b) total number density N; soot volume fraction source terms d fv/dt[s−1] for ( c)\nnucleation and condensation, ( d) surface growth, and ( e) oxidation (magnitude); and\ntotal number density source terms d N/dt[m−3s−1] for ( f) nucleation, ( g) coagulation\n(magnitude), and ( h) oxidation (magnitude). Computational results using HMOM (top\nrow) and BMMSM (12 sections, bottom row) are compared. The dotted line corresponds\nthe mean stoichiometric mixture fraction iso-contour.\n(fv), total number density ( N), and their source terms (d fv/dtand d N/dt,\nrespectively). The magenta dashed line indicates the mean stoichiometric\nmixture fraction iso-contour ⟨Z⟩=Zst, which represents the mean position\nof the flame front.\nThe fields of soot volume fraction in column ( a) show good agreement\nbetween BMMSM and HMOM, with a 20% disparity in the maximum values\nconsistent with the centerline profiles discussed above. The peak is located in\nthe mean rich region, as indicated by the Zstisocontour. However, in column\n34(b), HMOM overestimates the total number density compared to BMMSM,\nshowing a difference of about 40%, again consistent with the centerline pro-\nfiles discussed above. Additionally, the number density demonstrates a much\nmore significant presence of soot in regions where the mean mixture faction\nis fuel-lean in HMOM compared to BMMSM, even with the same location\nof the maximum.\nColumn ( c) shows the source term of the soot volume fraction resulting\nfrom the combined effect of nucleation and condensation. Very similar results\nare obtained for both HMOM and BMMSM, which occurs primarily in the\nrich region, since this a function only of the gas-phase soot precursors. On\nthe other hand, the surface growth (column ( d)) and oxidation (column ( e))\nsource terms exhibit higher rates of soot volume fraction production when\nusing HMOM. However, the explanation for this trend is simple: the soot\nvolume fraction is higher in HMOM compared to BMMSM, and the surface\ngrowth and oxidation rates scales with the amount of soot so are expected\nto be larger in magnitude.\nColumn ( f) shows the field of the rate of change of total soot number den-\nsity due to nucleation. Both HMOM and BMMSM show similar results, with\nBMMSM results radially spread further than HMOM; this minor difference is\nsimply due to lower number density predicted by BMMSM in regions closer\nto the mean Zstresulting in less condensation relative to nucleation. Fields of\ncoagulation in column ( g) show more significant differences between models.\nHMOM predicts the peak coagulation rate at the mean Zstcontour along\nthe centerline, while BMMSM predicts the peak coagulation also along the\nZstcontour but closer to the burner and away from the centerline. This ex-\n35plains the increased number density in HMOM compared to BMMSM along\nwith the centerline with a delay in coagulation until further downstream.\nLikewise, the fields of column ( h) for oxidation rates also exhibit qualita-\ntive differences. The magnitude of the oxidation rate for the number density\nis much higher for BMMSM (despite the volume fraction magnitude being\nlower) compared to HMOM. Clearly, even though the mean soot volume\nfraction is comparable between HMOM and BMMSM, the two approaches\nprovide fundamentally different evolutions of the total number density so,\nby extension, also mean particle size. The reason for this differences can be\nexplained by analyzing individual sections.\nThe evolution of the contribution of the odd sections i= 1 to 11 on\nthe soot volume fraction is shown in Fig. 8 (top). Particles of larger size\nare located preferentially downstream. A significant decrease of soot volume\nfraction is observed in the latest sections, which indicates few big particles\nexists and that the choice of the spacing factor was appropriate. Similarly,\nthe fields of number density by section are plotted in Fig. 8 (bottom). Al-\nthough the distribution by section looks qualitatively similar as the soot\nvolume fraction, quantitatively the number density fields maximizes in the\nfirst section and decreases as the local section number increases. By focusing\non specific axial and radial locations, bimodality can be identified. For in-\nstance, at x/D = 90 and r/D = 2.5, significant soot is formed from section\ni= 1 to 9 and peaks between sections i= 5 to 7. Similar behavior is observed\nfurther downstream, and the peak value moves toward larger sections. This\nis analogous to what it was observed in Fig. 6.\nTo understand better the behavior of number density source terms, ob-\n36Figure 8: Fields of mean soot volume fraction (top) and number density (bottom): contri-\nbution by odd sections i= 1 to 11. Computational results using BMMSM (12 sections) are\nshown. The dotted line corresponds the mean stoichiometric mixture fraction iso-contour.\nserved in columns ( g) and ( h) of Fig. 7, the fields of the contributions by odd\nsections i= 1 to 11 for coagulation (top) and oxidation (bottom) are plotted\nin Fig. 9. The rates due to coagulation indicate a transition from negative\nvalues to positive values as the local section number increases: the smallest\nparticles are only lost to form larger particles and the largest particles pre-\ndominantly formed from smaller, with the small-large transition increasing in\nsize with downstream distance as larger and larger particles are formed. The\ncoagulation and oxidation rates of small particles are greater in magnitude\nthan those of large particles, owing to their higher number density. Larger\nparticles tend to oxidize closer to the mean Zstcontour since this is where\n37the particles are located. Note that the coagulation dynamics governing the\nparticle sizes by downstream distance: the distribution of smaller particle\ndecreases as the coagulation rate turns negative as downstream distance in-\ncreases. On the other hand, the distribution of larger particles increases as\nthe coagulation rate tends to remain positive in the further downstream dis-\ntance, only decreasing in magnitude as for the largest particles until oxidation\ndominates and destroy the remaining soot particles. The oxidation rate is\nthen a reflection of where the particles are and the dependence of rate on\nsize. Basically, the smaller particles, which are located further upstream, are\noxidized quickly and disappear, while the larger particles, which are located\nfurther downstream, are oxidized more slowly so reach further downstream\ndistances. This is directly correlated to the observations in Fig. 6. The first\nmode peak decreases in both streamwise (by coagulation) and radial (both\ncoagulation and oxidation) directions. The second mode peak increases in\nthe streamwise direction more than in the radial direction due to the com-\npetition between coagulation and oxidation as the particles reach the flame\nfront and leaner regions. Overall, BMMSM gives a larger loss of particles\nsince first-order HMOM only has one large particle size rather than a distri-\nbution of large particle sizes so cannot capture this size-dependent oxidation\nprocess.\n4.7. Computational costs\nThe computational costs of simulations using BMMSM and HMOM are\nassessed in Table 1. The computational costs, expressed in time per time step,\nare averaged over 10,000 time steps using the same computational resources.\nSpecific costs to both solve the scalar transport equations and evaluate the\n38Figure 9: Fields of the rate of total soot number density change due to coagulation (top)\nand oxidation (bottom): contribution by odd sections i= 1 to 11. Computational results\nusing BMMSM (12 sections) are shown. The dotted line corresponds the mean stoichio-\nmetric mixture fraction iso-contour.\nsoot source terms are also examined. Remarkably, on average, BMMSM\nsimulations cost about 1.44 and 1.86 times more than HMOM in total using\n8 and 12 sections, respectively. Within these, the combined costs of scalar\ntransport and soot calculations are about 2 and 3.2 times more than HMOM,\nrespectively. Additionally, by varying the computational resources for each\nsimulation (i.e., more or less compute nodes), the cost of BMMSM compared\nto HMOM follows the same tendency (not shown), so the results in Table 1\ndo not seem to be sensitive to the balance between communication and com-\nputation between HMOM and BMMSM. The cost of scalar transport and\n39Table 1: Computational time comparison between soot models per time step. Costs units\nare in s.\nMethod NsNs\nvNfcs\nvTotal Scalar+Soot\nHMOM − 4 13 4.37 1.28\n24 BMMSM 8 24 33 6.31 2.57\n36 BMMSM 12 36 45 8.13 4.07\nsoot roughly scales linearly with the total transported variables ( Nfcs\nv), i.e.,\nconsidering all flow, combustion, and soot variables. These costs confirm\nthat BMMSM constitutes a “low-cost sectional model” with joint V-Schar-\nacterization of soot, allowing for detailed characterization of the soot size\ndistribution in turbulent reacting flows.\n5. Conclusions\nBased on the Multi-Moment Sectional Method (MMSM), a new bivariate\nformulation of MMSM was developed in this work, called BMMSM, account-\ning for a joint volume-surface formalism that can capture fractal aggregate\nmorphology of soot. The model derivation also included some features of\nprevious models such as HMOM.\nBMMSM was first implemented in a laminar flame framework. A burner-\nstabilized laminar premixed flame was simulated using both the univari-\nate and bivariate models and validated against experimental measurements.\nComputational results using BMMSM indicate an improvement in soot quan-\ntities, not only to capture the number density trends but also the bimodal\nparticle size distribution.\n40Then, BMMSM was integrated in an LES framework. PSDF results using\nBMMSM with 8 and 12 sections were compared against experimental data.\nConcerning global soot quantities, HMOM was observed to significantly over-\npredict the total number density fluctuations in the oxidation region com-\npared to BMMSM. This indicates that BMMSM outperforms HMOM in\nreproducing essential dynamics of the soot population, especially for oxi-\ndation. The new model is capable of reproducing particle size distribution\nevolution accurately, which is not a strong function of number of sections.\nFurther analyses suggest that the PSDF evolves from unimodal to bimodal in\nboth streamwise (further downstream than the available experimental mea-\nsurements) and radial directions, providing a better understanding of size\ndistribution in turbulent jet flames. Subsequently, soot source terms were\nanalyzed using HMOM and BMMSM in order to identify soot quantities be-\nhavior along the entire flame. Key differences are observed in aspects such\nas surface oxidation, as expected, and coagulation.\nFinally, the computational costs using BMMSM were evaluated and com-\npared against those of HMOM. An extraordinary outcome was observed:\nBMMSM with 8 and 12 sections respectively cost only 1.44 and 1.86 times\nmore than HMOM in total, indicating a nearly linear scaling with the total\nnumber of variables transported in each model. Overall, this methodology\nis very promising and noteworthy, allowing for detailed soot characteriza-\ntion in large-scale complex industrial configurations at limited increase in\ncomputational cost compared to widely used moment methods.\n41Acknowledgments\nThe authors gratefully acknowledge funding from the National Science\nFoundation, Award CBET-2028318. The simulations presented in this ar-\nticle were performed on computational resources supported by the Prince-\nton Institute for Computational Science and Engineering (PICSciE) and the\nPrinceton University Office of Information Technology Research Computing\ndepartment. The authors gratefully acknowledge the experimental data for\nthe KAUST flame provided by Dr. Wesley R. Boyette.\nReferences\n[1] R. P. Lindstedt, H. A. Michelsen, M. E. Mueller, Special issue and per-\nspective on the chemistry and physics of carbonaceous particle forma-\ntion, Combust. Flame (2023) 113042. doi: https://doi.org/10.1016/\nj.combustflame.2023.113042 .\n[2] M. Balthasar, M. Kraft, A stochastic approach to calculate the particle\nsize distribution function of soot particles in laminar premixed flames,\nCombust. Flame 133 (2003) 289–298.\n[3] M. Frenklach, S. J. Harris, Aerosol dynamics modeling using the method\nof moments, J. Colloid Interface Sci. 118 (1987) 252–261.\n[4] S. Park, S. Rogak, A novel fixed-sectional model for the formation and\ngrowth of aerosol agglomerates, J. Aerosol Sci. 35 (2004) 1385–1404.\n[5] A. Kazakov, M. Frenklach, Dynamic modeling of soot particle coagula-\ntion and aggregation: Implementation with the method of moments and\n42application to high-pressure laminar premixed flames, Combust. Flame\n114 (1998) 484–501.\n[6] J. Bhatt, R. Lindstedt, Analysis of the impact of agglomeration and sur-\nface chemistry models on soot formation and oxidation, Proc. Combust.\nInst. 32 (2009) 713–720.\n[7] K. Netzell, H. Lehtiniemi, F. Mauss, Calculating the soot particle\nsize distribution function in turbulent diffusion flames using a sectional\nmethod, Proc. Combust. Inst. 31 (2007) 667–674.\n[8] P. Rodrigues, B. Franzelli, R. Vicquelin, O. Gicquel, N. Darabiha, Cou-\npling an LES approach and a soot sectional model for the study of\nsooting turbulent non-premixed flames, Combust. Flame 190 (2018)\n477–499.\n[9] M. Schiener, R. Lindstedt, Transported probability density function\nbased modelling of soot particle size distributions in non-premixed tur-\nbulent jet flames, Proc. Combust. Inst. 37 (2019) 1049–1056.\n[10] Q. Zhang, H. Guo, F. Liu, G. Smallwood, M. Thomson, Modeling of\nsoot aggregate formation and size distribution in a laminar ethylene/air\ncoflow diffusion flame with detailed PAH chemistry and an advanced\nsectional aerosol dynamics model, Proc. Combust. Inst. 32 (2009) 761–\n768.\n[11] Q. Zhang, M. Thomson, H. Guo, F. Liu, G. Smallwood, A numerical\nstudy of soot aggregate formation in a laminar coflow diffusion flame,\nCombust. Flame 156 (2009) 697–705.\n43[12] C. Saggese, S. Ferrario, J. Camacho, A. Cuoci, A. Frassoldati, E. Ranzi,\nH. Wang, T. Faravelli, Kinetic modeling of particle size distribution of\nsoot in a premixed burner-stabilized stagnation ethylene flame, Com-\nbust. Flame 162 (2015) 3356–3369.\n[13] F. Sewerin, S. Rigopoulos, An LES-PBE-PDF approach for predicting\nthe soot particle size distribution in turbulent flames, Combust. Flame\n189 (2018) 62–76.\n[14] Z. Huo, M. J. Cleary, A. R. Masri, M. E. Mueller, A coupled MMC-LES\nand sectional kinetic scheme for soot formation in a turbulent flame,\nCombust. Flame 241 (2022) 112089.\n[15] S. Salenbauch, A. Cuoci, A. Frassoldati, C. Saggese, T. Faravelli,\nC. Hasse, Modeling soot formation in premixed flames using an ex-\ntended conditional quadrature method of moments, Combust. Flame\n162 (2015) 2529–2543.\n[16] A. Wick, T.-T. Nguyen, F. Laurent, R. O. Fox, H. Pitsch, Modeling\nsoot oxidation with the extended quadrature method of moments, Proc.\nCombust. Inst. 36 (2017) 789–797.\n[17] ¨O. H. Cokuslu, C. Hasse, K. P. Geigle, F. Ferraro, Soot prediction\nin a model aero-engine combustor using a quadrature-based method of\nmoments, in: AIAA Scitech 2022 Forum, 2022, p. 1446.\n[18] F. Ferraro, S. Gierth, S. Salenbauch, W. Han, C. Hasse, Soot particle\nsize distribution reconstruction in a turbulent sooting flame with the\n44split-based extended quadrature method of moments, Phys. Fluids 34\n(2022) 075121.\n[19] B. Franzelli, A. Vi´ e, N. Darabiha, A three-equation model for the pre-\ndiction of soot emissions in LES of gas turbines, Proc. Combust. Inst.\n37 (2019) 5411–5419.\n[20] S. Yang, M. E. Mueller, A Multi-Moment Sectional Method (MMSM)\nfor tracking the soot Number Density Function, Proc. Combust. Inst.\n37 (2019) 1041–1048.\n[21] M. Mueller, G. Blanquart, H. Pitsch, Hybrid method of moments for\nmodeling soot formation and growth, Combust. Flame 156 (2009) 1143–\n1155.\n[22] H. Maldonado Colm´ an, A. Attili, M. E. Mueller, Large Eddy Simula-\ntion of turbulent nonpremixed sooting flames: Presumed subfilter PDF\nmodel for finite-rate oxidation of soot, Combust. Flame (2023) 112602.\ndoi:https://doi.org/10.1016/j.combustflame.2022.112602 .\n[23] M. E. Mueller, G. Blanquart, H. Pitsch, A joint volume-surface model\nof soot aggregation with the method of moments, Proc. Combust. Inst.\n32 (2009) 785–792.\n[24] M. Frenklach, H. Wang, Detailed modeling of soot particle nucleation\nand growth, Symp. (Int.) Combust. 23 (1991) 1559–1566.\n[25] L. Waldmann, K. Schmitt, Thermophoresis and diffusiophoresis of\naerosols, in: Aerosol science, volume 137, Academic Press New York,\n1966.\n45[26] F. Bisetti, G. Blanquart, M. E. Mueller, H. Pitsch, On the formation\nand early evolution of soot in turbulent nonpremixed flames, Combust.\nFlame 159 (2012) 317–335.\n[27] M. Mueller, G. Blanquart, H. Pitsch, Modeling the oxidation-induced\nfragmentation of soot aggregates in laminar flames, Proc. Combust.\nInst. 33 (2011) 667–674.\n[28] G. Blanquart, H. Pitsch, A joint volume-surface-hydrogen multi-variate\nmodel for soot formation, Combustion generated fine carbonaceous par-\nticles (2009) 437–463.\n[29] S. E. Pratsinis, Simultaneous nucleation, condensation, and coagulation\nin aerosol reactors, J. Colloid Interface Sci. 124 (1988) 416–427.\n[30] S. J. Harris, I. M. Kennedy, The coagulation of soot particles with van\nder waals forces, Combust. Sci. Technol. 59 (1988) 443–454.\n[31] F. E. Kruis, K. A. Kusters, S. E. Pratsinis, B. Scarlett, A simple model\nfor the evolution of the characteristics of aggregate particles undergoing\ncoagulation and sintering, Aerosol Sci. Tech. 19 (1993) 514–526.\n[32] K. Neoh, J. Howard, A. Sarofim, Effect of oxidation on the physical\nstructure of soot, Symp. (Int.) Combust. 20 (1985) 951–957.\n[33] A. Kazakov, H. Wang, M. Frenklach, Detailed modeling of soot for-\nmation in laminar premixed ethylene flames at a pressure of 10 bar,\nCombust. Flame 100 (1995) 111–120.\n46[34] A. D. Abid, N. Heinz, E. D. Tolmachoff, D. J. Phares, C. S. Campbell,\nH. Wang, On evolution of particle size distribution functions of incipient\nsoot in premixed ethylene–oxygen–argon flames, Combust. Flame 154\n(2008) 775–788.\n[35] H. Pitsch, Flamemaster: A C++ computer program for 0d combustion\nand 1d laminar flame calculations, URL: http://www. itv. rwth-aachen.\nde/en/downloads/flamemaster 81 (1998).\n[36] G. Blanquart, P. Pepiot-Desjardins, H. Pitsch, Chemical mechanism for\nhigh temperature combustion of engine relevant fuels with emphasis on\nsoot precursors, Combust. Flame 156 (2009) 588–607.\n[37] K. Narayanaswamy, G. Blanquart, H. Pitsch, A consistent chemical\nmechanism for oxidation of substituted aromatic species, Combust.\nFlame 157 (2010) 1879–1898.\n[38] M. E. Mueller, H. Pitsch, LES model for sooting turbulent nonpremixed\nflames, Combust. Flame 159 (2012) 2166–2180.\n[39] S. Yang, J. K. Lew, M. E. Mueller, Large Eddy Simulation of soot\nevolution in turbulent reacting flows: Presumed subfilter PDF model\nfor soot–turbulence–chemistry interactions, Combust. Flame 209 (2019)\n200–213.\n[40] R. S. Barlow, A. N. Karpetis, J. H. Frank, J.-Y. Chen, Scalar profiles and\nno formation in laminar opposed-flow partially premixed methane/air\nflames, Combust. Flame 127 (2001) 2102–2118.\n47[41] G. L. Hubbard, C. L. Tien, Infrared mean absorption coefficients of\nluminous flames and smoke, J. Heat. Transf. 100 (1978) 287–305.\n[42] S. Yang, J. K. Lew, M. E. Mueller, Large Eddy Simulation of soot evo-\nlution in turbulent reacting flows: Strain-Sensitive Transport Approach\nfor Polycyclic Aromatic Hydrocarbons, Combust. Flame 220 (2020)\n219–234.\n[43] P. P. Duvvuri, H. M. Colm´ an, M. E. Mueller, Relative influence of\nsoot oxidation kinetics and subfilter soot-turbulence interactions on soot\nevolution in turbulent nonpremixed flames, Proc. Combust. Inst. 39\n(2023) 959–967.\n[44] H. Maldonado Colm´ an, P. P. Duvvuri, M. E. Mueller, Large eddy sim-\nulation of soot evolution in turbulent nonpremixed bluff body flames,\nProc. Combust. Inst. 39 (2023) 857–866.\n[45] M. E. Mueller, H. Pitsch, Large eddy simulation subfilter modeling of\nsoot-turbulence interactions, Phys. Fluids 23 (2011).\n[46] W. Boyette, S. Chowdhury, W. Roberts, Soot particle size distribution\nfunctions in a turbulent non-premixed ethylene-nitrogen flame, Flow\nTurbul. Combust. 98 (2017) 1173–1186.\n[47] S. Chowdhury, W. R. Boyette, W. L. Roberts, Time-averaged proba-\nbility density functions of soot nanoparticles along the centerline of a\npiloted turbulent diffusion flame using a scanning mobility particle sizer,\nJ. Aerosol Sci. 106 (2017) 56–67.\n48[48] J. Zhang, C. R. Shaddix, R. W. Schefer, Design of “model-friendly”\nturbulent non-premixed jet burners for C2+ hydrocarbon fuels, Rev.\nSci. Inst. 82 (2011) 074101.\n[49] O. Desjardins, G. Blanquart, G. Balarac, H. Pitsch, High order con-\nservative finite difference scheme for variable density low mach number\nturbulent flows, J. Comput. Phys. 227 (2008) 7125–7159.\n[50] J. F. MacArt, M. E. Mueller, Semi-implicit iterative methods for low\nmach number turbulent reacting flows: Operator splitting versus ap-\nproximate factorization, J. Comput. Phys. 326 (2016) 569–595.\n[51] M. Germano, U. Piomelli, P. Moin, W. H. Cabot, A dynamic subgrid-\nscale eddy viscosity model, Phys. Fluids 3 (1991) 1760–1765.\n[52] C. Meneveau, T. S. Lund, W. H. Cabot, A Lagrangian dynamic subgrid-\nscale model of turbulence, J. Fluid Mech. 319 (1996) 353–385.\n[53] W. R. Boyette, A. M. Bennett, E. Cenker, T. F. Guiberti, W. L. Roberts,\nEffects of pressure on soot production in piloted turbulent non-premixed\njet flames, Combust. Flame 227 (2021) 271–282.\n49"    },    {        "title": "2402.04577v1.The_Bondi_Sachs_formalism_for_the_Einstein_scalar_field_equations_with_the_zero_cosmological_constant.pdf",        "content": "arXiv:2402.04577v1  [gr-qc]  7 Feb 2024THE BONDI-SACHS FORMALISM FOR THE EINSTEIN\nSCALAR FIELD EQUATIONS WITH THE ZERO\nCOSMOLOGICAL CONSTANT\nJIALUE LI2AND XIAO ZHANG1,2,3\nAbstract. Inspiredbyinteractionofgravitational wavesanddarkmat -\nters, we study the Bondi-Sachs formalism for Einstein massl ess scalar\nfield with zero cosmological constant. We provide asymptoti c expan-\nsions for the Bondi-Sachs metrics as well as the scalar fields and prove\nthe peeling property. We also prove the positivity of the Bon di energy-\nmomentum under conditions ensuring certain asymptoticall y null hy-\npersurfaces are of order 2.\n1.Introduction\nIn general relativity, a spacetime is a 4-dimensional Loren tzian manifold\nwhose metric gsatisfies the Einstein field equations\nRµν−R\n2gµν+Λgµν=Tµν, (1.1)\nwhereRµνis the Ricci curvature, Ris the scalar curvature, Λ is the cos-\nmological constant and Tµνis the energy-momentum tensor of matter. In\nparticular, it is vacuum when Tµνvanishes. When it is wave-like and ra-\ndiates energy, the metric gis referred as gravitational waves, see, eg., [24].\nThe existence of gravitational waves was predicted by Einst ein theoretically\nand was first detected by LIGO and Virgo in September, 2015, wh ich is a\nblack hole event, named GW150914 [1]. Measured by solar mass , the initial\nblack hole masses are 36 units and 29 units, and they collide a nd form the\nfinal black hole whose mass is 62 units, with 3 units of mass rad iated away\nin gravitational waves.\nGravitational waves areassumedtobevacuumspacetimes. Wh enthecos-\nmological constant is zero, they were firstly studied by Bond i, van der Burg,\nMetzner for axially symmetric isolated spacetimes and by Sa chs for general\nasymptotically flat spacetimes in 1962 [3, 15]. They introdu ced so-called\nthe Bondi-Sachs metrics using retarded time together and po lar coordinates\nof 3-dimensional space and Bondi defined the Bondi energy and derived its\nnon-increasing property. Soon later, Penrose introduced s imple spacetimes\nto study gravitational waves in terms of conformal compacti fication and\n2000Mathematics Subject Classification. 53C50, 83C35.\nKey words and phrases. Bondi-Sachs metric; Scalar field; Peeling property; Bondi\nenergy-momentum.\n12 J LI AND X ZHANG\nshowed the peeling property of the Weyl curvatures, see, e.g . [10, 11, 12].\nThe Bondi energy is referred as the total energy after loss ca rried away by\ngravitational waves. It is a fundamental problem whether is olated gravita-\ntional systems can radiate away more energy than they initia lly have, i.e.,\nwhether Bondi’s Energy is nonnegative. The proofs of this po sitivity were\nclaimed by using both Schoen-Yau and Witten’s positive ener gy arguments,\nc.f. [16, 4] and references therein. The rigorous and comple te arguments\nwere provided under some extra conditions ensuring certain asymptotically\nnull hypersurfaces are of order 2 in [9, 22, 23], and, without these extra\nconditions, the arguments fail to work. The positivity of th e Bondi en-\nergy should require conservation of system’s total energy- momentum. But\nasymptotes of the Bondi-Sachs metrics are weak and not suffici ent for the\nADM total energy-momentum to be conserved in general.\nFor other issues related to the Bondi-Sachs spacetimes, we r efer to Wang\n[18] and the reference therein for study of the angular momen tum, and to\ne.g. [6, 19] for study of the case of nonzero cosmological con stant. We point\nout that, when the cosmological constant is nonzero, some co smological\nconstraints occur for the boundary conditions. This causes the situation on\nasymptotes is completely different, and the good definition fo r the Bondi\nenergy is still lack.\nIn this paper, we study the Bondi-Sachs formulism for the Ein stein scalar\nfield equations when the cosmological constant is zero. Phys ically, scalar\nfields can describe dark matter, see, e.g. [2]. Therefore the theory can be\nreferredas interaction of gravitational waves and dark mat ters. For massless\nscalar field Ψ, the energy-momentum tensor is given by\nTµν=∇µΨ∇νΨ−1\n2gµν∇σΨ∇σΨ. (1.2)\nThe Einstein scalar field equations are\nRµν=∇µΨ∇νΨ. (1.3)\nDenote/square=∇µ∇µthe wave operator. Twice contracted Bianchi identity\nimplies that\n/squareΨ = 0. (1.4)\nThis paper is organized as follows. In Section 2, we study the structure\nof the Einstein scalar field equations for Bondi-Sachs metri cs and separate\nthem into seven equations. In Section 3, we provide asymptot ic expansions\nfor the Bondi-Sachs metrics as well as the scalar fields. In Se ction 4, we\nprove the peeling property for the Einstein scalar field equa tions. In Section\n5, we derive the first and the second fundamental forms of asym ptotically\nnull spacelike hypersurfaces in Bondi-Sachs spacetimes an d show that they\nare asymptotically null of order 1. In Section 6, we prove the positivity\nof the Bondi energy-momentum for the Einstein scalar fields u nder certain\nconditions ensuring certain asymptotically null hypersur faces are of order\n2. In Appendix, we provide explicit formulas relating to the Einstein scalar\nfield equations.BONDI-SACHS FORMALISM 3\n2.Einstein scalar field equations for Bondi-Sachs metrics\nInthissection, westudythestructureoftheEinsteinscala rfieldequations\nfor Bondi-Sachs metrics. A Bondi-Sachs metric gon some open set of the\nMinkowski spacetime takes the following form\nds2=−/bracketleftbigge2βV\nr−r2/parenleftbigg\ne2γU2cosh(2δ)+2UWsinh(2δ)\n+e−2γW2cosh(2δ)/parenrightig/bracketrightbigg\ndu2−2e2βdudr\n−2r2/parenleftbigg\ne2γUcosh(2δ)+Wsinh(2δ)/parenrightbigg\ndudθ\n−2r2/parenleftbigg\ne−2γWcosh(2δ)+Usinh(2δ)/parenrightbigg\nsinθdudφ\n+r2/parenleftbigg\ne2γcosh(2δ)dθ2+2sinh(2δ)sinθdθdφ\n+e−2γcosh(2δ)sin2θdφ2/parenrightbigg(2.1)\nin coordinates {u, r, θ, φ }(uis retarded time)\n−∞<u<∞, r/greaterorequalslant0,0/lessorequalslantθ/lessorequalslantπ,0/lessorequalslantφ/lessorequalslant2π,\nwhereβ,γ,δ,U,VandWare smooth functions of u,r,θ,φwhich defined\nonS2for eachu, i.e., they and their derivatives take the same value at\nφ= 0,2π. We assume that (2.1) is asymptotically flat, i.e., for suffici ently\nlarger, it takes the form\nds2=−du2−2dudr+r2/parenleftbig\ndθ2+sin2θdφ2/parenrightbig\n+O/parenleftbigg1\nr/parenrightbigg\n. (2.2)\nProposition 2.1. LetΨbe any smooth scalar field over a Bondi-Sachs\nspacetime. Denote\nΩµν=Rµν−∇µΨ∇νΨ,Ω =gµνΩµν. (2.3)\nIt holds that\n∇µ/parenleftbigg\nΩµν−Ω\n2gµν/parenrightbigg\n=−/squareΨ∇νΨ. (2.4)\nProof :By the twice contracted Bianchi identity,\nLHS =∇µ/parenleftbigg\nRµν−R\n2gµν−∇µΨ∇νΨ+∇σΨ∇σΨ\n2gµν/parenrightbigg\n=−/squareΨ∇νΨ−∇µΨ∇µ∇νΨ+∇ν∇σΨ∇σΨ\n=−/squareΨ∇νΨ−∇µΨ/parenleftbig\n∇µ∇νΨ−∇ν∇µΨ/parenrightbig\n=−/squareΨ∇νΨ.\nTherefore the proposition holds. Q.E.D.4 J LI AND X ZHANG\nThe Einstein scalar field equations (1.3) give that\nΩµν= 0 (2.5)\nforµ, ν= 0,1,2,3. Same as [3, 15, 17], these ten equations separate into\nthree groups\n(1) six main equations\nΩ11= Ω12= Ω13= Ω22= Ω23= Ω33= 0, (2.6)\n(2) one trivial equation\nΩ01= 0, (2.7)\n(3) three supplementary equations\nΩ00= Ω02= Ω03= 0. (2.8)\nFor Bondi-Sachs metric (2.1), the six main equations (2.6) i mply that the\ntrivial equation (2.7) holds. Moreover, they imply that the three supple-\nmentary equations hold everywhere if they hold for at some r0>0. Indeed,\nby [3, 15, 17], the Christoffel symbols of the Bondi-Sachs metr ics give that\ngαǫΓ0\nαǫ=2e−2β\nr, (2.9)\nand, (1.3), (2.4) give that\ngαǫ/parenleftbigg∂Ωµα\n∂xǫ−1\n2∂Ωαǫ\n∂xµ−Γδ\nαǫΩµδ/parenrightbigg\n= 0 (2.10)\nforµ= 0,1,2,3. Thus, using (2.6), we get\n(1)µ= 1: (2.10) reduces to\n−gαǫΓ0\nαǫΩ01= 0. (2.11)\n(2)µ=A: using (2.7) and (2.9), (2.10) gives\ne−2β\nr2∂\n∂r/parenleftbig\nr2Ω0A/parenrightbig\n= 0, A= 2,3. (2.12)\n(3)µ= 0: using (2.7), (2.9) and Ω 02= Ω03= 0, (2.10) gives\ne−2β\nr2∂\n∂r/parenleftbig\nr2Ω00/parenrightbig\n= 0. (2.13)\nTherefore, (2.7) holds by (2.9), and (2.11), (2.8) hold ever ywhere if they\nhold at some r0>0 by (2.12), (2.13). So it only needs to study the six main\nequations, which separate into two groups.\n(1) four hypersurface equations\nΩ11= Ω12= Ω13= 0,\n/parenleftbig\ne−2γΩ22+e2γcsc2θΩ33/parenrightbig\ncosh(2δ)−2cscθΩ23sinh(2δ) = 0.BONDI-SACHS FORMALISM 5\n(2) two standard equations\ne−2γΩ22−e2γcsc2θΩ33= 0,\n/parenleftbig\ne−2γΩ22+e2γcsc2θΩ33/parenrightbig\nsinh(2δ)−2cscθΩ23cosh(2δ) = 0.\nIt concludes that the Einstein scalar field equations (1.3) a re equivalent to\nthe following seven equations\nβr=R1,(2.14)\n/parenleftig\nr4e−2β/parenleftbig\ne2γUrcosh(2δ) +Wrsinh(2δ)/parenrightbig/parenrightig\nr=R2,(2.15)\n/parenleftig\nr4e−2β/parenleftbig\nUrsinh(2δ) +e−2γWrcosh(2δ)/parenrightbig/parenrightig\nr=R3,(2.16)\nVr=R4,(2.17)\n(rγ)urcosh(2δ)+2r(γuδr+δuγr)sinh(2δ) =R5,(2.18)\n(rδ)ur−2rγuγrsinh(2δ)cosh(2δ) =R6,(2.19)\n(rΨu)r=R7,(2.20)\nwhere R1,...,R7are given in the appendix. The most important feature is\nthatR1,...,R7do not contain any derivatives with respect to x0=u.\nNull-timelike boundary problems for (2.14)-(2.20): Given initial data\nγ(u0), δ(u0),Ψ(u0)\non a null hypersurface {u=u0}, boundary values\nβ(r0), U(r0), Ur(r0), W(r0), Wr(r0), V(r0), γu(r0), δu(r0),Ψu(r0)\non a timelike hypersurface {r=r0>0}such that the constraint (2.8)\nΩ00(r0) = Ω02(r0) = Ω03(r0) = 0\nhold on{r=r0>0}, can one prove the local and global existence?\nWe refer to [7] for null-timelike boundary problems for line ar wave equa-\ntions on the Bondi-Sachs metrics. The question is still open for the Einstein\nfield equations. Heuristically, we can solve (2.14)-(2.20) as follows. On the\nnull hypersurface {u=u0}, integrating (2.14) with respect to r, we obtain\nβ=/integraldisplayr\nr0R1dr+B, (2.21)\nwhere\nB(u0,θ,φ) =β(u0,r0,θ,φ).6 J LI AND X ZHANG\nIntegrating (2.15), (2.16) with respect to r, we obtain\nr4e−2β/parenleftbig\ne2γUrcosh(2δ)+Wrsinh(2δ)/parenrightbig\n=/integraldisplayr\nr0R2dr−6N,(2.22)\nr4e−2β/parenleftbig\nUrsinh(2δ)+e−2γWrcosh(2δ)/parenrightbig\n=/integraldisplayr\nr0R3dr−6P,(2.23)\nwhere\nN=−1\n6/parenleftig\nr4e−2β/parenleftbig\ne2γUrcosh(2δ)+Wrsinh(2δ)/parenrightbig/parenrightig/vextendsingle/vextendsingle/vextendsingle\nu=u0,r=r0,\nP=−1\n6/parenleftig\nr4e−2β/parenleftbig\nUrsinh(2δ)+e−2γWrcosh(2δ)/parenrightbig/parenrightig/vextendsingle/vextendsingle/vextendsingle\nu=u0,r=r0\ndetermined by the initial data γ(u0),δ(u0) atr=r0and the boundary\nvaluesβ(r0),Ur(r0),Wr(r0) atu=u0. From (2.22), (2.23), we can solve Ur\nandWr, then integrate them with respect to r, we obtain\nU=/integraldisplayr\nr0e2βr−4/parenleftbig\ne−2γxcosh(2δ)−ysinh(2δ)/parenrightbig\ndr+Y, (2.24)\nW=/integraldisplayr\nr0e2βr−4/parenleftbig\ne2γycosh(2δ)−xsinh(2δ)/parenrightbig\ndr+X, (2.25)\nwhere\nY(u0,θ,φ) =U(u0,r0,θ,φ), X(u0,θ,φ) =W(u0,r0,θ,φ).\nIntegrating (2.17), we obtain\nV=/integraldisplayr\nr0R4dr−2M, (2.26)\nwhere\nM(u0,θ,φ) =−1\n2V(u0,r0,θ,φ).\nNote that (2.18) and (2.19) take the form\nfr+ζˆf=R5, (2.27)\nˆfr−ζf=R6. (2.28)\nwhere\nf=rγucosh(2δ),ˆf=rδu, ζ= 2γrsinh(2δ),\nThen we can solve γu,δuby integrating (2.27), (2.28) with respect to r,\nas well asγ,δby integrating the expressions of γu,δuwith respect to u,\nwhere integration constants are determined by the boundary valuesγu,δu\natr=r0and the initial data γ,δatu=u0. Finally, integrating (2.20) with\nrespect torandu, we obtain\nΨ =1\nr/integraldisplayu\nu0/integraldisplayr\nr0R7dr+r0\nrΨu(u0,r0,θ,φ)(u−u0)+Ψ(u0,r,θ,ψ).\nWe repeat this procedure to extend uby using the obtained γ(u,r,θ,φ),\nδ(u,r,θ,φ) and Ψ(u,r,θ,φ) as the new initial data. Then we get the local\nand global existence.BONDI-SACHS FORMALISM 7\n3.Asymptotic expansions\nIn this section, we study the power series solutions of the Ei nstein scalar\nfield equations (2.14)-(2.20) for asymptotically flat Bondi -Sachs metric (2.1)\nwith asymptotical condition (2.2). We follow the idea of [3, 15, 17] and\nassume that γ,δsatisfy the following outgoing radiating condition\nγ=c\nr+/parenleftbigg\n−1\n6c3−3\n2d2c+C/parenrightbigg1\nr3+O/parenleftbigg1\nr4/parenrightbigg\n, (3.1)\nδ=d\nr+/parenleftbigg\n−1\n6d3+1\n2c2d+D/parenrightbigg1\nr3+O/parenleftbigg1\nr4/parenrightbigg\n. (3.2)\nThe absence of r−2term in the above expansions prevent appearance of\nlnrterm in the expansions of unknown in vacuum Einstein field equ ations\n[3, 15, 17]. We refer to [5] for polyhomogeneity expansions i nvolvingr−2\nterm and ln rterm.\nDefinition 3.1. A 4-dimensional Lorentzian manifolds ( M,g) is called an\nasymptotically flat Bondi-Sachs spacetime if there exists a 4-dimensional\nLorentzian manifold Mcfoliated by compact 3-dimensional spacelike hyper-\nsurfaces and a diffeomorphism\nM\\Mc/ma√sto−→R3,1\\(BR0×R)\nfor someR0>0and the metric gtakes form (2.1)with asymptotic expan-\nsions(3.1),(3.2)onM\\Mc.\nAssume that the scalar field Ψ takes the following expansion\nΨ =I(u,θ,φ)+O/parenleftbigg1\nr/parenrightbigg\n.\nIt is easy to know that the ln rterm doesn’t appear in the power series\nexpansions of U,W,V,γuandδuif and only if\nIθ=Iφ= 0.\nOn the other hand, compare the left and right sides of (2.20), we obtain\nIu= 0.\nTherefore, in order to prevent the ln rterm in the power series expansions\nof (2.14)-(2.20), we need to choose Ias constant. As the Einstein scalar\nfield equations (1.3) are invariant by adding any constant in to Ψ, we choose\nI= 0 and take the following power series expansion for Ψ\nΨ =H\nr+K\nr2+L\nr3+O/parenleftbigg1\nr4/parenrightbigg\n. (3.3)\nUsing (2.21), (2.24), (2.25) and (2.26), we obtain the asymp totic form of\ng00=(X2+Y2)r2+r/parenleftbigg\n4dXY−e2BYcotθ8 J LI AND X ZHANG\n+4e2BYBθ+4e2BXBφcscθ−e2BXφcscθ\n−e2BYθ−2c(X2−Y2)/parenrightbigg\n+O(1).\nThe condition (2.2) concludes\nX=Y= 0.\nIn the following we show that, similar to [3], we can reduce B= 0 by a\ncoordinate transformation\n\nu=u0(ˆu,ˆθ,ˆφ),+u1(ˆu,ˆθ,ˆφ)\nˆr+O/parenleftbigg1\nˆr2/parenrightbigg\n,\nr= ˆr+r0(ˆu,ˆθ,ˆφ)+O/parenleftbigg1\nˆr/parenrightbigg\n,\nθ=ˆθ+θ1(ˆu,ˆθ,ˆφ)\nˆr+O/parenleftbigg1\nˆr2/parenrightbigg\n,\nφ=ˆφ+φ1(ˆu,ˆθ,ˆφ)\nˆr+O/parenleftbigg1\nˆr2/parenrightbigg\n.(3.4)\nDenote (ˆxα) = (ˆu,ˆr,ˆθ,ˆφ) and\nˆgµν=g/parenleftbigg∂\n∂ˆxµ,∂\n∂ˆxν/parenrightbigg\n.\nTransformation (3.4) yields\nˆg01=−(u0)ˆu(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)+O/parenleftbigg1\nˆr/parenrightbigg\n,\nˆg11=1\nˆr2/parenleftbigg\n2u1(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)+θ2\n1(ˆu,ˆθ,ˆφ)\n+φ2\n1(ˆu,ˆθ,ˆφ)sin2ˆθ/parenrightbigg\n+O/parenleftbigg1\nˆr3/parenrightbigg\n,\nˆg12=−(u0)ˆθ(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)−θ1(ˆu,ˆθ,ˆφ)+O/parenleftbigg1\nˆr/parenrightbigg\n,\nˆg13=−(u0)ˆφ(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)−φ1(ˆu,ˆθ,ˆφ)sin2ˆθ+O/parenleftbigg1\nˆr/parenrightbigg\n.\nTherefore\nˆg22ˆg33−ˆg2\n23−ˆr4sin2ˆθ\n=2ˆr3/parenleftbigg\n−2(u0)ˆφ(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)Bφ(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)csc2θ\n−2(u0)ˆθ(ˆu,ˆθ,ˆφ)e2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)Bθ(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ)\n+(φ1)ˆφ(ˆu,ˆθ,ˆφ)+(θ1)ˆθ(ˆu,ˆθ,ˆφ)+θ1(ˆu,ˆθ,ˆφ)cotˆθBONDI-SACHS FORMALISM 9\n+2r0(ˆu,ˆθ,ˆφ)/parenrightbigg\nsin2θ+O/parenleftbig\nˆr2/parenrightbig\n.\nChoosingu0such that\n(u0)ˆu(ˆu,ˆθ,ˆφ) = e−2B(u0(ˆu,ˆθ,ˆφ),ˆθ,ˆφ),\nwe obtain\nˆB= 0.\nTakeB=X=Y= 0. Denote\nl=cθ+2ccotθ+dφcscθ,ˆl=dθ+2dcotθ−cφcscθ.(3.5)\nSubstituting (3.1), (3.2), (3.3) into (2.21), (2.22), (2.2 3), (2.24) and (2.25),\nwe obtain the following asymptotic expansions\nβ=−2(c2+d2)+H2\n8r2−HK\n3r3+4\nβ\nr4+O/parenleftbigg1\nr5/parenrightbigg\n, (3.6)\nU=−l\nr2+2(2cl+2dˆl+3N)\n3r3+4\nU\nr4+O/parenleftbigg1\nr5/parenrightbigg\n, (3.7)\nW=−ˆl\nr2+2(2dl−2cˆl+3P)\n3r3+4\nW\nr4+O/parenleftbigg1\nr5/parenrightbigg\n, (3.8)\nV=r−2M−/parenleftbig\nlθ+lcotθ+ˆlφcscθ/parenrightbig\n+1\nV\nr+O/parenleftbigg1\nr2/parenrightbigg\n,(3.9)\nwhere\nM=M−lθ+lcotθ+ˆlφcscθ\n2, (3.10)\nand\n4\nβ=1\n8/parenleftbigg\n(c2+d2)2−6(cC+dD)−3HL−2K2/parenrightbigg\n,\n4\nU=1\n24/parenleftbigg\n−48c3cotθ−12c2(3cθ+2dφcscθ)\n−24d2cθ−6c(2dcφcscθ+8d2cotθ+2ddθ\n+12N−H2cotθ)+36Cθ+72Ccotθ\n−36d2dφcscθ−72dP+36Dφcscθ+3H2cθ\n+3H2dφcscθ+8HKθ−4KHθ/parenrightbigg\n,\n4\nW=1\n24/parenleftbigg\n−12c2(−3cφcscθ+2dθ+4dcotθ)\n−12d2(3dθ−2cφcscθ)−12c(dcθ−ddφcscθ\n−6P)+36(−Cφcscθ+Dθ+2Dcotθ)10 J LI AND X ZHANG\n−48d3cotθ−72dN−3H2(cφcscθ−dθ)\n−4KHφcscθ+8HKφcscθ+6dH2cotθ/parenrightbigg\n,\n1\nV=1\n12/parenleftbigg\n2(25cos2θ−1)c2csc2θ+2c(5cφφcsc2θ\n+37cθcotθ+5cθθ+24dφcotθcscθ)\n+2d(−24cφcotθcscθ+37dθcotθ+5dθθ\n+5dφφcsc2θ)−32cφdθcscθ+32cθdφcscθ\n+22c2\nθ+22c2\nφcsc2θ+2(24csc2θ−25)d2\n+22d2\nφcsc2θ+22d2\nθ−12Nθ−12Ncotθ\n−12Pφcscθ+3H2−3H(Hθθ+Hθcotθ\n+Hφφcsc2θ)+3H2\nθ+3H2\nφcsc2θ/parenrightbigg\n.\nUsing (2.18), (2.19), we obtain\nγu=cu\nr+3γu\nr3+O/parenleftbigg1\nr4/parenrightbigg\n, (3.11)\nδu=du\nr+3\nδu\nr3+O/parenleftbigg1\nr4/parenrightbigg\n, (3.12)\nwhere\n3γu=1\n48/parenleftbigg\n2c/parenleftbig\n11cθθ+21cθcotθ−11cφφcsc2θ\n−48dud+16dθφcscθ+8dφcotθcscθ+12M/parenrightbig\n−/parenleftbig\n10c2\nφ+10d2\nφ−3H2\nφ+3HHφφ−12Pφsinθ/parenrightbig\ncsc2θ\n+10/parenleftbig\nc2\nθ+d2\nθ/parenrightbig\n+2d/parenleftbig\n−16cθφcscθ−8cφcotθcscθ\n+11dθθ+21dθcotθ−11dφφcsc2θ/parenrightbig\n−8(3−cos(2θ))c2csc2θ−16d2(6cu+1+csc2θ)\n+3HHθθ−3Hθ(Hθ+Hcotθ)+12Ncotθ−12Nθ/parenrightbigg\n,\n3\nδu=1\n24/parenleftbigg\n2c/parenleftbig\n11cθφcscθ+5cφcotθcscθ+24cud−4dθθ\n−8dθcotθ+4dφφcsc2θ/parenrightbig\n+/parenleftbig\n10cθcφ+10dθdφ\n+3(HHθφ−HθHφ−HHφcotθ)−6Nφ/parenrightbig\ncscθ\n+2d/parenleftbig\n4cθθ+8cθcotθ−4cφφcsc2θ+11dθφcscθ\n+5dφcotθcscθ+6M/parenrightbig\n−6Pθ+6Pcotθ/parenrightbigg\n.BONDI-SACHS FORMALISM 11\nSubstituting (3.1) into (3.11) and (3.2) into (3.12), we obt ain\n−c2cu\n2−3dduc−3d2cu\n2+Cu=3γu, (3.13)\n−d2du\n2+ccud+c2du\n2+Du=3\nδu. (3.14)\nSubstituting the above power series expansions into (2.20) , we obtain\nΨu=Hu\nr−Hθθ+Hθcotθ+Hφφcsc2θ\n2r2+3\nΨu\nr3+O/parenleftbigg1\nr4/parenrightbigg\n,(3.15)\nwhere\n3\nΨu=1\n4/parenleftbigg\n4cθHθ−4cφHφcsc2θ+2Hcθθ+6Hcθcotθ\n−2Hcφφcsc2θ+2c/parenleftbig\nHθθ+3Hθcotθ−Hφφcsc2θ\n−2H/parenrightbig\n+4dθHφcscθ+4dφHθcscθ+4Hdθφcscθ\n+4Hdφcotθcscθ+4dHθφcscθ+4dHφcotθcscθ\n+2HM−Kθθ−Kθcotθ−Kφφcsc2θ−2K/parenrightbigg\n.\nSubstituting (3.3) into (3.15), we obtain\nKu=−Hθθ+Hθcotθ+Hφφcsc2θ\n2, Lu=3\nΨu.(3.16)\nFinally, comparing the coefficients of r−2term in three supplementary equa-\ntions (2.8), we obtain\nMu=−c2\nu−d2\nu−H2\nu\n2, (3.17)\n6Nu=4cudφcscθ+4dφcsc3θ−4ducφcscθ+4dcuφcscθ\n+2cθ−cθθθ+3cucθ−ccuθ−3cθθcotθ+3cθcsc2θ\n+3cθφφcsc2θ−4cduφcscθ+3dudθ−dduθ (3.18)\n+dφφφcsc3θ−3dθθφcscθ−3dθφcotθcscθ\n+3HuHθ\n2−HHuθ\n2−2Mθ,\n6Pu=−4cudθ+4cθdu−4dcuθ−8dcucotθ−dduφcscθ\n−3cθθφcscθ−ccuφcscθ−3cθφcotθcscθ\n+cφφφcsc3θ+3cucφcscθ+4cφcsc3θ+4cduθ\n+8cducotθ+3dudφcscθ−dθ+dθθθ+3dθθcotθ(3.19)\n−4dθcsc2θ−3dθφφcsc2θ+dθcot2θ\n+3HuHφ\n2cscθ−HHuφ\n2cscθ−2Mφcscθ.12 J LI AND X ZHANG\n4.Peeling property\nIn this section, we prove the peeling property for the Einste in scalar fields\nwhen spacetimes are asymptotically flat Bondi-Sachs and the scalar field Ψ\nsatisfies (3.3).\nPeeling property indicates that the Weyl curvatures of asym ptotically\nflat spacetimes have good asymptotic behaviors, which plays a significant\nrole for extracting gravitational waves from numerical sim ulation. When\nspacetimes are vacuum and asymptotically simple, Newman an d Penrose\nintroduced a complex null tetrad and proved this property [1 0, 11, 12]. This\nprocedureisreferredastheNewman-Penroseformalism. For vacuumBondi-\nSachsmetrics, peelingpropertyholdsalsowhencosmologic al constantiszero\n[3, 15], or nonzero [19]. In [8], He and Cao proved the peeling property for\nEinstein scalar field equations with zero cosmological cons tant via Newman-\nPenrose formalism. As it is unclear whether Newman-Penrose and Bondi-\nSachs coordinates are equivalent, the peeling property is a lso important via\nBondi-Sachs formalism.\nDenote i =√−1. The Bondi-Sachs metric (2.1) has the following null\ntetrad [19]\nˆe0=l= e−2β/parenleftbigg∂\n∂u−V\n2r∂\n∂r+U∂\n∂θ+Wcscθ∂\n∂φ/parenrightbigg\n,\nˆe1=k=∂\n∂r,\nˆe2=m=e−γ/parenleftbig\n1−isinh(2δ)/parenrightbig\nr/radicalbig\n2cosh(2δ)∂\n∂θ+ieγ/radicalbig\ncosh(2δ)cscθ√\n2r∂\n∂φ,\nˆe3=¯m=e−γ/parenleftbig\n1+isinh(2δ)/parenrightbig\nr/radicalbig\n2cosh(2δ)∂\n∂θ−ieγ/radicalbig\ncosh(2δ)cscθ√\n2r∂\n∂φ.(4.1)\nUnder this null tetrad, both the metric matrix (ˆ gµν) = (g(ˆeµ,ˆeν)) and its\ninverse matrix (ˆ gµν) are\n\n0−1 0 0\n−1 0 0 0\n0 0 0 1\n0 0 1 0\n.\nThe structure coefficients ˆCσ\nµνof the tetrad satisfy\n[ˆeµ,ˆeν] =ˆCσ\nµνˆeσ.\nDenote\nˆCµνσ=ˆCτ\nµνˆgτσ.\nUse the Koszul formula [13]\n2/a\\}b∇acketle{t∇XY,Z/a\\}b∇acket∇i}ht=X/a\\}b∇acketle{tY,Z/a\\}b∇acket∇i}ht+Y/a\\}b∇acketle{tZ,X/a\\}b∇acket∇i}ht−Z/a\\}b∇acketle{tX,Y/a\\}b∇acket∇i}htBONDI-SACHS FORMALISM 13\n+/a\\}b∇acketle{t[X,Y],Z/a\\}b∇acket∇i}ht−/a\\}b∇acketle{t[Y,Z],X/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t[Z,X],Y/a\\}b∇acket∇i}ht,\nwhere/a\\}b∇acketle{tX,Y/a\\}b∇acket∇i}ht=g(X,Y), we obtain connection coefficients\nˆΓµνσ=/a\\}b∇acketle{t∇ˆeµˆeν,ˆeσ/a\\}b∇acket∇i}ht=1\n2/parenleftig\nˆCµνσ−ˆCνσµ+ˆCσµν/parenrightig\n.\nWe introduce the spin coefficients defined in [14].\nDefinition 4.1. The spin coefficients of the null tetrad (4.1)are twelve\ncomplex-valued functions\nκ=−/a\\}b∇acketle{t∇kk,m/a\\}b∇acket∇i}ht=−ˆΓ112,\nρ=−/a\\}b∇acketle{t∇¯mk,m/a\\}b∇acket∇i}ht=−ˆΓ312,\nσ=−/a\\}b∇acketle{t∇mk,m/a\\}b∇acket∇i}ht=−ˆΓ212,\nτ=−/a\\}b∇acketle{t∇lk,m/a\\}b∇acket∇i}ht=−ˆΓ012,\nν=/a\\}b∇acketle{t∇ll,¯m/a\\}b∇acket∇i}ht=ˆΓ003,\nµ=/a\\}b∇acketle{t∇ml,¯m/a\\}b∇acket∇i}ht=ˆΓ203,\nλ=/a\\}b∇acketle{t∇¯ml,¯m/a\\}b∇acket∇i}ht=ˆΓ303,\nπ=/a\\}b∇acketle{t∇kl,¯m/a\\}b∇acket∇i}ht=ˆΓ103,\nε=1\n2/parenleftig\n−/a\\}b∇acketle{t∇kk,l/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t∇km,¯m/a\\}b∇acket∇i}ht/parenrightig\n=1\n2/parenleftig\n−ˆΓ110+ˆΓ123/parenrightig\n,\nγ=1\n2/parenleftig\n/a\\}b∇acketle{t∇ll,k/a\\}b∇acket∇i}ht−/a\\}b∇acketle{t∇ l¯m,m/a\\}b∇acket∇i}ht/parenrightig\n=1\n2/parenleftig\nˆΓ001−ˆΓ032/parenrightig\n,\nβ=1\n2/parenleftig\n−/a\\}b∇acketle{t∇mk,l/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t∇mm,¯m/a\\}b∇acket∇i}ht/parenrightig\n=1\n2/parenleftig\n−ˆΓ210+ˆΓ223/parenrightig\n,\nα=1\n2/parenleftig\n/a\\}b∇acketle{t∇¯ml,k/a\\}b∇acket∇i}ht−/a\\}b∇acketle{t∇ ¯m¯m,m/a\\}b∇acket∇i}ht/parenrightig\n=1\n2/parenleftig\nˆΓ301−ˆΓ332/parenrightig\n.\nIt is straightforward that the spin coefficients of Bondi-Sac hs metric (2.1)\nκ=0,\nρ=−1\nr,\nσ=−γr−tanh(2δ)δr+i/parenleftbigg\nsinh(2δ)γr−sech(2δ)δr/parenrightbigg\n,\nτ=1\n2√\n2r/radicalbig\ncosh(2δ)/parenleftbigg\ne−2β−γ/parenleftbig\n−2e2ββθ+e2γr2Urcosh(2δ)\n+r2Wrsinh(2δ)/parenrightbig\n+i/parenleftbig\n−2eγβφcosh(2δ)cscθ\n+2e−γβθsinh(2δ) +e−2β−γr2Wr/parenrightbig/parenrightbigg\n,\nν=e−2β−γVθ\n2√\n2r2/radicalbig\ncosh(2δ)+i/parenleftbigge−2β−γVθsinh(2δ)\n2√\n2r2/radicalbig\ncosh(2δ)14 J LI AND X ZHANG\n−e−2β+γVφcscθ/radicalbig\ncosh(2δ)\n2√\n2r2/parenrightbigg\n,\nµ=e−2β\n2r2/parenleftbigg\nr2Ucotθ+r2(Uθ+Wφcscθ)−V/parenrightbigg\n,\nλ=e−2β−2γ\n8r/parenleftigg\n8rtanh(2δ)(−Wcotθ+Wθ)\n+4e2γ/parenleftbigg\n−V(δrtanh(2δ)+γr)\n+rU(−cotθ+2δθtanh(2δ)+2γθ)\n+2rWcscθ(δφtanh(2δ)+γφ)\n+r(−Wφcscθ+Uθ+2δutanh(2δ) +2γu)/parenrightbigg\n+isech(2δ)/parenleftbigg\n−4e4γrcosh2(2δ)Uφcscθ\n+2r(3−cosh(4δ))(Wcotθ−Wθ)\n−2e2γrU(4δθ+sinh(4δ)cotθ−2sinh(4δ)γθ)\n−2e2γ/parenleftig\n2rWcscθ(2δφ−γφsinh(4δ))\n+rWφsinh(4δ)cscθ−2Vδr+4rδu\n+sinh(4δ)(Vγr−rUθ−2rγu)/parenrightig/parenrightbigg/parenrightigg\n,\nπ=e−2β−γ/radicalbig\ncosh(2δ)\n2√\n2r/parenleftbigg\n2e2ββθsech(2δ)+e2γr2Ur\n+r2Wrtanh(2δ)−i/parenleftbig\n2e2γ+2ββφcscθ\n−2e2ββθtanh(2δ)+r2Wrsech(2δ)/parenrightbig/parenrightbigg\n,\nε=βr+isech(2δ)\n4/parenleftbigg\nsinh(4δ)γr−2δr/parenrightbigg\n,\nγ=e−2β\n8r2/parenleftbigg\n−2V+2rVr\n−ie−2γrsech(2δ)/parenleftig\n2e4γrcosh2(2δ)cscθUφ\n+2rcosh2(2δ)Wcotθ−2rcosh2(2δ)Wθ\n+e2γ/parenleftbig\n4rWδφcscθ−2rWγφcscθsinh(4δ)\n+rUsinh(4δ)cotθ+4rUδθ−2rUγθsinh(4δ)\n−2Vδr+Vγrsinh(4δ)+rWφcscθsinh(4δ)BONDI-SACHS FORMALISM 15\n−rUθsinh(4δ) +4rδu−2rγusinh(4δ)/parenrightbig/parenrightig/parenrightbigg\n,\nβ=e−γ/radicalbig\ncosh(2δ)\n4√\n2r/parenleftig\n2βθsech(2δ)−2γθsech(2δ)\n+2cotθsech(2δ)−2δθtanh(2δ)sech(2δ)\n+r2e2γ−2βUr+e−2βr2Wrtanh(2δ)\n+2i/parenleftbig\ne2γβφcscθ−βθtanh(2δ)\n+e2γδφcscθtanh(2δ) +γθtanh(2δ)\n+e2γγφcscθ−2δθ+δθtanh2(2δ)\n−cotθtanh(2δ)+1\n2e−2βr2Wrsech(2δ)/parenrightbig/parenrightig\n,\nα=e−γ/radicalbig\ncosh(2δ)\n4√\n2r/parenleftig\n2βθsech(2δ)+2γθsech(2δ)\n−2cotθsech(2δ)+2δθtanh(2δ)sech(2δ)\n+r2e2γ−2βUr+e−2βr2Wrtanh(2δ)\n+2i/parenleftbig\n−e2γβφcscθ+βθtanh(2δ)\n+e2γδφcscθtanh(2δ) +γθtanh(2δ)\n+e2γγφcscθ−2δθ+δθtanh2(2δ)\n−cotθtanh(2δ)−1\n2e−2βr2Wrsech(2δ)/parenrightbig/parenrightig\n.\nIt is clearly\n¯µ=µ,α+¯β=π.\nDenoteWthe (0,4) type Weyl tensor. We introduce the following Weyl\nscalars [14]\nΦ0=Wkmkm,\nΦ1=Wklkm,\nΦ2=Wkm¯ml,\nΦ3=Wlkl¯m,\nΦ4=Wl¯ml¯m.\nTheorem 4.1. The peeling property\nΦk=fk\nr5−k+O/parenleftbigg1\nr6−k/parenrightbigg\n, k= 0,...,4,\nholds for asymptotically flat Bondi-Sachs spacetime, where fkare functions\ndepending only on u,θ,φ.\nProof :It is straightforward that\nΦ0=k(σ)−σ(2ρ+3ε−¯ε),16 J LI AND X ZHANG\nΦ1=k(β)−m(ε)−(α+π)σ−(ρ+ε−¯ε)β,\nΦ2=k(µ)−m(π)+l(Ψ)k(Ψ)\n6−m(Ψ)¯m(Ψ)\n6−ρµ\n−σλ−2βπ+(ε+¯ε)µ,\nΦ3=k(ν)−l(π)−µ(π+¯τ)−λ(¯π+τ)−π(γ−¯γ)\n+ν(3ε+¯ε)−1\n2l(Ψ)¯m(Ψ),\nΦ4=−l(λ)+¯m(ν)−(2µ+3γ−¯γ)λ+(2α+2π−¯τ)ν.\nUsing the asymptotic expansions, we obtain\nΦ0=−12C+cH2+i(12D+dH2)\n2r5+O/parenleftbigg1\nr6/parenrightbigg\n,\nΦ1=1\n4√\n2r4/parenleftig\n4dcφcscθ−10ccθ−4cdφcscθ\n−8c2cotθ−10ddθ−8d2cotθ+12N\n−HHθ+i/parenleftbig\n−4dcθ−10ccφcscθ+4cdθ\n−10ddφcscθ+12P−HHφcscθ/parenrightbig/parenrightig\n+O/parenleftbigg1\nr5/parenrightbigg\n,\nΦ2=1\n6r3/parenleftig\n−6ccu−6ddu−6M+HHu+6c\n−9cθcotθ−3cθθ+3cφφcsc2θ−6dθφcscθ\n−6dφcotθcscθ+i/parenleftbig\n−6dcu−6cθφcscθ\n−6cφcotθcscθ+6cdu+3dθθ+9dθcotθ\n−3dφφcsc2θ−6d/parenrightbig/parenrightig\n+O/parenleftbigg1\nr4/parenrightbigg\n,\nΦ3=−lu−iˆlu√\n2r2+O/parenleftbigg1\nr3/parenrightbigg\n,\nΦ4=−cuu−iduu\nr+O/parenleftbigg1\nr2/parenrightbigg\n.\nTherefore the theorem follows. Q.E.D.\n5.Asymptotically null spacelike hypersurfaces\nIn this section, we study the geometry of asymptotically nul l spacelike\nhypersurfaces. As metrics of null hypersurfaces degenerat e and many geo-\nmetric properties are lost, asymptotically null spacelike hypersurfaces play\nan important role in understanding null infinity.\nIn Minkowski spacetime, the hypersurface\nu=/radicalbig\n1+r2−rBONDI-SACHS FORMALISM 17\nis hyperbola H3equipped with the standard hyperbolic metric ˘ g. Let{˘ei}\nbe the frame\n˘e1=/radicalbig\n1+r2∂\n∂r,˘e2=1\nr∂\n∂θ,˘e3=1\nrsinθ∂\n∂Ψ.\nLet{˘ei}be the coframe. Denote ˘∇i=˘∇˘ei, where ˘∇is the Levi-Civita\nconnection of ˘ g. The connection 1-forms {˘ωij}are given by\n˘∇˘ei˘ej= ˘ωk\nj(˘ei)˘ek,˘ωkj= ˘gkl˘ωl\nj=−˘ωjk.\nIt gives that [20]\n˘ω12=−√\n1+r2\nr˘e2,˘ω13=−√\n1+r2\nr˘e3,˘ω23=−cotθ\nr˘e3.(5.1)\nLet˘Mbe an asymptotically flat Bondi-Sachs spacetime ( M,g), which is\ngiven by the inclusion:\ni:˘M−→M,\n(y1,y2,y3)/ma√sto−→(x0,x1,x2,x3),\nand, onM\\Mc,\nx0=u(y1,y2,y3), x1=y1=r, x2=y2=θ, x3=y3=φ.\nLet ˘g=i∗gbethe inducedmetric of ˘Mand˘∇bethe Levi-Civita connection\nof˘M. For any tangent vectors ˘Yi,˘Yj∈T˘M,i∗˘Yi,i∗˘Yjare the tangent\nvectors along ˘M. Letenbe the downward unit normal of ˘M. The second\nfundamental form is defined as\n˘h(˘Yi,˘Yj) =g/parenleftig\n∇i∗˘Yii∗˘Yj,en/parenrightig\n.\nNow it is straightforward that [9]\ni∗∂\n∂yi=∂xµ\n∂yi∂\n∂xµ=∂x0\n∂yi∂\n∂x0+∂\n∂xi,\nDenoteei=i∗˘ei,u,i=∂u\n∂yi. Then\ne1=/radicalbig\n1+r2/parenleftbigg\nu,1∂\n∂x0+∂\n∂x1/parenrightbigg\n,\ne2=1\nr/parenleftbigg\nu,2∂\n∂x0+∂\n∂x2/parenrightbigg\n,\ne3=1\nrsinθ/parenleftbigg\nu,3∂\n∂x0+∂\n∂x3/parenrightbigg(5.2)\nand\n˘g(˘ei,˘ej) =g(ei,ej), i, j= 1,2,3.18 J LI AND X ZHANG\nDefinition 5.1. A spacelike hypersurface ( ˘M,˘g,˘h) in an asymptotically flat\nBondi-Sachs spacetime is asymptotically null of order τ >0if, onM\\Mc\nwith sufficiently large r,\n˘g(˘ei,˘ej) =δij+aij,˘h(˘ei,˘ej) =δij+bij,\nwhereaij,bijsatisfy\n/braceleftig\naij,˘∇kaij,˘∇l˘∇kaij,bij,˘∇kbij/bracerightig\n=O/parenleftbigg1\nrτ/parenrightbigg\n.\nLet (˘M,˘g,˘h) be an asymptotically null spacelike hypersurface with the\ninduced metric ˘ gand the second fundamental form ˘hin asymptotically flat\nBondi-Sachs spacetime ( M,g), which is given by\nu=u0+/radicalbig\n1+r2−r+/parenleftbig\nc2+d2+λH2/parenrightbig\nu=u0\n12r3+a3(θ,φ)\nr4(5.3)\nfor sufficiently large rguaranteeing u>u0, whereλis a real constant. The\ninduced metric can be obtained by substituting d uinto (2.1). Let Xnbe\nthe downward normal vector\nXn=−∂\n∂x0−̺i∂\n∂xi.\nLeteibe given by (5.2). Since Xnis orthogonal to ei, we obtain\ng(ei,Xn) = 0.\nWe obtainXnby solving\nu,1g01+g11u,1g02+g12u,1g03+g13\nu,2g01+g21u,2g02+g22u,2g03+g23\nu,3g01+g31u,3g02+g32u,3g03+g33\n\n̺1\n̺2\n̺3\n=−\nu,1g00+g01\nu,2g00+g02\nu,3g00+g03\n.\nDenote the unit normal vector\nen=Xn/radicalbig\n−g(Xn,Xn).\nThen the second fundamental form is given by\n˘h(˘ei,˘ej) =g(∇eiej,en), i, j= 1,2,3.\nProposition 5.1. Asymptotically null spacelike hypersurface (5.3)in an\nasymptotically flat Bondi-Sachs spacetime is of order 1.\nProof :Forrsufficiently large on M\\Mc, we expand the following func-\ntions atu=u0by Taylor series\nc(u,θ,φ) =c(u0,θ,φ)+cu(u0,θ,φ)(u−u0)\n+cuu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nd(u,θ,φ) =d(u0,θ,φ)+du(u0,θ,φ)(u−u0)\n+duu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,BONDI-SACHS FORMALISM 19\nC(u,θ,φ) =C(u0,θ,φ)+Cu(u0,θ,φ)(u−u0)\n+Cuu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nD(u,θ,φ) =D(u0,θ,φ)+Du(u0,θ,φ)(u−u0)\n+Duu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nM(u,θ,φ) =M(u0,θ,φ)+Mu(u0,θ,φ)(u−u0)\n+Muu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nN(u,θ,φ) =N(u0,θ,φ)+Nu(u0,θ,φ)(u−u0)\n+Nuu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nP(u,θ,φ) =P(u0,θ,φ)+Pu(u0,θ,φ)(u−u0)\n+Puu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nH(u,θ,φ) =H(u0,θ,φ)+Hu(u0,θ,φ)(u−u0)\n+Huu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nK(u,θ,φ) =K(u0,θ,φ)+Ku(u0,θ,φ)(u−u0)\n+Kuu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n,\nL(u,θ,φ) =L(u0,θ,φ)+Lu(u0,θ,φ)(u−u0)\n+Luu(u0,θ,φ)\n2(u−u0)2+O/parenleftbig\n(u−u0)3/parenrightbig\n.\nWe obtain\n˘g(˘e1,˘e1) =1−(1−2λ)H2\n4r2+1\n12r3/parenleftbigg\n96a3+6M\n+3(lθ+lcotθ+ˆlφcscθ)−6(ccu+ddu)\n−3HHu−8HK/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘g(˘e1,˘e2) =−l\n2r2+1\n12r3/parenleftbigg\n12N−3lu−6(ccθ+ddθ)\n−4cdφcscθ+4dcφcscθ−8(c2+d2)cotθ\n−2λHHθ/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘g(˘e1,˘e3) =−ˆl\n2r2+1\n12r3/parenleftbigg\n12P−3ˆlu−6(ccφ+ddφ)cscθ20 J LI AND X ZHANG\n+4cdθ−4dcθ−2λHHφcscθ/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘g(˘e2,˘e2) =1+2c\nr+2(c2+d2)+cu\nr2+1\nr3/parenleftbigg\nc3+cd2\n+2C+2(ccu+ddu)+cuu\n4/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘g(˘e2,˘e3) =2d\nr+du\nr2+1\nr3/parenleftbigg\nc2d+d3+2D+duu\n4/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘g(˘e3,˘e3) =1−2c\nr+2(c2+d2)−cu\nr2+1\nr3/parenleftbigg\n−c3−cd2\n−2C+2(ccu+ddu)−cuu\n4/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘h(˘e1,˘e1) =1+8c2+8d2+(1+6λ)H2\n8r2+1\n6r3/parenleftbigg\n96a3−6M\n−3(lθ+lcotθ+ˆlφcscθ)+4HK/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘h(˘e1,˘e2) =l\n2r2+1\n12r3/parenleftbigg\n3lu+4(c2+d2)cotθ−24N\n+2(cdφ−dcφ)cscθ−6(ccθ+ddθ)−8λHHθ/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘h(˘e1,˘e3) =ˆl\n2r2+1\n12r3/parenleftbigg\n3ˆlu+2(dcθ−cdθ)−24P\n−6(ccφ+ddφ)cscθ−8λHHφcscθ/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘h(˘e2,˘e2) =1+c\nr+8cu+(1−2λ)H2\n8r2+1\n8r3/parenleftbigg\n6M−32a3\n−8C−4c(c2+d2)+10(ccu+ddu)+3cuu−lθ\n+3lcotθ+3ˆlφcscθ+(1−2λ)cH2+HHu\n+8\n3HK/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,\n˘h(˘e2,˘e3) =d\nr+du\nr2+1\n8r3/parenleftbigg\n(4cot2θ+4csc2θ+(1−2λ)H2)d\n−4d(c2+d2)−8cφcotθcscθ−2dφφcsc2θ\n−2dθcotθ−2dθθ−8D+3duu/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n,BONDI-SACHS FORMALISM 21\n˘h(˘e3,˘e3) =1−c\nr+(1−2λ)H2−8cu\n8r2+1\n8r3/parenleftbigg\n6M−32a3\n+8C+4c(c2+d2)+10(ccu+ddu)−3cuu+3lθ\n−lcotθ−ˆlφcscθ−(1−2λ)cH2+HHu\n+8\n3HK/parenrightbigg\n+O/parenleftbigg1\nr4/parenrightbigg\n.\nHere all functions in the right hand sides take value at u=u0. Therefore\n(˘M, ˘g,˘h) is asymptotically null of order 1. Q.E.D.\nIn [20] (see also [22, 23]), the second author proved the posi tive energy\ntheorem for asymptotically null infinity. Let ( ˘M, ˘g,˘h) be an asymptotically\nnull spacelike hypersurface of order τ >3\n2in an asymptotically flat Bondi\nspacetime which satisfies the dominant energy condition\nT00/greaterorequalslant/radicalig\nT2\n01+T2\n02+T2\n03, T00/greaterorequalslant|Tµν| (5.4)\nfor frame {eα}such thate0is timelike, eiis spacelike, 1 ≤i≤3. The total\nenergy-momentum of ( ˘M, ˘g,˘h) are given by\nEν(˘M) =1\n16πlim\nr→∞/integraldisplay\nSrEnνr˘e2∧˘e3(5.5)\nwhere\nE=˘∇ja1j−˘∇1tr˘g(a)+(a22+a33)+2(b22+b33).(5.6)\nIt holds that\nE0(˘M)/greaterorequalslant/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3Ei(˘M)2. (5.7)\nIfE0(˘M) = 0, then Mis flat along ˘M.\nFor the Einstein scalar fields\nT00=1\n2/parenleftig\n(e0(Ψ))2+/summationdisplay\n1/lessorequalslanti/lessorequalslant3(ei(Ψ))2/parenrightig\n,\nTii=1\n2/parenleftig\n(e0(Ψ))2+(ei(Ψ))2−/summationdisplay\n1/lessorequalslantj/lessorequalslant3,j/negationslash=i(ej(Ψ))2/parenrightig\n,\nTαi=eα(Ψ)ei(Ψ), α/\\e}atio\\slash=i.\nTherefore the dominant energy condition (5.4) holds.\n6.The Bondi energy-momentum and positivity\nIn this section we prove the positivity of the Bondi energy-m omentum\nfor the Einstein scalar fields when spacetimes are asymptoti cally flat Bondi-\nSachs and the scalar field Ψ satisfies (3.3). Following from §6 in [9], we22 J LI AND X ZHANG\ndefine the Bondi energy-momentum of null hypersurface as\nmν(u) =1\n4π/integraldisplay\nS2M(u,θ,φ)nνdS, (6.1)\nfor eachu, whereν= 0, 1, 2, 3, and\nn0= 1, n1= sinθcosφ, n2= sinθsinφ, n3= cosθ.\nBy (3.17), we obtain\ndmν\ndu=−1\n8π/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\nnνdS. (6.2)\nIn particular, taking ν= 0, (6.2) gives rise to the famous Bondi energy loss\nformula\ndm0\ndu=−1\n8π/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\ndS≤0 (6.3)\nfor the Einstein scalar fields. In this case, it is reasonable to referMas the\nmass aspect andcu,du,Huas thenews functions .\nRemark 6.1. The original definition of Bondi energy-momentum is\nmν(u) =1\n4π/integraldisplay\nS2M(u,θ,φ)nνdS\nfor vacuum, i.e., Ψ = 0. However, in order to prove (6.2), we need extra\nassumptions that\n/integraldisplay2π\n0c(u,0,φ)dφ=/integraldisplay2π\n0c(u,π,φ)dφ= 0\nfor eachu, see, e.g. [3, 15, 17, 21, 9] . The definition (6.1)is more natural.\nSimilar to [9], we prove the following Bondi energy-momentu m loss for-\nmula.\nProposition 6.1. On asymptotically flat Bondi-Sachs spacetimes, the Bondi\nenergy-momentum for the Einstein scalar fields with conditi on(3.3)satisfies\nd\ndu\nm0−/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3m2\ni\n/lessorequalslant0, (6.4)\nand the equality at some u0implies that\ncu/vextendsingle/vextendsingle\nu=u0=du/vextendsingle/vextendsingle\nu=u0=Hu/vextendsingle/vextendsingle\nu=u0= 0. (6.5)\nProof :We use the same argument as Proposition 2.1 [9] and provide th e\nproof here for the sake of completeness. Denote\n|m|=/radicalig\nm2\n1+m2\n2+m2\n3.\nWe assume |m| /\\e}atio\\slash= 0, otherwise the proof follows from (6.3). Using\n(n1)2+(n2)2+(n3)2= 1BONDI-SACHS FORMALISM 23\nand Cauchy-Schwarz inequality, we obtain\n/summationdisplay\n1/lessorequalslanti/lessorequalslant3/parenleftbigg/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\nnidS/parenrightbigg2\n=/summationdisplay\n1/lessorequalslanti/lessorequalslant3/parenleftbigg/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig1\n2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig1\n2nidS/parenrightbigg2\n≤/summationdisplay\n1/lessorequalslanti/lessorequalslant3/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\ndS/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig/parenleftbig\nni/parenrightbig2dS\n=/parenleftbigg/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\ndS/parenrightbigg2\n.\nTherefore\n/summationdisplay\n1/lessorequalslanti/lessorequalslant3mi/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\nnidS\n/lessorequalslant|m|/bracketleftigg/summationdisplay\n1/lessorequalslanti/lessorequalslant3/parenleftbigg/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\nnidS/parenrightbigg2/bracketrightigg1\n2\n/lessorequalslant|m|/integraldisplay\nS2/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig\ndS.\nThis together (6.2) yield\nd\ndu/parenleftbigg\nm0−|m|/parenrightbigg\n=dm0\ndu−1\n|m|3/summationdisplay\ni=1midmi\ndu≤0.\nIf equality holds at some u0in (6.4), then there exists constants ǫi, which\nare independent of θandφ, such that, for i= 1,2,3,\nǫi/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig1\n2=/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig1\n2ni,\nholds atu0. This implies that\n/parenleftbig\n2c2\nu+2d2\nu+H2\nu/parenrightbig/vextendsingle/vextendsingle\nu=u0= 0.\nTherefore (6.5) holds. Q.E.D.\nLemma 6.1. Let (M,g) be asymptotically flat Bondi-Sachs spacetimes and\n(˘M,˘g,˘h) be asymptotically null spacelike hypersurface given by (5.3)for\nsufficiently large r. If\nc(u0,θ,φ) =d(u0,θ,φ) = 0, (6.6)\nthen\nE=4M(u0,θ,φ)\nr3+O/parenleftbigg1\nr4/parenrightbigg\n. (6.7)24 J LI AND X ZHANG\nProof :Using the asymptotic expansions of ˘ g,˘hderived in Proposition\n5.1, we obtain, at u=u0,\na22+a33=4(c2+d2)\nr2+4(ccu+ddu)\nr3+O/parenleftbigg1\nr4/parenrightbigg\n,\nb22+b33=(1−2λ)H2\n4r2+3M−16a3+5(ccu+ddu)\n2r3\n+lθ+lcotθ+ˆlφcscθ\n4r3+H(3Hu+8K)\n12r3+O/parenleftbigg1\nr4/parenrightbigg\n.\nUsing the formula\n˘∇kaij= ˘ek(aij)−3/summationdisplay\nl=1ajl˘ωli(˘ek)−3/summationdisplay\nl=1ail˘ωlj(˘ek),\nwe obtain\nE=˘∇ja1j−˘e1(tr˘g(a))+a22+a33+2(b22+b33)\n=˘e1(a11)+ ˘e2(a12)+ ˘e3(a13)−3/summationdisplay\nj=1(aj2˘ω21(˘ej)+aj3˘ω31(˘ej))\n−a113/summationdisplay\nj=1˘ω1j(˘ej)−a123/summationdisplay\nj=1˘ω2j(˘ej)−a133/summationdisplay\nj=1˘ω3j(˘ej)\n−˘e1(a11)−˘e1(a22)−˘e1(a33)+a22+a33+2(b22+b33)\n=1\nr∂a12\n∂θ+1\nrsinθ∂a13\n∂φ−3/summationdisplay\nj=1aj2√\n1+r2\nr˘e2(˘ej)−3/summationdisplay\nj=1aj3√\n1+r2\nr˘e3(˘ej)\n+√\n1+r2\nra11˘e2(˘e2)+√\n1+r2\nra11˘e3(˘e3)−a12√\n1+r2\nr˘e2(˘e1)\n+cotθ\nra12˘e3(˘e3)−a13√\n1+r2\nr˘e3(˘e1)−cotθ\nra13˘e3(˘e2)\n−/radicalbig\n1+r2∂a22\n∂r−/radicalbig\n1+r2∂a33\n∂r+a22+a33+2(b22+b33)\n=1\nr∂a12\n∂θ+1\nrsinθ∂a13\n∂φ−√\n1+r2\nr(a22+a33)+2√\n1+r2\nra11\n+cotθ\nra12−/radicalbig\n1+r2∂(a22+a33)\n∂r+a22+a33+2(b22+b33)\n=8√\n1+r2\nr3/parenleftbig\nc2+d2/parenrightbig\n+12√\n1+r2\nr4(ccu+ddu)\n+M+16a3−ccu−ddu\nr3+3M−16a3+5(ccu+ddu)\nr3\n+lθ+lcotθ+ˆlφcscθ\n2r3+O/parenleftbigg1\nr4/parenrightbiggBONDI-SACHS FORMALISM 25\n=8(c2+d2)\nr2+8M+32(ccu+ddu)+lθ+lcotθ+ˆlφcscθ\n2r3+O/parenleftbigg1\nr4/parenrightbigg\n.\nIf (6.6) holds, we have, at u=u0,\nlθ+lcotθ+ˆlφcscθ=−2c(u0,θ,φ)+cθθ(u0,θ,φ)−cφφ(u0,θ,φ)csc2θ\n+2dθφ(u0,θ,φ)cscθ+3cθ(u0,θ,φ)cotθ\n+2dφ(u0,θ,φ)cotθcscθ= 0.\nTherefore (6.7) follows. Q.E.D.\nTheorem 6.1. Let (M,g) be an asymptotically flat Bondi-Sachs space-\ntime which satisfies the Einstein scalar field equations and t he scalar field Ψ\nsatisfies (3.3). Suppose that (6.6)holds for some u=u0.\n(i) For allu/lessorequalslantu0,\nm0(u)/greaterorequalslant/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3m2\ni(u). (6.8)\nIfm0(u0) = 0, the spacetime is flat and the scalar field Ψvanishes along\nany asymptotically null spacelike hypersurface given by (5.3)for sufficiently\nlarger.\n(ii) If there exists u1< u0such that the equality holds at u=u0, u1in\n(6.8), then the equality holds and\nc=d=Hu= 0 (6.9)\nfor allu∈[u1,u0].\n(iii) If there exists u1< u0such thatm0(u1) =m0(u0) = 0, then the\nspacetime is flat and the scalar field Ψvanishes for all u∈(u1,u0].\nProof :(i) Choose an asymptotically null spacelike hypersurface ˘Mgiven\nby (5.3) for sufficiently large r. By Lemma 6.1, we obtain, for ν= 0,1,2,3,\nEν(˘M) =1\n16πlim\nr→∞/integraldisplay\nSrEnνr˘e2∧˘e3\n=1\n16πlim\nr→∞/integraldisplay\nS2Enνrr2sinθdθdφ\n=1\n16πlim\nr→∞/integraldisplay\nS2Enνr3dS\n=1\n4πlim\nr→∞/integraldisplay\nS2M(u0,θ,φ)\nr3nνr3dS\n=mν(u0).\nAs the dominant energy condition holds, we can apply (5.7) an d obtain\nm0(u0) =E0(˘M)/greaterorequalslant/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3E2\ni(˘M) =/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3m2\ni(u0).26 J LI AND X ZHANG\nThen the Bondi energy-momentum loss formula (6.4) gives (6. 8).\nm0(u)−/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3m2\ni(u)/greaterorequalslantm0(u0)−/radicaligg/summationdisplay\n1/lessorequalslanti/lessorequalslant3m2\ni(u0)/greaterorequalslant0.\nThis gives (6.8) for all u/lessorequalslantu0. Ifm0(u0) = 0, then E0(˘M) = 0 and\nT00= 0 along any asymptotically null spacelike hypersurface gi ven by (5.3)\nfor sufficiently large r. This implies that the spacetime is flat and the scalar\nfield Ψ vanishes along these hypersurfaces.\n(ii) By the Bondi energy-momentum loss formula (6.4), we kno w that the\nequality and (6.5) hold for all u∈[u1, u0]. Then (6.9) holds as c=d= 0 at\nu=u0.\n(iii) By the Bondi energy loss formula (6.3), we know that m0(u) = 0 and\n(6.9) holds for all u∈[u1, u0]. By (i), the spacetime is flat and the scalar\nfield Ψ vanishes along any asymptotically null spacelike hyp ersurface given\nby\nu= ˇu0+/radicalbig\n1+r2−r+/parenleftbig\nc2+d2+λH2/parenrightbig\nu=ˇu0\n12r3+a3(θ,φ)\nr4\nfor sufficiently large rguaranteeing u>ˇu0, whereu1/lessorequalslantˇu0/lessorequalslantu0. This gives\nthe conclusion. Q.E.D.\nAcknowledgement The work is supported by the National Natural Science Founda tion\nof China 12326602, the special foundation for Junwu and Guan gxi Ba Gui Scholars.\nAppendix: Explicit formulas for R1,···,R7in(2.14)-(2.20)\nR1=1\n2r/parenleftig\nγ2\nrcosh2(2δ)+δ2\nr/parenrightig\n+1\n4rΨ2\nr,\nR2=2r2/parenleftig\nβrθ+2δrδθ−2r−1βθ−4γrδθsinh(2δ)cosh(2δ)\n−/parenleftbig\nγrθ−2γrγθ+2γrcotθ/parenrightbig\ncosh2(2δ)/parenrightig\n+2r2e2γcscθ/parenleftig\n−δrφ−2δrγφ+(γrφ+2γrγφ)sinh(2δ)cosh(2δ)\n+2γrδφ(1+2sinh2(2δ))/parenrightig\n+2r2ΨrΨθ,\nR3=2r2e−2γ/parenleftig\n−δrθ+2δrγθ−2δrcotθ−(γrθ−2γrγθ\n+2γrcotθ)cosh(2δ)sinh(2δ)−2γrδθ(1+2sinh2(2δ))/parenrightig\n+2r2cscθ/parenleftig\nβrφ+2δrδφ−2r−1βφ+4γrδφcosh(2δ)sinh(2δ)\n+(γrφ+2γrγφ)cosh2(2δ)/parenrightig\n+2r2ΨrΨφcscθ,BONDI-SACHS FORMALISM 27\nR4=2e2βcscθ/parenleftig\n(βθφ+βθβφ+2δθδφ)sinh(2δ)\n+(δθφ+δφcotθ+δθγφ−γθδφ+δθβφ+βθδφ)cosh(2δ)/parenrightig\n−e2β−2γ/parenleftig\n(βθθ+β2\nθ+βθcotθ+2γ2\nθ+2δ2\nθ−1−γθθ\n−3γθcotθ−2βθγθ)cosh(2δ) +(δθθ+3δθcotθ+2βθδθ\n−4γθδθ)sinh(2δ)/parenrightig\n−e2β+2γcsc2θ/parenleftig\n(βφφ+β2\nφ+2γ2\nφ+2δ2\nφ\n+γφφ+2βφγφ)cosh(2δ) +(δφφ+2βφδφ+4γφδφ)sinh(2δ)/parenrightig\n−1\n4r4e−2β/parenleftig\n(e2γU2\nr+e−2γW2\nr)cosh(2δ) +2UrWrsinh(2δ)/parenrightig\n+1\n2r(rUrθ+rUrcotθ+4Uθ+4Ucotθ)+1\n2rcscθ(rWrφ+4Wφ)\n−1\n2e2β/parenleftig\n(e−2γΨ2\nθ+e2γΨ2\nφcsc2θ)cosh(2δ)−2ΨθΨφsinh(2δ)cscθ/parenrightig\n,\nR5=1\n2(γrVr+γrrV+r−1γrV)cosh(2δ)+2γrδrVsinh(2δ)\n+1\n8r3e−2β(e2γU2\nr−e−2γW2\nr)+1\n2r−1e2β−2γ(βθθ+β2\nθ−βθcotθ)\n−1\n2r−1e2β+2γ(βφφ+β2\nφ)csc2θ+r−1e2β(βθδφ−βφδθ)cscθ\n+1\n4re2γcscθ/parenleftig\n(Urφ+2r−1Uφ)sinh(2δ) +4δrUφcosh(2δ)/parenrightig\n−1\n4re−2γ/parenleftig\n(Wrθ−Wrcotθ)sinh(2δ)+2r−1(Wθ−Wcotθ)sinh(2δ)\n+4δr(Wθ−Wcotθ)cosh(2δ)/parenrightig\n−1\n4r/parenleftbig\nUrθ+2r−1Uθ−Urcotθ\n−2r−1Ucotθ+4r−1γθU+4γrθU+2γθUr+2γrUθ\n+2γrUcotθ/parenrightbig\ncosh(2δ)−r(δrUθ+2γrδθU+2δrγθU\n−δrUcotθ)sinh(2δ)+1\n4rcscθ(Wrφ+2r−1Wφ−4r−1γφW\n−4γrφW−2γφWr−2γrWφ)cosh(2δ)+rcscθ(δrWφ−2δrγφW\n−2γrδφW)sinh(2δ) +1\n4re2β/parenleftbig\ne−2γΨ2\nθ−e2γΨ2\nφcsc2θ/parenrightbig\n,\nR6=1\n2/parenleftig\nδrVr+δrrV+r−1δrV−2γ2\nrVcosh(2δ)sinh(2δ)/parenrightig\n−1\n2re2β−2γ(βθθ\n+β2\nθ−βθcotθ)sinh(2δ)−1\n2re2β+2γcsc2(θ)(βφφ+β2\nφ)sinh(2δ)\n−1\nre2βcscθ/parenleftbig\n−βθφ−βθβφ+βφcotθ+βθγφ−γθβφ/parenrightbig\ncosh(2δ)\n+1\n8r3e−2β/parenleftig\n(e2γU2\nr+e−2γW2\nr)sinh(2δ) +2UrWrcosh(2δ)/parenrightig28 J LI AND X ZHANG\n−1\n2r/parenleftig\n2δrθU+2r−1δθU+δrUθ+δθUr+δrUcotθ−2(γrUθ\n−γrUcotθ+2γrγθU)cosh(2δ)sinh(2δ)/parenrightig\n−1\n2rcscθ/parenleftig\n2δrφW\n+2r−1δφW+δrWφ+δφWr+2(γrWφ−2γrγφW)cosh(2δ)sinh(2δ)/parenrightig\n−1\n4re−2γ/parenleftig\nWrθ−Wrcotθ+2r−1(Wθ−Wcotθ)−4γr(Wθ\n−Wcotθ)cosh2(2δ)/parenrightig\n−1\n4re2γcscθ/parenleftig\nUrφ+2r−1Uφ+4γrUφcosh2(2δ)/parenrightig\n−e2β\n4r/parenleftig\n(e−2γΨ2\nθ+e2γΨ2\nφcsc2θ)sinh(2δ)−2ΨθΨφcosh(2δ)cscθ/parenrightig\n,\nR7=1\n2VΨrr−rUΨrθ−rWcscθΨrφ+1\n2re2β−2γcosh(2δ)Ψθθ\n−1\nre2βΨθφcscθsinh(2δ)+1\n2re2β+2γΨφφcosh(2δ)csc2θ\n+r\n2/parenleftigV\nr2+Vr\nr−Ucotθ−Uθ−Wφcscθ/parenrightig\nΨr\n+/parenleftbigg1\n2re2β−2γ/parenleftig\ncosh(2δ)cotθ+2cosh(2δ)βθ+2sinh(2δ)δθ\n−2cosh(2δ)γθ/parenrightig\n−1\n2(2U+rUr)−1\nre2βcosh(2δ)cscθδφ\n−1\nre2βsinh(2δ)cscθβφ/parenrightbigg\nΨθ+/parenleftbigg1\n2re2β+2γcsc2θ/parenleftig\n2cosh(2δ)γφ\n+2sinh(2δ)δφ+2cosh(2δ)βφ/parenrightig\n−1\n2(rWr+2W)cscθ\n−1\nre2ββθsinh(2δ)cscθ−1\nre2βδθcosh(2δ)cscθ/parenrightbigg\nΨφ\nReferences\n[1] B. Abbott et al, Observation of gravitational waves from a binary black hole\nmerger, Phys. Rev. Lett. 116(2016), 061102.\n[2] M. Alcubierre, F. Guzm´ an, T. Matos et al, Galactic collapse of scalar field dark\nmatter, Class. Quantum Grav. 19(2002), 5017-5024.\n[3] H. Bondi, M. van der Burg, A. Metzner, Gravitational waves in general relativ-\nity VII. Waves from axi-symmetric isolated systems , Proc. Roy. Soc. London A.\n269(1962), 21-52.\n[4] P. Chru´ sciel, J. Jezierski, S. Leski, The Trautman-Bondi mass of initial data sets ,\nAdv. Theor. Math. Phys. 1(2004), 83-139.\n[5] P.Chru´ sciel, M.MacCallum, D.Singleton, Gravitational waves in general relativity\nXIV. Bondi expansions and the ‘polyhomogeneity’ of I, Phil. Trans. Roy. Soc. A\n350(1995), 113-141.\n[6] H. Ge, M. Luo, Q. Su, D. Wang, X. Zhang, Bondi-Sachs metrics and photon\nrockets, Gen Relativ Gravit. 43(2011), 2729-2742.BONDI-SACHS FORMALISM 29\n[7] Q. Han, L. Zhang, Asymptotics for null-timelike boundary problems for gener al\nlinear wave equations , Sci. China Math. 64(2021), 111-128.\n[8] X. He, Z. Cao, On the conserved quantities in an Einstein-scalar system , J. Math.\nPhys. 64(2023), 092502.\n[9] W.-L. Huang, S.T. Yau, X. Zhang, Positivity of the Bondi mass in Bondi’s radi-\nating spacetimes , Rend. Lincei Mat. Appl. 17(2006), 335-349.\n[10] E. Newman, R. Penrose, An approach to gravitational radiation by a method of\nspin coefficients , J. Math. Phys. 3(1962), 566-578.\n[11] E. Newman, R. Penrose, Errata: An approach to gravitational radiation by a\nmethod of spin coefficients , J. Math. Phys. 4(1963), 998-998.\n[12] E. Newman, W. Unti, Behavior of asymptotically flat empty spaces , J. Math. Phys.\n3(1962), 891-901.\n[13] B. O’Neill, Semi-Riemannian geometry with applications to general rel ativity, Aca-\ndemic Press. 1983.\n[14] B. O’Neill, The geometry of Kerr black holes , A K Peters, Ltd. 1995.\n[15] R. Sachs, Gravitational waves in general relativity VIII. Waves in as ymptotically\nflat space-time , Proc. Roy. Soc. London, A 270(1962), 103-126.\n[16] R. Schoen, S.T. Yau, Proof that the Bondi mass is positive , Phys. Rev. Lett.\n48(1982), 369-371.\n[17] M. van der Burg, Gravitational waves in general relativity IX. Conserved qu anti-\nties, Proc. Roy. Soc. London A 294(1966), 112-122.\n[18] M.-T. Wang, Angular momentum and supertranslation in general relativi ty,\narXiv:2303.02424.\n[19] F. Xie, X. Zhang, The peeling property of Bondi-Sachs metrics for nonzero cos -\nmological constants , Sci. China Math. 63(2020), 617-626.\n[20] X. Zhang, A definition of total energy-momenta and the positive mass th eorem on\nasymptotically hyperbolic 3-manifolds. I , Commun. Math. Phys. 249(2004), 529-\n548.\n[21] X. Zhang, On the relation between ADM and Bondi energy-momenta , Adv. Theor.\nMath. Phys. 10(2006), 261-282.\n[22] X. Zhang, Recent progress on the positive energy theorem , Int. J. Mod. Phys. A.\n30(2015), 1545018.\n[23] X. Zhang, The positive energy theorem in general relativity (in Chine se), Sci. Sin\nMath. 47(2017), 673–688.\n[24] X. Zhang, Bondi-Sachs metrics and gravitational waves (in Chinese) , Sci. Sin\nMath. 48(2018), 849–858.\n1Academy of Mathematics and Systems Science, Chinese Academ y of Sci-\nences, Beijing 100190, PR China\n2School of Mathematical Sciences, University of Chinese Aca demy of Sci-\nences, Beijing 100049, PR China\n3Guangxi Center for Mathematical Research, Guangxi Univers ity, Nan-\nning, Guangxi 530004, PR China\nEmail address :lijialue@amss.ac.cn2\nEmail address :xzhang@amss.ac.cn1,2, xzhang@gxu.edu.cn3"    },    {        "title": "2402.04604v1.Symmetric_bilinear_Forms_and_Galois_Theory.pdf",        "content": "arXiv:2402.04604v1  [math.NT]  7 Feb 2024SYMMETRIC BILINEAR FORMS AND GALOIS THEORY\nSUGATA MANDAL\nA/b.pc/s.pc/t.pc/r.pc/a.pc/c.pc/t.pc. Let/u1D43Ebe a field admitting a Galois extension /u1D43Fof degree/u1D45B, denoting the\nGalois group as /u1D43A=Gal(/u1D43F//u1D43E). Our focus lies on the space Symm/u1D43E(/u1D43F)of symmetric\n/u1D43E-bilinear forms on /u1D43F. We establish a decomposition of Symm/u1D43E(/u1D43F)into direct sum\nof/u1D43E-subspaces/u1D434/u1D70E/u1D456, where/u1D70E/u1D456∈/u1D43A. Notably, these subspaces /u1D434/u1D70E/u1D456exhibit nice\nconstant rank properties. The central contribution of this paper is a decomposition\ntheorem for Symm/u1D43E(/u1D43F), revealing a direct sum of(/u1D45B+1)\n2constant rank /u1D45B-subspaces,\neach having dimension of /u1D45B. This holds particularly when /u1D43Ais cyclic, represented\nas/u1D43A=Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht. For cyclic extensions of even degree /u1D45B=2/u1D45A, we present\nslightly less precise but analogous results. In this scenar io, we enhance and enrich\nthese constant results and show that,the component /u1D434/u1D70Eoften decomposes directly into\na constant rank subspaces. Remarkably, this decomposition is universally valid when\n−1∉/u1D43F2. Consequently, we derive a decomposition of Symm/u1D43E(/u1D43F)into subspaces of\nconstant rank under several situations. Moreover, leverag ing these decompositions,\nwe investigate the maximum dimension of an /u1D45B-subspace inside /u1D440(/u1D45B,/u1D43E)and/u1D446(/u1D45B,/u1D43E)\nfor various field /u1D43Ewhere/u1D440(/u1D45B,/u1D43E)and/u1D446(/u1D45B,/u1D43E)denote the vector spaces (/u1D45B×/u1D45B)\nmatrices and symmetric matrices over /u1D43E, respectively.\nKeywords. symmetric form, constant rank space, cyclic extension\n2020 Math. Subj. Class. : 12F05, 12F10, 15A63\n1.I/n.pc/t.pc/r.pc/o.pc/d.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc\nLet/u1D43Ebe a field of characteristic other than two and Symm/u1D43E(/u1D449)denote the space\nof all symmetric bilinear forms on a /u1D43E-space/u1D449of dimension /u1D45B. Suppose/u1D43Eadmits a\nGalois extension /u1D43Fof degree/u1D45B. Taking the /u1D45B-dimensional /u1D43E-space/u1D43Fas a model for\n/u1D449it is shown in in this paper that ideas from Galois Theory can b e fruitfully applied\nfor studying symmetric bilinear forms on /u1D449. Notably, this approach sheds light on the\nsubspaces of Symm/u1D43E(/u1D449)whose nonzero skew-forms all have the same rank equal to /u1D458,\nsay. Such “/u1D458-subspaces\" besides being interesting in their own right pl ay an important\nrole in coding theory (see [ 6],[5]). An essential consideration is given to /u1D45B-subspaces of\nSymm/u1D43E(/u1D449)-subspaces where all nonzero skew forms are non-degenerate . The Galois\nnature of the extension enhances the vector space structure , enabling constructions\nthat would be unachievable without the additional field-the oretic apparatus. The\ndevolopment in this paper is modelled on that of [ 1] and [ 2] for skew-forms. In\nparticular, we adapt many key results in these papers to our s ituation. In this paper,\n12 SUGATA MANDAL\nreplacing/u1D449by the/u1D43E-space/u1D43F, we intend to investigate symmetric bilinear forms\ndefined on/u1D43F×/u1D43Fand taking values in /u1D43E. Our particular interest lies in subspaces of\nsuch forms and their properties concerning rank.\nThe Galois-theoretic trace map Tr/u1D43F\n/u1D43E:/u1D43F→/u1D43Eis a central tool, defined by\nTr/u1D43F\n/u1D43E(/u1D44E)=/summationdisplay.1\n/u1D70E∈Gal(/u1D43F//u1D43E)/u1D70E(/u1D44E),∀/u1D44E∈/u1D43F.\nAs is well known, this form is non-identically zero and exhib its the Galois invariance\nproperty Tr (/u1D70F(/u1D465))=Tr(/u1D465)for all/u1D465∈/u1D43Fand all/u1D70F∈/u1D43A. It serves as the primary tool for\nconstructing and investigating subspaces of symmetric bil inear forms in our study.\n2.S/y.pc/m.pc/m.pc/e.pc/t.pc/r.pc/i.pc/c.pc /b.pc/i.pc/l.pc/i.pc/n.pc/e.pc/a.pc/r.pc /f.pc/o.pc/r.pc/m.pc/s.pc /a.pc/n.pc/d.pc G/a.pc/l.pc/o.pc/i.pc/s.pc /e.pc/x.pc/t.pc/e.pc/n.pc/s.pc/i.pc/o.pc/n.pc/s.pc\nFor each/u1D70E∈/u1D43A:=Gal(/u1D43F//u1D43E)and/u1D44F∈/u1D43Fwe may define the symmetric bilinear form\n/u1D719/u1D44F,/u1D70E(/u1D465,/u1D466)=Tr/u1D43F\n/u1D43E(/u1D44F(/u1D465/u1D70E(/u1D466) +/u1D70E(/u1D465)/u1D466)),∀/u1D465,/u1D466∈/u1D43F. (2.1)\nLemma 2.1. Let/u1D719=/u1D719/u1D44F,/u1D70Ebe a symmetric bilinear form defined above. Thus we have\n/u1D719(/u1D465,/u1D466)=Tr((/u1D70E−1(/u1D44F/u1D465) +/u1D44F/u1D70E(/u1D465)/u1D466))\nfor all/u1D465and/u1D466in/u1D43F.\nCorollary 2.1. An element/u1D465is in the radical of /u1D719/u1D44F,/u1D70Eif and if\n/u1D70E−1(/u1D44F/u1D465)=−/u1D44F/u1D70E(/u1D465). (2.2)\nRecall that if /u1D439is an intermediate subfield and /u1D44E∈/u1D43Fthen the/u1D43F//u1D439-norm/u1D441/u1D43F//u1D439(/u1D44E)\nof/u1D44Eis defined as /u1D441/u1D43F//u1D439(/u1D44E)=/productdisplay.1\n/u1D703∈Gal(/u1D43F//u1D439)/u1D703(/u1D44E). For the sake of convenience in what\nfollows we shall denote the fixed field of /u1D70E∈Gal(/u1D43F//u1D43E)by/u1D43F/an}bracketle{t/u1D70E/u1D456/an}bracketri}ht.\nLemma 2.2. Let/u1D719=/u1D719/u1D44F,/u1D70Ebe a symmetric bilinear form defined above, with /u1D44F∈/u1D43F×,\nthen/u1D719is degenerate if and only if −/u1D70E(/u1D44F)/u1D44F−1is expressible in the form /u1D70E2(/u1D450)/u1D450−1for\nsome/u1D450∈/u1D43F×that is,/u1D719is degenerate if and only if\n/u1D441/u1D43F//u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht(−/u1D70E(/u1D44F)//u1D44F)=1. (2.3)\nMoreover, in this case rk(/u1D719)=/u1D45B−/u1D45B\n[/u1D43F:/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht]\nProof. Let/u1D445denote the radical of /u1D719and let/u1D465be an element of /u1D445. By Corollary 2.1\nand applying /u1D70Eon (2.2), we obtain\n/u1D44F/u1D465=−/u1D70E(/u1D44F)/u1D70E2(/u1D465).SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 3\nConsequently if /u1D465≠0 then taking /u1D465=/u1D450−1, we have\n−/u1D70E(/u1D44F)/u1D44F−1=/u1D70E2(/u1D450)/u1D450−1.\nThe next assertion is now clear in view of the Hilbert Theorem 90.\nNow will compute the rank of /u1D719when/u1D719is degenerate. At first we claim that dim (/u1D445)=\n/u1D45B\n[/u1D43F:/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht]. Indeed let /u1D465∈/u1D445then it can be checked that /u1D465∈/u1D450−1/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht, also conversely\nif/u1D466∈/u1D450−1/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}htthen/u1D466∈/u1D445. Thus/u1D445=/u1D450−1/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht. Thus dim /u1D445=/u1D45B\n[/u1D43F:/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht], that is,\nrk(/u1D719)=/u1D45B−/u1D45B\n[/u1D43F:/u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht]. /square\nLemma 2.3. Suppose that the order of /u1D70Eis odd then for each non-zero element /u1D44F∈/u1D43F,\nthe symmetric bilinear form /u1D719/u1D44F,/u1D70Eis non degenerate.\nProof. If possible, let there be a /u1D44F∈/u1D43F×such that/u1D719/u1D44F,/u1D70Eis degenerate. Then by Lemma\n2.2we obtain a/u1D450∈/u1D43Fsatisfying\n−/u1D70E(/u1D44F)/u1D44F−1=/u1D70E2(/u1D450)/u1D450−1.\nAgain by [ 1, Lemma 3], /u1D453/u1D44F,/u1D70Eis degenerate, where /u1D453/u1D44F,/u1D70Eis a skew-form defined by\n/u1D453/u1D44F,/u1D70E(/u1D465,/u1D466)=Tr/u1D43F\n/u1D43E(/u1D44F(/u1D465/u1D70E(/u1D466) −/u1D70E(/u1D465)/u1D466)),∀/u1D465,/u1D466∈/u1D43F.\nConsequently there exists a /u1D451∈/u1D43Fsuch that\n/u1D70E(/u1D44F)/u1D44F−1=/u1D70E2(/u1D451)/u1D451−1,\nby [1, Lemma 2]. Thus\n/u1D70E(/u1D450/u1D451−1)=−/u1D450/u1D451−1,\nwhich is contrary to the fact that −1 is an eigenvalue of /u1D70E. /square\nObservation. In particular if /u1D70E=/u1D456/u1D451then/u1D719/u1D44F,/u1D70E≠0and/u1D719/u1D44F,/u1D70E(/u1D465,/u1D466)=Tr(2/u1D44F/u1D465/u1D466). It is\nnon-degenerate as it is well known fact that the Trbilinear form of a Galois extension\nis non-degenerate.\nLemma 2.4. Suppose that the order of /u1D70Eis even, say 2/u1D45Fthen there exist /u1D44F,/u1D44F′∈/u1D43F×such\nthat the symmetric bilinear forms /u1D719/u1D44F,/u1D70Eand/u1D719/u1D44F′,/u1D70Eare degenerate and non-degenerate\nrespectively.\nProof. Since/u1D70Ehas even multiplicative order therefore −1 is an eigenvalue of /u1D70E, let/u1D44F\nbe a corresponding eigenvector. Consequently, /u1D70E(/u1D44F)/u1D44F−1=−1. Then taking /u1D450=1, we\nmust have\n−/u1D70E(/u1D44F)/u1D44F−1=/u1D70E2(/u1D450)/u1D450−1.4 SUGATA MANDAL\nThus/u1D719/u1D44F,/u1D70Eis degenerate.\nNext we will show the existancy of a non-degenerate symmetri c bilinear form. At first\nwe assume that /u1D43Eis finite. Let |/u1D43F/an}bracketle{t/u1D70E/an}bracketri}ht|=/u1D45Ethen|/u1D43F|=/u1D45E2/u1D45F. It is straight forward to\nobserve that the number of elements of the form −/u1D70E(/u1D44F)/u1D44F−1is\n/u1D45E2/u1D45F−1\n/u1D45E−1\nwhile the number of elements of the form /u1D70E2(/u1D450)/u1D450−1is\n/u1D45E2/u1D45F−1\n/u1D45E2−1.\nAs\n/u1D45E2/u1D45F−1\n/u1D45E2−1</u1D45E2/u1D45F−1\n/u1D45E−1,\nour claim follows in this case.\nNow we suppose that /u1D43Eis infinite. If possible let /u1D719/u1D44F,/u1D70Ebe degenerate ∀/u1D44F∈/u1D43F. Then by\nLemma 2.2\n/u1D441/u1D43F//u1D43F/an}bracketle{t/u1D70E2/an}bracketri}ht(−/u1D70E(/u1D44F)//u1D44F)=1,\nfor all/u1D44F∈/u1D43F×. This condition is easily seen to be equivalent to\n(−1)/u1D45F/u1D44F/u1D70E2(/u1D44F).../u1D70E2/u1D45F−2(/u1D44F)=/u1D70E(/u1D44F)/u1D70E3(/u1D44F).../u1D70E2/u1D45F−1(/u1D44F),\nfor all/u1D44F∈/u1D43F×. In particular, if we take any element /u1D6FC∈/u1D43Eand replace/u1D44Fabove by\n/u1D6FC−/u1D44F, we also obtain\n(−1)/u1D45F(/u1D6FC−/u1D44F)/u1D70E2(/u1D6FC−/u1D44F).../u1D70E2/u1D45F−2(/u1D6FC−/u1D44F)=/u1D70E(/u1D6FC−/u1D44F)/u1D70E3(/u1D6FC−/u1D44F).../u1D70E2/u1D45F−1(/u1D6FC−/u1D44F).\nWe can choose an element /u1D44Ffor which the 2 /u1D45FGalois conjugates /u1D44F,/u1D70E(/u1D44F),···,/u1D70E2/u1D45F−1(/u1D44F)\nare all different and define the polynomials /u1D45D1and/u1D45D2over/u1D43Fby\n/u1D45D1(/u1D465)=(−1)/u1D45F(/u1D465−/u1D44F)(/u1D465−/u1D70E2(/u1D44F))...(/u1D465−/u1D70E2/u1D45F−2(/u1D44F)),\n/u1D45D2(/u1D465)=(/u1D465−/u1D70E(/u1D44F))(/u1D465−/u1D70E2(/u1D44F))...(/u1D465−/u1D70E2/u1D45F−1(/u1D44F)).\nLet/u1D45D(/u1D465)=/u1D45D1(/u1D465) −/u1D45D2(/u1D465). Since/u1D45D1,/u1D45D2have no roots in common so /u1D45Dis a non-zero\npolynomial and vanishes at every point in /u1D43E, a contradiction. Thus there exist /u1D44F′∈/u1D43F×\nsuch that/u1D719/u1D44F′,/u1D70Ein non-degenerate. /square\nRemark 2.1. Iford(/u1D70E)=2/u1D45Fthen as/u1D44Fruns over the elements in /u1D43F×, the symmetric\nbilinear form /u1D719/u1D44F,/u1D70Ehave rank either /u1D45B−/u1D45B//u1D45For/u1D45B.\nIn particular, if ord(/u1D70E)=2then the elements /u1D719/u1D44F,/u1D70Eare either 0 or else non-degenerate.\nIn this case /u1D719/u1D44F,/u1D70Eis zero if and only if /u1D44F∈/u1D438, where/u1D438be the eigenspace of /u1D70Ewith\nrespect to the eigenvalue −1.SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 5\nDefinition 2.1. Let/u1D70E∈Gal(/u1D43F//u1D43E). We set\n/u1D434/u1D70E:={/u1D719/u1D44F,/u1D70E:/u1D44F∈/u1D43F}.\nIt can be checked that /u1D434/u1D70Eis a subspace of Symm (/u1D43F)and/u1D434/u1D70E=/u1D434/u1D70E−1. We will\ninvestigate the properties of /u1D434/u1D70E.\nTheorem 2.1. Let/u1D70E∈Gal(/u1D43F//u1D43E). Then\ndim(/u1D434/u1D70E)=/braceleftBigg\n/u1D45B, iford(/u1D70E)≠2\n/u1D45B/2, iford(/u1D70E)=2.\nWhen/u1D70Ehas odd order then all the nonzero elements of /u1D434/u1D70Eare non-degenerate.\nWhen ord(/u1D70E)=2/u1D45Fthen the non-zero elements of /u1D434/u1D70Eeither non-degenerate or else\nhave rank/u1D45B−/u1D45B//u1D45F.\nProof. Let/u1D713:/u1D43F→/u1D434/u1D70Ebe a map define by\n/u1D44F→/u1D719/u1D44F,/u1D70E.\nClearly/u1D713is/u1D43E-linear. Let/u1D44F∈ker/u1D713. By Corollary 2.1,/u1D719/u1D44F,/u1D70E=0 implies\n−/u1D44F/u1D465=/u1D70E(/u1D44F)/u1D70E2(/u1D465),\nfor all/u1D465∈/u1D43F. Taking/u1D465=1 we obtain/u1D70E(/u1D44F)=−/u1D44F. Thus if/u1D44F≠0, we must have\n/u1D70E2(/u1D465)=/u1D465\nfor all/u1D465∈/u1D43F. Consequently ord (/u1D70E)=2 otherwise if /u1D70E=id then there is no non-\nzero/u1D44F∈/u1D43Fsuch that/u1D70E(/u1D44F)=−/u1D44F. Thus if ord (/u1D70E)≠2 then/u1D713is an isomorphism,\nconsequently dim (/u1D434/u1D70E)=/u1D45B. On the otherhand if ord (/u1D70E)=2 then[/u1D43F/an}bracketle{t/u1D70E/an}bracketri}ht:/u1D43E]=/u1D45B\n2.\nConsequently the dimension of the eigenspace correspondin g to the eigenvalue −1 is\nalso/u1D45B/2, thus dim (/u1D434/u1D70E)=/u1D45B\n2. /square\nNext we will show that when [/u1D43F:/u1D43E]=/u1D45Bis odd, the space Symm (/u1D43F)is the direct\nsum of(/u1D45B+1)/2,/u1D45B-dimensional subspaces of the form /u1D434/u1D70Ewhere/u1D70E∈Gal(/u1D43F//u1D43E).\nSuppose that /u1D45B=2/u1D45A+1 and Gal (/u1D43F//u1D43E)={1,/u1D70E1,/u1D70E−1\n1,...,/u1D70E/u1D45A,/u1D70E−1\n/u1D45A}.\nWe shall correspondingly write /u1D434/u1D456in place of/u1D434/u1D70E/u1D456and/u1D4340:={/u1D719/u1D44F,/u1D456/u1D451:/u1D44F∈/u1D43F}, so that\n/u1D434/u1D456={/u1D719/u1D44F,/u1D70E/u1D456:/u1D44F∈/u1D43F}.\nLet/u1D43F/u1D45A+1denote the set of ordered (/u1D45A+1)-tuples(/u1D4650,/u1D4651,...,/u1D465/u1D45A+1)of elements of /u1D43F.\n/u1D43F/u1D45A+1is naturally a vector space of dimension /u1D45B(/u1D45A+1)=/u1D45B(/u1D45B+1)/2 over/u1D43Eunder the\nusual rules of addition of /u1D45A-tuples and scaler multiplication.6 SUGATA MANDAL\nTheorem 2.2. Suppose that /u1D45B=[/u1D43F:/u1D43E]is odd and the Galois group\n/u1D43A={1,/u1D70E1,···,/u1D70E/u1D45A,/u1D70E−1\n1,···,/u1D70E−1\n/u1D45A}\nwhere/u1D45A=(/u1D45B−1)/2. Then there is a direct decomposition\nAlt/u1D43E(/u1D43F)=/u1D4340⊕/u1D4341⊕/u1D4342⊕ ··· ⊕/u1D434/u1D45A, (2.4)\nWhere each /u1D434/u1D456is an/u1D45B-dimensional /u1D45B-subspace of Symm(/u1D43F)for1≤/u1D456≤/u1D45A.\nProof. We define a/u1D43E-linear map\n/u1D713:/u1D43F/u1D45A+1→Symm(/u1D43F)\nby\n/u1D713(/u1D44F0,...,/u1D44F/u1D45A)=/u1D45A/summationdisplay.1\n/u1D456=0/u1D719/u1D44F/u1D456,/u1D70E/u1D456\nWe claim that /u1D713is an isomorphism. Indeed let\n/u1D713(/u1D44F0,/u1D44F1,···,/u1D44F/u1D45A)=0,\nthen\n/u1D713(/u1D44F0,/u1D44F1,...,/u1D44F/u1D45A)(/u1D465,/u1D466)=/u1D45A/summationdisplay.1\n/u1D456=0/u1D719/u1D44F/u1D456,/u1D70E/u1D456(/u1D465,/u1D466)=0\nfor all/u1D465and/u1D466. As trace form is linear it follows from Corollary 2.1that\n/u1D45A/summationdisplay.1\n/u1D456=0/u1D70E−1\n/u1D456(/u1D44F/u1D456/u1D465) −/u1D44F/u1D456/u1D70E/u1D456(/u1D465)=0\nfor all/u1D465∈/u1D43F. Thus\n/u1D45A/summationdisplay.1\n/u1D456=0/u1D70E−1\n/u1D456(/u1D44F/u1D456)/u1D70E−1\n/u1D456−/u1D44F/u1D456/u1D70E/u1D456=0.\nBy Artin’s theorem on the independence of characters we obta in\n/u1D44F0=/u1D44F1=...=/u1D44F/u1D45A=0.\nNow as/u1D43F/u1D45A+1and Symm (/u1D43F)have same dimension over /u1D43Ethus/u1D713is an isomorphism.\nCosequently we obtain the decomposition ( 2.4) of Alt/u1D43E(/u1D43F). Moreover, it follows from\nthe Lemma 2.3that each/u1D434/u1D456is an/u1D45B-subspace. /square\nTheorem 2.3. Suppose that /u1D45B=[/u1D43F:/u1D43E]is even and the Galois group\n/u1D43A={/u1D70F1,···,/u1D70F/u1D458,1,/u1D70E1,···,/u1D70E/u1D45A,/u1D70E−1\n1,···,/u1D70E−1\n/u1D45A},\nwhere{/u1D70F1,/u1D70F2,···,/u1D70F/u1D458}are the involutions of /u1D43A, then there is a direct decomposition\nAlt/u1D43E(/u1D43F)=/u1D4351⊕/u1D4352⊕ ··· ⊕/u1D435/u1D458⊕/u1D4340⊕/u1D4341⊕/u1D4342⊕ ··· ⊕/u1D434/u1D45A. (2.5)SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 7\nwhere/u1D435/u1D456:=/u1D434/u1D70F/u1D456is an/u1D45B-subspace of dimension /u1D45B/2for all 1≤/u1D456≤/u1D458and/u1D434/u1D457:=/u1D434/u1D70E/u1D457\n(1≤/u1D457≤/u1D45A) has dimension /u1D45B. Moreover, the rank of each element in /u1D434/u1D457is given in\naccordance with the results of Lemma 2.3and Remark 2.1\nProof. We define a/u1D43E-linear map\n/u1D713:/u1D43F/u1D458+1+/u1D45A→Symm(/u1D43F)\nby\n/u1D713(/u1D44F0,...,/u1D44F/u1D45A)=/u1D458/summationdisplay.1\n/u1D456=1/u1D719/u1D450/u1D456,/u1D70F/u1D456+/u1D45A/summationdisplay.1\n/u1D457=0/u1D719/u1D44F/u1D456,/u1D70E/u1D457.\nBy Theorem 2.1the kernel of /u1D719consists of all elements\n(/u1D44F1,...,/u1D44F/u1D458,0,..., 0),\nwhere/u1D44F/u1D456∈/u1D43F/an}bracketle{t/u1D70F/u1D456/an}bracketri}ht, 1≤/u1D456≤/u1D458. Consequently, dim ker /u1D713=/u1D458/u1D45B/2. Furthermore, since\ndim/u1D43F/u1D458+/u1D45A+1−dim ker/u1D713=(/u1D458+/u1D45A+1)/u1D45B−/u1D458/u1D45B\n2=/u1D45B(/u1D45B+1)/2=Symm(/u1D43F),\nit follows that /u1D713is surjective. /square\n3.S/o.pc/m.pc/e.pc R/e.pc/s.pc/u.pc/l.pc/t.pc/s.pc /o.pc/n.pc C/y.pc/c.pc/l.pc/i.pc/c.pc G/a.pc/l.pc/o.pc/i.pc/s.pc /e.pc/x.pc/t.pc/e.pc/n.pc/s.pc/i.pc/o.pc/n.pc\nIn the earlier section, we have decomposed the space Symm/u1D43E(/u1D43F)into a direct sum\nof constant-rank /u1D45B-subspaces/u1D434/u1D456when[/u1D43F:/u1D43E]is odd. Our aim in this section is to\nestablish the decomposition of Symm/u1D43E(/u1D43F)into constant-rank subspaces, particularly\nwhen/u1D43Ais cyclic and /u1D45Bis even.\nNotation 1. Throughout this section /u1D43F//u1D43Edenotes a cyclic extension with Galois\ngroup Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht=/u1D45B=2/u1D45F >1. For the sake of convenience in what follows we\nshall denote the subfield /u1D43F/an}bracketle{t/u1D70E/u1D456/an}bracketri}htas/u1D43F/u1D456.\nWe begin by noting the following restatement of the degenera cy criterion Lemma\n2.2.\nProposition 3.1. Let/u1D44F∈/u1D43F. Then the skew-form /u1D719/u1D44F,/u1D70Eis degenerate if and only if\n(−1)/u1D45F/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D44F)//u1D44F)=1, (3.1)\nthat is,/u1D453/u1D44F,/u1D70Eis degenerate if and only if\n/u1D441/u1D43F//u1D43F2(/u1D44F)=/u1D44F/u1D70E2(/u1D44F)···/u1D70E/u1D45B−2(/u1D44F) (3.2)\nis an eigenvector of /u1D70Ecorresponds to either the eigenvalue −1or1depending on\nwhether/u1D45Fis odd or even.8 SUGATA MANDAL\nProof. The first assertion is now clear in view of the Lemma 2.2. Moreover the\ncondition (−1)/u1D45F/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D44F)//u1D44F)=1 is easily seen to be equivalent to the condition\n(−1)/u1D45F/u1D441/u1D43F//u1D43F2(/u1D44F)=/u1D70E(/u1D441/u1D43F//u1D43F2(/u1D44F))\nwhere/u1D441/u1D43F//u1D43F2(/u1D44F)=/u1D44F/u1D70E2(/u1D44F)···/u1D70E/u1D45B−2(/u1D44F). /square\nSuppose that /u1D70E/u1D456is not an involution. By Lemma 2.2, the symmetric bilinear form\n/u1D719/u1D44F,/u1D70E/u1D456∈/u1D434/u1D456⊆Symm/u1D43E(/u1D43F)is degenerate if and only if −/u1D70E/u1D456(/u1D44F)//u1D44F=/u1D70E2/u1D456(/u1D450)//u1D450for\nsome/u1D450∈/u1D43F. As/u1D70E2/u1D456is a generator for Gal (/u1D43F//u1D43F2/u1D456), in view of Hilbert Theorem 90,\n/u1D719/u1D44F,/u1D70E/u1D456is degenerate if and only if /u1D441/u1D43F//u1D43F2/u1D456(−/u1D70E/u1D456(/u1D44F)//u1D44F)=1. A glance at Proposition\n3.1above shows that this is precisely the condition for the symm etric bilinear form\n/u1D719∼\n/u1D44F,/u1D70E/u1D456∈Symm/u1D43F/u1D456(/u1D43F)defined by\n/u1D719∼\n/u1D44F,/u1D70E/u1D456=Tr/u1D43F\n/u1D43F/u1D456(/u1D44F(/u1D465/u1D70E(/u1D466) +/u1D70E(/u1D465)/u1D466)),∀/u1D465,/u1D466∈/u1D43F.\nto be degenerate (we write /u1D719∼\n/u1D44F,/u1D70E/u1D456instead of/u1D719/u1D44F,/u1D70E/u1D456to emphasize the fact that we are now\nconsidering/u1D43Fas/u1D43F/u1D456-space).\nLet us write /u1D434∼1:={/u1D719∼\n/u1D44F,/u1D70E/u1D456|/u1D44F∈/u1D43F}. In view of Theorem 2.1we then have a /u1D43E-\nisomorphism /u1D434/u1D456/simequal/u1D43Fvia/u1D719/u1D44F,/u1D70E/u1D456/uni∈1A6.endl→/u1D44Fand an/u1D43F/u1D456-isomorphism /u1D43F/simequal/u1D434∼1via/u1D44F/uni∈1A6.endl→/u1D719∼\n/u1D44F,/u1D70E/u1D456.\nThe composition of these maps clearly yields a /u1D43E-isomorphism /u1D434/u1D456/simequal/u1D434∼1. The\nfollowing is then clear.\nRemark 3.1. With respect to the above isomorphism if an /u1D43F/u1D456-subspace W ≤/u1D434∼1\nhas all its non-zero symmetric bilinear forms non-degenera te (or all its non-zero\nsymmetric bilinear forms forms degenerate) then the same is true for the corresponding\n(K)-subspace in /u1D434/u1D456.\nRemark 3.2. If/u1D43F//u1D43Eis cyclic Galois extension of degree /u1D45Bwith/u1D43A=Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht\nwe define/u1D434/u1D456:=/u1D434/u1D70E/u1D456. Thus/u1D434/u1D456={/u1D453/u1D44F,/u1D70E/u1D456:/u1D44F∈/u1D43F}. If/u1D45Bis even then there is a unique\ninvolution/u1D70F1=/u1D70E/u1D45B/2and in this case we denote /u1D4351:=/u1D434/u1D70F1={/u1D453/u1D44F,/u1D70E/u1D45B/2:/u1D44F∈/u1D43F}. Then\nthe decomposition (2.5)becomes\nAlt/u1D43E(/u1D43F)=/u1D4351⊕/u1D4340⊕/u1D4341⊕/u1D4342⊕ ··· ⊕/u1D434/u1D45A. (3.3)\nNote that/u1D4351is an/u1D45B-subspace of dimension /u1D45Band by Lemma 2.3,/u1D434/u1D456is an/u1D45B-subspace\nof dimension /u1D45Bif/u1D70E/u1D456has order odd.\nTheorem 3.1. Let/u1D43Ebe a field and /u1D45B=2/u1D458, where/u1D458≥1is odd. Let/u1D43Fbe any cyclic\nextension of /u1D43Eof degree/u1D45Bwith Galois group /u1D43A=/an}bracketle{t/u1D70E/an}bracketri}ht. Then\n/u1D4341=U1⊕ V 1, (3.4)SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 9\nwhereU1is an/u1D45B-subspace of dimension /u1D458andV1is an(/u1D45B−2)-subspace of dimension\n/u1D458.\nProof. Let/u1D448:=/u1D43F/u1D458and 0≠/u1D462∈/u1D448. Clearly\n/u1D70E2(/u1D462),/u1D70E4(/u1D462),...,/u1D70E2/u1D458−2(/u1D462) ∈/u1D448.\nIt follows that\n/u1D441/u1D43F//u1D43F2(/u1D462) ∈/u1D43F2∩/u1D448=/u1D43F2∩/u1D43F/u1D458=/u1D43E.\nBy Proposition 3.1the skew-form /u1D453/u1D462,/u1D70Eis non degenerate, thus U1is an/u1D45B-subspace.\nNote that/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D462)//u1D462)=1. Let/u1D457is an eigenvector of /u1D70Ecorresponding to the\neigenvalue −1 then\n/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D457)//u1D457)=−1.\nBy Proposition 3.1/u1D719/u1D457,/u1D70Eis degenerate. Set /u1D449:=/u1D457/u1D448. Then for 0 ≠/u1D462∈/u1D448\n/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D457/u1D462)\n/u1D457/u1D462)=/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D457)\n/u1D457)/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D462)\n/u1D462)=(−1).1=−1.\nIt thus follows by proposition 3.1that all the nonzero skew-forms /u1D719/u1D44F,/u1D70Ewhere/u1D44Flies\nin the subspace /u1D449=/u1D457/u1D448(of dimension /u1D458) are degenerate. Clearly /u1D448∩/u1D449={0}\nso/u1D43F=/u1D448⊕/u1D449. By Theorem 2.1the subspace /u1D448of/u1D43Fcorresponds to a subspace\nU1of Symm/u1D43E(/u1D43F)with the same dimension defined by U1:={/u1D719/u1D44F,/u1D70E:/u1D44F∈/u1D448}.\nSimilarly/u1D449corresponds to V1≤Symm/u1D43E(/u1D43F)such that dim (/u1D449)=dim(V1). Then the\ndecomposition ( 3.4) follows. /square\nCorollary 3.1. Let/u1D43Ebe a field and /u1D45Bbe even. Suppose /u1D43Fis a cyclic Galois extension\nof a field/u1D43Eof degree/u1D45Bwith Galois group Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht. Iford(/u1D70E/u1D456) ≡2(mod 4)\nandord(/u1D70E/u1D456)≠2then\n/u1D434/u1D456=U/u1D456⊕V/u1D456,\nwhereU/u1D456is an/u1D45B-subspace of dimension /u1D45B/2andV/u1D456is an(/u1D45B−2/u1D45B/ord(/u1D70E/u1D456))-subspace\nof dimension /u1D45B/2.\nProof. This follows from Theorem A, noting Remark 3.1and the fact (Remark 2.1)\nthat a skew form in /u1D434/u1D456is either non-degenerate or has rank equal to /u1D45B−2/u1D45B/ord(/u1D70E/u1D456)./square\nConsequently we obtain the following.10 SUGATA MANDAL\nCorollary 3.2. Let/u1D43Ebe a field and /u1D45B=2/u1D458, where/u1D458≥1is odd. Let/u1D43Fbe any cyclic\nGalois extension of /u1D43Eof degree/u1D45Bwith Galois group /u1D43A=/an}bracketle{t/u1D70E/an}bracketri}ht. Then\nAlt/u1D43E(/u1D43F)=/u1D4351/circleplusdisplay.1\n/u1D4340/circleplusdisplay.1/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡0(mod 2)\nord(/u1D70E/u1D456)≠2/parenleftBig\nU/u1D456/circleplusdisplay.1\nV/u1D456/parenrightBig/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA/circleplusdisplay.1/parenlefttpA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡1(mod 2)/u1D434/u1D456/parenrighttpA/parenrightexA\n/parenrightbtA,\n(3.5)\nwhere/u1D4351,/u1D4340,/u1D434/u1D456(forord(/u1D70E/u1D456) ≡1(mod 2)) andU/u1D456are/u1D45B-subspace of dimension /u1D45B/2,\n/u1D45B,/u1D45Band/u1D45B/2respectively and V/u1D456is an(/u1D45B−2/u1D45B/ord(/u1D70E/u1D456))-subspace of dimension /u1D45B/2.\nProof. In light of Remark 3.2, the decomposition ( 3.3) becomes\nAlt/u1D43E(/u1D43F)=/u1D4351/circleplusdisplay.1\n/u1D4340/circleplusdisplay.1/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡0(mod 2)\nord(/u1D70E/u1D456)≠2/u1D434/u1D456/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA/circleplusdisplay.1/parenlefttpA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡1(mod 2)/u1D434/u1D456/parenrighttpA/parenrightexA\n/parenrightbtA.\nNow it is clear in view of Corollary 3.1and Lemma 2.3. /square\nIn view of Theorem 3.1in following theorems we focus on the case where /u1D45B=2/u1D45Fis\ndivisible by 4. Since in this case /u1D45Fis even thus by Proposition 3.1it follows that /u1D719/u1D44F,/u1D70E\nis degenerate if and only if\n/u1D441/u1D43F//u1D43F2(/u1D70E(/u1D44F)//u1D44F)=1,\nthat is,/u1D441/u1D43F//u1D43F2(/u1D44F)is an eigenvector of /u1D70Ecorresponds to eigenvalue 1. This degeneracy\ncriterion for /u1D719/u1D44F,/u1D70Eis identical to the degeneracy criterion for the skew form /u1D453/u1D44F,/u1D70E(see\n[2, Proposition 3.1]).\nProposition 3.2. ([2, Lemma 3.1]) Let/u1D45B=2/u1D6FC/u1D458where/u1D6FC≥2and/u1D458is odd. Suppose\nthat/u1D43Fis a cyclic extension of a field /u1D43Eof degree/u1D45Bwith Galois group Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht.\nThen the following hold.\n(i)For1≤/u1D456≤/u1D6FC−1the subspace /u1D438/u1D456:={/u1D44F∈/u1D43F:/u1D70E/u1D45B/2/u1D456(/u1D44F)=−/u1D44F} ≤/u1D43Fhas\ndimension/u1D45B/2/u1D456.\n(ii)Let/u1D4491:={/u1D44F∈/u1D43F:/u1D70E/u1D458(/u1D44F)=/u1D44F}and/u1D4492:={/u1D44F∈/u1D43F:/u1D70E/u1D458(/u1D44F)=−/u1D44F}. Then\ndim(/u1D4491)=dim(/u1D4492)=/u1D458.\nLetE/u1D456be the subspace of /u1D4341corresponding to /u1D438/u1D456:={/u1D44F∈/u1D43F:/u1D70E/u1D45B/2/u1D456(/u1D44F)=−/u1D44F}under\nthe isomorphism of Theorem 2.1, that is,E/u1D456={/u1D719/u1D44F,/u1D70E:/u1D44F∈/u1D438/u1D456}( 1≤/u1D456≤/u1D6FC−1 ).\nSimilarly, let V/u1D457correspond to the subspace /u1D449/u1D457of/u1D43F, that is,V/u1D457={/u1D719/u1D44F,/u1D70E(/u1D457=1,2)}.SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 11\nTheorem 3.2. ([2, Theorem B]) Suppose/u1D45B=2/u1D6FC/u1D458where/u1D6FC≥2and/u1D458is odd. Let/u1D43Ebe\nan algebraic number field such that −1is not a square in /u1D43E. Then there exists a cyclic\nextension/u1D43Fof/u1D43Eof degree/u1D45Bwith the Galois group /u1D43A=/an}bracketle{t/u1D70E/an}bracketri}htsuch that\n/u1D4341=E1⊕ ··· ⊕ E /u1D6FC−1⊕ V 1⊕V 2, (3.6)\nwhere\n(i)E/u1D456is an/u1D45B-subspace of dimension /u1D45B/2/u1D456for1≤/u1D456≤/u1D6FC−1,\n(ii)V/u1D457is an(/u1D45B−2)-subspace of dimension /u1D458for1≤/u1D457≤2.\nCorollary 3.3. ([2, Corollary 4.3]) In the situation of Theorem 3.2iford(/u1D70E/u1D456) ≡\n0(mod 4), sayord(/u1D70E/u1D456)=2/u1D6FD/u1D458′(/u1D6FD≥2)then\n/u1D434/u1D456=V/u1D456\n1⊕ V/u1D456\n2⊕ E/u1D456\n1⊕ ···E/u1D456\n/u1D6FD−1, (3.7)\nwhere\n(i)E/u1D456\n/u1D458is an/u1D45B-subspace of dimension /u1D45B/2/u1D456for1≤/u1D458≤/u1D6FD−1,\n(ii)V/u1D456\n/u1D457is an(/u1D45B−2)-subspace of dimension /u1D458′/u1D45B/ord(/u1D70E/u1D456)for1≤/u1D457≤2.\nCorollary 3.4. In the situation of Theorem 3.2there is direct-decomposition\nAlt/u1D43E(/u1D43F)=/u1D4351/circleplusdisplay.1\n/u1D4340/circleplusdisplay.1/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡2(mod 4)\nord(/u1D70E/u1D456)≠2/parenleftBig\nU/u1D456/circleplusdisplay.1\nV/u1D456/parenrightBig/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA/circleplusdisplay.1/parenlefttpA/parenleftexA\n/parenleftbtA/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡1(mod 2)/u1D434/u1D456/parenrighttpA/parenrightexA\n/parenrightbtA\n/circleplusdisplay.1\nord(/u1D70E/u1D456) ≡0(mod 4)/parenleftBig\nV/u1D456\n1/circleplusdisplay.1\nV/u1D456\n2/circleplusdisplay.1\nE/u1D456\n/u1D6FD−1/circleplusdisplay.1\n···/circleplusdisplay.1\nE/u1D456\n1/parenrightBig\n(3.8)\nProof. Using Corollaries 3.1,3.4and Lemma 2.3as well as the decomposition 3.3,\nwe can deduce the required decomposition. /square\nRemark 3.3. Let/u1D45B=2/u1D6FC/u1D458where/u1D6FC≥1and/u1D458is odd. Suppose that /u1D43Fis a cyclic\nextension of a field /u1D43Eof degree/u1D45Bwith Galois group Gal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/an}bracketri}ht. Iford(/u1D70E/u1D456)is\neven then there always exists an /u1D45B-subspace of dimension /u1D45B/2inside/u1D434/u1D456. If/u1D6FC=1this\nfollows from Corollary 3.2. Otherwise if /u1D6FC >1then it follows from Theorem 3.2that\nE1:={/u1D453/u1D44F,/u1D70E:/u1D44F∈/u1D4381}is the desired subspace for /u1D4341. The corresponding assertion for\n/u1D434/u1D456now follows in the light of Corollary 3.4.\nRemark 3.4. ([2, Remark 4.2]) As its proof shows, Theorem 3.2as well as its corol-\nlaries remain valid for an arbitrary cyclic extension /u1D43F//u1D43Eof degree/u1D45B=2/u1D6FC/u1D458(/u1D6FC≥2)12 SUGATA MANDAL\nsuch that −1is not a square in /u1D43F. Similarly, let /u1D43Ebe a field such that /u1D453(/u1D44B):=/u1D44B4+1\nis irreducible in /u1D43E[/u1D44B](it is not difficult to show that /u1D43Ehas this property if and only\nif none of −1,2and−2is a square in /u1D43E). Then Theorem /u1D435holds true for any cyclic\nextension/u1D43F//u1D43Eof degree/u1D45B=2/u1D6FC/u1D458. Indeed, if/u1D702/u1D456is a2/u1D456-root of unity for /u1D456≥1then the\nconditions −1∉/u1D43E2and/u1D70E(/u1D702/u1D456)=−/u1D702−1\n/u1D456mean that/u1D702/u1D456∉{−± 1,±/u1D456}, where/u1D456denotes a\nprimitive 4-th root of unity in /u1D43F. Thus/u1D702must have order 2/u1D460where/u1D460≥3. Since/u1D702∈/u1D43F2\nthis would mean that /u1D43F2contains an element of order 8and thus a root of /u1D453implying\n/u1D453has a quadratic factor in /u1D43E[/u1D44B].\nTheorem 3.3. ([2, Theorem C]) Let/u1D43Ebe a finite field with /u1D45Eelements such that −1is\nnot a square in /u1D43E. Let/u1D45E+1=2/u1D44E/u1D459(l odd) where /u1D44E≥1and/u1D45B=2/u1D6FC/u1D458(k odd) where\n/u1D6FC≥2. Suppose/u1D43Fis a cyclic extension of /u1D43Eof degree/u1D45BwithGal(/u1D43F//u1D43E)=/an}bracketle{t/u1D70E/u1D453/an}bracketri}htwhere\n/u1D70E/u1D453is the Frobenius map of /u1D43Fdefined by/u1D70E/u1D453:/u1D44F→/u1D44F/u1D45E.\n(1)If/u1D6FC≤/u1D44E+1then\n/u1D4341=V1⊕V 2⊕ E 1⊕ ··· ⊕ E /u1D6FC−1, (3.9)\nwhere\n(i)E/u1D456is an/u1D45B-subspace of dimension /u1D45B/2/u1D456for1≤/u1D456≤/u1D6FC−1,\n(ii)V/u1D457is an(/u1D45B−2)-subspace of dimension /u1D458for1≤/u1D457≤2.\n(2)If/u1D6FC>/u1D44E+1and/u1D459=1, that is,/u1D45E=2/u1D44E−1, then\n/u1D4341=V1⊕V 2⊕ E 1⊕ ··· ⊕ E /u1D6FC−1, (3.10)\nwhere\n(i)E/u1D456is an/u1D45B-subspace of dimension /u1D45B/2/u1D456for1≤/u1D456≤/u1D44Eand an(/u1D45B−2)-\nsubspace of dimension /u1D45B/2/u1D456for/u1D44E+1≤/u1D456≤/u1D6FC−1,\n(ii)V/u1D457is an(/u1D45B−2)-subspace of dimension /u1D458for1≤/u1D457≤2.\nRemark 3.5. ([2, Remark 5.1]) In Theorem 3.3when/u1D6FC > /u1D44E+1and/u1D459 >1thenE/u1D456is\nneither an/u1D45B-subspace nor an (/u1D45B−2)-subspace for /u1D44E+1≤/u1D456≤/u1D6FC−1.\nTheorem 3.4. ([2, Theorem D]) Let/u1D45Dbe a prime and /u1D43E=Q/u1D45Dbe the/u1D45D-adic completion\nofQsuch that −1is not a square in /u1D43E. Let/u1D45D+1=2/u1D44E/u1D459(l odd) where /u1D44E≥1and\n/u1D45B=2/u1D6FC/u1D458(k odd) where 2≤/u1D6FC≤/u1D44E+1. Then there exists a cyclic extension /u1D43Fof/u1D43Eof\ndegree/u1D45Bsuch that the decomposition (3.9)holds.SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 13\n4.A /s.pc/h.pc/o.pc/r.pc/t.pc /s.pc/u.pc/r.pc/v.pc/e.pc/y.pc /o.pc/n.pc /u1D45B-/s.pc/u.pc/b.pc/s.pc/p.pc/a.pc/c.pc/e.pc /o.pc/f.pc /u1D440/u1D45B(/u1D43E)/a.pc/n.pc/d.pc/u1D446/u1D45B(/u1D43E)\nFor every/u1D45B∈Nand every field /u1D43E, let/u1D440(/u1D45B,/u1D43E)be the vector space of the (/u1D45B×/u1D45B)\nmatrices over /u1D43E, let/u1D446(/u1D45B,/u1D43E)be the vector space of the symmetric (/u1D45B×/u1D45B)matrices over\n/u1D43E. We recall that a /u1D43E-subpace/u1D446of/u1D440(/u1D45B,/u1D43E)or/u1D446(/u1D45B,/u1D43E)is an/u1D45B-subspace if all of its\nnon-zero elements are invertible. Define\n/u1D70F/u1D45B(/u1D43E)=max{dim/u1D446:/u1D446is an/u1D45Bsubspace of/u1D440(/u1D45B,/u1D43E)},\n/u1D707/u1D45B(/u1D43E)=max{dim/u1D446:/u1D446is an/u1D45Bsubspace of/u1D446(/u1D45B,/u1D43E)}.\nIt is easy to observed that /u1D707/u1D45B(/u1D43E) ≤/u1D70F/u1D45B(/u1D43E). For an arbitrary field /u1D43Eexact determination\nof/u1D70F/u1D45B(/u1D43E)or/u1D707/u1D45B(/u1D43E)is a difficult problem. However /u1D70F/u1D45B(/u1D43E) ≤/u1D45B, as any subspace\n/u1D446⊂/u1D440/u1D45B(/u1D43E)of dimension greater than /u1D45Bmust intersect nontrivially with the subspace\nof matrices with the first column equal to zero. The invariant /u1D707/u1D45B(/u1D43E)is intimately\nrelate with the invariant /u1D70F/u1D45B(/u1D43E). In general if /u1D45Bis even then we can easily checked that\n/u1D707/u1D45B(/u1D43E) ≥/u1D70F/u1D45B/2(/u1D43E). Since if/u1D448be an/u1D45B/2-subspace of /u1D440(/u1D45B/2,/u1D43E)then the subspace of\nall/u1D45B×/u1D45Bsymmetric matrices of the form/parenleftbigg0/u1D434\n/u1D434/u1D4470,/parenrightbigg\nwhere/u1D434runs over all (/u1D45B/2×/u1D45B/2)matrices with entries in /u1D448and/u1D434/u1D447denotes the\ntranspose of /u1D434is an/u1D45B-subspace of /u1D446(/u1D45B,/u1D43E).\nIn the following, we will discuss the values of the invariant s/u1D70F/u1D45B(/u1D43E)and/u1D707/u1D45B(/u1D43E)across\nvarious fields.\n4.1.For algebraic closed field. Suppose/u1D43Eis an algebraic closed field. It can be\neasily checked that /u1D70F/u1D45B(/u1D43E)=/u1D707/u1D45B(/u1D43E)=1 as for any /u1D434,/u1D435∈/u1D434/u1D45B(/u1D43E),/u1D434−/u1D706/u1D435is singular\nwhere/u1D706is an eigen-value of /u1D434/u1D435−1.\n4.2.For real number field. Suppose/u1D43Eis real number field. Then it follows from\n[3, Theorem 1] /u1D70F/u1D45B(/u1D43E)=/u1D70C(/u1D45B), where/u1D70C(/u1D45B)denotes the Radon-Hurtwitz number and is\ndefined by/u1D70C(/u1D45B)=2/u1D450+8/u1D451, whenever/u1D45B=(2/u1D44E+1)2/u1D450+4/u1D451where/u1D44E,/u1D44F,/u1D450,/u1D451 are integers\nwith 0≤/u1D450≤3. Now if/u1D45Bis odd then/u1D70C(/u1D45B)=1, consequently /u1D70F/u1D45B(/u1D43E)=/u1D707/u1D45B(/u1D43E)=1. On\nthe otherhand if /u1D45Bis even then/u1D70F/u1D45B/2(/u1D43E) ≤/u1D707/u1D45B(/u1D43E) ≤/u1D70F/u1D45B(/u1D43E). Thus\n/u1D707/u1D45B(/u1D43E)=8/u1D451,if/u1D450=0\nand\n/u1D707/u1D45B(/u1D43E) ∈ \n[1+8/u1D451,2+8/u1D451] if/u1D450=1\n[2+8/u1D451,4+8/u1D451] if/u1D450=2\n[4+8/u1D451,8+8/u1D451] if/u1D450=3.14 SUGATA MANDAL\n4.3.For algebraic number field. Suppose/u1D43Eis an algebraic number field. Then by\n[4, Lemma 4] for each /u1D45Bthere exists a cyclic Galois extension /u1D43Fof/u1D43E. Let/u1D45D(/u1D465)be\na irreducible polynomial of degree /u1D45Bover/u1D43Eand let/u1D434∈/u1D440(/u1D45B,/u1D43E)be a matrix whose\ncharacteristic polynomial is /u1D45D(/u1D465). Let/u1D448⊂/u1D440(/u1D45B,/u1D43E)be the/u1D43E-subspace spanned by\nthe powers of /u1D434. Then/u1D448is an/u1D45B-subspace of /u1D440(/u1D45B,/u1D43E)of dimension /u1D45Bas/u1D448is a field\nisomorphic to /u1D43E[/u1D465]//parenleftbig/u1D45D(/u1D465)/parenrightbig. Thus/u1D70F/u1D45B(/u1D43E)=/u1D45B. Although/u1D434may not necessarily belong\nto/u1D446(/u1D45B,/u1D43E)but still we can still obtain an /u1D45B-dimensional /u1D45B-subspace inside /u1D446(/u1D45B,/u1D43E).\nThen by Theorem 2.2and Remark 3.2,/u1D4340is an/u1D45B-subspace inside Symm/u1D43E(/u1D43F). Since\nSymm/u1D43E(/u1D43F)is isomorphic to /u1D446(/u1D45B,/u1D43E), hence/u1D4340corresponds to an /u1D45B-subspace of\n/u1D434(/u1D45B,/u1D43E). Consequently /u1D707/u1D45B(/u1D43E)=/u1D45B.\n4.4.For finite field. Suppose/u1D43Ebe a finite field. Then for each /u1D45B∈Nthere exists a\ncyclic Galois extension /u1D43Fof/u1D43Eof degree/u1D45B. Then, employing a similar argument as in\nthe case of an algebraic number field, we deduce /u1D70F/u1D45B(/u1D43E)=/u1D707/u1D45B(/u1D43E)=/u1D45B.\n5.C/o.pc/n.pc/c.pc/l.pc/u.pc/s.pc/i.pc/o.pc/n.pc\nThe eigenspaces associated with elements of the Galois grou p yield constant rank\nsubspaces in Symm/u1D43E(/u1D43F). If ord(/u1D70E/u1D456) ≡0(mod 2), then in light of the Remark 3.3, an\n/u1D45B-subspace of dimension /u1D45B/2 can always be found inside /u1D434/u1D456. As mentioned earlier in\nthis paper, the degeneracy criterion for the symmetric form /u1D719/u1D44F,/u1D70Ealigns with that of the\nskew form/u1D453/u1D44F,/u1D70E(defined in [ 1]), consequently, the quest for determining the maximum\ndimension of an /u1D45B-subspace inside /u1D4341⊆Symm/u1D43E(/u1D43F)is analogous to solving the\nproblem of the maximum dimension of an /u1D45B-subspace within /u1D4341⊆Alt/u1D43E(/u1D43F). However,\nspaces derived from eigenvalues may not necessarily repres ent the maximum possible\ndimension of an /u1D45B-subspace in /u1D4341(as exemplified in [ 2, Section 5]), unless /u1D45B=2/u1D458with\n/u1D458odd (Theorem 3.1) or/u1D43Eis finite (or more generally, /u1D4361[4, Lemma 3] ). Furthermore,\nit is noted that if /u1D43Eadmits a cyclic Galois extension of degree /u1D45B, then/u1D434/u1D456forms an\n/u1D45B-subspace of dimension /u1D45B(when/u1D70E/u1D456has odd order). The question remains open\nwhether there exists an /u1D45B-subspace in Symm/u1D43E(/u1D43F)with exactly dimension /u1D45Bor greater\nthan/u1D70F/u1D45B\n2(/u1D43E)(when/u1D45Bis even) for a field /u1D43Elacking a Galois extension. Furthermore, as\nindicated by [ 1, Theorem 6] and [ 1, Theorem 7] for the cyclic extension /u1D43F//u1D43E, we can\ninfer that the rank of any non-zero symmetric form in the dire ct sum/u1D4341⊕/u1D4342⊕...⊕/u1D434/u1D458\nis at least/u1D45B−2/u1D458.SYMMETRIC BILINEAR FORMS AND GALOIS THEORY 15\nA/c.pc/k.pc/n.pc/o.pc/w.pc/l.pc/e.pc/d.pc/g.pc/e.pc/m.pc/e.pc/n.pc/t.pc/s.pc\nThe author gratefully acknowledges support from an NBHM res earch award.\nR/e.pc/f.pc/e.pc/r.pc/e.pc/n.pc/c.pc/e.pc/s.pc\n[1] R. Gow, R. Quinlan, Galois extensions and subspaces of al ternating bilinear forms with special\nrank properties, Linear Algebra And Its Applications 430pp. 2212-2224 (2008)\n[2] A.Gupta, S.Mandal, Constant rank subspaces of alternat ing bilinear forms from galois theory,\narXiv:2310.03340 v1(2023)\n[3] J. F. Adams, Peter D. Lax and Ralph S. Phillips, On Matrice s Whose Real Linear Combinations\nare Nonsingular, Proceedings of the American Mathematical Society 16pp. 318-322 (1965)\n[4] R. Gow, R. Quinlan, On the vanishing of subspaces of alter nating bilinear forms, Linear And\nMultilinear Algebra 54pp. 415-428 (2006)\n[5] P.Delsarte, Bilinear Forms over a Finite Field with Appl ications to Coding Theory. J. Combin.\nTheory Ser. A 25pp. 226-241 (1978).\n[6] P. Delsarte, J.M. Goethals, Alternating bilinear Forms over/u1D43A/u1D439(/u1D45E),J. Combin. Theory Ser. A 19\npp. 26-50 (1975).\nS/u.pc/g.pc/a.pc/t.pc/a.pc M/a.pc/n.pc/d.pc/a.pc/l.pc, D/e.pc/p.pc/a.pc/r.pc/t.pc/m.pc/e.pc/n.pc/t.pc /o.pc/f.pc M/a.pc/t.pc/h.pc/e.pc/m.pc/a.pc/t.pc/i.pc/c.pc/s.pc, R/a.pc/m.pc/a.pc/k.pc/r.pc/i.pc/s.pc/h.pc/n.pc/a.pc M/i.pc/s.pc /s.pc/i.pc/o.pc/n.pc V/i.pc/v.pc/e.pc/k.pc/a.pc/n.pc/a.pc/n.pc/d.pc/a.pc E/d.pc/u.pc/c.pc/a.pc-\n/t.pc/i.pc/o.pc/n.pc/a.pc/l.pc /a.pc/n.pc/d.pc R/e.pc/s.pc/e.pc/a.pc/r.pc/c.pc/h.pc I/n.pc/s.pc/t.pc/i.pc/t.pc/u.pc/t.pc/e.pc (B/e.pc/l.pc/u.pc/r.pc C/a.pc/m.pc/p.pc/u.pc/s.pc), H/o.pc/w.pc/r.pc/a.pc/h.pc, WB 71 1202, I/n.pc/d.pc/i.pc/a.pc\nEmail address :gmandal1961@gmail.com"    },    {        "title": "2402.04613v2.Wasserstein_Gradient_Flows_for_Moreau_Envelopes_of_f_Divergences_in_Reproducing_Kernel_Hilbert_Spaces.pdf",        "content": "Wasserstein Gradient Flows for Moreau Envelopes of\nf-Divergences in Reproducing Kernel Hilbert Spaces\nSebastian Neumayer†∗Viktor Stein‡∗Gabriele Steidl‡Nicolaj Rux‡\nMarch 12, 2024\nAbstract\nMost commonly used f-divergences of measures, e.g., the Kullback-Leibler diver-\ngence, are subject to limitations regarding the support of the involved measures. A\nremedy consists of regularizing the f-divergence by a squared maximum mean dis-\ncrepancy (MMD) associated with a characteristic kernel K. In this paper, we use the\nso-called kernel mean embedding to show that the corresponding regularization can be\nrewritten as the Moreau envelope of some function in the reproducing kernel Hilbert\nspace associated with K. Then, we exploit well-known results on Moreau envelopes\nin Hilbert spaces to prove properties of the MMD-regularized f-divergences and, in\nparticular, their gradients. Subsequently, we use our findings to analyze Wasserstein\ngradient flows of MMD-regularized f-divergences. Finally, we consider Wasserstein\ngradient flows starting from empirical measures. We provide proof-of-the-concept nu-\nmerical examples for f-divergences with both infinite and finite recession constant.\n1 Introduction\nIn variational inference [10, 30] and generative modeling [5, 22], a common task is to min-\nimize the f-divergence for a fixed target measure over some hypothesis space. For this,\nmany different f-divergences were deployed in the literature, such as the Kullback-Leibler\n(KL) divergence [35], Tsallis- αdivergences [44], power divergences [37], Jeffreys and Jensen-\nShannon divergences [39], and Hellinger distances [25]. If the recession constant is infinite,\nthen the hypothesis space in the above tasks reduces to reweighted target samples. To\neliminate this disadvantage, regularized f-divergences can be applied. The probably most\nwell-known case is the regularization of F= KL( · |ν)with target measure νby the squared\n‡Institute of Mathematics, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany, {stein,\nsteidl,rux}@math.tu-berlin.de\n†Institute of Mathematics, TU Chemnitz, Reichenhainer Straße 39, 09126 Chemnitz, Germany\nsebastian.neumayer@mathematik.tu-chemnitz.de\n∗Equal contribution.\n1arXiv:2402.04613v2  [stat.ML]  9 Mar 2024Wasserstein-2 distance. This regularization appears in the backward scheme for computing\nthe gradient flow of Fin the Wasserstein geometry [31]. It can be considered as a Moreau\nenvelope of Fin the Wasserstein-2 space.\nIn this paper, we discuss the regularization of generic f-divergences by a squared maxi-\nmummeandiscrepancy(MMD)thatisinducedbyacharacteristickernel K. Sincethespace\nof signed Borel measures embeds into any reproducing kernel Hilbert space (RKHS) with\nreproducing kernel K, the associated MMD can be rewritten as a distance in this RKHS. As\nour first contribution, we establish a link between our MMD-regularized f-divergences and\nMoreau envelopes of a certain functional on this RKHS. Here, two main challenges arise.\nFirst, covering f-divergences both with a finite and an infinite recession constant makes the\nanalysis more involved. Second, as the kernel mean embedding is not surjective, we must\nsuitably extend the f-divergence to the RKHS and have to prove that the corresponding\nfunctional is lower semicontinuous. Based on this link and the well-known properties of\nMoreau envelopes, we prove similar properties for our regularized f-divergences. As our\nsecond contribution, we analyze Wasserstein gradient flows of the regularized f-divergences\nand, in particular, the associated particle gradient flows. Using our theoretical insights from\nthe first part, we prove that the regularized f-divergences are λ-convex along generalized\ngeodesics in the Wasserstein space for sufficiently smooth kernels. Then, we show that their\nsubdifferential consists of just one element and use this to determine the Wasserstein gradi-\nent flow equation. Finally, we deal with particle flows, which are proven to be Wasserstein\ngradient flows that start in an empirical measure with an empirical target measure. For\nthe numerical simulations, we focus on the Tsallis- αdivergence, which is smoothed based\non the inverse multiquadric kernel. Even in the case of a finite recession constant, where a\nrepresenter theorem can not be applied to reduce the dual problem to a finite-dimensional\nproblem, we prove that we can still only solve a finite-dimensional problem if the regu-\nlarization parameter satisfies a certain lower bound. We simulate the gradient flows for\nthree different target measures from the literature. Since α= 1corresponds to the KL\ndivergence, we are especially interested in the behavior for different values of α. It appears\nthat choosing αmoderately larger than one improves the convergence of the gradient flow.\nA similar behavior was observed in [24] for differently regularized Tsallis-divergences.\n1.1 Related work\nOurworkisinspiredbythepaperofGlaseretal.[21](whichbuildson[6,43])onWasserstein\ngradient flows with respect to the MMD-regularized KL divergence. In contrast to their\npaper, we deal with arbitrary f-divergences and relate the functionals to Moreau envelopes\nin Hilbert spaces. This helps to streamline the proofs. Moreover, our simulations cover the\nmore general Tsallis- αdivergences.\nAn opposite point of view is to regularize MMDs by f-divergences [34]. The provided\nanalysis covers entropy functions fwith an infinite recession constant and probability mea-\nsuresthathavetofulfilladditionalmomentconditions. Thepaperfocusesonkernelmethods\n2of moments as an alternative to the generalized method of moments. It is different from\nour paper, where we deal with gradient flows.\nWasserstein-1 regularized f-divergences for measures on compact metric spaces are con-\nsidered in [66] and when choosing their exponent αto be equal to two, one obtains a Moreau\nenvelope. However, their definition of f-divergence is slightly more general than ours, since\nit accounts for negative measures, too.\nThe authors of [9] investigated the regularization of f-divergences using the infimal\nconvolution with general integral probability metrics. Again, only entropy functions f\nwith an infinite recession constant were considered. Choosing a MMD as the integral\nprobability metric leads, in contrast to our paper, to a regularization of f-divergences with\nthenon-squared MMD. For this setting, no interpretation as Moreau envelope is possible.\nWasserstein gradient flows of these regularized f-divergences and their discretization are\ndiscussed in [24], although neither existence nor uniqueness of the Wasserstein gradient flow\nis proven.\n1.2 Outline of the paper\nIn Section 2, we collect basic facts from convex analysis, especially about Moreau envelopes.\nFurther, we recall the notation of RHKSs and MMDs. Then, in Section 3, we discuss f-\ndivergences both with finite and an infinite recession constant. We introduce their MMD-\nregularized counterparts and establish the relation to Moreau envelopes of a specific proper,\nconvex and lower semicontinuous function on the RKHS associated with the kernel of the\nMMD. In Section 4, we recall the concept of Wasserstein gradient flow and prove the\nexistence and uniqueness of the Wasserstein gradient flow with respect to MMD-regularized\nf-divergences. We discuss the simulation of these Wasserstein gradient flows when both the\ntarget and the starting point of the flow are empirical measures in Section 5. Here, we show\nthat the representer theorem can be also applied for the constrained optimization setting\nrelated to f-divergences with finite recession constant. Then, we illustrate the behavior\nof such flows with three numerical examples in Section 6. Finally, we draw conclusions in\nSection 7. We collect examples of entropy functions, their conjugates, and their associated\nf-divergences in the A. Further ablation plots and implementation details are provided in\nthe supplementary material (Section B).\n2 Convex Analysis in Reproducing Kernel Hilbert spaces\nThis section contains the necessary preliminaries and notions. In Subsection 2.1, we start\nwith basic facts from convex analysis in Hilbert spaces, especially Moreau envelopes and\nproximal mappings can be found, e.g., in the textbook [7]. Our Hilbert spaces of choice\nwill be RKHSs, which are properly introduced in Subsection 2.2. In particular, we require\ntheir relation to the space of signed Borel measures in terms of the so-called kernel mean\nembedding. ThisembeddingalsorelatesthedistanceinRKHSswiththeMMDofmeasures.\n32.1 Moreau Envelopes in Hilbert Spaces\nLetHbe a separable Hilbertspace with inner product ⟨·,·⟩Hand corresponding norm ∥·∥H.\nThe domain of an extended function F:H → (−∞,∞]is defined by dom( F):={h∈ H:\nF(h)<∞}, and Fis proper if dom( F)̸=∅. By Γ0(H), we denote the set of proper,\nconvex, lower semi-continuous extended real-valued functions on H. TheFenchel conjugate\nfunction of a proper function F:H → (−∞,∞]is given by\nF∗(h) = sup\ng∈H{⟨h, g⟩H−F(g)}.\nIfFis proper and convex, then F∗∈Γ0(H). Thesubdifferential of a function F∈Γ0(H)\nath∈dom( F)is defined as the set\n∂F(h):=\b\np∈ H:F(g)≥F(h) +⟨p, g−h⟩H∀g∈ H\t\n.\nIfFis differentiable at h, then ∂F(h) ={∇F(h)}. Further, p∈∂F(h)implies h∈∂F∗(p).\nNext, recall that the Moreau envelope of a function G∈Γ0(H)is defined by\nGλ(h):= min\ng∈Hn\nG(g) +1\n2λ∥h−g∥2\nHo\n, λ > 0, (1)\nwhere the minimizer is unique. Hence, the proximal map proxλG:H → Hwith\nproxλG(h):= arg min\ng∈Hn\nG(g) +1\n2λ∥h−g∥2\nHo\n, λ > 0,\nis well-defined. The Moreau envelope has many advantageous properties, including the\nfollowing ones.\nTheorem 1. The Moreau envelope of a function G∈Γ0(H)has the following properties:\ni) The dual formulation of Gλ:H →Rreads as\nGλ(h) = max\np∈Hn\n⟨h, p⟩H−G∗(p)−λ\n2∥p∥2\nHo\n. (2)\nIfˆpis the maximizer in (2), then the minimizer in (1)isˆg=h−λˆpand vice versa.\nii) The function Gλis Fréchet differentiable with derivative given by\n∇Gλ(h) =λ−1\u0000\nh−proxλG(h)\u0001\n.\nIn particular, it holds that ∇Gλis1\nλ-Lipschitz and that Gλis continuous.\niii) For λ↘0, we have Gλ↗G, and for λ→ ∞thatGλ↘minx∈Rd{G(x)}pointwise.\n42.2 Reproducing Kernel Hilbert Spaces and MMDs\nA Hilbert space Hof real-valued functions on Rdis called a reproducing kernel Hilbert\nspace(RKHS), if the point evaluations h7→h(x),h∈ H, are continuous for all x∈\nRd. There exist various textbooks on RKHS from different points of view [63, 19, 60].\nBy [63, Thm. 4.20], every RKHS admits a unique symmetric, positive definite function\nK:Rd×Rd→Rwhich is determined by the reproducing property\nh(x) =⟨h, K(x,·)⟩Hfor all h∈ H. (3)\nIn particular, we have that K(x,·)∈ Hfor all x∈Rd. Conversely, for any symmetric,\npositive definite function K:Rd×Rd→R, there exists a unique RKHS, denoted by HK\nwith reproducing kernel K[63, Thm. 4.21].\nAssumption 1. In the following, we use the term „kernel” for symmetric, positive definite\nfunctions K:Rd×Rd→Rthat are\ni) bounded, i.e., supx∈RdK(x, x)<∞, and\nii)K(x,·)∈ C0(Rd)for all x∈Rd.\nThe properties (i)) and (ii)) are equivalent to the fact that HK⊂ C0(Rd). Further, the\nembedding is continuous: HK,→ C 0(Rd)[61, Cor. 3].\nRKHSs are closely related with the Banach space M(Rd)of finite signed Borel measures\nwith total variation norm ∥ · ∥ TV. Later, we also need its subset M+(Rd)of nonnegative\nmeasures. For any µ∈ M(Rd), there exists a unique function mµ∈ H Ksuch that\nEµ[h]:=⟨h, µ⟩C0×M=Z\nRdh(x) dµ(x) =Z\nRd⟨h, K(x,·)⟩HKdµ(x)\n=D\nh,Z\nRdK(x,·) dµ(x)E\nHK=⟨h, m µ⟩HK (4)\nfor all h∈ H K. The linear, bounded mapping m:M(Rd)→ H Kwith µ7→mµgiven by\nmµ(x) =⟨K(x,·), µ⟩=Z\nRdK(x, y) dµ(y), (5)\nis called kernel mean embedding (KME) [41, Sec. 3.1].\nAssumption 2. In this paper, we restrict our attention to so-called characteristic kernels\nK, for which the KME is injective.\nA kernel Kis characteristic if and only if HKis dense in (C0(Rd),∥ · ∥∞), see [61].\nHowever, the KME is not surjective and we only have (ran(m),∥ · ∥HK) =HK, see [64].\n5Themaximum mean discrepancy (MMD) [12, 23] dK:M(Rd)×M(Rd)→Ris defined\nas\ndK(µ, ν)2:=Z\nRd×RdK(x, y) d(µ(x)−ν(x)) d(µ(y)−ν(y))\n=Z\nRd×RdK(x, y) dµ(x) dµ(y)−2Z\nRd×RdK(x, y) dµ(x) dν(y)\n+Z\nRd×RdK(x, y) dν(x) dν(y).\nBy the reproducing property (3), see also [23, Lemma 4], this can be rewritten as\ndK(µ, ν) =∥mµ−mν∥HK.\nSince the KME is injective, dKis a metric on M(Rd)and, in particular, dK(µ, ν) = 0if\nand only if µ=ν. The widely used radial kernels are of the form K(x, y) =ϕ(∥x−y∥2\n2)\nfor some continuous function ϕ: [0,∞)→R.\nRemark 2. By Schoenberg’s theorem [72, Thm. 7.13], a radial kernel is positive definite\nif and only if ϕis completely monotone on [0,∞), that is ϕ∈ C∞((0,∞))∩ C([0,∞))\nand(−1)kϕ(k)(r)≥0for all k∈Nand all r >0. Hence, ϕandϕ′′are decreasing on\n(0,∞). Moreover, the Hausdorff–Bernstein–Widder theorem [72, Thm. 7.11] asserts that\nϕis completely monotone on [0,∞)if and only if there exists a nonnegative finite Borel\nmeasure νonB([0,∞))such that\nϕ(r) =Z∞\n0e−rtdν(t),∀r≥0.\nBy [62, Prop. 11], K(x, y) =ϕ(∥x−y∥2\n2)is characteristic if and only if supp( ν)̸={0}. Stan-\ndard examples of radial characteristic kernels are the Gaussians with ϕ(r) = exp( −1\n2σ2r),\nσ∈Rand the inverse multiquadric with ϕ(r) = (σ+r)−1\n2, where σ >0.\nWe have the following regularity result.\nLemma 3. For a radial kernel K(x, y) =ϕ(∥x−y∥2\n2)with ϕ∈ C2([0,∞))it holds that\nHK,→C2(Rd). Then, we have for any y,˜y∈Rdthat\n∥∂yiK(·, y)∥2\nHK=−2ϕ′(0)\nand\n∥∂yiK(·, y)−∂yiK(·,˜y)∥2\nHK≤4ϕ′′(0)\u0000\n2|yi−˜yi|2+∥y−˜y∥2\n2\u0001\n.\nFurthermore, we have for any h∈ H Kthat\n∥∇h(y)− ∇h(˜y)∥ ≤2∥h∥HKp\nϕ′′(0)(d+ 2)∥y−˜y∥. (6)\n6Proof.The continuous embedding follows from [63, Cor. 4.36]. Next, we obtain by straight-\nforward computation ∂xi∂yiK(x, y) =−4ϕ′′(∥x−y∥2\n2)(xi−yi)2−2ϕ′(∥x−y∥2\n2). By applying\n[63, Lem. 4.34] for the feature map y7→K(·, y), we get\n\n∂yiK(·, y), ∂yiK(·,˜y)\u000b\nHk=∂xi∂yiK(y,˜y) =−4ϕ′′(∥y−˜y∥2\n2)(yi−˜yi)2−2ϕ′(∥y−˜y∥2\n2)(7)\nand in particular ∥∂yiK(·, y)∥2\nHK=−2ϕ′(0). By (7) and since ϕ′is Lipschitz continuous\nwith Lipschitz constant ∥ϕ′′∥∞=ϕ′′(0), see Remark 2, it holds\n∥∂yiK(·, y)−∂yiK(·,˜y)∥2\nHK=−4ϕ′(0) + 8 ϕ′′(∥y−˜y∥2\n2)(yi−˜yi)2+ 4ϕ′(∥y−˜y∥2\n2)\n≤4ϕ′′(0)\u0000\n2(yi−˜yi)2+∥y−˜y∥2\n2\u0001\n.\nThe final assertion follows by\n|∂ih(y)−∂ih(˜y)|=|∂i⟨h, K(·, y)⟩ −∂i⟨h, K(·,˜y)⟩|=|⟨h, ∂yiK(·, y)−∂yiK(·,˜y)⟩|\n≤ ∥h∥HK∥∂yiK(·, y)−∂yiK(·,˜y)∥HK.\nThis finishes the proof.\nIn the rest of this paper, kernels always have to fulfill Assumptions 1 and 2.\n3 MMD-regularized f-Divergences and Moreau Envelopes\nLet us briefly describe the path of this section. First, we define f-divergences Dfof non-\nnegative measures for entropy functions fwith infinite or finite recession constant. Such\nf-divergences were first introduced by [17, 2], and we refer to [38] for a detailed overview.\nWe prove some of their properties in Subsection 3.1, where allowing a finite recession con-\nstantmakestheproofsmoreexpansive. Then, inSubsection3.2, wedealwiththeassociated\nfunctionals Df,ν:=Df(·|ν)with a fixed target measure ν. If the recession constant is infi-\nnite, then Df,νis only finite for measures that are absolutely continuous with respect to ν.\nThis disadvantage can be circumvented by using the MMD-regularized functional\nDλ\nf,ν(µ):= inf\nσ∈M +(Rd)n\nDf,ν(σ) +1\n2λdK(µ, σ)2\n|{z}\n∥mµ−mσ∥2\nHKo\n, λ > 0. (8)\nNow, our main goal is to show that (8) can be rewritten as the Moreau envelope of some\nGf,ν:HK→(−∞,∞]. Tothisend, weexploittheKME mandintroduce Gf,ν=Df,ν◦m−1\non ran (m)and set Gf,ν=∞otherwise. This function is proper and convex. We prove\nthatGf,νis lower semicontinuous, where we have to face the difficulty that ran (m)̸=HK.\nThen Gf,ν∈Γ0(HK)yields the desired Moreau envelope identification\nDλ\nf,ν(µ) =Gλ\nf,ν(mµ):= min\ng∈HKn\nGf,ν(g) +1\n2λ∥g−mν∥2\nHKo\n.\nwhich allows to exploit Theorem 1 to show various of its properties in Subsection 3.3.\n73.1 f-Divergences\nA function f:R→[0,∞]is called an entropy function , iff∈Γ0(R)with f(t) =∞for\nt <0andf(1) = 0. The corresponding recession constant is given by f′\n∞= lim t→∞f(t)\nt.\nThen, f∗∈Γ0(R)is non-decreasing and int (dom( f∗)) = (−∞, f′\n∞), see, e.g., [37]. Further,\nf∗is continuous on dom( f∗)andf∗(0) = 0, in particular 0∈dom( f∗). By definition of the\nsubdifferentialandsince f(1) = 0, itfollowsthat 0∈∂f(1), andthen 1∈∂f∗(0). Examples\nof entropy functions together with their recession constants and conjugate functions are\ncollected in Table 1 in A.\nLetfbe an entropy function and ν∈ M +(Rd). Recall that every measure µ∈ M +(Rd)\nadmits a Lebesgue decomposition µ=ρν+µs, where ρ∈L1(Rd, ν),ρ≥0andµs⊥ν, i.e.,\nthereexistsaBorelset A ⊆Rdsuchthat ν(Rd\\A) = 0andµs(A) = 0[37, Lemma2.3]. The\nf-divergence Df:M+(Rd)×M +(Rd)→[0,∞]between a measure µ=ρν+µs∈ M +(Rd)\nandν∈ M +(Rd)is defined by\nDf(ρν+µs|ν) =Z\nRdf◦ρdν+f′\n∞µs(Rd) = sup\ng∈Cb(Rd),\nf∗◦g∈Cb(Rd)\u001aZ\nRdgdµ−Z\nRdf∗◦gdν\u001b\n(9)\nwith the usual convention 0·(±∞) = 0, see [37, Eq. (2.35), Thm. 2.7, Rem. 2.8]. The\nfunction Dfis jointly convex and nonnegative, see [37, Cor. 2.9]. Moreover, we will need\nthe following lemma.\nLemma 4. Letf∈Γ0(R)be an entropy function with f′\n∞>0that has its unique minimizer\nat one.\ni) The function Dfis a divergence: for µ, ν∈ M +(Rd), the relation Df(µ, ν) = 0\nimplies that µ=ν.\nii) For all t > τ > 1we have f(t)>f(τ)\nτ−1(t−1)and thus limt→∞f(t) =∞.\nProof. i) Suppose we have ν∈ M +(Rd)andµ=ρν+µs∈ M +(Rd)such that Df(µ|\nν) = 0. Since both summands in the primal definition (9) of Dfare nonnegative,\nthey must be equal to zero. In particular, µs(Rd) = 0and since µs∈ M +(Rd), this\nimplies µs= 0. Hence, µ=ρνand\nDf(µ|ν) =Df(ρν|ν) =Z\nRd(f◦ρ)(x) dν(x) = 0 .\nSince fis nonnegative and ν∈ M +(Rd), we must have f◦ρ= 0ν-a.e. Since fonly\nhas one as its only minimizer and f(1) = 0, this implies ρ= 1ν-a.e., which means\nthatµ=ν.\n8ii) Fix τ >1. For any t > τwe write τasτ= 1−λ+λt, with λ=τ−1\nt−1∈(0,1). By the\nconvexity of fwe have\n0< f(τ)≤(1−λ)f(1) + λf(t) =λf(t) =τ−1\nt−1f(t).\nSince 1is the unique minimizer and τ >1this implies\nlim\nt→∞f(t)≥lim\nt→∞f(τ)\nτ−1(t−1) =∞.\nAssumption 3. From now on we assume that all entropy functions fulfill both assumptions\nin Lemma 4.\nExamples of f-divergences are contained in Table 2 in A. Note that the assumptions of\nLemma 4 are fulfilled for all f-divergences in Table 1 except for the Marton divergence and\nthe trivial zero divergence. However, it is not hard to check that the Marton divergence is\npositive definite on the space of probability measures P(Rd), too. Below is an example of a\nnon-trivial f-divergence that does not have this property.\nExample 5 (Rescaled Marton divergence) .Letf(t) = max(1\n2−x,0)2on[0,∞). Then\nf′\n∞= 0and we have for any absolutely continuous ν∈ P(Rd)that Df(1\n2ν+1\n2δ0|ν) =\nf(1\n2) = 0.\nRecall that M(Rd)is the dual space of C0(Rd). A sequence of measures (µn)n⊂ M (Rd)\nconverges weak* to a measure µ∈ M(Rd)if we have for all g∈ C0(Rd)that⟨g, µn⟩ → ⟨ g, µ⟩\nasn→ ∞. Moreover, if (µn)n⊂ M +(Rd)is bounded, then µ∈ M +(Rd), see [52,\nLemma 4.71, Cor. 4.74].\nThe following lemma also follows from the more general [3, Thm. 2.34], but we prefer\nto give a simpler proof for our setting to make the paper self-contained.\nLemma 6. For any fixed ν∈ M +(Rd), the f-divergence (9)can be rewritten as\nDf(ρν+µs|ν) = sup\ng∈C0(Rd;dom( f∗))\u001aZ\nRdgdµ−Z\nRdf∗◦gdν\u001b\n. (10)\nTherefore, Dfis jointly weak* lower semi-continuous.\nProof.In the proof, we have to be careful concerning the support of ν.\ni) (9) ≥(10): This direction is obvious since g∈ C0(Rd; dom( f∗)),0∈dom( f∗)and the\ncontinuity of f∗on its domain imply g∈ Cb(Rd)andf∗◦g∈ Cb(Rd).\n9ii) (9) ≤(10): Let g∈ Cb(Rd)with f∗◦g∈ Cb(Rd). Using continuous cutoff functions, we\ncan approximate gby a family of functions (gk)k∈N⊂ C0(Rd; dom( f∗))with supg≥\nmaxgk≥infgk≥min(inf g,0), which implies |f∗◦gk| ≤sup|f∗◦g|, for any k∈N,\nandlimk→∞gk(x) =g(x)for all x∈Rd. Further, the continuity of f∗on its domain\nimplies that (f∗◦gk)k∈Nsatisfies limk→∞(f∗◦gk)(x) = (f∗◦g)(x). Now, the claim\nfollows as in the first part using the dominated convergence theorem.\niii) Let (µn)nand(νn)nconvergeweak*to µandν, respectively. Since f∗iscontinuouson\nits domain, where f∗(0) = 0andg∈ C0(Rd; dom( f∗)), it follows that f∗◦g∈ C0(Rd).\nThen, we have for g∈ C0(Rd; dom( f∗))that\nlim\nn→∞{⟨g, µn⟩ − ⟨f∗◦g, νn⟩}=⟨g, µ⟩ − ⟨f∗◦g, ν⟩ (11)\nand by inserting (11) into (10) we get\nDf(µ|ν) = sup\ng∈C0(Rd;dom( f∗))n\nlim\nn→∞⟨g, µn⟩ − ⟨f∗◦g, νn⟩o\n≤lim inf\nn→∞sup\ng∈C0(Rd;dom( f∗))\b\n⟨g, µn⟩ − ⟨f∗◦g, νn⟩\t\n= lim inf\nn→∞Df(µn|νn).\nThus, Dfis jointly weak* lower semi-continuous.\n3.2 Regularized f-Divergences\nToovercomethedrawbackthat Df(µ, ν)requires µtobeabsolutelycontinuouswithrespect\ntoν, we introduce the MMD- regularized f-divergence Dλ\nf:M+(Rd)× M +(Rd)→[0,∞)\nas\nDλ\nf(µ|ν):= inf\nσ∈M +(Rd)n\nDf(σ|ν) +1\n2λdK(µ, σ)2o\n, λ > 0. (12)\nFor fixed ν∈ M +(Rd), we investigate the functional Df,ν:=Df(· |ν):M+(Rd)→[0,∞].\nSimilarly, we consider its regularized version Dλ\nf,ν:M(Rd)→[0,∞)that was already\nannounced in (8). Note that Dλ\nf,νis well-defined also for nonpositive measures. Further, it\nis no longer required that µis absolutely continuous with respect to νto keep the function\nvalue Dλ\nf,ν(µ)finite.\nNow, our aim is to reformulate (8) as the Moreau envelope of a certain function on HK.\nUsing the KME in (5), we see that\nDf,ν(µ) =Gf,ν(mµ), (13)\nwhere Gf,ν:HK→[0,∞]is given by\nGf,ν(h):=\u001aDf,ν(µ),if∃µ∈ M +(Rd)s.t.h=mµ,\n∞,else.(14)\n10Since m−1is linear and Dfis jointly convex, the concatenation Gf,νis convex. Further,\nGf,ν(mν) = 0, so that the function is also proper.\nWe now prove that even though ran(m)is not closed in HK, the function Gf,νis lower\nsemicontinuous. Hence, it is covered by the theory for Moreau envelopes on Γ0(HK).\nLemma 7. Iff′\n∞>0andfhas its unique minimizer at t= 1, then Gf,ν:HK→[0,∞]is\nlower semicontinuous.\nProof.Fixh∈ H Kand let (hn)n⊂ H Kwith hn→h. We need to show that Gf,ν(h)≤\nlim inf n→∞Gf,ν(hn). Iflim inf n→∞Gf,ν(hn) = +∞,wearedone. For lim inf n→∞Gf,ν(hn)<\n+∞, we pass to a subsequence which realizes the limes inferior. We choose this subsequence\nsuch that Gf,ν(hn)<∞for all n∈N. Then, there exist µn∈ M +(Rd)with m(µn) =hn.\nIn principle, two cases can occur.\nCase 1: The sequence (∥µn∥TV)ndiverges to ∞.\nSince Df,0=f′\n∞· ∥ · ∥ TV, this is impossible for ν= 0. Ifν∈ M +(Rd)\\ {0}, then Jensen’s\ninequality implies\nGf,ν(hn) =Df,ν(µn) =Z\nRdf◦ρndν+f′\n∞µs\nn(Rd)\n≥ ∥ν∥TVf\u0012Z\nRdρn\n∥ν∥TVdν\u0013\n+f′\n∞µs\nn(Rd)\n=∥ν∥TVf\u00121\n∥ν∥TV\u0010\nµn(Rd)−µs\nn(Rd)\u0011\u0013\n+f′\n∞µs\nn(Rd). (15)\nSince limn→∞Gf,ν(hn)<∞andf′\n∞>0, (15) implies that the sequence (µs\nn(Rd))nmust\nbe bounded. By Lemma 4 the same holds for the sequence (µn(Rd)−µs\nn(Rd))n. Hence,\nalso(µn(Rd))n= (∥µn∥TV)nis bounded, which contradicts our initial assumption and thus\nthis case cannot occur.\nCase 2: The sequence (µn)nsatisfies lim inf n→∞∥µn∥TV<∞.\nSinceC0(Rd)isseparableand C0(Rd)∗∼=M(Rd),theBanach-Alaoglutheorem[33,Cor.5.4.2]\nimplies that there exists a subsequence (µnk)kwhich converges weak* to some µ∈ M(Rd).\nSince K(x,·)∈ C0(Rd)for any x∈Rd, we thus have\nlim\nk→∞mµnk(x) = lim\nk→∞Z\nRdK(x, y) dµnk(y) =Z\nRdK(x, y)dµ(y) =mµ(x).(16)\nSince (mµnk)kconverges to hinHK, (16) entails\nh(x) = lim\nk→∞mµnk(x) =mµ(x)∀x∈Rd.\nThus, his represented by the measure µ. For any nonnegative test function φ∈ Cc(Rd),\nthe weak* convergence of (µnk)ktoµyieldsR\nRdφdµ = lim k→∞R\nφdµ nk≥0. By [52,\n11Lemma 4.71] we have µ∈ M +(Rd). The weak* lower semicontinuity of Df,νfrom Lemma\n6 yields\nGf,ν(h) =Df,ν(µ)≤lim inf\nk→∞Df,ν(µnk) = lim inf\nk→∞Gf,ν(hnk) = lim\nn→∞Gf,ν(hn).\nThe following lemma and its proof build upon the theory of convex integral functionals\ninitiated by [56, 57] and developed in, e.g., [13].\nLemma 8. The conjugate function of Gf,νin(14)is given by\nG∗\nf,ν(h) =Eν[f∗◦h] +ιC0(Rd,dom( f∗))(h) (17)\nProof.Using (4) and (13), we obtain for h∈ H Kthat\nG∗\nf,ν(h) = sup\ng∈HK\b\n⟨g, h⟩HK−Gf,ν(g)\t\n= sup\nµ∈M +(Rd)\b\n⟨mµ, h⟩HK−Df,ν(µ)\t\n= sup\nµ∈M +(Rd)\b\n⟨h, µ⟩C0×M−Df,ν(µ)\t\n.\nApplying [1, Prop. 4.2.6] (a similar argument can be found in the earlier paper [51]) with (in\ntheir notation) (Ω,F)beingRdequipped with itsBorel σ-algebra, X=M(Rd)andYequal\nto the space of bounded measurable functions on Rd, we get that (X, Y)isν-decomposable\nwith Ξ ={∅}, so that ess im Ξ= ran, and thus\nsup\nµ∈M +(Rd)\b\n⟨h, µ⟩C0×M−Df,ν(µ)\t\n=(\nEν[f∗◦h],ifran(h)⊂dom( f∗),\n∞, else.\nfor any h∈Y, so in particular for any h∈ H K.\nNext, we establish the link between Dλ\nf,νand the Moreau envelope of Gf,ν.\nCorollary 9. For any fixed ν∈ M +(Rd)letGf,νbe defined by (14). Then, it holds for\nµ∈ M(Rd)that\nDλ\nf,ν(µ) =Gλ\nf,ν(mµ). (18)\nProof.Since Gf,ν∈Γ0(Rd), we have that\nGλ\nf,ν(mµ) = min\ng∈HKn\nGf,ν(g) +1\n2λ∥g−mν∥2\nHKo\nis well-defined andclearly the unique minimizer fulfills g∈ran(m). Hence we can substitute\ng=mµforµ∈ M +(Rd)to obtain\nGλ\nf,ν(mµ) = min\nµ∈M +(Rd)\b\nDf,ν(µ) +1\n2λ∥mµ−mν∥2\nHK\t\n,\nwhich yields the assertion.\n123.3 Properties of MMD-regularized f-Divergences\nNow, wecombineTheorem9withthepropertiesofMoreauenvelopesinTheorem1toprove\nvarious properties of the MMD-regularized functional Dλ\nf,νin a sequence of corollaries. For\nthe KL divergence, these properties have been shown differently in [21] without using the\nMoreau envelope properties.\nCorollary 10 (Dual formulation) .Forν∈ M +(Rd)andµ∈ M(Rd), we have that\nDλ\nf,ν(µ) = max\np∈HK\np(Rd)⊆dom( f∗)n\nEµ[p]−Eν[f∗◦p]−λ\n2∥p∥2\nHKo\n. (19)\nIfˆp∈ H Kmaximizes (19), then ˆg=mµ−λˆpminimizes (18)and vice versa. Further, it\nholds\nλ\n2∥ˆp∥2\nHK≤Dλ\nf,ν(µ)≤ ∥ˆp∥HK(∥mµ∥HK+∥mν∥HK)and∥ˆp∥HK≤2\nλdK(µ, ν).(20)\nProof. i) From Theorem 9 and Theorem 1(i)), we obtain\nDλ\nf,ν(µ) =Gλ\nf,ν(mµ) = max\np∈HKn\n⟨p, m µ⟩HK−G∗\nf,ν(p)−λ\n2∥p∥2\nHKo\n.\nBy (4), we have ⟨p, m µ⟩HK=Eµ[p]and plugging this into (17) yields the first asser-\ntion.\nii) Using (18), we get\nDλ\nf,ν(µ) =Gf,ν(ˆg) +1\n2λ∥ˆg−mµ∥2\nHK≥1\n2λ∥ˆg−mµ∥2\nHK=λ\n2∥ˆp∥2\nHK,(21)\nwhich is the first lower estimate. For the upper one, we use that f(1) = 0, so that\nf∗(p)≥pfor all p∈R. Then, together with (4), the upper estimate follows by\nDλ\nf,ν(µ) =Eµ[ˆp]−Eν[f∗◦ˆp]−λ\n2∥ˆp∥2\nHK≤Eµ−ν[ˆp]≤ ∥ˆp∥HK(∥mµ∥HK+∥mν∥HK).\nConcerning the second statement in (20), we conclude by (21) and the above estimate\nthatλ\n2∥ˆp∥2\nHK≤Eµ[ˆp]−Eν[ˆp]≤ ∥ˆp∥HKdK(µ, ν).\nRemark 11. In [21, Eq. (2)], Glaser et al. introduced a so-called KALE functional. This is\nexactly the dual functional (19)multiplied by 1+λfor the Kullback-Leibler entropy function\nfKLin Table 1. As expected, the dual function of the KALE functional [21, Eqs. (6)]\n13coincides with our primal formulation (12). Indeed, we have Df(σ|ν)<∞if and only if\nthere exists a density ρ∈L1(Rd, ν;R≥0)such that σ=ρν. Thus,\nmin\nσ∈M +(Rd)Df(σ|ν) +1\n2λdK(µ, σ)2= min\nρ∈L1(Rd)\nρ≥0Z\nRdfKL◦ρdν+1\n2λ∥mρν−mµ∥2\nHK.\nCorollary 12 (Topological properties) .\ni) For every ν∈ M +(Rd), the function Dλ\nf,ν:M(Rd)→Ris weakly continuous.\nii) If Dfis a divergence, then Dλ\nfin(12)is a divergence as well.\niii) If Dfis a divergence and f∗is differentiable in 0, then Dλ\nfmetrizes the topology on\nthe balls Br(µ0) ={µ∈ M +(Rd) :dK(µ, µ0)≤r} ⊂(M+(Rd), dK)for any r >0\nand any µ0∈ M +(Rd).\nProof. i) Letthesequenceofmeasures (µn)n∈Nconvergeweaklyto µ,i.e., limn→∞⟨f, µn⟩=\n⟨f, µ⟩for all f∈ Cb(Rd). Then, we have by [61, Lemma 10] that mµn→mµinHK.\nSince Gλ\nf,νis continuous by Theorem 1(iii)), this implies together with Theorem 9\nthat\nDλ\nf,ν(µ) =Gλ\nf,ν(mµ) = lim\nn→∞Gλ\nf,ν(mµn) = lim\nn→∞Dλ\nf,ν(µn).\nii) The definiteness follows directly from definition (12) since both summands are non-\nnegative. We refer to [9, Thm. 75.4] for a more detailed proof in a slightly different\nsetting.\niii) Let µn, µ∈Br(µ0)for all n∈N. As Gλ\nf,νis continuous, the relation dK(µn, µ) =\n∥mµn−mµ∥HK→0implies Dλ\nf(µn|µ) =Gλ\nf,µ(mµn)→Dλ\nf(µ|µ) = 0.\nFor the reverse direction, assume that Dλ\nf(µn|µ)→0. For any n∈N, we define\ngn:=mµn−µ=mµn−mµ, for which it holds that\n∥gn∥C0≤C∥gn∥HK=CdK(µn, µ)≤C\u0000\ndK(µn, µ0) +dK(µ0, µ)\u0001\n≤2Cr,(22)\nwhere C >0is the embedding constant from HK,→ C 0(Rd). Hence, it holds for any\nf′\n∞\n2Cr> ϵ > 0thatϵgn(Rd)⊂dom( f∗)since ϵ|gn(x)|<f′\n∞\n2Cr|gn(x)| ≤f′\n∞for all x∈Rd.\nBy (19), we further get that\nDλ\nf(µn|µ)≥Eµn(ϵgn)−Eµ(f∗◦(ϵgn))−λϵ2\n2∥gn∥2\nHK\n=ϵ∥gn∥2\nHK−Eµ[f∗◦(ϵgn)−ϵgn]−λϵ2\n2∥gn∥2\nHK\n=\u0010\nϵ−λϵ2\n2\u0011\n∥gn∥2\nHK−Eµ[f∗◦(ϵgn)−ϵgn]. (23)\n14UsingTaylor’stheoremwiththePeanoformoftheremainder,thereexists h: (−∞, f′\n∞)→\nRwith limt→0h(t)\nt= 0such that f∗(t) =t+h(t)for all t∈(−∞, f′\n∞). Thus, we get\nEµ[f∗◦(ϵgn)−ϵgn] =Z\nRdh\u0000\nϵgn(x)\u0001\ndµ(x)≤\r\r\rh◦(ϵgn)\nϵgn\r\r\r\n∞∥ϵgn∥∞µ(Rd)(24)\nwith the conventionh◦(ϵgn(x))\nϵgn(x)= 0ifgn(x) = 0. Plugging (24) into (23) yields\nDλ\nf(µn|µ)≥ϵ∥gn∥HK\u0012\u0010\n1−λ\n2ϵ\u0011\n∥gn∥HK−C\r\r\rh◦(ϵgn)\nϵgn\r\r\r\n∞µ(Rd)\u0013\n.(25)\nSince limt→0h(t)\nt= 0andϵgnis uniformly bounded, equation (22) implies that the\nsecond factor in (25) converges to ∥gn∥HKforϵ↘0. Thus, we can pick ϵ > 0\nindependent of nsuch that Dλ\nf(µn|µ)≥ϵ\n2∥gn∥2\nHK=ϵ\n2dK(µn, µ)2. From this, we\ninfer dK(µn, µ)→0.\nNow, we investigate the two asymptotic regimes of Dλ\nf.\nCorollary 13 (Limits for λ→0andλ→ ∞).\ni) If 0∈int(dom( f∗)), then it holds limλ→0Dλ\nf(µ|ν) =Df(µ|ν)forµ, ν∈ M +(Rd).\nii) We have that Dλ\nf,νconverges to Df,νin the sense of Mosco: It holds for every µ∈\nM+(Rd), every monotonically decreasing sequence (λn)n∈N⊂(0,∞)with λn→0,\nand every sequence (µn)n∈N⊂ M +(Rd)withµn⇀ µthat\nDf,ν(µ)≤lim inf\nn→∞Dλn\nf,ν(µn).\nFurther, there exists a sequence (˜µn)n∈N⊂ M +(Rd)withµn→µsuch that\nDf,ν(µ) = lim\nn→∞Dλn\nf,ν(˜µn).\niii) If f∗is differentiable in 0, then it holds for any r >0and any µ0∈ M +(Rd)that\nlim\nλ→∞sup\nµ∈Br(µ0)\f\f\f(1 +λ)Dλ\nf(µ|ν)−1\n2dK(µ, ν)2\f\f\f= 0.\nProof.\ni) ByTheorem1(iii)), wegetpointwiseconvergenceto Gν,f. Then, thestatementfollows\nby the final part of Lemma 8 and Theorem 9.\n15ii) By Theorem 1(iii)), the sequences (Gλn\nf,ν(h))n∈Nare monotonically increasing with\nsupn∈NGλn\nf,ν(h) =Gf,ν(h)for any h∈ H K. As Gf,νis lower semicontinuous, [14,\nRem. 2.12] implies that (Gλn\nf,ν)n∈NΓ-converges to Gν,f. More precisely, it holds that\nGf,ν(h)≤lim inf n→∞Gλn\nf,ν(hn)for every h∈ H Kand any sequence (hn)n∈N⊂ H K\nwith hk→h. Further, for every h∈ H Kit holds that Gf,ν(h) = lim n→∞Gλn\nf,ν(h).\nThe statement now follows from the fact that Dλ\nf,ν=Gλ\nν,f◦mby Theorem 9 and\nsince µn⇀ µinM+(Rd)implies mµn→mµby [61, Lemma 10].\niii) From (12), we infer that (1 +λ)Dλ\nf(µ|ν)≤1+λ\n2λdK(µ, ν)2. To also get a lower bound,\nwe proceed as for Corollary 12(iii)). For gµ,λ:=1\nλmµ−νwith λ >C\nf′∞|µ−ν|(Rd)(note\nthat f′\n∞>0since f∗is differentiable at 0), we have that gµ,λ(Rd)⊂dom( f∗).\nAnalogously to (23) and (25) (using the same h), we get the lower bound\n(1 +λ)Dλ\nf(µ|ν)≥(1 +λ)\nλdK(µ, ν)\u00121\n2dK(µ, ν)−Cν(Rd)\r\r\rh◦gµ,λ\ngµ,λ\r\r\r\n∞\u0013\n≥1\n2dK(µ, ν)2−Cν(Rd)dK(µ, ν)1 +λ\nλ\r\r\rh◦gµ,λ\ngµ,λ\r\r\r\n∞.(26)\nCombining (26) with the above upper estimate, we get for any µ∈Br(µ0)that\n\f\f\f(1 +λ)Dλ\nf,ν(µ)−1\n2dK(µ, ν)2\f\f\f\n≤dK(µ, ν) max\u0012dK(µ, ν)\n2λ, Cν(Rd)1 +λ\nλ\r\r\rh◦gµ,λ\ngµ,λ\r\r\r\n∞\u0013\n≤\u0000\ndK(µ0, ν) +r\u0001\nmax\u0012dK(µ0, ν) +r\n2λ, Cν(Rd)1 +λ\nλ\r\r\rh◦gµ,λ\ngµ,λ\r\r\r\n∞\u0013\n.\nHere, the first term in the maximum converges to zero as λ→ ∞. As for Corol-\nlary 12(iii)), ∥gµ,λ∥∞≤C\nλ(dK(µ0, ν) +r)together with limt→01\nth(t) = 0yields that\nalso the second term in the maximum converges to zero. Thus, the claim follows.\nTheMoreauenvelopeinterpretationof Dλ\nf,νallowsthecalculationofitsgradientwithout\nthe implicit function theorem, which is used to justify the calculations for the particular\ncase of the KALE function in [21, Lem. 2].\nCorollary 14 (Gradient) .The function Dλ\nf,ν:M(Rd)→[0,∞)is Fréchet differentiable\nand∇Dλ\nf,ν(µ) = ˆ p, where ˆp∈ H Kis the maximizer in (19). Further, the mapping\n∇Dλ\nf,ν:M(Rd)→ C 0(Rd)is1\nλ-Lipschitz with respect to dK.\nProof.By Theorem 1(ii)) and Corollary 10, we obtain\n∇Gλ\nf,ν(mµ) =λ−1\u0000\nmµ−proxλGf,ν(mµ)\u0001\n= ˆp.\n16As the concatenation of a Fréchet differentiable and a continuous linear mapping also\nDλ\nf,ν=Gλ\nf,ν◦mis Fréchet differentiable. The Fréchet derivative of the KME m:M(Rd)→\nHKatµ∈ M(Rd)isdmµ=m∈L(M(Rd);HK). Using these computations together with\nthe chain rule, we get for any σ∈ M(Rd)that\ndDλ\nf,ν(µ)[σ] = d( Gλ\nf,ν◦m)(µ)[σ] = dGλ\nf,ν(mµ)[mσ] =⟨ˆp, σ⟩C0×M.\nHence, we can identify ∇Dλ\nf,ν(µ) = ˆp∈ C0(Rd). Finally, we have by Theorem 1(ii)) that\n∥∇Gλ\nf,ν(mµ)− ∇Gλ\nf,ν(mσ)∥∞≤λ−1∥mµ−mσ∥HK=λ−1dK(µ, σ),\nwhich completes the proof.\n4 Wasserstein Gradient Flows of Regularized f-Divergences\nNow, we are interested in gradient flows of Dλ\nf,νin the Wasserstein space. This requires\nsome preliminaries from [4, Secs. 8.3, 9.2, 11.2], which are adapted to our setting in the\nfollowing subsection.\n4.1 Wasserstein Gradient Flows\nWe consider the Wasserstein space P2(Rd)of Borel probability measures with finite second\nmoments, equipped with the Wasserstein distance\nW2(µ, ν)2:= min\nπ∈Γ(µ,ν)Z\nRd×Rd∥x−y∥2\n2dπ(x, y), (27)\nwhere Γ(µ, ν) ={π∈ P 2(Rd×Rd) : (P1)#π=µ,(P2)#π=ν}. Here, T#µ:=µ◦T−1\ndenotes the push-forward ofµvia the measurable map T, and Pi(x):=xi,i= 1,2, for\nx= (x1, x2)∈Rd×Rd.\nA curve γ: [0,1]→ P 2(Rd),t7→γtis ageodesic ifW2(γt1, γt2) =W2(γ0, γ1)|t2−t1|for\nallt1, t2∈[0,1]. The Wasserstein space is geodesic, i.e., any two measures µ0, µ1∈ P2(Rd)\ncan be connected by a geodesic. These are all of the form\nγt=\u0000\n(1−t)P1+tP2\u0001\n#ˆπ, t ∈[0,1],\nwhere ˆπ∈Γ(µ0, µ1)realizes W2(µ0, µ1)in (27).\nFor a function F:P2(Rd)→(−∞,∞], we set dom(F):={µ∈ P 2(Rd) :F(µ)<∞}.\nThe function Fis called M-convex along geodesics with M∈Rif, for every µ0, µ1∈\ndom(F), there exists at least one geodesic γ: [0,1]→ P 2(Rd)between µ0andµ1such that\nF(γt)≤(1−t)F(µ0) +tF(µ1)−M\n2t(1−t)W2(µ0, µ1)2, t ∈[0,1].\n17Frequently, we also need a more general notion of convexity. Based on the set of three-plans\nwith base σ∈ P2(Rd)given by\nΓσ(µ, ν):=\b\nα∈ P2(Rd×Rd×Rd) : (P1)#α=σ,(P2)#α=µ,(P3)#α=ν\t\n,\nageneralized geodesic γα: [0,1]→ P 2(Rd)joining µandνwith base σis defined as\nγα,t:=\u0000\n(1−t)P2+tP3\u0001\n#α, t ∈[0,1],\nwhere α∈Γσ(µ, ν)with (P1,2)#α∈Γopt(σ, µ)and(P1,3)#α∈Γopt(σ, ν). Here, Γopt(µ, ν)\ndenotes the set of optimal transport plans that minimize (27). The plan αis interpretable\nas transport from µtoνviaσ. A function F:P2(Rd)→(−∞,∞]isM-convex along\ngeneralized geodesics if, for every σ, µ, ν ∈dom(F), there exists at least one generalized\ngeodesic γα: [0,1]→ P 2(Rd)such that\nF(γα,t)≤(1−t)F(µ) +tF(ν)−M\n2t(1−t)W2\nα(µ, ν), t ∈[0,1],\nwhere\nW2\nα(µ, ν):=Z\nRd×Rd×Rd∥y−z∥2\n2dα(x, y, z ).\nEachfunctionthatis M-convexalonggeneralizedgeodesicsisalso M-convexalonggeodesics\nsince generalized geodesics with base σ=µare actual geodesics.\nThestrong Fréchet subdifferential ∂F(µ)of a proper, lower semi-continuous function\nF:P2(Rd)→(−∞,∞]atµconsists of all v∈L2(Rd, µ)such that for every η∈ P 2(Rd)\nand every π∈Γ(µ, η), it holds\nF(η)− F(µ)≥Z\nRd×Rd⟨v(x1), x2−x1⟩dπ(x1, x2) +o(C2(π)), (28)\nwhere C2(π):=R\nRd×Rd∥x1−x2∥2\n2dπ(x1, x2)and the asymptotic o(·)has to be understood\nwith respect to the W2metric.\nA curve γ: (0,∞)→ P 2(Rd)is called absolutely continuous if there exists a Borel\nvelocity field v:Rd×(0,∞)→Rd,(x, t)7→vt(x)withR∞\n0∥vt∥L2(dγt)dt <∞such that the\ncontinuity equation\n∂tγt+∇ ·(vtγt) = 0\nis fulfilled on (0,∞)×Rdin a weak sense, i.e.,\nZ∞\n0Z\nRd∂tφ(t, x) +⟨∇xφ(t, x), vt(x)⟩dγt(x) dt= 0for all φ∈ C∞\nc\u0000\n(0,∞)×Rd\u0001\n.\nAn absolutely continuous curve γ: (0,∞)→ P 2(Rd)with velocity field vtin the regular\ntangent space of P2(Rd)is called a Wasserstein gradient flow of F:P2(Rd)→(−∞,∞]if\nvt∈ −∂F(γt),for a.e. t >0. (29)\n18In principle, (29) involves the so-called reduced Fréchet subdifferential ∂F. We will show\nthat our functionals of interest Dλ\nf,νare Fréchet differentiable such that we can use the\nstrong Fréchet subdifferential from (28) instead. For this setting, we have the following\ntheorem from [4, Thm. 11.2.1], see also [26, Thm. 3].\nTheorem 15. Assume that F:P2(Rd)→(−∞,∞]is proper, lower semi-continuous, coer-\ncive and M-convex along generalized geodesics. Given µ0∈dom(F), there exists a unique\nWasserstein gradient flow of Fwithγ(0+) = µ0.\nRemark 16. By [4, Eq. (11.2.1b)] any proper M-convex functional that is bounded from\nbelow by some constant is coercive in the sense of [4, Eq. (2.1.2b)].\n4.2 Wasserstein Gradient Flow of Dλ\nf,ν\nNow, we show that Dλ\nf,νis locally Lipschitz, M-convex along generalized geodesics, and\nthat ∂Dλ\nf,ν(µ)is a singleton for every µ∈ P 2(Rd). To this end, we rely on [70, Prop. 2,\nCor. 3], which we adapt to our setting in the following lemma.\nLemma 17. Let the kernel Kfulfill K(x, x) +K(y, y)−2K(x, y)≤C2\nemb∥x−y∥2\n2with\nsome constant Cemb>0. Then, it holds\ndK(µ, ν)≤CembW2(µ, ν). (30)\nIfK(x, y) = Φ( x−y)is translation invariant, then we get Cemb=p\nλmax(−∇2Φ(0)). For\nradial kernels K(x, y) =ϕ(∥x−y∥2\n2), we have\n∇2Φ(x) = 4 ϕ′′(∥x∥2\n2)xxT+ 2ϕ′(∥x∥2\n2) id,\nso that Cemb=p\n−2ϕ′(0).\nNote that the authors in [70] found Cemb=p\nλmax(−∇2Φ(0))Φ(0) instead, which we\ncould not verify. Now, we prove the local Lipschitz continuity of Dλ\nf,νwith Theorem 1(ii))\nand Lemma 17.\nLemma 18. The function Dλ\nf,ν: (P2(Rd), W2)→[0,∞)is locally Lipschitz continuous.\nProof.By Theorem 1(ii)), we know that Gλ\nf,ν:HK→[0,∞)is continuously Fréchet dif-\nferentiable. Hence, it is locally Lipschitz continuous. Since m: (P2(Rd), dK)→ H Kis an\nisometry and Gλ\nf,ν◦m, the claim follows using (30).\nNext, we show the M-convexity of Dλ\nf,νalong generalized geodesics, where the Moreau\nenvelope interpretation allows a simpler proof than the one given in [6, Lem. 4].\nTheorem 19. LetK(x, y) =ϕ(∥x−y∥2\n2)withϕ∈ C2(Rd). Then, Dλ\nf,ν:P2(Rd)→[0,∞)\nis(−M)-convex along generalized geodesics with M:=8\nλp\n(d+ 2)ϕ′′(0)ϕ(0).\n19Proof.Letµ1, µ2, µ3∈ P 2(Rd)andγ: [0,1]→ P 2(Rd),t7→((1−t)P2+tP3)#αbe a\ngeneralized geodesic associated to a three-plan α∈Γµ1(µ2, µ3). Furthermore, ˜γ: [0,1]→\nP2(Rd),t7→(1−t)µ2+tµ3denotes the linear interpolation between µ2andµ3. Since Gλ\nf,ν\nandDλ\nf,νare linearly convex, it holds for t∈[0,1]that\nDλ\nf,ν(γt)≤(1−t)Dλ\nf,ν(µ2) +tDλ\nf,ν(µ3) +Dλ\nf,ν(γt)−Dλ\nf,ν(˜γt)\n≤(1−t)Dλ\nf,ν(µ2) +tDλ\nf,ν(µ3) +\n∇Gλ\nf,ν(mγt), mγt−m˜γt)\u000b\nHK.\nWe consider the third summand. Let ˆptmaximize the dual formulation (2) of Gλ\nf,ν(mγt).\nThen, we know by Theorem 1(ii)) that ∇Gλ\nf,ν(mγt) = ˆpt, so that\n\n∇Gλ\nf,ν(mγt), mγt−m˜γt\u000b\nHK\n=Z\nRd×Rd×Rdˆpt\u0000\n(1−t)y2+ty3)\u0001\n−\u0000\n(1−t)ˆpt(y2) +tˆpt(y3)\u0001\ndα(y1, y2, y3).\nDue to (6), ∇ˆptis Lipschitz continuous with Lip(∇ˆpt)≤2∥ˆpt∥HKp\nϕ′′(0)(d+ 2). Hence,\nthe descent lemma [42, Lemma 1.2.3, Eq. (1.2.12)] implies that ˆptis(−Lip(∇ˆpt))-convex\nand\n\n∇Gλ\nf,ν(mγt), mγt−m˜γt\u000b\nHK≤1\n2Lip(∇ˆpt)t(1−t)Z\nRd×Rd×Rd∥y2−y3∥2\n2dα(y1, y2, y3)\n=1\n2Lip(∇ˆpt)t(1−t)W2\nα(µ2, µ3).\nBy (20), we have ∥ˆpt∥HK≤2\nλ(∥mγt∥HK+∥mν∥HK). For any µ∈ P2(Rd), it holds that\n∥mµ∥2\nHK=Z\nRd×RdK(x, y) dµ(y) dµ(x) =Z\nRd×Rdϕ(∥x−y∥2\n2) dµ(y) dµ(x)≤ϕ(0),\nwhich implies\n∥ˆpt∥HK≤4\nλp\nϕ(0). (31)\nThus, for all t∈[0,1], we have Lip(∇ˆpt)≤M:=8\nλp\nϕ(0)ϕ′′(0)(d+ 2), and Dλ\nf,νis−M-\nconvex along generalized geodesics.\nThe following proposition determines the strong subdifferential of Dλ\nf,ν.\nLemma 20. LetK(x, y) =ϕ(∥x−y∥2\n2)be a radial kernel and ϕ∈ C2(Rd). Then, it holds\nfor any µ∈ P2(Rd)that∂Dλ\nf,ν(µ) ={∇ˆp}, where ˆp∈ H Kmaximizes (19).\n20Proof.First, we show that ∇ˆp∈∂Dλ\nf,ν(µ). By convexity of Gλ\nf,νand Lemma 14 , we obtain\nfor any η∈ P2(Rd)and any π∈Γ(µ, η)that\nDλ\nf,ν(η)−Dλ\nf,ν(µ) =Gλ\nf,ν(mη)−Gλ\nf,ν(mµ)≥ ⟨∇ Gλ\nf,ν(mµ), mη−mµ⟩HK\n=⟨ˆp, m η−mµ⟩HK=Z\nRdˆp(x2) dη(x2)−Z\nRdˆp(x1) dµ(x1)\n=Z\nRd×Rdˆp(x2)−ˆp(x1) dπ(x1, x2). (32)\nCombining (6) and (31), we get that ˆp∈ H Khas a Lipschitz continuous gradient with\nLipschitz constant L:=8\nλp\nϕ(0)ϕ′′(0)(d+ 2). Hence, we obtain by the descent lemma that\nDλ\nf,ν(η)−Dλ\nf,ν(µ)≥Z\nRd×Rd⟨∇ˆp(x1), x2−x1⟩ −L\n2∥x2−x1∥2dπ(x1, x2)\n=Z\nRd×Rd⟨∇ˆp(x1), x2−x1⟩dπ(x1, x2)−L\n2C2\n2(π).\nNext, we show that ∇ˆpis the only subdifferential. For some fixed v∈L2(Rd, µ), we define\nthe perturbation map At(x):=x+t(∇ˆp(x)−v(x)),t >0and the perturbed measures\nµt:= (At)#µwith an induced plan πt= (id , At)#µ∈Γ(µ, µt). For these choices, it holds\nby (32) that\nDλ\nf,ν(µt)−Dλ\nf,ν(µ)≥Z\nRd×Rd⟨∇ˆp(x1), x2−x1⟩dπt(x1, x2)−L\n2C2\n2(πt)\n=Z\nRd×Rd⟨v(x1), x2−x1⟩+⟨∇ˆp(x1)−v(x1), x2−x1⟩dπt(x1, x2)−L\n2C2\n2(πt)\n=Z\nRd×Rd⟨v(x1), x2−x1⟩dπt(x1, x2) +tZ\nRd∥v(x)− ∇ˆp(x)∥2dµ(x)−L\n2C2\n2(πt)\n=Z\nRd×Rd⟨v(x1), x2−x1⟩dπt(x1, x2) +tC2\n2(π1)−L\n2C2\n2(πt).\nSince tC2(π1) =C2(πt), we further have\nDλ\nf,ν(µt)−Dλ\nf,ν(µ)≥Z\nRd×Rd⟨v(x1), x2−x1⟩dπt(x1, x2) +C2(πt)C2(π1)\u0010\n1−L\n2t\u0011\n.\nUnless C2(π1) = 0, which is equivalent to ∇ˆp=v µ-a.e., this contradicts (28), which\nconcludes the proof.\nBased on Theorem 15 and Lemma 20, we have the following result.\nCorollary 21. LetK(x, y) =ϕ(∥x−y∥2\n2)be a radial kernel and ϕ∈ C2(Rd). Let ˆpγtbe\nthe maximizer in the dual formulation of Dλ\nf,ν. Then, for any µ0∈ P2(Rd)the equation\n∂tγt−div(γt∇ˆpγt) = 0 . (33)\nhas a unique weak solution γtfulfilling γ(0+) = µ0, where γ(0+) := lim t↘0γt.\n21Wasserstein Gradient Flows for Empirical Measures. Finally, we investigate the\nWasserstein gradient flow (33) starting in an empirical measure\nµ0:=1\nNNX\ni=1δx(0)\ni. (34)\nTo solve (33) numerically, we consider a particle functional DN:RdN→[0,∞)ofDλ\nf,ν\ndefined for x:= (x1, . . . , x N)as\nDN(x):=Dλ\nν,f\u00121\nNNX\nk=1δxk\u0013\n,\nand consider the particle flow x: [0,∞)→RdNwith\n˙x=−N∇DN(x), x(0):=\u0000\nx(0)\n1, . . . , x(0)\nN\u0001\n. (35)\nIn general, solutions of (35) differ from those of (33). However, for our setting, we can show\nthat the flow (35) induces a Wasserstein gradient flow.\nProposition 22. Assume that K(x, y) =ϕ(∥x−y∥2\n2)is a radial kernel, where ϕ∈ C2(Rd).\nLetx(t) = ( xi(t))N\ni=1⊂RdN,t∈[0,∞), be a solution of (35). Then, the corresponding\ncurve of empirical measures γN: [0,∞)→ P 2(Rd)given by\nγN(t) =1\nNNX\ni=1δxi(t)\nis a Wasserstein gradient flow of Dλ\nν,fstarting in µ0.\nProof.ForMN(t):=1\nNPN\ni=1˙xi(t)δxi(t)=−PN\ni=1∇xiDN(x)δxi(t)andφ∈C∞\nc((0,∞)×\nRd)it holds that\nZ∞\n0\u0012Z\nRd∂tφ(t, x) dγN(t) +Z\nRd∇xφ(t, x) dMN(t)\u0013\ndt\n=1\nNNX\ni=1Z∞\n0\u0000\n∂tφ(t, xi(t)) +∇xφ(t, xi(t))·˙xi(t)\u0001\ndt\n=1\nNNX\ni=1Z∞\n0d\ndtφ(t, xi(t)) dt= 0.\nHence, ∂tγN+ div( MN) = 0holds in weak sense, and it is left to show that MN(t) =\n−∇ˆpγN(t)γN(t). First, note that DN(x) =Gf,νλ(mγN)andmγN=1\nNPN\ni=1K(·, xi). For\n22the second term, we obtain by Lemma 3 that ∇ximγN=1\nN∇xiK(·, xi). Thus, we can apply\nthe chain rule, Corollary 14 and the derivative reproducing property of ∂jK(·, xi)to get\nN∇xiDN(x) =\u0010\nˆpγN, ∂jK(·, xi)\u000b\nHK\u0011d\nj=1=∇ˆpγN(xi).\nConsequently, we obtain ∂tγN−div(γN∇ˆpγN) = 0as required.\nRemark 23 (Consistentdiscretization) .An advantage of the regularized f-divergences Dλ\nν,f,\nν∈ P 2(Rd), is that Dλ\nν,f(µ)<∞for any µ∈ P 2(Rd)(even if Dν,f(µ) =∞). This is\nimportant when approximating the Wasserstein gradient flow of the original functional Dν,f\nstarting in µ0∈dom( Dν,f)numerically with gradient flows starting in empirical measures\nµ0,N=1\nNPN\ni=1δxi(0), which might not be in dom( Dν,f). To deal with this issue, we can\nchoose (λN)N∈Nwith limN→∞λN= 0such that supN∈NDλN\nν,f(µ0,N)<∞. Then, as shown\nin Corollary 13(i)), the functionals DλN\nν,fMosco converge to Dν,f. If we additionally have\na uniform lower bound on their convexity-modulus along generalized geodesics, then [4,\nThm. 11.2.1] implies that the Wasserstein gradient flows of DλN\nν,fstarting in µ0,Nconverge\nlocally uniformly in [0,∞)to the flow of Dν,fstarting in µ0. Whether such a lower bound\nexists is so far an open question. In Theorem 19, we only established a lower bound on the\nweak convexity modulus of DλN\nν,fwhich scales as 1/λN. A similar convergence result was\nrecently established in [36] for regularization with the Wasserstein distance W2instead of\nthe MMD dK. However, their proof cannot be directly extended to our setting.\n5 Computation of Flows for Empirical Measures\nIn this section, we are interested in the Wasserstein gradient flows γstarting in an empirical\nmeasure (34), where Dλ\nf,νhas an empirical target measure ν=1\nMPM\nj=1δyj. For the KL\ndivergence, such flows were considered in [21] under the name KALE flows. The Euler\nforward discretization of the Wasserstein gradient flow (35) with step size τ >0is given by\nthe sequence (γn)n∈N⊂ P 2(Rd)defined by\nγn+1:= (id−τ∇ˆpn)#γn,\nwhere ˆpnmaximizes the dual formulation (19) of Dλ\nν,f(γn). Since the push-forward of\nan empirical measure by a measurable map is again an empirical measure with the same\nnumber of particles, we have that\nγn=1\nNNX\ni=1δx(n)\ni,\nwhere\nx(n+1)\ni =x(n)\ni−τ∇ˆpn(x(n)\ni), i = 1, . . . , N. (36)\n23Since both γnandνare empirical measures, the dual problem (19) for ˆpnbecomes\nˆpn= arg max\np∈HK\np(Rd)⊆dom( f∗)\u001a1\nNNX\ni=1p(x(n)\ni)−1\nMMX\nj=1f∗(p(yj))−λ\n2∥p∥2\nHK\u001b\n. (37)\n5.1 Discretization Based on Representer Theorem\nIff′\n∞=∞or supp (ν) =Rd, then the constraint p(Rd)⊂dom( f∗)in (37) becomes\nsuperfluous. By the Representer Theorem [59], the solution of (37) is of the form\nˆpn=M+NX\nk=1b(n)\nkK(·, z(n)\nk), b(n):= (b(n)\nk)M+N\nk=1∈RM+N, (38)\nwhere\n(z(n)\n1, . . . , z(n)\nN+M) =\u0000\ny1, . . . , y M, x(n)\n1, . . . , x(n)\nN\u0001\n∈Rd×(M+N).\nWe denote the associated kernel matrix by K(n):= (K(z(n)\ni, z(n)\nj))M+N\ni,j=1.\nIff′\n∞<∞, we cannot drop the constraint p(Rd)⊂dom( f∗)in (19) to apply the\nRepresenter Theorem. The situation is similar for the primal problem (12), where the\ncondition f′\n∞<∞entails that we have to consider measures µthat are not absolutely\ncontinuous with respect to the (empirical) target ν. In the following, we provide a condition\nunder which computing ˆpnfor (37) reduces to a tractable finite-dimensional problem. To\nsimplify the notation, we omit the index nand use the shorthand L(p) =Eµ[p]−Eν[f∗◦\np]−λ\n2∥p∥2\nHKfor the objective in (19).\nLemma 24. Letfbe an entropy function with f′\n∞∈(0,∞). If µ, ν∈ M +(Rd)and\nλ >2dK(µ, ν)p\nϕ(0)/f′\n∞, then\narg max\np∈Hk\np(Rd)⊆supp( f∗)L(p) = arg max\n∥p∥HK<f′∞√\nϕ(0)L(p).\nIfµ, νare empirical measures with Z= supp( µ)∪supp( ν), then the solution ˆpof the discrete\nproblem(37)has a finite representation of the form (38).\nProof.First, we introduce shorthand notations for the feasible sets\nC:=\b\np∈ H K:p(Rd)⊂dom( f∗)\t\nand X:=\u001a\np∈ H K:∥p∥HK<f′\n∞p\nϕ(0)\u001b\n.\nIt holds for any p∈Xthat∥p∥∞≤p\nϕ(0)∥p∥HK< f′\n∞, which implies X⊆C. By (20),\nthe maximizer ˆpof (19) satisfies ∥ˆp∥HK≤2\nλdK(µ, ν). For λ >2dK(µ, ν)p\nϕ(0)/f′\n∞, we get\n24ˆp∈Xdue to ∥ˆp∥HK≤2\nλdK(µ, ν)< f′\n∞/p\nϕ(0). Since X⊆C, we get that ˆpmaximizes L\noverX. This proves the first claim.\nFor the second part, consider the parameterization Ψ:HK→Xwith Ψ(0) := 0and\nΨ(p):=p\n∥p∥ψ(∥p∥)otherwise, where ψ: [0,∞)→[0,f′\n∞\nϕ(0)]is given by\nψ(t) =2f′\n∞\nπp\nϕ(0)arctan( t).\nThe bijectivity of arctandirectly implies the bijectivity of Ψ. Hence, the reparameterized\nproblem arg maxq∈HKL(Ψ(q))has the unique maximizer ˆq= Ψ−1(ˆp). Since ˆq∈ H k, we can\ndecompose ˆqasˆq= ˆs+ ˆr, where ˆs∈span{K(·, z) :z∈Z}andˆr∈span{K(·, z) :z∈Z}⊥.\nThen, for all z∈Zwe get that\nˆq(z) =⟨ˆq, K(·, z)⟩=⟨ˆs+ ˆr, K(·, z)⟩=⟨ˆs, K(·, z)⟩= ˆs(z).\nHence, it holds that\nL(Ψ(ˆq)) =Eµ[Ψ(ˆq)]−Eν[f∗◦Ψ(ˆq)]−λ\n2∥Ψ(ˆq)∥2\nHK\n=1\n|supp( µ)|X\nz∈supp( µ)Ψ(ˆs(z))−1\n|supp( ν)|X\nz∈supp( ν)f∗◦Ψ(ˆs(z))−λ\n2∥Ψ(ˆq)∥2\nHK\n=Eµ[Ψ(ˆs)]−Eν[f∗◦Ψ(ˆs)]−λ\n2∥Ψ(ˆq)∥2\nHK\nNow,∥Ψ(ˆq)∥HK=ψ(∥ˆq∥HK),∥ˆq∥2\nHK=∥ˆs∥2\nHK+∥ˆr∥2\nHKand the strict monotonicity of Ψ2\nimply\nL(Ψ(ˆq)) =Eµ[Ψ(s)]−Eν[f∗◦Ψ(ˆs)]−λ2(f′\n∞)2\n(πp\nϕ(0))2arctan\u0010q\n∥ˆs∥2\nHK+∥ˆr∥2\nHK\u00112\n≤Eµ[Ψ(ˆs)]−Eν[f∗◦Ψ(ˆs)]−λ2(f′\n∞)2\n(πp\nϕ(0))2arctan( ∥ˆs∥HK)2=L(Ψ(ˆs)).\nHence, we must have ˆq= ˆs. This directly implies ˆp= Ψ(ˆ q)∈span{K(·, z) :z∈Z}, which\nmeans that we can write ˆp=P\nz∈ZbzK(·, z)andb∈RZ.\n5.2 Numerical Implementation Details\nRecall that both for infinite and finite recession constant (see Lemma 24), the solution ˆpn\nto the dual problem has the representation\nˆpn=N+MX\nk=1b(n)\nkK\u0000\n·, z(n)\nk\u0001\n.\n25To determine the coefficients b(n)= (b(n)\nk)M+N\nk=1, we look instead at the primal problem (18),\nwhich by Corollary 10 has a solution of the form\nˆgn=mγn−λˆpn=M+NX\nk=1β(n)\nkK\u0000\n·, z(n)\nk\u0001\n=mˆσn, ˆσn:=M+NX\nk=1β(n)\nkδz(n)\nk,\nwith the nonnegative coefficients\nβ(n)\nk:=(\n−λb(n)\nk,ifk∈ {1, . . . , M }\n1\nN−λb(n)\nk,ifk∈ {M+ 1, . . . , M +N}.\nHence, to compute b(n), we can minimize the primal objective only with respect to the\ncoefficients β(n), which results in the objective\nJ\u0000\nβ(n)\u0001\n=Df,ν(ˆσn) +1\n2λ∥mˆσn−mγn∥2\nHK. (39)\nWith the change of variables q(n):=−λMb(n)∈RN+M, we getdˆσn\ndν(z(n)\nk) = q(n)\nkfor\nk∈ {1, . . . , M }andˆσs\nn(Rd) = 1 +1\nMPM+N\nk=M+1q(n)\nk. Plugging this into (39) yields\nJ\u0000\nq(n)\u0001\n=Z\nRdf◦dˆσn\ndνdν+f′\n∞ˆσs\nn(Rd) +1\n2λM2(q(n))TK(n)q(n)\n=1\nMMX\nk=1f\u0000\nq(n)\nk\u0001\n+f′\n∞+f′\n∞\nMM+NX\nk=M+1q(n)\nk+1\n2λM2(q(n))TK(n)q(n).\nThe constraint β(n)≥0translates into the constraints\nq(n)\nk=−λb(n)\nkM(\n≥0, k ∈ {1, . . . , M },\n=β(n)\nk−M\nN≥ −M\nN, k∈ {M+ 1, . . . , M +N}.\nWe minimize the convex objective Jwith these box constraints using the L-BFGS-B lim-\nited memory approximation of the Broyden–Fletcher–Goldfarb–Shannon quasi-Newton al-\ngorithm, which is provided in the SciPy package [49]. For this, we require the gradient of\nJwith respect to q(n), which reads\n∇J\u0000\nq(n)\u0001\n=1\nM\"\u0002\nf′\u0000\nq(n)\nk\u0001\u0003M\nk=1\nf′\n∞1N#\n+1\nλM2K(n)q(n).\nTo summarize, with the optimal coefficients ˆq(n)\nk, the update step (36) becomes\nx(n+1)\nj =x(n)\nj+τ\nλMM+NX\nk=1ˆq(n)\nk∇K\u0000\n·, z(n)\nk\u0001\n(x(n)\nj), j ∈ {1, . . . , N }.(40)\n266 Numerical Results\nIn this section, we use three target measures from the literature to compare how fast the\ndiscrete Wasserstein gradient flow (40) for different Dλ\nf,νconverge∗. We always choose N=\nM= 900particles. Further, we use the inverse multiquadric kernel defined in Remark 2.\nWe focus on the Tsallis- αdivergence for α≥1because the corresponding entropy function\nhas an infinite recession constant and is differentiable in the interior of its domain. For\nα≥1, the Tsallis- αdivergence Dfα(µ|ν)between µ, ν∈ P(Rd)with µ≪νreads\nDfα(µ|ν) =1\nα−1\u0012Z\nRd\u0010dµ\ndν(x)\u0011α\ndν(x)−1\u0013\n.\nIn our experiments, the commonly used KL divergence (which corresponds to the limit for\nα↘1) is outperformed in terms of convergence speed by values of αthat are moderately\nlarger than one. In all our examples, we observed exponential convergence of the target,\nboth in terms of the MMD and the Wasserstein distance. Since by Corollary 13(iii)) the\nfunctional (1 +λ)Dλ\nf,νconverges to1\n2dK(·, ν)2forλ→ ∞, we always consider the gradient\nflow with respect to the functional (1 +λ)Dλ\nf,νinstead of Dλ\nf,ν.\nThree rings target First, we consider the three rings target from [21, Fig. 1]. Our\nsimulations are provided in Figure 1. The starting point µ0of the flow are samples from a\nnormal distribution with variance s2= 2×10−3around the leftmost point on the rightmost\ncircle. Further, we choose the kernel width σ2= 5×10−2, regularization parameter λ=\n10−2, and step size τ= 10−3.\nWe observe that for larger α, the particles advance faster towards the leftmost circle\nin the beginning, and accordingly, the MMD decreases the fastest; see also Figure 2. On\nthe other hand, for too large α, there are more outliers (points that are far away from the\nrings) at the beginning of the flow. Even at t= 50, some of them remain between the rings,\nwhich is in contrast to our observations for smaller α. This is also reflected by the fact that\nthe MMD and W2values plateau for these values of αbefore they finally converge to zero.\nThe “sweet spot” for αseems to be α∈[3,4]since the MMD and W2loss drop below 10−9\nthe fastest for these values. For the first few iterations, the MMD values are monotone\nwith respect to α: the lower curve is the one belonging to α= 7.5, the one above it belongs\ntoα= 5, and the top one belongs to α= 1. For the last time steps, this order is nearly\nidentical. We also observed that for even larger values of α, e.g., α∈ {10,50,100,500}the\nflow behaves even worse in the sense that the plateau phases becomes longer, i.e., both the\nW2and the squared MMD loss converge to 0 even slower.\nNeal’s cross target Inspired by [73, Fig. 1f], the target νcomprises four identical\nversions of Neal’s funnel, each rotated by 90 degrees about the origin, see Figure 3. We\ngenerate the samples {(x1,k, x2,k)}N\nk=1of Neal’s funnel by drawing normally distributed\n∗The code is available at https://github.com/ViktorAJStein/Regularized_f_Divergence_Particle_\nFlows , where every entropy function from Table 1 and many kernels are implemented.\n271\n 3\n 7.5\nt=0\n t=0.1\n t=0.2\n t=1\n t=10\n t=50\nFigure 1: Discretized Wasserstein gradient flow of the regularized Tsallis- αdivergence Dλ\nfα,ν\nforα∈ {1,3,7.5}, where νare the three rings.\nFigure 2: Comparison of squared MMD ( left) and W2distance ( right) along the flow for\ndifferent α, where νis the three circles target. Note the logarithmic axis.\nt=0\n t=0.1\n t=1\n t=10\n t=50\nFigure 3: Discretized Wasserstein gradient flow of the regularized Tsallis- αdivergence\nDλ\nf7.5,ν, where νis Neals cross.\n28Figure 4: Comparison of the flow for Dλ\nfα,ν, where νis Neal’s cross. We depict the squared\nMMD (left)and Wasserstein distance (right)toνfor different α.\nsamples x2,k∼ N(7.5,2)andx1,k∼ N(0, e1\n3x2,k), where N(m, s2). For our simulations, we\nchoose α= 7.5,λ= 10−2,τ= 10−3andσ2= 0.25.\nWe observe that the particles of the flow (blue) are mostly pushed toward the regions\nwith a high density of target particles (orange) and that the low-density regions at the ends\nofthefunnelarenotmatchedexactly. Inpractice, weoftenassumethattheempiricaltarget\nmeasure νis obtained by drawing samples from some underlying non-discrete distribution.\nHence, this behavior is actually acceptable. Figure 4 shows that for α= 1the gradient\nflow with respect to Dλ\nfα,νrecovers the target slower, both in terms of the MMD or the\nWasserstein metric and that α= 7.5performs best.\nBananas target The set-up of this last experiment is inspired by Aude Genevay’s\ntalk “Learning with Sinkhorn divergences: from optimal transport to MMD”†, see Figure\n5. The target νis multimodal and the “connected components” of its support are far apart.\nAdditionally, the initial measure is not chosen in a manner that takes the properties of\nsupp( ν)into account. Still, we observe the convergence of the particles to the target ν. We\nalso observe a mode-seeking behavior: the particles concentrate near the mean of the right\n“banana” first, which is even more visible for the left “banana” at later time points.\nSince the parameter λpenalizes the disjoint support condition, we choose λto be higher\nthan for the other targets, namely λ= 1, so that the particles are encouraged to “jump”\nfrom one mode to another. We further choose τ= 10−1,σ= 2×10−2andα= 3for the\nsimulations. Empirically, we observed that the value of αmakes no difference though.\nFinite recession constant and annealing Lastly, in Figures 6 and 7, we show that\nthe three rings target is also well recovered for entropy functions with finite recession con-\nstant. To illustrate the convergence of our procedure, we first choose the total variation\n†Talk given at MIFODS Workshop on Learning with Complex Structure 2020, see https://youtu.be/\nTFdIJib_zEA?si=B3fsQkfmjea2HCA5 .\n29t=0\n t=10\n t=100\nt=1000\n t=10000\n t=1000000\nFigure 5: Discretized Wasserstein gradient flow of the regularized Tsallis-3 divergence Dλ\nf3,ν,\nwhere νis the bananas target.\n30t=0\n t=1\n t=10\n t=50\n t=100\nFigure 6: Discretized Wasserstein gradient flow of the regularized TV divergence Dλ\nfTV,ν,\nwhere νis three rings target.\nt=0\n t=1\n t=10\n t=50\n t=100\nFigure7: DiscretizedWassersteingradientflowoftheregularized1\n2-Tsallisdivergence Dλ\nf1\n2,ν\nwithout annealing (top) and with annealing (bottom), where νis three rings target.\nentropy function, τ= 10−3and the kernel width σ= 5×10−2. In order for Lemma 24 to\napply, we choose λ= 2.25. As a second example, we choose the1\n2-Tsallis entropy function,\nτ= 10−3and the kernel width σ= 5×10−2as well as λ= 2.25(so that Lemma 24 applies).\nSince smaller λleads to better target recovery, we use an annealing heuristic (similar\nto [15]): Ultimately, we want to sample for the target distribution. Hence, we reduce the\nvalue of λalong the flow. This is justified as follows. For the Wasserstein gradient flow\n(γt)twith respect to Dλ\nf,νfor a fixed λ, we expect that Dλ\nf(γt, ν)→0ast→ ∞. Then,\nby Corollary 12(3), the same holds for dK(γt, ν), see also Figures 2 and 4 for empirical\nevidence. Based on Lemma 24, we can thus reduce the value of λalong the flow while\nmaintaining the finite-dimensional minimization problem. In this last example, we divide\nthe value of λby five three times, namely at t∈ {5,10,20}. We observe that this indeed\nimproves the convergence.\n7 Conclusions and Limitations\nWe considered interpolations between f-divergences and squared MMDs with characteristic\nkernels. For these interpolations, we have proven that they are M-convex along generalized\n31geodesics, and calculated their gradients. This allowed us to establish the existence and\nuniqueness of the associated Wasserstein gradient flows. Proving the empirically observed\nexponential convergence rate under reasonable assumptions is subject to future work.\nWhen considering particle flows for f-divergences with infinite recession constant or if\nthe regularization parameter is chosen in a certain way, the Euler forward scheme reduces\nto solving a finite-dimensional, strongly convex optimization problem in every iteration.\nFurther, we would like to extend this paper’s theory to non-differentiable kernels such\nas the Laplace kernel, i.e., the Matérn-1\n2kernel, and to non-bounded kernels like Coulomb\nkernels or Riesz kernels, which are of interest, e.g., in generative modeling [26, 27] and\nimage dithering [20].\nRecently, we became aware of the nice paper [16], where the authors considered several\ninfimal convolution functionals with two or three summands as smoother loss functions in\ngenerative adversarial networks. In particular, the infimal convolution of the MMD arising\nfrom the Gaussian kernel and convex functions appears to be a promising loss function.\nPotentially, our results can contribute in this direction, which is also in the spirit of [9].\nAcknowledgments\nV.S. and G.S. gratefully acknowledge funding from the BMBF project “VI-Screen” with\nnumber 13N15754. V.S. extends his gratitude to Jean-François Bercher for making the\npaper [69] available to him.\nReferences\n[1] R. Agrawal and T. Horel. Optimal bounds between f-divergences and Integral Prob-\nability Metrics. J. Mach. Learn. Res. , 22(1), jan 2021.\n[2] S.M.AliandS.D.Silvey. Ageneralclassofcoefficientsofdivergenceofonedistribution\nfrom another. J. R. Stat. Soc. Ser. B. Stat. Methodol. , 28(1):131–142, 1966.\n[3] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discon-\ntinuity problems . Oxford Mathematical Monographs. Oxford University Press, 2000.\n[4] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the\nspace of probability measures . Springer Science & Business Media, 2 edition, 2008.\n[5] A. Andrle, N. Farchmin, P. Hagemann, S. Heidenreich, V. Soltwisch, and G. Steidl.\nInvertible neural networks versus MCMC for posterior reconstruction in grazing in-\ncidence X-ray fluorescence. In International Conference on Scale Space and Varia-\ntional Methods in Computer Vision , pages 528–539, Virtual event, May 16—20 2021.\nSpringer.\n32[6] M. Arbel, L. Zhou, and A. Gretton. Generalized energy based models. In International\nConference on Learning Representations , Virtual event, May 3–7 2021.\n[7] H. Bauschke and P. Combettes. Convex Analysis and Monotone Operator Theory in\nHilbert Spaces . CMS Books in Mathematics. Springer New York, 2011.\n[8] F. Beier, J. von Lindheim, S. Neumayer, and G. Steidl. Unbalanced multi-marginal\noptimal transport. J. Math. Imaging Vision , 65(3):394–413, 2023.\n[9] J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis, and L. Rey-Bellet. (f,Γ)-\ndivergences: Interpolating between f-divergences and integral probability metrics. J.\nMach. Learn. Res. , 23(39):1–70, 2022.\n[10] D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for\nstatisticians. J. Amer. Statist. Assoc. , 112(518):859–877, 2017.\n[11] D. E. Boekee. A generalization of the Fisher information measure . PhD thesis, Delft\nUniversity, 1977.\n[12] K. M. Borgwardt, A. Gretton, M. J. Rasch, H.-P. Kriegel, B. Schölkopf, and A. J.\nSmola. Integrating structured biological data by kernel maximum mean discrepancy.\nBioinformatics , 22(14):e49–e57, 07 2006.\n[13] J. M. Borwein and A. S. Lewis. Partially-finite programming in L1and the existence\nof maximum entropy estimates. SIAM J. Optim. , 3(2):248–267, 1993.\n[14] A. Braides. A Handbook of Γ-convergence. In M. Chipot, editor, Handbook of Differ-\nential Equations: Stationary Partial Differential Equations , volume 3, pages 101–213.\nElsevier, 2006.\n[15] L. Chizat. Mean-field Langevin dynamics: Exponential convergence and annealing. J.\nMach. Learn. Res. , 2022, 2022.\n[16] C. Chu, K. Minami, and K. Fukumizu. Smoothness and stability in GANs. In In-\nternational Conference on Learning Representations , Virtual event, Apr 26 – May 1\n2020.\n[17] I. Csiszár. Eine informationstheoretische Ungleichung und ihre Anwendung auf den\nBeweis der Ergodizität von Markoffschen Ketten. Magyar. Tud. Akad. Mat. Kutato\nInt. Kozl. , 8:85–108, 1964.\n[18] I. Csiszár and J. Fischer. Informationsentfernungen im Raum der Wahrschein-\nlichkeitsverteilungen. Magyar. Tud. Akad. Mat. Kutato Int. Kozl. , 7:159–180, 1962.\n[19] F. Cucker and D. Zhou. Learning Theory: An Approximation Theory Viewpoint . Cam-\nbridge University Press, 2007.\n33[20] M.Ehler,M.Gräf,S.Neumayer,andG.Steidl. Curvebasedapproximationofmeasures\non manifolds by discrepancy minimization. Found. Comput. Math. , 21(6):1595–1642,\n2021.\n[21] P. Glaser, M. Arbel, and A. Gretton. KALE flow: A relaxed KL gradient flow for\nprobabilities with disjoint support. In Advances in Neural Information Processing\nSystems, volume 34, pages 8018–8031, Virtual event, 6–14 Dec 2021.\n[22] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair,\nA. Courville, and Y. Bengio. Generative adversarial nets. In Advances in Neural\nInformation Processing Systems , volume 27, Montréal, Canada, 8–13 Dec 2014.\n[23] A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. Smola. A kernel\ntwo-sample test. J. Mach. Learn. Res. , 13(25):723–773, 2012.\n[24] H. Gu, P. Birmpa, Y. Pantazis, L. Rey-Bellet, and M. A. Katsoulakis. Lipschitz regu-\nlarized gradient flows and latent generative particles. arXiv preprint arXiv:2210.17230,\n2022.\n[25] E. Hellinger. Neue Begründung der Theorie quadratischer Formen von unendlichvielen\nVeränderlichen. J. Reine Angew. Math. , 1909(136):210–271, 1909.\n[26] J. Hertrich, M. Gräf, R. Beinert, and G. Steidl. Wasserstein steepest descent flows of\ndiscrepancies with Riesz kernels. J. Math. Anal. Appl. , 531(1):127829, 2024.\n[27] J. Hertrich, C. Wald, F. Altekrüger, and P. Hagemann. Generative sliced MMD flows\nwith Riesz kernels. In International Conference on Learning Representations (ICLR) ,\nVienna, Austria, 7 – 11 May 2024.\n[28] H. Jeffreys. An invariant form for the prior probability in estimation problems. Proc.\nR. Soc., Lond., Ser. A , 186(1007):453–461, 1946.\n[29] H. Jeffreys. Theory of Probability . Oxford, at the Clarendon Press, 2 edition, 1948.\n[30] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to\nvariational methods for graphical models. Mach. Learn. , 37:183–233, 1999.\n[31] R. Jordan, D. Kinderlehrer, and F. Otto. The variational formulation of the Fokker–\nPlanck equation. SIAM J. Math. Anal , 29(1):1–17, 1998.\n[32] P.Kafka,F.Österreicher,andI.Vincze. Onpowersof f-divergencesdefiningadistance.\nStudia Sci. Math. Hungar. , 26(4):415–422, 1991.\n[33] S. Kesavan. Functional Analysis , volume 52. Springer Nature, 2 edition, 2023.\n34[34] H. Kremer, N. Y., B. Schölkopf, and J.-J. Zhu. Estimation beyond data reweighting:\nkernel methods of moments. In ICML’23: Proceedings of the 40th International Con-\nference on Machine Learning , volume 202, page 17745–17783, Honolulu, Hawaii, USA,\nJuly 23 - 29 2023.\n[35] S. Kullback and R. A. Leibler. On information and sufficiency. Ann. Math. Stat. ,\n22(1):79–86, 1951.\n[36] H. Leclerc, Q. Mérigot, F. Santambrogio, and F. Stra. Lagrangian discretization of\ncrowd motion and linear diffusion. SIAM J. Numer. Anal. , 58(4):2093–2118, 2020.\n[37] M. Liero, A. Mielke, and G. Savaré. Optimal entropy-transport problems and a new\nHellinger–Kantorovich distance between positive measures. Invent. Math. , 211(3):969–\n1117, 12 2017.\n[38] F. Liese and I. Vajda. Convex Statistical Distances . Teubner-Texte der Mathematik,\n1987.\n[39] J. Lin. Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory ,\n37(1):145–151, 1991.\n[40] B. G. Lindsay. Efficiency versus robustness: The case for minimum Hellinger distance\nand related methods. Ann. Statist. , 22(2):1081 – 1114, 1994.\n[41] K. Muandet, K. Fukumizu, B. Sriperumbudur, and B. Schölkopf. Kernel mean embed-\nding of distributions: A review and beyond. Found. Trends Mach. Learn. , 10(1-2):1–\n141, 2017.\n[42] Y. Nesterov. Introductory Lectures on Convex Optimization: A Basic Course , vol-\nume 87 of Applied Optimization . Springer New York, NY, 1 edition, 2003.\n[43] X. Nguyen, M. J. Wainwright, and M. Jordan. Estimating divergence functionals\nand the likelihood ratio by penalized convex risk minimization. In J. Platt, D. Koller,\nY. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems ,\nvolume 20. Curran Associates, Inc., 2007.\n[44] F. Nielsen and R.Nock. On Rényi andTsallis entropies and divergencesfor exponential\nfamilies. arXiv preprint arXiv:1105.3259, 2011.\n[45] F. Österreicher. The construction of least favourable distributions is traceable to a\nminimal perimeter problem. Studia Sci. Math. Hungar. , 17:341–351, 1982.\n[46] F. Österreicher. On a class of perimeter-type distances of probability distributions.\nKybernetika (Prague) , 32:389–393, 1996.\n35[47] F. Österreicher and I. Vajda. A new class of metric divergences on probability spaces\nand its applicability in statistics. Ann. Inst. Statist. Math. , 55(3):639–653, 2003.\n[48] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin,\nN. Gimelshein, L. Antiga, et al. PyTorch: An imperative style, high-performance deep\nlearning library. In Advances in Neural Information Processing Systems , volume 32,\n8–14 Dec 2019.\n[49] Pauli Virtanen et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in\nPython. Nat. Methods , 17:261–272, 2020.\n[50] K. Pearson. On the criterion that a given system of deviations from the probable in\nthe case of a correlated system of variables is such that it can be reasonably supposed\nto have arisen from random sampling. London Edinburgh Philos. Mag. & J. Sci. ,\n50(302):157–175, 1900.\n[51] A.-P. Perkkiö. Conjugates of integral functionals on continuous functions. J. Math.\nAnal. Appl. , 459(1):124–134, 2018.\n[52] G. Plonka, D. Potts, G. Steidl, and M. Tasche. Numerical Fourier Analysis . Applied\nand Numerical Harmonic Analysis. Birkhäuser, second edition, 2023.\n[53] Y. Polyanskiy and Y. Wu. Information theory: From coding to learning. Book draft.\n[54] M. L. Puri and I. Vincze. Measure of information and contiguity. Statist. Probab. Lett. ,\n9(3):223–228, 1990.\n[55] C. E. Rasmussen and C. K. I. Williams. Gaussian processes for Machine Learning ,\nvolume 1. MIT Press, 2006.\n[56] R. T. Rockafellar. Integrals which are convex functionals. Pacific J. Math. , 24(3):525–\n539, 1968.\n[57] R. T. Rockafellar. Integrals which are convex functionals. II. Pacific J. Math. ,\n39(2):439–469, 1971.\n[58] Rémi Flamary et al. POT: Python optimal transport. J. Mach. Learn. Res , 22(78):1–8,\n2021.\n[59] B. Schölkopf, R. Herbrich, and A. J. Smola. A generalized representer theorem. In\nD. Helmbold and B. Williamson, editors, Computational Learning Theory , pages 416–\n426, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg.\n[60] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis . Cambridge\nUniversity Press, fourth edition, 2009.\n36[61] C.-J. Simon-Gabriel and B. Schölkopf. Kernel distribution embeddings: Universal\nkernels, characteristic kernels and kernel metrics on distributions. J. Mach. Learn.\nRes, 19(1):1708–1736, 2018.\n[62] B. Sriperumbudur, K. Fukumizu, and G. Lanckriet. On the relation between univer-\nsality, characteristic kernels and RKHS embedding of measures. In Y. W. Teh and\nM. Titterington, editors, Proceedings of the Thirteenth International Conference on\nArtificial Intelligence and Statistics , volume 9 of Proceedings of Machine Learning Re-\nsearch, pages 773–780, Chia Laguna Resort, Sardinia, Italy, 13–15 May 2010. PMLR.\n[63] I.SteinwartandA.Christmann. Support Vector Machines . SpringerScience&Business\nMedia, 2008.\n[64] I. Steinwart and J. Fasciati-Ziegel. Strictly proper kernel scores and characteristic\nkernels on compact spaces. Appl. Comput. Harmon. Anal. , 51:510–542, 2021.\n[65] T. Séjourné, J. Feydy, F.-X. Vialard, A. Trouvé, and G. Peyré. Sinkhorn divergences\nfor unbalanced optimal transport. arXiv:1910.12958, 2019.\n[66] D. Terjék. Moreau-Yosida f-divergences. In International Conference on Machine\nLearning (ICML) , pages 10214–10224, Virtual event, Jul 18–24 2021. PMLR.\n[67] C. Tsallis. Generalized entropy-based criterion for consistent testing. Phys. Rev. E ,\n58(2):1442, 1998.\n[68] I. Vajda. On the f-divergence and singularity of probability measures. Period. Math.\nHungar., 2(1-4):223–234, 1972.\n[69] I. Vajda. χα-divergence and generalized Fisher information. In J. Kozesnik, editor,\nTransactions of the Sixth Prague Conference on Information Theory, Statistical De-\ncision Functions and Random Processes, Held at Prague, from September 19 to 25,\n1971, pages 873–886. Czechoslovak Academy of Sciences, Academia Publishing House,\n1973.\n[70] T. Vayer and R. Gribonval. Controlling Wasserstein distances by kernel norms with\napplication to compressive statistical learning. J. Mach. Learn. Res , 24(149):1–51,\n2023.\n[71] I. Vincze. On the concept and measure of information contained in an observation. In\nJ. Gani and V. Rohatgi, editors, Contributions to Probability , pages 207–214. Elsevier,\n1981.\n[72] H. Wendland. Scattered Data Approximation . Cambridge University Press, 2004.\n37[73] Z. Xu, N. Chen, and T. Campbell. Mixflows: principled variational inference via mixed\nflows. In ICML’23: Proceedings of the 40th International Conference on Machine\nLearning , page 38342–38376, Honolulu, Hawaii, USA, July 23 - 29 2023.\n38A Entropy Functions and Their f-Divergences\nIn Table 1 and 2, we give an extensive overview on entropy functions ftogether with their\nrecession constants, convex conjugates, and associated f-divergences. Here, ( ⋆) means that\nwe only know f∗in terms of the inverse of the regularized incomplete beta function. Let\nus briefly discuss some cases of the f-divergences below:\n•The Tsallis-2-divergence is also called χ2or Pearson divergence [50]. The Tsallis-1\n2-\ndivergence is equal to the Matusita-1\n2-divergence and equal to half of the squared\nHellinger distance [25]. In the limit α↗1, the Matusita entropy function converges\nto the TV entropy function.\n•Forα→1, both the power and the Tsallis divergence recover the Kullback-Leibler\ndivergence. However, only the power divergence recovers the reverse Kullback-Leibler\ndivergence for α→0, while the Tsallis entropy function converges to 0.\n•The1\n2-Lindsay divergence is called triangular discrimination or Vincze-Le Cam [71,\nEq. (2)] divergence. The Lindsay divergence interpolates between the χ2divergence\nand the reverse χ2divergence, which are recovered in the limits α↘1andα↗0,\nrespectively. In the limit α→1, the perimeter-type divergence recovers the Jensen-\nShannon divergence, and in the limit α→0, it recovers1\n2times the TV-divergence.\nThe special case α=1\n2already appears in [45, p. 342].\n•Forα↘1, Vadja’s χαdivergence becomes the TV-divergence - the only (up to\nmultiplicative factors) f-divergence that is a metric.\n•The Marton divergence plays an essential role in the theory of concentration of mea-\nsures; see [53, Rem. 7.15] and the references therein.\n39Table 1: Entropy functions with recession constants and conjugates.name entropy f(x) f′\n∞ f∗(y)\nTsallis [67],\nα∈(0,1)∪(1,∞)1\nα−1(xα−αx+α−1) ι(0,1)(α)−α\nα−1\u0000α−1\nαy+ 1\u0001α\nα−1\n+−1 +1(0,1)(α)·ι\u0010\n−∞,α\n1−α\u0011(y)\npower divergence,\nα∈R\\ {0,1}[37]1\nα(α−1)(xα−αx+α−1)(\n∞, α ≥1,\n1\n1−α, α < 1\n\n1\nα((α−1)y+ 1)α\nα−1\n+−1\nα,ifα >1,\ngα(y) +ι\u0010\n−∞,1\n1−α\u0011(y),if0< α < 1,\ngα(y) +ι\u0010\n−∞,1\n1−αi(y),ifα <0,,\nwhere gα(y):=1\nα((α−1)y+ 1)α\nα−1−1\nα\nKullback-Leibler\n[35]xln(x)−x+ 1 ∞ ey−1\nJeffreys [28, Eq. 1] (x−1) ln( x) ∞ y+ 2 + W0(e1−y) +1\nW0(e1−y)\nVajda’s χα,\nα >1[68, 69]|x−1|α∞\n\ny+ (α−1)\u0010|y|\nα\u0011α\nα−1,ify≥ −α,\n−1, else.\nequality indicator\n[8, 65]ι{1} ∞ y\nJensen-Shannon\n[39]xln(x)−(x+ 1) ln\u0000x+1\n2\u0001\nln(2) ι(ln(2) ,∞)(y)−ln(2−ey).\nα-Lindsay,\nα∈[0,1)[40](x−1)2\nα+(1−α)x1\n1−α\n\n∞ ify >1\n1−α,\n2−α\n(1−α)2, ify=1\n1−α,\nα(α−1)y−2√\n(α−1)y+1+2\n(α−1)2 else.\nPerimeter-type,\nα∈R\\ {0,1}\n[46, 47]sgn(α)\n1−α\u0010\n(x1\nα+ 1)α−2α−1(x+ 1)\u0011\n\nln(2), α = 1,\n1\n1−α(1−2α−1), α > 0,\n1\n2, α = 0,\n1\n1−α2α−1, α < 0.\n\ny−sgn(α)\n1−α\u0012\nhα(y)α\nα−1−2α−1\u0013\n\u0012\nhα(y)1\nα−1−1\u0013α +sgn(α)\n1−α2α−1, y < f′\n∞,\n1\n1−α(1−2α−1) +ι(−∞,0)(α), y =f′\n∞,\n∞, else,\nwhere hα(y):=1−α\nsgn(α)y+ 2α−1\nBurg [35] −ln(x) +x−1 1 −ln(1−y) +ι(−∞,1)(y)\nSymmetrized Tsallis,\ns∈(0,1)[18]1 +x−(xs+x1−s) 1 (⋆)\nMatusita α∈(0,1)\n[11, Eq. (4.1.30)]\n[29, p. 158]|1−xα|1\nα 1 (y− |y)1\n1−α\u0010\n1−sgn(y)|y|α\n1−α)\u0011−1\nα+ι(−∞,1)(y)\nKafka α∈(0,1]\n[54], [32]|1−x|1\nα(1 +x)α−1\nα 1 (⋆)\ntotal variation\n[8, 65]|x−1| 1(\nmax(−1, y)ify≤1,\n∞ otherwise\nMarton\n[53, Rem. 7.15]max(0 ,1−x)20\n\n∞, ify >0,\n1\n4y2+y,if−2≤y≤0,\n−1 else.\nzero [8, 65] ι(0,∞) 0 ι(−∞,0](y)\n40Table 2: The f-divergences belonging to the entropy functions from Table 1.divergence Df Df(µ|ν), where µ=ρν+µs\nTsallis- α,α∈(0,1)1\nα−1\u0002R\nRdp(x)α−1 dν(x)−α\u0000\nµ(Rd)−ν(Rd)\u0001\u0003\nTsallis- α,α >1(\n1\nα−1\u0002R\nRdρ(x)α−1 dν(x)−α\u0000\nµ(Rd)−ν(Rd)\u0001\u0003\n,ifµs= 0,\n∞, else.\npower divergence, α <1, α̸= 0R\nRdρ(x)α−1\nα(α−1)dν(x) +1\nα−1\u0000\nν(Rd)−µ(Rd)\u0001\npower divergence, α >1(R\nRdρ(x)α−1\nα(α−1)dν(x) +1\nα−1\u0000\nν(Rd)−µ(Rd)\u0001\n,ifµs= 0,\n∞, else.\nKullback-Leibler(R\nRdρ(x) ln\u0000\nρ(x)\u0001\ndν(x)−µ(Rd) +ν(Rd)ifµs= 0,\n∞ else.\nJeffreys(R\nRd\u0000\nρ(x)−1\u0001\nln(ρ(x)) dν(x),ifµs= 0andν({ρ= 0}) = 0 ,\n∞, else\nVajda’s χα,α >1(R\nRd|ρ(x)−1|αdν(x),ifµs= 0,\n∞, else.\nequality indicator(\n0ifµ=ν,\n∞otherwise\nJensen-ShannonR\nRdρ(x) ln\u0000\nρ(x)\u0001\n−\u0000\nρ(x) + 1\u0001\nln\u00001\n2\u0000\nρ(x) + 1\u0001\u0001\ndν(x) + ln(2) µs(Rd).\nα-Lindsay, α∈[0,1)R\nRd\u0000\nρ(x)−1\u00012\nα+(1−α)ρ(x)dν(x) +1\n1−αµs(Rd)\nPerimeter-type, α∈R\\ {0,1}sgn(α)\n1−αhR\nRd\u0010\nρ(x)1\nα+ 1\u0011α\ndν(x)−2α−1\u0000\nµ(Rd) +ν(Rd)\u0001\n+1(0,∞)(α)µs(Rd)i\nBurg(R\nRdln\u0000\nρ(x)\u0001\ndν(x) +µ(Rd)−ν(Rd),ifν({ρ= 0}) = 0 ,\n∞, else..\nSymmetrized Tsallis µ(Rd) +ν(Rd)−R\nRdρ(x)s+ρ(x)1−sdν(x)\nMatusitaR\nRd|1−ρ(x)α|1\nαdν(x) +µs(Rd)\nKafkaR\nRd\f\f1−ρ(x)\f\fα\u0000\n1 +ρ(x)\u0001α−1\nαdν(x) +µs(Rd)\ntotal variationR\nRd|ρ(x)−1|dν(x) +µs(Rd)\nMarton1\n2R\nRd|ρ(x)−1|dν(x) +1\n2\u0000\nν(Rd)−µ(Rd)\u0001\n.\nzero 0\n41B Supplementary Material\nHere, we provide more numerical experiments and give some implementation details.\nB.1 Implementation Details\nTo leverage parallel computing on the GPU, we implemented our model in PyTorch [48].\nFurthermore, we use the POTpackage [58] for calculating W2(µ, ν)2along the flow. Solving\nthe dual (37) turned out to be much more time-consuming than solving the primal problem\n(18), so we exclusively outlined the implementation for the latter. As a sanity check, we\ncalculate the “pseudo-duality gap”, which is the difference between the value of the primal\nobjective at the solution qand the value of the dual objective at the corresponding dual\ncertificate1\nλN[−q,1N]. Then, the relative pseudo-duality gap is computed as the quotient\nof the pseudo-duality gap and the minimum of the absolute value of the involved objective\nvalues. Since the particles get close to the target towards the end of the flow, we use\ndouble precision throughout (although this deteriorates the benefit of GPUs). With this,\nall quantities can still be accurately computed and evaluated.\nB.2 The Kernel Width\nFor the first ablation study, we use the parameters from Section 6, namely α= 5,λ= 10−2,\nτ= 10−3andN= 900. In Figure 8, we see that the kernel width has to be calibrated\ncarefully to get a sensible result. We observe that the width σ2= 10−3is too small. In\nparticular, the repulsion of the particles is too powerful, and they immediately spread out\nbefore moving towards the rings, but only very slowly. If σ2= 10−2, everything works out\nreasonably, and there are no outliers. For σ2= 10, the particles only recover the support\nof the target very loosely.\nThe Matern kernel with smoothness parameter ν=3\n2, see [55, Subsec. 4.2.1], reads\nkσ2,3\n2(x, y) = \n1 +√\n3\nσ2∥x−y∥2!\nexp \n−√\n3\nσ2∥x−y∥2!\n,\nThe shape of the flow for different kernel widths σ2is quite different. In Figure 9, we choose\nα= 3,λ= 10−2,τ= 10−3andN= 900. We can see that when the width σ2= 10−3is\ntoo small, then the particles barely move. The particles only spread on the two rightmost\nrings for σ2= 10−2. For σ2= 10−1, the particles initially spread only onto two rings but\nalso match the third one after some time. The width σ2= 1performs best, and there are\nno outliers. For σ2= 10, the behavior is very similar to the case σ2= 10in Figure 8.\nForσ2= 102, we observe an extreme case of mode-seeking behavior: the particles do not\nspread and only move towards the mean of the target distribution.\n4210−3\n10−2\n10−1\n 1\n 10\nt=0\n t=0.1\n t=1\n t=3\n t=10\n t=25\nFigure 8: Ablation study for the parameter σ2of the inverse multiquadric kernel.10−3\n10−2\n10−1\n 1\n 10\n 100\nt=0.1\n t=1\n t=5\n t=10\n t=50\n t=100\nFigure 9: Ablation study for the parameter σ2of the Matern kernel with ν=3\n2.\n4310−3\n10−2\n10−1\n 1\nt= 0.1\n t= 0.5\n t= 1\n t= 10\n t= 20\n t= 50\nFigure 10: Ablation study for the parameter λ, illustrated for the Jeffreys divergence.\nB.3 The Regularization Parameter λ\nFirst, we investigate the behavior of the flow if only the regularization parameter λvaries.\nFor this prupose, we choose the Jeffreys divergence. In our scheme, we choose τ= 10−3\nandN= 900and the inverse multiquadric kernel with width σ2= 5×10−1. Even though\nthe Jeffreys entropy function is infinite at x= 0, the flow still behaves reasonably well, see\nFigure 10. If λis very small, many particles get stuck in the middle and do not spread\ntowards the funnels. If, however, λis much larger than the step size τ, then the particles\nonly spread out slowly and in a spherical shape. This behavior is also reflected in the\ncorresponding distances to the target measure ν, see Figure 11. In our second example,\nwe use the compactly supported “spline” kernel k(x, y):= (1− ∥x−y∥2)3\n+(3∥x−y∥+ 1),\nthe 3-Tsallis divergence, τ= 10−3,N= 900and the three circles target from before, see\nFigure 12. Overall, the behavior is similar to before.\nTo find the combinations of λandτthat give the best results, we use the two “two\nbananas” target and evaluate each flow after 100 iterations of the forward Euler scheme.\nNote thatτ\nλis the prefactor in front of the gradient term in the update step (40). Hence,\nit influences how much gradient information is taken into consideration when updating\nthe particles. The results are provided in Figure 13. We observed that the behavior for\nλ∈ {102,103}is nearly identical to the behavior for λ= 10.\n44Figure 11: Comparison of the squared MMD ( left) and W2distance ( right) for Figure 10.\nFigure 12: Comparing the squared MMD ( left) and W2distance ( right) for different λ.\n45λ= 10−2\nλ= 10−1\n λ= 1\n λ= 10\nτ= 10−3\nτ= 10−2\nτ= 10−1\nτ= 1\nFigure 13: Here, we use α= 2, the inverse multiquadric kernel with width σ2= 5×10−2,\nandN= 900. We plot the configuration of the particles at t= 100. One should not choose\nτto be larger than λby orders of magnitude.\n46"    },    {        "title": "2402.04622v2.On_rigidity_of_hypersurfaces_with_constant_shifted_curvature_functions_in_hyperbolic_space.pdf",        "content": "arXiv:2402.04622v2  [math.DG]  22 Feb 2024ON RIGIDITY OF HYPERSURFACES WITH CONSTANT SHIFTED\nCURVATURE FUNCTIONS IN HYPERBOLIC SPACE\nWEIMIN SHENG, YINHANG WANG, AND JIE WU\nAbstract. In this paper, we first give some new characterizations of geodesic spheres\ninthehyperbolicspacebytheconditionthathypersurfacehascon stantweightedshifted\nmean curvatures, or constant weighted shifted mean curvature ratio, which generalize\nthe result of Hu-Wei-Zhou [ 25]. Secondly, we investigate several rigidity problems for\nhypersurfaces in the hyperbolic space with constant linear combina tions of weighted\nshifted mean curvatures as well as radially symmetric shifted mean c urvatures. As\napplications, we obtain the rigidity results for hypersurfaces with c onstant linear com-\nbinations of mean curvatures in a general form and constant Gaus s-Bonnet curvature\nLkunder weaker conditions, which extend the work of the third autho r and Xia [ 42].\n1.Introduction\nThe rigidity problem of hypersurfaces with constant curvature fu nctions is a funda-\nmental question in differential geometry. A classical theorem due t o Alexandrov [ 4]\nstates that any closed, embedded hypersurface in Euclidean spac e with constant mean\ncurvature is a round sphere. Alexandrov’s method is based on the m aximum principle\nfor elliptic equations and is now referred to as Alexandrov’s reflectio n method. This\nresult is remarkable in that it requires no assumptions about the top ology of the hyper-\nsurface, which improves previous results due to S¨ uss [ 39] and Hsiung [ 23]. Later, Reilly\n[36] provided a new proof of the Alexandrov’s theorem by using his famo us integral\nformula. Following the work of Reilly [ 36], Ros [37] generalized Alexandrov’s result to\nthe hypersurfaces with constant scalar curvature. Also, by usin g the classical Alexan-\ndrov’s reflection method, Korevaar [ 28] gave another proof to this result and indicated\nthat Alexandrov’s reflection method works as well for hypersurfa ces in the hyperbolic\nspace and the hemisphere. Later, Ros [ 38] extended his result to any constant k-mean\ncurvature, which was also proved by Montiel and Ros [ 34] using a direct integral method\ndue to Heintze-Karcher [ 20]. The argument in Montiel and Ros’s paper [ 34] also applies\nto the hyperbolic space and the hemisphere.\n2020Mathematics Subject Classification. 53C24, 53C42.\nKey words and phrases. Constant shifted curvature, Rigidity, Hyperbolic space, Gauss-B onnet\ncurvature.\n1After the work of Montiel and Ros, lots of extensions appeared on such rigidity topic.\nFor example, Koh [ 26,27] gave a new characterization of spheres in terms of the ratio of\ntwo mean curvatures. Aledo-Al´ ıas-Romero [ 3] extended the result to compact space-like\nhypersurfaces with constant higher order mean curvature in de S itter space. In [ 21],\nHe-Li-Ma-Ge investigated the compact embedded hypersurfaces with constant higher\norder anisotropic mean curvatures. In [ 9], Brendle showed that Alexandrov Theorem\nholds in general warped product manifolds, including the (Anti-)deS itter-Schwarzschild\nmanifolds as a typical example. Brendle and Eichmair [ 10] later extended Brendle’s\nresult to any closed, star-shaped convex hypersurface with con stant higher order mean\ncurvature. For other generalizations, see for instance [ 1,2,6,7,22,31,33] and references\ntherein.\nIn a different direction, the curvature quantity with weight Vappears naturally.\nHereV= coshr, whereris the hyperbolic distance to a fixed point in Hn+1. For\ninstance, the weighted mean curvature integral/integraltext\nΣVH1dµappears naturally in the def-\ninition of the quasi-local mass in Hn+1and the Penrose inequality for asymptotically\nhyperbolic graphs [ 14]. The Alexandrov-Fenchel inequalities with weight Valso hold\nin the hyperbolic space. For example, Brendle-Hung-Wang [ 11] and de Lima-Gir˜ ao [ 15]\nproved an Alexandrov-Fenchel-type inequality for the weighted me an curvature integral/integraltext\nΣVH1dµ. Ge, Wang and the third author [ 18] established an optimal inequality con-\ncerning/integraltext\nΣVHkdµ. See also [ 24,40]. In [41], the third author considered the weighted\nhigher-order mean curvature VHkand gave a new characterization of geodesic spheres\nin the hyperbolic space Hn+1.\nTheorem A ([41]).LetΣbe a closed embedded hypersurface in Hn+1. If either of the\nfollowing conditions holds on Σ:\n(i)VHkis constant for some 1≤k≤n,\n(ii)VHk\nHlis constant for some 0≤l < k≤nandHldose not vanish on Σ,\nthenΣis a centered geodesic sphere.\nThe proof of Theorem Aused the similar argument of Montiel and Ros as in [ 34,38],\nand the Heintze-Karcher-type inequality for mean convex hypers urface in Hn+1(due to\nBrendle) also plays an important role. Recently, Hu-Wei-Zhou [ 25] established a new\nHeintze-Karcher-type inequality for hypersurfaces with mean cu rvatureH > nin the\nhyperbolic space. See ( 2.19) for the precise statement. As applications, they obtained an\nAlexandrov type theorem for closed embedded hypersurfaces wit h constant shifted kth\nmean curvaturein hyperbolic space. For this, we recall thedefinitio n of shifted kthmean\ncurvature. Let Σ be a smooth hypersurface in Hn+1, its shifted principal curvatures are\n2defined by ˜ κ= (˜κ1,···,˜κn) = (κ1−1,···,κn−1), where κ= (κ1,···,κn) are the\nprincipal curvatures of Σ. Then the shifted kth mean curvature Hk(˜κ) of Σ is given by\nthe thenormalized kthelementary symmetric functionof ˜ κ. These definitions arise quite\nnaturally in the setting of horospherically convex (h-convex) geom etry of hypersurfaces\ninHn+1(see [5,16] for instance).\nTheorem B ([25]).LetΣbe a closed embedded hypersurface in Hn+1. If the shifted kth\nmean curvature Hk(˜κ)is constant for some k∈ {1,···,n}, thenΣis a geodesic sphere.\nFirstly, we give some generalizations of the above result for closed h ypersurfaces in\nthe hyperbolic space. We prove the following theorem.\nTheorem 1.1. Assume that χ(s)is a smooth, positive and monotone non-decreasing\nfunction defined on R+. LetΣbe a closed embedded hypersurface in Hn+1. If either of\nthe following conditions holds on Σ:\n(i)χ(V)Hk(˜κ)is constant for some 1≤k≤n,\n(ii)χ(V)Hk(˜κ)\nHl(˜κ)is constant for some 0≤l < k≤nandHl(˜κ)dose not vanish on Σ,\nthenΣis a geodesic sphere. Moreover, if χis strictly increasing, then Σis a centered\ngeodesic sphere.\nIfχ= 1, case ( i) of Theorem 1.1reduces to Hu-Wei-Zhou’s result [ 25]. Our second\nresult is the rigidity for hypersurfaces with constant linear combina tions of weighted\nshifted mean curvatures in Hn+1.\nTheorem 1.2. Letn≥2. Assume that χ(s)is a smooth, positive and monotone non-\ndecreasing function defined on R+. Let0≤k≤nbe an integer and Σbe a closed\nhypersurface in Hn+1with˜κ∈Γ+\nk. If one of the following holds:\n(i) 2≤l < k≤nand there are nonnegative constants {ai}l−1\ni=1and{bj}k\nj=l, at least\none of them not vanishing, such that\nl−1/summationdisplay\ni=1aiHi(˜κ) =k/summationdisplay\nj=lbj(χ(V)Hj(˜κ));\n(ii)there are nonnegative constants a0and{bj}k\nj=1, at least one of them not vanish-\ning, such that\na0=k/summationdisplay\nj=1bj(χ(V)Hj(˜κ));\n3(iii) Σis star-shaped, 1≤l < k≤n−1and there are nonnegative constants {ai}l−1\ni=0\nand{bj}k\nj=l, at least one of them not vanishing, such that\nl−1/summationdisplay\ni=0aiHi(˜κ) =k/summationdisplay\nj=lbj(χ(V)Hj(˜κ)),\nthenΣis a geodesic sphere.\nRemark 1.1. Theorem 1.2contains the case that the weighted shifted mean curvature\nratioχ(V)Hk(˜κ)\nHl(˜κ)are constant for k > l. We notice that the condition of ˜κ∈Γ+\nkis\nsuperfluous in this case since it is implied by the constancy o fχ(V)Hk(˜κ)\nHl(˜κ). Furthermore,\nwe also prove a similar rigidity result with weight χ(V−u)for h-convex hypersurfaces.\nSee Theorem 4.1below.\nTheorem 1.2will be proved by using the classical integral method due to Hsiung [ 23]\nand Reilly [ 36]. The main tools are a family of Newton-Maclaurin inequalities as well as\na New Minkowski type formula, Lemma 2.5.\nFollowing the idea of Theorem 1.2, we investigate the following general formof rigidity\nresult for linear combinations of shifted higher order mean curvatu res.\nTheorem 1.3. Letn≥2. Let0≤k≤nbe an integer and Σbe a closed hypersurface\ninHn+1with˜κ∈Γ+\nk. Letrbe the distance in Hn+1from a fixed point p0. If one of the\nfollowing holds:\n(i) 2≤l < k≤nand there are two families of nonnegative smooth functions\n{ai(r)}l−1\ni=1and{bj(r)}k\nj=l, which are monotone decreasing and monotone increas-\ning respectively, at least one of them not vanishing, such th at\nl−1/summationdisplay\ni=1ai(r)Hi(˜κ) =k/summationdisplay\nj=lbj(r)Hj(˜κ);\n(ii)there are nonnegative smooth monotone decreasing function sa0(r)and monotone\nincreasing functions {bj(r)}k\nj=1, at least one of them not vanishing, such that\n(1.1) a0(r) =k/summationdisplay\nj=1bj(r)Hj(˜κ);\n(iii) Σis star-shaped, 1≤l < k≤n−1and there are two families of nonnegative\nsmooth functions {ai(r)}l−1\ni=0and{bj(r)}k\nj=l, which are monotone decreasing and\nmonotone increasing respectively, at least one of them not v anishing, such that\nl−1/summationdisplay\ni=0ai(r)Hi(˜κ) =k/summationdisplay\nj=lbj(r)Hj(˜κ),\n4thenΣis a geodesic sphere.\nIn fact, the above equation is the equation of Krylov type which has been introduced\nand studied by Krylov in [ 29] in the following form\n(1.2)k−1/summationdisplay\ni=0αi(x)Hi/parenleftbig\nD2u/parenrightbig\n=Hk/parenleftbig\nD2u/parenrightbig\n, x∈Ω⊂Rn,\nwhere Ω is a ( k−1)-convex domain and uis a twice continuously differentiable function.\nIt’s an extension of the work investigated by Caffarelli et al. [ 12,13] on the Hessian\nequation. Krylov observed that if αi(x)≥0 for 0≤i≤k−1, the natural admissible\nconeto make theequation ( 1.2) elliptic is theΓ+\nk-conewhich is thesame asthe k-Hessian\nequation case.\nTheorem 1.3contains the case whereHk(˜κ)\nHl(˜κ)=η(r) for some monotone decreasing\nfunction ηandk > l. This result is the general form of Theorem 1.1(ii).\nNext, we investigate a rigidity result of non-linear form.\nTheorem 1.4. Letn≥2. LetΣbe a closed star-shaped hypersurface with ˜κ∈Γ+\nkand\nrbe the distance in Hn+1from a fixed point p0. If there are two families of nonnegative,\nsmooth, monotone increasing functions {aj(r)}k\nj=1and{bj(r)}k\nj=1, such that\n(1.3)k/summationdisplay\nj=1/parenleftbigg\naj(r)Hj(˜κ)+bj(r)H1(˜κ)Hj−1(˜κ)/parenrightbigg\n=η(r),\nfor some smooth positive radially symmetric function η(r)which is monotone decreasing\ninr, thenΣis a geodesic sphere.\nTheorem 1.4contains two special cases which are worth mentioning: (i) Hk(˜κ) =η(r)\nand(ii)H1(˜κ)Hk−1(˜κ) =η(r), whereη(r)isamonotonedecreasing function. Thesecond\ncase can be seen as a “non-linear” version of Theorem B.\nIn [30, Theorem 3], Kwong, Lee and Pyo proved a rigidity theorem for self- expanding\nsolitons to the weighted generalized inverse curvature flow in Rn+1\n(1.4)d\ndtX=/summationdisplay\n0≤i<j≤nai,j/parenleftbiggHi\nHj/parenrightbigg1\nj−i\nν,\nwhere the weight functions {ai,j(x)|0≤i < j≤n}are non-negative functions on the\nhypersurface satisfying/summationtext\n0≤i<j≤nai,j(x) = 1. We next extend it to the hyperbolic case.\nTheorem 1.5. Let{ai,j(x)|0≤i < j≤n}be non-negativefunctionson the hypersurface\nsatisfying/summationtext\n0≤i<j≤nai,j(x) = 1,k= max{j|ai,j>0for some 0≤i < j≤n}, andΣbe\n5a closed hypersurface with ˜κ∈Γ+\nkinHn+1. If there exists a constant β >0satisfying\n(1.5)/summationdisplay\n0≤i<j≤kai,j/parenleftbiggHi(˜κ)\nHj(˜κ)/parenrightbigg1\nj−i\n=βu\nV−u,\nwhereu=/an}b∇acketle{tsinhr∂r,ν/an}b∇acket∇i}htis the support function, then Σis a geodesic sphere.\nFor the first application of Theorem 1.3is a large class of the rigidity results for\nhypersurfaces with constant linear combinations of Hk.\nCorollary 1.1. Let1≤k≤nbe an integer and Σbe a closed hypersurface in Hn+1\nwith˜κ∈Γ+\nk. If there are nonnegative constants a0and{bj}k\nj=1, at least one of them not\nvanishing, such that\na0+k/summationdisplay\ni=1(−1)i+1bi=k/summationdisplay\nl=1k−l/summationdisplay\nj=0(−1)j/parenleftbiggl+j\nl/parenrightbigg\nbl+jHl(κ),\nthenΣis a geodesic sphere.\nRemark 1.2. The third author and Xia [42]proved the rigidity for k-convex hypersur-\nfaces with constant linear combinations of Hk, i.e.,\na0=k/summationdisplay\nj=1bjHj,\nwhere the coefficients a0andbjare nonnegative. In fact, Corollary 1.1obtains a larger\nclass of such rigidity results including linear combinatio ns ofHkwith negative coeffi-\ncients.\nThe second application of Theorem 1.3is about rigidity problems on some intrinsic\ncurvature functions of induced metric from that of the hyperbolic space. Let (Σ ,g) be a\nhypersurface in Hn+1. Thekth Gauss-Bonnet curvature Lkof the metric gis defined by\n(1.6) Lk(g) :=1\n2kδi1i2···i2k−1i2k\nj1j2···j2k−1j2kRi1i2j1j2···Ri2k−1i2kj2k−1j2k,\nwhereRijklis the Riemannian curvature tensor in the local coordinates with res pect to\nthe metric g, and the generalized Kronecker delta is defined by\nδi1i2···ir\nj1j2···jr= det\nδi1\nj1δi1\nj2···δi1\njr\nδi2\nj1δi2\nj2···δi2\njr............\nδir\nj1δir\nj2···δir\njr\n.\n6ThekthGauss-Bonnetcurvature Lkoftheinducedmetricofahypersurfaceinhyperbolic\nspace can be expressed in terms of the shifted kth mean curvatures (see Lemma 6.2\nbelow). Explicitly,\nLk=/parenleftbiggn\n2k/parenrightbigg\n(2k)!k/summationdisplay\nj=02j/parenleftbiggk\nj/parenrightbigg\nH2k−j(˜κ).\nNotice here all the coefficients are positive. Therefore, as a direct consequence of Theo-\nrem1.3(ii), we have the following\nCorollary 1.2. Let1≤k≤n\n2be an integer and Σbe a closed hypersurface embedded\nin the hyperbolic space Hn+1with˜κ∈Γ+\n2k. If there are nonnegative constants a0and\n{bj}k\nj=1, at least one of them not vanishing, such that\na0=k/summationdisplay\nj=1bjLj,\nthenΣis a geodesic sphere. In particular, if Lkis constant, then Σis a geodesic sphere.\nRemark 1.3. In[42], the third author and Xia proved the rigidity result on Lk, for\nhoroconvex hypersurfaces. Here, a hypersurface in Hn+1is horospherical convex if ˜κ∈\nΓ+n. Corollary 1.2generalizes their result to the hypersurfaces with ˜κ∈Γ+\n2k.\nThe rest of this paper is organized as follows. In Section 2, we first collect some basic\nfacts about the kth normalized elementary symmetric function Hk. Then we provide\nsome Minkowski-type formulas and inequalities, which are the most imp ortant tools of\nthis paper. In Section 3and4, we give the proof of the main results, Theorem 1.1\nand Theorem 1.2. Section 5is devoted to prove the general form of the rigidity results,\nTheorems 1.3–1.5. In Section 6, we focus on the rigidity problem on the intrinsic Gauss-\nBonnet curvatures and show Corollaries 1.1and1.2.\n2.Preliminaries\nIn this section, first let us recall some basic definitions and propert ies of higher order\nmean curvature.\nLetσkbe thekth elementary symmetry function σk:Rn→Rdefined by\nσk(λ) =/summationdisplay\ni1<···<ikλi1···λikforλ= (λ1,···,λn)∈Rn.\nFor a symmetric n×nmatrixA=/parenleftbig\nAj\ni/parenrightbig\n, we set\n(2.1) σk(A) =1\nk!δi1···im\nj1···jmAj1\ni1···Ajm\nim,\n7whereδi1···im\nj1···jmis the generalized Kronecker symbol. If λ(A) = (λ1(A),···,λn(A)) are the\nreal eigenvalues of A, then\nσk(A) =σk(λ(A)).\nThekth Newton transformation is defined as follows\n(Tk)j\ni(A) =∂σk+1\n∂Ai\nj(A) =1\nk!δjj1···jk\nii1···ikAi1\nj1···Aik\njk.\nIfAis a diagonal matrix with λ(A) = (λ1,···,λn), then\n(Tk)j\ni(A) =∂σk+1\n∂λi(λ)δj\ni.\nThe following formulae for the elementary symmetric functions are w ell-known.\nLemma 2.1 ([35]).We have\n(2.2)/summationdisplay\ni,j(Tk−1)j\ni(A)Ai\nj=kσk(A).\n(2.3)/summationdisplay\ni,j(Tk−1)j\ni(A)δi\nj= (n+1−k)σk−1(A).\nThekth G˚ arding cone Γ+\nkis defined by\n(2.4) Γ+\nk={λ∈Rn:σi(λ)>0,1≤i≤k}.\nAnd its closure is denoted by Γ+\nk. A symmetric matrix Ais said to belong to Γ+\nkif its\neigenvalues λ(A)∈Γ+\nk. Let\n(2.5) Hk=σk\nCk\nn,\nbe the normalized kth elementary symmetry function. As a convention, we take H0= 1,\nH−1= 0. We have the following Newton-Maclaurin inequalities.\nLemma 2.2 ([19]).For1≤l < k≤nandλ∈Γ+\nk, the following inequalities hold:\n(2.6) Hk−1(λ)Hl(λ)≥Hk(λ)Hl−1(λ).\n(2.7) Hl(λ)≥Hk(λ)l\nk.\nMoreover, equality holds in ( 2.6) or (2.7) atλif and only if λ=c(1,1,···,1).\nNext, we collect some well-known results on geometry of hypersurf aces in hyperbolic\nspace.\n8In this paper, we view hyperbolic space as the warped product spac eHn+1= [0,∞)×\nSnequipped with the metric\n¯g=dr2+λ(r)2gSn,\nwheregSnis the standard round metric of Snandλ(r) = sinh r. Let Σ be a smooth\nhypersurface in Hn+1. Denote ¯∇and∇as the Levi-Civita connection on Hn+1and Σ,\nrespectively. Let {ei}n\ni=1andνbe an orthonormal basis and the unit outward normal\nof Σ, respectively. Then the induced metric gof Σ isgij= ¯g(ei,ej), and the second\nfundamental form h= (hij) of Σ in Hn+1is given by\nhij=h(ei,ej) = ¯g/parenleftbig¯∇eiν,ej/parenrightbig\n.\nThe principal curvatures κ= (κ1,···,κn) of Σ are the eigenvalues of the Weingarten\nmatrix/parenleftbig\nhj\ni/parenrightbig\n=/parenleftbig\ngjkhki/parenrightbig\n, where ( gij) is the inverse matrix of ( gij). The mean curvature\nof Σ is defined as\nH=gijhij=n/summationdisplay\ni=1κi.\nA hypersurface Σ in Hn+1is called star-shaped if its support function\n(2.8) u=/an}b∇acketle{tλ(r)∂r,ν/an}b∇acket∇i}ht ≥0.\nNote that X=λ(r)∂ris a conformal vector field satisfying\n(2.9) ¯∇X=λ′¯g.\nWe first show that there exists at least one elliptic point such that all principal curva-\nturesκi>1 fori= 1,···,non any closed hypersurface in the hyperbolic space Hn+1,\nby following the argument as Lemma 2.1 in [ 31].\nLemma 2.3. LetΣbe a closed hypersurface in the hyperbolic space Hn+1. Then there\nexists at least one elliptic point xsuch that all principalcurvatures κi>1fori= 1,···,n\nonΣ.\nProof.Let{e1,···,en}be a local orthonormal frame on Σ, and assume that the second\nfundamental form hij=/angbracketleftbig¯∇eiν,ej/angbracketrightbig\nis diagonal with eigenvalues κ1,···,κn. Then\n(2.10) ∇ei∇r=∇ei/parenleftbigg1\nλ(r)λ(r)∂⊤\nr/parenrightbigg\n=−λ′\nλ(∇eir)∂⊤\nr+1\nλ∇ei/parenleftbig\nλ∂⊤\nr/parenrightbig\n.\nIt follows from ( 2.9) that\n(2.11)∇ei/parenleftbig\nλ∂⊤\nr/parenrightbig\n=∇ei(λ∂r−/an}b∇acketle{tλ∂r,ν/an}b∇acket∇i}htν) =/parenleftbig¯∇ei(λ∂r−/an}b∇acketle{tλ∂r,ν/an}b∇acket∇i}htν)/parenrightbig⊤\n=λ′ei−/an}b∇acketle{tλ∂r,ν/an}b∇acket∇i}htκiei.\n9Substituting ( 2.11) into (2.10), we get\n(2.12) ∇ei∇r=−λ′\nλ(∇eir)∂⊤\nr+1\nλ(λ′−/an}b∇acketle{tλ∂r,ν/an}b∇acket∇i}htκi)ei.\nNow we consider at the maximum point xofr, we have ∇r= 0,ν=∂rand∇2r≤0 at\nx. Then from ( 2.12), we get\nκi≥λ′\nλ>1, i= 1,···,n,\ni.e.xis an elliptic point of Σ such that all principal curvatures κi>1 fori= 1,···,n.\n/square\nWe need the following Minkowski type formula in hyperbolic space, whic h is included\nin the proof of [ 24, Lemma 2.6] or [ 25, Lemma 2.3]. For completeness, we involve the\nproof here.\nLemma 2.4. LetΣbe a closed hypersurface in Hn+1. Denote by V= coshrand\nu=/angbracketleftbig¯∇V,ν/angbracketrightbig\n=/an}b∇acketle{tλ∂r,ν/an}b∇acket∇i}ht. Then we have\n(2.13)/integraldisplay\nΣ(V−u)Hk−1(˜κ)dµ=/integraldisplay\nΣuHk(˜κ)dµ, k= 1,···,n.\nProof.It follows from the Gauss-Weingarten formula and ( 2.9) that\n(2.14) ∇i∇jV=/angbracketleftbig¯∇i(λ∂r),ej/angbracketrightbig\n−uhij=Vgij−uhij.\nDenote˜h= (hi\nj−δi\nj)n×nand note that Hk(˜κ) =Hk(˜h). Multiplying ( 2.14) by the kth\nNewton transform tensor ( Tk−1)j\ni(˜h) and summing over i,j, we obtain\n(2.15)n/summationdisplay\ni,j=1(Tk−1)j\ni(˜h)∇i∇jV=n/summationdisplay\ni,j=1(Tk−1)j\ni(˜h)/parenleftbig\nVδi\nj−uhi\nj/parenrightbig\n=n/summationdisplay\ni,j=1(Tk−1)j\ni(˜h)/bracketleftbig\n(V−u)δi\nj−u/parenleftbig\nhi\nj−δi\nj/parenrightbig/bracketrightbig\n=kCk\nn((V−u)Hk−1(˜κ)−uHk(˜κ)).\nwhere in the last equality we used ( 2.2) and (2.3). Notice that ˜hij=hij−gijis a Codazzi\ntensor, i.e., ∇ℓ˜hijis symmetric in i,j,ℓ. It follows that ( Tk−1)j\ni(˜h) is divergence free.\nThe equation ( 2.13) follows from integration by parts and the divergence free proper ty\nof (Tk−1)j\ni(˜h). /square\nFor later purpose to prove the rigidity result on weighted shifted cu rvature functions,\nwe need to extend the above lemma to the following type.\n10Lemma 2.5. Assume that χ(s)is a smooth function defined on R+. LetΣbe a closed\nhypersurface in Hn+1. We have\n(2.16)/integraldisplay\nΣχ(V)uHk(˜κ)dµ=/integraldisplay\nΣχ(V)(V−u)Hk−1(˜κ)dµ+1\nkCk\nn/integraldisplay\nΣχ′(V)(Tk−1)j\ni(˜h)∇iV∇jVdµ,\nwhere˜h= (hi\nj−δi\nj)n×n. Moreover, if χis non-decreasing and ˜κ∈Γ+\nk, then we have\n(2.17)/integraldisplay\nΣχ(V)uHk(˜κ)dµ≥/integraldisplay\nΣχ(V)(V−u)Hk−1(˜κ)dµ.\nProof.From (2.15), we arrive at\n(2.18) ( Tk−1)j\ni(˜h)∇i∇jV=kCk\nn((V−u)Hk−1(˜κ)−uHk(˜κ)).\nMultiplying above equation by the function χ(V) and integrating by parts, one obtains\nthe desired result ( 2.16). Under the assumption that ˜ κ∈Γ+\nk, the (k−1)th Newton\ntensorTk−1is semi-positively definite (see e.g. Guan [ 19]), hence\n(Tk−1)j\ni(˜h)∇iV∇jV≥0.\nTogether with assumption χ′≥0, (2.17) holds. /square\nFinally, we need the Heintze-Karcher-type inequality due to Hu, Wei and Zhou [ 25].\nProposition 2.1 ([25]).LetΩbe a bounded domain with smooth boundary Σ =∂Ω\nin hyperbolic space Hn+1(n≥2). Fix a point o∈Hn+1andV(x) = cosh r(x), where\nr(x) =d(o,x)is the distance to this point o. Assume that the mean curvatur e ofΣ =∂Ω\nsatisfiesH > n, then\n(2.19)/integraldisplay\nΣV−u\nH−ndµ≥n+1\nn/integraldisplay\nΩVdvol,\nwhereu=/angbracketleftbig¯∇V,ν/angbracketrightbig\n=/an}b∇acketle{tsinhr∂r,ν/an}b∇acket∇i}htis the support function of Σandνdenotes the unit\noutward normal of Σ. Equality holds in ( 2.19) if and only if Σis umbilic.\nWeremark that for thecase of n= 1 (the curve case), the inequality ( 2.19) was proved\nby Li and Xu [ 32].\n3.Proof of Theorem 1.1\nAfter all the preparation work, we are ready to prove our main the orems. We start\nwiththeweightedshiftedcurvatureandtheirquotients. TheMinko wski-typeinequalities\nand the Heintze-Karcher-type inequality play important roles.\n11Proof of Theorem 1.1.(i) Fixk∈ {1,···,n}. Sinceχ(V)Hk(˜κ) =cfor some constant\nc, the condition χ >0 and Lemma 2.3imply that c >0, which in turn implies that\nHk(˜κ)>0 on Σ. Then by the result of G˚ arding [ 17], we have ˜ κ∈Γ+\nk.\nIt follows from ( 2.17) that\n(3.1) χ(V)Hk(˜κ)/integraldisplay\nΣudµ=/integraldisplay\nΣχ(V)uHk(˜κ)dµ≥/integraldisplay\nΣχ(V)(V−u)Hk−1(˜κ)dµ.\nBy the Newton-Maclaurin inequality ( 2.7), we have\nHk−1(˜κ)≥Hk(˜κ)k−1\nk,\nthus\n(3.2)/integraldisplay\nΣχ(V)(V−u)Hk−1(˜κ)dµ≥/integraldisplay\nΣχ(V)(V−u)Hk(˜κ)k−1\nkdµ\n= (χ(V)Hk(˜κ))k−1\nk/integraldisplay\nΣχ(V)1\nk(V−u)dµ.\nHence (3.1) and (3.2) imply\n(3.3)/integraldisplay\nΣudµ≥(χ(V)Hk(˜κ))−1\nk/integraldisplay\nΣχ(V)1\nk(V−u)dµ,\nand equality holds in and only if Σ is a geodesic sphere. On the other han d, applying\nProposition 2.1and the Newton-Maclaurin inequality ( 2.7) we derive that\n(3.4)/integraldisplay\nΣudµ= (n+1)/integraldisplay\nΩVdvol≤/integraldisplay\nΣV−u\nH1(˜κ)dµ≤/integraldisplay\nΣV−u\nHk(˜κ)1\nkdµ\n= (χ(V)Hk(˜κ))−1\nk/integraldisplay\nΣχ(V)1\nk(V−u)dµ.\nFinally combining ( 3.3) and (3.4) together, we conclude that the equality holds in the\nNewton-MacLaurin inequality ( 2.7), so Σ is totally umbilical and then is a geodesic\nsphere. Moreover, suppose χ′>0. Since Σ is totally umbilical and χ(V)Hk(˜κ) is a\nconstant, we have Vis a constant. Therefore the distance of each point in Σ to the\norigin is a constant. So we conclude that Σ is a centered geodesic sph ere.\n(ii) The first step is more or less the same as above. Lemma 2.3andχ >0 imply\nthat the ratio χ(V)Hk(˜κ)\nHl(˜κ)is a positive constant. Since by the assumption Hl(˜κ) dose not\nvanish on Σ, Hl(˜κ) andHk(˜κ) are positive everywhere in Σ. From [ 17], we know that\n˜κ∈Γ+\nk.\nIfl= 0, it is reduced to the case of Theorem 1.1(i). In the following, we consider the\ncasel≥1. Denote the positive constant by c, namely,\nc:=χ(V)Hk(˜κ)\nHl(˜κ)>0.\n12Applying the Newton-Maclaurin inequality ( 2.6), we note that\nHk(˜κ)\nHk−1(˜κ)≤Hl(˜κ)\nHl−1(˜κ)\nwhich yields\n(3.5) χ(V)Hk−1(˜κ)\nHl−1(˜κ)≥c.\nIt follows from ( 2.17) and (2.13) that\n/integraldisplay\nΣχ(V)(V−u)Hk−1(˜κ)dµ≤/integraldisplay\nΣχ(V)uHk(˜κ)dµ=c/integraldisplay\nΣuHl(˜κ)dµ=c/integraldisplay\nΣ(V−u)Hl−1(˜κ)dµ.\nThis gives/integraldisplay\nΣ(V−u)(χ(V)Hk−1(˜κ)−cHl−1(˜κ))dµ≤0.\nNote that V−u >0. The above together with ( 3.5) imply\nχ(V)Hk−1(˜κ)\nHl−1(˜κ)=c,\neverywhere in Σ. By an iteration argument, we have\nχ(V)Hk−l(˜κ)\nH0(˜κ)=χ(V)Hk−l(˜κ) =c,\neverywhere in Σ. Finally, from Theorem 1.1(i), we complete the proof. /square\nFor the similar rigidity problem with weight V−u, the horoconvexity is necessary.\nThat isthefollowing theoremwhich isincluded in[ 32, Proposition8.1]and[ 25, Theorem\n1.3]. For the completeness, we involve the proof here.\nTheorem 3.1. Assume that χ(s)is a smooth, positive and monotone non-decreasing\nfunction defined on R+. LetΩbe a smooth bounded and uniformly h-convex (κi≥1+ǫ,\nfor some constant ǫ >0)domain in Hn+1. IfΣ =∂Ωsatisfies\n(3.6)Hk(˜κ)\nHl(˜κ)=χ(V−u),\nwhere0≤l < k≤nandHl(˜κ)dose not vanish on Σ, thenΣis a geodesic sphere.\nMoreover, if χis strictly increasing, then Σis a centered geodesic sphere.\nProof.As the hypersurface Σ is uniformly h-convex, we have ˜ κi>0,Hk(˜κ)>0 and\n∂Hk(˜κ)\n∂˜κi>0. For the case of l= 0, it has been proved in [ 25]. In the following, we only\nfocus on the case l≥1.\n13Applying the Newton-Maclaurin inequality ( 2.6), we note that\nHk(˜κ)\nHk−1(˜κ)≤Hl(˜κ)\nHl−1(˜κ)\nwhich implies\n(3.7)Hk−1(˜κ)\nHl−1(˜κ)≥χ(V−u).\nSimilar to the proof of ( 2.17), one can prove that\n(3.8)/integraldisplay\nΣχ(V−u)uHk(˜κ)dµ≤/integraldisplay\nΣχ(V−u)(V−u)Hk−1(˜κ)dµ,\nwhere we used ∇i(V−u) =−(hj\ni−δj\ni)∇jV. It follows from ( 3.8) and (2.13) that\n/integraldisplay\nΣχ(V−u)(V−u)Hl−1(˜κ)dµ≥/integraldisplay\nΣχ(V−u)uHl(˜κ)dµ=/integraldisplay\nΣuHk(˜κ)dµ=/integraldisplay\nΣ(V−u)Hk−1(˜κ)dµ.\nThis yields/integraldisplay\nΣ(V−u)(χ(V−u)Hl−1(˜κ)−Hk−1(˜κ))dµ≥0.\nNote that V−u >0. The above together with ( 3.7) imply\nHk−1(˜κ)\nHl−1(˜κ)=χ(V−u),\neverywhere in Σ. By an iteration argument, one obtains\nHk−l(˜κ)\nH0(˜κ)=Hk−l(˜κ) =χ(V−u),\neverywhere in Σ. It reduces to the case of Hu-Wei-Zhou’s result [ 25]. We complete the\nproof. /square\n4.Proof of Theorem 1.2\nNext, we show the rigidity result for constant linear combinations of weighted shifted\nmean curvatures in the hyperbolic space. This argument needs pay more attention to\nthe use of the Newton-Maclaurin inequality at the first step.\nProof of Theorem 1.2.(i) By Lemma 2.3and non-vanishing of at least one coefficient,\nwe know that/summationtextl−1\ni=1aiHi(˜κ)>0. Since ˜ κ∈Γ+\nk, we recall from ( 2.6) that\n(4.1) Hi(˜κ)Hj−1(˜κ)≥Hi−1(˜κ)Hj(˜κ),1≤i < j≤k,\n14where all equalities hold if and only if Σ is umbilical. Multiplying ( 4.1) byai,bjandχ\nand summing over iandj, we obtain\n(4.2)l−1/summationdisplay\ni=1aiHi(˜κ)k/summationdisplay\nj=lbj(χ(V)Hj−1(˜κ))≥l−1/summationdisplay\ni=1aiHi−1(˜κ)k/summationdisplay\nj=lbj(χ(V)Hj(˜κ)).\nBy the assumption\nl−1/summationdisplay\ni=1aiHi(˜κ) =k/summationdisplay\nj=lbj(χ(V)Hj(˜κ))>0,2≤l < k≤n,\nwe infer from ( 4.2) that\n(4.3)k/summationdisplay\nj=lbj(χ(V)Hj−1(˜κ))≥l−1/summationdisplay\ni=1aiHi−1(˜κ).\nOn the other hand, we obtain from ( 2.17) that\n(4.4)0 =/integraldisplay\nΣ/parenleftiggk/summationdisplay\nj=lbj(χ(V)Hj(˜κ))−l−1/summationdisplay\ni=1aiHi(˜κ)/parenrightigg\nudµ\n≥/integraldisplay\nΣ/parenleftiggk/summationdisplay\nj=lbj(χ(V)Hj−1(˜κ))−l−1/summationdisplay\ni=1aiHi−1(˜κ)/parenrightigg\n(V−u)dµ≥0.\nHere, the last inequality follows from ( 4.3) andV−u >0. We conclude that the\nequality holds in the Newton-MacLaurin inequality ( 2.6), which implies that Σ is totally\numbilical and thus a geodesic sphere.\n(ii) Making use of ( 2.6) and (2.17), we derive\na0/integraldisplay\nΣudµ=/integraldisplay\nΣu/parenleftiggk/summationdisplay\nj=1bjχ(V)Hj(˜κ)/parenrightigg\ndµ≥/integraldisplay\nΣ(V−u)/parenleftiggk/summationdisplay\nj=1bjχ(V)Hj−1(˜κ)/parenrightigg\nH1(˜κ)\nH1(˜κ)dµ\n≥/integraldisplay\nΣ(V−u)/parenleftiggk/summationdisplay\nj=1bjχ(V)Hj(˜κ)/parenrightigg\n1\nH1(˜κ)dµ=a0/integraldisplay\nΣV−u\nH1(˜κ)dµ\n≥(n+1)a0/integraldisplay\nΩVdvol =a0/integraldisplay\nΣudµ,\nwhere in the last inequality we used Proposition 2.1. Therefore, the equality in both\ncase yields that Σ is a geodesic sphere.\n(iii) The proof is essentially the same as above. One only needs to notic e the slight\ndifference regarding the value of indices. By Lemma 2.3and non-vanishing of at least\none coefficient, we have/summationtextl−1\ni=0aiHi(˜κ)>0. Since ˜ κ∈Γ+\nk, we recall from ( 2.6) that\n(4.5) Hi(˜κ)Hj+1(˜κ)≤Hi+1(˜κ)Hj(˜κ),0≤i < j≤k,\n15where all equalities hold if and only if Σ is umbilical. Multiplying ( 4.5) byai,bjandχ\nand summing over iandj, we get\n(4.6)l−1/summationdisplay\ni=0aiHi(˜κ)k/summationdisplay\nj=lbj(χ(V)Hj+1(˜κ))≤l−1/summationdisplay\ni=0aiHi+1(˜κ)k/summationdisplay\nj=lbj(χ(V)Hj(˜κ)).\nUsing the assumption\nl−1/summationdisplay\ni=0aiHi(˜κ) =k/summationdisplay\nj=lbj(χ(V)Hj(˜κ))>0,1≤l < k≤n−1,\nwe obtain from ( 4.6) that\n(4.7)k/summationdisplay\nj=lbj(χ(V)Hj+1(˜κ))≤l−1/summationdisplay\ni=0aiHi+1(˜κ).\nApplying ( 2.13) and (2.17) again,\n(4.8)0 =/integraldisplay\nΣ/parenleftiggl−1/summationdisplay\ni=0aiHi(˜κ)−k/summationdisplay\nj=lbj(χ(V)Hj(˜κ))/parenrightigg\n(V−u)dµ\n≥/integraldisplay\nΣ/parenleftiggl−1/summationdisplay\ni=0aiHi+1(˜κ)−k/summationdisplay\nj=lbjHj+1(˜κ)/parenrightigg\nudµ≥0.\nHere, the last inequality follows from ( 2.8) and (4.7). We finish the proof by examining\nthe equality case as before. /square\nIn a similar way, one can also prove the rigidity result for the weight V−u. We only\nstate the result here and leave the proof to readers.\nTheorem 4.1. Letn≥2. Assume that χ(s)is a smooth, positive and monotone non-\ndecreasing function defined on R+. LetΣbe a closed uniformly h-convex hypersurface\ninHn+1. If either of the following holds:\n(i) 2≤l < k≤nand there are nonnegative constants {ai}l−1\ni=1and{bj}k\nj=l, at least\none of them not vanishing, such that\nl−1/summationdisplay\ni=1ai(χ(V−u)Hi(˜κ)) =k/summationdisplay\nj=lbjHj(˜κ);\n(ii) Σis star-shaped, 1≤l < k≤n−1and there are nonnegative constants {ai}l−1\ni=0\nand{bj}k\nj=l, at least one of them not vanishing, such that\nl−1/summationdisplay\ni=0ai(χ(V−u)Hi(˜κ)) =k/summationdisplay\nj=lbjHj(˜κ),\n16thenΣis a geodesic sphere. Moreover, if χis strictly increasing, then Σis a centered\ngeodesic sphere.\n5.Proof of Theorems 1.3–1.5\nIn this section, we use the similar idea as in the work of Kwong-Lee-Py o [30]. The\nfollowing formulas will play an essential role in the proof.\nLemma 5.1. Letφbe a smooth function on a closed hypersurface ΣinHn+1. We have\n(5.1)/integraldisplay\nΣφuHk(˜κ)dµ=/integraldisplay\nΣφ(V−u)Hk−1(˜κ)dµ+1\nkCkn/integraldisplay\nΣ(Tk−1)j\ni(˜h)∇iφ∇jVdµ,\nwhere/tildewidehi\nj=hi\nj−δi\nj.\nProof.Similarly, by ( 2.15), we arrive at\n(Tk−1)j\ni(˜h)∇i∇jV=kCk\nn((V−u)Hk−1(˜κ)−uHk(˜κ)).\nMultiplying above equation by the function φand integrating by parts, one obtains the\ndesired result ( 5.1). /square\nProof of Theorem 1.3.(i) It follows from ( 5.1) and ˜κ∈Γ+\nkthat, for each iandj,\n(5.2)/integraldisplay\nΣai(r)((V−u)Hi−1(˜κ)−uHi(˜κ))\n=−1\niCin/integraldisplay\nΣλ(r)a′\ni(r)(Ti−1)q\np(˜h)∇pr∇qr≥0\nand\n(5.3)/integraldisplay\nΣbj(r)((V−u)Hj−1(˜κ)−uHj(˜κ))\n=−1\njCj\nn/integraldisplay\nΣλ(r)b′\nj(r)(Tj−1)q\np(˜h)∇pr∇qr≤0\nSumming ( 5.2) overiand (5.3) overj, and then taking the difference gives\n(5.4)0 =/integraldisplay\nΣ/parenleftiggk/summationdisplay\nj=lbj(r)Hj(˜κ)−l−1/summationdisplay\ni=1ai(r)Hi(˜κ)/parenrightigg\nudµ\n≥/integraldisplay\nΣ/parenleftiggk/summationdisplay\nj=lbj(r)Hj−1(˜κ)−l−1/summationdisplay\ni=1ai(r)Hi−1(˜κ)/parenrightigg\n(V−u)dµ.\nAs in the proof of Theorem 1.2(i), one can obtain the following inequalities:\n(5.5)k/summationdisplay\nj=lbj(r)Hj−1(˜κ)≥l−1/summationdisplay\ni=1ai(r)Hi−1(˜κ).\n17Combining this with ( 5.4), we conclude that all integrands in ( 5.4) are zero. This implies\n(5.5)isanequality andhence Σisageodesicsphere by theNewton-Maclau rin inequality.\n(ii) By dividing ( 1.1) bya0(r), it suffices to prove the result in the case where\n(5.6) 1 =k/summationdisplay\nj=1bj(r)Hj(˜κ).\nFrom (2.6) and (5.1), we have\n/integraldisplay\nΣudµ=/integraldisplay\nΣu/parenleftiggk/summationdisplay\nj=1bj(r)Hj(˜κ)/parenrightigg\ndµ≥/integraldisplay\nΣ(V−u)/parenleftiggk/summationdisplay\nj=1bj(r)Hj−1(˜κ)/parenrightigg\nH1(˜κ)\nH1(˜κ)dµ\n≥/integraldisplay\nΣ(V−u)/parenleftiggk/summationdisplay\nj=1bj(r)Hj(˜κ)/parenrightigg\n1\nH1(˜κ)dµ=/integraldisplay\nΣV−u\nH1(˜κ)dµ\n≥(n+1)/integraldisplay\nΩVdvol =/integraldisplay\nΣudµ\nwhere in the last inequality we used Proposition 2.1. Therefore, the equality in both\ncase yields that Σ is a geodesic sphere.\n(iii) Similarly, as in the proof of Theorem 1.2(iii), one can obtain\n(5.7)k/summationdisplay\nj=lbj(r)Hj+1(˜κ)≤l−1/summationdisplay\ni=0ai(r)Hi+1(˜κ).\nWe infer from ( 5.1) and ˜κ∈Γ+\nkthat, for each iandj,\n(5.8)/integraldisplay\nΣai(r)((V−u)Hi(˜κ)−uHi+1(˜κ))\n=−1\n(i+1)Ci+1n/integraldisplay\nΣλ(r)a′\ni(r)(Ti)q\np(˜h)∇pr∇qr≥0\nand\n(5.9)/integraldisplay\nΣbj(r)((V−u)Hj(˜κ)−uHj+1(˜κ))\n=−1\n(j+1)Cj+1\nn/integraldisplay\nΣλ(r)b′\nj(r)(Tj)q\np(˜h)∇pr∇qr≤0\nSumming ( 5.8) overiand (5.9) overj, and then taking the difference gives\n(5.10)0 =/integraldisplay\nΣ/parenleftiggl−1/summationdisplay\ni=0ai(r)Hi(˜κ)−k/summationdisplay\nj=lbj(r)Hj(˜κ)/parenrightigg\n(V−u)dµ\n≥/integraldisplay\nΣ/parenleftiggl−1/summationdisplay\ni=0ai(r)Hi+1(˜κ)−k/summationdisplay\nj=lbj(r)Hj+1(˜κ)/parenrightigg\nudµ≥0.\n18whereinthelastinequalityweused( 2.8)and(5.7). BytheNewton-Maclaurininequality,\nit implies that Σ is a geodesic sphere. /square\nProof of Theorem 1.4.By dividing ( 1.3) byη(r), it suffices to prove the theorem in the\ncase that\n(5.11)k/summationdisplay\nj=1/parenleftbigg\naj(r)Hj(˜κ)+bj(r)H1(˜κ)Hj−1(˜κ)/parenrightbigg\n= 1.\nByu≥0 and the Newton-Maclaurin inequality ( 2.6), we have\n(5.12)/integraldisplay\nΣudµ=/integraldisplay\nΣu/parenleftiggk/summationdisplay\nj=1(aj(r)Hj(˜κ)+bj(r)H1(˜κ)Hj−1(˜κ))/parenrightigg\ndµ\n=/integraldisplay\nΣk/summationdisplay\nj=1aj(r)uHj(˜κ)dµ+/integraldisplay\nΣk/summationdisplay\nj=1bj(r)uH1(˜κ)Hj−1(˜κ)dµ\n≥/integraldisplay\nΣk/summationdisplay\nj=1aj(r)uHj(˜κ)dµ+/integraldisplay\nΣk/summationdisplay\nj=1bj(r)uHj(˜κ)dµ.\nIt follows from ( 5.1) and ˜κ∈Γ+\nkthat\n(5.13)/integraldisplay\nΣaj(r)((V−u)Hj−1(˜κ)−uHj(˜κ))\n=−1\njCj\nn/integraldisplay\nΣλ(r)a′\nj(r)(Tj−1)q\np(˜h)∇pr∇qr≤0.\nSimilarly,\n(5.14)/integraldisplay\nΣbj(r)((V−u)Hj−1(˜κ)−uHj(˜κ))\n=−1\njCj\nn/integraldisplay\nΣλ(r)b′\nj(r)(Tj−1)q\np(˜h)∇pr∇qr≤0.\nSubstituting ( 5.13) and (5.14) into (5.12), using ( 2.6), we have\n/integraldisplay\nΣudµ≥/integraldisplay\nΣk/summationdisplay\nj=1aj(r)(V−u)Hj−1(˜κ)dµ+/integraldisplay\nΣk/summationdisplay\nj=1bj(r)(V−u)Hj−1(˜κ)dµ\n≥/integraldisplay\nΣk/summationdisplay\nj=1aj(r)(V−u)Hj(˜κ)\nH1(˜κ)dµ+/integraldisplay\nΣk/summationdisplay\nj=1bj(r)(V−u)Hj−1(˜κ)dµ\n=/integraldisplay\nΣV−u\nH1(˜κ)/parenleftiggk/summationdisplay\nj=1(aj(r)Hj(˜κ)+bj(r)H1(˜κ)Hj−1(˜κ))/parenrightigg\ndµ=/integraldisplay\nΣV−u\nH1(˜κ)dµ.\n19On the other hand, Hu-Wei-Zhou’s inequality (Proposition 2.1) is the reverse inequal-\nity/integraldisplay\nΣudµ≤/integraldisplay\nΣV−u\nH1(˜κ)dµ.\nThese two inequalities yield the equality in Hu-Wei-Zhou’s inequality. We c onclude that\nΣ is a geodesic sphere. /square\nProof of Theorem 1.5.The assumption ˜ κ∈Γ+\nksaysHj(˜κ)>0 for all j= 0,···,k. It\nfollows from ( 1.5) thatu >0.\nAssume first that k≥2. By Newton-Maclaurin inequality ( 2.6), we have for 0 ≤i <\nj≤k,\n(5.15)/parenleftbigg1\nH1(˜κ)/parenrightbiggj−i\n≤Hi(˜κ)\nHj(˜κ)=j−1/productdisplay\nm=iHm(˜κ)\nHm+1(˜κ)≤/parenleftbiggHj−1(˜κ)\nHj(˜κ)/parenrightbiggj−i\n.\nTherefore, by Newton-Maclaurin inequality ( 2.6) again,\n(5.16)\nβu\nV−u=/summationdisplay\ni<jai,j/parenleftbiggHi(˜κ)\nHj(˜κ)/parenrightbigg1\nj−i\n≤/summationdisplay\ni<jai,jHj−1(˜κ)\nHj(˜κ)≤/summationdisplay\ni<jai,jHk−1(˜κ)\nHk(˜κ)=Hk−1(˜κ)\nHk(˜κ),\nand\n(5.17) βu\nV−u=/summationdisplay\ni<jai,j/parenleftbiggHi(˜κ)\nHj(˜κ)/parenrightbigg1\nj−i\n≥/summationdisplay\ni<jai,j1\nH1(˜κ)=1\nH1(˜κ).\nThe inequality ( 5.16) gives\nβ/integraldisplay\nΣuHk(˜κ)dµ≤/integraldisplay\nΣ(V−u)Hk−1(˜κ)dµ,\nwhich in turn implies β≤1 by (2.13).\nOn the other hand, ( 5.17) yields\nβ/integraldisplay\nΣuH1(˜κ)dµ≥/integraldisplay\nΣ(V−u)dµ,\nand thus β≥1 again by ( 2.13).\nWe conclude that β= 1 and all the inequalities in ( 5.15) are all equalities. Therefore\nΣ is umbilical and thus is a geodesic sphere.\nWhenk= 1, (5.17) becomes an equality and hence β= 1 by (2.13). By the Newton-\nMaclaurin inequality, we get\n(5.18) H2(˜κ)u\nV−u=H2(˜κ)\nH1(˜κ)≤H1(˜κ)\nH0(˜κ)=H1(˜κ).\n20Integrating this inequality and comparing to the Minkowski type for mula (2.13) for\nk= 2, we again infer that ( 5.18) is an equality and hence Σ is a geodesic sphere. /square\n6.Proof of Corollaries 1.1and1.2\nAs in the proof of Lemma 2.4 in [ 5], we first derive the formula for Hk(˜κ) in terms of\nHi(κ),i= 0,···,k.\nLemma 6.1. For a hypersurface (Σ,g)inHn+1, its shifted mean curvature Hk(˜κ)of the\ninduced metric gcan be expressed as follows\n(6.1) Hk(˜κ) =k/summationdisplay\ni=0(−1)k−i/parenleftbiggk\ni/parenrightbigg\nHi(κ),\nwhere˜κ= (˜κ1,···,˜κn) = (κ1−1,···,κn−1)are the shifted principal curvatures of Σ.\nProof.By the definition of the elementary symmetric polynomials, we have\nn/productdisplay\ni=1(t+ ˜κi) =n/summationdisplay\nk=0σk(˜κ)tn−k.\nOn the other hand,\nn/productdisplay\ni=1(t+ ˜κi) =n/productdisplay\ni=1(t−1+κi) =n/summationdisplay\nl=0σl(κ)(t−1)n−l\n=n/summationdisplay\nl=0σl(κ)n−l/summationdisplay\ni=0/parenleftbiggn−l\ni/parenrightbigg\nti(−1)n−l−i\n=n/summationdisplay\nk=0/parenleftiggk/summationdisplay\ni=0/parenleftbiggn−i\nk−i/parenrightbigg\n(−1)k−iσi(κ)/parenrightigg\ntn−k.\nComparing the coefficients of tn−k, we have\nσk(˜κ) =k/summationdisplay\ni=0/parenleftbiggn−i\nk−i/parenrightbigg\n(−1)k−iσi(κ) =k/summationdisplay\ni=0/parenleftbiggn−i\nk−i/parenrightbigg\n(−1)k−i/parenleftbiggn\ni/parenrightbigg\nHi(κ)\n=/parenleftbiggn\nk/parenrightbiggk/summationdisplay\ni=0(−1)k−i/parenleftbiggk\ni/parenrightbigg\nHi(κ).\nThus\nHk(˜κ) =/parenleftbiggn\nk/parenrightbigg−1\nσk(˜κ) =k/summationdisplay\ni=0(−1)k−i/parenleftbiggk\ni/parenrightbigg\nHi(κ).\n/square\n21Next, we show that the Gauss-Bonnet curvature Lkcan be expressed as a linear\ncombination of the shifted mth mean curvatures with mranging from kto 2k.\nLemma 6.2. For a hypersurface (Σ,g)inHn+1, its Gauss-Bonnet curvature Lkof the\ninduced metric gcan be expressed as follows\n(6.2) Lk=/parenleftbiggn\n2k/parenrightbigg\n(2k)!k/summationdisplay\nj=02j/parenleftbiggk\nj/parenrightbigg\nH2k−j(˜κ).\nProof.The proof of this lemma can be found in [ 24, Lemma 9.1]. For convenience of the\nreaders, we include the proof here. Firstly, we recall the Gauss fo rmula for hypersurfaces\nin hyperbolic space:\n(6.3)Rijsl=/parenleftbig\nhs\nihl\nj−hl\nihs\nj/parenrightbig\n−/parenleftbig\nδs\niδl\nj−δl\niδs\nj/parenrightbig\n=/parenleftig\n˜hs\ni+δs\ni/parenrightig/parenleftig\n˜hl\nj+δl\nj/parenrightig\n−/parenleftig\n˜hl\ni+δl\ni/parenrightig/parenleftig\n˜hs\nj+δs\nj/parenrightig\n−/parenleftbig\nδs\niδl\nj−δl\niδs\nj/parenrightbig\n=/parenleftig\n˜hs\ni˜hl\nj+δs\ni˜hl\nj+˜hs\niδl\nj/parenrightig\n−/parenleftig\n˜hl\ni˜hs\nj+δl\ni˜hs\nj+˜hl\niδs\nj/parenrightig\n,\nwhere˜hj\ni=hj\ni−δj\ni. Substituting ( 6.3) into the definition ( 1.6) ofLkand noting ( 2.1),\nwe have\nLk=δi1i2···i2k−1i2k\nj1j2···j2k−1j2k/parenleftig\n˜hj1\ni1˜hj2\ni2+2˜hj1\ni1δj2\ni2/parenrightig\n···/parenleftig\n˜hj2k−1\ni2k−1˜hj2k\ni2k+2˜hj2k−1\ni2k−1δj2k\ni2k/parenrightig\n=k/summationdisplay\nm=0/parenleftbiggk\nm/parenrightbigg\n2mδi1i2···i2k\nj1j2···j2kδj2\ni2δj4\ni4···δj2m\ni2m˜hj1\ni1˜hj3\ni3···˜hj2m−1\ni2m−1˜hj2m+1\ni2m+1˜hj2m+2\ni2m+2···˜hj2k−1\ni2k−1˜hj2k\ni2k\n=k/summationdisplay\nm=0/parenleftbiggk\nm/parenrightbigg\n2m(n+1−2k)···(n+1−2k+m−1)(2k−m)!/parenleftbiggn\n2k−m/parenrightbigg\nH2k−m(˜κ)\n=/parenleftbiggn\n2k/parenrightbigg\n(2k)!k/summationdisplay\nm=0/parenleftbiggk\nm/parenrightbigg\n2mH2k−m(˜κ).\nHere in the first equality we used the symmetry of the generalized Kr onecker delta, and\nin the third equality we used the basic property of the generalized Kr onecker delta\nδi1i2···ip−1ip\nj1j2···jp−1jpδj1\ni1= (n+1−p)δi2i3···ip\nj2j3···jp.\n/square\nFrom above Lemmas and Theorem 1.3(ii) with constant coefficients, the proof of\nCorollaries 1.1and1.2are apparent.\nAcknowledgements. Sheng waspartiallysupportedbyNationalKeyR&DProgram\nof China(No. 2022YFA1005500) and Natural Science Foundation o f China under Grant\n22No. 12031017 and No. 11971424. Wu was partially supported by Nat ional Key R&D\nProgram of China(No. 2022YFA1005501) and Natural Science Fou ndation of China\nunder Grant No. 11731001.\nReferences\n[1] Luis J. Al´ ıas, Jorge H. S. de Lira, and J. Miguel Malacarne, Constant higher-order mean curvature\nhypersurfaces in Riemannian spaces , Journal of the Institute of Mathematics of Jussieu, 2006,\n5(4),: 527-562.\n[2] Luis J. Al´ ıas, Debora Impera, and Marco Rigoli, Hypersurfaces of constant higher order mean\ncurvature in warped products , Transactions of the American Mathematical Society, 2013, 365(2):\n591-621.\n[3] Juan A. Aledo, Luis J. Al´ ıas, and Alfonso Romero, Integral formulas for compact space-like hy-\npersurfaces in de Sitter space: applications to the case of c onstant higher order mean curvature ,\nJournal of Geometry and Physics, 1999, 31(2-3): 195-208.\n[4] A. D. Alexsandorov, Uniqueness theorem for surfaces in the large , Vestnik Leningrad Univ., 1956,\n11: 5-17.\n[5] B. Andrews, X. Chen and Y. Wei, Volume preserving flow and Alexandrov–Fenchel type inequal ities\nin hyperbolic space , Journal of the European Mathematical Society, 2021, 23(7): 2467-2509.\n[6] C.P. Aquino and H.F. de Lima, On the unicity of complete hypersurfaces immersed in a semi-\nRiemannian warped product , The Journal of Geometric Analysis, 2014, 24: 1126-1143.\n[7] Jo˜ ao Lucas Marques Barbosa and Antˆ onio Gervasio Colares, Stability of hypersurfaces with\nconstant-mean curvature , Annals of Global Analysis and Geometry, 1997, 15(3): 277-297.\n[8] Irl Bivens, Integral formulas and hyperspheres in a simply connected sp ace form , Proceedings of\nthe American Mathematical Society, 1983, 88(1): 113-118.\n[9] S. Brendle, Constant mean curvature surfaces in warped product manifol ds, Publications\nmath´ ematiques de l’IH ´ES, 2013, 117(1): 247-269.\n[10] S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschil d manifold ,\nJournal of Differential Geometry, 2013, 94(3): 387-407.\n[11] S. Brendle, P.-K. Hung, and M.-T. Wang, A Minkowski Inequality for Hypersurfaces in the Anti-de\nSitter-Schwarzschild Manifold , Communications on Pure and Applied Mathematics, 2016, 69(1):\n124-144.\n[12] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic\nequations I: Monge-Amp` ere equation , Communications on Pure and Applied Mathematics, 1984,\n37(3), 369-402.\n[13] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic\nequations, III: Functions of the eigenvalues of the Hessian , Acta Mathematica, 1985, 155(3-4),\n261-301.\n[14] M. Dahl, R. Gicquaud, A. Sakovich, Penrose type inequalities for asymptotically hyperbolic g raphs,\nAnnales Henri Poincar´ e, 2013, 14(5): 1135-1168.\n[15] L.L. de Lima and F. Gir˜ ao, An Alexandrov–Fenchel-type inequality in hyperbolic spac e with an\napplication to a Penrose inequality , Annales Henri Poincar´ e. Cham: Springer International Pub-\nlishing, 2016, 17(4): 979-1002.\n[16] Jos´ e M. Espinar, Jos´ e A. G´ alvez, and Pablo Mira, Hypersurfaces in Hn+1and conformally in-\nvariant equations: the generalized Christoffel and Nirenbe rg problems , Journal of the European\nMathematical Society, 2009, 11(4): 903-939.\n[17] Lars G˚ arding, An inequality for hyperbolic polynomials , Journal of Mathematics and Mechanics,\n1959,8: 957-965.\n23[18] Y.Ge, G. WangandJ.Wu, The GBC mass for asymptotically hyperbolic manifolds , Mathematische\nZeitschrift, 2015, 281: 257-297.\n[19] P. Guan, Topics in geometric fully nonlinear equations , Lecture Notes, http://www. math. mcgill.\nca/guan/notes. html, 2002.\n[20] E. Heintze and H. Karcher, A general comparison theorem with applications to volume es timates\nfor submanifolds , Annales Scientifiques ´Ecole Normale Sup´ erieure. 1978, 11(4): 451-470.\n[21] Y. He, H. Li, H. Ma, and J. Ge, Compact embedded hypersurfaces with constant higher order\nanisotropic mean curvatures , Indiana University Mathematics Journal, 2009, 58(2): 853-868.\n[22] Oussama Hijazi, Sebasti´ an Montiel, and Xiao Zhang, Dirac Operator on Embedded Hypersurfaces ,\nMathematical Research Letters, 2001, 8(2): 195-208.\n[23] C.C. Hsiung, Some integral formulas for closed hypersurfaces , Mathematica Scandinavica, 1954, 2:\n286-294.\n[24] Y.Hu, H.LiandY.Wei, Locally constrained curvature flows and geometric inequali ties in hyperbolic\nspace, Mathematische Annalen, 2022, 382(3-4): 1425-1474.\n[25] Y. Hu, Y. Wei and T. Zhou, A Heintze-Karcher type inequality in hyperbolic space , arXiv:\n2306.14591, 2023.\n[26] Sung-Eun Koh, A characterization of round spheres , Proceedings of the American Mathematical\nSociety, 1998, 126(12): 3657-3660.\n[27] Sung-Eun Koh, Sphere theorem by means of the ratio of mean curvature functi ons, Glasgow Math-\nematical Journal, 2000, 42(1): 91-95.\n[28] Nicholas J. Korevaar, Sphere theorems via Alexandrov for constant Weingarten cur vature hyper-\nsurfaces. Appendix to a note of A. Ros, Journal of Differential Geometry, 1988, 27(2): 221-223.\n[29] N. Krylov, On the general notion of fully nonlinear second-order ellip tic equations , Transactions of\nthe American Mathematical Society, 1995, 347(3), 857-895.\n[30] K.K. Kwong, H. Lee and J. Pyo, Weighted Hsiung-Minkowski formulas and rigidity of umbili cal\nhypersurfaces , Math. Res. Lett., 2018, 25(2): 597-616.\n[31] H. Li, Y. Wei and C. Xiong, A note on Weingarten hypersurfaces in the warped product man ifold,\nInternational Journal of Mathematics, 2014, 25(14): 1450121.\n[32] H. Li and B. Xu, Hyperbolic p-sum and horospherical p-Brunn-Minkowski the ory in hyperbolic case ,\narXiv: 2211.06875, 2022.\n[33] Sebasti´ an Montiel, Uniqueness of spacelike hypersurfaces of constant mean cur vature in foliated\nspacetimes , Mathematische Annalen, 1999, 314(3): 529-553.\n[34] Sebasti´ an Montiel and Antonio Ros, Compact hypersurfaces: the Alexandrov theorem for higher\norder mean curvatures , Pitman Monographs and Surveys in Pure and Applied Mathematics 52\n(1991) (in honor of M.P. do Carmo; edited by B. Lawson and K. Tenen blat), 279-296.\n[35] Robert C. Reilly, On the Hessian of a function and the curvatures of its graph , Michigan Mathe-\nmatical Journal, 1974, 20(4): 373-383.\n[36] Robert C. Reilly, Applications of the Hessian operator in a Riemannian manifo ld, Indiana Univer-\nsity Mathematics Journal, 1977, 26(3): 459-472.\n[37] Antonio Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem ,\nJournal of Differential Geometry, 1988, 27(2): 215-220.\n[38] Antonio Ros, Compact hypersurfaces with constant higher order mean curv atures, Revista\nMatem´ atica Iberoamericana, 1987, 3(3): 447-453.\n[39] W. S¨ uss, ¨Uber Kennzeichnungen der Kugeln und Affinsph¨ aren durch Herr n K.P. Grotemeyer ,\nArchiv der Mathematik, 1952, 3(4): 311-313.\n[40] J. Scheuer and C. Xia, Locally constrained inverse curvature flows , Transactions of the American\nMathematical Society, 2019, 372(10): 6771-6803.\n[41] J. Wu, A new characterization of geodesic spheres in the hyperboli c space, Proceedings of the\nAmerican Mathematical Society, 2016, 144(7): 3077-3084.\n24[42] J. Wu and C. Xia, On rigidity of hypersurfaces with constant curvature funct ions in warped product\nmanifolds , Annals of Global Analysis and Geometry, 2014, 46(1): 1-22.\nWeimin Sheng: School of Mathematical Sciences, Zhejiang Un iversity, Hangzhou\n310058, China.\nEmail address :weimins@zju.edu.cn\nYinhang Wang: School of Mathematical Sciences, Zhejiang Un iversity, Hangzhou\n310058, China.\nEmail address :22035021@zju.edu.cn\nJie Wu: School of Mathematical Sciences, Zhejiang Universi ty, Hangzhou 310058,\nChina.\nEmail address :wujiewj@zju.edu.cn\n25"    },    {        "title": "2402.04639v1.Macroscopic_Magnetic_Dynamics.pdf",        "content": "arXiv:2402.04639v1  [cond-mat.mes-hall]  7 Feb 2024Macroscopic Magnetic Dynamics\nChen Sun∗\nSchool of Physics and Electronics, Hunan University, Chang sha 410082, China.\nWayne M. Saslow†\nTexas A&M University, College Station, Texas, 77843, U.S.A .\n(Dated: February 8, 2024)\nFerromagnetic metals and spin-polarized3He are spin 1/2 systems with the same macroscopic\nsymmetry, and thus should have macroscopic magnetic dynami cs with the same structure. Using\nOnsager’s irreversible thermodynamics, we develop a theor y for these systems that contains two\nrelaxation times (one for the magnetization /vectorMand the other for the spin current /vectorJi), a magnetic\ncompressibility, andamean-fieldparameter. Currentlyspi ntronics dataon metallic ferromagnets are\nanalyzed using a complex decay length from a theory employin g a diffusion constant, a lifetime, and\namean-fieldparameter. The presenttheory leads toacomplex decay lengthwith thesame structure.\nOnneglecting decayof /vectorM, thepresent theoryapplies toliquidsand gases. For macros copic equations\nthe particle statistics is not relevant, so the theory also a pplies to bosons. The theory predicts a\nlongitudinal spin wave whose velocity we estimate for liqui d3He and for paramagnetic metals; but\nsuch a wave should also occur for ferromagnets and for gases.\nPACS numbers:\nI. INTRODUCTION\nThe theory of spin transport in conducting ferromag-\nnetsincludesstudyofhowferromagneticresonancedrives\nspin currents out of a ferromagnet (spin pumping) and\nof how incident spin currents enter and transfer their\nspin to a ferromagnet (spin transfer torque). A two-\nmagnetization model due to Zhang, Levy, and Fert,1,2\nwithacomplexdecaylengthinvolvingexchange-fieldpre-\ncession, magnetization decay, and spin current diffusion,\nhas been applied to these phenomena by Taniguchi3and\nothers to analyze such experiments.\nAssuming such a two-magnetization model we have\nstudied the irreversiblethermodynamics of such systems.\nIn the limit of small magnetization the theory should go\nto that of Leggett’s implementation of Landau’s Fermi\nliquid theory for (paramagnetic) spin-polarized3He,4\nwhich can be done by proper choice of macroscopic pa-\nrameters. However, in that limit we find that the two-\nmagnetization model yields only a single equation of\nmotion, for the magnetization /vectorM.5On the other hand,\nLeggett’s theory, intended to analyze spin-echo experi-\nments for the transverse degrees of freedom, also has an\nequation of motion for the spin current /vectorJi, with a term\nproportional to ∂i/vectorM. In the two-magnetization model,\ntaking the gradient ofthe non-equilibrium magnetization\nto obtain ∂t/vectorJidoes not give a ∂i/vectorMterm. Therefore the\ntheories are not equivalent.\nNevertheless, the decay length has so far only been\nstudied in the two-magnetization model. For systems\nwhere the magnetization is very nearly conserved, such\nasliquidsandgases,onecannotobtainadecaylengthlike\nthat found in the model of Ref. 1. Such a decay length\ncan be obtained in Fermi liquid theory by incorporating a\nfinite magnetization decay time τM.\nThe present work considers an irreversible thermody-namics version of Fermi liquid theory, retaining the same\ngeneral symmetry, but replacing Fermi liquid coefficients\nby other dimensionless parameters. It includes decay of\nboth/vectorJiand/vectorM. Note that, on including /vectorMle, the local\nequilibrium value of /vectorM, the form ∂i(/vectorM−/vectorMle) appears in\nthe Fermiliquid theoryfor ∂t/vectorJi.6This requiresrelaxation\nof/vectorM, either directly orby the much slowerprocess ofdif-\nfusion. We show that for the linearized equations the dc\nwavevector, which gives the decay length l, corresponds\nto oscillation and decay from surfaces, as in Ref. 1.\nThe analysis finds that for the spin current there is a\nvery general term proportional to the gradient of what\ncan be called a vector spin pressure, which would ap-\npear even for non-interacting but spin-polarized systems.\nThat term combines with the decay time τ/vectorJfor/vectorJito give\nan effective diffusion constant. With the decay time τM\nfor/vectorMthe resulting transverse wavevector, given in (42),\nhas the same structure as that in Ref. 1.\nRelated studies for paramagneticconductors, based on\nSilin’s implementation of Landau theory,7–11have been\nmade by Platzman and Wolff,12and by Fredkin and\nWilson.13From the Fermi liquid interaction parameters\nin liquid3He and paramagnetic metals, the longitudinal\nspin wave mode velocity is calculated. This mode is only\nrarely discussed in the literature.\nII. VARIABLES, THERMODYNAMICS,\nSYMMETRIES\nForspecificity wetakegyromagneticratio −γwithγ >\n0, as for electrons. Reversing the sign of γyields the case\nof liquid3He.\nVariables. We write the differential of the scalar en-\nergy density dεin terms of:\n(a) the scalarentropydensity sandits thermodynamic2\nconjugate, the temperature T;\n(b) the spin space vector magnetization /vectorMand its\nthermodynamic conjugate, the effective field/vectorB∗. This\nincludes the applied field /vectorBand an internal field that in\nequilibrium leads to /vectorB∗\neq=/vector0. On setting µ0= 1 we take\n/vectorB∗=/vectorB−1\nχ(/vectorM−/vectorM0), (1)\nso that/vectorMeq=/vectorM0+χ/vectorB; in the absence of anisotropy the\nremanent magnetization /vectorM0will be parallel to /vectorB. We\nmay also write\nδ/vectorM≡/vectorM−/vectorMeq,/vectorB∗=−χ−1(/vectorM−/vectorMeq) =−χ−1δ/vectorM.\n(2)\nFor this approach to be valid /vectorMmust be able to relax to\nequilibrium, so there must be a finite relaxation time τM.\n(c) the mixed tensor (one real space index and one\nspin space index) spin flux /vectorJiand its thermodynamic\nconjugate, /vector vi, both of which are zero in equilibrium.\nBecause /vectorMhas units of γ/planckover2pi1n, where nis a number\ndensity, /vectorJihas units of γ/planckover2pi1nv, wherevis a velocity. We\nalso include a factor of 1/2 for the spin.\nWe write a reactive (i.e., non-dissipative) term\n/vectorJi≈ρM/vector vi, (3)\nwhere for /vector vito have units of velocity, the material pa-\nrameter ρMmust have units of γ/planckover2pi1n. For/vectorJi·/vector vito be\nan energy density we multiply it by 2 m/γ/planckover2pi1, wheremis\nan appropriate particle mass. On including this factor,\nthe thermodynamics will be taken to include the term\n2m\nγ/planckover2pi1/vector vi·d/vectorJi.\nThermodynamics. Using these variables, the ther-\nmodynamics is specified in the differential of the energy\ndensity\ndε=Tds−/vectorB∗·d/vectorM+2m\nγ/planckover2pi1/vector vi·d/vectorJi. (4)\nSymmetries. For the energy to be invariant under\nvirtual rotations of all spin-space vectors by δ/vectorθ, as in\nδ/vectorM=δ/vectorθ×/vectorM, (to be modified when spin-orbit effects\nare present) we have\n−/vectorB∗×/vectorM+2m\nγ/planckover2pi1/vector vi×/vectorJi=/vector0, (5)\nThis implies\n/vector vi·(/vectorJi×/vectorB∗) = 0, /vector vi·(/vectorJi×/vectorM) = 0,(6)\nresults that will simplify some of the calculations to fol-\nlow. Additional relations follow from (5) but they will\nnot be needed.\nIn the absence of dissipation, under time-reversal /vectorB∗\nand/vectorMare odd, and /vector viand/vectorJiare even.III. EQUATIONS OF MOTION\nOnsager’s irreversible thermodynamics involves un-\nknown fluxes (generically, upper case J’s to avoid con-\nfusion with the lower case space index j) and unknown\nsources(generically R’s). However,insteadofthe flux /vectorJij\nwe use/vectorΠij– a spin-space stress tensor. The general ap-\nproach enforces the condition that (4) remains true at all\ntimes, and that the rate of entropy production is always\nnon-negative.\nIn terms of these quantities we take the equations of\nmotion to have the form\n∂tε+∂iJε\ni= 0. (7)\n∂ts+∂iJs\ni=Rs≥0. (8)\n∂t/vectorM+∂i/vectorJi=−γ/vectorM×/vectorB+/vectorRM. (9)\n∂t/vectorJi+∂j/vectorΠij=−γ/vectorJi×/vectorB−γ/vectorJi×λ/vectorM+/vectorR/vectorJ\ni.(10)\nBoth/vectorMand/vectorJiprecess around /vectorB, and we include the\nsymmetry-allowed precession of /vectorJiabout a mean-field\nalong/vectorM, as found in Fermi liquid theory.4,6The coef-\nficientλcan be found from Fermi liquid theory.\nWe now rewrite (4) using all of the above equations of\nmotion. Then TRssatisfies\n0≤TRs=T∂iJs\ni+T∂ts=∂i(TJs\ni)−Js\ni∂iT+∂tε+...\n(11)\nWritten out in detail TRsis a sum of terms that can be\nrewritten in the form of\n(a) a divergence involving fluxes,\n(b) a sum of products of fluxes and their thermodynamic\nforces, and\n(c)asumofproductsofsourcesandtheirthermodynamic\nforces.\nBecause of (6), the precession terms do not contribute to\nTRs.\nIn (4) the term in /vectorB∗·d/vectorMhas sign opposite to the\nother terms, a feature that will appear in the divergence\nterms, theflux-relatedentropyproductionterms, andthe\nsource-related entropy production terms.\n(a) Divergence terms. InTRsthe flux terms sum\nto\n−∂i/bracketleftbig\nJε\ni−TJs\ni+/vectorB∗·/vectorJi−2m\nγ/planckover2pi1/vector vj·/vectorΠji/bracketrightbig\n.(12)\nThis will be set to zero, thus determining Jε\niin terms\nof the other fluxes. These fluxes, in the linear response\nregime, are proportional to the thermodynamic forces\nwith coefficients perhaps associated with the order pa-\nrameter /vectorM. We will not actually need Jε\ni.\n(b) Product terms. InTRsthe terms due to prod-\nucts of fluxes and thermodynamic forces sums to\n−Js\ni∂iT+/vectorJi·∂i/vectorB∗−/vectorΠji·2m\nγ/planckover2pi1∂i/vector vj≥0.(13)\nWe will take the fluxes to be proportional to (as consis-\ntent with symmetry) a sum of thermodynamic forces.3\nReversible flux terms. Time-reversal symmetry\npermits a reversible term /vectorΠijthat is proportional to\n/vectorB∗δijand a reversible term in /vectorJiofρM/vector vi. With the\nremaining parts of /vectorΠjiand/vectorJigiven as /vectorΠ′\njiand/vectorJ′\ni, we\nwrite\n/vectorΠji=/vectorΠ′\nji+α/vectorB∗δij,/vectorJi=/vectorJ′\ni+ρM/vector vi.(14)\nThe reversible terms cannot contribute to the rate of en-\ntropy production; as a consequence, they are related.\nWe now use (14) in (13) to replace /vectorΠjiby/vectorΠ′\njiand/vectorJi\nby/vectorJ′\ni. This gives (13) the terms\nρM/vector vi·∂i/vectorB∗−2m\nγ/planckover2pi1α/vectorB∗·∂i/vector vi\n=∂i[ρM/vector vi·/vectorB∗]−(ρM+2m\nγ/planckover2pi1α)/vectorB∗·∂i/vector vi.(15)\nThe last two terms are proportional to /vectorB∗·∂i/vector vi, which is\nodd under time-reversal. They cannot contribute to the\nrate of entropy production, which is even under time-\nreversal, so they must sum to zero, so\nα=−2m\nγ/planckover2pi1ρM. (16)\nThis is a powerful constraint on the coefficient of /vectorB∗\nthat is independent of the specifics of the system other\nthanα.\n(c) Decay terms. The entropy production terms due\ntoproductsofsourcesandthermodynamicforcessum, on\nusing/vectorB∗·(/vectorM×/vectorB∗) = 0 and (6), to\n/vectorB∗·/vectorRM−2m\nγ/planckover2pi1/vector vi·/vectorRJ\ni≥0, (17)\nwhere/vectorRMand/vectorRJ\nimust be chosento makethe two terms\nequal and non-negative.\nIV. THERMODYNAMIC FLUXES AND\nFORCES\nThe thermodynamic forces are ∂iT,∂i/vectorB∗, and∂i/vector vj.\nRetaining only the diagonal terms, for the dissipative, or\nirreversible, thermodynamic fluxes we have\nJs\ni=−κ\nT∂iT, (18)\n/vectorJ′\ni=DMχ∂i/vectorB∗, (19)\n/vectorΠ′\nji=−D/vectorJρM∂i/vector vj. (20)\nIntheabove,themagneticsusceptibility χandthe“mag-\nnetic compressibility” ρMare non-negative equilibrium\nparameters, and DMandD/vectorJare new diffusion constants\nassociated with /vectorMand/vectorJi. To ensure non-negative en-\ntropy production, the parameters DM,D/vectorJand the ther-\nmal conductivity κmust be non-negative.Retaining only the diagonal terms, for the dissipative\nthermodynamic sources we write\n/vectorRM=−χ\nτM/vectorB∗, (21)\n/vectorRJ\ni=−1\nτ/vectorJ/vectorJi. (22)\nHereτMandτ/vectorJare new relaxation times associated with\n/vectorM, and/vectorJ. To ensure non-negative entropy production\nthese times are non-negative.\nThe right-hand-sides of eqs. (18-22) are zero in equi-\nlibrium, consistent with no flux or source terms in equi-\nlibrium.\nMagnetic Equations of Motion. We now use (14)\nand (16) to write /vectorΠij≈ −δijρM/vectorB∗and use (2) to write\n/vectorB∗=−χ−1δ/vectorM. We also use the collinearity of /vectorM0and\n/vectorBwhen there is no anisotropy. Then, with ∂t/vectorM=∂tδ/vectorM,\nthe equations of motion (9) and (10) read\n∂tδ/vectorM+∂i/vectorJi=−γδ/vectorM×/vectorB−1\nτMδ/vectorM.(23)\n∂t/vectorJi+ρM\nχ∂iδ/vectorM=−γ/vectorJi×(/vectorB+λ/vectorM)−1\nτ/vectorJ/vectorJi.(24)\nClearly the independent variables are δ/vectorMand/vectorJi. Rela-\ntive to the Fermi liquid theory of Ref. 4, Eq. (23) has δ/vectorM\nin place of /vectorM, and the decay term in τM, and Eq. (24)\nhas∂iδ/vectorMin place of ∂i/vectorM.\nWe now consider a time-dependence e−iωt. Then tak-\ning the time derivatives gives a factor of −iω, and we\nobtain\n∂i/vectorJi=−δ/vectorM×/vectorB−/parenleftbigg1\nτM−iω/parenrightbigg\nδ/vectorM. (25)\nρM\nχ∂iδ/vectorM=−γ/vectorJi×(/vectorB+λ/vectorM)−/parenleftbigg1\nτ/vectorJ−iω/parenrightbigg\n/vectorJi.(26)\nParamagnet. We write χin terms of N(0), the total\ndensity of states at the Fermi level:6,7,16\nN(0) =m∗pf\nπ2/planckover2pi13, (27)\nχ= (γ/planckover2pi1\n2)2N(0)\n1+1\n4Z0. (28)\nUse of Leggett’s results4then gives\nρM\nχ=1\n3v2\nF/parenleftbigg\n1+1\n4Z0/parenrightbigg/parenleftbigg\n1+1\n12Z1/parenrightbigg\n,(29)\nλ=1\nγ/planckover2pi11\nN(0)/parenleftbigg\nZ0−1\n3Z1/parenrightbigg\n, (30)\nwherevFistheFermivelocity, and Z0andZ1arethefirst\ntwo coefficients of the Legendre expansion of the Fermi\nliquid interaction. In the absence of such interactions, χ\nis the non-interacting value, ρM/χ=v2\nF/3, andλ= 0.4\nV. STEADY STATE SOLUTIONS\nAmong other things, spintronics requires the steady-\nstate response of a ferromagnet when a spin-polarized\ncurrententersorleavesit. Inpractice, forathin film (our\nconcern) the demagnetization field /vectorBdwill force /vectorMin the\nplane of the film, and the demagnetization field will be\npresent and different for the two transverse directions.14\nHowever, we will neglect this asymmetry.\nRecall that /vectorM=/vectorM0+δ/vectorM, soδ/vectorMand/vectorJiare de-\ntermined by their respective boundary conditions. We\nnow introduce G≡ρM/χ, given for the paramagnet by\n(29), which can be thought of as the spin analog of the\ncompressibility ∂P/∂ρ. Before tensorization of the pa-\nrameters of the theory, the time-independent equations\nof motion become\n∂i/vectorJi=−γδ/vectorM×/vectorB−1\nτMδ/vectorM. (31)\nG∂iδ/vectorM=−γ/vectorJi×(/vectorB+λ/vectorM0)−1\nτ/vectorJ/vectorJi. G≡ρM\nχ,(32)\nLet us now ‘tensorize’ in spin space, using\nG→G↔≡GLˆMˆM+GT(1↔−ˆMˆM).(33)\nTo be specific we take /vectorBand the equilibrium /vectorMto be\nalong ˆzand to be nearly uniform. Then the longitudinal\ncomponents satisfy\n∂iJLi=−1\nτMLδM, (34)\nGL∂iδM=−1\nτJLJLi. (35)\nTaking∂ion (35) and substituting (34) then gives\nGL∇2δM=1\nτJLτMLδM. (36)\nSettingδM,JMi∼ei/vectorkL·/vector r, we find that\nk2\nL=−1\nGLτJLτML. (37)\nRelative to Ref. 1, τMLis the same as τsf, andGLτJL\nis the same as 2 D0, whereD0is a spin diffusion constant.\nThus the effective diffusion constant is a combination of\nGL, associated with energy of nonuniformity (e.g., com-\npressibility), and τJL, associated with decay of /vectorJi.\nFollowing Refs. 1 and 3, for conducting ferromagnets,\nwe neglect /vectorB, so the exchange field dominates.\nTaking/vectorMto be nearly constant, and performing ∂i\non (31) while substituting (32), gives an equation in /vectorMT\nalone:\nGT∇2/vectorMT=1\nτMT[γλ/vectorMT×/vectorM0+1\nτJT/vectorMT].(38)Writing /vectorMT=Mxˆx+Myˆy∼ei/vectorkT·/vector r, and taking /vectorMfor\n/vectorM0, and along ˆ z, gives the two vector components\n(GTk2+1\nτJTτMT)Mx=−γλM\nτMTMy,(39)\n(GTk2+1\nτJTτMT)My= +γλM\nτMTMx.(40)\nCross-multiplication and factoring out MxMyleads to\n(GTk2+1\nτJTτMT)2=−(γλM\nτMT)2,(41)\nso the transverse wavevectors are\nk2\nT=−1\nGTτJTτMT±iγλM\nGTτMT. (42)\nThis has the same structure as the −(1/l)2equation\nfound by Zhang, Levy, and Fert,1,15wherelis the com-\nplex decay length. Thus the present approach, based\non the macroscopic properties of the magnetization /vectorM\nand its current /vectorJi, reproduces results that have been em-\nployed to analyze numerous spintronics experiments.3\nWhenBis included k2\nTbecomes:\nk2\nT=−1\nGTτJTτMT±iγ\nGT/bracketleftBig1\nτMT(B+λM)+1\nτJTB/bracketrightBig\n.\n(43)\nVI. FINITE FREQUENCY MODES\nWe now consider finite frequency modes by taking\nδ/vectorM,/vectorJi∼ei(/vectork·/vector r−ωt). Before tensorizing the parameters,\nthe time-independent equations of motion are given by\n(23) and(24). Toobtainfinite frequencyresults, consider\nthe time-dependence e−iωt, sotakingthe time derivatives\ngives a factor of −iω.\nTherefore the finite frequency modes can be obtained\nby making the following replacements to the decay rates\nin the steady state equations and solutions:\n1\nτM→1\nτM−iω, (44)\n1\nτ/vectorJ→1\nτ/vectorJ−iω. (45)\nPerforming these replacements in (37) gives the longi-\ntudinal wavevectors at finite frequency:\nk2\nL=−1\nGL/parenleftbigg1\nτJL−iω/parenrightbigg/parenleftbigg1\nτML−iω/parenrightbigg\n.(46)\nAt high frequency we find ω=±√GLk, corresponding\nto longitudinal magnetic sound.\nPerforming these replacements in (43) gives the trans-\nverse wavevectors at finite frequency:\nk2\nT=−1\nGT(1\nτJT−iω)(1\nτMT−iω)\n±iγ\nGT/bracketleftBig\n(1\nτMT−iω)(B+λM)+(1\nτJT−iω)B/bracketrightBig\n.(47)5\nAt high frequency we find ω=±√GTk, corresponding\nto transverse magnetic sound.\nVII. LONGITUDINAL SPIN WAVES FOR\nPARAMAGNETS\nThe Fermi liquid interaction coefficients are given in\nterms of dimensionless quantities expressed in terms of\nthe total density of states N(0).At least three dimension-\nless forms exist.\nThe Russian literature employs\nFl=N(0)fl, Z l=N(0)ζl. (48)\nLeggett uses Zl.\nBaym and Pethick employ\nFs\nl=Fl, Fa\nl=Zl\n4. (49)\nThese works emphasize3He.\nThe velocity of longitudinal spin waves vMfor small\nspin polarization, in the collision-dominated regime, is\ngiven by analogy to that for first sound. We express vM\nboth in terms of ZnandFn=Zn/4:\nv2\nM=G=1\n3v2\nF/parenleftbigg\n1+1\n4Z0/parenrightbigg/parenleftbigg\n1+1\n12Z1/parenrightbigg\n,\n=1\n3v2\nF(1+Fa\n0)/parenleftbigg\n1+1\n3Fa\n1/parenrightbigg\n. (50)\nFor liquid3He, we take m∗/m= 1.43 andpF//planckover2pi1=\nkF= 0.76×1010m−1,16which give vF= 112m/s. Then,\ntaking values at pressure P= 0 forZ0/4 andZ1/4,6,17–19\nwefindthat vMrangesfrom21m/sto37m/s. Toinclude\ndamping for liquid3He we may let τ/vectorJ→ ∞.\nPlatzman and Wolff, who expanded on Silin to\nstudy transverse spin resonance in paramagnetic metals,\nemploy12\nBl=fa\nlN(0)2\n2l+1=Fa\nl2\n2l+1=Zl1\n2(2l+1).(51)Eq. (50) also applies to simple metals. To evaluate\nthis for Na and K, we take B0andB1from Refs. 20\nand 21. For Na, we take vF= 1.07×106m/s and find\nvM= 5.8×105m/s. For K, we take vF= 0.86×106\nm/s, and find vM= 4.3×105m/s.\nVIII. SUMMARY AND CONCLUSIONS\nUsing the symmetry of a ferromagnet, which is inde-\npendent of particle statistics, we have applied Onsager’s\nirreversible thermodynamics to study the transverse and\nlongitudinal response of a magnet as a function of fre-\nquency. Two collision times, a spin stiffness, and an\nexchange field enter the theory. For the transverse re-\nsponse, the low frequency wavevectorhas the same struc-\nture as that of the distinct theory of Zhang, Levy, and\nFert, which has been used in previous analyses of spin-\ntronics experiments.\nThe longitudinal mode is similar, but without preces-\nsion effects. From the known Fermi liquid interaction\ncoefficients, and in the absence of damping, both liq-\nuid3He and the alkali metals should support longitu-\ndinal spin waves. To our knowledge, only transverse spin\nwaveshavebeen studied in these systems.20,21Longitudi-\nnal spin waves seem not to have received much attention\nin the literature.\nThe results that do not depend on having a finite\nτMshould also apply to atomic gases, both bosons and\nfermions.22–25\nAcknowledgements\nC.S. wassupportedbyNationalNaturalScienceFoun-\ndation of China under No. 12105094, and by the Funda-\nmental Research Funds for the Central Universities from\nChina.\n∗Electronic address: chensun@hnu.edu.cn\n†Electronic address: wsaslow@tamu.edu\n1S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002), “Mechanisms of Spin-Polarized Current-\nDriven Magnetization Switching”.\n2J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys.\nRev. Lett. 93, 256602 (2004), “Identification of Transverse\nSpin Currents in Noncollinear Magnetic Structures.”\n3T. Taniguchi, S. Yakata, H. Imamura, and Y. Ando, Appl.\nPhys. Express 1, 031302 (2008), “Determination of Pene-\ntration Depthof Transverse SpinCurrentin Ferromagnetic\nMetals by Spin Pumping”.\n4A. J. Leggett, J. Phys. C 3, 448-459 (1970), “Spin diffusionand spin echoes in liquid3He at low temperature”.\n5W. M. Saslow and C. Sun, unpublished.\n6G. Baym and C. Pethick, Chapter 1 of The Physics of Liq-\nuid and Solid Helium, Part II, edited by K. H. Bennemann\nand J. B. Ketterson (Wiley, New York, 1978).\n7L. D. Landau, Zh. Eksp. i Teor. Fiz. 30, 1058 (1956), So-\nviet Phys. JETP 3, 920 (1957), “The Theory of a Fermi\nLiquid”.\n8L. D. Landau, Zh. Eksp. Teor. Fiz. 32, 59 (1957) [Sov.\nPhys. JETP 5, 101 (1957)], “Oscillations in a Fermi Liq-\nuid.”\n9V. P. Silin, Zh. Eksp. i Teor. Fiz. 33, 127 (1957), Soviet\nPhys. JETP 6, 945 (1958), “Oscillations of a Fermi liquid6\nin a magnetic field.”\n10V. P. Silin, Zh. Eksp. i Teor. Fiz. 33, 495 (1957), Soviet\nPhys. JETP 6, 387 (1958), “Theory of a degenerate Fermi\nliquid.”\n11V. P. Silin, Zh. Eksp. i Teor. Fiz. 35, 1243 (1958), Soviet\nPhys. JETP 8, 870 (1958), “Oscillations of a Degenerate\nFermi liquid.”\n12P. M. Platzman and P. A. Wolff, Phys. Rev. Lett. 18, 280\n(1967). “Spin-Wave Excitation in Nonferromagnetic Met-\nals.”\n13A. B. Wilson and D. R. Fredkin, Phys. Rev. B 2, 4556,\n(1970), “Collective Oscillations in a Simple Metal. I. Spin\nWaves.”\n14C. Kittel, Phys. Rev. 73, 155 (1948), “On the Theory of\nFerromagnetic Resonance Absorption”.\n15In Ref. 1 the equation corresponding to k2\nTis−l−2, with\na real part that is equivalent to the inverse square of the\nspin flip length, and an imaginary part involving a term J\nthat is implicitly proportional to M.\n16A. A. Abrikosov and I. M. Khalatnikov, Rpts. Prog. Phys.\n22, 310 (1959), “Theory of a Fermi Liquid.”\n17L. R. Corruccini, D. D. Osheroff, D. M. Lee, and R. C.\nRichardson, Phys. Rev. Lett. 27, 650 (1971). “Spin Diffu-\nsion in Liquid3He: The Effect of Leggett and Rice.”\n18T. A. Alvesalo, T. Haavasoja, and M. T. Manninen, J. Low\nTemp. Phys. 45, 373 (1981). “Specific Heat of Normal andSuperfluid 3He.”\n19D. Candela, N. Masuhara, D. S. Sherell, and D. O. Ed-\nwards, J. Low Temp. Phys. 63, 369 (1986). “Collisionless\nspin waves in normal and superfluid3He.”\n20G. L. Dunifer, D. Pinkel, and S. L. Schultz, Phys. Rev. B\n10, 3159 (1974), “Experimental determintion of the Lan-\ndau Fermi-liquid-theory parameters: Spin waves in sodium\nand potassium.” See p. 3177 for the Fermi liquid coefficient\nvalues for Na and K.\n21D. Pinkel and S. Schultz, Phys. Rev. 8, 6639 (1978).\n“Experimental determination of the Landau Fermi-liquid-\ntheory parameters: Spin waves in rubidium.”\n22K. Miyake, W. J. Mullin, and P. C. E. Stamp, J. Physique\n46, 663-671. “Mean-field and spin-rotation phenomena in\nFermi systems: the relation between the Leggett-Rice and\nLhuillier-Lalo¨ e effects.”\n23M.¨O. Oktel and L. S. Levitov, Phys. Rev. Lett. 88, 230403\n(2002). “Internal Waves and Synchronized Precession in a\nCold Vapor.”\n24J. N. Fuchs, D. M. Gangardt, and F. Lalo¨ e, Phys. Rev.\nLett. 88, 230404 (2002). “Internal State Conversion in Ul-\ntracold Gases.”\n25J.E. Williams, T. Nikuni, and C. W. Clark, Phys. Rev.\nLett. 88, 230405 (2002). “Longitudinal Spin Waves in a\nDilute Bose Gas.”"    },    {        "title": "2402.04642v2.On_the_Mathematical_foundations_of_Diffusion_Monte_Carlo.pdf",        "content": "arXiv:2402.04642v2  [math.ST]  19 Feb 2024On the Mathematical foundations of Diffusion Monte Carlo\nMichel Caffarel1, Pierre Del Moral2and Luc de Montella2,3\n1University of Toulouse, Lab. de Chimie et Physique Quantiqu es, Toulouse , 31062, FR. E-Mail:\ncaffarel@irsamc.ups-tlse.fr\n2Centre de Recherche Inria Bordeaux Sud-Ouest, Talence, 334 05, FR. E-Mail:\npierre.del-moral@inria.fr,luc.de-montella@inria.fr\n2Naval Group, Bouguenais, 44340, FR. E-Mail: luc.demontella@naval-group.com\nFebruary 20, 2024\nAbstract\nThe Diffusion Monte Carlo method with constant number of walkers, a lso called\nStochasticReconfigurationaswellasSequentialMonteCarlo,isaw idelyusedMonte\nCarlo methodology for computing the ground-state energy and wa ve function of\nquantum systems. In this study, we present the first mathematic ally rigorous anal-\nysis of this class of stochastic methods on non necessarily compact state spaces,\nincluding linear diffusions evolving in quadratic absorbing potentials, yie lding what\nseems to be the first result of this type for this class of models. We p resent a\nnovel and general mathematical framework with easily checked Ly apunov stability\nconditions that ensure the uniform-in-time convergence of Diffusio n Monte Carlo\nestimates towards the top of the spectrum of Schr¨ odinger oper ators. For transient\nfree evolutions, we also present a divergence blow up of the estimat es w.r.t. the time\nhorizon even when the asymptotic fluctuation variances are unifor mly bounded. We\nalso illustrate the impact of these results in the context of generaliz ed coupled quan-\ntum harmonic oscillators with non necessarily reversible nor stable diff usive particle\nandaquadraticenergyabsorbingwellassociatedwith asemi-definit epositivematrix\nforce.\n1 Introduction\nThe many-body Schr¨ odinger equation describes interactin g quantum particles. Depend-\ning on the domain of application, these particles may repres ent electrons in solid-state\nphysics or quantum chemistry, nucleons in nuclear physics, atoms in quantum liquid\nphysics, or coupled modes of oscillators in molecular spect roscopy, among the main ap-\nplications. Except for trivial quantum systems, it is impos sible to solve this equation\nanalytically. The diffusion Monte Carlo method (abbreviated DMC) provides a power-\nful stochastic approach to numerically approximate the gro und state energy and wave\nfunction of Schr¨ odinger operators.\n1The DMC methodology has a long and rich history, dating back t o its first mention\nin 1949 by Ulam and Metropolis in [ 1]. The idea was first implemented by Donsker and\nKac [2], and by Kalos [ 3] in the early 1960s. Over the years, the physics community\nhas proposed numerous variants of Diffusion Monte Carlo, know n by various names\nsuch as Green’s function Monte Carlo,[ 3,4] Fixed-Node Diffusion Monte Carlo,[ 5], Pure\nDiffusion Monte Carlo, [ 6,7] Stochastic Reconfiguration Monte Carlo,[ 8,9,10,11] and\nReptation Monte Carlo,[ 12] to cite the main ones. Despite their apparent diversity, al l\nthese approaches are fundamentally based, in one way or anot her, on the stochastic\nsimulation of a specific implementation of the Feynman-Kac f ormula with importance\nsampling.\nFor a more detailed discussion on the origins and the applica tions of these Monte\nCarlo techniques in physics we refer the reader to the recent review article [ 13] as well\nas to [14,15] and references therein.\nThe version of interest employed here is the DMC method with a fixed number of\nwalkers, commonly known in physics as Stochastic Reconfiguration Monte Carlo ; see the\npioneering article by Hetherington [ 8], followed by Sorella and co-authors [ 9,10] and by\nthe first author and his co-workers in [ 11].\nIn mathematics, the methodology may also be referred to by di fferent names, such\nas genetic algorithm with selection and mutation, populati on Monte Carlo or sequential\nMonte Carlo [ 16,17,18,19,20]. For a more thorough discussion on these application\nmodel areas we refer to the books [ 21,22] and references therein.\nThese sequential Monte Carlo methods do not rely on biased va riational techniques.\nThey can be seen as a sophisticated genetic-type Monte Carlo methodology to simulate\ninteracting quantum many-body systems. Various asymptoti c results have been derived,\nincluding central limit theorems and large deviation princ iples, see for instance [ 23,24]\nand [25,26], as well as the books [ 27,21,22] for an overview.\nOur work concerns less studied non-asymptotic and time-uni form problems. Recall-\ning that the estimation of ground state energies relies on th e limiting behavior of the\nwalkers’ evolution in the DMC method, it is therefore crucia l to obtain uniform-in-time\nconvergence estimates. Despite its importance, there is a n otable gap in the literature\nand very few results have been proven in this respect. To the b est of our knowledge,\nsuch uniform controls are mainly valid for compact state spa ce models, see for instance\n[28,21,22] as well as [ 29]. Surprisingly, thetheoretical efficiency of the DMC method has\nnever been verified rigorously even in basic linear-Gaussia n scenarios such as the simple\nand well known harmonic oscillator. In this paper, we addres s this gap by establishing\nthe first uniform-in-time convergence estimates that apply to general state space models\nincluding the coupled harmonic oscillators presented in [ 30].\nOur approach is partly based on recent developments on the st ability of positive\nsemigroups presented in [ 31,32], see also the analysis of generalized coupled harmonic\noscillators presented in [ 30]. In the present article, we provide a natural Lyapunov con-\ndition that ensures the exponential stability of possibly t ime varying positive semigroups\non non necessarily compact state spaces (cf. ( 5) and the local conditions ( 6)). In the\ncontext of time homogeneous positive semigroups, these con ditions ensure the existence\n2of an unique leading eigen-triple (see for instance ( 11)). We underline that these results\ndo not rely on any reversibility-type condition, nor on some spectral theorem. They\ncan be seen as an extended version of Perron-Frobenius and Kr ein-Rutman theorems for\npossibly time varying positive operators.\nWe present a nonlinear Markov chain interpretation of the DM C methodologies. In\nthis interpretation, the genetic type evolution of the walk ers can be seen as a mean field\nparticle simulation of a nonlinear Markov chain (see Sectio n2.4). In this context, we\npresent an auxiliary Lyapunov condition that depends on the potential function and the\nfree evolution of the walkers that ensures the time uniform p erformance of the DMC\nmethodology (cf. condition ( 5), as well as Theorem 1and Corollary 2). We illustrate\nthis condition in the context of generalized coupled harmon ic oscillators for a linear\ndiffusive-type particle and a quadratic energy absorbing wel l associated with a semi-\ndefinite positive matrix force. In this context, we also show that the DMC methodology\nmay diverge when the free evolution of the walkers is unstabl efor any fixed number\nof walkers, even if the asymptotic variance of the Central Limi t Theorem is uniformly\nbounded with respect to the time parameter (see Proposition 1, as well as Section 5.3\nand Proposition 2).\nIn the context of absorbing wells centered at the origin, thi s study leads us to con-\njecture that stable free evolution transitions is a necessa ry and sufficient condition for\nthe DMC method to be uniformly convergent w.r.t. the time hor izon.\nAdditionally, we propose and, to some extent, establish the validity of an importance\nsamplingtransformation to overcome this difficulty. Thisty pe of technique is related but\nnot identical to the use of guiding wave functions in physics to direct the Monte Carlo\nmoves to improve the efficiency of the DMC method [ 33,34]. In contrast with con-\nventional guiding waves techniques our approach is based on conditional free evolutions\ntransitions and survival weight potential functions (cf. S ection2.4and Section 5.2).\nThe rest of the article is organized as follows: In Section 2, we provide a detailed\ndescription of the general framework in which our study is se t, as well as the theoretical\nfoundations on which our proof will be based.\nSection3is devoted to the presentation of our main results. Section 4is mainly\nconcerned the detailed proofs of time-uniform estimates.\nSection5is devoted to the application of our convergence result to ge neralized cou-\npledharmonicoscillators [ 30,35,36]. Thesemodels ariseinvarious fieldssuchasmolecu-\nlarspectroscopy[ 37], quantumoptics[ 38], quantumcryptography[ 39]andphotosynthesis\n[40]. In signal processing, the harmonic oscillator and the DMC methods coincides with\nthe Kalman and the particle filter [ 41,42].\n2 Description of the models\n2.1 Free evolution semigroups\nConsider a Markov chain Xnindexed by n∈Nand taking values in a locally compact\nPolish space ( E,E), where Eis the Borel σ-field onE. LetC(E) be the algebra of\n3continuous measurable functions on E. We also define Cb(E)⊂ C(E) as the sub-algebra\nof bounded measurable continuous functions endowed with th e supremum norm /⌊ard⌊l./⌊ard⌊l.\nWith a slight abuse of notation, we denote by 0 and 1 the null an d unit scalars as well\nas the null and unit functions on Eand we denote by I:x∈E/ma√sto→I(x) =xthe identity\nfunction on E.\nForn∈N∗, we consider the Markov transitions Pnassociated with Xn, and assume\nthat they are Feller; in the sense that for any f∈ Cb(E) we havePn(f)∈ Cb(E), with\nthe function Pn(f) defined for any x∈Eby the integral operator\nPn(f)(x) :=/integraldisplay\nEPn(x,dy)f(y) =E(f(Xn)|Xn−1=x).\nLetC∞(E)⊂ C(E) be the sub-algebra of uniformly positive continuous funct ions\nVthat grow at infinity; that is, for any r≥V⋆:= inf EV >0, ther-sub-level set\nV(r) :={V≤r} ⊂Eis a non-empty compact subset. We further assume that there\nexists aP-Lyapunov function V∈ C∞(E); in the sense that V(E)⊂[1,∞) and there\nexistsǫ∈[0,1) andc∈Rsuch that for any n∈N∗we have\nPn(V)≤ǫV+c. (1)\nLetCV(E)⊂ C(E) bethe sub-space of functions f∈ C(E) such that f/Vis bounded,\nequipped with the norm /⌊ard⌊lf/⌊ard⌊lV:=/⌊ard⌊lf/V/⌊ard⌊l. The Markov semigroup associated with the\nMarkov chain Xnis defined for any f∈ CV(E) by\nPk,n(f)(x) := E(f(Xn)|Xk=x).\nCondition ( 1) ensures that Pk,nisV-Feller in the sense that for f∈ CV(E) we have\nPk,n(f)∈ CV(E). To ensure the semigroup Pk,nis exponentially stable [ 31], we assume\nthe integral operator\nPn(x,dy) =pn(x,y)ν(dy)\nhas a density pnw.r.t. some Radon measure νsatisfying for some r1>0 and for any\nr≥r1the local minorization condition\n0<inf\nn∈N∗inf\nV(r)2pn≤sup\nn∈N∗sup\nV(r)2pn<∞and 0<ν(V(r))<∞. (2)\nTheV-norm semigroup contraction techniques developed in Secti on 8.2 in [ 43] (see\nalso Lemma 2.3 in [ 32] and Theorem 2.2 in [ 31]), ensure that for any µ∈ PV(E), there\nexists some parameters a,b >0 such that for any k≤n, and anyµ1,µ2∈ PV(E) we\nhave\n||µ1Pk,n−µ2Pk,n||V≤ae−b(n−k)||µ1−µ2||V. (3)\nNote that the r.h.s condition in ( 2) is met as soon as Vhas compact sub-level sets\nwith non empty interior and νis a Radon measure with full support; that is νis finite\non compact sets and strictly positive on non-empty open sets . For time-homogeneous\nmodels, the l.h.s. minorization condition is satisfied as so on as (x,y)∈(E◦)2/ma√sto→pn(x,y)\nis a continuous positive function on the interior E◦of the setE.\n42.2 Feynman-Kac semigroups\nWe associate with a sequence of strictly positive functions (Gn)n∈N∈ CV(E)Nthe dis-\ncrete generation Feynman-Kac semigroups\nQk,n(f)(x) =E\nf(Xn)n−1/productdisplay\np=kGp(Xp)|Xk=x\nand/hatwideQk,n(f) :=G−1\nkQk,n(Gnf),\nwithG−1\nk:= 1/Gk. To simplify notation, for k= (n−1) sometimes we write Qnand\n/hatwideQninstead ofQn−1,nand/hatwideQn−1,n. In this notation, we have\nQn(f)(x) =Gn−1(x)Pn(f)(x) and /hatwideQn(f)(x) =Pn(Gnf)(x).\nWe also use the convention Qn,n=Pn,n=Id, the identity operator.\nLetMb(E) be the set of bounded signed measures on E. Also, let P(E)⊂ Mb(E)\nbe the convex subset of probability measures on Eand denote by PV(E) the convex\nset of probability measures µ∈ P(E) such that µ(V)<+∞. The left action of Qnon\nPV(E) is given for any ( η,f)∈(PV(E),CV(E)) by the formula\n(ηQn)(f) :=η(Qn(f)) =/integraldisplay\nη(dx)Qn(f)(x) =/integraldisplay\nη(dx)Qn(x,dy)f(y).(4)\nBy Fubini’s theorem, the integration order doesn’t matter. Thus to simplify notation,\nwe sometimes write ηQn(f) instead of ( ηQn)(f) orη(Qn(f)).\nWe denote by C0(E) :={1/V:V∈ C∞(E)} ⊂ Cb(E) the sub-algebra of bounded\ncontinuous positive functions hthat vanish at infinity; that is, for any 0 <ǫ≤ /⌊ard⌊lh/⌊ard⌊l<∞\ntheǫ-super-level set {h≥ǫ} ⊂Eis a non empty compact subset.\nWe further assume the Lyapunov function Vintroduced in ( 1) is aQ-Lyapunov\nfunction in the sense that ( 1) holds and there exists Θ ∈ C0(E) and a compact subset\nK⊂Esuch that for any n≥1 we have\nQn(V)/V≤Θ and (Gn−1(x)−Gn−1(y))(1E\\K(x)V(x)−1E\\K(y)V(y))≤0.(5)\nNote that the l.h.s. condition in ( 5) holds as soon as there exists G∈ C0(E) such that\nfor anyGn≤G, for anyn≥0. This condition ensures that for any positive function\nf∈ CV(E) andn≥1 we have\nQn(f)/V≤Qn(V)/V≤Θ.\nBy (2) the integral operator Qn(x,dy) =qn(x,y)ν(dy) also has a density given by\nqn(x,y) =Gn−1(x)pn(x,y) and for any r≥r1we have the local condition\n0<inf\nV(r)2qn≤sup\nV(r)2qn<∞. (6)\n5Consider the normalized measure valued process ηn∈ PV(E) starting at η0∈ PV(E)\ndefined for any n≥1 by\nηn+1=φn+1(ηn) :=ψGn(ηn)Pn+1and/hatwideηn:=ψGn(ηn), (7)\nwith the updated Boltzmann-Gibbs transformations ψGnassociated with the potential\nfunctionGndefined by\nψGn(ηn)(dx) :=1\nη(Gn)Gn(x)η(dx).\nWe readily check that the evolution semigroup φk,n=φk+1,n◦φkassociated with the\nflow of measure ηnis given for any k≤nby the formula\nφk,n(ηk) =ηkQk,n\nηkQk,n(1).\nNote that for any µ∈ PV(E) we have the updating formula\nφk,n(ηk) =ψHµ\nk,n(ηk)¯Qk,n, (8)\nwith the Markov operator\n¯Qk,n(f) :=Qk,n(f)/Qk,n(1) andHµ\nk,n(x) :=Qk,n(1)(x)\nφ0,k(µ)Qk,n(1). (9)\nBy Theorem 4.2 in [ 32] (see also Theorem 1 in [ 44] in the context of nonlinear filtering),\nfor anyµ∈ PV(E), there exists some parameters a,b>0 such that for any k≤n, and\nanyµ1,µ2∈ PV(E) we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleµ1¯Qk,n−µ2¯Qk,n/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nV≤ae−b(n−k)||µ1−µ2||V/Hµ\nk,n. (10)\nNote that for constant potential functions Gn(x) =Gn(y) we have ¯Qk,n=Pk,nand the\nabove contraction estimates resume to ( 3).\nFor time homogeneous models Qn=Q, Theorem 4.4 in [ 32] ensures the existence of\na leading eigen triple ( h,E0,η∞)∈/parenleftbig\nCV(E)×R∗\n+×PV(E)/parenrightbig\n,such that\nQ(h) =E0h , η ∞Q=E0η∞andη∞(h) = 1. (11)\n2.3 Schr¨ odinger semigroups\nThe objects defined in the previous subsection are core to a va riety of physics problem.\nIndeed, consider an Hamiltonian differential operator Hgiven by the formula\nH:=−L+U,\nwhereUis a potential energy function from EtoR+, andLis a kinetic energy operator\nacting on a subset D(L) ofC(E). The time dependent Schr¨ odinger equation and the\n6imaginary time version associated with the hamiltonian Hare given, respectively, by the\nequations\ni∂tψt(x) =H(ψt)(x) and −∂tϕt(x) =H(ϕt)(x),\nwith prescribed initial conditions ( ψ0,ϕ0). In the above display, i∈Cstands for the\nimaginary unit. The right-hand side equation is obtained vi a a formal time change by\nsettingϕt(x) =ψ−it(x), and can be equivalently written in the following form\n∂tϕt(x) =L(ϕt)(x)−U(x)ϕt(x) with initial condition ϕ0. (12)\nFor a twice differentiable function ϕ0, the solution of ( 12) is given by the Feynman-\nKac path integral formula\nϕt(x) =Qt(ϕ0)(x) :=/integraldisplay\nQt(x,dy)ϕ0(y) (13)\n=E/parenleftbigg\nϕ0(Xt) exp/parenleftbigg\n−/integraldisplayt\n0U(Xs)ds/parenrightbigg\n| X0=x/parenrightbigg\n.\nIn the above display, Xtstands for a time homogeneous stochastic process XtonE, with\ngenerator L. To facilitate the interpretation of the theoretical and nu merical physics in\nthe measure theoretical framework used in this article, we n ote that the Feynman-Kac\npropagator defined by the integral operator ( 13) is sometimes written in terms of the\nexponential of the Hamiltonian operator with the exponenti al-type symbol\nQt:=e−tHor in the bra-kets formalism Qt(ϕ0) =|e−tH|ϕ0/an}⌊ra⌋ketri}ht.\nThe integral operator Qtis sometimes called the Feynman-Kac propagator. For any\ns,t≥0 the integral operators Qtsatisfy the semigroup property\nQs+t(x,dz) = (QsQt)(x,dy) :=/integraldisplay\nQs(x,dz)Qt(z,dy) =⇒ϕs+t=Qs(ϕt).\nIn terms of left action bra-kets, defining µϕ(dx) :=ϕ(x)dx, Fubini’s theorem yields\n/an}⌊ra⌋ketle{tϕ|e−sH|ϕt/an}⌊ra⌋ketri}ht=/integraldisplay\ndx ϕ(x)Qs(x,dy)ϕt(dy) = (µϕQs)(ψt)\n=µϕ((QsQt)(ϕ0)) =µϕQs+t(ϕ0) =/an}⌊ra⌋ketle{tϕ|e−(s+t)H|ϕ0/an}⌊ra⌋ketri}ht.\nThe exponential notation is compatible with finite space mod els and the matrix\nnotation of the continuous one-parameter semigroup for tim e homogeneous models. The\nbra-ket notation (a.k.a. Dirac notation) is also used to rep resents linear projection forms\nacting on Hilbert spaces associated with some reversible or some stationary measure,\nsuch as the Lebesgue measure for the harmonic oscillator.\nThe present article deals with different types of non necessar ily stationary stochastic\nprocesses, including the free evolution process Xtdiscussed in ( 13). Apart in the re-\nversible situation in which spectral theorems are stated on the Hilbert space associated\n7with a reversible measure, the use of the exponential symbol or the use of the bra-\nkets formalism is clearly not adapted to represent different e xpectations with respect to\ndifferent types of stochastic and non-necessarily reversibl e processes.\nTo analyze these general stochastic models, we have chosen t o only use elementary\nand standard measure theory notation such as ( 4). The integral actions of a given\nintegral operator Qton the right for functions and on the left for measures are cle arly\ncompatible with finite space models and matrix notation. The left actionµ/ma√sto→µQtmaps\nmeasures into measures, while the right action f/ma√sto→ Qt(f) maps functions into functions.\n(µQt)(dy) :=/integraldisplay\nEµ(dx)Qt(x,dy) andQt(f)(x) :=/integraldisplay\nEQt(x,dy)f(y).\nIn this context the normalized measures are defined for any s≤tby the flow of measures\nµt(f) := Φs,t(µs)(f) :=µsQt−s(f)/µsQt−s(1), (14)\nwhereµ0stands for the distribution of the initial random state X0. Note that\n∂tµ0Qt(1) =−E/parenleftbigg\nU(Xt)exp/parenleftbigg\n−/integraldisplayt\n0U(Xs)ds/parenrightbigg/parenrightbigg\n=−µ0Qt(U)\n=⇒∂tlogµ0Qt(1) =−µt(U) =⇒µ0Qt(1) = exp/parenleftbigg\n−/integraldisplayt\n0µs(U)ds/parenrightbigg\n.(15)\nThe operator Qtdefined in ( 13) is sometimes called a Feynman-Kac propagator.\nHowever, despite its mathematical elegance, it can rarely b e solved analytically. Under\nsome regularity conditions (cf. for instance [ 32]) the flow of measures µtconverge as\nt→ ∞to some limiting fixed point measure µ∞= Φs,t(µ∞) (a.k.a. quasi-invariant\nmeasure). In this case, choosing µ0=µ∞in (15) we have\nµ∞Qt(1) =e−λ0twithλ0:=µ∞(U).\nWhenever it exists, the ground state h0is the leading eigen-function associated with λ0;\nthat is, we have\nQt(h0) =e−λ0th0.\nOn a given time mesh tn=nδ, with time step tn−tn−1=δ>0 we clearly have\nµtn(f) =µtn−1Qδ(f)\nµtn−1Qδ(1)andQδ(x,dy) =Gδ(x)Pδ(x,dy),\nwith the function\nGδ(x) :=Qδ(1)(x) and the Markov transition Pδ(x,dy) :=Qδ(x,dy)\nQδ(1)(x).\nNote that\nµ∞(Gδ) =µ∞Qδ(1) =e−λ0δas well as Qδ(h0) =e−λ0δh0andµ∞Qδ=e−λ0δµ∞.\n8Inotherwords,choosing( G,P) = (Gδ,Pδ)inthetime-homogeneousFeynman-Kacmodel\ndiscussed in ( 7) and (11), we obtain the leading triple\n(η∞,E0,h) = (µ∞,e−λ0δ,h0).\nThis yields the formula\n−1\nδlogµ∞(Gδ) =λ0=µ∞(U).\nUnfortunately, with the notable exception of coupled harmo nic models (cf. [ 30] as\nwell as Proposition 7.1 in [ 31]), the potential function Gδcan rarely be evaluated and\nthe Markov transition Pδcannot be sampled. The Feynman-Kac measure ηnintroduced\nin (7) can also be interpreted as the solution of a discrete-time a pproximation of the\nformula ( 13). Indeed, consider a discrete time approximation Xtnof the process Xtnand\nlet\nQ(x,dy) =G(x)P(x,dy),\nwith\nG(x) = exp( −U(x)δ) andP(x,dy) =P(Xδ\ntn∈dy|Xδ\ntn−1=x).(16)\nIn this situation, choosing n=⌊t/δ⌋we have\nη0Q0,n(f) =E\nf(Xδ\ntn)/productdisplay\n0≤k<nG(Xδ\ntk)\n\n=E\nf(Xδ\ntn) exp\n\n−/summationdisplay\n0≤k<nU(Xδ\ntk)(tk+1−tk)\n\n\n≃δ↓0η0Qt(f).\nThe leading triple ( η∞,E0,h) associated with the discrete time Feynman-Kac approxi-\nmation model now depends on the time step δand we have\n(η∞,h)≃δ↓0(µ∞,h0) and −1\nδlogη∞(G) =−1\nδlogη∞(e−Uδ)≃δ↓0λ0=µ∞(U).\nIt is clearly out of the scope of this article to analyze the bi as introduced by the\ndiscrete time approximation discussed above.\n2.4 Diffusion Quantum Monte Carlo\nThe Diffusion Quantum Monte Carlo methodology relies on the fa ct that the flow of\nmeasuresηnintroduced in ( 7) can be interpreted as the probability distributions ηn=\nLaw(Xn) of the random states Xnof a nonlinear Markov chain Xn. The choice of the\nMarkov chain is far from unique. For instance, we have\nηn+1=φn+1(ηn) =ηnKn+1,ηn,\n9with the local Markov transition\nKn+1,ηn(x,dz) := (Sn,ηnPn+1)(x,dz)\n:=/integraldisplay\nSn,ηn(x,dy)Pn+1(y,dz) =P/parenleftbig\nXn+1∈dz|Xn=x/parenrightbig\n.\nIn the above display, Sn,ηnstands for the Markov transition\nSn,ηn(x,dy) :=ǫn(ηn)Gn(x)δx(dy)+(1−ǫn(ηn)Gn(x))ψGn(ηn)(dy).\nfor some tuning parameter ǫn(ηn)∈[0,1] chosen such that ǫn(ηn)Gn(x)∈[0,1]. For\ninstance, for ]0 ,1]-valued potential functions, we can choose ǫn(ηn) = 0 as well as\nǫn(ηn) = 1. For more general models, we can also choose the inverse o f theηn-essential\nsupremum of Gn.\nNote that the transition Xn/squigglerightXn+1depends on the probability distributions ηnof\nthe random states Xn. In reference with similar nonlinear Markov chain models ar ising\nin fluid mechanics, the Markov chain Xnis called a McKean interpretation of the flow\nof measures ( 7).\nThe mean field particle interpretation associated with a giv en McKean model is\ndefined by a discrete-time system of Nwalkersξn=/parenleftbig\nξi\nn/parenrightbig\n1≤i≤N. The system starts with\nNindependent copies of a random variable X0=X0with distribution η0. Given the\nsystemξnat some time n≥0, we sample Nconditionally independent walkers ξi\nn+1\nwith their respective distribution\nKn+1,ηNn(ξi\nn,dx) withηN\nn:=1\nN/summationdisplay\n1≤i≤Nδξin.\nIn other words, the DMC method consists of approximating the measureηnby using\nthe occupation measure ηN\nnassociated with a system of Nwalkers. The initial positions\nof the walkers are randomly chosen from the distribution η0. The evolution of each\nwalker follows then the following selection/mutation step s:\n•Selection: We evaluate the current position ξi\nnof a walker and its potential value\nGn(ξi\nn). With probability/parenleftbig\n1−ǫn(ηN\nn)Gn(ξi\nn)/parenrightbig\n,ξi\nnis killed and instantly replaced\nby another walker say /hatwideξi\nn=ξj\nnwith a probability proportional to Gn(ξj\nn) and\nj∈ {1,...,N}; otherwise we keep it and set /hatwideξi\nn:=ξi\nn.\n•Mutation: We move the selected walker /hatwideξi\nn=xto a new location ξi\nn+1=yusing\nthe transition kernel Pn+1(x,dy).\nThe selection transition associated with the choice ǫn(ηN\nn) = 0 coincides with the\nso-called proportional selection/reconfiguration. Note t hat the walker with the highest\npotential value is always selected when ǫn(ηN\nn) is the inverse of the ηN\nn-essential supre-\nmum ofGn.\nFor [0,1]-valued potential functions Gnwe can also choose ǫn(ηN\nn) = 1. In this\nsituation, particle are killed at a geometric clock that dep ends on the potential function.\n10Inthecontext ofthediscretetimeapproximatingFeynman-K acmodelsdiscussedin( 16),\nwhenthe time step δtends to 0, thesegeometric killing-rates converges to an ex ponential\nkilling rate. The limiting DMC scheme in continuous time con sists of a system of N\nwalkers. Between killing times, walkers explore the space w ith a free evolution with\ngenerator L; at rateUthe walkers are killed and instantly another walker in the po ol\nduplicates (see for instance [ 16,27,45,46,47]).We underline that without the geometric\nkilling rates, the variance of the proportional reconfigura tion associated with the choice\nǫn(ηN\nn) = 0blows up when the time step tends to 0.\nWe expect that the occupation measures of the system approxi mate the solution of\nthe measure-valued process ( 7); that is, in a sense to be given, for any time horizon\nn≥0 we have\nηN\nn=1\nNN/summationdisplay\ni=1δξin−→\nN→∞ηn,\nas well as\nψGn/parenleftbig\nηN\nn/parenrightbig\n=N/summationdisplay\ni=1Gn(ξi\nn)/summationtext\n1≤j≤NGn(ξj\nn)δξin−→\nN→∞ψGn(ηn) =/hatwideηn.\nThe Lyapunov drift condition ( 1) combined with the local minorization condition\n(2) ensures that the free evolution of the walkers is stable, in the sense that it forgets\nexponentially fast its initial condition (see ( 3)). This condition is not satisfied for linear\ngaussian processes with an unstable drift matrix. In this co ntext, as shown in Proposi-\ntion1and Proposition 2(see also Section 5.3)the DMC method diverges for any fixed\nnumber of walkers, even if the asymptotic variance of the Cent ral Limit Theorem stated\nin Lemma 7is uniformly bounded w.r.t. the time horizon .\nNote that the stability regions and the Lyapunov functions a re connected to the\npotential function by ( 5). Importance sampling techniques and twisted guiding wave s\ncan be used to design stable-like free evolutions. For insta nce, using the decomposition\n/hatwideQn+1(f)(x) =/hatwideGn/hatwidePn+1(f),\nwith the potential function\n/hatwideGn:=Pn+1(Gn+1) and /hatwidePn+1(f) :=Pn+1(Gn+1f)\nPn+1(Gn+1),\nwe readily check the evolution equation\n/hatwideηn+1=/hatwideφn+1(/hatwideηn) :=ψ/hatwideGn(/hatwideηn)/hatwidePn+1.\nThis shows that the updatedmeasures evolve as in ( 7) by replacing ( Gn,Pn) by (/hatwideGn,/hatwidePn).\nThe DMC associated with these objects is a genetic-type Mont e Carlo sampler with\nselection fitness functions /hatwideGnand mutation transitions /hatwidePn. From the mathematical\nviewpoint, this model coincides with the one discussed in ( 7). Nevertheless, the free\nevolution of the walkers associated with ( /hatwideGn,/hatwidePn) is now driven by the potential function.\n11Thislocal conditioning importancesamplingstrategy if of ten usedto turninfiniteenergy\nabsorbing wells (a.k.a. hard obstacles) into soft ones [ 17,48].\nMore generally, for any given time mesh kn≤kn+1we have\n/hatwideQkn,kn+1(f) =/hatwideGkn/hatwidePkn,kn+1(f),\nwith the potential function\n/hatwideGkn:=/hatwideQkn,kn+1(1) and /hatwidePkn,kn+1(f) :=/hatwideQkn,kn+1(f)\n/hatwideQkn,kn+1(1).\nThis yields the formula\n/hatwideηkn+1(f) =/hatwideφkn,kn+1(/hatwideηkn) =ψ/hatwideGkn(/hatwideηkn)/hatwidePkn,kn+1.\nThis shows that the updated measures /hatwideηknevolve as in ( 7) by replacing ( Gn,Pn) by\n(/hatwideGkn,/hatwidePkn,kn+1). The DMC associated with these objects is a genetic-type Mo nte Carlo\nsampler with selection fitness functions /hatwideGknand mutation transitions /hatwidePkn,kn+1.\nAs shown in Section 5.2(see also Corollary 4) in the context of coupled harmonic\noscillators there exists a time mesh for which the mutation t ransitions /hatwidePkn,kn+1and\nthe potential functions /hatwideGknsatisfy the required stability properties. For more genera l\nmodels, these objects do not have an analytic form. In this co ntext, we can use the\nunbiased Monte Carlo methodologies discussed in [ 49], see also Section 2.3.2 in [ 27],\nand Section 11.5 in [ 21].\n3 Statement of the main results\n3.1 Some regularity conditions\nForf∈ Cb(E), time-uniform Lp-convergence of the error made by the DMC method\nin estimating ηn(f) have been obtained (see for example [ 27,28], as well as Chapter\n4 in [21], Chapter 12 in [ 22] and the more recent article [ 32]) under the strong mixing\nassumption that there exist ǫP∈R∗\n+andǫG∈R∗\n+such that\n∀(x1,x2,n)∈E2×N, Gn(x1)≥ǫGGn(x2) andPn(x1,dy)≥ǫPPn(x2,dy).\nA significant consequenceofthisassumptionisatimeunifor mboundonthepotential\nfunction defined for some γ∈ PV(E) andk∈Nby\nGγ\nk,k+n:x∈E/ma√sto→Gγ\nk,k+n(x) := 1/Hγ\nk,k+n(x).\nUnfortunately, these uniform minorization and majorizati on conditions are rarely satis-\nfied whenEis non-compact.\nIn order to guarantee a time-uniform Lp-convergence for more general models in-\ncluding coupled harmonic oscillators, our framework requi res to estimate uniformly in\n12time the inverse moments of ηN\nq(Hγ\nq,n). To do this, we first assume that there exists\nγ∈ PV(E) such that\nsup\nn∈Nφ0,n(γ)(Gn)<+∞. (17)\nFor time-homogeneous models, without any further conditio ns on the potential, con-\ndition (17) is easily checked with γ=η∞. For this scenario, we will then consider in\nthe rest that γ=η∞. Moreover, this hypothesis trivially holds if the function sGnare\nbounded by some constant independent of n.\nWe assume that there exists W∈ CV(E),α∈]0,1] and aQ-Lyapunov function\n¯W∈ CV(E) such that\nQn(W)≥χ×WandW−α≤¯W, (18)\nwhere\nχ:= sup\n(n,x)∈(N∗×E)Pn(Gn)(x).\nFor time homogeneous models, this condition can be relaxed i nto the following\nQ(W)≥min{χ,E0}×WandW−α≤¯W. (19)\nNote that, for time-homogeneous models, the set of function sW∈ PV(E) such that\nQ(W)≥E0Wis non-empty as it contains at least the ground state h. It is also worth\nnoting that it is not necessary to know the exact value of χnor the one of E0in order\nto prove that ( 18) or (19) hold. Indeed, if one of these constants is less than some\nC∈R∪{+∞}, then it is sufficient to prove that for any c<C, there exist Wc∈ CV(E)\nandαc∈(0,1] such that\nQ(Wc)≥c WcandW−αcc≤V.\nFinally, we assume that there exists a Q-Lyapunov function ¯V∈ C(E) andλ∈2N\nsuch that\nVλ≤¯V. (20)\nWithout further mention, we assume that V,¯Vand¯Ware integrable with respect\ntoη0, i.e.,η0∈ PV(E)∩P¯W(E)∩P¯V(E). Under conditions ( 17) and (18) or (19), it is\nthen possible to obtain a time uniform bound on the random pot ential function Gγ\nk,n.\nLemma 1. Letγand(η,µ)be defined as in (17). There exists ¯β∈R∗\n+such that for\nanyβ≤¯βwe have\nsup\n(q,n,N)∈N3\nq≤nE/bracketleftBig\nφq(ηN\nq−1)(Hγ\nq,n)−β/bracketrightBig\n<+∞andsup\n(q,n,N)∈N3\nq≤nE/bracketleftBig\nηN\nq(Hγ\nq,n)−β/bracketrightBig\n<+∞.(21)\nThe proof of this pivotal Lemma is postponed to the appendix.\n133.2 A time-uniform convergence Theorem\nThe main goal of this paper is to establish that in the context we described, the Lp-norm\nof the error made by the DMC method in approximating the Feynm an-Kac measures\nηnremains bounded in time and converges to zero as the number of particles increases.\nOur main result can be stated as follows\nTheorem 1. For anyp∈N∗, there exists c∈Randβ∈(0,1]such that for any\nf∈ C\nVλ\n4p(E)we have\nsup\nn∈NE/parenleftbig\n|ηn(f)−ηN\nn(f)|p/parenrightbig1\np≤cN−β\n2.\nThe proof of this theorem is provided in subsection 4.1. For time-homogeneous\nmodels, adirect consequence of Theorem 1is acontrol over theestimation of thelimiting\nquasi-invariant measure η∞.\nCorollary 1. For anyp∈N∗, there exists (a,b,c)∈R∗3\n+andβ∈(0,1]such that for\nanyf∈ C\nVλ\n4p(E)we have\nsup\nn≥a+bln(N)E/parenleftbig\n|η∞(f)−ηN\nn(f)|p/parenrightbig1\np≤cN−β\n2.\nThe proof the above Corollary is provided in subsection 4.2.\nAssuming that there exists a measure µ∈ P(E) that is reversible for P, it becomes\npossible to obtain a re-normalized weak form of the ground st atehand its associated\neigenvalue from the limit measures η∞and/hatwideη∞ofηnand/hatwideηn.\nIndeed, referring to Section 9.5.5 in [ 43] (see also [ 16] as well as Section 12.4 in [ 21]),\nwe have\nη∞(G) =E0andη∞(f) =µ(P(h)f)\nµ(P(h))=µ(hP(f))\nµ(h)=ψh(µ)P(f),\nNote that\nη∞(f) =µ(Q(h)f/G)\nµ(h)=E0ψh(µ)(f/G).\nIn the reverse angle, we have the updated limiting measures\n/hatwideη∞(f) =ψG(η∞)(f) =ψh(µ)(f) =µ(hf)\nµ(h)=⇒η∞=/hatwideη∞P\nThe existence of a reversible measure is actually not requir ed to express the ground-\nstate energy using the limit measure; indeed, we always have\nη∞(h) =φ(η∞)(h) =η∞(GP(h))\nη∞(G)=E0η∞(h)\nη∞(G)⇒η∞(G) =E0.\nThose equalities, combined with the convergence stated in C orollary 1, guarantee\nthe efficiency of the DMC method for approximating the ground- state energy and wave\nfunction of quantum systems.\n14Corollary 2. Letp∈N∗, and assume that G∈ C\nVλ\n4p(E). There exists (a,b,c)∈R∗3\n+\nandβ∈(0,1]such that for any f∈ C\nVλ\n4p(E)we have\n\n\nsup\nn≥a+bln(N)E/parenleftbig\n|E0−ηN\nn(G)|p/parenrightbig1\np≤cN−β\n2\nsup\nn≥a+bln(N)E/parenleftbig/vextendsingle/vextendsingleψh(µ)(P(f))−ηN\nn(f)/vextendsingle/vextendsinglep/parenrightbig1\np≤cN−β\n2\nsup\nn≥a+bln(N)E/parenleftbig/vextendsingle/vextendsingleE0ψh(µ)(f/G)−ηN\nn(f)/vextendsingle/vextendsinglep/parenrightbig1\np≤cN−β\n2.\n3.3 Coupled harmonic oscillators\nTo illustrate the practical applications of Theorem 1, we carry out an in-depth study of\nthe generalized coupled harmonic oscillator [ 30]. First, we demonstrate its relevance by\nestablishing easily verifiable sufficient conditions for tim e-uniform control of the DMC\nmethod in a framework that includes the harmonic oscillator .\nWe consider E=Rdfor somed∈N∗. For any real definite positive d×dmatrices\nBandSand any a real d×dmatrixA, we denote by PA,BandGSthe Markov kernel\nand the potential function defined as follows\nPA,B(x,dy) =1\n(2π)k/2|B|1/2e−1\n2(Ax−y)⊤B−1(Ax−y)andGS(x) := exp/parenleftbigg\n−xTSx\n2/parenrightbigg\n.\nNote that the matrix A may not be symmetric nor a stable (also c alled Hurwitz) matrix.\nIn addition, the transition PA,Bis not necessarily reversible, unless AB=BAT, where\nATstands for the transposition of the matrix A.\nConsider a family of Feller Markov transitions ( Pn)n∈Nwith positive densities pn, an\ninitial distribution η0∈ P(E) and a family of positive functions ( Gn)n∈N∈ C0(Rd)Nthat\nis uniformly bounded in time.\nWe assume the following, where ( A,A′) are real matrices, ( B,B′,S) real positive\ndefinitematrices, pA,Bthedensityof PA,BandEA,B,Stheground-stateenergyassociated\nto the operators PA,BandGS.\n•There exists c1∈Rsuch that for any ( n,x,y)∈N×Rd2we have\npn(x,y)≤c1pA,B(x,y). (22)\n•For any (n,x,y)∈N×Rd2we have\nGn(x)pn+1(x,y)≥χ\nEA′,B′,S×GS(x)pA′,B′(x,y). (23)\n•There exists a compact K⊂Rdsuch that for any ( n,x,y)∈N×Rd2we have\n[GS(x)−GS(y)][G−1\nn(x)1Rd\\K(x)−G−1\nn(y)1Rd\\K(y)]≥0. (24)\n15•There exists c3∈R∗\n+such thatG−c3\nSis integrable with respect to η0.\nIfPnandGnare time-independent, it is possible to replace χin (23) by the ground\nstate energy associated with PandG.\nThese conditions hold trivially for the coupled harmonic os cillator, i.e if Pn=PA,B,\nGn=GSandη0is a normal distribution.\nIn this context, Theorem 1leads to a simple sufficient matrix condition which guar-\nantees theuniformconvergence of theDMC method. Theproofo f thefollowing corollary\ncan be found in subsection 5.1.\nCorollary 3. Assume that ATSA < S. In this situation, for any p∈N∗, there exist\n(β,α,c)∈(0,1]×R∗2\n+such that for any function f∈ CV(Rd)we have\nsup\nn∈NE/parenleftbig\n|ηn(f)−ηN\nn(f)|p/parenrightbig1\np≤cN−β\n2,withV:x∈Rd/ma√sto→exp/parenleftBigα\n2xTSx/parenrightBig\n.\nShifting the focus to the approximation of the measures ( /hatwideηn)n∈Nwithin the coupled\nharmonic oscillator framework, the convergence condition given in the previous corollary\ncan be overcome with a change of transition and selection in t he DMC method. Con-\nsideringP=PA,B,G=GSas well asη0∼ N(X0,P0) and assuming that A,BandS\ncan be diagonalized in the same basis, we recursively define f or anyk≥1 the function\n/hatwideG(k)∈ CV(E) and the Markov kernel P(k)onEsuch that for all f∈ CV(E) we have\n/hatwideG(k)=P(G/hatwideG(k−1)) and /hatwideP(k)(f) =/hatwideP(k−1)(P(fG))\n/hatwideP(k−1)(P(G)),\nwith the convention /hatwideG(0)= 1 and /hatwideP(0)=Idfork= 0. Equivalently, for any f∈ CV(E)\nandk≥1 we have the formula\n/hatwideQ0,k(f) =/hatwideG(k)/hatwideP(k)(f) with /hatwideG(k):=/hatwideQ0,k(1) and /hatwideP(k)(f) :=/hatwideQ0,k(f)//hatwideQ0,k(1).\n(25)\nFork∈N∗, consider a system of walkers /hatwideξ(k)\nn=/parenleftBig\nξ(k),i\nn/parenrightBig\n1≤i≤Nassociated to the DMC\nmethod with initial distribution ψG(η0), transitions /hatwideP(k)and selection function /hatwideG(k)as\nwell as the empirical measures\n/hatwideη(k),N\nn:=1\nN/summationdisplay\n1≤i≤Nδ/hatwideξ(k),i\nn. (26)\nThis system of walkers offers an approximation of the measures /hatwideηnfor anyn∈kN.\nThis type of change in the approximation, based on an importa nce sampling transfor-\nmation is analogous to using a guiding waves function to dire ct the Monte Carlo moves.\nWithout any additional condition, Theorem 1ensures the uniform convergence of the\nmodel. The details of the proof can be found in Subsection 5.1.\n16Corollary 4. Letp∈N∗, there exists ¯k∈Nsuch that for any k≥¯k, there exist\n(β,α,c)∈(0,1]×R∗2\n+satisfying for any f∈ CV(R)\nsup\nn∈NE/parenleftBig\n|/hatwideηnk(f)−η(k),N\nn(f)|p/parenrightBig1\np≤cN−β\n2,withV:x∈Rd/ma√sto→exp/parenleftbigα\n2xTSx/parenrightbig\n.\nAlthough the method does not provide an approximation for ev ery time step, several\nstrategies can be used to fill the gaps left by the approximati on. A simple approach,\nthough more computationally intensive, is to run independe nt systems of walkers for\neach time step in the interval /llbracket0,k−1/rrbracket. This method not only fills in the gaps, but also\nmaintains the convergence property.\nOur study concludes with a focus on the divergence of the DMC m ethod when ap-\nproximating the one-dimensional harmonic oscillator. Thi s confirms that the stability\ncondition stated in Corollary 3is necessary and that, in some cases, the set of assump-\ntions presented can closely approximate a sufficient and nece ssary condition. Addition-\nally, it emphasizes the significance of the importance sampl ing method introduced in the\nprevious corollary. Specifically, in the one-dimensional c ontext, the sufficient condition\nfor uniform convergence of the DMC method is expressed by A2<1. Proposition 1\nestablishes the divergence of the error made by the DMC metho d whenA2>1, leaving\nopen only the case A= 1.\nProposition 1. Assume that A2>1andP0>0. For anyp∈N∗we have\nsup\nn∈NE/parenleftbig\n|ηn(I)−ηN\nn(I)|p/parenrightbig1\np= +∞.\nThe proof of this proposition can be found in Subsection 5.3.\nNote that all corollaries in this subsection can be extended to a control on the\nestimation of the limit measures, ground state, and eigenva lue using the same approach\nas presented in Corollary 2.\n4 Stochastic interpolation\n4.1 Time varying semigroups\nIn this subsection, we focus on proving Theorem 1. To take advantage of the con-\nditional independence of the walkers, we structure our appr oach around the following\ndecomposition of the difference between the Feynman-Kac meas ure and its empirical\napproximation, using the convention ηN\n−1=η0. Following [ 27,28], we use the following\nstochastic interpolation formula\nηN\nn−ηn=n/summationdisplay\nq=0[φq,n(ηN\nq)−φq,n(φq(ηN\nq−1))]. (27)\n17Each term on the right-hand side represents the error that oc curs when using the\nDMC approximation instead of the real propagator for a singl e extra time step. Com-\nbining the uniform bound given in Lemma 1with the contraction property ( 10), the\nfollowing Lemma establishes an exponentially decreasing c ontrol for these local errors.\nLemma 2. For anyp∈N∗, there exists (c,ρ,β)∈R∗2\n+×(0,1]such that for any function\nf∈ C\nVλ\n4p(E)and any (N,q,n)∈N3withq≤nwe have\nE/bracketleftBig/vextendsingle/vextendsingle[φq,n(ηN\nq)−φq,n(φq(ηN\nq−1))](f)/vextendsingle/vextendsinglep/bracketrightBig1\np≤ce−(n−q)ρN−β\n2. (28)\nProof:\nLet (η,µ)∈ P(E) and letγ∈ PV(E) be defined as in ( 17). Consider Hq,n:=Hγ\nq,n\nas defined in ( 9). Applying the updating formula ( 8), we obtain\nφq,n(η)(f)−φq,n(µ)(f) = (ψHq,n(η)¯Qq,n−ψHq,n(µ)¯Qq,n)(f)\n=1\nη(Hq,n)(η−µ)/parenleftbig\nHq,n¯Qq,n[f−ψHq,n(µ)¯Qq,n(f)]/parenrightbig\n.\nThis yields the formula\nφq,n(η)(f)−φq,n(µ)(f) =1\nη(Hq,n)(η−µ)(Fµ\nq,n), (29)\nwith the function\nFµ\nq,n(x) :=Hq,n(x)/integraldisplay\nEψHq,n(µ)(dy)[¯Qq,n(f)(x)−¯Qq,n(f)(y)].\nThen, applying H¨ older’s inequality for any β∈[0,1), we obtain the estimate\nE/parenleftBig/vextendsingle/vextendsingleφq,n(ηN\nq)(f)−φq,n(φq(ηN\nq−1))(f)/vextendsingle/vextendsinglep/parenrightBig1\np\n≤E/parenleftBig\nηN\nq(Hq,n)−2p/vextendsingle/vextendsingle(ηN\nq−φq(ηN\nq−1))(FN\nq,n)/vextendsingle/vextendsingle2p(1−β)/parenrightBig1\n2pE/parenleftBig/vextendsingle/vextendsingle(ηN\nq−φq(ηN\nq−1))(FN\nq,n)/vextendsingle/vextendsingle2βp/parenrightBig1\n2p,\nwithFN\nq,n:=Fφq(ηN\nq−1)\nq,n. Using ( 29) this yields the estimate\nE/parenleftbig/vextendsingle/vextendsingleφq,n(ηN\nq)(f)−φq,n(φq(ηN\nq−1))(f)/vextendsingle/vextendsinglep/parenrightbig1\np\n≤E/parenleftBig\nηN\nq(Hq,n)−2βp/vextendsingle/vextendsingleφq,n(ηN\nq)(f)−φq,n(φq(ηN\nq−1))(f)/vextendsingle/vextendsingle2p(1−β)/parenrightBig1\n2p\n×E/parenleftBig/vextendsingle/vextendsingle(ηN\nq−φq(ηN\nq−1))(FN\nq,n)/vextendsingle/vextendsingle2βp/parenrightBig1\n2p.\n18Recalling that f∈ C\nVλ\n4p(E), this implies that\nE/parenleftbig/vextendsingle/vextendsingleφq,n(ηN\nq)(f)−φq,n(φq(ηN\nq−1))(f)/vextendsingle/vextendsinglep/parenrightbig1\np\n≤/bracketleftBig\nE/parenleftbig\nφq,n(ηN\nq)(Vλ)/parenrightbig1\n4p+E/parenleftbig\nφq,n(φq(ηN\nq−1))(Vλ)/parenrightbig1\n4p/bracketrightBig\n×E/parenleftbig\nηN\nq(Hq,n)−4βp/parenrightbig1\n4pE/parenleftBig/vextendsingle/vextendsingle(ηN\nq−φq(ηN\nq−1))(FN\nq,n)/vextendsingle/vextendsingle2βp/parenrightBig1\n2p.\nFrom Lemmas 1and9, we deduce that, to conclude, it is enough to prove that, for s ome\nconstantc∈R∗\n+independent of n,qandN, we have\nE/bracketleftBig/vextendsingle/vextendsingle/bracketleftbig\nηN\nq−φq(ηN\nq−1)/bracketrightbig/parenleftbig\nFN\nq,n/parenrightbig/vextendsingle/vextendsingle2βp/bracketrightBig1\n2p<ce−c2(n−q)N−β\n2. (30)\nLetβ′=2pβ\nλand assume βsmall enough so that β′<1/4.\nForq>0, the walkers ( ξi\nq)1≤i≤Nare independent conditionally to ηN\nq−1we have\n[ηN\nq−φq(ηN\nq−1)](f) =ηN\nq\n1\nN/summationdisplay\n1≤i≤Nhi\nwithhi:=f−Sq−1,ηN\nq−1Pq(f)(ξi\nq−1).\nMoreover, we have\nφq(ηN\nq−1) =1\nN/summationdisplay\n1≤i≤nµiwithµi=δξi\nq−1Sq−1,ηN\nq−1Pq.\nSince for any i∈/llbracket1,N/rrbracket, we haveµi(hi) = 0, we can apply Lemma 7.3.3 from [ 21],\nand deduce that there exists C∈Rsuch that\nE/bracketleftBig/vextendsingle/vextendsingle/bracketleftbig\nηN\nq−φq(ηN\nq−1)/bracketrightbig/parenleftbig\nFN\nq,n/parenrightbig/vextendsingle/vextendsingle2βp/bracketrightBig1\n2p=E/bracketleftBig\nE/parenleftBig/vextendsingle/vextendsingle/bracketleftbig\nηN\nq−φq(ηN\nq−1)/bracketrightbig/parenleftbig\nFN\nq,n/parenrightbig/vextendsingle/vextendsingleλβ′\n|ξq−1/parenrightBig/bracketrightBig1\n2p\n≤E/bracketleftbigg\nE/parenleftBig/vextendsingle/vextendsingle/bracketleftbig\nηN\nq−φq(ηN\nq−1)/bracketrightbig/parenleftbig\nFN\nq,n/parenrightbig/vextendsingle/vextendsingleλ|ξq−1/parenrightBigβ′/bracketrightbigg1\n2p\n≤C\nNβ/2E/bracketleftbigg\nφq(ηN\nq−1)/parenleftBig\n|FN\nq,n|λ/parenrightBigβ′/bracketrightbigg1/2p\n.\nForq= 0, the walkers are iid with common distribution η0. The previous reasoning\ntherefore holds using the convention E(X|ξ−1) =E(X).\nApplying the contraction property ( 10) withµ=δxandη=δywe get the existence\nof (a,ρ)∈R2\n+such that\n|¯Qq,n(f)(x)−¯Qq,n(f)(y)| ≤ae−ρ(n−q)/parenleftbigg\n1+V(x)\nHq,n(x)/parenrightbigg/parenleftbigg\n1+V(y)\nHq,n(y)/parenrightbigg\n.(31)\n19By substituting ( 31) into thedefinition of Fq,nandapplyingH¨ older’s inequality along\nwith Jensen’s inequality, we obtain, for some ( a′,ρ′)∈R∗2\n+\nE/bracketleftbigg\nφq(ηN\nq−1)/parenleftBig\n|FN\nq,n|λ/parenrightBigβ′/bracketrightbigg1/2p\n≤a′e−ρ′(n−q)E/bracketleftBig\nφq(ηN\nq−1){(Hq,n+V)λ}β′φq(ηN\nq−1){(Hq,n+V)}λβ′φq(ηN\nq−1)(Hq,n)−λβ′/bracketrightBig1\n2p\n≤a′e−ρ′(n−q)E/bracketleftBig\nφq(ηN\nq−1){(Hq,n+V)λ}/bracketrightBig1\n4pE/bracketleftBig\nφq(ηN\nq−1)(Hq,n)−2λβ′/bracketrightBig1\n4p.\nFrom our hypothesis on Qn, we deduce from Lemma 3.2 in [ 32] that there exists\na constant csuch that for any ( q′,n′)∈N2,Hq′,n′≤cV. We can then conclude by\nchoosing a small enough β′and using Lemmas 1and9.\nThe proof of Theorem 1is now relatively straightforward.\nProof of Theorem 1:\nLetf∈ C\nVλ\n4p(E). From the sub-additivity of the Lp-norm applied in ( 27), we have\nE(|ηN\nn(f)−ηn(f)|p)1\np≤n/summationdisplay\nq=0E/parenleftBig/vextendsingle/vextendsingleφq,n(ηN\nq)(f)−φq,n(φq(ηN\nq−1))(f)/vextendsingle/vextendsinglep/parenrightBig1\np.\nBy applying Lemma 2, we deduce that there exists ( C,ρ,β)∈R×R∗\n+×(0,1] such\nthat for any N∈N∗we have\nE(|ηN\nn(f)−ηn(f)|p)1\np≤C\nNβ\n2/summationdisplay\n0≤l≤ne−(n−l)ρ≤C\nNβ\n2(1−e−ρ).\nThis ends the proof of the theorem.\n4.2 Ground state estimates\nThissubsectionconcentratesonprovingCorollary 1. Weconsiderthusthetime-homogeneous\nmodel.\nLetf∈ C\nVλ\n4p(E). Notice that wecan decomposetheerrormadebytheDMC metho d\nin the following way:\nE(|ηN\nn(f)−η∞(f)|p)1\np≤E(|ηN\nn(f)−ηn(f)|p)1\np+|ηn(f)−η∞(f)|.\nTheorem 1implies that there exists ( C1,β)∈R∗×(0,1] such that\nsup\nn∈NE(|ηN\nn(f)−ηn(f)|p)1\np≤C1\nNβ/2.\n20According to Theorem 4.3 in [ 32], there exists ( C2,ω)∈R∗2\n+such that\n|ηn(f)−η∞(f)| ≤C2e−ωn.\nHence\nE(|ηN\nn(f)−η∞(f)|p)1\np≤C1\nNβ/2+C2e−ωn.\nThus, letting a=1\nωln(C2/C1) andb=β\n2ω, we deduce that there exists C∈Rsuch\nthat\nsup\nn≥a+bln(N)E(|ηN\nn(f)−η∞(f)|p)1\np≤C\nNβ/2.\nThis concludes the proof.\n5 Coupled harmonic oscillators\n5.1 Lyapunov functions\nThis subsection is dedicated to the proof of Corollary 3. Therefore we place ourselves\nwithin the framework associated with this corollary. We onl y consider the general case\nwherePandGdepend on a time parameter. If this is not the case, and χis replaced\nby the ground state energy in ( 23), then the demonstration is completely analogous.\nThe Markov transition kernels Pnconsidered are Feller. Moreover, it is clear from\nSubsection 2.2that proving the existence of a continuous P-Lyapunov function V∈\nC∞(Rd) makes ( 22) and (23) sufficient condition for ( 2) to hold. To guarantee the\nexistence of an appropriate Q-Lyapunov function, we need a result obtained by Kato in\n[50]. We present it here using the formulation provided in [ 51].\nLemma 3. Suppose that D⊂Ris an interval, and let Abe a continuous function from\nDto the space of real d×dmatrices. In this case, there exist deigenvalues (counted\nwith algebraic multiplicities) of A(t)which can be parameterized as continuous functions\nλ1(t), ...,λd(t)fromDtoR.\nWe can now ensure the existence of a Q-Lyapunov function under a simple matrix\ncondition.\nLemma 4. Assume that ATSA<S. There exists α∈R∗\n+such that the function\nV:x∈Rd/ma√sto→exp/parenleftBigα\n2xTSx/parenrightBig\n, (32)\nis aQ-Lyapunov function and it is integrable w.r.t η0.\n21Proof :\nFrom (24), we deduce that the r.h.s of ( 5) holds forVandGn. Then, together with\n(22), we deduce that it is enough to prove that Vis a Lyapunov function for PA,Bwith\nǫ<1/c. Let’s then compute PA,B(f) for any function of the form\nf:x∈Rd/ma√sto→exp/parenleftbigg1\n2xTFx/parenrightbigg\n,\nwhereFis an invertible matrix such that B−1−Fis positive definite.\nIn this setting, the Woodbury matrix identity provides the f ollowing equality:\n(B−F−1)−1=B−1−B−1/parenleftbig\nB−1−F/parenrightbig−1B−1.\nWe have then for any x∈Rd, with¯BF:= (B−1−F)−1:\nPA,B(f)(x) =1\n(2π)d/2det(B)1/2/integraldisplay\nRde−1\n2[(Ax−y)⊤B−1(Ax−y)−yTFy]dy\n=f(x)exp(1\n2xT(ATB−1¯BFB−1A−ATB−1A−F)x)\n(2π)d/2det(B)1/2\n×/integraldisplay\nRdexp/parenleftbigg\n−1\n2/bracketleftBig\n(¯BFB−1Ax−y)⊤¯B−1\nF(¯BFB−1Ax−y)/bracketrightBig/parenrightbigg\ndy.\nThis yields the formulae\nPA,B(f)(x)\n=/radicalBigg\ndet(¯BF)\ndet(B)f(x)exp/parenleftbigg1\n2xT(AT[B−1(B−1−F)−1B−1−B−1]A−F)x/parenrightbigg\n=1/radicalbig\ndet(Id−BF)f(x)exp/parenleftbigg\n−1\n2xT(F+AT(FB−Id)−1FA)x/parenrightbigg\n.(33)\nFrom those calculations, we deduce that Vis a Lyapunov function for PA,Bif the\nmatricesB−1−αSandS−AT(Id−αSB)−1SAare positive definite.\nLetλBbe the greatest eigenvalue of BandλSbe the greatest eigenvalue of S. It is\nclear that for α∈(0,1\nλBλS),B−1−αSis positive definite.\nConsider now the function\nψ:α∈/bracketleftbigg\n0,1\nλBλS/parenrightbigg\n/ma√sto→sp(S−AT(Id−αSB)−1SA)∈Rd.\n22Here,sp(M) represents the spectrum of a matrix Mwith multiplicity taken into\naccount.\nGiven the hypotheses on AandS, we can conclude that ψ(0)⊂R∗d\n+. Furthermore,\nby Lemma 2, it is clear that ψis a continuous function. Since R∗d\n+is an open set, there\nexists ¯α∈Rsuch that for any α∈(0,¯α),ψ(α)⊂R∗d\n+.\nBy choosing a sufficiently small value for αto ensure that Vis integrable w.r.t η0,\nwe can conclude.\nFrom this Lemma and the hypothesis on Gn, we deduce that the l.h.s of ( 5) holds as\nwell. We can now focus on verifying that ( 19) holds by proving the following Lemma.\nLemma 5. There exists a positive definite matrix Hsuch that\n∀n∈N, Qn(W)≥χ×WwithW(x) := exp( −1\n2xTHx).\nProof :\nFrom (23), we have\nQn(W)≥χ\nEA′,B′,S×GSPA′,B′(W).\nUsing (33), we derive the following expression for GS(x)PA′,B′(W)(x) withx∈Rd:\n1/radicalbig\ndet(Id+B′H)W(x)exp/parenleftbigg1\n2xT(H−A′T(HB′+Id)−1HA′−S)x/parenrightbigg\n.\nHence, chosing Has the solution to the Riccati equation\nH−A′T(HB′+Id)−1HA′−S= 0,\nwe deduce that\nEA′,B′,S=1/radicalbig\ndet(Id+B′H).\nThus:\nQn(W)≥χ×W.\nForαsufficiently small, W−αis lower than V. The right-hand side of ( 19) is then\nalso verified.\nUnder the condition ATSA < S, we have confirmed that all the assumptions con-\ncerningPn,Gn, andη0in Theorem 1are satisfied. We can therefore apply it to conclude\non the proof of Corollary 3.\n235.2 Conditional free evolutions\nThissubsectionfocusesonthestudyoftheimportancesampl ingdescribedinSection 2.4,\nSection3.3and on the proof of Corollary 4.\nWe consider the coupled harmonic oscillator, i.e, for some m atricesA,BandS, with\nBandSsymmetric definite positive, we consider P=PA,BandG=GS. Up to this\npoint, we have established that the Lp-norm of the error made by the DMC method is\nuniformly bounded in time, with a convergence rate of1\nNβ/2for someβ∈(0,1] when\nATSA<A.\nOur aim is now to use Theorem 1to prove that the approximation of the measures /hatwideηn\nmade by the DMC method - enhanced by the importance sampling s cheme described in\n(26) - remains uniformly bounded in time, regardless of the valu e ofAandS. However,\nthere’s a trade-off involved: we will only have access to the m easures at specific times.\nIndeed, despite the converging property that we are about to prove, the sequence of\nempirical measures /hatwideη(k)\nnonly approximates the measures /hatwideηlforl∈kN.\nTo proceed with our proof, we make the necessary assumption t hat the matrices A,\nB, andScan all be diagonalized in the same basis.\nBefore presenting the central corollary of this subsection , we lay the foundation with\na lemma that, usingrelation ( 25), proves that this scenario can beinterpreted as another\ninstanceofthecoupledharmonicoscillator approximatedb ytheusualDMCmethod. We\nthen proceed to compute the specific constants of this scenar io. Consider the parameters\nλ1:=1/radicalbig\ndet(I+BS)&S1:=AT(S−1+B)−1A,\nas well as\nA1:= (I+BS)−1A&B1:= (B−1+S)−1.\nFor anyn≥0 we also set\nλn+1:=λn/radicalbig\ndet(I+BS+BSn)&Sn+1:=AT(B+(S+Sn)−1)−1A,\nas well as\nAn+1:= (I+ABkATS+BS)−1AAn\nBn+1:= (I+ABnATS+BS)−1(ABnAT+B).\nLemma 6. For anyk≥0we have\n/hatwideG(k)(x) =λkexp(−1\n2xTSkx)andδx/hatwideP(k)∼ N(Akx,Bk).\nThe proof of this Lemma is relatively straightforward. Howe ver, it requires some\ntechnical calculations. It is therefore postponed to the Ap pendix.\nWe can now proceed to prove the central result of this subsect ion.\n24Proof of Corollary 4:Letk∈N. From Corollary 3, to prove that the DMC\nmethod associated with /hatwideG(k)and/hatwideP(k)is uniformly converging toward the Feynman-Kac\nmeasures, it is enough to prove that\nAT\nkSkAk<Sk.\nSince all these matrices can be diagonalized in the same basi s, proving that this\ncriterion holds for the matrices AkandSkis equivalent to proving that all eigenvalues\nofAkare in the interval ( −1,1).\nLetBa basis in which the matrices A,BandSare diagonal. We want to prove that\nfori∈/llbracket1,d/rrbracket, thei-th eigenvalue of Ais in the right interval. In the rest of the proof,\nwe denote by M(i)thei-th eigenvalue of a matrix Mthat can be diagonalized in B.\nUsing the expression derived in Lemma 6, we obtain:\nA(i)\nn=/productdisplay\n1≤i≤nA(i)\n1+S(i)tnwithtn:=/braceleftBigg\nA(i)2B(i)\nn+B(i)ifn≥2\nB(i)ifn= 1.\nWe will establish that as ntends towards infinity, the limit of/vextendsingle/vextendsingle/vextendsingleA(i)\n1+S(i)tn/vextendsingle/vextendsingle/vextendsingleis strictly\nless than 1. This result will consequently imply the converg ence of the sequence A(i)\nn\ntowards 0. For n∈N∗:\ntn+1=A(i)2B(i)\nn+1+B(i)=A(i)2tn\n1+S(i)tn+B(i):=ϕ(tn). (34)\nHere,ϕrepresents a Riccati operator defined as described in [ 52]. Using Equation\n(51) from the same article, we can derive that\nlim\nn→+∞tn>|A(i)|−1\nS(i). (35)\nHence lim\nn→+∞/vextendsingle/vextendsingle/vextendsingleA(i)\n1+S(i)tn/vextendsingle/vextendsingle/vextendsingle<1. Fornlarge enough, the approximation made by the\nDMC method enhanced by importance sampling is then the same a s the usual DMC\napproximation of an harmonic oscillator with a stable Marko v transition. We can thus\nconclude using Theorem 1.\n5.3 Divergence and fluctuation estimates\nIn the previous subsections, we presented a simple sufficient condition for controlling\nthe DMC method and introduced an importance sampling techni que that satisfies this\ncriterion. However, it is natural to question the robustnes s of this condition and whether\nitisnecessarytouseimportancesampling. Specifically, fo rtheuni-dimensionalharmonic\noscillator, the convergence condition reduced to A2<1, and we will prove divergence of\nthe DMC method when this stability condition is not met.\n25Within this framework, we can break down the evolution of the walkers into two\ndistinct steps, a mutation transition and a selection trans ition\n/parenleftbig\nξi\nn/parenrightbig\ni∈/llbracket1,N/rrbracket∈RNselection− −−−− →/parenleftBig\n/hatwideξi\nn/parenrightBig\ni∈/llbracket1,N/rrbracket∈RNmutation− −−−−− →/parenleftbig\nξi\nn+1/parenrightbig\ni∈/llbracket1,N/rrbracket.\nThe initial configuration/parenleftbig\nξi\n0/parenrightbig\ni∈/llbracket1,N/rrbracketis determined by sampling Nindependent ran-\ndom variables from the distribution η0. The selection transition involves the sampling\nofNindependent random variables/parenleftBig\n/hatwideξi\nn/parenrightBig\ni∈/llbracket1,N/rrbracketusing the weighted distributions\nǫn(ηN\nn)GS(ξi\nn)δξin+(1−ǫn(ηN\nn)GS(ξi\nn))/summationdisplay\nk∈/llbracket1,N/rrbrackete−S\n2ξk2\nn\n/summationtext\nj∈/llbracket1,N/rrbrackete−S\n2ξj2\nnδξkn.\nThe mutation transition is defined using a family of Gaussian random variables with\nzero-mean and unit variance ( Vi\nn)i∈/llbracket1,N/rrbracketsuch that\nξi\nn=A/hatwideξi\nn−1+√\nBVi\nn.\nThe measures ( ηn)n∈Ncan be described exhaustively using the Kalman filter equa-\ntions. It provides us with the mean and variances ( mn,σ2\nn) of the Gaussian random\nvariablesηnwith the recurrent equations\n\n\nmn+1=A\n1+Sσ2nmn\nσ2\nn+1=A2σ2\nn\n1+Sσ2n+B. (36)\nIn this scenario, when the condition A2>1 is met, it is possible to prove that the\nDMC’s error in approximating the Feynman-Kac measure does n ot admit a uniform-in-\ntime bound. It is properly stated in Property 1, and we can now conduct its proof.\nProof of Proposition 1:\nFor anyn≥2, we know from ( 36) that\nηn(I) :=mn=A2\n(1+Sσ2\nn−1)(1+Sσ2\nn−2)mn−2≤A2mn−2.\nFor anyn∈N∗, letξ∗\nn= min\ni∈/llbracket1,N/rrbracketξi\nnanddefinetherandomvariables V∗\nninthefollowing\nway:\nV∗\nn=−max\ni∈/llbracket1,N/rrbracket/vextendsingle/vextendsingleVi\nn/vextendsingle/vextendsingle.\nBy definition of the evolution of the walkers, there exits ( i,j)∈/llbracket1,N/rrbracketsuch that\n26ξ∗\n2n=A2ξj\n2n−2+√\nBVi\n2n−1+A√\nBVj\n2n−2≥A2ξ∗\n2n−2+√\nBV∗\n2n−1+|A|√\nBV∗\n2n−2.\nThus\nηN\n2n(I)−η2n(I)≥A2(ξ∗\n2(n−1)−m2(n−1))+√\nBV∗\n2n−1+|A|√\nBV∗\n2n−2.\nIterating the process, we obtain\nηN\n2n(I)−η2n(I)\nA2n≥(ξ∗\n0−m0)+√\nB/summationdisplay\n1≤k≤nV∗\n2k−1\nA2k+|A|√\nB/summationdisplay\n1≤k≤nV∗\n2(k−1)\nA2k.\nFor any sequence of Nindependent centred Gaussian random variables Uiwith unit\nvariance, we have\nE/bracketleftbigg\nmax\n1≤i≤N|Ui|/bracketrightbigg\n≤/radicalbig\n2log(2N).\nThisinequalityisobtainedbyusingJensen’sinequalityas follows, with t=/radicalbig\n2log(2N)\nexp/bracketleftbigg\ntE/parenleftbigg\nmax\n1≤i≤N|Ui|/parenrightbigg/bracketrightbigg\n≤E/bracketleftbigg\nexp/parenleftbigg\ntmax\n1≤i≤N|Ui|/parenrightbigg/bracketrightbigg\n≤N/summationdisplay\ni=1E[exp(t|Ui|)],\nand noticing that\nE[exp(t|Ui|)] = 2/integraldisplay+∞\n0exp/parenleftbigg\n−(x−t)2+t2\n2/parenrightbigg\ndx≤2exp(t2/2).\nThen, on the event\nΩǫ:=/braceleftBigg\nξ∗\n0≥ǫ+m0+2√\nB(1+|A|)\nA2−1/radicalbig\n2log(2N)/bracerightBigg\n, (37)\nwithǫ∈R∗\n+, we have\nE[ηN\n2n(I)−η2n(I)|ξ∗\n0]≥ǫA2nn→+∞− −−−− → +∞. (38)\nIntegrating over ξ∗\n0we deduce\nE[|ηN\n2n(I)−η2n(I)||]≥ǫA2nP(Ωǫ). (39)\nWe can then conclude by noticing\nP(Ωǫ) =η0/braceleftBigg/bracketleftBigg\nǫ+X0+2√\nB(1+|A|)\nA2−1/radicalbig\n2log(2N),+∞/parenrightBigg/bracerightBiggN\n>0.(40)\n27For the case A= 1, we are not able to assert whether or not a uniform bound exi sts.\nTo the best of our knowledge, the best divergence-type resul t proved to date is a linear\nbound on the variance of the unnormalized measure when R=S= 1 [53].\nThe divergence result in Proposition 1highlights the importance of studying non-\nasymptotic uniform convergence results rather than relyin g solely on central limit the-\norems (CLTs). Extensive research has been devoted to CLTs an d, under appropriate\nassumptions. We quote the first studies in this field [ 25,27] mainly based on uniformly\nbounded potential and test functions.\nMore general fluctuation theorems that apply to more general models including\ndiffusion-type processes with Lipschitz drift and diffusion fu nctions as well as test func-\ntions with at most polynomial growth are discussed in [ 54,55]. We can formulate the\nfollowing result :\nLemma 7. In the context of proportional selection/reconfiguration, f or anyn≥1, we\nhave the following convergence in law as N tends toward +∞\n√\nN[ηn(I)−ηN\nn(I)]L− −−− →\nN→∞N(0,σ2\nn),\nwith the asymptotic variance\nσ2\nn:=n/summationdisplay\np=0ηp/bracketleftBigg/parenleftbiggQp,n(1)\nηpQp,n(1)¯Qp,n(I−ηn(I))/parenrightbigg2/bracketrightBigg\n.\nRelated asymptotic variance formulae and comparisons are d iscussed in [ 48] in the\ncontext of random walks with absorbing barriers, including geometric killing rates and\nlocal reflection moves.\nIn [56], Nick Withley also obtains a uniform time bound on the asymp totic variance.\nHowever, our next proposition shows that, for a given system , the DMC method can\nhave a uniform time bound on its asymptotic variance, despit e the fact that its non-\nasymptotic variance is unbounded.\nProposition 2. LetG:x∈R/ma√sto→e−x2\n2S, andPsuch thatδxP∼ N(Ax,B)for some\n(A,B,S)∈(1,+∞]×R∗2\n+. There exists an intial distibution η0such that\nsup\nn∈Nσ2\nn<∞andsup\nn∈NE/parenleftbig\n|ηn(I)−ηN\nn(I)|p/parenrightbig1\np= +∞, (41)\nwhereσ2\nnis defined as in Lemma 7.\nProof :\nConsidering η0∼ N(m0,σ2\n0) and using the Kalman filter’s equations, we are able to\nfully describe the measures ηn. Forn∈N∗we have\n28ηn∼ N(mn,σ2\nn) with\n\nmn+1=A\n1+Sσ2nmn\nσ2\nn+1=A2\n1+Sσ2n+B.\nThe limit measure is given by η∞=N(0,σ2\n∞) whereσ2\n∞is the fixed point of the\nfunctionx∈R+/ma√sto→A2\n1+Sx+B. From now on, we assume that η0=η∞. We have then\nσ2\nn=n/summationdisplay\np=0η∞/parenleftbig\nQp,n(I)2/parenrightbig\n(η∞Qp,n(1))2.\nUsing the calculations and notations from Lemma 10, we have then\n\n\nη∞(Qp,n(1)) =λn−p/radicalbig\n2πσ2∞/integraldisplay\nRe−y2\n2σ2∞e−y2\n2qn−pdy=λn−p/radicalbig\nqn−pσ2∞+1\nη∞/parenleftbig\nQp,n(I)2/parenrightbig\n=µ2\nn−p/radicalbig\n2πσ2∞/integraldisplay\nRy2e−y2\n2σ2∞e−y2qn−pdy=µ2\nn−pσ2\n∞\n(2qn−pσ2∞+1)3/2,\nand\nσ2\nn=σ2\n∞n/summationdisplay\np=0µ2\nn−p\nλ2\nn−p(qn−pσ2\n∞+1)3/2\n(2qn−pσ2∞+1)3/2.\nAs a solution to a Riccati equation, qnconverges towards some q∞∈R∗\n+such that\nq∞=A2q∞\n1+q∞B+S.\nSinceA>1, we obtain that 1+ q∞B=A2+S/q∞>A. Thus, there exists C∈(0,1)\nsuch that, for nlarge enough, we have\nµn+1≤Cµn√1+qnB\nThus, comparing the defintion of λnand the previous bound, we deduce that there\nexistsα∈Rsuch that we have\nµ2\nn−p\nλ2\nn−p(qn−pσ2\n∞+1)3/2\n(2qn−pσ2∞+1)3/2≤αC2(n−p).\nWe can thus conclude on the right-hand side. of ( 41). The left-hand side is a direct\napplication of Proposition 1.\n29Acknowledgments\nMC thanks the European Research Council (ERC) under the Euro pean Union’s Horizon\n2020 research and innovation programme (grant agreement no . 863481) for financial\nsupport.\nLuc de Montella would like to thank Naval Group for its suppor t as part of a CIFRE\nfellowship.\nAppendix\nSome technical Lemmas\nLemma 8. Let(G,V)∈ C(E)×C∞(E)be positive functions and K⊂Ebe such that the\nright-hand side of (5)holds. There exists c∈R+such that for any probability measure\nµonE:\nψG(µ)(V)≤µ(V)+c.\nProof : Let¯V:=1R\\KV. For any (x,t)∈E2, we have:\n0≥(G(x)−G(y))(¯V(x)−¯V(y)) =G(x)¯V(x)+G(y)¯V(y)−G(y)¯V(x)−G(x)¯V(y).\nIntegrating with respect to the probability measure µover bothxandy, we obtain:\nµ(G¯V)−µ(G)µ(¯V)≤0.\nHence\nψG(µ)(¯V)−µ(¯V) =µ(G¯V)−µ(G)µ(¯V)\nµ(G)≤0.\nThus\nψG(µ)(V)≤ψG(µ)(¯V)+sup\nKV≤µ(¯V)+sup\nKV≤µ(V)+sup\nKV.\nLemma 9. LetV∈ C∞(E)be aQ-Lypaunov function, then\nsup\n(q,n,N)∈N3\nq≤nE/bracketleftbig\nφq,n(ηN\nq)(V)/bracketrightbig\n<+∞andsup\n(q,n)∈N2\nq≤nφq,n(ηq)(V)<+∞.(42)\n30Proof: First, let’s prove that for any γ∈ PV(E), we have, with ( ǫ,c)∈[0,1)×R\ndefined in ( 1),\nsup\n(q,n)∈N2\nq≤nφq,n(γ)(V)≤γ(V)+c\n1−ǫ. (43)\nTo do so, we begin by using the Q-Lyapunov property of Vas well as Lemma 8for\nsomel∈N∗to deduce that\nφq,n(γ)(V) =φq,n−1(γ)(GnPn(V))\nφq,n−1(γ)(Gn)≤ǫ φq,n−1(γ)(V)+c′.\nBy iterating the process, we obtain ( 43). We now prove that\nsup\nn∈NE/bracketleftbig\nηN\nn(V)/bracketrightbig\n≤η0(V)+c′\n1−ǫ. (44)\nNotice first that\nE/bracketleftbig\nηN\nn(V)|ηN\nn−1/bracketrightbig\n=1\nNN/summationdisplay\ni=1ηN\nn−1Sn−1,ηN\nn−1Pn(V) =ψGn−1(ηN\nn−1)(Pn(V)).\nThen, using as previously the Q-Lyapunov property of Vand Lemma 8we obtain\nψGn−1(ηN\nn−1)(Pn(V))≤ǫ ηN\nn−1(V)+c′.\nBy iterating the process, we obtain\nE/bracketleftbig\nηN\nn(V)/bracketrightbig\n≤ǫnη0(V)+c′n−1/summationdisplay\ni=0ǫi≤η0(V)+c′\n1−ǫ.\nThus, by combining ( 43) and (44), we can conclude regarding the first part of ( 42).\nThe second part is obtained by proceeding in a strictly analo gous way.\nProof of Lemma 1\nWe first consider the case where only ( 18) holds and prove that\nφ0,q(γ)(Qq,n(1))≤Cχn−q−1. (45)\nForq≤n+2, we have\n31Qq,n(1)(x) =Qq,n−2(Gn−1Pn(Gn))(x)\n≤χ Qq,n−2(Gn−1)(x)≤χ Qq,n−1(1)(x).\nIterating the process, we deduce\nQq,n(1)≤χn−q−1Gq−1.\nUsing (17) we deduce ( 45).\nForq≥1 andµ∈ {ηN\nq,φq(ηN\nq−1)}, we have then\n1\nµ(Hγ\nq,n)=φq,n(µ)(W)×φ0,q(γ)Qq,n(1)\nµ(Qq,n(W))≤Cχn−q−1φq,n(µ)(W)\nµ(Qq,n(W)).\nFrom Holder’s inequality, Jensen’s inequality and the hypo thesis onW, we get :\nE/bracketleftBig\nµ(Hq,n)−β/bracketrightBig\n≤CβE/bracketleftBig\nφq,n(µ)(W)2β/bracketrightBig1\n2E/bracketleftBig\nχ2β(n−q−1)µ(Qq,n(W))−2β/bracketrightBig1\n2\n≤Cβ\nχβE/bracketleftBig\nφq,n(µ)(V)2β/bracketrightBig1\n2E/bracketleftBig\nµ(W−2β)/bracketrightBig1\n2.\nThen, by choosing βsmall enough such that W−2β≤¯WandV2β≤¯V, where ¯W\nand¯VareQ-Lyapunov functions, we obtain\nsup\n(q,n,N)∈N3\nq≤nE/bracketleftBig\nµ(Hγ\nq,n)−β/bracketrightBig\n≤Cβ\nχβsup\n(q,n,N)∈N3\nq≤nE/bracketleftbig\nµ(¯V)/bracketrightbig1\n2×sup\n(q,n,N)∈N3\nq≤nE/bracketleftbig\nµ(¯W)/bracketrightbig1\n2.\nWe can conclude using Lemma 9.\nThe demostration for time-homogeneous models with E0< χis analogous. Indeed\nthe equivalent of ( 45) is obtained by noticing that\nφ0,q(η∞)Qq,n(1) =η∞Qq,n−1(G) =η∞Qq,n−1(1)η∞(G) =E0×η∞Qq,n−1(1) =En−q\n0.\nThe rest of the proof follows the same arguments, thus it is sk ipped. This ends the proof\nof the lemma.\nProof of Lemma 6\nLet us begin by proving the result for k= 1. Takex∈Rd:\n/hatwideG(1)(x) =1/radicalbig\ndet(I+BS)exp/parenleftbigg\n−1\n2xTAT(S−1+B)−1Ax/parenrightbigg\n,\n32Applying the Woodbury matrix identity we get:\n/radicalBig\n(2π)ddet([B−1+S]−1)×/hatwideP(1)(x,dy)\n= exp/parenleftbigg\n−1\n2/bracketleftbig\nyTSy+(y−Ax)TB−1(y−Ax)−xTAT(S−1+B)−1Ax/bracketrightbig/parenrightbigg\ndy\n= exp/parenleftbigg\n−1\n2(y−(I+BS)−1Ax)T(B−1+S)(y−(I+BS)−1Ax)/parenrightbigg\ndy.\nAssume the property true for k∈N∗. We let ¯Sk:=Sk+S. In this notation, we have\n/hatwideG(k+1)(x) =P(G/hatwideG(k))(x) =λkP/parenleftbigg\nz/ma√sto→exp(−1\n2zT¯Skz)/parenrightbigg\n(x).\nThis implies that\n/hatwideG(k+1)(x)\n=λk/radicalbig\n(2π)ddet(B)/integraldisplay\nRdexp/parenleftbigg\n−1\n2[z¯SkzT+(z−Ax)TB−1(z−Ax)/parenrightbigg\ndz\n=λk/radicalbig\n(2π)ddet(B)exp/parenleftbigg\n−1\n2[xTATB−1Ax−xTATB−1(¯Sk+B−1)−1B−1Ax]/parenrightbigg\n×/integraldisplayd\nRexp/parenleftbigg\n−1\n2(z−(¯Sk+B−1)−1B−1Ax)T(¯Sk+B−1)(z−(¯Sk+B−1)−1B−1Ax)/parenrightbigg\ndz\nfrom which we check that\n/hatwideG(k+1)(x) =λk/radicalbig\ndet(I+BS+BSk)exp/parenleftbigg\n−1\n2xTAT(B+(S+Sk)−1)−1Ax/parenrightbigg\n.\nMoreover, since δx/hatwideP(k)∼ N(Akx,B2\nk) we also have\nδx/hatwideP(k)P∼ N(AAkx,ABkAT+B).\nAccording the Gaussian update formula (see Proposition 4.5 .2 in [43] for example),\nifη=N(m,Σ), thenψGS/parenleftbig\nη) =N[(I+ΣS)−1m ,(I+ΣS)−1Σ/bracketrightbig\n. Then we have\nψG/parenleftBig\nδx/hatwideP(k+1)/parenrightBig\n∼ N/parenleftbig\n(I+ABkATS+BS)−1AAkx,(I+ABkATS+BS)−1(ABkAT+B)/parenrightbig\n.\nThis concludes the proof.\nLemma 10. LetG:x∈R/ma√sto→e−x2\n2S, andPsuch thatδxP∼ N(Ax,B)for some\n(A,B,S)∈R×R∗2\n+. Denoting by Qnthen-times composition of the operator Q, we have\nQn(1)(x) =λne−x2\n2qnandQn(I)(x) =µnx e−x2\n2qn, (46)\n33with the parameters (q0,λ0,µ0) = (S,1,A)and\n\n\nqn+1=A2qn\n1+qnB+S\nλn+1=λn√1+qnBandµn+1=Aµn\n(1+qnB)3/2.\nProof:\nForn= 0 the result is immediate. Assume that it holds for some n∈N. In this\nsituation, we have\nQn+1(1)(x) =Q(Qn(1))(x) =λnQ/bracketleftbigg\ny/ma√sto→e−y2\n2qn/bracketrightbigg\n(x)\n=λn√\n2πBe−x2\n2S/integraldisplay\nRe−y2\n2qn−(Ax−y)2\n2Bdy\n=λn√\n2πBe−x2\n2/parenleftBig\nA2\nB−A2\nB(1+qnB)+S/parenrightBig/integraldisplay\nRe−(1+qnB)(A\n1+qnBx−y)2\n2Bdy\n=λn√1+qnBe−x2\n2/parenleftbigg\nA2qn\n1+qnB+S/parenrightbigg\n.\nSimilarly, we have\nQn+1(I)(x) =Q(Qn(I))(x) =µnQ/bracketleftbigg\ny/ma√sto→ye−y2\n2qn/bracketrightbigg\n(x)\n=µn√\n2πBe−x2\n2/parenleftbigg\nA2qn\n1+qnB+S/parenrightbigg/integraldisplay\nRye−(1+qnB)(A\n1+qnBx−y)2\n2Bdy\n=Aµn\n(1+qnB)3/2xe−x2\n2/parenleftbigg\nA2qn\n1+qnB+S/parenrightbigg\n.\nThis ends the proof.\nReferences\n[1] Nicholas Metropolis and Stanislaw Ulam. The Monte Carlo method. Journal of the\nAmerican statistical association , 44(247):335–341, 1949.\n[2] Monroe D. Donsker and Mark Kac. A sampling method for dete rmining the lowest\neigenvalue and the principal eigenfunction of schr¨ odinge r’s equation. Journal of\nResearch of the National Bureau of Standards , 44(50):551–557, 1950.\n[3] Malvin H Kalos. Monte Carlo calculations of the ground st ate of three-and four-\nbody nuclei. Physical Review , 128(4):1791, 1962.\n34[4] Malvin H. Kalos D. M. Ceperley. Monte Carlo Methods in Statistical Physics .\nSpringer-Verlag Berlin, 1986.\n[5] Peter J. Reynolds, David M. Ceperley, Berni J. Alder, and Jr. Lester, William A.\nFixed-node quantum Monte Carlo for molecules. The Journal of Chemical Physics ,\n77(11):5593–5603, 12 1982.\n[6] Michel Caffarel and Pierre Claverie. Treatment of the Schr ¨ odinger equation through\na Monte Carlo method based upon the generalized Feynman-Kac formula. Journal\nof Statistical Physics , 43:797–801, 1986.\n[7] Michel Caffarel and Pierre Claverie. Development of a pure diffusion quantum\nMonte Carlo method using a full generalized Feynman-Kac for mula. i. formalism.\nThe Journal of chemical physics , 88(2):1088–1099, 1988.\n[8] Jack H. Hetherington. Observations on the statistical i teration of matrices. Physical\nReview A , 30(5):2713, 1984.\n[9] Sandro Sorella. Green function Monte Carlo with stochas tic reconfiguration. Phys-\nical review letters , 80(20):4558, 1998.\n[10] Sandro Sorella and Luca Capriotti. Green function Mont e Carlo with stochastic\nreconfiguration: An effective remedy for the sign problem. Physical Review B ,\n61(4):2599, 2000.\n[11] Roland Assaraf, Michel Caffarel, and Anatole Khelif. Diffu sion Monte Carlo meth-\nods with a fixed number of walkers. Physical Review E , 61(4):4566, 2000.\n[12] Baroni Stefano and Moroni Saverio. Reptation Quantum M onte Carlo: A method\nfor unbiased ground-state averages and imaginary-time cor relations. Phys. Rev.\nLett., 82:4745, 1999.\n[13] Michel Mareschal. The early years of quantum Monte Carl o (1): the ground state.\nThe European Physical Journal H , 46(1):11, 2021.\n[14] YaoMa, LuMeng, Yan-KeChen,andShi-LinZhu. Groundsta tebaryonsintheflux-\ntube three-body confinement model using diffusion Monte Carlo .Physical Review\nD, 107(5):054035, 2023.\n[15] Matthew WMC Foulkes, Lubos Mitas, RJ Needs, and Guna Raj agopal. Quantum\nMonte carlo simulations of solids. Reviews of Modern Physics , 73(1):33, 2001.\n[16] Pierre Del Moral and Laurent Miclo. Particle approxima tions of Lyapunov expo-\nnents connected to Schr¨ odinger operators and Feynman-Kac semigroups. ESAIM:\nProbability and Statistics , 7:171–208, 2003.\n[17] Pierre Del Moral and Arnaud Doucet. Particle motions in absorbing medium with\nhard and soft obstacles. Stochastic Analysis and Applications , 22(5):1175–1207,\n2004.\n35[18] Mathias Rousset. On the control of an interacting parti cle estimation of schr¨ odinger\nground states. SIAM J. Math. Anal. , 38:824–844, 2006.\n[19] Eric Canc´ es, Benjamin Jourdain, and Tony Leli` evre. Q uantum Monte Carlo sim-\nulations of fermions. A mathematical analysis of the fixed-n ode approximation.\nMathematical Models and Methods in Applied Sciences , 16, 11 2011.\n[20] Mohamed El Makrini, Benjamin Jourdain, andTony Leli` e vre. Diffusion montecarlo\nmethod: Numerical analysis in a simple case. ESAIM: Mathematical Modelling and\nNumerical Analysis , 41(2):189–213, 2007.\n[21] Pierre Del Moral. Feynman-Kac Formulae: Genealogical and Interacting Partic le\nSystems With Applications , volume 100. 05 2004.\n[22] Pierre Del Moral. Mean field simulation for Monte Carlo integration . CRC press,\n2013.\n[23] Pierre Del Moral and Alice Guionnet. Large deviations f or interacting particle sys-\ntems: applicationstonon-linearfiltering. Stochastic processes and their applications ,\n78(1):69–95, 1998.\n[24] Donald A Dawson and Pierre Del Moral. Large deviations f or interacting processes\nin the strong topology. In Statistical modeling and analysis for complex data prob-\nlems, pages 179–208. Springer, 2005.\n[25] Pierre Del Moral and Alice Guionnet. Central limit theo rem for nonlinear filtering\nand interacting particle systems. Annals of Applied Probability , pages 275–297,\n1999.\n[26] Nicolas Chopin. Central limit theorem for sequential M onte Carlo methods and its\napplication to bayesian inference. 2004.\n[27] P. Del Moral and L. Miclo. Branching and interacting particle systems approxi-\nmations of Feynman-Kac formulae with applications to non-l inear filtering , pages\n1–145. Springer Berlin Heidelberg, Berlin, Heidelberg, 20 00.\n[28] Pierre Del Moral and Alice Guionnet. On the stability of interacting processes\nwith applications to filtering and genetic algorithms. In Annales de l’Institut Henri\nPoincar´ e (B) Probability and Statistics , volume 37, pages 155–194. Elsevier, 2001.\n[29] Nick Whiteley. Sequential Monte Carlo samplers: error bounds and insensitivity to\ninitial conditions. Stochastic Analysis and Applications , 30(5):774–798, 2012.\n[30] P Del Moral and E Horton. Coupled quantum harmonic oscil lators and Feynman-\nKac path integrals for linear diffusive particles. Communications in Mathematical\nPhysics, 402(2):2079–2127, 2023.\n36[31] Marc Arnaudon, Pierre Del Moral, and El Maati Ouhabaz. A lyapunov approach to\nstability of positive semigroups: An overview with illustr ations.Stochastic Analysis\nand Applications , pages 1–80, 2023.\n[32] Pierre Del Moral, Emma Horton, and Ajay Jasra. On the sta bility of positive\nsemigroups. The Annals of Applied Probability , 33(6A):4424–4490, 2023.\n[33] Malvin H. Kalos. Monte Carlo calculations of the ground state of three- and four-\nbody nuclei. Phys. Rev. , 128:1791–1795, Nov 1962.\n[34] Malvin H. Kalos, Dominique Levesque, and Loup Verlet. H elium at zero tempera-\nture with hard-sphere and other forces. Phys. Rev. A , 9:2178–2195, May 1974.\n[35] Albert Messiah. Quantum Mechanics . Number vol. 2 in Quantum Mechanics.\nElsevier Science, 1961.\n[36] David J. Griffiths and Darrell F. Schroeter. Introduction to quantum mechanics .\nCambridge University Press, Cambridge ; New York, NY, third edition edition,\n2018.\n[37] GerhardHerzberg. Molecular Spectra and Molecular Structure: Infrared and Ram an\nof Polyatomic Molecules . D. Van Nostrand Company, Inc., 1945.\n[38] Dmitry N. Makarov. Quantum entanglement and reflection coefficient for coupled\nharmonic oscillators. Phys. Rev. E , 102:052213, Nov 2020.\n[39] Artur K. Ekert. Quantum cryptography based on bell’s th eorem.Phys. Rev. Lett. ,\n67:661–663, Aug 1991.\n[40] Elisabet Romero, Ramunas Augulis, Vladimir Novoderez hkin, Marco Ferretti, Jos\nThieme, Donatas Zigmantas, and Rienk van Grondelle. Quantu m coherence in\nphotosynthesis for efficient solar energy conversion. Nature Physics , 10, 07 2014.\n[41] Pierre Del Moral. Non linear filtering: Interacting par ticle solution. Markov Pro-\ncesses and Related Fields , 2:555–580, 03 1996.\n[42] RudolfE.K´ alm´ anandRichardS.Bucy. Newresultsinli nearfilteringandprediction\ntheory.Journal of Basic Engineering , 83:95–108, 1961.\n[43] Pierre Del Moral and Spiridon Penev. Stochastic Processes: From Applications to\nTheory. December 2016.\n[44] Nick Whiteley. Stability properties of some particle fi lters.The Annals of Applied\nProbability , 23(6):2500 – 2537, 2013.\n[45] P.Del Moral andL. Miclo. A Moran particle system approx imation of Feynman-Kac\nformulae. Stochastic Processes and their Applications , 86(2):193–216, 2000.\n37[46] Marc Arnaudon and Pierre Del Moral. A duality formula an d a particle Gibbs\nsampler for continuous time Feynman-Kac measures on path sp aces.Electronic\nJournal of Probability , 25(none):1 – 54, 2020.\n[47] Mathias Rousset. Onthecontrol of an interacting parti cle estimation of Schr¨ odinger\nground states. SIAM Journal on Mathematical Analysis , 38(3):824–844, 2006.\n[48] Pierre Del Moral and Ajay Jasra. A note on random walks wi th absorbing bar-\nriers and sequential monte carlo methods. Stochastic Analysis and Applications ,\n36(3):413–442, 2018.\n[49] Pierre Del Moral. Measure-valued processes and intera cting particle systems. appli-\ncation to nonlinear filtering problems. The Annals of Applied Probability , 8(2):438–\n495, 1998.\n[50] Tosio Kato. Perturbation Theory for Linear Operators . 1966.\n[51] Chi-Kwong Li and Fuzhen Zhang. Eigenvalue continuity a nd gersgorin’s theorem.\nElectronic Journal of Linear Algebra , 35:619–625, 12 2019.\n[52] Pierre Del Moral and Emma Horton. A theoretical analysi s of one-dimensional\ndiscrete generation Ensemble Kalman particle filters, 07 20 21.\n[53] Nick Whiteley, Nikolas Kantas, and Ajay Jasra. Linear v ariance bounds for particle\napproximations of time-homogeneous Feynman-Kac formulae .Stochastic Processes\nand their Applications , 122(4):1840–1865, 2012.\n[54] Pierre Del Moral and Jean Jacod. The Monte-Carlo method for filtering with\ndiscrete time observations. central limit theorems. In Proceedings Workshop on\nNumerical Methods and Stochastics. The Fields Institute 1999 , editors T.J. Lyons\nand T.S Salisbury . American Mathematical Society, 2002.\n[55] PierreDel Moral andJ.Jacod. Interacting particle filt eringwith discrete-time obser-\nvations: asymptotic behaviour in the gaussian case. In Kuni ta H. Rajput B. Watan-\nabe S. Hida T., Karandikar R. and Xiong J., editors, Stochastics in finite/infinite\ndimensions , pages 101–123. Trends in Mathematics, Birkhauser, 2001.\n[56] Nick Whiteley. Stability properties of some particle fi lters.The Annals of Applied\nProbability , 23(6), dec 2013.\n38"    },    {        "title": "2402.04654v1.Willmore_surfaces_and_Hopf_tori_in_homogeneous_3_manifolds.pdf",        "content": "arXiv:2402.04654v1  [math.DG]  7 Feb 2024WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS\n3-MANIFOLDS\nALMA L. ALBUJER AND F ´ABIO R. DOS SANTOS\nAbstract. Some classification results for closed surfaces in Berger sp heres are pre-\nsented. On the one hand, a Willmore functional for isometric ally immersed surfaces\ninto an homogeneous space E3(κ,τ) with isometry group of dimension 4 is defined\nand its first variational formula is computed. Then, we chara cterize Clifford and Hopf\ntori as the only Willmore surfaces satifying a sharp Simons- type integral inequality.\nOn the other hand, we also obtain some integral inequalities for closed surfaces with\nconstant extrinsic curvature in E3(κ,τ), becoming equalities if and only if the surface\nis a Hopf torus in a Berger sphere.\n1.Introduction\nA classical problem in the theory of isometric immersions is to classify immersed\nsurfaces into a space form of constant sectional curvature h aving either constant mean\ncurvature or constant Gaussian curvature. In this directio n, we can highlight the\nrigidity theorems due to Alexandrov [ 3], Liebmann [ 18] and Hilbert [ 14] on surfaces\nof constant curvature as the most celebrated results in the t heory of surfaces in the\nEuclidean space R3. For generalizations of these results we quote [ 2,21]. Besides\nthat, we cannot fail to highlight the classical Hopf’s theor em [15] which characterizes\ntotally umbilical spheres as the unique topological sphere s of constant mean curvature\nimmersed into a 3-dimensional space form of constant sectio nal curvature.\nA natural generalization of space forms are the so-called homogeneous spaces. A\nRiemannian manifold is said to be homogeneous if for any two p ointspandq, there\nexists an isometry that maps pintoq. Geometrically, an homogeneous manifold seems\nthe same everywhere. As it is well-known, simply connected 3 -dimensional Riemannian\nhomogeneous spaces areclassified. Suchmanifoldshave anis ometry groupof dimension\n6, 4 or 3. When the dimension is 6, they correspond to space for ms. When the\ndimension is 3, the manifold has the geometry of the Lie group Sol3. In the case where\nthe dimension of the isometry group is 4, such manifold fibers over a two-dimensional\nspaceformofconstant sectional curvature κ,M2(κ), andits fibersarethetrajectories of\na unit Killing vector field. These last manifolds are usually denoted by E3(κ,τ), where\nτis the constant bundle curvature of the natural projection π:E3(κ,τ)→M2(κ)\nandκ/ne}ationslash= 4τ2. According to the constants κandτwe can classify such spaces. When\nτ= 0,E3(κ,0) =M2(κ)×RwhereM2(κ) is the sphere S2(κ) of curvature κ >0 or\nthe hyperbolic plane H2(κ) of curvature κ <0. When τ/ne}ationslash= 0,E3(κ,τ) is a Berger\n2020Mathematics Subject Classification. 53C42, 53A10, 53C30.\nKey words and phrases. Willmore surface, homogeneous space, constant extrinsic c urvature, Clifford\ntorus, Hopf torus.\n1WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 2\nsphereS3\nb(κ,τ) ifκ >0, a Heisenberg group Nil 3(τ) ifκ= 0 or the universal cover of\nPSL(2,R) ifκ <0.\nIn the last years, the study of surfaces in homogeneous space s with 4-dimensional\nisometry group has attracted the attention of many geometer s. We can say that this\nattention is due to the studies of Abresch, Rosenberg and Mee ks which made possible\ngreat advances in the research in this area [ 1,20,25]. Indeed, Abresch and Rosen-\nberg [1] discovered an holomorphic quadratic differential for surfa ces with constant\nmean curvature in these spaces and solved the Hopf’s theorem for them. Moreover,\nthese spaces are also related to the eight geometries of Thur ston [28]. Furthermore,\nin [12], G´ alvez, Mart´ ınez and Miraconsidered thestudyof thecl assical Bonnet problem\nfor surfaces in the homogeneous 3-manifolds E3(κ,τ). Later on, Rosenberg and Tribuzy\nshowed in [ 26] a rigidity result for a family of complete surfaces in an hom ogeneous\nspace having the same positive extrinsic curvature and sati sfying a certain condition.\nSome years ago, Hu, Lyu and Wang developed in [ 16] a Simons-type integral in-\nequality for immersed minimal closed surfaces into the homo geneous space E3(κ,τ),\nthe equality being satisfied if and only if the surface has par allel second fundamental\nform. When the homogeneous space E3(κ,τ) is the Berger sphere S3\nb(κ,τ) (κ/ne}ationslash= 4τ2),\nthey showed that the equality holds if and only if the surface is the Clifford torus.\nWe recall that the Clifford torus is the only minimal Hopf torus in the Berger sphere.\nRecently, P´ ampano has considered in [ 24] a more general setting, where the ambient\nspace is the total space of a Killing submersion. Specificall y, he studies surface ener-\ngies depending on the mean curvature, which extend the class ical notion of Willmore\nenergy. Furthermore, the author constructs critical tori f or these energy functionals.\nConcerning product spaces, even more recently the second au thor has studied in [ 11]\nimmersed complete surfaces into a product space M2(κ)×Rwith nonnegative constant\nextrinsic curvature. In this setting, he has shown that thes e surfaces must be either\ncylinders when κ=−1, or slices when κ= 1. Our goal is, on the one hand, to present\na Willmore functional for immersed closed surfaces into E3(κ,τ), to obtain its Euler-\nLagrange equation, and as a consequence to present a charact erization result for closed\nWillmore surfaces in S3\nb(κ,τ) in terms of an integral inequality. On the other hand,\nwe extend the technics developed in [ 11] to the study of immersed closed surfaces with\nconstant extrinsic curvature into E3(κ,τ) (τ/ne}ationslash= 0).\nThe outline of the paper goes as follows. In Section 2we describe some basic facts\nabout surfaces in the homogeneous space E3(κ,τ) (τ/ne}ationslash= 0) with isometry group of di-\nmension 4, introducing some relevant families of surfaces i n such homogeneous spaces.\nLater on, working with the Cheng-Yau’s operator, we develop in Section 3a Simons-\ntype formula for these surfaces (cf. Proposition 3.2), as well as a divergence type\nformula involving the Cheng-Yau’s operator (cf. Lemma 3.4). In Section 4we com-\npute the Euler-Lagrange equation for the Willmore function al of an immersed closed\nsurface into an homogeneous space E3(κ,τ) (cf. Proposition 4.1). As an application,\nwe characterize Clifford and Hopf tori as the only Willmore sur faces satisfying a sharp\nSimons-type integral inequality (cf. Theorem 4.1). In the last section, we consider\nclosed surfaces with constant extrinsic curvature and we al so obtain integral inequal-\nities, becoming equalities if and only if the surface is a Hop f torus in a Berger sphere\nS3\nb(κ,τ) (cf. Theorems 5.1and5.2).WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 3\n2.Preliminaries\nIn this section, we will introduce some basic facts and notat ions that will appear\nalong the paper.\nLetκandτbe real numbers. The region Dof the Euclidean space R3given by\nD=/braceleftbigg\nR3, ifκ≥0\nD(2/√−κ)×R,ifκ <0\nand endowed with the homogeneous Riemannian metric\n/an}b∇acketle{t,/an}b∇acket∇i}htR=λ2(dx2+dy2)+(dz+τλ(ydx−xdy))2, λ=1\n1+κ\n4(x2+y2),\nis the so-called Bianchi-Cartan-Vranceanu space (BCV-space) which is usually denoted\nbyE3(κ,τ) := (D,/an}b∇acketle{t,/an}b∇acket∇i}htR).\nAs it is well-known, there exists a Riemannian submersion π:E3(κ,τ)→M2(κ),\nwhereM2(κ) is the 2-dimensional simply connected space form of consta nt curvature κ,\nsuch that πhas constant bundle curvature τand totally geodesic fibers. Furthermore,\nξ=E3is a unit Killing field on X(E3(κ,τ)) which is vertical with respect to π.\nTheBCV-spacesE3(κ,τ) are oriented, and then we can definea vectorial product ∧,\nsuch that if {e1,e2}are linearly independent vectors at a point p, then{e1,e2,e1∧e2}\ndetermines an orientation at p. Then the properties of ξimply (see [ 10]) that for any\nvector field XonX(E3(κ,τ)) the following relation holds\n(1) ∇Xξ=τ(X∧ξ),\n∇being the Levi-Civita connection of E3(κ,τ). Moreover, let us recall that the curva-\nture tensor of E3(κ,τ)1satisfies, (see [ 10]),\nR(X,Y)Z=(κ−3τ2)(/an}b∇acketle{tX,Z/an}b∇acket∇i}htY−/an}b∇acketle{tY,Z/an}b∇acket∇i}htX)\n+(κ−4τ2)/an}b∇acketle{tZ,ξ/an}b∇acket∇i}ht(/an}b∇acketle{tY,ξ/an}b∇acket∇i}htX−/an}b∇acketle{tX,ξ/an}b∇acket∇i}htY)\n+(κ−4τ2)(/an}b∇acketle{tY,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,ξ/an}b∇acket∇i}ht −/an}b∇acketle{tX,Z/an}b∇acket∇i}ht/an}b∇acketle{tY,ξ/an}b∇acket∇i}ht)ξ,(2)\nwhereX,Y,Z∈X(E3(κ,τ)).\nIn what follows, let Σ2be an isometrically immersed connected surface which we\nassume to be orientable and oriented by a globally defined uni t normal vector field N.\nLet us denote by Athe second fundamental form of the immersion with respect to N\nand by∇the Levi-Civita connection of Σ2. Then, the Gauss and Weingarten formulae\nare given by\n∇XY=∇XY+/an}b∇acketle{tA(X),Y/an}b∇acket∇i}htN\nand\nA(X) =−∇XN,\nfor every tangent vector fields X,Y∈X(Σ).\nFurthermore, we can consider a particular function natural ly attached to such a\nsurface Σ2, namely, C=/an}b∇acketle{tN,ξ/an}b∇acket∇i}ht. Let us observe that Cmeasures the cosinus of the\n1We adopt for the (1 ,3)-curvature tensor of the spacetime the following definiti on ([23, Chapter 3]),\nR(X,Y)Z=∇[X,Y]Z−[∇X,∇Y]Z.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 4\nangle determined by the vector fields Nandξ. A direct computation shows that the\nprojection of the vector field ξonX(Σ) is given by\n(3) T=ξ⊤=ξ−CN,\nwhere (·)⊤denotes the tangential component of a vector field in X(E3(κ,τ)) along Σ2.\nThus, we get\n(4) |T|2= 1−C2.\nBesides, from ( 1), (3) and the Gauss and Weingarten formulae we easily obtain the\nintegrability equations,\n(5) ∇XT=C(A−τJ)(X) and ∇C=−(A+τJ)(T),\nwhereJdenotes the (oriented) rotation of angle π/2 onTΣ given by J(X) =N∧X.\nIn particular,\n/an}b∇acketle{tJ(X),J(Y)/an}b∇acket∇i}ht=/an}b∇acketle{tX,Y/an}b∇acket∇i}htandJ2(X) =−X,\nfor every X,Y∈X(Σ). Therefore, from the first equation in ( 5) it easily follows that\n(6) div( T) = 2CH,\nwhere div denotes the divergence operator on Σ2andHstands for the mean curvature\nof Σ2, defined by H=1\n2tr(A). Furthermore, it is immediate to check that\n(7) 4 H2=|A|2+2Ke,\nwhere|A|2= tr(A2) andKe= det(A) denotes the extrinsic curvature of Σ2.\nAs it is well-known, the fundamental equations of Σ2are theGauss equation\nR(X,Y)Z=(κ−3τ2)(/an}b∇acketle{tX,Z/an}b∇acket∇i}htY−/an}b∇acketle{tY,Z/an}b∇acket∇i}htX)\n+(κ−4τ2)/an}b∇acketle{tZ,T/an}b∇acket∇i}ht(/an}b∇acketle{tY,T/an}b∇acket∇i}htX−/an}b∇acketle{tX,T/an}b∇acket∇i}htY)\n+(κ−4τ2)(/an}b∇acketle{tY,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,T/an}b∇acket∇i}ht −/an}b∇acketle{tX,Z/an}b∇acket∇i}ht/an}b∇acketle{tY,T/an}b∇acket∇i}ht)T\n+/an}b∇acketle{tA(X),Z/an}b∇acket∇i}htA(Y)− /an}b∇acketle{tA(Y),Z/an}b∇acket∇i}htA(X),(8)\nwhereRdenote the curvature tensor of Σ2andX,Y,Z∈X(Σ), and the Codazzi\nequation\n(9) ∇A(X,Y)−∇A(Y,X) = (κ−4τ2)C(/an}b∇acketle{tX,T/an}b∇acket∇i}htY−/an}b∇acketle{tY,T/an}b∇acket∇i}htX),\nwhere∇A:X(Σ)×X(Σ)−→X(Σ) denotes the covariant differential of A,\n∇A(X,Y) = (∇YA)(X) =∇YA(X)−A(∇YX),for allX,Y∈X(Σ).\nFrom the Gauss equation ( 8), jointly with ( 4) and (7) it holds\n(10) 2 K= 2τ2+2(κ−4τ2)C2+4H2−|A|2= 2τ2+2(κ−4τ2)C2+2Ke.\nLet us recall now some classical surfaces in E3(κ,τ) which can be constructed in the\nfollowing way. Given anyregular curve αinM2(κ),π−1(α) isanisometrically immersed\nsurface into E3(κ,τ) which is usually known as a Hopf cylinder . Hopf cylinders are flat\nsurfaces, which have ξas a parallel tangent vector field and they are characterized by\nC= 0. Furthermore, these cylinders satisfy\nH=kg/2, K= 0, Ke=−τ2and|Φ|2= 2H2+2τ2,\nwherekgis the geodesic curvature of α.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 5\nMoreover, if αis a closed curve and the Riemannian submersion πhas circular fibers,\nwhich happens just in the case where E3(κ,τ) is a Berger sphere S3\nb(κ,τ), thenπ−1(α)\nis a flat torus which is also called a Hopf torus .\nLet us remember at this point that the Berger sphere S3\nb(κ,τ) is isometric to the\nusual sphere S3={(z,w)∈C2;|z|2+|w|2= 1}endowed with the metric\n/an}b∇acketle{tX,Y/an}b∇acket∇i}ht=4\nκ/parenleftbigg\n/an}b∇acketle{tX,Y/an}b∇acket∇i}htS3+1\nκ/parenleftbig\n4τ2−κ/parenrightbig\n/an}b∇acketle{tX,V/an}b∇acket∇i}htS3/an}b∇acketle{tY,V/an}b∇acket∇i}htS3/parenrightbigg\n,\nwhere/an}b∇acketle{t,/an}b∇acket∇i}htS3stands for the usual metric on the sphere, V(z,w)=J(z,w) = (iz,iw) for\neach (z,w)∈S3andκ,τare real numbers with κ >0 andτ/ne}ationslash= 0. We note that if\nκ= 4τ2thenS3\nb(κ,τ) is, up to homotheties, the round sphere. The Hopf fibration\nπ:S3\nb(κ,τ)→S2(κ), defined by\nπ(z,w) =1√κ/parenleftbigg\nzw,1\n2/parenleftbig\n|z|2−|w|2/parenrightbig/parenrightbigg\n,\nis a Riemannian submersion whose fibers are geodesics. The ve rtical unit Killing vector\nfield is given by ξ=κ\n4τV. A particular Hopf torus in S3\nb(κ,τ) is theClifford torus given\nby\n{(z,w)∈S3\nb(κ,τ);|z|2=|w|2= 1/2}.\nIt is well-known that the Clifford torus is the only minimal Hop f torus in any Berger\nsphere (see for instance [ 30]).\nLet us finish this section by recalling a classification resul t for parallel surfaces in\nE3(κ,τ), proved by Belkhelfa, Dillen and Inoguchi in [ 6]. From now on, we will under-\nstand by a parallel surface a surface with parallel second fu ndamental form.\nLemma 2.1. [6, Theorem 8.2] LetΣ2be an isometrically immersed parallel surface\ninto the homogeneous space E3(κ,τ),κ−4τ2/ne}ationslash= 0. Then,\n(1) ifτ/ne}ationslash= 0 Σ2is a piece of a Hopf cylinder over a Riemannian circle in M2(κ),\nthat is, over a closed curve in M2(κ)with constant geodesic curvature.\n(2) ifτ= 0 Σ2is either a piece of a slice in M2(κ)×Ror of a Hopf cylinder over\na Riemannian circle in M2(κ).\n3.A Simons-type formula for the Cheng-Yau operator in E3(κ,τ)\nIn consideration of the foregoing we are going to compute the Laplacian of |A|2.\nFirst and foremost, we recall the following Weitzenb¨ ock fo rmula (see for instance [ 22])\n(11)1\n2∆|A|2=1\n2∆/an}b∇acketle{tA,A/an}b∇acket∇i}ht=|∇A|2+/an}b∇acketle{t∆A,A/an}b∇acket∇i}ht,\nwhere ∆ A:X(Σ)−→X(Σ) is the rough Laplacian of the second fundamental form,\nthat is,\n(12) ∆ A(X) = tr/parenleftbig\n∇2A(X,·,·)/parenrightbig\n=2/summationdisplay\ni=1∇2A(X,ei,ei),\n{e1,e2}being an orthonormal frame on X(Σ) and ∇2A(X,Y,Z) = (∇Z∇A)(X,Y)\nfor allX,Y,Z∈X(Σ). In this setting, on the one hand we obtain from the Codazz iWILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 6\nequation ( 9) and theintegrability equations ( 5) thefollowing symmetry in the two firsts\nvariables of ∇2A,\n∇2A(X,Y,Z) =∇2A(Y,X,Z)−(κ−4τ2)/an}b∇acketle{t(A+τJ)(T),Z/an}b∇acket∇i}ht(/an}b∇acketle{tX,T/an}b∇acket∇i}htY−/an}b∇acketle{tY,T/an}b∇acket∇i}htX)\n+(κ−4τ2)C2(/an}b∇acketle{tX,(A−τJ)(Z)/an}b∇acket∇i}htY−/an}b∇acketle{tY,(A−τJ)(Z)/an}b∇acket∇i}htX).(13)\nOn the other hand, it is not difficult to see that\n(14) ∇2A(X,Y,Z) =∇2A(X,Z,Y)+R(Y,Z)A(X)−A(R(Y,Z)X).\nMakingY=Z=eiin (13) and taking traces, we have\n2/summationdisplay\ni=1∇2A(X,ei,ei) =2/summationdisplay\ni=1∇2A(ei,X,ei)−(κ−4τ2)C2(2HX−(A+τJ)(X))\n−(κ−4τ2)(/an}b∇acketle{tX,T/an}b∇acket∇i}ht(A+τJ)(T)−/an}b∇acketle{tA(T),T/an}b∇acket∇i}htX).(15)\nFurthermore, from ( 14) it yields\n(16) ∇2A(ei,X,ei) =∇2A(ei,ei,X)+R(X,ei)A(ei)−A(R(X,ei)ei).\nObserve now that, using the Gauss equation ( 8), we get\n2/summationdisplay\ni=1R(X,ei)A(ei) =(κ−3τ2)(A(X)−2HX)−|A|2A(X)+A3(X)\n+(κ−4τ2)(/an}b∇acketle{tA(T),T/an}b∇acket∇i}htX−/an}b∇acketle{tX,T/an}b∇acket∇i}htA(T))\n+(κ−4τ2)(2H/an}b∇acketle{tX,T/an}b∇acket∇i}ht−/an}b∇acketle{tA(T),X/an}b∇acket∇i}ht)T\nand\n2/summationdisplay\ni=1A(R(X,ei)ei) =−(κ−3τ2)A(X)+A3(X)−2HA2(X)+(κ−4τ2)|T|2A(X).\nThus, inserting these two last equalities in ( 16),\n2/summationdisplay\ni=1∇2A(ei,X,ei) =2/summationdisplay\ni=1∇2A(ei,ei,X)+2(κ−3τ2)(A(X)−HX)+2HA2(X)\n+(κ−4τ2)/parenleftbig\n/an}b∇acketle{tA(T),T/an}b∇acket∇i}htX−/an}b∇acketle{tX,T/an}b∇acket∇i}htA(T)−|T|2A(X)/parenrightbig\n+(κ−4τ2)(2H/an}b∇acketle{tX,T/an}b∇acket∇i}ht−/an}b∇acketle{tA(T),X/an}b∇acket∇i}ht)T−|A|2A(X).(17)\nObserve now that, since the trace commutes with the Levi-Civ ita connection,\n2/summationdisplay\ni=1∇2A(ei,ei,X) = tr(∇X∇A) =∇X(tr(∇A)).\nWe claim that\n(18) tr( ∇A) = 2∇H+C(κ−4τ2)T.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 7\nIndeed, using the Codazzi equation ( 9),\n/an}b∇acketle{t∇A(ei,ei),X/an}b∇acket∇i}ht=/an}b∇acketle{t(∇eiA)(ei),X/an}b∇acket∇i}ht=/an}b∇acketle{tei,(∇eiA)(X)/an}b∇acket∇i}ht=/an}b∇acketle{tei,∇A(X,ei)/an}b∇acket∇i}ht\n=/an}b∇acketle{tei,∇A(ei,X)/an}b∇acket∇i}ht+(κ−4τ2)C(/an}b∇acketle{tX,T/an}b∇acket∇i}ht−/an}b∇acketle{tX,ei/an}b∇acket∇i}ht/an}b∇acketle{tT,ei/an}b∇acket∇i}ht),\nwhich implies that\n/an}b∇acketle{ttr(∇A),X/an}b∇acket∇i}ht=2/summationdisplay\ni=1/bracketleftbig\n/an}b∇acketle{tei,(∇XA)(ei)/an}b∇acket∇i}ht+C(κ−4τ2)(/an}b∇acketle{tX,T/an}b∇acket∇i}ht−/an}b∇acketle{tX,ei/an}b∇acket∇i}ht/an}b∇acketle{tT,ei/an}b∇acket∇i}ht)/bracketrightbig\n= 2/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht+(κ−4τ2)C/an}b∇acketle{tX,T/an}b∇acket∇i}ht,\nfor allX∈X(Σ), so the claim is proved. Hence, from ( 18) and (5), it holds\n(19)∇X(tr(∇A)) = 2∇X∇H−(κ−4τ2)/an}b∇acketle{t(A+τJ)(T),X/an}b∇acket∇i}htT+(κ−4τ2)C2(A−τJ)(X).\nTherefore, putting ( 15), (17) and (19) in (12),\n∆A(X) =2∇X∇H+2(κ−3τ2)(A(X)−HX)−|A|2A(X)+2HA2(X)\n+(κ−4τ2)/parenleftbig\n2/an}b∇acketle{tA(T),T/an}b∇acket∇i}htX−2/an}b∇acketle{tX,T/an}b∇acket∇i}htA(T)−|T|2A(X)/parenrightbig\n+(κ−4τ2)(2H/an}b∇acketle{tX,T/an}b∇acket∇i}ht−2/an}b∇acketle{tA(T),X/an}b∇acket∇i}ht)T+ 2(κ−4τ2)C2(A(X)−HX)\n−τ(κ−4τ2)(/an}b∇acketle{tJ(T),X/an}b∇acket∇i}htT+/an}b∇acketle{tX,T/an}b∇acket∇i}htJ(T)).\nConsequently,\n/an}b∇acketle{t∆A,A/an}b∇acket∇i}ht=2tr(A◦HessH)+2(κ−3τ2)(|A|2−2H2)+2(κ−4τ2)C2(|A|2−2H2)\n+2(κ−4τ2)/parenleftbig\n3H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht −2/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht−τ/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht/parenrightbig\n−(κ−4τ2)|T|2|A|2−|A|4+2Htr(A3).(20)\nNow, taking into account the characteristic polynomial of A, we observe that\n(21) 4 H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht −2/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht= 2|T|2Ke.\nBesides that, from ( 7) it holds\n2(κ−3τ2)(|A|2−2H2)+(κ−4τ2)(1−C2)(2Ke−|A|2)\n= 2(κ−3τ2)(|A|2−2H2)−2(κ−4τ2)(1−C2)(|A|2−2H2)\n= 2(|A|2−2H2)(τ2+(κ−4τ2)C2).(22)\nMoreover, it is easy to check that tr( A3) = 3H|A|2−4H3, so again from ( 7) we deduce\nthat\n(23) −|A|4+2Htr(A3) =−|A|4+6H2|A|2−8H4= 2(|A|2−2H2)Ke.\nHence, taking into account ( 10), (21), (22) and (23), (20) reads\n/an}b∇acketle{t∆A,A/an}b∇acket∇i}ht= 2tr(A◦HessH)+2(|A|2−2H2)K+2(κ−4τ2)C2(|A|2−2H2)\n+2(κ−4τ2)/parenleftbig\nH/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht−/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht−τ/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht/parenrightbig\n,WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 8\nso (11) yields\n1\n2∆|A|2=|∇A|2+2tr(A◦HessH)\n+2(|A|2−2H2)K+2(κ−4τ2)C2(|A|2−2H2)\n+2(κ−4τ2)/parenleftbig\nH/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht−/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht−τ/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht/parenrightbig\n.(24)\nRemark 3.1. Let usobserve that formula( 24) was already obtained in[ 16]. Infact, let\nusconsideralocalorthonormalframe {e1,e2}suchthat A(e1) =λ1e1andA(e2) =λ2e2.\nMoreover, by the definition of J, we must have J(e1) =e2andJ(e2) =−e1. Taking\ninto account these two facts and writing T=/an}b∇acketle{tT,e1/an}b∇acket∇i}hte1+/an}b∇acketle{tT,e2/an}b∇acket∇i}hte2, we have\n/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht= (λ2−λ1)/an}b∇acketle{tT,e1/an}b∇acket∇i}ht/an}b∇acketle{tT,e2/an}b∇acket∇i}ht,\nso we recover [ 16, Lemma 3.1]. However, we have included the proof for the sake of\ncompleteness, and because it represents an alternative rea soning based on tensorial\nanalysis.\nNevertheless, our aim in this section is to obtain a Simons-t ype formula for the\nCheng-Yau’s operator. To this respect, following [ 9] we introduce the Cheng-Yau’s\noperator /squareacting on any smooth function u: Σ2→Rgiven by\n/squareu= tr(P◦Hessu),\nwherePdenote the first Newton transformation of A, that is, P:X(Σ)→X(Σ) is the\noperator given by\n(25) P= 2HI−A,\nwhich is also a self-adjoint linear operator which commutes withAandsatisfies tr( P) =\n2H.\nTakingu= 2H, from equation ( 7) we obtain the following,\n/square(2H) = tr(P◦Hess(2H))\n= 2H∆(2H)−2tr(A◦HessH)\n=1\n2∆(2H)2−4|∇H|2−2tr(A◦HessH)\n=1\n2∆|A|2+∆Ke−4|∇H|2−2tr(A◦HessH).(26)\nInserting ( 24) in previous equality, we get\n/square(2H) =∆Ke+|∇A|2−4|∇H|2+2(|A|2−2H2)K+2(κ−4τ2)C2/parenleftbig\n|A|2−2H2/parenrightbig\n+2(κ−4τ2)/parenleftbig\nH/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht−/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht−τ/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht/parenrightbig\n.(27)\nFor our purpose, it will be more appropriate to deal with the t raceless part of A,\nwhich is given by Φ = A−HI, withIthe identity operator on X(Σ). Then, tr(Φ) = 0\nand\n(28) |Φ|2=|A|2−2H2≥0,WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 9\nwith equality at p∈Σ2if and only if pis an umbilical point. In contrast to the case\nwhere the ambient is a Riemannian product, it was proved in [ 27] that there does not\nexist any totally umbilical surface in E3(κ,τ) withτ/ne}ationslash= 0.\nNow, from the characteristic polynomial of Φ and identity ( 28), the following equal-\nities hold,\n−2/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht+2H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht=−|Φ|2|T|2−2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht\nand\n2C2|A|2−4H2C2= 2C2|Φ|2.\nBesides this, equations ( 4) and (10) give us\n2K+(κ−4τ2)/parenleftbig\n2C2−|T|2/parenrightbig\n= 2Ke+5(κ−4τ2)C2−κ+6τ2.\nTherefore, inserting these three last equations in ( 27), we have finally shown the fol-\nlowing Simons-type formula for the Cheng-Yau’s operator.\nProposition 3.2. LetΣ2be an isometrically immersed surface into an homogeneous\nspaceE3(κ,τ). Then,\n/square(2H) = ∆Ke+|∇A|2−4|∇H|2+|Φ|2/parenleftbig\n2Ke+(κ−4τ2)(5C2−1)+2τ2/parenrightbig\n−2(κ−4τ2)(H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht).\nRemark 3.3. Whenτ= 0, as it was said in the Introduction, the homogeneous space\nE3(κ,τ) is exactly the product space M2(κ)×R, whereM2(κ) is a space form with\nconstant sectional curvature κ. Thus, Proposition 3.2extends [ 11, Proposition 1.2].\nLet us finish this section by showing a nice divergence formul a involving the Cheng-\nYau’s operator.\nLemma 3.4. LetΣ2be an isometrically immersed surface into an homogoneous sp ace\nE3(κ,τ). Then,\n(29) div( P(2∇H)) =/square(2H)−2C(κ−4τ2)T(H).\nProof.Observe that by a standard tensor computation\n(30) div( P(2∇H)) =/square(2H)+2/an}b∇acketle{tdivP,∇H/an}b∇acket∇i}ht,\nwhere\ndiv(P) =2/summationdisplay\ni=1∇P(ei,ei)\nwith\n∇P(X,Y) = (∇YP)X=∇Y(PX)−P(∇YX),\nfor every X,Y∈X(Σ).\nIt remains to compute the last term of equation ( 30). Indeed, from ( 25),\n∇P(X,Y) = 2Y(H)X−∇A(X,Y),\nfor every X,Y∈X(Σ). Then, ( 18) implies that\n(31) div( P) = tr(∇P) = 2∇H−2∇H−(κ−4τ2)CT=−C(κ−4τ2)T,\nso finally ( 29) follows from ( 30) and (31). /squareWILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 10\n4.Willmore surfaces in S3\nb(κ,τ)\nLetx: Σ2→M3(κ) be an isometrically immersed orientable closed, i.e. comp act\nwithout boundary, surface into the Riemannian space form M3(κ) with constant sec-\ntional curvature κ. The Willmore functional is defined by\nW(x) =/integraldisplay\nΣ(H2+κ)dA,\nwheredAdenotes the area element of the induced metric on Σ2. Associated to this\nfunctional, there is the famous Willmore conjecture, solve d in 2012 by Marques and\nNeves [19], which guarantees that this integral is at least 2 π2when Σ2is an immersed\ntorus into R3. We say that Σ2is aWillmore surface if it is a stationary point for\nthe functional W. Moreover, it is well known that Wis a conformal invariant and its\nEuler-Lagrange equation is given by (see [ 7,31])\n∆H+|Φ|2H= 0.\nFor our interests, let x: Σ2→E3(κ,τ) be an isometrically immersed orientable\nclosed surface into the homogeneous 3-manifold E3(κ,τ). Following Weiner [ 31], we\nconsider the following Willmore functional,\nW(x) =/integraldisplay\nΣ(H2+K)dA,\nwhere at any p∈Σ2,Kdenotes the sectional curvature of TpΣ inE3(κ,τ), which\nfollowing ( 2)-(4) can be expressed as\n(32) K=τ2+(κ−4τ2)C2.\nIn the following result we obtain the Euler-Lagrange equati on ofW, extending the\nresult of Weiner [ 31, Theorem 2.2] for immersed surfaces into the homogeneous sp ace\nE3(κ,τ).\nProposition 4.1. Letx: Σ2→E3(κ,τ)be an isometrically immersed orientable closed\nsurface. Then xis a stationary point of Wif and only if\n∆H+/parenleftbig\n|Φ|2+(κ−4τ2)(1+C2)/parenrightbig\nH−2(κ−4τ2)/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht= 0.\nProof.Let us consider a variation of x, that is, a smooth map X: (−ε,ε)×Σ2→\nE3(κ,τ) satisfying that for each t∈(−ε,ε), the map Xt: Σ2→E3(κ,τ), given by\nXt(p) =X(t,p), is an immersion and X0=x. Then, we can compute the first variation\nofWalongX, that is,\nd\ndtW(Xt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=d\ndt/integraldisplay\nΣ(H2\nt+Kt)dAt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0\n=/integraldisplay\nΣ/parenleftbiggd\ndt(H2\nt+Kt)dAt+(H2\nt+Kt)d\ndt(dAt)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0,(33)\nwhere, for each t∈(−ε,ε),HtandKtstand, respectively, for the mean curvature of\nΣ2and the sectional curvature of TpΣ inE3(κ,τ) with respect to the metric induced\nbyXtanddAtdenotes its volume element.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 11\nObserve that, on the one hand, the following identity is well known (see for in-\nstance [5])\n2dHt\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0= ∆f+2/an}b∇acketle{t∇H,Y⊤/an}b∇acket∇i}ht+(Ric(N,N)+|A|2)f,\nRic being the Ricci curvature tensor of E3(κ,τ) andY=∂X\n∂t/vextendsingle/vextendsingle\nt=0the variational vector\nfield related to the variation X, which can be decomposed as Y=Y⊤+fNwith\nf=/an}b∇acketle{tY,N/an}b∇acket∇i}ht.\nOnthe other hand, denoting by Ntthe unit normal vector field along Σ2with respect\nto the metric induced by Xt, sinceN0=Nit holds\ndKt\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=d\ndt/parenleftbig\nτ2+(κ−4τ2)/an}b∇acketle{tNt,ξ/an}b∇acket∇i}ht2/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0= 2(κ−4τ2)/an}b∇acketle{tNt,ξ/an}b∇acket∇i}htd\ndt/an}b∇acketle{tNt,ξ/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0.\nSinceY=∂X\n∂t/vextendsingle/vextendsingle\nt=0is a coordinate field, there exists an orthonormal frame {e1,e2}in\nX(Σ) such that [ Y,ek] = 0, for any k= 1,2. Thus, a direct computation gives us\n∇f=−∇YN−A(Y⊤).\nThen, from the integrability equations ( 5) we get\ndKt\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=−2(κ−4τ2)C/parenleftBig\n/an}b∇acketle{t∇f,T/an}b∇acket∇i}ht+/an}b∇acketle{t(A+τJ)(T),Y⊤/an}b∇acket∇i}ht/parenrightBig\n.\nFurthermore, by using Lemma 4 .2 of [4] (see also [ 8, Lemma 5.4]), we have\nd\ndt(dAt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=/parenleftBig\n−2Hf+div(Y⊤)/parenrightBig\ndA.\nUsing the previous equalities we obtain\nd\ndt(H2\nt+Kt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0dA=2HdHt\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0dA+dKt\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0dA\n=H/parenleftbig\n∆f+(Ric(N,N)+|A|2)f/parenrightbig\ndA+/an}b∇acketle{t∇H2,Y⊤/an}b∇acket∇i}htdA\n−2(κ−4τ2)C/parenleftBig\n/an}b∇acketle{t∇f,T/an}b∇acket∇i}ht+/an}b∇acketle{t(A+τJ)(T),Y⊤/an}b∇acket∇i}ht/parenrightBig\ndA(34)\nand\n(35) ( H2+K)d\ndt(dAt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=−2H(H2+K)fdA+(H2+K)div(Y⊤)dA.\nLet us also observe that\ndiv(H2Y⊤) =H2div(Y⊤)+/an}b∇acketle{t∇H2,Y⊤/an}b∇acket∇i}ht.\nFrom (5) it also holds\ndiv(KY⊤) =Kdiv(Y⊤)−2(κ−4τ2)C/an}b∇acketle{t(A+τJ)(T),Y⊤/an}b∇acket∇i}ht.\nThen, it follows from ( 35) that\n(H2+K)d\ndt(dAt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=−2H(H2+K)fdA+div(H2Y⊤)dA−/an}b∇acketle{t∇H2,Y⊤/an}b∇acket∇i}htdA\n+div(KY⊤)dA+2(κ−4τ2)C/an}b∇acketle{t(A+τJ)(T),Y⊤/an}b∇acket∇i}htdA.(36)WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 12\nHence, replacing ( 34) and (36) in (33), we get\nd\ndtW(Xt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=/integraldisplay\nΣ/parenleftbig\nH∆f+H(Ric(N,N)+|A|2)f−2H(H2+K)f/parenrightbig\ndA\n−2(κ−4τ2)/integraldisplay\nΣC/an}b∇acketle{t∇f,T/an}b∇acket∇i}htdA\n=/integraldisplay\nΣ/parenleftbig\n∆H+(Ric(N,N)+|A|2)H−2H(H2+K)/parenrightbig\nfdA\n−2(κ−4τ2)/integraldisplay\nΣC/an}b∇acketle{t∇f,T/an}b∇acket∇i}htdA.\nBesides this, from ( 5) and (6),\ndiv(CfT) =Cfdiv(T)+C/an}b∇acketle{t∇f,T/an}b∇acket∇i}ht+f/an}b∇acketle{t∇C,T/an}b∇acket∇i}ht\n= 2HC2f+C/an}b∇acketle{t∇f,T/an}b∇acket∇i}ht−f/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht.\nTherefore\nd\ndtW(Xt)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0=/integraldisplay\nΣ/parenleftbig\nH∆f+H(Ric(N,N)+|A|2)f−2H(H2+K)f/parenrightbig\ndA\n+2(κ−4τ2)/integraldisplay\nΣ/parenleftbig\n2HC2−/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht/parenrightbig\nfdA\n=/integraldisplay\nΣ/parenleftbig\n∆H+(Ric(N,N)+|A|2)H−2H(H2+K)/parenrightbig\nfdA\n+2(κ−4τ2)/integraldisplay\nΣ/parenleftbig\n2HC2−/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht/parenrightbig\nfdA.\nConsequently xis a stationary point of the Willmore functional Wif and only if\n(37) ∆H+/parenleftbig\n|Φ|2+Ric(N,N)−2K+4(κ−4τ2)C2/parenrightbig\nH−2(κ−4τ2)/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht= 0.\nFinally, by an straightforward computation from ( 2) and (4) we easily obtain\nRic(N,N) =κ−2τ2−(κ−4τ2)C2,\nwhich jointly with ( 32) and (37) yields the desired result. /square\nRemark 4.2. It is not difficult to check that minimal tori and Hopf cylinder s over a\ncurveofgeodesiccurvature kg=/radicalbig\n2(2τ2−κ), forallκ,τ∈Rwithκ <2τ2satisfy (37).\nSo, they are stationary points of the Willmore functional W.\nBefore presenting our classification result for Willmore su rfaces in E3(κ,τ), we firstly\nneed the following lemma whose proof follows the ideas devel oped in [ 13, Lemma 2.1]\n(see also [ 17]).\nLemma 4.3. IfΣ2is an isometrically immersed orientable surface into the ho moge-\nneous space E3(κ,τ), then\n(38) |∇A|2≥3|∇H|2+2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht.\nProof.Given any a∈R,let us consider the following tensor\nF(X,Y,Z) =/an}b∇acketle{t∇A(X,Y),Z/an}b∇acket∇i}ht\n+a(/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht/an}b∇acketle{tY,Z/an}b∇acket∇i}ht+/an}b∇acketle{t∇H,Y/an}b∇acket∇i}ht/an}b∇acketle{tX,Z/an}b∇acket∇i}ht+/an}b∇acketle{t∇H,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,Y/an}b∇acket∇i}ht).WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 13\nA direct computation gives\nF(X,Y,Z)2=/an}b∇acketle{t∇A(X,Y),Z/an}b∇acket∇i}ht2+2aQ1(X,Y,Z)+a2Q2(X,Y,Z),\nwhere\nQ1(X,Y,Z) =/an}b∇acketle{t∇A(X,Y),Z/an}b∇acket∇i}ht(/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht/an}b∇acketle{tY,Z/an}b∇acket∇i}ht+/an}b∇acketle{t∇H,Y/an}b∇acket∇i}ht/an}b∇acketle{tX,Z/an}b∇acket∇i}ht+/an}b∇acketle{t∇H,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,Y/an}b∇acket∇i}ht)\nand\nQ2(X,Y,Z) =/parenleftbig\n/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht2/an}b∇acketle{tY,Z/an}b∇acket∇i}ht2+/an}b∇acketle{t∇H,Y/an}b∇acket∇i}ht2/an}b∇acketle{tX,Z/an}b∇acket∇i}ht2+/an}b∇acketle{t∇H,Z/an}b∇acket∇i}ht2/an}b∇acketle{tX,Y/an}b∇acket∇i}ht2/parenrightbig\n+2(/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht/an}b∇acketle{tY,Z/an}b∇acket∇i}ht/an}b∇acketle{t∇H,Y/an}b∇acket∇i}ht/an}b∇acketle{tX,Z/an}b∇acket∇i}ht+/an}b∇acketle{t∇H,X/an}b∇acket∇i}ht/an}b∇acketle{tY,Z/an}b∇acket∇i}ht/an}b∇acketle{t∇H,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,Y/an}b∇acket∇i}ht)\n+2/an}b∇acketle{t∇H,Y/an}b∇acket∇i}ht/an}b∇acketle{tX,Z/an}b∇acket∇i}ht/an}b∇acketle{t∇H,Z/an}b∇acket∇i}ht/an}b∇acketle{tX,Y/an}b∇acket∇i}ht.\nIn order to compute these last terms, let us take {e1,e2}an orthonormal frame on\nX(Σ). Then, it is not difficult to check that\n2/summationdisplay\ni,j,k/an}b∇acketle{t∇A(ei,ej),ek/an}b∇acket∇i}ht2=|∇A|2and2/summationdisplay\ni,j,kQ2(ei,ej,ek) = 12|∇H|2.\nBesides that, from Codazzi equation and ( 18), we have\n2/summationdisplay\ni,j,kQ1(ei,ej,ek) =2/summationdisplay\ni,j=1(/an}b∇acketle{t∇A(ei,ej),ej/an}b∇acket∇i}ht/an}b∇acketle{t∇H,ei/an}b∇acket∇i}ht+/an}b∇acketle{t∇A(ei,ej),ei/an}b∇acket∇i}ht/an}b∇acketle{t∇H,ej/an}b∇acket∇i}ht)\n+2/summationdisplay\ni=1/an}b∇acketle{t∇A(ei,ei),∇H/an}b∇acket∇i}ht\n=6|∇H|2+2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht.\nHence,\n|F|2=|∇A|2+2a/parenleftbig\n6|∇H|2+2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht/parenrightbig\n+12a2|∇H|2.\nTakinga=−1/2 we obtain ( 38). /square\nWe can finally present our first main result.\nTheorem 4.1. LetΣ2be an isometrically immersed orientable closed Willmore su rface\ninto an homogeneous space E3(κ,τ). Then,\n/integraldisplay\nΣ/parenleftbig\n|Φ|4−/parenleftbig\n2τ2−(κ−4τ2)(1−3C2)/parenrightbig\n|Φ|2/parenrightbig\ndA\n−(κ−4τ2)/integraldisplay\nΣ/parenleftbig\n|∇C|2+(Ke+τ2)(1−5C2)+2τ2(1−3C2)/parenrightbig\ndA≥0.\nwhere the equality holds if and only if Σ2is a parallel surface.\nIn particular, if κ <2τ2the equality holds if and only if E3(κ,τ) =S3\nb(κ,τ)and\nΣ2is either a Clifford torus or a Hopf torus over a closed curve of g eodesic curvature/radicalbig\n2(2τ2−κ)onS2(κ).WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 14\nProof.Firstly, taking into account ( 28), (26) can be written as follows,\n/square(2H) = 4H∆H−2tr(A◦HessH)\n= 2H∆H−1\n2∆|Φ|2−2|∇H|2+1\n2∆|A|2−2tr(A◦HessH),\nwhere ∆H2= 2H∆H+2|∇H|2has been used. Consequently, by ( 24),\n/square(2H) =2H∆H−1\n2∆|Φ|2+|∇A|2−2|∇H|2\n+|Φ|2/parenleftbig\n2Ke+(κ−4τ2)(5C2−1)+2τ2/parenrightbig\n−2(κ−4τ2)(H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht).(39)\nLet us observe now that from Lemma 4.3we get\n|∇A|2−2|∇H|2≥ |∇H|2+2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht\n≥2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht,(40)\nwhere the equality holds if and only if\n|∇A|2= 3|∇H|2+2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht= 0,\nthat is, if and only if Σ2is a parallel surface. Then, from Lemma 3.4and taking into\naccount ( 39) and (40) we obtain the following inequality,\ndiv(P(2∇H)) =/square(2H)−2(κ−4τ2)C/an}b∇acketle{t∇H,T/an}b∇acket∇i}ht\n≥2H∆H−1\n2∆|Φ|2+|Φ|2/parenleftbig\n2Ke+(κ−4τ2)(5C2−1)+2τ2/parenrightbig\n−2(κ−4τ2)(H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht).\nTherefore, the divergence theorem yields\n−2/integraldisplay\nΣH∆HdA≥/integraldisplay\nΣ|Φ|2/parenleftbig\n2Ke+(κ−4τ2)(5C2−1)+2τ2/parenrightbig\ndA\n−2(κ−4τ2)/integraldisplay\nΣ(H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht)dA.\nOn the one hand, from Proposition 4.1we can write\n2H∆H=−2H2/parenleftbig\n|Φ|2+(κ−4τ2)(1+C2)/parenrightbig\n+4(κ−4τ2)H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht\n=−/parenleftbig\n|Φ|2+2Ke/parenrightbig/parenleftbig\n|Φ|2+(κ−4τ2)(3C2−1)/parenrightbig\n+4(κ−4τ2)H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht\n=−|Φ|2/parenleftbig\n2Ke+|Φ|2+(κ−4τ2)(3C2−1)/parenrightbig\n−2(κ−4τ2)(3C2−1)Ke\n+4(κ−4τ2)H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht,(41)\nwhere we have used that\n(42) |Φ|2−2H2=−2Ke,\nwhich follows from ( 7) and (28). Hence,\n0≥/integraldisplay\nΣ|Φ|2/parenleftbig\n−|Φ|2+2(κ−4τ2)C2+2τ2/parenrightbig\ndA+2(κ−4τ2)/integraldisplay\nΣ(1−3C2)KedA\n+2(κ−4τ2)/integraldisplay\nΣ(H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht−τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht)dA.(43)WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 15\nOn the other hand, by using A2−2HA+KeI=A2−2HΦ−/parenleftbig\n|Φ|2+Ke/parenrightbig\nI= 0 and\nthe integrability equation ( 5), we easily obtain\n(44) |∇C|2= 2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+2τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht+/parenleftbig\n|Φ|2+Ke+τ2/parenrightbig\n|T|2.\nNow, let us consider the local orthonormal frame on X(Σ),{e1,e2}such that e1=T\n|T|\nande2=J(e1) we get\ndiv(J(T)) =−/an}b∇acketle{tJ(T),∇e1e1/an}b∇acket∇i}ht+e2(|T|),\nwhich using once more the integrability equations ( 5) yields\n(45) div( J(T)) = 2τC.\nSo, from ( 5) and (45),\ndiv(τCJ(T)) = 2τ2C2−τ/an}b∇acketle{t(A+τJ)T,J(T)/an}b∇acket∇i}ht=−τ/an}b∇acketle{tφ(T),J(T)/an}b∇acket∇i}ht−τ2(1−3C2).\nThus, by ( 44)\n2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht−2τ/an}b∇acketle{tφ(T),J(T)/an}b∇acket∇i}ht=|∇C|2+4div(τCJ(T))+τ2(3−11C2)\n−(|Φ|2+Ke)(1−C2)\nTherefore, by ( 43) we obtain\n0≥/integraldisplay\nΣ|Φ|2/parenleftbig\n−|Φ|2+(κ−4τ2)(3C2−1)+2τ2/parenrightbig\ndA\n+(κ−4τ2)/integraldisplay\nΣ/parenleftbig\n|∇C|2+(Ke+τ2)(1−5C2)+2τ2(1−3C2)/parenrightbig\ndA.(46)\nwhich is the desired inequality.\nMoreover, as we have remarked before, the equality holds in ( 46) if and only if Σ2is a\nclosed parallel surface. Then, from Lemma 2.1Σ2is either a Hopf torus (necessarily in\nS3\nb(κ,τ)) over a Riemannian circle in S2(κ), or a piece of a slice in M2(κ)×R. However,\nsince Σ2is closed this last case only occurs in the case τ= 0 and κ >0, which do not\nsatisfy the assumption κ <2τ2.\nConsequently, Σ2is a Hopf torus in S3\nb(κ,τ), so in particular C= 0 and Ke=−τ2.\nHence, (41) reads\n0 =/parenleftbig\n−|Φ|2+2τ2/parenrightbig\n(|Φ|2+κ−4τ2.\nThen, either |Φ|2= 2τ2, which implies that H= 0 and Σ2is the Clifford torus, or\n|Φ|2+κ−4τ2= 0. Thus, from ( 42) we get H=/radicalBig\n2τ2−κ\n2and, consequently, Σ2is\nisometric to a Hopf torus in S3\nb(κ,τ) over a curve of geodesic curvature/radicalbig\n2(2τ2−κ)\nonS2(κ), for 0< κ <2τ2. /square\n5.Classification results for constant extrinsic curvature c losed\nsurfaces\nLet us begin by obtaining some new interesting divergence fo rmulae, which will play\na fundamental role in the proof of the main results in this sec tion.\nLemma 5.1. LetΣ2be an isometrically immersed surface into the homogeneous s pace\nE3(κ,τ). Then the following divergence formulae hold on Σ2,WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 16\n(a)div(∇TT) =K|T|2+T(divT)+|∇T|2−4τ2C2.\n(b)div(div(T)T) =T(divT)+4H2C2.\n(c)div(|T|∇|T|) =K|T|2+T(divT)+|∇T|2−2τ2|T|2−2τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht.\nProof.Firstly, let usobservethat items ( a)and(b) havealready beenproved in[ 29](see\nalso [16, Lemma 3.2]). However, we will include the proofs for the sak e of completeness.\nFrom the integrability equations ( 5) it is immediate to check that\n(47)\ndiv(∇TT) = div(C(A−τJ)(T)) =−/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht+τ2|T|2+Cdiv(A(T))−τCdiv(J(T)).\nOn the one hand, given a local orthonormal frame {e1,e2}onX(Σ) diagonalizing A,\nfrom the Codazzi equation ( 9) it holds\ndiv(A(T)) =2/summationdisplay\ni=1/an}b∇acketle{t(∇eiA)(T),ei/an}b∇acket∇i}ht+2/summationdisplay\ni=1/an}b∇acketle{tA(∇eiT),ei/an}b∇acket∇i}ht\n= tr(∇TA)+C(κ−4τ2)|T|2+2/summationdisplay\ni=1/an}b∇acketle{t∇eiT,A(ei)/an}b∇acket∇i}ht\n= 2T(H)+C(κ−4τ2)|T|2+C|A|2,(48)\nwhere in the last equality we have used again ( 5) and the fact that the trace commutes\nwith the Levi-Civita connection.\nOn the other hand, ( 6) yields\n(49) T(div(T)) =−2H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht+2CT(H).\nThen, taking into account ( 45), (48) and (49), (47) reads\ndiv(∇TT) =K|T|2+T(div(T))+C2|A|2−2τ2C2,\nwhere we have used ( 10) and (21). Finally, item ( a) follows by observing that\n(50) |∇T|2=2/summationdisplay\ni,j=1/an}b∇acketle{t∇eiT,ej/an}b∇acket∇i}ht2=C2(|A|2+2τ2),\nfor any{e1,e2}local orthonormal frame on X(Σ).\nItem (b) follows directly from ( 6).\nWith respect to item ( c), a direct computation from ( 5) guarantees us that\n(51) |T|∇|T|=C(A+τJ)(T).\nThen, taking divergences in ( 51),\n(52) div( |T|∇|T|) = div(A(T))C+τdiv(J(T))C+/an}b∇acketle{t∇C,(A+τJ)(T)/an}b∇acket∇i}ht.\nIt is easy to check from ( 21) and from the integrability equations ( 5) that\n/an}b∇acketle{t∇C,(A+τJ)(T)/an}b∇acket∇i}ht=−/an}b∇acketle{tA2(T),T/an}b∇acket∇i}ht−2τ/an}b∇acketle{tA(T),J(T)/an}b∇acket∇i}ht −τ2|T|2\n=−2H/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht+Ke|T|2−2τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht −τ2|T|2.(53)\nThen, item ( c) follows by inserting ( 48)-(50) and (53) in (52). /square\nBringing all these formulae together we get the desired dive rgenge-type formulae,WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 17\nCorollary 5.2. LetΣ2be an isometrically immersed surface into an homogeneous\nspaceE3(κ,τ). Then the following divergence formulae hold,\ndiv(U) = ∆Ke+|∇A|2−4|∇H|2+2|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n−2(κ−4τ2)/parenleftbig\n2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+(Ke−τ2)(1−3C2)/parenrightbig\n,(54)\nwhereU=P(2∇H)+(κ−4τ2)(∇TT−|T|∇|T|+div(T)T)and\ndiv(V) = ∆Ke+|∇A|2−4|∇H|2+2|Φ|2/parenleftbig\nKe+3(κ−4τ2)C2+τ2/parenrightbig\n−2(κ−4τ2)/parenleftbig\n|∇C|2−2(Ke+τ2)C2/parenrightbig\n,(55)\nwhereV=P(2∇H)+(κ−4τ2)(|T|∇|T|+div(T)T−∇TT).\nProof.On the one hand, let U1=∇TT−|T|∇|T|+div(T)T, then from items ( a), (b)\nand (c) of Lemma 5.1we can compute\n(56) div( U1) =−2τ2(3C2−1)+2τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht+T(div(T))+4H2C2.\nThen, from ( 56), (49) and item ( d) in Lemma 5.1we get\ndiv(U) =/square(2H)−2H(κ−4τ2)/parenleftbig\n/an}b∇acketle{tA(T),T/an}b∇acket∇i}ht−2C2H/parenrightbig\n−2τ(κ−4τ2)/parenleftbig\nτ(3C2−1)−/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht/parenrightbig\n.\nTaking now into account Proposition 3.2jointly with ( 4) and the definition of Φ we\neasily deduce\ndiv(U) = ∆Ke+|∇A|2−4|∇H|2+2|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n+(κ−4τ2)/parenleftbig\n−4H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+(1−3C2)(2τ2+|Φ|2−2H2)/parenrightbig\n.\nThen (54) follows from ( 42).\nOn the other hand, let us observe that\nV −U= 2(κ−4τ2)(|T|∇|T|−∇TT).\nTherefore, from ( 54) and items ( a) and (c) of Lemma 5.1, it holds\ndiv(V) = ∆Ke+|∇A|2−4|∇H|2+2|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n+2(κ−4τ2)/parenleftbig\n4τ2C2−2τ2|T|2−2τ/an}b∇acketle{tΦ(T),J(T)/an}b∇acket∇i}ht−2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht−(Ke−τ2)(1−3C2)/parenrightbig\n.\nThen, the desired formula ( 55) follows from ( 4) and (44), so Corollary 5.2is proved. /square\nIn the next results we will approach the case in which the extr insic curvature is\nconstant and negative. For this, the following lemma is esse ntial.\nLemma 5.3. LetΣ2be an isometrically immersed orientable surface into the ho moge-\nneous space E3(κ,τ)with constant extrinsic curvature Ke<0. Then\n(57) |∇A|2≤4|∇H|2.\nIn particular, the equality holds if and only if Σ2is a parallel surface.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 18\nProof.Indeed, let {e1,e2}be a local orthonormal frame which diagonalizes A, that is,\nA(ei) =λiei,i= 1,2. Then |∇A|2=/summationtext2\ni,j=1(ei(λj))2, and by a direct computation we\nget\n4|∇H|2= (e1(λ1)+e1(λ2))2+(e2(λ1)+e2(λ2))2.\nHence,\n(58) |∇A|2−4|∇H|2=−2(e1(λ1)e1(λ2)+e2(λ1)e2(λ2)).\nOn the other hand, since Ke=λ1λ2is a negative constant, taking derivatives with\nrespect to e1ande2,\n(59) 0 = ei(Ke) =ei(λ1)λ2+λ1ei(λ2), i= 1,2.\nFurthermore, λ1,λ2/ne}ationslash= 0, so from ( 59) it holds\nei(λ1) =−λ1\nλ2ei(λ2), i= 1,2.\nTherefore, ( 58) reads\n(60) |∇A|2−4|∇H|2=2λ1\nλ2(e2\n1(λ2)+e2\n2(λ2)) =2Ke\nλ2\n2(e2\n1(λ2)+e2\n2(λ2))≤0\nas desired. The conclusion about the equality is immediate. /square\nCorollary 5.4. There exists no immersed surface into the homogeneous space E3(κ,τ)\nwithκ−4τ2/ne}ationslash= 0, satisfying the equality in (57)and having positive constant extrinsic\ncurvature.\nProof.Indeed, suppose there exists an immersed surface Σ2intoE3(κ,τ),κ−4τ2/ne}ationslash= 0,\nsatisfying the equality in ( 57) and having positive constant extrinsic curvature. Fol-\nlowing the same reasoning as in the proof of Lemma 5.3we obtain ( 60), so\n0 =|∇A|2−4|∇H|2=2Ke\nλ2\n2(e2\n1(λ2)+e2\n2(λ2))≥0.\nSinceKe>0, we must have e1(λ2) =e2(λ2) = 0. Therefore λ2is constant, so by the\nassumption on theextrinsic curvature λ1is also constant. Thus, Σ2should bea parallel\nsurface of E3(κ,τ). Hence, from Lemma 2.1Σ2is either isometric to a piece of a Hopf\ncylinder or of a slice, which is a contradiction since in both casesKe=−τ2≤0./square\nNow, we present our first result related to surfaces with cons tant extrinsic curvature\ninS3\nb(κ,τ).\nTheorem 5.1. LetΣ2be an isometrically immersed closed surface into the homoge -\nneous space E3(κ,τ),κ−4τ2/ne}ationslash= 0, with negative constant extrinsic curvature. Then\n/integraldisplay\nΣ|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\ndA≥(κ−4τ2)/integraldisplay\nΣQτ,KedA,\nwhere\n(61) Qτ,Ke= 2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+(Ke−τ2)(1−3C2).\nThe equality holds if and only if E3(κ,τ) =S3\nb(κ,τ)andΣ2is a Hopf torus over a\nRiemannian circle in S2(κ).WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 19\nProof.By Corollary 5.2,\ndiv(U) = ∆Ke+|∇A|2−4|∇H|2+2|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n−2(κ−4τ2)/parenleftbig\n2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+(Ke−τ2)(1−3C2)/parenrightbig\n.\nSincewearesupposingthattheextrinsiccurvatureisanega tiveconstant, fromLemma 5.3,\nwe can estimate the divergence in this way\ndiv(U)≤2|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n−2(κ−4τ2)/parenleftbig\n2H/an}b∇acketle{tΦ(T),T/an}b∇acket∇i}ht+(Ke−τ2)(1−3C2)/parenrightbig\n.\nTherefore, taking integrals and using the classical diverg ence theorem, we have\n(62)/integraldisplay\nΣ/braceleftbig\n|Φ|2/parenleftbig\nKe+(κ−4τ2)(4C2−1)+τ2/parenrightbig\n−(κ−4τ2)Qτ,Ke/bracerightbig\ndA≥0,\nwhereQτ,Keis defined as in ( 61), which is the desired inequality.\nFurthermore, the equality is satisfied if and only if the equa lity holds in ( 57). Since\nKe<0, Lemma 5.3guarantees that Σ2is a parallel surface in E3(κ,τ). Therefore,\nfrom Lemma 2.1, we conclude that Σ2is isometric to a piece of a Hopf cylinder or\nto a slice of M2(κ)×Rwhenτ= 0. Thus, by closedness and recalling that slices in\nM2(κ)×Rare totally geodesic surfaces, so consequently satisfy Ke= 0, the equality\nin (62) is only satified in the case where E3(κ,τ) =S3\nb(κ,τ) and Σ2is isometric to a\nHopf torus. /square\nWe can also obtain the following alternative characterizat ion result from ( 55).\nTheorem 5.2. LetΣ2be an isometrically immersed closed surface with negative c on-\nstant extrinsic curvature into the homogeneous space E3(κ,τ)such that κ−4τ2>0.\nThen\n(63)/integraldisplay\nΣ/braceleftbig/parenleftbig\n3(κ−4τ2)C2+Ke+τ2/parenrightbig\n|Φ|2+2(κ−4τ2)(Ke+τ2)C2/bracerightbig\ndA≥0.\nThe equality holds if and only if E3(κ,τ) =S3\nb(κ,τ)andΣ2is a Hopf torus over a\nRiemannian circle in S2(κ).\nProof.The proof of ( 63) follows immediately taking integrals in ( 55) and taking into\naccount Lemma 5.3. The conclusion regarding the equality follows as in Theore m5.1.\n/square\nAcknowledgements\nThe authors would like to heartily thank the referee for his/ her valuable remarks\nand comments. The first author is partially supported by MICI NN/FEDER project\nPGC2018-097046-B-I00, by the Regional Government of Andal usia ERDEF project\nPY20-01391 and Fundaci´ on S´ eneca project 19901/GERM/15, Spain. Her work is a\nresult of the activity developed within the framework of the Program in Support of\nExcellence Groups of the Regi´ on de Murcia, Spain, by Fundac i´ on S´ eneca, Science and\nTechnology Agency of the Regi´ on de Murcia. The second autho r is also partially\nsupported by CNPq, Brazil, under the grant 431976/2018-0.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 20\nReferences\n[1] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in S2×R\nandH2×R, Acta Math. 193(2004), 141–174.\n[2] J.A. Aledo, L.J. Al´ ıas and A. Romero, A new proof of Liebmann classical rigidity theorem for\nsurfaces in space forms , Rocky Mountain J. Math. 35(2005), 1811–1824.\n[3] A.D. Alexandrov, Uniqueness theorems for surfaces in the large I , Vestnik Leningrad Univ. 11\n(1956), 5–17.\n[4] J.L.M. Barbosa and A.G. Colares, Stability of hypersurfaces with constant r-mean curvature , Ann.\nGlobal Anal. Geom. 15(1997), 277–297.\n[5] J.L. Barbosa, M. doCarmoandJ. Eschenburg, Stability of hypersurfaces of constant mean curvature\nin Riemannian manifolds , Math. Z. 197, (1988) 123–138.\n[6] M. Belkhelfa, F. Dillen and J. Inoguchi, Surfaces with parallel second fundamental form in Bianchi-\nCartan-Vranceanu spaces , in: PDE’s, submanifolds and affine differential geometry, Ba nach Center\nPubl., Polish Acad. Sci. Inst. Math., Warsaw, 57(2002), 67–87.\n[7] R.L. Bryant, A duality theorem for Willmore surfaces , J. Differential Geom. 20(1984), 23–53\n[8] L. Cao and H.Li, r-minimal submanifolds in space forms , Ann. Global Anal. Geom. 32(2007),\n311-341.\n[9] S.Y. Cheng and S.T. Yau, Hypersurfaces with constant scalar curvature , Math. Ann. 225(1977),\n195–204.\n[10] B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds , Comment. Math.\nHelv.82(2007), 87–131.\n[11] F.R. dos Santos, Rigidity of surfaces with constant extrinsic curvature in t he Riemannian product\nspaces, Bull. Braz. Math. Soc, New Series 52(2021), 307–326.\n[12] J. G´ alvez, A. Mart´ ınez and P. Mira, The Bonnet problem for surfaces in homogeneous 3-manifolds ,\nComm. Anal. Geom. 16(2008), 907–935.\n[13] Z. Guo, Willmore submanifolds in the unit sphere Collect. Math. 55(2004), 279-287\n[14] D.Hilbert, ¨Uber Fl¨ achen von konstanter Gaußscher Kr¨ ummung , Trans.Amer.Math.Soc. 2(1901),\n87–99.\n[15] H. Hopf, Differential Geometry in the large , Lecture Notes in Math., 1000, Springer-Verlag, Berlin,\n1983.\n[16] Z. Hu, D. Lyu and J. Wang, On rigidity phenomena of compact surfaces in homogeneous 3-\nmanifolds , Proc. Amer. Math. Soc. 143(2015), 3097–3109.\n[17] G. Huisken, Flow by mean curvature of convex surfaces into spheres , J. Differential Geom. 20\n(1984), 237-266\n[18] H. Liebmann, Eine neue eigenschaft der kugel , Nach. Kgl. Ges. Wiss. G¨ ottingen, Math.-Phys.\nKlasse (1899), 44–55.\n[19] F.C. Marques and A. Neves, Min-max theory and the Willmore conjecture , Ann. of Math. 179\n(2014), 683–782.\n[20] W.H. Meeks and H. Rosenberg, Stable minimal surfaces in M×R, J. Differential Geom. 68(2004),\n515–534.\n[21] S. Montiel and A. Ros, Compact hypersurfaces: the Alexandrov theorem for higher o rder mean\ncurvatures , in Differential geometry, Pitman Monogr. Surveys Pure Appl . Math., vol. 52, Longman\nSci. Tech., Harlow, 1991, pp. 279–296.\n[22] K. Nomizu and B. Smyth, A formula of Simons’ type and hypersurfaces with constant me an\ncurvature , J. Differential Geom. 3(1969), 367–377.\n[23] B. O’Neill, Semi-Riemannian Geometry, with Applications to Relativit y, New York: Academic\nPress (1983).\n[24] A. P´ ampano, Critical tori for mean curvature energies in Killing submer sions, Nonlinear Anal.\n200(2020), 112092, 18 pp.\n[25] H. Rosenberg, Minimal surfaces in M2×R, Illinois J. Math 46(2002), 1177–1195.\n[26] H. Rosenberg and R. Tribuzy, Rigidity of convex surfaces in the homogeneous spaces , Bull. Sci.\nMath.136(2012), 892–898.WILLMORE SURFACES AND HOPF TORI IN HOMOGENEOUS 3-MANIFOLDS 21\n[27] R.SouamandE. Toubiana, Totally umbilic surfaces in homogeneous 3-manifolds , Comment.Math.\nHelv.84(2009), 673–704.\n[28] W.M. Thurston, Three-dimensional geometry and topology Vol. I , Princeton Mathematical Series,\nVol. 35. Princeton University Press, 1997.\n[29] F. Torralbo andF. Urbano, On the Gauss curvature of closed surfaces in homogeneous 3-m anifolds,\nProc. Amer. Math. Soc. 138(2010), 2561–2567.\n[30] F. Torralbo and F. Urbano, Compact stable constant mean curvature surfaces in homogen eous\n3-manifolds , Indiana Univ. Math. J. 61(2012), 1129–1156.\n[31] J.L. Weiner, On a problem of Chen, Willmore, et al , Indiana Univ. Math. J. 27(1978), 19–35.\nDepartamento de Matem ´aticas, Edificio Albert Einstein\nUniversidad de C ´ordoba, Campus de Rabanales,\n14071 C ´ordoba, Spain\nEmail address :aalbujer@uco.es\nDepartamento de Matem ´atica\nUniversidade Federal de Pernambuco\n50.740-560, Recife, Pernambuco, Brazil\nEmail address :fabio.reis@ufpe.br"    },    {        "title": "2402.04664v1.Momentum_resolved_resonant_photoelectron_spectroscopic_study_for_1T_TiSe__2___Observation_of_negative_q_in_the_Fano_resonance_due_to_inter_atomic_interaction_in_the_valence_band.pdf",        "content": "Momentum-resolved resonant photoelectron spectroscopic study for 1T-TiSe 2:\nObservation of negative qin the Fano resonance due to inter-atomic interaction in the\nvalence band\nShin-ichiro Tanaka∗and Shigemasa Suga\nSANKEN, The Institute of Scientific and Industrial Research, Osaka University, Ibaraki 567-0047, Japan\nKeiji Ueno\nGraduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan\nKeisuke Fukutani\nInstitute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan\nFumihiko Matsui\nUVSOR Synchrotron Facility, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan\n(Dated: February 8, 2024)\nThe remarkable properties of (1T-)TiSe 2among the transition metal dichalcogenides have at-\ntracted the attention of many researchers due to its peculiar behavior during the charge density\nwave (CDW) transition. Therefore, it is highly desirable to study its electronic structure down to\nthe atomic orbitals. In the present research, we applied momentum-resolved resonant photoelectron\nspectroscopy to study TiSe 2at the Ti2p →Ti3d absorption edge by using a momentum microscope,\nwhich can simultaneously detect the electronic states in a wide ( kx, ky) range. We have also used\nconstant initial state (CIS) spectroscopy and density functional theory (DFT) calculations to reveal\nthe hybridization between the Ti3d and Se4p orbitals within the valence band at the Γ point at\nroom temperature. In addition, an interesting result comes from our analysis of the CIS spectrum\nfor the energy band located at a binding energy of 2 eV at the M-point. This band, mainly com-\nposed of the Se4p orbital, exhibited a Fano line profile at the Ti2p edge, with a negative value of\nthe parameter ‘ q’. This is the first clear evidence of the inter-atomic interaction during the valence\nband photoelectron emission process. This behavior differs significantly from the standard reso-\nnant photoelectron emission, which usually involves intra-atomic interactions. It also differs from\nthe multi-atom resonant photoelectron emission (MARPE) observed in the core-level photoelectron\nemission, as we focus on the photoelectron emission from the valence band in this research.\nI. INTRODUCTION\nThe layered transition-metal-dichalcogenide (TMDC)\nTiSe 2has been extensively studied for decades due to its\ncharacteristic charge-density-wave (CDW) transition be-\nhavior [1]. The (2 ×2×2) superstructure, which forms\nbelow a critical temperature ( Tc) of approximately 200\nK, was first reported by Di Salvo et al. [2]. Debates\nhave arisen concerning the driving force behind the CDW\ntransition. One perspective suggests the involvement of\nthe electron-phonon interaction [3, 4], while another pro-\nposes the exciton condensation as the driving factor [5].\nAlthough the latter perspective is currently considered\nmore favorable, especially in light of critical measure-\nments using electron energy loss spectroscopy [6], the ori-\ngin and nature of the CDW transition in TiSe 2remains\ncontroversial.\n∗Correspondence email address: stanaka@sanken.osaka-u.ac.jpThe band structures, derived from our density func-\ntional theory (DFT) calculations, are presented along the\nM-Γ-M line (represented by the black dotted line) and the\nA-L-A line (depicted by the red dotted line) in Fig. 1(a)\nalong with the angle/momentum-resolved photoelectron\nspectroscopy (ARPES/MRPES) intensity map (recorded\nat room temperature with hν=80 eV) . The notation of\nthe symmetry points in the Brillouin zone is illustrated in\nFig. 1(d). Further details about the measurements and\ncalculations will be provided later.\nNote that the kzvalue in the ARPES/MRPES map,\ncalculated assuming the inner potential of 13 eV [7],\nvaries from −0.08˚A−1(BE=5 eV) to 0.06 ˚A−1(EF) at\nΓ(A) , and −0.18˚A−1(BE=5 eV) to −0.06˚A−1(EF)\nat M(L). The bands accountable for the CDW transition\nare schematically illustrated in Fig. 1(e). They consist of\nthe electron pocket (often referred to as the conduction\nband) situated at the M(L) point, clearly distinguish-\nable in Fig. 1(b), and the hole pocket (often referred to\nas the valence band) located at the Γ(A) point. ThearXiv:2402.04664v1  [cond-mat.mtrl-sci]  7 Feb 20242\nFigure 1. (a) Calculated band dispersions of TiSe 2along the Γ-M line (black dots) and the A-L line (red dots) together with\nthe experimentally obtained photoelectron intensity map in a color scale shown at the right side of the panel taken at photon\nenergy ( hν) of 80 eV and at room temperature. (b,c) Iso-energy photoelectron intensity maps at the binding energy (BE) of 0\nand 0.2 eV, respectively, under the same condition as (a). (d) the 3D Brillouin zones and high symmetry points. (e): Schematic\nof the electronic structure of TiSe 2near the Fermi level, where the electron and hole pockets are located at M (L) and Γ (A)\npoints, respectively.\nvalence band at the Γ (A) point is also observable in\nthe ARPES/MRPES map but slightly weaker than the\nelectron pocket in Fig. 1(c). In the proposed exciton con-\ndensation process [5], thermal excitations lead to the cre-\nation of electrons in the conduction band and holes in the\nvalence band, providing the electron-hole bound states\nknown as excitons. With a periodicity determined by\nthe spanning vector that connects the valence band max-\nimum to the conduction band minimum, this can cause\na transition to a coherent ground state of condensed ex-\ncitons and the superlattice formation [5]. Consequently,\nunderstanding the orbitals of these bands is critical to\na deeper understanding of the CDW transition. In the\nsimplest description, the electronic configuration of TiSe 2\nis (Se4p)6(Ti3d)0, where the valence band is commonly\nassumed to be derived from the Se4p orbital and the\nconduction band from the Ti3d orbital. However, some\nstudies [8–11] have proposed a hybridization between the\nTi3d and Se4p orbitals in the valence band located near\nthe Γ point in the CDW phase. Therefore, an experi-\nmental investigation is desirable to elucidate the orbital\ncharacter of these electronic states.\nIn order to access to this objective, resonant pho-\ntoelectron spectroscopy utilizing synchrotron radiation\nis an ideal tool and has been extensively employed to\nprobe the electronic structure of various materials [12–\n14]. In resonant photoelectron emission, the photon en-\nergy is scanned around the core-to-bound excitation en-\nergy, leading to multiple pathways of excitation and de-\ncay. In instances where the incident photon generates a\ncore hole (in the current case, it is Ti2p) and the bound\nelectron in the conduction band (illustrated as A in Fig.2), subsequent exchange and direct Auger decay [15] (de-\nnoted as B and B’ in Fig. 2) result in a hole within the\nvalence band.\nThe notations of the “exchange” and “direct” are ac-\ncording to Bambynek et al. [15] Since the final state of\nthis process is identical to that of the normal (direct)\nphotoelectron emission from the valence band, shown\nas C in Fig. 2, the photoelectron intensity from the\nvalence band is resonantly enhanced in the vicinity of\nthe core-level photo-absorption energy. Resonant en-\nhancement in the photoelectron emission is more pro-\nnounced when the interaction between the core-hole de-\ncay and the Auger-electron emission is stronger. This\ninteraction exhibits significantly greater strength when\nthe relevant core and valence/conduction bands are con-\ntributed from the atomic orbitals of the same atom,\nwhich is known as intra-atomic Auger decay, as opposed\nto when they are contributed by different atoms (inter-\natomic Auger decay). As a result, conducting resonant\nphotoelectron spectroscopy at the Ti2p excitation al-\nlows us to distinguish the Ti3d-derived band from the\nSe4p-derived band. Although some studies using reso-\nnant photoelectron spectroscopy on the Ti2p excitation\nin TiSe 2have been reported [16, 17], the measurements\nwere momentum-integrated and the bands near the Fermi\nlevel were not well resolved. Therefore, a detailed study\nof the momentum-resolved resonant photoelectron spec-\ntroscopy on TiSe 2is highly desirable. Here we adopt a\nnew approach, where the momentum-resolved CIS spec-\ntra are obtained and compared with the momentum-\nresolved DOS of specific atomic orbitals.\nFano resonance has been widely investigated in may3\nEnergyABB’C\nPhotoelectron\nFigure 2. The schematic model during the photoelectron\nemission process in the resonant photoelectron spectroscopy.\nSee the text for details.\nspectroscopic fields so far [18, 19] including the photo-\nelectron spectroscopy [12, 13, 20]. In the resonant pho-\ntoelectron spectroscopy, it arises from interference be-\ntween distinct energy levels during excitation, followed\nby the Auger decay (process A →B/B’), and excitation\ninto continuous levels (process C). Constant initial-state\nspectroscopy (CIS), in which the photoelectron intensity\nat a specific binding energy of interest is plotted as a\nfunction of photon energy, has been used to examin the\nFano shape. The Fano resonance [18, 19] is formulated\nas\nI=I0\n1 +q2(q+ϵ)2\n1 +ϵ2(1)\n, where qis the asymmetry parameter that determins the\nFano-type spectral shape, I0is the intensity, and ϵis the\nnormalized energy as\nϵ=x−x0\nΓ(2)\n, where xis the photon energy, x0is the core-absorption\nenergy, Γ is the Lorentzian width. With increasing |q|,\nthe peak shape becomes more symmetric and ultimately\nbecomes a Lorentzian curve when |q|=∞, implying that\nthe whole process is dominated by the distinct excitation\n(A→B/B′). In real cases, both the continuous and\ndistinct excitation are involved, and finit postive q’s have\nbeen observed reflecting the relative strength and phase\nof the two excitations. [12, 13, 20]\nMeanwhile, it has been highlighted that resonant pho-\ntoelectron emission can arise from the inter-atomic Auger\nprocess. When a core-level electron of one atom is ex-\ncited into an empty state at a higher energy than the\nFermi level, it may be possible to influence the neighbor-\ning atoms when they are bonded directly. Then, at the\nphoton energy corresponding to the absorption edge of\none atom, the photoelectron emission from the the neigh-\nboring atoms may be modulated via the inter-atomic in-\nteraction between two atoms due to their direct bond-\ning. This phenomenon is commonly referred to as multi-\natom resonant photoemission (MARPE) [21–23]. Thefirst reported case was the resonant behavior of the O1s\ncore-level photoelectron emission at the Mn2p absorption\nedge in MnO [21]. Subsequently, MARPE phenomena\nhave been claimed in various materials in the context of\nthe core-level photoelectron emission [24–26], although\nan ongoing debate persists regarding the observability of\nMARPE [27–29]. In the context of valence band photo-\nelectron emission, however, the resonant effect could eas-\nily be overshadowed by conventional resonant photoelec-\ntron emission because the inter-atomic interaction is sig-\nnificantly weaker than the intra-atomic interaction. Al-\nthough MARPE associated with the photoelectron emis-\nsion from the core level has been reported, we believe\nthat no MARPE from the valence band has been proved\nso far. It should be noted that inter-atomic Auger elec-\ntron decay involving the valenve band has been suggested\nin TiO 2based on the experimental results in electron-\nstimulated ion desorption, where the electron threshold\nenergy for O+ion desorption coincided with the Ti3p\nexcitation [30]. However, according to a later work on\nTiO 2using electron-ion coincidence spectroscopy, which\nis a more sophisticated and specific experimental tech-\nnique, along with systematic comparisons of ion desorp-\ntion from other metal oxides, their proposal was called\ninto question [31]. Therefore, we examined the possibil-\nity of the electron emission from the valence band due\nto the inter-atomic interaction for the first time by us-\ning the momentum-resolved experiment by photoelectron\nmomentum microscopy (PMM).\nII. DETAILS OF EXPERIMENTS AND\nCALCULATIONS\nAll the experiments were carried out at the soft\nX-ray beamline BL6U of the UVSOR synchrotron\nradiation facility of Institute for Molecular Science\n(IMS), Okazaki, Japan. The endstation of the beam-\nline consists of a load-lock chamber, a sample-prepare\nchamber and a measurement chamber which haused\na momentum microscope with a single hemispheri-\ncal electron energy analyzer manufactured by SPECS\nSurface Nano Analysis GmbH, which enabled us to\nachieve an Angle/Momentum-Resolved PhotoElectron\nSpectroscopy(ARPES/MRPES) [32–34]. The grazing-\nincidence monochromator with a varied-line-spacing\nplane grating (photon energy range was 40-700 eV) was\nused. The photon is always horizontally polarized, corre-\nsponding to the p-polarization with respect to the sample\nsurface. The momentum microscope was set as its objec-\ntive lens axis is normal to the sample surface.\n1T-TiSe 2single crystals were grown by the chemical\nvapor transport method [35, 36]. First, stoichiometric\namounts of Ti powder and Se lumps, and iodine as a\ntransport agent were vacuum-sealed in a quartz ampoule.\nNext, the quartz ampoule was placed in a horizontal tube\nfurnace and heated to 900◦C in the raw material zone and\n800◦C in the crystal growth zone for 1 week for the CVT4\ngrowth. Then, the sample was taken out from the quartz\nampoule, and inserted into the ultra-high vacuum con-\ndition. The sample was cleaved to obtain an atomically\nflat clean surface just before the measurements, which\ntypically lasted less than 12 hours. The condition of the\nsample surface was checked by the ARPES/MRPES and\ncore-level spectroscopy which showed no trace of any con-\ntamination.\nThe DFT calculation was carried out by using the\nopen-source Quantum Espresso package (ver. 6) [37, 38]\ninstalled at Research Center for Computational Science\nin IMS, Okazaki, Japan. The pseudopotential files for Ti\nand Se were Ti.rel-pbe-spn-kjpaw_psl.1.0.0.UPF\nand Se.rel-pbe-n-kjpaw_psl.0.2.UPF , both of which\nwere projector-augmented wave (PAW) type and gener-\nated using the “atomic” code produced by A. Dal Corso\nthrough fully-relativistic calculation. The energy cutoff\nof the plane wave and charge density were set to 52.0\nRy (707 eV) and 576 Ry ( 9197 eV), respectively. Be-\nfore calculating the self-consistent electron density, the\ncrystal structure was optimized using the variable cell\noptimization (vc-relax) method, which resulted in the\noptimized lattice parameters as a=b= 3.544, and\nc= 6.693˚A. The self-consistent field calculation was\nfollowed by a non-self-consistent field calculation using\ndense 64 ×64×12k-meshes in the Brillouin zone. Next,\nthe results were used to determine the band energies, den-\nsity of states, and projected density of states, wherein the\ncontribution of each individual atomic orbital was indi-\ncated.\nIII. RESULTS\nFigure 3(a) shows the electron yield curves at the Ti-\nL3edge, acquired using Ti LV V -Auger electron yield\n(solid black line) and photoelectron yield (solid red line)\nfrom the entire valence band (VB), where the binding en-\nergy and and |k|||are set within 5 eV and 5 ˚A−1, respec-\ntively, are displayed. These intensities were normalized\nto align the background and peak intensities. Notably,\nthese yields closely resemble the X-ray absorption spectra\nof TiSe 2previously reported [39]. The discernible peaks\nat 457 (III) and 459 eV (V) are attributed to the excita-\ntions from Ti2p 3/2to Ti3d- t2gand Ti3d- eglevels, respec-\ntively [39]. Figure 3(b) shows the momentum-resolved\nenergy-distribution curves (EDCs) at the Γ, M, and M∗\npoints (using a full window size of 0.2 ˚A−1forkxandkyin\nthek-space) without normalization, with the assumption\nthat the photon intensity in this region remains constant.\nFor M/M∗points, the data were averaged across three\nequivalent points in the first Brillouin zone to improve\nthe statistics. Photoelectron intensity maps at binding\nenergies (BE) of 0.1 ±0.1 eV and 2 ±0.1 eV are displayed\nin Figs. 3(c) and (d), respectively.\nFigures 3(e) shows a photoelectron intensity map\nwhere the horizontal axis is kxalong the M-Γ-M∗line\n(ky=0 using a full window size of 0.2 ˚A ) and the ver-\nBCA\nCʼ\n(d)(c)\nMM*\n(e)\nCC’\nCC’ABFigure 3. (a) The Ti- LV V Auger electron (black line)\nand valence band photoelectron (red line) yields of TiSe 2as\nfunctions of the photon energy at room temperature. (b)\nPhoton-energy dependent photoelectron spectra at M∗, M\nand Γ points. The used photon energies I-Vare given in\nFig. 3(a). (c,d) Iso-energy photoelectron intensity maps at\nBE=0.1 eV and 2 eV, respectively, (e) the photoelectron in-\ntensity map along the M∗-Γ-M line.\ntial axis shows the BE. In Figs. 3(c-e), we do not show\na detail of the resonant behavior of the photoelectron\nemission (this will be discussed later) but rather present\nthe overall band structures. Then, all the photoelectron\nintensities recorded at photon energies from I to V have\nbeen averaged to enhance the statistics and mitigate the\nresonance effect in Figs. 3(c-e). The first Brillouin zone\nand the positions of M and M∗points are depicted in\nFigs. 3(c) and (d), respectively. It is worth mentioning\nthat the notations of M (M∗) and Γ are not accurate but\napproximate, as the kzvalue is estimated to range from\n−0.15˚A−1(hν=455 eV) to −0.10˚A−1(hν=459 eV) at\nthe “M” point and from −0.1˚A−1(hν=455 eV) to −0.05\n˚A−1(hν=459 eV) at the “Γ” point. The determination\nof the kzvalues is based on the assumption as the inner\npotential V0of 13 eV [7].\nIn this analysis, we direct our attention towards three\nprominent peaks: Peak A, originating from the electron\npocket crossing the Fermi level at both the M and M∗\npoints; Peak B, associated with the hole pocket situated\njust below the Fermi level at the Γ point; and Peak C,\ncorresponding to the band with BE of 2 eV at the M\npoint as shown in Fig. 3(b). The peak A vanishes at the\nphoton energy below the Ti2p threshold (I) and becomes\nfaintly observable at photon energies II-IV, with partic-\nularly heightened intensity at V as clearly recognized in\nFig. 3(b). The peak A demonstrates roughly similar in-\ntensities at both the M and M∗points. The peak B5\n(d)\n(e)\nCCʼAB\nFigure 4. (a) Ti2p XAS spectra obtained by means of the\nphotoelectron-yield of the entire valence band (same as the\nspectrum in black line in Fig. 3(a)) (dots), and the result of\nfitting using two Lorentzian curves and linear background.\n(b,c) Constant Initial State (CIS) spectra (dots) and the re-\nsult of fitting using four Lorentzian curves and linear back-\nground for the BE=0.1 eV at the M point (peak A defined\nin Fig. 3(b) ) and BE=0.2 eV at the Γ point (peak B defined\nin Fig. 3(b) ), respectively. The peak height of the XAS and\nCIS spectra are normalized to the same value from (a) to (c).\n(d,e) Photoelectron intensity maps as functions of the bind-\ning energy and the photon energy at the M and M∗points,\nrespectively. (f) CIS spectra for BE=2 eV at M point (peak\nC defined in Fig. 3(b) ; red line) and M∗point (peak C’ de-\nfined in Fig. 3(b); black line) and their difference (blue line).\nThe intensities for the peaks C and C’ are averaged among\nthree equivalent M and M∗points, respectively. (g) the differ-\nence in the CIS intensity between the peaks C and C’ (dots)\ntogether with the result of the least-square fitting using two\nFano resonance curves with q=−0.5 and−0.8 (lines) .\nexhibits a comparable intensity trend to the peak A, as\nshown in Fig. 3 (b). The peak C, on the other hand,\nexhibits a somewhat different behavior. It is already ob-\nservable at the photon energy I (below the Ti2p X-ray\nabsorption threshold), and its intensity does not exhibit\nany resonant enhancement at the photon energy V where\nthe peaks A and B show clear enhancements in intensity.\nMoreover, the peak C’ at the M∗point consistently ex-\nhibits lower intensity compared to the peak C at the M\npoint across all the photon energies. The asymmetry\nin intensity between the peak C and C’ is further high-\nlighted in Fig. 3(d and e). Considering that M and M∗\nare energetically degenerate in the k-space, the asymme-\ntry in photoelectron intensity between them is attributed\nto the matrix element effect during the photoexcitation\nprocess. This discrepancy may arise when one takes into\naccount the light polarization vector, as depicted by the\nyellow allow in Fig. 3(d). Figure 4(a) shows the CIS\nspectra of the entire valence band, that mirrors Ti2p 3/2-\nto-bound excitation [Fig. 3(a)], together with the result\nof the least-square fitting using two Lorentzian peaks.\nThe peak positions (widths) in the photon energy for the\nTi3d- t2gand the Ti3d- egpeaks were estimated 456.9 and\n459.3 eV (0.8 and 1.6 eV), respectively. CIS spectra forthe peaks A and B [defined in Fig. 3(b)] are presented\nin Figs. 4(b) and (c), where the photoelectron inten-\nsity yields, with window sizes for electron energy and\nmomentum set at 0.2 eV and 0.4 ˚A−1, respectively, are\nplotted as dots. These spectra cannot be fitted using two\nLorentzian curves as was possible in Fig. 4(a), but two\nadditional components were required to achieve a rea-\nsonable agreement with the experimental results [Figs.\n4(b,c)]. Two components (blue and red lines) are fixed\nto the same energies and widths as the CIS spectrum\nof the total VB (relative intensities serve as fitting pa-\nrameters), while the other two (dashed lines in green\nand violet) are allowed to be independently adjusted.\nThe energy positions of additional components are set\nat 456.3 and 458.7 eV for the peak A at the M point\n[Fig. 4(b)], and 456.5 and 458.7 eV for the peak B at\nthe Γ point [Fig. 4(c)]. All the components observed in\nthe CIS spectra of the peaks A and B were represented\nby the symmetric Lorentzian curves and no asymmetric\nFano shaped curve was required. As mentioned in the in-\ntroduction, this suggests that the photoelectron process\nis largely governed by the cascade process following the\ncore excitation (processes A →B/B’ in Fig. 2) without an\ninterference with the direct photoelectron process (C in\nFig. 2). This implies that the probability of direct photo-\nelectron emission processes is considerably low [18, 19],\nwhich is consistent with the very low intensities of the\npeaks A and B in ARPES/MRPES at the photon energy\nbelow the Ti2p threshold [Fig. 3(b)].\nTwo additional peaks, distinct from the two X-ray ab-\nsorption peaks corresponding to the Ti2p 2/3→Ti3d- t2g\nand Ti2p 2/3→Ti3d- egexcitations[Fig. 4 (a)], are ob-\nserved in both the CIS of peak A (BE=0.1 eV) at the\nM-point and peak B (BE=0.2 eV) at the Γ point, re-\nspectively[Figs. 4 (b,c)]. These additional peaks in the\nCIS spectra exhibit shifts of 0.6eV from the Ti3d- t2gpeak\nand 0.5 eV from the Ti3d- egpeak in the XAS spetrum for\nthe peak A [Fig. 4 (b]], and 0.5 eV/0.5 eV from the Ti3d-\nt2g/Ti3d- egpeaks ifor the peak B [Fig. 4 (c]]. Plausibly,\nthese shifts can be attributed to surface sites (includ-\ning the surface defect site such as the Se-vacant at the\nsurface), considering the short escape depth of the pho-\ntoelectrons possessing a kinetic energy of several hundred\neV. This discovery represents, to the best of our knowl-\nedge, the first experimental evidence of a surface-energy\nshift in the X-ray absorption spectrum of the TiSe 2sur-\nface. It is important to note, for a detailed discussion,\nthat the excitation energy of the core-to-bound state is\ndetermined by three factors: a) the energy shift in the\nTi2p core-level, b) that in the conduction-band, and c)\nthe shit in the core-exciton binding energy.\nIn recent X-ray photoelectron spectroscopic works on\nthe TiSe 2crystal [40, 41], no surface energy shift in the\nTi2p core level has been observed on this surface. This\ncontradicts the factor (a). Moreover, there has been no\nreports on the surface peak in the valence band photo-\nelectron spectroscopy [4, 5, 7–9, 11],which may suggest\nthat there is no bulk-to-surface energy shift in the con-6\nduction band as well. Consequently, the factor (b) does\nnot seem to be the main reason for the observed phe-\nnomenon. Therefore, it’s plausible that the factor c),\ni.e., the energy-shift due to the formation of the surface\ncore exciton, dominates the surface energy-shift in the\nCIS spetra of TiSe 2. Although the surface core exciton\nhas seldom been investigated [42–44], it is crucial to be\naccounted, especially when a significant Coulomb inter-\naction may be expected between the core-hole and the\nbound electron in the conduction band. A recent study\non the CaF 2surface proposed the surface shift in the\nexciton binding energy using sophisticated DFT calcula-\ntions employing a many-body approach [45]. Our finding\nimplies a possible difference in the excitonic interaction\nbetween the Ti2p core hole and the Ti3d conduction band\nat the surface and the bulk. We note that the intensity\nratios between these additional peaks and those in the\nX-ray absorption spectrum differ substantially in Ti3d-\nt2gand Ti3d- egpeaks (peaks around 456.9 and 459.3 eV,\nrespectively) both in spectra in Fig. 4 (b) and (c). This\ndivergence indicates that a simple assumption of a single\nsite with a single energy cannot explain the spectrum.\nA more intricate scenario, involving multiple reconstruc-\ntions and/or defects at the surface, should be considered.\nA more systematic study, including complementary data\nregarding the surface condition of the sample, is required\nto reveal the true nature of the excitonic interaction at\nthe TiSe 2surface. This aspect will be investigated sepa-\nrately in the near future after improving the energy res-\nolution of the instrument.\nMeanwhile, Figs. 4 (d) and (e) show the photoelectron\nintensity maps as functions of the photon energy and the\nbinding energy at the M and M∗points, respectively. Fig-\nures 4 (d) and (e) illustrate the photoelectron intensity\nmaps as functions of photon energy and binding energy at\nthe M and M∗points, respectively. Notably, only at the\nM point[Fig. 4 (d)], the intensity at BE=2 eV is evident\nat all photon energies, consistent with the EDC spectra\n[Figs.3 (b,d,e)]. The CIS spectra of the peaks C and C’\nin Fig. 3 (b), along with their difference, are displayed in\nFig. 4(f), utilizing the same window sizes as those for the\npeaks A and B in Figs. 4(b and c). In both CIS spectra\nof the peaks C and C’[Fig. 4(f)], two peak structures are\nobserved around 456.9 eV and 459.3 eV (indicated by the\ndotted lines) similarly to the the X-ray absorption spec-\ntrum [Fig. 4(a)]. However, these peak structures may\nnot correspond to the true intensity changes in the peak\nC but rather to changes in the background. In the EDC\nspectra presented in Fig. 3(b), the peaks C and C’ are sit-\nuated on the shoulder of the peak centered around BE=3\neV, which exhibit a strong enhancement at the photon\nenergy of 459.3 eV as shown in the EDC spectrum V\nin Fig. 3 (b), and this enhancement may be contribut-\ning much to the apparent resonant enhancement of the\npeaks at 456.9 eV and 459.3 eV in the CIS spectra of the\npeaks C and C’ [Fig. 4 (f)]. The intensity of the peak\nat BE∼3eV, probably contributed mainly from the Ti3d\nstate, remains relatively consistent from M to M∗, as alsodepicted in Figs. 4(d) and (e). Consequently, their differ-\nence shown in Fig. 4(f) can be interpreted as an approx-\nimate true yield spectrum, effectively isolated from the\nbackground due to the Ti3d-related electron emission, of\nthe peak C. It should be noted that no adjustments in\nintensities between the CIS spectra for the peaks C (at\nM) and C’ (at M∗) were carried out before the subtrac-\ntion except averaging the intensity at three equivalent M\nand M∗points. The photon-energy dependence of the\npeak C is drastically different from the situation of the\npeaks A and B, where the resonance enhancements in\nsymmetric Lorentzian curves were observed at the Ti2p-\nto-bound excitation photon energy as shown in Figs. 4 (b\nand c). In contrast, it is clear that almost no “resonant\nenhancement” in the photoelectron emission is observed\nfor the peak C in the excitation photon energy region\nbetween 455 and 458 eV as seen in Fig. 3(b). Instead,\nthe largest peak intensity of the peak C is achieved when\nthe photon energy is just below the Ti2p photoexcita-\ntion threshold [case I in Fig. 3 (b)], and become smaller\nwhen the photon energy passed over 456.9 eV and 459.3\neV, which correspond to the excitation from Ti2p 3/2to\nTi3d- t2gand Ti3d- eg[Fig. 4(f)]. The disparities between\nthe CIS spectra at M and M∗points can be fitted using\nnon-conventional Fano-type asymmetric peak shapes. In\nFig. 4(g), two Fano-shaped curves (eqs. 1 and 2) are em-\nployed for fitting. In this fitting, x0(456.9 and 459.3 eV)\nand Γ (0.8 and 1.6 eV) are kept consistent with the values\nin the CIS spectrum of the entire VB [Fig. 4(a)], and only\ntwo sets of parameters of I0andqwere adjusted. The\nresults exhibit a remarkable alignment with the experi-\nmental outcomes. A noteworthy emphasis lies in the fact\nthat the obtained qparameters are both negative, with\nvalues of −0.8 and −0.5, a phenomenon hitherto unob-\nserved in resonant photoelectron spectroscopy to the best\nof our knowledge.\nFigure 5. (a) Momentum-resolved photoelectron spectrum\nat M(L) point taken at the photon energy indicated. (b) Two-\ndimensional photoelectron intensity maps at BE=2 eV taken\nat hν=455 eV (upper map) and 100 eV (lower map), (c)\nCalculated photoionization cross section for Ti3d and Se4p\norbitals [46].\nFor more comprehensive exploration of the photoelec-7\ntron emitting process and the peak interpretation, a\nstudy encompassing a wider range of photon energies\nwas conducted. Figure 5(a) presents EDC profiles at the\nM point (excluding M∗) across several photon energies.\nEach spectrum is normalized in intensity to the averaged\namplitude in the observed binding energy range. Mean-\nwhile, Fig. 5(b) displays an iso-energy photoelectron in-\ntensity maps at BE= 2 eV, recorded at photon energies\nof 455 and 100 eV.\nA clear observation emerges: the peak A near the\nFermi level exhibits pronounced strength at lower pho-\nton energies below 200 eV but disappears drastically at\nhν=455 eV (note that this value lies below the Ti2p\nthreshold which starts at around 456 eV). Conversely,\nthe peak C at BE=2 eV is almost absent at lower photon\nenergies below 100 eV but becomes prominent at hν=455\neV, both in the M-point EDC and the iso-energy intensity\nmaps at BE=2 eV [Figs. 5 (a,b)]. This divergence can\nbe attributed to the varying photoionization probability\nof atomic orbitals as a function of the photon energy.\nFigure 5 (c) displays the photoionization cross section\nof the Ti3d and Se4p orbitals, as calculated by Yeh and\nLindau [46]. Notably, the Se4p cross section exhibits a\nprominent dip around hν=100 eV and remains relatively\nconstant from 200 eV to 600 eV. Conversely, the Ti3d\ncross section decreased gradually and monotonically with\nthe photon energy up to 600 eV. The narrow thin lines\nindicate the photon energy range (455-459 eV) of our res-\nonant photoelectron spectroscopy measurements. Within\nthis range, the photoionization cross section of the Ti3d\nstate decreases significantly down to approximately 1%\nof that of the Se4p state. Consequently, (direct) photo-\nelectron emission from the bands predominated by the\nSe4p orbital occurs at the photon energies within 455-\n459 eV, while the dominance of the Ti3d orbital is evi-\ndent at photon energies at ∼80-100 eV. As a result, the\npeaks A and B in Fig. 3(b) can be attributed to the\nTi3d-derived band, whereas the peak C corresponds to\nthe Se4p-derived band.\nTo validate the aforementioned interpretation, we con-\nducted Density Functional Theory (DFT) calculations,\nwhich yielded wavefunction projections onto atomic or-\nbitals. The populations of Se4p and Ti3d orbitals in\nbands were determined using a 64 ×64×12k-mesh.\nConvolutions were applied in both momentum and en-\nergy, utilizing integration window sizes of ∆ k= 0.1˚A−1\nand ∆ E= 0.1 eV, respectively, across the Brillouin zone.\nThe subsequent results include EDC at specific k-points\nand iso-energy density of states at particular binding en-\nergies.\nFigures 6(a) and (b) respectively illustrate the inten-\nsity distribution of the Ti3d and Se4p bands at the Γ(A)\nand M(L) points. The calculated outcomes are integrated\nalong the kzaxis within the Brillouin zone, since there\nis a non-negligible dispersion along the kzaxis in some\nbands (refer to Fig. 1(a)) as the exact energy position\nof the band is not expounded and is not important here.\nOn analyzing the Γ (A) point [Fig. 6(a)], it becomes ev-\nFigure 6. Results of the DFT calculations: (a,b) Projected\nenergy-distribution curves for the Ti3d and Se4p orbitals at Γ-\nA (a) and M-L (b) points. The density of states are integrated\nalong the kzline in the Brillouin zone. (c,d) Projected density\nof states of Ti3d(c) and Se4p(d) orbitals in the kx-kyspace\nat BE=2 eV.\nident that the states around the Fermi level have the\nrather equivalent contributions from the Se4p and Ti3d\norbitals, although the absolute binding energy is some-\nwhat shifted to higher binding energy from the experi-\nmental result[Figs. 1(a,b) and Fig. 3(b)]. This calcu-\nlation signifies the hybridization between the Ti3d and\nSe4p orbitals at the top of the valence band at the Γ (A)\npoint. This correlates well with the CIS spectrum for the\npeak B [Fig. 4 (c)], where the resonant enhancement in\nthe photoelectron intensity is evident at the photon en-\nergy corresponding to the Ti2p →Ti3d excitation indicat-\ning the valence band top (the peak B) is contributed from\nthe Ti3d orbital. It is noteworthy that the hybridization\nof the valence band between the Ti3d and Se4p states at\nthe Γ(A) point in the CDW phase has been previously\npostulated [8–11], while our experiments and the DFT\ncalculation save been made on the non-CDW phase. The\nTi3d orbital contributes dominantly to the states crossing\nthe Fermi level at the M (L) point, as shown in Fig. 6(b).\nThis is consistent with the CIS spectrum of the peak A\nat the M (L) point in Fig. 4 (b).\nConversely, the electronic states of BE=1-2 eV at the\nM (L) point [Fig. 6(b)] are predominated by the Se4p or-\nbital, aligning with the interpretation as stated above.\nThis is further substantiated by comparing the calcu-\nlated results of the projected iso-energy two-dimensional\ndensity of states at the binding energy of 2 eV (Figs.\n6(c) and (d)) to the photoelectron intensity map taken\nat hν=455 eV (Se4p-dominated) and 100 eV (Ti3d-\ndominated)[Fig. 5(b)]. These photon energies are sen-\nsitive to Ti3d and Se4p, respectively, as predicted in Fig.\n4(c). Though the detailed structures might not be in per-\nfect agreement, possibly due to dispersion along the kz8\naxis, the overarching trend persists. Specifically, the Ti3d\nstates predominate in the vicinity of the Γ (A) point,\nwhile the Se4p states prevail around the M (L) point.\nIn accordance with the interpretation of the peak C\nbeing predominated by the Se4p orbital, its CIS spec-\ntrum [Fig. 4(g)] can be attributed to an Auger electron\nemission originating from the same band subsequent to\nthe Ti2p →Ti3d excitation. To further discuss the de-\ntail of this Auger electron emission proccess, two dis-\ntinct possibilities have to be checked. The first possibil-\nity is the CVV Auger decay, wherein C denotes the core\nlevel (Ti2p), and the two V’s represent the valence bands\n(Ti3d and Se4p). This scenario is depicted in Fig. 2. Al-\nternatively, a cascade (intra-atomic) CVV Auger decay\nmay transpire, where all pertinent levels (bands) are de-\nrived from the Se states following an inter-atomic CCC\n(Ti/Se/Se) Auger decay which yields the core hole at the\nSe atom. This inter-atomic CCC Auger process, in which\nthe Se-core holes are provided after the de-excitation of\nthe Ti2p core hole, is the same as the concept of the pre-\nvious MARPE. [21–23] In the latter process, the Fano\nline shape in the CIS spectrum of the peak C [Fig. 4(g)]\nis predominantly determined by the conventional core-\nlevel MARPE process. To investigate this possibility,\nangle-integrated photoelectron monitoring of the Se3d\ncore level was conducted while varying the photon energy\naround the Ti2p →Ti3d excitation threshold. Figure 7(a)\nillustrates the Se3d peak recorded at hν=450 eV, and\nFig. 7(b) showcases examples of the Se3d peak spectra\nat varying photon energies. The dots in the graphs de-\npict actual data, while the lines represent the results of a\nleast-square peak fitting calculation employing the Voigt\nfunction together with the Shirley background function\nalongside a linear background. The intensities of the peak\narea and the averaged background are plotted against the\nphoton energy in Fig. 7(c). The outcome is clear: Despite\nthe enhancement in background intensity, which likely\narises from secondary electrons, there is no resonance in\nSe3d photoelectron emission near the Ti2p →Ti3d exci-\ntation. In essence, we do not observe core-level MARPE\nin TiSe 2, and can deny the cascade process. Then, it\nleads to the conclusion that the Fano resonance observed\nin Fig. 4(g) arises from the interference between the\nSe4p photoelectron emission and the inter-atomic CV V\n(Ti2p/Ti3d/Se4p) Auger electron emission.\nIV. DISCUSSION\nIt appears evident that both the conduction band bot-\ntom at the M(L) point and the valence band top at\nthe Γ(A) point [the A and B peaks in the EDC spectra\nshown in Fig. 3(b)] are contributed by the Ti3d orbital.\nThis conclusion is supported by the noticeable increases\nin photoelectron intensities at the photon energy corre-\nsponding to the Ti2p →Ti3d photoexcitation [shown in\nFigs. 4(b,c)]. This interpretation gains further support\nfrom: a) A comparison of the photon energy dependence\nFigure 7. (a) Angle-integrated photoelectron spectrum\nrecorded at hν=450 eV in the region of the Se3d core level.\n(b) Series of the photoelectron spectra with changing the pho-\nton energy. The dots show the actual data and the lines show\nthe results of the peak fitting using the Voigt peak shape and\nlinear background. (c) Plots of the peak intensity and back-\nground intensity as a function of the photon energy.\nbetween ARPES/MRPES and the cross-section for pho-\ntoionization of atomic orbitals [depicted in Figs. 5], and\nb) Orbital-specific DFT calculations [illustrated in Figs.\n6 (a, c)]. The CIS spectra shown in Figs. 4(b, c) do\nnot consist of the asymmetric Fano-type curve that has\nbeen commonly observed in resonant photoelectron spec-\ntroscopy [12, 13, 20], but consist of symmetric Lorentzian\ncurves. This discrepancy can be explained by consider-\ning the very weak Ti3d photoionization cross-section at\nthe photon energies of around 455 eV [Figs. 4(c) and\n5(c)]. As mentioned already in the introduction, the\nconventional resonant photoelectron emission process is\ninduced by the interference between the direct photo-\nelectron emission process [process C in Fig. 2 ] and the\nAuger electron emission following the core-hole creation\n[processes A and B/B’ in Fig. 2 ]. Within the photon\nenergy range corresponding to the Ti2p →Ti3d photoex-\ncitation , the photoionization cross-section of the Ti3d\nstates is extremely small compared to that of the Se4p\nstates, leading to the predominance of the Auger elec-\ntron emission. Consequently, effective interference does\nnot occur, resulting in infinite value of the parameter q\nin the Fano resonance. As a consequence, the symmetric\nLorentzian curve is achieved as shown in Figs. 4 (b,c).\nThe evaluation of the contribution from the Se4p\norbital through the Ti2p-resonant photoelectron spec-\ntroscopy is not easy when the observed band has a sig-\nnificant contribution from the Ti3d orbitals which can\novershadow the weak resonance of the Se4p state. There-\nfore, our present results do not discount the possibility of\nhybridization between the Ti3d and Se4p orbitals. Con-\ncerning the top of the valence band [peak B in Fig.3(b)],\none expects the contribution from the Se4p state con-\nsidering the simple electronic configuration of TiSe 2as9\n(Se4p)6(Ti3d)0. Consequently, at the top of the valence\nband at the Γ(A) point, the Ti3d-Se4p hybridization\nis highly likely. It’s worth noting that the Ti3d-Se4p\nhybridization at the top of the valence band has been\npreviously proposed as a result of the CDW transition,\nwherein the M(L) point is folded into the Γ(A) point [8–\n11]. However, our current findings suggest that this hy-\nbridization occurs even in the absence of the CDW phase.\nTheoretical work based on our experimental results is\nhighly expected to elucidate the detailed mechanism of\nthe CDW transition.\nMeanwhile, the band corresponding to the peak C\n(BE∼2 eV at the M-point) in ARPES/MRPES spectra\nis predominated by the Se4p orbital, as proved by the\nphoton energy dependent ARPES/MRPES and DFT cal-\nculations. In conventional resonant photoelectron spec-\ntroscopy, the resonant enhancement occurs for the pho-\ntoelectron emission from a band consisting of the core-\nexcited atom’s orbitals, and a band originating from an\norbital not associated with the core-excited atom is typi-\ncally assumed to have no significant resonance. However,\nin the case of the peak C [BE ∼2 eV at the M point in\nFig. 3(b)], the CIS spectrum, after subtracting the CIS\nspectrum of C’ (at the M∗point) for the background\nelimination, can be successfully fitted using the Fano-\nresonance function as shown in Fig. 4(g) . This indicates\nthat there is some interference between the photoelec-\ntron emission process from the Se4p-derived band and the\ninter-atomic Auger electron emission following the Ti2p\ncore-level excitation[Figs. 2]. On the “exchange” Auger\ndecay (process B in Fig.2), the transition from the excited\nTi3d state to the Ti2p core hole provides the energy for\nthe electron emission from the occupied Se4p state via\nthe inter-atomic Coulomb interaction. In the case of the\n“direct” Auger decay (process B’ in Fig.2), meanwhile,\nthe inter-atomic transition from the Se4p state to the\nTi2p core hole provides the energy for the electron emis-\nsion from the Ti3d state. These “direct” and “exchange”\nAuger processes are energetically degenerate. It is impor-\ntant to note that the inter-atomic Coulomb interaction\nis generally much weaker than the intra-atomic interac-\ntion. As a result, the inter-atomic resonance effect is ex-\npected to be much smaller compared to the intra-atomic\nresonance effect observed in the conventional resonant\nphotoelectron spectrum. This likely explains why such\nan effect has not been previously reported. In the cur-\nrent scenario on TiSe 2, however, the direct photoelectron\nemission process from the Ti3d state is much weaker than\nthat for the Se4p sate[Fig. 4(c)]. This is the reason why\nthe Se4p resonance can be observed without being over-\nshadowed by the Ti3d resonance in the present case.\nAnother important result observed for the first time\nin this research is the negative qvalue in the Fano reso-\nnance in resonant photoelectron spectroscopy. Although\nnegative qhas often been reported in many researches\n(in fact, negative qwas reported in the very first applica-\ntion of the Fano formula to the 2 s2p1Pautoionized level\nat∼60 eV of double excitation of the He atom [18]),only positive qhas been reported in the resonant pho-\ntoelectron spectra [12, 13, 20]. In analogy to classical\nmechanics [47, 48], Fano resonance can be described as\nan interference between two coupled harmonic oscillators\nwith different frequencies and damping factors that de-\ntermine the spectral width. The damped oscillator with\na wider spectral width corresponds to the continuous lev-\nels, and the oscillator with a narrow width corresponds to\nthe distinct levels in the ordinal Fano resonance. The pa-\nrameter qis determined by the phase shift of the damped\noscillator at the resonant frequency. The sign of qde-\npends on the sign of the frequency difference between\nthe two oscillators [47, 48]. In the quantum mechanics,\nmeanwhile, qis also connected to ϕwhich is a phase-\nshift between two excitations (distinct and continuous)\nbyq=−cotϕ[19, 49]. Furthermore, it has been reported\nthatqcan be controlled from negative to positive by tun-\ning the temporal condition between the distinct and con-\ntinuous transitions using the ultrashort pulse lasers [49].\nTherefore, it is considered that the sign of qin the res-\nonant photoelectron emission is determined by the tem-\nporal characteristics of the core-excitation/Auger decay\nand the direct photoelectron emission processes. A fun-\ndamental distinction of the current situation, marked by\na negative q, from the conventional resonant photoelec-\ntron emission lies in the contrast between inter-atomic\nand intra-atomic interactions during the Auger decay. It\nseems reasonable to posit that these processes exhibit\ndiverse phase conditions concerning the interference be-\ntween direct photoelectron emission and Auger electron\nemission. It seems interesting to point out that only pos-\nitive qvalues have been reported in the resonant photo-\nelectron emission associated with the inter-atomic core-\ncore-core Auger decay known as MARPE [21–23]. The\ndiscernible contrast between conventional MARPE and\nour current investigation suggests a variance in the dy-\nnamics of excitation-decay during photoelectron/Auger\nelectron emission. This discrepancy becomes apparent\nwhen comparing scenarios involving solely localized core\nstates to those where delocalized valence states are in-\nvolved. To gain deeper insights into the sign of qduring\nresonant photoelectron emission, a more intricate theo-\nretical investigation focusing on the microscopic dynam-\nics of excitation and decay is required. Furthermore, con-\nducting similar experiments on magnetic compounds ex-\ncited by circularly polarized synchrotron radiation light\ncould potentially unlock avenues for a more comprehen-\nsive discussion of inter-atomic interactions, encompassing\nspin-spin and spin-orbit couplings, within resonant pho-\ntoelectron spectroscopy.\nV. CONCLUSION\nIn this study, we have conducted an investigation\nby means of momentum-resolved resonant photoelectron\nspectroscopy on 1T-TiSe 2at the Ti2p-edge, focusing un-\nder room temperature conditions. This investigation was10\nmade possible by the use of high performance momen-\ntum microscopy equipment. The investigation was fur-\nther facilitated by analyzing the comprehensive range\nof photon-energy dependence within the momentum-\nresolved photoelectron spectra. This analysis was com-\nplemented by the use of theoretical results of the atomic\norbital photoionization cross sections, together with den-\nsity functional theory (DFT) calculations. Through the\ncombined efforts of these experimental and theoretical\napproaches, we have uncovered the following significant\nfindings:\n1. The valence band located at the Γ(A) point exhibits\nthe intricate hybridization involving both the Ti3d\nand Se4p orbitals, even at room temperature. No-\ntably, this hybridization is not a consequence of the\ncharge-density-wave (CDW) transition.\n2. The CIS spectrum for the Ti3d-derived band at\nthe Ti2p edge is not characterized by the Fano-\ntype curve, but by the symmetric Lorentzian curve.\nThis can be understood by the fact that the pho-\ntoionization cross section of the Ti3d orbital is quite\nsmall at the photon energies corresponding to the\nTi2p edge. Therefore, the photoelectron emission\nin this photon energy region is almost dominated\nby the Auger electron emission following the ex-\ncitation from Ti2p to Ti3d states, and the Fano\ninterference is effectively suppressed.\n3. The band primarily characterized by the Se4p\norbital, located at the binding energy of ∼2\neV at the M-point, exhibits a fascinating res-onant phenomenon. This peculiar behavior is\nevident through the manifestation of a distinct\nFano line shape featuring a negative value of\nq. The occurrence of Fano resonance within\nthe Se4p-derived band can be attributed to the\nintricate interference between the direct photo-\nelectron emission originating from this band and\nthe inter-atomic CV V (Core-Valence-Valence, i.e.,\nTi2p/Ti3d/Se4p) Auger electron emission process.\nNotably, this observation of a negative qvalue is\nthe first occurrence in the field of resonant photo-\nelectron spectroscopy. It is also noteworthy that\nthe experimental evidence has demonstrated the\nunique nature of this result of the electron emis-\nsion from the valence band in contrast to the\nconventional multi atom resonant photoemission\n(MARPE), where only core-level states are involved\nin inter-atomic Auger electron emission.\nVI. ACKNOWLEDGEMENT\nWe would like to thank UVSOR staff for their kind\nsupport for the experiments. This work was performed\nat the BL6U of UVSOR Synchrotron Facility, Institute\nfor Molecular Science (IMS program: 20-212, 21-201).\nThe computation was performed using Research Cen-\nter for Computational Science, Okazaki, Japan (Project:\n21-IMS-C161,22-IMS-C162). This work was supported\nby JSPS KAKENHI Grant Numbers 21K04826 and\n22H05445.\n%apsrev4-2.bst 2019-01-14 (MD) hand-edited version\nof apsrev4-1.bst\n[1] G. Gr¨ uner, The Dynamics of Charge-Density Waves ,\nRev. Mod. Phys. 60, 1129 (1988).\n[2] F. J. Di Salvo, D. E. Moncton, and J. V. Waszczak,\nElectronic Properties and Superlattice Formation in the\nSemimetal TiSe 2, Phys. Rev. B 14, 4321 (1976).\n[3] M. Calandra and F. Mauri, Charge-Density Wave and\nSuperconducting Dome in TiSe 2from Electron-Phonon\nInteraction , Phys. Rev. Lett. 106, 196406 (2011).\n[4] A. Wegner, J. Zhao, J. Li, J. Yang, A. A. Anikin,\nG. Karapetrov, K. Esfarjani, D. Louca, and U. Chatter-\njee,Evidence for Pseudo–Jahn-Teller Distortions in the\nCharge Density Wave Phase of 1T-TiSe 2, Phys. Rev. B\n101, 195145 (2020).\n[5] H. Cercellier, C. Monney, F. Clerc, C. Battaglia, L. De-\nspont, M. G. Garnier, H. Beck, P. Aebi, L. Patthey,\nH. Berger, and L. Forr´ o, Evidence for an Excitonic In-\nsulator Phase in 1T-TiSe 2, Phys. Rev. Lett. 99, 146403\n(2007).\n[6] A. Kogar, M. S. Rak, S. Vig, A. A. Husain, F. Flicker,\nY. I. Joe, L. Venema, G. J. MacDougall, T. C. Chiang,\nE. Fradkin, J. van Wezel, and P. Abbamonte, Signatures\nof Exciton Condensation in a Transition Metal Dichalco-\ngenide , Science 358, 1314 (2017).[7] P. Chen, Y. H. Chan, X. Y. Fang, S. K. Mo, Z. Hussain,\nA. V. Fedorov, M. Y. Chou, and T. C. Chiang, Hidden\nOrder and Dimensional Crossover of the Charge Density\nWaves in TiSe 2, Sci. Rep. 6, 37910 (2016).\n[8] N. G. Stoffel, S. D. Kevan, and N. V. Smith, Experimental\nBand Structure of 1T-TiSe 2in the Normal and Charge-\nDensity-Wave Phases , Phys. Rev. B 31, 8049 (1985).\n[9] K. Rossnagel, L. Kipp, and M. Skibowski, Charge-\nDensity-Wave Phase Transition in 1T-TiSe 2:Exci-\ntonic Insulator versus Band-type Jahn-Teller Mechanism ,\nPhys. Rev. B 65, 235101 (2002).\n[10] M. Hellgren, J. Baima, R. Bianco, M. Calandra,\nF. Mauri, and L. Wirtz, Critical Role of the Exchange\nInteraction for the Electronic Structure and Charge-\nDensity-Wave Formation in TiSe 2, Phys. Rev. Lett. 119,\n176401 (2017).\n[11] M. D. Watson, O. J. Clark, F. Mazzola, I. Markovi´ c,\nV. Sunko, T. K. Kim, K. Rossnagel, and P. D. C. King,\nOrbital- and kz-Selective Hybridization of Se4p and Ti3d\nStates in the Charge Density Wave Phase of TiSe 2, Phys.\nRev. Lett. 122, 076404 (2019).\n[12] L. C. Davis, Photoemission from Transition Metals and\ntheir Compounds , J. Appl. Phys. 59, R25 (1986).11\n[13] A. Kotani and S. Shin, Resonant Inelastic X-ray Scatter-\ning Spectra for Electrons in Solids , Rev. Mod. Phys. 73,\n203 (2001).\n[14] C. Fadley, X-ray Photoelectron Spectroscopy: Progress\nand Perspectives , J. Electron Spectrosc. Relat. Phenom.\n178-179 2 (2010).\n[15] W. Bambynek, B. Crasemann, R. W. Fink, H. U. Freund,\nH. Mark, C. D. Swift, R. E. Price, and P. V. RAO, X-ray\nFluorescence Yields, Auger, and Coster-Kronig Transi-\ntion Probabilities , Rev. Mod. Phys. 44, 716 (1972).\n[16] A. S. Shkvarin, Y. M. Yarmoshenko, N. A. Skorikov,\nM. V. Yablonskikh, A. I. Merentsov, E. G. Shkvarina,\nand A. N. Titov, Resonance Photoelectron Spectroscopy\nof TiX 2(X = S, Se, Te) Titanium Dichalcogenides , J.\nExp. Theor. Phys. 115, 798 (2012).\n[17] C. W. Chuang, Y. Tanaka, M. Oura, K. Ross-\nnagel, and A. Chainani, Attractive Coulomb Interaction,\nTemperature-Dependent Hybridization, and Natural Cir-\ncular Dichroism in 1T-TiSe 2, Phys. Rev. B 102, 195102\n(2020).\n[18] U. Fano, Effects of Configuration Interaction on Intensi-\nties and Phase Shifts , Phys. Rev. 124, 1866 (1961).\n[19] M. F. Limonov, Fano Resonance for Applications , Adv.\nOpt. Photon. 13, 703 (2021).\n[20] A. Kakizaki, K. Sugeno, T. Ishii, H. Sugawara,\nI. Nagakura, and S. Shin, Resonant Photoemission in\nTransition-Metal Chlorides , Phys. Rev. B 28, 1026\n(1983).\n[21] A. Kay, E. Arenholz, S. Mun, F. J. G. de Abajo, C. S.\nFadley, R. Denecke, Z. Hussain, and M. A. V. Hove,\nMulti-Atom Resonant Photoemission: A Method for De-\ntermining Near-Neighbor Atomic Identities and Bonding ,\nScience 281, 679 (1998).\n[22] A. W. Kay, F. J. Garcia de Abajo, S.-H. Yang, E. Aren-\nholz, B. S. Mun, N. Mannella, Z. Hussain, M. A.\nVan Hove, and C. S. Fadley, Multiatom Resonant Pho-\ntoemission , Phys. Rev. B 63, 115119 (2001).\n[23] T. Fujikawa, K. Niki, and J. Kogo, Photoelectron Diffrac-\ntion, Multi-Atom Resonant Photoemission, Pioneered\nby Fadley , J. Electron Spectrosc. Relat. Phenom. 257,\n147202 (2022).\n[24] A. Kikas, E. N˜ ommiste, R. Ruus, A. Saar, and I. Martin-\nson, Multi-Atom Resonant Photoemission in Transition\nMetal Chlorides , Solid State Commun. 115, 275 (2000).\n[25] N. Mannella, S.-H. Yang, B. S. Mun, F. J. Garcia de\nAbajo, A. W. Kay, B. C. Sell, M. Watanabe, H. Ohldag,\nE. Arenholz, A. T. Young, Z. Hussain, M. A. Van Hove,\nand C. S. Fadley, Observation and Resonant X-ray Op-\ntical Interpretation of Multi-Atom Resonant Photoemis-\nsion Effects in O1s Emission from NiO , Phys. Rev. B 74,\n165106 (2006).\n[26] V. I. Grebennikov and T. V. Kuznetsova, Resonant Inter-\natomic Auger Transition in Chalcopyrite CuInSe 2, Phys-\nica Status Solidi (a) 216, 1800723 (2019).\n[27] M. Finazzi, G. Ghiringhelli, O. Tjernberg, L. Du` o,\nA. Tagliaferri, P. Ohresser, and N. B. Brookes, Mul-\ntiatomic Resonant Photoemission Spectroscopy on CuO\nand NiO: Observation of Antiresonant Behavior , Phys.\nRev. B 62, R16215 (2000).\n[28] D. Nordlund, M. G. Garnier, N. Witkowski, R. De-\nnecke, A. Nilsson, M. Nagasono, N. M˚ artensson, and\nA. F¨ ohlisch, Limits to the Quantitative Analysis of Mul-\ntiatom Resonant Photoemission: The Case of c(2×\n2)O/Ni(100), Phys. Rev. B 63, 121402(R) (2001).[29] X. Gao, W. Kuch, F. Offi, S. Kang, S. Imada, and\nJ. Kirschner, Search for Multi-Atom Resonant Photoe-\nmission in Magnetic Thin Films , J. Electron Spectrosc.\nRelat. Phenom. 123, 11 (2002).\n[30] M. L. Knotek and P. J. Feibelman, Ion Desorption by\nCore-hole Auger Decay , Phys. Rev. Lett. 40, 964 (1978).\n[31] S. Tanaka, K. Mase, and S. Nagaoka, Photostimulated Ion\nDesorption from the TiO 2(110) and ZnO( 10¯10) surfaces ,\nSurf. Sci. 572, 43 (2004).\n[32] Shigemasa Suga, Akira Sekiyama, Christian Tusche,\nPhotoelectron Spectroscopy: Bulk and Surface Electronic\nStructures , 2nd ed., Springer Series in Surface Sciences\n(Springer, 2021) pp.1-511.\n[33] F. Matsui, S. Makita, H. Matsuda, T. Yano, E. Naka-\nmura, K. Tanaka, S. Suga, and S. Kera, Photoelectron\nMomentum Microscope at BL6U of UVSOR-III Syn-\nchrotron , Jpn. J. Appl. Phys. 59, 067001 (2020).\n[34] F. Matsui, K. Hagiwara, E. Nakamura, T. Yano, H. Mat-\nsuda, Y. Okano, S. Kera, E. Hashimoto, S. Koh, K. Ueno,\nT. Kobayashi, E. Iwamoto, K. Sakamoto, S. Tanaka, and\nS. Suga, Soft X-ray Photoelectron Momentum Microscope\nfor Multimodal Valence Band Stereography , Rev. Sci. In-\nstrum. 94, 083701 (2023).\n[35] R. M. A. Lieth and J. C. J. M. Terhell, Transition Metal\nDichalcogenides , inPreparation and Crystal Growth of\nMaterials with Layered Structures , edited by R. M. A.\nLieth (Springer Netherlands, Dordrecht, 1977) pp. 141–\n223.\n[36] K. Ueno, Introduction to the Growth of Bulk Single\nCrystals of Two-dimensional Transition-Metal Dichalco-\ngenides , J. Phys. Soc. Jpn. 84, 121015 (2015).\n[37] P. Giannozzi et al. ,Quantum Espresso: A Modular and\nOpen-Source Software Project for Quantum Simulations\nof Materials , J. Phys.: Condens. Matt. 21, 395502 (2009).\n[38] P. Giannozzi et al. ,Advanced Capabilities for Materi-\nals Modelling with Quantum Espresso J. Phy.: Condens.\nMatt. 29, 465901 (2017).\n[39] B. Pal, Y. Cao, X. Liu, F. Wen, M. Kareev, A. T.\nN’Diaye, P. Shafer, E. Arenholz, and J. Chakhalian,\nAnomalous Orbital Structure in Two-Dimensional Tita-\nnium Dichalcogenides , Sci. Rep. 9, 1896 (2019).\n[40] A. S. Shkvarin, Y. M. Yarmoshenko, M. V. Yablonskikh,\nA. I. Merentsov, E. G. Shkvarina, A. A. Titov, Y. M.\nZhukov, and A. N. Titov, The Electronic Structure For-\nmation of Cu xTiSe 2in a Wide Range (0.04 <x<0.8)\nof Copper Concentration , J. Chem. Phys. 144, 074702\n(2016).\n[41] H. I. Starnberg, Interactions of Deposited Ca with TiSe 2\nand TiTe 2Surfaces , J. Phy. Commun. 7, 025006 (2023).\n[42] K. Ichikawa, O. Aita, M. Kamada, and K. Tsutsumi,\nSurface Core Exciton in LiCl Studied by Photoelectron\nSpectroscopy , Phys. Rev. B 43, 5063 (1991).\n[43] S. Tanaka, N. Takahashi, K.-P. Lee, and M. Kamada,\nSurface Core Exciton of Alkali Chloride Film on Si(100) ,\nJ. Electron Spectrosc. Relat. Phenom. 80, 205 (1996).\n[44] S. Tanaka, S. D. Mor´ e, and M. Kamada, Observation of\nthe Surface Core Exciton and its Decay on Solid Xe , J.\nElectron Spectrosc. Relat. Phenom. 114-116 , 295 (2001).\n[45] A. Sanz-Matias, S. Roychoudhury, X. Feng, F. Yang,\nL. C. Kao, K. R. Zavadil, J. Guo, and D. Prendergast,\nExcitonic Effects in X-ray Absorption Spectra of Fluoride\nSalts and Their Surfaces , Chem. Mat. 34, 9144 (2022).\n[46] J. Yeh and I. Lindau, Atomic Subshell Photoionization\nCross Sections and Asymmetry Parameters: 1 ⩽z⩽103,12\nAtomic Data and Nuclear Data Tables 32, 1 (1985).\n[47] Y. S. Joe, A. M. Satanin, and C. S. Kim, Classical Anal-\nogy of Fano Resonances , Physica Scripta 74, 259 (2006).\n[48] M. Iizawa, S. Kosugi, F. Koike, and Y. Azuma, The\nQuantum and Classical Fano Parameter q , Physica\nScripta 96, 055401 (2021).[49] C. Ott, A. Kaldun, P. Raith, K. Meyer, M. Laux, J. Ev-\ners, C. H. Keitel, C. H. Greene, and T. Pfeifer, Lorentz\nMeets Fano in Spectral Line Shapes: A Universal Phase\nand its Laser Control , Science 340, 716 (2013)."    },    {        "title": "2402.04675v1.Quantitative_isoperimetric_inequalities_for_classical_capillarity_problems.pdf",        "content": "arXiv:2402.04675v1  [math.AP]  7 Feb 2024QUANTITATIVE ISOPERIMETRIC INEQUALITIES\nFOR CLASSICAL CAPILLARITY PROBLEMS\nGIULIO PASCALE AND MARCO POZZETTA\nABSTRACT . We consider capillarity functionals which measure the per imeter of sets contained in a Euclidean half-\nspace assigning a constant weight /u1D706∈ (−1,1)to the portion of the boundary that touches the boundary of th e half-\nspace. Depending on /u1D706, sets that minimize this capillarity perimeter among those with fixed volume are known to be\nsuitable truncated balls lying on the boundary of the half-s pace.\nWe prove two quantitative isoperimetric inequalities for t his class of capillarity problems: a first sharp inequality\nestimates the Fraenkel asymmetry of a competitor with respe ct to the optimal bubbles in terms of the energy deficit; a\nsecond inequality estimates a notion of asymmetry for the pa rt of the boundary of a competitor that touches the boundary\nof the half-space in terms of the energy deficit.\nAfter a symmetrization procedure, the inequalities follow from a novel combination of a quantitative ABP method\nwith a selection-type argument.\nCONTENTS\n1. Introduction 1\n2. Preliminaries 5\n2.1. Sets of finite perimeter 5\n2.2. Preliminary results on the capillarity functional 5\n3. Isoperimetric inequality for /u1D443/u1D706, Fraenkel asymmetry and deficit 7\n3.1. Isoperimetric inequality via ABP method 7\n3.2. Asymmetry and deficit 10\n4. Reduction to bounded symmetric sets 11\n4.1. Reduction to bounded sets 11\n4.2. Reduction to (/u1D45B−1)-symmetric sets 16\n4.3. Reduction to Schwarz-symmetric sets 19\n5. First quantitative isoperimetric inequality 21\n5.1. Coupling 21\n5.2. Proof of the first quantitative isoperimetric inequali ty 27\n6. Second quantitative isoperimetric inequality 30\nAppendix A. Auxiliary results 34\nA.1. Regularity of (Λ,/u1D45F0)-minimizers 34\nA.2. Axially symmetric hypersurfaces 36\nReferences 36\n1. I NTRODUCTION\nThe classical isoperimetric problem in the Euclidean space ℝ/u1D45B, for/u1D45B≥2, aims at minimizing the (/u1D45B− 1)-\ndimensional area of boundaries of sets having fixed finite vol ume. More precisely, given /u1D463 >0, one aims to\ncharacterize minimizers to the problem\n(1.1) inf{/u1D443(/u1D438) ∶/u1D438 ⊂ℝ/u1D45B,/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u1D463},\nwhere/u1D443(/u1D438)denotes the perimeter of /u1D438(see Section 2.1) and/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜the Lebesgue measure of /u1D438. It is well-known\nthat balls (uniquely) minimize ( 1.1), cf. [ De 58 ] or [ Mag12 , Chapter 14], and this is encoded in the classical\nisoperimetric inequality\n(1.2) /u1D443(/u1D438)≥/u1D45B/u1D7141\n/u1D45B\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B,\nDate : February 8, 2024.\n2020 Mathematics Subject Classification. Primary: 49J40, 49Q10. Secondary: 49Q20,28A75, 26B30.\nKey words and phrases. Isoperimetric problem, isoperimetric inequality, capill arity, quantitative isoperimetric inequality, perimeter .\n12 GIULIO PASCALE AND MARCO POZZETTA\nwhere/u1D714/u1D45Bdenotes the measure of the unit ball in ℝ/u1D45B. To prove a quantitative version of ( 1.2) means to estimate the\ndistance of a competitor from the set of minimizers in terms o f the energy deficit of the competitor with respect\nto the infimum of the problem. The first quantitative isoperim etric inequality for ( 1.1) with sharp exponents was\nproved in [ FMP08 ], and it reads\n(1.3) /u1D6FC(/u1D438)2≤/u1D436(/u1D45B)/u1D437(/u1D438),\nwhere/u1D6FC(/u1D438)and/u1D437(/u1D438)are respectively the Fraenkel asymmetry and the isoperimet ric deficit of /u1D438, i.e.,\n/u1D6FC(/u1D438) ∶= inf/b⎜⎞ce⎨eft.⎟3/u1D438Δ/u1D435(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D465)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜∶/u1D465∈ℝ/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟3\n/u1D437(/u1D438) ∶=/u1D443(/u1D438)−/u1D443(/u1D435(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜))\n/u1D443(/u1D435(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)),\nwhere/u1D435(/u1D463,/u1D465)denotes the ball in ℝ/u1D45Bwith volume /u1D463centered at/u1D465, for/u1D463 >0and/u1D465∈ℝ/u1D45B, and/u1D435(/u1D463) ∶=/u1D435(/u1D463,0).\nThe inequality ( 1.3) improves the previous non-sharp inequality proved in [ Hal92 ], after [ HHW91 ;Fug89 ]. We\nalso mention [ FMP10 ;CL12 ;FI13 ;FJ14 ;IN15 ] for further quantitative isoperimetric inequalities for possibly\nanisotropic perimeters, [ BDS15 ;CMM19 ;CES23 ] for quantitative isoperimetric inequalities on manifold s, and\n[Cia+11 ;BDR12 ;BBJ17 ;Cin+22 ;FL23 ] about weighted quantitative isoperimetric inequalities .\nIn this paper we prove quantitative isoperimetric inequali ties for the following classical capillarity problems.\nIf/u1D438is a measurable set in the half-space {/u1D465/u1D45B>0}⊂ℝ/u1D45Band/u1D706∈ (−1,1), we define the weighted perimeter\nfunctional\n/u1D443/u1D706(/u1D438) ∶=/u1D443(/u1D438,{/u1D465/u1D45B>0})−/u1D706/u1D45B−1(/u1D715∗/u1D438∩{/u1D465/u1D45B= 0}),\nwhere/u1D458, for/u1D458≥0, denotes the /u1D458-dimensional Hausdorff measure in ℝ/u1D45B, and/u1D715∗/u1D438denotes the reduced boundary\nof/u1D438(see Section 2.1). Interpreting the perimeter as a measure of the surface ten sion of a liquid drop, the constant\n/u1D706basically represent the relative adhesion coefficient betwe en a liquid drop and the solid walls of the container\ngiven by{/u1D465/u1D45B>0}.\nIf/u1D463>0, we consider the isoperimetric capillarity problem\n(1.4) inf/b⎜⎞ce⎨eft.⎟1\n/u1D443/u1D706(/u1D438) ∶/u1D438 ⊂{/u1D465/u1D45B>0},/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u1D463/b⎜⎞ce⎜⎫g⎧t.⎟1\n.\nMinimizers for ( 1.4), below called isoperimetric sets for ( 1.4), are given by suitably truncated balls lying on the\nboundary of the half-space. More precisely, if /u1D435/u1D706= {/u1D465∈/u1D4351(0)⊂ℝ/u1D45B∶⟨/u1D465,/u1D452/u1D45B⟩>/u1D706}, and for/u1D463>0we set\n/u1D435/u1D706(/u1D463) ∶=/u1D4631\n/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B(/u1D435/u1D706−/u1D706/u1D452/u1D45B)\nthen minimizers for ( 1.4) are sets of the form\n(1.5) /u1D435/u1D706(/u1D463,/u1D465) ∶=/u1D435/u1D706(/u1D463)+/u1D465,\nfor/u1D465∈ {/u1D465/u1D45B= 0} , see [ Mag12 , Theorem 19.21].\nThe first variational results regarding capillarity proble ms go back to works by Giusti, Gonzalez, Massari and\nTamanini who established existence, symmetry and regulari ty results for the isotropic sessile drop problem, where\nan additional potential energy representing gravity is add ed to the minimization of /u1D443/u1D706(see [ Gon76 ;Gon77 ;GT77 ;\nGMT80 ;Giu80 ;Giu81 ]; see also [ Fin80 ] where uniqueness results for the symmetric sessile drop we re estab-\nlished). We refer to [ Fin86 ] and [ Mag12 , Chapters 19, 20] for a more complete treatment regarding cl assical\nresults. More recently, in [ Bae15 ] the shape of liquid drops and crystals, resting on a horizon tal surface and under\nthe influence of gravity, are described in the anisotropic se tting. The shape and the fine regularity of volume con-\ntrained minimizers of weighted perimeters like /u1D443/u1D706, where the weight on the interface touching the boundary of t he\ncontainer may be nonconstant and where an additional potent ial term is present, are addressed in [ MM16 ;DM15 ;\nCEL24 ]. In [ DM15 ] the perimeter functional measuring the area of the interfa ce that does not touch the container\nis also possibly anisotropic. Recently, the isoperimetric problem for the relative perimeter of sets contained in the\ncomplement of a convex set had been addressed in [ CGR07 ], where a sharp isoperimetric inequality is established,\nand in [ FM23 ], where the rigidity of the inequality is addressed in the ge nerality of measurable sets. Extensions\nof [CGR07 ] to higher codimension have been considered in [ LWW23 ;Kru17 ].\nThe minimality of sets /u1D435/u1D706(/u1D463,/u1D465)for (1.4) comes with an isoperimetric inequality for /u1D443/u1D706, see Theorem 3.5below.\nIn order to prove a quantitative isoperimetric inequality f or (1.4), we define the corresponding Fraenkel asymmetry\nand isoperimetric deficit by setting\n/u1D6FC/u1D706(/u1D438) ∶= inf/b⎜⎞ce⎨eft.⎟3/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜∶/u1D465∈ {/u1D465/u1D45B= 0}/b⎜⎞ce⎜⎫g⎧t.⎟3\n, /u1D437/u1D706(/u1D438) ∶=/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706(/u1D463))\n/u1D443/u1D706(/u1D435/u1D706(/u1D463)),\nfor any/u1D438 ⊂{/u1D465/u1D45B>0}. The infimum defining the asymmetry is, in fact, a minimum.\nThe first main result of the paper is the followingQUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 3\nTheorem 1.1. Let/u1D706∈ (−1,1)and/u1D45B∈ℕwith/u1D45B≥2. There exists a constant /u1D450iso=/u1D450iso(/u1D45B,/u1D706)>0such that for any\nmeasurable set /u1D438 ⊂ℝ/u1D45B∩{/u1D465/u1D45B>0}with finite measure there holds\n(1.6) /u1D6FC/u1D706(/u1D438)2≤/u1D450iso/u1D437/u1D706(/u1D438).\nAs for the classical ( 1.2), perturbing the boundary of an optimal bubble only inside t he container {/u1D465/u1D45B>0}, it is\npossible to check that exponents in ( 1.6) are sharp.\nIn the context of these capillarity problems it is also spont aneous to consider a notion of asymmetry for the part\nof the boundary of a set that touches the half-plane {/u1D465/u1D45B= 0} . For a measurable set /u1D438 ⊂{/u1D465/u1D45B>0}, we define\n/u1D6FD/u1D706(/u1D438) ∶= inf/b⎜⎞ce⎨eft.⎟4\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715∗/u1D438∩{/u1D465/u1D45B= 0} Δ/u1D715∗/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D465)∩{/u1D465/u1D45B= 0}/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715∗/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D465)∩{/u1D465/u1D45B= 0}/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1 ∶/u1D465∈ {/u1D465/u1D45B= 0}/b⎜⎞ce⎜⎫g⎧t.⎟4\n.\nThe previous quantity measures the asymmetry of the set /u1D715∗/u1D438∩{/u1D465/u1D45B= 0} with respect to (/u1D45B−1)-dimensional balls\nin{/u1D465/u1D45B= 0} having volume equal to the one of the trace of the optimal bubb le corresponding to the volume of /u1D438.\nWe establish the following second quantitative isoperimet ric inequality, that provides a quantitative estimate on /u1D6FD/u1D706.\nTheorem 1.2. Let/u1D706∈ (−1,1)and/u1D45B∈ℕwith/u1D45B≥2. There exists a constant /u1D450)uni2032(var\niso=/u1D450)uni2032(var\niso(/u1D45B,/u1D706)>0such that for any\nmeasurable set /u1D438 ⊂ℝ/u1D45B∩{/u1D465/u1D45B>0}with finite measure there holds\n(1.7) /u1D6FD/u1D706(/u1D438)≤/u1D450)uni2032(var\nisomax/b⎜⎞ce⎨eft.⎟2\n/u1D437/u1D706(/u1D438),/u1D437/u1D706(/u1D438)1\n2/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nStrategy of the proof and comments. Observing that, roughly speaking, the minimization proble m (1.4) is sym-\nmetric with respect to the first /u1D45B− 1axes, it is possible to adapt arguments in the spirit of [ FMP08 ] to see that,\nin order to prove Theorem 1.1, it is sufficient to prove ( 1.6) in the class of Schwarz-symmetric sets, see Corol-\nlary 4.9. Here a set /u1D438is said to be Schwarz-symmetric (Definition 3.1) if the intersection /u1D438∩ {/u1D465/u1D45B=/u1D461}is an\n(/u1D45B−1)-dimensional ball in {/u1D465/u1D45B=/u1D461}centered at (0,/u1D461), for any/u1D461. However, we point out that it seems not possible\nto push the strategy of [ FMP08 ] to the very end to prove Theorem 1.1. Indeed the arguments in [ FMP08 ] require\nto Schwarz-symmetrize a competitor with respect to a prefer red axis depending on the competitor, while in our\ncase it is only possible to symmetrize with respect to the /u1D45B-th axis. One finds an analogous obstruction also in a\npossible adaptation of the proof via symmetrization revise d in [ Mag08 ] (see also [ Fus15 ]); in [ Mag08 ] the quanti-\ntative isoperimetric inequality for Schwarz-symmetric se ts is eventually obtained performing a quantitative versio n\nof Gromov’s proof [ MS86 ] of the isoperimetric inequality, but again after having sy mmetrized a competitor with\nrespect to a convenient axis.\nThe proof of ( 1.6) in the class of Schwarz-symmetric sets is achieved here wit h a new combination of the so-\ncalled selection principle [ AFM13 ;CL12 ] with an Alexandrov–Bakelman–Pucci-type technique in the spirit of\n[Cin+22 ].\nIn the recent [ Cin+22 ], the authors prove sharp quantitative isoperimetric ineq ualities for a class of isoperimetric\nproblems in cones where volume and perimeter are weighted in terms of a function satisfying suitable homogeneity\nand concavity properties. The proof in [ Cin+22 ] stems from the fact that the isoperimetric inequality for t he\ncorresponding problem was proved in [ CRS16 ] by a so-called ABP argument. The methods that go under the\nname of ABP techniques were originally employed to derive re gularity estimates for second order elliptic equations\n[GT01 , Chapter 9] and they were applied to give a new direct proof of the classical isoperimetric inequality in\n[Cab00 ;Cab08 ] (see [ Cab17 ] for a detailed account on the method). More precisely, for t he classical isoperimetric\nproblem, if/u1D438 ⊂ℝ/u1D45Bis a smooth connected open set, one would consider a solution /u1D462to\n(1.8)/b⎜⎞ce⎨eft.⎟4\nΔ/u1D462=/u1D443(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜on/u1D438,\n/u1D715/u1D708/u1D462= 1 on/u1D715/u1D438,\nwhere/u1D715/u1D708/u1D462denotes outward normal derivative. It is immediate to check that∇/u1D462(/u1D438)uni2032(var)⊃ /u1D4351(0)where/u1D438)uni2032(var∶= {/u1D465∈\n/u1D438∶ ∇2/u1D462(/u1D465)≥0}, hence the area formula together with the arithmetic-geome tric mean inequality readily imply\nthe Euclidean isoperimetric inequality, indeed\n/u1D714/u1D45B=/u⎪⎫007C.v⎞⎜/u1D4351(0)/u⎪⎫007C.v⎞⎜≤/uni222B.dsp/u1D438)uni2032(vardet∇2/u1D462≤/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟2Δ/u1D462\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\n=/u1D443(/u1D438)/u1D45B\n/u1D45B/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1.\nNow the rough idea is that a control on the energy deficit shoul d control the \"asymmetry\" of the solution /u1D462with\nrespect to the solution corresponding to the optimal shape /u1D4351(0), that is the radially symmetric parabola /u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜2∕2.\nIn fact, this is achieved in [ Cin+22 ] by controlling the asymmetry of a coupling function which is defined as a\nsuitable convex envelope of /u1D462. Adapting arguments from [ FMP10 ], in [ Cin+22 ] the authors then show that it is4 GIULIO PASCALE AND MARCO POZZETTA\npossible to employ trace-type theorems to estimate the asym metry of a competitor set in terms of the asymmetry\nof the coupling function, which is in turn estimated by the en ergy deficit.\nIn Theorem 3.5below we will give an ABP proof of the isoperimetric inequali ty for the problem ( 1.4) by\nanalyzing an elliptic problem analogous to ( 1.8), see ( 3.4). We are then in position to consider a coupling function\nas done in [ Cin+22 ] and we can quantitatively estimate its asymmetry, which wi ll be achieved in Proposition 5.4.\nMoreover, Schwarz-symmetric sets that are sufficiently smal l/u1D4361-perturbations (in the sense of Definition 5.10) of\nan optimal bubble ( 1.5) readily verify the needed trace-type inequalities that re late asymmetry of the competitor\nwith the asymmetry of the coupling. Hence this establishes t he quantitative inequality ( 1.6) for Schwarz-symmetric\n/u1D4361-perturbations of optimal bubbles, see Corollary 5.11. Observe that, in our setting, isoperimetric sets are just\nLipschitz-regular and a set /u1D438is/u1D4361-close to an optimal bubble if just the relative boundary /u1D715/u1D438∩{/u1D465/u1D45B>0}is close\nto the relative boundary of an optimal bubble as /u1D4361-hypersurfaces with boundary.\nOnce ( 1.6) is proved for /u1D4361-perturbations of optimal bubbles (Corollary 5.11), we want apply a selection-type\nargument in the spirit of [ AFM13 ;CL12 ] in the class of Schwarz-symmetric sets in order to extend th e validity\nof the quantitative inequality to the whole class of Schwarz -symmetric sets. In this way we also avoid the imple-\nmentation of further technical results that in [ FMP10 ;Cin+22 ] allow to reduce to just consider sets that enjoy the\nrequired trace-type inequalities.\nRoughly speaking, in a selection-type argument one argues b y contradiction assuming existence of sets contradict-\ning the quantitative isoperimetric inequality and one uses such sets to define an auxiliary minimization problem, cf.\n(5.22). Minimizers to the previous problem still contradict the q uantitative isoperimetric inequality, but at the same\ntime they are shown to be small /u1D4361-perturbations of some isoperimetric set, contradicting t he inequality already\nproved for sets given by small perturbations of optimal ones .\nIn our case, we will prove that minimizers /u1D438to the auxiliary minimization problem are /u1D4361-perturbations of opti-\nmal bubbles up to the boundary of the half-space {/u1D465/u1D45B>0}as a consequence of the classical interior regularity of\n(Λ,/u1D45F0)-minimizers of the perimeter (Definition A.1), see [ Tam84 ] and [ Mag12 , Chapter 26], together with a simple\nvariational argument that allows us to propagate the regula rity up to the boundary of the half-space {/u1D465/u1D45B>0}. This\nessentially follows from the fact that a Schwarz-symmetric local(Λ,/u1D45F0)-minimizer/u1D438in{/u1D465/u1D45B>0}is locally of\nclass/u1D4361and has bounded mean curvature (in a generalized sense, see L emma A.2); hence a uniform bound on the\nwhole second fundamental form on a portion of boundary /u1D715/u1D438∩{0</u1D465/u1D45B</u1D700}follows just by showing that the set\n/u1D715/u1D438∩ {0< /u1D465/u1D45B< /u1D700}is far from the axis of revolution of /u1D438(see Lemma A.4), and the latter holds in the proof of\nTheorem 1.1by an energy estimate holding for minimizers to the auxiliar y minimization problems.\nWe stress that in our case it is not clear how to apply a Fuglede -type argument [ CL12 ;Fug89 ] to prove the\nquantitative inequality for sets given by small /u1D4361-perturbations of optimal ones. Indeed, the classical Fugl ede’s\nmethod relies on the precise knowledge of the eigenvalues of the Laplace–Beltrami operator, which is not available\nfor the operator on spherical caps corresponding to optimal bubbles ( 1.5) for generic /u1D706∈ (−1,1). Moreover,\nobserve that in our case it is not possible to globally parame trize/u1D4361-close boundaries one on the other as normal\ngraphs in general, introducing a further nontrivial techni cal difficulty in the implementation of a Fuglede-type\nargument.\nOnce Theorem 1.1is proved, for the proof of Theorem 1.2we argue as follows. First we establish a quantitative\ninequality that estimates the Hausdorff distance between th e relative boundary in {/u1D465/u1D45B>0}of a competitor /u1D438and\nthe relative boundary of some bubble in terms of the Fraenkel asymmetry of /u1D438, under the assumption that /u1D438is a so-\ncalled(/u1D43E,/u1D45F0)-quasiminimal set, see Definition 6.2and Lemma 6.4. This is achieved since quasiminimal sets enjoy\nuniform density estimates at boundary points, see Theorem 6.3. Exploiting Theorem 1.1, the previous quantitative\ninequality yields an inequality of the form ( 1.7) in the class of quasiminimal sets. Eventually, Theorem 1.2follows\nby applying again a selection-type argument where now /u1D6FD/u1D706plays the role of the Fraenkel asymmetry.\nOrganization. In Section 2we collect definitions and facts on sets of finite perimeter an d we prove some prelim-\ninary results on the capillarity functional. In Section 3we establish the sharp isoperimetric inequality for /u1D443/u1D706via\nABP method and we prove preliminary facts on the Fraenkel asy mmetry and on the deficit. Section 4is devoted to\na series of technical results that allow to reduce the analys is to Schwarz-symmetric sets. In Section 5we complete\nthe proof of Theorem 1.1by proving ( 1.6) on Schwarz-symmetric sets. Finally in Section 6we prove Theorem 1.2.\nIn Appendix Awe recall some facts about (Λ,/u1D45F0)-minimizers and we recall a formula for the mean curvature of\naxially symmetric hypersurfaces.\nAcknowledgments. The second author is partially supported by the PRIN Project 2022E9CF89 - PNRR Italia\nDomani, funded by EU Program NextGenerationEU. The authors are grateful to Nicola Fusco for many suggestions\nand for stimulating discussions.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 5\n2. P RELIMINARIES\nFrom now on and for the rest of the paper it is assumed that /u1D706∈ (−1,1)and/u1D45B∈ℕwith/u1D45B≥2are fixed.\nList of symbols.\n∙/u⎪⎫007C.v⎞⎜⋅/u⎪⎫007C.v⎞⎜denotes Lebesgue measure in ℝ/u1D45B.\n∙/u1D435/u1D45F∶=/u1D435/u1D45F(0)⊂ℝ/u1D45Bfor/u1D45F>0,/u1D435∶=/u1D4351.\n∙/u1D435/u1D706= {/u1D465∈/u1D435∶⟨/u1D465,/u1D452/u1D45B⟩>/u1D706}.\n∙/u1D435/u1D706(/u1D463) ∶=/u1D4631\n/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B(/u1D435/u1D706−/u1D706/u1D452/u1D45B), for any/u1D463>0.\n∙/u1D435/u1D706(/u1D463,/u1D465) ∶=/u1D435/u1D706(/u1D463)+/u1D465, for any/u1D465∈ {/u1D465/u1D45B= 0} . In particular /u1D435/u1D706(/u1D463) =/u1D435/u1D706(/u1D463,0).\n∙/u1D450(/u1D45B,/u1D706),/u1D436(/u1D45B,/u1D706)denote strictly positive constants, depending on /u1D45B,/u1D706only, that may change from line to line.\n∙/u1D451denotes Hausdorff distance in ℝ/u1D45B.\n∙/u1D43B∶= {/u1D465/u1D45B≤0}.\n∙/u1D451denotes/u1D451-dimensional Hausdorff measure in ℝ/u1D45B, for/u1D451≥0.\n∙/u1D443/u1D706(/u1D435/u1D706) ∶=/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)) =/u1D443(/u1D435,{/u1D465/u1D45B>/u1D706})−/u1D706/u1D45B−1(/u1D435∩{/u1D465/u1D45B=/u1D706}).\n∙/u1D444/u1D45F∶= [−/u1D45F,/u1D45F]/u1D45B⊂ℝ/u1D45B, for any/u1D45F>0.\n∙/u1D45F/u1D706∶= min/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2\n,/u1D445/u1D706∶= max/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\n2.1. Sets of finite perimeter. We recall basic definitions and properties regarding sets of finite perimeter, referring\nto [AFP00 ;Mag12 ] for a complete treatment on the subject. The perimeter of a m easurable set /u1D438 ⊂ℝ/u1D45Bin an open\nset/u1D434⊂ℝ/u1D45Bis defined by\n(2.1) /u1D443(/u1D438,/u1D434) ∶= sup/b⎜⎞ce⎨eft.⎟3\n/uni222B.dsp/u1D438div/u1D447(/u1D465)d/u1D465∶/u1D447∈/u1D4361\n/u1D450(/u1D434;ℝ/u1D45B),‖/u1D447‖∞≤1/b⎜⎞ce⎜⎫g⎧t.⎟3\n.\nDenoting/u1D443(/u1D438) ∶=/u1D443(/u1D438,ℝ/u1D45B), we say that /u1D438is a set of finite perimeter if /u1D443(/u1D438)<+∞. In such a case, the\ncharacteristic function /u1D712/u1D438has a distributional gradient /u1D437/u1D712/u1D438that is a vector-valued Radon measure on ℝ/u1D45Bsuch that\n/uni222B.dsp/u1D438div/u1D447(/u1D465)d/u1D465= −/uni222B.dspℝ/u1D45B/u1D447d/u1D437/u1D712/u1D438,∀/u1D447∈/u1D4361\n/u1D450(ℝ/u1D45B;ℝ/u1D45B).\nIt can be proved that the set function /u1D443(/u1D438,⋅)defined in ( 2.1) is the restriction of a nonnegative Borel measure to\nopen sets. The measure /u1D443(/u1D438,⋅)coincides with the total variation /u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜of the distributional gradient, and it is\nconcentrated on the reduced boundary\n/u1D715∗/u1D438∶=/b⎜⎞ce⎨eft.⎟3\n/u1D465∈ spt/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜∶ ∃/u1D708/u1D438(/u1D465) ∶= −lim\n/u1D45F→0/u1D437/u1D712/u1D438(/u1D435/u1D45F(/u1D465))\n/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438(/u1D435/u1D45F(/u1D465))/u⎪⎫007C.v⎞⎜with/u⎪⎫007C.v⎞⎜/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜= 1/b⎜⎞ce⎜⎫g⎧t.⎟3\n.\nIntroducing the sets of density /u1D461∈ [0,1]points for/u1D438defined by\n/u1D438(/u1D461)∶=/b⎜⎞ce⎨eft.⎟3\n/u1D465∈ℝ/u1D45B∶ lim\n/u1D45F→0/u⎪⎫007C.v⎞⎜/u1D438∩/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜=/u1D461/b⎜⎞ce⎜⎫g⎧t.⎟3\n,\nwe have that the reduced boundary coincides both with ℝ/u1D45B⧵(/u1D438(1)∪/u1D438(0))and with the set /u1D438(1∕2)up to/u1D45B−1-\nnegligible sets. The vector /u1D708/u1D438is called the generalized outer normal of /u1D438. Moreover/u1D443(/u1D438,⋅) =/u1D45B−1/u1D715∗/u1D438, and\nthe distributional gradient can be written as /u1D437/u1D712/u1D438= −/u1D708/u1D438/u1D45B−1/u1D715∗/u1D438.\n2.2. Preliminary results on the capillarity functional. In this section we prove some preparatory results on the\nfunctional/u1D443/u1D706. From now on, /u1D43Bdenotes the closed half-space /u1D43B∶= {/u1D465/u1D45B≤0}.\nRemark 2.1.Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a measurable set. We observe that\n/u1D443/u1D706(/u1D438) =/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B1−/u1D706/u⎪⎫27E8.⎟1\n/u1D452/u1D45B,/u1D708/u1D438/u⎪⎫27E9.⎟1\nd/u1D45B−1,\nwhere/u1D708/u1D438is the generalized outer normal to /u1D438. In particular, since /u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜<1, we have that /u1D443/u1D706(/u1D438)≥0. The previous\nidentity follows from the divergence theorem, indeed\n0 =/uni222B.dsp/u1D438div/u1D452/u1D45Bd/u1D465= −/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)+/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B/u⎪⎫27E8.⎟1/u1D452/u1D45B,/u1D708/u1D438/u⎪⎫27E9.⎟1d/u1D45B−1.\nWe now aim at proving an approximation result for sets of finit e perimeter contained in ℝ/u1D45B⧵/u1D43Bby sequences\nof sets having smooth boundary relative in ℝ/u1D45B⧵/u1D43B. We need the following lemma first.6 GIULIO PASCALE AND MARCO POZZETTA\nLemma 2.2. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with finite measure. For any /u1D461≥0define the\ndiffeomorphism /u1D711/u1D461∶ {/u1D465/u1D45B≥0}→{/u1D465/u1D45B≥0}given by/u1D711/u1D461(/u1D465)uni2032(var,/u1D465/u1D45B) ∶= (/u1D465)uni2032(var,/u1D465/u1D45B(1 +/u1D461/u1D452−/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜2)), where we wrote\n/u1D465= (/u1D465)uni2032(var,/u1D465/u1D45B) ∈ℝ/u1D45B−1×ℝfor any/u1D465∈ {/u1D465/u1D45B≥0}⊂ℝ/u1D45B. Then\n(1)for at most countably many /u1D461∈ [0,+∞) there holds\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/b⎜⎞ce⎨eft.⎟1/u1D465∈/u1D715∗(/u1D711/u1D461(/u1D438))⧵/u1D43B∶/u1D708/u1D711/u1D461(/u1D438)(/u1D465) = ±/u1D452/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1>0;\n(2)for at most countably many /u1D708∈/u1D54A/u1D45B−1there holds\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/b⎜⎞ce⎨eft.⎟1\n/u1D465∈/u1D715∗/u1D438∶/u1D708/u1D438(/u1D465) = ±/u1D708/b⎜⎞ce⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n>0.\nProof. We begin by proving (1). Define /u1D453/u1D461∶ℝ/u1D45B−1→ℝby/u1D453/u1D461(/u1D465)uni2032(var) ∶= 1+/u1D461/u1D452−/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜2, and let\n/u1D434/u1D461∶=/b⎜⎞ce⎨eft.⎟2\n(/u1D465)uni2032(var,/u1D465/u1D45B) ∈/u1D715∗/u1D438∩ℝ/u1D45B⧵/u1D43Bwith/u1D465)uni2032(var∈ℝ/u1D45B−1⧵{0} ∶/u1D708/u1D438(/u1D465)uni2032(var,/u1D465/u1D45B)is proportional to (/u1D465/u1D45B∇/u1D453/u1D461(/u1D465)uni2032(var),/u1D453/u1D461(/u1D465)uni2032(var))/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nWe claim that, if /u1D460,/u1D45F>0, with/u1D460≠/u1D45F, then/u1D434/u1D45F∩/u1D434/u1D460= )uni2205(var. Indeed, if/u1D434/u1D45F∩/u1D434/u1D460≠)uni2205(var, there exist (̄ /u1D465)uni2032(var,̄ /u1D465/u1D45B) ∈/u1D715∗/u1D438∩ℝ/u1D45B⧵/u1D43B\nand̄ /u1D6FC∈ℝ⧵{0}such that\n(̄ /u1D465/u1D45B∇/u1D453/u1D45F(̄ /u1D465)uni2032(var),/u1D453/u1D45F(̄ /u1D465)uni2032(var)) =̄ /u1D6FC(̄ /u1D465/u1D45B∇/u1D453/u1D460(̄ /u1D465)uni2032(var),/u1D453/u1D460(̄ /u1D465)uni2032(var)).\nSince/u1D465/u1D45B>0and/u1D465)uni2032(var≠0, then̄ /u1D465/u1D45B∇/u1D453/u1D45F(̄ /u1D465)uni2032(var) =̄ /u1D6FC̄ /u1D465/u1D45B∇/u1D453/u1D460(̄ /u1D465)uni2032(var)implies/u1D45F=̄ /u1D6FC/u1D460. Thus/u1D453/u1D45F(̄ /u1D465)uni2032(var) =̄ /u1D6FC/u1D453/u1D460(̄ /u1D465)uni2032(var)implies1 =̄ /u1D6FC,\nwhich in turn implies /u1D45F=/u1D460, contradiction.\nFor/u1D458∈ℕ≥1let us define\n/u1D440/u1D458∶=/b⎜⎞ce⎨eft.⎟2\n/u1D461>0 ∶/u1D45B−1(/u1D434/u1D461)>1\n/u1D458/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nWe want to prove that /u1D440/u1D458is finite. Let /u1D4611,…,/u1D461/u1D459∈/u1D440/u1D458, with/u1D461/u1D456≠/u1D461/u1D457if/u1D456≠/u1D457. Since/u1D434/u1D461/u1D456≠/u1D434/u1D461/u1D457= )uni2205(var, then\n/u1D443(/u1D438)≥/u1D459/u⎪⎫2211.⎟1\n/u1D456=1/u1D45B−1(/u1D434/u1D461/u1D456)≥1\n/u1D458/u1D459,\nand then#/u1D440/u1D458≤/u1D458/u1D443(/u1D438). Therefore the set\n/b⎜⎞ce⎨eft.⎟1\n/u1D461>0 ∶/u1D45B−1(/u1D434/u1D461)>0/b⎜⎞ce⎜⎫g⎧t.⎟1\n=/u⎪⎫22C3.⎟1\n/u1D458/u1D440/u1D458,\nis countable.\nSince the differential of /u1D711/u1D461is represented by the matrix\nd/u1D711/u1D461(/u1D465)uni2032(var,/u1D465/u1D45B) =/⎝⎞⎜e⎪⎨eft.⎟3\nIdℝ/u1D45B−10\n/u1D465/u1D45B∇/u1D453/u1D461(/u1D465)uni2032(var)/u1D453/u1D461(/u1D465)uni2032(var)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n,\nrequiring that /u1D708/u1D711/u1D461(/u1D438)(/u1D711/u1D461(/u1D465)) = ±/u1D452/u1D45Bfor some/u1D465= (/u1D465)uni2032(var,/u1D465/u1D45B)means that\n/u1D708/u1D438(/u1D465)uni2032(var,/u1D465/u1D45B)is proportional to (/u1D465/u1D45B∇/u1D453/u1D461(/u1D465)uni2032(var),/u1D453/u1D461(/u1D465)uni2032(var)),\nsee [Mag12 , Proposition 17.1] and [ AFP00 , Proposition 2.88]. Therefore for any /u1D461∉ ∪/u1D458/u1D440/u1D458there holds\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/b⎜⎞ce⎨eft.⎟1\n/u1D465∈/u1D715∗(/u1D711/u1D461(/u1D438))⧵/u1D43B∶/u1D708/u1D711/u1D461(/u1D438)(/u1D465) = ±/u1D452/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n= 0.\nThe proof of (2) is analogous. For /u1D458∈ℕ≥1let\n/u1D441/u1D458∶=/b⎜⎞ce⎨eft.⎟2\n/u1D708∈/u1D54A/u1D45B−1∶/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/b⎜⎞ce⎨eft.⎟1\n/u1D465∈/u1D715∗/u1D438∶/u1D708/u1D438=/u1D708/b⎜⎞ce⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n>1\n/u1D458/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nAs above one gets that ∪/u1D458/u1D441/u1D458is at most countable, and (2) follows. /square\nLemma 2.3 (Approximation with regular sets) .Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with finite measure.\nThen there exists a sequence of sets /u1D438/u1D456⊂ℝ/u1D45B⧵/u1D43Bsuch that\n(1)/u1D438/u1D456is a bounded set such that /u1D715/u1D438/u1D456⧵/u1D715/u1D43Bis a smooth hypersurface (possibly with smooth boundary) su ch\nthat either/u1D715/u1D438/u1D456∩/u1D715/u1D43B= )uni2205(varor/u1D715/u1D438/u1D456⧵/u1D715/u1D43Bintersects/u1D715/u1D43Borthogonally;\n(2)/u1D438/u1D456→/u1D438in/u1D43F1,/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵/u1D43B)→/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B),/u1D45B−1(/u1D715/u1D438/u1D456∩/u1D715/u1D43B)→/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B), as/u1D456→+∞.\nIn particular, /u1D443(/u1D438/u1D456)→/u1D443(/u1D438)and/u1D443/u1D706(/u1D438/u1D456)→/u1D443/u1D706(/u1D438)as/u1D456→+∞, for any/u1D706∈ (−1,1);\n(3)/u1D45B−1({/u1D465∈/u1D715∗/u1D438/u1D456∶/u1D708/u1D438/u1D456(/u1D465) = ±/u1D452/u1D457}) = 0 for any/u1D457= 1,…,/u1D45B−1;\n(4)/u1D45B−1({/u1D465∈/u1D715∗/u1D438/u1D456⧵/u1D43B∶/u1D708/u1D438/u1D456(/u1D465) = ±/u1D452/u1D45B}) = 0 .QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 7\nProof. Since/u1D443(/u1D438),/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜<+∞, by a diagonal argument we can assume without loss of general ity that/u1D438is bounded.\nStep 1. We first construct a sequence /u1D439/u1D456⊂ℝ/u1D45B⧵/u1D43Bsuch that 1. and 2. hold with /u1D439/u1D456in place of/u1D438/u1D456. Let us\ndenote by/u1D439the union of /u1D438with the reflection of /u1D438with respect to the hyperplane {/u1D465/u1D45B= 0} . There exists a\nsequence of smooth sets ̃/u1D439/u1D456⊂ℝ/u1D45B, symmetric with respect to {/u1D465/u1D45B= 0} , such that they converge to /u1D439in/u1D43F1(ℝ/u1D45B)\nand/u1D443(̃/u1D439/u1D456)→/u1D443(/u1D439)(see, e.g., [ AFP00 , Theorem 3.42]). The fact that ̃/u1D439/u1D456is symmetric with respect to {/u1D465/u1D45B= 0}\nfollows as̃/u1D439/u1D456can be obtained as superlevel set of a convolution of /u1D712/u1D439with a symmetric mollifier. In particular, if\n/u1D715̃/u1D439/u1D456∩/u1D715/u1D43B≠)uni2205(var, then/u1D715̃/u1D439/u1D456intersects/u1D43Borthogonally, and thus /u1D715̃/u1D439/u1D456∩/u1D715/u1D43Bis a smooth (/u1D45B−2)-dimensional manifold.\nLet/u1D439/u1D456∶=̃/u1D439/u1D456⧵/u1D43B. Then/u1D439/u1D456→/u1D438in/u1D43F1and\n/u1D443(/u1D439/u1D456,ℝ/u1D45B⧵/u1D43B) =1\n2/u1D443(̃/u1D439/u1D456)→1\n2/u1D443(/u1D439) =/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B).\nLet us define the function\n/u1D453∶ℝ/u1D45B→[0,+∞)\n/u1D453(/u1D463) =/u⎪⎫007C.v⎞⎜/u1D463/u⎪⎫007C.v⎞⎜−/u1D706⟨/u1D452/u1D45B,/u1D463⟩.\nNote that/u1D453is continuous; moreover /u1D453(/u1D461/u1D463) =/u1D461/u1D453(/u1D463)for any/u1D461≥0and/u1D453is convex. Let us set\n/u1D707/u1D456∶=/u1D708/u1D439/u1D456/u1D45B−1(/u1D715∗/u1D439/u1D456∩(ℝ/u1D45B⧵/u1D43B))\n/u1D707∶=/u1D708/u1D438/u1D45B−1(/u1D715∗/u1D438∩(ℝ/u1D45B⧵/u1D43B)).\nSince/u1D708/u1D439/u1D456/u1D45B−1/u1D715∗/u1D439/u1D456→/u1D708/u1D438/u1D45B−1/u1D715∗/u1D438weakly∗inℝ/u1D45B, then/u1D707/u1D456→/u1D707weakly∗inℝ/u1D45B⧵/u1D43B. Since we already know\nthat/u⎪⎫007C.v⎞⎜/u1D707/u1D456/u⎪⎫007C.v⎞⎜(ℝ/u1D45B⧵/u1D43B)→/u⎪⎫007C.v⎞⎜/u1D707/u⎪⎫007C.v⎞⎜(ℝ/u1D45B⧵/u1D43B), by Reshetnyak continuity theorem (see, e.g., [ AFP00 , Theorem 2.39]) we deduce\n/u1D443/u1D706(/u1D439/u1D456) =∫/u1D453(/u1D707/u1D456∕/u⎪⎫007C.v⎞⎜/u1D707/u1D456/u⎪⎫007C.v⎞⎜)d/u⎪⎫007C.v⎞⎜/u1D707/u1D456/u⎪⎫007C.v⎞⎜→∫/u1D453(/u1D707∕/u⎪⎫007C.v⎞⎜/u1D707/u⎪⎫007C.v⎞⎜)d/u⎪⎫007C.v⎞⎜/u1D707/u⎪⎫007C.v⎞⎜=/u1D443/u1D706(/u1D438).\nStep 2. Let us consider the flow /u1D711/u1D461∶ℝ/u1D45B⧵/u1D43B→ℝ/u1D45B⧵/u1D43Bsuch that\n/u1D711/u1D461(/u1D465)uni2032(var,/u1D465/u1D45B) ∶= (/u1D465)uni2032(var,/u1D465/u1D45B(1+/u1D461/u1D452−/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜2)),\nfor/u1D461≥0. By Lemma 2.2for a.e./u1D461we have that /u1D711/u1D461(/u1D439/u1D456)satisfies (4). Moreover, for any /u1D461≥0there exists a sequence\nof rotations /u1D457,/u1D461∶ (/u1D465)uni2032(var,/u1D465/u1D45B)↦(/u1D445/u1D457,/u1D461(/u1D465)uni2032(var),/u1D465/u1D45B)along the/u1D45B-th axis converging to the identity such that\n/u1D45B−1({/u1D465∈/u1D715∗(/u1D457,/u1D461(/u1D711/u1D461(/u1D439/u1D456)) ∶/u1D708/u1D457,/u1D461(/u1D711/u1D461(/u1D439/u1D456))(/u1D465) = ±/u1D452/u1D459}) = 0\nfor all/u1D459= 1,…,/u1D45B− 1. By a diagonal argument, since /u1D711/u1D461(/u1D439/u1D456)maintains orthogonal intersection with /u1D715/u1D43B, one\nextract the desired sequence /u1D438/u1D456. /square\nCorollary 2.4. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with finite measure. Then\n/u1D443/u1D706(/u1D438)≥1−/u1D706\n2/u1D443(/u1D438).\nProof. Let{/u1D438/u1D456}/u1D456be the sequence of smooth sets in ℝ/u1D45B⧵/u1D43Bgiven by Lemma 2.3. Since the orthogonal projection\non/u1D715/u1D43Bis a1-Lipschitz and surjective map from /u1D715/u1D438/u1D456∩(ℝ/u1D45B⧵/u1D43B)onto/u1D715/u1D438/u1D456∩/u1D715/u1D43B, then (see, e.g., [ AFP00 , Proposition\n2.49])\n/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵/u1D43B)≥ /u1D45B−1(/u1D715∗/u1D438/u1D456∩/u1D715/u1D43B).\nSince\n(2.2) /u1D443/u1D706(/u1D438/u1D456) =1+/u1D706\n2(/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵/u1D43B)−/u1D45B−1(/u1D715∗/u1D438/u1D456∩/u1D715/u1D43B))+1−/u1D706\n2/u1D443(/u1D438/u1D456),\nthe claim follows by passing to the limit /u1D456→∞. /square\n3. I SOPERIMETRIC INEQUALITY FOR /u1D443/u1D706, FRAENKEL ASYMMETRY AND DEFICIT\n3.1. Isoperimetric inequality via ABP method. In this section we give a proof of the isoperimetric inequali ty\nfor the capillarity functional /u1D443/u1D706exploiting an ABP method. We start by defining Schwarz-symme tric sets.\nDefinition 3.1. Let/u1D438 ⊂ℝ/u1D45Bbe a Borel set. Then its Schwarz symmetrization (with respec t to the/u1D45B-th axis) is the\nset\n/u1D438∗∶= {(/u1D465)uni2032(var,/u1D461) ∈ℝ/u1D45B−1×ℝ∶/u1D714/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜/u1D45B−1</u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461}), /u1D461∈ℝ}.\nA Borel set /u1D438 ⊂ℝ/u1D45Bis said to be Schwarz-symmetric if it coincides with its Schw arz symmetrization, up to\nnegligible sets. The profile function of a Schwarz-symmetri c set/u1D438is the a.e. defined map /u1D453∶ℝ→ℝgiven by\n/u1D453(/u1D461) ∶=/⎝⎞⎜e⎪⎨eft.⎟3/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})\n/u1D714/u1D45B−1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B−1\n.8 GIULIO PASCALE AND MARCO POZZETTA\nRemark 3.2.Let/u1D438 ⊂{/u1D465/u1D45B>0}be a set of finite perimeter. Let /u1D438be Schwarz-symmetric with profile function /u1D453.\nAssume that /u1D45B−1({/u1D465∈/u1D715∗/u1D438∶/u1D708/u1D438(/u1D465) = ±/u1D452/u1D45B} = 0 .\nThen/u1D453is differentiable almost everywhere (see, e.g., [ Mag12 , Theorem 18.11]) with derivative\n/u1D453)uni2032(var(/u1D461) = −/⎝⎞⎜e⎪⎨eft.⎟4\n1\n/u1D714/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u1D45B−2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟41\n/u1D45B−1\n/uni222B.dsp/u1D715∗/u1D438∩{/u1D465/u1D45B=/u1D461}/u⎪⎫27E8.⎟1/u1D708/u1D438(/u1D467,/u1D461),/u1D452/u1D45B/u⎪⎫27E9.⎟1\n/u⎪⎫007C.x/u⎪⎫007C.x/u1D708/u1D438(/u1D467,/u1D461)−⟨/u1D708/u1D438(/u1D467,/u1D461),/u1D452/u1D45B⟩/u1D452/u1D45B/u⎪⎫007C.x/u⎪⎫007C.xd/u1D45B−2(/u1D467),\nat a.e./u1D461∈ℝsuch that/u1D453(/u1D461)>0. The generalized outer normal to /u1D438can be written as\n/u1D708/u1D438(/u1D453(/u1D461)/u1D452,/u1D461) =1√\n1+(/u1D453)uni2032(var)2(/u1D452,−/u1D453)uni2032(var),\nfor any/u1D452= (/u1D4521,…,/u1D452/u1D45B−1) ∈ℝ/u1D45B−1with/u⎪⎫007C.v⎞⎜/u1D452/u⎪⎫007C.v⎞⎜= 1and for a.e./u1D461∈ℝsuch that/u1D453(/u1D461)>0.\nMoreover by area formula [ AFP00 , Theorem 2.71] we have\n/u1D443/u1D706(/u1D438) =/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B1−/u1D706/u⎪⎫27E8.⎟1/u1D452/u1D45B,/u1D708/u1D438/u⎪⎫27E9.⎟1d/u1D45B−1\n=/uni222B.dsp+∞\n0/⎝⎞⎜e⎪⎨eft.⎟3\n(/u1D45B−1)/u1D714/u1D45B−1√\n1+(/u1D453)uni2032(var)2/u1D453/u1D45B−2−/u1D706/uni222B.dsp/u1D715∗/u1D438∩{/u1D465/u1D45B=/u1D461}/u⎪⎫27E8.⎟1\n/u1D452/u1D45B,/u1D708/u1D438/u⎪⎫27E9.⎟1√\n1+(/u1D453)uni2032(var)2d/u1D45B−2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D461\n= (/u1D45B−1)/u1D714/u1D45B−1/uni222B.dsp+∞\n0/⎝⎞⎜e⎪⎨eft.⎟2√\n1+(/u1D453)uni2032(var)2/u1D453/u1D45B−2+/u1D706/u1D453)uni2032(var/u1D453/u1D45B−2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\nd/u1D461.\nSince the bubbles /u1D435/u1D706(/u1D463)are Schwarz-symmetric, we can exploit Remark 3.2to compute their energy.\nLemma 3.3. There holds /u1D443/u1D706(/u1D435/u1D706(/u1D463)) =/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B/u1D463/u1D45B−1\n/u1D45B, for any/u1D463≥0and/u1D706∈ (−1,1).\nProof. By scale invariance, it is sufficient to prove that /u1D443/u1D706(/u1D435/u1D706) ∶=/u1D443(/u1D435/u1D706,{/u1D465/u1D45B>/u1D706})−/u1D706/u1D45B−1(/u1D715/u1D435/u1D706∩{/u1D465/u1D45B=/u1D706})is\nequal to/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜.\nIf/u1D706= 0, we have/u1D4430(/u1D4350) =1\n2/u1D443(/u1D435) =1\n2/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u⎪⎫007C.v⎞⎜=/u1D45B/u⎪⎫007C.v⎞⎜/u1D4350/u⎪⎫007C.v⎞⎜. So if we prove that\n(3.1)d\nd/u1D706/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D443/u1D706(/u1D435/u1D706)−/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n= 0,\nfor any/u1D706∈ (−1,1), the claim follows. Let /u1D711(/u1D461) ∶= (1−/u1D4612)1\n2, for/u1D461∈ [−1,1], be the profile function of the standard\nunit ball in ℝ/u1D45B.\n∙By coarea formula, the volume of /u1D435/u1D706equals\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜=/uni222B.dsp1\n/u1D706/u1D714/u1D45B−1/u1D711/u1D45B−1d/u1D461.\n∙By Remark 3.2we get\n/u1D443/u1D706(/u1D435/u1D706) =/uni222B.dsp1\n/u1D706/⎝⎞⎜e⎪⎨eft.⎟2\n(/u1D45B−1)/u1D714/u1D45B−1√\n1+(/u1D711)uni2032(var)2/u1D711/u1D45B−2+/u1D706(/u1D45B−1)/u1D714/u1D45B−1/u1D711)uni2032(var/u1D711/u1D45B−2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\nd/u1D461\n=/uni222B.dsp1\n/u1D706(/u1D45B−1)/u1D714/u1D45B−1(1−/u1D706/u1D461)/u1D711/u1D45B−3d/u1D461.\n∙Since/u1D711)uni2032(var= −/u1D461∕/u1D711, the derivative of /u1D443/u1D706(/u1D435/u1D706)equals:\nd\nd/u1D706/u1D443/u1D706(/u1D435/u1D706) =/uni222B.dsp1\n/u1D706(/u1D45B−1)/u1D714/u1D45B−1(−/u1D461)/u1D711/u1D45B−3d/u1D461−(/u1D45B−1)/u1D714/u1D45B−1(1−/u1D7062)/u1D711/u1D45B−3(/u1D706)\n= (/u1D45B−1)/u1D714/u1D45B−1/uni222B.dsp1\n/u1D706/u1D711)uni2032(var/u1D711/u1D45B−2d/u1D461−(/u1D45B−1)/u1D714/u1D45B−1/u1D711/u1D45B−1(/u1D706)\n= −/u1D714/u1D45B−1/u1D711/u1D45B−1(/u1D706)−(/u1D45B−1)/u1D714/u1D45B−1/u1D711/u1D45B−1(/u1D706)\n= −/u1D45B/u1D714/u1D45B−1/u1D711/u1D45B−1(/u1D706).\n∙The derivative of the volume /u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜equals\nd\nd/u1D706/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜= −/u1D714/u1D45B−1/u1D711/u1D45B−1(/u1D706).\nPutting together the above computations we haved\nd/u1D706/u1D443/u1D706(/u1D435/u1D706) =/u1D45Bd\nd/u1D706/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, which is ( 3.1). /squareQUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 9\nRemark 3.4.Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a connected open set such that /u1D715/u1D438⧵/u1D43Bis a smooth hypersurface with boundary\nthat intersects /u1D715/u1D43Borthogonally. Then the Neumann problem\n(3.2)⎧\n⎪\n⎨\n⎪⎩Δ/u1D462=/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜in/u1D438,\n/u1D715/u1D462\n/u1D715/u1D708= 1 on/u1D715/u1D438⧵/u1D715/u1D43B,\n/u1D715/u1D462\n/u1D715/u1D708= −/u1D706 on/u1D715/u1D438∩/u1D715/u1D43B,\nhas a solution /u1D462∈/u1D4361(/u1D438)∩/u1D436∞(/u1D438).\nIndeed, existence of a weak solution of ( 3.2) follows by classical arguments exploiting the Riesz repre sentation\ntheorem. By [ Nit11 , Proposition 3.6] there exists /u1D6FE >0such that every weak solution is in /u1D4360,/u1D6FE(/u1D438). Hence we can\napply [ Lie88 , Theorem 1] to the equivalent problem\n⎧\n⎪\n⎨\n⎪⎩Δ/u1D462−/u1D462=/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜−/u1D462=∶/u1D453 in/u1D438,\n/u1D715/u1D462\n/u1D715/u1D708= 1 on/u1D715/u1D438⧵/u1D715/u1D43B,\n/u1D715/u1D462\n/u1D715/u1D708= −/u1D706 on/u1D715/u1D438∩/u1D715/u1D43B\ngetting that a weak solution is in fact /u1D4361(/u1D438).\nTheorem 3.5 (Isoperimetric inequality for /u1D443/u1D706).Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜∈ (0,+∞) .\nThen\n(3.3)/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B≥/u1D443/u1D706(/u1D435/u1D706)\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B=/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B.\nMoreover, equality occurs in (3.3)if and only if /u1D438=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)up to a translation and up to negligible sets.\nProof. We just give a proof of the inequality ( 3.3) here, referring to [ Mag12 , Theorem 19.21] for an alternative\nproof comprising the characterization of minimizers.\nBy the standard isoperimetric inequality, we can assume tha t/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)>0. By Lemma 2.3, we can\nfurther assume that /u1D438is a bounded set such that /u1D715/u1D438⧵/u1D715/u1D43Bis smooth and intersects /u1D715/u1D43Borthogonally.\nLet us further assume for the moment that /u1D438is connected. Let /u1D462be the solution of the Neumann problem\n(3.4)⎧\n⎪\n⎨\n⎪⎩Δ/u1D462=/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜in/u1D438,\n/u1D715/u1D462\n/u1D715/u1D708= 1 on/u1D715/u1D438⧵/u1D715/u1D43B,\n/u1D715/u1D462\n/u1D715/u1D708= −/u1D706 on/u1D715/u1D438∩/u1D715/u1D43B,\nwhere/u1D715/u1D462∕/u1D715/u1D708denotes the outer normal derivative of /u1D462on/u1D715/u1D438. Observe that such a solution exists and /u1D462∈/u1D4361(/u1D438)∩\n/u1D436∞(/u1D438)(see Remark 3.4). We consider the “lower contact set” of /u1D462defined by\n(3.5) Γ/u1D462∶=/b⎜⎞ce⎨eft.⎟2\n/u1D465∈/u1D438∶/u1D462(/u1D466)≥/u1D462(/u1D465)+⟨∇/u1D462(/u1D465),/u1D466−/u1D465⟩for all/u1D466∈/u1D438/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nWe claim that\n(3.6) /u1D435/u1D706⊆∇/u1D462(Γ/u1D462).\nTo show ( 3.6), take any/u1D45D∈/u1D435/u1D706. Let/u1D465∈/u1D438be a point such that\nmin\n/u1D466∈/u1D438{/u1D462(/u1D466)−⟨/u1D45D,/u1D466⟩}=/u1D462(/u1D465)−⟨/u1D45D,/u1D465⟩.\nIf/u1D465∈/u1D715/u1D438⧵/u1D715/u1D43Bthen the exterior normal derivative of /u1D462(/u1D466)−⟨/u1D45D,/u1D466⟩at/u1D465would be nonpositive and hence (/u1D715/u1D462∕/u1D715/u1D708)(/u1D465)≤\n/u⎪⎫007C.v⎞⎜/u1D45D/u⎪⎫007C.v⎞⎜<1, a contradiction with ( 3.4). Similarly, if /u1D465∈/u1D715/u1D438∩/u1D715/u1D43Bthen(/u1D715/u1D462∕/u1D715/u1D708)(/u1D465)≤⟨/u1D45D,−/u1D452/u1D45B⟩<−/u1D706, a contradiction\nwith ( 3.4). It follows that /u1D465∈/u1D438and, therefore, that /u1D465is an interior minimum of the function /u1D462(/u1D466)−⟨/u1D45D,/u1D466⟩over/u1D438.\nIn particular /u1D45D= ∇/u1D462(/u1D465)and/u1D465∈ Γ/u1D462, hence Claim ( 3.6) is now proved.\nFrom ( 3.6), since/u1D462∈/u1D436∞(/u1D438)andΓ/u1D462⊂/u1D438, we can apply the area formula on ∇/u1D462to deduce\n(3.7) /u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜∇/u1D462(Γ/u1D462)/u⎪⎫007C.v⎞⎜=/uni222B.dsp∇/u1D462(Γ/u1D462)d/u1D45D≤/uni222B.dspΓ/u1D462/u⎪⎫007C.v⎞⎜det∇2/u1D462(/u1D465)/u⎪⎫007C.v⎞⎜d/u1D465.\nSince points /u1D465∈ Γ/u1D462are interior minima for /u1D466↦/u1D462(/u1D466)−⟨∇/u1D462(/u1D465),/u1D466⟩, then∇2/u1D462(/u1D465)is positively semi-definite. Hence\nby the arithmetic-geometric mean inequality\n/u⎪⎫007C.v⎞⎜det∇2/u1D462/u⎪⎫007C.v⎞⎜=det∇2/u1D462≤/⎝⎞⎜e⎪⎨eft.⎟2Δ/u1D462\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\ninΓ/u1D462.10 GIULIO PASCALE AND MARCO POZZETTA\nHence\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜≤/uni222B.dspΓ/u1D462det∇2/u1D462d/u1D465≤/uni222B.dspΓ/u1D462/⎝⎞⎜e⎪⎨eft.⎟2Δ/u1D462\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\nd/u1D465≤/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟2Δ/u1D462\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\nd/u1D465,\nsinceΔ/u1D462≡/u1D443/u1D706(/u1D438)∕/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜. Plugging in the value of Δ/u1D462, the claimed inequality follows.\nIt remains to consider the case when /u1D438is not connected, hence when /u1D438is a disjoint union of finitely many bounded\nsets/u1D438/u1D456, for/u1D456= 1,…,/u1D458, such that/u1D715/u1D438/u1D456⧵/u1D715/u1D43Bis smooth and intersects /u1D715/u1D43Borthogonally. We can apply the isoperi-\nmetric inequality that we just proved for /u1D443/u1D706on each component /u1D438/u1D456. Summing the inequalities and exploiting the\nsubadditivity of /u1D461↦/u1D461/u1D45B−1\n/u1D45B, the final inequality follows. /square\n3.2. Asymmetry and deficit. We now define the Fraenkel asymmetry with respect to optimal b ubbles/u1D435/u1D706(/u1D463,/u1D465)\nand the deficit corresponding to the functional /u1D443/u1D706, proving some preliminary properties on these quantities.\nDefinition 3.6 (Fraenkel asymmetry) .Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a Borel set with measure /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u1D463∈ (0,+∞) . We define\n/u1D6FC/u1D706(/u1D438) ∶= inf/b⎜⎞ce⎨eft.⎟3/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜\n/u1D463∶/u1D465∈ {/u1D465/u1D45B= 0}/b⎜⎞ce⎜⎫g⎧t.⎟3\n.\nIt is readily checked that the Fraenkel asymmetry of /u1D438is a minimum.\nDefinition 3.7 (Isoperimetric deficit) .Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a Borel set with measure /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u1D463∈ (0,+∞) . We define\n/u1D437/u1D706(/u1D438) ∶=/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706(/u1D463))\n/u1D443/u1D706(/u1D435/u1D706(/u1D463)).\nLemma 3.8. There exists ̄ /u1D450=̄ /u1D450(/u1D45B,/u1D706)>0such that if a Borel set /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bsatisfies/u1D437/u1D706(/u1D438)< ̄ /u1D450then/u1D443(/u1D438,/u1D715/u1D43B)>\n0.\nProof. If/u1D438is a Borel set such that /u1D437/u1D706(/u1D438)< ̄ /u1D450 and/u1D443(/u1D438,/u1D715/u1D43B) = 0 , then the standard isoperimetric inequality\ntogether with Lemma 3.3imply\n/u1D45B/u1D7141\n/u1D45B\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B≤/u1D443(/u1D438) =/u1D443/u1D706(/u1D438)<(1+̄ /u1D450)/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)) =/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B(1+̄ /u1D450)/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B.\nSince/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜>0for̄ /u1D450small enough, we get a contradiction if ̄ /u1D450is sufficiently small. /square\nLemma 3.9. If{/u1D438/u1D456}/u1D456∈ℕand/u1D438are sets of finite perimeter in ℝ/u1D45B⧵/u1D43Bwith finite measure such that\n/u1D438/u1D456/u1D43F1\nloc/uni2190.x /uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x →/u1D438.\nThen\nliminf\n/u1D456→+∞/u1D443/u1D706(/u1D438/u1D456)≥/u1D443/u1D706(/u1D438),liminf\n/u1D456→+∞/u1D437/u1D706(/u1D438/u1D456)≥/u1D437/u1D706(/u1D438).\nProof. Exploiting Remark 2.1and Reshetnyak lower semicontinuity theorem (see, e.g., [ AFP00 , Theorem 2.38]),\none easily checks that /u1D443/u1D706is lower semicontinuous with respect to /u1D43F1\nlocconvergence (see the proof of Lemma 2.3).\n/square\nLemma 3.10. If{/u1D438/u1D456}/u1D456∈ℕand/u1D438are sets of finite perimeter in ℝ/u1D45B⧵/u1D43Bwith finite measure such that /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜>0and\n/u1D438/u1D456/u1D43F1\n/uni2190.x /uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x →/u1D438.\nThen\nlim\n/u1D456→+∞/u1D6FC/u1D706(/u1D438/u1D456) =/u1D6FC/u1D706(/u1D438)\nProof. Let/u1D463∶=/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜and/u1D463/u1D456∶=/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u⎪⎫007C.v⎞⎜, then/u1D463/u1D456→/u1D463.\nIf/u1D6FC/u1D706(/u1D438) = 2 , there exists /u1D466 >0such that /u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩< /u1D466}/u⎪⎫007C.v⎞⎜= 0. Let/u1D4660be the least upper\nbound of such /u1D466’s; then every spherical cap /u1D435/u1D706(/u1D463,/u1D465), with/u1D465∈/u1D715/u1D43B, is contained in {/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩</u1D4660}.\nBy/u1D43F1-convergence of {/u1D438/u1D456}/u1D456, for every/u1D467∈/u1D715/u1D43Band/u1D456sufficiently large we have\nsup\n/u1D467∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438/u1D456∩/u1D435/u1D706(/u1D463/u1D456,/u1D467)/u⎪⎫007C.v⎞⎜= sup\n/u1D467∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438/u1D456∩/u1D435/u1D706(/u1D463/u1D456,/u1D467)∩{/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩</u1D4660}/u⎪⎫007C.v⎞⎜+\n+/u⎪⎫007C.v⎞⎜/u1D438/u1D456∩/u1D435/u1D706(/u1D463/u1D456,/u1D467)∩{/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩>/u1D4660}/u⎪⎫007C.v⎞⎜\n≤/u⎪⎫007C.v⎞⎜/u1D438/u1D456∩{/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩</u1D4660}/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456,0)∩{/u1D465∈ℝ/u1D45B⧵/u1D43B∶⟨/u1D465,/u1D452/u1D45B⟩>/u1D4660}/u⎪⎫007C.v⎞⎜/uni2190.x /uni2190.x/uni2190.x/uni2190.x →\n/u1D4560,\nhence/u1D6FC/u1D706(/u1D438/u1D456)→2.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 11\nSuppose then that /u1D6FC/u1D706(/u1D438)<2, let/u1D6FC/u1D706(/u1D438)be attained by some /u1D435/u1D706(/u1D463,/u1D4650). Then\n/u1D6FC/u1D706(/u1D438) =/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u1D463,/u1D4650)/u⎪⎫007C.v⎞⎜\n/u1D463= lim\n/u1D456→+∞/⎝⎞⎜e⎪⎨eft.⎟3/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D435/u1D706(/u1D463,/u1D4650)/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456/u1D463/u1D456\n/u1D463/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n≥limsup\n/u1D456→+∞/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D435/u1D706(/u1D463/u1D456,/u1D4650)/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456,/u1D4650)Δ/u1D435/u1D706(/u1D463,/u1D4650)/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456\n≥limsup\n/u1D456→+∞/⎝⎞⎜e⎪⎨eft.⎟3\n/u1D6FC/u1D706(/u1D438/u1D456)−/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456,/u1D4650)Δ/u1D435/u1D706(/u1D463,/u1D4650)/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n= limsup\n/u1D456→+∞/u1D6FC/u1D706(/u1D438/u1D456).\nOn the other hand, let /u1D6FC/u1D706(/u1D438/u1D456)be attained by /u1D435/u1D706\n/u1D456(/u1D463/u1D456,/u1D465/u1D456). We claim that {/u1D465/u1D456}/u1D456is bounded. If {/u1D465/u1D456}/u1D456⊂ /u1D715/u1D43B were\nunbounded, then there would be a subsequence {/u1D465/u1D456/u1D457}/u1D457such that/u⎪⎫007C.v⎞⎜/u1D465/u1D456/u1D457/u⎪⎫007C.v⎞⎜→+∞. From the hypothesis we have that, for\nsufficiently large /u1D457,/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456/u1D457,/u1D4650)∩/u1D438/u1D456/u1D457/u⎪⎫007C.v⎞⎜>/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456/u1D457,/u1D465/u1D456/u1D457)∩/u1D438/u1D456/u1D457/u⎪⎫007C.v⎞⎜/uni2190.x /uni2190.x/uni2190.x/uni2190.x/uni2190.x →\n/u1D4570, in contradiction with the definition of asymmetry.\nTherefore {/u1D465/u1D456}/u1D456is bounded. Let {/u1D465/u1D456/u1D458}/u1D458be a subsequence such that lim/u1D458→+∞/u1D6FC/u1D706(/u1D438/u1D456/u1D458) = liminf/u1D456→+∞/u1D6FC/u1D706(/u1D438/u1D456). By the\nboundedness of {/u1D465/u1D456}/u1D456there is a subsequence {/u1D465/u1D456/u1D458/u1D459}/u1D459of/u1D465/u1D456/u1D458such that/u1D465/u1D456/u1D458/u1D459→/u1D465∈/u1D715/u1D43B. Then\n/u1D6FC/u1D706(/u1D438)≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜\n/u1D463≤lim\n/u1D456→+∞/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D438/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456= lim\n/u1D456→+∞/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456\n≤liminf\n/u1D459→+∞/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u1D458/u1D459Δ/u1D435/u1D706(/u1D463/u1D456/u1D458/u1D459,/u1D465/u1D456/u1D458/u1D459)/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u1D463/u1D456/u1D458/u1D459,/u1D465/u1D456/u1D458/u1D459)Δ/u1D435/u1D706(/u1D463,/u1D465)/u⎪⎫007C.v⎞⎜\n/u1D463/u1D456/u1D458/u1D459\n= liminf\n/u1D459→+∞/u1D6FC/u1D706(/u1D438/u1D456/u1D458/u1D459) = lim\n/u1D458→+∞/u1D6FC/u1D706(/u1D438/u1D456/u1D458) = liminf\n/u1D456→+∞/u1D6FC/u1D706(/u1D438/u1D456),\nwhich gives the needed reversed inequality. /square\nCorollary 3.11. If{/u1D438/u1D456}/u1D456∈ℕare sets of finite perimeter in ℝ/u1D45B⧵/u1D43Bsuch that /u⎪⎫007C.v⎞⎜/u1D438/u1D456⧵/u1D43E/u⎪⎫007C.v⎞⎜= 0for any/u1D456and for some\ncompact set /u1D43E ⊂ℝ/u1D45B, and if\nsup\n/u1D456∈ℕ/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u⎪⎫007C.v⎞⎜+/u1D443/u1D706(/u1D438/u1D456)<∞,\nthen there exists /u1D438of finite perimeter in ℝ/u1D45B⧵/u1D43Band/u1D456/u1D458→∞as/u1D458→∞such that\n/u1D438/u1D456/u1D458/u1D43F1\n/uni2190.x /uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x →/u1D438liminf\n/u1D458/u1D443/u1D706(/u1D438/u1D456/u1D458)≥/u1D443/u1D706(/u1D438).\nProof. By Corollary 2.4we have that /u1D443/u1D706(/u1D438/u1D456)≥1−/u1D706\n2/u1D443(/u1D438). Thensup/u1D456∈ℕ/u1D443(/u1D438/u1D456)<∞. Hence by classical pre-\ncompactness of sets of finite perimeter (see, e.g., [ Mag12 , Theorem 12.26]) and recalling Lemma 3.9the claim\nfollows. /square\n4. R EDUCTION TO BOUNDED SYMMETRIC SETS\nIn the following arguments, in order to prove Theorem 1.1, we will repeatedly reduce ourselves to consider sets\n/u1D438having isoperimetric deficit smaller than some chosen const ant. This reduction is always possible.\nIndeed, let/u1D6FF>0be some positive constant; if /u1D438is a set of finite perimeter such that /u1D437/u1D706(/u1D438)≥/u1D6FF, since/u1D6FC/u1D706(/u1D438)≤2,\nwe immediately get\n/u1D6FC2\n/u1D706(/u1D438)≤4\n/u1D6FF/u1D6FF≤4\n/u1D6FF/u1D437/u1D706(/u1D438).\nTherefore, if Theorem 1.1is proved on sets with deficit ≤/u1D6FF, then it is proved for any set.\nHence,\nwithin this section we will assume that /u1D437/u1D706(/u1D438)< ̄ /u1D450for any competitor /u1D438involved, where ̄ /u1D450is given by Lemma 3.8.\nIn particular /u1D443(/u1D438,/u1D715/u1D43B)>0.\n4.1. Reduction to bounded sets. In this section we prove that, in order to prove Theorem 1.1, it is sufficient to\nprove the quantitative isoperimetric inequality ( 1.6) among suitably uniformly bounded sets.\nFrom now on, we shall denote /u1D444/u1D459∶= [−/u1D459,/u1D459]/u1D45B⊂ℝ/u1D45B. We start by proving an estimate on the area of horizontal\nslices of a set in terms of its deficit.12 GIULIO PASCALE AND MARCO POZZETTA\nLemma 4.1. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a bounded set of finite perimeter such that /u1D715/u1D438∩ℝ/u1D45B⧵/u1D43Bis a smooth hypersurface\n(possibly with smooth boundary) with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and such that /u1D45B−1({/u1D465∈/u1D715∗/u1D438⧵/u1D43B∶/u1D708/u1D438(/u1D465) = ±/u1D452/u1D45B}) = 0 . Then\n(4.1) /u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B/⎝⎞⎜e⎪⎨eft.⎟3\n1−/u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B</u1D461}/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B−1\n/u1D45B\n−1−/u1D437/u1D706(/u1D438)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n,\nfor every/u1D461>0. In particular\n(4.2) /u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n−1−/u1D437/u1D706(/u1D438)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n.\nMoreover, if /u1D439 ⊂ℝ/u1D45B⧵/u1D43Bis a set of finite perimeter with /u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, then (4.1)holds with/u1D439in place of/u1D438for\nalmost every /u1D461>0, and (4.2)holds with/u1D439in place of/u1D438.\nThe estimates given by Lemma 4.1are clearly nontrivial only when the deficit is sufficiently sm all. On the other\nhand, if the deficit /u1D437/u1D706(/u1D438)is sufficiently small, since /u1D714/u1D45B∕/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜>1, (4.1) and ( 4.2) essentially yield a quantitative\nversion of Lemma 3.8, nontrivial also for slices /u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})with/u1D461>0as long as /u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B</u1D461}/u⎪⎫007C.v⎞⎜is small.\nProof of Lemma 4.1.Let/u1D463/u1D438(/u1D461) ∶=/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})for any/u1D461 >0, and let/u1D454(/u1D461) ∶=/u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B< /u1D461}/u⎪⎫007C.v⎞⎜∕/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. By\nthe standard isoperimetric inequality we have that\n/u1D443(/u1D438,{/u1D465/u1D45B>/u1D461})+/u1D463/u1D438(/u1D461) =/u1D443(/u1D438∩{/u1D465/u1D45B>/u1D461})≥/u1D45B/u1D7141\n/u1D45B\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B>/u1D461}/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B=/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n(1−/u1D454(/u1D461))/u1D45B−1\n/u1D45B, (4.3)\nfor any/u1D461>0. Moreover, for any /u1D461>0, we observe that for any /u1D465)uni2032(var∈/u1D715∗/u1D438∩/u1D715/u1D43B, the halfline [0,/u1D461] ∋/u1D465/u1D45B↦(/u1D465)uni2032(var,/u1D465/u1D45B)\neither intersects /u1D715∗/u1D438∩{0</u1D465/u1D45B≤/u1D461}or it intersects /u1D438∩{/u1D465/u1D45B=/u1D461}. Therefore\n/u1D443(/u1D438,{0</u1D465/u1D45B≤/u1D461})+/u1D463/u1D438(/u1D461)≥ /u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B), (4.4)\nfor any/u1D461>0. Hence we conclude that\n/u1D443/u1D706(/u1D435/u1D706)(1+/u1D437/u1D706(/u1D438)) =/u1D443/u1D706(/u1D438) =/u1D443(/u1D438,{/u1D465/u1D45B>/u1D461})+/u1D443(/u1D438,{0</u1D465/u1D45B≤/u1D461})−/u1D706/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)\n(4.4)\n≥/u1D443(/u1D438,{/u1D465/u1D45B>/u1D461})−/u1D463/u1D438(/u1D461)\n(4.3)\n≥/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n(1−/u1D454(/u1D461))/u1D45B−1\n/u1D45B−2/u1D463/u1D438(/u1D461),\nfor any/u1D461>0, which yields ( 4.1). By [ Mag12 , Theorem 18.11] (see also [ FMP08 , Theorem 6.1]), the function /u1D463/u1D438\nbelongs to/u1D44A1,1(0,+∞) , thus ( 4.2) follows by letting /u1D461→0+in (4.1).\nNow if/u1D439 ⊂ℝ/u1D45B⧵/u1D43Bis as in the assumptions, let /u1D438/u1D456be given by Lemma 2.3applied to/u1D439, and let̃/u1D438/u1D456∶=\n(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u⎪⎫007C.v⎞⎜)1\n/u1D45B/u1D438/u1D456. Hence the inequality ( 4.2) and the right hand side of ( 4.1) applied with /u1D438=̃/u1D438/u1D456pass to the limit\nas/u1D456→∞. Moreover\n/u⎪⎫007C.v⎞⎜̃/u1D438/u1D456Δ/u1D438/u⎪⎫007C.v⎞⎜=/uni222B.dsp+∞\n0/u1D45B−1(̃/u1D438/u1D456Δ/u1D438∩{/u1D465/u1D45B=/u1D461})d/u1D461≥/uni222B.dsp+∞\n0/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D45B−1(̃/u1D438/u1D456∩{/u1D465/u1D45B=/u1D461})−/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xd/u1D461.\nSince/u⎪⎫007C.v⎞⎜̃/u1D438/u1D456Δ/u1D438/u⎪⎫007C.v⎞⎜→0, the left hand side of ( 4.1) passes to the limit as well as /u1D456→∞, for a.e./u1D461>0. /square\nWe are ready to prove the claimed reduction to bounded sets. T he proof follows the line of [ FMP08 , Lemma\n5.1], essentially truncating a competitor with coordinate slabs having estimated width. To give a bound for the\ntruncation in the /u1D45B-th direction will need to modify the argument in [ FMP08 , Lemma 5.1] and we will exploit\nLemma 4.1.\nLemma 4.2 (Reduction to bounded sets) .There exist/u1D459=/u1D459(/u1D45B,/u1D706)>0and/u1D4361=/u1D4361(/u1D45B,/u1D706)>0such that, if/u1D438 ⊆ℝ/u1D45B⧵/u1D43B\nis a set of finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜∈ (0,+∞) , then there exists a set of finite perimeter /u1D438)uni2032(var⊆/u1D444/u1D459∩(ℝ/u1D45B⧵/u1D43B)such\nthat/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and\n(4.5) /u1D6FC/u1D706(/u1D438)≤/u1D6FC/u1D706(/u1D438)uni2032(var)+/u1D4361/u1D437/u1D706(/u1D438), /u1D437/u1D706(/u1D438)uni2032(var)≤/u1D4361/u1D437/u1D706(/u1D438).\nProof. By scale-invariance of the asymmetry and of the deficit, it is sufficient to prove the claim assuming also\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. First of all we observe that we may prove the claim assuming t hat/u1D715/u1D438∩ℝ/u1D45B⧵/u1D43Bis smooth and\n(4.6) /u1D45B−1({/u1D465∈/u1D715∗/u1D438∩ℝ/u1D45B⧵/u1D43B∶/u1D708/u1D438(/u1D465) = ±/u1D452/u1D456}) = 0\nfor all/u1D456= 1,…,/u1D45B. Indeed, if/u1D438is a generic set of finite perimeter, then by Lemma 2.3there exists a sequence\nof smooth sets {/u1D438/u1D456}/u1D456∈ℕconverging to /u1D438such that ( 4.6) holds. If we know that the claim holds for /u1D438/u1D456, we getQUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 13\nthe existence of /u1D438)uni2032(var\n/u1D456⊆ /u1D444/u1D459∩ (ℝ/u1D45B⧵/u1D43B)such that ( 4.5) holds with /u1D438,/u1D438)uni2032(varreplaced by /u1D438/u1D456,/u1D438)uni2032(var\n/u1D456. Hence we can apply\nCorollary 3.11 on the sequence /u1D438)uni2032(var\n/u1D456, and by Lemma 3.9and Lemma 3.10 the inequalities ( 4.5) pass to the limit.\nWithout loss of generality, we can further assume that\n(4.7) /u1D437/u1D706(/u1D438)<(21∕/u1D45B−1)∕4.\nLet us consider the axis /u1D4651first. Thanks to ( 4.6), by [ Mag12 , Theorem 18.11] (see also [ FMP08 , Theorem 6.1])\nwe deduce that\n/u1D463/u1D438(/u1D461) ∶=/u1D45B−1({/u1D465)uni2032(var∈ℝ/u1D45B−1∶ (/u1D461,/u1D465)uni2032(var) ∈/u1D438}) for/u1D461∈ℝ\nbelongs to/u1D44A1,1(ℝ), hence we may assume that /u1D463/u1D438is continuous. Setting\n/u1D438−\n/u1D461∶= {/u1D465∈/u1D438∶/u1D4651</u1D461}\nand\n/u1D443/u1D706(/u1D438,{/u1D4651</u1D461}) ∶=/u1D443(/u1D438,{/u1D465∈ℝ/u1D45B⧵/u1D43B∶/u1D4651</u1D461})−/u1D706/u1D45B−1({/u1D465∈/u1D715∗/u1D438∩/u1D715/u1D43B∶/u1D4651</u1D461})\nfor all/u1D461∈ℝ, by smoothness of /u1D438we have that\n(4.8) /u1D443/u1D706(/u1D438−\n/u1D461) =/u1D443/u1D706(/u1D438,{/u1D4651</u1D461})+/u1D463/u1D438(/u1D461), /u1D443/u1D706(/u1D438⧵/u1D438−\n/u1D461) =/u1D443/u1D706(/u1D438,{/u1D4651>/u1D461})+/u1D463/u1D438(/u1D461),\nwhere/u1D443/u1D706(/u1D438,{/u1D4651>/u1D461})is defined analogously. Let us now define the function /u1D454∶ℝ→[0,+∞) given by\n/u1D454(/u1D461) ∶=/u⎪⎫007C.v⎞⎜/u1D438−\n/u1D461/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜.\nHence/u1D454is a nondecreasing /u1D4361function with /u1D454)uni2032(var(/u1D461) =/u1D463/u1D438(/u1D461)∕/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Let−∞≤/u1D44E </u1D44F≤+∞ be such that {/u1D461∶ 0<\n/u1D454(/u1D461)<1} = (/u1D44E,/u1D44F). If/u1D461∈ (/u1D44E,/u1D44F), then by ( 3.3) we have\n/u1D443/u1D706(/u1D438−\n/u1D461)≥/u1D454(/u1D461)/u1D45B−1\n/u1D45B/u1D443/u1D706(/u1D435/u1D706).\nSimilarly,\n/u1D443/u1D706(/u1D438⧵/u1D438−\n/u1D461)≥(1−/u1D454(/u1D461))/u1D45B−1\n/u1D45B/u1D443/u1D706(/u1D435/u1D706).\nTherefore, from ( 4.6) and ( 4.8) we get that\n/u1D443/u1D706(/u1D438)+2/u1D463/u1D438(/u1D461)≥/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D454(/u1D461)/u1D45B−1\n/u1D45B+(1−/u1D454(/u1D461))/u1D45B−1\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\nfor all/u1D461∈ (/u1D44E,/u1D44F). Since by definition we have /u1D443/u1D706(/u1D438) =/u1D443/u1D706(/u1D435/u1D706)(1+/u1D437/u1D706(/u1D438)), we obtain\n(4.9) /u1D463/u1D438(/u1D461)≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D454(/u1D461)/u1D45B−1\n/u1D45B+(1−/u1D454(/u1D461))/u1D45B−1\n/u1D45B−1−/u1D437/u1D706(/u1D438)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n.\nLet us now define the concave function\n/u1D713∶ [0,1]→[0,+∞) /u1D713(/u1D461) ∶=/u1D461/u1D45B−1\n/u1D45B+(1−/u1D461)/u1D45B−1\n/u1D45B−1.\nNote that/u1D713(0) =/u1D713(1) = 0 and/u1D713achieves its maximum at /u1D713(1∕2) = 21∕/u1D45B−1, hence by concavity\n(4.10) /u1D713(/u1D461) =/u1D713/⎝⎞⎜e⎪⎨eft.⎟2\n2/u1D4611\n2+0/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≥2/u1D461/u1D713(1∕2)+0 = 2(21∕/u1D45B−1)/u1D461∀/u1D461∈/b⎜⎞c⎭et⎨eft.⎟2\n0,1\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n.\nRecall that by ( 4.7) there holds 2/u1D437/u1D706(/u1D438)< /u1D713(1∕2) . Let/u1D44E < /u1D4611< /u1D4612< /u1D44F be such that /u1D454(/u1D4611) = 1 −/u1D454(/u1D4612)and\n/u1D713(/u1D454(/u1D4611)) =/u1D713(/u1D454(/u1D4612)) = 2/u1D437/u1D706(/u1D438). Then\n(4.11) /u1D713(/u1D454(/u1D461))≥2/u1D437/u1D706(/u1D438) ∀/u1D461∈ (/u1D4611,/u1D4612)\nand, by ( 4.10),\n(4.12) /u1D454(/u1D4611) = 1−/u1D454(/u1D4612)≤/u1D437/u1D706(/u1D438)\n21∕/u1D45B−1.\nFor any/u1D4611≤/u1D461≤/u1D4612we have\n/u1D463/u1D438(/u1D461)(4.9)\n≥1\n2/u1D443/u1D706(/u1D435/u1D706)(/u1D713(/u1D454(/u1D461))−/u1D437/u1D706(/u1D438)) =1\n4/u1D443/u1D706(/u1D435/u1D706)/u1D713(/u1D454(/u1D461))+1\n4/u1D443/u1D706(/u1D435/u1D706)(/u1D713(/u1D454(/u1D461))−2/u1D437/u1D706(/u1D438))\n(4.11)\n≥/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n4/u1D713(/u1D454(/u1D461)).(4.13)\nSince/u1D463/u1D438(/u1D461) =/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D454)uni2032(var(/u1D461), we have\n(4.14) /u1D4612−/u1D4611(4.13)\n≤4\n/u1D45B/uni222B.dsp/u1D4612\n/u1D4611/u1D454)uni2032(var(/u1D461)\n/u1D713(/u1D454(/u1D461))d/u1D461=4\n/u1D45B/uni222B.dsp/u1D454(/u1D4612)\n/u1D454(/u1D4611)1\n/u1D713(/u1D460)d/u1D460≤4\n/u1D45B/uni222B.dsp1\n01\n/u1D713(/u1D460)d/u1D460=∶/u1D6FC,\nfor some/u1D6FC=/u1D6FC(/u1D45B)>0.14 GIULIO PASCALE AND MARCO POZZETTA\nLet\n/u1D70F1= max/b⎜⎞ce⎨eft.⎟3\n/u1D461∈ (/u1D44E,/u1D4611] ∶/u1D463/u1D438(/u1D461)≤/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438)\n2/b⎜⎞ce⎜⎫g⎧t.⎟3\n,\n/u1D70F2= min/b⎜⎞ce⎨eft.⎟3\n/u1D461∈ [/u1D4612,/u1D44F) ∶/u1D463/u1D438(/u1D461)≤/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438)\n2/b⎜⎞ce⎜⎫g⎧t.⎟3\n.\nNote that/u1D70F1and/u1D70F2are well defined since /u1D463/u1D438is continuous and /u1D463/u1D438(/u1D461)→0as/u1D461→/u1D44Eor/u1D461→/u1D44F; moreover, by ( 4.11)\nand ( 4.13),/u1D463/u1D438(/u1D70F1) =/u1D463/u1D438(/u1D70F2) =/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438)\n2. Moreover, from ( 4.12) and by definition of /u1D70F1, we have\n/u1D4611−/u1D70F1≤2\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438)/uni222B.dsp/u1D4611\n/u1D70F1/u1D463/u1D438(/u1D461)d/u1D461=2\n/u1D45B/u1D437/u1D706(/u1D438)/uni222B.dsp/u1D4611\n/u1D70F1/u1D454)uni2032(var(/u1D461)d/u1D461≤2/u1D454(/u1D4611)\n/u1D45B/u1D437/u1D706(/u1D438)≤2\n/u1D45B(21∕/u1D45B−1),\nand an analogous estimate holds for /u1D70F2−/u1D4612.\nWe consider the truncation ̃/u1D438∶=/u1D438∩ {/u1D465∶/u1D70F1< /u1D4651< /u1D70F2}. From the above estimate and ( 4.14), we have\nthat/u1D70F2−/u1D70F1< /u1D6FD for some/u1D6FD=/u1D6FD(/u1D45B)>0. Moreover by ( 4.8) and ( 4.12), by the definition of /u1D70F1,/u1D70F2, and since\n/u1D443(/u1D438,{/u1D4651< /u1D70F1,/u1D4651> /u1D70F2})≥/u1D706/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B∩ {/u1D4651< /u1D70F1,/u1D4651> /u1D70F2})(see the proof of Corollary 2.4) we can\nestimate\n(4.15) /u⎪⎫007C.v⎞⎜̃/u1D438/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D435/u1D706/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟3\n1−2/u1D437/u1D706(/u1D438)\n21∕/u1D45B−1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n, /u1D443/u1D706(̃/u1D438)≤/u1D443/u1D706(/u1D438)+/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438).\nWe finally define\n/u1D70E∶=/⎝⎞⎜e⎪⎨eft.⎟3/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜̃/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31∕/u1D45B\n, /u1D438)uni2032(var∶=/u1D70Ẽ/u1D438.\nClearly,/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and by ( 4.15) we get that /u1D438)uni2032(varis contained in a strip {/u1D70F)uni2032(var\n1</u1D4651</u1D70F)uni2032(var\n2}, with/u1D70F)uni2032(var\n2−/u1D70F)uni2032(var\n1≤/u1D70E(/u1D70F2−/u1D70F1)≤\n/u1D459)uni2032(var, where/u1D459)uni2032(var=/u1D459)uni2032(var(/u1D45B,/u1D706)>0. Let us now show that /u1D438)uni2032(varsatisfies ( 4.5) for a suitable constant /u1D4361=/u1D4361(/u1D45B,/u1D706)>0that\nmay change from line to line. To this aim, since we are assumin g/u1D437/u1D706(/u1D438)small by ( 4.7), from ( 4.15) we get that\n1≤/u1D70E≤1+/u1D4360/u1D437/u1D706(/u1D438), with/u1D4360=/u1D4360(/u1D45B). Thus, from ( 4.15) and ( 4.7), we get\n/u1D443/u1D706(/u1D438)uni2032(var) =/u1D70E/u1D45B−1/u1D443/u1D706(̃/u1D438)≤/u1D70E/u1D45B−1(/u1D443/u1D706(/u1D438)+/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438))\n=/u1D70E/u1D45B−1/u1D443/u1D706(/u1D435/u1D706)(1+2/u1D437/u1D706(/u1D438))≤/u1D443/u1D706(/u1D435/u1D706)(1+/u1D4361/u1D437/u1D706(/u1D438)).\nHence, the second inequality in ( 4.5) follows. To prove the first inequality, let us denote by /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D45D), with\n/u1D45D∈/u1D715/u1D43B, a spherical cap such that /u1D6FC/u1D706(/u1D438)uni2032(var) =/u⎪⎫007C.v⎞⎜/u1D438)uni2032(varΔ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D45D)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. From the first inequality in ( 4.15), recalling that\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, we then get\n/u1D6FC/u1D706(/u1D438)≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D45D∕/u1D70E)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n≤/u⎪⎫007C.v⎞⎜/u1D438Δ̃/u1D438/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜̃/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕/u1D70E/u1D45B,/u1D45D∕/u1D70E)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕/u1D70E/u1D45B,/u1D45D∕/u1D70E)Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D45D∕/u1D70E)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n=/u⎪⎫007C.v⎞⎜/u1D438⧵̃/u1D438/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜+/u1D6FC/u1D706(/u1D438)uni2032(var)\n/u1D70E/u1D45B+/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕/u1D70E/u1D45B)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n(4.15)\n≤/u1D4361/u1D437/u1D706(/u1D438)+/u1D6FC/u1D706(/u1D438)uni2032(var)+/u1D4361(/u1D70E−1)\n≤/u1D6FC/u1D706(/u1D438)uni2032(var)+/u1D4361/u1D437/u1D706(/u1D438).\nThus the set /u1D438)uni2032(varsatisfies ( 4.5) and points in /u1D438)uni2032(varhave first coordinate contained in an interval of length boun ded by\n/u1D459)uni2032(var.\nStarting from /u1D438)uni2032(var, we can repeat the same construction finitely many times with respect to the axes /u1D4652,…,/u1D465/u1D45B−1,\nthus getting a new set, still denoted by /u1D438)uni2032(var, satisfying ( 4.5).\nIt remains to adapt the construction with respect to the coor dinate axis/u1D465/u1D45B. In this case we eventually aim at\ntruncating the set /u1D438)uni2032(varin some controlled slab of the form {0</u1D465/u1D45B< ̄ /u1D70F2}. Define\n̄ /u1D463(/u1D461) ∶=/u1D45B−1({/u1D465)uni2032(var∈ℝ/u1D45B−1∶ (/u1D465)uni2032(var,/u1D461) ∈/u1D438)uni2032(var}),̄/u1D438−\n/u1D461∶= {/u1D465∈/u1D438)uni2032(var∶/u1D465/u1D45B</u1D461}, ̄ /u1D454(/u1D461) ∶=/u⎪⎫007C.v⎞⎜̄/u1D438−\n/u1D461/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,\nfor/u1D461>0. It is readily checked that, arguing as above, one estimates\n(4.16) ̄ /u1D463(/u1D461)≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D713(̄ /u1D454(/u1D461))−/u1D437/u1D706(/u1D438)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n,QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 15\nwhich is analogous to ( 4.9), for any/u1D461such that̄ /u1D454(/u1D461) ∈ (0,1). Similarly as before, we define 0<̄/u1D4611<̄/u1D4612such that\n̄ /u1D454(̄/u1D4611) = 1−̄ /u1D454(̄/u1D4612)and/u1D713(̄ /u1D454(̄/u1D4611)) =/u1D713(̄ /u1D454(̄/u1D4612)) = 2/u1D437/u1D706(/u1D438)uni2032(var). Therefore, using ( 4.16) and the concavity of /u1D713, arguing as\nbefore one estimates\n(4.17) ̄ /u1D463(/u1D461)≥/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n4/u1D713(̄ /u1D454(/u1D461)) ∀ /u1D461∈ [̄/u1D4611,̄/u1D4612],\nwhich is analogous to ( 4.13).\nLet\n/u1D434∶=1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n−1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n>0.\nWe claim that there exists ̄ /u1D700=̄ /u1D700(/u1D45B,/u1D706)>0such that if/u1D437/u1D706(/u1D438)< ̄ /u1D700then\n(4.18) ̄ /u1D463(/u1D461)≥/u1D434\n2for a.e./u1D461∈ (0,̄/u1D4611).\nIndeed, since /u1D437/u1D706(/u1D438)uni2032(var)≤/u1D4361/u1D437/u1D706(/u1D438)and/u1D713(̄ /u1D454(̄/u1D4611)) = 2/u1D437/u1D706(/u1D438)uni2032(var), then for any /u1D714>0there is̄ /u1D700=̄ /u1D700(/u1D45B,/u1D706)>0such that\n̄ /u1D454(̄/u1D4611)</u1D714 whenever/u1D437/u1D706(/u1D438)< ̄ /u1D700. Applying Lemma 4.1with/u1D439=/u1D438)uni2032(var, for almost every /u1D461∈ (0,̄/u1D4611)we find\n̄ /u1D463(/u1D461)≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n(1−̄ /u1D454(/u1D461))/u1D45B−1\n/u1D45B−1−/u1D437/u1D706(/u1D438)uni2032(var)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B/⎝⎞⎜e⎪⎨eft.⎟1\n1−̄ /u1D454(̄/u1D4611)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u1D45B−1\n/u1D45B−1−/u1D437/u1D706(/u1D438)uni2032(var)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n(1−/u1D714)/u1D45B−1\n/u1D45B−1−/u1D4361̄ /u1D700/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n≥/u1D434\n2,(4.19)\nprovided̄ /u1D700is small enough.\nTherefore, assuming without loss of generality that /u1D437/u1D706(/u1D438)< ̄ /u1D700, sincē /u1D454)uni2032(var(/u1D461) =̄ /u1D463(/u1D461)∕/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜we estimate\n̄/u1D4612(4.17)\n≤̄/u1D4611+4\n/u1D45B/uni222B.dsp̄/u1D4612\n̄/u1D4611̄ /u1D454)uni2032(var(/u1D461)\n/u1D713(̄ /u1D454(/u1D461))d/u1D461(4.18)\n≤2\n/u1D434/uni222B.dsp̄/u1D4611\n0̄ /u1D463(/u1D461)d/u1D461+4\n/u1D45B/uni222B.dsp1\n01\n/u1D713≤2/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n/u1D434+4\n/u1D45B/uni222B.dsp1\n01\n/u1D713=∶/u1D6FC)uni2032(var(/u1D45B,/u1D706). (4.20)\nThe rest of the construction follows analogously as above by defining\n̄ /u1D70F2∶= min/b⎜⎞ce⎨eft.⎟3\n/u1D461≥̄/u1D4612∶̄ /u1D454(/u1D461)<1, ̄ /u1D463(/u1D461)≤/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D437/u1D706(/u1D438)uni2032(var)\n2/b⎜⎞ce⎜⎫g⎧t.⎟3\n,\nestimatinḡ /u1D70F2−̄/u1D4612≤/u1D6FD)uni2032(var(/u1D45B,/u1D706), hence finally taking the set\n(4.21)/⎝⎞⎜e⎪⎨eft.⎟3/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var∩{/u1D465/u1D45B< ̄ /u1D70F2}/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B\n(/u1D438)uni2032(var∩{/u1D465/u1D45B< ̄ /u1D70F2}).\nUp to translation along /u1D715/u1D43B, the set defined in ( 4.21) yields the final one satisfying the claim of the lemma.\n/square\nCorollary 4.3 (Non-quantitative stability) .For anȳ /u1D700>0there exists̄/u1D6FF=̄/u1D6FF(/u1D45B,/u1D706,̄ /u1D700)>0such that if/u1D438 ⊆ℝ/u1D45B⧵/u1D43B\nis a Borel set such that /u1D437/u1D706(/u1D438)≤̄/u1D6FF, then/u1D6FC/u1D706(/u1D438)≤̄ /u1D700.\nProof. By scale-invariance of the asymmetry and of the deficit, it is sufficient to prove the claim assuming also\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. We argue by contradiction. Suppose there exist a number ̄ /u1D700 >0and a sequence of sets {/u1D438/u1D456}/u1D456, with\n/u1D438/u1D456⊆ℝ/u1D45B⧵/u1D43Band/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, such that/u1D437/u1D706(/u1D438/u1D456)<1\n/u1D456and/u1D6FC/u1D706(/u1D438/u1D456)> ̄ /u1D700for all/u1D456∈ℕ. Let us consider the sequence\nof sets{/u1D438)uni2032(var\n/u1D456}/u1D456, with/u1D438)uni2032(var\n/u1D456⊂ /u1D444/u1D459∩ (ℝ/u1D45B⧵/u1D43B)and/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var\n/u1D456/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, given by Lemma 4.2. Moreover the Lemma assures\nthat/u1D6FC/u1D706(/u1D438)uni2032(var\n/u1D456)> ̄ /u1D700∕2for large/u1D456, and/u1D437/u1D706(/u1D438)uni2032(var\n/u1D456)→0. Since each set /u1D438)uni2032(var\n/u1D456is contained in the same /u1D444/u1D459, by Corollary 3.11\nwe can assume, up to a subsequence, that /u1D438)uni2032(var\n/u1D456/u1D43F1\n/uni2190.x /uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x/uni2190.x →/u1D438)uni2032(varfor some set /u1D438)uni2032(varof finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. By the\nlower semicontinuity of the perimeters we get /u1D443/u1D706(/u1D438)uni2032(var)≤/u1D443/u1D706(/u1D435/u1D706), hence/u1D438)uni2032(var=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)for some/u1D465∈/u1D715/u1D43B∩/u1D444/u1D459\nby uniqueness of minimizers. The convergence of /u1D438)uni2032(var\n/u1D456to/u1D438)uni2032(varimplies that /u⎪⎫007C.v⎞⎜/u1D438)uni2032(var\n/u1D456Δ/u1D438)uni2032(var/u⎪⎫007C.v⎞⎜→0, against the assumption\n/u1D6FC/u1D706(/u1D438)uni2032(var\n/u1D456)>̄ /u1D700\n2. /square16 GIULIO PASCALE AND MARCO POZZETTA\nCorollary 4.4. There exist/u1D434/u1D706,/u1D447/u1D706,/u1D702 >0depending on /u1D45B,/u1D706such that for any set of finite perimeter /u1D438 ⊂ℝ/u1D45B⧵/u1D43B\nwith/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D702and/u1D437/u1D706(/u1D438)≤/u1D702there holds\n/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})≥/u1D434/u1D706,\nfor almost every /u1D461∈ (0,/u1D447/u1D706).\nProof. Let us prove the inequality assuming /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜first. Fix/u1D447)uni2032(var\n/u1D706>0such that\n/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B/⎝⎞⎜e⎪⎨eft.⎟4\n1−/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)∩{/u1D465/u1D45B</u1D447)uni2032(var\n/u1D706}/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4/u1D45B−1\n/u1D45B\n≥1+/u1D44E,\nfor some/u1D44E >0. By Corollary 4.3, for any/u1D714 >0there is/u1D702 >0such that if/u1D437/u1D706(/u1D438)< /u1D702then/u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B< /u1D447)uni2032(var\n/u1D706}/u⎪⎫007C.v⎞⎜≤\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)∩{/u1D465/u1D45B</u1D447)uni2032(var\n/u1D706}/u⎪⎫007C.v⎞⎜+/u1D714. For such a set /u1D438, Lemma 4.1implies that for almost every /u1D461∈ (0,/u1D447)uni2032(var\n/u1D706)there holds\n/u1D45B−1(/u1D438∩{/u1D465/u1D45B=/u1D461})≥1\n2/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B/⎝⎞⎜e⎪⎨eft.⎟3\n1−/u⎪⎫007C.v⎞⎜/u1D438∩{/u1D465/u1D45B</u1D461}/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B−1\n/u1D45B\n−1−/u1D437/u1D706(/u1D438)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n≥1\n2/u1D443/u1D706(/u1D435/u1D706)⎛\n⎜\n⎜⎝/⎝⎞⎜e⎪⎨eft.⎟3/u1D714/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟31\n/u1D45B/⎝⎞⎜e⎪⎨eft.⎟4\n1−/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)∩{/u1D465/u1D45B</u1D447)uni2032(var\n/u1D706}/u⎪⎫007C.v⎞⎜+/u1D714\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4/u1D45B−1\n/u1D45B\n−1−/u1D702⎞\n⎟\n⎟⎠.\nHence for sufficiently small /u1D702>0the right hand side in the previous estimate is bounded below by some constant\n/u1D434)uni2032(var\n/u1D706(/u1D45B,/u1D706)>0.\nFor a generic set /u1D438such that /u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D702and/u1D437/u1D706(/u1D438)≤/u1D702, the set/u1D438)uni2032(var=/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜1\n/u1D45B∕/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜1\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n/u1D438has measure equal\nto/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and deficit/u1D437/u1D706(/u1D438)uni2032(var)≤/u1D702. Up to decreasing /u1D702 >0, applying the first part of the proof to /u1D438)uni2032(var, the desired\nestimate holds on /u1D438for/u1D447/u1D706=/u1D447)uni2032(var\n/u1D706∕2and/u1D434/u1D706=/u1D434)uni2032(var\n/u1D706∕2. /square\n4.2. Reduction to (/u1D45B− 1)-symmetric sets. In this section we prove that, in order to prove Theorem 1.1, it is\nsufficient to further reduce to show ( 1.6) among(/u1D45B−1)-symmetric sets, i.e., sets which are symmetric with respec t\nto reflection across /u1D45B−1orthogonal hyperplanes, each one orthogonal to {/u1D465/u1D45B= 0} . The results are analogous to\n[Mag08 , Section 6].\nLemma 4.5. Let/u1D438 ⊆ℝ/u1D45B⧵/u1D43Bbe a Borel set with finite measure, symmetric with respect to /u1D458∈ {1,…,/u1D45B− 1}\northogonal half-hyperplanes /u1D43B/u1D457=/b⎜⎞ce⎨eft.⎟1/u1D465∈ℝ/u1D45B⧵/u1D43B∶/u⎪⎫27E8.⎟1/u1D465,/u1D708/u1D457/u⎪⎫27E9.⎟1= 0/b⎜⎞ce⎜⎫g⎧t.⎟1for1≤/u1D457≤/u1D458, where/u⎪⎫007C.v⎞⎜/u1D708/u1D457/u⎪⎫007C.v⎞⎜= 1and/u⎪⎫27E8.⎟1/u1D708/u1D457,/u1D452/u1D45B/u⎪⎫27E9.⎟1= 0\nfor any1≤/u1D457≤/u1D458. Then\n(4.22) min\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜≤min\n/u1D466∈/u1D715/u1D43B∩⋂/u1D458\n/u1D457=1/u1D43B/u1D457/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D466)/u⎪⎫007C.v⎞⎜≤3 min\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜.\nProof. We can suppose for simplicity that ∀/u1D457∈ {1,…,/u1D458}we have/u1D708/u1D457=/u1D452/u1D457. If/u1D4650= (/u1D4650\n1,…,/u1D4650\n/u1D45B−1,0)is such that\n/u1D6FC/u1D706(/u1D438)is achieved by /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650), then/u1D6FC/u1D706(/u1D438)is achieved also by /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜, ̄ /u1D4650), wherē /u1D4650= (−/u1D4650\n1,…,−/u1D4650\n/u1D458,/u1D4650\n/u1D458+1,…,\n/u1D4650\n/u1D45B−1,0). Since/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650)Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650)Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,̄ /u1D4650)/u⎪⎫007C.v⎞⎜we have\n/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜\n≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,̄ /u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.v⎞⎜\n≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\nΔ/u1D438/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,̄ /u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u⎪⎫007C.v⎞⎜\n= 3/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706/⎝⎞⎜e⎪⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D4650/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u⎪⎫007C.v⎞⎜. /square\nGiven a Borel set /u1D438 ⊆ℝ/u1D45B⧵/u1D43Bwith finite measure and a unit vector /u1D708with⟨/u1D708,/u1D452/u1D45B⟩= 0, we denote by /u1D43B+\n/u1D708=\n{/u1D465∈ℝ/u1D45B∶⟨/u1D465,/u1D708⟩>/u1D461}an open half-space orthogonal to /u1D708where/u1D461∈ℝis chosen in such a way that\n/u⎪⎫007C.v⎞⎜/u1D438∩/u1D43B+\n/u1D708/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜\n2.\nWe also denote by /u1D45F/u1D708∶ℝ/u1D45B⧵/u1D43B→ℝ/u1D45B⧵/u1D43Bthe reflection with respect to /u1D43B/u1D708∶=/u1D715/u1D43B+\n/u1D708, and by/u1D43B−\n/u1D708∶=/u1D45F/u1D708(/u1D43B+\n/u1D708)the\nopen half-space complementary to /u1D43B+\n/u1D708. Finally we write /u1D438±\n/u1D708∶=/u1D438∩/u1D43B±\n/u1D708.\nObserve that\n(4.23) /u1D437/u1D706(/u1D438±\n/u1D708∪/u1D45F/u1D708(/u1D438±\n/u1D708))≤2/u1D437/u1D706(/u1D438).QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 17\nIndeed\n/u1D443/u1D706(/u1D438±\n/u1D708∪/u1D45F/u1D708(/u1D438±\n/u1D708))−/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜))≤2/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D438∓\n/u1D708∪/u1D45F/u1D708(/u1D438∓\n/u1D708))−/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜))\n= 2(/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜)))+/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜))−/u1D443/u1D706(/u1D438∓\n/u1D708∪/u1D45F/u1D708(/u1D438∓\n/u1D708))\n≤2(/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜))),\nwhere in the last inequality we used the isoperimetric inequ ality of Theorem 3.5.\nLemma 4.6. There exist ̄/u1D4362,̄/u1D6FF2>0depending on /u1D45B,/u1D706such that, if/u1D438 ⊆ℝ/u1D45B⧵/u1D43Bis a Borel set with finite measure\nsuch that/u1D437/u1D706(/u1D438)≤̄/u1D6FF2, and if/u1D7081and/u1D7082are two orthogonal vectors, with ⟨/u1D708/u1D456,/u1D452/u1D45B⟩= 0, such that/u1D43B/u1D7081and/u1D43B/u1D7082divide\n/u1D438in four parts of equal measure, then there exist /u1D456∈ {1,2}and/u1D460∈ {+,−}such that, setting /u1D438)uni2032(var=/u1D438/u1D460\n/u1D708/u1D456∪/u1D45F/u1D708/u1D456(/u1D438/u1D460\n/u1D708/u1D456),\nthere holds\n(4.24) /u1D6FC/u1D706(/u1D438)≤̄/u1D4362/u1D6FC(/u1D438)uni2032(var).\nProof. By scale-invariance of Fraenkel asymmetry, it is sufficient t o prove the claim assuming also /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. If\n/u1D456∈ {1,2}and/u1D460∈ {+,−}, let/u1D438)uni2032(var/u1D460\n/u1D708/u1D456denote the sets obtained by reflecting /u1D438/u1D460\n/u1D708/u1D456along/u1D43B/u1D708/u1D456and let/u1D435/u1D706,/u1D460\n/u1D456=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465/u1D460\n/u1D456)\nbe four spherical caps such that\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u1D460\n/u1D708/u1D456Δ/u1D435/u1D706,/u1D460\n/u1D456/u⎪⎫007C.v⎞⎜= min\n/u1D465∈/u1D43B/u1D708/u1D456∩/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u1D460\n/u1D708/u1D456Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜.\nFor/u1D456= 1,2, by the triangular inequality we have\nmin\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706,+\n/u1D456/u⎪⎫007C.v⎞⎜\n=/u⎪⎫007C.v⎞⎜(/u1D438Δ/u1D435/u1D706,+\n/u1D456)∩/u1D43B+\n/u1D708/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜(/u1D438Δ/u1D435/u1D706,+\n/u1D456)∩/u1D43B−\n/u1D708/u1D456/u⎪⎫007C.v⎞⎜\n≤/u⎪⎫007C.v⎞⎜(/u1D438Δ/u1D435/u1D706,+\n/u1D456)∩/u1D43B+\n/u1D708/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜(/u1D438Δ/u1D435/u1D706,−\n/u1D456)∩/u1D43B−\n/u1D708/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜(/u1D435/u1D706,+\n/u1D456Δ/u1D435/u1D706,−\n/u1D456)∩/u1D43B−\n/u1D708/u1D456/u⎪⎫007C.v⎞⎜\n=1\n2/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D708/u1D456Δ/u1D435/u1D706,+\n/u1D456/u⎪⎫007C.v⎞⎜+1\n2/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D708/u1D456Δ/u1D435/u1D706,−\n/u1D456/u⎪⎫007C.v⎞⎜+1\n2/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n/u1D456Δ/u1D435/u1D706,−\n/u1D456/u⎪⎫007C.v⎞⎜.(4.25)\nOnce we show that if /u1D437/u1D706(/u1D438)is sufficiently small then there exists /u1D450/u1D45B,/u1D706>0such that at least one the following\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n1Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜≤2/u1D450/u1D45B,/u1D706/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7081Δ/u1D435/u1D706,+\n1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7081Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n2Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜≤2/u1D450/u1D45B,/u1D706/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7082Δ/u1D435/u1D706,+\n2/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7082Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2 (4.26)\nholds, then we soon conclude the proof. Indeed, assume for ex ample that the first inequality in ( 4.26) holds. Then,\nfrom ( 4.22) and ( 4.25) with/u1D456= 1, we get\nmin\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜(4.25)\n≤(/u1D450/u1D45B,/u1D706+1∕2)/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7081Δ/u1D435/u1D706,+\n1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7081Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n(4.22)\n≤3(/u1D450/u1D45B,/u1D706+1∕2)/⎝⎞⎜e⎪⎨eft.⎟2\nmin\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7081Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜+ min\n/u1D465∈/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7081Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n,\nthus proving ( 4.24) with̄/u1D4362= 6(/u1D450/u1D45B,/u1D706+1∕2) and/u1D438)uni2032(varequal to/u1D438)uni2032(var+\n/u1D7081or/u1D438)uni2032(var−\n/u1D7081.\nObserve that, given ̃ /u1D700 >0, Corollary 4.3, (4.22) and ( 4.23) imply that there exists ̄/u1D6FF2(/u1D45B,/u1D706)>0such that if\n/u1D437/u1D706(/u1D438)<̄/u1D6FF2then\n(4.27) max⎧\n⎪\n⎨\n⎪⎩/u1D6FC/u1D706(/u1D438),/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var±\n/u1D708/u1D456Δ/u1D435/u1D706,±\n/u1D456/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∶/u1D456= 1,2⎫\n⎪\n⎬\n⎪⎭< ̃ /u1D700.\nThanks to ( 4.27), we can show that the caps /u1D435/u1D706,±\n/u1D456get closer and closer to the optimal ones for /u1D438, as̄/u1D6FF2decreases.\nIndeed, let us assume by contradiction that there exists /u1D702 >0such that for every /u1D457∈ℕthere exist/u1D438/u1D457, with\n/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D437/u1D706(/u1D438/u1D457)<1\n/u1D457, with/u1D435/u1D706\n/u1D457∶=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465/u1D457)realizing the asymmetry of /u1D438/u1D457, but for/u1D456∈ {1,2}and\n/u1D460∈ {+,−}, if/u1D435/u1D706,/u1D460\n/u1D456,/u1D457∶=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465/u1D460\n/u1D456,/u1D457)is such that\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u1D460\n/u1D457,/u1D708/u1D457\n/u1D456Δ/u1D435/u1D706,/u1D460\n/u1D456,/u1D457/u⎪⎫007C.v⎞⎜= min\n/u1D465∈/u1D43B/u1D708/u1D457\n/u1D456∩/u1D715/u1D43B/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u1D460\n/u1D457,/u1D708/u1D457\n/u1D456Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)/u⎪⎫007C.v⎞⎜,\nwhere/u1D438)uni2032(var/u1D460\n/u1D457,/u1D708/u1D457\n/u1D456is given by reflections of truncations of /u1D438/u1D457along orthogonal subspaces /u1D43B/u1D708/u1D457\n1,/u1D43B/u1D708/u1D457\n2, then/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n/u1D456,/u1D457−/u1D465/u1D457/u⎪⎫007C.v⎞⎜>/u1D702\nfor some/u1D456∈ {1,2},/u1D460∈ {+,−}and any/u1D457. Without loss of generality we can assume that that /u1D456= 1and/u1D460= +.18 GIULIO PASCALE AND MARCO POZZETTA\nLet us translate every set in the above contradiction assump tion by−/u1D465/u1D457. Without relabeling the objects involved,\nup to subsequences, we have that /u1D438/u1D457→/u1D435/u1D706\n0∶=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)in/u1D43F1. We can show that /u1D438)uni2032(var+\n/u1D457,/u1D708/u1D457\n1→/u1D435/u1D706\n0as well. Indeed,\nup to a rotation we can further assume that\n/u1D43B/u1D708/u1D457\n1=/b⎜⎞ce⎨eft.⎟1(/u1D44E/u1D457,0,…,0)+/u1D452⟂\n1/b⎜⎞ce⎜⎫g⎧t.⎟1, /u1D708/u1D457\n1=/u1D4521,∀/u1D457,\nwith/u1D44E/u1D457∈ℝ. By the definition of /u1D43B/u1D708/u1D457\n1, we have\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n2=/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜\n2=/u⎪⎫007C.v⎞⎜/u1D438/u1D457∩{/u1D4651>/u1D44E/u1D457}/u⎪⎫007C.v⎞⎜.\nThen{/u1D44E/u1D457}is bounded and, up to subsequence, converges to /u1D44E∞= 0because, if/u1D44E∞≠0, then\n/u1D438/u1D457∩{/u1D4651>/u1D44E/u1D457}→/u1D435/u1D706\n0∩{/u1D4651>/u1D44E∞}\nwith/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D435/u1D706\n0∩{/u1D4651>/u1D44E∞}/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≠/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n2. In particular /u1D438)uni2032(var+\n/u1D457,/u1D708/u1D457\n1→/u1D435/u1D706\n0. Finally by ( 4.27)\ñ /u1D700/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜≥lim\n/u1D457/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D438)uni2032(var+\n/u1D457,/u1D708/u1D457\n1Δ/u1D435/u1D706,+\n1,/u1D457/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≥min\n/u1D465∈/u1D715/u1D43B,/u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜≥/u1D702/b⎜⎞ce⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)Δ/u1D435/u1D706\n0/b⎜⎞ce⎜⎫g⎧t.⎟1\n=∶/u1D436(/u1D702)>0.\nBut for/u1D457sufficiently large, since /u1D437/u1D706(/u1D438/u1D457)→0, by ( 4.27) we can choose ̃ /u1D700such that̃ /u1D700/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜< /u1D436(/u1D702)∕2, getting a\ncontradiction.\nWe observe that for ̃ /u1D700>0sufficiently small, that is for ̄/u1D6FF2>0sufficiently small, there exists /u1D450/u1D45B,/u1D706>0such that\nfor all possible choices of /u1D460,/u1D461∈ {+,−}there holds\n(4.28) /u⎪⎫007C.v⎞⎜(/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2)∩(/u1D43B/u1D460\n/u1D7081∩/u1D43B/u1D461\n/u1D7082)/u⎪⎫007C.v⎞⎜>/u⎪⎫007C.v⎞⎜/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2/u⎪⎫007C.v⎞⎜\n/u1D450/u1D45B,/u1D706.\nWe only sketch the argument for ( 4.28). Letting/u1D444∶= (/u1D43B/u1D460\n/u1D7081∩/u1D43B/u1D461\n/u1D7082)and/u1D4351(ℎ) ∶=/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,ℎ/u1D465/u1D460\n1),/u1D4352(ℎ) ∶=\n/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,ℎ/u1D465/u1D461\n2)forℎ∈ [0,1], one can compute\nd\ndℎ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x(/u1D4351(ℎ)Δ/u1D4352(ℎ))∩/u1D444/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/uni222B.dsp/u1D715/u1D4351(ℎ)∩/u1D4352(ℎ)∩/u1D444/u⎪⎫27E8.⎟1/u1D708/u1D4351(ℎ),/u1D465/u1D461\n2−/u1D465/u1D460\n1/u⎪⎫27E9.⎟1d/u1D45B−1+/uni222B.dsp/u1D715/u1D4352(ℎ)∩/u1D4351(ℎ)∩/u1D444/u⎪⎫27E8.⎟1/u1D708/u1D4352(ℎ),/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫27E9.⎟1d/u1D45B−1\n=/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/u1D715/u1D4351(ℎ)∩/u1D4352(ℎ)∩/u1D444/u⎪⎫27E8.⎟4\n/u1D708/u1D4351(ℎ),/u1D465/u1D461\n2−/u1D465/u1D460\n1\n/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜/u⎪⎫27E9.⎟4\n+/uni222B.dsp/u1D715/u1D4352(ℎ)∩/u1D4351(ℎ)∩/u1D444/u⎪⎫27E8.⎟4\n/u1D708/u1D4352(ℎ),/u1D465/u1D460\n1−/u1D465/u1D461\n2\n/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜/u⎪⎫27E9.⎟4/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜\n=/uni222B.dsp/u1D715(/u1D4351(ℎ)∩/u1D4352(ℎ))∩/u1D444/u⎪⎫27E8.⎟1\n/u1D708/u1D4351(ℎ)∩/u1D4352(ℎ),/u1D463/u1D460,/u1D461\n12/u⎪⎫27E9.⎟1\nd/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜\n≥/u1D450/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜,\nwhere/u1D463/u1D460,/u1D461\n12is obviously defined, provided ̃ /u1D700is small enough, for some /u1D450=/u1D450(/u1D45B,/u1D706)>0that will change from line to\nline. The last estimate follows since/u⎪⎫27E8.⎟1\n/u1D708/u1D4351(ℎ)∩/u1D4352(ℎ),/u1D463/u1D460,/u1D461\n12/u⎪⎫27E9.⎟1\n≥0pointwise and, for ̃ /u1D700small, centers /u1D465/u1D460\n1,/u1D465/u1D461\n2are so close\nthat/u⎪⎫27E8.⎟1\n/u1D708/u1D4351(ℎ)∩/u1D4352(ℎ),/u1D463/u1D460,/u1D461\n12/u⎪⎫27E9.⎟1\ncan be estimated from below by a positive constant on a set of /u1D45B−1-measure uniformly\nbounded from below away from zero. On the other hand one can es timate\n/u⎪⎫007C.v⎞⎜(/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2)/u⎪⎫007C.v⎞⎜≤/u1D450/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜.\nHence\n/u⎪⎫007C.v⎞⎜(/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2)∩(/u1D43B/u1D460\n/u1D7081∩/u1D43B/u1D461\n/u1D7082)/u⎪⎫007C.v⎞⎜=/uni222B.dsp1\n0d\ndℎ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x(/u1D4351(ℎ)Δ/u1D4352(ℎ))∩/u1D444/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xdℎ≥/u1D450/u⎪⎫007C.v⎞⎜/u1D465/u1D460\n1−/u1D465/u1D461\n2/u⎪⎫007C.v⎞⎜≥/u1D450/u⎪⎫007C.v⎞⎜(/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2)/u⎪⎫007C.v⎞⎜,\nand ( 4.28) follows.\nLetting\n/u1D4461= (/u1D435/u1D706,+\n1∩/u1D43B+\n/u1D7081)∪(/u1D435/u1D706,−\n1∩/u1D43B−\n/u1D7081), /u1D4462= (/u1D435/u1D706,+\n2∩/u1D43B+\n/u1D7082)∪(/u1D435/u1D706,−\n2∩/u1D43B−\n/u1D7082),\nwe deduce\n/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D4462/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜(/u1D4461Δ/u1D4462)∩(/u1D43B/u1D460\n/u1D7081∩/u1D43B/u1D461\n/u1D7082)/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜(/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2)∩(/u1D43B/u1D460\n/u1D7081∩/u1D43B/u1D461\n/u1D7082)/u⎪⎫007C.v⎞⎜>/u⎪⎫007C.v⎞⎜/u1D435/u1D706,/u1D460\n1Δ/u1D435/u1D706,/u1D461\n2/u⎪⎫007C.v⎞⎜\n/u1D450/u1D45B,/u1D706.\nIn particular we have\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n1Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n1Δ/u1D435/u1D706,+\n2/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n2Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜<2/u1D450/u1D45B,/u1D706/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D4462/u⎪⎫007C.v⎞⎜,\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n2Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n2Δ/u1D435/u1D706,+\n1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n1Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜<2/u1D450/u1D45B,/u1D706/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D4462/u⎪⎫007C.v⎞⎜.(4.29)QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 19\nIf by contradiction ( 4.26) were false, then\n(4.30) /u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7081Δ/u1D435/u1D706,+\n1/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7081Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜</u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n1Δ/u1D435/u1D706,−\n1/u⎪⎫007C.v⎞⎜\n2/u1D450/u1D45B,/u1D706and/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D7082Δ/u1D435/u1D706,+\n2/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D7082Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜</u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n2Δ/u1D435/u1D706,−\n2/u⎪⎫007C.v⎞⎜\n2/u1D450/u1D45B,/u1D706.\nHence\n/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D4462/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D438/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4462/u⎪⎫007C.v⎞⎜=1\n22/u⎪⎫2211.⎟1\n/u1D456=1/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+\n/u1D708/u1D456Δ/u1D435/u1D706,+\n/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−\n/u1D708/u1D456Δ/u1D435/u1D706,−\n/u1D456/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2(4.30)\n<1\n4/u1D450/u1D45B,/u1D7062/u⎪⎫2211.⎟1\n/u1D456=1/u⎪⎫007C.v⎞⎜/u1D435/u1D706,+\n/u1D456Δ/u1D435/u1D706,−\n/u1D456/u⎪⎫007C.v⎞⎜(4.29)\n≤/u⎪⎫007C.v⎞⎜/u1D4461Δ/u1D4462/u⎪⎫007C.v⎞⎜,\ngetting a contradiction. /square\nLemma 4.7 (Reduction to (/u1D45B− 1)-symmetric sets) .There exist/u1D4362,/u1D6FF2>0depending on /u1D45B,/u1D706such that, if /u1D438is a\nBorel set with /u1D438 ⊆ℝ/u1D45B⧵/u1D43B,/u1D438 ⊆/u1D444/u1D459,/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438)≤/u1D6FF2, there exists a Borel set /u1D439 ⊆ℝ/u1D45B⧵/u1D43B,/u1D439 ⊆/u1D4442/u1D459,\n/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, symmetric with respect to /u1D45B−1orthogonal half-hyperplanes (each orthogonal to /u1D715/u1D43B) and such that\n/u1D6FC/u1D706(/u1D438)≤/u1D4362/u1D6FC/u1D706(/u1D439), /u1D437/u1D706(/u1D439)≤2/u1D45B−1/u1D437/u1D706(/u1D438).\nProof. Let us define /u1D6FF2∶=̄/u1D6FF22−(/u1D45B−2), wherē/u1D6FF2is the constant appearing in Lemma 4.6. We can apply Lemma 4.6\n/u1D45B−2times to different pairs of orthogonal vectors in {/u1D4521,…,/u1D452/u1D45B−2}normal to corresponding pairs of affine hyper-\nplanes splitting the measure of /u1D438in two halves. Therefore, also recalling ( 4.23), we find an (/u1D45B−2)-symmetric set\n/u1D438)uni2032(varsuch that /u⎪⎫007C.v⎞⎜/u1D438)uni2032(var/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and\n/u1D6FC/u1D706(/u1D438)≤̄/u1D436/u1D45B−2\n2/u1D6FC/u1D706(/u1D438)uni2032(var), /u1D437/u1D706(/u1D438)uni2032(var)≤2/u1D45B−2/u1D437/u1D706(/u1D438).\nTo perform the last symmetrization, let us consider a half-h yperplane/u1D43B/u1D45B−1orthogonal to /u1D452/u1D45B−1and dividing /u1D438)uni2032(varinto\ntwo parts of equal measure. For simplicity let us assume that /u1D43B/u1D45B−1= {/u1D465/u1D45B−1= 0}⧵/u1D43B. We denote by /u1D438)uni2032(var+(resp.\n/u1D438)uni2032(var−) the set obtained by the union of /u1D438)uni2032(var∩{/u1D465/u1D45B−1>0}(resp./u1D438)uni2032(var∩{/u1D465/u1D45B−1<0}) with its reflection along /u1D43B/u1D45B−1. By\n(4.23) we have\n/u1D437/u1D706(/u1D438)uni2032(var±)≤2/u1D437/u1D706(/u1D438)uni2032(var)≤2/u1D45B−1/u1D437/u1D706(/u1D438).\nRegarding the asymmetry of /u1D438)uni2032(var±note that since /u1D438)uni2032(varis symmetric with respect to the first /u1D45B−2coordinate hyper-\nplanes,/u1D438)uni2032(var+and/u1D438)uni2032(var−are(/u1D45B−1)-symmetric. By Lemma 4.5we get\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D438)uni2032(var)≤/u⎪⎫007C.v⎞⎜/u1D438)uni2032(varΔ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜\n=/u⎪⎫007C.v⎞⎜(/u1D438)uni2032(varΔ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))∩{/u1D465/u1D45B−1>0}/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜(/u1D438)uni2032(varΔ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))∩{/u1D465/u1D45B−1<0}/u⎪⎫007C.v⎞⎜\n=1\n2/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var+Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438)uni2032(var−Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≤3/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜\n2/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D6FC/u1D706(/u1D438)uni2032(var+)+/u1D6FC/u1D706(/u1D438)uni2032(var−)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n.\nTherefore at least one of the sets /u1D438)uni2032(var+,/u1D438)uni2032(var−has asymmetry greater than1\n3/u1D6FC/u1D706(/u1D438)uni2032(var)and, denoting by /u1D439this set, we\nhave\n/u1D437/u1D706(/u1D439)≤2/u1D437/u1D706(/u1D438)uni2032(var)≤2/u1D45B−1/u1D437/u1D706(/u1D438)\n/u1D6FC/u1D706(/u1D438)≤̄/u1D436/u1D45B−2\n2/u1D6FC/u1D706(/u1D438)uni2032(var)≤3̄/u1D436/u1D45B−2\n2/u1D6FC/u1D706(/u1D439).\nFinally, the inclusion /u1D439 ⊂ /u1D4442/u1D459follows since /u1D439was obtained by performing reflections of /u1D438 ⊂ /u1D444/u1D459along affine\nhyperplanes of the form {/u1D465/u1D457=/u1D44E/u1D457}for/u1D457= 1,…,/u1D45B−1with/u1D44E/u1D457∈ (−/u1D459,/u1D459). /square\n4.3. Reduction to Schwarz-symmetric sets. In this section we observe that, in order to prove Theorem 1.1, it is\nsufficient to further reduce to show ( 1.6) just among Schwarz-symmetric sets. The proof is analogous to [Fus15 ,\nProposition 4.9].\nLemma 4.8 (Reduction to Schwarz-symmetric sets) .There exists/u1D4363=/u1D4363(/u1D45B,/u1D706)>0such that the following holds.\nLet/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Suppose that /u1D438is symmetric with respect to the\ncoordinate hyerplanes {/u1D4651= 0},…,{/u1D465/u1D45B−1= 0} and that\n/u1D437/u1D706(/u1D438)<1, /u1D438 ⊂/u1D4442/u1D459,\nwhere/u1D459=/u1D459(/u1D45B,/u1D706)is as in in Lemma 4.2. Then\n(4.31) /u⎪⎫007C.v⎞⎜/u1D438Δ/u1D438∗/u⎪⎫007C.v⎞⎜≤/u1D4363√\n/u1D437/u1D706(/u1D438) and/u1D437/u1D706(/u1D438∗)≤/u1D437/u1D706(/u1D438),\nwhere/u1D438∗denotes the Schwarz symmetrization of /u1D438with respect to the /u1D45B-th axis.20 GIULIO PASCALE AND MARCO POZZETTA\nProof. The second inequality in ( 4.31) follows from the fact /u⎪⎫007C.v⎞⎜/u1D438∗/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜and/u1D443/u1D706(/u1D438∗)≤/u1D443/u1D706(/u1D438). Exploiting Lemma\n2.3, we may assume\n(4.32) /u1D45B−1({/u1D465∈/u1D715∗/u1D438⧵/u1D43B∶/u1D708/u1D438(/u1D465) = ±/u1D452/u1D45B}) = 0,\nand thus that /u1D463/u1D438∈/u1D44A1,1(ℝ). Indeed, if/u1D438/u1D456is given by Lemma 2.3and the claim is proved for /u1D438/u1D456, by the contractivity\nof Schwartz rearrangement ([ Mag12 , Exercise 19.14]), for every ̃ /u1D700>0and with/u1D456sufficiently large, we get\n/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D438∗/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D438/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D438∗\n/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438∗\n/u1D456Δ/u1D438∗/u⎪⎫007C.v⎞⎜\n≤2/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D438/u1D456/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D438/u1D456Δ/u1D438∗\n/u1D456/u⎪⎫007C.v⎞⎜\n≤2̃ /u1D700+/u1D4363√\n/u1D437/u1D706(/u1D438/u1D456),\nand the last term tends to /u1D4363√\n/u1D437/u1D706(/u1D438)as/u1D456→∞.\nFor1-a.e./u1D461∈ (0,∞)denote\n/u1D463/u1D438(/u1D461) ∶=/u1D45B−1({/u1D465)uni2032(var∈ℝ/u1D45B−1∶ (/u1D465)uni2032(var,/u1D461) ∈/u1D438}),\n/u1D45D/u1D438(/u1D461) ∶=/u1D45B−2(/u1D715∗{/u1D465)uni2032(var∈ℝ/u1D45B−1∶ (/u1D465)uni2032(var,/u1D461) ∈/u1D438}),\nand employ analogous notation for /u1D438∗. Since/u⎪⎫007C.v⎞⎜/u1D715∗/u1D438∩/u1D715/u1D43B/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D715∗/u1D438∗∩/u1D715/u1D43B/u⎪⎫007C.v⎞⎜, we get\n/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706)≥/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D438∗) =/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B)−/u1D443(/u1D438∗,ℝ/u1D45B⧵/u1D43B).\nIt is therefore possible to reproduce the computation in [ Fus15 , Proposition 4.9] verbatim up to [ Fus15 , Eq. (4.29)]\nto estimate the right hand side in the last equation from belo w. We include the computation for the convenience of\nthe reader. By [ Mag12 , Theorem 19.11, (19.30)] one has\n/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706)≥/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B)−/u1D443(/u1D438∗,ℝ/u1D45B⧵/u1D43B)≥/uni222B.dsp∞\n0/⎝⎞⎜e⎪⎨eft.⎟3/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438+/u1D463)uni2032(var2\n/u1D438−/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438∗+/u1D463)uni2032(var2\n/u1D438/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D461\n=/uni222B.dsp∞\n0/u1D45D2\n/u1D438−/u1D45D2\n/u1D438∗/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438+/u1D463)uni2032(var2\n/u1D438+/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438∗+/u1D463)uni2032(var2\n/u1D438d/u1D461≥/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp∞\n0/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438−/u1D45D2\n/u1D438∗d/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟321\n∫∞\n0/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438+/u1D463)uni2032(var2\n/u1D438+/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438∗+/u1D463)uni2032(var2\n/u1D438d/u1D461\n≥/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp∞\n0/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438−/u1D45D2\n/u1D438∗d/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟321\n/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D438∗,ℝ/u1D45B⧵/u1D43B)\n≥/u1D450(/u1D706)/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp∞\n0/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438−/u1D45D2\n/u1D438∗d/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟321\n/u1D443/u1D706(/u1D438)+/u1D443/u1D706(/u1D438∗),\nwhere in the last inequality we used Corollary 2.4. Since/u1D437/u1D706(/u1D438)<1and/u1D45D/u1D438≥/u1D45D/u1D438∗, we have/u1D443/u1D706(/u1D438∗)≤/u1D443/u1D706(/u1D438)≤\n2/u1D443/u1D706(/u1D435/u1D706)and\n√\n/u1D437/u1D706(/u1D438)≥/u1D450/uni222B.dsp∞\n0/u⎪⎫221A.⎟1\n/u1D45D2\n/u1D438−/u1D45D2\n/u1D438∗d/u1D461=/u1D450/uni222B.dsp∞\n0√\n/u1D45D/u1D438+/u1D45D/u1D438∗√/u1D45D/u1D438∗/u⎪⎫221A.⎟2/u1D45D/u1D438−/u1D45D/u1D438∗\n/u1D45D/u1D438∗d/u1D461\n≥√\n2/u1D450/uni222B.dsp∞\n0/u1D45D/u1D438∗/u⎪⎫221A.⎟2/u1D45D/u1D438−/u1D45D/u1D438∗\n/u1D45D/u1D438∗d/u1D461,(4.33)\nfor some constant /u1D450=/u1D450(/u1D45B,/u1D706)>0changing from line to line. Note that (/u1D438∗)/u1D461is a(/u1D45B−1)-dimensional ball with the\nsame/u1D45B−1measure of/u1D438/u1D461. Then the quantity\n/u1D45D/u1D438(/u1D461)−/u1D45D/u1D438∗(/u1D461)\n/u1D45D/u1D438∗(/u1D461)\nis the classical isoperimetric deficit in ℝ/u1D45B−1of/u1D438/u1D461with respect to the standard perimeter. By the quantitative\nisoperimetric inequality in ℝ/u1D45B−1[FMP08 ], the fact that /u1D438/u1D461is/u1D45B−1symmetric with (/u1D438∗)/u1D461centered at the center of\nsymmetry of /u1D438/u1D461and Lemma 4.5, we have\n/u1D45B−1(/u1D438/u1D461Δ/u1D438∗\n/u1D461)\n/u1D45B−1((/u1D438∗)/u1D461)≤/u1D450(/u1D45B)/u⎪⎫221A.⎟3\n/u1D45D/u1D438(/u1D461)−/u1D45D/u1D438∗(/u1D461)\n/u1D45D/u1D438∗(/u1D461).\nBy (4.33) and the inclusion /u1D438 ⊂/u1D4442/u1D459we conclude\n√\n/u1D437/u1D706(/u1D438)≥/u1D450/uni222B.dsp∞\n0/u1D45D/u1D438∗(/u1D461)\n/u1D45B−1((/u1D438∗)/u1D461)/u1D45B−1(/u1D438/u1D461Δ/u1D438∗\n/u1D461)d/u1D461≥/u1D450\n/u1D459/uni222B.dsp∞\n0/u1D45B−1(/u1D438/u1D461Δ/u1D438∗\n/u1D461)d/u1D461=/u1D450\n/u1D459/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D438∗/u⎪⎫007C.v⎞⎜.\n/squareQUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 21\nPutting together Lemma 4.2, Lemma 4.7and Lemma 4.8, one immediately gets the following corollary.\nCorollary 4.9. There exist/u1D6FF4,̃/u1D459>0depending on /u1D45B,/u1D706such that for every 0</u1D6FF≤/u1D6FF4, if the quantitative isoperi-\nmetric inequality (1.6)holds (with some constant depending on /u1D45B,/u1D706) for every Schwarz-symmetric set /u1D439, with\n/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, contained in a cube /u1D444̃/u1D459and with/u1D437/u1D706(/u1D439)</u1D6FF, then (1.6)holds for any measurable set of finite measure\n(up to changing the multiplicative constant, depending on /u1D45B,/u1D706only).\n5. F IRST QUANTITATIVE ISOPERIMETRIC INEQUALITY\n5.1. Coupling. We start by recalling the general definition and properties o f the restricted envelopes introduced\nin [Cin+22 ].\nDefinition 5.1 ([Cin+22 , Definition 3.1]) .Let/u1D43E ⊂ℝ/u1D45Bbe a compact convex set, /u1D438 ⊂ℝ/u1D45Bbe a bounded open set,\nand/u1D462∈/u1D4360(/u1D438)∩/u1D4362(/u1D438). The/u1D43E-envelope of /u1D462is the function ̄ /u1D462/u1D43E∶ℝ/u1D45B→ℝgiven by\n̄ /u1D462/u1D43E(/u1D465) ∶= sup{/u1D44E+⟨/u1D709,/u1D465⟩∶/u1D709∈/u1D43E, /u1D44E+⟨/u1D709,/u1D466⟩≤/u1D462(/u1D466) ∀/u1D466∈/u1D438}.\nThe desired coupling corresponding to a competitor /u1D438will be essentially the /u1D43E-envelope of the solution to the\nelliptic problem ( 3.4), for/u1D43Eequal to the closure of the optimal bubble /u1D435/u1D706.\nDefinition 5.2 ([Cin+22 , Definition 3.9]) .Let/u1D43E ⊂ℝ/u1D45Bbe a compact convex set, /u1D438 ⊂ℝ/u1D45Bbe a bounded open set\nand/u1D462∈/u1D4360(/u1D438)∩/u1D4362(/u1D438). Given/u1D465∈ℝ/u1D45Band/u1D709∈/u1D43E, we define\n/u1D446/u1D709∶= argmin\n/u1D465∈/u1D438{/u1D462(/u1D465)−⟨/u1D709,/u1D465⟩}\nand\n/u1D43B(/u1D465,/u1D709,/u1D43E) ∶=⎧\n⎪\n⎨\n⎪⎩/u1D45A/u⎪⎫2211.⎟1\n/u1D456=1/u1D706/u1D456∇2/u1D462(/u1D460/u1D456) ∶1≤/u1D45A≤/u1D45B+1\n/u1D706/u1D456≥0,∑/u1D706/u1D456= 1\n/u1D460/u1D456∈/u1D446/u1D709∩/u1D438\n/u1D465−∑/u1D706/u1D456/u1D460/u1D456∈/u1D441(/u1D709,/u1D43E)⎫\n⎪\n⎬\n⎪⎭,\nwhere/u1D441(/u1D709,/u1D43E)is the normal cone of /u1D43Eat/u1D709, defined as\n/u1D441(/u1D709,/u1D43E) ∶= {/u1D463∈ℝ/u1D45B∶/u⎪⎫27E8.⎟1\n/u1D463,/u1D709)uni2032(var−/u1D709/u⎪⎫27E9.⎟1\n≤0for all/u1D709)uni2032(var∈/u1D43E}.\nWe will need the following\nProposition 5.3 ([Cin+22 , Proposition 3.10]) .Let/u1D43E ⊂ℝ/u1D45Bbe a compact convex set, /u1D438 ⊂ℝ/u1D45Bbe a bounded open\nset, and/u1D462∈/u1D4360(/u1D438) ∩/u1D4362(/u1D438). Assume that for any /u1D709∈/u1D43Eit holds/u1D446/u1D709⊂ /u1D438, then̄ /u1D462/u1D43E∶ℝ/u1D45B→ℝis a/u1D4361,1convex\nfunction such that\n∇̄ /u1D462/u1D43E(ℝ/u1D45B) = ∇̄ /u1D462/u1D43E(/u1D438) =/u1D43E.\nMoreover, for any /u1D465∈ℝ/u1D45Band any/u1D43B/u1D465∈/u1D43B(/u1D465,∇̄ /u1D462/u1D43E(/u1D465),/u1D43E)≠)uni2205(var, it holds∇2̄ /u1D462/u1D43E(/u1D465)≤/u1D43B/u1D465.\nWe will need to associate a coupling only to Lipschitz-regul ar connected competitors, having /u1D4361relative bound-\nary inℝ/u1D45B⧵/u1D43B. This is established in the next result, whose proof is analo gous to the one in [ Cin+22 ].\nProposition 5.4. There exists ̂/u1D436=̂/u1D436(/u1D45B,/u1D706)>0such that the following holds. Let /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a connected\nbounded open set. Suppose that /u1D438has Lipschitz boundary and that /u1D715/u1D438∩{/u1D465/u1D45B≥0}is a hypersurface of class /u1D4361\nwith boundary.\nThen there exists a 1-Lipschitz convex function Ψ ∶ℝ/u1D45B→ℝof class/u1D4361,1such that ΔΨ≤/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜and such that\n∇Ψ(ℝ/u1D45B) = ∇Ψ(/u1D438) =/u1D435/u1D706up to negligible sets. Moreover, if /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438)≤1, then\n(5.1)/uni222B.dsp/u1D438/u⎪⎫007C.v⎞⎜∇2Ψ−id/u⎪⎫007C.v⎞⎜d/u1D465≤̂/u1D436√\n/u1D437/u1D706(/u1D438)\n(5.2)/uni222B.dsp/u1D715∗/u1D438(1−/u⎪⎫007C.v⎞⎜∇Ψ/u⎪⎫007C.v⎞⎜)d/u1D45B−1≤̂/u1D436/u1D437/u1D706(/u1D438).\nProof. Let us assume first that /u1D715/u1D438⧵/u1D715/u1D43Bis a smooth hypersurface with smooth boundary intersecting /u1D715/u1D43Borthog-\nonally. Let/u1D462∶/u1D438→ℝbe a solution of ( 3.4) and(/u1D43E/u1D456)/u1D456∈ℕa sequence of compact convex sets such that /u1D43E/u1D456⊂⊂̊/u1D43E/u1D456+1\nand∪/u1D456∈ℕ/u1D43E/u1D456=/u1D435/u1D706. We showed in the proof of Theorem 3.5that for any /u1D709∈/u1D435/u1D706the minimum of /u1D462(/u1D465) −⟨/u1D709,/u1D465⟩\ncannot be achieved on the boundary of /u1D438. Moreover, at any point /u1D465∈/u1D438such that ∇2/u1D462(/u1D465)≥0, it holds\n0≤∇2/u1D462(/u1D465)≤Δ/u1D462(/u1D465)id =/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜id.22 GIULIO PASCALE AND MARCO POZZETTA\nTherefore, recalling Remark 3.4, by Proposition 5.3we get that̄ /u1D462/u1D43E/u1D456is a sequence of 1-Lipschitz functions, being\nsuprema of 1-Lipschitz functions, that is uniformly bounded in /u1D4361,1on compact sets; hence up to subsequence it\nconverges to a limit function Ψin/u1D4361\nloc(ℝ/u1D45B). Since∇̄ /u1D462/u1D43E/u1D456(ℝ/u1D45B) = ∇̄ /u1D462/u1D43E/u1D456(/u1D438) =/u1D43E/u1D456, then∇Ψ(ℝ/u1D45B) = ∇Ψ(/u1D438) =/u1D435/u1D706,Ψis\na convex function of class /u1D4361,1, and writing the inequality Δ̄ /u1D462/u1D43E/u1D456≤/u1D443/u1D706(/u1D438)\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜in the sense of distributions, one checks\nthat it readily passes to the limit as /u1D456→+∞for the function Ψ.\nTo prove that /u⎪⎫007C.v⎞⎜∇Ψ(/u1D438)Δ/u1D435/u1D706/u⎪⎫007C.v⎞⎜= 0, let/u1D44D ⊂/u1D438 be a compact set and notice that\n/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D438⧵/u1D44D)/u⎪⎫007C.v⎞⎜≤/u1D450(/u1D45B,/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D443/u1D706(/u1D438))/u⎪⎫007C.v⎞⎜/u1D438⧵/u1D44D/u⎪⎫007C.v⎞⎜,\n/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D44D)/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D438⧵(/u1D438⧵/u1D44D))/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D438)/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D438⧵/u1D44D)/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D43E/u1D456/u⎪⎫007C.v⎞⎜−/u1D450/u⎪⎫007C.v⎞⎜/u1D438⧵/u1D44D/u⎪⎫007C.v⎞⎜.\nPassing to the limit we find\n/u⎪⎫007C.v⎞⎜∇Ψ(/u1D44D)/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xlimsup\n/u1D456∇̄ /u1D462/u1D43E/u1D456(/u1D44D)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≥limsup\n/u1D456/u⎪⎫007C.v⎞⎜∇̄ /u1D462/u1D43E/u1D456(/u1D44D)/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜−/u1D450/u⎪⎫007C.v⎞⎜/u1D438⧵/u1D44D/u⎪⎫007C.v⎞⎜,\nhence letting /u1D44D↗/u1D438, we get that ∇Ψ(ℝ/u1D45B) = ∇Ψ(/u1D438) =/u1D435/u1D706up to negligible sets.\nSuppose now that /u1D438is a generic connected set as in the assumptions. If /u1D45B≥3we can apply the above argument\nto a sequence of sets /u1D438/u1D456approximating /u1D438given by Lemma 2.3, suitably modified connecting possibly disconnected\ncomponents with thin tubes vanishing in the limit. If /u1D45B= 2, then/u1D715/u1D438⧵/u1D715/u1D43Bis a union of /u1D4361curves, which thus can\nbe approximated by smooth ones touching /u1D715/u1D43Borthogonally preserving the connectedness of the set. Appl ying the\nfirst part of the proof on the approximating sequence /u1D438/u1D456we get a corresponding sequence of functions Ψ/u1D456uniformly\nbounded in/u1D4361,1on compact sets, hence converging in /u1D4361\nlocup to subsequence to a convex function Ψof class/u1D4361,1\nwithΔΨ≤/u1D443/u1D706(/u1D438)∕/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜. Moreover, since /u1D438has Lipschitz boundary, there exists a sequence of compact s ets/u1D44D/u1D457⊂/u1D438\nsuch that/u1D44D/u1D457⊂/u1D438/u1D456for any/u1D456≥/u1D456/u1D457and such that /u1D44D/u1D457↗/u1D438. Hence one can repeat the above argument with Ψ/u1D456,/u1D438,/u1D44D/u1D457\nin place of̄ /u1D462/u1D43E/u1D456,/u1D438,/u1D44D , respectively, to deduce that\n/u⎪⎫007C.v⎞⎜∇Ψ/u1D456(/u1D44D/u1D457)/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜−/u1D450(/u1D45B,/u⎪⎫007C.v⎞⎜/u1D438/u1D456/u⎪⎫007C.v⎞⎜,/u1D443/u1D706(/u1D438/u1D456))/u⎪⎫007C.v⎞⎜/u1D438/u1D456⧵/u1D44D/u1D457/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜−/u1D450(/u1D45B,/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜,/u1D443/u1D706(/u1D438))/u⎪⎫007C.v⎞⎜/u1D438/u1D456⧵/u1D44D/u1D457/u⎪⎫007C.v⎞⎜.\nLetting/u1D456→∞first, and then /u1D457→∞, we get that ∇Ψ(ℝ/u1D45B) = ∇Ψ(/u1D438) =/u1D435/u1D706up to negligible sets.\nWe now prove ( 5.1) and ( 5.2). The symbol ̂/u1D436shall denote a positive constant depending on /u1D45B,/u1D706changing from\nline to line. By the area formula, the arithmetic-geometric mean inequality and the properties of Ψwe get\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜∇Ψ(/u1D438)/u⎪⎫007C.v⎞⎜≤/uni222B.dsp/u1D438det(∇2Ψ)d/u1D45B≤/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟2ΔΨ\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\nd/u1D465≤/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\nd/u1D465=/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\nd/u1D465\n=/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D443/u1D706(/u1D435/u1D706)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜= (1+/u1D437/u1D706(/u1D438))/u1D45B/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜.(5.3)\nHence\n(5.4)/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟3/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\n−det(∇2Ψ)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D465≤/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜(1+/u1D437/u1D706(/u1D438))/u1D45B−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜≤̂/u1D436/u1D437/u1D706(/u1D438).\nBy [Cin+22 , Lemma A.1] applied with /u1D45A=/u1D45B,/u1D7061= … =/u1D706/u1D45B= 1,(/u1D4651,…,/u1D465/u1D45B)equal to the eigenvalues of ∇2Ψ\nand/u1D450=/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u1D443/u1D706(/u1D438)\n/u1D443/u1D706(/u1D435/u1D706)≥1, we obtain\n/u⎪⎫007C.v⎞⎜∇2Ψ−id/u⎪⎫007C.v⎞⎜2≤2/u⎪⎫007C.v⎞⎜∇2Ψ−/u1D450id/u⎪⎫007C.v⎞⎜2+2/u⎪⎫007C.v⎞⎜(/u1D450−1)id/u⎪⎫007C.v⎞⎜2≤̂/u1D436/⎝⎞⎜e⎪⎨eft.⎟3/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\n−det(∇2Ψ)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n+2/u1D45B/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)−/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟32\n≤̂/u1D436/⎝⎞⎜e⎪⎨eft.⎟3/⎝⎞⎜e⎪⎨eft.⎟3/u1D443/u1D706(/u1D438)\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B\n−det(∇2Ψ)+/u1D437/u1D706(/u1D438)2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\n.(5.5)\nTherefore by ( 5.4) and ( 5.5) we get\n/uni222B.dsp/u1D438/u⎪⎫007C.v⎞⎜∇2Ψ−id/u⎪⎫007C.v⎞⎜2d/u1D465≤̂/u1D436/u1D437/u1D706(/u1D438),\nwhich implies ( 5.1).\nArguing as in ( 5.3), by the divergence theorem we get\n/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜≤/uni222B.dsp/u1D438/⎝⎞⎜e⎪⎨eft.⎟2ΔΨ\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\nd/u1D465≤/u1D443/u1D706(/u1D438)/u1D45B−1\n/u1D45B/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1/uni222B.dsp/u1D438ΔΨd/u1D465=/u1D443/u1D706(/u1D438)/u1D45B−1\n/u1D45B/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1/uni222B.dsp/u1D715∗/u1D438/u⎪⎫27E8.⎟1∇Ψ,/u1D708/u1D438/u⎪⎫27E9.⎟1d/u1D45B−1\n≤/u1D443/u1D706(/u1D438)/u1D45B\n/u1D45B/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1−/u1D443/u1D706(/u1D438)/u1D45B−1\n/u1D45B/u1D45B/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u1D45B−1/uni222B.dsp/u1D715∗/u1D438(1−/u⎪⎫007C.v⎞⎜∇Ψ/u⎪⎫007C.v⎞⎜)d/u1D45B−1.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 23\nRearranging terms, since /u1D437/u1D706(/u1D438)≤1and/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, we obtain\n/uni222B.dsp/u1D715∗/u1D438(1−/u⎪⎫007C.v⎞⎜∇Ψ/u⎪⎫007C.v⎞⎜)d/u1D45B−1≤/u1D443/u1D706(/u1D438)/u1D45B−/u1D443/u1D706(/u1D435/u1D706)/u1D45B\n/u1D443/u1D706(/u1D438)/u1D45B−1≤̂/u1D436/u1D437/u1D706(/u1D438).\n/square\nWe now want to translate the quantitative estimates obtaine d on the coupling in Proposition 5.4into quantitative\nestimates on the asymmetry of a competitor. We will need some technical results first.\nThe next lemma is analogous to [ Cin+22 , Lemma 6.2], but with a varying range of parameters that here must\ndepend on/u1D706.\nLemma 5.5. Let/u1D438 ⊂[0,∞)be a1-dimensional set of locally finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜<∞and set\n/u1D45F/u1D706∶= min/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2\n, /u1D445/u1D706∶= max/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nThere exists ̂ /u1D450=̂ /u1D450(/u1D45B,/u1D706)>0such that for any7\n8/u1D45F/u1D706≤/u1D459≤9\n8/u1D445/u1D706there holds\n(5.6)/uni222B.dsp/u1D438Δ[0,/u1D459]/u1D461/u1D45B−1d/u1D461≤̂ /u1D450/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461+/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n.\nProof. It holds/u1D438Δ[0,/u1D459] = ((/u1D459,∞)∩/u1D438)∪([0,/u1D459]⧵/u1D438). We claim that\n(5.7) max/b⎜⎞ce⎨eft.⎟4\n/uni222B.dsp[/u1D459,∞)∩/u1D438/u1D461/u1D45B−1d/u1D461,/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461/b⎜⎞ce⎜⎫g⎧t.⎟4\n≤̂ /u1D450(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461+/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n.\nWe will estimate the two terms on the left-hand side separate ly.\nWithout loss of generality, suppose [/u1D459,∞)∩/u1D438≠)uni2205(var. Since/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜<∞, then/u1D715∗/u1D438∩[/u1D459,∞)is nonempty and we can\nassume it has finite supremum ̄/u1D461(otherwise the right hand side in ( 5.7) equals+∞). In particular the right-hand\nside in ( 5.7) is finite. It holds\n/uni222B.dsp[/u1D459,∞)∩/u1D438/u1D461/u1D45B−1d/u1D461≤/uni222B.dsp̄/u1D461\n/u1D459/u1D461/u1D45B−1d/u1D461≤̄/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜̄/u1D461−/u1D459/u⎪⎫007C.v⎞⎜≤/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461),\nand the first term in the left-hand side of ( 5.7) is bounded as wished.\nLet us now consider ∫/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461. Its value is a priori bounded by/⎝⎞⎜e⎪⎨eft.⎟2\n9\n8/u1D445/u1D706/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B\n. If/u1D715∗/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,3\n4/u1D45F/u1D706/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n≠)uni2205(varand/u1D70Fis\none of its elements, then\n/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461)≥/u1D70F/u1D45B/u⎪⎫007C.v⎞⎜/u1D459−/u1D70F/u⎪⎫007C.v⎞⎜≥/⎝⎞⎜e⎪⎨eft.⎟2/u1D45F/u1D706\n4/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B1\n8/u1D45F/u1D706≥̂ /u1D450(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟29\n8/u1D445/u1D706/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x9\n8/u1D445/u1D706−/u1D45F/u1D706\n2/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≥̂ /u1D450(/u1D45B,/u1D706)/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461.\nSo from now on we can assume that /u1D715∗/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,3\n4/u1D45F/u1D706/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n= )uni2205(var. If/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,3\n4/u1D45F/u1D706/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438≠)uni2205(var, then/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,3\n4/u1D45F/u1D706/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n= )uni2205(varand\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461+/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461)≥/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2/u1D461/u1D45B−1d/u1D461=̂ /u1D450(/u1D45B,/u1D706)≥̂ /u1D450(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟29\n8/u1D445/u1D706/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/u1D45B−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x9\n8/u1D445/u1D706−/u1D45F/u1D706\n2/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n≥̂ /u1D450(/u1D45B,/u1D706)/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461.\nSo we can further assume/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,3\n4/u1D45F/u1D706/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⊂ /u1D438. If/u1D715∗/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n= )uni2205(var, then/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⊂/u1D438 and there is nothing to prove.\nFinally, if/u1D715∗/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n≠)uni2205(var, let us denote by /u1D461the infimum of /u1D715∗/u1D438∩/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n4,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n. Then\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461≤/uni222B.dsp/u1D459\n/u1D461/u1D461/u1D45B−1d/u1D461≤/u1D459/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜=/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎨eft.⎟3\n/u1D459\n/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/u1D45B−1\n≤̂ /u1D450(/u1D45B,/u1D706)/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜\n≤̂ /u1D450(/u1D45B,/u1D706)/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461).24 GIULIO PASCALE AND MARCO POZZETTA\nThis concludes the proof of the claim ( 5.7). Hence\n/uni222B.dsp/u1D438Δ[0,/u1D459]/u1D461/u1D45B−1d/u1D461≤max/b⎜⎞ce⎨eft.⎟4\n/uni222B.dsp[/u1D459,∞)∩/u1D438/u1D461/u1D45B−1d/u1D461,/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2/u1D45F/u1D706\n2,/u1D459/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461/b⎜⎞ce⎜⎫g⎧t.⎟4\n+/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461\n(5.7)\n≤̂ /u1D450(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461+/uni222B.dsp/u1D715∗/u1D438/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D459−/u1D461/u⎪⎫007C.v⎞⎜d0(/u1D461)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n+/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D461/u1D45B−1d/u1D461,\nand the proof follows. /square\nWe shall also need the following standard technical result s tating that a Vol’pert property holds for the intersec-\ntions of a set of finite perimeter with rays from the origin, cf . [Vol67 ]. The proof follows, for example, by adapting\nthe proof of [ Fus04 , Theorem 3.21] working in polar coordinates rather than in C artesian coordinates.\nLemma 5.6. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜<+∞. If/u1D717∈/u1D54A/u1D45B−1∩(ℝ/u1D45B⧵/u1D43B), we define\n/u1D438/u1D717∶= {/u1D461≥0 ∶/u1D461/u1D717∈/u1D438}.\nThen, for /u1D45B−1-almost every /u1D717∈/u1D54A/u1D45B−1∩(ℝ/u1D45B⧵/u1D43B),/u1D438/u1D717is a1-dimensional set of locally finite perimeter such that\n/u1D715∗/u1D438/u1D717∩{/u1D461>0} = {/u1D461>0 ∶/u1D461/u1D717∈/u1D715∗/u1D438}.\nMoreover, if /u1D702∈/u1D43F1(/u1D715∗/u1D438)is nonnegative, we have\n(5.8)/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B/u1D702d/u1D45B−1≥/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp/u1D715∗/u1D438/u1D717/u1D461/u1D45B−1/u1D702(/u1D461/u1D717)d0(/u1D461)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D45B−1(/u1D717).\nCombining Lemma 5.5with Lemma 5.6we get the following result that estimates the symmetric diffe rence of\na competitor with a bubble that is just close to a standard bub ble/u1D435/u1D706(/u1D463,/u1D465). The result is analogous to [ Cin+22 ,\nProposition 6.1].\nLemma 5.7. There exist/u1D700,̃ /u1D450 >0depending on /u1D45B,/u1D706such that the following holds. Let /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a bounded\nset of finite perimeter. If/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D438∩/u1D435/u1D45F/u1D706\n2(0)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≥1\n2/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D435/u1D45F/u1D706\n2(0)⧵/u1D43B/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x, then\n/u⎪⎫007C.v⎞⎜/u1D438Δ(/u1D4351(/u1D4650)⧵/u1D43B)/u⎪⎫007C.v⎞⎜≤̃ /u1D450/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D465−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜d/u1D45B−1(/u1D465),\nfor any/u1D4650∈ℝ/u1D45Bsuch that /u⎪⎫007C.v⎞⎜/u1D4650−(0,…,0,−/u1D706)/u⎪⎫007C.v⎞⎜</u1D700.\nProof. If/u1D700is sufficiently small, depending only on /u1D45B,/u1D706, then for any /u1D717∈/u1D54A/u1D45B−1∩ (ℝ/u1D45B⧵/u1D43B)the set{/u1D461 >0 ∶\n/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜<1}is an open segment (0,/u1D461(/u1D717)), with/u1D461(/u1D717)close to the number\n/u1D447/u1D717∈/b⎜⎞c⎭et⎨eft.⎟2\nmin/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2\n,max/b⎜⎞ce⎨eft.⎟2√\n1−/u1D7062,1−/u1D706/b⎜⎞ce⎜⎫g⎧t.⎟2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n=∶ [/u1D45F/u1D706,/u1D445/u1D706]\nsuch that/u⎪⎫007C.v⎞⎜/u1D447/u1D717/u1D717+/u1D706/u1D452/u1D45B/u⎪⎫007C.v⎞⎜= 1. In particular, if /u1D700is sufficiently small, then9\n8/u1D445/u1D706≥/u1D461(/u1D717)≥7\n8/u1D45F/u1D706for any/u1D717∈/u1D54A/u1D45B−1∩(ℝ/u1D45B⧵/u1D43B).\nAs before, for any /u1D717∈/u1D54A/u1D45B−1⧵/u1D43Blet\n/u1D438/u1D717∶= {/u1D461≥0 ∶/u1D461/u1D717∈/u1D438}.\nBy coarea formula we get\n(5.9) /u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4351(/u1D4650)∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜=/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp/u1D438/u1D717Δ[0,/u1D461(/u1D717)]/u1D461/u1D45B−1d/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D45B−1(/u1D717).\nBy Lemma 5.5we obtain\n(5.10) /u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4351(/u1D4650)⧵/u1D43B/u⎪⎫007C.v⎞⎜≤/u1D450(/u1D45B,/u1D706)/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D717/u1D461/u1D45B−1d/u1D461+/uni222B.dsp/u1D715∗/u1D438/u1D717/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D461(/u1D717)−/u1D461/u⎪⎫007C.v⎞⎜d0/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\nd/u1D45B−1(/u1D717).\nFor every/u1D461>0, we claim that\n(5.11) /u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜≥/u1D450(/u1D45B,/u1D706)/u⎪⎫007C.v⎞⎜/u1D461−/u1D461(/u1D717)/u⎪⎫007C.v⎞⎜.\nNote that there exists /u1D6FF=/u1D6FF(/u1D45B,/u1D706,/u1D700) ∈ (0,/u1D45F/u1D706∕8)such that for /u1D461∈ [/u1D461(/u1D717)−/u1D6FF,/u1D461(/u1D717)+/u1D6FF]there holds\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xd\nd/u1D461/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u⎪⎫007C.x/u⎪⎫007C.x⟨(/u1D461/u1D717−/u1D4650)∕/u⎪⎫007C.x/u⎪⎫007C.x/u1D461/u1D717−/u1D4650/u⎪⎫007C.x/u⎪⎫007C.x,/u1D717⟩/u⎪⎫007C.x/u⎪⎫007C.x≥/u1D450(/u1D45B,/u1D706,/u1D700)>0.\nThen, for/u1D461∈ [/u1D461(/u1D717)−/u1D6FF,/u1D461(/u1D717)+/u1D6FF],\n/u⎪⎫007C.v⎞⎜/u1D461(/u1D717)−/u1D461/u⎪⎫007C.v⎞⎜≤/u1D450(/u1D45B,/u1D706)/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜,QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 25\nand in this case the claim follows. Regarding the remaining c ases, note that\n/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D461(/u1D717)−/u1D461/u⎪⎫007C.v⎞⎜→1 as/u⎪⎫007C.v⎞⎜/u1D461/u⎪⎫007C.v⎞⎜→+∞\nand the claim follows for /u1D461≥/u1D445=/u1D445(/u1D45B,/u1D706,/u1D700)>0big enough. Finally, if 0</u1D461</u1D445 and/u1D461∉ [/u1D461(/u1D717)−/u1D6FF,/u1D461(/u1D717)+/u1D6FF], then\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.x/u⎪⎫007C.x≥/u1D450(/u1D45B,/u1D706,/u1D6FF)>0\n/u⎪⎫007C.v⎞⎜/u1D461(/u1D717)−/u1D461/u⎪⎫007C.v⎞⎜≤/u1D450(/u1D45B,/u1D706,/u1D445)\nhence the claim follows as well.\nTherefore\n(5.12)\n/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/uni222B.dsp/u1D715∗/u1D438/u1D717/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u1D461−/u1D461(/u1D717)/u⎪⎫007C.v⎞⎜d0(/u1D461)d/u1D45B−1(/u1D717)(5.11)\n≤/u1D450(/u1D45B,/u1D706)/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/uni222B.dsp/u1D715∗/u1D438/u1D717/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜d0(/u1D461)d/u1D45B−1(/u1D717).\nBy Lemma 5.6we deduce\n/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D465−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜d/u1D45B−1(/u1D465)≥/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp/u1D715∗/u1D438/u1D717/u1D461/u1D45B−1/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D461/u1D717−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜d0(/u1D461)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D45B−1(/u1D717). (5.13)\nSince/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D465−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜≥/u1D450(/u1D45B,/u1D706)>0in/u1D435/u1D45F/u1D706\n2(0), by coarea formula and relative isoperimetric inequality w e get\n/uni222B.dsp/u1D54A/u1D45B−1⧵/u1D43B/⎝⎞⎜e⎪⎨eft.⎟4\n/uni222B.dsp/b⎜⎞c⎭et⎨eft.⎟2\n0,/u1D45F/u1D706\n2/b⎜⎞c⎭et⎜⎫g⎧t.⎟2\n⧵/u1D438/u1D717/u1D461/u1D45B−1d/u1D461/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\nd/u1D45B−1(/u1D717) =/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D435/u1D45F/u1D706\n2(0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n⧵/u1D438/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D435/u1D45F/u1D706\n2(0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n⧵/u1D438/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D45B−1\n/u1D45B/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D435/u1D45F/u1D706\n2(0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n⧵/u1D438/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x1\n/u1D45B\n≤/u1D450(/u1D45B,/u1D706)/uni222B.dsp/u1D715∗/u1D438∩/⎝⎞⎜e⎪⎨eft.⎟3\n/u1D435/u1D45F/u1D706\n2(0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3d/u1D45B−1\n≤/u1D450(/u1D45B,/u1D706)/uni222B.dsp/u1D715∗/u1D438⧵/u1D43B/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D465−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜d/u1D45B−1(/u1D465).(5.14)\nPutting together ( 5.10), (5.12), (5.13) and ( 5.14), the proof follows. /square\nWe can finally show that if a suitably regular Schwarz-symmet ric set satisfies a trace inequality, then the quan-\ntitative estimates in Proposition 5.4imply a quantitative isoperimetric inequality.\nProposition 5.8. There exists /u1D6FF5=/u1D6FF5(/u1D45B,/u1D706)>0such that for any /u1D450/u1D447>0there exists/u1D6FE=/u1D6FE(/u1D45B,/u1D706,/u1D450/u1D447)>0such\nthat the following holds. Let /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a bounded connected open set with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Suppose that /u1D438\nhas Lipschitz boundary and that /u1D715/u1D438∩ {/u1D465/u1D45B≥0}is a hypersurface of class /u1D4361with boundary. Assume that /u1D438is\nSchwarz-symmetric with respect to the /u1D45B-th axis and that there exists a constant /u1D450/u1D447such that for every function\n/u1D453∈/u1D435/u1D449(ℝ/u1D45B)∩/u1D43F∞(ℝ/u1D45B)there is a constant /u1D450∈ℝsuch that the following holds\n(5.15)/uni222B.dsp/u1D438d/u⎪⎫007C.v⎞⎜/u1D437/u1D453/u⎪⎫007C.v⎞⎜(/u1D465)≥/u1D450/u1D447/uni222B.dsp/u1D715∗/u1D438∩(ℝ/u1D45B⧵/u1D43B)tr/u1D438(/u⎪⎫007C.v⎞⎜/u1D453−/u1D450/u⎪⎫007C.v⎞⎜)d/u1D45B−1(/u1D465).\nIf/u1D437/u1D706(/u1D438)</u1D6FF5, then\n/u1D6FC2\n/u1D706(/u1D438)≤/u1D6FE/u1D437/u1D706(/u1D438).\nProof. LetΨbe given by Proposition 5.4. By ( 5.15) and ( 5.1) we get\n/uni222B.dsp/u1D715∗/u1D438∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜(∇Ψ−/u1D465)+/u1D4650/u⎪⎫007C.v⎞⎜d/u1D45B−1(/u1D465)≤/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438),\nwhere/u1D4650= (/u1D4651\n0,…,/u1D465/u1D45B\n0)is the vector whose /u1D456-th component is the constant /u1D450of (5.15) corresponding to the /u1D456-th\ncomponent of ∇Ψ−/u1D465. Therefore\n/uni222B.dsp/u1D715∗/u1D438∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D465−/u1D4650/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.x/u⎪⎫007C.xd/u1D45B−1(/u1D465)≤/uni222B.dsp/u1D715∗/u1D438∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜∇Ψ−(/u1D465−/u1D4650)/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜1−/u⎪⎫007C.v⎞⎜∇Ψ/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜d/u1D45B−1(/u1D465)\n(5.2)\n≤/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438).(5.16)\nWe observe that if /u1D700=/u1D700(/u1D45B,/u1D706)is given by Lemma 5.7, then for/u1D6FF5small enough depending on /u1D700, we ensure that\n/u⎪⎫007C.v⎞⎜/u1D4650−(0,…,0,−/u1D706)/u⎪⎫007C.v⎞⎜</u1D700. Indeed, if for every /u1D456∈ℕthere were/u1D438/u1D456satisfying the hypotheses of Proposition 5.8such\nthat/u1D437/u1D706(/u1D438)<1\n/u1D456with corresponding /u1D4650,/u1D456verifying /u⎪⎫007C.v⎞⎜/u1D4650,/u1D456− (0,…,0,−/u1D706)/u⎪⎫007C.v⎞⎜≥/u1D700, passing to limit in ( 5.16) we would\nget a contradiction with the fact that /u1D438/u1D456converges to /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜).26 GIULIO PASCALE AND MARCO POZZETTA\nHence we can apply Lemma 5.7. Since/u1D438is Schwarz-symmetric, we get\n/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4351(0,…,0,/u1D465/u1D45B\n0)∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4351(/u1D4650)∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜\n≤/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438).(5.17)\nArguing as above, up to taking a smaller /u1D6FF5, we can assume that /u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D4650−(0,…,0,−/u1D706)/u⎪⎫007C.v⎞⎜is so small that\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xd\nd/u1D461/u⎪⎫007C.v⎞⎜/u1D4351(0,…,0,/u1D461)⧵/u1D43B/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≥1\n2/u1D714/u1D45B−1(1−/u1D7062)/u1D45B−1\n2,\nfor any/u1D461∈ [−/u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜,/u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜]. Hence\n/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438)(5.17)\n≥/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D4351(0,…,0,/u1D465/u1D45B\n0)⧵/u1D43B/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n=/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D4351(0,…,0,/u1D465/u1D45B\n0)⧵/u1D43B/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D4351(0,…,0,−/u1D706)⧵/u1D43B/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n≥1\n2/u1D714/u1D45B−1(1−/u1D7062)/u1D45B−1\n2/u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜,\nwhich implies\n(5.18) /u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜≤/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438),\nfor a suitable constant. Therefore\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D4351(0,…,0,/u1D465/u1D45B\n0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\nΔ/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D4351(0,…,0,−/u1D706)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D450(/u1D45B,/u1D706)/u⎪⎫007C.v⎞⎜/u1D465/u1D45B\n0+/u1D706/u⎪⎫007C.v⎞⎜\n(5.18)\n≤/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438),(5.19)\nwhere in the first inequality we used that /u1D461↦/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟1/u1D4351(0,…,0,/u1D461)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1Δ/⎝⎞⎜e⎪⎨eft.⎟1/u1D4351(0,…,0,−/u1D706)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xis Lipschitz for\nsome Lipschitz constant /u1D450(/u1D45B,/u1D706)>0.\nFinally\n/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D4351(0,…,0,/u1D465/u1D45B\n0)∩(ℝ/u1D45B⧵/u1D43B)/u⎪⎫007C.v⎞⎜≥\n≥/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)Δ/u1D438/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D4351(0,…,0,/u1D465/u1D45B\n0)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\nΔ/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D4351(0,…,0,−/u1D706)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n(5.19)\n≥/u1D6FC/u1D706(/u1D438)−/u1D450(/u1D45B,/u1D706,/u1D450/u1D447)√\n/u1D437/u1D706(/u1D438).\n/square\nIn the next lemma we observe that optimal bubbles do satisfy t race inequalities.\nLemma 5.9. There exists ̄ /u1D450=̄ /u1D450(/u1D45B,/u1D706)>0such that for every function /u1D453∈/u1D435/u1D449(ℝ/u1D45B)∩/u1D43F∞(ℝ/u1D45B)there is a constant\n/u1D450∈ℝsuch that the following holds\n(5.20)/uni222B.dsp/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)d/u⎪⎫007C.v⎞⎜/u1D437/u1D453/u⎪⎫007C.v⎞⎜(/u1D465)≥̄ /u1D450/uni222B.dsp/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43Btr/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)(/u⎪⎫007C.v⎞⎜/u1D453−/u1D450/u⎪⎫007C.v⎞⎜)d/u1D45B−1(/u1D465).\nProof. The proof follows combining the classical Poincaré inequal ity [AFP00 , Theorem 3.44] with the boundary\ntrace theorem [ AFP00 , Theorem 3.87].\n/square\nWe now introduce a notion of /u1D4361-distance from /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)for sets in the half-space ℝ/u1D45B⧵/u1D43B, and we deduce that\nSchwarz-symmetric sets sufficiently close in /u1D4361to/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)enjoy a quantitative isoperimetric inequality.\nDefinition 5.10. Let/u1D711/u1D706∶/u1D715/u1D4351⧵/u1D43B→/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43Bbe such that /u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B= {/u1D711/u1D706(/u1D465)/u1D465∶/u1D465∈/u1D715/u1D4351⧵/u1D43B}.\nLet/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a bounded open set. Suppose that /u1D438has Lipschitz boundary and that /u1D715/u1D438∩ {/u1D465/u1D45B≥0}is a\nhypersurface of class /u1D4361with boundary. Assume that /u1D438is Schwarz-symmetric. Suppose that there exists a /u1D4361\nfunctions\n/u1D711∶/u1D715/u1D4351⧵/u1D43B→/u1D715/u1D438⧵/u1D43B\nwhose graph parametrizes the boundary of /u1D438inℝ/u1D45B⧵/u1D43B, that is\n/u1D715/u1D438⧵/u1D43B= {/u1D711(/u1D465)/u1D465∶/u1D465∈/u1D715/u1D4351⧵/u1D43B}.\nWe define the /u1D4361distance of/u1D438to/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)byd/u1D4361(/u1D438,/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)) ∶=‖/u1D711−/u1D711/u1D706‖/u1D4361(/u1D715/u1D4351∩ℝ/u1D45B⧵/u1D43B).\nA sequence of sets /u1D438/u1D457as above is said to converge to /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)in/u1D4361ifd/u1D4361(/u1D438/u1D457,/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))→0as/u1D457→+∞.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 27\nCorollary 5.11. There exist̂ /u1D700,̂ /u1D6FE >0depending only on /u1D45B,/u1D706such that the following holds. Let /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe as\nin Definition 5.10. Ifd/u1D4361(/u1D438,/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))≤̂ /u1D700, then\n/u1D6FC2\n/u1D706(/u1D438)≤̂ /u1D6FE(/u1D45B,/u1D706)/u1D437/u1D706(/u1D438).\nProof. Let/u1D711,/u1D711/u1D706be as in Definition 5.10. If/u1D712∶ [0,+∞)→[0,1]is a smooth cut-off function such that /u1D712(/u1D461) = 0\nfor/u1D461<1\n4min{√\n1−/u1D7062,1−/u1D706}and such that /u1D712(/u1D461) = 1 for/u1D461>1\n2min{√\n1−/u1D7062,1−/u1D706}, we define the diffeomorphism\n/u1D713∶ℝ/u1D45B⧵/u1D43B→ℝ/u1D45B⧵/u1D43B /u1D713 (/u1D465) =⎛\n⎜\n⎜\n⎜⎝1−/u1D712(/u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜)+/u1D712(/u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜)/u1D711/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D465\n/u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n/u1D711/u1D706/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D465\n/u⎪⎫007C.v⎞⎜/u1D465/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2⎞\n⎟\n⎟\n⎟⎠/u1D465.\nNote that\n‖/u1D713−id‖/u1D4361≤/u1D450̂ /u1D700,\n/u1D713(/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B) =/u1D715/u1D438⧵/u1D43B,\nfor some/u1D450=/u1D450(/u1D45B,/u1D706,/u1D712), ifd/u1D4361(/u1D438,/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))≤̂ /u1D700<1.\nLet/u1D454∈ Lip/u1D450(ℝ/u1D45B)and define/u1D453∶=/u1D454◦/u1D713. If/u1D450is the constant in ( 5.20) corresponding to /u1D453, then by area formula\nand ( 5.20) we get\n/uni222B.dsp/u1D715∗/u1D438∩⧵/u1D43B/u⎪⎫007C.v⎞⎜/u1D454−/u1D450/u⎪⎫007C.v⎞⎜d/u1D45B−1≤/u1D436(/u1D45B,/u1D706)/uni222B.dsp/u1D715/u1D435/u1D706⧵/u1D43B/u⎪⎫007C.v⎞⎜/u1D453−/u1D450/u⎪⎫007C.v⎞⎜d/u1D45B−1≤/u1D436(/u1D45B,/u1D706)/uni222B.dsp/u1D435/u1D706/u⎪⎫007C.v⎞⎜∇/u1D453/u⎪⎫007C.v⎞⎜d/u1D465\n≤/u1D436(/u1D45B,/u1D706)/uni222B.dsp/u1D438/u⎪⎫007C.v⎞⎜∇/u1D454/u⎪⎫007C.v⎞⎜d/u1D465.\nTherefore, if ̂ /u1D700is small enough, we can apply Proposition 5.8with/u1D450/u1D447therein depending on /u1D45B,/u1D706only, and we get\n/u1D6FC/u1D706(/u1D438)≤̂ /u1D6FE(/u1D45B,/u1D706)/u1D437/u1D706(/u1D438).\n/square\n5.2. Proof of the first quantitative isoperimetric inequality. We are ready to prove the main quantitative isoperi-\nmetric inequality. Let us recall the following immediate re sult, completely analogous to [ Fus15 , Lemma 5.3].\nLemma 5.12. The standard bubble /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)is the unique solution, up to translations along /u1D715/u1D43B, of\nmin/b⎜⎞ce⎨eft.⎟2\n/u1D443/u1D706(/u1D439)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x∶/u1D439 ⊂ℝ/u1D45B⧵/u1D43B/b⎜⎞ce⎜⎫g⎧t.⎟2\n,\nfor anyΛ>/u1D45B.\nProof of Theorem 1.1.Let̂ /u1D6FE(/u1D45B,/u1D706)be the constant given by Corollary 5.11 and let/u1D6FF4,̃/u1D459be given by Corollary 4.9.\nBy Corollary 4.9it is sufficient to prove that there exists /u1D6FF∈ (0,/u1D6FF4)such that, if /u1D438is a Schwarz-symmetric set\ncontained in /u1D444̃/u1D459such that /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438)</u1D6FF, then/u1D6FC/u1D706(/u1D438)≤2̂ /u1D6FE(/u1D45B,/u1D706)√\n/u1D437/u1D706(/u1D438).\nWe argue by contradiction. Let {/u1D438/u1D457}/u1D457be a sequence of Schwarz-symmetric sets contained in /u1D444̃/u1D459such that\n/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜, with/u1D443/u1D706(/u1D438/u1D457)→/u1D443/u1D706(/u1D435/u1D706)and\n(5.21) /u1D6FC/u1D706(/u1D438/u1D457)>2̂ /u1D6FE(/u1D45B,/u1D706)/u⎪⎫221A.⎟1\n/u1D437/u1D706(/u1D438/u1D457).\nFor every/u1D457we consider a minimizer /u1D439/u1D457of the problem\n(5.22) min{/u1D443/u1D706(/u1D439)+/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D439)−/u1D6FC/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜+Λ/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜∶/u1D439Schwarz-symmetric contained in /u1D444̃/u1D459},\nforΛ>0to be chosen large. Up to subsequence, /u1D439/u1D457converges in /u1D43F1to a minimizer of /u1D439↦/u1D443/u1D706(/u1D439) +/u1D6FC/u1D706(/u1D439) +\nΛ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x, hence, taking Λ>/u1D45B, we have that /u1D439/u1D457converges to /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)by Lemma 5.12. Also, by comparison\nwith with/u1D438/u1D457, we have that /u1D443/u1D706(/u1D439/u1D457)→/u1D443/u1D706(/u1D435/u1D706).\nWe prove that /u1D439/u1D457is a local (Λ1,/u1D45F0)-minimizer in ℝ/u1D45B⧵/u1D43B, for some Λ1,/u1D45F0>0and/u1D457large. Let us consider a\nball/u1D435/u1D45F(/u1D465)⊂⊂ℝ/u1D45B⧵/u1D43B, with/u1D45F<min{/u1D45F0,/u1D451(/u1D465,/u1D715/u1D43B)}, and a set/u1D43Asuch that/u1D439/u1D457Δ/u1D43A⊂⊂/u1D435/u1D45F(/u1D465). Denoting by (⋅)∗the\nSchwarz symmetrization with respect to the /u1D45B-th axis and by /u1D44D∶=/u1D43A∩/u1D444̃/u1D459, we have\n/u1D443(/u1D439/u1D457,ℝ/u1D45B⧵/u1D43B)≤/u1D443(/u1D44D∗,ℝ/u1D45B⧵/u1D43B)+/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D44D∗)−/u1D6FC/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D439/u1D457)−/u1D6FC/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜+Λ[/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜]\n≤/u1D443(/u1D44D,ℝ/u1D45B⧵/u1D43B)+/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D44D∗)−/u1D6FC/u1D706(/u1D439/u1D457)/u⎪⎫007C.v⎞⎜+Λ/u⎪⎫007C.v⎞⎜/u1D44DΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜\n≤/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D44D∗)−/u1D6FC/u1D706(/u1D439/u1D457)/u⎪⎫007C.v⎞⎜+Λ/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜.28 GIULIO PASCALE AND MARCO POZZETTA\nFor/u1D45F0small enough and /u1D457sufficiently large we have that /u⎪⎫007C.v⎞⎜/u1D43A/u⎪⎫007C.v⎞⎜≥/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜≥/u1D450(/u1D45B,/u1D706)>0. Assume for instance that\n/u1D6FC/u1D706(/u1D44D∗)≥/u1D6FC/u1D706(/u1D439/u1D457)(the opposite case being symmetric), then\n/u1D6FC/u1D706(/u1D44D∗)−/u1D6FC/u1D706(/u1D439/u1D457)≤/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−1/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D44D∗Δ/u1D439/u1D457/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.v⎞⎜/u1D439/u1D457Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜+/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−1/u⎪⎫007C.v⎞⎜/u1D439/u1D457Δ/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜\n≤/u1D450(/u1D45B,/u1D706)/u⎪⎫007C.v⎞⎜/u1D44DΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜+/u1D450(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟1\n/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n+/u⎪⎫007C.v⎞⎜/u1D44DΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜\n≤/u1D450(/u1D45B,/u1D706)/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜.\nArguing analogously in case /u1D6FC/u1D706(/u1D44D∗)</u1D6FC/u1D706(/u1D439/u1D457), we deduce\n/u1D443(/u1D439/u1D457,ℝ/u1D45B⧵/u1D43B)≤/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+Λ1/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u1D457/u⎪⎫007C.v⎞⎜,\nfor someΛ1= (Λ,/u1D45B,/u1D706).\nBy Theorem A.3we know that /u1D715∗/u1D439/u1D457∩ {/u1D465/u1D45B>0}is a/u1D4361,1\n2manifold and /u1D715/u1D439/u1D457∩ {/u1D465/u1D45B>0}⧵/u1D715∗/u1D439/u1D457has Hausdorff\ndimension ≤/u1D45B−8. Since/u1D439/u1D457is Schwarz-symmetric, if there exists a point (/u1D45F/u1D717,/u1D461) ∈/u1D715/u1D439/u1D457∩{/u1D465/u1D45B>0}⧵/u1D715∗/u1D439/u1D457for some\n/u1D45F,/u1D461>0,/u1D717∈/u1D54A/u1D45B−2, then(/u1D45F/u1D717)uni2032(var,/u1D461) ∈/u1D715/u1D439/u1D457∩{/u1D465/u1D45B>0}⧵/u1D715∗/u1D439/u1D457for any/u1D717)uni2032(var∈/u1D54A/u1D45B−2. Hence/u1D715/u1D439/u1D457∩{/u1D465/u1D45B>0}⧵/u1D715∗/u1D439/u1D457⊂{/u1D461/u1D452/u1D45B∶\n/u1D461>0}. However by Theorem A.3for every/u1D700>0the set/u1D715/u1D439/u1D457∩{/u1D465/u1D45B≥/u1D700}converges to /u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜) ∩{/u1D465/u1D45B≥/u1D700}in\n/u1D4361,/u1D6FCfor any0</u1D6FC<1\n2. Also, for/u1D457large we can apply Corollary 4.4which implies that /u1D45B−1(/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461})≥/u1D434/u1D706\nfor a.e./u1D461∈ (0,/u1D447/u1D706), for some/u1D434/u1D706,/u1D447/u1D706>0depending on /u1D45B,/u1D706. Then points /u1D461/u1D452/u1D45Bfor/u1D461∈ (0,/u1D447/u1D706∕2)are points of density\n1for/u1D439/u1D457, hence they belong to the interior of /u1D439/u1D457. Therefore, for /u1D457large enough, /u1D715/u1D439/u1D457∩ {/u1D465/u1D45B>0}⧵/u1D715∗/u1D439/u1D457must be\nempty and/u1D715/u1D439/u1D457∩{/u1D465/u1D45B>0}is an axially symmetric hypersurface of class /u1D4361,1\n2.\nBy the minimality of the /u1D439/u1D457, (5.21) and Lemma 5.12 we observe that\n/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x+/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D6FC/u1D706(/u1D439/u1D457)−/u1D6FC/u1D706(/u1D438/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D443/u1D706(/u1D438/u1D457)\n≤/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))+/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))\n4̂ /u1D6FE2(/u1D45B,/u1D706)/u1D6FC2\n/u1D706(/u1D438/u1D457)≤/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x+/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))\n4̂ /u1D6FE2(/u1D45B,/u1D706)/u1D6FC2\n/u1D706(/u1D438/u1D457).(5.23)\nTherefore, we have that\n/u⎪⎫007C.v⎞⎜/u1D6FC/u1D706(/u1D439/u1D457)−/u1D6FC/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜≤/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))\n4̂ /u1D6FE2(/u1D45B,/u1D706)/u1D6FC2\n/u1D706(/u1D438/u1D457).\nSince/u1D6FC/u1D706(/u1D438/u1D457)→0we get that\n/u1D6FC/u1D706(/u1D439/u1D457)\n/u1D6FC/u1D706(/u1D438/u1D457)→1.\nLet{̂/u1D706/u1D457}⊂(0,∞)such that, setting ̃/u1D439/u1D457∶=̂/u1D706/u1D457/u1D439/u1D457, then/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Clearlŷ/u1D706/u1D457→1since/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜→/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Since\n/u1D443/u1D706(/u1D439/u1D457)→/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))andΛ>/u1D45B, for/u1D457sufficiently large we have /u1D443/u1D706(/u1D439/u1D457)<Λ/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜and\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D443/u1D706(̃/u1D439/u1D457)−/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B−1\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜= Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x.\nHence, by definition of ̂/u1D706/u1D457and by ( 5.23) we get\n/u1D443/u1D706(̃/u1D439/u1D457)≤/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n(5.23)\n≤/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))+/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))\n4̂ /u1D6FE2(/u1D45B,/u1D706)/u1D6FC2\n/u1D706(/u1D438/u1D457).(5.24)\nSince/u1D6FC/u1D706(/u1D439/u1D457)∕/u1D6FC/u1D706(/u1D438/u1D457)→1as/u1D457→∞we have/u1D6FC/u1D706(/u1D438/u1D457)2<2/u1D6FC/u1D706(̃/u1D439/u1D457)2for/u1D457sufficiently large. Hence from ( 5.24) we\nfinally obtain\n(5.25) /u1D6FC/u1D706(̃/u1D439/u1D457)>√\n2̂ /u1D6FE(/u1D45B,/u1D706)/u⎪⎫221A.⎟1\n/u1D437/u1D706(̃/u1D439/u1D457).\nFor/u1D461>0let\n/u1D711−\ñ/u1D439/u1D457(/u1D461) ∶=/b⎜⎞ce⎨eft.⎟4\nmin/u1D465∈/u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461}/b⎜⎞ce⎨eft.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u1D465−/u1D461/u1D452/u1D45B/u⎪⎫007C.x/u⎪⎫007C.x/b⎜⎞ce⎜⎫g⎧t.⎟1\nif/u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461}≠)uni2205(var,\n0 if /u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461} = )uni2205(var.\nbe the function measuring the distance of /u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461}from the/u1D45B-th axis, set to zero in case /u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B=/u1D461} = )uni2205(var .\nFor/u1D457large we can apply Corollary 4.4again to deduce that there exists /u1D447/u1D706,/u1D434/u1D706>0such that /u1D45B−1(̃/u1D439/u1D457∩ {/u1D465/u1D45B=\n/u1D461})≥/u1D434/u1D706for almost every /u1D461∈ (0,/u1D447/u1D706). Sincẽ/u1D439/u1D457is Schwarz-symmetric and its relative boundary in {/u1D465/u1D45B>0}is/u1D4361\nregular, then we can write that /u1D711−\ñ/u1D439/u1D457(/u1D461)≥/u1D434)uni2032(var\n/u1D706>0for/u1D457large and for any /u1D461∈ (0,/u1D447/u1D706).QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 29\nRecalling that /u1D439/u1D457is a local (Λ1,/u1D45F0)-minimizer, by Lemma A.2its boundary has generalized mean curvature\nbounded by Λ1for any/u1D457. Sincẽ/u1D439/u1D457=̂/u1D706/u1D457/u1D439/u1D457witĥ/u1D706/u1D457→1, then/u1D715̃/u1D439/u1D457∩(ℝ/u1D45B⧵/u1D43B)is a hypersurface of class /u1D4361,1\n2with\ngeneralized mean curvature /u1D43B/u1D715̃/u1D439/u1D457bounded by 2Λ1for any/u1D457. Observe that if we locally parametrize /u1D715̃/u1D439/u1D457∩(ℝ/u1D45B⧵/u1D43B)\nwith the graph of a function Φ/u1D457, thenΦ/u1D457weakly solves the mean curvature equation\ndiv⎛\n⎜\n⎜\n⎜⎝∇Φ/u1D457/u⎪⎫221A.⎟1\n1+/u⎪⎫007C.v⎞⎜∇Φ/u1D457/u⎪⎫007C.v⎞⎜2⎞\n⎟\n⎟\n⎟⎠=/u⎪⎫27E8.⎟2\n/u1D43B/u1D715̃/u1D439/u1D457,/u1D441Φ/u1D457/u⎪⎫27E9.⎟2\n,\nwhere/u1D441Φ/u1D457is the unit normal corresponding to Φ/u1D457and/u1D43B/u1D715̃/u1D439/u1D457is evaluated along the graph of Φ/u1D457. Since/u1D43B/u1D715̃/u1D439/u1D457is\nbounded, we get that Φ/u1D457is of class/u1D44A2,/u1D45Dfor every/u1D45D<∞(see [ GT01 ]).\nFix/u1D45D0∈/u1D715̃/u1D439/u1D457∩ {/u1D4651>0,0< /u1D465/u1D45B≤/u1D447/u1D706} ∩ span{/u1D4521,/u1D452/u1D45B}. Since/u1D715̃/u1D439/u1D457∩ (ℝ/u1D45B⧵/u1D43B)is/u1D4361,1\n2, there exists a curve\n/u1D6FE/u1D457= (/u1D6FC/u1D457,0,…,0,/u1D6FD/u1D457) ∶ (/u1D44E,/u1D44F)→span{/u1D4521,/u1D452/u1D45B}⧵/u1D43Bsuch that the map /u1D54A/u1D45B−2× (/u1D44E,/u1D44F) ∋ (/u1D717,/u1D461)↦(/u1D6FC/u1D457(/u1D461)/u1D717,/u1D6FD/u1D457(/u1D461))\nparametrizes /u1D715̃/u1D439/u1D457in a neighborhood of /u1D45D0. We claim that /u1D6FC/u1D457,/u1D6FD/u1D457∈/u1D44A2,/u1D45D, up to reparametrization.\nIndeed, we can also parametrize /u1D715̃/u1D439/u1D457in a neighborhood /u1D448of/u1D45D0as the graph of a function Φ/u1D457with domain contained\nin some affine hyperplane of the form /u1D45D0+/u1D449, and without loss of generality we can assume that either /u1D449= {/u1D4651= 0}\nor/u1D449= {/u1D465/u1D45B= 0} . If/u1D45B= 2, then the claimed regularity immediately follows from the r egularity of Φ/u1D457. Then assume\n/u1D45B≥3, and suppose for example that /u1D449= {/u1D4651= 0} . The image of the curve /u1D6FE/u1D457in/u1D448can be parametrized as the\ngraph of a function /u1D461↦(/u1D453(/u1D461),0,…,0,/u1D461). Writing as (/u1D465)uni2032(var,/u1D465/u1D45B) ∈/u1D45D0+/u1D449the variable for Φ/u1D457, the fact that the distance\nfrom the/u1D45B-th axis is constant on the intersection of /u1D715̃/u1D439/u1D457with any horizontal hyperplane yields the identity\n/⎝⎞⎜e⎪⎨eft.⎟2\nΦ/u1D457(/u1D465)uni2032(var,/u1D465/u1D45B)+dist/u1D465/u1D45B(/u1D45D0)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟22\n+/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜2=/u1D453(/u1D465/u1D45B)2,\nwheredist/u1D465/u1D45B(/u1D45D0)denotes distance of /u1D45D0from the/u1D45B-th axis. Since Φ/u1D457is of class/u1D4361and/u1D44A2,/u1D45Danddist/u1D465/u1D45B(/u1D45D0)>\n0because/u1D711−\ñ/u1D439/u1D457≥/u1D434)uni2032(var\n/u1D706>0, inverting the above identity we find that /u1D453is of class/u1D44A2,/u1D45D, hence so is /u1D6FE/u1D457, up to\nreparametrization. In case /u1D449= {/u1D465/u1D45B= 0} , the observation follows analogously relating Φ/u1D457with a parametrization\nfor/u1D6FE/u1D457.\nWe further observe that, for /u1D6FC/u1D457,/u1D6FD/u1D457∶ (/u1D44E,/u1D44F)→(0,∞)as above, since /u1D6FC/u1D457,/u1D6FD/u1D457are of class/u1D44A2,/u1D45D\nloc, up to reparametriza-\ntion by arclength we can apply Lemma A.4to get that\n/u1D43B/u1D715̃/u1D439/u1D457/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x(/u1D6FC/u1D457(/u1D461)/u1D717,/u1D6FD/u1D457(/u1D461))=/⎝⎞⎜e⎪⎨eft.⎟4/u⎪⎫27E8.⎟2\n/u1D458/u1D6FE/u1D457,/u1D708/u⎪⎫27E9.⎟2\n−(/u1D45B−2)/u1D6FD)uni2032(var\n/u1D457\n/u1D6FC/u1D457/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n(−/u1D6FD)uni2032(var\n/u1D457/u1D717,/u1D6FC)uni2032(var\n/u1D457),\nin the notation of Lemma A.4. Recalling that /u1D711−\ñ/u1D439/u1D457≥/u1D434)uni2032(var\n/u1D706>0on(0,/u1D447/u1D706), we have that /u⎪⎫007C.v⎞⎜/u1D6FC/u1D457/u⎪⎫007C.v⎞⎜≥/u1D434)uni2032(var\n/u1D706and thus\n(5.26) /u⎪⎫007C.v⎞⎜/u1D458/u1D6FE/u1D457/u⎪⎫007C.v⎞⎜≤2Λ1+/u1D45B−2\n/u1D434)uni2032(var\n/u1D706.\nObserve that the upper bound in ( 5.26) is independent of /u1D457and of the initially chosen point /u1D45D0.\nFix now/u1D45E0∈/u1D715̃/u1D439/u1D457∩{/u1D4651>0,/u1D465/u1D45B=/u1D447/u1D706}∩span{/u1D4521,/u1D452/u1D45B}, let/u1D6FE0\n/u1D457∶ [0,/u1D4590)→span{/u1D4521,/u1D452/u1D45B}be part of a curve defined\nas before, parametrized by arclength, such that/u⎪⎫27E8.⎟2\n/u1D6FE0\n/u1D457(/u1D461),/u1D452/u1D45B/u⎪⎫27E9.⎟2\n≤/u1D447/u1D706for any/u1D461. Iflim/u1D461→/u1D459−\n0/u1D6FE0\n/u1D457(/u1D461) ∉/u1D715/u1D43B, the curve can\nbe extended to a longer one, parametrized by arclength, by jo ining/u1D6FE0\n/u1D457with a curve defined as before for the choice\n/u1D45D0= lim/u1D461→/u1D459−\n0/u1D6FE0\n/u1D457(/u1D461). Hence we can consider /u1D70E/u1D457∶ [0,/u1D43F/u1D457)→span{/u1D4521,/u1D452/u1D45B}the maximal extension of /u1D6FE0\n/u1D457parametrized\nby arclength that parametrizes ̃/u1D439/u1D457∩{/u1D4651>0,0</u1D465/u1D45B≤/u1D447/u1D706}∩span{/u1D4521,/u1D452/u1D45B}. Since the perimeter /u1D443(̃/u1D439/u1D457,ℝ/u1D45B⧵/u1D43B)is\nuniformly bounded, then sup/u1D457/u1D43F/u1D457<+∞. Obviously lim/u1D461→/u1D43F/u1D457/u1D70E/u1D457(/u1D461) ∈/u1D715/u1D43B, for otherwise the curve could be further\nextended. By construction, the uniform bound in ( 5.26) holds pointwise for the curvature of /u1D70E/u1D457. Therefore/u1D70E/u1D457can\nbe extended to a curve /u1D6FE/u1D457∶ [0,/u1D43F/u1D457]→span{/u1D4521,/u1D452/u1D45B}such that\n(5.27) ‖/u1D6FE/u1D457‖/u1D4361,1([0,/u1D43F/u1D457])≤/u1D436,\nwith/u1D436independent of /u1D457, depending only on /u1D45B,/u1D706,̃/u1D459and the upper bound on the curvature given by ( 5.26).\nUp to a subsequence, since ̃/u1D439/u1D457→/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)and we already know that for every /u1D700 >0the set/u1D715̃/u1D439/u1D457∩{/u1D465/u1D45B≥/u1D700}\nconverges to /u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)∩{/u1D465/u1D45B≥/u1D700}in/u1D4361,/u1D6FCfor any0</u1D6FC <1\n2, the bound ( 5.27) implies that ̃/u1D439/u1D457converges in /u1D4361\nsense to/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)in the sense of Definition 5.10. Hence, by Corollary 5.11, for/u1D457sufficiently large there holds\n/u1D6FC2\n/u1D706(̃/u1D439/u1D457)≤̂ /u1D6FE(/u1D45B,/u1D706)/u1D437/u1D706(̃/u1D439/u1D457),30 GIULIO PASCALE AND MARCO POZZETTA\nin contradiction with ( 5.25). /square\n6. S ECOND QUANTITATIVE ISOPERIMETRIC INEQUALITY\nWe will need the following technical lemma, proving that if t he energy/u1D443/u1D706(/u1D438/u1D456)a sequence of sets /u1D438/u1D456converges\nto the energy of the limit, then the sequence strictly conver ges in the sense of /u1D435/u1D449functions. The proof essentially\nfollows by analyzing the equality case in the Reshetnyak low er semicontinuity theorem, see [ AFP00 , Theorem\n2.38].\nLemma 6.1. Let{/u1D438/u1D456}/u1D456∈ℕbe a sequence of sets of finite perimeter in ℝ/u1D45B⧵/u1D43Bsuch that/u1D438/u1D456→/u1D438in/u1D43F1, for some set\nof finite perimeter /u1D438with/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜<+∞. If/u1D443/u1D706(/u1D438/u1D456)→/u1D443/u1D706(/u1D438), then\nlim\n/u1D456/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵/u1D43B) =/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B),lim\n/u1D456/u1D45B−1(/u1D715∗/u1D438/u1D456∩/u1D715/u1D43B) =/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B).\nProof. Let/u1D453(/u1D463) ∶=/u⎪⎫007C.v⎞⎜/u1D463/u⎪⎫007C.v⎞⎜−/u1D706⟨/u1D452/u1D45B,/u1D463⟩, for any/u1D463∈ℝ/u1D45B, and let/u1D708/u1D456∶=/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜⊗/u1D6FF/u1D708/u1D438/u1D456be a measure on ℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1.\nSince/u⎪⎫007C.v⎞⎜/u1D708/u1D456/u⎪⎫007C.v⎞⎜(ℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1)≤/u1D443(/u1D438/u1D456)≤2/u1D443/u1D706(/u1D438/u1D456)∕(1−/u1D706)by Corollary 2.4, up to subsequence, /u1D708/u1D456weakly* converges to\na finite measure /u1D708. Up to subsequence, also /u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜weakly* converges to a finite measure /u1D707onℝ/u1D45B⧵/u1D43B. Denoting\nby/u1D70B∶ℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1→ℝ/u1D45B⧵/u1D43Bthe natural projection, we have /u1D70B♯/u1D708/u1D456=/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜→/u1D707=/u1D70B♯/u1D708. Moreover,/u1D707≥/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜\nby lower semicontinuity. By the disintegration theorem [ AFP00 , Theorem 2.28], we can write /u1D708=/u1D707⊗/u1D708/u1D465, for a\n/u1D707-measurable map ℝ/u1D45B⧵/u1D43B∋/u1D465↦/u1D708/u1D465, where/u1D708/u1D465is a probability measure on /u1D54A/u1D45B−1. Analogously to [ AFP00 , Eq.\n(2.30)], we observe that\n(6.1)/uni222B.dsp/u1D54A/u1D45B−1/u1D463d/u1D708/u1D465(/u1D463) =/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜\n/u1D707(/u1D465),\nat/u1D707-a.e./u1D465∈ℝ/u1D45B⧵/u1D43B. Indeed, for any continuous function /u1D454withspt(/u1D454)⊂⊂ℝ/u1D45B⧵/u1D43Bwe find\n/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D454(/u1D465)/uni222B.dsp/u1D54A/u1D45B−1/u1D463d/u1D708/u1D465(/u1D463)d/u1D707(/u1D465) =/uni222B.dspℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1/u1D454(/u1D465)/u1D463d/u1D708(/u1D465,/u1D463) = lim\n/u1D456/uni222B.dspℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1/u1D454(/u1D465)/u1D463d/u1D708/u1D456(/u1D465,/u1D463)\n= −lim\n/u1D456/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D454(/u1D465)d/u1D437/u1D712/u1D438/u1D456(/u1D465) = −/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D454(/u1D465)d/u1D437/u1D712/u1D438(/u1D465) =/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D454(/u1D465)/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜\n/u1D707(/u1D465)d/u1D707(/u1D465).\nSince/u1D453is nonnegative, convex and continuous, by Remark 2.1we find\nlim\n/u1D456/u1D443/u1D706(/u1D438/u1D456) = lim\n/u1D456/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453(/u1D708/u1D438/u1D456)d/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜= lim\n/u1D456/uni222B.dspℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1/u1D453(/u1D463)d/u1D708/u1D456(/u1D465,/u1D463)≥/uni222B.dspℝ/u1D45B⧵/u1D43B×/u1D54A/u1D45B−1/u1D453(/u1D463)d/u1D708(/u1D465,/u1D463)\n=/uni222B.dspℝ/u1D45B⧵/u1D43B/uni222B.dsp/u1D54A/u1D45B−1/u1D453(/u1D463)d/u1D708/u1D465(/u1D463)d/u1D707(/u1D465)≥/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp/u1D54A/u1D45B−1/u1D463d/u1D708/u1D465(/u1D463)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D707(/u1D465)\n(6.1)=/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453/⎝⎞⎜e⎪⎨eft.⎟3\n/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜\n/u1D707(/u1D465)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D707(/u1D465) =/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453(/u1D708/u1D438(/u1D465))d/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜(/u1D465) =/u1D443/u1D706(/u1D438),(6.2)\nwhere in the second inequality we applied Jensen inequality , and where the last equality follows since /u1D453is positively\n1-homogeneous. Since lim/u1D456/u1D443/u1D706(/u1D438/u1D456) =/u1D443/u1D706(/u1D438)by assumption and since /u1D453is not affine, equality in Jensen inequality\nimplies that the identity map /u1D54A/u1D45B−1∋/u1D463↦/u1D463is constant/u1D708/u1D465-a.e., for/u1D707-a.e./u1D465∈ℝ/u1D45B⧵/u1D43B. This means that /u1D708/u1D465=/u1D6FF/u1D463/u1D465\nfor some/u1D463/u1D465∈/u1D54A/u1D45B−1for/u1D707-a.e./u1D465∈ℝ/u1D45B⧵/u1D43B. Hence ( 6.1) implies\n/u1D463/u1D465=/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜\n/u1D707(/u1D465),\nat/u1D707-a.e./u1D465∈ℝ/u1D45B⧵/u1D43B, and since /u⎪⎫007C.v⎞⎜/u1D463/u1D465/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D708/u1D438(/u1D465)/u⎪⎫007C.v⎞⎜= 1, then/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜∕/u1D707(/u1D465) = 1 at/u1D707-a.e./u1D465∈ℝ/u1D45B⧵/u1D43B, and/u1D463/u1D465=/u1D708/u1D438(/u1D465)\n/u1D707-almost everywhere. Inserting in ( 6.2) we deduce\n/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D708/u1D438(/u1D465)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\nd/u1D707(/u1D465) =/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453/⎝⎞⎜e⎪⎨eft.⎟3\n/uni222B.dsp/u1D54A/u1D45B−1/u1D463d/u1D708/u1D465(/u1D463)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3\nd/u1D707(/u1D465) =/uni222B.dspℝ/u1D45B⧵/u1D43B/u1D453(/u1D708/u1D438(/u1D465))d/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜(/u1D465).\nSince/u1D453(/u1D708/u1D438(/u1D465))>0and/u1D707≥/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜, we deduce that /u1D707=/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜, and then /u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜weakly* converges to /u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜.\nWe can now fix an increasing sequence of Lipschitz bounded ope n setsΩ/u1D457⊂⊂ℝ/u1D45B⧵/u1D43Bsuch that∪/u1D457Ω/u1D457=ℝ/u1D45B⧵/u1D43B\nand/u1D443(/u1D438/u1D456,/u1D715Ω/u1D457) =/u1D443(/u1D438,/u1D715Ω/u1D457) = 0 for every/u1D456,/u1D457. Hencelim/u1D456/u1D443(/u1D438/u1D456,Ω/u1D457) =/u1D443(/u1D438,Ω/u1D457)for any/u1D457. Moreover\n/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵(/u1D43B∪Ω/u1D457))≤1\n1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎨eft.⎟4\n/u1D443/u1D706(/u1D438/u1D456)−/uni222B.dspΩ/u1D457/u1D453(/u1D708/u1D438/u1D456)d/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u1D456/u⎪⎫007C.v⎞⎜/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4\n,QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 31\nfor any/u1D456,/u1D457. Applying Reshetnyak continuity theorem [ AFP00 , Theorem 2.39] on Ω/u1D457we get\nlimsup\n/u1D456/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵(/u1D43B∪Ω/u1D457))≤1\n1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜/uni222B.dspℝ/u1D45B⧵(/u1D43B∪Ω/u1D457)/u1D453(/u1D708/u1D438)d/u⎪⎫007C.v⎞⎜/u1D437/u1D712/u1D438/u⎪⎫007C.v⎞⎜≤1+/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜\n1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜/u1D443(/u1D438,ℝ/u1D45B⧵(/u1D43B∪Ω/u1D457)),\nfor any/u1D457. Therefore\nlimsup\n/u1D456/u1D443(/u1D438/u1D456,ℝ/u1D45B⧵/u1D43B)≤/u1D443(/u1D438,Ω/u1D457)+1+/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜\n1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜/u1D443(/u1D438,ℝ/u1D45B⧵(/u1D43B∪Ω/u1D457)),\nfor any/u1D457. Letting/u1D457→∞, the proof follows. /square\nWe will also exploit the concept of (/u1D43E,/u1D45F0)-quasiminimal set.\nDefinition 6.2. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a set of finite perimeter with finite measure, and let /u1D43E≥1,/u1D45F0>0. We say\nthat/u1D438is a(/u1D43E,/u1D45F0)-quasiminimal set (relatively in ℝ/u1D45B⧵/u1D43B) if\n/u1D443(/u1D438,ℝ/u1D45B⧵/u1D43B)≤/u1D43E/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B),\nfor any/u1D439 ⊂ℝ/u1D45B⧵/u1D43Bsuch that/u1D438Δ/u1D439 ⊂⊂/u1D435/u1D45F(/u1D465), for some ball /u1D435/u1D45F(/u1D465)⊂ℝ/u1D45Bwith/u1D45F≤/u1D45F0and/u1D465∈ {/u1D465/u1D45B≥0}.\nQuasiminimal sets have well-known topological regularity properties following from uniform density estimates\nat boundary points. We recall these facts in the following st atement. The proof follows, for example, by repeatedly\napplying [ Kin+13 , Theorem 4.2] with /u1D44B= {/u1D465/u1D45B≥0}in domains Ω =/u1D44B∩/u1D435/u1D45F0(/u1D465)for/u1D465∈/u1D44B, in the notation of\n[Kin+13 , Theorem 4.2]. Observe that in [ Kin+13 ], the perimeter functional coincides with the relative per imeter in\nℝ/u1D45B⧵/u1D43B, hence the definition of quasiminimal set in [ Kin+13 , Definition 3.1] coincides with our Definition 6.2. Al-\nternatively, the proof follows by adapting the proof of [ Mag12 , Theorem 21.11] working with (/u1D43E,/u1D45F0)-quasiminimal\nsets instead of (Λ,/u1D45F0)-minimizers.\nTheorem 6.3. Let/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bbe a(/u1D43E,/u1D45F0)-quasiminimal set, for some /u1D43E≥1,/u1D45F0>0. Then there exist /u1D45A=\n/u1D45A(/u1D45B,/u1D43E,/u1D45F0) ∈ (0,1)and/u1D45F)uni2032(var\n0=/u1D45F)uni2032(var\n0(/u1D45B,/u1D43E,/u1D45F0) ∈ (0,/u1D45F0]such that\n/u1D45A≤/u⎪⎫007C.v⎞⎜/u1D438∩/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜\n/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)⧵/u1D43B/u⎪⎫007C.v⎞⎜≤1−/u1D45A∀/u1D465∈/u1D715/u1D438⧵/u1D43B,∀/u1D45F∈ (0,/u1D45F)uni2032(var\n0].\nIn particular the set /u1D438(1)of points of density 1for/u1D438is an open representative for /u1D438.\nWe will identify a (/u1D43E,/u1D45F0)-quasiminimal set with its open representative /u1D438(1). In order to prove Theorem 1.2we\nneed two preparatory lemmas.\nLemma 6.4. For any/u1D43E≥1,/u1D45F0>0there exist/u1D6FF6,/u1D4365,/u1D4366>0depending on /u1D45B,/u1D706,/u1D43E,/u1D45F0such that the following\nholds. If/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bis a bounded (/u1D43E,/u1D45F0)-quasiminimal set with /u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438)≤/u1D6FF6, then\n(6.3) /u1D451/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D715/u1D438⧵/u1D43B,/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≤/u1D4365/u1D6FC/u1D706(/u1D438)1\n/u1D45B,\nwhere/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,/u1D465)is a bubble realizing the asymmetry of /u1D438. Moreover\n(6.4) /u1D6FD/u1D706(/u1D438)≤/u1D4366/u1D437/u1D706(/u1D438)1\n2/u1D45B.\nProof. Up to translation, we can assume that /u1D465= 0. Also, letting /u1D45A,/u1D45F)uni2032(var\n0be given by Theorem 6.3, up to decreasing\n/u1D45F0we can assume that /u1D45F0=/u1D45F)uni2032(var\n0. Let/u1D45D∈/u1D715/u1D438⧵/u1D43Bbe such that\n/u1D4510∶= dist/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D45D,/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n= max/b⎜⎞ce⎨eft.⎟2\ndist/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D466,/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n∶/u1D466∈/u1D715/u1D438⧵/u1D43B/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nHence/u1D435/u1D4510(/u1D45D)∩/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B= )uni2205(var. Then either /u1D435/u1D4510(/u1D45D)⧵/u1D43B ⊂/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)or/u1D435/u1D4510(/u1D45D)⧵/u1D43B ⊂ℝ/u1D45B⧵(/u1D43B∪/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)).\nIn the first case Theorem 6.3implies\n/u1D45A/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)⧵/u1D43B/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)⧵(/u1D43B∪/u1D438)/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D438/u⎪⎫007C.v⎞⎜=1\n2/u1D6FC/u1D706(/u1D438) ∀/u1D45F∈ (0,min{/u1D4510,/u1D45F0}),\nwhile in the second case Theorem 6.3implies\n/u1D45A/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)⧵/u1D43B/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D435/u1D45F(/u1D465)∩/u1D438/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D438⧵/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)/u⎪⎫007C.v⎞⎜=1\n2/u1D6FC/u1D706(/u1D438) ∀/u1D45F∈ (0,min{/u1D4510,/u1D45F0}).\nSince/u⎪⎫007C.v⎞⎜/u1D435/u1D445(/u1D465)⧵/u1D43B/u⎪⎫007C.v⎞⎜≥/u1D436/u1D45F/u1D45B, thenmin{/u1D4510,/u1D45F0}/u1D45B≤/u1D436/u1D6FC/u1D706(/u1D438), for/u1D436=/u1D436(/u1D45B,/u1D706,/u1D43E,/u1D45F0). So by Corollary 4.3, choosing/u1D6FF6\nsmall enough we have that /u1D6FC/u1D706(/u1D438)is so small that min{/u1D4510,/u1D45F0} =/u1D4510and then\n/u1D451/u1D45B\n0≤/u1D436/u1D6FC/u1D706(/u1D438).32 GIULIO PASCALE AND MARCO POZZETTA\nSince density estimates as those in Theorem 6.3hold for/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜), repeating the above argument exchanging the\nroles of/u1D438and/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜), (6.3) follows.\nFrom ( 6.3), we deduce that\n/u1D715/u1D438⧵/u1D43B ⊂/b⎜⎞ce⎨eft.⎟2\n/u1D466∈ {/u1D465/u1D45B≥0} ∶ dist/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D466,/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)⧵/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≤/u1D4365/u1D6FC/u1D706(/u1D438)1\n/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nHence\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715∗/u1D438Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n≤ /u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟2/b⎜⎞ce⎨eft.⎟2\n(/u1D465)uni2032(var,0) ∈ℝ/u1D45B∶ (1−/u1D7062)1\n2−/u1D4365/u1D6FC/u1D706(/u1D438)1\n/u1D45B≤/u⎪⎫007C.v⎞⎜/u1D465)uni2032(var/u⎪⎫007C.v⎞⎜≤(1−/u1D7062)1\n2+/u1D4365/u1D6FC/u1D706(/u1D438)1\n/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟2/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≤/u1D436/u1D6FC/u1D706(/u1D438)1\n/u1D45B≤/u1D436/u1D437/u1D706(/u1D438)1\n2/u1D45B,\nfor some/u1D436=/u1D436(/u1D45B,/u1D706,/u1D43E,/u1D45F0), where we used Theorem 1.1in the last inequality. Hence ( 6.4) follows. /square\nLemma 6.5. There exists /u1D6FF7,/u1D4367>0depending on /u1D45B,/u1D706such that for any measurable set /u1D438 ⊂ℝ/u1D45B⧵/u1D43Bwith\n/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438)≤/u1D6FF7there holds\n(6.5) /u1D6FD/u1D706(/u1D438)≤/u1D4367/u1D437/u1D706(/u1D438)1\n2/u1D45B.\nProof. FixΛ>/u1D45B. Let/u1D444⊂ℝ/u1D45Bbe a large cube whose interior contains the closure of /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜), and let/u1D439 ⊂/u1D444⧵/u1D43B\nbe such that /u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕2≤/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜≤2/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Let/u1D43A ⊂ℝ/u1D45B⧵/u1D43Bbe such that /u1D43AΔ/u1D439 ⊂⊂ /u1D435/u1D45F0(/u1D465), for/u1D465∈ {/u1D465/u1D45B≥0}and\n/u1D45F0∈ (0,1)to be chosen small. Let /u1D44D∶=/u1D43A∩/u1D444. Observe that\n/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D44DΔ/u1D439/u⎪⎫007C.v⎞⎜≤/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u⎪⎫007C.v⎞⎜1\n/u1D45B/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B≤/u1D7141\n/u1D45B\n/u1D45B/u1D45F/⎝⎞⎜e⎪⎨eft.⎟2\n/u⎪⎫007C.v⎞⎜/u1D43A/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B+/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜/u1D45B−1\n/u1D45B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2\n≤/u1D436(/u1D45B)/u1D45F0(/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B)),(6.6)\nwhere in the last inequality we used the relative isoperimet ric inequality in a half-space (see [ CGR07 ] for the sharp\ninequality). Let /u1D466,/u1D467∈/u1D715/u1D43Bbe such that\n/u1D6FD/u1D706(/u1D439) =/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D439Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1, /u1D6FD/u1D706(/u1D44D) =/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D44DΔ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D467)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D467)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1,\nObserve that /u1D466,/u1D467exist since/u1D439,/u1D44D ⊂/u1D444 . Suppose, for instance, that /u1D6FD/u1D706(/u1D44D)≥/u1D6FD/u1D706(/u1D439). Then\n/u1D6FD/u1D706(/u1D44D)−/u1D6FD/u1D706(/u1D439)≤/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D44DΔ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1−/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D439Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n≤/u1D45B−1(/u1D715∗/u1D44DΔ/u1D715∗/u1D439∩/u1D715/u1D43B)+/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D439Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n+/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1 +\n−/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D439Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n=/u1D45B−1(/u1D715∗/u1D44DΔ/u1D715∗/u1D439∩/u1D715/u1D43B)\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1+/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1−/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1 +\n+/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D439Δ/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎨eft.⎟4\n1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1−1\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/⎝⎞⎜e⎪⎜⎫g⎧t.⎟4(6.7)\nBy a trace inequality [ AFP00 , Theorem 3.87] we estimate\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715∗/u1D44DΔ/u1D715∗/u1D439∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n≤/u1D436(/u1D45B)(/u⎪⎫007C.v⎞⎜/u1D44DΔ/u1D439/u⎪⎫007C.v⎞⎜+/u1D443(/u1D44D,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B))\n≤/u1D436(/u1D45B)(/u⎪⎫007C.v⎞⎜/u1D43AΔ/u1D439/u⎪⎫007C.v⎞⎜+/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B))\n≤/u1D436(/u1D45B)(/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B)),\nwhere the last inequality follows as in ( 6.6), and/u1D436denotes a constant depending on suitable parameters that ch anges\nfrom line to line. For /u1D45F0small, depending only on /u1D45B,/u1D706, we can ensure that\n/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n≥/u1D436(/u1D45B,/u1D706)>0.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 33\nFinally\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n−/u1D45B−1/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜,/u1D466)∩/u1D715/u1D43B/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D43F/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u⎪⎫007C.v⎞⎜/u⎪⎫007C.v⎞⎜\n(6.6)\n≤/u1D436(/u1D45B,/u1D706)(/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B)),\nfor a suitable Lipschitz constant /u1D43F=/u1D43F(/u1D45B,/u1D706). Therefore ( 6.7) becomes\n/u1D6FD/u1D706(/u1D44D)−/u1D6FD/u1D706(/u1D439)≤/u1D436(/u1D45B,/u1D706)(/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B)).\nIn case/u1D6FD/u1D706(/u1D44D)</u1D6FD/u1D706(/u1D439), the very same argument leads to an analogous estimate. Henc e\n(6.8) /u⎪⎫007C.x/u⎪⎫007C.x/u1D6FD/u1D706(/u1D44D)−/u1D6FD/u1D706(/u1D439)/u⎪⎫007C.x/u⎪⎫007C.x≤/uni0303.s1/u1D436(/u1D45B,/u1D706)(/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439,ℝ/u1D45B⧵/u1D43B)).\nUp to taking a smaller /u1D45F0, we fix/u1D45F0=/u1D45F0(/u1D45B,/u1D706,Λ) ∈ (0,1)and/u1D7000=/u1D7000(/u1D45B,/u1D706,Λ) ∈ (0,1)such that 1 −/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜−\n/u1D7000/uni0303.s1/u1D436(/u1D45B,/u1D706)−Λ/u1D436(/u1D45B)/u1D45F0>0, and we define\n/u1D43E∶=1+/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜+/u1D7000/uni0303.s1/u1D436(/u1D45B,/u1D706)+Λ/u1D436(/u1D45B)/u1D45F0\n1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜−/u1D7000/uni0303.s1/u1D436(/u1D45B,/u1D706)−Λ/u1D436(/u1D45B)/u1D45F0>1.\nLet/u1D6FF6,/u1D4366be given by Lemma 6.4corresponding to the parameters /u1D43E,/u1D45F0∕2. We want to prove that if /u1D6FF7is\nsufficiently small, then\n/u1D6FD/u1D706(/u1D438)≤2/u1D4366/u1D437/u1D706(/u1D438)1\n2/u1D45B.\nWe argue by contradiction assuming that there exist sets /u1D438/u1D457⊂ℝ/u1D45B⧵/u1D43Bwith/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D437/u1D706(/u1D438/u1D457)≤1∕/u1D457such\nthat\n(6.9) /u1D6FD/u1D706(/u1D438/u1D457)>2/u1D4366/u1D437/u1D706(/u1D438/u1D457)1\n2/u1D45B,\nfor any/u1D457. Up to translation, /u1D438/u1D457→/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)and/u1D443/u1D706(/u1D438/u1D457)→/u1D443/u1D706(/u1D435/u1D706). Since the trace operator is continuous\nwith respect to strict convergence of /u1D435/u1D449functions, see [ AFP00 , Theorem 3.88], by Lemma 6.1we deduce that\n/u1D6FD/u1D706(/u1D438/u1D457)→0.\nLet/u1D439/u1D457be a minimizer of the problem\n(6.10) min/b⎜⎞ce⎨eft.⎟2\n/u1D443/u1D706(/u1D438)+/u1D7000/u⎪⎫007C.v⎞⎜/u1D6FD/u1D706(/u1D438)−/u1D6FD/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x∶/u1D438 ⊂/u1D444/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\nBy Corollary 3.11, up to subsequence /u1D439/u1D457converges to a limit set /u1D439in/u1D43F1. If by contradiction /u1D6FD/u1D457(/u1D439/u1D457) ̸→0, by\nLemma 5.12 for large/u1D457we would have that\n/u1D443/u1D706(/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜))+/u1D7000/u1D6FD/u1D706(/u1D438/u1D457)</u1D443/u1D706(/u1D439/u1D457)+/u1D7000/u⎪⎫007C.v⎞⎜/u1D6FD/u1D706(/u1D439/u1D457)−/u1D6FD/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D438/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x,\ncontradicting the minimality of /u1D439/u1D457. Hence/u1D6FD/u1D706(/u1D439/u1D457)→0. It follows that, up to translation, /u1D439/u1D457converges to /u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)\nin/u1D43F1. Comparing with /u1D438/u1D457, we also see that /u1D443/u1D706(/u1D439/u1D457)→/u1D443/u1D706(/u1D435/u1D706).\nWe want to show that /u1D439/u1D457is(/u1D43E,/u1D45F0)-quasiminimal for /u1D457large. Indeed, /u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜∕2≤/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜≤2/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜for/u1D457large. Hence\nwe can apply ( 6.6) and ( 6.8) with/u1D439=/u1D439/u1D457. Letting/u1D43A ⊂ℝ/u1D45B⧵/u1D43Bsuch that/u1D43AΔ/u1D439 ⊂⊂ /u1D435/u1D45F0(/u1D465), for/u1D465∈ {/u1D465/u1D45B≥0},\ndenoting/u1D44D∶=/u1D43A∩/u1D444, by minimality of /u1D439/u1D457for (6.10) we find\n(1−/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜)/u1D443(/u1D439/u1D457,ℝ/u1D45B⧵/u1D43B)≤(1+/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜)/u1D443(/u1D44D,ℝ/u1D45B⧵/u1D43B)+/u1D7000/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D6FD/u1D706(/u1D44D)−/u1D6FD/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D44D/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x\n≤(1+/u⎪⎫007C.v⎞⎜/u1D706/u⎪⎫007C.v⎞⎜)/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/⎝⎞⎜e⎪⎨eft.⎟2\n/u1D7000/uni0303.s1/u1D436(/u1D45B,/u1D706)+Λ/u1D436(/u1D45B)/u1D45F0/⎝⎞⎜e⎪⎜⎫g⎧t.⎟2/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D443(/u1D43A,ℝ/u1D45B⧵/u1D43B)+/u1D443(/u1D439/u1D457,ℝ/u1D45B⧵/u1D43B)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n,\nproving that /u1D439/u1D457is(/u1D43E,/u1D45F0)-quasiminimal.\nBy minimality of /u1D439/u1D457, we have\n/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x+/u1D7000/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D6FD/u1D706(/u1D439/u1D457)−/u1D6FD/u1D706(/u1D438/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D443/u1D706(/u1D438/u1D457)\n≤/u1D443/u1D706(/u1D435/u1D706)+/u1D443/u1D706(/u1D435/u1D706)\n(2/u1D4366)2/u1D45B/u1D6FD2/u1D45B\n/u1D706(/u1D438/u1D457)≤/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x+/u1D443/u1D706(/u1D435/u1D706)\n(2/u1D4366)2/u1D45B/u1D6FD2/u1D45B\n/u1D706(/u1D438/u1D457)(6.11)\nTherefore\n/u⎪⎫007C.v⎞⎜/u1D6FD/u1D706(/u1D439/u1D457)−/u1D6FD/u1D706(/u1D438/u1D457)/u⎪⎫007C.v⎞⎜≤/u1D443/u1D706(/u1D435/u1D706)\n/u1D7000(2/u1D4366)2/u1D45B/u1D6FD2/u1D45B\n/u1D706(/u1D438/u1D457),\nand then/u1D6FD/u1D706(/u1D439/u1D457)\n/u1D6FD/u1D706(/u1D438/u1D457)→1.34 GIULIO PASCALE AND MARCO POZZETTA\nNext we select {̂/u1D706/u1D457}⊂(0,∞)such that, setting ̃/u1D439/u1D457∶=̂/u1D706/u1D457/u1D439/u1D457, then/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜. Clearlŷ/u1D706/u1D457→1since/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜→/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜.\nSince/u1D443/u1D706(/u1D439/u1D457)→/u1D443/u1D706(/u1D435/u1D706))andΛ>/u1D45B, for/u1D457sufficiently large we have /u1D443/u1D706(/u1D439/u1D457)<Λ/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜and\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D443/u1D706(̃/u1D439/u1D457)−/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B−1\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤/u1D443/u1D706(/u1D439/u1D457)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x̂/u1D706/u1D45B\n/u1D457−1/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜= Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x.\nHence, by definition of ̂/u1D706/u1D457and by ( 6.11) we get\n/u1D443/u1D706(̃/u1D439/u1D457)≤/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜̃/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x=/u1D443/u1D706(/u1D439/u1D457)+Λ/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.v⎞⎜/u1D439/u1D457/u⎪⎫007C.v⎞⎜−/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x(6.11)\n≤/u1D443/u1D706(/u1D435/u1D706)+/u1D443/u1D706(/u1D435/u1D706)\n(2/u1D4366)2/u1D45B/u1D6FD2/u1D45B\n/u1D706(/u1D438/u1D457). (6.12)\nSince/u1D6FD/u1D706(/u1D439/u1D457)∕/u1D6FD/u1D706(/u1D438/u1D457)→1as/u1D457→∞and/u1D6FD/u1D706is scale invariant, we have /u1D6FD/u1D706(/u1D438/u1D457)2/u1D45B<2/u1D6FD/u1D706(̃/u1D439/u1D457)2/u1D45Bfor/u1D457sufficiently\nlarge. Hence from ( 6.12) we obtain\n/u1D6FD/u1D706(̃/u1D439/u1D457)2/u1D45B≥22/u1D45B−1/u1D4362/u1D45B\n6/u1D437/u1D706(̃/u1D439/u1D457),\nthat is/u1D6FD/u1D706(̃/u1D439/u1D457)≥21−1\n2/u1D45B/u1D4366/u1D437/u1D706(̃/u1D439/u1D457)1\n2/u1D45B. On the other hand, for /u1D457large,̃/u1D439/u1D457is(/u1D43E,̂/u1D706/u1D457/u1D45F0)-quasiminimal. As ̂/u1D706/u1D457→1, then\ñ/u1D439/u1D457is(/u1D43E,/u1D45F0∕2)-quasiminimal for /u1D457large. Moreover /u1D437/u1D706(̃/u1D439/u1D457)→0. By the choice of /u1D4366above, Lemma 6.4implies\nthat\n/u1D6FD/u1D706(̃/u1D439/u1D457)≤/u1D4366/u1D437/u1D706(̃/u1D439/u1D457)1\n2/u1D45B,\ngiving a contradiction. /square\nProof of Theorem 1.2.By Lemma 6.5it follows that for any /u1D434 >0there exists/u1D436/u1D434>0such that for any set\n/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bwith/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)≤/u1D434there holds\n(6.13) /u1D6FD/u1D706(/u1D438)≤/u1D436/u1D434/u1D437/u1D706(/u1D438)1\n2/u1D45B.\nIndeed, if/u1D437/u1D706(/u1D438)≤/u1D6FF7, for/u1D6FF7as in Lemma 6.5, then ( 6.13) follows with /u1D436/u1D434=/u1D4367. Otherwise we just have\n/u1D6FD/u1D706(/u1D438)≤/u1D436(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟1/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)+/u1D45B−1(/u1D715∗/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)∩/u1D715/u1D43B)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1≤/u1D436(/u1D45B,/u1D706,/u1D434)/u1D6FF1\n2/u1D45B\n7\n/u1D6FF1\n2/u1D45B\n7≤/u1D436(/u1D45B,/u1D706,/u1D434)/u1D4371\n2/u1D45B\n/u1D706.\nNext we observe that, letting /u1D436/u1D706such that/u1D443/u1D706(/u1D435/u1D706)≤/u1D436/u1D706/u1D45B−1(/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜) ∩/u1D715/u1D43B), then for any set /u1D438 ⊂ℝ/u1D45B⧵/u1D43B\nwith/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜and/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)≥2/u1D436/u1D706\n1−/u1D706/u1D45B−1(/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)∩/u1D715/u1D43B)there holds\n(6.14) /u1D6FD/u1D706(/u1D438)≤/u1D4368/u1D437/u1D706(/u1D438),\nfor a constant /u1D4368=/u1D4368(/u1D45B,/u1D706)>0.\nIndeed\n/u1D443/u1D706(/u1D438)−/u1D443/u1D706(/u1D435/u1D706)≥(1−/u1D706)/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)−/u1D436/u1D706/u1D45B−1(/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)∩/u1D715/u1D43B)≥1−/u1D706\n2/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B),\nand\n/u1D6FD/u1D706(/u1D438)≤/u1D436(/u1D45B,/u1D706)/⎝⎞⎜e⎪⎨eft.⎟1\n/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B)+/u1D45B−1(/u1D715∗/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜,0)∩/u1D715/u1D43B)/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n≤/u1D436(/u1D45B,/u1D706,/u1D436/u1D706)/u1D45B−1(/u1D715∗/u1D438∩/u1D715/u1D43B).\nSetting now /u1D434∶=2/u1D436/u1D706\n1−/u1D706/u1D45B−1(/u1D715/u1D435/u1D706(/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜)∩/u1D715/u1D43B)in (6.13), taking into account ( 6.14) we conclude that for any set\n/u1D438 ⊂ℝ/u1D45B⧵/u1D43Bwith/u⎪⎫007C.v⎞⎜/u1D438/u⎪⎫007C.v⎞⎜=/u⎪⎫007C.v⎞⎜/u1D435/u1D706/u⎪⎫007C.v⎞⎜there holds\n/u1D6FD/u1D706(/u1D438)≤max{/u1D436/u1D434,/u1D4368} max/b⎜⎞ce⎨eft.⎟2\n/u1D437/u1D706(/u1D438),/u1D437/u1D706(/u1D438)1\n2/u1D45B/b⎜⎞ce⎜⎫g⎧t.⎟2\n.\n/square\nAPPENDIX A. A UXILIARY RESULTS\nA.1. Regularity of (Λ,/u1D45F0)-minimizers. We recall definitions and basic properties of local (Λ,/u1D45F0)-minimizers of\nthe perimeter. A detailed account on the theory of (Λ,/u1D45F0)-minimizers can be found in [ Mag12 ].\nDefinition A.1. LetΩ⊂ℝ/u1D45Bbe an open set and let /u1D438 ⊂ℝ/u1D45Bbe a set of finite perimeter. We say that /u1D438is a local\n(Λ,/u1D45F0)-minimizer of the perimeter in Ω, withΛ,/u1D45F0>0, if\n/u1D443(/u1D438,/u1D435/u1D45F(/u1D465))≤/u1D443(/u1D439,/u1D435/u1D45F(/u1D465))+Λ/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D439/u⎪⎫007C.v⎞⎜,\nwhenever/u1D438Δ/u1D439 ⊂⊂/u1D435/u1D45F(/u1D465)⊂⊂Ωand/u1D45F≤/u1D45F0.\nIt is well-known that local (Λ,/u1D45F0)-minimizers have bounded mean curvature in a generalized se nse. We could\nnot find an explicit reference in the literature, hence we pro vide a proof in the following result.QUANTITATIVE ISOPERIMETRIC INEQUALITIES FOR CLASSICAL CA PILLARITY PROBLEMS 35\nLemma A.2. LetΩ⊂ℝ/u1D45Bbe an open set and let /u1D438 ⊂ℝ/u1D45Bbe a local (Λ,/u1D45F0)-minimizer of the perimeter in Ω. Then\nthere exists/u1D43B∈/u1D43F∞(/u1D443(/u1D438,⋅),ℝ/u1D45B)such that ‖/u1D43B‖/u1D43F∞≤Λand\n/uni222B.dsp/u1D715∗/u1D438div/u1D447/u1D44B= −/uni222B.dsp/u1D715∗/u1D438⟨/u1D44B,/u1D43B⟩∀/u1D44B∈/u1D4361\n/u1D450(Ω,ℝ/u1D45B),\nwherediv/u1D447/u1D44Bis the tangential divergence of /u1D44Balong the ( /u1D45B−1-a.e. defined) tangent space of /u1D715∗/u1D438. We shall refer\nto/u1D43Bas to the (generalized) mean curvature of/u1D438.\nProof. Let/u1D44B∈/u1D4361\n/u1D450(/u1D435/u1D45F(/u1D465)), with/u1D435/u1D45F(/u1D465)⊂⊂Ωand/u1D45F≤/u1D45F0. Let{/u1D454/u1D461}/u1D461</u⎪⎫007C.v⎞⎜/u1D702/u⎪⎫007C.v⎞⎜be the flow of the vector field /u1D44B, and define\n/u1D439/u1D461=/u1D454/u1D461(/u1D438). Then/u1D438Δ/u1D439/u1D461⊂⊂/u1D435/u1D45F(/u1D465). Let/u1D453/u1D461=/u1D454−1\n/u1D461=/u1D454−/u1D461. If/u1D462∈/u1D4361(Ω)then\n/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜/u1D462(/u1D453/u1D461(/u1D466))−/u1D462(/u1D466)/u⎪⎫007C.v⎞⎜d/u1D466=/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/uni222B.dsp/u1D461\n0/u1D715/u1D460[/u1D462(/u1D453/u1D460(/u1D466))]d/u1D460/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.xd/u1D466\n≤/uni222B.dsp/u1D435/u1D45F(/u1D465)/uni222B.dsp/u1D461\n0/u⎪⎫007C.v⎞⎜∇/u1D462(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜ /u⎪⎫007C.v⎞⎜/u1D44B(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜d/u1D460d/u1D466\n=/uni222B.dsp/u1D461\n0/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜∇/u1D462(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜ /u⎪⎫007C.v⎞⎜/u1D44B(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜/u1D43D/u1D453/u1D460(/u1D466)\n/u1D43D/u1D453/u1D460(/u1D466)d/u1D466d/u1D460\n≤(1+/u1D45C(1))/uni222B.dsp/u1D461\n0/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜∇/u1D462(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜ /u⎪⎫007C.v⎞⎜/u1D44B(/u1D453/u1D460(/u1D466))/u⎪⎫007C.v⎞⎜/u1D43D/u1D453/u1D460(/u1D466)d/u1D466d/u1D460\n= (1+/u1D45C(1))/uni222B.dsp/u1D461\n0/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜∇/u1D462(/u1D467)/u⎪⎫007C.v⎞⎜ /u⎪⎫007C.v⎞⎜/u1D44B(/u1D467)/u⎪⎫007C.v⎞⎜d/u1D467d/u1D460\n= (1+/u1D45C(1))/u1D461/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜∇/u1D462(/u1D467)/u⎪⎫007C.v⎞⎜ /u⎪⎫007C.v⎞⎜/u1D44B(/u1D467)/u⎪⎫007C.v⎞⎜d/u1D467,(A.1)\nwhere/u1D45C(1)→0as/u1D461→0.\nSetting/u1D462=/u1D462/u1D700=/u1D712/u1D438⋆/u1D71A/u1D700, then/u⎪⎫007C.v⎞⎜∇/u1D462/u1D700/u⎪⎫007C.v⎞⎜/u1D45B→/u1D443(/u1D438,⋅)as/u1D700→0by [Mag12 , Proposition 12.20]. Also\n/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D462/u1D700(/u1D453/u1D461(/u1D466))−/u1D712/u1D453−/u1D461(/u1D438)(/u1D466)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u1D43D/u1D453/u1D461\n/u1D43D/u1D453/u1D461d/u1D466≤2/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜/u1D462/u1D700(/u1D467)−/u1D712/u1D438(/u1D467)/u⎪⎫007C.v⎞⎜d/u1D467.\nHence setting /u1D462=/u1D462/u1D700in (A.1) and letting /u1D700→0implies\n/u⎪⎫007C.v⎞⎜/u1D438Δ/u1D439/u1D461/u⎪⎫007C.v⎞⎜=/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜/u1D712/u1D439/u1D461−/u1D712/u1D438/u⎪⎫007C.v⎞⎜≤(1+/u1D45C(1))/u1D461/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜/u1D44B/u⎪⎫007C.v⎞⎜d/u1D443(/u1D438,⋅).\nBy(Λ,/u1D45F0)-minimality we deduce\n/u1D443(/u1D438,/u1D435/u1D45F(/u1D465))−/u1D443(/u1D439/u1D461,/u1D435/u1D45F(/u1D465))≤(1+/u1D45C(1))Λ/u1D461/uni222B.dsp/u1D435/u1D45F(/u1D465)/u⎪⎫007C.v⎞⎜/u1D44B/u⎪⎫007C.v⎞⎜d/u1D443(/u1D438,⋅).\nDividing by /u1D461>0and letting/u1D461→0+we get\n−/uni222B.dsp/u1D715∗/u1D438div/u1D447/u1D44Bd/u1D443(/u1D438,⋅)≤Λ/uni222B.dsp/u1D715∗/u1D438/u⎪⎫007C.v⎞⎜/u1D44B/u⎪⎫007C.v⎞⎜d/u1D443(/u1D438,⋅).\nUp to changing /u1D44Bwith−/u1D44Bwe obtain\n/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/uni222B.dsp/u1D715∗/u1D438div/u1D447/u1D44Bd/u1D443(/u1D438,⋅)/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x/u⎪⎫007C.x≤Λ/uni222B.dsp/u1D715∗/u1D438/u⎪⎫007C.v⎞⎜/u1D44B/u⎪⎫007C.v⎞⎜d/u1D443(/u1D438,⋅),\nthat implies the existence of the generalized mean curvatur e/u1D43B∈/u1D43F∞(/u1D443(/u1D438,⋅)/u1D435/u1D45F(/u1D465))for/u1D438in/u1D435/u1D45F(/u1D465)with\n‖/u1D43B‖/u1D43F∞≤Λ. Since/u1D435/u1D45F(/u1D465)was arbitrary in Ω, by a partition of unity argument the claim follows. /square\nLet us further recall the following fundamental regularity properties of local (Λ,/u1D45F0)-minimizers.\nTheorem A.3 ([Tam84 ], [Mag12 , Theorem 26.3, Theorem 26.6]) .LetΩ⊂ℝ/u1D45Bbe an open set. Let /u1D438 ⊂Ωbe\na local(Λ,/u1D45F0)-minimizer in Ω. Then the set /u1D438(1)of points of density 1for/u1D438is an open representative for /u1D438.\nMoreover, representing /u1D438with/u1D438(1), we have that /u1D715∗/u1D438∩Ωis a/u1D4361,1\n2manifold and /u1D451(/u1D715/u1D438∩Ω⧵/u1D715∗/u1D438) = 0 for any\n/u1D451 >/u1D45B−8.\nLet/u1D438/u1D456⊂Ωbe a sequence of local (Λ,/u1D45F0)-minimizers in Ωthat converges to /u1D438in/u1D43F1(Ω). If/u1D715/u1D438∩/u1D435/u1D45F(/u1D465)is of class\n/u1D4362, for some/u1D435/u1D45F(/u1D465)⊂Ω, then/u1D715/u1D438/u1D456∩/u1D435/u1D45F∕2(/u1D465)is of class/u1D4361,1\n2for large/u1D456and converges to /u1D715/u1D438∩/u1D435/u1D45F∕2(/u1D465)in/u1D4361,/u1D6FCfor\nany/u1D6FC∈ (0,1∕2).36 GIULIO PASCALE AND MARCO POZZETTA\nThanks to Theorem A.3, we will always identify a local (Λ,/u1D45F0)-minimizer/u1D438with the open set /u1D438(1).\nA.2. Axially symmetric hypersurfaces. We recall a formula for the mean curvature of axially symmetr ic hyper-\nsurfaces of class /u1D44A2,/u1D45D.\nLemma A.4. Let/u1D44E < /u1D44F . Let/u1D6FC,/u1D6FD∶ (/u1D44E,/u1D44F)→(0,∞)be/u1D44A2,/u1D45Dfunctions, with /u1D45D∈ (1,∞], parametrizing the\ncurve/u1D6FE∶ (/u1D44E,/u1D44F)→span{/u1D4521,/u1D452/u1D45B}⊂ℝ/u1D45Bgiven by/u1D6FE(/u1D461) = (/u1D6FC(/u1D461),0,…,0,/u1D6FD(/u1D461)), and assume that /u⎪⎫007C.v⎞⎜/u1D6FE)uni2032(var(/u1D461)/u⎪⎫007C.v⎞⎜= 1and that\ninf(/u1D44E,/u1D44F)/u1D6FC>0. Let/u1D446be the axially symmetric hypersurface around the /u1D45B-th axis parametrized by\n/u1D711/u1D446∶/u1D54A/u1D45B−2×(/u1D44E,/u1D44F)→ℝ/u1D45B\n/u1D711/u1D446(/u1D717,/u1D461) = (/u1D6FC(/u1D461)/u1D717,/u1D6FD(/u1D461)).\nThen the vector\n/u1D43B=/⎝⎞⎜e⎪⎨eft.⎟3/u⎪⎫27E8.⎟1\n/u1D458/u1D6FE,/u1D708/u⎪⎫27E9.⎟1\n−(/u1D45B−2)/u1D6FD)uni2032(var\n/u1D6FC/⎝⎞⎜e⎪⎜⎫g⎧t.⎟3/⎝⎞⎜e⎪⎨eft.⎟1\n−/u1D6FD)uni2032(var/u1D717,/u1D6FC)uni2032(var/⎝⎞⎜e⎪⎜⎫g⎧t.⎟1\n,\nfor every/u1D717and a.e./u1D461, where/u1D458/u1D6FEis the curvature of /u1D6FEand/u1D708(/u1D461) = (−/u1D6FD)uni2032(var,0,…,0,/u1D6FC)uni2032(var), is the (generalized) mean\ncurvature of /u1D446. More precisely\n(A.2)/uni222B.dsp/u1D446div/u1D447/u1D44B= −/uni222B.dsp/u1D446⟨/u1D44B,/u1D43B⟩,\nfor any/u1D44B∈/u1D4361\n/u1D450(ℝ/u1D45B,ℝ/u1D45B)such that spt/u1D44B∩/u1D715/u1D446= )uni2205(var, wherediv/u1D447/u1D44Bis the tangential divergence of /u1D44Balong/u1D446.\nProof. If/u1D6FC,/u1D6FDare smooth, the claimed formula follows by a direct computat ion. The statement then follows by\napproximating /u1D6FC,/u1D6FDin/u1D44A2,/u1D45D. The approximating hypersurfaces /u1D446/u1D456converge to/u1D446in/u1D4361and the measures /u1D43B/u1D456/u1D45B−1\n/u1D446/u1D456converge to/u1D43B/u1D45B−1/u1D446in duality with compactly supported continuous fields. Henc e (A.2) passes to the limit.\n/square\nREFERENCES\n[AFM13] E. Acerbi, N. Fusco, and M. Morini. “Minimality via s econd variation for a nonlocal isoperimetric\nproblem”. In: Comm. Math. Phys. 322.2 (2013), pp. 515–557.\n[AFP00] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity prob -\nlems. Oxford Mathematical Monographs. Oxford University Press , Oxford, 2000, pp. xviii+434.\n[Bae15] E. Baer. “Minimizers of Anisotropic Surface Tensio ns Under Gravity: Higher Dimensions via Sym-\nmetrization”. In: Arch. Rational Mech. Anal. 215 (2015), pp. 531–578.\n[BBJ17] M. Barchiesi, A. Brancolini, and V. Julin. “Sharp di mension free quantitative estimates for the Gauss-\nian isoperimetric inequality”. In: Ann. Probab. 45.2 (2017), pp. 668–697.\n[BDS15] V. Bögelein, F. Duzaar, and C. Scheven. “A sharp quan titative isoperimetric inequality in hyperbolic\n/u1D45B-space”. In: Calc. Var. Partial Differential Equations 54.4 (2015), pp. 3967–4017.\n[BDR12] L. Brasco, G. De Philippis, and B. Ruffini. “Spectral o ptimization for the Stekloff-Laplacian: the sta-\nbility issue”. In: J. Funct. Anal. 262.11 (2012), pp. 4675–4710.\n[Cab00] X. Cabré. “Partial differential equations, geometry and stochastic control”. In: Butl. Soc. Catalana\nMat. 15.1 (2000), pp. 7–27.\n[Cab08] X. Cabré. “Elliptic PDE’s in probability and geomet ry: symmetry and regularity of solutions”. In:\nDiscrete Contin. Dyn. Syst. 20.3 (2008), pp. 425–457.\n[Cab17] X. Cabré. “Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci\nmethod: a survey”. In: Chinese Ann. Math. Ser. B 38.1 (2017), pp. 201–214.\n[CRS16] X. Cabré, X. Ros-Oton, and J. Serra. “Sharp isoperim etric inequalities via the ABP method”. In: J.\nEur. Math. Soc. (JEMS) 18.12 (2016), pp. 2971–2998.\n[CMM19] F. Cavalletti, F. Maggi, and A. Mondino. “Quantitat ive isoperimetry à la Levy-Gromov”. In: Comm.\nPure Appl. Math. 72.8 (2019), pp. 1631–1677.\n[CEL24] O. Chodosh, N. Edelen, and C. Li. Improved regularity for minimizing capillary hypersurfac es. 2024.\narXiv:2401.08028 .\n[CES23] O. Chodosh, M. Engelstein, and L. Spolaor. “The Riem annian quantitative isoperimetric inequality”.\nIn:J. Eur. Math. Soc. (JEMS) 25.5 (2023), pp. 1711–1741.\n[CGR07] J. Choe, M. Ghomi, and M. Ritoré. “The relative isope rimetric inequality outside convex domains in\n/u1D411/u1D45B”. In: Calc. Var. Partial Differential Equations 29.4 (2007), pp. 421–429.\n[Cia+11] A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli. “On the isoperimetric deficit in Gauss space”. In:\nAmer. J. Math. 133.1 (2011), pp. 131–186.REFERENCES 37\n[CL12] M. Cicalese and G. P. Leonardi. “A selection principl e for the sharp quantitative isoperimetric inequal-\nity”. In: Arch. Ration. Mech. Anal. 206.2 (2012), pp. 617–643.\n[Cin+22] E. Cinti, F. Glaudo, A. Pratelli, X. Ros-Oton, and J . Serra. “Sharp quantitative stability for isoperimet-\nric inequalities with homogeneous weights”. In: Transactions of the American Mathematical Society\n375.3 (Mar. 2022), pp. 1509–1550.\n[De 58] E. De Giorgi. “Sulla proprietà isoperimetrica dell’ ipersfera, nella classe degli insiemi aventi frontiera\norientata di misura finita”. In: Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat., Sez. I 8 (1958),\npp. 33–44.\n[DM15] G. De Philippis and F. Maggi. “Regularity of Free Boun daries in Anisotropic Capillarity Problems\nand the Validity of Young’s Law”. In: Arch. Rational Mech. Anal. 216 (2015), pp. 473–568.\n[FI13] A. Figalli and E. Indrei. “A sharp stability result fo r the relative isoperimetric inequality inside convex\ncones”. In: J. Geom. Anal. 23.2 (2013), pp. 938–969.\n[FMP10] A. Figalli, F. Maggi, and A. Pratelli. “A mass transp ortation approach to quantitative isoperimetric\ninequalities”. In: Invent. Math. 182.1 (2010), pp. 167–211.\n[Fin80] R. Finn. “The sessile liquid drop. I. Symmetric case .” In: Pac. J. Math. 88.2 (1980), pp. 541–587.\n[Fin86] R. Finn. Equilibrium Capillary Surfaces . Vol. 284. Die Grundlehren der mathematischen Wissen-\nschaften. Springer-Verlag New York Inc., 1986.\n[Fug89] B. Fuglede. “Stability in the isoperimetric proble m for convex or nearly spherical domains in /u1D411/u1D45B”. In:\nTrans. Amer. Math. Soc. 314.2 (1989), pp. 619–638.\n[Fus04] N. Fusco. “The classical isoperimetric theorem”. I n:Rend. Accad. Sci. Fis. Mat. Napoli (4) 71 (2004),\npp. 63–107.\n[Fus15] N. Fusco. “The quantitative isoperimetric inequal ity and related topics”. In: Bull. Math. Sci. 5.3 (2015),\npp. 517–607.\n[FJ14] N. Fusco and V. Julin. “A strong form of the quantitati ve isoperimetric inequality”. In: Calc. Var.\nPartial Differential Equations 50.3-4 (2014), pp. 925–937.\n[FL23] N. Fusco and D. A. La Manna. “Some weighted isoperimet ric inequalities in quantitative form”. In:\nJ. Funct. Anal. 285.2 (2023), Paper No. 109946, 24.\n[FMP08] N. Fusco, F. Maggi, and A. Pratelli. “The sharp quant itative isoperimetric inequality”. In: Ann. of\nMath. (2) 168.3 (2008), pp. 941–980.\n[FM23] N. Fusco and M. Morini. “Total positive curvature and the equality case in the relative isoperimetric\ninequality outside convex domains”. In: Calc. Var. Partial Differential Equations 62.3 (2023), Paper\nNo. 102, 32.\n[GT01] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order . Classics in\nMathematics. Reprint of the 1998 edition. Springer-Verlag , Berlin, 2001, pp. xiv+517.\n[Giu80] E. Giusti. “The pendent water drop. A direct approac h”. In: Bollettino UMI 17-A (1980), pp. 458–465.\n[Giu81] E. Giusti. “The equilibrium configuration of liquid drops”. In: J. reine angew Math. 321 (1981), pp. 53–\n63.\n[Gon76] E. Gonzalez. “Sul problema della goccia appoggiata ”. In: Rend. Sem. Mat. Univ. Padova 55 (1976),\npp. 289–302.\n[Gon77] E. Gonzalez. “Regolarità per il problema della gocc ia appoggiata”. In: Rend. Sem. Mat. Univ. Padova\n58 (1977), pp. 25–33.\n[GMT80] E. Gonzalez, U. Massari, and I. Tamanini. “Existenc e and regularity for the problem of a pendent\nliquid drop”. In: Pacific Journal of Mathematics 88.2 (1980), pp. 399–420.\n[GT77] E. Gonzalez and I. Tamanini. “Convessità della gocci a appoggiata”. In: Rend. Sem. Mat. Univ. Padova\n58 (1977), pp. 35–43.\n[Hal92] R. R. Hall. “A quantitative isoperimetric inequali ty in/u1D45B-dimensional space”. In: J. reine angew Math.\n428 (1992), pp. 161–176.\n[HHW91] R. R. Hall, W. K. Hayman, and A. W. Weitsman. “On asymm etry and capacity”. In: J. Analyse Math.\n56 (1991), pp. 87–123.\n[IN15] E. Indrei and L. Nurbekyan. “On the stability of the po lygonal isoperimetric inequality”. In: Adv. Math\n276 (2015), pp. 62–86.\n[Kin+13] J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugali ngam. “Regularity of sets with quasiminimal\nboundary surfaces in metric spaces”. In: J. Geom. Anal. 23.4 (2013), pp. 1607–1640.\n[Kru17] B. Krummel. Higher codimension relative isoperimetric inequality out side a convex set . 2017. arXiv:\n1710.04821[math.OC] .38 REFERENCES\n[Lie88] G. M. Lieberman. “Oblique derivative problems in Li pschitz domains II. Discontinuous boundary\ndata”. In: J. reine angew Math.. 389 (1988), pp. 1–21.\n[LWW23] L. Liu, G. Wang, and L. Weng. “The relative isoperime tric inequality for minimal submanifolds with\nfree boundary in the Euclidean space”. In: J. Funct. Anal. 285.2 (2023), Paper No. 109945, 22.\n[Mag08] F. Maggi. “Some methods for studying stability in is operimetric type problems”. In: Bull. Amer. Math.\nSoc. (N.S.) 45.3 (2008), pp. 367–408.\n[Mag12] F. Maggi. Sets of finite perimeter and geometric variational problems . Vol. 135. Cambridge Studies in\nAdvanced Mathematics. An introduction to geometric measur e theory. Cambridge University Press,\nCambridge, 2012, pp. xx+454.\n[MM16] F. Maggi and C. Mihaila. “On the shape of capillarity d roplets in a container”. In: Calc. Var. Partial\nDifferential Equations 55.5 (2016), Art. 122, 42.\n[MS86] V. D. Milman and G. Schechtman. Asymptotic theory of finite-dimensional normed spaces . Vol. 1200.\nLecture Notes in Mathematics. With an appendix by M. Gromov. Springer-Verlag, Berlin, 1986,\npp. viii + 156.\n[Nit11] R. Nittka. “Regularity of solutions of linear secon d order elliptic and parabolic boundary value prob-\nlems on Lipschitz domains”. In: J. Differential Equationa 251 (2011), pp. 860–880.\n[Tam84] I. Tamanini. “Regularity results for almost minima l oriented hypersurfaces in ℝ/u1D441”. In: Quaderni del\nDipartimento di Matematica dell’Università del Salento (L ecture Notes) 1 (1984), pp. 1–92.\n[Vol67] A. I. Vol’pert. “Spaces BVand quasilinear equations”. In: Mat. Sb. (N.S.) 73(115) (1967), pp. 255–\n302.\nDIPARTIMENTO DI MATEMATICA E APPLICAZIONI \"RENATO CACCIOPPOLI \", U NIVERSITÁ DEGLI STUDI DI NAPOLI \"FEDERICO II\",\nVIACINTIA - M ONTE SANT’ ANGELO , 80126 N APOLI , ITALY\nEmail address :giulio.pascale@unina.it\nDIPARTIMENTO DI MATEMATICA E APPLICAZIONI \"RENATO CACCIOPPOLI \", U NIVERSITÁ DEGLI STUDI DI NAPOLI \"FEDERICO II\",\nVIACINTIA - M ONTE SANT’ ANGELO , 80126 N APOLI , ITALY\nEmail address :marco.pozzetta@unina.it"    },    {        "title": "2402.04684v1.Towards_a_Parallel_Summation_Algorithm.pdf",        "content": "arXiv:2402.04684v1  [math.CO]  7 Feb 2024Towards a ParallelSummationAlgorithm\nShaoshiChen/u1D44E,/u1D44F, Ruyong Feng/u1D44E,/u1D44F,ManuelKauers/u1D450, Xiuyun Li/u1D44E,/u1D44F,/u1D450\n/u1D44EKLMM,AcademyofMathematicsandSystems Science,Chinese Acad emyofSciences,Beijing 100190,China\n/u1D44FSchoolofMathematicalSciences, University of Chinese Academ yof Sciences,Beijing 100049,China\n/u1D450Institute forAlgebra,JohannesKepler University, Linz, A4040, Austria\nschen@amss.ac.cn,ryfeng@amss.ac.cn,manuel.kauers@jku.at, lixiuyun@amss.ac.cn\nABSTRACT\nWe propose a summation analog of the paradigm of parallel int e-\ngration.Usingthisparadigm,wemakesomefirststepstoward san\nindefinitesummationalgorithmapplicabletosummandsthat ratio-\nnally depend onthesummationindex and a P-recursive sequen ce\nand itsshifts.Under theassumptionthatthecorresponding differ-\nence field has no unnatural constants, we are able to compute a\nboundonthenormalpartofthedenominatorofapotentialclo sed\nform. We can also handle the numerator. Our algorithm is inco m-\nplete so far as we cannot predict the special part of the denom i-\nnator. However, we do have some structural results about spe cial\npolynomials forthesettingunder consideration.\nCCS CONCEPTS\n•Computingmethodologies →Algebraic algorithms .\nKEYWORDS\nsymbolic summation;difference rings\nACMReference Format:\nShaoshi Chen/u1D44E,/u1D44F, Ruyong Feng/u1D44E,/u1D44F, Manuel Kauers/u1D450, Xiuyun Li/u1D44E,/u1D44F,/u1D450. 2024.\nTowards a Parallel Summation Algorithm. In .ACM, New York, NY, USA,\n9 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn\n1 INTRODUCTION\nThe main difference between the first and the second edition of\nManuel Bronstein’s classicaltextbookonsymbolicintegra tion[8]\nis an additional tenth chapter about parallel integration, which is\nbasedonhislastpaper[9]onthesubject.Parallelintegrat ionisan\nalternative approach to the more widely known Risch algorit hm\nforindefiniteintegration,whosecarefuldescriptiondomi natesthe\nremainder of Bronstein’s book. Parallel integration is als o known\nS. Chen was partially supported by the National Key R&D Progr am of China (No.\n2023YFA1009401), the NSFC grant (No. 12271511), CAS Projec t for Young Scientists\nin Basic Research (Grant No. YSBR-034), and the CAS Fund of th e Youth Innovation\nPromotion Association (No. Y2022001). R. Feng was partiall y supported by the Na-\ntional Key R&D Program of China (No. 2023YFA1009401) and the National Key Re-\nsearch and Development Project 2020YFA0712300. M. Kauers w as supported by the\nAustrianFWF grants PAT 9952223 and I6130-N. X. Li was partia llysupported by the\nLand Oberösterreichthrough the LIT-AILab.\nPermission to make digital or hard copies of all or part of thi s work for personal or\nclassroomuseisgrantedwithoutfeeprovidedthatcopiesar enotmadeordistributed\nfor profit or commercial advantage and that copies bear this n otice and the full cita-\ntiononthefirstpage.Copyrightsforcomponents of thiswork owned byothersthan\nACMmustbehonored.Abstractingwithcreditispermitted.T ocopyotherwise,orre-\npublish,topostonserversortoredistributetolists,requ irespriorspecificpermission\nand/or afee. Request permissionsfrompermissions@acm.or g.\n, ,\n© 2024 Associationfor Computing Machinery.\nhttps://doi.org/10.1145/nnnnnnn.nnnnnnnastheRisch-Normanalgorithm[12–14,19,20,23,31]andasp oor-\nman’s integrator [50].\nAlthough the technique is not complete, i.e., it fails to find a\nclosed form of certain integrals, it is an attractive altern ative to a\nfull implementation of the Risch algorithm, which is guaran teed\ntofindaclosedformwhenever thereisone.Oneadvantageisth at\nitismucheasiertoprogram.Indeed,Bronstein’sMapleimpl emen-\ntation [50] barely needs 100 lines of code. A second advantag e is\nthat it extends more easily to integrals of non-elementary f unc-\ntions.Forexample, it canfind theevaluation\n/uni222B.dsp/u1D4652+ (/u1D4652+2)/u1D44A(/u1D4652)\n/u1D465(1+/u1D44A(/u1D4652))2/u1D451/u1D465=1\n2/u1D4652\n/u1D44A(/u1D4652)+log(1+/u1D44A(/u1D4652))\ninvolvingtheLambert /u1D44Afunction[11].Thisisnotonlyinteresting\nbecause/u1D44Ais defined by a nonlinear equation, but also because\nthere is a factor in the denominator of the closed form that is not\nalreadypresent intheintegrand.\nIn a seminar talk that never led to a formal publication, Zim-\nmermannobservedthatparallelintegrationcanbecombined with\ntheconceptofcreativetelescoping[49]inordertohandled efinite\nintegralsinvolving aparameter,similarasdonebyRaab[36 ]with\nRisch’s algorithm. A version of parallel integration for in tegrals\ninvolving algebraic functions was presented byBöttnerin[ 6]and\nforintegrals ofAiryfunctions byDu and Raab in[15].\nTo our knowledge, the idea of parallel integration has not ye t\nbeentranslatedtothesetting ofsymbolicsummation.Thego alof\nthepresentpaperistodoso.Asummationexamplethatissimi lar\ntotheaboveintegralcanbegivenintermsofthelogisticseq uence\n/u1D461/u1D45B[17,Example1.9,Chapter1]satisfyingthenonlinearrecur rence\nequation/u1D461/u1D45B+1=/u1D461/u1D45B(1−/u1D461/u1D45B)with/u1D4610∈ (0,1). Here we have the\nsummationidentity\n/u1D45B−1/summationdisplay.1\n/u1D458=01\n1−/u1D461/u1D458=/u1D45B/summationdisplay.1\n/u1D458=1/parenleftbigg1\n/u1D461/u1D458+1−1\n/u1D461/u1D458/parenrightbigg\n=1\n/u1D461/u1D45B−1\n/u1D4610,\nand again, the denominator of the closed form contains a fact or\nthatis notalready present inthesummand.\nOntheotherhand,thedenominatorofaclosedformisnotcom-\npletely unpredictable. Like in parallel integration, we ca n distin-\nguish the specialand thenormalpart of a denominator. Based on\nthis distinction, we show in Section 2 how the normal part of t he\ndenominator of a closed form depends on the normal part of the\ndenominator of the corresponding summand. Unfortunately, we\ndo not have a complete understanding of the special part, but we\ndohavesomeresultsthatlimitthenumberofspecialpolynom ials\n(Sect. 2.2). Morecan besaid if wefocusonamorespecificsett ing.\nSect. 2 is about the general paradigm of parallel summation,\nwhich in principle could be applied to many different specific set-\ntings. In Sect. 3 we restrict the attention to one such settin g. Weconsider summationproblems oftheform\n/u1D45B/summationdisplay.1\n/u1D458=0rat(/u1D458,/u1D453(/u1D458),/u1D453(/u1D458+1),...,/u1D453(/u1D458+/u1D45F−1)),\nwhere rat is a multivariate rational function and /u1D453is defined by a\nlinearrecurrenceoforder /u1D45Fwithpolynomialcoefficients.Thetask\nis to decide whether a given sum of this type can be written as a\nrationalfunctionin /u1D45B,/u1D453(/u1D45B),...,/u1D453(/u1D45B+/u1D45F−1).Whilewearenot(yet)\nable to solve this task in full generality, the idea of parall el sum-\nmation provides a significant step towards such an algorithm .We\ncandescribemorepreciselythestructureofspecialpolyno mialsin\nthiscase(Sect.3.1),andwecaneffectivelysolvethe /u1D70E-equivalence\nproblem(Sect.3.2),which implies thatwecancompletelyid entify\nthenormalpartofthedenominatorof anyclosedform.Simila r as\ninthedifferential case[8,9],thespecialparthastobedete rmined\nheuristically, unless we impose further restrictions on th e setting\n(Sect.4).\n2 PARALLEL SUMMATION\nSimilar toparallelintegration, thegeneral idea ofparall el summa-\ntionistoavoidtherecursivenatureofsummationalgorithm ssuch\nas Karr’s algorithm by viewing the summand as an element of a\nfieldofmultivariaterationalfunctionsoveragroundfield. Wenow\nsetupthegeneralalgebraicfoundationforparallelsummat ionand\nlistsomerelatedproblems.Likeinthesituationofparalle lintegra-\ntion,these problemsare ingeneral far from beingsolved.\nLet/u1D434be a ring and /u1D70E:/u1D434→/u1D434be an automorphism of /u1D434. We\ncall the pair (/u1D434,/u1D70E)adifference ring and adifference filed if/u1D434is a\nfield. Note that the set {/u1D44E∈/u1D434|/u1D70E(/u1D44E)=/u1D44E}forms a subring of /u1D434\nwhich is called the constant subring of(/u1D434,/u1D70E), denoted by /u1D436/u1D434. A\ndifference ring (/u1D434∗,/u1D70E∗)is called a difference extension of(/u1D434,/u1D70E)if\n/u1D446⊆/u1D434∗and/u1D70E∗|/u1D434=/u1D70E. By abuse of notation, we will often write /u1D70E\nfor theextended automorphism /u1D70E∗of/u1D434∗.\nP/r.sc/o.sc/b.sc/l.sc/e.sc/m.sc1(I/n.sc/d.sc/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/e.scS/u.sc/m.sc/m.sc/a.sc/t.sc/i.sc/o.sc/n.scP/r.sc/o.sc/b.sc/l.sc/e.sc/m.sc). Let(/u1D434∗,/u1D70E)bea\nspecific difference extension of (/u1D434,/u1D70E). Given/u1D453∈/u1D434, decide whether\nthereexists/u1D454∈/u1D434∗suchthat/u1D453=/u1D70E(/u1D454)−/u1D454.Ifsucha/u1D454exists,/u1D453issaid\ntobesummable in/u1D434∗.\nBothAbramov’salgorithm[1–3]andPaule’salgorithm[33]s olved\ntheindefinitesummationproblemforrationalfunctions.Th eindef-\ninite hypergeometricsummationproblemwassolvedbyGospe r’s\nalgorithmin[21]andthemoregeneralP-recursivecasewith outde-\nnominators was solved by Abramov-van Hoeij’s algorithm [4, 5].\nAs a discrete analogue of Risch’s algorithm for elementary i nte-\ngration, Karr’s algorithm [26,27] solves the indefinite sum mation\nproblem in a so-called ΠΣ-extension of a given difference field.\nKarr’s algorithm has been implemented and improved by Schne i-\nder in [37, 40, 41] with applications in physics [42]. The ide as of\nKarr have also been extended to higher order equations [7, 24 , 38,\n39].\nKarr’s structuraltheorem shows that themost interesting s um-\nmation problem in ΠΣ-extensions is the in-field summation prob-\nlemwhere/u1D453,/u1D454are in the same field. This is quite different from\nthe situation in symbolic integration where computing the l oga-\nrithmic part is the most difficult step. In this paper, we will o nly\nfocusonthein-field summationproblem.Let/u1D436beafield ofcharacteristiczero and /u1D43E:=/u1D436(/u1D465)bethefield\nofrationalfunctionsin /u1D465over/u1D436.Wedefinetheusual shiftoperator\n/u1D70E:/u1D43E→/u1D43Eas a/u1D436-automorphism of /u1D43Esuch that/u1D70E(/u1D465)=/u1D465+1. So\n(/u1D43E,/u1D70E)becomesadifference fieldanditsconstantsubfieldis /u1D436.Let\n/u1D445:=/u1D43E[/u1D4610,...,/u1D461/u1D45B−1]and/u1D439:=/u1D43E(/u1D4610,...,/u1D461/u1D45B−1). Thecentral problem\nofparallelsummationis as follows.\nP/r.sc/o.sc/b.sc/l.sc/e.sc/m.sc 2. Given/u1D453∈/u1D439, decidewhether there exists /u1D454∈/u1D439such\nthat/u1D453=/u1D70E(/u1D454) −/u1D454.\nFor a general /u1D436-automorphism /u1D70Eof/u1D439, the following example\nshows that the difference field (/u1D439,/u1D70E)may contain new constants\nthatis notin /u1D436.\nE/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc 3. Let/u1D439=/u1D436(/u1D465,/u1D4610,/u1D4611)with the/u1D436-automorphism /u1D70Esat-\nisfying/u1D70E(/u1D465)=/u1D465+1,/u1D70E(/u1D4610)=/u1D4611, and/u1D70E(/u1D4611)=/u1D4610+/u1D4611. Then/u1D45D=\n(/u1D4612\n1−/u1D4612\n0−/u1D4610/u1D4611)2isa newconstant in /u1D439.\nIn general, deciding the existence of new constants is a very\nhard problem. The following example shows that in general, t he\ndenominator of /u1D454may have factors that are not related to any of\nthefactorsofthedenominator of /u1D453.\nE/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc 4. Let/u1D439=/u1D436(/u1D465,/u1D4610,/u1D4611)and/u1D70Eis the/u1D436-automorphism\ndefinedby/u1D70E(/u1D465)=/u1D465+1,/u1D70E(/u1D4610)=2/u1D4610+/u1D465/u1D4611and/u1D70E(/u1D4611)=2/u1D4611.Then\n/u1D70E/parenleftbigg/u1D4610\n/u1D4611/parenrightbigg\n−/u1D4610\n/u1D4611=2/u1D4610+/u1D465/u1D4611\n2/u1D4611−/u1D4610\n/u1D4611=/u1D465\n2.\nTomaketheproblemmoretractable,wewillimposethefollow -\ninghypothesis throughouttheremaining partof this paper.\nH/y.sc/p.sc/o.sc/t.sc/h.sc/e.sc/s.sc/i.sc/s.sc5. The constant fieldof (/u1D439,/u1D70E)isthe field/u1D436and/u1D70Eis\nalsoa/u1D436-automorphismof /u1D445.\nTosolveProblem2,onefirstneedstoestimatethepossibleir re-\nduciblepolynomialsin thedenominator of /u1D454. Tothis end, wenow\nextend thenotionof specialpolynomialsinparallelintegr ationto\nthesummationsetting.\nD/e.sc/f.sc/i.sc/n.sc/i.sc/t.sc/i.sc/o.sc/n.sc 6. A polynomial /u1D443∈/u1D445is said to be specialif there\nexist/u1D456∈Z\\ {0}such that/u1D443|/u1D70E/u1D456(/u1D443)and it is said to be normalif\ngcd(/u1D443,/u1D70E/u1D456(/u1D443))=1for all/u1D456∈Z\\ {0}. A polynomial /u1D443∈/u1D445is said\nto befactor-normal if all of its irreducible factors are normal. Two\npolynomials /u1D443,/u1D444∈/u1D445issaid to be /u1D70E-equivalent if there exist /u1D45A∈Z\nand/u1D462∈/u1D43Esuch that/u1D443=/u1D462·/u1D70E/u1D45A(/u1D444).\nBy the above definition, any nonzero element in /u1D43Eis both spe-\ncialandnormalandanirreduciblepolynomialin /u1D445iseitherspecial\nornormal.Theproductofspecialpolynomialsisalsospecia l.Iftwo\nnormalpolynomialsarenot /u1D70E-equivalent,thentheirproductisstill\nnormal.\nConcerningspecialandnormalpolynomials,therearetwoba sic\nand natural questions: firstly, how to decide a given irreduc ible\npolynomialisspecialornormal?secondly,howtodecidewhe ther\ntwo polynomials are /u1D70E-equivalent or not? We will answer these\nquestions in next section for the difference field generated b y P-\nrecursivesequences.\n2.1 Local dispersions anddenominator bounds\nAbramov in [1] introduced the notion of dispersions for rati onal\nsummation that is a discrete analogue of the multiplicity. W e de-\nfinealocalversionofAbramov’sdispersionsin /u1D445atanirreduciblenormal polynomial that was first used in [10]. Let /u1D45D,/u1D444∈/u1D445with\n/u1D45Dbeing an irreducible normal polynomial. If /u1D70E/u1D456\n/u1D461(/u1D45D) |/u1D444for some\n/u1D456∈Z, thelocal dispersion of/u1D444at/u1D45D, denoted by disp/u1D45D(/u1D444), is de-\nfinedasthemaximalintegerdistance |/u1D456−/u1D457|with/u1D456,/u1D457∈Zsatisfying\n/u1D70E/u1D456\n/u1D461(/u1D45D) |/u1D444and/u1D70E/u1D457\n/u1D461(/u1D45D) |/u1D444; otherwise we define disp/u1D45D(/u1D444)=−∞.\nConventionally, we set disp/u1D45D(0)=+∞. The (global) dispersion of\n/u1D444,denoted bydisp (/u1D444),is defined as\nmax{disp/u1D45D(/u1D444) |/u1D45Dis anirreduciblenormal polynomialin /u1D445}.\nNote that disp (/u1D444)=−∞if/u1D444∈/u1D445\\ {0}has no irreducible normal\nfactor. For a rational function /u1D453=/u1D44E//u1D44F∈/u1D439with/u1D44E,/u1D44F∈/u1D445and\ngcd(/u1D44E,/u1D44F)=1, we also define disp/u1D45D(/u1D453)=disp/u1D45D(/u1D44F)and disp(/u1D453)=\ndisp(/u1D44F). The set {/u1D70E/u1D456(/u1D45D) |/u1D456∈Z}is called the /u1D70E-orbit at/u1D45D, denoted\nby[/u1D45D]/u1D70E. Note that disp/u1D45D(/u1D444)=disp/u1D45E(/u1D444)if/u1D45E∈ [/u1D45D]/u1D70E. So we can\ndefine thelocaldispersionand dispersionofa rationalfunc tionat\na/u1D70E-orbit.\nThe following lemma shows how the local dispersions and dis-\npersionschangeundertheactionofthedifferenceoperator Δ,which\nis defined by Δ(/u1D453)=/u1D70E(/u1D453) −/u1D453for any/u1D453∈/u1D439.\nL/e.sc/m.sc/m.sc/a.sc 7. Let/u1D453=/u1D44E//u1D44F∈/u1D439with/u1D44E,/u1D44F∈/u1D445andgcd(/u1D44E,/u1D44F)=1and\nlet/u1D45D∈/u1D445be an irreduciblenormal factor of /u1D44F. Thendisp/u1D45D(Δ(/u1D453))=\ndisp/u1D45D(/u1D453) +1anddisp(Δ(/u1D453))=disp(/u1D453) +1.\nP/r.sc/o.sc/o.sc/f.sc.Let/u1D451=disp/u1D45D(/u1D44F). Without loss of generality, we may\nassume that /u1D45D|/u1D44Fbut/u1D70E/u1D456(/u1D45D)∤/u1D44Ffor any/u1D456<0. Since gcd (/u1D44E,/u1D44F)=1\nand/u1D70Eis a/u1D436-automorphismof /u1D43E[/u1D4610,...,/u1D461/u1D45B−1],gcd(/u1D70E/u1D456(/u1D44E),/u1D70E/u1D456(/u1D44F))=\n1forany/u1D456∈Z.Wenow write\n/u1D70E(/u1D453) −/u1D453=/u1D70E(/u1D44E)/u1D44F−/u1D44E/u1D70E(/u1D44F)\n/u1D44F/u1D70E(/u1D44F)=/u1D434\n/u1D435,\nwhere/u1D434,/u1D435∈/u1D43E[/u1D4610,...,/u1D461/u1D45B−1]and gcd(/u1D434,/u1D435)=1. Since/u1D45D|/u1D44Fbut\n/u1D45D∤/u1D44E/u1D70E(/u1D44F), we have/u1D45D∤(/u1D70E(/u1D44E)/u1D44F−/u1D44E/u1D70E(/u1D44F))and then/u1D45D∤/u1D434. By\nthe definition of local dispersions, /u1D70E/u1D451(/u1D45D) |/u1D44Fbut/u1D70E/u1D451+1(/u1D45D)∤/u1D44F.\nSince gcd (/u1D44E,/u1D44F)=1, we have/u1D70E/u1D451(/u1D45D)∤/u1D44Eand then/u1D70E/u1D451+1(/u1D45D)∤/u1D70E(/u1D44E).\nThen/u1D70E/u1D451+1(/u1D45D)∤/u1D70E(/u1D44E)/u1D44F, which implies /u1D70E/u1D451+1(/u1D45D)∤(/u1D70E(/u1D44E)/u1D44F−/u1D44E/u1D70E(/u1D44F))\nand also/u1D70E/u1D451+1(/u1D45D)∤/u1D434. So/u1D45D|/u1D435and/u1D70E/u1D451+1(/u1D45D) |/u1D435, which implies\nthat disp/u1D45D(/u1D435) ≥/u1D451+1. Since/u1D435|/u1D44F/u1D70E(/u1D44F), we have disp/u1D45D(/u1D435) ≤\ndisp/u1D45D(/u1D44F/u1D70E(/u1D44F))=/u1D451+1.Therefore, disp/u1D45D(/u1D435)=/u1D451+1.Sincetheequal-\nity disp/u1D45D(/u1D435)=/u1D451+1 holds for all irreducible normal factors, we\nhave disp (Δ(/u1D453))=disp(/u1D453) +1.\nBy the above lemma, we get that /u1D453is not/u1D70E-summable in /u1D439if\ndisp(/u1D453)=0. If we know how to detect the /u1D70E-equivalence in /u1D445,\nthen wecan writea given polynomial /u1D443∈/u1D445as/u1D443=/u1D443/u1D460·/u1D443/u1D45B,where\n/u1D443/u1D45Bismonic,allirreduciblefactorsof /u1D443/u1D460∈/u1D445arespecialandallirre-\nducible factors of /u1D443/u1D45B∈/u1D445are normal. We call (/u1D443/u1D460,/u1D443/u1D45B)thesplitting\nfactorization of/u1D443and/u1D443/u1D45Bthenormal part of/u1D443.\nT/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 8. Let/u1D453∈/u1D439and/u1D463/u1D45B∈/u1D445be the normal part of the\ndenominator of /u1D453. If/u1D453=/u1D70E(/u1D454) −/u1D454for some/u1D454∈/u1D439, then the normal\npartof thedenominatorof /u1D454dividesthepolynomial\ngcd/parenleftBigg/u1D451/productdisplay.1\n/u1D456=0/u1D70E/u1D456(/u1D463/u1D45B),/u1D451/productdisplay.1\n/u1D456=0/u1D70E−/u1D456−1(/u1D463/u1D45B)/parenrightBigg\n,\nwhere/u1D451:=disp(/u1D454)=disp(/u1D453) −1.P/r.sc/o.sc/o.sc/f.sc.Write/u1D453=/u1D462//u1D463∈/u1D439with/u1D462,/u1D463∈/u1D445and/u1D454/u1D450/u1D451(/u1D462,/u1D463)=1.\nAssume that the splitting factorization of /u1D463is(/u1D463/u1D460,/u1D463/u1D45B) ∈/u1D4452. If/u1D453=\n/u1D70E(/u1D454) −/u1D454for some/u1D454∈/u1D439, we also write /u1D454=/u1D45D//u1D45Ewith/u1D45D,/u1D45E∈/u1D445and\n/u1D454/u1D450/u1D451(/u1D45D,/u1D45E)=1 and let (/u1D45E/u1D460,/u1D45E/u1D45B)bethesplitting factorizationof /u1D45E.By\nLemma 7,we have /u1D451:=disp(/u1D45E/u1D45B)=disp(/u1D463/u1D45B) −1.Wenow show\n/u1D45E/u1D45B|gcd/parenleftBigg/u1D451/productdisplay.1\n/u1D456=0/u1D70E/u1D456(/u1D463/u1D45B),/u1D451/productdisplay.1\n/u1D456=0/u1D70E−/u1D456−1(/u1D463/u1D45B)/parenrightBigg\n. (1)\nWe first show that /u1D45E/u1D45B|/producttext.1/u1D451\n/u1D456=0/u1D70E/u1D456(/u1D463/u1D45B). The equality /u1D453=/u1D70E(/u1D454) −/u1D454\nimplies that\n/u1D454=/u1D463/u1D70E(/u1D454) −/u1D462\n/u1D463(2)\nApplying/u1D70Etobothsides of theaboveequationyields\n/u1D70E(/u1D454)=/u1D70E(/u1D463)/u1D70E2(/u1D454) −/u1D70E(/u1D462)\n/u1D70E(/u1D463).\nSubstituting /u1D70E(/u1D454)intheequation(2) yields\n/u1D454=1\n/u1D463/parenleftbigg\n/u1D463·/u1D70E(/u1D463)/u1D70E2(/u1D454) −/u1D70E(/u1D462)\n/u1D70E(/u1D463)−/u1D462/parenrightbigg\nAfter/u1D451repetitions oftheabove process,weget\n/u1D454=/u1D44E·/u1D70E/u1D451+1(/u1D454) −/u1D44F\n/u1D463/u1D70E(/u1D463) ···/u1D70E/u1D451(/u1D463)\nforsome/u1D44E,/u1D44F∈/u1D445. Thedenominator of the /u1D454is/u1D45E=/u1D45E/u1D460/u1D45E/u1D45B,whilethe\ndenominator of theright-hand sideof theabove equalityis a divi-\nsorof/u1D449:=/u1D463/u1D70E(/u1D463) ···/u1D70E/u1D451(/u1D463)/u1D70E/u1D451+1(/u1D45E).Then/u1D45E/u1D45B|/u1D449.Let(/u1D449/u1D460,/u1D449/u1D45B)bethe\nsplittingfactorizationof /u1D449.Then/u1D449/u1D45B=/u1D463/u1D45B/u1D70E(/u1D463/u1D45B)···/u1D70E/u1D451(/u1D463/u1D45B)/u1D70E/u1D451+1(/u1D45E/u1D45B)\nand/u1D45E/u1D45B|/u1D449/u1D45B. Since/u1D451/u1D456/u1D460/u1D45D(/u1D45E/u1D45B)=/u1D451, we have gcd (/u1D45E/u1D45B,/u1D70E/u1D451+1(/u1D45E/u1D45B))=1.\nHence we have /u1D45E/u1D45B|/u1D463/u1D45B/u1D70E(/u1D463/u1D45B)···/u1D70E/u1D451(/u1D463/u1D45B). The proofof thedivisibil-\nity/u1D45E/u1D45B|/producttext.1/u1D451\n/u1D456=0/u1D70E−/u1D456−1(/u1D463/u1D45B)is analogous. So the divisibility (1) holds.\nE/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc9. Let/u1D439=/u1D436(/u1D465)(/u1D4610,/u1D4611)witha/u1D436-automorphismdefined\nby/u1D70E(/u1D465)=/u1D465+1,/u1D70E(/u1D4610)=/u1D4611and/u1D70E(/u1D4611)=−6/u1D4610+5/u1D4611. Consider the\nequation\n/u1D453=636/u1D4613\n0+443/u1D4612\n0/u1D4611−1428/u1D4610/u1D4612\n1+565/u1D4613\n1\n2592(3/u1D4610−2/u1D4611)2(/u1D4610−/u1D4611)2(2/u1D4610−/u1D4611)(/u1D4610+/u1D4611)=/u1D70E(/u1D466.alt) −/u1D466.alt.\nWenow decidewhetherthisequationhasa solution in /u1D439.Firstly,we\ncandetectthattheirreduciblefactor 2/u1D4610−/u1D4611isspecialandotherirre-\nduciblefactorsarenormal.Thenthenormalpartofthedenom inator\nof/u1D453is/u1D435:=(3/u1D4610−2/u1D4611)2(/u1D4610−/u1D4611)2(/u1D4610+/u1D4611)with dispersion /u1D451=2.\nByTheorem8,the normalpartof thedenominatorofany soluti on/u1D454\ndivides the polynomial (/u1D4611+/u1D4610)3/u1D70E(/u1D4611+/u1D4610)2. Then we can make an\nansatz for/u1D454as\n/u1D454=/u1D448\n(/u1D4611+/u1D4610)3/u1D70E(/u1D4611+/u1D4610)2(2/u1D4610−/u1D4611),\nwhere/u1D448∈/u1D436(/u1D465)[/u1D4610,/u1D4611]satisfying therecurrenceequation\n(/u1D4611+/u1D4610)3/u1D70E(/u1D448) −/u1D70E(/u1D4611+/u1D4610)/u1D70E2(/u1D4611+/u1D4610)2/u1D448=/u1D44F,\nwhere/u1D44F=(/u1D4611+/u1D4610)2/u1D70E(/u1D4611+/u1D4610)(636/u1D4613\n0+443/u1D4612\n0/u1D4611−1428/u1D4610/u1D4612\n1+565/u1D4613\n1).\nWecanboundthedegreeof /u1D448whichis3.Thenweget /u1D448=/u1D4613\n0+4/u1D4612\n0+\n5/u1D4610/u1D4612\n1+2/u1D4613\n1bysolvingalineardifferencesystemforrationalsolutions .\nSowe have the rationalsolution\n/u1D454=2/u1D4611+/u1D4610\n36(/u1D4611−2/u1D4610)(/u1D4610+/u1D4611)(/u1D4611−/u1D4610)2.We will discuss how to estimate special factors in the denom-\ninator of/u1D454in next section for the difference fields generated by\nP-recursive sequences.\n2.2 Number ofirreducible specialpolynomials\nInSection2.1,wehavealreadyoutlinedtheprocedureforco mput-\ningthenormalpartofthedenominatorof /u1D454satisfying/u1D70E(/u1D454)−/u1D454=/u1D453.\nThechallengethatremainsistohandlethespecialpart.Asd emon-\nstratedinExample4,apeculiarsituationarises where thed enom-\ninator of/u1D454contains aspecialpolynomialthatdoesnotalready ap-\npear in the denominator of /u1D453. Hence, to determine the denomi-\nnator of/u1D454, it becomes necessary to identify all irreducible special\npolynomials.However,thecomputationofallspecialpolyn omials\nremains anunresolved issueatpresent. Inthissubsection, weaim\nto establish that there are at most /u1D45Birreducible special polynomi-\nals that do not pairwise differ by elements of /u1D43E∗. This provides a\ncrucialinsightintothelimiteddiversityofirreducibles pecialpoly-\nnomials. In Section 3.1,under certain assumptions, we will unveil\nthestructureofirreduciblespecialpolynomials.Webegin withthe\nfollowinglemmathatisadirectconsequenceofTheorem2.1. 12on\npage 114of[30].\nL/e.sc/m.sc/m.sc/a.sc 10. Suppose that Fis a/u1D70E-field with algebraically closed\nfield/u1D436of constants, and /u1D453∈ Fsatisfying that /u1D70Eℓ(/u1D453)=/u1D453for some\nℓ>0.Then/u1D453∈/u1D436.\nC/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc 11. Suppose that /u1D45D1,/u1D45D2,...,/u1D45D/u1D45Aare special polyno-\nmials that are linearly independent over /u1D43E,/u1D6FC2,...,/u1D6FC/u1D45A∈/u1D43E. Then\n/u1D45D1+/u1D6FC2/u1D45D2+ ···+/u1D6FC/u1D45A/u1D45D/u1D45Ais a special polynomial if and only if /u1D6FC2=\n···=/u1D6FC/u1D45A=0.\nP/r.sc/o.sc/o.sc/f.sc.Itsufficestoshowthenecessarypart.Supposethat /u1D45D1+\n/u1D6FC2/u1D45D2+···+/u1D6FC/u1D45A/u1D45D/u1D45Aisaspecialpolynomial.Thenthereisapositive\nintegerℓand/u1D6FE,/u1D6FD1,...,/u1D6FD/u1D45A∈/u1D43Esuch that/u1D70Eℓ(/u1D45D/u1D456)=/u1D6FD/u1D456/u1D45D/u1D456and\n/u1D70Eℓ(/u1D45D1+/u1D6FC2/u1D45D2+···+/u1D6FC/u1D45A/u1D45D/u1D45A)=/u1D6FE(/u1D45D1+/u1D6FC2/u1D45D2+···+/u1D6FC/u1D45A/u1D45D/u1D45A).\nA straightforward calculation reveals that /u1D6FD1=/u1D6FE, and/u1D70Eℓ(/u1D6FC/u1D456)/u1D6FD/u1D456=\n/u1D6FD1/u1D6FC/u1D456for all 2≤/u1D456≤/u1D45A. This implies that /u1D70Eℓ(/u1D6FC/u1D456/u1D45D/u1D456)=/u1D6FD1/u1D6FC/u1D456/u1D45D/u1D456and\nconsequently, /u1D70Eℓ(/u1D6FC/u1D456/u1D45D/u1D456//u1D45D1)=/u1D6FC/u1D456/u1D45D/u1D456//u1D45D1,forall2 ≤/u1D456≤/u1D45A.According\nto Lemma 10, /u1D6FC/u1D456/u1D45D/u1D456//u1D45D1∈/u1D436. For each 2 ≤/u1D456≤/u1D45A, as/u1D45D/u1D456and/u1D45D1are\nlinearly independent over /u1D43E,itfollowsthat /u1D6FC/u1D456=0.\nP/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc 12. Suppose that /u1D45D1,...,/u1D45D/u1D45Aare irreduciblespecial\npolynomials that are not pairwise shift equivalent. Denote byℓ/u1D456the\nsmallest positive integer such that /u1D45D/u1D456|/u1D70Eℓ/u1D456(/u1D45D/u1D456). Then the/u1D70E/u1D457(/u1D45D/u1D456),/u1D456=\n1,...,/u1D45A,/u1D457 =0,...,ℓ/u1D456−1are algebraicallyindependentover /u1D43E.\nP/r.sc/o.sc/o.sc/f.sc.Set/u1D441=lcm(ℓ1,...,ℓ/u1D45A).Then/u1D70E/u1D441(/u1D70E/u1D457(/u1D45D/u1D456))=/u1D6FC/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)\nfor some/u1D6FC/u1D456,/u1D457∈/u1D43E. Suppose on the contrary that the /u1D70E/u1D457(/u1D45D/u1D456),/u1D456=\n1,...,/u1D45A,/u1D457 =0,...,ℓ/u1D456−1 are algebraically dependent over /u1D43E. Due\nto thedifference analogueof Kolchin-Ostrowski theorem (se e [22,\n32]), there are integers /u1D451/u1D456,/u1D457,/u1D456=1,...,/u1D45A,/u1D457 =0,...,ℓ/u1D456−1, not all\nzero, and/u1D6FD∈/u1D43E∗such that\n/u1D45A/productdisplay.1\n/u1D456=1ℓ/u1D456−1/productdisplay.1\n/u1D457=0/u1D6FC/u1D451/u1D456,/u1D457\n/u1D456,/u1D457=/u1D70E/u1D441(/u1D6FD)\n/u1D6FD.Since/u1D70E/u1D441(/producttext.1\n/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457)=/producttext.1\n/u1D456,/u1D457/u1D6FC/u1D451/u1D456,/u1D457\n/u1D456,/u1D457/producttext.1\n/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457,wehavethat\n/u1D70E/u1D441/parenleftBigg/producttext.1\n/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457\n/u1D6FD/parenrightBigg\n=/producttext.1\n/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457\n/u1D6FD.\nDue to Lemma 10,/producttext.1\n/u1D456,/u1D457/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457=/u1D450/u1D6FDfor some/u1D450∈/u1D436. Denote\n/u1D4461={(/u1D456,/u1D457) |/u1D451/u1D456,/u1D457>0}and/u1D4462={(/u1D456,/u1D457) |/u1D451/u1D456,/u1D457<0}.Since the/u1D451/u1D456,/u1D457are\nnot all zero and /u1D4610,...,/u1D461/u1D45B−1are algebraically independent over /u1D43E,\nneither/u1D4461nor/u1D4462is empty.Thisleads to\n/productdisplay.1\n(/u1D456,/u1D457)∈/u1D4461/u1D70E/u1D457(/u1D45D/u1D456)/u1D451/u1D456,/u1D457=/u1D450/u1D6FD/productdisplay.1\n(/u1D456,/u1D457)∈/u1D4462/u1D70E/u1D457(/u1D45D/u1D456)−/u1D451/u1D456,/u1D457.\nChoose(/u1D4561,/u1D4571) ∈/u1D4461. Then/u1D70E/u1D4571(/u1D45D/u1D4561)divides/producttext.1\n(/u1D456,/u1D457)∈/u1D4462/u1D70E/u1D457(/u1D45D/u1D456)−/u1D451/u1D456,/u1D457\nandthusthereexists (/u1D4562,/u1D4572) ∈/u1D4462suchthat/u1D70E/u1D4571(/u1D45D/u1D4561)divides/u1D70E/u1D4572(/u1D45D/u1D4562).\nAs both/u1D70E/u1D4571(/u1D45D/u1D4561)and/u1D70E/u1D4572(/u1D45D/u1D4562)are irreducible, /u1D70E/u1D4571(/u1D45D/u1D4561)=/u1D6FE/u1D70E/u1D4572(/u1D45D/u1D4562)\nfor some/u1D6FE∈/u1D43E. If/u1D4561=/u1D4562then 0≤/u1D4571≠/u1D4572≤ℓ/u1D4561−1. Without\nlossofgenerality,assume /u1D4572>/u1D4571.Then/u1D70E/u1D4572−/u1D4571(/u1D45D/u1D4561)=/u1D45D/u1D4561//u1D70E−/u1D4571(/u1D6FE),\nwhich contradicts the minimality of ℓ/u1D4561. If/u1D4561≠/u1D4562then/u1D45D/u1D4561and/u1D45D/u1D4562\nareshift equivalent,contradictingtheinitial assumptio n.\nSince tr.deg(/u1D439//u1D43E)=/u1D45B,we have thefollowingcorollary.\nC/o.sc/r.sc/o.sc/l.sc/l.sc/a.sc/r.sc/y.sc 13. Suppose that /u1D45D1,...,/u1D45D/u1D45Aare irreducible special\npolynomialsthatarenotpairwiseshiftequivalent.Then/summationtext.1/u1D45A\n/u1D456=1ℓ/u1D456≤/u1D45B,\nwhereℓ/u1D456isthe smallestpositive integersuch that /u1D45D/u1D456|/u1D70Eℓ/u1D456(/u1D45D/u1D456).\n3 THE P-RECURSIVECASE\nP-recursive sequences, introduced by Stanley [45], satisf y linear\nrecurrenceequationswithpolynomialcoefficients.Thegene rating\nfunctionofaP-recursivesequenceisaD-finitefunction,wh ichsat-\nisfies a linear differential equations with polynomial coeffic ients.\nThis class of sequences has been extensively studied in comb ina-\ntorics[43,48]andsymboliccomputation[28,29]togetherw ithits\ngeneratingfunctions.Inthissection,wewillfocusonpara llelsum-\nmationin difference fields generated byP-recursive sequenc es.\nLet/u1D439be the field /u1D436(/u1D465)(/u1D4610,...,/u1D461/u1D45B−1)with a/u1D436-automorphism /u1D70E\nsatisfying that /u1D70E(/u1D465)=/u1D465+1,/u1D70E(/u1D4610)=/u1D4611,...,/u1D70E(/u1D461/u1D45B−2)=/u1D461/u1D45B−1, and\n/u1D70E(/u1D461/u1D45B−1)=/u1D44E0/u1D4610+···+/u1D44E/u1D45B−1/u1D461/u1D45B−1,\nwhere/u1D44E0,...,/u1D44E/u1D45B−1∈/u1D436(/u1D465)and/u1D44E0≠0. So/u1D70Eis a/u1D436-automorphism\nof the ring/u1D445=/u1D436(/u1D465)[/u1D4610,...,/u1D461/u1D45B−1]. We still assume in this section\nthattheconstant fieldof (/u1D439,/u1D70E)is thefield/u1D436.Inorder tostudythe\nindefinitesummationproblemin /u1D439,wefirstaddresstwobasicques-\ntions onspecialand normal polynomials.InSection 3.1,we p rove\nsome structural properties on special polynomials under ce rtain\nassumptions. In Section 3.2, we will answer the question of d ecid-\ning whether two irreduciblepolynomials in /u1D445are/u1D70E-equivalent or\nnot.\n3.1 Degrees ofirreducible specialpolynomials\nIntheP-recursivecase,byLemma14below,computingallspe cial\npolynomialsof degree /u1D45Ais equivalent tocomputingall hypergeo-\nmetricsolutionsof the /u1D45Athsymmetric power ofthesystem\n/u1D70E(/u1D44C)=/u1D434/u1D44C (3)or/u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cforsome/u1D460>1,where\n/u1D434=/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA0 1\n0 1\n......\n0 1\n/u1D44E0/u1D44E1/u1D44E2... /u1D44E/u1D45B−1/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA\nand/u1D434(/u1D460)=/u1D70E/u1D460−1(/u1D434).../u1D70E(/u1D434)/u1D434. Algorithms for computing all hy-\npergeometric solutions of a given linear difference equatio n are\nknown, for example, refer to [34]. Corollary 13 establishes the ex-\nistence of a degree bound for all irreducible special polyno mials.\nHowever, by the absence of a known degree bound, the computa-\ntion of all irreducible special polynomials remains an unre solved\nchallenge. Inthissubsection,weaimtoprovethatwhen/summationtext.1/u1D45A\n/u1D456=1ℓ/u1D456=\n/u1D45Bwithℓ/u1D456as defined in Corollary 13, all irreducible special poly-\nnomials are linear in /u1D4610,/u1D4611,...,/u1D461/u1D45B−1. Consequently, in this specific\ncase, the degree of all irreducible special polynomials is e xactly\nequalto1andthuswecancomputeallirreduciblespecialpol yno-\nmials.\nL/e.sc/m.sc/m.sc/a.sc14. Allspecialpolynomialsare homogeneous.\nP/r.sc/o.sc/o.sc/f.sc.Suppose that /u1D45Dis a special polynomial and is not ho-\nmogeneous. Write /u1D45D=/summationtext.1/u1D45A\n/u1D456=0/u1D45D/u1D456where/u1D45D/u1D456is the/u1D456-th homogeneous\npart of/u1D45Dand/u1D45D/u1D45A≠0. Assume that /u1D70Eℓ(/u1D45D)=/u1D6FC/u1D45Dfor some nonzero\n/u1D6FC∈/u1D436(/u1D465). Then/u1D70Eℓ(/u1D45D)=/summationtext.1/u1D45A\n/u1D456=0/u1D70Eℓ(/u1D45D/u1D456)=/u1D6FC/u1D45D=/summationtext.1/u1D45A\n/u1D456=0/u1D6FC/u1D45D/u1D456. Note that\n/u1D70Eℓ(/u1D45D/u1D456)isalsohomogeneousofdegree /u1D456.Wehavethat /u1D70Eℓ(/u1D45D/u1D456)=/u1D6FC/u1D45D/u1D456\nfor all 0≤/u1D456≤/u1D45A. Since/u1D45Dis not homogeneous, there is an /u1D4560such\nthat/u1D45D/u1D4560≠0.Hence/u1D70Eℓ(/u1D45D/u1D4560//u1D45D/u1D45A)=/u1D45D/u1D4560//u1D45D/u1D45A.AccordingtoLemma10,\n/u1D45D/u1D4560//u1D45D/u1D45A∈/u1D436, which contradicts the fact that the numerator and\ndenominator of /u1D45D/u1D4560//u1D45D/u1D45Ahave different degrees.\nWestartwiththe /u1D436-finitecase.Inthiscase,wewilldemonstrate\nthat the degree of all irreducible special polynomials is al ways\nequalto1,withoutrequiring theassumptionthat/summationtext.1/u1D45A\n/u1D456=1ℓ/u1D456=/u1D45B.\nP/r.sc/o.sc/p.sc/o.sc/s.sc/i.sc/t.sc/i.sc/o.sc/n.sc 15. Supposethat /u1D434∈GL/u1D45B(/u1D436).Thenall irreducible\nspecialpolynomialsare linearin /u1D4610,/u1D4611,...,/u1D461/u1D45B−1.\nP/r.sc/o.sc/o.sc/f.sc.Let/u1D435∈GL/u1D45B(/u1D436)such that/u1D435/u1D434/u1D435−1=diag(/u1D43D1,/u1D43D2,...,/u1D43Dℓ),\nwhere/u1D43D/u1D456is a Jordan blockof order /u1D45B/u1D456. We claim that /u1D45B/u1D456=1 for all\n1≤/u1D456≤ℓ. Without loss of generality, assume that /u1D45B1>1 and/u1D6FC1\nis the eigenvalue of /u1D43D1. Set¯/u1D447=(¯/u1D4610,...,¯/u1D461/u1D45B−1)/u1D461=/u1D435(/u1D4610,...,/u1D461/u1D45B−1)/u1D461.\nThen/u1D70E(¯/u1D447)=/u1D435/u1D434/u1D435−1¯/u1D447. Therefore /u1D70E(¯/u1D461/u1D45B1−1)=/u1D6FC1¯/u1D461/u1D45B1−1+¯/u1D461/u1D45B1and\n/u1D70E(¯/u1D461/u1D45B1)=/u1D6FC1¯/u1D461/u1D45B1. From these, it follows that /u1D70E(¯/u1D461/u1D45B1−1\n¯/u1D461/u1D45B1)=¯/u1D461/u1D45B1−1\n¯/u1D461/u1D45B1+1\n/u1D6FC1.\nConsequently,\n/u1D70E/parenleftbigg¯/u1D461/u1D45B1−1\n¯/u1D461/u1D45B1−/u1D465\n/u1D6FC1/parenrightbigg\n=¯/u1D461/u1D45B1−1\n¯/u1D461/u1D45B1−/u1D465\n/u1D6FC1.\nIn other words, ¯/u1D461/u1D45B1−1/¯/u1D461/u1D45B1−/u1D465//u1D6FC1∈/u1D436, leading to a contradiction.\nThisprovesourclaim.Therefore /u1D435/u1D434/u1D435−1=diag(/u1D6FC1,...,/u1D6FC/u1D45B),where\n/u1D6FC/u1D456∈/u1D436.Finally,supposethat /u1D45Disanirreduciblespecialpolynomial.\nNote that/u1D45Dcan be expressed as a polynomial in ¯/u1D4610,¯/u1D4611,...,¯/u1D461/u1D45B−1.\nCorollary11impliesthat /u1D45Disamonomialin ¯/u1D4610,¯/u1D4611,...,¯/u1D461/u1D45B−1.Hence\n/u1D45D=/u1D6FD¯/u1D461/u1D456for some 0 ≤/u1D456≤/u1D45B−1 and/u1D6FD∈/u1D43E, and so it is linear in\n/u1D4610,/u1D4611,...,/u1D461/u1D45B−1.R/e.sc/m.sc/a.sc/r.sc/k.sc16. IntheproofofProposition15,thespecialpolynomials\n¯/u1D4610,...,¯/u1D461/u1D45B−1are not pairwise shift equivalent and then the condition/summationtext.1/u1D45A\n/u1D456=1ℓ/u1D456=/u1D45Bis automatically satisfied. In fact, suppose /u1D70E(¯/u1D461/u1D4561)=/u1D6FD¯/u1D461/u1D4562\nforsome0≤/u1D4561≠/u1D4562≤/u1D45B−1and/u1D6FD∈/u1D43E.Since/u1D70E(¯/u1D461/u1D4561)=/u1D6FC¯/u1D461/u1D4561forsome\n/u1D6FC∈/u1D43E, it follows that ¯/u1D4611,¯/u1D4612are linearly dependent over /u1D43E, which\ncontradicts the fact that ¯/u1D4610,...,¯/u1D461/u1D45B−1are algebraically independent\nover/u1D43E.\nBefore proceeding to the general case, let’s recall some fun da-\nmentalresultsfromdifference Galoistheory.Fordetailedi nforma-\ntion, readers can refer to Chapter 1 of [46]. Let Rbe the Picard-\nVessiotringfor /u1D70E(/u1D44C)=/u1D434/u1D44Cover/u1D43E,where/u1D434isgivenasin(3).In R,\nthereexist idempotents /u1D4520,/u1D4521,...,/u1D452/u1D460−1such that\nR=R0⊕ R1⊕ ···⊕ R /u1D460−1,\nwhereR/u1D456=/u1D452/u1D456RandR/u1D456is a domain. Moreover, R/u1D456serves as the\nPicard-Vessiot ringfor /u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cover/u1D43Ewith\n/u1D434(/u1D460)=/u1D70E/u1D460−1(/u1D434).../u1D70E(/u1D434)/u1D434.\nLet/u1D43AbetheGaloisgroupof /u1D70E(/u1D44C)=/u1D434/u1D44Cover/u1D43Eand/u1D43BbetheGalois\ngroup of/u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cover/u1D43E. By Corollary 1.17 on page 13 of\n[46],[/u1D43A:/u1D43B]=/u1D460and consequently, /u1D43Bcontains/u1D43A◦, the identity\ncomponentof /u1D43A.Ontheotherhand,duetoProposition1.20, R/u1D456is\na trivial/u1D43B-torsor which implies that /u1D43Bis connected since R/u1D456is a\ndomain.Hence /u1D43B=/u1D43A◦.\nL/e.sc/m.sc/m.sc/a.sc17. Supposethat /u1D45D1,...,/u1D45D/u1D45Aarespecialpolynomials.Then\nthereexists a fundamentalmatrix Z ∈GL/u1D45B(R)of/u1D70E(/u1D44C)=/u1D434/u1D44Csuch\nthat/u1D45D/u1D456(Z/u1D457)is invertible in Rfor all1≤/u1D456≤/u1D45A,1≤/u1D457≤/u1D45B, where\nZ/u1D457denotesthe/u1D457thcolumn of Z.\nP/r.sc/o.sc/o.sc/f.sc.Let/u1D441be a positive integer such that /u1D45D/u1D456|/u1D70E/u1D441(/u1D45D/u1D456)for\nall 1≤/u1D456≤/u1D45Aand let/u1D45E=(/producttext.1/u1D45A\n/u1D456=1/u1D45D/u1D456)/u1D70E(/producttext.1/u1D45A\n/u1D456=1/u1D45D/u1D456)···/u1D70E/u1D441−1(/producttext.1/u1D45A\n/u1D456=1/u1D45D/u1D456).\nThen/u1D70E(/u1D45E)=/u1D6FC/u1D45Eforsome/u1D6FC∈/u1D43E.Wefirstshow thelemma for /u1D45E.\nLetZ ∈GL/u1D45B(R)be a fundamental matrix of /u1D70E(/u1D44C)=/u1D434/u1D44C. We\nclaim that there exists an /u1D440∈GL/u1D45B(/u1D436)such that/u1D45E((Z/u1D440)/u1D457)≠\n0. Letu=(/u1D4621,...,/u1D462/u1D45B)/u1D461be a vector with indeterminate entries.\nSinceZis invertible, as a polynomial in R[u],/u1D45E(Zu)≠0. Write\n/u1D45E(Zu)=/summationtext.1/u1D451\n/u1D457=1/u1D453/u1D457(u)m/u1D457, where/u1D453/u1D457(u) ∈/u1D436[u]andm1,...,m/u1D451∈ R\nare linearly independent over /u1D436. As/u1D45E(Zu)≠0, at least one of\n/u1D4531(u),...,/u1D453/u1D451(u)is not zero, say /u1D453/u1D4571(u)≠0. Set/u1D448tobethe Zariski\nopen subset of /u1D436/u1D45Bconsisting of all ain/u1D436/u1D45Bsuch that/u1D453/u1D4571(a)≠0.\nThen/u1D448×···×/u1D448isanon-emptyZariskiopensubsetof /u1D436/u1D45B×/u1D45B,where\nthedirectproducttakes /u1D45Btimes.Let/u1D440∈/u1D448×···×/u1D448besuchthat\ndet(/u1D440)≠0.Such/u1D440exists because /u1D448×···×/u1D448is Zariski dense in\n/u1D436/u1D45B×/u1D45B.Thenforeach column cof/u1D440,itfollowsthat /u1D453/u1D4571(c)≠0,and\nthus/u1D45E(Zc)≠0. Since/u1D440is invertible, Z/u1D440is also a fundamental\nmatrixof/u1D70E(/u1D44C)=/u1D434/u1D44C.This proves ourclaim.\nLetcbeacolumnof /u1D440. Then\n/u1D70E(/u1D45E(Zc))=/u1D45E/u1D70E(/u1D434Zc)=/u1D70E(/u1D45E)(Zc)=/u1D6FC/u1D45E(Zc)\nwhere/u1D45E/u1D70Edenotes the polynomial obtained by applying /u1D70Eto the\ncoefficients of /u1D45E. Hence,/u1D45E(Zc)generates a nonzero /u1D70E-ideal inR.\nSinceRis/u1D70E-simple, this ideal mustbeequalto Rand thus/u1D45E(Zc)\nis invertible in R. Finally, for each 1 ≤/u1D456≤/u1D45A, since/u1D45E(Zc)=\n/u1D45D/u1D456(Zc)ℎ/u1D456forsomeℎ/u1D456∈ R,/u1D45D/u1D456(Zc)is invertible in R.\nL/e.sc/m.sc/m.sc/a.sc 18. Suppose that there exist irreducible special polyno-\nmials/u1D45D1,...,/u1D45D/u1D45Athat are not pairwise shift equivalent, satisfyingthat/summationtext.1/u1D45A\n/u1D456=1ℓ/u1D456=/u1D45B, whereℓ/u1D456is the smallest positive integer such that\n/u1D45D/u1D456|/u1D70Eℓ/u1D456(/u1D45D/u1D456).Thendim(/u1D43A)=/u1D45B.\nP/r.sc/o.sc/o.sc/f.sc.Itsuffices toshow that dim (/u1D43A◦)=/u1D45B.Set\n/u1D45Eℓ0+ℓ1+···+ℓ/u1D456+/u1D457=/u1D70E/u1D457−1(/u1D45D/u1D456+1)\nwhere 0 ≤/u1D456≤/u1D45A−1, 1≤/u1D457≤ℓ/u1D456, andℓ0=0. By Lemma 17,\nthere exists a fundamental matrix Z ∈GL/u1D45B(R)of/u1D70E(/u1D44C)=/u1D434/u1D44C\nsuch that/u1D45E/u1D456(Z/u1D457)is invertible in Rfor all 1 ≤/u1D456,/u1D457≤/u1D45B, where\nZ/u1D457denotes the /u1D457th columnof Z. Consider R0, the Picard-Vessiot\nring for/u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cover/u1D43E. LetFbe the field of fractions\nofR0. Since/u1D43A◦is the Galois group of /u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cover/u1D43E,\ndim(/u1D43A◦)=tr.deg(F//u1D43E). Note that/u1D4520Zis a fundamental matrix\nof/u1D70E/u1D460(/u1D44C)=/u1D434(/u1D460)/u1D44Cand/u1D4520/u1D45E/u1D456(Z/u1D457)=/u1D45E/u1D456(/u1D4520Z/u1D457)isinvertiblein R0.Set\n/u1D441=lcm(ℓ1,...,ℓ/u1D45A). Then/u1D45E/u1D456|/u1D70E/u1D460/u1D441(/u1D45E/u1D456)for all 1≤/u1D456≤/u1D45B.Suppose\nthat/u1D70E/u1D460/u1D441(/u1D45E/u1D456)=/u1D6FC/u1D456/u1D45E/u1D456with/u1D6FC/u1D456∈/u1D43E.Thenfor each 1 ≤/u1D456≤/u1D45B,it holds\nthat/u1D70E/u1D460/u1D441(/u1D45E/u1D456(/u1D4520Z/u1D457))=/u1D6FC/u1D456/u1D45E/u1D456(/u1D4520Z/u1D457)for all1≤/u1D457≤/u1D45B.\nWe claim that for each 1 ≤/u1D457≤/u1D45B,/u1D45E1(/u1D4520Z/u1D457),...,/u1D45E/u1D45B(/u1D4520Z/u1D457)are\nalgebraically independent over /u1D43E. Assume on the contrary that\n/u1D45E1(/u1D4520Z/u1D457),...,/u1D45E/u1D45B(/u1D4520Z/u1D457)arealgebraicallydependentover /u1D43E.Using\nan argument similar to that in the proof of Proposition 12, we ob-\ntain integers /u1D451/u1D456,notall zero,and a nonzero /u1D6FD∈/u1D43Esuchthat\n/u1D70E/u1D460/u1D441/parenleftBigg/producttext.1\n/u1D456/u1D45E/u1D451/u1D456\n/u1D456\n/u1D6FD/parenrightBigg\n=/producttext.1\n/u1D456/u1D45E/u1D451/u1D456\n/u1D456\n/u1D6FD.\nDue to Lemma 10,/producttext.1\n/u1D456/u1D45E/u1D451/u1D456\n/u1D456=/u1D450/u1D6FDfor some/u1D450∈/u1D436. In other words,\n/u1D45E1,...,/u1D45E/u1D45Barealgebraicallydependentover /u1D43E.Thiscontradictsthe\nconclusionof Proposition12. Theclaim is established.\nNow,for 1 ≤/u1D4571≠/u1D4572≤/u1D45B,wehave that\n/u1D70E/u1D460/u1D441(/u1D45E/u1D456(/u1D4520Z/u1D4571)//u1D45E/u1D456(/u1D4520Z/u1D4572))=/u1D45E/u1D456(/u1D4520Z/u1D4571)//u1D45E/u1D456(/u1D4520Z/u1D4572).\nBy Lemma 10 (replacing /u1D70Ewith/u1D70E/u1D460),/u1D45E/u1D456(/u1D4520Z/u1D4571)=/u1D450/u1D456,/u1D4571,/u1D4572/u1D45E/u1D456(/u1D4520Z/u1D4572)\nfor all 1 ≤/u1D456≤/u1D45B, where/u1D450/u1D456,/u1D4571,/u1D4572∈/u1D436. Denote by ˜Fthe subfield\nofFgenerated by all /u1D45E/u1D456(/u1D4520Z/u1D457)over/u1D43E. Then the previous discus-\nsion implies that tr .deg(˜F//u1D43E)=/u1D45B.Note that for each 1 ≤/u1D457≤/u1D45B,\nevery entry of /u1D4520Z/u1D457is algebraic over /u1D43E(/u1D45E1(/u1D4520Z/u1D457),...,/u1D45E/u1D45B(/u1D4520Z/u1D457))\n(and thus algebraic over ˜F), because/u1D45E1(/u1D4520Z/u1D457),...,/u1D45E/u1D45B(/u1D4520Z/u1D457)are\nalgebraically independent over /u1D43Eand they are polynomial in the\nentries of/u1D4520Z/u1D457. HenceFis a finite algebraic extension of ˜F, as\nF=/u1D43E(/u1D4520Z). So tr.deg(F//u1D43E)=/u1D45Band then dim (/u1D43A◦)=/u1D45B. Conse-\nquently,dim (/u1D43A)=/u1D45B.\nT/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 19. Under the same assumption as in Lemma 18, all\nirreduciblespecialpolynomialsare linearin /u1D4610,/u1D4611,...,/u1D461/u1D45B−1.\nP/r.sc/o.sc/o.sc/f.sc.Let/u1D45E/u1D456,/u1D6FC/u1D456beasintheproofofLemma 18.Wefirstshow\nthat/u1D43A◦isatorus.DuetoLemma18,dim (/u1D43A◦)=/u1D45B.Hencetherank\nof/u1D44B(/u1D43A◦), the group of characters of /u1D43A◦(which is a free abelian\ngroup), is at most /u1D45B. As/u1D70E/u1D460/u1D441(/u1D45E/u1D456(/u1D4520Z1))=/u1D6FC/u1D456/u1D45E/u1D456(/u1D4520Z1), for each\n/u1D454∈/u1D43A◦,/u1D70E/u1D460/u1D441(/u1D454(/u1D45E/u1D456(/u1D4520Z1)))=/u1D6FC/u1D456/u1D454(/u1D45E/u1D456(/u1D4520Z1)).Hence/u1D454(/u1D45E/u1D456(/u1D4520Z1))=\n/u1D712/u1D456(/u1D454)/u1D454(/u1D45E/u1D456(/u1D4520Z1)), where/u1D712/u1D456(/u1D454) ∈/u1D436. In other words, /u1D45E/u1D456(/u1D4520Z1)in-\nduces a character /u1D712/u1D456∈/u1D44B(/u1D43A◦). Suppose that there are integers /u1D451/u1D456,\nnotallzero,suchthat/producttext.1\n/u1D456/u1D712/u1D451/u1D456\n/u1D456=id,whereidistheunitaryof /u1D44B(/u1D43A◦).\nThenfor all /u1D454∈/u1D43A◦,\n/u1D454/parenleftBigg/productdisplay.1\n/u1D456/u1D45E/u1D456(/u1D4520Z1)/u1D451/u1D456/parenrightBigg\n=/productdisplay.1\n/u1D456/u1D712/u1D451/u1D456\n/u1D456(/u1D454)/productdisplay.1\n/u1D456/u1D45E/u1D456(/u1D4520Z1)/u1D451/u1D456=/productdisplay.1\n/u1D456/u1D45E/u1D456(/u1D4520Z1)/u1D451/u1D456.TheGaloiscorrespondence(see,forexample,Lemma1.28onp age20\nof[46])impliesthat/producttext.1\n/u1D456/u1D45E/u1D456(/u1D4520Z1)/u1D451/u1D456∈/u1D43E,whichcontradictsthefact\nthat/u1D45E1(/u1D4520Z1),...,/u1D45E/u1D45B(/u1D4520Z1)arealgebraicallyindependentover /u1D43E.\nThereforetherankof /u1D44B(/u1D43A◦)isexactlyequalto /u1D45B.Let ˜/u1D7121,...,˜/u1D712/u1D45Bbe\nabase of/u1D44B(/u1D43A◦), as a freeabelian group.Consider themorphism\n/u1D719:/u1D43A◦−→GL1(/u1D436)/u1D45B\n/u1D454/u\\i∈∞A6.e\\dl−→ (˜/u1D7121(/u1D454),...,˜/u1D712/u1D45B(/u1D454)).\nThen/u1D719issurjectivewithfinitekernelbecausedim (/u1D43A◦)=/u1D45B.More-\nover,duetoLemmaB.20of[18],ker (/u1D719)isgeneratedasanalgebraic\ngroup by all unipotent elements of /u1D43A◦. Notethat if ℎ∈GL/u1D45B(/u1D436)is\nunipotent then ℎis of finite order if and only if ℎ=/u1D43C,the identity\nmatrix.Soker (/u1D719)={/u1D43C}and/u1D719isanisomorphism.Thisprovesthat\n/u1D43A◦is a torus.\nByTheorem2.1 of [24],there exists a /u1D447∈GL/u1D45B(/u1D43E)suchthat\n/u1D70E/u1D460(/u1D447)/u1D434(/u1D460)/u1D447−1=diag(/u1D44F1,...,/u1D44F/u1D45B),\nwhere/u1D44F/u1D456∈/u1D43E.Set(¯/u1D4610,...,¯/u1D461/u1D45B−1)/u1D461=/u1D447(/u1D4610,...,/u1D461/u1D45B−1)/u1D461.Then/u1D70E/u1D460(¯/u1D461/u1D456)=\n/u1D44F/u1D456¯/u1D461/u1D456for all 0 ≤/u1D456≤/u1D45B−1. In other words, ¯/u1D4610,...,¯/u1D461/u1D45B−1are spe-\ncial polynomials. Finally, using an argument similar to the proof\nof Proposition 15, it follows that every irreducible specia l polyno-\nmialis linear in /u1D4610,/u1D4611,...,/u1D461/u1D45B−1.\n3.2/u1D70E-Equivalence Problem\nWe now present a method for deciding whether two irreducible\npolynomials /u1D45D,/u1D45E∈/u1D445=/u1D43E[/u1D4610,...,/u1D461/u1D45F−1]are/u1D70E-equivalent ornot.\nIf one of/u1D45Dand/u1D45Eis special, say /u1D45D, then there exists a minimal\npositive integer /u1D45A(not greater than /u1D45Bby Corollalry 13) such that\n/u1D45D|/u1D70E/u1D45A(/u1D45D). To decide whether /u1D45Dand/u1D45Eare/u1D70E-equivalent, it suffices\nto check whether /u1D45Eis associate over /u1D43Eto one of elements in the\nset{/u1D45D,/u1D70E(/u1D45D),...,/u1D70E/u1D45A−1(/u1D45D)}.Itremainstoconsiderthecaseinwhich\nboth/u1D45Dand/u1D45Eare normal.\nWe now assume that /u1D45D,/u1D45E∈/u1D445are irreducible and normal in /u1D445.\nWewant todecidewhether there exist /u1D456∈Zand/u1D462∈/u1D43E\\{0}such\nthat/u1D70E/u1D456(/u1D45D)=/u1D462/u1D45E.Observe firstthattherecanbeatmostonesuch /u1D456.\nFor,if/u1D456,/u1D456′and/u1D462,/u1D462′aresuchthat /u1D70E/u1D456(/u1D45D)=/u1D462/u1D45Eand/u1D70E/u1D456′(/u1D45D)=/u1D462′/u1D45E,then\n/u1D70E/u1D456(/u1D45D)//u1D70E/u1D456′(/u1D45D)=/u1D462//u1D462′,so/u1D45D|/u1D70E/u1D456′−/u1D456(/u1D45D),andsince/u1D45Disnotspecial,we\nmusthave/u1D456=/u1D456′.Observealsothatforagivencandidate /u1D456∈Z,itis\neasytocheckwhetherthereexists a /u1D462with/u1D70E/u1D456(/u1D45D)=/u1D462/u1D45E.Therefore,\nitsuffices todetermine a finitelistof candidates for /u1D456.\nLet/u1D43F,/u1D440∈/u1D43E[/u1D446]be the (unique) monic minimal order anni-\nhilating operators of /u1D45Dand/u1D45E, respectively. Let /u1D460be their order.\nNote that/u1D45Dand/u1D45Ecannot be shift equivalent if the orders of /u1D43F\nand/u1D440are distinct. Write /u1D43F=/u1D446/u1D460+ℓ/u1D460−1/u1D446/u1D460−1+ ··· +ℓ0and/u1D440=\n/u1D446/u1D460+/u1D45A/u1D460−1/u1D446/u1D460−1+···+/u1D45A0.By theminimality oftheorder of /u1D43Fand\n/u1D440,wehaveℓ0,/u1D45A0≠0.\nForevery/u1D456∈Z,themonicminimal orderannihilating operator\nof/u1D70E/u1D456(/u1D45D)is\n/u1D43F(/u1D456):=/u1D446/u1D460+/u1D70E/u1D456(ℓ/u1D460−1)/u1D446/u1D460−1+···+/u1D70E/u1D456(ℓ0),\nand for every /u1D462∈/u1D43E\\ {0}, the monic minimal order annihilating\noperatorof1\n/u1D462/u1D45Eis\n1\n/u1D70E/u1D460(/u1D462)/u1D440/u1D462=/u1D446/u1D460+/u1D45A/u1D460−1/u1D70E/u1D460−1(/u1D462)\n/u1D70E/u1D460(/u1D462)/u1D446/u1D460−1+···+/u1D45A0/u1D462\n/u1D70E/u1D460(/u1D462).\nA necessary condition for a pair (/u1D456,/u1D462)to be a solutionto the shift\nequivalence problem is that /u1D43F(/u1D456)=1\n/u1D70E/u1D460(/u1D462)/u1D440/u1D462.Therefore, for everysuch pair wemusthave\n/u1D70E/u1D456(ℓ/u1D458)\n/u1D45A/u1D458=/u1D70E/u1D458(/u1D462)\n/u1D70E/u1D460(/u1D462)\nsimultaneouslyforall /u1D458∈ {0,...,/u1D460}.\nObserve that /u1D43Fmust have at least three terms. If it had only\ntwo terms, we would have /u1D43F=/u1D446/u1D460+ℓ0. This means /u1D70E/u1D460(/u1D45D)=−ℓ0/u1D45D,\nandthisisacontradictionto /u1D45Dnotbeingspecial.Wecantherefore\nassumethat /u1D43Fhasatleastthreeterms.Wemayfurtherassumethat\nthecoefficientof /u1D446/u1D458in/u1D43Fisnonzeroifandonlyifthecoefficientof\n/u1D446/u1D458in/u1D440is nonzero, because if this is not the case, then the shift\nequivalence problemhas no solution.\nL/e.sc/m.sc/m.sc/a.sc20. Underthesecircumstances,we have\n/u1D70E/u1D456/parenleftBigℓ/u1D458//u1D70E/u1D460(ℓ/u1D458)\n/u1D70E/u1D458(ℓ0)//u1D70E/u1D460(ℓ0)/parenrightBig\n=/u1D45A/u1D458//u1D70E/u1D460(/u1D45A/u1D458)\n/u1D70E/u1D458(/u1D45A0)//u1D70E/u1D460(/u1D45A0)(4)\nfor every/u1D458∈ {1,...,/u1D460−1}such thatℓ/u1D458≠0.\nP/r.sc/o.sc/o.sc/f.sc.From\n(/u1D44E)/u1D70E/u1D456(ℓ0)\n/u1D45A0=/u1D462\n/u1D70E/u1D460(/u1D462)and(/u1D44F)/u1D70E/u1D456(ℓ/u1D458)\n/u1D45A/u1D458=/u1D70E/u1D458(/u1D462)\n/u1D70E/u1D460(/u1D462)\nweobtain\n(/u1D450)/u1D70E/u1D456(ℓ0)\n/u1D45A0/u1D45A/u1D458\n/u1D70E/u1D456(ℓ/u1D458)=/u1D462\n/u1D70E/u1D458(/u1D462)\nApply/u1D70E/u1D458to(/u1D44E)and/u1D70E/u1D45Fto(/u1D450)toobtain\n(/u1D44E′)/u1D70E/u1D456(/u1D70E/u1D458(ℓ0))\n/u1D70E/u1D458(/u1D45A0)=/u1D70E/u1D458(/u1D462)\n/u1D70E/u1D460+/u1D458(/u1D462)and\n(/u1D450′)/u1D70E/u1D456(/u1D70E/u1D460(ℓ0))\n/u1D70E/u1D460(/u1D45A0)/u1D70E/u1D460(/u1D45A/u1D458)\n/u1D70E/u1D456(/u1D70E/u1D460(ℓ/u1D458))=/u1D70E/u1D460(/u1D462)\n/u1D70E/u1D460+/u1D458(/u1D462).\nDividing (/u1D44E′)by(/u1D450′)gives\n/u1D70E/u1D456(/u1D70E/u1D458(ℓ0))/u1D70E/u1D460(/u1D45A0)/u1D70E/u1D456(/u1D70E/u1D460(ℓ/u1D458))\n/u1D70E/u1D458(/u1D45A0)/u1D70E/u1D456(/u1D70E/u1D460(ℓ0))/u1D70E/u1D460(/u1D45A/u1D458)=/u1D70E/u1D458(/u1D462)\n/u1D70E/u1D460(/u1D462).\nFinally, divide (/u1D44F)bythis equationtoobtain\n/u1D70E/u1D458(/u1D45A0)/u1D70E/u1D456(/u1D70E/u1D460(ℓ0))/u1D70E/u1D460(/u1D45A/u1D458)/u1D70E/u1D456(ℓ/u1D458)\n/u1D70E/u1D456(/u1D70E/u1D458(ℓ0))/u1D70E/u1D460(/u1D45A0)/u1D70E/u1D456(/u1D70E/u1D460(ℓ/u1D458))/u1D45A/u1D458=1.\nTheclaim followsfrom here.\nUnless both sides of Equation (4) are constant, we get at most\none candidate for /u1D456and are done. It remains to consider the case\nwhenbothsidesareconstantforevery /u1D458withℓ/u1D458≠0(and/u1D45A/u1D458≠0).\nIn this case, the constant can only be 1, because ℓ0,ℓ/u1D458,/u1D45A0,/u1D45A/u1D458are\nrational functions and /u1D70Edoesnotchange leading terms.\nIf bothsides of (4) are equalto1 then\n/u1D70E/u1D460(/u1D45A0)/u1D4401\n/u1D45A0=/u1D460/summationdisplay.1\n/u1D458=0/u1D45A/u1D458\n/u1D70E/u1D458(/u1D45A0)//u1D70E/u1D460(/u1D45A0)/u1D446/u1D458=/u1D460/summationdisplay.1\n/u1D458=0/u1D70E/u1D460(/u1D45A/u1D458)/u1D446/u1D458=/u1D440(/u1D460).\nTherefore, if /u1D456∈Zand/u1D462∈/u1D43Eare such that /u1D43F(/u1D456)=1\n/u1D70E/u1D460(/u1D462)/u1D440/u1D462, then\nwealsohave /u1D43F(/u1D456+/u1D460)=1\n/u1D70E2/u1D460(/u1D462)/u1D440(/u1D460)/u1D70E/u1D460(/u1D462)=1\n/u1D70E/u1D460(/u1D70E/u1D460(/u1D462)/u1D45A0)/u1D440/u1D70E/u1D460(/u1D462)/u1D45A0.\nThis means that in terms of operators, the shift equivalence\nproblem may have more than one solution in the situation unde r\nconsideration. In the former cases, where there was at most o ne/u1D456\nwith/u1D43F(/u1D456)=1\n/u1D70E/u1D460(/u1D462)/u1D440/u1D462, this/u1D456is then the only candidate for which\nwe can possibly have /u1D70E/u1D456(/u1D45D)=/u1D462/u1D45E. In the present situation, wherethereareinfinitely many /u1D456’s thatsolvetheproblemonthelevel of\noperators, it remains to determine which of them (if any) sol ves\ntheoriginal problemin termsof /u1D45Dand/u1D45E.\nL/e.sc/m.sc/m.sc/a.sc21. If/u1D70E/u1D460(/u1D45A0)/u1D440=/u1D440(/u1D460)/u1D45A0,thenthereisan operator /u1D447∈\n/u1D436[/u1D446]with constant coefficients such that /u1D440is a right factor of the\nsymmetricproduct /u1D447⊗ (/u1D446/u1D460−/u1D45A0).\nP/r.sc/o.sc/o.sc/f.sc.The condition /u1D70E/u1D460(/u1D45A0)/u1D440=/u1D440(/u1D460)/u1D45A0means that for any\nsolution/u1D45Eof/u1D440,also1\n/u1D45A0/u1D70E/u1D460(/u1D45E)is asolutionof /u1D440.Butthesolutions\nof/u1D440form a/u1D436-vector space of dimension at most /u1D460, so for every\nsolution/u1D45Eof/u1D440,theelements\n/u1D45E,1\n/u1D45A0/u1D70E/u1D460(/u1D45E),1\n/u1D45A0/u1D70E/u1D460(/u1D45A0)/u1D70E2/u1D460(/u1D45E), ...,/parenleftbigg/u1D460−1/productdisplay.1\n/u1D456=01\n/u1D70E/u1D460/u1D456(/u1D45A0)/parenrightbigg\n/u1D70E/u1D4602(/u1D45E)\narelinearly dependent over /u1D436.\nTherefore, every solutionof /u1D440is also a solutionof an operator\noftheform\n/u1D446/u1D4602+/u1D450/u1D460(/u1D460−1)/u1D70E/u1D460(/u1D460−1)(/u1D45A0)/u1D446/u1D460(/u1D460−1)+···\n···+/u1D4501/parenleftbigg/u1D460−1/productdisplay.1\n/u1D456=1/u1D70E/u1D460/u1D456(/u1D45A0)/parenrightbigg\n/u1D446/u1D460+/u1D4500/parenleftbigg/u1D460−1/productdisplay.1\n/u1D456=0/u1D70E/u1D460/u1D456(/u1D45A0)/parenrightbigg\nforcertainconstants /u1D4500,/u1D450/u1D460,...,/u1D450/u1D460(/u1D460−1).\nThis operator can be factored as a symmetric product. Up to\n(irrelevant) left-multiplicationbyan element of /u1D43E,itis equalto\n(/u1D446/u1D4602+/u1D450/u1D460(/u1D460−1)/u1D446/u1D460(/u1D460−1)+···+/u1D4501/u1D446/u1D460+/u1D4500) ⊗ (/u1D446/u1D460−/u1D45A0).\nThis completes theproof.\nThismeansthateverysolutionof /u1D440,inparticular /u1D45E,canbeinter-\npretedasaproductofaC-finiteandan /u1D460-hypergeometricquantity.\n(We donotclaim thatthese factorsareelements of /u1D445.)\nWe can reason analogously for /u1D43Fand find that every solution\nof/u1D43F, in particular /u1D45D, can be interpreted as a product of a C-finite\nandan/u1D460-hypergeometricquantity,withthe /u1D460-hypergeometricpart\nannihilated by /u1D446/u1D460−ℓ0.\nIf the/u1D460-hypergeometric factors are not also C-finite, then their\ncomparisonleadstoatmostonecandidate /u1D456∈Zsuchthat/u1D70E/u1D456(/u1D45D)//u1D45E∈\n/u1D43E. In this comparison, we must take into account that in the fac -\ntorization of /u1D45Dand/u1D45Einto a C-finite and a hypergeometric part,\nexponential terms /u1D706/u1D465and polynomials in /u1D465can be freely moved\nfromone factortotheother.\nFor thecomparison, weuse theGosper-Petkovšek form [35] of\nℓ0and/u1D45A0:\nℓ0=/u1D706/u1D70E(/u1D44E)\n/u1D44E/u1D44F\n/u1D450, /u1D45A 0=˜/u1D706/u1D70E(˜/u1D44E)\n˜/u1D44E˜/u1D44F\n˜/u1D450.\nWeignore/u1D706,˜/u1D706,/u1D44E,˜/u1D44E,astheycorrespondtotheexponentialandpoly-\nnomial part, respectively, and check if there is an /u1D456∈Zsuch that\n/u1D70E/u1D456(/u1D44F//u1D450)=˜/u1D44F/˜/u1D450.If so,then this /u1D456is theonlycandidate forwhich we\nmayhave/u1D70E/u1D456(/u1D45D)//u1D45E∈/u1D43E.\nIt remains to consider the case when /u1D45Dand/u1D45Eboth are C-finite.\nByTheorem4.1in[44],thereexistpairwisedistinct /u1D7061,...,/u1D706/u1D461∈/u1D436,\npairwisedistinct /u1D7071,...,/u1D707/u1D461′∈/u1D436andpolynomials /u1D44E1,...,/u1D44E/u1D461,/u1D44F1,...,/u1D44F/u1D461′∈\n/u1D436[/u1D465]such that\n/u1D45D=/u1D44E1(/u1D45B)/u1D706/u1D45B\n1+···+/u1D44E/u1D461(/u1D45B)/u1D706/u1D45B\n/u1D461\n/u1D45E=/u1D44F1(/u1D45B)/u1D707/u1D45B\n1+···/u1D44F/u1D461′(/u1D45B)/u1D707/u1D45B\n/u1D461′,Therequirement /u1D70E/u1D456(/u1D45D)=/u1D462/u1D45Etranslates into\n/u1D70E/u1D456(/u1D44E1)/u1D706/u1D456+/u1D45B\n1+···+/u1D70E(/u1D44E/u1D460)/u1D706/u1D456+/u1D45B\n/u1D460=/u1D462/u1D44F1/u1D707/u1D45B\n1+···+/u1D462/u1D44F/u1D461/u1D707/u1D45B\n/u1D461.\nThereisnosolutionunless {/u1D7061,...,/u1D706/u1D461}={/u1D7071,...,/u1D707/u1D461′},sowemay\nassume that /u1D461=/u1D461′and/u1D706/u1D458=/u1D707/u1D458for all1≤/u1D458≤/u1D461.Thenwe need\n/u1D706/u1D456\n/u1D458/u1D70E/u1D456(/u1D44E/u1D458)=/u1D462/u1D44F/u1D458\nforall/u1D458.Fromanytwosuchequations,saythe /u1D458thandtheℓth,we\nget theconstraint\n(/u1D706/u1D458\n/u1D706ℓ)/u1D456/u1D70E/u1D456(/u1D44E/u1D458\n/u1D44Eℓ)=/u1D44F/u1D458\n/u1D44Fℓ\nIf/u1D44E/u1D458//u1D44Eℓisnotaconstantor /u1D706/u1D458//u1D706ℓisnotarootofunity,thenthere\nisatmostonesolution /u1D456.If/u1D44E/u1D458//u1D44Eℓisaconstantand /u1D706/u1D458//u1D706ℓisarootof\nunityforeverychoiceof /u1D458andℓ,then/u1D45Dcanbewrittenasaproduct\nof/u1D44E1and/u1D706/u1D465\n1anda/u1D436-linearcombinationofpowersofrootsofunity.\nThisimpliesthat /u1D45D//u1D44E1isaspecialpolynomial,whichconflictswith\ntheassumptionthat /u1D45Dis normal.\nIn conclusion, wehave proven the correctness of thefollowi ng\nalgorithm.\nA/l.sc/g.sc/o.sc/r.sc/i.sc/t.sc/h.sc/m.sc22. INPUT:/u1D45D,/u1D45E∈/u1D445=/u1D436(/u1D465)[/u1D4610,...,/u1D461/u1D45F−1]irreducible\nand normal\nOUTPUT:/u1D456∈Zsuch that either /u1D70E/u1D456(/u1D45D)//u1D45E∈/u1D436(/u1D465)or/u1D45Dand/u1D45Eare not\nshift-equivalent.\n1Compute monic minimal annihilating operators /u1D43F,/u1D440∈/u1D436(/u1D465)[/u1D446]\nof/u1D45Dand/u1D45E.\n2If there is a /u1D458such that the coefficient of /u1D446/u1D458is zero in one of the\ntwo operatorsbut nonzerointheother,return 0.\n3Let/u1D460betheorder of /u1D43F(and/u1D440).\n4For every/u1D458∈ {1,...,/u1D460−1}withℓ/u1D458≠0,do:\n5If at least one of the two rational functions /u1D44E:=ℓ/u1D458//u1D70E/u1D460(ℓ/u1D458)\n/u1D70E/u1D458(ℓ0)//u1D70E/u1D460(ℓ0)\n/u1D44F:=/u1D45A/u1D458//u1D70E/u1D460(/u1D45A/u1D458)\n/u1D70E/u1D458(/u1D45A0)//u1D70E/u1D460(/u1D45A0)isnot in/u1D436\n6Return/u1D456∈Zsuch that/u1D70E/u1D456(/u1D44E)=/u1D44F,or0ifnosuch/u1D456exists\n7ComputetheGosper-Petkovšekform /u1D706/u1D70E(/u1D44E)\n/u1D44E/u1D44F\n/u1D450ofℓ0andtheGosper-\nPetkovšek form ˜/u1D706/u1D70E(˜/u1D44E)\n˜/u1D44E˜/u1D44F\n˜/u1D450of/u1D45A0.\n8If/u1D44F//u1D450≠1or˜/u1D44F/˜/u1D450≠1then\n9Return/u1D456∈Zsuch that/u1D70E/u1D456(/u1D44F//u1D450)=˜/u1D44F/˜/u1D450,or0ifnosuch/u1D456exists\n10Computetheconstants /u1D7061,...,/u1D706/u1D460∈/u1D436,polynomials /u1D44E1,...,/u1D44E/u1D460,\n/u1D44F1,...,/u1D44F/u1D460∈/u1D436[/u1D465],as above.\n11For/u1D458=1,...,/u1D460,do:\n12Forℓ=1,...,/u1D458−1,do:\n13If/u1D44E/u1D458//u1D44Eℓisnot aconstant or /u1D706/u1D458//u1D706ℓisnot aroot ofunitythen\n14 Return/u1D456∈Zsuch that (/u1D706/u1D458\n/u1D706ℓ)/u1D456/u1D70E/u1D456(/u1D44E/u1D458\n/u1D44Eℓ)=/u1D44F/u1D458\n/u1D44Fℓ, or0if nosuch\n/u1D456exists\n4 THE C-FINITE CASE\nTheproblemofindefinitesummationintheC-finitecasehasbe en\ninvestigatedin[16,25,47]underspecificassumptions.Let /u1D434bede-\nfinedasin(3)andassumethat /u1D434∈GL/u1D45B(/u1D436).In[25],theauthorsin-\ntroduceamethodforcomputingrationalsolutionsoftheequ ation\n/u1D462/u1D70E(/u1D466.alt)−/u1D463/u1D466.alt=/u1D464,where/u1D462,/u1D463,/u1D464∈/u1D445,undertheassumptionthat /u1D45B=2\nand/u1D434has two eigenvalues /u1D7061,/u1D7062such that/u1D7061//u1D7062is not a root of\nunity.However,theirmethodisnotcompleteastheyareunab letoboundthemultiplicitiesofirreduciblespecialpolynomia lsappear-\ning in the denominator of solutions. In [16], assuming that /u1D434is a\ndiagonalizable matrix, the authors characterize all new co nstants\nin/u1D439andpresentanalgorithmforcomputingonerationalsolutio n\noftheequation /u1D450/u1D70E(/u1D466.alt) −/u1D466.alt=/u1D453,where/u1D450∈/u1D436∗and/u1D453∈/u1D439.According\ntotheproofofProposition15,if /u1D439containsnonew constant,then\n/u1D434will always be diagonalizable, and thus the C-finite case und er\nourassumptionis reducedtothecaseconsidered in[16].\nAcknowledgement. Theauthorsthank Michael Singer formany\ndiscussions aboutsummationinfinite terms.\nREFERENCES\n[1] Sergei A. Abramov. The summation of rational functions. Ž. Vyčisl. Mat i Mat.\nFiz.,11:1071–1075,1971.\n[2] Sergei A. Abramov. The rational component of the solutio n of a first order lin-\near recurrence relation with rational right hand side. Ž. Vyčisl. Mat. i Mat. Fiz. ,\n15(4):1035–1039, 1090,1975.\n[3] SergeiA.Abramov. Indefinite sumsof rationalfunctions . InISSAC ’95:Proceed-\ningsofthe1995InternationalSymposiumonSymbolicandAlg ebraicComputation ,\npages303–308, NewYork,NY, USA, 1995. ACM.\n[4] SergeiA.Abramovand MarkVanHoeij. Integration ofsolu tions of linearfunc-\ntional equations. Integral Transformsand Special Functions ,8(1-2):3–12,1999.\n[5] Sergei A. Abramov and Mark van Hoeij. A method for the inte gration of solu-\ntions of ore equations. In ISSAC ’97: Proceedings of the 1997 International Sym-\nposium on Symbolic and Algebraic Computation , pages 172–175, New York, NY,\nUSA, 1997.ACM.\n[6] StefanT. Boettner. Mixed Transcendental and AlgebraicExtensionsfor the Risc h-\nNormanAlgorithm . Phd thesis,Tulane University,NewOrleans,USA, 2010.\n[7] Manuel Bronstein. Onsolutions oflinearordinarydiffer enceequations intheir\ncoefficient field. Journal of SymbolicComputation , 29:841–877,2000.\n[8] Manuel Bronstein. Symbolic Integration I: Transcendental Functions . Springer-\nVerlag, Berlin,2005.\n[9] ManuelBronstein. Structuretheoremsforparallelinte gration.J.SymbolicCom-\nput., 42(7):757–769, 2007. Paper completed byManuel Kauers.\n[10] ShaoshiChen,RuyongFeng,PingchuanMa,andMichaelF. Singer. Separability\nproblems in creativetelescoping. In ISSAC ’21: Proceedings of the 2021 on Inter-\nnational Symposium onSymbolicand AlgebraicComputation ,pages83–90,New\nYork,NY, USA, 2021. Associationfor ComputingMachinery.\n[11] R.M.Corless,G.H.Gonnet, D.E. G.Hare,D.J.Jeffrey,an dD.E. Knuth. Onthe\nLambert/u1D44Afunction. Adv.Comput.Math. ,5(4):329–359, 1996.\n[12] J.H.DavenportandB.M.Trager.OntheparallelRischal gorithm.II. ACMTrans.\nMath.Software , 11(4):356–362, 1985.\n[13] James H. Davenport. On the parallel Risch algorithm (II I): use of tangents.\nSIGSAM Bull. , 16(3):3–6,aug1982.\n[14] JamesH.Davenport. TheparallelRischalgorithm.I. In Computeralgebra (Mar-\nseille,1982) ,volume144of LectureNotesinComput.Sci. ,pages144–157.Springer,\nBerlin-NewYork, 1982.\n[15] Hao Du and Clemens G. Raab. Complete reduction systems f or airy functions.\nBulletin ofChinese Applied Mathematics ,1(1):10–21, 2023.\n[16] LixinDuandYarongWei. Solutionstothefirstorderdiffe renceequationsinthe\nmultivariatedifferencefield,2024. arXiv:2401.13933.\n[17] Saber Elaydi. An introduction to difference equations . Undergraduate Texts in\nMathematics.Springer,New York,thirdedition, 2005.\n[18] RuyongFeng.Hrushovski’salgorithmforcomputingthe Galoisgroupofalinear\ndifferentialequation. Adv. inAppl. Math. ,65:1–37,2015.\n[19] John Fitch. User-based integration software. In Proceedings of the Fourth ACM\nSymposiumonSymbolicandAlgebraicComputation ,SYMSAC’81,page245–248,\nNewYork, NY, USA, 1981. AssociationforComputing Machiner y.\n[20] K.O.GeddesandL.Y.Stefanus.OntheRisch-Normaninte grationmethodandits\nimplementationinmaple. In ISSAC’89:ProceedingsoftheACM-SIGSAM1989In-\nternationalSymposiumonSymbolicandAlgebraicComputati on,ISSAC’89,page\n212–217, NewYork,NY, USA, 1989. Associationfor Computing Machinery.\n[21] Ralph William Gosper, Jr. Decision procedure for indefi nite hypergeometric\nsummation. Proceedings of theNational Academyof Sciences ,75(1):40–42, 1978.\n[22] Charlotte Hardouin. Hypertranscendance des systèmes aux différences diago-\nnaux.Compos.Math. ,144(3):565–581, 2008.\n[23] S. J.Harrington. A new symbolic integration systemin r educe*.The Computer\nJournal, 22(2):127–131, 01 1979.\n[24] Peter A. Hendriks and Michael F. Singer. Solving differe nce equations in finite\nterms.J.SymbolicComput. ,27(3):239–259,1999.\n[25] Qing-Hu Hou and Yarong Wei. Rational solutions to the fir st order difference\nequations inthe bivariatedifferencefield,2024. arXiv:240 1.11387.[26] MichaelKarr.Summationinfiniteterms. J.Assoc.Comput.Mach. ,28(2):305–350,\n1981.\n[27] Michael Karr. Theory of summation in finite terms. J. Symbolic Comput. ,\n1(3):303–315,1985.\n[28] ManuelKauers. D-Finite Functions . Springer-Verlag,Cham,2023.\n[29] ManuelKauersand Peter Paule. The ConcreteTetrahedron . Springer,2011.\n[30] Alexander Levin. Difference Algebra , volume 8 of Algebra and Applications .\nSpringer,NewYork,2008.\n[31] Arthur C Norman and PMA Moore. Implementing the new Risc h integration\nalgorithm. In Proceedingsof the4thInternational Colloquium onAdvanced Com-\nputing Methods inTheoretical Physics ,pages99–110, 1977.\n[32] HiroshiOgawara. Oncriteriaforalgebraicindependen ce of collections offunc-\ntions satisfying algebraic difference relations. Opuscula Math. , 37(3):457–472,\n2017.\n[33] Peter Paule. Greatest factorial factorization and sym bolic summation. J. Sym-\nbolicComput. ,20(3):235–268, 1995.\n[34] MarkoPetkovšek. Hypergeometricsolutions oflinearr ecurrenceswith polyno-\nmialcoefficients. J.SymbolicComput. ,14(2-3):243–264,1992.\n[35] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger ./u1D434=/u1D435. A. K. Peters\nLtd., Wellesley,MA, 1996.\n[36] Clemens G. Raab. Generalization of Risch’s algorithm t o special functions.\nInComputer algebra in quantum field theory , Texts Monogr. Symbol. Comput.,\npages285–304.Springer,Vienna,2013.\n[37] Carsten Schneider. Symbolic Summation in Difference Fields . PhD thesis, RISC-\nLinz,Johannes KeplerUniversitätLinz,2001.\n[38] Carsten Schneider. A collection of denominator bounds to solve parameter-\nizedlineardifferenceequations in ΠΣ-extensions. In Proceedings ofSYNASC’04 ,pages269–282, 2004.\n[39] Carsten Schneider. Degree bounds to find polynomial sol utions of parameter-\nizedlineardifferenceequations in ΠΣ-fields.Applicable AlgebrainEngeneering,\nCommunicationand Computing ,16(1):1–32, 2005.\n[40] Carsten Schneider. A refined difference field theory for s ymbolic summation.\nJournal ofSymbolic Computation ,43:611–644,2008.\n[41] CarstenSchneider.Adifferenceringtheoryforsymboli csummation. J.Symbolic\nComput.,72:82–127,2016.\n[42] CarstenSchneider andJohannes Blümlein. Computer Algebrain QuantumField\nTheory. Springer,2016.\n[43] Richard P. Stanley. Enumerative Combinatorics: Volume 2 , volume 62 of Cam-\nbridge Studies in Advanced Mathematics . CambridgeUniversityPress,1999.\n[44] Richard P. Stanley. Enumerative Combinatorics: Volume 1 , volume 49 of Cam-\nbridgeStudies inAdvanced Mathematics . CambridgeUniversityPress,USA, 2nd\nedition, 2011.\n[45] R.P. Stanley. Differentiably finite power series. European Journal of Combina-\ntorics,1(2):175–188, 1980.\n[46] Mariusvan der Put and Michael F. Singer. Galois theory of difference equations ,\nvolume1666 of LectureNotesin Mathematics . Springer-Verlag,Berlin, 1997.\n[47] Yarong Wei. Polynomial solutions to the first order diffe rence equations in the\nbivariatedifferencefield,2024. arXiv:2401.11388.\n[48] DoronZeilberger. A holonomic systemsapproachto spec ialfunctionidentities.\nJournal ofComputational and Applied Mathematics ,32:321–368,1990.\n[49] DoronZeilberger.Themethodofcreativetelescoping. JournalofSymbolicCom-\nputation, 11(3):195–204, 1991.\n[50] https://www-sop.inria.fr/cafe/Manuel.Bronstein/ pmint/, 2005."    },    {        "title": "2402.04686v1.The_Influence_of_Autofocus_Lenses_in_the_Camera_Calibration_Process.pdf",        "content": "IEEE TRANSACTIONS ON  JOURNAL NAME,  MANUSCRIPT ID  1 \n The influence of the focus in the camera \ncalibration process  \nCarlos Ricolfe- Viala  \nAbstract —— Camera calibration is a crucial step in robotics and computer vision. Accurate camera parameters are necessary \nto achieve robust applications. Nowadays, camera calibration process consists of adjusting a set of data to a pin- hole model, \nassuming that with a reprojection error close to cero, camera parameters are correct. Since all camera parameters are \nunknown, computed results are considered true. However, the pin- hole model does not represent the camera behavior \naccurately if the focus is considered. Real cameras change the focal length slightly to obtain sharp objects in the image and this feature skews the calibration result if a unique pin- hole model is computed with a constant focal length.  \nIn this paper, a deep analysis of the camera calibration pr ocess is done to detect and strengthen its weaknesses. The camera is \nmounted in a robot arm to known extrinsic camera parameters with accuracy and to be able to compare computed results with the true ones. Based on the bias that exist between computed results and the true ones, a modification of the widely accepted \ncamera calibration method using images of a planar template is presented. A pin- hole model with distance dependent focal \nlength is proposed to improve the calibration process substantially.  \nIndex  Terms — camera calibration, camera parameters coupling, 2D calibration template  \n——————————       ——————————  \n1 INTRODUCTION\nAMERA calibration is an essential issue in robotics \nand computer vision because it establishes the geomet-\nric relation between 2D image coordinates and 3D world \ncoordinates [1, 2, 3 , 4]. Many published papers explain \nhow to obtain the  correct mapping between 3D space and \nthe 2D camera plane. Most of  the work is based on the pin-\nhole camera model using 3D [ 5, 6], 2D [ 7, 8] or 1D [ 9, 10, \n11, 12] templates or doing self -calibration [ 13,14]. Photo-\ngrammetric methods use precise coordinates of calibration points in 3D space, arranged in a predesigned 3D, 2D or 1D template.  Self -calibration methods assume that several \nimages of a rigid scene with fixed camera parameters is enough to compute them.  \nMost existing methods propose a nonlinear minimiza-\ntion step that computes the correct camera parameters by iteratively minimizing the difference between the detected \ncontrol points in images and their computed projections in \nthe image plane. This difference is called the reprojection \nerror [ 15]. A closed -form linear transformation solution in-\nitializes the nonlinear  minimization step. The closed -form \nsolution obtains an approximation of all linear camera pa-rameters and the nonlinear minimization improves this \napproximation, together with nonlinear camera parame-\nters such as lens distortion. The nonlinear minimization \nprocess ends when the reprojection error is close to zero. If \nthe reprojection error is zero, the difference between the \ndetected points in the image and the control points pro-\njected by the computed model is also zero. In consequence, \nit is assumed that the computed model is correct.  \nThe problem arise s when the distance from the camera to the calibration template vary in each image and the cam-era focus changes the lens position to obtain a focused im-\nage. In this case, images with different focal length are used \nto compute a unique model with a constant focal length. The reprojection error verifies that the computed model is \nsatisfied with the input data, but it does not confirm that \nthe computed model represents the real camera and the \ntrue camera location w hen the calibration images were \ntaken.  \nIf the camera -template distance varies in each image \nand a unique model with a constant focal lenght is com-\nputed, defocused images should be used . Several authors \npropose different methods to detect template control \npoints in defocused images accurately [16 - 22]. Th is is out \nof the scope of this paper .  \nThis aim of this paper  is to analyse the influence of the \ncamera focus in the results of the camera calibration pro-\ncess and to define a set of rules that will help to  improve \nthe outcomes when a camera is calibrated  using an autofo-\ncus lens. The camera is a system in which intrinsic and ex-\ntrinsic camera parameters are tightly coupled. When pa-\nrameters are interdependent, a nonlinear minimization process that simultaneously computes all parameters to-\ngether may not be the best choice. It is assumed that ran-\ndomly acquired images reduce computed camera parame-ters bias but this is not always true as pointed out by Hu \nand Kantor [16].  Ryusuke and Y. Yasushi [17] proposed a \nseparate calibration method for intrinsic camera parame-ters, but special equipment and a controlled environment \nare required. Alturki [1 8] and Lu [19] compute the princi-\npal point singularly by finding orthogonal projections of \nthe camera optical axis on the image plane.  \nThis paper proposes a calibration process of a camera \nwith autofocus lenses modifiying a well- known camera \nxxxx-xxxx/0x/$xx.00 © 200x IEEE         Published by the IEEE Computer Society  ————————————————  \n• Carlos Ricolfe -Viala is with the Institute of Industrial Informatics and Au-\ntomat ic Control in Polythechic University of Valencia (UPV), Spain +34 \n963 877007; e -mail: cricolfe@upv.es).  C 2 IEEE TRANSACTIONS ON JOURNAL NAME,  MANUSCRIPT ID  \n calibration process with a set of rules that help to improve \nthe method . Section 2 analyses the camera calibration pro-\ncess using a two -dimensional template proposed by Zhang \n[4]. This analysis demonstrates that the interdependence of camera parameters means that the computed model can-\nnot accurately represent the real camera in some cases. \nThis inaccuracy arises in some conditions when if the final nonlinear minimization process computes all camera pa-\nrameters at the same time. To demonstrate the inaccuracy \nof the camera calibration process, the camera is assembled on a robot arm in order to be able to accurately measure \nthe extrinsic camera parameters. Computed camera pa-\nrameters with a reprojection error close to zero are com-pared to the real ones so as to note the any discrepancy be-\ntween them. Section 3 proposes a set of tests to calibrate \ncameras with guaranties of accurate results. First, intrinsic camera parameters are computed depending on the dis-\ntance of camera to the calibration template and second, ex-\ntrinsic parameters are computed alone to avoid the cou-pling between intrinsic and extrinsic camera parameters. \nSection 4 shows results that demonstrate the efficiency of \nthe proposed method. Upon testing the proposed method, more accurate results are obtained because intrinsic cam-\nera parameters are isolated and they are constant in the it-\nerative nonlinear minimization process. Paper ends wit h \nconclusions . \n2 A NALYSIS OF THE EXISTING CAMERA \nCALIBRATION PROCESS USING A 2D TEMPLATE  \nMany current methods compute both camera parameters and lens distortion models. The most widely implemented \nis the method proposed by Zhang in his 2000 paper on \ncamera  calibration [4]. This method uses several images of \na 2D chessboard template to compute the camera pin-hole \nmodel represented as:  \np t R A p sw c]· ·[ · =  (1) \nwhere cp = [p u, pv, 1]T is the 2D coordinates in the camera \nframe of the 3D point in the scene represented as wp = [px, \npy, pz, 1]T . As the camera model works with projective ge-\nometry, s is an arbitrary scale factor. Rotation matrix R  and \ntranslation vector t  are extrinsic parameters that relate to the camera and world frames. Extrinsic camera parameters represent the location of the camera in the world. Matrix A  \ncontains intrinsic parameters as follows: \n\n\n\n\n=\n1 0 0000\nvu\nAβγ α\n (2) \nα and β are the scale factors in the u  and v camera axis. Both \nscale factors are defined with the focal length f  of the cam-\nera and the size of the camera sensor along the u  and v axis. \n𝛾𝛾 is the skewness of axis u  and v in the case they are not \northogonal. u0 and v0 are the coordinates of the  principal \npoint in the image plane. Figure 1 shows a diagram of the \npin-hole camera model parameters.  \nThe calibration process proposed by Zhang consists of \nseveral steps. First, n  images of a flat chessboard template \nare captured from different locations. At each location where the images wer e captured, the coordinates of the \ntemplate points and their projection in the image are used to compute homographies between the template plane and camera plane. Second, combining elements of all homog-\nraphies, intrinsic camera parameters are computed. Third, \nextrinsic camera parameters are computed for every loca-tion in which an image was taken. The final step of calibra-\ntion consists of an iterative nonlinear minimization process \nto improve the computed intrinsic and extrinsic parame-ters, minimizing the reprojection error given by : \n( )2\n1 1, , , ^ ∑∑\n= =−n\nim\njjw\ni i ijcp t R A p p  (3) \nwhere n is the number of images, m  is the number of points \nin the chessboard template and p^ (A,Ri,ti,wpj) is the projec-\ntion with the estimated camera parameters A, Ri, ti, of cal-\nibration template point wpj in the image i according with \nequation (1). The minimization of the reprojection error shown by (3) is an iterative nonlinear minimization prob- \nFig. 1. Pin-hole camera model parameters.  \n \nFig. 2. Robot arm ABB IRB 140 with an EoSens® 12CXP+ camera of \n4,096 × 3,072 pixels and sensor size of 23.04 × 23.04 mm active area \nwith 18 mm lens with manual focus .  \nAUTHOR ET AL.:  TITL E 3 \n lem that is solved using the Levenberg -Marquardt algo-\nrithm. The aim is to reduce the distance that exists between \ndetected points in images and projected points in images \nusing the computed model.  \n2.1 Empirical test of the calibration process  \nNormally, intrinsic and extrinsic camera parameters are \nunknown and the result of the calibration process is con-\nsidered valid because the reprojection error is close to zero. To verify the accuracy of computed camera parameters, an \nempirical experiment is performed. Figure 2 shows the \nsetup of this experiment. It uses an EoSens® 12CXP+ cam-era o f 4,096×3,072 pixels, sensor size of 23.04×23.04 mm ac-\ntive area and an 18mm manual focus lens. This camera is assembled on a robot arm ABB IRB 140 for calibration pur-poses. As the camera is on a robot arm, extrinsic camera \nparameters can be accurately known when an image is \ntaken to calibrate it. This allows a comparison of the cali-bration algorithm results with the true values given by the \nlocation of the robot arm. Calibrating the camera with the \nwidely accepted method, proposed by Zhang, which uses a 2D template, gives a valid solution because the reprojec-\ntion error is close to zero. However, when computed ex-\ntrinsic camera parameters are compared with the true real ones, they are consistently revealed to be incorrect.  \nThe erro r in computed parameters is depicted in fig-\nures 3 and 4. The reprojection error defined in equation (3) has a mean value of 1.2092×e -09 with a standard deviation \nof 2.3201 pixels. The computed model projects all the tem-plate points in the images using equation (1) and 98% of distances between the projected and detected points in im-ages is in a range of ±4.64 pixels. Since the image has a res-\nolution of 4,096×3,072, 4.64 pixels represent an error of \n0.1%, which is small enough to validate the computed model. In this case, since the camera location is known for \neach image, computed extrinsic parameters are compared with the true ones to validate the calibration result. Figure 3 shows the calibration stage, including the location of \ncameras and template points, both in the capturing stage \nand after the calibration process. Locations with an asterisk denote the true locations when images were captured with \nthe camera on the robot arm. Locations without an asterisk \nshow the location of the camera given by the calib ration \nprocess. All computed camera locations are 100mm closer \nto the calibration template than the true location. Figure 4 \nshows the resulting calibration errors in location for coor-dinates x, y and z. Errors are computed using the true cam-\nera location as the reference frame. Errors in the x and y -\naxis are positive and negative in a range from -75mm to 75 \nmm. However, errors in Z coordinates show that the com-puted camera location is always closer to the calibration \ntemplate than the true real location by a  mean value of \n100mm, within a range from 60 mm to 160 mm. This means \nthat, in all cases, the computed camera location is closer to \nthe calibration template than the true location. Conse-quently, values for scale factors α and β are biased . \nTo summarize, the calibrated model is considered valid \nbecause the reprojection error is close to cero, but empirical experiments show that there are consistent bias in camera \nlocation. Bias in camera location means bias in intrinsic  \nFig. 3. Camera locations for 10 images. True camera locations have an asterisk. They are known because the camera is assembled on the \nrobot arm. Computed camera locations are denoted with the number only. Computed camera locations are all closer to the templa te by about \n100 mm .  \n4 IEEE TRANSACTIONS ON JOURNAL NAME,  MANUSCRIPT ID  \n camera parameters if r eprojection error is close to cero. If \nthe camera is calibrated using the currently well- estab-\nlished method, it is possible to arrive at a seemingly valid \nsolution, but it does not represent the real camera. \n2.2 Why a re the camera parameters biased?  \nCalibr ating a pin -hole model of a camera with focus lenses \nis liable to failure. This is because the pin -hole model does \nnot represent the focus camera behaviour accurately. A focus \ncamera system adjusts lenses to focus on the chessboard tem-\nplate with every imag e. In consequence, focal length f  (ex-\npressed with the scale factors α and β in the camera model) is \nnot constant and a systematic error arises if the pin -hole \nmodel with a constant focal length is adjusted for different camera- chessboard distances.  \nTo perform a precise camera calibration process, the com-\nputed model is only valid for a range of distances from the \ncamera to the calibration object. Given a fixed focal length, \nthe range of distances in front of and behind the focal plane within which objects a ppear sharp is called the depth of field. \nAssuming that applications need sharp images for a given cir-cle of confusion, objects appear acceptably sharp in images \nwhen they are in the depth of field zone. If an object is out of \nfocus, the focal length shoul d change to obtain a sharp image \nof the object. In consequence, for a specific application, it is necessary to know the required range of distances between the \ncamera and the object to perform a precise calibration. Out of \nthis zone, the sharpness of objec ts will be poor and focal \nlength will have to be changed to improve it.  \nThe depth of field is controlled using the camera diaphragm \naperture and the focal length. Given a constant diaphragm ap-erture, the rule is that the closer the object is, the smaller the depth of field is; conversely, the further the object is, the greater the depth of field is until it is infinite. The hyperfocal \ndistance is considered when the object is far away from the \ncamera and the depth of field is infinite. Therefore, if the cal-ibration is performed with the template located at different distances from the camera, it could obtain biased results, as the focal length varies in each image to obtain sharp objects. \nFigure 5 illustrates this effect.   \n2.3 Where is the hyperfocal distance?  \nIn figure 6 an analysis of rays that form the image passing through the camera lens is performed. The ideal projective \nray that represents the pin- hole model is the projection of \npoint p in the scene passing through the pin- hole of the \ncamera. It is denoted in red. The ray going through the edge of the lens is deviated according with the curvature \nof the lens and it is denoted in green. The ratio between the \nangle of the incoming ray φ and the angle of outcoming ray \nω is constant for a distance to the centre of the lens. As the \ndistance of the object to the lens varies, so do the angles φ \nand ω, as is shown in figure 6 for two distances. The sharp-\nest image arises when the intersection of the projective ray \nwith the ray coming from the edge of the lens coincides \nwith the sensor plane, as is shown in figure 6. This point is represented as point q  in that figure.  \nAn analysis of the variation of the point q with the dis-\ntance of the object to the camera will give the hyperfocal distance. The variation of point q gives different values of \nthe focal length f  depending on the distance of the object to \nthe camera. Two cases are shown in figure 6. Figure 6(a) \nshows the point q  when the object is far away from the \ncamera. In figure 6(b) the object is close to the camera.  De-\npending on the distance of the object to the camera the an-gle of the incoming ray φ  varies until a point that this var-\niation is insignificant for a circle of confusion. This point is where the hyperfocal distance is defined. \nFrom a mathematical point of view, considering a coordi-\nnate system XY  with the origin in the intersection of the opti-\ncal axis with the lens plane, coordinates of point p in figure 6 \nare (d, a) where d is the distance of point p to the lens, and a \nis its distance to the optical axis. The equation of the ideal \nprojective ray that represents the projection of point p  in the \nscene going through the pin- hole of the camera is:  \nxday ·− =  (4) \nThe equation of the projective ray of point p  that goes \nthrough the edge of the lens is:  \nD xda Dy +−= ·  (5) \nwhere D represents the radius of the lens. The angle of the \nincoming ray to the lens φ is defined as:  \n\n−=−\na Dd1tanϕ  (6) \nThe angle of the outcoming ray ω is proportional to the   \nFig. 4.  Bias in computed camera  locations. The reference frame is the true camera location obtained with the robot arm. Bias in X and Y -axis \nare positives and negatives, ranging from -75mm to 75 mm. However, bias in Z coordinates show that the computed camera location is always \ncloser t o the calibration template than the real location in 100mm as mean value within a range from 60 mm to 160 mm.  \nAUTHOR ET AL.:  TITL E 5 \n angle of the  incoming ray φ  as ω = k·φ where k depends on \nthe lens. The equation of the outcoming ray of the lens is:  \nD x y +− − = ·2tanϖπ (7) \nThe intersection of line defined in (4) and line defined in \n(7) gives the point q  where the sensor plane must be posi-\ntioned to obtain the sharpest image. Coordinates of point q are \n(f, b), where b  represents the projection of the point p in the \nimage plane and f  is the distance of the sensor to the lens de-\nfined as the focal length. Lines (4) and (7) intersection is \ngiven by:  \nD x xda+− − = − ·2tan · ϖπ (8) \nThe value of  x which solves (8) represents the focal length \nf. Therefore, the focal length that gives the sharpest image is \ncomputed as:  \ndaDf\n−−=\nϖπ\n2tan (9) \nExpression (9) demonstrates that the focal length varies to \nobtain sharp images when the distance of the object to the camera changes. Moreover, the focal length depends on \nthe sensor size. The size of the sensor is represented by the \nradius of the lens D that illuminates it. The bigger the sen-\nsor is, the larger the value of D. Figure 7 represents how focal length varies with the distance of the object to the \ncamera defined in equation (9). When the object is far away from the camera, variation s of f to obtain a sharp image are \ninsignificant. Considering that the sharpness of the image is defined up to a circle of confusion, objects that are far away from the hyperfocal distance of the camera will be \nfocused . \n2.4 Why is the reprojection error close to zero and \nparameters are biased ? \nIt’s widely accepted that the result of the calibration pro-\ncess is valid if the reprojection error defined in equation (3), gives an error close to zero. However, an exhaustive analysis of the pin- hole camera model reveals that this as-\nsumption is not  definitive. \nA camera is a tightly coupled system in which errors of \nintrinsic parameters are compensated with incorrect val-ues in the extrinsic parameters, and vice versa, to keep the reprojection error close to zero.  \nOn the one hand, the  camera focal length parameter f is \ntightly coupled with the distance between the camera and the calibration template, as can be observed in figure 8. Ac-\ncording to figure 8 the focal length and the distance of the \nobject to the camera p\nz, are related as:  \nz ypf\npv=  (10) \nUsing coordinates v and py to compute camera parame-\nters f and pz, the results are infinitely varied because the \nparameters are interdependent. According to equation (10) any pair f and p\nz, that satisfy the ratio v/ py is valid. In figure \n8(a) and (b), given fixed v and cpx, both different solutions \nof f and cpz are correct according to the criteria of the repro-\njection error, because both solutions satisfy it properly.  \nOn the other hand, the camera location, the focal length \nand the principal point of the camera are also tightly cou-pled. Figure 9 shows this coupling. Both images show \nvalid solutions for the calibration problem using the same coordinate points in the template p  (p\nx, py) and the image q  \n(u, v) as the input data of the calibration process. In both \ncases, camera parameters and the calibrating data give a repro jection error equal to zero. Errors in the principal \npoint are compensated by incorrect values in the camera \nlocation and vice versa. Moreover, biased values in the \nprincipal point and camera location have a direct effect on the camera focal length.  \nAccording to figure 9, given an optical axis perpendicular (a) \n \n(b) \n \n \n \n \n \n \n \n \n \n \nFig. 6.  The sharpest image appears when the intersection of the ideal \npin-hole ray and the ray coming from the edge of the lens is in the cam-\nera sensor plane. The angle of the incoming ray φ and the outcoming \nray ω varies with the distance of the object to the lens. Camera optics \nsystem adjusts lenses to focus the object in the scene.   \nFig. 5.  Camera optics system adjusts lenses to focus the object in the \nscene. In consequence, focal length f and scale factors α and β are not \nconstant and depend on the distance of the camera to the object. As-\nsuming that applications need sharp images f or a given circle of confu-\nsion, objects appear acceptably sharp in images when they are in the \ndepth of field zone. Given a fixed focal length, the range of distances in \nfront of and behind the focal plane is called depth of field, and objects in \nthis zone  appear sharp with no variation of focal length. If an object is \nout of focus, focal length should be changed so as to obtain a sharp \nimage of the object .  \n6 IEEE TRANSACTIONS O N JOURNAL NAME,  MANUSCRIPT ID  \n to the image plane, all camera parameters are represented by \nthis optical axis. In this figure, two different focal lengths, \nwhich use the same calibration data, are considered valid be-cause other camera parameters compensated this variation. Both solutions arise changing the optical axis , which  is per-\npendicular to the image plane. There is a plane that represents \nall candidates to camera optical axis given a pair of points p \nand q to calibrate the camera. This plane is called the Sophia \nplane and it is represented in figure 9(b). This plane contains \nthe ray that goes through  the point p(p\nx, py) to the point q(u, \nv) and all optical axis that represent different camera parame-\nters. Depending on the chosen optical axis, camera parame-ters vary.  \nCamera parameters coupling has been demonstrated \nusing just one pair of points p  and q in the scene and in the \nimage. When several points are used to calibrate the cam-\nera, several Sophia plane s arise and the camera parameters \nare defined with the best optical axis that satisfy all Sophia \nplanes. It is considered , that using several images of a cali-\nbration template with several calibration points,  the true \nsolution of camera parameters will be obtain. However, \nthis assumption fails if the empirical experiment described \nin subsection 2.1 is performed. Results showed in figures 3 and 4 are biased. This empirical experiment demonstrates \nthat using several calibration points in several images do \nnot guarantee unbiased camera parameters. Moreover, this  \nbias is not detected if camera parameters  are validated us-\ning the reprojection error tool.  \n3 PROPOSED CAMERA CALIB RATION PROCESS \nUSING A 2D TEMPLATE  \nExisting camera calibr ation methods perform two steps. \nFirst, an approximation of camera parameters is computed \nbased on an algebraic solution. Second, an iterative nonlin-ear minimization problem improves the algebraic solution \naccording to the criterion of the reprojection err or, with the \naim of computing the correct parameter values. Intrinsic \nand extrinsic parameters are updated in every iteration. It \nis assumed that using several images taken from different  \nFig. 9.  Coupling between the principal point ( u0,v0), focal length f  and \nthe extrinsic camera parameters tx and ty. Changes in intrinsic camera \nparameters ( u0,v0) are compensated with changes in the location o f the \ncamera tx and ty and focal length f . Figure (a) shows a solution for cam-\nera parameters and figure (b)  shows another solution for camera param-\neters using the same data to compute camera parameters (coordinates \npoints in the template ( px, py) and coordinate points in the image ( u, v))  \nAny optical axis that belongs to the Sophia plane gives a valid solution \nfor the camera parameters. This is shown in figure (b). Using reprojec-\ntion error as the tool to determine true camera parameters, all optical \naxis that belongs to the Sophia plane will be a valid solution.    \nFig. 7.  Variation of the focal length with the distance of the object to the \ncamera  according with expression (9) .   \nFig. 8.  Coupling between the camera focal length parameter f and the \ndistance of the camera to the calibration template. Using coordinates v \nand cpy to compute camera parameters f and cpz has infinite solutions \nbecause both parameters are tightly coupled. In this figure, starting with \nthe same value of v and cpy both f and cpz solutions (a) and (b) are valid .  \nAUTHOR ET AL.:  TITL E 7 \n locations will condition the nonlinear minimization algo-\nrithm to achieve the right solution [4, 10].  \nHowever, when a camera is calibrated using images \nfrom several locations, this assumption fails, as it was shown in the empirical experiment described in the previ-\nous section. As was shown in subsection 2.2.1, different \ndistances from the camera to the calibration object produce \nchanges in the focal length parameter to obtain sharp im-\nages. These variations in the focal length depend on the \ncamera sensor size also, as was demonstrated in equation (9). Using the method proposed by Zhang, since the cam-\nera calibration process computes only one camera model \nwith a fixed focal length, the interdependence between the camera parameters leads the nonlinear minimization pro-\ncess to get zero reprojection error that camouflages a bi-\nased result . \nTo improve the solution of the camera calibration pro-\ncess, it is necessary to define previously the range of dis-tances from the camera to the calibration object where the focal length parameter will remain constant when getting \na sharp image. This analysis will give the limits of the dis-\ntances where the focal length varies in each image and the distance where the focal length keeps constant (i. e., where \nthe hyperfocal distance starts).  \nFigure 7 represents these limits. The hyperfocal distance \nis the starting point from where focal length is constant alt-\nhough the distance between the object and the camera var-\nies. If the distance between the object and the camera is \nsmaller than the hyperfocal distance, the focal length is ad-\njusted in each image to o btain sharp images.  \nUsing the method proposed by Zhang in [4], if the cali-\nbration is done with images where the focal length varies (below the hyperfocal distance), all calibration images \nshould be taken with similar distance between the calibra-\ntion template and the camera in order to guarantee that the \ncomputed parameters are correct. If images taken at sev-\neral distances in the range where the focal length varies are \nused, computed result using the method proposed by \nZhang in [4] will be biased although the reprojection error \nis zero. In this case, since images are taken at several dis-\ntances in the range where the focal length varies, several \nfocal lengths should be computed depending on the dis-\ntance of the camera to the calibration template. From the point  of view of the application that uses the calibration \nresults, calibration will be valid only for images that are taken in the same range of distances defined in the calibra-tion process.  \nThe aim of the proposed method is to identify this hy-\nperfocal distance to ensure that the calibration process is \ncorrect. In the case where the calibration images are taken \nin a distance shorter than the hyperfocal distance, a \nmethod to compute several focal lengths that depends on the distance of the camera to the c alibration template is \nproposed . \n \n3.1 Identifying the hyperfocal distance and \ncomputing the scale factors α and β \nThe hyperfocal distance starts when the scale factors α and \nβ are constant. As it was described in equation (2) scale fac-\ntors α and β are defined with the focal length f of the camera \nand the size of the camera sensor along the u  and v axis. \nUsing the projective geometry property described in figure \n8 the coordinate v of a point in the image is:   \nβ·\nzy\nppv=  (11) \nwhere β corresponds to the scale factor in the v axis de-\nnoted with f in figure 8. Two points in a template plane in \nthe scene that is parallel to the image plane have the same \npz and py1, py2, respectively. These two points will project \nthe image in coordinates v 1 and v2. The increment between \nthe two coordinates in the v  axis is:  \n()\nzy y\nzy\nzy\npp ppp\nppvvβββ · · ·2 12 1\n2 1 −=−=−  (12) \npz is equal at both points because the template scene \nplane is parallel to the image plane. Several images of the same two points in the calibration template at different dis-\ntances will result in different increments in the image and \ndifferent scale factors.  \nii\nidtvβ·∆=∆  (13) \nwhere the camera distance p z is denoted with d i, and \nΔvi=(v 1-v2), i=1...n where n is the number of images. β i cor-\nresponds to the scale factor in each distance of the camera \nto the calibration template. Δ t is the constant increment be-\ntween adjacent points in a scene. Arranging the data in a \nmatrix form, the following expression arises:  \n\n\n\n\n∆∆∆\n∆=\n\n\n\n\nn n n dvdvdv\nt\n·...··\n·1\n...2 211\n21\nβββ\n (14) \nIn the case that α  corresponds to the u  axis, the deduc-\ntion is similar and the equation (14) becomes:  \n\n\n\n\n∆∆∆\n∆=\n\n\n\n\nn n n dududu\nt\n·...··\n·1\n...2 21 1\n21\nααα\n (15) \nIf elements of vector α i or βi are plotted, a figure similar \nto the figure 7 appears that is very useful to decide where  \nFig. 10.  Example of the image for the proposed method for calibrating \nthe scale factors α and β. Resulting image and values Δ u and Δ v.  \n8 IEEE TRANSACTIONS O N JOURNAL NAME,  MANUSCRIPT ID  \n is the hyperfocal distance.  \nTo compute the scale factors αi or βi of the camera axis  \nu and v using expressions (14) and (15), we propose a sim-\nple empirical experiment. Several images of a planar chess-\nboard template are taken, in which the image plane is as \nparallel as possible to the chessboard plane. It is important to know the distance from the camera to the chessboard \nplane when the image is captured.  \nFigure 10 shows an example of the resulting image. \nHere, Δ u and Δv are defined. The increment between adja-\ncent points of the chessboard calibration template is Δ t and \nd\ni is the distance from the camera to the chessboard . The \ncorners of the black and white boxes in the images are de-\ntected and the increments between the adjacent points are \ncomputed. The increments  along the horizontal axis u are \ndenoted with Δ ui,j and the ones in the vertical axis v are \ndenoted with Δ vi,j., where i, j represent the increment j of \nadjacent points in the image i. If the image has no distor-\ntion, the values of Δu i,j will be similar and their mean val-\nues will be represented with Δ ui. Likewise,  in the v  axis, Δ vi \ndenotes the value of the increment along the v axis in im-\nage i. Further, if the pixels are square, Δu i will be equal to \nΔvi and the scale factors α i or βi will be equal.  \nIn the case of distorted images, Δ ui and Δv i is computed \nusing the detected points in the centre of the image only, \ninstead of using points all over the image. In images with \nradial distortion, points close to the centre of the image \nhave an insignificant deviation from the undistorted posi-\ntion. If all points in the image are used, distortion could be \ncorrected before computing the increments of Δ ui and Δ vi \nusing the methods proposed by Ricolfe -Viala, [ 20, 21] or \nZhu [ 22]. However, improvement on the results when dis-\ntortion is corrected is negligible and therefore unnecessary . \n \n3.2 Computing camera model with several scale \nfactors  \nOnce the experiment described in previous subsection \nis performed, it is possible to define the hyperphocal dis-\ntance that establishes the limit where the focal length var-ies or it is constant. If calibration is done with images taken \nat distances where the scale factors are constant, the \nmethod proposed by Zhang in [4] could work properly. In case that the calibration proce ss is done with images taken \nat distances where the scale factors vary, it is necessary to compute extrinsic parameters considering that the scale factors are different in each image.  \nThis section describes a variation of the method pro-\nposed by Zhang in [4] to compute the extrinsic camera pa-rameters using images taken at distances where the scale \nfactors vary. The process consists of a nonlinear minimiza-\ntion process to adjust camera parameters to a set of data but in this case, computing only the principal point ( u\n0, v0), \nthe skewness and the extrinsic camera parameters and con-sidering the scale factors as constant valued computed with the method described in previous subsection.  \nTo perform a nonlinear minimization process, an initial \nguess is necessary. The scale factors α\ni and βi for several \ndistances di are computed with the specific trial described in previous subsection. The image principal point parame-\nters ( u0, v0) is initialized to the centre of the image. This in-\nitialization does not disturb the final result because the \nnonlinear minimization process will obtain the right solu-\ntion taking into account that the focal length is constant. It \nwill find out the correct optical axis that satisfy the focal length in each image according with the figure 9. As it was \nsaid before, a set of camera parameters are defined with \nand optical axis and in this case, the constant value of the focal length will force the nonlinear minimization to com-\npute the best optical axis for the given focal length. The \nskewness γ is initialized to zero and the extrinsic camera \nparameters are obtained as follows.  \nUsing initial values of intrinsic camera parameters, the \nmatrix A defined in (2) has a specific value for each camera \ndistance d\ni. Moreover, coordinates of calibration template \npoints have z =0 and the relation between the camera plane \nand the calibration template plane is defined with an homography H. Equation (1) is rewritten as:  \nptRApw\ni i ic]· ·[=  (16) \n[]\n\n\n\n\n=\n\n\n\n\n\n\n\n\n=\n\n\n\n\n1· ·\n10· ·\n12 1\n332313\n322212\n312111\nyx\ntrrAyx\nttt\nrrr\nrrr\nrrr\nA vu\ni\nzyx\ni (17) \n[ ] p Hph hhpw\niw c· ·3 2 1 = =  (18) \nTo compute the homography H i, a technique based on \nnonlinear least squares proposed in [4] is used. Let c=[h1 h2 \nh3 ]T, the equation (18) can be rewritten as:  \n0·· 0· 0=\ncpv ppu p\nT w T w TT w T T w\n (19) \nWith n points, n equation (19) arise which can be writ-\nten in matrix equation as L ·c=0 where L is a 2· n×12 matrix. \nThe solution is the right singular vector of L  associated \nwith the smallest singular value, or the eigenvector, of LT·L \nassociated with the smallest eigenvalue. Matrix L  is poorly \nconditioned because some elements are a constant 1, others \nare in pixels and others are in millimetres. Better results are \nobtained by performing a simple data normalization pro-\ncess, as proposed in Hartley [ 24].  \nOnce the homography H i is computed and using the \nintrinsic camera parameters arranged in a matrix A i, extrin-\nsic camera parameters are computed as:  \ni i ii i ii i ii i i\nhA trr rhA rhA r\n312 1 321\n211\n1\n······\n−−−\n=×===\nρρρ\n (20) \nBecause of the noise, computed elements of R will not \nsatisfy t he properties of a rotation matrix. To approximate \nthe computed 3×3 matrix Q  to the best rotation matrix R , \nthe singular value decomposition of Q is necessary. Given \nthe singular value decomposition Q=U·S·V, the best rota-\ntion matrix is R=U·VT. For further reference to matrix com-\nputation, see Golub and Loan [ 25]. \nThe maximum likelihood estimation is the nonlinear \nminimization process that improves computed results, re-AUTHOR ET AL.:  TITL E 9 \n ducing the geometric error between detected point coordi-\nnates in images cpij and projected points with the com-\nputed camera model defined as cpij^(Ai, Ri, ti, wpj). The \nmaximum likelihood estimation is computed by minimiz-ing the following function:  \n( )2\n1 1, , , ^ ∑∑\n= =−n\nim\njjw\ni i i ijcp t R A p p  (21) \nRotation matrix R i is expressed with three parameters \nusing the Rodrigues formula [26]. This nonlinear minimi-zation problem is solved with the Levenberg -Marquardt \nalgorithm using the values of A\ni, Ri and ti computed pre-\nviously as the searching starting point. Notice that several matrices A\ni of intrinsic camera parameters exist that de-\npend on the distance di. In this case, searching parameters \nare extrinsic camera parameters R i and ti, the principal \npoint ( u0, v0) and the skewness γ. Scale factors α i and  βi in \nmatrices Ai remain constant to avoid the intrinsic and ex-\ntrinsic camera parameters influencing each other which re-\nsult in a biased solution at the end of the minimization pro-\ncess. Moreover, constant values of scale factors α i and  βi \nforce the searching process to find out the finest optical \naxis in the Sophia plane that best represents the camera \nmodel according to figure 9. When the minimization pro-\ncess is finished, reprojection error will be close to cero but in this case, it has been computed with a constant value of \nthe focal length or scale factors α\ni and  βi. \n3.3 Camera lens distortion  \nMost of the authors of papers about camera lens distor-\ntion report that distortion function is dominated by radial components if the image distortion is small [ 21, 22, 2 6]. A \nsecond order radial distortion model between the distorted \npoint in the image  \ncp* and the correct one cp is defined as \nδ+ =*p pc c (22) \nsuch as  ( )\n( )4\n22\n14\n22\n1\n· · ·· · ·\nv v d vu u d u\nr k r k vr k r k u\n+ ∆ =+ ∆ =\nδδ (23) \nwhere r2 is the distance between the distorted point coor-\ndinate and the principal point. Δu d=ud-u0, Δvd=vd-u0. r is \ncomputed as r2=Δud2+Δvd2. \nSimilar to the method proposed by Zhang in [4], the \ncamera lens distortion model defined in (23) is included in the maximum likelihood estimation and equation (21) is extended taking into account radial distortion parameters: \n( )2\n1 12 1 , , , , , ^ ∑∑\n= =−n\nim\njjw\ni i i ijcp t R k k A p p  (24) \nAs before, sc ale factors α i and  βi of matrices A i remain \nconstant to avoid the intrinsic and extrinsic camera param-eters influencing each other. Maximum likelihood estima-\ntion computes extrinsic camera parameters R\ni and ti, and \nthe principal point ( u0, v0), the skewness γ  and the distor-\ntion parameters k 1, k2 as intrinsic camera parameters. k1, k2 \nare initialized to zero. \n4 E XPERIMENTAL RESULTS  \nTo demonstrate the influence of the focus in the camera \ncalibration process, two set of images are captured using \ntwo cameras with different features assembled on two ro-\nbot arms. The main difference between both cameras is the sensor size to verify that in bigger sensors, the hyperfocal \ndistance is further away from the camera than in smaller \nones as was defined in (9) . Therefore, the risk of capturing \nimages with different focal lengths for calibration pur-\nposes increases in cameras with bigger sensors. In conse-\nquence, if only one focal length is computed in the calibra-tion process, biased parameters will be computed. In these \ncases, several camera models with different focal lengths \nare necessary to represent the focusing process of the cam-era when the distance of the camera to the calibration tem-\nplate varies under the hyperfocal distance .  \nFig. 11.  Robot arm ABB IRB 140 with the measuring too l defining 3D coordinates of template control points. Robotiq wrist camera mounted in \ncollaborative robot Universal Robot UR3 .  \n10 IEEE TRANSACTIONS ON JOURNAL NAME,  MANUSCRIPT ID  \n In our experiment, one camera is a Robotiq wrist camera, \n1279×724 pixels with electrically actuated focus, assembled \non a collaborative industrial robot UR3. The other one is an EoSens® 12CXP+ of 4,096 × 3,072 pixels, 23.04 × 23.04 mm active area with an 18 mm lens with manual focus, mounted on an ABB IRB 140. Both cameras are assembled on robot \narms so as to establish the extrinsic camera parameters and to \ncompare the computed values with the known true ones. It is assumed that better extrinsic parameters will result in more accurate intrinsic parameters with similar reprojection errors. \nThe reprojection error is also computed to check if model is correct from a geometric point of view. \nCamera location is obtained by using the location of the \nend of the robot arm that is provided by its control unit. \nWith this location, it is possible to compute a transfor-mation matrix that transforms the location of the robot arm \ninto the location of the camera image frame or the measur-\ning tool. Figure 11 shows an im age of the robot arm ABB \nIRB 140 with the measuring tool, obtaining 3D coordinates \nof the calibration template points. The same operation is \nperformed with the collaborative industrial robot UR3. Camera and template locations are in the robot base coor-\ndinate frame. No simulated data is used because it does not \nrepresent the real world. With computer simulations, data is generated with models that have  a constant focal length \nwith independence of the distance of the camera to the cal-ibration template. Under the simulation umbrella, every-thing fits perfectly (even with Gaussian noise), because \nreal camera behaviours such as focus features, are not con-\nsidered in the theoretical model. In this case, only real data is used to demonstrate the influence of the came ra focus in \nthe calibration process. \n4.1 Computing the scale factors αi and βi. \nTo compute the scale factors α i and  βi using the method \nproposed in subsection III.A., several images of the calibra-\ntion chessboard template are captured, taking into account \nthat the image plane should be as parallel as possible to the \ntemplate plane. In addition, the distance of the camera to \nthe chessboard template is known for each image. Figure \n12 shows selected images that were captured to compute the scale factors α\ni and  βi of both cameras. Chessboard cor-\nners were detected using the Harris corner detection algo-rithm implemented in openCV. Increments Δu\ni and Δv i \nwere computed as described in subsection III.A.  \nScale factors α i and  βi for each camera -template distance \nare compute d with equations (14) and (15). As it was as-\nsumed, the focal length is not constant in all distances as it is shown in figure 13. Black crosses represent the values of the elements of the vector α\ni in (15). With a real camera, fo-\ncal length is adjusted when the distance of the camera to the template changes to capture sharp objects. In conse-quence, scale factors α\ni or βi do not have a constant value, \nexcept when the focus is set in the hyperfocal distance. When the focus is in the hyperfocal distance, the depth of field is at its maximum and the focal length does not \nchange to capture sharp images. This effect is illustrated \nwith the real values computed with the cameras. Figures 13(a) and 13(b) use crosses to show the elements of vector \nα\ni for both the Robotiq wrist camera and the EoSens® \n12CXP+ camera respectively. Analysing the results, three \nzones appear when the scale factor is plotted versus the \ndistance of the camera to the template. Zone 1 is d efined \nby a variation of the focal length with each image to obtain \nsharp images. Zone 2 is defined by the constant value of \nthe focal length: the focus is set in the hyperfocal distance \nand without changing the focal length, images are sharp in the image although the distance of the camera to the object \nchanges. Zone 3 is defined by variations in the scale factor \nbecause the template is far away from the camera and the poor quality of the calibration template in the image does \nnot allow accurate detection of the image’s control points.  \nWith this experiment, it is possible to obtain the range \nof distances at which the camera will work properly with a constant focal length defined using Zone 2. In addition, \nif the camera is closer to the object and it is working in Zone 1, different values of the focal length will help to obtain ac-\ncurate camera parameters. Moreover, Zone 3 will give the \ndistance of the camera to objects in which the accuracy is reduced because the image resolution is not enough to de-\ntect objec t details with precision. Zone 3 will be condi-\ntioned by the size of the chessboard calibration template. \nThe details of the objects than can be detected will be de-\nfined by the size of the squares in the chessboard template \n(in this experiment, one size measures 25 mm).  \nZone 1 of the Robotiq wrist camera is between 0 and 150 \nmm. Zone 2, where the focal length is constant, is when objects are between 150 mm to 1200 mm away from the camera. If objects of 25 mm are more than 1200 mm away \nfrom the camera, the de tection of their details is noisy, \nmeaning that the measurements are not accurate. On the \nother side, with the EoSens® 12CXP+ camera, zone 1 is \nfrom 0 mm to 700 mm and zone 2 is from 700 mm to \n2700mm, approximately.  \nFigures 13(a) and 13(b) show the compute d values of \nthe scale factor α\ni using different methods. As mentioned \nbefore, black crosses represent the values of the elements  \nFig. 12.  Images to calibrate the scale factors and principal point with the proposed method with the Robotiq wrist camera, 100mm, 700mm, \n1300mm, distance between camera and calibration template .  \nAUTHOR ET AL.:  TITL E 11 \n of the vector α i in (15). The black line is the mean value of \nthe elements of the vector α i in zone 2, which corresponds \nto the  scale factor α  computed with no variation of focal \nlength because the lens is focusing in the hypefocal dis-\ntance. The red line shows the computed value using the \nmethod based on the 2D calibration template proposed by \nZhang[4] when the maximum likelihood estimation (MLE) ends. The blue line shows the value computed with the al-\ngebraic solution from the Zhang method before it is im-\nproved with the maximum likelihood estimation. Calibra-tion with Zhang method is performed with 15 images \ntaken from several locations. Similar results are computed \nfor the scale factor β . Values calculated using each method \nare shown in table 2.  \nIf a continuous camera model were necessary to com-pute the focal length of the camera according with its dis-tance to the object, a function can be empirically adjusted \nto data in figure 13 as follows:   \n0 2α α+ − =dkf (25) \nwhere α is the scale factor for a distance camera- object de-\nfined with d and α0 is the scale factor when the focus is set \nto the hyperphocal distance. Parameters k f and α0 can be \nadjusted using the least squares technique. Figure 14 \nshows the results of adjusting scale factor data of both cam-eras to the function defined in (26). Similar expression is \ncomputed for β . \n4.2 Computing the complete camera model . \nUsing the scale factors α i or βi computed in previous sub-\nsection, it is possible to compute a complete set of camera  \nFig. 13.  (a). Results for the Robotiq wrist camera assembled on a collaborative industrial robot UR3. (b) Results for the EoSens® 12CXP + \ncamera assembled on an industrial robot arm ABB IRB 140. Black crosses represent the result of equation (15). Each black cros s is the scale \nfactor αi computed with the increments of each image Δ ui and the distance di from which it was taken. The black li ne represents the mean value \nof all values in zone 2 (equations (14) and (15)). The red line shows the computed value using Zhang ’s method when the maximum likelihood \nestimation (MLE) ends. The blue line shows the computed value of the algebraic solution o f the Zhang method before it is improved with the \nmaximum likelihood estimation .  \n \nFig. 14. Adjusting the camera scale factor to the function defined in (25 ) (a). Results for the Robotiq wrist camera. (b) Results for the EoSens® \n12CXP+ camera .  \n12 IEEE TRANSACTIONS ON JOURNAL NAME,  MANUSCRIPT ID  \n models with several scale factors using the method de-\nscribed in subsection III.B. An algebraic solution is com-\nputed with equation (20) that is improved with a nonlinear minimization problem defined in (21) or in (24) that in-\ncludes the camera lens distortion parameters.  \nThe aim is to minimize the reprojection error that rep-\nresents the distance between the detected points in the im-\nage and the chessboard template points projected with the \ncomputed model. The Zhang method tries to minimize this index by modifying all camera parameters at the same \ntime. With the proposed method, the scale factors α  and  β of the matrix A remain constant to avoid the coupling be-\ntween intrinsic and extrinsic camera parameters. The \nsearched parameters are the extrinsic ( R\ni and ti) and the \nintrinsic ones (the principal point ( u0, v0), skewness γ and \ndistortion, represented by k 1, k2). \nThe mean values and the standard deviation of the \nreprojection error, with different solutions, are summa-\nrised in table 1. Table 2 shows the computed camera pa-\nrameters with the proposed method and the Zhang method, including the algebraic and the MLE results.  \nFigure 15 compares several solutions of the position \nvector t\ni computed w ith both methods and both cameras. \nSince camera true location is given by the robot arm, bias \nerrors are obtained using the true camera location as the \npoint of reference. The blue line corresponds to the bias with the algebraic solution computed with the method \nproposed in subsection 3.2. The green line corresponds to \nthe MLE solution, computing only the subset of parame-ters proposed in this paper and assuming a constant focal \nlength. The red line corresponds to the algebraic solution \nproposed by Zhang, and the black line corresponds to Zhang’s MLE solution. Figure 15(a) corresponds to the er-\nrors in the location computed with the Robotiq wrist cam-\nera and figure 15(b) shows errors in location with the the EoSens® 12CXP+ camera. \n4.3 Discussion  \nBias in the position vector t i will give the accuracy of the \ncomputed camera parameters with each method. Analys-ing position vector t\ni bias in figure 15, with both cameras, \nthe camera positions computed with the proposed method are closer to the true positions than the positions comput ed (a)\n(b)\n \nFig. 15. (a). Error in locations for the Robotiq wrist camera. (b) Error in locations for the EoSens® 12CXP+ camera. In both cameras, the \nproposed method computes camera locations closer to the real one than the method proposed by Zhang. Analysis of the Zhang sol ution shows \nthat, although the reprojection error is bigger, the algebraic result is closer to the real location than the MLE .  \n \nTABLE  1 \nMEAN VALUES AND STANDARD DEVIATION OF THE REPROJECTION \nERROR WITH DIFFERENT  SOLUTIONS  \n \nMethod  Median Stand-\nard de-\nviation  \nRobotiq \nwrist cam-\nera Proposed method  0.3362  2.1709  \nMLE Proposed method  9.0165× e -11 1.5626 \nZhang algebraic solution  0.0472  1.8667  \nZhang MLE solution  1.8347× e -10 1.0487  \nEoSens® \n12CXP+  Proposed method  -2.4143  14.4151  \nMLE Proposed method  -1.5807×e -10 3.1542  \nZhang algebraic solution  0.4644  9.0036 \nZhang MLE solution  1.2092×e -09 2.3201  \nThe reprojection error with MLE is smaller because parameters are adjusted to \nthe data accurately. However, the differences between reprojection error vary-\ning all parameters and varying extrinsic parameters, distortion and skewness \nonly is not significant.   \n \n AUTHOR ET AL.:  TITL E 13 \n by Zhang method. With the proposed method, MLE ad-\njusts the computed model to the input data, changing only \nthe extrinsic camera parameters, distortion, skewness and \nprincipal point, and assuming that scale factors are accu-rate as they have been computed  by specific trials. MLE \ndoes not significantly modify the result solution. However, in the case of Zhang’s result solution, the difference be-tween the algebraic solution and MLE is significant . Anal-\nysis of the Zhang solution shows that the algebraic result is closer to the true location than the MLE. MLE obtains worse results than the algebraic solution. Even though the \nreprojection error is bigger with algebraic methods, esti-mated locations are closer to the real ones. This is ex-plained with the coupling between all camera parameters. \nThe algebraic solution computes the camera parameters separately and the MLE solution tries to find out the best solution for all camera parameters at the same time. Solv-\ning the nonlinear minimization problem based on repro-\njection error by computing all the camera parameters is not \na very good practice.  \nAnalysing table 1, reprojection error values with MLE \nare smaller because the parameters are accurately adjusted to the data. However, the differences between the reprojec-\ntion error computing all the parameters at the same time \nand computing all parameters assuming a constant focal \nlength are not significant. This is because the parameters \nare adjusted to data but in one case all parameters change \nin every iteration of the optim ization process and in the \nother case the focal length remains constant during the op-\ntimization process. A constant focal length sets a reference \nin the nonlinear optimization process that helps to reach \nan unbiased solution. Since all the parameters are t ightly \ncoupled, MLE is able to finish with a solution where the \nparameters satisfy the model for the given data but differ-\nent of the true one. Indeed, the computed solution is rep-\nresented by an optical axis in the Sophia plane which does \nnot correspond to the true one. With the proposed method, \nintrinsic parameters, such as scale factors are estimated with specific tests, and MLE is used to compute extrinsic \nparameters, distortion, skewness and the principal point. \nCamera parameters are divided into two gro ups to avoid \nincorrect solutions due to the coupling between them . The \nconstant focal length sets the reference to compute the un-biased optical axis in the Sophia plane.  \nAs can be seen in table 2, with the Robotiq wrist cam-\nera, the algebraic solution that results from the Zhang method is closer to the one computed with the lens focus-ing in the hypefocal distance. However, the MLE solution \nis notably different. Analysing the results from the \nEoSens® 12CXP+ camera, the algebraic solution is differ-ent from that of the lens focusing in the hypefocal distance, \nand the MLE solution even more so. To explain these re-\nsults, it is necessary to observe the range of distances that cove r zones 1, 2 and 3, respectively, with each camera. \nZone 2 of the Robotiq wrist camera, in which focal length \ndoes not change, is delimited by distances of 150mm and 1200mm. Capturing images of the chessboard closer to \n150mm to the camera is not possible if the chessboard is \nprinted in an A4 sheet. Consequently, since all images are taken in Zone 2, the focal length is constant and the alge-\nbraic solution of the Zhang method is similar to that of the \nlens focusing in the hypefocal distance, supposed as unbi-ased. However, when the algebraic solution is improved \nwith the maximum likelihoold estimation, the tightly cou-\npling between intrinsic and extrinsic parameters means \nthat, although the final solution has a very small reprojec-\ntion error, the computed paramet ers are biased.  \nGoing deeper , if the chessboard were smaller than an \nA4 sheet because the application needs images closer to the camera, most of the calibration images would be in the \nzone 1 and camera parameters would be different to the \nones computed wit h the lens focusing in the hypefocal dis-\ntance. Moreover, if camera parameters computed with the \nZhang method were used in a camera -object distance \ncloser than 150 mm, results would be biased.  \nAnalysing distances in Zones 1, 2 and 3 for the \nEoSens® 12CXP+ c amera, Zone 2 (where the focal length \nis constant), starts at 700 mm and ends at approximately \n2700 mm. With this camera, it is possible to put the A4 \nsheet calibration template in the image completely when \nthe camera is 300 mm away from the calibration te mplate. \nIn consequence, images that are taken within a range of 300 mm and 700 mm are in Zone 1, where the focal length var-ies in order to obtain a sharp image. Therefore, using these \nimages with the Zhang method disrupts the results be-\ncause they were take n with a different focal length. The al-\ngebraic solution from the Zhang method does not compute \ngood parameters similar to the ones computed when the \nlens is focusing in the hypefocal distance. Similar to the computed results with the Robotiq wrist camera, the MLE \nobtains an even more biased solution. When starting the \nMLE with biased parameters, the MLE will end producing more biased results due to the coupling between intrinsic \nand extrinsic camera parameters.  \nAs the experiments show, the proposed method g ives \na stable solution to obtain an accurate camera calibration. The method proposed by Zhang could lead to incorrect re-\nsults because of the camera parameters coupling and as TABLE  2 \nCAMERA PARAMETERS COM PUTED USING THE PROP OSED \nMETHOD IN ZONE 2 AND USING THE ZHANG METHOD  \n Param.  Proposed \nmethod Zhang alge-\nbraic solution  Zhang MLE \nsolution  \nRobotiq \nwrist \ncamera α 1370,8  1362,8  1319,7  \nβ 1373,8   1369,9 1329,7  \nu0 645,8  651,7  711,2  \nv0 359,3 370,5  375,9  \nγ 0,0001  -0,002  0,0008  \nk1 0.0087  0.0048    -0.016  \nk2 -0.072  -0.056  -0.003  \nEoSens® \n12CXP+  α 4755,8  4487,0  4289,8  \nβ 4789,1  4442,8  4295,1  \nu0 2052,3  2046,8  2127,5  \nv0 1548,5  1448,1  1566,7  \nγ 0,0012  0,0239  0,0280  \nk1 0.0022  0.0421  -0.1703  \n k2 0.0570  -0.134 0.1389  \n 14 IEEE TRANSACTIONS O N JOURNAL NAME,  MANUSCRIPT ID  \n images are taken at any distance and it does not define \nwhich set of images obtain the best solution. In most cam-\nera calibration papers an analysis of the number of images is performed, but the camera location is not defined . \n5 CONCLUSION  \nIn this paper, the influence of the focus in the calibration process has been analysed in depth. T o obtain the sharpest \nobjects in the image, lenses are adjusted depending on the \ndistance of the camera to the object varying the focal length \nin each image. Since the pin- hole model does not consider \nthis variation, computing the camera model using images \ncaptured from different distances to the calibration tem-\nplate could obtain incorrect results. Moreover, an exhaus-\ntive study of the interdependence between intrinsic and \nextrinsic parameters has demonstrated that they are tightly \ncoupled. The focal length of the camera is linked to the dis-\ntance of the camera to the object and the principal point is \nlinked to the location X and Y of the camera in the scene. \nIn consequence, computing all the camera parameters to-\ngether in an iterative nonlinear minimization pr oblem can-\nnot be considered a good practice in general. It gives a \nvalid result based on the reprojection error but the result \ndoes not represent the real camera.  \nThe camera calibration process has been improved in \nthree aspects. First, a camera model is pr oposed where sev-\neral values of scale factors are used depending on the dis-\ntance of the camera to the object in the scene. Secondly, the \nscale factors are computed with a specific, separate exper-iment, to avoid incorrect results due to the interference be-\ntween intrinsic and extrinsic camera parameters. Thirdly, \na nonlinear minimization problem is solved by computing only extrinsic parameters, the principal point, skewness \nand lens distortion instead of computing all camera pa-\nrameters at the same time.  \nThe proposed improved method accounts for the exist-\nence of several set points of the camera lens to obtain sharp images. In consequence, it is necessary a camera model with several values of scale factors to represent camera be-\nhaviour accurately. If a model wit h a constant value of \nscale factor is used, it is necessary to know the range of dis-\ntances between the camera and the objects where these \nscale factors camera parameters are valid or conversely, to \ncapture images of the calibration template in a range of d is-\ntances valid for the application. In addition, to avoid the \ncoupling effects between intrinsic and extrinsic camera pa-\nrameters, it is advisable to conduct a specific test to sepa-rately compute each camera parameter instead of trying to \nsolve the nonlinear minimization problem by computing \nall parameters at the same time. Computing all camera pa-rameters at the same time in a nonlinear minimization pro-\ncess computes biased results. An improved method to \ncompute the camera principal point with more accuracy, could be considered in a future work. At present, the pro-\nposed method is a step forward in the field of camera cali-\nbration that will help in any application where the camera parameters represent a crucial step.  REFERENCES  \n[1] L. Kang, L. Wu, Y. Wei, S. Lao, Y.  Yang, Two -view underwa-\nter 3D reconstruction for cameras with unknown poses under \nflat refractive interfaces Pattern Recognition, Volume 69, \n(2017).  \n[2] M. Alemán -Flores, L. Alvarez, L. Gomez, P. Henriquez, L. \nMazorra, Camera calibration in sport event scenari os Pattern \nRecognition, Volume 47, Issue 1, (2014) . \n[3] S. Yi, S.Min, A Practical Calibration Method for Stripe Laser \nImaging System, IEEE Transactions on Instrumentation and \nMeasurement , (2020) DOI 10.1109/TIM.2020.3024336,  \n[4] T Jiang, H. Cui ,  X. Cheng, Accurate calibration for large- scale \ntrackingbased visual measurement system, IEEE Transactions \non Instrumentation and Measurement ( 2020) DOI \n10.1109/TIM.2020.3016412,  \n[5] H. Yu, W. Zhen, W. Yang, S. Scherer , Line -Based 2 -D–3-D \nRegistration and Came ra Localization in Structured Environ-\nments. IEEE Transactions on Instrumentation and Measure-\nment.  Vol. 68, No 11 ( 2020)  \n[6] J. Weng, P. Cohen, and M. Herniou, Camera calibration with \ndistortion models and accuracy evaluation, IEEE Transactions \non Pattern Analy sis & Machine Intelligence, no. 10, pp. 965 –\n980. (1992)  \n[7] Z. Zhang, A flexible new technique for camera calibration, IEEE \nTransactions on Pattern Analysis and Machine Intelligence, vol. 22, \npp. 1330- 1334. (2000) . \n[8] D. Yang, B. He, M. Zhu, J. Liu, An Extrinsic Calibration Method \nWith Closed -Form Solution for Underwater Opti- Acoustic Imaging \nSystem . IEEE Transactions on Instrumentation and Measurement, \nVol.: 69, Issue: 9 (2020)  \n[9] B. Halloran, P. Premaratne, P. James, Robust one -dimensional \ncalibration and localisat ion of a distributed camera sensor \nnetwork. Pattern Recognition, Volume 98, (2020) . \n[10] E. Peng, L. Li, Camera calibration using one -dimensional in-\nformation and its applications in both controlled and uncon-\ntrolled environments. Pattern Recognition, Volume 43, Issue 3, (2010) . \n[11] J. de França, M. Stemmer, M. Bernadete, J. Chico, A new robust al-\ngorithmic for multi- camera calibration with a 1D object under gen-\neral motions without prior knowledge of any camera intrinsic param-\neter. Pattern Recognition, Volume 45, (2012 ). \n[12] J. Zhang, H. Yu, H. Deng, Z. Chai, M. Ma, X. Zhong, A Robust and \nRapid Camera Calibration Method by One Captured Image, IEEE \nTransactions on Instrumentation and Measurement, Vol.: 68, Issue: \n10 (2019)|  \n[13] Q.-T. Luong and Faugeras, O. Self -Calibration of a Moving Camera \nfrom Point Correspondences and Fundamental Matrices, Interna-\ntional Journal of Computer Vision, vol. 22, pp. 261- 289. (1997) . \n[14] D. Zhan, D. Jing, M. Wu, D. Zhang, L. Yu, T. Chen, An Accurate \nand Efficient Vision Measurement Approach for Railway Catenary \nGeometry Parameters, IEEE Transactions on Instrumentation and \nMeasurement, Volume: 67, Issue: 12 (2018)  \n[15] R. Hartley and Zisserman A. Multiple View Geometry in computer \nvision. Cambridge University Press. ISBN 0 -521-54051 -8 (2003) . \n[16] T. Bell, J. Xu, S  Zhang, Method for out- of-focus camera cali-\nbration. Applied Optics Vol. 55, No. 9  (2016)  \n[17] Y. Wang, X. Chen, J. Tao, K. Wang, M. Ma, Accurate feature \ndetection for out- of-focus  camera calibration . Applied Optics \nVol. 55, No. 28 ( 2016)   \n[18] P. Wang,  J. Wang,  J. Xu, Y. Guan,  G. Z hang,  K. Chen, Calibra-\ntion method for a large -scale structured  light measurement \nsystem . Applied Optics Vol. 56, No. 14 (2017 ) \n[19] H. Hyowon, P. Michal, A. Hatem, K. In So , S. Yaser , Deltille \ngrids for geometric camera calibration. Proceedings of the \nIEEE International Conference on Computer Vision. (2017)  \n[20] P. Songyou, and P.  Sturm. \"Calibration Wizard: A guidance \nsystem for camera calibration based on modeling geometric \nand corner uncertainty.\" Proceedings of the IEEE Interna-\ntional Conference on  Computer Vision. (2019)  AUTHOR ET AL.:  TITL E 15 \n [21] H. Haifei, H . Zhang, and Y . Cheung. The common self -polar \ntriangle of concentric circles and its application to camera cal-\nibration. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. (2015)  \n[22] A Robust Detection Method of Control Points for Calibration and Measurement With Defocused Images  \n[23] W. Ding, X. Liu, D. Xu, D. Zhang,  Z. Zhang IEEE Transac-\ntions on Instrumentation and Measurement, Vol .: 66, Issue: \n10, (2017)  \n[24] H. Hu, G. Kantor,  Instance Selec tion for Efficient and Reliable \nCamera Calibration. Proceedings of the IEEE International \nConference on Robotics and Automation (ICRA). (2016) . \n[25] S. Ryusuke and Y.  Yasushi, Accurate calibration of intrinsic \ncamera parameters by observing parallel light pairs ,\" IEEE In-\nternational Conference on Robotics and Automation, pp. 1390- 1397. (2008) . \n[26] A. Alturki, Loomis, J., Camera principal point estimation \nfrom vanishing points. Proceedings of the IEEE National Aer-\nospace and Electronics Conference (NAECON) and Ohio In-\nnovation Summit (OIS) (2016) . \n[27] M. Lu, J. Chuang, Fully Automatic Camera Calibration for \nPrincipal Point Using Flat Monitors. Proceedings of the 25th IEEE International Conference on Image Processing (ICIP). \n(2018) . \n[28] C. Ricolfe- Viala,  A. Sánchez -Salmerón, Robu st metric calibra-\ntion of non- linear camera lens distortion.  Pattern Recogni-\ntion, Volume 43 Issue 4 Pages 1688- 1699 (2010) . \n[29] C. Ricolfe- Viala, , A. Sánchez -Salmerón, Lens distortion mod-\nels evaluation. Applied optics. Volume 49 Issue 30 Pages \n5914- 5928 (2010) . \n[30] Q. Zhu, Estimating principal point and nonlinear parameters \nof camera from a planar calibration image.  International Conference on Neural Information Processing. pp 16 -24 \n(2012) .\n \n[31] A. Ruiz, Lopez -de-Teruel, P. E., García- Mateos, G., A note on prin-\ncipal p oint estimability. Proceedings 16th International Conference \non IEEE Pattern Recognition. (2002) . \n[32] R. Hartley, In defence of the 8 -point algorithm. Proceedings of the \n5th International Conference on Computer Vision. IEEE Computer \nSociety Press . (1995)  \n[33] G. Golub, Loan, C., Matrix Computations. The John Hopkins Uni-\nversity Press. (1996) . \n[34] O. Faugeras, Three -Dimensional Computer Vision: A Geometric \nViewpoint. MIT Press. (1993) .  \n[35] T. Rahman, & Krouglicof, N. An Efficient Camera Calibration Tech-nique Offering Robustness and Accuracy Over a Wide Range of Lens \nDistortion. IEEE Transactions on Image Processing, 21, 626- 637. \n(2012) . \n Carlos Ricolfe- Viala was born in  Valencia, Spain in 1973. She received the \nM. Sc. Degree in Industrial Electronics and Automatic Control Engineering in 2000 and his Ph.D. degree on Computer Vision , Robotics and Automatic \nControl in 2006 from the Polythechic University of Valencia (UPV), Spain . \nHe is lecturer in the Polythechic University of Valencia since 2001 at the Systems Engineering  and Control Department. He has participated in several \nnational and international research projects. His research interests are image processing and computer vision, especially in motion estimation, feature de-\ntection and matching, camera calibration, 3D computer vision and intelligent \nsystem robot. He cooperates with the Institute of Industrial Informatics and \nAutomatic Control of Valencia.  \n "    },    {        "title": "2402.04690v1.Near_equilibrium_growth_of_vapor_bubbles.pdf",        "content": "Near-equilibrium growth of vapor bubbles\nOrr Avni, Eran Sher, and Yuval Dagan∗\nFaculty of Aerospace Engineering, Technion - Israel Institute of Technology, Haifa, 320003, Israel.\nAbstract\nThis study delves into the near-equilibrium dynamics of vapor bubbles. Utilizing a regular perturbation\nmethod, we derive analytical solutions to capture the bubble’s initial growth stages, wherein surface tension\nand viscous dissipation forces impede the bubble’s growth. The accuracy of these solutions is validated with\nnumeric solutions of the complete Rayleigh-Plesset equation. Although limited to near-critical bubble radii,\nour analytical solutions predict the initial delay period as a function of two nondimensional parameters,\nsignifying the initial deviation from the critical radius and the surface tension to viscosity ratio. We found\na transition in the dominant delay mechanism, marking a shift from surface tension to viscous damping\ndominance. Our findings emphasize the significance of considering the surface tension delay, particularly for\nshort timescales, and highlight its crucial role in accurately modeling the bubble’s initial growth dynamics.\nThe derived analytical solutions and the obtained correlation for surface-tension-induced delay may prove a\npractical tool and could be integrated into existing models of vapor bubble growth.\n∗yuvalda@technion.ac.il\n1arXiv:2402.04690v1  [physics.flu-dyn]  7 Feb 2024I. INTRODUCTION\nNucleation of vapor bubbles within a superheated liquid is a fundamental yet complex and\nelusive phase-change phenomenon. The ubiquity of vapor bubbles in natural processes [1–3],\nwide-ranging industrial [4–8], and medical [9–11] applications poses a solid scientific research\nmotivation. Embryonic vapor nucleons constantly form spontaneously in superheated liquids due\nto the liquid’s inherent local density fluctuations; the contrasting effects of the inward-directed\nsurface tension and outward-directed pressure acting on the nucleated bubble interface cause small\nnucleons to collapse immediately. However, nucleons surpassing the unstable equilibrium radius\ninitiate localized liquid-vapor phase change. The subsequent growth of a single vapor in an\nunbounded superheated liquid includes three discernible stages [12, 13]. Surface tension forces\npredominate the initial growth stage, impeding significant growth for a – usually short – delay\nperiod. After the nucleus grows, surface tension influence diminishes, and inertia forces dominate\nthe growth. The constant growth is driven by a steady pressure gradient across the interface,\noriginating from the liquid’s superheat degree [14]. Heat consumed by evaporation at the interface\ncools the liquid boundary layer around it, reducing the vapor pressure and transitioning to the last\nstage of the bubble’s expansion. As the pressure gradient tends to zero, the growth is driven solely\nby heat supplied to the interface via thermal diffusion; hence, during this asymptotic stage, the\ngrowth rate is proportional to the square root of time [15].\nMost current analytic models focus on the latter stages of the growth; notably, the Mi-\nkic–Rohsenow–Griffith (MRG) model [16] pioneered the interpolation between the inertial and\nthermal regimes. Following works [17–19] have since modified the interpolation and improved\nits accuracy in the transitional regime. On the other hand, numerical studies [20–22] solved the\ncoupled inertial-thermal problem and successfully captured the entirety of the growth process,\nincluding the surface tension regime. Recently, Sullivan et al. [23] presented an improved inter-\npolated semi-analytical model capable of approximating the growth of the bubble across all three\nregimes. Near-equilibrium growth was studied numerically and subsequently interpolated with\nthe broader analytic model for the bubble’s growth. Another approach to studying the growth of\na post-nucleation vapor bubble is molecular dynamics simulations. Using Lennard-Jones large\ncomputational volume simulations of liquid-to-vapor nucleation, Ang ´elilet al. [24] managed to\nmeasure the properties of bubbles from their inception as stable, critically sized bubbles to their\ncontinued growth. Crucially, the study reported good agreement up to the molecular scale between\n2molecular dynamics simulations and results obtained via the simplified Rayleigh-Plesset model\nused in most analytic and numeric models. However, both numerical and molecular-dynamics-\nbased works are limited in their generality and applicability compared with analytical models,\nwhich yield immediate predictions without requiring significant computational effort.\nOn the other hand, studying the early dynamics of vapor bubbles faces experimental challenges.\nThe vapor bubble critical radius may reach extremely small scales [25], especially when considering\nhomogeneous nucleation in highly superheated liquids. Therefore, this study focuses on analytical\nmodeling of the near-equilibrium, surface-tension-dominated growth regime, aiming to bridge\nthe existing theory gap. We adopt a regular perturbation method to derive an approximate near-\nequilibrium solution. Using this method, one may explore the underlying physics governing the\ndynamics of vapor bubbles following nucleation inception, aiming to provide a better understanding\nof their initial growth.\nSection II outlines the methodology and mathematical modeling approach used to derive an\nanalytical solution for near-equilibrium bubble growth. Section III explores the growth of a vapor\nbubble wherein viscous dissipation effects are negligible, comparing several analytical and numeric\nsolutions. The analysis is extended in Section IV to damped growth, where the influence of viscous\ndamping is investigated. Finally, the implications and outlook of the present analysis are discussed\nin Section V.\nII. NEAR-EQUILIBRIUM SOLUTION\nThe immediate aftermath of a spontaneous nucleation event is considered here: the phase-\nchange-driven growth of a vapor bubble within a superheated liquid. Such vapor-liquid systems\nare mechanically unstable; the equilibrium state, i.e., the stable bubble radius, is predicted by the\nYoung-Laplace relation\n𝑅𝑐=2𝜎\nΔ𝑝0, (1)\nwhere𝑅𝑐is the critical bubble radius, 𝜎is the vapor-liquid surface tension, and the Δ𝑝0is the\npressure gradient acting on the interface. Thus, we consider initially still, expanding bubble\nwhose initial radius is perturbed with respect to the critical radius, ˜𝑅(˜𝑡=0)=𝑅𝑐+˜𝜀. This\nmathematical perturbation may be associated with the thermodynamic density fluctuations that\nlead to the bubble’s nucleation. The model assumes a perfectly spherical bubble immersed in a\nsemi-infinite, incompressible liquid medium; thus, any potential interactions between nucleated\n3bubbles are discounted. Our analysis is restricted to near-nucleation growth scales, wherein thermal\ndiffusion effects are insignificant and constant temperature difference across the sharp interface is\nmaintained. Finally, we assume the bubble does not contain dissolved gases and consists solely\nof vapor molecules, maintaining thermodynamic equilibrium with the outer superheated liquid.\nGiven a constant vapor temperature, a saturated state implies that the growth process is driven by\na steady pressure gradient across the bubble interface Δ𝑝0. The steady driving force distinguishes\nthe growth process of a vapor bubble from the dynamics of a gas bubble, wherein the inner pressure\nis coupled to the bubble’s volume.\nThe simplifying assumptions yield a single ordinary differential equation for the bubble radius\n˜𝑅, the Rayleigh-Plesset (RP) equation [15];\n˜𝑅¥˜𝑅+3\n2¤˜𝑅2=Δ𝑝0\n𝜌𝑙−2\n𝜌𝑙˜𝑅\u0010\n𝜎+2𝜇𝑙¤˜𝑅\u0011\n, (2)\nwhere𝜌𝑙is the liquid’s density and 𝜇𝑙is its viscosity. The resulting bubble oscillator experiences\nexternal forcing from a constant pressure gradient, countered by surface tension spring-like rigidity\nand the damping effect of viscosity. The inertia terms of the oscillator are nonlinear, originating\nfrom the integration of the unsteady (first LHS term) and radially convective (second LHS term)\ncomponents of the Navier-Stokes equation. Using the critical radius 𝑅𝑐and the inertial regime\nvelocity\n¤˜𝑅𝐼𝑛𝑒𝑟=√︄\n2Δ𝑝0\n3𝜌𝑙=√︄\n4𝜎\n3𝜌𝑙𝑅𝑐, (3)\nwe normalize Eq. (2) and its initial conditions, yielding the following initial value problem (IVP)\n2\n3𝑅¥𝑅+¤𝑅2−1+1\n𝑅\u0012\n1+4√\n3𝑂ℎ𝑐¤𝑅\u0013\n=0, 𝑅(0)=1+𝜀,¤𝑅(0)=0, (4)\nwhere𝑂ℎ𝑐is the nucleation Ohnesorge number , defined as\n𝑂ℎ𝑐=𝜇𝑙\n𝜎√︄\nΔ𝑝0\n2𝜌𝑙=√︄\n𝜇2\n𝑙\n𝜎𝜌𝑙𝑅𝑐. (5)\nThe nucleation Ohnesorge number 𝑂ℎ𝑐signifies the ratio between viscous damping and surface\ntension forces acting on the critically sized bubbles, i.e., the balance between the two at – and\nshortly after – nucleation inception. Since the surface tension forces are coupled to the bubble’s\ncharacteristic radius, the nucleation Ohnesorge number is dictated by both liquid medium properties\nand the thermodynamic conditions at nucleation.\n4Due to the inherent nonlinearity of the RP equation, obtaining a fully analytical solution is\npossible only for reduced, limiting cases. Hence, we seek an approximate solution for the initial,\nnear-equilibrium growth of the vapor bubble by substituting a regular perturbation series of the\nbubble radius 𝑅=𝑅0+𝜀𝑅1+𝜀2𝑅2+𝜀3𝑅3+...in the original IVP. The leading order solution is a\ntrivial one𝑅0=1, betokening the stability of the bubble in the absence of any perturbation. The\nfirst-order approximation yields a linearized, homogeneous IVP\nL[𝑅1]=2\n3¥𝑅1+4√\n3𝑂ℎ𝑐¤𝑅1−𝑅1=0, 𝑅1(0)=1,¤𝑅1(0)=0, (6)\nand the next terms of the perturbation series yield\nL[𝑅2]=−4\n3𝑅1¥𝑅1−¤𝑅12, 𝑅2(0)=¤𝑅2(0)=0 (7)\nL[𝑅3]=−4\n3\u0000𝑅1¥𝑅2+𝑅2¥𝑅1\u0001−2¤𝑅1¤𝑅2−𝑅1¤𝑅12−2\n3𝑅2\n1¥𝑅1, 𝑅3(0)=¤𝑅3(0)=0. (8)\nThe particular solution for the first-order term is\n𝑅1(𝑡)=𝑒−3𝑡\n4𝑂ℎ𝑐cosh \n3𝑡\n4√︂\n8\n3+𝑂ℎ2\n𝑐!\n+𝑂ℎ2\n𝑐sinh\u0012\n3𝑡\n4√︃\n8\n3+𝑂ℎ2\n𝑐\u0013\n√︃\n8\n3+𝑂ℎ2\n𝑐; (9)\nwhile higher-order analytical solutions are also obtainable. The approximated solution is, by\ndefinition, limited to near-equilibrium bubble radii. Thus, the following sections investigate\nthe solution accuracy and validity region by comparing them against numerical solutions of the\noriginal, complete RP oscillator Eq. (4). First, Section III examines a limiting case where the\nnucleation Ohnesorge number tends to zero, i.e., the bubble’s initial growth is not damped by\nviscous dissipation.\nIII. UNDAMPED GROWTH\nThe initial, near-equilibrium vapor bubble growth may be considered undamped when the\nviscous dissipation effects are negligible compared with the pressure difference and the surface\ntension forces. Undamped growth occurs in two distinct cases: when the liquid viscosity tends\nto zero and when nucleation conditions dictate a large critical radius. Large critical radii could\nprompt undamped bubble growth even in a highly viscous liquid, whereas a high superheating\ndegree prior to nucleation could result in a small nucleated bubble, for which viscous damping\n5FIG. 1. Dominant balance analysis of the normalized RP equation using a numeric solution for Eq. (4), given\ninitial radius perturbation of 𝜀=0.01. (a) Dotted lines depict the results for undamped growth 𝑂ℎ 𝑐=0,\nwhereas dashed lines depict a case model 𝑂ℎ 𝑐=1 for a damped growth. (b) Close-up of the initial growth\nstages. Changes due to increased damping are highlighted with colored arrows.\nis significant even when the surrounding liquid’s viscosity is low. Thus, the 𝑂ℎ𝑐→0 limit may\nrepresent different physical systems and serve as an important limiting case.\nFig. 1 presents a dominant balance analysis of the nonlinear oscillator leading terms while\nutilizing an RK4 numeric scheme for the solution of Eq. (4). Each term of the normalized RP\nequation is plotted individually, allowing one to assess the dominant physical terms throughout\nthe growth. The analysis is conducted for two distinct case models: undamped growth, wherein\n𝑂ℎ𝑐=0, and damped growth, wherein 𝑂ℎ𝑐=1. The latter case will be discussed in Section IV.\nAs our model postulates, the normalized pressure term remains constant in both cases; the surface\ntension and convective inertia terms switch roles as the second dominant terms. The linearized\nEq. (6) and Fig. 1 suggest the initial growth is governed by the balance between pressure and surface\ntension forces, justifying the omission of the nonlinear convective inertia term on the linearized IVP.\nHowever, as the bubble interface accelerates, the nonlinear term becomes dominant, breaking the\nassumptions of our linear analysis and inducing a shift in the bubble growth regime. Viscous effects\nalso influence the dominant terms; they delay the surge in the inertia term, playing a significant\nrole immediately post-nucleation and up until the bubble grows beyond the critical radius order\nof magnitude. Nevertheless, a distinct velocity threshold emerges in both cases, delineating a\ntransition from a surface-tension-dominated to an inertia-dominated growth regime.\n6FIG. 2. Comparison between a numeric solution of RP equation, represented by a dashed line, and different\norders of the approximated perturbation solution, represented by solid lines, given initial radius perturbation\nof𝜀=0.01 and undamped growth 𝑂ℎ 𝑐=0. (a) Bubble radius as a function of time and (b) Bubble interface\nvelocity as a function of bubble radius.\nFig. 2 compares the obtained approximated solutions with a numeric solution of Eq. (4) for initial\nperturbation of 𝜀=0.01. The approximated and numeric solutions converge and are indiscernible at\nthe initial growth stages. As the bubble grows, the approximated solutions diverge; the hyperbolic\nterms in each approximation asymptotically approach ±∞. Increased order of approximation\nyields improved precision and a marginally expanded region of validity. Nevertheless, a common\ndivergence among all solutions emerges when the velocity attains a magnitude of ¤𝑅≈0.1, i.e., when\nthe velocity reaches the inertial velocity order of magnitude. This common divergence accentuates\nthe critical influence of the nonlinear convective inertia term ¤𝑅2on the near-equilibrium bubble\ndynamics and the transition between the distinctly different growth regimes, as suggested by Fig. 1.\nThe surface tension delay may be evaluated using each approximated solution. Although the least\naccurate, Fig. 2 suggests that the first-order solution captures the essence of the initial delay while\nyielding a simple relation for the delay,\n𝜏𝑆𝑇≈√︂\n2\n3sinh−1\u00121\n5√\n6𝜀\u0013\n. (10)\nDominant balance analysis also indicates that the unsteady inertia term influence is compar-\natively small in the overall undamped system behavior. This observation allows for extracting a\nreduced outer solution, approximating the bubble’s behavior in the transition to the fully inertial\nregime. By excluding the unsteady inertia term and setting 𝑂ℎ𝑐=0, Eq. (4) simplifies into an\n7autonomous, first-order ODE,\n𝑅\u0010\n¤𝑅2−1\u0011\n+1=0. (11)\nThis ODE is amenable to inverse solution for 𝑅> 1, yielding\n𝑡(𝑅)=sinh−1\u0010√\n𝑅−1\u0011\n+√︁\n𝑅(𝑅−1)+𝐶1. (12)\nHowever, the remaining degree of freedom 𝐶1cannot be independently resolved; regular matching\nwith any inner solution is unobtainable since they tend to ±∞. Nevertheless, we may extract\na uniform solution for the undamped bubble growth based on the first-order inner solution by\nimposing continuity of both radius and velocity terms. Velocity continuity necessitates that the\nsolutions are tangent at their intersection 𝑅𝑡𝑎𝑛, resulting in an implicit relation dependent on the\ninitial perturbation\n1√︃\n1+(𝑅𝑡𝑎𝑛−1)2+2𝑅𝑡𝑎𝑛−1\n2√︁\n𝑅𝑡𝑎𝑛(𝑅𝑡𝑎𝑛−1)=√︄\n2\n3\u0002\n(𝑅𝑡𝑎𝑛−1)2−𝜀2\u0003. (13)\nSubstituting the extracted tangent intersection point allows the determination of the integration\nconstant\n𝐶1=√︂\n2\n3cosh−1\u0012𝑅𝑡𝑎𝑛−1\n𝜀\u0013\n−sinh−1\u0010√︁\n𝑅𝑡𝑎𝑛−1\u0011\n−√︁\n𝑅𝑡𝑎𝑛(𝑅𝑡𝑎𝑛−1), (14)\nand resolves the uniform solution.\nThe obtained solution is presented in Fig. 3, elucidating three stages in the bubble’s undamped\ngrowth for an initial perturbation of 𝜀=0.01. The bubble growth is initially delayed due to\nthe dominance of surface tension forces. The velocity threshold ¤𝑅≤0.1 marks the end of the\nsurface-tension-dominated regime as the system transitions into the inertial, linear growth regime.\nThe unified solution provides a new approximation for the surface-tension-dominated and the\nintermediate regime, wherein both surface tension and inertial forces influence the dynamics. The\nend of the transitional regime is marked by a second threshold, after which the bubble’s velocity\nis practically constant, as expected for inertially limited growth. This threshold is defined based\non the bubble interface acceleration, setting the criterion ¥𝑅≤0.01 to demarcate the transition\ninto this regime. We offer this criterion to provide a quantifiable indicator, facilitating a distinct\nseparation between the growth regimes.\nFig. 4 provides a comparative analysis of different solutions for the undamped growth of a\nvapor bubble. Standard analytic models such as the MRG model often overlook the initial surface-\ntension-induced delay and assume the growth begins from the linear regime – as depicted by the\n8FIG. 3. Unified approximated solution illustrating the undamped growth of a vapor bubble with an initial\nperturbation of 𝜀=0.01. The inner solution, given by Eq. (9), is represented by the blue line, while the outer\nsolution, given by Eq. (12), is depicted by the yellow line. Dashed lines demarcate the transitions between\ngrowth regimes: (a) transitional to the linear regime, defined using acceleration threshold ¥𝑅< 0.01 and (b)\nsurface-tension-dominated to the transitional regime, defined using velocity threshold ¤𝑅> 0.1.\nFIG. 4. Comparison of different solutions for an undamped bubble growth. The numeric (dashed black\nline), inner (blue line), and outer (blue line) are obtained for an initial perturbation of 𝜀=0.01. The green\nline represents estimates from common models, such as MRG [16], where the surface tension delay is not\nconsidered. The purple line incorporates the surface tension delay time, retarding the linear growth models\nto account for the initial delay.\n9green dashed line. We propose two enhancements for the current models. The first involves\nincorporating the surface tension delay time, defined by the velocity threshold, to retard the onset\nof linear growth. This simple retardation promptly improves the model’s agreement with the\nnumeric result. However, as illustrated in Fig. 4, the extracted uniform solution offers a markedly\nmore accurate approximation for short time scales. The deviation between the uniform solution\nand the numeric result stems from excluding the unsteady inertia term, notably impacting the\nintermediate regime. Fig. 4 findings emphasize the significance of considering the surface tension\ndelay, particularly during short timescales, and highlight its crucial role in accurately modeling\nthe dynamic behavior of the bubble. The analysis is extended in the following section to a general\ncase where the nucleation Ohnesorge number 𝑂ℎ𝑐does not necessarily approach zero, allowing\nfor investigation of viscous effects influence on near-equilibrium vapor bubble growth.\nIV. DAMPED GROWTH\nViscous effects could play a significant role in the near-equilibrium vapor bubble growth: it may\ndampen the bubble’s expansion, extending the surface-tension-dominated regime and retarding the\ntransition to the linear growth regime. The potential delay induced by viscous effects becomes\ndominant where the phase change is abrupt and characterized by short timescales. The delay\nperiod could be substantial compared to the entirety of the violent and fast growth process. As the\ncritical bubble radius reduces, the viscous effects could dominate the surface tension’s spring-like\nforce, as encapsulated by the definition of nucleation Ohnesorge number 𝑂ℎ𝑐. Revisiting the\ndominant balance analysis presented in Fig. 1, one may note that viscous damping affects both\nthe surface tension and transitional regime. However, its effect diminishes as the bubble radius\nincreases, not influencing the inertial regime. Thus, the analysis underscores that a reduced analytic\nsolution used for the undamped scenario cannot effectively describe the transitional regime. Such a\nsolution cannot be extracted since the viscous term is significant, resulting in unresolvable reduced\nIVP. However, as for undamped growth, the surface tension regime and delay can still be identified\nusing the velocity threshold, providing valuable insights into the near-equilibrium growth dynamics\ninfluenced by viscosity.\nFig. 5 presents the surface-tension-induced delay as a function of the nucleation Ohnesorge\nnumber𝑂ℎ𝑐. The delay values were obtained for an initial radius perturbation of 𝜀=0.01 while\nutilizing the previously defined velocity threshold. We evaluate here the sensitivity of the delay\n10FIG. 5. Surface-tension-induced delay as a function of the nucleation Ohnesorge number 𝑂ℎ 𝑐, given\ninitial radius perturbation of 𝜀=0.01. Comparison between results obtained using the numeric solution,\nrepresented by black crosses, and results obtained using different orders of the approximated perturbation\nsolution, represented by solid lines.\nto the chosen model order; low nucleation Ohnesorge values exhibit greater sensitivity to the used\napproximation order, whereas the results converge for 𝑂ℎ𝑐>1. A distinct transition of the initial\ndelay behavior as a function of nucleation Ohnesorge number is illuminated by Fig. 5. For small\nvalues of nucleation Ohnesorge, the surface tension delay remains unaffected by changes in its\nvalue. As the nucleation Ohnesorge tends to 𝑂ℎ𝑐=0.1, the delay begins to increase, marking a\nshift to a linear relation between the two for 𝑂ℎ𝑐 > 1. This transition indicates a change in the\ndominant delaying mechanism: for 𝑂ℎ𝑐<0.1, the initial delay is primarily governed by surface\ntension, rendering it independent of the nucleation Ohnesorge. Contrarily, for 𝑂ℎ𝑐>1, viscous\ndamping becomes the dominant factor setting the initial delay, manifested by the obtained linear\ncorrelation between 𝑂ℎ𝑐and𝜏𝑆𝑇.\nThe surface-tension-induced delay is set by both the nucleation Ohnesorge number and the\ninitial perturbation from the stable radius, with a shorter delay expected for larger deviations from\nequilibrium. However, the relation between the two is not trivial, as illustrated by Fig. 6, presenting\nthe initial perturbation 𝜀for various values of nucleation Ohnesorge number 𝑂ℎ𝑐. While the\n11FIG. 6. Surface-tension-induced delay as a function of the initial radius perturbation 𝜀for various values\nof nucleation Ohnesorge number 𝑂ℎ 𝑐. Solid lines demarcated the delay as extracted using the third-order\nperturbation solution, whereas dashed lines correspond to the prediction of the correlation Eq. (15).\ndelay indicated a relatively modest sensitivity to the solution order in Fig. 5, we opted to use the\nthird-order solution for enhanced precision. In the undamped case, we successfully derived an\nanalytical correlation for the delay, Eq. (10). However, in the damped scenario, obtaining a direct\nclosed-form expression for 𝜏𝑆𝑇(𝜀,𝑂ℎ𝑐)proved unattainable. Nevertheless, we opted to extend the\npreliminary correlation for surface tension delay in undamped growth using data extracted from\nthe perturbation solution, yielding the following relation:\n𝜏𝑆𝑇≈\u0010\n1+3.4𝑂ℎ𝑐𝜀0.01\u0011√︂\n2\n3sinh−1©\n«√︃\n8\n3+𝑂ℎ2𝑐\n20𝜀ª®®\n¬. (15)\nThis correlation is depicted in Fig. 6, providing a comparative analysis with predictions from the\nperturbation solution.\nFig. 6 reveals that delay values for low Onesorge values ( 𝑂ℎ𝑐=0–10−1) converge as expected,\nindicating that the delay is primarily controlled by surface-tension forces dictated by the initial\nperturbation. In this range, the delay exhibits higher relative dependence on 𝜀– an order of\nmagnitude change. The transition in dominant delaying mechanisms around 𝑂ℎ𝑐=10−1–100also\nalters the sensitivity to the initial perturbation, with a small relative change at higher Ohnesorge\nvalues (𝑂ℎ𝑐=101–102). Yet significant absolute changes in the delay are still expected at these\n12values, highlighting that perturbation magnitude could not be overlooked when viscosity is the\ndominant delaying mechanism. The extended correlation, Eq. (15), proves accurate for high\nOhnesorge values but overestimates the delay, especially for small initial perturbations. Since the\ndefinition of the delay is not deterministic and aims to evaluate only the phenomenon’s order of\nmagnitude, the correlation could still provide a prompt estimation of the delay. Thus, we believe\nFig. 6 and Eq. (15) may serve as applicative tools for the preliminary evaluation of the near-\nequilibrium growth, encapsulating the essence of the initial delay. Using two nondimensional, one\nmay readily estimate the delay between nucleation and violent, rapid growth onset. One, 𝑂ℎ𝑐, is\nderived directly from vapor-liquid properties, and the other, 𝜀, is evaluated through thermodynamic\nstatistical models, correlating with the physical mechanisms triggering nucleation. Comparing the\nestimated delay with the entire phase-change characteristic time scale will provide insights into the\nsignificance of this delay, assisting one in analyzing the process accurately.\nV. CONCLUDING REMARKS\nIn this study, we derived analytical solutions to explore the near-equilibrium growth of vapor\nbubbles. Regular perturbation methods were used to isolate the physical mechanisms controlling\nthe surface-tension-dominated regime, offering a closed mathematical formulation for the initial\ndelay of the bubble growth for the first time. Using the obtained solution, we studied two gener-\nalized growth scenarios discerned by the ratio between surface tension and viscous forces. The\nanalytical solutions were validated with direct numeric solutions of the Rayleigh-Plesset equation,\nsuccessfully capturing the initial growth delay.\nOur analysis predicts the initial delay period as a function of two nondimensional parameters: the\nnormalized perturbation from criticality and the nucleation Ohnesorge number, encapsulating the\ninterplay between viscous damping and surface tension forces near criticality. For low nucleation\nOhnesorge numbers, 𝑂ℎ𝑐<1, the delay is governed solely by surface tension, while higher values,\n𝑂ℎ𝑐>1, lead to a linear correlation with viscous damping. The derived analytical solutions,\nparticularly the correlation for surface-tension-induced delay, may be integrated into existing\nmodels for vapor bubble growth. This incorporation could refine the accuracy of such models,\nespecially during the early stages of bubble evolution. The obtained results highlight the necessity\nof further experimental efforts to study post-nucleation vapor bubble dynamics targeted at capturing\nthe predicted growth delay. Finally, this approach may be extended to include bubble curvature-\n13dependent and dynamic surface tension models. Such an extension broadens the applicability and\nenhances model predictions, especially applicable to bubbles nucleating within highly superheated\nliquids; an ongoing study is dedicated to unraveling the influence of this phenomenon on near-\nequilibrium dynamics.\nACKNOWLEDGMENTS\nThis research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1762/20).\nEran Sher acknowledges the financial support of Minerva Research Center (Max Plank Society\nContract No. AZ5746940764).\n[1] A. M. Smith, Negative pressure generated by octopus suckers: A study of the tensile strength of water\nin nature, Journal of Experimental Biology 157, 257 (1991).\n[2] J. E. Gardner, Surface tension and bubble nucleation in phonolite magmas, Geochimica et Cosmochim-\nica Acta 76, 93 (2012).\n[3] O. Vincent, P. Marmottant, P. A. Quinto-Su, and C.-D. Ohl, Birth and growth of cavitation bubbles\nwithin water under tension confined in a simple synthetic tree, Physical Review Letters 108, 184502\n(2012).\n[4] R. E. Apfel, The superheated drop detector, Nuclear Instruments and Methods 162, 603 (1979).\n[5] R. R. Allen, J. D. Meyer, and W. R. Knight, Thermodynamics and hydrodynamics of thermal ink jets.,\nHewlett-Packard Journal 36, 21 (1985).\n[6] A. D. Fulvio, C. Domingo, M. D. S. Pedro, E. D’ Agostino, M. Caresana, L. Tana, and F. D’Errico, Su-\nperheated emulsions and track etch detectors for photoneutron measurements, Radiation Measurements\n57, 19 (2013).\n[7] O. Avni, T. Bar-Kohany, and E. Sher, Flash boiling atomization triggered and driven by intensive\nradiation, Thermal Science and Engineering Progress 32, 101334 (2022).\n[8] T. Bar-Kohany, D. V. Antonov, P. A. Strizhak, and S. S. Sazhin, Nucleation and bubble growth during\npuffing and micro-explosions in composite droplets, Fuel 340, 10.1016/j.fuel.2022.126991 (2023).\n[9] A. D. Maxwell, T.-Y. Wang, C. A. Cain, J. B. Fowlkes, O. A. Sapozhnikov, M. R. Bailey, and Z. Xu,\nCavitation clouds created by shock scattering from bubbles during histotripsy, The Journal of the\n14Acoustical Society of America 130, 1888 (2011).\n[10] K. B. Bader, E. Vlaisavljevich, and A. D. Maxwell, For whom the bubble grows: Physical principles\nof bubble nucleation and dynamics in histotripsy ultrasound therapy., Ultrasound in medicine biology\n45, 1056 (2019).\n[11] L. Mancia, M. Rodriguez, J. Sukovich, Z. Xu, and E. Johnsen, Single–bubble dynamics in histotripsy\nand high–amplitude ultrasound: Modeling and validation, Physics in Medicine and Biology 65, 225014\n(2020).\n[12] M. S. Plesset and A. Prosperetti, Bubble dynamics and cavitation, Annual Review of Fluid Mechanics\n9, 145 (1977).\n[13] A. Prosperetti, Vapor bubbles, Annual Review of Fluid Mechanics 49, 221 (2017).\n[14] L. Rayleigh, Viii. on the pressure developed in a liquid during the collapse of a spherical cavity, The\nLondon, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 34, 94 (1917).\n[15] M. S. Plesset and S. A. Zwick, The growth of vapor bubbles in superheated liquids, Journal of Applied\nPhysics 25, 493 (1954).\n[16] B. Mikic, W. Rohsenow, and P. Griffith, On bubble growth rates, International Journal of Heat and\nMass Transfer 13, 657 (1970).\n[17] S. J. Board and R. B. Duffey, Spherical vapour bubble growth in superheated liquids, Chemical\nEngineering Science 26, 263 (1971).\n[18] T. G. Theofanous and P. D. Patel, Universal relations for bubble growth, International Journal of Heat\nand Mass Transfer 19, 425 (1976).\n[19] A. Prosperetti and M. S. Plesset, Vapour-bubble growth in a superheated liquid, Journal of Fluid\nMechanics 85, 349 (1978).\n[20] H. S. Lee and H. Merte, Spherical vapor bubble growth in uniformly superheated liquids, Int. J. Heat\nMass Transfer 39, 24272447 (1996).\n[21] Y. Hao and A. Prosperetti, The dynamics of vapor bubbles in acoustic pressure fields, Physics of Fluids\n11, 2008 (1999).\n[22] A. Robinson and R. Judd, The dynamics of spherical bubble growth, International Journal of Heat and\nMass Transfer 47, 5101 (2004).\n[23] P. Sullivan, D. Dockar, M. K. Borg, R. Enright, and R. Pillai, Inertio-thermal vapour bubble growth,\nJournal of Fluid Mechanics 948, A55 (2022).\n[24] R. Ang ´elil, J. Diemand, K. K. Tanaka, and H. Tanaka, Bubble evolution and properties in homogeneous\n15nucleation simulations, Physical Review E 90, 063301 (2014).\n[25] O. Avni, Y. Dagan, T. Bar-Kohany, and E. Sher, Bubble dynamics under negative pressures: A missing\nlink?, Thermal Science and Engineering Progress 46, 102162 (2023).\n16"    },    {        "title": "2402.04712v1.A_note_on_Weyl_gauge_symmetry_in_gravity.pdf",        "content": "arXiv:2402.04712v1  [hep-th]  7 Feb 2024A note on Weyl gauge symmetry in gravity\nN. Mohammedi∗\nInstitut Denis Poisson (CNRS - UMR 7013),\nUniversit´ e de Tours,\nFacult´ e des Sciences et Techniques,\nParc de Grandmont, F-37200 Tours, France.\nAbstract\nA scale invariant theory of gravity, containing at most two d erivatives, requires, in\naddition to the Riemannian metric, a scalar field and (or) a ga uge field. The gauge\nfield is usualy used to construct the affine connection of Weyl g eometry. In this note,\nwe incorporate both the gauge field and the scalar field to buil d a generalised scale\ninvariant Weyl affine connection. The Ricci tensor and the Ric ci scalar made out of\nthis generalised Weyl affine connection contain, naturally, kinetic terms for the scalar\nfield and the gauge field. This provides a geometric interpret ation for these terms.\nIt is also shown that scale invariance in the presence of a cos mological constant and\nmass terms is not completely lost. It becomes a duality trans formation relating various\nfields.\n∗e-mail: noureddine.mohammedi@univ-tours.fr\n11 Introduction\nIt is plausible to think that at very high energies, when gravity was st rong, there were only\nmassless fields in the Standard Model of particle physics. Hence the Standard Model coupled\ntogravitymighthave enjoyed ascale symmetry. This symmetry wou ldthenbeofimportance\nto the physics of the early Universe and its effects could be seen in co smological observations.\nThis explains also the renewed research activity in this direction. A his tory and a modern\nreview of scale invariant theories of gravity could be found in [1] (se e also [2], [3], [4], [5]).\nSome issues regarding the use of scale invariant gravity in cosmology , inflation and other\nastrophysical observations are treated in [6–32]. Various studies concerning the coupling of\nthe Standard Model of particle physics to gravity in a scale invariant manner are accounted\nfor in [33–45]. Classical solutions in the context of scale invariant gra vity could be found\nin [46], [47], [48], [43].\nAs it is well-known, in order to render Einstein theory of gravity invar iant under the\nrescaling of the metric tensor\ngµν−→eσ(x)gµν (1.1)\none needs the introduction of a scalar fiels ϕhaving the transformation\nϕ−→ϕ+1\nqσ(x). (1.2)\nThe scale invariant theory of gravity is then described by the action1\nS1=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−R−/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ/parenrightbigg\n+κe−qϕ/bracerightBigg\n.(1.3)\nThe last term is independently scale invariant and ξ,κandqare constants.\nThe action (1.3) can be cast in the form\nS=ξ/integraldisplay\nd4x√\n−G/braceleftBigg\n−R(G)+κ/bracerightBigg\n, (1.4)\nwhere the scale invariant metric Gµνis defined as\nGµν=e−qϕgµν (1.5)\nandR(G) is the Ricci scalar constructed out of this metric. Hence, the field ϕis simply the\nconformal factor of the metric Gµν.\nIn the action (1.3), the propagating scalar field is in fact\nχ= exp/parenleftBig\n−q\n2ϕ/parenrightBig\n. (1.6)\n1Our conventions are such the Riemann tensor is given by Rµ\nνρσ=∂ρΓµ\nνσ+Γµ\nραΓα\nνσ−(ρ↔σ), the Ricci\ntensor is Rµν=Rα\nµανand the Ricci scalar is R=gµνRµν. The signature of the metric gµνis mostly minus\n(+,−,−,−). The covariant derivative is denoted ∇µsuch that ∇µVν=∂µVν+Γν\nµσVσ, for a four-vector\nVν, and∇2=∇µ∇µ.\n2For convenience, we still work with the field ϕthrough out this note. The kinetic term (after\nintegration by parts) for the scalar field χis\n−6ξ√−g∂µχ∂µχ . (1.7)\nAsξis a positive constant, this kinetic terms is negative and leads to the in terpretation that\nthe scalar field χis a ghost.\nThere is in fact another manner to obtain a scale invariant theory of gravity. It consists\nin using a gauge field Cµhaving the transformation [1]\nCµ−→Cµ+1\nq′∂µσ(x). (1.8)\nHereq′is a constant. The scale invariant action in this case is given by\nS2=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−R−/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n+κe−qϕ/bracerightBigg\n.(1.9)\nNotice that one still needs the scalar field ϕ. It has the scale transformation (1.2).\nThe two action S1andS2are in fact related. The gauge field Cµin (1.9) is a non-\npropagating field whose equation of motion is\nCµ=q\nq′∂µϕ . (1.10)\nInjecting this in (1.9), we recover the action S1in (1.3).\nThe two scale invariant theories described by the actions S1andS2have two drawbacks.\nThe first is that thepropagating field χinS1isa ghost field having a negative kinetic term as\nshown in (1.7). The second is that the gauge field CµinS2does not propagate. Fortunately,\nthe remedy to this two problems is known in the literature. It consist s in considering the\nscale invariant actions [34]\nS(a)\nscale=Sa+σa/integraldisplay\nd4x√−ge−qϕDµϕDµϕ\n−1\n4/integraldisplay\nd4x√−gCµνCµν, (1.11)\nwhere\nCµν=∂µCν−∂νCµ (1.12)\nis the fields strenght corresponding to the gauge field Cµand\nDµϕ=∂µϕ−q′\nqCµ. (1.13)\nis the scale invariant covariant derivative.\n3In the actions (1.11) the index atakes the values 1 or 2. The two constants σ1and\nσ2are adjusted such that the kinetic terrm for the scalar field χis of the canonical form\n+1\n2√−g∂µχ∂µχ. The desired values for σ1andσ2are\nσ1=q2/parenleftbigg1\n8+3\n2ξ/parenrightbigg\n, σ 2=1\n8q2. (1.14)\nWith these two values, we have\nS(1)\nscale=S(2)\nscale (1.15)\nup to total derivative terms.\nAs explained in the next section, the two actions S1andS2have a natural interpretation\nin terms of Weyl geometry. However, the scale invariant actions in ( 1.11) lack a geometrical\ninterpretation. It is the aim of this note to provide such an interpre tation.\nThe paper is organised as follows: In the next section we quickly revie w Weyl geometry\nin connection with the two actions S1andS2. It is then followed by a subsection in which we\nrecall that the gauge kinetic term stems also from Weyl geometry. In section 3, we present\na method for scale invariance of gravity relying simultaneously on the gauge field Cµand\nthe scalar field ϕ. This has a natural interpretation in the context of a generalised W eyl\ngeometry as explained in section 4. This constitutes the main result o f this note.\nSections 5 and 6 are dedicated to the scale invariant Standard Mode l as an application of\ntheformalism. Insection7, weobtainacuriousresultintheformofa dualitytransformations\nresembling scale transformations. Finally, our conclusions and outlo ok are summarised in\nsection 8.\n2 The usual Weyl geometry\nThe actions S1andS2of the previous section can be given a geometrical interpretation. The\nChristoffel symbols\nΓα\nµν=1\n2gατ(∂µgτν+∂νgτµ−∂τgµν) (2.1)\ntransform under the scale symmetry gµν−→eσgµνas\nΓα\nµν−→Γα\nµν+1\n2/parenleftbig\nδα\nµ∂νσ+δα\nν∂µσ−gµνgατ∂τσ/parenrightbig\n. (2.2)\nOne can then, with the help of the gauge field Cµ, build the scale invariant symbols /tildewideΓα\nµνas\n/tildewideΓα\nµν= Γα\nµν−q′\n2/parenleftbig\nδα\nµCν+δα\nνCµ−gµνgασCσ/parenrightbig\n, (2.3)\nAs a consequence, the scale invariant Riemann tensor /tildewideRµ\nνρσis defined as\n/tildewideRµ\nνρσ=∂ρ/tildewideΓµ\nνσ−∂σ/tildewideΓµ\nνρ+/tildewideΓµ\nρα/tildewideΓα\nνσ−/tildewideΓµ\nσα/tildewideΓα\nνρ. (2.4)\nFinally, the scale invariant Ricci tensor /tildewideRµνand the Ricci scalar /tildewideRare\n/tildewideRµν=/tildewideRα\nµαν,/tildewideR=gµν/tildewideRµν. (2.5)\n4The expression of the Ricci scalar /tildewideRis given by\n/tildewideR=R+/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n. (2.6)\nUnder the local transformation gµν−→eσgµν, the Ricci scalar /tildewideRscales as\n/tildewideR−→e−σ/tildewideR . (2.7)\nThis suggests that the scale invariant theory should be given by the action\nSWeyl=/integraldisplay\nd4x√−g/braceleftBigg\n−ξ\n4κ/tildewideR2/bracerightBigg\n. (2.8)\nThis last action can be cast in the form\nSWeyl=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−/tildewideR+κe−qϕ/bracerightBigg\n. (2.9)\nThe fieldϕenters as a non-propagating auxiliary field whose equations of motio n is\ne−qϕ=/tildewideR\n2κ. (2.10)\nThe action SWeylin (2.8) is obtained when using this equation of motion.\nUsing the explicit expression of /tildewideRas written (2.6), the action (2.9) is precisely S2of the\nprevious section. The elimination of the gauge field Cµthrough its equation of motion gives\nthen the action S1.\n2.1 The gauge field kinetic term\nThe scale invariant Ricci tensor /tildewideRµνas defined in (2.5) is not symmetric in its indices. Its\nanti-symmetric part is given by\n/tildewideR[µν]=1\n2/parenleftBig\n/tildewideRµν−/tildewideRνµ/parenrightBig\n=−q′Cµν, C µν=∂µCν−∂νCµ. (2.11)\nTherefore, one could add to (2.8) the scale invariant action\nSadd=1\nq′2/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4/tildewideR[µν]/tildewideR[µν]/bracerightBigg\n=/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4CµνCµν/bracerightBigg\n. (2.12)\nThis known observation assigns a geometric origin to the kinetic term of the gauge field Cµ.\n53 An alternative way\nIt is now clear that scale symmetry of gravity requires a scalar field ϕand a gauge field Cµ.\nUnder the scale transformations\ngµν−→eσ(x)gµν,\nϕ−→ϕ+1\nqσ ,\nCµ−→Cµ+1\nq′∂µσ (3.1)\none obtains the changes\nR−→e−σ(x)(R−∆R),\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ−→e−σ(x)/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ+∆R/parenrightbigg\n,\n3q′∇µCµ−3\n2q′2CµCµ−→e−σ(x)/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ+∆R/parenrightbigg\n,(3.2)\nwhere\n∆R= 3∇2σ+3\n2∂µσ∂µσ . (3.3)\nNotice that the cancelation of the variation ∆ Rcoming from the Ricci scalar Rcan be\nachieved by including either the scalar field ϕor the gauge field Cµor both. It is this last\noption that we adopt.\nThe scale invariant gravitational theory we propose is given by the a ction\nSε=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−R−ε/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ/parenrightbigg\n−(1−ε)/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n+κe−qϕ/bracerightBigg\n. (3.4)\nHereεis a constant. In particular, the two actions S1andS2are obtained, respectively, for\nthe special values ε= 1 andε= 0.\nThis last action can be written as\nSε=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−R−/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n+κe−qϕ/bracerightBigg\n−3\n2q2εξ/integraldisplay\nd4x√−ge−qϕDµϕDµϕ . (3.5)\nThe total derivative term we discarded is −3ξεq/integraltext\nd4x√−g∇µ(e−qϕDµϕ) and the covariant\nderivative Dµϕis given in (1.13). Furthermore, we see that the action (3.5) is\nSε=S2−3\n2q2εξ/integraldisplay\nd4x√−ge−qϕDµϕDµϕ . (3.6)\n6This is the sum of two scale invariant parts.\nIn a similar way, one can show that the action (3.4) can also be written as\nSε=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−R−/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ/parenrightbigg\n+3\n2q2(1−ε)ξ/integraldisplay\nd4x√−ge−qϕDµϕDµϕ . (3.7)\nThis up to the total derivative term 3 q(1−ε)ξ/integraltext\nd4x√−g∇µ(e−qϕDµϕ). Hence, we have\nalso\nSε=S1+3\n2q2(1−ε)ξ/integraldisplay\nd4x√−ge−qϕDµϕDµϕ . (3.8)\nTherefore, the action proposed in (3.4) is the parent theory for S1andS2. These are now\nequivalent up the a scale invariant term proportional to/integraltext\nd4x√−ge−qϕDµϕDµϕ.\nWe notice that a positive canonical kinetic term\n+1\n2√−g∂µχ∂µχ , (3.9)\nwhere the scalar field χis as defined in (1.6), is obtained for\nε=−1\n12ξ. (3.10)\n4 A general Weyl geometry\nThe variation, due to the rescaling of the metric, of the Christoffel symbols as given in (2.2)\ncan be cancelled by taking the scale invariant symbols /tildewideΓα\nµνto be given by\n/tildewideΓα\nµν= Γα\nµν−q\n2ω/parenleftbig\nδα\nµ∂νϕ+δα\nν∂µϕ−gµνgασ∂σϕ/parenrightbig\n−q′\n2(1−ω)/parenleftbig\nδα\nµCν+δα\nνCµ−gµνgασCσ/parenrightbig\n, (4.1)\nwhereωis a constant. This can be also written as\n/tildewideΓα\nµν= Γα\nµν−q′\n2/parenleftbig\nδα\nµCν+δα\nνCµ−gµνgασCσ/parenrightbig\n−q\n2ω/parenleftbig\nδα\nµDνϕ+δα\nνDµϕ−gµνgασDσϕ/parenrightbig\n. (4.2)\nThe scale invariant Riemman tensor /tildewideRµ\nνρσis still as defined in (2.4) and the scale invariant\nRicci tensor /tildewideRµνand the Ricci scalar /tildewideRare as given in (2.5).\nAn explicit calculation gives\n/tildewideR=R+3qω∇µDµϕ+3q′∇µCµ\n−3\n2/parenleftBig\nq2ω2DµϕDµϕ+2ωqq′DµϕCµ+q′2CµCµ/parenrightBig\n. (4.3)\n7This allows us to write the two useful relations\ne−qϕ/tildewideR=e−qϕ/bracketleftbigg\nR+3q′∇µCµ−3\n2q′2CµCµ+3\n2q2ω(2−ω)DµϕDµϕ/bracketrightbigg\n+ 3qω∇µ/parenleftbig\ne−qϕDµϕ/parenrightbig\n(4.4)\nand\ne−qϕ/tildewideR=e−qϕ/bracketleftbigg\nR+3q∇2ϕ−3\n2q2∂µϕ∂µϕ−3\n2q2(1−ω)2DµϕDµϕ/bracketrightbigg\n−3q(1−ω)∇µ/parenleftbig\ne−qϕDµϕ/parenrightbig\n. (4.5)\nBy making the identification\nε=ω(2−ω) (4.6)\nand using the two relations (4.4) and (4.5), the action Sεin (3.5) or in (3.7) can be both\nwritten as\nSε=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−/tildewideR+κe−qϕ/bracerightBigg\n. (4.7)\nThis is up to total derivative terms. In this action, the term e−qϕ/tildewideRis as given in (4.4) or\n(4.5).\nIn the action (4.7), the scalar field ϕis no longer a Lagrange multiplier as /tildewideRcontains\nderivatives of this field. The kinetic term +1\n2√−g∂µχ∂µχ, for the scalar field χof (1.6),\nstems from the Ricci scalar /tildewideRwhenε=−1\n12ξ.\nThe anti-symmetric part of the scale invariant Ricci tensor /tildewideRµνas defined in (2.5), but\nwith the scale invariant symbols /tildewideΓα\nµνnow given in (4.2), is found to be given as\n/tildewideR[µν]=1\n2/parenleftBig\n/tildewideRµν−/tildewideRνµ/parenrightBig\n=−q′(1−ω)Cµν. (4.8)\nWe could supplement the gauge field Cµwith a kinetic term by including the additional\naction\nSadd=1\nq′2(1−ω)2/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4/tildewideR[µν]/tildewideR[µν]/bracerightBigg\n=/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4CµνCµν/bracerightBigg\n. (4.9)\nWe are of course assuming that ω/negationslash= 1 (or, according to (4.6), ε/negationslash= 1). Ifω= 1, then the\ngauge field Cµdoes not take part in the construction of /tildewideΓα\nµνas can be seen from (4.1).\nInsummary, ascaleinvariant theoryofgravitycouldbedescribed b ythegeometricaction\nSscale=Sε+Sadd\n=ξ/integraldisplay\nd4x√−ge−qϕ/braceleftBigg\n−/tildewideR+κe−qϕ/bracerightBigg\n+1\nq′2(1−ω)2/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4/tildewideR[µν]/tildewideR[µν]/bracerightBigg\n. (4.10)\n8The expressions of /tildewideRand/tildewideR[µν]are given, respectively, in (4.3) and (4.8). Notice that\n(1−ω)2= (1−ε), as can be seen from (4.6), and for a positive kinetic term for the s calar\nfieldχ, we takeε=−1\n12ξ.\nThe action (4.10), when unpacked, is also given in ref. [34] where diffe rent terms were\nintroduced by ’hand’. Here we provide a geometric origin to the scale in variant theory.\n5 Application: scale invariance of the Standard Model\ncoupled to gravity\nIn this section we supplement the Standard Model with the scalar fie ldϕand the gauge field\nCµand couple it to gravity in a scale invariant manner. The scale transfo rmations of the\nfields of the Standard Model are\nH−→e−1\n2σ(x)H ,\nψ(a)−→e−3\n4σ(x)ψ(a),\nA(i)\nµ−→A(i)\nµ, (5.1)\nwhereH,ψ(a)andA(i)\nµrepresent, respectively, the Higgs doublet, any fermion field and a ny\ngauge field in the Standard Model.\nThe scale invariant Standard Model coupled to gravity could be desc ribed by the action\nS=ξ/integraldisplay\nd4x√−g/parenleftbig\ne−qϕ+ζH†H/parenrightbig/braceleftBigg\n−R−ε/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ/parenrightbigg\n−(1−ε)/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n+κe−qϕ/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4CµνCµν−γ\n2CµνBµν/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n(DµH)†(DµH)−λ/parenleftbig\nH†H/parenrightbig2+.../bracerightBigg\n. (5.2)\nWe have included a mixing term, with a parameter γ, between the field strength Cµνand the\nU(1) field strength Bµν=∂µBν−∂νBµof the Standard Model. The last contribution in this\naction is the Standard Model action with the Higgs mass set to zero2and the dots stand for\nthe rest of the Standard Model Lagrangian (written in a covariant form). The non-minimal\ncoupling of the Higgs field to the scalar curvature, with parameter ζ, is permitted by the\nscale symmetry. Of course, we take ε=−1\n12ξ, as in (3.10), for a positive kinetic term for the\nscalar field χin (1.6).\n2In the absence of gravity and with a mass term for the Higgs field, th is theory represents a St¨ uckelberg\nextension of the Standard Model (see, for instance, [49,50]).\n9The gauge covariant derivative acting on the complex doublet His now given by3\nDµH=/bracketleftbigg\n∂µ−i\n2g′Bµ−igWµ+1\n2q′Cµ/bracketrightbigg\nH . (5.3)\nTheSU(2) and the U(1) gauge couplings are, respectively, gandg′.\nSince, the gauge fields do not scale, all the gauge kinetic terms in the Standard Model\nLagrangian are scale invariant. The kinetic terms corresponding to the fermions as well as\nthe Yukawa interaction terms are also scale invariant (see appendix A). We assume that the\nfermions are not charged with respect to the additional gauge field Cµ.\n6 A gauge choice : The Higgs non-minimaly coupled\nto gravity\nThe extra scalar field transforms as ϕ−→ϕ+1\nqσ(x). This calls for the obvious gauge fixing\nϕ= 0. (6.1)\nIn this gauge, the action (5.2) becomes\nS=/integraldisplay\nd4x√−g/braceleftBigg\n−ξ/parenleftbig\n1+ζH†H/parenrightbig\nR+ξκ\n−ξζ(1−ε)H†H/bracketleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/bracketrightbigg/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4CµνCµν−γ\n2CµνBµν+1\n2m2CµCµ/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n(DµH)†(DµH)−M2H†H−λ/parenleftbig\nH†H/parenrightbig2+.../bracerightBigg\n.(6.2)\nWe have discarded the total derivative -3 ξq′(1−ε)/integraltext\nd4x√−g∇µCµand made the identifi-\ncation\nM2=−ζξκ\nm2= 3ξq′2(1−ε). (6.3)\nWe notice that the Higgs field Hacquires a mass upon the breaking of the scale symmetry.\nSimilarly, the gauge field Cµbecomes massive. The two constants ξandζshould satisfy the\nrelation\nξ/parenleftbig\n1+ζv2/parenrightbig\n=M2\nP\n2, (6.4)\n3We could have taken DµH=/bracketleftbig\n∂µ−i\n2g′Bµ−igWµ+τ\n2q′Cµ+1\n2(1−τ)q∂µϕ/bracketrightbig\nH. However, if the real\nparameter τis different from 1 then the theory is not renormalisable in the absenc e of gravity.\n10whereMPisthereducedPlanckmassand v2=−M2\n2λistheHiggsexpectationvalue(assuming\nthatM2is negative). This relation means that the Planck mass is generated b y the Higgs\nexpectation value.\nThe action (6.2) when the gauge field Cµis absent and4\nξκ+λ/parenleftbiggM2\n2λ/parenrightbigg2\n= 0 (6.5)\nis exactly that studied in the context of inflation based on a non-minim ally coupled Higgs\nfield to gravity [51–53].\nIndeed, the coupling between the Higgs field and the Ricci scalar Rcan be absorbed by\nexpressing the action (6.2) in the Einstein frame. This is achieved by r escaling the metric as\ngµν−→e−αρgµν, (6.6)\nwhereαis a constant (to be fixed later) and the scalar field ρ(x) is such that\ne−αρ/parenleftbig\n1+ζH†H/parenrightbig\n=M2\nP\n2ξ. (6.7)\nThe fieldρ(x) will be identified with the inflaton field [54].\nIn the unitary gauge in which the Higgs doublet His given by\nH=/parenleftbigg\n0\nh(x)/parenrightbigg\n(6.8)\nthe action (6.2), in the Einstein frame, takes the form\nS=/integraldisplay\nd4x√−g/braceleftBigg\n−M2\nP\n2R+3\n4α2M2\nP/bracketleftBigg\n1+1\n61\nζξ1/parenleftBig\n1−2ξ\nM2\nPe−αρ/parenrightBig/bracketrightBigg\ngµν∂µρ∂νρ\n−V(ρ)+.../bracerightBigg\n. (6.9)\nWe have maintained only the terms relevant to inflation and the poten tialV(ρ) is given by\nV(ρ) =e−2αρ/braceleftBigg\nλ\nζ2/parenleftbiggM2\nP\n2ξ/parenrightbigg2/bracketleftBigg\neαρ−1/bracketrightBigg2\n−/bracketleftBigg\nξκ+λ/parenleftbiggM2\n2λ/parenrightbigg2/bracketrightBigg/bracerightBigg\n. (6.10)\nThe relation in (6.4) has been used in obtaining this potential.\nWhen the condition (6.5) holds and\nα=/radicalbigg\n2\n31\nMP,2ξ\nM2\nPe−αρ≪1,1\n61\nζξ≪1 (6.11)\ninflationis driven by the scalar field ρ[54] withapproximate kinetic term1\n2gµν∂µρ∂νρ. Inthis\nlimit the potential is flat (slow-roll condition) and is dominated by a cos mological constant\nequal toλ\nζ2/parenleftBig\nM2\nP\n2ξ/parenrightBig2\n.\n4ThisrelationexcludesacosmologicalconstanttermandgivestheHig gsfieldthe potential λ/parenleftbig\nH†H−v2/parenrightbig2\nwithv2=−M2\n2λ.\n117 A duality transformation\nWe will consider the same action as in (5.2) supplemented with a cosmolo gical constant Λ\nand mass terms for the complex doublet Hand the extra field ϕ. More precisely, we take\nthe action\nS=ξ/integraldisplay\nd4x√−g/parenleftbig\ne−qϕ+ζH†H/parenrightbig/braceleftBigg\n−R−ε/parenleftbigg\n3q∇2ϕ−3\n2q2∂µϕ∂µϕ/parenrightbigg\n−(1−ε)/parenleftbigg\n3q′∇µCµ−3\n2q′2CµCµ/parenrightbigg\n+κe−qϕ/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4CµνCµν−γ\n2CµνBµν/bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n(DµH)†(DµH)−λ/parenleftbig\nH†H/parenrightbig2+.../bracerightBigg\n+/integraldisplay\nd4x√−g/braceleftBigg\n−Λ−M2H†H−1\n2m2\nχe−qϕ/bracerightBigg\n. (7.1)\nOf course, the additional last term in this theory breaks scale invar iance and we would like\nto see to what extent this scale symmetry is broken.\nIndeed, the last term transforms under the scale symmetry gµν−→eσgµν,H−→e−1\n2σH\nandϕ−→ϕ+1\nqσas\n√−g/braceleftBigg\n−Λ−M2H†H−1\n2m2\nχe−qϕ/bracerightBigg\n−→√−g/braceleftBigg\n−Λe2σ−M2H†Heσ−1\n2m2\nχe−qϕeσ/bracerightBigg\n.\n(7.2)\nLet us now demand that/braceleftBigg\n−Λe2σ−M2H†Heσ−1\n2m2\nχe−qϕeσ/bracerightBigg\n=/braceleftBigg\n−Λ−M2H†H−1\n2m2\nχe−qϕ/bracerightBigg\n.(7.3)\nThis last relation is satisfied if σis given by\neσ=−1−1\nΛ/parenleftbigg\nM2H†H+1\n2m2\nχe−qϕ/parenrightbigg\n. (7.4)\nTo summarise, the action (7.1) remains unchanged under the trans formations\ngµν−→eσgµν,\nϕ−→ϕ+1\nqσ ,\nCµ−→Cµ+1\nq′∂µσ ,\nH−→e−1\n2σH ,\nψ(a)−→e−3\n4σψ(a),\nA(i)\nµ−→A(i)\nµ. (7.5)\n12The scale factor σis a function of the two fields Handϕas given in (7.4). This is a duality\ntransformation since σis not an arbitrary function but depends on the fields Handϕof the\ntheory. As usual, ψ(a)andA(i)\nµare the not-shown fermions and gauge fields of the Standard\nModel Lagrangian. This duality transformation assumes that the c osmological constant Λ\nis different from zero and is valid in a domain where\n−1−1\nΛ/parenleftbigg\nM2H†H+1\n2m2\nχe−qϕ/parenrightbigg\n≥0. (7.6)\n8 Conclusions\nA scale invariant theory of gravity necessitates, beside the metric tensor field gµν, a scalar\nfieldϕand a gauge field Cµ. We have in this note assembled these three fields together and\nconstructed a scale invariant Christoffel symbols /tildewideΓα\nµνas given in (4.2). The scalar field ϕ\nis then a physical field having a positive kinetic term. As an application o f the formalism,\nwe have coupled the Standard Model to gravity in a scale invariant ma nner. The widely\ninvestigated subject of the non-minimally coupled Standard Model t o gravity is obtained as\na consequence of the breaking of this scale symmetry. However, t he theory contains an extra\ngauge field Cµ. It would be of great interest to explore the issues brought by the presence of\nthis gauge field in the context of inflation in the early Universe.\nThe analyses carried out in this article could be easily generalised to th e case ofNscalar\nfieldsϕiwithi= 1,...,N. The scale invariant Christoffel symbols /tildewideΓα\nµνare then written as\n/tildewideΓα\nµν= Γα\nµν−q′\n2/parenleftbig\nδα\nµCν+δα\nνCµ−gµνgασCσ/parenrightbig\n−vi\n2/parenleftbig\nδα\nµDνϕi+δα\nνDµϕi−gµνgασDσϕi/parenrightbig\n, (8.1)\nwhereDµϕi=∂µϕi−q′\nqiCµandviandqiare constants. Each scalar field transforms as\nϕi−→ϕi+1\nqiσ.\nThe Ricci scalar constructed out of the scale invariant Christoffel symbols/tildewideΓα\nµνin (8.1) is\nthen given by\n/tildewideR=R+3vi∇µDµϕi+3q′∇µCµ\n−3\n2/parenleftBig\nvivjDµϕiDµϕj+2viq′DµϕiCµ+q′2CµCµ/parenrightBig\n. (8.2)\nUnder the local transformation gµν−→eσgµν, the Ricci scalar /tildewideRscales as /tildewideR−→e−σ/tildewideR.\nSimilarly, the anti-symmetric part of the scale invariant Ricci tensor /tildewideRµνis\n/tildewideR[µν]=1\n2/parenleftBig\n/tildewideRµν−/tildewideRνµ/parenrightBig\n=−q′/parenleftbigg\n1−vi\nqiω/parenrightbigg\nCµν. (8.3)\nA sum over the repeated index iis understood. The scale invariant theory is described by\n13the action\nSscale=ξ/integraldisplay\nd4x√−ge−qiϕi\nN/braceleftBigg\n−/tildewideR+κe−qjϕj\nN/bracerightBigg\n+1\nq′2/parenleftBig\n1−vi\nqi/parenrightBig2/integraldisplay\nd4x√−g/braceleftBigg\n−1\n4/tildewideR[µν]/tildewideR[µν]/bracerightBigg\n. (8.4)\nThe expressions of /tildewideRand/tildewideR[µν]are as given in (8.2) and (8.3), respectively.\nFinally, we have shown in section 5, that there is a kind of a duality tran sformation in the\ntheory written in (7.1). This duality holds only in the presence of a cos mological constant.\nAn extension of the present study would be a further exploration o f the consequences of this\nduality transformation.\n14A Scale invariance of a fermionic kinetic term and a\nYukawa coupling\nThe kinetic part of a Lagrangian for a fermion field ψis given by\nLfermion=i√−g¯ψγaEµ\naDµψ , (A.1)\nwhere\nDµψ=/parenleftbigg\n∂µ+1\n2ωbcΣbc+.../parenrightbigg\nψ . (A.2)\nThe dots in Dµψstand for possible coupling to gauge fields. Here we use the notation\ngµν=ηabea\nµeb\nµ, ea\nµEµ\nb=δa\nb, ea\nµEν\na=δν\nµ. (A.3)\nThe vielbeins scale as ea\nµ−→eσ/2ea\nµand their inverses as Eµ\na−→e−σ/2Eµ\na. Latin indices\nare raised and lowered with the flat metric ηaband its inverse ηab. The spin connection is\ngiven by\nωa\nµb=−Eν\nb/parenleftbig\n∂µea\nν−Γα\nµνea\nα/parenrightbig\n. (A.4)\nWe ave also\nΣab=1\n4[γa, γb], γaγb+γbγa= 2ηab. (A.5)\nThe spin connection transforms under the scaling symmetry gµν−→eσgµνas\nωa\nµb−→ωa\nµb+1\n2/parenleftbig\nEν\nbea\nν−ηadEν\ndec\nµηcb/parenrightbig\n∂νσ . (A.6)\nThe covariant derivative then changes as\nDµψ−→e−3σ/4/bracketleftbigg\nDµψ−3\n4∂µσψ+1\n4/parenleftbig\nηbcEν\ncea\nν−ηadEν\ndeb\nµ/parenrightbig\n∂νσΣabψ/bracketrightbigg\n.(A.7)\nNext, the Lagrangian transforms as\nLfermion−→Lfermion+i√−g¯ψEµ\nc/bracketleftbigg\n−3\n4γc∂µσ+1\n4/parenleftbig\nηbcγdΣdb−ηacγdΣad/parenrightbig\n∂µσ/bracketrightbigg\nψ .(A.8)\nUsing the properties of the Dirac matrices as in (A.5) and recalling tha tγa=ηabγb, one\nfinds that\nηbcγdΣdb−ηacγdΣad= 3γc. (A.9)\nThis means that the terms involving ∂µσin (A.8) cancel and we have\nLfermion−→Lfermion. (A.10)\nFinally, a typical Yukawa interaction in the Standard Model is of the f orm\nLYukawa=√−g/parenleftbig¯lH†L+¯LHl/parenrightbig\n, (A.11)\nwhereLis a fermionic doublet and lis a fermionic singlet. It is clear that this interaction\nterm is invariant under gµν−→eσgµν,L−→e−3\n4σL,l−→e−3\n4σlandH−→e−1\n2σH.\n15References\n[1] E. Scholz, The Unexpected Resurgence of Weyl Geometry in late 20th-Cen tury Physics ,\nEinstein Stud. 14(2018) 261-360, arXiv:1703.03187 [math.HO].\n[2] N. Rosen, Weyl’s geometry and physics , Foundations of Physics 12, 213 (1982).\n[3] J. T. Wheeler, Weyl geometry , Gen. Rel. Grav. 50(2018) 80, arXiv:1801.03178 [gr-qc].\n[4] Alfredo Iorio, L. O’Raifeartaigh and I. Sachs, Weyl gauging and conformal invariance ,\nNucl. Phys. B 495(1997) 433-450, hep-th/9607110 [hep-th].\n[5] LochlainnO’RaifeartaighandNorbert Straumann, Gauge theory: Historical origins and\nsome modern developments , Rev. Mod. Phys. 72(2000) 1-23.\n[6] M. Israelit, A Weyl-Dirac cosmological model with DM and DE , Gen. Rel. Grav. 43\n(2011) 751-775, arXiv:1008.0767 [gr-qc].\n[7] D. M. Ghilencea, Gauging scale symmetry and inflation: Weyl versus Palatini g ravity,\nEur. Phys. J. C 81, 510 (2021), arXiv:2007.14733 [hep-th].\n[8] D. M. Ghilencea, Non-metric geometry as the origin of mass in gauge theories o f scale\ninvariance , Eur. Phys. J. C 83, 176 (2023), arXiv:2203.05381 [hep-th].\n[9] P.Burikham, T.Harko,K.Pimsamarn, andS.Shahidi, Dark matter as a Weyl geometric\neffect, Phy. Rev. D 107, 064008 (2023), arXiv:2302.08289 [gr-qc].\n[10] Marius A. Oancea1 and Tiberiu Harko, Weyl geometric effects on the propagation of\nlight in gravitational fields , (2023), arXiv:2305.01313 [gr-qc].\n[11] Maria Crˇ aciuna and Tiberiu Harkob, Testing Weyl geometric gravity with the SPARC\ngalactic rotation curves database , Phys. Dark Univ. 43(2024) 101423, arXiv:2311.16893\n[gr-qc].\n[12] P. D. Mannheim and J. G. O’Brien, Fitting galactic rotation curves with conformal\ngravity and a global quadratic potential ,Phys. Rev. D 85(2012)124020,arXiv:1011.3495\n[astro-ph.CO]\n[13] J. G. O’Brien and P. D. Mannheim, Fitting dwarf galaxy rotation curves with conformal\ngravity, Mon. Not. R. Astron. Soc. 421(2) (2012) 1273–1282, arXiv:1107.5229 [astro-\nph.CO].\n[14] Cemsinan Deliduman, Oguzhan Kasikci and Baris Yapiskan, Flat galactic rotation\ncurves from geometry in Weyl gravity , Astrophys. Space Sci. 365 (3) (2020) 51,\narXiv:1511.07731 [gr-qc].\n[15] M. Hobson and A. Lasenby, Conformally-rescaled Schwarzschild metrics do not predic t\nflat galaxy rotation curves , Eur. Phys. J. C 82(7) (2022) 585, arXiv:2206.08097 [gr-qc].\n16[16] Z. Haghani and T. Harko, Compact stellar structures in Weyl geometric gravity , Phys.\nRev.D 107(2023) 064068, arXiv:2303.10339 [gr-qc].\n[17] M. A. Oancea, T. Harko, Weyl geometric effects on the propagation of light in gravita -\ntional fields , (2023), arXiv:2305.01313 [gr-qc].\n[18] P.K.Aluri, P.JainandN.K.Singh, Dark Energy and Dark Matter in General Relativity\nwith local scale invariance , Mod. Phys. Lett. A 24(2009) 1583-1595, arXiv:0810.4421\n[hep-ph].\n[19] P. K. Aluri, P. Jain, S. Mitra, S. Panda and N. K. Singh, Constraints on the Cos-\nmological Constant due to Scale Invariance , Mod. Phys. Lett. A 25(2010) 1349-1364,\narXiv:0909.1070 [hep-ph].\n[20] D. M. Ghilencea and T. Harko, Cosmological evolution in Weyl conformal geometry ,\n(2021), arXiv:2110.07056 [gr-qc].\n[21] Y. Tang and Y. L. Wu, Weyl Symmetry Inspired Inflation and Dark Matter , Phys. Lett.\nB 803(2020) 135320, arXiv:1904.04493 [hep-ph].\n[22] R. Kallosh and A. Linde, Universality Class in Conformal Inflation , JCAP07(2013)\n002, arXiv:1306.5220 [hep-th].\n[23] R. Jackiw and S. Y. Pi, Fake Conformal Symmetry in Conformal Cosmological Models ,\nPhys. Rev. D 91(2015) 067501, arXiv:1407.8545 [gr-qc].\n[24] R. Jackiw and S. Y. Pi, New Setting for Spontaneous Gauge Symmetry Breaking? ,\nFundam. Theor. Phys. 183(2016) 159, arXiv:1511.00994 [hep-th].\n[25] A. Paliathanasis, G. Leon and J. D. Barrow, Inhomogeneous spacetimes in Weyl inte-\ngrable geometry with matter source ,” Eur. Phys. J. C 80(2020) 731, arXiv:2006.01793\n[gr-qc].\n[26] J. Miritzis, Acceleration in Weyl integrable spacetime , Int. J. Mod. Phys. D 22(2013)\n1350019, arXiv:1301.5696 [gr-qc].\n[27] E. Scholz, MOND-like acceleration in integrable Weyl geometric gravi ty, Found. Phys.\n46(2016) 176-208, arXiv:1412.0430 [gr-qc].\n[28] R.Aguila,J.E.MadrizAguilar,C.MorenoandM.Bellini, Presentaccelerated expansion\nof the universe from new Weyl-Integrable gravity approach , Eur. Phys. J. C 74(2014)\n3158, arXiv:1408.4839 [gr-qc].\n[29] P. G. Ferreira, C. T. Hill and G. G. Ross, Weyl Current, Scale-Invariant Inflation and\nPlanck Scale Generation , Phys. Rev. D 95(2017) 043507, arXiv:1610.09243 [hep-th].\n[30] P. G. Ferreira, C. T. Hill and G. G. Ross, Inertial Spontaneous Symmetry Breaking and\nQuantum Scale Invariance , Phys. Rev. D 98(2018) 116012, arXiv:1801.07676 [hep-th].\n17[31] P. G. Ferreira, C. T. Hill and G. G. Ross, No fifth force in a scale invariant universe ,\nPhys. Rev. D 95(2017) 064038, arXiv:1612.03157 [gr-qc].\n[32] P. G. Ferreira, C. T. Hill and G. G. Ross, Scale-Independent Inflation and Hierarchy\nGeneration , Phys. Lett. B 763(2016) 174-178, arXiv:1603.05983 [hep-th].\n[33] D. M. Ghilencea, Spontaneous breaking of Weyl quadratic gravity to Einstein action and\nHiggs potential , JHEP03(2019) 049, arXiv:1812.08613 [hep-th].\n[34] D. M. Ghilencea and H. M. Lee, Weyl gauge symmetry and its spontaneous breaking in\nthe standard model and inflation , Phys. Rev. D 99, 115007 (2019), arXiv:1809.09174\n[hep-th].\n[35] D. M. Ghilencea, Stueckelberg breaking of Weyl conformal geometry and appli cations to\ngravity, Phys. Rev. D 101, 045010 (2020), arXiv:1904.06596 [hep-th].\n[36] D. M. Ghilencea and C. T. Hill, Standard Model in conformal geometry: local vs gauged\nscale invariance , Annals Phys. 460(2024) 169562, arXiv:2303.02515 [hep-th].\n[37] I. Bars, P. J. Steinhardt and N. Turok, Cyclic cosmology, conformal symmetry and the\nmetastability of the Higgs , Phys. Let. B 726, 50-55 (2013), arXiv:1307.8106 [gr-qc].\n[38] D. M. Ghilencea, Standard model in Weyl conformal geometry , Eur. Phys. J. C 82\n(2022) 23, arXiv:2104.15118 [hep-ph].\n[39] M. de Cesare, J. W. Moffat and M. Sakellariadou, Local conformal symmetry in non-\nRiemannian geometry and the origin of physical scales ,” Eur. Phys. J. C 77(2017) 605,\narXiv:1612.08066 [hep-th].\n[40] H.NishinoandS.Rajpoot, Implication of Compensator Field and Local Scale Invarianc e\nin the Standard Model , Phys. Rev. D 79(2009) 125025, arXiv:0906.4778 [hep-th].\n[41] E. I. Guendelman, H. Nishino and S. Rajpoot, Local scale-invariance breaking in the\nstandard model by two-measure theory , Phys. Rev. D 98(2018) 055022.\n[42] H. C. Ohanian, Weyl gauge-vector and complex dilaton scalar for conformal symmetry\nand its breaking , Gen. Rel. Grav. 48(2016) 25, arXiv:1502.00020 [gr-qc].\n[43] I. Quiros, On the physical consequences of a Weyl invariant theory of gr avity, (2020),\narXiv:1401.2643 [gr-qc].\n[44] P. Jain, S. Mitra and N. K. Singh, Cosmological Implications of a Scale Invariant Stan-\ndard Model , JCAP03(2008) 011, arXiv:0801.2041 [astro-ph].\n[45] J. Garcia-Bellido, J. Rubio, M. Shaposhnikov and D. Zenhausern ,Higgs-Dilaton Cos-\nmology: From the Early to the Late Universe , Phys. Rev. D 84(2011) 123504,\narXiv:1107.2163 [hep-ph].\n18[46] J.-Z. Yang, S. Shahidi, and T. Harko, Black hole solutions in the quadratic Weyl con-\nformal geometric theory of gravity , Eur. Phys. J. C 82(2022) 1171, arXiv:2212.05542\n[gr-qc]\n[47] P. D. Mannheim and D. Kazanas, Exact Vacuum Solution to Conformal Weyl Gravity\nand Galactic Rotation Curves , Astrophys. J. 342(1989) 635-638.\n[48] M. Hohmann, C. Pfeifer, M. Raidal and H. Veerm¨ a¨ ae, Wormholes in conformal gravity\n, JCAP10(2018) 003, arXiv:1802.02184 [gr-qc].\n[49] Boris Kors and Pran Nath, A Stueckelberg Extension of the Standard Model , Phys. Lett.\nB586(2004) 366-372, arXiv:hep-ph/0402047.\n[50] Daniel Feldman, Zuowei Liu and Pran Nath, Stueckelberg Z′extension with kinetic\nmixing and millicharged dark matter from the hidden sector , Phys. Rev. D75(2007)\n115001, arXiv:hep-ph/0702123.\n[51] David Kaiser, Primordial Spectral Indices from Generalized Einstein The ories, Phys.\nRev.D 52(1995) 4295-4306, arXiv:astro-ph/9408044.\n[52] Fedor L. Bezrukov and Mikhail Shaposhnikov, The Standard Model Higgs boson as the\ninflaton, Phys. Lett. B 659(2008) 703, arXiv:0710.3755 [hep-th].\n[53] FedorBezrukov, The Higgs field as an inflaton , Class. QuantumGrav. 30(2013)214001,\narXiv:1307.0708 [hep-ph].\n[54] N. Mohammedi, On Higgs inflation in non-minimally coupled models of gravit y, Phys.\nLett.B 831(2022) 137180, arXiv:2202.05696 [hep-th].\n19"    },    {        "title": "2402.04726v1.Phase_stability_and_mechanical_property_trends_for_MAB_phases_by_high_throughput_ab_initio_calculations.pdf",        "content": "Phase stability and mechanical property trends for\nMAB phases by high-throughput ab initio calculations\nNikola Koutn ´a1,2,*, Lars Hultman2, Paul H. Mayrhofer1, and Davide G. Sangiovanni2\n1Institute of Materials Science and Technology, TU Wien, Getreidemarkt 9, A-1060 Vienna, Austria\n2Department of Physics, Chemistry, and Biology (IFM), Link ¨oping University, SE-58183 Link ¨oping, Sweden\n*nikola.koutna@tuwien.ac.at; nikola.koutna@liu.se\nABSTRACT\nMAB phases (MABs) are atomically-thin laminates of ceramic/metallic-like layers, having made a breakthrough in the devel-\nopment of 2D materials. Though theoretically offering a vast chemical and phase space, relatively few MABs have yet been\nsynthesised. To guide experiments, we perform a systematic high-throughput ab initio screening of MABs that combine group\n4–7 transition metals (M); Al, Si, Ga, Ge, or In (A); and boron (B) focusing on their phase stability trends and mechanical\nproperties. Considering the 1:1:1, 2:1:1, 2:1:2, 3:1:2, 3:1:3, and 3:1:4 M:A:B ratios and 10 phase prototypes, possible\nstabilisation of a single-phase compound for each elemental combination is assessed through formation energy spectra of the\ncompeting mechanically and dynamically stable MABs. Based on the volumetric proximity of energetically-close phases, we\nidentify systems in which volume-changing deformations may facilitate transformation toughening. Subsequently, chemistry-\nand phase-structure-related trends in the elastic stiffness and ductility are predicted using elastic-constants-based descriptors.\nThe analysis of directional Cauchy pressures and Y oung’s moduli allows comparing mechanical response parallel and normal to\nM–B/A layers. Among the suggested most promising MABs are Nb 3AlB 4, Cr 2SiB 2, Mn 2SiB 2or the already synthesised MoAlB.\nKeywords : MAB phase; Ab initio; Phase stability; Elastic\nconstants; Ductility\n1 Introduction\nMAB phases (MABs) are atomically-laminated borides in\nwhich hard ceramic-like transition metal M–B layers alter-\nnate with relatively softer metallic-like mono- or bilayers of\nan A-element (typically Al, Si, Ga or In)1, 2. Though dis-\ncovered already in the 60s3, MABs have recently made a\nbreakthrough in the development of 2D materials for new-\ngeneration nanodevices4–6. Offering an interesting combina-\ntion of mechanical, magnetocaloric, and catalytic properties,\nhigh-temperature oxidation resistance, as well as damage and\nradiation tolerance, MABs are prominent candidates for appli-\ncations in the fields of protective and wear-resistant coatings,\nmagnetic cooling, electrocatalysis or electrochemical sensing,\nor radiation shielding1, 7–9.\nWith typical formula M n+1AB 2n(n={1,2,3})10and\npossible structures with hexagonal2or orthorhombic1sym-\nmetry, MABs theoretically provide a vast chemical and\nphase space. However, relatively few material systems\nhave been achieved experimentally: mostly bulk poly-\ncrystals (Ti 2InB 22, Cr 2AlB 211–13, Cr 3AlB 412, 13, Cr 4AlB 613,\nMoAlB9, 14, WAlB9, 15, Fe 2AlB 216, Mn 2AlB 217), and only\none thin film (MoAlB18–20). Thus, further development of\nboth bulk and thin film MAB phases calls for a systematic\ncomputational screening across a relevant subspace of the\nperiodic table, in particular, predicting trends in the phase\nstability and structure–property relationships for various M\nand A combinations.\nIn terms of phase stability predictions, solid work has al-\nready been done using the ab initio density functional the-ory (DFT) framework (see, e.g., Refs.7, 10, 21–25), or machine-\nlearning based approaches26, 27. In particular, Khazaei et\nal.22studied stability of the orthorhombic M 2AlB 2, MAlB,\nM3AlB 4, and M 4AlB 6systems with M from the group 3–6\ntransition metals, considering the decomposition into compet-\ning M–B, M–Al, and M–Al–B compounds. Siriwardane et\nal.7suggested that stability of a MAB phase for a given M\ndecreases for A changing from Al →Ga→In→Tl, i.e., with\nincreasing atomic number of the A element. This observation\nwas correlated with increased M–A and B–A bond lengths,\ncausing a decreased ionicity. Later, Carlsson et al.10screened\northorhombic and hexagonal MAB, M 2AB2, M3AB4, M4AB4,\nand M 4AB 6phases with M from the group 3–6 transition\nmetals or Mn, Fe, Co, and A ={Al, Ga, In}, confirming\nthermodynamic stability of 7 previously synthesised MABs,\npredicting 3 additional ones to be stable and 23 nearly sta-\nble. Furthermore, the authors hypothesised on preferential\northorhombic/hexagonal symmetry for MAB phases contain-\ning Al and {Ga, In}, respectively.\nMost ab initio investigations have been directed to identifi-\ncation of new stable MABs, while only few aimed on system-\natic predictions of materials’ properties and their relationships\nto phase prototypes and/or elemental composition27–30. For\npurposes of this work, we focus on studies addressing mechan-\nical behaviour28, 29, 31–33. Within the DFT framework, this is\ntypically realised by calculating the elastic constants ( Ci j) and\nderiving phenomenological strength and ductility indicators:\nthe Young’s modulus, shear-to-bulk modulus ratio, or Cauchy\npressure (widely accepted trend-givers in the family of ce-\nramics34–38). Employing Ci j-based indicators, Liu et al.29\nproposed TcAlB, NbAlB, WAlB, Tc 2AlB 2, Co 2AlB2, and\nNi2AlB 2to be the most ductile and Mo 2AlB 2with W 2AlB 2arXiv:2402.04726v1  [cond-mat.mtrl-sci]  7 Feb 2024to be the stiffest MABs of the MAB and M 2AB2phase proto-\ntypes, considering M from the group 3–5 transition metals and\nA=Al. Going beyond calculations of elastic constants, Dai\net al.33modelled (0 K) shear deformation of (CrB 2)nCrAl,\nn={1,2,3}, suggesting that tiltable B–A–B bonds can re-\nlease shear strain in weaker A layers, hence, contribute to high\nfracture toughness and damage tolerance.\nExperimental investigations on MABs’ mechanical re-\nsponse have most often concerned Young’s modulus mea-\nsurements (e.g., MoAlB18, 19and Mn 2AlB 217) whereas\ntoughness-related quantities have been significantly less re-\nsearched. For example, Fe 2AlB 239exhibited K1C=5.4±\n0.2MPa√m, which exceeds typical values for transition metal\ndiborides40–42). Recently, crack deflection and crack healing\nbehaviour have been reported for MoAlB43, 44and Fe 2AlB 245,\nfurther motivating the need for theory-based understanding\nof MABs’ mechanical response in relation to their chemistry\nand phase structure.\nOur study uses high-throughput DFT calculations to map\nphase stability trends, structural and mechanical properties of\nMAB phases containing group 4–7 transition metals, M= (Ti,\nZr, Hf; V , Nb, Ta; Cr, Mo, W; Mn, Tc, Re), A= (Al, Ga,\nIn, Si, Ge), and boron (B). For each elemental combination,\n10 phase prototypes with various M:A:B ratios are consid-\nered. These include experimentally known MAB and MAX\n(X=C, N) phases1, 2, 46, or are inspired by common transition\nmetal (di)boride structures, exhibiting intrinsically layered\ncharacter but not yet regarded as possible MAB structures.\nOur initial screening concerns formation energy, mechani-\ncal, and dynamical stability calculations, identification of the\nmost favourable structure, and prediction of a single-phase\nMABs synthesisability. For systems passing our stability cri-\nteria, chemistry- and phase-structure-related trends in elastic\nstiffness and ductility are predicted, including both the poly-\ncrystalline approximates and directional values parallel and\nnormal to the metal/ceramic layers. Finally, the most promis-\ning candidates for the synthesis of novel MABs are suggested.\n2 Methods\nDensity Functional Theory (DFT) calculations were per-\nformed using the Vienna Ab-initio Simulation Package\n(V ASP)47, 48together with the projector augmented plane-\nwave (PAW) method49and the Perdew-Burke-Ernzerhof\n(PBE) generalised gradient approximation50. Following con-\nvergence tests, the plane-wave cutoff energy was set to 600 eV\nand the Γ–centred k-point mesh of the Brillouin-zone was au-\ntomatically generated with a length parameter ( Rk) of 60 Å\n(i.e., k-points separated by 1 /60 Å−1along each bvector).\nIn total 60 atomically-laminated M–A–B systems were\nmodelled, including all combinations of M= (Ti, Zr, Hf; V ,\nNb, Ta; Cr, Mo, W; Mn, Tc, Re), A= (Al, Ga, In, Si, Ge) and\nboron (B). For each elemental combination, we considered\nthe 1:1:1, 2:1:1, 2:1:2, 3:1:2, 3:1:3, and 3:1:4 ratio between\nM, A, and boron, and 10 MAB phase prototypes given below\nand schematically depicted in Fig. 1 (the thereby establishednotation will be used throughout this work):\n•For the 1:1:1 chemistry :MAB (orthorhombic with a\nspace group (s.g.) Cmcm ; a 12-atom simulation cell).\n•For the 2:1:1 chemistry :M2AB(hexagonal with a s.g.\nP63/mmc ; an 8-atom simulation cell).\n•For the 2:1:2 chemistry :M2AB 2(orthorhombic with\na s.g. Cmmm ; a 10-atom simulation cell), α-M2AB 2\n(hexagonal with a s.g. P6m2; a 10-atom simulation cell),\nω-M2AB 2(hexagonal with a s.g. P6 3/mmc, a 20-atom\nsimulation cell), and ω’-M 2AB 2(hexagonal with a s.g.\nP63/mmc, a 20-atom simulation cell), γ-M2AB2(hexag-\nonal with a s.g. P6 3/mmc, a 10-atom simulation cell).\n•For the 3:1:2 chemistry :M3AB2(hexagonal with a s.g.\nP63/mmc ; a 12-atom simulation cell).\n• For the 3:1:3 chemistry :M3AB 3(orthorhombic with a\ns.g. Pnma; a 14-atom simulation cell).\n• For the 3:1:4 chemistry :M3AB 4(orthorhombic with a\ns.g. Pmmm; an 8-atom simulation cell).\nThe above-described phase prototypes were fully relaxed\nfor all combinations of M and A elements (in total 600 MABs),\nuntil forces on ions did not exceed 0.005 eV/Å and total\nenergies were converged up to 10−5eV/supercell. Cr- and\nMn-based MAB phases have been considered in their non-\nmagnetic state. Subsequently, relative chemical stability was\nestimated by the energy of formation\nEf=1\n∑sns \nEtot−∑\nsnsµs!\n, (1)\nwhere Etotis the total energy of the simulation cell (from the\nlast ionic step of a structure relaxation), nsandµsare the\nnumber of atoms and the chemical potential, respectively, of\na species s. Chemical potentials, µs, were conventionally set\nto the total energy per atom of the ground-state structure for\nthe respective element, from Material’s project55that is fcc-\nAl; bcc-V , -Nb, -Ta, -Cr, -Mo, -W, -Mn; diamond-Si, -Ge;\ntetragonal-In; orthorhombic-Ga; rhomboedral-B; and hcp-Ti,\n-Zr, -Hf, -Tc, -Re.\nThe stress-strain method56–58was used to calculate fourth\norder elasticity tensors (according to Hooke’s law), which\nwere projected onto symmetric 6×6matrices of elastic con-\nstants, Ci j, using the V oigt notation. Positive definiteness of\ntheCi jmatrix was verified in order to determine mechanical\nstability of the corresponding structure59. For mechanically\nstable MABs, their dynamical stability was assessed based\non the corresponding phonon spectra: by checking for no\nimaginary phonon modes (i.e. non-zero phonon density of\nstates only in the positive frequency region). The phonon\nspectra were obtained with the aid of the Phonopy package60,\nusing the finite displacement method with the default displace-\nment of 0.01 Å and 2 ×2×2 replicas of the fully relaxed MAB\nstructures (supercells with 64–160 atoms). The supercells size\n2Figure 1. Snapshots of the here considered MAB phase\nprototypes . (a–c) The MAB ,M2AB2,M3AB4are known MAB\nphase prototypes1experimentally achieved for, e.g., MoAlB18,\nMn2AlB 217, and Cr 3AlB 412, 13. (d–e) The M2ABandM3AB2are\nexperimentally known MAX ( X=C, N) phase prototypes46, where\nthe M 2AB structure has been considered for MABs in a recent DFT\nstudy51. (f–j) The α-M2AB2,ω-M2AB2,ω’-M 2AB2,γ-M2AB2,\nandM3AB3are hand-designed based on common structures of\ntransition metal borides (TMBs). The α-M2AB2is based on the\nhexagonal α-AlB 2type phase ( P6/mmm ; typical for the group 4–6\nTMB 2s52) and has been reported for a bulk Ti 2InB 2MAB phase2.\nTheω-M2AB2,ω’-M 2AB2, and γ-M2AB2are based on the\nω-WB 2type phase (P6 3/mmc; ABBA stacking) and the γ-ReB 2\ntype phase (P6 3/mmc; BABA stacking), see Ref.53. The difference\nbetween ω-M2AB2andω’-M 2AB2is that in the former (latter), Al\nreplaces the flat (puckered) B sheets while the puckered (flat) sheets\nremain B and form the ceramic M–B layer. The M 3AB3prototype is\nbased on the structure of TiB (Pnma)54, 55. Other structures of\ncommon boride-based ceramics (e.g., Ti 3B4and Ti 2B) inspired\nhypothetical phase prototypes of additional MABs, however, were\nfound irrelevant due to their high formation energies, thus excluded.\neffects have been tested for several cases, indicating that the\nchosen supercells are sufficient to verify dynamical stability.\nTheCi jmatrices of MABs fulfilling conditions for mechan-\nical and dynamical stability were further post-processed to\nestimate mechanical properties. Imposing the macroscopic\nsymmetry, the matrices were projected on those of a hexago-\nnal or an orthorhombic system, yielding 5 and 9 independent\nelastic constants, respectively ( C11,C33C12,C13,C44for the\nhexagonal symmetry, and additional C22,C23, and C55for the\northorhombic symmetry). The polycrystalline Young’s modu-\nlus,E=9BG/(3B+G)was calculated using Hill’s average\nof the bulk, B, and shear modulus, G61, 62. The polycrystalline\nPoisson’s ratio, ν, and the directional Young’s moduli, E⟨001⟩,\nE⟨010⟩, and E⟨100⟩, were calculated following Ref.61. Conse-quently, the out-of-plane and the in-plane Young’s moduli, i.e.,\nnormal and parallel to the metal/ceramic layers, respectively,\nwere calculated as\nE⊥=E⟨001⟩, (2)\nE∥=1\n2(E⟨010⟩+E⟨100⟩). (3)\n3 Results and discussion\n3.1 Stability trends\nThe searched chemical and phase space of hypothetical MAB\nphases contains all combinations of the group 4–7 transition\nmetals (M elements) with Al, Si, Ga, Ge or In (A elements)\nand 10 phase prototypes (Fig. 1) for each elemental combi-\nnation. Though Tc-based MABs have expectably low appeal\nfor applications63, they are included for completeness. Our\nfirst aim is predicting stability trends, preferential phase pro-\ntotypes, and energetically-close competing MABs.\nThe energy of formation, Ef, serves as a basic chemi-\ncal stability indicator, allowing to pre-select hypothetically\n(meta)stable MABs further tested for stability with respect\nto “small” elastic deformations and phonon vibrations, i.e.,\nmechanical and dynamical stability. For each (M, A) com-\nbination, we identify (i) the lowest-energy phase, and (ii)\nenergetically-close phases, i.e., within an energy threshold,\nEthr\nf, from the lowest-energy phase (here Ethr\nf:=0.25eV/at.).\nFor thereby selected MABs, we verify positive definiteness\nof the corresponding elastic constants matrix (mechanical\nstability condition59) and the absence of imaginary phonon\nmodes in the phonon spectra (dynamical stability condition).\nNote that MABs passing these stability conditions may be\nstill metastable against the decomposition to any competing\nbinary or ternary non-MAB compounds10, 22not considered\nin this work.\nThe predicted Eftrends are depicted in Fig. 2. Out of\nall hypothetical MABs (total 600), about 50% (317 MABs)\nfulfil the above selection criteria (energetic, mechanical and\ndynamical stability), and are visualised by coloured symbols.\nTogether with the analysis of the corresponding structural\nproperties—per-atom volumes, Vper-at. (Suppl. Fig. S1), and\ndensities, ρ(Suppl. Fig. S2)—results in Fig. 2 lead to the\nfollowing observations:\n(3.1a) Trends in the energetic ( Ef) stability are mainly\ndriven by M . Typically, the MABs’ formation en-\nergy increases for M from the group 4 →5→6→7.\nExemplarily, Efof the M 2AlB 2phase gradually in-\ncreases from −0.93to−0.41eV/at. for M changing as\nTi→V→Cr→Mn. Generally, the Efincrease is accom-\npanied by the volume (Vper-at. )decrease andthe density\n(ρ)increase . Changing the M’s period (4 →5→6) has a\nrelatively minor effect on Ef,Vper-at. , and ρ.\n(3.1b) For a given M, Al-containing (In-containing) MABs\ntypically exhibit the lowest (highest) Efamong the\n3here-considered phase prototypes. In-containing MABs\nalso show the highest volume and the lowest density ,\npossibly due to the large atomic radius of In (compared\nto Al, Ga, Si, and Ge), which is closer to that of M ele-\nments (see the atomic difference ratio analysis in Suppl.\nFig. S3). Most often, Efincreases for A changing as\nAl→Si→Ga→Ge→In. The choice of A more signifi-\ncantly impacts stability for MABs that contain M from\ngroups 6–7 (compared with M from groups 4–5). There\nare no stable MABs combining Re and In.\n(3.1c) The M element’s group also notably influences which\nphase prototypes are energetically favourable and\nhow many energetically-close MABs exists . The\nM3AB 4prototype always yields the lowest volume and\nthe highest density, while the M 2AB and M 3AB 2are\ntypically the least dense.\n•For M from the group 4 ( =Ti, Zr, Hf) , the lowest-\nenergy MABs are M 3AB 4(A={Al, Ga, In}) and\nα-M2AB 2(A={Si, Ge}), followed by M 2AB 2\nandω-M2AB 2. The Ti–Al–B and Ti–Si–B sys-\ntems exhibit the highest number of possible MABs:\nM3AB 4,α-M2AB 2, M 2AB 2,ω-M2AB 2, M 3AB 3,\nand MAB, where the last two exhibit the highest\nEf, thus are the least likely to form.\n•For M from the group 5 ( =V, Nb, Ta) , the lowest-\nenergy MABs are M 3AB 4,α-M2AB 2or M 3AB 2,\nwhere the last one only concerns Ta–A–B systems\nwith A={Si, Ge, In}. Compared to M from the\ngroup 4, there are more competing MABs. The\nMAB phase prototype becomes a competing phase,\nparticularly for A={Al, Si}. The M 2AB 2and\nω-M2AB 2prototypes are competing phases in the\nV–A–B and Nb–A–B systems for A ={Ga, In}.\n•For M from the group 6 ( =Cr, Mo, W) , the lowest-\nenergy MABs are M 3AB 4,α-M2AB 2, M 2AB 2,\nMAB (Mo–Al–B and W–Al–B systems), ω′-\nM2AB 2(W–Si–B and W–Ga–B systems), and\nM3AB 3(W–Ge–B system). While nearly all the\nhere-considered phase prototypes fall within the Ef\nthreshold, some are dynamically unstable.\n•For M from the group 7 ( =Mn, Tc, Re) , the\nlowest-energy MABs are ω′-M2AB2, M2AB2(Mn–\nA–B system for A={Al, Si, Ga}), α-M2AB2(Mn–\nGe–B system), M 3AB 4(Mn–In–B system). Mn–\nAl–B and Re–Al–B exhibit the highest number of\ncompeting MABs.\nCalculated with respect to energies of the constituting ele-\nments (i.e. their chemical potentials), absolute Efvalues in\nFig. 2 may change under (highly) non-equilibrium synthesis\nconditions, reflected by changes of the reference chemical\npotentials (see a case-study of TaN64). Consequently, this may\nalter relative stability order of energetically-close MABs. Theactual Efdistribution of all dynamically stable MABs, how-\never, will be less affected by “small” variations of chemical\npotentials. In particular, a system in which the lowest-energy\nphase exhibits a “large” Efseparation from competing phases\nwill likely enable single-phase MAB formation (supposing\nit will not decompose into binary or ternary non-MAB com-\npounds), contrarily to a system with many energetically-close\nphases. To quantify this intuitive idea, we take inspiration\nin the EFA (entropy forming ability) descriptor by Curtarolo\nand co-workers65, 66and introduce a single-phase indicator ,\nSPI. Considering the Efdistribution of competing MABs in a\ngiven M–A–B system, SPI is calculated as\nSPI(n) ={σ[spectrum (Ef(n))]}−1, (4)\nwhere σis a standard deviation of energies, Ef. A “low” SPI\nsuggests a high propensity to form a single-phase MAB, simi-\nlar to Curtarolo’s “low” EFA pointing towards formation of a\nsingle-phase high-entropy ceramic65. An example Efspectra\ntogether with the resulting SPI are given in Fig. 3(a,b,c). The\nSPIs evaluated for all elemental combinations (Fig. 3d) render\nthe following hypotheses:\n•Hf and Zr combined with Si or Ge are likely to form\nsingle-phase MABs ( SPI≈6–7 (eV/at.)−1), if not de-\ncomposed into other non-MAB compounds.\n•Contrarily, Re in combination with Si ( SPI≈\n98(eV/at.)−1) or Ge ( SPI≈56(eV/at.)−1), and Mn and\nW in combination with In ( SPIof≈37and 45 (eV/at.)−1,\nrespectively) do not provide a suitable basis for stabilisa-\ntion of single-phase MABs.\n•For A =Al—the most typical A element in experimen-\ntally reported MABs—there is no extremely high or\nlow propensity to single-phase MABs formation. Ex-\namples of M elements yielding relatively “lower” and\n“higher” SPI are M ={Ti, Cr, Mo, Mn, Tc, Re}, with\nSPI≈10–13 (eV/at.)−1, and M ={Zr, Hf, Ta} with ≈21–\n26 (eV/at.)−1, respectively.\nThe SPI reflects energetic aspects only (here, zero Kelvin\nformation energies of dynamically stable MABs). In practice,\nhowever, single-phase MAB formation will be driven also by\nkinetic factors and specific experimental setup (e.g. sputter-\ning from a ternary vs. elemental targets). A simple add-on\nto the SPI may be considering the volumetric proximity of\ncompeting phases based on the Vper-at data in Suppl. Fig. S1.\nSupposing the MABs’ layered character enables a rela-\ntively “easy” transformation pathway from one phase proto-\ntype to another, the availability of energetically-close phases\n(intermediate-to-high SPI) with similar volumes as the energet-\nically most preferred phase may actually be beneficial: facili-\ntating transformation toughening during volume-changing me-\nchanical deformation. For example, cubic Ti 0.5Al0.5N subject\nto [001] tension undergoes local lattice transformations to the\nenergetically-close wurtzite structure, thus improving tough-\nness67. An additional important condition for the phase trans-\nformation may be that the M:A:B ratio remains unchanged.\n4●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●●\n●●a) M=Ti\n−1−0.8−0.6−0.4−0.200.2Formation energy [eV/at.]M from Group 4\n●●●\n●d) M=VM from Group 5\n●●●\n●●g) M=CrM from Group 6\n●●●\n●j) M=MnM from Group 7\n●●●b) M=Zr\n−1−0.8−0.6−0.4−0.200.2Formation energy [eV/at.]●●●e) M=Nb\n●●●h) M=Mo\n●k) M=Tc\n●●●c) M=Hf\n−1−0.8−0.6−0.4−0.200.2Formation energy [eV/at.]High E f (>Efmin+Efthr), and/or,\nmechanically unst. , and/or,\ndynamically unst.\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=GeEfminEfmin+Efthr(=0.25eV/at.) ●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure 2. Trends in the phase stability of MABs , as quantified by the formation energy, Ef(see Eq. (1)). For each (M, A) combination,\nthe black solid lines guide the eye for MABs energetically close to the lowest-energy phase, i.e. within the Efthreshold, Ethr\nf=0.25 eV/at.\nMABs that are above Ethr\nf, and/or, mechanically, and/or dynamically unstable are depicted in grey. All MABs marked by colour are\nmechanically and dynamically stable. Trends in the corresponding per-atom volumes, densities, and atomic difference ratios are shown in\nSuppl. Fig. S1, Suppl. Fig. S2, and Suppl. Fig. S3, respectively.\n5a) Hf−Ge−B MABs:\nSPI=6.1 (eV/at.)−1\nb)Dynamically stable MABsTa−Al−B MABs\nSPI=21.3 (eV/at.)−1\n−1.0−0.8−0.6−0.4−0.20.0c) Mo−In−B MABs\nSPI=18.2 (eV/at.)−1\nEf [eV/at.]102050100\n●\n●●●●\n●●●●●\n●SPI [eV/at.]−1\n Single−phase       Competing phasesd)\nM=Ti\nM=Zr\nM=Hf\nM=V\nM=Nb\nM=Ta\nM=Cr\nM=Mo\nM=W\nM=Mn\nM=Tc\nM=Re●A=AlA=GaA=InA=SiA=GeFigure 3. Prediction of the MABs’ propensity to form a single-phase compound , as quantified by the single-phase indicator , SPI, see\nEq.(4). (a,b,c) Examples of formation energy ( Ef) spectra used to derive SPI. The spectra contain all dynamically stable phases (identified in\nFig. 2) for the respective (M, A) combination. (d) The SPI descriptor for all (M, A) combinations. Low SPI values indicate tendency to form\nas single-phase MAB, whereas high SPIs are a sign of several energetically-close competing MABs. Mind the log scale of the y-axis (SPI).\nOur Efand volumetric analysis points towards Cr 2AlB 2,\nRe2AlB 2, Cr 2SiB 2, W 2SiB 2, and Mn 2SiB 2as to MAB phases\nwith intermediate SPI≈11–25 (eV/at.)−1, favouring the 2:1:2\nstoichiomtery and exhibiting energetically-close 2:1:2-type\nphases with volumes by 1%–10% larger (smaller in case of\nMn2SiB 2), thus possibly forming under tensile (compressive)\nstrains.\n3.2 Trends in the electronic structure\nOur stability predictions revealed in total 317 (meta)stable\nMABs (see coloured symbols in Fig. 1). Due to various crystal\nsymmetries, chemistry, and elemental composition, however,\nunderstanding trends in their mechanical properties may be\ndifficult. To shed light on their most fundamental similarities\nand differences, we first investigate their electronic density of\nstates (DOS) near the Fermi level.\nFig. 4 presents DOS of selected representative MABs, il-\nlustrating general trends observed also for other phase proto-\ntypes and (M, A) combinations. The energy range in focus\nis≈[−15,5]eV , where 0 corresponds to the Fermi level, EF.\nFrom a qualitative DOS analysis, we infer the following:\n(3.2a) The DOS of all MABs has metallic character and the\nFermi level vicinity is dominated by the transition metal\nd-electrons (Fig. 4a–o), suggesting that the M element\nsignificantly influences mechanical properties.\n(3.2b) The most decisive factor for the general shape of DOS\nis the phase prototype, whereas the EFposition with\nrespect to the nearest peaks is mainly governed by the M\nelement’s group (Fig. 4a–l). Exceptions (e.g. Cr 3B4in\nFig. 4i) may be rationalised by high formation energy of\nthe corresponding phase prototype for given M.(3.2c) For the same phase prototype and M from the same\ngroup in the periodic table (i.e. with the same valence\nelectron concentration), both the shape of DOS and EF\nare typically nearly the same (Fig. 4d–f).\n(3.2d) Changing the A element influences the general shape of\nDOS, however, to a lesser extent than changing the phase\nprototype (Fig. 4m–o). The relative EFposition with\nrespect to the neighbouring peaks is nearly uninfluenced.\nOur results indicate that the main features of DOS near EF\nare dictated by the phase prototype, which may be the most\nnatural mean of sorting MABs when searching for trends in\nmechanical properties. The group of the M element—in other\nwords, the number of M’s valence electrons—influences the\nFermi level position with respect to the closest DOS mini-\nmum or peak, thus, may be a crucial factor for optimisation\nof mechanical properties. Shifting the EFfor different phase\nprototypes, however, will likely lead to filling different states\nand an in-depth analysis (out of our scope) would be neces-\nsary to understand how these impact the MABs’ mechanical\nresponse.\n3.3 Stiffness and ductility indicators\nThis section focuses on predicting trends in mechanical prop-\nerties of MAB phases via phenomenological elastic-constants-\nbased descriptors. Specifically, Young’s modulus ( E), shear-\nto-bulk modulus ratio ( G/B), and Cauchy pressure ( CP) are\nused to compare all dynamically stable MAB candidates (all\ncoloured symbols in Fig. 1, comprising 317 MABs) in terms\nof theoretical stiffness and ductility. Although ductility is a\ncomplex property—dictated by structure, density, and mobil-\nity of extended crystallographic defects over different length\nand time scales— Ci j-based indicators have served as common\n6total  s(M)  p(M)  d(M)  s(A)  p(A)  s(B)  p(B)\n(a−c) M 2AB2 phase prototype, A=Al, M from Group 4−6 & Period 4\nTi2AlB2a)DOS [a.u.]V2AlB2b)Cr2AlB2c)\n(d−f) M 2AB2 phase prototype, A=Al, M from Group 5 & Period 4−6\nV2AlB2d)DOS [a.u.]Nb2AlB2e)Ta2AlB2f)\n(g−i) M 3AB4 phase prototype, A=Al, M from Group 4−6 & Period 4\nTi3AlB4g)DOS [a.u.]V3AlB4h)Cr3AlB4i)\n(j−l) MAB, phase prototype, A=Al, M from Group 4−6 & Period 4\nTiAlBj)DOS [a.u.]VAlBk)CrAlBl)\n(m−o) M 3AB4 phase prototype, M=Ti, A={Al, Si, Ga} \nTi3AlB4m)DOS [a.u.]\n−10 −5 0 5\nE−EF [eV]Ti3SiB4n)\n−10 −5 0 5\nE−EF [eV]Ti3GaB4o)\n−10 −5 0 5\nE−EF [eV]Figure 4. Electronic density of states (DOS) for representative MABs illustrating general DOS shape depending on the phase\nprototype and elemental composition . The grey-shaded area denotes the total DOS, while the red, blue,and yellow lines are partial\ncontributions from the M, A, and B element (dotted line: s-electrons, dashed line: p-electrons, solid line: d-electrons). The zero energy\nalways denotes the Fermi level ( EF). (a–c) The role of the group from which M is chosen , exemplified by the M2AB2phase prototype.\nThe DOS shapes are very similar, while the EFshifts. (d–f) The role of the period from which M is chosen , exemplified by the M2AB2\nphase prototype. Both the DOS shape and EFare very similar. (g–i) The role of the group from which M is chosen , exemplified by the\nM3AB4phase prototype. The DOS shape is very similar (slightly differing for the eneregtically least stable Cr 3AlB 4), while the EFshifts.\n(j–l)The role of the group from which M is chosen , exemplified by the MAB phase prototype. The DOS shape is very similar, while the\nEFshifts. (m-l) The role of the A element , exemplified by the M3AB4phase prototype. The DOS shape changes slightly, while the EF\nposition relative to the nearest DOS peak remains nearly constant.\ntrend-givers in the family of refractory ceramics34–37, 68, 69\nwith reasonably similar crystal and electronic structures.\nDue to the non-cubic crystal symmetry and layered archi-\ntecture, MAB phases should generally exhibit an anisotropic\nelastic response. The degree of anisotropy can be estimated\nby the universal anisotropy index, AU70, identically zero for\nlocally isotropic single crystals. In our case, AUvaries be-\ntween 0.01 ( α-Nb 2InB 2) and 6.35 (Ta 3InB 4), where nearly80% of all MABs exhibit AU≤0.5(Suppl. Fig. S8). The most\nenergetically preferred phase for a given (M, A) combination\nyields AU=0.01–1.43, comparable to AU=0.01–1.88 of\ntransition metal diboride ceramics (MB 2, M from the group 4–\n7;AUwas evaluated based on data from Ref.53). Interestingly,\nAUof the α-M2AB2phase prototype is fairly independent of\nelemental composition ( AU=0.11±0.14), while others, e.g.,\nthe M 2AB 2and M 3AB 4phase prototype, exhibit larger AU\n7variations ( AU=0.54±0.49andAU=0.66±1.16, respec-\ntively).\nIn the first step (Fig. 5), we disregard the MABs’ elas-\ntic anisotropy and evaluate “effective” strength and ductility\nindicators using the polycrystalline moduli ( B,G,E) and\naveraging the directional Cauchy pressure values,\nCPeff=CP⊥+CP∥\n[1]+CP∥\n[2]=\n=C12−C66\n3+C13−C44\n3+C23−C55\n3, (5)\nwhere CP⊥(CP∥\n[1],CP∥\n[2]) are out-of-plane (in-plane) Cauchy\npressure, i.e. orthogonal and parallel to the M–B/A layers.\nThe obtained ductility map ( CPeffvs.G/Bin Fig. 5a) in-\ndicates an important role of the M element. The strongest\ntrend observed is a ductility increase for M from the group\n4→5→6→7. Among the most ductile MABs containing com-\nmon elements (excluding Ga, Ge, In, Mn, Tc) are V 2SiB,\nNb2SiB, Ta 2AlB 2, Ta 3AlB 4, Ta 2SiB, Cr 3AlB 3, Mo 2AlB 2,\nMo 2SiB 2, Mo 3AlB 3, Mo 3SiB 3, W 3SiB 2, Re 2AlB 2, and Re-\nAlB. There are, however, several outliers. For example,\nCr2AlB 2and MoAlB (M from the group 6) are predicted\nto be surprisingly brittle ( CPeff<−30GPa), while Ti 3GeB 4\nand ZrGaB (M from the group 4) would be surprisingly duc-\ntile (CPeff>30GPa). These outliers may be explained by (i)\nenergetic reasons ( Ef“high” above that of the lowest-energy\nphase, e.g., ZrGaB with Ef0.24 eV/at above Efof the most\nstable Zr 3GaB 4), by (ii) differences between the phase pro-\ntotypes, i.e. different optimal position of the Fermi level\n(inducing more ductile/brittle response to deformation), by\n(iii) effects of the A element (recall that In-containing MABs\nwere always the least energetically stable and the least dense),\nor by (iv) elastic anisotropy. To support the energetic argu-\nment (i), all data points in Fig. 5 are scaled based on their\nEfdifference from the lowest-energy phase for a given (M,\nA) combination, so that smaller symbol sizes correspond to\nlarger Efdifferences.\nThe so far experimentally reported MABs most often crys-\ntallised in the M 3AB 4, M 2AB 2, and MAB type phase, de-\npicted in Fig. 5b–d. Here, Ta 3AlB 4, Ta 2AlB 2, Re 2AlB 2, and\nReAlB stand out in terms of theoretical ductility.\nFig. 5e–h shows the relationship between the elastic stiff-\nness and ductility, estimated by the polycrystalline Young’s\nmodulus, E, and the shear-to-bulk modulus ratio, G/B, re-\nspectively. To prevent failure during mechanical loads, one\nseeks a compromise between high Eand low G/B, providing\nan atomic-level basis for initially hard but then reasonably\nplastic response to deformation. For the here-studied MABs,\nEvaries significantly—between 96 GPa ( ω-Mo 2GeB 2) and\n441 GPa (Cr 3AlB 4)—and seems to be less controlled by the\nM element than ductility.\nSuggested already by low density of In-containing MABs\n(Section 3.1, Suppl. Fig. S2), their Emoduli are generally low,\n232±50 GPa, where the standard deviation represents values\nfrom various M elements and phase prototypes. Al- and Si-\ncontaining MABs, in contrast, posses relatively high Evalues,312±60 GPa. There is eleven MABs with E>400GPa.\nOnly one contains a group 4 M element (Ti 3AlB 4) and the top\nthree are Cr 3AlB 4, TaSiB, and VSiB possessing G/Bof 0.73,\n0.80 0.84, respectively, thus illustrating the typical inverse\nrelationship between stiffness and ductility. A combination of\nrelatively high Young’s modulus ( E>350GPa) and low G/B\nratio ( G/B<0.55) is shown by Cr 2SiB 2, Cr 3SiB 4, W 2AlB 2,\nα-W2SiB 2, and γ-Re 2GeB 2. Among them, Cr 3SiB 4is the\nlowest-energy phase in the Cr–Al–B system, and the other\nones are energetically close to their most favourable phases\n(∆Ef=0.01–0.05 eV/at.).\nAs noted at the beginning of this section, elastic response\nof MABs is strongly directional. This is illustrated in Fig. 6\npresenting two Cauchy pressure and Young’s modulus val-\nues: in-plane, i.e., parallel to the metal/ceramic layers (de-\nnoted by CP∥,E∥), and out-of-plane, i.e. orthogonal to\nthe metal/ceramic layers (denoted by CP⊥,E⊥). The lat-\nter is aligned with the most typical growth direction. Ac-\ncording to Fig. 6a, about 20% of all MABs can be seen as\nstrongly Cauchy-pressure anisotropic, with |CP∥−CP⊥|>\n100GPa. Interestingly, the ratio of MABs with CP∥>CP⊥\nandCP∥<CP⊥is nearly 1:1. Examples of extreme cases in-\nclude Mn 2SiB 2andα-Re 2AlB 2(CP∥−CP⊥>150GPa, indi-\ncating superior in-plane ductility), or Ta 2SiB and γ-Mo 2GaB 2\n(CP⊥−CP∥>150GPa, indicating superior out-of-plane duc-\ntility). Furthermore, the data show that (i) almost all MABs\ncontaining group 4 transition metals exhibit CP∥<CP⊥, and\n(ii) the ω′-M2AB 2,γ-M2AB 2and M 2AX prototypes show\nCP∥≪CP⊥for the group 6 M elements.\nFocusing on the most common phase prototypes and (M,\nA) combinations, the M 3AB 4(Fig. 6b) and the M 2AB 2\n(Fig. 6c) type phases are predicted to be more ductile in-plane,\nwith extreme cases ( CP∥≫CP⊥) being Nb 3AlB 4, Ta 3AlB 4,\nTa2AlB 2, and Re 2AlB 2. The MAB phase prototype (Fig. 6d)\ncan be seen as rather Cauchy-pressure isotropic.\nConcerning Young’s moduli, their in-plane and out-of-plane\nvalues Fig. 6e–h do not show a simply reversed trend with\nrespect to the directional Cauchy pressures. Besides Ti 3SiB 4,\nTi2SiB 2, and ZrAlB, all MABs containing group 4 transi-\ntion metals and most MABs containing group 5 transition\nmetals are stiffer in-plane compared to their [001] direction.\nThis is intuitively expected due to relatively weak bonding\nbetween the ceramic (M–B) and metallic (A) layers. Simi-\nlar to Fig. 6a, the ω′-M2AB 2,γ-M2AB 2and M 2AX proto-\ntypes tend to cluster for the group 6 M elements and can be\nseen as outliers with significantly higher out-of-plane stiffness\n(E⊥−E∥>150 GPa).\nAmong the most common phase prototypes and (M, A)\ncombinations, the M 3AB4(Fig. 6f) and the M 2AB2(Fig. 6g)\ntype phase typically exhibit E∥>E⊥. In particular, Nb 3AB4,\nTa3AB4, and Re 2AlB 2yield E∥−E⊥>100GPa. The MAB\nphase prototype (Fig. 6h) shows small differences between\nin-plane and out-of-plane Young’s moduli, with the most\nanisotropic MABs being CrAlB, ZrSiB, and the energetically\nrather unlikely (smaller symbol sizes) TiAlB and V AlB.\n8●●● ●●● ●●● ●●●M from Group 4                   M from Group 5                    M from Group 6                     M from Group 7\nTiZrHf VNbTa CrMoW MnTcRe● Phase\nprototypes:M3AB4\nM2AB2α−M2AB2\nω−M2AB2ω'−M2AB2\nγ−M2AB2MAB\nM3AB3M2AB\nM3AB2\n●●●\n●●\n●●●●●●●●\n●●\n●●\n●●\n●●\n●●●\n●●●●●●●\n●●●\n−100−50050100150CPeff [GPa]a)\nPettifor's ductility                                                          \ncriterion                                                                        \nPugh's ductility                                       \ncriterion                                                 INCREASING\nDUCTILITYAll phase prototypes and elements\n●\n●\n●●●●●\n●●\n●\n●●\n●●\n●●\n●●●\n●●●●\n●●●●●\n●●\n●●●\n●\n100200300400500E [GPa]e) All phase prototypes and elements\nINCREASING\nSTIFFNESS\n●●\n●●\n●●\n●●●●\n●\n−100−50050100150CPeff [GPa]b)\n(Cr,Al)(Cr,Si)\n(Hf,Al)(Mo,Si)\n(Nb,Al)(Ta,Al)\n(Ti,Al)(Ti,Si)(V,Al)(V,Si)\n(Zr,Al)M3AB4, M!={Mn,Tc} & A={Al,Si}\n●\n●●\n●●\n●●\n●●\n● ●\n100200300400500E [GPa]f)\n(Cr,Al)\n(Cr,Si)(Hf,Al)\n(Mo,Si)(Nb,Al)(Ta,Al)(Ti,Al)\n(Ti,Si)(V,Al)\n(V,Si)\n(Zr,Al)M3AB4, M!={Mn,Tc} & A={Al,Si}\n−100−50050100150CPeff [GPa]c)\n(Cr,Al)(Cr,Si)\n(Hf,Al)(Mo,Al)(Re,Al)(Ta,Al)\n(Ti,Al)(Ti,Si)(V,Al)(W,Al)\n(Zr,Al)M2AB2, M!={Mn,Tc} & A={Al,Si}\n100200300400500E [GPa]g)\n(Cr,Al) (Cr,Si)\n(Hf,Al)\n(Mo,Al)(Re,Al)\n(Ta,Al)(Ti,Al)\n(Ti,Si)(V,Al)(W,Al)\n(Zr,Al)M2AB2, M!={Mn,Tc} & A={Al,Si}\n−100−50050100150\n0.20.30.40.50.60.70.80.9\nG/BCPeff [GPa]d)\n(Cr,Al)(Cr,Si)\n(Mo,Al)(Mo,Si)\n(Nb,Al)(Nb,Si)(Re,Al)\n(Ta,Al)\n(Ta,Si)(Ti,Al)\n(Ti,Si)(V,Al)(V,Si) (W,Al)(Zr,Si)MAB, M!={Mn,Tc} & A={Al,Si}\n100200300400500\n0.20.30.40.50.60.70.80.9\nG/BE [GPa]h)\n(Cr,Al)(Cr,Si)\n(Mo,Al)(Mo,Si)\n(Nb,Al)(Nb,Si)\n(Re,Al)(Ta,Al)(Ta,Si)\n(Ti,Al)(Ti,Si)(V,Al)(V,Si)(W,Al)\n(Zr,Si)MAB, M!={Mn,Tc} & A={Al,Si}Figure 5. Trends in theoretical ductility (a–d) and stiffness (e–h) of MAB phases estimated via elastic-constants-based descriptors :\neffective Cauchy pressure ( CPeff, Eq. 5), polycrystalline shear-to-bulk modulus ratio ( G/B), and polycrystalline Young’s modulus ( E). The\ndashed lines in (a–d) guide the eye for Pettifor’s71and Pugh’s72semi-empirical ductility criteria (commonly used for ceramics34–37, 68, 69but\noriginally developed for metals, these criteria should be treated only on a qualitative level). Panels (a) and (e) show results for all stable\nphases (marked by colour in Fig. 2), while panels (b–c) and (f–h) focus on the most common phase prototypes and elements (excluding Mn,\nTc, Ga, Ge, In). The underlying (phase-, M-element-, and A-element-resolved) B,G, and Evalues for all MABs are shown in Suppl. Fig. S4,\nSuppl. Fig. S5, and Suppl. Fig. S6, respectively. The corresponding Poisson’s ratios are shown in Suppl. Fig. S7.\n9●●● ●●● ●●● ●●●M from Group 4                   M from Group 5                    M from Group 6                     M from Group 7\nTiZrHf VNbTa CrMoW MnTcRe● Phase\nprototypes:M3AB4\nM2AB2α−M2AB2\nω−M2AB2ω'−M2AB2\nγ−M2AB2MAB\nM3AB3M2AB\nM3AB2\n●●●\n●●\n●●●●●●●●\n●●\n●●\n●●\n●●\n●●●\n●●●●●●●\n●●●\n−100−50050100150CPin−plane [GPa]a) All phase prototypes and elements\nIncreasing ductility out−of−planeIncresing\nductility\nin−plane●\n●●●●\n●●\n●●\n●●●●●●\n●\n●●●●\n●●●\n●●●●\n●●●\n●●●\n●\n100200300400500Ein−plane [GPa]e) All phase prototypes and elements\nIncreasing stiffness out−of−planeIncresing\nstiffness\nin−plane\n●●\n●●\n●●\n●●●●\n●\n−100−50050100150CPin−plane [GPa]b)\n(Cr,Al)(Cr,Si)\n(Hf,Al)(Mo,Si)\n(Nb,Al)(Ta,Al)\n(Ti,Al)(Ti,Si)(V,Al)(V,Si)\n(Zr,Al)M3AB4, M!={Mn,Tc} & A={Al,Si}\n●●\n●●\n●●●\n●●\n●\n●\n100200300400500Ein−plane [GPa]f)\n(Cr,Al)(Cr,Si)\n(Hf,Al)\n(Mo,Si)(Nb,Al)\n(Ta,Al)(Ti,Al)(Ti,Si)(V,Al)\n(V,Si)\n(Zr,Al)M3AB4, M!={Mn,Tc} & A={Al,Si}\n−100−50050100150CPin−plane [GPa]c)\n(Cr,Al)(Cr,Si)\n(Hf,Al)(Mo,Al)(Re,Al)\n(Ta,Al)\n(Ti,Al)(Ti,Si)(V,Al)(W,Al)\n(Zr,Al)M2AB2, M!={Mn,Tc} & A={Al,Si}\n100200300400500Ein−plane [GPa]g)\n(Cr,Al)(Cr,Si)\n(Hf,Al)(Mo,Al)(Re,Al)\n(Ta,Al)(Ti,Al)\n(Ti,Si)(V,Al)(W,Al)\n(Zr,Al)M2AB2, M!={Mn,Tc} & A={Al,Si}\n−100−50050100150\n−100−50050100150CPin−plane [GPa]\nCPout−of−plane [GPa]d)\n(Cr,Al)(Cr,Si)\n(Mo,Al)(Mo,Si)\n(Nb,Al)\n(Nb,Si)(Re,Al)\n(Ta,Al)(Ta,Si)(Ti,Al)\n(Ti,Si)(V,Al)\n(V,Si)(W,Al)(Zr,Si)MAB, M!={Mn,Tc} & A={Al,Si}\n100200300400500\n100200300400500Ein−plane [GPa]\nEout−of−plane [GPa]h)\n(Cr,Al)(Cr,Si)(Mo,Al) (Mo,Si)\n(Nb,Al)(Nb,Si)\n(Re,Al) (Ta,Al)(Ta,Si)\n(Ti,Al)(Ti,Si)\n(V,Al)(V,Si)\n(W,Al)\n(Zr,Si)MAB, M!={Mn,Tc} & A={Al,Si}Figure 6. Trends in theoretical in-plane vs. out-of-plane ductility (a–d) and stiffness (e–h) of MAB phases estimated via\nelastic-constants-based descriptors : directional Cauchy pressure ( CPin-plane =CP∥,CPout-of-plane =CP⊥) and directional Young’s modulus\n(Ein-plane =E∥,Eout-of-plane =E⊥). The dashed diagonal lines guide the eye for the case of equal in-plane and out-of-plane values. Panels (a)\nand (e) show results for all stable phases (marked by colour in Fig. 2), while panels (b–c) and (f–h) focus on the most common phase\nprototypes and elements (excluding Mn, Tc, Ga, Ge, In).\n103.4 Suggestions for the most promising MABs\nIn Fig. 7 we present M–A–B systems suitable for the develop-\nment of MAB phases with favourable combination of stiffness\nand ductility. These are identified based on Cauchy pressure\nand Young’s modulus values of all mechanically and dynam-\nically stable MABs, weighted according to their formation\nenergy difference from the energetically most stable phase.\nConsequently, the following elemental combinations are\nproposed:\n•A=AlandM={Nb, Mo, W, Mn, (Tc), Re} (Fig. 7a).\nThe most energetically stable compounds—listed in\nTab. 1—are Nb 3AlB 4, Mo 2AlB 2, MoAlB, W 2AlB 2,\nWAlB, Mn 2AlB 2,γ-Re 2AlB 2, and ω′-Re 2AlB 2(Tab. 1).\nBulk WAlB9, 15, bulk Mn 2AlB 215, 17, and thin film and\nbulk MoAlB9, 14, 18–20have already been synthesised.\nThe experimental Young’s modulus of MoAlB19,E=\n379±30GPa, compares well with our DFT value,\nE=346GPa (Tab. 1). Furthermore, the here-predicted\nMo 2AlB 2was observed in HRTEM after topochemical\ndeintercalation of Al from the single crystalline MoAlB4.\nWith both Cauchy pressures positive ( CP∥≈36GPa,\nCP⊥≈16GPa) and the lowest G/B(≈0.53), W 2AlB 2\nstands out in terms of theoretical ductility. Mn- and Re-\ncontaining MABs exhibit the highest elastic stiffness.\nMn2AlB 2is significantly stiffer in-plane, E∥≈419GPa,\nE⊥≈292GPa (comparable to room-temperature ex-\nperimental value of 243 GPa17), whereas γ-Re 2AlB 2is\nnotably more isotropic, E∥≈373 GPa, E⊥≈371 GPa).\n•A=SiandM={V, Nb, Ta, Cr, Mo, W, Mn, (Tc),\n(Re)} (Fig. 7b). The most energetically stable com-\npounds are α-V2SiB 2, V3SiB 4,α-Nb 2SiB 2, Ta 3SiB 2,\nα-Ta2SiB 2, Ta 2SiB, Cr 2SiB 2, Cr 3SiB 4,α-Cr2SiB 2,α-\nMo 2SiB 2,ω′-W2SiB 2,γ-W2SiB 2, W 2SiB, α-W2SiB 2,\nW3SiB 3, Mn 2SiB 2,α-Mn 2SiB 2,γ-Mn 2SiB 2, and ω′-\nMn 2SiB 2(Tab. 1). Re exhibits many competing phases.\nContrarily to Al-containing MABs, their Si counterparts\nare experimentally mostly unexplored. Considering out-\nstanding oxidation properties of Si-alloyed boride ceram-\nics73–75, we envision that also Si-containing MABs will\nattract attention.\nAs suggested by Cauchy pressure and Poisson’s ratio\nvalues in Tab. 1, Si-based MABs are slightly less brittle\nthan Al-based MABs. With the 2:1:2 M:A:B chemistry,\nW exhibits three energetically- and volumetrically-close\nphase prototypes ( ∆Ef=0.001–0.045 eV/at., ∆V=0–\n3.4%) providing a basis to optimise mechanical proper-\nties. Similarly, Mn offers four energetically-close phases\nof the 2:1:2 chemistry. The α-Mn 2SiB 2exhibits the high-\nest ductility indicators among all Si-containing MABs\n(CP∥≈102GPa, CP⊥≈27GPa, G/B≈0.4,ν≈0.32),\nbut rather low Young’s moduli ( <300 GPa).\n•A=GaandM={V, Cr, W} (Fig. 7c). The most en-\nergetically stable compounds are α-V2GaB 2, V3GaB 4,Cr3GaB 4,α-Cr2GaB 2, Cr 2GaB 2,ω′-W2GaB 2, and γ-\nW2GaB 2(Tab. 1). Although no Ga-containing MABs\nhave been reported, several Ga-containing MAX phases\n(Ti2GaC, Ti 4GaC 3, Cr 2GaC) have been explored76, 77.\nThe here-suggested Ga-based MABs are significantly\nYoung’s-modulus- and Cauchy-pressure anisotropic. For\nillustration, W-containing MABs show CP⊥≥110GPa\nandCP∥≤ −56 GPa.\n•A=GeandM={V, Mn, (Re)} (Fig. 7d). The most\nenergetically stable compounds are α-V2GeB 2andα-\nMn2GeB 2(Tab. 1). Re exhibits many competing phases.\nSimilar to Ge, also Ge-based MAB phases are currently\na theoretical concept. Nonetheless, there are reports\non Ge-based MAX-phase thin films (Ti 2GeC, Ti 3GeC 2,\nTi4GeC 3, Cr 2GeC78, 79), which may also inspire the de-\nvelopment Ge-containing MABs.\nThe here-suggested Ge-based MABs have generally\nrather low bulk moduli. The α-Mn 2GeB 2stand out in\nterms of theoretical ductility ( CP∥≈84GPa, CP⊥≈\n45 GPa, G/B≈0.39, and ν≈0.33).\n4 Summary and conclusions\nHigh-throughput ab initio calculations were employed to fa-\ncilitate rational material selection for the synthesis of novel\nternary borides with MAB-phase structures. The most promis-\ning MAB candidates were identified based on the predicted\nphase stability trends, energetics and volumetric proximity of\ncompeting MABs, as well as elastic-constants-based indica-\ntors of intrinsic stiffness and ductility parallel and normal to\nthe metal/ceramic layers.\nThe searched chemical and phase space contained all com-\nbinations of the group 4–7 transition metals (M elements) and\nAl, Si, Ga, Ge or In (A elements), with 10 possible phase\nprototypes for each elemental combination. Representing the\n1:1:1, 2:1:1, 2:1:2, 3:1:2, 3:1:3, and 3:1:4 M:A:B ratios, the\nprototypes included experimentally known MAB and MAX\nphases, or were inspired by typical transition metal (di)boride\nstructures interlayered with an A layer. The main predictions\nare as follows:\n1.The MABs’ formation energy typically increases (indi-\ncating lower chemical stability) for M from the group\n4→5→6→7. The group of M also most notably influ-\nences the preferred phase prototype and the number of\nenergetically- and volumetrically-close MABs, which is\nthe highest for the group 5–6 transition metals. Com-\npared to M, the A element’s effect is lower, most often\ndecreasing stability as Al →Si→Ga→Ge→In, where In-\ncontaining MABs are also the least dense.\n[Supporting data and discussion in Sec. 3.1 ]\n2.Consistently with qualitative analysis of the electronic\ndensity of states, the M element significantly influences\nelastic properties. The strongest trend observed is a\n11●\n●\n●●\n●\n●●\n●\n●●\n●\n●M element in M−A−B\nTi\nZr\nHfV\nNb\nTaCr\nMo\nWMn\nTc\nRe\nSymbol size scales\ninversely with SPI:\nlarger −> tendency to form\na single−phase MABa) A=Al\n●●●●●●\n●●●\n●●●\n●●●●●●\n●●●\n●●●\n00.20.40.60.81\n00.20.40.60.81\nRelative stiffness indicatorRelative ductility indicatorM=NbM=MoM=W\nM=MnM=TcM=Re\nDuctility &\nstiffness\nincreaseSelectedb) A=Si\n●●●●●●●●\n●●\n●\n●\n●●●●●●●●\n●●\n●\n●\n00.20.40.60.81\n00.20.40.60.81\nRelative stiffness indicatorRelative ductility indicatorM=VM=NbM=Ta\nM=CrM=MoM=WM=Mn\nM=Tc\nM=Re\nc) A=Ga\n●●\n●●●●\n●●\n●●●\n●●\n●●●●\n●●\n●●●\n00.20.40.60.81\n00.20.40.60.81\nRelative stiffness indicatorRelative ductility indicatorM=VM=CrM=Wd) A=Ge\n●●●●●●●●●\n●●\n●\n●●●●●●●●●\n●●\n●\n00.20.40.60.81\n00.20.40.60.81\nRelative stiffness indicatorRelative ductility indicatorM=VM=MnM=Ree) A=In\n●●●●●●●●●●●\n●●●●●●●●●●●\n00.20.40.60.81\n00.20.40.60.81\nRelative stiffness indicatorRelative ductility indicatorFigure 7. Suggestion of suitable M–A–B systems for the development of MAB phases with favourable combination of stiffness and\nductility. The relative stiffness (ductility) indicator is calculated as a weighted average of the polycrystalline Young’s moduli, E, (effective\nCauchy pressures, CPeff) of all stable MABs in the respective M–A–B system (identified in Fig. 2). The weights are based on the formation\nenergy difference from the lowest-energy phase—having a weight of 1—and the values are then normalised with respect to the global maxima\n(the Mn–Si–B and Re–Si–B system for ductility and stiffness, respectively). The symbol size scales with the single-phase indicator (SPI,\nEq. (4)): systems that tend to form single-phase MABs are depicted by larger symbols, while systems with energetically-close competing\nphases are depicted by smaller symbols. The dashed lines guide guide the eye for top 60%-ranking material systems (top-right corner).\nductility increase for M from the group 4 →5→6→7. Al-\nand Si-containing MABs typically possess the highest\nYoung’s moduli. The energetically most stable phases\ntend to show the lowest degree of elastic anisotropy,\ncomparable to that of transition metal diborides.\n[Supporting data and discussion in Sec. 3.2 and 3.3 ]\n3.The suggested most promising MAB candidates combine\ngroup 5–6 transition metals and Al or Si, most often with\nthe 2:1:2 chemistry. Based on the Cauchy pressures\nand Poisson’s ratio, Si-based MABs are predicted to be\nslightly less brittle. Among them, W 2SiB 2and Mn 2SiB 2\nexhibit energetically- and volumetrically-close phases of\nthe same chemistry, possibly facilitating transformation\nplasticity upon loading.\n[Supporting data and discussion in Sec. 3.4 ]\nIn projection, our study may guide experimental develop-\nment of laminated borides with optimised structure–property\nrelationships. Additionally, the here-produced coherent and\naccurate ab initio dataset can serve to train machine-learning\nmodels (e.g., for formation energy and elastic constants pre-\ndictions) or to fit machine-learning interatomic potentials.Possible next steps on the computational side include (i) cal-\nculations of decomposition energies with respect to non-MAB\ncompounds (e.g. intermetallics), (ii) the impact of point de-\nfects on phase stability and elastic properties, (iii) in-depth\nstudies on Cr 2SiB 2and Mn 2SiB 2(which we suggested as\npromising but treated as non-magnetic), and (iv) transforma-\ntion pathways between specific prototypes (e.g. for various\n2:1:2 types, to assess the possibility of transformation plastic-\nity under mechanical loads as suggested here).\n12Table 1. Properties of the most energetically favourable MABs for promising (M, A) combinations (as identified in Fig. 7). The\ncolumns “M”, “A”, “Phase”, ∆Ef, and∆Vspecify the elements, M and A, the phase prototype, the formation energy and volume difference\nfrom the most stable phase in the respective M–A–B system (phases with ∆Ef<0.05 eV/at. are shown). Furthermore, the universal\nanistotropy index ( AU70), the polycrystalline bulk, shear, and Young’s modulus ( B,G,E), the in-plane and out-of-plane Young’s modulus ( E∥,\nE⊥), the effective Cauchy pressure ( CPeff), the in-plane and out-of-plane Cauchy pressure ( CP∥,CP⊥), the B/Gratio, and the Poisson’s ratio\n(ν) are presented.\nM A Phase ∆Ef ∆V AUB G E E∥E⊥CPeff CP∥CP⊥G/B ν\n[eV/at.] [%] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa] [GPa]\nNb Al M 3AB4 0 0 0.21 227 127 322 351 228 8 37 −49 0.56 0.26\nAl α-M2AB2 0.011 −15.6 0.03 182 130 315 297 268 −36 −42 −23 0.71 0.21\nMo Al M 2AB2 0 0 0.06 237 134 338 379 284 13 25 −9 0.56 0.26\nAl MAB 0.006 −4.6 0.16 211 141 346 276 252 −32 −45 −6 0.67 0.23\nW Al MAB 0 0 0.18 229 146 361 269 276 −23 −41 12 0.64 0.24\nAl M 2AB2 0.02 4 0.06 260 138 352 375 287 30 36 16 0.53 0.27\nMn Al M 2AB2 0 0 0.09 237 166 404 419 292 −45 −17 −102 0.70 0.22\nRe Al γ-M2AB2 0 0 0.02 255 162 402 373 371 −17 −23 −5 0.64 0.24\nAl ω′-M2AB2 0.002 0 0.02 253 159 394 373 362 −12 −19 1 0.63 0.24\nV Si α-M2AB2 0 0 0.09 218 156 379 338 310 −48 −70 −4 0.72 0.21\nSi M 3AB4 0.003 12.1 0.26 256 136 346 390 448 18 50 −44 0.53 0.28\nNb Si α-M2AB2 0 0 0.23 223 136 339 268 279 −15 −45 45 0.61 0.25\nTa Si M 3AX 2 0 0 0.36 217 115 294 216 211 10 −15 60 0.53 0.27\nSi α-M2AB2 0.006 12 0.36 239 133 336 243 297 0 −43 87 0.56 0.27\nSi M 2AX 0.044 −5.5 0.92 205 89 233 131 206 27 −23 129 0.43 0.31\nCr Si M 2AB2 0 0 0.15 269 143 365 402 472 23 52 −32 0.53 0.27\nSi M 3AB4 0.022 6 0.13 279 151 383 424 478 22 45 −23 0.54 0.27\nSi α-M2AB2 0.045 −7.1 0.11 237 152 375 351 256 −22 −27 −14 0.64 0.24\nMo Si α-M2AB2 0 0 0.09 253 147 369 324 296 4 −8 27 0.58 0.26\nW Si ω′-M2AB2 0 0 0.32 264 151 381 284 474 −3 −51 94 0.57 0.26\nSi γ-M2AB2 0.001 0 0.37 264 147 373 255 486 1 −54 111 0.56 0.26\nSi M 2AX 0.038 −13.3 0.62 247 128 328 223 171 6 −15 49 0.52 0.28\nSi α-M2AB2 0.045 3.4 0.09 273 147 373 321 318 24 11 49 0.54 0.27\nSi M 3AB3 0.049 −5.1 0.25 213 114 290 304 259 12 −1 39 0.54 0.27\nMn Si M 2AB2 0 0 0.26 282 122 320 369 281 68 123 −42 0.43 0.31\nSi α-M2AB2 0.033 −8.7 0.11 237 94 249 292 250 77 102 27 0.40 0.32\nSi γ-M2AB2 0.045 −9.6 0.14 237 158 388 306 366 −32 −49 0 0.67 0.23\nSi ω′-M2AB2 0.048 −9.6 0.2 236 148 366 284 316 −19 −35 13 0.63 0.24\nV Ga α-M2AB2 0 0 0.02 183 146 346 367 295 −61 −56 −70 0.8 0.18\nGa M 3AB4 0.011 13.5 0.14 223 143 353 387 261 −20 10 −82 0.64 0.24\nCr Ga M 3AB4 0 0 0.09 243 132 335 398 314 20 42 −22 0.54 0.27\nGa α-M2AB2 0.028 −14.9 0.05 202 148 357 381 265 −47 −33 −74 0.73 0.2\nGa M 2AB2 0.032 −9 0.15 224 119 304 372 268 20 47 −34 0.53 0.27\nW Ga ω′-M2AB2 0 0 0.24 233 131 331 255 410 3 −56 121 0.56 0.26\nGa γ-M2AB2 0 0.1 0.23 234 135 339 255 411 −2 −59 110 0.58 0.26\nV Ge α-M2AB2 0 0 0.05 204 144 349 328 289 −38 −55 -4 0.70 0.22\nMn Ge α-M2AB2 0 0 0.17 218 85 226 283 259 71 84 45 0.39 0.33\nGe M 3AB3 0.048 −1.7 0.46 169 89 226 301 192 8 −3 29 0.53 0.28\nCRediT authorship contribution statement\nNK: Conceptualisation, Data curation, Formal analysis, In-\nvestigation, Methodology, Visualisation, Writing – original\ndraft. LH: Resources, Writing – review & editing. PHM : Re-\nsources, Writing – review & editing. DGS : Conceptualisation,\nMethodology, Resources, Writing – review & editing.\nDeclaration of Competing Interests\nThe authors declare no competing interests.\nData Availability\nThe data presented in this study are available from the corre-\nsponding author upon reasonable request.Acknowledgements\nNK acknowledges the Austrian Science Fund, FWF, (T-1308).\nLH acknowledges financial support from the Swedish Govern-\nment Strategic Research Area in Materials Science on Func-\ntional Materials at Linköping University SFO-Mat-LiU No.\n2009 00971. Support from Knut and Alice Wallenberg Foun-\ndation Scholar Grants KAW2016.0358 and KAW2019.0290\nis also acknowledged by LH. DGS acknowledges financial\nsupport from the Swedish Research Council (VR) through\nGrant N ºVR-2021-04426 and the Competence Center Func-\ntional Nanoscale Materials (FunMat-II) (Vinnova Grant No.\n2022-03071). The computations handling were enabled by\nresources provided by the National Academic Infrastructure\nfor Supercomputing in Sweden (NAISS) and the Swedish\n13National Infrastructure for Computing (SNIC) at the National\nSupercomputer Center (NSC) partially funded by the Swedish\nResearch Council through grant agreements no. 2022-06725\nand no. 2018-05973, as well as by the Vienna Scientific Clus-\nter (VSC) in Austria.\nReferences\n1.Kota, S., Sokol, M. & Barsoum, M. W. A progress re-\nport on the MAB phases: atomically laminated, ternary\ntransition metal borides. Int. Mater. Rev. 65, 226–255\n(2020).\n2.Wang, J. et al. Discovery of hexagonal ternary phase\nTi2InB 2and its evolution to layered boride TiB. Nat.\nComm. 10, 2284 (2019).\n3.Aronsson, B., Engström, I., Åselius, J., Refn, S. & Westin,\nG. X-ray Investigations on Me-Si-B Systems (Me =Mn,\nFe, Co). Acta Chem. Scand. 14(1960).\n4.Alameda, L. T., Moradifar, P., Metzger, Z. P., Alem, N. &\nSchaak, R. E. Topochemical deintercalation of Al from\nMoAlB: stepwise etching pathway, layered intergrowth\nstructures, and two-dimensional MBene. J. Am. Chem.\nSoc.140, 8833–8840 (2018).\n5.Zhou, J. et al. Boridene: Two-dimensional Mo 4/3B2−x\nwith ordered metal vacancies obtained by chemical exfo-\nliation. Science 373, 801–805 (2021).\n6.Yang, X., Shang, C., Zhou, S. & Zhao, J. MBenes:\nemerging 2D materials as efficient electrocatalysts for\nthe nitrogen reduction reaction. Nanoscale Horizons 5,\n1106–1115 (2020).\n7.Siriwardane, E., Joshi, R. P., Kumar, N. & Çakır,\nD. Revealing the formation energy–exfoliation energy–\nstructure correlation of MAB phases using machine learn-\ning and DFT. ACS Appl. Mater. & interfaces 12, 29424–\n29431 (2020).\n8.Rosenkranz, A., Zambrano, D., Przyborowski, A., Shah,\nR. & Jastrz˛ ebska, A. M. Mab-phases and beyond—a\ntribological success story? Adv. Mater. Interfaces 9,\n2200869 (2022).\n9.Zhang, D. et al. Experimental and theoretical inves-\ntigation of the damage evolution of irradiated MoAlB\nand WAlB MAB phases. J. Alloy. Compd. 942, 169099\n(2023).\n10.Carlsson, A., Rosen, J. & Dahlqvist, M. Theoretical\npredictions of phase stability for orthorhombic and hexag-\nonal ternary MAB phases. Phys. Chem. Chem. Phys. 24,\n11249–11258 (2022).\n11.Berastegui, P., Riekehr, L. & Jansson, U. Magnetron\nsputtering of nanolaminated Cr 2AlB 2.Coatings 10, 735\n(2020).\n12.Kota, S. et al. Magnetic properties of Cr 2AlB 2, Cr 3AlB 4,\nand CrB powders. J. Alloy. Compd. 767, 474–482 (2018).13.Ade, M. & Hillebrecht, H. Ternary borides Cr 2AlB 2,\nCr3AlB 4, and Cr 4AlB 6: The first members of the series\n(CrB 2)nCrAl with n=1, 2, 3 and a unifying concept\nfor ternary borides as MAB-phases. Inorg. chemistry 54,\n6122–6135 (2015).\n14.Chen, Y ., Kota, S., Barsoum, M. W. & Radovic, M. Com-\npressive deformation of MoAlB up to 1100◦C.J. Alloy.\nCompd. 774, 1216–1222 (2019).\n15.Roy, C., Mondal, S., Banerjee, P. & Bhattacharyya, S.\nLow temperature atmospheric synthesis of WAlB and\nMn 2AlB 2mab phases by modified molten salt shielded\nsynthesis method. Adv. Powder Technol. 34, 103983\n(2023).\n16.Liu, J. et al. Rapid synthesis and characterization of a\nnanolaminated Fe 2AlB 2compound. J. Alloy. Compd.\n766, 488–497 (2018).\n17.Kota, S. et al. Synthesis and characterization of the\natomic laminate Mn 2AlB 2.J. Eur. Ceram. Soc. 38, 5333–\n5340 (2018).\n18.Achenbach, J.-O. et al. Synthesis and properties of or-\nthorhombic MoAlB coatings. Coatings 9, 510 (2019).\n19.Evertz, S., Pöllmann, P., Holzapfel, D. M., Mayer, E. &\nSchneider, J. M. Low temperature synthesis of dense\nMoAlB thin films. J. Eur. Ceram. Soc. 41, 6302–6308\n(2021).\n20.Sahu, R., Bogdanovski, D., Achenbach, J.-O., Schneider,\nJ. M. & Scheu, C. Defects in an orthorhombic MoAlB\nMAB phase thin film grown at moderate synthesis tem-\nperature. Nanoscale 14, 2578–2585 (2022).\n21.Dahlqvist, M. et al. Theoretical prediction and synthesis\nof a family of atomic laminate metal borides with in-plane\nchemical ordering. J. Am. Chem. Soc. 142, 18583–18591\n(2020).\n22.Khazaei, M. et al. Novel MAB phases and insights into\ntheir exfoliation into 2D MBenes. Nanoscale 11, 11305–\n11314 (2019).\n23.Shen, C. et al. Designing of magnetic mab phases for\nenergy applications. J. Mater. Chem. A 9, 8805–8813\n(2021).\n24.Dahlqvist, M. & Rosen, J. Predictions of attainable com-\npositions of layered quaternary i-mab phases and solid so-\nlution mab phases. Nanoscale 13, 18311–18321 (2021).\n25.Dahlqvist, M. & Rosen, J. Chemical order or disorder–a\ntheoretical stability expose for expanding the composi-\ntional space of quaternary metal borides. Mater. Adv. 3,\n2908–2917 (2022).\n26.Sun, Y . et al. Accelerating the discovery of transition\nmetal borides by machine learning on small data sets.\nACS Appl. Mater. & Interfaces (2023).\n27.Li, S. et al. High-throughput study and machine learning\non max and mab phases: new materials and fingerprints\n14of superior lattice thermal conductivities. Acta Materialia\n254, 119001 (2023).\n28.Lind, H., Dahlqvist, M. & Rosen, J. In-plane ordered\nquaternary phases (i-mab): electronic structure and me-\nchanical properties from first-principles calculations. J.\nPhysics: Condens. Matter 33, 255402 (2021).\n29.Liu, Y ., Jiang, Z., Jiang, X. & Zhao, J. New refractory\nmab phases and their 2d derivatives: insight into the\neffects of valence electron concentration and chemical\ncomposition. RSC advances 10, 25836–25847 (2020).\n30.Qi, X. et al. Stability trend, weak bonding, and magnetic\nproperties of the al-and si-containing ternary-layered\nborides mab phases. J. Am. Ceram. Soc. 106, 1513–1530\n(2023).\n31.Zhou, Y ., Xiang, H., Dai, F.-Z. & Feng, Z. Electrical con-\nductive and damage-tolerant nanolaminated mab phases\ncr2alb2, cr3alb4 and cr4alb6. Mater. Res. Lett. 5, 440–\n448 (2017).\n32.Dai, F.-Z., Feng, Z. & Zhou, Y . First-principles investiga-\ntion on the chemical bonding, elastic properties and ideal\nstrengths of moalb and walb nanolaminated mab phases.\nComput. Mater. Sci. 147, 331–337 (2018).\n33.Dai, F.-Z., Feng, Z. & Zhou, Y . Easily tiltable BAlB\nlinear chain: The origin of unusual mechanical properties\nof nanolaminated MAB phases (CrB 2)nCrAl. J. Alloy.\nCompd. 723, 462–466 (2017).\n34.Moraes, V . et al. Ab initio inspired design of ternary\nboride thin films. Sci. reports 8, 1–9 (2018).\n35.Koutná, N., Brenner, A., Holec, D. & Mayrhofer, P. H.\nHigh-throughput first-principles search for ceramic super-\nlattices with improved ductility and fracture resistance.\nActa Materialia 206, 116615 (2021).\n36.Balasubramanian, K., Khare, S. V . & Gall, D. Valence\nelectron concentration as an indicator for mechanical\nproperties in rocksalt structure nitrides, carbides and car-\nbonitrides. Acta Materialia 152, 175–185 (2018).\n37.Sangiovanni, D. G., Chirita, V . & Hultman, L. Electronic\nmechanism for toughness enhancement in Ti xM1−xN\n(M=Mo and W). Phys. Rev. B 81, 104107 (2010).\n38.Gu, X., Liu, C., Guo, H., Zhang, K. & Chen, C. Sorting\ntransition-metal diborides: New descriptor for mechani-\ncal properties. Acta Materialia 207, 116685 (2021).\n39.Li, N. et al. Rapid synthesis, electrical, and mechanical\nproperties of polycrystalline Fe 2AlB 2bulk from elemen-\ntal powders. J. Am. Ceram. Soc. 100, 4407–4411 (2017).\n40.Fuger, C. et al. Tissue phase affected fracture toughness\nof nano-columnar TiB 2+zthin films. Mater. Res. Lett. 11,\n613–622 (2023).\n41.Fuger, C. et al. Influence of Tantalum on phase stability\nand mechanical properties of WB 2.MRS Commun. 9,\n375–380 (2019).42.Hahn, R. et al. Unraveling the superlattice effect for\nhexagonal transition metal diboride coatings. Scripta\nMaterialia 235, 115599 (2023).\n43.Lu, X. et al. Crack healing behavior of a MAB phase:\nMoAlB. J. Eur. Ceram. Soc. 39, 4023–4028 (2019).\n44.Lu, X., Li, S., Zhang, W., Yu, W. & Zhou, Y . Ther-\nmal shock behavior of a nanolaminated ternary boride:\nMoAlB. Ceram. Int. 45, 9386–9389 (2019).\n45.Bai, Y . et al. High-temperature mechanical properties and\nthermal shock behavior of ternary-layered MAB phases\nFe2AlB 2.Int. J. Refract. Met. Hard Mater. 80, 151–160\n(2019).\n46.Sokol, M., Natu, V ., Kota, S. & Barsoum, M. W. On the\nchemical diversity of the MAX phases. Trends Chem. 1,\n210–223 (2019).\n47.Kresse, G. & Furthmüller, J. Efficient iterative schemes\nforab initio total-energy calculations using a plane-wave\nbasis set. Phys. Rev. B 54, 11169–11186, DOI: 10.1103/\nPhysRevB.54.11169 (1996).\n48.Kresse, G. & Joubert, D. From ultrasoft pseudopotentials\nto the projector augmented-wave method. Phys. Rev. B 59,\n1758–1775, DOI: 10.1103/PhysRevB.59.1758 (1999).\n49.Kohn, W. & Sham, L. J. Self-Consistent Equations In-\ncluding Exchange and Correlation Effects. Phys. Rev.\n140, A1133–A1138, DOI: 10.1103/PhysRev.140.A1133\n(1965).\n50.Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gra-\ndient Approximation Made Simple. Phys. Rev. Lett. 77,\n3865–3868, DOI: 10.1103/PhysRevLett.77.3865 (1996).\n51.Aydin, S. 211-MAX borides: The stable boron-\nsubstituted 211-max compounds by first-principles.\nMater. Today Commun. 25, 101689 (2020).\n52.Magnuson, M., Hultman, L. & Högberg, H. Review of\ntransition-metal diboride thin films. Vacuum 196, 110567,\nDOI: 10.1016/j.vacuum.2021.110567 (2022).\n53.Leiner, T. et al. On energetics of allotrope transformations\nin transition-metal diborides via plane-by-plane shearing.\nVacuum 215, 112329 (2023).\n54.Panda, K. & Chandran, K. R. First principles determina-\ntion of elastic constants and chemical bonding of titanium\nboride (TiB) on the basis of density functional theory.\nActa Materialia 54, 1641–1657 (2006).\n55.Jain, A. et al. Commentary: The Materials Project: A\nmaterials genome approach to accelerating materials in-\nnovation. APL Mater. 1, 011002 (2013).\n56.Le Page, Y . & Saxe, P. Symmetry-general least-squares\nextraction of elastic data for strained materials from ab\ninitio calculations of stress. Phys. Rev. B 65, 104104\n(2002).\n1557.Le P., Y . & Saxe, P. Symmetry-general least-squares ex-\ntraction of elastic coefficients from ab initio total energy\ncalculations. Phys. Rev. B 63, 174103 (2001).\n58.Yu, R., Zhu, J. & Ye, H. Q. Calculations of single-crystal\nelastic constants made simple. Comput. Phys. Commun.\n181, 671–675, DOI: 10.1016/j.cpc.2009.11.017 (2010).\n59.Mouhat, F. & Coudert, F.-X. Necessary and sufficient\nelastic stability conditions in various crystal systems.\nPhys. Rev. B 90, 224104 (2014).\n60.Togo, A., Oba, F. & Tanaka, I. First-principles calcu-\nlations of the ferroelastic transition between rutile-type\nand CaCl 2-type SiO 2at high pressures. Phys. Rev. B 78,\n134106 (2008).\n61.Nye, J. F. Physical properties of crystals: their represen-\ntation by tensors and matrices (Oxford University Press,\n1985).\n62.Hill, R. The elastic behaviour of a crystalline aggregate.\nProc. Phys. Soc. Sect. A 65, 349 (1952).\n63.Wildung, R., McFadden, K. & Garland, T. Technetium\nsources and behavior in the environment. J. Environ.\nQual. 8, 156–161 (1979).\n64.Stampfl, C. & Freeman, A. J. Metallic to insulating nature\nof TaN x: Role of Ta and N vacancies. Phys. Rev. B 67,\n064108 (2003).\n65.Sarker, P. et al. High-entropy high-hardness metal car-\nbides discovered by entropy descriptors. Nat. Comm. 9,\n4980 (2018).\n66.Kaufmann, K. et al. Discovery of high-entropy ceramics\nvia machine learning. Npj Comput. Mater. 6, 42 (2020).\n67.Sangiovanni, D. G., Tasnádi, F., Johnson, L. J. S.,\nOdén, M. & Abrikosov, I. A. Strength, transformation\ntoughening, and fracture dynamics of rocksalt-structure\nTi1−xAlxN (0≤x≤0.75) alloys. Phys. Rev. Mater. 4,\n033605 (2020).\n68.Kindlund, H., Sangiovanni, D., Petrov, I., Greene, J. E.\n& Hultman, L. A review of the intrinsic ductility and\ntoughness of hard transition-metal nitride alloy thin films.\nThin Solid Films 688, 137479 (2019).\n69.Kretschmer, A. et al. High-entropy alloy inspired de-\nvelopment of compositionally complex superhard (Hf,\nTa, Ti, V, Zr)-BN coatings. Mater. & Des. 218, 110695\n(2022).\n70.Ranganathan, S. I. & Ostoja-Starzewski, M. Universal\nelastic anisotropy index. Phys. Rev. Lett. 101, 055504\n(2008).\n71.Pettifor, D. G. Theoretical predictions of structure and\nrelated properties of intermetallics. Mater. Sci. Technol.\n8, 345–349 (1992).\n72.Pugh, S. F. Relations between the elastic moduli and\nthe plastic properties of polycrystalline pure metals. TheLondon, Edinburgh, Dublin Philos. Mag. J. Sci. 45, 823–\n843 (1954).\n73.Aschauer, E. et al. Ultra-high oxidation resistance of\nnano-structured thin films. Mater. & Des. 201, 109499\n(2021).\n74.Kiryukhantsev-Korneev, P. V ., Sytchenko, A., Potanin,\nA. Y ., V orotilo, S. & Levashov, E. Mechanical proper-\nties and oxidation resistance of Mo–Si–B and Mo–Hf–\nSi–B coatings obtained by magnetron sputtering in DC\nand pulsed dc modes. Surf. Coat. Technol. 403, 126373\n(2020).\n75.Glechner, T. et al. Influence of Si on the oxidation be-\nhavior of TM-Si-B 2±zcoatings (TM =Ti, Cr, Hf, Ta, W).\nSurf. Coat. Technol. 434, 128178 (2022).\n76.Etzkorn, J., Ade, M., Kotzott, D., Kleczek, M. & Hille-\nbrecht, H. Ti 2GaC, Ti 4GaC 3and Cr 2GaC—synthesis,\ncrystal growth and structure analysis of Ga-containing\nMAX-phases M n+1GaC nwith M =Ti, Cr and n=1, 3. J.\nSolid State Chem. 182, 995–1002 (2009).\n77.Siebert, J. P., Hajra, D., Tongay, S. & Birkel, C. S.\nThe synthesis and electrical transport properties of\ncarbon/Cr 2GaC MAX phase composite microwires.\nNanoscale 14, 744–751 (2022).\n78.Högberg, H., Eklund, P., Emmerlich, J., Birch, J. & Hult-\nman, L. Epitaxial Ti 2GeC, Ti 3GeC 2, and Ti 4GeC 3MAX-\nphase thin films grown by magnetron sputtering. J. Mater.\nRes.20, 779–782 (2005).\n79.Eklund, P. et al. Epitaxial growth and electrical transport\nproperties of Cr 2GeC thin films. Phys. Rev. B 84, 075424\n(2011).\n80.Zhang, Y . et al. The role of Hume-Rothery’s rules play\nin the MAX phases formability. Materialia 12, 100810\n(2020).\n16SUPPLEMENTARY DATA\n•Supplementary figures Fig. S1, Fig. S2, and Fig. S3 complement the discussion of stability trends in Section 3.1 of the\nmain text.\n•Supplementary figures Fig. S4, Fig. S5, Fig. S6, Fig. S7, and Fig. S8 complement the discussion of mechanical properties\nin Section 3.3 of the main text.\ni●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●●\n●●a) M=Ti\n8101214161820Volume [A°3\n/at.]M from Group 4\n●●●\n●d) M=VM from Group 5\n●●●\n●●g) M=CrM from Group 6\n●●●\n●j) M=MnM from Group 7\n●●●b) M=Zr\n8101214161820Volume [A°3\n/at.]\n●●●e) M=Nb\n●●●h) M=Mo\n●k) M=Tc\n●●●c) M=Hf\n8101214161820Volume [A°3\n/at.]\nHigh E f (>Efmin+Efthr), and/or,\nmechanically unst. , and/or,\ndynamically unst.\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S1. Trends in per-atom volume, Vper-at. , for MABs depicted in Fig. 1 in the main text. MABs lying above the Efthreshold\n(Ethr\nf=0.25 eV/at, described in the main text), and/or, mechanically, and/or dynamically unstable MABs are marked by grey colour. All\nMABs marked by colour are mechanically and dynamically stable. The symbol sizes scale with energetic stability quantified by the Ef\ndifference from the lowest- Efphase for a given (M, A) combination.\nii●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●\n●●●a) M=Ti\n0.120.140.160.180.200.22Density [g/cm3]M from Group 4\n●●\n●●d) M=VM from Group 5\n●●\n●●\n●g) M=CrM from Group 6\n●●\n●●j) M=MnM from Group 7\n●●\n●b) M=Zr\n0.120.140.160.180.200.22Density [g/cm3]\n●●\n●e) M=Nb\n●●●h) M=Mo\n●k) M=Tc\n●●\n●c) M=Hf\n0.120.140.160.180.200.22Density [g/cm3]High E f (>Efmin+Efthr), and/or,\nmechanically unst. , and/or,\ndynamically unst.\nA=Al\nA=GaA=In\nA=Si\nA=Ge●\n●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S2. Trends in density, ρ, for MABs depicted in Fig. 1 in the main text. Denoting Vvolume, the density is calculated as ρ=m/V,\nwith m= (mMnM+mAnA+mBnB)/NA, where miandniare the mass and number of atoms of element type i(i={M, A, B}), respectively,\nandNAis the Avogadro number. MABs lying above the Efthreshold ( Ethr\nf=0.25 eV/at, described in the main text), and/or, mechanically,\nand/or dynamically unstable MABs are marked by grey colour. All MABs marked by colour are mechanically and dynamically stable. The\nsymbol sizes scale with energetic stability quantified by the Efdifference from the lowest- Efphase for a given (M, A) combination.\niii●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●\n●●\n●a) M=Ti\n0.000.040.080.120.160.200.240.280.320.36 At. diff. ratioM from Group 4\n●●\n●●\n●d) M=VM from Group 5\n●●●●\n●g) M=CrM from Group 6\n●●●\n●\n●j) M=MnM from Group 7\n●●\n●●\n●b) M=Zr\n0.000.040.080.120.160.200.240.280.320.36 At. diff. ratio●●\n●●\n●e) M=Nb\n●●\n●●\n●h) M=Mo\n●●\n●●\n●k) M=Tc\n●●\n●●\n●c) M=Hf\n0.000.040.080.120.160.200.240.280.320.36 At. diff. ratio\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●\n●●\n●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●\n●●\n●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●\n●●\n●l) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S3. Trends in atomic difference ratio (alternatively called the “size factor”), ∆r, for MABs depicted in Fig. 1 in the main text.\nFollowing Zhang et al.80,∆ris calculated as ∆r=|rM−rA|\nrM, where rM(rA) represent atomic radius of the M (A) element in a MAB phase.\nFrom the definition, ∆r, is the same irrespective of the phase prototype, therefore, all points for a given (M, A) combination overlap.\niv●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●\n●●\n●a) M=Ti\n100150200250300B [GPa]M from Group 4\n●●\n●●d) M=VM from Group 5\n●●\n●●\n●g) M=CrM from Group 6\n●\n●\n●●j) M=MnM from Group 7\n●●\n●b) M=Zr\n100150200250300B [GPa]●●\n●e) M=Nb\n●●\n●h) M=Mo\n●k) M=Tc\n●●\n●c) M=Hf\n100150200250300B [GPa]\nA=Al\nA=GaA=In\nA=Si\nA=Ge●\n●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S4. Polycrystalline bulk modulus, B, calculated for all MABs fulfilling all stability criteria ( Ef“close” above that of the\nlowest-energy phase, mechanical and dynamical stability). The symbol sizes scale with energetic stability quantified by the Efdifference\nfrom the lowest- Efphase for a given (M, A) combination.\nv●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●\n●●\n●a) M=Ti\n50100150200G [GPa]M from Group 4\n●\n●\n●●d) M=VM from Group 5\n●\n●\n●●\n●g) M=CrM from Group 6\n●\n●\n●●j) M=MnM from Group 7\n●●\n●b) M=Zr\n50100150200G [GPa]●\n●●e) M=Nb\n●●●h) M=Mo\n●k) M=Tc\n●●\n●c) M=Hf\n50100150200G [GPa]\nA=Al\nA=GaA=In\nA=Si\nA=Ge●\n●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S5. Polycrystalline shear modulus, G, calculated for all MABs fulfilling our stability criteria ( Ef“close” above that of the\nlowest-energy phase, mechanical and dynamical stability). The symbol sizes scale with energetic stability quantified by the Efdifference\nfrom the lowest- Efphase for a given (M, A) combination.\nvi●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●\n●●\n●a) M=Ti\n50100150200250300350400450500E [GPa]M from Group 4\n●●\n●●d) M=VM from Group 5\n●\n●\n●●\n●g) M=CrM from Group 6\n●\n●\n●●j) M=MnM from Group 7\n●●\n●b) M=Zr\n50100150200250300350400450500E [GPa]●\n●●e) M=Nb\n●●●h) M=Mo\n●k) M=Tc\n●●\n●c) M=Hf\n50100150200250300350400450500E [GPa]\nA=Al\nA=GaA=In\nA=Si\nA=Ge●\n●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S6. Polycrystalline Young’s modulus, E, calculated for all MABs fulfilling our stability criteria ( Ef“close” above that of the\nlowest-energy phase, mechanical and dynamical stability). The symbol sizes scale with energetic stability quantified by the Efdifference\nfrom the lowest- Efphase for a given (M, A) combination.\nvii●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●●●●a) M=Ti\n0.150.200.250.300.350.400.45 νM from Group 4\n●●●●d) M=VM from Group 5\n●●●●●g) M=CrM from Group 6\n●●●\n●j) M=MnM from Group 7\n●●●b) M=Zr\n0.150.200.250.300.350.400.45 ν\n●●\n●e) M=Nb\n●\n●\n●h) M=Mo\n●k) M=Tc\n●●●c) M=Hf\n0.150.200.250.300.350.400.45 ν\nA=Al\nA=GaA=In\nA=Si\nA=Ge●●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S7. Polycrystalline Poisson’s ratio, ν, calculated for all MABs fulfilling our stability criteria ( Ef“close” above that of the\nlowest-energy phase, mechanical and dynamical stability). The symbol sizes scale with energetic stability quantified by the Efdifference\nfrom the lowest- Efphase for a given (M, A) combination.\nviii●314−type MABs\nM3AB4 (o; Pmmm) 212−type MABs\nM2AB2 (o; Pmm2) \nα−M2AB2 (h; P−6m2) \nω−M2AB2 (o; Cmc2 1)\nω'−M2AB2 (o; Cmcm) \nγ−M2AB2 (m; C1m1) 222−type MABs\nMAB (o; Cmcm)Other−type MABs\nM3AB3 (m; P2 1/m)\nM2AB (h; P6 3/mmc) \nM3AB2 (h; P6 3/mmc) \n●●●\n●●a) M=Ti\n0.00.10.20.30.40.50.60.70.80.91.0AUM from Group 4\n●●●\n●d) M=VM from Group 5\n●●●\n●●g) M=CrM from Group 6\n●●●\n●j) M=MnM from Group 7\n●●●\nb) M=Zr\n0.00.10.20.30.40.50.60.70.80.91.0AU\n●●\n●e) M=Nb●\n●\n●h) M=Mo\n●k) M=Tc\n●●●\nc) M=Hf\n0.00.10.20.30.40.50.60.70.80.91.0AU\nA=Al\nA=GaA=In\nA=Si\nA=Ge●f) M=Ta\nA=Al\nA=GaA=In\nA=Si\nA=Ge●i) M=W\nA=Al\nA=GaA=In\nA=Si\nA=Gel) M=Re\nA=Al\nA=GaA=In\nA=Si\nA=GeFigure S8. Universal anisotropy index, AUcalculated for all MABs fulfilling our stability criteria ( Ef“close” above that of the\nlowest-energy phase, mechanical and dynamical stability). The AUis calculated as AU=5GV\nGR−BV\nBR−6(Eq. 9 in Ref.70), where GandBare\nthe shear and bulk moduli and the upper indexes, VandR, denote the V oigt and Reuss estimates. The symbol sizes scale with energetic\nstability quantified by the Efdifference from the lowest- Efphase for a given (M, A) combination.\nix"    },    {        "title": "2402.04728v1.Detection_Schemes_with_Low_Resolution_ADCs_and_Spatial_Oversampling_for_Transmission_with_Higher_Order_Constellations_in_the_Terahertz_Band.pdf",        "content": "1\nDetection Schemes with Low-Resolution ADCs and\nSpatial Oversampling for Transmission with\nHigher-Order Constellations in the Terahertz Band\nChristian Forsch, Graduate Student Member, IEEE, Peter Zillmann, Osama Alrabadi, Stefan Brueck,\nand Wolfgang Gerstacker, Senior Member, IEEE\nAbstract —In this work, we consider Terahertz (THz) commu-\nnications with low-resolution uniform quantization and spatial\noversampling at the receiver side. We compare different analog-\nto-digital converter (ADC) parametrizations in a fair manner by\nkeeping the ADC power consumption constant. Here, 1-, 2-, and\n3-bit quantization is investigated with different oversampling fac-\ntors. We analytically compute the statistics of the detection vari-\nable, and we propose the optimal as well as several suboptimal\ndetection schemes for arbitrary quantization resolutions. Then,\nwe evaluate the symbol error rate (SER) of the different detectors\nfor a 16- and a 64-ary quadrature amplitude modulation (QAM)\nconstellation. The results indicate that there is a noticeable\nperformance degradation of the suboptimal detection schemes\ncompared to the optimal scheme when the constellation size is\nlarger than the number of quantization levels. Furthermore, at\nlow signal-to-noise ratios (SNRs), 1-bit quantization outperforms\n2- and 3-bit quantization, respectively, even when employing\nhigher-order constellations. We confirm our analytical results by\nMonte Carlo simulations. Both a pure line-of-sight (LoS) and a\nmore realistically modeled indoor THz channel are considered.\nThen, we optimize the input signal constellation with respect to\nSER for 1-bit quantization. The results show that the minimum\nSER can be lowered significantly for 16-QAM by increasing\nthe distance between the inner and outer points of the input\nconstellation. For larger constellations, however, the achievable\nreduction of the minimum SER is much smaller compared to\n16-QAM.\nIndex Terms —Low-resolution quantization, oversampling, Ter-\nahertz communications, maximum-likelihood detection, symbol\nerror rate, constellation optimization.\nI. I NTRODUCTION\nFUTURE wireless communication systems are expected to\nprovide ultra-high data rates with extremely low latency\nin order to enable a plethora of new applications [2]. The large\nbandwidths and high symbol rates which are required for such\napplications and which can be realized in the Terahertz (THz)\nband necessitate very high sampling frequencies of analog-to-\ndigital converters (ADCs), causing a high power consumption.\nThis problem can be tackled by decreasing the quantization\nThis article was presented in part at the 2022 IEEE Latin-American\nConference on Communications (LATINCOM) [1]. This work was supported\nby a gift from Qualcomm Technologies, Inc.\nChristian Forsch and Wolfgang Gerstacker are with the Institute for Digital\nCommunications, Friedrich-Alexander-Universit ¨at Erlangen-N ¨urnberg, Erlan-\ngen, Germany (e-mail: christian.forsch@fau.de; wolfgang.gerstacker@fau.de).\nPeter Zillmann, Osama Alrabadi, and Stefan Brueck are\nwith Qualcomm CDMA Technologies, N ¨urnberg, Germany (e-\nmail: pzillman@qti.qualcomm.com; osamaa@qti.qualcomm.com;\nsbrueck@qti.qualcomm.com).resolution of the ADC [3], arriving at low-resolution quan-\ntization such as 1-, 2-, or 3-bit quantization. However, such\nquantization with only few bit limits the spectral efficiency to\n1-3 bit per channel use (bpcu) per real dimension for Nyquist-\nrate sampling [4]. Oversampling provides a remedy to increase\nthe achievable rate by retrieving some of the information\nlost due to quantization [5]–[8]. This can be achieved in the\ntemporal domain by increasing the sampling frequency or in\nthe spatial domain by receiving the useful signal via multiple\nantennas/channels. Therefore, in this work, we consider the\ncase of low-resolution quantization with spatial oversampling\nat the receiver side. Furthermore, we focus on higher-order\nconstellations, implying that the number of quantization in-\ntervals of the ADC is typically smaller than the number of\npossible transmit symbols in our scenarios.\nCommunication systems with oversampled low-resolution\nquantization have been already previously studied in the\nliterature. In particular, the case of 1-bit quantization was\nconsidered frequently in previous works due to its relevance\nand simplicity. In [9], a spread spectrum system with 1-bit\nquantization was considered which has similarities to our\nsystem model. It was shown that binary phase-shift keying\n(PSK) is not optimal anymore with respect to the channel\ncapacity when observing the transmitted symbol multiple\ntimes, i.e., employing oversampling. In [10] and [11], it was\ndemonstrated that oversampling can increase the capacity\nof communication systems with 1-bit quantization to more\nthan 1 bpcu per real dimension. In [12], the results of [10]\nand [11] were extended to a more realistic system model\nwith intersymbol interference (ISI) which is even beneficial\nfor transmitting with higher-order constellations, especially\nat high signal-to-noise ratios (SNRs). In [13], the achievable\nrate for different modulation schemes including higher-order\nmodulation was investigated and the benefit of oversampled\n1-bit quantization was confirmed. The authors of [14] showed\nthat higher-order constellations can be detected with 1-bit\nquantizers in a spatially oversampled system in case the\nnumber of receive antennas is sufficiently high and the SNR\nis appropriate. Thus, these previous works motivate the usage\nof higher-order constellations for 1-bit quantization.\nFurthermore, there are also results on multi-bit quantiza-\ntion. In [15], the achievable rate of massive multiple-input\nmultiple-output (MIMO) systems, which correspond to spatial\noversampling, with 1- and 2-bit quantization was analyzed.\nThe results show that the use of higher-order constellations isarXiv:2402.04728v1  [cs.IT]  7 Feb 20242\nfeasible. In [16], it was also demonstrated that the achievable\nrate can be increased for 2-bit quantization with higher-order\nconstellations such as 64-ary quadrature amplitude modulation\n(QAM), and it was shown that the performance in terms of\nachievable rate for 3-bit quantization is very close to the\nunquantized case.\nIn addition to the achievable rate analysis of low-resolution\nquantization with oversampling, which was mainly conducted\nin the above referenced works, the performance in terms of\nerror rate for higher-order constellations is of interest. In [17],\nthe symbol error rate (SER) for different symbol sources\nwas analyzed for 1-bit quantization with oversampling. The\nresults indicate that 4-ary amplitude-shift keying (ASK) with\nindependent and uniformly distributed symbols results in a\nhigh error floor for the utilized detector. In [18], it was shown\nvia numerical evaluation that the SER for quantization with 1-3\nbit for higher-order constellations can be reduced by inducing\nartificial noise in a maximum ratio combining receiver. The\nperformance for PSK-modulated symbols was investigated\nin [19] for a massive MIMO scenario with 1-bit quantization.\nAccording to the presented SER results, the spatial oversam-\npling enables a reliable detection even for higher-order PSK\nconstellations. There are various further works which deal with\nmassive MIMO systems with low-resolution quantization and\nhigher-order constellations, e.g., [20]–[24]. In these works,\ndifferent detection schemes are developed and their viability\nis shown via Monte Carlo simulations.\nThe above referenced works do not provide any exact\nanalytical results on the SER of higher-order QAM or ASK\nconstellations. Some corresponding results were presented\nin [25] and [26]. Here, the SER for 1-bit quantization with\ndifferent oversampling factors and a 4-ASK constellation was\nanalyzed, and the optimal maximum-likelihood (ML) detec-\ntor was developed. However, multi-bit quantization was not\nstudied in [25] and [26].\nMore recently, low-resolution quantization has been consid-\nered particularly for THz communications. In [27], a transmis-\nsion with zero crossing modulation (ZXM) signals specifically\ntailored to 1-bit quantization and temporal oversampling at the\nreceiver is studied in the sub-THz band. Finite-state machines\nare developed for an efficient demodulation of the ZXM\ntransmit signals. Similarly as in our work, a single-antenna\ntransmitter and a receiver with multiple antennas are assumed\nin a line-of-sight (LoS) scenario. Conventional linear modula-\ntion and higher-order constellations are not considered. In [28],\na deep learning-assisted THz receiver is designed for a single-\ninput single-output (SISO) transmission with 1-bit quantiza-\ntion, temporal oversampling, and THz device imperfections.\nAn optimum ML detector is derived for quadrature PSK, and\nthe device imperfections are combatted via a twin-phase train-\ning strategy and a neural network based demodulator. In [29],\na downlink multi-user indoor THz communication system with\ndistance-aware multi-carrier modulation, an array-of-subarrays\narchitecture with hybrid precoding, and low-resolution digital-\nto-analog converters (DACs) and ADCs is investigated. The\nachievable rate is analyzed, and it is shown via numerical\nresults that with moderate resolution DACs and ADCs with 3-\n5 bit almost the same performance as in the infinite resolutioncase can be obtained. No specific modulations are considered,\nand no error rate analysis is conducted.\nIn this work, based on [1], we extend the results in [25]\nand [26] by providing a more detailed analysis on the 1-bit\nquantization case and considering multi-bit quantization. The\ngeneralization to arbitrary quantization resolution is also an\nextension of our previous work [1]. We consider a simplified\nchannel model as well as a more realistic channel model than\nin our previous work and show that the developed optimal\nsymbol detector performs well also for realistically modeled\nTHz band indoor channels. We compare different quantization\nresolutions in a fair manner by constraining the total ADC\npower consumption to remain constant. Such approach was\nalso adopted in some existing works, e.g., [24] and [30],\nhowever, not for higher-order constellations. Moreover, we\nperform a constellation optimization which was also conducted\nin [4] and [11]. Here, our approach is different since we do\nnot focus on capacity-achieving constellations but construct\nconstellations which achieve a minimum SER for a given con-\nstellation size and SNR. Compared to our previous work [1],\nwe investigate not only 4-ASK but also larger constellations.\nOur contributions can be summarized as follows:\n•We derive the optimal ML detector for an arbitrary\nquantization resolution when performing the detection\nbased on an average of the oversampled observations in\na frequency-flat single-path LoS THz channel.\n•We compare the performance of the optimal ML detector\nto two suboptimal detection schemes. From the SER\nresults, it can be observed that there is a noticeable\nperformance degradation of the suboptimal detectors\nwhen the constellation size is larger than the number of\nquantization levels of the ADC.\n•Analytical SER results are presented which are favorable\nin terms of computation time compared to exhaustive\nMonte Carlo simulations. Hence, SER curves can be\nobtained for a fine SNR grid revealing some interesting\nproperties when employing low-resolution quantization\nin conjunction with higher-order constellations. It can be\nobserved that the SER curves are not smooth and exhibit\nmultiple local minima.\n•We compare the performance of quantization with 1-3\nbit in a fair manner by keeping the total ADC power\nconsumption constant. Thereby, we show that 1-bit quan-\ntization outperforms 2- and 3-bit quantization at low\nSNRs, even for higher-order constellations.\n•We show that our proposed optimal detector also per-\nforms well in a realistic indoor THz channel with multi-\npath propagation and ISI.\n•We optimize the transmit symbol constellation for 1-\nbit quantization with respect to the SER. Here, we gain\nsome interesting insights into optimizing higher-order\nconstellations. In particular, the minimum achievable SER\ncan be decreased by increasing the distance of the outer-\nmost symbols to the remaining inner symbols of a QAM\nconstellation. The optimization yields very high gains in\nterms of the minimum achievable SER.\nThis paper is organized as follows. In Section II, we3\nDetectionVGA\nVGA\nVGA\nFig. 1. SIMO LoS THz channel with analog phase shifters, variable gain\namplifiers, uniform quantizers, linear filter, and detector at the receiver side.\nintroduce and motivate the system model. In Section III, the\nADC power consumption model is presented, and we specify\nADC parametrizations for a prescribed power consumption. In\nSection IV, we compute the statistics of the considered detec-\ntion variable and present corresponding detectors, including\nthe optimal ML detector. In Section V, we present numerical\nresults for the SER of the presented detectors. In Section VI,\nwe optimize the transmit constellation with respect to the\nSER of 1-bit quantization. Some conclusions are drawn in\nSection VII.\nNotation: a,a, andArepresent a scalar, a column vector,\nand a matrix, respectively. aiis the ithelement of the vector a.\n(·)∗and(·)Tdenote the complex conjugate and the transposi-\ntion operation, respectively. diag{·}is a diagonal matrix with\nthe elements in brackets on the main diagonal. The Hadamard\nproduct is given by ⊙.|A|stands for the cardinality of the\nsetA.E{·}is the expectation operator and ⌊·⌋represents the\nfloor function. The set of natural numbers including zero and\nthe set of real numbers are denoted by N0andR, respectively.\n0N,1N, andINrepresent an all-zero column vector of length\nN, an all-ones column vector of length N, and an N×N\nidentity matrix, respectively. R{·} andI{·} return the real\nand imaginary part of the argument, respectively. N(µ,Σ)and\nCN(µ,Σ)denote a real and complex multivariate Gaussian\ndistribution with mean vector µand covariance matrix Σ,\nrespectively. Q(x) =1√\n2πR∞\nxe−t2\n2dtis the Q-function.\nII. S YSTEM MODEL\nWe consider a LoS THz channel with uniform quantiza-\ntion and oversampling at the receiver side. Oversampling is\napplied in the spatial domain, i.e., we consider a single-\ninput multiple-output (SIMO) system with multiple receive\nantennas. The corresponding system model is illustrated in\nFig. 1. Here, the input symbol xc∈ X is chosen from\na complex finite alphabet Xwith cardinality |X| =M\nand power σ2\nx=Exc\b\n|xc|2\t\n.1More specifically, an M-\nary square QAM constellation with equiprobable symbols is\nconsidered. For the following developments, we assume a\nfrequency-flat single-path LoS THz channel which is a valid\nassumption since non-line-of-sight (NLoS) paths are heavily\nattenuated due to the use of directional antennas and the\nadditional reflection and scattering loss in the THz band [31].\nFurthermore, we consider transmission windows which are\n1We use the superscript (·)cto denote complex-valued variables. When\ndealing with real-valued quadrature components later on, we omit this\nsuperscript for better readability.not heavily affected by molecular absorption which renders\nthem practically frequency flat. However, these assumptions\nare discarded in Subsection V-B where we study a realistic\nindoor THz channel, modeled based on ray tracing. The SIMO\nchannel with Nreceive antennas can be described as a vector\nhc= [hc\n1···hc\nN]Twith channel coefficients [32]\nhc\ni=c\n4πfdi·e−1\n2κ(f)di·e−j2πfdi\nc, (1)\nwhere cis the speed of light, fis the operating frequency, di\ndenotes the distance between transmitter and receive antenna\ni, and κ(f)stands for the frequency-dependent molecular\nabsorption coefficient. Thus, the first term in (1) accounts for\nthe spreading loss, the second term stands for the molecular\nabsorption loss, and the last term indicates the phase shift the\nwave experiences when traveling over a distance di. At the\nreceiver, the signal is corrupted by additive white Gaussian\nnoise (AWGN) which yields the receive vector\nyc=hcxc+nc, (2)\nwithnc∼ CN (0N, σ2\nnIN). We define the SNR as the\nratio of the average received symbol power and the noise\npower, SNR =||hc||2\n2σ2\nx\nNσ2n. Before quantization, the noisy receive\nsamples yc\niare fed into analog phase shifters and variable\ngain amplifiers (VGAs) which compensate for the channel’s\nphase shift and attenuation. For this, perfect knowledge of the\nchannel coefficients hc\niis assumed at the receiver. The analog\nsignal processing yields the vector\n˜ yc=˜hc∗⊙yc=xc1N+˜ nc, (3)\nwith ˜hc=h\nhc\n1\n|hc\n1|2···hc\nN\n|hc\nN|2iT\nand ˜ nc∼\nCN\u0010\n0N,diagn\nσ2\nn\n|hc\n1|2, . . . ,σ2\nn\n|hc\nN|2o\u0011\n. For co-located receive\nantennas, the different path gains are approximately the\nsame, i.e., |hc\n1|2≈ ··· ≈ | hc\nN|2=:|hc|2. Hence, the noise in\ndifferent branches is independent and identically distributed\n(i.i.d.), ˜ nc∼ CN\u0000\n0N,˜σ2\nnIN\u0001\nwith ˜σ2\nn=σ2\nn\n|hc|2. This yields N\nindependent AWGN channels with the same noise variance.\nThen, the vector ˜ ycis quantized element-wise by a b-bit\nquantizer with law Qc\nb{˜yc\ni}=Qb{R{˜yc\ni}}+ jQb{I{˜yc\ni}},\ni.e., real and imaginary part are quantized separately. The\nquantizer is chosen as a uniform midrise quantization law\nwith step size ∆,\nQb{˜yi}=\n\nsign(˜yi)·\u0010j\n|˜yi|\n∆k\n∆ +∆\n2\u0011\nfor|˜yi|<2b−1∆\nsign(˜yi)·(2b−1)∆\n2otherwise,\n(4)\nas illustrated in Fig. 2. The resulting vector zcis processed by\na linear filter fc∈RNwhich returns the detection variable\ndc=fcTzc=fcTQc\nb{˜ yc}=fcTQc\nb{xc1N+˜ nc}.(5)\nWe aim at analyzing the performance of the system for a\nspecific detection filter fc=1\nN1N, constituting a simple\naveraging filter. Finally, the detector returns the estimated input\nsymbol ˆxc.4\nFig. 2. Law Qb{·}ofb-bit uniform midrise quantizer with step size ∆.\nNote that the spatial-wideband effect [33] is not relevant\nin our scenario since the sampling time of each ADC can be\nadjusted according to the path delay of the respective antenna.\nEstimating these delays as well as the channel coefficients\nin (1) based on quantized observations is beyond the scope of\nthis paper. However, there exist already approaches for such\nestimation in millimeter-wave channels under low-resolution\nquantization, cf. e.g., [34]–[37], based on compressive sens-\ning techniques, the expectation-maximization (EM) algorithm,\ndeep generative networks, and a combined ML and least-\nsquares approach, respectively. It is expected that correspond-\ning channel estimation approaches can be also designed for\nthe THz channels considered in this work.\nFurthermore, our system model is also applicable to hybrid\nreceivers which are often discussed in the literature. Here,\nthe signal after analog combining in a receiver subarray via\nphase shifters and an adder would be fed into the VGA before\nquantization. It should be noted that especially in the case of\nhigher-order modulation, the availability of a sufficient number\nof quantized observations is crucial, i.e., analog combining\nshould provide a sufficient number of output streams. Finally,\nwe note that VGAs are not necessary in the case of 1-bit\nquantization. However, the knowledge of the channel gains is\nstill important here for detection.\nIII. ADC P OWER CONSUMPTION\nIn this section, we introduce the adopted model for the\npower PADC consumed in the ADC based on the models\nin [38]–[41] which can be unified as\nPADC=γ·N·2ζq·fν\ns, (6)\nwhere γdenotes a constant which depends on the utilized\nADC technology. Here, one common choice for γis the\nWalden figure of merit [42]. Furthermore, Nstands for the\noversampling factor as introduced in the previous section, q\ndenotes either the total number of bits bused in quantization\nor the effective number of bits (ENOB) of a given ADC, the\nparameter ζ∈ {1,2}depends on the quantization resolution,fsis the sampling frequency, and the parameter ν∈ {1,2}is\nrelated to the sampling frequency. For lower-resolution ADCs,\nζ= 1 is usually chosen, whereas ζ= 2 is used for moderate-\nto-higher-resolution ADCs, and ν= 1 is utilized for small\nsampling frequencies while ν= 2 for larger sampling fre-\nquencies. According to [3] and [43], a doubling of the power\nconsumption when increasing the quantization resolution by\n1, i.e., ζ= 1, occurs for ADCs with a signal-to-noise-and-\ndistortion ratio (SNDR) of SNDR <50 dB which is equivalent\nto ENOB <8 bit. Larger sampling frequencies, i.e., ν= 2,\ncorrespond to the range fs>100 MHz [3], [43].\nIn this work, we analyze the performance of low-resolution\nADCs which are primarily of interest for very high sampling\nfrequencies. Hence, we consider the parametrization ζ= 1and\nν= 2. Furthermore, we use q=bwhich facilitates the power\nconsumption computation of the b-bit quantizer introduced in\nthe last section.2As a consequence of this parametrization,\nincreasing the quantization resolution by 1 bit has the same\neffect as doubling the oversampling factor N. Therefore, for\nexample, 1-bit quantization ( b= 1) with an oversampling\nfactor of N= 64 results in the same power consumption as\nusing a 2-bit ADC with N= 32 or a 3-bit ADC with N= 16 ,\nrespectively. These three cases will be analyzed in Section V\nin more detail. It should be noted that the linear scaling of the\npower consumption with respect to the oversampling factor\ndoes not contradict the quadratic increase with respect to the\nsampling frequency fsaccording to (6) with ν= 2. Here, the\nlinear scaling with respect to Ncan be motivated by time-\ninterleaved ADC architectures or by spatial oversampling.\nIV. O PTIMAL SYMBOL DETECTION\nThe optimal ML detector maximizes the probability mass\nfunction (PMF) of the detection variable dcgiven the input\nsymbol xc, i.e., pdc|xc(dc|xc). The resulting ML estimate of\nthe transmitted symbol is\nˆxc= arg max\nxc∈Xpdc|xc(dc|xc). (7)\nIn order to derive the likelihood function pdc|xc(dc|xc), the\nconditional probability distributions of the unquantized and\nquantized received signals have to be determined first. For\nthis, we make the assumption of co-located receive antennas\nwhich yields approximately i.i.d. noise samples. Furthermore,\nwe consider the real and the imaginary part separately which\nis justified since both components are independent and have\nthe same noise variance according to (3). In the following,\nwe express the fact that we consider only one quadrature\ncomponent by omitting the superscript (·)c, e.g., xandd\ndenote either the in-phase or the quadrature component of the\ninput symbol xcand the detection variable dc, respectively.\nThe ML estimate of the respective quadrature components is,\nhence, given by\nˆx= arg max\nx∈X′pd|x(d|x), (8)\n2ENOB depends on the actual hardware realization of the ADC. Since we\ndo not consider any specific ADC realization, we use q=b.5\nwith the M′-ary quadrature component constellation X′=\nR{X} =I{X} of the original M-ary square QAM constel-\nlationXwhere M′:=√\nM.\nA. Probability Distributions of Received Signals and Detec-\ntion Variable\nFor a fixed input symbol x, the unquantized vector ˜yfollows\nthe same probability distribution as the noise vector except\nfor a shift of the mean by x, i.e., ˜y∼ N (x1N,˜σ2\nn\n2IN)\nwhich implies that the unquantized samples are i.i.d. Gaussian\ndistributed with mean xand variance˜σ2\nn\n2. We denote the\nrandom variable corresponding to one received sample ˜ynas\n˜y. Due to the independent unquantized observations and the\nelement-wise quantization, the quantized vector zalso consists\nof independent entries if conditioned on x. Therefore, the\ncorresponding likelihood function can be written as\npz|x(z|x) =NY\nn=1pz|x(zn|x), (9)\nwhere pz|x(zn|x)is the PMF of the nthentry of zgiven\nx,n∈ {1, . . . , N }. Here, the quantized samples znare\nrealizations of the discrete random variable zwhich follows\na multinoulli distribution with K= 2bdifferent values\nz(k):=\u0000\nk−K+1\n2\u0001\n∆,k∈ {1, . . . , K }. Due to the normally\ndistributed elements in the unquantized vector, the conditional\nprobabilities of the possible values of the random variable z\ncan be computed as\nPk(x):=pz|x(z(k)|x) =py|x(y∈ Dy(z(k))|x)\n=Q\u0012τlow(z(k))−x\n˜σn/√\n2\u0013\n−Q\u0012τup(z(k))−x\n˜σn/√\n2\u0013\n,(10)\nwhere the quantizer decision region and corresponding thresh-\nolds are given by\nDy(z(k)) ={y∈R|τlow(z(k))≤y < τ up(z(k))},(11)\nτlow(z(k)) =(\nz(k)−∆\n2forz(k)>−(K−1)∆\n2\n−∞ otherwise, (12)\nand\nτup(z(k)) =(\nz(k) +∆\n2forz(k)<(K−1)∆\n2\n+∞ otherwise. (13)\nNow, the mean and the variance of zgiven xcan be calculated\nas\nµz(x) =Ez|x{z|x}\n= ∆· K−1X\nk=1Q\u0012τup(z(k))−x\n˜σn/√\n2\u0013\n−K−1\n2!\n,(14)\nσ2\nz(x) =Ez|x\b\n(z−µz(x))2|x\t\n= ∆2· K−1X\nk=1(2k−K)·Q\u0012τup(z(k))−x\n˜σn/√\n2\u0013\n+\u0012K−1\n2\u00132!\n−µ2\nz(x),(15)where we have used the fact that τlow(z(k+ 1)) = τup(z(k))\nfork∈ {1, . . . , K −1}.\nAs mentioned above, the random variable zfollows a\nmultinoulli distribution with Kpossible values and is observed\nNtimes. We denote the number of observations of the kth\nvalue z(k)asκk∈ {0, . . . , N }withPK\nk=1κk=N. Hence,\nthe collection of the numbers of observations is multinomially\ndistributed,\npκ|x(κ|x) =\u0012N\nκ1, . . . , κ K\u0013KY\nk=1Pκk\nk(x), (16)\nwith the vector of numbers of observations κ= [κ1···κK]\nand the multinomial coefficient\u0000N\nκ1,...,κ K\u0001\n=N!\nκ1!...κK!. Since\nthe detection variable dis the average of all Nobservations,\nit can be expressed in terms of the numbers of observations\nκk,\nd=d(κ) =1\nNKX\nk=1κkz(k) =1\nNKX\nk=1κk\u0012\nk−K+ 1\n2\u0013\n∆.\n(17)\nEquation (17) can be rearranged as\nKX\nk=1κkk=N\u0012d\n∆+K+ 1\n2\u0013\n. (18)\nTherefore, all vectors κwhich satisfy (18) for fixed dyield\nthe same value dof the detection variable. Hence, the PMF\nof the detection variable is given by a sum of multinomial\ndistributions,\npd|x(d=d(κ)|x) =X\nκ∈Kd\u0012N\nκ1, . . . , κ K\u0013KY\nk=1Pκk\nk(x),(19)\nwith the set of vectors of numbers of observations correspond-\ning to detection variable value d,\nKd=(\nκ∈NK\n0\f\f\f\f\fKX\nk=1κk=N∧KX\nk=1κkk=N\u0012d\n∆+K+ 1\n2\u0013)\n.\n(20)\nNote that the detection variable can only take on discrete\nvalues from z(k= 1) toz(k=K)with a spacing of\n∆\nN, resulting in N(K−1) + 1 possible different values.\nThe set of all possible vectors of numbers of observations\nK={κ∈NK\n0|PK\nk=1κk=N}consists of |K|=\u0000N+K−1\nK−1\u0001\ndifferent elements [44]. If b >1andN > 1, there are more\nelements in Kthan possible detection variable values. Hence,\nthere is no one-to-one mapping between κanddfor multi-\nbit quantization under oversampling which is exemplified in\nTable I for b= 2 andN= 2. This means that the receiver is\nnot able to unambiguously determine the vector of numbers\nof observations κwhen inspecting the detection variable d,\ncontrary to 1-bit quantization where always a one-to-one\nmapping between κanddexists. This has an influence on\nthe complexity and the performance of the ML detector based6\nTABLE I\nALL POSSIBLE VECTORS OF NUMBERS OF OBSERVATIONS AND\nCORRESPONDING DETECTION VARIABLE VALUES FOR b= 2 ANDN= 2.\nκ1κ2κ3κ4 d\n2 0 0 0−3/2 ∆\n1 1 0 0 −∆\n0 2 0 0−1/2 ∆\n1 0 1 0−1/2 ∆\n1 0 0 1 0\n0 1 1 0 0\n0 1 0 1 1/2 ∆\n0 0 2 0 1/2 ∆\n0 0 1 1 ∆\n0 0 0 2 3/2 ∆\n00.050.10.150.2\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nFig. 3. PMF pd|x(d|x)(19) and (scaled) continuous approximation\nfd|x,CLT(d|x)(23) for b= 1 ,∆ = 2 ,N= 64 , andX′={±1,±3}\nat SNR =0 dB.\nondwhich will be discussed later on in more detail. The mean\nand the variance of dcan be computed using (14) and (15),\nµd(x) =Ed|x{d|x}=1\nNNX\nn=1µz(x) =µz(x), (21)\nσ2\nd(x) =Ed|x\b\n(d−µd(x))2|x\t\n=1\nN2NX\nn=1σ2\nz(x) =1\nNσ2\nz(x).\n(22)\nThus, a higher oversampling factor leads to a smaller variance\nof the detection variable. With the above calculated mean and\nvariance, via the central limit theorem (CLT), we can approx-\nimate the PMF of dfor large Nwith a continuous Gaussian\nprobability density function (PDF), d∼ N(µd(x), σ2\nd(x)),\nfd|x,CLT(d|x) =1√\n2πσd(x)e−(d−µd(x))2\n2σ2\nd(x). (23)\nIn Fig. 3, the PMF of das well as the continuous CLT\napproximation which is scaled with∆\nNare illustrated for b= 1,\n∆ = 2 ,N= 64 , and a 4-ASK constellation, corresponding\nto 16-QAM when considering both quadrature components. It\ncan be seen that the approximation and the true distribution\nmatch very well for the given oversampling factor, input\nconstellation, and SNR. However, there are some limitations\nto the continuous approximation of the discrete system which\nwill be pointed out later. Furthermore, the variance of dfor\ninput symbols with different magnitudes is not equal, resulting\nin consequences for the ML detector which will be discussed\nin the next subsection. In Fig. 4, the PMF of dis shown for\nb= 3,∆ = 1 ,N= 16 , and otherwise the same parameters\n00.010.020.030.040.050.060.07\n-4 -3 -2 -1 0 1 2 3 4Fig. 4. PMF pd|x(d|x)(19) and (scaled) continuous approximation\nfd|x,CLT(d|x)(23) for b= 3 ,∆ = 1 ,N= 16 , andX′={±1,±3}\nat SNR =0 dB.\nas in Fig. 3. Here, the variance of dfor different input\nsymbols varies as well. However, the relative difference in the\nvariances is smaller than for 1-bit quantization which makes\n3-bit quantization with N= 16 closer to the unquantized case\nwhere the variances are equal for all input symbols.\nB. Symbol Detectors\nWe can use the PMF in (19) to specify the ML detector\naccording to (8),\nˆxML= arg max\nx∈X′X\nκ∈Kd\u0012N\nκ1, . . . , κ K\u0013KY\nk=1Pκk\nk(x). (24)\nThe complexity of the detector according to (24) is quite high\ndue to the summation of |Kd|terms, each comprising a multi-\nnomial coefficient and a product of Kpowers. However, for\n1-bit quantization, the detector simplifies significantly because\n|Kd|= 1∀ddue to the one-to-one mapping between κand\nd, and, therefore, the multinomial coefficient is irrelevant for\nthe maximization with respect to x. Hence, only the product\nofK= 2 powers needs to be computed which was already\nindicated in [1] and [26]. Moreover, we consider an alternative\ndetector based on the CLT approximation which is given by\nˆxCLT= arg max\nx∈X′1\nσd(x)e−(d−µd(x))2\n2σ2\nd(x). (25)\nIn our case, we cannot employ the classical ML approach for\nan unquantized channel and minimize the Euclidean distance\nbetween dandµd(x)in order to realize (25) because the\nvariance σ2\nd(x)depends on x. Therefore, in general, the\noptimal decision thresholds do not lie exactly in the middle\nbetween the mean values µd(x)of adjacent input symbols.\nNevertheless, we investigate the performance of such subop-\ntimal approach as well with decision rule\nˆxminDist = arg min\nx∈X′(d−µd(x))2. (26)\nThe three detectors according to (24), (25), and (26) result\nin three different sets of decision thresholds. In Fig. 5, the\nthresholds between decision regions for x= +1 andx= +3\nare depicted exemplarily for 1-bit quantization.7\n00.020.040.060.080.10.120.140.16\n0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8\nFig. 5. Thresholds between the decision regions for x= +1 andx= +3\naccording to the detectors (24), (25), and (26) for b= 1,∆ = 2 ,N= 64 ,\nandX′={±1,±3}at SNR =0 dB. The colored discrete markers and\ncontinuous lines represent the PMF pd|x(d|x)(19) and (scaled) continuous\napproximation fd|x,CLT(d|x)(23), respectively.\nThe fully optimum ML detector is directly based on the\nquantized vector z, i.e., no filter fcis applied, and can be\nexpressed via the likelihood function in (9), resulting in\nˆxnoFilt= arg max\nx∈X′pz|x(z|x) = arg max\nx∈X′KY\nk=1Pκk\nk(x).(27)\nAccording to (27), the exact vector zis not relevant for\ndetection but the numbers of observations κk,k∈ {1, . . . , K },\nare a set of sufficient statistics which was also mentioned\nin [9] for 1-bit quantization. Due to the one-to-one mapping\nbetween dandκin case of 1-bit quantization, which makes\nthe averaging filtering an invertible operation, it can be easily\nshown via the data processing theorem [45] that the detector\nin (24) is equivalent to the fully optimum detector in (27).\nAs a consequence, the averaging filter fcis optimal for the\nconsidered system with b= 1 . For b > 1andN > 1,\nhowever, this is not the case anymore due to the many-\nto-one mapping between the numbers of observations and\nthe detection variable, exemplified in Table I. Therefore, for\nmulti-bit quantization with oversampling, the detector in (24)\nperforms worse than the optimal detector in (27). However,\nthe performance loss is negligible as will be observed later\nfrom the numerical results.\nC. Symbol Error Rate Computation\nIn order to compute the SER analytically for a given\ndetector, M′−1decision boundaries [b1b2···bM′−1]per\nquadrature component have to be determined first. Without\nloss of generality, an ordering of the constellation symbols\nxi∈ X′is assumed, i.e., x1< x2< . . . < x M′. For the ML\ndetector in (24) and the CLT-based detector in (25), the deci-\nsion threshold bican be found by comparing the PMF (19) and\nPDF (23), respectively, for the two adjacent input symbols xi\nandxi+1, and choosing the smallest argument value of dwhere\nthe function value corresponding to the greater symbol xi+1\nis higher than or equal to the function value corresponding to\nthe smaller symbol xi. For the suboptimal detector (26), the\nithdecision threshold is given by bi=µd(xi)+µd(xi+1)\n2with\nµd(x)from (21). In general, the decision boundaries dependon the noise variance and, therefore, have to be recalculated\nin case the SNR changes.\nFor computing the SER analytically via the detection vari-\nabled, we define for each possible quadrature input symbol xi\na decision region Difor the detection variable which depends\non the utilized detector. Given the transmitted symbol is x1,\na detection error occurs if the detection variable dis equal\nto or above the boundary b1. Thus, the decision region for\nx1isD1= [z(k= 1) , b1). For the last symbol xM′, a\nwrong symbol will be detected if dis smaller than bM′−1, thus\nDM′= [bM′−1, z(k=K)]. The decision region for any inner\nsymbol xi,i∈ {2, . . . , M′−1}, is given by Di= [bi−1, bi).\nCorrespondingly, the SER of one quadrature component for\nall detectors based on dcan be expressed as\nSER ASK=1\nM′X\nxi∈X′X\nd/∈Dipd|x(d|xi), (28)\nusing the likelihood function pd|x(d|x)in (19).\nIn order to compute the SER analytically for the detector\nutilizing z, we can evaluate the likelihood function in (9) for\nall possible quantized vectors zand all possible input symbols\nxand then find suitable decision sets for all x. However, there\nareKNpossible quantized vectors which might be too many\nfor an efficient SER computation. Instead, we can compute\nthe conditional probabilities of all possible vectors of numbers\nof observations κ∈ K for each input symbol x,pκ|x(κ|x),\naccording to (16). Then, we find the most probable symbol\nxifor each vector κbased on the computed probabilities\nand include the considered vector κin the decision set of\nthe corresponding symbol xi,Ki={κ∈ K | pκ|x(κ|xi)≥\npκ|x(κ|xj)∀j̸=i}, where we do not allow a vector κto\nbelong to multiple decision sets. The resulting SER of one\nquadrature component is given by\nSER ASK,noFilt =1\nM′X\nxi∈X′X\nκ/∈Kipκ|x(κ|xi). (29)\nDue to the orthogonality of in-phase and quadrature com-\nponents, the SER of the complex-valued input symbols x∈ X\nis given by\nSER= 1−(1−SER′)2= 2·SER′−SER′2, (30)\nwhere SER′is the SER of one quadrature component, obtained\nfrom (28) or (29).\nV. N UMERICAL RESULTS\nA. Frequency-Flat Single-Path LoS THz Channel\nIn the following, SER results for the frequency-flat single-\npath LoS THz channel with co-located receive antennas, as\nintroduced in Section II, are presented for the derived detectors\nand the three ADC parametrizations mentioned in Section III.\nHere, the exact choice of the transmit power, the carrier\nfrequency, the distance between transmitter and receiver, or\nthe noise variance is not important since the performance\nfinally depends on the SNR. Therefore, we do not specify these\nindividual parameters here. The following results are valid\nfor any combination of the aforementioned parameters which\nyield a given SNR value. A realistic choice of parameters8\nfor THz communications is made in the next subsection.\nMoreover, we have computed the SER of an unquantized\nsystem with the same oversampling factor as ultimate bench-\nmark which is equivalent to a single-antenna receiver with\nnoise variance˜σ2\nn\nN. In this case, the presented suboptimal\ndecision boundaries become optimal since the variance of the\ndetection variable for different input symbols become equal\nand minimizing the Euclidean distance is optimal. Besides,\nwe have conducted Monte Carlo simulations with 108random\nsymbol transmissions in each case in order to verify the\nanalytical results. Furthermore, for 2- and 3-bit quantization,\nthe choice of the step size ∆is important. We will not optimize\nthe step size analytically. However, we heuristically choose\ndifferent step sizes for b= 2 andb= 3 , depending on\nthe transmit symbol constellation. For 16-QAM transmissions\nwithX′={±1,±3}, we choose the step size to be ∆ = 2 for\nb= 2and∆ = 1 forb= 3such that the constellation symbols\nare evenly spread over the quantization region. For 64-QAM\ntransmissions with X′={±1,±3,±5,±7}, we choose ∆ = 4\nforb= 2 and∆ = 2 forb= 3 for the same reason.\nAt first, we consider a 16-QAM constellation with X′=\n{±1,±3}. The corresponding SER curves are illustrated in\nFig. 6. For 1-bit quantization (Fig. 6a), we observe that the\nSER curves are not smooth. This behavior stems from the\nstrong nonlinearity in the system and the discrete nature of\nthe detection variable and can be only observed when the\nresolution with respect to the SNR-axis, based on which the\ncurves are drawn, is sufficiently high. Using our analysis,\nthe SER results can be computed very fast for arbitrary\nSNR values which is not feasible for exhaustive Monte Carlo\nsimulations. Furthermore, it can be observed that the SER\ndegrades for high SNR and converges to 0.75 as expected\ndue to the fact that at high SNR the four symbols in each\nquadrant, e.g., 1+j,1+3j ,3+j and3+3j in the first quadrant,\ncannot be distinguished anymore since they are all mapped to\nthe same quantization level due to the negligible influence\nof noise. Furthermore, one can observe from Fig. 6a that\nthe decision rule based on the continuous CLT approximation\nleads to a significant performance degradation compared to the\noptimal ML detector. Besides, the other suboptimal approach\nbased on minimizing the Euclidean distance also leads to\na significant performance degradation compared to the ML\ndetector. Furthermore, it can be observed that the ML detector\nbased on the detection variable din (24) shows exactly the\nsame performance as the fully optimum ML detector based on\nthe quantized vector zin (27) which was already mentioned\nin Subsection IV-B. Finally, the performance gap between 1-\nbit quantization and the unquantized case seems acceptable\nwhen considering the benefit of possible power savings in a\nreal system. Here, the loss in SNR equals 2.3 dB, 3.2 dB, and\n6.0 dB at an error rate of SER = 10−1, SER = 10−2, and\nSER= 10−3, respectively.\nFor 2-bit quantization (Fig. 6b), a performance deviation\nbetween the optimal ML detector based on dand both subop-\ntimal detectors can be still observed even though the number\nof quantization levels is equal to the number of possible input\nsymbols. Only for 3-bit quantization (Fig. 6c), there is almost\n-20 -15 -10 -5 0 5 10 15 2010-610-410-2100(a)b= 1,N= 64 .\n-20 -15 -10 -5 0 5 10 15 2010-610-410-2100\n(b)b= 2,N= 32 ,∆ = 2 .\n-20 -15 -10 -5 0 5 10 15 2010-610-410-2100\n(c)b= 3,N= 16 ,∆ = 1 .\nFig. 6. Analytical SER (solid lines) and simulated SER (markers) for the\nML detector (24), CLT-based detector (25), minimum distance detector (26),\nand fully optimum detector without filtering (27) under 16-QAM transmission\nwithX′={±1,±3}. Furthermore, the ML detection performance for the\nunquantized case with the same oversampling factor is shown.\nno noticeable difference between the three detectors. Zooming\nin the curves for 2- and 3-bit quantization (not shown here),\none observes a small performance loss of the ML detector\nin (24) compared to the fully optimum detector in (27). This\nloss is related to the nonexistent one-to-one mapping between\nthe vectors of numbers of observations κand the detection\nvariable values dwhich was discussed in Subsection IV-A.\nHowever, the loss is negligibly small, thus, the combination\nof the quantized observations via an averaging filter does\npractically not degrade the system performance and yields\nnear-optimum results. Besides, it can be noticed for 2- and 3-\nbit quantization that the SER goes to zero for increasing SNR\nbecause the number of quantization levels is larger or equal\nto the number of possible input symbols. Nevertheless, there\nis still a performance loss compared to the unquantized case,9\n-20 -15 -10 -5 0 5 10 15 2010-610-410-2100\nFig. 7. Analytical SER for the ML detector (24) and different ADC\nparametrizations under 16-QAM transmission with X′={±1,±3}.\neven for 3-bit quantization. Finally, it can be observed that the\nanalytically obtained SER results match exactly with the SER\nresults obtained by Monte Carlo simulations. This confirms\nthat our analysis is accurate. In Fig. 7, the performance of\nthe different ADC parametrizations is compared for the ML\ndetector (24). Here, 1-bit quantization is superior to multi-\nbit quantization over a relatively large SNR region up to\n2.9 dB. For higher SNR, 2-bit quantization always yields the\nsmallest SER. Hence, it seems to be not useful to increase the\nquantization resolution beyond log2(M′)bit.\nNext, we analyze the SER performance for a 64-QAM\nconstellation with X′={±1,±3,±5,±7}in Fig. 8. The min-\nimum SER for 1-bit quantization is quite high at approximately\n0.24 due to the large constellation and low resolution. Here,\nthe performance could be enhanced by increasing the number\nof receive antennas N. The receiver with 2-bit quantization\nyields a minimum SER of SER = 10−2. Besides, the SER\nincreases for b= 1 andb= 2 at high SNR because there\nare more constellation symbols than quantization levels. For\n3-bit quantization, the SER decays to zero for high SNR as\nexpected. Furthermore, we observe a performance gap be-\ntween the optimal detector based on dand the two suboptimal\ndetectors as in the 16-QAM transmission case. Moreover, it\ncan be observed again that the analytically obtained SER\nresults match exactly with the SER results obtained by Monte\nCarlo simulations. Finally, when comparing the results for the\ndifferent ADC parametrizations in Fig. 9, we deduce that 1-\nbit quantization performs best up to an SNR of approximately\n2.2 dB. Then, there is an SNR window up to approximately\n10.8 dB for which 2-bit quantization yields the minimum SER.\nHowever, for SNR >10.8 dB, 3-bit quantization performs\nbest. For both 16-QAM and 64-QAM transmission, at low\nSNRs below a certain threshold, 1-bit quantization yields the\nsmallest SER when constraining the ADC power consumption\nto be constant.\nB. Realistic Indoor THz Channel\nIn this subsection, we consider a realistic frequency-\nselective indoor THz channel with multipath propagation in-\ncluding reflected and scattered rays and adopt a corresponding\nray tracing based channel model [46]. We show that our\nproposed ML detector (24) performs similarly well for this\n-20 -15 -10 -5 0 5 10 15 2010-310-210-1100(a)b= 1 andN= 64 .\n-20 -15 -10 -5 0 5 10 15 2010-310-210-1100\n(b)b= 2,N= 32 ,∆ = 4 .\n-20 -15 -10 -5 0 5 10 15 2010-310-210-1100\n(c)b= 3,N= 16 ,∆ = 2 .\nFig. 8. Analytical SER (solid lines) and simulated SER (markers) for the\nML detector (24), CLT-based detector (25), minimum distance detector (26),\nand fully optimum detector without filtering (27) under 64-QAM transmission\nwithX′={±1,±3,±5,±7}. Furthermore, the ML detection performance\nfor the unquantized case with the same oversampling factor is shown.\nrealistic scenario as for the simplified scenario of a frequency-\nflat single-path LoS channel, analyzed in Subsection V-A. We\nconsider data transmission at a carrier frequency of 300 GHz\nwith a bandwidth of 20 GHz in a large rectangular room\nwith a length, width, and height of 10 m, 20 m, and 2.5 m,\nrespectively. The walls of the room are made of plaster\n”sample s2” from [47] which has a high roughness. We\nassume a transmit power of 13 dBm [48] and employ a root-\nraised-cosine (RRC) filter with a roll-off factor of 0.25 at\nboth transmitter and receiver side. This results in a data rate\nof 64 Gbps and 96 Gbps for 16- and 64-QAM transmission,\nrespectively. Furthermore, we consider perfectly aligned horn\nantennas at both the transmitter and the receiver. Here, we\nuse the Gaussian beam model for the radiation pattern of the\nhorn antenna with a gain of 18.9 dBi [49]. Besides, we utilize10\n-20 -15 -10 -5 0 5 10 15 2010-310-210-1100\nFig. 9. Analytical SER for the ML detector (24) and different ADC\nparametrizations under 64-QAM transmission with X′={±1,±3,±5,±7}.\nTABLE II\nTHZ SIMULATION PARAMETERS .\nParameter Value\nCarrier frequency 300 GHz\nBandwidth 20 GHz\nDistance 1-17 m\nWall material Plaster\nTransmit power 13 dBm\nTransmit pulse and receive filter RRC filter with roll-off factor 0.25\nTransmit and receive antennas Horn antenna with 18.9 dBi gain\nNoise power density −174 dBm/Hz\nNoise figure 15 dB\na uniform planar array (UPA) with λ/2antenna spacing at\nthe receiver with 8×8,8×4, and 4×4antennas in the\nhorizontal and vertical direction for N= 64 ,N= 32 , and\nN= 16 , respectively. Finally, the noise power density is set\nto −174 dBm/Hz, and a noise figure of 15 dB is selected. The\nsimulation parameters are summarized in Table II.\nThe transmitter is placed at one side of the room at the\ncoordinates 5 m, 1.5 m, 2.4 m as illustrated in Fig. 10. The\nreceiver is located at a height of 1.5 m at nine different\ndistances from the transmitter, ranging from 1 m to 17 m.\nFor each distance, we consider three different positions of\nthe receiver: one in the middle of the room, one close to\nthe wall, and one in between as can be also observed from\nFig. 10. For each receiver position, we transmit 100 inde-\npendent blocks, each consisting of 104consecutive symbols.\nISI occurs only between the symbols within one block due\nto protection intervals between the blocks. Then, the SER\nresults corresponding to the three different positions with\na given distance are averaged. Hence, we consider in total\n3·106symbol tranmissions for each distance. Finally, in order\nto guarantee a good detection performance even for small\ndistances, i.e., for high SNR, we add artificial noise in front of\nthe ADCs for 16-QAM and 64-QAM transmission in case of 1-\nbit quantization and for 64-QAM transmission in case of 2-bit\nquantization, respectively. The optimum power of the artificial\nnoise is determined by inspecting the analytical results from\nSubsection V-A and identifying the SNR value which yields\nthe smallest SER. According to Figs. 6a and 8a, this optimum\nSNR value is given by approximately 5.6 dB and 3.8 dB for\n16-QAM and 64-QAM transmission, respectively, in case of\n1-bit quantization. For 2-bit quantization and 64-QAM, the\n20\n15 012z\n0\ny10\nx55\n10 01mTx3m5m7m9m11m13m15m17mFig. 10. Indoor environment with transmitter and receiver positions.\n0 2 4 6 8 10 12 14 16 1810-410-310-210-1100\nFig. 11. Simulated SER versus distance in a realistic indoor THz channel\nfor the ML detector (24) and different ADC parametrizations under 16-QAM\ntransmission with X′={±1,±3}.\noptimum SNR value is 12.2 dB according to Fig. 8b. Note\nthat these optimum SNR values can be only reached when the\nactual SNR is larger than the optimum value which is the case\nfor relatively small distances. When the actual SNR is lower\nthan the optimum value, i.e., for large distances, we cannot\nenhance the performance by adding artificial noise.\nIn Fig. 11, SER versus the communication distance is shown\nfor 16-QAM. Here, we can observe that 2-bit quantization\nperforms best at small distances. This is in agreement with the\nresults in Fig. 7 for high SNR. For distances larger than 13 m,\n1-bit quantization performs best. Furthermore, we observe\nthat adding artificial noise before 1-bit quantization enhances\nthe performance for small distances and keeps the error rate\nconstant up to a distance of approximately 9 m. In this case,\nthe SER is as small for sufficiently small distances as for the\nsimplified frequency-flat single-path LoS channel in Fig. 6a.\nHence, we conclude that the simplifying assumptions are\njustified for the considered indoor THz channel, i.e., ISI can\nbe neglected, and all channel gains are approximately equal.\nHence, our proposed detector is still (close-to-)optimal in this\ncase. SER versus distance for 64-QAM transmission is shown\nin Fig. 12. Here, similar observations can be made as in case of\n16QAM before. 1- and 2-bit quantization benefit from adding\nartificial noise, whereas 3-bit quantization only performs best\nat small distances. A comparison with the results of Fig. 8\nagain confirms the validity of the simplifying assumptions\nadopted for detector design.\nIt should be noted that a similar performance can be also\nattained for larger distances by increasing the SNR via, e.g.,\nincreasing the transmit power, using more directional antennas,\nor decreasing the noise level at the receiver.11\n0 2 4 6 8 10 12 14 16 1810-410-310-210-1100\nFig. 12. Simulated SER versus distance in a realistic indoor THz channel\nfor the ML detector (24) and different ADC parametrizations under 64-QAM\ntransmission with X′={±1,±3,±5,±7}.\nVI. C ONSTELLATION OPTIMIZATION FOR 1-BIT\nQUANTIZATION\nOne main difference of the statistics of the detection variable\ndcompared to those of the detection variable y=x+nof\nan unquantized Nyquist-rate sampling transmission is that the\nvariance of ddepends on the transmitted symbol, whereas\nthe variance of yis identical for all input symbols. Due\nto the equal variance in the unquantized case, it is optimal\nto place the input symbols equidistantly, i.e., |x1−x2|=\n|x2−x3|=. . .=|xM′−1−xM′|. However, for the\nquantized system, this is not valid anymore. Especially for\n1-bit quantization, the variance can vary significantly for\ndifferent input symbols as can be seen in Fig. 3. Therefore,\nit is not obvious what is the optimum constellation which\nyields a minimum SER. In the following, we aim at de-\ntermining the optimum constellation for 1-bit quantization.\nWe focus on square 16-, 36-, and 64-QAM constellations\nwith one, two, and three degrees of freedom, respectively,\nfor constellation optimization. Here, the inner four symbols\nare fixed to ±1±j, whereas the outer symbols are variable.\nTherefore, the quadrature component constellations are given\nbyX′={±1,±a1}for 16-QAM, X′={±1,±a1,±a2}\nfor 36-QAM, and X′={±1,±a1,±a2,±a3}for 64-QAM.\nWe find the optimal constellations empirically in each case\nby comparing SER for different choices of the available free\nparameters. Similar analyses can be done for 2- and 3-bit\nquantization. However, here, the choice of the quantizer step\nsize∆has to be considered as well.\nIn Fig. 13, analytically obtained SER curves are shown for\nthe ML detector (24) and various 16-QAM constellations with\ndifferent values of a1. It can be observed that the minimum\nSER can be decreased significantly by increasing a1. For\nexample, the minimum SER for a1= 3is9.8·10−4, while the\nconstellation with a1= 8yields a minimum SER of 4.5·10−7.\nThis emphasizes the importance of choosing an optimized con-\nstellation. However, the minimum SER for a1= 8is achieved\nat a higher SNR than that for a1= 3. Another interesting\nobservation can be made for a1>8. Here, the location of the\nminimum SER moves to higher SNR for increasing a1while\nthe minimum SER value itself does not decrease anymore.\nThis suggests that the minimum SER, illustrated in Fig. 14,\nconverges to a fixed value for increasing a1, and a1≈8is\n-20 -15 -10 -5 0 5 10 15 2010-610-410-2100Fig. 13. Analytical SER for the ML detector (24) and different 16-QAM\nconstellations with X′={±1,±a1},b= 1, and N= 64 .\n2 4 6 8 10 12 14 1610-610-410-2\nFig. 14. Minimum SER for the ML detector (24) and different 16-QAM\nconstellations with X′={±1,±a1},b= 1, and N= 64 .\nthe smallest outer quadrature component constellation point\nfor which this value can be reached closely. However, the\nouter constellation point with this property depends on the\noversampling factor. Hence, the constellation optimization has\nto be carried out individually for each oversampling factor of\ninterest. Furthermore, we note that the results indicate that for\nincreasing SNR the optimal outer points will become larger or\nequivalently the inner points shrink when including a power\nnormalization into the constellation. For SNR → ∞ , this will\nresult in a 9-QAM constellation which was also proven to be\noptimal in [11]. Some further results on optimized 16-QAM\nconstellations can be found in [1].\nNext, we investigate the SER performance for different\n36-QAM constellations, illustrated in Fig. 15. Here, a large\nnumber of constellations is considered with integer values for\na1anda2, and 2≤a1< a 2≤17. As for the 16-QAM\nconstellations, we observe different minimum SER values for\ndifferent constellations, while multiple constellations yield the\nsame minimum SER at different SNR. The minimum SER\nversus the constellation parameters a1anda2is shown in\nFig. 16. Here, also non-integer values of a1anda2have\nbeen included. From the analytically obtained SER values,\nit can be observed that, similarly to the 16-QAM case, the\nminimum SER converges for a1= 3.3anda2>11. Hence,\nwe can conclude that the position of the optimum middle\nquadrature component constellation point is almost identical\nto the classical case with a1= 3, but the position of the outer\nconstellation point is moved to much higher values compared\nto the standard constellation. Also, we can observe from12\n-20 -15 -10 -5 0 5 10 15 2010-210-1100\nFig. 15. Analytical SER for the ML detector (24) and different 36-QAM\nconstellations with X′={±1,±a1,±a2},b= 1, and N= 64 .\n2 4 6 8 10 12 14 16246810121416\n10-210-1\nFig. 16. Minimum SER for the ML detector (24) and different 36-QAM\nconstellations with X′={±1,±a1,±a2},b= 1, and N= 64 .\nFig. 16 that selecting a large value for a1or choosing a2close\ntoa1always results in a poor performance. For the classical\nconstellation with a1= 3 anda2= 5, a minimum SER of\n8.4·10−2is achieved, while for the optimized constellation\nwitha1= 3.3anda2>11a SER of approximately 1.0·10−2\ncannot be undercut.\nNow, we study 64-QAM constellations with three degrees\nof freedom, a1,a2, and a3. The corresponding SER versus\nSNR is depicted in Fig. 17 for different constellations with\ninteger constellation points with 2≤a1< a 2< a 3≤17.\nA similar behavior as for 16- and 36-QAM constellations can\nbe recognized. The same holds for the minimum SER which\nis illustrated in Fig. 18 for integer values of a1,a2, and a3\nfor simplicity. However, we also conducted a full search over\na finer grid, showing that a minimum SER of approximately\n1.0·10−1is achieved for a1= 3.1,a2= 5.7, and a3>14,\nwhereas the classical constellation yields a minimum SER of\n2.4·10−1.\nFinally, we motivate the reduction of SER for the optimized\nconstellations. For 16-, 36-, and 64-QAM transmission, we\nobserved that the minimum SER can be decreased by in-\ncreasing the distance of the outermost constellation points to\nthe remaining inner constellation points and simultaneously\nkeeping the inner points almost equidistantly. This behavior\ncan be explained by comparing the PMFs pd|x(d|x)(19)\nconditioned on the symbols of the classical constellation\nand the optimized constellation at the SNR which yields\nthe minimum SER, respectively, shown in Fig. 19. For 16-\nQAM, the main difference between the two PMFs is that for\nFig. 17. Analytical SER for the ML detector (24) and different 64-QAM\nconstellations with X′={±1,±a1,±a2,±a3},b= 1, and N= 64 .\n155\n1010\n515\n14 12 10 8 6 4 2\n10-1\nFig. 18. Minimum SER for the ML detector (24) and different 64-QAM\nconstellations with X′={±1,±a1,±a2,±a3},b= 1, and N= 64 .\nthe optimized constellation (Fig. 19b) the probability mass\nfor the outer symbols is concentrated in a single detection\nvariable value d=±1, whereas for the classical constellation\n(Fig. 19a) the probability mass for the outer symbols is spread\nover essentially three detection variable values. Therefore, the\noverlap between the probability mass for the outer constella-\ntion point and the probability mass for the inner constellation\npoint is smaller for the optimized constellation, and, hence, the\nSER is reduced. The same holds for 64-QAM transmission,\ncf. Figs. 19c and 19d. In Table III, the SERs of single\nquadrature component constellation symbols are collected. It\ncan be observed that for a given classical constellation the error\nrates of all symbols are of same order of magnitude. However,\nthis is not true anymore for the optimized constellations. Here,\nthe error rate of the outer constellation points is negligible\nwhich is in accordance with the previously presented PMFs.\nHence, SER is determined by the remaining inner symbols.\nAlso, the reduction in error rate compared to the classical\nconstellation is smaller for the inner symbols than for the outer\nsymbols. This explains why optimizing the constellation yields\na higher gain for 16-QAM than for 64-QAM.\nVII. C ONCLUSION\nIn this work, we have investigated the SER performance of\na THz band transmission with uniform multi-bit quantization\nand spatial oversampling at the receiver. At first, we consid-\nered the ADC power consumption in general and determined\ndifferent low-resolution reception schemes with equal ADC13\nTABLE III\nERROR RATES OF SINGLE QUADRATURE COMPONENT CONSTELLATION SYMBOLS x∈ X′AT OPTIMUM SNR FOR THE ML DETECTOR (24) WITH b= 1\nANDN= 64 .\nClassical constellation\nx∈ X′−7 −5 −3 −1 +1 +3 +5 +7\n16-QAM / / 3.9·10−46.0·10−46.0·10−43.9·10−4/ /\n36-QAM / 4.6·10−27.1·10−21.3·10−21.1·10−27.1·10−24.6·10−2/\n64-QAM 1.0·10−12.9·10−19.0·10−23.7·10−22.8·10−29.0·10−22.9·10−11.0·10−1\nOptimized constellation\nx∈ X′−a3 −a2 −a1 −1 +1 +a1 +a2 +a3\n16-QAM / / 7.6·10−415.6·10−73.3·10−77.6·10−41/ /\n36-QAM / 7.6·10−129.2·10−37.4·10−35.1·10−39.2·10−37.6·10−12/\n64-QAM 2.2·10−57.6·10−27.4·10−27.1·10−25.4·10−27.4·10−27.6·10−22.2·10−5\n00.20.40.60.81\n-1 -0.5 0 0.5 1\n(a) Classical 16-QAM.\n00.20.40.60.81\n-1 -0.5 0 0.5 1 (b) Optimized 16-QAM.\n00.20.40.60.81\n-1 -0.5 0 0.5 1\n(c) Classical 64-QAM.\n00.20.40.60.81\n-1 -0.5 0 0.5 1 (d) Optimized 64-QAM.\nFig. 19. PMF pd|x(d|x)(19) for b= 1,∆ = 2 ,N= 64 , and different\nconstellations at optimum SNR, respectively.\npower consumption for a fair comparison. Then, the statistics\nof the detection variable which is generated by combining the\nquantized observations have been analyzed. We have formu-\nlated the ML detector as well as several suboptimal detection\nschemes. It has been shown that the ML detector for 1-bit\nquantization has a significantly reduced complexity compared\nto multi-bit quantization. Furthermore, our results indicate that\nthe suboptimal detection schemes perform noticeably worse\nthan the optimal ML detector when the constellation size is\nlarger than the number of quantization levels of the ADC. We\nhave also shown that 1-bit quantization outperforms 2- and\n3-bit quantization at low SNRs even in case of a transmission\nwith more than 1 bpcu per real dimension. Then, we have\ndetermined SER-optimized 16-, 36-, and 64-QAM symbol\nconstellations for 1-bit quantization. Here, the minimum SER\ncan be decreased significantly by increasing the distance ofthe outermost symbols to the remaining inner symbols. The\neffect of this outer point movement is especially strong for\nsmall constellations, resulting in a SER reduction by several\norders of magnitude.\nIn future work, we want to generalize our findings on\noptimum constellations to multi-bit quantization and arbitrary\noversampling factors and constellation sizes. Furthermore,\nchannel estimation with low-resolution ADCs and the con-\nsequences of imperfect channel state information on the de-\ntection performance of our proposed detectors are of interest.\nAnother future topic of interest is channel coding for quantized\nreception and the exploitation of soft information available due\nto oversampling in the decoding process.\nREFERENCES\n[1] C. Forsch, P. Zillmann, O. Alrabadi, S. Brueck, and W. Gerstacker, “Per-\nformance analysis of 1-bit quantization with oversampling for higher-\norder constellations,” in Proc. of IEEE Latin-American Conference on\nCommunications (LATINCOM) , 2022, pp. 1–6.\n[2] I. F. Akyildiz, A. Kak, and S. Nie, “6G and beyond: The future of\nwireless communications systems,” IEEE Access , vol. 8, pp. 133 995–\n134 030, 2020.\n[3] B. Murmann, “The race for the extra decibel: A brief review of current\nADC performance trajectories,” IEEE Solid-State Circuits Magazine ,\nvol. 7, no. 3, pp. 58–66, 2015.\n[4] J. Singh, O. Dabeer, and U. Madhow, “On the limits of communication\nwith low-precision analog-to-digital conversion at the receiver,” IEEE\nTransactions on Communications , vol. 57, no. 12, pp. 3629–3639, 2009.\n[5] E. Gilbert, “Increased information rate by oversampling,” IEEE Trans-\nactions on Information Theory , vol. 39, no. 6, pp. 1973–1976, 1993.\n[6] S. Shamai, “Information rates by oversampling the sign of a bandlimited\nprocess,” IEEE Transactions on Information Theory , vol. 40, no. 4, pp.\n1230–1236, 1994.\n[7] T. Koch and A. Lapidoth, “Increased capacity per unit-cost by oversam-\npling,” in Proc. of IEEE 26-th Convention of Electrical and Electronics\nEngineers in Israel , 2010, pp. 000 684–000 688.\n[8] W. Zhang, “A general framework for transmission with transceiver dis-\ntortion and some applications,” IEEE Transactions on Communications ,\nvol. 60, no. 2, pp. 384–399, 2012.\n[9] O. Dabeer, J. Singh, and U. Madhow, “On the limits of communication\nperformance with one-bit analog-to-digital conversion,” in Proc. of IEEE\n7th Workshop on Signal Processing Advances in Wireless Communica-\ntions , 2006, pp. 1–5.\n[10] S. Krone and G. Fettweis, “Achievable rate with 1-bit quantization and\noversampling at the receiver,” in Proc. of IEEE Communication Theory\nWorkshop , 2010, pp. 1–2.\n[11] ——, “Capacity of communications channels with 1-bit quantization and\noversampling at the receiver,” in Proc. of 35th IEEE Sarnoff Symposium ,\n2012, pp. 1–7.\n[12] ——, “Communications with 1-bit quantization and oversampling at the\nreceiver: Benefiting from inter-symbol-interference,” in Proc. of IEEE\n23rd International Symposium on Personal, Indoor and Mobile Radio\nCommunications - (PIMRC) , 2012, pp. 2408–2413.14\n[13] L. T. N. Landau, M. D ¨orpinghaus, and G. P. Fettweis, “1-bit quantization\nand oversampling at the receiver: Sequence-based communication,”\nEURASIP Journal on Wireless Communications and Networking , vol.\n2018, no. 1, Apr. 2018.\n[14] S. Jacobsson, G. Durisi, M. Coldrey, U. Gustavsson, and C. Studer,\n“One-bit massive MIMO: Channel estimation and high-order modula-\ntions,” in Proc. of IEEE International Conference on Communications\nWorkshops (ICCW) , 2015, pp. 1304–1309.\n[15] J. Zhang, L. Dai, S. Sun, and Z. Wang, “On the spectral efficiency of\nmassive MIMO systems with low-resolution ADCs,” IEEE Communi-\ncations Letters , vol. 20, no. 5, pp. 842–845, 2016.\n[16] S. Jacobsson, G. Durisi, M. Coldrey, U. Gustavsson, and C. Studer,\n“Throughput analysis of massive MIMO uplink with low-resolution\nADCs,” IEEE Transactions on Wireless Communications , vol. 16, no. 6,\npp. 4038–4051, 2017.\n[17] T. Halsig, L. Landau, and G. Fettweis, “Spectral efficient communica-\ntions employing 1-bit quantization and oversampling at the receiver,” in\nProc. of IEEE 80th Vehicular Technology Conference (VTC2014-Fall) ,\n2014, pp. 1–5.\n[18] U. Ugurlu and R. Wichman, “Enabling low-resolution ADC with high-\norder modulations for millimeter-wave systems,” in Proc. of IEEE\nInternational Conference on Communications (ICC) , 2016, pp. 1–6.\n[19] B. Sun, Y . Zhou, J. Yuan, Y .-F. Liu, L. Wang, and L. Liu, “High order\nPSK modulation in massive MIMO systems with 1-bit ADCs,” IEEE\nTransactions on Wireless Communications , vol. 20, no. 4, pp. 2652–\n2669, 2021.\n[20] J. Choi, D. J. Love, D. R. Brown, and M. Boutin, “Quantized distributed\nreception for MIMO wireless systems using spatial multiplexing,” IEEE\nTransactions on Signal Processing , vol. 63, no. 13, pp. 3537–3548, 2015.\n[21] S. Wang, Y . Li, and J. Wang, “Multiuser detection in massive MIMO\nwith quantized phase-only measurements,” in Proc. of IEEE Interna-\ntional Conference on Communications (ICC) , 2015, pp. 4576–4581.\n[22] J. Choi, J. Mo, and R. W. Heath, “Near maximum-likelihood detector\nand channel estimator for uplink multiuser massive MIMO systems with\none-bit ADCs,” IEEE Transactions on Communications , vol. 64, no. 5,\npp. 2005–2018, 2016.\n[23] S.-N. Hong, S. Kim, and N. Lee, “A weighted minimum distance de-\ncoding for uplink multiuser MIMO systems with low-resolution ADCs,”\nIEEE Transactions on Communications , vol. 66, no. 5, pp. 1912–1924,\n2018.\n[24] A. B. ¨Uc ¸¨unc¨u, E. Bj ¨ornson, H. Johansson, A. ¨O. Yılmaz, and E. G.\nLarsson, “Performance analysis of quantized uplink massive MIMO-\nOFDM with oversampling under adjacent channel interference,” IEEE\nTransactions on Communications , vol. 68, no. 2, pp. 871–886, 2020.\n[25] Y . Nakashima, T. Yamazato, Y . Tadokoro, and S. Arai, “A stochastic\nresonance receiver for 4-PAM signals,” in Proc. of IEICE International\nSymposium on Nonlinear Theory and Its Applications (NOLTA2017) ,\n2017, pp. 291–294.\n[26] Y . Nakashima, T. Yamazato, S. Arai, H. Tanaka, and Y . Tadokoro,\n“Noise-aided demodulation with one-bit comparator for multilevel pulse-\namplitude-modulated signals,” IEEE Wireless Communications Letters ,\nvol. 7, no. 5, pp. 848–851, 2018.\n[27] P. Neuhaus, M. D ¨orpinghaus, H. Halbauer, S. Wesemann, M. Schl ¨uter,\nF. Gast, and G. Fettweis, “Sub-THz wideband system employing 1-bit\nquantization and temporal oversampling,” in Proc. of IEEE International\nConference on Communications (ICC) , 2020, pp. 1–7.\n[28] D. He, Z. Wang, T. Q. S. Quek, S. Chen, and L. Hanzo, “Deep learning-\nassisted Terahertz QPSK detection relying on single-bit quantization,”\nIEEE Transactions on Communications , vol. 69, no. 12, pp. 8175–8187,\n2021.\n[29] Y . Zhang, D. Li, D. Qiao, and L. Zhang, “Analysis of indoor THz com-\nmunication systems with finite-bit DACs and ADCs,” IEEE Transactions\non Vehicular Technology , vol. 71, no. 1, pp. 375–390, 2022.\n[30] X. Cheng, B. Xia, K. Xu, and S. Li, “Bayesian channel estimation\nand data detection in oversampled OFDM receiver with low-resolution\nADC,” IEEE Transactions on Wireless Communications , vol. 20, no. 9,\npp. 5558–5571, 2021.[31] H. Sarieddeen, M.-S. Alouini, and T. Y . Al-Naffouri, “An overview of\nsignal processing techniques for Terahertz communications,” Proceed-\nings of the IEEE , vol. 109, no. 10, pp. 1628–1665, 2021.\n[32] J. M. Jornet and I. F. Akyildiz, “Channel modeling and capacity analysis\nfor electromagnetic wireless nanonetworks in the Terahertz band,” IEEE\nTransactions on Wireless Communications , vol. 10, no. 10, pp. 3211–\n3221, 2011.\n[33] B. Wang, F. Gao, S. Jin, H. Lin, and G. Y . Li, “Spatial- and frequency-\nwideband effects in millimeter-wave massive MIMO systems,” IEEE\nTransactions on Signal Processing , vol. 66, no. 13, pp. 3393–3406, 2018.\n[34] I.-S. Kim and J. Choi, “Spatial wideband channel estimation for\nmmWave massive MIMO systems with hybrid architectures and low-\nresolution ADCs,” IEEE Transactions on Wireless Communications ,\nvol. 20, no. 6, pp. 4016–4029, 2021.\n[35] E. Vlachos, A. Kaushik, and M. Z. Shakir, “Millimeter wave channel\nestimation for lens based hybrid MIMO with low resolution ADCs,” in\nProc. of IEEE International Conference on Communications Workshops\n(ICC Workshops) , 2023, pp. 1789–1793.\n[36] A. Doshi and J. G. Andrews, “One-bit mmWave MIMO channel estima-\ntion using deep generative networks,” IEEE Wireless Communications\nLetters , vol. 12, no. 9, pp. 1593–1597, 2023.\n[37] T. Mewes, S. Zeitz, P. Neuhaus, M. D ¨orpinghaus, and G. Fettweis,\n“Channel estimation for two-wave with diffuse power fading channels\nunder 1-bit quantization,” in Proc. of IEEE Wireless Communications\nand Networking Conference (WCNC) , 2023, pp. 1–6.\n[38] A. Mezghani and J. A. Nossek, “How to choose the ADC resolution for\nshort range low power communication?” in Proc. of IEEE International\nSymposium on Circuits and Systems , 2010, pp. 1025–1028.\n[39] S. Krone and G. Fettweis, “Energy-efficient A/D conversion in wideband\ncommunications receivers,” in Proc. of IEEE Vehicular Technology\nConference (VTC2011-Fall) , 2011, pp. 1–5.\n[40] P. Neuhaus, M. Schl ¨uter, C. Jans, M. D ¨orpinghaus, and G. Fettweis,\n“Enabling energy-efficient Tbit/s communications by 1-bit quantization\nand oversampling,” in Proc. of Joint European Conference on Networks\nand Communications & 6G Summit (EuCNC/6G Summit) , 2021, pp.\n84–89.\n[41] O. Casta ˜neda, S. H. Mirfarshbafan, S. Ghajari, A. Molnar, S. Jacob-\nsson, G. Durisi, and C. Studer, “Resolution-adaptive all-digital spatial\nequalization for mmWave massive MU-MIMO,” in Proc. of IEEE 22nd\nInternational Workshop on Signal Processing Advances in Wireless\nCommunications (SPAWC) , 2021, pp. 386–390.\n[42] R. Walden, “Analog-to-digital converter survey and analysis,” IEEE\nJournal on Selected Areas in Communications , vol. 17, no. 4, pp. 539–\n550, 1999.\n[43] B. Murmann, “ADC performance survey 1997-2022.” [Online].\nAvailable: http://web.stanford.edu/ ∼murmann/adcsurvey.html\n[44] W. Feller, An Introduction to Probability Theory and Its Applications ,\n3rd ed. John Wiley & Sons, 1968, vol. 1.\n[45] T. M. Cover and J. A. Thomas, Elements of Information Theory , 2nd ed.\nJohn Wiley & Sons, 2006.\n[46] A. Moldovan, M. A. Ruder, I. F. Akyildiz, and W. H. Gerstacker, “LOS\nand NLOS channel modeling for terahertz wireless communication with\nscattered rays,” in Proc. of IEEE Globecom Workshops (GC Wkshps) ,\n2014, pp. 388–392.\n[47] R. Piesiewicz, C. Jansen, D. Mittleman, T. Kleine-Ostmann, M. Koch,\nand T. Kurner, “Scattering analysis for the modeling of THz commu-\nnication systems,” IEEE Transactions on Antennas and Propagation ,\nvol. 55, no. 11, pp. 3002–3009, 2007.\n[48] L. John, A. Tessmann, A. Leuther, P. Neininger, T. Merkle, and T. Zwick,\n“Broadband 300-GHz power amplifier MMICs in InGaAs mHEMT\ntechnology,” IEEE Transactions on Terahertz Science and Technology ,\nvol. 10, no. 3, pp. 309–320, 2020.\n[49] S. Priebe, M. Jacob, and T. K ¨urner, “The impact of antenna directivities\non THz indoor channel characteristics,” in Proc. of 6th European\nConference on Antennas and Propagation (EUCAP) , 2012, pp. 478–\n482."    },    {        "title": "2402.04740v1.Non_Parametric_Estimation_of_Multi_dimensional_Marked_Hawkes_Processes.pdf",        "content": "Non-Parametric Estimation of Multi-dimensional\nMarked Hawkes Processes\nSobin Joseph\nDepartment of Management Studies\nIndian Institute of Science\nBangalore 560012\nsobinjoseph@iisc.ac.inShashi Jain\nDepartment of Management Studies\nIndian Institute of Science\nBangalore 560012\nshashijain@iisc.ac.in\nAbstract\nAn extension of the Hawkes process, the Marked Hawkes process distinguishes\nitself by featuring variable jump size across each event, in contrast to the constant\njump size observed in a Hawkes process without marks. While extensive literature\nhas been dedicated to the non-parametric estimation of both the linear and non-\nlinear Hawkes process, there remains a significant gap in the literature regarding\nthe marked Hawkes process. In response to this, we propose a methodology for\nestimating the conditional intensity of the marked Hawkes process. We introduce\ntwo distinct models: Shallow Neural Hawkes with marks - for Hawkes processes\nwith excitatory kernels and Neural Network for Non-Linear Hawkes with Marks -\nfor non-linear Hawkes processes. Both these approaches take the past arrival times\nand their corresponding marks as the input to obtain the arrival intensity. This\napproach is entirely non-parametric, preserving the interpretability associated with\nthe marked Hawkes process. To validate the efficacy of our method, we subject the\nmethod to synthetic datasets with known ground truth. Additionally, we apply our\nmethod to model cryptocurrency order book data, demonstrating its applicability\nto real-world scenarios.\nKEYWORDS: Marked Hawkes processes, non-parametric estimation, online learning, cryptocurrency\norder book\n1 Introduction\nHawkes process proposed by A.G. Hawkes is a self or mutually exciting multivariate point process\nwith the intensity function dependent on time steps of past events Hawkes [1971]. Marked Hawkes\nprocess is an extended version of the Hawkes process, where the conditional intensity function\nconsiders the past timestamps of events and the associated mark with each timestamp. These marks\ncan represent various factors, aside from event time, that directly affect the intensity of the events.\nConsider the scenario of market order arrivals in a exchange; the time of the next market order is not\nonly determined by the history of order arrivals but is also influenced by the history of past order\nvolumes. This illustrates a marked Hawkes process where, in an arrival process, the occurrence of the\nnext event depends on both the arrival history and the mark history (in this example, order volume\nserves as the mark component). The marked Hawkes process has found applications in diverse\nfields such as seismology where magnitude and position of the earthquake are the mark variables\n(Ogata [1998], Zhuang et al. [2004], and Fox et al. [2016]), in social networks where the number of\nfollowers serves as a mark category (Kobayashi and Lambiotte [2016], and Chen and Tan [2018]), in\nfinance which has volume (Chavez-Demoulin and McGill [2012] and Fauth and Tudor [2012]) and\nthe occurrence of extreme events (Embrechts et al. [2011], and Stindl and Chen [2019]) as the mark\nvariable and, in criminology with fraud transactions as marks (Yuan et al. [2019], and Narayanan\nPreprint. Under review.arXiv:2402.04740v1  [stat.ML]  7 Feb 2024et al. [2022]). In a Hawkes process without marks, when an event occurs, it can excite or inhibit more\nevents. The size of each excitation or decay, referred to as the jump size, remains constant across all\nevents. In contrast, in a marked Hawkes process, an event’s impact on future events is characterized\nby varying jump sizes. Figure 1 illustrates the distinction between a Hawkes process and a marked\nHawkes through an intensity-time plot. In this plot, we observe a constant intensity jump for Hawkes\nand a variable jump for the marked Hawkes process.\n(a) Hawkes Process\n (b) Marked Hawkes Process\nFigure 1: Comparison between Hawkes process and Marked Hawkes process\nFor a marked Hawkes process, similar to a Hawkes process, the intensity function combines exogenous\nbackground rate and endogenous kernels. This combination can take linear or non-linear forms. A\nHawkes process, which models the self-excitability of events based on their past occurrences, can\nbe limited in explaining dependencies related to the characteristics of the event, such as volume,\nprice, magnitude, number of followers or any other relevant features. Consider the earlier example of\nmarket order arrivals in an exchange; even though a Hawkes process without marks can explain the\ndependency of an order arrival with its past arrival, it won’t be able to explain the effects of order\nvolume on market order evolution. But considering it as a marked Hawkes process allows the model\nto capture not only the temporal dependencies between successive orders but also the impact of the\nvolume of past orders on the evolution of the market. This is essential in understanding how the\narrival of market orders is influenced by past order volumes. Take another example of earthquake\naftershocks: the magnitude of an earthquake serves as a crucial mark that influences the likelihood\nof further aftershocks. A marked Hawkes process enables the modelling of the arrival intensity of\naftershocks, considering both temporal dependencies between the aftershocks and the influence of\ntheir magnitudes.\nThe kernels in a marked Hawkes process intensity function play a pivotal role in capturing the\ntemporal and mark dependencies. They quantify the influence of past events and their marks on\nthe likelihood of future events. Thus understanding the form of these kernels is crucial for accurate\nmodelling and estimation in the marked Hawkes process.\nHence here, we propose a novel feed-forward neural network-based approach to model the marked\nHawkes process kernel function. Our method utilizes a 2-layered feed-forward neural network to\ncapture the Hawkes kernel using the history of event times and their associated marks. The network\ndesign includes a hidden layer and an output layer for approximating the kernel function that receives\nboth the historical data of past arrivals and their corresponding marks as the input. This modelling\napproach extends the neural network-based models, specifically the Shallow Neural Hawkes (SNH)\n(Joseph et al. [2022]) and the Neural Network for Non-Linear Hawkes (NNNH) (Joseph and Jain\n[2023]), which were initially designed for Hawkes processes without marks.\n1.1 Literature Review of Marked Hawkes Process\nWhile there has been considerable research on estimating the Hawkes kernels and base intensity,\nthe literature focused specifically on the marked Hawkes process is relatively scarce. Most of the\navailable studies on estimating the marked Hawkes process primarily revolve around parametric\nassumptions, with only a few exceptions exploring non-parametric methods.\nThe early studies of the marked Hawkes process primarily focused on predicting earthquake after-\nshocks, known as ETAS (Epidemic-Type Aftershock Sequence). The ETAS model was developed in\nOgata [1988] and Ogata [1999], considering past earthquake event times as the temporal variable\n2and earthquake magnitude as the mark variable. Subsequently, Zhuang et al. [2004] modified this\napproach to incorporate the event location as an additional spatial mark factor. It’s worth noting that\nboth methods rely on parametric assumptions for the kernel and mark function. Embrechts et al.\n[2011] leans towards parametric assumptions, applying the marked Hawkes process to model extreme\nprice moves in stock market index data. Similarly, Chen and Tan [2018] adopts a parametric marked\nHawkes process to model retweet cascading on Twitter data.\nTransitioning to non-parametric approaches for the marked Hawkes process, Bacry and Muzy [2016]\nuses the Wiener-Hopf integral, while Fox et al. [2016] employs the EM algorithm for estimating the\nmarked Hawkes kernel and base intensity. Although both models are non-parametric, they model the\ntemporal and mark part of the kernel as distinct functions with no interdependence. Furthermore,\nseveral non-parametric techniques based on neural networks have been proposed. For instance,\nMei and Eisner [2017], Du et al. [2016] and Shchur et al. [2019] utilize recurrent neural network\n(RNN)-based models with maximum likelihood estimation (MLE) as the loss function to estimate\nthe marked Hawkes process. Another approach, as proposed by Xiao et al. [2017], adopts the RNN-\nbased Wasserstein generative adversarial network (WGAN) to learn the marked Hawkes process.\nHowever, it’s worth noting that RNNs may face challenges in capturing extended dependencies and\nencounter issues like gradient vanishing and gradient explosion, as highlighted by Pascanu et al.\n[2013]. Introducing a different perspective, Zuo et al. [2020] presents the transformer Hawkes,\nwhich utilizes self-attention modules combined with feed-forward networks to learn the marked\nHawkes process. Fabre and Toke [2024] utilize the physics-informed neural network (PINN) to\nestimate the kernels of a marked Hawkes process. This model incorporates the first and second-order\ncharacteristics equation of Bacry and Muzy [2016] to facilitate the training of the neural network.\n1.2 Contribution and Organization of the paper\nThe aforementioned methods employ neural networks to directly model the conditional intensity\nfunction, making it challenging to deduce the causal relationships between each event and its historical\ncontext. Even though neural network-based methods have conducted joint analyses of marks and time,\nthose approaches typically handle discrete marks (marks in Z+). In the approaches that consider\nmarks as continuous, such as Fabre and Toke [2024], the marked Hawkes kernel function is taken as\na decoupled function of time and mark.\nThus, as our first contribution, we propose a novel approach based on neural networks, which can\napproximate the marked Hawkes process kernel instead of the intensity function. This allows for\ninferring causal relationships between events across each dimension using the estimated kernels. The\ntwo-layered feed-forward network accommodates continuous marks, enabling a more comprehensive\nmodelling of the marked Hawkes process. The parameters of the model are estimated by maximizing\nthe log-likelihood function. Since the log-likelihood function is non-convex for a Hawkes process,\neven for a parametric setting (Joseph et al. [2022]), we use the stochastic gradient descent (SGD)\nmethod to derive the network parameters. The model represents a novel approach capable of jointly\nmodelling a marked Hawkes kernel’s temporal and mark dependency.\nThe application of the proposed method to the high-frequency cryptocurrency trading data is our\nnext contribution. The dataset comprises the arrival times and their associated volumes for market\norders in Bitcoin-US Dollar and Ethereum-US Dollar pairs. From the proposed method, we obtain\nthe underlying kernels, providing insights into the relationship of an event concerning both the history\nof time and the mark. This improves predictions of future arrivals and uncovers connections between\narrival patterns, historical data, and associated order volumes. This insight can provide a clearer\nunderstanding of the underlying dynamics of the market’s microstructure.\nThe remainder of the paper is structured as follows. Section 2 provides the definition of the marked\nHawkes process and its associated log-likelihood function. In Section 3, we establish the neural\nnetwork models of the SNH with marks and the NNNH with marks. Section 4 is dedicated in\nvalidating the performance of our proposed model, with evaluations conducted on simulated datasets\nwhere the ground truth is known. Furthermore, we apply the estimation method to a real-life\ncryptocurrency market order arrivals dataset, considering order volume as the mark component.\nFinally, Section 5 presents a concise conclusion of the paper.\n32 Preliminary Definitions\nDefinition of the Marked Hawkes Process: As per the definition provided by Daley and Vere-\nJones [2007], a marked point process N(X × M ), with event time in the space Xand marks in the\nspaceM, is a point process {(tn, mn)}n≥1onX × M with the additional property that ground\nprocess Ng(.)(Ngconstitutes the collection of the event times {tn}) is itself a point process; i.e. for\na bounded set A∈BX(where BXis the Borel set of the space X),Ng(A) =N(A×M)<∞.\nA D-dimensional marked point process defined as {N1(X × M ), . . . , N D(X × M )}, where each\nNd(X × M ) =Ng\nd(t)ford= 1, . . . , D represents a marked point process. We introduce H≡\nHt:t≥0as the internal history, where Htis the history of the process up to and including time\nt, andHt−is the history of the process up to but not including time t.For the given marked point\nprocess N:={Nd(X × M )}D\nd=1, the mark space Mcan take various forms: it can be discrete, as\nin the case where marks represent the number of fraudulent transactions (Narayanan et al. [2022]\nor the number of Twitter followers (Kobayashi and Lambiotte [2016]), positive continuous when\nmarks represent energy (Ogata [1998]), volume (Chavez-Demoulin and McGill [2012]) or price\n(Lee and Seo [2017]), or even multidimensional Euclidean space for space-time processes. In this\npaper, without loss of generality, we explore examples where Mis positively continuous, denoted as\nM ∈R+, andX ∈[0, T).\nThedth dimensional conditional intensity of the marked point process Ndefined on [0, T)× M ,\nwith respect to its internal history Hrepresented by λd(t, m)is given by,\nλd(t, m)dt dm ≈E[Nd(dt×dm)|Ht−]. (1)\nThe given intensity λd(t, m)can also be represented as,\nλd(t, m) =λg\nd(t|Ht−)fd(m|t,Ht−), (2)\nwhere λg\nd(t|Ht−),is the conditional intensity of the ground process (from this point onwards, for\nease of notation we will denote λg\nd(t|Ht−)asλg\nd(t)), and fd(m|t,Ht−)is the conditional density of\na mark at tgivenHt−.This emphasizes the significance of the distribution of marks in defining the\nHawkes intensity function. Consider the following definitions from Daley and Vere-Jones [2007]:\n•Independent marks: Nd:=Nd([0, T)× M )has independent marks if, for a given set of\nevent times {td\nn}n≥1,the marks {md\nn}n≥1are mutually independent random variables such\nthat the distribution of md\nndepends only on the corresponding event time td\nn.\n•Unpredictable marks: Ndhas unpredictable marks if the distribution of mark at tis inde-\npendent of its history Ht−.\nIn the context of a marked counting process, the independence of marks implies that both marks and\nthe event times are independent of each other, i.e. the marks don’t influence event times and vice-versa.\nAdditionally, unpredictable marks occur when the distribution of marks can influence the subsequent\nevolution of the event times, but event times don’t influence the distribution of marks. This work’s\nproposed models are suitable for independent and unpredictable marks. For both the independent and\nunpredictable marks case, the conditional mark density is i.i.d. with fd(m|t,Ht−) =fd(m).\nThe conditional intensity, λd(t, m),for the d-th dimension of a Ddimensional non-linear marked\nHawkes process can be expressed as:\nλd(t, m) = Ψ d\nµd(t) +DX\nj=1X\n{∀k|(tj\nk<t)}ϕdj(t−tj\nk, mj\nk)\nfd(m), (3)\nwhere Ψd:R→R+,is a non-negative non-linear function, which is required to be Lipschitz\ncontinuous (Brémaud and Massoulié [1996]), µd(t) :R+→R+,is the exogenous base intensity for\nthed-th dimension, and ϕdj(t−tj\nk, mj\nk),1≤d, j≤D,are the kernels that quantify the magnitude\nof excitation or decay of the intensity for the d-th dimension at time tdue to the past arrivals tj\nk,and\npast marks mj\nk,{∀k|tj\nk< t}in the jth dimension. These kernel functions have their support in R.\n4The intensity function provided in Equation 3 serves as a comprehensive representation of the marked\nHawkes process, wherein the Hawkes kernel ϕdjis influenced by both the histories of marks and\nevent times. However, alternative modelling approaches exist for capturing the impact of marks on\nthe process.\nA commonly employed marked intensity function in the literature is expressed as follows:\nλd(t, m) = Ψ d\nµd(t) +DX\nj=1X\n∀k|(tj\nk<t)ψdj(mk\nj)ϕdj(t−tj\nk)\nfd(m), (4)\nSimilar to Equation 3, this formulation involves the conditional ground intensity, which is a function\nof past event times and corresponding mark values. However, unlike Equation 3, it assumes that\nthe excitation kernel can be decomposed into a function of marks and a function of past event time\nstamps.\nIn cases where marks are independent and do not influence the evolution of events, the conditional\nintensity for the marked version is written as:\nλd(t, m) = Ψ d\nµd(t) +DX\nj=1X\n∀k|(tj\nk<t)ϕdj(t−tj\nk)\nfd(m). (5)\nHere, the marks contribute nothing to the Hawkes kernel, and consequently, they do not impact\nthe ground intensity function, λg\nd(t). Equation 3, 4, and 5 represent different definitions of marked\nHawkes processes in terms of mark dependencies. From a model-building perspective, Equations 4\nand 5 can be generalized as Equation 3. Therefore, it is necessary to consider only Equation 3 for\nestimation. To the best of our knowledge, the proposed model is the first non-parametric approach to\nmodel the process defined by Equation 3.\nEstimating a marked Hawkes process requires obtaining the base intensity function µd, and its kernel\nfunctions, ϕdj,either by assuming a parametric or non-parametric form for the kernels. Similary for\nestimating the mark density function, fd(m), one can adopt either a parametric or non-parametric\napproach. The prevalent method for estimating a Hawkes process typically entails maximizing the\nlog-likelihood function of the process.\nThe Log-Likelihood Function of Marked Hawkes Process The categorization of methods used\nfor estimating marked Hawkes processes can be broadly classified into two groups:\n•Intensity-based approach: The most commonly used approach for estimating a marked\nHawkes process involves deriving the parameters using the conditional intensity function,\nλd(t, m). The maximum likelihood estimation (MLE) is the prevailing method in this\nregard. Notably, this MLE approach has been employed by various researchers, including\nDu et al. [2016], Mei and Eisner [2017], and Zuo et al. [2020], among others. However,\nit’s worth noting that MLE methods can be computationally expensive, with their time\ncomplexity increasing quadratically with the number of event arrivals. This can pose chal-\nlenges when dealing with large datasets or complex models. To address this computational\ncost, alternative approaches like the Expectation-Maximization (EM) method have been\nused by researchers such as Fox et al. [2016] and Yuan et al. [2019]. EM also utilizes\nthe likelihood function as the loss function, but it offers more computationally efficient\noptimization strategies compared to traditional MLE.\n•Intensity free approach: The use of intensity free approach for estimating marked Hawkes is\nvery scarce. Among them, Xiao et al. [2017] and Shchur et al. [2019] employ the Wasserstein\ndistance and conditional distribution function, respectively, to estimate the Marked Hawkes\nprocess. While Bacry and Muzy [2016] and Fabre and Toke [2024] use the second-order\ncharacteristics equation of a marked Hawkes kernel to estimate the base intensity and kernel\nof a marked Hawkes process.\nIn this paper, we use the maximum log-likelihood(MLE) approach to estimate the base intensity\nintensity and the optimal kernels of the intensity function.\n5In aD-dimensional non-linear marked Hawkes process, {Nd[0, T)× M} with the parameters\nµ= [µd(t)]D×1,Φ= [ϕdj(t, m)]D×Dandf= [fd(m)]D;1≤d, j≤Dbe the base intensity\nintensity, kernel function, and mark density function respectively. Let θ= [θ1, . . . θ l. . . θ L]be the\nset of parameters used to model µ,Φandf. These parameters can be estimated by maximizing\nthe log-likelihood function over the events, Hsampled from the process. The log-likelihood (LL)\nfunction, Lcorresponding to Hfor the non-linear marked Hawkes process is given by (see for\ninstance Daley and Vere-Jones [2007]),\nL(θ,H) =DX\nd=1\nX\n{(tdn,mdn)}∈Hlog(λd(td\nn, md\nn))−ZT\n0Z\nMλd(s, m)dsdm\n, (6)\napplying the λd(t, m)from the Equation 2 in to the equation,\nL(θ,H) =DX\nd=1\nX\n{(tdn,mdn)}∈H\b\nlog(λg\nd(td\nn)) + log( fd(md\nn))\t\n−ZT\n0Z\nMλg\nd(s)fd(m)dsdm\n.\nAs the sum of mark density function across the marked space Mis unity, i.e.R\nMfd(m)dm= 1, the\nabove equation can be written as,\nL(θ,H) =DX\nd=1\nX\n{(td\nn,md\nn)}∈Hlog(λg\nd(td\nn))−ZT\n0λg\nd(s)ds+X\n{md\nn}∈Hfd(md\nn)\n.\nThe provided LL function Lcan be expressed as a sum of two terms, L=Lλ+Lf, where Lλis the\nlog-likelihood function for the ground intensity and Lfis the log-likelihood function for the marks\ndensity function.\nLλ(θλ,H) =DX\nd=1\nX\n{(tdn,mdn)}∈Hlog(λg\nd(td\nn))−ZT\n0λg\nd(s)ds\n, (7)\nLf(θf,H) =DX\nd=1\nX\n{mdn}∈Slog(fd(md\nn))\n, (8)\nWhere (θλ,θf)∈θand,θλandθfare the parameter sets for the intensity function and density\nfunction, respectively. There are no parameters that appear in both LλandLf. Thus, both terms can\nbe maximized separately.\nFrom Equation 3, the λg\nd(t)inLλcan be written as a combination of a base intensity rate µd(t)\nand kernel function ϕdj(t, m). Even for the most common parametric form of the ϕdj(t, m), theLλ\nfunction may or may not be convex. For example, take the simplest case of the exponential kernel\nwith multiplicative marks, ϕdj(t, m) =me−βt,Lλwon’t be a convex function. Hence, the Stochastic\nGradient Descent (SGD) method is employed to find the local optimum of the Lλfunction in the\nparametric space. To estimate the parameters θusing SGD, an unbiased estimator of the gradient of\nLλwith respect to each parameter θp∈θλis required and is given by,\n∇θp(Lλ(θλ,H)) =DX\nd=1\nX\n{(tdn,mdn)}∈H∇θp (\nlog(λg\nd(td\nn))−Ztd\nn\ntd\nn−1λg\nd(s)ds)!\n.\nFollowing the above equation, for each event points (td\nn, md\nn), an unbiased estimator of the gradient\nofLλwith respect to each of the parameters θpwill be,\n∇θpbLλ(θλ, td\nn, md\nn) :=∇θp \nlog(λg\nd(td\nn))−Ztd\nn\ntd\nn−1λg\nd(s)ds!\n, (9)\n6Figure 2: Flowchart showing process for estimation of Marked Hawkes process\nwhere td\nnandmd\nnare sampled from H.In this context, we aim to maximize bLλby adopting a non-\nparametric approach for the kernel function and base intensities. The detailed model is elaborated in\nSection 3.\nBecause the log-likelihood is separable, we can independently fit the conditional mark density. In our\napproach, we employ the Gaussian mixture model (GMM) as a non-parametric method to estimate\nthe mark density. This entails identifying the GMM parameters, denoted as θf,that optimize the\nlog-likelihood function Lf.Further details on this process can be found in Section 3.1. By combining\nthe non-parametric estimation of the kernel function and base intensities along with the GMM-based\napproach for Lf, we aim to comprehensively and accurately estimate the parameters for the marked\nHawkes process model. Figure 2 provides an overview of the marked Hawkes process estimation for\neach of the dth dimensions. On the left side of the chart, marks {md\nn} ∈Hserve as input to the mark\ndensity likelihood function, Lfdfrom Equation 8, which is then processed by the GMM method to\nestimate the mark density function, fd. This estimation results in the prediction of subsequent marks\nfor each dimension d. On the right side of the chart, both marks and historical time stamps Htdn−are\nutilized as inputs for the intensity likelihood function, Lλdgiven in Equation 7. The SNH with marks,\nor the NNNH with marks estimation approaches are employed to estimate the network parameters of\n7the ground intensity function, λg\nd. The obtained functional form of λg\ndis used to generate future event\ntimes, td\nn.\n3 Proposed Model\nIn the context of the internal history H, we estimate the Hawkes conditional intensity function λd(t)\nby separately modelling the base intensity µd(t)and the kernel ϕdj(t, m). Estimating the Hawkes\nkernel is crucial in understanding the underlying temporal and mark dependency with the conditional\nintensity (as explained in Section 1). Our approach employs a two-layered neural network with a\nsingle hidden layer for approximating the Hawkes kernel ϕdj. The choice of using a two-layered\nneural network is motivated by the universal approximation theorem established by Hornik et al.\n[1989], which proves that a multilayered neural network with at least one single hidden layer within\na certain group of activation functions can approximate any continuous function in a compact set\narbitrarily well. This theorem has been extended to include a wider range of activation functions,\nincluding exponential and Rectified Linear Units (ReLu) by Leshno et al. [1993]. We refrain from\nemploying a deeper network because estimating the network parameters through maximizing the\nlog-likelihood involves evaluating the time integral of the kernels. Using a shallow network allows for\nefficient computation of this integral using analytical expressions. In contrast, with deeper networks,\nthe computation of the integral becomes computationally intensive.\nThis paper proposes two distinct models based on neural networks for estimating the marked Hawkes\nprocess –Shallow Neural Hawkes with marks (SNH with Marks) and Neural Network for Non-linear\nHawkes with marks (NNNH with marks). The proposed models are the extensions of the SNH\n(Shallow Neural Hawkes) model Joseph et al. [2022] and the NNNH (Neural Network for Non-linear\nHawkes) model Joseph and Jain [2023]. While both the SNH and NNNH use only event times for\nestimating the Hawkes kernel, the extended marks model incorporates both event times and their\ncorresponding marks, making it a more versatile estimation of the kernel function.\nShallow Neural Hawkes with Marks In their research, Joseph et al. [2022] proposed the Shallow\nNeural Hawkes (SNH) method, which is a neural network-based approach designed for estimating\nthe kernels of a linear Hawkes process, specifically for process with excitation or positive kernels. Ex-\npanding upon the SNH model, the Shallow Neural Hawkes with marks (SNH with marks) incorporates\nboth event times and their corresponding marks in the Hawkes intensity function. This enhancement\nenables the method to handle Hawkes processes involving temporal information (event times) and\nmark information.\nThe SNH with marks is specifically designed for the linear Hawkes process, which makes it suitable\nonly for positive or excitatory kernels, constraining the kernel function ˆϕdj(t, m)to be bounded\ninR+. A Linear Hawkes process implies linearity in intensity function, λg\nd(t). Therefore, in the\nexpression of λg\nd(t)in Equation3, Ψdis an identity function. Thus, the estimated ground intensity\nfunction ˆλg\nd(t)can be expressed as follows:\nˆλg\nd(t) = ˆµd(t) +DX\nj=1X\n{∀k|(tj\nk<t)}ˆϕdj(t−tj\nk, mj\nk),\nwhere each ˆµd(t),andˆϕdj(t−tj\nk, mj\nk), d= 1, . . . , D, are approximated using a neural network.\nGivenH, the kernel ˆϕdj(t, m) :R+×R→R+is modelled as,\nˆϕdj(t, m) = Π◦A2◦φ◦A1, (10)\nwhere A1is the hidden layer, A1:R+×R→RPandA2is the output layer, A2:RP→R+. More\nprecisely,\nA1(x) = W1x+b1forx∈R2×1, W1∈RP×2andb1∈RP,\nA2(y) = W2y+b2fory∈RP, W2∈R1×Pandb2∈R,\nwhere Pis the number of neurons. Also, φ:R→R+,1≤i≤Pis the ReLU activation function\ngiven by,\nφ(yi) = max( yi,0),\n8and function Π:R→R+, is the exponential function. We take W1=\u0014\na1\n1a2\n1. . . aP\n1\na1\n2a2\n2. . . aP\n2\u0015T\n,\nW2=\u0002\na1\n3, a2\n3, . . . , aP\n3\u0003\n, andb1=\u0002\nb1\n1, b2\n1, . . . , bP\n1\u0003\n. Thus, the kernel function ˆϕdjcan be written as,\nˆϕdj(t, m) =exp \nb2+pX\ni=1ai\n3max\u0000\nai\n1t+ai\n2m+bi\n1,0\u0001!\n. (11)\nNeural Network for Non-Linear Hawkes with Marks To overcome the limitation of the SNH\nwith marks, which is confined to linear Hawkes processes with only excitation kernel functions,\nwe propose a novel model called Neural Network for Non-linear Hawkes with Marks (NNNH with\nmarks). Adapted from the Neural Network for Non-linear Hawkes (NNNH) method introduced by\nJoseph and Jain [2023], the NNNH with marks can model both excitatory and inhibitory kernels\nin a non-linear marked Hawkes process. This means that the kernel function ˆϕdjcan take negative\nvalues with the range of the function mapping to R. To guarantee that the intensity function λg\nd(t)\nremains positive, the function Ψdin Equation 3 is selected as the ReLU function. By employing the\nRELU function, the estimated Hawkes intensity ˆλg\nd(t)is guaranteed to remain positive throughout the\nestimation, thus adhering to the necessary constraint. The framework is flexible enough to consider\nthe base intensity µd(t)as a function of time. Thus the estimated Hawkes ground intensity function\nˆλg\nd(t)for the NNNH with marks is given by,\nˆλg\nd(t) = max\nˆµd(t) +DX\nj=1X\n{∀k|(tj\nk<t)}ˆϕdj(t−tj\nk, mj\nk),0\n, (12)\nwhere each ˆµd(t),andˆϕdj(t−tj\nk, mj\nk), d= 1, . . . , D, are approximated using a neural network.\nThe network utilized to represent ˆϕdj(t, m)shares similarities with the one outlined in Equation 11.\nHowever, in the case of the NNNH network, the function Πtakes the form of an identity function\ninstead of an exponential one. This selection guarantees that ˆϕdjeffectively maps from R+×Rto\nR. With this modification in ˆϕdj, the NNNH with mark can effectively accommodate excitation and\ninhibition kernels, making it more versatile for modelling a broader range of processes involving\ndiverse dependencies between past arrivals and their corresponding marks. The kernel function\nˆϕdj(t, m)givenHis modelled as,\nˆϕdj(t, m) =b2+pX\ni=1ai\n3max\u0000\nai\n1t+ai\n2m+bi\n1,0\u0001\n. (13)\nRemark 3.1. This paper employs two methodologies to estimate the marked Hawkes process: SNH\nwith marks for linear marked Hawkes and NNNH with marks for both linear and non-linear marked\nHawkes. Despite NNNH with marks being capable of estimating the linear marked Hawkes process,\nSNH with marks is preferred due to its lower computational cost. When there is ambiguity in\ndetermining whether an event results in excitation or inhibition of the ground intensity in a dataset\nH, it is recommended to use NNNH with marks. However, if it is certain that the arrival of an event\nleads to the excitation of the ground intensity process, then SNH with marks is the preferable choice.\nThe neural network used for both estimation models consists of a hidden layer of Pneurons and\na single output layer, resulting in a parameter dimension of 4P+ 1for a one-dimensional marked\nHawkes setup. For a D-dimensional marked Hawkes process with a constant base intensity, the\nnumber of parameters can go up to (4P+ 1)( D2). If we also consider the case of non-constant base\nintensity, µd(t), then the number of parameters to be estimated will be (4P+ 1)( D2+D).\nTo estimate the parameter set θλencompassing both the kernels and the base intensity parameters\n((4P+ 1)( D2+D)parameters), we use the ground intensity log-likelihood function Lλgiven\nin Equation 7 across the observed dataset H. The computation of Lλusing the neural network\napproximation of λg\nd(t)in both the models necessitates obtaining the integrated intensity function\nand log intensity function in Lλ, the details of which are given in Appendix A.\nPost obtaining the log-likelihood function Lλ, in order to find the optimal network parameter set ˆθλ,\nwhich provides a local maximum for the Lλ, we employ batch stochastic gradient descent (SGD)\n9combined with Adam optimization algorithm introduced by Kingma and Ba [2014]. The gradients of\ntheLλwith respect to each of the parameters in the θλcan be computed using Equation 9. These\ngradients acquired through the batch SGD iteratively update the parameters in θλ, leading to the\nidentification of the optimal parameter set, ˆθλmaximizing the Lλ, providing the optimal estimations\nfor the kernels and the base intensity.\n3.1 Gaussian Mixture Model\nGaussian mixture model (GMM) represents a class of mixture models used to estimate density\nfunctions. According to Murphy [2012] and McLachlan and Basford [1988], mixture models\nare probabilistic models where a distribution can be represented as the weighted sum of two or\nmore base distributions. GMM specifically uses multivariate Gaussian distribution as the base\ndistribution. GMM with a sufficiently large number of mixture components (Gaussian distributions)\ncan approximate any continuous density function defined on R(Murphy [2012]). This paper utilizes\na GMM density function to approximate the fd(m),mark density function in the dth dimension.\nWhile the marks, represented as [m1, . . . , mD], may exhibit correlations, our specific assumption is\nthat marks within a dimension are independent of those in other dimensions. For instance, the trade\nvolumes for buy and sell market orders are assumed to be independent in the context of trade orders.\nConsequently, the marks within each dimension are treated as independent and identically distributed\n(i.i.d.). The estimated GMM density function ˆfd(m|θf),corresponding to the dth dimensional marks\nm={md\nn}n≥1sampled from H, given the parameter set θf=\b\n(zj, µj, σ2\nj)\t\n, is expressed as\nfollows:\nˆfd(m|θf)≈kX\nj=1zjF(m|µj, σ2\nj), (14)\nwhere F(m|µj, σ2\nj)is the density function of the jth Gaussian distribution given by,\nF(m|µj, σ2\nj) =1q\n2πσ2\njexp− \n(m−µj)2\n2σ2\nj!\n.\nwith parameters (µj, σ2\nj),kis the number of Guassian distributions and zj,0≤zj≤1is the weight\nparameter. The weight parameter, zjshould satisfyPk\nj=1zj= 1. The parameters of the GMM, θf\nare estimated by the expectation maximization (EM) algorithm explained by Dempster et al. [1977].\n4 Experiments and Results\nThe evaluation process commences with a comprehensive analysis of synthetic datasets, encom-\npassing both one-dimensional and multi-dimensional Hawkes processes with continuous marks.The\neffectiveness of our neural network-based approaches, SNH with marks and NNNH with marks is\nscrutinized against the vanilla models of SNH and NNNH1.\nThe evaluation extends to practical applications, specifically utilizing our neural network approach\non cryptocurrency trading data. This data includes detailed tick-by-tick sell and buy market orders\nfor Bitcoin US Dollars (BTC-USD) and Ethereum US Dollar (ETH-USD) pairs. The neural network-\nbased approach is applied to unveil the causal relationships concerning arrival times and marks within\nthe specified pairs.\nTo ensure the robustness of our methodology, we initiate the evaluation with data preprocessing steps\nbefore directly implementing the estimation technique. Moreover, we adopt a scaling mechanism\nfor the dataset, which will be elaborated upon in the subsequent sections. The choice of initial\nparameters and hyperparameter selection are also addressed, including factors such as the number\nof hidden neurons, learning rates, and the stopping criteria. These aspects collectively contribute\nto appropriately evaluating our neural network-based approach’s performance and its practical\napplicability in different scenarios.\n1The code and the data that has been used in this section is given in https://github.com/sobin-joseph/Marked-\nHawkes-Estimation\n104.1 Simulation of Marked Hawkes Process\nTo generate a synthetic dataset for a multivariate marked Hawkes process, it is crucial to define\na simulation algorithm for the process. For simulating a Hawkes process, the most widely used\nalgorithm is Ogata’s modified thinning algorithm proposed by Ogata [1981]. Ogata’s modified\nthinning algorithm builds upon the thinning algorithm initially introduced by Lewis and Shedler\n[1979] to simulate the point process. Thinning algorithms rely on the conditional intensity function\nto simulate successive intervals. For the simulation of a marked point process Daley and Vere-\nJones [2007] extended the thinning algorithm of Lewis and Shedler [1979] to incorporate the\nmarks. The resulting thinning algorithm for the marked Hawkes process given in Algorithm 1, is\nemployed to simulate the marked Hawkes process. Similar to the aforementioned thinning algorithms,\nthis algorithm for the marked point process follows a sequential approach. It utilizes the ground\nconditional intensity, λg\nd(t)to select the next time point. Subsequently, after selecting a new event\ntime, the corresponding mark is chosen using the density function of the marks, fd(m).\nAlgorithm 1: Thinning algorithm for marked Hawkes process\nInput : base intensity {µd}D\nd=1, kernel function {ϕdj}D\nd,j=1, market density function {fd}D\nd=1, end\ntimeT;\nOutput : Event times {τd}D\nd=1and marks {md}D\nd=1;\nInitialize t←0,{τd}D\nd=1← ∅,{md}D\nd=1← ∅;\nwhile t <Tdo\nInitialize M(t)←0;\nford= 1 to D do\nλg\nd(t)←max\nµd, µd+dX\nj=1X\nt≤τjϕdj(t−τj,mj)\n;\nM(t)←M(t) +λg\nd(t);\nend\nD∼U(0,1);\ns←t−log(D)\nM(t);\nU∼U(0,1);\nInitialize λg(s)←0;\nford= 1 to D do\nλg\nd(s)←max\n0, µd+dX\nj=1X\ns<τjϕdj(s−τj,mj)\n;\nλg(s)←λg(s) +λg\nd(s);\nifλg(s)\nM(t)< U then\nt←s;\nτd∪s;\nF∼U(0,1);\nm←f−1\nd(F);\nmd∪m;\nelse\nend\nend\npass;\nend\n4.2 Data preprocessing and choice of hyper-parameters:\nBefore conducting the experiments, we partition the dataset Hinto training, validation, and test\nsets following a 70:15:15 ratio. It is important to ensure that the partition preserves the event times’\nchronological order. For both the validation and test sets, all the historical information leading\n11up to the current event is considered, starting from the inception of the data. Prior to the dataset\ndivision, we apply scaling to enhance the data’s suitability for analysis. This scaling procedure is\naimed at transforming the dataset into a more canonical form, which, in turn, reduces the inherent\nvariability. By reducing variability, we achieve a twofold effect: diminishing the generalization error\nand decreasing the model’s size, which, in our case, pertains to the number of neurons required to\neffectively train the model (Goodfellow et al. [2016]).\nThis preprocessing procedure involves both event times and their corresponding marks. Given\ndataset H={(td\nn, md\nn)}n≥1,where td\nn∈[0, T)andmd\nn∈ M denote the ordered arrivals and their\ncorresponding marks for the dth dimension ,the following definitions are relevant:\ntd\nmax= max\b\ntd\nn\t\nn≥1, Tmax= max\b\ntd\nmax\tD\nd=1, N=DX\nd=1Nd(Tmax)andmd\nmean=P\nmdn∈Hmd\nn\nN\nConsequently, each event time td\nnand its corresponding mark md\nnare scaled as follows:\nˆtd\nn=td\nnN\nTmax,\nˆmd\nn=md\nn1\nmdmean.\nThe scaled dataset ˆH={(ˆtd\nn,ˆmd\nn)}n≥1,will be the input to the proposed model.\nInitialization of network parameters and choice of batch size and hyperparameters: Proper\ninitialization the parameters of the proposed models is of paramount importance, as it significantly\nimpacts the performance of most algorithms. According to Goodfellow et al. [2016], the choice of\ninitial values plays a critical role in determining whether an algorithm converges successfully or not.\nIn certain cases, unstable initial inputs can even lead to the complete failure of the model.\nTo address this concern, we adopt distinct initialization strategies for SNH with marks and NNNH\nwith marks. For SNH with marks, the parameters outlined in Equation 11 are initialized as follows:\nai\n1∼U(−0.5,0.5), ai\n2∼U(−0.2,0.2), ai\n3∼U(−1.0,0.0), bi\n1∼U(0.0,0.03),andbi\n2∼\nU(−0.1,0),\nwhere U(a, b),denotes uniform distribution between aandb.\nFor the NNNH with marks, the parameters given in Equation 13 are initialized as:\nai\n1∼U(−0.7,0), ai\n2∼U(−0.2,0.2), ai\n3∼U(0,1.0), bi\n1∼U(0.0,0.25),andbi\n2= 0.\nThe initial base intensity for both the models is set to be initialised µ= 1 (For cases of varying\nbase intensity in the NNNH with marks approach, the parameters of base intensity are initialised as\nexplained in Joseph and Jain [2023]). We maintain a fixed hidden layer size of sixty-four neurons in\nboth the SNH with marks and NNNH with marks. This choice of the number of neurons results from a\ncareful balance between computational time requirements and achieving the optimum log-likelihood.\nOur stochastic gradient calculations are performed using a batch size of one hundred. We employ\ndistinct learning rates for updating parameters in the hidden and output layers, as we have observed\nfaster convergence with this configuration. The specific learning rates adopted for our experiments\nare as follows:\n•For the SNH with marks model, we use a learning rate of 2×10−2for updating network\nparameters in the output layer and 2×10−3for updating parameters in the hidden layer.\n•For the NNNH with marks, 5×10−3and5×10−4are used to update the parameters in the\noutput and hidden layers, respectively.\nIn both models, the learning rate for updating µis1×10−3. These learning rate choices have\ncontributed to efficient and effective convergence during the training process for both the SNH with\nmarks and NNNH with marks.\n12Stopping criteria: We employ a stopping criterion to prevent the algorithm from underfitting or\noverfitting the dataset. For both the SNH with marks and NNNH with marks, we utilize the number\nof iterations during which the validation log-likelihood did not show improvement as the stopping\ncriterion. In this context, we set the number of iterations for the stopping criteria to a fixed value of\n10. This ensures that the optimization process halts after 10consecutive iterations without observable\nenhancements in the validation log-likelihood.\n4.3 Synthetic Data Experiments\nThis section assesses the effectiveness of the proposed models on simulated datasets of a marked\nHawkes process. We explore three distinct scenarios: a one-dimensional linear marked Hawkes\nprocess with a combined (or coupled) kernel function, a one-dimensional non-linear marked Hawkes\nprocess with decoupled kernel function (separable function of time and marks), and a multivariate\nmarked Hawkes with decoupled kernel functions.\nFor generating synthetic datasets for each scenario, we employ the thinning algorithm for marked\nHawkes process outline in Algorithm 1. The evaluation of the proposed approaches involves the\nfollowing measures: as an initial performance metric, we compare the error between the known theo-\nretical kernel and the kernel estimated using the proposed methods- both visually and quantitatively\nthrough an error plot. We analyze the absolute difference between the theoretical kernel and the\nestimated kernel at all regions in the error plot. Next, we compare the predictions of the estimated\nmarked Hawkes process for arrival times with the vanilla models. This comparison is facilitated using\nQuantile-Quantile (QQ) plots for the test dataset.\nQQ plots validate the accuracy of interarrival time predictions from the estimation method. When\npredicting the next arrival time with a 90% confidence level based on the current history, ideally, 90%\nof the observed next arrivals should occur within the predicted time, and at least 10% should occur\nbeyond it. If all subsequent arrivals fall within the predicted time, the algorithm is overly conservative,\npredicting too far into the future. If fewer than 90% of subsequent arrivals align with the predicted\nhorizon, the predictions are too soon. A well-performing prediction model should exhibit Q% correct\npredictions at a Q% confidence level. A correctly fitted model is represented by an ideal QQ plot,\nwhich is a 45-degree line running through the origin.\n4.3.1 One-dimensional Marked Hawkes Process\nMarked Hawkes with coupled kernel function: This example uses the coupled (or combined)\nkernel function of the generalized marked Hawkes conditional intensity, as described in Equation 3.\nSpecifically, we focus on a scenario representing a linear marked Hawkes process. In this context, we\ninvestigate situations where ϕ(t, m)is not a separable function of time tand marks m. To the best\nof our knowledge, no other non-parametric models can estimate such kernels. The marked Hawkes\nprocess is simulated using the following functions:\n• Kernel function, ϕ(t, m) =me−t(1+5m),\n• Base intensity, µ= 0.7, and\n• Mark distribution, f(m) =1\n0.5m√\n2πe−(logm)2\n2×0.52.\nUsing Algorithm 1 with specified parameters, we conducted a simulation for the time interval\n(0,2000] , resulting in 1743 arrivals. The dataset is generated using a linear Hawkes process, and\ntherefore, for estimation, we can consider either SNH with marks or NNNH with marks. SNH\nwith marks is preferred due to its lower computational demand, as explained in Remark 3.1. The\nSNH estimation reaches the stopping criteria in approximately 690 seconds. Figure 3 displays the\ntheoretical, estimated, and error kernels. Despite using a limited number of data points for fitting,\nthe proposed model effectively estimates the kernel. The error plot in Figure 3 depicts the absolute\ndifference between theoretical and estimated kernels ( |ϕ−ˆϕ|), demonstrating the model’s capability\nto capture excitation in both time and mark space.\nFigure 4 presents a QQ plot analysis comparing the performance of SNH with marks and vanilla SNH.\nThe plot indicates that vanilla SNH tends to predict shorter interarrival times more often, whereas\nSNH with marks accurately captures the distribution of interarrival times. This comparison suggests\n13(a) Estimated and theoretical kernel\n (b) Error plot\nFigure 3: (a) describes the theoretical and estimated kernel obtained using SNH with marks for a\none-dimensional Linear Hawkes process, while (b) presents the absolute error between the theoretical\nand estimated kernel expressed as |ˆϕ−ϕ|\nFigure 4: QQ plot providing a comparative analysis between SNH and SNH with marks estimation\nthat the SNH with marks model outperforms the vanilla SNH model in predicting the linear marked\nHawkes process.\nMarked Hawkes process with decoupled kernel function: In our second example, we consider a\nnon-linear marked Hawkes process with decoupled (or separable) kernel function, outlined by the\nintensity function in Equation 5. A decoupled kernel function implies that the kernel has distinct\nfunctions for its temporal and mark components. For the estimation of non-linear marked Hawkes,\nit is necessary to use the NNNH with marks, as the SNH with marks is unsuitable for non-linear\nHawkes. We use the following parameters for the functions given in the intensity Equation 5:\n• Kernel function, ϕ(t, m) =ψ(m)ϕ(t) = log( m)(−0.4e−2t),\n• base intensity, µ= 0.9, and\n• Mark distribution, f(m) =1\nm√\n2πe−(logm−0.5)2\n2×12.\nFor the specified parameters, simulation is conducted over a period of (0,8000] , generating a dataset\nof length 3000 . Despite the theoretical kernel having a separate time and mark function, estimation\nis performed by approximating the kernel as a combined function of time and mark, following the\nmodel given in Equation 10. The estimation took 1160 seconds to converge to the optimal network\n14(a) Theoretical and estimated kernel\n (b) Error plot\nFigure 5: (a) shows the theoretical kernel and kernel estimated using NNNH with Hawkes for a\none-dimensional Non-linear Hawkes process and (b) depicts the absolute error between the theoretical\nand estimated kernel\nparameters based on the stopping criteria. Figure 5 displays the theoretical kernel, and the estimated\nkernel using the NNNH with marks and the corresponding error plot. The kernel and the error plot,\ndepicted in 5, suggest that the NNNH with marks can accurately approximate the given kernel. A\ncomparison is also made with the vanilla NNNH estimation for the test data using the QQ plot\npresented in Figure 6. From the QQ plot, it is evident that both the NNNH with marks and vanilla\nNNNH effectively predict the interarrival time for the test dataset.\nFigure 6: QQ plot comparing the interarrival times obtained from NNNH and the NNNH with marks\nEstimation.\n4.3.2 Multi-dimensional Hawkes Estimation\nWe aim to investigate a 2-dimensional marked Hawkes process with distinct kernel functions for\ntemporal and mark components, as defined in the intensity Equation 5. This specific example,\nborrowed from Dassios et al. [2013], explores a scenario where each kernel possesses its unique\nmarks generating function fdj(m), signifying that each kernel is associated with its own set of marks.\nThe simulation is conducted for a 2 dimensional Hawkes process for a period of [0,5000) with the\nkernel function,\nϕdj(t, m) =\u0014\nme−2tme−50t\nme−4.5tme−3t\u0015\n,\n15the mark density function,\nfdj(m) =\u0014\ne−2.5me−6m\ne−3me−1.5m\u0015\n,\nand base intensity, µ= [0.3,0.3]. The simulation generated a dataset of size of 2082 . The dataset\nis subjected to the estimation process using multi-dimensional SNH with marks, and the kernels\nare subsequently obtained. Figure 7(a) presents a comparison between the estimated kernels and\ntheir theoretical counterparts, while Figure 7(b) illustrates the absolute error between the true and\nestimated kernels. These plots collectively affirm that multidimensional SNH with marks has the\ncapacity to recover the theoretical kernel with precision. Furthermore, a QQ plot is provided for both\nmultidimensional SNH with marks and the vanilla SNH in Fig 8. The QQ plot illustrates that both\nmethods, despite vanilla SNH not considering marks, can predict interarrival times within the given\nconfidence interval.\n(a) Kernel\n (b) Error\nFigure 7: (a) The theoretical and the estimated kernel obtained using SNH with marks for a 2−d\nmarked Hawkes process with decoupled kernel function. (b) illustrates the error between the\ntheoretical and estimated kernels\nFigure 8: QQ plot demonstrating the precision of prediction for multivariate SNH and SNH with\nmarks estimation.\n4.4 Real Data\nThis section discusses the application of the proposed estimation method in the real-world dataset,\nwhere we take the case of cryptocurrency trade in an exchange, specifically the market order book\ndata of Bitcoin-US Dollar(BTC-USD) and Ethereum-US Dollar(ETH-USD) trades in the Binance\n16exchange. The application uses the volume of trade as the mark component, thus analysing the impact\nof the traded size on the market order arrival rate. This application gives a glimpse of the ability of\nour method to model real-world events with attached marked components.\n4.4.1 Cryptocurrency Dataset\nData and Preprocessing: We utilize the high-frequency dataset from cryptocurrency trading\ninvolving two of the most commonly traded cryptocurrencies- Bitcoin (BTC) and Ethereum (ETH).\nThe dataset consists of timestamps of buy and sell market orders for Bitcoin-US Dollars(BTC-USD)\nand Ethereum-US Dollars (ETH-USD) pairs on the Binance exchange. The Binance exchange was\nselected because it is a major exchange for various cryptocurrencies with high trading volumes. The\ndataset details arrival timestamps, volume, price, buyer ID and seller ID for each market order. The\ntime horizon considered is from 16th July 2022, from 02 : 00 : 00 UTC to 02 : 30 : 00 UTC. The\nselection of Bitcoin and Ethereum currencies is based on the fact that they are the most commonly\nand heavily traded cryptocurrencies in the global cryptocurrency market.\nFor the cryptocurrency dataset, we adopt the volume of trade as the mark component in the marked\nHawkes process. This aligns with the approach of Chavez-Demoulin and McGill [2012], Fauth and\nTudor [2012], Rambaldi et al. [2017] and Fabre and Toke [2024], which explore the impact of volume\nof trade on the clustering of market order. This justifies the use of volume as the mark component. In\nthis context, the trade volume represents the quantity of bitcoins (or Ethereum) traded.\nBefore applying the estimation method, some preprocessing is done to the dataset. There are instances\nin which a single sell/buy market order is fulfilled by multiple buyers (sellers) in the market. These\norders are considered as a single sell (buy) market order, combining their respective trading volumes.\nTable 1 shows the descriptive statistics of the dataset after preprocessing. The table shows that the\ntrade volume for ETH-USD currency (both buy and sell) is 10 times greater than the BTC-USD\ncurrency trade despite the higher number of BTC-USD trades. This discrepancy is primarily due\nto the higher price of Bitcoin compared to Ethereum, resulting in a greater quantity (volume) of\nETH-USD in a single transaction compared to BTC-USD.\nTable 1: Descriptive statistics of the cryptocurrency market orders\nCryptocurrency Trade-Type Number of Events V olume of trade\nBTC-USD sell 7512 463.23\nBTC-USD buy 9046 734.72\nETH-USD sell 6989 9139.5\nETH-USD buy 6270 8101.3\nTotal 30388 18438 .75\nResult and Summary: Figure 9 illustrates the kernels, ϕdj(t, m)obtained after applying SNH with\nmarks model to the cryptocurrency dataset. From Figure 9 the following can be observed regarding\nthe trade arrivals of BTC-USD and ETH-USD trades:\n•All the cryptocurrency trades(both buy and sell) exhibit self-excitatory behaviour but the\namount of self-excitation varies for each currency. The self-excitation remains constant\nwith the volume of trade for sell BTC-USD trade but decreases with the volume for buy\nBTC-USD trades. While, for buy and sell ETH-USD trades, the intensity of trade arrival\ndue to its trades increases with the volume of trade.\n•Cross excitatory behaviour is observed in sell BTC-USD and buy ETH-USD. The occurrence\nof trades in buy ETH-USD triggers sell BTC-USD trades. This triggering intensity gradually\nincreases with an increase in trading volume.\n•Trades in sell ETH-USD cross excite more trades in buy BTC-USD. This excitation is\nobserved only for a small volume of ETH-USD trade and diminishes with an increasing\nvolume.\nThe above observations give the microstructure in a cryptocurrency market, i.e. Bitcoin trades\n(buy/sell) are influenced by Ethereum (sell/buy) trades, but not the other way around. There is also\n17a clear dependency on the volume of trade with the intensity of trade. This statement will also be\nsupported by the QQ plot given in 10.\nThe QQ plot compares the performance of SNH with marks, vanilla SNH and EM.2The QQ pots\nfor sell BTC-USD and buy ETH-USD indicate a volume dependency, suggesting that the volume of\ntrade plays a pivotal role in the arrival of the next trade, while there is no observable dependency for\nthe volume of trade in the other two dimensions (buy BTC-USD and sell ETH-USD).\nFigure 9: Kernels for the given cryptocurrency dataset estimated using SNH with marks\nIn the context of market microstructure, understanding market dynamics in relation to various\nfactors is crucial. The application of SNH with marks to the cryptocurrency dataset contributes to\nunderstanding the change in trade intensity concerning both time and volume. Figure 11 illustrates\nthe self-excitatory kernel of ETH-USD estimated using SNH with marks ( ϕ4,4(t, m)in Figure 9)\nand using Vanilla SNH. Figure 11a shows a clear mark dependency for the kernel with the intensity\nincreasing as the volume of trade increases. In contrast, kernels from vanilla SNH (Figure 11b) don’t\nconsider the volume, providing an average intensity across all the volumes at a given time t. Thus\nusing SNH with marks one can deduce the dependency of the volume on kernel intensity, a feature\nabsent in vanilla SNH.\n4.5 Prediction of Marked Hawkes process\nThe prediction of marked Hawkes is closely tied to the simulation process. The key quantities of\ninterest for prediction would be the time to the next event (interarrival time) and the probability of\n2EM is Expectation Maximization algorithm for Hawkes process estimation proposed by Lewis and Mohler\n[2011].\n18Figure 10: QQ plot comparing the performance of SNH with marks, vanilla SNH and EM estimation\nmethods.\n(a) SNH with marks\n (b) Vanilla SNH\nFigure 11: Estimated self-excitatory kernel of buy ETH-USD indicating the difference with marks\nand without marks estimation\nan event occurring within a given interval of time (exponential of the integrated intensity). These\nquantities can be derived if the conditional intensity is known, and the functional form of the\nconditional intensity is obtained through the discussed estimation methods. If the structure of the\nconditional intensity is known, conditional intensity can be obtained with any arbitrary history of\ntime and mark. Thus it is straightforward to see prediction as an application and extension of the\npreceding procedures. A step-by-step detail of the prediction process is given as,\n1.Select a time period of the dataset (0, A]to estimate the marked Hawkes process, thereby\nobtaining the functional form of conditional intensity function.\n2.Simulate the process beyond (0, A]using Algorithm 1 leveraging the known structure of the\nconditional intensity function and the known history HA.\n3.Extract from the simulation the quantity required to predict (interarrival time or probability\nof arrival).\n4.Repeat steps 2 and 3 a sufficient number of times to achieve the desired precision for the\nprediction quantity.\n195.The output is the empirical distribution of the quantity obtained from multiple successive\nsimulations.\n5 Conclusion\nThis paper introduces a novel non-parametric estimation technique tailored for the marked Hawkes\nprocess. This is the first known approach capable of non-parametrically estimating the kernel of\nthe Hawkes process as a joint function of time and marks. Drawing inspiration from the vanilla\nSNH method and the NNNH method, the proposed model employs a two-layer feed-forward neural\nnetwork to approximate the Hawkes kernel, treating the kernel as a joint function of event times\nand their associated marks. This model is applied to the log-likelihood function and the optimal\nneural network parameters are derived using batch SGD combined with Adam. The use of batch\nSGD renders the method suitable for online learning applications, facilitating the updation of network\nparameters based on the arrival of new data points.\nHere we explain two approaches SNH with marks- tailored for the linear marked Hawkes process\nand NNNH with marks- for the non-linear marked Hawkes process. Even though the NNNH with\nmarks possesses the capability to model linear Hawkes models, we limit its use to scenarios suspected\nof negative dependencies as clarified in Remark 3.1. We apply the estimation methods to synthetic\nand cryptocurrency datasets. We compare all the results obtained with the corresponding estimation\nmethods without marks, and QQ plots are obtained. Although QQ plots for estimation with marks and\nwithout marks appear unique (in some datasets), using estimation with marks proves advantageous\nwhen understanding the dependency of event arrivals on marks is crucial. In the cryptocurrency\nsetting, estimation with marks reveals the relationship of the market order arrival intensity with the\nvolume of the trade apart in addition to the history of trade arrivals. This provides a comprehensive\ninsight into the dynamics of the cryptocurrency market microstructure.\nWhile the proposed methods prove robust in estimating Hawkes process with marks, determining the\nappropriate mark component in real-life applications remains a relevant factor, which can be taken as\na future area of focus. Additionally, incorporating spatial coordinates as marked components could\nbe a potential focus for future directions. Including spatial elements in the model has the potential to\nenhance our understanding and prediction capabilities in earthquake modelling and related fields.\nReferences\nEmmanuel Bacry and Jean-François Muzy. First-and second-order statistics characterization of\nhawkes processes and non-parametric estimation. IEEE Transactions on Information Theory , 62\n(4):2184–2202, 2016.\nPierre Brémaud and Laurent Massoulié. Stability of nonlinear hawkes processes. The Annals of\nProbability , pages 1563–1588, 1996.\nValérie Chavez-Demoulin and JA McGill. High-frequency financial data modeling using hawkes\nprocesses. Journal of Banking & Finance , 36(12):3415–3426, 2012.\nFeng Chen and Wai Hong Tan. Marked self-exciting point process modelling of information diffusion\non twitter. The Annals of Applied Statistics , 12(4):2175–2196, 2018.\nDaryl J Daley and David Vere-Jones. An introduction to the theory of point processes: volume II:\ngeneral theory and structure . Springer Science & Business Media, 2007.\nAngelos Dassios, Hongbiao Zhao, et al. Exact simulation of hawkes process with exponentially\ndecaying intensity. Electronic Communications in Probability , 18, 2013.\nArthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data\nvia the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological) , 39(1):\n1–22, 1977.\nNan Du, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez-Rodriguez, and Le Song.\nRecurrent marked temporal point processes: Embedding event history to vector. In Proceedings\nof the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining ,\npages 1555–1564, 2016.\n20Paul Embrechts, Thomas Liniger, and Lu Lin. Multivariate hawkes processes: an application to\nfinancial data. Journal of Applied Probability , 48(A):367–378, 2011.\nTimothée Fabre and Ioane Muni Toke. Neural hawkes: Non-parametric estimation in high dimension\nand causality analysis in cryptocurrency markets. arXiv preprint arXiv:2401.09361 , 2024.\nAlexis Fauth and Ciprian A Tudor. Modeling first line of an order book with multivariate marked\npoint processes. arXiv preprint arXiv:1211.4157 , 2012.\nEric Warren Fox, Frederic Paik Schoenberg, and Joshua Seth Gordon. Spatially inhomogeneous\nbackground rate estimators and uncertainty quantification for nonparametric hawkes point process\nmodels of earthquake occurrences. 2016.\nIan Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning . MIT press, 2016.\nAlan G Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika , 58\n(1):83–90, 1971.\nK Hornik, M Stinchcombe, and H White. Multilayer feedforward networks are universal approxima-\ntors. Neural Networks , 2(5):359–366, 1989.\nSobin Joseph and Shashi Jain. A neural network based model for multi-dimensional nonlinear hawkes\nprocesses. arXiv preprint arXiv:2303.03073 , 2023.\nSobin Joseph, Lekhapriya Dheeraj Kashyap, and Shashi Jain. Shallow neural hawkes: Non-parametric\nkernel estimation for hawkes processes. Journal of Computational Science , page 101754, 2022.\nDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint\narXiv:1412.6980 , 2014.\nRyota Kobayashi and Renaud Lambiotte. Tideh: Time-dependent hawkes process for predicting\nretweet dynamics. In Tenth International AAAI Conference on Web and Social Media , 2016.\nKyungsub Lee and Byoung Ki Seo. Marked hawkes process modeling of price dynamics and volatility\nestimation. Journal of Empirical Finance , 40:174–200, 2017.\nMoshe Leshno, Vladimir Ya Lin, Allan Pinkus, and Shimon Schocken. Multilayer feedforward\nnetworks with a nonpolynomial activation function can approximate any function. Neural networks ,\n6(6):861–867, 1993.\nErik Lewis and George Mohler. A nonparametric EM algorithm for multiscale hawkes processes.\nJournal of Nonparametric Statistics , 1(1):1–20, 2011.\nPA W Lewis and Gerald S Shedler. Simulation of nonhomogeneous poisson processes by thinning.\nNaval research logistics quarterly , 26(3):403–413, 1979.\nGeoffrey J McLachlan and Kaye E Basford. Mixture models: Inference and applications to clustering ,\nvolume 38. M. Dekker New York, 1988.\nHongyuan Mei and Jason M Eisner. The neural hawkes process: A neurally self-modulating\nmultivariate point process. In Advances in Neural Information Processing Systems , pages 6754–\n6764, 2017.\nKevin P Murphy. Machine learning: a probabilistic perspective . MIT press, 2012.\nSanthosh Narayanan, Carsten Maple, and Mark Hooper. A point process model for rare event\ndetection. arXiv preprint arXiv:2209.04792 , 2022.\nYosihiko Ogata. On lewis’ simulation method for point processes. IEEE Transactions on Information\nTheory , 27(1):23–31, 1981.\nYosihiko Ogata. Statistical models for earthquake occurrences and residual analysis for point\nprocesses. Journal of the American Statistical association , 83(401):9–27, 1988.\nYosihiko Ogata. Space-time point-process models for earthquake occurrences. Annals of the Institute\nof Statistical Mathematics , 50:379–402, 1998.\n21Yosihiko Ogata. Seismicity analysis through point-process modeling: A review. In Seismicity patterns,\ntheir statistical significance and physical meaning , pages 471–507. Springer, 1999.\nRazvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural\nnetworks. In International conference on machine learning , pages 1310–1318. Pmlr, 2013.\nMarcello Rambaldi, Emmanuel Bacry, and Fabrizio Lillo. The role of volume in order book dynamics:\na multivariate hawkes process analysis. Quantitative Finance , 17(7):999–1020, 2017.\nOleksandr Shchur, Marin Biloš, and Stephan Günnemann. Intensity-free learning of temporal point\nprocesses. arXiv preprint arXiv:1909.12127 , 2019.\nTom Stindl and Feng Chen. Modeling extreme negative returns using marked renewal hawkes\nprocesses. Extremes , 22(4):705–728, 2019.\nShuai Xiao, Mehrdad Farajtabar, Xiaojing Ye, Junchi Yan, Le Song, and Hongyuan Zha. Wasserstein\nlearning of deep generative point process models. Advances in neural information processing\nsystems , 30, 2017.\nBaichuan Yuan, Hao Li, Andrea L Bertozzi, P Jeffrey Brantingham, and Mason A Porter. Multivariate\nspatiotemporal hawkes processes and network reconstruction. SIAM Journal on Mathematics of\nData Science , 1(2):356–382, 2019.\nJiancang Zhuang, Yosihiko Ogata, and David Vere-Jones. Analyzing earthquake clustering features\nby using stochastic reconstruction. Journal of Geophysical Research: Solid Earth , 109(B5), 2004.\nSimiao Zuo, Haoming Jiang, Zichong Li, Tuo Zhao, and Hongyuan Zha. Transformer hawkes process.\nInInternational conference on machine learning , pages 11692–11702. PMLR, 2020.\nA Approximation of the Log-likelihood Function\nThe ground intensity part of the log-likelihood function, Lλgiven in the Equation 7 can be partitioned\ninto two components: the log intensity part,log(λg\nd(td\nn))and the integrated intensity part,RT\n0λg\nd(s)ds.\nFor both the introduced models- SNH with marks and NNNH with marks, it is necessary to obtain\nboth the integrated intensity and log intensity functions to obtain the gradients from Equation 9.\nAlthough obtaining the log intensity part is straightforward, the integrated intensity part is not so\ndirect for both the SNH with marks and NNNH with marks.\nA.1 Integrated Intensity of SNH with marks\nFor the SNH with marks, as the intensity function, λd(t)is linear, it is written as the arithmetic\ncombination of base intensity, µ(taking the baseline intensity as constant) and the kernel functions,\nϕdj(t),∀1≤j≤D. According to Joseph et al. [2022], the integrated intensity function for a linear\nHawkes process without marks can be written as,\nZT\n0λd(s)ds=X\ntdn∈HZtd\nn\ntd\nn−1µdds+DX\nj=1X\ntj\nn∈HZT−tj\nn\n0ϕdj(s)ds.\nThe above equation can be modified for a marked Hawkes process. Thus, the integrated ground\nintensity with modification in ϕdjto accommodate marks is written as\nZT\n0λg\nd(s)ds=X\ntdn∈HZtd\nn\ntd\nn−1µdds+DX\nj=1X\n{tdn,mdn}∈HZT−tj\nn\n0ϕdj(s, mj\nn)ds. (15)\nFor the SNH with marks, the above kernel can be approximated according to Equation 11,\nˆϕdj(s, m) =exp \nb2+PX\ni=1ai\n3max\u0000\nai\n1s+ai\n2m+bi\n1,0\u0001!\n.\n22Each of the integrated kernel functionRT−tj\nn\n0ˆϕdj(s, mj\ni)dsin Equation 15 involves integrating over\nthe neural network model given in Equation 11. As the model consists of the RELU function, it\nis necessary to integrate over max function for each of the ith neuron, ∀1≤i≤P. This involves\nobtaining the zero-crossings and integrating over non-zero paths of each ith neuron over the interval\n[0, T−tj\nn). For the ith neuron, the zero-crossing ,yiis given by,\nai\n1yi+ai\n2mj\nn+bi\n1= 0\nyi=−ai\n2mj\nn+bi\n1\nai\n1.\nLetY= (y1, . . . , y P)be the zero crossings across all the Pneurons. Arranging the Yin the\nincreasing order and within the bounds of [0, T−tj\nn)given by Y∗= (0, y∗\n1, . . . , y∗\nu, T−tj\nn), where\nu≤P. Thus the integrated kernel function across each of [0, T−tj\nn)is given as,\nZT−tj\nn\n0ˆϕdj(s, mj\nn)dt=Zy∗\n1\n0ˆϕdj(s, mj\nn)ds+Zy∗\n2\ny∗\n1ˆϕdj(s, mj\nn)ds+. . .+ZT−tj\nn\ny∗uˆϕdj(s, mj\nn)ds.\n(16)\nThe obtained approximated integrated intensity function is substituted to the Lλin Equation 7 and is\nmaximized to obtain the network parameters.\nA.2 Integrated Intensity of NNNH with marks\nFor the NNNH with marks method, the calculation of the integrated intensity function is complex\ncompared to SNH with marks due to the presence of two non-linear functions ( max functions). The\nintegrated intensity function,RT\n0λg\nd(s)dscan be written as,\nZT\n0λg\nd(s)ds=X\n{tdn,mdn}∈HZtd\nn\ntd\nn−1λg\nd(s)ds,\nthe sum of all distinct arrival times, as it is more convenient to calculate the intensity function across\nall distinct intervals. From the non-linear marked Hawkes intensity function given in Equation 12,\nλg\nd(s)across each distinct interval is,\nZtd\nn\ntd\nn−1λg\nd(s)ds=Ztd\nn\ntd\nn−1max\n\nµd+DX\nj=1X\ntj\nk<sϕdj(s−tj\nk, mj\nk)\n,0\nds,\n=Ztd\nn\ntd\nn−1max\n\nµd+DX\nj=1X\ntj\nk<sb2+PX\ni=1ai\n3max\u0010\nai\n1(s−tj\nk) +ai\n2mj\nk+bi\n1,0\u0011\n,0\nds,\nthe second equality substitutes neural network model from Equation 13 in to ϕdj(s−tj\nk, mj\nk). For\nfinding outRtd\nn\ntd\nn−1λg\nd(s)ds, it is necessary to integrate the function over non-zero paths. This is done\nby determining the zero-crossings across each [td\nn−1, td\nn]pairs. For finding out zero-crossings for an\ngiven interval first, we have to find the zero-crossings for the inner max function layer and then the\nouter max layer. Let Y= (y1, . . . , y P)be the zero-crossings for the entire neural network with P\nneurons (inner max layer), yifor each ith neuron is given by,\nai\n1(yi−tj\nk) +ai\n2mj\nk+bi\n1= 0,\nyi=ai\n1tj\nk−ai\n2mj\nk−bi\n1\nai\n1.\nThus for each of theRtd\nn\ntd\nn−1λg\nd(s)ds, there will be F=PDl(Hs−)(l(Hs−)denotes the length of\nhistory up to s) number of zero crossings for the inner max layer. Arranging Yin chronological\norder and in the bounds of\u0002\ntd\nn−1, td\nn\u0003\ngiven by Y′= (td\nn−1, y1, . . . , y F, td\nn). Next, we found\n23out the zero-crossings across the outer max layer. This is done by finding the zero-crossings\nin each of the pairs in Y′. For each of the pair in Y′, there will be at most one zero-crossings ,\ndenoted by y∗\ni. Thus the modified Y′with all the zero-crossings across the interval\u0002\ntd\nn−1, td\nn\u0003\n,\nY∗= (td\nn−1, y∗\n1, y1, . . . , y F, td∗\nn, td\nn)(Y∗includes zero-crossings of both inner and outer max\nfunction)\nFinally, the integrated intensity,Rtd\nn\ntd\nn−1λg\nd(s)dsis given by,\nZ\nY∗λg\nd(s)ds=Zy∗\n1\ntd\nn−1λg\nd(s)ds+Zy1\ny∗\n1λg\nd(s)ds+. . .+Ztd∗\nn\nyFλg\nd(s)ds+Ztd\nn\ntd∗nλg\nd(s)ds, (17)\nwhere λg\nd(s) =µd+PD\nj=1P\ntj\ni<TRtd\nn−1\ntdnϕdj(s, m).\n24"    },    {        "title": "2402.04743v1.Temperature_evolution_in_the_Early_Universe_and_freeze_in_at_stronger_coupling.pdf",        "content": "Temperature evolution in the Early Universe\nand freeze-in at stronger coupling\nCatarina Cosme1, Francesco Costa2and Oleg Lebedev3\n1Univ Coimbra, Faculdade de Ciˆ encias e Tecnologia da Universidade de Coimbra\nand CFisUC, Rua Larga, 3004-516 Coimbra, Portugal\n2Institute for Theoretical Physics, Georg-August University G¨ ottingen,\nFriedrich-Hund-Platz 1, G¨ ottingen D-37077, Germany\n3Department of Physics and Helsinki Institute of Physics,\nGustaf H¨ allstr¨ omin katu 2a, FI-00014 Helsinki, Finland\nAbstract\nDark matter freeze-in at stronger coupling is operative when the Standard Model (SM) bath\ntemperature never exceeds the dark matter mass. An attractive feature of this scenario is\nthat it can be probed by direct detection experiments as well as at the LHC. In this work, we\nshow how the mechanism can be realized in a simple UV complete framework, emphasizing\nthe role of the maximal temperature of the SM thermal bath. We demonstrate that the\nmaximal temperature can coincide with the reheating temperature or be close to it such\nthat dark matter production is always Boltzmann–suppressed. This possibility is realized,\nfor example, if the inflaton decays primarily into feebly interacting right-handed neutrinos,\nwhich subsequently generate the SM thermal bath. In this case, the SM sector temperature\nremains constant over cosmological times prior to reheating.arXiv:2402.04743v1  [hep-ph]  7 Feb 2024Contents\n1 Introduction 1\n2 Evolution of the Standard Model sector temperature 2\n2.1 Example: SM sector production via νRdecay . . . . . . . . . . . . . . . . . . 4\n3 Dark matter freeze-in production at stronger coupling 7\n3.1 Constant T(a) =Tmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2 T(a)∝a−l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.3 Total particle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.4 Tmax> T Rcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n4 Conclusion 12\n1 Introduction\nThe dark matter (DM) puzzle remains one of the outstanding questions in modern physics.\nWhile thermal dark matter in the form of a weakly-interacting-massive-particle (WIMP)\nemerges naturally in many extensions of the Standard Model, the lack of observational\nevidence in support of WIMPs [1] motivates us to explore further options. In particular, dark\nmatter can be non-thermal and produced in a variety of ways. One popular mechanism is\nknown as “freeze-in”, whereby DM is gradually produced by the SM thermal bath [2, 3, 4, 5].\nIts conventional version requires a feeble coupling between the SM sector and dark matter, as\nwell as a high enough reheating temperature. This scenario assumes a zero DM abundance\nbefore the thermal production starts, which appears quite unlikely in the world with gravity\n[6]. The freeze-in mechanism is also notoriously difficult to test, if possible at all.\nThe recently proposed “freeze-in at stronger coupling” [7] addresses some of the prob-\nlematic aspects of traditional freeze-in models. It requires a low reheating temperature such\nthat DM production is Boltzmann-suppressed. In this case, the coupling between DM and\nthe SM sector is allowed to be up to O(1) without affecting the non-thermal nature of dark\nmatter. This makes DM potentially observable in direct detection experiments as well as at\nthe LHC.\nThe low reheating temperature allows for dilution of gravitationally produced dark mat-\nter, which otherwise is problematic in most models [6]. Specifically, all particles are produced\nby gravity or via gravity-induced interactions both during and after inflation. This is par-\nticularly problematic for scalar DM, whose field fluctuations build up during inflation [8]\nresulting in DM overproduction, unless a suppression or dilution mechanism is invoked.\nEven if DM production gets suppressed during inflation, e.g. via a large Hubble-induced\ndark matter mass, it gets produced during preheating all the more efficiently. Also, classical\nand quantum gravitational effects induce higher dimensional operators, which couple dark\nmatter sto the inflaton ϕ[9], e.g.\nCϕ4s2\nM2\nPl, (1)\n1where Cis a Wilson coefficient. During the inflaton oscillation epoch, this operator produces\ndark matter very efficiently in traditional (large field) inflation models, normally leading to\na totally dark Universe.\nThe resulting abundance can be made consistent with observations only if the Universe\nundergoes a very long period of inflaton matter-domination, which dilutes all relativistic\nrelics, or if the Wilson coefficient happens to be tiny, which cannot be guaranteed in the\nabsence of a UV complete quantum gravity framework. The constraint can be put in the\nform [6]\n∆NR≳106C2 ϕ8\n0\nH5/2\nendM11/2\nPlms\nGeV, (2)\nwhere ∆ NR≡\u0010\nHend\nHreh\u00111/2\ncharacterizes the duration of the matter-dominated expansion pe-\nriod. Here HendandHrehare the Hubble rates at the end of inflation and at reheating,\nrespectively; ϕ0is the inflaton field value at the end of inflation and msis the DM mass. Tak-\ningϕ0∼MPlandHend∼1014GeV, one obtains a very strong bound ∆ NR≳1017C2ms\nGeV,\nwhich implies a low reheating temperature ∆ NR≫1 (unless dark matter is super-light,\nms≪eV, or C2happens to be vanishingly small). Similar considerations apply to fermionic\ndark matter, although the constraints are weaker [10].\nFreeze-in at stronger coupling operates in models with low reheating temperature TR,\nthereby avoiding the problem of gravitational DM overproduction. The dark relics generated\nduring and immediately after inflation are diluted away such that one may assume their\nnegligible abundance at the onset of thermal production. If dark matter is heavier than TR,\nits production is Boltzmann-suppressed and its coupling to the SM fields can be significant.\nOn the other hand, the DM abundance becomes very sensitive to the thermal history of\nthe Universe. In particular, it is produced most efficiently when the SM bath temperature\nreaches its maximum, which generally happens before reheating.\nIn our previous work [7], we have resorted to the low energy effective description and, in\nparticular, to the instant reheating approximation, which assumes that dark matter is mostly\nproduced at TRand the preexisting abundance can be neglected. The goal of the current\nwork is to study under what circumstances this assumption is adequate and, more generally,\nexplore cosmological frameworks where freeze-in at stronger coupling can be realized. We find\nbroad classes of models in which the SM bath temperature stays constant over cosmological\ntimes such that the maximal and reheating temperature are naturally close. These allow for\na reliable calculation of freeze-in DM abundance, without the problem of initial conditions\ninherent in high TRmodels, and justify our previous results.\n2 Evolution of the Standard Model sector temperature\nLittle is known as to how exactly the SM fields were produced after inflation. They could\nbe generated through their direct couplings to the inflaton [11] or result from interactions\nwith a secondary field, e.g. decay thereof. The SM sector temperature exhibits very different\nevolution in these cases.\nLet us consider a general case of the SM particle production from decay of a field χ,\nwhich does not necessarily dominate the energy density of the Universe. The SM quanta are\n2typically produced in the relativistic regime, so we will assume that the SM energy density\nρscales as radiation. The energy density of the χfield is ρχ, whose scaling is a−n, while that\nof the Hubble rate Hisa−m, with abeing the scale factor. The values of nandmare, in\ngeneral, unrelated and depend on further details. Denoting the decay width of χby Γ χ, we\narrive at the following system:\n˙ρ+ 4Hρ= Γ χρχ,\nH=H0/am,\nρχ=ρ0\nχ/an, (3)\nThe label 0 refers to the initial moment a0= 1 corresponding to the end of inflation and\nρ(1) = 0. We assume here that the SM sector does not contribute significantly to the energy\nbalance of the Universe and that n, m are constant in the regime of interest.\nThe solution for ρ(a) is easily found as follows: ˙ ρ+ 4Hρ=1\na4d\ndt(a4ρ) and dt=da/(aH),\nso that\na4ρ= Γ χρ0\nχZda\naHa4−n, (4)\nand our boundary condition ρ(1) = 0 requires\nρ(a) =Γχρ0\nχ\n(4−n+m)H0\u00141\nan−m−1\na4\u0015\n→Γχρ0\nχ\n(4−n+m)H01\nan−m(5)\nata≫1 since n−m < 4 for all cases of interest. In practice, this approximation becomes\nsensible already at a=O(1).\nThe above scaling is different from the usual 1 /a4law applicable when the SM radiation\ndominates. In fact, n−mcan be of either sign or zero. The SM energy density can thus\ngrow in time or stay constant before reheating. Since thermalization in the SM sector is fast,\none can associate the SM thermal bath temperature with ρ1/4, i.e.\nT≃\u001230\ng∗π2\u00131/4\nρ1/4, (6)\nwhere g∗is the number of the SM degrees of freedom. Therefore, the SM sector temperature\nis allowed to grow in time or stay constant before reheating. The latter is defined as the\ntime when the SM energy density starts dominating the energy balance of the Universe and\nis associated with temperature TR.\nIf the SM fields are produced directly by the inflaton, n= 2mandρ∝a−m. This\ncorresponds to the a−3/2anda−2scaling for the ϕ2andϕ4inflaton potentials, respectively.\nIn this case, the SM temperature decreases before reheating and the maximal temperature\nTmaxwould typically be much larger than the reheating temperature TR[11, 12, 13].\nIf the SM sector is produced by decay of a field other than the inflaton, the temperature\nscaling can be a0ora+|k|meaning that the maximal and reheating temperatures may be\nclose and even coincide,\nTmax≃TR. (7)\n3A natural candidate for such a subdominant field is the right-handed neutrino νR. It can be\nproduced via inflaton decay and subsequently decay into SM states. The above calculation\n(5) applied to the neutrino production ϕ→νRνRimplies\nρν\f\f\f\nϕ4=3\n2ΓϕH0M2\nPl1\na2, (8)\nρν\f\f\f\nϕ2=6\n5ΓϕH0M2\nPl1\na3/2, (9)\n(10)\nfor the ϕ4andϕ2inflaton potentials, respectively. ρνis now the source for the SM radiation,\nso using νRasχin (3), one finds\nn−m= 0 (11)\nas long as νRremains subdominant in the energy balance. As a result, the SM bath tem-\nperature stays constant . In what follows, we study this scenario in more detail.\nWe note that the SM temperature can also increase before reheating. This happens if\nn−m < 0, which can be realized in models where the SM sector is produced via decay\nof a sub-subdominant component in the energy density of the Universe. Indeed, the above\nconsiderations show that ncan be zero while m > 0.\n2.1 Example: SM sector production via νRdecay\nConsider the possibility that the inflaton decays primarily into feebly interacting right-\nhanded neutrinos, ϕ→νRνR. The relevant terms in the Lagrangian are\n∆L=yϕϕ νRνR+yνHc¯ℓνR+ h.c. , (12)\nwhere the Yukawa coupling are small, yϕ, yν≪1, and Handℓare the Higgs and lepton\ndoublet fields, respectively. Here we focus on the parameter regime where the direct coupling\nbetween the inflaton and the SM states is tiny such that ϕ→hhis insignificant. This can\nbe enforced by an approximate Z4symmetry of the type ϕ→ −ϕ; (ℓ, νR)→i(ℓ, νR) [14].\nThe reheating process proceeds in two stages,\nϕ→νRνR, ν R→SM, (13)\nwhere the final state in the νRdecay can be Hℓor lighter states, if the Higgs channel is\nforbidden kinematically. We assume that νRis feebly coupled such that both decays take\nplace at late times and the inverse processes can be neglected. We also take the effective\ninflaton mass to be far above the νRmass, hence the energy density of νRscales as radiation\nuntil late times. Given the feebleness of the right-handed neutrino interactions, it does not\nform a thermal bath.\nThe energy density components satisfy the following equations\n˙ρϕ+ 3Hρϕ=−Γϕρϕ, (14)\n˙ρν+ 4Hρν= Γ ϕρϕ−Γνρν, (15)\n˙ρ+ 4Hρ= Γ νρν, (16)\nρϕ+ρν+ρ= 3H2M2\nPl, (17)\n4where ρϕ, ρν, ρare the inflaton, νRand SM energy densities, respectively; Γ ϕand Γ νare the\ndecay rates of the corresponding energy densities.1The initial values for ρνandρare zero,\nand 3 H2\n0M2\nPl=ρϕ(0) = V(ϕ0), where “0” refers to the end of inflation. Here we assume for\ndefiniteness a non-relativistic inflaton, i.e. V(ϕ0) = 1 /2m2\nϕϕ2\n0with ϕ0∼MPl,H0∼10−5MPl.\nAs long as the decay widths are small and Γ ϕρϕ≫Γνρν, the energy densities are hierar-\nchical and we recover the scaling behaviour of the solution described earlier:\nρϕ∝a−3, ρν∝a−3/2, ρ∝a0(18)\nata >O(1). Furthermore, even if Γ νis time-dependent, the SM energy density can be\ncomputed via\nρ(a) =1\na4Za\n1da a3Γνρν\nH. (19)\nThis expression makes it explicit that if ρνscales as Hand Γ ν= const, the SM energy\ndensity remains constant. The latter can also increase in time if Γ νρν/Hgrows, which\ncould be due to the increase of ρν/Hor broadening of the neutrino width.\nAt late times, the different energy densities start approaching each other and their hi-\nerarchical structure gets lost. In this case, the full numerical solution is necessary (Fig. 1).\nBelow, we distinguish the cases of Γ ϕand Γ νbeing vastly different, and being of similar size.\nΓϕ≫Γν.The inflaton field decays into the right-handed neutrinos and drops out of the\nenergy balance when H∼Γϕ. Until SM reheating, the dominant role is taken by relativistic\nνR. In this period, the SM temperature decreases as T∝a−1/2. Reheating occurs when\nH∼Γν.\nΓϕ≪Γν.As long as Γ ϕρϕ≫Γνρν, the energy components evolve as in the previous case.\nThe transition occurs when Γ ϕρϕ∼Γνρν. The neutrino energy density ρνstarts decreasing\nfaster ( ρν∝ρϕ∝a−3), although it does not decay exponentially quickly due to continuous\nreplenishment from the inflaton sector. The SM sector takes over the subleading role in\nthe energy balance until H∼Γϕ, when both νRandϕdecay exponentially quickly. The\nSM temperature decreases in this period as T∝a−3/8, so again the maximal temperature\nexceeds the reheating temperature.\nΓϕ∼Γν.This parameter choice realizes the possibility Tmax≃TR. Since the inflaton and\nneutrinos decay at the same time, the SM sector takes over the energy balance immediately\nthereafter. As Fig. 1 (bottom right) demonstrates, the temperature evolution can be approx-\nimated by a constant followed by a power law decrease T∝a−1.\nWe therefore conclude that, in the simple framework where the SM sector is produced by the\nright-handed neutrino decay, the maximal and reheating temperatures do not differ signifi-\ncantly if the inflaton and νRdecay widths are similar. Their approximate equality Tmax≃TR\nis realized for Γ ϕ≃Γν.\n1The energy density decay widths Γ iare such that possible 1 + ωifactors, where ωiis the equation of\nstate coefficient, are absorbed in their definition.\n51 10410810121016102010-3010-10101010301050\naρ[GeV4]Γϕ=10-11GeV,Γ ν=3x10-14GeV,T R=100 GeV\nϕ\nνR\nSM\n1 10410810121016102010-3010-10101010301050\naρ[GeV4]Γϕ=3.5x10-14GeV,Γ ν=10-11GeV,T R=100 GeV\nϕ\nνR\nSM\n1 10410810121016102010-3010-10101010301050\naρ[GeV4]Γϕ=7x10-14GeV,Γ ν=3.5x10-14GeV,T R=100 GeV\nϕ\nνR\nSM\n10121013101410151016101710185075100125150175200\naT[GeV]Γϕ=6.8x10-14GeV,Γ ν=3x10-14GeV\nSMFigure 1: Evolution of the energy density components and the SM sector temperature. Top\nleft:Γϕ≃3.3×102Γν;top right: Γϕ≃3.5×10−3Γν;bottom left: Γϕ≃2 Γν;bottom right:\nSM temperature evolution for Γ ϕ≃2 Γν.\nOne should also note that certain thermal effects can play a role in flattening the T(a)\ndependence prior to reheating. In particular, if Tis far above the neutrino mass Mν, which we\nkeep as a free parameter, the neutrino decay into the SM final states is blocked kinematically.\nThis can be accounted for by replacing Γ ν→Γνθ(ρ⋆−ρ) in the evolution equations. Here\nρ⋆is proportional to M4\nνand for values of ρ⋆∼T4\nR, the SM temperature stays constant till\nreheating even if Γ ϕ≫Γν.\nFinally, let us estimate the size of the couplings required for a low reheating temperature.\nThe relation Γ ϕ∼Γνimplies\nyν\nyϕ∼rmϕ\nMν, (20)\nwhere mϕandMνare the inflaton and right-handed neutrino masses, respectively. This is\nbecause both ϕ→νRνRandνR→Hℓdecay widths scale as the initial particle mass times the\ncoupling squared. The typical parameter values in Fig. 1 are TR∼100 GeV and Γ i∼10−14\nGeV, which for mϕ∼1013GeV implies yϕ∼10−13. Assuming Mν∼1 TeV, the neutrino\nYukawa coupling is then yν∼10−8, consistent with its non-thermalization [14]. (For light νR,\nits decay can be loop-suppressed.) The inflaton coupling appears particularly small, although\n6it is not surprising in inflationary cosmology given the flatness of the inflaton potential. The\nsmallness of the couplings in the neutrino sector is also not unexpected, hence one may find\nthe above scenario plausible, although the parameter choice can only be motivated in a more\nfundamental theory.\nThese estimates also show that the direct decay ϕ→SM would be highly suppressed\nin this model. The inflaton coupling to the SM fields is generated at one loop, e.g. the\ncoupling to the Higgs bilinear H†His proportional to the loop factor ×yϕy2\nνMν, where the\nlast factor is required by the Z4symmetry [14] breaking. This makes the decay rate of\nϕ→SM suppressed by about 56 orders of magnitude compared to that of ϕ→νRνR. More\ngenerally, the SM energy density generated via the cascade ϕ→νR→SM scales as Γ ϕΓν,\nwhereas that from the direct decay ϕ→SM scales as Γ ϕΓ2\nν. Therefore, the latter can be\nmade highly suppressed by decreasing Γ ν.\n3 Dark matter freeze-in production at stronger cou-\npling\nLet us consider freeze-in dark matter production in the class of models described in the\nprevious section. The SM sector temperature has never been very high in this framework.\nThe maximal and reheating temperatures are of similar size and possibly coincide. Before\nreheating, the Universe undergoes a long period of inflaton matter-dominated expansion\nwhich dilutes the relativistic relics generated right after inflation. Hence, one may focus\nentirely on thermal production of dark matter.\nWe are interested in freeze-in at larger couplings [7], so we take the DM mass to be\nmuch larger than the SM temperature at any stage of the Universe evolution. On the other\nhand, the temperature profile is model-dependent and can be highly non-trivial in a multi-\ncomponent Universe. We thus focus on a simplified case motivated in the previous section.\nSpecifically, we assume that the relevant stage in the Universe evolution can be approximated\nby the period of constant SM temperature, followed by a power-law decrease T−las displayed\nin Fig. 1 (bottom right). We focus on the parameter regime in which the DM production via\ndirect decay of the inflaton or the right-handed neutrino is highly suppressed or forbidden.2\nConsider, for definiteness, real scalar dark matter sof mass mswhich couples to the SM\nsector through the Higgs portal [15, 16, 17],\nV(s) =1\n2λhss2H†H+1\n2m2\nss2, (21)\nwhere λhsis the Higgs portal coupling. For T≪ms, dark matter is produced via scattering of\nenergetic SM quanta from the “Boltzmann tail”. The leading process is the Higgs and vector\nboson pair annihilation into pairs of s. For ms≫mh, all these modes can be accounted\nfor by using four effective Higgs degrees of freedom, according to the Goldstone equivalence\ntheorem.\n2In the example of the previous section, the inflaton decay into DM is highly suppressed by powers of\nsmall couplings as well as Mν/mϕ, mh/mϕ, while the νRdecay into DM plus leptons would be forbidden\nkinematically if Mν<2mDM.\n7The dark matter number density nsatisfies the Boltzmann equation\n˙n+ 3Hn= Γ hihi→ss−Γss→hihi, (22)\nwhere hirepresents 4 Higgs d.o.f. at high energies of order ms≫mh; Γhihi→ssand Γ ss→hihi\nare the DM production and annihilation rates per unit volume, respectively.\nConsider first the pure freeze-in regime, in which the annihilation term can be neglected.\nSince the initial state particles have energies far above the temperature, their distribution is\ngiven by the “Boltzmann tail”. Then, the production rate per unit volume is [7]\nΓhihi→ss≃λ2\nhsT3ms\n27π4e−2ms/T, (23)\nfor 4 Higgs d.o.f. Integrating the Boltzmann equation, we get\na3n=H−1\n0Z\nda am+2Γhihi→ss(a). (24)\nHere we use the scaling H=H0/am, as before. Note that one cannot trade the integration\nvariable aforTsince T(a) can be constant. Using the explicit form of the reaction rate, we\nfind\na3n=λ2\nhsms\n27π4H0Z\nda am+2T3(a)e−2ms/T(a). (25)\nConsider now two representative T(a) functions.\n3.1 Constant T(a) =Tmax\nThis scaling applies when the SM sector is a sub-subdominant component in the energy\ndensity of the Universe. The integral is trivial and the total particle number is\na3n=λ2\nhsms\n27π4H0am+3\nm+ 3T3\nmaxe−2ms/Tmax(26)\nfora≫a0= 1. We see that the particle number (and the density) grows with aand is\ndominated by the late time contributions at largest a.3\n3.2 T(a)∝a−l\nDuring and after reheating, the SM temperature decreases. We may approximate it by a\npower law T(a)∝a−l, with ltime-dependent but approximately constant during each period\nof interest. Let us take\nT(a) =Tmax\u0010amax\na\u0011l\n, (27)\nwhere amaxcorresponds to the point where the temperature starts decreasing, and compute\nthe particle number generated from amaxon. Using the approximation\nZ∞\nz0dz zke−z≃zk\n0e−z0(28)\n3A similar result applies to the case when Tincreases in time: the particle number is dominated by the\nlargest acontribution.\n8forz0≫1, obtained using integration by parts, we get\na3n≃λ2\nhsms\n27π4H0am+3\nmax\nlTmax\n2msT3\nmaxe−2ms/Tmax. (29)\nThis expression assumes 2 ms/Tmax≫1. In contrast to the previous case, the particle number\nis dominated by the early time contribution. This is due to the exponential suppression of\nparticle production at lower temperatures.\nBoth (26) and (29) can be rewritten in terms of H(a). The time dependence of the\nparticle density is captured by\nn(a)\f\f\f\nT=const∝1\nH(a), n(a)\f\f\f\nT∝a−l∝1\na3H(amax). (30)\n3.3 Total particle number\nLet us now combine the two contributions. Defining amaxas the transition point from one\nscaling law to the other, we obtain the total DM particle number produced by the SM\nthermal bath,\na3n\f\f\f\ntot=a3\nmaxn(amax)\f\f\f\nT=const×\u0012\n1 +m+ 3\nlTmax\n2ms\u0013\n, (31)\nwhere a3\nmaxn(amax)|T=const is found from (26). DM production is most efficient in the vicinity\nofa=amaxwith the flat part of the temperature evolution giving the main contribution\nsinceTmax\n2ms≪1. We note that the l-dependence of the result is very mild: lonly affects the\ncorrection.\nThe consequent constraint on the model is formulated in terms of the relic abundance of\nthes-particles,\nY=n\nsSM, s SM=2π2g∗\n45T3, (32)\nwhere g∗is the number of the SM d.o.f. and nis the number density of the s-quanta. The\nobservational constraint on the DM abundance is Yobs= 4.4×10−10\u0010\nGeV\nms\u0011\n[18].\nAssuming that Tmax=TR, the Hubble rate at a=amaxis determined by the SM tem-\nperature, Hmax∝T2\nmaxandl= 1. We thus get\nY≃√\n90 45\n28π7g3/2\n∗λ2\nhsmsMPle−2ms/Tmax\nT2\nmax\u00121\nm+ 3+Tmax\n2ms\u0013\n. (33)\nTo obtain this expression, we have estimated YatT=Tmax, after which the abundance re-\nmains approximately constant in the pure freeze-in regime. The correct DM relic abundance\nis reproduced for λhs∼10−11ems/Tmaxp\nms/GeV. The coupling can be as large as order\none if ms≫Tmax. On the other hand, models with a high reheating temperature require\nλhs∼10−11[19, 20].\nIn our previous work [7], we resorted to a low energy effective approach without specifying\nthe full cosmological history. In particular, we assumed zero particle abundance before\nreheating, which amounts to computing particle production from the T∝a−lregion only.\n9λhs⋁ fog\n10 153060120300LZ 2022\n⋁ fog⋁ fog\n200 500 1000 5000 10410-210-11\nms[GeV]Figure 2: Parameter space of the Higgs portal DM model. Along the colored lines, the\ncorrect relic density is reproduced for a given TR=Tmax(in GeV). The thin lines correspond\nto the calculation presented in [7], while the thick lines are obtained in the UV complete\nmodel of the current work. The purple line corresponds to the thermal DM abundance. The\nshaded area is excluded by direct DM detection (LZ 2022) [1] and the dashed line shows the\n“neutrino fog”.\nTaking m= 3/2 corresponding to inflaton matter domination prior to reheating, we find that\nignoring “prehistory” leads to underestimating the particle number by about a factor of 10.\nTo reproduce the correct DM relic abundance, the effective approach required 2 ms/TR∼40\nfor typical parameter values. Since Yscales approximately as e−2ms/TR, the 10-fold increase\ninYcan be compensated by a 5% shift in TR. Hence, the full computation leads to a TR\ncorrection of order 5%,\nTR→0.95×TR. (34)\nEquivalently, for a fixed TR, the DM mass msincreases by about 5%. Hence, the parameter\nspace plot remains largely unchanged. This is shown in Fig. 2. The thin and thick colored\nlines correspond to the correct relic density computed within the effective approach of our\nprevious work [7] and using the UV complete model presented here, respectively.\nAt larger couplings, the annihilation effect becomes significant. The corresponding reac-\ntion rate per unit volume is given by [7]\nΓ(ss→hihi) =σ(ss→hihi)vrn2, σ(ss→hihi)vr= 4×λ2\nhs\n64πm2\ns(35)\nform2\ns≫m2\nhand 4 Higgs d.o.f. Here vris the relative particle velocity. As one increases\nthe coupling further, the system thermalizes [7, 21] and the resulting DM relic abundance is\ngiven by the purple line in Fig. 2.\nFig. 3 illustrates the process of thermalization with our T(a) function. At low couplings,\nthe DM particle number differs from its equilibrium value at all stages, apart from one\npoint. For larger couplings, the DM density reaches the equilibrium value somewhat before\n10a=amax, which is followed by DM freeze-out. For convenience, we redefine the scale factor\nin the plot as a/a max→asuch that the SM temperature starts decreasing at a= 1 and\nTR=T(1).\nneqa3\n1.0 1.2 1.4 1.6 1.86.0×10-77.0×10-78.0×10-79.0×10-71.0×10-61.1×10-61.2×10-6\nana3[GeV]3\nneqa3\n0.01 0.05 0.10 0.50 1 510-910-810-710-610-5\nana3[GeV]3\nFigure 3: Dark matter particle number evolution. neqrepresents the thermal particle\ndensity at the SM sector temperature T, which stays constant before a= 1 and decreases as\n1/aafter that. Left: pure freeze-in regime ( λhs= 10−3,ms= 602 .5 GeV, Tmax= 30 GeV).\nRight: DM thermalization and freeze-out ( λhs= 0.195,ms= 651 GeV, Tmax= 30 GeV).\n3.4 Tmax> T Rcase\nMore generally, TmaxandTRcan differ. For instance, if Γ ϕ≫Γνor Γ ϕ≪Γν, the maximal\ntemperature exceeds the reheating temperature. In this case, the flat T= const stage,\nduring which the inflaton field dominates, is followed by a period of a mild temperature\ndecrease T∝a−lwith l <1. During this stage, the right-handed neutrinos take over the\nenergy balance. As these decay, the SM bath comes to dominate and lbecomes 1 signifying\nreheating. Thus, TRcan be significantly lower than Tmax.\nConsider the pure freeze-in regime. It is clear from the above calculations that dark\nmatter is mostly produced at the point a=amax. Hence, Eq. 31 applies to this case as well.\nHowever, the resulting DM abundance changes since the SM entropy gets produced until\nreheating, after which Yremains approximately constant. The result can be conveniently\nexpressed as a correction factor to (33). Let us parametrize the scaling of the Hubble rate\nbetween amaxandaRasH∝a−m′, such that Hevolves with aaccording to\n1a−m\n−→ amaxa−m′\n−→ aRa−2\n−→ ∞ . (36)\nSince the DM particle number does not change between amaxandaR, we obtain the following\ncorrection factor to the result (33):\nY≃Y\f\f\f\nTmax=TR×\u0012TR\nTmax\u0013m′+3\nl−5\n. (37)\n11For example, in the case of radiation domination, m′= 2 and l= 1/2. Hence, the correc-\ntion factor is ( TR/Tmax)5. Although this factor can be substantial, it does not change the\nparameter space picture drastically: taking Tmax= 10 TRresults in about a 25% shift in ms.\nFreeze-in production at stronger coupling implies that the maximal temperature is far\nbelow the DM mass. In this case, particle production is always Boltzmann-suppressed and\nof freeze-in type. This condition is typically violated if the inflaton decays directly into\nthe SM states, implying a high Tmax. The dynamics of particle production would be more\ncomplicated in this case: depending on further details, DM may thermalize and freeze out\nbefore Boltzmann-suppressed freeze-in becomes operative.\nMany of our results also apply if the SM sector is produced by a sub-subdominant com-\nponent in the energy density of the Universe.4In this case, the SM sector temperature\nincreases according to a power law until the energy balance changes and the evolution of T\nflattens, followed by its decrease. DM production is most efficient at the end of the flat T\nperiod, hence the above considerations remain largely valid.\nIn summary, the main results of our work are presented in Fig. 2. The direct detection\nexperiments already probe freeze-in DM up to 3 TeV. As their sensitivity increases, they\nwill cover further consistent freeze-in models with higher reheating temperature or lower\ncouplings. The difference between the full model results with TR=Tmaxand those using the\ninstant reheating approximation is small in terms of the {λhs, ms}parameter space. If Tmax\nis significantly above TR, the{λhs, ms}relation receives a correction due to (37):\nλhs→λhs×\u0012Tmax\nTR\u0013m′+3−5l\n2l\n, (38)\nas long as ms≫Tmax. Due to the exponential mass-dependence, this results in a modest\nshift of the {λhs, ms}curve at fixed TRto smaller masses.\nAt low ms, the direct detection sensitivity is normally superseded by that of the LHC via\nthe constraint on the invisible Higgs decay. This also applies to the Higgs portal model with\nlowTR[22]. Indirect detection constraints are weak in the Higgs portal models [17]. Let us\nnote that there exist other examples of freeze-in models, although with a more complicated\ndark sector, that can potentially be probed by a combination of experiments [23, 24, 25].\n4 Conclusion\nDark matter freeze-in at stronger coupling is a well motivated scenario, which addresses the\nproblem of initial conditions in conventional freeze-in models. It assumes a low reheating\ntemperature allowing for dilution of gravitationally produced relics and making DM produc-\ntion Boltzmann-suppressed. As a result, the DM coupling to the SM fields is allowed to be\nO(1) rendering it potentially observable in direct detection experiments [7]. On the other\nhand, the DM relic abundance in this framework is sensitive to the thermal history of the\nStandard Model sector and, in particular, to the relation between the maximal and reheating\ntemperatures.\n4This refers to the energy balance immediately after inflation, i.e. n−m < 0 in (5). Shortly before\nreheating, this component may dominate the energy density.\n12In this work, we have studied a class of models, in which the maximal and reheating\ntemperatures are in the same ballpark, and can even coincide. This is the case when the\nSM sector is produced via decay of a subdominant component in the energy density of\nthe Universe. The role of this component can be played by feebly interacting right-handed\nneutrinos νR. If these couple to the inflaton much stronger than the SM fields do, the inflaton\ndecays predominantly into pairs of νR, which subsequently produce the SM sector. In this\ncase, the SM bath temperature stays constant over cosmological times prior to reheating,\nexplaining the proximity of the maximal and reheating temperatures.\nSince the temperature evolution is known in this framework, the dark matter abundance\ncan be calculated reliably. We find that the resulting constraint on the coupling vs mass is\nvery close to that computed in the instant reheating approximation [7] as long as TR≃Tmax.\nIf the maximal and reheating temperatures differ, the coupling receives a rescaling factor\n(38).\nAn attractive feature of freeze-in models with stronger coupling is that they are cur-\nrently being probed by direct DM detection experiments. As the search sensitivity improves\n[26, 27], further models with higher TRor lower couplings will be explored. In addition, the\nLHC can place a complementary constraint via precise measurements of the Higgs invisible\ndecay. All in all, dark matter freeze-in at stronger coupling provides us with an interesting\nalternative to traditional freeze-in or freeze-out models.\nAcknowledgements. C.C. is supported by the FCT project Grant No. IN1234CEECINST\n/00099/2021, and partially by the FCT project Grant No. CERN/FIS-PAR/0027/2021.\nThe work of C.C. was also supported by FCT - Funda¸ c˜ ao para a Ciˆ encia e Tecnologia,\nI.P. through the projects UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers\n10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively.\nReferences\n[1] J. Aalbers et al. [LZ], [arXiv:2207.03764 [hep-ex]].\n[2] S. Dodelson and L. M. Widrow, Phys. Rev. Lett. 72, 17-20 (1994).\n[3] J. McDonald, Phys. Rev. Lett. 88, 091304 (2002).\n[4] A. Kusenko, Phys. Rev. Lett. 97, 241301 (2006).\n[5] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, JHEP 03, 080 (2010).\n[6] O. Lebedev, JCAP 02, 032 (2023).\n[7] C. Cosme, F. Costa and O. Lebedev, “Freeze-in at stronger coupling,” [arXiv:2306.13061\n[hep-ph]].\n[8] A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357-6368 (1994).\n[9] O. Lebedev and J. H. Yoon, JCAP 07, no.07, 001 (2022).\n13[10] F. Koutroulis, O. Lebedev and S. Pokorski, “Gravitational production of sterile neutri-\nnos,” [arXiv:2310.15906 [hep-ph]].\n[11] E. W. Kolb and M. S. Turner, Front. Phys. 69, 1-547 (1990).\n[12] G. F. Giudice, E. W. Kolb and A. Riotto, Phys. Rev. D 64, 023508 (2001).\n[13] M. A. G. Garcia, K. Kaneta, Y. Mambrini and K. A. Olive, Phys. Rev. D 101, no.12,\n123507 (2020).\n[14] V. De Romeri, D. Karamitros, O. Lebedev and T. Toma, JHEP 10, 137 (2020).\n[15] V. Silveira and A. Zee, Phys. Lett. B 161, 136-140 (1985).\n[16] B. Patt and F. Wilczek, [arXiv:hep-ph/0605188 [hep-ph]].\n[17] O. Lebedev, Prog. Part. Nucl. Phys. 120, 103881 (2021).\n[18] P. A. R. Ade et al. [Planck], Astron. Astrophys. 594, A13 (2016).\n[19] C. E. Yaguna, JHEP 08, 060 (2011).\n[20] O. Lebedev and T. Toma, Phys. Lett. B 798, 134961 (2019).\n[21] J. Silva-Malpartida, N. Bernal, J. Jones-P´ erez and R. A. Lineros, JCAP 09, 015 (2023).\n[22] T. Bringmann, S. Heeba, F. Kahlhoefer and K. Vangsnes, JHEP 02, 110 (2022).\n[23] T. Hambye, M. H. G. Tytgat, J. Vandecasteele and L. Vanderheyden, Phys. Rev. D 98,\nno.7, 075017 (2018).\n[24] H. An and D. Yang, Phys. Lett. B 818, 136408 (2021).\n[25] P. N. Bhattiprolu, G. Elor, R. McGehee and A. Pierce, JHEP 01, 128 (2023).\n[26] E. Aprile et al. [XENON], JCAP 11, 031 (2020).\n[27] J. Aalbers et al. [DARWIN], JCAP 11, 017 (2016).\n14"    },    {        "title": "2402.04752v2.Homogenization_of_Lévy_type_operators_with_random__ergodic_coefficients.pdf",        "content": "arXiv:2402.04752v2  [math.PR]  8 Feb 2024Homogenization of L´ evy type operators with random, ergodi c\ncoefficients\nTomasz Klimsiak1,2Tomasz Komorowski1, /A0Lorenzo Marino3\nFebruary 9, 2024\n1Institute of Mathematics, Polish Academy of Sciences,\n´Sniadeckich 8, 00-656 Warsaw, Poland, email: tkomorowski@impan.pl , corresponding author\n2Faculty of Mathematics and Computer Science, Nicolaus Cope rnicus University,\nChopina 12/18, 87-100 Toru´ n, Poland, email: tomas@mat.umk.pl\n3Unit´ e de Math´ ematiques Appliqu´ ees (UMA), ENSTA Paris,\n828 Boulevard des Mar´ echaux, F-91120 Palaiseau, France, e mail:lorenzo.marino@ensta-paris.fr\nAbstract\nWe consider the problem of homogenization of a family of Rd-valued L´ evy type processes\n(Xε,x;ω\nt)t≥0,ε∈(0,1], starting at xand whose (random) Fourier symbols equal qε(x,ξ;ω) =\n1\nεαq/parenleftBig\nx\nε,εξ;ω/parenrightBig\n, where\nq(x,ξ;ω) =−ib(x;ω)·ξ+ξ·a(x;ω)ξ+/integraldisplay\nRd(1−eiy·ξ+iy·ξ1{|y|≤1})n(x,dy;ω),\nfor (x,ξ,ω)∈R2d×Ω. Here α∈(0,2] and the L´ evy triplet/parenleftBig\nb(x;ω),a(x;ω),n(x,·;ω)/parenrightBig\nx∈Rdis\na stationary ergodic random field over some probability space (Ω ,G,µ). Our main assumptions\nare that: 1) for any ω∈Ω the operator −q(·,D;ω), defined on the space of compactly supported\nC2functions, is closable in the space of continuous functions vanishing at infinity and its closure\ngenerates a Feller semigroup, 2) there exist constants cQ,CQ>0 independent of ( x,ξ,ω), such\nthat Req(x,ξ;ω)≥cQ|ξ|αfor allξ∈Rdand|q(x,ξ;ω)| ≤CQ|ξ|αfor|ξ| ≤1, and 3) qε(0,ξ;ω)≈\nqL(ξ;ω), asε↓0, inµ-probability, where qL(ξ;ω) is the Fourier symbol of some L´ evy process for\neachω. Under some additional technical assumptions concerning bounde dness of the coefficients\nand irreducibility of the processes we prove the weak convergence of the laws of ( Xε,x;·\nt)t≥0in the\nSkorokhod space, as ε↓0, to a L´ evy process whose Fourier symbol ¯ q(ξ) is given by/integraltext\nΩqL(ξ)Φ∗dµ,\nwhere Φ ∗is a strictly positive density w.r.t. measure µ. Our result has an analytic interpretation\nin terms of the convergence, as ε↓0, of the solutions of random integro-differential equations\n∂tuε(t,x;ω) =−qε(x,D;ω)uε(t,x;ω), with the initial condition uε(0,x;ω) =f(x), where fis a\nbounded and continuous function.\nKey words: Martingale problem, Feller process, homogenization, stat ionary and ergodic coeffi-\ncients, Alexandrov-Bakelman-Pucci estimates.\n2020 Mathematics Subject Classification: primary 35B27; secondary 60H15, 60H20.\n11 Introduction\nHomogenization of diffusions with stationary and ergodic coe fficients is a classical topic in the theory\nof random media. It started with seminal papers of [27, 33] an d has been developed since by\nmany authors. We refer an interested reader to monographs [2 , 3, 21, 24, 26, 32, 36, 47] and the\nreferences therein. Recently, there has been a growing inte rest in showing possible scaling limits\nfor solutions of stochastic differential equations driven by general L´ evy processes, with stationary\nand ergodic coefficients. Generically, such limits require n on-diffusive scalings and the result of the\nhomogenization is a L´ evy process. We mention in this contex t, papers [11, 13, 18, 19, 22, 37, 38, 39,\n40].\nIn the present paper, we consider the problem of homogenizat ion for a class of L´ evy type pro-\ncesses. More precisely, we suppose that (Ω ,G,µ) is a probability space and the L´ evy triplet\n/parenleftBig\nb(x;ω),a(x;ω),n(x,·;ω)/parenrightBig\nx∈Rd(1.1)\nis a stationary and ergodic random field that takes values in t he space Rd×S+\nd×ML(Rd). Here,\nS+\ndis the space of all d×dnon-negative definite, symmetric matrices and ML(Rd) is the space of\nall L´ evy measures on Rd, i.e. Borel measures νsatisfying ν({0}) = 0 and/integraltext\nRd(1∧|z|2)ν(dz)<+∞.\nFor a given ω∈Ω, we consider an Rd-valued L´ evy-type process ( Xx;ω\nt)t≥0, defined over some\nprobability space (Θ ,A,P), satisfying Xx;ω\n0=x,P-a.s. and whose random Fourier symbol equals\nq(x,ξ;ω) =−ib(x;ω)·ξ+1\n2ξ·a(x;ω)ξ\n+/integraldisplay\nRd(1−eiz·ξ+iz·ξ1{|z|≤1})n(x,dz;ω),(x,ξ,ω)∈R2d×Ω,(1.2)\nand is bounded for each ω∈Ω, i.e.\nsup\n|ξ|≤1sup\nx∈Rd|q(x,ξ;ω)|<∞, ω∈Ω. (1.3)\nThe respective generator of the process is defined by Lωu(x) =−q(x,D;ω)u(x) for any u∈C2\nc(Rd)\n- the space of all the compactly supported, C2-class functions on Rd. Here\n−q(x,D;ω)u(x) :=b(x;ω)·∇u(x)+1\n2Tr/parenleftbig\na(x;ω)∇2u(x)/parenrightbig\n+/integraldisplay\nRd(u(x+z)−u(x)−∇u(x)·z1{|z|≤1})n(x,dz;ω).(1.4)\nOur principal assumption is that the L´ evy triplet (1.1) is s uch that for any ω∈Ω the operator\n(−q(·,D;ω),C2\nc(Rd)) is closable in C0(Rd) and its closure generates a Feller semigroup ( Pω\nt) on\nC0(Rd) - the space of all continuous functions vanishing at infinit y, see Hypothesis 2.1 below. Under\nthis assumption, the martingale problem corresponding to Lω, see Section 2.2 below, is well-posed\non the Skorokhod space DofRd-valued c` adlag paths equipped with the J1-topology, see [4, Section\n12]. Furthermore, we assume that the associated semigroup ( Pω\nt)t≥0isirreducible for each ω∈Ω,\ni.e. for any Borel subset A⊂Rdof positive Lebesgue measure, its 1-resolvent is strictly p ositive:\nRω\n11A(x) :=/integraldisplay+∞\n0e−tPω\nt1A(x)dt >0x∈Rd.\n2Givenα∈(0,2] we consider the scaled processes Xε,x;ω\nt(ζ) :=εXx/ε;ω\ntε−α(ζ),t≥0, (ω,ζ)∈Ω×Θ,\nwhereε >0. These processes are defined over the product probability s pace (Ω×Θ,G⊗A,µ⊗P).\nFor anyω∈Ω, the process ( Xε,x;ω\nt)t≥0is of L´ evy-type with the associated Fourier symbol\nqε(x,ξ;ω) =1\nεαq/parenleftBigx\nε,εξ;ω/parenrightBig\n. (1.5)\nInformally, our main result can be formulated as follows. Se e Theorem 2.8 below for the precise\nstatement.\nTheorem 1.1. Letα∈(0,2]be such that there exist constants cQ,CQ>0independent of (x,ξ,ω),\nfor which Req(x,ξ;ω)≥cQ|ξ|αfor allξ∈Rdand|q(x,ξ;ω)| ≤CQ|ξ|αfor|ξ| ≤1andqε(0,ξ;ω)→\nqL(ξ;ω), asε↓0, inµ-probability, where qL(ξ;ω)is the Fourier symbol of a certain L´ evy process for\neachω∈Ω. Then, under some additional assumptions concerning bounde dness of the coefficients\nand irreducibility of the processes , the laws of (Xε,x;·\nt)t≥0converge weakly in D, asε↓0, to the law\nof a L´ evy process whose Fourier symbol equals ¯q(ξ) =/integraltext\nΩqL(ξ;ω)Φ∗(ω)µ(dω), whereΦ∗is a certain\ndensity with respect to µwhich is strictly positive.\nOur result has an obvious analytic interpretation in terms o f solutions to random integro-\ndifferential equations of the form\n∂tuε(t,x;ω) =−qε(x,D;ω)uε(t,x;ω),whereuε(0,x;ω) =f(x) (1.6)\nandfis a bounded continuous function. In Theorem 3.2 below we sho w the convergence in µ-\nprobability, as ε↓0, ofuε(t,x;ω) to the solution ¯ u(t,x) to the following “homogenized” equation:\n∂t¯u(t,x) =−¯q(D)¯u(t,x),¯u(0,x) =f(x). (1.7)\nThe present paper generalizes the result of [34], where the c ase of diffusions with no jumps\nhas been considered, i.e. b≡0 and n ≡0. In this case qε(x,ξ;ω) =1\n2a(x/ε;ω)ξ·ξ, therefore\nqL(ξ;ω) =1\n2a(0;ω)ξ·ξ. Concerning the method of the proof, we extend the technique used in\n[34] by showing in Theorem 3.6 the existence of a strictly pos itive invariant density Φ ∗∈L1(µ)\nfor the process of the environment viewed from the trajector y of the process ( Xx\nt)t≥0, see (3.1)\nbelow for the definition. The invariant density is in fact Lp′integrable for any p >1+d/α. Here\n1/p+1/p′= 1. The conclusions of Theorem 3.2 then follow from an applic ation of suitable criteria\nfor the convergence in law of Feller processes formulated in [15]. To prove Theorem 3.6 we formulate\nin Theorem A.1 an extension of the Alexandrov-Bakelman-Puc ci estimates in the case of non-local\noperators. This result could be of independent interest. An analogous result has been shown for\ncertain classes of L´ evy processes (the coefficients in (1.2) are independent of x) in [5].\nConcerning the organization of the paper, in Section 2, we in troduce the notation and formulate\nthe main hypotheses used in the article (Sections 2.1-2.4). Our main result is formulated in Theorem\n2.8. Section 2.6 contains examples of random L´ evy-type pro cesses that can be homogenized using\nour main result, see Theorems 2.9, 2.12 and 2.14.\nIn Section 3 we state an abstract homogenization result, see Theorem 3.2, that allows us to\nconclude our main result, provided we can prove the existenc e of an invariant and ergodic measure,\n3equivalent with µ, for the so called environment process taking values in Ω. Th e results concerning\nthe existence of such a measure are formulated in Theorems 3. 6 and 3.11. Section 4 is devoted\nto the proof of these theorems. The proof of Aleksandrov-Bak elman-Pucci-type estimates is given\nin Section A of the Appendix. Section B is devoted to the proof of Proposition 2.4, establishing\nirreducibility for a certain class of L´ evy-type processes . Finally, in Section C we prove Propositions\n2.11 and 2.13 that are crucial in establishing applicabilit y of our homogenization result formulated\nin Theorems 2.9, 2.12 and 2.14.\nAcknowledgments\nT. Komorowski acknowledges the support of the NCN grant 2020 /37/B/ST1/00426, T. Klimsiak\nacknowledges the support of the NCN grant 2022/45/B/ST1/01 095.\n2 Preliminaries and the formulation of the homogenization r esult\n2.1 Some generalities\nWe denote by B(y,r) the open ball of radius r >0 centered at y∈Rd, with respect to the euclidean\nmetric on Rd. We denote by Cb(Rd) (resp.C0(Rd)) the space of all bounded (resp. vanishing at\ninfinity) and continuous functions on Rd. For a given k∈N, we denote by Ck\nb(Rd) (resp.Ck\nc(Rd))\nthe space of all the k-times differentiable functions with continuous and bounded (resp. compactly\nsupported)derivatives. Wealsoconsiderthespace Cc(Rd)ofall thecompactly supported,continuous\nfunctions on Rd. ByBb(Rd), we denote the space of all bounded and Borel measurable fun ctions on\nRd.\nLetS+\ndbe the space of all d×dsymmetric and non-negative definite matrices. For a given λ >0,\nwe denote by S+\nd(λ) those matrices afor which a−λId∈S+\nd, whereIdis thed×didentity matrix.\nFor anyd×dmatrixa= [ai,j], we also define its norm /ba∇dbla/ba∇dbl:=/summationtextd\ni,j=1|ai,j|.\nLetMfin(Rd) be the space of all finite (positive) Borel measures on Rd. It is a closed subset of\nthe Banach space/parenleftbig\nMsign(Rd),/ba∇dbl·/ba∇dblTV/parenrightbig\nof signed Borel measures ν, with a finite total variation\n/ba∇dblν/ba∇dblTV:= sup\n/bardblf/bardbl∞≤1/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRdfdν/vextendsingle/vextendsingle/vextendsingle<+∞.\nHere/ba∇dblf/ba∇dbl∞:= supx∈Rd|f(x)|.Letr(z) := 1∧ |z|2,z∈Rd. Define the mapping R:ML(Rd)→\nMfin(Rd) given by dR(ν) :=rdν. It is 1−1 and onto. The space of L´ evy measures ML(Rd) equipped\nwith the metric\ndR,TV(ν1,ν2) :=/ba∇dblν1−ν2/ba∇dblTV,R=/ba∇dblR(ν1)−R(ν2)/ba∇dblTV, ν1,ν2∈ML(Rd),\nis complete and non-separable.\nWe shall also consider the Kantorovich-Rubinstein metric\ndR,KR(ν1,ν2) :=/ba∇dblR(ν1)−R(ν2)/ba∇dblKR, ν1,ν2∈ML(Rd)\n4generated by the norm\n/ba∇dblµ/ba∇dblKR:= sup/braceleftBigg/integraldisplay\nRdf dµ:/ba∇dblf/ba∇dbl∞+sup\nx/ne}ationslash=y|f(x)−f(y)|\n|x−y|≤1/bracerightBigg\n, µ∈Msign.\nThe space/parenleftBig\nML(Rd),dR,KR/parenrightBig\nin turn is complete and separable, and its topology coincide s with the\ntopology of weak convergence of measures.\n2.2 Probability space with a group of measure preserving tra nsformations\nLet (Ω,d) be a Polish metric space, Gits Borel σ-algebra of sets and µa probability measure on\nG. We suppose further that τx: Ω→Ω,x∈Rdis a group of transformations that preserves the\nmeasure µ, i.e.\n(τx)♯µ(A) =µ(A) for any A∈G, x∈Rd. (2.1)\nHere and in what follows, given two measure spaces (Σ i,Ai,mi),i= 1,2 and a measurable mapping\nS: Σ1→Σ2, we denote by S♯m1the push-forward of m1through S, i.e. the measure on (Σ 2,A2)\ngiven by S♯m1(A) =m1/parenleftbig\nS−1(A)/parenrightbig\nfor anyAinA2.\nWe denote by B(Ω) (resp. Bb(Ω)) the space of all (resp. all bounded) Borel measurable fu nctions\non Ω. Let Cb(Ω) be the space of all bounded and continuous (in metric d) fu nctions on Ω.\nWe suppose that the action of the group is ergodic, i.e. ifA∈GsatisfiesτxA=Afor allx∈Rd,\nthenµ(A) = 0 or 1 and continuous , i.e. for any f∈Cb(Ω) we have\nlim\nx→0f/parenleftbig\nτxω) =f(ω), ω∈Ω. (2.2)\nThe above condition implies in particular that for any f∈B(Ω) and δ >0, we have\nlim\nx→0µ/parenleftbig\n|f/parenleftbig\nτxω)−f(ω)| ≥δ/parenrightbig\n= 0. (2.3)\n2.3 Random Feller processes\nLetDbe the space of all the functions ζ: [0,+∞)→Rdwhich are right continuous and possess left\nlimits at any time t≥0 (c` adl` ag), equipped with the J1-Skorokhod topology. For a precise definition\nof such a topology, see e.g. [4, Section 12]. Define the canoni cal process Xt(ζ) :=ζ(t),ζ∈Dand\nits natural filtration ( Ft)t≥0, withFt:=σ(Xs,0≤s≤t). Then, F:=σ(Xt,0≤t) is the Borel\nσ-algebra on D. We also introduce the canonical shift θs:D→Dgiven by θs(ζ)(t) :=ζ(t+s),\nt,s≥0.\nGiven random elements b: Ω→Rd,a: Ω→S+\ndandn: Ω→ML(Rd), we introduce the corre-\nsponding symbol\nq(ξ;ω) :=−ib(ω)·ξ+1\n2ξ·a(ω)ξ+/integraldisplay\nRd(1−eiz·ξ+iz·ξ1{|z|≤1})n(dz;ω),(ξ,ω)∈Rd×Ω.(2.4)\nWe shall consider as well the respective stationary random fi elds\nb(x;ω) :=b(τxω), a(x;ω) :=a(τxω),n(x,dz;ω) :=n(dz;τxω) (2.5)\n5and\nq(x,ξ;ω) :=q(ξ;τxω),(x,ξ,ω)∈Rd×Rd×Ω. (2.6)\nWe mention here that any sufficiently regular stationary rand om field/parenleftBig\nb(x),a(x),n(x,·)/parenrightBig\nx∈Rdis\nstatistically equivalent with the field of the form (2.5), se e Remark 3.5 below for details.\nAssuming that condition (1.3) holds, we define the operator q(·,D;ω) onC2\nc(Rd) by\nq(x,D;ω)u(x) :=/integraldisplay\nRdq(x,ξ;ω)ˆu(ξ)eix·ξdξ, x ∈Rd.\nThroughout the paper we shall frequently use the following c ondition on the boundedness of the\nsymbolq(x,ξ;ω) and the existence of an associated Feller generator.\nHypothesis 2.1. There exists Λ>0such that\n|q(x,ξ;ω)| ≤Λ(1+|ξ|2),(x,ξ,ω)∈Rd×Rd×Ω. (2.7)\nFurthermore, for any ω∈Ω, the operator (−q(·,D;ω),C2\nc(Rd))is closable in C0(Rd)and its closure\ngenerates a Feller semigroup (Pω\nt)onC0(Rd).\nLet us mention that (2.7) implies that there exists a constan tC(Λ,d), depending only on the\nindicated parameters, such that\n|b(x;ω)|+/ba∇dbla(x;ω)/ba∇dbl+/ba∇dbln(x,·;ω)/ba∇dblTV,R≤C(Λ,d), x,ξ∈Rd, ω∈Ω. (2.8)\nConversely, if (2.8) holds for some constant C >0 then (2.7) is satisfied with a constant Λ depending\nonly onC,d, see e.g. [43, Lemma 6.2].\nUnder Hypothesis 2.1 the martingale problem corresponding to the operator Lωgiven in (1.4) is\nwell posed for each ω∈Ω, see [31, Theorem 3.1]. It means, cf. [45], that for every pr obability Borel\nmeasure monRd, there exists a unique Borel probability measure Pm;ωonD, called a solution to\nthe martingale problem forLωwith initial distribution m, such that\ni)Pm;ω(X0∈A) =m(A) for any Borel measurable A⊂Rd,\nii) for any f∈C∞\nc(Rd), the process\nMω\nt[f] :=f(Xt)−f(X0)−/integraldisplayt\n0Lωf(Xr)dr, t≥0 (2.9)\nis a (c` adlag) martingale under measure Pm;ω, with respect to the natural filtration ( Ft)t≥0.\nAs usual, we write Px;ω:=Pδx;ω,x∈Rd. The expectations with respect to Pm;ωandPx;ωshall be\ndenoted by Em;ωandEx;ω, respectively.\nFrom (2.9), one easily gets that\nPω\ntf(x) =Ex;ωf/parenleftbig\nXt/parenrightbig\n, f∈C0(Rd), x∈Rd, t≥0. (2.10)\n6For anyβ >0 andω∈Ω, we introduce the β-resolvent operator Rω\nβ:C0(Rd)→C0(Rd) as\nRω\nβf(x) :=/integraldisplay+∞\n0e−βtPω\ntf(x), x∈Rd. (2.11)\nRecall from [6, Theorem 2.36] that\nq(x,ξ;ω) = (2π)dlim\nt↓01−Ex;ωei(Xt−x)·ξ\nt, x,ξ∈Rd. (2.12)\nSinceq(x,ξ;ω) satisfies (2.6), one can easily show that\nPx+y;ω= (sy)♯Px;τyω, x,y∈Rd,ω∈Ω, (2.13)\nwheresy:D→Dis given by sy(ζ)(t) :=y+ζ(t),t≥0. In what follows for a given Borel probability\nmeasure µon Ω, we let\nPx;µ(A) =/integraldisplay\nΩµ(dω)/integraldisplay\nD1A(ω,ζ)Px;ω(dζ), A∈G⊗F. (2.14)\n2.4 Regularity of the coefficients\nLet us fix α∈(0,2] and denote by Ldthe Lebesgue measure on Rd. The following conditions shall\nbe used extensively throughout the paper.\nHypothesis 2.2. There exist two constants CQ,cQ>0such that for any ωinΩ,\n(Q1)|q(ξ;ω)| ≤CQ|ξ|α, for any ξ∈Rdsuch that |ξ| ≤1;\n(Q2)Req(ξ;ω)≥cQ|ξ|α, for any ξ∈Rd.\nHypothesis 2.3. For any ωinΩthe1-resolvent Rω\n1(and thus also any β-resolvent) is irreducible,\ni.e.Rω\n11A(x)>0for anyx∈Rdand any Borel set A⊆Rdsuch that Ld(A)>0.\nAccording to [43, Lemma 6.2 (c)] condition (Q1) implies (2.7 ). Furthermore, under Hypothesis\n2.1 for each ω∈Ω, there exists a Feller process ( Px;ω) with symbol q(x,ξ;ω) given by (1.2). Hy-\npothesis 2.1 together with condition (Q2) imply that each Px;ωis in fact strongly Feller and admits\na transition density function pω(t,x,y) which is bounded on [ δ,∞)×Rd×Rd×Ω for any δ >0 (see\n[44, Theorem 1.2]). The irreducibility condition formulat ed in Hypothesis 2.3 can be guaranteed by\none of the following sufficient conditions.\nProposition 2.4. Let(Px)x∈Rdbe a Feller process with bounded coefficients (b,a,n). Denote by\n(Pt)t≥0the corresponding transition probability semigroup on C0(Rd). Then, each of the following\nconditions implies Hypothesis 2.3:\nC1) the matrix abelongs to S+\nd(λ)for some λ >0;\nC2) Hypothesis 2.2 holds and\n7i)suppn(x,·) =Rdfor anyx∈Rd,\nii) for any ϕ∈Cc(Rd), compact set A⊆Rdsuch that Ld(A)∈(0,+∞), we have\nlim\nt↓0/integraldisplay\nRd/parenleftbig\nPt1A(x)−1A(x)/parenrightbig\nϕ(x)dx= 0.\nThe proof of the above proposition is postponed to Section B i n the Appendix.\nRemark 2.5. Observe that C2.ii) holds provided that the dual semigroup P∗\nt:Msign(Rd)→Msign(Rd)\nmapsCc(Rd)intoL1(Ld), andP∗\ntη(x)→η(x)ast→0+forLda.e.x∈Rdand any η∈Cc(Rd).\nHere we treat Cc(Rd),L1(Ld)as subspaces embedded into Msign(Rd). Condition C2.ii) also easily\nfollows when we know that for any bounded open set Dthe semigroup (PD\nt), corresponding to the\nprocess killed upon exiting D, is strongly continuous in Lp(Ld)for some p≥1(see Section A.2 for\nthe definition of the semigroup (PD\nt)).\n2.5 Homogenization result\nSuppose that α∈(0,2]. Under Hypothesis 2.1 for each ε∈(0,1) andω∈Ω, we consider the scaled\nprocess ( Px;ω\nε)x∈Rdsuch that Px;ω\nε(X0=x) = 1 and whose Fourier symbol equals\nqε(x,ξ;ω) =1\nεαq/parenleftBigx\nε,εξ;ω/parenrightBig\n,(x,ξ,ω)∈Rd×Rd×Ω. (2.15)\nNote that Px;ω\nε=/parenleftbig\nTε/parenrightbig\n♯Px/ε;ω, whereTε:D→Dis the mapping given by Tε(ζ)(t) :=εζ(t/εα).\nWe denote by ( Pω\nt,ε)t≥0the Feller semigroup associated with the process ( Px;ω\nε)x∈Rdand byLω\nε\nthe corresponding generator. Let us introduce now the proba bility measure Px;µ\nεon (Ω×D,G⊗F)\ngiven by\nPx;µ\nε(A) =/integraldisplay\nΩµ(dω)/integraldisplay\nD1A(ω,ζ)Px;ω\nε(dζ), A∈G⊗F. (2.16)\nWe shall assume the following averaging property of the Four ier symbol q.\nHypothesis 2.6. There exists qL:Rd×Ω→Csuch that for each ω∈Ω,ξ/mapsto→qL(ξ;ω)is the\nsymbol of some L´ evy process, corresponding to bL: Ω→Rd,aL: Ω→S+\ndandnL: Ω→ML(Rd)via\nformula (2.4), and for any δ >0,\nlim\nε↓0µ/parenleftbig/braceleftbig\nω∈Ω :/vextendsingle/vextendsingleε−αq(εξ;ω)−qL(ξ;ω)/vextendsingle/vextendsingle≥δ/bracerightbig/parenrightbig\n= 0, ξ∈Rd. (2.17)\nRemark 2.7. Sinceq(0;ω)≡0, we obviously have qL(0;ω)≡0. Thanks to condition (Q1) of\nHypothesis 2.2, it also holds that /ba∇dblε−αq(εξ)/ba∇dblL∞(µ)≤Cfor some constant C >0andε∈(0,1].\nTherefore, /ba∇dblqL(ξ)/ba∇dblL∞(µ)≤C. According to the remark made after (2.7), we finally have\n/ba∇dblbL/ba∇dblL∞(µ)+/ba∇dblaL/ba∇dblL∞(µ)+µ-esssup\nω∈Ω/ba∇dblnL(ω)/ba∇dblTV,R<+∞. (2.18)\nRecall that µis a probability measure. In the remainder of the paper, we le tEµdenote expec-\ntation with respect to the probability µ. Our homogenization result can be formulated as follows.\n8Theorem 2.8. Assume that Hypotheses 2.1–2.3 and 2.6 are in force. Further more, suppose that\none of the following conditions hold:\n(i)There exists a closed and separable subset M(s)\nL(Rd)ofML(Rd)(with metric dR,TV) such that\nthe function Rd∋x/mapsto→(b(τxω),a(τxω),n(τxω))∈Rd×S+\nd×M(s)\nL(Rd)is continuous for each\nω∈Ω.\n(ii)α∈(0,2)and the mapping Rd∋x/mapsto→(b(τxω),a(τxω),n(τxω))∈Rd×S+\nd×ML(Rd)is\ncontinuous with the Kantorovich-Rubinstein metric dR,KRonML(Rd)for each ω∈Ω.\n(iii) (ξ,x)/mapsto→q(ξ;τxω)is real valued and continuous for each ω∈Ω.\nLetxbe inRd. Then, the measures (Px;µ\nε)ε∈(0,1)defined in (2.16)converge weakly over D, asε↓0,\nto a measure ¯Qx. In particular, there exists Φ∗: Ω→(0,+∞)satisfying EµΦ∗= 1,Φ∗∈Lp′(µ)for\nanyp >1+d/αand such that ¯Qxis the law of the L´ evy process/parenleftbig\nx+¯Z(t)/parenrightbig\nt≥0whose L´ evy symbol\nequals\n¯q(ξ) =Eµ[qL(ξ)Φ∗], ξ∈Rd. (2.19)\nFurthermore, for each f∈Cb(Rd), we have\nlim\nε↓0Eµ/vextendsingle/vextendsinglePt,εf(x)−¯Ptf(x)/vextendsingle/vextendsingle= 0, t > 0, x∈Rd, (2.20)\nwhere/parenleftbig¯Pt/parenrightbig\nt≥0is the transition probability semigroup of/parenleftbig¯Z(t)/parenrightbig\nt≥0.\nTheorem 2.8 is a direct consequence of Theorems 3.2 and 3.6 fo rmulated below.\n2.6 Some examples\nIn what follows we formulate some sufficient conditions on the coefficients b,aandnwhich allow\nus to apply the conclusion of Theorem 2.8.\n2.6.1 The diffusive scaling\nFixedλ >0, leta: Ω→S+\nd(λ),n: Ω→ML(Rd) be bounded random elements such that the\nstationary field ( a(x;ω),n(x;ω))x∈Rdis continuous for each ω∈Ω. As before, ML(Rd) is considered\nwith the metric d R,TV. Moreover, we assume that for each ω∈Ω,\n/integraldisplay\nRd|z|2n(dz;ω)<+∞. (2.21)\nLet us introduce the corresponding symbol\nq(ξ;ω) =1\n2ξ·a(ω)ξ+/integraldisplay\nRd(1−eiz·ξ+iz·ξ)n(dz;ω),(ξ,ω)∈Rd×Ω.(2.22)\nFurthermore, we assume that q(ξ;ω) satisfies condition (Q1) in Hypothesis 2.2.\nIt follows from [45, Section 4] (but see also [42, Theorem 3.2 3]) that the martingale problem\ncorresponding to q(x,ξ;ω) is well-posed. In particular condition (Q1) implies that ξ/mapsto→q(x,ξ;ω) is\n9continuous at ξ= 0, uniformly in x∈Rd. Thus, by virtue of [42, Lemma 3.26 and Theorem 3.25],\nthe transition probability semigroup ( Pω\nt) given in (2.10) is Feller and Hypothesis 2.1 is satisfied.\nOne can also easily verify condition (Q2) in Hypothesis 2.2 a nd Hypothesis 2.3. Finally, thanks to\nProposition 2.4, Hypothesis 2.6 holds with\nqL(ξ;ω) =1\n2ξ·(a(ω)+a1(ω))ξ,\nwhere\na1(ω)ξ:= 2/integraldisplay\nRdz(ξ·z)n(dz;ω)\nSummarizing, our main result in the diffusive case is the follo wing.\nTheorem 2.9. Letx∈Rdandα= 2. Under the assumptions made in the present section, the\nmeasures (Px;µ\nε)ε∈(0,1)converge weakly over D, asε↓0, to the law of a d-dimensional Brownian\nmotion/parenleftbig\nx+¯B(t)/parenrightbig\nt≥0with a non-degenerate diffusivity. In addition, the respecti ve transition proba-\nbility semigroups converge, according to (2.20), where/parenleftbig¯Pt/parenrightbig\nt≥0is the transition probability semigroup\nof(B(t))t≥0.\n2.6.2 The fractional diffusion limit\nAs in the previous section, we assume that the random element sa: Ω→S+\nd(λ) (for some λ >0)\nandn: Ω→ML(Rd) are bounded and the stationary field ( a(x;ω),n(x;ω))x∈Rdis continuous for\neachω∈Ω. Moreover, we suppose that\nsup\nx∈Rd|b(x;ω)| ≤K ω ∈Ω, (2.23)\nwhere\nb(x;ω) =b(τxω) and b(ω) =−p.v./integraldisplay\n{|z|≥1}zn(dz;ω). (2.24)\nThe principal value of the above integral is understood as th e limit of the integrals over {1≤\n|z| ≤N}, asN→+∞. We assume that q(ξ;ω), given by (2.4), satisfies Hypothesis 2.2 for some\nα∈(0,2). Hypothesis 2.3 holds as well, thanks to part i) of Proposi tion 2.4. As in Section 2.6.1, we\nconclude that Hypothesis 2.1 is satisfied. Below we formulat e some conditions which allow to verify\nHypothesis 2.6.\nThe case of non-symmetric jumps\nIn the present section, we suppose that for each ω∈Ω,\n/integraldisplay\n{|z|≥1}|z|n(dz;ω)<+∞. (2.25)\nFurthermore, we assume that for any δ >0\nlim\nM→+∞sup\nε∈(0,1)µ/parenleftBigg\nε1−α/integraldisplay\n{|εz|≥M}|z|n(dz;ω)≥δ/parenrightBigg\n= 0, (2.26)\nlim\nκ↓0sup\nε∈(0,1)µ/parenleftBigg\nε2−α/integraldisplay\n{|εz|≤κ}|z|2n(dz;ω)≥δ/parenrightBigg\n= 0. (2.27)\n10For any a= (a1,...,a d),b= (b1,...,bd)∈Rd, we shall write a/p∇ecedesequalb, ifai≤bi,i= 1,...,d.\nIfa/p∇ecedesequalb, we call ∆( a,b) := [a1,b1]×...×[ad,bd] a box. We suppose that there exists a random\nfunction s:Rd×Ω→[0,+∞) together with a Borel measure ¯ νonRdsuch that for any δ >0 and\nanya/p∇ecedesequalbsuch that 0 /\\e}atio\\slash∈∆(a,b), we have\nlim\nε↓0µ/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleε−αn/parenleftBig\nz∈Rd:εz∈∆(a,b);ω/parenrightBig\n−/integraldisplay\n∆(a,b)s(z;ω)¯ν(dz)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥δ/parenrightBigg\n= 0.(2.28)\nand for each ω∈Ω, /integraldisplay\n∆(a,b)s(z;ω)¯ν(dz)<+∞. (2.29)\nLetχ: [0,+∞)→[0,+∞) be given by\nχ(r) =/braceleftBigg\nr2,forr≤1,\nr,forr >1.(2.30)\nIn Section C.1 of the Appendix we prove the following result.\nProposition 2.10. Under the assumptions made in the present section, it holds t hat\n/integraldisplay\nRdχ/parenleftbig\n|z|)s(z;·) ¯ν(dz)<+∞, µ-a.s. (2.31)\nLet us define\nqL(ξ;ω) =/integraldisplay\nRd(1−eiz·ξ+iz·ξ)s(z;ω)¯ν(dz),(ξ,ω)∈Rd×Ω. (2.32)\nThe proof of the following proposition is presented in Secti on C.2 of the Appendix.\nProposition 2.11. Under the assumptions made in the present section, Hypothes is 2.6 holds with\nsymbolqLdefined in (2.32).\nAs an immediate corollary, we can conclude the following.\nTheorem 2.12. Under the assumptions made above, the measures (Px;µ\nε)ε∈(0,1]defined in (2.16)\nsatisfy the conclusions of Theorem 2.8.\nThe case of symmetric jumps\nIf we make an additional assumption on the structure of jumps , the hypotheses on the integrability\nof the random L´ evy measure n(dz;ω) made in (2.25) and (2.26) can be omitted. Suppose that\nn(−dz;ω) =n(dz;ω), ω∈Ω. (2.33)\nAs a result, we have b(ω)≡0 in (2.24). Assume as well that (2.27) and (2.28) hold. Inste ad of\n(2.26), we suppose that for any δ >0, it holds\nlim\nM→+∞sup\nε∈(0,1)µ/parenleftbig\nε−αn/parenleftbig\nz:|εz| ≥M;ω/parenrightbig\n≥δ/parenrightbig\n= 0. (2.34)\n11Let us introduce now the following symbol\nqL(ξ;ω) =/integraldisplay\nRd(1−eiz·ξ)s(z;ω)¯ν(dz),(ξ,ω)∈Rd×Ω. (2.35)\nThe following result holds, see Section C of the Appendix for a proof.\nProposition 2.13. Under the assumptions made in the present section, Hypothes is 2.6 holds with\nsymbolqLdefined in (2.35).\nAs a result we conclude the following.\nTheorem 2.14. Under the assumptions made in the present section, the measu res(Px;µ\nε)ε∈(0,1]\ndefined in (2.16)satisfy the conclusions of Theorem 2.8.\n2.6.3 Random α-stable kernels\nExample 2.15 (Isotropic α-stable kernels) .Suppose that α∈(0,2)andn(dz,ω) =c(ω)dz\n|z|d+α. Here\nc: Ω→Rdis a random variable over (Ω,F,µ)such that\nc1≤c(ω)≤c2, z∈Rd(2.36)\nfor some deterministic positive constants c1,c2and such that c(x;ω) :=c(τxω)has continuous real-\nizations for all ω∈Ω.\nOne can check that all the hypotheses made in Theorem 2.14 are s atisfied. In this case ¯ν(dz) =\ndz\n|z|d+αands(z;ω)≡c(ω)in(2.28).\nExample 2.16 (Homogeneous α-stable kernels) .Letα∈(1,2). Example 2.15 can be generalized to\nthe case when the random field {c(z;ω)}z∈Rdsatisfies\nc1≤c(z;ω)≤c2, z∈Rd, µa.s. (2.37)\nand there exists ρ∗>0for which\nc(z;ω) =c(ˆz;ω),|z| ≥ρ∗.\nHereˆz:=z/|z|. We let n(dz,ω) =c(z;ω)dz\n|z|d+α. One can verify the assumptions made in Theorem 2.12,\nwith¯ν(dz) =dz\n|z|d+αands(z) =c(ˆz;ω)in(2.28).\n3 More general formulation of the homogenization result\n3.1 The process of an environment as seen from the particle\nWe introduce an Ω-valued process ( ηt)t≥0over the probability space ( D,F,P0;ω), sometimes referred\nto as the environment process, defined by\nηt(ω) :=τXtω, t≥0. (3.1)\nIn what follows, we denote by Cb(Ω) the space of all bounded and continuous (with respect to t he\nmetric d) functions F: Ω→R.\n12Proposition 3.1. For each ω∈Ω, the process (ηt(ω))t≥0is(Ft)t≥0-Markovian under measure P0;ω.\nIts transition semigroup is given by\nPtF(ω) =E0,ωF/parenleftbig\nηt(ω)/parenrightbig\n=Pω\nt˜F(·;ω)(0), F∈Bb(Ω), (3.2)\nwhere˜F(y;ω) :=F(τyω).\nTheproof of theabove resultcan beobtained following thesa me arguments as in[26, Proposition\n9.7].\n3.2 The homogenization result\nTheorem 3.2. Letx∈Rd. Assume that there exists an invariant ergodic probability measure µ∗\nfor the process (ηt)t≥0which is equivalent to µ. Furthermore, suppose that Hypothesis 2.1, condition\n(Q1) in Hypothesis 2.2 and Hypothesis 2.6 are in force for som eα∈(0,2]. Then, the measures\n(Px;µ\nε)ε∈(0,2]defined in (2.16)converge weakly over D, asε↓0, to¯Qx, the law of the L´ evy process/braceleftbig\nx+¯Z(t)/bracerightbig\nt≥0with L´ evy symbol Eµ∗qL(ξ). Furthermore, for any f∈Cb(Rd),t >0andx∈Rd, the\nrandom variables Pt,εf(x)converge, as ε↓0, in the sense of (2.20).\nProof.First we prove that ( Px;µ\nε)ε∈(0,1]is tight. By [43, Lemma 6.2], Hypothesis 2.1 implies that\n|bε(x;ω)|+/ba∇dblaε(x;ω)/ba∇dbl+/ba∇dblnε(x,·;ω)/ba∇dblTV,R≤C(Λ,d), x,ξ∈Rd, ω∈Ω, (3.3)\nwhere (bε,aε,nε) is the L´ evy triplet corresponding to qεgiven by (1.5). There exists then a constant\nC >0 such that for any u∈C2\nb(Rd),ε∈(0,1) andω∈Ω,\n/ba∇dblLω\nεu/ba∇dbl∞≤C(/ba∇dblu/ba∇dbl∞+/ba∇dblDu/ba∇dbl∞+/ba∇dblD2u/ba∇dbl∞). (3.4)\nHence, tightness of ( Px;µ\nε)ε∈(0,1]follows, see the proof of [25, Theorem 4.9.2], or [45, Theore m A.1].\nAccording to [15, Theorem 5], in order to prove the weak conve rgence it suffices only to show\nthat for any t >0,ξ∈Rdandδ >0\nlim\nε↓0Px;µ\nε/parenleftbigg/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplayt\n0q/parenleftBigXs\nε,εξ/parenrightBig\nds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/parenrightbigg\n= 0. (3.5)\nSinceµis invariant under ( τx), the law Px;µ\nεcoincides with the law P0;µ\nε(cf (2.13)). To prove the\nresult, it thus suffices to show the weak convergence of P0;µ\nε, asε→0+. Clearly, we may equivalently\nreplace measures P0;µ\nεbyP0;µ∗ε,ε∈(0,1]. Notice that P0;µ∗εcoincides with the law of the process\nt/mapsto→εXs/εαunder measure P0;µ∗(see (2.14) and the comment following (2.15)). Consequentl y,\nP0;µ∗ε/parenleftbigg/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplayt\n0q/parenleftBigXs\nε,εξ/parenrightBig\nds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/parenrightbigg\n=P0;µ∗/parenleftbigg/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplayt\n0q(Xs/εα,εξ)ds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/parenrightbigg\n.(3.6)\nUsing the environment process ( ηt)t≥0defined in (3.1), we can further write that\nq(Xs/εα,εξ) =q(εξ;ηs/εα).\n13Therefore, the right-hand side of (3.6) equals\nP0;µ∗/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/integraldisplayt/εα\n0q(εξ;ηs)ds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/parenrightBigg\n≤P0;µ∗/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/integraldisplayt/εα\n0q(εξ;ηs)ds−εα/integraldisplayt/εα\n0qL(ξ;ηs)ds/vextendsingle/vextendsingle/vextendsingle> δ/2/parenrightBigg\n+P0;µ∗/parenleftBigg/vextendsingle/vextendsingle/vextendsingleεα/integraldisplayt/εα\n0qL(ξ;ηs)ds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/2/parenrightBigg\n(3.7)\nThe first term on the right hand side can be estimated using the Markov inequality by\n2\nδEµ∗/bracketleftBigg\nεα/integraldisplayt/εα\n0/vextendsingle/vextendsingleε−αq(εξ;ηs)−qL(ξ;ηs)/vextendsingle/vextendsingleds/bracketrightBigg\n=2t\nδEµ∗/vextendsingle/vextendsingleε−αq(εξ)−qL(ξ)/vextendsingle/vextendsingle→0,asε↓0.\nThe first equality holds due to the stationarity of measure µ∗while the limit is a consequence of\nHypothesis 2.6. Recall that µ∗is equivalent with µ,/ba∇dblqL(ξ)/ba∇dblL∞(µ)<+∞andε/mapsto→ε−α/ba∇dblq(εξ)/ba∇dblL∞(µ),\nε∈(0,1], is bounded, thanks to condition (Q1) in Hypothesis 2.2. B y the Birkhoff ergodic theorem,\nwe have\nlim\nε↓0εα/integraldisplayt/εα\n0qL(ξ;ηs)ds=tEµ∗qL(ξ),P0;µ∗-a.s.\nTherefore, also the second term on the right hand side of (3.7 ) vanishes, as ε↓0. Summarizing, we\nconclude that (3.5) holds, finishing in this way the proof of t he weak convergence of ( Px;µ\nε)ε∈(0,2].\nChoosef∈Cb(Rd) andx∈Rd. To prove (2.20) it suffices to show that for any sequence εn→0\none can choose a subsequence ( εnk) such that\nlim\nk→+∞Eµ|Pt,εnkf(x)−¯Ptf(x)|= 0. (3.8)\nWe have Pω\nt,εnkf(x) =Pτ−xω\nt,εnkf◦Tx(0), where Tx:Rd→Rdis given by Tx(y) =x+y,x,y∈Rd.\nTherefore, to show (3.8) it suffices only to prove that\nlim\nk→+∞Eµ/vextendsingle/vextendsingle/vextendsingleE0;·f/parenleftBig\nεnkXt/εαnk/parenrightBig\n−¯Ptf(0)/vextendsingle/vextendsingle/vextendsingle= 0. (3.9)\nfor anyf∈Cb(Rd). The fact that ( P0;ω\nε)ε∈(0,1]is tight for any ω∈Ω follows from (3.4). Choose a\nsequence εn→0. By the argument from the previous part of the proof we can co nclude that there\nexists Ω 0∈Gs.t.µ(Ω0) = 1 and such that for some subsequence ( εnk)\nlim\nk→+∞P0;ω\nεnk/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0qεnk(Xs,ξ)ds−tEµ∗qL(ξ)/vextendsingle/vextendsingle/vextendsingle> δ/parenrightbigg\n= 0 (3.10)\nfor allω∈Ω0,t∈[0,+∞)∩Qandξ∈Qd, whereQis the set of all rationals. Thanks to (2.18) and\n(3.3) we conclude that the function ( t,ξ)/mapsto→tEµ∗qL(ξ) is continuous. By the same token (see also\n[17, Proposition 2.1] or [6, Proposition 2.17]), Ω 0can be chosen so that for each ω∈Ω0,R,T >0\nthe family of functions [0 ,T]×[|ξ| ≤R]∋(t,ξ)/mapsto→/integraltextt\n0qεnk(Xs,ξ;ω)ds, indexed by k= 1,2,..., is\nequi-continuous a.s. Therefore (3.10) holds for all ω∈Ω0,t >0 andξ∈Rd. Thus, we conclude\n14thatP0;ω\nεnkconverge weakly, as k→+∞, to¯Q0, as in the statement of the theorem for each ω∈Ω0.\nFormula (3.9) then follows from the definition of the weak con vergence and the Lebesgue dominated\nconvergence theorem.\n3.3 On the existence of an ergodic invariant measure: Theore m I\nLet us introduce the space W:=Bb(Rd;Rd×S+\nd×ML(Rd)) of all bounded and Borel measurable\nfunctions t= (b,a,n):Rd→Rd×S+\nd×ML(Rd), where ML(Rd) is equipped with the metric d R,KR.\nFurthermore, let\nQ:={q∈B(Rd×Rd;C):q(x,·) is non-negative definite for any x∈Rd,\nq(x,0)≡0,sup\n|ξ|≤1,x∈Rd|q(x,ξ)|<+∞},(3.11)\nand\nQ1,2:={q∈Q:qsatisfies Hypothesis 2 .2}. (3.12)\nSuppose that ( b,a,n): Ω→Rd×S+\nd×ML(Rd) andq:Rd×Ω→Care random elements as in\n(2.4) and (2.5). They induce the maps CW: Ω→WandCQ: Ω→Qgiven by\nCW(ω)(x) := (b(τxω),a(τxω),n(τxω)) and CQ(ω)(x,ξ) :=q(ξ;τxω) (x,ξ)∈R2d.\nGiven any L´ evy triplet ( b,a,n), we can assign to it the Fourier symbol q(b,a,n) using formula\n(1.2). It satisfies q(b,a,n)(0) = 0. According to [42, part f) of Theorem 2.15], having a n egative-\ndefinite function q:Rd→Cthat satisfies q(0) = 0 one can find a unique L´ evy triplet ( b,a,n) such\nthatq(b,a,n) =q. We let\nW1,2:={k∈W:q(k)∈Q1,2}.\nDenoted C:=C2\nc(Rd), we introduce now the set\nA:={Lis a linear operator on Cthat is closable and its closure Lis a Feller generator }.(3.13)\nWe also let\nAc:={L:L∈A},Ac,1,2:={L∈Ac: Fourier symbol of Lsatisfies Hypothesis 2 .2}.(3.14)\nSuppose that q:Rd×Ω→Cis as in (2.4). If −q(D;τ·ω)∈Afor each ω∈Ω, we can define the\nmapCAc: Ω→Acby letting\nCAc(ω) :=−q(D;τ·ω)|C. (3.15)\nRecall that for any x∈Rdwe have introduced in the foregoing the operator Tx:Rd→Rdgiven by\nTx(y) =x+y,y∈Rd.\nHypothesis 3.3. There exist a set KW⊆W1,2and a metric DWonKWsuch that\nµ({ω∈Ω :CW(ω)∈KW}) = 1,\nand\n15(F0)CW: Ω→KWis measurable, with Borel σ-field on KW,\n(F1) The pair (KW,DW)is a Polish metric space. Moreover, −q(k)(·,D)∈A, for any k∈KW;\n(F2) If{k,k1,k2,...} ⊆KWare such that DW(kn,k)→0, whenn→ ∞, then for any η∈Cwe\nhave\nlim\nn→+∞q(kn)(·,D)η=q(k)(·,D)ηinC0(Rd); (3.16)\n(F3) For any k∈KW, we have k◦Tx∈KW, x∈Rd, and the mapping x/mapsto→k◦Txis continuous;\n(F4) For any k∈KW, there exist a increasing sequences (M′\nN)N≥1,(MN)N≥1⊆Nand/parenleftbig\nkN/parenrightbig\nN≥1⊂\nKWsuch that\n- eachkNisMN-periodic in every direction of the variable x;\n-M′\nN< MN,N= 1,2,...andlimN→+∞M′\nN\nMN= 1;\n-limN→+∞supy∈[−M′\nN/2,M′\nN/2]dDW(kN◦Ty,k◦Ty) = 0.\nRemark 3.4. It is possible to formulate a condition analogous to Hypothes is 3.3 starting with a\nsubsetKQ⊂Q1,2defined in (3.12), introducing the respective complete and separable metric DQand\nreplacing the set KWand mapping CWin Hypothesis 3.3 by KQandCQ, respectively.\nSimilarly, using as a starting point a subset KAcofAc,1,2defined in (3.14), one can also formulate\nHypothesis 3.3 (in this case for k∈KActhe operator k◦Tyshould be understood as [(k◦Ty)(η)](x) :=\n[k(η◦T−1\ny)](x+y)).\nRemark 3.5. The setup presented in the foregoing is convenient when deali ng with stationary ran-\ndom fields. Suppose that e:= (b,a,n) :Rd×Θ→Rd×S+\nd×ML(Rd)is a stationary random field, de-\nfined over some complete probability space (Θ,F,P)and such that e(·;θ)∈KW, forPa.s.θ. Station-\narity of the field means that the laws of the random elements/parenleftbig\ne(x1+h),...,e(xn+h)/parenrightbig\n,x1,...,x n,h∈\nRddo not depend on h. Letµbe the law of the field over the space KW. Define the group of transfor-\nmations (τx)x∈RdonKWby letting τx(k) :=k◦Tx,x∈Rd. The stationarity of (e(x))x∈Rdimplies\nthat the group (τx)x∈Rdleaves measure µinvariant. Let E:KW→Rd×S+\nd×ML(Rd)be given by\nE(k) = (b,a,n) :=k(0). The field ˜e:= (˜b,˜a,˜n) :Rd×KW→Rd×S+\nd×ML(Rd)defined over the\nprobability space/parenleftBig\nKW,B/parenleftbig\nKW/parenrightbig\n,µ/parenrightBig\nby letting\n(˜b(x;k),˜a(x;k),˜n(x;k)) = (b(τxk),a(τxk),n(τxk))\nhas the law that is identical with that of the field (e(x))x∈Rd. In fact, then we could carry on our\nconsideration taking Ω =KWwith the respective group (τx)x∈Rd. A similar setup can be made in\nthe case of spaces KQ,KAc.\nTheorem 3.6. Letp > d/α +1and assume that Hypotheses 2.2 and 3.3 hold. Then, there exist s\nΦ∗∈Lp′(µ)such that\n(i)/ba∇dblΦ∗/ba∇dblL1(µ)= 1;\n16(ii)the measure µ∗on(Ω,G)such that Φ∗=dµ∗\ndµis invariant under the dynamics of the environ-\nment process (ηt)t≥0, i.e.\n/integraldisplay\nΩPtF dµ∗=/integraldisplay\nΩF dµ∗, F∈Bb(Ω), t≥0;\n(iii) Φ∗satisfies\n/ba∇dblΦ∗/ba∇dblLp′(µ)≤C∗,where\nC∗=cdG1/p(α,cQ)\n1−exp{−C(d)CQ}/bracketleftBigg\n1+/parenleftbigg1\np−1−d/α/parenrightbigg1/p/bracketrightBigg\n<∞;(3.17)\n(iv)if additionally we assume that Hypothesis 2.3 holds, then Φ∗>0µ-a.s. onΩand the measure\nµ∗is ergodic, i.e. if F∈L∞(µ∗)satisfies PtF=Ffor anyt >0, thenFis constant µ∗-a.s.\ninΩ.\nAbove,cd,C(d)>0are constants depending only on the dimension d,cQandCQare the constants\nappearing in Hypothesis 2.2 and\nG(α,cQ) :=Γ(d/α)\nαcd/α\nQ. (3.18)\nHereΓ(·)is the Euler gamma function.\nThe proof of Theorem 3.6 is postponed to Section 4.\nExample 3.7. Letσbe a bounded measure on Rdand letM(σ)\nL(Rd)denote the set of all µ∈ML(Rd)\nthat satisfy µ≪σ. It is a closed and separable subset of ML(Rd)in the metric dR,TV. For any\nΛ,λ >0, we introduce the space KWof continuous functions k= (k(1),k(2),k(3)):Rd→Rd×\nS+\nd(λ)×M(σ)\nL(Rd)satisfying\nsup\nx∈Rd\nd/summationdisplay\ni=1|k(1)\ni(x)|+d/summationdisplay\ni,j=1|k(2)\ni,j(x)|+/integraldisplay\nRd(1∧|z|)2k(3)(x,dz)\n≤Λ,\nwith the symbol qofk, i.e.\nq(x,ξ;k) =−ik(1)(x)·ξ+1\n2ξ·k(2)(x)ξ+/integraldisplay\nRd(1−eiz·ξ+iz·ξ1{|z|≤1})k(3)(x,dz),(3.19)\nsatisfying Hypothesis 2.2. We equip the space KWwith the Fr´ echet metric\nDW(k1,k2) :=+∞/summationdisplay\nK=11\n2K·/ba∇dblk1−k2/ba∇dbl∞,K\n1+/ba∇dblk1−k2/ba∇dbl∞,K, (3.20)\nforkj:= (k(1)\nj,k(2)\nj,k(3)\nj)∈KW,j= 1,2. Here, for a given K >0\n/ba∇dblk/ba∇dbl∞,K:= sup\n|x|≤K/parenleftBig\n|k(1)(x)|+|k(2)(x)|+/vextenddouble/vextenddoublek(3)(x)/vextenddouble/vextenddouble\nTV,R/parenrightBig\n.\n17Measurability of the mapping CWis clear. Using [45], [6, Theorem 3.25, Lemma 3.26], and [23,\nTheorem 19.25] one can verify conditions (F1) and (F2) of Hypo thesis 3.3. Condition (F3) is a\nconsequence of the definition of the Fr´ echet metric. To see t hat (F4) holds, consider an increasing\nsequence of integers (aN)N≥1and letMN:= (2N+8)aN. For each N∈N, one can find a smooth\nfunction βN:R→[0,1]such that /ba∇dblβ′\nN/ba∇dbl∞≤1and\nβN(r)≡\n\n1,ifr∈[−(MN−2)/2,(MN−2)/2]\n0ifr∈[−MN/2,MN/2]c.\nGivenk∈KW, one can then define\nkN(x) :=k/parenleftbig\nβN(x1)x1,...,β N(xd)xd/parenrightbig\n, x∈[−MN/2,MN/2]d,\nand then periodically extend it to the whole Rd. Condition (F4) in Hypothesis 3.3 then follows, with\nM′\nN:= 2NaN, directly from the definition of the set KWand metric DW.\nExample 3.8. Letα∈(0,2). FixΛ,λ >0. LetKWconsist of continuous mappings k=\n(k(1),k(2),k(3)):Rd→Rd×S+\nd(λ)×ML(Rd)such that Dk(2)∈C(Rd),\nsup\nx∈Rd\nd/summationdisplay\ni=1|k(1)\ni(x)|+d/summationdisplay\ni,j=1|k(2)\ni,j(x)|+d/summationdisplay\ni,j=1|Dk(2)\ni,j(x)|+/integraldisplay\nRd(1∧|z|)2k(3)(x,dz)\n≤Λ,\nand the symbol qofk, see(3.19), satisfies Hypothesis 2.2.\nWe equip the space KWwith the Fr´ echet metric\nDW(k1,k2) :=+∞/summationdisplay\nK=11\n2K·/ba∇dblk1−k2/ba∇dbl∞,K\n1+/ba∇dblk1−k2/ba∇dbl∞,K, (3.21)\nforkj:= (k(1)\nj,k(2)\nj,k(3)\nj)∈KW,j= 1,2. Here, for a given K >0\n/ba∇dblk/ba∇dbl∞,K:= sup\n|x|≤K/parenleftBig\n|k(1)(x)|+|k(2)(x)|+|Dk(2)(x)|+/vextenddouble/vextenddoublek(3)(x)/vextenddouble/vextenddouble\nTV,R/parenrightBig\n.\nThe pair (KW,DW)satisfies (F0)–(F4) again.\nMeasurability of the mapping CWis clear. Applying [29, Theorem 1.1] and [23, Theorem 19.25]\none can conclude that conditions (F1) and (F2) of Hypothesis 3.3 hold (see also [29, Proposition\n7.1] and the top of the page 595 in [41]). Condition (F3) is a con sequence of the definition of the\nFr´ echet metric. For the periodic approximation formulate d in (F4) one can utilize the sequence (kN)\nconstructed in Example 3.7.\nExample 3.9. Letνbe a L´ evy measure and λ1,λ2>0. For given C1regular function f:Rd→R\nandK≥1, we denote\n/ba∇dblf/ba∇dbl1,∞,K:= max{/ba∇dblf/ba∇dbl∞,K,/ba∇dbl∂x1f/ba∇dbl∞,K,...,/ba∇dbl∂xdf/ba∇dbl∞,K}.\nHere/ba∇dblf/ba∇dbl∞,K= sup|x|≤K|f(x)|. We write /ba∇dbl·/ba∇dbl1,∞in caseK= +∞.\n18LetMdbe the space of all d×dmatrices equipped with the metric corresponding to the maxi mum\nnorm. Consider the space Ethat is the completion of the space E0made of elements e= (b,σ,h),\nwhereb:Rd→Rd,σ:Rd→Mdandh:R2d→RdareC∞smooth and compactly supported, in the\nmetric\ndE(e1,e2) :=+∞/summationdisplay\nK=11\n2K·/ba∇dblb1−b2/ba∇dbl1,∞,K+/ba∇dblσ1−σ2/ba∇dbl1,∞,K+/ba∇dblh1−h2/ba∇dbl∞,K+[h1−h2]\n1+/ba∇dblb1−b2/ba∇dbl1,∞,K+/ba∇dblσ1−σ2/ba∇dbl1,∞,K+/ba∇dblh1−h2/ba∇dbl∞,K+[h1−h2],\nwithej= (bj,σj,hj),j= 1,2. Here\n[h] := max\n\nsup\nx∈Rd/braceleftBigg/integraldisplay\n|z|≤1|h(x,z)|2ν(dz)/bracerightBigg1/2\n,sup\nx∈Rd/braceleftBigg/integraldisplay\n|z|≤1|∂xih(x,z)|2ν(dz)/bracerightBigg1/2\n, i= 1,...,d\n\n.\nThe space Eis separable. Let K(λ1,λ2)⊂Ebe the set consisting of those efor which the following\nconditions hold:\ni) functions b,σ,hsatisfy\nmax/braceleftbig\n/ba∇dblb/ba∇dbl1,∞,/ba∇dblσ/ba∇dbl1,∞,[h]/bracerightbig\n≤λ1,\nii)\n|h(x,z)| ≤λ2(|z|+1), x,z∈Rd.\niii) Fourier symbol q(·,·)ofesatisfies Hypothesis 2.2.\nWe denote by KActhe set of all operators of the form\nLeu(x) =b(x)·∇u(x)+1\n2Tr/parenleftbig\na(x)∇2u(x)/parenrightbig\n+/integraldisplay\nRd/parenleftBig\nu/parenleftbig\nx+h(x,z)/parenrightbig\n−u(x)−∇u(x)·h(x,z)1{|z|≤1}/parenrightBig\nν(dz), u∈C,\nwithe= (b,σ,h)∈K(λ1,λ2). Herea=σ2:Rd→S+\nd. ForLej∈KAc,j= 1,2we define\nDAc(Le1,Le2) := d E(e1,e2).The set K(λ1,λ2)is closed as the convergence of operators in metric\nDAcimplies the convergence of their respective Fourier symbol s. Thus, the space/parenleftbig\nKAc,DAc/parenrightbig\nis\ncomplete and separable.\nMeasurability of the mapping CAcis clear. By virtue of the results of [1, Section 6.2] and [28,\nTheorem 2.3, p. 125] the martingale problem corresponding to anyL∈KAcis well posed. Thanks\nto Hypothesis 2.2 and [44, Theorem 3.25, Lemma 3.26] we have KAc⊂Acand condition (F1)\nis therefore satisfied. Verification of the remaining condit ions (F2)-(F3) is routine (we use [23,\nTheorem 19.25] again). A construction of a sequence of period ic approximations for any k∈KAc\ncan be done analogously to the previous example, i.e. for e= (b,σ,h)∈K(λ1,λ2)we define eN(x) :=\ne(βN(x)), x∈[−MN/2,MN/2]and then we periodically extend it to the whole Rd.\nExample 3.10. Fixδ >0. We let ld,α:= 2[d\nα∨2]+3+dand\nKQ:={q∈Q1,2: Req=q, ∂β\nxq∈C(R2d),∀M >0∃δM>0 :|∂β\nxq(x,ξ)| ≤δM(1+|ξ|α),\n|β| ≤ld,α,|x| ≤M, ξ∈Rd}.\n19Hereβ= (β1,...,β d)is a non-negative integer valued multi-index, ∂β\nx=∂β1x1...∂βdxdand|β|=/summationtextd\nj=1βj. Forq1,q2∈KQwe set\nDQ(q1,q2) :=∞/summationdisplay\nn=12−n/summationdisplay\n|β|≤ld,αmin{sup\n|x|,|ξ|≤n|∂β\nxq1(x,ξ)−∂β\nxq2(x,ξ)|,1}.\nMeasurability of the mapping CQis clear. Using [17, Theorem 4.2] (see also [17, Theorem 4.1]), [6,\nTheorem 3.25, Lemma 3.26] and [23, Theorem 19.25] one can concl ude that conditions (F1) and (F2)\nof Hypothesis 3.3 hold. Condition (F3) is a consequence of the definition of the Fr´ echet metric. To see\nthat (F4) holds, let us consider a smooth function η:R→[0,1]such that η(x) = 1, x∈[−1,1]and\nη(x) = 0, x /∈[−2,2]. Letan≥1be an increasing sequence of natural numbers and MN:= (2n+8)an,\nM′\nN:= 2nan. Set\nαN(r) :=\n\n0, ifr /∈[−MN/2,MN/2]\n1 ifr∈[−MN/2+1,MN/2−1]\nη(r−2+MN/2)ifr∈[−MN/2,−MN/2+1]\nη(r+2−MN/2)ifr∈[MN/2−1,MN/2],\nand˜αN(x) :=/producttextd\nj=1αN(xj), x= (x1,...,x d)∈Rd. Now, one easily shows that the periodic extension\nof the function\nkN(x) := ˜αN(x)k(x)+(1−˜αN(x))k(0), x∈[−MN/2,MN/2]d\nsatisfies (F4).\n3.4 On the existence of an ergodic invariant measure: Theore m II\nIn the following result, we shall dispense with Hypothesis 3 .3.\nTheorem 3.11. Assume that each Fourier symbol (x,ξ)/mapsto→q(ξ;τxω), corresponding to the L´ evy\ntripletx/mapsto→(b(τxω),a(τxω),n(τxω))satisfies Hypotheses 2.1, 2.2, and the respective operator s atisfies\nHypothesis 2.3. Then each of the following conditions implie s the conclusions of Theorem 3.6.\n(i)There exists a closed and separable subset M(s)\nL(Rd)ofML(Rd)such that the function Rd∋\nx/mapsto→(b(τxω),a(τxω),n(τxω))∈Rd×S+\nd×M(s)\nL(Rd)is continuous for each ω∈Ω, in the total\nvariation metric dR,TV.\n(ii)α∈(0,2)and the mapping Rd∋x/mapsto→(b(τxω),a(τxω),n(τxω))∈Rd×S+\nd×ML(Rd)is\ncontinuous in the Kantorovich-Rubinstein metric dR,KRfor each ω∈Ω.\n(iii) (x,ξ)/mapsto→q(ξ;τxω)is real valued and continuous for each ω∈Ω.\nThe proof of Theorem 3.11 is presented in Section 4.4.\n204 Proof of Theorem 3.6: the existence of an invariant measure\nWe assume that Hypotheses 2.2 and 3.3 hold. Let\n˜τx:KW→KW,˜τx(k) :=k(x+·), x∈Rd.\nFor each k∈KW, We also denote by ˜q(·,k) the Fourier symbol corresponding to the L´ evy triplet\nkand\n˜a(k) :=k(1)(0),˜n(dz;k) :=k(2)(0,dz),˜b(k) :=k(3)(0),\n˜a(x;k) :=˜a(˜τxk),˜ n(x,dz;k) :=˜n(dz;˜τxk),˜b(x;k) :=˜b(˜τxk),˜q(x,ξ;k) :=q(ξ,˜τxk).(4.1)\nConsequently, ˜ µ:=/parenleftbig\nCW/parenrightbig\n♯µis a well-defined Borel probability measureon KW. Inaddition, (˜ τx)♯˜µ=\n˜µ,x∈Rdis ergodic w.r.t. the group action of (˜ τx)x∈Rd.\nThe conclusion of Theorem 3.6 follows, once we prove that the re exists a Borel probability\nmeasure ˜ µ∗onKW, that is absolutely continuous w.r.t. ˜ µ, with˜Φ∗:=d˜µ∗\nd˜µ, which is an invariant and\nergodic for the semigroup\n˜PtF(k) :=E0;kF(˜τXtk), F∈Bb(KW),k∈KW.\nIn addition, we show that the remaining assertions of Theore m 3.6 hold, with suitable replacement\nof the objects appearing in the statement by ˜Φ∗,˜µ,KW,B(KW),˜Pt. Then, dµ∗:= Φ∗dµ, with\nΦ∗:=˜Φ∗◦CWsatisfies the conclusion of Theorem 3.6.\nIn light of the above, we may and shall assume that Ω = KWthroughout the remainder of the\npresent section. We shall also omit the superscript tilde an d subscript Wfrom our notation.\n4.1 Ergodic theorem\nOur first result is a version of the ergodic theorem somewhat a nalogous to the one that can be found\nin [35], see also [34, Lemma 3.2]. Before its formulation we n eed some notation. Let Tbe the one\ndimensional unit torus, i.e. the interval [ −1/2,1/2], whose endpoints −1/2 and 1/2 are identified.\nGivenM >0, we can then denote by Td\nM:= (MT)dthed-dimensional torus of length Mand by\nℓ(dy) :=M−ddythe normalized Lebesgue measure on Td\nM. Furthermore, let e i,i= 1,...,dbe the\ncanonical basis of Rd. For notational simplicity, let us denote QK:= [−K/2,K/2]dfor anyK >0.\nLemma 4.1. There exists ¯ω∈Ωsuch that the sequence (¯µM)M∈Nof Borel measures on Ω, given by\n¯µM(A) :=/integraldisplay\nTd\nM1A(τy¯ω)ℓ(dy), A∈G,\nweakly converges, as M→+∞, toµ.\nProof.Note that there exist a metric ¯D on Ω which is equivalent to D and a countable family\nZ:= (Fn)n∈Nof bounded functions on Ω such that Zis dense in U¯D(Ω) in the supremum norm.\nHereU¯D(Ω) is the space of all real valued, uniformly continuous in m etric¯D functions on Ω. This\ncan be seen as follows. Since Ω is Polish, it is well-known tha t Ω is homeomorphic to a subset of a\n21compact metric space - the Hilbert cube. Therefore, there ex ists an equivalent metric ¯D on Ω such\nthat/parenleftbig\nΩ,¯D/parenrightbig\nis totally bounded. It then follows that the completion of Ω u nder¯D, denoted by ¯Ω,\nis compact. Now, the space U¯D(Ω) is isometrically isomorphic with the space C(¯Ω) of continuous\nfunctions on ¯Ω, both equipped with the topology of uniform convergence. S ince it is known (cf. [8,\nLemma VI.8 .4]) that the latter is separable, so is U¯D(Ω).\nBy the individual ergodic theorem, we can choose Ω 1⊂Ω such that µ(Ω1) = 1 and for any\n¯ω∈Ω1andF∈Z\nlim\nM→+∞/integraldisplay\nTd\nMF(τy¯ω)ℓ(dy) = lim\nM→+∞1\nMd/integraldisplay\nQMF(τy¯ω)dy=/integraldisplay\nΩF dµ. (4.2)\nA density argument implies that (4.2) holds forany function FinU¯D(Ω) and ¯ω∈Ω1. Theconclusion\nof the lemma follows then from [46, Theorem 1 .1.1].\nWe can now state our version of the ergodic theorem.\nTheorem 4.2. There exist a sequence (ωn)n∈NinΩand an increasing sequence of positive numbers\n(Mn)n∈Nsuch that Mn→+∞and\n(i)eachωnisMn-periodic in each variable, i.e. τMneiωn=ωn, forn∈Nandi= 1,...,d;\n(ii)the following sequence of Borel probability measures on Ω\nµn(A) :=/integraldisplay\nTd\nMn1A(τyωn)ℓ(dy), A∈G, (4.3)\nweakly converges to µ, asn→+∞, i.e.\nlim\nn→+∞/integraldisplay\nΩF dµn=/integraldisplay\nΩF dµ, F ∈Cb(Ω). (4.4)\nProof.Thanks to Lemma 4.1, there exists ¯ ω∈Ω such that\nlim\nM→+∞/integraldisplay\nTd\nMF(τy¯ω)ℓ(dy) =/integraldisplay\nΩF dµ, F ∈Cb(Ω). (4.5)\nWe fix an arbitrary ρ >0. LetF∈UD(Ω). Let us now consider increasing sequences of integers\n(M′\nn)n≥1, (Mn)n≥1and L´ evy triplets ωn= (ω(1)\nn,ω(2)\nn,ω(3)\nn) satisfying condition (F4) of Hypothesis\n3.3. Thanks to (4.5), we have\nlim\nn→+∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTd\nMnF(τy¯ω)ℓ(dy)−/integraldisplay\nΩF dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0. (4.6)\nThanks to (F4), it then follows that\n|F(τyωn)−F(τy¯ω)|< ρ, y ∈QM′n, n≥n0 (4.7)\n22Recalling the definition of the measures µnin (4.3), we infer that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF dµn−/integraldisplay\nΩF dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTd\nMnF(τyωn)ℓ(dy)−/integraldisplay\nΩF dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤M−d\nn/integraldisplay\nQMn|F(τyωn)−F(τy¯ω)|dy+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTd\nMnF(τy¯ω)ℓ(dy)−/integraldisplay\nΩFdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(4.8)\nThe second integral tends to 0, as n→+∞, thanks to (4.6). The first integral on the other hand\ncan be estimated from (4.7) by\nρ+1\nMdn/integraldisplay\nQMn/integerdivideQM′n|F(τyωn)−F(τyω)|dy≤ρ+2/ba∇dblF/ba∇dbl∞/bracketleftBigg\n1−/parenleftbiggM′\nn\nMn/parenrightbiggd/bracketrightBigg\n.\nSinceM′\nn\nMn→1, asn→+∞(condition (F4)), the above estimate together with (4.8) im ply that\nlimsup\nn→+∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF dµn−/integraldisplay\nΩF dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ρ\nfor any arbitrary ρ >0, which in turn implies (4.4) for any F∈UD(Ω). Finally, another application\nof [46, Theorem 1 .1.1] allows us to conclude the proof of Theorem 4.2.\n4.2 Construction of an invariant density\nLet (ωn)n∈Nand (Mn)n∈Nbe as in the statement of Theorem 4.2. Fix n∈N. With each L´ evy\ntripletωnwe associate the operator Lωn:C2\nc(Rd)→C0(Rd) as in (1.4). Let πn:Rd→Td\nMnbe\nthe canonical projection of RdontoTd\nMn. Then, the process ˜Px\nn:= (πn)♯Px;ωn,x∈Td\nMn, is strongly\nMarkovian and Feller on Td\nMn, with transition probability densities ˜ pn,t(x,y) given by\n˜pn,t(x,y) :=/summationdisplay\nm∈Zdpωn\nt(x,y+Mnm), t≥0, x,y∈Td\nMn.\nHere,pωn\nt(x,y) are the transition probability densities corresponding t o the path measure Px;ωn. In\naddition, for any ˜f∈C(Td\nMn) we have\n˜Ex\nn˜f(˜Xt) =Ex;ωnf(Xt), x∈Td\nMn, (4.9)\nwheref∈Cb(Rd) is the Mn-periodic extension of ˜f,˜Xt=πn(Xt) and˜Ex\nnis the expectation\ncorresponding to ˜Px\nn. Let\n˜ nn(x,dz) :=/summationdisplay\nm∈Zdn(dz+Mnm;τxωn) ˜an(x) :=a(τxωn),and˜bn(x) :=b(τxωn) (4.10)\nfor (x,z)∈T2d. The path measure ˜Px\nnis a strong Markovian, Feller solution to the martingale\nproblem associated with the operator\n˜Lnu(x) =−˜bn(x)·∇u(x)+1\n2Tr/parenleftbig\n˜an(x)∇2u(x)/parenrightbig\n+/integraldisplay\nTd\nMn(u(x+z)−u(x)−∇u(x)·z1{|z|≤1})˜ nn(x,dz), u∈C2(Td\nMn).\n23Theorem 4.3. Assume that Hypotheses 2.2 and 3.3 are in force. Then, for each n∈Nthere exists\nan invariant density φnfor the process (˜Px\nn)x∈Td\nMn, i.e.φn≥0,ℓn-a.e. inTd\nMn,/ba∇dblφn/ba∇dblL1(Td\nMn)= 1and\n/integraldisplay\nTd\nMn/bracketleftBig\n˜Ex\nnf(˜Xt)/bracketrightBig\nφn(x)ℓn(dx) =/integraldisplay\nTd\nMnf(x)φn(x)ℓn(dx), t≥0, f∈Bb(Td\nMn).(4.11)\nHere,ℓnis the normalized Lebesgue measure on Td\nMn. Moreover, we have\n/ba∇dblφn/ba∇dblLp′(Td\nMn)≤C∗, n≥1, (4.12)\nwhereC∗is given by (3.17). If, in addition Hypothesis 2.3 holds the invariant density i s unique and\nstrictly positive.\nThe proof of Theorem 4.3 is presented in Section 4.3 below. It is based on a generalization of\nthe Alexandrov-Bakelman-Pucci estimates to solutions of i ntegro-differential equations, see Theorem\nA.4 below. Here we shall use the result to finishthe proof of Th eorem 3.6. Let ( νn)n∈Nbea sequence\nof Borel probability measures on Ω defined as follows\nνn(A) :=/integraldisplay\nTd\nMn1A(τxωn)φn(x)ℓn(dx), A∈G, (4.13)\nwithφnas in Theorem 4.3.\nLemma 4.4. Assume that the Hypotheses 2.2 and 3.3 of Theorem 3.6 are in for ce. Then, the\nsequence (νn)n∈Nis tight. Moreover, any weak limiting point ν∗is absolutely continuous with respect\ntoµ, the corresponding density Φ∗:=dν∗\ndµbelongs to Lp′(µ)and satisfies inequality (3.17).\nProof.We recall that µnis defined in (4.3). Obviously from the definition, µn(A) = 0 implies that\nνn(A) = 0 for any AinGand any n∈N. Therefore, νnis absolutely continuous w.r.t. µn. Let\nΦn:=dνn\ndµnbe the corresponding density. From (4.3) and (4.13), we conc lude that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩΦnF dµn/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF dνn/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTd\nMnF(τxωn)φn(x)ℓ(dx)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/parenleftBigg/integraldisplay\nTd\nMn|F(τxωn)|pℓ(dx)/parenrightBigg1\np/parenleftBigg/integraldisplay\nTd\nMn|φn(x)|p′ℓ(dx)/parenrightBigg1\np′\n≤C∗/ba∇dblF/ba∇dblLp(µn)\nfor anyF∈Bb(Ω). Above, C∗is the constant appearing in (4.12). Using (4.12), we obtain\n/ba∇dblΦn/ba∇dblLp′(µn)≤C∗, n∈N.\nAccording to Theorem 4.2, the sequence ( µn)n∈Nweakly converges to µ. It is therefore tight: for\nanyε >0, there exists a compact set K⊂Ω such that µn(Kc)< εfor anyn∈N. Hence, by the\nH¨ older inequality\nνn(Kc) =/integraldisplay\nΩeΦn1Kcdµn≤ /ba∇dblΦn/ba∇dblLp′(µn)µ1/p\nn(Kc)≤C∗ε1/p,\n24which proves tightness of the sequence ( νn)n∈N. Let us consider now a weak limiting point ν∗for a\nsubsequence of ( νn)n∈N, which, for simplicity, we denote by the same symbol. For FinCb(Ω), we\ncan then write\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF dν∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim\nn→+∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nΩF dνn/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C∗lim\nn→+∞/parenleftbigg/integraldisplay\nΩ|F|pdµn/parenrightbigg1\np\n=C∗/ba∇dblF/ba∇dblLp(µ).\nSince Ω is Polish, by the Ulam theorem ν∗is Radon, see e.g. [41, Theorem 9], and therefore Cb(Ω)\nis dense in Lp(µ), see e.g. [9, Proposition 7.9]. In consequence, the linear functional Cb(Ω)∋F/mapsto→/integraltext\nΩF dν∗extends uniquely to a bounded linear functional on Lp(µ). The conclusion of the lemma is\na consequence of the Riesz representation theorem for such f unctionals.\nIn order to conclude the proof of Theorem 3.6, we need the foll owing lemma, ensuring the Cb-\nFeller property for the semigroup generated by the environm ent process.\nLemma 4.5. The semigroup (Pt)t≥0given by (3.2)isCb-Feller, i.e.\nPt(Cb(Ω))⊆Cb(Ω), t > 0.\nProof.Let (ωn)n∈N⊂Ω andω∈Ω such that D( ωn,ω)→0, asn→+∞. It is an easy consequence\nof condition (F2) in Hypothesis 3.3 and [23, Theorem 19.25] t hatP0,ωnconverge to P0,ω, weakly over\nD. Tightness of the laws of XtunderP0;ωnthen implies that for any ε >0 there exists a compact\nsetK⊂Rdsuch that\nP0;ωn(Xt/\\e}atio\\slash∈K)< ε, n≥1 and P0,ω(Xt/\\e}atio\\slash∈K)< ε.\nFixedF∈Cb(Ω), it then follows that\n|PtF(ωn)−PtF(ω)| ≤/vextendsingle/vextendsingleE0;ωn[F(τXtωn), Xt∈K]−E0;ω[F(τXtω), Xt∈K]/vextendsingle/vextendsingle+2ε/ba∇dblF/ba∇dbl∞.\nBy condition (F3) in Hypothesis 3.3, the function F, when restricted to the compact set H(K) :=\n{τy(ω):y∈K, ω∈H}, whereH:={ω,ω1,ω2,...}, is uniformly continuous. Since D( ωn,ω)→0,\nwe have\nlim\nn→+∞/vextendsingle/vextendsingleE0;ωn[F(τXtωn), Xt∈K]−E0;ωn[F(τXtω), Xt∈K]/vextendsingle/vextendsingle= 0. (4.14)\nThe function y/mapsto→F(τyω) is bounded and continuous on Rd(see condition (F3) in Hypothesis 3.3).\nSinceP0;ωnconverge weakly over DtoP0;ω, it holds\nlim\nn→+∞/vextendsingle/vextendsingleE0;ωn[F(τXtω)]−E0;ω[F(τXtω)]/vextendsingle/vextendsingle= 0. (4.15)\nHence,\nlimsup\nn→+∞/vextendsingle/vextendsingleE0;ωn[F(τXtω), Xt∈K]−E0;ω[F(τXtω), Xt∈K]/vextendsingle/vextendsingle≤2ε/ba∇dblF/ba∇dbl∞. (4.16)\nSummarizing, we have shown\nlimsup\nn→+∞|PtF(ωn)−PtF(ω)| ≤4ε/ba∇dblF/ba∇dbl∞,\nfor anyε >0. Thus, |PtF(ωn)−PtF(ω)| →0, asn→+∞, and the conclusion of the lemma\nfollows.\n25We finish now the proof of Theorem 3.6 by showing that any weak l imiting point ν∗for the\nsequence {νn}n∈Nis invariant for {ηt}t≥0defined in (3.1). Let us fix FinCb(Ω). We have\n/integraldisplay\nΩPtF(ω)νn(dω) =/integraldisplay\nΩE0;ωF(ηt(ω))νn(dω) =/integraldisplay\nTd\nMnE0;τxωnF(τXtτxωn)φn(x)ℓn(dx), n∈N.\nBy virtue of (2.13), (4.9) we can then rewrite the utmost righ t-hand side as\n/integraldisplay\nTd\nMnEx;ωnF(τXtωn)φn(x)ℓn(dx) =/integraldisplay\nTd\nMn˜Ex\nnF/parenleftBig\nτ˜Xtωn/parenrightBig\nφn(x)ℓn(dx).\nUsing (4.11), we conclude that the right hand side equals\n/integraldisplay\nTd\nMnF(τxωn)φn(x)ℓ(dx) =/integraldisplay\nΩF(ω)νn(dω).\nBy virtue of Lemma 4.5, we have PtF∈Cb(Ω). By the weak convergence of ( νn)n≥1toν∗and the\nabove argument we get\n/integraldisplay\nΩPtF dν∗= lim\nn→+∞/integraldisplay\nΩPtF dνn= lim\nn→+∞/integraldisplay\nΩF dνn=/integraldisplay\nΩFdν∗,\nwhich proves the invariance of any weak limiting point ν∗.\nSuppose that Hypothesis 2.3 holds. We have already establis hed that ν∗- any weak limiting\npoint (νn)n≥1- is absolutely continuous w.r.t. µ. We claim that Φ ∗=dν∗\ndµ>0,µ-a.s. in Ω. Indeed,\nletA:={ω∈Ω: Φ∗(ω) = 0}. Suppose that µ(A)>0. It follows that µ(A)<1, as Φ ∗is a density\nw.r.t.µ. Moreover, see (2.10), we have\n/integraldisplay∞\n0e−tdt/parenleftbigg/integraldisplay\nΩ1AΦ∗dµ/parenrightbigg\n=/integraldisplay∞\n0e−tdt/parenleftbigg/integraldisplay\nΩPt1AΦ∗dµ/parenrightbigg\n=/integraldisplay\nΩΦ∗(ω)µ(dω)/integraldisplay∞\n0e−tPω\ntφA(·;ω)dt=/integraldisplay\nΩRω\n1φA(0;ω)Φ∗(ω)µ(dω),\nwhereφA(y;ω) =1A(τyω). This leads to a contradiction, as Rω\n1φA(0;ω)>0 for all ω∈Ω, due to\nHypothesis 2.3.\nThanks to (2.1), it follows that for any y∈Rd, Φ∗(τ−yω) = 0, µ-a.s. inA. Therefore,\nµ(τy(A)∆A) = 0 for any yinRd, where ∆ denotes the symmetric difference of sets. This in turn\nimplies that µ(A)∈ {0,1}, due to ergodicity of µ, which contradicts the fact that 0 < µ(A)<1.\nHence, we have shown that Φ ∗>0,µ-a.s. The above argument in fact proves that a density of any\ninvariant measure absolutely continuous with respect to µhas to be strictly positive µa.s. Ergodic-\nity ofν∗is then an easy consequence of this fact. Indeed, if there exi stsAsuch that ν∗(A)∈(0,1)\nand1A(ηt(ω)) =1A(ω),ν∗-a.s. in Ω, then both measures ν∗(A)−11AΦ∗dν∗andν∗(Ac)−11AcΦ∗dν∗\nwould have been invariant and of disjoint supports, which le ads to a contradiction.\nAs a consequence of the above argument and Lemma 4.4, we concl ude the following.\nCorollary 4.6. Assume that Hypotheses 2.2, 3.3 and 2.3 are in force. The seque nce(νn)n∈Ndefined\nby(4.13)converges weakly. Its limit is an invariant measure for the en vironment process (ηt)t≥0that\nis absolutely continuous with respect to µ. The corresponding invariant density is strictly positive\nand belongs to Lp′(µ)for anyp > d/2+1.\n264.3 Proof of Theorem 4.3\nLet (ωn)n∈N, (Mn)n∈Nand (˜Px\nn) be the sequences appearing in Theorem 4.2. We let\nˆPx\nn:= (TM−1\nn)♯˜PxMnn\nwhereTεhas been defined in (2.15). We see that process ( ˆPx\nn) onTdis the scaled process ( ˜Px\nn) on\nTd\nMn. Notice that\nˆPx\nn=π♯Px;ωn\nM−1\nn,\nwhere the family ( Px;ω\nε) has been introduce in (2.15) and π:Rd→Tdis the canonical projection of\nRdontoTd. Letqn(·,·) be the Fourier symbol of the process ( Px;ωn\nM−1\nn) (see (2.15)). By Hypothesis\n2.2, we have\nsup\nx∈Rdsup\n|ξ|≤1|qn(x,ξ)|= sup\nx∈Rdsup\n|ξ|≤1Mα\nn|q(x,ξ/M n;ωn)| ≤CQ, n≥1,\nand\nϕn(t) :=/integraldisplay\nRdexp/parenleftbigg\n−t\n16inf\nx∈RdReqn(x,ξ)/parenrightbigg\ndξ=/integraldisplay\nRdexp/parenleftbigg\n−tMα\nn\n16inf\nx∈RdReq(x,ξ/M n;ωn)/parenrightbigg\ndξ\n≤/integraldisplay\nRdexp/parenleftbigg\n−tcQ|ξ|α\n16/parenrightbigg\ndξ.\nAn elementary calculation shows that/integraltext\nRde−a|x|bdx=cdΓ(d/b)\nbad/bfor anya,b >0. Herecd>0 is some\nconstant depending only on the dimension d. Therefore,\nϕn(t)≤cdG(α,cQ)\ntd/α, t >0, (4.17)\nwhereG(α,cQ) is defined in (3.18). Let ˆP(n)\ntf(x) :=ˆEx\nnf/parenleftbigˆXt/parenrightbig\n,t≥0,f∈Bb(Td) be the transition\nprobability semigroup corresponding to/parenleftbigˆPx\nn/parenrightbig\nx∈Td(hereˆX=π(X)). It is Cb-Feller, thanks to\ncondition (Q2) in Hypothesis 2.2 and [6, Theorem 1.9]. Let\nˆRn\n1f(x) =/integraldisplay+∞\n0e−tˆP(n)\ntf(x)dt, x∈Td, f∈Bb(Td) (4.18)\nbe the respective 1-resolvent operator. Since the transiti on probabilities of ˆPx\nnadmit densities, we\nhaveˆRn\n1:L∞(Td)→L∞(Td). Theorem A.4 now implies that\n/ba∇dblˆRn\n1f/ba∇dblL∞(Td)≤C∗/ba∇dblf/ba∇dblLp(Td), f∈Lp(Td), (4.19)\nwith the same p >1+d/αas in Theorem 3.6 and C∗given by (3.17).\nSinceTdis compact and ( ˆPx\nn)x∈TdisCb-Feller, the Krylov-Bogoliubov theorem (cf. [7, Theorem\n3.1.1, p.20]) ensures the existence of an invariant probabi lity measure ˆ µnfor the process. Due to the\nfact that the transition probability functions of ˆPx\nnhave densities, ˆ µnis absolutely continuous with\nrespect to the Lebesgue measure on Tdanddµn=ˆφndℓ, where/ba∇dblˆφn/ba∇dblL1(Td)= 1. If Hypothesis 2.3\nis satisfied then ˆRn\n1f(x)>0,x∈Td, for any non-negative fthat is non-trivial. As a consequence\nˆφn>0 a.e. and an invariant density has to be unique. We can write\n/integraldisplay\nTd/bracketleftBig\nˆEx\nnf(ˆXt)/bracketrightBig\nˆφn(x)ℓ(dx) =/integraldisplay\nTdf(x)ˆφn(x)ℓ(dx), t≥0, f∈Bb(Td).(4.20)\n27We also have ˆRn,⋆\n1ˆφn=ˆφn, where ˆRn,⋆\n1:L1(Td)→L1(Td) is dual to ˆRn\n1. Thanks to (4.19), we\nconclude that /ba∇dblˆφn/ba∇dblLp′(Td)≤C∗, for any n∈N. Letφn(x) :=ˆφn(x/Mn),x∈Td\nMn. We have\n/ba∇dblˆφn/ba∇dblLp′(Td)=/ba∇dblφn/ba∇dblLp′(Td\nMn), therefore (4.12) is satisfied. We claim that φn(x) is an invariant density\nfor the process ( ˜Px\nn)x∈Td\nMn. Indeed, let f∈C(Td\nMn). By (4.20), with the notation jMn(x) :=x/Mn,\nwe have\n/integraldisplay\nTd\nMnf(x)φn(x)ℓn(dx) =/integraldisplay\nTdf◦j−1\nMn(x)ˆφn(x)ℓ(dx)\n=/integraldisplay\nTdˆEx\nxf◦j−1\nMn(ˆXtM−α\nn)ˆφn(x)ℓ(dx) =/integraldisplay\nTd\nMn˜Ex\nnf(˜Xt)φn(x)ℓn(dx),\nand thus, we have concluded the proof of Theorem 4.3.\n4.4 Proof of Theorem 3.11\nWe shall only provide a detailed proof of (i) and then we sketc h the proofs of (ii) and (iii) since they\nare analogous to (i).\nFix anyρ∈(0,1). We define C(ρ)\nW: Ω→Wgiven by\n/parenleftbig\nC(ρ)\nW(ω)/parenrightbig\n(x) := (b(τxω),ρId+a(τxω),n(τx)), x∈Rd.\nConsider Polish space ( KW,DW) of Example 3.7. Observe that Hypothesis 3.3 holds with CW\nreplaced by C(ρ)\nW. Furthermore, the Fourier symbol of the process ( Px;ω\nρ) - the Feller processes\nassociated with C(ρ)\nW(ω) - admits the form q(ρ)(ξ;τxω) =1\n2ρ|ξ|2+q(ξ;τxω), where qis the Fourier\nsymbol of the L´ evy triplet ( b(τ·ω),a(τ·ω),n(τ·ω)) (this symbol is given by (2.4)), which as a result\nimpliesthat Hypothesis2.2 holdsfor( Px;ω\nρ) withthesameconstant in(Q2) andconstant CQreplaced\nby 2CQin (Q1). Consequently, by Theorem 3.6 there exists an invari ant, ergodic measure µρ\n∗on Ω\nfor the semigroup ( Pρ\nt) anddµρ\n∗= Φρ\n∗dµ, where\nPρ\ntF(ω) =E0;ω\nρF(τXtω), F∈Bb(Ω).\nHereE0;ω\nρis the expectation of the process ( Px;ω\nρ). By (iii) of Theorem 3.6 we have an estimate\n/ba∇dblΦρ\n∗/ba∇dblLp′(Ω;µ)≤CandCisindependentof ρ∈(0,1). Thus, uptoasubsequence,Φρ\n∗→Φ∗, asρ→0+,\nweakly in Lp′(Ω;µ) for some Φ ∗∈Lp′(Ω;µ). Letdµ∗:= Φ∗dµ. Observe that q(ρ)(D;τ·ω)η→\nq(D;τ·ω)ηfor anyη∈Candω∈Ω. By [23, Theorem 19.25] we have P0;ω\nρ⇒P0;ω, asρ↓0, weakly\nin the sense of convergence of measures on D. Consequently, by continuity of the group action/parenleftbig\nτx/parenrightbig\nx∈Rd(see (2.2)), we get\nPρ\ntF(ω) =E0;ω\nρF(τXtω)→E0;ωF(τXtω) =PtF(ω),asρ→0+\nfor anyF∈Cb(Ω) and ω∈Ω. As a result, for any F∈Cb(Ω),\n/integraldisplay\nΩPtF dµ∗=/integraldisplay\nΩPtFΦ∗dµ= lim\nρ↓0/integraldisplay\nΩPρ\ntFΦρ\n∗dµ= lim\nρ↓0/integraldisplay\nΩFΦρ\n∗dµ=/integraldisplay\nΩFΦ∗dµ=/integraldisplay\nΩF dµ∗.\n28The proof that µ∗is ergodic follows from Hypothesis 2.3 and can be conducted i n the same way as\nin Section 4.2.\nIn the proof of (ii) one has to consider metric space ( KW,DW) of Example 3.8 and mapping\nC(ρ)\nW(ω) := (jρ∗b(τ·ω),ρId+jρ∗a(τ·ω),jρ∗n(τ·)),\nwherejρis a standard smooth mollifier (notice that q(ρ)(ξ;τ·ω) =jρ∗q(ξ;τ·ω)).\nIn the proof of (iii) in turn one has to consider the metric spa ce (KQ,DQ) of Example 3.10 and\nthe mapping C(ρ)\nQgiven by\nC(ρ)\nQ(ω)(x,ξ) :=/parenleftbig\njρ∗q(ξ,τ·ω)/parenrightbig\n(x).\nA Aleksandrov-Bakelman-Pucci-type estimates for Feller p rocesses\nA.1 Some preliminaries\nLet (Tt) be a Feller semigroup on C0(Rd) that generates the operator ( L,D(L)), with C∞\nc(Rd)⊆\nD(L). By the Courr´ ege-von Waldenfels theorem, it can be repres ented for any u∈C2\n0(Rd) as\nLu(x) =b(x)·∇u(x)+1\n2Tr/parenleftbig\na(x)∇2u(x)/parenrightbig\n+/integraldisplay\nRd/parenleftbig\nu(x+z)−u(x)−1{|z|≤1}z·∇u(x)/parenrightbig\nν(x,dz)−c(x)u(x), x∈Rd.\nHere,c:Rd→[0,+∞),b:Rd→Rd,a:Rd→Rd⊗Rdare measurable functions, a(x) is a non-\nnegative definite d×dmatrix valued function and ν(x,dz) is a non-negative, σ-finite kernel on\nRd×B(Rd) such that for every x∈Rd,ν(x,·) is a L´ evy measure on Rd. The Fourier symbol\nq:Rd×Rd→Cassociated with the operator Lis given by\nq(x,ξ) :=c(x)−ib(x)·ξ+1\n2a(x)ξ·ξ+/integraldisplay\nRd/bracketleftBig\n1−e−iz·ξ−iξ·z1{|z|≤1}(z)/bracketrightBig\nν(x,dz),(x,ξ)∈Rd×Rd.\nWe will always assume in this section that\ncq:= sup\n|ξ|≤1sup\nx∈Rd|q(x,ξ)|<+∞andq(x,0) = 0, x∈Rd. (A.1)\nThis obviously implies c(x)≡0. According to [43, Lemma 6.2], condition (A.1) implies tha t\nsup\nx∈Rd|q(x,ξ)| ≤2cq(1+|ξ|2), ξ∈Rd.\nFurthermore, we suppose that there exists p >1 such that\nC(p,q) := [ϕ(1)]1/p+/braceleftbigg/integraldisplay1\n0[ϕ(t)]1/(p−1)dt/bracerightbigg1−1/p\n<+∞, (A.2)\n29whereϕ: [0,+∞)→(0,+∞) is defined by\nϕ(t) := (4π)−d/integraldisplay\nRdexp/parenleftbigg\n−t\n16inf\nx∈RdReq(x,ξ)/parenrightbigg\ndξ. (A.3)\nNote that\nReq(x,ξ) =1\n2a(x)ξ·ξ+2/integraldisplay\nRdsin2/parenleftbiggz·ξ\n2/parenrightbigg\nν(x,dz)≥0,(x,ξ)∈Rd×Rd.\nLet (Px)x∈Rddenote the Feller process corresponding to the semigroup ( Tt)t≥0, i.e. for any f∈\nC0(Rd),\nTtf(x) =Exf(Xt), t≥0, x∈Rd.\nA.2Lp-estimates of the resolvent\nLetD⊂Rdbe an open set. Define the exit time τD:D→[0,+∞] of the canonical process ( Xt)t≥0\nfromDas\nτD:= inf{t >0:Xt/\\e}atio\\slash∈D}.\nIt is a stopping time, i.e. {τD≤t} ∈Ftfor anyt≥0. We then define for any f∈Bb(D), the\ntransition semigroup on Dwith the null exterior condition as\nTD\ntf(x) :=Ex[f(Xt), t < τ D]t≥0, x∈D.\nFurthermore, for any β >0, we introduce the β-resolvent of LonDfor anyf∈Bb(D) by letting\nRD\nβf(x) :=/integraldisplay∞\n0e−βtTD\ntf(x)dt=Ex/bracketleftbigg/integraldisplayτD\n0e−βtf(Xt)dt/bracketrightbigg\n, x∈D.\nIf we suppose that the exit time from Dis a.s. finite, i.e.\nPx(τD<+∞) = 1, x∈D,\nthen, we can define the resolvent also for β= 0 as\nRDf(x) :=/integraldisplay∞\n0TD\ntf(x)dt=Ex/bracketleftbigg/integraldisplayτD\n0f(Xt)dt/bracketrightbigg\n, x∈D. (A.4)\nNow we are ready to formulate the main result of the present se ction.\nTheorem A.1. Under the hypotheses made in Section A.1, the following asse rtions are true.\n(i) The1-resolvent RD\n1extends to an operator from Lp(D)toL∞(D). In addition,\n/ba∇dblRD\n1f/ba∇dblL∞(D)≤C(p,q)/ba∇dblf/ba∇dblLp(D), f∈Lp(D), (A.5)\nwhereC(p,q)is given by (A.2).\n30(ii) If\ntD:= sup\nx∈RdExτD<+∞,\nthen, the 0-resolvent RDextends to an operator from Lp(D)toL∞(D). Furthermore,\n/ba∇dblRDf/ba∇dblL∞(D)≤˜C(p,q,D)/ba∇dblf/ba∇dblLp(D), f∈Lp(D), (A.6)\nwhere\n˜C(p,q,D) =/braceleftbigg\n[2p′]!/parenleftbigg2p′\n[2p′]/parenrightbigg/bracerightbigg1/p′\nϕ1/p(1)t[2p′]/p′\nD(1+tD)(2p′−[2p′])/p′+/braceleftbigg/integraldisplay1\n0[ϕ(t)]1/(p−1)dt/bracerightbigg1/p′\n.\nRemark A.2. Formally, we can treat u=RD\nβfas a solution of the Poisson equation with the null\nexternal boundary condition \n\nβu(x)−Lu(x) =f(x)onD;\nu= 0 onDc.\nThe result formulated in Theorem A.1 can be treated as a L´ evy-t ype generalisation of Alexandrov-\nBakelman-Pucci estimates for elliptic diffusive operators [16, Theorem 9.1, p. 220].\nProof of Theorem A.1. To show (A.5) we need to prove that\nsup\nx∈D/vextendsingle/vextendsingle/vextendsingle/vextendsingleEx/bracketleftbigg/integraldisplayτD\n0e−tf(Xt)dt/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(p,q)/ba∇dblf/ba∇dblLp(D). (A.7)\nIt follows from [44, Theorem 1.2] that under our assumptions , (Px)x∈Rdadmits a transition density\npt(x,·) onRd. The the process killed upon leaving Dhas a transition density given by the Dynkin-\nHunt formula, see [12, equation (4.1.6), p. 154], or [30, equ ation (5.11), p. 130], namely\npD(t,x,y) :=p(t,x,y)−Ex[p(t−τD,XτD,y);t≥τD],\nThe expression under the absolute value in (A.7) equals\n/integraldisplay+∞\n0e−tEx[f(Xt), t < τ D]dt=I1+I2, (A.8)\nwhereI1andI2correspond to integrations from 0 to 1 and from 1 to ∞. By H¨ older inequality, we\nthen get\n|I1|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0/integraldisplay\nDe−tpD(t,x,y)f(y)dydt/vextendsingle/vextendsingle/vextendsingle/vextendsingle(A.9)\n≤/parenleftbigg/integraldisplay1\n0/integraldisplay\nD|f(y)|pdydt/parenrightbigg1/p/parenleftbigg/integraldisplay1\n0/integraldisplay\nD|pD(t,x,y)|p′dydt/parenrightbigg1/p′\n≤ /ba∇dblf/ba∇dblLp(D)/parenleftBigg/integraldisplay1\n0/bracketleftBigg\nsup\ny∈D/parenleftBig\npp′−1\nD(t,x,y)/parenrightBig/integraldisplay\nDpD(t,x,y)dy/bracketrightBigg\ndt/parenrightBigg1/p′\n≤ /ba∇dblf/ba∇dblLp(D)/parenleftBig/integraldisplay1\n0sup\ny∈D/parenleftbig\npp′−1\nD(t,x,y)/parenrightbig\ndt/parenrightBig1/p′\n.\n31The second contribution can be estimated in the following wa y:\n|I2| ≤/integraldisplay∞\n1e−t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDpD(t,x,y)f(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingledt≤/integraldisplay∞\n1e−t/parenleftbigg/integraldisplay\nDpD(t,x,y)|f(y)|pdy/parenrightbigg1/p\ndt\n≤sup\nt≥1,y∈Dp1/p\nD(t,x,y)/ba∇dblf/ba∇dblLp(D).(A.10)\nBy [44, Theorem 1.2], we finally have\nsup\nx,y∈RdpD(t,x,y)≤sup\nx,y∈Rdp(t,x,y)≤(4π)−d/integraldisplay\nRdexp/parenleftBig\n−t\n16inf\nz∈RdRep(z,ξ)/parenrightBig\ndξ=ϕ(t).\nThis combined with (A.9) and (A.10) gives (i).\nFor (ii), we start with the following fact.\nLemma A.3. For any γ≥1, it holds\nsup\nx∈RdExτγ\nD≤γtDsup\nx∈RdExτγ−1\nD. (A.11)\nProof.Suppose that γ >1. Using the Markov property and then the Fubini theorem we ca n write\nEx/bracketleftbigg/integraldisplayτD\n0sγ−2EXsτDds/bracketrightbigg\n=/integraldisplay+∞\n0Ex/bracketleftbigg\nsγ−21{s<τD}/integraldisplays+τD◦θs\nsdρ/bracketrightbigg\nds\n=Ex/bracketleftbigg/integraldisplayτD\n0sγ−2ds/integraldisplayτD\nsdρ/bracketrightbigg\n=Ex/bracketleftbigg/integraldisplayτD\n0dρ/integraldisplayρ\n0sγ−2ds/bracketrightbigg\n=Exτγ\nD\nγ(γ−1).\nEstimate (A.11) follows easily from the above identity.\nWe can split the integral on the right-hand side of (A.4) as in (A.8) (without e−ton the left-\nhand side). Observe that (A.9) also holds true without the fa ctore−t. We only need to modify the\nestimate of |I2|. Note that\n|I2| ≤/integraldisplay∞\n1/parenleftbig\nEx(f(Xt)1{t<τD})p/parenrightbig1/p/parenleftbig\nEx1{t<τD}/parenrightbig1/p′\ndt\n≤/integraldisplay∞\n1/parenleftbigg/integraldisplay\nDpD(t,x,y)|f(y)|pdy/parenrightbigg1/p/parenleftbig\nEx1{t<τD}/parenrightbig1/p′\ndt\n≤sup\nt≥1,x,y∈Dp1/p\nD(t,x,y)/ba∇dblf/ba∇dblLp(D)sup\nx∈Rd/integraldisplay∞\n1/parenleftBigExτ2p′\nD\nt2p′/parenrightBig1/p′\ndt\n≤sup\nt≥1,x,y∈Dp1/p\nD(t,x,y)/ba∇dblf/ba∇dblLp(D)sup\nx∈Rd/parenleftbig\nExτ2p′\nD/parenrightbig1/p′\n.\nEstimate (A.6) follows from an application of (A.11).\nA.3 Periodic case\nLetπ:Rd→Tdbe the canonical projection of RdontoTd. LetCper(Rd) be the space of all\ncontinuous functions which are 1-periodic in each variable . There is a one-to-one correspondence\n32between Cper(Rd) andC(Td), i.e. for every ˜f∈C(Td), there exists a unique f∈Cper(Rd) such that\nf(x) =˜f◦π(x),x∈Rd.\nLet us suppose that the transition probability semigroup ( Tt)t≥0given in Section A.1 has the\nfollowing property\nTt/parenleftBig\nCper(Rd)/parenrightBig\n⊆Cper(Rd), t > 0.\nThe semigroup ( Tt)t≥0induces a strongly continuous semigroup ( ˜Tt)t≥0onC(Td), by virtue of [42,\nLemma 1.18]. The respective process ˜Px:= (π)♯Px,x∈Td, is strongly Markovian and Feller on Td,\nwith transition probability densities given by\n˜pt(x,y) =/summationdisplay\nm∈Zdpt(x,y+m), t >0, x,y∈Td.\nHere,pt(x,y) are the transition probability densities corresponding t o the path measure Px. In\naddition, for any ˜f∈C(Td), we have\n˜Ex˜f(˜Xt) =Exf(Xt), t≥0, x∈Td,\nwheref∈Cper(Rd) is the 1-periodic extension of ˜f,˜Xt=π(Xt) and˜Exis the expectation corre-\nsponding to ˜Px. The 1-resolvent corresponding to ( ˜Tt)t≥0is then given for any ˜f∈Bb(Td) by\n˜R1˜f(x) :=/integraldisplay+∞\n0e−t˜Tt˜f(x)dt, x ∈Td.\nWe finally denote\nq∗:= sup\nx∈Tdsup\ny∈B(x,1)sup\n|ξ|≤1|q(y,ξ)|.\nTheorem A.4. Suppose that the hypotheses made in Section A.1 are in force. Letp >1be as in\n(A.2). Then, the 1-resolvent ˜R1extends to an operator from Lp(Td)toL∞(Td). In addition,\n/ba∇dbl˜R1˜f/ba∇dblL∞(Td)≤¯C(p,q∗)/ba∇dbl˜f/ba∇dblLp(Td),˜f∈Lp(Td),\nfor the constant\n¯C(p,q∗) :=2·3dC(p,q)\n1−exp{−(C(d)q∗)−1},\nwhereC(d)>0depends only on the dimension dandC(p,q)is as in Theorem A.1.\nProof.Recall that B(x,r) denotes a ball centered at xwith radius r >0. Let D:={x∈\nRd: dist(x,Q1)<1}. Suppose also that f∈Cper(Rd) is such that f=˜f◦π. Using the strong\nMarkov property, we can then write for any x∈Td:\n˜R1˜f(x) =Ex/bracketleftbigg/integraldisplayτD\n0e−tf(Xt)dt/bracketrightbigg\n+Ex/bracketleftbigg\ne−τD/integraldisplay∞\n0e−tEXτD[f(Xt)]dt/bracketrightbigg\n.\nBy Theorem A.1 we then have\nsup\nx∈Td|˜R1˜f(x)| ≤sup\nx∈TdEx/bracketleftbigg/integraldisplayτD\n0e−s|f(X(s))|ds/bracketrightbigg\n+ sup\nx∈TdEx/bracketleftbig\ne−τD/bracketrightbig\nsup\nx∈Td/vextendsingle/vextendsingle/vextendsingle/vextendsingleEx/integraldisplay∞\n0e−sf(X(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C(p,q)/ba∇dblf/ba∇dblLp(D)+ sup\nx∈TdEx/bracketleftbig\ne−τD/bracketrightbig\nsup\nx∈Td|˜R1˜f(x)|.\n33SinceD⊆Q3, we can now use periodicity of fto conclude that\nγsup\nx∈Td|˜R1˜f(x)| ≤3dC(p,q)/ba∇dbl˜f/ba∇dblLp(Td), (A.12)\nwhere\nγ:= 1−sup\nx∈TdEx/bracketleftbig\ne−τD/bracketrightbig\n. (A.13)\nSinceB(x,1)⊆Dfor anyx∈Td, by [44, Proposition 4.3] we infer that for each t∈(0,1]\n1−Ex/bracketleftbig\ne−τD/bracketrightbig\n≥1−Ex/bracketleftbig\ne−τB(x,1)/bracketrightbig\n≥(1−e−t)Px(τB(x,1)> t)\n≥(1−e−t)(1−C(d)tq∗).\nHere a positive constant C(d) depends only on the dimension d. Finally, if one chooses t:=\n(2C(d)q∗)−1, then\nγ >1\n2/parenleftBig\n1−exp/braceleftbig\n−(2C(d)q∗)−1/bracerightbig/parenrightBig\nand the conclusion of the theorem follows.\nB Proof of Proposition 2.4\nThe fact that C1) implies the assertion of the proposition fo llows from [10, Theorem 4.2].\nBelow we show that also C2) suffices for the irreducibility of R1. Thanks to assumption i), by [44,\nTheorem1.2], ( Px)isstronglyFeller, thusitstransitionsemigroup( Pt)satisfies Pt(Bb(Rd))⊂Cb(Rd),\nwhich in turn implies that Rβ(Bb(Rd))⊂Cb(Rd) for any β >0. We let R=R0.\nWe show first that\nR11B(y,r)(x)>0,for anyx,y∈Rd, r >0. (B.1)\nThanks to the right continuity of paths of the process condit ion (B.1) follows, provided we can prove\nthat\nPx(σB(y,r)<∞)>0, x∈Rd, (B.2)\nwhereσB(y,r):= inf{t >0 :Xt∈B(y,r)}. Ifx∈B(y,r) (B.2) trivially holds. Suppose that\nx /∈B(y,r). Let 0< δ < r/ 4. By using the L´ evy system, see [20], we have\nEx/summationdisplay\n0<s≤t1B(x,δ)(Xs−)1B(y,r/2)(Xs) =Ex/integraldisplayt\n01B(x,δ)(Xs)n(Xs,B(y−Xs,r/2))ds. (B.3)\nBy assumption ii), we have n/parenleftbig\nXs,B(y−Xs,r/2)/parenrightbig\n>0 for all s∈[0,t]. Thanks to (B.1) with\nsome positive Px-probability the set of those times for which 1B(x,δ)(Xs)>0 is of positive Lebesgue\nmeasure. Therefore the right hand side of (B.3) is positive. In consequence,\nPx/parenleftbig\nσB(y,r)<∞/parenrightbig\n≥Px(∃0<s≤tXs∈B(y,r/2))≥Px\n/summationdisplay\n0<s≤t1B(x,δ)(Xs−)1B(y,r/2)(Xs)>0\n>0\n34and (B.2) follows, which in turn implies (B.1). As a corollar y we conclude that\nR1f(x)>0,for allx∈Rd, (B.4)\nfor anyf∈Cb(Rd) that is non-negative and not trivially equal to 0.\nNow we proceed with proving that\nη(x) :=R11A(x)>0, x∈Rd, (B.5)\nwhenAis an arbitrary Borel set such that Ld(A)∈(0,+∞). By regularity of the Lebesgue measure,\nwe may assume without loss of generality that Ais compact. Note that in order to prove (B.5) it\nsuffices to show that η=R11Ais not trivially equal to 0. This is due to the fact that by the\nstrong Feller condition ηis continuous. Using this fact, (B.4) and the resolvent iden tity, see e.g. [14,\nformula (8.10), p. 41] yields that ˜ η=R1Asatisfies\n0< R1η(x)≤Rη(x)≤˜η(x),for allx∈Rd. (B.6)\nClearly ˜η(x)>0 if and only if η(x)>0. To see that ηis non-trivial it suffices to prove that\nlim\nβ→+∞/integraldisplay\nRd/parenleftbig\nβRβ1A(x)−1A(x)/parenrightbig\nϕ(x)dx= 0 (B.7)\nfor anyR >0 andϕ∈Cc(Rd) (notice that η(x)>0⇔ ∀β>0Rβ1A(x)>0). Choose an arbitrary\nε >0, and let K:=supp[ϕ]. By part iii) of condition C2) we can find δ >0 such that\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRd/parenleftbig\nPt1A(x)−1A(x)/parenrightbig\nϕ(x)dx/vextendsingle/vextendsingle/vextendsingle< ε, t∈[0,δ]. (B.8)\nWe can write\n/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRd/parenleftbig\nβRβ1A(x)−1A(x)/parenrightbig\nϕ(x)dx/vextendsingle/vextendsingle/vextendsingle≤β/vextendsingle/vextendsingle/vextendsingle/integraldisplayδ\n0e−βt/integraldisplay\nK/parenleftbig\nPt1A(x)−1A(x)/parenrightbig\nϕ(x)dx/vextendsingle/vextendsingle/vextendsingle\n+β/ba∇dblϕ/ba∇dbl∞/integraldisplay+∞\nδe−βt/integraldisplay\nK/vextendsingle/vextendsinglePt1A(x)−1A(x)/vextendsingle/vextendsingledx.\nThe second integral on the right hand side can be estimated by 2e−βδ/ba∇dblϕ/ba∇dbl∞|K| →0, asβ→+∞.\nThe first integral can be estimated, using (B.8), by ε/ba∇dblϕ/ba∇dbl∞. Thus (B.7) follows.\nC Proof of Propositions 2.10, 2.11 and 2.13\nWe start with the following result that is a direct consequen ce of (2.28). Its proof, obtained by\napproximating a continuous function by a linear combinatio n of indicator functions of intervals, is\nleft to a reader.\nLemma C.1. Letg:Rd→Rbe a continuous function which is compactly supported in Rd\\{0}and\nδ >0. Then,\nlim\nε↓0µ/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplay\nRdg(εz)n(dz;ω)−/integraldisplay\nRdg(z)s(z;ω) ¯ν(dz)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥δ/parenrightbigg\n= 0. (C.1)\n35C.1 Proof of Proposition 2.10\nLet us fix arbitrary ρ >0 andδ >0. From (2.27), we know that there exists 0 < κ <1 such that\nµ/parenleftBigg\nε−α/integraldisplay\n{|εz|≤κ}/parenleftbig\n|εz|2/parenrightbig\nn(dz;ω)≥δ\n2/parenrightBigg\n≤ρ\n6(C.2)\nfor anyε∈(0,1). Let us introduce now an increasing sequence fn: [0,+∞)→Rof continuous\nfunctionswhicharecompactlysupportedin Rd/integerdivide{0}andsuchthat fn(r)→1{r≤κ}(r)r2, asn→+∞.\nThen, using Lemma C.1 and (C.2), we obtain\nµ/parenleftBigg/integraldisplay\n{|z|≤κ}fn(|z|)s(z;ω)¯ν(dz)≥δ/parenrightBigg\n≤ρ\n3.\nLettingn→+∞, Fatou lemma implies that\nµ/parenleftBigg/integraldisplay\n{|z|≤κ}|z|2s(z;ω)¯ν(dz)≥δ/parenrightBigg\n≤ρ\n3. (C.3)\nExploiting a similar argument, Lemma C.1 and (2.26) show the existence of a constant M >1 such\nthat\nµ/parenleftBigg/integraldisplay\n{|z|≥M}|z|s(z;ω)¯ν(dz)≥δ/parenrightBigg\n≤ρ\n3. (C.4)\nThanks to (2.29), we conclude also that for any ρ >0, there exists δ >0 such that\nµ/parenleftBigg/integraldisplay\n{κ≤|z|≤M}χ(|z|)s(z;ω)¯ν(dz)≥δ/parenrightBigg\n≤ρ\n3. (C.5)\nFrom (C.3), (C.4) and (C.5) we get that for any ρ >0 there exists δ >0 such that\nµ/parenleftbigg/integraldisplay\nRdχ(|z|)s(z;ω)¯ν(dz)≥δ/parenrightbigg\n≤ρ, (C.6)\nwhere the function χwas defined in (2.30). Hence, the assertion of Proposition 2. 10 follows.\nC.2 Proof of Propositions 2.11 and 2.13\nThe conclusions of Propositions 2.11 and 2.13 are consequen ces of the following extension of Lemma\nC.1.\nLemma C.2. Letg:Rd→Rbe a continuous function such that there exist C >0for which\n|g(z)| ≤Cχ(|z|), z∈Rd, (C.7)\nwhere the function χwas defined in (2.30). Then, for any δ >0andgas above, formula (C.1)\nholds.\n36Proof.Let us firstly suppose that gis compactly supported in Rdand\n|g(z)| ≤C|z|2,|z| ≤1. (C.8)\nWe fix arbitrary ρ,δ >0. By virtue of (2.27) and (2.31), one can find 0 < κ <1 such that\nµ/parenleftBigg\nε−α/integraldisplay\n{|εz|≤κ}|εz|2n(dz;ω)≥δ\n3C/parenrightBigg\n+µ/parenleftBigg/integraldisplay\n{||z|≤κ}|z|2s(z;ω)¯ν(dz)≥δ\n2C/parenrightBigg\n<ρ\n2.(C.9)\nWe can then write g=g1+g2, whereg1is continuous, supported in B(0,κ/2) and satisfies |g1(z)| ≤\nC0|z|2,z∈Rdwhileg2is compactly supported in Rd\\{0}. Using (C.9), we get\nµ/parenleftbigg\nε−α/integraldisplay\nRd/vextendsingle/vextendsingleg1(εz)/vextendsingle/vextendsinglen(dz;ω)≥δ\n2/parenrightbigg\n+µ/parenleftbigg/integraldisplay\nRd|g1(z)|s(z;ω)¯ν(dz)≥δ\n2/parenrightbigg\n<ρ\n2\nand, by Lemma C.1,\nlim\nε↓0µ/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplay\nRsg2(εz)n(dz;ω)−/integraldisplay\nRdg2(z)s(z;ω)¯ν(dz)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥δ\n2/parenrightbigg\n= 0.\nHence,\nlimsup\nε↓0µ/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleε−α/integraldisplay\nRsg(εz)n(dz;ω)−/integraldisplay\nRdg(z)s(z;ω)¯ν(dz)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥δ/parenrightbigg\n≤ρ\n2\nand, since δ >0, has been arbitrary, we conclude (C.1).\nIn the same way one can show that the conclusion of the lemma is valid for any function gsuch\nthat\n0/\\e}atio\\slash∈suppgand satisfying |g(z)| ≤C|z|,|z| ≥1. (C.10)\nSinceanarbitraryfunctionsatisfying (C.7)canbereprese ntedasasum g1+g2, whereg1iscompactly\nsupported and satisfies (C.8), and g2satisfies (C.10), the conclusion of the lemma follows.\nReferences\n[1] D. Applebaum, Levy Processes and stochastic calculus , Cambridge Univ. Press (2004).\n[2] Armstrong, S.; Kuusi, T.; Mourrat, J.-Ch. Quantitative stochastic homogenization and large-\nscale regularity Grundlehren Math. Wiss., 352[Fundamental Principles of Ma thematical Sci-\nences] Springer, Cham, 2019,\n[3] Bensoussan, A.; Lions, J.-L., Papanicolaou, G. Asymptotic analysis for periodic structures.\nStudies in Mathematics and its Applications, 5. North-Holl and Publishing Co., Amsterdam-\nNew York, 1978.\n[4] Billingsley, P. (1999). Convergence of probability mea sures. Second edition. Wiley Series in\nProbability and Statistics: Probability and Statistics. W iley-Interscience Publication. John Wi-\nley & Sons, Inc., New York.\n37[5] Biswas, A.; Lorinczi, J.: Maximum principles and Aleksa ndrov-Bakelman-Pucci type estimates\nfor nonlocal Schr¨ odinger equations with exterior conditi ons. SIAM J. Math. Anal. 51 (2019),\nno. 3, 1543-1581.\n[6] B¨ ottcher, B., Schilling, R., Wang, J.: L´ evy Matters II I. L´ evy-Type Processes: Construction,\nApproximation and Sample Path Properties. Lecture Notes in Math. 2099Springer, Cham\n(2013).\n[7] Da Prato, G.; Zabczyk, J. Ergodicity for infinite-dimensional systems. London Mathematical\nSociety Lecture Note Series, 229. Cambridge University Pre ss, Cambridge, 1996.\n[8] Dunford, N.; Schwartz, J. T. Linear operators. Part I. Ge neral theory. With the assistance of\nWilliam G. Bade and Robert G. Bartle. Reprint of the 1958 orig inal. Wiley Classics Library. A\nWiley-Interscience Publication. John Wiley & Sons, Inc., N ew York, 1988.\n[9] Folland, G. B., Real Analysis: Modern Techniques and Their Applications , 2nd ed. Wiley &\nSons, 1999\n[10] Foondun, M.: Harmonic Functions for a Class of Integro-differential operat ors.Potential Anal-\nysis,31, 21–44 (2009).\n[11] B. Franke, A functional non-central limit theorem for jump-diffusions with periodic coefficients\ndriven by stable Levy-noise, J. Theoret. Probab. 20 (2007) 1087–1100.\n[12] Fukushima, M., Oshima, Y., Takeda, M. Dirichlet forms and symmetric Markov processes. De\nGruyter Studies in Mathematics, 19, Berlin-New York, 2011.\n[13] T. Fujiwara, M. Tomisaki, Martingale approach to limit theorems for jump processes , Stoch.\nStoch. Rep. 50 (1–2) (1994) 35–64.\n[14] R. K. Getoor, Markov processes: Ray processes and right processes, Lecture Notes in Math.,\nVol. 446, Springer-Verlag, Berlin, Heidelberg and New York , 1975.\n[15] Grigelionis, B.; Mikulyavichyus, R. On the weak convergence of semimartingales. Litovsk. Mat.\nSb.21(1981), no.3, 9–24.\n[16] Gilbarg, D.; Trudinger, N. S. Elliptic partial different ial equations of second order. Reprint of\nthe 1998 edition. Classics in Mathematics. Springer-Verla g, Berlin, 2001.\n[17] Hoh, W.: Pseudo differential operators with negative definite symbol s and the martingale prob-\nlem. Stoch. Stoch. Rep. 55, 225–252 (1995)\n[18] Horie, M., Inuzuka, T. and Tanaka, H. Homogenization of certain one-dimensional discontinu-\nous Markov processes . Hiroshima Math. J. 7 (1977) 629–641.\n[19] Huang, Q., Duan, J., Song R., Homogenization of nonlocal partial differential equations related\nto stochastic differential equations with L´ evy noise. Bernoulli 28(3), 2022, 1648–1674\n38[20] Ikeda, N., Watanabe, S.: On some relations between the harmonic measure and the L´ evy measure\nfor a certain class of Markov processes . J. Math. Kyoto Univ. 2, 79-95 (1962)\n[21] V. V. Jikov S. M. Kozlov O. A. Oleinik, Homogenization of Differential Operators and Integral\nFunctionals . Springer, 1994.\n[22] Kassmann, M., Piatnitski, A. and Zhizhina, E. Homogenization of L´ evy-type operators with\noscillating coefficients. SIAM J. Math. Anal. 51 (2019) 3641–3665.\n[23] Kallenberg, O.: Foundations of modern probability. Se cond edition. Probability and its Appli-\ncations (New York) Springer-Verlag, New York (2002).\n[24] Jikov, V. V.; Kozlov, S. M.; Oleinik, O. A. Homogenization of differential operators and integral\nfunctionals. Springer-Verlag, Berlin, 1994,\n[25] Kolokoltsov, Vassili N.: Markov processes, semigroups and generators. De Gruyter Stud. Math.,\n38 Walter de Gruyter & Co., Berlin, 2011, xviii+430 pp.\n[26] Komorowski, T.; Landim, C.; Olla, S., Fluctuations in Markov processes. Time symmetry\nand martingale approximation. Grundlehren der mathematischen Wissenschaften [Fundamen tal\nPrinciples of Mathematical Sciences], 345. Springer, Heid elberg, 2012.\n[27] Kozlov, S. M. The averaging of random operators.(Russian) Mat. Sb. (N.S.)109(151)(1979),\nno.2, 188-202, 327.\n[28] Kurtz, T., Equivalence of Stochastic Equations and Martingale Proble ms, in Stochastic Analysis\n2010 pp 113–130, D. Crisan edt. Springer.\n[29] K¨ uhn, F., Kunze, M.: Feller generators with measurable lower order terms Positivity (2022)\n26:85\n[30] Ma, Z.-M., R¨ ockner, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Form s.\nSpringer-Verlag, 1992.\n[31] Okitaloshima, O; van Casteren, J.A.: On the uniqueness of the martingale problem. Internat.\nJ. Math. 7(1996) 775–810.\n[32] Olla, S., Homogeniztion of Diffusion Processes in Random Fields , Sentre de Math. Appl.,\nPalaiseau, 1994.\n[33] Papanicolaou, G. C.; Varadhan, S. R. S.(1-NY-X) Boundary value problems with rapidly oscil-\nlating random coefficients. Random fields, Vol. I, II (Esztergom, 1979), 835–873,\n[34] Papanicolaou, George C.; Varadhan, S. R. S. Diffusions with random coefficients. Statistics\nand probability: essays in honor of C. R. Rao, pp. 547–552 Nor th-Holland Publishing Co.,\nAmsterdam, 1982\n39[35] Parthasarathy, K. R. On the category of ergodic measure s. Illinois J. Math. 5 1961 648–656.\n[36] Pavliotis, G. A.; Stuart, A. Multiscale methods. Averaging and homogenization . Texts Appl.\nMath., 53 Springer, New York, 2008.\n[37] Piatnitski, A. and Zhizhina, E. Periodic homogenization of nonlocal operators with a\nconvolution-type kernel. SIAM J. Math. Anal. 49 (2017) 64–81.\n[38] Piatnitski, A. and Zhizhina, E. Stochastic homogenization of convolution type operators. J.\nMath. Pures Appl. 134 (2020) 36–71\n[39] Rhodes. R., Vargas V., Scaling limits for symmetric Itˆ o–L´ evy processes in random medium.\nStoch. Proc. and their Appl. 119(2009) 4004–4033,\n[40] Sandric, N. (2016). Homogenization of periodic diffusion with small jumps . J. Math. Anal. Appl.\n435 551–577.\n[41] Schilling, R.L.: Growth and H¨ older conditions for the sample paths of Feller processes. Probab.\nTheory Relat. Fields 112, 565–611 (1998)\n[42] B. B¨ ottcher, R. Schilling, J. Wang L´ evy Matters III: L´ evy-Type Processes: Construction, Ap-\nproximation and Sample Path Properties , Springer 2013.\n[43] R. L. Schilling, A. Schnurr, The Symbol Associated with the Solution of a Stochastic Differ ential\nEquation , Vol. 15 (2010), Paper no. 43, pages 1369–1393.\n[44] R. L. Schilling and J. Wang: Some theorems on Feller proc esses: transience, local times and\nultracontractivity. (2013)\n[45] Stroock, D.W.: Diffusion processes associated with L´ ev y generators. Probab. Theory Relat.\nFields32, 209–244 (1975)\n[46] Stroock, Daniel W.; Varadhan, S. R. Srinivasa Multidimensional diffusion processes. Reprint of\nthe 1997 edition. Classics in Mathematics. Springer-Verla g, Berlin, 2006.\n[47] Tartar, Luc, On homogenization and Γ-convergence , GAKUTO Internat. Ser. Math. Sci. Appl.,\n18 Gakk¯ otosho Co., Ltd., Tokyo, 2003, 191–211.\n40"    },    {        "title": "2402.04784v1.Equidistribution_of_cusp_points_of_Hecke_triangle_groups.pdf",        "content": "EQUIDISTRIBUTION OF CUSP POINTS\nOF HECKE TRIANGLE GROUPS\nLAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nAbstract. In the framework of infinite ergodic theory, we derive equidistribu-\ntion results for suitable weighted sequences of cusp points of Hecke triangle\ngroups encoded by group elements of constant word length with respect to a set\nof natural generators. This is a generalization of the corresponding results for\nthe modular group, for which we rely on advanced results from infinite ergodic\ntheory and transfer operator techniques developed for AFN-maps.\nContents\n1. Introduction and statement of main results 1\n1.1. Main results 3\n1.2. Key elements of proofs, and structure of article 6\n1.3. Related results 7\n2. Fundamental properties of the generalized Farey maps 7\n2.1. Origin of the generalized Farey maps 8\n2.2. Topologically mixing 9\n2.3. AFN-maps 10\n2.4. Transfer operators and invariant measure 12\n2.5. Tail probabilities 15\n3. Proof of Theorem 1.1 18\n4. Proof of Theorems 1.2 and 1.3 19\nReferences 27\n1.Introduction and statement of main results\nThe classical Stern–Brocot sequence ( Sn)n∈N0partitions the rational numbers\nin the interval [0 ,1] into subsets by sorting them by means of iterated mediants\nstarting from the two seminal reduced fractions 0 /1 and 1 /1. Here, a mediant of two\nreduced fractions a/bandc/dmeans the fraction ( a+c)/(b+d). In other words,\nwith a view to the (modified) Stern–Brocot tree of iterated mediants as indicated in\nFigure 1, the set Snconsists of the (reduced) fractions that are new at tree level n.\nThus\nS0=\u001a0\n1,1\n1\u001b\n,S1=\u001a1\n2\u001b\n,S2=\u001a1\n3,2\n3\u001b\n,S3=\u001a1\n4,2\n5,3\n5,3\n4\u001b\n, . . . .\nThe number-theoretical sequence of the Stern–Brocot sets ( Sn)n∈N0is well studied\nand subject of a correspondingly vast body of literature. See, e.g., [BBT12; K˚ ur12;\n2020 Mathematics Subject Classification. Primary: 11J70, 37F32, 37E05; Secondary: 37D40,\n37A44.\nKey words and phrases. Equidistribution results, Hecke triangle groups, cusp points, infinite\nergodic theory, transfer operators, AFN-maps, generalized Farey maps, generalized Stern–Brocot\nsequences.\n1arXiv:2402.04784v1  [math.DS]  7 Feb 20242 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\n1\n2\n1\n3\n1\n4\n1\n52\n72\n5\n3\n83\n72\n3\n3\n5\n4\n75\n83\n4\n5\n74\n5\nFigure 1. The Stern–Brocot tree restricted to [0 ,1], starting at tree level 1.\nBat14; BV18; Len19; Reu19; AK22]. Also its connection to Minkowski’s question\nmark function has gained some interest in the last decades [KS08; DVA15; Arr15;\nMan17; FST17; MT19; Pum20]. Starting from intensive investigations in the\ntheoretical physics literature, the closely connected Farey fractional spin chain has\nbeen an important model for thermodynamic systems with first and second order\nphase transitions showing also intermittent behavior as a dynamical system [KO99;\nFKO03; FK04; PFK06; Ban+10; Bai11; Tec24]. Enumerating the elements of Sn\nforn≥1 in increasing order, say\nSn={rn\nℓ:ℓ= 1, . . . , 2n−1},\nallows us to define the family of even Stern–Brocot intervals\nEn:=\b\u0002\nrn+1\n2ℓ−1, rn+1\n2ℓ\u0003\n:ℓ= 1, . . . , 2n−1\t\n, n∈N,\ne.g.,\nE1=\u001a\u00141\n3,2\n3\u0015\u001b\n, E 2=\u001a\u00141\n4,2\n5\u0015\n,\u00143\n5,3\n4\u0015\u001b\n, . . . .\nAt the center of the investigations of the Farey fractional spin chain is the even\nStern–Brocot partition function\nZn(t):=X\nI∈En|I|t,\nwhich was first given in this form by Feigenbaum, Procaccia and T´ el in [FPT89],\nand the corresponding pressure function\nP(t):= lim sup\nn→∞1\nnlogZn(t).\nHere,|I|denotes the length of the interval I∈En. For the connection of ZnandP\nto the topological pressure of the Farey map on [0 ,1] (for a definition see Section 1.1,\ncaseq= 3) with geometric potential, the connection to the multifractal Lyapunov\nspectrum, and a detailed study of the phase transition we refer to [KS04; PFK06;\nKS07b]. From [FKO03; KS07b] it follows that Phas a phase transition of the second\nkind in 1, i.e., the pressure function Pis differentiable in 1 with derivative 0, but the\nsecond derivative of Phas a discontinuity there. From the multifractal viewpoint\nthis means that with respect to the Farey map there exists no invariant finite Gibbs\nmeasure with Hausdorff dimension 1 and Lyapunov exponent 0. In fact, there exists\naninfinite absolutely continuous invariant measure with density x7→1/x. By\nthis observation we enter the domain of infinite ergodic theory, which will also be\ncentral to this paper. To better understand the nature of the phase transition of P\nin 1, Kleban and Fiala in [FK04] posed the question of the asymptotic behavior\nofZn(1) for n→ ∞ . It turned out that the complete answer to this question lies\nat the center of infinite ergodic theory [KS12b; KS12a; KMS16; Hee16], which ledEQUIDISTRIBUTION OF CUSP POINTS 3\nto (equi-)distribution results that are also the subject of this paper and which are\ndiscussed in more detail in Section 1.1.\nAs already indicated above, the Stern–Brocot sequence is intimately related to\nthe Farey map and the geodesic flow on the modular surface PSL 2(Z)\\H. Here,\nHrefers to the hyperbolic plane. This relation between the a priori arithmetic\nStern–Brocot sequence and the dynamics of the geodesic flow triggers the question\nas to which degree results on the Stern–Brocot sequence, such as the distribution\nresults alluded to above, are intrinsically of arithmetic nature (i.e., are inherent to\nthe arithmeticity of PSL 2(Z)) or generalize to non-arithmetic but somewhat similar\nsettings.\nWith this article we contribute to this question in the following way: For the family\nof cofinite Hecke triangle groups Γ qwith odd angle parameter q(see Sections 1.1\nand 2 for details), we consider generalizations of the Farey map and the Stern–\nBrocot sequence that arise from a discretization of the geodesic flow on the Hecke\ntriangle orbisurface Γ q\\H. As Γ 3=PSL 2(Z), and Γ qis non-arithmetic for q≥5, we\nhave here a one-parameter sequence of Fuchsian groups that proceeds immediately\nfrom our initial, arithmetic setting of PSL 2(Z) into a non-arithmetic regime while\nremaining close to each other from a dynamical point of view. We study certain\n(equi-)distribution results of the generalized Stern–Brocot sequences and further\ngeneralizations as presented in Section 1.1 below and observe that our findings\ngeneralize the corresponding distribution results for the classical, arithmetic Stern–\nBrocot sequence (and its generalizations within the modular group Γ 3=PSL 2(Z))\nwithout any limitations. Thus, for this type of results, the arithmeticity of PSL 2(Z)\nis not relevant; these are purely dynamical results.\n1.1.Main results. The Hecke triangle group Γ qwith angle parameter q∈N≥3,\nor cusp width λq:= 2cos(π/q), is the subgroup of PSL 2(R) generated by the two\nelements\nTq:=\u00141λq\n0 1\u0015\nand S:=\u00140 1\n−1 0\u0015\n.\nHere, and throughout this article, we denote an element in PSL 2(R) (or, more gen-\nerally, in PGL 2(R)) by a representing matrix in SL2(R) (more generally, in GL2(R))\nbut with square brackets. From now on, we restrict the considerations to odd values\nofq.\nIn order to state the generalized Farey map Fqassociated with Γ qin explicit\nterms, we set\nUq:=TqS=\u0014λq−1\n1 0\u0015\nand, for any k∈Z,\n(1) gq,k:=\u0000\nUk\nqS\u0001−1=\u0014s(k)−s(k+ 1)\n−s(k−1) s(k)\u0015\nwith\ns(x):=sin\u0010\nx\nqπ\u0011\nsin\u0010\nπ\nq\u0011.\nSince Uq\nq=id, the sequence ( gq,k)k∈Zis indeed periodic (with minimal period q).\nThus gq,k+q=gq,kfor all k∈Z. The element\nQ:=\u00140 1\n1 0\u0015\nofPGL 2(R) constitutes an outer symmetry of Γ q, i.e., QΓqQ= Γq. In particular,\nQgq,k=gq,q−kQfor all k∈Z. The generalized Farey map Fqis the selfmap on [0 ,1]4 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\n0 1\nλ+11\nλλ\n2101\nFigure 2. Graph of the generalized Farey map associated to q= 5\nthat is piecewise given by the bijections\n[g−1\nq,k.0, g−1\nq,k.1]→[0,1], x7→gq,k.x , (2)\nand\n[(Qgq,k)−1.1,(Qgq,k)−1.0]→[0,1], x7→Qgq,k.x , (3)\nwith k∈ {(q+ 1)/2, . . . , q −1}. Here, the action of Γ qandQonP1(R) =R∪ {∞}\nis given by fractional linear transformation. Thus\n\u0014a b\nc d\u0015\n.x=ax+b\ncx+d\nwith the mnemonic 1 /0 =∞(i.e., continuous extension to ∞and potential singu-\nlarities), for any x∈P1(R) and any element\u0002a b\nc d\u0003\nin the group generated by Γ q\nandQ. As\n(Qgq,k+1)−1.0 =g−1\nq,k+1.∞=g−1\nq,k.0\nand\n(Qgq,k)−1.1 =g−1\nq,k.1\nfor any k∈Z, the map Fqis indeed well-defined. We note that 0 is a fixed point\nofFq. We refer to Figure 2 for an indication of the graph of Fq. The generalized\nStern–Brocot sequence (Sn,q)n∈N0is then defined by\nSn,q:=eSn,q\\ Sn−1,q forn∈N0,\nwhere\neSn,q:=F−n\nq(0)∪F−n\nq(1),S−1,q:=∅.\nIn Figure 3, an indication of the location of the first few elements for the generalized\nStern–Brocot sequence for q= 5 is shown. For each odd q≥3, the union of\nStern–Brocot elements is Γ q.∞ ∩ [0,1]. We have Γ 3.∞ ∩ [0,1] =Q∩[0,1], and\nΓ5.∞∩[0,1] =Q(√\n5)∩[0,1]. For some more values of qthis set is known explicitly,\nsee [Leu74, Satz 2]. In Theorems 1.2 and 1.3 we will consider even more general\nsequences than Stern–Brocot; the Stern–Brocot sequence is related to the case x= 1\nin Theorem 1.2 and v/w= 1 in Theorem 1.3.\nOur first main result concerns the shrinking properties of compact subsets of [0 ,1]\nthat are bounded away from the fixed point 0 under backwards propagation via Fq\nwith respect to the Lebesgue measure mon [0 ,1]. We endow [0 ,1] with its BorelEQUIDISTRIBUTION OF CUSP POINTS 5\nn= 1\n0 1\nλ+11\nλλ\n21\nn= 2\n0 1\n2λ+11\n22λ+1\n3λ+12λ+2\n3λ+11\nn= 3\n0 1\nFigure 3. First elements of generalized Stern–Brocot sequence\n(Sn,5), omitting S0,5={0,1}\nσ-algebra B[0,1], and we denote by ⋆-limthe weak-star limit of measures on [0 ,1],\ni.e., a sequence of measures ( νn)non [0 ,1] (or B[0,1]) converges to the measure ν\non [0,1] if and only if for all continuous functions fon [0,1] we have\nνn(f)→ν(f).\nWe note that weak-star convergence of measures is sometimes called “weak conver-\ngence of measures,” omitting “star.” We further note that each continuous function\non [0,1] is of course bounded due to the compactness of [0 ,1].\nTheorem 1.1. Letq≥5odd. For all 0< α≤β≤1we have\n⋆-lim\nn→∞\u0012\nlog(n)·m|F−n\nq\u0000\n[α,β]\u0001\u0013\n= log ( β/α)m .\nAn immediate consequence of Theorem 1.1 is the observation that for αandβas\nabove,\n(4) m\u0000\nF−n\nq([α, β])\u0001\n∼log (β/α)\nlog(n)asn→ ∞ ,\nwhich provides a precise shrinking rate of the Lebesgue volume along the se-\nquence\u0000\nF−n\nq([α, β])\u0001\nn. It shows that this rate is structurally the same for all\nsuch intervals [ α, β] with positive length (i.e., α̸=β), but its scale depends logarith-\nmically on β/α. For singletons, i.e., if α=β, the expressions on both sides of (4)\nare identically vanishing.\nOur proof technique for Theorem 1.1 does not apply to the case q= 3 as the\nclassical Farey map F3is not an AFN-map. See Section 2.3 for further details. For\nq= 3, Theorem 1.1 is established in [KS12a] with a different proof, which however\ndoes not apply to q >3. Nevertheless, as Theorem 1.1 is valid for all odd q≥3, we\nwill use it in this generality to establish some of its consequences.\nTheorem 1.1 (for any odd q≥3) is crucial for the two results stated in The-\norems 1.2 and 1.3 below, both of which can be understood as a equidistribution\nresult for weighted Dirac combs . For these results, we set\n(5) Λ q:=n\ng−1\nq,k,(Qgq,k)−1:k∈ {(q+ 1)/2, . . . , q −1}o\n,\nwhich collects the inverses of the elements acting in the branches of the generalized\nFarey map Fq(see(2)-(3)). The semigroup in PGL 2(R) generated by Λ qis free (as\nwe will discuss in Section 2.1), and hence every element in this semigroup can be\nunderstood as a unique word over the alphabet Λ q. For n∈N, we let Wq,ndenote\nthe set of words of length nover Λ q.\nTheorem 1.2. Letq≥3odd. For each x∈(0,1]we have\n⋆-lim\nn→∞xlog(n)X\nh∈Wq,n|h′(x)|δh.x=m .6 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nWe note that the statement of Theorem 1.2 obviously does not extend to x= 0.\nRecall that λq:= 2cos(π/q) is the cusp width of Γ q. We call a pair ( a, c)∈\nZ[λq]×Z[λq] areduced fraction if there exists an element in Γ qof the form\n\u0014a∗\nc∗\u0015\n.\nClearly, if ( a, c) is a reduced fraction, then also ( −a,−c) is a reduced fraction. For\nthis reason, we identify ( a, c) and ( −a,−c) and, by slight abuse of notion, continue to\ndenote the equivalence class [( a, c)] ={(a, c),(−a,−c)}by (a, c) and also continue\nto call it a reduced fraction. We further identify the (equivalence class of the)\nreduced fraction ( a, c) with the value a/c∈R∪ {∞} . Clearly, each point in Γ q.∞\ncan be represented by a reduced fraction. The representing reduced fraction is\nindeed unique, which is another justification for descending from the original pairs in\nZ[λq]2to the equivalence classes (see Section 4 for details). For q= 3, this notion of\nreduced fraction is the classical notion of reduced fractions from integer arithmetic.\nFor any q≥3,n∈Nand reduced fraction v/w∈Γq.∞ ∩[0,1], we let\nRFq,n(v, w):=n\n(r, s) reduced fraction :r\ns∈F−n\nq\u0010v\nw\u0011o\ndenote the preimages of v/wunder Fn\nqin reduced fraction representation representing\ncusp points of level nseen from v/w. We emphasize that the level is indeed uniquely\ndetermined as soon as v/w̸= 0. A rather immediate consequence of Theorem 1.2 is\nthe following equidistribution result for weighted Dirac combs supported on cusp\npoints of increasing level.\nTheorem 1.3. Letq≥3odd. For each reduced fraction (v, w)∈Γq.∞ ∩(0,1]we\nhave\n⋆-lim\nn→∞cv/wvwlog(n)X\n(r,s)∈RFq,n(v,w)1\ns2δr/s=m ,\nwhere c1:= 2andcx:= 1forx̸= 1.\nWe emphasize that the values vwands2in Theorem 1.3 are independent of the\nchoice of the representing pair in Z[λq]2for the respective reduced fraction and\nhence are indeed well defined.\nAs Theorem 1.2, Theorem 1.3 obviously does not extend to 0. Further, [KS12a,\nTheorem 1.2] is covered by Theorem 1.3 as the special case with q= 3.\n1.2.Key elements of proofs, and structure of article. Central for the proofs\nof Theorems 1.1–1.3 are methods from infinite ergodic theory and transfer operator\ntechniques. For Theorem 1.1 we will take advantage of a result by Melbourne and\nTerhesiu [MT12] on the asymptotic behavior of iterates of the transfer operator of\na topologically mixing AFN-map. We restate this result in a specialized version\nas Theorem 2.1 in Section 2. The main bulk of work for establishing Theorem 1.1\nis therefore to show that the generalized Farey maps are eligible for this result\nand to determine all necessary entities in quite some detail. For that reason we\nprove—in Section 2—that the generalized Farey maps Fqfor odd q≥5 are indeed\ntopologically mixing and AFN-maps. At this point we observe the subtlety that\nthe Farey map F3is not an AFN-map. We further provide the Perron–Frobenius\noperator, the invariant measure, and the transfer operator in explicit form. Moreover,\nwe undertake a detailed study of the tail probabilities and their summation functions.\nSection 3 is devoted to combining the results from Section 2 with [MT12] to a proof\nof Theorem 1.1.\nThe proof of Theorem 1.2 makes crucial use of Theorem 1.1 in combination with\na detailed comparison of the weighted sums of Dirac measures in the statement of\nTheorem 1.2 and suitably restricted and rescaled Lebesgue measures. We provideEQUIDISTRIBUTION OF CUSP POINTS 7\nthe details in Section 4. Theorem 1.3 is a rather straightforward consequence of\nTheorem 1.2; a proof is discussed in Section 4.\n1.3.Related results. As mentioned above, the generalized Farey maps as well as\nthe Perron–Frobenius operators arose in [Poh14; MP13; Poh16]. See also Section 2.1.\nWe here restrict the considerations to Hecke triangle groups with oddangle parameter.\nHowever, we expect that similar results are valid for the Hecke triangle groups with\neven angle parameter, which are discussed alongside with the odd angle parameter\ngroups in [Poh16]. In the even angle parameter situation, the generalized Farey\n“map” is not a genuine map but a relation with weights, and the associated Perron–\nFrobenius operator features additional weights. See [Poh16]. The investigations\nin [MP13; Poh16] show an intimate relation between eigenfunctions with eigenvalue 1\nof the Perron–Frobenius operator and Maass cusp forms for the Hecke triangle groups,\nas well as between the Perron–Frobenius operator and the Selberg zeta function,\nthereby also providing a generalized Gauss map. See also [Poh12; AP20; FP20]. It\nwould be interesting to understand if these relations can be used to improve on our\nresults, probably in regard to higher order asymptotics as in [Hee16]. Moreover, it\nwould be interesting to see if our results can be generalized to Hecke triangle groups\nof infinite co-area or other classes of Fuchsian groups. The necessary discretizations\nand Perron–Frobenius operators with similar properties are already available. See,\ne.g., [BP23; Poh14; PW; Wab].\nIn [KS12a] the equidistribution results for q= 3 (i.e., Theorems 1.1 and 1.3\nfor PSL 2(Z)) were compared with the famous equidistribution results of the Farey\nsequence according to Franel and Landau [Lan24]. It would be interesting to\nconsider analogous results for Hecke triangle groups for the distribution of cusp\npoints determined by the size of the denominators of their defining reduced fractions.\nWe emphasize that our results are closely related to the notion of measure-theoretical\nmixing for dynamical systems. Indeed, the following consequence of Theorem 2.1\nbelow applied to our situation lies at the heart of the proof of our main theorem\nand reveals the following mixing property: For all intervals U, V⊂(0,1] that are\nbounded away from 0,\nlog(n)µ\u0000\nF−n(U)∩V\u0001n→∞−→µ(U)µ(V).\nWith Theorem 1.1 we obtain\nlog(n)m\u0000\nF−n(U)∩V\u0001n→∞−→µ(U)m(V).\nFor the first situation, Aaronson [Aar77] introduced the concept of ratio mixing\nfor this type of mixing behavior in the context of infinite ergodic theory. See\nalso [KS07a; Dai+15; BGL18]. Moreover, the same discretizations giving rise to the\ngeneralized Farey maps also give rise to reduction theories for indefinite quadratic\nforms associated to Hecke triangle groups [PS]. An investigation of potential relations\nbetween our equidistribution results and properties of these reduction theories are\nof interest.\nAcknowledgement. The research of LB and AP is funded by the Deutsche Forschungs-\ngemeinschaft (DFG, German Research Foundation) – project no. 441868048 (Priority\nProgram 2026 “Geometry at Infinity”).\n2.Fundamental properties of the generalized Farey maps\nIn this section we will establish several fundamental properties of the generalized\nFarey map and of related objects such as transfer operators and tail probabilities,\nwhich we will need for the proof of Theorem 1.1. A crucial role in our proof of\nTheorem 1.1 will be played by the following result by Melbourne and Terhesiu,\nwhich we state here initially without further explanation and refer to the discussions8 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nfurther below for details and definitions. Throughout let X:= [0,1] denote the\nclosed unit interval and endow it with its Borel σ-algebra, which we denote B=BX.\nTheorem 2.1 (Special case of [MT12, Theorem 1.1]) .LetH:X→Xbe a\ntopologically mixing AFN-map and let X′denote the complement of the indifferent\nfixed points of HinX. Suppose that µis aσ-finite H-invariant measure that\nis absolutely continuous to the Lebesgue measure on X, and let hbe the density\nofµ. Further suppose that (X,B, H, µ )has regularly varying tail probabilities with\nexponent 1. LetbHdenote the transfer operator associated to Handµ, and let w\ndenote the tail probabilities summation function. Then\nlim\nn→∞w(n)bHn(v) =Z\nXv dµ\nuniformly on compact subsets of X′for all v:X→Rof the form v=u/h, where u\nis Riemann integrable on X.\nIn the following subsections we will show that the generalized Farey map Fq\nsatisfies the hypotheses of Theorem 2.1 for odd q≥5. In Section 3 we will combine\nour results with Theorem 2.1 to a proof of Theorem 1.1. For q= 3, the map Fqis\nnot eligible for Theorem 2.1 because, as we will see below, F3is not an AFN-map.\nNevertheless we will consider throughout all odd q≥3. If there is no need to\ndistinguish between different values for qor if the range for qis clear from the\ncontext, we will omit qfrom the subscripts to simplify the notation.\n2.1.Origin of the generalized Farey maps. The generalized Farey map F\ndeduces from a symbolic dynamics for the geodesic flow on the Hecke triangle\norbisurface Γ \\H, which is developed in [Poh14]. Alternatively, it can be developed\nfrom a symbolic dynamics for the billiard flow on the triangle surface eΓ\\H, where\neΓ:=⟨Γ, Q⟩is the triangle group underlying Γ, i.e., the subgroup of PGL 2(R)\ngenerated by Γ and Q. See [Poh16]. We provide a few more details.\nFor any subset U⊆Rwe set\nUst:=U\\Γ.∞.\nHere, the subscript strefers to “strong,” a notion which is explained in detail\nin [Poh14] but is not essential for our discussions. The construction from [Poh14]\n(for Γ\\Hsee in particular the sequence of examples therein) provides a conjugation\n(more precisely, an almost-conjugation; see below) of the geodesic flow on Γ \\Hto\nthe bijective map\nH: (0,∞)st×(−∞,0)st→(0,∞)st×(−∞,0)st\nthat is determined by\n(g−1\nk.0, g−1\nk.∞)st×(−∞,0)st→(0,∞)st×(gk.(−∞), gk.0)st,\n(x, y)7→(gk.x, g k.y)\nfork∈ {1, . . . , q −1}. The relation between the geodesic flow on Γ \\Hand the\nmap His as follows: using the upper half plane model for H, for any geodesic bγ\non Γ\\Hwe fix a representing geodesic γonHwhich satisfies γ(+∞)∈(0,∞) and\nγ(−∞)∈(−∞,0) (this is indeed possible). This is equivalent to asking that γ\nintersects the imaginary axis iR>0, say at time t0(i.e., γ(t0)∈iR>0), and travels\nfrom left to right (in the euclidean coordinates of H). Then bγintersects Γ \\iR>0also\nat time t0. Suppose that ( tj)j∈Jwith J⊆Zis the sequence of intersection times of\nbγwith Γ \\iR>0. Each intersection time is corresponding to a (unique) representing\ngeodesic on Hintersecting iR>0at that time. Let ( γj)j∈Jbe the corresponding\nsequence of geodesics on Hwith endpoints sequence\u0000\n(γj(+∞), γj(−∞))\u0001\nj∈J. NoteEQUIDISTRIBUTION OF CUSP POINTS 9\nthatγ=γ0. Applying Hiteratively to ( γ(+∞), γ(−∞)) moves along this sequence\nin positive index direction; applying H−1allows us to move in negative index\ndirection. In other words, the application of Hcorresponds to moving along bγfrom\ntime tjtotj+1. In the case that γ(+∞)∈Γ.∞orγ(−∞)∈Γ.∞, the map His\nnot defined initially. In these cases, J̸=Z. However, if γ(+∞) is not of the form\ng−1\nk.0 org−1\nk.∞for some k∈ {1, . . . , q −1}, then we can continuously extend H\ntoγ(+∞). Analogously, for H−1andγ(−∞) of the form gk.(−∞) orgk.0. In the\nremaining cases, there are no future (or past) intersection times after t0(before t0).\nFor a detailed discussion we refer to [Poh14].\nFor our purposes, we restrict to the “forward direction” of H, meaning to the\nmap\nH+: (0,∞)st→(0,∞)st\nthat is determined by the bijections\n(g−1\nk.0, g−1\nk.∞)st→(0,∞)st, x7→gk.x ,\nfork∈ {1, . . . , q −1}. This map enjoys a symmetry under the action of Q, i.e.,\nQ.H +(x) =H+(Q.x). Factoring out this symmetry results in the map\nF+: (0,1)st→(0,1)st\nthat is given by the bijections\n\u0000\ng−1\nk.0, g−1\nk.1\u0001\nst→(0,1)st, x7→gk.x ,\nand\n\u0000\n(Qgk)−1.1,(Qgk)−1.0\u0001\nst→(0,1)st, x7→Qgk.x ,\nfork∈ {(q+ 1)/2, . . . , q −1}. The continuous extension of F+to all of X= [0,1] is\nthe generalized Farey map F.\nThe conjugation relation between Hand the geodesic flow implies that the\nsubsemigroup of Γ generated by {g−1\nk:k= 1, . . . , q −1}is free. This property\ngets inherited under the Q-symmetry and shows that the semigroup in PGL 2(R)\ngenerated by Λ (see (5)) is indeed free.\n2.2.Topologically mixing. With the following proposition we show that the Farey\nmap Fis topologically mixing. Essentially, this is a consequence of the bijections\nin(2)and(3)being sufficiently expanding and having full images. We provide here\nanother proof, taking advantage of the origin of F(see Section 2.1) and results\nestablished in [Poh14].\nProposition 2.2. The Farey map Fis topologically mixing.\nProof. We set\nη:=\u001a\u0002\ng−1\nk.0, g−1\nk.1\u0003\n,\u0002\n(Qgk)−1.1,(Qgk)−1.0\u0003\n:k=q+ 1\n2, . . . , q −1\u001b\nand consider, for n∈N0, the refinement\n(6) ηn\n0:=n_\nj=0F−j(η) =\n\nn\\\nj=0F−j(Aj) :Aj∈η, j∈ {0, . . . , n }\n\n.\nThis family consists of intervals which are iterative refinements of the initial intervals\ninη. For any choice of Aj∈η,j∈ {0, . . . , n }, the evaluation of (2)and(3)shows\nthat\n(7)n\\\nj=0F−j(Aj)→[0,1], x7→Fn+1(x),10 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nis a bijection. Further, from the origin of Fand the results proved in [Poh14,\nProposition 8.28, Remark 8.32] (initially for the map H+from Section 2.1 and then\ninherited to FbyQ-symmetry) we know that refinement intervals in the family (6)\nbecome arbitrarily small as n→ ∞ .\nLetB1, B2be non-empty, open subsets of X. Then we find N∈N0such that B1\ncontains some interval from ηN\n0. Hence, using the bijectivity of (7), we obtain that\nFn+1(B1)∩B2̸=∅for all n≥N, showing that Fis topologically mixing. □\n2.3.AFN-maps. In this section we show that the generalized Farey map F=Fq\nfor odd q≥5 is an AFN-map, but not for q= 3. We further discuss that for\nany odd q≥3, the map Fis ergodic and conservative for any measure on Xthat\nis absolutely continuous to the Lebesgue measure m. We start by recalling the\ndefinition of AFN-maps and the auxiliary notion of piecewise monotonic systems,\nwhich both were introduced in [Zwe98].\nDefinition 2.3. Apiecewise monotonic system is a triple ( X, T, ξ ) where Xis the\nunion of some finite family of disjoint bounded open intervals, ξis a collection of\nnon-empty, pairwise disjoint open subintervals with the following two properties:\nWith\nXξ:=[\nI∈ξI\nwe have\nm(X\\Xξ) = 0 .\nFurther, T:X→Xis a map such that T|Iis continuous and strictly monotonic\nfor each I∈ξ.\nDefinition 2.4. Let (X, T, ξ ) be a piecewise monotonic system. Suppose that T|I\nis twice differentiable for each I∈ξ. Then Tis called an AFN-map if it satisfies\nthe following properties:\n(A)Adler’s condition: The map T′′/(T′)2is bounded on Xξ.\n(F)Finite image condition: The set T(ξ) ={T(I) :I∈ξ}is finite.\n(N)Non-uniformly expanding: There is a finite set ξ0⊆ξsuch that each\nI∈ξ0has a (unique) element xIin the boundary of Iwith the following\nproperties:\n•We have\nlim\nx∈I\nx→xIT(x) =xI\nand\nlim\nx∈I\nx→xIT′(x) = 1 .\n•The map T′decreases on ( −∞, xI)∩Ior on ( xI,∞)∩I, depending\non which of these sets is non-empty.\n•The map Tis uniformly expanding on sets bounded away from {xI:\nI∈ξ0}. That is, for all ε >0 there exists ϱ >1 such that for all\nx∈X\\S\nI∈ξ0((xI−ε, xI+ε)∩I) we have\n|T′(x)| ≥ϱ .\nThe set ξ0is indeed unique. The elements xI,I∈ξ0, are called the indifferent fixed\npoints ofT. The complement of the set of indifferent fixed points of Tis denoted X′.\nIn regard to the generalized Farey map F, we set throughout\nξ:={g.(0,1) :g∈Λ},EQUIDISTRIBUTION OF CUSP POINTS 11\nwhere Λ is the set defined in (5). With respect to the Lebesgue measure mand\nhence any measure absolutely continuous to m, the system ξis a measure-theoretic\npartition of X= [0,1]. One easily checks that 0 is the only indifferent fixed point\nofF. The proof of Proposition 2.5 below shows that for F=Fqwith q≥5 odd, we\nhave\nξ0=\b\u0000\ng−1\nq−1.0, g−1\nq−1.1\u0001\t\n=\u001a\u0012\n0,1\nλ+ 1\u0013\u001b\nand\n(8) X′= (0,1].\nProposition 2.5. For every odd q≥5, the Farey map F=Fqis an AFN-map.\nProof. The union of all elements in the collection ξclearly has full Lebesgue measure\ninX. On each interval in ξ, the Farey map Fcorresponds to a fractional linear\ntransformation associated to some g=\u0002a b\nc d\u0003\n∈PSL(2 ,R). As the derivative of the\naction of gis given by g′(x) =det(g)(cx+d)−2for all x∈Rsuch that g.x̸=∞, it\nfollows that ( X, F, ξ ) is piecewise monotonic. Since ξis a finite set, it is obvious\nthatFsatisfies the finite image condition. In particular, each branch (i.e., F|Ifor\nI∈ξ) is surjective onto (0 ,1).\nThe second derivative of a fractional linear transformation g=\u0002a b\nc d\u0003\n∈PSL(2,R)\nis given by g′′(x) =−2cdet(g)(cx+d)−3. It follows that\ng′′(x)\n(g′(x))2=−2cdet(g)(cx+d).\nSince Fconsists of finitely many branches, there is a maximum value which bounds\nthis quotient uniformly, yielding the Adler property for F.\nIt remains to show that Fis non-uniformly expanding. We find that\nξ0=\b\u0000\ng−1\nq−1.0, g−1\nq−1.1\u0001\t\nas the point 0 in the boundary of the interval in ξ0satisfies\nlim\nx→0F(x) = lim\nx→0gq−1.x= lim\nx→0x\n−λx+ 1= 0\nas well as\nlim\nx→0F′(x) = lim\nx→0g′\nq−1(x) = lim\nx→01\n(−λx+ 1)2= 1.\nIn particular, for each ε >0 and x∈g−1\nq−1.(0,1)\\(0, ε) we have\n(9) |F′(x)|=g′\nq−1(x)≥1\n(−λε+ 1)2>1.\nFor the last inequality we note that for ε >1/(λ+ 1), the set g−1\nq−1.(0,1)\\(0, ε)\nis empty. Consider the matrix representation of gkgiven in (1). Then we have\ncx+d >0 for any g=\u0002a b\nc d\u0003\n∈Λ and x∈g.(0,1). Let k∈ {(q+1)/2, . . . , q −2}and\nx∈g−1\nk.(0,1). Then F(x) =gk.xand the coefficient cis negative, which implies\ng′′\nk(x) =−2c(cx+d)−3>0.\nSince g−1\nkis monotone increasing, it follows that\n(10) |F′(x)|=g′\nk(x)≥g′\nk(g−1\nk.0) =\u0012sin(kπ/q)\nsin(π/q)\u00132\n≥λ2.\nLetx∈g−1\nkQ.(0,1) for k∈ {(q+ 1)/2, . . . , q −1}. The coefficient cofQgkis\npositive, which impliesd\ndx|(Qgk)′(x)|=−2c(cx+d)−3<0. Consequently,\n(11) |F′(x)|=|(Qgk)′(x)| ≥ |(Qgk)′(g−1\nkQ.0)|=\u0012sin((k−1)π/q)\nsin(π/q)\u00132\n≥λ2.12 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nForq≥5, we have λ > 1. Thus, the combination of (9)–(11) yields that Fis\nnon-uniformly expanding. This completes the proof. □\nWe saw in the proof above that |F′\nq| ≥λqon the partition elements in ξ\\ξ0. For\nq= 3 we have λ3= 1, and indeed |F′\n3(1)|= 1 which means that F3does not satisfy\nCondition (N) of an AFN-map.\nWe now turn to establish that the generalized Farey maps are ergodic and\nconservative with respect to any measure on Xthat is absolutely continuous to the\nLebesgue measure m. For the classical Farey map, i.e., for F3, this is well-known.\nSee [KMS16]. In order to treat Fqfor odd q >3, we first prove a general result on\nergodicity and conservativity for topologically mixing AFN-maps. For any measure ν\non (X,B) and any measurable set B1, B2∈ Bwe write B1ν=B2ifB1andB2differ\nonly by a ν-null set, i.e., if B1△B2= (B1\\B2)∪(B2\\B1) is a ν-null set.\nLemma 2.6. Any topologically transitive AFN-map on Xis ergodic and conservative\nwith respect to the Lebesgue measure mand hence any measure absolutely continuous\ntom.\nProof. LetTbe a topologically transitive AFN-map on X= [0,1]. It is proven in\n[Zwe98, Theorem 1] that there exists a finite number of pairwise disjoint open sets\nX1, . . . , X msuch that\n•T(Xi)m=XiandT|Xiis ergodic and conservative with respect to m.\n•Each Xiadmits a finite partition into sets that are finite unions of open\nintervals.\nWe will now show that Xm=X1, which immediately yields that Tis ergodic and\nconservative with respect to m. To that end we first note that the property (N)\nof non-uniform expansiveness of Tfrom Definition 2.4 implies that for each open,\nnon-empty interval I⊆Xand each n∈Nthere exist an open interval J⊆Iand\nsome ϱ >1 such that\f\f\f\u0000\nTn\u0001′|J\f\f\f≥ϱ .\nIn particular, m(Tn(I))>0. We now set E:=X\\X1and suppose m(E)>0,\nwith the goal to establish a contradiction. As Eis a finite union of closed intervals\n(among which might be singletons), there exists an open, non-empty interval B⊆E.\nSince Tis topologically transitive, we find n∈Nsuch that\nTn(X1)∩B̸=∅.\nAsX1decomposes into a finite union of open, non-empty intervals and Tis piecewise\ncontinuous, the stability of connectedness under continuous maps in combination\nwith Bbeing an open (connected) interval shows that there is an open interval\nI⊆X1such that Tn(I)⊆B. By our previous argumentation, m(Tn(I))>0. This\ncontradicts to T(X1)m=X1. In turn, m(E) = 0. □\nAs topologically mixing maps are a fortiori topologically transitive, the combina-\ntion of Proposition 2.5 with Lemma 2.6 yields the following corollary for odd q≥5.\nForq= 3, it is in [KMS16].\nCorollary 2.7. For all odd q≥3, the Farey map F=Fqis ergodic and conservative\nwith respect to any measure on Xthat is absolutely continuous to the Lebesgue\nmeasure.\n2.4.Transfer operators and invariant measure. In this section, we establish\nthe existence of an essentially unique σ-finite F-invariant measure on ( X,B) that\nis absolutely continuous to the Lebesgue measure mand we provide its density in\nexplicit form. We further provide an explicit expression for the transfer operatorEQUIDISTRIBUTION OF CUSP POINTS 13\nassociated to Fand this measure. We emphasize that throughout this section we\nconsider all odd q≥3.\nFor any measure νon (X,B), the transfer operator\nL:L1(X, ν)→L1(X, ν)\nassociated to ( F, ν) is determined by the identity\n(12) ∀B∈ B ∀ f∈L1(X, ν):Z\nBL(f) dν=Z\nF−1(B)fdν .\nForνbeing the Lebesgue measure m, we call the transfer operator associated\nto (F, m)Perron–Frobenius operator and denote it by P. It has the explicit\nexpression\nP=q−1X\nk=q+1\n2τ(gk) +τ(Qgk),\nwhere\nτ(g)f(x):=\f\f(g−1)′(x)\f\ff\u0000\ng−1.x\u0001\nforg∈PGL 2(R) and f:R\\ {g.∞} → C. See also [Poh16], where it arises as the\ntransfer operator with parameter 1 for the triangle group ⟨Γ, Q⟩. The essentially\nunique eigenfunction of the Perron–Frobenius operator Pwith eigenvalue 1 is the\ndensity of an F-invariant σ-finite measure. In Proposition 2.8 we provide this\neigenfunction, in Proposition 2.9 we show the invariance of the associated measure\n(among other results).\nProposition 2.8. The map\nh: (0,1]→C, x7→1\nx\nis a real analytic eigenfunction of Pwith eigenvalue 1.\nProof. We define the two operators\nL0,L1: Fct( R>0,C)→Fct(R>0,C)\nby\nL0f(x):=q−1X\nk=1f\u0000\ng−1\nk.x\u0001\nand\nL1f(x):=q−1X\nk=1τ(gk)f(x) =q−1X\nk=1\u0000\ng−1\nk\u0001′(x)f\u0000\ng−1\nk.x\u0001\n.\nWe set f, g:R>0→R,f(x):=logxandg(x):= 1/x=f′(x). We recall from (1)\nthat, for k∈Z,\ng−1\nk=\u0014s(k) s(k+ 1)\ns(k−1) s(k)\u0015\n.\nThen\nL0f(x) =q−1X\nk=1log\u0012xs(k) +s(k+ 1)\nxs(k−1) +s(k)\u0013\n=q−1X\nk=1\u0002\nlog\u0000\nxs(k) +s(k+ 1)\u0001\n−log\u0000\nxs(k−1) +s(k)\u0001\u0003\n= log\u0000\nxs(q−1) +s(q)\u0001\n−log\u0000\nxs(0) + s(1)\u0001\n= log\u0000\nxs(1)\u0001\n−log\u0000\ns(1)\u000114 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\n= log x=f(x).\nThus, fis an eigenfunction of L0with eigenvalue 1. Further, for any x∈R>0,\ng(x) =1\nx=d\ndxf(x) =d\ndxL0f(x) =q−1X\nk=1(g−1\nk)′(x)g\u0000\ng−1\nk.x\u0001\n=L1g(x).\nThus, gis an eigenfunction with eigenvalue 1 of L1. We now combine this property\nwith the facts that gisτ(Q)-invariant (i.e., τ(Q)g=g), that τis an equivariance\nwhen restricted to the group elements and function spaces considered here, and that\nQgkQ=gq−kfor all k∈Z. We obtain\nPg=q−1X\nk=q+1\n2τ(gk)g+τ(Qgk)g\n=q−1X\nk=q+1\n2τ(gk)g+q−1X\nℓ=q+1\n2τ(QgℓQ)g\n=q−1X\nk=q+1\n2τ(gk)g+q−1\n2X\nℓ=1τ(gℓ)g\n=L1g=g .\nNoting that h=g|(0,1]completes the proof. □\nWe let µdenote the measure on ( X,B) which is absolutely continuous to the\nLebesgue measure mwith density being the P-eigenfunction h:x7→1/xfrom\nProposition 2.8, i.e.,\n(13) d µ=hdm .\nWe further let\nbF:L1(X, µ)→L1(X, µ)\ndenote the transfer operator associated to ( F, µ). The following proposition discusses\nthe relation between the Perron–Frobenius operator Pand the transfer operator bF,\nthereby providing a convenient explicit formula for bF. Moreover, it shows that µis\nindeed F-invariant.\nProposition 2.9. The measure µis infinite, σ-finite and Fq-invariant. The transfer\noperator bF:L1(µ)→L1(µ)ofFwith respect to µsatisfies\n(14) bF(f) =P(fh)h−1.\nProof. It is clear that µisσ-finite and infinite. Recall that W1= Λ is the alphabet\ngenerated by the inverse branches of F. The pre-image of a measurable set B⊆X\nis given by the µ-almost disjoint union of g.Bfor all g∈W1. We have for all\nf∈L1(µ),\nZ\nF−1(B)fdµ=X\ng∈W1Z\ng.Bfhdm=X\ng∈W1Z\nB(fh)(g.x)|g′(x)|dm(x).\nBy the definition of Pthis expression is equal toZ\nBP(fh) dm=Z\nBP(fh)h−1dµ .\nIt follows that bF(f) =P(fh)h−1. Finally, as his an eigenfunction of eigenvalue 1\nofP, it holds that\nbF(1) =h−1P(h) =1,EQUIDISTRIBUTION OF CUSP POINTS 15\nwhich implies that µisF-invariant. □\n2.5.Tail probabilities. For any subset Y⊆Xwe let φYdenote the first return\ntime map ofY, that is,\n(15) φY:Y→N∪ {∞} , φ Y(x):= inf{n∈N:Fn(x)∈Y},\nwith the convention that inf∅=∞. The tail probabilities of (X,B, F, µ )associated\ntoYare the values µ(φY> n) for n∈Nor, more precisely, the sequence of these\nvalues. This sequence is said to be regularly varying with exponent 1 if\nµ(φY> n) =ℓY(n)n−1\nfor a positive sequence ( ℓY(n))nsatisfying\nℓY(⌊αn⌋)\nℓY(n)n→∞−→1\nfor all α >0. (In this case, ( ℓY(n))nis called slowly varying.) The associated tail\nprobabilities summation function is\nwY(n):=nX\nj=1µ(φY> j) for n∈N.\nIn this section we will show a rather uniform result for the tail probabilities\nof (X,B, F, µ), namely that for each compact subset CofX′= (0 ,1] we find\na measurable set Y(C)⊆(0,1] such that the tail probabilities\u0000\nµ(φY(C)> n)\u0001\nnare regularly varying with exponent 1 and the tail probabilities summation func-\ntionwY(C)is asymptotic to lognasn→ ∞ . We emphasize that the asymptotics of\nthe tail probabilities summation functions will indeed be found to be independent\nofC, which is the uniformity needed for Theorem 2.1. We refer to [BGT87] for\nfurther information on the concept of regularly varying, which is much more general\nthan needed here.\nOur proof for providing the sets Y(C) in Proposition 2.11 below is constructive,\nfollowing an approach by [Zwe00]. A crucial role will be played by the set\n(16) A:= (g−1\nq−1.1,1] =\u00121\nλ+ 1,1\u0015\n,\nwhich has nicely structured propagation properties under F. See Lemma 2.10. In\nparticular, Ais asweep-out set for (F, µ), i.e.,\n(17)[\nn∈N0F−n(A)µ=X ,\nmeaning that the sets on both sides equal up to a µ-null set. As ( X,B, F, µ) is\nconservative and ergodic (see Corollary 2.7), every set of positive and finite measure,\nsuch as A, is a sweep-out set by [KMS16, Lemma 2.4.3]. However, for the set Aa\nhands-on proof of the sweep-out property is straightforward and explicitly provides\ntheµ-null set in the equality (17), for which reason we include it into the proof of\nLemma 2.10.\nLemma 2.10. For all n∈Nwe have\n(18)n−1[\nk=0F−k(A) = (g−n\nq−1.1,1] =\u00121\nnλ+ 1,1\u0015\n.\nIn particular,\n(19)∞[\nk=0F−k(A) = (0 ,1],\nand the set A= (g−1\nq−1.1,1]is a sweep-out set for (F, µ).16 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nProof. We start by showing the first statement by induction. For n= 1, (18)is\nimmediate by the choice of A. For n >1 we first note that for each B⊆(0,1] we\nhave\nF−1(B) =q−1[\nk=(q+1)/2g−1\nk.B∪(Qgk)−1.B ,\nwhere, because of A= (g−1\nq−1.1,1],\ng−1\nk.B⊆A fork=q+ 1\n2, . . . , q −2\nas well as\n(Qgk)−1.B⊆A fork=q+ 1\n2, . . . , q −1.\nThus, if B⊆Athen\n(20) F−1(B)∪A=g−1\nq−1.B∪A .\nWe suppose now that (18)is valid for some n∈N. Then, taking advantage of (20),\nwe obtain\nn[\nk=0F−k(A) =F−1 n−1[\nk=0F−k(A)!\n∪A\n=F−1\u0000\n(g−n\nq−1.1,1]\u0001\n∪A\n=g−1\nq−1.\u0000\n(g−n\nq−1.1,1]\u0001\n∪(g−1\nq−1.1,1]\n= (g−(n+1)\nq−1.1,1],\nwhich establishes (18)forn+ 1. Hence (18)is indeed valid for all n∈N. For the\nother two statements of the lemma we use that\ng−n\nq−1.1 =1\nnλ+ 1forn∈N\nand hence\n∞[\nk=0F−k(A) = (0 ,1].\nBecause {0}is aµ-null set, it follows immediately that Ais a sweep-out set for\n(F, µ). □\nFor any subset Y⊆X, we let eφYdenote the first hitting time map ofY, that is,\n(21) eφY:X→N∪ {∞} ,eφY(x):= inf{n∈N:Fn(x)∈Y},\nwith the convention that inf∅=∞. We emphasize that the domains of φY(see(15))\nandeφYdiffer. Restricted to Y, both maps are identical. The sweep-out property\nfor (F, µ) yields that the first hitting time of Yis almost surely finite if Yhas\npositive and finite measure. The first hitting time map will simplify the proof of\nProposition 2.11 below. We emphasize that the proof of this proposition is indeed\nconstructive.\nFor any positive sequences ( an)n,(bn)nwe write an∼bnif they are asymptotic\nforn→ ∞ , i.e., if\nlim\nn→∞an\nbn= 1.\nProposition 2.11. The first return time map φA:A→N∪ {∞} satisfies\nµ(φA> n)∼n−1asn→ ∞ .EQUIDISTRIBUTION OF CUSP POINTS 17\nMoreover, for every compact set C⊆(0,1], there exists a measurable set Y(C)⊆\n(0,1]that contains Cand whose first return time map φY(C)satisfies\nµ(φY(C)> n)∼n−1asn→ ∞ .\nProof. In what follows we will provide a construction of the measurable sets Y(C)\nin the second statement. These sets will be using the set Ain such a way that the\nproof will show that the first statement follows immediately from a special case\nin the proof of the second statement. For this reason, we will discuss the second\nstatement only.\nLetCbe a compact subset of (0 ,1]. By Lemma 2.10 we find N∈N0such that\n(22) C⊆N[\nk=0F−k(A) =:Y(C).\nTo simplify notation, let Y:=Y(C). We reformulate the mass of {φY> n}by\nusing the F-invariance of µand the disjointness of level sets of the first return time\nmap for different times as follows:\nµ\u0010\nY∁∩\b\neφY=n\t\u0011\n=µ\u0010\nF−1\u0010\nY∁∩\b\neφY=n\t\u0011\u0011\n=µ\u0000\nφY=n+ 1\u0001\n+µ\u0010\nY∁∩\b\neφY=n+ 1\t\u0011\n=µ\u0000\nφY=n+ 1\u0001\n+µ\u0000\nφY=n+ 2\u0001\n+µ\u0010\nY∁∩\b\neφY=n+ 2\t\u0011\n=∞X\nk=1µ\u0000\nφY=n+k\u0001\n=µ\u0000\nφY> n\u0001\n,\nwhere we used the sweep-out property of Aand induction in the penultimate equality.\nFurther we have, using Lemma 2.10 for the last equality,\nY∁∩\b\neφY=n\t\n=F−n(Y)\\n−1[\nj=0F−j(Y)\n=\nn[\nj=0F−n(Y)\n\\\nn−1[\nj=0F−j(Y)\n\n=\nn[\nj=0N[\nk=0F−(j+k)(A)\n\\\nn−1[\nj=0N[\nk=0F−(j+k)(A)\n\n=N+n[\nℓ=0F−ℓ(A)\\N+n−1[\nm=0F−m(A)\n=\u0000\ng−(N+n+1)\nq−1 .1, g−(N+n)\nq−1.1\u0003\n.\nCombining this set equality with the reformulation of the mass above, and recalling\nthe definition of µin (13), we obtain\nµ\u0000\nφY> n\u0001\n=µ\u0000\u0000\ng−(N+n+1)\nq−1 .1, g−(N+n)\nq−1.1\u0003\u0001\n= log\u0012\n1 +λ\n(N+n)λ+ 1\u0013\n.\nUsing the bounds x/(1 +x)≤log(1 + x)≤x, valid for x >−1, we conclude that\nλ\n(N+n+ 1)λ+ 1≤µ\u0000\nφY> n\u0001\n≤λ\n(N+n)λ+ 1.\nThus,\nµ\u0000\nφY> n\u0001\n∼n−1asn→ ∞ ,18 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nwhich is the second statement of the proposition. Considering N= 0 in (22)(and\npicking, e. g., C=∅), we obtain the first statement as well. □\nIn the following proposition we provide asymptotics for the summation functions\nof the tail probabilities in Proposition 2.11. As all of these tail probabilities enjoy\nthe same asymptotics for n→ ∞ , i.e., independent of the specific set Y(C), we omit\nthe subscript Y(C).\nProposition 2.12. For the tail probabilities summation functions associated to the\nsetsY(C)in Proposition 2.11 we have\nw(n)∼logn asn→ ∞ .\nProof. From Proposition 2.11 we obtain\nw(n) =nX\nj=1µ(φY(C)> j)∼nX\nj=11\njasn→ ∞ .\nObserving that\nnX\nj=11\nj∼logn asn→ ∞\ncompletes the proof. □\n3.Proof of Theorem 1.1\nThroughout this proof section we consider odd q≥5. In particular, q̸= 3. We\nconsider all measures as operators on the space of continuous functions. Let [ α, β]\nbe any compact subinterval of (0 ,1] and let g∈C(X;R) be any continuous function\nonX= [0,1]. Then gis Riemann-integrable as Xis compact and gcontinuous.\nThe Farey map Fis a topologically mixing AFN-map by Propositions 2.2 and 2.5.\nThe set of indifferent fixed points of ( X, F) is{0}, and hence X′= (0,1]. See (8).\nThe tail probabilities of ( X,B, F, µ ) are regularly varying with exponent 1 and the\ntail probabilities summation functions are asymptotic to log(n) by Propositions 2.11\nand 2.12. Hence [MT12, Theorem 1.1] (see Theorem 2.1) provides the convergence\nlog(n)bFn\u0010g\nh\u0011n→∞−→Z\nXg\nhdµ\nand establishes that this convergence is uniform on [ α, β]. Applying the defining\nrelation of the transfer operator bFfrom (12), Lebesgue’s dominated convergence\ntheorem and the relation d µ=hdm(in this order), we obtain that\nlim\nn→∞log(n)m|F−n([α,β])(g) = lim\nn→∞Z\nF−n([α,β])log(n)gdm\n= lim\nn→∞Z\n[α,β]log(n)bFn\u0010g\nh\u0011\ndµ\n=µ([α, β])Z\nXg\nhdµ=µ([α, β])Z\nXgdm\n=µ([α, β])m(g).\nObserving that µ([α, β]) = log( β/α) now completes the proof. □EQUIDISTRIBUTION OF CUSP POINTS 19\n4.Proof of Theorems 1.2 and 1.3\nIn this final section we provide detailed proofs for Theorems 1.2 and 1.3. As\nmentioned in the Introduction (Section 1), Theorem 1.3 is essentially only a special-\nization of Theorem 1.2 to reduced fractions, for which reason we start by presenting\nthis argumentation. The rather longish proof of Theorem 1.2 follows further below.\nThroughout this section we consider all odd q≥3.\nLemma 4.1. The cusp points Γ.∞and the reduced fractions (understood as equiva-\nlence class) in Z[λ]×Z[λ]are in bijection.\nProof. Clearly, each point in Γ .∞can be represented by a reduced fraction. We\nshow that the representation is unique. Let ( a, c) be reduced and let g=[a∗\nc∗].\nSuppose that h∈Γ is such that h.∞=a/c=g.∞. Then g−1h.∞=∞which\nmeans that g−1his an element of the stabilizer Stab Γ(∞) of∞in Γ. The generating\nelement T=[1λ\n0 1]also stabilizes ∞. All stabilizers in a Fuchsian group are cyclic,\nand hence the Poincar´ e Theorem on fundamental polyhedrons yields that\nStab Γ(∞) =⟨T⟩.\nTherefore, there exists some n∈Nsuch that\nh=gTn=\u0014a∗\nc∗\u0015\u00141nλ\n0 1\u0015\n=\u0014a∗\nc∗\u0015\n.\nHence, the reduced fraction representing a/cis unique. □\nLemma 4.2. The pair (a, c)∈Z[λ]×Z[λ]is a reduced fraction if and only if there\nexists g∈ ⟨Γ, Q⟩such that\ng=\u0014a∗\nc∗\u0015\n.\nProof. Since Qis an outer symmetry, we have\n⟨Γ, Q⟩= Γ∪QΓ.\nIn particular, if ( a, c) is reduced then there is an element in Γ ⊆ ⟨Γ, Q⟩with a matrix\nrepresentation of the form [a∗\nc∗]. For the other direction, let g=\u0002a b\nc d\u0003\n∈ ⟨Γ, Q⟩. If\ng∈Γ, then ( a, c) is reduced by definition. Suppose that g=Qhfor some h∈Γ.\nThen gQ∈Γ, and for the generator Sof Γ as considered in Section 1.1 it holds\ngQS =\u0014\na−b\nc−d\u0015\n∈Γ,\nwhich yields that ( a, c) is reduced. □\nThe key observation needed for establishing Theorem 1.3 (in addition to the\nvalidity of Theorem 1.2) is a (natural) correspondence between words of length n\n(i.e., elements of Wn) and reduced fractions of level nrelated to a given reduced\nfraction v/w∈Γ.∞ ∩(0,1) (i.e., elements of RFn(v, w)), as stated in the following\nlemma.\nLemma 4.3. For each reduced fraction v/w∈Γ.∞ ∩(0,1)and each n∈N, the\nsetsWnandRFn(v, w)are in bijection via the map\n(23) Wn→RFn(v, w),\u0014a b\nc d\u0015\n7→(av+bw, cv +dw).\nForv/w= 1, the map in (23) is2 : 1, i.e., each element in RFn(v, w)arises from\nexactly two elements in Wn.20 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nWe emphasize that under the identification of any reduced fraction ( v, w) with\nthe point v/winP1(R), the map in Lemma 4.3 reads\nWn→RFn(v, w), h7→h.v\nw.\nIndeed, the presentation of h.(v/w) for h∈Wn,h=\u0002a b\nc d\u0003\n, as a reduced fraction is\n(av+bw, cv +dw).\nProof of Lemma 4.3. Letv/w∈Γ.∞ ∩ (0,1]. We show that the map in (23) is\nwell-defined. For each h∈Wngiven by h=\u0002a b\nc d\u0003\nwe have\nh.v\nw=a(v/w) +b\nc(v/w) +d=av+bw\ncv+dw,\nwhich is an element in F−n(v/w). Since ( v, w) is a reduced fraction, there exists\n[v∗\nw∗]∈Γ and therefore\n\u0014a b\nc d\u0015\u0014v∗\nw∗\u0015\n=\u0014av+bw∗\ncv+dw∗\u0015\n∈ ⟨Γ, Q⟩.\nTaking advantage of Lemma 4.2 in case that deth=−1, this shows that the\npair ( av+bw, cv +dw) is the reduced fraction representing h.(v/w) and that it is\ncontained in RF n(v, w).\nForv/w̸= 1, let h1=\u0002a1b1\nc1d1\u0003\n, h2=\u0002a2b2\nc2d2\u0003\n∈Wnbe such that\n(a1v+b1w, c1v+d1w) = (a2v+b2w, c2v+d2w).\nThen a1v+b1w=a2v+b2wandc1v+d1w=c2v+d2w. Since ( v, w)̸= (0,0) and\nv̸=wit follows that h1=h2, hence the map in (23) is injective.\nForv/w= 1, as Q.1 = 1, we have\ng−1\nk.1 = ( Qgk)−1.1 for k∈\u001aq+ 1\n2, . . . , q −1\u001b\nandg−1\nk.1̸=g−1\nℓ.1 for k, ℓ∈ {(q+ 1)/2, . . . , q −1},k̸=ℓ. Therefore, the map from\nΛ to Λ .1 is 2 : 1. As any point in Λ .1 is different from 1 and Wn=Wn−1Λ, the\nargument from above for v/w̸= 1 implies that the map in (23)is indeed 2 : 1 for\nv/w= 1.\nFor all v/w∈Γ.∞ ∩ (0,1], surjectivity of the map follows by definition of\nRFn(v, w). □\nWith these preparations we can now provide a short proof of Theorem 1.3.\nProof of Theorem 1.3. Using Theorem 1.2 for x=v/wand combining with the\nmap from Lemma 4.3 we obtain\nm=⋆-lim\nn→∞v\nwlog(n)X\nh∈Wn\f\f\fh′\u0010v\nw\u0011\f\f\fδh.(v/w)=⋆-lim\nn→∞cv/wvwlog(n)X\n(r,s)∈RFn(v,w)1\ns2δr/s,\nwhere c1= 2 and cx= 1 for x̸= 1. For the second equality, we use that for each\nh∈Wnwe have\nh′\u0010v\nw\u0011\n=w2\ns2\nwith r/sdenoting be the reduced form of h.(v/w). □\nWe now turn to the discussion of Theorem 1.2 and start with some preparations.\nAs we will establish Theorem 1.2 first for x̸= 1 and then use this result to show it\nforx= 1, we first let x∈(0,1). We emphasize that almost all objects defined inEQUIDISTRIBUTION OF CUSP POINTS 21\nwhat follows depend on x. Anyhow, this dependence will not be reflected in the\nnotation in favor of simplicity. For n∈Nset\nϱn:=xlog(n)X\nh∈Wn|h′(x)|δh.x,\nwhich is a Borel measure on (0 ,1]. Theorem 1.2 is the statement that\n(24) ⋆-lim\nn→∞ϱn=m .\nIn order to establish (24), we will take advantage of Theorem 1.1.\nForε >0 we set\nVε:=Vε(x):=\u0010\nx−ε\n2, x+ε\n2\u0011\n,\nthe open ε/2-neighborhood of xinR. For n∈N0we further set\nVε,n:=F−n\u0000\nVε\u0001\nand\nmε,n:=log(n)\nµ\u0000\nVε\u0001m|Vε,n,\nwhich is a Borel measure on (0 ,1]. By Theorem 1.1 we know that\n(25) ⋆-lim\nn→∞mε,n=m .\nIn what follows, we will show that, for ε >0 sufficiently small, the measures mε,n\nandϱnare sufficiently near to each other such that their weak-star limits (as\nn→ ∞ andε→0, in this order) are the same. For this, we will proceed via their\ndistribution functions. We let ∆m\nε,ndenote the distribution function of mε,n, and\n∆ϱ\nnthe distribution function of ϱn. That is, for all y∈(0,1],\n∆m\nε,n(y):=mε,n\u0000\n(0, y]\u0001\nand ∆ϱ\nn(y):=ϱn\u0000\n(0, y]\u0001\n.\nFrom now on we suppose that ε >0 is sufficiently small such that Vε⊆(0,1). Then,\nfor all n∈N,\nF−n(Vε) =[\nh∈Wnh.Vε\nis a disjoint union due to the pairwise disjointness of the sets h.(0,1),h∈Wn. In\nparticular,\nm\u0000\nF−n(Vε)\u0001\n=X\nh∈Wnm(h.Vε).\nFor each y∈(0,1] we now obtain that\n\f\f∆m\nε,n(y)−∆ϱ\nn(y)\f\f\n= log( n)\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nF−n\u0000\nVε\u0001\n∩(0, y]\u0001\n−xX\nh∈Wn|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f\n= log( n)\f\f\f\f\fX\nh∈Wn1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f\n≤log(n)X\nh∈Wn\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f. (26)\nFor each h∈Wn, the set h.Vεis an open interval and hence we either have\nh.Vε⊆(0, y] orh.Vε∩(0, y] =∅ory∈h.Vε. For the former two situations we22 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nimmediately obtain the following simplification of the summand for hin (26):\n\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f\n=(\f\f\fm(h.Vε)\nµ(Vε)−x|h′(x)|\f\f\fifh.Vε⊆(0, y],\n0 if h.Vε∩(0, y] =∅.\nBy disjointness of the sets h.Vε,h∈Wn, the situation y∈h.Vεcan materialize for\nat most one h∈Wn, in which case we denote this element by hy,ε,n. Combining\nthese arguments with (26), we obtain the following, a priori rather rough estimate\nfor the difference of the two distribution functions at y:\n\f\f∆m\nε,n(y)−∆ϱ\nn(y)\f\f\n≤log(n)\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nhy,ε,n.Vε∩(0, y]\u0001\n−x|h′\ny,ε,n(x)|δhy,ε,n.x\u0000\n(0, y]\u0001\f\f\f\f\f\n+ log( n)X\nh∈Wn\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε\u0001\n−x|h′(x)|\f\f\f\f\f, (27)\nwhere the summand with hy,ε,n is to be understood as 0 if hy,ε,n does not exist.\nIn what follows we will prove that the summand with hy,ε,nas well as the sum\nover Wnbecomes small as n→ ∞ andε→0. Before we start with a rigorous\ndiscussion, we provide an indication why this might be expected. To that end we\nnote that\n(28) h.Vε=(\u0000\nh.\u0000\nx−ε\n2\u0001\n, h.\u0000\nx+ε\n2\u0001\u0001\nif det h >0\u0000\nh.\u0000\nx+ε\n2\u0001\n, h.\u0000\nx−ε\n2\u0001\u0001\nif det h <0 .\nThus,\nm\u0000\nh.Vε\u0001\n=\f\f\fh.\u0010\nx−ε\n2\u0011\n−h.\u0010\nx+ε\n2\u0011\f\f\f\nand hence\nlim\nε→0m\u0000\nh.Vε\u0001\nε= lim\nε→0\f\fh.\u0000\nx−ε\n2\u0001\n−h.\u0000\nx+ε\n2\u0001\f\f\nε=|h′(x)|.\n(We note that his indeed differentiable on all of (0 ,1), for each h∈Wn.) Further,\n(29) µ\u0000\nVε\u0001\n=x+ε/2Z\nx−ε/21\nxdm= log\u0012x+ε/2\nx−ε/2\u0013\nand hence\n(30) lim\nε→0ε\nµ\u0000\nVε\u0001= lim\nε→0ε\nlog\u0010\nx+ε/2\nx−ε/2\u0011=x .\nThus,\nlim\nε→01\nµ\u0000\nVε\u0001m\u0000\nh.Vε\u0001\n= lim\nε→0ε\nµ\u0000\nVε\u0001m\u0000\nh.Vε\u0001\nε=x|h′(x)|.\nThis shows that each summand in the Wn-sum in (27)converges to 0 as ε→0.\nHowever, for showing that the same is true for the full sum in the limit n→ ∞ , a\nmore subtle argumentation is necessary. The consideration above further gives an\nindication that the hy,ε,n-summand becomes small for ε→0 and, in a certain sense,\nalso for n→ ∞ .EQUIDISTRIBUTION OF CUSP POINTS 23\nIn the remainder of this section, we complete these heuristics with rigorous proofs.\nWe start with two auxiliary results in Lemmas 4.4 and 4.5 below, finding the largest\nderivative among the elements in Wn.\nLemma 4.4. For every h∈W1andx∈[0,1]we have g−1\nq−1.x≤h.x.\nProof. We recall the matrix representation of gkin(1). Since s(k)≤s(k−1) for\nevery k∈ {(q+ 1)/2, . . . , q −1}, we see that for ( q+ 1)/2≤k1< k2≤q−1 and\nx∈[0,1] it holds\n(s(k2)x+s(k2+ 1))( s(k1−1)x+s(k1))≤(s(k2−1)x+s(k2))(s(k1)x+s(k1+ 1))\nwhich is equivalent to\ng−1\nk2.x≤g−1\nk1.xfor all x∈[0,1],\nand in particular g−1\nq−1.x≤g−1\nk−1.x. Since g−1\nkfor any k∈ {(q+ 1)/2, . . . , q −1}is\nmonotone increasing and Q.[0,1] = [1 ,∞], it follows that g−1\nk.x≤g−1\nkQ.xfor all\nx∈[0,1]. □\nFor convenience in the proof of the following lemma, we note that for each k∈Z,\n(31) Qgk=\u0014\n−s(k−1) s(k)\ns(k)−s(k+ 1)\u0015\n.\nLemma 4.5. For each n∈Nandh∈Wn,|h′|is monotone decreasing. Further,\u0000\n(g−1\nq−1)n\u0001′≥ |h′|.\nProof. From (1)and(31) we obtain that each element in the alphabet Λ = W1\nhas a matrix representation with purely non-negative coefficients of which the two\nbottom-row coefficients are positive and the two top-row coefficients cannot vanish\nsimultaneously. Thus, also h∈Wnhas such a matrix representation as being a\ncomposition of elements in W1, say\nh=\u0014a b\nc d\u0015\nwith c, d > 0 (and a, b≥0). Since |h′(ξ)|= (cξ+d)−2, we immediately see that |h′|\nis monotone decreasing.\nIt remains to establish the second statement of the lemma. For each k∈\n{(q+ 1)/2, . . . , q −1}we obtain from (1) and (31) that\n|(g−1\nk)′(x)|=1\n\u0000\ns(k−1)x+s(k)\u00012,\nand\n|(g−1\nkQ)′(x)|=1\n\u0000\ns(k)x+s(k−1)\u00012\nfor all x∈[0,1]. For every k∈ {(q+ 1)/2, . . . , q −1}we have s(k)≤s(k−1), and\nhence, for all x∈[0,1],\ns(k)x+s(k−1)≥s(k−1)x+s(k)>0.\nThus, for x∈[0,1],\n|(g−1\nk)′(x)| ≥ |(g−1\nkQ)′(x)|.\nTherefore the element in W1with the largest absolute derivative is of the form gk\nfork∈ {(q+ 1)/2, . . . , q −1}. Using again s(k)≤s(k−1), we deduce\ns(k−1)x+s(k)≥s(q−2)x+s(q−1) = λx+ 1\nfork∈ {(q+ 1)/2, . . . , q −1}andx∈[0,1]. It follows that for h∈W1,\n(32) |h′| ≤ |(g−1\nq−1)′|24 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\non [0,1]. Further, for h∈W1andx∈[0,1] we have\n(33) g−1\nq−1.x≤h.x .\nSee Lemma 4.4. For any h=h1. . . h n∈Wn, combining (32),(33)and the fact that\n|h′|is decreasing, the product rule for derivatives yields\n|h′| ≤\f\f\f\u0000\n(g−1\nq−1)n\u0001′\f\f\f\non all of [0 ,1]. Finally, because\ng−n\nq−1=\u00141 0\nnλ 1\u0015\nand hence\u0000\ng−n\nq−1\u0001′(x) =1\n(nλx+ 1)2,\nthe derivative of g−n\nq−1is positive on [0 ,1]. This completes the proof. □\nWe can now establish estimates for the hy,ε,n-summand and the summands of\ntheWn-sum in (27) that are sufficient for our purposes.\nLemma 4.6. We have the following estimates:\n(i)For all x∈(0,1), all ε >0such that Vε(x)⊆(0,1), all n∈Nand all\nh∈Wnwe have\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε\u0001\n−x|h′(x)|\f\f\f\f\f≤ \f\f\f\f\fε\nµ\u0000\nVε\u0001−x\f\f\f\f\f+ε\nx!\nm\u0000\nh.Vε\u0001\nε,\nwhere Vε:=Vε(x).\n(ii)Letx∈(0,1)andε >0such that Vε(x)⊆(0,1). Then there exists M > 0\n(depending on xandε) such that for all y∈(0,1]and all n∈Nwe have\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nhy,ε,n.Vε∩(0, y]\u0001\n−x|h′\ny,ε,n(x)|δhy,ε,n.x\u0000\n(0, y]\u0001\f\f\f\f\f≤M\nn2,\nwhere Vε:=Vε(x).\nProof. Letx∈(0,1),ε >0 such that Vε=Vε(x)⊆(0,1),n∈Nandh∈Wn. In\norder to establish (i), we first show that\n(34)\f\f\f\f|h′(x)|ε\nm(h.Vε)−1\f\f\f\f≤ε\nx2.\nTo that end we note that\nm(h.Vε) =\f\f\fh.\u0010\nx−ε\n2\u0011\n−h.\u0010\nx+ε\n2\u0011\f\f\f=ε|h′(ξ)|\nfor some ξ∈Vε(depending on x), by the mean value theorem. We pick the\nrepresentative of hwith positive bottom-row entries, say\nh=\u0014a b\nc d\u0015\nwith c, d > 0. Then\n|h′(x)|ε\nm(h.Vε)−1 =|h′(x)|\n|h′(ξ)|−1 =(cξ+d)2\n(cx+d)2−1 =c2(ξ+x) + 2cd\n(cx+d)2(ξ−x). (35)\nFrom x, ξ∈(0,1) we obtain 0 < x+ξ <2, and ξ∈Vε(x) implies |ξ−x|< ε/2.\nUsing these estimates in (35) yields\n\f\f\f\f|h′(x)|ε\nm(h.Vε)−1\f\f\f\f≤εc2+cd\n(cx+d)2=ε\nx2c2+cd\n\u0000\nc+d\nx\u00012<ε\nx2c2+cd\n(c+d)2<ε\nx2,EQUIDISTRIBUTION OF CUSP POINTS 25\nwhich shows (34). For the penultimate inequality we used that d >0 and x∈(0,1)\nand hence d/x > d , and further c >0. We conclude further that\n\f\f\f\f1\nµ(Vε)m(h.Vε)−x|h′(x)|\f\f\f\f=\f\f\f\fε\nµ(Vε)m(h.Vε)\nε−x|h′(x)|\f\f\f\f\n=\f\f\f\f\u0012ε\nµ(Vε)−x\u0013m(h.Vε)\nε+x\u0012m(h.Vε)\nε− |h′(x)|\u0013\f\f\f\f\n≤\f\f\f\fε\nµ(Vε)−x\f\f\f\fm(h.Vε)\nε+x\f\f\f\fm(h.Vε)\nε− |h′(x)|\f\f\f\f\n≤\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013m(h.Vε)\nε,\nwhere we used (34) for the last inequality. This establishes (i).\nIn order to show (ii)lety∈(0,1] and set h:=hy,ε,n. We note that in this case\ny∈h.Vε. Then\n\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\n≤1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n+x|h′(x)|δh.x\u0000\n(0, y]\u0001\n≤1\nµ\u0000\nVε\u0001m\u0000\nh.Vε\u0001\n+x|h′(x)|. (36)\nBy Lemma 4.5, |h′|is monotone decreasing. Thus,\n(37) |h′(x)| ≤\f\f\fh′\u0010\nx−ε\n2\u0011\f\f\f\nand further, applying the mean value theorem,\nm\u0000\nh.Vε\u0001\n=\f\f\fh.\u0010\nx−ε\n2\u0011\n−h.\u0010\nx+ε\n2\u0011\f\f\f≤ε\f\f\fh′\u0010\nx−ε\n2\u0011\f\f\f. (38)\nUsing (37) and (38) in (36), we obtain that\n(39)\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f≤ \nε\nµ\u0000\nVε\u0001+x!\f\f\fh′\u0010\nx−ε\n2\u0011\f\f\f.\nBy(29)and(30), the first factor of this bound is near 2 xfor the whole range of\nadmissible values for ε. I.e., there exists C >0 (depending on x) such that\nε\nµ\u0000\nVε\u0001+x≤(C+ 1)x≤C+ 1\nfor all ε∈(0,min{2x,2−2x}). (The bounds on εdeduce from the hypothesis\nVε(x)⊆(0,1).) Regarding the second factor of the bound in (39), we recall from\nLemma 4.5 that\ng−n\nq−1=\u00141 0\nnλ 1\u0015\nis the element of Wnwith largest absolute derivative. Therefore\n\f\f\fh′\u0010\nx−ε\n2\u0011\f\f\f≤\u0000\ng−n\nq−1\u0001′\u0010\nx−ε\n2\u0011\n=1\n\u0000\nnλ\u0000\nx−ε\n2\u0001\n+ 1\u00012≤1\nλ2\u0000\nx−ε\n2\u00012n2.\nUsing the latter two estimates in (39) we obtain\n\f\f\f\f\f1\nµ\u0000\nVε\u0001m\u0000\nh.Vε∩(0, y]\u0001\n−x|h′(x)|δh.x\u0000\n(0, y]\u0001\f\f\f\f\f≤M\nn226 LAURA BREITKOPF, MARC KESSEB ¨OHMER, AND ANKE POHL\nwith\nM:=C+ 1\nλ2\u0000\nx−ε\n2\u00012.\nWe note that x−ε/2̸= 0 by the choice of ε. We further note that Mis indeed\nindependent of yandn. □\nWith these preparations we can now provide a proof of Theorem 1.2, for which it\nnow suffices to show that the difference of the distribution functions in (27)vanishes\nasn→ ∞ andε→0 (in this order).\nProof of Theorem 1.2. We first prove Theorem 1.2 for x∈(0,1). To that end, we\nresume the notation from above and continue the estimate in (27)by combining it\nwith the bounds from Lemma 4.6, indicating the dependency of MonεasMε. We\nhave\f\f∆m\nε,n(y)−∆ϱ\nn(y)\f\f\n≤\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013µ(Vε)\nεlogn\nµ(Vε)X\nh∈Wnm(h.Vε) +logn\nn2Mε\n=\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013µ(Vε)\nεmε,n\u0000\n(0,1]\u0001\n+logn\nn2Mε.\nAsn→ ∞ , this bound converges to\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013\nm\u0000\n(0,1]\u0001µ(Vε)\nε=\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013µ(Vε)\nε.\ndue to the weak convergence of ( mε,n)ntom. Thus,\n\f\f∆m\nε,n(y)−∆ϱ\nn(y)\f\f≤\u0012\f\f\f\fε\nµ(Vε)−x\f\f\f\f+ε\nx\u0013µ(Vε)\nε.\nForε→0, the term on the left hand side vanishes due to (30). This completes the\nproof of Theorem 1.2 for x∈(0,1).\nFor the proof of Theorem 1.2 for x= 1, we note that for any p∈Λ,p.1̸= 1, and\nthat for all n∈N,Wn+1=WnΛ. We obtain\nlog(n+ 1)X\nh∈Wn+1|h′(1)|δh.1= log( n+ 1)X\ng∈WnX\np∈Λ|(gp)′(1)|δgp.1\n= log( n+ 1)X\np∈Λ|p′(1)|X\ng∈Wn|g′(p.1)|δg.(p.1)\n=X\np∈Λ|p′(1)|1\np.1p.1·log(n+ 1)X\ng∈Wn|g′(p.1)|δg.(p.1)\nn→∞−→X\np∈Λ|p′(1)|1\np.1m\nby applying Theorem 1.2 for p.1∈(0,1),p∈Λ. Further\nX\np∈Λ|p′(1)|1\np.1=Ph(1) = h(1) = 1\nby Proposition 2.8. Thus,\nlog(n+ 1)X\nh∈Wn+1|h′(1)|δh.1n→∞−→m ,\nwhich completes the proof. □REFERENCES 27\nReferences\n[Aar77] J. Aaronson. Rational ergodicity and a metric invariant for Markov shifts.\nIsrael J. Math. 27.2 (1977), 93–123. doi:10.1007/BF02761661 .\n[AK22] D. Aiylam and T. Khovanova. Stern-Brocot sequences from weighted mediants.\nJ. Number Theory 238 (2022), 313–330. doi:10.1016/j.jnt.2021.08.014 .\n[AP20] A. Adam and A. Pohl. A transfer-operator-based relation between Laplace\neigenfunctions and zeros of Selberg zeta functions. Ergodic Theory Dyn. Syst.\n40.3 (2020), 612–662. doi:10.1017/etds.2018.51 .\n[Arr15] A. Arroyo. Generalised L¨ uroth expansions and a family of Minkowski’s\nquestion-mark functions. C. R. Math. Acad. Sci. Paris 353.10 (2015), 943–946.\ndoi:10.1016/j.crma.2015.08.008 .\n[Bai11] Z.-Q. Bai. A transfer operator approach to random Fibonacci sequences. J.\nPhys. A 44.11 (2011), 115002, 24. doi:10.1088/1751-8113/44/11/115002 .\n[Ban+10] O. F. Bandtlow, J. Fiala, P. Kleban, and T. Prellberg. Asymptotics of the\nFarey fraction spin chain free energy at the critical point. J. Stat. Phys. 138.1-3\n(2010), 447–464. doi:10.1007/s10955-009-9900-4 .\n[Bat14] B. Bates. The Stern-Brocot continued fraction. Integers 14 (2014), Paper No.\nA39, 23.\n[BBT12] B. Bates, M. Bunder, and K. Tognetti. Child’s addition in the Stern-Brocot tree.\nEuropean J. Combin. 33.2 (2012), 148–167. doi:10.1016/j.ejc.2011.09.038 .\n[BGL18] C. Bonanno, P. Giulietti, and M. Lenci. Infinite mixing for one-dimensional\nmaps with an indifferent fixed point. Nonlinearity 31.11 (2018), 5180–5213.\ndoi:10.1088/1361-6544/aadc04 .\n[BGT87] N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation . Encyclo-\npedia of mathematics and its applications v. 27. Cambridge [Cambridgeshire];\nNew York: Cambridge University Press, 1987.\n[BP23] R. Bruggeman and A. Pohl. Eigenfunctions of transfer operators and automor-\nphic forms for Hecke triangle groups of infinite covolume . English. Vol. 1423.\nMem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS),\n2023. doi:10.1090/memo/1423 .\n[BV18] P. Bundschuh and K. V¨ a¨ an¨ anen. Note on the Stern-Brocot sequence, some\nrelatives, and their generating power series. J. Th´ eor. Nombres Bordeaux 30.1\n(2018), 195–202. doi:10.5802/jtnb.1022 .\n[Dai+15] I. Dai, X. Garcia, T. P˘ adurariu, and C. E. Silva. On rationally ergodic and\nrationally weakly mixing rank-one transformations. Ergodic Theory Dynam.\nSystems 35.4 (2015), 1141–1164. doi:10.1017/etds.2013.96 .\n[DVA15] Z. Dresse and W. Van Assche. Orthogonal polynomials for Minkowski’s\nquestion mark function. J. Comput. Appl. Math. 284 (2015), 171–183. doi:\n10.1016/j.cam.2014.07.013 .\n[FK04] J. Fiala and P. Kleban. Thermodynamics of the Farey fraction spin chain. J.\nStatist. Phys. 116.5-6 (2004), 1471–1490. doi:10.1023/B:JOSS.0000041727.\n66775.d3 .\n[FKO03] J. Fiala, P. Kleban, and A. ¨Ozl¨ uk. The phase transition in statistical models\ndefined on Farey fractions. J. Statist. Phys. 110.1-2 (2003), 73–86. doi:10.\n1023/A:1021014627403 .\n[FP20] K. Fedosova and A. Pohl. Meromorphic continuation of Selberg zeta functions\nwith twists having non-expanding cusp monodromy. Sel. Math., New Ser. 26.1\n(2020). Id/No 9, 55. doi:10.1007/s00029-019-0534-3 .\n[FPT89] M. J. Feigenbaum, I. Procaccia, and T. T´ el. Scaling properties of multifractals\nas an eigenvalue problem. Phys. Rev. A (3) 39.10 (1989), 5359–5372. doi:\n10.1103/PhysRevA.39.5359 .\n[FST17] J. Fern´ andez S´ anchez and W. Trutschnig. A note on singularity of a recently\nintroduced family of Minkowski’s question-mark functions. C. R. Math. Acad.\nSci. Paris 355.9 (2017), 956–959. doi:10.1016/j.crma.2017.09.009 .\n[Hee16] B. Heersink. An effective estimate for the Lebesgue measure of preimages of\niterates of the Farey map. Adv. Math. 291 (2016), 621–634. doi:10.1016/j.\naim.2016.01.003 .28 REFERENCES\n[KMS16] M. Kesseb¨ ohmer, S. Munday, and B. O. Stratmann. Infinite ergodic theory of\nnumbers . De Gruyter Graduate. De Gruyter, Berlin, 2016, pp. xiii+191. doi:\n10.1515/9783110439427 .\n[KO99] P. Kleban and A. E. ¨Ozl¨ uk. A Farey fraction spin chain. Comm. Math. Phys.\n203.3 (1999), 635–647. doi:10.1007/s002200050629 .\n[KS04] M. Kesseb¨ ohmer and B. O. Stratmann. Stern-Brocot pressure and multifractal\nspectra in ergodic theory of numbers. Stoch. Dyn. 4.1 (2004), 77–84. doi:\n10.1142/S0219493704000948 .\n[KS07a] M. Kesseb¨ ohmer and M. Slassi. Limit laws for distorted critical return time\nprocesses in infinite ergodic theory. Stoch. Dyn. 7.1 (2007), 103–121. doi:\n10.1142/S0219493707001962 .\n[KS07b] M. Kesseb¨ ohmer and B. O. Stratmann. A multifractal analysis for Stern-\nBrocot intervals, continued fractions and Diophantine growth rates. J. Reine\nAngew. Math. 605 (2007), 133–163. doi:10.1515/CRELLE.2007.029 .\n[KS08] M. Kesseb¨ ohmer and B. O. Stratmann. Fractal analysis for sets of non-\ndifferentiability of Minkowski’s question mark function. J. Number Theory\n128.9 (2008), 2663–2686. doi:10.1016/j.jnt.2007.12.010 .\n[KS12a] M. Kesseb¨ ohmer and B. O. Stratmann. A dichotomy between uniform distri-\nbutions of the Stern-Brocot and the Farey sequence. Unif. Distrib. Theory\n7.2 (2012), 21–33.\n[KS12b] M. Kesseb¨ ohmer and B. O. Stratmann. On the asymptotic behaviour of the\nLebesgue measure of sum-level sets for continued fractions. Discrete Contin.\nDyn. Syst. 32.7 (2012), 2437–2451. doi:10.3934/dcds.2012.32.2437 .\n[K˚ ur12] P. K˚ urka. The Stern–Brocot graph in M¨ obius number systems. Nonlinearity\n25.1 (2012), 57–72. doi:10.1088/0951-7715/25/1/57 .\n[Lan24] E. Landau. Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel.\nNachr. Ges. Wiss. G¨ ottingen, Math.-Phys. Kl. 1924 (1924), 202–206.\n[Len19] H. Lennerstad. The n-dimensional Stern-Brocot tree. Int. J. Number Theory\n15.6 (2019), 1219–1236. doi:10.1142/S1793042119500672 .\n[Leu74] A. Leutbecher. ¨Uber die Heckeschen Gruppen G(λ). II. Math. Ann. 211 (1974),\n63–86. doi:10.1007/BF01344143 .\n[Man17] G. Mantica. Minkowski’s question mark measure. J. Approx. Theory 222\n(2017), 74–109. doi:10.1016/j.jat.2017.06.003 .\n[MP13] M. M¨ oller and A. Pohl. Period functions for Hecke triangle groups, and the\nSelberg zeta function as a Fredholm determinant. Ergodic Theory Dyn. Syst.\n33.1 (2013), 247–283. doi:10.1017/S0143385711000794 .\n[MT12] I. Melbourne and D. Terhesiu. Operator renewal theory and mixing rates for\ndynamical systems with infinite measure. Invent. Math. 189.1 (2012), 61–110.\ndoi:10.1007/s00222-011-0361-4 .\n[MT19] G. Mantica and V. Totik. Regularity of Minkowski’s question mark measure,\nits inverse and a class of IFS invariant measures. J. Lond. Math. Soc. (2) 99.3\n(2019), 707–732. doi:10.1112/jlms.12197 .\n[PFK06] T. Prellberg, J. Fiala, and P. Kleban. Cluster approximation for the Farey\nfraction spin chain. J. Stat. Phys. 123.2 (2006), 455–471. doi:10.1007/s10955-\n006-9034-x .\n[Poh12] A. Pohl. A dynamical approach to Maass cusp forms. J. Mod. Dyn. 6.4 (2012),\n563–596. doi:10.3934/jmd.2012.6.563 .\n[Poh14] A. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hy-\nperbolic good orbifolds. English. Discrete Contin. Dyn. Syst. 34.5 (2014),\n2173–2241. doi:10.3934/dcds.2014.34.2173 .\n[Poh16] A. Pohl. Odd and even Maass cusp forms for Hecke triangle groups, and the\nbilliard flow. English. Ergodic Theory Dyn. Syst. 36.1 (2016), 142–172. doi:\n10.1017/etds.2014.64 .\n[PS] A. Pohl and V. Spratte. A geometric reduction theory for indefinite binary\nquadratic forms over Z[λ]. arXiv:1512.08090.REFERENCES 29\n[Pum20] U. Pumerantz. Word metric, stationary measure and Minkowski’s question\nmark function. Unif. Distrib. Theory 15.2 (2020), 23–38. doi:10.2478/udt-\n2020-0009 .\n[PW] A. Pohl and P. Wabnitz. Selberg zeta functions, cuspidal accelerations, and\nexistence of strict transfer operator approaches . arXiv:2209.05927.\n[Reu19] C. Reutenauer. On the Stern-Brocot expansion of real numbers. J. Th´ eor.\nNombres Bordeaux 31.3 (2019), 697–722. doi:10.5802/jtnb.1104 .\n[Tec24] M. Technau. Remark on the Farey fraction spin chain. Proc. Amer. Math.\nSoc. 152.1 (2024), 63–69. doi:10.1090/proc/16520 .\n[Wab] P. Wabnitz. Strict transfer operator approaches for non-compact developable\norbisurfaces . arXiv:2209.06601.\n[Zwe00] R. Zweim¨ uller. Ergodic properties of infinite measure-preserving interval maps\nwith indifferent fixed points. Ergodic Theory and Dynamical Systems 20.5\n(2000), 1519–1549. doi:10.1017/S0143385700000821 .\n[Zwe98] R. Zweim¨ uller. Ergodic structure and invariant densities of non-Markovian\ninterval maps with indifferent fixed points. Nonlinearity 11.5 (1998), 1263–\n1276. doi:10.1088/0951-7715/11/5/005 .\nUniversity of Bremen, Department 3 – Mathematics, Institute for Dynamical Systems,\nBibliothekstr. 5, 28359 Bremen, Germany\nEmail address :{breitlau, mhk, apohl }@uni-bremen.de"    },    {        "title": "2402.04790v1.On_the_feasibility_of_a_component_based_approach_to_predict_aerodynamic_noise_from_high_speed_train_bogies.pdf",        "content": "On the feasibility of a component-based approach to predict aerodynamic \nnoise from high-speed train bogies \nEduardo Latorre Iglesias1*, David Thompson2, Jorge Muñoz Paniagua3, Javier García García3  \n1 Escuela Técnica Superior de Ingeniería y Sistemas de Telecomunicación, Universidad Politécnica de Madrid,  \nC/ Nikolas Tesla s/n (Campus Sur), 28031, Madrid, Spain \n2 Institute of Sound and Vibration Research, University of Southampton, Southampton, SO17 1BJ, UK. \n3 Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid,  \nC/ José Gutiérrez Abascal, 2, 28006, Madrid, Spain \n*Corresponding author : \ne-mail address : eduardo.latorre.iglesias@upm.es (E. Latorre Iglesias) \nAbstract. At speeds above 300 km/h, aerodynamic noise becomes a significant source of \nrailway noise. In a high-speed train, the bogie area is one of the most important \naerodynamic noise sources. To predict aerodynamic noise, semi-empirical component-\nbased models are attractive as they allow fast and cheap calculations compared with \nnumerical methods. Such component-based models have been previously applied \nsuccessfully to predict the aerodynamic noise from train pantographs which consist of \nvarious cylinders. In this work, the feasibility is evaluated of applying them to the \naerodynamic noise from bogies, which consist of various bluff bodies. To this end, noise \nmeasurements were carried out in an anechoic wind tunnel for different flow speeds using \nsimple shapes representing various idealised components of a bogie. These results have \nbeen used to adjust the empirical constants of the component-based prediction model to \nenable its application to the bogie case. In addition, the noise from a 1:10 scale simplified \nbogie mock-up was measured and used for comparison with the prediction model. The \nresults show good agreement between measurements and predictions for the bogie alone. \nWhen the bogie is located between two ramps approximating a simplified bogie cavity the \nnoise is underpredicted at low frequencies. This is attributed to the noise from the cavity \nwhich is not included in the model. \nKeywords:  Aerodynamic noise, train bogie, prediction model, wind tunnel tests. \n1 Introduction \nHigh-speed trains operate at speeds of 300 km/h or greater in a number of countries. Above about \n300 km/h, aerodynamic noise becomes a significant source of railway noise [1]. The aerodynamic noise \nis produced by the interaction of the incoming air flow with different components or areas of the train,  \n2 in particular the bogies, pantograph, inter-coach cavities, and the train nose. In the case of the bogies, \nthe aerodynamic noise is influenced by the complex turbulent flow recirculation around the bogie area \n[2]. The leading bogie of the train contributes more aerodynamic noise than the others, as it is exposed \nto higher incoming air flow speeds [3-6]. For the remaining bogies, although the presence of upstream \nbogies or equipment significantly reduces the incident flow speed, their contribution can still be \nsignificant, as there are many bogies along the train [7-9]. The estimated contribution of the \naerodynamic noise from all the bogies to the overall aerodynamic noise of a high-speed train during \npass-by is around 15 dB greater than the contribution from the pantograph [2]; the pantograph produces \na significant noise peak during the pass-by but it has a limited duration. In terms of frequency content, \nthe aerodynamic noise from the bogie region is broadband. Results reported by Mellet et al. [9] from \nmicrophone array measurements on French high-speed trains indicated that bogie aerodynamic noise \nhas a significant contribution to the A-weighted sound pressure between 315 Hz and 2 kHz, and \nespecially below 1 kHz [9]. From measurements on Japanese trains at 320 km/h, Uda et al. [10] found \nthat bogie aerodynamic noise played a greater role than the rolling and equipment noise between 125 \nand 500 Hz. From scale model tests, the noise from the bogie cavity was found to be predominant for \nfrequencies below 100 Hz (at full scale) [11, 12].  \nThe bogie aerodynamic noise can be assessed by measurements or predictions (an extensive review \nof assessment methods is given in [13]). If an experimental approach is used, it can be based on field \nmeasurements or wind tunnel tests. Field tests allow the noise from a real train in operation to be \nmeasured but they have drawbacks such as the availability of the train and track, and the difficulty to \nseparate aerodynamic noise from rolling noise in the bogie region. On the other hand, anechoic wind \ntunnel tests have several advantages: only the aerodynamic noise is present, so there is no need for \nsource separation; different configurations can be quickly tested using prototypes to evaluate low-noise \ndesigns; and by testing different configurations it is possible to assess the contribution of the various \nbogie components to the overall noise. However, due to the limitation in the wind tunnel size, scaled \nmodels have to be used and the Reynolds numbers obtained might not be representative of those of the \nreal case, in which case the aerodynamic phenomena will not scale correctly. Moreover, not all facilities \ncan reach the required flow speed to represent the real conditions. In addition, these tests are expensive. \nExamples are found in the literature of anechoic wind tunnel tests to assess bogie aerodynamic noise \n[14-16], to estimate the contribution of different bogie components [12, 17] and bogie configurations \n[18] and to assess the noise reduction obtained when applying counter measures [19, 20]. Field tests \nand anechoic wind tunnel measurements can also be combined with prediction models to assess bogie \naerodynamic noise. For instance, Kitagawa et al. used the prediction tool TWINS [21] to calculate the \nrolling noise produced by a Shinkansen (Japanese high-speed train) and wind tunnel tests with a scaled \ntrain to estimate the aerodynamic noise produced by the bogies [22, 23].  \nNumerical predictions using computational fluid dynamics (CFD) allow the bogie aerodynamic noise \nto be estimated without the need for a real train or expensive test facilities. Examples of using large  \n3 eddy simulations (LES) and delayed detached-eddy simulations (DDES) together with suitable acoustic \nanalogy for calculating the noise generated by a simplified bogie are found in [24, 25]. However, the \napplication of numerical models to large domains with complex geometry and flow behaviour is still \nextremely costly. The computational cost can be significantly reduced by the application of semi-\nempirical component-based prediction models. In these models, a database is built from experiments or \nnumerical simulations of the noise produced by simple geometries that are used to approximate \ncomponents of the whole system (e.g. pantograph, bogie). This allows the empirical constants of the \nmodel to be adjusted. The overall noise is calculated as the incoherent sum of the noise produced by \neach component. Models of this kind have been applied to predict the aerodynamic noise produced by \naircraft landing gears [26] and train pantographs [27, 28], which are mainly produced by vortex \nshedding phenomena from cylinders. Promising results have been obtained, allowing a quick evaluation \nof different configurations and the assessment of the contribution of each component to the overall \nnoise. However, there are additional challenges for its application to predict aerodynamic noise from a \nhigh-speed train bogie because the noise generation phenomena for more compact bluff bodies are \ndifferent and the noise generated is not tonal due to vortex shedding, but broadband [29]. \nThis work aims to investigate the feasibility of the application to a train bogie of a semi-empirical \ncomponent-based model originally developed for train pantograph aerodynamic noise. The model for \nthe pantograph case presented in [27] is applied to the bogie case. To this end, results are presented \nfrom noise measurements carried out in an anechoic wind tunnel for flow speeds ranging between 20 \nand 50 m/s. During these tests, noise was measured using simple geometries to represent bogie \ncomponents and the results are used to adjust the empirical constants of the model. Measurements were \nalso carried out using a 1:10 scale simplified bogie and the results are compared with predictions from \nthe model. In practice, the effect of the (moving) ground beneath the bogie will modify the incoming \nflow conditions and will have a significant impact on the radiated noise [30]. For practical reasons, \nhowever, this could not be considered in the current experimental set-up.  \nThe paper is organized as follows: the existing component-based model based on that applied to the \npantograph case is summarised in Section 2. In Section 3, details are given about the wind tunnel tests; \nthe experimental set-up is described, and the measurement results obtained using simple shapes to \napproximate bogie components are presented. The empirical constants used in the prediction model are \nadjusted based on these noise measurements and are also presented in Section 3. Section 4 includes the \nset-up and results obtained during the anechoic wind tunnel tests made using a 1:10 scale simplified \nbogie mock-up, alone and inside a simplified cavity. The comparison between predictions and \nmeasurements is included in Section 5. Finally, the conclusions are given in Section 6.  \n4 2 Component-based prediction model  \nThe semi-empirical component-based prediction model is based on the assumption that the mean square \nsound pressure spectrum radiated by a structure exposed to an incoming flow can be expressed as the \nincoherent sum of the spectra of each individual component, as follows [28]: \n𝑝ଶതതത(𝑥,𝑓)=(𝜌𝑐ଶ)ଶ\n16𝑀ఈ𝜂𝑆𝐹(𝑓)\n𝑅ଶ𝐷ௗ(𝜓,𝜙)\n(1−𝑀cos(𝜃))ସ\n (1) \nwhere ρ0 is the air density, c0 is the speed of sound, 𝑀=𝑈/𝑐 is the Mach number (where 𝑈 is the \nincident flow speed), 𝑅 is the distance between component and receiver, 𝛼 is the speed exponent, 𝜂 \nis the amplitude factor, 𝑆 is the total surface area of the bogie component, 𝐹(𝑓) is a normalized \nfrequency spectrum obtained using empirical relations and the subscript i refers to each component. \nThe directivity function 𝐷ௗ(𝜓,𝜙) depends on the angles between the lift vector acting on the \ncomponent and the position vector of the receiver with respect to the component position; ψ is the angle \nin the x-y plane and ϕ is the angle in the y-z plane, where x is the airflow direction, y is the lateral \ndirection and z is the height. The convective amplification effect for a noise source depending on the \nobserver position [31] is accounted for through the factor (1−𝑀cos(𝜃))ସ, where θ is the angle between \nthe flow direction and the observer position. \nThe amplitude factor 𝜂 for each component i is chosen to fit the noise level to a reference spectrum. \nThe normalized reference spectrum is obtained by fitting the shape of a given function to that of the \nreference measurement. Two different functions, 𝐹(𝑓) and 𝐹(𝑓)  are used to model the normalized \nspectrum so 𝐹(𝑓) is obtained as the sum of these two. If vortex shedding noise is produced by the \ncomponent, a “haystack-like” spectrum is used to approximate the noise from the vortex shedding peak, \ngiven by [28] \n𝐹(𝑓)=𝑎ଵ\n൬ቀ𝑓𝑓ൗቁଶ\n−ቀ𝑓\n𝑓ൗቁଶ\n൰ଶ\n+𝑎ଶ (2) \nwhere 𝑓 is the vortex shedding frequency that will also be the frequency of the peak of the normalized \nfunction, and 𝑎ଵ is a constant used as normalization factor to force the integral of 𝐹(𝑓) over frequency \nto be unity,  \n𝑎ଵ=1\n∫𝐹(𝑓)𝑑𝑓 (3) \nThe constant 𝑎ଶ is used to control the bandwidth of the function and is obtained from  \n𝑎ଶ=1\n2.25൬𝐵\n𝑓൰ଶ\n (4)  \n5 where the relative bandwidth, defined as 𝐵/𝑓, is obtained as the relative width of the function at a \nlevel of 10 dB below the peak value. \nA second normalized spectrum is required to model the broadband noise. This function is also used \nfor components producing broadband noise without a distinguishable tonal peak. The following \nfunction is chosen to provide a broader “haystack-like” spectral shape [28]: \n𝐹(𝑓)=𝑎ଵ\n1+ቀ𝑓\n𝑓ൗቁమ\n+ቀ𝑛ଶ\n𝑛ଵቁቀ𝑓\n𝑓ൗቁିభ (5) \nThis function is arranged to have its maximum value at the same frequency 𝑓 as the vortex shedding, \nwhere present. The value of the constant 𝑎ଵ is chosen to ensure that the integral of 𝐹(𝑓) over \nfrequency is equal to unity, as explained above for 𝑎ଵ. For broadband noise it is useful to control the \nslope of the function separately for frequencies below and above 𝑓. This is achieved by modifying the \nvalues of the constant 𝑛ଵ for frequencies below 𝑓 and 𝑛ଶ for frequencies above 𝑓. The ratio 𝑛ଶ/𝑛ଵ is \npresent in the denominator to ensure that the maximum of the function occurs at the frequency 𝑓 when \n𝑛ଵ≠𝑛ଶ. \n3 Simple shapes: experiments and model adjustment \nA series of measurements has been carried out in the anechoic wind tunnel of the Institute of Sound and \nVibration Research (ISVR) in the UK [32]. A smooth air flow with low turbulence is delivered through \na rectangular nozzle with dimensions 0.5 × 0.35 m. The test samples consisted of various simple shapes, \nrepresenting simplified bogie components, as well as a simplified 1:10 scale bogie. Figure 1 shows the \ntest set-up, which is common for all the test cases. The test samples were attached to a stiff plywood \npanel surrounded by a baffle that was used to minimize the noise generated by flow interaction with the \nedges of the panel. The baffle was aligned flush with the edge of the nozzle to ensure smooth flow \ndelivery and avoid flow separation. An end view of the set-up is shown in Figure 1(a) and a side view \nis shown in Figure 1(b). The nozzle is located on the left side (Figures 1(b) and 1(c)) with air flowing \nfrom left to right. In the configuration shown in Figure 1(a) a bogie model is attached to the rear of the \nbaffle (see Section 4). For the tests with the simple shapes, the baffle is relocated to align with the far \nside of the nozzle (right-hand side in Figure 1(a)) and the shapes are attached on the other side of the \nbaffle, as indicated in Figure 1(b). A microphone (circled in Figure 1(a)) was attached to the arc-shaped \nmetal structure above the baffle in a direction normal to the flow opposite the centre of the test sample. \nFigure 1(c) shows the distance from the centre of the test samples to the nozzle outlet (0.7 m) and to the \nmicrophone (1.4 m). This microphone is used for model adjustment and comparison with predictions \n(as this is placed on the radiation axis the convective amplification and shear layer refraction effects \ncan be neglected).   \n6  \nFig. 1. Experimental set-up. (a) End view; (b) side view; (c) sketch showing the relative position \nbetween the centre of the sample, the nozzle and the microphones. \nThe measured noise was recorded over 10 s at a sampling frequency of 48 kHz. To obtain the narrow \nband frequency spectra, a Hanning window of 8192 samples was used with overlap of 50%, yielding a \nfrequency resolution Δ𝑓 = 5.85 Hz. \n3.1 Anechoic wind tunnel noise tests  \nMeasurements were performed first on various simple shapes made of solid plywood, used to \napproximate bogie components. A disc was used to represent wheels and disc brakes, and two \nrectangular cuboids with different dimensions, one to represent the motor and the other, with rounded \ncorners, to represent the gearbox; all of them had dimensions of approximately 1/5 of the size of real \nbogie components. A cube with sharp edges was also used. These objects are shown in Figure 2 together \nwith their main dimensions. Holes were introduced as shown to allow a strut to be inserted to support \nthe objects. The characteristic dimension D is defined in each case as that perpendicular to the flow \ndirection and parallel to the radiation axis. These idealised geometrical shapes are used to simplify the \nexperiments while keeping the main aerodynamic features of the corresponding objects in a bogie. It is \nexpected that the lack of small details will lead to underestimates of the high-frequency noise in \n \n7 comparison with that produced by the real bogie components. The noise spectra measured with these \nsimple shapes is used to adjust the value of the empirical constants in the prediction model.  \n \nFig. 2. Simple shapes used to approximate the main bogie components. Dimensions are given in mm. \n(a) Cube; (b) long rectangular cuboid representing a gearbox; (c) disc representing a wheel or disc \nbrake; (d) rectangular cuboid representing a motor. In the centre, coordinates used during the \nexperiments. The flow direction corresponds to the x-axis in each case. \n3.1.1 Experimental set-up  \nFigure 3 shows the front view of the specific experimental set-up used for the simple shapes. These \nwere supported using a long horizontal strut with diameter 12 mm attached to the baffle at one end and \nsupported by a stand outside the flow at the other end. The strut was wrapped with porous foam of \nthickness 6 mm to avoid vortex shedding and minimize the background noise. To ensure clean incident \nflow, the different shapes were installed with a 50 mm gap between them and the baffle. This ensures \nthat their location is not inside the turbulent boundary layer developed along the baffle. The nozzle \nprovided flow with different flow speeds (20, 25, 31.5, 40 and 50 m/s) covering a Reynolds number \nrange 3.42×10ହ≤𝑅𝑒≤ 8.56×10ହ based on D = 0.025 m (for the wheel), and 2.05×10≤𝑅𝑒≤\n5.14×10 based on D = 0.150 m (for the cube, see Figure 2). The Reynolds number is given by: \n𝑅𝑒= 𝑈ஶ𝐷\n𝜈 (6) \nwhere 𝑈ஶ is the mean flow velocity and 𝜈= 1.46×10ିହ m2/s is the kinematic viscosity. \n \n8  \n \nFig. 3. Front view of the experimental set-up used for the noise measurements using simple shapes. (a) \nSet-up using a cube; (b) set-up using a disc; (c) set-up using a rectangular cuboid; (d) set-up using a \nlong rectangular cuboid. Flow is from left to right. \nThe orientation of the simple shapes with respect of the incoming flow and the location of the \nmicrophone were chosen to be equivalent to the situation during a train pass-by; the exception to this \nwas the rectangular cuboid which was installed with its larger faces parallel to the baffle. This is \nillustrated in Figure 4 which shows a sketch of a train body and bogie, indicating the location of a \nmicrophone at the side of the track during a train pass-by. The orientation with respect to the \nmicrophone of the objects used to represent the wheel (disc), the motor (rectangular cuboid) and the \ngearbox (long rectangular cuboid) is also shown.  \nFigure 5 shows the noise radiated by the circular strut used to hold the objects, for a flow speed of \n31.5 m/s, with and without the addition of the foam. The results show that wrapping porous foam around \nthe strut has suppressed the peak in the noise spectrum due to vortex shedding, in this case at 500 Hz. \nAlthough the foam is very efficient to suppress the vortex shedding noise (and hence reduce the \nbackground noise for most frequencies) its effect on the flow around the simple shapes was not \naccounted for. This configuration is considered as the background noise in the experiments. \n (a) (b) \nFlow \nFlow \n Flow \nFlow \n(c) (d) \nFlow \n Flow \n(a) (b)  \n9  \nFig. 4. Sketch showing the microphone position during pass-by noise measurements. The relative \norientation of the simplified bogie components representing the train wheel (disc), motor (rectangular \ncuboid) and gearbox (long rectangular cuboid) is also shown. \nFigure 6(a) compares the one-third octave band noise spectra measured with the supporting strut \nwrapped with porous foam and attached to the baffle and when the cube is also included. The signal-\nto-noise ratio (SNR), defined as the difference between the noise of test configuration being measured \nand the background noise, is in this case between 3.0 and 10.5 dB in the whole frequency range, except \nfor the frequency band of 10 kHz, with a SNR of 2.5 dB. A SNR between 3.0 and 10.0 dB is sufficient \nto allow the contribution of the cube to be identified but requires a correction to be applied to allow for \nthe effect of background noise. Figure 6(b) shows the level difference SNR of different simple shapes. \nFor the rectangular cuboid, this is high for frequencies between 315 Hz and 2 kHz, decreasing at lower \nand higher frequencies, and is lower than 3 dB for frequencies below 125 Hz and above 8 kHz. For the \ncube, the SNR is between 2.5 and 10 dB in the whole frequency range, while for the long rectangular \ncuboid it is between 3 and 6 dB for frequencies between 250 Hz and 6.3 kHz, and below 3 dB at lower \nand higher frequencies. Finally, for the disc the SNR is only greater than 3 dB at the vortex shedding \npeak, and is lower than this for all other frequencies.  \nThresholds of 3 and 10 dB are shown in Figure 6(b). A background noise correction was applied to \nthe frequency bands with a SNR lower than 10 dB, while the frequency bands with a SNR less than 3 \ndB were discarded (a SNR of 3 dB indicates that the background noise and the noise from object have \nthe same magnitude). For the disc, only the peak noise has a SNR above 3 dB, so the measured data are \nnot reliable for determining the broadband noise from the disc. In any case, this component of the noise \ncan be neglected as it will have no influence on the total noise produced by the bogie. The background \nnoise correction consists of the energetic subtraction of the background noise from the noise measured \nfor the different test configurations. The same procedure was applied for all flow speeds. \n \n10  \nFig. 5. Noise spectra radiated by the circular support strut when it is wrapped with porous foam and \nwithout the foam. Flow speed of 31.5 m/s, frequency resolution Δ𝑓 = 5.85 Hz.  \n \nFig. 6. (a) One-third octave noise spectra radiated by the baffle and supporting strut (considered as \nbackground noise) alone and with the addition of a single cube. (b) Level difference between the noise \nspectra measured using different simple shapes and the background noise. Flow speed of 31.5 m/s. \n3.1.2 Results  \nFigure 7 shows the measured 1/3 octave band noise spectra for all simple shapes at different flow \nspeeds after applying the background noise correction. For each object, the spectral shape varies with \nthe flow speed: it is relatively flat at low frequencies (except for the disc) and decreases with a nearly \nconstant slope above a specific frequency that is dependent on the flow speed. For the disc, a clear peak \ncan be distinguished, probably due to vortex shedding. Considering the thickness of the disc as the \ncharacteristic dimension D (as shown in Figure 2), this peak corresponds to a Strouhal number St = \n0.23. The measured values with SNR less than 3 dB (1 dB for the disc) are not included in Figure 7. As \n(a) (b)  \n11 the SNR depends on the flow speed and object type, the frequency range covered by the different figure \ntraces is not always the same.   \n \n \nFig. 7. One-third octave band noise spectra measured for different flow speeds. (a) Cube; (b) long \nrectangular cuboid; (c) disc; (d) rectangular cuboid.  \nFigure 8 shows the speed dependence of the overall sound pressure level (OSPL) for the various \nsimple shapes. As described above, the noise spectra have first been corrected for the background noise. \nIn the case of the long rectangular cuboid only flow speeds of 31.5, 40 and 50 m/s were used, as for 20 \nand 25 m/s a significant number of frequency bands had a SNR below 3 dB. The OSPL is calculated by \nsumming over those frequency bands that are retained. The speed exponent 𝛼 is obtained from fitting \nthe OSPL measured for five different flow speeds: 20, 25, 31.5, 40 and 50 m/s against the logarithm of \nthe speed. In each case this value is close to 6.0. \nIn Figure 9, the measured noise spectra for each flow speed are collapsed in amplitude by subtracting \na factor 10logଵ(𝑈ఈ) and by plotting the results against Strouhal number  𝑆𝑡=𝑓𝐷/𝑈ஶ. The values \nused for the exponents 𝛼 are those from Figure 7. The reference spectrum is obtained by averaging the \ncollapsed spectra. This will be used to determine the empirical constants of the prediction model.  \n(a) (b) \n(c) (d)  \n12  \nFig. 8. Speed dependence of the measured noise for simple shapes: cube (C), disc (D), rectangular \ncuboid (R) and long rectangular cuboid (LR). A speed exponent, 𝛼, of 5.9 was obtained for the cube, \n6.0 for the long rectangular cuboid, 5.9 for the disc and 5.9 for the rectangular cuboid. \n \n  \nFig. 9. Collapsed noise spectra plotted against Strouhal number. (a) Cube ( U5.9); (b) long rectangular \ncuboid (U6.0); (c) disc ( U5.9); (d) rectangular cuboid ( U5.9). \n(a) (b) \n(c) (d)  \n13 3.2 Adjustment of the component-based prediction model \nThe value of the amplitude factor, 𝜂, and the empirical constants required for the normalized \nspectrum, 𝐹(𝑓), are chosen to fit the models in Eqs (2-5) to the reference spectra shown in Figure 9. \nTables 1 and 2 show the values of the empirical constants obtained from the fitting. The peak Strouhal \nnumber 𝑆𝑡 is obtained from the frequency at which the peak of the reference spectrum is located, even \nwhere no clear vortex shedding peak is seen. The amplitude factor 𝜂 and the empirical constants of the \nnormalized spectrum function are obtained independently for the peak and broadband noise spectra. In \naddition, the amplitude factor is obtained for an incident flow speed 𝑈 = 31.5 m/s, using the speed \nexponents  𝛼 shown in Table 1. The specific speed exponent for each of the simple shapes is also \nintroduced in Eq. (3). \nThe fitting is performed first for the broadband noise spectrum. The amplitude factor 𝜂 and the \nexponents 𝑛ଵ and 𝑛ଶ are adjusted to fit the peak amplitude and width of the predicted spectrum to the \nbroadband part of the reference spectrum. Subsequently, the parameters for the peak noise spectra were \nchosen: the amplitude factor 𝜂 is first chosen to fit the value of the predicted peak to that of the \nreference spectrum, and subsequently the empirical constant 𝑎ଶ is chosen to obtain an appropriate \nbandwidth. This procedure was made by eye. \nTable 1. Values of the empirical constants obtained from the fitting of the predictions with the \nreference spectrum for each of the simple shapes. Peak noise.  \nParameter  Disc Long rect. Rectangle Cube \n𝑆𝑡 0.23 0.56 * 2.40 \n𝛼 5.9 6.0 * 5.9 \n𝑎ଶ 0.041 0.725 * 0.150 \n𝜂 0.24×10ିହ 2.6×10ିହ ∗ 0.02×10ିହ \n𝑆 (m2) 0.072 0.113 0.115 0.135 \nTable 2. Values of the empirical constants obtained from the fitting of the predictions with the \nreference spectrum for each of the simple shapes. Broadband noise. \nParameter  Disc Long rect. Rectangle Cube \n𝑛ଵ 1.9 2.5 2.25 2.0 \n𝑛ଶ 1.8 2.0 1.75 2.0 \n𝑆𝑡 0.23 0.56 2.25 2.40 \n𝜂 0.09×10ିହ 0.16×10ିହ 0.96×10ିହ 0.20×10ିହ \n𝛼 5.9 6.0 5.9 5.9 \n𝑆 (m2) 0.072 0.113 0.115 0.135  \n14 Figure 10 shows a comparison between predicted and measured spectra for the different simple \nshapes at a flow speed of 31.5 m/s. There is good agreement between the measurements corrected for \nbackground noise and the fitted curves. For the rectangular cuboid, the peak noise component can be \nneglected as the broadband noise spectrum matches the measurement. The differences in overall level \nbetween the predictions and measurements are mostly less than 1 dB for all objects and all flow speeds.  \n \n \n   \nFig. 10. Comparison between experiments and predictions for different simple shapes at flow speed \n31.5 m/s. (a) Cube; (b) long rectangular cuboid; (c) disc; (d) rectangular cuboid. \nOnly a single microphone was used in the measurements, which was placed along the line \nperpendicular to the flow direction from the centre of the simple shapes, as shown in Figure 1. \nMeasurements of the radiated noise from different angles would be required to estimate the directivity \nfunction 𝐷ௗ(𝜓,𝜙), so this is beyond the scope of this work and could be addressed in future \nexperiments. \n(a) (b) \n(c) (d)  \n15 4 Experiments and predictions for 1:10 scale bogie mock-up \nA second set of experiments was carried out with a 1:10 scale simplified bogie with removable \ncomponents made of high-density stiff foam. Some components were added or removed to assess their \neffect on the noise. This also allowed a trailer and a motor version of the simplified bogie to be tested. \nFigure 11 shows a sketch of the motor and trailer bogie configurations, in which the main components \nare identified. The use of a scale factor of 1:10 implies that the frequency of the noise spectra is 10 \ntimes higher than those for a full-scale bogie, while the sound pressure amplitudes are reduced. For \ncomparison with full-scale quantities, Eq.(1) can be used to adjust the amplitude.  \n \nFig. 11. Sketch of the bogie mock-ups used. The geometry of the bogie components is approximated \nby simple objects. (a) Motor bogie; (b) trailer bogie. \n \n4.1 Experimental set-up \nThe bogie was installed on the same plate in the baffle, see Figure 1. It was installed in two \nconfigurations, shown in Figure 12: in one it was simply mounted on the flat plate (Figure 12(a)) and \nin the other it was installed between two curved ramps, used to approximate a simplified bogie cavity \n(Figure 12(b)). In both cases, the noise from both the motor and trailer bogie configurations was \nmeasured.  \n \nFig. 12. 1:10 scale simplified bogie attached to the baffle. (a) Without ramps; (b) with ramps. \n(a) (b) \nFlow (a) (b)  \n16 Figure 13 shows the test set-up with the bogie and ramps attached to the baffle. The upstream ramp \nwas wider than the nozzle to avoid separation at the edges and to ensure the flow incident on the bogie \nmock-up is laminar.  \n \nFig. 13. Experimental set-up showing the 1:10 scale simplified bogie inside a simplified bogie cavity. \n4.2. Results \nFigure 14(a) shows examples of the noise spectra measured for a flow speed of 50 m/s. Results are \nshown for the baffle alone, the baffle with the ramps, the motor bogie on the baffle with ramps, and the \ntrailer bogie on the baffle with ramps. Compared with the result for the baffle alone, the signal-to-noise \nratio is greater than 7 dB in the whole frequency range for both the motor and trailer bogie. However, \ncompared with the case of the baffle and ramps, the signal-to-noise ratios are smaller. Figure 14(b) \nshows the level difference between the noise measured with the motor or trailer bogie installed and for \nthe case with the ramps alone at different flow speeds. In all cases, except at low frequency, the \ndifferences are greater than 3 dB. From this it can be inferred that the noise from the bogie is higher \nthan that from the ramps.  \nFigure 15 shows the noise measured for both the motor and trailer bogie configurations without the \nramps, for flow speeds of 25, 40 and 50 m/s. Both axles are present in each case, but the components \nattached to them (motor and gearbox or brake discs) are removed from each axle in turn. Results are \ncompared for configurations in which the bogie components are installed on both axles, only on the \nfront axle (bare rear axle) and only on the rear axle (bare front axle). When the front axle is exposed, \ntwo prominent peaks appear, at a frequency that increases with flow speed, probably due to vortex \nshedding from the axle. These peaks disappear when the components are installed on the front axle. No \nobvious peaks are found when the rear axle is exposed. \n \n \n \n17  \nFig. 14. (a) Noise spectra measured at 50 m/s for: baffle alone (background noise); baffle and ramps; \nbaffle, ramps and motor bogie; and baffle, ramps and trailer bogie. (b) Difference between the noise \nspectra measured with ramps alone and with ramps and the motor or trailer bogie at different flow \nspeeds. \nFigure 16 shows the corresponding results when the two ramps are installed, meaning that the bogie \nis located in a simplified cavity. Compared with the results in Figure 15, the sound levels are generally \nabout 5 dB lower across most of the frequency range. No strong tonal peaks are apparent when the front \naxle is exposed. For frequencies below 1250 Hz, the noise when the components are installed on both \naxles is slightly lower than when they are installed only on one axle. Above this frequency, the noise \nwith the front axle exposed is lower than the other two configurations, these giving very similar results.  \n \n \n \nFig. 15. Noise measured for the bogie with the components installed on both axles, just on the front \naxle (bare rear axle) and just on the rear axle (bare front axle), for flow speeds of 25, 40 and 50 m/s \nand no ramps. (a) Motor bogie; (b) trailer bogie. \n(a) (a) (b) \n(b)  \n18   \nFig. 16. Noise measured for the bogie with the components installed on both axles, only on the \nfront axle or only on the rear axle, for flow speeds of 25, 40 and 50 m/s and with ramps. (a) Motor \nbogie; (b) trailer bogie. \n5 Comparison between measurements and predictions \nIn this section the measurement results for the 1:10 scale bogie mock-up are compared with predictions \nin which the bogie components are represented by the simple shapes according to their geometry. Cases \nwith and without the upstream and downstream ramps are considered. In the case with the ramps, two \neffects have to be considered: the self-noise from the ramps themselves and the flow screening effect \ndue to the upstream ramp. Flow speeds of 25, 40 and 50 m/s are considered. As in the previous section, \nthe frequency scale shown in all the figures in this section is as it was measured, which is 10 times \nhigher than at full scale. \nFor simplicity, only the front components of the bogie are included in the prediction model, those \ncomponents being highlighted in Figure 17. This simplification is based on the observation that the \nnoise from the complete bogie and the bogie with exposed rear axle is nearly the same, as shown in \nFigure 15 for the bogie alone and Figure 16 for the bogie with ramps. This observation implies that the \nupstream components are shielding the flow from the downstream components, reducing the incident \nflow speed, especially for the case without ramps. All the comparisons are made for the microphone \nposition perpendicular to the flow direction from the bogie centreline so the effects of the directivity \nand convective amplification can be neglected.  \n(a) (b)  \n19  \nFig. 17. Sketch of the bogie mock-ups. The components included in the prediction model are \nhighlighted. (a) Motor bogie; (b) trailer bogie. \n5.1 Bogie alone \nIn the case of the bogie alone, with no ramps, the front components are exposed to the incoming flow. \nThe incident flow speed for all these components is assumed to be equal to the mainstream flow speed \n𝑈ஶ and the incoming turbulence Iu is neglected, to approximate the conditions during the experiments. \nThe dimensions of each component, and the objects used to approximate their geometry, are given in \nTable A.1 in the Appendix.  \nFigure 18 shows a comparison between the predicted and measured noise spectra for the motor and \ntrailer bogie configurations for a flow speed of 50 m/s. For the motor bogie (Figure 18(a)), the difference \nbetween predictions and measurements is 0.6 dB in terms of OSPL, while for flow speeds of 25 and 40 \nm/s (not shown) differences of 0.8 and 0.5 dB were obtained. The predicted spectral shape is also in \ngood agreement with the measurements apart from a prominent peak at 5 kHz in the measurements that \nis not predicted by the model. The frequency of this peak is found to be independent of the flow speed, \nso it is probably not related to the noise produced by any of the components of the bogie. According to \nthe predictions, the noisiest component for the motor bogie is the motor, which has the largest frontal \narea. The axle box also contributes to the overall noise, mainly at high frequencies, and the noise from \nthe gearbox is significant at mid frequencies. Several other components are important at low frequencies \nincluding the suspension, gearbox and axle.  \nIn the case of the trailer bogie, a good agreement is again found between the predicted and measured \nOSPL, with a maximum difference of 0.8 dB for a flow speed of 50 m/s. However, the differences in \nthe spectral shape are more significant, as shown in Figure 16(b). In this case the noise source with the \nhighest contribution at low frequencies is the brake system, while for mid and high frequencies the \ncontribution from the axle boxes is again significant. \n(a) (b)  \n20  \nFig. 18. Comparison between the measured and predicted sound pressures at 50 m/s when the \nramps are not attached. (a) Motor bogie; (b) trailer bogie. Frequency range as measured. \nFigure 19 shows the level differences between the predicted and measured noise spectra. Results are \nshown for the motor and trailer bogie without ramps at flow speeds of 25, 40 and 50 m/s. For frequencies \nbelow 2 kHz the noise is slightly overpredicted, while for frequencies above this the noise is \nunderpredicted. The noise difference is between -4 and +5 dB in the whole frequency range. The \ndifferences in the OSPL are between 0 and +1 dB for all cases. The consistency of these results indicates \nthat the speed dependence used in the predictions also fits the experimental results. \n \nFig. 19. Differences between the predicted and measured noise spectra for the bogie without ramps at \nflow speeds of 25, 40 and 50 m/s. (a) Motor bogie; (b) trailer bogie. \n5.2 Bogie in cavity \nWhen the bogie is installed in the cavity between the ramps, the size and geometry of the components \nincluded in the prediction model is based on the parts of the motor and trailer bogie configurations that \nare exposed to the incident flow. Figure 20 shows the relative position of the bogie components and the \n(a) \n(a) (b) \n(b)  \n21 ramps, indicating the parts of the components exposed to the incoming air flow and the parts shielded \nby the ramps. Only the parts that are exposed to the flow are included in the model. \nDue to the inclusion of the ramps, it may be appropriate to use different normalised spectra for some \nof the components. However, without further evidence no adjustments were made. The size of the \ncomponents and the objects chosen to represent them are given in Table A.2 in the Appendix. For \nsimplicity the incident flow speed for all these components is assumed to be the mainstream flow speed \n𝑈ஶ and the incoming turbulence Iu is neglected. Further work is required to generalise the approach for \nmodelling the components based on their relative position in the bogie cavity so it can be applied to \ndifferent bogie geometry. \n \nFig. 20. Sketch showing the position of the different components of the bogie with respect to the \nupstream ramp. The bogie model has a maximum length of 344.0 mm and maximum width of 277.9 \nmm. The upstream ramp has a length of 300.0 mm, a width of 700.0 mm and a height of 60.0 mm. \nThe components of the bogie that are not shielded by the upstream ramp can be distinguished from \nthose which are shielded. (a) Trailer bogie; (b) motor bogie. \nThe results obtained for the motor bogie configuration are shown in Figure 21(a). The predicted noise \nis slightly higher than the measurements in the mid frequencies. The motor remains the component with \nthe highest contribution to the overall noise. The hump at 5 kHz is not predicted by the model, similar \nto the result found when the ramps were not included. However, unlike the case without ramps, the \npredicted noise at low frequencies is significantly lower than the measurements. The measured noise \nfor the case without the bogie (empty cavity) is also shown and is significant for frequencies below \n315 Hz. As the noise from the cavity is not included in the model, this seems to be the main reason for \nthe disagreement between prediction and measurement at low frequencies.  \nThe same trend is found for the trailer bogie, Figure 21(b). The noise at low frequencies is under-\npredicted due the noise produced by the cavity. In the measured data a prominent peak appears at 2.5 \nkHz, which is not found in the predictions. This frequency remains unchanged with the flow speed and \nis believed to be unrelated to the bogie components. The main noise source from the bogie at low \nfrequencies is the brake system, the axle boxes also being significant at mid and high frequencies.  \n \n(a) (b)  \n22  \nFig. 21. Comparison between measured and predicted sound pressures at 50 m/s for the bogie \nconfigurations with ramps. (a) Trailer bogie; (b) motor bogie. Frequency range as measured. \n \nFigure 22 shows the differences between the predicted and measured noise spectra for flow speeds \nof 25, 40 and 50 m/s. For frequencies above 630 Hz, in the case of the motor bogie, the results show \ngood agreement between predictions and measurements (differences less than 2.5 dB), while for the \ntrailer bogie the noise is underpredicted, with a maximum difference of 4 dB. For frequencies below \n630 Hz the noise is underpredicted for both bogie configurations, the discrepancy increasing as the \nfrequency decreases. Nevertheless, the differences in the OSPL are between -1 and 0 dB for all cases. \n \n \nFig. 22. Differences between the predicted and measured noise spectra for the motor and trailer bogie \nwith ramps at. flow speeds of 25, 40 and 50 m/s. (a) Motor bogie; (b) trailer bogie. \n5.3 Discussion \nFrom these results it can be concluded that the model is working well to predict the noise of the bogie \nalone, that is, directly exposed to the incoming flow. However, for the case of the bogie inside a cavity \nthe model underpredicts the noise in the low frequency region. However, this region corresponds to \n(a) (b) \n(a) (b)  \n23 frequencies below around 50 Hz at full scale; for speeds up to 100 m/s it could extend up to 100 Hz. It \nis believed that this disagreement is due to the noise produced by the cavity itself, which is not predicted \nby the model (see Fig. 21).  \nSeveral recommendations are given that could improve the prediction model accuracy: the geometry \nof each component in the model should take account of its relative position with respect to the bogie \ncavity. Where components are partially shielded by the upstream bogie cavity step, the geometry \nexposed to the flow will differ from that of the whole component. Moreover, the effect of incoming \nturbulence, as well as the interaction between bogie components should be considered. In addition, for \nthe application of the model to a practical case, omission of the small details could lead to an \nunderprediction of the high-frequency noise. \nIn addition, further investigations are required to include the effect of the ground in the measurement \nset-up to obtain flow conditions more representative of those in a real train bogie cavity. The distance \nbetween the different simple shapes and the ground and their relative position inside the bogie cavity \nshould also be considered as they can have a non-negligible effect on the radiated noise. The estimation \nof the directivity function 𝐷ௗ(𝜓,𝜙) of the different simple shapes could also be addressed in future \nexperiments by adding additional microphones at different angles with respect the shape axis \nperpendicular to the flow direction. \n6. Conclusions \nA semi-empirical component-based model originally developed for the aerodynamic noise from train \npantographs, consisting of cylinders, has been extended for application to train bogies consisting of \nbluff bodies. Anechoic wind tunnel noise measurements were carried out using simple shapes \nrepresenting bogie components. The empirical constants in the model were adjusted for application to \nthe bogie case by fitting the model to reference spectra obtained after collapsing in amplitude the \nmeasured noise spectra for different flow speeds ranging from 20 to 50 m/s.  \nPredictions from this model have been compared with noise measurements on a 1:10 scale bogie \nmock-up in motor and trailer bogie configurations. This was installed on a baffle plate on its own and \nbetween two ramps representing a simplified bogie cavity. In the case of the bogie alone, predictions \nshowed good agreement with measurements, with overall sound pressure levels within less than 1.2 dB \nof the measurements for all the flow speeds tested and with spectral differences between -5 and 5 dB in \nthe whole frequency range. When the bogie was installed between the ramps, the OSPL differences \nbetween predictions and measurements were less than 1.0 dB for all flow speeds but the spectral \ndifferences increased to 7 dB at low frequencies, showing that the low-frequency noise is \nunderpredicted. This low-frequency noise is attributed to the cavity and will be concentrated below 100 \nHz on a full-scale train.  \n24 The results obtained show the feasibility of the application of the semi-empirical component-based \nmodel to the bogie case. The prediction can be improved if the definition of the objects used in the \nmodel is extended to consider the screening effect of the cavity. Account should also be taken of the \neffect of the incoming turbulence, variations in the incident flow speed, interactions between \ncomponents and the inclusion of small geometrical details. \n \nFunding: This work was supported by the Minsterio de Ciencia e Innovación, the Agencia Estatal de \nInvestigación and the FEDER [grant number PID2021-122237OA-I00]. In addition, J. García would \nlike to thank the Programa de Excelencia para el Profesorado Universitario de la Comunidad de Madrid \nfor their financial support. \nReferences \n1. Talotte C. Aerodynamic noise: a critical survey. J Sound Vib 2000;231(3):549-62. \ndoi:10.1006/jsvi.1999.2544 \n2. Thompson DJ. Railway noise and vibration: mechanisms, modelling and means of control: Elsevier; 2008. \n3. Poisson F, Gautier P, Letourneaux F. Noise sources for high speed trains: a review of results in the TGV case.  \nNoise and vibration mitigation for rail transportation systems: Springer; 2008. p. 71-7. doi:10.1007/978-3-\n540-74893-9_10 \n4. He B, Xiao XB, Zhou Q, Li ZH, Jin XS. Investigation into external noise of a high-speed train at different \nspeeds. J Zhejiang Univ Sci A. 2014;15(12):1019-33. doi:10.1631/jzus.A1400307 \n5. Yang H-X, Liu D-M. Numerical study on the aerodynamic noise characteristics of CRH2 high-speed trains. \nJ Vibroengineering 2017;19(5):3953-67. doi:10.21595/jve.2017.18427 \n6. Zhang J, Xiao X, Wang D, Yang Y, Fan J. Source contribution analysis for exterior noise of a high-speed \ntrain: experiments and simulations. Shock Vib 2018;2018:5319460 doi:10.1155/2018/5319460 \n7. Kitagawa T, Nagakura K. Aerodynamic noise generated by Shinkansen cars. J Sound Vib 2000;231(3):913-\n24. doi:10.1006/jsvi.1999.2639 \n8. Lu W-T, Wang Y, Zhang C-Q. Numerical simulation of aerodynamic noises in the far field of the high-speed \ntrain with considering bogies and connection windshields. J Vibroengineering 2017;19(3):2262-79. \ndoi:10.21595/jve.2017.18097 \n9. Mellet C, Létourneaux F, Poisson F, Talotte C. High speed train noise emission: Latest investigation of the \naerodynamic/rolling noise contribution. J Sound Vib 2006;293(3-5):535-46. doi:10.1016/j.jsv.2005.08.069 \n10. Uda T, Yamazaki N, Kitagawa T, Nagakura K, Wakabayashi Y. Estimation of aerodynamic bogie noise \nthrough field and wind tunnel tests.  Noise and Vibration Mitigation for Rail Transportation Systems: \nSpringer; 2018. p. 377-87. doi:10.1007/978-3-319-73411-8_28 \n11. Uda T, Kitagawa T, Saito S, Wakabayashi Y. Low frequency aerodynamic noise from high speed trains. \nQuarterly Report of RTRI. 2018;59(2):109-14. doi;10.11345/nctam.56.195  \n25 12. Yamazaki N, Uda T, Kitagawa T, Wakabayashi Y. Influence of bogie components on aerodynamic bogie \nnoise generated from Shinkansen trains. Quarterly Report of RTRI. 2019;60(3):202-7. \ndoi:10.2219/rtriqr.60.3_202 \n13. Thompson DJ, Latorre Iglesias E, Liu X, Zhu J, Hu Z. Recent developments in the prediction and control of \naerodynamic noise from high-speed trains. Int J Rail Transp 2015;3(3):119-50. \ndoi;10.1080/23248378.2015.1052996 \n14. Yamazaki N, Ido A. Evaluation methods for aerodynamic noise from a high-speed train bogie in a wind tunnel \ntest. In: 40th International Congress and Exposition on Noise Control Engineering (INTERNOISE), Osaka, \nJapan ; 2011. \n15. Yamazaki N, Kitagawa T, Uda T, Nagakura K, Iwasaki M, Wakabayashi Y. Evaluation method for \naerodynamic noise generated from the lower part of cars in consideration of the characteristics of under-floor \nflows on Shinkansen trains. Quarterly Report of RTRI. 2016;57(1):61-8. doi:10.2219/rtriqr.57.1_61 \n16. Yamazaki N, Uda T, Nakayama M, Nishiura T. Acoustic properties of aerodynamic bogie noise generated \nfrom Shinkansen train. In: 46th International Congress and Exposition on Noise Control Engineering \n(INTERNOISE), Hong Kong, China; 2017. \n17. Sawamura Y, Kitagawa T, Yokoyama H, Iida A. Measurement and reduction of the aerodynamic bogie noise \ngenerated by high-speed trains in terms of wind tunnel testing. Proceedings of the 13th International \nWorkshop of Railway Noise Ghent, Belgium (2019). doi:10.1007/978-3-030-70289-2_5 \n18. Latorre Iglesias E, Thompson DJ, Smith M, Kitagawa T, Yamazaki N. Anechoic wind tunnel tests on high-\nspeed train bogie aerodynamic noise. Int J Rail Transp 2017;5(2):87-109. \ndoi:10.1080/23248378.2016.1274685 \n19. Yamazaki N, Takaishi T. Wind tunnel tests on reduction of aeroacoustic noise from car gaps and bogie \nsections. Quarterly Report of RTRI. 2007;48(4):229-35. doi;10.2219/rtriqr.48.229 \n20. Sawamura, Y, Uda, T, Kitagawa, T, Yokoyama, H, Iida, A. Measurement and Reduction of the Aerodynamic \nBogie Noise Generated by High-Speed Trains in Terms of Wind Tunnel Testing. In: Degrande G., et al. Noise \nand Vibration Mitigation for Rail Transportation Systems. Notes on Numerical Fluid Mechanics and \nMultidisciplinary Design, vol 150. Springer, Cham. 2021. https://doi.org/10.1007/978-3-030-70289-2_5. \ndoi:10.1007/978-3-030-70289-2_5 \n21. Thompson DJ, Hemsworth B, Vincent N. Experimental validation of the TWINS prediction program for \nrolling noise, part 1: description of the model and method. J Sound Vib 1996;193(1):123-35. \ndoi:10.1006/jsvi.1996.0252 \n22. Kitagawa T, Nagakura K, Kurita T. Contribution of rolling noise and aerodynamic noise to the total noise \ngenerated from the lower part of Shinkansen cars running at high-speed. Quarterly Report of RTRI. \n2013;54(4):214-21. doi:10.2219/rtriqr.54.214 \n23. Yamazaki N, Uda T, Kurita T, Wakabayashi Y, Nishiura T. Estimation methods for aerodynamic noise \ngenerated from the bogie section of Shinkansen trains by using wind tunnel test. Transactions of the JSME \n(in Japanese). 2017;83(851):17-00146. doi:10.1299/transjsme.17-00146 \n24. Lan J, Han J. Research on the radiation characteristics of aerodynamic noises of a simplified bogie of the \nhigh-speed train. J Vibroengineering. 2017;19(3):2280-93. doi:10.21595/jve.2017.18229  \n26 25. Zhu, J, Hu, Z, Thompson DJ. Flow behaviour and aeroacoustic characteristics of a simplified high-speed train \nbogie. Proc Inst Mech Eng, Part F: JRail Rapid Transit 2016;230(7):1642-1658. \ndoi:10.1177/0954409715605129 \n26. Smith MG, Chow L. Prediction method for aerodynamic noise from aircraft landing gear. In: 4th AIAA/CEAS \nAeroacoustics Conference, Toulouse, France; 1998. \n27. Latorre Iglesias E, Thompson DJ, Smith M. Component-based model to predict aerodynamic noise from high-\nspeed train pantographs. J Sound Vib 2017;394:280-305. doi:10.1016/j.jsv.2017.01.028 \n28. Liu, X, Zhang, J, Thompson, DJ., Latorre Iglesias, E, Squicciarini, G, Hu, Z, Toward, M, Lurcock, D. \nAerodynamic noise of high-speed train pantographs: Comparisons between field measurements and an \nupdated component-based prediction model. App Acoust 2021;175:107791. \ndoi:10.1016/j.apacoust.2020.107791 \n29. Wang Y, Thompson DJ, Hu, Z. Effect of wall proximity on the flow over a cube and the implications for the \nnoise emitted, Phys Fluids 31(7), 077-101, 2019. doi:10.1063/1.5096072 \n30. Zhu JY, Hu ZW, Thompson DJ. The effect of a moving ground on the flow and aerodynamic noise behaviour \nof a simplified high-speed train bogie. Int J Rail Transp 2017;5(2):110-125. \ndoi:10.1080/23248378.2016.1212677 \n31. Dowling AP, Ffowcs Williams JE. Sound and sources of sound: Ellis Horwood; 1983. \n32. Chong, TP, Joseph PF, Davies P. Design and performance of an open jet wind tunnel for aero-acoustic \nmeasurement, App Acoust 2009;70:605-614. doi:10.1016/j.apacoust.2008.06.011 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n27 Appendix A \nThe dimensions of each component and the simple shape that approximates its geometry for the bogie \nalone are given in Table A.1. MB stands for motor bogie and TB stands for trailer bogie. \nTable A.1. Main dimensions and type of normalized spectrum chosen for each of the 1:10 scale bogie \ncomponents included in the prediction model. Dimensions in mm. Bogie without ramps. The \ninclusion of the component in the trailer bogie (TB), motor bogie (MB) or both of them, is shown \nusing the marker X. For the length of the wheel axle the value between brackets corresponds to the \ncase of the trailer bogie where the axle is considered as two independent shorter cylinders to account \nfor the effect of the brake discs. \n \n D \n(mm) B \n(mm) L \n(mm) Norm. spectrum MB TB \nWheel shaft 17.5 - 115.0 (2 x 45) Circular cylinder X X \nCircular support 27.0 - - Cube X X \nAxle box 35.0 40.0 27.0 Rectangle X X \nLateral frames 25.0 14.0 64.0 Long rectangle X X \nWheels 12.5 92.0 - Disc X X \nMotor 53.0 64.0 94.0 Rectangular X - \nGearbox 28.0 140.0 61.0 Long rectangular X - \nBrake system 114.0 96.0 36.0 Rectangular - X \nBrake discs 8.0 64.0 - Discs - X \nLateral damper 25.0 16.0 4.0 Long rectangular X X \nVertical damper 9.0 - 62.0 Circular cylinder X X \nSuspension 70.0 62.0 - Disc X X \n \nTable A.2. Main dimensions and type of normalized spectrum chosen for each of the 1:10 scale bogie \ncomponents included in the prediction model. Dimensions in mm. Bogie with ramps. The dimension \nB is expressed using fractions to indicate the part of the component that is not shielded by the \nupstream ramp in relation with the total size of the component. \n D (mm) B (mm) L (mm) Norm. spectrum \nAxle box 35.0 40.0/4 27.0 Rectangle  \nWheels 12.5 92.0/2 - Disc \nMotor 53.0 64.0/3 94.0 Rectangular  \nGearbox 28.0 140.0/3 61.0 Long rectangular  \nBrake system 114.0 96.0/3 36.0 Rectangular  \nBrake discs 8.0 64.0/2 - Discs \n "    },    {        "title": "2402.04795v1.Stability_under_dwell_time_constraints__Discretization_revisited.pdf",        "content": "1\nStability under dwell time constraints:\nDiscretization revisited\nThomas Mejstrik, and Vladimir Yu. Protasov\nAbstract — We decide the stability and compute the Lya-\npunov exponent of continuous-time linear switching sys-\ntems with a guaranteed dwell time. The main result as-\nserts that the discretization method with step size hap-\nproximates the Lyapunov exponent with the precision C h2,\nwhere Cis a constant. Let us stress that without the dwell\ntime assumption, the approximation rate is known to be\nlinear in h. Moreover, for every system, the constant Ccan\nbe explicitly evaluated. In turn, the discretized system can\nbe treated by computing the Markovian joint spectral radius\nof a certain system on a graph. This gives the value of\nthe Lyapunov exponent with a high accuracy. The method\nis efficient for dimensions up to, approximately, ten; for\npositive systems, the dimensions can be much higher, up\nto several hundreds.\nIndex Terms — discretization, dynamical system on\ngraphs, extremal norm, joint spectral radius, linear switch-\ning system, stability, Lyapunov exponent, multinorm,\n49M25, 93C30, 37C20, 15A60\nI. INTRODUCTION\nWe consider a linear switching system of the form\n\u001a˙x(t) =A(t)x(t), t∈[0,+∞), A (t)∈ A\nx(0) = 0(1)\nwith a positive dwell time restriction. This is a linear ODE on\nthe vector-function x:R+→Rdwith a matrix function A(t)\ntaking values from a given finite set A={A1, . . . , A n}called\ncontrol set of matrices ( regimes, modes ). The control function ,\nor the switching law is an arbitrary piecewise constant function\nA:R+→ A with the lengths of every stationary interval\n(switching interval ) at least m, where m > 0is a given\ndwell-time constraint. This dwell time assumption has the\npractical meaning that the switches between regimes are not\ninstantaneous but take some positive time.\nFor the sake of simplicity we consider only the case of finite\ncontrol sets Aand of the same dwell time for all regimes Aj.\nAll our results are easily extended to general conditions, see\nRemark 1.\nA. Statement of the problem\nSystems (1) regularly arise in engineering applications, see,\nfor instance, [2], [16], [17], [32] and references therein. One\nSubmitted: 22. January 2024. This work was supported by the Aus-\ntrian Science Foundation (FWF) under Grant P33352-N.\nThomas Mejstrik, University of Vienna;\nthomas.mejstrik@gmx.at\nVladimir Yu. Protasov, University of L ’Aquila, Italy;\nvladimir.protasov@univaq.itof the main problems is to find or estimate the fastest possible\ngrowth rate of trajectories, in particular, to decide about the\nstability of the system. The stability problem under the dwell\ntime restrictions have been studied in numerous work [4], [5],\n[7], [8], [9], [31], [33].\nThe Lyapunov exponent σ=σ(A)of a linear switching\nsystem is the infimum of the numbers αsuch that for every\ntrajectory x(·), we have ∥x(t)∥ ≤ Ceαt,t≥0, for some\nconstant C=C(x). Clearly, if σ < 0, then the system\nis asymptotically stable, i.e., all its trajectories tend to zero\nast→+∞. The converse is also true, although less triv-\nial [4], [21]. Thus, the asymptotic stability is equivalent to the\ninequality σ < 0. If, in addition, the system is irreducible,\ni.e., the matrices from Ado not share common nontrivial\ninvariant subspaces, then the inequality σ≤0is equivalent\nto the (usual) stability, when all trajectories are bounded. Let\nus note that if the matrices of the control set Ahave a common\ninvariant subspace, then the computation of the Lyapunov\nexponent is reduced to problems in smaller dimensions by a\ncommon matrix factorization [17]. Therefore, in what follows\nwe additionally assume that the system is irreducible.\nMost of results on the stability of a linear switching\nsystems have been obtained without the dwell time assump-\ntion. Usually it is done either by constructing a Lyapunov\nfunction [3], [11], [12], [18], [21], or by approximating of\nthe trajectories [23], [29], [30]. The latter includes the dis-\ncretization approach, which is, of course, the most obvious\nway to analyse ODEs. Replacing the derivative ˙x(t)by the\ndivided differencex(t+h)−x(t)\nh, we get the Euler piecewise-\nlinear approximation. Another way of discretization is to\nreplace the switching law by a piecewise-constant function\nwith intervals being multiples of a given step size h > 0.\nSolving the corresponding ODE in each interval we obtain\nx(t+h) =ehAjx(t), which gives a piecewise-exponential\napproximation of the trajectory. Both of those methods lead to\na discrete time switching system of the form yk+1=B(k)yk,\nk≥0. For the Euler discretization, the control set Bconsists of\nmatrices Bj=I+hAj, for the second approach, Bj=ehAj.\nThe Lyapunov exponent of the discrete system (we denote\nit by σh) can be efficiently computed by evaluating the joint\nspectral radius of the matrix family B[1]. The recent progress\nin the joint spectral radius problem [13], [19] allows us to\ncalculate it with a good accuracy or, in most cases, even\nto find it precisely. Thus, the Lyapunov exponent σ(A)of\nsystem (1) can be efficiently computed, provided it is close\nenough to the Lyapunov exponent σhof its discretization.\nThe crucial problem is to estimate the precision, i.e., thearXiv:2402.04795v1  [math.DS]  7 Feb 20242\ndifference |σ(A)−σh|, depending on the step size h.\nThe discretization method in the stability problem has drawn\nmuch attention in the recent literature and several estimates\nfor the precision |σ(A)−σh|have been obtained [12], [28],\n[29], [30]. For both aforementioned methods, it is linear in h,\ni.e.,|σ(A)−σh| ≤Ch, where, for the constant C, usually\nonly rough upper bounds are known.\nB. Overview of the main results\nWe establish lower and upper bounds which localize the\nLyapunov exponent of the system (1) to an interval of length\nat most C h2, and moreover, show that Ccan be explicitly\nfound.\nThe main result, Theorem 5, states that the second method\nof discretization, when Bj=ehAj, leads to the double\ninequality σh≤σ(A)≤σh+C h2, where the constant Cis\nexpressed by means of the so-called extremal multinorm . This\nmultinorm is constructed simultaneously with the computation\nofσh.\nThe estimate from Theorem 5 gives a very precise method of\ncomputation of the Lyapunov exponent σ(A). The numerical\nresults in dimensions d≤9are given in Section 6. In\ndimension d= 6the precision is usually between 0.1and0.15.\nThe computation time on our PC (5 cores, 3.6 GHz) is about\none hour. For positive systems, the method performs much\nbetter even in high dimensions. For d≤200, the precision\ndoes not exceed 0.01. For d≤100, the computation takes a\nfew seconds.\nThe use of the new estimate, however, is complicated by\nthe fact that the stability of a discrete system under the dwell\ntime constraint cannot be analysed by the traditional scheme.\nFor such systems, the Lyapunov function may not exist at\nall [10]. The computation of the Lyapunov exponent requires\nthe concept of restricted orMarkovian joint spectral radius [6],\n[15], [24], [26], which are special cases of the recent theory\nof dynamical system on graphs developed in [10], [22], [25].\nThat is why we need to do preliminary work to introduce the\nsystem on graphs and the concept of Lyapunov multinorm.\nRemark 1: Our estimates for the approximation rate do not\ninclude the number of matrices, which makes them applicable\nfor arbitrary compact control set A. They are also easily\ngeneralized to mode-dependent constraints on the dwell time,\nwhen mis a function of the matrix A∈ A. Finally, the results\ncan be extended to mixed (discrete-continuous) systems with\nhybrid control. In particular, when every switching from the\nregime AitoAjis realized by a given linear operator Eji\nthat can be different from emAj.\nC. The structure of the paper\nIn Section II we introduce auxiliary facts and notation\nsuch as Markovian joint spectral radius, dynamical systems,\nand Lyapunov multinorms. The fundamental theorem and\ncorollaries are formulated and discussed in Sections III, the\nproofs are given in Section IV. Sections V and VI present the\nalgorithm and analyse numerical properties.\nThroughout the paper we denote vectors by bold letters, Iis\nthe identity matrix, ρ(X)is the spectral radius of the matrix X,\nwhich is the maximum modulus of its eigenvalues.II. P RELIMINARY FACTS . DYNAMICAL SYSTEMS ON\nGRAPHS\nThe discretization with step size happroximates the sys-\ntem (1) with the dwell time parameter m > 0by the discrete-\ntime system yk+1=B(k)yk, where for each k≥0, the\nmatrix B(k)is either ehAjoremAj,Aj∈ A, see Definition 4\nbelow. The dwell time assumption imposes a firm restriction\nto the switching law: It must be a sequence of blocks of the\nform B(k+N)···B(k+ 1)B(k) = ( ehAj)NemAj, where\nj∈ {1, . . . , n }, the length N≥0depends on the block,\nand two neighbouring blocks must have different modes j.\nThus, each block has to begin with emAjfollowed by a power\nofehAj, and then switch to the next block with a different j.\nThe general theory of discrete systems with constraints on\nswitching laws has been developed in [6], [10], [15], [26] and\nthen extended to general dynamical systems on graphs. We\nbegin with necessary definitions and notation.\nConsider a directed graph Gwith vertices v1, . . . , v nand\nedges ℓji, which may include loops ℓii. Each vertex viis\nassociated to a finite-dimensional linear space Vi. To every\nedge ℓji, if it exists, we associate a linear operator Eji:Vi→\nVjand a positive number hji, which is referred to as the time\nof action ofEji.\nDefinition 2: The dynamical system on the graph Gis the\nequation xk+1=E(k)xk, on the sequence {xk}k≥0, where\nfor each k, the operator E(k)is chosen from the set {Eji}n\nj=1\nifxk∈Vi. The point xkcorresponds to the time tkand\ntk+1=tk+hji.\nThe usual discrete-time linear switching system xk+1=\nB(k)xk,B(k)∈ B ={Bj}n\nj=1, corresponds to the case\nwhen Gis a complete graph, Vjare all equal to Rd, every\nvertex vjhasnincoming edges ℓji, i= 1, . . . , n , with the\noperators Ejiequal to the same operator Bj, and all time\nintervals hjiare equal to 1.\nLet us have an arbitrary dynamical system on a graph.\nWe assume that each space Vjis equipped with a certain\nnorm ∥·∥j. The collection of those norms {∥·∥ j}n\nj=1is\ncalled a multinorm . We use the short notation ∥x∥meaning\nthat∥x∥=∥x∥jforx∈Vj.\nEvery trajectory {xk}∞\nk=0of the system on Gcorresponds\nto an infinite path vi0→vi1→ ··· , where x0∈Vi0is a\nstarting point and xk+1=Eik+1ikxk,k≥0. Thus, xk∈Vik\nfor every k. Every point xkcorresponds to the time tk=Pk\ns=1hisis−1, which is the total time of the way from vi0\ntovik.\nThe (Markovian) joint spectral radius ˆρ= ˆρ(A)of the\nsystem is\nˆρ= lim\nk→∞max\n{xs}k\ns=0∥xk∥1/tk,\nwhere the maximum is computed over all trajectories {xs}s≥0\nwith∥x0∥= 1. Thus, for every trajectory, we have ∥xk∥ ≤\nCˆρtk,k∈N. The joint spectral radius is the rate of the\nfastest growth of trajectories. See [10] for the correctness of\nthe definition and for basic properties of this notion.\nLet us now consider an arbitrary cycle of the graph G:\nvi0→vi1→ ··· → vik=vi0. Denote by Πthe product of\noperators Eikik−1···Ei1i0along the cycle. We have Πx0=3\nxk. For the spectral radius ρ(Π), which is the maximal\nmodulus of eigenvalues of Π, we have [ρ(Π)]1/tk≤ˆρ. Indeed,\nthe left-hand side is the rate of growth of a periodic trajectory\ngoing along that cycle, and it does not exceed the maximal rate\nof growth ˆρover all trajectories. On the other hand, there are\ncycles for which the left-hand side is arbitrarily close to ˆρ[10],\n[15].\nDefinition 3: A multinorm is called extremal for a system\non a graph if for each i, j∈ {1, . . . , n }and for every x∈Vi,\nwe have ∥Ejix∥j≤ˆρhji∥x∥i, provided the edge ℓjiexists.\nFor the extremal multinorm, for every trajectory, we have\n∥xk∥ ≤ˆρtk∥x0∥,k∈N.\nThe extremal multinorm always exists, provided the system\nis irreducible. The reducibility means the existence of sub-\nspaces V′\nj⊂Vj,i= 1, . . . , n , where at least one subspace\nis nontrivial and at least one inclusion is strict, such that\nEjiV′\ni⊂V′\nj, whenever the edge ℓjiexists.\nTheorem A [10] An irreducible dynamical system on a graph\npossesses an extremal multinorm .\nTheorem A implies, in particular, that for every irreducible\nsystem, we have ˆρ >0. Otherwise, Ejix= 0 for all x∈Vi,\ni.e., all Ejiare equal to zero, in which case the system is\nclearly reducible. Therefore, one can always normalize the\noperators as ˜Eji= ˆρ−hjiEji, after which the new dynamical\nsystem (on the same graph and with the same time intervals)\nhas the joint spectral radius equal to one. Moreover, it pos-\nsesses the same extremal norm, for which ∥˜Ejix∥j≤ ∥x∥i,\nhence, for every trajectory ˜xk, the sequence ∥˜xk∥is non-\nincreasing in k.\nNow we are going to define the discretization of the dwell\ntime constrained system (1) and present it as a system on a\nsuitable graph. Then we apply Theorem A and use an extremal\nmultinorm to estimate the approximation rate.\nDefinition 4: Theh-discretization of the system Awith step\nsizehis a system Ahon a complete graph with nvertices, in\nwhich Vj=Rd,Ejj=ehAj,hjj=hfor all j= 1, . . . , n ,\nandEji=emAj,hji=mfor all pairs (i, j),i̸=j.\nThus, the h-discretization Ahis a dynamical system on a\ncomplete graph, where every vertex vjhasn−1incoming\nedges from all other vertices, all associated to the opera-\ntoremAjwith the time of action m, and also has a loop\nassociated to ehAjwith time of action h. Respectively, a\nmultinorm ∥·∥ ={∥·∥ j}n\nj=1can now be interpreted as a\ncollection of norms in Rd, each associated to the correspond-\ning regime Aj. See Figure 1 for an example of a system A\nwith three matrices.\nTheh-discretization can also be presented as a discrete-time\nlinear switching system in Rd:xk+1=B(k)xk, k≥0, where\nthe sequence B(k)has values from the control set {emAj,\nehAj}n\nj=1with the following restriction:\nEvery element B(k)equal to emAjorehAjcan be followed\nby either ehAjoremAs,s̸=j.\nThe operators emAj, ehAjact during the time intervals\nof lengths mandhrespectively. We denote by ˆρ(Ah)the\n(Markovian) joint spectral radius of the system Ah. By The-\norem A, if the family Ais irreducible, then for every h\nitsh-discretization possesses an extremal norm. Note that this\nnorm can be different for different h.V1V2\nV3emA\n1emA\n2\nemA1\nemA3emA2\nemA3\nehA1ehA2\nehA3\nFig. 1. The dynamical system on graph, n= 3.\nIII. T HE FUNDAMENTAL THEOREM\nNow we are formulating the main result. We consider a\nlinear switching system Agiven by (1) and its h-discretiza-\ntionAh. The joint spectral radius of Ahis denoted by ˆρ(Ah).\nTheorem 5: LetAbe an irreducible continuous-time linear\nswitching system with the dwell time constraint m. Then for\nevery discretization step h >0, we have\nσh≤σ(A)≤σh−1\nmln\u0012\n1−∥(A −σhI)2∥\n8h2\u0013\n,(2)\nwhere σh= ln ˆ ρ(Ah),∥·∥ ={∥·∥ j}n\nj=1is an extremal\nmultinorm of Ah, and\n∥(A −σhI)2∥= max\nj=1,...,n∥(Aj−σhI)2∥j.\nRemark 6: The estimate (2)uses two values: the joint\nspectral radius of the h-discretization ˆρ(Ah)and the operator\nnorms of (Aj−σhI)2in the jth component of the extremal\nmultinorm {∥·∥ j}forAh. They are both found by the invari-\nant polytope algorithm [10]. We give a brief description in\nSection 5.\nWriting the Taylor expansion of the logarithm up to the\nsecond order, we obtain the following\nCorollary 7: Under the assumptions of Theorem 5, we have\nσh≤σ(A)≤σh+∥(A −σhI)2∥\n8mh2+O(h4),ash→0.\n(3)\nRemark 8: If h=m, then we have basically the discretiza-\ntion of an unrestricted system, and (3)becomes\nσh≤σ(A)≤σh+∥(A −σhI)2∥\n8h+O(h3),ash→0,\nwhich again reveals the linear dependence on the discretiza-\ntion step hfor unrestricted systems.\nRemark 9: If ||·|| is an approximation of an extremal\nmultinorm of Ahup to a factor of 1 +εi.e. (in the notation\nof Definition 3)\nˆρhji∥x∥i≤ ∥Ejix∥j≤(1 +ε)ˆρhji∥x∥i,\nthen (3)becomes\nσ−\nh≤σ(A)≤σ+\nh−1\nmln\u0012\n1−∥(A −σ−\nhI)2∥\n8h2\u0013\n,\nwhere σ−\nh= ln ˆρ(Ah), and σ+\nh= ln(1 + ε)ˆρ(Ah).4\nThe quadratic rate of approximation in formula (3) is quite\nunexpected since the trajectories of the continuous-time sys-\ntem are approximated by the trajectories of its h-discretization\nonly with the linear rate. Nevertheless, the approximation of\nthe Lyapunov exponent is quadratic.\nRemark 10: The performance of the estimate (2)can be\nspoiled in two cases: either the dwell time mis too small,\nor the operator norm of (Aj−σhI)2is too large. Note also\nthat (2)is an a posteriori estimate since the operator norm\ndepends on hand is not known in advance.\nTheorem 5 yields the following stability conditions for\nsystem (1):\nCorollary 11: If an h-discretization Ahis unstable, then\nso is the system A. IfAhis stable and ˆρ≤\u0000\n1−\nh2∥(A−(lnρ)I)2∥\n8\u00011/m, then Ais stable.\nProof: The first inequality in (2) implies that if ˆρ >1\nand hence σh>0, then σ(A)>0. The converse is established\nsimilarly.\nIV. P ROOFS OF THE MAIN RESULTS\nThe proof of Theorem 5 is based on a simple geometrical\nargument. If a curve connects two ends of a segment of\nlength h, then the distance from this curve to the segment\ndoes not exceed h2multiplied by the curvature and by a\ncertain constant. The nontrivial moment is that we need this\nproperty in an arbitrary norm in Rdand need to evaluate the\nconstant depending on this norm. Then we apply this fact to\neach component ∥·∥jof the extremal norm of the system Ah\nand estimate the growth of trajectory of the system A.\nLemma 12: Let∥·∥ be an arbitrary norm in Rdandx:\n[0, h]→Rdbe a C2-curve. Then, for every τ∈[0, h], the\ndistance from the point x(τ)to the segment [x(0),x(h)]does\nnot exceedh2\n8∥¨x∥C[0,h].\nProof: Denote by ∥·∥∗the dual norm in Rd, thus∥y∥∗=\nsup∥x∥=1(y,x). Let x(0) = 0andx(h) =a. Let x(ξ)be\nthe most distant point of the arc {x(τ), τ∈[0, h]}to the\nsegment [0,a]and let this maximal distance be equal to r.\nSuppose yis the closest to x(ξ)point of that segment and\ndenote s=x(ξ)−y. Thus, ∥s∥=r. The segment [0,a]\ndoes not intersect the interior of the ball of radius rcentred\natx(ξ). Therefore, by the convex separation theorem, there\nexists a linear functional p∈Rd,∥p∥∗= 1, which is non-\npositive on that segment, non-negative on the ball, and such\nthat(p,s) =∥s∥. Since the point ybelongs to the ball, it\nfollows that (p,y) = 0 . We have (p,s) = (p,x(ξ))−(p,y) =\n(p,x(ξ)), therefore, (p,x(ξ)) =r, see Figure 2.\nDefining the function f(t) =\u0000\np,x(t)\u0001\nwe obtain\nf(ξ)−f(0) =\u0000\np,x(ξ)−x(0)\u0001\n=\u0000\np,x(ξ)\u0001\n=r. (4)\nWithout loss of generality we assume that ξ≤1\n2h, otherwise\none can interchange the ends of the segment [0, h]. The Taylor\nexpansion of fat the point ξgives f(t) =f(ξ) +f′(ξ)(t−\nξ) +1\n2f′(η)(t−ξ)2, where η∈[t, ξ]. The maximum of f(t)\nis attained at t=ξ, hence, f′(ξ) = 0 . For t= 0, this yields\nf(0) = f(ξ) +1\n2f′′(η)ξ2. Combining with (4), we obtain\nr=−1\n2f′′(η)ξ2=1\n2\u0000\np,−¨x(η)\u0001\nξ2≤1\n2∥¨x∥ξ2≤h2\n8∥¨x∥.s\nx(0) = 0yx(ξ)\nx(h) =ap≤0p≥0\nr≤h2\n8∥¨x∥\nFig. 2. Construction in proof of Lemma 12.\nwhich completes the proof.\nTheorem 13: Letx(t)be a solution of the differential\nequation ˙x=Axwith a constant d×dmatrix A,∥·∥ be an\narbitrary norm in Rd, and h∈\u0000\n0,p\n8/∥A2∥\u0001\nbe a number.\nThen for every τ∈[0, h], we have\n\r\rx(τ)\r\r≤1\n1−h2\n8∥A2∥maxn\n∥x(0)∥,∥x(h)∥o\n.(5)\nProof: It suffices to consider the vector x(τ)with\nthe maximal norm over all τ∈[0, h]. Since ¨x=A˙x=\nA2x, it follows that ∥¨x∥C[0,h]= max t∈[0,h]∥A2x(t)∥ ≤\n∥A2∥ · ∥x(τ)∥. Hence, by Lemma 12, the distance from the\npoint x(τ)to the closest point yof the segment [x(0),x(h)]\ndoes not exceedh2\n8∥A2∥ · ∥x(τ)∥. On the other hand, this\ndistance is not less than ∥x(τ)∥ − ∥ y∥. It remains to note\nthat∥y∥ ≤ max{∥x(0)∥,∥x(h)∥}, which follows from the\nconvexity of the norm. Thus,\nh2\n8∥A2∥ · ∥x(τ)∥ ≥ ∥ x(τ)∥ −maxn\n∥x(0)∥,∥x(h)∥o\n.\nExpressing ∥x(τ)∥we arrive at (5).\nProof: [Proof of Theorem 5] The lower bound fol-\nlows trivially. To prove the upper bound, we first assume\nthatˆρ(Ah) = 1 . By Theorem A, the system Ahpossesses\nan extremal multinorm ∥·∥={∥·∥ j}n\nj=1. For each j≤n,\nwe denote αj=−1\nmln\u0010\n1−∥A2\nj∥j\n8h2\u0011\n. Let us show that for\nevery i̸=jand for every point z0∈Vi, the trajectory z(t)\ngenerated in Vjby the ODE ˙z=Ajzand starting at z0\npossesses the property: ∥z(t)∥j≤eαjt∥z0∥jfor all t≥m.\nSince the multinorm is extremal, it follows that for every in-\nteger k≥0, one has\r\rz(m+kh)\r\r\nj=\r\remAj\u0000\nehAj\u0001kz0\r\r\nj≤\n∥z0∥j. Let kbe the maximal integer such that t≥m+kh.\nThus, t=m+kh+τ,τ∈[0, h). Consider the arc of the\ntrajectory z(·)on the time interval [m+kh, m +(k+1)h]and\ndenote x(0) = z(m+kh),x(h) =z(m+kh+h). Observe\nthat both ∥x(0)∥and∥x(h)∥do not exceed ∥z0∥. Applying\nTheorem 13 we obtain\n∥z(t)∥j=∥x(τ)∥j≤1\n1−h2\n8∥A2\nj∥j∥z0∥j\n=eαjm∥z0∥j≤eαjt∥z0∥j.\nUsing the multinorm notation and setting α= maxn\nj=1αj,\nwe conclude that for every trajectory z(·)generated by one\nregime it holds that ∥z(t)∥ ≤eαt∥z(0)∥for all t≥m.\nConsider now an arbitrary trajectory x(·)of the system A.\nIf it does not have switches, then it is generated by some5\nregime Aj, and the proof follows immediately from the in-\nequality above. Let it have the switching points t0< t1< . . . .\nThis set can be infinite or finite. Applying the inequality above\nto arbitrary t∈[tk, tk+1]and denoting z(0) = x(tk), we\nobtain ∥x(t)∥ ≤eα(t−tk)∥x(tk)∥. Now take a time interval\n[0, T]and let tNbe the largest switching point on it. Applying\nour inequality successively for all switching intervals, we get\n∥x(T)∥ ≤eα(t−tN)N−1Y\nk=0eα(tk+1−tk)∥x(t0)∥\n≤eα(T−t0)∥x(t0)∥.\nThus, for every trajectory x(·), we have ∥x(T)∥=O(eαt)as\nT→ ∞ . Let us remember that the norm ∥·∥ can be different\non different switching intervals. Nevertheless, they all belong\nto the finite set of norms {∥·∥ j}n\nj=1which are all equivalent.\nTherefore, σ(A)≤α.\nThis concludes the proof for the case ˆρ= 1. The general\ncase follows from this one by normalization: We replace the\nfamily Aby˜A=A −σhI. Then ˆρ(eh˜A) = 1 andσ(˜A) =\nσ(A)−σh. Finally, applying the theorem for the system ˜A\nand substituting to (2), we complete the proof.\nNow, to put Theorem 5 into practise, we need to compute\nthe value ˆρ(Ah)and construct an extremal multinorm. We will\ndo it in Section 5 for a general system on a graph.\nV. C OMPUTING THE JOINT SPECTRAL RADIUS AND AN\nEXTREMAL MULTINORM FOR A SYSTEM ON A GRAPH\nTheorem 5 gives a recipe to compute the Lyapunov expo-\nnent of a linear switching system (1) with sufficiently high\naccuracy. Due to the quadratic dependence of hin (3) one\ncan make the distance between the upper and lower bounds\nsmall by choosing an appropriate discretization step h. This\nplan requires solving two problems: (i) Compute the value\nofˆρ(Ah). (ii) Construct an extremal multinorm for Ah. Both\nare solved simultaneously by the invariant polytope algorithm\n(ipa) derived in [13]. The ipa computes the joint spectral radius\nof several matrices by constructing an extremal norm. In [10]\nthe invariant polytope algorithm was extended to discrete time\nsystems with restrictions and to systems of graphs. We present\nthe main idea of the algorithm in this section; for details\nsee [13], [19], [10], [14]. The reference implementation of\nthe algorithm can be found at gitlab.com/tommsch/ttoolboxes .\nThe (Markovian) invariant polytope algorithm ( ipa) finds\nthe joint spectral radius ˆρ(Ah)and an extremal multinorm. It\nconsists of two steps.\na) Step 1.: We fix a number N(not very large) and\nexhaust all cycles vi0→vi1→ ··· → viL=vi0ofG\nof length at most N. To each cycle we denote by Π =\nEiLiL−1···Ei1i0the product of the linear operators associated\nto its edges and by T=PL\nk=1hikik−1its total time. Recall\nthatEjj=ehAj,hjj=hfor all jandEji=emAj,hji=m\nfor all pairs (i, j),i̸=j.\nWe choose a cycle with the biggest value r=ρ(Π)1/Tand\ncall it leading cycle and respectively leading product Π. We\nset˜Aj=Aj−(lnr)I,j= 1, . . . , n . For the new system ˜Ah,\nwe have ρ(˜Π) = ρ(˜EiLiL−1···˜Ei1i0) = 1 .In many cases to find the leading cycle we can avoid\nthe exhaustion and use the auxiliary Algorithm 18. It is an\nadaptation of the modified Gripenberg algorithm from [19] to\nour setting. Some of its numerical properties are assessed in\nAppendix A.\nFor the sake of simplicity of exposition, we assume that the\nleading eigenvector of ˜Πis positive and thus equal to one. The\ngeneral case is considered similarly, see [10], [20].\nDenoting by x0the leading eigenvector of ˜Π, it follows\nthatx0→x1→ ··· → xL−1→x0is the periodic trajectory\ncorresponding to that cycle.\nb) Step 2.: We try to prove that actually ˆρ(Ah) =rand,\nif so, to find an extremal norm.\nFor every j, we denote by V(0)\njthe set of points x(s),\ns= 0, . . . , L −1that belong to Vj. If there are no such\npoints, then V(0)\nj=∅. Suppose after kiterations we have\nfinite sets V(k)\nj⊂Vj,j= 1, . . . , n . Denote by cosV(j)\njthe\nconvex hull of the set V(k)\nj∪(−V(k)\nj). Now for every j, we\nadd to V(k)\njall points of the sets em˜AjV(k)\nsfor all s̸=jand\nof the set eh˜AjV(k)\nj. We add only those points that do not\nbelong to cosV(j)\nj, the others are redundant and we discard\nthem. This way we obtain the sets V(k+1)\nj ,j= 1, . . . , n .\nWe do this until V(k+1)\nj =V(k)\njfor all j, in which case the\nalgorithm terminates. We conclude that ˆρ(Ah) =ρ(Π)1/Tand\nthe Minkowski norms of the polytopes {cosV(j)\nj}n\nj=1form an\nextremal multinorm for Ah.\nRemark 14: To find the Lyapunov exponent σ(A), we need\nto compute the operator norms ∥Aj−σhI∥j. Since the unit\nballs of the norms ||·||jare polytopes, this can be done\nefficiently by solving an LP problem [13].\nTo achieve a good precision one needs to choose an ap-\npropriate step size hwhich, however, cannot be too small.\nOtherwise, the matrices ehAjwill be close to the identity\ncomplicating the computation of the joint spectral radius [12].\nIn particular, the length of the leading cycle may get too\nlarge to be handled efficiently or cannot be found at all, see\nExample 17. In most cases hcannot be chosen less than 0.1\n(as from our numerical tests).\nA. Positive systems\nA continuous-time linear switching system is called positive\nif every trajectory x(t)starting in the positive orthant Rd\n+\nremains in Rd\n+for all t. Positive systems have been studied\nwidely in literature due to many applications.\nThe positivity of the system is equivalent to that all the\nmatrices Ajare Metzler, i.e. all off-diagonal entries are\nnonnegative. In this case all the matrices ehAj, emAjof the\ndiscrete-time system Ahare nonnegative. Hence, the matrix Π\nin the invariant polytope algorithm is also nonnegative, and\ntherefore, the Perron-Frobenius theorem implies the nonneg-\nativity of the leading eigenvector x0. Consequently, all the\nsetsV(k)\njare nonnegative, i.e., the algorithm runs entirely\ninRd\n+. It is then possible to replace the polytopes cosV(j)\nj\nby the positive polytopes co+V(j)\nj={x≥0 :x≤cosV(j)\nj},\nwhere the inequalities are understood element wise.6\nTABLE I\nCOMPUTATION OF THE LYAPUNOV EXPONENT FOR ARBITRARY\nSYSTEMS AND FOR POSITIVE SYSTEMS\nRandom matrices\ndim ub −lb time\n2 0 .016512 4 s\n3 0 .020846 58 s\n4 0 .036887 780 s\n5 0 .108899 2300 s\n6 0 .133852 4000 s\n7 0 .234899 3300 s\n8 0 .314734 4900 s\n9 0 .409434 3500 sRandom Metzler matrices\ndim ub −lb time\n2 0 .018851 1 s\n5 0 .008385 1 s\n13 0 .007760 2 s\n34 0 .007225 4 s\n89 0 .009330 14 s\n144 0 .006306 41 s\n233 0 .005352 150 s\n377 0 .005544 560 s\nFig. 3. Polytopes from Example 16.\nSince the positive polytopes Q(k)\njare in general much larger\nthan P(k)\nj, they absorb more points in each iteration. This\nreduces significantly the number of vertices of the polytopes\nand, respectively, the complexity of each iteration. In fact, this\nmodification of the invariant polytope algorithm for positive\nsystems works very efficiently even for large dimensions d.\nVI. N UMERICAL RESULTS\nWe demonstrate the performance of estimate (2) from The-\norem 5 to the computation of the Lyapunov exponent. Given\npairs of random matrices and random dwell time m∈(0,1),\nTable I presents the performance of the Markovian ipa. For\neach dimension d, we conducted about 15 tests and we give\nthe median of the best achieved accuracies i.e. the maximal\nlower bound from Equation (3). As test matrices we used (lhs)\nmatrices with normally distributed entries, and (rhs)Metzler\nmatrices with random integer entries in [−9,9]. To make the\nexamples more interesting, we omit simple cases when one\nmatrix dominates others (which often occurs). To this end, the\nmatrices got normalized such that the 2-norm is equal to 1.\nOne can see that the Markovian ipa can compute bounds\nin reasonable time (although the needed time depends very\nstrongly on the matrices) up to dimension ≈10for general\nmatrices, and up to dimension ≈500for Metzler matrices.\nRemark 15: The scripts used to obtain the experimen-\ntal results can be found at gitlab.com/tommsch/ttoolboxes/-\n/tree/master/demo/dwelltime .\nA. Examples\nWe begin with a simple two-dimensional example illustrat-\ning the Lyapunov exponent computation by the estimates of\nTheorem 5 and the Markovian ipa (Section 5).TABLE II\nRESULTS FOR EXAMPLE 17: THE BOUNDS FOR THE LYAPUNOV\nEXPONENT AND THE LEADING CYCLES DEPENDING ON h.\nh lb ub leading cycle\n0.400 0 .0762 13 .2624 (1 .30; 1.70)\n0.330 0 .0698 8 .6828 (1 .16; 1.82; 1.49; 1.49)\n0.300 0 .0751 7 .1247 (1 .40; 1.70)\n0.250 0 .0742 4 .7571 (1 .25; 1.75)\n0.200 0 .0762 3 .0066 (1 .30; 1.70)\n0.125 0 .0762 1 .1888 (1 .25; 1.625)\n0.100 0 .0000 0 .7298 ?\n0.050 0 .0000 0 .3659 ?\n0.040 0 .0000 0 .3796 ?\n0.025 0 .0000 0 .3125 ?\nExample 16: Given dwell time m= 1, two matrices\nA1=1√\n2 + 2\u00140 0\n1 0\u0015\n, A 2=1√\n2 + 2\u0014−2−2\n−1−2\u0015\n,\nand discretization step h= 0.2. The product Π = ( emA 2)1\n(ehA2)7(emA 1)1(ehA1)180(emA 2)1(ehA2)7(emA 1)1(ehA1)181\n(emA 2)1(ehA2)7(emA 1)1(ehA1)183is a leading cycle. The\nipa gives ˆρ(Ah) = 1 .0331..., or respectively σh= 0.0325...,\nand polytopes P1, P2⊆R2such that e(m−σh)A1P2⊆P1,\ne(h−σh)A1P1⊆P1,e(h−σh)A2P2⊆P2, and e(m−σh)A2P1⊆\nP2.\nThe relevant norms compute to ∥(A1−σhI)2∥P1=\n0.0083..., and ∥(A2−σhI)2∥P2= 2.8361.... Summing up\nwe obtain the bounds 0.0325 < σ(A)<0.0469 .\nIn Figure 3 the polytopes P1(blue-thick-dashed line) and\nP2(red-thick-dot-dashed line) are plotted, as well the images\nunder the operators e(m−σh)A1, . . . (thin-blue line and thin-red-\ndot-dashed line).\nExample 17: Let\nA1=\n−1−1 1 −1\n1−1−1−1\n1 1 −1−1\n1−1 1 −1\n, A2=\n−1−1−1−1\n1−1 1 1\n−1 1 −1−1\n1−1 1 1\n,\nand be the dwell time m= 0.5.\nIn Table II we report the discretization length (h), the lower\nbound σhofσ(A)given in (2) (lb), the upper bound σh−\n1\nmln\u0010\n1−∥(A−σhI)2∥\n8h2\u0011\nofσ(A)(ub), and a leading cycle ,\nwhich is a periodic switching law of the discretized system\nthat produces the fastest growth of trajectories. The leading\ncycle is denoted as (a1\n1;a2\n1;. . .;a1\nk;a2\nk), where a1\n·anda2\n·\nare the time of actions of the modes A1andA2respectively.\nFor example, the leading cycle (1.3; 1.7)in the first line of\nTable II (the case h= 0.4) is the mode A2acting the time 1.3\nfollowed by the time 1.7of the mode A1. From the table we\nsee that the leading cycle seems to stabilize around (1.3; 1.7)\nashincreases.\nThe character “?” denotes that no leading cycle could\nbe found. On can see, that for too small values of h, the\nleading cycle cannot be determined, and thus, the lower bound\nbecomes meaningless. Nevertheless, small discretization steps\nmay still give good upper bounds.7\nAPPENDIX\nA. Markovian modified Gripenberg algorithm\nThe Markovian modified Gripenberg algorithm, is an adap-\ntion of the modified Gripenberg algorithm [19]. The main\ndifference is that in each iteration, still, all possible products\nhave to be considered, but just admissible cycles (w.r.t. to the\ngraph) are used to compute the intermediate bounds.\nAlgorithm 18 (Markovian modified Gripenberg Algorithm):\nThe algorithm searches for a leading cycle Πmaxof a graph\nGand linear operators E.\nρmax= 0,Π0=I\nfork= 1, . . . , K\nEk= Π k−1× E // all possible products in iteration k\nρk= max {ρ(E)1/T(E):E∈ EkandEis a cycle }\n// where T(E)is the time of action of E\nΠk={E∈ Ek:ρ(E)1/T(E)=ρk}\nifρk> ρ maxthen\nΠmax= Π k, ρ max=ρk\nelse\nΠmax= Π k∪Πmax\nΠk= Π+\nk∪Π−\nk={E∈ Ek:||E||1/T(E)≥ρ+\nk} ∪\n{E∈ Ek:||M||1/T(M)≤ρ−\nk}\n// where ρ±\nk≥ρmaxandρ±\nkare such that\f\f\fΠ±\nk\f\f\f=N/2\nreturn Πmax, ρmax\nExample 19: Given pairs of matrices of various dimensions,\nrandom dwell time m∈(0,1)and random discretization\nsteph∈(0, m). Table III presents the performance of the\nMarkovian modified Gripenberg algorithm (mg) compared to\na brute force method (bf). For each dimension dwe conducted\n15 tests and we report in how many cases the Gripenberg\nlike algorithm, or the brute force algorithm worked better\nFurthermore we give the average runtime of the algorithm\n(Note though that the Gripenberg like algorithm in general\nfound the final result after approximately 2s).\nAs test matrices we used (a)matrices with random normally\ndistributed entries, and (b)Metzler matrices with random\ninteger entries in [−9,9], always normalized such that the\n2-norm equals 1.\nOne can see that the Gripenberg like algorithm performs in\ngeneral faster and better than a brute force method.\nACKNOWLEDGMENT\nREFERENCES\n[1] N. Barabanov, Lyapunov indicators of discrete inclusions i–iii , Autom.\nRemote Control, 49 (1988), 152–157.\n[2] C. Basso, Switch-mode power supplies spice simulations and practical\ndesigns , McGraw-Hill, Inc., New York, NY , USA, 1 edition, 2008.\n[3] F. Blanchini and S. Miani, A new class of universal Lyapunov functions\nfor the control of uncertain linear systems , IEEE Trans. Automat.\nControl, 44 (1999) 3, 641–647.\n[4] F. Blanchini, D. Casagrande and S. Miani, Modal and transition\ndwell time computation in switching systems: a set-theoretic approach ,\nAutomatica J. IFAC, 46 (2010) 9, 1477–1482.TABLE III\nRESULTS FOR EXAMPLE 19: A SSESSMENT OF MARKOVIAN MODIFIED\nGRIPENBERG ALGORITHM\n(a)Random Gaussian matrices\ndim mg better bf better tmg tbf\n2 33% 7% 2 .5s13.6s\n3 20% 0% 2 .5s12.7s\n4 33% 7% 2 .4s16.4s\n5 13% 13% 2 .6s16.1s\n7 13% 0% 2 .6s12.9s\n8 20% 0% 2 .6s15.7s\n11 20% 0% 2 .6s13.3s\n13 0% 0% 2 .6s15.9s\n17 0% 0% 2 .6s14.0s\n21 13% 0% 2 .6s17.2s\n(b)Random Metzler matrices\ndim mg better bf better tmg tbf\n2 0% 0% 4 .2s27.3s\n3 0% 0% 2 .6s27.5s\n4 40% 0% 2 .6s18.6s\n5 0% 0% 2 .6s26.9s\n7 20% 0% 2 .6s17.6s\n8 40% 0% 2 .5s17.2s\n11 20% 0% 2 .6s13.5s\n13 0% 0% 2 .6s13.4s\n17 40% 0% 2 .5s13.7s\n21 20% 0% 2 .6s14.2s\n[5] C. Briat and A. Seuret, Affine characterizations of minimal and mode-\ndependent dwell times for uncertain linear switched systems , IEEE\nTrans. Automat. Control 58 (2013) 5, 1304–1310.\n[6] X. Dai, Robust periodic stability implies uniform exponential stability\nof Markovian jump linear systems and random linear ordinary differ-\nential equations , J. Franklin Inst., 351 (2014), 2910–2937.\n[7] G. Chesi and P. Colaneri. Homogeneous rational Lyapunov functions\nfor performance analysis of switched systems with arbitrary switching\nand dwell time constraints , IEEE Trans. Automat. Control, 62 (2017)\n10, 5124–5137.\n[8] G. Chesi, P. Colaneri, J. C. Geromel, R. Middleton, and R. Shorten. A\nnonconservative LMI condition for stability of switched systems with\nguaranteed dwell time , IEEE Trans. Automat. Control, 57 (2012) 5,\n1297–1302.\n[9] Y . Chitour, N. Guglielmi, V . Yu. Protasov, and M. Sigalotti, Switching\nsystems with dwell time: computing the maximal Lyapunov exponent ,\nNonlinear Anal. Hybrid Syst. 40 (2021), Paper No. 101021.\n[10] A. Cicone, N. Guglielmi, and V . Y . Protasov, Linear switched\ndynamical systems on graphs , Nonlinear Anal. Hybrid Syst., 29 (2018),\n165–186.\n[11] J.C. Geromel and P. Colaneri, Stability and stabilization of continuous-\ntime switched linear systems , SIAM J. Control Optim. 45 (2006) 5,\n1915–1930.\n[12] N. Guglielmi, L. Laglia, and V . Yu. Protasov, Polytope Lyapunov\nfunctions for stable and for stabilizable LSS . Found. Comput. Math.,\n17 (2017) 2, 567–623.\n[13] N. Guglielmi and V . Yu. Protasov, Exact computation of joint spectral\ncharacteristics of linear operators , Found. Comput. Math., 13 (2013),\n37–97.\n[14] V . Yu. Protasov, and R. Kamalov, Stability of Continuous Time Linear\nSystems with Bounded Switching Intervals , SIAM J. Control Optimiz.,\n61 (2023) 5, 3051–3075.\n[15] V . Kozyakin, The Berger-Wang formula for the Markovian joint\nspectral radius , Linear Alg. Appl., 448 (2014), 315–328.\n[16] T. Kr ¨oger, On-line trajectory generation in robotic systems: basic\nconcepts for instantaneous reactions to unforeseen (sensor) events ,\nSpringer Berlin Heidelberg, Berlin, Heidelberg, 2010.\n[17] D. Liberzon, Switching in systems and control , Systems & Control:\nFoundations & Applications. Birkh ¨auser Boston, Inc., Boston, MA,\n2003.\n[18] D. Liberzon and A. S. Morse, Basic problems in stability and design of\nswitched systems , IEEE Control Systems Magazine, 19 (1999), 59–70.\n[19] T. Mejstrik, Improved invariant polytope algorithm and applications ,8\nACM Trans. Math. Softw., ACM Transactions on Mathematical Soft-\nware 46 (2020) 3, 1–26.\n[20] T. Mejstrik and V . Yu. Protasov, Elliptic polytopes and invariant norms\nof linear operators , Calcolo 60 (2023) 56.\n[21] A. Molchanov and Y . Pyatnitskiy, Criteria of asymptotic stability of\ndifferential and difference inclusions encountered in control theory ,\nSyst. Control Lett., 13 (1989), 59–64.\n[22] P. Pepe, Converse Lyapunov theorems for discrete-time switching\nsystems with given switches digraphs, IEEE Trans. Autom. Control,\n64 (2019) 6, 2502–2508.\n[23] A. Pietrus and V . Veliov, On the discretization of switched linear\nsystems , Syst. Cont. Letters, 58 (2009) 6, 395–399.\n[24] M. Philippe, R. Essick, R. Dullerud, and R. M. Jungers, Stability of\ndiscrete-time switching systems with constrained switching sequences ,\nAutomatica, 72 (2016), 242–250.\n[25] M. Philippe and R. M. Jungers, A sufficient condition for the bounded-\nness of matrix products accepted by an automaton , Proceedings of the\n18th International Conference on Hybrid Systems: Computation and\nControl, ACM New York, NY , USA (2015), 51–57.\n[26] M. Philippe, G. Millerioux, and R. M. Jungers, Deciding the bound-\nedness and dead-beat stability of constrained switching systems , Non-\nlinear Anal. Hybr. Syst., 23 (2017), 287–299.\n[27] V . Yu.Protasov, Generalized Markov-Bernstein inequalities and stability\nof dynamical systems , Proc. Steklov Inst., 319 (2022), 237–252.\n[28] V . Yu. Ptotasov and R. M. Jungers, Analysing the stability of linear\nsystems via exponential Chebyshev polynomials, IEEE Trans. Autom.\nControl 61 (2016) 3, 795–798.\n[29] F. Rossi, P. Colaneri, and R. Shorten, Pad´e discretization for linear\nsystems with polyhedral lyapunov functions , IEEE Trans. Autom.\nControl, 56 (2011) 11, 2717–2722.\n[30] R. Shorten, M. Corless, S. Sajja and S. Solmaz, On Pad ´e approxima-\ntions, quadratic stability and discretization of switched linear systems ,\nSyst. & Contr. Letters, 60 (2011) 9, 683–689.\n[31] M. Souza, A. Fioravanti, and R. Shorten, Dwell-time control of\ncontinuous-time switched linear systems , In Proceedings of the 54th\nIEEE Conference on Decision and Control (2015), 4661–4666.\n[32] A. van der Schaft and H. Schumacher, An introduction to hybrid dy-\nnamical systems , vol. 251 of Lecture Notes in Control and Information\nSciences, Springer-Verlag London, Ltd., London, 2000.\n[33] W. Xiang, On equivalence of two stability criteria for continuous-time\nswitched systems with dwell time constraint , Automatica J. IFAC, 54\n(2015), 36–40."    },    {        "title": "2402.04807v1.Long_term_dynamics_around_the_Didymos_Dimorphos_binary_asteroid_of_boulders_ejected_after_the_DART_impact.pdf",        "content": "Astronomy &Astrophysics manuscript no. main ©ESO 2024\nFebruary 8, 2024\nLong term dynamics around the Didymos-Dimorphos binary\nasteroid of boulders ejected after the DART impact\nK. Langner1,2,3, F. Marzari1,2, A. Rossi1, and G. Zanotti4\n1IFAC-CNR, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy\n2Dipartimento di Fisica, Università di Padova, Italy\n3Institute Astronomical Observatory, Faculty of Physics, Adam Mickiewicz University, Pozna ´n, Poland\n4Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Milano, Italy\nFebruary 8, 2024\nABSTRACT\nContext. In 2022 the DART mission spacecraft impacted the asteroid Dimorphos, the secondary body of the binary Didymos system,\nejecting a large number of dust particles, rocks and boulders. The ESA Hera mission will reach the system in 2026 for post–impact\nstudies and possible detection of orbiting fragments.\nAims. We investigate the long term dynamics of the large boulders ejected by DART to test if any of these objects survive in orbit\nuntil the arrival of the Hera mission.\nMethods. To model the dynamics of the boulders we use a numerical model which includes the gravity of non-spherical Didymos\nand Dimorphos, the solar gravity and the radiation pressure. The SPICE kernels are used to define the correct reference frame for the\nintegrations.\nResults. The dynamics of the boulders is highly chaotic and 1% of the initial boulders survive at least for 4 years on quasi–stable\norbits. These orbits are characterised by wide oscillations in eccentricity in antiphase with those in inclination (including spin flips),\na mechanism similar to the Kozai one. This behaviour may protect these bodies from close encounters with both asteroids. We also\ncompute the distribution on the surfaces of the asteroids of sesquinary impacts which may influence the dust emission, after the initial\nDART impact, and the surface composition of the asteroids.\nConclusions. The probability of observing boulders by the mission Hera is small but not negligible and an almost constant flux of\nescaping boulders is expected in the coming years since their lifetime after the DART impact covers a large time interval. Most of\nre–impacts on Dimorphos occur in the hemisphere opposite to the impact site, preferentially close to the equatorial plane.\nKey words. Minor planets, asteroids: individual: Didymos /Minor planets, asteroids: individual: Dimorphos /Celestial mechanics\n1. Introduction\nOn September 26, 2022 the DART probe impacted Dimorphos,\nthe small moon of the binary 65803 Didymos asteroid system\n(Daly et al. 2023; Cheng et al. 2023). The event generated a large\nejecta plume that was imaged first be the small LICIACube cube-\nsat (Dotto et al. 2021; Capannolo et al. 2021), released by DART\n15 days before the impact, during its fly-by. The images revealed\na complex plume with a cone-like structure, crammed with dust,\nclumps of objects, filaments and larger boulders (Dotto & al.\n2023; Deshapriya et al. 2023). The event and the resulting ejecta\nwere also observed by several space and ground based observa-\ntories which allowed the characterisation of the plume evolution\nand, in particular, the formation of a long tail of debris, pushed\nby the solar radiation pressure, stretching for thousands of km\n(Li et al. 2023). Further observations by the Hubble Space Tele-\nscope revealed the presence of a large population of dozens of\nboulders moving in the surroundings of the system (Jewitt et al.\n2023). Assuming a geometric albedo of 0.15, these large objects\nspan a range of dimensions between about 4 to ≈7 m in diam-\neter. Jewitt and co-authors estimated sky-plane velocities in the\nrange between 10 cm /s to more than 67 cm /s, without a signif-\nicant correlation between the size and the velocity of the boul-\nders. Since the escape velocity from the binary system is about\n24 cm /s, these velocities are consistent with objects that do not\nre-impact Dimorphos right after the ejection and that can staywithin the system for a comparatively long time span on highly\nperturbed chaotic orbits.\nIn (Rossi et al. 2022) a theoretical analysis of the behaviour\nof these long survivors was performed highlighting the complex\ndynamical environment and the possible fate of the ejected large\nparticles (of the order of tens of cm). In this work we concentrate\non a synthetic population of objects similar in size to the ob-\nserved boulders. By means of a comprehensive model, described\nin Sect. 2, we explore the long term dynamics of this population\nwith the aim of characterising, also from the statistical point of\nview, its behaviour with respect to the possible end-states, in-\ncluding re-impact against the two asteroids or escape from the\nsystem.\nWithin the international collaboration for planetary defence,\nthe Hera probe, from the European Space Agency, will be\nlaunched in October 2024 and will reach the Didymos system\nin late 2026 (Michel et al. 2022). Hera will rendezvous the sys-\ntem and will perform a detailed in-situ analysis of the asteroids\nto fully characterise the outcomes of the DART impact. The pos-\nsibility that some of the ejected boulders could survive up to the\narrival of Hera poses some caveats both on the safety of the probe\nand on the best way to possibly observe them. In Sect. 4 some\nconsiderations on this subject are presented.\nArticle number, page 1 of 12arXiv:2402.04807v1  [astro-ph.EP]  7 Feb 2024A&A proofs: manuscript no. main\n2. Dynamical Model\nTo investigate the long term dynamics of the ejecta fragments,\nonce they leave the surface of Dimorphos, we have numerically\nintegrated their trajectories. During their evolution, the frag-\nments may have frequent close encounters with either compo-\nnents of the binary asteroid, making their orbits highly chaotic.\nThe 15th-order Radau numerical integrator (Everhart 1985),\nwhich has a variable step size and is very accurate in modelling\nclose approaches, is very well suited to handle this problem.\nThe gravitational attraction of both Didymos and Dimorphos\nis computed taking into account their non-spherical shape. In the\nmodel two computational methods are considered: a less CPU\ntime-consuming analytical approach based on the MacCullagh\nformula (Murray & Dermott 2000) or a more cumbersome poly-\nhedral approach (Werner 1994; Werner & Scheeres 1997). In\nshort, the MacCullagh formula approximates the gravitational\npotential of an irregular body as\nV=−GM\nr−G(A+B+C−3I)\n2r3(1)\nwhere A,B,Care the inertia moments of the body along the\nprincipal axes, x,y,zare coordinates measured along those axes,\nris the distance from the center of a body and\nI=(Ax2+By2+Cz2)\nr2(2)\nFor a triaxial ellipsoid, the inertia moments are related to the\nprincipal semi-axes a,b,cthrough the equations:\nA=4\n15πρabc(b2+c2)\nB=4\n15πρabc(a2+c2)\nC=4\n15πρabc(a2+b2)\nEven if this algorithm is slightly inaccurate close to the body\nsurface, where the polyhedron model would be more precise\neven if significantly slower, we have adopted it because of the\nstrong chaotic nature of the trajectories. The values of a,b,cfor\nboth Didymos and Dimorphos are derived from the shape of the\nobjects as observed by DART (Daly et al. 2023) and obtained\nfrom the SPICE kernels. We recall that the \"SPICE\" (Space-\ncraft, Planet, Instrument, C-matrix, Events), maintained by the\nNASA’s Navigation and Ancillary Information Facility (NAIF),\nis an observation geometry information system designed to as-\nsist scientists in planning and interpreting scientific observations\nfrom space-based instruments aboard robotic planetary space-\ncraft (https: //naif.jpl.nasa.gov /naif/data.html). As an example,\ntypically the SPICE provide positions and velocities of planets,\nsatellites, comets, asteroids and spacecraft in several di fferent\nreference systems.\nThe numerical integration of the fragments’ orbits is per-\nformed in a body-fixed reference frame centred on Dimorphos,\nrotating with the asteroid’s angular velocity. This reference sys-\ntem is coded within the SPICE kernels as IAU_DIMORPHOS .\nTherefore the apparent forces due to the frame rotation are in-\ncluded in the force equation giving the acceleration:\naapp=−ω×(ω×r)−2ω×v, (3)whereωis the rotation vector of the reference frame and rand\nvare the position and velocity vectors in this rotating frame.\nThe solar tide force is computed in the model as di fference\nbetween the Sun gravitational force acting on the fragment and\nthat on Dimorphos. The radial vectors with respect to the Sun are\nderived from the heliocentric orbit of the binary asteroid, once\nagain as provided by the SPICE kernels in the appropriate inte-\ngration reference system.\nThe solar radiation pressure is also included in the trajectory\ncomputation with the standard formulation\nF=S A\ncQPRs (4)\nwhere Sis the solar radiation flux density at the heliocen-\ntric distance of the body, Ais the fragment geometrical cross\nsection, and QPRis a dimensionless coe fficient determining the\namount of radiation that is either reflected, absorbed and re-\nemitted (Burns et al. 1979). The versor sis directed in the anti-\nsolar direction.\nThe fragments can be modelled as ellipsoidal rotating parti-\ncles, therefore posing a time-varying cross section for the com-\nputation of the solar radiation pressure (see Rossi et al. (2022)\nfor details). We note that in the current study, focused on large\nboulders with a low area-to-mass ratio, the e ffect of the parti-\ncles shape is not relevant and therefore spherical particles were\nconsidered. The algorithm allows also to treat collisions between\nthe fragments. These collisions are mostly e ffective for small ob-\njects in the first phases of the plume evolution (see e.g. Fahne-\nstock et al. (2022)), therefore, since we are interested in the dy-\nnamics of boulders following the first stormy post-collision mo-\nments, we assume they are not significantly perturbed by im-\npacts against the smaller fragments and do not consider this ef-\nfect here.\nThe accurate heliocentric orbit of the Didymos system and\nthe binary orbital dynamics is based on the JPL Ephemeris, as\nderived from the post DART impact SPICE kernels.\n3. Long Term Dynamical Behaviour of the Boulders\n3.1. Initial conditions\nIn trying to understand the general behaviour of the ejected boul-\nders we have performed a simulation including 10000 synthetic\nlarge objects over an interval of four years after the DART im-\npact, to cover the Hera arrival time–span. The diameters of the\nsimulated objects range from 4 to 10 m and their number, for\neach size bin in which the size range has been divided, is de-\nrived by using a power law distribution with slope q=−3.9.\nThis distribution is derived from the Hubble images taken by Je-\nwitt et al. (2023). The boulders are all started near the location\nof the DART impact site (Daly et al. (2023), retrieved within the\nmodel from the SPICE kernels) and a small delay in their release\nfrom the crater is assumed (Raducan et al. (2023)), within an in-\nterval of time of about 10 minutes after the impact. The initial\nvelocities with respect to the surface of Dimorphos are encom-\npassed between 6 and 13 cm /s with a uniform random distri-\nbution. The directions of the velocity vectors is defined by the\nejection angle, measured from the impact site plane. Such angle\nspans the range between 72◦and 44◦(Li et al. 2023; Dotto &\nal. 2023; Deshapriya et al. 2023), assuming a linear distribution\nfrom the innermost ejecta with the steeper angles to the outer-\nmost ones, including also the e ffects of an oblique impact (Zan-\notti & Lavagna 2020; Cintala et al. 1999; Anderson et al. 2003),\nArticle number, page 2 of 12K. Langner et al.: Long term dynamics around the Didymos-Dimorphos\nbased on values estimated from numerical experiments (Radu-\ncan et al. 2023). The narrow range of velocities and the large\nsize adopted for the bodies were chosen with the intent of max-\nimising the number of particles potentially surviving for a long\ntime. Note that the number of boulders we simulate is at least\ntwo orders of magnitude larger than the expected number. The\nreason of this choice is to have a significant statistics of bodies\nwhich are trapped in the gravity field of the binary asteroid for a\nlong time.\n3.2. General statistics\nThe boulders ejected after DART are divided into three groups\ndepending on their final state in the simulation. The first group\nincludes the objects which escape from the system ending in\na heliocentric orbit. To determine if an object is outside the\nDidymos-Dimorphos gravitational influence we took a simple\napproach based on the classical definition of a Hill radius:\nrH=aD(1−eD) 1\n3M⊙!1\n3\n≈68km, (5)\nwhere aDis the heliocentric Didymos semi-major axis, eDis its\neccentricity and M⊙is the mass of the Sun. Because of the high\neccentricity of Didymos we found 1 rHtoo restrictive, so to add\nextra margin we decided to define the escaping object as and\nobject that is at a distance greater than 5 rHfrom the Didymos-\nDimorphos barycenter. When a particle reaches that distance, it\nis considered definitively escaped and removed from the sim-\nulation. This is an empirical criterion adopted to define a large\nenough distance beyond which we can safely claim that the body\nis not gravitationally tied to the two asteroids. For this reason,\neven if the Hill’s sphere changes with the heliocentric distance,\nthis does not a ffect in any way our results. Boulders colliding\neither with Didymos or Dimorphos belong to group two. Nu-\nmerically, an object collides with one of the two asteroids if it\nenters the ellipsoid representing the surface of the asteroid. Fi-\nnally, group three includes all objects that survive until the end\nof the simulation i.e. objects that do not escape nor impact the\nasteroids during the four-year time span of the simulation.\nFigure 1 illustrates the boulders lifetime vs the initial ejec-\ntion velocity from Dimorphos’ surface. The red dots show the\nfragments belonging to group two which re-impact the asteroid\nafter some time, the green dots represent the bodies which escape\nfrom the system (group one) while the purple points are the boul-\nders surviving until the end of the simulation (4 years) therefore\nbelonging to group three. According to Fig. 1 most of the low ve-\nlocity boulders re-impact Dimorphos on a short timescale while\nonly few escape. On the other hand, the higher velocity frag-\nments are more prompt to escape after timescales longer than\nabout 100 days and a few either impact Dimorphos or, prefer-\nentially, Didymos. The long surviving bodies which accumulate\nin the rightmost part of the plot, do not show a significant de-\npendence of the ejection velocity. The statistics of the final state\nof the boulders is shown in Table 1. It is worth noticing that the\npercentages of objects colliding and escaping changes over time,\nwhere the re-impacts with Dimorphos dominate the early stages\nof the simulation, while escapes and impacts with Didymos be-\ncome more frequent later. Most of the particles are removed dur-\ning the first 100 days and only 1565 of the original 10000 re-\nmained. At the end of the simulation, after 4 years, 92 boulders\nsurvived which is slightly less than 1% of the initial objects. In\nthe next section, we describe the evolution in each of the groups\nof boulders in more detail.Table 1. The state of the particles in simulation.\nStatus Total t<100d t>100d\nImpact Didymos 1863 759 225\nImpact Dimorphos 5429 5285 144\nEscape 3417 2313 1104\nSurvived 92 1565 92\n3.3. Escaping boulders\nThe images in Jewitt et al. (2023) show a population of boul-\nders left within the Didymos system after the DART impact. As\nmentioned before, in our simulation we restricted the initial ve-\nlocities to a narrow range which is slightly lower than escape ve-\nlocity from Dimorphos up to slightly greater than escape veloc-\nity from Didymos-Dimorphos system. Most of our bodies have\nvelocities between this two limits, since we are interested in the\nlong surviving bodies. Therefore our escaping population might\nfollow a di fferent dynamical path, possibly dominated by close\nencounters with the two asteroids, with respect to the some of the\npopulation observed by Hubble which partly consists of boulders\nejected with a higher initial velocity.\nIf we look at the total number of escaping particles (Fig. 2),\nwe can see the peak in the first 100 days of simulation, then the\nnumber of escape events decreases. The initial peak consists in\nboth objects that initially have high enough velocity to escape\nthe system and the slower ones that achieved the required ve-\nlocity due to gravitational interaction with asteroids (see Fig. 1).\nThe latter escapes are a product of the dynamical evolution of\nboulders in the Didymos-Dimorphos system. The number of es-\ncape events decreases exponentially as a result of the decrease\nin the total number of remaining particles. In the simulation, we\nnotice an increase in the number of escapes and in the velocity of\nescaping particles with a local maximum around 800 days after\nthe Dart impact. This coincides with the moment when Didy-\nmos is in its perihelion. The lower Sun-Didymos distance causes\nstronger Solar tides, and results in boulders on weakly bounded\norbits to escape more likely and with higher relative velocity.\nWe note that this result might suggest to have a dedicated obser-\nvational campaign at the time of the next perihelion passage to\ncheck if some escaping boulders could be detected.\n3.4. Re-impacting boulders\nThe investigation of re-impacting boulders on the surface of ei-\nther Didymos or Dimorphos is relevant under many aspects.\nThey may perform some orbits in the system before colliding\nand, therefore, they can be classified as sesquinary. It has been\nshown that this type of impacts may significantly a ffect the sur-\nface of Phobos producing low velocity crater chains (Nayak &\nAsphaug 2016). In the case of boulders re–impacting the surface\nof Didymos /Dimorphos, the collision velocity is low, of the order\nof tens of cm per second. It is possibly too low to produce sig-\nnificant secondary craters on the surface of the bodies. However,\nthe sesquinary impacts can lift dust from the surface of Dimor-\nphos and Didymos which will be subsequently removed by so-\nlar radiation pressure. This second generation dust may explain\nthe extended period of time during which a tail behind Didy-\nmos has been observed (see also Moreno et al. (2023); Ferrari\n& al. (2023)). They may also change the surface composition\nby exposing, after the dust lift, deeper portion of the surface.\nThese boulders would also increase the overall boulder count-\ning in the densely re-impacted regions, giving an additional con-\nArticle number, page 3 of 12A&A proofs: manuscript no. main\nFig. 1. The status of all simulated objects. The time shows the moment when the object left the simulation by re-impacting or escaping.\ntribution to the surface composition alteration. These boulders\nare in fact produced by the DART impact which excavated and\npartly melted material from deeper regions of the asteroid body\nand when they land on the surface of the asteroids they bring\nmaterial with a composition which di ffers from that of the sur-\nroundings. Finally, re-impacts can be relevant for the evolution\nof the mutual orbit of the binary components (e.g., see Richard-\nson et al. (2022)). For this reason it is important not only to es-\ntimate the frequency of this events but also to find the regions\non the surface of both asteroids where most of the primary frag-\nments impact.\nA first group of re-impacting particles consists of those\nwhich are too slow to escape the gravity of Dimorphos and\ncollide with it after a few hours from the DART impact. They\nre-impact on Dimorphos with a velocity close to their ejection\nspeed. A second group of bodies leave Dimorphos and orbit\nDidymos for a while before colliding with either of the two as-\nteroids later in the simulation. As shown in Fig. 3 in the first 40\ndays most of the boulders impact with Dimorphos while later on\nthe impacts with Didymos become more likely.\nFor each of the impacts recorded in the simulation we cal-\nculate the velocity (Fig. 4). The collisions with Dimorphos are\ngenerally slower with mean velocity equal to 8.9 cm /s. Most of\nthe slowest impacts occur early and if we consider only impacts\nafter 10 days the mean impact velocity increases to 10.9 cm /s.The second peak in the histogram of the collision speed on Di-\nmorphos is due to boulders which were injected in a prograde\norbit around Didymos and re-impact Dimorphos at a later time,\nwith the earliest one registered 65 days after the beginning of\nthe simulation. This second smaller group of fragments collides\nwith the asteroid with an average higher velocity, about 40 cm /s.\nThe population of boulders colliding with Didymos has a\nhigher mean velocity, about 26.6 cm /s, due to the stronger grav-\nity of Didymos. In contrast to Dimorphos the re-impact velocity\nis not correlated with the orbit being prograde or retrograde.\nThe statistical distribution of impacts on the surface of either\nasteroids is shown in Fig. 5. To define the latitude and longitude\nwe assumed an ellipsoidal shape of both asteroids and used pos-\nitive rotation axis direction as the North pole (note that both ro-\ntations are retrograde with respect to the ecliptic plane or Didy-\nmos heliocentric orbital plane). The longitude angle increases in\nthe direction of rotation, and for Dimorphos the prime meridian\nis directed towards Didymos. On Didymos the impact locations\nare uniformly scattered around the equator with a slightly higher\nprobability of impacting on the northern hemisphere. The equa-\ntorial preference is related to the initial orbits of most boulders\nwhich share an inclination similar to that of Dimorphos which\nis close to the equatorial plane of Didymos. A small number of\nbodies re-impact Didymos at later times during which their or-\nArticle number, page 4 of 12K. Langner et al.: Long term dynamics around the Didymos-Dimorphos\n0 200 400 600 800 1000 1200 1400\ntime [d]103\n102\n101\nFraction of escaping objects\nFig. 2. Objects escaping from Didymos system. The top plot shows the\ntotal number of escaping boulders in the simulation (normalised, where\n1 is the total number of escapes). In the bottom plot each point repre-\nsents a single boulder showing its escape time and velocity relative to\nDidymos barycenter at the escape time.\nbits had time to evolve in inclination due to the perturbation in\nthe system.\nTo compute the distribution of re-impacts on the surface of\nDimorphos we assumed that its rotation is tidally locked to its\norbital motion. However, it is possible that after DART Dimor-\nphos entered a state of tumbling (e.g. Agrusa et al. (2021); Meyer\net al. (2023)). However, since most of the re-impacts occur on a\nvery short timescale after the DART collision, our impact map is\nstill roughly reliable even in the case of the switch to tumbling\nrotation. Note that, even in the case of a principal axis rotation,\nDimorphos could still experience a libration with an amplitude\non the order of tens of degrees (Agrusa et al. 2021; Meyer et al.\n2023) that might slightly alter the simulated distribution on our\nimpact map. The re-impact distribution on Dimorphos signifi-\ncantly di ffer from that on Didymos as expected since most of\nre-impacting bodies do not come from fragments that have ex-\nperienced a significant orbital evolution, such as in the case of\nDidymos. The latitude of the impacts is no longer limited to\nnear equatorial regions and the distribution of re-impact longi-\ntudes becomes non-uniform with a strong preference for positive\nvalues. The re-impacts mostly occur on the opposite hemisphere\nwith respect to the DART impact site. To explain this we use\n10 20 30 40 50 60 70 80 90 100\ntime [d]0.0000.0020.0040.0060.0080.0100.0120.014number of re-impactsDimorphos\nDidymos\n0 200 400 600 800 1000 1200\ntime [d]103\n102\n101\n100number of re-impactsDimorphos\nDidymosFig. 3. Bottom panel: total number of re-impact events in the whole\nsimulation time-span (logarithmic scale on the y-axis). Top panel: de-\ntail of the period from 10 to 100 days after the DART impact, using a\nlinear scale on the y-axis. The blue line shows the number of re-impacts\nwith Dimorphos while the orange line represents the re-impacts with\nDidymos. Both plots are normalised such as 1 is the total number of\nre-impacts on both asteroids.\nthe fact that the initial orientation of the orbit of a boulder is\nthe same as the orientation of Dimorphos orbit. When the colli-\nsion happens, a boulder is usually near a pericenter of its orbit\nso its velocity vector is near perpendicular to their position vec-\ntor (w.r.t. Didymos), so it should have similar direction to the\nDimorphos velocity vector. The faster object, in this case a boul-\nder, must then hit the slower one from behind i.e. from the direc-\ntion opposite to velocity vector of the slower object. Since the\nboulders are not exactly in their pericenter and their orbital ori-\nentation evolves over time, the impact locations are heavily scat-\ntered with maximum number of re-impacts around the ( +90◦,0◦)\npoint.\nThe distribution of re-impacting bodies may be useful in the\nanalysis of Hera images since some of the observed craters will\ndate back to the time of DART impact. These fresh craters may\nmask some of the older features on the surface of both asteroids\nand, in the case of crater counting for age determination, they\nshould be excluded from the crater distribution. Therefore, our\nmaps of re–impactors can be a useful tool for interpreting the\ncrater distribution of the asteroid surfaces.\nArticle number, page 5 of 12A&A proofs: manuscript no. main\n0 10 20 30 40 50 60 70 80\nre-impact velocity [cm/s] (Didymos)0.0000.0250.0500.0750.1000.1250.1500.1750.200fraction of re-impacts\n0 10 20 30 40 50 60 70 80\nre-impact velocity [cm/s] (Dimorphos)103\n102\n101\n100fraction of re-impacts\nFig. 4. The velocity of boulders re-impacting with Didymos (top) and\nDimorphos (bottom). For Dimorphos the logarithmic scale is used for\nbetter readability.\n3.5. Quasi-stable orbits\nThe previous dynamical analyses and simulations of the parti-\ncle orbits (Rossi et al. 2022) show that stable orbits can exist\nin the Didymos-Dimorphos system. The stable orbits could be\neither circumbinary – orbiting both asteroids outside of the or-\nbit of Dimorphos, or circumprimary – orbiting around Didymos\ninside the orbit of Dimorphos. Those orbits exist only if the so-\nlar radiation pressure is weak enough, which means that only\nlarger bodies with low area-to-mass ratio should remain in the\nsystem. The existence of such orbital solutions is not a su fficient\ncondition, there also must be a mechanism placing the boulders\nreleased from Dimorphos on this kind of orbits. Therefore, it\nis required to perform simulations where boulders are released\nfrom the surface of Dimorphos to test whether some of these\nbodies are placed in these almost stable orbits.\nThe large initial population of the boulders in the simulation\nexponentially decays over time as shown on Fig. 6. The rate of\ndecay slowly decreases over time, but about 700 days after the\nDART impact the rate has a temporary increase. This temporary\nrise occurs when Didymos is near its perihelion around the Sun.\nAfter the perihelion, approximately after 1000 days from the be-\nginning of the simulation, the decay rate slows down again. A\ntotal of 92 objects survived until the end of the simulation with 6\nFig. 5. Location of re–impacts on Didymos (top) and on Dimorphos\n(bottom). Only the re-impacts later than 24 hours after DART impact\nare included. On the bottom the green dot marks the DART impact lo-\ncation and the (0◦,0◦) point is the direction towards Didymos and north\npole ( +90◦latitude) is in the direction of the rotation axis, while the\n(−90◦,0◦) is the direction of Dimorphos orbital motion. Most of the\nreimpacts concentrates near the point (90◦,0◦) which is opposite to the\ndirection of orbital motion of Dimorphos and close to opposite to the\nDART impact location.\nof them on escape trajectories that have not yet reached the limit-\ning distance of 5 Hill radii from Didymos but are going to escape\nwithin the following 30 days. If we exclude these objects and a\nfew others on very distant orbits which are heliocentric rather\nthan circumbinary, 80 objects survive locked in orbit around the\nbinary asteroid, about 0.8 percent of initial number of particles\nin the simulation.\nThe histograms of Fig. 7 show the distribution of the oscu-\nlating orbital elements of the surviving particles at the end of\nthe simulation. The angular elements are measured in the eclip-\ntic J2000 reference frame. It is noteworthy that all the surviving\nobjects are on circumbinary orbits suggesting that there are no\ndynamical pathways connecting the boulders from their ejection\nto an almost stable circumprimary orbit around Didymos lasting\nat least a few hundreds days. Also there are no particles orbiting\nDimorphos in the simulation. The typical orbit of long term sur-\nviving objects have high eccentricity (only 14 of the final orbits\nhave e<0.7) and are on a retrograde orbits with low inclination.\nArticle number, page 6 of 12K. Langner et al.: Long term dynamics around the Didymos-Dimorphos\n0 200 400 600 800 1000 1200 1400\nTime [days]102\n101\n100Number of boulders\nFig. 6. The total number of particles in the simulation over time (where\n1 is the normalised initial population size)\n. Objects that escaped or re-impacted are removed from the\nsimulation.\nPrograde and inclined orbits are also present but they are less fre-\nquent. All except one of the orbits have the pericenter distance\nlarger than the orbit of Dimorphos around Didymos. Low peri-\ncenter distances significantly reduce the survival chances due to\nthe high probability of close encounters with the asteroids.\nThe histograms of the pericenter arguments and nodal lon-\ngitudes (bottom panels of Fig. 7) show that the ascending nodes\nconcentrates around 120 degrees while the pericenters are lo-\ncated close to one of the nodes i.e. ωclose to 0oor 180o.\nA more detailed analysis of the distribution of the orbital\nangles (Fig. 8 top) show that the boulders with nodes concen-\ntrated around 120oare on highly inclined orbits. Almost all bod-\nies (with the exception of two) with inclination close to 90ohave\ntheir periapsis argument concentrated around 0◦. The evolution\nof the angular elements (Fig. 8 bottom) shows that this might\nbe a kind of unstable equilibrium point, where the pericenters of\nhighly inclined orbits1are clustering and angular elements are\nevolving very slowly. There are also orbits moving away from\nthis equilibrium point, lowering in the process their inclination.\nThe direction of this equilibrium at first glance looks quite ar-\nbitrary, but it is located perpendicular to the Sun-Didymos peri-\ncenter line, which has a longitude ≈30◦. The exact dynamical\nmechanism of this pericenter clustering requires further more de-\ntailed analysis.\nIn the Fig. 9 we show the trajectories in the inclination-\neccentricity plane and in the inclination- ωplane for all particles\nduring the last two years of the simulation. The di fferent colours\non the plots are used to distinguish between di fferent objects.\nThe first plot clearly shows the connection between an increase\nin eccentricity and inclination and also that all low eccentricity\norbits are near the ecliptic plane. Both plots shows a behaviour\nsimilar to that of orbits a ffected by the Kozai mechanism with\nthe Sun as the outer perturbing body. However, the orbital flips\nobserved for many particles in the simulation are quite rapid and\noccur mostly near the epoch of Didymos perihelion, while in\nthe classical Kozai mechanism the time scale should be longer\n1The terms \"high\" and \"low\" for inclination are used to describe the\nangle between the orbital plane and the Ecliptic. The retrograde orbits\nnear the Ecliptic plane with inclination close to 180◦are then considered\nto have low inclination.than a few periods of the perturbing body. This means that the\nrapid orbital changes observed here are the e ffect of perturba-\ntions near the perihelion of the eccentric Didymos orbit. The\nother source of the changes in the inclination are the perturba-\ntions from Dimorphos, which may be creating an inner Kozai\nmechanism (Naoz et al. (2017)), that a ffects orbits regardless of\nthe Sun distance. The coupling of e ffects by inner and outer per-\nturbing bodies together with other dynamical e ffects (like obliq-\nuity of the asteroids and radiation pressure) creates a compli-\ncated dynamical system, which will be explored theoretically in\nmore detail in future works.\nThe vast majority of boulders have orbits entirely between\nthe orbit of Dimorphos and the Hill radius of the Didymos sys-\ntem. Only one boulder at the end of simulation has pericenter\ncloser than a Didymos orbit radius, and there are a few boul-\nders with apocenter more distant than Hill sphere. The particles\nwith closer apocenter have relatively strong gravitational bound\nto Didymos and usually are on regular elliptical orbit slowly pro-\ncessing due to gravitational perturbations. For those objects the\nmore rapid changes of osculating orbits are present only as the\neffects of close flybys. The second group contains the high in-\nclination orbits discussed in the previous paragraph undergoing\nthe increase in inclination phenomena. In the final results we\nalso find some surviving particles ejected on orbits with apocen-\nter at very large distance (more than two times the Hill radius)\nfrom Didymos where Solar tide perturbations are considerably\nstronger. Those objects are loosely bound with Didymos, but for\nsome reason they did not reach enough distance to get to our es-\ncape distance condition. Their trajectory often could have more\nexotic shape than weakly perturbed Keplerian ellipses and can\nbe considered as to be on heliocentric orbit close to Didymos\nrather than perturbed orbit around the asteroid.\nObserving di fferent particle trajectories in the simulation we\ncan propose the following typical scenario for long time sur-\nviving particles: first the boulder is ejected from the surface of\nDimorphos by the DART impact. If its velocity is too low it\nquickly re-impacts, if it is too large, the boulder escapes the sys-\ntem within few days or weeks. If the boulder lies in the correct\nvelocity range it remains in the system on initial orbit with peri-\ncenter close to the DART impact location. This initial orbit can\nbe very chaotic due to possible close encounters with Dimor-\nphos and sooner or later the particle is either removed (escape or\ncollision) by one of these encounters or sent into an orbit with\nmore distant apocenter. At those larger distances Solar tide have\ngreater e ffect on the orbit and could possibly increase the peri-\ncenter distance, allowing the particle to reduce the probability\nof close flybys near Dimorphos. This mechanism could possi-\nbly place the boulder on quasi-stable orbits where some of them\ncan last for hundreds of days. Those quasi-stable orbits are per-\nturbed by the Sun and asteroids gravity resulting in an increase\nin eccentricity and inclination or orbital flips from retrograde to\nprograde and vice-versa. This orbital evolution may cause the\nescape or re-impact of the boulder tens or hundreds of days after\nit was released from Dimorphos.\n3.6. Size dependency\nThe size of the ejecta is relevant when the solar radiation pres-\nsure becomes an important force which may significantly per-\nturb gravity. Previous simulations (Yu et al. 2017; Yu & Michel\n2018; Rossi et al. 2022) have shown that radiation pressure can\nquickly destabilise the orbits of small particles around Didymos–\nDimorphos and then reduce their lifetime. To test the relevance\nof the solar radiation pressure for larger bodies we have per-\nArticle number, page 7 of 12A&A proofs: manuscript no. main\n0 50 100 150 200\na [km]0510152025303540number of particles\n0.0 0.2 0.4 0.6 0.8 1.0\ne0510152025303540number of particles\n0 50 100 150\ni0510152025303540number of particles\n0 100 200 300\n0510152025303540number of particles\n0 100 200 300\n0510152025303540number of particles\n0 5 10 15 20\nq [km]0510152025303540number of particles\nFig. 7. Osculating elements distribution at the end of the simulation. Only elliptic orbits are included. In the last plot showing the distribution of\npericenter distance we excluded 5 particles with q>24. for better readability (orange line shows the distance of 1.2 km)\nformed an additional simulation where the size of the particles\nranges from 10 cm to 10 m. In randomly creating the parti-\ncle population we have used a uniform distribution of sizes to\nequally represent the particles in each size range.\nIn Fig. 10 we plot the mean particle lifetime in the simu-\nlation vs the particle size. Each point is the mean value of the\nlifetime for all particles encompassed within a size range of ±5\ncm. The moving average with range ±50 cm (solid line in the\nplot) have a shape similar to a logarithmic function with a rapid\ninitial growth for lower diameters and slower asymptotic growth\nfor larger boulders. In the simulation the smallest particles have\na lifetime less than 40 days and it raising to about 100 days for\nparticles a few metre in size. It must be noted that the average\nmay depend on the initial velocities and on the condition for a\nparticle to be considered escaped from the system. For example,\nparticles with low initial velocity which are re-impacting Dimor-\nphos a few hours after the DART impact can decrease the mean\nvalue. In addition, in our simulation the particles require about\none month to reach the escape distance of 5 rH, increasing in this\ncase the lifetime. However, the analysis of the data shows that\nthese e ffects a ffect ’democratically’ the estimates of the average\nlifetime.\nThe number of objects lasting until the end of the simulation\n(4 Earth years) also shows a dependence on the particle size (see\nFig. 11). For particles with diameter around 4 metres the frac-\ntion of surviving boulders is close to 0.8% in agreement with\nour previous computations. For larger boulders the percentage\ngrows up to 1.2–1.4 % while for smaller boulders it reduces toapproximately 0.4 % showing that the solar radiation pressure\nacts always to destabilise orbits.\n3.7. Stability of the solutions\nTo estimate the stability of the long term orbits described in\nSect. 3.5 we performed another set of simulations where we used\nthe orbits of 92 long term surviving boulders to create a large\npopulation of clones with slightly altered initial conditions.\nIn the first simulation we generated 100 clones for each long\nsurviving particle by randomly altering the initial position of the\noriginal particle by a factor of 10−4of the original value in each\nof the Cartesian coordinates. Therefore, the absolute change was\nsmaller than a few millimetres. These cloned particles are inte-\ngrated starting at the moment of the DART impact.\nThe very small change in initial conditions has a significant\nimpact on the resulting particle dynamics and their lifetime. Dur-\ning the first 30 days more than 3000 clones (i.e., 33 % of the\ntotal) re-impacted with one of the asteroid. Further on, during\nthe first 100 days 5322 objects are removed from the simulation\ndue to re-impacting or escaping. This means that the rate of sur-\nvival in the clone population is only slightly higher than in the\noriginal simulation and still most of the particles are removed.\nThe number of particles surviving until the end of the simula-\ntion is 1%, just slightly higher than the global statistics in the\noriginal simulation with random initial conditions. The reason\nof this small di fference is that in the simulation with the clones\nArticle number, page 8 of 12K. Langner et al.: Long term dynamics around the Didymos-Dimorphos\nwe only include objects with an initial velocity large enough to\navoid immediate re-impact against Dimorphos.\nTo test if a specific set of initial conditions could be poten-\ntially more prone to stability, for each of the 92 long lasting ini-\ntial boulders we counted the number of clones which survived\nover a long time–scale. For about half (47) of the original 92\nquasi–stable boulders we find that none of their clones survived\nat least for the same time–span of the original cloned boulder.\nFor 25 original bodies only one clone survived, while for the\nremaining 67 cases more than 1 clone lasted till the end of the\nsimulation. In a single case, 8 clones (out of 100) of the original\nbody survived, while 6 clones survived in other two cases. This\nsuggests that the dynamics is strongly chaotic and a small change\n(few millimetres as stated above) in the initial position can dra-\nmatically alter the lifetime of a particle in the system. In addition,\nwe can state that, within the assumptions of our model and the\nexplored simulation cases, there is not a specific extended set of\ninitial conditions leading to quasi–stable orbits.\nTo test whether the particles, after avoiding the initial highly\nchaotic evolution, can be injected into stable orbits, we per-\nformed an additional set of simulations where the clones are cre-\nated 100 days after the DART impact. For each of the 92 original\nquasi–stable objects we generated 100 clones by altering the co-\nordinates of their position vector again by up to 10−4of its nom-\ninal value (corresponding in this case to an absolute change of\na few metres). Then we numerically integrated this new popula-\ntion of 9200 clones over a time-span of 4 years (100 days beyond\nthe end of the original integration). Out of this initial population,\n3285 particles survived the entire simulation (about 36%), 909\nimpacted Didymos, 646 re-impacted Dimorphos and 4360 es-\ncaped from the system. A small number of original objects give\nrise to a line of clones quite stable, with most of them surviving\nuntil the end of the simulation. Indeed, 7 of them have all 100\nclones surviving. On the other hand, about half of the original\nobjects have less than 25% of their clones lasting till the end of\nthe simulation and 3 have all of their clones removed by escapes\nand re-impacts.\nThe higher percentage of survivors (36% vs 1%) in this sec-\nond population of clones implies that, after 100 days, the 92\noriginal quasi–stable boulders have reached a zone of the phase\nspace where it is more easy to survive in orbit for a long interval\nof time. This also points to the possible existence of wide re-\ngions in the phase space where quasi–stable orbits can be found,\nbut they are not necessarily accessible by boulders ejected after\na collision on the surface of Dimorphos.\n4. Discussion\nAn important question concerning the environment around the\nDidymos–Dimorphos system is the long term survival of boul-\nders ejected after the DART impact. If they stay long enough\non quasi–stable orbits they may be observed by the Hera mis-\nsion but, at the same time, they would represent a threat for the\nspacecraft. To answer this question we have performed detailed\nnumerical simulations of a large number of bodies in order to\nhave a glimpse on the dynamics of the system. We have devel-\noped an accurate numerical model able to include all the relevant\nphysics and explore a reasonable range of initial conditions for\nthe ejecta. We have also implemented in the code the correct\ngeometry of ejection by using the SPICE kernels to have the ex-\nact orientation of the binary system in the appropriate reference\nframes.\nIn general the dynamics near Didymos-Dimorphos can be\nviewed as a restricted four body problem. For particles closer tothe asteroid pair the most relevant forces are the gravity of Didy-\nmos and Dimorphos while for the more distant ones the grav-\nity of the Sun plays an important role. A significant aspect of\nthe the combination of all the gravitational perturbations is that\nthey force a relation between the eccentricity and the inclination\nof the boulders similar to a Kozai state but with a di fferent be-\nhaviour. An increase in inclination is connected with an increase\nin eccentricity (however not all highly eccentric objects increase\ntheir inclination). Paradoxically, this may act as a sort of protec-\ntion mechanism against instability since the high inclination re-\nduces the probability of close encounter with Dimorphos when\nthe orbits are eccentric. In addition, these perturbations enable\norbital flips leading to polar and prograde orbits.\nIn our study we show that a fraction of the ejected boulders\ncan indeed survive and orbit around the binary asteroid for an ex-\ntended period of time. To estimate the fraction of the initial boul-\nders that are injected in these quasi–stable orbits we have inte-\ngrated a large number of bodies in order to explore in more detail\nthe phase space and to derive a statistical probability of survival.\nHowever, since we do not know from observations which is the\nreal number of boulders ejected by the impact, we cannot pre-\ndict even an approximate number of bodies that will orbit the\nbinary at the time of the Hera arrival. The Hubble Space Tele-\nscope (HST) observed over 60 large boulders and several boul-\nders were also observed by the LICIACube. Whereas the boul-\nders observed by HST are mostly on escape trajectories, these\nobservations allows us to assume that a significant number of\nboulders were ejected, possibly also with a low initial velocity.\nSome of them may end up on the quasi–stable orbits which we\noutline in our paper. The fraction of surviving objects in our sim-\nulation is small, about 1% of the initial sample, but not negligi-\nble. Given the strong chaotic nature of the evolution it is hard\nto postulate if the use of more precise initial conditions or of an\nimproved physical model (e.g., including the post-impact binary\neccentricity or Dimorphos’s asynchronous rotation state) could\nlead to more realistic results. In any case, from our statistics, it\ncan be safely inferred that there is a very limited risk of impacts\nbetween the arriving Hera spacecraft and a large object trapped\nwithin the system.\nSince the long term orbits are not stable, there is an almost\ncontinuous leak of bodies from the population of boulders mov-\ning in the system. The escaping rate grows when Didymos ap-\nproaches the Sun at pericenter due to the increased solar tide.\nThese boulders might be observed from the ground in particular\nin 2025 when Didymos passes through the pericenter.\nAn additional outcome of our modelling is the distribution\nof re–impacts on the surface of Dimorphos and impacts on that\nof Didymos which can be classified as sesquinary impacts. The\nknowledge of their distribution is useful to evaluate the extended\nemission of dust from the surface of the two bodies which refills\nthe dust tail even at later times respect to the DART impact. It is\nalso relevant when interpreting the images taken by Hera since\nthese sesquinary impacts can slightly change the local features\nby increasing the density of boulders and by a ffecting in di fferent\nways the local surface composition.\n5. Acknowledgments\nThis work is supported by the Italian Space Agency (ASI) within\nthe Hera agreement (ASI-UNIBO, n. 2022-8-HH.0). The authors\nwish to thank the anonymous reviewer for the useful and con-\nstructive remarks that helped to improve the paper.\nArticle number, page 9 of 12A&A proofs: manuscript no. main\nReferences\nAgrusa, H. F., Gkolias, I., Tsiganis, K., et al. 2021, Icarus, 370, 114624\nAnderson, J. L., Schultz, P. H., & Heineck, J. T. 2003, Journal of Geophysical\nResearch: Planets, 108\nBurns, J. A., Lamy, P. L., & Soter, S. 1979, Icarus, 40, 1\nCapannolo, A., Zanotti, G., Lavagna, M., et al. 2021, Acta Astronautica, 182,\n208\nCheng, A. F., Agrusa, H. F., Barbee, B. W., et al. 2023, Nature, 616, 457\nCintala, M. J., Berthoud, L., & Hörz, F. 1999, Meteoritics & Planetary Science,\n34, 605\nDaly, R. T., Ernst, C. M., Barnouin, O. S., et al. 2023, Nature, 616, 443\nDeshapriya, J. D. P., Hasselmann, P. H., Gai, I., et al. 2023, The Planetary Science\nJournal, 4, 231\nDotto, E. & al. 2023, Nature, in press\nDotto, E., Della Corte, V ., Amoroso, M., et al. 2021, Planetary and Space Sci-\nence, 199, 105185\nEverhart, E. 1985, Astrophysics and Space Science Library, V ol. 115, An ef-\nficient integrator that uses Gauss-Radau spacings (Cambridge University\nPress), 185\nFahnestock, E. G., Cheng, A. F., Ivanovski, S., et al. 2022, The Planetary Science\nJournal, 3, 206\nFerrari, F. & al. 2023, submitted\nJewitt, D., Kim, Y ., Li, J., & Mutchler, M. 2023, ApJ, 952, L12\nLi, J.-Y ., Hirabayashi, M., Farnham, T. L., et al. 2023, Nature, 616, 452\nMeyer, A. J., Agrusa, H. F., Richardson, D. C., et al. 2023, The Planetary Science\nJournal, 4, 141\nMichel, P., Küppers, M., Bagatin, A. C., et al. 2022, Planetary Science J., 3, 160\nMoreno, F., Bagatin, A. C., Tancredi, G., et al. 2023, The Planetary Science\nJournal, 4, 138\nMurray, C. D. & Dermott, S. F. 2000, Solar System Dynamics (Cambridge Uni-\nversity Press)\nNaoz, S., Li, G., Zanardi, M., de Elía, G. C., & Di Sisto, R. P. 2017, The Astro-\nnomical Journal, 154, 18\nNayak, M. & Asphaug, E. 2016, Nature Communications, 7, 12591\nRaducan, S., Jutzi, M., Cheng, A., et al. 2023, Nature Astronomy, in press\nRichardson, D. C., Agrusa, H. F., Barbee, B., et al. 2022, The Planetary Science\nJournal, 3, 157\nRossi, A., Marzari, F., Brucato, J. R., et al. 2022, The Planetary Science Journal,\n3, 118\nWerner, R. A. 1994, Celestial Mechanics and Dynamical Astronomy, 59, 253\nWerner, R. A. & Scheeres, D. J. 1997, Celestial Mechanics and Dynamical As-\ntronomy, 65, 313\nYu, Y . & Michel, P. 2018, Icarus, 312, 128\nYu, Y ., Michel, P., Schwartz, S. R., Naidu, S. P., & Benner, L. A. M. 2017, Icarus,\n282, 313\nZanotti, G. & Lavagna, M. 2020, in 71st International Astronautical Congress\n(IAC 2020), 1–15\nArticle number, page 10 of 12K. Langner et al.: Long term dynamics around the Didymos-Dimorphos\nFig. 8. The final distribution (top) and evolution during last 365 days of simulation (bottom) of angular osculating elements: longitude of ascending\nnode, argument of periapsis and inclination (as color). On the bottom figure all objects present in the last 365 days are plotted and those which are\npresent at the end are marked by black dot at the end of their track.\nArticle number, page 11 of 12A&A proofs: manuscript no. main\nFig. 9. The trajectories on eccentricity-inclination and inclination-\nargument of pericenter planes. The plots included orbits of all objects\nfor the last two years of simulation time span. Di fferent colours are used\nto distinguish di fferent particles, but due to large number of objects the\nsame colour may be used by more than one particle.\nFig. 10. The mean lifetime of particles in the simulation depending on\ntheir diameter. The dots are the mean value for all the particles with size\nrange ±5 cm (about 100 particles per dot) and the solid line is the mean\nvalue with moving average for particles in a wider range of sizes ±50\ncm.\nFig. 11. The percentage of boulders that survived in the Didymos Sys-\ntem until the end of the simulation (4 years after DART impact) depend-\ning on size range.\nArticle number, page 12 of 12"    },    {        "title": "2402.04846v1.Investigation_of_photosensitive_and_photodetector_characteristics_of_n_TPA_IFA_p_Si_heterojunction_structure.pdf",        "content": " 1 Investigation of photosensitive and photodetector characteristics of n-TPA-IFA/p-Si heterojunction structure Şükrü Çavdar1,*, Pınar Oruç1, Serkan Eymur2, and Nihat Tuğluoğlu2 1Faculty of Science, Department of Physics, Gazi University, Ankara, Turkey  2Faculty of Engineering, Department of Energy Systems Engineering, Giresun University, Giresun, Turkey Address correspondence to e-mail: cavdar@gazi.edu.tr  Abstract n-TPA-IFA organic material was synthesized and deposited on p-Si by spin coating method to produce n-TPA-IFA/p-Si heterojunction diode. We determined that the dielectric constant and energy band gap of TPA-IFA organic material were 3.91 and 3.37 eV by DFT/B3LYP/6-311G(d,p) method using on Gaussian 09 W, respectively and the carrier type of TPA-IFA organic semiconductor material was also n-type at room temperature from temperature-dependent hall effect measurements. Using forward and reverse bias 𝐼–𝑉 measurements in the dark and under various light intensities, we examined the key electrical characteristics of the n-TPA-IFA/p-Si heterojunction diode, including 𝛷! and 𝑛. It was determined that the rectification ratio (RR) was approximately 104. The reverse current's observed increasing behavior with increasing light indicates that the produced heterojunction diode can be utilized as a photo-diode, detector, or sensor. The photodetector properties of n-TPA-IFA/p-Si heterostructure were explored at different light intensities, and the photoresponsivity (R), photosensitivity (PS), specific detectivity (𝐷∗), and linear dynamic range (LDR) of the heterojunction found to be changed with reverse voltage and light intensity. It was found that as light intensity increased, the linear dynamic range (LDR), a crucial characteristic for image sensors, increased as well (10.15 dB and 18.84 dB for 20 and 100 mW/cm2). Ultimately, the findings confirmed that the TPA-IFA-based heterojunction diode could be obtained for the photodetector application.  Keywords: Heterojunction diode, current-voltage, photodiode, photodetector, organic semiconductor            2 1. Introduction  The potential primary advantages of organic material-based electronic and photoelectrical devices, such as their low cost and easy preparation processes, versatility, and ability to cover large areas, have led to a significant increase in research interest in recent decades. Due to their superior photoconductive capabilities and exceptional thermal and chemical stability, organic semiconducting compounds have been used extensively in photoelectric and electronic devices are used in optoelectronic devices. Organic D-π-A conjugated systems, composed of donor (D) and acceptor (A) groups linked by π-bridges, have received significant attention in various scientific fields due to their unique photophysical properties. Because of their remarkable optical properties, these hybrid D-π-A skeletons have been widely used in materials chemistry for applications such as organic light-emitting diodes (OLEDs), dye-sensitized solar cells (DSSCs), nonlinear optical materials, fluorescence imaging, and memory [1-3]. Thus far, the literature has documented the utilization of organic dyes, specifically D-(π-A), D- π-A, D-D- π-A, D-A- π-A, and (D- π-A). One of the dyes that stands out is the D-A- π-A conjugate system, which offers several significant benefits [4-6]. These include broad absorption and emission bands in the UV-visible region, effective intramolecular charge transfer, easily adjustable molecular orbital energy levels, optoelectronic properties, and narrow bandgap energy. Therefore, utilizing \"D-A-π-A\" organic sensitizers has been considered a potential strategy for enhancing cell efficiency and optimizing photovoltaic performance in optoelectronic devices, as opposed to conventional \"D-π-A\" sensitizers. Organic-inorganic (OI) heterojunctions are one of the electronic and photoelectrical device classes. They are widely studied to use the advantages of both organic and inorganic materials in a single structure [2–20]. A metal/organic material/semiconductor (MOmS) heterojunction is created by adding a thin organic layer in between the metal/Si contacts [7-9]. A thin organic substance between the metal and semiconductor can form a dipole layer, allowing the Schottky barrier height of the junctions to be controlled [10-12]. It has been reported in many articles in the literature that organic semiconductor films on Si semiconductor substrate improve the electronic and optoelectronic properties of metal-semiconductor contacts [13-17].  Triphenylamine (TPA) derivatives, which serve as hole transfer materials, have gained popularity in fabricating electroluminescent devices across multiple industries [18-21]. TPA compounds offer numerous benefits, including their photoconductive and light-emitting characteristics, adjustable electronics, and straightforward processing. Due to their exceptional thermal and optical stability, they are well-suited for altering donor-π-acceptor systems [22-25]. Moreover, TPA derivatives' impressive hole drift mobility enhances their potential for optoelectronic devices [26-30]. Triphenylamine-derived compounds have also been synthesized for several optoelectronic applications. Few of these, however, have had their structural and optical characteristics examined about applying TPA compound as an interfacial layer on metal/p-Si or metal/n-Si thin films. Schiff base compounds have become a topic of interest as organic semiconductors due to their unique electrical and optical properties. Schiff base molecules are often used to create Schottky barrier diodes, but their use is limited due to hydrolysis during device fabrication [31, 32]. The use of Schiff-base compounds in creating Schottky barrier diodes is greatly appreciated due to their straightforward synthesis, low cost, and minimal time requirement. However, while triphenylamine and its derivatives have been used as linkers to create hybrid materials, limited research has been conducted on the  3 impact of TPA-based Schiff base compounds as organic interfacial layers on the electronic parameters of Schottky devices. Therefore, more research on their optical and electrical properties is required to explore the potential applications of TPA-based Schiff-base compounds in optoelectronics. In this work, we report for the first-time n-type TPA-IFA (\"D-A-π-A\") compound, in which triphenylamine is used as a donor, the C=N imine part is an acceptor, benzene is a π-spacer, and ester is an acceptor/anchor. Then, n-TPA-IFA was used as a new synthesized organic interfacial layer to modify or improve the photodiode parameters and device performance by generating dipoles at the interface. To the best of the author's knowledge, TPA-IFA thin film has not yet been employed as an interfacial layer in heterojunction and optoelectronic applications in the literature. For this reason, the TPA-IFA organic molecule interface heterojunction devices, n-TPA-IFA/p-Si heterojunction diode, were created, and the optoelectronic and electronic characteristics of the fabricated devices were investigated by using current-voltage (I-V) measurements under both different light intensities and dark conditions. 2. Experimental  2.1 Synthesis and characterization of the TPA-IFA The synthesis pathway for TPA-IFA is shown in Scheme 1. Compound 2 was synthesized from triphenylamine (1) by using Vilsmeier Haack’s formylation reaction [33]. To a solution of compound 2 (1.0 mmol) in THF (10 mL), dimethyl 5-aminoisophthalate (3) (2.1 mmol) was added. Then, the mixture was refluxed at 80 oC for 12 h, then the THF was removed via a rotary evaporator. The obtained solid was subjected to filtration, drying, and subsequent recrystallization using ethanol, yielding a pure compound. 1H NMR (400 MHz, CDCl3) δ: 8.54 (s, 2H), 8.48 (s, 1H), 8.08-8.06 (m, 4H), 7.64 (d, J = 8.5 Hz, 4H), 7.40-7.38 (m, 2H), 7.28-7.21 (m, 5H), 3.99 (s, 12H).  13C NMR (100 MHz, CDCl3) δ: 166.5, 161.6, 152.0, 149.9, 146.7, 131.4, 130.4, 130.1, 126.7, 126.2, 125.8, 124.2, 52.3. 2.2. The device fabrication  Silicon wafer with a 2-inch diameter p-type, orientation (100), 400 μm thickness, 1-10 ohm.cm was cut to suitable sizes with a diamond-tipped pen. First, the p-Si wafer was cleaned with acetone and methanol for 10 and 5 minutes, respectively an ultrasonic bath at 50°C. RCA1 and RCA2 procedures were used for p-Si wafer cleaning. H2O2:H2O solutions were prepared with a mixing ratio of 10:6. The RCA1 cleaning process was used to remove an organic layer; in this process, 5 ml of NH3 (%35) and 25 ml of H20 was mixed and heated to 70 oC. Then, 5 ml of H2O2:H2O solution was added to this solution. The Si substrate was put in the solution and kept for 15 minutes. RCA2 process was used to remove an inorganic layer; for this cleaning process, 5 ml of HCl (%27) and 25 ml H2O were mixed and heated to 70° C. Then 5 ml of H2O2:H2O solution was added to this solution. The Si substrate was put in the solution and kept for 15 minutes. To remove the oxide layer, the Si wafer was placed in a 2% HF solution for 3 minutes. Then it was rinsed with H2O. The Al ohmic contact was deposited with the help of a tungsten crucible under 10-6 bar pressure with a thermal evaporation system for the p-Si wafer. After, the Si wafer was annealed for five minutes at 570 °C in a pure argon atmosphere. Also, a soda-lime glass substrate was cleaned with chloroform, deionized water, methyl alcohol, acetone, and propanol for 10 minutes in an ultrasonic bath for the Hall measurement. 40 mg TPA-IFA was dissolved in 10 ml N,N-Dimethylformamide (DMF), and the solution was mixed for 10 minutes on a magnetic stirrer. A yellowish-clear solution was obtained. TPA-IFA solution was grown  4 on the p-Si wafer and glass substrate by using the spin-coating method at 1000 rpm for 30 seconds. 1.15 mm diameter Al dots contacts were deposited with the same conditions as ohmic contact with a thermal evaporation system for p-Si wafer for electrical I-V measurements (Fig. 1). The samples grown on a glass substrate for Hall measurement were cut into a square of 7 mm by 7 mm, and their contacts were done from each corner with the help of silver paste.  Current-voltage measurements were taken between -3V and +3V step by 0.01V at dark and under different light illumination conditions such as 20 mW.cm-2, 40 mW.cm-2, 60 mW.cm-2, 80 mW.cm-2, and 100 mW.cm-2. Hall measurements were taken by van der Pauw technique at 0.5 T magnetic field between 100K and 400K step by 50K (Fig. 2). \n Fig. 1. Schematic diagram of n-TPA-IFA/p-Si heterojunction diode \n Fig. 2 Schematic diagram of Hall measurements  3. Results and discussion 3.1 Theoretical studies To obtain comprehensive knowledge about the compound's electronic structure, conductivity, and optimal molecular geometry in its ground state, the synthesized TPA-IFA was first computationally evaluated using DFT.  The Gaussian 09 program suite software was used to obtain various properties of our compound, including the optimal structural geometry, analysis of the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO), MEP surface analysis, molecular static isotropic polarizability volume, \n 5 and molar volume [34]. Figure 3 shows the simulated HOMO and LUMO drawings of the molecule at the gas phase using the DFT-B3LYP/6-311G(d,p) computational level. The computed results of the aforementioned molecular descriptors were given in Table 1 based on these values. As seen, the energy band gap in our molecule was calculated theoretically to be 3.37 eV. As is well known from the literature, ∆E offers details about the compound's conductivity. Since semiconductors' ∆E range is between 0.5 to 3.5 eV, TPA-IFA compound may exhibit semiconducting properties. Moreover, the Clausius-Mossotti equation, which provides a connection between a material's molar volume and polarizability and its dielectric constant, is known to be stated as [35]: #!$%#!&'=()*\"+#,-$                                                                                                                     (1) where 𝜀. is the dielectric constant, VM is the molar volume of material, NA is Avogadro’s constant. Also, the 𝛼/ is the polarizability volume in terms of  a/4p𝜀0. For TPA-IFA at the isolated gas phase state, the compound's dielectric constant was found as 3.91. Table 1 Some computed molecular descriptors depending on HOMO and LUMO energy values of the compound. Parameters (eV) Value ELUMO -2.207 EHOMO -5.577 Energy band gap |EHOMO\u0000ELUMO| 3.370 Ionization potential (I = \u0000EHOMO) 5.577 Electron affinity (A = \u0000ELUMO) 2.207 Chemical hardness (\u0000 = (I\u0000A)/2) 1.685 Chemical softness (\u0000 = 1/2\u0000) 0.297 Electronegativity (χ = (I+A)/2) 3.892 Chemical potential (μ = \u0000(I+A)/2) -3.892 Electrophilicity index (\u0000 = μ2/2\u0000\u0000 4.496  \n  EHOMO = -5.577 eV ELUMO = -2.207 eV ΔE = |EHOMO-ELUMO| = 3.370 eV  \n 6 Fig. 3 Simulated HOMO and LUMO drawings of the TPA-IFA at the gas phase using the DFT-B3LYP/6-311G(d,p) computational level. 3.2 Synthesis of TPA-IFA Scheme 1 displays the chemical routes used to synthesize the TPA-IFA compound. Firstly, 4,4'-(phenylazanediyl)dibenzaldehyde (2) was obtained through Vilsmeier formylation of the triphenylamine [33].  For this, triphenylamine (1), which was commercially available, was treated with a POCl3 in DMF for 12 h to give compound 2 in good yield (98%). In a further reaction step, compound 2, treated with the two equimolar amounts of dimethyl 5-amino isophthalate, formed TPA-IFA as a major product with a yield of 55%. \n Scheme 1  Synthetic route for TPA-IFA The 1H NMR spectrum of compound TPA-IFA in CDCl3 is presented in Fig. 4. The structural assignment of TPA-IFA was based on the appearance and disappearance of the 1H NMR signals at δ 8.54 for the CH=N protons peaks of the imine groups and at δ 9.88 for the aldehyde protons of compound 2, respectively. As expected, the characteristic CH3 protons (12H) peaks were observed at 3.99 ppm. In addition, other aromatic ring protons (12H) resonate as singlets at 8.48 (2H) and 8.08 (4H), doublet at 7.64 (d, 4H), and multiplets at 7.38-7.40 (m, 2H), and 7.21-7.28 (m, 5H). In the 13C NMR spectra of compound TPA-IFA, the signals at 166.5 ppm were assigned to carbons of C=O groups, whereas the imine carbons (C=N-) were observed at 161.6 ppm. The NMR data were in very good agreement with the structure of the TPA-IFA molecule. 3.3 The Hall Measurement Fig. 5 shows a color mapping plot of mobility for TPA-IFA. Hall measurements were taken at 0.5 T magnetic field between 100K and 400K step by 50K. The sample showed p-type characterization at 100K and 150 K, while it showed n-type characterization between 200 K and 400 K. The carrier charge density type of the sample changed between 150K and 200K. The mobility of TPA-IFA varied depending on temperature. The mobility increased with increasing temperature between 100K and 250 K. Then it is decreased for 300K and 350K. Last, it increased to 400K. The temperature with the highest mobility value is 250K. According to these results, the TPA-IFA sample shows n-type behavior at room temperature.  NNCO2MeCO2MeNMeO2CCO2MeTHF, 80 0C, 48 hTPA-IFANOHHOCO2MeNH2MeO2CNOHHONPOCl3DMF, 0 0C, 12 h12\n23 7  Fig. 4 1H-NMR spectrum of TPA-IFA \n Fig. 5 Results of hall measurement of the TPA-IFA thin film. 3.4. Current-voltage characterization of n-TPA-IFA/p-Si heterojunction Fig. 7 illustrates the current-voltage (I-V) characteristics of the n-TPA-IFA/p-Si heterostructure under dark and different light intensities. The plots clearly demonstrate that all measurement curves exhibit behavior similar to that of a diode. Rectification behavior can be seen from all curves in Fig. 7, and the rectification rate in the dark is very high at ∓3V, and its value is 1.095x104. The rectification rates for dark and different light intensities at ∓3V \n 8 are given in Table 2. The same graphs additionally demonstrate strong linear behavior at intermediate bias voltages that have distinct slopes, but at sufficiently large forward biases, they deviate from linearity, primarily due to the interface layer. Thermionic emission theory can analyze the I-V characteristics of this type of rectifying contact, and the parameters of the heterojunction can be calculated with the help of the following equation (to understand the formation of the n-TPA-IFA/p-Si heterojunction, see Fig. 6): [36-38]: 𝐼=𝐼0,exp-1(-$34%)678.−11+(-$34%)4%&                                                                                                                    (2) where the reverse saturation current (𝐼0) is also given by the following equation [37]: 𝐼0=𝐴𝐴∗𝑇'𝑒𝑥𝑝-−19'78.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t     (3) where 𝐴∗is the effective Richardson constant, 𝐴 is the active device area, 𝑛 is the ideality factor, 𝛷! is the barrier height, 𝑇 is the temperature, 𝑞 is the electronic charge, 𝑉\tis the applied voltage, 𝑘 is the Boltzmann constant, and 𝑅:;\tis the saturation current.   The ideality factor (𝑛) and barrier height (𝛷!) values of the fabricated heterojunction diode can be calculated by the following equations [39]: 𝑛=178<-<(=63)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t                                                           (4)  𝛷!=781𝑙𝑛->>∗8)3*.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t                                                                                (5)  \n Fig. 6 Layout of structure formation after contact \n 9  Fig. 7 The forward and reverse bias lnI-V characteristics of n-TPA-IFA/p-Si heterojunction diode in dark and under different light intensities. (inset: schematic diagram of circuit model for n-TPA-IFA/p-Si heterojunction diode) From thermionic emission (for V≥3kT⁄q) theory, the intercept and slope of the ln I-V plots of Fig. 7 are used to determine the 𝐼0 ,𝛷! and 𝑛, respectively. Table 2 lists the computed values of  𝐼0 ,𝛷! and 𝑛 values for the n-TPA-IFA/p-Si heterojunction diode in response to varying light intensities and dark. Table 2 illustrates that the barrier height rises with light intensity, and the ideality factor falls comparatively. For the fabricated n-TPA-IFA/p-Si heterojunction diode, the calculated experimental values of \t𝐼0,𝛷! and 𝑛 were 121 nA, 0.716 eV, and 3.01 in the dark and 26.1 nA, 0.755 eV, and 1.63 under 100 mW/cm2. The ideality factor value from Table 2 is greater than the ideal value of 1, indicating that the diode is not behaving ideally. The greater ideality factor value indicates the existence of interface states, an organic TPA-IFA film on the silicon crystal, and series resistance in addition to the inhomogeneities of the Schottky barrier height [40, 41]. As seen in Fig. 7, the photocurrent rises as illumination intensity rises in the reverse bias voltages, confirming that the fabricated heterojunction diode exhibits both photovoltaic and photoconductive activity [42-44]. Also, the circuit diagram of the photodiode is given in Fig. 7 inset. The circuit consists of a parallel shunt resistance (Rsh)- diode connected in series resistance (Rs). Leakage current appeared under light illumination and was shown in the circuit diagram.    \n 10 Table 2 The calculated fundamental experimental parameters of the n-TPA-IFA/p-Si heterojunction diode at dark and under different light intensities.   I-V 𝑅?−𝑉 Light intensity (mW/cm2) RR x103 𝐼0 (nA) 𝑛 𝛷! (eV)     𝑅:                    (Ω)                 𝑅:; (kΩ)  Dark 10.95 121 3.01 0.716 49.3 753 20  3.36 82.1 2.55 0.726 49.9 211 40  2.29 60.6 2.24 0.734 50.6 129 60  1.73 43.9 1.97 0.742 51.2 90 80  1.41 36.0 1.83 0.747 51.7 69 100  1.18 26.1 1.63 0.755 52.2 58  The resistance found in the metal-interfacial interface is called the series resistance in the heterojunction diode. The heterojunction diode's functionality and its capacity to regulate the flow of electrical current may be impacted by this resistance [29]. Numerous elements, including imperfections in the interfacial layer, metal or semiconductor layer contaminants, and variations in the interfacial layer's thickness and quality, can contribute to series resistance. Multiple ways are available for calculating series resistance in a heterojunction diode [40, 45]. The resistance (𝑅?) values of the heterojunction diode can be determined from Ohm’s Law (𝑅?=𝑑𝑉?𝑑𝐼?⁄) both in dark and under different illumination intensities. 𝑅? versus voltage graphs of the n-TPA-IFA/p-Si heterostructure diode for varying the measurement light intensity are shown in Fig. 8. For sufficient lower and higher bias voltages, the values of 𝑅? approach nearly to a constant value; hence, the values of 𝑅? refer to the correct values of the shunt resistance (𝑅:;) and series resistance (𝑅:), respectively, both in the dark and under various levels of light. These values were calculated for the fabricated heterojunction diode and are given in Table 2. Because of the increasing current, particularly at reverse biases, the 𝑅?\tvalues declined almost regularly as the light intensity increased (Fig. 8, Table 2). For a certain type of light sensor application, this outcome is significant. These values are suitable for a heterojunction diode: 58-753 kΩ and 49-52 Ω are the levels of 𝑅:;\tand 𝑅:values.  11  Fig. 8 𝑅?−𝑉 under different light intensities for the n-TPA-IFA/p-Si heterojunction diode.    Photocurrent (𝐼@;), a reverse current, rises with increasing illumination intensities, as shown in Fig. 9. Photodiodes are devices that convert optical signals to electrical current. They have been the subject of extensive research in recent years. Based on light intensity, current measurements verify that the electricity produced photodiode demonstrates photovoltaic activity [46-48]. Through the process of the photovoltaic effect, light energy is transformed into electrical energy by photovoltaic materials. In this context, organic materials such as TPA-IFA thin-film heterojunction can provide several benefits. Photons that penetrate the n-TPA-IFA/p-Si heterojunction diode when it is exposed to light intensity cause electron-hole pairs to form in the semiconductor structures. As a result, a reverse bias voltage is applied to separate the produced electron-hole pairs, and the interface layer's injected holes provide a large photocurrent [49-51]. The following formula yields the photocurrent (𝐼@;): 𝐼@;=𝐼=?A;B−𝐼<C.7                                                                                                               (6) where 𝐼<C.7 and 𝐼=?A;B represent the reverse bias current values at dark and under different light intensities, respectively. This indicates that because higher energy photons are more likely to overcome the energy barrier, at greater light intensities, the photocurrent likewise increases. The drift velocity of photogenerated electrons and holes at the TPA-IFA/p-Si film interface may cause a rise in photocurrent [52, 53].  Additionally, the interaction between the light intensity (𝑃) and the photodiode's photocurrent (𝐼@;) is shown below [52]: 𝐼@;=𝐴𝑃D\t\t;\t\t\t\t\t\t𝑙𝑛𝐼@;=𝑚𝑙𝑛𝑃+𝐴                                                                                      (7) \n 12 where 𝑚 is an exponent determining the photoconduction process and A is the proportionality constant. The n-TPA-IFA/p-Si heterojunction photodiode's log(𝐼@;)-log(𝑃) plot is shown in Fig. 9, and the magnitude 𝑚 was calculated from its slope, which was 0.621 at –3.0 V. The photoconductive transient mechanism of the n-TPA-IFA/p-Si heterostructure was verified by the linearity seen in Fig. 9, making it a promising option for photodetection applications [54]. The value of the 𝑚 can be used to identify the nature of the photo-conducting mechanism. If this value falls between 0.5 and 1, it indicates that localized trap levels are continuously distributed [55, 56].  \n Fig. 9 log(𝐼@;)-log(𝑃) plot for the n-TPA-IFA/p-Si heterojunction photodiode. Transient photocurrent measurements of the diode were carried out for additional photoresponse analysis, and the results are displayed in Fig. 10. The transient current measurements provide insight into the current conduction mechanism and aid in the determination of several photodetector properties, including photosensitivity, responsivity, and detectivity, for variations in light intensity [50, 57]. As seen in Fig. 10, when the diode is illuminated, electrons contribute to the current, and the quantity of photogenerated charge carriers increases with illumination. Both the diode's current and the quantity of free electrons decrease when the light is turned off. As a result, the trap centers found in the TPA-IFA organic material determine the photoconducting characteristics of the diode.     \n 13  Fig. 10 Transient photocurrent measurements of n-TPA-IFA/p-Si heterojunction photodiode depending on light intensity Through the utilization of semiconductors, photovoltaic devices can transform solar energy into electrical energy. Because of the photoelectric effect, these semiconductor structures can take in photons from the sun and then emit electrons as a result of the process. When photons have energies higher than or equal to the energy gap (hv ≥ Eg), they can cause additional electrons in the valence band (Ev) and trap levels to move to the conduction band (Ec) or trap levels, thus absorbing the photons. This process is known as photocurrent (𝐼@;). Depending on the intensity of the light, this electron mobility results in the formation of the open-circuit voltage (𝑉EF) and short-circuit current (𝐼:F) at I =0 A and V=0 V, respectively. The characteristics of current density (𝐽) against voltage of the n-TPA-IFA/p-Si heterojunction diode are shown in Fig. 11 under different light intensities (from 20 to 100 mW/cm2). The inset of Fig. 11 shows the characteristics of short circuit current density (𝐽:F) and open circuit voltage (𝑉EF) against the light intensity of the n-TPA-IFA/p-Si heterojunction diode. Also, the calculated values of short circuit current density (𝐽:F) and open circuit voltage (𝑉EF) with light intensity are illustrated in Table 3. It presents a notable rise in open-circuit voltage (𝑉EF) and short-circuit current (𝐽:F) values with increasing light intensity, as seen in Fig. 11. This is because a heterojunction diode will absorb more light and produce more photon carriers when the light intensity is increasing. Because the n-TPA-IFA/p-Si structure has photovoltaic characteristics, it can, therefore, be employed as a photodiode. Consequently, the research findings demonstrated that when light intensity increased, the photovoltaic characteristics increased.  \n 14  Fig. 11 Current density-voltage (J-V) plots of the n-TPA-IFA/p-Si heterojunction photodiode under different light intensities. The inset shows the changing of 𝐽:F and 𝑉EF with light intensity (mW/cm2) for the heterojunction photodiode.  The following equations for photoresponsivity (R), photosensitivity (PS), specific detectivity (𝐷∗), and linear dynamic range (LDR) were used to determine the detector characteristics of the n-TPA-IFA/p-Si heterostructure diode [58, 59]. 𝑅=3+,-&.$3/0!1G.>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(8) 𝑃𝑆(%)=3+,-&.$3/0!13/0!1\tx100                                                                                                     (9) 𝐷∗=𝑅H>'13/0!1                                                                                                                      (10) LDR=20log-3+,-&.3/0!1.\t                                                                                                             (11) Here, 𝐴\trepresents the detector's active area, and 𝑃 represents the incident light density. One important metric that shows how a photodiode reacts to incoming light is photoconductive photoresponsivity (𝑅) [60]. Its definition is as follows: it is the ratio of photocurrent generated to the  optical power of incoming light. The variation of 𝑅 for the photodiode with an applied reverse bias voltage at light intensities ranging from 20 to 100 mW.cm−2 is depicted in Fig. 12. The change of 𝑅 depending on the light power applied at -3V is given in the inset of Fig. 12. The various values that were found range from 11.20 to 20.59 mA/W, which is advantageous for high-speed diode operation [61, 62]. 𝑅 values depending on light intensity are also given in Table 3. \n 15  \n Fig. 12 Plot of 𝑅 vs. 𝑉 of the n-TPA-IFA/p-Si heterojunction diode. The inset shows the plot 𝑅 vs. 𝑃 of the heterojunction at -3V. The ratio of the photocurrent to the dark one for the heterojunction diode is known as photosensitivity (𝑃𝑆(%)). Measurements of photosensitivity characteristics as a function of light intensities are also made to determine whether the device is suitable for use with photodiodes. This crucial variable can also be employed to maintain the level of light-to-current conversion. Figures 13 display the reverse bias voltage profile of the photosensitivity (𝑃𝑆(%)) of the n-TPA-IFA/p-Si heterostructure. The change of 𝑃𝑆(%) depending on the light power (𝑃) applied at -3V is given in the inset of Fig. 13. From the inset of Fig. 13, the values of 𝑃𝑆(%) were found to range from 321.76 to 875.44 for 20 and 100 mW/cm2, respectively. 𝑃𝑆(%) values depending on light intensity are also given in Table 3.  \n 16  Fig. 13 Plot of 𝑃𝑆(%) vs. 𝑉 of the n-TPA-IFA/p-Si heterojunction diode. The inset shows the plot 𝑃𝑆(%)  vs. 𝑃 of the heterojunction at -3V. The primary parameter for a photodetector is its specific detectivity (D*), which is the minimum light intensity that a diode can detect. In other words, 𝐷∗describes a photodetector's capacity to detect low-light signals. This parameter is based on the produced photodiode's noise and responsivity. Figure 14 displays the reverse bias voltage profile of the specific detectivity (𝐷∗) of the n-TPA-IFA/p-Si heterostructure. The change of 𝐷∗depending on the light power (𝑃) applied at -3V is given in the inset of Fig. 14. By increasing the light intensity from 20 to 100 mW/cm2, the computed values of 𝐷∗at -3V decreased from 3.22x109 Jones to 1.75x109 Jones. Fig. 18 and the inset of Fig. 18 show how the 𝐷∗\tvaries when the reverse bias voltage and light intensity change. Also, 𝐷∗ values depending on light intensity are also given in Table 3. The produced n-TPA-IFA/p-Si photodetector's predicted LDR values increase from 10.15 to 18.84 dB when light intensity is raised from 20 to 100 mW/cm2. LDR is essential because image sensors must function over various intensities. For example, if the LDR is sufficiently wide, a clear picture can be created in any situation [63, 64]. LDR values rise as the light intensity rises. Considering the findings, the n-TPA-IFA/p-Si heterojunction diode can be used as an image sensors and photo-devices. LDR values depending on light intensity are also given in Table 3.   \n 17  Fig. 14 Plot of 𝐷∗vs. 𝑉 of the n-TPA-IFA/p-Si heterojunction diode. The inset shows the plot  𝐷∗  vs. 𝑃 of the heterojunction at -3V. \n Fig. 15 Plot of LDR vs. 𝑉 of the n-TPA-IFA/p-Si heterojunction diode. The inset shows the plot LDR vs. 𝑃 of the heterojunction at -3V.  \n 18 Table 3 Photovoltaic and photodetector parameters of n-TPA-IFA/p-Si heterojunction under different light intensities. Power (mW/cm2) Voc (V) Jsc (𝜇A) R  PS(%)  D* (Jones)x109 LDR (dB) Iph (𝜇A) at −3.0 V 20  0.0185   0.083    20.60    321.77      3.22      10.15        5.14 40  0.0307   0.149    14.93    466.59      2.33      13.38        7.95 60  0.0395   0.208   12.99    609.17      2.03      15.69       10.95 80  0.0446   0.253   11.87    741.97      1.86      17.41       13.79 100  0.0499   0.293   11.21   875.44      1.75      18.84       16.53   4. Conclusions In conclusion, we report for the first-time n-type TPA-IFA (\"D-A-π-A\") compound, in which triphenylamine is used as a donor, C=N imine part is an acceptor, benzene is a π-spacer, and ester is an acceptor/anchor. Also, The HOMO and LUMO energies of the TPA-IFA were calculated by DFT/B3LYP/6-311G(d,p) method using Gaussian 09 W. The hall measurements showed that the TPA-IFA sample shows n-type behavior at room temperature. Then, the spin coating method was used to create a heterojunction device based on an n-type TPA-IFA thin film on the p-type Si substrate. We investigated the photodiode and photodetector parameters of the n-TPA-IFA/p-Si heterojunction diode in both dark and different light conditions, considering its possible optoelectronic applications. The n-TPA-IFA/p-Si heterojunction diode's ideality factor and Schottky barrier height were found to be, respectively, 0.716 eV and 3.01 in the dark and 0.755 eV and 1.63 in the 100 mW/cm2. In this phenomenon, the ideality factor decreases as the Schottky barrier height increases with increasing light intensity. A heterojunction with a high rectification ratio of 104 and a low series resistance of 49 Ω was produced. Its lower series resistance value demonstrates the heterostructure's capability for effective photovoltaic cells. A large rectification ratio is seen for heterostructure contact. A decrease in tunneling resistance and space charge modulation across the interface may be the reason for enhanced electrical properties of heterostructures, such as an increase in barrier height and an ideality factor that deviates from unity.    To further investigate the heterojunction diode's response to light, measurements of its photoresponsivity (𝑅), photosensitivity (𝑃𝑆(%)), specific detectivity (𝐷∗), and linear dynamic range (LDR) were made. The photodiode's 𝑅, 𝑃𝑆(%), 𝐷∗, and LDR values were found to be 11.20 mA/W, 321.76, 3.22x109 Jones, and 10.15 dB at -3 V and 20 mW/cm2, respectively.  According to all the findings, the TPA-IFA organic material is a strong candidate for optoelectronic applications. From the results obtained, it was concluded that the newly synthesized TPA-IFA organic material is a photoactive n-type material and improves device performance, and as a result, it can be used as low-cost optoelectronic devices.   19 Acknowledgments The authors would like to thank Dr. Sait Malkondu from Giresun University for his help in the synthesis of the TPA-IFA compound. The authors wish to acknowledge the financial support of the Scientific Research Projects Foundation of Gazi University. Author Contributions ŞC: Investigation, Methodology, Data Analysis, Review and editing, Methodology. PO: Conceptualization, Experimentation, Data Analysis, Writing and editing. SE: Conceptualization, Experimentation, Data Analysis, Writing and editing. NT: Conceptualization, Experimentation, Data Analysis, Writing and editing. Funding This work was financially supported by the Scientific Research Projects Foundation of Gazi University (BAP- FOA-2022-7389) Data availability The authors of this study declare that the data that supports their findings is available within the paper. The datasets that were generated and/or analyzed during the study are also available from the corresponding author upon reasonable request. Declarations Competing interests Şükrü Çavdar, Pınar Oruç, Serkan Eymur, and Nihat Tuğluoğlu declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  References 1. L. Alibabaei, J.H. Kim, M. Wang, N. Pootrakulchote, et al., Molecular design of metal-free D–π-A substituted sensitizers for dye-sensitized solar cells, Energy & Environmental Science 3 (11), 1757-1764 (2010). htttp://doi.org/10.1039/C0EE00218F 2. Y.J. Chang and T.J. Chow, Dye-sensitized solar cell utilizing organic dyads containing triarylene conjugates, Tetrahedron 65 (24), 4726-4734 (2009). https://doi.org/10.1016/j.tet.2009.04.024 3. A. Mishra, M.K.R. Fischer, and P. Bäuerle, Metal-Free Organic Dyes for Dye-Sensitized Solar Cells: From Structure: Property Relationships to Design Rules, Angew. Chem. Int. Ed. 48 (14), 2474-2499 (2009). https://doi.org/10.1002/anie.200804709 4. M. Erdoğan, A novel dibenzosuberenone bridged D-A-π-A type dye: Photophysical and photovoltaic investigations, J. Mol. Struct. 1232, 130056 (2021). https://doi.org/10.1016/j.molstruc.2021.130056 5. Y. Li, J. Liu, D. Liu, X. Li, et al., D-A-π-A based organic dyes for efficient DSSCs: A theoretical study on the role of π-spacer, Computational Materials Science 161, 163-176 (2019). https://doi.org/10.1016/j.commatsci.2019.01.033  20 6. J. Yang, X. Wang, W.-L. Yim, and Q. Wang, Computational Study on the Intramolecular Charge Separation of D-A-π-A Organic Sensitizers with Different Linker Groups, The Journal of Physical Chemistry C 119 (47), 26355-26361 (2015). http://doi.org/10.1021/acs.jpcc.5b09117 7. M.F. Şahin, E. Taşcı, M. Emrullahoğlu, H. Gökce, et al., Electrical, photodiode, and DFT studies of newly synthesized π-conjugated BODIPY dye-based Au/BOD-Dim/n-Si device, Physica B 614, 413029 (2021). https://doi.org/10.1016/j.physb.2021.413029 8. Ş. Çavdar, S. Izmirli, H. Koralay, N. Turan, et al., Optoelectronic Properties of Triphenylamine Organic Thin Film Layered Al/p-Si /TPA/Al Heterojunction for Photodiode Application, ECS J. Solid. State. Sci. Technol. 12 (4), 045001 (2023). https://doi.org/10.1149/2162-8777/acc68c 9. B. Topaloğlu Aksoy, G. Keşan, E. Özcan, E. Tanrıverdi Eçik, et al., Solution-processable BODIPY decorated triazine photodiodes and their comprehensive photophysical evaluation, New J. Chem. 44 (5), 2155-2165 (2020). https://doi.org/10.1039/C9NJ05662A 10. D. Yang and D. Ma, Development of Organic Semiconductor Photodetectors: From Mechanism to Applications, Adv. Opt. Mater. 7 (1), 1800522 (2019). https://doi.org/10.1002/adom.201800522 11. S. Demirezen, M. Ulusoy, H. Durmuş, H. Cavusoglu, et al., Electrical and Photodetector Characteristics of Schottky Structures Interlaid with P(EHA) and P(EHA-co-AA) Functional Polymers by the iCVD Method, ACS Omega 8 (49), 46499-46512 (2023). https://doi.org/10.1021/acsomega.3c04935 12. A. Kocyigit, S. Aydogan, I. Ü, and M. Yılmaz, The Electrical Characteristics of the Co/Giemsa/n-Si Heterostructure Depending on Measurement Temperatures and Frequencies, IEEE Sens. J. 23 (8), 8184-8191 (2023). https://doi.org/10.1109/JSEN.2023.3255180 13. N. Kaplan, E. Tasci, M. Emrullahoglu, H. Gökce, et al., Analysis of illumination dependent electrical characteristics of α- styryl substituted BODIPY dye-based hybrid heterojunction, J. Mater. Sci.: Mater. Electron. 32 (12), 16738-16747 (2021). https://doi.org/10.1007/s10854-021-06231-8 14. H.H. Gullu, D.E. Yildiz, A. Kocyigit, and M. Yildirim, Electrical properties of Al/PCBM:ZnO/p-Si heterojunction for photodiode application, J. Alloys Compd. 827 (2020). https://doi.org/10.1016/j.jallcom.2020.154279 15. S. Eymur, N. Tugluoglu, A. Apaydin, Ü. Akin, et al., Illumination Dependent Electrical and Photovoltaic Properties of Au/n-Type Si Schottky Diode with Anthracene-Based NAMA Interlayer, ECS J. Solid. State. Sci. Technol. 10 (5) (2021). https://doi.org/10.1149/2162-8777/abf9ec 16. S. Özden, N. Avci, O. Pakma, and I.A. Kariper, Influence of illumination intensity on the electrical properties of Al/NOA65/p-Si/Al heterojunction MPS device, J. Mater. Sci.: Mater. Electron. 33 (16), 12796-12807 (2022). https://doi.org/10.1007/s10854-022-08225-6 17. K.A. Hussain, G.M. Ali, A. Boubaker, and A. Kalboussi, Structural, Electrical, and Photodetection Characterization of a Pentacene Interdigitated MSM Photodiode, J. Electron. Mater. (2023). https://doi.org/10.1007/s11664-023-10826-8 18. M. Jin, Y. Chen, W. Song, H. Lian, et al., Synthesis, characterization, and electroluminescent properties of indazole, pyrazole, and triazole/triphenylamine-based compounds, Dyes and Pigments 173, 106912 (2020). https://doi.org/10.1016/j.dyepig.2018.07.058  21 19. T. Keawin, N. Prachumrak, S. Namuangruk, S. Pansay, et al., Efficient bifunctional materials based on pyrene- and triphenylamine-functionalized dendrimers for electroluminescent devices, RSC Adv. 5 (90), 73481-73489 (2015). https://doi.org10.1039/C5RA07161E 20. S.-L. Lai, Q.-X. Tong, M.-Y. Chan, T.-W. Ng, et al., Distinct electroluminescent properties of triphenylamine derivatives in blue organic light-emitting devices, J. Mater. Chem. 21 (4), 1206-1211 (2011). https://doi.org/10.1039/C0JM02550J 21. S. Tao, Y. Zhou, C.-S. Lee, S.-T. Lee, et al., A triphenylamine derivative as a single-emitting component for highly-efficient white electroluminescent devices, J. Mater. Chem. 18 (33), 3981-3984 (2008). https://doi.org10.1039/B800985F 22. P. Agarwala and D. Kabra, A review on triphenylamine (TPA) based organic hole transport materials (HTMs) for dye sensitized solar cells (DSSCs) and perovskite solar cells (PSCs): evolution and molecular engineering, J Mater. Chem. 5 (4), 1348-1373 (2017). https://doi.org/10.1039/C6TA08449D 23. F. Bahrani, R. Hameed, S. Resan, and M. Al-Anber, Impact of Torsion Angles to Tune Efficient Dye-Sensitized Solar Cell/Donor-π-Acceptor Model Containing Triphenylamine: DFT/TD-DFT Study, Acta Physica Polonica A 141 (6), 561-568 (2022). https://doi.org10.12693/APhysPolA.141.561 24. S. Bibi, R. Ahmad Khera, A. Farhat, and J. Iqbal, Triphenylamine based donor-acceptor-donor type small molecules for organic solar cells, Computational and Theoretical Chemistry 1198, 113176 (2021). https://doi.org/10.1016/j.comptc.2021.113176 25. G. Deogratias, O.S. Al-Qurashi, N. Wazzan, T. Pogrebnaya, et al., Effect of substituent in the acceptor on optical and electronic properties of triphenylamine based dyes: A density functional theory/time-dependent density functional theory investigation, Materials Science in Semiconductor Processing 150, 106935 (2022). https://doi.org/10.1016/j.mssp.2022.106935 26. A.E. Bejan, C.P. Constantin, and M.D. Damaceanu, Evidence of diimide structure variation on overall performance of electro(fluoro)chromic devices integrating versatile triphenylamine-based polyimides, Mater. Today Chem. 26, 101100 (2022). https://doi.org/10.1016/j.mtchem.2022.101100 27. M. Erdoğan, Z. Orhan, and E. Daş, Synthesis of electron-rich thiophene triphenylamine based organic material for photodiode applications, Opt. Mater. 128, 112446 (2022). https://doi.org/10.1016/j.optmat.2022.112446 28. T.K. Singh, R.A. Grandy, E.S. Dennis, A.S.B. Schouten, et al., Exploring Structure-Property Relationships in a Family of Ferrocene-Containing, Triphenylamine-Based Hybrid Organic Dyes, Applied Sciences 12 (12), 6001 (2022). https://doi.org/10.3390/app12126001 29. Y.-K. Wang, Z.-C. Yuan, G.-Z. Shi, Y.-X. Li, et al., Dopant-Free Spiro-Triphenylamine/Fluorene as Hole-Transporting Material for Perovskite Solar Cells with Enhanced Efficiency and Stability, Adv. Funct. Mater. 26 (9), 1375-1381 (2016). https://doi.org/10.1002/adfm.201504245 30. M. Yahya, A. Suvitha, and A. Bouziani, Structural, photophysical and optoelectronic activity of triphenylamine-based push–pull chromophores: a theoretical study, Opt. Quantum Electron. 54 (12), 816 (2022). https://doi.org/10.1007/s11082-022-04239-w 31. B. Dutta and S. Halder, Fabrication of Schottky Barrier Diodes Utilizing Schiff Base Compounds and Metal Complexes with Schiff Base Ligands, ChemistrySelect 8 (24), e202301586 (2023). https://doi.org/10.1002/slct.202301586  22 32. G.D. Sharma, Electrical and photoelectrical properties of Schottky barrier devices using the chloro aluminium phthalocyanines, Synth. Met. 74 (3), 227-234 (1995). https://doi.org/10.1016/0379-6779(95)03360-V 33. G. Lai, X.R. Bu, J. Santos, and E.A. Mintz, Reinvestigation of the Vilsmeier-Haack Formylation of Triphenylamine, Synlett 1997 (11), 1275-1276 (1997). https://doi.org/10.1055/s-1997-1024 34. M.J. Frisch, G.W. Trucks, H.B. Schlegel, and G.E.e.a. Scuseria, Gaussian 16 Rev. C.01. 2016: Wallingford, CT. 35. E. Talebian and M. Talebian, A general review on the derivation of Clausius–Mossotti relation, Optik 124 (16), 2324-2326 (2013). https://doi.org/10.1016/j.ijleo.2012.06.090 36. N. Tugluoglu, Ö. Yüksel, H. Safak, and S. Karadeniz, The double Gaussian distribution of inhomogeneous barrier heights in the organic-on-inorganic Schottky devices, Physica Status Solidi a-Applications and Materials Science 209 (11), 2313-2316 (2012). https://doi.org/10.1002/pssa.201228163 37. A.O. Tezcan, S. Eymur, E. Tasci, M. Emrullahoglu, et al., Investigation of electrical and photovoltaic properties of Au/n-Si Schottky diode with BOD-Z-EN interlayer, Journal of Materials Science-Materials in Electronics 32 (9), 12513-12520 (2021). https://doi.org/10.1007/s10854-021-05886-7 38. S. Çavdar, Y. Sahin, N. Turan, H. Koralay, et al., Structural and electrical characterization of Cd-doped ZnO thin films produced on p-type Si substrate by SILAR technique, J. Mater. Sci.: Mater. Electron. 34 (25) (2023). https://doi.org/10.1007/s10854-023-11134-x 39. O. Ongun, E. Tasci, M. Emrullahoglu, Ü. Akin, et al., Fabrication, illumination dependent electrical and photovoltaic properties of Au/BOD-Pyr/n-Si/In schottky diode, J. Mater. Sci.: Mater. Electron. 32 (12), 15707-15717 (2021). https://doi.org/10.1007/s10854-021-06122-y 40. A. Barkhordari, H.R. Mashayekhi, P. Amiri, S. Altindal, et al., Role of graphene nanoparticles on the electrophysical processes in PVP and PVP:ZnTiO3polymer layers at Schottky diode (SD), Semiconductor Science and Technology 38 (7) (2023). https://doi.org/10.1088/1361-6641/acd2fa 41. K. Moraki, S. Bengi, S. Zeyrek, M.M. Bülbül, et al., Temperature dependence of characteristic parameters of the Au/C20H12/n-Si Schottky barrier diodes (SBDs) in the wide temperature range, J. Mater. Sci.: Mater. Electron. 28 (5), 3987-3996 (2017). https://doi.org/10.1007/s10854-016-6011-2 42. C.A. Canbay, A. Tataroğlu, A. Dere, A.G. Al-Sehemi, et al., Electrical, kinetic and photoelectrical properties of CuAlMnMg shape memory alloy/n-Si Schottky diode, J. Alloys Compd. 888, 161600 (2021). https://doi.org/10.1016/j.jallcom.2021.161600 43. A.G. Al-Sehemi, K. Ocakoglu, M. Ince, A. Karabulut, et al., The electronic and optoelectronic properties of Al/hydroxymethyl functionalized Zn(II)Pc/p-Si photonic device, Polymer Bulletin (2023). https://doi.org/10.1007/s00289-023-04906-2 44. D. Wood, Optoelectronic Semiconductor Devices. 1994, New York: Prentice Hall. 45. S. Altindal, Y. Azizian-Kalandaragh, M. Ulusoy, and G. Pirgholi-Givi, The illumination effects on the current conduction mechanisms of the Au/(Er2O3:PVC)/n-Si (MPS) Schottky diodes, Journal of Applied Polymer Science 139 (27) (2022). https://doi.org/10.1002/app.52497 46. E. Özcan, B. Topaloğlu Aksoy, E. Tanrıverdi Eçik, A. Dere, et al., Fabrication of hybrid photodiode systems: BODIPY decorated cyclotriphosphazene covalently grafted graphene oxides, Inorganic Chemistry Frontiers 7 (16), 2920-2931 (2020). https://doi.org/10.1039/D0QI00468E  23 47. H. Seymen and Ş. Karataş, Investigation of electrical, photodiode and photovoltaic properties of Au/SiO2/n-Si structures with GO and P3C4MT interface, Mater. Chem. Phys. 310, 128449 (2023). https://doi.org/10.1016/j.matchemphys.2023.128449 48. A.G. Al-Sehemi, A. Tataroglu, A. Dere, A. Karabulut, et al., The structural, electrical, and photoelectrical properties of Al/Cu2CdSnS4 chalcogenide film/p-Si Schottky-type photodiode, J. Mater. Sci.: Mater. Electron. 34 (30) (2023). https://doi.org/10.1007/s10854-023-11465-9 49. A.A. Noroozi and Y. Abdi, A graphene/Si Schottky diode for the highly sensitive detection of protein, RSC Adv. 9 (34), 19613-19619 (2019). https://doi.org/10.1039/C9RA03765A 50. D.E. Yıldız, A. Kocyigit, and M. Yıldırım, Investigation of Ag/ZnO/p-Si heterostructure for diode and photodiode applications in visible spectrum, Physica Scripta 99 (1), 015913 (2024). https://doi.org/10.1088/1402-4896/ad0d6e 51. S.A. Halim, A.-S. Badran, N. Roushdy, E.M. Ahmed, et al., A new hybrid structure based Pyranoquinoline-Pyridine derivative: Synthesis, optical properties, theoretical analysis, and photodiode applications, J. Mol. Struct. 1293, 136233 (2023). https://doi.org/10.1016/j.molstruc.2023.136233 52. A.A.M. Farag, A. Ashery, E.M.A. Ahmed, and M.A. Salem, Effect of temperature, illumination and frequency on the electrical characteristics of Cu/p-Si Schottky diode prepared by liquid phase epitaxy, J. Alloys Compd. 495 (1), 116-120 (2010). https://doi.org/10.1016/j.jallcom.2010.01.098 53. A. Alshahrie, A.A. Al-Ghamdi, M.S. Abdel-wahab, and W.E. Mahmoud, The structural and optoelectronic properties of Cu1-xTixO (0≤x≤0.05) diodes prepared via a co-sputtering technique, Micro and Nanostructures 164, 107115 (2022). https://doi.org/10.1016/j.spmi.2021.107115 54. M.M. Furchi, D.K. Polyushkin, A. Pospischil, and T. Mueller, Mechanisms of Photoconductivity in Atomically Thin MoS2, Nano Letters 14 (11), 6165-6170 (2014). https://doi.org/10.1021/n1502339q 55. S. Demirezen, H.G. Çetinkaya, M. Kara, F. Yakuphanoğlu, et al., Synthesis, electrical and photo-sensing characteristics of the Al/(PCBM/NiO: ZnO)/p-Si nanocomposite structures, Sensors and Actuators A: Physical 317, 112449 (2021). https://doi.org/10.1016/j.sna.2020.112449 56. B.A. Gozeh, A. Karabulut, C.B. Ismael, S.I. Saleh, et al., Zn-doped CdO effects on the optical, electrical and photoresponse properties of heterojunctions-based photodiodes, J. Alloys Compd. 872, 159624 (2021). https://doi.org/10.1016/j.jallcom.2021.159624 57. A. Karabulut, A. Dere, O. Dayan, A.G. Al-Sehemi, et al., Silicon based photodetector with Ru(II) complexes organic interlayer, Materials Science in Semiconductor Processing 91, 422-430 (2019). https://doi.org/10.1016/j.mssp.2018.11.035 58. M. Ashtar, K. Yao, M.A. Marwat, J. Yang, et al., High performance self-powered photodetector based on CuBi2O4/MAPbI3 heterostructure, Vacuum 219, 112759 (2024). https://doi.org/10.1016/j.vacuum.2023.112759 59. S. Demirezen, A. Dere, H.G. Çetinkaya, A.G. Al-Sehemi, et al., Hybrid photonic device based on Graphene Oxide (GO) doped P3HT-PCBM/p-Silicon for photonic applications, Physica Scripta 98 (11), 115916 (2023). https://doi.org/10.1088/1402-4896/acfce2 60. F. Sarcan, U. Doğan, A. Althumali, H.B. Vasili, et al., A novel NiO-based p-i-n ultraviolet photodiode, J. Alloys Compd. 934, 167806 (2023). https://doi.org/10.1016/j.jallcom.2022.167806  24 61. T. Ji, Q. Liu, R. Zou, Y. Zhang, et al., Enhanced UV-visible light photodetectors with a TiO2/Si heterojunction using band engineering, J. Mater. Chem. C . 5 (48), 12848-12856 (2017). https://doi.org/10.1039/C7TC04811D 62. V. Balasubramani, J. Chandrasekaran, V. Manikandan, T.K. Le, et al., Upgraded photosensitivity under the influence of Yb doped on V2O5 thin films as an interfacial layer in MIS type Schottky barrier diode as photodiode application, J. Solid State Chem. 301, 122289 (2021). https://doi.org/10.1016/j.jssc.2021.122289 63. S. Demirezen, A.G. Al-Sehemi, A. Yüzer, M. Ince, et al., Electrical characteristics and photosensing properties of Al/symmetrical CuPc/p-Si photodiodes, J. Mater. Sci.: Mater. Electron. 33 (26), 21011-21021 (2022). https://doi.org/10.1007/s10854-022-08906-2 64. H. Ko, S. Park, H.J. Son, and D.S. Chung, Wide-Linear-Dynamic-Range Polymer Photodiode with a New Benzo[1,2-b:4,5-b′]dithiophene-Copolymer: The Role of Crystalline Orientation, Chemistry of Materials 32 (7), 3219-3228 (2020). https://doi.org/10.1021/acs.chemmater.0c00347  "    },    {        "title": "2402.04908v3.Explicit_lower_bounds_for_the_height_in_Galois_extensions_of_number_fields.pdf",        "content": "arXiv:2402.04908v3  [math.NT]  23 Feb 2024Explicit lower bounds for the height in Galois\nextensions of number fields∗\nJonathan Jenvrin\nAbstract\nAmoroso and Masser proved that for every real ǫ >0, there is a\nconstant c(ǫ)>0, with the property that, for every algebraic number\nαsuch that Q(α)/Qis a Galois extension, the height of αis either 0\nor at least c(ǫ)[Q(α) :Q]−ǫ. In this article, we establish an explicit\nversion of this theorem.\n1 Introduction\nIn this article, we let Qbe a fixed algebraic closure of Q. For an algebraic\nnumberα, we denote by h(α)≥0its absolute logarithmic Weil height.\nIt’s well know by Kronecker’s theorem that h(α) = 0 if and only if α= 0\norαis a root of unity. Lehmer’s conjecture predicts the existen ce of a positive\nconstant csuch that\nh(α)≥cd−1\nwhenever α/\\e}atio\\slash= 0has degree doverQand is not a root of unity.\nThis is obviously true for αnot a unit with c= log(2) . The conjecture\nhas been established for various classes of α. Forαnon-reciprocal (which\nis always the case for dodd) the result holds true with c= 3h(θ) = log(θ),\nwhereθis the real root >1ofX3−X−1(see [Sch73]). If αbelongs to an\nabelian extension and h(α)/\\e}atio\\slash= 0, we have h(α)≥log(5)\n12(see [AD00]). For αin\na CM-field with |α| /\\e}atio\\slash= 1, we have h(α)≥1\n2log(1+√\n5\n2)(see [Sch73]). The best\nunconditional result we have to Lehmer’s conjecture is Dobr owolski’s result\n∗This work has been partially supported by the Institut Fouri er, Université Grenoble1,\nUMR 5582 duCNRS, 100 rue des mathématiques, BP 74, 38402 St Ma rtin d’hères.\n1[Dob79], which implies that for any ǫ >0, there is c(ǫ)>0such that either\nh(α) = 0 orh(α)≥c(ǫ)d−1−ǫ.\nAmoroso and David proved Lehmer’s conjecture when Q(α)/Qis a Galois\nextension, see [AD99, Corollaire 1.7]. Later, Amoroso and M asser [AM16]\nproved the following stronger result.\nTheorem 1.1. For anyǫ >0, there is a positive effective constant c(ǫ)with\nthe following property. If α∈Q∗is of degree doverQ,αis not a root of\nunity and Q(α)/Qis Galois, then\nh(α)≥c(ǫ)d−ǫ.\nThe goal of our article is to give a explicit version of this th eorem. Our\nmain result is the following:\nTheorem 1.2. Supposeα∈Q∗is of degree doverQ. Ifαis not a root of\nunity and Q(α)/Qis Galois, then\nh(α)≥exp/parenleftbig\n−500log(3 d)3/4log(log(3 d))/parenrightbig\n.\nTheorem 1.2 is proved in Section 3. Our proof strategy relies on that of\nAmoroso and Masser in [AM16, Theorem 3.3]. There will be two c ases. They\nare related to the comparison of the multiplicative rank ρof the conjugates\nofα, andd. Ifρis big in terms of d, then they originally use the main\nresult of [AD99]. It says that for any α1,...,α rmultiplicatively independent\nalgebraic number and any ǫ >0, there exists C(ǫ)such that\nmax\nih(αi)≥C(ǫ)D−1/r−ǫ\nwhereD= [Q(α1,...,α r) :Q]. We will instead use a more precise and\nexplicit result from [AV12] (see Theorem 2.1). If ρis small in terms of d,\nthen they use the relative Dobrowolski lower bound of [AZ10] . Again, we will\ninstead apply a more precise and explicit result from [AD07] (see Theorem\n2.2). We moreover need an explicit estimate for the Euler Tot ient function\n(see Lemma 2.2).\n2 Auxiliary Results\nWe state two important results that will be the key ingredien ts in our proof.\nThe first was proved by Amoroso and Viada in [AV12, Corollary 1 .6], where,\n2more generally, they give an explicit version of a generaliz ed Dobrowolski\nresult on Lehmer’s problem.\nTheorem 2.1 ([AV12, Corollary 1.6]) .Letα1,...,α nbe multiplicatively in-\ndependent algebraic numbers in a number field K. Then\nh(α1)...h(αn)≥[K:Q]−1(1050n5log(3[K:Q]))−n2(n+1)2.\nThe second result was proved by Amoroso and Delsinne in [AD07 , Théorème\n1.3]. We recall a special case of this theorem, which is enoug h for our pur-\nposes.\nTheorem 2.2 ([AD07, Théorème 1.3]) .Letα∈Q∗be not a root of unity\nand letLbe a number field. Then if L/Qis a finite abelian extension and\nD= [L(α) :L], we have\nh(α)≥D−1loglog(5D)3\nlog(2D)4.\nThe following lemma is a general version of [AM16, Lemma 2.2] , with an\nimprovement on the value of n(ρ).\nLemma 2.1. LetF/Kbe a finite Galois extension and α∈F\\{0}. Letρbe\nthe multiplicative rank of the conjugates α1,...,αdofαoverK, and suppose\nρ≥1. Then there exists a subfield L⊂Fwhich is Galois over Kof degree\n[L:K]≤n(ρ)and an integer k≥1, such that Fcontains a primitive k-th\nroot of unity ζkandαk∈L. We can take\nn(ρ) =ρ!2ρforρ= 1,3,5andρ >10.\nOtherwise we can take\nn(ρ) = 135ρ!2ρ−1.\nProof. Letkbe the order of the group of roots of unity in F, in particular\nFcontains K(ζk). Define βi=αk\nifor1≤i≤dandL=K(β1,...,β d). We\nhaveL⊂FbecauseF/Kis Galois, and we easily check that L/Kis Galois.\nTheZ−module\nM={βa1\n1...βad\nd|a1,...,a d∈Z}\nis torsion free by the choice of kand so, by the Classification Theorem for\nabelian group, is free, of rank ρ. This shows that the action of Gal(L/K)\n3overMdefines an injective representation from Gal(L/K)toGLρ(Z), hence\nGal(L/K)identifies with a finite subgroup of GLρ(Z). To conclude, we use a\ntheorem stated by Feit in 1996, and proved by Rémond in [Rém20 , Théorème\n7.1], which, in particular, gives an explicit bound for the o rder of the maximal\nfinite subgroups of GLρ(Z).\nWe end this section with the following two lemmas.\nLemma 2.2. Letφbe the Euler’s totient function. For every positive integer\nn, we have1\nn\nφ(n)≤3log(log(3 n))\n2log(log(3))\nand\nφ(n)≥/radicalbiggn\n2.\nProof. Forn≥3, let\nan=φ(n)log(log( n))\nn.\nWe are going to show that an≥a3for alln≥3. It’s easy to check that\na4≥a3, so we can suppose n≥5. We have (see [BS96, Theorem 8.8.7]) that\nfork≥3\nφ(k)>k\neγlog(log(k))+3\nlog(log(k))\nwhereγis the Euler’s constant. Thus for all k≥3we have\nak≥log(log(k))\neγlog(log(k))+3\nlog(log(k)).\nConsider the function\nf(x) =log(log(x))\neγlog(log(x))+3\nlog(log(x)).\nIts derivative is\nf′(x) =6log(log( x))\nxlog(x)(eγ(log(log(x)))2+3)2.\n1The factor 3inlog(log(3 n))is here for the cases n= 1andn= 2.\n4So the function fis increasing on the interval [3,+∞[. We can check that\nf(5)≥2log(log(3))\n3. As a result, for n≥5\nan≥f(n)≥f(5)≥2log(log(3))\n3=a3.\nHence, for n≥3we have\nn\nφ(n)≤3log(log( n))\n2log(log(3)).\nThe second inequality is well known, and, for instance, it ca n be deduce\neasily from the previous one for n≥503, and checked for small values of\nn.\nLemma 2.3. For every positive integer n, we have\n2nn!\nnn−5≤21818!\n1818−5<80601.\nProof. One can check that the result hold true for n≤18. Forn≥18, let\nUn=2nn!\nnn−5. Then\nUn+1\nUn= 2/parenleftbiggn+1\nn/parenrightbigg5/parenleftbiggn\nn+1/parenrightbiggn\n≤2/parenleftbigg19\n18/parenrightbigg5/parenleftbiggn\nn+1/parenrightbiggn\n≤/parenleftbigg19\n18/parenrightbigg5/parenleftbigg18\n19/parenrightbigg18\n<1,\nwhere we used the fact that/parenleftbig/parenleftbign\nn+1/parenrightbign/parenrightbig\nn≥1is a decreasing sequence. Hence the\nsequence (Un)n≥18is decreasing, which proves the result.\n3 Proof of Theorem 1.2\nWe can now establish the proof of Theorem 1.2. Firstly, we wil l demonstrate a\nlower bound for h(α)depending on the multiplicative rank ρof the conjugates\nofα, and on the degree dofQ(α)overQ.\n5Lemma 3.1. Letα∈Q∗of degree doverQand not a root of unity. Let ρ\nbe the multiplicative rank of the conjugates of α. Define\ng1(ρ,d) = min\n1≤r≤ρ/parenleftBig\nd1/r(1050r5log(3d))r(r+1)2/parenrightBig\nand\ng2(ρ,d) = 1020ρρlog/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig5.\nThen:\nh(α)≥min(g1(ρ,d),g2(ρ,d))−1.\nProof. By Theorem 2.1 applied to Q(α), and given that all conjugates of α\noverQare inQ(α)(sinceQ(α)/Qis Galois), we have, for every 1≤r≤ρ,\nthat\nh(α)≥d−1/r(1050r5log(3d))−r(r+1)2,\nsoh(α)≥g1(ρ,d)−1.\nBy Lemma 2.1, there exists L⊂Q(α)andk≥1such that L/Qis Galois,\n[L:Q]≤135ρ!2ρ−1,Q(ζk)⊂Q(α)andαk∈L. SetM=Q(ζk). We have\nM(α) =Q(α) =L(α). Notice that\n[M(α) :M] = [L(α) :L][L:Q]\n[M:Q].\nSinceαis a root of Xk−αk∈L[X], we have [L(α) :L]≤k. We also\nhaveφ(k) = [M:Q]. Therefore, we obtain\n[M(α) :M]≤k[L:Q]\n[M:Q]≤k\nφ(k)[L:Q].\nBy Lemma 2.2, we have\nk\nφ(k)≤3log(log(3 k))\n2log(log(3)),\nso\n[M(α) :M]≤405log(log(3 k))\nlog(log(3))ρ!2ρ−2.\nWe notice that φ(k)≤d. By the last statement of Lemma 2.3, we have/radicalbig\nk/2≤φ(k).Hence, we have k≤2d2, and so\n[M(α) :M]≤405log(log(6 d2))\nlog(log(3))Dρ!2ρ−2≤104ρ!2ρlog/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig\n.\n6By Lemma 2.3 we have\n[M(α) :M]≤109ρρ−5log/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig\n.\nBy Theorem 2.2, applied to Q(ζk)/Q, we have\nh(α)≥[M(α) :M]−1log(log(5[ M(α) :M]))3\nlog(2[M(α) :M])4.\nSince the function\nf(x) =1\nxlog(log(5 x))3\nlog(2x)4\nis decreasing on the interval [1,+∞[, setting\nX(ρ,d) = 109ρρ−5log/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig\nwe deduce that\nh(α)−1≤X(ρ,d)log(2X(ρ,d))4≤1020ρρlog/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig5.\nHenceh(α)≥g2(ρ,d)−1. This proves the lemma.\nProof of Theorem 1.2. Now, we are ready to prove Theorem 1.2. Suppose,\nfirst, that ρ≤log(3d)1/4. By Lemma 3.1, we get\nh(α)−1≤g2(ρ,d) = 1020/parenleftbig\nlog(3d)1/4/parenrightbiglog(3d)1/4\nlog/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig5\n= exp/parenleftbig\n20log(10)+log(3 d)1/4log(log(3 d)1/4)+5log(log/parenleftbig\nlog/parenleftbig\n6d2/parenrightbig/parenrightbig\n)/parenrightbig\n.\nSo, we have\nh(α)−1≤exp/parenleftbig\n20log(10)+6log(3 d)1/4log(log(3 d)))/parenrightbig\n. (1)\nNow, we suppose that ρ≥log(3d)1/4. We choose r∈Zsuch that\nlog(3d)1/4+1> r≥log(3d)1/4. In particular r≤ρ. Then, thanks again to\nLemma 3.1, we obtain\nh(α)−1≤g1(ρ,d) =d1/r(1050r5log(3d))r(r+1)2\n≤d1/log(3d)1/4/parenleftbig\n1050(log(3 d)1/4+1)5log(3d)/parenrightbig(log(3d)1/4+1)(log(3 d)1/4+2)2\n≤d1/log(3d)1/4/parenleftbig\n216log(3d)9/4/parenrightbig23log(3d)3/4\n= exp/parenleftbig\nlog(3d)3/4+23log(3d)3/4log(216log(3d)9/4)/parenrightbig\n.\n7So, we have\nh(α)−1≤exp/parenleftbig\nlog(3d)3/4+400log(3 d)3/4log(log(3 d))/parenrightbig\n. (2)\nFrom (1) and (2), we obtain\nh(α)−1≤exp/parenleftbig\n500log(3 d)3/4log(log(3 d))/parenrightbig\n.\nAcknowledgments\nI am grateful to Gaël Rémond for comments and suggestions on a n earlier\ndraft of this article, and for pointing out the reference [Ré m20].\nReferences\n[AD99] Francesco Amoroso and Sinnou David. The higher-dime nsional\nLehmer problem. J. Reine Angew. Math. , 513:145–179, 1999.\n[AD00] Francesco Amoroso and Roberto Dvornicich. A lower bo und for\nthe height in abelian extensions. J. Number Theory , 80(2):260–272,\n2000.\n[AD07] Francesco Amoroso and Emmanuel Delsinne. An explici t relative\nlower bound for the height in an extension of an abelian exten sion.\nDiophantine geometry , 1–24, 2007.\n[AM16] Francesco Amoroso and David Masser. Lower bounds for the height\nin Galois extensions. Bull. Lond. Math. Soc. , 48(6):1008–1012, 2016.\n[AV12] Francesco Amoroso and Evelina Viada. Small points on rational\nsubvarieties of tori. Comment. Math. Helv. , 87(2):355–383, 2012.\n[AZ10] Francesco Amoroso and Umberto Zannier. A uniform rel ative Do-\nbrowolski’s lower bound over Abelian extensions. Bull. Lond. Math.\nSoc., 42(3):489–498, 2010.\n[BS96] Eric Bach and Jeffrey Shallit. Algorithmic number the ory, Vol. 1:\nEfficient algorithms. Cambridge, MA: The MIT Press , 1996.\n8[Dob79] Edward Dobrowolski. On a question of Lehmer and the n umber of\nirreducible factors of a polynomial. Acta Arith. , 34:391–401, 1979.\n[Rém20] Gaël Rémond. Degré de définition des endomorphismes d’une var-\niété abélienne. J. Eur. Math. Soc. (JEMS) , 22(9):3059–3099, 2020.\n[Sch73] Andrzej Schinzel. On the product of the conjugates o utside the unit\ncircle of an algebraic number. Acta Arith. , 24:385–399, 1973.\nJONATHAN JENVRIN: Univ. Grenoble Alpes, CNRS, IF, 38000 Gre no-\nble, France\nE-mail adress : jonathan.jenvrin@univ-grenoble-alpes.fr\n9"    },    {        "title": "2402.04917v1.Maximal_displacement_of_a_time_inhomogeneous_N_T__particles_branching_Brownian_motion.pdf",        "content": "MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-PARTICLES\nBRANCHING BROWNIAN MOTION\nALEXANDRE LEGRAND AND PASCAL MAILLARD\nAbstract. TheN-particles branching Brownian motion ( N-BBM) is a branching Markov process which\ndescribes the evolution of a population of particles undergoing reproduction and selection. It shares many\nproperties with the N-particles branching random walk ( N-BRW), which itself is strongly related to physical\np-spin models, or to Derrida’s Random Energy Model [ 25,26]. The N-BRW can also be seen as the\nrealization of an optimization algorithm over hierarchical data, which is often called beam search [15].\nMore precisely, the maximal displacement of the N-BRW (or N-BBM) can be seen as the output of the\nbeam search algorithm; and the population size Nis the “width” of the beam, and (almost) matches the\ncomputational complexity of the algorithm.\nIn this paper, we investigate the maximal displacement at time Tof an N-BBM, where N=N(T)\nis picked arbitrarily depending on Tand the diffusion of the process σ(·) is inhomogeneous in time. We\nprove the existence of a transition in the second order of the maximal displacement when logN(T) is of\norder T1/3. When logN(T)≪T1/3, the maximal displacement behaves according to the Brunet-Derrida\ncorrection [ 21,6] which has already been studied for Na large constant and for σconstant. When\nlogN(T)≫T1/3, the output of the algorithm (i.e. the maximal displacement) is subject to two phenomena:\non the one hand it begins to grow very slowly (logarithmically) in terms of the complexity N; and on\nthe other hand its dependency in the time-inhomogeneity σ(·) becomes more intricate. The transition at\nlogN(T)≈T1/3can be interpreted as an “efficiency ceiling” in the output of the beam search algorithm,\nwhich extends previous results from [ 1] regarding an algorithm hardness threshold for optimization over the\nContinuous Random Energy Model.\nContents\n1. Introduction and main results 1\n2. Motivations and comments 8\n3. Construction and couplings of the N-BBM 14\n4. Preliminaries on the BBM with barriers 21\n5. Moment estimates on the BBM with barriers 27\n6. Lower bound on the maximum of the N-BBM 44\n7. Upper bound on the maximum of the N-BBM 52\n8. Proofs of complementary results 61\nAcknowledgements 70\nReferences 71\n1.Introduction and main results\nThe branching Brownian motion (BBM) can be described as follows. At time t= 0 we consider a\n(non-empty) initial configuration of particles on the real line, which all start moving as standard Brownian\nmotions until some exponentially distributed random times with parameter β0. All those movements and\n2020 Mathematics Subject Classification. Primary: 60J80, 68Q17, 82C21 ; Secondary: 60J70, 92D25, 60K35.\nKey words and phrases. Branching Brownian motion, branching random walk, time-inhomogeneous diffusion, algorithmic\nlower bounds, selection, beam search, Airy functions.\n1arXiv:2402.04917v1  [math.PR]  7 Feb 20242 A. LEGRAND AND P. MAILLARD\nexponential “clocks” are taken independently from one another. When one of the exponential clocks rings, the\ncorresponding particle splits into a random number ξ≥2 of new ones at its location. Then, those particles\nstart evolving the same way, independently, with their own exponential clocks. Following a generalization\nfirst introduced in [ 29], in this paper we will be interested in BBM’s with time-inhomogeneous motion. More\nprecisely, let σ∈ C2([0,1]) be a positive function: then for some fixed final time T >0, we assume that, at\ntime t∈[0, T], the infinitesimal variance of all the Brownian motions involved in the construction is given by\nσ2(t/T).\nFrom this process, one can construct the (time-inhomogeneous) N-particles branching Brownian motion\n(N-BBM) by adding the following selection mechanism: we start from an initial configuration containing at\nmost Nparticles, N∈N. At any time of a splitting event, we remove all particles which are not among the\nNhighest of the whole population. We denote by XN\nTthe particle configuration of the N-BBM at time T,\nseen as a (finite) counting measure on R(full formal notation and construction of the N-BBM are presented\nmore extensively below). We write max(XN\nT) for the maximal displacement of the process at time T, i.e. the\nposition of the highest living particle from the N-BBM. Similarly, XTdenotes the particle configuration of\nthe BBM (without selection) at time T, and max( XT) the maximal displacement of the BBM.\nOn the one hand, the maximum displacement of the BBM has been extensively investigated in the literature,\nboth for the time-homogeneous and inhomogeneous cases, see [ 19,20,29,48] or more recently [ 2,46,47]\namong other works. On the other hand, studying the maximum of the homogeneous N-BBM is a more\ndifficult matter: it was first done in [ 21] with heuristic methods, and later in [ 43] (see also [ 6]). The goal of\nthis paper is to study the maximum of the N-BBM in the time-inhomogeneous case. Moreover, we will be\ninterested in the case where Ndepends on T, with N=N(T)→ ∞ asT→ ∞ . More precisely, we define\n(1.1) L(T) := log N(T),\nand we shall consider the regime where 1 ≪logN(T)≪T. Note that the original BBM formally corresponds\ntoN= +∞.\nIn the remainder of this section, we state the mathematical results of this paper. The most important\ntakeaway from this article is that the maximal displacement of the N(T)-BBM undergoes a phase transition\nat a critical scale L(T)≍T1/3, and we fully determine the asymptotics at, below and above this scale.\nWe also provide results for variants of the model including the continuous random energy model (CREM) .\nThis is related to the topic of hardness thresholds for optimization algorithms on random instances. Such\na threshold was already established in [ 1] for the specific case of the CREM—see Figure 1 for simulations\nwhich illustrate this phenomenon for the N-BBM considered here. In this context our results amount to\n“zooming in” at the threshold. We further discuss this in Section 2.1 below. We also provide more refined\nnumerical experiments on a discrete-space variant in Section 2.3.\n1.1.Statement of results on the N-BBM.\nNotations. Throughout this paper, C2([0,1]) denotes the set of C2functions from [0 ,1] to (0 ,+∞) (in\nparticular they are positive). Let the infinitesimal variance of the N(T)-BBM be given by σ2(t/T),t∈[0, T],\nfor some σ∈ C2([0,1]). Define\n(1.2) v(s) =Zs\n0σ(s) ds , s ∈[0,1],\nwhich is called the natural speed of the BBM. Recall that ξdenotes the offspring distribution of particles,\nandβ0denotes the branching rate: in this paper we assume E[ξ2]<+∞, and, using the Brownian scaling\nproperty, one can assume without loss of generality that β0= (2(E[ξ]−1))−1.\nLetCdenote the set of all finite particle configurations, i.e. all finite counting measures on R; and for\nN≥1, let CN:={µ∈C;µ(R)≤N}. For µ∈C(resp. µ∈CN), the law of the BBM (resp. N-BBM)\nstarting from the initial configuration µwill be denoted Pµ(we use the same notation for both, and what\nbranching process is considered at any given time will always be clear from context).MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 3\nFigure 1. Running maximum of simulations of time-inhomogeneous N-BBM with varying N.\nParameters: T= 1000, σ(t) = 0 .125+ t2. The dashed curve corresponds to the theoretical position\nof the running maximum of the time-inhomogeneous BBM without selection ( N=∞), which is\nattained up to random O(1) fluctuations, see [17].\nFor any µ∈C, define\nQT(µ) := sup\b\nq∈R\f\f∃κ∈[0,1], µ\u0000\n[q−κσ(0)L(T),+∞)\u0001\n≥N(T)κ\t\n(1.3)\n= inf\b\nq∈R\f\f∀κ∈[0,1], µ\u0000\n[q−κσ(0)L(T),+∞)\u0001\n< N (T)κ\t\n.\nWe shall see below that this term determines how the choice of the initial configuration µ∈CN(T)reverberates\non the displacement of the N(T)-BBM after a long time (see Section 3.3 for more details). For instance, one\ncan check for y∈RthatQT(δy) =yandQT(N(T)δy) =y+σ(0)L(T).\nLet Ai and Bi denote respectively the Airy functions of first and second kind, and define\n(1.4) Ψ( q) :=q2/3\n21/3sup\b\nλ≤0 ; Ai( λ)Bi(λ+ (2q)1/3) = Ai( λ+ (2q)1/3)Bi(λ)\t\n<0, q > 0,\nand Ψ( −q) :=q+ Ψ( q) for−q <0; and finally Ψ(0) := −π2\n2. It is proven in [ 47, Lemmata 1.7, A.4] that\nq7→Ψ(q) is well-posed and convex (hence continuous) on R. Moreover, let a 1denotes the absolute value of\nthe largest root of Ai (i.e. a 1= 2.33811 . . .); then one has Ψ( q)∼ −a1q2/3\n21/3asq→+∞.\nLet us denote the positive and negative parts of a real number with\n(1.5) ( x)+:=x1{x≥0}, (x)−:=−x1{x≤0},∀x∈R,\nand, for a function f:R→R, let us write similarly ( f)+(x) := ( f(x))+and ( f)−(x) := ( f(x))−,x∈R.\nFurthermore, oP(f(T)) denotes a random quantity which, when divided by f(T), converges to 0 in PµT-\nprobability as T→+∞. Finally, we say that a function σ∈ C2([0,1])changes its monotonicity finitely\nmany times if there exists u0= 0< u 1< . . . < u p= 1 such that σis monotonic (i.e. non-increasing or\nnon-decreasing) on each [ ui−1, ui], 1≤i≤p.4 A. LEGRAND AND P. MAILLARD\nMain result: asymptotic of the maximum. We now state the main result of this paper.\nTheorem 1.1. Letσ∈ C2([0,1]), letL(T) =logN(T)→+∞asT→+∞, and denote with XN(T)\nT the\nempirical measure on Rat time T >0of aN(T)-BBM with infinitesimal variance σ2(·/T), started from\nsome initial configuration µT∈CN(T),T≥0. Let max(XN(T)\nT)denote the maximal displacement of the\nprocess at time T. We have the following.\n(i) (Sub-critical regime) Assume 1≪L(T)≪T1/3. Then,\n(1.6) max( XN(T)\nT) =QT(µT) +v(1)T\u0012\n1−π2\n2L(T)2\u0013\n+oP\u0012T\nL(T)2\u0013\n.\n(ii) (Critical regime) Assume L(T)∼αT1/3for some α∈(0,+∞). Then,\n(1.7) max( XN(T)\nT) =QT(µT) +v(1)T+\u0014Z1\n0σ(u)\nα2Ψ\u0010\n−α3σ′(u)\nσ(u)\u0011\ndu\u0015\nT1/3+oP\u0000\nT1/3\u0001\n.\n(iii) (Super-critical regime) Assume T1/3≪L(T)≪T, and that σchanges its monotonicity finitely\nmany times. Then,\n(1.8) max( XN(T)\nT) =QT(µT) +v(1)T+\u0014Z1\n0(σ′)+(u) du\u0015\nL(T) +oP\u0000\nL(T)\u0001\n.\nIn the super-critical regime (1.8), ifσis non-increasing, Theorem 1.1 gives little information. However,\nwe complete it with the following result.\nProposition 1.2. Letσ∈ C2([0,1])be strictly decreasing, let T1/3≪L(T)≤+∞, and consider the initial\nparticle configuration δ0(i.e. one particle at the origin). Then as T→+∞, one has,\n(1.9) max( XN(T)\nT) =v(1)T−a1\n21/3\u0014Z1\n0σ(u)1/3|σ′(u)|2/3du\u0015\nT1/3+oP\u0000\nT1/3\u0001\n.\nSome assumptions in this proposition (namely, constraining the initial configuration to be δ0, and assuming\nσisstrictly decreasing) are not as general as one would expect when compared to Theorem 1.1: we further\ndiscuss them in the proof, see Section 8.1 below.\nRemark 1.1. Let us point out that Proposition 1.2 also holds for L(T) = +∞, that is for the maximum of\nthe BBM without selection when σis decreasing: this has already been proven in [46]. However, when σis\nnot decreasing, the maximum of the BBM without selection is vmaxT+o(T)for some vmax> v(1)(this is a\nwell-known result, which follows e.g. from direct adaptations of [18]or[47]). This is in line with (1.8), since\none can let L(T)be arbitrarily close to T, hence it implies that the maximum of the BBM without selection\nis larger than any v(1)T+o(T)forσnot decreasing.\nNotational convention and rephrasing of the main result. In order to write Theorem 1.1 and upcoming\nstatements in a more condensed form, let us denote the three regimes (i.e. T1/3≪L(T)≪T,L(T)∼αT1/3\nand 1 ≪L(T)≪T1/3) respectively with the superscripts “sup”, “crit” and “sub”. We will also occasionally\nconsider the regime L(T)≫T1/3in the case of non-increasing σ, which we denote with the superscript\n“sup-d”. In what follows, we will always assume that Ndepends on Taccording to one of the four regimes.\nFurthermore, in the “sup” regime we always implicitly assume that σchanges its monotonicity finitely many\ntimes (this technical restriction is further discussed below) and in the “ sup-d ” regime we assume that σis\ndecreasing. Then, we define the error scaling terms for each regime with\n(1.10) bsup\nT:=L(T), bcrit\nT=bsup-d\nT :=T1/3, and bsub\nT:=T\nL(T)2,MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 5\nforT≥0; and the limiting terms with,\n(1.11)msup\nT:=v(1)T+\u0014Z1\n0(σ′)+(u) du\u0015\nL(T),\nmcrit\nT:=v(1)T+\u0014Z1\n0σ(u)\nα2Ψ\u0010\n−α3σ′(u)\nσ(u)\u0011\ndu\u0015\nT1/3,\nmsub\nT:=v(1)T\u0012\n1−π2\n2L(T)2\u0013\n,\nand msup-d\nT :=v(1)T−a1\n21/3\u0014Z1\n0σ(u)1/3|σ′(u)|2/3du\u0015\nT1/3.\nTheorem 1.1 and Proposition 1.2 can then be summarized as follows:\nTheorem 1.3 (Rephrasing of Theorem 1.1 and Proposition 1.2) .Let the assumptions of Theorem 1.1 (in\nthe regimes “ sup”, “crit” or “ sub”) or of Proposition 1.2 (in the regime “ sup-d”) hold. Then, for every\n∗ ∈ { sup,crit,sub,sup-d}, we have as T→+∞,\n(1.12) max( XN(T)\nT) =QT(µT) +m∗\nT+oP\u0000\nb∗\nT\u0001\n.\nComplementary result: empirical measure and diameter. We complement the main result with a statement,\nin the critical and super-critical regimes, about the empirical measure of the particles below the asymptotic\nmaximum and the diameter of the configuration at the final time. We do not expect the very same claim to\nhold in general in the subcritical regime: especially for (1.13) , a random centering would be required, see\nRemark 8.2. We write log+(x) = log( x∨1).\nProposition 1.4. Suppose the assumptions of Theorem 1.1 hold and that we are in the critical or super-\ncritical regime, i.e. ∗ ∈ { crit,sup}. Then, as T→ ∞ ,\n(1.13) sup\ny∈[0,1]\f\f\f\f\flog+XN(T)\nT\u0000\n[QT(µT) +m∗\nT−yσ(1)L(T),+∞)\u0001\nL(T)−y\f\f\f\f\f−→ 0, inPµT-probability.\nAdditionally, if min(XN(T)\nT)denotes the position of the minimum of the N(T)-BBM at time T, then\n(1.14) max( XN(T)\nT)−min(XN(T)\nT) =σ(1)L(T) +oP(L(T)).\nRemark 1.2. We can rephrase the first part of Proposition 1.4 as follows: in the critical and super-critical\nregimes, one has for Tlarge,\nXN(T)\nT([QT(µT) +m∗\nT−yσ(1)L(T),+∞)) = N(T)y+oP(1),\nuniformly in y≤1not too close to zero.\n1.2.Results on variants of the N-BBM. In this section, we introduce two variants and one extension of\ntheN-BBM and state results analogous to Theorem 1.1 and Propositions 1.2 and 1.4.\nN-BBM with deterministic branching times. Our results also apply to a variant of the N-BBM in which\nparticles branch simultaneously at deterministic times on a time grid aN, for some a >0. Let ξandσbe as\nabove. The process is then defined as follows: Given T >0, particles diffuse independently according to time\ninhomogeneous branching Brownian motions with infinitesimal variance σ2(t/T) at time t. Furthermore, at\neach time which is a multiple of a= 2logE[ξ], each individual is replaced independently from the others by\na random number of particles with the same distribution as ξ. In the following, the BBM with deterministic\nbranching times will be called “BBMdb”, and its counterpart with selection will be called “ N-BBMdb”.\nProposition 1.5. LetN(T)→+∞asT→+∞, then the results of Theorem 1.1, Propositions 1.2 and 1.4\nalso apply to the N(T)-BBMdb.\nRemark 1.3. Note that we have chosen the value of aabove for convenience, but we can handle any value\nofaby a time-change and using a different function σ.6 A. LEGRAND AND P. MAILLARD\nContinuous random energy model (CREM). In the same vein as Proposition 1.5, the results presented in\nthis paper can be applied to a class of discrete-time branching random walks with Gaussian increments.\nHere, we focus on the specific case of the Continuous Random Energy Model (CREM). This model was\nintroduced in [ 18] as a generalization of Derrida’s Generalized Random Energy Model (GREM) [ 27], and\ncan be constructed as follows.\nConsider a function Afrom [0 ,1] to [0 ,1] which is C1and satisfies A(0) = 0, A(1) = 1 and A′(s)>0 for\nalls∈[0,1]. Let TT:={∅} ∪ST\ni=1{0,1}idenote the rooted binary tree of depth T∈N, and for 0 ≤i≤T,\nwrite Vifor the set of vertices u∈TTwith depth |u|=i. Let XCREM\n∅:= 0. For u∈TT−1, and uione of its\ntwo offspring, i∈ {0,1}, letXCREM\nui−Xube a centered Gaussian random variable with variance\nT\u0000\nA\u0000|u|+1\nT\u0001\n−A\u0000|u|\nT\u0001\u0001\n,\nand assume the XCREM\nui−XCREM\nu,u∈TT−1,i∈ {0,1}are independent. Then the (Hamiltonian of the) CREM\nwith parameters A(·) and Tis given by the values of the process Xon the leaves of TT, that is ( XCREM\nu)u∈VT.\nThe CREM can also be constructed from a variant of the time-inhomogeneous BBMdb presented above.\nIndeed, consider ( eXt)t∈[0,T]a BBMdb with infinitesimal variance σ2(s) := A′(s),s∈[0,1], and where\nall particles branch simultaneously into ξ≡2 offspring each at each time of the grid aN, where we take\na:= 2log2. We assume T∈aN, but the process ends at time Tbefore branching. Assuming eXstarts from\ntwo particles at the origin (or that it branches instantly at time 0), one obtains the following equality in law:\n(1.15) eXT(d)=\u0000p\n2 log 2 XCREM\nu\u0001\nu∈VT/(2 log 2).\nLetN∈N, and let ( XCREM\nu)u∈TTbe the construction of the CREM on the whole tree TTas presented\nabove. We may construct an “ N-CREM” with the following procedure: perform a breadth-first exploration\nof the tree TT, noting encountered values of XCREMat depth kwith Xk,1, Xk,2, . . .Then, remove from the\ntree all vertices (as well as the sub-tree they support) from Vkwhich are not associated with one of the\nNhighest values from the sequence Xk,i,i≥1. Repeat that procedure at depth k+ 1, considering only\nthe offspring of vertices which were not removed. When that procedure ends, it yields a family which we\ndenote ( XN-CREM\nT,i )i≤N(T)⊂(XCREM\nu)u∈VT: this can be seen as an optimization algorithm on the CREM, with\ncomplexity —that is, the number of queries throughout the procedure— of order O(TN). In particular,\nchoosing a specific sequence N=N(T) allows for any complexity in Tfor that algorithm. One can also\ninterpret ( XN-CREM\nT,i )i≤Nas the final values of a (discrete time) branching random walk with selection.\nRecall that we defined the N-BBMdb above, i.e. the BBM with selection and deterministic branching\ntimes. Then the BBM–CREM correspondence (1.15) also apply to the N-particles variants: more precisely,\none has\n(1.16) eX2N\nT(d)=\u0000p\n2 log 2 XN-CREM\nT/(2 log 2) ,i\u0001\n1≤i≤N.\nRemark 1.4. The presence of a 2Nin the l.h.s. of (1.16) comes from the fact that, in the N-CREM, one has\nto consider 2NGaussian increments, then select the Nhighest before having the particles reproduce; whereas\nin the N-BBMdb, the selection happens just after the reproduction event. Notice that, if N=N(T)→+∞\nasT→+∞, one has log(2N)∼logN=:L(T): so the maximal displacement of the N-BBM and (2N)-BBM\nhave the same asymptotics in Theorem 1.1.\nLetmax(XN-CREM\nT ) :=max{XN-CREM\nT,i ; 1≤i≤N(T)}, and recall (1.10) and(1.11) . We have the following\nresult, which is an immediate corollary of Proposition 1.5 and (1.16).\nTheorem 1.6. Consider the CREM with parameters A(·)andT∈N. Let N=N(T)→+∞asT→+∞,\nand consider the associated N-CREM. Let ∗ ∈ { sup,crit,sub,sup-d}denote the regime satisfied by L(T) :=\nlogN(T), and σ2(·) :=A′(·)∈ C2([0,1]). Then as T→+∞, one has,\n(1.17) max( XN-CREM\nT ) = (2 log 2)−1/2m∗\n(2 log 2) T+oP\u0000\nb∗\n(2 log 2) T\u0001\n.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 7\nMoreover, assume ∗ ∈ { crit,sup}; then one has as T→+∞,\n(1.18)\nsup\ny∈[0,1]\f\f\f\f\flog+#\b\ni≤N;XN-CREM\nT,i ≥(2 log 2)−1/2[m∗\n(2 log 2) T−yσ(1)L(T)]\t\nL(T)−y\f\f\f\f\f−→0,inPµT-probability,\nand\n(1.19) max( XN-CREM\nT )−min(XN-CREM\nT ) = (2 log 2)−1/2σ(1)L(T) +oP(L(T)).\nN-BBM with time-inhomogeneous selection. It is natural to extend the N-BBM by allowing the selection\nmechanism to be time-inhomogeneous as well. That is, starting from a (time-inhomogeneous) BBM over\nthe time horizon T >0, at any time s∈[0, T], keep only the particles at the N(s, T) highest of the whole\npopulation, for some function N(·, T) fixed beforehand. Let us call this model the N(·, T)-BBM. The results\npresented above—namely Theorem 1.1 and Proposition 1.4—can be extended to a class of N(·, T)-BBM\nin which the selection does not vary too much: more precisely, the selection remains in the “same regime”\nthroughout the time interval [0 , T].\nConsider some growing function bL(T)→+∞asT→+∞, and a positive function ℓ∈ C1([0,1]), note\nthat this implies that ℓis bounded away from 0 and + ∞. Define for 0 ≤s≤T,\n(1.20) L(s, T) = log N(s, T) := ℓ(s/T)bL(T),\nthe log-population size at time s∈[0, T] of the N(·, T)-BBM. We say that,\n—bL(T) issub-critical if 1≪bL(T)≪T1/3,\n—bL(T) iscritical ifbL(T)∼T1/3,\n—bL(T) issuper-critical ifT1/3≪bL(T)≪T.\nLet us adapt the notation from (1.10–1.11) by defining for T≥0,\n(1.21) bbsup\nT:=bL(T),bbcrit\nT:=T1/3, and bbsub\nT:=T\nbL(T)2,\nas well as,\n(1.22)bmsup\nT:=v(1)T+\u0014Z1\n0ℓ(u)(σ′)+(u) du\u0015\nbL(T),\nbmcrit\nT:=v(1)T+\u0014Z1\n0σ(u)\nℓ(u)2Ψ\u0010\n−ℓ(u)3σ′(u)\nσ(u)\u0011\ndu\u0015\nT1/3,\nand bmsub\nT:=v(1)T \n1−π2\n2bL(T)2Z1\n01\nℓ(u)2du!\n.\nRecall also the definitions of bsup-d\nT,msup-d\nT. Then we have the following.\nTheorem 1.7. Letσ∈ C2([0,1]). LetbL(T)→+∞asT→+∞, letℓ∈ C1([0,1])and define N(·, T)as\nin(1.20) . Denote with XN(·,T)\nT the empirical measure on Rat time T >0of aN(·, T)-BBM with infinitesimal\nvariance σ2(·/T), started from some initial configuration µT∈CN(0,T),T≥0. Let max(XN(·,T)\nT )denote the\nmaximal displacement of the process at time T. Let∗ ∈ { sub,crit,sup}denote the regime satisfied by bL(T).\nThen as T→+∞, one has\n(1.23) max( XN(·,T)\nT ) = QT(µT) +bm∗\nT+oP\u0000\nb∗\nT\u0001\n.\nMoreover, if σis strictly decreasing, µT≡δ0andT1/3≪bL(T)≤+∞, then\n(1.24) max( XN(·,T)\nT ) = msup -d\nT +oP\u0000\nbsup -d\nT\u0001\n.8 A. LEGRAND AND P. MAILLARD\nFinally if the regime satisfies ∗ ∈ { crit,sup}, one also has\n(1.25) sup\ny∈[0,1]\f\f\f\f\flog+XN(·,T)\nT ([QT(µT) +bm∗\nT−yσ(1)ℓ(1)bL(T),+∞))\nℓ(1)bL(T)−y\f\f\f\f\f−→ 0,inPµT-probability,\nand\n(1.26) max( XN(·,T)\nT )−min(XN(·,T)\nT ) =σ(1)ℓ(1)bL(T) +oP(L(T)).\nLet us stress that the result (1.24) in the regime sup-d does notdepend on bL(T) and ℓ(·), but matches\nProposition 1.2 for homogeneous selection.\nOrganization of the paper. In Section 2 we motivate the results obtained in this article and compare\nthem with previous works from the literature. Then, the vast majority of this paper focuses on the proof of\nTheorem 1.1: in particular, branching epochs of particles are assumed random (exponentially distributed),\nandN(T),L(T) are defined as in (1.1). Even though Theorem 1.1 is a weaker statement than Theorem 1.7,\nthe authors believe that this organization of the paper makes the proof more understandable. Moreover,\nmost complementary results, notably Proposition 1.5 and Theorem 1.7, are consequences of Theorem 1.1, or\ndirect adaptations of arguments presented in its proof.\nIn Section 3 we present the time-inhomogeneous BBM and N(T)-BBM. Then in Proposition 3.2 we\nintroduce the main coupling argument that is used throughout this paper. In Section 3.3, we state two\nkey results in Propositions 3.3 and 3.4, which provide respectively a lower bound and an upper bound on\nthe maximal displacement of the N(T)-BBM for some specific initial configurations. Using these and the\ncoupling from Proposition 3.2, we deduce Theorem 1.1 for any initial configuration.\nThe four next sections of the paper are dedicated to the proofs of Propositions 3.3 and 3.4. The core idea\nof the proof is to approximate the N(T)-BBM with a different selection mechanism on the BBM, namely\nkilling particles when they reach certain well-chosen barriers. In Section 4 we introduce the relevant barriers\ndepending on the regime: super-critical, sub-critical and critical; as well as some preliminary results. Then\nin Section 5 we compute moment estimates on some functions of the BBM between barriers in each of those\nregimes.\nWith these moment estimates, the proofs of Propositions 3.3 and 3.4 are finally displayed in Sections 6\nand 7 respectively. The comparison between N(T)-BBM and BBM with barriers passes through couplings\nwith some intermediary processes, the N−-BBM and N+-BBM respectively, which have already been\nintroduced in [ 43] for the time-homogeneous N-BBM with constant N. Let us mention that the proofs\npresented in these sections rely almost solely on the moment estimates proven in Section 5, in particular\nmost arguments apply simultaneously to all three regimes. Nevertheless, the sub-critical regime requires\nmore work than the other two, both for the lower and upper bound, because the moment estimates from\nSection 5 are slightly weaker in that case. These sections complete the proof of Theorem 1.1.\nAfterwards, Section 8 presents the proofs of all remaining statements from Section 1, that is Proposi-\ntions 1.2, 1.4, 1.5 (from which one deduces Theorem 1.6), and Theorem 1.7. All of these rely on Theorem 1.1,\nor arguments from its proof presented in previous sections.\n2.Motivations and comments\n2.1.Motivation: algorithmic hardness threshold for the CREM. The initial motivation for this\nwork stems from a large body of literature on algorithmic hardness thresholds for combinatorial optimization\nproblems on random instances. This has been a very active research area in the last two decades, drawing\nextensively on results and methods from the theory of spin glasses in statistical mechanics. See e.g. [ 30,33]\nand the references therein. A stylized model of a spin glass is the continuous random energy model (CREM),\ndefined in Section 1. The algorithmic hardness of optimizing the Hamiltonian of the CREM has been studied\nby Addario-Berry and Maillard [1]. We recall their main result:MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 9\nTheorem 2.1 (from [ 1]).Consider the CREM with parameters A(·)andT∈N, letσ2(·) :=A′(·)∈ C2([0,1])\nandvc:=√2 log 2 v(1) =√2 log 2R1\n0σ(s)ds. Let ε >0, then the following holds\n(i)There exists a linear-time algorithm that finds a vertex u∈VTsuch that XCREM\nu≥(vc−ε)Twith high\nprobability.\n(ii)There exists γ=γ(A, ε)>0such that for Tsufficiently large, for any algorithm, the number of\nqueries performed before finding a vertex u∈VTsuch that XCREM\nu≥(vc+ε)Tis stochastically bounded from\nbelow by a geometric random variable with parameter exp(−γT).\nIn other words, Theorem 2.1 proves the existence of an algorithmic hardness threshold for the CREM:\nfinding a vertex with a value greater than ( vc+ε)Tfor a given ε >0 typically requires a number of queries\nexponential in T, whereas values smaller than ( vc−ε)Tcan be obtained in linear time.\nIn light of Theorem 2.1, it is natural to ask about the complexity of finding vertices in the CREM with\nvalue near the threshold value vcT. The present paper aims to provide a partial answer to this question.\nIndeed, we study in detail the efficiency of a particular algorithm—the N-CREM—which has complexity\n(i.e. number of queries) O(TN). The interesting regime is when the complexity is stretched exponential\ninT, i.e. N=N(T) =exp(O(Tκ)) for some κ∈(0,1). Our results can then be summarized as follows\n(see Theorem 1.6): when κ < 1/3, then with high probability, the value found by the algorithm is far\nbelow the threshold, more precisely, at vcT−O(T1−2κ). On the other hand, if κ >1/3, then the value\nfound by the algorithm is with high probability above the threshold and of order vcT+O(Tκ)—unless σ\nis non-increasing, in which case the value found by the algorithm is o(T1/3) close to the maximum value,\nwhich is of order vcT−O(T1/3) with high probability. Furthermore, we precisely describe the transition\nbetween the subcritical regime κ <1/3 and the super-critical regime κ >1/3.\nTheN-CREM considered in this article can be viewed as a particular greedy-type algorithm, in a similar\nspirit as the algorithm studied in [ 1]. More precisely, it may be regarded as a beam search algorithm [ 15],\nwith the parameter Nbeing the width of the beam . Our main result (Theorem 1.6) then precisely describes\nhow the output of the algorithm depends on the width of the beam N, with a phase transition happening\natlogN≈T1/3. Apractical takeaway might be the following: for the beam search algorithm, increasing\nthe width Nof the beam substantially improves the output in the subcritical regime logN≪T1/3, due to\nthe singular second order term T/(logN)2in the value of the output; whereas in the super-critical regime\nlogN≫T1/3, increasing the width of the beam comes with little improvement of the output, which only\ngrows logarithmically in N.\nThe efficiency of beam search algorithms is still an active research area, see e.g. [ 41] and the references\ntherein. The beam search algorithm considered here is quite special, due to the nature of the CREM. For\nexample, the output of the algorithm is a non-decreasing function of the width of the beam, which is in\ngeneral not the case [ 41]. Nevertheless, we hope that our results shed light on the behavior of general beam\nsearch algorithms for hard optimization problems on random instances, as the width of the beam grows to\ninfinity.\nThe efficiency of more general optimization algorithms for the CREM is an interesting open problem.\nWe believe that the N-CREM considered here is close to optimal within the class of algorithms of a given\ncomplexity. Indeed, due to the particular structure of the CREM (in particular, the branching property),\nit is always favorable (on average) to explore subtrees of vertices of large values as opposed to subtrees of\nvertices of smaller values. The N-CREM is therefore a very natural candidate for an asymptotically optimal\nalgorithm for this model.\n2.2.Comparison with previous results.\nThe Brunet-Derrida behavior in the sub-critical regime. Results similar to (1.6) have already been obtained\nfor some (time-homogeneous) branching processes with selection, see e.g. [ 6,43] respectively for the N-\nparticles branching random walk ( N-BRW) and the N-BBM. In those papers, the authors prove for fixed\nN∈Nthe existence of an asymptotic speed vN:=limT→+∞max(XN\nT)/T, where ( XN\nT) denotes either the\nN-BRW or N-BBM. Then, when N→+∞, this asymptotic speed converges very slowly —like ( logN)−2—10 A. LEGRAND AND P. MAILLARD\ntov∞:=limT→+∞max(XT)/T, the asymptotic speed of the corresponding (time-homogeneous) branching\nprocess without selection. This slow convergence has been called Brunet-Derrida behavior : it was first\nobserved in [ 21] with heuristic methods and numerical simulations, and it is expected to hold for many\nmodels that fall under the universality class of the FKPP equation (see [ 23]). The phrasing of Theorem 1.1\nabove differs from previous results on N-particles branching processes such as [ 6], where the authors first\ntake the limit T→+∞forNfinite, then N→+∞. In fact, equation (1.6) in Theorem 1.1 can be seen as\nan expansion of the Brunet-Derrida behavior to all diverging sequences ( T, N) such that T1/3≫logN≫1.\nTo emphasize this, we claim the following.\nCorollary 2.2. Let(XN\nT)T≥0anN-BBM (or N-BBMdb) with infinitesimal variance σ2(·/T). For any\nε >0and sequence (µT)T≥0inC, one has,\n(2.1) lim sup\nN→+∞lim sup\nT→+∞PµT\u0012(logN)2\nT\f\f\f\fmax(XN\nT)−QT(µT)−v(1)T\u0012\n1−π2\n2(log N)2\u0013\f\f\f\f> ε\u0013\n= 0.\nThis corollary follows naturally from (1.6) and a diagonal argument: if (2.1) does not hold, then one\ncan construct a sequence ( Nk, Tk)k≥1for which the probability above remains large. However, one can\nfreely choose ( Tk)k≥1such that Tk≫(logNk)3for large k, and this directly contradicts the sub-critical\nresult from Theorem 1.1 (we leave the details of the proof to the reader). Nevertheless, let us mention that\nCorollary 2.2 does not directly imply the almost-sure existence of an asymptotic speed for finite N, i.e.\nvN=limT→+∞max(XN\nT)/T(e.g., compare with [ 6, Proposition 2]), which would require a little more work.\nAs a side note, let us point out that one could consider an N-BBM with fixed N∈N, choose a time-\nhorizon T:=T(N), then let N→+∞. Our result can directly be adapted to that convention, where\nthe three regimes (sub-critical, critical and super-critical), respectively match a time-horizon which is long\n(T≫(logN)3), critical ( T≈(logN)3) or short ( T≪(logN)3). Moreover, let us mention that this critical\ntime scale T≈(logN)3already appeared in the study of the (time homogeneous) N-BRW [ 6,47] and\nN-BBM [43].\n1:3 space-time scaling in branching Brownian motion and branching random walks. The 1:3 space-time\nscaling has appeared many times in the study of branching Brownian motion and branching random walks.\nFor the time-homogeneous versions of these processes, it appears in the N-particle process mentioned above\nas well as in the process with absorption at a linear space-time barrier, with the earliest appearance being,\nto our knowledge, in Kesten [ 39] and later developments by many authors [ 28,50,34,7,9,8,10,11,44,45].\nPemantle [ 50] is motivated by algorithmic aspects, inspired by Aldous [ 3,4]. The 1:3 scaling also appears\nin the study of the particles in BBM or BRW without selection remaining close to the running maximum\nthroughout their trajectory, which is usually called a “consistent(ly) maximal displacement” [ 31,37,32,51].\nA heuristic explanation of the appearance of the 1:3 space-time scaling is the following: Suppose we\nconsider particles whose trajectory is confined to a region in space whose size is of order L≪T1/2. Forcing\na particle to stay within such a region incurs a probability cost of order exp(−O(T/L2))— this comes\nfrom a calculation involving a single Brownian motion. On the other hand, the density of particles decays\nexponentially fast when moving away from the bulk, since we enter a large deviation regime. Hence, when\ncalculating the number of particles staying in this space-time region, we can gain a factor exp(O(L)) by\nshifting the reference frame towards the bulk by an amount of order L, which we can do without leaving\nthe region we are considering. We can make the two factors match if L≍T1/3. Of course, determining the\nprecise constants is a non-trivial task and amounts to finding the optimal shape of the space-time region.\nIn the context of time-inhomogeneous BBM or BRW, the 1:3 scaling has been considered to our knowledge\nonly in the study of extremal particles, in the regimes where their trajectories stay close to the running\nmaximum during a macroscopic time [ 47,46]. Relatedly, it appears in the time-inhomogeneous Fisher-KPP\nequation [49], due to a duality relation with the time-inhomogeneous BBM.\n2.3.Generic branching random walk and numerical simulations. In Theorem 1.6 we provided an\nasymptotic of max(XN-CREM\nT ) asT→+∞for the N-CREM, N=N(T), and we commented in Section 2.1MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 11\nthat it may be seen as an optimization algorithm on realizations of the CREM. We conjecture that such\nresults adapt to more general (i.e. non-Gaussian) branching random walks (BRW). In this section we present\nthe conjectured formulae that would extend Theorem 1.1 to a generic BRW, and then we display numerical\nsimulations in the case of a Bernoulli BRW.\nConjecture for the general BRW. We introduce some notation in the vein of [ 47], which we only use in this\nsection. Let ( Ls)s∈[0,1]be a family of laws of point processes. Then, the BRW with offspring distributions\n(Ls)s∈[0,1]is constructed until time T∈N, starting from some initial configuration µT∈C, by induction: at\ngeneration t < T , an individual u∈ N tlocated in x∈Rgenerates |Lu\nt/T|children located respectively in\nx+YforY∈Lu\nt/T, where the point processes Lu\nt/T∼ Lt/Tare independent in u,t. We write ξt/T∼ |Lu\nt/T|\nfor the law of the number of children at generation t, and we assume E[ξ2\ns]<+∞andms:=E[ξs]≥1 for\nalls∈[0,1]. We write,\n(2.2) κs(θ) := log E\"X\nY∈LseθY#\n, θ ≥0, s∈[0,1],\nfor the log-Laplace transform of the offspring point processes. Consider its Fenchel-Legendre transform, that\nis\n(2.3) κ∗\ns(v) = sup\nθ>0[vθ−φs(θ)], v ∈R, s∈[0,1].\nMorally, if a= (as)s∈[0,1]denotes a “speed profile”, the number of particles from the BRW that remain close\ntoat/TTat all time t∈[0, T], is roughly exp(−PT\nt=1κ∗\nt/T(at/T)), see e.g. [ 12,4] or more recently [ 47]. In\nparticular, letting vmaxbe the first order of the speed of the maximum of the BRW (without selection), it\nsatisfies,\nvmax= sup\u001aZ1\n0asds; (as)s∈[0,1]c` adlag s.t. ∀s≤1,Zs\n0κ∗\nu(au)du≤0\u001b\n.\nAssume that for all s∈[0,1], there exists a greatest root vsofκ∗\ns(v) = 0, and that κ∗\nsis finite in a\nneighborhood of vs; then the “natural speed” of the process is defined by\n(2.4) vnat= sup\u001aZ1\n0asds; (as)s∈[0,1]c` adlag s.t. ∀s≤1, κ∗\ns(as)≤0\u001b\n=Z1\n0vsds≤vmax.\nFinally, one defines ( θs)s∈[0,1], (σs)s∈[0,1]with,\n(2.5) ∀s≤1, θ s:=∂vκ∗\ns(vs) = (∂θκs)−1(vs),and σ2\ns=∂2\nθκs(θs) = 1 /∂2\nvκ∗\ns(vs).\nRemark 2.1. In the case of a centered Gaussian BRW (i.e. the variables Y∈Lsare independent with law\nN(0, σ2(s))), then one has κs(θ) =logms+θ2σ2(s)/2. In particular one has vs:=σ(s)√2 logms,s∈[0,1];\nandvs,σ(s)do satisfy (2.5) withθs:=√2 logms/σ(s). In particular for ms=mconstant for all s∈[0,1],\nvnat=v(1)√2 logmmatches the definition of v(1)from Section 1 (up to a scaling factor coming from our\ninitial choice of branching rate β0).\nConjecture 2.3. With the notation above, let N(T) =eL(T)and consider XN(T)\nT the configuration at\ngeneration Tof an N(T)-BRW started from a single particle at the origin (i.e. µT=δ0) and with offspring\ndistributions (Lt/T),t≤T. Assume that (vs)s∈[0,1]is well-defined. Then, if L(T)∼αT1/3for some α∈R,\none has as T→+∞,\n(2.6) max\u0000\nXN(T)\nT\u0001\n=vnatT+\"Z1\n0θsσ2\ns\nα2Ψ \nα3˙θs\nθ3sσ2s!\nds#\nT1/3+oP\u0000\nT1/3\u0001\n.\nIf1≪L(T)≪T1/3, then\n(2.7) max\u0000\nXN(T)\nT\u0001\n=vnatT−\u0012π2\n2Z1\n0θsσ2\nsds\u0013T\nL(T)2+oP\u0012T\nL(T)2\u0013\n.12 A. LEGRAND AND P. MAILLARD\nIfT1/3≪L(T)≪T, then\n(2.8) max\u0000\nXN(T)\nT\u0001\n=vnatT+\"Z1\n0(˙θs)−\nθ2sds#\nL(T) +oP\u0000\nL(T)\u0001\n,\nwhere (·)−denotes the negative part; and if additionally ˙θs≥0for all s∈[0,1], then\n(2.9) max\u0000\nXN(T)\nT\u0001\n=vnatT−a1\n21/3\"Z1\n0(˙θsσs)2/3\nθsds#\nT1/3+oP\u0000\nT1/3\u0001\n.\nOn different matter, recall that we left the case L(T)≍T(that is N(T) =eγTfor some γ >0) completely\nopen. Considering the definitions above, we may write the following conjecture, which matches [ 1, (5.1)] in\nparticular.\nConjecture 2.4. For the N-BRW with N(T) =eγT,γ >0, one has max(XN(T)\nT) =vγ\nmaxT+oP(T)as\nT→+∞, where\nvγ\nmax:= sup\u001aZ1\n0asds; (as)s∈[0,1]c` adlag s.t. ∀s≤1,−γ≤Zs\n0κ∗\nu(au)du≤0\u001b\n.\nExample: Bernoulli BRW. We now turn to the case of Bernoulli increments: for p∈[0,1], we write\nY∼Ber(p) ifP(Y= +1) = 1 −P(Y= 0) = p. In particular E[Y] =pandVar(Y) =p(1−p). Let\np: [0,1]→(0,1) aC2function (we write ps:=p(s)), and assume that Ls,s∈[0,1] is such that, for Ls∼ Ls,\nthen the variables Y∈Lsare independent with law Ber(ps). Then the log-Laplace and Fenchel-Legendre\ntransforms from (2.2–2.3) can be written,\nκs(θ) = log ms+ log(1 + ps(eθ−1)),\nand\nκ∗\ns(v) =−logms+DKL(Ber( a)||Ber(ps)) =−logms+ (1−v) log1−v\n1−ps+vlogv\nps,\nwhere DKL(·||·) denotes the Kullback–Leibler divergence. Moreover, the speed profile ( vs)s∈[0,1]and the\nnatural speed vnatin (2.4) are well defined if msps<1 for all s∈[0,1].\nIn the following, we take ξs≡2 =msfor all s∈[0,1] (i.e. branching is binary), and ps<1/2. Then, the\nfunctions vs,θsandσs,s∈[0,1] are well defined and can be explicitly expressed in terms of each other.\nOne can numerically calculate vsas the greatest root of κ∗\ns(v) = 0, and express θsandσsin terms of vsas\nfollows:\nθs=∂vκ∗\ns(vs) = log\u0012(1−ps)vs\n(1−vs)ps\u0013\n, (2.10)\nσ2\ns= 1/∂2\nvκ∗\ns(vs) =vs(1−vs). (2.11)\nWe conducted extensive numerical simulations of maximal (and minimal) displacements of this particular\nBernoulli N-BRW. These simulations were made for two different choices of ps,s∈[0,1] and various values\nofTandN=N(T, α), the latter being chosen such that ( logN)/T1/3takes a predetermined value α >0,\nwith α∈ {0.5,1,2,4}. For every simulation, we start with Nparticles in 0, and we plot\n(2.12) maxN(T)\nT:=max\u0000\nXN(T)\nT\u0001\n−vnatT\nT1/3,minN(T)\nT :=min\u0000\nXN(T)\nT\u0001\n−vnatT\nT1/3\nthe position of the maximum and minimum, recentered by the first-order term provided by Conjecture 2.3\nand rescaled by T1/3, as a function of α. We further compare the output with the theoretical result as\nT→ ∞ , with fixed α. The results of the simulations are presented in Figure 2.\nThe simulations use a trick from Brunet and Derrida [ 22], which consists of storing the number of particles\nat each site, instead of the position of every particle individually. This allows for an algorithm with a\ncomplexity of O(αL(T)T) arithmetical operations, since only O(αL(T)) sites are occupied at every time,MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 13\n(a)ps= 0.4−0.3s(θsdecreasing in s)\n(b)ps= 0.1 + 0 .3s(θsincreasing in s)\nFigure 2. Numerical simulations of the recentered, rescaled maximum and minimum positions\nmaxN(T)\nT andminN(T)\nT (see(2.12) ) of the binary, Bernoulli N-BRW and comparison with the\ntheoretical values. See text for details.\nwith very high probability. The code, written in Julia, took several hours to run on a 2020 MacBook Pro\nwith M1 chip.\n2.4.Perspectives. Let us discuss several ways in which our work could be expanded.\nTechnical restrictions. There are a few technical assumptions in Theorem 1.1 which we do not expect to be\noptimal. On the one hand we take an offspring distribution per individual ξ≥2 a.s. for the sake of simplicity,\nbut we also expect our results to hold for ξ≥0,E[ξ]>1 and E[ξ2]<+∞. Notice that if P(ξ= 0) >0, then14 A. LEGRAND AND P. MAILLARD\nthe BRW/BBM has a positive probability of extinction, hence such results should hold conditionally to the\nsurvival of the process.\nOn the other hand, we take σ(·) inC2([0,1]), and in the super-critical regime we assume that it changes\nits monotonicity finitely many times. We expect the results to hold for all C1function, however proving\nthis is not a trivial endeavor (for example, the termR\n(σ′)+(u)duin(1.8), or equivalently the total variation\nofσ, may be arbitrarily large even for σinC1([0,1]) and close to a constant for the uniform norm). Most\nimportantly, our results also assume that σis bounded from above and below: this assumption is largely\nneeded throughout our proof, but it does rule out many functions of physical relevance (see e.g. [40]).\nEmpirical distribution in the sub-critical regime. Estimates on the empirical distribution of the process in\nProposition 1.4 are expected to hold in the sub-critical regime, however it is expected that msub\nTshould be\nreplaced with a random centering (see Remark 8.2 for more details). We also expect (1.14) to hold in that\ncase, but it is not an immediate consequence of our results so we leave it to further work.\nVery large population. In the super-critical regime we left out the case L(T)∼cT,c >0. As stated in\nConjecture 2.4, we expect the first order of max(XN\nT) to transition from v(1)TtovmaxTascincreases, when\nσis not decreasing. Even though our methodology is still expected to yield the correct estimates with an\nappropriate choice of barriers in Section 4, removing the assumption L(T)≪Tfrom our proofs requires\nseveral additional arguments and technical estimations, that we leave to further work.\nGenealogy of the N-BBM. In [9], the authors study the genealogy of a sample of particles in a (time\nhomogeneous) BBM with drift and adsorption, and prove that it converges to the genealogy of the Bolthausen-\nSznitman coalescent. More specifically, they choose a near critical drift depending on some constant L >0,\nsuch that the process contains roughly eLparticles throughout a time interval of length L3, and they remain\nin a space interval of length L; moreover the adsorption only kills the bottom-most particles of the process.\nComparing these properties with those of the N(T)-BBM, we therefore expect the same convergence to\nhold for the genealogy of the N(T)-BBM in the critical regime logN≈T1/3, up to a time-change of the\ncoalescent due to the inhomogeneity in time. Regarding the sub-critical (resp. super-critical) regime, a\nsimilar convergence should hold towards the (time-changed) law of the Bolthausen-Sznitman coalescent,\nrunning on a very long (resp. very short) time interval.\nComparison between “beam search” N-BBM/BRW and other algorithms. As mentioned at the end of\nSection 2.1 for the CREM, we conjecture that other optimization algorithms on BRW trajectories do not\nfare much better than the beam search/ N-BRW for the same complexity. Moreover, we believe that the\nphenomenon of “algorithmic hardness threshold” that we observed at logN≈T1/3for the N-BRW, may\nextend beyond the sole case of beam search to more general optimization algorithms on random instances.\nGeneral time-inhomogeneous BRW. Our current results only apply to the N-BBM and Gaussian N-BRW,\nbut we conjecture that they can be extended to more general BRW laws, as presented in Conjecture 2.3.\nMoreover, let us stress that the largest part of Sections 6 and 7 does notrely on the Gaussian distribution\n(nor the random branching times, see Section 8.3). Therefore, most of the required work should come from\nobtaining moment estimates as in Section 5, which we expect to be technically involved.\n3.Construction and couplings of the N-BBM\n3.1.Definition of the time-inhomogeneous BBM and N-BBM. Let us start this section by recalling\nelementary facts on time-inhomogeneous Brownian motions, and introducing some notation. Throughout\nthis paper, the standard, time-homogeneous Brownian motion on Rwill be denoted with ( Wt)t≥0. Let T >0\nandσ∈ C2([0, T]): then the time-inhomogeneous Brownian motion on [0 , T], with infinitesimal variance\nσ2(·/T) and started from 0, is the centered Gaussian process ( Bt)t∈[0,T]such that,\nE0[BsBt] =Zs∧t\n0σ2(u/T) du , ∀s, t∈[0, T].MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 15\nIt satisfies\n(3.1) ( Bt)t∈[0,T](d)= (WJ(t))t∈[0,T], where J(t) :=Zt\n0σ2(s/T) ds ,\nandJ(·) is a time-change C1-diffeomorphism. Notice that the later identity also holds for W,Bstarted\nfrom some x∈R, i.e. W0=B0=xPx-a.s.. In this paper, the law and expectation of a (non-branching)\nBrownian motion started from x∈Rwill always be denoted with PxandExrespectively. Similarly, the law\nof the Brownian motion started from time-space location ( s, x)∈[0, T]×R(i.e. shifted in time by s) will be\ndenoted P(s,x). It will always be clear from context whether the process considered is the time-homogeneous\n(W) or inhomogeneous ( B) variant.\nThe branching Brownian motion (BBM). We now turn to the branching Brownian motion. In this paper\nwe do not expand to much on precise definitions of branching Markov processes, but the reader can refer\nto [35, 36] for a very complete and general construction, or e.g. [5] for a more accessible presentation.\nThe (time-inhomogeneous) branching Brownian motion (BBM) on [0 , T] can be described with some\nrandom families, ( Nt)t∈[0,T]and ( Xu(t))u∈Nt,t∈[0,T], where the (finite) set Ntdenotes the labels of particles\nalive at time t∈[0, T], and Xu(t) denotes the position at time t∈[0, T] of a particle u∈ N t; which satisfies\nthe following properties:\n— each individual u∈ N t,t∈[0, T] dies at rate β0≥0, and is immediately replaced by a random number\nof descendants with law ξ≥2 at the same position,\n— for u∈ N t,t∈[0, T], the function ( Xu(s))s∈[0,t]denotes the positions of uand its ancestors throughout\n[0, t]: it has same law as a time-inhomogeneous Brownian motion ( Bs)s∈[0,T]started from Xu(0),\n— the evolution of particles (lifespan, number of descendants and infinitesimal displacement) are indepen-\ndent.\nRemark 3.1. Throughout the remainder of this paper, unless stated otherwise, the branching processes\nwe consider have offspring distribution ξ≥2withE[ξ2]<+∞, and branching rate β0= (2(E[ξ]−1))−1;\nin particular, if the process is started from a single particle at x∈R, a standard computation yields\nEδx[|Nt|] =et/2for all t∈[0, T](see e.g. [5]). We do not write those assumptions again.\nRecall that Cdenotes the set of all finite counting measures on R. Then, letting Xt:=P\nu∈NtδXu(t), the\nfamily ( Xt)t∈[0,T]defines a Markov process on C, which completely describes the particle configurations of\nthe BBM —in the following, we only write the sets of labels ( Nt)t∈[0,T]explicitly if they are needed. Since\nwe assumed E[ξ2]<+∞, the total population of the process does not blow up on [0 , T] with probability 1\n(see e.g. [ 52] for a proof). For µ∈C, the law and expectation of the (time-inhomogeneous) BBM started\nfrom the initial configuration X0=µwill be denoted PµandEµrespectively throughout this paper. When\nit is started from a single particle at the origin (i.e. µ=δ0), we shall sometimes omit the subscript and\nwriteP,E.\nWith a slight abuse of notation, any finite counting measure µ∈Ccan be written as a finite subset of\nR, with possible repetition of its elements. In particular, for µ, ν∈C, one may write µ⊂νif all atoms in\nthe counting measure µare also present in ν. Regarding the BBM, one has max(Xt) =max u∈NtXu(t) with\nthat notation. Finally, let us mention that one can consider a time-homogeneous BBM very similarly by\nreplacing ( Bs)s∈[0,T]with ( Ws)s∈[0,T]in the definition above; but unless specified otherwise, we shall only\nconsider time-inhomogeneous BBM’s throughout this paper.\nThe N-BBM. Recall that CNdenotes the set of counting measures on Rwith total mass at most N. The\nN-particles branching Brownian motion (N-BBM) started from µ∈CNcan be defined from the original\nBBM ( Xt)t∈[0,T]by only keeping its Nhighest particles at all time, killing (i.e. removing from the process)\nthe others as well as their offspring. For convenience, we allow the N-BBM to start with fewer than N\nparticles. Its particle configuration and set of (living) particles at time t∈[0, T] are respectively denoted with\nXN\ntandNN\nt(the positions of particles are still denoted Xu(t),u∈ NN\nt,t∈[0, T]). A rigorous construction\nof the N-BBM is presented in Proposition 3.1 below.16 A. LEGRAND AND P. MAILLARD\n3.2.Monotonous couplings. Forµ, ν∈C, we write µ≺νifµ([x,+∞))≤ν([x,+∞)) for all x∈R; in\nparticular this implies µ(R)≤ν(R) and max(µ)≤max(ν). Moreover, for two random counting measures X,\nYonR, we say that “ Xis stochastically dominated by Y” if there exists a coupling between XandYsuch\nthatP(X ≺ Y ) = 1. In this section, we are interested in couplings between BBM’s and/or N-BBM’s which\npreserve the comparison ≺through time. Those are quite standard properties, which we reproduce here for\nthe sake of completeness. Recall that, with an abuse of notation, any counting measure µ∈Ccan be seen as\na finite subset of R(with possible repetition of its elements).\nProposition 3.1. ForN∈N,µ∈CN, there exists a coupling between a BBM and an N-BBM both\nstarted from µ, such that, with probability 1, one has XN\nt⊂ X tfor all t∈[0, T]. In particular, one has\nPµ(∀t∈[0, T],XN\nt≺ X t) = 1 .\nProof. Consider a BBM without selection ( Xt)t∈[0,T]started from µ, and recall that the trajectory of an\nindividual u∈ N tthroughout [0 , t] in the BBM is written ( Xu(s)),s∈[0, t]. Let us construct an N-BBM\nwhich satisfies XN\nt⊂ X tfor all t∈[0, T]. First we let XN\n0=µ; then, let ( tk)1≤k≤K,t1< t2< . . . < t K\ndenote the (random) epochs of branching events in the BBM X, and let t0:= 0, tK+1:=T(since the\nbranching process does not explode in finite time, that sequence is a.s. well-defined and finite). For s < t 1,\nletNN\ns:=NsandXN\ns:=Xs.\nAssume that NN\nt,XN\ntare defined for t∈[0, tk), 1≤k≤K, and satisfy XN\nt⊂ X tfor all t∈[0, tk);\nand let us extend their definition to [ tk, tk+1). Let Mk−1≤Ndenote the number of living particles in the\nN-BBM between times tk−1andtk. Let us assume (without loss of generality) that for s∈[tk−1, tk), the set\nof particle labels can be written NN\ns={uk−1\n1, . . . , uk−1\nMk−1} ⊂ N s.\nAt time tk, one individual w0∈ N tkbranches and is immediately replaced with ξk∼ξdescendants,\nwhich we label w1, . . . , w ξk. Ifw0/∈ NN\ntk−1, then we let NN\ns:=NN\ntk−1andXN\ns:=PMk−1\ni=1δXuk−1\ni(s)for all\ns∈[tk, tk+1). Otherwise, we consider the set\nL:={w1, . . . , w ξk}[\nNN\ns\\ {w0}.\nForu∈ L, define Yu:=limt↑tkXu(t) ifu∈ NN\ns, and Yu:=limt↑tkXw0(t) ifu∈ {w1, . . . , w ξk}. Let us order\nthe family Y= (Yu)u∈L, and for all s∈[tk, tk+1), letNN\ns⊂ L denote the Nlabels associated with the\nhighest values of Y(if|L|< N, letNN\ntk:=L). Finally, let XN\ns:=P\nu∈NNsδXu(s)for all s∈[tk, tk+1).\nTherefore, we have extended the definition of ( NN\ns,XN\ns) to the interval [ tk, tk+1); and one can check that\nit has the same law as an N-BBM, and that XN\ns⊂ X sfor all s < t k+1with probability 1. By iterating\nthis construction up to time T(upon which there is no reproduction event with probability 1), we obtain a\ncoupling between the BBM and N-BBM such that, with probability 1, XN\ns⊂ X sthroughout [0 , T]. □\nFurthermore, we claim that for N1≤N2it is possible to couple an N1- and an N2-BBM such that, if\ntheir respective initial configurations µ1,µ2satisfy µ1≺µ2, then the stochastic domination between the\nprocesses is maintained throughout [0 , T] with probability 1. We also provide a similar result for the BBM\nwithout selection.\nProposition 3.2. (i)Letµ1, µ2∈Csuch that µ1≺µ2. There exists (N1,t,X1,t)t∈[0,T],(N2,t,X2,t)t∈[0,T]\ntwo BBM’s on the same probability space such that X1,0=µ,X2,0=νandX1,t≺ X 2,tfor all t∈[0, T]with\nprobability 1.\n(ii)LetN1, N2∈N,N1≤N2. Let µ1∈CN1andµ2∈CN2which satisfy µ1≺µ2: then there also\nexists (NN1\nt,XN1\nt)t∈[0,T]and(NN2\nt,XN2\nt)t∈[0,T]respectively an N1- and an N2-BBM, such that XN1\n0=µ1,\nXN2\n0=µ2andXN1\nt≺ XN2\ntfor all t∈[0, T]with probability 1.\n(iii)Under the assumptions of (ii), let0≤β1≤β2; then the processes XN1,XN2above may be taken\nwith respective branching rates β1andβ2instead of β0, and the monotonous coupling also holds.\nProposition 3.2.( iii) is the only statement in this paper where we consider branching rates different from\nβ0. It will only be applied with β1= 0 and β2=β0>0: since a branching rate equal to 0 means that thereMAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 17\nis no reproduction event (hence no selection) throughout [0 , T], in that case the process ( XN1\nt)t∈[0,T]is a\ncollection of independent, non-branching Brownian motions, started from the atoms of µ1.\nProof. (i) The construction of a coupling for processes without selection is very straightforward. Let\nk=µ1(R),K=µ2(R) (so k≤K), and let ( M1\nt,Z1\nt)t∈[0,T], . . . , (MK\ntZK\nt)t∈[0,T]beKi.i.d. copies of a\n(time-inhomogeneous) BBM, all starting from the initial configuration δ0∈ C—more precisely, the Mi\nt\ndenote the sets of particle labels, and Zi\nt∈Cthe counting measure describing the particles’ positions from\nthei-th BBM at time t∈[0, T]. For any x∈Rand fixed i≤K, we write\nx+Zi\nt:=X\nu∈Mi\ntδx+Xu(t),\nfort∈[0, T]: then ( Mi\nt, x+Zi\nt)t∈[0,T]defines a BBM started from δx∈ C. Writing µ1=Pk\ni=1δxifor some\nx1≥. . .≥xk∈R, we let\nN1,t:=G\ni≤kMi\nt, andX1,t:=kX\ni=1(xi+Zi\nt), t ∈[0, T],\nso (N1,t,X1,t)t∈[0,T]is a BBM started from µ1. Moreover, there exists y1≥. . .≥yKsuch that µ2=PK\ni=1δyi;\nthus, we define\nN2,t:=G\ni≤KMi\nt, andX2,t:=KX\ni=1(yi+Zi\nt), t ∈[0, T],\nand (N2,t,X2,t)t∈[0,T]is a BBM started from µ2. Finally, the assumption µ1≺µ2implies xi≤yifor all\n1≤i≤k; so for any t∈[0, T] and z∈R,\nX1,t([z,+∞)) =kX\ni=1X\nu∈Mi\nt1{xi+Xu(t)≥z}≤kX\ni=1X\nu∈Mi\nt1{yi+Xu(t)≥z}≤KX\ni=1X\nu∈Mi\nt1{yi+Xu(t)≥z}\n=X2,t([z,+∞)),\nwhich concludes the proof.\n(ii) In order to couple two processes undergoing selection, one needs a little more caution than in the\nproof of ( i).\nLetN:=N2≥N1: for the convenience of the proof, we extend the counting measures µ1,µ2to\nR:=R∪ {−∞} by setting µi({−∞} ) := N−µi(R),i∈ {1,2}. Then, µ1(R) =µ2(R) =N, and in the\nfollowing any counting measure on Rwith mass at most Nwill be extended similarly to a counting measure\nonRwith mass exactly N. Moreover, we may write µ1=PN\nn=1δx0nfor some x0\n1≥. . .≥x0\nN≥ −∞ , and\nµ2=PN\nn=1δy0nfor some y0\n1≥. . .≥y0\nN≥ −∞ . Since we assumed µ1≺µ2, one has x0\nn≤y0\nnfor 1≤n≤N.\nWe consider the following, which we take independently from each other:\n— (Bn,k\nt)t≥0, 1≤n≤N,k≥0, an i.i.d. family of (time-inhomogeneous) Brownian motions,\n—{tk,1≤k≤K}the atoms of a homogeneous Poisson point process on [0 , T] with intensity β0N(in\nparticular K∼Pois( β0NT)); we also write t0:= 0 and tK+1:=T.\n— (Uk)k≥1an i.i.d. family of uniform random variables on {1, . . . , N },\n— (ξk)k≥1an i.i.d. family of random variables with same law as ξ.\nLet us introduce some notation. For 0 ≤s < t 1, letNN\ns:={u0\n1, . . . , u0\nN}a set of Narbitrary labels; and\nfor 1≤n≤N,s∈[0, t1),\nX1\nu0n(s) := x0\nn+Bn,0\ns, and X2\nu0n(s) := y0\nn+Bn,0\ns.\nThen, we may define\nZ1\ns:=NX\nn=1δX1\nu0n(s), andZ2\ns:=NX\nn=1δX2\nu0n(s),18 A. LEGRAND AND P. MAILLARD\nwhich are counting measures on Rwith total mass N. From these, we construct the Ni-BBM for i∈ {1,2},\nby letting XNisbe the restriction of Zi\nstoR, and NNis:={u0\ni;Xu0\ni(s)∈R}: since we assumed that µi\ncontains at most Nireal points, so does the counting measure Zi\ns(or equivalently XNis) for all s∈[0, t1).\nLet us assume that the processes ( LN\ns,Zi\ns)s∈[0,tk)and (NNis,XNis)s∈[0,tk),i∈ {1,2}are defined for some\n1≤k≤K, and that they satisfy the following:\n(a) for s∈[tk−1, tk), the set of labels is given by LN\ns:={uk−1\n1, . . . , uk−1\nN},\n(b) for i∈ {1,2}, the number of realparticles in the i-th process at time tk−1isMk−1\ni:=Zi\ntk−1(R)≤Ni;\nand for s∈[tk−1, tk), the labels of realparticles are given by NNis={uk−1\nn;n≤Mk−1\ni}.\n(c) one has Z1\ntk−1=PN\nn=1δxk−1\nnfor some xk−1\n1≥. . .≥xk−1\nN≥ −∞ , andZ2\ntk−1=PN\nn=1δyk−1\nnfor some\nyk−1\n1≥. . .≥yk−1\nN≥ −∞ . Moreover, one has Z1\ns≺ Z2\nsfor all s∈[tk−1, tk), in particular xk−1\nn≤yk−1\nnfor\n1≤n≤N.\n(d) one has Zi\ns=PN\nn=1δXi\nuk−1n(s)fori∈ {1,2}ands∈[tk−1, tk), where\nX1\nuk−1\nn(s) := xk−1\nn+Bn,k−1\ns−Bn,k−1\ntk−1, and X2\nuk−1\nn(s) := yk−1\nn+Bn,k−1\ns−Bn,k−1\ntk−1;\nin particular, XNisis the restriction of Zi\nstoR. Moreover, one has XN1s≺ XN2sfor all s < t k.\nLet us show that those objects can be extended to the interval [ tk, tk+1) in a way that satisfies the same\nproperties.\nFirst, let us reorder the labels depending on their position when s↑tk. For i∈ {1,2}, and 1 ≤n≤N,\ndefine Yi\nn:=limt↑τkXi\nuk−1\nn(t). Hence for i∈ {1,2}, there exists a permutation πi∈SNsuch that\n(Yi\nπi(n))1≤n≤Nis non-increasing in n. Moreover, the assumption ( c) above ensures us that Y1\nn≤Y2\nnfor all\n1≤n≤N.\nThen, recall the definitions of Ukandξk. At time tk, let us have the processes Zi\n(·),i∈ {1,2}realize a\nbranching event, such that in both processes the Uk-th particle counting from the top at time tk(it has\nlabel uk−1\nπ−1\ni(Uk)) dies and is replaced with ξkdescendants. Notice that there a three possibilities:\n— the reproducing particle is at −∞in both Zi\ns,i∈ {1,2},s∈[tk−1, tk), in particular it is in neither\nXN1sorXN2s,\n— the reproducing particle is at −∞inZ1\ns, but is real in Z2\ns,s∈[tk−1, tk),\n— the reproducing particle is real in both processes.\nLetLN\ntk={uk\nn; 1≤n≤N}a new set of labels. For i∈ {1,2}, if the “reproducing” particle in Zi\nsis at\n−∞, we simply forgo the reproduction and write Mk\ni:=Mk−1\ni,NNi\ntk={uk\nn;n≤Mi}. If a real particle\nreproduces at time tk(so it has label uk−1\nπ−1\ni(Uk)), it is removed from the process Zi\n(·), and replaced with ξk\noffspring: hence, we let Mk\ni:=max(Ni, Mk−1+ξk−1) and Ni\ntk:={uk\nn;n≤Mi}. In both cases, we let\nXi\nukn(tk) :=Yi\nπi(n)for 1≤n≤N(recall that the Y’s have been reordered), and,\nZi\ntk:=NX\nn=1δXi\nukn(tk),andXNi\ntk:=Mk\niX\nn=1δXi\nukn(tk),\nthe latter being the restriction of Zi\ntktoRby construction. Moreover, one notices that Z1\ntk≺ Z2\ntk, and\nsimilarly XN1\ntk≺ XN2\ntk.\nThis completely defines the particle configurations at time tk: for i∈ {1,2},s∈(tk, tk+1), we naturally\nextend the definitions of Zi\ns,XNi\ntkabove by letting\nXi\nukn(s) := Yi\nπi(n)+Bn,k\ns−Bn,k\ntk,\nand since one has Y1\nn≤Y2\nnfor all 1 ≤n≤N, the comparison XN1s≺ XN2sstill holds for all s < t k+1.\nIterating this construction until time tK+1=T(upon which there is no reproduction), this defines processes\n(NNis,XNis)s∈[0,T],i∈ {1,2}, with XN1s≺ XN2sfor all s∈[0, T]. Moreover, it follows naturally from theMAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 19\nconstruction that each process has the same law as a time-inhomogeneous Ni-BBM, finishing the proof of\nthe proposition.\n(iii) The proof is very similar to that of ( ii). Recall that, when the branching rates were both assumed\nequal to β0, the sequence of branching epochs of the coupled processes can be described with a Poisson\npoint process (PPP) {tk, k≥1}with intensity β0Non [0 , T]. Thus, let us define A1:={t1\nk, k≥1},\nA2:={et2\nk, k≥1}two independent PPP on [0 , T] with respective intensities β1Nand ( β2−β1)N; and let\n{t2\nk, k≥1}:=A1∪ A 2: this defines a PPP with intensity β2Non [0 , T]. Then, one can reproduce the\nconstruction from ( ii) with the following modification: construct the two N-BBM’s XN1\n(·),XN2\n(·)by induction\non the intervals [ t2\nk−1, t2\nk),k≥1. Then:\n— if t2\nk∈ A 1, define the two N-BBM’s on [ t2\nk, t2\nk+1) exactly as in Proposition 3.2.( ii),\n— ift2\nk∈ A 2, only apply the reproduction and selection procedure to the N2-BBM, and let the configuration\nof the N1-BBM be unchanged at t2\nk. Since the reproduction and selection mechanism only “increases” the\nmeasure XN2\nt2\nkat time t2\nk, one still has XN1\nt2\nk≺ XN2\nt2\nk.\nApplying these changes to the proof of ( ii), one deduces ( iii) straightforwardly. For the sake of conciseness,\nwe do not reproduce all the details here but leave them to the reader. □\n3.3.Main propositions and proof of Theorem 1.1. LetN=N(T)→+∞asT→+∞, and define\nL(T) =logN(T). Using the coupling propositions presented above, notably Proposition 3.2.( ii), we claim\nthat it is sufficient to prove Theorem 1.1 for some specific initial configurations, and the main result follows.\nIn order to condense all upcoming statements, let us recall the following notation: the three regimes\n(L(T)≪T1/3,L(T)≫T1/3andT∼αT1/3) are respectively denoted with the abbreviations sub,sup\nandcrit. Recall the definitions of the scaling and limiting terms in all regimes from (1.10) and(1.11) . In\nthe remainder of this paper, we shall write “let ∗ ∈ { sup,sub,crit}” instead of “let 1 ≪L(T)≪Twhich\nsatisfies either L(T)≪T1/3,L(T)≫T1/3orL(T)∼αT1/3for some α >0 asT→+∞”; and the symbol\n∗shall denote the regime corresponding to the choice of L(T). In particular, many upcoming statements are\nformulated in terms of b∗\nT,m∗\nTinstead of bsup,bsub. . . mcrit\nT, (and similarly for any upcoming notation).\nLet us introduce two specific families of initial configurations. We will estimate the maximal displacement of\ntheN(T)-BBM when started from one of those, then we shall deduce the general case with the coupling result\nfrom Proposition 3.2. On the one hand, for κ∈[0,1] we shall consider the measure ⌊Nκ⌋δ−κσ(0)L(T)∈CN.\nOn the other hand we define for ε∈(0,1),\n(3.2) µε:=⌈ε−1⌉X\nk=0l\nNkε+ε\n2m\nδ−kεσ(0)L(T)∈C.\nNotice that µεcontains more than Nparticles: when starting an N-BBM from µε, we instantaneously kill\nall particles which are not in the Nhighest. This measure can be seen as an almost-exponential distribution\nof (roughly) Nparticles over the interval [ −σ(0)L(T),0]. Furthermore, recall (1.3): then one can show that,\nforκ∈[0,1] and ε∈(0,1),\nQT(⌊Nκ⌋δ−κσ(0)L(T)) =σ(0) log\u0000\n⌊Nκ⌋N−κ\u0001\n=o(1), and 0 ≤QT(µε)≤εσ(0)L(T).\nIn particular, assuming εis arbitrarily small, these quantities are of order o(L(T)) when Tis large.\nFurthermore, it will be convenient in the remainder of this paper to formulate statements which hold\nuniformly on some class of variance functions σ(s/T),s∈[0, T],σ∈ C2([0,1]). Therefore, we define for\nη >0 small,\n(3.3) Sη:=\b\nσ∈ C2([0,1]) ;∀u∈[0,1],|σ′(u)| ≤η−1,|σ′′(u)| ≤η−1andη≤σ(u)≤η−1\t\n,\nin particular C2([0,1]) =S\nη>0Sη, and σ∈ Sηimplies\n(3.4) ∀0≤u, v≤1,\f\f\f\fσ(u)\nσ(v)−1\f\f\f\f≤η−2|u−v|.20 A. LEGRAND AND P. MAILLARD\nFor the convenience of the notation, we also define\nSsup\nη:=\b\nσ∈ Sη;∃u0= 0< u1< . . . < u ⌈η−1⌉= 1 ; σis monotonic on [ ui−1, ui],1≤i≤ ⌈η−1⌉\t\n, (3.5)\nScrit\nη=Ssub\nη:=Sη,\nWith those definitions, we have the following results.\nProposition 3.3. Let∗ ∈ { sup,crit,sub}, and λ, η > 0. Then,\n(3.6) lim\nT→+∞sup\nκ∈[0,1]sup\nσ∈S∗ηPNκδ−κσ(0)L(T)\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≤ −λ\u0013\n= 0.\nProposition 3.4. Let∗ ∈ { sup,crit,sub}, and λ, η > 0. Then,\n(3.7) lim\nε→0lim sup\nT→+∞sup\nσ∈S∗ηPµε\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≥λ\u0013\n= 0.\nPropositions 3.3 and 3.4 may be seen as particular cases of Theorem 1.1 (with some added uniformity\ninσ(·)). Their proofs are contained in Sections 6 and 7 respectively, and rely on moment estimates from\nSection 5. In the remainder of this section, we deduce Theorem 1.1 from these propositions.\nProof of Theorem 1.1 subject to Propositions 3.3 and 3.4. LetµT∈CN(T). Recall the definition of QT(·)\nfrom (1.3) and notice that, for any x∈R,\n(3.8) QT(µT(· −x)) = x+QT(µT)).\nHence, by shifting the process and initial configuration by −QT(µT),T≥0, we may assume without loss of\ngenerality that QT(µT) = 0. Let λ >0, and let us write with an union bound,\nPµT\u00121\nb∗\nT\f\f\fmax(XN(T)\nT)−m∗\nT\f\f\f≥λ\u0013\n≤PµT\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≤ −λ\u0013\n+PµT\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≥λ\u0013\n. (3.9)\nThen we treat both terms separately.\nLetε >0. Since QT(µT) = 0, there exists κ=κ(ε, T)∈[0,1] such that\nµT\u0000\n[−ε−κσ(0)L(T),+∞)\u0001\n≥N(T)κ.\nIn particular, this implies µT≻Nκδ(−ε−κσ(0)L(T)). Therefore, Proposition 3.2 and a shift by εyield,\nPµT\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≤ −λ\u0013\n≤PNκδ(−κσ(0)L(T))\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−ε−m∗\nT\u0011\n≤ −λ\u0013\n,\nand since ε/b∗\nT< λ/ 2 for Tsufficiently large, we deduce from Proposition 3.3 that the first term in (3.9)\nvanishes as T→+∞.\nOn the other hand, for ε >0 the definition of QT(µT) implies\n∀κ∈[0,1], µ T\u0000\n[ε−κσ(0)L(T),+∞)\u0001\n< Nκ.\nRecall the definition of µεfrom (3.2). In particular, one notices for all κ∈[0,1],\nµε\u0000\n[−κσ(0)L(T),+∞)\u0001\n≥N(⌊κε−1⌋+1\n2)ε≥Nκ.\nRecalling that µT(R)≤N(T) by assumption, one obtains that µT≺µε(· −ε). Therefore, Proposition 3.2\nand a shift by −εyield,\nPµT\u00121\nb∗\nT\u0010\nmax(XN(T)\nT)−m∗\nT\u0011\n≥λ\u0013\n≤Pµε\u00121\nb∗\nT\u0010\nmax(XN(T)\nT) +ε−m∗\nT\u0011\n≥λ\u0013\n.\nAssuming εwas taken sufficiently small and letting Tbe large, we deduce from Proposition 3.4 that the\nsecond term in (3.9) can be arbitrarily small, which concludes the proof of the theorem. □MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 21\n4.Preliminaries on the BBM with barriers\nLet us put aside the N-BBM for now, and consider the branching Brownian motion between barriers , a\nvariant of the BBM which is the cornerstone of the proof of Theorem 1.1. This section assembles all our\nnotation on the BBM killed at certain barriers, as well as preliminary results. We first introduce some\nnotation which are used in Sections 5 through 7, then we present the main ideas and tools for the proofs of\nPropositions 3.3 and 3.4.\n4.1.Preliminaries and notation. Recall (3.3–3.4), where we fix η >0 sufficiently small so that σ∈ Sη. In\nthe following, for any function θ:R+→R∗\n+such that θ(T)→0 asT→+∞, we write for f, g:R+→R+,\nandT≥0,\n(4.1) f(T)≤θg(T) if f(T)≤θ(T)g(T),\nand, symmetrically, g(T)≥θf(T) ifg(T)≥θ(T)−1f(T). In particular, having (4.1) for some θ(·) and all T\nlarge implies f(T)≪g(T); and, conversely, having f(T)≪g(T) implies that there exists some θ(·) such\nthatf(T)≤θg(T) for Tsufficiently large.\nIn the following we fix 1 ≪L(T)≪Tsuch that L(T)≪T1/3,L(T)≫T1/3orL(T)∼αT1/3,α >0 as\nT→+∞; and let ∗ ∈ { sup,sub,crit}denote the matching regime. Then, let θ(·) be an (arbitrary) vanishing\nfunction, which may depend on L(T), such that, for Tsufficiently large, one has\n(4.2)(\n1≤θL(T)≤θT1/3, if∗= sub ,\nT1/3≤θL(T)≤θT , if∗= sup .\nA pair of barriers , which we usually write ( γ∗\nT(·),γ∗\nT(·)) in the remainder of this paper, is a pair of (smooth)\nfunctions from [0 , T] toR, depending on L(T), which satisfy the following for some h > x > 0:\n(4.3) γ∗\nT(0) := −xσ(0)L(T)<0, and γ∗\nT(r)−γ∗\nT(r) := hσ(r/T)L(T),∀r∈[0, T].\nWe refer to γ∗\nT(·) (resp. γ∗\nT(·)) as the lower (resp. upper ) barrier. Therefore, throughout the article and all\nregimes, the parameter hdenotes (up to a scaling term) the gap in-between the two barriers, and xdenotes\nthe distance from the origin to the lower barrier at time t= 0. When we want to explicit the parameters\nh > x > 0 for which the barriers satisfy (4.3), we shall add them as superscripts by writing γ∗,h,x\nT,γ∗,h,x\nT\n(when they are clear from context we shall not write them, to lighten formulae).\nRecall that Eµ,Pµdenote the expectation and law of a BBM started from some configuration µ∈C. For\ny, w∈R, 0≤s≤t≤T, we let\nG∗(y, w, s, t ) dy (4.4)\n:=Eδ(s,γ∗\nT(s)+yσ(s/T)L(T))\u0014\f\f\fn\nu∈ N t;Xu(r)∈\u0002\nγ∗\nT(r),γ∗\nT(r)\u0003\n,∀r∈[s, t],Xu(t)−γ∗\nT(t)\nσ(t/T)L(T)∈dwo\f\f\f\u0015\n,\ndenote the expected number of descendants in the BBM of a single particle at time-space location ( s, γ∗\nT(s) +\nyσ(s/T)L(T)), whose path remain between the barriers γ∗\nT(·),γ∗\nT(·) until time t, at which point it reaches an\ninfinitesimal neighborhood of γ∗\nT(t) +wσ(t/T)L(T) (notice that it is zero unless y, w∈[0, h]). Furthermore,\nfort∈[0, T], we denote the set of particles which remained between the barriers throughout [0 , t] and ended\nat time tin some interval γ∗\nT(t) +σ(t/T)L(T)·I,I⊂[0, h], with\n(4.5) A∗\nT,I(t) :=n\nu∈ N t\f\f\fXu(s)∈\u0002\nγ∗\nT(s),γ∗\nT(s)\u0003\n,∀s∈[0, t] ;Xu(t)−γ∗\nT(t)\nσ(t/T)L(T)∈Io\n.\nIn particular, one has Eδ0[|A∗\nT,I(t)|] =R\nIG∗(x, w, 0, t)dw. To lighten notation, we shall also write A∗\nT(t) :=\nA∗\nT,[0,h](t) and A∗\nT,z(t) := A∗\nT,[z,h](t) respectively for the specific cases I= [0, h] (no constraint on the\nfinal height within the barriers) and I= [z, h], for some z∈[0, h) (lower constraint only). Finally, for22 A. LEGRAND AND P. MAILLARD\n0≤s≤t≤T, letR∗\nT(s, t)∈Ndenote the number of particles which remain above γ∗\nTuntil they get killed\nbyγ∗\nTat some time r∈[s, t]: more precisely,1\n(4.6) R∗\nT(s, t) :=\f\f\f\f\f\f[\nr∈[s,t]n\nu∈ N r\f\f\fXu(v)∈\u0000\nγ∗\nT(v),γ∗\nT(v)\u0001\n∀v < r ;Xu(r) =γ∗\nT(r)o\f\f\f\f\f\f.\nWith the above notation at hand, we may present the main ideas of the remainder of the proof. Recall\nthe definitions of v(·) and Ψ( ·) from (1.2) and (1.4) respectively, and let wh,T∈ C1([0,1]) be defined by2\n(4.7) wh,T(r) :=−Zr\n0σ(u)\nα3h2Ψ\u0012\nα3h3σ′(u)\nσ(u)\u0013\ndu≥0, r ∈[0,1].\nLeth > x > 0. Then, depending on L(T) and its regime ∗ ∈ { sup,sub,crit}, we define a pair of barriers by\nsetting, for t∈[0, T],\nγsup\nT(t) := v(t/T)T+h L(T)Zt/T\n0(σ′)−(u) du−x σ(0)L(T), (4.8)\nγsub\nT(t) := v(t/T)Ts\n1−π2\nh2L(T)2−x σ(0)L(T), (4.9)\nγcrit\nT(t) := v(t/T)T−wh,T(t/T)L(T)−x σ(0)L(T), (4.10)\nand we let γ∗\nT(t) :=γ∗\nT(t) +hσ(t/T)L(T) in each regime, so that (4.3) holds for h > x > 0 fixed. One of\nthe core ideas used in Sections 5 through 7 is that, when started from a single particle, the N-BBM is\nquite similar to a BBM whose particles are killed when reaching the barriers γ∗\nT(·),γ∗\nT(·), as soon as their\nparameters h > x > 0 are both close to 1. In particular, recall (1.10–1.11), and let us point out the following\nconvergence.\nLemma 4.1. Let∗ ∈ { sup,sub,crit}. Then,\n(4.11) lim sup\n(h,x)→(1,1),\nh>x> 0lim sup\nT→+∞1\nb∗\nT\f\f\fm∗\nT−γ∗,h,x\nT(T)\f\f\f= 0.\nProof. The proof is straightforward in all three regimes. In the super-critical case, one has\nγsup,h,x\nT (T) =v(1)T+hL(T)Z1\n0\u0002\n(σ′)+−σ′\u0003\n(u) du−xσ(0)L(T) +hσ(1)L(T)\n=v(1)T+hL(T)Z1\n0(σ′)+(u) du+ (h−x)σ(0)L(T),\nso1\nL(T)|msup\nT−γsup,h,x\nT (T)| ≤η−1(|1−h|+|h−x|) for all T≥0, which yields the expected result. In the\nsub-critical regime, one has\nγsub,h,x\nT (T) =v(1)Ts\n1−π2\nh2L(T)2+O(L(T)),\nwhere O(L(T)) is locally uniform in h > x > 0. Letting hclose to 1 and writing the Taylor expansion√1−y= 1−y\n2+o(y) asy→0, this yields (4.11) . Regarding the critical regime, recall that Ψ satisfies\n1One can check with standard branching processes theory that R∗\nT(s, t) is a measurable, almost surely finite random variable;\nand that, with probability 1, two particles do not reach the upper barrier at the same time. For the sake of conciseness we do\nnot develop on that in this paper.\n2Let us point out that, compared to (1.7), we added an αin the denominator: this is because it will be more convenient in\nupcoming computations to express the second order of the critical regime (4.10) in terms of L(T)∼αT1/3instead of T1/3.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 23\nΨ(−q) =q+ Ψ(q) for all q∈R. Therefore, wh,Tsatisfies\nwh,T(1) = −Z1\n0σ(u)\nα3h2Ψ\u0012\nα3h3σ′(u)\nσ(u)\u0013\ndv\n=−Z1\n0σ(u)\nα3h2Ψ\u0012\n−α3h3σ′(u)\nσ(u)\u0013\ndv+σ(1)−σ(0).\nPlugging this into (4.10) and recalling that L(T)∼αT1/3in this regime, this straightforwardly concludes\nthe proof. □\nIn Section 5 below, we prove that the BBM between the barriers γ∗\nT,γ∗\nTsatisfies the following moment\nestimates:\n(4.12)\nEδ0\u0002\n|A∗\nT,I(t)|\u0003\n≈e(x−infI)L(T),Eδ0\u0002\n|A∗\nT,z(t)|2\u0003\n≲e(x+h−2z)L(T),and Eδ0\u0002\nR∗\nT(0, t)\u0003\n≲e−(h−x)L(T).\nforTsufficiently large, where precise assumptions on t, z, I and statements are formulated in Section 5.\nThereafter, Sections 6 and 7 make use of these estimates to state rigorous comparisons between the N-BBM\nand the BBM between the barriers γ∗\nT,γ∗\nT, from which we finally deduce Propositions 3.3 and 3.4.\n4.2.Toolbox for moment estimates. In order to prove the moment estimates from (4.12) , we introduce\nsome technical tools which are useful throughout Section 5 and for similar computations below (e.g.\nLemma 7.2).\nMost of the first moment estimates below rely on the first moment formula for branching Markov\nprocesses [ 36, Theorem 4.1], often called “Many-to-one lemma”, as well as Girsanov’s theorem: we condense\nthem in the following statement.\nLemma 4.2. Let∗ ∈ { sup,sub,crit}. Let t≤T,γ∗\nT,γ∗\nT∈ C2([0, T])which satisfy (4.3) for some h > x > 0,\nandI∈Bor([0 , h]). Then, one has\nEδ0\u0002\n|A∗\nT,I(t)|\u0003\n=et\n2Exσ(0)L(T)\"\n1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤t;Bt\nσ(t/T)L(T)∈I}(4.13)\n×exp \n−γ∗\nT′(t)Bt\nσ2(t/T)+γ∗\nT′(0)\nσ(0)xL(T) +Zt\n0∂\n∂u\u0012γ∗\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds−Zt\n0(γ∗\nT′(s))2\n2σ2(s/T)ds!#\n.\nMoreover, letting H0(Y) := inf {t≥0, Yt= 0}for(Yt)t≥0∈ C0(R+), one has\nEδ0\u0002\nR∗\nT(0, t)\u0003\n=E(h−x)σ(0)L(T)\"\neH0(B)/21{H0(B)≤t}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}(4.14)\n×exp \n−γ∗\nT′(0)\nσ(0)(h−x)L(T)−ZH0(B)\n0∂\n∂u\u0012γ∗\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds−ZH0(B)\n0(γ∗\nT′(s))2\n2σ2(s/T)ds!#\n.\nLet us mention that (4.14) has an analogous formulation in terms of γ∗\nTinstead of γ∗\nT, which involves the\nhitting time of the curve ( hσ(s/T)L(T))s∈[0,T]. We decided to stick with the expression (4.14) in the lemma,\nsince it is used more often in this paper.24 A. LEGRAND AND P. MAILLARD\nProof. Let us start with (4.13). The Many-to-one lemma [36, Theorem 4.1] and Girsanov’s theorem yield,\nEδ0\u0002\n|A∗\nT,I(t)|\u0003\n=Eδ0\"X\nu∈Nt1\bXu(s)−γ∗\nT(s)\nσ(s/T)L(T)∈[0,h],∀s≤t;Xu(t)−γ∗\nT(t)\nσ(t/T)L(T)∈I\t#\n=Eδ0\u0002\n|Nt|\u0003\n×P0\u0012Bs−γ∗\nT(s)\nσ(s/T)L(T)∈[0, h],∀s≤t;Bt−γ∗\nT(t)\nσ(t/T)L(T)∈I\u0013\n=Eδ0\u0002\n|Nt|\u0003\n×E−γ∗\nT(0)\"\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s≤t;Bt\nσ(t/T)L(T)∈I\te−Rt\n0γ∗\nT′(s)\nσ2(s/T)dBs−Rt\n0(γ∗\nT′(s))2\n2σ2(s/T)ds#\n. (4.15)\nRecall that Eδ0\u0002\n|Nt|\u0003\n=et\n2under our assumptions, and write with an integration by parts, for t∈[0, T],\n(4.16)Zt\n0γ∗\nT′(s)\nσ2(s/T)dBs=γ∗\nT′(t)Bt\nσ2(t/T)−γ∗\nT′(0)B0\nσ2(0)−Zt\n0∂\n∂u\u0012γ∗\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds .\nSince (4.3) implies γ∗\nT(0) =−xσ(0)L(T), plugging this into (4.15) yields (4.13).\nRegarding (4.14), the Many-to-one lemma gives\nEδ0\u0002\nR∗\nT(0, t)\u0003\n=E0\u0014\neH0(γ∗\nT−B)/21{H0(γ∗\nT−B)≤t}1\bγ∗\nT(s)−Bs\nσ(s/T)L(T)≤h,∀s≤H0(γ∗\nT−B)\t\u0015\n.\nThen, applying Girsanov’s theorem and recalling that ( Bs)s≤tunder Pxas the same law as ( −Bs)s≤tunder\nP−x,x∈R, one obtains\nEδ0\u0002\nR∗\nT(0, t)\u0003\n=Eγ∗\nT(0)\u0014\neH0(B)/21{H0(B)≤t}1\b\nBs\nσ(s/T)L(T)≤h,∀s≤H0(B)\teRH0(B)\n0γ∗\nT′(s)\nσ2(s/T)dBs−RH0(B)\n0(γ∗\nT′(s))2\n2σ2(s/T)ds\u0015\n.\nRecalling that γ∗\nT(0) = ( h−x)σ(0)L(T) by(4.3), and replacing γ∗\nTwith γ∗\nTin(4.16) , this yields (4.14) and\nfinishes the proof of the lemma. □\nRemark 4.1. The reader may notice that, in the super-critical regime, Lemma 4.2 requires to differentiate\n(σ′)+or(σ′)−: since we assume σ∈ Ssup\nη⊂ C2([0,1])in that regime, there is only finitely many points where\nthose derivatives are ill-defined —and even at those points, (σ′)+and(σ′)−admit left- and right-derivatives,\nwhich are bounded (in absolute value) by η−1. Therefore, all estimations involving these can be rendered\nrigorous by splitting the interval [0, T], and we may write with an abuse of notation ∥((σ′)+)′∥∞≤η−1,\n∥((σ′)−)′∥∞≤η−1. In order not to overburden the presentation of the proofs, we omit those details in the\nremainder of this paper.\nLet us also quote the “Many-to-two lemma” [ 36, Theorem 4.15] below, which will be used in all second\nmoment computations.\nLemma 4.3 (Many-to-two lemma) .Let∗ ∈ { sup,sub,crit}. Let t≤T,γ∗\nT,γ∗\nT∈ C1([0, T])which sat-\nisfy(4.3), and z∈[0, h). Then, one has\n(4.17) Eδ0\u0002\n|A∗\nT,z(t)|2\u0003\n=Eδ0\u0002\n|A∗\nT,z(t)|\u0003\n+β0E[ξ(ξ−1)]Zt\n0dsZh\n0G∗(x, y,0, s)\u0012Zh\nzG∗(y, w, s, t ) dw\u00132\ndy .\nThen we quote a sharp result on the survival probability of the standard Brownian motion between\nbarriers, see e.g. [ 16, Part II.1, Eq. 1.15.8] or more recently [ 43, (7.8–7.10)]. Recall that the standard,\ntime-homogeneous Brownian motion is denoted ( Ws)s≥0.\nLemma 4.4 (Brownian motion in an interval) .Fort >0,h >0, and x, z∈[0, h], one has,\n(4.18)Px\u0000\nWs∈[0, h],∀s≤t;Wt∈dz\u0001\n=2\nh∞X\nn=1exp\u0010\n−π2\n2h2n2t\u0011\nsin\u0010πnx\nh\u0011\nsin\u0010πnz\nh\u0011\ndz\n=2\nhexp\u0010\n−π2\n2h2t\u0011\nsin\u0010πx\nh\u0011\nsin\u0010πz\nh\u0011\u0000\n1 +o(1)\u0001\ndz,MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 25\nwhere o(1)is a term vanishing as t/h2→+∞, uniformly in x, z(see[43, (7.9)] for an explicit expression).\nFinally, we provide a technical lemma on our choice of barriers γ∗\nT,γ∗\nT, for∗ ∈ { sup,sub,crit}. Recall their\ndefinition from (4.8–4.10), and the notation γ∗\nT=γ∗,h,x\nT andγ∗\nT=γ∗,h,x\nT. We claim that we may “tighten”\nthe barriers on a short time interval (i.e. shorter than L(T)3), by modifying the parameters h > x > 0.\nRecall from (4.1) that θ(·) denotes a function vanishing at + ∞.\nLemma 4.5. Let∗ ∈ { sup,sub,crit}, and let t=t(T)such that 0≤t(T)≤θL(T)3∧TforTsufficiently\nlarge. Let h > x > 0andh′> x′>0such that\n(4.19) x′< x , and h′−x′< h−x .\nThen, there exists T0such that, for T≥T0ands∈[0, t(T)], one has\n(4.20) γ∗,h,x\nT(s)≤γ∗,h′,x′\nT (s)≤γ∗,h′,x′\nT (s)≤γ∗,h,x\nT(s).\nMoreover, T0is uniform in σ∈ Sη, and locally uniform in h > x > 0,h′> x′>0which satisfy (4.19) .\nProof. Notice that the assumptions also imply h′< h. We prove this claim separately for each ∗ ∈\n{sup,sub,crit}.\nIn the super-critical case, since ( h′−h)<0, one has for all s∈[0, T],\nγsup,h′,x′\nT (s)−γsup,h,x\nT (s) = ( h′−h)L(T)Zt/T\n0(σ′)−(u) du−(x′−x)σ(0)L(T)\n≥(h′−h)η−1θ(T)L(T) + (x−x′)ηL(T)≥0,\nwhere the last inequality holds for Tlarger than some T0locally uniform in x, x′, h, h′. Moreover, one can\neasily check that\n(4.21) γsup,h,x\nT (s) =v(t/T)T+h L(T)Zt/T\n0(σ′)+(u) du+ (h−x)σ(0)L(T),\nfor all s∈[0, T],h > x > 0. Hence, one has for all s∈[0, T].\nγsup,h,x\nT (s)−γsup,h′,x′\nT (s) = ( h−h′)L(T)Zt/T\n0(σ′)+(u) du+ [(h−x)−(h′−x′)]σ(0)L(T)≥0,\nwhich concludes the proof.\nRegarding the sub-critical case, on the one hand we have for s∈[0, T],\n(4.22) γsub,h′,x′\nT (s)−γsub,h,x\nT (s) = ( x−x′)σ(0)L(T) +\"s\n1−π2\nh′2L(T)2−s\n1−π2\nh2L(T)2#\nv(s/T)T ,\nand a direct Taylor expansion gives as L(T)→+∞,\n(4.23)s\n1−π2\nh′2L(T)2−s\n1−π2\nh2L(T)2=−π2(h2−h′2)\n2(hh′L(T))2+O(L(T)−4).\nRecall that t(T)≤θ(T)L(T)3≤Tin the sub-critical regime: thus, one has for s∈[0, t(T)],\nv(s/T)T=TZs/T\n0σ(u) du≤η−1t(T)≤η−1θ(T)L(T)3.\nSince x > x′, we deduce that for Tsufficiently large, the second term in the r.h.s. of (4.22) is larger than\n−1\n2(x−x′)ηL(T), uniformly in σ∈ Sηands∈[0, t(T)], locally uniformly in x, x′, h, h′; which is one of the26 A. LEGRAND AND P. MAILLARD\nexpected results. On the other hand we have for all s∈[0, T],\nγsub,h,x\nT (s)−γsub,h′,x′\nT (s)\n= (h−h′)σ(s/T)L(T) + (x′−x)σ(0)L(T) +\"s\n1−π2\nh2L(T)2−s\n1−π2\nh′2L(T)2#\nv(s/T)T\n= [(h−x)−(h′−x′)]σ(s/T)L(T) + (x−x′)[σ(s/T)−σ(0)]L(T)−O(θ(T)L(T)),\nwhere we used (4.23) . Recalling (3.4) and that t(T)≤θ(T)T, the second term above is larger than\n−O(θ(T)L(T)) for Tlarge. Therefore, the r.h.s. above is larger than1\n2[(h−x)−(h′−x′)]ηL(T) for T\nsufficiently large: this concludes the proof in the sub-critical regime.\nWe finally turn to the critical case. Recall (4.7) and that Ψ is continuous, hence\nsup\nr∈[h′,h]sup\ns∈[0,t(T)]|wr,T(s/T)| ≤t(T)\nTη−1\nα3h′2supn\n|Ψ(q)|;q∈[−α3h′3η−2, α3h3η−2]o\n.\nSince we assumed t(T)≤θ(T)(L(T)3∧T) for Tlarge, there exists C > 0, uniform in σ∈ Sηand locally\nuniform in h, h′, such that for Tsufficiently large, one has |wr,T(s/T)| ≤Cθ(T) for all r∈[h′, h] and\ns∈[0, t(T)]. In particular, this implies\nγcrit,h′,x′\nT (s)−γcrit,h,x\nT (s)≥(x−x′)σ(0)L(T)−2Cθ(T)L(T),\nand γcrit,h,x\nT (s)−γcrit,h′,x′\nT (s)≥(h−h′)σ(s/T)L(T) + (x′−x)σ(0)L(T)−2Cθ(T)L(T),\nfor all s∈[0, t(T)]. Assuming Tis sufficiently large and reproducing the arguments from the sub-critical\ncase (we do not write them again), this completes the proof of the lemma. □\n4.3.Two auxiliary particle systems. In order to compare the N-BBM and the BBM between barriers,\nwe use in Sections 6 and 7 the following variants of the N-BBM, which were already introduced in [ 43] for\nthe time-homogeneous N-BBM. For some γ∈ C1([0, T]), the N−-BBM and the N+-BBM are constructed as\ntwo systems of BBM undergoing selection with respective mechanisms:\n— In the N−-BBM, a particle uis killed at time tifeither it reaches the lower barrier γ(i.e.Xu(t)≤γ(t))\northere are at least Nother particles v∈ N twith larger displacement Xv(t);\n— In the N+-BBM, a particle uis killed at time tif it is simultaneously located below the lower barrier\n(Xu(t)≤γ(t))andbelow at least Nother particles v∈ N twith larger displacement Xv(t).3\nLet (XN−\nt)t≥0(resp. ( XN+\nt)t≥0) denote the empirical measure of an N−-BBM (resp. N+-BBM). Then\nwe have the following result.\nLemma 4.6 (Lemma 2.9 in [ 43]).Letγ∈ C1([0, T]). Let ν−\n0,νN\n0andν+\n0denote (possibly random) finite\ncounting measures on Rwith M(νN\n0)≤Nandν−\n0≺νN\n0≺ν+\n0. Then there exists a coupling between\ntheN−-BBM, N-BBM and N+-BBM started respectively from ν−\n0,νN\n0andν+\n0such that (XN−\nt)t≥0≺\n(XN\nt)t≥0≺(XN+\nt)t≥0with probability 1.\nRemark 4.2. The results in Maillard [43]concern the time-homogeneous N-BBM, but the proofs can be\nextended verbatim to the time-inhomogeneous N-BBM. Since we already presented a complete construction\nof a coupling of time-inhomogeneous processes in Proposition 3.2, we decided not to reproduce that proof in\nthis paper. However, let us mention that there are two simple improvements to [43, Lemma 2.9] which can be\nmade with a direct check of the proof therein (see [43, Section 2.3] ):\n(i)The proof of the coupling between N+- and N-BBM in [43, Section 2.3] relies on the fact that the lower\nbarrier can only kill the bottommost particles of the process N+. However this assumption can be relaxed for\nthe coupling between the N−-BBM and N-BBM, since killing any other particle than the bottommost in the\nN−-BBM only lowers its empirical measure by that much. Therefore, in the N−-BBM one can additionally\nkill at some upper barrier γand the coupling of Lemma 4.6 still holds.\n3In both cases, whenever the N-th lowest particle is above γand branches, we arbitrarily kill only one of its two descendants.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 27\n(ii)One can also consider a multi-type N+-BBM (or N−-BBM): let Idenote a (finite) set of particle\ntypes and define some barriers γi∈ C1([0, T]),i∈I; then, define a multi-type N+-BBM by killing a particle\nof type i∈Iat time t∈[0, T]if it is below γi(t) and there are Nparticles with higher displacements. Then\nthe proof of [43, Lemma 2.9] can directly be adapted to this multi-type N+-BBM, yielding a monotonic\ncoupling with a (typeless) N-BBM.\n5.Moment estimates on the BBM with barriers\nIn this section we state and prove precise versions of (4.12) in the three regimes: super-critical, sub-critical\nand critical. Let us warn the reader that the upcoming proofs contain a lot of bookkeeping (especially for\nthe first moment estimates), mainly due to the fact that we need to obtain bounds which are uniform in\ncertain ranges of values for t,σ(·) and other parameters.\n5.1.Super-critical case. In this section we assume T1/3≪L(T)≪T, and σ∈ Ssup\nηfor some η >0\n(recall (3.3–3.5)); in particular, there exists u0= 0< u1< . . . < u η−1such that σis monotonic on [ ui−1, ui],\n1≤i≤η−1(we assume w.l.o.g. that η−1∈Nto lighten notation). Recall that γsup\nTis defined in (4.8),\nand that γsup\nTis such that (4.3) holds. Recall also (4.4–4.5): we begin this section by computing first\nmoment estimates for Asup\nT,I(t),t∈[0, T],I⊂[0, T]. Finally, recall (4.1), and that we fixed some arbitrary\nθ:R+→R∗\n+that vanishes at + ∞.\nProposition 5.1 (First moment, Super-critical) .Leth >0. As T→+∞, one has\n(5.1) E\u0002\n|Asup\nT,I(t)|\u0003\n≤e(x−infI)L(T)+o(L(T)),\nuniformly in σ∈ Ssup\nη,x∈[0, h]andI⊂[0, h]a non-trivial sub-interval (i.e. which contains at least two\npoints). Moreover for any h >0,T0≥0fixed, one has as T→+∞,\n(5.2) E\u0002\n|Asup\nT,I(t)|\u0003\n≥e(x−infI)L(T)+o(L(T)),\nlocally uniformly in x∈(0, h),infI∈[0, h]with supI−infI >0, uniformly in σ∈ Ssup\nη, and uniformly in\nt=t(T)∈[0, T]which satisfies L(T)≤θt(T)≤TforT≥T0.\nRemark 5.1. (i)Here, “locally uniformly” means that for any ε >0, there exists o(L(T))∈Rsuch that\n|o(L(T))| ≪L(T)asT→+∞, and the lower bound holds uniformly in x∈[ε, h−ε],infI∈[0, h−4ε]and\nsupI−infI >4ε. Moreover, let us stress that “uniformly” in σ(·)andt(T)means that this error term does\nnot depend on those, as long as η,θ(·)andT0are fixed. In the remainder of the paper we will not write T0\nin similar statements, but rather “ L(T)≤θt(T)≤TforTsufficiently large”, to lighten the phrasing.\n(ii)Finally, let us mention that, in (5.2), the assumption t≥θL(T)is necessary (but maybe not optimal),\nsince a branching process (without selection) started from one individual needs a time ≈L(T)to grow to a\npopulation of size eO(L(T)).\nThe core of the proof of Proposition 5.1 is contained in the following lemma. Its proof is displayed\nafterwards.\nLemma 5.2. Assume T1/3≪L(T)≪T, and let h, η > 0,ε∈(0, h/2). Define,\nΥ1:=\b\n(x, t, σ, I )\f\ft∈[θ−1(T)L(T), T], σ∈ Sη, I= [a, b]⊂[0, h], b−a≥2ε, x∈[ε, h−ε]\t\n,\nand\nΥ2:=\b\n(x, t, σ, I )\f\ft∈[0, T], σ∈ Sη, I= [a, b]⊂[0, h], b−a≥2ε, x∈[ε, h−ε]∩[a+ε2−η−1, b−ε2−η−1]\t\n,\nandΥ := Υ 1∪Υ2. Then, as T→+∞, one has\n1\nL(T)inf\n(x,t,σ,I )∈ΥlogExσ(0)L(T)\u0014\nexp\u0012\n−1\nTZt\n0|σ′(s/T)|\nσ2(s/T)Bsds\u0013\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈I\t\u0015\n(5.3)\n−→\nT→+∞0.28 A. LEGRAND AND P. MAILLARD\nLet us comment briefly on this statement: this proves that the expectation in (5.3) is larger than eo(L(T))\nuniformly in x, t, σ, I , under one of the following assumptions: either tis not too small (see the definition of\nΥ1), or the starting point of the process xis not too far from the target interval I(see Υ 2: for the sake\nof simplicity we restrict ourselves to x∈I◦). These assumptions are necessary (but maybe not optimal):\nindeed, the reader can check with (3.4) and standard Gaussian estimates on Btthat, if tis too small and\nd(x, I) too large, the expectation in (5.3) may be much smaller than e−L(T).\nProof of Proposition 5.1. Letσ∈ Ssup\nη; in particular, there exists u0= 0< u1< . . . < u η−1= 1 such that\nσ, for each 1 ≤i≤η−1,σis either non-increasing or non-decreasing on [ ui−1, ui]. Let us assume h > x > 0,\notherwise the l.h.s. of (5.1) is zero and the proof is immediate. Recall Lemma 4.2, in particular (4.13) .\nNotice that (4.8) implies,\nγsup\nT′(s) =σ(s/T) +hL(T)\nT(σ′(s/T))−,∀s∈[0, T],\nso one has for t∈[0, T],\nE\u0002\n|Asup\nT,I(t)|\u0003\n=et\n2Exσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈I\t\n×exp\u0012\n−γ∗\nT′(t)Bt\nσ2(t/T)+γ∗\nT′(0)\nσ(0)xL(T) +Zt\n0∂\n∂u\u0012γ∗\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds−Zt\n0(γ∗\nT′(s))2\n2σ2(s/T)ds\u0013\u0015\n=exL(T)+O(L(T)2/T)Exσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈I\t(5.4)\n×exp\u0012\n−Bt\nσ(t/T)−1\nTZt\n0σ′(s/T)\nσ2(s/T)Bsds−hL(T)\nTZt\n0(σ′(s/T))−\nσ(s/T)ds\u0013\u0015\n.\nUpper bound. One has −Bt\nσ(t/T)≤ −(infI)L(T) in(5.4), and for u∈[0,1],σ′(u) = ( σ′)+(u)−(σ′)−(u).\nThus, (5.4) yields\nE\u0002\n|Asup\nT,I(t)|\u0003\n≤e(x−inf(I))L(T)+O(L(T)2/T)Exσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈I\t\n×exp\u0012\n−1\nTZt\n0(σ′)+(s/T)\nσ2(s/T)Bsds\u0013\u0015\n,\nand the latter expectation is bounded by 1, which proves the upper bound.\nLower bound. Letε >0 such that inf( I) + 4ε <sup(I), and define,\n(5.5) Iη−1:= [inf( I),inf(I) + 4ε]⊂I .\nBy constraining ( Bs/σ(s/T)L(T))s≥0to end in Iη−1at time t, one deduces from (5.4) that,\nE\u0002\n|Asup\nT,I(t)|\u0003\n≥e(x−infI+4ε)L(T)+O(L(T)2/T)Exσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈Iη−1\t(5.6)\n×exp\u0012\n−1\nTZt\n0σ′(s/T)\nσ2(s/T)Bsds−hL(T)\nTZt\n0(σ′(s/T))−\nσ(s/T)ds\u0013\u0015\n.\nLet us prove that the expectation in the r.h.s. is larger than exp(−η−1εL(T)) for Tlarge, uniformly in\nσ∈ Ssup\nηandL(T)≤θt≤T. Then, letting ε→0 this gives the expected result.\nRecall the definition of ( ui)0≤i≤η−1: letJ1(resp. J2) denote the indices of the intervals on which σ(·) is\nnon-decreasing (resp. decreasing), more precisely,\nJ1:={1≤i≤η−1;σ′(u)≥0,∀u∈[ui−1, ui]}, and J2:={1, . . . , η−1} \\J1.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 29\nThen, a direct computation yields,\n(5.7) −1\nTZt\n0(σ′(s/T))−\nσ(s/T)ds=X\ni∈J2log\u0012σ(ui∧(t/T))\nσ(ui−1∧(t/T))\u0013\n.\nLetimax=imax(t) :=max{i≤η−1;t > u iT}. Finally, define ( Ii)1≤i≤η−1a sequence of intervals such that\nI1has length at least 2 ε, and\n∀2≤i≤η−1, Ii−1(ε2−i)⊂Ii and Ii−1⊂[ε, h−ε],\nwhere Iη−1⊂Iwas defined in (5.5) andA(ε2−i)denotes the closed ( ε2−i)-neighborhood of a set A⊂R. We\nalso write I0= [ε, h−ε], and assume εsmall enough so that x∈I0. Constraining the process to be in Iiat\ntime uiT, 1≤i≤η−1, and applying Markov’s property at times ( uiT)i≤imax, one obtains,\n(5.8) Exσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈Iη−1\texp\u0012\n−1\nTZt\n0σ′(s/T)\nσ2(s/T)Bsds\u0013\u0015\n≥η−1Y\ni=1Ei,\nwhere Ei:= 1 for i > i max, and else\nEi:= inf\nyi∈Ii−1E(ui−1T,yiσ(ui−1)L(T))\"\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[ui−1T,uiT∧t] ;BuiT∧t\nσ(ui∧(t/T))L(T)∈Ii\t (5.9)\n×exp \n−1\nTZuiT∧t\nui−1Tσ′(s/T)\nσ2(s/T)Bsds!#\n.\nWe treat those factors differently depending on i. Recall Lemma 5.2 and that t≥θL(T). Ifi∈J1, then\nσ′(u)≥0 for all u∈[ui−1T, uiT∧t]. Moreover, one notices for yi∈Ii−1andTsufficiently large that,\n— if i= 1, then ( y1, t∧(u1T), σ(·), I1)∈Υ1,\n— if i >1 and t≥ui−1T, then ( yi, t∧(uiT)−ui−1T, σ(· −ui−1), Ii)∈Υ2.\nTherefore, we deduce from Lemma 5.2 that, for Tlarger than some T1>0, one has Ei≥e−εL(T).\nMoreover, that T1does not depend on i∈J1orσ∈ Ssup\nη.\nLet us now consider i∈J2: in order to lighten notation, let us assume i= 1 without loss of generality. Then\nwe apply Girsanov’s theorem to the shifted process ( Bs−hσ(s/T)L(T))s≤u1T. Letting ρ(s) :=hσ(s/T)L(T),\none notices that, on the event {|Bs| ≤hη−1L(T),∀s≤u1T}, one has\n(5.10)\f\f\f\f\fZu1T∧t\n0ρ′(s)\nσ2(s/T)dBs\f\f\f\f\f≤3h2η−4L(T)2\nT, and\f\f\f\f\fZu1T∧t\n0(ρ′(s))2\n2σ2(s/T)ds\f\f\f\f\f≤h2η−4L(T)2\n2T,\n(this follows from an integration by parts and computations similar to those of (4.13) and(5.4), we do not\ndetail them again). Both those terms are o(L(T)) uniformly in σ∈ Ssup\nη; therefore, Girsanov’s theorem\nyields for y1∈[0, h],\nEy1σ(0)L(T)\"\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,u1T∧t] ;B(u1T)∧t\nσ(u1∧(t/T))L(T)∈I1\texp \n−1\nTZu1T∧t\n0σ′(s/T)\nσ2(s/T)Bsds!#\n=eo(L(T))E(y1−h)σ(0)L(T)\"\n1\b\nBs\nσ(s/T)L(T)∈[−h,0],∀s∈[0,u1T∧t] ;B(u1T)∧t\nσ(u1∧(t/T))L(T)+h∈I1\t\n×exp \n−1\nTZu1T∧t\n0σ′(s/T)\nσ2(s/T)\u0002\nBs+hσ\u0000\ns/T\u0001\nL(T)\u0003\nds!#\n,30 A. LEGRAND AND P. MAILLARD\nand, by symmetry of the Brownian motion, we obtain\nE1=eo(L(T))inf\ny1∈I0E(h−y1)σ(0)L(T)\"\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,u1T∧t] ;B(u1T)∧t\nσ(u1∧(t/T))L(T)∈(h−I1)}\n×exp \n1\nTZu1T∧t\n0σ′(s/T)\nσ2(s/T)Bsds−hL(T) log\u0012σ(u1∧(t/T))\nσ(0)\u0013!#\n.\nRecall that we assumed i= 1∈J2, soσ′(u) =−|σ′(u)|for all u∈[0, u1]; moreover for any y1∈I0, one has\n(y1, t∧(u1T), σ(·), I1)∈Υ1. Therefore, we may apply Lemma 5.2 to deduce that, for Tlarger than some T0,\none has\nE1≥exp\u0012\n−εL(T)−hL(T) log\u0012σ(u1∧(t/T))\nσ(0)\u0013\u0013\n.\nMoreover the same lower bound holds for all i∈J2,i >1, where we use that ( yi, t∧(uiT)−ui−1T, σ(· −\nui−1), Ii)∈Υ2fort≥ui−1T,yi∈Ii−1. Plugging this into (5.8), we finally obtain\nExσ(0)L(T)\u0014\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈Iη−1\texp\u0012\n−1\nTZt\n0σ′(s/T)\nσ2(s/T)Bsds\u0013\u0015\n≥exp \n−η−1εL(T)−X\ni∈J2hL(T) log\u0012σ(ui∧(t/T))\nσ(ui−1∧(t/T))\u0013!\n.\nRecollecting (5.6–5.7) and taking ε→0, this concludes the proof of the lower bound. □\nProof of Lemma 5.2. Since the expectation is lower than 1, we only have to prove that the infimum is\nbounded from below by some o(L(T)). Let x∈[ε, h−ε],t:=t(T)∈[0, T],σ∈ SηandI⊂[0, h] with length\nat least 2 ε: for convenience we restrict ourselves to [ z−ε, z+ε]⊂Ifor some z∈[ε, h−ε]. Moreover, if\n(x, t, σ, I )∈Υ2, then one can always choose zsuch that\n(5.11) |x−z| ≤ε′:=ε(1−2−η−1).\nRecall from (3.1) the definition of the time-change C1-diffeomorphism J(·): in particular, ( Wr)r≥0denotes\nthe standard, time-homogeneous Brownian motion, and ( WJ(s))s∈[0,T]has the same law as ( Bs)s∈[0,T]. Hence,\nfort∈[0, T], one has\nExσ(0)L(T)\u0014\nexp\u0012\n−1\nTZt\n0|σ′(s/T)|\nσ2(s/T)Bsds\u0013\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s∈[0,t] ;Bt\nσ(t/T)L(T)∈[z−ε,z+ε]\t\u0015\n=Exσ(0)L(T)\u0014\nexp \n−1\nTZJ(t)\n0|σ′(J−1(s)/T)|\nσ4(J−1(s)/T)Wsds!\n1\b\nWs\nσ(J−1(s)/T)L(T)∈[0,h],∀s∈[0,J(t)] ;WJ(t)\nσ(t/T)L(T)∈[z−ε,z+ε]\t\u0015\n≥Exσ(0)L(T)\u0014\nexp \n−η−5\nTZJ(t)\n0Wsds!\n1\b\nWs\nσ(J−1(s)/T)L(T)∈[0,h],∀s∈[0,J(t)] ;WJ(t)\nσ(t/T)L(T)∈[z−ε,z+ε]\t\u0015\n,\nwhere we used that σ∈ Sη.\nRecall that T1/3≪L(T)≪t≤Tby assumption. Let ( τT)T>0, which satisfies, as T→+∞,\n(5.12) max\u0000\nL(T),T\nL(T)\u0001\n≪τT≪min(T, L(T)2).\nFort, Tsuch that J(t)>2τT, we constrain the trajectory ( Ws)s∈[0,J(t)]to be valued in [√τT,2√τT] at times\nτTandJ(t)−τT, and to remain below 3√τTin-between; then we apply Markov’s property at times τTand\nJ(t)−τT. Therefore, for any t, Tsuch that J(t)>2τT, one has\n(5.13)\nExσ(0)L(T)\u0014\nexp \n−η−5\nTZJ(t)\n0Wsds!\n1\b\nWs\nσ(J−1(s)/T)L(T)∈[0,h],∀s∈[0,J(t)] ;WJ(t)\nσ(t/T)L(T)∈[z,z+ε]\t\u0015\n≥B1×B2×B3,MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 31\nwhere\nB1:=Exσ(0)L(T)\u0014\nexp\u0012\n−η−5\nTZτT\n0Wsds\u0013\n1\b\nWs\nσ(J−1(s)/T)L(T)∈[0,h],∀s≤τT;WτT∈[√τT,2√τT]\t\u0015\n,\nB2:= inf\ny∈[√τT,2√τT]Ey\u0014\nexp \n−η−5\nTZJ(t)−2τT\n0Wsds!\n1\b\nWs∈[0,3√τT],∀s≤J(t)−2τT;WJ(t)−2τT∈[√τT,2√τT]\t\u0015\n,\nB3:= inf\ny∈[√τT,2√τT]Ey\u0014\nexp\u0012\n−η−5\nTZτT\n0Wsds\u0013\n1\b\nWs\nσ3(s/T)L(T)∈[0,h],∀s≤τT;WτT\nσ(t/T)L(T)∈[z−ε,z+ε]\t\u0015\n,\nand where we wrote σ3(s/T) :=σ(J−1(s+J(t)−τT)/T),s∈[0, τT] to lighten notation in B3. Let us bound\nfrom below the three factors B1B2andB3separately, and then detail how the case J(t)≤2τTis handled.\nCase J(t)>2τT.Let us start with B2. On the event {Ws∈[0,3√τT],∀s∈[0, J(t)−2τT]},(3.1) and(5.12)\nimply that,\n1\nTZJ(t)−2τT\n0Wsds≤3√τT(J(t)−2τT)\nT≤3η−2√τT≪L(T),\nasT→+∞, uniformly in tandσ∈ Sη. Hence one has,\n(5.14) B2≥eo(L(T))inf\ny∈[√τT,2√τT]Py\u0010\nWs∈[0,3√τT],∀s≤J(t)−2τT;WJ(t)−2τT∈[√τT,2√τT]\u0011\n,\nasT→+∞. Recalling Lemma 4.4, notice that there exists a constant K > 0 such that,\n∀t′≥K , inf\ny∈[1,2]Py\u0000\nWs∈[0,3],∀s≤t′;Wt′∈[1,2]\u0001\n≥sin2(π/3)\n3exp\u0012\n−π2\n18t′\u0013\n.\nIn particular, if t(T) satisfiesJ(t)−2τT\nτt≥K, then (5.14) and the Brownian scaling property yield\n(5.15) B2≥eo(L(T))×sin2(π/3)\n3exp\u0012\n−π2\n18J(t)−2τT\nτt\u0013\n≥eo(L(T)),\nwhere the last inequality follows the observation thatJ(t)−2τT\nτt=O(T/τT) =o(L(T)) by (3.1) and(5.12) ,\nwhich does not depend on t. On the other hand, if tsatisfiesJ(t)−2τT\nτt≤K, then the Brownian scaling\nproperty gives\ninf\ny∈[√τT,2√τT]Py\u0010\nWs∈[0,3√τT],∀s≤J(t)−2τT;WJ(t)−2τT∈[√τT,2√τT]\u0011\n= inf\ny∈[1,2]PyqτT\nJ(t)−2τT\u0010\nWs∈q\nτT\nJ(t)−2τT[0,3],∀s≤1 ;W1∈q\nτT\nJ(t)−2τT[1,2]\u0011\n≥ inf\ny∈[1,3/2]PyqτT\nJ(t)−2τT\u0010\nWs∈q\nτT\nJ(t)−2τT[0,3],∀s≤1 ;W1∈q\nτT\nJ(t)−2τT[y, y+ 1/2]\u0011\n≥P0\u0010\n|Ws| ≤q\nτT\nJ(t)−2τT,∀s≤1 ;W1∈q\nτT\nJ(t)−2τT[0,1/2]\u0011\n≥P0\u0010\n|Ws| ≤1√\nK,∀s≤1 ;W1∈h\n0,1\n2√\nKi\u0011\n>0, (5.16)\nwhere the first inequality is obtained by splitting the interval [1 ,2] into [1 ,3/2] and [3 /2,2] in the infimum,\nthen applying the Brownian symmetry property to the second term. Recollecting (5.14) and taking the\nlargest lower bound from (5.15–5.16), this finally proves that B2≥eo(L(T))uniformly in t.\nLet us turn to B1. Since τT≪T, one has on the event {Ws≤hσ(0)L(T),∀s≤τT}that,\n(5.17)1\nTZτT\n0Wsds≤hη−1L(T)τT\nT≪L(T),32 A. LEGRAND AND P. MAILLARD\nasT→+∞, uniformly in tandσ∈ Sη. Hence,\n(5.18) B1≥eo(L(T))Pxσ(0)L(T)\u0010\nWs∈[0, hσ(0)L(t)],∀s≤τT;WτT∈[√τT,2√τT]\u0011\n,\nforTlarge. Let us provide the following standard estimates, which are obtained with the Brownian reflection\nprinciple (see e.g. [ 16, Part 1, Ch. 4]) and some direct computations with the Gaussian distribution (details\nare left to the reader):\nPxσ(0)L(T)\u0000\nWτT∈[√τT,2√τT]\u0001\n≥1√\n2πexp\u0010\n−(xσ(0)L(T)−√τT)2\n2τT\u0011\n,\nPxσ(0)L(T)\u0000\nWτT∈[√τT,2√τT],inf\ns≤τTWs≤0\u0001\n≤1√\n2πexp\u0010\n−(xσ(0)L(T)+√τT)2\n2τT\u0011\n,\nPxσ(0)L(T)\u0000\nWτT∈[√τT,2√τT],sup\ns≤τTWs≥hσ(0)L(T)\u0001\n≤1√\n2πexp\u0010\n−((2h−x)σ(0)L(T)−2√τT)2\n2τT\u0011\n.\nTherefore, a union bound and (5.12) yield as T→+∞,\nPxσ(0)L(T)\u0010\nWs∈[0, hσ(0)L(t)],∀s≤τT;WτT∈[√τT,2√τT]\u0011\n≥1√\n2πexp\u0010\n−(xσ(0)L(T)−√τT)2\n2τT\u0011\n(1−o(1))≥eo(L(t)), (5.19)\nsoB1≥eo(L(T))asT→+∞, uniformly in tandσ∈ Sη.\nRegarding B3, it is handled very similarly to B1by reproducing the estimates (5.17–5.19): more precisely,\none can additionally constrainWτT\nσ(t/T)L(T)to be lower than z+ε/2≤h−ε/2, then use the reflection principle\n(we leave the details to the reader). Recollecting (5.13) , this finally gives a lower bound of order eo(L(T)),\nwhose expression does not depend on tandσ, and holds as soon as J(t)>2τT.\nCase J(t)≤2τT.Fort, Tsatisfying J(t)≤2τT, one obtains on the event {Ws∈[0, hL(T)σ(J−1(s)/T)],∀s∈\n[0, J(t)]}that,\n1\nTZJ(t)\n0Wsds≤2τT×hη−1L(T)\nT≪L(T),\nso one may bound the l.h.s. of (5.13) from below with\n(5.20) eo(L(T))Pxσ(0)L(T)\u0012Ws\nσ(J−1(s)/T)L(T)∈[0, h],∀s≤J(t) ;WJ(t)\nσ(t/T)L(T)∈[z−ε, z+ε]\u0013\n.\nNotice that (3.1), (5.12) and the fact that J(t)≤2τTimply,\n(5.21) t≤η−2J(t)≤2η−2τT≪T ,\nforTsufficiently large; in particular (3.4) yields that the probability in (5.20) is larger than\n(5.22) Pxσ(0)L(T)\u0010\nWs∈[0, h(1−ε′2)σ(0)L(T)],∀s≤J(t) ;WJ(t)∈σ(0)L(T)[z−ε′, z+ε′]\u0011\n,\nforTsufficiently large, where ε′< εis defined in (5.11) . Here again we provide standard estimates on\nthe Brownian motion, which are obtained with the reflection principle and direct computations with the\nGaussian distribution. However in the following we have to distinguish whether ( x, t, σ, I ) is in Υ 1or Υ 2. InMAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 33\ngeneral, one has\nPxσ(0)L(T)\u0010WJ(t)\nσ(0)L(T)∈[z−ε′, z+ε′]\u0011\n≥2ε′\np\n2πJ(t)exp\u0010\n−(|x−z|+ε′)2σ2(0)L(T)2\n2J(t)\u0011\n,\nPxσ(0)L(T)\u0010WJ(t)\nσ(0)L(T)∈[z−ε′, z+ε′],inf\ns≤J(t)Ws≤0\u0011\n≤2ε′\np\n2πJ(t)exp\u0010\n−(x+z)2σ2(0)L(T)2\n2J(t)\u0011\n,\nand Pxσ(0)L(T)\u0010WJ(t)\nσ(0)L(T)∈[z−ε′, z+ε′],sup\ns≤J(t)Ws\nσ(0)L(T)≥h(1−ε′2)\u0011\n≤2ε′\np\n2πJ(t)exp\u0010\n−(2h(1−ε′2)−x−z−ε′)2σ2(0)L(T)2\n2J(t)\u0011\n.\nLet us first assume ( x, t, σ, I )∈Υ1, that is t(T)≥θ(T)−1L(T): since (5.12) and(5.21) implyL(T)2\nJ(t)≥\nL(T)2\n2τT→+∞, we have with a union bound that,\nPxσ(0)L(T)\u0010\nWs∈[0, h(1−ε′2)σ(0)L(T)],∀s≤J(t) ;WJ(t)∈σ(0)L(T)[z−ε′, z+ε′]\u0011\n≥2εp\n2πJ(t)exp\u0010\n−(|x−z|+ε′)2σ2(0)L(T)2\n2J(t)\u0011\n(1−o(1))\n≥ε′\n4√πτTexp\u0010\n−h2η−4\n2θ(T)L(T)\u0011\n=eo(L(T)), (5.23)\nforTsufficiently large, where we used that t(T)≥θ(T)−1L(T) and (5.21) implyL(T)2\nJ(t)≤η−2θ(T)L(T). Let\nus now assume ( x, t, σ, I )∈Υ2. Then,\nPxσ(0)L(T)\u0010WJ(t)\nσ(0)L(T)∈[z−ε′, z+ε′]\u0011\n≥P0\u0010WJ(t)\nσ(0)L(T)∈[0, ε′]\u0011\n≥P0\u0010\nW1∈h\n0,ε′σ(0)L(T)√2τTi\u0011\n≥ε′σ(0)L(T)\n4√πτTe−(ε′σ(0)L(T))2\n4τT , (5.24)\nwhere the first inequality follows from (5.11) and the symmetry property of the Brownian motion, the second\nfrom the Brownian scaling and (5.21) , and the third from a standard Gaussian estimation. By (5.12) , this is\nlarger than eo(L(T)).\nTo conclude, consider the minimum between the lower bounds computed in (5.13–5.19), (5.20–5.23)\nand(5.24) . This yields a term of order eo(L(T))which is uniform in L(T)≤θt≤T; and it bounds from\nbelow the l.h.s. of (5.13) regardless of the sign of J(t)−2τTand whether ( x, t, σ, I ) is in Υ 1or Υ 2; therefore,\nthis concludes the proof of the lower bound. □\nWe now provide an upper bound on the second moment of |Asup\nT,I(t)|when I= [z, h] for some z∈[0, h)\nandt∈[0, T].\nProposition 5.3 (Second moment, Super-critical) .Leth >0. One has as T→+∞,\n(5.25) E\u0002\n|Asup\nT,z(t)|2\u0003\n≤e(x+h−2z)L(T)+o(L(T)),\nuniformly in t∈[0, T],σ∈ Sη,x∈[0, h]andz∈[0, h].\nRemark 5.2. The proof of Proposition 5.3 relies mostly on the Many-to-two lemma (Lemma 4.3) and\nthe upper bound (5.1) from Proposition 5.1. Noticeably, the second moment estimates in the sub-critical\nand critical cases will be obtained very similarly, by using respectively the first moment upper bounds from\nPropositions 5.5 and 5.9 below.\nProof. The Many-to-two lemma (Lemma 4.3) states that\nE\u0002\n|Asup\nT,z(t)|2\u0003\n−E\u0002\n|Asup\nT,z(t)|\u0003\n=β0E[ξ(ξ−1)]Zt\n0dsZh\n0Gsup(x, y,0, s)\u0012Zh\nzGsup(y, w, s, t ) dw\u00132\ndy .34 A. LEGRAND AND P. MAILLARD\nNotice thatRh\nzGsup(y, w, s, t )dw=E[|eAsup\nT,z(t−s)|], where eAsup\nT,zis defined similarly to Asup\nT,zby replacing\nthe diffusion coefficient σbyσ(· −s/T). Recall also the upper bound (5.1) from Proposition 5.1, which is\nuniform in x∈[0, h],I⊂[0, h],σ∈ Sηandt∈[0, T]. Therefore, we have for any s∈[0, t],\nZh\n0Gsup(x, y,0, s) dy\u0012Zh\nzGsup(y, w, s, t ) dw\u00132\n≤eo(L(T))Zh\n0Gsup(x, y,0, s)e2(y−z)L(T)dy ,\nwhere we used that the error term from (5.1) is uniform in y, z, s, t . Let K > 0 a large constant, and split\n[0, h] into Kintervals of lengthh\nK. For any 0 ≤i < K , one deduces from Proposition 5.1,\nZ(i+1)h\nK\nih\nKGsup(x, y,0, s)e2(y−z)L(T)dy≤e2((i+1)h\nK−z)L(T)×e(x−ih\nK)L(T)+o(L(T))\n≤e((2+i)h\nK+x−2z)L(T)+o(L(T)).\nTherefore, summing over 0 ≤i < K and integrating over s∈[0, t], one obtains\nE\u0002\n|Asup\nT,z(t)|2\u0003\n−E\u0002\n|Asup\nT,z(t)|\u0003\n≤eo(L(T))×e2h\nKL(T)×t×K×e(x+h−2z)L(T).\nTaking K→+∞, and recalling that t≤T≤L(T)3andE[|Asup\nT,z(t)|]≤e(x−z)L(T)+o(L(T)), this yields the\nexpected upper bound uniformly in t,σ(·),xandz. □\nWe conclude this section with an estimate on the number of particles killed at the upper barrier. Recall\nthe definition of Rsup\nT(s, t), 0≤s≤t≤Tfrom (4.6).\nProposition 5.4 (Killed particles, Super-critical) .Leth >0. Then as T→+∞, one has\n(5.26) E[Rsup\nT(0, T)]≤e−(h−x)L(T)+o(L(T)),\nuniformly in x∈[0, h]andσ∈ Sη.\nProof. Recall that we may write γsup\nTas in (4.21): in particular, one has for s∈[0, T],\n(γsup\nT)′(s) =σ(s/T) +h(σ′)+(s/T)L(T)\nT.\nRecollect (4.14) from Lemma 4.2. On the one hand, one has for Tsufficiently large, uniformly in s∈[0, T]\nandσ∈ Sη, that\n∂\n∂u\u0012γsup\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=s≥ −1\nTσ′(s/T)\nσ2(s/T)−hη−7L(T)\nT2,\nso, on the event {Bs≤hσ(s/T)L(T),∀s≤H0(B)}, one obtains\n−ZH0(B)\n0∂\n∂u\u0012γsup\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds≤1\nTZH0(B)\n0σ′(s/T)\nσ2(s/T)Bsds+h2η−8L(T)2\nT.\nOn the other hand, one has\nZH0(B)\n0(γsup\nT′(s))2\n2σ2(s/T)ds≥H0(B)\n2+hL(T)\nTZH0(T)\n0(σ′)+(s/T)\nσ(s/T)ds .\nTherefore, (4.14) eventually yields\nE[Rsup\nT(0, T)]≤e−(h−x)L(T)+o(L(T))E(h−x)σ(0)L(T)\u0014\n1{H0(B)≤T ,Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}\n×exp\u00121\nTZH0(B)\n0σ′(s/T)\nσ2(s/T)Bsds−hL(T)\nTZH0(T)\n0(σ′)+(s/T)\nσ(s/T)ds\u0013\u0015\n.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 35\nRecall that σ′(u) = ( σ′)+(u)−(σ′)−(u) for all u∈[0,1]. Thus, on the event {Bs≤hσ(s/T)L(T),∀s≤\nH0(B)}, one obtains\nE[Rsup\nT(0, T)]≤e−(h−x)L(T)+o(L(T))E(h−x)σ(0)L(T)\u0014\n1{H0(B)≤T ,Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}\n×exp\u0012\n−1\nTZH0(B)\n0(σ′)−(s/T)\nσ2(s/T)Bsds\u0013\u0015\n,\nand the latter expectation is bounded by 1, which concludes the proof. □\n5.2.Sub-critical case. In this section we assume ∗=sub, that is 1 ≪L(T)≪T1/3, and recall from (4.9)\nand (4.3) that we defined the lower and upper barriers, for t∈[0, T], by\nγsub\nT(t) :=s\n1−π2\nh2L(T)2×v(t/T)T−x σ(0)L(T),\nandγsub\nT(t) := γsub\nT(t) +hσ(t/T)L(T) for some h > x > 0. Recall (4.4–4.5): in particular, the set of\ndescendants remaining between the barriers and reaching an interval I⊂[0, h] at time tis denoted\n(5.27) Asub\nT,I(t) :=n\nu∈ N t\f\f\fXu(s)∈\u0002\nγsub\nT(s),γsub\nT(s)\u0003\n,∀s∈[0, t] ;Xu(t)−γsub\nT(t)\nL(T)σ(t/T)∈Io\n.\nRecall also (3.3) and(4.1), where we fixed some η >0 and θ:R+→R∗\n+such that θ(T)→0 asT→+∞\nand (4.2) holds.\nProposition 5.5 (First moment, Sub-critical) .Leth >0. As T→+∞, one has\n(5.28) E\u0002\n|Asub\nT,I(t)|\u0003\n≤e(x−infI)L(T)+o(L(T)),\nuniformly in x∈[0, h],I⊂[0, h]a non-trivial sub-interval, σ∈ Sη, and t=t(T)such that 0≤t(T)≤θp\nTL(T)3forTsufficiently large. Moreover, as T→+∞one also has\n(5.29) E\u0002\n|Asub\nT,I(t)|\u0003\n≥e(x−infI)L(T)+o(L(T)),\nlocally uniformly in x∈(0, h),infI∈[0, h)andsupI−infI >0; and uniformly in σ∈ Sηandt=t(T)\nsuch that L(T)≤θt(T)≤θp\nTL(T)3forTsufficiently large.\nThis proposition should be compared with Proposition 5.1 for the super-critical regime. The definitions of\n“uniformly” and “locally uniformly” are the same as in Remark 5.1. Conversely to the super-critical and\ncritical regimes, we are not able to derive sharp moment estimates throughout the whole time interval [0 , T],\nbut only up to times smaller thanp\nTL(T)3≪T. This inconvenience actually has some consequences on\nour strategy for the proof: to obtain (1.6) in Theorem 1.1, we have to decompose the full interval [0 , T] into\nblocks of length o(p\nTL(T)3), on which the comparison between N-BBM and BBM with barriers holds.\nProof. Recall Lemma 4.2, in particular (4.13). Notice that (4.9) implies that for all s∈[0, T],T≥0,\nγsub\nT′(s)\nσ(s/T)=s\n1−π2\nh2L(T)2,\nwhich does not depend on s. On the one hand, this yields for all s∈[0, T],\n(5.30)\u0000\nγsub\nT′(s)\u00012\n2σ2(s/T)=1\n2−π2\n2h2L(T)2,\nOn the other hand, on the event {Bs\nσ(s/T)L(T)∈[0, h],∀s≤t}, one has\n(5.31)Zt\n0∂\n∂u\u0012γsub\nT′(u)\nσ2(u/T)\u0013\f\f\f\f\nu=sBsds≤η−3\nT×hη−1L(T)×θ(T)p\nTL(T)3=o(L(T)),36 A. LEGRAND AND P. MAILLARD\nuniformly in t≤θp\nTL(T)3andσ∈ Sη. Therefore, (4.13) becomes\nE\u0002\n|Asub\nT,I(t)|\u0003\n= exp\u0012π2\n2h2t\nL(T)2+o(L(T))\u0013\nExσ(0)L(T)\"\nexp\u0012\n−Bt\nσ(t/T)+xL(T)\u0013\n(5.32)\n×1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤t;Bt\nσ(t/T)L(T)∈I}#\n.\nWe now focus on the latter expectation. It can be bounded from above with\n(5.33) e(x−infI)L(T)Pxσ(0)L(T) \nBs\nσ(s/T)L(T)∈[0, h],∀s≤t!\n;\nand, letting any εT→0 asT→+∞and adding the constraintBt\nσ(t/T)L(T)≤infI+εT, it can be bounded\nfrom below with\n(5.34) e(x−εT−infI)L(T)Pxσ(0)L(T) \nBs\nσ(s/T)L(T)∈[0, h]∀s≤t;Bt\nσ(t/T)L(T)∈(infI,infI+εT)!\n.\nTherefore, writing z:=infIto lighten notation, both statements of the proposition are obtained by showing\nthat (5.33–5.34) are of order exp((x−z)L(T)−π2\n2h2t\nL(T)2+o(L(T))). Once again, we achieve this via\na comparison with the standard, time-homogeneous Brownian motion ( Ws)s≥0. In the following, recall\nLemma 4.4.\nUpper bound. Using (3.4), the Brownian scaling property and a time change, we have\nPxσ(0)L(T)\u0010\nBs∈[0, h σ(s/T)L(T)],∀s≤t\u0011\n≤Pxσ(0)L(T)\u0010\nBs∈[0, h(σ(0) + η−2t/T)L(T)],∀s≤t\u0011\n=Pxσ(0)\nσ(0)+η−2t/T\u0010\nBs∈[0, h],∀s≤t\nL(T)2(σ(0)+η−2t/T)2\u0011\n=Pxσ(0)\nσ(0)+η−2t/T\u0010\nWs∈[0, h],∀s≤t\nL(T)2(σ(0)+η−2t/T)2t\n∫\n0σ2(u/T)du\u0011\n≤Pxσ(0)\nσ(0)+η−2t/T\u0010\nWs∈[0, h],∀s≤t\nL(T)2(σ(0)−η−2t/T)2\n(σ(0)+η−2t/T)2\u0011\n.\nThen, (3.4) implies that, for Tsufficiently large,(σ(0)−η−2t/T)2\n(σ(0)+η−2t/T)2≥1\n2uniformly in σ∈ Sηandt≤θp\nTL(T)3.\nMoreover Lemma 4.4 implies that there exists a constant K > 0 such that,\n∀t′≥K, sup\ny∈[x/2,x]Py(Ws∈[0, h],∀s≤t′)≤8\nπexp\u0012\n−π2\n2h2t′\u0013\n,\nwhere we also used thatRh\n0sin(πz/h )dz= 2h/π. For any tsuch that t≥2KL(T)2, this yields\nPx\u0010\nWs∈[0, h σ(t/T)/σ(0)],∀s≤t/L(T)2\u0011\n≤8\nπexp\u0012\n−π2\n2h2t\nL(T)2(σ(0)−η−2t/T)2\n(σ(0)+η−2t/T)2\u0013\n≤8\nπexp\u0012\n−π2\n2h2t\nL(T)2+2π2η−3\nh2t2\nL(T)2T\u0013\n, (5.35)\nand, by assumption, we havet2\nL(T)2T≤θ(T)2L(T) =o(L(T)) for Tlarge. On the other hand if t≤2KL(T)2,\nthen one may write |π2\n2h2t\nL(T)2| ≤Kπ2\nh2in(5.32) , and the probability in (5.33) is bounded by 1. Finally,\ntaking the maximum of this and (5.35) , we obtain the expected upper bound uniformly in σ∈ Sηandt(T)\nsuch that 0 ≤t≤θp\nTL(T)3.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 37\nLower bound. Similarly to the upper bound, we have to distinguish the cases tsmaller or larger than KL(T)2\nfor some constant Kwhich is determined below. We begin with the case tlarge.\nLetεTsuch that 1 ≫εT≫max(θ(T)p\nL(T)3T−1, L(T)−1) as T→+∞, and recall (5.34) . For T\nsufficiently large, this and (3.4) imply z+εT< hand [z+εT/3, z+ 2εT/3]σ(0)⊂[z, z+εT]σ(t/T) uniformly\ninσ∈ Sηand locally uniformly in z. Thus we have the lower bound\n(5.36)Pxσ(0)L(T)\u0012\nBs∈[0, h σ(s/T)L(T)],∀s≤t;Bt∈[z, z+εT]σ(t/T)L(T)\u0013\n≥Pxσ(0)L(T)\u0012\nBs∈[0, h σ(0)L(T)],∀s≤t;Bt∈[z+εT/3, z+ 2εT/3]σ(0)L(T)\u0013\n.\nDefine\n(5.37) τ=τ(t) :=1\nσ2(0)L(T)2Zt\n0σ2(u/T) du ,\nin particular (3.4) and t≤θp\nTL(T)3imply that\n(5.38)\f\f\fτ−t\nL(T)2\f\f\f=T\nL(T)2\f\f\f\f\fZt/T\n0\u0012σ(u)2\nσ(0)2−1\u0013\ndu\f\f\f\f\f=O\u0012θ(T)2L(T)\nT\u0013\n=o(L(T)).\nuniformly in σ∈ Sηandt. Using the Brownian scaling property and a time change, we have\nPxσ(0)L(T)\u0012\nBs∈[0, h σ(0)L(T)],∀s≤t;Bt∈[z+εT/2, z+εT]σ(0)L(T)\u0013\n=Px\u0010\nWs∈[0, h],∀s≤τ;Wτ∈[z+εT/3, z+ 2εT/3]\u0011\n. (5.39)\nBy Lemma 4.4, there exists a large constant K > 0 such that, for Tsufficiently large, if t≥KL(T)2then\nτ≥K/2, and,\nPx\u0010\nWs∈[0, h],∀s≤τ;Wτ∈[z+εT/3, z+ 2εT/3]\u0011\n≥1\nhexp\u0010\n−π2\n2h2τ\u0011\nsin\u0010πx\nh\u0011Zz+2εT/3\nz+εT/3sin\u0010πy\nh\u0011\ndy . (5.40)\nRecalling εT≫L(T)−1and that sin(πw)≥1\n2(w∧1−w),∀w∈[0,1], we have as soon as Tis sufficiently\nlarge,\n(5.41)Zz+2εT/3\nz+εT/3sin\u0010πy\nh\u0011\ndy≥εT\n6\u0012z+εT/3\nh∧\u0010\n1−z+ 2εT/3\nh\u0011\u0013\n≥ε2\nT\n18h= exp( o(L(T))), .\nPlugging this and (5.38) into(5.40) , this finally yields the lower bound uniformly in t≥KL(T)2andσ∈ Sη.\nFort≤KL(T)2, we have to reproduce the computation (5.36–5.39) with a different choice of εTsatisfying\nL(T)−1≪εT≪L(T)−1\n2, then we only need to prove that (5.39) is larger than exp(o(L(T))) (since one has\nπ2\n2h2t\nL(T)2≥0 in (5.32)). Using standard computations on the Gaussian density, one has\n(5.42) Px\u0000\nWτ∈[z+εT/3, z+ 2εT/3]\u0001\n≥εT\n31√\n2πτexp\u0012\n−(|z−x|+εT)2\n2τ\u0013\n≥exp(o(L(T))),\nwhere the second inequality follows from the observation that (3.4),(5.37) andt≥θL(T) imply τ−1≤\n2θ(T)L(T) =o(L(T)) for Tsufficiently large. Moreover, we claim that\n(5.43) lim inf\nT→+∞Px\u0010\nWs∈[0, h],∀s≤τ\f\f\fWτ∈[z+εT/3, z+ 2εT/3]\u0011\n>0,\nlocally uniformly in x, z. Thus, one only needs to take the minimum between (5.40) and (5.42–5.43), then\nplug it into (5.34), to obtain the announced lower bound uniformly in t(T) and σ(·).38 A. LEGRAND AND P. MAILLARD\nTo prove (5.43), we write that t≤KL(T)2implies τ≤2KforTlarge, and\n(5.44)\nPx\u0010\nWs∈[0, h],∀s≤τ\f\f\fWτ∈[z+εT/3, z+ 2εT/3]\u0011\n≥Px\u0010\nτ−1\n2\u0000\nWuτ−uz−(1−u)x\u0001\n∈[−x∧z√\n2K,h−x∨z√\n2K],∀u≤1\f\f\fτ−1\n2(Wτ−z)∈τ−1\n2[εT/3,2εT/3]\u0011\n.\nMoreover, (5.37) implies τ≥(2θ(T)L(T))−1forTlarge, so τ−1\n2εT→0 as T→+∞. Therefore, the\nGaussian process ( τ−1\n2(Wuτ−uz−(1−u)x))u∈[0,1]conditioned to W0−x= 0, Wτ−z∈[εT/3,2εT/3]\nconverges in law to a standard Brownian bridge ( X0,0\nu)u∈[0,1]asT→+∞(see [ 14, Sect. 9]); more precisely,\nthe r.h.s of (5.44) converges to P(X0,0\nu∈[−x∧z√\n2K,h−x∨z√\n2K],∀u≤1)>0 asT→+∞, and this convergence is\nuniform in L(T)≤θt≤KL(T)2. Since the latter probability is positive, this concludes the proof. □\nWe now provide an upper bound on the second moment of |Asub\nT,I(t)|when I= [z, h] for some z∈[0, h).\nLet us point out that, conversely to the super-critical case (recall Proposition 5.3), the statement below\ninvolves an error factor O(t): in general one cannot guarantee that t≤eo(L(T))in the sub-critical regime,\nespecially when tis close top\nTL(T)3andL(T) grows very slowly in T. However this will not be an issue in\nthis paper, since in Sections 6–7 we shall consider interval lengths t(T) that are at most polynomial in L(T).\nProposition 5.6 (Second moment, Sub-critical) .Leth >0. As T→+∞, one has\n(5.45) E\u0002\n|Asub\nT,z(t)|2\u0003\n≤e(x+h−2z)L(T)+o(L(T))×O(t),\nuniformly in x∈[0, h],z∈[0, h],σ∈ Sηandt=t(T)such that 0≤t(T)≤θp\nTL(T)3forTsufficiently\nlarge.\nProof. This proposition is very similar to Proposition 5.3. Reproducing all arguments from its proof but\nusing (5.28) instead of (5.1), one obtains for K∈NandTsufficiently large,\nE\u0002\n|Asub\nT,z(t)|2\u0003\n−E\u0002\n|Asub\nT,z(t)|\u0003\n≤e2h\nKL(T)+o(L(T))×t×K×e(x+h−2z)L(T),\nuniformly in x, z∈[0, h],σ∈ Sηand 0≤t≤θp\nTL(T)3. Taking Klarge, this yields the expected result. □\nWe conclude this section with an estimate on the number of particles killed at the upper barrier. Recall\nthe definition of Rsub\nT(s, t), 0≤s≤t≤Tfrom (4.6).\nProposition 5.7 (Killed particles, Sub-critical) .Leth >0. Then as T→+∞, one has\n(5.46) E\u0002\nRsub\nT(0, t)\u0003\n≤e−(h−x)L(T)+o(L(T))×O\u0000\ntL(T)−2∨1\u0001\n,\nuniformly in x∈[0, h],σ∈ Sη, and t=t(T)such that 0≤t≤θp\nTL(T)3forTsufficiently large.\nNotice that, similarly to Proposition 5.6, we have an additional error term O(tL(T)−2∨1), which vanishes\nas soon as t(T) is at most polynomial in L(T).\nProof. This proof relies on a comparison with the time-homogeneous case, which has already been studied\nin [42,43]. Recollect (4.14) from Lemma 4.2. Notice that ( γsub\nT)′(s) = (γsub\nT)′(s) +O(L(T)/T) uniformly in\ns∈[0, T],σ∈ Sη. Combining this with (5.30–5.31), we deduce from (4.14) that,\nE[Rsub\nT(0, t)] =e−(h−x)L(T)+o(L(T))E(h−x)σ(0)L(T)\"\nexp\u0012π2\n2h2H0(B)\nL(T)2\u0013\n1{H0(B)≤t}1{Bs\nσ(s/T)L(T)≤h,∀s≤H0(B)}#\n.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 39\nTherefore, it only remains to bound from above the latter expectation. Using the Brownian scaling property,\nwe have\nE(h−x)σ(0)L(T)\"\nexp\u0012π2\n2h2H0(B)\nL(T)2\u0013\n1{H0(B)≤t}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}#\n=E(h−x)\"\nexp\u0012π2\n2h2σ2(0)H0(B)\u0013\n1{H0(B)≤t/(L(T)2σ2(0))}1{Bs∈[0,h σ(s/T)/σ(0)],∀s≤H0(B)}#\n.\nLet (Ws)s≥0denote the standard, time-homogeneous Brownian motion, recall the definition of J(s) from (3.1),\nand recall (3.4). In particular, we have for s∈[0, T],\ns σ2(0)\u0000\n1−η−2s\nT\u00012≤J(s)≤s σ2(0)\u0000\n1 +η−2s\nT\u00012.\nThus, applying the time change (3.1) gives,\nE(h−x)\"\nexp\u0012π2\n2h2σ2(0)H0(B)\u0013\n1{H0(B)≤t/(L(T)2σ2(0))}1{Bs∈[0,h σ(s/T)/σ(0)],∀s≤H0(B)}#\n=E(h−x)\"\nexp\u0012π2\n2h2σ2(0)J−1(H0(W))\u0013\n1{H0(W)≤J(t/[L(T)2σ2(0)])}1{Ws∈[0,h σ(s/T)/σ(0)]∀s≤H0(W)}#\n≤E(h−x)\"\nexp\u0012π2\n2h2\u0000\n1−η−2t\nT\u0001−2H0(W)\u0013\n1\b\nH0(W)≤t\nL(T)2(1+η−2t/T)2\t1{Ws∈[0,h(1+η−2t/T)]∀s≤H0(W)}#\n.\nMoreover, on the event H0(W)≤2t\nL(T)2, one has\u0000\n1−η−2t\nT\u0001−2H0(W) =\u0000\n1 +η−2t\nT\u0001−2H0(W) +o(L(T))\n(recall that t≤θp\nTL(T)3). Then, the expectation above has already been estimated in [ 42, Lemma 2.2.1]\n(see also [43, Lemma 7.1]): therefore, we have for some universal C >0,\nE(h−x)\"\nexp\u0012π2\n2h2\u0000\n1 +η−2t\nT\u0001−2H0(W)\u0013\n1\b\nH0(W)≤t\nL(T)2(1+η−2t/T)2\t1{Ws∈[0,h(1+η−2t/T)]∀s≤H0(W)}#\n≤πt\nh2L(T)2sin\u0010\nπh−x\nh\u0000\n1 +η−2t\nT\u0001−1\u0011\n+Ch−x\nh\u0000\n1 +η−2t\nT\u0001−1=O\u0000\ntL(T)−2∨1\u0001\n,\nuniformly in σ∈ Sηand 0 ≤t(T)≤θp\nTL(T)3, which concludes the proof. □\n5.3.Critical case. In this section we assume there exists α >0 such that L(T)∼αL(T)1/3asT→+∞.\nRecall the definition and properties of Ψ from (1.4). We recollect the following result from [47].\nLemma 5.8 ([47, Lemma 2.4]) .LetWa standard time-homogeneous Brownian motion, q∈R,0< a < b < 1\nand0< a′< b′<1. Then,\n(5.47)lim\nt→∞1\ntsup\nx∈[0,1]logEx\u0014\ne−qRt\n0Wsds1{Ws∈[0,1]∀s≤t}\u0015\n= lim\nt→∞1\ntinf\nx∈[a,b]logEx\u0014\ne−qRt\n0Wsds1{Ws∈[0,1]∀s≤t, Wt∈[a′,b′]}\u0015\n= Ψ( q).\nIn that regime, recall from (4.10) and(4.3) that the lower and upper barriers are defined, for t∈[0, T], by\nγcrit\nT(t) := v(t/T)T−wh,T(t/T)L(T)−x σ(0)L(T),\nandγcrit\nT(t) := γcrit\nT(t) +hσ(t/T)L(T) for some h > x > 0, and where wh,T∈ C1([0,1]) is defined in (4.7).40 A. LEGRAND AND P. MAILLARD\nRemark 5.3. Let us point out that wh,T(u)∼π2\n2α3h2v(u)asα→0, so that choice of upper barrier\nmatches the sub-critical asymptotics of the N-BBM when αis small (recall (1.11) , Lemma 4.1 and that\nL(T) =o(T/L(T)2)in that case). Moreover, the relation Ψ(−q) =q+ Ψ(q),q∈Ryields\nα\u0012\nwh,T(u)−hZu\n0(σ′)−(s) ds\u0013\n−→\nα→+∞a1\n21/3Zu\n0σ(v)1/3|σ′(v)|2/3dv ,\nfor all u∈(0,1]. Thus, multiplying these terms by T1/3∼α−1L(T), we see that the upper barrier matches\nthe super-critical asymptotics of the N-BBM when αis large; and when σis decreasing, one also recovers\nthe asymptotics of Proposition 1.2.\nRecalling (4.5), the set of descendants remaining between the barriers and ending in a (rescaled) interval\nI⊂[0, h] at time tis denoted\n(5.48) Acrit\nT,I(t) :=n\nu∈ N t\f\f\fXu(s)∈\u0002\nγcrit\nT(s),γcrit\nT(s)\u0003\n,∀s∈[0, t] ;Xu(t)−γcrit\nT(t)\nL(T)σ(t/T)∈Io\n.\nProposition 5.9 (First moment, Critical) .Leth >0. One has, as T→+∞,\n(5.49) E\u0002\n|Acrit\nT,I(t)|\u0003\n≤e(x−infI)L(T)+o(T1/3),\nuniformly in t∈[0, T],σ∈ Sη,x∈[0, h], and I⊂[0, h]a non-trivial sub-interval. Moreover, one also has,\nasT→+∞,\n(5.50) E\u0002\n|Acrit\nT,I(t)|\u0003\n≥e(x−infI)L(T)+o(T1/3),\nlocally uniformly in x∈(0, h),infI∈[0, h)andsupI−infI >0; and uniformly in σ∈ Sηandt=t(T)\nsuch that L(T)≤θt(T)≤TforTsufficiently large.\nHere again, this proposition should be compared with Propositions 5.1 and 5.5 for the super- and sub-critical\nregimes; and the definitions of “locally uniformly” and “uniformly” are the same as in Remark 5.1.\nProof. Recall (4.13) from Lemma 4.2. Notice that (4.10) implies\n(5.51) γcrit\nT′(s) =σ(s/T) +w′\nh,T(s/T)L(T)\nT=σ(s/T) +O(T−2/3),\nuniformly in s∈[0, T] and σ∈ Sη; more precisely, one can check that the function Ψ( ·) is bounded on\n[−α3h3η−2, α3h3η−2], as well as wh,T(·) and w′\nh,T(·) on [0 ,1], uniformly in σ∈ Sη. In the following, we let\nΨ := sup {|Ψ(q)|, q∈[−α3h3η−2, α3h3η−2]}, Ψ′:= sup {|Ψ′(q)|, q∈[−α3h3η−2, α3h3η−2]},\nwhich are well defined since Ψ is convex on R. Therefore, on the event {Bs\nσ(s/T)L(T)∈[0, h],∀s≤t}, one\ndeduces from (4.13) and a straightforward computation that,\n(5.52)E\u0002\n|Acrit\nT,I(t)|\u0003\n= exp\u0012\nxL(T) +L(T)\nTZt\n0w′\nh,T(s/T)\nσ(s/T)ds+O(T−1/3)\u0013\n×Exσ(0)L(T)\u0014\nexp\u0012\n−Bt\nσ(t/T)−1\nTZt\n0σ′(s/T)\nσ2(s/T)Bsds\u0013\n1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤t;Bt\nσ(t/T)L(T)∈I}\u0015\n.\nUpper bound. Letz:=infI. Let t∈[0, T], and assume first that ( t, T) satisfies t≤T5/6. Then, (5.52) gives\nE\u0002\n|Acrit\nT,I(t)|\u0003\n≤e(x−z)L(T)+O(T1/6)Pxσ(0)L(T)\u0010\nBs\nσ(s/T)L(T)∈[0, h],∀s≤t;Bt\nσ(t/T)L(T)∈I\u0011\n≤e(x−z)L(T)+O(T1/6), (5.53)\nuniformly in σ∈ Sηandt∈[0, T5/6].MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 41\nFor (t, T) such that t≥T5/6, let us split [0 , t] into intervals of length KL(T)2, where Kis a large constant\nwhich is determined below. Writing imax:=⌊t/(KL(T)2)⌋, we let ti:=iKL(T)2for 0≤i < i max, and\ntimax:=t(notice that ( timax−timax−1)∈[KL(T)2,2KL(T)2]). Define for 1 ≤i≤imax,\n(5.54)σi:= sup\n[ti−1,ti]σ , σi:= inf\n[ti−1,ti]σ ,\nand σ′\ni:= sup\n[ti−1,ti]σ′, σ′\ni:= inf\n[ti−1,ti]σ′,\nandσ0=σ0:=σ(0). Using the Markov property at times ti, 1≤i < i max, one has\nExσ(0)L(T)\u0014\nexp\u0012\n−Bt\nσ(t/T)−1\nTZt\n0Bsσ′(s/T)\nσ2(s/T)ds\u0013\n1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤t;Bt\nσ(t/T)L(T)∈I}\u0015\n≤e−zL(T)imaxY\ni=1sup\ny∈[0,hσiL(T)]E(ti−1,y)\"\nexp \n−1\nTZti\nti−1Bsσ′\ni\nσ2\nids!\n1{Bs∈[0,hσiL(T)]∀s∈[ti−1,ti]}#\n. (5.55)\nTo lighten notation, let us focus on the factor i= 1, but the following proof holds for other blocks as well\n(including [ timax−1, timax]). Let ε >0. Recalling the time-change J(s) :=Rs\n0σ2(r/T)dr,s≤Tfrom (3.1),\nand using the Brownian scaling property, we have for y∈[0, hσ1L(T)],\nEy\u0014\nexp\u0012\n−1\nTσ′\n1\nσ2\n1Zt1\n0Bsds\u0013\n1{Bs∈[0,hσ1L(T)]∀s≤t1}\u0015\n(5.56)\n=Ey\"\nexp \n−1\nTσ′\n1\nσ2\n1ZJ(t1)\n0Ws(J′(J−1(s)))−1ds!\n1{Ws∈[0,hσ1L(T)]∀s≤J(t1)}#\n≤Ey\"\nexp \n−1\nTσ′\n1\nσ4\n1Zσ2\n1t1\n0Wsds!\n1{Ws∈[0,hσ1L(T)]∀s≤σ2\n1t1}#\n≤Ey/(hσ1L(T))\"\nexp \n−1\nTσ′\n1\nσ4\n1(hσ1L(T))3Zσ2\n1t1/(hσ1L(T))2\n0Wsds!\n1{Ws∈[0,1]∀s≤σ2\n1t1/(hσ1L(T))2}#\n≤Ey/(hσ1L(T))\"\nexp \n−σ′\n1\nσ1α3h3(1−ε)Zσ2\n1t1/(hσ1L(T))2\n0Wsds!\n1{Ws∈[0,1]∀s≤σ2\n1t1/(hσ1L(T))2}#\n,\nwhere the last inequality holds for Tsufficiently large, by recalling (3.4) and that L(T)∼αT1/3. Let ε >0,\nand recall that ti−ti−1≥KL(T)2for all i. Assume w.l.o.g that L(T)>0 for all T: then t1→+∞as\nK→+∞uniformly in T: applying Lemma 5.8, there exists Kε>0 such that for K > K ε, one has\nsup\ny∈[0,1]Ey\"\nexp \n−σ′\n1\nσ1α3h3(1−ε)Zσ2\n1t1/(hσ1L(T))2\n0Wsds!\n1{Ws∈[0,1]∀s≤σ2\n1t1/(hσ1L(T))2}#\n≤exp\u0012t1\nh2L(T)2σ2\n1\nσ2\n1\u0014\nΨ\u0012σ′\n1\nσ1α3h3(1−ε)\u0013\n+ε\u0015\u0013\n,\nfor all T≥0. Then, for Tsufficiently large, (3.4) and the definitions of Ψ,Ψ′yield,\nσ2\n1\nσ2\n1\u0014\nΨ\u0012σ′\n1\nσ1α3h3(1−ε)\u0013\n+ε\u0015\n≤Ψ\u0012σ′\n1\nσ1α3h3\u0013\n+c1ΨKL(T)2\nT+εc1(Ψ′+ 1),42 A. LEGRAND AND P. MAILLARD\nfor some constant c1>0. Therefore, there exists T0(K, ε)>0 such that, for Ksufficiently large and\nT≥T0(K, ε), (5.55) becomes\nExσ(0)L(T)\u0014\nexp\u0012\n−Bt\nσ(t/T)−1\nTZt\n0Bsσ′(s/T)\nσ2(s/T)ds\u0013\n1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤t;Bt\nσ(t/T)L(T)∈I}\u0015\n≤e−zL(T)exp imaxX\ni=1(ti−ti−1)\nh2L(T)2\u0014\nΨ\u0012σ′\ni\nσiα3h3\u0013\n+c1ΨKL(T)2\nT+εc1(Ψ′+ 1)\u0015!\n.\nUsing a Riemann sum approximation, there exists T0(K, ε)>0 such that, for Kvery large and T≥T0(K, ε),\nand for t≥T5/6, one has\n(5.57)imaxX\ni=1(ti−ti−1)\nL(T)2Ψ\u0012σ′\ni\nσiα3h3\u0013\n≤1\nL(T)2Zt\n0Ψ\u0012σ′(r/T)\nσ(r/T)α3h3\u0013\ndr+εt\nL(T)2.\nMoreover, one has\n(5.58)imaxX\ni=1(ti−ti−1)\nh2L(T)2\u0014\nΨKL(T)2\nT+ε(Ψ′+ 1)\u0015\n=O(1) + ε O(T1/3).\nRecollect (5.52) and that L(T)∼αT1/3. Recalling the definition of wh,Tfrom (4.7) and that1\nL(T)2∼α−3L(T)\nT,\nwe finally obtain that there exist some C, C′>0 (depending only on η, εandα) such that for Tlarge enough,\none has\n(5.59) E\u0002\n|Acrit\nT,z(t)|\u0003\n≤exp\u0010\n(x−z)L(T) +C+C′εT1/3\u0011\n,\nfor all t≥T5/6. Taking the maximum of (5.53) and(5.59) , and letting ε→0, this finally yields the expected\nupper bound uniformly in σ(·) and t∈[0, T].\nLower bound. Similarly to the upper bound, we distinguish the cases tsmaller or larger than T5/6. We\nfirst consider the case t≥T5/6, using the same time split with intervals of length KL(T)2and notation as\nabove (recall (5.54) ). Let 0 < a < b < h such that x∈(a, b), and ε >0 such that z+ε < h . We bound the\nexpectation in (5.52) from below by constraining the trajectory to pass through the intervals [ a, b]·σiL(T)\nat times ti, 1≤i≤imax−1. Hence, the Markov property gives\nExσ(0)T1/3\"\nexp\u0012\n−Bt\nσ(t/T)−1\nTZt\n0Bsσ′(s/T)\nσ2(s/T)ds\u0013\n1\u001a\nBs∈[0,hσ(s/T)L(T)]∀s≤t,\nBt≥zσ(t/T)L(T)\u001b#\n≥e−(z+ε)L(T)imax−1Y\ni=1inf\ny∈[a,b]·σi−1L(T)E(ti−1,y)\"\nexp \n−1\nTσ′\ni\nσ2\niZti\nti−1Bsds!\n1\u001aBs∈[0,hσiL(T)]∀s∈[ti−1,ti],\nBti∈[aσiL(T),bσiL(T)]\u001b#(5.60)\n× inf\ny∈[a,b]·σimax−1L(T)E(timax−1,y)\"\nexp \n−1\nTσ′\nimax\nσ2\nimaxZtimax\ntimax−1Bsds!\n1\u001aBs∈[0,hσimaxL(T)]∀s∈[timax−1,t],\nBt∈[zσ(t/T)L(T),(z+ε)σ(t/T)L(T)]\u001b#\n.\nLet us focus on the factor i= 1 here again (others are handled similarly, including the last one). Reproducing\nthe time-change argument from (5.56), one has for y∈[a, b]·σ(0)L(T),\nEy\"\nexp\u0012\n−1\nTσ′\n1\nσ2\n1Zt1\n0Bsds\u0013\n1\u001aBs∈[0,hσ1L(T)]∀s≤t1,\nBt1∈[aσ1L(T),bσ1L(T)]\u001b#\n≥Ey\"\nexp \n−1\nTσ′\n1\nσ4\n1ZJ(t1)\n0Wsds!\n1\u001aWs∈[0,hσ1L(T)]∀s≤J(t1),\nWJ(t1)∈[aσ1L(T),bσ1L(T)]\u001b#\n.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 43\nRecall (3.4). Applying Lemma 5.8 and the Brownian scaling property, one obtains similarly to the upper\nbound (we do not write the details again),\ninf\ny∈[a,b]·σ(0)L(T)Ey\"\nexp \n−1\nTσ′\n1\nσ4\n1ZJ(t1)\n0Wsds!\n1\u001aWs∈[0,hσ1L(T)]∀s≤J(t1),\nWJ(t1)∈[aσ1L(T),bσ1L(T)]\u001b#\n≥exp\u0014t1\nh2L(T)2Ψ\u0012σ′\n1\nσ1α3h3\u0013\n−O(L(T)2/T)−ε O(1)\u0015\n.\nfor some Kε>0,K≥KεandT0(K, ε)>0,T≥T0(K, ε). Reproducing this lower bound for all factors\nof(5.60) , using a Riemann sum approximation similar to (5.57) and recollecting (5.52) , we may conclude\nsimilarly to the upper bound in the case t≥T5/6.\nLet us now consider the case t≤T5/6. Recall the proof of the first moment lower bound in the sub-critical\ncase (Proposition 5.5), that is (5.34, 5.36–5.39). We define as in (5.37),\nτ=τ(T) :=1\nσ2(0)L(T)2Zt\n0σ2(u/T) du ,\nand ( εT)T≥0such that T−1/3≪εT≪T−1/6asT→+∞. Then, the Brownian scaling property and a time\nchange yield, similarly to the sub-critical regime,\n(5.61) E\u0002\n|Acrit\nT,I(t)|\u0003\n≥e(x−z−εT)L(T)+o(T1/3)Px\u0010\nWs∈[0, h],∀s≤τ;Wτ∈[z+εT/3, z+ 2εT/3]\u0011\n.\nIt remains to prove that the probability above is larger than eo(T1/3). Recall Lemma 4.4: proceeding similarly\nto (5.40–5.41), there exists R >0 such that,\n∀t′≥R , Px\u0000\nWs∈[0, h],∀s≤t′;Wt′∈[z+εT/3, z+ 2εT/3]\u0001\n≥1\nhε2\nT\n18hsin\u0010πx\nh\u0011\nexp\u0012\n−π2\n2h2t′\u0013\n.\nIft, Tsatisfy 2 RL(T)2≤t≤T5/6, then (3.4) implies R≤τ≤2α−2T1/6forTsufficiently large, uniformly\ninσ∈ Sη. In particular the r.h.s. above, evaluated at t′=τ, is larger than eO(T1/6), uniformly in σ∈ Sη\nand locally uniformly in x, z. On the other hand, if t, Tsatisfy t≤2RL(T)2, then τ≤4RforTsufficiently\nlarge, uniformly in σ∈ Sη. Moreover, one notices that t≥θL(T) and εT≪T−1/6imply that τ−1/2εT→0\nasT→+∞. Recalling (5.42–5.44) from the sub-critical case, we have already proven by introducing a\nBrownian bridge that, under this assumption, the probability in (5.61) is larger than eo(L(T))forTlarge: we\ndo not replicate the details of the proof here.\nTherefore, we derived three lower bounds which hold respectively in the cases t≥T5/6, 2RL(T)2≤t≤\nT5/6andL(T)≤θt≤2RL(T)2: taking the minimum of these, we obtain a lower bound that holds uniformly\ninL(T)≤θt≤Tandσ∈ Sη, which fully concludes the proof of the proposition. □\nProposition 5.10 (Second moment, Critical) .Leth >0. As T→+∞, one has\n(5.62) E\u0002\n|Acrit\nT,z(t)|2\u0003\n≤e(x+h−2z)L(T)+o(T1/3).\nuniformly in x∈[0, h],z∈[0, h],σ∈ Sηandt∈[0, T].\nThis statement is analogous to Propositions 5.3 and 5.6 in the super- and sub-critical cases respectively.\nSince its proof is identical (using the upper bound (5.49) , and observing that t≤T=eo(T1/3)in the critical\nregime), we do not reproduce it here.\nWe now state the analogue of Proposition 5.7 regarding the number of particles killed by the upper barrier.\nRecall from (4.6) thatRcrit\nT(s, t), 0≤s≤t≤Tdenotes the expected number of particles killed by the upper\nbarrier on the time interval [ s, t].\nProposition 5.11 (Killed particles, Critical) .Leth >0. Then as T→+∞, one has\n(5.63) E[Rcrit\nT(0, T)]≤e−(h−x)L(T)+o(T1/3),44 A. LEGRAND AND P. MAILLARD\nuniformly in x∈[0, h]andσ∈ Sη.\nProof. Recollect (4.14) from Lemma 4.2. Notice that ( γcrit\nT)′(·) = (γcrit\nT)′(·) +hσ′(·/T)L(T)\nT, and recall that\nγcrit\nT(0) = ( h−x)σ(0)L(T). Combining this with (5.51) , we deduce from (4.14) with a straightforward\ncomputation that\nE[Rcrit\nT(0, T)] = e−(h−x)L(T)+O(T−1/3)E(h−x)σ(0)L(T)\"\n1{H0(B)≤T}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}\n×exp \nL(T)\nTZH0(B)\n0w′\nh,T(s/T)−hσ′(s/T)\nσ(s/T)ds+1\nTZH0(B)\n0σ′(s/T)\nσ2(s/T)Bsds!#\n,\nand it remains to show that the latter expectation is of order exp(o(T1/3)). Let us apply Girsanov’s theorem\nand the Brownian symmetry property to the process ( hσ(s/T)L(T)−Bs)s≥0: recalling estimates from (5.10)\nand setting eH(B) :=H0(B−hσ(·/T)L(T)) to lighten notation, this yields,\nE(h−x)σ(0)L(T)\"\n1{H0(B)≤T}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤H0(B)}eL(T)\nTRH0(B)\n0w′\nh,T(s/T)−hσ′(s/T)\nσ(s/T)ds+1\nTRH0(B)\n0σ′(s/T)\nσ2(s/T)Bsds#\n=eO(T−1/3)Exσ(0)L(T)\"\n1{eH(B)≤T}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤eH(B)}eL(T)\nTRfH(B)\n0w′\nh,T(s/T)\nσ(s/T)ds−1\nTRfH(B)\n0σ′(s/T)\nσ2(s/T)Bsds#\n.\nNotice that the latter formula is equivalent to rewriting (4.14) in terms of γ∗\nTinstead of γ∗\nT(details are left to\nthe reader). Then we split [0 , T] into intervals of length T5/6, setting ti:=iT5/6for 0≤i < i max:=⌊T1/6⌋\nandtimax:=T. Thus we have,\nExσ(0)L(T)\"\n1{eH(B)≤T}1{Bs\nσ(s/T)L(T)∈[0,h]∀s≤eH(B)}eL(T)\nTRfH(B)\n0w′\nh,T(s/T)\nσ(s/T)ds−1\nTRfH(B)\n0σ′(s/T)\nσ2(s/T)Bsds#\n≤imax−1X\ni=0Exσ(0)L(T)\"\n1{ti<eH(B)≤ti+1}1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s≤eH(B)\teL(T)\nTRfH(B)\n0w′\nh,T(s/T)\nσ(s/T)ds−1\nTRfH(B)\n0σ′(s/T)\nσ2(s/T)Bsds#\n≤imax−1X\ni=0eO(T1/6)Exσ(0)L(T)\"\n1\b\nBs\nσ(s/T)L(T)∈[0,h],∀s≤iT5/6\teL(T)\nTRiT5/6\n0w′\nh,T(s/T)\nσ(s/T)ds−1\nTRiT5/6\n0σ′(s/T)\nσ2(s/T)Bsds#\n,\nuniformly in σ∈ Sη. Recalling (5.55–5.57) from the proof of Proposition 5.9, we have already proven that\nthe latter expectation is of order eo(T1/3)asT→+∞uniformly in 1 ≤i≤imax=O(T1/6) (we do not\nreproduce the details). This concludes the proof of the proposition. □\n6.Lower bound on the maximum of the N-BBM\nIn this section we prove Proposition 3.3 by considering a BBM between well chosen barriers and showing\nthat it is equal to an N−-BBM with large probability. Then, the lower bound is obtained by showing that\nthe maximum of the BBM between barriers is close to the upper barrier via a moment method. The following\nresults hold for all three regimes ∗ ∈ { sub,sup,crit}, where we use our notation from Section 4.\n6.1.Estimates for the BBM between barriers. We first focus on the case κ∈(0,1), and consider\na BBM between barriers starting with Nκparticles. In order to lighten upcoming formulae, we assume\nthat they are initially located at the origin: then, results on the process started from the initial measure\nNκδ−κσ(0)L(T)are obtained through a direct shift of upcoming estimates.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 45\nSince the moment estimates from Section 5.2 do not hold on the whole interval [0 , T] in the sub-critical\ncase, let us define\n(6.1) t∗\nT:=(\nmin\u0000\nL(T)4, L(T)2T1/3\u0001\nif∗= sub ,\nTif∗ ∈ { sup,crit},\nwhich is the time horizon we consider below: we compute a lower bound on the N-BBM at time t∗\nT,\n∗ ∈ { sub,sup,crit}; then, in the subcritical regime, we extend the lower bound at time Twith additional\narguments.\nLetη >0, and consider a vanishing function θ(T)→0 asT→+∞(depending on L(T)), such that for T\nsufficiently large, one has (4.2), as well as\n(6.2) L(T)3≤θtsub\nT≤(L(T)3)2/3T1/3≤θp\nTL(T)3, if∗= sub .\nLet 0 < ε < 1 be small, and fix\n(6.3) h= 1−ε , x = (1−κ)(1−ε)∈(0, h).\nThen, define the barriers γ∗\nT,γ∗\nTfor each regime ∗ ∈ { sup,sub,crit}according to (4.8),(4.9) and(4.10)\nrespectively, with these parameters h > x > 0; in particular they satisfy (4.3).\nRecall the definitions of A∗\nT,I,R∗\nT(s, t) from (4.5),(4.6), and that N=N(T) =eL(T). Then, moment\nestimates from Propositions 5.1, 5.3, 5.5, 5.6, 5.9 and 5.10 yield for all regimes ∗ ∈ { sup,sub,crit},\nEδ0\u0002\n|A∗\nT,I(t)|\u0003\n≤Nx−infI+o(1), (6.4)\nEδ0\u0002\n|A∗\nT,I(t)|\u0003\n=Nx−infI+o(1)if, additionally, t≥θL(T), (6.5)\nEδ0\u0002\n|A∗\nT,z(t)|2\u0003\n≤Nx+h−2z+o(1), (6.6)\nfor some vanishing terms o(1) as T→+∞: in (6.4, 6.6), these error terms are uniform in σ∈ S∗\nη, 0≤t≤t∗\nT\nandx, z∈[0, h],I⊂[0, h] a non-trivial sub-interval; and in (6.5), it is uniform in σ∈ S∗\nη,L(T)≤θt≤t∗\nT\nand locally uniform in x, I. Let us point out that the additional error factors from Propositions 5.6 and 5.7\nin the sub-critical case are absorbed into the No(1), since tsub\nT≤L(T)4=eo(L(T)).\nWith those parameters and initial condition, let us first prove that the BBM between the barriers γ∗\nT,γ∗\nT\ndoes not contain more than Nparticles at any time in [0 , t∗\nT] with high probability.\nProposition 6.1. Letε,h,xas in (6.3), and κ∈(0,1). Then there exist constants c1, c2>0such that,\nforTsufficiently large, one has\n(6.7) PNκδ0\u0000\n∃s≤t∗\nT;|A∗\nT(s)|> N\u0001\n≤c1N−c2,\nwhere c1, c2are uniform in σ∈ S∗\nη, and locally uniform in ε, κ∈(0,1).\nProof. Define λ:= 1−ε\n2(1−κ)∈(x+κ,1). With a union bound, we write\nPNκδ0\u0000\n∃s≤t∗\nT;|A∗\nT(s)|> N\u0001\n≤t∗\nT−1X\nk=0PNκδ0\u0000\n∃s∈[k, k+ 1] ;|A∗\nT(s)|> N\u0001\n≤t∗\nT−1X\nk=0PNκδ0\u0000\n|A∗\nT(k)|> Nλ\u0001\n+PNκδ0\u0000\n|A∗\nT(k)| ≤Nλ;∃s∈[k, k+ 1],|A∗\nT(s)|> N\u0001\n, (6.8)\nwhere we wrote t∗\nT−1 instead of ⌈t∗\nT−1⌉to lighten notation. On the one hand, the additivity of |A∗\nT(k)|in\nthe initial measure implies that\nENκδ0[|A∗\nT(s)|] =NκEδ0[|A∗\nT(s)|].\nOne deduces from Markov’s inequality and (6.4) that for all 0 ≤k≤t∗\nT−1,\n(6.9) PNκδ0\u0000\n|A∗\nT(k)|> Nλ\u0001\n≤NκN−λEδ0\u0002\n|A∗\nT(k)|\u0003\n=N−(λ−x−κ)+o(1),\nforTlarge, where o(1) does not depend on korσ∈ S∗\nη.46 A. LEGRAND AND P. MAILLARD\nOn the other hand, let ( Zt)t≥0denote the population size of a BBM without selection (in particular its\npopulation size is non-decreasing in time), and let Adenote the set of counting measures on Rwith mass at\nmost Nλ. Using Markov’s property and a straightforward coupling argument, one has for 0 ≤k≤t∗\nT−1,\nPNκδ0\u0000\n|A∗\nT(k)| ≤Nλ;∃s∈[k, k+ 1],|A∗\nT(s)|> N\u0001\n≤sup\nµ∈APµ(Z1> N).\nSince Eµ[Z1] =e1/2µ(R) for any µ∈ A, one has by Markov’s inequality,\nPµ(Z1> N)≤N−(1−λ)e1/2.\nRecollecting (6.8, 6.9) and (6.1), we finally obtain in all regimes,\nPNκδ0\u0000\n∃s≤t∗\nT;|A∗\nT(s)|> N\u0001\n≤L(T)4×N−ε\n2(1−κ)+o(1)=N−ε\n2(1−κ)+o(1)‘,\nwhich concludes the proof. □\nWe now bound from below the number of particles located at a given height, at a time tnot too small.\nProposition 6.2. There exist c1, c2>0such that, for Tsufficiently large, one has\n(6.10) PNκδ0\u0000\n|A∗\nT,[z,z+ε](t)| ≥N1−ε−z\u0001\n≥1−c1N−c2,\nwhere c1, c2are uniform in σ∈ S∗\nη,t∈[θ(T)−1L(T), t∗\nT]andz∈[0,1−2ε], and locally uniform in\nε, κ∈(0,1).\nNotice that, taking z= 1−2ε, this proposition implies that, with large probability, there are at least Nε\nparticles located in the vicinity of the upper barrier γ∗\nTat time t∗\nT.\nProof. Applying Paley-Zygmund’s inequality, we have\n(6.11) PNκδ0\u0000\n|A∗\nT,[z,z+ε](t)| ≥N1−ε−z\u0001\n≥\u0012\n1−N1−ε−z\nENκδ0[|A∗\nT,[z,z+ε](t)|]\u00132\n×ENκδ0[|A∗\nT,[z,z+ε](t)|]2\nENκδ0[|A∗\nT,[z,z+ε](t)|2].\nFor all M∈N,I⊂[0, h] and t≥0, recall that EMδ0[|A∗\nT,I(t)|] =MEδ0[|A∗\nT,I(t)|]. Regarding the second\nmoment, by splitting the sum over pairs of particles u, v∈ N tdepending on whether they come from the\nsame ancestor or two distinct ancestors at time 0, we obtain for any I⊂[0, h],t≥0,\n(6.12) EMδ0\u0002\n|A∗\nT,I(t)|2\u0003\n=MEδ0\u0002\n|A∗\nT,I(t)|2\u0003\n+M(M−1)Eδ0[|A∗\nT,I(t)|]2.\nRecalling (6.4–6.6) and the definitions of x, hfrom (6.3), we obtain\nPNκδ0\u0000\n|A∗\nT,[z,z+ε](t)| ≥1\u0001\n≥\u0010\n1−N(1−ε−z)−κ−(x−z)+o(1)\u00112\u0010\n1 +Nx+h−2z−κ−2(x−z)+o(1)\u0011−1\n=\u0010\n1−N−εκ+o(1)\u00112\u0010\n1 +N−εκ+o(1)\u0011−1\n≥1−N−εκ+o(1),\nasT→+∞, uniformly in σ∈ S∗\nηandt∈[θ(T)−1L(T), t∗\nT], which concludes the proof. □\n6.2.Proof of Proposition 3.3. We finally prove Proposition 3.3. This is achieved by coupling the BBM\nbetween barriers with an N-BBM, through the introduction of an N−-BBM and the application of Lemma 4.6.\nRecall that the point measure of the N-BBM throughout time is denoted ( XN\nt)t≥0. We first claim the\nfollowing, which is a consequence of Propositions 6.1 and 6.2.\nLemma 6.3. Let∗ ∈ { sub,sup,crit}. There exists c1, c2>0such that for Tsufficiently large, one has\n(6.13) PNκδ0\u0010\nXN\nt\u0002\nγ∗\nT(t) + (1 −2ε)σ(t/T)L(T),+∞\u0001\n< Nε\u0011\n≤c1N−c2,\nand\n(6.14) PNκδ−κσ(0)L(T)\u0010\nmax(XN\nt∗\nT)≤γ∗\nT(t∗\nT) + (1 −2ε)σ(t∗\nT/T)L(T)−κσ(0)L(T)\u0011\n≤c1N−c2,\nwhere c1, c2are uniform in σ∈ S∗\nη,t∈[θ(T)−1L(T), t∗\nT], and locally uniform in ε, κ∈(0,1).MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 47\nProof of Lemma 6.3. Let us first prove (6.13) . Denote with ( XN−\nt)t≥0the point process of an BBM which\nundergoes two selection mechanism: particles which are not in the Nhighest orthat hit one of the barriers\nγ∗\nT,γ∗\nTare killed. Recalling Section 4.3 and Remark 4.2.( i), notice that ( XN−\nt)t≥0is an N−-BBM; so\nLemma 4.6 gives for any y∈R,\n(6.15) PNκδ0\u0000\nXN\nt∗\nT([y,+∞))< Nε\u0001\n≤PNκδ0\u0000\nXN−\nt∗\nT([y,+∞))< Nε\u0001\n.\nLet denote ( XB\nt)t≥0the point process of a BBM killed at the barriers γ∗\nT,γ∗\nTbut without any other selection\nmechanism. By Proposition 6.1, it has a large probability under PNκδ0to contain fewer than Nparticles at\nall time on [0 , t∗\nT], in which case its trajectory is equal to that of the process ( XN−\nt)t∈[0,t∗\nT]. Therefore, for T\nsufficiently large and y∈R,\n(6.16) PNκδ0\u0000\nXN−\nt∗\nT([y,+∞))< Nε\u0001\n≤PNκδ0\u0000\nXB\nt∗\nT([y,+∞))< Nε\u0001\n+c1N−c2.\nFinally, letting t∈[θ(T)−1L(T), t∗\nT] and y=γ∗\nT(t) + (1−2ε)σ(t/T)L(T), and applying Proposition 6.2 with\nz= 1−2ε, this yields\nPNκδ0\u0010\nXN\nt\u0002\nγ∗\nT(t) + (1 −2ε)σ(t/T)L(T),+∞\u0001\n< Nε\u0011\n≤PNκδ0\u0000\n|A∗\nT,1−2ε(t∗\nT)|< Nε\u0001\n+c1N−c2≤2c1N−c2,\nuniformly in σ,t, from which (6.13) follows.\nRegarding (6.14), one deduces straightforwardly from (6.13) that\nPNκδ0\u0010\nmax(XN\nt∗\nT)≤γ∗\nT(t∗\nT) + (1 −2ε)σ(t∗\nT/T)L(T)\u0011\n≤c1N−c2,\nand shifting this estimate by −κσ(0)L(T) concludes the proof. □\nWith this at hand, we resume the proof of Proposition 3.3. To that end, we first claim that it suffices\nto prove the following, slightly weaker statement in which the uniformity in κ∈[0,1] is replaced by local\nuniformity in κ∈(0,1). Recall (1.10, 1.11).\nLemma 6.4. Let∗ ∈ { sup,sub,crit}. Let λ >0,κ∈(0,1)andη >0. Then as T→+∞, one has\n(6.17) PNκδ−κσ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\u0013\n−→ 0,\nand the convergence is uniform in σ∈ S∗\nη, and locally uniform in κ∈(0,1).\nProof of Proposition 3.3 subject to Lemma 6.4. Recall Proposition 3.2. We start with the case κclose to 1.\nLet∗ ∈ { sup,sub,crit}andλ >0. Notice that there exists ε >0 such that\n(6.18) lim sup\nT→+∞εσ(0)L(T)\nb∗\nT≤λ\n3.\nIndeed, in the sub-critical regime this follows from the fact that L(T)≪bsub\nT, and in the other cases\nthis holds as soon as ε≤(λη/3)limT→+∞(L(T)/b∗\nT) (the latter limit being equal to 1, resp. α, in the\nsuper-critical, resp. critical regime). Moreover, one has Nκδ−κσ(0)L(T)≻N1−εδ−σ(0)L(T)forκ∈[1−ε,1],\nso Proposition 3.2 implies that,\nPNκδ−κσ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\u0013\n≤PN1−εδ−σ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\u0013\n≤PN1−εδ−(1−ε)σ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\n3\u0013\n,\nwhere the last inequality is obtained for Tsufficiently large by shifting the process upward by εσ(0)L(T),\nand by recalling (6.18) . Applying Lemma 6.4 with κ= 1−ε, this proves that (6.17) holds uniformly in\nκ∈[1−ε,1].\nRegarding the case κsmall, let ε >0, and recall that ( Nt)t≥0denotes the set of particles in the BBM\nthroughout time; and let Zt:=|NT|,t≥0. We claim the following.48 A. LEGRAND AND P. MAILLARD\nLemma 6.5. Letε, ε′>0. One has for Tsufficiently large:\n(i)Pδ0(ZεL(T)≤Nε/4)≤ε′,\n(ii)Pδ0(∃s≤εL(T), Zs≥N)≤ε′,\n(iii)Pδ0(∃u∈ N εL(T);Xu(εL(T))≤ −2εη−1L(T))≤ε′.\nThose statements follow from classical results on the BBM and birth processes, we postpone their proof for\nnow. For ε >0 and κ∈[0, ε], notice that Nκδ−κσ(0)L(T)≻δ−εσ(0)L(T); so we deduce from Proposition 3.2\nand a shift that, for ε >0 and κ∈[0, ε],\nPNκδ−κσ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\u0013\n≤Pδ0\u00121\nb∗\nT\u0000\nmax(XN\nT)−εσ(0)L(T)−m∗\nT\u0001\n≤ −λ\u0013\n.\nLetε′>0. We apply the Markov property at time εL(T), noticing that Lemma 6.5 implies for Tsufficiently\nlarge,\nPδ0\nXN\nεL(T)\u0010\u0000\n− ∞,−2εη−1L(T)\u0003\u0011\n= 0,\nXN\nεL(T)\u0010\u0002\n−2εη−1L(T),+∞\u0001\u0011\n≥Nε/4\n≥1−3ε′,\n(indeed, on the event {∀s≤εL(T), Zs< N}, the particle configurations of the BBM and N-BBM at time\nεL(T) are the same). In particular, on that event we have XN\nεL(T)≻Nε/4δ−2εη−1L(T). Applying Markov’s\nproperty at time εL(T), then Proposition 3.2 again and a shift, we obtain\nPNκδ−κσ(0)L(T)\u00121\nb∗\nT\u0000\nmax(XN\nT)−m∗\nT\u0001\n≤ −λ\u0013\n≤3ε′+PNε/4δ(εL(T),−ε\n4σ(0)L(T))\u00121\nb∗\nT\u0000\nmax(XN\nT)−ε\u0000\n3σ(0)/4 + 2 η−1\u0001\nL(T)−m∗\nT\u0001\n≤ −λ\u0013\n.\nRecall that (3 σ(0)/4 + 2 η−1)L(T) =O(b∗\nT); hence, choosing ε′andε=ε(λ, η) small enough, we conclude\nthe proof of Proposition 3.3 by applying Lemma 6.4 at time T−εL(T) with Nε/4initial particles. □\nWe now prove Lemma 6.5. Let us first recall the following classical result on pure birth processes (see\ne.g. [ 5, Ch. III]). In the following we consider a pure birth process ( Zt)t≥0with same rate β0and offspring\ndistribution ξas our time-inhomogeneous BBM ( Xt)t∈[0,T], and with Z0:= 1; in particular, when restricted\nto [0, T], it has the same distribution as the BBM’s population size ( |Nt|)t∈[0,T]under Pδ0.\nProposition 6.6. [5, Theorems III.7.1–2] LetFt:=σ(Zs, s≤t),t≥0. Then, under Pδ0, the process\n(e−t/2Zt)t≥0is a(Ft)t≥0-martingale, is non-negative and converges a.s. and in L1to some random variable\nW. Moreover, E[W] = 1 andPδ0(W > 0) = 1 .\nProof of Lemma 6.5. Let us start with Lemma 6.5.(i–ii). We prove them for a birth process ( Zt)t≥0by using\nProposition 6.6; then these statements also hold for ( |Nt|)t∈[0,T], assuming Twas taken sufficiently large.\nOn the one hand, for any x >0, one has for tsufficiently large\nP(Zt≤et/4)≤P(e−t/2Zt≤x)≤P(W≤2x) +o(1),\nwhere the second inequality comes from the L1convergence of the martingale, i.e. P(|W−e−t/2Zt|> x)→0\nast→+∞. Assuming xwas chosen sufficiently small and replacing twith εL(T), this proves Lemma 6.5.(i)\nforTsufficiently large. On the other hand, we have by Doob’s martingale inequality that, for t≥0,\nP(∃s≤t;e−s/2Zs≥et/2)≤e−t/2.\nAgain, letting t=εL(T) and assuming Tlarge enough, this implies Lemma 6.5.(ii).4\n4Actually this proves a much stronger result, but we do not need it in this paper.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 49\nWe now turn to Lemma 6.5.(iii), where we let Zt:=|Nt|fort∈[0, T]. Let ( Yi\nt)t∈[0,T],i∈Ndenote a\nsequence of i.i.d. time-inhomogeneous Brownian motions with infinitesimal variance σ(·/T). As a consequence\nof Slepian’s lemma [53], one has for t∈[0, T] and k∈N,\nPδ0\u0000\n∃u∈ N t;Xu(t)≤ −2η−1t\f\fZt=k\u0001\n≤P0\u0000\n∃i≤k;Yi\nt≥2η−1t\u0001\n= 1−\u0002\n1−P0(Y1\nt≥2η−1t)\u0003k,\nwhere we also used the Brownian symmetry property. Recall that for i∈N,\nE0\u0002\n(Yi\nt)2\u0003\n=Zt\n0σ2(s/T) ds≤η−2t,\nand recall the standard Gaussian tail estimate, P0(W1≥x)≤1\nx√\n2πe−x2/2forx >1. Thus, for tsufficiently\nlarge, one has\nP0(Y1\nt≥2η−1t)≤P0(W1≥2√\nt)≤1\n2√\n2πte−2t.\nLett=εL(T) in the above, and recall from Proposition 6.6 that Pδ0(ZεL(T)≥N(T)3ε/2)→0 asT→+∞\n(this is similar to Lemma 6.5, we do not write the details again). Therefore, we deduce for Tlarge,\nPδ0\u0000\n∃u∈ N εL(T);Xu(εL(T))≤ −2εη−1L(T)\u0001\n≤o(1) + 1 −\"\n1−1\n2p\n2πεL(T)N(T)−2ε#N(T)3ε/2\n=o(1),\nwhich concludes the proof. □\nIt only remains to prove Lemma 6.4, which is a consequence of Lemmata 6.3 and 4.1. We proceed\ndifferently depending on the regime satisfied by L(T).\nProof of Lemma 6.4, N(T)super-critical or critical. In the super-critical and critical regimes, the proof\nis instantaneous. Indeed, in both cases one has t∗\nT=T; recalling (6.3) and the notation γ∗,h,x\nT(·) from\nSection 4.1, one has\nγ∗,1−ε,(1−ε)(1−κ)\nT (T) + (1 −2ε)σ(1)L(T)−κσ(0)L(T)\n=γ∗,1−ε,1−ε(1−κ)\nT (T)−εσ(1)L(T). (6.19)\nLetting εbe arbitrarily small and applying Lemmata 4.1 and 6.3 (more precisely (6.14) ), this gives exactly\nLemma 6.4 in both regimes. □\nWe now turn to the proof of Lemma 6.4 in the sub-critical regime ( L(T)≪T1/3), which requires a little\nmore care. Since our estimates do not directly hold at time Tin that regime, we split the interval [0 , T] into\nblocks whose lengths are of order tsub\nT, apply our estimates on each of those, then use them to reconstruct a\nprocess on [0 , T] which is dominated by the N-BBM.\nFirst, we may always bound the N-BBM from below by having it start with fewer particles, recall\nProposition 3.2: hence, in (6.13) and in the following, we assume that εis small and that κ=εwithout loss\nof generality. Moreover, it will be very convenient to work with quantiles instead of the maximal displacement.\nFor some particle configuration µ∈C, define\n(6.20) qκ(µ) := inf {q∈R;µ([q,+∞))< Nκ}.\nNotice that qκis non-decreasing, i.e. for µ≺ν, one has qκ(µ)≤qκ(ν). Moreover, we clearly have\nmax(XN\nT)≥qκ(XN\nT), so it suffices to prove the lemma with max( XN\nT) replaced by qκ(XN\nT).\nWe now introduce an auxiliary process ( XN\nt)0≤t≤T. Let us define K:=⌊2T/tsub\nT⌋, and for 0 ≤k≤K−1,\nlettk:=k\n2tsub\nT, and tK=T(sotK−tK−1∈[1\n2tsub\nT, tsub\nT]). The process ( XN\nt)0≤t≤Tis defined as follows:\nstarting from ⌊Nκ⌋δ0(we omit the integer part in the following), it evolves between times tkandtk+1as the\nprocess XN. Then, at every time tk, 1≤k < K , all but the Nκtop-most particles are removed, and the\nremaining particles are all set to the lowest among their positions. In other words, a configuration µ∈CNis\nreplaced by the configuration Nκδqκ(µ). By a coupling argument (recall Proposition 3.2) and an induction,50 A. LEGRAND AND P. MAILLARD\none obtains straightforwardly a coupling such that XN\nT≺ XN\nTwith probability 1. In particular, the law of\nqκ(XN\nT) stochastically bounds from below the law of qκ(XN\nT), which is lower than max(XN\nT); hence it suffices\nto prove (6.17) with max( XN\nT) replaced by qκ(XN\nT).\nFor 1≤k≤K, write\nXk=qκ\u0000\nXN\ntk\u0001\n−qκ\u0000\nXN\ntk−1\u0001\n,\nso that qκ(XN\nT) =X1+···+XK(recall that XN\n0=Nκδ0). By the definition of the process XNand the\nfact that a translation of the N-BBM is again an N-BBM started from a translated initial configuration, the\nrandom variables X1, . . . , X Kare independent.\nUsing that notation, we may finally prove Lemma 6.4 in the sub-critical regime. However, the proof relies\non two different methods, depending on the speed at which N(T) diverges.\nProof of Lemma 6.4, N(T)sub-critical and super-polynomial. Assume that N(T) is super-polynomial, i.e.\nL(T)≫log(T). Recall (6.13), which implies for 1 ≤k≤Kthat\n(6.21) P\u0000\nXk≥γsub\nT(tk)−γsub\nT(tk−1)−η−1L(T)\u0001\n≥1−c1N−c2,\nfor some c1, c2locally uniform in ε=κ∈(0,1). Recalling the definition of the process ( XN\nt)0≤t≤T, that the\nXk, 1≤k≤Kare independent and that qκ(XN\nT) =X1+···+XK, one deduces through a direct induction\nthat\nPNκδu0\u0010\nqκ(XN\nT)≥γsub\nT(T)−Kη−1L(T)\u0011\n≥(1−c1N−c2)K.\nNotice that KN−c2≤θ(T)T\nL(T)3e−c2L(T); so, under the assumption L(T)≫log(T), the latter probability goes\nto 1 as T→+∞, locally uniformly in κ∈(0,1). Recalling that qκ(XN\nT) stochastically bounds from below\nmax(XN\nT), this finally implies\nPNκδu0\u0010\nmax(XN\nT)≤γsub\nT(T)−Kη−1L(T)\u0011\n−→0\nasT→+∞, locally uniformly in κ. Moreover, Kη−1L(T)≤η−1θ(T)T\nL(T)2=o(T/L(T)2) and L(T) =\no(T/L(T)2). Hence, applying a shift −κσ(0)L(T) to the estimate above, we deduce from Lemma 4.1 and\nthe same computation as (6.19) that, with εarbitrarily small, the lower bound (6.17) holds in the case N(T)\nsub-critical, super-polynomial. □\nIfL(T) is too small, the decomposition above involves so many blocks that, with large probability, the\nauxiliary process XNdoes not satisfy the event (6.21) on some of them. However, having a small L(T) allows\nus to derive sharp moment estimates on Xk, 1≤k≤K, then to estimate the final maximal displacement\nwith a second moment method.\nProof of Lemma 6.4, N(T)sub-critical, L(T)very small. In the following, we assume that L(T)≪T1/8for\nTsufficiently large. In particular, we are still in the sub-critical regime, and (6.1) yields tsub\nT=L(T)4. We\nclaim the following: there exist c1, c2>0 such that for Tsufficiently large, for every k∈ {1, . . . , K }, we have\nE[Xk]≥(γsub\nT(tk)−γsub\nT(tk−1))·(1−c1N−c2)−c1L(T), (6.22)\nVar\u0000\nXk\u0001\n≤c1(tsub\nT)2. (6.23)\nLet us see how equations (6.22) and (6.23) imply the lemma. First note that using (6.22), we have\nE[X1+. . .+XK]≥γsub\nT(T)(1−c1N−c2)−c1L(T)×K,\nand, using that K≤2T/tsub\nT= 2T/L(T)4, we get that\nE[qκ(XN\nT)] =E[X1+. . .+XK]≥γsub\nT(T) +o\u0012T\nL(T)2\u0013\n.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 51\nFurthermore, by the independence of the random variables X1, . . . , X k, we get using (6.23) that\nVar(qκ(XN\nT)) =KX\nk=1Var(Xk)≤2c1tsub\nTT=o \u0012T\nL(T)2\u00132!\n,\nwhere for the last equality we used that tsub\nT=L(T)4andL(T)≪T1/8. Using the Bienaym´ e-Chebychev\ninequality, this yields that\nqκ(XN\nT)≥γsub\nT(T) +oP\u0012T\nL(T)2\u0013\n,\nwith large probability, where we recall the notation oP(·) from Theorem 1.1.. Here again, applying a shift\n−κσ(0)L(T) =o(T/L(T)2) to the estimate above and letting κ, εbe small, we deduce from Lemma 4.1 and\nthe same computation as (6.19) that the lower bound (6.17) holds in the case L(T)≪T1/8.\nIt remains to prove (6.22) and(6.23) . We start with (6.22) . Noting that the process evolves between\ntimes tkandtk+1as aN-BBM with variance profile σ(tk/T+·), it is enough to show that\n(6.24) ENκδ0\u0002\nqκ(XN\nt)\u0003\n≥γsub\nT(t) (1−c1N−c2).\nwhere c1, c2are uniform in σ∈ S∗\nηandt∈[1\n2tsub\nT, tsub\nT]. Most hard work has been done in Lemma 6.3, which\nyields that, with the same notation,\n(6.25) PNκδ0\u0000\nqκ(XN\nt)< γsub\nT(t)\u0001\n≤c1N−c2.\nAssuming from now on that Tis large enough, so that γsub\nT(t)≥ctsub\nT≥0 for some c > 0 and all\nt∈[1\n2tsub\nT, tsub\nT], we deduce from (6.25) that\n(6.26) ENκδ0h\nqκ(XN\nt)1{qκ(XN\nt)≥0}i\n≥γsub\nT(t)PNκδ0\u0000\nqκ(XN\nt)≥γsub\nT(t)\u0001\n≥γsub\nT(t) (1−c1N−c2),\nuniformly in t∈[1\n2tsub\nT, tsub\nT]. On the other hand, recalling from Proposition 3.1 that we can couple XNand\nXin such a way that the particles of XNform a subset of the particles of X, we have\nENκδ0h\nqκ(XN\nt)1{qκ(XN\nt)≤0}i\n≥ −ENκδ0h\n(min(Xt))−1{qκ(XN\nt)≤0}i\n,\nand, using the Cauchy-Schwarz inequality and the symmetry of the Gaussian distribution, we get\n(6.27) ENκδ0h\nqκ(XN\nt)1{qκ(XN\nt)≤0}i\n≥ −r\nENκδ0h\n(max(Xt))2\n+i\n×PNκδ0\u0000\nqκ(XN\nt)≤0\u0001\n.\nLet us recall the following standard result on Gaussian random variables. Let us mention that we are not\naiming for optimal constants or bounds in this statement. The proof is postponed to the end of this section.\nLemma 6.7. Lett∈(0, T],M≥1and(gi)1≤i≤Ma centered Gaussian vector, such that each gihas variance\nρ2≥0. Then, for M≥2, one has\n(6.28) E\u0002\n(max{gi,1≤i≤M})2\n+\u0003\n≤4ρ2logM .\nWe wish to apply Lemma 6.7 to bound ENκδ0h\n(max(Xt))2\n+i\n. To do this, condition on the branching\ntimes and denote by Ztthe number of particles at time t. Then apply the lemma with M=Ztandρ2=\nσ2(t/T)×tand recall that σ∈ Sη. This gives\nENκδ0h\n(max(Xt))2\n+i\n≤4η−2×t×ENκδ0[logZt].\nBut we have ENκδ0[Zt] =Nκet/2andZt̸= 0 almost surely, so that by Jensen’s inequality,\nr\nENκδ0h\n(max(Xt))2\n+i\n≤p\n4η−2×t×(κlogN+t/2)≤Ctsub\nT,52 A. LEGRAND AND P. MAILLARD\nfor some C >0, using that t≤tsub\nTandlogN=L(T) =o(tsub\nT). Plugging this into (6.27) and using (6.25) ,\nwe get, possibly modifying the values of c1andc2,\n(6.29) ENκδ0h\nqκ(XN\nt)1{qκ(XN\nt)≤0}i\n≥ −γsub\nT(t)c1N−c2.\nCombining (6.26) and (6.29) yields (6.24) and therefore (6.22).\nThe proof of (6.23) is much simpler: it suffices to prove E[X2\nk]≤c1(tsub\nT)2for some c1>0, which\nis obtained via the embedding of the N-BBM into a BBM without selection used above, together with\nLemma 6.7. Details are omitted. □\nProof of Lemma 6.7. This result is standard, but we provide a proof for completeness. For u0>0, bound\nE\u0002\n(max{gi,1≤i≤M})2\n+\u0003\n=Z+∞\n02uP(max{gi,1≤i≤M} ≥u) du\n≤u2\n0+Z+∞\nu02uP(max{gi,1≤i≤M} ≥u) du .\nA union bound and a standard estimation on the Gaussian tail imply that, for u > ρ ,\n(6.30) P(max{gi,1≤i≤M} ≥u)≤Mr\n2\nπρ\nuexp(−u2/2ρ2)≤Mr\n2\nπexp(−u2/2ρ2).\nUsing (6.30), one has for u0> ρ,\nE\u0002\nmax{gi,1≤i≤M}2\u0003\n≤u2\n0+ 2r\n2\nπMρ2exp(−u2\n0/2ρ2).\nLetting u0=ρ√2 logM, and M≥2, this concludes the proof. □\nRemark 6.1. With this, the proof of Lemma 6.4 is finally completed in every regime for L(T). Indeed, the\nonly case we have not treated is when the sub-critical regime alternates between the sub-cases N(T)super-\npolynomial and L(T)smaller than T1/8. This situation can be handled e.g. by letting L1(T) :=L(T)∧(logT)2,\nL2(T) :=L(T)∨(logT)2, then applying (6.17) to both cases (for which we have displayed complete proofs).\nThen the l.h.s. in (6.17) for the “oscillating” L(T)is bounded from above by the maximum of two vanishing\nsequences: we leave the details to the reader.\n7.Upper bound on the maximum of the N-BBM\nLet∗ ∈ { sub,sup,crit}. In this section, we prove Proposition 3.4: in a similar manner to Section 6\nand the proof of Proposition 3.3, we show that the trajectory of a BBM between well-chosen barriers is\nsimilar to that of an N+-BBM with large probability, and deduce an upper bound on the N-BBM with the\ncoupling argument from Lemma 4.6 and the definition of the upper barrier. However in this section we face a\nnoticeable complication: since we are considering a quite long time interval [0 , t∗\nT] (especially for the critical\nand sub-critical cases: for ∗ ∈ { sub,crit}, one has t∗\nT≳L(T)3) the total population between the barriers\nmay grow significantly over time, and some individuals may eventually realize very large displacements\nwith positive probability. Thus, one cannot simultaneously guarantee that, with large probability, at least\nNparticles of the process remain between the barriers throughout [0 , t∗\nT] , and that our choice of upper\nbarrier does not kill any particle —which is necessary for the coupling with the N+-BBM, as opposed to the\nN−-BBM (recall Lemma 4.6 and Remark 4.2). This observation is made more precise in Remark 7.1 below.\nTo circumvent that issue, we choose the parameters of the barriers so that the number of particles\nin-between is at least Nat all time with large probability, and we apply some additional “control” on the\npopulation to prevent it from growing too much. The key idea for this purpose is the following: particles\nthat contribute the most to the growth of the population are those which realize “peaking” events in their\ntrajectory; that is, they rise close to the upper barrier at some time, then generate a large offspring in the\ntime span before they fall near the lower barrier. This was observed in particular in [ 9] when studying aMAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 53\n(homogeneous) BBM with adsorption and near-critical drift: the authors introduce a killing barrier at height\nL >0, and choose a drift such that the first moment of the population size remains roughly eLfor a time\nL3; and in that regime, they observe that removing “peaking” particles that reach a fixed height strongly\naffects the survival probability of the process on the time interval [0 , L3]. Therefore in our setting, we mark\nparticles that rise close to the upper barrier; and, in order to control the population, we apply a slightly\nstronger selection to their offspring: more precisely, we kill their children at a shifted-upward version of the\nlower barrier. In this way, our choice of barriers is still lax enough to keep Nsurviving particles in-between\nat all time, but the offspring of any peaking particle is kept in check. Overall, this ensures us a better control\non the population of the BBM between barriers, which prevents with large probability the birth of particles\nthat outreach the upper barrier: in turn, this finally enables the comparison with a multi-type N+-BBM,\nwhich itself can be coupled to the N-BBM.\nLetε∈(0,1) a small parameter, and fix\n(7.1) h= 1 + ε , x = 1 +2\n3ε .\nThen, define once again the barriers γ∗\nT,γ∗\nTfor each regime ∗ ∈ { sup,sub,crit}according to (4.8),(4.9)\nand (4.10) respectively, with those h, x; in particular they satisfy (4.3).\nWe define the “red” barriers for s∈[0, T] with\n(7.2) γred\nT(s) :=γ∗\nT(s) +ε\n3σ(s/T)L(T),and γred\nT(s) :=γ∗\nT(s) +ε\n3σ(s/T)L(T),\nwhich are slightly shifted versions of γ∗\nT,γ∗\nT, so that γ∗\nT(s)≤γred\nT(s)≤γ∗\nT(s)≤γred\nT(s) for all s∈[0, T]. We\nalso define the sets of “white” and “red” particles with\n(7.3) Nwhite\ns :=\b\nu∈ N s\f\fXu(r)∈\u0000\nγ∗\nT(r),γ∗\nT(r)\u0001\n,∀r≤s\t\n,\nand\n(7.4) Nred\ns:=\n\nu∈ N s\f\f\f\f\f\f\f∃v≤s,s.t.Xu(v) =γ∗\nT(v),\nXu(r)∈\u0000\nγ∗\nT(r),γ∗\nT(r)\u0001\n,∀r≤v,\nXu(r)∈\u0000\nγred\nT(r),γred\nT(r)\u0003\n,∀v≤r≤s\n\n.\nIn other words, we start the process with only white particles, which are killed at γ∗\nT. When they reach γ∗\nT,\nthey are “colored” red (instead of being killed) and keep evolving; however, red particles and their offspring\nare thereafter killed at the barriers γred\nTandγred\nT.\nRecall t∗\nTandθ(·) from (6.1–6.2). Recall the definitions of A∗\nT,I,R∗\nT(s, t) from (4.5),(4.6), which are\nexpressed in terms of γ∗\nTandγ∗\nT. Let us rewrite all estimates from Propositions 5.1, 5.3, 5.4, 5.5, 5.6, 5.7,\n5.9, 5.10 and 5.11 for an initial condition (or, equivalently, both barriers) shifted by −yσ(0)L(T),y∈[0, x)\n(respectively shifted by + yσ(0)L(T)): this formulation is more convenient to handle all atoms from the\ninitial distribution µε(recall (3.2)). For all regimes ∗ ∈ { sup,sub,crit}, one has\nEδ−yσ(0)L(T)\u0002\n|A∗\nT,I(t)|\u0003\n≤Nx−y−infI+o(1), (7.5)\nEδ−yσ(0)L(T)\u0002\n|A∗\nT,I(t)|\u0003\n=Nx−y−infI+o(1)if, additionally, t≥θL(T), (7.6)\nEδ−yσ(0)L(T)\u0002\n|A∗\nT,z(t)|2\u0003\n≤Nx+h−y−2z+o(1), (7.7)\nEδ−yσ(0)L(T)\u0002\nR∗\nT(0, t∗\nT)\u0003\n≤N−(h+y−x)+o(1), (7.8)\nfor some vanishing terms o(1) as T→+∞: more precisely, in (7.5, 7.7, 7.8), these error terms are uniform in\nσ∈ S∗\nη, 0≤t≤t∗\nTandx, z∈[0, h],I⊂[0, h] a non-trivial sub-interval; and in (7.6), it is uniform in σ∈ S∗\nη,\nL(T)≤θt≤t∗\nTand locally uniform in x, I.54 A. LEGRAND AND P. MAILLARD\nForε >0,s∈[0, T], let us generalize the counting measure µεdefined in (3.2), by letting\n(7.9) µε,s:=⌈ε−1⌉X\nk=0l\nNkε+ε\n2m\nδ−kεσ(s/T)L(T), and µε:=µε,0,\nwhich is supported on [ −σ(0)L(T),0]⊂(γred\nT(0),γ∗\nT(0)), and notice that δ0≺µεandµε([−σ(0)L(T),0])≥\nN1+ε\n2. In the remainder of this section we will assume that ε−1∈Nand omit all integer parts, in order to\nlighten all formulae.\n7.1.Estimates for the BBM between barriers. We start the branching-selection process with only\nwhite particles distributed according to µε, and prove that, with large probability, the following three claims\nhold:\n(C-1) there are at least N(white) particles above γred\nTat all time t∈[0, t∗\nT],\n(C-2) no particle reaches γred\nTthroughout [0 , t∗\nT],\n(C-3) the distribution of particles between the barriers at time t≲t∗\nTis close to µε,t.\nLet us mention that the third claim (C-3) is actually not needed to prove Theorem 1.1; however it is\nneeded for Proposition 1.4, and it relies on moment methods similar to (C-1) and (C-2), so we included its\nproof in this section.\nLower bound on the number of living particles. Let us first prove (C-1), which ensures us that no particle\nfrom ( Nwhite\ns∪ Nred\ns)s∈[0,t∗\nT]killed either by γ∗\nTorγred\nTis among the Nhighest of the process at any time.\nProposition 7.1. Letε∈(0,1)andh,xas in (7.1). Then there exist constants c1, c2>0such that, for T\nsufficiently large, one has\n(7.10) Pµε\u0000\n∃s≤t∗\nT;|A∗\nT,ε\n3(s)|< N\u0001\n≤c1N−c2,\nwhere c1, c2are uniform in σ∈ S∗\nη, and locally uniform in ε∈(0,1).\nThe proof uses the following lemma, which bounds from below the survival probability, for a time shorter\nthan L(T)3, of a single particle between γred\nTandγ∗\nT. Recall that θ(T) is defined in (6.1–6.2).\nLemma 7.2. Lett′(T) :=θ(T)(L(T)3∧T). Then\nPδ−σ(0)L(T)\u0010\n∃u∈ N t′;∀s≤t′, Xu(s)∈[γred\nT(s),γa\nT(s)]\u0011\n≥e−o(L(T)),\nasT→+∞, uniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1).\nProof of Lemma 7.2. Recall Lemma 4.5, which ensures us that we may tighten the barriers on a short time\ninterval. More precisely let K > 3 a large constant, and recall Lemma 4.5 and the notation γ∗,h′,x′\nT ,γ∗,h′,x′\nT\nforh′> x′>0. Then, the lemma and a vertical shift of the process yield\nPδ−σ(0)L(T)\u0010\n∃u∈ N t′;∀s≤t′, Xu(s)∈[γred\nT(s),γ∗\nT(s)]\u0011\n≥Pδ0\u0010\n∃u∈ N t′;∀s≤t′, Xu(s)∈h\nγ∗,ε\nK,ε\n2K\nT (s),γ∗,ε\nK,ε\n2K\nT (s)i\u0011\n,\nRecall the definition of A∗\nT(·) from (4.5), and let us extend it to a generic pair of barriers γ∗,h′,x′\nT ,γ∗,h′,x′\nT ,\nforh′> x′>0, by writing for s∈[0, T],\nA∗,h′,x′\nT (s) :=n\nu∈ N t\f\f\fXu(s)∈h\nγ∗,h′,x′\nT (s),γ∗,h′,x′\nT (s)i\n,∀s∈[0, t]o\n.\nThen, Paley-Zygmund’s inequality gives\nPδ0\u0010\n∃u∈ N t′;∀s≤t′, Xu(s)∈h\nγ∗,ε\nK,ε\n2K\nT (s),γ∗,ε\nK,ε\n2K\nT (s)i\u0011\n=Pδ0\u0000\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f≥1\u0001\n≥ \n1−1\nEδ0\u0002\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f\u0003!\nEδ0\u0002\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f\u00032\nEδ0\u0002\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f2\u0003.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 55\nRecalling the moment estimates (7.5–7.7) for the triplet of parameters ( h′, x′, y) = (ε\nK,ε\n2K,0), one obtains\nasT→+∞,5\nEδ0\u0002\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f\u0003\n=Nε\n2K+o(1), and Eδ0\u0002\f\fA∗,ε\nK,ε\n2K\nT (t′(T))\f\f2\u0003\n≤N3ε\n2K+o(1).\nTherefore, one finally deduces that\nPδ−σ(0)L(T)\u0010\n∃u∈ N t′;∀s≤t′, Xu(s)∈[γred\nT(s),γ∗\nT(s)]\u0011\n≥N−ε\nK+o(1),\nasT→+∞; letting K→+∞, this finishes the proof of the lemma. □\nProof of Proposition 7.1. Since one has µε≻N1+ε\n2δ−σ(0)L(T), we only have to prove (7.10) for the latter\ninitial measure; then the proposition follows from a direct monotonic coupling argument.\nWe prove this proposition with a union bound, splitting the time interval [0 , t∗\nT] into a first part of length\nt′(T) :=θ(T)(L(T)3∧T), and the remainder into intervals of length 1. Thus,\nPN1+ε\n2δ−σ(0)L(T)\u0000\n∃s≤t∗\nT;|A∗\nT,ε\n3(s)|< N\u0001\n≤PN1+ε\n2δ−σ(0)L(T)\u0000\n∃s≤t′(T) ;|A∗\nT,ε\n3(s)|< N\u0001\n+t∗\nT−1X\nk=t′(T)PN1+ε\n2δ−σ(0)L(T)\u0000\n∃s∈[k, k+ 1] ;|A∗\nT,ε\n3(s)|< N\u0001\n, (7.11)\nwhere, again, we respectively wrote t′(T),t∗\nT−1 instead of ⌊t′(T)⌋,⌈t∗\nT−1⌉to lighten notation. Notice that\nthe sum may be empty in the super-critical regime.\nWe start with the first term in (7.11) . Let Mdenote the number of individuals from the initial population\nwhich have at least one descendant surviving between γred\nTandγ∗\nTuntil time t′:=t′(T). For a single initial\nparticle in −σ(0)L(T), we write,\npT:=Pδ−σ(0)L(T)\u0000\n∃u∈ N t′;∀s≤t′, Xu(s)∈[γred\nT(s),γ∗\nT(s)]\u0001\n≥e−o(L(T)),\nwhere the last inequality is the content of Lemma 7.2. In particular, one has,\npT=Pδ−σ(0)L(T)(M= 1) = 1 −Pδ−σ(0)L(T)(M= 0).\nStarting from an initial population of N1+ε\n2particles, recall that they have independent offspring. Therefore,\nunder PN1+ε\n2δ−σ(0)L(T), one has that Mis a binomial random variable with parameters ( N1+ε\n2, pT). Moreover,\nbounding the first term in (7.11) from above by killing particles at γred\nT, we have,\nPN1+ε\n2δ−σ(0)L(T)\u0000\n∃s≤t′(T) ;|A∗\nT,ε\n3(s)|< N\u0001\n≤PN1+ε\n2δ−σ(0)L(T)(M < N ).\nThen, Paley-Zygmund’s inequality yields,\nPN1+ε\n2δ−σ(0)L(T)(M≥N)≥ \n1−N\nEN1+ε\n2δ−σ(0)L(T)[M]!2EN1+ε\n2δ−σ(0)L(T)[M]2\nEN1+ε\n2δ−σ(0)L(T)[M2]\n=\u0000\n1−N−ε\n2p−1\nT\u00012 N1+ε\n2pT\n1−pT+N1+ε\n2pT.\nThis implies PN1+ε\n2δ−σ(0)L(T)(M < N )≤N−ε\n2+o(1)asT→+∞, uniformly in σ∈ S∗\nη, which is the announced\nupper bound for the first term in (7.11).\nWe now turn to the second term in (7.11) . Recall (4.5): in particular, let D(s) denotes the set of white\nparticles that end in the interval [ γred\nT(s) +ε\n2σ(s/T)L(T),γ∗\nT(s)−ε\n2σ(s/T)L(T)] at time s, that is,\nD(s) := A∗\nT,[5ε\n6,h−ε\n2](s).\n5Recall that we can ensure L(T)≤θt′(T) (which is required to apply (7.6)), by choosing θ(·) decaying arbitrarily slowly\nin (6.1–6.2), so that L(T)≤θt′(T)≤θT.56 A. LEGRAND AND P. MAILLARD\nThen, we bound each term of the sum with an union bound, for k≥t′(T),\n(7.12)\nPN1+ε\n2δ−σ(0)L(T)\u0000\n∃s∈[k, k+ 1] ;|A∗\nT,ε\n3(s)|< N\u0001\n≤PN1+ε\n2δ−σ(0)L(T)\u0000\n|D(k)|< N1+ε\n4\u0001\n+PN1+ε\n2δ−σ(0)L(T)\u0000\n|D(k)| ≥N1+ε\n4;∃s∈[k, k+ 1] ;|A∗\nT,ε\n3(s)|< N\u0001\nWe handle those two terms separately. Applying Paley-Zygmund’s inequality, we have\n(7.13) PN1+ε\n2δ−σ(0)L(T)\u0010\n|D(k)| ≥N1+ε\n4\u0011\n≥ \n1−N1+ε\n4\nEN1+ε\n2δ−σ(0)L(T)\u0002\n|D(k)|\u0003!2EN1+ε\n2δ−σ(0)L(T)\u0002\n|D(k)|\u00032\nEN1+ε\n2δ−σ(0)L(T)\u0002\n|D(k)|2\u0003.\nThen, since k≥t′(T)≥θL(T), (7.6) implies that\nEδ−σ(0)L(T)[|D(k)|] =N−ε\n6+o(1),\nasT→+∞. Moreover, (7.7) yields\nEδ−σ(0)L(T)\u0002\n|D(k)|2\u0003\n≤N1+o(1),\nasT→+∞. Noticing that the first moment of |D(k)|is additive in the initial measure, applying the\ndevelopment (6.12) to its second moment and plugging these estimates into (7.13), we finally obtain\n(7.14) PN1+ε\n2δ−σ(0)L(T)\u0010\n|D(k)| ≥N1+ε\n4\u0011\n≥\u0010\n1−N−ε\n12+o(1)\u00112\u0010\n1 +N−ε\n6+o(1)\u0011−1\n= 1−N−ε\n12+o(1),\nwhere o(1) denotes a term vanishing as T→+∞uniformly in t′(T)≤k≤t∗\nTandσ∈ S∗\nη.\nRegarding the second term in (7.12) , letAdenote the set of counting measures supported on [ γred\nT(k) +\nε\n2σ(k/T)L(T),γ∗\nT(k)−ε\n2σ(k/T)L(T)] with total mass at least N1+ε\n4. Using the Markov property at time k,\nwe have\nPN1+ε\n2δ−σ(0)L(T)\u0010\n|D(k)| ≥N1+ε\n4;∃s∈[k, k+ 1],|A∗\nT,ε\n3(s)|< N\u0011\n≤sup\nµ∈APµ\u0000\n∃s≤1,|eA∗\nT,ε\n3(s)|< N\u0001\n,\nwitheA∗\nT,ε\n3(·) being defined similarly to the event A∗\nT,ε\n3(·) for barriers γ∗\nT,γ∗\nTshifted in time by k. Since\nadding particles to the initial measure µonly decreases the probability in the r.h.s. above (this is follows\nfrom a direct coupling argument), on can restrict the supremum to measures µwith total mass exactly N1+ε\n4.\nIf the total population decreases from N1+ε\n4toNat some time s≤1, this implies that, among the initial\nparticles, at least N1+ε\n4−Nof them have no living descendant at time 1. Hence, a union bound yields\n(7.15) sup\nµ∈APµ\u0000\n∃s≤1,|eA∗\nT,ε\n3(s)|< N\u0001\n≤\u0012N1+ε\n4\nN1+ε\n4−N\u0013\n×\u0012\nsup\nyPδy\u0000\n|eA∗\nT,ε\n3(1)|= 0\u0001\u0013N1+ε\n4−N\n,\nwhere the supremum is taken over y∈[γred\nT(k) +ε\n2σ(k/T)L(T),γ∗\nT(k)−ε\n2σ(k/T)L(T)]. For any such y, we\nmay couple the N-BBM starting from ywith a Brownian motion ( Bs)s≥0without reproduction, by looking\nat an arbitrary descendant of y. If the N-BBM starting from δygoes extinct, the coupled Brownian motion\ncrosses one of the barriers γred\nT,γ∗\nTon the time interval [0 ,1]. To do so, it must travel a distance at least\nε\n4σ(0)L(T), uniformly in y: therefore, there exists c, C > 0 such that,\nsup\nyPδy\u0000\n|eA∗\nT,ε\n3(1)|= 0\u0001\n≤sup\nyPy\u0000\n∃s≤1,|Bs−y|>ε\n4σ(0)L(T)\u0001\n≤Ce−cL(T)2,\nwhere the last inequality follows from standard computations on the Brownian motion. Plugging this\ninto(7.15) and using that\u0000a\nb\u0001\n≤aa−bfor any a, b∈N, the second term in (7.12) is bounded from above by\nN−O(L(T)N). Recollecting (7.14) and summing over O(t∗\nT) terms in (7.11) , we finally obtain the announced\nupper bound. □MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 57\nNumber of particles killed by the upper barrier. Let us now turn to the second claim (C-2). We provide an\nestimate on the number of particles reaching the upper barriers before time t∗\nT: first we bound from above\nthe number of white particles that reach γ∗\nT(i.e. white particles which are colored red); and then, for each\nparticle reaching γ∗\nTfor the first time, the number of (red) particles from its offspring that reach γred\nT. Recall\nthe definition of R∗\nT(0, t∗\nT) from (4.6) as well as (7.8).\nClaim 7.3. One has, as T→+∞,\n(7.16) Eµε[|R∗\nT(0, t∗\nT)|]≤Nε\n6+o(1),\nuniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1).\nClaim 7.4. Lett0∈[0, t∗\nT), and consider a BBM starting from one (red) particle in time-space location\n(t0,γ∗\nT(t0)). Then one has, as T→+∞,\n(7.17) Eδ(t0,γ∗\nT(t0))\n\f\f\f\f\f\f[\nt0≤s≤t∗\nT(\nu∈ N r\f\f\f\f\fXu(r)∈(γred\nT(r),γred\nT(r))∀t0≤r < s,\nXu(s) =γred\nT(s))\f\f\f\f\f\f\n≤N−ε\n3+o(1),\nwhere o(1)vanishes as T→+∞uniformly in t0∈[0, t∗\nT)andσ∈ S∗\nη, and locally uniformly in ε∈(0,1).\nProof of Claims 7.3, 7.4. The first result is a direct corollary of (7.8) and the additivity in the initial measure:\nindeed, one has for any 0 ≤k≤ε−1,\nENkε+ε\n2δ−kεσ(0)L(T)[R∗\nT(0, t∗\nT)] =Nkε+ε\n2Eδ−kεσ(0)L(T)[R∗\nT(0, t∗\nT)]≤N(kε+ε\n2)−(kε+ε\n3)+o(1)=Nε\n6+o(1),\nso Claim 7.3 follows by summing over k. Regarding Claim 7.4, it also follows from (7.8) applied to a time-\nand space-shifted BBM on the time interval [ t0, t∗\nT], killed at the barriers γred\nT,γred\nT, and starting from a\nsingle particle at distanceε\n3σ(t0/T)L(T) from the upper barrier γred\nT(we do not write the details again). □\nRemark 7.1. Let us point out that the upper bound in Claim 7.3 is larger than 1, and it is expected that\nthere exist white particles which reach γ∗\nTbefore time t∗\nTwith positive probability. The reader can check\nthat, in the sub-critical and critical regimes, there does not exist a choice of barrier parameters (h, x)and\ninitial configuration µεwhich yields simultaneously Proposition 7.1, and a moment estimate lower than 1 in\nClaim 7.3. Nonetheless let us mention that, in the super-critical regime, The proof of Proposition 7.1 can be\nsimplified (especially (7.11) ), because the time interval length is “short” tsup\nT=T≪L(T)3; and this allows\nfor a proof of Proposition 3.4 which does not use a multi-type N+-BBM.\nFollowing these observations, we may finally prove that γred\nTdoes not kill any particle with high probability.\nLetRred\nT(0, t) denote the number of particles killed by γred\nT. In particular, they remained above γ∗\nTuntil some\ntime τu≤t∗\nTupon which they reached γ∗\nTand were colored red; then they remained above γred\nTthroughout\n[τu, t∗\nT] and reached γred\nTat some time r∈[τu, t] (upon which they are killed).\nProposition 7.5. One has, as T→+∞,\n(7.18) Eµε\u0002\nRred\nT(0, t∗\nT)\u0003\n≤N−ε\n6+o(1),\nuniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1).\nWith this proposition at hand, claim (C-2) is obtained by applying Markov’s inequality to Rred\nT(0, t∗\nT).\nProof. This result is a consequence of Claims 7.3, 7.4 and a bit of stopping lines theory6, which extends\nMarkov stopping time theory to branching Markov processes. In our setting, define\nL:=\b\n(u, s) ;Xu(r)∈(γ∗\nT(r),γ∗\nT(r))∀r < s andXu(s) =γ∗\nT(s)\t\n,\n6For conciseness we do not write every detail here, but the reader can refer to [13, 24, 38] to learn more about this.58 A. LEGRAND AND P. MAILLARD\nwhich is a (random) stopping line . Let\nFL:=σ\n\n\n∀r≤s, Xu(r)> γ∗\nT(r),\n∀r < s, X u(r)<γ∗\nT(r),\nXu(s)∈A\n\n;s≥0, u∈ N s, A∈Bor(R)\n.\nInformally, FLis the sigma-algebra containing all information about white particles. Then the strong\nbranching property [38, Theorem 4.14] states that, conditionally on FL, the sub-trees of the process rooted\nat the pairs ( u, s)∈ Lare independent with respective distributions Pδ(s,Xu(s))=Pδ(s,γ∗\nT(s)).\nNotice that Claim 7.3 implies |L|=R∗\nT(0, t∗\nT)<+∞,Pµε-almost surely. Moreover, any particle in\nRred\nT(0, t∗\nT) almost surely has a single ancestor ( u, τ)∈ L—in particular this ancestor uwas colored red at\ntime τ. Therefore, we obtain by conditioning with respect to FLand applying the strong branching property,\nEµε\u0002\nRred\nT(0, t∗\nT)\u0003\n=Eµε\nEµε\nX\n(u,τ)∈L\f\f\f\f\f[\nτ≤r≤t∗\nT(\nv∈ N r, v≽u\f\f\f\f\fXv(s)∈(γred\nT(s),γred\nT(s))∀s∈[τ, r),\nXv(r) =γred\nT(r))\f\f\f\f\f\f\f\f\f\f\fFL\n\n\n=Eµε\nX\n(u,τ)∈LEδ(τ,γsub\nT(τ))\n\f\f\f\f\f\f[\nτ≤t≤t∗\nT(\nv∈ N r\f\f\f\f\fXv(s)∈(γred\nT(s),γred\nT(s))∀s∈[τ, r),\nXv(r) =γred\nT(r))\f\f\f\f\f\f\n\n,\nwhere v≽umeans that v∈ N ris a descendant of u∈ N τ,τ≤r. Plugging Claims 7.3, 7.4 into this, we\nfinally obtain\nEµε\u0002\nRred\nT(0, t∗\nT)\u0003\n≤Eµεh\nN−ε\n3+o(1)×\f\fL\f\fi\n=N−ε\n3+o(1)×Nε\n6+o(1)=N−ε\n6+o(1),\nwhich concludes the proof. □\nParticle distribution at the final time. Finally, we prove the third statement (C-3). We first provide the\nfollowing two estimates, respectively for white and red particles.\nLemma 7.6. The following statements hold uniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1).\n(i)For0≤j, k≤ε−1, one has, as T→+∞,\n(7.19) inf\ns∈[θ(T)−1L(T),t∗\nT]PNkε+ε\n2δ−kεσ(0)L(T)\u0010\nN(j+1)ε≤\f\f\fA∗\nT,[h−(j+1)ε,h−jε](s)\f\f\f≤N(j+2)ε\u0011\n≥1−N−ε\n6+o(1).\n(ii)Lett0∈[0, t∗\nT), and consider a BBM starting from one (red) particle in time-space location (t0,γ∗\nT(t0)).\nThen for 0≤j≤ε−1, one has, as T→+∞,\n(7.20) sup\ns∈[t0,t∗\nT]Eδ(t0,γ∗\nT(t0))\"\f\f\f\f\f(\nu∈ N s\f\f\f\f\fXu(r)∈(γred\nT(r),γred\nT(r))∀t0≤r < s,\nXu(s)−γ∗\nT(s)\nσ(s)L(T)∈[h−(j+ 1)ε, h−jε])\f\f\f\f\f#\n≤N(j+1)ε+o(1).\nFrom these results, the claim (C-3) follows naturally: let ( Xwhite−red\ns )s∈[0,t∗\nT]denote the empirical mass\nmeasure on Rof the process defined by white and red particles, that is\n(7.21) Xwhite−red\ns :=X\nu∈Nwhites∪NredsδXu(s), s ∈[0, t∗\nT].\nRecall (7.9). In order to have a condensed statement, let us write µ(y)\nε,sfor the counting measure µε,sshifted\nupward by y, that is µ(y)\nε,s(·) :=µε,s(· −y), for s∈[0, T],y∈R.\nProposition 7.7. Letε >0. Then one has as T→+∞,\n(7.22) inf\ns∈[θ(T)−1L(T),t∗\nT]Pµε\u0010\nµ(γ∗\nT(s)−εσ(s)L(T))\nε,s ≺ Xwhite−red\ns ≺N5\n2εµ(γ∗\nT(s))\nε,s\u0011\n≥1−N−ε\n6+o(1),\nuniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1).MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 59\nProof of Lemma 7.6. (i) Recall (7.1) and (7.5–7.7). On the one hand, Markov’s inequality gives\nPNkε+ε\n2δ−kεσ(0)L(T)\u0010\f\f\fA∗\nT,[h−(j+1)ε,h−jε](s)\f\f\f> N(j+2)ε\u0011\n≤Nkε+ε\n2N−(j+2)εEδ−kεσ(0)L(T)h\f\f\fA∗\nT,[h−(j+1)ε,h−jε](s)\f\f\fi\n≤Nkε+ε\n2N−(j+2)εNx−kε−h+(j+1)ε+o(1)=N−5\n6ε+o(1),\nfor 0≤k, j≤ε−1,s≤t∗\nTandTlarge. On the other hand, Paley-Zygmund’s inequality and (7.5),(7.7)\nyield for s∈[θ(T)−1L(T), t∗\nT], (we leave the details to the reader),\nPNkε+ε\n2δ−kεσ(0)L(T)\u0010\f\f\fA∗\nT,[h−(j+1)ε,h−jε](s)\f\f\f≥N(j+1)ε\u0011\n≥\u0010\n1−N−ε\n6+o(1)\u00112\u0010\n1 +N−ε\n6+o(1)\u0011−1\n≥1−N−ε\n6+o(1),\nfor some c1, c2>0 and Tsufficiently large, which concludes the proof of (7.19).\n(ii) Similarly to Claim 7.4, this follows from (7.5) applied to a time- and space-shifted BBM on the\ntime interval [ t0, t∗\nT], killed at the barriers γred\nT,γred\nT, and starting from a single particle at distance\n(h−ε\n3)σ(t0/T)L(T) from the lower barrier γred\nT(we leave the details to the reader). □\nProof of Proposition 7.7. Recall Claim 7.3 and (7.20) . Using the strong branching property [ 38, Theo-\nrem 4.14], one deduces uniformly in 0 ≤j≤ε−1,s∈[θ(T)−1L(T), t∗\nT],\nEµεh\f\f\fn\nu∈ Nred\ns;Xu(s)−γ∗\nT(s)\nσ(s)L(T)∈\u0002\n−(j+ 1)ε,−jε\u0003o\f\f\fi\n≤Nε\n6+(j+1)ε+o(1).\nHence, a union bound and Markov’s inequality yield,\nPµε\u0010\n∃0≤j≤ε−1;\f\f\fn\nu∈ Nred\ns;Xu(s)−γ∗\nT(s)\nσ(s)L(T)∈[−(j+ 1)ε,−jε]o\f\f\f> N(j+2)ε\u0011\n≤N−5\n6ε+o(1).\nMoreover, a union bound and (7.19) yield that, uniformly in s∈[θ(T)−1L(T), t∗\nT],\nPµε\u0010\n∃0≤j≤ε−1;\f\f\fn\nu∈ Nwhite\nt∗\nT;Xu(t∗\nT)−γ∗\nT(t∗\nT)\nσ(t∗\nT)L(T)∈\u0002\n−(j+ 1)ε,−jε\u0003o\f\f\f/∈(ε−1+ 1)\u0002\nN(j+1)ε, N(j+2)ε\u0003\u0011\n≤ε−1X\nj=0ε−1X\nk=0sup\ns∈[1\n2t∗\nT,t∗\nT]PNkε+ε\n2δ−kεσ(0)L(T)\u0010\f\f\fA∗\nT,[h−(j+1)ε,h−jε](s)\f\f\f/∈\u0002\nN(j+1)ε, N(j+2)ε\u0003\u0011\n≤N−ε\n6+o(1),\nforTlarge. Therefore, there exists Tε>0 such that for T≥Tε, with large Pµε-probability, for all\n0≤j≤ε−1, there are between N(j+1)εandN(j+3)εparticles (white or red) ending in the interval\nγ∗\nT(s) +σ(s)L(T)[−(j+ 1)ε,−jε] at time s, uniformly in s∈[θ(T)−1L(T), t∗\nT]. Recalling (7.9), this directly\nimplies the proposition. □\n7.2.Proof of Proposition 3.4. We may finally display the proof of Proposition 3.4. It shares several\nsimilarities to that of Lemma 6.4, notably with the sub-critical case needing additional work (since our\nmoment estimates do not hold until time T). Recall that the point measure of the N-BBM throughout time\nis denoted ( XN\nt)t≥0. Let ε∈(0,1),∗ ∈ { sub,sup,crit}, and recall (6.1),(7.1) and the definitions of γ∗\nT,γ∗\nT\nfrom (4.8–4.10).\nLemma 7.8. Letε∈(0,1)and∗ ∈ { sub,sup,crit}. There exists c1, c2>0such that for Tsufficiently large,\none has\n(7.23) inf\nt∈[0,t∗\nT]Pµε\u0000\nmax(XN\nt)≤γred\nT(t)\u0001\n≥1−c1N−c2,\nand\n(7.24) inf\nt∈[θ(T)−1L(T),t∗\nT]Pµε\u0010\nXN\nt≺µ(γ∗\nT(t)+3εσ(t/T)L(T))\nε,t\u0011\n≥1−c1N−c2,\nwhere c1, c2are uniform in σ∈ S∗\nηand locally uniform n ε∈(0,1).60 A. LEGRAND AND P. MAILLARD\nLet us mention that the proof of Theorem 1.1 only requires (7.23) ; however (7.24) is obtained with the\nsame method and is needed to prove Proposition 1.4 in Section 8.\nProof of Lemma 7.8. Let us start with (7.23) . Recall from (7.3–7.4) and (7.21) that (Xwhite−red\ns )s≥0denotes\nthe empirical mass measure of the process containing both white and red particles. Notice that Proposition 7.5\nand Markov’s inequality imply that, as T→+∞,\n(7.25) Pµε\u0010\n∃s∈[0, t∗\nT],∃u∈ Nwhite\ns∪ Nred\ns;Xu(s) =γred\nT(s)\u0011\n≤c1N−c2,\nfor some c1, c2>0, uniform in σ∈ S∗\nηand locally uniform in ε∈(0,1). Furthermore, Proposition 7.1\nimplies that, with probability larger than 1 −c1N−c2, there are Nparticles above γred\nT(·) at all time t≤t∗\nT.\nRecalling Lemma 4.6 and Remark 4.2, on the intersection of these two events, ( Xwhite−red\ns )s≥0matches the\ntrajectory of a (multi-type) N+-BBM with large Pµε-probability. This implies that, for some c1, c2>0,\nPµε\u0000\nmax(XN\nt)>γred\nT(t)\u0001\n≤Pµε\u0000\nmax(Xwhite−red\nt )>γred\nT(t)\u0001\n+ 2c1N−c2= 2c1N−c2,\nforTsufficiently large, uniformly in t≤t∗\nT; this proves (7.23).\nFurthermore, the same coupling argument and Proposition 7.7 yield\ninf\ns∈[1\n2t∗\nT,t∗\nT]Pµε\u0010\nXN\ns≺N5\n2εµ(γ∗\nT(s))\nε,s\u0011\n≥1−3c1N−c2,\nfor large T, uniformly in σ∈ S∗\nηand locally uniformly in ε∈(0,1). Moreover, XN\nscontains at most N\nparticles by definition, whereas µε,scontains strictly more. Hence, recalling the definition of µε,s(7.9) and\nshifting it upward by 3 εσ(s/T)L(T), this finally implies (7.24). □\nProof of Proposition 3.4, N(T)super-critical or critical. In the super-critical and critical regimes, Proposi-\ntion 3.4 follows directly from Lemma 7.8 (more precisely (7.23) ) and Lemma 4.1. Indeed, recall (7.1–7.2)\nand that t∗\nT=Tin those regimes (see (6.1)). Recalling the notation γ∗,h,x\nT,h > x > 0, one has\nγred\nT(T) =γ∗,1+ε,1+2ε\n3\nT (T) +ε\n3σ(1)L(T).\nLetting εarbitrarily small and applying Lemmata 7.8 and 4.1, this implies Proposition 3.4 in both regimes. □\nWe now turn to the sub-critical regime. In a similar manner to Section 6.2, we split the interval [0 , T]\ninto blocks of length1\n2tsub\nT: more precisely, let K:=⌊2T/tsub\nT⌋, and for 0 ≤k≤K−1, let tk:=k\n2tsub\nT,\nandtK=T(sotK−tK−1∈[1\n2tsub\nT, tsub\nT]). However, conversely to Section 6.2, a first moment method is\nsufficient to prove Proposition 3.4 (no second moment estimate is required), and this proof holds throughout\nthe sub-critical regime (so there is no need to handle the super-polynomial case separately).\nProof of Proposition 3.4, N(T)sub-critical. We define an auxiliary process ( bXN\nt)t∈[0,T]as follows: it starts\nfrom Nδ0and evolves as the process XNbetween times tkandtk+1. Then at each time tk, all particles are\ndisplaced to the highest among their positions: in other words, a configuration µis replaced with Nδmax( µ)\n(notice that bXNalways contains exactly Nparticles). By a coupling argument (recall Proposition 3.2) and\nan induction, one may construct a coupling such that XN\nT≺bXN\nTwith probability 1. In particular, it is\nsufficient to prove the proposition with max( bXN\nT) instead of max( XN\nT).\nFor 1≤k≤K, let us define\nYk:= max( bXN\ntk)−max(bXN\ntk−1),\nso that max( bXN\nT) =Y1+···+YK. Let us prove that, for 1 ≤k≤K, one has\n(7.26) E[Yk]≤γsub\nT(tk)−γsub\nT(tk−1) +c1L(T),\nfor some c1>0, and we claim that the Proposition 3.4 follows. Indeed, this implies\nE[max(bXN\nT)]≤γsub\nT(T) +c1L(T)×K .MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 61\nRecalling that K≤2T/tsub\nTand that tsub\nT≫L(T)3(see (6.2)), Markov’s inequality yields that\nmax(bXN\nT)≤γsub\nT(T) +oP\u0012T\nL(T)2\u0013\n.\nRecall (4.9) and that max(bXN\nT) dominates stochastically max(XN\nT): hence, letting ε→0 and applying\nLemma 4.1, this finally yields (3.7).\nLet us prove (7.26) . Notice that the process bXNevolves between times tkandtk+1as the N-BBM with\nvariance profile σ(tk/T+·), and that bXN\ntk=Nδmax(bXN\ntk)for all 0 ≤k < K . Thus it is enough to show that\n(7.27) ENδ0[maxXN\nt]≤Eµε[maxXN\nt] +η−1L(T)≤γsub\nT(t) +c1L(T),\nfor some c1>0 uniform in σ∈ Sηandt∈[1\n2tsub\nT, tsub\nT]. The first inequality in (7.27) is obtained by writing\nµε≻Nδ−η−1L(T)and translating the N-BBM by η−1L(T), so we only have to prove the second one. Recall\nfrom Proposition 3.1 that there exists a coupling between XNand a BBM ( Xt)t∈[0,T]without selection, such\nthatXN\nt⊂ X ta.s. for all t∈[0, T]. Thus for t∈[1\n2tsub\nT, tsub\nT], one has\n(7.28) Eµε\u0002\nmaxXN\nt\u0003\n≤Eµε\u0002\nmaxXN\nt1{maxXN\nt≤γred\nT(t)}\u0003\n+Eµε\u0002\nmaxXt1{maxXN\nt>γred\nT(t)}\u0003\n.\nThe first term from (7.28) is clearly bounded by γred\nT(t)≤γsub\nT(t) +η−1L(T), so it remains to bound the\nsecond term. Let C >0 a large constant: then one has\nEµε\u0002\nmaxXt1{maxXN\nt>γred\nT(t)}\u0003\n≤C(tsub\nT)2Pµε\u0000\nmaxXN\nt>γred\nT(t)\u0001\n+Eµε\u0002\nmaxXt,1{maxXt>C(tsub\nT)2}\u0003\n.\nRecalling Lemma 7.8, more precisely (7.23) , the first term in the r.h.s. above is bounded by C(tsub\nT)2×c1N−c2\nuniformly in t∈[1\n2tsub\nT, tsub\nT]. Moreover, (6.1) implies that tsub\nT≤L(T)4, so this term vanishes when T\nbecomes large.\nFinally, let us prove that Eµε[maxXt1{maxXt>C(tsub\nT)2}] also vanishes as T→+∞, uniformly in t∈\n[1\n2tsub\nT, tsub\nT] and locally uniformly in ε. Recalling Proposition 3.2, it is sufficient to prove this for a BBM\nstarting from the configuration N2δ0≻µε. Moreover, we claim that for any M≥2,A≥1, and ( gi)1≤i≤Ma\ncentered Gaussian vector such that each gihas variance ρ2>0, one has\n(7.29) E\u0002\nmax( gi,1≤i≤M)1{max( gi,1≤i≤M)≥A}\u0003\n≤Mρ√\n2π\u0000\n1 +ρ2/A2\u0001\ne−A2/2ρ2.\nThis result is standard, and can be proven similarly to Lemma 6.7 by writing for A≥0 and any real random\nvariable Y,\nE[Y1{Y≥A}] =AP(Y≥A) +Z+∞\nAP(Y≥x) dx .\nFor the sake of conciseness we leave the details to the reader. Therefore, conditioning Xwith respect to its\nbranching epochs and letting Ztdenote its population size at time t∈[1\n2tsub\nT, tsub\nT], we obtain\nEN2δ0\u0002\nmaxXt1{maxXt>C(tsub\nT)2}\u0003\n≤EN2δ0\u0002\nmaxXt1{maxXt>C(tsub\nT)2}\u0003\n≤c1×EN2δ0\u0002\nZt\u0003\n×tsub\nTe−c2(tsub\nT)2,\nfor some c1, c2>0, where the first inequality follows from Proposition 3.2. Moreover, one has EN2δ0[Zt] =\nN2et/2≤N2etsub\nT, so this concludes the proof of (7.27). □\n8.Proofs of complementary results\nIn this section we complete the proofs of all remaining statements from Section 1, by showing Propo-\nsitions 1.2, 1.4, 1.5 (the latter and (1.16) implying Theorem 1.6), and finally Theorem 1.7. All of these\nare either obtained through refinements of arguments from the proof of Theorem 1.1 presented before; or\nthey are direct applications of Theorem 1.1, coupling propositions or other results from previous sections.\nTherefore, let us warn the reader that most of the upcoming proofs are not presented in full details. Indeed,62 A. LEGRAND AND P. MAILLARD\nthe authors believe that understanding the pivotal arguments from the proof of Theorem 1.1, specifically\nProposition 3.2 and the main ideas from Sections 6–7, is vital before moving to other results. Hence, the\nfocus of this section is put on additional, new arguments which are to be combined with those presented\nbefore.\n8.1.Super-critical L(T), decreasing variance (Proposition 1.2). Letσ∈ C2([0,1]) be strictly decreas-\ning, let L(T) =logN(T)≫T1/3(so the regime is super-critical), and consider the initial configuration δ0.\nIn [46], the authors study the speed of a time-inhomogeneous BBM ( Xt)t∈[0,T], without selection and with\nstrictly decreasing variance. Recall that a 1denotes the absolute value of the largest zero of Ai. Starting\nfrom a single particle in 0 (so QT(δ0) = 0), they prove in [46, Theorem 1.1] that, under those assumptions,\n(8.1)\u0012\nmax(XT)−v(1)T+a1\n21/3T1/3Z1\n0σ(u)1/3|σ′(u)|2/3du+σ(1) log( T)\u0013\nT≥0is tight.\nThus, Proposition 1.2 can be deduced from the couplings from Propositions 3.1 and 3.2, by comparing\nthe super-critical N-BBM with a critical BBM on the one hand, and with a BBM without selection on the\nother hand. For α∈R, recall the definitions of mcrit\nT=mcrit\nT(α) and msup-d\nT from (1.11) . Then, recalling the\nasymptotic properties of Ψ( ·) (see (1.4)) and that σ′(·) is negative, one notices that,\nlim\nα→+∞lim sup\nT→+∞T−1/3\f\f\fmcrit\nT(α)−msup-d\nT\f\f\f= 0.\nDefine Mα(T) :=exp(αT1/3): for Tsufficiently large, one has Mα(T)≤N(T). By Propositions 3.1–3.2, for\nTlarge, one can construct two couplings with an Mα(T)-BBM and a BBM without selection (all started\nfrom δ0), such that\nXMα(T)\ns ≺ XN(T)\ns ≺ X s,∀s∈[0, T].\nRecall that, for λ >0, Proposition 3.3 yields\nlim\nT→+∞Pδ0\u0010\nT−1/3\u0010\nmax\u0000\nXMα(T)\nT\u0001\n−mcrit\nT(α)\u0011\n≤ −λ\u0011\n= 0.\nMoreover, the result (8.1) directly implies,\n(8.2) lim\nT→+∞Pδ0\u0010\nT−1/3\u0000\nmax(XT)−msup-d\nT\u0001\n≥λ\u0011\n= 0.\nCombining these estimates with the coupling above, and assuming that αwas chosen sufficiently large\n(depending on λ), we conclude that T−1/3\u0000\nmax(XN(T)\nT)−msup-d\nT\u0001\nconverges to 0 in Pδ0-probability as\nT→+∞. □\nRemark 8.1. Let us mention that [46]makes strong assumptions ( σstrictly decreasing and initial con-\nfiguration δ0) in order to obtain a sharp result in (8.1). If the convergence (8.2) were to be proven with σ\nnon-increasing and a generic initial configuration, then Proposition 1.2 and its proof would immediately\nextend to those more general assumptions. This is further discussed in the proof of Proposition 1.5 below.\n8.2.Final-time distribution for the critical and super-critical regimes (Proposition 1.4). Let\n∗ ∈ { crit,sup}(in particular t∗\nT=T) and η >0 throughout this section. We first prove (1.13) , then we use\nit to deduce (1.14).\nLetλ >0, recall the couplings from Propositions 3.1, 3.2, and recall from Section 3.3 how they imply\nTheorem 1.1 subject to Propositions 3.3, 3.4. The same coupling method can be used to deduce (1.13) from\nthe following lemma.\nLemma 8.1. Let∗ ∈ { crit,sup}, and λ, η > 0. Then, one has\n(8.3) lim\nε→0lim sup\nT→+∞sup\nσ∈S∗ηPµε\u0010\n∃y∈[0,1] ;XN(T)\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n≥N(T)y+λ\u0011\n= 0,MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 63\nand, for any fixed r∈(0,1),\n(8.4) lim\nT→+∞sup\nκ∈[0,1]sup\nσ∈S∗ηPNκδ−κσ(0)L(T)\u0010\n∃y∈[r,1] ;XN(T)\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n≤N(T)y−λ\u0011\n= 0.\nProof of (1.13) subject to Lemma 8.1. Using arguments very similar to the ones from the proof of Theo-\nrem 1.1 in Section 3.3, we obtain from Lemma 8.1 that for every fixed r∈(0,1), as T→ ∞\n(8.5) sup\ny∈[r,1]\f\f\f\f\flogXN(T)\nT([QT(µT) +m∗\nT−yσ(1)L(T),+∞))\nL(T)−y\f\f\f\f\f−→ 0, inPµT-probability.\nIn particular, we have XN(T)\nT([QT(µT) +m∗\nT−rσ(1)L(T),+∞))≥1 with high probability, so that we can\nreplace log by log+. It remains to consider y∈[0, r]. We have\nsup\ny∈[0,r]\f\f\f\f\flog+XN(T)\nT([QT(µT) +m∗\nT−yσ(1)L(T),+∞))\nL(T)−y\f\f\f\f\f\n≤log+XN(T)\nT([QT(µT) +m∗\nT−rσ(1)L(T),+∞))\nL(T)+r,\nso that (8.5) implies that the latter goes to 2 rinPµT-probability. Letting r→0, we obtain the result. □\nProof of Lemma 8.1. We first prove (8.3), i.e. the upper bound on the final-time distribution of the process.\nLetε∈(0,1),λ >0 and recall the definition of µε,s,s∈[0, T] from (7.9), and of m∗\nTfrom (1.11) . By\nLemma 4.1, one has for εsufficiently small and Tlarge that,\n(8.6)1\nL(T)\f\fγ∗,1+ε,1+2\n3ε\nT (T)−m∗\nT\f\f≤λ\n2,\nwhere we wrote the parameters of the barrier γ∗\nTexplicitly. Then, recall Proposition 7.7, and that we showed\nin (7.24) that it implies for Tsufficiently large,\ninf\nσ∈S∗ηPµε\u0010\nXN\nT≺µ(γ∗\nT(T)+3εσ(1)L(T))\nε,T\u0011\n≥1−c1N−c2,\nwhere c1,c2are constants depending locally uniformly on ε; and γ∗\nT(·) is defined in (4.8) and(4.10) (with\nparameters h= 1 + ε,x= 1 +2\n3ε). In particular, for λ >0,\nsup\nσ∈S∗ηPµε\u0000\n∃y∈[0,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n≥N(T)y+λ\u0001\n(8.7)\n≤sup\nσ∈S∗ηPµε\u0010\n∃y∈[0,1] ;µ(γ∗\nT(T)+3εσ(1)L(T))\nε,T\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n≥N(T)y+λ\u0011\n+c1N−c2,\nFinally, for y∈[0,1], one has\nµ(γ∗\nT(s)+3εσ(s/T)L(T))\nε,s\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n=ε−1X\nk=0Nkε+ε\n21{γ∗\nT(s)+3εσ(s/T)L(T)−kεσ(1)L(T)≥m∗\nT−yσ(1)L(T)}\n≤yε−1+3η−2+ε−1λ\n2 X\nk=0Nkε+ε\n2≤(ε−1(1 +λ/2) + 3 η−2)Ny+4η−2ε+λ\n2,\nwhere we used (8.6). Assuming εis sufficiently small above (depending on λandη), this implies that the\nr.h.s. of (8.7) is equal to c1N−c2forTsufficiently large: this completes the proof of (8.3) in the super-critical\nand critical regimes.64 A. LEGRAND AND P. MAILLARD\nWe now turn to (8.4). Quite similarly to Proposition 3.3, let us first prove that the convergence holds\nlocally uniformly in κ∈(0,1), then extend it to κ∈[0,1] via coupling arguments (recall Lemma 6.4 from\nSection 6.2). Let ε∈(0,1),h= 1−ε,x= (1−ε)(1−κ), and recall Proposition 6.2. Replicating the coupling\narguments from Section 6.2, more precisely (6.15–6.16), one obtains for 1 ≤j≤ε−1andTsufficiently large,\nPNκδ0\u0010\nXN\nT\u0000\u0002\nγ∗\nT(T)−jεσ(1)L(T),+∞\u0001\u0001\n≥Njε\u0011\n≥1−c1N−c2,\nwhere c1, c2>0 are constants uniform in σ∈ S∗\nη, 1≤j≤ε−1, and locally uniform in ε, κ∈(0,1) (we do\nnot reproduce the details). Writing a union bound, this yields\n(8.8) PNκδ0\u0010\n∀1≤j≤ε−1,XN\nT\u0000\u0002\nγ∗\nT(T)−jεσ(1)L(T),+∞\u0001\u0001\n≥Njε\u0011\n≥1−ε−1c1N−c2.\nLetε′>0: recalling (1.11) and Lemma 4.1, one has for εsufficiently small and Tlarge that,\n(8.9)1\nL(T)\f\fγ∗,1−ε,(1−ε)(1−κ)\nT (T)−κσ(0)L(T)−m∗\nT\f\f≤ε′,\nwhere we used that γ∗,1−ε,(1−ε)(1−κ)\nT (T)−κσ(0)L(T) =γ∗,1−ε,1−ε(1−κ)\nT (T). Moreover, for y∈[ε+ε′η,1],\nthere exists 1 ≤j≤ε−1such that ( y−ε′η−1)ε−1∈[jy, jy+ 1); thus we deduce that,\nn\n∃y∈[ε+ε′η−1,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n< Ny−λo\n⊂n\n∃1≤j≤ε−1;XN\nT\u0000\u0002\nγ∗,1−ε,(1−ε)(1−κ)\nT (T)−κσ(0)L(T)−jεσ(1)L(T),+∞\u0001\u0001\n< N(j+1)ε+ε′η−1−λo\n.\nAssume that ε′, εare sufficiently small so that ε+ε′η−1≤min(r, λ). Then, shifting the initial distribution\nin (8.8) by −κσ(0)L(T), this implies\n(8.10) PNκδ−κσ(0)L(T)\u0010\n∃y∈[r,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n< Ny−λ\u0011\n≤ε−1c1N−c2,\nuniformly in σ∈ S∗\nηand locally uniformly in κ∈(0,1), which is the expected result.\nTo finish the proof of Lemma 8.1, it remains to extend (8.10) toκclose to 0 or 1, which is achieved very\nsimilarly to the proof of Proposition 3.3 subject to Lemma 6.4 in Section 6.2. For the case κlarge, assume\nwithout loss of generality that 0 < r < ε < λ , then recall Proposition 3.2, and that for any κ∈[1−εη−2,1],\none has N1−εη−2δ−σ(0)L(T)≺Nκδ−κσ(0)L(T). Hence,\nPNκδ−κσ(0)L(T)\u0010\n∃y∈[r,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n< Ny−λ\u0011\n≤PN(1−εη−2)δ−(1−εη−2)σ(0)L(T)\u0010\n∃y∈[r,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T) +εη−2σ(0)L(T),+∞\u0001\u0001\n< Ny−λ\u0011\n≤PN(1−εη−2)δ−(1−εη−2)σ(0)L(T)\u0010\n∃y∈[r−ε,1] ;XN\nT\u0000\u0002\nm∗\nT−yσ(1)L(T),+∞\u0001\u0001\n< Ny−(λ−ε)\u0011\n,\nwhere we also shifted the process upward by εη−2σ(0)L(T) in the first inequality. Applying (8.10) to the\nlatter (with κ= 1−εη−2), this proves that the convergence also holds uniformly in κclose to 1.\nFinally, it only remains to treat the case κsmall. Recall Lemma 6.5, which implies that, for an N-BBM\nstarted from δ0, one has XN\nεL(T)≻Nε/4δ−2εη−1L(T)withPδ0-probability close to 1. Letting κ∈[0, ε], writing\nδ−εσ(0)L(T)≺Nκδ−κσ(0)L(T)and applying the Markov property at time εL(T), one can again extend the\nconvergence uniformly to κ∈[0, ε] through Proposition 3.2: since this is a carbon copy of arguments\npresented in Section 6.2, we leave the details to the reader. □\nRemark 8.2. Let us mention that the proof above can be applied to the sub-critical regime, yielding an\nestimate on the distribution of the process at any time t∈[θ(T)−1L(T), tsub\nT]. Unfortunately, the error term\no(T/L(T)2)in our maximal displacement result renders this obsolete at time T. If one manages to compute\nsharper estimates on the maximal displacement of the N-BBM with sub-critical population (with an error at\nmost o(L(T))), the arguments above can be adapted straightforwardly. However, it seems very unlikely that\nsuch a sharp limit estimate would be non-random: instead we expect it to be obtained through martingaleMAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 65\nmethods or similar arguments, with a non-trivial dependency on the realization of the process near time t= 0\nand on the “peaking” events throughout the trajectory (i.e. when a particle rises and reproduces significantly:\nsee[43]for a study of those in the case of an N-BBM with constant N, time-homogeneous variance and in a\nmeta-stable regime).\nProof of (1.14) .We start with the lower bound on the diameter. Let δ >0, and notice that Theorem 1.1\nimplies for ∗ ∈ { crit,sup},\nPµT\u0000\nmax(XN(T)\nT)−min(XN(T)\nT)≤(1−δ)σ(1)L(T)\u0001\n≤PµT\u0000\nmin(XN(T)\nT)≥m∗\nT−(1−δ/2)σ(1)L(T)\u0001\n+o(1)\n=PµT\u0000\nXN(T)\nT\u0000\n[m∗\nT−(1−δ/2)σ(1)L(T),+∞)\u0001\n≥N\u0001\n+o(1),\nand by (1.13), the latter goes to 0 as Tgoes to + ∞, for any δ >0.\nRegarding the upper bound on the diameter, it suffices to prove that\n(8.11) PµT\u0010\nmin(XN(T)\nT)≥m∗\nT−(1 +δ)σ(1)L(T)\u0011\n=o(1)\nasT→+∞for any δ >0, and the result follows similarly from Theorem 1.1. Recall the couplings from\nPropositions 3.1 and 3.2.\nLemma 8.2. Let∗ ∈ { crit,sup}, and consider XN(T)\nT−δ3L(T)the particle configuration of the N-BBM at time\nT−δ3L(T). Ifδis sufficiently small, then with probability close to 1, no particle below m∗\nT−(1 +δ)σ(1)L(T)\nat time T−δ3L(T)has a living descendant in XN(T)\nT.\nWith this lemma at hand, the proof of (8.11) is straightforward. Write\n(8.12) A:=XN(T)\nT−δ3L(T)∩\u0002\nm∗\nT−(1 +δ)σ(1)L(T),+∞\u0001\n.\nApplying the Markov property at time T−δ3L(T) and Lemma 8.2, one has\nPµT\u0010\nmin(XN(T)\nT)≥m∗\nT−(1 + 2 δ)σ(1)L(T)\f\fXN(T)\nT−δ3L(T)\u0011\n=P(T−δ3L(T),XN(T)\nT−δ3L(T))\u0010\nmin(XN(T)\nT)≥m∗\nT−(1 + 2 δ)σ(1)L(T)\u0011\n=PA\u0010\nmin(eXN(T)\nδ3L(T))≥m∗\nT−(1 + 2 δ)σ(1)L(T)\u0011\n+o(1),\nwhere eXN(T)is an N-BBM with diffusion σ(s/T+ 1−δ3L(T)/T),s∈[0, δ3L(T)].\nIn Proposition 3.2.( iii), we proved that the N-BBM started from Adominates stochastically a collection\n(Bi\nt)t≥0, 1≤i≤#Aof (non-branching) independent Brownian motions started from Bi\n0=m∗\nT−(1 +\nδ)σ(1)L(T) with the same diffusion function. Notice that in general, having µ≺ν,µ, ν∈Cdoes not allow\nfor a comparison between min(µ) and min(ν); however, with a direct adaptation of the coupling therein,\none can prove that min(eXN(T)\nδ3L(T)) started from Adominates stochastically min(Bi\nδ3L(T), i≤N) started from\nNδm∗\nT−(1+δ)σ(1)L(T)(we leave the details to the reader). By standard Gaussian estimates, one has\nP\u0000\n∃i≤N(T), Bi\nδ3L(T)−Bi\n0≤ −δL(T)\u0001\n≤N(T)P(Bi\nδ3L(T)−Bi\n0≤ −δL(T)) (8.13)\n≤eL(T)×cp\nδ/L(T)e−cδ−1L(T),\nfor some c >0 uniform in σ∈ Sη, and this vanishes as T→+∞as soon as δ >0 is small enough. This\nconcludes the proof of (8.11), and thus of (1.14) subject to Lemma 8.2. □\nProof of Lemma 8.2. Let us define, similarly to (8.12),\nA1:=XN(T)\nT−δ3L(T)∩\u0002\nm∗\nT−(1 +δ/3)σ(1)L(T),+∞\u0001\n,\nand A2:=XN(T)\nT−δ3L(T)∩\u0000\n− ∞, m∗\nT−(1 +δ)σ(1)L(T)\u0003\n,66 A. LEGRAND AND P. MAILLARD\nand let us consider two BBMs eX1,eX2without selection started respectively from A1andA2. Then we\nprove that the following claims hold with large probability:\n(i) Throughout [0 , δ3L(T)] the line m∗\nT−(1 + 2 δ/3)σ(1)L(T) is crossed by no particle from eX1oreX2.\n(ii) At time δ3L(T),eX1contains more than Nparticles.\nThen Lemma 8.2 follows naturally. Indeed, recall the coupling between the N-BBM and BBM from\nProposition 3.1: reproducing that construction procedure, it is clear from ( i) and ( ii) that, when the process\neX1reaches a population of Nparticles, all descendants from eX2have been selected out of the N-BBM in\nthe procedure.\nLet us first prove ( i) foreX2(and the proof for eX1is obtained by symmetry). Recalling Proposition 3.2.( i),\nit is sufficient to prove it for an initial configuration Nδm∗\nT−(1+δ)σ(1)L(T)≻ A 2. To lighten notation, let us\nshift the initial configuration by m∗\nT−(1 + 2 σ/3)(1)L(T), and let R0denote the number of particles from the\nBBMeX2that reach 0 before δ3L(T). Then, using a union bound, Markov’s property and the Many-to-one\nlemma [36, Theorem 4.1], one has\nPNδ−δσ(1)L(T)/3(∃s≤δ3L(T),max(eX2\ns)≥0)≤NE−δσ(1)L(T)/3[R0]\n≤N eδ3L(T)/2P−δσ(1)L(T)/3(∃s≤δ3L(T), Bs≥0)\n=N eδ3L(T)/2P0(|Bδ3L(T)| ≥δσ(1)L(T)/3),\nwhere Bis a time-inhomogeneous Brownian motion with diffusion σ(s/T+ 1−δ3L(T)/T),s∈[0, δ3L(T)].\nReproducing standard Gaussian estimates as in (8.13) (we do not write the details again), the latter\nprobability is smaller than N(T)−2forTlarge, provided δwas chosen small enough. This finishes the proof\nof (i).\nRegarding ( ii), notice that there exists c >0 such that, for Tsufficiently large and all σ∈ Sη,\n|m∗\nT−m∗\nT−δ3L(T)| ≤c δ3L(T),and σ(1)−σ(1−δ3L(T)/T)≤cδ3L(T)/T.\nTherefore, Lemma 8.1 (more specifically (8.4)) implies that, with large probability, A1contains at least N1−δ4\nparticles. Recalling Proposition 6.6, one deduces from standard arguments on birth processes that, with\nlarge probability, eX1\nδ3L(T)contains much more than Nparticles: this yields ( ii) and finishes the proof. □\n8.3.Adaptation of the proof to deterministic branching times (Proposition 1.5). In this section we\ndo not present a full proof of Proposition 1.5: instead we detail how all our previous results and proofs can be\nadapted to the time-inhomogeneous BBMdb. Throughout this section, we let ( eXt)t∈[0,T](resp. ( eXN\nt)t∈[0,T])\ndenote the particle configurations of the BBMdb (resp. N-BBMdb). We first focus on adapting Theorem 1.1,\nmore precisely Sections 3 through 7.\nSection 3. We start with the N-BBMdb construction and the coupling statements. Recall Section 3.1, as\nwell as Proposition 3.2 and its proof. The two main differences which appear with deterministic branching\ntimes are:\n— one has to replace the exponential variables in the definitions of the BBMdb and N-BBMdb, as well\nas the Poisson point process ( tk)k≥1on [0 ,1] in the proof of Proposition 3.2.( ii), with a fixed sequence of\nbranching epochs (2 klogEξ)k≥1.\n— at each branching epoch, all individuals have to branch simultaneously. This only affects which particles\nare kept in the N-BBMdb after the branching epoch, and the selection procedure is still well-posed (breaking\nties arbitrarily, as we did with the standard N-BBM).\nFrom this, we may construct the N-BBMdb rigorously, and we conclude that Propositions 3.1 and 3.2\nalso hold for the BBMdb and N-BBMdb. Thus, it remains to prove that Propositions 3.3 and 3.4 also hold\nfor the deterministic-branching variants, then the same arguments as in Section 3.3 show that Theorem 1.1\nalso holds for the N-BBMdb.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 67\nSections 4 and 5. All definitions and notation from Section 4.1 are unchanged. The main differences are in\nSection 4.2, more precisely in the statements of the Many-to-one and Many-to-two lemmas. On the one hand,\nrecall that |Nt|denotes the population size of the BBM at time t∈[0, T], and let |eNt|denote that of the\nBBMdb. Then the Many-to-one lemma still holds for the BBMdb, subject to replacing Eδ0[|Nt|] =et/2with,\nEδ0[|eNt|] =\u0000\nEξ\u0001⌊t/2 logEξ⌋,\nin Lemma 4.2 (the equality above is proven by induction). Notice that this is of order et/2+O(1), so that\nchange does not affect the resulting first moment estimates up to constant factors. On the other hand, we\nprovide a new version of the Many-to-two lemma for deterministic branching (it is obtained by decomposing\npairs of individuals according to their most recent common ancestor). Let eG(·),eA∗\nT,z(·) be defined similarly\ntoG(·),A∗\nT,z(·) respectively in (4.4–4.5), where we replace the BBM with a BBMdb.\nLemma 8.3 (Many-to-two lemma, deterministic branching) .Let∗ ∈ { sup,sub,crit}. Let t≤T,γ∗\nT,γ∗\nT∈\nC1([0, T])which satisfy (4.3), and z∈[0, h). Then, one has,\nEδ0\u0002\n|eA∗\nT,z(t)|2\u0003(8.14)\n=Eδ0\u0002\n|eA∗\nT,z(t)|\u0003\n+E[ξ(ξ−1)]⌊t/(2 log Eξ)⌋−1X\nk=0Zh\n0eG∗(x, y,0,(2 logEξ)k)\u0012Zh\nzeG∗(y, w, (2 logEξ)k, t)dw\u00132\ndy.\nThis lemma follows naturally from the Markov property (we leave the details to the reader). Replacing\nLemma 4.3 with Lemma 8.3 in the proofs of Propositions 5.3, 5.6 and 5.10, and using the adaptation of\nLemma 4.2 discussed above in all the other propositions from Section 5, we obtain exactly the same moment\nestimates for the BBMdb and BBM between barriers, in all regimes.\nFinally, it remains to adapt Lemma 4.6 from Section 4.3. An N−-BBMdb (resp. N+-BBMdb) can be\ndefined very similarly to the N−-BBM (resp. N+-BBM), by applying its selection mechanism to a BBMdb\ninstead of a BBM. Then, we can reproduce the adaptations discussed above for Section 3 to the proof of the\ncoupling in Lemma 4.6 (from [43]). In order not to overburden this paper, we do not further detail it.\nSections 6 and 7. All statements in these sections immediately hold for the N-BBMdb: indeed, the proofs in\nthese sections are entirely based on the moment estimates from Section 5, as well as some technical results\non the barriers (such as Lemma 4.5) which are not affected by the choice of branching mechanism. Therefore,\nthis finishes the adaptation of Propositions 3.3 and 3.4 to the N(T)-BBMdb, proving that the results of\nTheorem 1.1 also hold for that process.\nSuper-critical regime, decreasing variance. Recall the proof of Proposition 1.2 from Section 8.1. To the\nauthors knowledge, there is currently no equivalent to (8.1) for the BBMdb (or BRW) in the literature,\nhowever [ 47] has proven that (8.2) holds for a large class of discrete-time, time-inhomogeneous BRW’s,\namong which Gaussian BRW’s, see [ 47, Theorem 1.3]. Therefore, replacing Propositions 3.1 and 3.2 with the\ncoupling arguments discussed above for the ( N-)BBMdb, and applying [ 47, Theorem 1.3] to the BBMdb at\nits final time, the adaptation of Proposition 1.2 to deterministic branching times is straightforward (we do\nnot replicate the proof).\nRemark 8.3. Let us mention that [47, Theorem 1.3] also holds for σnon-increasing, hence so does (1.12) for\nthe superscript sup-din the N-BBMdb. This provides a slightly more complete statement for the N-BBMdb\nandN-CREM than Proposition 1.2 (where we assumed σstrictly decreasing), however one still requires an\ninitial configuration δ0when quoting [47].\nFinal-time distribution Regarding (1.13) , it is sufficient to prove that Lemma 8.1 also holds for the N-BBMdb\neXN. A thorough inspection of the proof shows that all arguments therein can be adjusted to deterministic\nbranching times, with no more work than what has already been presented for the adaptation Theorem 1.168 A. LEGRAND AND P. MAILLARD\nabove. In order not to overburden this paper, we do not repeat those arguments here. In the proof of (1.14) ,\nthere is one occurrence of the Many-to-one lemma which has to be replaced with the N-BBMdb version as\nabove, and the rest of the proof is unchanged. This fully concludes the proof of Proposition 1.5. □\n8.4.N-BBM with time-inhomogeneous selection (Theorem 1.7). Letℓ∈ C1([0,1]),bL(T)→+∞\nasT→+∞andL(·, T) defined as in (1.20) . In this section we first consider the super-critical and critical\nregimes ( bL(T)≫T1/3orbL(T) =T1/3): we prove (1.23) and(1.25) simultaneously, then (1.24) and(1.26) .\nFinally, we prove (1.23) in the sub-critical regime ( bL(T)≪T1/3).\nBefore that, let us comment on the definition and construction of the branching Brownian motion with\ntime-inhomogeneous selection ( N(·, T)-BBM). In this system, a particle is killed at time t∈[0, T] if it is not\nin the N(t, T) highest ones. This may happen in one of two ways:\n—tis the epoch of a branching event,\n— the integer part of N(t, T) realizes a negative jump at t.\nOn the one hand, the t’s corresponding to branching events are described by the very same point process on\n[0, T] that is involved in the construction of the N(T)-BBM in Section 3.1. On the other hand, the t’s for\nwhich ⌊N(t, T)⌋changes are deterministic for Tfixed; and since we assumed that ℓisC1, there are finitely\nmany of them for almost all T≥0. Therefore, the construction of the N(T)-BBM from Section 3.1 can be\nadapted very directly to the process with time-inhomogeneous selection.\nMoreover, recall Propositions 3.1 and 3.2: these couplings may also be constructed for processes with\ninhomogeneous selection. More precisely, in the proof of Proposition 3.2.( ii) it suffices to augment the\nPoisson point process with the (finite) sequence of deterministic epochs when ⌊N(t, T)⌋decreases: then the\nthe rest of the proof follows from arguments that we discussed above, so we do not write the details here.\nProposition 8.4. LetT≥0fixed. Let N1(·, T),N2(·, T)two positive functions on [0, T]such that\nN1(t, T)≤N2(t, T)for all 0≤t≤T. Assume that their integer parts change finitely many times in\nt∈[0, T]. Let µ1∈CN1(0,T)andµ2∈CN2(0,T)which satisfy µ1≺µ2: then there exists (XN1(·,T)\nt )t∈[0,T]and\n(XN2(·,T)\nt )t∈[0,T], respectively an N1(·, T)- and an N2(·, T)-BBM, such that XN1(·,T)\n0 =µ1,XN2(·,T)\n0 =µ2\nandXN1(·,T)\nt ≺ XN2(·,T)\nt for all t∈[0, T]with probability 1.\nMoreover, there also exists a coupling with a time-inhomogeneous BBM without selection started from µ1,\nsuch that XN1(·,T)\nt ≺ X tfor all t∈[0, T]with probability 1.\nLet us now return to the proof of Theorem 1.7, starting with (1.23) and(1.25) in the critical and\nsuper-critical regimes.\nProof of (1.23) and(1.25) , super-critical and critical regimes. Recall the definition of QT(·) from (1.3), and\nlet us extend it into\n(8.15) Qs\nT(µ) := sup\b\nq∈R\f\f∃κ∈[0,1], µ\u0000\n[q−κσ(s/T)L(T),+∞)\u0001\n≥N(T)κ\t\n, s ∈[0, T].\nRecall Proposition 1.4: then we have the following.\nLemma 8.5. Let∗ ∈ { crit,sup}andη >0. Let µT∈CN, and denote with XN(T)anN(T)-BBM (with\ntime-homogeneous selection) started from µT,T≥0, with infinitesimal variance σ2(·/T),σ∈ S∗\nη. Then as\nT→+∞, one has\n(8.16) sup\nσ∈S∗η1\nL(T)\f\f\fQT\nT(XN(T)\nT)−QT(µT)−m∗\nT\f\f\f−→0, inPµT-probability.\nProof of Lemma 8.5. This is a direct corollary of Proposition 1.4 —more precisely Lemma 8.1— and the\ndefinition (8.15) . Indeed, the upper bound on QT\nT(XN(T)\nT ) follows from the fact that (8.3) provides an upper\nbound on XN(T)\nT\u0000\n[m∗\nT−yσ(1)L(T),+∞)\u0001\nholding for ally∈[0,1] with large probability; the lower bound\nis obtained through a direct application of (8.4). Finally, as stated in Lemma 8.1, the result is uniform in\nσ∈ S∗\nη. □MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 69\nWith Proposition 8.4 and Lemma 8.5 at hand, Theorem 1.7 is obtained quite naturally in the critical\nand super-critical regimes, by writing a block decomposition of the process. Indeed, let us first consider\nthe critical regime, i.e. bL(T) =T1/3. Recall the definition of bmcrit\nTfrom (1.22) . Let ε >0 small (assume\nε−1∈Nfor the sake of simplicity), and part [0 , T] into ε−1intervals of length εT. Define for 1 ≤i≤ε−1,\nℓi:= inf {ℓ(u) ;u∈[(i−1)ε, iε]}, and ℓi:= sup {ℓ(u) ;u∈[(i−1)ε, iε]},\nand define similarly σi,σi,σ′\niandσ′\ni. Define also,\nNi(T) := exp( ℓiT1/3), and Ni(T) := exp( ℓiT1/3).\nFinally, we let for 1 ≤i≤ε−1,\nmi,ε:=TZiεT\n(i−1)εTσ(u)du+ε T1/3inf\u001aσ\nℓΨ\u0012\n−ℓ3σ′\nσ\u0013\n;σ∈[σi,σi], σ′∈[σ′\ni,σ′\ni], ℓ∈[ℓi,ℓi]\u001b\n,\nmi,ε:=TZiεT\n(i−1)εTσ(u)du+ε T1/3sup\u001aσ\nℓΨ\u0012\n−ℓ3σ′\nσ\u0013\n;σ∈[σi,σi], σ′∈[σ′\ni,σ′\ni], ℓ∈[ℓi,ℓi]\u001b\n.\nUsing a Riemann sum approximation and that σ∈ S∗\nη,ℓ∈ C1([0,1]), one observes that\n(8.17) lim\nε→0T−1/3\f\f\f\f\fbmcrit\nT−ε−1X\ni=1mi,ε\f\f\f\f\f= lim\nε→0T−1/3\f\f\f\f\fbmcrit\nT−ε−1X\ni=1mi,ε\f\f\f\f\f= 0.\nLet (Fs)s∈[0,T]denote the natural filtration of the N(·, T)-BBM. Following from Proposition 8.4, there exists\nanNi(T)- and a Ni(T)-BBM, both starting from the initial measure XN(·,T)\n(i−1)εTat time ( i−1)εT, such that,\n(8.18) PµT\u0010\nXNi(T)\niεT ≺ XN(·,T)\niεT ≺ XNi(T)\niεT\f\f\fF(i−1)εT;XNi(T)\n(i−1)εT=XN(·,T)\n(i−1)εT=XNi(T)\n(i−1)εT\u0011\n= 1.\nMoreover, Lemma 8.5 states that, for λ >0, there exists T0>0 (depending only on η, ε, λ ) such that for\nT≥T0,\n(8.19)PµT\u00121\nT1/3\u0012\nQiεT\nT\u0000\nXNi(T)\niεT\u0001\n−Q(i−1)εT\nT\u0000\nXN(i−1)(T)\n(i−1)εT\u0001\n−mi,ε\u0013\n<−ελ\f\f\f\fF(i−1)εT\u0013\n≤ε2,\nand PµT\u00121\nT1/3\u0012\nQiεT\nT\u0000\nXNi(T)\niεT\u0001\n−Q(i−1)εT\nT\u0000\nXN(i−1)(T)\n(i−1)εT\u0001\n−mi,ε\u0013\n> ελ\f\f\f\fF(i−1)εT\u0013\n≤ε2.\nRecalling Theorem 1.1, one obtains similarly for T≥T0,\n(8.20)PµT\u00121\nT1/3\u0012\nmax\u0000\nXNi(T)\niεT\u0001\n−Q(i−1)εT\nT\u0000\nXN(i−1)(T)\n(i−1)εT\u0001\n−mi,ε\u0013\n<−ελ\f\f\f\fF(i−1)εT\u0013\n≤ε2,\nand PµT\u00121\nT1/3\u0012\nmax\u0000\nXNi(T)\niεT\u0001\n−Q(i−1)εT\nT\u0000\nXN(i−1)(T)\n(i−1)εT\u0001\n−mi,ε\u0013\n> ελ\f\f\f\fF(i−1)εT\u0013\n≤ε2.\nApply the Markov property at times ( i−1)εT, 1≤i≤ε−1−1, one deduces with a union bound and (8.18) ,\nPµT\n1\nT1/3\u0012\nmax\u0000\nXN(·,T)\nT\u0001\n−QT(µT)−ε−1X\ni=1mi,ε\u0013\n<−λ\n\n≤ε−1−1X\ni=1PµT\u00121\nT1/3\u0012\nQiεT\nT\u0000\nXNi(T)\niεT\u0001\n−Q(i−1)εT\nT\u0000\nXN(i−1)(T)\n(i−1)εT\u0001\n−mi,ε\u0013\n<−ελ\f\f\f\fXNi(T)\n(i−1)εT=XN(·,T)\n(i−1)εT\u0013\n+PµT\u00121\nT1/3\u0012\nmax\u0000\nXNε−1(T)\nT\u0001\n−Q(1−ε)T\nT\u0000\nXN(ε−1−1)(T)\n(1−ε)T\u0001\n−mε−1,ε\u0013\n<−ελ\f\f\f\fXNε−1(T)\n(1−ε)T=XN(·,T)\n(1−ε)T\u0013\n,\nforT≥T0(η, λ, ε ); and by (8.19–8.20), the latter sum is bounded by ε. Recalling (8.17) and letting ε→0,\nthis finally proves the lower bound in (1.23) ; and the upper bound is obtained similarly. Moreover, (1.25) is70 A. LEGRAND AND P. MAILLARD\nobtained with an analogous computation, by applying Proposition 1.4 instead of (8.20) to the last block\n(i=ε−1) of the decomposition above. This concludes the proof of Theorem 1.7 in the critical regime.\nRegarding the super-critical case, it is proven by replicating the proof above mutatis mutandis : that\nis, replacing T1/3withbL(T)≫T1/3,bmcrit\nTwithbmsup\nT, and adapting the definitions of mi,ε,mi,εabove\naccordingly. This is very straightforward, so we leave the details to the reader. □\nProof of (1.24) .This is immediate: recall that, in the super-critical regime with decreasing variance, the\nconvergence (1.24) does not depend on the specifics of ℓ(·),bL(T). Hence the super-critical N(·, T)-BBM can\nstill be coupled with, on the one hand a critical Mα(T)-BBM with population size Mα(T) :=exp(αT1/3)≪\ninf{N(s, T),0≤s≤T},α > 0; and on the other hand a BBM without selection (see Proposition 8.4).\nTherefore, the proof of Proposition 1.2 above fully accommodates to super-critical, time-inhomogeneous\nselection. We leave the details to the reader. □\nProof of (1.26) .This is also straightforward: recalling the proof of (1.23, 1.25), it suffices to estimate\nmin(XN(·,T)\nT ) in the last block of the decomposition. One can estimate both min(XNε−1\nT ) and min(XNε−1\nT )\nby using (1.14) and Theorem 1.1; and reproducing the coupling arguments from the proof of (1.14) (we\ndo not write the details again, but let us recall that they do not immediately follow from the definition of\nstochastic domination), this gives the expected estimate on min( XN(·,T)\nT ). □\nProof of (1.23) , sub-critical regime. In that regime there is no equivalent to Proposition 1.4 or Lemma 8.5,\nso one cannot split the interval [0 , T] into blocks of length εT; however, the proofs of Lemma 6.4 (for the lower\nbound) and Proposition 3.4 (upper bound), in the sub-critical regime, already rely on a block decomposition:\nso Theorem 1.7 can be obtained by approximating the N(·, T)-BBM by a process with piecewise-constant\nselection, and reproducing the arguments from the proof of Theorem 1.1.\nMore precisely, let us first discuss the lower bound. Let K=⌊2T/tsub⌋and for 0 ≤k≤K−1,tk=k\n2tsub\nT,\nandtK=T. Let Nk=inf{N(s, T), tk−1≤s≤tk}, and recall the construction of the auxiliary process XN\nfrom Section 6.2: let us tweak it such that (1) at times tk, 1≤k≤K, all particles but the ⌊Nk⌋top-most\nones are removed, and the remaining particles are set to the lowest among their positions; and (2) on the\ninterval [ tk−1, tk], the process XNevolves like an Nk-BBM, where Nkis constant. Therefore, this process is\nstill stochastically dominated by the N-BBM, and the remainder of the proof from Section 6.2 still holds for\nthis process (since the selection is constant on each interval [ tk−1, tk]), both for the super-polynomial N(T)\nand small L(T) cases. We deduce from this a lower bound on the maximal displacement of the N(·, T)-BBM,\nand a Riemann sum approximation (analogous to (8.17) ) finally shows that it is of order bmsub\nT−o(T/L(T)2)\nforTlarge. Since all these adaptations are very straightforward, and rely on arguments that were already\napplied in other parts of the paper, we leave the remaining details to the reader. Finally, the upper bound is\nobtained with analogous arguments. □\nRemark 8.4. Let us point out that Theorem 1.7 assumes that the “regime” for the selection (i.e. bL(T))\nremains the same throughout [0, T]. If the regime changes finitely many times, e.g. at some tk,1≤k≤K,\none can derive similar results by induction: indeed, (1.25) and Lemma 8.5 provide an estimate on Qtk\nT(XN(·,T)\ntk)\nif the regime is critical or super-critical on [tk−1, tk]; and in the sub-critical case bLk(T)≪T1/3, one has,\nQtk\nT(XN(·,T)\ntk) =Qtk\nT(δmaxXN(·,T)\ntk) +O(bLk(T)).\nThen, one can apply Theorem 1.7 on each interval [tk−1, tk]iteratively. We do not write any statement or\nproof for this fact, since this is a direct replication of arguments presented above.\nAcknowledgements\nWe thank Marc Lelarge for mentioning the relation of our work with the beam-search algorithm.MAXIMAL DISPLACEMENT OF A TIME-INHOMOGENEOUS N(T)-BBM 71\nA. Legrand acknowledges support from the ANR projects “REMECO”, ANR-20-CE92-0010 and “LOCAL”,\nANR-22-CE40-0012. P. Maillard acknowledges support from the ANR-DFG project “REMECO”, ANR-20-\nCE92-0010 and Institut Universitaire de France (IUF).\nReferences\n[1]L. Addario-Berry and P. Maillard. The algorithmic hardness threshold for continuous random energy models. Math. Stat.\nLearn. , 2(1):77–101, 2019.\n[2] E. A¨ ıd´ ekon. Convergence in law of the minimum of a branching random walk. Ann. Probab. , 41(3A):1362–1426, 2013.\n[3] D. Aldous. Greedy search on the binary tree with random edge-weights. Combin. Probab. Comput. , 1(4):281–293, 1992.\n[4] D. Aldous. A metropolis-type optimization algorithm on the infinite tree. Algorithmica , 22(4):388–412, Dec 1998.\n[5]K. B. Athreya and P. E. Ney. Branching processes . Die Grundlehren der mathematischen Wissenschaften, Band 196.\nSpringer-Verlag, New York-Heidelberg, 1972.\n[6]J. B´ erard and J.-B. Gou´ er´ e. Brunet-Derrida behavior of branching-selection particle systems on the line. Comm. Math.\nPhys. , 298(2):323–342, 2010.\n[7]J. B´ erard and J.-B. Gou´ er´ e. Survival probability of the branching random walk killed below a linear boundary. Electronic\nJournal of Probability , 16(14):396–418, 2011.\n[8]J. Berestycki, N. Berestycki, and J. Schweinsberg. Survival of near-critical branching Brownian motion. Journal of Statistical\nPhysics , 143(5):833–854, may 2011.\n[9]J. Berestycki, N. Berestycki, and J. Schweinsberg. The genealogy of branching brownian motion with absorption. The\nAnnals of Probability , 41(2), mar 2013.\n[10]J. Berestycki, N. Berestycki, and J. Schweinsberg. Critical branching Brownian motion with absorption: survival probability.\nProbability Theory and Related Fields , 160(3-4):489–520, 2014.\n[11]J. Berestycki, N. Berestycki, and J. Schweinsberg. Critical branching Brownian motion with absorption: Particle configura-\ntions. Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques , 51(4):1215–1250, 2015.\n[12] J. D. Biggins. The growth and spread of the general branching random walk. Ann. Appl. Probab. , 5(4):1008–1024, 1995.\n[13] J. D. Biggins and A. E. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. , 36(2):544–581, 2004.\n[14]P. Billingsley. Convergence of probability measures . Wiley Series in Probability and Statistics: Probability and Statistics.\nJohn Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication.\n[15]R. Bisiani. Beam search. In C. S. Shapiro, editor, Encyclopedia of Artificial Intelligence , pages 56–58. John Wiley & Sons,\nInc., 1987.\n[16]A. N. Borodin and P. Salminen. Handbook of Brownian motion—facts and formulae . Probability and its Applications.\nBirkh¨ auser Verlag, Basel, second edition, 2002.\n[17]A. Bovier and L. Hartung. Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime.\nAlea, 12(1):261–291, mar 2015.\n[18]A. Bovier and I. Kurkova. Derrida’s generalized random energy models 2: models with continuous hierarchies. Annales de\nl’Institut Henri Poincare (B) Probability and Statistics , 40(4):481–495, 2004.\n[19]M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. ,\n44(285):iv+190, 1983.\n[20] M. D. Bramson. Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. , 31(5):531–581, 1978.\n[21] E. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E (3) , 56(3):2597–2604, 1997.\n[22] ´E. Brunet and B. Derrida. Microscopic models of traveling wave equations. Computer Physics Communications , 121-\n122:376–381, sep 1999.\n[23]E. Brunet, B. Derrida, A. H. Mueller, and S. Munier. Phenomenological theory giving the full statistics of the position of\nfluctuating pulled fronts. Phys. Rev. E , 73:056126, May 2006.\n[24]B. Chauvin. Product martingales and stopping lines for branching Brownian motion. Ann. Probab. , 19(3):1195–1205, 1991.\n[25] B. Derrida. Random-energy model: Limit of a family of disordered models. Phys. Rev. Lett. , 45:79–82, Jul 1980.\n[26]B. Derrida. Random-energy model: An exactly solvable model of disordered systems. Phys. Rev. B , 24:2613–2626, Sep\n1981.\n[27]B. Derrida. A generalization of the Random Energy Model which includes correlations between energies. Journal de\nPhysique Lettres , 46(9):401–407, 1985.\n[28]B. Derrida and D. Simon. The survival probability of a branching random walk in presence of an absorbing wall. Europhysics\nLetters (EPL) , 78(6):60006, jun 2007.\n[29]B. Derrida and H. Spohn. Polymers on disordered trees, spin glasses, and traveling waves. Journal of Statistical Physics ,\n51(5):817–840, Jun 1988.\n[30]A. El Alaoui, A. Montanari, and M. Sellke. Optimization of mean-field spin glasses. The Annals of Probability , 49(6), nov\n2021.\n[31]M. Fang and O. Zeitouni. Consistent Minimal Displacement of Branching Random Walks. Electronic Communications in\nProbability , 15:no. 11, 106–118, mar 2010.72 A. LEGRAND AND P. MAILLARD\n[32]G. Faraud, Y. Hu, and Z. Shi. Almost sure convergence for stochastically biased random walks on trees. Probability Theory\nand Related Fields , 154(3-4):621–660, dec 2012.\n[33]D. Gamarnik. The overlap gap property: A topological barrier to optimizing over random structures. Proceedings of the\nNational Academy of Sciences of the United States of America , 118(41), 2021.\n[34]N. Gantert, Y. Hu, and Z. Shi. Asymptotics for the survival probability in a killed branching random walk. Annales de\nl’Institut Henri Poincar´ e (B) Probabilit´ es et Statistiques , 47(1):111–129, feb 2011.\n[35] N. Ikeda, M. Nagasawa, and S. Watanabe. Branching Markov processes. I. J. Math. Kyoto Univ. , 8:233–278, 1968.\n[36] N. Ikeda, M. Nagasawa, and S. Watanabe. Branching Markov processes. III. J. Math. Kyoto Univ. , 9:95–160, 1969.\n[37]B. Jaffuel. The critical barrier for the survival of branching random walk with absorption. Ann. Inst. Henri Poincar´ e\nProbab. Stat. , 48(4):989–1009, 2012.\n[38] P. Jagers. General branching processes as Markov fields. Stochastic Process. Appl. , 32(2):183–212, 1989.\n[39]H. Kesten. Branching Brownian motion with absorption. Stochastic Processes and their Applications , 7(1):9–47, mar 1978.\n[40]A. Klimovsky. High-dimensional Gaussian fields with isotropic increments seen through spin glasses. Electron. Commun.\nProbab. , 17:no. 17, 14, 2012.\n[41]S. Lemons, C. L. L´ opez, R. C. Holte, and W. Ruml. Beam Search: Faster and Monotonic. Proceedings International\nConference on Automated Planning and Scheduling, ICAPS , 32:222–230, 2022.\n[42] P. Maillard. Branching Brownian motion with selection . Theses, Universit´ e Pierre et Marie Curie - Paris VI, Oct. 2012.\n[43]P. Maillard. Speed and fluctuations of N-particle branching Brownian motion with spatial selection. Probab. Theory Related\nFields , 166(3-4):1061–1173, 2016.\n[44]P. Maillard and J. Schweinsberg. Yaglom-type limit theorems for branching Brownian motion with absorption. Annales\nHenri Lebesgue , 5:921–985, 2022.\n[45] P. Maillard and J. Schweinsberg. The all-time maximum for branching Brownian motion with absorption conditioned on\nlong-time survival. arXiv:2310.00707 , page 23pp, 2023.\n[46]P. Maillard and O. Zeitouni. Slowdown in branching brownian motion with inhomogeneous variance. Annales de l’Institut\nHenri Poincar´ e, Probabilit´ es et Statistiques , 52(3), Aug 2016.\n[47]B. Mallein. Maximal displacement of a branching random walk in time-inhomogeneous environment. Stochastic Processes\nand their Applications , 125(10):3958–4019, 2015.\n[48]H. P. McKean. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl.\nMath. , 28(3):323–331, 1975.\n[49]J. Nolen, J.-M. Roquejoffre, and L. Ryzhik. Power-Like Delay in Time Inhomogeneous Fisher-KPP Equations. Communi-\ncations in Partial Differential Equations , 40(3):475–505, dec 2014.\n[50] R. Pemantle. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. , 19(4):1273–1291, 2009.\n[51]M. I. Roberts. Fine asymptotics for the consistent maximal displacement of branching brownian motion. Electronic Journal\nof Probability , 20:1–26, sep 2015.\n[52] T. H. Savits. The explosion problem for branching Markov process. Osaka Math. J. , 6:375–395, 1969.\n[53] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. , 41:463–501, 1962.\nUniversite Claude Bernard Lyon 1, ICJ, CNRS UMR 5208, Ecole Centrale de Lyon, INSA Lyon, Universit ´e Jean\nMonnet, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France\nInstitut de Math ´ematiques de Toulouse, CNRS UMR 5219, Universit ´e de Toulouse, 118 route de Narbonne,\n31062 Toulouse Cedex 9, France and Institut Universitaire de France.\nEmail address :legrand@math.univ-lyon1.fr; pascal.maillard@math.univ-toulouse.fr"    },    {        "title": "2402.04923v1.Jet_transport_coefficients_by_elastic_and_radiative_scatterings_in_the_strongly_interacting_quark_gluon_plasma.pdf",        "content": "Jet transport coefficients by elastic and radiative scatterings in the strongly interacting\nquark-gluon plasma\nIlia Grishmanovskii,1,∗Olga Soloveva,2, 1Taesoo Song,3Carsten Greiner,1, 2and Elena Bratkovskaya4, 1, 2\n1Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at,\nMax-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany\n2Helmholtz Research Academy Hesse for FAIR (HFHF),\nGSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt, 60438 Frankfurt, Germany\n3GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH,Planckstrasse 1, D-64291 Darmstadt, Germany\n4GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, Planckstrasse 1, D-64291 Darmstadt, Germany\n(Dated: February 8, 2024)\nWe extend the investigation on jet transport coefficients within the effective Dynamical QuasiPar-\nticle Model (DQPM) – constructed for the description of non-perturbative QCD phenomena of the\nstrongly interacting quark-gluon plasma (sQGP) in line with the lattice QCD equation-of-state – by\naccounting for inelastic 2 →3 reactions with gluon radiation additionally to the elastic scattering of\npartons. The elastic and inelastic reactions are calculated explicitly within leading-order Feynman\ndiagrams with effective propagators and vertices from the DQPM by accounting for all channels and\ntheir interferences. We present the results for the jet transport coefficients such as the transverse\nmomentum transfer squared ˆ qper unit length as well as the energy loss ∆ E=dE/dx per unit\nlength in the sQGP and investigate their dependence on the temperature Tand momentum of the\njet parton depending on the choice of the strong coupling constant αsin thermal, jet parton and\nradiative vertices. For the latter we consider different scenarios used in the literature and find a\nvery strong dependence of ˆ qand ∆ Eon the choice of αs. Moreover, we explore the relation of ˆ q/T3\nto the ratio of specific shear viscosity to entropy density η/sand show that the ratio T3/ˆqtoη/s\nhas a strong Tdependence – especially when approaching to Tc– on the choice of αsin scattering\nvertices.\nI. INTRODUCTION\nJets are considered as one of the penetrating probes that\nallow to obtain information about the strong interactions in\nthe QGP medium created in heavy-ion collisions. As ob-\nserved experimentally at RHIC [1, 2] and at LHC [3, 4]\njets, produced in these A+A collisions, are modified com-\npared to those produced in proton-proton (p+p) collisions\ndue to the scattering of partons from the jet shower with\nthe partons from the QGP environment. Theoretical un-\nderstanding of the experimental data on jet attenuation in\nheavy-ion collisions stimulated a lot of a theoretical efforts\non an understanding of jet properties in both equilibrium\nand non-equilibrium cases [5–25]. The jet energy loss in the\nQGP medium occurs due to elastic 2 →2 partonic scat-\nterings as well as by radiative 2 →3 processes with the\nemission of a gluon. The soft gluon radiation is screened\nin the medium due to the coherence effect called Landau-\nPomeranchuk-Migdal (LPM) effect [26, 27] which has to be\naccounted for the interpretation of experimental data. The\nfirst calculations of the gluon Bremsstrahlung processes with\nthe non-Abelian LPM effect in the context of perturbative\nQCD [12, 28–31] showed a suppression of soft gluon radia-\ntion due to the destructive interference of the emitted glu-\nons in the medium contrary to the vacuum, cf. also Refs.\n[13, 14, 32–37]. A further formulation of radiative energy loss\n- as a dynamical process in the QGP medium of the ther-\nmal scattering centers - has been derived by Arnold, Moore,\nand Yaffe (AMY) within thermal field theory [16, 38, 39] and\ndeveloped further in recent years by going beyond the limita-\ntions of soft-emission approximations [40, 41]. Also progress\nhas been achieved in the resummation of multiple scatterings\n∗grishm@itp.uni-frankfurt.de[24, 42, 43].\nSince the jets are multi-scale objects, the microscopic sim-\nulation of jet propagation and interactions with the QCD\nmedium is notoriously difficult. The description of partonic\nenergy loss in the QGP is widely studied in the literature\nin terms of transport coefficients such as the transverse mo-\nmentum transfer squared ˆ qper unit length to characterize\nthe jet transverse momentum broadening in the medium as\nwell as the energy loss ∆ E=dE/dx per unit length to ac-\ncount for the modification of jet energy during propagation\nin the medium.\nThe ˆqrepresents a measure of the interaction between a\nhigh-energetic jet and the QGP medium, which is defined\nas the amount of momentum transfer that the hard parton\nexperiences per unit length as it travels through the dense\nQGP medium. The value of this coefficient depends on sev-\neral factors, including the coupling strength of the interac-\ntion, the nature of the plasma (whether it is dominated by\nquasi-particles or not), and the underlying micro-physics of\nmany-body QCD matter.\nRecently, in Ref. [44] the jet transport coefficients ˆ qand\n∆Ehave been studied for elastic 2 →2 interactions of\npartons within the effective dynamical quasi-particle model\n(DQPM) constructed to describe the non-perturbative prop-\nerties of the sQGP at finite temperature Tand baryon chemi-\ncal potential µBin terms of strongly interacting off-shell par-\ntons (quarks and gluons) with dynamically generated spec-\ntral functions whose properties are adjusted to reproduce\nthe lQCD EoS for the QGP in thermodynamic equilibrium.\nIt has been shown that ˆ qand ∆ Eshow a strong tempera-\nture dependence due to the growing strong coupling αs(T)\nwhen approaching the critical temperature Tc. However, ad-\nditionally, to elastic scattering, the gluon radiative processes\n2→3 are also important for an understanding of the high\nenergetic jet attenuation in heavy-ion collisions, whose con-\ntribution grows with increasing energy of jet partons - cf.arXiv:2402.04923v1  [hep-ph]  7 Feb 20242\n[23, 25, 45] and Refs. therein.\nIn Ref. [46] the massive gluon radiation processes from the\noff-shell quark-quark ( q+q) and quark-gluon ( q+g) scatter-\nings have been calculated explicitly (for the first time) within\nthe DQPM based on leading order Feynman diagrams with\neffective propagators and vertices from the DQPM without\nany further approximations. There the total and differen-\ntial radiative cross sections have been evaluated versus the\ncollision invariant energy√s, temperature T, and baryon\nchemical potential µBand compared to the corresponding\nelastic cross sections. While in the limit of zero masses and\nwidths of quasiparticles, the DQPM reproduces the results\nof pQCD for 2 →3 cross sections [46], the results for the\nradiation of heavy gluons from quasiparticle scattering may\ndiffer substantially from the pQCD calculations. Thus, one\nexpects an influence of the radiative process on the inter-\naction of fast partons propagated through the equilibrated\nsQGP medium of quasiparticles.\nIn the present work we aim to understand how impor-\ntant gluon radiative reactions are for the jet transport coef-\nficient in the case of massive quarks and gluons and explore\nthe temperature and momentum dependence of the trans-\nport coefficients. The latter can be of great interest in the\ncase of jet quenching models such as JEWEL [47], JET [48],\nJETSCAPE [49] etc. It is important to note that we do\nnot use the assumption of perturbative QCD (pQCD) that\nthe interaction between an energetic parton and the medium\nis dominated by small-angle scattering and induced gluon\nradiation. In this work, we aim to evaluate jet transport co-\nefficients taking into account properly all diagrams and chan-\nnels involved in 2 →2 and 2 →3 partonic scattering with\nmassive medium partons as described by DQPM, i.e. the\nin-medium modification of jet partons occurs by sequences\nof independent elastic and inelastic reactions with thermal\npartons from the medium at given temperature T.\nThe framework of quasiparticle models facilitates its im-\nplementation in the transport model for the dynamical evo-\nlution of the QGP matter. In particular, the off-shell\nquasi-particle model DQPM is implemented in the Parton-\nHadron-String Dynamics (PHSD) transport approach [50,\n51], whereas the on-shell QPM [52] is implemented in the\nCatania transport approach [53].\nIn this study we also investigate the dependence of elas-\ntic and inelastic transport coefficients on the choice of the\nstrong coupling constant αsused in thermal, jet parton and\ngluon radiative vertices, which could differ from the coupling\nconstant for the thermalized partons since the jet parton is\nnot in equilibrium with the QGP medium [54–56]. For that\ngoal we select 4 models for αs– used in the literature – and\ncompare the results with default the DQPM αs(T). Also we\nexplore the dependence of elastic and inelastic ˆ qanddE/dx\non the mass of the emitted gluon. Some model cases we com-\npare to the results from the BAMPS (Boltzmann Approach\nto Multi-Parton Scatterings) pQCD cascade which explicitly\nincludes 2 →2 and 2 →3 (as well as backward) reactions\n[57, 58] and the LPM effect [37].\nFurthermore, we investigate the relation of ˆ q/T3to the ra-\ntio of specific shear viscosity to entropy density η/sdiscussed\nin [59].\nThis paper is organized as follows. In Section II we recall\nthe main ideas of the DQPM, in Section III we describe the\nframework for the calculation of transport coefficients andpresent the scenarios for the coupling constant. In Sections\nIV, V and VI we report on the results for the radiative dif-\nferential cross sections and transport coefficients. We sum-\nmarize our study in Section VII.\nII. DYNAMICAL QUASIPARTICLE MODEL\nThe Dynamical Quasiparticle Model (DQPM) [51, 60–\n64] is an effective model which describes the Quark-Gluon\nPlasma (QGP) in terms of strongly interacting quarks and\ngluons. This approach involves fitting the properties of\nthese quasiparticles to match the results of lattice QCD cal-\nculations in thermal equilibrium and at vanishing chemi-\ncal potential. Here we briefly recall the basic ideas of the\nDQPM. The quasi-particles in the DQPM are characterized\nby ”dressed” propagators, i.e. single-particle (two-point)\nGreen’s functions, which have the form\nGR\nj(ω,p) =1\nω2−p2−M2\nj+ 2iγjω(1)\nfor quarks, antiquarks, and gluons ( j=q,¯q, g), using ω=p0\nfor energy, the widths γjand the masses Mjand the complex\nself-energies for gluons Π = M2\ng−2iωγgand for (anti)quarks\nΣq=M2\nq−2iωγq, where the real part of the self-energies is\nassociated with dynamically generated thermal masses, while\nthe imaginary part provides information about the lifetime\nand reaction rates of the particles.\nThe spectral function of off-shell quasiparticles in the\nDQPM are parametrized in Lorenzian form with a finite\nwidth γj[65]:\nρj(ω,p) =γj\n˜Ej \n1\n(ω−˜Ej)2+γ2\nj−1\n(ω+˜Ej)2+γ2\nj!\n≡4ωγj\u0000\nω2−p2−M2\nj\u00012+ 4γ2\njω2(2)\nwith ˜E2\nj(p) =p2+M2\nj−γ2\nj. The spectral function is anti-\nsymmetric in ωand normalized as\n∞Z\n−∞dω\n2πω ρj(ω,p) =∞Z\n0dωω\nπρj(ω,p) = 1 . (3)\nThe DQPM involves a model ansatz for the masses\nMj(T, µq) and widths γj(T, µq) as functions of the tempera-\ntureTand the quark chemical potential µq. By comparison\nof the entropy density – computed within the DQPM frame-\nwork – to the lQCD data, one can fix the few parameters\nused in the ansatz for the quasiparticle masses and widths.\nThe following ansatz is used in the DQPM for the definition\nof the quasiparticle properties (masses and widths) as func-\ntions of Tandµq. The dynamical quasiparticle pole masses\nare given by the HTL thermal mass in the asymptotic high-\ntemperature regime - cf. [65, 66]\nM2\ng(T, µq) =g2(T, µq)\n6 \u0012\nNc+1\n2Nf\u0013\nT2+Nc\n2X\nqµ2\nq\nπ2!\n,\n(4)3\nand for quarks (antiquarks) by\nM2\nq(¯q)(T, µq) =N2\nc−1\n8Ncg2(T, µq) \nT2+µ2\nq\nπ2!\n, (5)\nwhere Nc(= 3) stands for the number of colors and Nf(=\n3) denotes the number of light flavors. Eq. (5) determines\npole masses for the ( u, d) quarks, the strange quark has a\nlarger bare mass for controlling the strangeness ratio in the\nQGP. Empirically, we find Ms(T, µB) =Mu/d(T, µB)+∆ M,\nwhere ∆ M≃30 MeV has been fixed once in comparison to\nexperimental data [51].\nThe effective quarks, antiquarks, and gluons in the DQPM\nacquire sizable widths γj, which are taken in the form [65]\nγj(T, µ B) =1\n3Cjg2(T, µ B)T\n8πln\u00122cm\ng2(T, µ B)+ 1\u0013\n.(6)\nHere cm= 14.4 is related to a magnetic cutoff, which is an\nadditional parameter in the DQPM, while Cq=N2\nc−1\n2Nc=\n4/3 and Cg=Nc= 3 are the QCD color factors for quarks\nand gluons, respectively. We also assume that all (anti-\n)quarks have the same T-dependence for the width.\nThe coupling constant g2= 4παsessentially defines the\nmasses and widths of quasiparticles - cf. Eqs. (5), (6). The\nthermal properties of quasiparticles and their interactions\n(observed via transport coefficients) thus strongly depend\non the coupling constant g.\nIn the DQPM g2is extracted from lattice Quantum Chro-\nmodynamics (lQCD) data on the entropy density sby a\nparametrization method introduced in Ref. [67]. There\nit has been found that for a given value of g2, the ra-\ntios(T, g2)/T3is almost constant for different tempera-\ntures, i.e.∂\n∂T(s(T, g2)/T3) = 0. Therefore the entropy\ndensity sand the dimensionless equation of state in the\nDQPM is a function of the effective coupling only, i.e.\ns(T, g2)/sSB(T) = f(g2) where sQCD\nSB = 19 /9π2T3is the\nStefan-Boltzmann entropy density. Thus, by inverting the\nf(g2) function, the coupling constant g2can be directly ob-\ntained from the parametrization of lQCD data for the en-\ntropy density s(T, µB= 0) at zero baryon chemical poten-\ntial:\ng2(T, µB= 0) = d\u0010\n(s(T,0)/sQCD\nSB)e−1\u0011f\n. (7)\nHere d= 169 .934, e=−0.178434 and f= 1.14631 are the\ndimensionless parameters obtained by adjusting the quasi-\nparticle entropy density s(T, µB= 0) to the lQCD data pro-\nvided by the BMW Collaboration [68, 69].\nThe DQPM αsaccounts for nonperturbative effects which\nis larger compared to the analytical 2- or 1-loop running\nconstant [70] approaching low temperatures.\nThe extension of the coupling constant to finite baryon\nchemical potential µBis realized using a scaling hypothesis\n[71] that works up to µB≈500 MeV.\nThus, the DQPM provides the quasiparticle properties,\ndressed propagators, and coupling constant, which can be\nused to evaluate the scattering amplitudes as well as the\ncross sections and the transport coefficients of quarks and\ngluons in the QGP as a function of the temperature and the\nchemical potential - cf. Refs. [46, 51, 72].III. METHODOLOGY\nThe propagation of a fast jet parton through the thermal-\nized medium can be characterized by jet transport coeffi-\ncients that can be calculated within kinetic transport theory\nby accounting for the sequence of elastic and inelastic inter-\nactions of the ”jet” parton per unit length x(or time using\nx=ct, where cis the speed of light. We use the units c= 1\nthroughout this work). We note that we do not consider in\nthis study a suppression of the radiation of soft gluons in\nfinite medium due to the LPM effects.\nWe start with the general expression for a transport coeffi-\ncient in kinetic theory [18, 73–75] for elastic 2→2reactions :\n⟨O⟩el=1\n2EjetX\ni=q,¯q,gZd3pi\n(2π)32EidifiZd3p1\n(2π)32E1\n×Zd3p2\n(2π)32E2(1±f1)(1±f2)\n× O | ¯M2→2|2\njet+i(2π)4δ(4)(pjet+pi−p1−p2),(8)\nwhere piis the 4-momentum of the incoming medium par-\nton,p1andp2are the outgoing jet and medium parton 4-\nmomenta, respectively, diis the medium parton’s degeneracy\nfactor for spin and color (2 Ncfor quarks and 2( N2\nc−1) for\ngluons); fi=fi(Ei, T, µ q) are the Fermi distribution func-\ntions for quarks and fi=fi(Ei, T) are the Bose distribution\nfunctions for gluons.\nIn case of the inelastic 2→3reaction the expression for\nthe jet transport coefficients takes the following form:\n⟨O⟩inel=1\n2EjetX\ni=q,¯q,gZd3pi\n(2π)32EidifiZd3p1\n(2π)32E1\n×Zd3p2\n(2π)32E2Zd3p3\n(2π)32E3(1±f1)(1±f2)(1±f3)\n× O | ¯M2→3|2\njet+i(2π)4δ(4)(pjet+pi−p1−p2−p3),(9)\nwhere p3denotes the momentum of the emitted gluon.\nHere the matrix elements squared, i.e. |¯M2→2|2\njet+ifor\nelastic 2 →2 parton scattering and |¯M2→3|2\njet+ifor inelastic\n2→3 reactions of gluon radiation from the off-shell quark-\nquark ( q+q) and quark-gluon ( q+g) scatterings, are cal-\nculated explicitly within the DQPM based on leading order\nFeynman diagrams with the effective propagators and ver-\ntices from the DQPM without any further approximations.\nThe corresponding Feynman diagrams for the t-channel of\ntheq+q→q+q+gandq+g→q+g+gprocesses are\nillustrated in Figure 1. For the details of the evaluation of\nthe 2→2 transport coefficients and for the inelastic 2 →3\ncross sections we refer the reader to Refs. [44, 46].\nVarious transport coefficients can be calculated by select-\ning different operators Oin equations (8) and (9):\n•O= 1 – scattering rate Γ,\n•O=|pT−p′\nT|2– jet transport coefficient ˆ q,\n•O=E−E′– energy loss ∆ E=dE/dx per unit length,\n•O=pL−p′\nL– drag coefficient A.\nHere E, E′,pT, p′\nTandpL, p′\nLdenote initial and final en-\nergy, transverse and longitudinal momenta of the jet parton,\nrespectively.4\n(1)pa\npbp1\np2p3\n(3)pa\npbp1\np2p3\n(2)pa\npbp1\np2\np3(4)pa\npbp1\np2\np3\n(5)pa\npbp1\np2p3\n(1)pa\npbp1\np2p3\n(3)pa\npbp1\np2p3\n(2)pa\npbp1\np2\np3(4)pa\npbp1\np2\np3\n(5)pa\npbp1\np2p3 (6)pa\npbp1\np2\np3\nFIG. 1. Feynman diagrams for the t-channel of the q+q→q+q+g(left) and q+g→q+g+g(right) processes. The green dots\nindicate the vertices corresponding to thermal partons. The blue dots indicate the vertices corresponding to the jet quark while the\nred dots denote the vertices corresponding to the emitted gluon. For the case of the 4-gluon point interaction (diagram 6 in the right\nfigure) the medium parton vertex and the emitted gluon vertex are merged.\nA. Mass of the radiated gluon\nAn important feature of the DQPM - contrary to the\npQCD-based models - is the non-zero mass of the emitted\ngluon in the 2 →3 reactions. In a thermal DQPM medium\nthe mass of the radiated gluon in the on-shell case is assumed\nto be the gluon pole mass (obtained from Eq. (4)). The jet\nparton, however, is not part of the QGP medium and should\nhave a non-thermal mass. In turn, it is not clear whether the\nemitted gluon mass should still be thermal since the gluon\ncan be emitted from a non-thermal jet parton. Therefore,\nin this work we also investigate the dependence of transport\ncoefficients on the emitted gluon mass.\nWe note that the influence of the mass of the emitted gluon\non the radiative energy loss of heavy quarks scattering with\nmassless quarks and gluons from the QGP medium has been\nstudied within scalar pQCD in Refs. [76–78], where it has\nbeen shown that the emission rate decreases with increasing\ngluon mass.\nMoreover, we indicate that all calculations below are pre-\nsented for a quark jet with mass M= 0.01 GeV. As shown\nin our previous study [46], at high collision energies the in-\nelastic cross sections only slightly depend on the mass of jet\nparton.\nB. Effective coupling constants for jet elastic and\ninelastic scattering\nIn this section we consider five scenarios for different strong\ncouplings for every possible vertex. As illustrated in Fig. 1,\nfor every possible interaction channel for the 2 →3 reaction\nin the t-channel of the q+q→q+q+g(left) and q+g→q+g+g(right) processes, one can define 3 different strong\ncouplings: one associated with the thermal parton (green\ndot), one associated with the jet parton (blue dot) and one\nassociated with the emitted gluon (red dot). We note that\nfor the elastic 2 →2 reactions the thermal (green) and jet\n(blue) vertices are the same in the corresponding Feynman\ndiagrams as for the inelastic case.\nDifferent scenarios (defined below) for the strong coupling\ngare illustrated in Fig. 2 as a function of the temperature\nT(upper plot) and jet energy E(lower plot).\n1. Scenario 0: DQPM thermal g(T)\nThis scenario corresponds to the default version of the\nDQPM where in all vertices the thermal DQPM coupling\nconstant is taken, i.e.\ngDQPM(T) =gDQPM(T, µB= 0), (10)\nwhich is defined by Eq. (7).\nSince the DQPM coupling depends strongly on the\nmedium temperature T(cf. blue line in Fig. 2) and grows\nrapidly in the vicinity of the critical temperature Tc, we ex-\npect a large value and strong T-dependence for the temper-\nature dependence of the transport coefficients for inelastic\nreactions similar to the elastic one, as shown in our previous\nstudy [44]. The DQPM coupling does not depend on the jet\nmomentum or momentum transfer, so we expect a strong de-\npendence for the inelastic transport coefficients on the initial\njet energy Esimilar to elastic case [44].5\n2. Scenario I: g=const\nIn this scenario, we investigate the transport coefficients\nwithout taking into account the direct influence of the cou-\npling constant. For this we consider a constant value of the\nstrong coupling gfor all vertices in 2 →2 and 2 →3 dia-\ngrams. The parton masses, however, are kept with the same\ntemperature dependence as in the default DQPM.\nThe value of gis chosen to match the commonly taken\nvalue αs= 0.3 (cf. the pQCD BAMPS model [58]), i.e.,\ng=√\n4π·0.3≈1.94. (11)\nAs one can see in Fig. 2, although this value of g(gray dot-\nted lines) is smaller than the thermal gDQPM(T) for temper-\natures up to T≈0.6 GeV, it is still larger than 1, so it must\ngive a significant contribution to inelastic amplitudes where\nit is accounted in each vertex. For this scenario we expect\na smaller value of ˆ qcompared to ”Scenario 0”, as well as\na different form of the temperature dependence and energy\ndependence of transport coefficients.\n3. Scenario II: g(Q2)from the Zakharov model\nMotivated by Refs. [31, 79], in this scenario we consider\na momentum-dependent strong coupling frozen at low mo-\nmenta at some value αfr\ns:\ng2(Q2) =(\n4παfr\ns ifQ≤Qfr,\n48π2\n(11Nc−2Nf) ln(Q2/Λ2\nQCD)ifQ > Q fr,(12)\nwith Λ QCD= 0.2 GeV, Qfr= Λ QCDexp(2 π/9αfr\ns) while αfr\ns\nis a free parameter. In this work we consider αfr\ns= 1.05 (vac-\nuum value) and αfr\ns= 0.42 (in medium value) taken from\nRef. [31]. It is important to note that here thermal effects en-\ncompassed in Q2-dependence suppress the in-medium QCD\ncoupling.\nFor the jet vertex for both elastic and radiative collisions\ntheQin Eq. (12) is defined as the momentum transfer be-\ntween the jet and the medium parton. The coupling for the\nthermal vertex remains to be gDQPM(T). For the emitted\ngluon in 2 →3 reaction the value of Qis suggested to be kT\n– the transverse momentum of the emitted gluon.\nSumming up, the choice of the couplings for the inelastic\nprocesses for Scenario II is the following:\n•thermal vertex – gDQPM(T).\n•jet vertex – g(Q2) with Qbeing the momentum transfer\nbetween the jet and the medium parton.\n•radiative vertex – g(k2\nT), where kTis the transverse\nmomentum of the emitted gluon.\n4. Scenario III: QLBT model\nAnother way to define strong couplings has been intro-\nduced in the QLBT model [80, 81], where the couplings as-\nsociated with the jet parton and the emitted gluon vertices\nare assumed to have the following parametric form:\ng2(E) =48π2\n(11Nc−2Nf) ln [( AE/T c+B)2], (13)where Eis the jet parton energy in the rest frame, (check!:\nTc= 150 MeV) is the critical temperature, and the parame-\ntersA, B are determined from the heavy quark observables\n(such as RAAandv2). The coupling for the thermal vertex\nremains to be gDQPM(T). The coupling g2(E) is shown in\nFig. 2 by the green lines: dashed-dotted for E= 10 GeV\nand dashed for E= 100 GeV on the upped plot showing the\ntemperature dependence and by the dash-dotted line in the\nlower plot showing the E-dependence.\nThe choice of the couplings for Scenario III is the following:\n•thermal vertex – gDQPM(T).\n• •jet/radiative vertices – g(E),Eis the jet energy.\n5. Scenario IV: DREENA framework\nThe last scenario is motivated by the couplings used in the\nDREENA framework [82, 83] where the jet interaction with\nthe thermal QCD medium is computed within the two loop\nHTL model [84, 85] and implies the following form of the\ncoupling:\ng2(t) =48π2\n(11Nc−2Nf)1\nln (t\nΛ2), (14)\nwith Λ = 0 .2 GeV.\nThe choice of the couplings for Scenario IV is the following:\n•thermal vertex – gDQPM(T).\n•jet vertex – g(ET) according to Eq.(14) with t=ET,\nwhere Eis the jet energy.\n•radiative vertex – g(Q2) according to Eq.(14) with t=\nQ2, where Qis the virtuality of the intermediate parton\nbefore(after) the gluon emission.\nThe orange lines in Fig. 2 show the coupling g2(E): dot-\ndashed for E= 10 GeV and dashed for E= 100 GeV on the\nupped plot for the T-dependence and by the dot-dashed line\nforT= 0.2 GeV and by the dashed line for T= 0.4 GeV in\nthe lower plot for the E-dependence.\nSummary table for Scenarios 0 - IV\nAn overview of the presented scenarios is shown in Table\nI.\nIV. RESULTS: RADIATIVE DIFFERENTIAL\nCROSS SECTIONS\nWe recall that the 2 →3 cross sections and interaction\nrates for q+q→q+q+gandq+g→q+g+gscatter-\ning have been calculated within the DQPM in Ref. [46] and\ncompared to the elastic one. It has been shown that the in-\nelastic reactions contribute only little to the interaction rate\nΓ of the QGP in thermal equilibrium. However, their contri-\nbution is expected to be important for the calculation of the\nenergy loss of the off-equilibrium fast jet parton propagating\nthrough the thermal QGP medium, since the inelastic q+q6\nVertex\nModel •medium parton •jet parton •emitted gluon\nScenario 0 gDQPM(T)\nScenario I g=√\n4π·0.3\nScenario II gDQPM(T) g(Q2), Eq. (12) g(k2\nT), Eq. (12)\nScenario III gDQPM(T) g(E), Eq. (13) g(E), Eq. (13)\nScenario IV gDQPM(T) g(ET), Eq. (14) g(Q2), Eq. (14)\nTABLE I. Scenarios for the effective coupling.\n0.2 0.3 0.4 0.5 0.6\nT [GeV]123456gDQPM\nDREENA E=10 GeV\nDREENA E=100 GeV\nQLBT E=10 GeV\nQLBT E=100 GeV\n/radicalbig\n40.3\n20 40 60 80 100\nE [GeV]1.52.02.53.03.54.04.5gDQPM T=0.2 GeV\nDQPM T=0.4 GeV\nDREENA T=0.2 GeVDREENA T=0.4 GeV\nQLBT/radicalbig\n40.3\nFIG. 2. Strong coupling gas a function of the temperature\nT(upper) and jet energy E(lower). Blue lines correspond to\nthe DQPM coupling defined by Eq. (7), orange lines show the\nDREENA coupling defined by Eq. (14), green lines display the\nQLBT coupling defined by Eq. (13), and the gray dotted line\nshows g=√\n4π·0.3.\nandq+gcross sections grow with increasing collision energy\n[46].\nAs mentioned above, while the DQPM considers the prop-\nerties of all colliding partons to be in thermal equilibrium, a\nfast equilibration of the emitted gluon after the first inelas-\ntic collision is questionable. Thus, it is interesting to study\nhow the differential cross sections – related to the energy\nloss of the jet parton – and the momentum distribution ofthe emitted gluon are sensitive to the mass of the radiated\ngluon.\nIn Fig. 3 we present the different types of differential\nDQPM cross sections for the case of thermal massive –\nwith the pole DQPM mass Mg(T) defined by Eq. (4) –\n(blue lines) and massless (orange lines) emitted gluon for\ntheq+q→q+q+gscattering at√s= 10 GeV and T= 0.3\nGeV. The upper left plot shows the differential cross sec-\ntions dσ/dk tas a function of the transverse momentum kt\nof the emitted gluon. The upper right plot indicates the dif-\nferential cross section dσ/dq tas a function of the transverse\nmomentum qtof the jet (upper right). The lower left plot\ndisplays the differential cross section dσ/dE gas a function\nof the energy Egof the emitted gluon. The lower left plot\ndemonstrates the distribution x dσ/dx as a function of the\nlongitudinal ratio x=pg\nL/pj\nL, where pg\nLandpj\nLare the longi-\ntudinal momenta of gluon and initial jet parton, respectively.\nOne can see from Fig. 3 all DQPM differential cross sec-\ntions for radiation of massless gluons at small kt, qt, Egor\nxare larger than for the massive cases. This is due to the\nfact that the kinematically available phase space is strongly\nreduced for the emission of massive heavy gluons compared\nto the massless ones. Therefore, it is expected to obtain a\nsimilar increase of transport coefficients for massless emit-\nted gluons versus massive ones. Our result is consistent with\nthe findings in Refs. [76–78] for the radiative energy loss of\nheavy quarks in the QGP medium described by scalar pQCD,\nwhere it has been shown that the gluon emission cross section\ndecreases with increasing gluon mass.\nIn the upper left plot we compare the DQPM dσ/dk tto the\npQCD differential cross section within the BAMPS frame-\nwork for αs= 0.3 and massless partons (green line) which\ndiverges at kt→0 without LPM effect [37]. Thus, the pQCD\ndσ/dk tis much larger at low ktand decreases much faster\nwith increasing ktrelative to the DQPM cross sections.\nV. RESULTS: ˆqAND ENERGY LOSS dE/dx\nHere we present the T- and p-dependence of the jet trans-\nport coefficients – ˆ qand energy loss dE/dx – for elastic and\ninelastic reactions for different scenarios for the strong cou-\npling as presented in Section III B and compare our results\nwith previous estimations from the literature. Also we study\nthe dependence of transport coefficients on the mass of emit-\nted gluons within the DQPM.\nWe will show the results for the ’elastic’ and ’inelastic’ (or\nradiative) ˆ qas well as for the sum of both elastic and inelastic\ncontributions:\nˆq= ˆqelastic+ ˆqinelastic;\nsimilar holds for the energy loss:\ndE/dx = (dE/dx )elastic+ (dE/dx )inelastic.\nA. Dependence of transport coefficients ˆqand dE/dx\non the mass of the emitted gluon\nIn this section we study the influence on the mass of emit-\nted gluons from inelastic scattering of the jet parton with the7\n0.5 1.0 1.5 2.0 2.5\nkt [GeV/c]101\n100101102dqq/dkt [mb/GeV]\n/radicalbig\ns=10 GeV, T=0.3 GeV\nMg=0.7 GeV\nMg=0\nBAMPS\n1 2 3 4\nqt [GeV/c]101\n100dqq/dqt [mb/GeV]\n/radicalbig\ns=10 GeV, T=0.3 GeV\nMg=0.7 GeV\nMg=0\n0 1 2 3 4 5\nEg [GeV]101\n100dqq/dEg [mb/GeV]\n/radicalbig\ns=10 GeV; T=0.3 GeV\nMg=0.7 GeV\nMg=0\n0.0 0.2 0.4 0.6 0.8\nx101\n100xd/dx [mb]\n/radicalbig\ns=10 GeV, T=0.3 GeV\nMg=0.7 GeV\nMg=0\nFIG. 3. DQPM differential cross sections dσ/dk tas a function of the transverse momentum ktof the emitted gluon (upper left),\ndσ/dq tas a function of the transverse momentum qtof the jet (upper right), dσ/dE gas a function of the energy Egof the emitted gluon\n(lower left), and x dσ/dx as a function of the longitudinal ratio x=pg\nL/pj\nL(lower right) for q+q→q+q+gscattering at√s= 10\nGeV and T= 0.3 GeV in case of massive (thermal) emitted gluon (blue lines) and massless emitted gluon (orange lines). The green\nline on the upper left plot shows dσ/dk tcalculated with the pQCD BAMPS model for αs= 0.3 and massless partons without LPM cutoff.\nthermal sQGP medium – described by the DQPM quasipar-\nticles – on the transport coefficients ˆ qand energy loss dE/dx .\nFor this study we consider the pure DQPM model (Scenario\n0).\nFig. 4 shows the temperature dependence of the scaled\nˆqcoefficient (left plots) and the scaled energy loss dE/dx\n(right plots) for elastic (blue lines), inelastic scattering (or-\nange lines) with radiation of a massive gluon (solid lines) and\nmassless ( Mg= 0) gluon (dashed lines) for two quark jet\nmomenta: p= 10 GeV /c(upper plots) and p= 100 GeV /c\n(lower plots). Similar to differential cross sections, the ˆ q\nanddE/dx strongly increase as the gluon mass decreases.\nMoreover, for the massless emitted gluon the inelastic ˆ qand\ndE/dx become significant even for small jet momenta of 10\nGeV/c.\nFig. 5 displays the inelastic ˆ q/T3(left plot) and energy loss\n(1/T)2dE/dx (right plot) as functions of the emitted gluon\nmass Mg. For all displayed temperatures and jet momenta\nthe values of ˆ qand energy loss dE/dx monotonically decrease\nwith increasing of the emitted gluon mass.\nB. Comparison of jet transport coefficients with\ndifferent scenarios for the effective coupling constant\nHere we investigate the influence of the effective coupling\ngin thermal, jet, and radiative vertices on the temperatureand energy dependence of ˆ qanddE/dx within the different\nScenarios 0-IV as described in Section III B. We stress that all\ncalculations of Feynman diagrams are presented within the\n”standard” DQPM propagators and masses of quasiparticles\ntaken at the pole mass of the spectral function, i.e. only the\ncoupling constants in the vertices are varied – cf. Table I\nand Fig. 2.\nIn Figs. 6 and 7 we show the temperature dependence\nof the scaled ˆ qanddE/dx , respectively, for Scenarios 0-IV\n(from top to bottom) for two momenta of the jet parton of\np= 10 GeV /c(left plots) and p= 100 GeV /c(right plots).\nThe blue lines represent the scaled transport coefficients for\nelastic processes only, while the orange lines stand for the\nresults of the sum of elastic and inelastic contributions.\n•The upper rows in Figs. 6 and 7 show the standard\nDQPM results (”Scenario 0”) where for all vertices the ther-\nmalg(T) is used. As seen from these plots, accounting\nfor the inelastic scattering substantially increases ˆ q/T3and\n(1/T2)dE/dx , especially at low temperatures due to the\nstrong T-dependence of the strong coupling g(T) – cf. Fig.\n2.\n•The second row in Figs. 6 and 7 show the results for the\n”Scenario I” where all vertices (thermal, jet parton, emitted\ngluon) have been taken as a constant g=√\n4π·0.3, i.e.\nαs= 0.3. This scenario allows to illustrate explicitly the\ninfluence of the choice of the coupling gon ˆqanddE/dx .\nOne can see that the scaled ˆ qdecreases with decreasing T, so8\n0.2 0.3 0.4 0.5 0.6\nT [GeV]100101102q/T3\np=10 GeV/c\nelastic\ninelastic, Mg=Mg(T)\ninelastic, Mg=0\n0.2 0.3 0.4 0.5 0.6\nT [GeV]100101102(1/T2)dE/dxp=10 GeV/c\nelastic\ninelastic, Mg=Mg(T)\ninelastic, Mg=0\n0.2 0.3 0.4 0.5 0.6\nT [GeV]100101102103q/T3\np=100 GeV/c\nelastic\ninelastic, Mg=Mg(T)\ninelastic, Mg=0\n0.2 0.3 0.4 0.5 0.6\nT [GeV]100101102103(1/T2)dE/dxp=100 GeV/c\nelastic\ninelastic, Mg=Mg(T)\ninelastic, Mg=0\nFIG. 4. Temperature dependence of ˆ q/T3(left plots) and of the energy loss (1 /T2)dE/dx (right plots) for elastic (blue lines) and\ninelastic (orange lines) scattering for different momenta of the jet parton: p= 10 GeV /c(upper plots) and 100 GeV /c(lower plots).\nThe inelastic cross sections for the emission of a massive gluon are shown by solid lines while for a massless ( Mg= 0) gluon by dashed\nlines.\n0.0 0.2 0.4 0.6 0.8 1.0\nMg [GeV]101102qinel/T3\nT=0.2 GeV\nT=0.4 GeVp=10 GeV/c\np=100 GeV/c\n0.0 0.2 0.4 0.6 0.8 1.0\nMg [GeV]101102(1/T2)dEinel/dxT=0.3 GeV\nT=0.4 GeV\np=10 GeV/c\np=100 GeV/c\nFIG. 5. Scaled inelastic jet transport coefficient ˆ q/T3(left plot) and the energy loss (1 /T2)dE/dx (right plot) as functions of the\nemitted gluon mass Mgfor different momenta of jet p= 10 GeV /c(solid lines) and 100 GeV /c(dashed lines) for two temperatures\nT= 0.2 GeV (blue lines) and T= 0.4 GeV (orange lines).\nthe strong rise of ˆ qfor ”Scenario 0” at low Tcomes from the\nincrease of the DQPM strong coupling g(T) in the vicinity\nofTcas illustrated in Fig. 2. The same holds for the energy\nlossdE/dx , however, the relative contribution of inelastic\nreactions is larger than that for ˆ qand the p-dependence is\nstronger.•The third row in Figs. 6 and 7 shows the ”Scenario II”\nwhere the coupling constants in the jet parton and emitted\ngluon vertices are taken from the Zakharov model [31, 79]\nand depend on the transverse momenta squared in the cor-\nresponding vertex – g(Q2) org(k2\nt). The scaled ˆ qfor this\nscenario shows a small rise with decreasing T. The dashed9\nT [GeV]101\n100101102q/T3\nScenario 0: DQPMp=10 GeV/c\nelastic\ninelastic\nelastic + inelastic\nT [GeV]101\n100101102p=100 GeV/c\nT [GeV]101\n100101102q/T3\nScenario I\nT [GeV]101\n100101102\nT [GeV]101\n100101102q/T3\nScenario II\nT [GeV]101\n100101102\nT [GeV]101\n100101102q/T3\nScenario III\nT [GeV]101\n100101102\n0.2 0.3 0.4 0.5 0.6\nT [GeV]101\n100101102q/T3\nScenario IV\n0.2 0.3 0.4 0.5 0.6\nT [GeV]101\n100101102\nFIG. 6. Temperature dependence of the scaled ˆ qcoefficient for Scenarios 0-IV (from top to bottom) for the strong coupling for two\nmomenta of the jet parton of p= 10 GeV /c(left plots) and p= 100 GeV /c(right plots). Blue lines represent the results for elastic\nprocesses only, while orange lines represent the results for the sum of elastic and inelastic contributions.\nlines show the dependence of ˆ qon the choice of the param-\neterαfr\nsin Eq. (12): the solid lines stand for αfr\ns= 1.05\n(vacuum value) while the dashed line indicates the results\nforαfr\ns= 0.42 (in medium value). One can see that the\nin-medium modification of αfr\nsonly slightly reduces ˆ q. This\nreduction is more prominent for inelastic reactions depicted\nby the green lines. Again, similar observations are valid for\ndE/dx with stronger T- and p- dependencies.\n•The fourth row in Figs. 6 and 7 represent the ”ScenarioIII” with an energy dependent g(E) – in line with the QLBT\nmodel [80, 81] – in the jet and radiative vertices. This leads\nto a quite flat ˆ qof smaller value as well as only a small in-\ncrease of ˆ qwith jet energy (cf. left and right plots) compared\nto the other scenarios due to the fact the g(E) is smaller than\ngfrom the other scenarios; it doesn’t depend on Tand de-\ncrease with Eas demonstrated in Fig. 2 by the green lines.\nFordE/dx we observe a similar trend, however, the rela-\ntive contribution of inelastic reactions is smaller than for the10\nT [GeV]101\n100101102(1/T2)dE/dxScenario 0: DQPMp=10 GeV/c\nelastic\ninelastic\nelastic + inelastic\nT [GeV]101\n100101102p=100 GeV/c\nT [GeV]101\n100101102(1/T2)dE/dxScenario I\nT [GeV]101\n100101102\nT [GeV]101\n100101102(1/T2)dE/dxScenario II\nT [GeV]101\n100101102\nT [GeV]101\n100101102(1/T2)dE/dxScenario III\nT [GeV]101\n100101102\n0.2 0.3 0.4 0.5 0.6\nT [GeV]101\n100101102(1/T2)dE/dxScenario IV\n0.2 0.3 0.4 0.5 0.6\nT [GeV]101\n100101102\nFIG. 7. Temperature dependence of the scaled energy loss dE/dx for different Scenarios 0-IV for two momenta of the jet parton with\np= 10 GeV /c(left side) and p= 100 GeV /c(right side). Blue lines represent results for elastic processes only, while orange lines\nrepresent results for the sum of elastic and inelastic contributions.\nother scenarios.\n•The fifth row in Figs. 6 and 7 displays the ”Scenario\nIV” in line with the couplings used in the DREENA model\n[82–85] implying the HTL interaction of the jet quark with\nthe thermal QCD medium. Here the jet vertex is replaced by\ntheg(ET) coupling and the radiative vertex by g(Q2) where\nQ2stands for the virtuality of the intermediate parton be-\nfore(after) the gluon emission. In this scenario the behaviour\nof ˆqanddE/dx is similar to ”Scenario III” and dominatedby the energy dependence of the coupling g(ET) – cf. orange\nlines in Fig. 2.\nThe results for the temperature dependence of the elastic\n+ inelastic ˆ q/T3for Scenarios 0-IV (for the momentum of a\njet parton of p= 10 GeV /c) are combined now in Fig. 8 in\nterms of blue lines (cf. legend) and compared to the different\nmodels: the pink dash-dotted line represents the LBT results\nforNf= 3 and p= 10 GeV /c[86], while the red (upper)\nand purple (lower) areas represent lQCD estimates [87] for11\npure SU(3) gauge theory and (2+1) flavour QCD, respec-\ntively, in the limit of an infinitely hard jet parton. The gray\narea corresponds to the results from the JETSCAPE Col-\nlaboration ( p= 100 GeV /c) [49]. The black dots show the\nphenomenological extraction by the JET Collaboration pre-\nsented for p= 10 GeV /c[48]. The yellow and red dots rep-\nresent results from the phenomenological extraction within\nthe BDMPS-Z quenching formalism using data of inclusive\nparticle suppression at RHIC and LHC energies for the two\ndistinct hydro frameworks from Ref. [88] for dynamical evo-\nlution – ”Hirano” and KLN. The vertical gray dashed line\nindicates the critical temperature Tc= 0.158 GeV.\n0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nT [GeV]0123456789q/T3\nDQPM\nScen. I\nScen. II\nScen. III\nScen. IV\n \n Lattice [Pure SU(3)]\nLattice [(2+1)-flavor]\nJETSCAPE\nLBT [Nf=3]\nJET\nC. Andres et. al, Hirano\nC. Andres et. al, KLN\nFIG. 8. Temperature dependence of the scaled jet transport co-\nefficient ˆ q/T3. Blue lines describe the DQPM results for a quark\njet with mass M= 0.01 GeV and energy E= 10 GeV for the\nscenarios described in Sec. III B. The pink dash-dotted line rep-\nresents the LBT results for Nf= 3 and p= 10 GeV /c[86], while\nthe red (upper) and purple (lower) areas show lQCD estimates\n[87] for pure SU(3) gauge theory and (2+1) flavor QCD, respec-\ntively, in the limit of an infinitely hard jet parton. The gray area\ncorresponds to the results from the JETSCAPE Collaboration\n(p= 100 GeV /c) [49]. The black dots represent the phenomeno-\nlogical extraction by the JET Collaboration presented for p= 10\nGeV/c[48]. The yellow and red dots display results from [88].\nThe vertical gray dashed line indicates the critical temperature\nTc= 0.158 GeV.\nAs follows from Fig. 8 the behaviour of the scaled ˆ qis\ndominated by the choice of the strong coupling gand devi-\nates very strongly especially at low temperatures T. While\n”Scenario I” with a fixed αs= 0.3 shows a decrease of ˆ qat\nlowT, the DQPM (”Scenario 0”) shows a strong rise due to\nthe thermal g(T) for all vertices.\nWe stress that the difference between Scenario 0 and Sce-\nnario I is related only to the ratio ( g(T)/√\n4π·0.3)6which is\nchanging the T-dependence of ˆ q(T) drastically. Other Sce-\nnarios II-IV include the momentum/energy dependence of\nthe strong coupling in jet and radiative vertices that modifies\nT- as well as E−dependencies of ˆ qand suppresses the low T-\nrise of Scenario 0: ”thermal” DQPM. We note that ”Scenario\nII” (motivated by the QLBT model) can not be associated\ndirectly with the LBT calculations [86] since ”Scenario II”is based on the DQPM model for computing ˆ qwhere only\njet and radiative vertices have been replaced in line with the\nQLBT model. We recall that a similar finding of the strong\ndependence of the drag coefficient A(T) of a heavy quark\non the strong coupling gand the parton masses has been\npointed out in Ref. [89].\nAs seen from Fig. 8 the spread of the different model\nresults for ˆ qis very large and strongly depends on the model\nassumptions as well on the method used for the extraction\nof ˆqfrom the heavy-ion data on the ratio RAA(the nuclear\nmodification factor of high pTtransverse spectra of hadrons\ninA+Acollisions versus p+pcollisions scaled with Ncoll–\nthe number of binary collisions in A+A) and flow coefficients\nvn.\nIn Figs. 9 and 10 we present the jet momentum depen-\ndence of ˆ q/T3and (1 /T2)dE/dx , respectively, for differ-\nent scenarios for a jet parton with a medium temperature\nT= 0.2 GeV (left side) and T= 0.4 GeV (right side). The\nblue lines represent results for elastic processes only and the\norange lines represent the results for the sum of elastic and\ninelastic contributions. Here one can see that the scaled ˆ q\nanddE/dx for elastic and elastic + inelastic collisions grow\nwith increasing of jet momentum pfor all Scenarios for the\ncoupling constants. The p-dependence of the elastic ˆ qand\ndE/dx is weaker than for the inelastic one.\nAs follows from Figs. 9 and 10 the absolute values of ˆ q/T3\nanddE/dx depend strongly on the choice of the strong cou-\npling in the interaction vertices. The assumption of a fast\nthermalisation of the jet parton and emitted gluon in the\nsQGP leads to a larger energy loss at low Tand a transverse-\nmomentum broadening (Scenario 0) compared to the other\nscenarios with a strong coupling that depends on the en-\nergy/momentum transfer.\nThe different scenarios for the jet parton interaction in\nthe sQPG can be tested in future studies with realistic cal-\nculations of RAAandv2based on the PHSD microscopic\ntransport approach [51, 71] by incorporating the radiative\nprocesses and accounting for the LPM effects. We note that\nthe sensitivity of the RAAandv2to the choice of the strong\ncoupling has been addressed within the BAMPS model by\ncomparing the results for constant αs= 0.3 with those for\na temperature dependent αs(T) [37]. This sensitivity might\nbe larger within the DQPM based description of jet attenua-\ntion in the sQGP medium due to the additional temperature\ndependence of the DQPM parton masses and widths.\nVI. RELATION BETWEEN η/sAND T3/ˆq\nFinally, we explore the relation of ˆ qto the ratio of the\nspecific shear viscosity to entropy density η/s. It’s important\nto recognize that the jet quenching parameter ˆ q/T3serves as\na direct measure of the parton coupling strength within the\nmedium. Furthermore, generally one can anticipate that a\nhigher coupling strength will correspond to a lower value of\nη/s[90].\nIn Ref. [59] it was proposed that in the weakly coupled\nlimit η/sis proportional to T3/ˆq:\nη/s≈1.25T3\nˆq. (15)\nThis approximation can be tested within more flexible quasi-12\np [GeV/c]101\n100101102q/T3\nScenario 0: DQPMT=0.2 GeV\nelastic\ninelastic\nelastic + inelastic\np [GeV/c]101\n100101102T=0.4 GeV\np [GeV/c]101\n100101102q/T3\nScenario I\np [GeV/c]101\n100101102\np [GeV/c]101\n100101102q/T3\nScenario II\np [GeV/c]101\n100101102\np [GeV/c]101\n100101102q/T3\nScenario III\np [GeV/c]101\n100101102\n20 40 60 80 100\np [GeV/c]101\n100101102q/T3\nScenario IV\n20 40 60 80 100\np [GeV/c]101\n100101102\nFIG. 9. Momentum dependence of ˆ q/T3for different scenarios for a jet parton with medium temperature T= 0.2 GeV (left side) and\nT= 0.4 GeV (right side). Blue lines represent results for elastic processes only and orange lines represent results for the sum of elastic\nand inelastic contributions.\nparticle model frameworks, where we can explore properties\nof the QGP beyond the weakly interacting limit and grasp\nunderstanding towards the regime described by lQCD pre-\ndictions for thermodynamic observables. The violation of the\nlimit and more rigorous description of the QGP medium is\nespecially relevant for moderate jet energies. It is generally\nexpected that the assertion from Ref. [59], which posits the\ntemperature dependence of η/sas a linearly growing function\nofT3/ˆq, holds predominantly in the high-temperature, weakcoupling regime. Conversely, for temperatures T≤2Tc, the\nratio T3/ˆqis expected to decrease rapidly, significantly devi-\nating from the linear trajectory described by 1 .25T3/ˆq(for\nmore details, see discussion in [83]). Thus, there are two\nregimes of a weakly and strongly coupled medium which lead\nto different jet attenuation in the medium. The considered\nquasi-particle description differs from our model, therefore\nhere we aim to provide our findings employing different cou-\npling scenarios. Moreover, we include elastic and radiative13\np [GeV/c]101\n100101102(1/T2)dE/dxScenario 0: DQPMT=0.2 GeV\nelastic\ninelastic\nelastic + inelastic\np [GeV/c]101\n100101102T=0.4 GeV\np [GeV/c]101\n100101102(1/T2)dE/dxScenario I\np [GeV/c]101\n100101102\np [GeV/c]101\n100101102(1/T2)dE/dxScenario II\np [GeV/c]101\n100101102\np [GeV/c]101\n100101102(1/T2)dE/dxScenario III\np [GeV/c]101\n100101102\n20 40 60 80 100\np [GeV/c]101\n100101102(1/T2)dE/dxScenario IV\n20 40 60 80 100\np [GeV/c]101\n100101102\nFIG. 10. Momentum dependence of the energy loss dE/dx for different scenarios for a jet parton with medium temperature T= 0.2\nGeV (left side) and T= 0.4 GeV (right side). Blue lines represent results for elastic processes only and orange lines represent results\nfor the sum of elastic and inelastic contributions.\nscatterings.\nIn Fig. 11 we show the specific shear viscosity η/sas a\nfunction of the scaled temperature T/Tc(upper plot) and\nthe ratio of η/stoT3/ˆqas a function of the scaled tempera-\ntureT/Tc(lower plot). The red solid line in the upper plot\nshows η/scomputed by the RTA (relaxation-time approxi-\nmation) approach within the DQPM model [91] in compari-\nson to the lQCD data (symbols) for gluodynamics ( Nf= 0).\nThe blue area indicates the DQPM results (Scenario 0) for1.25T3/ˆqcomputed for the jet momentum range of 3 −10\nGeV/c(following the DREENA-A selection) which defines\nthe width of the spread results. Here ˆ qincludes the elastic\nand inelastic parts. The DQPM result is compared to the es-\ntimates from the DREENA model [82, 83] (green area). As\nexpected, the scaled results 1 .25T3/ˆqandη/sagree only at\nlarge T≈2.5−3.5Tcwhile at low Tthe ratio T3/ˆqdecreases\nwith Tstronger than η/s– similar to the DREENA model.\nThe deviation from the weak-coupling scaling (15) is14\ndemonstrated in the lower part of Fig. 11 which shows the\nratio of ( η/s)(T) toT3/ˆqas a function of the scaled tem-\nperature T/Tc. The dashed gray line corresponds to the\nweak interaction limit 1.25 from Ref. [59]. The blue area\nstands for the DQPM results for Scenario 0 with thermal\ncoupling g(T) and the thermal mass of the emitted gluon\nMq(T), which shows a strong deviation from 1.25 for lower\nT. The DQPM result is compared to the estimates from the\nDREENA model [82, 83] (green area). Both models show a\nsimilar behavior of η/s, however, the DQPM rises faster with\nT. On the low plot we present the ratio ( η/s)(T) toT3/ˆqfor\nthe DQPM results for different scenarios of strong coupling\nseparately: the blue area stands for the DQPM results for\nScenario 0 from the upper plot; the red area illustrates the\nScenario I – the DQPM with αs= 0.3 and thermal mass\nof the emitted gluon Mg(T), which decreases with decreas-\ningT/Tc; the orange area stands for Scenario II (Zakharov\ncouplings), which shows a much flatter distribution.\nThus, our results demonstrate that the ratio η/stoT3/ˆq\nhas a strong dependence – especially when approaching Tc\n– on the choice of the strong coupling gin the scattering\nvertices. For the models with a temperature-dependent g(T)\nas in the DQPM, the ratio deviates from a constant at low\nTwhich corresponds to the strong interaction regime and\nit is approaching the scale of 1.25 at high Tas predicted\nin Ref. [59], where the strong coupling g(T) decreases to a\nsmall value of 0.3.\nVII. CONCLUSIONS\nWe have studied the jet transport coefficients such as ˆ q–\nthe squared average transverse momentum exchange between\nthe medium and the fast parton per unit length – as well as\n∆E=dE/dx – the energy loss per unit length, including the\nelastic and inelastic reactions of jet partons with the thermal\nsQGP described by DQPM in terms of strongly interacting\nquasiparticles which properties are extracted from the com-\nparison to the lQCD equation-of-state. The transport coef-\nficients have been calculated within kinetic transport theory\nby accounting for the independent elastic or inelastic colli-\nsions of a jet parton with a thermal parton. The contribu-\ntions of elastic ( q+q→q+qandq+g→q+g) and inelastic\n(q+q→q+q+gandq+g→q+g+g) reactions have been cal-\nculated by evaluating the leading order Feynman diagrams\nwith effective propagators and vertices from the DQPM by\naccounting for all channels and their interferences without\napproximations [46].\nWe have studied different scenarios for elastic and inelas-\ntic scattering of jet parton with a parton from the sQGP\nmedium: we start with the ”default” DQPM calculations\nwhere one assumes that all scattered partons, as well as the\nemitted gluon for inelastic reactions, are in thermal equilib-\nrium. However, since the scattered jet parton and the emit-\nted gluon might be out of equilibrium, we additionally have\ninvestigated the influence of different choices for the strong\ncoupling constant αsin thermal, jet parton, and radiative\nvertices as considered in the literature.\nOur findings can be summarized as following:\n•We have found a strong dependence of transport coeffi-\ncients on the choice of the strong coupling gused in thermal,\njet parton and emitted gluon vertices (cf. Table I). This de-\n1.0 1.5 2.0 2.5 3.0 3.5\nT/Tc0.00.20.40.60.81.0#\n/s (DQPM)\n1.25×T3/q (DQPM)\n1.25×T3/q (DREENA)\n1/(4)\nJHEP 04 101\nPRL 94 072305\nPoS LATTICE2007 221\nPRD 76 101701\n1.0 1.5 2.0 2.5 3.0 3.5\nT/Tc0123456/s×q/T3\nDQPM\nDQPM, s=0.3\nDQPM Zakharov\n1.25FIG. 11. Upper plot: Specific shear viscosity η/sobtained from\nEq. (15) as a function of the scaled temperature T/Tc: the blue\narea corresponds to the Scenario 0 – the DQPM with thermal\ng(T) and thermal mass of the emitted gluon Mq(T); the green area\nshows the estimates from the DREENA model [83]. The widths of\nthe areas correspond to the jet momentum range of 3 −10 GeV /c.\nThe dashed gray line shows the infinitely strong coupling limit or\nKovtun-Son-Starinets bound [92] ( η/s)KSS= 1/4π. The symbols\ndisplay lQCD data for pure SU(3) gauge theory taken from Refs.\n[93] (blue circles), [94] (orange circles), [95] (green circles), [96]\n(red circles). Lower plot: product of η/sand ˆq/T3as a function\nof the scaled temperature T/Tc. The dashed gray lines show the\nweak interaction limit [59]. The blue area corresponds to the\nScenario 0 (from the upper plot), the red area shows the Scenario\nI – the DQPM with αs= 0.3 and thermal mass of the emitted\ngluon Mg(T); the orange area indicates the Scenario II (Zakharov\ncouplings) and thermal mass of the emitted gluon Mg(T).\npendence is stronger for the inelastic reactions compared to\nelastic ones due to the extra g2in the gluon emission vertex.\n•A strong rise of ˆ qfor elastic and inelastic collisions and\nan increase of dE/dx for inelastic reactions in the vicinity of\nT→Tcfor the default DQPM scenario is related to the rapid\nrise of the DQPM coupling g(T) at low T– cf. Fig. 2. The\nscenario with constant αs= 0.3 shows a decrease of trans-\nport coefficients at low Twhich is related to the structure\nof the matrix element squared of the DQPM. Other scenar-\nios with different momentum-dependent jet parton and gluon\nemitted vertices show a flatting of ˆ qand a less steep decrease\nofdE/dx at low T.\n•We have found a rise of ˆ qanddE/dx with jet momentum\np, which is stronger for inelastic reactions compared to the15\nelastic ones.\n•The contribution of inelastic reactions to the total ˆ qis the\nstrongest for the default DQPM scenario due to the larger\nvalue of the strong coupling g(T) compared to other scenar-\nios.\n•We have observed a strong dependence of inelastic ˆ qon the\nmass of the emitted gluon – ˆ qis larger for Mg= 0 compared\nto the thermal mass of the emitted gluon Mg=Mg(T) for\nall jet momenta p; ˆqdecreases with increasing mass of the\nemitted gluon; similar observations hold for dE/dx .\n•We have checked the validity of the scaling η/s≈1.25T3\nˆq\nproposed in Ref. [59] for the DQPM model. We have demon-\nstrated that the ratio T3/ˆqtoη/sstrongly depends on the\nchoice of αsin the scattering vertices. It rises steeply with a\nlowering of Tfor the case of a temperature-dependent strong\ncoupling g(T), only at large Tthe ratio falls to the predicted\nscaling value of 1.25 [59], while for constant αsand transverse\nmomentum dependent gthe ratio approaches a constant al-\nready at T1.5Tc. Thus, in line with the findings in Ref. [83]\nfor the HTL-based model, this scaling is valid only in the\nweak coupling regime of large Tand violated for the strong\ncoupling regime of small T.\nOur findings are relevant for the interpretation of exper-imental observables on jet attenuation in the medium and\nextraction of the transport coefficients ˆ qanddE/dx as well\nas their relation to η/sfrom heavy-ion data. This will be\naddressed in future study within the PHSD transport ap-\nproach.\nACKNOWLEDGEMENTS\nThe authors acknowledge inspiring discussions with J.\nAichelin, W. Cassing, M. Djordjevic, P.-B. Gossiaux, O.\nKaczmarek and F. Senzel. Furthermore, we acknowledge\nsupport by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) through the grant CRC-TR\n211 ”Strong-interaction matter under extreme conditions” –\nproject number 315477589 – TRR 211. I.G. also acknowl-\nedges support from the ”Helmholtz Graduate School for\nHeavy Ion Research”. This work is supported by the Euro-\npean Union’s Horizon 2020 research and innovation program\nunder grant agreement No 824093 (STRONG-2020). The\ncomputational resources have been provided by the Goethe-\nHLR Center for Scientific Computing.\n[1] J. Adams et al. (STAR), Phys. Rev. Lett. 91, 072304 (2003),\narXiv:nucl-ex/0306024.\n[2] C. Adler et al. (STAR), Phys. Rev. Lett. 90, 082302 (2003),\narXiv:nucl-ex/0210033.\n[3] K. Aamodt et al. (ALICE), Phys. Lett. B 696, 30 (2011),\narXiv:1012.1004 [nucl-ex].\n[4] G. Aad et al. (ATLAS), Phys. Rev. Lett. 105, 252303 (2010),\narXiv:1011.6182 [hep-ex].\n[5] M. H. Thoma and M. Gyulassy, Nucl. Phys. B 351, 491\n(1991).\n[6] Y. L. Dokshitzer, V. A. Khoze, and S. I. Troian, J. Phys. G\n17, 1481 (1991).\n[7] S. Mrowczynski, Phys. Lett. B 269, 383 (1991).\n[8] M. Gyulassy and X.-N. Wang, Nucl. Phys. B 420, 583 (1994),\narXiv:nucl-th/9306003.\n[9] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne,\nand D. Schiff, Nucl. Phys. B 483, 291 (1997), arXiv:hep-\nph/9607355.\n[10] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne,\nand D. Schiff, Nucl. Phys. B 484, 265 (1997), arXiv:hep-\nph/9608322.\n[11] R. Baier, Y. L. Dokshitzer, A. H. Mueller, and D. Schiff,\nNucl. Phys. B 531, 403 (1998), arXiv:hep-ph/9804212.\n[12] B. G. Zakharov, JETP Lett. 63, 952 (1996), arXiv:hep-\nph/9607440.\n[13] M. Gyulassy, P. Levai, and I. Vitev, Nucl. Phys. B 571, 197\n(2000), arXiv:hep-ph/9907461.\n[14] U. A. Wiedemann, Nucl. Phys. B 588, 303 (2000), arXiv:hep-\nph/0005129.\n[15] X.-f. Guo and X.-N. Wang, Phys. Rev. Lett. 85, 3591 (2000),\narXiv:hep-ph/0005044.\n[16] P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 06, 030\n(2002), arXiv:hep-ph/0204343.\n[17] N. Armesto, C. A. Salgado, and U. A. Wiedemann, Phys.\nRev. D 69, 114003 (2004), arXiv:hep-ph/0312106.\n[18] G. D. Moore and D. Teaney, Phys. Rev. C 71, 064904 (2005),\narXiv:hep-ph/0412346.\n[19] M. Djordjevic and U. W. Heinz, Phys. Rev. Lett. 101, 022302\n(2008), arXiv:0802.1230 [nucl-th].[20] K. Zapp, G. Ingelman, J. Rathsman, J. Stachel, and U. A.\nWiedemann, Eur. Phys. J. C 60, 617 (2009), arXiv:0804.3568\n[hep-ph].\n[21] A. Majumder, Phys. Rev. D 85, 014023 (2012),\narXiv:0912.2987 [nucl-th].\n[22] S. Caron-Huot and C. Gale, Phys. Rev. C 82, 064902 (2010),\narXiv:1006.2379 [hep-ph].\n[23] S. Stojku, B. Ilic, M. Djordjevic, and M. Djordjevic, Phys.\nRev. C 103, 024908 (2021), arXiv:2007.07851 [nucl-th].\n[24] J. a. Barata and Y. Mehtar-Tani, JHEP 10, 176 (2020),\narXiv:2004.02323 [hep-ph].\n[25] S. Cao and X.-N. Wang, Rept. Prog. Phys. 84, 024301 (2021),\narXiv:2002.04028 [hep-ph].\n[26] L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk Ser.\nFiz.92, 535 (1953).\n[27] A. B. Migdal, Phys. Rev. 103, 1811 (1956).\n[28] R. Baier, Y. L. Dokshitzer, S. Peigne, and D. Schiff, Phys.\nLett. B 345, 277 (1995), arXiv:hep-ph/9411409.\n[29] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne,\nand D. Schiff, Nucl. Phys. B 478, 577 (1996), arXiv:hep-\nph/9604327.\n[30] X.-N. Wang, M. Gyulassy, and M. Plumer, Phys. Rev. D 51,\n3436 (1995), arXiv:hep-ph/9408344.\n[31] B. G. Zakharov, J. Phys. G 48, 055009 (2021),\narXiv:2007.09772 [hep-ph].\n[32] P. B. Arnold, Phys. Rev. D 79, 065025 (2009),\narXiv:0808.2767 [hep-ph].\n[33] K. C. Zapp, J. Stachel, and U. A. Wiedemann, JHEP 07,\n118 (2011), arXiv:1103.6252 [hep-ph].\n[34] N. Armesto et al. , Phys. Rev. C 86, 064904 (2012),\narXiv:1106.1106 [hep-ph].\n[35] J. Casalderrey-Solana and E. Iancu, JHEP 08, 015 (2011),\narXiv:1105.1760 [hep-ph].\n[36] W. Ke, Y. Xu, and S. A. Bass, Phys. Rev. C 100, 064911\n(2019), arXiv:1810.08177 [nucl-th].\n[37] F. Senzel, The Landau-Pomeranchuk-Migdal effect within\na partonic transport approach , Ph.D. thesis, Goethe U.,\nFrankfurt (main), https://inspirehep.net/literature/1816215\n(2020).16\n[38] P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 01, 030\n(2003), arXiv:hep-ph/0209353.\n[39] P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 11, 001\n(2000), arXiv:hep-ph/0010177.\n[40] P. Arnold, T. Gorda, and S. Iqbal, JHEP 11, 053 (2020),\n[Erratum: JHEP 05, 114 (2022)], arXiv:2007.15018 [hep-ph].\n[41] P. Arnold, T. Gorda, and S. Iqbal, JHEP 04, 085 (2022),\narXiv:2112.05161 [hep-ph].\n[42] C. Andres, F. Dominguez, A. V. Sadofyev, and C. A. Sal-\ngado, Phys. Rev. D 106, 074023 (2022), arXiv:2207.07141\n[hep-ph].\n[43] C. Andres, L. Apolin´ ario, F. Dominguez, and M. G. Mar-\ntinez, (2023), arXiv:2307.06226 [hep-ph].\n[44] I. Grishmanovskii, T. Song, O. Soloveva, C. Greiner,\nand E. Bratkovskaya, Phys. Rev. C 106, 014903 (2022),\narXiv:2204.01561 [nucl-th].\n[45] M. Djordjevic, Phys. Rev. C 74, 064907 (2006), arXiv:nucl-\nth/0603066.\n[46] I. Grishmanovskii, O. Soloveva, T. Song, C. Greiner, and\nE. Bratkovskaya, (2023), arXiv:2308.03105 [hep-ph].\n[47] K. Zapp, J. Stachel, and U. A. Wiedemann, PoS High-pT\nphysics09 , 022 (2009), arXiv:0904.4885 [hep-ph].\n[48] K. M. Burke et al. (JET), Phys. Rev. C 90, 014909 (2014),\narXiv:1312.5003 [nucl-th].\n[49] S. Cao et al. (JETSCAPE), Phys. Rev. C 104, 024905 (2021),\narXiv:2102.11337 [nucl-th].\n[50] W. Cassing and E. L. Bratkovskaya, Nucl. Phys. A 831, 215\n(2009), arXiv:0907.5331 [nucl-th].\n[51] P. Moreau, O. Soloveva, L. Oliva, T. Song, W. Cassing,\nand E. Bratkovskaya, Phys. Rev. C 100, 014911 (2019),\narXiv:1903.10257 [nucl-th].\n[52] S. Plumari, W. M. Alberico, V. Greco, and C. Ratti, Phys.\nRev. D 84, 094004 (2011), arXiv:1103.5611 [hep-ph].\n[53] F. Scardina, S. K. Das, V. Minissale, S. Plumari, and\nV. Greco, Phys. Rev. C 96, 044905 (2017), arXiv:1707.05452\n[nucl-th].\n[54] B. G. Zakharov, (2023), arXiv:2311.18607 [hep-ph].\n[55] K. Boguslavski, A. Kurkela, T. Lappi, F. Lindenbauer, and\nJ. Peuron, (2023), arXiv:2312.00447 [hep-ph].\n[56] S. Shi, R. Modarresi Yazdi, C. Gale, and S. Jeon, Phys. Rev.\nC107, 034908 (2023), arXiv:2212.05944 [hep-ph].\n[57] J. Uphoff, F. Senzel, O. Fochler, C. Wesp, Z. Xu,\nand C. Greiner, Phys. Rev. Lett. 114, 112301 (2015),\narXiv:1401.1364 [hep-ph].\n[58] O. Fochler, J. Uphoff, Z. Xu, and C. Greiner, Phys. Rev. D\n88, 014018 (2013), arXiv:1302.5250 [hep-ph].\n[59] B. M¨ uller, Phys. Rev. D 104, L071501 (2021),\narXiv:2107.14775 [hep-ph].\n[60] A. Peshier and W. Cassing, Phys. Rev. Lett. 94, 172301\n(2005), arXiv:hep-ph/0502138.\n[61] W. Cassing, Nucl. Phys. A 795, 70 (2007), arXiv:0707.3033\n[nucl-th].\n[62] W. Cassing, Nucl. Phys. A 791, 365 (2007), arXiv:0704.1410\n[nucl-th].\n[63] H. Berrehrah, E. Bratkovskaya, T. Steinert, and W. Cassing,\nInt. J. Mod. Phys. E 25, 1642003 (2016), arXiv:1605.02371\n[hep-ph].\n[64] O. Soloveva, D. Fuseau, J. Aichelin, and E. Bratkovskaya,\nPhys. Rev. C 103, 054901 (2021), arXiv:2011.03505 [nucl-th].\n[65] O. Linnyk, E. L. Bratkovskaya, and W. Cassing, Prog. Part.\nNucl. Phys. 87, 50 (2016), arXiv:1512.08126 [nucl-th].\n[66] M. L. Bellac, Thermal Field Theory , Cambridge Mono-\ngraphs on Mathematical Physics (Cambridge University\nPress, 2011).\n[67] H. Berrehrah, W. Cassing, E. Bratkovskaya, and T. Steinert,\nPhys. Rev. C 93, 044914 (2016), arXiv:1512.06909 [hep-ph].[68] S. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg,\nC. Ratti, and K. K. Szabo, JHEP 08, 053 (2012),\narXiv:1204.6710 [hep-lat].\n[69] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and\nK. K. Szabo, Phys. Lett. B 730, 99 (2014), arXiv:1309.5258\n[hep-lat].\n[70] W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974).\n[71] W. Cassing, Eur. Phys. J. ST 168, 3 (2009), arXiv:0808.0715\n[nucl-th].\n[72] H. Berrehrah, E. Bratkovskaya, W. Cassing, P. B. Gossiaux,\nJ. Aichelin, and M. Bleicher, Phys. Rev. C 89, 054901 (2014),\narXiv:1308.5148 [hep-ph].\n[73] B. Svetitsky, Phys. Rev. D 37, 2484 (1988).\n[74] M. G. Mustafa, D. Pal, and D. K. Srivastava, Phys. Rev. C\n57, 889 (1998).\n[75] H. Berrehrah, P.-B. Gossiaux, J. Aichelin, W. Cassing,\nand E. Bratkovskaya, Phys. Rev. C 90, 064906 (2014),\narXiv:1405.3243 [hep-ph].\n[76] P. B. Gossiaux, J. Aichelin, T. Gousset, and V. Guiho, J.\nPhys. G 37, 094019 (2010), arXiv:1001.4166 [hep-ph].\n[77] J. Aichelin, P. B. Gossiaux, and T. Gousset, Phys. Rev. D\n89, 074018 (2014), arXiv:1307.5270 [hep-ph].\n[78] P. B. Gossiaux, J. Aichelin, T. Gousset, M. Nahrgang,\nV. Ozvenchuk, and K. Werner, Nucl. Phys. A 931, 581\n(2014), arXiv:1409.0900 [hep-ph].\n[79] B. G. Zakharov, JETP Lett. 112, 681 (2020),\narXiv:2011.01526 [hep-ph].\n[80] F.-L. Liu, W.-J. Xing, X.-Y. Wu, G.-Y. Qin, S. Cao, and\nX.-N. Wang, (2021), arXiv:2107.11713 [hep-ph].\n[81] F.-L. Liu, X.-Y. Wu, S. Cao, G.-Y. Qin, and X.-N. Wang,\nPhys. Lett. B 848, 138355 (2024), arXiv:2304.08787 [hep-ph].\n[82] D. Zigic, I. Salom, J. Auvinen, P. Huovinen, and M. Djord-\njevic, Front. in Phys. 10, 957019 (2022), arXiv:2110.01544\n[nucl-th].\n[83] B. Karmakar, D. Zigic, I. Salom, J. Auvinen, P. Huovinen,\nM. Djordjevic, and M. Djordjevic, Phys. Rev. C 108, 044907\n(2023), arXiv:2305.11318 [hep-ph].\n[84] M. Djordjevic and M. Djordjevic, Physics Letters B 734, 286\n(2014).\n[85] M. Djordjevic, Phys. Rev. C 80, 064909 (2009),\narXiv:0903.4591 [nucl-th].\n[86] Y. He, T. Luo, X.-N. Wang, and Y. Zhu, Phys. Rev. C 91,\n054908 (2015), [Erratum: Phys.Rev.C 97, 019902 (2018)],\narXiv:1503.03313 [nucl-th].\n[87] A. Kumar, A. Majumder, and J. H. Weber, “Jet trans-\nport coefficient ˆ qin (2+1)-flavor lattice qcd,” (2020),\narXiv:2010.14463 [hep-lat].\n[88] C. Andr´ es, N. Armesto, M. Luzum, C. A. Salgado, and\nP. Zurita, Eur. Phys. J. C 76, 475 (2016), arXiv:1606.04837\n[hep-ph].\n[89] H. Berrehrah, M. Nahrgang, T. Song, V. Ozvenchuck, P. B.\nGossiaux, K. Werner, E. Bratkovskaya, and J. Aichelin,\nFIAS Interdisc. Sci. Ser. , 151 (2017), arXiv:1604.02343 [hep-\nph].\n[90] A. Majumder, B. Muller, and X.-N. Wang, Phys. Rev. Lett.\n99, 192301 (2007), arXiv:hep-ph/0703082.\n[91] O. Soloveva, P. Moreau, and E. Bratkovskaya, Phys. Rev. C\n101, 045203 (2020), arXiv:1911.08547 [nucl-th].\n[92] P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett.\n94, 111601 (2005), arXiv:hep-th/0405231 [hep-th].\n[93] N. Astrakhantsev, V. Braguta, and A. Kotov, JHEP 04, 101\n(2017), arXiv:1701.02266 [hep-lat].\n[94] A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, 072305\n(2005), arXiv:hep-lat/0406009.\n[95] S. Sakai and A. Nakamura, PoS LATTICE2007 , 221 (2007),\narXiv:0710.3625 [hep-lat].\n[96] H. B. Meyer, Phys. Rev. D 76, 101701 (2007),\narXiv:0704.1801 [hep-lat]."    },    {        "title": "2402.04932v1.Growth_in_the_universal_cover_under_large_simplicial_volume.pdf",        "content": "arXiv:2402.04932v1  [math.DG]  7 Feb 2024GROWTH IN THE UNIVERSAL COVER UNDER\nLARGE SIMPLICIAL VOLUME\nHANNAH ALPERT\nAbstract. Consider a closed manifold Mwith two Riemannian metrics: one\nhyperbolic metric, and one other metric g. What hypotheses on gguarantee\nthat for a given radius r, there are balls of radius rin the universal cover of\n(M,g) with greather-than-hyperbolic volumes? We show that this conclusion\nholds for all r≥1 if (Vol( M,g))2is less than a small constant times the\nhyperbolic volume of M. This strengthens a theorem of Sabourau and is\npartial progress toward a conjecture of Guth.\n1.Introduction\nLetMbe a closed Riemannian manifold. For each r >0 we let Vr(M) denote\nthe supremal volume of a ball of radius rinM. We let /bardblM/bardbl∆denote the Gromov\nsimplicial volume of M; the definition appears in Section 3. The relevant properties\narethat/bardblM/bardbl∆dependsonlyonthefundamentalgroupof M, andifMishyperbolic\nthen/bardblM/bardbl∆is equal to the hyperbolic volume of Mtimes a constantdepending only\non the dimension of n. The purpose of this paper is to prove the following theorem.\nTheorem 1 (Main theorem) .For every n≥2, there exists δn>0such that if\nMis a closed, oriented, connected n-dimensional manifold admitting a hyperbolic\nmetric, and gis another Riemannian metric on Mwith(Vol(M,g))2\n/bardblM/bardbl∆< δn, then for\nallr≥1we have\nVr(/tildewiderM)≥Vr(Hn),\nwhere/tildewiderMdenotes the universal cover of (M,g), andHndenotes the hyperbolic n-\nspace.\nTheorem 1 strengthens the following theorem of Sabourau and also uses a large\npart of its proof.\nTheorem 2 ([Sab22]) .For every n≥2, there exists δn>0such that if Mis a\nclosed, oriented, connected n-dimensional manifold admitting a hyperbolic metric,\nandgis another Riemannian metric on MwithVol(M,g)< δn, then for all r≥1\nwe have\nVr(/tildewiderM)≥Vr(Hn).\nSabourau’s proof uses Papasoglu’s method of area-minimizing separ ating sets,\nintroduced in [Pap20]. The part of the proof of Theorem 1 that is diffe rent from the\nproof of Theorem 2 is contained mainly in [Alp22, Theorem 5], which comb ines Pa-\npasoglu’s method with simplicial volume to prove the following theorem, originally\ndue to Guth.\n12 HANNAH ALPERT\nTheorem 3 (Theorem 2 of [Gut11]) .For every n≥2, there exists δn>0such that\nifMis a closed, oriented, connected n-dimensional manifold admitting a hyperbolic\nmetric, and gis another Riemannian metric on MwithVol(M,g)\n/bardblM/bardbl∆< δn, then we have\nV1(/tildewiderM)≥V1(Hn).\nThe most ambitious conjecture along these lines is [Gut11, Conjectu re 2], which\nsays that if Vol( M,g)<Vol(M,hyp), then Vr(/tildewiderM)> Vr(Hn) for allr. This state-\nment strengthens the theorem of Besson, Courtois, and Gallot wh ich proves the\nconclusion for all sufficiently large r[BCG95]. A more feasible-sounding next step,\nsuggested by Guth, would be to combine the conclusion of Theorems 1 and 2 with\nthe hypothesis of Theorem 3.\nConjecture 4 ([Gut11]) .For every n≥2, there exists δn>0such that if Mis a\nclosed, oriented, connected n-dimensional manifold admitting a hyperbolic metric,\nandgis another Riemannian metric on MwithVol(M,g)\n/bardblM/bardbl∆< δn, then for all r≥1\nwe have\nVr(/tildewiderM)≥Vr(Hn).\nKaram has proved the n= 2 case of Conjecture 4 in [Kar15]. In Section 6 we\nspeculate on what would be needed to prove Conjecture 4 if combinin g Karam’s\nmethods with those of the present paper.\nSection 2 summarizes the proof of Theorem 1 and explains which part s are taken\nfrom [Sab22]. Section 3 presents the property of simplicial volume ne eded for the\nproof. Section 4 presents the use of Papasoglu’s method from [Alp 22]. Section 5\ncompletes the proof of Theorem 1, and Section 6 discusses open qu estions.\n2.Relationship with Sabourau’s proof\nIn this section we give an overview of the ingredients needed for the proof of\nTheorem 1. This includes stating the parts of [Sab22] that we use un altered, and\ncommenting on the changes in the remainder of the proof.\nTogetthecomparison Vr(/tildewiderM)≥Vr(Hn)forallr≥1,weneedtoshowthat Vr(/tildewiderM)\nhas at least the same exponential growth rate as Vr(Hn). Sabourau’s argument\nrelies on the following theorem.\nTheorem 5 ([DKL19]) .For each n, there exists κsuch that the fundamental group\nGof every closed, connected, hyperbolic n-dimensional manifold satisfies the κ-\nTits alternative: for every symmetric subset SofGcontaining the identity, either\nSgenerates a group of subexponential growth, or there exist t wo elements in Sκ\ngenerating a nonabelian free subgroup.\nOn any Riemannian manifold M, theMargulis function µ:M→R∪ {∞}\nexpresses the threshold between subexponential growth and ex ponential growth.\nAt each point x∈M, the function value µ(x) is defined to be the supremal value µ\nsuch that the subgroup of π1(M,x) generated by the loops based at xwith length\nat mostµis a group of subexponential growth. The function µ(x) is greater than\nor equal to the systole ofM, denoted by sys M, which is the infimal length of\na homotopically nontrivial loop in M. Forρ <1\n2µ(x), the image in π1(M) of\nthe ball B(x,ρ) has subexponential growth and thus is amenable; for ρ >1\n2µ(x),\nTheorem 5 exhibits two loops at xof length at most 2 ρκthat generate a nonabelianGROWTH AND SIMPLICIAL VOLUME 3\nfree subgroup of π1(M), and so the volumes of balls in /tildewiderMhave at least a certain\nexponential growth rate. Sabourau uses this idea to prove the fo llowing lemma.\nLemma 6 ([Sab22]) .For every n≥2, there exists a small number αn>0such that\nthe following holds. Let Mbe a closed, oriented, connected n-dimensional manifold\nadmitting a hyperbolic metric, and let gbe another Riemannian metric on M.\nSuppose that there exists x0∈Msuch that1\n2µ(x0)< αnandVolB/parenleftbig\nx0,1\n2µ(x0)/parenrightbig\n≥/parenleftbig1\n2µ(x0)/parenrightbign\nn!. Then for all r≥1we have Vr(/tildewiderM)≥Vr(Hn).\nBecause Sabourau does not state it in this way, we include the followin g notes\non how to deduce this statement from the proofs of [Sab22, Theor em 3.7, Lemma\n3.8, Corollaries 3.9 and 3.10].\nProof summary. As in [Sab22, Corollary 3.10], Theorem 5 implies that Msatisfies\ntheκ-Tits alternative for some κdepending only on n. Letµ0=µ(x0), and let/tildewidex0\ndenote any lift of x0to/tildewiderM. In our notation, [Sab22, Lemma 3.8] says that because\nof our hypothesis Vol B/parenleftbig\nx0,1\n2µ0/parenrightbig\n≥(1\n2µ0)n\nn!we have\nVolB/tildewiderM(/tildewidex0,r)≥/parenleftbig1\n2µ0/parenrightbign\nn!·/parenleftBigg\n3/floorleftbigg\nr−µ0\n2\n2κµ0/floorrightbigg\n−1/parenrightBigg\nfor allr≥1\n2µ0. The proof is unchanged, depending only on the definition of κ-Tits\nalternative. For sufficiently small αnwe have µ0<5κµ0<1, so the proof of Case\n1 of [Sab22, Theorem 3.7] gives a constant Cn,κsuch that\nVolB/tildewiderM(/tildewidex0,r)≥VolBHn(Cn,κr)\nfor allr≥1. Then as in [Sab22, Corollary 3.9] we can use Cn,κto find a large\nconstant λn,κsuch that applying the above inequality to λn,κMimplies the desired\nconclusion\nVolB/tildewiderM(/tildewidex0,r)≥VolBHn(r)\nfor allr≥1, using the value of λn,κto select αnsufficiently small. /square\nThe remainder of Theorem 2 is proved by showing in [Sab22, Theorem 2 .7 and\nProposition 3.3] that the assumption Vol( M,g)< δnimplies the existence of a\npointx0with the two specified properties. We prove Theorem 1 by showing th at\nthe assumption(Vol(M,g))2\n/bardblM/bardbl∆< δnis sufficient to guarantee such a point x0.\nWeinformallysummarizetheproofofTheorem1asfollows. Insteado fproducing\nonly one point x0, for each ε >0 we produce a finite set of points Z0(ε)⊆Mwith\nthe following properties:\n(1) VolB(x,1\n8µ(x))/greaterorsimilarµ(x)n−εfor each x∈Z0(ε), by Lemma 9. This plays\nthe role of the second property of x0in Lemma 6, and the proof is the same\nas Sabourau uses.\n(2) VolM≥VolB/parenleftbig\nx,1\n2µ(x)/parenrightbig\n/greaterorsimilarµ(x)n·#/parenleftbig\nZ0∩B/parenleftbig\nx,1\n4µ(x)/parenrightbig/parenrightbig\n−εfor each\nx∈Z0(ε), by Lemma 9. This property compensates for the fact that the\nballs around various points of Z0(ε) may overlap.\n(3) The number of points in Z0(ε) is at least1\n2n· /bardblM/bardbl∆, by Theorem 7 and\nLemma 8. In contrast, Sabourau’s proof only guarantees that th e number\nof points in Z0(ε) is nonzero. This statement that a large value of /bardblM/bardbl∆4 HANNAH ALPERT\nimplies a large number of points in Z0(ε), and not just one point, is the\nmain new idea in our proof.\nThe proof of Theorem 1, given in Section 5, consists of summing the v olume esti-\nmates over a suitable subset of Z0(ε) and taking the limit as εapproaches zero. To\nget the idea, the reader may safely ignore all instances of εon a first read.\n3.Amenable Reduction Lemma\nThe simplicial norm was introduced by Gromov in [Gro82]. Let z=/summationtext\niaiσi\nbe a singular d-cycle on a space P, with real coefficients ai∈Rand simplices\nσi: ∆d→P. Theℓ1norm of z, denoted by |z|1, is/summationtext\ni|ai|, and the simplicial\nnormof a given homology class is the infimum of |z|1over all cycles zrepresenting\nthe homology class. The simplicial norm of the class [ z] is denoted by /bardbl[z]/bardbl∆.\nThesimplicial volume of a closed, oriented, connected manifold M, denoted\nby/bardblM/bardbl∆, is the simplicial norm of the fundamental homology class of Mwith\nR-coefficients.\nThepropertyofsimplicialnormthatweuseinthispaperisTheorem7, Gromov’s\nAmenable Reduction Lemma. To understand the relationship betwee n simplicial\nnorm and amenable groups, it may be helpful to start with Gromov’s V anishing\nTheorem, found in [Gro82, Section 3.1] or [Iva87, Corollary 6.3]. The V anishing\nTheorem can be viewed as a special case of the Amenable Reduction L emma.\nTo state Theorem 7, we need to define what it means to color the ver tices of\na singular cycle. Given an arbitrary singular d-cyclez=/summationtext\niaiσiin a space P,\nwe form a ∆-complex in the following way. We take one Euclidean d-simplex for\neachσi, and identify the ( d−1)-dimensional faces of these simplices whenever the\ncorresponding singular ( d−1)-simplices are equal. Because ∂z= 0, every ( d−1)-\ndimensional face is identified with at least one other face. This comple x has a\nsimplicial d-cycle in which the coefficient of each simplex equals the coefficient in\nzof the corresponding singular simplex. We can think of zas the image of this\nsimplicial cycle under the map from the ∆-complex into P, and we can talk about\nthe vertices and edges of z, meaning the vertices and edges of the ∆-complex,\nequipped with their maps into P.\nIfPis path-connected, we define a amenable vertex coloring ofzto be a way\nto assign colors to the vertices of zsuch that for each color, if we take the union\nof all edges of zfor which both vertices are that color, then the subgroup of π1(P)\ngenerated by this 1-complex is amenable. In an amenable vertex colo ring, we also\nrequire that zdoes not contain any edges from a vertex to itself. (In this case,\ncoloring every vertex a different color gives, trivially, an amenable ve rtex coloring.)\nWe define a rainbow simplex of such a coloring to be any simplex in zfor which\nalld+1 vertices are different colors. The following theorem is from [Gro09 , Section\n3.2] (or see [AK16]).\nTheorem 7 (Amenable Reduction Lemma) .Letz=/summationtext\niaiσibe a singular cycle\non a path-connected cell complex P, with an amenable vertex coloring. Then the\nsimplicial norm of the homology class of zsatisfies\n/bardbl[z]/bardbl∆≤/summationdisplay\nrainbow σi|ai|.GROWTH AND SIMPLICIAL VOLUME 5\n4.Separating filtrations\nThis sectioncontainslemmasfrom[Alp22]that arebasedonPapaso glu’smethod\nof area-minimizing separating sets, which was introduced in [Pap20]. T o learn the\nmethod, anotherusefulexpositionisbyNabutovskyin [Nab22, The orem1.1], which\nhas as a corollary Gromov’s Systolic Inequality, first proven in [Gro83 , Theorem\n0.1.A]\nLetMbe a closed n-dimensional Riemannian manifold, and let ρ:M→Rbe a\npositive, bounded function. A ρ-separating filtration ofMconsists of sets\nM=Zn⊇Zn−1⊇ ··· ⊇Z1⊇Z0⊇Z−1=∅,\nsuch that for each i= 0,1,...,n, the setZi−1is aρ-separating subset ofZi; that\nis, foreachconnectedcomponentof Zi\\Zi−1, thereissome x∈Msuchthatthe ball\nB(x,ρ(x)) contains this connected component. (Henceforth we do not inc ludeZ−1\nin the notation.) We also require smoothness and transversality con ditions, given\nin [Alp22], which guarantee in particular that Madmits a triangulation in which\neachZiis ani-dimensional subcomplex. These conditions ensure the following\nlemma.\nLemma 8 (Lemma 4 of [Alp22]) .LetMbe a closed, connected n-dimensional\nRiemannian manifold, and let ρ:M→(0,∞)be a bounded function with the\nproperty that for every ball B(x,ρ(x))inM, its image in π1(M)is amenable.\nThen for every ρ-separating filtration\nM=Zn⊇Zn−1⊇ ··· ⊇Z1⊇Z0\nthere is a triangulation of Mwith an amenable vertex coloring, such that the number\nof rainbow simplices is 2n·#Z0.\nWe omit the proof, which can be obtained from the proof in [Alp22] by replacing\nthe constant radius Rby the varying radius ρand by requiring the contributions\ntoπ1(M) to be amenable instead of trivial. The process for constructing th e trian-\ngulation from the filtration is the same in both proofs.\nThe following lemma shows that it is possible to choose a ρ-separating filtration\nsuch that the balls around the points of Z0satisfy lower bounds on their volumes.\nLemma 9 (Lemma 7 of [Alp22]) .LetMbe a closed n-dimensional Riemannian\nmanifold. For every positive, bounded function ρ:M→Rand every ε >0, there\nexists aρ-separating filtration\nM=Zn⊇Zn−1⊇ ··· ⊇Z1⊇Z0,\nsuch that for all x∈Mand allr1,r2with0< r1< r2< ρ(x)we have\n#(Z0∩B(x,r1))·(r2−r1)n\nn!≤VolB(x,r2)+ε.\nWe omit the proof, which can be obtained from the proof in [Alp22] by replacing\nthe constant radius Rby the varying radius ρ, both in this lemma and in the\npreceding Lemma 6 of [Alp22]. When computing the choice of how close e achZi\nshould be to the infimal possible area, we use sup ρinstead of R.6 HANNAH ALPERT\n5.Main proof\nThe proof of Theorem 1 is obtained by combining Lemma 6 with the follow ing\nTheorem 10, which is similar both in statement and in proof to [Alp22, Th eorem\n5], but where the role of1\n2sysMis replaced by r0(M), defined as follows. Given a\nclosed, connected n-dimensionalRiemannian manifold M, wedefine the ball growth\nthreshold radius r0(M), orgrowth radius for short, to be the infimum of1\n2µ(x),\ntaken over only those x∈Mthat satisfy the property that for all r≤1\n2µ(x), we\nhaveVol B(x,r)≥rn\nn!. Hereµdenotesthe Margulisfunction asdefined in Section2.\nSabourau’s proof of Theorem 2 shows that if /bardblM/bardbl∆>0, thenr0(M) is finite.\nTheorem 10. LetMbe a closed, oriented, connected n-dimensional Riemannian\nmanifold. Then we have\n/bardblM/bardbl∆≤16n(n!)2·/parenleftbiggVolM\nr0(M)n/parenrightbigg2\n.\nAssuming Theorem 10, we can quickly finish the proof of Theorem 1.\nProof of Theorem 1. Using the αnguaranteed by Lemma 6, we choose δnto be\nδn=(αn)2n\n16n(n!)2.\nThen if(VolM)2\n/bardblM/bardbl∆< δn, using Theorem 10 we have\nr0(M)2n\n16n(n!)2≤(VolM)2\n/bardblM/bardbl∆< δn=(αn)2n\n16n(n!)2,\nand thus r0(M)< αn. Applying Lemma 6 completes the proof. /square\nProof of Theorem 10. We select a sequence εk→0 and apply Lemma 9 with ε=εk\nandρ=1\n2µ. In the resulting ρ-separating filtration, we define ρkby\nρk= min\nx∈Z0ρ(x).\nWe select a subset SofZ0in the following way. We consider the points of Z0in\norder from greatest to least value of ρ, and for each point x∈Z0, we include it in\nSif the ball B(x,1\n4ρ(x)) is disjoint from all balls B(s,1\n4ρ(s)) assranges over the\npoints already selected to be in S. Then the balls {B(s,1\n4ρ(s)) :s∈S}are disjoint\nand the balls {B(s,1\n2ρ(s)) :s∈S}coverZ0.\nApplying Lemma 9 with r1=1\n2ρandr2→ρ, for each s∈Swe have\n#/parenleftbigg\nZ0∩B/parenleftbigg\ns,1\n2ρ(s)/parenrightbigg/parenrightbigg\n≤(VolB(s,ρ(s))+εk)·n!/parenleftbig1\n2ρ(s)/parenrightbign≤(VolM+εk)·n!\n(1\n2ρk)n.\nApplying Lemma 9 with r1→0 andr2=1\n4ρ, for each s∈Swe have\n1≤VolB/parenleftbig\ns,1\n4ρ(s)/parenrightbig\n/parenleftbigg\n(1\n4ρ(s))n\nn!−εk/parenrightbigg≤VolB/parenleftbig\ns,1\n4ρ(s)/parenrightbig\n/parenleftbigg\n(1\n4ρk)n\nn!−εk/parenrightbigg.GROWTH AND SIMPLICIAL VOLUME 7\nCombining these inequalities with Lemma 8 and Theorem 7, we have\n/bardblM/bardbl∆≤2n·#Z0≤\n≤2n·/summationdisplay\ns∈S#/parenleftbigg\nZ0∩B/parenleftbigg\ns,1\n2ρ(s)/parenrightbigg/parenrightbigg\n≤\n≤2n·/summationdisplay\ns∈S(VolM+εk)·n!\n(1\n2ρk)n≤\n≤2n·(VolM+εk)·n!\n(1\n2ρk)n·/summationdisplay\ns∈SVolB/parenleftbig\ns,1\n4ρ(s)/parenrightbig\n/parenleftbigg\n(1\n4ρk)n\nn!−εk/parenrightbigg≤\n≤2n·(VolM+εk)·n!\n(1\n2ρk)n·VolM/parenleftbigg\n(1\n4ρk)n\nn!−εk/parenrightbigg.\nLetxkbe a point in Z0withρ(xk) =ρk. Because Mis compact, by passing to\na subsequence we may assume that the sequence xkconverges to a point x∈M.\nThe function ρ=1\n2µis continuous; in fact it is 1-Lipschitz. Thus ρk→ρ(x) and\ntaking the limit we have\n/bardblM/bardbl∆≤16n(n!)2·/parenleftbiggVolM\n(ρ(x))n/parenrightbigg2\n.\nWhat remains is to show that ρ(x)≥r0(M). For each xk, for allr≤ρ(xk), by\nLemma 9 we have\nrn\nn!≤VolB(xk,r)+εk.\nWe need to show that for all r≤ρ(x), we have\nrn\nn!≤VolB(x,r).\nIf so, then ρ(x)≥r0(M) by definition.\nFor allr, we have\nB(xk,r−d(x,xk))⊆B(x,r).\nBy the 1-Lipschitz property of ρ, ifr≤ρ(x), thenr−d(x,xk)≤ρ(xk). Thus we\nhave\n(r−d(x,xk))n\nn!≤VolB(xk,r−d(x,xk))+εk≤VolB(x,r)+εk,\nand taking the limit gives\nrn\nn!≤VolB(x,r),\nas desired. /square\n6.Open questions\nIn this open questions section, first we propose some possible stre ngthenings\nof Theorem 10, and then we propose some possible statements tha t could prove\nConjecture 4.\nTheorem10isstrongerthanthesamestatementwith r0(M)replacedby1\n2sysM.\nA different (asymptotic) strengthening of that statement about sysMis the follow-\ning theorem of Gromov.8 HANNAH ALPERT\nTheorem 11 (Theorem 6.4.D’ of [Gro83]) .There exists Cn>0such that if Mis\na closed, oriented, connected n-dimensional Riemannian manifold, then we have\n/bardblM/bardbl∆≤CnVolM\n(sysM)n·/parenleftbigg\nlog/parenleftbigg\nCnVolM\n(sysM)n/parenrightbigg/parenrightbiggn\n.\nWe can ask whether there is an analogous further strengthening o f Theorem 10.\nQuestion 12. Does Theorem 11 remain true if sysMis replaced by r0(M)?\nThe following further strengthening of Theorem 11 would imply Conje cture 4.\nQuestion 13. Is the following statement true? There exists Cn>0such that if\nMis a closed, oriented, connected n-dimensional Riemannian manifold, then for\nallr≤1\n2sysM, we have\n/bardblM/bardbl∆≤CnVolM\nrn·/parenleftbigg\nlog/parenleftbigg\nCnVr(M)\nrn/parenrightbigg/parenrightbiggn\n.\nThe proof of Theorem 11 constructs a metric minimizing the value ofVolM\n(sysM)n\nand then proves the statement above for this special metric. Thu s, it is reasonable\nto hope that the same statement may be true of an arbitrary metr ic. If so, then we\nwould have\nCnVr(M)\nrn≥exp/parenleftBigg\nr/parenleftbigg/bardblM/bardbl∆\nCnVolM/parenrightbigg1\nn/parenrightBigg\n,\nmeaning that ifVolM\n/bardblM/bardbl∆is small, then Vr(M) has a high exponential growth rate as\nlong asr≤1\n2sysM. Taking a covering space in which sys Mis arbitrarily large,\nthis statement would imply Conjecture 4.\nA second potential approach to Conjecture 4 is to try to extend K aram’s proof\nof then= 2 case to higher dimensions, as follows.\nQuestion 14. Is the following statement true? There exist Cn,εn>0such that\nifMis a closed, oriented, connected n-dimensional Riemannian manifold, then M\nhas a finite-sheeted covering space /hatwiderMsuch that the following holds. Suppose that\nin the(1\n2sys/hatwiderM)-separating filtration\n/hatwiderM=Zn⊇Zn−1⊇ ··· ⊇Z1⊇Z0,\nfor every i= 0,...,n−1, the set Ziis within εnof the infimal i-dimensional area\nof(1\n2sys/hatwiderM)-separating subsets of Zi+1. Then\nLengthZ1≤Cn·/parenleftBig\nVol/hatwiderM+/bardbl/hatwiderM/bardbl∆/parenrightBig\n.\nWe quickly sketch a proof, using the method of [Kar15], that an affir mative\nanswer to this question would imply Conjecture 4. By a rescaling argu ment similar\nto [Sab22, Corollary 3.9], it suffices to show that there are constant san,Rn,δ′\nn>0\nsuch that ifVolM\n/bardblM/bardbl∆< δ′\nn, thenVr(/tildewiderM)≥an·2rfor allr≥Rn. To do this, in a cover\n/hatwiderMwith1\n2sys/hatwiderM > rwe find a subset Z′\n1ofZ1that, when viewed as a graph, has\nminimum vertex degree at least 3, has no null-homotopic cycles, and h as at least\n/bardbl/hatwiderM/bardbl∆\n2nedges. Thus, ifVolM\n/bardblM/bardbl∆≤/parenleftBig\n1\n12·2nCn/parenrightBign\n, then the average edge length in Z′\n1isGROWTH AND SIMPLICIAL VOLUME 9\nat most1\n6. Then [Kar15, Theorem C] implies that there is a point x0inZ′\n1such\nthat for all rwith 1≤r≤1\n2sys/hatwiderM, we have\nLengthZ′\n1∩B(x0,r)≥3\n4·2r.\nUsing the near-minimizing property of each Zias in [Alp22, Lemma 6], this implies\nby induction that the i-dimensional area of Zi∩B(x0,r) is at least3\n42r−(i−1)−\n(i−1)εnifi≤r≤1\n2sys/hatwiderM, and in particular, that the volume of B(x0,r) is at\nleast3\n2n+12r−(n−1)εnifn≤r≤1\n2sys/hatwiderM. Taking εn→0 gives the desired lower\nbound for Vr(/tildewiderM).\nReferences\n[AK16] Hannah Alpert and Gabriel Katz, Using simplicial volume to count multi-tangent trajec-\ntories of traversing vector fields , Geom. Dedicata 180(2016), 323–338.\n[Alp22] Hannah Alpert, Simplicial volume and 0-strata of separating filtrations , arXiv e-prints\n(2022), arXiv:2211.06362, accepted for publication in Journal of Topology and Analysis .\n[BCG95] G. Besson, G. Courtois, and S. Gallot, Entropies et rigidit´ es des espaces localement\nsym´ etriques de courbure strictement n´ egative , Geom. Funct. Anal. 5(1995), no. 5, 731–\n799.\n[DKL19] Subhadip Dey, Michael Kapovich, and Beibei Liu, Ping-pong in Hadamard manifolds ,\nM¨ unster J. Math. 12(2019), no. 2, 453–471.\n[Gro82] Michael Gromov, Volume and bounded cohomology , Inst. Hautes ´Etudes Sci. Publ. Math.\n(1982), no. 56, 5–99 (1983).\n[Gro83] Mikhael Gromov, Filling Riemannian manifolds , J. Differential Geom. 18(1983), no. 1,\n1–147.\n[Gro09] Mikhail Gromov, Singularities, expanders and topology of maps. I. Homology versus\nvolume in the spaces of cycles , Geom. Funct. Anal. 19(2009), no. 3, 743–841.\n[Gut11] Larry Guth, Volumes of balls in large Riemannian manifolds , Ann. of Math. (2) 173\n(2011), no. 1, 51–76.\n[Iva87] N. V. Ivanov, Foundations of the theory of bounded cohomology , Journal of Soviet Math-\nematics 37(1987), no. 3, 1090–1115.\n[Kar15] Steve Karam, Growth of balls in the universal cover of surfaces and graphs , Trans. Amer.\nMath. Soc. 367(2015), no. 8, 5355–5373.\n[Nab22] Alexander Nabutovsky, Linear bounds for constants in Gromov’s systolic inequalit y and\nrelated results , Geom. Topol. 26(2022), no. 7, 3123–3142.\n[Pap20] Panos Papasoglu, Uryson width and volume , Geom. Funct. Anal. 30(2020), no. 2, 574–\n587.\n[Sab22] St´ ephane Sabourau, Macroscopic scalar curvature and local collapsing , Ann. Sci. ´Ec.\nNorm. Sup´ er. (4) 55(2022), no. 4, 919–936.\nAuburn University, 221 Parker Hall, Auburn, AL 36849\nEmail address :hcalpert@auburn.edu"    },    {        "title": "2402.04950v1.Global_hypoellipticity_for_a_class_of_first_order_evolution_equations_on_compact_Lie_groups.pdf",        "content": "arXiv:2402.04950v1  [math.AP]  7 Feb 2024GLOBAL HYPOELLIPTICITY FOR A CLASS OF FIRST ORDER EVOLUTION\nEQUATIONS ON COMPACT LIE GROUPS\nWAGNER A. A. DE MORAES\nAbstract. We present necessary and sufficient conditions to have global hypoellipticity for a class of\ncomplex-valued coefficient first order evolution equations d efined on T1×G, whereGis a compact Lie\ngroup. First, we show that the global hypoellipticity of the constant coefficient operator related to this\noperator is a necessary condition, but not a sufficient condit ion. Under certain hypothesis, we show\nthat the global hypoellipticity of this class of operator is completely characterized by Nirenberg-Treves’\ncondition ( P).\n1.Introduction\nIn this note, we study the global hypoellipticity of a class of first ord er evolution equations on T1×G,\nwhereGis a compact Lie group. Precisely, we consider the operator L:D′(T1×G)→ D′(T1×G)\ndefined\nL:=∂t+c(t)X+q,\nwhereXis a vector field on G,c∈C∞(T1),c(t) =a(t) +ib(t), andq∈Cand we are interested to\nestablish conditions on candqto ensure that u∈C∞(T1×G) whenever Lu∈C∞(T1×G).\nWe can summarize the results that we present in this work as follows:\nTheorem 1.1. LetGbe a compact Lie group and consider the operator L:D′(T1×G)→ D′(T1×G)\ndefined as\nL:=∂t+c(t)X+q,\nwhereX∈g,c∈C∞(T1),c(t) =a(t)+ib(t), andq∈C. Assume that\na)b/\\e}atio\\slash≡0;\nb)L0:=∂t+c0X+qis globally hypoelliptic, where c0is the average of c.\nThenLis globally hypoelliptic if and only if bdoes not change sign.\nThe case where b≡0 was completely characterized in [ 11] and it was obtained that Lis globally\nhypoelliptic if and only if L0is globally hypoelliptic, where L0=∂t+a0X+q, witha0the average of a\nonT1.\nIn [5], Greenfield and Wallach considered the case where G=T1,cis a constant, and q= 0, in which\nthey obtained that Lis globally hypoelliptic if, and only if, either Im( c)/\\e}atio\\slash= 0 orcis a non-Liouville\nirrational. When cis not constant and q= 0, J. Hounie showed in [ 8] that in the case where b≡0,\nthe operator Lis globally hypoelliptic if, and only if, L0=∂t+a0∂xis globally hypoelliptic, where a0\n2010Mathematics Subject Classification. Primary 35F05, 58D25; Secondary 35B10, 43A77.\nKey words and phrases. compact Lie groups, global hypoellipticity, Fourier serie s, complex-valued coefficient.\n12 WAGNER A. A. DE MORAES\nis the average of a. On the other hand, when b/\\e}atio\\slash≡0 the global hypoellipticity of Lis characterized by\nNierember–Treves’ condition ( P), which in this case translates to the property of the function bdoes not\nchange sign.\nIn[6], GreenfieldandWallachproposedthefollowingconjecture: ifaclose dsmoothorientablemanifold\nadmits a globally hypoelliptic vector field, then this manifold is C∞−diffeomorphic to a torus, and\nthis vector field is C∞−conjugated to a constant vector field whose coefficients satisfy a Diophantine\ncondition. The validity of this conjecture was proved for some part icular cases, including in the same\narticle the proof for the case where the manifold is a compact Lie gro up. In light of this result, it is\nnatural to assume that q/\\e}atio\\slash= 0 when considering Gas a non-commutative compact Lie group, as for\nexample in [ 9] and [11].\nStill in the toroidal case, in [ 1] Bergamasco studied perturbations of globally hypoelliptic vector fi elds\nby zeroth order terms. One of the results obtained was that in the case where bdoes not change sign,\nthen for any q, the operator remains globally hypoelliptic. It is worth noting here th at this phenomenon\ndoes not extend when considering Gas any compact Lie group. As stated in Theorem 1.1, the fact\nthatbdoes not change sign is not sufficient for the global hypoellipticity of L, and will be illustrated in\nExample 5.2.\nThe paper is organized as follow. In Section 2, we present some preliminaries results and the notation\nused throughout the paper is fixed. In Section 3we introduce the operator in question, accompanied\nby a discussion of the previously results related to it. In Section 4we give necessary conditions and in\nSection5we present the sufficient conditions of the Theorem 1.1.\n2.Preliminaries Results\nIn this section, we recall most of the notations and preliminary resu lts necessary for the development\nof this study. The references [ 4] and [12] contain a very detailed discussion of these concepts as well as\nproofs of all of the results presented here.\nLetGbe a compact Lie group of dimension dim G=dand let Rep( G) be the set of continuous\nirreducible unitary representations of G. SinceGis compact, every continuous irreducible unitary repre-\nsentationηis finite dimensional and it can be viewed as a matrix-valued function η:G→Cdη×dη, where\ndη= dimη. We say that η∼ψif there exists an unitary matrix A∈Cdη×dηsuch thatAη(x) =ψ(x)A,\nfor allx∈G. We will denote by /hatwideGthe quotient of Rep( G) by this equivalence relation.\nForf∈L1(G) the group Fourier transform of fatη∈Rep(G) is\n/hatwidef(η) =/integraldisplay\nGf(x)η(x)∗dx,\nwheredxis the normalized Haar measure on G. By the Peter-Weyl theorem, we have that\nB:=/braceleftBig/radicalbig\ndηηrs;η= (ηrs)dη\nr,s=1,[η]∈/hatwideG/bracerightBig\n,\nis an orthonormal basis for L2(G), where we pick only one matrix unitary representation in each class\nof equivalence, and we may write\nf(x) =/summationdisplay\n[η]∈/hatwideGdηTr(η(x)/hatwidef(η)).GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 3\nMoreover, the Plancherel formula holds:\n/bardblf/bardblL2(G)=\n/summationdisplay\n[η]∈/hatwideGdη/bardbl/hatwidef(η)/bardbl2\nHS\n1/2\n=:/bardbl/hatwidef/bardblℓ2(/hatwideG),\nwhere\n/bardbl/hatwidef(η)/bardbl2\nHS= Tr(/hatwidef(η)/hatwidef(η)∗) =dη/summationdisplay\nr,s=1/vextendsingle/vextendsingle/hatwidef(η)rs/vextendsingle/vextendsingle2.\nThe group Fourier transform of u∈ D′(G) at a matrix unitary representation ηis the matrix /hatwideu(η)∈\nCdη×dη, whose components are given by\n/hatwideu(η)rs=/a\\}bracketle{tu,ηsr/a\\}bracketri}ht,\nwhere/a\\}bracketle{t·,·/a\\}bracketri}htdenotes the distributional duality.\nLetLGbetheLaplace-Beltramioperatorof G. Foreach[η]∈/hatwideG, itsmatrixelementsareeigenfunctions\nofLGcorrespondent to the same eigenvalue that we will denote by −ν[η], whereν[η]≥0. Thus\n(2.1) −LGηrs(x) =ν[η]ηrs(x),for all 1≤r,s≤dη,\nand we will denote by\n/a\\}bracketle{tη/a\\}bracketri}ht:=/parenleftbig\n1+ν[η]/parenrightbig1/2\nthe eigenvalues of ( I−LG)1/2.\nForx∈G,X∈gandf∈C∞(G), define\nLXf(x) :=d\ndtf(xexp(tX))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0,\nand when there is no possibility of ambiguous meaning, we write only Xinstead ofLX.\nLetP:C∞(G)→C∞(G) be a continuous linear operator. The symbol of the operator Pinx∈G\nandη∈Rep(G),η= (ηrs)dη\nr,s=1is\nσP(x,η) :=η(x)∗(Pη)(x)∈Cdη×dη,\nwhere (Pη)(x)rs:= (Pηrs)(x), for all 1 ≤r,s≤dη, and we have\nPf(x) =/summationdisplay\n[η]∈/hatwideGdim(η)Tr/parenleftBig\nη(x)σP(x,η)/hatwidef(η)/parenrightBig\n,\nfor allf∈C∞(G) andx∈G.\nWhenP:C∞(G)→C∞(G) is a continuous linear left-invariant operator we have that σPis inde-\npendent of x∈Gand\n/hatwiderPf(η) =σP(η)/hatwidef(η),\nfor allf∈C∞(G) and [η]∈/hatwideGand, by duality, this remains true for all f∈ D′(G).\nWe have that Xis a left-invariant operator on GandiXis a symmetric operator on L2(G). Thus,\nfor each [η]∈/hatwideGwe can choose a representative η∈Rep(G) such that\n(2.2) σX(η)rs=iµr(η)δrs,1≤r,s≤dη,4 WAGNER A. A. DE MORAES\nwithµr(η)∈Rfor all [η]∈/hatwideG, and we have\n(2.3) |µr(η)| ≤C/a\\}bracketle{tη/a\\}bracketri}ht,\nfor all [η]∈/hatwideG, 1≤r≤dη, whereC >0 is a constant which depends only on X. Moreover, as iXis\nnot bounded from L2\ns(G) toL2\ns−1\n2(G), one can verify that there must exist a sequence ([ ηj])j∈N⊂/hatwideGand\nC >0 such that\n(2.4) µr(ηj)≥C/a\\}bracketle{tηj/a\\}bracketri}ht1\n2,\nfor some 1 ≤r≤dηjand everyj∈N. We may assume without loss of generality that r=dηj, for all\nj∈N. For a detailed discussion about these properties see [ 2], [11] and [12].\nGivenf∈L1(T1×G) andη∈Rep(G), the partial Fourier coefficient of fwith respect to the second\nvariable at ηis defined by\n/hatwidef(t,η) :=/integraldisplay\nGf(t,x)η(x)∗dx∈Cdη×dη, t∈T1,\nwith components\n/hatwidef(t,η)rs=/integraldisplay\nGf(t,x)η(x)srdx,1≤r,s≤dη.\nGivenu∈ D′(T1×G),η∈Rep(G) and 1 ≤r,s≤dη. Thers-component of the partial Fourier\ncoefficient of uwith respect to the second variable at ηis the linear functional defined by\n/hatwideu(·,η)rs:C∞(T1)−→C\nψ/mapsto−→ /a\\}bracketle{t/hatwideu(·,η)rs,ψ/a\\}bracketri}ht:=/a\\}bracketle{tu,ψ×ηsr/a\\}bracketri}ht,\nwhereψ×ηsr(t,x) :=ψ(t)ηsr(x).\nWe have the following characterization of C∞(T1×G) andD′(T1×G) through their partial Fourier\ncoefficients presentend on [ 10].\nTheorem 2.1 (Theorem 3.5 of [ 10]).LetGbe a compact Lie group, and let {/hatwidef(·,η)rs}be a sequence\nof functions on T1. Define\nf(t,x) :=/summationdisplay\n[η]∈/hatwideGdηdη/summationdisplay\nr,s=1/hatwidef(t,η)rsηsr(x).\nThenf∈C∞(T1×G)if and only if /hatwidef(·,η)rs∈C∞(T1), for all [η]∈/hatwideG,1≤r,s≤dηand for every\nβ∈N0andℓ>0there existCβℓ>0such that\n/vextendsingle/vextendsingle∂β\nt/hatwidef(t,η)rs/vextendsingle/vextendsingle≤Cβℓ/a\\}bracketle{tη/a\\}bracketri}ht−ℓ,∀t∈T1,[η]∈/hatwideG,1≤r,s≤dη.\nTheorem 2.2 (Theorem 3.6 of [ 10]).LetGbe a compact Lie groups and/braceleftbig\n/hatwideu(·,η)rs/bracerightbig\nbe a sequence of\ndistributions on T1. Define\nu=/summationdisplay\n[η]∈/hatwideGdηdη/summationdisplay\nr,s=1/hatwideu(·,η)rsηsr\nThenu∈ D′(T1×G)if and only if there exist K∈NandC >0such that\n(2.5)/vextendsingle/vextendsingle/a\\}bracketle{t/hatwideu(·,η)rs,ϕ/a\\}bracketri}ht/vextendsingle/vextendsingle≤CpK(ϕ)/a\\}bracketle{tη/a\\}bracketri}htK,\nfor allϕ∈C∞(T1)and[η]∈/hatwideG, wherepK(ϕ) :=/summationtext\nβ≤K/bardbl∂β\ntϕ/bardblL∞(T1).GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 5\n3.A class of first order evolution equations\nLetGbe a compact Lie group and consider the operator L:D′(T1×G)→ D′(T1×G) defined\n(3.1) L:=∂t+c(t)X+q,\nwhereXis a vector field on G,c(t) =a(t)+ib(t), witha,b∈C∞(T1;R) andq∈C.\nThe global hypoellipticity of ( 3.1) in the case where either cis a constant function or Im( c) =b≡0\nwas completely characterized in [ 11] in a more general context that can be adapted to our setting as\nfollow.\nTheorem 3.1. [Theorems 3.3 and 5.1 of [ 11]] LetGbe a compact Lie group, X∈gandc,q∈C. The\noperatorL=∂t+cX+qis globally hypoelliptic if and only if the following condit ions are satisfied:\n1. The set\nN={(k,[η])∈Z×/hatwideG;k+cµr(η)−iq= 0,for some 1≤r≤dη}\nis finite.\n2.∃C, M > 0such that\n(3.2) |k+cµr(η)−iq| ≥C(|k|+/a\\}bracketle{tη/a\\}bracketri}ht)−M,\nfor allk∈Z,[η]∈/hatwideG,1≤r≤dη,wheneverk+cµr(η)−iq/\\e}atio\\slash= 0.\nTheorem 3.2. [Propositions 6.6 and 6.12 of [ 11]] LetGbe a compact Lie group, X∈g,q∈Canda(t)\nbe a smooth real-valued function on T1. The operator L=∂t+a(t)X+qis globally hypoelliptic if and\nonly if the operator L0=∂t+a0X+qis globally hypoellitpic, where a0=/integraltext\nT1a(s)ds.\nHence, we will assume throughout the paper that b(t) is not a constant function.\nNotice that in Theorem 3.2the global hypoellipticity of Lwhenb≡0 is strictly related to the global\nhypoellipticity of the constant coefficient operator L0=∂t+a0X+qbecause these two operator are\nconjugated by the automorphism\n(3.3) Ψ au(t,x) :=/summationdisplay\n[η]∈/hatwideGdηdη/summationdisplay\nr,s=1eiµr(η)A(t)/hatwideu(t,η)rsηsr(x),\nwhereA(t) is a primitive for a(t), that is, Ψ a◦L=L0◦Ψa(for more details, see Section 6 of [ 11]).\nBy the same approach, in the case where bis not a constant function, we could define an operator\nΨcthat conjugates the operators LandL0, but we would not have the growth control of the term\ne−µr(η)B(t)that would appear in the definition of Ψ c, which implies that Ψ cis not an automorphism in\nthe space of smooth functions and therefore the global hypoellipt icity ofLis not equivalent to the global\nhypoellipticity of L0. Although these properties are not equivalent, we will prove in Prop osition4.1that\nthe global hypoellipticity of L0is a necessary condition for the global hypoellipticity of Land we give\nan example where this is not a sufficient condition.6 WAGNER A. A. DE MORAES\nFirst, observe that by the automorphism Ψ agiven in ( 3.3) we may assume that a(t) is a constant\nfunction. Henceforth we will study the operator,\n(3.4) L=∂t+(a0+ib(t))X+q,\nwherebis a real-valued smooth function on T1,Xis a vector field on Gandq∈C.\nConsider the equation Lu=f∈C∞(T1×G). Taking the partial Fourier coefficient with respect to\nthe second variable we obtain\n/hatwidef(t,η) =/hatwiderLu(t,η) =∂t/hatwideu(t,η)+c(t)σX(η)/hatwideu(t,η)+q/hatwideu(t,η),[η]∈/hatwideG,\nwhere the representation ηsatisfies ( 2.2). Hence, for each [ η]∈/hatwideGwe obtain the following system of\nequations\n(3.5) /hatwidef(t,η)rs=/hatwiderLu(t,η)rs=∂t/hatwideu(t,η)rs+i(µr(η)c(t)−iq)/hatwideu(t,η)rs,\nwith 1≤r,s≤dη.\nLet\nC(t) =/integraldisplayt\n0c(τ)dτ−c0t,wherec0=1\n2π/integraldisplay2π\n0c(τ)dτ.\nMultiplying by eiµr(η)C(t), we obtain\n∂t/hatwideu(t,η)rseiµr(η)C(t)+i(µr(η)c(t)−iq)/hatwideu(t,η)rseiµr(η)C(t)=/hatwidef(t,η)rseiµr(η)C(t)\nThen\n∂t/bracketleftbig\n/hatwideu(t,η)rseiµr(η)C(t)/bracketrightbig\n+i(µr(η)c0−iq)/hatwideu(t,η)rseiµr(η)C(t)=/hatwidef(t,η)rseiµr(η)C(t),\nthat is, for each η∈/hatwideGand 1≤r,s≤dη, we have that /hatwideu(t,η)rseiµr(η)C(t)is a solution of\n(3.6) ∂tv(t,η)rs+i(µr(η)c0−iq)v(t,η)rs=g(t,η)rs,\nwhereg(t,η)rs=/hatwidef(t,η)rseiµr(η)C(t). The solution of this differential equation on T1is given by:\nLemma 3.3. Letλ∈Cand consider the equation\n(3.7)d\ndty(t)+λy(t) =h(t),\nwhereh∈C∞(T1).\nIfλ /∈iZthen the equation (3.7)has a unique solution that can be expressed by\n(3.8) y(t) =1\n1−e−2πλ/integraldisplay2π\n0e−λsh(t−s)ds,\nor equivalently,\n(3.9) y(t) =1\ne2πλ−1/integraldisplay2π\n0eλrh(t+r)dr.\nIfλ∈iZand/integraltext2π\n0eλsf(s)ds= 0then we have that\n(3.10) y(t) =e−λt/integraldisplayt\n0eλsh(s)ds\nis a solution of the equation (3.7).GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 7\nIt follows from Lemma 3.3that (3.6) has a unique solution given by\nv(t,η)rs=1\n1−e−2πi(µr(η)c0−iq)/integraldisplay2π\n0e−i(µr(η)c0−iq)τg(t−τ,η)rsdτ,\nwheneverµr(η)c0−iq /∈Z, or equivalently by\nv(t,η)rs=1\ne2πi(µr(η)c0−iq)−1/integraldisplay2π\n0ei(µr(η)c0−iq)τg(t+τ,η)rsdτ.\nTherefore, we obtain\n(3.11) /hatwideu(t,η)rs=1\n1−e−2πi(µr(η)c0−iq)/integraldisplay2π\n0e−qτe−iµr(η)(c0τ−C(t−τ)+C(t))/hatwidef(t−τ,η)rsdτ,\nor equivalently,\n(3.12)/hatwideu(t,η)rs=1\ne2πi(µr(η)c0−iq)−1/integraldisplay2π\n0eqτe−iµr(η)(−c0τ−C(t+τ)+C(t))/hatwidef(t+τ,η)rsdτ.\nTo study the global properties of the operator Lwe have to control the behavior of the numerical\nsequence that precedes the integral in the expression above. Th us, we will use the following technical\nlemma which proof is similar to the toroidal case studied in [ 3].\nLemma 3.4. Are equivalent:\n1.:There exist C,M >0such that\n|k+c0µr(η)−iq| ≥C(|k|+/a\\}bracketle{tη/a\\}bracketri}ht)−M,\nfor allk∈Z,[η]∈/hatwideG,1≤r≤dη, whenever k+c0µr(η)−iq/\\e}atio\\slash= 0.\n2.:There exist C,M >0such that\n(3.13)/vextendsingle/vextendsingle/vextendsingle1−e±2πi(c0µr(η)−iq)/vextendsingle/vextendsingle/vextendsingle≥C/a\\}bracketle{tη/a\\}bracketri}ht−M,\nfor all[η]∈/hatwideG,1≤r≤dη, whenever c0µr(η)−iq /∈Z.\n4.Necessary Conditions\nIn this section, we will prove the necessary conditions stated in The orem1.1about the global hypoel-\nlipticity of the operator L(3.4). First, let us prove that the global hypoellipticity of the correspo nding\nconstant coefficient operator L0is a necessary condition.\nProposition 4.1. IfLis globally hypoelliptic, then L0is globally hypoelliptic.\nProof.Assume that L0is not globally hypoelliptic. By Theorem 3.1we have two cases to consider:\n(i) The set\nN={(k,[η])∈Z×/hatwideG;k+c0µr(η)−iq= 0,for some 1 ≤r≤dη}\nhas infinitely many elements or;\n(ii) for allM >0, there exists kM∈Zand [ηM]∈/hatwideGsatisfying\n0<|kM+c0µr(ηM)−iq| ≤(|k|+/a\\}bracketle{tηM/a\\}bracketri}ht)−M,\nfor some 1 ≤r≤dηM.8 WAGNER A. A. DE MORAES\nCase (i): Assumethat thereexistsasequence[ ηk]∈/hatwideGsuchthatc0µr(ηk)−iq∈Z, forsome1 ≤r≤dηk.\nfor allk∈N. We may assume without loss of generality that r= 1 for all [ ηk]∈/hatwideG. For eachk∈N, let\ntk∈[0,2π] such that\nmk:= max\nt∈[0,2π]/integraldisplayt\n0(Req−µ1(ηk)b(s))ds=/integraldisplaytk\n0(Req−µ1(ηk)b(s))ds.\nSet\n/hatwideu(t,η)rs=\n\nemkexp/braceleftBig\n−/integraltextt\n0(iµ1(ηk)c(s)+q)ds/bracerightBig\n,if [η] = [ηk] andr=s= 1,\n0, otherwise.\nSincec0µ1(ηk)−iq∈Z, for allk∈N, the sequence of functions {/hatwideu(t,η)rs}is well-defined on T1. Notice\nthat\n|/hatwideu(t,ηk)11|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleemkexp/braceleftbigg\n−/integraldisplayt\n0(Req−µ1(ηk)b(s))ds/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1,\nby the definition of mk. By Theorem 2.2, we have that u∈ D′(T1×G). Moreover, we have\n|/hatwideu(tk,ηk)11|= 1,\nfor allk∈N. By Theorem 2.1we conclude u /∈C∞(T1×G). Since each element of the sequence\n{/hatwideu(t,η)rs}satisfies\n∂t{/hatwideu(t,η)rs}+i(µr(η)c(t)−iq){/hatwideu(t,η)rs}= 0,\nfor all [η]∈/hatwideG, 1≤r,s≤dη, we conclude that Lu= 0, which implies that Lis not globally hypoelliptic.\nCase (ii): By the equivalence given in Lemma 3.4, we can construct a sequence [ ηk] satisfying for all\nk∈N\n(4.1) 0 <|1−e−2πi(c0µr(ηk)−iq)|</a\\}bracketle{tηk/a\\}bracketri}ht−k,\nfor some 1 ≤r≤dηk. We may assume r= 1 for convenience of notation and c0µ1(ηk)−iq /∈Zfor all\nk∈Z, because Nis finite.\nFor eachk∈N, choosetk∈[0,2π] such that\nmax\nt∈[0,2π]/integraldisplayt\n0(µ1(ηk)b(s)−Req)ds=/integraldisplaytk\n0(µ1(ηk)b(s)−Req)ds.\nNotice that with this choice we have\n(4.2)/integraldisplayt\ntk(µ1(ηk)b(s)−Req)ds≤0,for allt∈[0,2π].\nBy the compactness of the torus, we may assume, by passing to a s ubsequence, that there exists t0∈\n[0,2π] such that tk→t0, ask→ ∞.\nLetϕ∈C∞(T1) be a real-valued smooth function satisfying supp( ϕ)⊆I, 0≤ϕ(t)≤1, and\n/integraltext2π\n0ϕ(s)ds>0, whereIis a closed interval in (0 ,2π) such that t0/∈I.\nConsider\n/hatwidef(t,η)rs=\n\n(1−e−2πi(c0µ1(ηk)−iq))exp/braceleftBig\n−i/integraltextt\ntk(µ1(ηk)c(w)−iq)dw/bracerightBig\nϕ(t),if [η] = [ηk],r=s= 1,\n0, otherwise,GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 9\nfort∈[0,2π]. Since supp( ϕ)⊆I, the sequence {/hatwidef(t,η)rs}is well-defined on T1. Let us show that\n{/hatwidef(t,η)rs}defines a smooth function on T1×G. Forα∈Nwe have\n/vextendsingle/vextendsingle/vextendsingle∂α\nt/hatwidef(t,ηk)11/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle1−e−2πi(c0µ1(ηk)−iq)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nβ≤α/parenleftBigg\nα\nβ/parenrightBigg\n∂β\ntexp/braceleftbigg\n−i/integraldisplayt\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\n∂α−βϕ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nBy Fa` a di Bruno’s formula we have\n∂β\ntexp/braceleftbigg\n−i/integraldisplayt\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\n=/summationdisplay\nγ∈∆(β)β!\nγ!exp/braceleftbigg\n−i/integraldisplayt\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\n×β/productdisplay\nj=1/parenleftBigg\n−i∂j\nt/integraltextt\ntk(µ1(ηk)c(w)−iq)dw\nj!/parenrightBiggγj\n,\nwhere ∆(β) =/braceleftbigg\nγ∈Nβ\n0;β/summationtext\nj=1jγj=β/bracerightbigg\n.\nSince for all k∈Nwe have |µ1(ηk)| ≤ /a\\}bracketle{tηk/a\\}bracketri}htand/integraltextt\ntk(Req−µ1(ηk)b(w))dw≤0, for allt∈[0,2π], we\nobtain\n∂β\ntexp/braceleftbigg\n−i/integraldisplayt\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\n≤Cβ/a\\}bracketle{tηk/a\\}bracketri}htβ,\nfor someCβ>0. By (4.1) we obtain\n/vextendsingle/vextendsingle/vextendsingle∂α\nt/hatwidef(t,ηk)11/vextendsingle/vextendsingle/vextendsingle≤Cα/a\\}bracketle{tηk/a\\}bracketri}htα−k,\nfor someCα>0. By Theorem 2.1we conclude that f∈C∞(T1×G).\nLet us construct now a distribution u∈ D′(T1×G)\\C∞(T1×G) satisfying Lu=f. By (3.11), set\n/hatwideu(t,ηk)11=1\n1−e−2πi(c0µ1(ηk)−iq)/integraldisplay2π\n0exp/braceleftbigg\n−i/integraldisplayt\nt−τ(c(w)µ1(ηk)−iq)dw/bracerightbigg\n/hatwidef(t−τ,ηk)11dτ\nand/hatwideu(t,η)rs= 0 for all the other cases. Notice that for t−τ <0 we need to use the 2 π–periodic\nextension of fon de definition of /hatwideu(t,ηk)11. Hence,\n/hatwideu(t,ηk)11=1\n1−e−2πi(c0µ1(ηk)−iq)/integraldisplay2π\n0exp/braceleftbigg\n−i/integraldisplayt\nt−τ(c(w)µ1(ηk)−iq)dw/bracerightbigg\n/hatwidef(t−τ,ηk)11dτ\n=/integraldisplayt\n0exp/braceleftbigg\n−i/integraldisplayt\nt−τ(c(w)µ1(ηk)−iq)dw−i/integraldisplayt−τ\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\nϕ(t−τ)dτ\n+/integraldisplay2π\ntexp/braceleftbigg\n−i/integraldisplayt\nt−τ(c(w)µ1(ηk)−iq)dw−i/integraldisplayt−τ+2π\ntk(µ1(ηk)c(w)−iq)dw/bracerightbigg\nϕ(t−τ+2π)dτ\n=/integraldisplayt\n0exp/braceleftbigg\n−i/integraldisplayt\ntk(c(w)µ1(ηk)−iq)dw/bracerightbigg\nϕ(t−τ)dτ (4.3)\n+/integraldisplay2π\ntexp/braceleftbigg\n−i/integraldisplayt\ntk(c(w)µ1(ηk)−iq)dw−i2π(µ1(ηk)c0−iq)/bracerightbigg\nϕ(t−τ+2π)dτ10 WAGNER A. A. DE MORAES\nWe have 0 ≤ϕ(t)≤1, so by ( 4.2) we obtain\n|/hatwideu(t,ηk)11|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0exp/braceleftbigg\n−i/integraldisplayt\ntk(c(w)µ1(ηk)−iq)dw/bracerightbigg\nϕ(t−τ)dτ\n+/integraldisplay2π\ntexp/braceleftbigg\n−i/integraldisplayt\ntk(c(w)µ1(ηk)−iq)dw−i2π(µ1(ηk)c0−iq)/bracerightbigg\nϕ(t−τ+2π)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplayt\n0exp/braceleftbigg/integraldisplayt\ntk(µ1(ηk)b(s)−Req)/bracerightbigg\n|ϕ(t−τ)|dτ\n+/integraldisplay2π\ntexp/braceleftbigg/integraldisplayt\ntk(µ1(ηk)b(s)+Req)−2π(Req−b0µ1(ηk))/bracerightbigg\n|ϕ(t−τ+2π)|dτ\n≤t+(2π−t)exp{2π(Req−b0µ1(ηk))} ≤4π\nfor sufficiently large k, where the last inequality comes from the fact that by the relation ( 4.1) we have\n|e−2πi(c0µ1(ηk)−iq)| →1, whenk→ ∞. By Theorem 2.2, we haveu∈ D′(T1×G).\nNotice that if t0>supI, thentk>supI, forksufficiently large, which implies that tk≥τ, for every\nτ∈supp(ϕ). By (4.3) we obtain\n|/hatwideu(tk,ηk)11|=/integraldisplay2π\n0ϕ(tk−τ)dτ=/bardblϕ/bardblL1(T1)>0.\nOn the other hand, if t0<infI, we have for ksufficiently large. that tk<τ, for everyτ∈supp(ϕ). By\n(4.3) we have\n|/hatwideu(tk,ηk)11|= exp{2π(Req−b0µ1(ηk))}/integraldisplay2π\n0ϕ(tk−τ+2π)dτ >1\n2/bardblϕ/bardblL1(T1)>0.\nBy Theorem 2.1we conclude that u /∈C∞(T1×G). Therefore Lis not globally hypoelliptic. /square\nTheorem 4.2. Assume that b/\\e}atio\\slash≡0. IfL=∂t+(a0+ib(t))X+qis globally hypoelliptic then bdoes not\nchange sign.\nProof.Suppose that bchange sign and b0>0. Consider\nG(t,τ) =/integraldisplayt+τ\nt(a0+ib(w))dw=a0τ+i/integraldisplayt+τ\ntb(w)dw, t,τ ∈[0,2π]\nand define\nB= min\n0≤t,τ≤2πImG(t,τ) = ImG(t0,τ0) =/integraldisplayt0+τ0\nt0b(w)dw.\nSincebchange sign, we have B <0. Moreover, we can consider t0,τ0∈(0,2π) andb(0)/\\e}atio\\slash= 0. It can be\nshown that b(t0+τ0) = 0, which implies that t0+τ0∈(0,2π).\nLetϕ∈C∞(T1) such that supp( ϕ)⊂[t0+τ0−δ,t0+τ0+δ]⊂(0,t0) withϕ(t)≡1 fort∈\n[t0+τ0−δ/2,t0+τ0+δ/2] and 0 ≤ϕ(t)≤1.\nLet us construct a distribution u∈ D′(T1×G)\\C∞(T1×G) such that Lu=f∈C∞(T1×G). From\n(2.4), there exist C >0 and a sequence {[ηj]}j∈Nin/hatwideGsuch that\n(4.4) C/a\\}bracketle{tηj/a\\}bracketri}ht1\n2≤µdηj(ηj),\nfor allj∈N. By Proposition 4.1we have that L0is globally hypoelliptic. In particular, the set\nN={[η]∈/hatwideG;µr(η)c0−iq∈Z,for some 1 ≤r≤dη}GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 11\nis finite and we may assume that [ ηj]/∈Z, for allj∈N. Define\n/hatwidef(t,η)rs=\n\n˜ckeBµr(ηj)ϕ(t)e−iµr(ηj)a0(t−t0),if [η] = [ηj],for somej∈N, andr=s=dηj,\n0, otherwise,\nwhere ˜ck:=e2πi(µr(ηj)c0−iq)−1. In order to provethat the sequence {/hatwidef(t,η)rs}defines a smooth function\ninT1×G, it is enough to consider the representations [ ηj] and the components r=s=dηj.\nNotice that\n|∂α\nt/hatwidef(t,ηj)dηjdηj|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(e2πi(µdηj(ηj)c0−iq)−1)eBµdηj(ηj)/summationdisplay\nβ≤α/parenleftbiggα\nβ/parenrightbigg\n∂β\nte−iµdηj(ηj)a0(t−t0)∂α−β\ntϕ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsinglee2πi(µdηj(ηj)c0−iq)−1/vextendsingle/vextendsingle/vextendsingleeBµdηj(ηj)/summationdisplay\nβ≤α/parenleftbiggα\nβ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµdηj(ηj)a0(t−t0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α−β\ntϕ(t)/vextendsingle/vextendsingle/vextendsingle.\nObserve that\n/vextendsingle/vextendsingle/vextendsinglee2πi(µdηj(ηj)c0−iq)−1/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsinglee2πi(µdηj(ηj)c0−iq)/vextendsingle/vextendsingle/vextendsingle+1≤e2π(−µdηj(ηj)b0+Req)+1≤C,\nfor someC >0, becauseb0≥0 andµdηj(ηj)→ ∞. Notice that\n/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµdηj(ηj)a0(t−t0)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle(−iµdηj(ηj)a0)βe−iµdηj(ηj)a0(t−t0)/vextendsingle/vextendsingle/vextendsingle≤Cβ/a\\}bracketle{tηj/a\\}bracketri}htβ.\nMoreover, since B <0, we obtain from ( 4.4)\neBµdηj(ηj)≤eCB/angbracketleftηj/angbracketright1\n2.\nHence,\n|∂α\nt/hatwidef(t,ηj)dηjdηj| ≤CαeCB/angbracketleftηj/angbracketright1\n2/a\\}bracketle{tηj/a\\}bracketri}htα.\nSinceB <0, for anyN >0 there exists CαNsuch that\n|∂α\nt/hatwidef(t,η)dηdη| ≤CαN/a\\}bracketle{tη/a\\}bracketri}ht−N.\nBy Theorem 2.1, we conclude that f∈C∞(T1×G). IfLu=f, then\n(4.5) /hatwidef(t,η)rs=/hatwiderLu(t,η)rs=∂t/hatwideu(t,η)rs+iµr(η)(a0+ib(t)−iq)/hatwideu(t,η)rs,\nfor 1≤r,s,≤dη. Sinceµdηj(ηj)c0−iq /∈Z, by (3.12) we obtain\n/hatwideu(t,ηj)dηjdηj=1\ne2πi(µdηj(ηj)c0−iq)−1/integraldisplay2π\n0eqτeiµdηj(ηj)G(t,τ)/hatwidef(t+τ,ηj)rsdτ\n=e−iµdηj(ηj)a0(t−t0)/integraldisplay2π\n0eqτeµdηj(ηj)(B−ImG(t,τ))ϕ(t+τ)dτ.\nIn all the other cases set /hatwideu(t,η)rs= 0. First, let us show that the sequence {/hatwideu(t,η)rs}defines a\ndistribution u∈ D′(T1×G), where\nu=/summationdisplay\n[η]∈/hatwideGdηdη/summationdisplay\nr,s=1/hatwideu(t,η)rsηsr.\nIn order to apply the Theorem 2.2, it is enough to consider the case where [ η] = [ηj] for somej∈N\nandr=s=dηjbecause the other cases are well-controlled.12 WAGNER A. A. DE MORAES\nLetψ∈C∞(T1), then\n|(/hatwideu(t,ηj)rs,ψ)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay2π\n0e−iµdηj(ηj)a0(t−t0)/integraldisplay2π\n0eqτeµdηj(ηj)(B−ImG(t,τ))ϕ(t+τ)dτψ(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay2π\n0/integraldisplay2π\n0eReqτeµdηj(ηj)(B−ImG(t,τ))|ϕ(t+τ)||ψ(t)|dτdt\n≤(2π)2/bardblϕ/bardbl∞/bardblψ/bardbl∞\n≤Kp1(ψ)/a\\}bracketle{tηj/a\\}bracketri}ht.\nNotice that here we have used the fact that µdηj(ηj)(B−ImG(t,τ))≤1. Therefore u∈ D′(T1×G).\nConsider the function\nθ(τ) =B−ImG(t0,τ) =B−/integraldisplayt0+τ\nt0b(w)dw.\nWe may consider δsmall enough in the properties of ϕsuch that either cos(Im q) or sin(Imq) does not\nchange sign on ( τ0+δ,τ0+δ). Assume without loss of generality that sin(Im q)≥0 on (τ0+δ,τ0+δ).\nThus\n|/hatwideu(t0,ηj)dηjdηj|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay2π\n0eqτeµdηj(ηj)θ(τ)ϕ(t0+τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥/integraldisplayτ0+δ\nτ0−δeReqτsin(Imqτ)eµdηj(ηj)θ(τ)ϕ(t0+τ)dτ\n≥/integraldisplayτ0+δ/2\nτ0−δ/2eReqτsin(Imqτ)eµdηj(ηj)θ(τ)dτ\n≥K/integraldisplayτ0+δ/2\nτ0−δ/2e/angbracketleftηj/angbracketrightθ(τ)dτ,\nwhere we use the fact that θ(τ)≤0, for allτ∈[0,2π],µdηj(ηj)≤ /a\\}bracketle{tηj/a\\}bracketri}htfor all [ηj]∈/hatwideG, and there exists\nK >0 such that eReqτsin(Imqτ)≥Kon [τ0−δ/2,τ0+δ/2].\nLet us analyze the behavior of the function\nJ(ηj) =/integraldisplayτ0+δ/2\nτ0−δ/2e/angbracketleftηj/angbracketrightθ(τ)dτ\nwhen/a\\}bracketle{tηj/a\\}bracketri}ht → ∞. We have\nθ(τ0) =B−/integraldisplayt0+τ0\nt0b(w)dw=B−B= 0\nand\nθ′(τ0) =−b(t0+τ0) = 0.\nThus by Taylor’s formula, we have\nθ(τ0+h) =θ(τ0)+θ′(τ0)h+1\n2θ′′(τ0+θ(h))h2=1\n2θ′′(τ0+θ(h))h2,\nforh∈(τ0−δ/2,τ0+δ/2) andθ(h)∈[τ0−δ/2,τ0+δ/2]. Let\nM:= sup\nτ0−δ≤y≤τ0+δ/vextendsingle/vextendsingle/vextendsingle/vextendsingleθ′′(y)\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nIfM= 0 thenθ≡0 in [τ0−δ/2,τ0+δ/2]. Thus\n/integraldisplayτ0+δ/2\nτ0−δ/2e/angbracketleftηj/angbracketrightθ(τ)dτ=/integraldisplayτ0+δ/2\nτ0−δ/2e0dτ=δ≥C1/radicalbig\n/a\\}bracketle{tηj/a\\}bracketri}ht,\nfor someC1>0.GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 13\nIfM >0, then\n−θ(τ0+h) =−1\n2θ′′(τ0+θ(h))h2≤Mh2.\nSo\n/a\\}bracketle{tηj/a\\}bracketri}htθ(τ0+h)≥ −M/a\\}bracketle{tηj/a\\}bracketri}hth2.\nThereby/integraldisplayτ0+δ/2\nτ0−δ/2e/angbracketleftηj/angbracketrightθ(τ)dτ=/integraldisplayδ/2\n−δ/2e/angbracketleftηj/angbracketrightθ(τ0+h)dh≥/integraldisplayδ/2\n−δ/2e−M/angbracketleftηj/angbracketrighth2dh≥C2/radicalbig\n/a\\}bracketle{tηj/a\\}bracketri}ht,\nfor someC2>0. Considering C= max{KC1,KC2}, we have\n|/hatwideu(t0,ηj)dηjdηj| ≥C/radicalbig\n/a\\}bracketle{tηj/a\\}bracketri}ht,\nfor all [ηj]∈/hatwideG. Therefore u /∈C∞(T1×G).\nThe case where b0<0 is analogous to the previous one, but needs some adaptions. Here we take\n/tildewideB:= max\n0≤t,τ≤2πImH(t,τ) = ImH(t1,τ1) =/integraldisplayt1\nt1−τ1b(w)dw.\nSincebchange sign, then /tildewideB >0. Forr=s=dηj, define\n/hatwidef(t,ηj)rs= (1−e−2πi(µr(ηj)c0−iq))e−/tildewideBµr(ηj)/tildewideϕ(t)e−iµr(ηj)a0(t−t1),\nwhere/tildewideϕ∈C∞(T1) satisfies similar properties of ϕ. One can shows that f∈C∞(T1×G) and there\nexistsu∈ D′(T1×G)\\C∞(T1×G) such that Lu=f. For this, define for r=s=dηj\n/hatwideu(t,ηj)rs=e−iµr(ηj)a0(t−t1)/integraldisplay2π\n0eµr(ηj)(ImH(t,τ)−/tildewideB)/tildewideϕ(t−τ)dτ.\nThe proof that u∈ D′(T1×G)\\C∞(T1×G) is similar to the previous case and it will be omitted. /square\n5.Sufficient Conditions\nIn this section, we will prove the sufficient conditions on Theorem 1.1regarding the global hypoel-\nlipticity of L. Because of Proposition 4.1, from now we will assume that L0is global hypoelliptic. By\nTheorem 3.1, this assumption implies that the set\n(5.1) N={(k,[η])∈Z×/hatwideG;k+c0µr(η)−iq= 0,for some 1 ≤r≤dη}\nis finite and there exist C,M >0 such that\n(5.2) |k+c0µr(η)−iq| ≥C(|k|+/a\\}bracketle{tη/a\\}bracketri}ht)−M,\nfor allk∈Z, [η]∈/hatwideG, 1≤r≤dη, whenever k+c0µr(η)−iq/\\e}atio\\slash= 0.\nTheorem 5.1. Assume that L0is globally hypoelliptic and b/\\e}atio\\slash≡0. Ifbdoes not change sign then Lis\nglobally hypoelliptic.14 WAGNER A. A. DE MORAES\nProof.Assume that b(t)≥0 for allt∈T1. By hypothesis, b0/\\e}atio\\slash= 0, where c0=a0+ib0. Notice that\nthe global hypoellipticity of L0implies that µr(η)c0−iq∈Zfor only finitely many representations. So\nthere is no loss of generality to assume that µr(η)c0−iq /∈Z. Letf∈C∞(T1×G) such that Lu=f,\nfor someu∈ D′(T1×G). Let us show that u∈C∞(T1×G).\nDefine\nH(τ,t) =c0τ−C(t−τ)+C(t).\nForµr(η)<0, consider the solution ( 3.11):\n(5.3) /hatwideu(t,η)rs=1\n1−e−2πi(µr(η)c0−iq)/integraldisplay2π\n0e−qτe−iµr(η)H(τ,t)/hatwidef(t−τ,η)rsdτ\nand forµr(η)≥0, consider the solution ( 3.12):\n(5.4) /hatwideu(t,η)rs=1\ne2πi(µr(η)c0−iq)−1/integraldisplay2π\n0eqτe−iµr(η)H(−τ,t)/hatwidef(t+τ,η)rsdτ.\nNotice that\nH(τ,t) =c0τ−C(t−τ)+C(t)\n=c0τ−/integraldisplayt−τ\n0c(w)dw+(t−τ)c0+/integraldisplayt\n0c(w)dw−tc0\n=/integraldisplayt\nt−τc(w)dw\nSo, using the fact that b(t)≥0, for allt∈T1, we obtain\n(5.5) Im H(τ,t) = Im/integraldisplayt\nt−τc(w)dw=/integraldisplayt\nt−τb(w)dw≥0.\n(5.6) Im H(−τ,t) = Im/integraldisplayt\nt+τc(w)dw=/integraldisplayt\nt+τb(w)dw≤0.\nNotice that there exist K >0 such that\n/vextendsingle/vextendsinglee±qτ/vextendsingle/vextendsingle≤K,\nbecause 0 ≤τ≤2π. Letα∈N0andµr(η)c0−iq /∈Z. Ifµr(η)<0, by (5.3) we have\n/vextendsingle/vextendsingle∂α\nt/hatwideu(t,η)rs/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n1−e−2πi(µr(η)c0−iq)/integraldisplay2π\n0∂α\nt/bracketleftbig\ne−qτe−iµr(η)H(τ,t)/hatwidef(t−τ,η)rs/bracketrightbig\ndτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n1−e−2πi(µr(η)c0−iq)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay2π\n0|e−qτ|α/summationdisplay\nβ=0/parenleftBigg\nα\nβ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµr(η)H(τ,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α−β\nt/hatwidef(t−τ,η)rs/vextendsingle/vextendsingle/vextendsingledτ.\nBy the assumption of the global hypoellipticity of L0, we obtain from Lemma 3.4constantsC,M >0\nsatisfying\n(5.7) |1−e−2πi(µr(η)c0−iq)|−1≤C/a\\}bracketle{tη/a\\}bracketri}htM,\nfor all [η]∈/hatwideG, 1≤r≤dη, whenever c0µr(η)−iq /∈Z. By Fa` a di Bruno’s Formula, we have\n∂β\nte−iµr(η)H(t,t)=/summationdisplay\nγ∈∆(β)β!\nγ!(−iµr(η))|γ|e−iµr(η)H(τ,t)β/productdisplay\nj=1/parenleftBigg\n∂j\ntH(τ,t)\nj!/parenrightBiggγj\n,GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 15\nwhere ∆(β) =/braceleftbigg\nγ∈Nβ\n0;β/summationtext\nj=1jγj=β/bracerightbigg\n. Hence,\n/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµr(η)H(t,t)/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay\nγ∈∆(β)β!\nγ!|µr(η)||γ|eµr(η)ImH(τ,t)β/productdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j\ntH(τ,t)\nj!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγj\n.\nNotice that by ( 2.3) we have\n|µr(η)||γ|≤ /a\\}bracketle{tη/a\\}bracketri}ht|γ|≤ /a\\}bracketle{tη/a\\}bracketri}htβ,\nfor all [η]∈/hatwideG, 1≤r≤dη, andγ∈∆(β). Moreover, by ( 5.5) we have\neµr(η)ImH(τ,t)≤1.\nThus,\n/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµr(η)H(t,t)/vextendsingle/vextendsingle/vextendsingle≤ /a\\}bracketle{tη/a\\}bracketri}htβ/summationdisplay\nγ∈∆(β)β!\nγ!β/productdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j\ntH(τ,t)\nj!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγj\n.\nBy the continuity of the function Hand the compactness of T1, for allβ∈N0there exists Cβ>0 such\nthat\n/summationdisplay\nγ∈∆(β)β!\nγ!β/productdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j\ntH(τ,t)\nj!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγj\n≤Cβ,\nfor all 0≤t,τ≤2π.\nLetN >0. Sincef∈C∞(T1×G), by Theorem 2.1for everyβ≤αthere exists CβN>0 such that\n|∂α−β\nt/hatwidef(t,η)rs| ≤CβN/a\\}bracketle{tη/a\\}bracketri}ht−(N+β+M),\nfor allt∈T1, withMas in (5.7). Therefore,\n/vextendsingle/vextendsingle∂α\nt/hatwideu(t,η)rs/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n1−e−2πi(µr(η)c0−iq)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay2π\n0|e−qτ|α/summationdisplay\nβ=0/parenleftBigg\nα\nβ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle∂β\nte−iµr(η)H(τ,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α−β\nt/hatwidef(t−τ,η)rs/vextendsingle/vextendsingle/vextendsingledτ.\n≤KC/angbracketleftη/angbracketrightM/integraldisplay2π\n0α/summationdisplay\nβ=0/parenleftBigg\nα\nβ/parenrightBigg\nCβ/angbracketleftη/angbracketrightβCβN/angbracketleftη/angbracketright−(N+β+M)dτ\n≤CαN/angbracketleftη/angbracketright−N.\nWe can obtain the same type of estimate when µr(η)≥0. In this case, it is enough to consider the\nexpression ( 5.4) to take the derivatives. We can adjust CαN, if necessary, to obtain\n/vextendsingle/vextendsingle∂α\nt/hatwideu(t,η)rs/vextendsingle/vextendsingle≤CαN/a\\}bracketle{tη/a\\}bracketri}ht−N,\nfor every [η]∈/hatwideG, 1≤r,s≤dη. By Theorem 2.1we conclude that u∈C∞(T1×G). The case b(t)≤0,\nfor allt∈T1, is totally analogue, just use ( 5.3) forµr(η)≥0 and (5.4) forµr(η)<0. /square\nExample 5.2. LetGbe a compact Lie group and q∈i(R\\Z). The operator\nL:=∂t+(eit+i)X+q\nis globally hypoelliptic. Indeed, we have Imeit+i= sin(t)+1/\\e}atio\\slash≡0and the operator L0=∂t+iX+qis\nglobally hypoelliptic by Theorem 3.1because in this case we have\nN={[η]∈/hatwideG;iµr(η)−iq∈Z}=∅,16 WAGNER A. A. DE MORAES\nand\n|k+iµr(η)−iq| ≥ |k−iq| ≥C,\nfor someC >0, for allk∈Z,[η]∈/hatwideG,1≤r≤dη. SinceImeit+i= sin(t)+1does not change sign,\nby Theorem 5.1we conclude that Lis globally hypoelliptic. On the other hand, the operator\nL:=∂t+(2eit+i)X+q.\nis not globally hypoelliptic. Indeed, notice that Im2eit+i= 2sin(t) + 1/\\e}atio\\slash≡0and the operator L0=\n∂t+iX+q=L0is globally hypoelliptic and since Im2eit+i= 2sin(t)+1changes sign, we conclude by\nTheorem 4.2thatLis not globally hypoelliptic.\nFinally, let us see that bdoes not change sign is insufficient for the global hypoellipt icity ofL. Consider\nG=S3andX=i∂0(see [9] and [12] for the precise Fourier analysis on S3) . We may identify/hatwiderS3∼1\n2N\nand in this case we have that σX(ℓ)rs=irδrs, for allℓ∈1\n2N,r−ℓ,s−ℓ∈Z. The operator\nL=∂t+(eit+i)X+in,\nwheren∈Z, is not globally hypoelliptic. Indeed, we have the correspo nding constant coefficient operator\nL0is given by L0=∂t+iX+inand\nN=/braceleftbigg\nℓ∈1\n2N;−r+n∈Z/bracerightbigg\n=N,\nwhich fails the condition 1. of Theorem 3.1for the global hypoellipticity of L0. Hence, by Proposition 4.1\nwe conclude that Lis not globally hypoelliptic. We point out that this phenome non occurs because when\nGis not a torus, the fact that bdoes not change sign does not imply the global hypoelliptici ty ofL0.\nAcknowledgments\nThisstudy wasfinanced bythe NationalCouncilfor Scientific andTe chnologicalDevelopment –CNPq\n[423458/2021-3].\nReferences\n[1] A. P. Bergamasco. Perturbations of globally hypoellipt ic operators. J. Differential Equations , 114(2):513–526, 1994.\n[2] D. Cardona, A. Kirilov, A. P. Kowacs, and W. A. A. de Moraes . On the Sobolev boundedness of vector fields on closed\nmanifolds. preprint, 2024.\n[3] F. de ´Avila Silva, R. B. Gonzalez, A. Kirilov, and C. de Medeira. Gl obal hypoellipticity for a class of pseudo-differential\noperators on the torus. J. Fourier Anal. Appl. , 25(4):1717–1758, 2019.\n[4] V. Fischer and M. Ruzhansky. Quantization on nilpotent Lie groups , volume 314 of Progress in Mathematics .\nBirkh¨ auser/Springer, [Cham], 2016.\n[5] S. J. Greenfield and N. R. Wallach. Global hypoellipticit y and Liouville numbers. Proc. Amer. Math. Soc. , 31:112–114,\n1972.\n[6] S. J. Greenfield and N. R. Wallach. Globally hypoelliptic vector fields. Topology , 12:247–254, 1973.\n[7] S. J. Greenfield and N. R. Wallach. Remarks on global hypoe llipticity. Trans. Amer. Math. Soc. , 183:153–164, 1973.\n[8] J. Hounie. Globally hypoelliptic and globally solvable first-order evolution equations. Trans. Amer. Math. Soc. ,\n252:233–248, 1979.\n[9] A. Kirilov, W. A. A. de Moraes, and R. Paleari. Global anal ytic hypoellipticity for a class of evolution operators on\nT1×S3.J. Differential Equations , 296:699–723, 2021.GLOBAL HYPOELLIPTICITY FOR A FIRST ORDER OPERATOR ON COMPAC T LIE GROUPS 17\n[10] A. Kirilov, W. A. A. de Moraes, and M. Ruzhansky. Partial Fourier series on compact Lie groups. Bull. Sci. Math. ,\n160:102853, 27, 2020.\n[11] A. Kirilov, W. A. A. de Moraes, and M. Ruzhansky. Global h ypoellipticity and global solvability for vector fields on\ncompact Lie groups. J. Funct. Anal. , 280(2):108806, 2021.\n[12] M. Ruzhansky and V. Turunen. Pseudo-differential operators and symmetries , volume 2 of Pseudo-Differential Oper-\nators. Theory and Applications . Birkh¨ auser Verlag, Basel, 2010. Background analysis and advanced topics.\nUniversidade Federal do Paran ´a, Departamento de Matem ´atica, C.P.19096, CEP 81531-990, Curitiba, Brazil\nEmail address :wagnermoraes@ufpr.br"    },    {        "title": "2402.04998v1.Revealing_the_KH2PO4_soft_mode_coupling_mechanism_with_infrared_spectroscopy_under_pressure.pdf",        "content": "Revealing the KH2PO4soft-mode coupling mechanism with infrared spectroscopy\nunder pressure\nD. Santos-Cottin,1S. Nasrallah,1F. Capitani,2P. Simon,3C. C. Homes,4A. Akrap,1and R. P. S. M. Lobo5,∗\n1Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland\n2Synchrotron Soleil, L’Orme des Merisiers - D´ epartementale 128 - F-91190 Saint-Aubin, France\n3CNRS, CEMHTI UPR3079, Univ. Orl´ eans, F-45071 Orl´ eans, France\n4National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n5LPEM; ESPCI Paris, PSL Research University; CNRS; Sorbonne Universit´ e, F-75005 Paris, France\n(Dated: February 8, 2024)\nWe measured the far-infrared reflectivity of a KH 2PO 4single crystal up to pressures of 2 GPa in the\nferroelectric and paraelectric phases. We find that the ν4vibrational mode of the PO 4tetrahedron\nis strongly affected by the applied pressure. At ambient pressure this phonon is destabilized by the\npresence of the H ions and hence shows a highly damped character, beyond the phonon propagation\nthreshold. Applying a pressure close to 0 .6 GPa makes this phonon clearly underdamped. Its\nbehavior closely follows the soft-mode behavior observed in Raman spectroscopy. Our results solve\na long standing open problem, demonstrating that the ν4mode is the excitation mediating the\ncoupling of the hydrogen network to the lattice modes that create the ferroelectic polarization in\nKH 2PO 4.\nThe riches of ferroelectrics and new phenomena aris-\ning from their large dielectrict constant are closely tied\nto the soft mode mechanism in these materials [1]. Soft-\nmode-driven remarkable properties have recently brought\nlong-known materials with lattice instabilities into the\nlimelight. The soft-mode and the proximity to the ferro-\nelectric instability are the key mechanism for the extreme\ndilute superconductivity in SrTiO3[2, 3]. KTaO3is an\nincipient ferroelectric [4–6] where the soft-mode under-\npins new opportunities in spintronics [7] and 2D super-\nconductivity [8]. In this paper we study a classic ferro-\nelectric and find a missing link in its peculiar soft-mode.\nKH2PO4(KDP) is a ferroelectric material with a Curie\ntemperature of TC≃123 K [9]. It has a tetragonal ( I42d)\nparaelectric phase, which becomes orthorhombic ( Fdd2)\nin the ferroelectric state. Each H atom in KDP connects\ntwo O atoms. Slater [10] proposed that the proton sit-\nting in the middle of the O-H-O bonding would be an\naverage position. In his model the H atom stays closer\nto one of the O atoms in the PO4tetrahedron. He found\nthat 6, out of the 16 possible configurations, are ener-\ngetically more stable. Looking at the free energy of the\nsystem he calculated an electrical polarization that re-\nmains constant below a critical temperature TC. Slater’s\norder-disorder theory gives an overall good description of\nKDP but fails in two important points. The first problem\nis the constant, temperature independent, polarization\nbelow TC. The second is the lack of an explanation for\nwhyTCincreases to 222 K when deuterium substitutes\nfor hydrogen.\nPirenne [11] incorporated the kinetic energy of the pro-\nton vibrations into Slater’s picture, thus explaining the\ndifference in TCbetween H and D materials. Elaborat-\ning on Pirenne’s ideas, Blinc [12] placed the proton in a\ndouble well potential and proposed that the proton could\ntunnel between the two wells. He assumed a symmetric\nFigure 1. (color online) (a) KDP structure, with K atoms\nsuppressed, showing the proton position with respect to\nPO 4tetrahedra in the orthorhombic ferroelectric phase. (b)\nAtomic displacements for the ν4vibration mode of the PO 4\ntetrahedron.\npotential in the high temperature phase and asymmetric\nwells in the ferroelectric phase. The tunneling probabil-\nity of D being much smaller than that for H, TCis higher\nfor the former material. Within this formalism, KDP\nis no longer a simple order-disorder ferroelectric but it\nrather has a mixed character with a soft-mode composed\nof a proton collective mode strongly coupled to an optical\nphonon and to an acoustic mode.\nCuriously, according to this description, ferroelectric-\nity in KDP comes from a Hamiltonian dominated by the\nproton network with heavy ions entering as a pertur-\nbative term. Nevertheless, the ferroelectric polarization\ncomes from the displacement of heavier ions. Figure 1 (a)\nshows the proton network in KDP and its connections\nwith the PO4tetrahedron in the ferroelectric phase. Re-\nmarkably, the ferroelectric polarization forms along the\nc-axis — a direction perpendicular to the ab-plane, which\ncontains the proton tunneling mode.\nTypeset by REVT EXarXiv:2402.04998v1  [cond-mat.mtrl-sci]  7 Feb 20242\nGradually, Blinc’s theory settled as the accepted model\nfor the ferroelectric transition in KDP. Some experimen-\ntal evidence supporting this picture started to emerge\nwith Raman measurements of an overdamped soft mode\n[13]. Later, neutron structural studies showed the ex-\nistence of the proton double-well potential [14]. More\nrecently, neutron Compton scattering [15] measured the\nmomentum distribution of hydrogen in KH2PO4. This\ndata, in conjuction with ab-initio calculations [16] high-\nlight the importance of the coupling between the proton\ntunneling and the PO4tetrahedron [17, 18].\nThe importance of the H coupling to the PO4group\nhad already been put forward by Simon and Gervais\n[19, 20]. They showed that the infrared reflectivity of\nRbH2PO4has an overdamped soft mode, seen by Raman,\nand that the same mechanism occurs in KH2PO4and\nKH2AsO4. Their spectra also shows a highly-damped\nν4mode of the PO4tetrahedra. Figure 1 (b) shows the\natomic displacements of this mode. They proposed that\nthe nearby proton tunneling between their two equilib-\nrium positions destabilizes the ν4mode, which softens its\nfrequency and increases its damped character. As this vi-\nbration has a dipole moment, they conjectured that this\nmode would be the missing link coupling the proton net-\nwork to the external translation overdamped mode.\nHere, we utilized infrared measurements under pres-\nsure to give a definite proof that this phonon is indeed\ncoupled to the soft mode. Peercy showed that the over-\ndamped soft mode measured by Raman scattering be-\ncomes underdamped under an applied pressure close to\n1 GPa [21]. We find that the damping of the ν4mode also\nstrongly decreases under similar pressures in both the fer-\nroelectric and paraelectric phases. This shows that the\ntwo modes are coupled and that the ν4vibration of the\nPO4group is the excitation responsible for coupling the\nhydrogen network to the soft mode of KH2PO4.\nWe measured the infrared reflectivity of a KDP sin-\ngle crystal in Bruker IFS 113v and 66v/S interferome-\nters. The absolute value of the reflectivity was obtained\nin an ARS Helitran cryostat utilizing an in situ gold\noverfilling technique [22]. We obtained data in the fer-\nroelectric and paraelectric phases with the electric field\nof light polarized along the ferroelectric axis between 10\nand 6000 cm−1.\nHigh pressure infrared reflectivity data, up to 2 .7 GPa,\nwere collected at the SMIS beamline, SOLEIL Syn-\nchrotron, in a home-made horizontal microscope with\ncustom objectives. The microscope is coupled to a\nThermo-Fisher iS50 interferometer with synchrotron ra-\ndiation as its light source; a solid substrate beamsplitter;\nand a 4K Si:B bolometer to cover the 100 to 600 cm−1\nregion. We employed a membrane-driven diamond anvil\ncell (DAC) with a 600 µm diameter culet, a 60 µm thick\npre-indented stainless steel gasket with a 250 µm diame-\nter hole filled with polyethylene as hydrostatic medium\n[23]. The sample was placed in contact with one of the\n05 001 0001 5000.00.20.40.60.81.01.21.41.6W\nave number [cm-1]Reflectivity \n0 GPa \n2 GPaT\n = 82 KT = 140 K \n0 GPa \n2 GPa01 002 000.00.51.00\n.05 GPa0.91\n.02.02.7 GPaFigure 2. (color online) Infrared reflectivity polarized along\nthe ferroelectric c-axis direction in the paraelectric (140 K)\nand ferroelectric (82 K) phases. The ambient pressure data\nwere recorded in the full shown spectral range, whereas data\nat 2 GPa were taken between 100 and 600 cm−1. The thicker,\nlight-colored lines are fits taking a dielectric function of the\nform given by Eq. (1) and the thinner lines are the measure-\nments. Note that the reflectivity of the 140K data is shifted\nupwards by 0.6, for clarity. The inset highlights the soft-mode\nresponse with low pressure data taken in a different set-up\ndown to 40 cm−1(see text).\nculets and we employed a ruby sphere for the in situ\npressure measurement. The DAC was attached to the\ncold head of a ℓN2flow cryostat. We utilized the dia-\nmond table as a reference. The thus-obtained reflectivity\nin the DAC at the lowest pressure was then corrected\nto match the measurement taken outside the cell at the\nsame temperature. We applied this correction function\nto the other higher pressure measurements.\nThe data below 1 GPa were measured at BNL within\na clamped pressure cell using kerosene as the pressure\ntransmitting medium; the entire arrangement sits at the\nbottom of a compact open-flow cryostat. This cell em-\nploys a wedged diamond window ( ≃7◦) with a working\ndiameter of about 1.5 mm, allowing measurements down\nto≃20 – 30 cm−1on a Bruker 66v/S using conventional\ninfrared sources. The reflectivity unit uses toroidal mir-\nrors with a long focal length. Rotations allow the reflec-\ntivity from the sample in the cell to be separated from\nthe reflectivity obtained from the front of the diamond\nwindow, which, similar to the approach used with the\nDAC, is then used as an optical reference [24].\nFigure 2 shows the infrared reflectivity of KDP in\nthe paraelectric and ferroelectric phases. We show the\ndata for the full phonon energy range at zero pressure\nand for a more restricted range at 2 GPa. The thin\nlines are the data and the thicker lines are fits utiliz-\ningR=|(√ε−1)/(√ε+ 1)|2and the factorized form of3\n0123450\n12345Im(ε)T\n = 82 Kν4ν\n2(a)2\n.0 GPa1\n.751\n.51\n.251\n.00\n.750\n.50\n.40\n.30\n.20\n.1 GPaIm(ε)T\n = 140 Kν4ν2(b)0\n.1 GPa0.250.50.751.01.252.01\n.51.752.7 GPa2\n.354\n004 505 0001234Re(ε)W\nave number (cm-1)0.1 GPa2.0 GPaT\n = 82 K(c)ν\n2ν43\n504 004 5001234Re(ε)0\n.1 GPa2.75 GPaT\n = 140 K(d)ν\n2ν4\nFigure 3. (color online) Kramers-Kronig obtained dielectric\nfunction around the ν2andν4vibrational modes of the PO 4\ntetrahedron. Panels (a) and (b) show the imaginary part of\nthe dielectric function. Data in panel (a) is taken at 82 K and\nin panel (b) at 140 K. For clarity, the baseline of each curve\nis increased by 0.2 upon increasing pressure. It is worth not-\ning that in panel (b) ν2has a strong asymetric Fano profile\nat low pressures and becomes a more symmetric conventional\nphonon at high pressures. Panels (c) and (d) show the cor-\nresponding real part of the dielectric function. Here, again,\neach spectrum is shifted upwards by 0.2 with respect to the\nprevious pressure.\nthe dielectric function [25]:\nε(ω)\nε∞=Y\nkω2\nLOk−ω2−iγLOkω\nω2\nTOk−ω2−iγTOkω. (1)\nωTOandωLOare the transverse optical (TO) and lon-\ngitudinal optical (LO) resonance frequencies for each\nphonon k. As this model allows for different dampings\n(γTOandγLO) at each of these frequencies, it can bet-\nter describe anharmonic effects than a Lorentz oscillator\nmodel. In order to properly fit the data under pressure,\nwe utilized the 0 GPa spectra below 100 cm−1and above\n600 cm−1. We kept the same extension of the data range\nto perform Kramers-Kronig analysis of the reflectivity.\nWe are interested in the behavior under pressure of the\nPO4tetrahedra ν4mode, situated around 400 cm−1. Fig-\nure 3 shows the Kramers-Kronig obtained imaginary part\nof the dielectric function in the ν4vicinity as a function\nof external pressure for the ferroelectric and paraelectric\nphases, at 82 K [panel (a)] and 140 K [panel (b)]. Thesefigures also show the ν2mode, which is not related to the\nferroelectric polarization but that will help us to under-\nstand the behaviour of the ν4mode.\nAt both temperatures, the ν4mode gets narrower upon\nincreasing pressure. At 82 K and 0 .1 GPa one can barely\ndiscern a broad peak around 450 cm−1, which becomes\nincreasingly sharper when moving towards 2 GPa. In the\nparaelectric phase one observes a similar effect. The ν4\nmode is hardly visible at 0 .1 GPa and, upon increasing\npressure, it develops a sharp step creating a prominent\ndip around 400 cm−1. At this temperature, the ν2mode\nalso shows a particular behavior. At 0 .1 GPa it has a\nvery asymmetric line shape and at 2 GPa it recovers a\nregular phonon symmetric profile. As we will discuss fur-\nther, this can be understood as a Fano coupling to the\nhighly-damped ν4mode broad background, that disap-\npears when the ν4vibration becomes less damped.\nPanels (c) and (d) show the real part of the dielectric\nfunction corresponding to the data in panels (a) and (b).\nAt 82 K the ν4mode develops a sharper profile. At 140 K\nwe can only devise a single excitation at low pressures,\nwhereas two clear peaks for the ν2andν4modes appear\nat high pressures.\nIn order to quantify our results, let us take a look at\nthe fitting parameters at 140 K, where the effect is more\nspectacular. For this, besides an analysis utilizing the\nfactorized form of the dielectric function from Eq. (1),\nwe also looked at the aforementioned asymmetric line-\nshapes. We looked at the imaginary part of the dielectric\nsusceptibility χ′′\nFwith a Fano profile [26]:\n∆χ′′\nF=R\u0014(ωF−ω−γFq)2\n(ωF−ω)2+γ2\nF−1\u0015\n. (2)\nHere, ωFis the renormalized phonon frequency, γFits\ndamping, Ra transition rate, and qthe Fano-Wigner-\nBreit parameter. As q−2approaches zero, the line shape\nbecomes more symmetric. The ∆ symbol emphasizes\nthat a Fano oscillator does not exist by itself. It must\nsit on top of a larger background as a differential correc-\ntion to the dielectric function. In our case, we utilized\ntwo large Lorentzian oscillators in the background to de-\nscribe surrounding phonons. The Lorentzians, within un-\ncertainties, do not change as a function of pressure.\nThe thicker lines in Fig. 4(a) show the Kramers-Kronig\nobtained imaginary part of the dielectric function at\n140 K. For each pressure we show two fits. The thin\ndashed line utilizes Eq. (1). It gives an acceptable overall\ndescription of the data but fails to describe the sharper\nline shapes. The thin solid lines are from Eq. (2) and\nfollow the data almost perfectly.\nWe begin with the imperfect fits coming from the fac-\ntorized form of the dielectric function. It is useful to\ndefine the criterion√γTOγLO>(ωLO−ωTO). This is\nthe limit where, in the absence of other phonons, the\nreal part of Eq. (1) becomes positive for all frequencies4\n01 2 3 0.51.01.52.0 \n140 K ν4 \n140 K Soft(γTO x γLO)1/2 / (/s119TO - /s119LO)P\nressure [GPa](b)4004 500123Im(ε)W\nave number (cm-1)0.1 GPa2.0 GPaT\n = 140 K(a)ν\n2ν43\n80390400410/s119F [cm-1]T\n = 140 K(c)ν\n2ν40\n204060γF [cm-1]T = 140 K(d)ν\n2ν40\n1 2 3 0.00.51.0P\nressure [GPa]1 / q2T\n = 140 K(e)ν\n2\nFigure 4. (color online) (a) Imaginary part of the dielectric\nfunction for the ν2andν4modes at 0.1 and 2 GPa. The\nthick lines come from a Kramers-Kronig transformation of the\nreflectivity. The solid thin lines are fits utilizing Fano profiles\n[Eq. (2)] for these modes. The dashed lines fits come from the\nfactorized dielectric function [Eq. (1)]. (b) Geometric average\nof TO and LO dampings normalized by the TO–LO frequency\nsplitting, extracted from the factorized form of the dielectric\nfunction, for the ν4mode (solid circles and solid triangles).\nThe stars are the same quantity for the soft-mode from data\ntaken at Brookhaven (solid) and Soleil (open). The soft-mode\ndata was re-scaled by a factor 1.4 in order to compare with\ntheν4damping. (c)–(e) Fano parameters for the ν2andν4\nmodes.\nbetween ωTOandωLO, killing the Reststrahlen phonon\nscattering. Figure 4 (b) shows that at 140 K this quan-\ntity drops sharply in the pressure region where Peercy\n[21] observed the underdamping of the Raman soft mode.\nIn particular the ratio between the averaged damping\nand the TO-LO splitting crosses the critical value of 1.\nWe also show this normalized damping value for the soft\nmode. As the Brookhaven data stops at 40 cm−1and\nthe Soleil data at 100 cm−1, we have a large error bar for\nthis quantity. Nevertheless, the temperature evolution\nfor the soft-mode damping shows the same trend as the\none observed for the ν4mode.\nFigures 4 (c), (d) and (e) show the results of Fano fit-\ntings to ν2andν4at 140 K. The resonance frequency\nand linewidth of the ν4change remarkably at a pressure\nlower than 1 GPa. The ν4frequency [Fig. 4 (c)] hardens\nsignificantly at the same pressure that this mode narrows\nby almost one order of magnitude [Fig. 4 (d)]. This high-lights the predicted effect of this phonon destabilization\nby the proton tunneling [19, 20], which goes away with\npressure.\nLooking at the ν2phonon, we observe that both its\nfrequency and linewidth are essentially pressure indepen-\ndent. However, q−2[Fig. 4 (e)] for this mode changes dra-\nmatically around 0 .6 GPa. It shows ν2going from a very\nasymmetric to a fully-symmetric line shape. As ν2andν4\nare vibrational modes of the PO4group, they are natu-\nrally coupled when they have close resonance frequencies\nand hence the Fano asymmetric profile. Close to am-\nbient pressure, the resonance frequency of ν4is smaller\nthan that of ν2. The ν4frequency crosses ν2at a pressure\nclose to 0 .6 GPa, at the same value where its Fano damp-\ning decreases by an order of magnitude. This means that\nν4gets narrow and moves away from ν2upon increasing\npressure. This better separation of the modes is what\nmakes the ν2more symmetric. Note that ν2has no par-\nticular role in the ferroelectric transition. Here it serves\nas a probe to the narrowing of ν4.\nIn summary, we measured the infrared reflectivity\nof KH2PO4under pressure up to 2 .7 GPa. We found\nthat the ν4vibration mode of the PO4tetrahedron is\nstrongly damped at ambient pressure and it becomes un-\nderdamped upon increasing pressure. Its behavior closely\nfollows the underdamping observed by Peercy for the\nKDP soft-mode in Raman scattering, indicating a cou-\npling between these two modes. The ν4mode, which\nhas an electric dipole and involves only heavier P and\nO ions, is strongly destabilized by the presence of the\nnearby protons. It is the missing piece of KDP soft-mode\nas it represents the excitation responsible for coupling the\nhydrogen network to the lattice modes. In particular it\nexplains how the in-plane proton dynamics affects the\nout-of-plane phonon at the origin of the electric polariza-\ntion.\nMeasurements at Soleil Synchrotron were carried out\nat the SMIS beamline under proposal 20220159. A.A.\nacknowledges funding from the Swiss National Science\nFoundation through project PP00P2 202661. Work at\nBrookhaven National Laboratory was supported by the\nOffice of Science, U.S. Department of Energy under Con-\ntract No. DE-SC0012704.\n∗lobo@espci.fr\n[1] S. Kamba, Soft-mode spectroscopy of ferroelectrics and\nmultiferroics: A review, APL Materials 9, 020704 (2021).\n[2] X. Lin, G. Bridoux, A. Gourgout, G. Seyfarth,\nS. Kr¨ amer, M. Nardone, B. Fauqu´ e, and K. Behnia, Criti-\ncal Doping for the Onset of a Two-Band Superconducting\nGround State in SrTiO 3-δ, Phys. Rev. Lett. 112, 207002\n(2014).\n[3] C. W. Rischau, D. Pulmannov´ a, G. W. Scheerer,\nA. Stucky, E. Giannini, and D. van der Marel, Isotope5\ntuning of the superconducting dome of strontium ti-\ntanate, Phys. Rev. Res. 4, 013019 (2022).\n[4] G. Shirane, R. Nathans, and V. J. Minkiewicz, Tem-\nperature Dependence of the Soft Ferroelectric Mode in\nKTaO 3, Phys. Rev. 157, 396 (1967).\n[5] V. ˘Zelezn´ y, A. Pashkin, J. Petzelt, M. Savinov,\nV. Trepakov, and S. Kapphan, Soft-Mode Study in Li-\nDoped KTaO 3, Ferroelectrics 302, 195 (2004).\n[6] Y. Ichikawa, M. Nagai, and K. Tanaka, Direct observa-\ntion of the soft-mode dispersion in the incipient ferroelec-\ntric KTaO 3, Phys. Rev. B 71, 092106 (2005).\n[7] A. Gupta, H. Silotia, A. Kumari, M. Dumen, S. Goyal,\nR. Tomar, N. Wadehra, P. Ayyub, and S. Chakraverty,\nKTaO 3– The New Kid on the Spintronics Block, Ad-\nvanced Materials 34, 2106481 (2022).\n[8] C. Liu, X. Zhou, D. Hong, B. Fisher, H. Zheng,\nJ. Pearson, J. S. Jiang, D. Jin, M. R. Norman, and\nA. Bhattacharya, Tunable superconductivity and its ori-\ngin at KTaO 3interfaces, Nature Communications 14, 951\n(2023).\n[9] G. Busch, J. Helv. Physics. Acta 11, 269 (1938).\n[10] J. C. Slater, Theory of the Transition in KH 2PO 4, The\nJournal of Chemical Physics 9, 16 (1941).\n[11] J. Pirenne, On the ferroelectricity of KH 2PO 4and\nKD 2PO 4crystals, Physica 15, 1019 (1949).\n[12] R. Blinc, On the isotopic effects in the ferroelectric be-\nhaviour of crystals with short hydrogen bonds, Journal\nof Physics and Chemistry of Solids 13, 204 (1960).\n[13] I. P. Kaminow and T. C. Damen, Temperature Depen-\ndence of the Ferroelectric Mode in KH 2PO 4, Phys. Rev.\nLett. 20, 1105 (1968).\n[14] R. J. Nelmes, Recent structural studies of the KDP-type\ntransition: A review, Ferroelectrics 53, 207 (1984).\n[15] G. F. Reiter, J. Mayers, and P. Platzman, Direct Ob-\nservation of Tunneling in KDP using Neutron Compton\nScattering, Phys. Rev. Lett. 89, 135505 (2002).\n[16] Q. Zhang, F. Chen, N. Kioussis, S. G. Demos, and H. B.Radousky, Ab initio study of the electronic and structural\nproperties of the ferroelectric transition in KH 2PO 4,\nPhys. Rev. B 65, 024108 (2001).\n[17] H. Sugimoto and S. Ikeda, Isotope effects in hydrogen-\nbonded crystal KH 2PO 4, Phys. Rev. Lett. 67, 1306\n(1991).\n[18] A. Bussmann-Holder and K. H. Michel, Bond Geome-\ntry and Phase Transition Mechanism of H-Bonded Fer-\nroelectrics, Phys. Rev. Lett. 80, 2173 (1998).\n[19] P. Simon and F. Gervais, Phase-transition mechanism\nin RbH 2PO 4-type ferroelectrics, Phys. Rev. B 32, 468\n(1985).\n[20] P. Simon, F. Gervais, and E. Courtens, Paraelectric-\nferroelectric phase transitions of KH 2PO 4, RbH 2PO 4,\nand KH 2AsO 4studied by infrared reflectivity, Phys. Rev.\nB37, 1969 (1988).\n[21] P. S. Peercy, Observation of an Underdamped “Soft”\nMode in Potassium Dihydrogen Phosphate, Phys. Rev.\nLett. 31, 379 (1973).\n[22] C. C. Homes, M. Reedyk, D. A. Crandles, and T. Timusk,\nTechnique for measuring the reflectance of irregular,\nsubmillimeter-sized samples, Appl. Opt. 32, 2976 (1993).\n[23] A. Celeste, F. Borondics, and F. Capitani, Hydrostaticity\nof pressure-transmitting media for high pressure infrared\nspectroscopy, High Pressure Research 39, 608 (2019).\n[24] I. K´ ezsm´ arki, R. Ga´ al, C. C. Homes, B. S´ ıpos, H. Berger,\nS. Bord´ acs, G. Mih´ aly, and L. Forr´ o, High-pressure in-\nfrared spectroscopy: Tuning of the low-energy excitations\nin correlated electron systems, Phys. Rev. B 76, 205114\n(2007).\n[25] T. Kurosawa, Polarization waves in solids, J. Phys. Soc.\nJapan 16, 1298 (1961).\n[26] L. C. Davis and L. A. Feldkamp, Interaction of many\ndiscrete states with many continua, Phys. Rev. B 15,\n2961 (1977)."    },    {        "title": "2402.05031v1.Realization_of_Wess_Zumino_Witten_transitions_with_levels__k_6__and__k_4__in_a_frustrated_spin_3_chain.pdf",        "content": "Realization of Wess-Zumino-Witten transitions with levels k= 6and k= 4in a\nfrustrated spin-3 chain\nNatalia Chepiga1\n1Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: February 8, 2024)\nWe study dimerization transitions in a frustrated spin-3 chain with next-nearest neighbor and\nthree-site interactions. We show that two independent coupling constants of the model are sufficient\nto fine-tune the system to the critical point in the Wess-Zumino-Witten SU(2) 6universality class.\nThis critical point appears as the end point of an extended SU(2) 4critical line. This implies that the\nrenormalization group flow lead to the critical theory with the largest level ksuch that the number\nof relevant operators is reduced by one and the parity of the level is preserved. We also report the\nappearance of non-magnetic Ising transition between the topologically trivial uniform and dimerized\nphases. This transition takes place within the singlet sector, while magnetic gap remains open.\nI. INTRODUCTION\nAntiferromagnetic Heisenberg spin chains have at-\ntracted a lot of attention over the years. Competing in-\nteractions induce frustration and are known to lead to\nnew phases and quantum phase transitions. For exam-\nple, the J1−J2spin-1/2 chain undergoes a Kosterlitz-\nThouless transition[1] into a spontaneously dimerized\nphase[2, 3]. By contrast, phases realized in the J1−J2\nspin-1 chain are non-dimerized and gapped: for J2/J1≲\n0.75 the chain is in the topologically non-trivial Haldane\nphase[4]; beyond this point the ground-state corresponds\nto a pair of intertwined Haldane chains[5–7]. The dimer-\nized phase can be realized in spin-1 chain in the presence\nof biquadratic interaction Jb(Si·Si+1)2or due to the\nthree-site interaction J3[(Si−1·Si)(Si·Si+1) + h .c.][8–\n14]. The latter is a generalization of the Majumdar-\nGhosh point and realizes exact dimerized state at J3=\nJ1/[4S(S+ 1)−2] for any value of spin S[12]. In fact,\nfor not too large next-nearest-neighbor coupling J2, the\nexact dimerized state remains the ground-state along the\nline[15]:\nJ3\nJ1−2J2=1\n4S(S+ 1)−2, (1)\nmaking the J1−J2−J3model an ideal play-ground\nto study dimerization transitions in spin-S chains. The\nmodel is defined by the following microscopic Hamilto-\nnian:\nH=J1X\niSi·Si+1+J2X\niSi−1·Si+1\n+J3X\ni[(Si−1·Si)(Si·Si+1) + h.c.].(2)\nwhere without loss of generality we fix J1= 1; in this\npaper we restrict ourselves to positive coupling constants\nJ2, J3≥0.\nNumerical investigation of this model already lead to\nan impressive list of exotic quantum critical phenom-\nena. Among them a non-magnetic Ising phase transition\nbetween the fully-dimerized phase and the next-nearestneighbor (NNN-) Haldane phase in the spin-1 chain at\nwhich the singlet-triplet gap remains open[16]; partially-\ndimerized phases in spin-3 /2 and 5 /2 chains[17, 18] sepa-\nrated by Kosterlitz-Thouless transitions[1] from the criti-\ncal phases; and the emergence of magnetic floating phase\n- critical phase with incommensurate quasi-long-range\ncorrelations[17, 18]. Another unusual feature revealed\nin these systems was a termination of the Wess-Zumino-\nWitten (WZW) SU(2) 2Scritical lines due to the pres-\nence of marginal operators in spin-1 and spin-3 /2 chains.\nAbove the end point the transition to the fully dimer-\nized phase is first order - a very counter-intuitive conclu-\nsion for for half-integer chain given that on one side of\nthis transition the phase is critical[19]. In both cases -\nin spin-1 and spin-3/2 chains - the marginal operators do\nnot change the nature of the transition and the end point\nbelongs to the same universality class as the critical line\nit terminates.\nThe situation is radically different for spin chains\nwith S≥5/2 due to appearance of additional rele-\nvant operators[20]. Realization of higher levels WZW\nSU(2) kuniversality classes is traditionally attributed to\nintegrable spin- Schains with a microscopic Hamiltonian\ngiven by a polynomial of degree 2 Sin (Si·Si+1)[21–\n23] that artificially fine-tunes all relevant operators to\nvanish. However, for WZW SU(2) kcritical theory with\n5≤k < 10 there are only two relevant operators.\nThis means that two independent parameters, as in the\nJ1−J2−J3model, might be sufficient to fine-tune the sys-\ntem to the WZW SU(2) 2Scritical point. Recently, this\nhas been reported in spin-5 /2 chain where SU(2) 5critical\npoints appear at the end of the extended SU(2) 3critical\nline[18]. First relevant operator is responsible for the ap-\npearance of the dimerized phase; the corresponding cou-\npling constant changes its sign at the transition. The sec-\nond relevant operator controls the type of this transition:\nwhen its coupling constant is positive, the transition is\nfirst order; when it is zero both operators vanish and\nthe critical theory is fine-tuned to WZW SU(2) 5; when\nit is negative the underlying critical theory renormalizes\nto the lower level, SU(2) 3. For small values of J2andJ3\nwhen the coupling constant of both relevant operators arearXiv:2402.05031v1  [cond-mat.str-el]  7 Feb 20242\nnegative, the extended critical phase is described by the\nWZW SU(2) 1[19] critical theory. According to the recent\nfield theory prediction in the presence of both SU(2) and\na discrete Z2symmetry a renormalization-group flow is\nonly possible between WZW SU(2) ktheories if the par-\nity of the level index kdoes not change[24]. Spectacular\nsequence of WZW criticalities with odd levels k= 1,3\nand 5 reported in spin-5 /2 chain confirms this theory\nprediction[18]. At the same time possible renormaliza-\ntion between even levels remains unexplored.\nIn the present paper we aim to fill this gap by looking\nat the critical properties of the spin-3 J1−J2−J3chain.\nIn addition, we will explore the situation when there are\nseveral possible candidates with lower levels kthat sat-\nisfies the parity constraints. We will show that at the\nisolated point the transition is fine-tuned to the WZW\nSU(2) 6that appears as an end point of WZW SU(2) 4\ncritical line. We address this problem numerically with\na state-of-the-art density matrix renormalization group\n(DMRG) algorithm[25–28]. Throughout the paper, un-\nless explicitly stated otherwise, we use a chain with an\neven number of sites and open boundary conditions. In\ntwo-site DMRG we typically keep up to 1500 states, per-\nform 6 sweeps and discard singular values smaller than\n10−8.\nII. PHASE DIAGRAM\nWe present a basic phase diagram of J1−J2−J3model\nfor spin-3 chain obtained numerically in Fig.1. Accord-\ning to the Haldane’s conjecture[4] integer-spin chain at\nJ2=J3= 0 is in the uniform and gapped phase. It\nis instructive to visualize this phase in terms of valence\nbond singlets (VBS) - distribution of effective spin-1/2\nsinglets. To match the total spin S= 3 six VBS have to\nterminate at each lattice site. The phase at J2=J3= 0\ncorresponds to a uniform VBS covering of the lattice with\n3 spin-1/2 singlets per nearest-neighbor bond. In this\nrepresentation it becomes obvious that there are three\neffective spins-1/2 at each end of the chain that remain\nunpaired and form spin-3/2 edge states, however only\nspin-1/2 edge states are topologically protected. At large\nJ2by analogy with spin-1 case one might expect the\nVBS singlets to occupy next-nearest-neighbor bonds - the\nspin-3 analogue of the NNN-Haldane phase. This phase\nis topologically trivial and therefore the system under-\ngoes at least one topological transition upon increasing\nJ2. In fact, Berry phase calculations on system sizes up\ntoN= 10 sites has shown that spin-3 J1−J2chain\nundergoes a sequence of three topological transitions[29].\nVerification of whether all of these transitions will be\npresent in the thermodynamic limit and for finite values\nofJ3is an extremely challenging numerical task that we\nleave for future investigations. But it is important to\nkeep in mind that all these phases are uniform and show\nno sign of dimerization.[30]. From the Berry phase cal-\nculations we can expect the trivial uniform phase to be\n0 0.005 0.01 0.015 0.0200.20.40.60.81\ndisorder lineDimerized phase\nUniform phases\nexactly dimerized6 6 6 66 6 6 6 6\nor\n3 3 3 3 33 3 3 33 3 3 33 3 3 3\n IsingFigure 1. Phase diagram of the S= 3 chain with next-nearest-\nneighbor J2and three-site interactions J3. There are at least\ntwo topologically distinct uniform phases; according to Ref.29\nthere might be four of them separated by topological transi-\ntions; their exact location are outside of the scope of this\npaper. Fully dimerized phase spontaneously breaks trans-\nlation symmetry and has two-fold degenerate ground-states.\nThe fully dimerized phase is separated from the (topologically\nnon-trivial) uniform phase by continuous WZW SU(2) 4tran-\nsition along the blue line that terminates at the end point\n(red star) located at J2≈0.08, beyond which the transition\nis first order. For large values of J2the transition between\n(topologically trivial) uniform phase and the fully-dimerized\none is consistent with Ising universality class. The sketches\nare visualizations of the corresponding phases in terms of va-\nlence bond singlets (VBS): numbers near black lines state for\nthe total number of VBS singlets at the corresponding bond.\nstable at least for J2≳0.9[29].\nFor large J3the system is in the fully dimerized phase\nwhere every other nearest-neighbor bond is occupied by\nsix VBS singlets leading to two-fold degenerate ground-\nstates. The line of exact dimerization defined in Eq.1\nterminates at J2≈0.16 - at this point the dimerization\ndrops at once from S(S+ 1) = 12 to zero signaling the\npresence of the first order transition, similar to the sce-\nnario reported in the spin-1 chain[16]. The transition be-\ntween the topologically non-trivial uniform phase and the\ndimerized one is continuous for J2≲0.08 in the WZW\nSU(2) 4universality class, at J2≈0.08 this critical line\nterminates with the SU(2) 6end point. At large J2the\ntransition is between the two topologically trivial phases\nand is consistent with a non-magnetic Ising transition.\nIn the next two sections we provide details on the nature\nof these continuous transitions.\nPrevious investigations of the J1−J2−J3model also\nrevealed that a large portion of the phase diagram is char-\nacterized by incommensurate spin-spin correlations[6, 7,\n17, 31, 32]. In spin-3 chain incommensurability appears\nonly as a short-range order. Dotted line inside the uni-\nform phases in Fig.1 marks the disorder line along which\nincommensurability develops. Examples of the connected\nspin-spin correlation function Ci,j=⟨Si·Sj⟩−⟨Si⟩·⟨Sj⟩\non both sides of the disorder line are provided in Fig.2.3\n0 10 20 30 40 50 6010-1010-810-610-410-2100102\nFigure 2. Scaling of the connected correlation functions\nCi,j=⟨Si·Sj⟩ − ⟨Si⟩ · ⟨Sj⟩with distance between spins for\nJ3= 0.005 and various values of J3. Starting from J2≈0.24\nsystem demonstrates the presence of incommensurate short-\nrange correlations.\nInside the dimerized phase the disorder line coincides\nwith the exact dimerized line.\nIII. FROM WZW SU(2) 6TO SU(2) 4\nLet us now focus on the continuous transition at small\nvalues of J2. In order locate the transition we look at\nthe finite-size scaling of the middle-chain dimerization\nDmid=|⟨SN/2−1·SN/2⟩ − ⟨SN/2·SN/2+1⟩|. In a log-\nlog scale convex curves signals finite dimerization in the\nthermodynamic limit and therefore are associated with\nthe dimerized phase. In contrast, concave curves point\ntowards the non-dimerized uniform phase. Quantum crit-\nical point between these two phases shows up as a sepa-\nratrix on the finite-size scaling plot. By keeping track of\nthe change of curvature we can narrow down the inter-\nval in which the quantum phase transition takes place.\nIn Fig.3(a) we provide an example of finite-size scal-\ning for J2= 0.08; the transition takes place between\nJ3= 0.017423 where the scaling curves a bit downwards\nandJ3= 0.017424 with data that slightly curve upwards.\nBy fitting the main slope of these two finite-size scalings\nwe estimate the upper and lower bounds of the critical\nscaling dimension dfor a given value of J2. The scal-\ning dimensions extracted along the transition line are\nsummarized in Fig.3(b). According to the conformal\nfield theory (CFT), the scaling dimension of the WZW\nSU(2) kis given by d=3\n2(2+k), this implies dk=4=1\n4and\ndk=6=3\n16. For small values of J2the scaling dimension\ndis in a reasonable agreement with dk=4, we believe the\ndiscripancy between the exact value and the measured\none is due to logarithmic corrections that appear when\none of the relevant operators renormalizes to zero. It\n0 0.02 0.04 0.06 0.08 0.10.150.20.250.3\n3.5 4 4.5 51.551.61.651.71.751.81.851.91.95 (a) (b)Figure 3. (a) Example of finite-size scaling of the middle-\nchain dimerization for J2= 0.08. The critical point is asso-\nciated with the separatrix at 0 .017423 < Jc\n3<0.017424, and\nthe slope gives an effective scaling dimension 0 .185≲deff≲\n0.198. Dashed lines are guide to the eyes; solid lines are results\nof the fit. (b) Resulting value of the effective scaling dimen-\nsiondeffas a function of J2along the transition between the\nuniform and the fully dimerized phases. We associate the end\npoint with the crossing points of the resulting curve and the\nhorizontal line dk=6= 3/16 (dashed red). The dashed blue\nline stands for dk=4= 1/4.\nis very important to stress, that our results would not\nbe consistent with another candidate satisfying Furuya\nand Oshikawa’s parity rule[24] - the scaling dimension\nof WZW SU(2) 2isd=3\n8that significantly exceeds the\nscaling dimension exgtracted numerically.\nFollowing the procedure described in Ref.18 we asso-\nciate the point J2≈0.08 where the scalind dimension\nis consistent with WZW SU(2) 6and perform additional\nchecks to confirm the universality class at the end points.\nFirst, we extract two critical exponents: βandνthat\nmeasures the scaling of the order parameter (dimeriza-\ntion) and the correlation length with the distance to the\ntransition. The results are presented in Fig.4. For both\ncritical exponents we see excellent agreement between nu-\nmerical data (filled symbols) and CFT predictions for\nWZW SU(2) 6ν=2+k\n2kandβ=3\n4k(red lines). For com-\nparison we also present results for J2= 0 that agree well\nwith CFT predictions for SU(2) 4. Despite the fact that\nthe two pairs of critical exponents are very close (0.125\nvs 0.1875 for βand 0.75 vs 0.667 for ν) one can clearly\ndistinguish these two cases in Fig.4.\nTo the best of our knowledge the realization of the\nWZW SU(2) 6phase transition in a non-integrable spin\nchain is reported for the first time and therefore desrves\nan additional check by extracting conformal the towers\nof states. Conformal tower is a very specific structure\nof excitation energy spectra that system takes at the\nconformal critical point and under specific cponformally-\ninvariant boundary conditions[33]. Here we probe only\nthe lowest level of each magnetization sector - the ”en-\nvelop” of the conformal tower. The ground-state of the\nsystem with N-even is a singlet and thus has total spin\nj= 0; chain with odd number of sites have ground-state4\n-13 -12 -11 -10 -91.31.41.51.61.71.81.922.12.2\n-13 -12 -11 -10 -9 -81.522.533.544.5(a)\n(b)\nFigure 4. Scaling (a) of the middle chain dimerization D\nand (b) of the correlation length ξupon approaching the\nphase transition from the fully dimerized phase to the uniform\nphase. Open symbols state for the horizontal cut at J2= 0,\nfilled symbols for the horizontal cut through the end point at\nJ2≈0.08; the data are in excellent agreement with the CFT\npredictions for the critical exponents for WZW SU(2) 4(blue\nlines) and SU(2) 6(red lines).\nwith total spin j= 3. We restrict ourselves to Sz\ntot≤12.\nWe construct the envelop of these two towers closely fol-\nlowing Ref.[18, 34]. The predictions for the envelops are\nsummarized in Table I.\nj=0\nSz\ntot 0123456789101112\n(E−E0)N/πv 012345691215182124\nj=3\nSz\ntot 3456789101112\n(E−E0)N/πv ,024681012162024\nTable I. Lowest excitation energy with spin Sz\ntotfor both j= 0\nandj= 3 WZW SU(2) 6conformal towers.\nNumerically we extract excitation energies for system\nsizes up to N= 160 for N-even and up to N= 91 for\nN-odd. We extract a non-universal value of the sound\nvelocity by fitting two lowest exchitation energy levels for\nN-even; the extracted value is v≈3.34±0.05. In Fig.5\nwe assume this value to draw CFT predictions without\nany further fitting or adjustments. The agreement with\nnumerically extracted levels of the towers is excllent.\n0 0.01 0.02 0.03 0.0400.511.522.533.50 0.01 0.02 0.03 0.040510152025\n0 0.01 0.02 0.03 0.0400.511.522.530 0.01 0.02 0.03 0.040510152025(a)(b)\n(c) (d)Figure 5. Conformal towers of states (top) and finite-size\nscaling of excitation spectrum (bottom) at the WZW SU(2) 6\nend point. Symbols are DMRG data, lines are CFT predic-\ntions summarized in TableI, sound velocity v≈3.34±0.05\nhas been extracted by fitting the lowest two excitation en-\nergies with N-even. For Neven, the ground state is in the\nsinglet sector ( j= 0 tower), for Nodd, the ground state has\ntotal spin j=3. The towers show only the lowest state within\neach sector of total magnetization up to Sz\ntot= 12.\nIV. ISING TRANSITION AT LARGE J2\nWe have shown so far that for small values of J2the\ntransition to the dimerized phase is continuous and ends\nat the end point at J2≈0.08. Beyond this point the tran-\nsition is first order. This conclusion is supported by the\nobservation that at the point where exact dimerized line\nhits the transition the order parameter jumps discontin-\nuously. Now, we are going to ask ourselves whether this\nfirst order transition will eventually turn (again) into a\ncontinuous transition at large values of J2. As a dis-\nclaimer, let us mention here that we leave detailed inves-\ntigation of the uniform phase(s) and phase transition(s)\nout of them within the range 0 .2≲J2≲0.8 outside of\nthe scope of this work. However, we want to share with\nour readers one very interesting observation.\nIn the region where the uniform phase can be ex-5\npected to be topologically trivial, the domain wall be-\ntween the uniform and the dimerized phases are non-\nmagnetic. This transition is therefore expected to take\nplace entirely in the singlet sector. Since Z2symmetry\nis broken at the transition it is natural to expect the\nunderlying critical theory to be Ising CFT with scaling\ndimension d= 1/8. Our numerical results for the finite-\nsize scaling of the order parameter presented in Fig.6 is in\nexcellent agreement with this prediction. Non-magnetic\nnature of this transition is supported by finite-size scaling\nof the singlet-triplet gap presented in the inset of Fig.6\nthat shows no tendency to close in the thermodynamic\nlimit.\n3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.82.32.42.52.62.7\n0 0.01 0.02 0.03 0.04 0.0500.10.20.30.40.50.6\nFigure 6. Finite-size scaling of the middle-chain dimerization\nforJ2= 1. Separatrix is associated with the phase transition;\nthe slope gived the effective scaling dimension that is in the\nexcellent agreement with the CFT prediction d= 1/8. In-\nset: Finite-size scaling of the singlet-triplet gap at the critical\npoint Jc\n3≈0.00605. The magnetic gap remains open at the\ntransition.\nV. DISCUSSION\nTo summarize, in the present paper we have studied\ndimerization transitions in a frustrated spin-3 chain. We\nidentified the end point in the WZW SU(2) 6universality\nclass into which the J1−J2−J3model with two inde-\npendent parameters is fine tuned. The universality class\nof the underlying critical theory has been verified with\nthe scaling dimension dand conformal towers of statescomputed at the transitions as well as with the critical\nexponents βandνcontrolling the scaling of the order\nparameter and the divergence of the correlation length\nupon approaching the transition.\nAway from the fine-tuned end point we detect that\nthe critical theory due to the presence of second relevant\noperator re-normalizes to a critical theory of the lower\nlevel of the same parity - SU(2) 4. We see no signature\nof possible renormalization to the level k= 2. This sug-\ngests that renormalization group flow will lead towards\nthe highest level with the right number of relevant op-\nerators. In other words, for frustrated spin model with\n5/2≤S <5 and two independent parameters one might\nexpect SU(2) 2Sto be realized as the end points termi-\nnating the WZW SU(2) 4critical line for integer- and the\nSU(2) 3line for half-integer spin chains - both indepen-\ndent on the value of spin S. The case of spin S= 5\nis less obvious due to the appearance of an additional\nmarginal operator. Beyond that for S > 5 a generic\ntwo-dimensional parameter space would not suffice to\nfine-tune the system into WZW SU(2) 2Scritical points\nand we expect the critical lines for integer (half-integer)\nchains to be in SU(2) 4(SU(2) 3) terminated with SU(2) 10\n(SU(2) 9) end points. This hypothesis can be checked nu-\nmerically with S >3, though, of course, computationally\nthis is very challenging. One has to admit a possibility\nthat for a given model, e.g. for J1−J2−J3, the interval\nof continuous transition might shrink for large Sto the\npoint when it can eventually disappear.\nIn addition, for large value of J2we report an ap-\npearance of non-magnetic Ising transition, previously de-\ntected in frustrated spin-1 chain. This supports previous\nconjecture that the appearance of the Ising transition\nat the boundary of the dimerized phase and large J2is\ngeneric for any integer spin chains.\nVI. ACKNOWLEDGMENTS\nI am indebted to Frederic Mila and Ian Affleck for\nmany inspiring discussions on dimerization transitions.\nThis research has been supported by Delft Technology\nFellowship. Numerical simulations have been performed\nwith the Dutch national e-infrastructure with the support\nof the SURF Cooperative and at the DelftBlue HPC.\n[1] J. M. Kosterlitz and D. J. Thouless, Journal of Physics\nC: Solid State Physics 6, 1181 (1973).\n[2] C. K. Majumdar and D. K. Ghosh, Journal of Mathe-\nmatical Physics 10, 1388 (1969).\n[3] K. Okamoto and K. Nomura, Physics Letters A 169, 433\n(1992).\n[4] F. D. M. Haldane, Physics Letters A 93, 464 (1983).\n[5] A. K. Kolezhuk and U. Schollw¨ ock, Phys. Rev. B 65,\n100401 (2002).[6] A. Kolezhuk, R. Roth, and U. Schollw¨ ock, Phys. Rev.\nLett.77, 5142 (1996).\n[7] A. Kolezhuk, R. Roth, and U. Schollw¨ ock, Phys. Rev. B\n55, 8928 (1997).\n[8] M. N. Barber and M. T. Batchelor, Phys. Rev. B 40,\n4621 (1989).\n[9] A. Kl¨ umper, Europhysics Letters 9, 815 (1989).\n[10] Y. Xian, Physics Letters A 183, 437 (1993).\n[11] A. L¨ auchli, G. Schmid, and S. Trebst, Phys. Rev. B 74,\n144426 (2006).6\n[12] F. Michaud, F. Vernay, S. R. Manmana, and F. Mila,\nPhys. Rev. Lett. 108, 127202 (2012).\n[13] F. Michaud, S. R. Manmana, and F. Mila, Phys. Rev. B\n87, 140404 (2013).\n[14] N. Chepiga and F. Mila, Phys. Rev. B 100, 104426\n(2019).\n[15] Z.-Y. Wang, S. C. Furuya, M. Nakamura, and R. Ko-\nmakura, Phys. Rev. B 88, 224419 (2013).\n[16] N. Chepiga, I. Affleck, and F. Mila, Phys. Rev. B 93,\n241108(R) (2016).\n[17] N. Chepiga, I. Affleck, and F. Mila, Phys. Rev. B 101,\n174407 (2020).\n[18] N. Chepiga, I. Affleck, and F. Mila, Phys. Rev. B 105,\n174402 (2022).\n[19] I. Affleck and F. D. M. Haldane, Phys. Rev. B 36, 5291\n(1987).\n[20] P. Lecheminant and H. Nonne, Phys. Rev. B 85, 195121\n(2012).\n[21] P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin, Let-\nters in Mathematical Physics 5, 393 (1981).\n[22] L. A. Takhtajan, Physics Letters A 87, 479 (1982).\n[23] H. M. Babujian, Nuclear Physics B 215, 317 (1983).[24] S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 118,\n021601 (2017).\n[25] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).\n[26] U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005).\n[27] S. ¨Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537\n(1995).\n[28] U. Schollw¨ ock, Annals of Physics 326, 96 (2011), january\n2011 Special Issue.\n[29] S. Fubasami, T. Mizoguchi, and Y. Hatsugai, Phys. Rev.\nB100, 014438 (2019).\n[30] In our finite-size calculations the dimerization measured\nin the middle of the chain in the uniflrm phases is Dmid∝\nO(10−2). For comparison, the mid-chain dimerization in\nthe partially dimerized phase in spin-5/2 chain is Dmid∝\nO(1).\n[31] R. Roth and U. Schollw¨ ock, Phys. Rev. B 58, 9264\n(1998).\n[32] N. Chepiga, I. Affleck, and F. Mila, Phys. Rev. B 94,\n205112 (2016).\n[33] P. Di Francesco, P. Mathieu, and D. S´ en´ echal, Conformal\nField Theory , Graduate Texts in Contemporary Physics\n(Springer, New York, 1997).\n[34] I. Affleck, D. Gepner, H. J. Schulz, and T. Ziman, Journal\nof Physics A: Mathematical and General 23, 4725 (1990)."    },    {        "title": "2402.05076v1.Markovian_Analysis_of_Information_Cascades_with_Fake_Agents.pdf",        "content": "arXiv:2402.05076v1  [cs.SI]  7 Feb 2024Markovian Analysis of Information Cascades with\nFake Agents\nYuming Han\nNorthwestern University, Evanston, USA\nEmail: yuminghan2021@u.northwestern.cedu\nAbstract —People often learn from other’s actions when they\nmake decisions while doing online shopping. This kind of obs erva-\ntional learning may lead to information cascades, which mea ns\nagents might ignore their own signals and follow the ”trend”\ncreated collectively by the actions of their predecessors. It is\nwell-known that with rational agents, such a cascade model c an\nresult in either correct or incorrect cascades. In this pape r, we\nadditionally consider the presence of fake agents who alway s take\nfixed actions and we investigate their influence on the outcom e of\nthese cascades. We propose an infinite Markov Chain sequence\nstructure and a tree structure to analyze how the fraction an d\nthe type of such fake agents impacts behavior of the upcoming\nagents. We show that an increase in the fraction of fake agent s\nmay reduce the chances of their preferred outcome, and also\nthere is a certain lower bound for the probability of a wrong\ncascade. In particular, we discuss the probability of an age nt\nbeing fake tends to 1 and the effect of a constant portion of fa ke\nagents.\nIndex Terms —information cascades, fake agents, Bayesian\nlearning model, Markovian analysis.\nI. I NTRODUCTION\nIn this paper, we analyze a recommendation-based market\nwhere agents sequentially arrive and decide whether to buy a\nnew product that is up for sale based on their own prior knowl-\nedge of its quality and by observing the decisions of their\npredecessors. Every agent’s decision is recorded as a refer ence\nfor later agents to a common database so that later agents are\nable to obtain information. This kind of model was proposed\nin [1]- [3]as a Bayesian learning model where a sequence of\nagents take optimal actions. In this model, every agent come s\nwith his own opinion or signal about this new product followe d\nby a known probbibilty distribution. Besides, there is a giv en\ncommon database contains all previous desicions made by\npredecessors as a reference. Each newly arrived agent makes\na one-time Bayesian rational decision based on his own signa l\nand these observations from the common database, and then\nhis action is recorded.\nWe emphasize on two main results for such Bayesian model.\nFirst, there is learning. It comes from agents make an optima l\ndecision by observing the database to achieve the true value of\nthe product. Second, there is information cascade. Informa tion\ncascade is an agent ignores his own knowledge and follows\nthe decisions made by his predecessors. The cascade situati on\nwould also prevent the upcoming agents from learning from\nsubsequent observations, i.e., once a wrong cascade (socia lly\nsub-optimal) occurred, this market would be trapped in thewrong cascade forever. [1]- [3]assumes agents have discret e\nknown distribution private signals, which leads to a cascad e\nhappens in a finite number of agents with probability one.\nThat gives us a great opportunity to study on the probability\nof wrong casesde with different model settings.\nIn this paper, we consider a similar model as in [1]- [3].\nIn addition to rational agents, we introduce a certain type\nof agents named ”fake agents” who always make a fixed\naction. Regardless of their own opinion or the actions of pri or\nagents, fake agents will only buy (Y-type) or not buy (N-type )\na product to influence the probability of the outcome of a\ncascade. For example, when people go to Amazon to buy an\nitem, even if this item has a poor quality, a Y-type fake agent\nwould continually buy it and record that this item has good\nquality on the website (five stars). This information will ha ve\nan impact for the future agents. On the other hand a N-type\nfake agent would act in the opposite way to always give a bad\nreview (one star). We study on the influences of these fake\nagents on the probabilites of wrong information cascades an d\nsome certain thresholds appeared in the simulation resutls . An\ninfinite Markov chain model, similar to [4], is proposed. In\nour model, we mainly concentrate on the impact of varying\nthe portion of fake agents on the probability of correct and\nincorrect cascades. Our work is based on [5], Poojary and\nBerry discuss the scenario of one type of fake agents in the\nnetwork.\nThere are some other related work similar to our model,\nsuch as [7], where the agent’s observations depend on an\nunderlying network and [8], where the crazy and stubborn\nagents are invloved. The work of Smith and Sorensen [9] re-\nlaxed both the assumptions of binary signals and homogeneou s\nagents. Another aspect of related work is that which conside rs\ndifferent model structure. In [11] a non-Bayesian model was\nconsidered, and an information structure with selected cos t\nwas introduced in [12].\nIn Section II, we propose our model. We analyze our model\nand discuss the cascade results in Section III. Next, in Sect ion\nIV , we introduce the Markov Chain process. We talk about\nsome certain thresholds, approximations and bounds in Sect ion\nV . We conclude in Section VI with some directions for future\nwork on modeling and analyzing for realistic settings.II. S YSTEM MODEL\nWe consider a model, similar to [1], in which with a\ncountable sequence of agents, indexed i= 1,2,... where the\nindex represents both the time and the order of arrival actio ns\nfor each agent. Every agent i chooses an action Aiof either\nbuying (Y) or not buying (N) one item which has a true value\n(V) that could be either good (G) or bad (B). We consider\nV=GandV=Bto be equiprobable. The prior distribution\nof the item value is assumed to be common knowledge and\nthe realized value of the item is assumed to be the same for\nagents.\nFurther, each agent ireceives a private signal Si∈\n{H(High),L(Low)}that represents its prior knowledge of\nthe product’s quality. This private signal Sipartially re-\nveals the true value Vthrough a binary symmetric channel\n(BSC) with crossover probability 1−p, depicted in Figure\n1. Here,p∈ {0,1}denotes the private signal quality. Agent\nithen takes a rational action Aibased on his private signal\nSiand the past observations {O1,O2,...,O i−1}of actions\n{A1,A2,...,A i−1}. Next, we modify the model in [1] by\nconsidering that at each time instant, an agent could be a Y\n-type fake agent (w.p.) ǫ∈[0,1)or a N -type fake agent\n(w.p.)β∈[0,1)or an ordinary agent w.p. 1−γ. Here,\nγ=ǫ+βdenotes the total probability of an agent being\nfake. An ordinary agent is honest in reporting its actions,\ni.e.,Oi=Ai. Whereas, a Y-type (N-type) fake agent always\nreports a Y (N).\nVG\nBH\nLSiAi OiY\nNY\nNp\n1−p\n1−p\np1−β\nβ\nǫ\n1−ǫ\nFig. 1: The Binary Symmetric Chnnel though which agents\nreceive signals\nConsider the first agent, as there are no past observations,\nhis optimal action is to follow S1, i.e. to buy the product\nonly if he receives a H(High) signal. Now, for the nth\nagent’s decision, let the prior observation history be deno ted\nbyHn−1={O1,O2,...,O n−1}. Each agent then takes the\nBayesian optimal action given Hn−1andSnwhich is defined\nas follows.\nLetγn=P(G|Sn,Hn−1)denotes the posterior probability\nfor the true value of item is good, V=G.\nThen the Bayesian optimal action taken by agent nis\ndefined as:\nAn=\n\nY, ifγn>1/2\nN, ifγn<1/2\nfollowsS n,ifγn= 1/2(1)\nHence, from now on, we assume that each agent n\ntakes action Anbased on his private signal Snand noisy\nobservations O1,O2,...,O n−1of all past agents’ actionsA1,A2,...,A n−1. In the next chapter, we will discuss several\nscenarios based on this model.\nIII. I NFORMATION CASCADE\nDefinition 1. The public likelihood ratio is defined as\nln(Hn) =P(Hn|B)/P(Hn|G). The private likelihood ratio\nof agentn,δn, is defined as δn(Sn) =P(Sn|B)/P(Sn|G).\nUsing the above definition, we have γn= 1/(1+δnln−1).\nCombine with 1, in a decision-making process, a Y is the\nchoice whenever δnln−1<1. We name this case as informa-\ntion cascade, which is defined as follows:\nDefinition 2. An information cascade is said to occur\nwhen a new agent takes a fixed action regardless of his private\nsignal.\nNow, if agent nhappens to be in an information cascade,\nthen his optimal decision as described in (1) implies that γn>\n1/2(<1/2)for eachSn∈ {H,L}and thenthagent is said\nto be in a Y(N) cascade. The probability is when γ≥1/2for\nSn=High andγ≤1/2forSn=Low in which case, An\nfollowsSn.\nLemma 1: Agent ncascades to the action Y(N) when\nln−1<1−p\np(ln−1>1−p\np) and otherwise follows its private\nsignalSn.\nAnd once a cascade happens, the public likelihood ratio\nlnwould stay as the same because the observation Ondoes\nnot bring any new information about the true value Vto the\nsuccessors. Therefore, ln+i=ln−1for alli= 0,1,2,...,\nwhich is the following property.\nProperty 1: Once a cascade happens, it stays forever.\nNow, considering that agent nis not in a cascade, Property\n1 implies that AifollowSifor alli= 1,2,...,n−1. Thus,\nthe public likelihood ratio for the history Hncan be expressed\nasln= (1−p\np)hn, wherehn=ηYnY−ηNnNis the weighted\ndifference between the number of Y’s( kY) and the number\nof N’s (kN) inHn. The weights ηY=loga\n1−b/logp\n1−pand\nηY=logb\n1−a/logp\n1−pwhere for all i= 1,2,...,n−1.\na=P(Oi=Y|V=G)andb=P(Oi=N|V=B).(2)\nDenote the probabilities when Oifollows the true value V\ngiven that AifollowsSi. From Figure 1, we have\na=p(1−β)+ǫ(1−p)andb=p(1−ǫ)+β(1−p)(3)\nThus, until a cascade occurs, the likelihood ratio lndepends\nonly on the number of Y’s and N’s present in the history\nthroughhnwhich thereby serves as a sufficient statistic of the\ninformation contained in the past observations, which give us\nthe following property.\nProperty 2: agents will follow their own private signal\nuntil cascade occurs. Now, substituting the above obtained\nexpression for lnin Lemma 1, we see that agent ncascades to\na Y (N) only when hn>1(hn<−1). Until then, the process\n{hn}updates as per the rule : information cascade happenswhenhn<−1orhn>1, and starting at 0, the update rule\nforhnis:\nhn=/braceleftBigg\nhn−1+ηY,ifOn=Y\nhn−1−ηN,ifOn=N(4)\nand thereafter stops updating (Property 1).\nIV. M ARKOVIAN ANALYSIS OF CASCADES\nIt follows from last section that the random process {hn}\nis a countable state space DTMC that starts at 0and takes\nvalues in [-1,1] until it ends up in one of the two absorption\nstates: the left wall (−∞,−1)and the right wall (1,∞)that\ncorrespond to a N cascade and Y cascade respectively. Each\nstate represents the agent’s observation history. Moreove r, until\na cascade occurs, the update rule for {hn}in (3.3) describes it\nas a random walk that either takes a forward (rightward) step\nbyηYwith probability pfor takes a backward (leftward) step\nbyηNwith probability 1−pf. Here,pf:=P(On=Y|V)\nrefers to the ”forward jump probability”, i.e, the probabil ity\nof a Y observation given the true state V. From (1) we have\npf=aforV=Gandpf= 1−bforV=B. (5)\n1−pfpf\n-1−ηN 0ηY 1\nFig. 2: Transition Diagram of the Random Walk\nFigure 2 depicts the random walk.\nIn [4], the noise is unbiased towards each of the possible\nactions, which leads to ηY=ηNand finite states on Markov\nchain. Furthermore, in [5], the noise is biased only towards\nto one preferred action that represented an uncountable sta t-\nspace Markov chain. Note that in our case, we mainly consider\nthe non-integer values of 1/ηYand1/ηN, which means the\nhnneeds to take uncountable steps to get one of absorbing\nstates. Because when ηYandηNsatisfies1/ηYand1/ηN=r\nfor some r= 1,2,3,... the random walk is a sample\nMarkov Chain with finite state and having two absorbing state s\nrepresented Y and N cascade.\nFor the random walk {hn}, given the true value V∈\n{G,B}, letPV\nYdenote the probability of a Y cascade. The\nN cascade probability denoted by PV\nNis simply 1−PV\nY, as a\ncascade occurs almost surely.\nA. Y cascade probability\nIn this subsection, we plot the probability of a wrong\ncascade under V=B, i.e.,PB\nYforǫvarying from 0 to 1−β\nfor a fixed β. Refer to Fig. 1 which compares two plots of PB\nY\nfor values: β= 0andβ= 0.5. Recall that β= 0corresponds\nto the single fake agent-type scenario of [5]. Compared to on e\ntype of fake agent scenario, the occurrence of two types of fa ke\nagents shows that a certain decrease of wrong cascade. When\nwe enroll the second type of fake agents, they can be treated a sa compensation to the negative effect caused by the first type of\nfake agents. Counter to basic knowledge, the probability of Y\ncascade is not a monotonically increasing function with res pect\nofǫ. There is a significant decrease at some certain ǫvalue.\nThis character will be discussed in the next section. Furthe r,\nfrom the point of view of the fake agents, they would not\nbenefit from their portion increasing as ǫcontinue increasing.\nThat is because as the probability of an agent being fake tend s\nto 1, the information included in a Y observation is negligib le.\nThere is less and less information being considered from an\nobserved Y .\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.10.150.20.250.30.350.40.45\nεprob[hit the wall]wrong cascade\n  \nβ=0\nβ=0.05\nFig. 3: Probability of Y Cascade as a function of ǫfor V=B,\np=0.7 and β= 0,β= 0.05\nNext, we compare PB\nYforβ= 0 with the plots for PB\nY\nwhereβ= 0.1,0.2in Figure. 4 respectively. We observe a\ngeneral trend over most values of ǫthat an increase in the\nfraction of N type fake agents βleads to a reduction in the\nwrong cascade prob. under V=B.\nAs the value of βincrease, the presence of second type\nof fake agents reduces the probability of wrong cascade. The\nfractional decrease at some certain ǫshrinks as well. Figure. 4\nshows that the wrong cascade curve becomes more and more\nsmooth, which we will discuss in next section.\nB. Error Thresholds\nIn [5], the author discussed the fractional decrease at some\ncertainǫthresholds. At first when there is no fake agent, the\nsequence needs to observe one action two times to have a\ncascade. As the portion of fake agents increase, the number\nof consecutive Y needed to start a cascade increases because\neach observation provides less information. It can be prove d\nin the following lemma.\nLemma 2: Letα=p/(1−p). Forr= 0,1,2,... define\nthe sequence of thresholds {ǫr}∞\nr=1. By using Bayes’ rule, the\ndenotes the posterior probability γnis given as:0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.10.150.20.250.30.350.40.45\nεProb[hit the wall]wrong cascade\n  \nβ=0\nβ=0.1\n(a)β= 0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.10.150.20.250.30.350.40.45\nεProb[hit the wall]wrong cascade\n  \nβ=0\nβ=0.2\n(b)β= 0.2\nFig. 4: Probability of Y Cascade as a function of ǫfor V=B\nand p=0.7 with different β\nγn=(1−p)ar\n(1−p)ar+p(1−b)r(6)\nAgentndoes not have a cascade if γn≤1\n2, i.e.(1−p)ar≤\np(1−a)r. Therththreshold ǫris given as:\nǫr= (1−β)α−α1\nr\nα1\nr+1−1(7)\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.10.150.20.250.30.350.40.45\nεProb[hit the wall]wrong cascade\n  \nβ=0\nBayesian Threshold\nβ=0.05\nBayesian Threshold\nFig. 5: Bayesian Thresholds\nWe observe in Figure. 5 that the relatively larger drops in\nPB\nY(ǫ) (marked by *) occur exactly at the threshold points\n{ǫr}∞\nr=1. Here, a slight increase in ǫbeyondǫrcauses a\nsignificant decrease in the probability of a Y cascade. An\nintuitive reasoning for this reduction the different numbe r of\nconsecutive Ys are needed to start a cascade when there is onl y\none N in the sequence. That leads to we wonder what would\nhappen if the sequence had more than one N and how many\nconsecutive Ys are needed. Following that idea, the cascade\nthresholds are defined as calculating the probability of wro ng\ncascade based on multiple Ns in the sequence. As shows in\nthe Figure. 6, Bayesian thresholds, second and third cascad e\nthresholds capture most of discontinuities in the curve.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.180.20.220.240.260.280.30.320.34\nεProb[hit the wall]wrong cascade\n  \nβ=0.1\nBayesian Threshold\nCascade Threshold\n(a)β= 0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.180.20.220.240.260.280.30.32\nεProb[hit the wall]wrong cascade\n  \nβ=0.2\nBayesian Threshold\nCascade Threshold\n(b)β= 0.2\nFig. 6: Bayesian Thresholds and Cascade Thresholds\nC. Tree and Sequence Structure\nIn this subsection, we outline two approaches for estimatin g\nthe wrong cascade probability. A tree structure is proposed to\nestimate PB\nYand a sequence structure to discuss the lower\nbound.\n1) Tree Structure: Let us consider a way to estimate the\nprobability of a Y cascade. Because our focus is non-integer\nvalues of 1/ηresulting in uncountable values of the number\nof needed consecutive Ys. Besides, due to two types of fake\nagents, sequence would be allowed multiple Ns without hitti ng\nthe left side wall. It leads to infinite long sequence. To have\nan approximation of our model, we proposed a tree structure\nto describe our model. We firstly defined the comfort zone for\nevery sequence with different ηN. The comfort zone is given\nas:[ηN−1,1−ηN]. Then, we divide the whole region [-1,1]\ninto several sub-regions, such as [−1,ηN−1],[ηN−1,0],\n[0,1−ηN]and[1−ηN,1]. The following iterative process\ndepicted in Figure. 7 describes all possible sequences that can\nlead to a Y cascade.\nx -1 ηN−10comfort zone\n1−ηN 1\nFig. 7: Comfort Zone\nThe iteration process shows in Figure. 7, r0=⌈1\nηY⌉,\nr1=⌈ηN−1\nηY⌉,r2=⌈1−ηN\nηY⌉. Let M denotes the number of\niterations. The sequence will be starting at 0, moving forwa rd\nwith probability Pfand backward with probability 1−Pf. For\nevery time the sequence come back to the comfort zone, a new\niteration will begin until the sequence hit one of walls. Not e\nthat the comfort zone is not the same as before for a new\niteration process. Because it is unlikely the sequence woul d\nland exactly at 0 when it comes back to the comfort zone.\nHence, for the accuracy of our approximation, a new strating\npoint and comfort zone will be defined before the next iterati on\nprocess.\nIn Figure. 8, we draw the upper and lower approximation\nof our model based on this iteration rule. The plot uses\nM=10 which gives a close approximation. Our approximation\npefectly captures many discontinuities at Bayesian and Cas -\ncade Thresholds. Note that the simulation is outside of our\napproximation when ǫis small. It did not expect a fractional\ndecrease there.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.050.10.150.20.250.30.350.4\nεProb[hit the wall]wrong cascade\n  \nβ=0.1\nBayesian Threshold\nCascade Threshold\nUpper approximation\nLower approximation\nFig. 8: Upper and Lower Approximation with Thresholds\n2) Sequence Structure: In order to have an accurate lower\nbound for the probability of wrong cascade, we proposed\nanother sequence structure to represent all possible seque nces\nas Figure.10 shown. In this case, we divide positive region[ 0,1]\ninto (K+2) sub-regions based on ηY. Once the sequence land\ninto the(K+ 1)thstage, it only needs one Y to have a Y\ncascade. The way we count all possible sequences is starting\nat 0 and moving forward stage by stage.\n0ηY2ηY3ηY 1−ηN 1(1) (2) (3) ( K+1)\nFig. 10: Stage of Sequence Strcuture\nWe assume ηN> ηY. The counting process starts at 0. Let\nr1=⌈1\nηY⌉, denotes the number of consecutive Y needed to\nhit the right wall starting at 0, t1=⌈ηN\nηY⌉, denotes the number\nof consecutive Y needed to compensate from a N to come\nback. The relationship between these two parameters is give n\nas:\n1−ηN= (K+1)ηY⇒K+1 =r1−t1 (8)\nEach stage includes all possible sequences landed in this\nregion, including multiple Ns cases. We can calculate the\nwhole stage probability of all the possible sequences at one\ntime. After the counting process arrives (K+1)thstate, theiteration begins. Here the (K+1)thstage is much similar as\nthe comfort zone we talked about in the last section.\nFigure. 11 shows both the lower bound and lower approx-\nimation with Bayesian Thresholds. The plot uses M= 10\nwhich gives an error of less than 10−4.\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.050.10.150.20.250.30.350.4\nεProb[hit the wall]wrong cascade\n  \nβ=0.05\nBayesian Threshold\nlower approximation\nlower bound\nFig. 11: Lower Bound and Lower Approximation with\nBayesian Thresholds\nV. C ONCLUSIONS\nIn this paper, We studied the effect of randomly arriving\ntwo types of fake agents that seek to influence the outcome\nof an information cascade. We concentrated on the impact of\nvarying one type of fake agents on the probability of their\npreferred cascade. Using a Markov chain based analysis we\ndetermined two types of thresholds, i.e. Bayesian and Casca de\nthresholds. In addition, we employed a tree structure and\nsequence structure to have an approximation and a lower\nbound of the probability of wrong cascade. In future work\nwe plan on a non-Bayesian rationality network, and the limit\nwhen there is a small amount of fake agents.\nREFERENCES\n[1] S. Bikhchandani, D. Hirshleifer, and I. Welch, “A theory of fads, fashion,\ncustom, and cultural change as informational cascades,” Journal of\npolitical Economy , vol. 100, no. 5, pp. 992–1026, 1992.\n[2] A. V . Banerjee, “A simple model of herd behavior,” The quarterly\njournal of economics , vol. 107, no. 3, pp. 797–817, 1992.\n[3] I. Welch, “Sequential sales, learning, and cascades,” The Journal of\nfinance , vol. 47, no. 2, pp. 695–732, 1992.\n[4] T. N. Le, V . G. Subramanian, and R. A. Berry, “Information cascades\nwith noise,” IEEE Transactions on Signal and Information Processing\nover Networks , vol. 3, no. 2, pp. 239–251, 2017.\nYcascade\nYYr1\nN Yt1\nY N Yt1−1\nY2N Yt1−2\nYt1N\nStage(1)YN Yt1+1\nY N Yt1\nY2N Yt1−1\nYt1+1N\nStage(2)...N Yt1+K\nY N Yt1+K−1\nY2N Yt1+K−2\nYt1+KN\nStage(K+1)Ycascade\nYt1ort1+1\nN Yt1ort1−1\nY N Yt1−1ort1−2\nY2N Yt1−2ort1−3\nYt1ort1−1N\nStage(K+1)...\nFig. 9: An enumeration of all possible sequences that would lead to a Ycascade.[5] P. Poojary and R. Berry, ”Observational Learning with Fa ke Agents,”\n2020 IEEE International Symposium on Information Theory (I SIT),\n2020, pp. 1373-1378.\n[6] P. Poojary and R. Berry, ”Observational Learning with Fa ke Agents,”\n2020 IEEE International Symposium on Information Theory (I SIT),\n2020, pp. 1373-1378.\n[7] D. Acemoglu, M. A. Dahleh, I. Lobel, and A. Ozdaglar, “Bay esian\nlearning in social networks,” The Review of Economic Studies , vol. 78,\nno. 4, pp. 1201–1236, 2011.\n[8] L. Smith and P. Sørensen, “Pathological outcomes of obse rvational\nlearning,” Econometrica , vol. 68, no. 2, pp. 371–398, 2000.\n[9] D. Acemoglu, G. Como, F. Fagnani, and A. Ozdaglar, “Opini on fluctua-\ntions and disagreement in social networks,” Mathematics of Operations\nResearch , vol. 38, no. 1, pp. 1–27, 2013.\n[10] L. Smith, P. Sorensen, ”Pathological Outcomes of Obser vational Learn-\ning”, Econometrica , vol. 68, pp. 371-398, 200\n[11] Y . Wang, P. Djuric, ”Social learning with Bayesian agen ts and random\ndecision making”, IEEE Trans Sig. Process , vol. 63, no. 12, 2015.\n[12] Y . Song, ”Social Learning with Endogenous Network Form ation”,\nsubmitted to J. Econ. Theory, 2015.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.10.150.20.250.30.350.40.45\nεProb[hit the wall]wrong cascade\n  \nβ=0\nβ=0.15"    },    {        "title": "2402.05077v2.Cycle_factors_in_oriented_graphs.pdf",        "content": "Cycle-factors in oriented graphs∗\nZhilan Wang, Jin Yan†, Jie Zhang\nSchool of Mathematics, Shandong University, Jinan 250100, China\nAbstract\nLetkbe a positive integer. A k-cycle-factor of an oriented graph is a set of\ndisjoint cycles of length kthat covers all vertices of the graph. In this paper,\nwe prove that there exists a positive constant csuch that for nsufficiently large,\nany oriented graph on nvertices with both minimum out-degree and minimum in-\ndegree at least (1 /2−c)ncontains a k-cycle-factor for any k≥4. Additionally,\nunder the same hypotheses, we also show that for any sequence n1, . . . , n twithPt\ni=1ni=nand the number of the niequal to 3 is αn, where αis any real number\nwith 0 < α < 1/3, the oriented graph contains tdisjoint cycles of lengths n1, . . . , n t.\nThis conclusion is the best possible in some sense and refines a result of Keevash\nand Sudakov.\nKeywords: Cycle-factor; minimum semi-degree; oriented graph\nMathematics Subject Classifications: 05C70, 05C20, 05C38\n1 Introduction\nResearchers have made a fruitful study on tilings in graphs. Among them, the most\noutstanding result, Dirac’s Theorem [2], asserts that any graph Gonn≥3 vertices\nwith minimum degree at least n/2 contains a Hamiltonian cycle , which is a cycle pass-\ning through every vertex of G. This theorem raises the crucial question of determining\nwhat minimum degree is needed to identify a specific structure in a graph. The Hajnal-\nSzemer´ edi theorem [6] is a significant result in this regard, which shows that for every\npositive integer t, ifGis a graph on nvertices with n≡0 (mod t) such that the minimum\ndegree of Gis at least (1 −1/t)n, then Gcontains a Kt-factor.\nWe naturally ask if these results have some similarities to oriented graphs , which are\norientations of simple graphs. As any transitive tournament (with large minimum degree)\ncontains no a cycle, we consider the minimum semi-degree δ0(D) = min {δ+(D), δ−(D)}\nof an oriented graph Dinstead of the minimum degree condition, where δ+(D) (resp.,\nδ−(D)) is the smallest value of the out-degrees (resp., in-degrees) of the vertices in D.\n∗The author’s work is supported by NNSF of China (No.12071260)\n†Corresponding author. E-mail address: yanj@sdu.edu.cn\n1arXiv:2402.05077v2  [math.CO]  2 Mar 2024Thomassen [14] posed a question about the analogue of Dirac’s theorem in oriented\ngraphs. He inquired about what minimum semi-degree forces a Hamiltonian cycle in an\noriented graph. Since the question was raised, a series of improved bounds have been\nachieved in [4, 5, 10, 15, 16]. Subsequently, Keevash, K¨ uhn and Osthus [9] provided\na precise minimum semi-degree condition to prove the existence of Hamiltonian cycles,\nwhich answers Thomassen’s question for oriented graphs with large order.\nTheorem 1.1. (Keevash et al. [9]) There exists a number n0such that any oriented graph\nDonn≥n0vertices with δ0(D)≥ ⌈(3n−4)/8⌉contains a Hamiltonian cycle.\nWe further study cycle structure in oriented graphs. A number of (directed) graphs\nare said to be vertex-disjoint (briefly disjoint ) if no two of them have any common vertex.\nLetTandDbe directed graphs. A T-factor ofDrefers to a collection of disjoint copies\nofTinD, covering all vertices of D. In particular, we call it a cycle-factor ifTis a\ncycle. Additionally, if Tis ak-cycle, a cycle of length k(≥3), then we say it a k-cycle-\nfactor . Notably, a Hamiltonian cycle can be considered as a cycle-factor with exactly one\ncycle. Furthermore, the following theorem establishes the existence of cycle-factors with\nthe same length in oriented graphs.\nTheorem 1.2. Letkbe a positive integer with k≥4. There exist positive constants c\nandn0such that for every n≥n0andn≡0 (mod k ), the following holds. If Dis an\noriented graph on nvertices with δ0(D)≥(1/2−c)n, then Dcontains a k-cycle-factor.\nSpecially, Li and Molla [12] proved that an oriented graph Dhas a 3-cycle-factor\nif and only if Ddoes not have a divisibility barrier under the same hypotheses as in\nTheorem 1.2. Here a divisibility barrier ofDis a partition P={V1, V2, V3}ofV(D) with\nthe following properties: (i) there is no arc from VjtoVj−1for any j∈[3] and V0=V3;\nand (ii) |V1|,|V2|and|V3|are not all equivalent modulo 3.\nA tournament is regular if every vertex has the same in-degree and out-degree. To-\ngether with the result of Li and Molla and Theorem 1.2, we can establish the following\ncorollary.\nCorollary 1.3. There exists a positive integer n0such that for any k≥3and every odd\ninteger n≥n0with n≡0 (mod k ), any regular tournament on nvertices contains a\nk-cycle-factor.\nFurther, we also present the following significant finding.\nTheorem 1.4. There exists a positive constant csuch that for nsufficiently large, if Dis\nan oriented graph on nvertices with δ0(D)≥(1/2−c)n, then for any sequence n1, . . . , n t\nsatisfyingPt\ni=1ni=nand the number of the niequal to 3is less than αn, where αis\nany real number with 0< α < 1/3,Dcontains tdisjoint cycles of lengths n1, . . . , n t.\nNotice that the value of αis best possible in Theorem 1.4 in some sense since,\nifDcontains a divisibility barrier, then Ddoes not have a 3-cycle-factor whatever the\nminimum semi-degree of Dis. The above theorem refines a result of Keevash and Sudakov.\n2Theorem 1.5. (Keevash and Sudakov [11]) There exist constants c′, C > 0such that for\nnsufficiently large, if Dis an oriented graph on nvertices with δ0(D)≥(1/2−c′)nand\nn1, . . . , n tare numbers withPt\ni=1ni≤n−C, then Dcontains disjoint cycles of length\nn1, . . . , n t.\nCombining with Corollary 1.3 and Theorem 1.4, we can obtain the result as follows.\nCorollary 1.6. There is n0such that for any positive integer partition n=n1+···+nt,\nwhere ni≥3for each i∈[t], every regular tournament on n≥n0vertices contains t\ndisjoint cycles of lengths n1, . . . , n t.\nThe organization of the rest of the paper is as follows. In Section 2, we prove Theorem\n1.4 by utilizing Theorem 1.2. In Section 3, we introduce some terminology and notation in\nSubsection 3 .1. Then we provide a brief outline of the proof of Theorem 1.2 in Subsection\n3.2, and present some results used in the proof of Theorem 1.2 in Subsection 3 .3. Finally,\nin Section 4, we give the detailed proof of Theorem 1.2.\n2 Proof of Theorem 1.4\nIn this section, we apply Theorem 1.2 to prove Theorem 1.4. Further, our proof requires\nthe lemma and theorems below. Lemma 2.1 is derived from the “Chernoff bound” approx-\nimation: P(|X−EX|> aEX)<2e−a2\n3EX, where 0 < a < 3/2 and Xis a hypergeometric\nrandom variable with expectation EX. Additionally, let Dbe an oriented graph and U\nbe a vertex set of D, we use D[U] to represent the subdigraph of Dinduced by U. For a\nvertex uinD, we also denote by d+(u, U) and d−(u, U) the out-degree and the in-degree\nofuinU, respectively.\nLemma 2.1. (Keevash and Sudakov [11] - Lemma 3 .2)For any α, β > 0there exists\na number n0such that the following holds. Suppose Dis an oriented graph on n≥n0\nvertices with δ0(D)≥αnandβn≤s≤(1−β)n. Then there exists a partition of\nV(D)asA∪Bwith|A|=sand|B|=n−ssuch that δ0(D[A])≥(α−n−1/3)sand\nδ0(D[B])≥(α−n−1/3)(n−s).\nKelly, K¨ uhn and Osthus showed a precise lower bound on the minimum semi-degree\nrequired to guarantee the existence of an l-cycle for all 3 ≤l≤ |V(D)|in a sufficiently\nlarge oriented graph D.\nTheorem 2.2. (Kelly et al. [10]) There is an integer n0such that every oriented graph\nDonn≥n0vertices with δ0(D)≥(3n−4)/8contains a k-cycle for each k∈ {3, . . . , n }.\nKeevash and Sudakov stated a result of the almost 3-cycle-factor under the same\nminimum semi-degree condition as in Theorem 1.4.\nTheorem 2.3. (Keevash and Sudakov [11]) There is some real c1>0so that for suf-\nficiently large n, any oriented graph Donnvertices with δ0(D)≥(1/2−c1)ncontains\ndisjoint 3-cycles covering all but at most 3vertices.\n3They also gave a precise minimum semi-degree bound to force some cycle-factors in\nan oriented graph.\nTheorem 2.4. (Keevash and Sudakov [11]) For any δ >0there exist numbers Mand\nn0such that if Dis an oriented graph on n > n 0vertices with δ0(D)≥(3/8 +δ)nand\nn1, . . . , n tare numbers satisfying ni≥Mfor any i∈[t]andPt\ni=1ni=n, then there is a\ncycle-factor along with cycle lengths n1, . . . , n tinD.\nProof of Theorem 1.4. Choose Msuch that Theorem 2.4 applies with δ= 1/10. Then,\nchoose c′with 1 /81> c′>4(M−2)n−1/3such that Theorem 1.2 holds for all k≥4 and\nsufficiently large nwith the parameter c′playing the role of c. Select the real number c1\nwith c1≥c′/2 ensuring the conditions for Theorem 2.3 are satisfied.\nSuppose that n=n1+···+ntis large enough, in which the number of the niequal\nto 3 is αn, where αis any real number with 0 < α < 1/3. Consider Dto be an oriented\ngraph on nvertices with δ0(D)≥(1/2−c)n, where c < c′/2−n−1/3. For 3 ≤k≤M, let\nNkbe the number of the niequal to k, and assume NL=P\nni>Mni. By Lemma 2.1, there\nexists a partition of V(D) asX∪Y, where |X|= 3N3+3 and Y=n−(3N3+3), such that\nδ0(D[X])≥(1/2−c−n−1/3)(3N3+ 3) and δ0(D[Y])≥(1/2−c−n−1/3)(n−(3N3+ 3)).\nThen Theorem 2.3 suggests that D[X] contains N3disjoint 3-cycles (denoted by C3) with\nreplacing c1in Theorem 2.3 with c+n−1/3since c+n−1/3< c′/2≤c1. Let{x1, x2, x3}=\nX\\V(C3) and Y′=Y∪ {x1, x2, x3}. Let c2be a real number with c2>c+3α(c+n−1/3)\n1−3α.\nClearly c2<1/80. Then we have\ndσ(xi, Y′)≥(1/2−c)n−(1/2 +c+n−1/3)(3N3+ 3)≥(1/2−c2)|Y′|\nfor any i∈[3] and σ∈ {+,−}. Also, for any y∈Y,\ndσ(y, Y′)≥(1/2−c−n−1/3)(n−(3N3+ 3))≥(1/2−c2)|Y′|.\nIn the following, we will get all the other cycles of length more than 3 in D[Y′]. For the\nsake of convenience, let D′=D[Y′] and n′=|Y′|.\nIf there exists a ksuch that Nk<c2n′\n3M2orNL<c2n′\n4, then we can repeatedly use\nTheorem 2.2 to select Nkdisjoint k-cycles or all cycles of lengths more than Mas desired,\nrespectively. This is because by c2<1/80, we have that\n(1/2−c2)n′−MX\nk=4k·c2n′\n3M2−c2n′\n4≥(1/2−c2)n′−c2n′\n4−c2n′\n4>(3n′−4)/8.(2.1)\nAlso, after picking all cycles for any kwith Nk<c2n′\n3M2orNL<c2n′\n4, we will be left with\nminimum semi-degree at least (1\n2−3c2\n2)n′by (2.1). By applying Lemma 2.1 repeatedly, We\nmay get a partition of the remaining vertices into M−2 disjoint vertex sets V4, . . . , V M, VL\nsuch that for any i∈ {4, . . . , M, L }\n|Vi|=\n\niNi,ifNi≥c2n′\n3M2,fori∈ {4, . . . , M },\nNL,ifNL≥c2n′\n4,fori=L,\n0,otherwise ,\n4andδ0(D[Vi])≥(1/2−3c2/2−(M−2)(n′)−1/3)|Vi| ≥(1/2−2c2)|Vi| ≥(3/8 +δ)|Vi|.\nThen, according to Theorem 1.2, we can embed Nidisjoint i-cycles in D[Vi] for any\ni∈ {4, . . . , M }. Also, we may embed all “long” cycles in D[VL] by using Theorem 2.4. □\nRemark. If the number of 3-cycles is a constant, that is, it is independent of the\ncardinality of the oriented graph D, then we can first greedily find N3disjoint 3-cycles by\nTheorem 2.2 and the bound of δ0(D). Then in the remaining oriented subgraph, we use\nthe proof of Theorem 1.4 to find all the other cycles.\n3 Preparation for Theorem 1.2\n3.1 Terminology and notation\nLetD= (V, A) be an oriented graph and let randsbe two distinct vertices of D. The\nnotation rsis defined as an arc from rtos. The out-neighbour (resp., in-neighbour ) ofs\nis defined by N+(s) ={r∈V:sr∈A}(resp., N−(s) ={w∈V:ws∈A}). The out-\ndegree (resp., in-degree ) ofs, which is denoted by d+(s) (resp., d−(s)), is the cardinality\nofN+(s) (resp. N−(s)) that is, d+(s) =|N+(s)|(resp. d−(s) =|N−(s)|). Further, the\ndegree ofsinDis given by dD(s) =d+(s) +d−(s). We define the minimum total degree\nofDto be δ(D) = min s∈VdD(s).\nFor any subset R⊆V, define Nσ(r, R) =Nσ(r)∩Randdσ(r, R) =|Nσ(r, R)|for\nanyσ∈ {− ,+}. We use the notation e(R) to present the number of arcs in D[R]. We\nrefer to Ras an i-setif its cardinality |R|is equal to i. Let D−R=D[V\\R] and\nR=V\\R. For another vertex set SinVthat is not necessarily disjoint to R, we denote\nbye+(R, S) (resp., e−(R, S)) the number of arcs from RtoS(resp., from StoR). If\nR={s}, we simply write sinstead of {s}throughout the paper.\nTheγ-out-neighborhood (resp., γ-in-neighborhood ) ofR, denoted by N+\nγ(R) (resp.,\nN−\nγ(R)), to be the set of vertices rinRsuch that d+(R, r)≥ |R| −γn(resp., d−(R, r)≥\n|R| −γn}), that is, N+\nγ(R) ={r:r∈Randd+(R, r)≥ |R| −γn}(resp., N−\nγ(R) =\n{r:r∈Randd−(R, r)≥ |R| −γn}). In addition, for a vertex set SinR, we define\nN+\nγ,S(R) =N+\nγ(R)∩S(resp., N−\nγ,S(R) =N−\nγ(R)∩S).\nThe following definition gives a special partition of the vertex set S⊆V, which is a\ncrucial structure obtained in the proof of Theorem 1.2. Let dbe a positive integer, η, δ\nandβbe positive real numbers, and n=|V|.\nDefinition 3.1. (The δ-extremal partition ) A partition P={V1, . . . , V d}ofSisδ-\nextremal if|Vi| ≥ ηnfor every i∈[d], and there exists A∈ P such that the new\npartition {A∪R, S−, S+}ofSsatisfies the following statements, where S−=N−\nγ,S(A),\nS+=N+\nγ,S(A) and R=S\\(A∪S−∪S+).\n(i) (1 /3−100k3δ)n≤ |A∪R|,|S−|,|S+| ≤(1/3 + 50 k3δ)n, and\n(ii) the number of arcs from A∪RtoS−, from S−toS+and from S+toA∪Rare each\nat most δn2.\n5We define the number of vertices of a path as its length and a k-path is a path of length\nk. We often represent a k-path (-cycle) Casv1v2···vk(v1) when V(C) ={v1, v2, . . . , v k}.\nForV1, . . . , V k⊆V,k-cyc(V1, . . . , V k) counts the number of k-cycles with the vertex set\n{v1, . . . , v k}such that vi∈Vifor each i. We can abbreviate k-cyc(V1, . . . , V k) as k-\ncyc(Vt\n1, Vt+1, . . . , V k) when V1=V2=. . .=Vt, where t∈ {1, . . . , k }. In particular, if\nt=k, then we write k-cyc(V1) instead of k-cyc(Vk\n1). For a positive integer s, we simply\nwrite [ s] for{1, . . . , s}.\nThe proof of Theorem 1.2 is done with the help of hypergraphs. For an integer k≥2,\nak-uniform hypergraph is defined as a pair H= (V(H), E(H)) in which V(H) is a finite\nset of vertices and E(H) ={e⊆V(H) :|e|=k}is a set of k-element subsets of V(H),\nwhose members are called the edges ofH. Further, for a vertex r∈V(H), let N(r)\ndenote the neighbour set of rinH, that is, N(r) ={S∈[V(H)]k−1:S∪ {r} ∈E(H)}.\nThedegree ofrinHis denoted as d(r) =|N(r)|. The minimum degree ofHisδ1(H) =\nmin r∈V(H)d(r).\nIn the following, we construct a k-uniform hypergraph of an oriented graph.\nDefinition 3.2. (Thek-uniform hypergraph of D) For every oriented graph D, the k-\nuniform hypergraph H(D) is a k-uniform hypergraph with the vertex set V(D) and the\nk-set{v1, . . . , v k}is an edge in H(D) if and only if it spans a k-cycle in D.\nIt is evident that there is a perfect matching in H(D) if and only if the oriented\ngraph Dhas a k-cycle-factor.\nWe also need to introduce the following definitions that mainly come from the works of\nHan [7], which build on the absorbing method initiated by R¨ odl, Ruci´ nski, and Szemer´ edi\n[13]. Let H= (V(H), E(H)) be a k-uniform hypergraph. For a positive integer r, we\ncall an additive subgroup of Zralattice . LetP={V1, . . . , V r}be a partition of V(H).\nThe index vector iP(R)∈Zrof a subset R⊆V(H) with respect to Pis denoted by\niP(R) = (|R∩V1|, . . . ,|R∩Vr|). We define IP(H) ={iP(e) :e∈E(H)}andLP(H) as\nthe lattice generated by IP(H). For any µ >0, we call Iµ\nP(H) to be the set of all i∈Zr\nsuch that there are at least µnkedges e∈E(H) satisfying iP(e) =i. Similarly, we denote\nbyLµ\nP(H) the lattice in Zrgenerated by Iµ\nP(H). The ith unit vector , denoted by ui, is\nthe vector in which the coordinate iis 1 and all other coordinates are 0. We call a non-\nzero difference ui−ujfor distinct i, j∈[r] atransferral inZr. Further, a 2 -transferral\nv∈Lµ\nP(H) is a transferral for which there exist v1,v2∈Iµ\nP(H) such that v=v1−v2.\nIn addition, Lµ\nP(H) is called 2 -transferral-free if it contains no a 2-transferral.\nLetα, β > 0 be real numbers and k, l, r > 0 be integers. We define two vertices u\nandvinV(H) to be ( H, β, l )-reachable if the number of ( kl−1)-sets Rsuch that both\nH[R∪ {u}] and H[R∪ {v}] have perfect matchings, is at least βnkl−1. We also define\neNH,α,1(v) as the set of vertices that are ( H, α, 1)-reachable to v. A vertex set U⊆V(H) is\nconsidered ( H, β, l )-closed if for any two distinct vertices uandvinU, they are ( H, β, l )-\nreachable. Moreover, a partition P={V1, . . . , V r}ofV(H) is called a ( H, β, l )-closed\npartition ifViis (H, β, l )-closed for every i∈[r].\n6In this paper, we use the notation α≪βto indicate that αis chosen to be sufficiently\nsmall relative to βsuch that all calculations needed in our proof are valid. For the real\nnumbers aandb, we use a±bto represent an unspecified real number in the interval\n[a−b, a+b].\n3.2 Proof overview of Theorem 1.2\nIn this subsection, we provide a brief outline of the proof of Theorem 1.2 in order to give\nreaders a general understanding of the proof of Theorem 1.2.\nLetDbe an oriented graph as stated in Theorem 1.2 and H(D) be the k-uniform\nhypergraph of D. By utilizing Lemmas 3.6-3.7 in Subsection 3 .3, we will obtain a vertex\nsetS⊆V(D) such that it covers all vertices in V(D) expect for the vertices in V0and\nsatisfies Lemma 3.8. Additionally, combining with Lemma 3.6 again, we could get a\npartition of SintoV1, . . . , V rsuch that for any i∈[r],|Vi| ≥ηnandViis (H(D), β, l)-\nclosed in H(D), where the integers r, l≥1. Let P′={V0, V1, . . . , V r}. We divide the\nproof in Theorem 1.2 into two cases based on whether Sis (H(D), β, l)-closed.\nThe proof in the case when Sis (H(D), β, l)-closed mainly relies on the lattice-based\nabsorbing method developed recently by Han [3]. The absorbing technique initiated by\nR¨ odl, Ruci´ nski, and Szemer´ edi [13] has been shown to be efficient on finding spanning\nstructures in graphs and hypergraphs. The lemma below serves as our absorbing lemma.\nLemma 3.3. (Ding et al. [3] - Lemma 2 .9)Letβandηbe real numbers with 0< β, η < 1\nand let landkbe positive integers. There exists ω >0such that the following holds for\nthe sufficiently large integer n. Assume that His ak-uniform hypergraph on nvertices\nwith characteristics as follows.\n(1) For any v∈V(H), there are at least ηnk−1distinct edges containing it.\n(2) There exists V0⊆V(H)with|V0| ≤η2nsuch that V(H)\\V0is(H, β, l )-closed.\nThen there exists a vertex set Uwith V0⊆U⊆V(H)and|U| ≤ηnsuch that for any\nvertex set W⊆V(H)\\Uwith|W| ≤ωnand|W| ≡0 (modk), both H[U]andH[U∪W]\ncontain perfect matchings.\nOtherwise, Sis not ( H(D), β, l)-closed. In this case, if Lµ\nP′(H(D)) contains a 2-\ntransferral ui−ujfor distinct i, j∈[r], then merge the partite sets ViandVjand\nthink about the new partition P′−Vi−Vj+ (Vi∪Vj). By applying Lemma 3.9 that\nis stated below in Subsection 3 .3, continuously merge the partite sets corresponding to\n2-transferrals until we obtain a final partition P={V0, V1, . . . , V d}ofV(D) such that P\ncontains no 2-transferral ui−ujfor distinct i, j∈[d] and Viis closed for any i∈[d]. We\ncan assume that d≥2.\nIn the subsequent section, we will demonstrate that there exists a partite set A∈S\nwith k-cyc(Ak−1, S\\A)≤αnkin Claim 4.2 in Subsection 4.2. Additionally, it will\nfollow from Lemma 3.13 that P \\V0forms a δ-extremal partition of Sin Claim 4.3.\nLeta1= (k−1)/3,a2= (k−2)/3 and a3=k/3 for convenience. Then by good\nproperties of this δ-extremal partition, we will prove that there exist disjoint partite\n7subsets A1, A2, A0∈ P \\ V0that satisfy the following conclusions.\n\n\nk-cyc(Aa1+1\n1, Aa1\n2, Aa1\n0), k-cyc(Aa1\n1, Aa1+1\n2, Aa1\n0)≥µnk, when k≡1 (mod 3) ,\nk-cyc(Aa2+2\n1, Aa2\n2, Aa2\n0), k-cyc(Aa2+1\n1, Aa2+1\n2, Aa2\n0)≥µnk, when k≡2 (mod 3) ,\nk-cyc(Aa3\n1, Aa3\n2, Aa3\n0)≥µnk, k-cyc(Aa3+1\n1, Aa3\n2, Aa3−1\n0)≥µnk,when k≡0 (mod 3) .\nBased on these conclusions, we can conclude that when k≡1, or 2 (mod 3), we can merge\nA1andA2. Similarly, when k≡0 (mod 3), we can merge A1andA0. This leads to a\ncontradiction and hence Theorem 1.2 holds when Sis not ( H(D), β, l)-closed.\n3.3 Tools when Sis not ( H(D), β, l)-closed\nLemma 3.4. Assume that cis a real number less than 1/2. Let Dbe an oriented graph\nonlvertices with δ(D)≥(1−2c)l. Then for any positive integer s,Dhas at least s\nvertices of out-degree at least (l−s−2cl)/2inDand at least svertices of in-degree at\nleast (l−s−2cl)/2inD. Moreover, there exists a vertex vinDsatisfying l/4−cl≤\nd+(v)≤3l/4 +cl.\nProof. Letv1, . . . , v lbe a vertex ordering of Dsuch that d+(v1)≥ ··· ≥ d+(vl). Set\nD′=D[{vs, . . . , v l}]. By the degree condition of D, we obtain that in D′, every vertex is\nnot joined to at most 2 clvertices. Since d+(vs)≥ ··· ≥ d+(vl), we get that\nd+(vs)· |D′| ≥lX\ni=sd+(vi)≥e(D′) =lX\ni=sd+\nD′(vi)≥|D′| ·(|D′| −2cl)\n2,\nwhich implies d+(vs)≥(|D′| −2cl)/2≥(l−s−2cl)/2. So Dcontains svertices of out-\ndegree at least ( l−s−2cl)/2 inD. Similarly, Dcontains svertices of in-degree at least\n(l−s−2cl)/2 inD. This means that if let s=⌊l/2⌋, then we get that there are at least\n⌊l/2⌋vertices with out-degree at least ( l−⌊l/2⌋−2cl)/2≥l/4−cl, and if let s=⌈l/2⌉+1,\nthen there are ⌈l/2⌉+1 vertices with in-degree at least ( l−⌈l/2⌉−1−2cl)/2≥l/4−1−cl.\nThis yields that Dcontains a vertex vwith out-degree at least l/4−cland in-degree at\nleast l/4−1−cl, which implies l/4−cl≤d+(v)≤(l−1)−(l/4−1−cl) = 3 l/4 +cl.\nLemma 3.5. Letcbe a positive constant with c <1\n2200, and Dbe an oriented graph on n\nvertices with δ0(D)≥(1/2−c)n. For any U⊆V(D), the following statements hold.\n(i)e+(U,U), e−(U,U) =|U|·|U|\n2±c|U|n;\n(ii)e(U)≥|U|(|U|−2cn)\n2;\n(iii)LetS⊆V(D), where SandUare not necessarily disjoint. If |S|,|U| ≥(1/2−c)n,\nthen there are at least n2/41arcs from StoU.\nProof. (i) The proof for (i) has been previously provided in [12]. However, for the com-\npleteness of the paper, we present a straightforward proof of it here.\nOn the one hand, by the lower bound of δ0(D), for any σ∈ {+,−}, we have that\neσ(U,U) =P\nu∈Udσ(u)−dσ(u, U)≥\n8P\nu∈Udσ(u)−\u0000|U|\n2\u0001\n≥ |U| ·(1/2−c)n−|U|2\n2=|U|·|U|\n2−c|U|n.\nOn the other hand, together with the fact that e+(U,U) +e−(U,U)≤ |U| · |U|, we get\nthateσ(U,U)≤|U|·|U|\n2+c|U|nfor any σ∈ {+,−}. Hence (i) holds.\n(ii) is easily derived from the minimum semi-degree condition of D.\n(iii) If necessary, we may assume that |S|=|U|= (1/2−c)nby deleting the vertices.\nOn the one hand, suppose that |S∩U|> n/ 5, then by (ii) and c <1\n2200, it is obtained\nthat\ne+(S, U)≥e(S∩U)≥|S∩U| ·(|S∩U| −2cn)\n2≥n\n5·\u0010n\n5−2cn\u0011\n>n2\n41,\nOtherwise, |S∩U| ≤n/5. Then by (i) of this lemma, e+(S, U)≥e+(S,S)− |S| · |S∪U|,\n|S∪U|=n− |S∪U|=n−(|S|+|U| − |S∩U|) and|S|=|U|= (1/2−c)n, we have\ne+(S, U)≥(1/2−c)2n2\n2−(1/2−c)n·(2cn+|S∩U|)≥\u00121\n40+5c2\n2−13c\n10\u0013\nn2≥n2\n41,\nwhere the last inequality is due to c <1\n2200. This complete the proof of the lemma.\nAdler, Alon and Ross [1] proved that any tournament of order nhas at mostn3/2(n−1)!\n2n\ndistinct Hamiltonian cycles. This implies that any oriented graph on kvertices contains at\nmostk3/2(k−1)!\n2k distinct k-cycles. we will use this fact to prove the lemma in what follows.\nWe say that two cycles C1andC2arestrongly distinct ifV(C1)̸=V(C2).\nLemma 3.6. Letcbe a constant with c <1/2andDbe an oriented graph on nvertices\nwithδ0(D)≥(1/2−c)n. Let H(D)be the k-uniform hypergraph of D. Then the following\nstatements hold.\n(i)Anyx∈V(D)belongs to at leastnk−1\n6k1/2k!strongly distinct k-cycles. This implies that\nδ1(H(D))≥nk−1\n6k1/2k!≥1\n6k3/2·\u0000n−1\nk−1\u0001\n.\n(ii)Every set of 6k3/2+ 1vertices in H(D)contains two vertices that are (H(D), α,1)-\nreachable, where αis a real number with 0< α <1\n108k4·k!.\nProof. We first give the proof of (i). To construct k-cycles containing the vertex x,\nwe start by selecting as many ( k−2)-paths as passible that begin at x, which gives\nat least (1 /2−c)n·Qk−5\ni=0((1/2−c)n−i) such ( k−2)-paths. For any one of these\n(k−2)-paths P=x···y, we may choose an arc from N+(y) toN−(x) without using\nany vertices in V(P) to form a k-cycle. Obviously there are at most ( k−2)(n−1) arcs\nthat repeat vertices in V(P). Therefore, according to Lemma 3.5-(iii), there exist at least\nn2\n41−(k−2)(n−1) arcs from N+(y) toN−(x) that are disjoint from P. So there are at\nleast (n2\n41−(k−2)(n−1))·(1/2−c)n·Qk−5\ni=0((1/2−c)n−i)≥nk−1\n48·2k−3k-cycles through x.\nRecall that xis in at mostk3/2(k−1)!\n2k distinct k-cycles. Therefore, the number of strongly\ndistinct k-cycles through xis at least2k\nk3/2(k−1)!·nk−1\n48·2k−3≥nk−1\n6k1/2k!.\nSecondly, we prove that (ii) holds. Suppose not, let Xbe a vertex set with |X|=\n6k3/2+ 1 such that for any two distinct vertices u, v∈X, they are not ( H(D), α,1)-\nreachable. This yields |N(u)∩N(v)|< αnk−1inH(D). Let X={u1, . . . , u6k3/2+1}, then\n9by (i) of this lemma, in H(D) we have that\n\u0012n−1\nk−1\u0013\n≥ |N(u1)∪N(u2)∪ ··· ∪ N(u6k3/2+1)|\n≥X\n1≤i≤6k3/2+1|N(ui)| −X\n1≤i<j≤6k3/2+1|N(ui)∩N(uj)|\n≥(6k3/2+ 1)·1\n6k3/2·\u0012n−1\nk−1\u0013\n−αnk−1·\u00126k3/2+ 1\n2\u0013\n,\na contradiction since α <1\n108k4·k!.\nThe following lemma states that if every set of vertices of constant size in a hyper-\ngraph contains two reachable vertices, then we can find a very large set Sin which all\nvertices have large reachable neighborhoods.\nLemma 3.7. (Ding et al. [3] - Lemma 2 .6)Assume that δ, α > 0and integers f, k≥2.\nIfHis ak-uniform hypergraph on nvertices such that every set of f+1vertices contains\ntwo vertices that are (H, α, 1)-reachable, then there exists S⊆V(H)with cardinality\n|S| ≥(1−fδ)nsuch that for any v∈S,|eNH,α,1(v)∩S| ≥δn.\nThe following lemma is utilized to obtain a partition of a vertex set Sof the oriented\ngraph that possesses desirable properties.\nLemma 3.8. (Han and Treglown [8] - Lemma 6 .3)Suppose that δ > 0and integers\nf, k≥2. Let αbe a positive number with α≪δ. Then there exists a constant β > 0\nsuch that the following statement holds for all sufficiently large n. Assume that His a\nk-uniform hypergraph on nvertices and S⊆V(H)satisfying |eNH,α,1(v)∩S| ≥δnfor any\nv∈S. Additionally, suppose that every set of f+ 1vertices in Scontains two vertices\nthat are (H, α, 1)-reachable in H. Then we can partition Sinto subsets V1, . . . , V rwith\nr≤min{f,⌊1/δ⌋}such that for every i∈[r], we have that |Vi| ≥(δ−α)nandViis\n(H, β, 2f−1)-closed in H.\nThe lemma below states that for a closed partition P={V1, . . . , V r}of the subset S\nof an oriented graph D, the partition of Sformed by merging ViandVjis closed if ui−uj\nis a 2-transferral in Lµ\nP∪V0(H(D)) for 1 ≤i < j≤d, where V0=V(D)\\S.\nLemma 3.9. (Han [7] - Lemma 3 .4)Assume that 1/n≪β≪β′, µ≪1/r,1/l. Let\nDbe an oriented graph and let S⊆V(D)andV0=V(D)\\S. Suppose that P=\n{V1, . . . , V r}is a partition of S. IfPis(H(D), β′, l)-closed and there exists a 2-transferral\nui−uj∈Lµ\nP∪V0(H(D))fori, j∈[r], then the partition of Sformed by merging ViandVj\nis(H(D), β,(k+ 1)l+ 1)-closed.\nIn the proof of Theorem 1.2, in order to find as many distinct k-cycles as possible, we\nwill frequently utilize the following lemma and some of the ideas presented in its proof.\nBefore introducing the lemma, we provide the definition as follows.\n10Figure 1: Definition 3.10 for σ= +.\nDefinition 3.10. Assume that ϵis a real number with ϵ≪1. For any vertex subset X\nof an oriented graph and every positive integer i, we denote that\nXiσ={x∈X(i−1)σ:d−(x, X(i−1)σ)≥ϵn}for any σ∈ {+,−},\nwhere X0σ=Xand write X1σasXσfor abbreviation.\nLemma 3.11. Suppose that 1/n≪c≪ϵ≪η≪1. Let Dbe an oriented graph on n\nvertices with δ0(D)≥(1/2−c)n. Then for any X⊆V(D)with|X|> ηn , we have that\n|Xiσ| ≥ |X(i−1)σ|/2−ϵnand|Xiσ| ≥ |X|/2i−2ϵnfor any σ∈ {+,−}.\nProof. We only give the proof of the case when σ= +, because the proof of the case when\nσ=−is similar. We first prove |Xi+| ≥ |X(i−1)+|/2−ϵn. Let X′=X(i−1)+\\Xi+, then\nby Lemma 3.5-(ii), we get\n|X′| ·ϵn+ (|X(i−1)+| − |X′|)· |X(i−1)+|>X\nx∈X(i−1)+d−(x, X(i−1)+)\n≥|X(i−1)+|(|X(i−1)+| −2cn)\n2,\nwhich implies that |X′|<|X(i−1)+|(|X(i−1)+|/2+cn)\n|X(i−1)+|−ϵn≤|X(i−1)+|\n2+ϵn. So|Xi+| ≥|X(i−1)+|\n2−ϵn.\nThen|Xi+| ≥|X|\n2i−ϵn·Pi−1\nj=01\n2j≥|X|\n2i−2ϵn. Hence this proves the lemma.\nLemmas 3.12-3.15 play a key role in the search for the δ-extremal partition of S.\nLemma 3.12. (Li and Molla [12] - Lemma 20) Suppose that 1/n≪c≪γ2< γ 1/2, γ/2≪\n1, and that Dis an oriented graph on nvertices with δ0(D)≥(1/2−c)n. Let AandB\nbe two disjoint vertex sets of V(D)ande+(A, B)≤γ2n2. Then |N−\nγ,B(A)| ≥ |B| −γ1n\nand|N+\nγ,A(B)| ≥ |A| −γ1n.\nLemma 3.13. Suppose that 1/n≪c≪α≪γ≪δ <1\n216k3(k!)2. Let Dbe an oriented\ngraph on nvertices with δ0(D)≥(1/2−c)nand let S⊆V(D)with|S| ≥(1−6k3/2δ)n.\nIfAis a subset of Swith|A| ≥n\n7k3/2andk-cyc(Ak−1, S\\A)≤αnk, then the following\nhold.\n(i)|N+\nγ,S(A)|,|N−\nγ,S(A)|=|S\\A|/2±50k3δn, and\n(ii)|A| ≤(1/3 + 50 k3δ)n.\nProof. Suppose that α≪ϵ≪γ2≪γ1≪γ≪η <1\n216k3(k!)2. Let S+=N+\nγ,S(A),\nS−=N−\nγ,S(A) and R=S\\(S+∪S−∪A). Set V0=V(D)\\S. We first declare that\n11Claim 3.14. |R| ≤γ1n.\nProof. For any vertex xinR, letX=N+(x, A) and Y=N−(x, A). Then by the definition\nofR, we have that |X|,|Y| ≤ |A| −γn. Since d+(x), d−(x)≥δ0(D)≥(1/2−c)nand\n|A| ≥n\n7k3/2, it follows that\n|X|,|Y| ≥γn−2cn≥γn/2 and |X|+|Y| ≥ |A| −2cn≥4ηn. (3.1)\nBased on the number of arcs from XtoY, we consider two cases as follows.\nCase 1. For any vertex x∈R, it satisfies that e+(X, Y )≥γ2n2.\nBy the pigeonhole principle, there exist at least |R|/2 vertices xsuch that |X| ≥ |Y|\nor|X| ≤ |Y|. Without loss of generality, assume that there are |R|/2 vertices xsatisfying\n|X| ≥ |Y|. Then by (3.1), we get that |X| ≥(|A|−2cn)/2. Let R′be the set of these |R|/2\nvertices x. In the following, for any vertex xinR′, we will calculate the lower bound of\nk-cyc(x, Ak−1), which is an integer multiple of |R|. Together with k-cyc(Ak−1,A)≤αnk,\nwe can get |R| ≤γ1nas requested.\nDefine X+={y∈X:d+(y, Y)≥ϵn}, and then γ2n2≤e+(X, Y )<|X+| · |Y|+\n(|X|−|X+|)·ϵn,which suggests that |X+| ≥γ2ndue to |X| ≥ |Y|andγ2≫ϵ. Recall that\nby Definition 3.10 with σ= +, we have Xi+={x∈X(i−1)+:d−(x, X(i−1)+)≥ϵn}, which\nimplies that there are at leastQk−4\ni=0(ϵn−i)· |X(k−2)+|distinct ( k−2)-paths from X+to\nX(k−2)+. Further, by Lemma 3.11, we get |Xi+| ≥|X(i−1)+|\n2−ϵnand|X(k−2)+| ≥|X+|\n2k−3−2ϵn.\nThen we obtain (see Figure 2 (a))\nk-cyc(x, Ak−1)≥Qk−4\ni=0(ϵn−i)|X(k−2)+| ·ϵn≥ϵk−2·\u0000γ\n2k−3−3ϵ\u0001\n·nk−1.\n(a) Case 1 when e+(X, Y )≥γ2n2for any vertex x∈R. (b) Case 2 when e+(X, Y )< γ 2n2for some vertex x∈R.\nFigure 2: In the figure, the blue arcs form a k-cycle.\nFurthermore, by k-cyc(Ak−1, S\\A)≤αnkandA∩R=∅, we have\nαnk≥k-cyc(S\\A, Ak−1)≥k-cyc(R, Ak−1)≥X\nx∈R′k-cyc(x, Ak−1)\n≥|R|\n2·ϵk−2·\u0010γ\n2k−3−3ϵ\u0011\n·nk−1.\n12By simplifying the above inequality and due to α≪ϵ≪γ2≪γ1, we get that |R| ≤γ1n.\nCase 2. There exists a vertex xinRsuch that e+(X, Y )< γ 2n2.\nWithout loss of generality, we assume that ϵn=⌊ϵn⌋. In this case, we will show that\nthere exist ϵndistinct vertices y1, y2, . . . , y ϵninYand for any yiwith i∈[ϵn], there are\nϵndistinct vertices xi,1, . . . , x i,ϵninXsuch that k-cyc(S\\A, y i, Ak−3, xi,j)≥ηγϵk−4\n2k·nk−2.\nSuppose that γ′is a real number with γ2≪γ′≪γ. By Lemma 3.12, we have that\n|N−\nγ′,Y(X)| ≥ |Y| −γ′nand|N+\nγ′,X(Y)| ≥ |X| −γ′n. (3.2)\nThis implies that there exists a subset Y′⊆Ywith|Y′| ≥ |Y| −γ′nsuch that, for any\nvertex yiinY′, we have d+(yi, X)≥ |X| −γ′n, and we define Xi=N+(yi, X).\nStep 2.1. Find ϵnvertices y1, y2, . . . , y ϵninY′such that for any i∈[ϵn], we have that\nd−(yi, S\\A)≥|S\\A|+|X|\n2−4k3/2δn.\nTo see this, for any i∈[ϵn], byδ0(D)≥(1/2−c)nand Lemma 3.5-(ii), we can choose\nyi∈Y′\\ {y1, . . . , y i−1}such that d+(yi, Y′)≥|Y′|−(i−1)−2cn\n2. Then by (3.1), we have that\n(1/2−c)n≤d−(yi)≤d−(yi, S\\A)+|V0|+d−(yi, X)+d−(yi, Y)+2cn. Since d−(yi, X)≤\n|X| −d+(yi, X),d−(yi, Y)≤(|Y| − |Y′|) +|Y′| −d+(yi, Y′),d+(yi, X)≥ |X| −γ′nand\nd+(yi, Y′)≥|Y′|−(i−1)−2cn\n2, we obtain (1 /2−c)n≤d−(yi, S\\A)+|V0|+|Y′|\n2+2γ′n+3cn+i−1\n2.\nTogether with |V0| ≤6k3/2δnandY′⊆Y, this yields that d−(yi, S\\A)≥|S\\A|+|X|\n2−\n4k3/2δn.\nStep 2.2. For any given i∈[ϵn], find ϵnvertices xi,1, . . . , x i,ϵninXisuch that for any\nj∈[ϵn], we have that d+(xi,j, S\\A)≥|S\\A|+|Y|\n2−4k3/2δn.\nRecall that by (3.1), |Xi| ≥ |X|−γ′n≥γn/2−γ′n > γn/ 3. Further, by (3.2), for any\ngiven i∈[ϵn], we can obtain that |N+\nγ′(Y)∩Xi| ≥ |Xi|−γ′n. Then by δ0(D)≥(1/2−c)n\nand Lemma 3.5-(ii) again, there is a vertex xi,j∈(N+\nγ′(Y)∩Xi)\\ {xi,1, . . . , x i,j−1}with\nd−(xi,j, Xi)≥|N+\nγ′(Y)∩Xi| −(j−1)−2cn\n2≥|Xi| −γ′n−(j−1)−2cn\n2. (3.3)\nBecause xi,j∈N+\nγ′(Y), it is easily seen that\nd−(xi,j, Y)≥ |Y| −γ′n. (3.4)\nHence by (3.1) and the lower bound of δ0(D), we have that (1 /2−c)n≤d+(xi,j)≤\nd+(xi,j, S\\A)+|V0|+d+(xi,j, X)+d+(xi,j, Y)+2cn. Also, due to (3.3)-(3.4), |V0| ≤6k3/2δn,\nd+(xi,j, X)≤(|X| − |Xi|) +|Xi| −d−(xi,j, Xi),d+(xi,j, Y)≤ |Y| −d−(xi,j, Y), and the\nproperty of vertex xi,j, we obtain that\nd+(xi,j, S\\A)≥(1/2−c)n−6k3/2δn−\u0012|Xi|\n2+5γ′n\n2+ 3cn+j−1\n2\u0013\n≥|S\\A|+|Y|\n2−4k3/2δn.\nSo Step 2 .2 is completed.\n13Note that it follows from xi,j∈Xithat yixi,j∈A(D) for any i, j∈[ϵn]. Set\nAi,j=N+(xi,j, S\\A)∩N−(yi, S\\A) for any i, j∈[ϵn]. Then by Steps 2 .1-2.2 and\n(3.1) and |A| ≥n\n7k3/2, we have |Ai,j| ≥|X|+|Y|\n2−8k3/2δn≥|A|−2cn\n2−8k3/2δn≥ηn. Let\nX−\ni,j=N−(xi,j, Xi) for any i, j∈[ϵn]. By (3.3), we get that |X−\ni,j| ≥|Xi|−γ′n−(j−1)−2cn\n2.\nAlso by Lemma 3.11-(ii), we have that |Xs−\ni,j| ≥|X(s−1)−\ni,j|\n2−ϵnfor any s∈ {2,3, . . . , k −3}\nand|X(k−3)−\ni,j| ≥|X−\ni,j|\n2k−4−2ϵn. Recall that X(k−3)−\ni,j ⊆X−\ni,j=N−(xi,j, Xi)⊆Xi=N+(yi, X).\nFurther, due to |Ai,j| ≥ηn,α≪ϵ≪γ≪ηandk-cyc(S\\A, Ak−1)≥k-cyc(S\\\nA, y i, Ak−3, xi,j), we have that (see Figure 2 (b))\nk-cyc(S\\A, Ak−1)≥(ϵn)2· |Ai,j| ·(|X(k−3)−\ni,j| ·k−5Y\ni=0(ϵn−i))\n≥(ϵn)2·ηn·(|X−\ni,j|\n2k−4−2ϵn)·k−5Y\ni=0(ϵn−i)\n≥(ϵn)2·ηγϵk−4\n2k·nk−2> αnk.\nThis contradicts the fact that k-cyc(Ak−1, S\\A)≤αnk. Hence Claim 3.14 holds.\nIn the following, we first prove (i) by using the fact that |R| ≤γ1n. According to the\ndefinition of S−σforσ∈ {+,−}(i.e., S−σ=S−ifσ= +, and S−σ=S+ifσ=−), we\nhave that\neσ(A, S\\A)≤ |A| · |Sσ|+ (|A| −γn)· |R|+γn· |S−σ| ≤ |A| · |Sσ|+ 2γn2.\nSoeσ((A, S\\A) =|A|·|Sσ|±2γn2. Together with Lemma 3.5-(i), we get that e+(A,A) =\ne−(A,A)±2γn2. Since A= (S\\A)∪V0and|V0| ≤6k3/2δn, we have e+(A, S\\A) =\ne−(A, S\\A)±7k3/2δn. It follows from eσ(A, S\\A) =|A| · |Sσ| ±2γn2that\n|A| · |S+|=e−(A, S\\A)±(2γ+ 7k3/2δ)n2=|A| · |S−| ±(2γ+ 14k3/2δ)n2,\nTherefore, because |A| ≥n\n7k3/2, it is not hard to check that |S+|=|S−| ±99k3δn. In\naddition, together with Claim 3.14, we obtain\n|S\\A|=|S+|+|S−|+|R|= 2|Sσ| ±100k3δn, for any σ∈ {+,−}.\nHence Lemma 3.13-(i) is proved.\nNext, we will give the proof of (ii). Suppose, to the contrary, that |A|>(1/3 +\n50k3δ)n, which implies |S\\A| ≤ | A|<(2/3−50k3δ)n. By (i), we have that |S+| ≥\n|S\\A|\n2−50k3δn. Clearly, there exists a vertex x∈S+satisfying d+(x, S+)≤ |S+|/2. Then\nd+(x, A)≤γndue to x∈S+. Further, combining with d+(x) =d+(x,A) +d+(x, A),\nd+(x,A)≤ |S−|+|R|+|V0|+|S+|/2≤ |S\\A| − |S+|/2 + 6 k3/2δn, and c≪δ, we have\nthat\nd+(x)≤ |S\\A| −|S\\A|\n4+ 25k3δn+ 6k3/2δn+γn\n<3\n4·(2/3−50k3δ)n+ 32k3δn < (1/2−c)n.\n14This contradicts the fact that δ0(D)≥(1/2−c)n, and so Lemma 3.13-(ii) is true. Thus,\nwe complete the proof of Lemma 3.13.\nLemma 3.15. Assume that 1/n≪α, c≪γ≪η, δ <1\n216k3(k!)2. Let Dbe an oriented\ngraph on nvertices with δ0(D)≥(1/2−c)nand let S⊆V(D)with|S| ≥(1−6k3/2δ)n.\nForA⊆Ssuch that |A| ≥ηnandk-cyc(A)≤αnk, the following statements hold.\n(i)Forσ∈ {− ,+}, there is xσ∈Asatisfying dσ(xσ, A)≥ |A| −γn.\n(ii)For every partition {A, B, C }ofSwith k-cyc(Ak−1, C)≤αnk, there are disjoint\nsubsets C+, C−⊆Csuch that for σ∈ {+,−},\n(⋆)e−σ(A, Cσ)≤γn2and|Cσ| ≥ |C|/2 + (|A| − |B|)/2−7k3/2δn.\nProof. Suppose that ϵ, γ1,γ2andγ3be real numbers with α≪ϵ, γ3≪γ1, γ2≪γ.\nWe first prove (i). It suffices to prove that there exists a vertex x−∈Asatisfying\nd−(x−, A)≥ |A|−γn, and the existence of the vertex x+is a similar argument. Let Xbe\nthe set of vertices xinAsatisfying k-cyc(x, Ak−1)≤α1/2nk−1. Since ( |A|−|X|)·α1/2nk−1≤P\nx∈X\\Ak-cyc(x, Ak−1)≤k·k-cyc(A)≤αknk, we have that\n|X| ≥ |A| −α1/2kn. (3.5)\nPick x−∈Xwith d−(x−, X) maximal. Set X+=N+(x−, X) and X−=N−(x−, X). In\nthe following, we will show that |X−| ≥ |A| −γn, which indicates that x−is the vertex\nwith d−(x−, A)≥ |A| −γnas required.\nAccording to δ0(D)≥(1/2−c)n, it is not hard to see that |X+| ≥ |X|−|X−|−2cn. On\nthe one hand, due to x−∈X, we have k-cyc(x−, Ak−1)≤α1/2nk−1. On the other hand, we\ncan estimate the lower bound of k-cyc(x−, Ak−1). Recall that by Definition 3.10, we have\nXi+={x∈X(i−1)+:d−(x, X(i−1)+)≥ϵn}. This implies that for any xk−2∈X(k−2)+,\nwe obtain that d−(xk−2, X(k−3)+)≥ϵn. Similarly, by the definition of X(k−3)+, for any\nxk−3∈X(k−3)+∩N−(xk−2, X(k−3)+), we have that d−(xk−3, X(k−4)+)≥ϵn. Repeating the\nprocess above, this gives at leastQk−4\ni=0(ϵn−i) distinct ( k−2)-paths from X+toX(k−2)+.\nSo we get that k-cyc(x−, Ak−1)≥e+(X(k−2)+, X−)·Qk−4\ni=0(ϵn−i). Hence\nα1/2nk−1≥k-cyc(x−, Ak−1)≥e+(X(k−2)+, X−)·Qk−4\ni=0(ϵn−i),\nwhich implies that e+(X(k−2)+, X−)≤2\nϵk−3·α1/2n2≤γ3n2. It follows from Lemmas\n3.11-3.12 that\n|N+\nγ1(X−)∩X(k−2)+| ≥ |X(k−2)+| −γ2n≥|X+|\n2k−3−2ϵn−γ2n≥|X+|\n2k−3−2γ2n.\nFurther, by δ0(D)≥(1/2−c)nand the pigeonhole principle, there exists w∈\nN+\nγ1(X−)∩X(k−2)+such that d−(w, X(k−2)+)≥|N+\nγ1(X−)∩X(k−2)+|−2cn\n2≥|X+|\n2k−2−2γ2n. Since\nw∈X(k−2)+⊆X+⊆X, and x−has the maximum in-degree in X, and d−(w, X−)≥\n|X−| −γ1ndue to w∈N+\nγ1(X−), we obtain that\n|X−| ≥d−(w, X(k−2)+) +d−(w, X−)≥|X| − |X−| −2cn\n2k−2−2γ2n+|X−| −γ1n.\n15This implies that |X−| ≥ |X| −2cn−2k−2(2γ2+γ1)n. Combining with (3.5) and c≪\nγ1, γ2≪γ, we deduce that d−(x−, A)≥ |X−| ≥ |A| −γn.\nWe secondly prove (ii). It will be proved with γ/2 taking on the role of γ. In\nparticular, if there are subsets C+andC−satisfying ( ⋆) but C+∩C−is not empty, then\n(|A| −2cn)· |C+∩C−| ≤e+(A, C+∩C−) +e−(A, C+∩C−)≤γn2.\nTogether with |A| ≥ηn, this means |C+∩C−| ≤γn/2. Therefore, in this case, the\ndisjoint vertex sets C+\\C−andC−\\C+satisfy ( ⋆).\nIn the following, we will only prove the existence of the vertex set C+satisfying ( ⋆).\nUsing the similar derivation, we can also prove that the desired vertex set C−exists.\nLetYbe the set of vertices yinAsatisfying k-cyc(y, Ak−2, C)≤α1/2nk−1. Then by\nk-cyc(Ak−1, C)≤αnk, we have\n(|A| − |Y|)·α1/2nk−1≤P\ny∈A\\Xk-cyc(y, Ak−2, C)≤k·k-cyc(Ak−1, C)≤αknk.\nThis implies that |Y| ≥ |A| −α1/2kn. Due to k-cyc(Y)≤k-cyc(A)≤αnk, if follows from\n(i) that there is y∈Ysatisfying\nd−(y, A)≥d−(y, Y)≥ |Y| −γn/4≥ |A| −γn/3.\nLetC+=N+(y, C) ={y1, . . . , y |C+|},A1=N−(y, A) and Yj=N+(yj, A1) for any\nj∈[|C+|]. Clearly |A\\A1| ≤γn/3. Next, we will give an upper bound of e+(C+, A) by\nestimating the upper and lower bounds of k-cyc(y, Ak−2, C+).\nWe first count k-cyc(y, yi, Ak−2) for any yi∈C+. By Definition 3.10-(i), we know\nYi+\nj={y∈Y(i−1)+\nj :d−(y, Y(i−1)+\nj )≥ϵn}, which gives at leastQk−4\ns=0(ϵn−s) distinct\n(k−3)-paths from YjtoY(k−4)+\nj . Also, for any j∈[|C+|], by Lemma 3.11-(i), we have\nthat|Yi+\nj| ≥|Y(i−1)+|\n2−ϵnfor any i∈[k−3] and |Y(k−3)+\nj| ≥|Yj|\n2k−3−2ϵn. Hence\nk-cyc(y, yj, Ak−2)≥(|Y(k−3)+\nj| −(k−3))·k−4Y\ns=0(ϵn−s)\n≥\u0012|Yj|\n2k−3−2ϵn−(k−3)\u0013\n·k−4Y\ns=0(ϵn−s).\nFurthermore, we have that\nk-cyc(y, C+, Ak−2) =|C+|X\nj=1k-cyc(y, yj, Ak−2)≥(|C+|X\nj=1|Yj|\n2k−3−(2ϵn+k−3)|C+|)k−4Y\ns=0(ϵn−s).\nIn addition, we deduce that\n|C+|X\nj=1|Yj|=e+(C+, A1) =e+(C+, A)−e+(C+, A\\A1)≥e+(C+, A)−γn2/3.\n16Together with k-cyc(y, Ak−2, C+)≤k-cyc(y, Ak−2, C)≤α1/2nk−1since y∈Y, we can\nobtain that\nα1/2nk−1≥k-cyc(y, C+, Ak−2)≥\u0012e+(C+, A)−γn2/3\n2k−3−(2ϵn+k−3)|C+|\u0013k−4Y\ns=0(ϵn−s)\n≥\u0012e+(C+, A)\n22k−6−γn2\n22k−4\u0013\nϵk−3nk−3. (3.6)\nHence, due to α≪ϵ≪γ, this implies e−(A, C+) =e+(C+, A)≤γn2/2.\nFinally, since V(D) =A∪B∪C∪V0,|C+| ≥d+(y)−d+(y, A)− |B| − |V0|, and\n|V0| ≤6k3/2δn, and d+(y, A) =|A| −d−(y, A)≤γn/3, we also have that\n|C+| ≥ |C|/2 + (|A| − |B|)/2−γn/2−6k3/2δn≥ |C|/2 + (|A| − |B|)/2−7k3/2δn,\nsince d+(y)≥δ0(D)≥(1/2−c)nandn≥ |A|+|B|+|C|. Therefore, the lemma holds.\n4 The proof of Theorem 1.2\nChoose a constant c′with 1 /2> c′>1\n6k1/2k!such that Theorem 1.5 holds. Let n0,n,l\nanddbe positive integers with n≥n0, and suppose that c≤c′−1\n6k1/2k!and\n1/n≪c≪β′≪β≪µ≪α′≪ϵ≪α≪γ3≪γ2≪γ1≪\nγ≪γ′≪η, δ < 1/(216k3(k!)2)< δ 1<1/(20k2),1/d,1/l\nLetDbe an oriented graph on nvertices with δ0(D)≥(1/2−c)n, and let H(D) be\nthek-uniform hypergraph of D. For each v∈V(D), recall that eNH(D),α,1(v) is the set\nof vertices that are ( H(D), α,1)-reachable to v. Together with Lemmas 3.6-3.7 in which\nwe replace fwith 6 k3/2, we get that there exists S⊆V(D) with |S| ≥(1−6k3/2δ)n\nsuch that |eNH(D),α,1(v)∩S| ≥δnfor any v∈S. Then by Lemma 3.8, we can find a\npartition of SintoV1, . . . , V rwith r≤min{6k3/2,⌊1/δ⌋}such that |Vi| ≥(δ−α)n > ηn\nandViis (H(D), β,26k3/2−1)-closed in H(D) for any i∈[r]. Let V0=V(D)\\Sand clearly\n|V0| ≤6k3/2δn. SetP′={V0, V1, . . . , V r}.\nClaim 4.1. IfSis(H(D), β, l)-closed, then Theorem 1.2 holds.\nProof. By Lemma 3.6, we have that H(D) satisfies Lemma 3.3 with1\n6k1/2k!playing the\nrole of η. Let Ube the set guaranteed by Lemma 3.3 and D′=D−U. Notice that\nδ0(D′)≥δ0(D)−U≥(1/2−c)n−n\n6k1/2k!≥(1/2−c′)|D′|.\nThus Theorem 1.5 implies that there exists a constant C > 0 such that D′contains disjoint\nk-cycles covering all but at most Cvertices of D′. Let Wbe the vertex set without being\ncovered. Clearly, |W| ≤C≤ε|D′| ≤εnand|W| ≡0 (mod k) due to |V(D′)| ≡0 (mod\nk). Again using Lemma 3.3, we able to deduce that there is a perfect matching M′in\nH(D)[U∪W], which implies there is a k-cycle-factor in D[W∪U]. Therefore, it can be\nconcluded that Dhas a k-cycle-factor, which completes the proof of Claim 4.1.\n17In the following, we will only treat the case when Sis not ( H(D), β, l)-closed. If\nLµ\nP′(H(D)) contains a 2-transferral ui−ujfor distinct i, j∈[r], then we merge the\npartite sets ViandVjand consider the new partition P′−Vi−Vj+ (Vi∪Vj) ofV(D). By\nLemma 3.9, we continue to merge the partite sets corresponding to 2-transferrals until we\nget a final partition P={V0, V1, . . . , V d}ofV(D) such that Pcontains no 2-transferral\nui−ujfor distinct i, j∈[d], and for some integer l′with l′≥26k3/2−1,P1\\V0is a\n(H(D), β′, l′)-closed partition of Swith|Vi| ≥ηnfor every i∈[d]. For convenience, let\nP1=P \\V0. If|P1|= 1, then Sis (H(D), β, l)-closed which contradicts the assumptions.\nTherefore, we assume that |P1| ≥2.\n4.1 Properties of the partition P1\nIn this part, we present two claims that highlight the favorable properties of the partition\nP1ofSand are frequently used in proving Theorem 1.2. For simplicity, we write D[S] as\nSthroughout the following discussion.\nClaim 4.2. There is a partite set A∈ P 1with|A| ≥n\n7k3/2such that k-cyc(Ak−1, S\\A)≤\nαnk.\nProof. For a contradiction, suppose that\n(O1)k-cyc(Vk−1\ni, S\\Vi)≥αnkfor all Vi∈ P 1={V1, . . . , V d}.\nBecause |Vi| ≥ηnfor every i∈[d], we get η≤1/d. This means that\n(O2)for any J⊆[d] and every i∈[d], there exists j′∈Jwith\nk-cyc(Vk−1\ni, Vj′)≥η·k-cyc(Vk−1\ni,S\nj∈JVj).\nLetA=Vxbe the largest part of P1. It follows by d≤6k3/2that|A| ≥n−|V0|\n6k3/2≥n\n7k3/2.By\n(O1) and ( O2) with J= [d]−x, there is Vy=B∈ P 1−Asuch that k-cyc(Ak−1, B)≥α′nk.\nLetC=S\\(A∪B). Similarly, by ( O1) and ( O2) with J= [d]−y, there is Vz=Q∈ P 1−B\nsuch that k-cyc(Bk−1, Q)≥α′nk. Let F=S\\(B∪Q). In the following, we first declare\n(O3)k-cyc(A),k-cyc(Ak−1, C),k-cyc(B),k-cyc(Bk−1, F)< α′nk.\nWe only prove k-cyc(A),k-cyc(Ak−1, C)< α′nk, since the upper bounds of k-cyc(B)\nandk-cyc(Bk−1, F) can be similarly proved.\nIt is obvious that v1= (k−1)ux+uyandv2= (k−1)uy+uzbelong to Iµ\nP1(H(S)). If k-\ncyc(A)≥α′nk, then v3=kux∈Iµ\nP1(H(S)). Hence v1−v3is a 2-transferral in Lµ\nP(H(S)).\nAlso, if k-cyc(Ak−1, C)≥α′nk, then by ( O2) with J= [d]−x−y, there is Vw∈ P 1−A−B\nsuch that k-cyc(Ak−1, Vw)≥α′nk. This implies that v4= (k−1)ux+uw∈Iµ\nP1(H(S)). So\nv1−v4is a 2-transferral in Lµ\nP1(H(S)). With the fact that Lµ\nP1(H(S)) is 2-transferral-free,\nthese imply that ( O3) holds.\nBy Lemma 3.15 and ( O3), there are disjoint subsets C+andC−ofCsuch that, for\nσ∈ {− ,+}\ne−σ(A, Cσ)≤γn2and|Cσ| ≥ |C|/2 + (|A| − |B|)/2−7k3/2δn. (4.1)\n18Through the choice of A, we have that |A| ≥ |B|. Hence (4.1) suggests that\n|Cσ| ≥ |C|/2−7k3/2δn. (4.2)\nSince C+andC−are disjoint, we get that min {|C+|,|C−|} ≤ | C|/2. Together with (4.1),\nwe have that\n|B| ≥ |A| −14k3/2δn. (4.3)\nAnalogously, By Lemma 3.15 again, with BandFtaking on the roles of AandC\nrespectively, there exist disjoint subsets F+, F−⊆Fsuch that, for σ∈ {− ,+}\ne−σ(B, Fσ)≤γn2and|Fσ| ≥ |F|/2 + (|B| − |Q|)/2−7k3/2δn. (4.4)\nBy the choice of Aand (4.3), it is obtained that |B|+ 14k3/2δn≥ |A| ≥ |Q|. So|Fσ| ≥\n|F|/2−14k3/2δn, and\n|F+|+|F−| ≥ |F| −28k3/2δn. (4.5)\nAs follows, we will think about two cases based on whether sets AandQare same.\nCase 1. A̸=Q.\nIn this case, we will search for a vertex subset B′⊆Bsuch that there is a vertex\nx∈B′with d+(x)>(1/2+c)n, which will imply that d−(x)<(1/2−c)n, a contradiction.\nSince A̸=Q, we have that A⊆F. Fix σ′∈ {− ,+}satisfying |F−σ′∩A| ≥ |Fσ′∩A|.\nWithout loss of generality, assume σ′= +. Then by (4.5), we have\n|F−∩A| ≥(|F−∩A|+|F+∩A|)/2≥(|A| −28k3/2δn)/2 =|A|/2−14k3/2δn. (4.6)\nBy (4.1) and Lemma 3.12, we get that |A\\N−\nγ′(C+)| ≤γ′n\n2. It follows by Lemma 3.12,\n(4.4) and (4.6) that |N−\nγ′(B)∩F−∩A| ≥ |A|/2−15k3/2δn. Let A′=N−\nγ′(B)∩N−\nγ′(C+)∩A,\nthen we have that\n|A′| ≥ |N−\nγ′(B)∩F−∩A| − |A\\N−\nγ′(C+)| ≥ |A|/2−16k3/2δn.\nSince k-cyc(A′)≤k-cyc(Ak)≤αnk, by Lemma 3.15-(i), there is x∈A′such that\nd+(x, A′)≥ |A′| −γ′n≥ |A|/2−17k3/2δn.\nClearly x∈N−\nγ′(B)∩N−\nγ′(C+), which suggests that d+(x, B)≥ |B| −γ′nand\nd+(x, C+)≥ |C+| −γ′n≥ |C|/2−7k3/2δn−γ′n≥ |C|/2−8k3/2δn.\nHence, combining with (4.3), |A| ≥n\n7k3/2andc≪δ <1/(216k3(k!)2), we obtain that\nd+(x)≥d+(x, A′) +d+(x, B) +d+(x, C+)\n≥(|A|/2−17k3/2δn) + (|B| −γ′n) + (|C|/2−8k3/2δn)\n=n/2 +|B|/2−γ′n−25k3/2δn\n≥n/2 +|A|/2−33k3/2δn > (1/2 +c)n.\nSo,d−(x)<(1/2−c)n, which contracts with the fact that δ0(D)≥(1/2−c)n.\n19Case 2. A=Q.\nIn this case, we will show that either k-cyc(( F+∩C+)k−2, F−∩C−, A)≥αnkand\nk-cyc(( F+∩C+)k−2, F−∩C−, B)≥αnkin Subcase 2.1, or k-cyc(Ak−2, B, R )≥µnkfor\nsome R∈ P 1\\ {A, B}andk-cyc(Ak−2, B, A )≥α′nkin Subcase 2.2. This will lead to a\n2-transferral in Lµ\nP1(H(S)) in each subcase, a contradiction.\nBy (4.1) and Lemma 3.12, we have that\n|N−\nγ′(C+)∩A| ≥ |A| −γ′n/2,|N+\nγ′(A)∩C+| ≥ |C+| −γ′n/2,\n|N+\nγ′(C−)∩A| ≥ |A| −γ′n/2,|N−\nγ′(A)∩C−| ≥ |C−| −γ′n/2.(4.7)\nAlso, by (4.4) and Lemma 3.12, we get that\n|N−\nγ′(F+)∩B| ≥ |B| −γ′n/2,|N+\nγ′(B)∩F+| ≥ |F+| −γ′n/2,\n|N+\nγ′(F−)∩B| ≥ |B| −γ′n/2,|N−\nγ′(B)∩F−| ≥ |F−| −γ′n/2.(4.8)\nSubcase 2.1. |F+∩C−| ≤2γ′nor|F−∩C+| ≤2γ′n.\nWithout loss of generality, assume that |F+∩C−| ≤2γ′n. Together with |Fσ| ≥\n|F|/2−14k3/2δn=|C|/2−14k3/2δnwith σ∈ {− ,+}and|C−σ| ≤ |C|/2 + 7 k3/2δnby\n(4.2), we obtain that\n|F+∩C+|=|F+∩C| − |F+∩C−| − |F+∩(C\\ {C+, C−})|\n≥(|C|/2−14k3/2δn)−2γ′n−14k3/2δn≥ |C+| −36k3/2δn. (4.9)\nThis implies |F−∩C+| ≤36k3/2δnsince F+andF−are disjoint, and then\n|F−∩C−|=|F−∩C| − |F−∩C+| − |F−∩(C\\ {C+, C−})|\n≥(|C|/2−14k3/2δn)−36k3/2δn−14k3/2δn≥ |C−| −71k3/2δn. (4.10)\nAccording to Lemma 3.4, we have that D[F+∩C+] contains at least |F+∩C+|/2 vertices\nof out-degree at most 3 |F+∩C+|/4 +cninD[F+∩C+]. For convenience, we denote the\nset of these vertices by X. Let Y=N+\nγ′(A)∩N+\nγ′(B)∩Xand then |Y| ≥ |X| −γ′nby\n(4.7) and (4.8). By definitions of F+,C+andY, we can infer that for any y∈Y,\nd+(y, F−∩C−)≥δ0(D)−d+(y, F+∩C+)−d+(y, A∪B)−127k3/2δn\n≥(1/2−c)n−(3|F+∩C+|/4 +cn)−2γ′n−127k3/2δn≥n/8.(4.11)\nWhere the last inequality is derived from |F+∩C+| ≤ |C|/2+7k3/2δn,n≥ |A|+|B|+|C|,\n|A| ≥n\n7k3/2andc≪γ′≪δ <1\n216k3(k!)2.\nOn the other hand, by Lemma 3.11-(i), |Yi+| ≥|Y(i−1)+|\n2−ϵnand|Y(k−3)+| ≥|Y|\n2k−3−\n2ϵn. According to (4.7)-(4.8), in F−∩C−, with the exception of at most γ′nvertices, all\nother vertices zsatisfy d+(z, A)≥ |A| −γ′n. Combining with the definition of Y, we get\n20thatd−(y, N+(z, A))≥ |A| −2γ′nfor any y∈Y. Therefore, we obtain that\nk-cyc(Y, Y+, . . . , Y(k−3)+, F−∩C−, A)\n≥ |Y(k−3)+| ·(ϵn)·k−4Y\ni=0(ϵn−i)·(n/8−γ′n)·(|A| −2γ′n)\n≥|C| −86k3/2δn−2γ′n\n2k−2·(ϵn)k−3·(n/8−γ′n)·ηn\n2≥αnk, (4.12)\nwhere the first inequality is due to (4.11) and the second inequality is from (4.9) and (4.2)\nand|A| ≥n\n7k3/2.\nSimilarly, except for at most γ′nvertices, all other vertices z′inF−∩C−satisfy\nd+(z, B)≥ |B| −γ′n. By the definition of Y, we get d−(y′, N+(z, B))≥ |B| −2γ′nfor\nanyy′∈Y. Similar to (4.12), we can also get that\nk-cyc(Y, Y+, . . . , Y(k−3)+, F−∩C−, B)≥αnk. (4.13)\nTogether with (4.12), (4.13) and µ≪α, this implies Lµ\nP1(H(S)) contains a 2-transferral,\nwhich contradicts our assumption.\nSubcase 2.2. Both|F+∩C−| ≥2γ′nand|F−∩C+| ≥2γ′n.\nDue to (4.7)-(4.8), except for at most γ′nvertices, all other vertices xinF+∩C−\nsatisfy that d−(x, B)≥ |B| −γ′nandd+(x, A)≥ |A| −γ′n. Similarly, except for at\nmost γ′nvertices, all other vertices yinF−∩C+meet that d+(y, B)≥ |B| −γ′nand\nd−(y, A)≥ |A| −γ′n. For the sake of convenience, for any such vertices xandy, let\nU1=N−(x, B)∩N+(y, B) and U2=N+(x, A)∩N−(y, A), and then\n|U1| ≥ |B| −2γ′nand|U2| ≥ |A| −2γ′n. (4.14)\nAlso let U1,1={u∈U1:d+(u, U 2)≥ |U2|/2−2cn}andU1,2={u∈U1:d−(u, U 2)≥\n|U2|/2−2cn}. It is not hard to get that either |U1,1| ≥ |U1|/2 or|U1,2| ≥ |U1|/2.\n(a) The case when |U1,1| ≥ |U1|/2. (b) The case when |U1,2| ≥ |U1|/2.\nFigure 3: In the figure, the blue arcs form a k-cycle.\nIf|U1,1| ≥ |U1|/2, then for any ui∈U1,1, letU2,i=N+(ui, U2) and then\n|U2,i| ≥ |U2|/2−2cn≥ |A|/2−2γ′n. (4.15)\n21Recall that Us+\n2,i={u∈U(s−1)+\n2,i :d−(u, U(s−1)+\n2,i )≥ϵn}for any s∈[k−3]. Then by\nLemma 3.11, we have that |Us+\n2,i| ≥|U(s−1)+\n2,i|\n2−ϵnand|U(k−3)+\n2,i| ≥|U2,i|\n2k−3−2ϵn. Thus by\n(4.14)-(4.15), and α≪ϵ≪γ′≪η, we can get that (see Figure 3 (a))\nk-cyc(F−∩C+, B, Ak−2)\n≥(|F−∩C+| −γ′n)·(|U1|/2−γ′n)·(ϵn)·k−4Y\ni=0(ϵn−i)·(|U(k−3)+\n1,i| −γ′n)\n≥γ′n·(|B|/2−2γ′n)·(ϵn)k−3·\u0012|A| −4γ′n\n2k−2−γ′n\u0013\n≥γ′ηϵk−3\n2k−1·nk≥αnk, (4.16)\nIf|U1,2| ≥ |U1|/2, similar to the above process, for any vj∈U1,2, letU2,j=N−(vj, U2).\nThen we deduce that\n|U2,j| ≥ |U2|/2−2cn≥ |A|/2−2γ′n. (4.17)\nBy Definition 3.10, we have that Ui−\n2,j={v:v∈U(i−1)−\n2,j andd+(v, U(i−1)−\n2,j)≥ϵn}. Then\nby Lemma 3.11, we get that |Ui−\n2,j| ≥|U(i−1)+\n2,j|\n2−ϵnand|U(k−3)+\n2,j| ≥|U2,j|\n2k−3−2ϵn.Similarly,\nwe can deduce that (see Figure 3 (b))\nk-cyc(F+∩C−, Ak−2, B)≥(|F+∩C−| −γ′n)·(ϵn)k−3·(|U(k−3)+\n2,j| −γ′n)·(|U1|/2−γ′n)\n≥γ′n·(ϵn)k−3·\u0012|A| −4γ′n\n2k−2−γ′n\u0013\n·(|B|/2−2γ′n)≥αnk.\nTogether with (4.16), this implies that there exists some partite set RofP1such that\nk-cyc(R∩(F−∩C+), B, Ak−2)≥µnkork-cyc(R∩(F+∩C−), Ak−2, B)≥µnk. Recall that\nk-cyc(Ak−1, B)≥α′nk, so we get a 2-transferral vA−vRinLµ\nP1(H(S)), which contradicts\nour assumption. Hence this claim holds.\nClaim 4.3. P1is aδ-extremal partition of S.\nProof. It suffices to verify that P1satisfies Definition 3.1. Combining with Claim 4.2, we\nknow that there exists A∈ P 1with|A| ≥n\n7k3/2such that k-cyc(Ak−1, S\\A)≤αnk. For\n(not necessarily disjoint) subsets U1, U2, . . . , U k, Uk+1, we denote L(U1, U2, . . . , U k, Uk+1)\nto be the collection of ( k+ 1)-sets {u1, u2, . . . , u k, uk+1}such that ui∈Uifori∈[k+ 1]\nand both u1u2···uku1andu2···ukuk+1u2arek-cycles. We declare that\n|L(A, V, . . . , V, S \\A)| ≤γ3nk+1. (4.18)\nOtherwise, due to |P| ≤ 1/η, there exist U2, . . . , U k∈ P andUk+1∈ P 1−Asuch that\n|L(A, U 2, . . . , U k, Uk+1)| ≥ηk·γ3nk+1≥kµnk+1. This implies that k-cyc(A, U 2, . . . , U k), k-\ncyc(U2, . . . , U k, Uk+1)≥µnk, which contradicts our conclusion that Lµ\nP1(H(S)) is 2-\ntransferral-free.\n22SetSσ=Nσ\nγ2,S(A) for σ∈ {+,−}andR=S\\(S+∪S−∪A). By Lemma 3.13, we\nhave that\n|R| ≤2γ3nand|A| ≤(1/3 + 50 k3δ)n. (4.19)\nWe first prove that Definition 3.1-(i) holds. Given σ∈ {− ,+}, suppose that there is\nak-cycle C=x···yzxwith V(C)\\ {z} ⊆ Sσandz∈S−σ. Let w∈Asuch that\nV(C)\\ {y} ∪ { w}spans a k-cycle. Since y∈Sσandz∈S−σ, we get that d−σ(y, A)≥\n|A| −γ2nanddσ(z, A)≥ |A| −γ2n. With (4.18), we see that\n(|A| −2γ2n)·k-cyc(( Sσ)k−2, Sσ, S−σ)≤ |L(A,(Sσ)k−2, S−σ, Sσ)| ≤γ3nk+1.\nTogether with |A| ≥n/(7k3/2), we get that\nk-cyc(( Sσ)k−2, Sσ, S−σ)≤γ2nk.\nFurther, k-cyc(( Sσ)k−2, Sσ, A)≤2γ2n·\u0000|Sσ|\nk−1\u0001\n≤γ2nk, and due to (4.19), we have that\nk-cyc(( Sσ)k−2, Sσ, R)≤2γ2nk. Hence,\nk-cyc(( Sσ)k−2, Sσ, S\\Sσ) =k-cyc(( Sσ)k−2, Sσ, S−σ∪A∪R)≤4γ2nk.\nApplying Lemma 3.13 with Sσand 4 γ2taking on the roles of Aandα, respectively, we\nget that\n|S−|,|S+| ≤(1/3 + 50 k3δ)n. (4.20)\nSetA′=A∪R, and then {A′, S−, S+}is a partition of S. It follows from (4.19) and\n(4.20) that |A′|,|S−|,|S+| ≤(1/3 + 50 k3δ)n. Thus we can deduce that\n(1/3−100k3δ)n≤ |A′|,|S−|,|S+| ≤(1/3 + 50 k3δ)n. (4.21)\nHence (i) of Definition 3.1 holds. Now we show that Definition 3.1-(ii) also holds. Since\nA′=A∪Rand|R| ≤2γ3n, for each v∈Sσwith σ∈ {− ,+}, we get that d−σ(v, A′)≥\n|A| −γ2n≥ |A′| −ηn. This implies that\neσ(Sσ, A′)≤ |Sσ| ·γ1n≤γ1n2. (4.22)\nIt follows by Lemma 3.5-(i) that e+(S+,S+)≥e−(S+,S+)−2cn2≥e−(S+, A′)−2cn2.\nBy 4.21, this implies\ne+(S+, S−) =e+(S+,S+)−e+(S+, A′)−e+(S+, V0)\n≥(|A′| −γ1n)|S+| −2cn2−γ1n2−50k3δn·6k3/2δn≥ |S+| · |S−| −δn,\nwhere the last inequality is derived from the fact that δ <1\n216k3(k!)2. Thus e+(S−, S+)≤\nδn. Together with (4.21)-(4.22), we get that Pis aδ-extremal partition of S.\n234.2 Completion of Theorem 1.2\nBy Claim 4.2, there exists a partite set A∈ P 1with|A| ≥n\n7k3/2andk-cyc(Ak−1, S\\A)≤\nαnk. Set Sσ=Nσ\nγ2,S(A) for σ∈ {+,−}. Let A′\n2(resp., A′\n0) be the partite set of P1such\nthat|A′\n2∩S+|(resp., |A′\n0∩S−|) is maximum and define A2=A′\n2∩S+andA0=A′\n0∩S−.\nSince (1 /3−100k3δ)n≤ |S+|,|S−| ≤ (1/3 + 50 k3δ)nandd≤6k3/2, we have that\n|A2|,|A0| ≥n\n20k3/2. Further, by Claim 4.3, A2⊆S+andA0⊆S−, we get that,\nd−(x, A)≥ |A| −γ2nfor any x∈A2,ande+(A2, A)≤δn2,and\nd+(y, A)≥ |A| −γ2nfor any y∈A0, e−(A, A 0)≤δn2ande+(A0, A2)≤δn2.\nThen by Lemma 3.12 with δ,A2andA(resp., δ,A0andA2) taking on the roles of γ2,A\nandB, respectively, we obtain that\n|N−\nδ1(A2)∩A| ≥ |A| −δ1n\n2and|N+\nδ1(A)∩A2| ≥ |A2| −δ1n\n2,and\n|N−\nδ1(A0)∩A2| ≥ |A2| −δ1n\n2and|N+\nδ1(A2)∩A0| ≥ |A0| −δ1n\n2.(4.23)\nFor consistency, we define A1:=A. Combining with the definitions of A2andA0, we get\nthat for every i∈ {0,1,2},\n\f\f\f\f\u001a\nz∈Ai:d−(z, A i−1)≥ |Ai−1| −δ1n\n2andd+(z, A i+1)≥ |Ai+1| −δ1n\n2\u001b\f\f\f\f≥ |Ai| −δ1n,\n(4.24)\nhere and subsequently, the subscripts of Ai−1andAi+1take the remainder modulo 3.\nWithout loss of generality, assume that |A2| ≥ |A0|. In the following, we consider three\ncases in terms of the value of the remainder of kmodulo 3. In each case, we will get that\nLµ\nP1(H(S)) contains a 2-transferral.\n(a)k= 4 for Case 1. (b) k= 5 for Case 2. (c) k= 6 for Case 3.\nFigure 4: Examples for Cases 1-3.\nCase 1. k≡1 (mod 3).\nIn this case, let C1\n1be the family of k-cycles u1···uku1such that for every j∈[k],\nuj∈Aiwhen j≡i(mod 3) . (4.25)\n24Similarly, let C2\n1be the family of k-cycles u1···uku1, where for j∈[k],ujsatisfies that\nuj∈Ai+1when j≡i(mod 3) . (4.26)\nWe first calculate the size of C1\n1. By Lemma 3.4, there exists A′\n1⊆A1with|A′\n1| ≥ |A1|/2\nsuch that d−(x, A 1)≥ |A1|/4−cnfor any x∈A′\n1. Pick any u1∈A′\n1. To construct\nk-cycles through u1, we pick paths u1u2···uk−1that starting at u1andujmeeting (4.25)\nforj∈[k−1]. By (4.24), the number of such ( k−1)-paths is at least\n(|A2| −δ1n)(|A0| −δ1n)·(k−4)/3Y\ni=1((|A1| −δ1n−i)(|A2| −δ1n−i)(|A0| −δ1n−i))\n≥nk−2\n3kkk−1, (4.27)\nwhere the inequality is due to |A1|=|A| ≥n\n3−100k3δn,|A2|,|A0| ≥n\n20k3/2andδ1<1\n20k2.\nGiven such a ( k−1)-path u1···uk−1, we may make it a k-cycle by selecting the vertex uk\ninN−(u1, A1)∩N+(uk−1, A1). Clearly there exist at least |A1|/4−cn−δ1n−(k−1)/3\nsuch vertices uk. Hence by (4.27) we can get that the number of k-cycles satisfying (4.25)\nis at least\n1\nk·(|A1| −δ1n)·nk−2\n3kkk−1·\u0012|A1|\n4−cn−δ1n−k−1\n3\u0013\n≥µnk. (4.28)\nNext we calculate the size of C2\n1. By Lemma 3.4, there exists vertex subset A′\n2⊆A2\nwith|A′\n2| ≥ |A2|/2 so that for any y∈A′\n2, we obtain that d−(y, A 2)≥ |A2|/4−cn. With\nA2,A3,A1andA′\n2taking on the roles of A1,A2,A3andA′\n1respectively in the process\nof calculating C1\n1described above, we can get that |C2\n1| ≥µnk. Together with |C1\n1| ≥µnk,\nthis suggests that Lµ\nP1(H(S)) contains a 2-transferral, which contradicts our assumption.\n(We give an example when k= 4 for Case 1 in Figure 4 (a).)\nCase 2. k≡2 (mod 3).\nLetC1\n2be the family of k-cycles u1···uku1such that\nu1∈A1and for any j∈ {2,3, . . . , k }, uj∈Ai−1when j≡i(mod 3) . (4.29)\nLikewise, let C2\n2be the family of k-cycles u1···uku1such that\nforj∈[2], uj∈Aj,and for any j∈ {3,4, . . . , k }, uj∈Ai−1when j≡i(mod 3) .(4.30)\nWe first prove that |C1\n2| ≥µnk. Again by Lemma 3.4, there exists A′\n1⊆A1with|A′\n1| ≥\n|A1|/2 such that d−(x, A 1)≥ |A1|/4−cnfor any x∈A′\n1. Let A′′\n1={x∈A′\n1:d−(x, A′\n1)≥\nϵn}, and then by Lemma 3.11, we can get that |A′′\n1| ≥ | A′\n1|/2−ϵn. Similar to the\ncalculation process for C1\n1, by (4.24), the size of C1\n2is at least\n1\nk·\u0012|A1|\n4−cn−δ1n−2−k−5\n3\u0013\n·ϵn·(|A′′\n1| −δ1n)·(|A2| −δ1n)(|A0| −δ1n)·\n(k−5)/3Y\ni=1(|A1| −2δ1n−1−i)(|A2| −δ1n−i)(|A0| −δ1n−i),\n25which is larger than µnksince|A1|=|A| ≥n\n3−100k3δn,|A2|,|A0| ≥n\n20k3/2,|A′′\n1| ≥\n|A′\n1|/2−ϵn≥ |A1|/4−ϵn, and c≪µ≪δ1<1\n20k2.\nWe next estimate the size of C2\n2. For any u1∈A′\n1, let A′\n2=N+(u1, A2), and then\n|A′\n2| ≥ | A2| −δ1n/2 by (4.23). Define A′′\n2={y∈A′\n2:d−(y, A′\n2)≥ϵn}, and then\n|A′′\n2| ≥ |A′\n2|/2−ϵnvia Lemma 3.11. In addition, there exist at least |A′′\n2| −δ1n/2 vertices\nin|A′′\n2|such that each of these vertices u3satisfying d+(u3, A0)≥ |A0| −δ1n/2, and\nthere exist at least |A0| −δ1nvertices in A0so that each of these vertices u4satisfying\nd+(u4, A1)≥ |A1| −δ1n/2. Greedily, we can get the path u1···uk−1and the number of\nvertices ukinN−(u1, A1) with uk−1uk∈Ais at least |N−(u1, A1)| −δ1n/2−(k−5)/3.\nHence the size of C2\n2is at least\n1\nk·(|A1|/4−cn−δ1n−(k−5)/3)·(|A′\n1| −δ1n)·ϵn·(|A′′\n2| −δ1n)·(|A0| −δ1n)·\n(k−5)/3Y\ni=1((|A1| −δ1−i)(|A2| −δ1n−1−i)(|A0| −δ1n−i)),\nwhich is more than µnksince|A1| ≥n\n3−100k3δn,|A2|,|A0| ≥n\n20k3/2,|A′′\n1| ≥ |A1|/4−ϵn,\n|A′′\n2| ≥ | A′\n2|/2−ϵn≥ |A2|/2−δ1n/4−ϵn, and c≪µ≪δ1<1\n20k2. Combining with\n|C1\n2| ≥µnk, we can get a 2-transferral, which contradicts that Lµ\nP1(H(S)) is 2-transferral-\nfree (we give an example when k= 5 for Case 2 in Figure 4 (b)).\nCase 3. k≡0 (mod 3).\nLetC1\n3be the family of k-cycles u1···uku1such that for any j∈[k]\nuj∈Aiwhen j≡i(mod 3) . (4.31)\nSimilarly, C2\n3is the family of k-cycles u1···uku1, where u1, u2∈A1, u3, u4∈A2and\nfor any j∈ {5, . . . , k }, uj∈Ai+1when j≡i(mod 3) . (4.32)\nIt is not hard to get from (4.24) that\n|C1\n3| ≥(|A1| −δ1n−(k−3)/3)·(|A2| −δ1n)·(|A0| −δ1n)·\n(k−3)/3Y\ni=1(|A1| −δ1n−i)(|A2| −δ1n−1−i)(|A0| −δ1n−i)≥µnk.\nAlso, similar to the estimations of |C1\n2|and|C2\n2|, there exist A′\n1, A′′\n1⊆A1with|A′\n1| ≥ |A1|/2\nand|A′′\n1| ≥ |A′\n1|/2−ϵnsuch that d−(x, A 1)≥ |A1|/4−cnandd−(y, A′\n1)≥ϵnfor any\nx∈A′\n1andy∈A′′\n1, respectively. At the same time, For any u2∈A′′\n1, there exist\nA′\n2, A′′\n2⊆A2with|A′\n2| ≥ |A2| −δ1n/2 and |A′′\n2| ≥ |A′\n2|/2−ϵnsuch that A′\n2=N+(u2, A2)\nandA′′\n2={y∈A′\n2:d−(y, A′\n2)≥ϵn}. Clearly, there exist at least |A′′\n1| −δ1n/2 vertices u4\nin|A′′\n1|such that d+(u4, A0)≥ |A0| −δ1n/2. Further, similar to |C2\n2|, we can proceed by\ngreedily finding vertices u5, . . . , u kuntil we obtain a k-cycle u1···uku1. Hence, we get\n|C2\n3| ≥ϵn·(|A′\n1|/2−ϵn−δ1n/2)·ϵn·(|A′\n2|/2−ϵn−δ1n/2)·(|A0| −δ1n)·\n(k−6)/3Y\ni=1(|A1| −δ1n−1−i)(|A2| −δ1n−1−i)(|A0| −δ1n−i)·\n((|A1|/4−cn−δ1n−(k−6)/3))≥µnk.\n26Together with |C2\n3| ≥µnk, this contradicts what we already know about that Lµ\nP1(H(S))\nis 2-transferral-free (we give an example when k= 6 under this case in Figure 4 (c)).\nThus we complete the proof of Theorem 1.2. □\nAcknowledgments\nWe would especially like to thank Jie Han for helpful discussion and valuable comments.\nWe also thank the referee for the careful readings and suggestions that improve the pre-\nsentation.\nReferences\n[1] I. Adler, N. Alon, and S. M. Ross, On the maximum number of Hamiltonian paths\nin tournaments , Random Struct. Alg. 18(2001), no. 3, 291–296.\n[2] G.-A. Dirac, Some theorems on abstract graphs , Proc. London Math. Soc. 3-2(1952),\nno. 1, 69–81.\n[3] L. Ding, J. Han, S. Sun, G. Wang, and W. Zhou, F-factors in quasi-random hyper-\ngraphs , J. London Math. Soc. 106(2022), no. 3, 1810–1843.\n[4] R. H¨ aggkvist, Hamilton cycles in oriented graphs , Combin. Probab. Comput. 2\n(1993), no. 1, 25–32.\n[5] R. H¨ aggkvist and A. Thomason, Oriented Hamilton cycles in oriented graphs , in:\nCombinatorics, Geometry and Probability, Cambridge University Press, (1997), 339–\n353.\n[6] A. Hajnal and E. Szemer´ edi, Proof of a conjecture of P. Erd˝ os , Combinatorial Theory\nand Its Application 2(1970), 601–623.\n[7] J. Han, Near perfect matchings in k-uniform hypergraphs II , SIAM J. Discrete Math.\n30(2016), no. 3, 1453–1469.\n[8] J. Han and A. Treglown, The complexity of perfect matchings and packings in dense\nhypergraphs , J. Combin. Theory Ser. B 141(2020), 72–104.\n[9] P. Keevash, D. K¨ uhn, and D. Osthus, An exact minimum degree condition for Hamil-\nton cycles in oriented graphs , J. London Math. Soc. 79(2009), 144–166.\n[10] L. Kelly, D. K¨ uhn, and D. Osthus, A Dirac-type result on Hamilton cycles in oriented\ngraphs , Combin. Probab. Comput. 17(2008), no. 5, 689–709.\n[11] P. Keevash and B. Sudakov, Triangle packings and 1-factors in oriented graphs , J.\nCombin. Theory Ser. B 99(2009), no. 1, 709–727.\n27[12] L. Li and T. Molla, Cyclic triangle factors in regular tournaments , Electron. J. Com-\nbin.26(2019), no. 4, 4–24.\n[13] V. R¨ odl, A. Ruci´ nski, and E. Szemer´ edi, A Dirac-type theorem for 3-uniform hyper-\ngraphs , Combin. Probab. Comput. 15(2006), no. 1-2, 229–251.\n[14] C. Thomassen, Long cycles in digraphs with constraints on the degrees , in: B. Bollobs\n(Ed.), Surveys in Combinatorics, in: London Math. Soc. Lecture Notes, vol. 38,\nCambridge University Press, (1979), 211–228.\n[15] C. Thomassen, Long cycles in digraphs , Proc. London Math. Soc. 3-42 (1981), no.\n2, 231–251.\n[16] C. Thomassen, Edge-disjoint Hamiltonian paths and cycles in tournaments , Proc.\nLondon Math. Soc. 45(1982), 151–168.\n28"    },    {        "title": "2402.05093v1.Moduli_Parameters_of_Complex_Singularities_with_Non_Degenerate_Newton_Boundary.pdf",        "content": "MODULI PARAMETERS OF COMPLEX SINGULARITIES WITH\nNON-DEGENERATE NEWTON BOUNDARY\nJANKO BÖHM, MAGDALEEN S. MARAIS, AND GERHARD PFISTER\nAbstract. Our recent extension of Arnold’s classification includes all singularities\nof corank ≤2equivalent to a germ with a non-degenerate Newton boundary, thus\nbroadening the classification’s scope significantly by a class which is unbounded with\nrespect to modality and Milnor number. This method is based on proving that all\nright-equivalence classes within a µ-constant stratum can be represented by a single\nnormal form derived from a regular basis of a suitably selected special fiber. While\nboth Arnold’s and our preceding work on normal forms addresses the determination\nof a normal form family containing the given germ, this paper takes the next natural\nstep: We present an algorithm for computing for a given germ the values of the moduli\nparameters in its normal form family, that is, a normal form equation in its stable\nequivalence class. This algorithm will be crucial for understanding the moduli stacks of\nsuch singularities. The implementation of this algorithm, along with the foundational\nclassification techniques, is implemented in the library arnold.lib for the computer\nalgebra system Singular .\n1.Introduction\nOur recent extension Böhm, Marais, and Pfister (2020) of Arnold’s classification of\nisolated hypersurface singularities (see Arnold (1976); Arnold et al. (1985)) includes all\nsingularities of corank ≤2which are equivalent to a germ with non-degenerate Newton\nboundary in the sense of Kouchnirenko. This broadens the scope of the classification\nby a class of singularities, which is unbounded both in terms of modality and Milnor\nnumber. We establish that there is a single polynomial normal form that contains rep-\nresentatives (at least one, but only finitely many) of all right equivalence classes within\na given µ-constant stratum of a given input germ. Based on this result, and algorithmic\nmethods developed in Böhm, Marais, and Pfister (2016a,b), an algorithm for effectively\ndetermining a normal form from a regular basis of a suitably chosen special fiber is given.\nWhile both Arnold’s and our preceding work on normal forms only addresses the\ndetermination of a normal form family containing the given germ, this continuation takes\nthe next natural step: determining the moduli parameters in the normal form families\nassociated with these input germs. That is, we find for a given input germ an element\nin the normal form associated to its µ-constant stratum such that this element is right\nequivalent to the input germ, hence determining exactly its stable and right equivalence\nclass. While one could argue that, with the normal form known, such a germ could be\nfoundusinganAnsatzfortherightequivalencebytakingfinitedeterminacyintoaccount,\nthis is practically inefficient and will not yield a result except for trivial input. Taking our\n2020Mathematics Subject Classification. Primary 14B05; Secondary 32S25, 14Q05.\nKey words and phrases. Hypersurface singularities, Newton non-degenerate germs, moduli parameter,\nalgorithmic classification, normal forms.\nThis research was supported by Project B5 of SFB-TRR 195. Gefördert durch die Deutsche Forschungs-\ngemeinschaft (DFG) - Projektnummer 286237555 - TRR 195 (Funded by the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) - Project-ID 286237555 - TRR 195).\n1arXiv:2402.05093v1  [math.AG]  7 Feb 2024MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 2\nclue from some case-by-case studies in Marais and Steenpaß (2015, 2016); Böhm, Marais\nand Steenpaß (2019); Böhm, Marais, and Pfister (2016a) we thus develop an iterative\nmethod for eliminating the terms of the germ which are not present in the normal form\nfamily. This methods has some similarities the algorithm finding the normal form. The\nmain challenge arises here from the fact that singularities with non-degenerate Newton\nboundary in general do not satisfy Condition A. In an iterative process we thus have\nto control higher order contributions in the binomial expansion when applying a right\nequivalence to the germ (since those can be of lower piecewise degree). Applications of\nour result could occur in the context of the study of Baikov polynomials, see Böhm et\nal. (2018); Lee and Pomeransky (2013).\nOur paper is structured as follows:\nIn Section 2 we give a review of the foundational concepts and preliminary results on\nsingularities and classification.\nIn Section 3, we recall the main results on the determination of normal forms for\nsingularities of corank ≤2equivalent to a germ with non-degenerate Newton boundary.\nWe also recall the algorithmic framework determining the normal form.\nIn Section 4, we address the determination of the moduli parameters in the normal\nform corresponding to a given input germ.\nAcknowledgements. Our thanks go to Gert-Martin Greuel, Hans Schönemann for valu-\nable discussions and insights. We also thank Alexander Mathis for work towards an\nexperimental implementation in the computer algebra system OSCAR .\n2.Definitions and Preliminary Results\nIn this section, fundamental definitions, theorems, and notations relevant to our dis-\ncussion are presented. We use C{x1, . . . , x n}to denote the ring of convergent power\nseries, that is, power series that converge in open neighborhoods of the point (0, . . . , 0).\nWe use mfor the maximal ideal of C{x1, . . . , x n}.\nNotation 2.1. We denote by mon( x1, . . . , x n)the monoid of monomials in x1, . . . , x n.\nForf∈C{x1, . . . , x n}and a monomial m∈mon( x1, . . . , x n), the coefficient of minfis\ndenoted by coeff( f, m).\nDefinition 2.2. Ifw= (c1, . . . , c n)∈Nnis a weight for the variables (x1, . . . , x n), the\nw-weighted degree on mon( x1, . . . , x n)is defined by the expression\nw- deg nY\ni=1xsi\ni!\n=nX\ni=1cisi.\nIn the case that the weight of all variables is one, the weighted degree of a monomial m\nis called its standard degree, denoted by deg(m). This notation is also used for terms in\npolynomials.\nDefinition 2.3. Consider a finite family of weights w= (w1, . . . , w s)∈(Nn)sfor\n(x1, . . . , x n). For a term m∈C[x1, . . . , x n], itspiecewise weight with respect to w\nis defined as\nw- deg( m) := min\ni=1,...,swi- deg( m).\nDefinition 2.4. Fix a (piecewise) weight wonmon( x1, . . . , x n).MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 3\n(1) Suppose\nf=∞X\ni=0fi\nis the decomposition of f∈C{x1, . . . , x n}into weighted homogeneous components\nfiwith w-degree of i. In the case that fi= 0fori > dandfd̸= 0is the lowest\nnon-zero component, we set w- deg( f) =d. The(piecewise) weighted j-jetoff,\ndenoted by w- jet(f, j), is given by\nw- jet(f, j) :=jX\ni=0fi.\nThe sum of terms of fwith the lowest w-degree is called the principal part off\nwith respect to w. Theorderoffwith respect to wis defined as the degree of its\nprincipal part, and is denoted by w- ord( f).\n(2) A power series in C{x1, . . . , x n}is said to have filtration d∈Nwith respect to\nwif all its monomials have a w-weighted degree ≥d. By Ew\ndwe denote the sub-\nvector space of C{x1, . . . , x n}of power series of filtration dwith respect to w. The\nsub-spaces Ew\nd, for varying d∈N, form a filtration on C{x1, . . . , x n}.\nDefinition 2.5. A piecewise homogeneous germ f0of degree dsatisfies Condition A, if\nfor every germ gof filtration d+δ > din the ideal spanned by the derivatives of f0, there\nis a decomposition\ng=X\ni∂f0\n∂xivi+g′,\nwhere the vector field vhas filtration δandg′has filtration bigger than d+δ.\nDefinition 2.6. We say that f∈m2⊂C{x1, . . . , x n}isk-determined if\nf∼jet(f, k) +gfor all g∈Ek+1,\nwith respect to right-equivalence. The determinacy off, denoted by dt(f), is the small-\nest integer kfor which fisk-determined.\nDefinition 2.7. Forf∈C{x1, . . . , x n}, theJacobian ideal\nJac(f) =\u001c∂f\n∂x1, . . . ,∂f\n∂xn\u001d\nis the ideal of C{x1, . . . , x n}generated by the partial derivatives of f. Thelocal algebra\nQf=C{x1, . . . , x n}/Jac(f)\noffis the quotient of C{x1, . . . , x n}by the Jacobian ideal. The Milnor number off\nis the dimension of Qfas aC-vector space.\nRemark 2.8. If the germ fdefines an isolated singularity, then fisk-determined if\nk≥µ(f)+1, hence fis finitely determined. So an isolated singularity can be represented\nby a polynomial.\nDefinition 2.9. Theannihilator of a germ f, denoted by ann(f), is the ideal of all\nelements of C{x1, . . . , x n}that yield zero when multiplied with f.\nDefinition 2.10. Suppose that ϕis aC-algebra automorphism of C{x1, . . . , x n}, and w\nis a single weight on mon( x1, . . . , x n).MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 4\n(1) For any positive integer j, the automorphism w- jet(ϕ, j) :=ϕw\nj, is defined by\nϕw\nj(xi) :=w- jet(ϕ(xi), w- deg( xi) +j)fori= 1, . . . , n.\nIfw= (1, . . . , 1), we use the notation ϕjforϕw\nj.\n(2) We say that ϕhasfiltration dif\n(ϕ−id)Ew\nλ⊆Ew\nλ+d\nfor all λ∈N.\nRemark 2.11. We note that ϕ0(xi) = jet( ϕ(xi),1)fori= 1, . . . , n. Moreover, note that\nϕw\n0has filtration ≤0. For j >0,ϕw\njhas filtration jifϕw\nj−1= id.\nDefinition 2.12. Forf=P\ni1,...,i nai1,...,i nxi1\n1···xinn∈C{x1, . . . , x n}, write\nmon( f) := {xi1\n1···xinn|ai1,...,i n̸= 0}\nfor the set of monomials off, and\nsup(f) := {i1···in|ai1,...,i n̸= 0}\nfor thesupport off. We set\nΓ+(f) :=[\nxi1\n1···xinn∈supp( f)((i1, . . . , i n) +Rn\n+)\nand define Γ(f)as the boundary of the convex hull of Γ+(f)inRn\n+. The set Γ(f)is called\ntheNewton boundary off. Then:\n(1) The compact segments of Γ(f)are referred to as facets.1If∆is a facet, we write\nsupp( f,∆)for the set of monomials of fwith exponent vector on ∆. The sum\nof the terms lying on ∆is denoted by jet(f,∆). Moreover, we write supp(∆) for\nthe set of monomials corresponding to the lattice points of ∆. Considering the\nmonomials lying on a union of facets, we use the same terminology for a set of\nfacets.\n(2) To a facet ∆we associate a weight w(∆)on the monomials in mon( x1, . . . , x n)as\nfollows: If −(wx1, . . . , w xn)is the normal vector of ∆in lowest terms with integers\nwx1, . . . , w xn>0, we define\nw(∆) - deg( x1) =wx1, . . . , w (∆) - deg( xn) =wxn.\n(3) Now suppose that w1, . . . , w sare the weight vectors of the facets of Γ(f)ordered by\nincreasing slope. Then there are uniquely determined minimal integers λ1, . . . , λ s≥\n1with the property that the piecewise weight with respect to\nw(f) := ( λ1w1, . . . , λ sws)\nis constant on the Newton boundary Γ(f). We refer to this constant by d(f).\n(4) Suppose that ∆1and∆2are adjacent facets with weights w1andw2, respectively, w\nis the piecewise weight defined by w1andw2, and dis the w-degree of the monomials\non∆1and∆2. We then write span(∆ 1,∆2)for the Newton polygon associated to\nthe sum of all monomials of (w1, w2)-degree d.\n1In convex geometry, codimension 1faces of the convex hull Γ+(f)are referred to as facets.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 5\n(5) If Γ(f)has at least one facet, we say that a monomial mis strictly below, on\nor above Γ(f), if the w(f)-degree of mis less than, equal to or larger than d(f),\nrespectively.\n(6) We write jet(f,Γ(f))for the expansion of fup to w(f)-order d(f).\nDefinition 2.13. Assume that fhas finite Milnor number. A basis {e1, . . . , e µ}of the\nlocal algebra of fconsisting out of homogeneous elements is regular with respect to the\nfiltration given the piecewise weight w, if for each D∈N, the basis elements of degree D\nwith respect to ware independent modulo the sum vector space Jac(f) +Ew\n>Dof germs\nof filtration larger than D.\nRemark 2.14. Arnold has proven in Arnold (1974) that for each germ f∈C{x,. . . , x n}\nthere exists a regular basis consisting out of monomials.\nDefinition 2.15. For a union of right-equivalence classes K⊂C{x1, . . . , x n}anormal\nformforKis a smooth map\nΦ :B −→ C[x1, . . . , x n]⊂C{x1, . . . , x n}\nof a finite-dimensional C-linear space Binto the space of polynomials such that:\n(1)Φ(B)intersects all equivalence classes of K,\n(2) for each equivalence class the inverse image in Bis finite\n(3)Φ−1(Φ(B)\\K)is contained in a proper hypersurface in B.\nWe denote the elements of the image of Φasnormal form equations . A normal form\nis called a polynomial normal form if the map Φis polynomial.\nExample 2.16. For the germ f=x4+y4of Arnold’s type X9, the µ-constant stratum\noffis covered by the normal form Φ :C→C[x, y],Φ(a) =x4+ax2y2+y4. For instance,\nthefunction g=x4+ϵx3y+y4, withafixedvalueof ϵ, liesinthesame µ-constantstratum\nasf. Hence, there is a C-algebra isomorphism ϕ1transforming gintox4+ax2y2+y4\nwith some ainC. As a result, there is also a C-algebra isomorphism ϕ2that maps gto\nx4−ax2y2+y4.\nDefinition 2.17. (Arnold et al. (1985)) Let f∈m2⊂C{x, y}and let kbe an upper\nbound for the determinacy of f. Themodality of a germ f∈m2⊂C{x, y}is the least\nnumber such that a sufficiently small neighborhood of jet(f, k)in the k-jet space can be\ncovered by a finite number of m-parameter families of orbits under the right-equivalence\naction.\nDefinition 2.18. (Arnold, 1974) Let f∈m2⊂C{x, y}be a germ with a non-degenerate\nNewton boundary. The inner modality is the number of monomials in a regular basis\nforQflying on or above Γ(f).\nRemark 2.19. In the subsequent section, we will recall that the inner modality of a\ngerm f∈C{x, y}is equal to the number of parameters in the normal form of the germ.\nMoreover it is shown in Böhm, Marais, and Pfister (2020), using results from Gabriélov\n(1974), that the inner modality and modality of a germ with a non-degenerate Newton\nboundary coincide.\n3.Classification of Singularities with Non-Degenerate Newton\nBoundary\nIn this section we recall the results from Böhm, Marais, and Pfister (2020), where it\nis, in particular, shown thatMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 6\n(1) the µ-constant stratum of a germ f∈C{x, y}with a non-degenerate Newton\nboundary can be covered, up to right-equivalence, by a single normal form.\n(2) there is a normalization condition for the Newton boundary of a germ with a\nnon-degenerate Newton boundary, that ensures the following: In a µ-constant\nstratum which contains a germ with non-degenerate Newton boundary, all germs\nwith normalized non-degenerate Newton boundary have the same Newton poly-\ngon. Hence, the Newton polygon can be considered as a name of the µ-constant\nstratum, replacing Arnolds notation of a type.\n(3) there is an effective algorithm to compute the normal form (satisfying the nor-\nmalization condition) for a given input germ f∈C{x, y}which is equivalent to\ngerm a non-degenerate Newton boundary.\nNote that, in this section, we only consider germs in the bivariante convergent power\nseries ring C{x, y}.\nThe following result from Greuel et al. (2007), Corollary 2.71, and Böhm, Marais, and\nPfister (2020), Theorem 3.9, gives a local description of the µ-constant stratum of a germ\nwith a non-degenerate Newton boundary.\nTheorem 3.1. Letf∈C{x, y}be a germ with a non-degenerate Newton boundary at\nthe origin. A miniversal, equisingular unfolding is given by\nF(x, y, t ) =f+mX\ni=1tigi,\nwhere mis the modality of f, and g1, . . . , g mrepresent a regular basis for Qfon and\nabove Γ(f).\nA first step to find a normal form for the entire µ-constant stratum of a germ with a\nnon-degenerate Newton boundary is to investigate how a regular basis of the germs in\nthe stratum change, while moving through the stratum.\nProposition 3.2. (Böhm, Marais, and Pfister (2020), Proposition 3.12) Let f0be a\ngerm with a non-degenerate Newton boundary Γ(f0)and let fbe a germ with the same\nNewton polygon as f0and non-degenerate Newton boundary. Then for fsufficiently\nclose to f0with respect to the Euclidean distance in the (µ+ 1)-jet space, the monomials\ninmon( x, y)representing a regular basis for f0with respect to the filtration defined by\nΓ(f0) = Γ( f)also represent a regular basis for fwith respect to the same filtration.\nNext, it is important to observe that all the germs in the µ-constant stratum of a germ\nwith a non-degenerate Newton boundary have the same topological type (see Böhm,\nMarais, and Pfister (2020), Remark 3.16). Since germs with a non-degenerate Newton\nboundary has the same topological type if and only if their characteristic exponents\nand intersection numbers coincide (see Brieskorn, Knörrer (1986),Theorem 15), and the\ncharacteristic exponents and intersection numbers of a germ in C{x, y}determines the\nnon-degenerate Newton boundaries of a germ that is equivalent to a germ with a non-\ndegenerate Newton boundary (see Böhm, Marais, and Pfister (2020), Proposition 4.17\nand Corollary 4.18), the next result follows:\nTheorem 3.3. (Böhm, Marais, and Pfister (2020), Theorem 3.18) Suppose f∈C{x, y}\nis a convenient germ with non-degenerate Newton boundary Γ. Then all the germs in the\nµ-constant stratum of fare equivalent to a germ with the same Newton polygon Γand\nnon-degenerate Newton boundary.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 7\nLetfbe a germ with a non-degenerate Newton boundary Γ. Taking the previous\nresult into account, the next result shows that there exists a set of monomials that is\na regular basis for at least one germ, a germ with a non-degenerate Newton boundary\nand a Newton polygon that coincide with that of Γ, in each right-equivalence class of the\ngerms in the µ-constant strantum of f.\nLemma 3.4. (Böhm, Marais, and Pfister (2020), Lemma 3.20) Let fbe a convenient\ngerm with non-degenerate Newton boundary. Define f0as the sum of the monomials of\nflying on the vertex points of Γ(f). Then any regular basis of f0is also a regular basis\nfor every germ with a non-degenerate Newton boundary in the µ-constant stratum of f.\nCorollary 3.5. (Böhm, Marais, and Pfister (2020), Corollary 3.21) Suppose fis a\nconvenient germ with non-degenerate Newton boundary. Define f0as the sum of the\nmonomials of flying on the vertex points of Γ(f), and f′\n0as the sum of the terms of f\non the vertex points of Γ(f). Then any regular basis for f0is also a regular basis for f′\n0\nand for f.\nBy the next theorem, every germ in the µ-constant with non-degenerate Newton\nboundary can be written in terms of its Newton boundary and a regular basis of the\ngerm.\nProposition3.6. (Boubakri et al. (2011), Corollary 4.6) Let f∈C{x, y}be a convenient\ngerm with a non-degenerate Newton boundary. Let f0be the principal part of fand let\n{e1, . . . , e n}be the set of all monomials in a regular basis for f0lying above Γ(f0). Then\nthere are αisuch that\nf∼f0+nX\ni=1αiei.\nUsing Theorem 3.1, Proposition 3.2, Theorem 3.3, Lemma 3.4 and Proposition 3.6 the\nfollowing theorem can be proved (see Böhm, Marais, and Pfister (2020), Theorem 3.22).\nTheorem 3.7. Suppose fis a convenient germ with non-degenerate Newton boundary.\nDefine f0as the sum of the monomials of flying on the vertex points of Γ(f), and let\n{e1, . . . , e n}be the set of all monomials in a regular basis for f0lying on or above Γ(f).\nThen the family\nf0+nX\ni=1αiei,\ndefines a normal form of the µ-constant stratum containing f. Restricting the parameters\nα1, . . . , α nto values such that every germ f0+Pn\ni=1αieihas a non-degenerate Newton\nboundary and the same Newton polygon as that of f, we obtain all germs in the µ-constant\nstratum of f.\nRemark 3.8. Using Theorem 3.7 a normal form can be constructed for any germ f∈\nC{c, y}with a non-degenerate Newton boundary. Note that the Newton polygon of\nffixes the Newton polygon of all the germs in the constructed normal form. Since the\nnormal form constructed in Theorem 3.7 is a normal form for the full µ-constant stratum,\nit follows that if f′is a germ with a non-degenerate Newton boundary and a different\nNewton polygon than f, then the normal form constructed for f′describes the same\nµ-constant stratum as that for f. In fact, by Theorem 3.3, fis equivalent to a germ\nwith a non-degenerate Newton boundary and same Newton polygon as f′. Hence, the\nnormal form for the µ-constant stratum of fas constructed using Theorem 3.7, dependsMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 8\non the choice of non-degenerate Newton boundary of germs in the equivalence class of\nfand the choice of regular basis for the chosen f0. In his lists of normal forms, Arnold\nassociate a type Tto each µ-constant stratum. He then fixes a Newton polygon and\na choice of moduli monomials (which boils down to the choice of regular basis for f0\nin Theorem 3.7). For distinguishing between different types, it is sufficient to know the\nNewton polygon of the normal form.\nTo achieve uniqueness of the Newton polygon associated to a fixed type (in order to\nlabel types by Newton polygons), a normalization condition on the Newton boundary\nof a germ with a non-degenerate Newton boundary is needed. Such a condition ensures\nthat the same Newton polygon for any germ in the µ-constant stratum is consistently\nchosen in order to construct a normal form by using Theorem 3.7.\nIt is important to distinguish between smooth and non-smooth facets:\nDefinition 3.9. A facet of the Newton polygon of a germ is called a smooth if the\nsaturation of its jet is smooth.\nDefinition 3.10. Suppose f∈m2⊂C{x, y}is a convenient germ with non-degenerate\nNewton boundary. Let ∆be a facet of Γ(f), and write w=w(∆). Then jet(f,∆)\nfactorizes in C[x, y]as\njet(f,∆) = xa·yb·g1···gn·eg,\nwhere a,bare integers, g1, . . . , g nare linear homogeneous polynomials not associated to x\nory, andegis a product of non-associatedirreducible non-linear homogeneous polynomials.\nWe say that fisnormalized with respect to the facet ∆, if\nw(x) =w(y) = ⇒a, b̸= 0\nw(x)> w(y)and a= 0 = ⇒ n= 0\nw(x)< w(y)and b= 0 = ⇒ n= 0\nA germ for which all the facets satisfy the above normalization condition is not nec-\nessarily convenient. We can transform such a germ to a convenient germ in the same\nright-equivalence class by adding the terms xdorydwith d=µ(f) + 2, if needed. This\nwill create non-normalized smooth facets that cut the coordinate axes. We address this\nin the following definition:\nDefinition 3.11. A germ f∈m2⊂C{x, y}satisfies the normalization condition\nif all of its facets, except smooth facets cutting the coordinate axes, are normalized, and\neach of its smooth facets that cut a coordinate axis, cut the axis in standard degree d,\nwhere d=µ(f) + 1.\nIt directly follows from Theorem 3.7 that two germs with the same normalized Newton\nboundary have the same normal form and hence lie in the same µ-constant stratum. The\nfollowing theorem states that the normalization condition is reasonable:\nTheorem 3.12. In a µ-constant stratum which contains a germ with a non-degenerate\nNewton boundary, every right-equivalence class contains a normalized germ, and all germs\nin the µ-constant stratum satisfying the normalization condition have the same Newton\npolygon (up to permutation of the variables).\nIn Böhm, Marais, and Pfister (2020), algorithms are given to determine a normal form\nfor the µ-constant stratum of a germ which is equivalent to a germ with a non-degenerate\nNewton boundary. This algorithm also detects if the given input germ is not equivalent\nto one with a non-degenerate Newton boundary.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 9\nAs a first step, this algorithm uses Algorithm 4 of Böhm, Marais, and Pfister (2020) to\ntransform an input germ f∈m2⊂C{x, y}to a germ which is right-equivalent and has\na normalized non-degenerate Newton boundary (this step will detect non-degeneracy).\nKnowing the normalized non-degenerate Newton boundary, Algorithm 6 of Böhm,\nMarais, and Pfister (2020) is used to find a regular basis for the sum of the vertex\nmonomials of the non-degenerate Newton polygon.\nFinally, Algorithm7appliesTheorem3.7toconstructanormalformforthe µ-constant\nstratum of the input germ f.\nBuilding on this construction and its implementation in Böhm, Marais, and Pfister\n(2024), the subsequent section will address finding the moduli parameters corresponding\nto the given input germ in the normal form.\n4.Determining the Values of the Moduli Parameters in the Normal\nFrom of a germ with a Non-Degenerate Newton Boundary\nAfter finding the normal form as discussed in Section 3, the values of the moduli\nparameters can, in theory, be computed via an Ansatz for the right equivalence mapping\nthegivengermunderconsiderationtoanelementofthenormalform(makinguseoffinite\ndeterminacy). However, this is not practicable except for very small examples. In this\nsection, we discuss an efficient algorithm for this problem. For brevity of presentation,\nwe introduce the following shorthand notation: if ∆is a set of facets of the Newton\npolygon of f, we write f∆= jet( f,∆)as introduced in Definition 2.12.\nNow, let f∈m2be a polynomial with a normalized non-degenerate Newton boundary,\nwith w(f) = (w1, . . . , w n)the induced weight on the Newton polygon. Let f0be the sum\nof the vertex monomials of Γ(f)and write f′\n0=w(f) - jet( f, d(f))for the sum of the\nterms of fonΓ(f).\nRecall from Lemma 3.4 and Corollary 3.5 that a regular basis Bforf0is also a regular\nbasis for f′\n0and for f. Assume that fhas a term above Γ(f)not in B, and let d′be the\nlowest w(f)-degree occurring among these terms. Let tbe a term of piecewise degree d′\ninf. Note that by the properties of a regular basis we can write\nt=g∂f\n∂x+h∂f\n∂y+terms of w(f)-degree d′inB+terms of w(f)-degree > d′, (1)\nwhere g, h∈C[x, y]. Define the right equivalence ϕ:C[[x, y]]→C[[x, y]]by\nϕ(x) =x−g, ϕ(y) =y−h, (2)\nwhere g, hare as in equation (1). Note that\nϕ(f) =f−\u0012∂f\n∂xg+∂f\n∂yh\u0013\n|{z }\nfirst order of the binomial expansion of ϕ(f)+1\n2\u0012∂2f\n∂2yh2+∂2f\n∂x∂ygh+∂2f\n∂2xg2\u0013\n+···\n| {z }\nhigher order of the binomial expansion of ϕ(f)(3)\nAgermwithanon-degenerateNewtonboundarydoesnotnecessarilysatisfyCondition\nA. Hence we cannot be certain that terms in the higher-order terms in the binomial\nexpansion of ϕ(f)are of w(f)-degree larger than d′. Thus the method introduced in\nBöhm, Marais, and Pfister (2020) for finding a normal form equation in general cannot\nbe applied. Algorithm 1 provides a method applicable to any germ with a non-degenerate\nNewton boundary.\nWe rely on the following lemma to formulate the algorithm. Fix a set Γ′of connected\nfacets of the Newton polygon of f.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 10\nLemma 4.1. Suppose that aandbare the maximal exponents such that the monomial\nxaybdivides both∂fΓ′\n∂xand∂fΓ′\n∂y, and define the exponents a′andb′byfΓ′=xa′yb′sat(fΓ′).\nIf0̸=g, h∈C[x, y]satisfy\ng·∂fΓ′\n∂x+h·∂fΓ′\n∂y= 0,\nthen\u0012g\nxs,h\nyt\u0013\nis a syzygy of\u0012\nxi·sat\u0012∂fΓ′\n∂x\u0013\n, yν·sat\u0012∂fΓ′\n∂y\u0013\u0013\n,\nwhere s, i, t, νare given by\ns=(\nmax{α|xαdivides∂fΓ′\n∂y}ifa= 0,\n1 ifa̸= 0,i=(\nmax{α|xαdivides∂fΓ′\n∂x}ifa= 0,\n0 ifa̸= 0,\nt=(\nmax{β|yβdivides∂fΓ′\n∂x}ifb= 0,\n1 ifb̸= 0,ν=(\nmax{β|yβdivides∂fΓ′\n∂y}ifb= 0,\n0 ifb̸= 0,\nMoreover, the vector\u0010\ng\nxs,h\nyt\u0011\nis a polynomial multiple of the Koszul syzygy of\n\u0012\nxi·sat\u0012∂fΓ′\n∂x\u0013\n, yν·sat\u0012∂fΓ′\n∂y\u0013\u0013\n.\nWe postpone the proof of the lemma to the end of this section. We only remark at\nthe current point that xisat\u0010\n∂fΓ′\n∂x\u0011\n,yνsat\u0010\n∂fΓ′\n∂y\u0011\nis a regular sequence, hence the vector\u0010\ng\nxs,h\nyt\u0011\nis a polynomial multiple of the mentioned Koszul syzygy.\nDefinition 4.2. Letf∈C[x, y]and suppose m=m1\nm2∈Quot(C[x, y])with monomial\nm1, m2∈C[x, y], that is, mis aLaurent monomial inx, y. Then fmis defined as the\nsum of all terms toffsuch that m·t∈C[x, y].\nUsing the next result, we will be able to show in the proof of Algorithm 1 that higher\norder terms t′of the binomial expansion of ϕ(f), where ϕis defined in (2), can be written\nas\nt′=g′∂f\n∂x+h′∂f\n∂y+terms of w(f)-degree d′inB+terms of w(f)-degree > d′,\nwhere g′, h′∈C[x, y]. Moreover we will show that the transformation ϕnew:C[x, y]→\nC[x, y]defined by ϕnew(x) =x−g′,ϕnew(y) =y−h′has a higher filtration than ϕ.\nTheorem 4.3. LetIbe the ideal generated by all the monomials of w-degree d′with\nd′≥d(f)fixed. If (a, b)is a syzygy of (∂f\n∂x,∂f\n∂y)overC[x, y]/I, then there exist Laurent\nmonomials zjinx, ysuch that\na−kX\nj=1zj∂f\n∂yzj\n∈Ann\u0012∂f\n∂x\u0013\nand b+kX\nj=1zj∂f\n∂xzj\n∈Ann\u0012∂f\n∂y\u0013\noverC[x, y]/I.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 11\nFurthermore if no yα,α >0, is a monomial of∂f\n∂yzj, then xdivides all terms of zj∂f\n∂yzj\nthat are not in Ann(∂f\n∂x)and do not get cancelled in the sumPk\nj=1zj∂f\n∂yzj. Similarly, if\nnoxβ,β >0, is a monomial of∂f\n∂xzj, then ydivides all terms of zj∂f\n∂xzjthat are not in\nAnn(∂f\n∂y)and do not get cancelled in the sumPk\nj=1zj∂f\n∂xzj.\nProof.Let(a, b)be a syzygy of\u0010\n∂f\n∂x,∂f\n∂y\u0011\ninC[x, y]/I, where Iis the ideal generated by\nall the monomials of w-degree d′. If the syzygy equation g1=a∂f\n∂x+b∂f\n∂y= 0has no\ncancellation below w-degree d′then we are finished. Now suppose g1has cancellation\nbelow degree d′. Suppose in particular g1has cancellation below w1-degree d′. Let\nl1= min\u0012\nw1- ord( a∂f\n∂x), w1- ord( b∂f\n∂y)\u0013\n.\nFor a face ∆of the Newton polygon, we write\nf∆,x=∂jet(f,∆)\n∂xand f∆,y=∂jet(f,∆)\n∂y.\nThen\nw1- jet(g1, l1) =m(1)\nxs1yν1sat (f∆1,y)| {z }\nw1- jet(a,l1−d(f))f∆1,x−yt1xi1sat (f∆1,x)| {z }\nw1- jet(b,l1−d(f))f∆1,y\n,\nwhere m(1)is a monomial, ∆1is the face with the smallest slope of Γ(f), and i1,ν1,s1\nandt1is as in Lemma 4.1.\nNow consider the syzygy\u0010\n∂f\n∂y,−∂f\n∂x\u0011\nof\u0010\n∂f\n∂x,∂f\n∂y\u0011\ninC[x, y], with syzygy equation\ng0=∂f\n∂y∂f\n∂x−∂f\n∂x∂f\n∂y= 0. Let l0=w1- ord(∂f\n∂y∂f\n∂x), then\nw1- jet(g0, l0) =mw1\u0000\nxs1yν1sat (f∆1,y)f∆1,x−yt1xi1sat (f∆1,x)f∆1,y\u0001\n,\nwhere mw1is the product of the maximal power of xand the maximal power of y\ndividing both f∆1,xandf∆1,y. Let n(1)=lcm(mw1, m(1)). Then the lowest nonzero\nw1-jet ofn(1)\nm(1)(a, b)andn(1)\nmw1\u0010\n∂f\n∂y,∂f\n∂x\u0011\ncoincide.\nThen\n(a(1), b(1)) = \nn(1)\nm(1)a−n(1)\nmw1∂f\n∂y,n(1)\nm(1)b+n(1)\nmw1∂f\n∂x!\nis a syzygy of\u0010\n∂f\n∂x,∂f\n∂y\u0011\ninC[x, y]/I(1), where I(1)=n(1)\nm(1)I.\nLetd′\n2=d′\u0010\nw1- deg\u0010\nn(1)\nm(1)\u0011\u0011\n. Now, if the equation g2=a(1)∂f\n∂x+b(1)∂f\n∂yhas any terms\nofw1-degree less than d′\n2, then the lowest non-zero w1-jet of (a(1), b(1))is\nm(2)\u0000\nxs2yν2sat (f∆1,y),−yt2xi2sat (f∆1,x)\u0001\n.\nLetn(2)=lcm\u0000\nm(2), mw1\u0001\n, thenthelowestnon-zero w1-jetofthesyzygiesn(2)\nm(2)\u0000\na(1), b(1)\u0001\nandn(2)\nmw1\u0010\n∂f\n∂y,∂f\n∂x\u0011\ncoincide. Similarly, as before, we can create the syzygy\n\u0010\na(2), b(2)\u0011\n= \nn(2)\nm(2)a(1)−n(2)\nmw1∂f\n∂y,n(2)\nm(2)b(1)+n(2)\nmw1∂f\n∂x!MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 12\ninC[x, y]/I(2), where I(2)=n(2)\nm(2)I(1).\nWe can go on with this process until the syzygy equation gk=a(k−1)∂f\n∂x−b(k−1)∂f\n∂y\nhas no terms of w1-degree less than w1-degree d′\nk=d′\n(k−1)\u0010\nw1- deg\u0010\nn(k−1)\nm(k−1)\u0011\u0011\n. We\nshow now that this will eventually happen. Now, n(j)= gcd( c, d, e, h )andm(j+1)be\nthe the monomial with the maximal x- and y-power dividing c−dande−h, where\nc=n(j+1)\nm(j+1)a(j),d=n(j+1)\nmw1∂f\n∂y,e=n(j+1)\nm(j+1)b(j)andh=n(j+1)\nmw1∂f\n∂x. Since the lowest non-zero\nw1-jet of candd, and eandh, cancel, respectively, in c−dande−h, it follows that\nw1- deg\u0000\nm(j+1)\u0001\n> w 1- deg\u0000\nn(j)\u0001\n. Now,\nw1- deg \nn(j+1)\nmw1!\n·w1- deg\u0012∂f\n∂y\u0013\n−d′\nj+1\n=w1- deg \nn(j+1)\nmw1!\n·w1- deg\u0012∂f\n∂y\u0013\n−w1- deg \nn(j+1)\nm(j+1)!\nd′\nj\n=w1- deg \nm(j+1)\nn(j+1)n(j+1)\nmw1!\n·w1- deg\u0012∂f\n∂y\u0013\n−d′\nj\n=w1- deg \nm(j+1)\nmw1!\n·w1- deg\u0012∂f\n∂y\u0013\n−d′\nj\n< w 1- deg \nn(j)\nmw1!\n·w1- deg\u0012∂f\n∂y\u0013\n−d′\nj.\nThis implies that the difference in the degree of the lowest non-zero w1-jet of (a(j), b(j))\nand the degree of the lowest order elements in d′\nkbecomes smaller. Hence eventually the\nw1-degree of (a(j), b(j))will be ≥d′\nj.\nSuppose now that (a(j−1), b(j−1))is such that gjhas no terms below w1-degree d′\nj−1.\nWe now consider the w2-degree of (a(j−1), b(j−1)). We follow the same strategy. Suppose\ngjhas terms of w2-degree less than d(2)\nj−1, where\nd(r)\nk=wr- deg \nn(1)n(2)···n(k)\nm(1)m(2)···m(k)!\nd′,\nthen let\nlj= min\u0012\nw2- ord( a(j−1)∂f\n∂x), w2- ord( b(j−1)∂f\n∂y)\u0013\n.\nThen\nw2- jet(gj, lj) =m(j)\u0000\nxsjyνjsat (f∆2,y)f∆2,x−ytjxijsat (f∆2,x)f∆2,y\u0001\n,\nwhere m(j)is a monomial, ∆2is the face with the second smallest slope of Γ(f), and sj,\nνj,ijandtjis as in Lemma 4.1. Furthermore\nw2- jet(g0, l0) =mw2\u0000\nxsjyνjsat (f∆2,y)f∆2,x−ytjxijsat (f∆2,x)f∆2,y\u0001\n,\nwhere mw2= gcd ( f∆2,x, f∆2,y). With\nn(j):=lcm(mw2, m(j))MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 13\nthe lowest nonzero w2-jets of\nn(j)\nm(j)\u0010\na(j−1), b(j−1)\u0011\nandn(j)\nmw2\u0012∂f\n∂y,∂f\n∂x\u0013\ncoincide. Similarly, as before, we can create the syzygy\n\u0010\na(j), b(j)\u0011\n= \nn(j)\nm(j)a(j−1)−n(j)\nmwj∂f\n∂y,n(j)\nm(j)b(j−1)+n(j)\nmwj∂f\n∂x!\ninC[x, y]/I(j), where\nI(j)=n(j)\nm(j)I(j−1).\nNote that the syzygy equation gjhas no terms below w2-degree\nd′\nj:=w2- deg \nn(j)\nm(j)!\nd(2)\nj−1\nandw1-degree\nd′′\nj:= \nw1- deg(n(j)\nm(j))!\n·d′\nj−1.\nIf this would not be the case, this would imply that\n−n(j)\nmw2∂f\n∂y∂f\n∂x+n(j)\nmw2∂f\n∂x∂f\n∂y\nhas terms below w1-degree d′′\nj.\nLetl′\n(x,w1)andl′\n(y,w1)be the lowest non-zero w1-orders of∂f\n∂xand∂f\n∂y, respectively. Let\nl′\n(x,w2)andl′\n(y,w2)be the lowest w2-orders of∂f\n∂xand∂f\n∂y, respectively. We observe that\nw1- jet\u0012∂f\n∂x, l′\n(x,w1)\u0013\nandw2- jet\u0012∂f\n∂x, l′\n(x,w2)\u0013\nhave coinciding terms at a vertex monomial. The same holds true for\nw1- jet\u0012∂f\n∂y, l′\n(y,w1)\u0013\nandw2- jet\u0012∂f\n∂y, l′\n(y,w2)\u0013\n.\nAll the terms of\nw1- jet\u0012∂f\n∂x, l′\n(x,w1)\u0013\nhave the same w1-degree. The same holds true for\nw1- jet\u0012∂f\n∂y, l′\n(y,w1)\u0013\n.\nSimilarly, all terms of\nw2- jet\u0012∂f\n∂x, l′\n(x,w2)\u0013\nhave the same w2-degree, and the same holds true for the terms of w2- jet\u0010\n∂f\n∂y, l′\n(y,w2)\u0011\n.\nHence, we conclude that\nw2- jet \n−n(j)\nmw2∂f\n∂y∂f\n∂x+n(j)\nmw2∂f\n∂x∂f\n∂y, lj!MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 14\nalso has terms below w1-degree d′′\nj, which means that\nw2- jet \nn(j)\nm(j)a(j−1)∂f\n∂x+n(j)\nm(j)b(j−1)∂f\n∂y, lj!\nhas terms below w1-degree d′′\nj. This again implies that w2- jet(a(j−1)∂f\n∂x−b(j−1)∂f\n∂y, lj−1)\nhas terms below w1-degree d′\nj−1. We conclude that a(j−1)∂f\n∂x−b(j−1)∂f\n∂yhas terms below\nw1-degree d′\nj−1, a contradiction.\nContinuing as above, we can construct a syzygy (a(k), b(k))of(∂f\n∂x,∂f\n∂y)inC[x, y]/I(k),\nwhere the syzygy equation gkhas no terms below wi-degree d(i)\nk, fori= 1,2, . . . , n. Let\nr0=n(1)...n(k)\nm(1)···m(k)andrj=n(j)...n(k)\nm(j+1)···m(k)mwj, for j={1, . . . , k −1}, and rk=n(k)\nmwik. By\nconstruction\nwi- jet\u0012\nr0a, d(i)\nk−(wi- ord(∂f\n∂x))\u0013\n=wi- jet\nkX\nj=1rj∂f\n∂y, d(i)\nk−(wi- ord(∂f\n∂x))\n.(4)\nSince r0divides all the terms on the right side of the equation, it follows that ifrj\nr0t̸∈\nC[x, y], for some term tof∂f\n∂y, then teither gets cancelled on the righthand side of the\nequation or rjtis ofwi-degree higher than d(i)\nk−(wi- ord(∂f\n∂x))for all i, that is rjt∂f\n∂xis\nofwi-order higher than d′for all i. Hence rjt∂f\n∂xis contained in I(k). Therefore\nwi- jet\u0012\nr0a, d(i)\nk−(wi- ord(∂f\n∂x))\u0013\n=wi- jet\nkX\nj=1rj∂f\n∂yzj\n, d(i)\nk−(wi- ord(∂f\n∂x))\n,(5)\nwhere\nzj=m(1)···m(j)\nn(1)···n(j−1)mwj.\nTherefore\na−kX\nj=1zj∂f\n∂yzj\n∈Ann(∂f\n∂y)and b+kX\nj=1zj∂f\n∂xzj\n∈Ann(∂f\n∂y)\noverC[x, y]/I.\nWe will now show that if yαis not a term of∂f\n∂yzj, then xdivides all terms of zj∂f\n∂yzj\nthat are not in Ann(∂f\n∂x)and do not get cancelled in the sumPk\nj=1zj∂f\n∂yzj.\nSo suppose that∂f\n∂yzj, has only mixed terms. Let cxαyβ,c∈Cbe such a term of∂f\n∂yzj\nsuch that zjcxαyβ̸∈Ann(∂f\n∂x)andzjcxαyβis not cancelled in the sum\nkX\nj=1zj∂f\n∂yzj\n.\nClearly xα−1yβ+1is a monomial of∂f\n∂x, and there exists a monomial lxηyγ,l∈Cof∂f\n∂x\nsuch that\nzjcxαyβxηyγ̸∈I.\nBut then lxη+1yγ−1is a monomial of∂f\n∂y. Furthermore, notice that rjcα\nβ+1xα−1yβ+1\nis cancelled inPk\nj=1rj∂f\n∂xif and only if rjcxαyβis cancelled inPk\nj=1rj∂f\n∂y. Since, inMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 15\naddition, zjxα−1yβ+1xη+1yγ−1̸∈I, it follows that xα−1yβ+1is a monomial of∂f\n∂xzj. On\nthe other hand, since fis convenient, it follows that yα,α > 0, is a monomial of∂f\n∂y,\nfor some α >0. Furthermore, since fhas a non-degenerate Newton boundary, it follows\nthat if yβis a monomial of∂f\n∂y,β < α. Hence, since yαis not a monomial of∂f\n∂xzj,\nzjyα̸∈C[x, y]. This implies that zjyβ̸∈C[x, y]. Therefore yβ, for any β >0is not a\nmonomial of∂f\n∂xzj. This again implies that α >1. Suppose zj=aj\nbj, then bj|xαyβand\nbj|xα−1yβ. Hence x|zjxαyβ. In a similar way, we can prove that if xβ̸∈∂f\n∂xzj, then\ny|zj∂f\n∂xzj,β >0.\nAnalogously we can show that, if xβis not a term of∂f\n∂xzjfor all j, then ydivides all\nterms of zj∂f\n∂xzjthat are not in Ann(∂f\n∂y)and do not get cancelled. □\nIn line 20 of Algorithm 1, we rely on Lemma 4.4 below. Since its statement is quite\nobvious, we postpone the proof of the lemma to the end of the chapter.\nLemma 4.4. LetΓbe the Newton polygon of a germ with normalized, non-degenerate\nNewton boundary. Let f0be the sum of the monomials corresponding to the vertices of\nΓ, and let Bbe a regular basis for f0. Then\n(1) any monomial corresponding to a lattice point of Γis a monomial of f0or an\nelement of B, and\n(2) at most two monomials of f0of degree ≤dt(f0)are not in B.\nThe above lemma shows that for a germ with a normalized non-degenerate Newton\nboundary every monomial corresponding to a lattice point of its Newton polygon is a\nmonomial occurring in any normal form of the germ with the same Newton polygon.\nFurthermore it shows that only two vertex monomials under the determinacy and on\nthe Newton boundary are not parameter monomials in any such normal form. This\nimplies that the Newton boundary of the given germ can be transformed to the Newton\nboundary of any of its normal form equations by scaling xandy, except for terms above\nthe determinacy.\nWe now prove correctness and termination of Algorithm 1.\nProof of Algorithm 1 . In the transformation ϕin line 8 we know that terms coming from\nfirst partial derivatives with degree ≤d′, not in B, cancel, as described in (3). We\nnow discuss the effect of higher order terms in the binomial expansion after applying ϕ.\nNow, it follows from Theorem 4.3 that ϕ(x) =x+P\nizi∂f\n∂yziandϕ(y) =y+P\nizi∂f\n∂xzi.\nNote that, since applying any transformation of filtration <2tofwill change its non-\ndegenerate Newton boundary to a degenerate boundary, ϕhas filtration ≥2. That is\nord\u0010\nzi∂f\n∂yzi\u0011\n≥2andord\u0010\nzi∂f\n∂xzi\u0011\n≥2. We systematically consider the terms coming\nfrom the higher order binomial expansions of ϕ(f). In higher order binomial expansions\nthe terms are of the following form:\n∂nf\n∂xs∂yn−s· \nzi1∂f\n∂yzi1!\n··· \nzis∂f\n∂yzis!\n· \nzis+1∂f\n∂xzis+1!\n··· \nzin∂f\n∂xzin!\n,(6)MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 16\nAlgorithm 1 Determining the moduli parameters in the normal Form of a germ with a\nnormalized non-degenerate Newton boundary\nInput:A polynomial germ f∈Q[x, y],f∈m3of corank 2with a normalized non-\ndegenerate Newton boundary; f0, the sum of the vertex monomials of Γ(f); a set of\nmonomials B={b1, . . . , b m}that is the set of all monomials of w(f)-degree ≥d(f)\nin a regular basis for f0\nOutput: A normal form of fand a normal form equation equivalent to fsuch that the\nnormal form equation is a member of the normal form.\n1:Letd′′be a determinacy bound for f.\n2:Letw:= (w1, . . . , w n) :=w(f).\n3:F:=f0+Pm\ni=1αi·bi.\n4:S:= mon( f−w(f) - jet( f, d(f))).\n5:while S̸⊂Bdo\n6:Letd′be the lowest w(f)-degree above d(f)with non-zero terms in f.\n7:Write the sum qof the terms of w(f)-degree d′as\nq=g∂f\n∂x+h∂f\n∂y+terms in Bofw(f)-degree d′+terms of higher w(f)-degree than d′.\n8:Define ϕ:C[x, y]→C[x, y],ϕ(x) =x+g,ϕ(y) =y+h.\n9:Letlbe the filtration of ϕ.\n10: q:=w(j) - jet( ϕ(f)−(f+g∂f\n∂x+h∂f\n∂y), d′).\n11: f:= jet( ϕ(f), d′′).\n12:while q̸= 0do\n13: Write\nq=ϕx·∂f\n∂y+ϕy·∂f\n∂x,where ϕx, ϕy∈ ⟨x, y⟩l+1.\n14: ϕx=w- jet(ϕx, d′), ϕ y=w- jet(ϕy, d′)\n15: Define ϕ:C[x, y]→C[x, y]by\nϕ(x) :=x−w- jet(ϕx, d′), ϕ(y) :=y−w- jet(ϕy, d′).\n16: q:=ϕ(f)−(f+q).\n17: f:= jet( ϕ(f), d′′).\n18: l= min(ord( ϕx),ord(ϕy)).\n19: S:= mon( f−w(f) - jet( f, d(f))).\n20:Normalize two terms of fofw(f)-degree d(f), of degree ≤d′′, and not in Bto\ncoefficient 1.\n21:ifΓ(f)does not intersect the x-axisthen f:=f+xawith a=µ(f) + 2.\n22:ifΓ(f)does not intersect the y-axisthen f:=f+yawith a:=µ(f) + 2.\n23:return F,f\nwhere n > 1,ij’s∈Z, as well as terms in J, where Jis the ideal generated by all\nmonomials of degree d′+ 1. Let zij=aij\nbij. We distinguish between the following types of\nbij’s:\n(i)bij=cxαyβ,α, β > 0;MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 17\n(ii)bij=cyβ,β >0;\n(iii)bij=cxα,α >0;\n(iv)bij= 1.\nWe consider the following cases:\n(1)bij= 1for some j≤s;\n(2)bij= 1for some j > s;\n(3) all the bij’s are of type (i), (ii) and (iii), and the number of bij’s of type (i) and\n(iii) is < s.\n(4) all the bij’s are of type (i), (ii) and (iii), the number of bij’s of type (ii) < n−s,\nand the number of bij’s of type (iii) is < s.\n(5) all the bij’s are of type (i), (ii) and (iii), and the number of bij’s of type (i) and\n(ii) is < n−s.\nNote that if zk=ak\nbkandbk= 1, then\nak\nbk\u0010\nxs·yn−(s+1) ∂nf\n∂xs∂yn−szk\u0011\nxs·yn−(s+1)=ak\u0010\nxs·yn−(s+1) ∂nf\n∂xs∂yn−s\u0011\nxs·yn−(s+1)=zk \n∂nf\n∂xs∂yn−szk!\n.(7)\nSimilarly\nak\nbk\u0010\nxs−1·yn−s∂nf\n∂xs∂yn−szk\u0011\nxs−1·yn−s=ak\u0010\nxs−1·yn−s∂nf\n∂xs∂yn−s\u0011\nxs−1·yn−s=zk \n∂nf\n∂xs∂yn−szk!\n.\nFurthermore, note that the monomials of\u0010\nxs−1·yn−s∂nf\n∂xs∂yn−szk\u0011\n∈C[x, y]is a subset\nof the monomials of∂f\n∂xzk. Noticing that for all terms tin∂f\n∂y−∂f\n∂yzk, it follows from the\nproof of Theorem 4.3 that ak·t∈Ann R/J′(∂f\n∂x), where J′is the ideal generated by all\nthe terms of w-degree (d′+w- deg( bk) + 1), and zk=ak\nbk, it follows that\n\u0012\nxs−1·yn−s∂nf\n∂xs∂yn−s\u0013\n·zk∂f\n∂yzk\n=zk \nxs−1·yn−s∂nf\n∂xs∂yn−szk!\n·∂f\n∂y+J.(8)\nIn case (1), it hence follows from (7) and (8) that\n∂nf\n∂xs∂yn−s·sX\nk=1 \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n=1\nxs−1·yn−s·\u0012\nxs−1·yn−s∂nf\n∂xs∂yn−s\u0013sX\nk=1 \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n= \nzij∂nf\n∂xs∂yn−szij!\n·X\nk=1...s, k̸=j \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n| {z }\n∈⟨x,y⟩l+1⊂C[x,y]·∂f\n∂y.\nNote that\u0010\nzik∂f\n∂xzik\u0011\n,\u0010\nzik∂f\n∂yzik\u0011\n∈ ⟨x, y⟩l⊂C[x, y], for all k=s+1, . . . , n, where l≥2.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 18\nIn case (2) it similarly follows that\n∂nf\n∂xs∂yn−s·sX\nk=1 \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n= \nzij∂nf\n∂xs∂yn−szij!\n·sX\nk=1 \nzik∂f\n∂yzik!\n·X\nk=s+1,...n, k ̸=j \nzik∂f\n∂xzik!\n| {z }\n∈⟨x,y⟩l+1⊂C[x,y]·∂f\n∂x.\nNext we consider case (3). Note that\n\u0012\nxs−1yn−s∂nf\n∂xs∂yn−s\u0013\n·zk∂f\n∂yzk\n·zl∂f\n∂xzl\n=zk \nxs−1yn−s∂nf\n∂xs∂yn−szk!\n·∂f\n∂y·zl∂f\n∂xzl\n=zk \nxs−1yn−s∂nf\n∂xs∂yn−szk!\n·zl∂f\n∂yzl\n·∂f\n∂x\nUsing the above method, it follows that\n∂nf\n∂xs∂yn−s·sX\nk=1 \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n=zij\nxs−1yn−s \nxs−1yn−s∂nf\n∂xs∂yn−szij!X\nk=1,...,s, k ̸=j \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n∂f\n∂y,\nwhere all the bij’s are ordered that first, bij’s of type (i) and (iii) occur and then those of\ntype (ii). In other words the last n−s+ 1b′\nijs are of type (ii). This means by Theorem\n4.3 that\n∂nf\n∂xs∂yn−s·sX\nk=1 \nzik∂f\n∂yzik!\n·nX\nk=s+1 \nzik∂f\n∂xzik!\n=1\nxs−1·zis \nxs−1·yn−s∂nf\n∂xs∂yn−szis!X\nk=1,...,s−1 \nzik∂f\n∂yzik!\n·nX\nk=s+1\u0010\nzik∂f\n∂xzik\u0011\ny|{z}\n∈C[x,y]∂f\n∂y,\nSince the number of b′\nijsthat are of type (ii) is > n−s,bisis of type (ii). But then,\narguing similar as in (7), xs−1|zis\u0010\nxs−1·yn−s∂nf\n∂xs∂yn−szis\u0011\n. Hence (6) can be expressed\nas\nq′zk∂f\n∂xzk\ny|{z}\n∈⟨x,y⟩l+1∂f\n∂y,with q′∈ ⟨x, y⟩2, (9)\nandbkis of type (ii). In case (4) it follows similarly that (6) can be expressed as in (9),\nwhere bkis of type (i) or (ii).\nIn case (5), exchanging xandyin (9), we express (6) as\nq′zk∂f\n∂yzk\nx|{z}\n∈⟨x,y⟩l+1∂f\n∂x,with q′∈ ⟨x, y⟩2,MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 19\nwhere bkis of type (iii).\nWe conclude that qcan be expressed as\nq=ϕx·∂f\n∂x+ϕy∂f\n∂y, (10)\nwhere ϕx, ϕy∈ ⟨x, y⟩l+1. Then the higher orders of the binomial expansion of ϕ(f)of\ndegree ≤d′can be removed by the first order terms of the transformation ϕ:C[x, y]→\nC[x, y]defined by\nϕnew(x) :=x−ϕx, ϕnew(y) :=y−ϕy.\nIn an analogous way one can show that the sum, qnew, of higher order terms of the\nbinomial expansion of degree ≤d′ofϕnew(ϕ(f))can be written in terms of the same\nformulas, now in terms of ϕnew(x) =x−P\niz′\ni∂f\n∂yz′\niandϕnew(y) =y−P\niz′\ni∂f\n∂xz′\ni, as\nabove. Note that the filtration of ϕnewis higher than that of ϕ. Hence eventually there\nwill be no terms, not in B, ofw-degree ≤d′. □\nFor many examples the transformations arising from line 7 can be chosen in such a\nway that the higher orders of the binomial expansion of ϕ(f)are of degree larger than\nd′. Choosing gandhin such a way, the while-loop from line 12 to 18 is redundant. The\nnext example proves that this is unfortunately not in general the case.\nExample 4.5. Letf=y28+xy7+x2y3+ 11x2y4+x22. Then∂f\n∂x= 2xy3+ 22xy4+\ny7+ 22x21and∂f\n∂y= 3x2y2+ 44x2y3+ 7xy6+ 28y27, and a regular basis for fis\nx22y, x21y, x20y, x19y, x18y, x17y, x16y, x2y3. Note that x2y4is the only monomial above\nthe Newton Boundary, with w(f)-degree 1008. We can express −11x2y4as follows in\nterms of the first partial derivatives.\n−11x2y4=−(7xy+ 28y22)∂f\n∂x+y2∂f\n∂y.\nThe corresponding transformation ϕis given by\nϕ(x) = x−(7xy+ 28y22) =x−X\nizi∂f\n∂y=x−1\ny5∂f\n∂y1\ny5\n+px\nϕ(y) = y+y2=y−X\nizi∂f\n∂x=y−(−1\ny5∂f\n∂x1\ny5\n) +py,\nwhere px∂f\n∂xandpy∂f\n∂yare of w(f)-degree 1008or higher. Note that the filtration of ϕis\nl= 2. Now let\nq=w- jet(ϕ(f)−f−ϕ(x)∂f\n∂x−ϕ(y)∂f\n∂y,1008) .\nThen qis the contribution of the orders >1in the binomial expansion of ϕin the 1008-jet\nofϕ(f).\nq=−28xy9+ 182 y30\nTo write qas in line 13 in Algorithm 1, we are computing ϕxandϕy. We do this by\nforming an ideal Igenerated by the set\n\u001a\nxiyj∂f\n∂x, xiyj∂f\n∂y\f\f\f\fi+j=l+ 1 = 3\u001b\n,\nand all monomials of higher piecewise degree than 1008. We then write qin terms of\n∂f\n∂x,∂f\n∂y, where the coefficients are of order 3or higher, using a Gröbner basis of IwithMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 20\na global ordering, and then lifting the result back to the original generators using the\ncommand liftstd() inSingular . It turns out that\nq=\u0000\n(−28−364y17−4004y18)xy2+ 182 y23\u0001\n·∂f\n∂x\nBy abuse of notation, let f=ϕ(f). We now form a new ϕas described in line 15 of\nAlgorithm 1. We set\nϕx= 28xy2−182y23andϕy= 0.\nWe define ϕ:C[x, y]→C[x, y]byϕ(x) =x−ϕxandϕ(y) =y−ϕy=y.Hence the first\norder terms of the binomial expansion of ϕ(f)up to degree 1008cancel q. In this case\nhigher order terms of the binomial expansion of ϕ(f)does not have terms of w(f)-degree\n1008 or lower. In fact w- jet(ϕ(f),1008) = y28+xy7+x2y3+x22.\nIn the next example more than one iteration of the while-loop in Algorithm 1 is needed\nto transform the germ fto its normalform equation up to w(f)-degree 700.\nExample 4.6. Forf=x2y4+x4y2+x20+y40+60x21y14, we have∂f\n∂x= 4x3y2+2xy4+\n20x19+ 1260 x20y14and∂f\n∂y= 2x4y+ 4x2y3+ 840 x21y13+ 40y39, and a regular basis is\nx4y4,x4y3,x3y4,x4y2,x3y3,x2y4. Note that x21y14is the only monomial above the\nNewton boundary with w(f)-degree 700. Now 60x21y14can be written as\n60x21y14= (2 x4y12+ 4x2y14+ 40y50−10x20y10)∂f\n∂x−(4x3y13+ 2xy15)∂f\n∂y.\nThe corresponding transformation ϕis given by\nϕ(x) = x−(2x4y12+ 4x2y14+ 40y50−10x20y10) =x−y11∂f\n∂yy11\n−5x16y9∂f\n∂yx16y9\n+px\nϕ(y) = y+ (4x3y13+ 2xy15) =y−(−y11∂f\n∂xy11\n−5x16y9∂f\n∂xx16y9\n) +py,\nwhere px∂f\n∂xandpy∂f\n∂yis ofw-degree 700or higher. Note that the filtration of ϕisl= 16.\nNow let\nq=w- jet(ϕ(f)−f−ϕ(x)∂f\n∂x−ϕ(y)∂f\n∂y,700).\nThen qis the contribution of the orders >1in the binomial expansion of ϕin the 700-jet\nofϕ(f).\nq=−24x10y26−12x8y28−12x6y30−24x4y32−224x7y44−32x5y46+ 144 x6y60\n+12160 x6y64+ 12640 x4y66+ 2800 x2y68+ 384 x7y74+ 952320 x7y78\n+472320 x5y80+ 79680 x3y82+ 256 x8y88+ 11704320 x6y94+ 1467360 x4y96\n+9600 x2y102+ 1600 y104+ 21065216 x5y110+ 38400 x5y114−12800 x3y116\n+12800 xy118+ 245661440 x6y124+ 38400 x4y130+ 38400 x2y132+ 51200 x3y146\n−256000 xy152+ 25600 x4y160+ 2560000 y202.\nNote that the order of qis36.\nTo write qas in line (13) in Algorithm 1, we are computing ϕxandϕy. We do this by\nforming an ideal Igenerated by\n\u001a\nxiyj∂f\n∂x, xiyj∂f\n∂y\f\f\f\fi+j=l+ 1 = 17\u001b\n,MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 21\nand all monomials of higher piecewise degree than 700. We write qin terms of∂f\n∂xand\n∂f\n∂y, by using a Gröbner basis of Iwith a global ordering, and then lifting the result back\nto the original generators using the command liftstd() . We now form ϕas described\nin line 15 of Algorithm 1. Let ϕxandϕybe as defined in line(14) in Algorithm 1. It\nturns out that\nϕx= 256 x21y10+ 512 x19y12+ 128 x17y14+ 448 x15y16+ 288 x13y18−144x11y20\n+72x9y22−42x7y24+ 18x5y26−12x3y28+12650\n567x34y10−320x10y36\n+160 x8y38−80x6y40−16x4y42+ 288 x9y52−144x7y54+ 72x5y56−4800x9y56\n+2400 x7y58−1200x5y60+ 3640 x3y62+ 1320 xy64−384x8y68+ 192 x6y70\n−326400 x8y72+ 163200 x6y74+ 156480 x4y76+ 39840 x2y78+ 128 x7y84\n−8769600 x7y88+ 4384800 x5y90+ 733680 x3y92−38400 x7y92+ 19200 x5y94\n−9600x3y96+ 4800 xy98−21065216 x6y104+ 10532608 x4y106−115200 x6y108\n+57600 x4y110−19200 x2y112+ 6400 y114+ 122830720 x5y120+ 38400 x5y124\n−19200 x3y126+ 19200 xy128−51200 x4y140+ 25600 x2y142−512000 x4y144\n+256000 x2y146−128000 y148+ 12800 x3y156+ 192000 x3y160−128000 xy162\nand\nϕy=−512x20y11−256x18y13−256x16y15−512x14y17−2560x36y9+ 40y65+ 64000 y163.\nNote that order of ϕxandϕy, and hence the filtration of ϕ, is31, respectively. Unfortu-\nnately the higher order binomial expansions of ϕ(f)again contributes terms of w-degree\n≤700. Note that this time the order of these terms is 66>36. Now let\nq=w- jet(ϕ(f)−f−ϕ(x)∂f\n∂x−ϕ(y)∂f\n∂y,700).\nThen\nq= 1296 x8y58−1008x6y60−3072x7y74−7488x8y88+ 736800 x6y94+ 189120 x4y96\n+6049280 x5y110+ 136922880 x6y124−98264000 x4y130−12833600 x2y132\n−681446400 x3y146−23950380800 x4y160−77184000 x2y166−5408000 y168\n−94208000 xy182+ 33001344000 x2y196+ 13038080000 y232.\nLetf=ϕ(f). Repeating the process, we again write qin terms of\n\u001a\nxiyj∂f\n∂x, xiyj∂f\n∂y\f\f\f\fi+j=l+ 1 = 32\u001b\n,\nand all terms of piecewise degree higher than 700to compute ϕxandϕy. It turns out\nthat\nϕx=−3312x9y52+ 1656 x7y54−504x5y56+ 3072 x8y68−1536x6y70−3744x7y84\n−358560 x7y88+ 179280 x5y90+ 94560 x3y92−6049280 x6y104+ 3024640 x4y106\n+68461440 x5y120+ 73408000 x5y124−36704000 x3y126−6146400 xy128\n+681446400 x4y140−340723200 x2y142−11975190400 x3y156+ 77184000 x3y160\n−38592000 xy162+ 94208000 x2y176−47104000 y178+ 15848768000 xy192\nand\nϕy=−135200 y129+ 325952000 y193.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 22\nApplying ϕ, formed as in line (15),\nw- jet(ϕ(f),700) = x4y2+x2y4+x20+y40.\nWe now turn to the proof of our two main lemmata, and begin with Lemma 4.1,\nProof.We say that terms of fx:=∂fΓ′\n∂xandfy:=∂fΓ′\n∂yare in correspondence if they\noriginate from the same term of fΓ′.\n•Ifa̸= 0, then xais the largest power of xdividing fx, and xa+1is the largest power of\nxdividing fy. To see this, we first note that, excluding a pure x-power term in fx, all\nterms of fxandfyare in one-to one correspondence. If fxwould be divisible by xa+1\nthen, since all terms of fyare in correspondence to a term in fx, all terms of fywould\nbe divisible by xa+2, contradicting the minimal choice of a. Thus xais the largest\npower of xdividing fx, and fyis divisible by xa+1. Ifxa+2would divide fy, then xa+1\nwould also divide every corresponding term of fx. Since also a possible xα-term of fx\nis then divisible by xa+2(since it is the term with the largest x-exponent), this again\ncontradicts the minimal choice of x. Moreover the argument above implies that xa+1\nis the highest power of xdividing f.\n•Ifa= 0, then x∤fyorx∤fx.\n–Ifx∤fythen fΓ′has a term that is a pure y-power. Hence x∤f.\n–Ifx|fy, then x∤fx, which implies that fΓ′has a term of the form xylwith l≥1.\nHence, in this case, if xsdivides fy(and hence f),s= 1.\nNow, let xa′yb′be the product of the highest power of xand the highest power of y\ndividing jet(f,Γ′).\n•Then a′=a+ 1, ifa̸= 0.\n•Ifa= 0, we have two cases:\n–Ifx∤fy, then a′= 0.\n–Ifx|fy, then 1is the highest power of xdividing fΓ′. Hence a′=a+ 1.\nSimilarly b′=b+ 1, ifb̸= 0. Ifb= 0andy∤fx, then b′= 0. Ifb= 0,y|fx(implying\nthaty∤fy, then 1is the highest power of ydividing fΓ′. Hence b′=b+ 1.\nWe assume that\ng·fx+h·fy= 0.\nThen\nxayb(g·yt·xi·sat(fx) +h·xs·yν·sat(fy)) = 0 ,\nwhere s,t,iandνare defined as above. To see this, consider first the case a̸= 0,b̸= 0,\nthen\nxayb(g·y·sat(fx) +h·x·sat(fy)) =xa′yb′\u0012g\nx·sat(fx) +h\ny·sat(fy)\u0013\n= 0,\nwhich can be verified by the arguments above. The other cases can be similarly verified.\nIfa̸= 0, or if a= 0, and x|fy, then s= 1anda′=a+ 1. In both these cases i= 0. If\na= 0andx∤fy, then a′=a= 0ands= 0. Therfore we have in general.\nxa′yb(g·yt·xi·sat(fx) +h·xs·yν·sat(fy)) = 0 .\nSimilarly, if b̸= 0, or if b= 0, and y|fx, then ν= 1andb′=b+ 1. In both these cases\nt= 0. Ifb= 0andy∤fx, then b′=b= 0andt= 0. Therfore we have in general.\nxa′yb′(g\nxs·xi·sat(fx) +h\nyt·yν·sat(fy)) = 0 .MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 23\nTherefore\ng\nxs·xi·sat(fx) +h\nyt·yν·sat(fy) = 0 .\nHence x|g, ifa̸= 0, or if a= 0andx|fy, and y|h, ifb̸= 0, or if b= 0andy|fx.\nWe now show that xi·sat(fx)andyν·sat(fy)have no common monomial factor. To\ndo that, first note that sat(fΓ′)is nondegenerate by definition. This implies that sat(fΓ′)\ndoes not have a multiple monomial factor, hence sat(fΓ′)and∂satfΓ′\n∂xhave no common\nmonomial factor, and sat(fΓ′)and∂satfΓ′\n∂yhave no common monomial factor. Thus,\nxa′yb′accounts for the only multiple factors in fΓ′.\nConsider the following equations for the partial derivatives of fΓ′:\nfx=a′xa′−1yb′(sat(fΓ′)) +xa′yb′(∂satfΓ′\n∂x)\nfy=b′xa′yb′−1(sat(fΓ′)) +xa′yb′(∂satfΓ′\n∂y).\nIf the saturations of the partial derivatives of fΓ′with respect to xandyshare a com-\nmon factor, then this factor would also be a factor of sat(fΓ′). However, this contradicts\nthe previous equations. Furthermore y∤sat(fx)ifν >0andx∤sat(fy)ifi >0.\nSince xisat\u0010\n∂fΓ′\n∂x\u0011\n,yνsat\u0010\n∂fΓ′\n∂y\u0011\nis a regular sequence, the vector\u0010\ng\nxs,h\nyt\u0011\nis a poly-\nnomial multiple of the Koszul syzygy of (xi·sat(∂fΓ′\n∂x), yν·sat(∂fΓ′\n∂y)). This completes the\nproof.\n□\nWe turn now to the proof of Lemma 4.4, where we use the following observation:\nRemark 4.7. With the notation of Lemma 4.4, let mbe a monomial corresponding to a\nlattice point of Γ. Suppose gis a germ with non-degenerate Newton boundary Γ(g) = Γ\nand with monomial m, such that the corresponding term cannot be removed from gby\na right equivalence. Then Theorem 3.20 implies that mis a monomial of f0orm∈B.\nWe now prove Lemma 4.4:\nProof.Weprovethefirstpartofthelemma: Itissufficienttoshowthatforanymonomial\nmin the relative interior of a face ∆ofΓ, there exists a germ gas in Remark 4.7. For\nany germ gwith Γ(g) = Γ, we can write\njet(g,∆) = xa·yb·g1···gn·eg\nwhere a,bare integers, g1, . . . , g nare linear homogeneous polynomials not associated to\nxory, andegis a product of non-associated irreducible non-linear homogeneous polyno-\nmials. Note that aandbare the distances of ∆from the y- and x-coordinate axes. Note\nalso that smooth faces meeting the coordinate axes do not contain interior lattice points.\nAfter possibly exchanging xandy, we may assume that for the weight wof∆we have\nw(x)≥w(y). First consider the case that w(x)> w(y):\n(1.a) Suppose that a≥2. Let gbe a germ non-degenerate Newton boundary Γ. Then\nany right-equivalence which does not act only as a rescaling of variables on jet(g,∆)\ngenerates terms on the coordinate axes below Γ, hence, changes the Newton poly-\ngon. The claim follows directly (choosing a non-degenerate g=f0+c·mwith\nc∈C∗).MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 24\n(1.b) Consider now the case a= 0. Since Γcorresponds to a normalized germ with\nrespect to ∆, we have n= 0andeg̸= 1. This implies that w(y)∤w(x), hence\nthere does not exist a w-weighted homogeneous right-equivalence except rescaling\nof variables.\n(1.c) Finally, suppose a= 1. Ifw(y)∤w(x), then, as above, n= 0and˜g̸= 0which\nimplies that rescalings are the only right-equivalences on the face. If w(y)|w(x),\nthen, writing τ=w(x)/w(y), anyright-equivalencewhichdoesnotcreateanyterms\nof lower w-weight than that of ∆is of the form\nx7→c1x+c2yτ, y7→c3y,\nwhere c1, c3∈C∗andc2∈C, hence acts on yas a rescaling. We may therefore\nassume that b= 0. The vertices of ∆correspond to monomials of the form xp\nandxy(p−1)·τwith p≥2. Any monomial in the interior of ∆is of the form m=\nxsyτ·(p−s)with 0< s < p . For g=f0+c·mwith c∈C∗, the jet with regard\nto∆is then jet(g,∆) = xp+c·m+xy(p−1)·τ. We now show that there is no\nright-equivalence which keeps Γ(g)and, hence, is of the above form, that removes\nm. Keeping the face ∆and removing mamounts to the conditions\u0012p\np−s\u0013\ncs\n1cp−s\n2+c·cs\n1cτ·(p−s)\n3 = 0\ncp\n2+c·cs\n2cτ·(p−s)\n3 +c2c(p−1)·τ\n3 = 0\nwhich correspond to the vanishing of the coefficients of mandyp·τ. Using c1̸= 0a\nsolution of the first equation for c2is of the form\nc2= ˜c·cτ\n3\nwhere ˜cp−s=−c/\u0000p\ns\u0001\n. Inserting this into the second equation, leads to the equation\n0 = ˜cp·cτ·p\n3+c·˜cscτ·p\n3+ ˜c·cτ·p\n3\n=cτ·p\n3·(˜cp+c·˜cs+ ˜c)\n=cτ·p\n3·\u0012\n˜cp−\u0012p\ns\u0013\n·˜cp+ ˜c\u0013\n=cτ·p\n3·˜c·\u0012\u0012\n1−\u0012p\ns\u0013\u0013\n·˜cp−1+ 1\u0013\nSince\u0000p\ns\u0001\n̸= 1, there is a Zariski open set of values of ˜c, equivalently of c, such that\nthe expression in the bracket does not vanish and gis non-degenerate. For such a\nchoice of cand thus of g, it follows that c3= 0, a contradiction.\nNow consider the case that w(x) = w(y). Since jet(g,∆)is homogeneous, hence\nfactorizes into linears, in this case we have a≥1,b≥1and˜g= 0.\n(1.d) If a, b≥2, and gis a germ with non-degenerate Newton boundary and Newton\npolygon Γ, then any right-equivalence which does not only act as a rescaling of\nvariables on jet(g,∆)changes the Newton polygon (with the same argument as in\nthe case w(x)> w(y),a≥2).\n(1.e) The case a≥2andb= 1, can be handled in the same way as the case w(x)> w(y),\na= 1,w(y)|w(x)above. An analoguos argument also applies to the case b≥2\nanda= 1.\n(1.f) In the case a=b= 1, a Gröbner basis calculation shows directly that any monomial\ncorresponding to a lattice point of ∆is either a monomimal of f0or an element of\nB(note that smooth faces do not contain any interior lattice points): The germMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 25\nxpy+xyp,p≥2, is right-equivalent to a germ h(applying for instance the right-\nequivalance x7→x+y,y7→y+ 2x) with vertex monomials xp+1andyp+1, hence,\nby Theorem has Milnor number µ= (p+ 1)2−2(p+ 1) + 1 = p2. The germ\ng=xpy+xyp+xp2+2+yp2+2is right-equivalent to h, hence also has Milnor\nnumber µ=p2. This implies that ⟨x, y⟩p2+2⊂Jac(g). We determine a standard\nbasis of\nJac(g) =⟨gx, gy⟩+⟨x, y⟩p2+2\nwhere\ngx=p·xp−1y+yp+ (p2+ 2)·xp2+1,gy=xp+p·xyp−1+ (p2+ 2)·yp2+1\nwith regard the local degree reverse lexicographic ordering. The S-polynomial of gx\nandgyleads to the standard basis element xypafter reducing the tail by ⟨x, y⟩p2+2.\nThe S-polynomial of xypwith gxleads to the standard basis element y2p−1after\nreducing the tail by ⟨x, y⟩p2+2. The S-polynomial of xypwith gyreduces to zero,\nwhile all remaining ones vanish. Hence, the classes of the monomials\nyp+1, x2yp−1, . . . , xp−2y3,\nform a basis of the Milnor algebra in degree p+ 1. Using the relation gxthese\nmonomials are equivalent to\nx2yp−1, . . . , xp−2y3, xp−1y2\nwhich form a regular basis in degree p+1. From the standard basis of the Jacobian\nit is clear that this is the only option for a regular basis, which proves the claim.\nWe now prove the second part of the lemma. Let m1, . . . , m nbe the monomials of\nf0and let g=P\nikimi. For a face ∆ofΓ, we denote again its weight by w. In the\nfirst part of the proof we have seen that for all faces ∆, except those where a= 1,\nw(x)≥w(y),w(y)|w(x), and b≥2ifw(x) =w(y), or those where b= 1,w(x)≤w(y),\nw(x)|w(y), and b≥2ifw(x) =w(y), there does not exist a w-weighted homogeneous\nright-equivalence except rescaling of variables.2In order to describe in these two cases\nthew-homogeneous transformations on jet(g,∆)keeping the face, by symmetry, it is\nsufficient to consider the first of the two. Here, writing τ=w(x)/w(y), the jet is of\nthe from jet(g,∆) = yq·\u0000\nkixp+kjxy(p−1)·τ\u0001\n,ki, kj̸= 0,p≥2. All homogeneous\ntransformations are of the form x7→c1x+c2yτ, y7→c3y, and keeping ∆amounts to the\ncondition\nkicp\n2+kjc2c(p−1)·τ\n3 = 0,\nwhich implies that either c2= 0(which corresponds to a rescaling of variables), or that\nbetween c2andc3there is an algebraic relation c2=k·cτ\n3(with kp−1=−kj/ki).\nWenowshowthatthereisagerm gwiththesamemonomialsas f0andnon-degenerate\nNewton boundary, such that only two terms of gcan be normalized to coefficient one\nby a right-equivalence keeping the Newton polygon. Since this gis right-equivalent to a\ngerm in the normal form described in Theorem 3.7, this then implies that at most two\nmonomials of f0are not in B.\nIn case there are only two monomials of f0of degree ≤dt, the claim is trivial choosing\ng=f0.3Otherwise, let ms, mtandmlbe three distinct monomials of f0of degree\n2Note that these cases correspond to the second part of (1.c) and to (1.e), which are the only settings\nwhere there may exist (and, in fact, exist) transformations on the jet of the respective face, which are\nnot just rescalings of variables.\n3Note that this includes the case where there is a face with w(x) =w(y)anda=b= 1.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 26\n≤dt. We prove that there is a Zariski open set of germs gsuch that not all three\nmonomials ms, mt, mlcan be normalized to coefficient one. To see this, it is enough to\nprove that, for any gwith the same monomials as f0, after choosing two monomials out\nofms, mt, mland normalizing their coefficients to one, when restricting the action of\nthe right-equivalence group to the Newton boundary and stabilizing the two normalized\ncoefficients, the stabilizer acts as a finite group.\n(2.a) If all three monomials lie on faces of Γwhich permit only rescaling of variables,\nthen the claim is obvious.\n(2.b) If exactly two of the three monomials, say ms, mt, lie on faces of Γwhich permit\nonly rescaling of variables, then, without loss of generality, we may assume that\nmllies on a face with a= 1,w(x)≥w(y),w(y)|w(x). Ifw(x) =w(y), then it\nis sufficient to consider the case b≥2. After normalization of the coefficients of\nmsandmtto value one via rescaling of variables, stabilizing the two normalized\ncoefficients together with the above condition c2= 0orc2=k·cτ\n3admits only\nfinitely many solutions.\n(2.c) Suppose that exactly one of the three monomials, say ms, lies on a face of Γwhich\npermits only rescaling of variables.\nIfmtandmllie on the same face ∆, we may assume that the jet of ∆is of the\nform jet(g,∆) = yq·\u0000\nklxp+ktxy(p−1)·τ\u0001\n. Then coefficients of the monomials ms\nandmlcan only be changed via rescaling of variables. Then, similar to the previous\ncase, normalization of the coefficients of msandmlto value one together with the\ncondition c2= 0orc2=k·cτ\n3admits only finitely many solutions.\nIfmtandmllie on different faces, we may assume that mtlies on a face ∆with\nweight wanda= 1,w(x)≥w(y),w(y)|w(x). Moreover, in case w(x) =w(y),\nwe can assume that b≥2, since the case b= 1has already been discussed. Then\njet(g,∆)is of the form\njet(g,∆) = yq·\u0010\nkixp+kjxy(p−1)·τ\u0011\n,\nwith τ=w(x)/w(y)andt∈ {i, j}, and right-equivalences keeping the Newton\npolygon act on the jet as\nx7→c1x+c2yτ,y7→c3y\nsatisfy the condition c2= 0orc2=k·cτ\n3with kp−1=−kj/ki. Similarly, we may\nassume that mllies on a face Πwith weight vwith b= 1,v(x)≤v(y),v(x)|v(y).\nAgain, in case v(x) =v(y), we may assume that a≥2. Then jet(g,Π)is of the\nform\njet(g,Π) = xq′·\u0010\nkryp′+ksyx(p′−1)·τ′\u0011\n,\nwith τ′=v(y)/v(x)andl∈ {r, s}, and right-equivalences keeping the Newton\npolygon act on the jet as\nx7→c1x,y7→c3y+c′\n2xτ′\nsatisfying the condition c′\n2= 0orc′\n2=k′·cτ′\n1with (k′)p−1=−ks/kr. Ifmtis\nthe monomial of jet(g,∆)of larger x-degree (that is, mt=yqxpandt=i), or if\nmlis the monomial of jet(g,Π)of larger y-degree (that is, ml=yq′xp′andl=r),\nthen the coefficient of the respective monomial can only be changed via rescaling\nof variables, and we can argue as in previous case. Suppose now that mtandml\nare the x-linear monomials of the respective jets. Normalizing the coefficients of mtMODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 27\nandmlthen amounts to the relations\nc1cq\n3·(ktc(p−1)·τ\n3 +kicp−1\n2) = 1\nc3cq′\n1·(klc(p′−1)·τ′\n1 +kr(c′\n2)p′−1) = 1,\nAfter inserting c2=k·cτ\n3with kp−1=−kt/kithe first equation implies that\nc2= 0, a contradiction. Similarly, inserting c′\n2=k′·cτ′\n1with (k′)p−1=−kl/kr\ninto the second equation, yields a contradiction. Hence, c2=c′\n2= 0, that is, right-\nequivalences keeping the Newton polygon act on the jets as rescaling of variables.\nSo after normalizing the coeffcients of two monomials, the coefficient of the third\none can take only finitely many values.\n(2.d) If none of the three monomials ms, mtandmllies on a face of Γwhich permits only\nrescaling of variables, then we may assume that ms, mtlie on a face with a= 1,\nw(x)≥w(y),w(y)|w(x), and b≥2ifw(x) =w(y), and mllies on a face with b= 1,\nw(x)≤w(y),w(x)|w(y), and a≥2ifw(x) =w(y). We can thus argue as previous\ncases to obtain only finitely many solutions for the action of the right-equivalence\ngroup on the Newton boundary.\n□\n5.Next steps\nWe have also developed an algorithm to enumerate all normal form families up to a\nspecified Milnor number or modality. These algorithms will be presented in an up-coming\npaper.\nReferences\nArnold, V.I., 1976. Local Normal Forms of Functions. Inventiones. Math. 35, 87–109.\nArnold, V.I., 1974. Normal forms of functions in neighbourhoods of degenerate critical\npoints. Russ. Math. Surv. 29(2), 10–50.\nArnold, V.I., Gusein-Zade, S.M., Varchenko, A.N., 1985. Singularities of Differential\nMaps, Vol. I. Birkhäuser, Boston.\nArtin, M., 1969. Algebraic approximation of structures over complete local rings. Publi-\ncations Mathematiques de l’IHÉS (36): 23–58.\nBöhm, J., Georgoudis, A., Larsen, K.J., Schönemann, H., Zhang, Y.: Complete\nintegration-by-parts reductions of the non-planar hexagon-box via module intersec-\ntions. J. High Energ. Phys. 09 (2018) 24, 30 pp.\nBöhm, J., Marais, M., Pfister, G., 2016. A Classification Algorithm for Complex Sin-\ngularities of Corank and Modality up to Two. Singularities and Computer Algebra\n- Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday, Springer,\n21–46.\nBöhm, J., Marais, M., Pfister, G., 2016. classify2.lib .ASingular libraryforclassify-\ningisolatedhypersurfacesingularitiesofcorankandmodalityupto2, andtodetermine\nthe moduli parameters. Singular distribution.\nBöhm, J., Marais, M., Pfister, G., 2020. Classification of Complex Singularities with\nNon-Degenerate Newton Boundary. arXiv:2010.10185.\nBöhm, J., Marais, M., Pfister, G., 2024. arnold.lib . ASingular library for classifying\nisolated hypersurface singularities with non-degenerate Newton boundary. Singular\ndistribution.\nBöhm, J., Marais, M., Steenpaß, A., 2020. The classification of real singularities using\nSingular Part III: Unimodal Singularities of Corank 2. J. Symb. Comp. 99, 250–282.MODULI PARAMETERS OF NON-DEGENERATE NEWTON BOUNDARY SINGULARITIES 28\nBöhm, J., Marais, M., Steenpaß, A., 2019. realclassify.lib . ASingular library for\nclassifying isolated hypersurface singularities over the reals. Singular distribution.\nBoubakri, Y., Greuel, G.-M., Markwig, T., 2011. Normal forms of hypersurface singular-\nities in positive characteristic. Moscow Mathematical Journal, 11(4), 657–683.\nBrieskorn, E., Knörrer, H., 1986. Plane Algebraic Curves. Birkhäuser Verlag, Stuttgart.\nDecker, W., Greuel, G.-M., Pfister, G., Schönemann, H., 2024. Singular 4-3-2 – A com-\nputeralgebrasystemforpolynomialcomputations. http://www.singular.uni-kl.de\nDe Jong, T., Pfister, G., 2000. Local Analytic Geometry. Vieweg, Braunschweig.\nGabriélov, A.M., 1974. Bifurcations, Dynkin diagrams and the modality of isolated sin-\ngularities. Funct. Anal. Appl. 8, 94–98.\nGreuel, G.-M., 2018. Singularities in positive characteristic: equisingularity, classifica-\ntion, determinacy. In: Greuel, Gert-Martin (ed.) et al., Singularities, algebraic geom-\netry, commutative algebra, and related topics. Festschrift for Antonio Campillo on\nthe occasion of his 65th birthday, University of Valladolid, Spain, June 19–23, 2017,\nSpringer, 37–53.\nGreuel, G.-M., Lossen, C., Shustin, E., 2007. Introduction to Singularities and Deforma-\ntions. Springer, Berlin.\nGreuel, G.-M., Lossen, C., Shustin, E., 2018. Singular Algebraic Curves. Springer, Berlin.\nGreuel, G.-M., Pfister, G., 2008. A Singular Introduction to Commutative Algebra, sec-\nond ed. Springer, Berlin.\nKouchnirenko, K., 1976. Polyedres de Newton et nombres de Milnor. Inventiones Math-\nematicae. 32, 1–31.\nKrüger, K., 1997. classify.lib . ASingular library for classifying isolated hypersur-\nface singularities w.r.t. right equivalence, based on the determinator of singularities by\nV.I. Arnold. Singular distribution.\nLee, R., Pomeransky, A. A., 2013. Critical points and number of master integrals . Journal\nof High Energy Physics. 165.\nMarais, M., Steenpaß, A., 2015. The classification of real singularities using Singular\nPart I: Splitting Lemma and Simple Singularities. J. Symb. Comput. 68, 61-71.\nMarais, M., Steenpaß, A., 2016. The classification of real singularities using Singular\nPart II: The Structure of the Equivalence Classes of the Unimodal Singularities. J.\nSymb. Comput. 74, 346–366.\nJanko Böhm, Department of Mathematics, University of Kaiserslautern, Erwin-Schrö-\ndinger-Str., 67663 Kaiserslautern, Germany\nEmail address :boehm@mathematik.uni-kl.de\nMagdaleen S. Marais, Department of Mathematical Sciences, Stellenbosch University,\nand African Institute for Mathematical Sciences, P/Bag X1, Matieland 7602, Stellen-\nbosch, South Africa,\nEmail address :msmarais@sun.ac.za\nGerhard Pfister, Department of Mathematics, University of Kaiserslautern, Erwin-\nSchrödinger-Str., 67663 Kaiserslautern, Germany\nEmail address :pfister@mathematik.uni-kl.de"    },    {        "title": "2402.05161v1.Approximate_Keys_and_Functional_Dependencies_in_Incomplete_Databases_With_Limited_Domains_Algorithmic_Perspective.pdf",        "content": "arXiv:2402.05161v1  [cs.DB]  7 Feb 2024https://doi.org/10.31449/inf.v42i3.0000 Informatica 45page 501–yyy 1\nApproximate Keys and Functional Dependencies in Incomplet e\nDatabases With Limited Domains–Algorithmic Perspective\nMunqath Alatar\nITRDC, University of Kufa, Iraq\nmunqith.alattar@uokufa.edu.iq\nAttila Sali\nAlfr´ ed R´ enyi Institute of Mathematics\nand Department of Computer Science,\nBudapest University of Technology and Economics\nBudapest, Hungary\nsali.attila@renyi.hu\nKeywords: Strongly possible functional dependencies, Strongly poss ible keys, incomplete databases,\napproximate functional dependencies, approximate keys.\nReceived: February 1, 2023\nA possible world of an incomplete database table is obtained by imputing values from\nthe attributes (infinite) domain to the place of NULLs. A table satisfies a possible key or\npossiblefunctional dependencyconstraint ifthereexists apossibleworldof thetablethat\nsatisfies the given key or functional dependency constraint . A certain key or functional\ndependency is satisfied by a table if all of its possible world s satisfy the constraint.\nRecently, an intermediate concept was introduced. A strong ly possible key or functional\ndependency is satisfied by a table if there exists a strongly p ossible world that satisfies\nthe key or functional dependency. A strongly possible world is obtained by imputing\nvalues from the active domain of the attributes, that is from the values appearing in\nthe table. In the present paper, we study approximation meas ures of strongly possible\nkeys and FDs. Measure g3is the ratio of the minimum number of tuples to be removed\nin order that the remaining table satisfies the constraint. W e introduce a new measure\ng5, the ratio of the minimum number of tuples to be added to the ta ble so the result\nsatisfies the constraint. g5is meaningful because the addition of tuples may extend the\nactive domains. We prove that if g5can be defined for a table and a constraint, then\ntheg3value is always an upper bound of the g5value. However, the two measures are\nindependent of each other in the sense that for any rational n umber0≤p\nq<1there\nare tables of an arbitrarily large number of rows and a consta nt number of columns that\nsatisfyg3−g5=p\nq. A possible world is obtained usually by adding many new valu es not\noccurring in the table before. The measure g5measures the smallest possible distortion\nof the active domains. We study complexity of determining th ese approximate measures.\nPovzetek:\n1 Introduction\nThe information in many industrial and research\ndatabases may usuallybeincompleteduetomany\nreasons. For example, databases related to in-\nstrument maintenance, medical applications, and\nsurveys [8]. This makes it necessary to handlethe cases when some information missing from a\ndatabase and are required by the user. Impu-\ntation (filling in) is one of the common ways to\nhandle the missing values [13].\nA new approach for imputing values in place\nof the missing information was introduced in [3],\nto achieve complete data tables, using only in-2Informatica 45page 501–yyy M. Alatar et al.\nformation already contained in the SQL table at-\ntributes (which are called the active domain of an\nattribute). Any total table obtained in this way\nis called a strongly possible world . We use only\nthe data shown on the table to replace the miss-\ning information because in many cases there is no\nproper reason to consider any other attribute val-\nues than the ones that already exist in the table.\nUsing this concept, new key and functional de-\npendencyconstraints called strongly possiblekeys\n(spKeys) and strongly possible functional depen-\ndencies (spFDs) were defined in [5, 4] that are\nsatisfied after replacing any missing value ( NULL)\nwith a value that is already shown in the corre-\nsponding attribute. In Section 2, we provide the\nformal definitions of spKeys and spFDs.\nThe present paper continues the work started\nin [5], where an approximation notion was intro-\nduced to calculate how close any given set of at-\ntributes can be considered as a key. Tuple re-\nmoval may be necessary because the active do-\nmains do not contain enough values to be able to\nreplacethe NULLvaluessothatthetuplesarepair-\nwise distinct on a candidate key set of attributes\nK. In the present paper, we study approxima-\ntion measures of spKeys and spFDs by adding\ntuples. Adding a tuple with new unique values\nwill add more values to the attributes’ active do-\nmains, thus some unsatisfied constraints may get\nsatisfied.\nFor example, CarModelandDoorNo is de-\nsigned to form a key in the Cars Types table\nshown in Table 1 but the table does not satisfy\nthe spKey sp/a\\}⌊ra⌋ketle{tCarModel,DoorNo /a\\}⌊ra⌋ketri}ht. Two tuples\nwould need to be removed, but adding a new tu-\nple with distinct door number value to satisfy\nsp/a\\}⌊ra⌋ketle{tCarModel,DoorNo /a\\}⌊ra⌋ketri}htis better than removing\ntwo tuples. In addition to that, we know that the\ncarmodelanddoornumberdeterminestheengine\ntype, then the added tuple can also have a new\nvalue in the DoorNo attribute so that the table\nsatisfy (CarModel,DoorNo )→spEngineType\nrather than removing other two tuples.\nTable 1: Cars Types Incomplete Table\nCarModel Door No Engine Type\nBMW I3 4 doors ⊥\nBMW I3 ⊥ electric\nFord explorer ⊥ V8\nFord explorer ⊥ V62 Definitions\nLetR={A1,A2,...An}be a relation schema.\nTheset ofall thepossiblevaluesforeach attribute\nAi∈Ris called the domain of Aiand denoted as\nDi=dom(Ai) fori= 1,2,...n. Then, for X⊆R,\nthenDX=/producttext\n∀Ai∈KDi.\nAn instance T= (t1,t2,...ts) overRis a list\nof tuples such that each tuple is a function t:\nR→/uniontext\nAi∈Rdom(Ai) andt[Ai]∈dom(Ai) for all\nAiinR. By taking a list of tuples we use the\nbag semantics that allows several occurrences of\nthe same tuple. Usage of the bag semantics is\njustified by that SQL allows multiple occurrences\nof tuples. Of course, the order of the tuples in an\ninstance is irrelevant, so mathematically speaking\nwe consider a multiset of tuples as an instance.\nFor a tuple tr∈TandX⊂R, lettr[X] be the\nrestriction of trtoX.\nIt is assumed that ⊥is an element of each at-\ntribute’s domain that denotes missing informa-\ntion.tris called V-total for a set Vof attributes\nif∀A∈V,tr[A]/\\e}atio\\slash=⊥. Also,tris a total tuple\nif it isR-total.t1andt2areweakly similar on\nX⊆Rdenoted as t1[X]∼wt2[X] defined by\nK¨ ohler et.al. [12] if\n∀A∈X(t1[A] =t2[A] ort1[A] =⊥ort2[A] =⊥).\nFurthermore, t1andt2arestrongly similar on\nX⊆Rdenoted by t1[X]∼st2[X] if\n∀A∈X(t1[A] =t2[A]/\\e}atio\\slash=⊥).\nFor the sake of convenience we write t1∼wt2if\nt1andt2are weakly similar on Rand use the\nsame convenience for strong similarity. Let T=\n(t1,t2,...ts) be a table instance over R. Then,\nT′= (t′\n1,t′\n2,...t′\ns) is apossible world ofT, ifti∼w\nt′\nifor alli= 1,2,...sandT′is completely NULL-\nfree. Thatis, wereplacetheoccurrencesof ⊥with\na value from the domain Didifferent from ⊥for\nall tuples and all attributes. Active domain of an\nattribute is the set of all the distinct values shown\nunder the attribute except the NULL. Note that\nthis was called the visible domain of the attribute\nin papers [3, 4, 5, 2].\nDefinition 1 Theactive domain of an attribute\nAi(VDT\ni) is the set of all distinct values exceptApproximate Keys and Functional Dependencies... Informatica 45page 501–yyy 3\n⊥that are already used by tuples in T:\nVDT\ni={t[Ai] :t∈T}\\{⊥}forAi∈R.\nTo simplify notation, we omit the upper index T\nif it is clear from the context what instance is\nconsidered.\nWhile a possible world is obtained by using the\ndomain values instead of the occurrence of NULL,\na strongly possible world is obtained by using the\nactive domain values.\nDefinition 2 A possible world T′ofTis called a\nstrongly possibleworld (spWorld) ift′[Ai]∈VDT\ni\nfor allt′∈T′andAi∈R.\nThe concept of strongly possible world was intro-\nduced in [3]. Strongly possible worlds allow us to\ndefinestrongly possible keys (spKeys) andstrongly\npossible functional dependencies (spFDs) .\nDefinition 3 A strongly possible functional de-\npendency, in notation X→spY, holds in table\nTover schema Rif there exists a strongly pos-\nsible world T′ofTsuch that T′|=X→Y.\nThat is, for any t′\n1,t′\n2∈T′t′\n1[X] =t′\n2[X]im-\npliest′\n1[Y] =t′\n2[Y]. The set of attributes Xis\na strongly possible key, if there exists a strongly\npossible world T′ofTsuch that Xis a key in T′,\nin notation sp/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}ht. That is, for any t′\n1,t′\n2∈T′\nt′\n1[X] =t′\n2[X]impliest′\n1=t′\n2.\nIfT={t1,t2,...,tp}andT′={t′\n1,t′\n2,...,t′\np}is\nan spWorld of it with ti∼wt′\ni, thent′\niis called an\nsp-extension or in short an extension ofti. Let\nX⊆Rbe a set of attributes and let ti∼wt′\ni\nsuch that for each A∈R:t′\ni[A]∈VD(A), then\nt′\ni[X] is anstrongly possible extension of tionX\n(sp-extension)\n3 Related Work\nKivinen et. al. [11] introduced the measure g3\nfor total tables. Giannella et al. [9] measure the\napproximate degree of functional dependencies.\nThey developed the IFD approximation measure\nand compared it with the other two measures: g3\n(minimum number of tuples need to be removed\nso that the dependency holds) and τ(the prob-\nability of a correct guess of an FD satisfaction)\nintroduced in [11] and [10] respectively. They de-\nveloped analytical bounds on the measure differ-\nences and compared these measures analysis onfive datasets. The authors show that when mea-\nsures are meant to define the knowledge degree\nofXdetermines Y(prediction or classification),\nthenIFDandτmeasures are more appropriate\nthang3. On the other hand, when measures are\nmeant to define the number of ”violating” tuples\nin an FD, then, g3measure is more appropriate\nthanIFDandτ.\nIn [15], Jef Wijsen summarizes and discusses\nsome theoretical developments and concepts in\nConsistent query answering CQA (when a user\nqueries a database that is inconsistent with re-\nspect to a set of constraints). Database repairing\nwas modeled by an acyclic binary relation ≤dbon\nthe set of consistent database instances, where r1\n≤dbr2means that r1is at least as close to dbas\nr2. One possible distance is the number of tuples\nto be added and/or removed. In addition to that,\nBertossi studied the main concepts of database\nrepairs and CQA in [6], and emphasis on tracing\nback the origin, motivation, and early develop-\nments. J. Biskup and L. Wiese present and ana-\nlyze an algorithm called preCQE that is able to\ncorrectly compute a solution instance, for a given\noriginal database instance, that obeys the formal\nproperties of inference-proofness and distortion\nminimality of a set of appropriately formed con-\nstraints in [7].\n4 Approximation of strongly\npossible integrity constraints\nDefinition 4 Attribute set Kis an approximate\nstrongly possible key of ratio ain tableT, in no-\ntationasp−\na/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht, if there exists a subset Sof the\ntuplesTsuch that T\\Ssatisfies sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht, and\n|S|/|T| ≤a. The minimum asuch that asp−\na/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht\nholds is denoted by g3(K).\nThe measure g3(K) has a value between 0 and\n1, and it is exactly 0 when sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}htholds in T,\nwhich means we don’t need to remove any tuples.\nFor this, we used the g3measure introduced in\n[11], to determine the degree to which ASPkey\nis approximate. For example, the g3measure of\nsp/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}hton Table 2 is 0.5, as we are required to\nremove two out of four tuples to satisfy the key\nconstraint as shown in Table 3.\nTheg3approximation measure for spKeys was\nintroduced in [5]. In this section, we introduce a4Informatica 45page 501–yyy M. Alatar et al.\nnew approximation measure for spKeys.\nDefinition 5 Attribute set Kis an add-\napproximate strongly possible key of ratio bin\ntableT, in notation asp+\nb/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht, if there exists a\nset of tuples Ssuch that the table TSsatisfies\nsp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht, and|S|/|T| ≤b. The minimum bsuch\nthatasp+\nb/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}htholds is denoted by g5(K).\nThe measure g5(K) represents the approximation\nwhich is the ratio of the number of tuples needed\nto be added over the total number of tuples so\nthatsp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}htholds. The value of the measure\ng3(K) ranges between 0 and 1, and it is exactly 0\nwhensp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}htholds in T, which means we do not\nhave to add any tuple. For example, the g5mea-\nsure ofsp/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}hton Table 2 is 0.25, as it is enough\nto add one tuple to satisfy the key constraint as\nshown in Table 4.\nDefinition 6 For the attribute sets XandY,\nσ:X→spYis a remove-approximate strongly\npossible functional dependency of ratio ain a ta-\nbleT, in notation\nT|=≈−\naX→spY, if there exists a set of tuples\nSsuch that the table T\\S|=X→spY, and\n|S|/|T| ≤a. Then, g3(σ)is the smallest asuch\nthatT|=≈−\naσholds.\nDefinition 7 For the attribute sets XandY,σ:\nX→spYis an add-approximate strongly possible\nfunctional dependency of ratio bin a table T, in\nnotation T|=≈+\nbX→spY, if there exists a set\nof tuples Ssuch that the table T∪S|=X→spY,\nand|S|/|T| ≤b. Then, g5(σ)is the smallest b\nsuch that T|=≈+\nbσholds.\nLetTbe a table and U⊆Tbe the set of the\ntuples that we need to remove so that the spKey\nholdsinT, i.e, weneedtoremove |U|tuples, while\nby adding a tuple with new values, we may make\nmore than one of the tuples in Usatisfy the sp-\nKey using the new added values for their NULLs.\nIn other words, we may need to add a fewer num-\nberof tuplesthanthenumberoftuplesweneedto\nremove to satisfy an spKey in the same given ta-\nble. For example, Table 2 requires removing two\ntuples to satisfy sp/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}ht, while adding one tuple is\nenough.Table 2: Incomplete Table to measure sp/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}ht\nX\nX1X2\n⊥1\n2⊥\n2⊥\n2 2\nTable 3: The table after removing ( asp−\na/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}ht)\nX\nX1X2\n⊥1\n2 2\n4.1 Relation between g3andg5\nmeasures\nResults together with their proofs of this subsec-\ntion were reported in the conference volume [1],\nso the proofs are not included here. The following\nProposition is used to prove Proposition 2.\nProposition 1 LetTbe an instance over schema\nRand letK⊆R. If theK-total part of the ta-\nbleTsatisfies the key sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht, then there exists a\nminimum set of tuples Uto be removed that are\nall non-K-total so that T\\Usatisfies sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht.\nProposition 2 For anyK⊆Rwith|K| ≥2, we\nhaveg3(K)≥g5(K).\nApart form the previous inequality, the two mea-\nsures are totally independent for spKeys.\nTable 4: The table after adding ( asp+\nb/a\\}⌊ra⌋ketle{tX/a\\}⌊ra⌋ketri}ht)\nX\nX1X2\n⊥1\n2⊥\n2⊥\n2 2\n3 3Approximate Keys and Functional Dependencies... Informatica 45page 501–yyy 5\nTheorem 1 Let0≤p\nq<1be a rational num-\nber. Then there exist tables over schema {A1,A2}\nwith arbitrarily large number of rows, such that\ng3({A1,A2})−g5({A1,A2}) =p\nq.\nUnfortunately, the analogue of Proposition 1 is\nnot true for spFDs, so the proof of the following\ntheorem is quiet involved.\nTheorem 2 LetTbe a table over schema R,σ:\nX→spYfor some X,Y⊆R. Then g3(σ)≥\ng5(σ).\nTheorem 3 can be proven by a construction simi-\nlar to the proof of Theorem 1.\nTheorem 3 For any rational number 0≤p\nq<1\nthere exists tables with an arbitrarily large num-\nber of rows and bounded number of columns that\nsatisfyg3(σ)−g5(σ) =p\nqforσ:X→spY.\n4.2 Complexity problems\nDefinition 8 TheSPKey problem is the follow-\ning.\nInputTableTover schema RandK⊆R.\nQuestion Is it true that T|=sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht?\nTheSPKeySystem problem is the following.\nInputTableTover schema RandK ⊆2R.\nQuestion Is it true that T|=sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht?\nTheSPFD problem is the following.\nInputTableTover schema RandX,Y⊆R.\nQuestion Is it true that T|=X→spY?\nThe following was shown in [4].\nTheorem 4 SPKey∈P, SPkeySystem and SPFD\nare NP-complete\nHowever, the approximation measures raise new,\ninteresting algorithmic questions.\nDefinition 9 TheSpKey-g3 problem is the\nfollowing.\nInputTableTover schema R,K⊆Rand\n0≤q <1.\nQuestion Is it true that g3(K)≤qin tableT?\nTheSpKey-g5 problem is the following.\nInputTableTover schema R,K⊆Rand\n0≤q <1.\nQuestion Is it true that g5(K)≤qin tableT?\nProposition 3 The decision problem SpKey-g5\nis in P.Proof: Let us assume that tuples si:i=\n1,2,...,pover schema Rare such that T∪\n{s1,s2,...sp}is optimal, so g5(K) =p\nm. Then\nclearly wemayreplace sibys′\ni= (zi,zi,...,zi)for\nalli= 1,2,...,pwherezi’s are pairwise distinct\nnew values not appearing in the (extended) ta-\nbleT∪{s1,s2,...sp}so thatT∪{s′\n1,s′\n2,...s′\np} |=\nsp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht. Thus,if g5(K)≤qisneededtobechecked\nforatable Tofmtuples, onemayadd ⌊q·m⌋com-\npletely new tuples to obtain table T′and check\nwhether T′|=sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}htin polynomial time by The-\norem 4. /square\nTheorem 5 Decision problem SpKey-g3 is in P.\nProof: Consider the relational schema Rand\nK⊆R. Furthermore, let Tbe an instance over R\nwithNULLs. LetT′be the set of total tuples T′=\n{t′∈ΠA∈KVDT(A):∃t∈Tsuch that t[K]∼w\nt′[K]}, furthermore let G= (T,T′;E) be the bi-\npartite graph, called the K-extension graph of T,\ndefined by {t,t′} ∈E⇐⇒t[K]∼wt′[K]. Find-\ning a matching of Gthat covers all the tuples in\nT(if exists) provides the set of tuples in T′to\nreplace the incomplete tuples in Twith, to verify\nthatKis an spKey.\nIt was shown in [5] that the g3approximation\nmeasure for strongly possible keys satisfies\ng3(K) =|T|−ν(G)\n|T|.\nwhereν(G) denotes the maximum matching size\nin theK-extension graph G. However, the size of\nGis usually exponential function of thesize of the\ninput of the decision problem SpKey-g3, as T′is\nusually exponentially large.\nIn order to make our algorithm run in poly-\nnomial time we only generate part of T′.\nLetT={t1,t2...tm}andℓ(ti) =|{t⋆∈\nΠA∈KVDT(A):t⋆∼wti[K]}|. Note that ℓ(ti) =/producttext\nA:ti[A]=⊥|VDT(A)|, hence these values can be\ncalculated by scanning Tonce and using appro-\npriate search tree data structures to hold values\nof active domains of each attribute. Sort tu-\nples ofTin non-decreasing ℓ(ti) order, i.e. as-\nsume that ℓ(t1)≤ℓ(t2)≤...≤ℓ(tm). Let\nj= max{i:ℓ(ti)< i}andTj={t1,t2,...tj},\nfurthermore T⋆\nj={t⋆:∃t∈Tj:t⋆∼wt[K]} ⊆\nΠA∈KVDT(A). Note that |T⋆\nj| ≤1\n2j(j−1). If\n∀i= 1,2,...,m:ℓ(ti)≥i, then define j= 0 and\nT⋆\nj=∅. LetG⋆= (Tj,T⋆\nj;E⋆) be the induced6Informatica 45page 501–yyy M. Alatar et al.\nsubgraph of Gon the vertex set Tj∪T⋆\nj. Note\nthat|T⋆\nj| ≤1\n2j(j−1).\nClaimν(G) =ν(G⋆)+|T\\Tj|.\nProof of Claim: The inequality ν(G)≤ν(G⋆) +\n|T\\Tj|is straightforward. On the other hand,\na matching of size ν(G⋆) inG⋆can greedily be\nextended to the vertices in |T\\Tj|, asti∈T\\Tj\nhas at least ineighbours (which can be generated\nin polynomial time).\nThus it is enough to determine ν(G⋆) in order\nto calculate g3(K), and that can be done in poly-\nnomial time using Augmenting Path method [14].\n/squareNote that the proof above shows that the exact\nvalueofg3(K) can be determined in polynomial\ntime. This gives the following corollary.\nDefinition 10 The decision problem SpKey-g3-\nequal-g5 is defined as InputTableTover schema\nR,K⊆R.\nQuestion Isg3(K) =g5(K)?\nCorollary 1 The decision problem SpKey-g3-\nequal-g5 is in P.\nExample LetR={A1,A2,A3},K1=\n{A1,A2},K2={A2,A3}.\nT=A1A2A3\nt11⊥1\nt21 2 2\nt32 1 1\nt42 1 1\nT\\ {t4} |=sp/a\\}⌊ra⌋ketle{tK1/a\\}⌊ra⌋ketri}htandT\\ {t4} |=sp/a\\}⌊ra⌋ketle{tK2/a\\}⌊ra⌋ketri}ht, but\nthe spWorlds are different. In particular, this im-\nplies that for K={K1,K2}we have g3(K)>\nmax{g3(K):K∈ K}On the other hand, trivially\ng3(K)≥max{g3(K):K∈ K}holds. This moti-\nvates the following definition.\nDefinition 11 The problem Max-g3defined as\nInputTableTover schema R,K ⊆2R.\nQuestion Isg3(K) = max{g3(K):K∈ K}?\nTheorem 6 Let Table Tover schema Rand\nK ⊆2R. The decision problem Max-g3 is NP-\ncomplete.\nProof: The problem is in NP, a witness con-\nsists of a set of tuples Uto be removed, an index\nj:|U|\n|T|=g3(Kj), also an spWorld T′ofT\\Usuch\nthat each Kiis a key in T′. Verifying the witness\ncan be done in three steps.1.g3(Kj)/\\e}atio\\slash≤|U|−1\n|T|is checked in polynomial time\nusing Theorem 5.\n2. For all i/\\e}atio\\slash=jcheck that g3(Ki)≤|U|\n|T|using\nagain Theorem 5.\n3. Using standard database algorithms check\nthat∀i:Kiis a key in T′.\nOn the other hand, the SPKeySystem problem\ncan be Karp-reduced to the present question as\nfollows. First check for each Ki∈ Kseparately\nwhether sp/a\\}⌊ra⌋ketle{tKi/a\\}⌊ra⌋ketri}htholds, this can be done in poly-\nnomial time. If ∀i:T|=sp/a\\}⌊ra⌋ketle{tKi/a\\}⌊ra⌋ketri}htthen give K\nandTas input for Max-g3. It will answer Yes\niffT|=sp/a\\}⌊ra⌋ketle{tK/a\\}⌊ra⌋ketri}ht. However, if ∃i:T/\\e}atio\\slash|=sp/a\\}⌊ra⌋ketle{tKi/a\\}⌊ra⌋ketri}ht,\nthen give the example above as input for Max-\ng3. Clearly both problems have No answer. /square\nAccording to Theorem 4, it is NP-complete to de-\ncide whether a given SpFD holds in a table. Here\nwe show that approximations are also hard.\nDefinition 12 TheSPFD-g3 (SPFD-g5) prob-\nlems are defined as follows.\nInputA tableTover schema R,X,Y⊆R, and\npositive rational number q.\nQuestion Isg3(X→spY)≤q? (g5(X→spY)≤\nq?)\nTheorem 7 Both decision problems SPFD-g3\nand SPFD-g5 are NP-complete.\nProof: To show that SPFD-g3 ∈NP one may\ntake a witness consisting of a subset U⊂T, an\nspWorld T⋆ofT\\Usuch that T⋆|=X→Y\nand|U|/|T| ≤q. The validity of the witness can\neasily be checked in polynomial time. Similarly,\nto show that SPFD-g5 ∈NP one may take a set of\ntuplesSoverRand an spWorld T⋆ofT∪Ssuch\nthatT⋆|=X→Yand|S|/|T| ≤q.\nOn the other hand, if |T|=mandq <1/m,\nthen both SPFD-g3 and SPFD-g5 are equivalent\nwith the original SPFD problem, since the small-\nest non-zero approximation measure is obtained\nif one tuple is needed to be deleted or added.\nAccording to Theorem 4, SPFD problem is NP-\ncomplete, thus so are SPFD-g3 and SPFD-g5. /square\nAcknowledgement\nResearch of the second author was partially sup-\nported by the National Research, Development\nand Innovation Office (NKFIH) grants K–116769Approximate Keys and Functional Dependencies... Informatica 45page 501–yyy 7\nand SNN-135643. This work was also supported\nby the BME- Artificial Intelligence FIKP grant\nof EMMI (BME FIKP-MI/SC) and by the Min-\nistry of Innovation and Technology and the Na-\ntional Research, Development and Innovation Of-\nfice within the Artificial Intelligence National\nLaboratory of Hungary.\nReferences\n[1] M. Al-Atar and A. Sali. Approximate keys\nand functional dependencies in incomplete\ndatabases with limited domains. In Foun-\ndations of Information and Knowledge Sys-\ntems 12th International Symposium, FoIKS\n2022 Helsinki, Finland, June 20–23, 2022\nProceedings , volume 13388 of LNCS, pages\n147–167. Springer Nature Switzerland AG,\n2022.\n[2] M. Al-Atar and A. Sali. Strongly possible\nfunctional dependencies for sql. Acta Cyber-\nnetica, 2022.\n[3] M. Alattar and A. Sali. Keys in rela-\ntional databases with nulls and bounded\ndomains. In European Conference on Ad-\nvances in Databases and Information Sys-\ntems, pages 33–50. Springer, 2019.\n[4] M. Alattar and A. Sali. Functional depen-\ndencies in incomplete databases with lim-\nited domains. In International Symposium\non Foundations of Information and Knowl-\nedge Systems , pages 1–21. Springer, 2020.\n[5] M. Alattar and A. Sali. Strongly possible\nkeys for sql. Journal on Data Semantics ,\n9(2):85–99, 2020.\n[6] L. Bertossi. Database repairs and consis-\ntent query answering: Origins and further\ndevelopments. In Proceedings of the 38th\nACM SIGMOD-SIGACT-SIGAI Symposium\non Principles of Database Systems , pages 48–\n58, 2019.\n[7] J. Biskup and L. Wiese. A sound\nand complete model-generation procedure\nfor consistent and confidentiality-preserving\ndatabases. Theoretical Computer Science ,\n412(31):4044–4072, 2011.[8] A. Farhangfar, L. A. Kurgan, and\nW. Pedrycz. A novel framework for\nimputation of missing values in databases.\nIEEE Transactions on Systems, Man, and\nCybernetics-Part A: Systems and Humans ,\n37(5):692–709, 2007.\n[9] C. Giannella and E. Robertson. On approx-\nimation measures for functional dependen-\ncies.Information Systems , 29(6):483–507,\n2004.\n[10] L. A. Goodman and W. H. Kruskal. Mea-\nsures of association for cross classifications.\nMeasures of association for cross classifica-\ntions, pages 2–34, 1979.\n[11] J. Kivinen and H. Mannila. Approximate\ninference of functional dependencies from\nrelations. Theoretical Computer Science ,\n149(1):129–149, 1995.\n[12] H. K¨ ohler, U. Leck, S. Link, and X. Zhou.\nPossible and certain keys for sql. The VLDB\nJournal, 25(4):571–596, 2016.\n[13] W. Lipski Jr. On databases with incomplete\ninformation. Journal of the ACM (JACM) ,\n28(1):41–70, 1981.\n[14] L.Lov´ aszandM.Plummer. Matching theory ,\nvolume 367. American Mathematical Soc.,\n2009.\n[15] J. Wijsen. Foundations of query answering\non inconsistent databases. ACM SIGMOD\nRecord, 48(3):6–16, 2019."    },    {        "title": "2402.05164v1.A_Resource_Model_For_Neural_Scaling_Law.pdf",        "content": "Under review as a workshop paper at ICLR 2024\nA R ESOURCE MODEL FOR NEURAL SCALING LAWS\nJinyeop Song1,∗, Ziming Liu1,2,∗, Max Tegmark1,2& Jeff Gore1\n1Department of Physics, MIT, Cambridge, MA, USA\n2IAIFI, Cambridge, MA 02134, USA\n{yeopjin, zmliu, tegmark, gore }@mit.edu\nABSTRACT\nNeural scaling laws characterize how model performance improves as the model\nsize scales up. Inspired by empirical observations, we introduce a resource model\nof neural scaling: A task is usually composite hence can be decomposed into many\nsubtasks, which “compete” for resources (measured by the number of neurons\nallocated to subtasks). On toy problems, we empirically find that: (1) The loss\nof a subtask is inversely proportional to its allocated neurons. (2) When multiple\nsubtasks are present in a composite task, the resources acquired by each subtask\nuniformly grow as models get larger, keeping the ratios of acquired resources\nconstants. We hypothesize these findings to be generally true and build a model\nto predict neural scaling laws for general composite tasks, which successfully\nreplicates the neural scaling law of Chinchilla models reported in (Hoffmann et al.,\n2022). We believe that the notion of resource used in this paper will be a useful\ntool for characterizing and diagnosing neural networks.\n1 I NTRODUCTION\nNeural scaling laws (NSL) refer to empirically observed power law relations in large language model\nbetween the loss and model size, dataset size, and compute cost (Kaplan et al., 2020; Henighan\net al., 2020; Gordon et al., 2021; Hestness et al., 2017; Sharma & Kaplan, 2020; Bahri et al., 2021).\nSeveral theories have been proposed to explain NSL, including the data manifold dimension (Kaplan\net al., 2020; Sharma & Kaplan, 2020), quantization of a task (Michaud et al., 2023), random feature\nmodels (Bahri et al., 2021), and lottery ticket ensembling (Liu & Tegmark, 2023). However, the\nreal mechanism of the neural scaling law remains unclear. The difficulty in verifying these theories\nstem from complex nature of both language tasks and neural networks.\nRecently, there are attempts to mechanistically interpret state-of-art artificial intelligence models to\nelucidate their structures and functions. One key finding in this direction is the identification of\nmodules or subnetworks (Conmy et al., 2023b; Michaud et al., 2023; Chen et al., 2023; Gokhale,\n2023). Since a complicated task (e.g., language modelling) involves many subtasks, a neural net-\nwork is expected to be disentangled into subnetworks or modules; each module is responsible for a\ncorresponding subtask. Examples of modules include induction heads (Olsson et al., 2022b), short-\ncut for sequential operations (Liu et al., 2023), and learned algorithms for mathematical operations\n(Zhong et al., 2023; Nanda et al., 2023; Gromov, 2023). Throughout this paper, we will call such a\nsubnetwork a module1.\nWhile people have discovered the existence of modules in neural networks, we are interested in the\nresource perspective of these modules, i.e., how many neurons are allocated to modules? The idea\nis that neurons are the resource, serving as building blocks for forming any module in a network.\nSince a composite task involves many subtasks, neural networks must allocate reasonable resources\nto each of the subtasks according to their importance. Describing the picture in anthropomorphic\nterms: Training of models is analogous to evolution of life. A species (in this analogy, a model) is\nevolved to optimally use limited nutrients (neurons) to build organs (modules for subtasks) that can\nbest fit its habitat.\n∗Equal contributions\n1People interchangeably use circuit (Olah et al., 2020) or motif (Alon, 2019) depending on contexts.\n1arXiv:2402.05164v1  [cs.LG]  7 Feb 2024Under review as a workshop paper at ICLR 2024\nFigure 1: Overview of the resource model . (Top left) Neurons in a neural network play the role of\nresources, while (sub)tasks are consumers competing for these resources. (Top right) An example\nof a composite task consisting of subtasks combined in parallel and in series. (Bottom) The homo-\ngeneous growth hypothesis: when the network grows wider, each task will acquire more resources\n(allocated neurons), while the ratios of their acquired resources are kept constant.\nIn our study, we conduct a series of toy experiments to investigate how neural networks allocate\nresources to single/multiple subtasks, in the presence of pressure towards sparsity. We will use the\nnumber of allocated neurons as a metric to quantify the resources assigned to each subtask, and use\nthese two terms interchangeably throughout the paper. Below are key findings of the paper:\n•For a single task: More resource allocations lead to lower losses as a power law (Section 2) :\nWe observe that networks can adjust the number of neurons for a module based on the importance\nof a subtask relative to the fixed sparsity regularization. Empirically, the relationship between the\nloss of a corresponding subtask (mean squared error for regression tasks or cross-entropy for clas-\nsification tasks) and the number of allocated neurons Nsubtask for the subtask is found to be roughly\ninversely proportional, i.e., lsubtask∝N−1\nsubtask .\n•For multiple subtasks: Homogeneous growing of allocated resources (Section 3) : We empir-\nically observe that the number of neurons allocated to each subtask increases at a constant ratio as\nthe total number of available neurons grows. Based on this observation, we propose the following\nconjecture: the trained states of artificial neural networks preserve constant ratios in the allocation\nof neurons across different subtasks. This conjecture might be equivalent to the optimal allocation\nof neurons subject to the finite model capacity, specifically the number of neurons in each layer.\nCombining the single task scaling law and the homogeneous growth conjecture, we can derive (and\nindeed observe) that the scaling law of the overall composite task can emerge, i.e., the total loss ℓtotal\nis inversely proportional to the total number of neurons Ntotal, i.e., ℓtotal∝N−1\ntotal.\n•The implication for neural scaling laws in large language models (Section 4) : For a finite-\nwidth neural network trained for complex tasks which contains many subtasks, the optimal loss\nscales with the total resource is ℓ∝N−1. Under a few reasonable assumptions we have Np∝N3,\nsoℓ∝N−1/3\np (Npis the number of model parameters), agreeing with the compute-optimal scaling\nlaw of Chinchilla models where they observed ℓ∝N−0.34\np (Hoffmann et al., 2022).\n2Under review as a workshop paper at ICLR 2024\nFigure 2: (A) Toy experiment : a single x2regression task experiment. (B)(top) The weight and\nbias map of fixed seeds are plotted for four representative values of α= 3.5,9.0,19.3,128. Dots\nwhich have nonzero weights are classified as allocated to x2module and colored as red. Number of\nallocated neurons Nand task loss ℓare plotted relative α(bottom). (C) The N−1scaling between\nnumber of allocated neurons and task loss.\n2 R ESOURCE MODEL FOR SINGLE TASK\nThis section investigates how more resources can lead to lower losses of a subtask, predictably\nas a power law with a scaling exponent of −1. We have the following hypothesis and use a toy\nexperiment to support the hypothesis.\nResource model hypothesis 1 (scaling law of a single subtask) : For\na single subtask, if a network allocates Nneurons for that subtask, the\nnetwork can achieve a loss that scales as ℓ∼N−1.\nExperimental setup Let us consider a simple regression task in figure 2-A. The fully connected\nnetwork has a single hidden layer with 1000 neurons, with a scalar input x∼Uniform [a, b]and\naims to predict y=x2. The network is parametrized as\nfNN(x) =NX\ni=1viσ(wix+bi) +c,\nwhere σis the SiLU activation function, and θ={vi, wi;i= 1,2,···, N}denote all weight\nparameters. The task loss lof our experiment is MSE of the network output ˆy=fNN(x)compared\nto the target function y=x2,\nl=⟨(ˆy−y)2⟩,\nwhere ⟨·⟩is averaging over data samples. We also add penalty pressure to encourage sparsity and\nsmall norms, so the total training objective is\nL=αl+λ1θ1+λ2θ2,\nwhere αis a hyperparameter emphasizing the relative importance of task loss compared to regu-\nlarization. We sweep α∈[1,512].θ1≡ ∥θ∥1andθ2≡ ∥θ∥2are L1 and L2 norms of weights,\nrespectively.We fix λ1= 0.0001 andλ2= 0.0005 . Such dual regularization approach aims to re-\nduce training variability and foster a clear module structure. The Adam optimizer is used to train\nthe neural network with total epochs set to 100000. The learning rate is initially started at 0.01 and\nreduced by a factor of 0.3 every 20000 epochs. The batch size is set to 500.\nOur toy experiment is designed to answer two questions regarding resources (the number of allocated\nneurons): (1) How many neurons are used to do the task (while others are useless)? (2) How do more\n3Under review as a workshop paper at ICLR 2024\nresources contribute to reducing task losses? Hypothetically the balance between the task loss and\nsparsity pressure determine the amount of resources used to perform the task, which we verify below.\nResults We train a network with a fixed seed, varying α∈[1,512] and plot the number of allocated\nneurons, and task loss in figure 2-B. Allocated neurons were identified by a nonzero (>10−3)\nvariance of activation over various inputs xand annotated as red dots. we are able to observe that\nhighαindeed promotes formation of more allocated neurons. Next, We look into how the task\nloss decreases as the number of allocated neurons increases. Notably, we observe a steep decline in\ntask loss exactly at the same time as the number of allocated neurons suddenly jumps. This pattern\nindicates that the increase in resources might be crucial for reducing task loss in this model.\nWe train networks over many random seeds ( nseeds= 16 for each αvalues) and plot the relationship\nof the number of allocated neurons Nand the task loss is shown in figure 2-C. Notably, we observe\naN−1scaling relation between task losses and the number of allocated neurons. The summary of\nthe toy experiment is: the task loss is roughly N−1, where Nis the number of allocated neurons.\nRemark (Scaling laws of various regression tasks) : We further investigated whether ℓ∝N−1\nscaling law holds across regression tasks with various target functions(see Appendix A for details).\nWe observe that majority of chosen functions roughly scales with exponent of -1, but with a case of\nexception where loss scales faster than −1at the initial phase.\nRemark (Mechanism behind the ℓ∝N−1scaling): One might question the underlying mechanism\nfor the observed ℓ∝N−1scaling law. In this regard, we investigated if ensembling mechanism,\nwhich indicates ensembling independent and orthogonal outputs can reduce an error, could account\nfor the observed scaling relation(see Figure 6 in Appendix for details). However, our analysis re-\nvealed that the ensembling does not occur in the current experiment.\n3 N EURON REDUNDANCY IN COMPOSITE TASKS\nPractical tasks are often far more complex than x2regression. It is common for certain task to\nbe a (possibly complex) composition of multiple subtasks. There are two types of basic composi-\ntions: composition in parallel or composition in series. An example of composition in parallel is\na computer vision model tasked with inferring the overall context of an image. The model is ex-\npected to detect various classes of objects together, such as cars, animals and human. An example\nof composition in series is a model expected to build the flow of logical reasoning: the model first\nrecognizes the objects, an then interprets the relationships between objects, and finally deduces the\noverall context.\nOur question is: when a task is a composition of multiple subtasks, how does a neural network\nallocate its resources to accommodate all subtasks? In particular, we are curious how do resource\nallocations change when the neural network is scaled up? We have the following hypothesis:\nResource model hypothesis 2 ( Homogeneous growth of resource al-\nlocation ): If a neural network with Nneurons allocates resources Nito\nsubtask i, then if the network size is scaled up (along width) such that\nN→aN, all the resources are scaled up by the same factor a, i.e.,\nNi→aNi.\nWe will use toy experiments below to show that the homogeneous growth of resource allocation is\ntrue in two composite scenarios: tasks in parallel and tasks in series.\n3.1 T WO TASKS IN PARALLEL\nExperiment setup We start with the two tasks connected in parallel as in figure 3-B. The network\nhas a single hidden layer with 1000 neurons, with a two dimensional vector (x1, x2)s.tx1, x2∼\nUniform [−1,1]2and aims to predict (y1, y2) = (x2\n1, x2\n2). Note that the task is basically the parallel\ncomposition of two x2regression tasks. We set the different relative weights for the first and second\ntask, so the task loss lfrom the network output fNN(x1, x2) = ( ˆy1,ˆy2)is:\n4Under review as a workshop paper at ICLR 2024\nFigure 3: (A) A neural network with one hidden layer is tasked to perform two independent squared\nregression tasks in parallel. We vary hyperparameters αfor overall task intensity and βfor relative\nweights of two tasks. We annotated neurons allocated for the first and the second tasks red and blue\ncolors, respectively. (B) Resources of the first and second tasks increase together as αincreases.\nSurprisingly, we observe the constant ratios of allocated neurons across a range of α, which serve\nas the basis for the homogeneous growth of resource allocation hypothesis. (C) The ratios of MSE\nand ratios of allocated neurons are insensitive to αbut depend heavily on β. (D) We observe the\nemergence of N−1scaling for the composite task of two tasks combined in parallel.\nl=β⟨( ˆy1−y1)2⟩+ (1−β)⟨( ˆy2−y2)2⟩,\nwhere ⟨·⟩means averaging over data samples, βis relative intensity of subtasks sampled within a\nrange [0,1]. The training loss function is defined as\nL=αl+λ1θ1+θ2ℓ2.\nWe use the same experimental conditions as in our previous toy experiment.\nResults We train a network with fixed seeds and β= 0.75, while varying αwithin the range [1,512].\nWe then plot the weight and bias information of each neurons toward two inputs. Each neuron was\nallocated to either the first or the second regression task, characterized by their nonzero weights in\nthe first layer connected to the input neurons. Notably, we did not observe the occurrence of super-\nposition in the experiments, i.e., a single neuron is either allocated completely to xor completely to\nyor useless2. We plot the ratio of number of neurons allocated to the first task respect to that of the\nsecond task in figure 3-B. We observe that this ratio remains constant at 2.35 ±0.24 ( nseeds = 8)\n2Superposition, the phenomenon where a neuron represents two or more distinct concepts, is often observed\nin practical models (Elhage et al. (2022)). However, we did not observe superposition in our experiments.\nThis may be attributed to the model size being sufficiently large relative to the complexity of the given tasks,\ndiffering from the typical conditions for superposition occurrence, where the representation dimension exceeds\nthe model size (Elhage et al. (2022)).\n5Under review as a workshop paper at ICLR 2024\nacross a wide range of αvalues. Based on these observations, we hypothesize that the ratio of allo-\ncation for each task is determined by the relative task intensity rather than the absolute intensity. To\ninvestigate this hypothesis, we now trained a network many times with various seeds( nseeds = 16 ),\nvarying α= 32 ,64,128,256,512 andβ= 0.5,0.75,0.85,0.9,0.95. We observe that both the\nratio of number of allocated neurons and MSE of the first task in respect to the second tasks are\nmaintained constant3. This is consistent with the homogeneous growth hypothesis.\nTheresource model enables to establish the scaling relation of tasks in parallel. According to Re-\nsource model hypothesis 1 , the loss associated to each subtask inversely scales with the number of\nallocated neurons, i.e., MSE i∼Ni−1. If it is assumed that the proportion of neurons allocated to\neach subtask remains constant as N, the total number of allocated neurons ( N=N1+N2), varies,\nthen the normalized composite task loss is given by:\nlglobal =βMSE 1+ (1−β)MSE 2∼β\nn1+1−β\nn2∼1\nN\nThis scaling between task loss and the total allocated neuron count is consistent with our observation\nfor various values of β= 0.5,0.75,0.8,0.9,0.95, as shown in Figure 1-D. Based on the Resource\nmodel hypothesis 1 and2, combined with the linear additivity of losses, we have\nTheorem 1 (Scaling of Tasks Combined in Parallel) .Consider a set of target functions fi:x→\nfi(x)fori= 1,2, . . . , Q , and corresponding number of allocated neurons Ni. Consider a compos-\nite task, which is a linear and parallel combination of these regression tasks i.e composite task loss\nisl=P\ni∈QαiMSE i. Assuming a fixed relative ratio of resources for each subtask (the homoge-\nneous growth hypothesis), the composite loss scales as ∼N−1\ntotal, where Ntotal=P\niNiis the total\nnumber of neurons allocated to the composite task.\nRemark (Origin of homogeneous growth of resource allocation )Homogeneous growth of resource\nallocation may naturally arise from the optimal solution to resource allocation problem. We explore\nthis possibility through a constraint-optimization problem, detailed in the Appendix C.\n3.2 COMPOSITION OF TASKS IN SERIES\nNow we consider tasks arranged in series. A simple example is the regression task of\na composite function. We aim to perform regression on the function F(x1, x2, x3, x4) =p\n(x1−x2)2+ (x3−x4)2. This function is a composition of two distinct functions f◦g, where\nf(x) =√xandg(x1, x2, x3, x4) = (x1−x2)2+ (x3−x4)2.\nToy experiment setup Consider a network with four hidden layers, each consisting of 30 neurons.\nThe network processes a four-dimensional input vector (x1, x2, x3, x4), where each xiis indepen-\ndently sampled from a Uniform distribution U[−1,1]. The task loss lfrom the network output\nfNN(x1, x2, x3, x4) = (ˆy)is\nl=⟨(ˆy−p\n(x1−x2)2+ (x3−x4)2)2⟩,\nwhere ⟨·⟩denotes averaging over data samples. The training objective is defined as\nL=αl+λ1θ1+λ2θ2.\nWe use the same experimental conditions as in our previous experiments, except that we sample α\nfrom the range [20,10240] and set the number of training epochs to 3×105. This adjustment in the\nexperimental setup is intended to speed up convergence.\nResults In our experiment, we train a model with fixed seeds while varying the parameter α. As\nαincreases, the network’s density also increases, a trend depicted in Figure 4-B. Notably, in the\nnetwork’s third layer, we identify a specific set of neurons. These neurons approximate the function\n(x1−x2)2+(x3−x4)2. Within this set, we consistently observed two distinct classes of neurons: one\n3At low alpha discrete nature of number of neurons are making the ratio sudden jumps to constant value.\nThis phenomena is obvious particularly when alpha is low and beta is high, so that the second task is extremely\nlow and corresponding neurons are kept 2 or 4.\n6Under review as a workshop paper at ICLR 2024\nFigure 4: Composition of tasks in series (A) A neural network with four hidden layer is tasked to\nperform the regression of composite function of fandg. We vary hyperparameters αfor overall task\nintensity. (B) (top) Weight structures of trained model are visualized for α= 80,1280 . (bottom)\nNeurons in the third layers of both model exhibit high correlation with the g(x), green for postiive\nand purple for negative correlation. We found the evidence that the first three layers perform g(x)\nand the last layer perform f(x). (C) We observe the N−1scaling for the composite task loss of two\ntasks combined in series.\nclass positively correlated (marked in green) and the other negatively correlated (marked in purple).\nThis pattern indicates that the model processes the f◦gcomputation in a sequential manner. We\npostulated that the first three layers seem to represent the module performing g, while the final layer\ncorresponds to the module performing f.\nNext, we investigate the relationship between the number of allocated neurons and task loss, varying\nαin Figure 4-C. Initially, the task loss exhibits a rapid scaling, but with increasing α, it adheres to an\nN−1scaling. Notably, the transition to the scaling of N−1is consistent with Figure 4-B, where the\nfraction of nonzero neuronal redundancy get less than 50% around α= 640 . We propose that our\ndataset within the chosen αregime reveals two distinct phases. In the first phase, where loss steeply\ndecrease faster than N−1, possibly indicating a qualitative change in neuronal strategies, such as\nthe introduction of neurons with unprecedented functions as αincreases. In the second regime,\nwhere loss scales as N−1, redundant neurons are introduced to reduce task loss. Investigating this\nhypothesis further could elucidate the conditions under which N−1scaling emerges across various\nregression tasks.\nFurthermore, we checked if the resource model can establish the observed scaling relation of tasks\nin series. First, we demonstrated that, under certain assumptions (errors of subtasks should be\northogonal and independent)4, composite task loss of sequential tasks is linearly seperable to sum\nof subtask losses(see Appendix C for details). Next, according to Resource model hypothesis\n1, the loss associated to each subtask inversely scales with the number of allocated neurons, i.e.,\nMSE f∼Nf−1andMSE g∼Ng−1. It allows to establish the scaling law as follows :\nl=∥f(g(x))−f(G(x))∥2+∥f(G(x))−F(G(x))∥2≃A\nNg+B\nNf,\nwhere AandBare constants.\nFurthermore, we examined the validity of the Homogeneous Growth of hypothesis for experiments\ninvolving tasks in series. We analyzed how the number of live neurons in each layer varies with\nrespect to α, as depicted in Figure 8 in Appendix. We indeed observe a coherent increase in the\nnumber of active neurons across all layers. Therefore, under the assumption of Resource model\nhypothesis 2 , the loss scales with the total number of allocated neurons\n4However, it remains to be uncertain whether these prerequisites apply to the composite function in the toy\nexperiment.\n7Under review as a workshop paper at ICLR 2024\nl≃A\nNg+B\nNf∼1\nN.\nSummarizing the above argument, we have established the following theorem :\nTheorem 2 (scaling of tasks combined in series) .: Consider a set of target functions fi:x→\nfi(x) (i= 1,2, . . . , Q ). Consider a composite task, which is a serial combination of such regression\ntasks i.e. l=MSE (ˆy,(f1◦f2◦f3◦. . .◦fQ)(x)). Assuming a fixed relative ratio of resources for\neach subtask (homogeneous growth of resource allocation) and a linear seperability of the composite\ntask loss function, then the total loss scales ∼N−1\ntotal, where Ntotal=P\niNiis the total number of\nneurons allocated to the composite task.\n3.3 C ONJECTURE FOR GENERAL COMPOSITE TASKS\nWe have shown that, for simple composition cases (in parallel and in series), the loss function\nis linearly separable for subtasks, i.e., the loss function can be written as the sum of functions\nof resources for each subtask. We conjecture this linear separability of loss can apply to general\ncomposite tasks.\nResource model hypothesis 3 (Linear additivity of losses) : The loss of\na composite task can be decomposed as a linear summation of the losses\nof its subtasks.\nCombining all three hypothesis, we can derive that\nThe scaling law of general composite tasks : Given three hypothesis of\nthe resource model being true, we can derive that the loss ℓof a general\ncomposite task scales as ℓ∝N−1, where Nis the number of neurons\ncontributing to the overall task.\n4 E MERGENT NEURAL SCALING OF MODEL SIZE\nIn previous sections, we demonstrated that ℓ∝N−1, i.e., the loss ℓis a scaling law against the\nnumber of allocated neurons N. Can we use this result as a starting point to derive the more useful\nneural scaling laws of large language models (LLMs)? Notice that there are two major gaps between\nour resource model and the LLM setup: (1) In LLM, the number of parameters Npis measured\nagainst instead of the number of allocated neurons N; (2) In LLM, the depth is usually scaled with\nwidth, while our resource model (hypothesis 2) assumes only scaling up width. Below we aim to\naddress these two gaps and derive a meaningful scaling law for LLMs.\nIn LLMs the empirical neural scaling law we care about is ℓ∝N−α\npwhere Npis the number of\nmodel parameters. How is Nprelated to N? We aim to show that Np∝N3and finally derive\nα= 1/3, with a number of reasonable assumptions5:\nAssumption 1: The number of modules is proportional to the network width, i.e., N∝Nwidth .\nThis is aligned with the homogeneous growth conjecture. Redundant representations have been\nempirically observed in Liu & Tegmark (2023); Doimo et al. (2021).\nAssumption 2: The depth is scaled as the same factor as the width, i.e., Ndepth∝Nwidth . The\ntheoretical derivation of the optimal depth-width ratio (Levine et al., 2020) has been the foundation\nfor best practices in LLM where the depth and width of the network are scaled by the same factor,\ne.g., in Hoffmann et al. (2022).\n5we denote Nas effectively represents the overall resources (number of allocated neurons) of various sub-\ntasks. The homogeneous growing of resource conjecture enables to regard the resources of various tasks as a\nsingle quantity.\n8Under review as a workshop paper at ICLR 2024\nAssumption 3: The number of parameters can be roughly estimated as N=N2\nwidthNdepth . Most\nneurons of the neural network are in MLPs weights, while the number of other parameters are nearly\nnegligible. The number of MLP weight parameters is simply N2\nwidthNdepth .\nCombining all three assumptions, we have\nNp=N2\nwidthNdepth∝N3\nwidth∝N3, (1)\ntogether with ℓ∝N−1immediately implies that ℓ∝N−1/3\np . Pleasantly, this agrees with the em-\npirical scaling law of Chinchilla models (Hoffmann et al., 2022) where they observed ℓ∝N−0.34\np .\nSince our theory has no hyperparameter at all hence does not have the ability to “fine-tune” to any\nempirical data, it is indeed surprising that our theory can predict (Hoffmann et al., 2022) amazingly\nwell. However, it could be a pure coincidence.\nAlso, we would like to point out something counterintuitive when assumption 1 and 2 are considered\ntogether, which assumes that Nis independent of Ndepth . We conjecture: (1) there exists a critical\ndepth such that above the critical depth, further increasing depth does not help; (2) current LLMs\nare beyond the critical depth. If these conjectures are indeed true, they would then give a testable\nhypothesis that the better way to scale up current LLMs is by scaling up the width but keeping the\ndepth constant, which will give a faster scaling ℓ∝N−1\n2p.\n5 R ELATED WORKS\nNeural Scaling laws (NSL) is the phenomenon where model test losses decreases as power laws\nas model size, compute, or data scale up (Kaplan et al., 2020; Henighan et al., 2020; Gordon et al.,\n2021; Hestness et al., 2017; Sharma & Kaplan, 2020; Bahri et al., 2021). The origin of neural scaling\nlaws is still debatable, but there are a few plausible theories from data manifold dimension (Kaplan\net al., 2020; Sharma & Kaplan, 2020), “quantization” of a task into subtasks (Michaud et al., 2023),\nor random feature models (Bahri et al., 2021). Our paper proposes yet another neural scaling mech-\nanism from module multiplicity.\nNeuronal redundancy is refering to phenomenon that there are more neurons in brains than are\nstrictly necessary for basic functioning. Recently, it is actively explored the similar concept of\nneuronal redundancy in artificial neural network in terms of overparameterization Allen-Zhu et al.\n(2019) and their roles Liu et al. (2021); Zou & Gu (2019).\nEmergence refers to the phenomenon where a model has abrupt improvements in performance\nwhen its size is scaled up [13], although this can depend on specific metrics [14]. Our discussion\non ensembling versus synergy may further provide a useful tool to categorize types of emergence:\ntrivial (pure ensembling) or non-trivial (with synergy).\nMechanistic interpretability is a research field aiming to mechanistically understand internal com-\nputations and representations of neural networks (Nanda et al., 2023; Wang et al., 2022; Zhong\net al., 2023). Works in mechanistic interpretability focus on identifying semantically meaningful\nneurons (Gurnee et al., 2023), heads (Olsson et al., 2022a) or circuits (Conmy et al., 2023a). Our\nwork is an example of module (circuit) level analysis, suggesting that the apparent complexity of\nneural networks may come from many copies of the same module.\nOptimization in artificial intelligence often involves heuristic and empirical methods. Our resource\nmodel suggests that optimal allocation of neurons to each subtasks are crucial for enhancing perfor-\nmance. In this sense, our model validates the efficacy of Normalization (Ioffe & Szegedy, 2015; Ba\net al., 2016) and dropout (Srivastava et al., 2014), which encourage the comprehensive utilization of\nneurons.\n6 C ONCLUSION\nIn this work, we introduced a novel resource model : When a task is a (possibly complex) compo-\nsition of subtasks, each subtask corresponds to a module in the neural network consuming certain\namount of resources. We found the resource model can explain neural scaling laws in our well-\ncontrolled toy setups: a single task, tasks composed in parallel and tasks composed in series. Our\n9Under review as a workshop paper at ICLR 2024\nempirical results show good evidence for the effectiveness of the resource model, leading to many\nphenomenon which could be interesting on their own sake, including lack of superposition and neu-\nron redundancy. Although this resource model is humbly framed as mainly hypothesis, the agree-\nment from toy experiments give the hope that this simple yet effective framework can be extended\nto realistic models, e.g., large language models.\nACKNOWLEDGEMENT\nWe would like to thank Yizhou Liu for helpful discussion. J.S. and J.G. acknowledges support from\nthe Alfred P. Sloan Foundation and Schmidt Polymath Award. Z.L. and M.T. are supported by the\nRothberg Family Fund for Cognitive Science and IAIFI through NSF grant PHY-2019786.\nREFERENCES\nZeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via over-\nparameterization. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the\n36th International Conference on Machine Learning , volume 97 of Proceedings of Machine\nLearning Research , pp. 242–252. PMLR, 09–15 Jun 2019.\nUri Alon. An introduction to systems biology: design principles of biological circuits . CRC press,\n2019.\nJimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization, 2016.\nYasaman Bahri, Ethan Dyer, Jared Kaplan, Jaehoon Lee, and Utkarsh Sharma. Explaining neural\nscaling laws. arXiv preprint arXiv:2102.06701 , 2021.\nMayee F Chen, Nicholas Roberts, Kush Bhatia, Jue Wang, Ce Zhang, Frederic Sala, and Christopher\nR´e. Skill-it! a data-driven skills framework for understanding and training language models. arXiv\npreprint arXiv:2307.14430 , 2023.\nArthur Conmy, Augustine N Mavor-Parker, Aengus Lynch, Stefan Heimersheim, and Adri `a Garriga-\nAlonso. Towards automated circuit discovery for mechanistic interpretability. arXiv preprint\narXiv:2304.14997 , 2023a.\nArthur Conmy, Augustine N. Mavor-Parker, Aengus Lynch, Stefan Heimersheim, and Adri `a\nGarriga-Alonso. Towards automated circuit discovery for mechanistic interpretability, 2023b.\nDiego Doimo, Aldo Glielmo, Sebastian Goldt, and Alessandro Laio. Redundant representations\nhelp generalization in wide neural networks. arXiv preprint arXiv:2106.03485 , 2021.\nNelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec,\nZac Hatfield-Dodds, Robert Lasenby, Dawn Drain, Carol Chen, Roger Grosse, Sam McCandlish,\nJared Kaplan, Dario Amodei, Martin Wattenberg, and Christopher Olah. Toy models of superpo-\nsition. Transformer Circuits Thread , 2022.\nShreyas Gokhale. The semantic landscape paradigm for neural networks. arXiv preprint\narXiv:2307.09550 , 2023.\nMitchell A Gordon, Kevin Duh, and Jared Kaplan. Data and parameter scaling laws for neural\nmachine translation. In Proceedings of the 2021 Conference on Empirical Methods in Natural\nLanguage Processing , pp. 5915–5922, 2021.\nAndrey Gromov. Grokking modular arithmetic, 2023.\nWes Gurnee, Neel Nanda, Matthew Pauly, Katherine Harvey, Dmitrii Troitskii, and Dimitris Bert-\nsimas. Finding neurons in a haystack: Case studies with sparse probing. arXiv preprint\narXiv:2305.01610 , 2023.\nTom Henighan, Jared Kaplan, Mor Katz, Mark Chen, Christopher Hesse, Jacob Jackson, Heewoo\nJun, Tom B Brown, Prafulla Dhariwal, Scott Gray, et al. Scaling laws for autoregressive generative\nmodeling. arXiv preprint arXiv:2010.14701 , 2020.\n10Under review as a workshop paper at ICLR 2024\nJoel Hestness, Sharan Narang, Newsha Ardalani, Gregory Diamos, Heewoo Jun, Hassan Kianinejad,\nMd Mostofa Ali Patwary, Yang Yang, and Yanqi Zhou. Deep learning scaling is predictable,\nempirically. arXiv preprint arXiv:1712.00409 , 2017.\nJordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza\nRutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, et al. Train-\ning compute-optimal large language models. arXiv preprint arXiv:2203.15556 , 2022.\nSergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by\nreducing internal covariate shift. CoRR , abs/1502.03167, 2015. URL http://arxiv.org/\nabs/1502.03167 .\nJared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child,\nScott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language\nmodels. arXiv preprint arXiv:2001.08361 , 2020.\nYoav Levine, Noam Wies, Or Sharir, Hofit Bata, and Amnon Shashua. The depth-to-width interplay\nin self-attention. arXiv preprint arXiv:2006.12467 , 2020.\nBingbin Liu, Jordan T. Ash, Surbhi Goel, Akshay Krishnamurthy, and Cyril Zhang. Transformers\nlearn shortcuts to automata, 2023.\nShiwei Liu, Lu Yin, Decebal Constantin Mocanu, and Mykola Pechenizkiy. Do we actually need\ndense over-parameterization? in-time over-parameterization in sparse training. In Marina Meila\nand Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning ,\nvolume 139 of Proceedings of Machine Learning Research , pp. 6989–7000. PMLR, 18–24 Jul\n2021.\nZiming Liu and Max Tegmark. A neural scaling law from lottery ticket ensembling. arXiv preprint\narXiv:2310.02258 , 2023.\nEric J Michaud, Ziming Liu, Uzay Girit, and Max Tegmark. The quantization model of neural\nscaling. arXiv preprint arXiv:2303.13506 , 2023.\nNeel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. Progress measures\nfor grokking via mechanistic interpretability. arXiv preprint arXiv:2301.05217 , 2023.\nChris Olah, Nick Cammarata, Ludwig Schubert, Gabriel Goh, Michael Petrov, and Shan Carter.\nZoom in: An introduction to circuits. Distill , 5(3):e00024–001, 2020.\nCatherine Olsson, Nelson Elhage, Neel Nanda, Nicholas Joseph, Nova DasSarma, Tom Henighan,\nBen Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Dawn Drain, Deep Ganguli,\nZac Hatfield-Dodds, Danny Hernandez, Scott Johnston, Andy Jones, Jackson Kernion, Liane\nLovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish,\nand Chris Olah. In-context learning and induction heads. Transformer Circuits Thread , 2022a.\nCatherine Olsson, Nelson Elhage, Neel Nanda, Nicholas Joseph, Nova DasSarma, Tom Henighan,\nBen Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Dawn Drain, Deep Ganguli,\nZac Hatfield-Dodds, Danny Hernandez, Scott Johnston, Andy Jones, Jackson Kernion, Liane\nLovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish,\nand Chris Olah. In-context learning and induction heads, 2022b.\nUtkarsh Sharma and Jared Kaplan. A neural scaling law from the dimension of the data manifold.\narXiv preprint arXiv:2004.10802 , 2020.\nNitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.\nDropout: A simple way to prevent neural networks from overfitting. Journal of Machine\nLearning Research , 15(56):1929–1958, 2014. URL http://jmlr.org/papers/v15/\nsrivastava14a.html .\nKevin Wang, Alexandre Variengien, Arthur Conmy, Buck Shlegeris, and Jacob Steinhardt. Inter-\npretability in the wild: a circuit for indirect object identification in gpt-2 small. arXiv preprint\narXiv:2211.00593 , 2022.\n11Under review as a workshop paper at ICLR 2024\nZiqian Zhong, Ziming Liu, Max Tegmark, and Jacob Andreas. The clock and the pizza: Two stories\nin mechanistic explanation of neural networks. arXiv preprint arXiv:2306.17844 , 2023.\nDifan Zou and Quanquan Gu. An improved analysis of training over-parameterized deep neural\nnetworks, 2019.\n12Under review as a workshop paper at ICLR 2024\nA S CALING LAW ACROSS VARIOUS REGRESSION TASKS\nFigure 5: Scaling relationship between the number of allocated neurons and task loss across various\nregression tasks. Simulations were conducted for three different network depths, with hidden layer\nconfigurations of [1000], [200, 200], and [150, 150, 150], which are represented by the colors red,\nyellow, and blue, respectively.\nWe investigated the that N−1scaling law between task loss and number of allocated neurons holds\nacross various simple function regression tasks, as shown in. Experiments were performed with\nidentical settings in [Section 2] except changing the target functions. We also trained the model with\ndepth = 1,2,3to address the possibility that some target functions might need more than a single\nlayer to form a module. We observe that majority of functions shows -1 scaling law asymtotically,\nafter the faster decrease phase, except for sin2(π(x)).\nB D ISCUSSION ON MECHANISM OF ℓ∝N−1SCALING LAW\nB.1 E NSEMBLING MECHANISM\nIn this section, we describe the ensembling mechanisms that ensembling of multiple independent and\nidentical outputs can reduce uncorrelated errors. lets focus on regerssion tasks, while classification\ntasks are described in the later section. In order for network to have nmultiple independent and\nidentical outputs, there should be nseperated subgraphs of the similar topology but with slightly\ndifferent parameters in the network. We will call such subgraphs as regressors. The total number\nof allocated neurons are proportional to number of regressors, i.e N=nk, where kis a number of\nneurons to form a regressor.\nThe following Lemma suggests a regressor has zero mean of error.\nLemma (Zero Mean Error) Consider a simple regression task x→f(x)and corresponding re-\ngressor that performs F:x→F(x)to minimize l=Ex∈U∥f(x)−F(x)∥2, where Uis input\ndistribution. Then, the expectation of error is zero.\nEx∈U[f(x)−F(x)] = 0\nProof Suppose that f=E(f(x))andF=E(F(x)). Then\nl=Ex∈U∥f(x)−F(x)∥2\n=Ex∈U∥f(x)−f(x) +f(x)−F(x)∥2\n=Ex∈U∥f(x)−f(x)∥2+Ex∈U∥f−F(x)∥2\n=Ex∈U∥f(x)−f∥2+Ex∈U∥F(x)−F+F−F(x)∥2\n=Ex∈U∥f(x)−f∥2+Ex∈U∥F(x)−F∥2+∥f−F∥2\n13Under review as a workshop paper at ICLR 2024\nThe last term Ex∈U∥f−F∥2denotes the deviation of mean between of regressor output to true func-\ntion. The neural network with bias terms has capability to reduce the last term to zero. Therefore,\nf=FIn summary, bias term is now can be ignored.\nFrom now on, We assume that f=F= 0without loss of generality.\nDefinition : Independent and Identical Regressors Consider a simple regression task x→f(x)\nand a model that have multiple regressors Mi(i∈1,2, ..N M)performs Fi:x→Fi(x). The\nregressors are considered to be independent and identical distributed if their error functions ei(x) =\nf(x)−F(x)satisfy two conditions : firstly, Ex∈U[ei(x)] = 0 and secondly, Ex∈U[ei(x)ej(x)] =\nˆe2δij, where ˆe2as mean squared error of each individual module.\nEnsembling mechanisms Consider a regression task f:x→f(x). Assume that a model Mcom-\nprises nindependent and identical regressors M1, M2, . . . , M n, where each regressor performs a\nfunction Fi:x→Fi(x) =f(x) +ei(x)with a mean squared error (MSE) of var(e(x)) = ˆ e2\nin a specific layer. We further assume that error is negligible compared to the function norm i.e\nˆe <<∥f∥and error is orthogonal. Then, the model is able to perform the regression task with MSE\nofˆe2/Nin the subsequent layer through ensembling of M1, M2, . . . , M n.\nProof Consider a neuron in the subsequent layer receives outputs from regressors and perform sum-\nmation. These modules are associated with weights w1, w2, . . . , w N, so that the activation of the\nneuron is F∗:x7→F∗(x) =P\niwiFi(x)Then the neuron has the following MSE.\nl=Ex∈U∥f(x)−F∗(x)∥2\n=Ex∈U∥f(x)−X\niwiFi(x)∥2\n=Ex∈U[1−X\niwi]2∥f(x)|2+ 2Ex∈U[1−X\niwif(x) +b∗][X\njej(x)] +Ex∈UX\niX\njwiwj[ei(x)ej(x)]\n=Ex∈UX\ni=1,2,..nX\nj=1,2,..nwiwj[ei(x)ej(x)] =X\niw2\niei(x)≥ˆe2\nn\nThe equality holds when w1=w2=..=wn\nB.2 M ATCHING ENSEMBLING MECHANSIM WITH x2REGRESSION EXPERIMENT\nWe revealed that the ensembling mechanism, suggested in Liu & Tegmark (2023), does not con-\nvincingly account for loss scaling observed in a toy experiment for single x2regression task. In\nthe experiment, ensembling is expected to occur between outputs from several x2regressors, pairs\nof symmetric neurons that have the same bias and the opposite weight in the first layer Liu &\nTegmark (2023). We identified multiple x2regressors through matching pair of neurons by mini-\nmizing relative distances when projected onto their counterpart as in Figure 6(left). We ploted the\nerror functions(center) across the input distribution and their correlations(right). Correlation values\nare±1in most cases, indicating these error function are highly correlated(anticorrelated). Therefore,\nwe concluded that ensembling does not occur for a set of x2regressors in our single2regression toy\nexperiment.\nC O RIGIN OF HOMOGENEOUS GROWTH OF RESOURCE ALLOCATION\nPlausible scenario about Homogeneous growth of resource allocation It is insightful to note that\ntheHomogeneous growth of resource allocation could be derived by assuming optimal solution\nof resource allocation. For instance, consider the following optimization problem in our two-task\nexperiments:\nminimize l(n1, n2) =a\nn1+b\nn2, (2)\nsubject to 2(n1+n2) =N. (3)\n14Under review as a workshop paper at ICLR 2024\nFigure 6: Error functions of x2regressors in the toy experiment. alpha = 3.53,9.07,19.33,128\nrespectivley. from top to bottom (left) Plots show weight and bias of the first layer. Neurons exhibit-\ning symmetry are identified and paired based on their relative distances when projected onto their\ncounterparts, annotated by colored lines. (center) the error function profiles for the regressors with\nrespect to the input x. (right) The correlations between error functions of each regressors are plotted\nThe optimal solution results in a constant ratio between n1andn2. This principle can be extended to\nany composite task where the loss function is linearly separable to sum of subtasks loss. Therefore,\nit implies that once Resource model hypothesis 1 andResource model hypothesis 3 hold, then\nResource model hypothesis 2 naturally follows assuming the optimal solution.\nD S CALING LAW IN COMPOSITE TASKS\nD.1 N EURON REDUNDANCY MEASURE\nIn the toy experiment of sequential task, We observe that neurons have quite similar activations,\nwhich turns the analysis of resources to the analysis of neuron redundancy. We measure how much\nredundancy lies in the model by quantifying the correlation between neuron activations within lay-\ners. To be specific, for each neuron, we count the “neuronal redundancy”, analogous to neuroscience,\na number of other neurons in a layer that has highly correlated activation patterns (corr >0.75). In\nFigure 4-B, we observe that a higher fraction of neurons in the hidden layer exhibit non-zero neu-\nronal redundancy as αincreases. The increase of redundant neurons suggests that the network gains\n15Under review as a workshop paper at ICLR 2024\nFigure 7: Neuron redundancy, defined by a number of other neurons in a layer that has highly\ncorrelated activation patterns, are plotted as respect to α.\nredundancy overall regardless of specific functions each neuron serves in performing a composite\nfunction regression.\nThe observed patterns of neuron redundancy imply two distinct regimes in how the model adapts to\nthe task with the addition of neurons. In the first regime (where α <640), the model assigns novel\nfunctions to newly acquired neurons, leading to changes in the module’s structure. Conversely, in\nthe second regime (where α >640), the addition of neurons enhances previously existing functions,\nresulting in an increase in the module’s size while its structure remains unchanged. Our resource\nmodel is predicting -1 exponent only when module is increasing in size, not their structure. Such\nconceptual picture can explain two different regimes of scaling observed in Figure 4-C.\nD.2 S EPERABILITY OF SEQUENTIAL TASK LOSS\nWe propose that the nature of the composite task loss, which is linearly separable into the sum\nof individual subtasks loss, accounts for observed scaling. Consider a model tasked to calculate a\ncomposite function f◦g. Let us assume that the model performs this calculation in a composite\nmanner, such that it first performs ˆg:x→ˆg(x)withmodules ofgand then ˆf:x→ˆf(x)with\nmodules off. The Mean Squared Error (MSE) for the regression task for f◦gcan be expressed as\nfollows:\n16Under review as a workshop paper at ICLR 2024\nl=\r\r\rf(g(x))−ˆf(ˆg(x))\r\r\r2\n=\r\r\rf(g(x))−f(ˆg(x)) +f(ˆg(x))−ˆf(ˆg(x))\r\r\r2\n=∥f(g(x))−f(ˆg(x))∥2+\r\r\rf(ˆg(x))−ˆf(ˆg(x))\r\r\r2\n+ 2\r\r\r(f(g(x))−f(ˆg(x))) (f(ˆg(x))−ˆf(ˆg(x)))\r\r\r2\nOnce we assume that each error function of ˆfandˆg, such that ef(x) =ˆf(x)−f(x)andeg(x) =\nˆg(x)−g(x), are independent and orthogonal for the given input distribution, the last term vanishes.\nMeanwhile, the first term ∥f(g(x))−f(ˆg(x))∥2=∥f′(g(x)∗)eg(x)∥2(by Taylor series) represents\nthe regression error of the gpart. Assuming f′(g(x)∗)andeg(x)is independent and orthogonal,\nthe first term is proportional to the ∥eg(x)∥2. Similarly, the second term\r\r\rf(ˆg(x))−ˆf(ˆg(x))\r\r\r2\nrepresents the regression error of the fpart and is proportional to ∥ef(x)∥2under few assuptions.\nIn summary, the task loss can be seperated to linear sum of MSE of eg(x)and MSE of ef(x)as\nfollows:\nl=A∥eg(x)∥2+B∥ef(x)∥2, (4)\nThe above derivation of linear seperability can be applied to any number of subtasks combined in\nseries.\nD.3 A LLOCATED NEURONS BY LAYERS IN THE SEQUENTIAL TASK\nFigure 8: Number of allocated neurons in each layer are plotted as respect to αin toy experiment\nfor composition of tasks in series.\n17"    },    {        "title": "2402.05166v1.Hot_spaces_with_positive_cosmological_constant_in_the_canonical_ensemble__de_Sitter_solution__Schwarzschild_de_Sitter_black_hole__and_Nariai_universe.pdf",        "content": "Hot spaces with positive cosmological constant in the canonical ensemble:\nde Sitter solution, Schwarzschild-de Sitter black hole, and Nariai universe\nJos´ e P. S. Lemos\nCenter for Astrophysics and Gravitation - CENTRA, Departamento de F´ ısica,\nInstituto Superior T´ ecnico - IST, Universidade de Lisboa - UL,\nAvenida Rovisco Pais 1, 1049-001, Portugal, email: joselemos@ist.utl.pt\nOleg B. Zaslavskii\nDepartment of Physics and Technology, Kharkov V. N. Karazin National University,\n4 Svoboda Square, Kharkov 61022, Ukraine, email: zaslav@ukr.net\nIn a space with fixed positive cosmological constant Λ, we consider a system with a black hole sur-\nrounded by a heat reservoir at radius Rand fixed temperature T, i.e., we analyze the Schwarzschild-\nde Sitter black hole space in a cavity. We use results from the Euclidean path integral approach to\nquantum gravity to study, in a semiclassical approximation, the corresponding canonical ensemble\nand its thermodynamics. We give the action for the Schwarzschild-de Sitter black hole space and\ncalculate expressions for the thermodynamic energy, entropy, temperature, and heat capacity. The\nreservoir radius Rgauges the other scales. Thus, the temperature T, the cosmological constant Λ,\nthe black hole horizon radius r+, and the cosmological horizon radius rc, are gauged to RT, ΛR2,\nr+\nR, andrc\nR. The whole extension of Λ R2, 0≤ΛR2≤3, can be split into three ranges. The first\nrange, 0 ≤ΛR2<1, includes York’s pure Schwarzschild black holes. The other values of Λ R2\nwithin this range also have black holes. The second range, Λ R2= 1, opens up a folder containing\nNariai universes, rather than black holes. The third range, 1 <ΛR2≤3, is unusual. One striking\nfeature here is that it interchanges the cosmological horizon with the black hole horizon. The end of\nthis range, Λ R2= 3, only existing for infinite temperature, represents a cavity filled with de Sitter\nspace inside, except for a black hole with zero radius, i.e., a singularity, and with the cosmological\nhorizon coinciding with the reservoir radius. For the three ranges, for sufficiently low temperatures,\nwhich for quantum systems involving gravitational fields can be very high when compared to normal\nscales, there are no black hole solutions and no Nariai universes, and the space inside the reservoir\nis hot de Sitter. The limiting value RTthat divides the nonexistence from existence of black holes\nor Nariai universes, depends on the value of Λ R2. For each Λ R2different from one, for sufficiently\nhigh temperatures there are two black holes, one small and thermodynamically unstable, and one\nlarge and stable. For Λ R2= 1, for any sufficiently high temperature there is the small unstable\nblack hole, and the neutrally stable hot Nariai universe. Phase transitions can be analyzed, the\ndominant phase has the least action. The transitions are between Schwarzschild de Sitter black hole\nand hot de Sitter phases and between Nariai and hot de Sitter. For small cosmological constant, the\naction for the stable black hole equals the pure de Sitter action at a certain black hole radius and\ntemperature, and so the phases coexist equally. For 0 <ΛR2<1 the equal action black hole radius\nis smaller than the Buchdahl radius, the radius for total collapse, and the corresponding Buchdahl\ntemperature is greater than the equal action temperature. So above the Buchdahl temperature, the\nsystem collapses and the phase is constituted by a black hole. For Λ R2≥1 a phase analysis is also\nmade.arXiv:2402.05166v1  [hep-th]  7 Feb 20242\nI. INTRODUCTION\nOne of the most fascinating aspects of an event hori-\nzon consists in the fact that it possesses entropy as\nwell as other quantum and thermodynamic properties.\nThese understandings emerged through the initial works\nof Bekenstein [1] and Hawking [2], and were endorsed\nby Gibbons and Hawking [3] within a Euclidean path\nintegral approach that led to the statistical mechanics\ncanonical ensemble formalism for black holes. The Eu-\nclidean path integral approach was extended by Gibbons\nand Hawking [4] and Ginsparg and Perry [5] to include\ncosmological horizons, in particular the de Sitter one,\nthat further permitted the analysis of semiclassical ef-\nfects in such spaces. The path integral and the ensemble\ntheory were put on a firm basis by York [6] who realized\nthat proper boundary conditions on the walls of a heat\nreservoir that encloses a cavity with a black hole inside\nis a well posed problem. This idea was implemented by\nWhiting and York [7] by advancing the correct manner\nto constraining the problem. Hayward [8] examined the\napproach in spaces with cosmological de Sitter horizons,\nBraden, Brown, Whiting, and York [9] included electri-\ncally charged black holes in the formalism, Zaslavskii\nenlarged it further to ensembles with arbitrary config-\nurations of self-gravitating systems [10], Lemos applied\nthe approach to the two-dimensional black hole of the\nTeitelboim-Jackiw theory [11], Zaslavskii studied the ex-\ntreme state of a charged black hole in a grand canonical\nensemble [12], and also analyzed the geometry of nonex-\ntreme black holes near the extreme state [13]. Pe¸ ca and\nLemos implemented the formalism to the grand canonical\nensemble of electrically charged black holes in anti-de Sit-\nter spaces [14], Andr´ e and Lemos extended the results to\nddimensions [15, 16], Fernandes and Lemos constructed\nthe grand canonical ensemble of the electric charged d-\ndimensional case [17], and Lemos and Zaslavskii studied\nthe interaction between black holes and matter in the\ncanonical ensemble [18]\nThe de Sitter space with its cosmological horizon is in\nitself fascinating, and its study is now of central impor-\ntance since it is realized as the asymptotic solution for\nour real expanding universe, as well as forming the basis\nfor the inflationary models of the early universe. It is\nthus of significance to learn not only the classical aspects\nrelated to it, but also its quantum and thermodynamic\nproperties. A key feature of this space is that its cosmo-\nlogical horizon radiates through quantum processes, but\nsince this radiation is due to a fixed cosmological con-\nstant, it radiates on and on, and so, unlike a black hole\nhorizon, the de Sitter horizon does not evanesce. In this\nsense, the de Sitter cosmological horizon is quantum sta-\nble. One might want to add a central black hole to the\nde Sitter space, obtaining thus the Schwarzschild-de Sit-\nter space, which is a solution of general relativity. The\nSchwarzschild-de Sitter solution has an appeal of its own\nsince it has two horizons, namely, the black hole hori-zon and the cosmological horizon. These two horizons\ngenerically have different temperatures, and so there is\nno possible thermodynamic equilibrium solution. This\nmeans that in this setting the problem should be treated\nas a nonequilibrium case and no thermodynamics can\nbe properly devised. However, some insights to bypass\nthis obstacle to a thermodynamic formulation have been\ngiven. One way to have a proper thermodynamics is\nto apply a reservoir kept at some temperature and with\nboundary at some radius, and use York’s Euclidean path\nintegral formalism. Developments in this direction can\nbe mentioned. Wang and Huang [19, 20] made a study\nof the thermodynamics of the Schwarzschild-de Sitter\nspace in York’s formalism and also extended to the ther-\nmodynamics of Reissner-Nordstr¨ om-de Sitter. Ghezel-\nbash and Mann [21] analyzed the action and entropy of\nSchwarzschild-de Sitter black holes. Saida [22] treated\nsome aspects of the Schwarzschild-de Sitter thermody-\nnamics in the canonical ensemble, and Draper and Farkas\n[23] discussed de Sitter black holes in the Euclidean path\nintegral approach. Banihashem and Jacobson [24] consid-\nered thermodynamic ensembles with cosmological hori-\nzons, Banihashemi, Jacobson, Svesko, and Visser [25] ex-\nplored further the minus sign that enters the thermody-\nnamic energy in the first law of thermodynamics for de\nSitter horizons, Jacobson and Visser [26, 27] built the\npartition function for a volume of space as well as the\npartition function and the entropy of causal diamond en-\nsembles, and Morvan, van der Schaa, and Visser, exam-\nined the Euclidean action of de Sitter black holes [28].\nIn this work, we want to further understand the ther-\nmodynamics of the Schwarzschild-de Sitter black hole\nspace in the canonical ensemble within York’s formalism.\nIn using it one has to choose whether the heat reservoir,\nput in-between the two horizons, is a reservoir for the\ninside, i.e., is a reservoir for the black hole horizon re-\ngion, or is a reservoir for the outer universe, i.e., for the\nregion that includes the cosmological horizon. Here we\nare interested in the first situation, and will study the\nthermodynamics of the black hole region in a cavity with\na heat reservoir outside. The black hole horizon and its\ntemperature together with the reservoir and its temper-\nature play a principal role in the thermodynamic analy-\nsis, indeed the equilibrium situations are established by\nthem. The cosmological horizon has no major role in this\nsetting, actually its place is a function of the black hole\nhorizon location, and it is directly determined once the\nlocation off this latter has been found through thermo-\ndynamic computation. In this setup there are three main\nscales, the scale set by the size of the reservoir, the scale\nset by the temperature of the reservoir, and the scale\nset by the cosmological constant, which in turn yield the\nscale set by the size of the black hole horizon and the\nscale set by the size of the cosmological horizon. It is\nthus expected that the existence of these various scales\nyields new, interesting, and important properties of the\nsystem. One that we can advance now, is that the set of3\nensembles is comprised not only of the Schwarzschild-de\nSitter black hole but also of the Nariai universe, which\narises naturally when the cosmological length scale and\nthe reservoir length scale are equal. This intermediate\ncase divides the ensembles into a set of ensembles with\nlow cosmological constant that has familiar properties,\nand another set with high cosmological constant that is\nnew and in which the black hole and cosmological hori-\nzons exchange roles. Other thermodynamic properties\nbecome quite unusual as compared with the no cosmo-\nlogical constant black hole case in the canonical ensemble.\nWe would like to mention several other different at-\ntempts devised to understand the quantum and thermo-\ndynamic nature of black holes in de Sitter space that\nare interesting on themselves but that do not have a di-\nrect bearing with our work here. Davies [29] studied the\nblack hole mechanics and thermodynamics of the Kerr-\nNewman-de Sitter family of solutions, and Romans [30]\nperformed an important analysis of the de Sitter black\nholes, classifying them as temperature goes in cold, luke-\nwarm, and warm. Bousso and Hawking [31] put for-\nward the interesting possibility of having evaporation and\nanti-evaporation of Schwarzschild–de Sitter black holes,\nMaeda, Koike, Narita, and Ishibashi [32] found an up-\nper bound for the entropy of an asymptotically de Sit-\nter spacetime, Wu [33] worked out the entropy of black\nholes with different surface gravities with applications\nto Schwarzschild-de Sitter black holes, Yueqin, Lichun,\nand Ren [34] used the brick wall method of ´t Hooft\nto calculate the entropy of Schwarzschild-de Sitter black\nholes, Bousso [35] has defined causal diamonds in de Sit-\nter black hole spaces and inspected their entropy, Cai\n[36] considered the Cardy-Verlinde formula in connection\nto thermodynamics of de Sitter black holes, Shankara-\nnarayanan [37] attempted to set a scheme where the dif-\nferent temperatures of the two Schwarzschild-de Sitter\nhorizons could be made consistent, and Teitelboim and\nGomberoff [38, 39] examined the de Sitter black holes\nwith either one of the two horizons working as a thermo-\ndynamic boundary. Dias and Lemos [40] analyzed pair\ncreation of de Sitter black holes on a cosmic string back-\nground and the associated entropy, Cardoso, Dias, and\nLemos [41] displayed the Schwarzschild-de Sitter, Nariai,\nBertotti-Robinson, and anti-Nariai solutions in higher di-\nmensions, in particular their temperatures including the\nlukewarm cases, Sekiwa [42] investigated Schwarzschild-\nde Sitter spaces with the cosmological constant as a ther-\nmodynamic variable, Choudhury and Padmanabhan [43]\ninvoked a new concept of temperature in spaces with sev-\neral horizons, Myung [44] inspected the thermodynamics\nof the Schwarzschild-de Sitter black hole and the Nariai\nsolution in five dimensions, Pappas and Kanti [45] treated\nSchwarzschild–de Sitter spaces and the role of tempera-\nture in the emission of Hawking radiation, Simovic and\nMann [46] exhibited critical phenomena of certain types\nof de Sitter black holes in cavities, Qiu and Traschen [47]\nobtained new results related to thermodynamics of blackpair production in Schwarzschild-de Sitter spaces, Singha\n[48] developed further the thermodynamics of spaces with\nseveral horizons, Volovik [49] suggested a double Hawk-\ning temperature ansatz to explain the thermodynamics\nof a black hole in de Sitter space, and Akhmedov and\nBazarov [50] made an analysis of the backreaction issue\nfor a black hole in de Sitter space.\nAn important concept for finite self-gravitating sys-\ntems, as the ones we want to consider, is the Buchdahl\nbound that sets a maximum mass or a maximum gravi-\ntational radius for the energy that can be enclosed in a\ncavity before the system turns singular and presumably\nsuffers total gravitational collapse. Usually, the Buchdahl\nradius concerns the mechanical structure of balls or stars\nin general relativity, but it should also appear somehow\nin connection with thermodynamics and thermodynamic\nphases, since when, in a cavity, there is energy in the form\nof matter or radiation with gravitational radius larger\nthan the gravitational radius permitted by the Buchdahl\nbound for a given cavity size, that energy should collapse.\nThus, a thermodynamic system that has too much ther-\nmodynamic energy for a given cavity size must collapse.\nSince here, we are interested in self-gravitating systems\nin a positive cosmological constant background in a gen-\neral relativistic context, the Buchdahl bound of interest\nis the one found by Andr´ easson and B¨ ohmer [51].\nThe paper is organized as follows. In Sec. II we state\nthe main general thermodynamic results, derived from\nthe canonical ensemble set by the Euclidean path in-\ntegral approach, for a cavity containing the black hole\nhorizon region of the Schwarzschild-de Sitter space in-\nside a heat reservoir. In Sec. III we give specific re-\nsults for the thermodynamics of Schwarzschild-de Sitter\nblack holes with small values of the cosmological con-\nstant, Λ R2<1, also studying thermodynamic phases\nand phase transitions. In Sec. IV we give specific results\nfor the thermodynamics of Schwarzschild-de Sitter black\nholes wth the intermediate value of the cosmological con-\nstant, Λ R2= 1, which is found to be the Nariai universe,\nand also studying thermodynamic phases and phase tran-\nsitions. In Sec. V we give specific results for the ther-\nmodynamics of Schwarzschild-de Sitter black holes wth\nlarge values of the cosmological constant, Λ R2>1, and\nalso studying thermodynamic phases and phase transi-\ntions. In Sec. VI we present important plots and make a\nthorough analytic study of all the cases. In Sec. VII we\ndraw our conclusions. In the Appendix A we state the\nbasic geometric elements of the Schwarzschild-de Sitter\nand Nariai spaces. In the Appendix B the Nariai limit\nfrom the Schwarzschild-de Sitter space in a cavity in a\nthermodynamic setting is presented in all detail. In the\nAppendix C we derive explicitly some expressions of the\nmain text.4\nII. THERMODYNAMICS OF THE\nSCHWARZSCHILD-DE SITTER SPACE IN THE\nCANONICAL ENSEMBLE: GENERAL RESULTS\nFOR THE BLACK HOLE HORIZON REGION\nINSIDE A HEAT RESERVOIR\nA. Setup and Euclidean metric\nPut, at some radius R, the boundary of a spherical cav-\nity with a black hole in a positive cosmological constant\nbackground inside a heat reservoir. At this boundary, one\nspecifies the data that determine the ensemble, see Fig. 1.\nWe fix the temperature at the boundary at R, so we are\nrc\nreservoir heat\nhotr+TR\nde SitterSchwarzschild\nFIG. 1: A drawing of a black hole in a cavity within a\nheat reservoir at temperature Tand radius Rin a space\nwith positive cosmological constant. Outside the black\nhole radius r+the geometry is a Schwarzschild-de Sitter\ngeometry. The cosmological radius rcis beyond the\nheat reservoir. The Euclideanized space and its\nboundary have R2×S2andS1×S2topologies,\nrespectively, where the S1subspace with proper length\nβ=1\nTis not displayed. See text for more details.\nworking with the canonical ensemble. Furthermore, we\nconsider that the heat reservoir is a heat reservoir for\nthe inner region of the Schwarzschild-de Sitter space and\nthermodynamically discard the region r > R . This is the\napproach first suggested by York for the Schwarzschild\nmetric [6–9], that was generalized further to include mat-\nter [10].\nIn order to implement the Euclidean path integral\napproach and work out canonical ensemble results for\na black hole in a positive constant background we use\nthe Schwarzschild-de Sitter solution in general relativity.\nThen, for the Schwarzschild-de Sitter black hole metric,\none Euclideanizes time and fixes its period βat radius\nRto be the heat reservoir inverse temperature, β≡1\nT.\nThe Schwarzschild-de Sitter space is characterized by two\nparameters, namely, the mass mand the positive cosmo-\nlogical constant Λ. Instead of working with mand Λ, it is\nsometimes preferable to use the black hole horizon radius\nr+and the cosmological horizon radius rc, the two sets oftwo parameters are interchangeable by precise formulas.\nIn the case we are working, the reservoir is a reservoir\nfor the inner region that contains a black hole, so that\nr+is inside Randrcis outside R. The Euclidean line\nelement of the Schwarzschild-de Sitter space in spheri-\ncal coordinates ( t, r, θ, ϕ ) is obtained by Euclideanizing\ntime, t→it, to get the Euclidean Schwarzschild-de Sit-\nter space,\nds2=V(r)dt2+dr2\nV(r)+r2(dθ2+ sin2θ dϕ2),\n0≤t < βH\n+, r+≤r≤R , (1)\nwhere the metric potential V(r) has the form\nV(r) = 1−2m\nr−Λr2\n3, (2)\nand 0 ≤θ≤π, 0≤ϕ <2π. The range of coordinates of\nEuclidean time tis 0≤t < βH\n+, where βH\n+is the period of\nthe time coordinate such that the line element given by\nEqs. (1) and (2) has no conical singularities. The relation\nwith the Hawking temperature TH\n+isβH\n+=1\nTH\n+. Due to\nthe reservoir at radius Rthe range of the radial coordi-\nnate is r+≤r≤R. The line element provided in Eqs. (1)\nand (2) is the Euclideanized form of the Schwarzschild-\nde Sitter spacetime, see Appendix A, and its topology is\nR2×S2.\nThe black hole horizon radius r+is one of the two real\nzeros of V(r) in Eq. (2) and so obeys\nr+\n2\u0012\n1−Λr2\n+\n3\u0013\n=m . (3)\nThen mcan be traded for r+to give V(r) of Eq. (2) as\nV(r) = 1−r+\nr−Λr2\n3\u0010\n1−\u0000r+\nr\u00013\u0011\n, i.e.,\nV(r) =\u0010\n1−r+\nr\u0011\u0012\n1−Λr2\n3\u0012\n1 +r+\nr+\u0010r+\nr\u00112\u0013\u0013\n.\n(4)\nThe cosmological horizon radius is the other zero of V(r)\nin Eq. (2). It can be found once r+is known through the\nequation\nrc=−r+\n2+r+\n2s\n12−3Λr2\n+\nΛr2\n+, (5)\norrc=−r+\n2+1\n2q\n3\nΛq\n4−Λr2\n+. Since the heat reservoir\nenvelopes the inside, the primary horizon radius is the\nblack hole horizon r+, which has to be found from ther-\nmodynamic considerations. The cosmological horizon ra-\ndiusrchas a secondary role, being determined once r+\nis known.\nNow, the radius of the heat reservoir Rsets a scale for\nour problem. It is then meaningful to gauge all the length\nscales involved in the problem to R. Thus, the heat reser-\nvoir temperature T, the cosmological constant Λ, the5\nblack hole horizon radius r+, and the cosmological hori-\nzon radius rc, are gauged to quantities without units as\nRT, ΛR2,r+\nR, andrc\nR. The extensions of these quantities\nare important. They are: 0 ≤RT < ∞, 0≤ΛR2≤3,\n0≤r+\nR≤1, and 1 ≤rc\nR<∞. It is advisable to sep-\narate the whole extension of Λ R2, 0≤ΛR2≤3, into\nthree cases, namely, small values of the cosmological con-\nstant which means Λ R2<1, more exactly 0 ≤ΛR2<1,\nthe intermediate value of the cosmological constant which\nmeans Λ R2= 1, and large values of the cosmological con-\nstant which means Λ R2>1, more exactly 1 <ΛR2≤3.\nNote that the cosmological constant, which has units of\ninverse length square, has associated to it a natural cos-\nmological length scale ℓgiven by ℓ2=1\nΛ.\nB. Action, energy, entropy\nWe list here the most relevant general formulas that\ncome out of the path integral approach, one can look\nelsewhere to pick up these formulas, see, e.g., [6–18]. The\nEuclidean action Iis\nI=βR\u0010\n1−p\nV(R)\u0011\n−πr2\n+, (6)\nwhere β=1\nTis the inverse temperature of the ensemble,\ni.e., at the boundary of the heat reservoir, and where\nV(R) is given by Eq. (2) at r=R, i.e., V(R) = 1−2m\nR−\nΛR2\n3or\nV(R) =\u0010\n1−r+\nR\u0011\u0012\n1−ΛR2\n3\u0012\n1 +r+\nR+\u0010r+\nR\u00112\u0013\u0013\n.\n(7)\nNote that Iin Eq. (6) is I=I(β, R, Λ;r+). The statis-\ntical mechanics ensemble is characterized by Λ which is\nfixed for each space, by T=1\nβandRwhich are fixed\nfor each ensemble, and by r+which can vary, there are\nparticular r+solutions for which Iis stationary,dI\ndr+= 0,\nyielding the thermodynamic solutions of the problem.\nThe Euclidean action and the free energy are related\nbyI=βF, so that F=R\u0010\n1−p\nV(R)\u0011\n−Tπr2\n+. Now,\nF=E−TS, so the thermodynamic or quasilocal energy\nhere also the thermal energy at Rat this order can be\nfound to be\nE=R\u0010\n1−p\nV(R)\u0011\n. (8)\nThe entropy of the system is\nS=πr2\n+, (9)\nwhich is the Bekenstein-Hawking entropy.\nTo find the thermodynamic stability one has to com-\npute the heat capacity at constant reservoir area A,CA.\nThis is given by\nCA=\u0012dE\ndT\u0013\nA. (10)IfCA<0 the system is thermodynamically unstable, if\nCA≥0 the system is thermodynamically stable, with\nthe equality giving the marginal case. Since A= 4πR2,\nEq. (10) is equivalent to CR=\u0000dE\ndT\u0001\nR.\nC. Temperature and solutions\nIn order that the whole formalism be meaningful, the\nline element of Eqs. (1) and (2) should have no conical\nsingularities in the Euclidean r×tplane. This implies\nthat the time coordinate thas to have period βH\n+given by\nβH\n+=4π\n(dV(r)\ndr)r+. It turns out that this period is related\nto the Hawking temperature TH\n+through βH\n+=1\nTH\n+, so\nTH\n+=(dV(r)\ndr)r+\n4π. From Eq. (2) we obtain\nTH\n+=1\n4πr+\u0000\n1−Λr2\n+\u0001\n. (11)\nUsing Eq. (3) this can be also put in the form TH\n+=\n1\n2πr+\u0010\n3m\nr+−1\u0011\n.\nThe stationary points of the action Eq. (6) are found\nthroughdI\ndr+= 0, which yields the equation\nT=TH\n+p\nV(R), (12)\nwith TH\n+being the Hawking temperature given in\nEq. (11) and V(R) is given in Eq. (7). This is\nthe Tolman relation for the temperature at the reser-\nvoir and the Hawking temperature. Since Thas the\nexpression given in Eq. (12) one finds explicitly for\nthis case, using Eq. (11) and Eq. (7), that T=\n1\n4πr+(1−Λr2\n+)\nq\n1−2m\nR−ΛR2\n3, i.e., T=1\n4πr+(1−Λr2\n+)\nq\n1−r+\nR−Λ\n3R(R3−r3\n+). Thus,\n4πRT =1\nr+\nR1−ΛR2(r+\nR)2\nr\n1−r+\nR−ΛR2\n3\u0010\n1−(r+\nR)3\u0011, which can be put in\nthe form\n4πRT =1\nr+\nR1−ΛR2\u0000r+\nR\u00012\nq\n1−r+\nRr\u0010\n1−ΛR2\n3\u0010\n1 +r+\nR+\u0000r+\nR\u00012\u0011\u0011.\n(13)\nWe want to find r+that obey Eq. (13), and so generically\none hasr+\nR(ΛR2, RT). For fixed RTone hasr+\nR(ΛR2),\nand for fixed Λ R2one hasr+\nR(RT).\nThus, for a given fixed Tat the boundary, for the en-\nsemble, one can look for solutions r+. Indeed, depending\non the parameters ( T, R, Λ), Eq. (13) can have no solu-\ntion, one solution, or two solutions r+. When it has two\nsolutions we denote these by\nr+1=r+1(R,Λ, T), (14)\nand\nr+2=r+2(R,Λ, T), (15)6\nwith r+1≤r+2, say. The functions r+1andr+2have\nto be worked out in some way or another. The fact that\nfor given TandRthere exist two roots r+1andr+2is\nsimilar to that for the Schwarzschild space [6], here it is\nfor a given TandR, and for a given Λ. Moreover, given\nthe solutions r+1of Eq. (14) and r+2of Eq. (15), one\nfinds from Eq. (5) the two corresponding cosmological\nhorizon radii, namely,\nrc1=rc1(r+1(R,Λ, T)), (16)\nand\nrc2=rc2(r+2(R,Λ, T)), (17)\nrespectively, with rc1≥rc2.\nIn brief, we have already generic results, though not\nexplicit. We now treat separately the three cases, 0 ≤\nΛR2<1, ΛR2= 1, and 1 <ΛR2≤3. In general there\nare no analytical solutions. In the first case, analytical\nexpressions for particular ranges of Λ R2can be found,\nin the second case one finds that one is in the presence\nof a Nariai universe which has exact thermodynamic so-\nlutions, and in the third case expressions for particular\nranges of Λ R2can also be found. The high temperature\nlimit in the three cases yields analytical solutions. Ther-\nmodynamic phases and phase transitions can be analyzed\nin all the three cases.\nIII. THERMODYNAMICS OF\nSCHWARZSCHILD-DE SITTER BLACK HOLES\nIN THE CANONICAL ENSEMBLE: SMALL\nVALUES OF THE COSMOLOGICAL CONSTANT,\nΛR2<1\nA. Solutions\nWe treat here the small positive cosmological constant,\nΛR2<1, problem. Again, we put the boundary of a\nspherical cavity with a black hole in a positive cosmolog-\nical constant background inside a heat reservoir, at some\nradius R, where it is also specified a fixed temperature\nT, see Fig. 1 anew. Small positive cosmological constant\nmeans exactly in this context that\n0≤ΛR2<1. (18)\nWithin this range, for fixed RTand generic Λ R2, it is\nhard to find solutions of Eq. (13) for black hole horizon\nradii r+analytically. However, for very small Λ R2or for\nΛR2very near one, one can make some progress.\nFor very small Λ R2, i.e., for Λ R2≪1, one finds from\nEq. (13) that there are no black hole solutions for\nRT <√\n27\n8π\u0012\n1−5\n54ΛR2\u0013\n, ΛR2≪1,(19)\nonly hot de Sitter space is possible. Still for very small\nΛ, i.e., for Λ R2≪1, there are two black hole solutionsfor\nRT≥√\n27\n8π\u0012\n1−5\n54ΛR2\u0013\n, ΛR2≪1.(20)\nOne of the two solutions is the small black hole\nr+1(R,Λ, T), and the other solution is the large black\nholer+2(R,Λ, T). For zero cosmological constant, Λ R2=\n0, one has a pure Schwarzschild black hole and one re-\ncovers York’s result of RT≥√\n27\n8πto have black hole solu-\ntions. The minus sign inside the parenthesis in Eq. (20)\nis what one expects really. The two solutions merge into\none sole solution when the equality sign in Eq. (20) holds.\nIn this case the coincident double solution has horizon ra-\ndius given by\nr+1\nR=r+2\nR=2\n3\u0012\n1 + +17\n81ΛR2\u0013\n,ΛR2≪1.(21)\nThe corresponding cosmological radius can then be found\nfrom Eq. (5) to be given by\nrc1\nR=rc2\nR=r\n3\nΛR2 \n1−1\n3r\nΛR2\n3!\n,ΛR2≪1.\n(22)\nFor Λ R2very near unity, i.e., for (1 −ΛR2)≪1, one\nfinds from Eq. (13) that there are no black hole solutions\nfor\nRT <1\n2π \n1 +\u00123\n8\u0000\n1−ΛR2\u0001\u00131\n3!\n,(1−ΛR2)≪1,(23)\nonly hot de-Sitter space is possible. Still for small\nΛR2−1, i.e., for (1 −ΛR2)≪1, there are two black\nhole solutions for\nRT≥1\n2π \n1 +\u00123\n8\u0000\n1−ΛR2\u0001\u00131\n3!\n,(1−ΛR2)≪1.(24)\nWhen the equality holds the coincident double solution\nhas horizon radius given by\nr+1\nR=r+2\nR= 1−\u00123\n8(1−ΛR2)2\u00131\n3\n,(1−ΛR2)≪1.(25)\nThe corresponding cosmological radius can then be found\nfrom Eq. (5) to be given by\nrc1\nR=rc2\nR= 1 +\u00123\n8(1−ΛR2)2\u00131\n3\n,(1−ΛR2)≪1.(26)\nOne could work out in both regimes, i.e., Λ R2≪1\nand (1 −ΛR2)≪1, the action I, the thermodynamic\nenergy E, the entropy S, and the heat capacity CA, given\nthrough Eqs. (6)-(10). Apart from the entropy expression\nS= 4πr2\n+, valid for each of the two black hole solutions,\nthe calculation of the other quantities is not practical and\nnot be particularly illuminating. However, an instance\nwhere all quantities can be worked out, in particular the\nheat capacity CA, is the hight temperature limit to which\nwe now turn.7\nB. High temperature limit: Analytical solutions\nFor the range of values of the cosmological constant\nconsidered in this section, 0 ≤ΛR2<1, one can find\nsolutions in the limit in which RTgoes to infinity, see\nEqs. (12) and (13). Since Ris the quantity that we con-\nsider as the gauge, RTgoing to infinity is the same in\nthis context as Tgoing to infinity. Let us then find ex-\nplicit results by taking the limit of high temperature. In\nthis case the equations can be solved.\nFor a given Tthere are two black hole solutions, the\nsmall black hole solution r+1and the large black hole\nsolution r+2. We set the heat reservoir temperature T\nfixed but very high, in the sense that T→ ∞ . From\nEq. (12) there are two possibilities. Either TH\n+→ ∞\nwhich corresponds to the small black hole solution having\na very small r+1, orV(R)→0 which corresponds to the\nlarge black hole solution r+2approaching the reservoir\nradius. Let us work one at a time for Tfixed and very\nhigh.\nThe first solution for a very high heat reservoir tempera-\nture, T→ ∞ , isr+=r+1→0 with TH\n+→ ∞ . It is clear\nfrom Eqs. (11) together with Eq. (12), or directly from\nEq. (13), that this requires that the black hole solution\nis of the form r+=r+1→0. So, in this limit one has\nTH\n+1=1\n4πr+1, (27)\nwhere the equality sign is valid within the approxima-\ntion taken. From Eq. (13) one finds the small black hole\nsolution r+1to be of the form\nr+1\nR=1\n4πRTq\n1−ΛR2\n3, (28)\nwhere the equality sign is valid within the approxima-\ntion taken. The expression inside the square root of\nEqs. (28) is clearly positive. As a by-product, we also\nfind from Eq. (3) that in this limit one has m1=r+1\n2.\nThe corresponding cosmological radius rc1can then be\nfound directly from Eq. (5), yielding a correspondingly\nfar away value for rc1, see Eq. (5). One could work out\nin this order, i.e., T→ ∞ , the action I, the energy E,\nthe entropy S, and the heat capacity CA, given through\nEqs. (6)-(10). The most interesting quantity is the heat\ncapacity CA, which yields the criterion for thermody-\nnamic stability, indeed when CA<0 the solution is ther-\nmodynamically unstable, when CA≥0 the solution is\nthermodynamically stable. We thus find an explicit ex-\npression for CA. From Eq. (10), i.e., CA=\u0000dE\ndT\u0001\nAor\nequivalently, CA=\u0000dE\ndT\u0001\nR, we find from Eq. (8) that\nCA=1\n2√\nV(R)\u0010\ndr+1\ndT\u0011\nR, which upon using Eq. (28) yields\nCA+1=−1\n8πT2\u0000\n1−ΛR2\n3\u0001<0, (29)so that CAfor the small black hole r+1is negative. The\nsmall black hole r+1solution is thus unstable. Note that\nactually, the black hole is surrounded by quantum fields.\nWe neglect their backreaction on the metric. However,\nifTH→ ∞ , the corresponding energy density and other\ncomponents of the stress-energy tensor diverge. To avoid\nthis, we restrict r+1in the the sense that is has to be\nlarger than the Planck length scale lpl, i.e., r+1> lpl.\nThe second solution for a very high heat reservoir temper-\nature, T→ ∞ hasV(R)→0. It is clear from Eqs. (12)\nor (13) that the condition V(R)→0, implies, in the case\n0≤ΛR2<1, that r+2=Rminus a small quantity.\nNow, from Eq. (11) one has in this limit\nTH\n+2=1−ΛR2\n4πR, (30)\nwhere the equality sign is valid within the approximation\ntaken. In first order, we can perform a Taylor expansion,\nand write V(R) =\u0000dV\ndr\u0001\nr+2(R−r+2) plus higher order\nterms. Since\u0000dV\ndr\u0001\nr+2= 4πTH\n+2, one can write V(R) =\n4πTH\n+2(R−r+2). Using Eq. (12), or Eq. (13), we have\nr+2\nR= 1−1−ΛR2\n(4πRT)2, (31)\nwhere the equality is valid within the approxima-\ntion taken. As a by-product, we also find from\nEqs. (3) and (31) that in this limit one has m2=\nR\n2h\n1−ΛR2\n3−(1−ΛR2)2\n(4πRT)2i\n. The corresponding cosmolog-\nical radius rc2can then be found directly from Eq. (5),\nwe refrain from showing the explicit formula here, not-\ning nevertheless that Λ R2can be small of order of zero\nin which case the cosmological horizon is very far away,\nor of order one in which case the cosmological horizon\nis very near the reservoir and the black hole horizon.\nWe could work out in this order, i.e., T→ ∞ , the\naction I, the energy E, the entropy S, and the heat\ncapacity CA, given through Eqs. (6)-(10). Again, the\nmost interesting one is the heat capacity CA. For the\nheat capacity CA, given in Eq. (10), i.e., CA=\u0000dE\ndT\u0001\nA,\nequivalently, CA=\u0000dE\ndT\u0001\nR, we find from Eq. (8) that\nCA=1\n2√\nV(R)\u0000dm2\ndT\u0001\nR, where it was used the expres-\nsion V(R) = 1 −2m2\nR−ΛR2\n3given in Eq. (2). Thus,\nusing the expression for m2just found above we have\nCA=1\n2√\nV(R)1\n2(1−ΛR2)2\n16π2T3Rand sincep\nV(R) =1−ΛR2\n4πRTit\ngives\nCA+2=1−ΛR2\n4πT2>0, (32)\nso that CA+2is small and positive. The large black hole\nr+2solution is thus stable.8\nC. Thermodynamic phases and phase transitions\nbetween hot Schwarzschild-de Sitter and hot de\nSitter in the ΛR2<1case\nWe now work out the thermodynamic phases and\nphase transitions for the Λ R2<1 case. The discus-\nsion is valid for the thermodynamically stable black\nhole, the black hole r+2, since the unstable one r+1\nhas at most a fleeting existence and could not count\nfor a phase. From Eq. (6) we get that the action I\nfor a hot Schwarzschild-de Sitter r+2phase is ISdS=\nβR\u0010\n1−p\nV(R)\u0011\n−πr2\n+2, where from Eq. (7) we have\nV(R) =\u0000\n1−r+2\nR\u0001\u0010\n1−ΛR2\n3\u0010\n1 +r+2\nR+\u0000r+2\nR\u00012\u0011\u0011\n, with\n0≤ΛR2<1 here. The free energy F=I\nβ=ITfor hot\nSchwarzschild-de Sitter is then\nFSdS=R\u0010\n1−p\nV(R)\u0011\n−πTr2\n+2, 0≤ΛR2<1.\n(33)\nAnother phase that might exist is hot de Sitter, in which\ncaser+= 0, V(R) = 1−ΛR2\n3and the action is IHdS=\nβR\u0012\n1−q\n1−ΛR2\n3\u0013\n. The free energy is then\nFHdS= \n1−r\n1−ΛR2\n3!\nR , 0≤ΛR2<1.\n(34)\nIn the canonical ensemble, for systems characterized by\nthe size and the temperature of the heat reservoir, the\nphase that has lowest Fis the phase that dominates. So,\nthe hot Schwarzschild-de Sitter black hole phase domi-\nnates over hot de Sitter, or the two phases coexist equally,\nwhen\nFSdS≤FHdS, 0≤ΛR2<1, (35)\ni.e.,\u0010\n1−p\nV(R)\u0011\nR−πTr2\n+2≤\u0012\n1−q\n1−ΛR2\n3\u0013\nR,\ni.e.,\nRT≥q\n1−ΛR2\n3−p\nV(R)\nπr2\n+2\nR2, 0≤ΛR2<1.(36)\nWe see that Eq. (35) is an implicit equation, because\nr+2=r+2(R, T). For each Λ R2in the interval above,\nand for each RTone gets an r+2, which can then be\nput into Eq. (36) to see whether the Schwarzschild-de\nSitter phase dominates over the de Sitter phase or not.\nIn the case it dominates then a black hole can nucleate\nthermodynamically from hot de Sitter space.\nFor Λ R2≪1 we can find some interesting numbers.\nOne finds equality between the actions, i.e., that FSdS=\nFHdS, see Eqs. (35) and (36), when RT= (RT)eqwith\n(RT)eq=27\n32π\u0012\n1−13\n486ΛR2\u0013\n, ΛR2≪1,(37)valid in first order, as all equations in the discussion\nbelow will be valid in this order. It is of interest\nto put this value in decimal notation, i.e., ( RT)eq=\n0.269\u0000\n1−0.027Λ R2\u0001\n, approximately. Forr+2 eq\nRone has\nin this case that\nr+2 eq\nR=8\n9\u0012\n1 +77\n729ΛR2\u0013\n, ΛR2≪1.(38)\nIn decimal notation, this can be put asr+2 eq\nR=\n0.889\u0000\n1 + 0 .106Λ R2\u0001\n, approximately. Thus, FSdS≤\nFHdS, see Eqs. (35) and (36), when\nRT≥(RT)eq, , (39)\nand when\nr+2\nR≥r+2 eq\nR. (40)\nSo, when the inequalities given in Eqs. (39) and (40) hold,\nthen the black hole phase dominates, but nevertheless the\nhot de Sitter phase has some probability of turning up.\nThere is another radius of interest here, which although\nnot strictly thermodynamic, it appears through dynam-\nical arguments, and is important in this discussion of\nphases and phase transitions. For matter or energy en-\nclosed in a box, which one can consider that it config-\nures a star, there is a mass, or energy, above which the\nstar cannot support its self gravity and tends to col-\nlapse. This is called the Buchdahl limit which for spaces\nwith positive cosmological constant has been calculated\nin [51]. Here one should envisage this limit as giving,\nfor a given Rfixed, the mass mBuch above which the\nenergy within the system is so large that the system col-\nlapses. For a given Rand Λ, mBuch ismBuch\nR=2\n9+\n2\n9√\n1 + 3Λ R2−ΛR2\n3. Since m=r+\n2\u0010\n1−Λr2\n+\n3\u0011\n, see Eq. (3)\none has the Buchdahl limit is given by the equation\nr+Buch\nR\u0010\n1−\u0000r+Buch\nR\u00012ΛR2\n3\u0011\n=4\n9+4\n9√\n1 + 3Λ R2−2ΛR2\n3,\nwhich is a cubic equation forr+Buch\nRthat can in principle\nbe solved. Givenr+Buch\nRone can then work out what is\nthe temperature ( RT)Buch that yields the related black\nhole with radiusr+2\nR. Here we are dealing with small\nΛR2. in this case one gets\n(RT)Buch=27\n32π\u0012\n1 +985\n486ΛR2\u0013\n, ΛR2≪1.\n(41)\nOne further has\nr+Buch\nR=8\n9\u0012\n1 +64\n81ΛR2\u0013\n, ΛR2≪1.(42)\nThese two values can be put in decimal notation\nas (RT)Buch = 0.269\u0000\n1 + 2 .027 Λ R2\u0001\nandr+Buch\nR=\n0.889\u0000\n1 + 0 .790 Λ R2\nBuch\u0001\n, approximately. There is col-\nlapse when for a given Λ and a given RandTone has\nRT≥(RT)Buch. (43)9\nand\nr+2\nR≥r+Buch\nR(44)\nSo, there is collapse if for a given Λ, RandTEqs. (43)\nand (44) are obeyed. Now, interestingly enough, com-\nparing Eq. (37) with (41) and Eq. (38) with (42) we see\nthat\n(RT)Buch>(RT)eq. (45)\nand\nr+Buch\nR>r+ eq\nR. (46)\nThus, one has that for sufficiently high temperatures the\nblack hole is a dominant phase but not the unique, hot\nde Sitter might pop up, and for even higher temperatures\nthen the black hole is the unique phase as the system\ntends to collapse. A comment is in order. The Buch-\ndahl bound applies to a self-gravitating mechanical sys-\ntem consisting of a ball of radius Rcontaining matter.\nFor a fixed R, the bound determines the maximum value\nof the gravitational radius r+2, i.e., the maximum mass or\nenergy within R, above which the system collapses. The\nsystem we are working with is a thermodynamic system,\nwith a boundary that has radius Rand temperature T\nfixed. In the approximation we are using, the system\ncontains no matter with a black hole appearing as a re-\nsult of thermodynamically imposed data in the ensemble,\nnot as a result of a dynamic process. Nevertheless, one\ncan think in going to the next order of approximation,\nwhere now the system contains lumps of energy or par-\nticles. In this case, for fixed R, there is a maximum\nvalue for the energy within R, above which the gravita-\ntional radius r+2is higher than the value permitted by\nthe Buchdahl bound, and one can infer that the system\nmust collapse. This reasoning is plausible, however it\ncomes from dynamical arguments, and as such is outside\nthe thermodynamic approach we are using.\nSo, we have the following picture for fixed and tiny\nΛR2. For 0 ≤RT <√\n27\n8π\u0000\n1−5\n54ΛR2\u0001\n, see Eq. (19)\nthere is only hot de Sitter space. For√\n27\n8π\u0000\n1−5\n54ΛR2\u0001\n≤\nRT <27\n32π\u0000\n1−13\n486ΛR2\u0001\n, see Eqs. (20) and (37), hot\nde Sitter space is a phase that dominates over the\nSchwarzschild-de Sitter black hole phase, where the black\nhole is the large one, the small one being unstable is\nof no interest in this context. For27\n32π\u0000\n1−13\n486ΛR2\u0001\n=\nRT, the Schwarzschild-de Sitter black hole and the\npure de Sitter phases coexist equally, see Eq. (37).\nFor27\n32π\u0000\n1−13\n486ΛR2\u0001\n< RT <27\n32π\u0000\n1 +985\n486ΛR2\u0001\n, the\nSchwarzschild-de Sitter black hole phase dominates over\nthe pure de Sitter phase. For27\n32π\u0000\n1 +985\n486ΛR2\u0001\n≤\nRT < ∞, see Eqs. (41) and (45), there is only the\nSchwarzschild-de Sitter black hole phase, the system suf-\nfers total gravitational collapse. Note that in the phase\ntransition from hot de Sitter to the Schwarzschild-de Sit-\nter black hole phase there is topology change, since herethe Euclidean topology of hot de Sitter is S1×R3, and the\nEuclidean topology of the Schwarzschild-de Sitter black\nhole is R2×S2.\nThe case Λ R2= 0 is a particular case of the Λ R2≪1\ncase just shown. Nevertheless, there are interesting as-\npects worth mentioning. In this case the thermodynamic\nphases are hot flat space and the Schwarzschild black\nhole. The Schwarzschild black hole phase dominates, or\nthe two phases coexist equally, when FSchw≤FHFS. We\ncan take it directly to be RT≥1−√\nV(R)\nπr2\n+2\nR2, where here\nV(R) = 1−r+2\nR. From the calculations above one finds\nthat for 0 ≤RT <√\n27\n8πthere is only hot flat space. For√\n27\n8π≤RT <27\n32π, hot de Sitter space is a phase that dom-\ninates over the Schwarzschild-de Sitter black hole phase.\nFor27\n32π=RT, the Schwarzschild-de Sitter black hole and\nthe pure de Sitter phases coexist equally. Now, note that\nr+2\nR=8\n9is the radius where the two actions, for hot flat\nand Schwarzschild spaces, have the same value, which is\nzero in this case [6]. But this value is in fact equal to\nthe Buchdahl limiting radius in general relativity, as was\nfound for dimensions d≥4 in [15–17]. The fact that\nthe thermodynamic existence of a black hole phase and\nthe Buchdal limit coincide in this case is an interesting\nand unexpected property, and it can help to compare\nboth processes, thermodynamic an dynamic, of forming\na black hole. Then, for slightly higher temperatures, one\ncan infer that when the Schwarzschild black hole phase\ndominates, it actually has sufficient energy to collapse\nitself to a black hole, i.e., when the two phases, hot flat\nspace and Schwarzschild black hole, start to coexist, the\nblack hole phase actually dominates completely, since the\nsystem has sufficient energy to collapse to a black hole.\nThus, here, for27\n32π< RT < ∞, the Schwarzschild black\nhole phase should be the only phase that exists, as the\nsystem must suffer total gravitational collapse.\nFor generic Λ R2in the range Λ R2<1 a similar anal-\nysis can be made, but we do not do it here.\nD. Comments on the ΛR2<1case\nWe note that although the analysis made in Eqs. (19)-\n(22) is valid for a very small cosmological constant Λ R2,\nand the analysis made in Eqs. (23)-(26) is valid for a\ncosmological constant Λ R2near unity from below, one\ncan have a good idea of the behavior of the solutions as\nΛR2is increased from zero up to a value less than unity.\nThis is also done with the help with the results of the\nhigh temperature limit Eqs. (27)-(32).\nFor Λ R2= 0, i.e., for zero cosmological constant, we re-\ncover York’s results for a heat reservoir in a Schwarzschild\nblack hole space. In this case, when RT <√\n27\n8πthere are\nno black hole solutions, only hot flat space, and when\nRT≥√\n27\n8πthere are two solutions, the small black hole\nr+1which is unstable, and the large black hole r+2, which\nis stable, the two solutions are the same when RT=√\n27\n8π10\nwith horizon radiusr+1\nR=r+2\nR=2\n3. Still for Λ R2= 0,\nwhen RTis very high, i.e., the temperature Tof the\nheat reservoir is very high, thenr+1\nRis small tending to\nzero andr+2\nRis large tending to one. The behavior of\nthe Λ R2= 0 can be heuristically explained through the\nthermal wavelength of the radiation λand the size of the\nreservoir R. For small T, one has that the correspond-\ning thermal wavelength λ≡1\nTis high relative to R. In\nfact, RTsmall meansR\nλsmall, i.e.,λ\nRhigh, so that the\nwavelength of the thermal energy packets is stuck to the\nwalls of the reservoir, and the packets cannot collapse to\nform a black hole. For higher T,λis small relative to R.\nThus, RTlarge meansR\nλlarge, i.e.,λ\nRlow, so that the\nwavelength of the thermal packets is sufficiently small,\nthe packets are free inside the reservoir, and eventually\ncollapse to form a black hole.\nFor Λ R2fixed and tiny we can spell the results as\nwell. Now the space of black hole solutions is over a two-\ndimensional domain, specifically, RTand Λ R2. When\nRTis less than some number, which itself is smaller than√\n27\n8π, then there are no solutions, only hot de Sitter space,\nand when RTis larger than this same number, which it-\nself is smaller than√\n27\n8π, then there are two solutions, the\nsmall black hole r+1, which is unstable, and the large\nblack hole r+2, which is stable. When RTis very high,\ni.e., when the temperature Tof the heat reservoir is very\nhigh, thenr+1\nRis small tending to zero andr+2\nRis large\ntending to one. The two solutions are the same solu-\ntion when RT=√\n27\n8π\u0000\n1−5\n54ΛR2\u0001\n, ΛR2being the fixed\nand tiny value, with coincident horizon radius given by\nr+1\nR=r+2\nR=2\n3\u0000\n1 +17\n81ΛR2\u0001\n. Moreover, from the re-\nsult that the coincident horizon radius has the value just\ngiven, one can deduce that as one goes along increasing\nΛ, specifically, increasing Λ R2, for fixed RT, the radii\nr+1\nRandr+2\nRincrease. This behavior for spaces with tiny\nΛR2can be heuristically explained through the thermal\nwavelength, the size of the reservoir, and the cosmologi-\ncal length. For small T, one has that the corresponding\nthermal wavelength λ≡1\nTis high relative to R. In fact,\nRTsmall meansR\nλsmall, i.e.,λ\nRhigh, so that the wave-\nlength of the thermal energy packets is stuck to the walls\nof the reservoir, and the packets cannot collapse to form a\nblack hole. But now, due to the new cosmological length\nscale ℓset by Λ, ℓ=1√\nΛ, the space inside the reservoir\nis more curved and, so to speak, a bit larger, and thus a\nlower T, i.e., a higher λ, is allowed so that they are free\nto collapse inside the reservoir and form a black hole in\nthis case.\nFor Λ R2fixed, not tiny and less than one, we can de-\nduce several results. Solutionsr+1\nRandr+2\nRstart to ap-\npear at a certain RTwhich is ever decreasing as Λ R2is\nincreasing, and when Λ R2is close to one then one finds\nthatRTis close to and a bit higher than1\n2π. Thus, when\nRTis small one can find black hole solutions for spaces\nwith cosmological constant near one, but there are no\nblack holes for spaces with any other lower cosmological\nconstant. Moreover, for RTclose to1\n2π, thenr+1\nRandr+2\nRare very near one and merge at1\n2π. In addition, for\nfixed RTas one goes along increasing Λ, i.e., increasing\nΛR2, then the radiir+1\nRandr+2\nRincrease. The solution\nr+1\nRis the small solution and the solutionr+2\nRis the large\nsolution that for Λ R2close to one yieldsr+2\nRclose to one.\nWhen RTis very high, i.e., when the temperature Tof\nthe heat reservoir is very high, thenr+1\nRis small tending\nto zero andr+2\nRis large tending to one. This behavior\nof ΛR2fixed, not tiny and less than one, can be heuris-\ntically explained through the thermal wavelength, the\nsize of the reservoir, and the cosmological length. As the\ntemperature gets lower and lower, the associated thermal\nwavelength λ=1\nTgets higher and higher, and to have\nblack hole solutions the space needs to be more curved,\nand so larger, to accommodate those wavelengths λand\nallow the corresponding thermal energy packets to col-\nlapse. For very low temperatures, in the limit RT→1\n2π,\ni.e.,λ\nR= 2π, energy packets can only build a black hole\nhorizon for sufficiently high Λ, i.e., for Λ R2→1. This\nfirst black hole that appears at RT→1\n2πand Λ R2→1\nhas large horizon radius given byr+1\nR=r+2\nR→1. For\nhigher T, i.e., higher RT, then the wavelength of the en-\nergy packets is sufficiently small that allows for black hole\nsolutionsr+1\nRandr+2\nR.\nTo sum up, in the Λ R2<1 case, for sufficiently high\ntemperatures there are two solutions, the small mass\nbranch with black hole horizon radius r+1which is un-\nstable, and the massive branch with black hole horizon\nradius which is stable, a fact the holds for any temper-\nature RTand any Λ R2<1. This is similar to what\nhappens in the thermodynamics of pure Schwarzschild,\ni.e., Λ R2= 0. As Λ R2is increased from zero, black\nholes can form with less and less temperatures RT, and\nfor Λ R2near one, can form black holes with the least\ntemperature, namely, RTtending to1\n2π.\nThe case with Λ R2= 1 precisely has to be dealt as a\nseparate case, as an intermediate value case for Λ R2. As\nwe will show it reserves interesting surprises.\nIV. THERMODYNAMICS OF\nSCHWARZSCHILD-DE SITTER BLACK HOLES\nIN THE CANONICAL ENSEMBLE:\nINTERMEDIATE VALUE OF THE\nCOSMOLOGICAL CONSTANT, ΛR2= 1, THE\nNARIAI UNIVERSE INSIDE THE HEAT\nRESERVOIR\nA. Solutions and the Euclidean metric\nWe treat here the intermediate positive cosmological\nconstant, Λ R2= 1, problem. Again, we put the bound-\nary of a spherical cavity with a black hole in a positive\ncosmological constant background inside a heat reservoir,\nat some radius R, where it is also specified a fixed tem-\nperature T, see Fig. 1 anew. The intermediate value of11\nthe cosmological constant is precisely\nΛR2= 1. (47)\nFor this value of Λ R2, and for fixed RTone can find\nsolutions of Eq. (13) for black hole horizon radii r+ana-\nlytically. However, when Λ R2= 1 the problem has to be\ntreated with care. There are still two solutions, r+1and\nr+2.\nThe solution r+1, the small black hole solution, can\nbe taken directly from Eq. (13) putting Λ R2= 1 which\nis then 4 πRT =1\nr+\nR1−(r+\nR)2\nq\n1−r+\nRr\u0010\n1−1\n3\u0010\n1+r+\nR+(r+\nR)2\u0011\u0011. This\nequation can be transformed into a quartic equation in\nr+\nRyielding then the solution r+1, the small and unstable\nsolution.\nThe solution r+2, the large black hole solu-\ntion, needs special attention. One cannot sim-\nply put Λ R2= 1 into Eq. (13), i.e., 4 πRT =\n1\nr+\nR1−ΛR2(r+\nR)2\nq\n1−r+\nRr\u0010\n1−ΛR2\n3\u0010\n1+r+\nR+(r+\nR)2\u0011\u0011, which gives directly\nthe solutionr+2\nR= 1 with RT=1\n2π. The correct way\nis to take the limit Λ R2→1 andr+2\nR→1. Then, RT\ncan have a broad range of values. One then also finds\nthat, forr+2\nR→1, the cosmological radius solution can\nbe taken from Eq. (5) to giverc2\nR→1. This limit takes\nus to the Nariai universe, which we now find. The Nariai\nuniverse is to be seen as Schwarzschild-de Sitter black\nhole with maximal mass, it is an extremal case.\nThe Nariai solution can be found from the\nSchwarzschild-de Sitter solution in the limit that the two\nhorizons r+2andrc2coincide, see also Appendix A. Here,\nwe have a heat reservoir at Rthat acts as a reservoir\nfor the inside region, the region containing a black hole.\nThis heat reservoir, at R, is in between r+2andrc2,\nand thus the limit we want to take is such that r+2,R,\nandrc2coincide, see Appendix B for detail. We drop\nthe subscript 2in the following analysis. Now, if we\ndor+\nR→1, then Eq. (12), T=TH\n+√\nV(R), together with\nEq. (4), V(r) =\u0000\n1−r+\nr\u0001\u0010\n1−Λr2\n3\u0010\n1 +r+\nr+\u0000r+\nr\u00012\u0011\u0011\n,\ngives at face value that the heat reservoir is at very\nhigh temperature T. But, there is a way to have Tfi-\nnite withr+\nR→1. From Eq. (12) we see that if we\ndor+\nR→1 concomitantly with TH\n+→0 then Tis fi-\nnite. Since TH\n+=1\n4πr+\u0000\n1−Λr2\n+\u0001\n, see Eq. (11), TH\n+→0\nmeans 1 −Λr2\n+→0, but sincer+\nR→1 this also means\n1−ΛR2→0. In brief, in this limit we haver+\nR→1\nand Λ R2→1, both from below and both of the same\ninfinitesimal order. Then, Eq. (12), T=TH\n+√\nV(R), gives\nin this limit that T=1\n2πR(1−r+\nR)+(1−√\nΛR2)q\n1−r+\nRq\n(1−r+\nR)+2(1−√\nΛR2).\nSince 1 −√\nΛR2and 1−r+\nRare infinitesimal in this limit,\nwe see that Tis finite and can have a range of values de-\npending on the precise infinitesimal values of 1 −√\nΛR2and 1 −r+\nR. One more thing. We have deduced that\nin this limit r+→R→1√\nΛ, so that from Eq. (5) one\nhasrc→1√\nΛ, and since1√\nΛ→R, then rc→R. So, in\nthe limit we have r+=R=rc. To see that this is the\nNariai limit with a reservoir Rin the middle, we have to\ndo some work on the original line element, Eqs. (1) and\n(2). We present the results below, see Appendix B for a\ndetailed derivation.\nLet us start. The reservoir temperature Tis also the\nlocal Tolman temperature TatRand has an associated\nexpression given by T=TH\n+√\nV(R), with TH\n+being the tiny\nHawking temperature as we have just found. Expand-\ning the metric potential V(r) of Eq. (2) near r+in a\nTaylor series gives V(r) = 4 πTH\n+(r−r+)−1\nR2(r−r+)2,\nplus higher order terms. Make now the transformations\n(t, r)→(¯t, z) as r−r+= 4πTH\n+R2sin2\u00001\n2arcos(z\nR)\u0001\nandt=¯t\n2πTH\n+Rwith 0 ≤t≤1\nTH\n+corresponding to\n0≤¯t≤2πRand r+≤r≤Rcorresponding to\n−R≤z≤Z. Then, since V(r) is actually a V(r, r+),\nwe have here V(r) =V(r, r+) =V(r−r+) =V(z) =\n(2πTH\n+R)2sin2\u0000\narcos(z\nR)\u0001\n= (2πTH\n+R)2\u0010\n1−z2\nR2\u0011\n. From\nthe original Schwarzschild-de Sitter line element, Eqs. (1)\nand (2), together with Eq. (3), and dropping the bar in ¯t\nwhich is now meaningless, we obtain then the Nariai line\nelement, i.e.,\nds2=V(z)dt2+dz2\nV(z)+R2\u0000\ndθ2+ sin2θ dϕ2\u0001\n,\n0≤t≤2πR , −R < z < Z , (48)\nwhere Zis now the heat reservoir boundary in the z-\ndirection, the other coordinates are in the range 0 ≤θ≤\nπ, 0≤ϕ <2π, and the metric potential V(z) being given\nby\nV(z) = 1−z2\nR2. (49)\nThe line element given in Eqs. (48) and (49) corresponds\nto the Nariai universe, which can be seen to be decom-\nposable into a two-dimensional de Sitter space times a\nsphere. So, the ensemble with its boundary data, Tand\nR, provide automatically the range of coordinates of the\nsolution. Note also that the range of values for the heat\nreservoir boundary Zis−R≤Z≤R. From Eq. (49)\nwe see that there are two horizons, one is z+=−R, the\nother is zc=R, but the subscripts now are just names,\nsince the two horizons are of the same type. The topology\nof the Nariai universe in Euclideanized form is R2×S2\nand its boundary has S1×S2topology, where the S1\nsubspace has proper length1\nT.\nNow, the temperature Tis\nT=TH\n+p\nV(Z), (50)\nwith TH\n+being the Hawking temperature given by TH\n+=\nκ\n2π, and κbeing the surface gravity of the black hole12\nhorizon. For the metric (48) one has κ=1\n2V′(z+), and\nso the Hawking temperature for the + horizon is TH\n+=\n1\n4π\u0000dV\ndz\u0001\nz+. Using Eq. (48) we get\nTH\n+=1\n2πR. (51)\nAs well, from Eq. (49) we have that the metric potential\nat the heat reservoir boundary Zis\nV(Z) = 1−Z2\nR2. (52)\nSo, Eqs. (50)-(52) give that the reservoir temperature is\ngiven by T=1\n2πRq\n1−Z2\nR2, where Zis the boundary on\nthezcoordinate, see Fig. 2 for a representation of the\nNariai universe in a heat reservoir. For a given Tat the\nboundary, for the ensemble, there are two solutions of\nthis equation, in general. Namely, one solution is for Z\nbetween −Rand 0 and the other for Zbetween 0 and\nR. These two solutions yield different physical situations\nof course, as the reservoir boundary Zis put in different\npositions relatively to z+. Note that in Schwarzschild-\nde Sitter space, the two solutions were for the horizon\nr+, one small r+1the other large r+2, both relative to\nthe reservoir R. Here, the two solutions are not for the\nhorizons but instead for the boundary Z, one Z1, the\nother Z2, with\nZ1=−Rs\n1−1\n(2πRT)2, (53)\nand\nZ2=Rs\n1−1\n(2πRT)2, (54)\nnow both relative to the horizon z+. Within the two\nchoices for the boundary, Z1orZ2, we can pick the one\nZ\nidentify identify\nz+R R\nhot NariaiTheat reservoircz\nFIG. 2: A drawing of a Nariai horizon z+within a heat\nreservoir at temperature T, with cylindrical radius R,\nand situated at Z. The cosmological horizon zcis\nsituated beyond the heat reservoir. The Euclideanized\nspace and its boundary have R2×S2andS1×S2\ntopologies, respectively, where the S1subspace with\nproper length β=1\nTis not displayed. See text for more\ndetails.we wish. In addition, since z+andzcare indistinguish-\nable, in the sense they are of the same type, we can also\ninterchange z+with zc, in which case the situation would\nbe the same. The boundary has always area radius R,\nwhich together with T, forms the data for the canonical\nensemble.\nFrom Eqs. (53) and (54) we see that for\nRT <1\n2π, (55)\nthere are no solutions for Z1orZ2, so in this case the\nboundary Zdoes not exist, a reservoir does not exist, one\ncannot define a temperature Tanywhere, and so there is\nno thermodynamic Nariai solution. Presumably one has\nsimply hot de Sitter space inside R. In decimal notation\nEq. (55) is RT < 0.159, approximately.\nFrom Eqs. (53) and (54) we see that for\nRT≥1\n2π, (56)\nthere are two Nariai solutions, one with one horizon z+=\n−Rand boundary Z1, the other with one horizon z+=\n−Rand boundary Z2, both boundaries can be picked\nup. When the equality sign holds in Eq. (56) there is one\nsolution with Z1=Z2= 0, so in this case the boundary\nZpops up in the middle, at Z= 0, and so −R≤z≤0.\nIn this case, The reservoir is at Z= 0, has radius R, and\nthe horizon is at z+=−R.\nThe generic Tolman temperature formula in the Nariai\nspace is T(z) =1\n2πRq\n1−z2\nR2for−R≤z < Z , with the\nreservoir temperature Tbeing expressed as T≡T(Z).\nSo,T(z=−R) =∞as expected since z=−Ris a\nhorizon, it is the horizon z+. Increasing zfrom−Rone\nsees that T(z) decreases and stops if one picks Z1, and if\none picks Z2it decreases up to z= 0, and then increases\nback up to Z2. In case Z2=R, then T(Z2=R) =∞\nas is expected since z=Ris a horizon, it is the horizon\nzc. Clearly, the horizons are given by z+=−Rand\nzc=R, so they do not depend on T. The dependence on\nTis transferred to the boundary Z, so the structure has\nchanged from that of the Schwarzschild-de Sitter.\nWe now list the most relevant general formulas for the\nthermodynamics of Nariai. These can be taken directly\nfrom the equations provided in Sec. II. The action Iis\nnow\nI=βR−πR2. (57)\nThe action and the free energy are related by I=βF, so\nF=R−TπR2.Now, F=E−TS, so the thermodynamic\nor quasilocal energy here also the thermal energy at Ris\nE=R. (58)\nThe entropy is\nS=πR2, (59)13\nand is independent of z+. The heat capacity CA=\u0000dE\ndT\u0001\nAis\nCA= 0, (60)\nso there is neutral thermodynamic equilibrium in the\nNariai universe. That CA= 0 can be taken directly from\nEq. (58) which shows that the energy Ehas no depen-\ndence on the temperature T.\nB. High-temperature limit\nFor the value of the cosmological constant considered\nin this section, Λ R2= 1, one can work out the limit in\nwhich RTgoes to infinity, see Eqs. (53) and (54). Since\nRis the quantity that we consider as the gauge, RT\ngoing to infinity is the same in this context as Tgoing\nto infinity. Let us then find explicit results by taking the\nlimit of high temperature.\nFor very high T, or very high TR, one has from Eq.(53)\nthe possibility\nZ1\nR=−1 +1\n21\n(2πRT)2, (61)\nwith1\n21\n(2πRT)2≪1. In this case Z1is very near the hori-\nzonz+=−Rand one finds that the space in-between\nthe horizon and the reservoir is essentially a Rindler\nspace. To see this, note that in this limit 1 −z2≪1\nfor 0 ≤z≤Z1. So with a ¯ zcoordinate defined by\n1−z2= ¯z2one obtains from Eqs. (48) and (49) the line\nelement ds2= ¯z2dt2+d¯z2+R2\u0000\ndθ2+ sin2θ dϕ2\u0001\n, i.e.,\nthe Euclidean Rindler line element. For very high T, or\nvery high TR, one has from Eq.(54) the other possibility\nZ2\nR= 1−1\n21\n(2πRT)2, (62)\nwith1\n21\n(2πRT)2≪1. In this case the boundary Z2is very\nnear the horizon zc=R, but since −R≤z≤Z2< zc\nthe space in-between the horizon and the reservoir is not\ngenerically a Rindler space, it approximates the Rindler\nline element only within the region near Z2.\nC. Thermodynamic phases and phase transitions\nbetween hot Nariai and hot de Sitter in the ΛR2= 1\ncase\nWe now work out thermodynamic phases and phase\ntransitions for the Λ R2= 1 case. The discussion is valid\nfor the thermodynamically stable black hole, the black\nholer+2, since the unstable one r+1has at most a fleeting\nexistence and could not count for a phase. Here the black\nholer+2is in fact a Nariai universe.\nFrom Eq. (57) we see that the action Ifor Nariai is\nINariai =βR−πR2, where we have used that in Nariaione has r+2=R. So the free energy F=I\nβ=ITfor a\nhot Nariai phase is\nFNariai =R−πR2T , ΛR2= 1. (63)\nAnother phase that might exist is hot de Sitter, in which\ncaser+= 0, V(R) = 1−ΛR2\n3=2\n3as ΛR2= 1, and the\naction is IHdS=βR\u0010\n1−q\n2\n3\u0011\n. The free energy is then\nFHdS= \n1−r\n2\n3!\nR , ΛR2= 1.(64)\nIn the canonical ensemble the phase that has lowest Fis\nthe phase that dominates. So, the Nariai universe domi-\nnates over hot de Sitter space, or the two phases coexist\nequally, when\nFNariai≤FHdS, ΛR2= 1. (65)\nOne finds equality between the two actions when R−\nπR2T=\u0010\n1−q\n2\n3\u0011\nR, i.e., when\n(RT)eq=√\n2√\n3π, ΛR2= 1. (66)\nIn decimal notation, this is ( RT)eq= 0.260, approxi-\nmately. So Nariai prevails over hot de Sitter, or the two\nphases coexist equally, when\nRT≥√\n2√\n3π, ΛR2= 1. (67)\nRecall that for RT <1\n2π, there are no Nariai solutions\nonly hot de Sitter, see Eq. (55), and for RT≥1\n2π, see\nEq. (56), two possible cases pop up, the unstable black\nhole which is of no interest here, and the Nariai universe\nwhich is neutrally stable and of interest here. So, for\nΛR2= 1 we have the following picture. For 0 ≤RT <1\n2π\nhot de Sitter is mandatory. For1\n2π≤RT <√\n2√\n3πhot de\nSitter prevails as a thermodynamic phase over Nariai, so\nthat if the phase is a Nariai one, it will probably tran-\nsition to a hot the Sitter phase. For√\n2√\n3π=RThot de\nSitter and Nariai coexist as thermodynamic phases. For√\n2√\n3π≤RT < ∞Nariai prevails as a thermodynamic\nphase over hot de Sitter. Note that in the phase transi-\ntion from hot de Sitter to hot Nariai or from hot Nariai\nto hot de Sitter there is topology change, since the Eu-\nclidean topology of hot de Sitter is S1×R3, and the\nEuclidean topology of Nariai is R2×S2.\nD. Comments on the ΛR2= 1case\nIt is really interesting that the resulting metric inside\nthe heat reservoir is described by the Nariai metric. The\nprocedure of obtaining it in our context is completely14\ndifferent from the usual procedure. The heat reservoir\nradius Rand the temperature Tplay a crucial role here,\nand so the limit to the the Nariai universe is naturally\nrelated to thermodynamics. As we have just seen, the\nNariai solution is of utmost importance in any analysis\nof the Schwarzschild-de Sitter space in the canonical en-\nsemble.\nA feature of great importance in the overall picture\nis the minimum temperature T, i.e., RT, of the ensem-\nble above which there are Nariai solutions for a given Λ,\ni.e., Λ R2. For RT <1\n2πthere are no black hole solu-\ntions whatsoever for any Λ R2, specifically, there are no\nsmall black hole solutions with horizon radiusr+1\nRnei-\nther Nariai universes withr+2\nR= 1. Indeed, from the\nequations found above for the heat reservoir boundary of\nthe Nariai universe, namely, Z1=−Rq\n1−1\n(2πRT)2and\nZ2=Rq\n1−1\n(2πRT)2, we see that for RT <1\n2π, there are\nno solutions for Z1orZ2. So in this case the boundary\nZdoes not exist, a reservoir does not exist, one cannot\ndefine a temperature Tanywhere, and so there is no ther-\nmodynamic Nariai solution. Presumably one has simply\nhot de Sitter space inside R. The reason is clear if one\nthinks in thermal wavelengths. For very small tempera-\ntures, the associated thermal wavelength is very long and\nthere is no boundary Zthat can accommodate such cor-\nresponding thermal energy packets. For RT=1\n2πthere\nis one solution only, it has Λ R2= 1 precisely. There\nare no solutions at this temperature of any other Λ R2.\nSo, a Nariai solution is the first solution to pop up as\none increases the temperature from absolute zero. As\none increases RTabove1\n2π, solutions with Λ R2different\nfrom one start to appear, first for Λ R2near one, than for\nΛR2far from one as RTis further increased as we have\ndiscussed in the previous section for Λ R2<1.\nThe high temperature limit for the Nariai universe, i.e.,\nfor Λ R2= 1, connects with the cases Λ R2<1. Indeed,\nwhen Λ R2<1, for RTvery large, there are two solutions,\nthe very small oner+1\nRtending to zero, and the very large\noner+2\nRtending to one. When Λ R2= 1 only the very\nlarge one,r+2\nR, satisfies the conditionr+2\nR= 1, necessary\nto get a Nariai universe.\nIt remains to be found the spectrum of solutions for\nΛR2>1. We turn to this problem now, and it happens\nthat there are unexpected results.\nV. THERMODYNAMICS OF\nSCHWARZSCHILD-DE SITTER BLACK HOLES\nIN THE CANONICAL ENSEMBLE: LARGE\nVALUES OF THE COSMOLOGICAL CONSTANT,\nΛR2>1\nA. Solutions\nWe treat here the large positive cosmological constant,\nΛR2>1, problem. Again, we put the boundary of a\nspherical cavity with a black hole in a positive cosmolog-ical constant background inside a heat reservoir, at some\nradius R, where it is also specified a fixed temperature\nT, see Fig. 1 anew. Large positive cosmological constant\nmeans exactly in this context that\n1<ΛR2≤3. (68)\nWithin this range, for fixed RTand generic Λ R2, we can\ndraw the result that looking at Eq. (13) we can ascertain\nwith surprise that for Λ R2>1 there are solutions with\nr+\nR<1. Nonetheless, it is hard to find analytic solutions\nof Eq. (13) for black hole horizon radii r+. However,\nfor Λ R2very near one from above, one can make some\nprogress.\nFor Λ R2a tiny bit larger than one, i.e., for (Λ R2−1)≪\n1, there are no black hole solutions for\nRT <1\n2π \n1 +\u00123\n8\u0000\nΛR2−1\u0001\u00131\n3!\n,(ΛR2−1)≪1,(69)\nonly hot de-Sitter space, valid in this order of approxi-\nmation, as all equations in this context below are valid to\nthis order. Still for small Λ R2−1, i.e., for (Λ R2−1)≪1,\nthere are two black hole solutions for\nRT≥1\n2π \n1 +\u00123\n8\u0000\nΛR2−1\u0001\u00131\n3!\n,(ΛR2−1)≪1,\n(70)\nOne of the two solutions is the small black hole\nr+1(R,Λ, T), and the other solution is the large black\nhole r+2(R,Λ, T). The plus sign inside the parenthesis\nin Eq. (70) is what one expects really. The two solutions\nmerge into one sole solution when the equality sign in\nEq. (20) holds. In this case the coincident double solu-\ntion has horizon radius given by\nr+1\nR=r+2\nR= 1−\u00123\n8(ΛR2−1)2\u00131\n3\n,(ΛR2−1)≪1.(71)\nThe corresponding cosmological radius can then be found\nfrom Eq. (5) to be given by\nrc1\nR=rc2\nR= 1 +\u00123\n8(ΛR2−1)2\u00131\n3\n,(ΛR2−1)≪1.(72)\nOne can work out in this order in Λ R2−1 the action\nI, the energy E, the entropy S, and the heat capacity\nCA, given through Eqs. (6)-(10). Apart from the entropy\nS= 4πr2\n+for each of the two black hole solutions, the\nother quantities would not be particularly illuminating.\nAn instance where these quantities can be worked out, in\nparticular the heat capacity CA, is the hight temperature\nlimit to which we now turn.\nB. High temperature limit: Analytical solution\nFor the range of high values of the cosmological con-\nstant considered in this section, 1 <ΛR2≤3, one can15\nfind solutions in the limit in which RTgoes to infinity,\nsee Eqs. (12) and (13). Since Ris the quantity that we\nconsider as the gauge, RTgoing to infinity is the same\nin this context as Tgoing to infinity. Let us then find\nexplicit results by taking the limit of high temperature.\nIn this case the equations can be solved.\nFor a given reservoir temperature Tthere are two black\nhole solutions, the small black hole solution r+1and the\nlarge black hole solution r+2. We set Tfixed but very\nhigh, in the sense T→ ∞ . From Eq. (12) there are\ntwo possibilities. Either TH\n+→ ∞ which corresponds to\nthe small black hole solution having a very small r+1,\norV(R)→0 which corresponds to the large black hole\nsolution r+2but now not approaching the reservoir radius\nin this range of Λ R2. Let us work one solution at a\ntime for a fixed very high value of T,T→ ∞ . Here we\npresent the expressions at zeroth order, not displaying\nthe corrections in1\nT.\nThe first solution for a very high heat reservoir tem-\nperature, T→ ∞ , isr+=r+1→0 with TH\n+→ ∞ .\nIt is clear from Eqs. (11), together with Eqs. (12) and\n(13), that this requires that the black hole solution is\nof the form r+=r+1→0. So, in this limit one has\nagain TH\n+1=1\n4πr+1, and then from Eq. (13) one finds\nthe small black hole r+1solution to be of the form\nr+1\nR=1\n4πRTq\n1−ΛR2\n3. The expression inside the square\nroot is clearly positive. So, in the T→ ∞ limit we have,\nr+1\nR= 0, (73)\nplus higher order corrections. As a by-product, we also\nfind from Eq. (3) that in this limit one has m1=\nr+1\n2. For the heat capacity CA, given in Eq. (10), i.e.,\nCA=\u0000dE\ndT\u0001\nA, equivalently, CA=\u0000dE\ndT\u0001\nR, we find from\nEq. (8) that CA+1=1\n2√\nV(R)\u0010\ndr+1\ndT\u0011\nR, which upon using\nEq. (28) yields CA+1=−1\n8πT2\u0010\n1−ΛR2\n3\u0011≤0, so in the\nlimit\nCA+1= 0−, (74)\nhere 0 −means that CA+1tends to zero from negative val-\nues, so that CA+1is nonpositive. Having negative heat\ncapacity, the small black hole r+1solution is thus unsta-\nble.\nThe second solution for a very high heat reservoir tem-\nperature, T→ ∞ , is some r+2but now not neces-\nsarily near R. Indeed, from Eq. (13), i.e., 4 πRT =\n1\nr+\nR1−ΛR2(r+\nR)2\nq\n1−r+\nRr\u0010\n1−ΛR2\n3\u0010\n1+r+\nR+(r+\nR)2\u0011\u0011, one sees that, when\nΛR2>1,RT→ ∞ for 1−ΛR2\n3\u0010\n1 +r+2\nR+\u0000r+2\nR\u00012\u0011\n= 0.\nThis is a quadratic forr+2\nRand it has as solution\nr+2\nR=1\n2 r\n3\nΛR2p\n4−ΛR2−1!\n. (75)This is the curve traced byr+2\nRas a function of Λ R2\nwhen RT =∞. So, from Eq. (75) we find that\nwhen Λ R2= 1 one hasr+2\nR= 1 as it should, and\nwhen Λ R2= 3 one hasr+2\nR= 0. In this latter\ncase, the solutionr+2\nRjoins the small black hole solu-\ntionr+1\nR= 0, so that Λ R2= 3 is the turning point\nofRT=∞,r+1\nR=r+2\nR= 0. For the heat capacity\nCA, given in Eq. (10), i.e., CA=\u0000dE\ndT\u0001\nA, equivalently,\nCA=\u0000dE\ndT\u0001\nR, we find that CA+2=√\nΛR2\n8πT2\u0010\n2 +√\nΛR2−\n√\n12−3ΛR2\u0001\u0010\n2−√\nΛR2+√\n12−3ΛR2\u0011\n≥0, which\nafter reworking can also be written as CA+2=\n√\nΛR2\n4πT2\u0010\nΛR2+√\nΛR2√\n12−3ΛR2−4\u0011\n≥0, so in the in-\nfinite temperature limit\nCA+2= 0+, (76)\nwhere 0 +means that CA+2tends to zero from positive\nvalues, so that CA+2is essentially positive, and the large\nblack hole r+2solution is stable. For Λ R2= 3 the heat\ncapacity is CA+2=√\n3\nπT2, and in the infinite temperature\nlimit one recovers Eq. (76). The cosmological radius can\nalso be found from Eq. (75), yielding for any Λ R2in the\nrange in question that\nrc2\nR= 1, (77)\nwhere Eq. (5) has been used. Note from the Tolman for-\nmula that the whole region between r+2andRis at infi-\nnite temperature. Indeed, T(r) =T\nV(R), forr+2\nR≤r\nR≤1.\nSince T=∞andV(R) is finite one has that T(r) is in-\nfinite in the region. The temperature at rnormalized to\nthe heat reservoir temperature TisT(r)\nT=1\nV(R)which\nis finite everywhere except at r+2where it is infinite. So\nthe radius r+2yields a doubly infinite temperature. For\nΛR2= 3, the space is pure de Sitter at infinite temper-\nature with Λ R2= 3 from the reservoir at Rup to the\ncenter where there is a singular black hole with horizon\nradius given byr+2\nR= 0.\nC. Thermodynamic phases and phase transitions\nbetween hot Schwarzschild-de Sitter and hot de\nSitter in the ΛR2>1case\nWe now mention thermodynamic phases and phase\ntransitions for the Λ R2>1 case. The discussion\nis valid for the thermodynamically stable black hole,\nthe black hole r+2, since the unstable one r+1has\nat most a brief existence that could not count as a\nphase. From Eq. (6) we get that the action Ifor\na hot Schwarzschild-de Sitter r+2phase is ISdS =\nβR\u0010\n1−p\nV(R)\u0011\n−πr2\n+2, where from Eq. (7) we have\nV(R) =\u0000\n1−r+2\nR\u0001\u0010\n1−ΛR2\n3\u0010\n1 +r+2\nR+\u0000r+2\nR\u00012\u0011\u0011\n, with\n1<ΛR2≤3 here. The free energy F=I\nβ=ITfor hot16\nSchwarzschild-de Sitter is then\nFSdS=\u0010\n1−p\nV(R)\u0011\nR−πTr2\n+2, 1<ΛR2≤3,\n(78)\nAnother phase that might exist is hot de Sitter, in which\ncaser+= 0, V(R) = 1−ΛR2\n3and the action is IHdS=\nβR\u0012\n1−q\n1−ΛR2\n3\u0013\n. The free energy is then\nFHdS= \n1−r\n1−ΛR2\n3!\nR , 1<ΛR2≤3.\n(79)\nIn the canonical ensemble the phase that has lowest Fis\nthe phase that dominates. So, the hot Schwarzschild-de\nSitter black hole dominates over hot de Sitter, or the two\nphases coexist equally, when\nFSdS≤FHdS, 1<ΛR2≤3, (80)\ni.e.,\u0010\n1−p\nV(R)\u0011\nR−πTr2\n+2≤\u0012\n1−q\n1−ΛR2\n3\u0013\nR,\ni.e.,\nRT≥q\n1−ΛR2\n3−p\nV(R)\nπr2\n+\nR2, 1<ΛR2≤3.(81)\nAs before, Eq. (35) is an implicit equation because r+2=\nr+2(R, T). For each Λ R2in the interval above, and for\neach RTone gets an r+2, which can then be put into the\nexpression just found to see whether the Schwarzschild-\nde Sitter phase dominates over the de Sitter phase or not.\nIn the case it dominates than a black hole can nucleate\nthermodynamically from hot de Sitter space.\nHere, we just comment on the limiting case, Λ R2= 3,\nwhich can be done directly. In this case the only solution\nisT=∞, with r+= 0, i.e., de Sitter space with a singu-\nlarity at the center. Thus, as T→ ∞ , one has FSdS→R\nfrom below. The de Sitter free energy for Λ R2= 3 is\nFSdS=R, exactly. Although both free energies are equal\ntoRin the limit, one free energy tends to zero, the other\nis identically zero. So for Λ R2→3,FSdS≤FHdSand one\ncan say that singular Schwarzschild de Sitter phase pre-\nvails. In the limit both phases have the same free energy\nand coexist in the ensemble in equal quantities. The dif-\nference between the two phases is that one has a singular\nblack hole at the center, i.e., a naked massless singularity,\nand the other does not. Note that in the phase transition\nfrom hot de Sitter to the Schwarzschild-de Sitter black\nhole phase, and vice-versa, there is topology change, since\nthe Euclidean topology of hot de Sitter is S1×R3, and the\nEuclidean topology of the Schwarzschild-de Sitter black\nhole is R2×S2.\nD. Comments on the ΛR2>1case\nWe note that although the analysis made for Eqs. (69)-\n(72) is valid for a very small value of Λ R2−1, one canhave a good idea of the behavior of the solutions as Λ R2\nis increased up to the value 3. This is also done with the\nhelp of the results of the high temperature limit Eqs. (73)-\n(77).\nFor Λ R2near one from above, we recover the previ-\nous result that for RT <1\n2πthere are no solutions. So\nRT=1\n2πis the minimum temperature to have solutions\nat all. As in the case Λ R2<1 which has solutions for\nRThigher than1\n2π, the case Λ R2>1 also has solutions\nforRThigher than1\n2π. For fixed RTgreater than the\nminimum value, i.e., RT≥1\n2π, two solutions exist up to\na maximum value of Λ R2, where at this value the two\nsolutions merge, r+1=r+2.\nForRT=∞one could find the exact dependence\nofr+2\nRin terms of Λ R2. Here, the maximum value is\nΛR2= 3, where the solutions merge with horizon ra-\ndiusr+1=r+2= 0. This behavior for RT=∞can be\nheuristically explained through the thermal wavelength,\nthe size of the reservoir, and the cosmological length.\nTgoing to infinity means that the associated thermal\nwavelength is zero and so the small black hole as radius\nr+1= 0, i.e., a black hole solution of zero size can be\nformed. The understanding of the large black hole with\nhorizon radius r+2very small when compared to R, in-\ndeed tending to zero, is here not so straightforward. The\ncosmological scale ℓ≡1√\nΛhas now the minimum possible\nvalue, ℓ=R√\n3.Tgoing to infinity implies that the asso-\nciated thermal wavelength is vanishingly small, and the\nresult implies that this wavelength only fits within the\nscale allowed by ℓso that r+2is also vanishingly small.\nVI. DIAGRAMS FOR THE\nSCHWARZSCHILD-DE SITTER AND NARIAI\nTHERMODYNAMIC SOLUTIONS AND\nANALYSIS\nA. Diagrams for the Schwarzschild-de Sitter and\nNariai thermodynamic solutions\n1. Preliminaries\nWe now draw some diagrams that help in the\nunderstanding of the thermodynamic solution of the\nSchwarzschild-de Sitter and Nariai horizons in a cavity.\nThere are two different sets of diagrams. The first set\ncontains six diagrams. In each diagram, it is plotted, for\na fixed value of 4 πRT, the values ofr+\nRthat are solution\nof the thermodynamic problem, as a function of√\nΛR2,\nsee Fig. 3. The second set contains also six diagrams.\nIn each diagram, it is plotted, for a fixed value of Λ R2,\nthe values ofr+\nRthat are solution of the thermodynamic\nproblem, as a function of 4 πRT, see Fig. 4. We use the\nvariable 4 πRT rather than RTbecause it is in a sense\nmore natural in this analysis.17\n2. Diagrams with RTfixed\nThe first set of six diagrams is shown in Fig. 3. It gives\na snapshot for each 4 πRT of how the black hole horizon\nradiir+\nRbehave in relation to√\nΛR2.\nThe case 0 ≤4πRT < 2 is not represented since thereare no black hole solutions. One has either absolute zero\npure de Sitter space when 4 πRT = 0, and hot, say, de\nSitter space when 0 <4πRT < 2. For a heat reservoir\nradius Rof the order of the size of a neutron, typical tem-\nperatures would be of the order of 1011K, so the term hot\neven when 4 πRT < 2 can be considered as appropriate.\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 3: Plots ofr+\nRas a function of√\nΛR2for six different values of 4 πRT. For plotting purposes it is defined\nx≡√\nΛR2,y≡r+\nR, and w≡4πRT. (a) A plot of the horizon solution for temperature w= 2, i.e., RT=1\n2π= 0.16,\nthe later equality being approximate. The only solution is the Nariai solution with x= 1 and y1=y2= 1. (b) A\nplot of the two horizon solutions y1andy2for temperature w= 2.5, i.e., RT= 0.20 approximately. (c) A plot of the\ntwo horizon solutions y1andy2for temperature w=√\n27\n2= 2.60, i.e., RT=√\n27\n8π= 0.21, the decimal equalities being\napproximate. (d) A plot of the two horizon solutions y1andy2for temperature w= 3, i.e., RT= 0.24\napproximately. (e) A plot of the two horizon solutions y1andy2for temperature w= 100, i.e., RT= 8.0\napproximately. (f) A plot of the two horizon solutions y1andy2for temperature w= 10000, i.e., RT= 796\napproximately. Note that 10000 → ∞ in this context. See text for details.\nThe case 4 πRT = 2, i.e., RT=1\n2π= 0.159, the last\nequality being approximate, see Fig. 3a, is the case with\nminimum temperature that yields a solutionr+\nR. Thisfirst solution is a solution with Λ R2= 1, and no other\nΛR2. It hasr+1\nR=r+2\nR= 1, and no other radius. It is\na Nariai solution, the coldest one. Indeed, it is the first18\nsolution that pops out when 4 πRT, i.e., T, increases from\nzero. This means that Nariai is easier to manufacture\nthermodynamically than Schwarzschild-de Sitter. The\ncorresponding cosmological horizon radius obeyrc1\nR=\nrc2\nR= 1.\nThe case 4 πRT = 2.5, i.e., RT= 0.199 approximately,\nsee Fig. 3b, shows that for each Λ R2there are two so-\nlutions, the small oner+1\nR, unstable, and the large one\nr+2\nR, stable, but these solutions only exist for a narrow\nrange of the abscissas, namely, 0 .60≤ΛR2≤1.10, where\nthe numbers are approximate, i.e., there are solutions for\nΛR2<1 and for Λ R2>1, but all solutions are still near\nΛR2= 1. The Nariai solution is also included in this\ncase, now with a temperature higher than the previous\ncase. The cosmological horizon radiirc1\nRandrc2\nRcan then\nbe found directly from the corresponding horizon radii.\nThe case 4 πRT =√\n27\n2= 2.598, i.e., RT=√\n27\n8π=\n0.207, the equalities in decimal notation being approxi-\nmate, see Fig. 3c], displays for the first time a solution\nwith zero cosmological constant, Λ R2= 0, which is the\npure Schwarzschild solution, the one with zero cosmolog-\nical constant, found first by York. This solution is a co-\nincident horizon solution withr+1\nR=r+2\nR=2\n3. For other\nlarger Λ R2there are two solutions, the small oner+1\nR,\nunstable, and the large oner+2\nR, stable, and these solu-\ntions only exist for some range of the abscissas, namely,\n0≤ΛR2≤1.12, where the latter number is approxi-\nmate. The Nariai solution is also included in this case,\nnow with a temperature higher than the previous case.\nThe cosmological horizon radiirc1\nRandrc2\nRcan then be\nfound directly from the corresponding horizon radii using\nthe appropriate equation.\nThe case 4 πRT = 3, i.e., RT= 0.239 approximately,\nsee Fig. 3d, shows that for zero cosmological constant,\nΛR2= 0, there are two pure Schwarzschild solutions with\nr+1\nR<2\n3andr+2\nR>2\n3, the first unstable, the second sta-\nble. For other larger Λ R2there are also two solutions,\nthe small oner+1\nR, unstable, and the large oner+2\nR, stable,\nand these solutions only exist for some range of the ab-\nscissas, namely, 0 ≤ΛR2≤1.20, the latter number being\napproximate. The Nariai solution is also included in this\ncase, now with a temperature higher than the previous\ncase. The cosmological horizon radiirc1\nRandrc2\nRcan then\nbe found directly from the corresponding horizon radii.\nThe case 4 πRT = 100, i.e., RT= 7.958 approximately,\nsee Fig. 3e, is a case where the temperature is high, but\nnot divergingly high. For zero cosmological constant,\nΛR2= 0, the two pure Schwarzschild solutions are, one\nwithr+1\nRnow very small approaching zero, which is un-\nstable, and the otherr+2\nRnow large and approaching one,\nwhich is stable. For other larger Λ R2there are also two\nsolutions, the small one approaching zeror+1\nR, unstable,\nand the large one approaching oner+2\nR, stable, and these\nsolutions exist for a larger range of the abscissas, namely,\n0≤ΛR2≤2.70, the latter number being approximate.\nThe Nariai solution is also included in this case, now with\na temperature higher than the previous case. The cos-\nmological horizon radiirc1\nRandrc2\nRcan then be founddirectly from the corresponding horizon radii.\nThe case 4 πRT =∞, precisely 4 πRT = 10000, i.e.,\nRT= 795 .8 approximately, see Fig. 3f, displays the max-\nimum spectrum of solutions. This case has been analyzed\nabove in some detail and exact expressions forr+1\nRand\nr+2\nRhave been found. There are solutions in the range\n0≤ΛR2≤3. The small horizon solution hasr+1\nR= 0 all\nthe way and is unstable. The large horizon solution has\nr+2\nR= 1 up to Λ R2= 1, and then decreases to zero at\nΛR2= 3 where it joins ther+1\nRsolution, and it is a stable\nsolution. The Nariai solution is also included in this case,\nnow with a temperature tending to infinity, 4 πRT→ ∞ .\nThe cosmological horizon radiirc1\nRandrc2\nRcan then be\nfound directly from the corresponding horizon radii and\nthere are exact expressions for them.\n3. Diagrams with ΛR2fixed\nThe second set of six diagrams is shown in Fig. 4. It\ngives a snaphot for each Λ R2of how the black hole hori-\nzon radiir+\nRbehave in relation to 4 πRT.\nThe case Λ R2= 0, see Fig. 4a, is the case with min-\nimum Λ R2in this context. For this case, in the range\n0≤4πRT <√\n27\n2, there are no radiir+\nRthat are solution\nof the thermodynamic problem, so presumably the in-\nside of the reservoir is filled with hot de Sitter space. At\n4πRT =√\n27\n2the coincident solutionr+1\nR=r+2\nR=2\n3ap-\npears. For larger 4 πRT,r+1\nRtends to zero andr+2\nRtends\nto one. Since Λ R2= 0 means zero cosmological constant,\ni.e., Λ = 0, this case is York’s solution, specifically, the\npure Schwarzchild case.\nThe case Λ R2= 0.64, see Fig. 4b, is a case with an\nintermediate value of Λ R2. At some definite value of\n4πRT, smaller than the value of the previous case, the\ncoincident solutionr+1\nR=r+2\nRappears. For larger 4 πRT,\nr+1\nRtends to zero andr+2\nRtends to one.\nThe case Λ R2= 1, see Fig. 4c, is the case where the\nNariai universe exists. For this case, in the range 0 ≤\n4πRT < 2, there are no radiir+\nRthat are solution of the\nthermodynamic problem, so presumably the inside of the\nreservoir is filled with hot de Sitter space. At 4 πRT = 2\nthe coincident solution appears withr+1\nR=r+2\nR= 1. This\nis the coldest Nariai solution. For larger 4 πRT,r+1\nRtends\nto zero andr+2\nRtends to one, indeed the solutionr+2\nR= 1\nfor any 4 πRT≥2 is a hot Nariai universe.\nThe case Λ R2= 1.21, see Fig. 4d, is a case typical of\nΛR2>1. At some definite 4 πRT the coincident solution\nr+1\nR=r+2\nRappears. For larger 4 πRT,r+1\nRtends to zero\nandr+2\nRtends to some value value that is less than one.\nThe case Λ R2= 2.25, see Fig. 4e, is also a case typical\nof ΛR2>1, but the new features are more evident. At\nsome definite 4 πRT the coincident solutionr+1\nR=r+2\nR\nappears, now with greater 4 πRT than the previous case.\nFor larger 4 πRT,r+1\nRtends to zero andr+2\nRtends to\nsome value value less than one. This value less than one\ndecreases rapidly with increasing Λ R2.19\nThe case Λ R2= 3, see Fig. 4f, is the last possible case,\nis a limit case. In this case, there is only one solution,\nwhich is the coincident solutionr+1\nR=r+2\nR= 0. In the\nplot this solution is represented as a point in the highestshown 4 πRT, which is meant to be 4 πRT =∞. There\nis divergently hot de Sitter space in the cavity at radius\nR, apart from a central singularity at zero radius,r+1\nR=\nr+2\nR= 0.\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 4: Plots ofr+\nRas a function of 4 πRT for six different values of Λ R2. For plotting purposes it is defined\nx≡√\nΛR2,y≡r+\nR, and w≡4πRT. (a) A plot of the two horizon solutions y1andy2forx2= 0. At temperature\nw=√\n27\n2, i.e., RT=√\n27\n8π, the coincident solution y1=y2=2\n3appears. For larger w,y1tends to zero and y2tends to\none. (b) A plot of the two horizon solutions y1andy2forx2= 0.64. (c) A plot of the two horizon solutions y1and\ny2forx2= 1. At temperature w= 2, i.e., RT=1\n2π, the solution y2= 1 appears, which in this case is a coincident\nsolution, indeed y1=y2= 1. It is a Nariai solution. The solution y2= 1 for any higher temperature wis Nariai. (d)\nA plot of the two horizon solutions y1andy2forx2= 1.21. (e) A plot of the two horizon solutions y1andy2for\nx2= 2.25. (f) A plot of the two horizon solutions y1andy2forx2= 3. Here, there is only the coincident solution,\nwith values y1=y2= 0 and temperature w=∞. See text for details.\n4. Additions\nIt is important to make additional comments to the\nthe plots that have been displayed in Figs. 3 and 4. In\nwhat follows the discussion will be qualitative.\nStacking with interpolation Figs. 3a-3f, one can\nglimpse the correctness of Figs. 4a-4f, and stacking with\ninterpolation Figs. 4a-4f, one can glimpse, in turn, thecorrectness of Figs. 3a-3f.\nOne striking feature, that can be deduced from the\nplots, is that the space of black hole horizon radius solu-\ntions is enlarged as the reservoir temperature T, or rather\n4πRT, is increased. In fact, for very low temperatures\nthere are no solutions for any Λ, or rather, for any Λ R2.\nAt the temperature 4 πRT = 2 there is only one solution,\nthe coldest possible Nariai universe. For higher 4 πRT20\nthere are solutions for some values of Λ R2, but not all.\nFor instance, for 4 πRT =√\n27\n2, there appears a solution\nwith Λ = 0, i.e., Λ R2= 0, and there are solutions up\nto ΛR2= 1.12, approximately, so that the range of Λ R2\nis 0≤ΛR2≤1.12 for this temperature. Finally, for in-\nfinite temperature, 4 πRT =∞, there are solutions for\nall possible Λ R2, namely 0 ≤ΛR2≤3. To understand\nother features of the solutions it is perhaps advisable to\nincorporate the case Λ R2= 1 in both the low and high\ncosmological constant cases and thus divide the whole\nrange into 0 ≤ΛR2≤1 and 1 ≤ΛR2≤3.\nIn the range 0 ≤ΛR2≤1, now with the help of Figs. 3\nand 4, one can summarize the qualitative explanation for\nthe reason of why black hole solutions with lower cosmo-\nlogical constant appear only for higher T, i.e., higher RT.\nThus, let us start with Λ = 0, so that the cosmological\nlength scale ℓ=1√\nΛis infinite, ℓ=∞. In this case there\nis no coupling of this length scale with the other two,\nλ=1\nTandR. In this situation, we see that for low T,\nhigh λ, one has λ≫R, so that, since the thermal wave-\nlength is very large compared to the reservoir radius R,\nthen this wavelength is stuck to the reservoir and the cor-\nresponding energy cannot collapse to form a black hole\nin any circumstances. When Tis sufficiently increased,\ni.e.,RTis larger than about one, the wavelength is suf-\nficiently small, and the corresponding energy can travel\nfreely inside the reservoir and can collapse, so that for-\nmation of black holes is possible. The value 4 πRT =√\n27\n2\ndivides no black hole from black hole solutions. Now, let\nus do ℓfinite. This is the third scale. For low enough\nℓ, say a bit larger than R, the space inside the reservoir\ngets higher curvature due to the high cosmological con-\nstant, and so in some manner this inner space has more\nlength along the radius, so that although the reservoir\narea radius is still R, the radial length is large, and so\nthe volume is also larger. This means that energy pack-\nets with higher λ, relatively to the cases with lower ℓ,\ncan continue to travel freely in the inside and can form\nblack holes. The limiting situation is when Randℓare\nequal, i.e., Λ R2= 1, orR2\nl2= 1, so that energy packets\nwith the highest possible λ, actually λ= 2πR, can give\na solution, which in this case is a Nariai solution.\nIn the range 1 ≤ΛR2≤3, now with the help of Figs. 3\nand 4, one can also s give a qualitative explanation for\nthe reason of why now black hole solutions with higher\ncosmological constant, i.e., higher Λ R2, appear only for\nhigher T, i.e., higher RT. Within this range, the cosmo-\nlogical constant is very high, i.e., the cosmological length\nscale ℓis very short, and so determines and dominates\nthe processes. Indeed, now ℓ < R . Let us now start with\nthe situation when Randℓare equal, i.e., Λ R2= 1, or\nR2\nl2= 1, so that the energy packets with the highest pos-\nsible λ, actually λ= 2πR, can give a solution, a Nariai\nsolution in this case. Still, forR2\nl2= 1 and higher temper-\natures, i.e, lower wavelengths λ, there is smaller unstable\nr+1black hole solution and the Nariai solution. ForR2\nl2\nlarger than one, i.e., ℓa bit smaller, the temperatureshave to be a higher, and so the wavelength λof the en-\nergy packets has to be a bit smaller, to have the two black\nblack hole solutions, as in the Λ R2<1 case. The small\nr+1solution forms in the same manner. The new feature\nis with the large black hole solution r+2. Now, for fixed\nΛR2<1, the r+2solution is always less then Reven\nwhen Tis very large. This means that the corresponding\nsmall wavelengths λnow are constrained by the scale ℓ\nso that the interplay is between r+2,λ, and ℓand not\nanymore between r+2,λ, and R. The final case is when\nℓ=R√\n3. The space inside is pure de Sitter, except for\na singular black horizon at the center, with the energy\npackets having a zero thermal wavelength λ= 0, with\nthe temperature Tof the reservoir being infinite. Only\nthose λ= 0 energy packets can collapse to form a black\nhole, packets with a higher λ, corresponding to a lower\nreservoir Tcannot fit to the scale set by ℓ.\nAnother characteristic radius that is part of the\nSchwarzschild-de Sitter solution is the cosmological ra-\ndius rc. In the setting we are working, where the heat\nreservoir is for the inside that harbors a possible black\nhole, the cosmological radius rchas only a secondary\nrole. This characteristic radius rccan be calculated once\nthe black hole horizon radius is found on thermodynamic\ngrounds.\nB. Mathematical analysis of the plots: Black hole\nhorizons\n1. Nomenclature\nWe now obtain through a mathematical analysis some\nimportant features displayed in the plots above, Figs. 3\nand 4. We repeat here Eq. (13), i.e., 4 πRT =\n1\nr+\nR1−ΛR2(r+\nR)2\nq\n1−r+\nRr\u0010\n1−ΛR2\n3\u0010\n1+r+\nR+(r+\nR)2\u0011\u0011. The natural vari-\nables without units are Λ R2andr+\nR. In this context it is\nperhaps preferable to work with√\nΛR2rather than with\nΛR2, so to shorten the notation we define the variables\nxandyas\nx≡√\nΛR2, (82)\ny≡r+\nR, (83)\nwith the range being 0 ≤x≤√\n3, or 0 ≤x2≤3, and\n0≤y≤1. In these variables, Eq. (13) is 4 πTR =\n1−x2y2\ny√1−yq\n1−x2\n3(1+y+y2). Define in addition the variable w\nas\nw≡4πRT . (84)\nThen, with these definitions Eq. (13) is\nw=1−x2y2\ny√1−yq\n1−x2\n3(1 +y+y2), (85)21\nwith 2 ≤w <∞. Solutions exist only for w≥2.\nNow, for a fixed temperature T, more properly for a\nfixed RT, i.e., fixed w, one has dw= 0, and sody\ndx=−∂w\n∂x\n∂w\n∂y.\nAfter some calculations we obtain\ndy\ndx=−2xy(1−y)Q(y)\n3R(y), (86)\nwhere\nQ(y)≡(1 +y+y2)(1 + x2y2)−6y2, (87)\nand\nR(y)≡ −2\n3(1+x2y2)(1−y)[3−x2(1+y+y2)]+(1−x2y2)2y .\n(88)\nIn addition, we need in the analysis∂w\n∂y. Obtain from\nEq. (85) that\n∂w\n∂y=√\n3S\n2y2(1−y)3/2h\n3−x2(1 +y+y2)i3/2(89)\nwhere we have defined\nS(x, y) =x4[2y2(1−y3)+3y5]+2x2(−y3−3y2+1)−6+9y .\n(90)\nWe have seen that the point Λ R2= 1, i.e., x2= 1,\nis important, as it gives the Nariai solution. So, let us\nconsider below the limit when x→1 from below and\nfrom above. We will see now that the result depends on\nhow the limit is taken. We recall that for each Λ R2there\nare two solutions, r+1, the small solution, and r+2, the\nlarge solution, which change as RTis changed, i.e., for\neachx, there are y1andy2, which change as wis changed.\n2. Analysis\nThere are general results here that we can mention.\nLet us analyzed the three ranges separately, namely, the\nregimes x2<1,x2= 1, and x2>1.\nx2<1:\nThis range of x2is specifically 0 ≤x2<1. In this range\nofx, the range of yis\n0≤y <1. (91)\nWe now find y1, then y2, and then we analyze the coin-\ncident solutions y1=y2.\nFirst we find y1, i.e., r+1, when xvaries within this\nrange, 0 ≤x2<1. For that, we fix y < 1 and move\nalong the positive xdirection. In the space ( x, y) the\ncorresponding point moves along a horizontal line. Then,\nfrom Eq. (85) one finds w=1−x2y2\ny√1−yq\n1−x2\n3(1+y+y2), i.e., y\nobeys an equation of the type\ny1p\n1−y1r\n1−x2\n3(1 +y1+y2\n1)w= 1−x2y2\n1,0≤x2<1,(92)\nfor a given xfixed. One of the solutions of this equation\nisy1, with wfixed and wwith a value obeying w > 2,\nand with y1(x)<1 always. The limit of x→1 for y1is\nsmooth, see below. Another property is that for any x,\nw→ ∞ when y1→0. This is the solution y1= 0, i.e.,\nr+1\nR= 0, when the temperature goes to infinity. Another\ninteresting property is the point wheredy1\ndx= 0, if there\nis one. This is when r+1attains a minimum value in\nrelation to Λ R2. This is the root of equation Q(y) =\n0, and happens for the small solution, i.e., y1. From\nEq. (87), Q(y1) = 0 gives\nx2=5y2\n1−y1−1\ny2\n1(1 +y1+y2\n1). (93)\nNow, the lowest y1is given when x= 0 by the solution\nof 5y2\n1−y1−1 = 0, which is1+√\n21\n10. The highest y1is\ny1→1 which yields x→1. Thus,dy1\ndx= 0 in the range\n0≤x2<1 happens in the range\n1 +√\n21\n10≤y1<1, (94)\ni.e., 0 .56≤y1<1, the first number in the left in-\nequality being approximate, with the range of xbe-\ning 0 ≤x2<1. The corresponding range of wfrom\nEq. (85) is 2 < w≤10√\n10\n(1+√\n21)√\n9−√\n21, or in round num-\nbers 2 < w≤2.70, the last number being approximate.\nWhy there is an xfor whichdy\ndx= 0 in the y1solution\nis not clear on physical, heuristic, terms, but possibly is\na nonlinear interplay between the length scale ℓ≡1√\nΛ\nand the length scales Randλ=1\nT. Further properties\nfor the small back hole depend on the specific xand the\nspecific wthat one picks, but there is nothing else more\ngeneral that we can mention.\nSecond we find y2, i.e., r+2, when xvaries within this\nrange, 0 ≤x2<1. From Eq. (85), one has\ny2p\n1−y2r\n1−x2\n3(1 +y2+y2\n2)w= 1−x2y2\n2,\n0≤x2<1,(95)\nfor a given xfixed, and one finds that there is a solution\ny2(x) for each x. The maximum of the curve is at x=\n1 and y2= 1, and at this point the derivative can be\nany of those found above, so it is not well defined. It\nis important to find the behavior and the properties of\ny2when xis near 1, i.e., x→1 with y2→1. A direct\nproperty is that for any x,w→ ∞ when y2→1. This is\nthe solution y2= 1, i.e.,r+2\nR= 1, when the temperature\ngoes to infinity. Other important properties are related\nto the derivative of y2, in particular,dy2\ndxforx→1. The\ncalculations ofdy2\ndxforx→1 are done in detail in the\nAppendix C, here we state the result, namely,\ndy2\ndx=w√\nw2−4−1, w ≥2, x →1.(96)22\nTherefore, taking the two limits for the temperature, i.e.,\nforw, we have\ndy2\ndx→ ∞ for w→2,dy2\ndx→0w→ ∞ ,\nx→1.(97)\nThus, the derivativedy2\ndxobeysdy2\ndx≥0, and can have\nvalues from 0 to infinity when one approaches x= 1\nfrom x <1. Another possible interesting property is the\npoint wheredy2\ndx= 0, if there is one. For the solution y2\none finds that there is no point with zero derivative.\nNow we find y1=y2forwfixed, i.e., the point xwhere\ny1=y2. There can exist two points at whichdy\ndx=∞for\nfixed w, one for x <1, the other for x >1. Here we work\noutx <1. It is given by the root of equation R(y) = 0,\nsee Eq. (88), and it defines the bifurcation point where\ny1=y2forw= constant. The equation is2\n3(1+x2y2)(1−\ny)[3−x2(1 + y+y2)]−(1−x2y2)2y= 0, which can\nbe put in the form x4\u00022\n3y2(1−y)(1 + y+y2) +y5\u0003\n−\nx2\u0002\n2y2−2\n3(1−y)(1 + y+y2)\u0003\n−(2−3y) = 0. This is a\nquadratic in x2of the form x4a(y)−x2b(y)−c(y) = 0,\nwhere a(y) =2\n3y2(1−y)(1 + y+y2) +y5,b(y) =\n2y2−2\n3(1−y)(1 + y+y2), and c(y) = 2 −3y. The\nsolution is\nx2=b(y)−p\nb2(y) + 4a(y)c(y)\n2a(y), 0< x2≤1,\n(98)\nInverting Eq. (98) one finds y(x), i.e., the solution y1=\ny2. For 0 ≤x2<1 there are solutions for y >2\n3, which\nfrom Eq. (85) means in turn w <√\n27\n2, i.e., for RT <√\n27\n8π,\nwith RT=√\n27\n8πatx= 0 being marginal. For higher w,\ni.e., higher RT, there are no solutions, y1andy2never\ncoincide for any xin this range 0 ≤x2<1.\nNow, we find y1=y2, the coincident solution for fixed\nx, and wvarying. This is when∂w\n∂y= 0, see Eq. (89),\ni.e.,S(x, y) = 0. From Eq. (90) we have then that the\nbifurcation points where y1=y2forx= constant are\ngiven by the equation x4[2y2(1−y3) + 3y5] + 2x2(−y3−\n3y2+1)−6+9y= 0, i.e., x4[2y2(1−y3)+3y5]−x2[2(y3+\n3y2−1)]−3(2−3y) = 0. This is a quadratic in x2,\nwhich can be written as x4d(y)−x2e(y)−f(y) = 0, with\nd(y) = 2 y2+y5,e(y) = 2( y3+3y2−1),f(y) = 3(2 −3y),\nand with solution\nx2=e(y)−p\ne2(y) + 4d(y)f(y)\n2d(y), 0< x2≤1.\n(99)\nInverting Eq. (99) one has y(x), for which y1=y2. To\nmake progress and find an analytical result we take x→1\nandy→1. The calculations are involved and we leave\nthem to the Appendix C, the result is that the coincident\nsolution takes the form\ny1=y2= 1−\u00123\n8\u0000\n1−x2\u00012\u00131\n3\n, (100)for (1−x2)≪1. Then, returning to the original variables\none hasr+1\nR=r+2\nR= 1−\u00003\n2(1−ΛR2)2\u00011\n3for 1−ΛR2\ntiny, which is Eq. (25).\nx2=1 :\nThis specific value of x,x= 1, is important as it hides a\nconsiderable structure, indeed, the whole Nariai solution\nis inside it. For this value of x, the range of yis\n0≤y≤1. (101)\nWe now find y1, then y2, an then we analyze the coinci-\ndent solutions y1=y2.\nFirst we find y1, i.e., r+1, when x= 1. For that, we\nfixy < 1 and move along the positive xdirection. In\nthe space ( x, y) the corresponding point moves along a\nhorizontal line. Then, from Eq. (85) one finds that for\nx= 1 one has w=√\n3√1−y(1+y)\ny√\n2−y−y2. Since 2 −y−y2=\n(1−y)(2 + y), one has\n\u0010\ny1p\n2 +y1\u0011\nw=√\n3(1 + y1), x2= 1,(102)\nand, from Eq. (89), one also findsdw\ndy<0. Now, Eq. (102)\nhas one solution y1(x= 1), with wfixed and wwith a\nvalue obeying w > 2. For any w, with w > 2 one has\ny1(x= 1) <1 always. It is clear that point ( x, y1) =\n(1, y1) lies on the branch of the curve that corresponds\nto the small root, to the small black hole.\nSecond, we find y2, i.e., r+2. From Eq. (85) one finds\nthat for x= 1 one has\u0010\ny2p\n2 +y2\u0011\nw=√\n3(1 + y2), x2= 1.(103)\nForx= 1, the large black hole always has y2= 1 but\nwcan be any as we have seen. So, Eq. (103) in the\ncontext of y2is deceiving. Indeed, if we put y2= 1 into\nEq. (102) one finds w= 2, i.e., RT=1\n2π. But we know,\nfrom our calculation above that at x= 1, wcan be any,\nindeed 2 ≤w <∞. So, Eq. (103) only gives one of the\ninfinite number of solutions, which all correspond to the\nNariai universe. Since this has been and will be further\ndiscussed we refrain to take further comments.\nNow we find y1=y2forwfixed. Here xis fixed, x= 1,\ni.e.,x2= 1. The solution is\nx2=y1=y2= 1, (104)\nwith w= 2. Then, in the original variables we have\nΛR2=r+1\nR=r+2\nR= 1 with RT=1\n2π. It is the first\nNariai universe solution as far as increasing temperature\ngoes.\nNow we find y1=y2forxfixed and wvarying. It is\ngiven by Eq. (104) since wis fixed as we saw.\nx2>1:\nThis range of x2is specifically 1 < x2≤3. In this range\nofx, the range of yis\n0< y≤ye, y e=−1\n2+r\n3\nx2−3\n4,(105)23\nwhere ye=ye(x) is the edge, or maximum, value that y\ncan have for each xin this range. The expression for yeis\ngiven by the root of the equation y2+y+ 1−3\nx2= 0, see\nEq. (85), with solution ye=−1\n2+q\n3\nx2−3\n4. Note that\nye=−1\n2+q\n3\nx2−3\n4<1\nx≤1. When y→yewe have\nfrom Eq. (85) that w=1−x2y2\ne\nye√1−ye√\n(ye−y)(y∗+y)with y∗≡\n1\n2+√\n3q\n1\nx2−1\n4. Since y∗=ye+1 this can also be written\nasw=1−x2y2\ne\nye√1−ye√\n(ye−y)(ye+y+1). So, for each xin the\nrange 1 < x2≤3, one finds yefrom Eq. (105), and then\none finds the corresponding w, i.e., the corresponding\nRT, which for y→yeisw→ ∞ . We now find y1, then\ny2, and then we analyze the coincident solutions y1=y2.\nFirst we find y1, i.e., r+1, when xvaries within the\nrange we are working, namely, 1 < x2≤3. For that, we\ngive some w, then we fix y < y eand move along the pos-\nitive xdirection. In the space ( x, y) the corresponding\npoint moves along a horizontal line. Then, from Eq. (85)\none finds w=1−x2y2\ny√1−yq\n1−x2\n3(1+y+y2), i.e., yobeys an equa-\ntion of the type\ny1p\n1−y1r\n1−x2\n3(1 +y1+y2\n1)w= 1−x2y2\n1,\n1< x2≤3,(106)\nfor a given xfixed. One of the solutions of this equation\nisy1, with wfixed and wwith a value obeying w > 2,\nand with y1(x)< yealways. The limit of y1forx→1 is\nsmooth. Also, for any xin the range 1 < x2≤3, when\nw→ ∞ one has y1→0. This is the solution y1= 0, i.e.,\nr+1= 0, when the temperature goes to infinity. Another\npossible interesting property is the point wheredy1\ndx= 0.\nFrom Eq. (86) this happens when Q(y1) = 0. Then,\nfrom Eq. (87) one finds that there are none in the range\n1< x2≤3. Moreover, y1(x) is a smooth curve, infinitely\ndifferentiable, in the whole range 0 ≤x2≤3, there is\nno discontinuity in any derivative at x= 1. Further\nproperties for the small back hole depend on the specific\nxand the specific wthat one picks, but there is nothing\nelse general that we can mention.\nSecond we find y2, i.e., r+2. From Eq. (85), one finds\ny2p\n1−y2r\n1−x2\n3(1 +y2+y2\n2)w= 1−x2y2\n2,\n1< x2≤3,(107)\nand one finds that there is a solution y2(x) for each x.\nThe maximum of the curve is at x= 1, y2= 1, and\nthe derivative can be any of those derived above, so it is\nnot well defined in this sense. It is important to discuss\nthe behavior and the properties of y2, when x→1 from\nabove. A direct property is that for any x,w→ ∞\nwhen y2→ye. This is the solution y2=yewhen the\ntemperature goes to infinity. Other important propertiesare related tody2\ndx, in particular,dy2\ndxwhen x→1 from\nabove. The calculations for findingdy2\ndxwhen x→1 from\nabove, are done in detail in the Appendix C, here we\nstate the result, namely\ndy2\ndx=−w√\nw2−4−1, w ≥2, x →1,(108)\nwhere the limit is from above. Therefore, taking the two\nlimits for the temperature, i.e., for w, we have\ndy2\ndx→ −∞ forw→2,dy2\ndx→ −2 for w→ ∞ ,\nx→1, (109)\nwhere the limit is from above. Thus, the derivativedy2\ndx\nobeysdy\ndx≤0, and it can be from 0 to minus infinity when\none approaches x= 1 from x >1. We see again that the\npoint ( x, y) = (1 ,1) is very rich, in fact it corresponds\nto the Nariai limit, a universe full of structure as we\nhave studied. Another possible interesting property is\nthe point wheredy2\ndx= 0, if there is one. For the solution\ny2one finds that there is no point with zero derivative.\nNow we find y1=y2forwfixed, i.e., the point xwhere\ny1=y2. There exist two points at whichdy\ndx=∞for\nfixed w, one for x <1, the other for x >1. Here we work\noutx >1. It is given by the root of equation R(y) = 0,\nsee Eq. (88), and it defines the bifurcation points where\ny1=y2forw= constant. The equation is2\n3(1+x2y2)(1−\ny)[3−x2(1 + y+y2)]−(1−x2y2)2y= 0, which can\nbe put in the form x4\u00022\n3y2(1−y)(1 + y+y2) +y5\u0003\n−\nx2\u0002\n2y2−2\n3(1−y)(1 + y+y2)\u0003\n−(2−3y) = 0. This is a\nquadratic in x2. Writing it as x4a(y)−x2b(y)−c(y) = 0,\nwhere a(y) =2\n3y2(1−y)(1 + y+y2) +y5,b(y) =\n2y2−2\n3(1−y)(1+ y+y2), and c(y) = 2−3y, the solution\nis\nx2=b(y) +p\nb2(y) + 4a(y)c(y)\n2a(y), 1< x2≤3.\n(110)\nInverting Eq. (110) one obtains y(x) for which y1=y2.\nFor 1 < x2≤3 there are solutions for y >0, i.e., 0 ≤\ny < y e, which means in turn w >2, i.e., for 2 < w < ∞.\nForw=∞, one has x2= 3, and y1=y2= 0.\nNow, we find y1=y2, the coincident solution for fixed\nx, and wvarying. This is when∂w\n∂y= 0, see Eq. (89),\ni.e.,S(x, y) = 0. From Eq. (90) we have then that the\nbifurcation points where y1=y2forx= constant are\ngiven by the equation x4[2y2(1−y3) + 3y5] + 2x2(−y3−\n3y2+1)−6+9y= 0, i.e., x4[2y2(1−y3)+3y5]−x2[2(y3+\n3y2−1)]−3(2−3y) = 0. This is a quadratic in x2, which\ncan be written as x4d(y)−x2e(y)−f(y) = 0, with d(y) =\ny5+ 2y2>0,e(y) = 2( y3+ 3y2−1),f(y) = 3(2 −3y),\nand with solution\nx2=e(y) +p\ne2(y) + 4d(y)f(y)\n2d(y), 1< x2≤3.\n(111)24\nOne can show that the solution of the quadratic equation\nwith the minus sign before radical is inconsistent with\nthe condition x > 1, so it is not considered. Inverting\nEq. (111) one has y(x) for which y1=y2. To make\nprogress and find an analytical result we take x→1\nandy→1. The calculations are involved and we leave\nthem to the Appendix C, the result is that the coincident\nsolution takes the form\ny1=y2= 1−\u00123\n8\u0000\nx2−1\u00012\u00131\n3\n, (112)\nforx2−1≪1. Then, returning to the original variables\none has,r+1\nR=r+2\nR= 1−\u00003\n8(ΛR2−1)2\u00011\n3for Λ R2−1\ntiny, which is Eq. (71).\n3. Synopsis\nWhen looking for roots y=y(x, w) of the equation\nw(x, y) =w, we have found that, depending on the fixed\nvalue of w, one can have no roots, one root, or two roots.\nThe no root situation means that the space inside the\nheat reservoir is hot de Sitter space. The one root situa-\ntion means that there is one possible black hole solution\ny1=y2inside the reservoir. The two roots situation\nmeans that there are two possible black hole solutions\ninside the reservoir, one small y1and unstable, the other\nlarge y2and stable. The case x2= 1 is important. When\nx= 1 there is the sole solution y1=y2= 1 for w= 2,\nbut for any other w,w >2, there are two solutions, one\nisy1, corresponding to the small black hole, the other is\ny2= 1, corresponding to the larger black holes, which for\nx2= 1 have transubstantiated into the Nariai cosmolog-\nical universes with all the allowed values of w,w≥2.\nC. Further analysis: Cosmological horizons\nIn the Schwarzschild-de Sitter space there is also the\nradius of the cosmological horizon rc. However, since the\nradius of the heat reservoir Ris inside rc, this latter has\nno role in the thermodynamics. The cosmological horizon\nradius, is just a parameter, which has only a coadjuvant\nrole in the whole setting. It is found once r+has been\nfound from the thermodynamics.\nThe radius of the cosmological horizon rcis the largest\nroot of the equation V(r) = 0, see Eq. (2) for the ex-\npression of V(r). As we have seen, it is related to the the\nblack hole horizon radius according to r2\nc+rcr++r2\n+−3\nΛ=\n0, whence rc=−r+\n2+r+\n2r\n12−3Λr2\n+\nΛr2\n+, see Eq. (5), or\nrc=−r+\n2+q\n3\nΛ−3\n4r2\n+. Let us define\nu≡rc\nR, (113)\nas the cosmological radius in units of R. Then, from\nEq. (5) we have that uis given in terms of xandyofEqs. (82) and (83), respectively, by\nu=−y\n2+r\n3\nx2−3\n4y2. (114)\nCalculatingdu\ndyone finds\ndu\ndy≤0, (115)\ni.e.,drc\ndr+≤0. This means that as the horizon radius r+\nincreases the cosmological horizon decreases, in general.\nWe restrict here to calculate some of the properties of\nthe cosmological radius u.\nWe want to calculate the properties of uat the neigh-\nborhood of the point x= 1, y= 1, u= 1, and find\ndu\ndx. Since this pertains to the large black hole, we use,\nas usual, the subscript 2 to refer to that solution, see\nAppendix C for details. One finds that\n\u0012du2\ndx\u0013\nx=1−=\u0012dy2\ndx\u0013\nx=1+(116)\nand\n\u0012du2\ndx\u0013\nx=1+=\u0012dy2\ndx\u0013\nx=1−(117)\nwhere 1 −and 1 +mean that one is taking the limit x→1\nfrom below, i.e., x <1, and from above, i.e., x >1, re-\nspectively. Using Eqs. (96), (108), (116), and (117) we\ncan deduce some further specific properties of the cos-\nmological horizon. They are that\u0000du2\ndx\u0001\nx=1−→ −∞ for\nw→2, and\u0000du2\ndx\u0001\nx=1+→ − 2 for w→ ∞ , and that\u0000du2\ndx\u0001\nx=1−→ ∞ forw→2, and\u0000du2\ndx\u0001\nx=1+→0 for\nw→ ∞ .\nThis interchange of equations as displayed in\nEqs. (116) and (117), is mostly clearly seen in the case\nw=∞, i.e., RT→ ∞ . Indeed, the case w→ ∞ can be\nsolved exactly as we have seen in Eq. (114). Thus, for\n0≤x2≤1, one has y2= 1 and u2=−1\n2+q\n3\nx2−3\n4,\nwhereas for 1 ≤x2≤3,y2=−1\n2+q\n3\nx2−3\n4andu2= 1,\nsee Fig. 5.\nIn brief, the black hole horizon equation for Λ R2<1\nturns into the cosmological horizon equation for Λ R2>1,\nand the cosmological horizon equation for Λ R2<1 turns\ninto the black hole horizon equation for Λ R2>1.25\nFIG. 5: Plots ofr+2\nRandrc2\nRas a function of√\nΛR2for\nRT=∞, i.e., essentially infinite reservoir temperature.\nFor plotting purposes it is defined x≡√\nΛR2,y2≡r+2\nR,\nu2≡rc2\nR, and w≡4πRT. So this is the case w=∞.\nNote that y2andu2exchange character at the\nbifurcation point x= 1. See text for details.\nVII. CONCLUSIONS\nWe have taken on the problem of understanding the\nSchwarzschild-de Sitter black hole thermodynamically.\nFor that we have put the black hole in a cavity of ra-\ndiusRsurrounded by a heat reservoir and kept at tem-\nperature T. This structure allowed the thermodynamic\nproblem to be solved within the canonical formalism and\nthe Euclidean path integral to quantum gravity. More-\nover, it led naturally into two other spaces, namely, hot\nde Sitter and Nariai cosmological spaces, besides the ini-\ntial Schwarzschild-de Sitter black hole.\nThere are new results. One is that the range of the\nrelevant parameter Λ R2is extendable up to 3, i.e., 0 ≤\nΛR2≤3. It was also found that to properly treat the\nproblem one has to divide this range into three ranges,\n0≤ΛR2<1, ΛR2= 1, 1 <ΛR2≤3.\nOn the lower side of Λ R2, 0≤ΛR2<1, York’s solu-\ntion for pure Schwarzschild is automatically incorporated\nwhen Λ R2= 0, appearing first for RT=√\n27\n8π, with a\ncoincident black hole horizon radius r+1=r+2=2\n3R.\nFor higher Λ R2the coincident black hole horizon radius\ngets increased values for some lower RT. A heuristic un-\nderstanding of this behavior has been given. Changing\nthe values and Λ R2andRTone obtains either two ther-\nmodynamics solutions r+1small and r+2large, the first\nthermodynamically unstable and the second stable, or no\nsolutions in which case one is in the presence of hot de\nSitter.\nAt the intermediate value of Λ R2, ΛR2= 1, the small\nr+1unstable black hole exists. More interestingly, the\nlarge r+2black hole is now the solution r+2=Rand\nopens out into a spectrum of a beautiful set of Nariai\nuniverses that can have all temperatures in the range1\n2π≤RT≤ ∞ . The ensemble data RandTnow de-\ntermine the location of the boundary Zin the Nariai\nuniverse rather than the black hole radius solution. The\nNariai universe is thermodynamically neutrally stable.\nOn the higher side of Λ R2, 1<ΛR2≤3, unexpected\nblack hole solutions also arise. The small r+1unstable\nblack hole exists without changing character. The large\nr+2stable solutions interchange the role of black hole r+2\nand cosmological rc2horizons, and the maximum value\nr+2can have is attained for infinite temperature, i.e.,\nRT=∞, and is less than one changing as Λ R2changes\nup to 3. The case at the end of the range, Λ R2= 3, only\nexists for infinite temperature, and represents a reservoir\nfilled with de Sitter space inside, except at the center,\nwhere there is a black hole with zero horizon radius r+=\n0, i.e., a naked singularity.\nAnother important result is that increasing the tem-\nperature from zero, i.e., increasing RTfrom zero, one\nfinds that the first solution that appears for the whole\nrange 0 ≤ΛR2≤3 has temperature RT=1\n2πand is a\nsolution with Λ R2= 1, and radius r+1=r+2=R, i.e.,\nit is the coincident Nariai solution. As one increases RT\nsolutions with lower and higher Λ R2peal out.\nYet another interesting finding is related to phase\ntransitions. In the Λ R2<1 case, in particular for\nΛR2≪1, the possible thermodynamic phases that ap-\npear as one increases the temperature from zero are hot\nde Sitter only, hot de Sitter favored in relation to the\nSchwarzschild-de Sitter black hole, hot de Sitter coexist-\ning equally with the Schwarzschild-de Sitter black hole,\nthe Schwarzschild-de Sitter black hole favored in relation\nto hot de Sitter, and the Schwarzschild-de Sitter black\nhole alone. This latter case comes out when one con-\nsiders a high enough temperature so that there is suffi-\ncient energy in the cavity to surpass the Buchdahl bound\nand presumably the system collapses. There are topol-\nogy changes when the system performs a phase transition\nfrom hot de Sitter to Schwarzschild-de Sitter and vice\nversa as it is allowed in this formalism, since it is in a\nsemiclassical approximation to quantum gravity, and in\nquantum gravity topology changes of the psce can hap-\npen. In the Λ R2= 1 case, i.e., the Nariai universe, one\nhas that the possible thermodynamic phases that appear\nas one increases the temperature from zero are hot de\nSitter only, hot de Sitter favored in relation to the Nariai\nuniverse, hot de Sitter coexisting equally with the Nariai\nuniverse, the Nariai universe favored in relation to hot\nde Sitter. Here there is no phase with only the Nariai\nuniverse. There are topology changes when the system\nperforms a phase transition from hot de Sitter to the\nNariai universe and vice versa as it is allowed in this for-\nmalism. In the Λ R2>1 case, phase transitions between\nhot de Sitter and the Schwarzschild-de Sitter black hole,\ncan also be explored.\nThus, we have given a full thermodynamic description\nof the Schwarzschild-de Sitter black hole space in a finite\nsize cavity.26\nAcknowledgments\nWe thank conversations with Tiago Fernandes and\nFrancisco Gandum on the Euclidean path integral ap-\nproach to quantum gravity. We acknowledge financial\nsupport from Funda¸ c˜ ao para a Ciˆ encia e Tecnologia -\nFCT through the project No. UIDB/00099/2020.\nAppendix A: The Schwarzschild-de Sitter and Nariai\nspacetimes: Basics\n1. The Schwarzschild-de Sitter spacetime\nThe line element of the Schwarzschild-de Sitter space-\ntime in spherical coordinates ( t, r, θ, ϕ ) is given by\nds2=−V(r)dt2+dr2\nV(r)+r2(dθ2+ sin2θ dϕ2),(A1)\nwhere metric potential V(r) has the form\nV(r) = 1−2m\nr−Λr2\n3, (A2)\nwith mbeing the spacetime mass and Λ the cosmologi-\ncal constant which we consider positive, Λ >0, see also\nFig. A1. The coordinate ranges are −∞ < t < ∞,\nr+< r < r c, 0≤θ≤π, and 0 ≤ϕ <2π, where r+and\nrcare the black hole and cosmological horizons of the\nspacetime, respectively. The spacetime has topology R4.\nThese coordinates can be further extended, e.g., through\na Kruskal-Szekeres extension, but it is not necessary here\nto do so.\nThe equation V(r) = 0 is\nr−2m−Λr3\n3= 0, (A3)\nwhich can be written as m=r\n2\u0010\n1−Λr2\n3\u0011\n. Note that\nEq. (A3) has at most two positive real roots. When it\nhas roots, one root corresponds to the black hole horizon\nr+, with\nr+=r+(m,Λ). (A4)\nThe radius of the black hole horizon obeys the inequality\n0≤r+≤1√\nΛ. The explicit form of r+(m,Λ),can be\ngiven since Eq. (A3) is a cubic equation for r, but it is\ncumbersome and there is no need to present it explicitly.\nGiven r+, then V(r) in Eq. (A2) can be written as V(r) =\n1−r+\u0012\n1−Λr2\n+\n3\u0013\nr−Λr2\n3orV(r) = 1−r+\nr−Λ\n3r\u0000\nr3−r3\n+\u0001\n,\nwhere use of Eq. (A3) has been made. The other root\ncorresponds to the cosmological horizon rc,\nrc=rc(m,Λ), (A5)\nwith rc≥r+. The radius of the cosmological horizon\nobeys the inequality1√\nΛ≤rc≤q\n3\nΛ. The explicit formofrc(m,Λ),can be given since Eq. (A3) is a cubic equa-\ntion for r, but it is cumbersome and there is no need to\npresent it explicitly.\nr+\nSchwarzschild\nde Sitter spacerc\nFIG. A1: A drawing of the black hole with its event\nhorizon r+and of the cosmological horizon rcin the\nSchwarzschild-de Sitter spacetime. The topology of the\n3-space is R3.\nOn the other hand, since there are two roots of\nEq. (A3), r+andrc, one can write\nV(r) =Λ\n3r(r−r+)(rc−r)(r+r++rc), (A6)\nwith\nr2\nc+rcr++r2\n+=3\nΛ, (A7)\nand\nrcr+(rc+r+) =6m\nΛ. (A8)\nThus Λ and mcan be swapped for r+andrc. Moreover,\nEq. (A7) can be written as r2\nc+rcr++r2\n+−3\nΛ= 0 which\nis a quadratic either for rcin terms of r+or vice versa.\nThe solution is\nrc=−r+\n2+r+\n2s\n12−3Λr2\n+\nΛr2\n+, (A9)\norrc=−r+\n2+q\n3\nΛ−3\n4r2\n+. Of course, the equation r2\nc+\nrcr++r2\n+−3\nΛ= 0 is also a quadratic for r+which gives\nr+in terms of rcin the same form of Eq. (A9) with the\nroles reversed. Another way of obtaining this is that since\nthere are two solutions of Eq. (A3), r+andrc, one has\nfrom Eq. (A3) that rc−2m−Λr3\nc\n3= 0 and r+−2m−\nΛr3\n+\n3= 0. Subtracting one equation from the other one\neliminates 2 mto get ( rc−r+)−Λ\n3\u0000\nr3\nc−r3\n+\u0001\n= 0. Since\u0000\nr3\nc−r3\n+\u0001\n= (rc−r+)(r2\nc+rcr++r2\n+), one finds that\n1−Λ\n3(r2\nc+rcr++r2\n+) = 0, i.e., r2\nc+rcr++r2\n+−3\nΛ= 0.\nOne can then write the solution of rcin term of r+as27\nrc=−r+\n2+q\n3\nΛ−3\n4r2\n+, which is Eq. (A9). In addition,\nEq. (A8) with the help of Eq. (A7) can be written as\nr2\nc+rcr+−2mr2\n+\nr+−2m= 0, which is a quadratic either for rc\nin terms of r+or vice versa. The solution is\nrc=−r+\n2+r+\n2s\nr++ 6m\nr+−2m, (A10)\norrc=r+\n2\u0010q\nr++6m\nr+−2m−1\u0011\n. Another way of obtaining\nthis is that since there are two solutions of Eq. (A3), r+\nandrc, one has from Eq. (A3) that rcr3\n+−2mr3\n+−Λr3\ncr3\n+\n3=\n0 and r+r3\nc−2mr3\nc−Λr3\n+r3\nc\n3= 0. Subtracting one equation\nfrom the other one eliminates Λ to get −r+rc(r2\nc−r2\n+) +\n2m(r3\nc−r3\n+) = 0. Thus, −r+rc(rc−r+)(rc+r+)+2m(rc−\nr+)(r2\nc+rcr++r2\n+) = 0, i.e., −r+rc(rc+r+) + 2m(r2\nc+\nrcr++r2\n+) = 0. So, r2\nc+rcr+−2mr2\n+\nr+−2m= 0, with solution\nrc=−r+\n2+r+\n2q\nr++6m\nr+−2mwhich is Eq. (A10).\nNote that rc=r+forrc=r+=1√\nΛ= 3m, and\nthis happens when 9 m2Λ = 1. In this limit one can\nhave either an extremal Schwarzschild-de Sitter space-\ntime, where the two regions off the extremal horizon,\nthe inside and the outside regions, are time dependent,\nor the Nariai spacetime where the topology of the space\nchanges, see below. For 9 m2Λ>1 there are no horizons,\nthe spacetime is asymptotically de Sitter, with a massive\nnaked singularity at the center.\n2. The Nariai spacetime\nThe line element of the Nariai spacetime in spherical\ncoordinates ( t, z, θ, ϕ ), is given by\nds2=−V(z)dt2+dz2\nV(z)+1\nΛ\u0000\ndθ2+ sin2θ dϕ2\u0001\n,(A11)\nwhere the metric potential V(z) has the form\nV(z) = 1−Λz2, (A12)\nwith Λ being the cosmological constant which we con-\nsider positive, Λ >0, see also Fig. A2. The coordinate\nranges are −∞ < t < ∞,z+< z < z c, where z+and\nzcare the horizons of the spacetime, 0 ≤θ≤π, and\n0≤ϕ < 2π. Note that the spacetime has topology\nR2×S2. These coordinates can be further extended,\ne.g., through a Kruskal-Szekeres extension, but it is not\nnecessary here to do so.\nThe equation V(z) = 0 is\n1 = Λ z2. (A13)\nIn general Eq. (A13) has two roots. One root corresponds\nto the black hole horizon z+, by convention, with\nz+=−1√\nΛ. (A14)\ncz\nNariai universeidentify identify\n+zR RFIG. A2: A drawing of the Nariai universe with its\nevent horizon z+and its cosmological horizon zc. The\ncylindrical character of the Nariai space is made clear\nafter identifying the two vertical end lines of the\ndiagram. The radius Ris the radius of the cylinder.\nThe labelling of the two horizons z+andzcis\nconvention as they are of the same type. The topology\nof the 3-space is R×S2.\nThe other root corresponds to the cosmological horizon\nzc, by convention, with\nzc= +1√\nΛ, (A15)\nwith zc≥z+.\nNote that when\nΛ =∞, (A16)\nboth roots coincide,\nz+=zc= 0, (A17)\nand in this extremal limit the spacetime disappears.\nNote also that when Λ = 0, the z+andzchave roots\nz+=−∞, z c= +∞, (A18)\nand in this limit there are no horizons, one is in the\npresence of simply a Minkowski space with two coor-\ndinates possibly wrapped around, i.e., one can have\na torus, a cylinder, or an infinite plane. This can\nbe seen from the angular part of Eq. (A11) by doing\nθ=√\nΛx,ϕ=√\nΛy, and then doing Λ →0 to give\nds2=−dt2+dz2+dx2+dy2, where the range of the\ncoordinates xandyis 0≤x≤x1and 0 ≤y≤y1, re-\nspectively, with x1andy1having any value one chooses\nfrom a finite value to infinite.\nAppendix B: The Nariai limit from the\nSchwarzschild-de Sitter space in a thermodynamic\nsetting\nIn order to understand the limiting thermodynamic\nprocess of obtaining a Nariai space from a Schwarzschild-\nde Sitter space in the limit Λ R2= 1,r+\nR= 1, andrc\nR= 1,\nit is useful to resort to Fig. B1.28\nTo be complete we give again the Schwarzschild-de Sit-\nter Euclidean line element\nds2=V(r)dt2+dr2\nV(r)+r2(dθ2+ sin2θ dϕ2),\n0≤t < βH\n+, r+≤r≤R , (B1)\nwhere the metric potential V(r) has the form\nV(r) = 1−2m\nr−Λr2\n3. (B2)\nNote that the range of coordinates of Euclidean time tis\n0≤t < βH\n+, where βH\n+is the period of the coordinate such\nthat the line element given by Eqs. (B1) and (B2) has\nno conical singularities. The relation with the Hawking\ntemperature TH\n+isβH\n+=1\nTH\n+. In addition, due to the\nreservoir at radius Rthe range of coordinates of ris now\nr+≤r≤R. Note now that this is Euclidean space\nand the topology of this space is R2×S2. There are two\nhorizon for V(r) = 0, see Eq. (B2), the black hole horizon\nr+and the cosmological horizon rc.The local Tolman temperature Tof the heat reservoir\natRhas to be treated with care. It is\nT=TH\n+p\nV(R), (B3)\nwith TH\n+being the Hawking temperature given by TH\n+=\nκ+\n2π, and κ+being the surface gravity of the black hole\nhorizon and V(R) is the potential at R. For the line ele-\nment of Eq. (B1) one has κ+=1\n2V′(r+), where a prime\nmeans derivative with respect to r, and so the Hawking\ntemperature for the black hole is TH\n+=1\n4π\u0000dV\ndr\u0001\nr+. Using\nEq. (B2) we get\nTH\n+=1\n4πr+\u0000\n1−Λr2\n+\u0001\n. (B4)\nThe potential at Ris\nV(R) = 1−2m\nR−ΛR2\n3. (B5)\nr RT+rc\nheatreservoirinside a heat reservoirde Sitter black hole horizon r+\ncz\nZ\nz+heat\nreservoirTRinside a heat reservoirNariai horizon z +\nFIG. B1: Drawings of a black hole horizon inside a heat reservoir in the Schwarzschild-de Sitter black hole space on\nthe left, and of the Nariai universe with one of its horizons inside a heat reservoir on the right. Specifically, on the\nleft the drawing shows a slant view of a t= constant and θ= constant space of the black hole horizon r+inside a\nheat reservoir at temperature Tand radius Rin the Schwarzschild-de Sitter geometry. Outside Rthere is the\ncosmological horizon rc. The Euclideanized space and its boundary have R2×S2andS1×S2topologies,\nrespectively, with the S1subspace having proper length β=1\nT. On the right, the drawing shows a slant view of a\nt= constant and θ= constant space of the horizon z+inside a heat reservoir at temperature T, with cylindrical\nradius R, and situated at Zin the Nariai universe geometry. Outside Zthere is the cosmological horizon zc. The\nEuclideanized space and its boundary have R2×S2andS1×S2topologies, respectively, with the S1subspace\nhaving proper length β=1\nT. In pictorial terms it is clear how the Schwarzschild-de Sitter black hole space originates\nthe Nariai universe, with the slant view of the Schwarzschild-de Sitter space helping in the visualization of the\nprocess. Indeed, if the two Schwarzschild-de Sitter different horizon radii, r+andrc, are squeezed into the heat\nreservoir radius R, then the two horizons pinch off to form the Nariai universe with the heat reservoir still at\ntemperature T, with cylindrical radius R, and situated now at some Zin-between the two displaced distinct\nhorizons z+andzc. See text for more details.\nThe Nariai solution can be found from the\nSchwarzschild-de Sitter solution in the limit that the two\nhorizons r+andrccoincide. Here, we have a heat reser-voir at Rthat acts for the inside region which is a cav-\nity with the black hole. This heat reservoir at Ris in-\nbetween r+andrc, and thus the limit is such that r+,R,29\nandrccoincide. Now, if we do r+→Rthen Eq. (B3),\nT=TH\n+√\nV(R), gives at face value that the heat reservoir is\nat very high temperature since V(R)→0. But, there is\na way to have Tfinite with r+→R, and this is to do\nconcomitantly TH\n+→0. Since TH\n+=1\n4πr+\u0000\n1−Λr2\n+\u0001\n, see\nEq. (B4), TH\n+→0 means 1 −Λr2\n+→0. Since r+→R,\nwe put,\nr+\nR= 1−ε , (B6)\nε≪1, which implies\n√\nΛR2= 1−δ , (B7)\nwhere δ≡1\n2(1−Λr2\n+)−ε,δ≪1, andε\nδis of order one.\nThus, Rand the length scale1√\nΛare equal at zeroth\norder. Since r+→R→1√\nΛthen from Eq. (5) one has\nrc\nR→1, i.e., one has rc→1√\nΛ, and since1√\nΛ→R, then\nrc→R. From the expansion of Eqs. (B6) and (B7) one\nhas that Eq. (B4) gives\nTH\n+=δ+ε\n2πR, (B8)\nin first order, as required. To understand the behavior of\nV(r) near Rin this limit we use Eq. (B5). It gives\nV(R) =ε(ε+ 2δ), (B9)\nin first order. Thus, Eq. (B3), i.e., T=TH\n+√\nV(R)gives\nT=1\n2πRε\nδ+ 1q\nε\nδ\u0000ε\nδ+ 2\u0001, (B10)\nwhich given a RT, i.e., given a T, implies someε\nδ, and so\nis finite and consistent.\nBut we have not finished. The metric potential V(R)\ngiven in Eq. (B1) vanishes in these coordinates, and\nthe line element of Eq. (B1) looses sense. So, we have\nto pay attention to this limit indeed with care. It is\nthe Nariai limit with a reservoir Rin the middle, see\nFig. (B1), an interesting case. Expanding the metric\npotential V(r) near r+in a Taylor series gives V(r) =\u0000dV\ndr\u0001\nr=r+(r−r+) +1\n2\u0010\nd2V\ndr2\u0011\nr=r+(r−r+)2plus higher\norder terms. Recall that\u0000dV\ndr\u0001\nr=r+= 4πTH\n+and find\u0010\nd2V\ndr2\u0011\nr=r+=−2\nr2\n+. Now, in this limit r+=R, so\n\u0010\nd2V\ndr2\u0011\nr=r+=−2\nR2. So,\nV(r) = 4 πTH\n+(r−r+)−1\nR2(r−r+)2, (B11)\nplus higher order terms. Make the coordinate transfor-\nmations ( t, r)→(¯t,¯z) as\nr−r+= 4πTH\n+R2sin2\u00121\n2¯z\nR\u0013\n, t =¯t\n2πTH\n+R,(B12)with 0 ≤t≤1\nTH\n+corresponding to 0 ≤¯t≤2πR, and\nr+≤r≤Rcorresponding to 0 ≤¯z≤¯Z. Then obtain\nfrom Eq. (B2) that\nV(r) =V(r−r+) =V(¯z) = (2 πTH\n+R)2sin2\u0010¯z\nR\u0011\n,\n(B13)\nand from Eq. (B1) obtain the line element,\nds2= sin2\u0010¯z\nR\u0011\nd¯t2+d¯z2+R2dΩ2, (B14)\nwhich is a form of the Nariai line element. Note now\nfrom Eq. (B13) that V(R) = V(R−r+) = V(¯Z) =\n(2πTH\n+R)2sin2\u0010¯Z\nR\u0011\n, and so from Eq. (B3), i.e., T=\nTH\n+√\nV(R), one has\nT=1\n2πRsin\u0010¯Z\nR\u0011. (B15)\nThis means that the ensemble boundary values TandR\nspecify automatically the maximum value for ¯ z, namely\n¯Z. So, for the ensemble with boundary data one has\n0< t < 2πR, 0≤¯z≤¯Z, 0≤θ≤π, 0≤ϕ <2π. In turn\n0≤¯Z≤Rπ. Note that ¯ z= 0 and ¯ z=Rπare horizons.\nWe can continue further. Let us make another coordi-\nnate transformation, z=Rcos¯z\nR, Then\nds2=V(z)dt2+dz2\nV(z)+R2dΩ2, (B16)\nwhere we have dropped the bar in ¯t. Then, the metric\npotential V(z) is\nV= 1−z2\nR2, (B17)\nthe one given in Eq. (A12). So, for the ensemble with\nboundary data one has 0 < t < 2πR,−R < z < Z ,\n0≤θ≤π, 0≤ϕ <2π. In turn −R≤Z≤R. Note that\nz=−Randz=Rare horizons. The line element given\nin Eq. (B16) and (B17) corresponds to a two-dimensional\nde Sitter space times a sphere, the topology is R2×S2.\nIt has two horizons, we label z+=−Randzc=R, but\nnow they are just labels, since the two horizons have the\nsame character.\nWe have that the heat reservoir temperature Tis now\nT=TH\n+ Nariaip\nV(Z), (B18)\nwith TH\n+ Nariai being the Hawking temperature given by\nTH\n+ Nariai =κ+\n2π, and κ+being the surface gravity of the\nblack hole horizon. For the metric given in Eq. (B16)\none has κ+=1\n2V′(z+), and so the Hawking temperature\nfor + horizon is TH\n+ Nariai =1\n4π\u0000dV\ndz\u0001\nz+. Using Eqs. (B16)\nand (B17) we get\nTH\n+ Nariai =1\n2πR. (B19)30\nSo, from Eqs. (B18) and (B19) one finds\nT=1\n2πRq\n1−Z2\nR2, (B20)\nwhere Zis the boundary of Z. For a given Tat the\nboundary, for the ensemble, there are two solutions of\nEq. (B20) in general, namely, one for Zbetween −R\nand 0 and the other for Zbetween 0 and R. These two\nsolutions may be thought of yielding different physical\nsituations. Note also that in Schwarzschild-de Sitter the\ntwo solutions were for the horizon r+, one small r+1the\nother large r+2, here the two solutions are not for the\nhorizons but instead for the boundary Z, one Z1, the\nother Z2, with\nZ1=−Rs\n1−1\n(2πRT)2, (B21)\nand\nZ2=Rs\n1−1\n(2πRT)2, (B22)\nWe are free to choose where to put the boundary, we\nhave two choices, either Z1orZ2, noting that z+and\nzccan also be exchanged if one wants. Exchanging Z1\nwith Z2and concomitantly exchanging z+with zcre-\nverses to the original situation. The boundary is at Z1\norZ2, has cylindrical radius R, and acts as a reservoir\nfor the inner region, i.e., z≤Z1orz≤Z2depending on\nwhere we choose to put the boundary. Note, in addition\nthat the reservoir instead of being two-dimensional with\nspace topology S2as in the de Sitter space, it is still two\ndimensional but now with topology R×S1, i.e., it is a\ncylinder.\nFrom Eqs. (B21) and (B22) we see that for\nRT <1\n2π, (B23)\nthere are no solutions for Z1orZ2, so in this case the\nboundary in Zdoes not exist, and so there is no Nariai\nsolution for the thermodynamic problem in the canonical\nensemble. The reason is clear if one thinks in thermal\nwavelengths, indeed when the temperatures are relatively\nvery small, the associated thermal wavelengths are very\nlong and there is no boundary Z that can accommodate\nthem.\nFrom Eqs. (B21) and (B22) we see that for\nRT≥1\n2π, (B24)\nthere are two Nariai solutions, one with one horizon z=\n−Rand boundary Z1, the other with with one horizon\nz=−Rand boundary Z2, both boundaries can be picked\nup. When the equality sign holds there is one solution\nwith Z1=Z2= 0, so in this case the boundary in Zpopsup in the middle, at z= 0, and so −R≤z≤0. The\nreservoir is at Z1=Z2= 0 and has cylindrical radius R\nand the horizon is at z=−R.\nThree comments are in order. The first comment is\nthat the resulting metric inside the reservoir is in this\ncase described by the Nariai universe as we have seen.\nThe procedure of obtaining the Nariai metric as a limit\nfrom the Schwarzschild-de Sitter metric is known and\nhas different versions, e.g., see [5, 41]. However, we\nhave done it here in a completely different manner and\nin a completely different context that is related natu-\nrally to thermodynamics. The procedure we used is\nsimilar to that described in [13] for the extremal limit\nof nonextremal electrically charged black holes. Here,\nin Eq. (B11), the potential expansion takes the form\nV(r) = 4 πTH\n+(r−r+)−1\nR2(r−r+)2instead of V(r) =\n4πTH\n+(r−r+) +1\nR2(r−r+)2, i.e., a plus instead of the\nminus sign of [13] appears in the second term of the ex-\npansion. As a result, we have arrived at the Nariai metric\ninstead of obtaining the Bertotti-Robinson metric as in\n[13]. The second comment, is that in the above con-\nsiderations, we have discussed two completely different\nlimits, one is the high-temperature limit and the other\nis the extremal Nariai limit. However, they can be com-\nbined if the boundary Z1→ − R. The third comment\nis that we have done the Nariai limit from below, i.e.,\nfrom Λ R2<1. If we do the limit Λ R2→1 from above,\ni.e., from Λ R2>1, we get the same result, namely, the\nNariai universe inside a heat reservoir in the canonical\nensemble.\nAppendix C: Deduction of formulas of Sec. VI\nHere, we deduce in detail some equations found in\nSec. VI. Specifically, we want to deduce Eqs. (96), (100),\n(108), and (112), and study Eqs. (116) and (117).\nFor the derivation of Eqs. (96), (100), (108), and (112)\nit is convenient to shorten the notation and define the\nvariables√\nΛR2andr+\nRasxandyvariables, namely,\nx≡√\nΛR2, (C1)\ny≡r+\nR. (C2)\nWith these definitions we want to find Eqs. (96) and\n(100) of the x2<1 case, and Eqs. (108) and (112) of\nthex2>1 case.\nx2<1:\nLet us deduce Eq. (96). We have to find the behavior of\ny2, i.e., r+2, atx2near 1, i.e., Λ R2near 1. From Eq. (85),\nor from Eq. (92), one finds that there is a solution y2(x)\nfor each x. The maximum of the curve is at x= 1 and\ny2= 1, and the derivative there is not well defined. So,31\nit is important to discuss the behavior and the proper-\nties of y2when x→1. For that we define infinitesimal\nquantities δandε, related to xandy, respectively, by\nx= 1−δ , y = 1−ε , (C3)\nwith δ≪1 and ε≪1, both positive, and withε\nδbeing a\nquantity of order one. Indeed, δ >0 for x <1, and ε≥0\nalways. Then, from Eq. (85) one gets\nw=2(ε\nδ+ 1)pε\nδ(ε\nδ+ 2), (C4)\nvalid in first order, in the order we are working. One sees\nthat for wfiniteε\nδis finite, so generically here in this\ncalculation εis indeed of the order δ. From Eq. (86) one\nalso gets\ndy\ndx=ε\nδ, (C5)\nalso valid in the order we are working. We obtain from\nEq. (C4) thatε\nδ=w√\nw2−4−1 with w≥2. Thus, since\nheredy\ndx=ε\nδ, Eq. (C5), one finds that\ndy\ndx=w√\nw2−4−1, w ≥2, (C6)\nwhich is Eq. (96), the equation we wanted to deduce.\nLet us deduce now Eq. (100). We want to find y1=y2,\nthe coincident solution for fixed x, and wvarying. This\nis whendw\ndy= 0, see Eq. (89), i.e., S(x, y) = 0. From\nEq. (90) we have then that the bifurcation points where\ny1=y2forx= constant are given by Eq. (99). We\nare interested in finding y1=y2when x→1. Instead\nof working with Eq. (99) we work with Eq. (90) which\nis more direct. We find from x= 1−δof Eq. (C3),\nthat Eq. (90) in this limit is S= (y−1)3(y2+ 3y+\n4)−4δ(y−1)2(y+ 1)(1 + y+y2) + 4 δ2(2y2+y5). In\naddition, from y= 1−εof Eq. (C3), Eq. (90) becomes\nthen S= 12 δ2−24δε2−8ε3. As we have seen, the\ncoincident root is given whendw\ndy= 0, i.e., S= 0. Then,\n12δ2−24δε2−8ε3= 0, i.e., δ2−2δε2−2\n3ε3= 0. This is a\nquadratic in δ, with solution given by δ=q\n2\n3ε3+ε4+ε2,\ni.e.δ=ε3\n2q\n2\n3+ε+ε2. Since ε≪1 the dominant term\ngives\nδ=\u00122\n3ε3\u00131\n2\n. (C7)\nThus, δgoes with ε3\n2or, if one prefers, εgoes with δ2\n3,\nindeed ε=\u00003\n2δ2\u00011\n3. Recall that x= 1−δ, and so x2=\n1−2δin this approximation, and that y= 1−ε. Then,\nsubstituting δforx2andεforyin Eq. (C7), one has\nx2= 1−\u00128\n3(1−y)3\u00131\n2\n. (C8)We see that Eq. (C8) is Eq. (99) in the limit x→1 and\ny→1. Inverting Eq. (C8) one has that the coincident\nsolution takes the form\ny1=y2= 1−\u00123\n8\u0000\n1−x2\u00012\u00131\n3\n, (C9)\nvalid for 1 −x2≪1, which is Eq. (100), the equation we\nwanted to deduce.\nx2>1:\nLet us deduce Eq. (108). We have to find y2, i.e., r+2.\nFrom Eq. (85), or from Eq. (106), one finds that there is\na solution y2(x) for each x. The maximum of the curve\nis at x= 1 and y2= 1, and the derivative there it is not\nwell defined. So, it is important to discuss the behavior\nand the properties of y2when x→1 from above. For\nthat we define infinitesimal quantities ¯δand ¯ε, related to\nxandy, respectively, by\nx= 1 + ¯δ , y = 1−¯ε , (C10)\nwith ¯δ≪1 and ¯ ε≪1, both positive, and with¯ε\n¯δbeing a\nquantity of order one. Indeed, ¯δ >0 for x >1, and ¯ ε >0\nalways. Then, from Eq. (85) one gets\nw=2(¯ε\n¯δ−1)q\n¯ε\n¯δ(¯ε\n¯δ−2), (C11)\nvalid in first order, in the order we are working. One sees\nthat for wfinite¯ε\n¯δis finite, so generically here in this\ncalculation ¯ εis of the order of ¯δ. From Eq. (86) one also\ngets\ndy\ndx=−¯ε\n¯δ, (C12)\nalso valid in the order we are working. Since ¯δ >0 and\n¯ε >0, we obtain from Eq. (C11)¯ε\n¯δ=w√\nw2−4+ 1 with\nw≥2. Thus, we have from Eq. (C12) that\ndy\ndx=−w√\nw2−4−1, w ≥2, (C13)\nwhich is Eq. (108), the equation we wanted to deduce.\nLet us deduce now Eq. (112). We want to find y1=y2,\nthe coincident solution for fixed x, and wvarying. This\nis whendw\ndy= 0, see Eq. (89), i.e., S(x, y) = 0. From\nEq. (90) we have then that the bifurcation points where\ny1=y2forx= constant are given by Eq. (111). We are\ninterested in finding y1=y2when x→1 from above.\nInstead of working with Eq. (111) we work with Eq. (90)\nwhich is more direct. We find Then, from Eq. (105) one\nhasye= 1−2¯δ, plus higher order. For that, let x= 1+ ¯δ\nas in Eq. (C10). Then, from Eq. (105) one has ye= 1−2¯δ,\nplus higher order terms. Now we have to work out the\nvicinity of ye. Then put y=ye−ϵwhere again yeis the32\nroot of the equation y2+y+ 1−3\nx2= 0, with solution\ngiven by Eq. (105), i.e., ye=−1\n2+√\n3q\n1\nx2−1\n4, and\nnote that ϵand ¯εare different quantities. Then, these\ntwo equations yield y= 1−2¯δ−ϵ. From x= 1 + ¯δof\nEq. (C10) we find that Eq. (90) is S= (y−1)3(y2+3y+\n4) + 4 ¯δ(y−1)2(y+ 1)(1 + y+y2) + (2 y2+y5)4¯δ2. From\ny= 1−2¯δ−ϵwe have just found, we find that Eq. (90) is\nnowS= 12¯δ2+ 24¯δϵ2−8ϵ3. The coincident root is given\nwhen∂w\n∂y= 0, i.e., S= 0. Then, 12 ¯δ2+ 24¯δϵ2−8ϵ3, i.e.,\n¯δ2+2¯δϵ2−2\n3ϵ3= 0. The solution is ¯δ=−ϵ2+q\nϵ4+2\n3ϵ3.\nIt turns out that equation∂w\n∂y= 0 gives a self-consistent\nsolution if ¯δ≪¯ϵ, see below. Since ϵ≪1 the dominant\nterm gives ¯δ= (2\n3ϵ3)1\n2. We defined ¯ εasy= 1−¯εand\nalso had y= 1−2¯δ−ϵ, so ¯ε= 2¯δ+ϵ. But ¯δgoes with\nϵ3\n2, so in this order ¯ ε=ϵ, so that\n¯δ=\u00122\n3¯ε3\u00131\n2\n. (C14)\nThus, ¯δgoes with ¯ ε3\n2or, if one prefers, ¯ εgoes with ¯δ2\n3,\nindeed ¯ ε=\u00003\n2¯δ2\u00011\n3. Recall that x= 1 + ¯δ, and so x2=\n1−2¯δin this approximation, and that y= 1−¯ε. Then,\nsubstituting ¯δforx2and ¯εforyin Eq. (C14), one has\nx2= 1 +\u00128\n3(1−y)3\u00131\n2\n. (C15)\nWe see that Eq. (C15) is Eq. (111) in the limit x→1\nfrom above and y→1. Inverting Eq. (C15) one has that\nthe coincident solution takes the form\ny1=y2= 1−\u00123\n8\u0000\nx2−1\u00012\u00131\n3\n, (C16)\nvalid for x2−1≪1, which is Eq. (112), the equation we\nwanted to deduce.Finally, we calculate the implications of Eqs. (116) and\n(117). We have defined uas\nu≡rc\nR, (C17)\nas the cosmological radius in units of R, which from\nEq. (A9) means that uobeys\nu=−y\n2+r\n3\nx2−3\n4y2. (C18)\nCalculatingdu\ndyat the neighborhood of the point x= 1,\ny= 1, u= 1, and recalling that we are interested in the\nlarge black hole, i.e., the one with subscript 2, one finds\nfrom Eq. (C18), with the help of Eqs. (C5) and (C12) ,\nthat\n\u0012du2\ndx\u0013\nx=1−=\u0012dy2\ndx\u0013\nx=1+(C19)\nand\n\u0012du2\ndx\u0013\nx=1+=\u0012dy2\ndx\u0013\nx=1−(C20)\nwhere 1 −and 1 +means that one is taking the limit\nx→1 from below, i.e., x < 1, and from above, i.e.,\nx > 1, respectively. We can now deduce some fur-\nther specific properties of the cosmological horizon. In\nEq. (C13) we found\u0010\ndy2\ndx\u0011\nx=1+=−w√\nw2−4−1 so that\nfrom Eq. (C19) we have\u0000du2\ndx\u0001\nx=1−=−w√\nw2−4−1. Thus,\u0000du2\ndx\u0001\nx=1−→ −∞ forw→2, and\u0000du2\ndx\u0001\nx=1+→ −2 for\nw→ ∞ . In Eq. (C6) we found\u0010\ndy2\ndx\u0011\nx=1−=w√\nw2−4−1\nso that from Eq. (C20) we have\u0000du2\ndx\u0001\nx=1+=w√\nw2−4−1.\nThus,\u0000du2\ndx\u0001\nx=1+→ ∞ forw→2, and\u0000du2\ndx\u0001\nx=1+→0\nforw→ ∞ . This was stated in the main text.\n[1] J. D. Bekenstein, “Black holes and entropy”, Phys. Rev.\nD7, 2333 (1973).\n[2] S. W. Hawking, “Particle creation by black holes”, Com-\nmun. Math. Phys. 43, 199 (1975).\n[3] G. W. Gibbons and S. W. Hawking, “Action integrals\nand partition functions in quantum gravity”, Phys. Rev.\nD15, 2752 (1977).\n[4] G. W. Gibbons and S. W. Hawking, “Cosmological event\nhorizons, thermodynamics, and particle creation”, Phys.\nRev. D 15, 2738 (1977).\n[5] P. Ginsparg and M. J. Perry, “Semiclassical perdurance\nof the de Sitter space”, Nuclear Physics B 222, 215\n(1983).\n[6] J. W. York, “Black hole thermodynamics and the Eu-\nclidean Einstein action”, Phys. Rev. D 33, 2092 (1986).\n[7] B. F. Whiting and J. W. York, “Action principle and\npartition function for the gravitational field in black holetopologies”, Phys. Rev. Lett. 61, 1336 (1988).\n[8] G. Hayward, “Euclidean action and the thermodynamics\nof manifolds without boundary”, Phys. Rev. D 41, 3248\n(1990).\n[9] H. W. Braden, J. D. Brown, B. F. Whiting, and J. W.\nYork, “Charged black hole in a grand canonical ensem-\nble”, Phys. Rev. D 42, 3376 (1990).\n[10] O. B. Zaslavskii, “Canonical ensemble for arbitrary con-\nfigurations of self-gravitating systems”, Phys. Lett. A\n152, 463 (1991).\n[11] J. P. S. Lemos, “Thermodynamics of the two-dimensional\nblack hole in the Teitelboim-Jackiw theory”, Phys. Rev.\nD54, 6206 (1996); arXiv:gr-qc/9608016.\n[12] O. B. Zaslavskii, “Extreme state of a charged black hole\nin a grand canonical ensemble”, Phys. Rev. Lett. 76, 2211\n(1996).\n[13] O. B. Zaslavskii, “Geometry of nonextreme black holes33\nnear the extreme state”, Phys. Rev. D 56, 2188 (1997);\narXiv:gr-qc/9707015.\n[14] C. S. Pe¸ ca and J. P. S. Lemos, “Thermodynamics\nof Reissner-Nordstr¨ om-anti-de Sitter black holes in the\ngrand canonical ensemble”, Phys. Rev. D 59, 124007\n(1999); arXiv:gr-qc/9805004.\n[15] R. Andr´ e and J. P. S. Lemos, “Thermodynamics of\nfive-dimensional Schwarzschild black holes in the canon-\nical ensemble”, Phys. Rev. D 102, 024006 (2020);\narXiv:2006.10050 [hep-th].\n[16] R. Andr´ e and J. P. S. Lemos, “Thermodynamics of\nd-dimensional Schwarzschild black holes in the canon-\nical ensemble”, Phys. Rev. D 103, 064069 (2021);\narXiv:2101.11010 [hep-th].\n[17] T. V. Fernandes and J. P. S. Lemos, “Grand canonical\nensemble of a d-dimensional Reissner-Nordstr¨ om black\nhole in a cavity”, Phys. Rev. D 108, 084053 (2023);\narXiv:2309.12388 [hep-th].\n[18] J. P. S. Lemos and O. B. Zaslavskii, “Black holes and hot\nshells in the Euclidean path integral approach to quan-\ntum gravity”, Classical Quantum Gravity 40, 235012\n(29023); arXiv:2304.06740 [hep-th].\n[19] B.-B. Wang and C.-G. Huang, “Thermodynamics of de\nSitter spacetime in York’s formalism”, Mod. Phys. Lett.\nA16, 1487 (2001); arXiv:2304.06740 [hep-th].\n[20] B.-B. Wang and C.-G. Huang, “Thermodynamics of\nReissner–Nordstr¨ om-de Sitter black hole in York’s for-\nmalism”, Classical Quantum Gravity 19, 2491 (2002).\n[21] A. M. Ghezelbash and R. B. Mann, “Action, mass and\nentropy of Schwarzschild-de Sitter black holes and the\nde Sitter/CFT correspondence”, J. High Energy Phys.\nJHEP 01 (2002) 005; arXiv:hep-th/0111217 [hep-th].\n[22] H. Saida, “De Sitter thermodynamics in the canoni-\ncal ensemble”, Prog. Theor. Phys. 122, 1239 (2009);\narXiv:0908.3041 [gr-qc].\n[23] P. Draper and S. Farkas, “De Sitter black holes as con-\nstrained states in the Euclidean path integral”, Phys.\nRev. D 105, 126022 (2022); arXiv:2203.02426 [hep-th].\n[24] B. Banihashem and T. Jacobson, “Thermodynamic en-\nsembles with cosmological horizons”, J. High Energy\nPhys. JHEP 07 (2022) 042; arXiv:2204.05324 [hep-th].\n[25] B. Banihashemi, T. Jacobson, A. Svesko, and M. R.\nVisser, “The minus sign in the first law of de Sitter\nhorizons”, J. High Energy Phys. JHEP 01 (2023) 054;\narXiv:2208.11706 [hep-th].\n[26] T. Jacobson and M. R. Visser, “Partition function for a\nvolume of space”, Phys. Rev. Lett. 130, 221501 (2023);\narXiv:2212.10607 [hep-th].\n[27] T. Jacobson and M. R. Visser, “Entropy of causal\ndiamond ensembles”, SciPost Phys. 15 (2023) 023;\narXiv:2212.10608 [hep-th].\n[28] E. K. Morvan, J. P. van der Schaa, and M. R. Visser,\n“On the Euclidean action of de Sitter black holes and\nconstrained instantons”, SciPost Phys. 14, 022 (2023);\narXiv:2203.06155 [hep-th].\n[29] P. C. W. Davies, “Thermodynamic phase transitions of\nKerr-Newman black holes in de Sitter space”, Classical\nQuantum Gravity 6, 1909 (1989).\n[30] L. J. Romans, “Supersymmetric, cold and lukewarm\nblack holes in cosmological Einstein-Maxwell theory”,\nNuc. Phys. B 383, 395 (1992).\n[31] R. Bousso and S. W. Hawking, “(Anti-)evaporation of\nSchwarzschild–de Sitter black holes”, Phys. Rev. D 57,\n2436 (1998); arXiv:hep-th/9709224.[32] K. Maeda, T. Koike, M. Narita, and A. Ishibashi,\n“Upper bound for entropy in asymptotically de Sitter\nspace-time”, Phys. Rev. D 57, 3503 (1998); arXiv:gr-\nqc/9712029.\n[33] Z. C. Wu, “Entropy of a black hole with distinct sur-\nface gravities”, General Relativ. Gravit. 32, 1823 (2000);\narXiv:gr-qc/9911078.\n[34] W. Yueqin, Z. Lichun, and Z. Ren, “Black hole and cos-\nmic entropy for Schwarzschild–de Sitter spacetime”, Int.\nJ. Theor. Phys. 40, 1001 (2001).\n[35] R. Bousso, “Positive vacuum energy and the N-bound”,\nJ. High Energy Phys. JHEP 0011 (2000) 038; arXiv:hep-\nth/0010252.\n[36] R.-G. Cai, “Cardy-Verlinde formula and thermodynam-\nics of black holes in de Sitter spaces”, Nucl. Phys. B 628,\n375 (2002); arXiv:hep-th/0112253.\n[37] S. Shankaranarayanan, “Temperature and entropy of\nSchwarzschild–de Sitter spacetime”, Phys. Rev. D 67,\n084026 (2003); arXiv:gr-qc/0301090.\n[38] C. Teitelboim, “Gravitational thermodynamics of\nSchwarzschild-de Sitter space”, in Strings and Grav-\nity: Tying the Forces Together , Proceedings of the Fifth\nFrancqui Colloquium in Brussels in 2001 (de Boek Uni-\nversit´ e, Bruxelles, 2003), p. 291; hep-th/0203258.\n[39] A. Gomberoff and C. Teitelboim, “De Sitter black holes\nwith either of the two horizons as a boundary”, Phys.\nRev. D 67, 104024 (2003); hep-th/0302204.\n[40] O. J. C. Dias and J. P. S. Lemos, “Pair creation of de\nSitter black holes on a cosmic string background”, Phys.\nRev. D 69, 084006 (2004); arXiv:hep-th/0310068.\n[41] V. Cardoso, O. J. C. Dias, and J. P. S. Lemos, “Nariai,\nBertotti-Robinson and anti-Nariai solutions in higher di-\nmensions”, Phys. Rev. D 70, 024002 (2004): arXiv:hep-\nth/0401192.\n[42] Y. Sekiwa, “Thermodynamics of de Sitter black holes:\nThermal cosmological constant”, Phys. Rev. D 73,\n084009 (2006); arXiv:hep-th/0602269.\n[43] T. R. Choudhury and T. Padmanabhan, “Concept of\ntemperature in multi horizon spacetimes: Analysis of\nSchwarzschild–de Sitter metric”, Gen. Relativ. Gravit.\n39, 1789 (2007); arXiv:gr-qc/0404091.\n[44] Y. S. Myung, “Thermodynamics of Schwarzschild-de Sit-\nter black hole: Thermal stability of Nariai black hole”,\nPhys. Rev. D 77, 104007 (2008); arXiv:0712.3315 [gr-qc].\n[45] T. Pappas and P. Kanti, “Schwarzschild–de Sitter space-\ntime: The role of temperature in the emission of\nHawking radiation”, Phys. Lett. B 775, 140 (2017);\narXiv:1707.04900 [hep-th].\n[46] F. Simovic and R. B. Mann, “Critical phenomena of\nBorn-Infeld-de Sitter black holes in cavities”, J. High En-\nergy Phys. JHEP 05 (2019) 136; arXiv:1904.04871 [gr-qc].\n[47] Y. Qiu and J. Traschen, “Black hole and cosmological\nparticle production in Schwarzschild-de Sitter”, Classical\nQuantum Gravity 37, 135012 (2020); arXiv:1908.02737\n[hep-th].\n[48] C. Singha, “Thermodynamics of multi horizon\nspacetimes”, Gen. Relativ. Gravit. 54, 38 (2022);\narXiv:2108.11704 [gr-qc].\n[49] G. E. Volovik, “Double Hawking temperature: From\nblack hole to de Sitter”, Universe 8, 639 (2022),\narXiv:2205.06585 [gr-qc].\n[50] E. T. Akhmedov and K. V. Bazarov, “On the backre-\naction issue for the black hole in de Sitter spacetime”,\nPhys. Rev. D 107, 105012 (2023); arXiv:2212.06433.34\n[51] H. Andr´ easson and C. G. B¨ ohmer, “Bounds on M/R\nfor static objects with a positive cosmological con-\nstant”, Classical Quantum Gravity 26, 195007 (2009);arXiv:0904.2497 [gr-qc]."    },    {        "title": "2402.05167v1.The_2d_Free_Boson_Minkowski_CFT_with_Asymptotic_Charges.pdf",        "content": "arXiv:2402.05167v1  [hep-th]  7 Feb 2024The 2d Free Boson Minkowski CFT with Asymptotic\nCharges\nNicholas Agia and Daniel L. Jafferis\nCenter for the Fundamental Laws of Nature, Harvard Universi ty, Cambridge, MA, USA\nE-mail:nagia@g.harvard.edu ,jafferis@g.harvard.edu\nAbstract: We explicitly construct the states of the 2d free boson on the infinite line\nwith specified asymptotic charges. The Minkowski CFT states derive from (the analytic\ncontinuation of) the shrinking limit of Euclidean angular q uantization, wherein the super-\nselection sector for fixed asymptotic charges corresponds t o specific endpoint operators for\nthe angular quantization on the sphere. When applied to stri ng theory, we obtain a new\nclass of fundamental strings that are neither open nor close d and which we dub “stretched\nstrings”. We describe constant-time snapshots of the world sheets of the different types of\nstretched strings, and as a special case we find a heuristic pi cture of Maldacena’s “long\nstrings” in the c= 1 string theory.Contents\n1 Introduction 1\n2 Review of Angular Quantization 2\n2.1 Noncompact Free Boson with Neumann-Class Boundary Cond itions 4\n2.2 Noncompact Free Boson with Dirichlet-Class Boundary Co nditions 6\n2.3 Compact Free Boson 7\n3 The Shrinking Limit: Asymptotic Conditions 9\n4 Special Case: Zero Net Charge 11\n4.1 Oscillator States 12\n4.2 Noncompact Boson on Rand the Equivalence of Neumann and Dirichlet 17\n4.3 Compact Boson on R 21\n5 General Case: Nonzero Net Charge 23\n5.1 Infinitely Excited Oscillator State in the Spectrum 28\n5.2 The Other Oscillator States 30\n5.3 Exponential Operator Normalization 33\n5.4 Equivalence of Neumann and Dirichlet 35\n6 Prelude to String Theory 36\n6.1 Pure Momentum Charges 37\n6.2 Pure Winding Charges 39\n7 Summary 40\nA Identities 41\n1 Introduction\nThe 2d free boson is one of the simplest quantum field theories , so it is perhaps surprising\nthat there is still more to learn about it. Similarly, a great deal is known about conformal\nfield theories in states on a sphere due to the famous state/op erator correspondence. Far\nless is understood about CFT states in Minkowski space, wher e the spectrum is continuous\nand gapless. In this paper, we describe the 2d free boson CFT i n Minkowski space by\ntaking the “shrinking limit” of the angular quantization in troduced in [ 1]. This procedure\nconstructs a wide class of states with nontrivial asymptoti c conditions at spatial infinity,\nwhich here corresponds to injecting momentum or winding int o the system. While we\nfocus on the free boson CFT for simplicity and concreteness, the same methodology allows\n– 1 –one to study asymptotic charges in any CFT with a WZW-like sec tor. A generalization of\nthese ideas should also be able to describe higher-dimensio nal Minkowski CFT states with\narbitrary asymptotic conditions.\nOne main line of motivation to understand asymptotic condit ions in CFTs and QFTs\nmore generally stems from the prevalence of such considerat ions in applications to hologra-\nphy. Our more immediate motivation, however, is describing string theory states obtained\nfrom angular quantization between operators with Euclidea n time winding. It was argued\nin [2] why such a quantization scheme is required when there are op erators with nonzero\ntime winding present. This paper is thus a precursor for futu re work showing that this\nnew class of string provides a string-theoretic descriptio n of the microstates of the cigar\nblack hole. Along the way, we shall show that these strings fr om angular quantization also\naccount for Maldacena’s “long strings” in the c= 1 string theory [ 3]. While this paper\nis concerned only with CFT, we shall already find a heuristic p icture reminiscent of the\nlatter.\nThe organization of this paper is as follows. In Section 2, we review the basics of\nangular quantization together with the explicit results fo r the noncompact and compact\nfree bosons. In Section 3, we provide the asymptotic conditions that the states which\nsurvive the shrinking limit of angular quantization must ob ey. In Section 4, we explicitly\nperform the shrinking limit to obtain the free boson Minkows ki CFT states for the special\ncase in which the sum of the asymptotic charges vanishes. Sec tion5tackles the general\ncase of nonvanishing net asymptotic charge. Finally, Secti on6contains a short nod to\nstring theory where we describe the Lorentzian worldsheets of the different basic types\nof “stretched strings” defined by angular quantization, of w hich the long folded string of\nMaldacena is a special case. To this end, Figure 3represents the most important finding\nof this paper.\n2 Review of Angular Quantization\nAngular quantization in 2d CFT is a technique developed in [ 1] to decompose sphere\ncorrelators as a trace over states living on a meridian conne cting two antipodal points. At\neach of the two endpoint poles lies a local operator. In order to obtain a Hilbert space of\nstates with an associated evolution operator, the endpoint operators must be regularized\nby excising holes around them on which suitable boundary con ditions are placed. These\nboundary conditions must have the property that shrinking t he excised holes down to a\npoint reproduces the original operator, up to a constant. If one such hole has radius εon\nthe sphere with stereographic coordinate z, then a boundary condition Biplaced at |z|=ε\nwhich shrinks to an operator Oi(0) at the origin may be thought of as a state |Bi/a\\}bracketri}ht/a\\}bracketri}htεin\nradial quantization with the property\n|Bi/a\\}bracketri}ht/a\\}bracketri}htεε→0−→bii−siε∆iOi(ε). (2.1)\nHere, ∆ iis the scaling dimension of the operator Oi,biis a numerical constant determined\nby the specific boundary condition chosen and the phase i−siwheresiis the spin of the\noperator Oiwas factored out of bifor convenience; moreover, placing the operator at z=ε\n– 2 –asε→0 gives a precise meaning to the normalization and phase of bi. Likewise, there is an\noperator Oj(∞) at infinity, which is tobethought of as theorigin of theothe r stereographic\npatch with transition function z′=1\nz. The other boundary condition Bjplaced at |z|=1\nε\nmay then be thought of as the BPZ conjugate state ε/a\\}bracketle{t/a\\}bracketle{tBj|with the property\nε/a\\}bracketle{t/a\\}bracketle{tBj|ε→0−→bjisjε−∆jOj(1\nε). (2.2)\nIn the regulated set-up, all sphere correlators can then be s liced along constant-angle rays,\ndefining the angular quantization Hilbert space HBiBj\nij(ε) with a Hamiltonian HBiBj\nR(ε)\nwhich generates counterclockwise rotations. By construct ion, the “thermal” traces over\nthe angular quantization Hilbert space on the annulus are pr oportional to the original\nsphere correlators in the shrinking limit ε→0. That is,\nTrHBiBj\nij/bracketleftbigg\nO1(z1,z1)···On(zn,zn)e−2πHBiBj\nR(ε)/bracketrightbigg\nS2\nε→0−→bibjisj−siε∆i−∆j/angbracketleftbig\nOi(ε)O1(z1,z1)···On(zn,zn)Oj(1\nε)/angbracketrightbig\nS2.(2.3)\nOf course, the sphere correlator appearing explicitly in ( 2.3) decays as ε2∆j, so traces in\nthe shrinking limit of angular quantization always become ε∆i+∆jtimes the finite physical\nquantity. The equality in ( 2.3) for arbitrary bulk operator insertions encapsulates the\nrequired locality of the endpoint boundary conditions.\nThe Hamiltonian HBiBj\nRis labeled with ‘R’ because it becomes the Rindler Hamil-\ntonian if the angular coordinate on the sphere is analytical ly continued to a real time\ncoordinate. When originally defining angular quantization , it is preferable to work on the\nsphere because of the natural interpretation of the shrinki ng limit as the insertion of two\nendpoint local operators. Nevertheless, the physical inte rpretation of the angular quantiza-\ntion Hilbert space descends from the shrinking limit in the c ylinder conformal frame. This\ntransformation is effected as usual by z=eiτ+σ, whereτis the Euclidean time coordinate\non the cylinder and σis the spatial coordinate. At finite regulator, the cylinder has length\n2L, whereL=−lnε, and the relation ( 2.3) in this frame becomes1\nTrHBiBj\nij/bracketleftbigg\nO1(τ1,σ1)···On(τn,σn)e−2πHBiBj\nR(L)/bracketrightbigg\nS1×R\nL→∞−→bibj/a\\}bracketle{tOi(0,−L)O1(τ1,σ1)···On(τn,σn)Oj(0,L)/a\\}bracketri}htS1×R.(2.4)\nThe states |ψ/a\\}bracketri}ht ∈ HBiBj\nij(L) live on the spatial line parametrized by σ, with−L/lessorequalslantσ/lessorequalslantL,\nobeying boundary condition Biatσ=−Land obeying boundary condition Bjatσ=L.\nThe time coordinate τin (2.4) was compactified with periodicity β= 2π, but now we may\nconsider the theory on the semi-infinite strip with τdecompactified for which HBiBj\nij(L)\nandHBiBj\nR(L) are unchanged. Finally, the shrinkinglimit is obtained by takingL→ ∞. In\nthis limit, the vector space HBiBj\nij(L) becomes the angular quantization vector space Hij;\n1Note that the thermal trace in angular quantization in a give n conformal frame is defined by the\npartition function in that frame, so it picks up the same Weyl anomaly that the correlator does. In this\ncase,Zflat plane=εc/6Zflat cylinder , wherecis the central charge.\n– 3 –the only reason we have indicated possible L-dependence is that the limit Hijmay be a\nquotient of HBiBj\nij(L) if the inner product develops a null subspace. Moreover, th e angular\nquantization state space Hijshould only depend on the original endpoint operators Oiand\nOj, not on which specific boundary conditions BiandBjare chosen (as long as they are\nlocal and shrink to the operators OiandOj, respectively). This independence of boundary\ncondition is not obvious, and below we shall demonstrate the equivalence of Neumann-class\nand Dirichlet-class boundary conditions for the free boson .\nThe angular quantization Hilbert space Hijdescribes a state space of the CFT on\nMinkowski space, i.e. the states live on Rand are evolved in real time via the Hamiltonian\nHij= limL→∞[HBiBj\nR(L)−a(L)], wherea(L) is a possible c-number offset to render the\neigenvalues of Hijfinite. In particular, the Minkowski spectrum of states for a ny CFT\nis continuous and gapless, and there is no state/operator co rrespondence. Instead, the\nangular quantization associated to Hijimparts asymptotic conditions on any state |ψ/a\\}bracketri}ht\nwhich describe its behavior as σ→ −∞(from theL→ ∞limit ofBi) and its behavior\nasσ→ ∞(from theL→ ∞limit ofBj). In this sense, angular quantization and hence\nMinkowski CFT itself possesses an asymptotics/operator co rrespondence, meaning that to\neachpairof local operators in the CFT corresponds a state space Hijwith asymptotic\nconditions determined by that pair. As such, the full Hilber t space of the Minkowski CFT\nshould be regarded as\nH=/circleplusdisplay\ni,jHij, (2.5)\nwhere each pair of primaries labels a superselection sector of the theory (because no finite-\nenergy operation can change the asymptotic conditions). If the CFT possesses a global\nsymmetry, and if the endpoint operators OiandOjare charged under that symmetry, then\nthe asymptotic conditions associated to the states in Hijcan be thought of as injecting\nthose charges at the boundary of space. It is in this sense tha t angular quantization\nwith nontrivial endpoint operators constructs the Minkows ki space CFT with specified\nasymptotic charges. As the example considered in this paper , the shrinking limit of the\nfinite-regulator angular quantization descriptions revie wed below describes the free boson\nMinkowski CFT with asymptotic momentum and winding charges .\n2.1 Noncompact Free Boson with Neumann-Class Boundary Conditions\nConsider a noncompact free boson Xwhose endpoint local operators are the exponential\nprimaries Oi=:eik1X: andOj=:eik2X:. The Neumann-class boundary conditions shrink-\ning to these operators fix the normal derivative of Xat the boundaries. Specifically, it was\nproven in [ 1] that the boundary conditions\n/parenleftbig\nz∂X+z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=ε=−ik1 (2.6)\n/parenleftbig\nz∂X+z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=1\nε= +ik2 (2.7)\n– 4 –on the two-holed sphere have the desired property. On the reg ulated cylinder with bound-\naries atσ=±L, these conditions read\n∂σX/vextendsingle/vextendsingle/vextendsingle\nσ=−L=−ik1 (2.8)\n∂σX/vextendsingle/vextendsingle/vextendsingle\nσ=+L= +ik2. (2.9)\nThe finite regulator HN\nk1k2(L) andHN\nR(L) are then obtained by canonical quantization.\nSpecifically, the mode expansion of the free boson on the semi -infinite strip is2\nX(τ,σ) =x−πip\nLτ−i(k1−k2)\n2σ−i(k1+k2)\n4L(τ2−σ2)\n+i√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n,(2.10)\nwhere the nonvanishing commutators are [ x,p] =iand [αℓ,αℓ′] =nδℓ,−ℓ′. Then, the\nRindler Hamiltonian associated to the Neumann-class bound ary conditions is given by\nHN\nR(L) =πp2\n2L−i(k1+k2)\n2πx+π\n2L∞/summationdisplay\nℓ=1α−ℓαℓ+(k1−k2)2\n8πL+(k1+k2)2\n12πL−π\n48L.(2.11)\nNote that the constant mode xappears explicitly in the Hamiltonian so that the cylinder\nevolution operator e−2πHN\nRcontains the factor ei(k1+k2)xone expects from the endpoint\noperators:eik1X: and:eik2X:. Thefinite-regulator angular quantization Hilbertspace HN\nk1k2\nis then built up from the oscillator vacuum |p;{0}/a\\}bracketri}htk1k2, which is an eigenstate of the\ncenter-of-mass momentum operator p, by acting with all possible combinations of creation\noperatorsα−ℓ,ℓ>0. That is, the Fock basis of HN\nk1k2consists of the states\n|p;{Nℓ}/a\\}bracketri}htk1k2≡/parenleftBigg∞/productdisplay\nℓ=11/radicalbig\nℓNℓNℓ!αNℓ\n−ℓ/parenrightBigg\n|p;{0}/a\\}bracketri}htk1k2; (2.12)\nwe use the scripts on the states to denote explicitly the fact orHk1k2in the total Hilbert\nspaceHin which they reside. Using ( 2.10), the mode expansion of an exponential operator\non the semi-infinite strip is\n:eikX(τ,σ): =/bracketleftBigπ\nLeπτ\nLcos/parenleftBigπσ\n2L/parenrightBig/bracketrightBigk2/2\ne(k1−k2)k\n2σ+(k1+k2)k\n4L(τ2+σ2)eikxeπkτ\nLp\n×exp/bracketleftBigg\n√\n2k∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\n×exp/bracketleftBigg\n−√\n2k∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2Lcos/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightBigg\n. (2.13)\n2We use a slightly different convention for the constant mode h ere compared to that used in [ 1], namely\nxhere=xthere−i(k1+k2)L\n12.\n– 5 –Using these expressions, the thermal one-point function of a bulk exponential primary\nOk3(τ,σ) on the cylinder at finite regulator is computed to be\nTrHN\nk1k2/bracketleftBig\nOk3(τ,σ)e−2πHN\nR(L)/bracketrightBig\nS1×R\n=δ(k1+k2+k3)√\n2η(2iL\nπ)e−(k2\n1+k2\n2)L/2\ne(k2\n1−k2\n2)σ/2/bracketleftbigg\neL\n6η/parenleftbigg2iL\nπ/parenrightbigg\nϑ4/parenleftbigg\n−iσ\nπ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2iL\nπ/parenrightbigg/bracketrightbiggk2\n3/2\n.(2.14)\nThe term in brackets approaches unity as L→ ∞.\n2.2 Noncompact Free Boson with Dirichlet-Class Boundary Conditions\nWe again consider the angular quantization of the noncompac t free boson Xwith the\nsame endpoint operators : eik1X(0): and :eik2X(∞):, but now with Dirichlet-class boundary\nconditions which fix the tangential derivative of Xat the boundaries. The appropriate\nDirichlet-class boundary conditions are\n/parenleftbig\nz∂X−z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=ε=/parenleftbig\nz∂X−z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=1\nε= 0 (2.15)\non the two-holed sphere, or equivalently\n∂τX/vextendsingle/vextendsingle/vextendsingle\nσ=±L= 0 (2.16)\non the regulated cylinder or strip. Of course, these boundar y conditions do not depend on\nk1ork2and so cannot by themselves shrink to the desired endpoint op erators :eik1X: and\n:eik2X:. Instead, the above Dirichlet-class boundary conditions are imposed by boundary\nLagrange multipliers, and the parameters k1andk2appear in the equations of motions for\nthese Lagrange multipliers; for the details, see [ 1]. The mode expansion of the free boson\non the semi-infinite strip is3\nX(τ,σ) =x+/parenleftbiggxs\nL−i(k1−k2)\n2/parenrightbigg\nσ+√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n,(2.17)\nwhere the nonvanishing mode commutators are [ x,p] = [xs,ps] =iand [αℓ,αℓ′] =ℓδℓ,−ℓ′.\nThe momenta pandpscanonically conjugate to xandxs, respectively, appear in the\nmode expansions of the boundary Lagrange multipliers; sinc e the latter are not needed\nin anything that follows, we shall not write them here. Note t hat nowXhas two zero-\nmomentum-modes, the center-of-mass mode xand also the ‘slope’ mode xs. The constant\n−i(k1−k2)\n2was separated out in the definition of xsin (2.17) for convenience, essentially to\nmimic the analogous term appearing in ( 2.10). The Rindler Hamiltonian associated to the\nDirichlet-class boundary conditions is given by\nHD\nR(L) =−i(k1+k2)\n2πx+1\n2πLx2\ns+π\n2L∞/summationdisplay\nℓ=1α−ℓαℓ+(k1−k2)2\n8πL−π\n48L.(2.18)\n3We have rescaled xs→xs\nLandps→psLcompared to the conventions of [ 1].\n– 6 –The Fock basis of HD\nk1k2again consists of the states |p′,xs;{Nℓ}/a\\}bracketri}htk1k2, which we take to\nbe simultaneous eigenstates of the center-of-mass momentu mpand slope mode xs. A\npeculiarity about the slope mode is that its eigenstates, de spite forming a continuum,\nare normalized to /a\\}bracketle{txs|xs/a\\}bracketri}ht= 1; this unusual property is due to a proper treatment of the\nboundary Lagrange multipliers, as described in [ 1]. Using ( 2.17), the mode expansion of\nan exponential operator on the semi-infinite strip is\n:eikX(τ,σ): =/bracketleftbiggπ\n4Lcos(πσ\n2L)/bracketrightbiggk2\n2\ne(k1−k2)k\n2σeikxeikσ\nLxsexp/bracketleftBigg\ni√\n2k∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\n×exp/bracketleftBigg\ni√\n2k∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2Lsin/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightBigg\n.(2.19)\nUsing these expressions, the thermal one-point function of a bulk exponential primary\nOk3(τ,σ) on the cylinder at finite regulator is computed to be\nTrHD\nk1k2/bracketleftBig\nOk3(τ,σ)e−2πHD\nR(L)/bracketrightBig\nS1×R=δ(k1+k2+k3)\n4√\n2π2η(2iL\nπ)e−(k2\n1+k2\n2)L/2\ne(k2\n1−k2\n2)σ/2/bracketleftBigg\neL\n2η3(2iL\nπ)\nϑ4/parenleftbig\n−iσ\nπ/vextendsingle/vextendsingle2iL\nπ/parenrightbig/bracketrightBiggk2\n3/2\n.\n(2.20)\nThe term in brackets again approaches unity as L→ ∞.\n2.3 Compact Free Boson\nNow consider a compact free boson Xof radiusR, i.e.X∼X+2πR. We take the endpoint\noperators to beexponential primaries On1,w1(0) andOn2,w2(∞), wherenandware integers\ndenoting momentum and winding, respectively. For the excis ion regulator we shall employ\nDirichlet-class boundary conditions, suppressing the lab el ‘D’ since it is the only regulator\nfor the compact boson used in this paper. The relevant Dirich let-class boundary conditions\nare\n/parenleftbig\nz∂X−z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=ε=−iw1R (2.21)\n/parenleftbig\nz∂X−z∂X/parenrightbig/vextendsingle/vextendsingle/vextendsingle\n|z|=1\nε= +iw2R (2.22)\non the two-holed sphere, or equivalently\n∂τX/vextendsingle/vextendsingle/vextendsingle\nσ=−L= +w1R (2.23)\n∂τX/vextendsingle/vextendsingle/vextendsingle\nσ=+L=−w2R (2.24)\non the regulated cylinder or strip. As with the noncompact bo son, the momenta n1andn2\nof the endpoint operators do not appear in the boundary condi tions; as before, the above\nrelations are imposed via boundary Lagrange multipliers, a ndn1andn2appear in their\nequations of motion. The mode expansion of the compact boson on the semi-infinite strip\nis\n– 7 –X(τ,σ) =x+/parenleftbiggxs\nL−i(n1−n2)\n2R/parenrightbigg\nσ+(w1−w2)R\n2τ−(w1+w2)R\n2Lτσ\n+√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n,(2.25)\nwhere once again [ x,p] = [xs,ps] =iand [αℓ,αℓ′] =ℓδℓ,−ℓ′, withxandxsbeing the two\nindependent zero-momentum-modes. The associated Rindler Hamiltonian for the compact\nboson is\nHR(L) =−i(n1+n2)\n2πRx+i(w1−w2)R\n2p−i(w1+w2)R\n2ps+1\n2πLx2\ns+π\n2L∞/summationdisplay\nℓ=1α−ℓαℓ\n+(n1−n2)2\n8πR2L+(w1+w2)2R2\n24πL+(w1−w2)2R2\n8πL−π\n48L.(2.26)\nThe Fock basis of the finite-regulator Hilbert space Hn2,w2n1,w1(L) similarly consists of the\nstates|m,xs;{Nℓ}/a\\}bracketri}htn2,w2n1,w1, for which p|m,xs;{Nℓ}/a\\}bracketri}htn2,w2n1,w1=m\nR|m,xs;{Nℓ}/a\\}bracketri}htn2,w2n1,w1for any given\nm∈Z. As before, the slope mode eigenstates obey /a\\}bracketle{txs|xs/a\\}bracketri}ht= 1. The mode expansion of a\nbulk exponential primary On,w, heuristically of the form: ei(kLXL+kRXR): withkL=n\nR+wR\nandkR=n\nR−wR, on the semi-infinite strip is given precisely by\nOn,w(τ,σ) =/bracketleftbiggπ\n4Lcos(πσ\n2L)/bracketrightbiggn2\n2R2/bracketleftBigπ\nLcos/parenleftBigπσ\n2L/parenrightBig/bracketrightBigw2R2\n2e(w1+w2)wR2\n12L+πi\n2nw−πiσ\n2Lnw+πτ\n2Lw2R2\n×e[(n1−n2)n\n2R2+(w1−w2)wR2\n2]σei[(n1−n2)w+n(w1−w2)]\n2τe(w1+w2)R\n4L[wR(τ2−σ2)−2in\nRτσ]\n×ein\nRxe−πiwRpeπiwRp se1\nL(in\nRσ−wRτ)xs\n×exp/braceleftBigg\ni√\n2∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2L/bracketleftbiggn\nRsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n+iwRcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg\n×exp/braceleftBigg\ni√\n2∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2L/bracketleftbiggn\nRsin/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg\n−iwRcos/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg\n.\n(2.27)\nUsing these expressions, the thermal one-point function of a bulk exponential primary\nOn3,w3(τ,σ) on the cylinder at finite regulator is computed to be\nTrHn2,w2n1,w1/bracketleftBig\nOn3,w3(τ,σ)e−2πHR(L)/bracketrightBig\nS1×R=Rδn1+n2+n3,0δw1+w2+w3,0\n4√\n2π2\n×(−1)n3w1e−(∆1+∆2)L\nei(n1w1−n2w2)τe(∆1−∆2)σ1\nη(2iL\nπ)[eL\n6η(2iL\nπ)]1\n2(3n2\n3\nR2+w2\n3R2)\nϑ4/parenleftbig\n−iσ\nπ/vextendsingle/vextendsingle2iL\nπ/parenrightbig1\n2(n2\n3\nR2−w2\n3R2),(2.28)\nwhere ∆ j=1\n2(n2\nj\nR2+w2\njR2).\n– 8 –3 The Shrinking Limit: Asymptotic Conditions\nNow we would like to describe in detail the shrinking limit L→ ∞for the free boson\nangular quantizations briefly reviewed above. This limit is not trivial, as many expressions\nhave parts that blow up or decay as L→ ∞, which can obfuscate the meaningful finite\nquantities. Nevertheless, some aspects of this limit are st raightforward, for instance the\nmanner in which the discrete oscillator modes become a conti nuum in the infinite-volume\nlimit, a familiar feature from introductory quantum mechan ics. However, one of the most\nimportant aspects of angular quantization at finite regulat or is the treatment of the bound-\nary conditions, which in the shrinking limit transmute into the asymptotic conditions, so\nthe behavior of the fields near spatial infinity will be of para mount importance.\nLet us be clear about the end goal of the shrinking limit. Ulti mately, we are construct-\ning the CFT on R1,1as a collection of superselection sectors corresponding to specific\nasymptotic conditions. For each superselection sector, th ere is a vector space Hijof states\nwith corresponding Hamiltonian Hij, and each state |ψ/a\\}bracketri}htij∈ Hijobeys the asymptotic con-\nditions specified by the original primaries OiandOj. Moreover, for each pair of states in\nHijthe norm of the Hamiltonian Hijdiffers by a finite amount. Finally, there must exist\nan inner product pairing on the total state vector space/circleplustext\ni,jHij. In short, we aim for an\nenumeration of finite-energy states together with a vector p airing thereof.\nIn the free boson examples studied here, the existence of a Fo ck construction for the\nangular quantization at finite regulator greatly simplifies the task of enumeration, allowing\nus to describe the resulting states concretely and explicit ly. Likewise, that the CFT has\na field description allows us to express the asymptotic condi tions directly in terms of the\nfield itself.\nThe question remains: what are the asymptotic conditions co rresponding to the free\nboson exponential primary endpoint operators? As we have al ready mentioned, these\nasymptotic conditions cannot depend on the specific finite-r egulator boundary conditions\nused, as long as they shrink to the desired operators. For ins tance, the Neumann-class\nand Dirichlet-class boundary conditions described in the p revious section implement the\nboundaryconditionsquitedifferently(theformerfixestheno rmalderivativewhilethelatter\nfixes the tangential derivative), but they both must fix the sa me asymptotic conditions.\nFortunately, these asymptotic conditions are simple to obt ain from the same source which\nmotivated theboundaryconditions —theOPE.Specifically, t hesingularpartsof theOPEs\nof∂Xand∂Xwith an exponential primary : eikX: are\n∂X(z):eikX(0):∼ −ik\n2z:eikX(0): (3.1)\n∂X(z):eikX(0):∼ −ik\n2z:eikX(0):. (3.2)\nThe first OPE means that ∂X(z) behaves, to leading order, like the function −ik\n2zwhen in\nthe presence of an exponential operator : eikX: at the origin, and the second OPE means\nthat∂X(z) similarly behaves like −ik\n2z. As usual, these statements are operatorial, meaning\nthey apply inside any correlator as long as no other operator is closer to the exponential\noperator in question. Thus, both z∂X(z) andz∂X(z) behave like the constant −ik\n2near\n– 9 –the operator : eikX(0):. In fact, these statements uniquely characterize the expo nential\nprimary operator, and hence they arethe asymptotic conditions. Previously, we chose to\nusez∂X(z)±z∂X(z) instead since these are the more natural linear combinatio ns on the\ncylinder; we make this transition again here since the Minko wski space CFT is obtained\nupon analytic continuation of the L→ ∞limit on the strip. Therefore, the asymptotic\nconditions in the sector Hk1k2for the noncompact free boson are\nnoncompact free boson: ∂σX(τ,σ)σ→−∞−→ −ik1 (3.3)\n∂τX(τ,σ)σ→−∞−→0 (3.4)\n∂σX(τ,σ)σ→+∞−→+ik2 (3.5)\n∂τX(τ,σ)σ→+∞−→0, (3.6)\nfor all Euclidean τ∈R. These relations must hold in arbitrary correlators, but it suffices to\ncheck the asymptotic conditions by computing the expectati on value /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htk1k2in any\nstate inHk1k2. Likewise, the asymptotic conditions in the sector Hn2,w2n1,w1for the compact\nfree boson are\ncompact free boson: ∂σX(τ,σ)σ→−∞−→ −in1\nR(3.7)\n∂τX(τ,σ)σ→−∞−→+w1R (3.8)\n∂σX(τ,σ)σ→+∞−→+in2\nR(3.9)\n∂τX(τ,σ)σ→+∞−→ −w2R, (3.10)\nagain for all Euclidean τ∈R.\nSeveral comments are in order. The first is that the asymptoti c conditions ( 3.3)-(3.6)\nand (3.7)-(3.10) fix both the behavior of the spatial derivative andthat of the temporal\nderivative of the free boson. In contrast, the boundary cond itions used in the finite-\nregulator angular quantization fixed either the spatial der ivativeorthat of the temporal\nderivative. The reason is because the temporal derivative o fXis always related to its\nconjugate momentum, and placing such conditions on both fiel ds simultaneously would\nviolate the classical Poisson algebra, leading to an incons istent canonical quantization. It\nis thusnontrivial toensureall asymptotic conditions areo beyed despitestartingfromstates\nwhich initially satisfy half of them and only at one point.\nThe second comment is that the asymptotic conditions are far simpler in the special\ncase that the net asymptotic charge vanishes, i.e. for k1+k2= 0 in the noncompact case\nand forn1+n2=w1+w2= 0 in the compact case. While this scenario does not look much\ndifferent than the generic one in ( 3.3)-(3.6) or (3.7)-(3.10), the simplification arises because\nthese asymptotic conditions can be satisfied by letting the e xpectation values of ∂σX(τ,σ)\nand∂τX(τ,σ) be constants for all τandσ. An exactly linear configuration automatically\nsolves the classical equation of motion, and hence the asymp totic conditions can be obeyed\nwithout any contribution from the oscillators. On the other hand, when the sum of the\nasymptotic charges does not vanish, it is impossible to sati sfy the asymptotic conditions\n– 10 –without contributions from the oscillators. In fact, any fin ite oscillator contribution cannot\nchange the asymptotics, so the nonzero net charge case requi res infinitely-excited oscillator\nstates. Due to this sharp difference, the net vanishing and net nonzero charge scenarios\nare treated separately in the following two sections.\nLastly, the third comment is that ( 3.3)-(3.6) and (3.7)-(3.10) apply only for real τ,\ni.e. in Euclidean signature. Indeed, the asymptotic condit ions in angular quantization\narise because the shrinking limit in Euclidean signature is always conformally equivalent\nto the shrinking of holes on the sphere into local endpoint op erators. Obviously there is\nno analog of shrinking a noncompact Lorentzian time directi on to a point and recovering\na local operator. Instead, the Euclidean data is enough to co mpletely specify the states of\nthe theory, from which we may simply derive the asymptotic be havior in the Lorentzian\ntheory by performing the analytic continuation. For this re ason, all expressions below will\nbe evaluated for arbitrary complex τ; computing /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htijand restricting to real τwill\nverify the asymptotic conditions, and subsequently restri cting to pure imaginary τwill\nreveal the Lorentzian consequences of injecting asymptoti c charges.\n4 Special Case: Zero Net Charge\nAs discussed in Section 3, the case wherein the net asymptotic charge vanishes (namel y\nk1+k2= 0andn1+n2=w1+w2= 0for thenoncompactandcompact boson, respectively)\nhas the simplest shrinking limit because the oscillators me rely need to have vanishing\ncontributions in the asymptotic limit. We shall first treat t he oscillator sectors separately\nbefore including the zero-momentum-modes.\nWe already know that the oscillator ground state is a part of t he angular quantization\nHilbert space when the net asymptotic charge vanishes becau se/a\\}bracketle{t{0}|αℓ|{0}/a\\}bracketri}ht= 0. It is\nsimple to see that the entire Fock space built on top of |{0}/a\\}bracketri}htis also part of this space by\nshowing that thetotal contribution to /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htfromthesumover αℓvanishesfor σ→ ±∞\nasL→ ∞. Specifically, consider the oscillator state |{Nℓ}/a\\}bracketri}ht. From\n/a\\}bracketle{t{Nℓ′−sgn(ℓ)δ|ℓ|,ℓ′}|αℓ|{Nℓ′}/a\\}bracketri}ht=/a\\}bracketle{t{0}|αN|ℓ|−sgn(ℓ)\n|ℓ|αℓαN|ℓ|\n−|ℓ||{0}/a\\}bracketri}ht\n/radicalBig\n|ℓ|2N|ℓ|−sgn(ℓ)(N|ℓ|−sgn(ℓ))!N|ℓ|!(4.1)\n=/radicalBig\n|ℓ|(N|ℓ|+θℓ), (4.2)\nwhereθℓis unity ifℓ>0 and vanishes otherwise, the oscillator contribution to th e matrix\nelement ofX(τ,σ) is\n/a\\}bracketle{t{Nℓ′−sgn(ℓ)δ|ℓ|,ℓ′}|Xosc(τ,σ)|{Nℓ′}/a\\}bracketri}ht\n=√\n2∞/summationdisplay\nℓ=1/radicalbigg\nNℓ\nℓ/bracketleftbigg/radicalbigg\n1+1\nNℓe−πℓτ\n2L∓eπℓτ\n2L/bracketrightbigg/braceleftbiggicos\nsin/bracerightbigg/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n,(4.3)\nwhere the top line is for Neumann-class boundary conditions and the bottom line is for\nDirichlet-class boundaryconditions. Thesurvivingmomen tumstates intheshrinkinglimit,\nconstructed explicitly in the subsection below, are those f or which the mode number ℓis\n– 11 –scaled with LasL→ ∞withℓ\nLheld finite. For such states, the summand is proportional\nto1√\nℓ, and there is no possible enhancement from having infinitely many terms in the sum\nbecause only finitely many of the occupation numbers Nℓcan be nonzero by definition of a\nFock space, so it may seem like the above contribution vanish es. However, the sum over ℓ\nthen also contributes a factor of Lwhen being turned into an integral, so it seems like the\nabove matrix element is infinite in the shrinking limit. This infinity is simply due to the\nfact that the basis oscillator momentum states are not thems elves normalizable. In forming\nnormalizable sums over these states, the above matrix eleme nt necessarily vanishes as\nσ→ ±∞sincethewavepacket hasnosupportthere. Finally, therear ethetermsfrommode\nnumbersℓwhich do not scale with LasL→ ∞. All such terms certainly remain finite,\nbut they are also independent of position since the dependen ce on the latter occurs only in\nthe formℓτ\nLandℓσ\nL, both of which vanish in the shrinking limit. These contribu tions from\nfinite mode numbers are hence constants, which do not alter th e derivative asymptotics; in\nfact, we never have to consider states involving finite mode n umbers in the shrinking limit\nbecause their effects can always be absorbed into a redefinitio n of the constant mode of X.\nTherefore, we have proven that the simple Fock space structu re of the oscillators persists\nafter the shrinking limit in the case of vanishing net asympt otic charge.\n4.1 Oscillator States\nThis subsection details the continuum limit of momentum mod es for a free field in a box\nand hencemay beskippedwithoutloss of continuity; it is inc ludedmerely for completeness.\nThe pure oscillator sectors of the free theories discussed i n this paper are all of the same\nform, consistingofquantizedmomentumsolutionsoftheKle in-Gordonequationwitheither\npure Neumann or pure Dirichlet boundary conditions. In this way,Lacts as an infrared\nregulator by replacing the infinite spatial volume of Minkow ski space R1,1with the finite\nvolume 2L. In the shrinking limit, the discrete oscillator states |{Nℓ}/a\\}bracketri}htthus become the\ncontinuum of momentum states in the usual way. However, even though|{Nℓ}/a\\}bracketri}htis the most\nconvenient oscillator basis to use at finite regulator, it is a rather inconvenient basis in\nwhich to consider the shrinking limit. For fixed ℓ, the state |Nℓ= 1/a\\}bracketri}htis a superposition of\nwaves with spatial momenta ±πℓ\n2Lon the strip, so the finite-momentum states are obtained\nby writing the mode number as ℓ=qL\nπand taking the limit L→ ∞while keeping qfinite;\nthe resulting continuum state is a standing-wave superposi tion involving both momentum\nqand momentum −q. Let us note that states with multiple finite-momentum excit ations\nin the shrinking limit arise from scaling multiple mode numb ers to infinity in this way,\nand as such it is highly atypical for two modes to be scaled to l and on the same finite\nmomentum q. To reflect the phase space suppression of such coincident-m omenta states,\nit is best to change bases from |{Nℓ}/a\\}bracketri}ht, whereNℓis any natural number for each ℓ, to a\nbasis consisting of a finite number of modes excited with no as sumption about different\nmode numbers being related. That is, we change to a new basis o fj-particle states for any\nj= 0,1,2,..., spanned by states of the form |{ℓ1,ℓ2,...,ℓj}/a\\}bracketri}htdefined by\n|{ℓ1,ℓ2,...,ℓj}/a\\}bracketri}ht ≡/parenleftBiggj/productdisplay\ni=1iℓi\n√ℓiα−ℓi/parenrightBigg\n|{0}/a\\}bracketri}ht, (4.4)\n– 12 –where|{0}/a\\}bracketri}htis the oscillator vacuum; the choice of phase iℓiwill be convenient in the\nshrinking limit below. These states are orthonormalized wh en all of the ℓiare distinct but\nare missing the symmetry factors when some of the mode number s coincide. Nevertheless,\nthis lack of normality allows for the simple completeness re lation\n1=∞/summationdisplay\nj=01\nj!∞/summationdisplay\nℓ1=1···∞/summationdisplay\nℓj=1|{ℓ1,...,ℓj}/a\\}bracketri}ht/a\\}bracketle{t{ℓ1,...,ℓj}|, (4.5)\nwhere all terms in the inner sums with distinct {ℓi}are overcounted by j!, and the terms\nwith some of the mode numbers coincident are less overcounte d but compensated by their\nlarger norms.\nStart with a fixed large Land parametrize the shrinking limit by writing L(λ) where\nL(0) =LandL(1) =∞, for example. Then we consider sequences of states by writin g\neach mode number as\nℓ(λ) =q/parenleftbig\nL(λ)/parenrightbig\nπL(λ), (4.6)\nwhereq(∞) =qis the desired oscillator momentum in the continuum limit; t he subleading\norder1\nLcorrection to q(L) is important because this is what actually forces ℓ(λ) to be an\ninteger as a function of λ. There are of course infinitely many ways to choose q(L), but\nthe one-particle states in the shrinking limit are equivale nce classes thereof that form a\ncomplete Dirac-orthonormal basis of L2(R), so we must also be careful not to overcount.\nFor generic qandL,qL\nπis irrational, so two seemingly easy options are to define the\nsubleading piece in q(L) so thatℓ(λ) is either the floor or the ceiling ofqL\nπ. However, it\nis not obvious that these two limits actually lead to indepen dent states because they are\nnot orthonormal. A far more natural choice is to define the sub leading piece in q(L) so\nthatℓ(λ) is either always even or always odd, because the oscillator s in the Neumann or\nDirichlet mode expansions, for example in ( 2.10) and (2.17) respectively, involve either\ncos(πℓσ\n2L+πℓ\n2) or sin(πℓσ\n2L+πℓ\n2). For any of the Neumann and Dirichlet oscillator expansion s\non the very wide strip, we may then rewrite\ni√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n=i√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0/parenleftbiggαL\nℓ\nℓeπiℓ\nL(σ+iτ)+αR\nℓ\nℓe−πiℓ\nL(σ−iτ)/parenrightbigg\n(4.7)\n√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n=i√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0/parenleftbigg\n−αL\nℓ\nℓeπiℓ\nL(σ+iτ)+αR\nℓ\nℓe−πiℓ\nL(σ−iτ)/parenrightbigg\n,(4.8)\nwhere we have defined the strange-looking operators\nαL\nℓ≡(−1)ℓ\n2/bracketleftBigg\nα2ℓ−i/parenleftBigg\nℓe−sgn(ℓ)πi\n2L(σ+iτ)\n|ℓ|−1/2/parenrightBigg\nα2ℓ−sgn(ℓ)/bracketrightBigg\n(4.9)\nαR\nℓ≡(−1)ℓ\n2/bracketleftBigg\nα2ℓ+i/parenleftBigg\nℓesgn(ℓ)πi\n2L(σ−iτ)\n|ℓ|−1/2/parenrightBigg\nα2ℓ−sgn(ℓ)/bracketrightBigg\n, (4.10)\n– 13 –wheresgn(ℓ)≡ℓ\n|ℓ|andwesuppressthe τandσdependenceinthenotation. Theseoperators\nsatisfy\n[αL\nℓ,αL\nℓ′] = [αR\nℓ,αR\nℓ′] =ℓ/parenleftbigg\n1+1\n2(2|ℓ|−1)/parenrightbigg\nδℓ,−ℓ′ (4.11)\n[αL\nℓ,αR\nℓ′] =ℓ\n2/parenleftBigg\n1−e−sgn(ℓ)πi\nLσ\n1−1/2|ℓ|/parenrightBigg\nδℓ,−ℓ′. (4.12)\nMoreover, the oscillator part of any of the Hamiltonians ( 2.11), (2.18) or (2.26) is universal,\nand is rewritten as\nπ\n2L∞/summationdisplay\nℓ=1α−ℓαℓ=π\n2L∞/summationdisplay\nℓ=1/braceleftBigg/bracketleftBigg\n1+/parenleftbigg\n1−1\n2ℓ/parenrightbigg2/bracketrightBigg\nsec2/parenleftBigπσ\n2L/parenrightBig/bracketleftbig\nαL\n−ℓαL\nℓ+αR\n−ℓαR\nℓ/bracketrightbig\n+/bracketleftBigg\n1−1\n4ℓ\nℓsec2/parenleftBigπσ\n2L/parenrightBig\n−2tan2/parenleftBigπσ\n2L/parenrightBig/bracketrightBigg\n/bracketleftbig\nαL\n−ℓαR\nℓ+αR\n−ℓαL\nℓ/bracketrightbig\n−2itan/parenleftBigπσ\n2L/parenrightBig/bracketleftbig\nαL\n−ℓαR\nℓ−αR\n−ℓαL\nℓ/bracketrightbig/bracerightBigg\n,\n(4.13)\nwhich is of course independent of τandσ. For any fixed τandσ, the operators αL\n−ℓ\nandαR\n−ℓcreate a valid one-particle basis, though such a descriptio n at finite regulator\nis complicated because these are obviously not eigenstates . Now write ℓ=q(L)\nπLwith\nq(L)>0 understood as above, and define\n/radicalbiggq\n2πa(−q)≡lim\nL→∞αL\nq(L)\nπL/radicalbiggq\n2πa(−q)†≡lim\nL→∞αL\n−q(L)\nπL(4.14)\n/radicalbiggq\n2πa(q)≡lim\nL→∞αR\nq(L)\nπL/radicalbiggq\n2πa(q)†≡lim\nL→∞αR\n−q(L)\nπL. (4.15)\nFrom the distributional limit\n2Lδℓ1,ℓ2L→∞−→2πδ(q1−q2), (4.16)\nwe immediately see that the limits of the finite-regulator co mmutators ( 4.11) and (4.12)\nare\n[a(q1),a(q2)†] = 2πδ(q1−q2) (4.17)\n[a(q1),a(q2)] = [a(q1)†,a(q2)†] = 0 (4.18)\nfor allq1,q2∈R. Moreover, the shrinking limits of any of the Neumann and Dir ichlet\noscillator expansions on the very wide strip are\ni√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\nL→∞−→i√\n2/integraldisplay∞\n−∞dq/radicalbig\n2π|q|/bracketleftBig\na(q)e−|q|τ−iqσ−a(q)†e|q|τ+iqσ/bracketrightBig\n(4.19)\n√\n2/summationdisplay\nℓ∈Z\nℓ/ne}ationslash=0αℓ\nℓe−πℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\nL→∞−→i√\n2/integraldisplay∞\n−∞dq/radicalbig\n2π|q|sgn(q)/bracketleftBig\na(q)e−|q|τ−iqσ−a(q)†e|q|τ+iqσ/bracketrightBig\n.\n(4.20)\n– 14 –Therefore, a(q)†indeed creates a properly normalized plane-wave wavefunct ioneiqσfor\nany real value of the momentum q, so that the set of |{q}/a\\}bracketri}ht=a(q)†|{0}/a\\}bracketri}htmanifestly forms\na complete Dirac-orthonormal basis of the one-particle Hil bert space L2(R), as required.\nAt finite regulator, the left- and right-moving oscillators αL\nℓandαR\nℓmay be expanded in\nterms ofa(±q), receiving both corrections to the overall normalization due to (4.11) as\nwell as couplings between a(q) anda(−q) due to ( 4.12). The latter is of course expected,\nbecause the Neumann or Dirichlet boundary conditions on a lo ng but finite line couple\nthe left- and right-moving modes due to reflection at the boun dary. For example, taking\nthe finite-regulator basis to consist of αL\nℓandαR\nℓevaluated at τ=σ= 0, the subleading\nexpansions read\nαL\nℓ>0=/radicalbiggq\n2πa(−q)−1\n16L/radicalbigg\n2π\nq[a(q)−a(−q)]+O/parenleftbigg1\nL2/parenrightbigg\n(4.21)\nαR\nℓ>0=/radicalbiggq\n2πa(q)+1\n16L/radicalbigg\n2π\nq[a(q)−a(−q)]+O/parenleftbigg1\nL2/parenrightbigg\n, (4.22)\ntogether with their conjugates. The leading contribution t o the universal oscillator part of\nthe Rindler Hamiltonians as well as its order1\nLcorrection is then given by\nπ\n2L∞/summationdisplay\nℓ=1α−ℓαℓL→∞−→/integraldisplay∞\n−∞dq\n2π|q|a(q)†a(q)−π\n4L/integraldisplay∞\n−∞dq\n2π/bracketleftBig\na(q)†a(q)−a(−q)†a(q)/bracketrightBig\n+O/parenleftbigg1\nL2/parenrightbigg\n.\n(4.23)\nThe one-particle states with finite qthus all have finite energy relative to the oscillator\nvacuum and form normalizable states via superposition in th e usual way.\nHavingdetailedtheshrinkinglimitoftheone-particle sta tes, itisnowsimpletoanalyze\nthe multiparticle states |{q1,...,qj}/a\\}bracketri}ht ≡a(q1)†···a(qj)†|{0}/a\\}bracketri}ht, where each qi∈R\\ {0},\nnormalized as\n/angbracketleftbig\n{q′\n1,q′\n2,...,q′\nj′}/vextendsingle/vextendsingle{q1,q2,...,qj}/angbracketrightbig\n=δj′j(2π)j/summationdisplay\nσ∈Sjj/productdisplay\ni=1δ(q′\ni−qσ(i)),(4.24)\nwhereSjis the group of permutations on jelements; these states obey the completeness\nrelation\n1=∞/summationdisplay\nj=01\nj!/integraldisplay∞\n−∞dq1\n2π···/integraldisplay∞\n−∞dqj\n2π|{q1,...,qj}/a\\}bracketri}ht/a\\}bracketle{t{q1,...,qj}|. (4.25)\nWe know that the shrinking limits of the oscillators are ( −1)ℓα−2ℓ→/radicalBig\nq\n2π[a(q)†+a(−q)†]\nand−i(−1)ℓα−2ℓ+1→/radicalBig\nq\n2π[a(q)†−a(−q)†], whereqis the limit ofπℓ\nL. The shrinking limit\nof a finite-regulator oscillator state, scaling each ℓi=qiL\nπwith finite qi>0 and choosing\neachℓito be either even or odd throughout the limit, is then\n|{ℓ1,ℓ2,...,ℓj}/a\\}bracketri}htL→∞−→1\n(2L)j/2/summationdisplay\n/vector s∈Zj\n2(−1)s1+s2+...+sj|{(−1)s1q1,(−1)s2q2,...,(−1)sjqj}/a\\}bracketri}ht,\n(4.26)\n– 15 –where we sum over the 2jchoices of signs in the momenta. The unconventional phase in\nthe definition of |{ℓ1,...,ℓj}/a\\}bracketri}htwas chosen to cancel the factors of ( −1)ℓand−i(−1)ℓin\nchanging basis from circular polarization to linear polari zation. The states |{q1,...,qj}/a\\}bracketri}ht\nwith some {qi}coincident have divergent normalizations (whose regulari zed form is 2 Lfor\neach coincidence) but are correspondingly higher codimens ion in phase space.\nFinally, we can confirm we have properly constructed the norm alizable Hilbert space.\nThe generic oscillator state in the finite-regulator Hilber t space is\n|ψJ(L)/a\\}bracketri}ht=J/summationdisplay\nj=0∞/summationdisplay\nℓ1,...,ℓj=1fj({ℓ1,...,ℓj},L)|{ℓ1,...,ℓj}/a\\}bracketri}ht, (4.27)\nwhich has finite norm for all fj({ℓ},L)∈L2(Nj); by definition this Hilbert space is the\n(closure ofthe) direct sumof the j-particle subspaces, sothestates consistofarbitrarily b ut\nfinitely manyparticles. Considera j-particle wavefunction fj({ℓ},L) that is sharplypeaked\nat some{ℓ0}and has compact support of width σ, where Vol[ Bj(1)]σj≃µ[suppfj({ℓ},L)];\na basis ofL2(Nj) may be formed out of such functions. For this j-particle state to remain\na nontrivial j-particle state in the shrinking limit, we must take {ℓ0}={q0L\nπ}as well as\nσ=L∆qfor finite and positive q0and ∆q; we need not consider the case where only\na subset of the mode numbers is scaled with L, since they only affect the dressings of\nthe finite-energy states at finite IR regulator. To preserve n ormalizability, fj({ℓ},L) must\nscale as order 1 /(2L)j/2, so define/tildewidefj({q},L)≡(2L)j/2fj({qL\nπ},L) which is peaked at\n{|q|}={q0}, has width ∆ qwhere Vol[Bj(1)](∆q)j≃µ[supp/tildewidefj({q},L)] and scales as order\n(2L)0. Hence,\n/tildewidefj({q})≡lim\nL→∞/tildewidefj({q},L)∈L2/parenleftbig\nRj/parenrightbig\n. (4.28)\nThe shrinking limit of these states is therefore\nJ/summationdisplay\nj=0∞/summationdisplay\nℓ1,...,ℓj=1fj({ℓ1,...,ℓj},L)|{ℓ1,...,ℓj}/a\\}bracketri}ht\nL→∞−→J/summationdisplay\nj=0/summationdisplay\n/vector s∈Zj\n2/parenleftbigg\n(2L)j/integraldisplay∞\n−∞dq1\n2π···/integraldisplay∞\n−∞dqj\n2π/parenrightbigg/parenleftBigg/tildewidefj({(−1)sk})\n(2L)j/2/parenrightBigg/parenleftBigg\n(−1)/summationtexts\n(2L)j/2|{(−1)sk}/a\\}bracketri}ht/parenrightBigg\n(4.29)\n=J/summationdisplay\nj=0/summationdisplay\n/vector s∈Zj\n2(−1)/summationtexts/integraldisplay∞\n−∞dq1\n2π···/integraldisplay∞\n−∞dqj\n2π/tildewidefj({(−1)sk})|{(−1)s1k1,...,(−1)sjkj}/a\\}bracketri}ht,(4.30)\nwhich is indeed a normalizable state in the continuum theory . Note that this result also\nholds in the generalized basis sense for fj({ℓ}) =δ{ℓ},{ℓ0}which does not follow the as-\nsumptions (it has σ= 0 and the limit of (2 L)j/2δ{ℓ},{ℓ0}is not in Lebesgue space); its\nlimit produces |{ℓ0}/a\\}bracketri}ht →/summationtext\n/vector s∈Zj\n2(−1)/summationtexts\n(2L)j/2|{(−1)sq0}/a\\}bracketri}htthat we concluded above. Thus we have\nconstructed the complete oscillator sector of the angular q uantization Hilbert spaces of\nfinite-energy normalizable states for the free boson with ze ro net charge. The Neumann\nand Dirichlet oscillators are manifestly equivalent in the shrinking limit, as the latter differ\nonly by a phase for the left-moving states compared to the for mer.\n– 16 –4.2 Noncompact Boson on Rand the Equivalence of Neumann and Dirichlet\nNow we incorporate the shrinking limit of the non-oscillato r sectors. The state space must\nconsist of all (possibly Dirac-)normalizable states of fini te energy. The guiding princi-\nple we use is that of the Reeh-Schlieder theorem [ 4], which asserts that any state may\nbe arbitrarily-well-approximated by acting on a “vacuum” s tate by the algebra of local\noperators in a finite (Lorentzian) spacetime region. Since w e have already treated the os-\ncillators, consider the finite-regulator state |p;{0}/a\\}bracketri}htk,−kfor the Neumann-class noncompact\nboson angular quantization, where pis a fixed constant not scaled with L. From Section 2,\nthe actions of the free boson, the Hamiltonian and an exponen tial primary on it are given\nby\nX(τ,σ)|p;{0}/a\\}bracketri}htk,−k=/bracketleftbigg\nx−πip\nLτ−ikσ/bracketrightbigg\n|p;{0}/a\\}bracketri}htk,−k+oscillators (4.31)\nHN\nR(L)|p;{0}/a\\}bracketri}htk,−k=/bracketleftbiggπp2\n2L+k2\n2πL−π\n48L/bracketrightbigg\n|p;{0}/a\\}bracketri}htk,−k (4.32)\nOk′(τ,σ)|p;{0}/a\\}bracketri}htk,−k=/bracketleftBigπ\nLeπτ\nLcos/parenleftBigπσ\n2L/parenrightBig/bracketrightBigk′2/2\nekk′σeπk′τ\nLp\n×exp/bracketleftBigg\n√\n2k′∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\n|p+k′;{0}/a\\}bracketri}htk,−k.(4.33)\nThe first expression above shows that the expectation values of the free boson derivatives\nin the oscillator vacuum satisfy\n/a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htk,−kL→∞−→ −ik (4.34)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htk,−kL→∞−→0, (4.35)\nshowing that the asymptotic conditions ( 3.3)-(3.6) are indeed obeyed. Note that the state\nin which we are computing expectation values is a properly no rmalizable state obtained by\nsmearing |p;{0}/a\\}bracketri}htk,−kinp. To show that the exponential operator above is normalizabl e,\nwe compute\n−k,k/a\\}bracketle{t{0};p|Ok′′(τ2,σ2)Ok′(τ1,σ1)|p;{0}/a\\}bracketri}ht−k,k\n= 2πδ(k′+k′′)/bracketleftbiggπ2\nL2eπ(τ1+τ2)\nLcos/parenleftBigπσ1\n2L/parenrightBig\ncos/parenleftBigπσ2\n2L/parenrightBig/bracketrightbiggk′2/2\neπ\nL[(τ1−τ2)p−k′τ2]ekk′(σ1−σ2)\n×exp/bracketleftBigg\n2k′2∞/summationdisplay\nℓ=11\nℓeπℓ(τ1−τ2)\n2Lcos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg/bracketrightBigg\n.(4.36)\nIt is simple to evaluate the sum remaining in the last line, an d taking its L→ ∞limit\ngives\n∞/summationdisplay\nℓ=11\nℓeπℓτ12\n2Lcos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\nL→∞−→ −1\n8ln/bracketleftbiggπ4[(|τ12|2+σ2\n12)2−(2σ12Imτ12)2]\nL4/bracketrightbigg\n+i\n4/bracketleftbigg\ntan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg/bracketrightbigg\n+πi\n8[sgn(σ12+Imτ12)−sgn(σ12−Imτ12)],\n(4.37)\n– 17 –whereτij≡τi−τjandσij≡σi−σj. The shrinking limit of the above matrix element is\nthen\n−k,k/a\\}bracketle{t{0};p|Ok′′(τ2,σ2)Ok′(τ1,σ1)|p;{0}/a\\}bracketri}ht−k,kL→∞−→2πδ(k′+k′′)ekk′σ12\n[(|τ12|2+σ2\n12)2−(2σ12Imτ12)2]k′2/4\n×exp/braceleftbiggik′2\n2/bracketleftbigg\ntan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+πθ(|Imτ12|−|σ12|)sgn(Imτ12)/bracketrightbigg/bracerightbigg\n,\n(4.38)\nandhencetheexponentialoperatorisnormalizable. Weconc ludethatthestates |p;{0}/a\\}bracketri}htk,−k\nfor allp∈Rmust be included because any one of them is obtained from any o ther\nby applying a normalizable local operator. Therefore, the H ilbert space Hk,−kfor the\nnoncompact boson on Rhas the basis |p;{q1,...,qj}/a\\}bracketri}htk,−kforp,qi∈Randj∈N, and the\nmode expansion and Hamiltonian are\nX(τ,σ) =x−ikσ+i√\n2/integraldisplay∞\n−∞dq/radicalbig\n2π|q|/bracketleftBig\na(q)e−|q|τ−iqσ−a(q)†e|q|τ+iqσ/bracketrightBig\n(4.39)\nHk,−k= lim\nL→∞/bracketleftbigg\nHN\nR(L)−k2\n2πL/bracketrightbigg\n=/integraldisplay∞\n−∞dq\n2π|q|a(q)†a(q), (4.40)\nwhere [x,p] =iand [a(q),a(q′)†] = 2πδ(q−q′). The center-of-mass xis now a true zero-\nmode. The inner product is that computed from the commutatio n relations among the\nmode operators.\nNow we obtain the shrinking limit for the Dirichlet-class no ncompact boson angu-\nlar quantization, demonstrating the equivalence of the Neu mann-class and Dirichlet-class\nboundary conditions. So consider the finite-regulator stat e|p,xs;{0}/a\\}bracketri}htk,−kinHD\nk,−k. From\nSection2, the actions of the free boson, the Hamiltonian and an expone ntial primary on it\nare given by\nX(τ,σ)|p,xs;{0}/a\\}bracketri}htk,−k=/bracketleftBig\nx+/parenleftBigxs\nL−ik/parenrightBig\nσ/bracketrightBig\n|p,xs;{0}/a\\}bracketri}htk,−k+oscillators (4.41)\nHD\nR(L)|p,xs;{0}/a\\}bracketri}ht=/bracketleftbigg1\n2πLx2\ns+k2\n2πL−π\n48L/bracketrightbigg\n|p,xs;{0}/a\\}bracketri}ht (4.42)\nOk′(τ,σ)|p,xs;{0}/a\\}bracketri}htk,−k=/bracketleftbiggπ\n4Lcos(πσ\n2L)/bracketrightbiggk′2\n2\nekk′σeik′σ\nLxs\n×exp/bracketleftBigg\ni√\n2k′∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\n|p+k′,xs;{0}/a\\}bracketri}htk,−k.(4.43)\nIn order for the energy between states to remain finite, the di fference in values of xscannot\nscale faster than√\nL. It is also required that any one value ofxs\nLvanish asL→ ∞, so that\nthe free boson again satisfies the asymptotic conditions\n/a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htk,−kL→∞−→ −ik (4.44)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htk,−kL→∞−→0. (4.45)\n– 18 –For the normalizability of the exponential operators, we co mpute\n−k,k/a\\}bracketle{tp,xs;{0}|Ok′′(τ2,σ2)Ok′(τ1,σ1)|p,xs;{0}/a\\}bracketri}htk,−k= 2πδ(k′+k′′)ekk′(σ1−σ2)eik′\nL(σ1−σ2)xs\n[16L2\nπ2cos(πσ1\n2L)cos(πσ2\n2L)]k′2\n2\n×exp/bracketleftBigg\n2k′2∞/summationdisplay\nℓ=11\nℓeπℓ(τ1−τ2)\n2Lsin/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg/bracketrightBigg\n.(4.46)\nOnce again, the remaining sum is easy to compute, and its limi t asL→ ∞is\n∞/summationdisplay\nℓ=1eπℓτ12\n2L\nℓsin/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\nL→∞−→1\n8ln/bracketleftBigg\n(4L\nπ)4\n(|τ12|2+σ2\n12)2−(2σ12Imτ12)2/bracketrightBigg\n+i\n4/bracketleftbigg\ntan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+π\n2sgn(σ12+Imτ12)−π\n2sgn(σ12−Imτ12)/bracketrightbigg\n.\n(4.47)\nHence, we have computed\n−k,k/a\\}bracketle{tp,xs;{0}|Ok′′(τ2,σ2)Ok′(τ1,σ1)|p,xs;{0}/a\\}bracketri}htk,−kL→∞−→2πδ(k′+k′′)ekk′σ12\n[(|τ12|2+σ2\n12)2−(2σ12Imτ12)2]k′2\n4\n×exp/braceleftbiggik′2\n2/bracketleftbigg\ntan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+πθ(|Imτ12|−|σ12|)sgn(Imτ12)/bracketrightbigg/bracerightbigg\n,\n(4.48)\nwhich again proves that the exponential operators are norma lizable. Now, however, we see\nthat neither the mode operator xsnor its conjugate psappear anywhere in the shrinking\nlimit of the algebra of local normalizable operators. Hence , there is no finite-energy process\nby which the eigenvalue of xsin a state |p,xs;{0}/a\\}bracketri}htcan change. As such, the eigenvalue of\nxseffectively ‘freezes’ and no longer labels states in the L→ ∞limit. More properly, the\n(xs,ps)-sector of HD\nk,−k(L) decouples in this limit, by which we should quotient. There -\nfore, the shrinking limit of the Dirichlet-class noncompac t free boson angular quantization\nconsists of a Hilbert space Hk,−kspanned by states of the form |p,{q1,...,qj}/a\\}bracketri}htk,−k, where\np,qi∈Randj∈N, for which the mode expansions of the free boson and the Hamil tonian\nare\nX(τ,σ) =x−ikσ+i√\n2/integraldisplay∞\n−∞dq/radicalbig\n2π|q|sgn(q)/bracketleftBig\na(q)e−|q|τ−iqσ−a(q)†e|q|τ+iqσ/bracketrightBig\n(4.49)\nHk,−k= lim\nL→∞/bracketleftbigg\nHD\nR(L)−k2\n2πL/bracketrightbigg\n=/integraldisplay∞\n−∞dq\n2π|q|a(q)†a(q), (4.50)\nwhere [x,p] =iand [a(q),a(q′)†] = 2πδ(q−q′). The sole zero-mode subspace is the ( x,p)-\nsector, and the inner product follows from the commutation r elations among the mode\noperators as usual.\nFrom the above constructions, it is almost obvious that the s hrinking limits of the\nNeumann-class and theDirichlet-class approaches areequi valent. Both obtain a Fock space\n– 19 –structurewithbasisstatesoftheform |p;{q1,...,qj}/a\\}bracketri}htk,−ktogether withthemodeoperators\nx,panda(q),q∈R\\{0}. Theequivalenceisthenprovenbyexhibitinganisometrybe tween\nthese Hilbert spaces which maps one Hamiltonian to the other . Momentarily scripting the\noperators with ‘N’ or ‘D’ based on the regulator used, it is im mediately clear from ( 4.39)\nand (4.40) as well as ( 4.49) and (4.50) that the desired isometry is effected by\n|p,{0}/a\\}bracketri}htN\nk,−k/ma√sto−→ |p,{0}/a\\}bracketri}htD\nk,−k (4.51)\nxN/ma√sto−→xD (4.52)\npN/ma√sto−→pD (4.53)\naN(q)/ma√sto−→sgn(q)aD(q). (4.54)\nThis isometry merely flips the sign of the left-moving excita tions relative to the right-\nmoving ones, which of course is exactly what one expects — the difference between a\nwave reflected from a Neumann boundary and a Dirichlet bounda ry is precisely a phase\nshift byπ. There is but one subtle point which is worthy of addressing. The above\nisometry manifestly maps X(τ,σ) andHk,−kbetween their respective expressions, but\nthe exponential operators require more care. We found that t he exponentials in the two\napproaches have the mode expansions\nON\nk′(τ,σ) =ekk′σeik′xVN\nk′[aN(q)](τ,σ) (4.55)\nOD\nk′(τ,σ) =ekk′σeik′xVD\nk′[aD(q)](τ,σ) (4.56)\nwhere\nVN\nk′≡lim\nL→∞/braceleftBigg/parenleftBigπ\nL/parenrightBigk′2\n2exp/bracketleftBigg\n√\n2k′∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\nexp/bracketleftBigg\n−√\n2k′∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2Lcos/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightBigg/bracerightBigg\n(4.57)\nVD\nk′≡lim\nL→∞/braceleftBigg/parenleftBigπ\n4L/parenrightBigk′2\n2exp/bracketleftBigg\ni√\n2k′∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightBigg\nexp/bracketleftBigg\ni√\n2k′∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2Lsin/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightBigg/bracerightBigg\n.\n(4.58)\nFor the oscillator exponentials Vk′, there is no simple way to express the shrinking limit\nthat is independent of L. Indeed, the operators Vk′(τ,σ) have finite norm; the diverging\nnorms of the individual exponentials are canceled by the exp licit factors of1\nLk′2/2. In these\nexpressions, the limit L→ ∞is understood to mean that all matrix elements are to be\ncomputed by using the finite-regulator sums and only afterwa rd taking the L→ ∞of the\nc-number result. Such a procedure is required not only to make sense of the Lk′2/2factors\nin the denominator but also to resolve a curiosity in the isom etry between the Neumann-\nclass and Dirichlet-class approaches. If one were blindly t o apply the leading shrinking\nlimits of the sums in the above expressions for VN\nk′andVD\nk′, then one would find that\nthe mapaN(q)/ma√sto→sgn(q)aD(q) indeed maps the exponentials to each other, since after all\nthey simply descend from ( 4.39) and (4.49). However, the prefactors involved are (π\nL)k′2/2\nand (π\n4L)k′2/2, and so one might be tempted to conclude VN\nk′[sgn(q)aD(q)]?= 2k′2VD\nk′[aD(q)].\n– 20 –Nevertheless, it is in fact true that\nVN\nk′[sgn(q)aD(q)] =VD\nk′[aD(q)]! (4.59)\nThis perhaps surprising result emphasizes the need to perfo rm the oscillator sums before\ntaking the limit L→ ∞. The easiest way to verify ( 4.59) is by noting that the exponential\nnorms computed in ( 4.38) and (4.48) are manifestly equal. It is simple to see that the\ndiscrete sums must be performed first whenever there are summ ations which must be\ncommuted past each other, where the corresponding integral s would have a divergence;\nthe sums may of course be performed asymptotically in the lim itL→ ∞since only the\ncontributions diverging and constant in Lare required. Then, the magical factor of 2 which\nmakes all Neumann-class and Dirichlet-class computations equal is due to\n∞/summationdisplay\nℓ=11\nℓeπℓτ\n2L/bracketleftbigg\ncos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\n−sin/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg/bracketrightbigg\n=∞/summationdisplay\nℓ=1(−1)ℓ\nℓeπℓτ\n2Lcos/parenleftbiggπℓ(σ1+σ2)\n2L/parenrightbigg\nL→∞−→∞/summationdisplay\nℓ=1(−1)ℓ\nℓ=−ln2.(4.60)\n4.3 Compact Boson on R\nHaving presented the shrinking limit of angular quantizati on in great detail for the non-\ncompact boson, we may easily adapt the results to the compact boson of radius R, which\nwe reviewed in Section 2exclusively for Dirichlet-class boundary conditions. We a re still\nworking in the special case where the total asymptotic charg e vanishes, i.e. here we con-\nstructHn2,w2n1,w1for (n1,w1) = (n,w) and (n2,w2) = (−n,−w). The operators of interest\nacting on a finite-regulator oscillator vacuum |m,xs;{0}/a\\}bracketri}ht−n,−w\nn,w(for which we temporarily\nomit thenandwlabels) are\nX(τ,σ)|m,xs;{0}/a\\}bracketri}ht=/bracketleftbigg\nx+/parenleftbiggxs\nL−in\nR/parenrightbigg\nσ+wRτ/bracketrightbigg\n|m,xs;{0}/a\\}bracketri}ht+oscillators (4.61)\nHR(L)|m,xs;{0}/a\\}bracketri}ht=/bracketleftbigg\nimw+1\n2πLx2\ns+n2\n2πR2L+w2R2\n2πL−π\n48L/bracketrightbigg\n|m,xs;{0}/a\\}bracketri}ht(4.62)\nOn′,w′(τ,σ)|m,xs;{0}/a\\}bracketri}ht=/bracketleftbiggπ\n4Lcos(πσ\n2L)/bracketrightbiggn′2\n2R2/bracketleftBigπ\nLcos/parenleftBigπσ\n2L/parenrightBig/bracketrightBigw′2R2\n2eπi\n2n′w′−πiσ\n2Ln′w′+πτ\n2Lw′2R2\n×e[nn′\nR2+ww′R2]σei[nw′+n′w]τe−πimw′e1\nL(in′\nRσ−w′Rτ)xs\n×exp/braceleftBigg\ni√\n2∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ′\n2L/bracketleftbiggn′\nRsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n+iw′Rcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg\n|m+n′,xs−πw′R;{0}/a\\}bracketri}ht.(4.63)\nAs before, finite energy difference and the asymptotic conditi ons require thatxs\nL→0 and\n∆xs√\nL→0 for any fixed sloped eigenvalues xsasL→ ∞. Then, the expectation values of\nthe free boson derivatives are\n/a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htL→∞−→ −in\nR(4.64)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htL→∞−→wR, (4.65)\n– 21 –showing that the compact boson asymptotic conditions ( 3.7)-(3.10) are indeed satisfied in\nthis case. For normalizability of the exponential operator s, we compute\n/a\\}bracketle{tm,xs;{0}|On′′,w′′(τ2,σ2)On′,w′(τ1,σ1)|m,xs;{0}/a\\}bracketri}ht\n=δn′,−n′′δw′,−w′′/bracketleftbiggπ2\n16L2cos(πσ1\n2L)cos(πσ2\n2L)/bracketrightbiggn′2\n2R2/bracketleftbiggπ2\nL2cos/parenleftBigπσ1\n2L/parenrightBig\ncos/parenleftBigπσ2\n2L/parenrightBig/bracketrightbiggw′2R2\n2\n×e−πi(σ1+σ2)\n2Ln′w′+π(τ1+τ2)\n2Lw′2R2e[nn′\nR2+ww′R2]σ12ei[nw′+n′w]τ12e1\nL(in′\nRσ12−w′Rτ12)xseπ\nL(in′\nRσ2−w′Rτ2)w′R\n×exp/braceleftBigg\n2∞/summationdisplay\nℓ=1eπℓτ12\n2L\nℓ/bracketleftbiggn′\nRsin/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\n+iw′Rcos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg/bracketrightbigg/bracketleftbiggn′\nRsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\n−iw′Rcos/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg\n.\n(4.66)\nAs before, the remaining sum is straightforward to evaluate , and taking the L→ ∞limit\nresults in\n/a\\}bracketle{tm,xs;{0}|On′′,w′′(τ2,σ2)On′,w′(τ1,σ1)|m,xs;{0}/a\\}bracketri}ht\nL→∞−→δn′,−n′′δw′,−w′′(−1)n′w′e[nn′\nR2+ww′R2]σ12ei[nw′+n′w]τ12\n[(|τ12|2+σ2\n12)2−(2σ12Imτ12)2]∆n′,w′/parenleftbiggσ12−iτ12\nσ12+iτ12/parenrightbiggn′w′\n×exp/braceleftbigg\n2i∆n′,w′/bracketleftbigg\ntan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+πθ(|Imτ12|−|σ12|)sgn(Imτ12)/bracketrightbigg/bracerightbigg\n,\n(4.67)\nwhere ∆ n′,w′=1\n4(n′2\nR2+w′2R2), and hence these operators are indeed normalizable. As in\nthe noncompact case, the operator xsdoes not appear in the mode expansion of any local\nnormalizable operator in the shrinking limit. However, in c ontrast with the noncompact\ncase, its conjugate momentum psdoes appear in the mode expansions of the exponential\noperators. The entire effect of the slope mode is thus that acti ng with On′,w′shifts the\nstate|xs/a\\}bracketri}htto|xs−πw′R/a\\}bracketri}ht. As a result, the ( xs,ps)-sector for the compact boson angu-\nlar quantization does not get eliminated completely in the s hrinking limit, but rather ps\nbecomes compactified (at radius1\nπR), rendering the spectrum of xsdiscrete. As before,\nshifting the entire spectrum of xsby a constant does nothing, so we might as well set one\nof its eigenvalues to be xs= 0, followed by the operator rescaling xs/ma√sto→πRxsandps/ma√sto→ps\nπR.\nThen, the oscillator vacuum states in the shrinking limit ar e written |m,ms;{0}/a\\}bracketri}ht, where\nm,ms∈Z. The mode expansion of an exponential operator then becomes\nOn′,w′(τ,σ) =eπi\n2n′w′e[nn′\nR2+ww′R2]σei[nw′+n′w]τein′\nRxe−πiw′Rpeiw′psVn′,w′[a(q)](τ,σ),(4.68)\nwhere\nVn′,w′≡lim\nL→∞/bracketleftBigg/parenleftBigπ\n4L/parenrightBign′2\n2R2/parenleftBigπ\nL/parenrightBigw′2R2\n2exp/braceleftBigg\ni√\n2∞/summationdisplay\nℓ=1α−ℓ\nℓeπℓτ\n2L/bracketleftbiggn′\nRsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n+iw′Rcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg\n×exp/braceleftBigg\ni√\n2∞/summationdisplay\nℓ′=1αℓ′\nℓ′e−πℓ′τ\n2L/bracketleftbiggn′\nRsin/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg\n−iw′Rcos/parenleftbiggπℓ′(σ+L)\n2L/parenrightbigg/bracketrightbigg/bracerightBigg/bracketrightBigg\n.(4.69)\n– 22 –Acting with On′,w′on a zero-mode state |m,ms/a\\}bracketri}htthus yields the state |m+n′,ms−w′/a\\}bracketri}ht.\nTherefore, the Hilbert space H−n,−w\nn,wfor the compact boson on Rhas the basis states\n|m,ms;{q1,...,qj}/a\\}bracketri}ht−n,−w\nn,wform,ms∈Z,qi∈R\\0 andj∈N, and the mode expansion\nand Hamiltonian are\nX(τ,σ) =x−in\nRσ+wRτ+i√\n2/integraldisplay∞\n−∞dq/radicalbig\n2π|q|sgn(q)/bracketleftBig\na(q)e−|q|τ−iqσ−a(q)†e|q|τ+iqσ/bracketrightBig\n(4.70)\nH−n,−w\nn,w= lim\nL→∞/bracketleftbigg\nHR(L)−n2\n2πR2L−w2R2\n2πL/bracketrightbigg\n=iwRp+/integraldisplay∞\n−∞dq\n2π|q|a(q)†a(q),(4.71)\nwhere [x,p] =iand [a(q),a(q′)†] = 2πδ(q−q′); the integer msis the eigenvalue of xs, and\nits conjugate momentum psis compactified at unit radius. The compact boson CFT on\nMinkowski space thus has two zero-modes; the first is from the obvious shift symmetry,\nwhile the second does not affect the value of Xat all but whose entire role is to enforce\nwinding conservation of correlators.\n5 General Case: Nonzero Net Charge\nIn Section 4, we constructed the noncompact and compact free boson Minko wski CFTs in\nthespecialcasethatthenetasymptoticchargevanishes. Th ere, wefoundasimplestructure\nto each result, essentially given by a free oscillator Fock s pace built on top of a classical\nsolution to the equations of motion. We must now turn to the ge neral case of arbitrary\nendpoint exponential operators in angular quantization. T he corresponding vector spaces\nHk1k2andHn2,w2n1,w1describe states with unbalanced asymptotic conditions whi ch cannot be\nsatisfied by any classical saddle, which is the origin of the g reater technical difficulty. At\nfinite regulator, the boundary conditions on the strip are so lved in (2.10) and (2.25) by a\nquadratic term whose derivatives at σ=±Lare asymmetric. However, in the shrinking\nlimitL→ ∞, all such quadratic terms vanish due to the explicit factor o f1\nL, making\nit impossible to then satisfy the asymptotic conditions ( 3.3)-(3.6) or (3.7)-(3.10) without\nhelp from the oscillators. In the noncompact Dirichlet-cla ss approach, the finite regulator\nmode expansion ( 2.17) does not contain a quadratic term, but it also does not obey t he\nasymptotic conditions anywhere. Fortunately, the zero-mo mentum-mode sectors remain\nsimilar in form to those of Section 4, so we are tasked with finding a suitable replacement\nfor the “oscillator vacuum” on which to construct all other s tates.\nThe first point we must make is that the putative state spaces Hk1k2andHn2,w2n1,w1would\nnot be Hilbert spaces. Specifically, an inner product would n ot be defined between two\nstates in Hk1k2but rather would be a pairing between an element of Hk1k2and an element\nofH−k1,−k2, or likewise between Hn2,w2n1,w1andH−n2,−w2\n−n1,−w1. To demonstrate why, we recall\nthat angular quantization is defined so that it reproduces al l Euclidean correlators on the\nsphere. In particular, at finite regulator in the cylinder fr ame, this requirement is nothing\nother than the Cardy condition but applied to non-conformal ly-invariant boundary states\nand conditions. The cylinder equivalence of matrix element s between boundary states and\nthermal traces bestows upon angular quantization a notion o f Hermitian conjugation that\nis inherited from that of radial quantization. At finite regu lator, the cylinder thermal trace\n– 23 –Hij H¯ı¯=⇒\nconjugate\n|Bi/a\\}bracketri}ht/a\\}bracketri}ht/a\\}bracketle{t/a\\}bracketle{tBj|\n|Bj/a\\}bracketri}ht/a\\}bracketri}ht/a\\}bracketle{t/a\\}bracketle{tBi|\nFigure 1 . The Hermitian conjugate of a state in angular quantization belongs to the sector with\nconjugated endpoint operators. The blue line represents a state in angular quantization, with the\narrow indicating orientation. The red arrow shows Euclidean time evo lution by the associated\nRindler Hamiltonian.\nover the angular quantization space Hijis equal to the cylinder matrix element between\nboundary states |Bi/a\\}bracketri}ht/a\\}bracketri}htand|Bj/a\\}bracketri}ht/a\\}bracketri}htin radial quantization, as shown on the left in Figure 1.\nThen, applying radial quantization Hermitian conjugation inverts the cylinder, with the\ntwo boundary conditions being both interchanged and replac ed by their respective BPZ\nconjugates |Bi/a\\}bracketri}ht/a\\}bracketri}htand|Bj/a\\}bracketri}ht/a\\}bracketri}ht, as shown on the right in Figure 1. The conjugated cylinder\nmatrix element then equals the trace over the Hermitian-con jugated angular quantization\nspace, each of whose states obeys the boundary condition Biat the extreme left of the\nspatial slice and the boundary condition Bjat the extreme right. The boundary conditions\nBiandBjon the sphere shrink to the BPZ-conjugate operators Oi(0) andOj(∞), and\nhence the Hermitian conjugate of a state in the Hijsector of angular quantization is a bra\nin theH¯ı¯sector. In the examples studied here, the endpoint operator s are exponential\noperators Ok= :eikX: whose conjugates are Ok= :e−ikX: =O−k. Therefore, the inner\nproduct pairing on the Minkowski free boson CFT would be betw een a state in Hk1k2\nand a state in H−k1,−k2, or analogously between Hn2,w2n1,w1andH−n2,−w2\n−n1,−w1. Of course, there\nwould be a canonical isomorphism between Hk1k2andH−k1,−k2simply given by negating\nall appearances of the external momenta.\nThe second, and far more important, point we must make is that the putative state\nspacesHk1k2andHn2,w2n1,w1do not exist, at least in the familiar sense. When one speaks o f a\nvector space of states, its elements are pure states of the th eory, and local operators map\neach vector in the space to another vector. In the preceding S ection4, we were able to\nconstruct such vector spaces of pure states directly from th e shrinking limit of the finite-\nregulator Hilbert spaces. In the present case, on the other h and, it is simple to see that\nnone of the finitely-excited pure states satisfy the asympto tic conditions in the shrinking\nlimit. Indeed, from the expressions given in Section 2, the shrinking limit of the free boson\n– 24 –expectation values in the oscillator vacua |p;{0}/a\\}bracketri}htor|p,xs;{0}/a\\}bracketri}htare\nnoncompact: /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htosc.vac.=/a\\}bracketle{tx/a\\}bracketri}ht−i(k1−k2)\n2σ (5.1)\ncompact: /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htosc.vac.=/a\\}bracketle{tx/a\\}bracketri}ht−i(n1−n2)\n2Rσ+(w1−w2)R\n2τ,(5.2)\nfrom which it is obvious that the asymptotic conditions ( 3.3)-(3.6) or (3.7)-(3.10) are not\nobeyed. Then, since we already showed that any finite oscilla tor excitation does not alter\ntheσ→ ±∞behavior of expectation values, we conclude that none of the states|p;{Nℓ}/a\\}bracketri}ht\nor|p,xs;{Nℓ}/a\\}bracketri}htsatisfy the asymptotic conditions. Of course, this failure is closely related\nto the fact that the angular quantization bare thermal trace s on the cylinder vanish. The\nonly nonvanishing traces are those for which bulk operators are inserted which provide\nthe compensating momenta for momentum conservation to be sa tisfied. It is only the\ninsertion of the correct bulk operators which yields finite a nswers, so in the logic of angular\nquantization one would expect that the insertion of such ope rators should be the key in\nobtaining the correct asymptotic behavior as well.\nJust as with a boundary condition, the meaning of an asymptot ic condition lies in the\nbehavior of correlators of local operators as the position o f one of the operators approaches\nsome limit point, here spatial infinity. Thus, the quantity /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htwhich must obey the\nasymptotic conditions must really be defined as\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}ht ≡/a\\}bracketle{tX(τ,σ)O1(τ1,σ1)···On(τn,σn)/a\\}bracketri}ht\n/a\\}bracketle{tO1(τ1,σ1)···On(τn,σn)/a\\}bracketri}ht, (5.3)\nwhere/a\\}bracketle{tO1(τ1,σ1)···On(τn,σn)/a\\}bracketri}htisanynonvanishingcorrelatorofoperatorswhoseposition s\nare held finite. Of course, the quantity ( 5.3) depends on all these other operators being\ninserted, which we suppress in the notation. The point of an a symptotic condition is\nthat the quantity /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htin the limit σ→ ±∞is completely independent of the other\noperator insertions in ( 5.3) and moreover equals a specific function of τandσ. Now, as |σ|\ngrows large, the n-point OPE converges, which we use to replace O1(τ1,σ1)···On(τn,σn)\nwith a sumof local operators at some fixed coordinate ( τ0,σ0). By linearity, the asymptotic\nconditionholdsgenerally ifandonlyifitholdsforeachloc al operatorinthissumseparately.\nThus we need only consider\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}ht=/a\\}bracketle{tX(τ,σ)Ok(τ0,σ0)/a\\}bracketri}ht\n/a\\}bracketle{tOk(τ0,σ0)/a\\}bracketri}ht, (5.4)\nwhereOkis an exponential primary, with k=−(k1+k2) for the noncompact case and\nk= (n,w) =−(n1+n2,w1+w2) for the compact case, or one of its descendants.\nThis discussion of asymptotic conditions involves only cor relation functions of local\noperators. It is thus far more useful to use the more general d efinition of ‘state’ — a state\nis a linear map from the algebra of operators to the complex nu mbers. We take a state\nto be normalized so that it maps the identity operator to unit y; many sources also require\nthat the map be Hermitian, but we somewhat relax this conditi on here. This definition of\na stateρis the one preferred in algebraic quantum field theory, which generalizes the map\n– 25 –O →Tr[Oρ] to a von Neumann algebra of operators (for which a trace does not exist if it\nis of Type III). For a review of the role of von Neumann algebra s in quantum field theory,\nsee for instance [ 5] and references therein.\nWe may now explicitly construct these states by using the equ ality of correlators with\nthe shrinkinglimit of traces taken in angular quantization . We know we need only consider\nonelocal operator Ok(τ0,σ0); firstweconsidertheexponential primary, andinthefollo wing\nsubsection we take into account all descendants by acting wi th arbitrary combinations of\ncreation and annihilation operators. Note that the exponen tial operators Ocan all be split\ninto a zero-momentum mode piece O(0)times an oscillator creation exponential and an\noscillator annihilation exponential. Then, using cyclici ty of the trace, the nonvanishing\ncylinder traces in angular quantization take the form\nTrH/bracketleftbig\nOk(τ0,σ0)e−2πHR/bracketrightbig\nS1×R\n= TrH/bracketleftbigg\ne/summationtext∞\nℓ=1αℓ\nℓf−k\nℓ(−τ0,σ0)∗O(0)\nk(τ0,σ0)e−2πHRe/summationtext∞\nℓ′=1α−ℓ′\nℓ′fk\nℓ′(τ0,σ0)/bracketrightbigg\nS1×R,(5.5)\nwhere\nnoncompact (N): fk\nℓ(τ,σ) =√\n2keπℓτ\n2Lcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n(5.6)\nnoncompact (D): fk\nℓ(τ,σ) =i√\n2keπℓτ\n2Lsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n(5.7)\ncompact: fn,w\nℓ(τ,σ) =i√\n2eπℓτ\n2L/bracketleftbiggn\nRsin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n+iwRcos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg/bracketrightbigg\n.(5.8)\nMoreover, the asymptotic conditions dictate the behavior o f the trace ( 5.5) with an extra\nX(τ,σ) inserted, Tr H[Ok(τ0,σ0)X(τ,σ)e−2πHR]S1×R, asσ→ ±∞. Then, we define a state\nρ(τ0,σ0) such that\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}ht(τ0,σ0) = Tr[ρ(τ0,σ0)X(τ,σ)]. (5.9)\nFrom the explicit expressions for the operators involved, w e construct this state as\nρ(τ0,σ0) =ρ(0)⊗∞/circlemultiplydisplay\nℓ=1ρosc\nℓ(τ0,σ0), (5.10)\nwhere the zero-momentum-mode part of the state is\nnoncompact: ρ(0)\np=|p/a\\}bracketri}ht/a\\}bracketle{tp| (5.11)\ncompact: ρ(0)\nm,ms=|m,ms/a\\}bracketri}ht/a\\}bracketle{tm,ms|, (5.12)\nandρosc\nℓis the density matrix\nρosc\nℓ(τ0,σ0) =1\nNℓ∞/summationdisplay\nNℓ=0e−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)|Nℓ/a\\}bracketri}ht/a\\}bracketle{tNℓ|eαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓ.(5.13)\nThis definition is very roughly a Poisson distribution for th e system to be in the state\neα−ℓ\nℓfk\nℓ(τ0,σ0)|Nℓ/a\\}bracketri}ht, but this characterization should not be taken literally si nce these states\n– 26 –are not orthogonal. Here, the normalization constant Nℓis determined by the condition\nTr[ρosc\nℓ(τ0,σ0)] = 1, namely\nNℓ=∞/summationdisplay\nNℓ=01\nℓNℓNℓ!∞/summationdisplay\nm=0e−π2ℓ\nL(Nℓ+m)\nm!2/parenleftbigg1\nℓ2f−k\nℓ(−τ0,σ0)∗fk\nℓ(τ0,σ0)/parenrightbiggm\n/a\\}bracketle{t0|αNℓ+m\nℓαNℓ+m\n−ℓ|0/a\\}bracketri}ht(5.14)\n=1\n1−e−π2ℓ\nLexp\ne−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k\nℓ(−τ0,σ0)∗fk\nℓ(τ0,σ0)\n. (5.15)\nSeveral comments are in order before proceeding. The definit ion (5.13) may seem strange\nbecause its matrix eigenvalues are not all positive, wherea s probabilities should be positive;\nequivalently, the density operator is not generally Hermit ian. However, there is no problem\nhere because eα−ℓ\nℓfk\nℓ(τ0,σ0)|Nℓ/a\\}bracketri}htand (/a\\}bracketle{tNℓ|eαℓ\nℓf−k\nℓ(−τ0,σ0)∗)†actually belong to different finite-\nregulator vector spaces, namely Hk1k2orHn2,w2n1,w1for the former but H−k1,−k2orH−n2,−w2\n−n1,−w1\nfor the latter. Thus, the matrix elements need not satisfy a p ositivity condition except in\nthe special case that the external momenta are pure imaginar y, for which ρosc\nℓis manifestly\nHermitian, and it is simple to check that all the eigenvalues are indeed positive. Next, note\nthat the original trace from which we defined this state satis fies the appropriate asymptotic\nconditions on the Euclidean cylinder, but the definition ( 5.13) is supposed to be a valid\nstate in the full angular quantization and so should not care whether or not τis compact.\nThat is, ( 5.13) will also satisfy the asymptotic conditions on the whole Eu clidean strip\n(i.e. forany τ∈R) becausetheconditions ( 3.3)-(3.6) or (3.7)-(3.10) donot dependon time.\nNevertheless, thedefinition ( 5.13) always retains theconnection to theoriginal cylinder due\nto thefactors e−π2\n2Lα−ℓαℓwhicharosefromtheevolution operator e−βHRwithβ= 2π. Thus,\nall states in this superselection sector are thermal-like w ith inverse temperature exactly 2 π,\nand generically there are no pure states at all. This behavio r is very similar to Rindler\nquantum field theory, where all states are mixed with an inver se temperature 2 πas one\napproaches the Rindler horizon to reproduce the original Mi nkowski pure states.\nLastly, a state like ( 5.13) already takes into account the presence of exponential op-\nerators with the correct net charge, and so the nonvanishing expectation values in it are\nstill those for which the total charge vanishes. We already w rote the zero-mode part of\nthe state as depending on one continuous parameter for the no ncompact case and on two\nintegral parameters for the compact case because the slope- mode sector gets eliminated\nor discretized in exactly the same way as described in Sectio n4. We must also note that\nthere is some ambiguity in how one defines the zero-mode part o f the density matrix. The\nalgebra of operators is graded by the total charge, and we cho seρ(τ0,σ0) to act nontrivially\nonly on the charge zero sector. We can also define the zero-mod e part to be given by |p/a\\}bracketri}ht/a\\}bracketle{tp′|\nor|m,ms/a\\}bracketri}ht/a\\}bracketle{tm′,m′\ns|instead, for which the state acts nontrivially only on the ch argep−p′or\n(m′−m,m′\ns−ms) sector. We stick with the charge zero definition so that the e xpectation\nvalue/a\\}bracketle{tX/a\\}bracketri}ht= Tr[ρX] is nonvanishing.\nIt remains to verify the above claims by actually computing t he expectation values in\nthe state ( 5.13), where we keep the parameters τ0andσ0arbitrary for now.\n– 27 –5.1 Infinitely Excited Oscillator State in the Spectrum\nIn order to compute /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htin the full state ρ(τ0,σ0), we first calculate the expectation\nvalues/a\\}bracketle{tα±ℓ/a\\}bracketri}htℓof one oscillator in the state ρosc\nℓ(τ0,σ0) given in ( 5.13). These fixed mode\nnumber expectation values are\n/a\\}bracketle{tα±ℓ/a\\}bracketri}htℓ= Tr[ρosc\nℓ(τ0,σ0)α±ℓ]\n=1\nNℓ∞/summationdisplay\nNℓ=0/a\\}bracketle{tNℓ|eαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓα±ℓe−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)|Nℓ/a\\}bracketri}ht\n=e−π2ℓ\n2L\nNℓ/braceleftbiggfk\nℓ(τ0,σ0)\nf−k\nℓ(−τ0,σ0)∗/bracerightbigg∞/summationdisplay\nNℓ=0e−π2ℓ\nLNℓ\nNℓ!∞/summationdisplay\nm=0(Nℓ+m+1)!\nm!(m+1)!\ne−π2ℓ\nL\nℓf−k\nℓ(−τ0,σ0)∗fk\nℓ(τ0,σ0)\nm\n=e−π2ℓ\n2L\nNℓ/parenleftbig\n1−e−π2ℓ\nL/parenrightbig2/braceleftbiggfk\nℓ(τ0,σ0)\nf−k\nℓ(−τ0,σ0)∗/bracerightbigg\nexp\ne−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k\nℓ(−τ0,σ0)∗fk\nℓ(τ0,σ0)\n\n=e−π2ℓ\n2L\n1−e−π2ℓ\nL/braceleftbiggfk\nℓ(τ0,σ0)\nf−k\nℓ(−τ0,σ0)∗/bracerightbigg\n. (5.16)\nThe corresponding free boson oscillator expectation value s are then\n(N):/a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}ht=i√\n2∞/summationdisplay\nℓ=1e−π2ℓ\n2L[e−πℓτ\n2Lf−k1−k2\nℓ(τ0,σ0)−eπℓτ\n2Lfk1+k2\nℓ(−τ0,σ0)∗]\nℓ(1−e−π2ℓ\nL)cos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n(5.17)\n(D):/a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}ht=√\n2∞/summationdisplay\nℓ=1e−π2ℓ\n2L[e−πℓτ\n2Lf−k1−k2\nℓ(τ0,σ0)+eπℓτ\n2Lfk1+k2\nℓ(−τ0,σ0)∗]\nℓ(1−e−π2ℓ\nL)sin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n(5.18)\ncompact: /a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}ht=√\n2∞/summationdisplay\nℓ=1e−π2ℓ\n2L[e−πℓτ\n2Lf−n1−n2,−w1−w2\nℓ(τ0,σ0)+eπℓτ\n2Lfn1+n2,w1+w2\nℓ(−τ0,σ0)∗]\nℓ(1−e−π2ℓ\nL)sin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n.\n(5.19)\nWith the expressions given above for the functions fk\nℓ(τ0,σ0), these sums can be performed\nin closed form involving q-hypergeometric functions, but this result is not illumina ting.\nFortunately, we need only the leading terms which do not vani sh in the shrinking limit, so\nwe may perform an asymptotic expansion in1\nL. Doing so eliminates all factors of e−π2ℓ\nL,\nreducing the above expressions to\n(N):/a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}htL→∞−→ −2i(k1+k2)L\nπ2∞/summationdisplay\nℓ=1(e−πℓ(τ−τ0)\n2L+eπℓ(τ−τ0)\n2L)\nℓ2cos/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ0+L)\n2L/parenrightbigg\n(5.20)\n(D):/a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}htL→∞−→ −2i(k1+k2)L\nπ2∞/summationdisplay\nℓ=1(e−πℓ(τ−τ0)\n2L+eπℓ(τ−τ0)\n2L)\nℓ2sin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ0+L)\n2L/parenrightbigg\n(5.21)\ncompact: /a\\}bracketle{tXosc(τ,σ)/a\\}bracketri}htL→∞−→ −2iL\nπ2∞/summationdisplay\nℓ=1e−πℓ(τ−τ0)\n2L[n1+n2\nRsin(πℓ(σ0+L)\n2L)+i(w1+w2)Rcos(πℓ(σ0+L)\n2L)]\nℓ2sin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n−2iL\nπ2∞/summationdisplay\nℓ=1eπℓ(τ−τ0)\n2L[n1+n2\nRsin(πℓ(σ0+L)\n2L)−i(w1+w2)Rcos(πℓ(σ0+L)\n2L)]\nℓ2sin/parenleftbiggπℓ(σ+L)\n2L/parenrightbigg\n,(5.22)\n– 28 –which are now of dilogarithmic form. The sums that we need her e are\n∞/summationdisplay\nℓ=11\nℓ2cosh/parenleftbiggπℓτ12\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\n=π2\n8/bracketleftbigg1\n3−θ(|σ12|−|Imτ12|)|σ12|\nL+θ(|Imτ12|−|σ12|)sgn(Imτ12)iτ12\nL+σ2\n1+σ2\n2−τ2\n12\n2L2/bracketrightbigg\n(5.23)\n∞/summationdisplay\nℓ=11\nℓ2cosh/parenleftbiggπℓτ12\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\n=π2\n8/bracketleftbigg\n1−θ(|σ12|−|Imτ12|)|σ12|\nL+θ(|Imτ12|−|σ12|)sgn(Imτ12)iτ12\nL−σ1σ2\nL2/bracketrightbigg\n(5.24)\n∞/summationdisplay\nℓ=11\nℓ2sinh/parenleftbiggπℓτ12\n2L/parenrightbigg\ncos/parenleftbiggπℓ(σ1+L)\n2L/parenrightbigg\nsin/parenleftbiggπℓ(σ2+L)\n2L/parenrightbigg\n=π2i\n8/bracketleftbigg\n−θ(|Imτ12|−|σ12|)sgn(Imτ12)σ12\nL+θ(|σ12|−|Imτ12|)sgn(σ12)iτ12\nL+2iσ2τ12\nL2/bracketrightbigg\n.(5.25)\nThe final term in each expression above vanishes in the shrink ing limit. Adding these\nresults to the zero-momentum-mode contributions ( 5.1) and (5.2), we conclude that the\nfree boson expectation values in the shrinking limit of the i nfinitely excited oscillator state\nρ(τ0,σ0) are\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=/a\\}bracketle{tx/a\\}bracketri}ht−i(k1−k2)\n2σ−i(k1+k2)L\n6\n+i(k1+k2)\n2θ/parenleftBig\n|σ−σ0|−|Im(τ−τ0)|/parenrightBig\n|σ−σ0|\n+k1+k2\n2θ/parenleftBig\n|Im(τ−τ0)|−|σ−σ0|/parenrightBig\nsgn/parenleftbig\nIm(τ−τ0)/parenrightbig\n(τ−τ0) (5.26)\nfor the noncompact Neumann-class angular quantization,\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=/a\\}bracketle{tx/a\\}bracketri}ht−i(k1−k2)\n2σ−i(k1+k2)L\n2\n+i(k1+k2)\n2θ/parenleftBig\n|σ−σ0|−|Im(τ−τ0)|/parenrightBig\n|σ−σ0|\n+k1+k2\n2θ/parenleftBig\n|Im(τ−τ0)|−|σ−σ0|/parenrightBig\nsgn/parenleftbig\nIm(τ−τ0)/parenrightbig\n(τ−τ0) (5.27)\nfor the noncompact Dirichlet-class angular quantization a nd finally\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=/a\\}bracketle{tx/a\\}bracketri}ht−i(n1−n2)\n2Rσ+(w1−w2)R\n2τ−i(n1+n2)L\n2R\n+θ/parenleftBig\n|σ−σ0|−|Im(τ−τ0)|/parenrightBig/bracketleftbiggi(n1+n2)\n2R|σ−σ0|−(w1+w2)R\n2sgn(σ−σ0)(τ−τ0)/bracketrightbigg\n+θ/parenleftBig\n|Im(τ−τ0)|−|σ−σ0|/parenrightBig\nsgn/parenleftbig\nIm(τ−τ0)/parenrightbig/bracketleftbiggn1+n2\n2R(τ−τ0)−i(w1+w2)R\n2(σ−σ0)/bracketrightbigg\n(5.28)\nfor the compact boson angular quantization. At last we may ea sily verify that these\nexpectation values do actually obey the requisite asymptot ic conditions. Restricting to\n– 29 –Euclidean time ( τ,τ0∈R), the derivative expectation values in the shrinking limit are\nnoncompact (N): /a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=−i(k1−k2)\n2+i(k1+k2)\n2sgn(σ−σ0) (5.29)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)= 0 (5.30)\nnoncompact (D): /a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=−i(k1−k2)\n2+i(k1+k2)\n2sgn(σ−σ0) (5.31)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)= 0 (5.32)\ncompact: /a\\}bracketle{t∂σX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=−i(n1−n2)\n2R+i(n1+n2)\n2Rsgn(σ−σ0) (5.33)\n/a\\}bracketle{t∂τX(τ,σ)/a\\}bracketri}htρ(τ0,σ0)=(w1−w2)R\n2−(w1+w2)R\n2sgn(σ−σ0),(5.34)\nwhere we do not write the delta function terms for brevity sin ce they play no role here.\nTherefore, we immediately see that the noncompact boson exp ectation values ( 5.26) and\n(5.27) satisfy theasymptotic conditions ( 3.3)-(3.6) and that thecompact boson expectation\nvalue (5.28) satisfies the asymptotic conditions ( 3.7)-(3.10) for arbitrary τ0andσ0, showing\nthat the infinitely excited oscillator state ρ(τ0,σ0) is indeed in the final spectrum of angular\nquantization. Moreover, the full expectation values ( 5.26) and (5.27) are equal (up to an\nadditive constant), which almost completes the proof that t he Neumann- and Dirichlet-\nclass boundary conditions are equivalent in angular quanti zation.\n5.2 The Other Oscillator States\nHaving constructed one oscillator state in the Minkowski sp ectrum, we must now deduce\nhow to obtain all remaining oscillator states. The most natu ral guess is that a Fock-like\nstructure still prevails but now built on top of a different sta te than the vacuum; the most\nimportant difference is that the base state is already infinite ly excited compared to the\noriginal vacuum, so one should act on it with all combination s of creation andannihilation\noperators. One would then expect that no finite combinations of such oscillators could\nalter the asymptotic conditions, which is what we shall esta blish in this subsection.\nNote that the creation and annihilation operators map state s to states via conjugation\nof the density operator. Starting from the ρ(τ0,σ0) defined by ρosc\nℓ(τ0,σ0) in (5.13), we\ndefine a new oscillator state ρ{Nℓ}(τ0,σ0) by using\nρosc\nℓ,Nℓ(τ0,σ0)≡1\nc(Nℓ)αNℓ\n−ℓρosc\nℓ(τ0,σ0)αNℓ\nℓ(5.35)\nin place of ρosc\nℓ(τ0,σ0). As usual only finitely many of the ‘occupation’ numbers Nℓcan\nbe nonzero, and c(Nℓ) are normalization constants. In the definition of ρosc\nℓ,Nℓ, we now let\nNℓ∈Z, whereα−ℓto a power Nℓ<0 is defined to be αℓto the power −Nℓ, i.e.\nαNℓ\n−ℓ≡α|Nℓ|\n−sgn(Nℓ)ℓ. (5.36)\nIt is clear that, with this definition, all such states ρ{Nℓ}are independent but they are\nnot orthogonal. The normalization constants c(Nℓ) ensuring Tr[ ρosc\nℓ,Nℓ] = 1 must still be\n– 30 –computed; in our conventions, c(0) = 1 because the state ρosc\nℓis already normalized. From\nTr[ρosc\nℓ] =1\nc(Nℓ)Nℓ∞/summationdisplay\nN′\nℓ=0/a\\}bracketle{tN′\nℓ|eαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓαNℓ\nℓαNℓ\n−ℓe−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)|N′\nℓ/a\\}bracketri}ht,\n(5.37)\nwe compute\nNℓ/greaterorequalslant0:c(Nℓ) =1\nNℓ∞/summationdisplay\nN′\nℓ=0ℓNℓ(Nℓ+N′\nℓ)!e−π2ℓ\nLN′\nℓ\nN′\nℓ!ee−π2ℓ\nL\nℓf−k∗\nℓfk\nℓLNℓ+N′\nℓ\n−e−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ\n\n=ℓNℓNℓ!\n(1−e−π2ℓ\nL)NℓLNℓ\n−e−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ\n (5.38)\nNℓ<0:c(Nℓ) =ℓ|Nℓ|\nNℓee−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ∞/summationdisplay\nN′\nℓ=0e−π2ℓ\nLN′\nℓ\nN′\nℓ!|Nℓ|/summationdisplay\nn=0(−1)n+Nℓ|Nℓ|!2\nn!2(|Nℓ|−n)!(N′\nℓ+n)!LN′\nℓ+n\n−e−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ\n\n=ℓ|Nℓ||Nℓ|!\ne−π2ℓ\nL\n1−e−π2ℓ\nL\n|Nℓ|\nL|Nℓ|/parenleftBigg\n−1\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightBigg\n, (5.39)\nwhere it is understood that fk\nℓalways means fk\nℓ(τ0,σ0) and that f−k∗\nℓalways means\nf−k\nℓ(−τ0,σ0)∗. Hence, these normalization constants are given by the sing le expression\nc(Nℓ) =ℓ|Nℓ||Nℓ|!\n(1−e−π2ℓ\nL)|Nℓ|e−π2ℓ\nL|Nℓ|θ(−Nℓ)L|Nℓ|\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ\n.(5.40)\nNote thatc(Nℓ) andc(−Nℓ) become equal in the shrinking limit L→ ∞.\nNow let us show that ρ{Nℓ}(τ0,σ0) also belongs to the Minkowksi CFT state space by\nverifying the asymptotic conditions of X(τ,σ) in it. To do so, we must first compute\n/a\\}bracketle{tα±ℓ/a\\}bracketri}htρ{Nℓ}=1\nc(Nℓ)Tr/bracketleftBigg\nα±ℓ/parenleftBigg∞/productdisplay\nℓ′=1αNℓ′\n−ℓ′/parenrightBigg∞/circlemultiplydisplay\nℓ′′=1ρosc\nℓ′′(τ0,σ0)/parenleftBigg∞/productdisplay\nℓ′′′=1αNℓ′′′\nℓ′′′/parenrightBigg/bracketrightBigg\n=1\nc(Nℓ)Nℓ∞/summationdisplay\nN′\nℓ=0/a\\}bracketle{tN′\nℓ|eαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓαNℓ\nℓα±ℓαNℓ\n−ℓe−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)|N′\nℓ/a\\}bracketri}ht,\n(5.41)\nwhere all the other states |{N′\nℓ′}/a\\}bracketri}htwithℓ′/\\e}atio\\slash=ℓdo not contribute since they have already\nbeen normalized. For Nℓ/greaterorequalslant0, we have\n– 31 –1\nNℓ∞/summationdisplay\nN′\nℓ=01\nℓN′\nℓN′\nℓ!/a\\}bracketle{t0|αN′\nℓ\nℓeαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓαNℓ\nℓα±ℓαNℓ\n−ℓe−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)αN′\nℓ\n−ℓ|0/a\\}bracketri}ht\n=ℓNℓe−π2ℓ\n2L\nNℓ/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbigg∞/summationdisplay\nN′\nℓ=0e−π2ℓ\nLN′\nℓ\nN′\nℓ!∞/summationdisplay\nm=0(Nℓ+N′\nℓ+m+1)!\nm!(m+1)!\ne−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ\nm\n=/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbiggℓNℓNℓ!e−π2ℓ\n2L\n(1−e−π2ℓ\nL)Nℓ+1L(1)\nNℓ\n−e−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ\n.(5.42)\nLikewise, for Nℓ<0, we have\n1\nNℓ∞/summationdisplay\nN′\nℓ=01\nℓN′\nℓN′\nℓ!/a\\}bracketle{t0|αN′\nℓ\nℓeαℓ\nℓf−k\nℓ(−τ0,σ0)∗e−π2\n2Lα−ℓαℓα|Nℓ|\n−ℓα±ℓα|Nℓ|\nℓe−π2\n2Lα−ℓαℓeα−ℓ\nℓfk\nℓ(τ0,σ0)αN′\nℓ\n−ℓ|0/a\\}bracketri}ht\n=(−1)|Nℓ|ℓ|Nℓ|e−π2ℓ\n2L\nNℓ/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbigg∞/summationdisplay\nN′\nℓ=0e−π2ℓ\nLN′\nℓ\nN′\nℓ!|Nℓ|/summationdisplay\nn=0(−1)n|Nℓ|!2\nn!2(|Nℓ|−n)!\n×\n∞/summationdisplay\nm=0(N′\nℓ+m+n+1)!\nm!(m+1)!\ne−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ\nm\n−n∞/summationdisplay\nm=0(N′\nℓ+m+n)!\nm!(m+1)!\ne−π2ℓ\nL\nℓf−k∗\nℓfk\nℓ\nm\n\n=ℓ|Nℓ||Nℓ|!e−π2ℓ\n2L\n1−e−π2ℓ\nL/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbigg\ne−π2ℓ\nL\n1−e−π2ℓ\nL\n|Nℓ|\nL(1)\n|Nℓ|/parenleftBigg\n−1\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightBigg\n.(5.43)\nDividing by the normalization constant ( 5.40), we have thus computed\n/a\\}bracketle{tα±ℓ/a\\}bracketri}htρ{Nℓ}=e−π2ℓ\n2L\n1−e−π2ℓ\nL/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbiggL(1)\n|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg\nL|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg. (5.44)\nNoting that the large- Llimit of the Laguerre polynomial ratio above is\nL(1)\n|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg\nL|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbiggL→∞−→1, (5.45)\nwe conclude that the single-oscillator expectation value i n the shrinking limit is\n/a\\}bracketle{tα±ℓ/a\\}bracketri}htρ{Nℓ}(τ0,σ0)L→∞−→e−π2ℓ\n2L\n1−e−π2ℓ\nL/braceleftbiggfk\nℓ\nf−k∗\nℓ/bracerightbigg\n=/a\\}bracketle{tα±ℓ/a\\}bracketri}htρ(τ0,σ0). (5.46)\nTherefore, allsuchfiniteoscillator excitations alsosati sfythecorrectasymptoticconditions.\nThe basis of nontrivial oscillator excitations in the shrin king limit are then formed in\nthe same way as before, namely by scaling ℓ=qL\nπwithqfinite. The resulting state is\n– 32 –writtenρ{ˆq1,...,ˆqi,qi+1,...,qi+j}(τ0,σ0) and consists of jparticles and iholes on top of the basic\nstateρ(τ0,σ0), where a hole of the corresponding momentum is indicated by a circumflex.\nNote that acting with all possible combinations of creation and annihilation operators\nautomatically constructs states with any possible values o fτ0andσ0as well as states\ncorresponding to arbitrary descendants of Ok(τ0,σ0), thus generating all correlators with\narbitrary insertions of local operators via the previous OP E argument.\n5.3 Exponential Operator Normalization\nFor future calculations, it is useful to compute the expecta tion values /a\\}bracketle{tαm\n±ℓ/a\\}bracketri}htρ{Nℓ}and\n/a\\}bracketle{tαm\n−ℓαm′\nℓ/a\\}bracketri}htρ{Nℓ}for any non-negative integers mandm′. For the former, we find\n/a\\}bracketle{tαm\n±ℓ/a\\}bracketri}htρ{Nℓ}=\ne−π2ℓ\n2L\n1−e−π2ℓ\nL\nm/braceleftbigg(fk\nℓ)m\n(f−k∗\nℓ)m/bracerightbiggL(m)\n|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg\nL|Nℓ|/parenleftbigg\n−e−π2ℓ\nLθ(Nℓ)\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg.(5.47)\nNote that, in the shrinking limit, this expectation value fa ctorizes as\n/a\\}bracketle{tαm\n±ℓ/a\\}bracketri}htρ{Nℓ}L→∞−→/parenleftBig\n/a\\}bracketle{tα±ℓ/a\\}bracketri}htρ{Nℓ}/parenrightBigm\n. (5.48)\nFor the latter, define m+≡max(m,m′) andm−≡min(m,m′). Then, the expectation\nvalue forNℓ/greaterorequalslant0 is\n/a\\}bracketle{tαm\n−ℓαm′\nℓ/a\\}bracketri}htρ{Nℓ}=(−1)m−ℓm−m+!m−!\nNℓ!(1−e−π2ℓ\nL)m+−m−/braceleftbigg(e−π2ℓ\n2Lf−k∗\nℓ)m+−m−\n(e−π2ℓ\n2Lfk\nℓ)m+−m−/bracerightbigg\n×m−/summationdisplay\nj=0(−1)j(Nℓ+j)!\nj!(j+m+−m−)!(m−−j)!1\n(1−e−π2ℓ\nL)jL(m+−m−)\nNℓ+j/parenleftbigg\n−e−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg\nLNℓ/parenleftbigg\n−e−π2ℓ\nL\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg,(5.49)\nwhere the top line applies for m/greaterorequalslantm′and the bottom line applies for m < m′, and for\nNℓ<0 this expectation value is\n/a\\}bracketle{tαm\n−ℓαm′\nℓ/a\\}bracketri}htρ{Nℓ}=ℓm−(|Nℓ|+m−)!\n|Nℓ|!(1−e−π2ℓ\nL)m+−m−/braceleftbigg(e−π2ℓ\n2Lf−k∗\nℓ)m+−m−\n(e−π2ℓ\n2Lfk\nℓ)m+−m−/bracerightbigg\n×\ne−π2ℓ\nL\n1−e−π2ℓ\nL\nm−L(m+−m−)\n|Nℓ|+m−/parenleftbigg\n−1\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg\nL|Nℓ|/parenleftbigg\n−1\nℓ(1−e−π2ℓ\nL)f−k∗\nℓfk\nℓ/parenrightbigg.(5.50)\nWe now obtain the normalization of the exponential operator in thegenerating state ( 5.13).\nSettingNℓ= 0 above, the expectation value of αm\n−ℓαm′\nℓis\n/a\\}bracketle{tαm\n−ℓαm′\nℓ/a\\}bracketri}htρ=ℓm′m′!(f−k∗\nℓ)m−m′e−π2ℓ\n2L(m+m′)\n(1−e−π2ℓ\nL)mL(m−m′)\nm′/parenleftBigg\n−f−k∗\nℓfk\nℓ\nℓ(1−e−π2ℓ\nL)/parenrightBigg\n,(5.51)\n– 33 –which works for both m/greaterorequalslantm′andm < m′. We are evaluating the expectation value\nofOk′′(τ2,σ2)Ok′(τ1,σ1) in the state ρ(τ0,σ0). The zero-mode part of this expectation\nvalue provides the familiar δ(k′+k′′) for the noncompact case and δn′+n′′,0δw′+w′′,0for the\ncompact case. Letting fk′\nj,ℓbe short for fk′\nℓ(τj,σj), for instance, the pure oscillator part of\nthe expectation value is then the product over ℓof\n/angbracketleftbigg\neα−ℓ\nℓf−k′\n2,ℓeαℓ\nℓfk′∗\n2,ℓeα−ℓ\nℓfk′\n1,ℓeαℓ\nℓf−k′∗\n1,ℓ/angbracketrightbigg\nρosc\nℓ=e1\nℓfk′∗\n2,ℓfk′\n1,ℓ/angbracketleftbigg\neα−ℓ\nℓ(fk′\n1,ℓ+f−k′\n2,ℓ)eαℓ\nℓ(f−k′∗\n1,ℓ+fk′∗\n2,ℓ)/angbracketrightbigg\nρosc\nℓ\n=e1\nℓfk′∗\n2,ℓfk′\n1,ℓexp\ne−π2ℓ\n2L[f−k∗\n0,ℓ(fk′\n1,ℓ+f−k′\n2,ℓ)+(f−k′∗\n1,ℓ+fk′∗\n2,ℓ)fk\n0,ℓ]+e−π2ℓ\nL(f−k′∗\n1,ℓ+fk′∗\n2,ℓ)(fk′\n1,ℓ+f−k′\n2,ℓ)\nℓ(1−e−π2ℓ\nL)\n,\n(5.52)\nwhere the functions fk′\nℓ(τj,σj) are given explicitly in ( 5.6), (5.7) and (5.8). For the first\nexponential e1\nℓfk′∗\n2,ℓfk′\n1,ℓ, the infinite product over ℓis easily performed in the three cases ‘N’\n(noncompact Neumann-class), ‘D’ (noncompact Dirichlet-c lass) and ‘c’ (compact) via the\nexact summations\n∞/summationdisplay\nℓ=11\nℓfk′∗\n2,ℓfk′\n1,ℓN=k′2\n2ln/bracketleftBigg\n(L\nπ)2eiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightBigg\n(5.53)\n∞/summationdisplay\nℓ=11\nℓfk′∗\n2,ℓfk′\n1,ℓD=k′2\n2ln/bracketleftBigg\n(4L\nπ)2eiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightBigg\n(5.54)\n∞/summationdisplay\nℓ=11\nℓfk′∗\n2,ℓfk′\n1,ℓc=1\n2/parenleftbiggn′2\nR2+w′2R2/parenrightbigg\nln/bracketleftBigg\n(L\nπ)2eiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightBigg\n+n′2\nR2ln4\n−n′w′/bracketleftbigg\nln/parenleftbiggσ2\n12+|τ12|2−2σ12Imτ12\nσ2\n12+|τ12|2+2σ12Imτ12/parenrightbigg\n+iφ12/bracketrightbigg\n, (5.55)\nwhere we have defined the angles\nϕ12≡tan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n−tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+πθ(|Imτ12|−|σ12|)sgn(Imτ12) (5.56)\nφ12≡tan−1/parenleftbiggReτ12\nσ12+Imτ12/parenrightbigg\n+tan−1/parenleftbiggReτ12\nσ12−Imτ12/parenrightbigg\n+πθ(|σ12|−|Imτ12|)sgn(σ12).(5.57)\nThe infinite product over the lengthier exponential seems mu ch more daunting. Fortu-\nnately, as before, we care only about the shrinking limit and so we may perform an asymp-\ntotic expansion in1\nL. The leading contributions in this expansion are then due to replacing\ne−π2ℓ\naL\n1−e−π2ℓ\nLin the exponential withL\nπ2ℓ. The resulting sums are then easy to perform. It is\nreadily verified that in all three cases the shrinking limits of these sums are\nL∞/summationdisplay\nℓ=1(f−k′∗\n1,ℓ+fk′∗\n2,ℓ)(fk′\n1,ℓ+f−k′\n2,ℓ)\nℓ2L→∞−→0 (5.58)\n– 34 –and\nL∞/summationdisplay\nℓ=1f−k∗\n0,ℓ(fk′\n1,ℓ+f−k′\n2,ℓ)+(f−k′∗\n1,ℓ+fk′∗\n2,ℓ)fk\n0,ℓ\nℓ2L→∞−→0. (5.59)\nThe vanishing of the leading terms simply arises because eac h such contribution is indepen-\ndent of all positions, and the terms in fk′\n1,ℓ+f−k′\n2,ℓorf−k′∗\n1,ℓ+fk′∗\n2,ℓhave opposite signs. The\nvanishing of the subleading terms is then because the nontri vial position dependence in all\nthe sums is quadratic as a result of the linear pieces canceli ng. Therefore, including the\nzero-mode contributions and the numerical prefactors, the exponential operator two-point\nfunction for the Neumann-class noncompact boson is\n/a\\}bracketle{tOk′′(τ2,σ2)Ok′(τ1,σ1)/a\\}bracketri}htρ(τ0,σ0)\nL→∞−→2πδ(k′+k′′)e(k1−k2)k′\n2σ12/bracketleftbiggeiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightbiggk′2\n2\n,(5.60)\nfor the Dirichlet-class noncompact boson is\n/a\\}bracketle{tOk′′(τ2,σ2)Ok′(τ1,σ1)/a\\}bracketri}htρ(τ0,σ0)\nL→∞−→2πδ(k′+k′′)e(k1−k2)k′\n2σ12/bracketleftbiggeiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightbiggk′2\n2\n(5.61)\nand finally for the compact boson is\n/angbracketleftbig\nOn′′,w′′(τ2,σ2)On′,w′(τ1,σ1)/angbracketrightbig\nρ(τ0,σ0)\nL→∞−→2πδn′+n′′,0δw′+w′′,0e[(n1−n2)n′\n2R2+(w1−w2)w′R2\n2]σ12ei[(n1−n2)w′+n′(w1−w2)]\n2τ12\n×/bracketleftbiggeiϕ12\n(σ2\n12+|τ12|2)2−(2σ12Imτ12)2/bracketrightbigg1\n2(n′2\nR2+w′2R2)/parenleftbigg\n−σ2\n12+|τ12|2+2σ12Imτ12\nσ2\n12+|τ12|2−2σ12Imτ12e−iφ12/parenrightbiggn′w′\n.\n(5.62)\nThese results are independent of ( τ0,σ0) and are manifestly finite, proving that the ex-\nponentials are indeed (Dirac-)normalizable operators in t he infinitely excited Minkowski\nCFT states generated by ρ(τ0,σ0). The elimination or quantization of the slope-mode\nsector then follows from the same reasoning as in Section 4.\n5.4 Equivalence of Neumann and Dirichlet\nWe are finally in the position to provide the remaining detail proving that the Neumann-\nclass and Dirichlet-class regulators are equivalent in ang ular quantization. As in Section 4,\nwesimplyneedtoprovidetheappropriateisometrybetweent hestatesandoperators, where\nthe only difference lies in the zero-momentum-mode sector. To wit, due to the underlying\nshift symmetry of the theory, we must be careful to ensure tha t the constant-mode xis\ndefined in the same way in the two approaches.\n– 35 –Sincexappears explicitly in the Hamiltonian as the term −i(k1+k2)\n2πx, constant shifts\ninxchange the energy. For this subsection, denote the constant -modes in the Neumann-\nclass and Dirichlet-class approaches as xNandxD, respectively. We shall deduce the map\nbetweenxNandxDby comparing the expectation values of the Hamiltonian in bo th cases.\nUsing the previous result for /a\\}bracketle{tαm\n−ℓαm′\nℓ/a\\}bracketri}htρwithm=m′= 1, the expectation value of the\noscillator part of the Hamiltonian is\nπ\n2L∞/summationdisplay\nℓ=1/a\\}bracketle{tα−ℓαℓ/a\\}bracketri}htρ=π\n2L∞/summationdisplay\nℓ=1ℓe−π2ℓ\nL\n1−e−π2ℓ\nL/bracketleftBigg\n1+f−k∗\nℓfk\nℓ\nℓ(1−e−π2ℓ\nL)/bracketrightBigg\n(5.63)\nL→∞−→e−π2\nL\n2π(1−e−π2\nL)+L\n2π3∞/summationdisplay\nℓ=11\nℓ2f−k∗\nℓfk\nℓ. (5.64)\nThe remaining sums for the noncompact Neumann-class and Dir ichlet-class cases are\n∞/summationdisplay\nℓ=11\nℓ2f−k∗\nℓfk\nℓN=−π2k2\n4/parenleftbigg1\n3+σ2\n0\nL2/parenrightbigg\n(5.65)\n∞/summationdisplay\nℓ=11\nℓ2f−k∗\nℓfk\nℓD=−π2k2\n4/parenleftbigg\n1−σ2\n0\nL2/parenrightbigg\n, (5.66)\nwhere we recall that k=−(k1+k2). Thus, the expectation values of the full Hamiltonians\nare\n/a\\}bracketle{tHN\nR(L)/a\\}bracketri}htρL→∞−→ −i(k1+k2)\n2π/a\\}bracketle{txN/a\\}bracketri}ht+(k1−k2)2\n8πL+(k1+k2)2\n24πL+L\n2π3−1\n4π(5.67)\n/a\\}bracketle{tHD\nR(L)/a\\}bracketri}htρL→∞−→ −i(k1+k2)\n2π/a\\}bracketle{txD/a\\}bracketri}ht+(k1−k2)2\n8πL−(k1+k2)2\n8πL+L\n2π3−1\n4π.(5.68)\nFrom these expressions, we see that the appropriate map betw een the zero-momentum-\nmodes is\nxN/ma√sto−→xD−i(k1+k2)\n3L. (5.69)\nUnder this map, we immediately see from ( 5.26) and (5.27) that\n/a\\}bracketle{tX(τ,σ)/a\\}bracketri}htN\nρ(τ0,σ0)/ma√sto−→ /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htD\nρ(τ0,σ0),\nshowing that the free boson expectation values now agree exa ctly including the constant\nterm. Under the isometry defined by ( 5.69) andaN(q)/ma√sto→sgn(q)aD(q), we have shown\nthat the actions of all states on the operator algebra manife stly agree in the two cases.\nTherefore, the Neumann-class and Dirichlet-class regulat ors are exactly equivalent in the\nshrinking limit of angular quantization and hence define ide ntical Minkowski CFTs.\n6 Prelude to String Theory\nIn Section 5, we computed the expectation value /a\\}bracketle{tX(τ,σ)/a\\}bracketri}htof the free boson in the base\nstateρin angular quantization for the noncompact case and for the c ompact case in ( 5.26)\n– 36 –and (5.28), respectively, from which it trivially followed that the E uclidean asymptotic\nconditions are satisfied. Those computations were performe d for arbitrary complex τ, and\nhence they may be applied to the physically relevant Lorentz ian theory for which τ=it\nwitht∈R. In the context of string theory, /a\\}bracketle{tXµ(t,σ)/a\\}bracketri}htrepresents the embedding of a string\nworldsheet in target spacetime. These worldsheets belong t o a new class of fundamental\nstringwhichisdefinedfromangularquantization. Thesestr ingsarecharacterized byhaving\nasymptotic boundaries and are hence neither open nor closed , which have finite boundaries\nand no boundaries, respectively. We dub this new class “stre tched strings”, and in this\nsection we provide snapshot descriptions of the different typ es of stretched strings. We\nfocus on the basic explicit states ρ(τ0,σ0) which will describe classical string worldsheets of\nsingle stretched strings in isolation. The full quantum-me chanical treatment of stretched\nstrings will later be presented in [ 6].\nFrom (5.26) and (5.28), the Lorentzian noncompact free boson expectation value i n\nthe stateρ(t0,σ0) with asymptotic charges k1andk2is\n/a\\}bracketle{tX(t,σ)/a\\}bracketri}htρ(t0,σ0)=/a\\}bracketle{tx/a\\}bracketri}ht−i(k1−k2)\n2σ+i(k1+k2)\n2θ/parenleftBig\n|σ−σ0|−|t−t0|/parenrightBig\n|σ−σ0|\n+i(k1+k2)\n2θ/parenleftBig\n|t−t0|−|σ−σ0|/parenrightBig\n|t−t0|,(6.1)\nand the Lorentzian compact free boson expectation value in t he stateρ(t0,σ0) is\n/a\\}bracketle{tX(t,σ)/a\\}bracketri}htρ(t0,σ0)=/a\\}bracketle{tx/a\\}bracketri}ht−i(n1−n2)\n2Rσ+i(w1−w2)R\n2t\n+θ/parenleftBig\n|σ−σ0|−|t−t0|/parenrightBig/bracketleftbiggi(n1+n2)\n2R|σ−σ0|−i(w1+w2)R\n2sgn(σ−σ0)(t−t0)/bracketrightbigg\n+θ/parenleftBig\n|t−t0|−|σ−σ0|/parenrightBig/bracketleftbiggi(n1+n2)\n2R|t−t0|−i(w1+w2)R\n2sgn(t−t0)(σ−σ0)/bracketrightbigg\n.(6.2)\nIn angular quantization, the free boson is defined on a comple xified contour essentially\ndefined by its mode expansion. Here, we interpret Xas a target space coordinate and plot\nconstant-time snapshots in the coordinate iX. In the actual string theory applications of\ninterest, the field Xis either a time-like boson or a linear dilaton (more general ly, Liouville\nfield), for which the corresponding string profiles will inde ed be real-valued. Furthermore,\nwe shall take iXas noncompact, as even the compact boson is generally noncom pact when\ncontinued to the imaginary direction. As such, we need only p lot (6.2), as it subsumes\n(6.1), and we set /a\\}bracketle{tx/a\\}bracketri}ht= 0 since we already understand the shift symmetry. We shall p lot\nthe snapshots as a function of the worldsheet time; it is unde rstood that in any slicing\nof an actual string worldsheet, a gauge choice is made to fix th e relation between tand\nX0. Finally, we consider the pure momentum and pure winding cas es separately in what\nfollows. The most general stretched string worldsheet exhi bits a combination of the two\neffects.\n6.1 Pure Momentum Charges\nFirst consider w1=w2= 0 so that only asymptotic momentum charges are present. The re\nare two distinct cases. The generic case is n1/\\e}atio\\slash=n2, for which the linear term −i(n1−n2)\n2Rσin\n– 37 –t\nt0\niX\nFigure 2 . The pure-momentum stretched string in the generic case n1/\\e}atio\\slash=n2. All snapshots are\nidentical, consisting of a space-filling string.\nt\nt0\niXσ=σ0\n|n|\nR∆t\nFigure 3 . The pure-momentum stretched string in the special case n1=n2. Each snapshot is a\nhalf-space-filling string doubled back on itself. Over time, the folding point (σ=σ0) evenly travels\ninto space, reaches a maximum (at t=t0) and then recedes back to infinity.\n(6.2) automatically creates a stretched string from one end of sp ace to the other, as shown\nin Figure 2. Of course, which value of the worldsheet coordinate σgives rise to which value\nofiXdoes change over time (as long as n1/\\e}atio\\slash=−n2), but the point is that all values of iX\nare always reached. Since a fundamental string has no width, the coordinate σitself is not\nphysical.\nThe far more interesting scenario is the special case n1=n2for which the linear term\n−i(n1−n2)\n2Rσis absent. Snapshots of this stretched string are plotted in Figure3. Unlike\nthe generic case, for n1=n2both asymptotic ends of the stretched string emerge from\nthe same spatial infinity. While the string itself is infinite ly long, there is a distinguished\n– 38 –t\nt0\niXσ=σ0slope(w2−w1)R\n2\n|w1+w2|R\n2∆t\nFigure 4 . The pure-winding stretched string. Such a string has finite length at all times, which\ncontracts to a point (at t=t0) before expanding again all at a constant rate.\npoint where the string folds and doubles back on itself. This folding point travels into the\nbulk of space at a constant speed, reaches a maximum and then r ecedes again at the same\nspeed, which is controlled by the asymptotic momentum|n|\nR. This heuristic picture agrees\nwith that of the “long strings” in the c= 1 string theory discovered by Maldacena in [ 3].\nIn that work, however, the “long string” was obtained less di rectly as a scaling limit of an\ninfinitely-energetic open string on a receding FZZT-brane. For more appearances of long\nand folded strings, see for instance [ 7] and [8]. We propose that the proper string-theoretic\nconstruction of such long string states is via the angular qu antization procedure which\ndefines stretched strings more generally. We shall have more to say about this connection\nwhen we present the full BRST quantization in [ 6].\n6.2 Pure Winding Charges\nNow we consider n1=n2= 0 so that only asymptotic winding charges are present. From\n(6.2), we note that the point σ=σ0simply moves at a constant speed set by w1−w2. At\nt=t0, the entire string is at the same location as σ=σ0. As time elapses, all points with\n|σ−σ0|>|t−t0|are at the same distance from σ=σ0, set byw1+w2. The remaining\npoints with |σ−σ0|<|t−t0|simply fill in the interval between σ=σ0and this maximum\ndistance. The resulting stretched string is shown in Figure 4. Since in applications the\n– 39 –compact boson will be that of Euclidean time, the result of Fi gure4is to be interpreted\nthat there exists a one-to-one map between tandX0(for instance given by the line σ=σ0)\nso that there indeed exists a well-defined global gauge relat ing the worldsheet and target\ntime coordinates. To wit, in the usual case of interest the an gular quantization endpoint\noperators will have opposite Euclidean time winding, in whi ch Figure 4already collapses\nto a single line.\n7 Summary\nIn this work, we detailed the states of the 2d free boson on the infinite line with speci-\nfied asymptotic charges at spatial infinity, corresponding t o a superselection sector in the\nfull Minkowski CFT. These states are constructed by taking t he shrinking limit of an-\ngular quantization with exponential endpoint operators, w hich in particular retains only\nthose states obeying the appropriate asymptotic condition s. In the special case that the\nnet asymptotic charge vanishes, we found a simple vector spa ce structure of pure states\ninherited from the finite-regulator Fock space constructio n. In the general case that the\nnet asymptotic charge does not vanish, we do not find a basis of pure states and instead\nmust work with mixed states originating from the Euclidean c ylinder having the correct\nperiodicity. While working with the purely algebraic defini tion of states may be more cum-\nbersome, we found a similar structure as the previous case — a one-parameter continuous\nzero-mode (noncompact theory) or a two-parameter discrete zero-mode (compact theory)\ncombined with an oscillator sector which is generated from a basic state by acting with all\ncombinations of creation and annihilation operators. One n ovelty of this construction is\nthat the resulting “Fock space” is bi-directional. We also s howed that the Neumann-class\nand Dirichlet-class regulators for angular quantization, which are seemingly quite different\nin the finite-regulator theory, are in fact entirely equival ent after the shrinking limit is\ntaken.\nAs a basic motivating example, we also briefly discussed the r amifications of defining\nCFTs in Minkowski space via angular quantization on string t heory. This leads to a\nnew type of fundamental string which we called “stretched st rings”, which will be the\npurview of a future paper. We provided snapshots of the stret ched string worldsheets\nthat are obtained from a free boson sector of the worldsheet C FT. While the general\nconstruction of strings defined by angular quantization is n ew, some examples of stretched\nstrings have been discussed in disguise over the years. One o f the most famous such\nexamples is Maldacena’s “long strings” in c= 1 string theory, and one special case of the\nstretched strings heuristically matches this example. Thi s briefest of connections is meant\nas an appetizer for the next paper in the installment, in whic h we plan to detail the angular\nquantization of “long strings” in the actual framework of st ring theory.\nAcknowledgments\nThis work is supported in part by DOE grant DE-SC0007870.\n– 40 –A Identities\nThe following identities are useful for the calculations in volved in this paper.\nαm′\n−ℓαm\nℓ=m/summationdisplay\nj=0(−1)jℓjm′!m!\nj!(m′−j)!(m−j)!αm−j\nℓαm′−j\n−ℓ(A.1)\n1F1(a;b|z) =ez1F1(b−a;b|−z) (A.2)\n1F1(−N;α+1|x) =N!α!\n(N+α)!L(α)\nN(x) (A.3)\n∞/summationdisplay\nN=0tNL(α)\nN(x) =e−tx\n1−t\n(1−t)α+1(A.4)\n(−x)m\nm!L(m−n)\nn(x) =(−x)n\nn!L(n−m)\nm(x) (A.5)\nMoreover, the following identities are found in [ 9] —\nn/summationdisplay\nm=0tm\n(n−m)!(m+α)!L(α)\nm(x) =(1+t)n\n(n+α)!L(α)\nn/parenleftbiggxt\n1+t/parenrightbigg\n(A.6)\n∞/summationdisplay\nm=0(n+m)!\nn!m!tmL(α)\nn+m(x) =e−xt\n1−t\n(1−t)n+α+1L(α)\nn/parenleftbiggx\n1−t/parenrightbigg\n(A.7)\n∞/summationdisplay\nk=01\nk!tkL(k−n)\nn(x) =1\nn!et(t−x)n, (A.8)\nrespectively as identity 4.4.1.7, identity 5.11.2.8 and id entity 5.11.4.2.\nReferences\n[1] N. Agia and D. L. Jafferis, “Angular Quantization in CFT,” [ arXiv:2204.11872 ]\n[2] D. L. Jafferis and E. Schneider, “Stringy ER = EPR,” JHEP 10, 195 (2022)\n[arXiv:2104.07233 ]\n[3] J. M. Maldacena, “Long strings in two dimensional string theory a nd non-singlets in the\nmatrix model,” JHEP 09, 078 (2005) [ arXiv:hep-th/0503112 ]\n[4] H. Reeh and S. Schlieder, “Bemerkungen zur unit¨ ar¨ aquivalenz von lorentzinvarianten\nfeldern,” Nuovo Cim. 22, no.5, 1051-1068 (1961)\n[5] E. Witten, “A Background Independent Algebra in Quantum Grav ity,” [arXiv:2308.03663 ]\n[6] N. Agia and D. L. Jafferis, work in progress\n[7] B. Balthazar, V. A. Rodriguez and X. Yin, “Long String Scatterin g in c = 1 String Theory,”\nJHEP01, 173 (2019) [ arXiv:1810.07233 ]\n[8] A. Giveon, N. Itzhaki and U. Peleg, “Instant Folded Strings and Black Fivebranes,” JHEP\n08, 020 (2020) [ arXiv:2004.06143 ]\n[9] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Volume 2 —\nSpecial Functions , Gordon and Breach Science Publishers, 1992\n– 41 –"    },    {        "title": "2402.05228v1.Weight_Reduced_Stabilizer_Codes_with_Lower_Overhead.pdf",        "content": "Weight Reduced Stabilizer Codes with Lower Overhead\nEric Sabo,1,∗Lane G. Gunderman,1,∗Benjamin Ide,1,∗Michael Vasmer,1, 2, 3and Guillaume Dauphinais1\n1Xanadu, Toronto, Ontario M5G 2C8, Canada\n2Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada\n3Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada\n(Dated: February 9, 2024)\nStabilizer codes are the most widely studied class of quantum error-correcting codes and form the\nbasis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set\nof parity-check operators, which are measured in order to infer information about errors that may\nhave occurred. In typical settings, measuring these operators is itself a noisy process and the noise\nstrength scales with the number of qubits involved in a given parity check, or its weight. Hastings\nproposed a method for reducing the weights of the parity checks of a stabilizer code, though it has\npreviously only been studied in the asymptotic regime. Here, we instead focus on the regime of\nsmall-to-medium size codes suitable for quantum computing hardware. We provide both a fully\nexplicit description of Hastings’s method and propose a substantially simplified weight reduction\nmethod that is applicable to the class of quantum product codes. Our simplified method allows us\nto reduce the check weights of hypergraph and lifted product codes to at most six, while preserving\nthe number of logical qubits and at least retaining (in fact often increasing) the code distance.\nThe price we pay is an increase in the number of physical qubits by a constant factor, but we find\nthat our method is much more efficient than Hastings’s method in this regard. We benchmark\nthe performance of our codes in a photonic quantum computing architecture based on GKP qubits\nand passive linear optics, finding that our weight reduction method substantially improves code\nperformance.\nI. INTRODUCTION\nQuantum error correction (QEC) is believed to be necessary in order to run large-scale quantum algorithms [1, 2].\nRecent years have seen remarkable progress in realizing QEC codes on various hardware platforms [3–6], with most\nexperiments thus far demonstrating a particular QEC code called the surface code [7–9].\nIn a stabilizer code (a widely-studied class of QEC codes), one measures parity-check operators that give partial\ninformation about errors that may have occurred on the physical qubits of the code. However, in a realistic setting\nthe act of measuring the parity-check operators is itself a noisy process. In many hardware platforms, the noise\nassociated with measuring a parity check scales with the weight of the check (the number of qubits that the check\nacts on non-trivially), as higher check weights correspond to deeper measurement circuits. Hence, one way to find\npractically useful stabilizer codes is to search for stabilizer codes with low-weight parity-checks, which are known as\nquantum low-density parity-check (qLDPC) codes [10].\nThe surface code is an example of a qLDPC code, as all of its checks are weight four. However, the surface code\nsuffers from the drawback that it always encodes a single logical qubit (no matter its size). This low encoding rate\nmeans that for realistic noise rates we expect to have approximately 1,000physical qubits per logical qubit, which\nleads to estimates of millions of physical qubits being required for large-scale quantum algorithms [11, 12]. There exist\nother families of qLDPC codes with higher encoding rates than the surface code [10], though this often comes with\nslightlyhigherparity-checkweightsandtherequirementoflong-rangeconnectivity[13–15]. Nevertheless, thepotential\nof these qLDPC codes has lead to recent proposals for implementing them in a variety of hardware platforms [16, 17],\nand is particularly well suited to photonic architectures based on GKP qubits where there are minimal constraints on\nthe locality of qubit connectivity [18, 19].\nGiven a qLDPC code with favorable parameters (e.g. high encoding rate) but parity-check weights that are high\nenough to limit its performance under realistic noise assumptions, one can ask if there is any way to reduce the\nweights of the parity-checks while (mostly) retaining the favorable parameters of the code. Hastings [20, 21] provided\na method known as weight reduction, which takes an input CSS code and can output a qLDPC code with O(1)parity-\ncheck weights, while increasing the number of physical qubits by a constant factor and reducing the code distance\nby a constant factor. This method has thus far been applied mostly in the asymptotic regime [21, 22] where one is\n∗These authors contributed equally. Corresponding author: eric.sabo@xanadu.ai.arXiv:2402.05228v1  [quant-ph]  7 Feb 20242\nprimarily interested in the scaling of properties such as the encoding rate of a family of codes as a function of the\ncode size.\nHere, we investigate the utility of weight reduction techniques for modifying qLDPC codes of small-to-medium size\nthat could potentially be implemented on hardware in the short-to-medium term. We first provide a self-contained\npresentation of Hastings’s weight reduction procedure, providing an alternative, algorithmic perspective while also\noptimizing the procedure to improve its overhead. Next, we propose a new weight reduction technique for classical\ncodes, the outputs of which can be used as input to quantum product code constructions [23–26]. We use these two\ntechniques to construct qLDPC codes with low-weight parity-checks, finding that our method gives codes with better\nparameters in the finite-size regime of interest. In particular, for an input hypergraph product code with parameters\n[ [45,9,3] ], weight seven checks, and qubit degree four, our method gives a [ [65,9,4] ]code with weight six checks and\nqubit degree three, whereas Hastings’s method gives a [ [2892 ,9,5] ]code with weight six checks and qubit degree six.\nThe default circuit for measuring a parity check in a circuit-based quantum computer is to introduce an ancilla\nqubit, apply controlled Pauli gates from the ancilla to the qubits in the support of the check, and then measure the\nancilla [27]. If all the operations in the circuit are noisy, then the noise in this process scales with the weight of\nthe parity check. In a measurement-based quantum computer the situation is similar: for each measurement of a\nweight wparity check we introduce a degree- wnode in the cluster state1, where the node represents a qubit prepared\nin the |+⟩state and each edges represents a control- Zbetween this qubit and another qubit [28–30]. In Xanadu’s\narchitecture, cluster states are constructed from two-qubit entangled GKP states [31] and N-body continuous-variable\nGHZ measurements [18, 19, 32]. In this architecture, the strength of the effective noise acting on a qubit in the cluster\nstate scales (to first order) as 1−erfh\nc/√\n2σ2Ni\n, where σ2is the variance of a single phase space peak of a GKP state’s\nWigner function (which can be related to the total transmissivity of the system [19]), Nis the number of neighbors\nof the qubit, cis a positive constant, and erfis the error function. Therefore, if we reduce the check weights, then we\nreduce the effective qubit-level noise in the cluster state for a fixed GKP state quality (or amount of optical loss).\nWe benchmark the performance of our weight-reduced codes using Monte Carlo simulations of the architecture\ndescribed above as a quantum memory. We find that our weight reduction technique substantially improves the\nperformance for cluster states constructed from hypergraph product codes and lifted product codes, in terms of both\nthe logical error rates and the break-even point. Although we benchmarked our codes using a specific noise model\nrelevant to photonic hardware based on GKP qubits, we would expect that similar results hold for other hardware\nplatforms where the noise associated with measuring a parity-check scales with its weight.\nThe remainder of this paper is structured as follows. In Section II, we review the relevant background in classical\nand quantum coding theory, and we provide additional background material on homological algebra in Appendix A. In\nSection III, we describe the steps of Hastings’s weight reduction method, discuss optimizations to reduce its overhead,\nand comment on its implications for iterative decoding. In Section IV, we present our alternative weight reduction\nmethod for classical codes and show how it can be applied to reduce the stabilizer weights of quantum product codes.\nIn Section V, we present examples of weight-reduced codes and the results of our numerical simulations. We conclude\nin Section VI.\nII. BACKGROUND & NOTATION\nThroughout this paper, we assume familiarity with the stabilizer formalism [33] of QEC codes and with the basics of\nclassical coding theory [34]. For simplicity, we exclusively work with F2vector spaces, although many of the ideas are\neasily extended to higher fields and do not depend on this choice. Missing entries in matrices are assumed to be zero,\nand horizontal and vertical dividing lines within matrices are added throughout solely for visual convenience. Let H\nbe a full-rank, (n−k)×nmatrix. The [n, k, d ](classical) linear code C=C(H)associated with His the subspace\northogonal to the row space of Hwith respect to the standard (Euclidean) inner product, C={v∈Fn\n2|HvT= 0},\nwhere v= (v1, . . . , v n)is thought of as a row vector and superscript ‘ T’ is the standard matrix transpose. The support\nofvis the set supp( v) ={i|vi̸= 0}. In this context, His called the parity-check matrix of Cand serves a role\nsimilar to the stabilizers in QEC. The minimum distance of the code is given by d= min {wt(v)|0̸=v∈ C}, where\nwt =|supp( v)|denotes the Hamming weight.\nAn[ [n, k, d ] ]Pauli stabilizer code is described numerically by its stabilizer matrix in symplectic form whose first n\ncolumns denote Pauli Xoperators and subsequent ncolumns Pauli denote Zoperators. CSS codes [35, 36] can be\ndescribed by two matrices HXandHZofncolumns comprising of the XandZstabilizer generators, respectively,\nsuch that the stabilizer matrix is of the form HX⊕HZ. The inputs of a procedure acting on a set of stabilizers\n1We note that the degree of the node may be greater than wif the check comprises more than one type of Pauli operator.3\nappear as HXandHZand the outputs appear with tildes, ˜HXand ˜HZ. Following Refs. [21, 37], let nXandnZ\nbe the number of XandZstabilizers, respectively, wXandwZbe the maximum Hamming weight of the XandZ\nstabilizer generators, respectively, and qXandqZbe the maximum Hamming weight of the columns of HXandHZ,\nrespectively. The parameters qXandqZare sometimes known as the X- and Z-qubit degrees, respectively, as, for\nexample, qXdenotes the maximum number of stabilizer generators that have nontrivial support on the same qubit.\nNote that these parameters are not inherent to the code but are relative to the specific form of the generators chosen.\nHypergraph product codes [23], HGP( H1, H2), are CSS codes constructed from two parity-check matrices H1and\nH2with stabilizers\nHX=\u0000\nH1⊗I I⊗HT\n2\u0001\n, H Z=\u0000\nI⊗H2HT\n1⊗I\u0001\n. (1)\nIfC(Hi)has parameters [ni, ki, di]andC(HT\ni)has parameters [mi, kT\ni, dT\ni], where kT\nianddT\niare the dimension and\ndistance of C(HT), respectively, then HGP( H1, H2)has parameters\n[ [n1n2+m1m2, k1k2+kT\n1kT\n2,min(d1, d2, dT\n1, dT\n2)] ]. (2)\nLetRℓ=F2[x]/(xℓ−1)be a polynomial quotient ring. A circulant matrix is a square matrix specified by the first\nrow or column, where each subsequent row (column) is cyclically shifted to the right (down) by one index. An element\ng(x) =g0+g1x+. . .+gℓ−1xℓ−1∈Rℓis associated with the ℓ×ℓcirculant matrix, B(g(x)), whose first column is\ngiven by the coefficients of g(x).2This is called the lift of g(x). The lift of a matrix A∈Mm×n(Rℓ)with elements in\nRℓis the matrix B(A)∈Mmℓ×nℓconstructed by replacing each element of Awith its lift. The matrix Ais called the\nbase (or weight or protograph) matrix of the lift and ℓis the lift size. For example, for ℓ= 2,\nA=\u0012\n1x\n0 1 + x\u0013\n,B(A) =\u0012\nB(1)B(x)\nB(0)B(1 +x)\u0013\n=\n1001\n0110\n0011\n0011\n. (3)\nAlthough typically defined by the form of its generator matrix, here we define a quasi-cyclic code [38] to be the\nlinear code defined by the parity-check matrix H=B(A). Quasi-cyclic lifted product codes [39] are a generalization\nof hypergraph product codes based on quasi-cyclic codes. Let A1∈Mm1×n1(Rℓ)andA2∈Mm2×n2(Rℓ)be base\nmatrices and define\nAX=\u0000A1⊗I I⊗A2\u0001\n, A Z=\u0000\nI⊗AT\n2AT\n1⊗I\u0001\n, (4)\nwhere the transpose of g(x)is determined by the transpose of its lift: gT(x) =g0+gℓ−1x+. . .+g1xℓ−1. The lifted\nproduct code LP(A1, A2)is the CSS code with parity-check matrices HX=B(AX)andHZ=B(AZ). The length\nof the code is n=ℓ(n1m2+n2m1)but there are currently no general formulas for kandd; however, lifted product\ncodes often have superior parameters to hypergraph product codes of similar size [39–41].\nWe will often use the Tanner graph representation of linear codes and stabilizer codes. Check nodes will be denoted\nby rectangles and variable nodes by circles. For CSS codes, open rectangles will denote Xstabilizers and filled-in\nrectangles Zstabilizers.\nA chain complex Cis, for the sake of our purposes, an ordered sequence of vector spaces {Ci}overF2with maps\n∂ibetween each ordered pair {Ci−1, Ci}such that ∂i◦∂i+1= 0:\n··· → Ci+1∂i+1− − − → Ci∂i− →Ci−1→ ··· .\nWe will only be interested in chain complexes with a finite number of vector spaces. A chain complex with ℓspaces\nis called an ℓ-term chain complex. Since ∂i◦∂i+1= 0, we have im (∂i+1)⊆ker (∂i). A sequence is said to be exact\natCiifim (∂i+1) = ker ( ∂i)and is said to be exact if it is exact at each Ci. The ith homology group is defined as\nHi(·) = ker ( ∂i)/im (∂i+1). The letter Hwill also be used for parity-check and stabilizer matrices, but there should\nbe no confusion as to the context. The dual of a chain complex is a cochain complex\n··· ← Ci+1δi+1← − − − Ciδi← −Ci−1← ··· .\n2Assigningthecoefficientstothecolumninsteadofthefirstrowhasprecedentintheclassicalerrorcorrectionandmathematicalliteratures\nbut is opposite of recent convention used in the quantum literature. The difference comes down to left versus right multiplication in the\nring of circulants.4\nThe corresponding dual of the homology groups are the cohomology groups Hi(·) = ker δi+1/imδi. In this work,\nδi=∂T\ni, since we assume the standard basis.\nThere is a natural correspondence between codes and chain complexes. Let H∈Fn−k×n\n2be a parity-check matrix\nof an [n, k, d ]-linear code. Then we can express this code as a chain complex\nFn\n2H− →Fn−k\n2. (5)\nAny diagram consisting of a single map vacuously satisfies the definition of a chain complex.\nConsider an [ [n, k, d ] ]CSS code generated by nZindependent Zstabilizers given by the matrix HZandnXX\nstabilizers given by HX. Since HT\nZHX= 0, we can treat this as the chain complex\nFnZ\n2HT\nZ− − →Fn\n2HX− − →FnX\n2 (6)\nwith the qubits in the center.3Conversely, we can derive a CSS code from any two consecutive boundary maps,\nsetting HT\nZ=∂i+1andHX=∂i. The Zlogical operators commute with the Xstabilizers ( kerHX) and are not Z\nstabilizers ( imHT\nZ= rowspace( HZ)), which make them elements of the first homology group H1. The dual problem\nis the cochain\nFnZ\n2HZ← − −Fn\n2HT\nX← − −FnX\n2,\nwhich gives the Xlogicals H1(·) = ker HZ/imHT\nX.4\nA detailed discussion of the tensor product of chain complexes, the mapping cone, and their respective homologies\nis given in Appendix A.\nIII. QUANTUM WEIGHT REDUCTION\nReference [22] provides a good summary of Hastings’s quantum weight reduction method [21]. Rather than du-\nplicating this work, we aim to complement it by providing a description of the method using diagrams and matrices\nwithout algebraic topology. Our notation is roughly aligned with [22], which is a simplification of [21]. In addition to\nreviewing previous work, we provide new insight into the method and discuss the subtleties of its implementation [42].\nThe four steps of quantum weight reduction method are:\n1. Copying — reduces qX,\n2. Gauging — reduces wX,\n3. Thickening and choosing heights — reduces qZ,\n4. Coning — reduces wZ.\nWe will examine each step independently, although they must be applied in this order as some care is required to\navoid undoing the progress achieved in previous steps. While parameters from previous steps are indeed kept O(1)\nin subsequent steps, the exact constants can be significant for constructing codes that are compatible with realistic\narchitectures. Therefore, in this section we will focus on the explicit constants rather than on asymptotic results. The\nexamples and figures provided are contrived to demonstrate a specific concept and are not meant to represent good\nstabilizer codes. Applying these operations to real codes produce large matrices and complicated Tanner graphs from\nwhich we believe it is difficult to discern the underlying structure.\nCopying\nThe goal of copying is to reduce qXto at most three. Start by making qXcopies of each qubit. By this we mean\nto add qX−1new qubits (initialized to zero) per original qubit; the value of each original qubit is not copied to the\nnew qubits:\n\u0000v1v2. . . v n\u0001\n7→\u0000v1,1. . .v1,qXv2,1. . .v2,qX. . .vn,1. . .vn,qX\u0001\n.\n3It is often assumed that the qubits are at C1; this is true for this work but is not strictly necessary.\n4There is a one-to-one correspondence between the usual set difference definition of logical operators and equivalence classes of the\nquotient.5\nFor every Xstabilizer of length n, make a new stabilizer of length qXnsuch that for every viin the support of the\nstabilizer one of {vi,1, . . . , v i,qX}receives the value of vi. If a stabilizer uses the qubit vi,j, another stabilizer cannot\nuse it. For example, valid copies of the stabilizer\u00001 1 1 1 1 1\u0001\nwith qX= 3include\n\u0000100100100100100100\u0001\n(7)\nand\n\u0000010001100001100010\u0001\n.\nAs vectors, these are equivalent up to qubit permutations. For simplicity, this work always fills the columns from left\nto right, as in Eq. (7). Suppose Eq. (7) is used and\u00001 1 0 0 1 1\u0001\nis another stabilizer. Then\n\u0000100010000000010010\u0001\nis not a valid copy because the qubit v1,1is already used by the first stabilizer. Note that every original Xstabilizer\nis kept, although now in a permuted form.\nIn addition to the copied stabilizers, (qX−1)nnewXstabilizers are added to link the copies of visuch that they\ncollectively “behave” like the original, single qubit. These are weight two and of the form vi,jvi,j+1for1≤j≤qX−1.\n(These are not constrained by the “validity” concept required for the previous stabilizers.) For example,\n\u0000110000000000000000\u0001\n,\u0000011000000000000000\u0001\n,\u0000000110000000000000\u0001\n,\u0000000011000000000000\u0001\n,\n...\nThis imposes a classical repetition code on the copies; see Fig. 1 below.\nThe commutativity with the Zstabilizers is maintained on every copy by putting the value at viatvi,jfor all\n1≤j≤qX. For example, if\u00001 0 1 0 1 0\u0001\nis aZstabilizer, then for qX= 3, the new Zstabilizer is\n\u0000111000111000111000\u0001\n.\nThis comes at the price of increasing wz, which will be dealt with in subsequent steps.\nExample 1. Copying the stabilizers\nHX=\n1 1 1 0 0 0\n1 1 0 0 1 1\n1 0 1 1 1 0\n1 0 0 0 0 1\n\nHZ=\u00001 0 1 0 0 1\u00016\nv1 v2 v3\n(a)\n (b)\nFigure 1. Open squares represent X stabilizers, filled squares Z stabilizers, and circles qubits. (a) The maximum column weight\nisqX= 4at vertex v2. (b) The result of applying the copying procedure to (a). The copied variables in the repetition codes\nwould have labels v1,1, v1,2, v1,3, v1,4, v2,1, . . ., and so on from left-to-right.\ngives\n˜HX=\n1 1 1\n1 1 1 1\n1 1 1 1\n1 1\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n11\n(8)\n˜HZ=\u00001111 1111 1111\u0001\n. (9)\nThe parameters have transformed from (wX= 4, qX= 4, wZ= 3, qZ= 1)to( ˜wX= 4,˜qX= 3,˜wZ= 12,˜qZ= 1).\nIt follows from above that ˜n=qXn,˜nX=nX+ (qX−1)n,˜nZ=nZ,˜wX=wX,˜qX= min {qX,3},˜wZ=qxwZ,\nand˜qZ=qZ. It follows that ˜k= ˜n−˜nX−˜nZ=n−nx−nz=k. The dimension is traditionally computed by\ncounting the number of logical operators, but since we know the dimension already, we will write down kindependent\noperators that commute with the stabilizers and are therefore logical operators. A Z-logical operator must commute\nwith both the old and new Xstabilizers. The old Xstabilizers are supported on qubits vi,1for1≤i≤n. Fix an iin\nthe overlap between an old Xstabilizer and old Zlogical. This does not commute with the new Xstabilizer vi,1vi,2.\nThe only way to fix this is to extend the support of the Zstabilizer to all of the copied qubits {vi,1, . . . , v i,qX}. New\nZlogicals are thus of the form z⊗\u00001. . .1\u0001\n, where zis aZ-logical operator of the input code and the all-ones vector\nhas length qX. The Xlogicals of the input code still commute with the new Zstabilizers on the bits vi,1. This gives\n˜dX=dXand˜dZ=qXdZ.\nGraphically, copying replaces each variable node in the Tanner graph with a repetition code of Xstabilizers\nprotecting against phase errors. For example, the Tanner graph of Fig. 1a is transformed to that of Fig. 1b. The\nmanner in which the edges from the check nodes are attached to the repetition codes correspond to the choice in\nplacing viin its copies. In particular, the ordering of the original stabilizers induces a potential permutation of the\nedges. We will discuss implications of this in Section IIIC.7\nGauging\nThe goal of gauging is to reduce wXto less than or equal to three without increasing qX. Consider an Xstabilizer\nof weight w > 3with support on qubits labeled by {v1, . . . , v w}. For each such stabilizer, gauging introduces w−3\nnew qubits, {v′\n1, . . . , v′\nw−3}. These are in addition to any new qubits introduced by copying. The input Xstabilizer\nis replaced by new Xstabilizers with supports\n{v1, v2, v′\n1},{v3, v′\n1, v′\n2}, . . . ,{vw−2, v′\nw−4, v′\nw−3},{vw−1, v′\nw−3, vw}.\nTo see this in matrix form, assume without loss of generality that the support of the stabilizer is permuted to the first\nwqubits. Then,\n\u0000v1···vw\n1···10···0\u0001\n7→\nv1v2v3···vw−2vw−1vw v′\n1v′\n2···v′\nw−2v′\nw−3\n11 ···1\n1 ···11\n... ···...\n1 ··· 1 1\n1 1··· 1\n, (10)\nwhere the column of dots represent qubits not in the support of the current stabilizer. Note that the right-hand side\nisHT\nw−2, where the transpose\nHℓ=\n1 1\n1 1\n...\n1 1\n1 1\n(11)\nis a parity-check matrix for the [ℓ,1, ℓ]classical repetition code. The left side of the matrix has support on the original\nqubits. The new (primed) qubits are not used by any other Xstabilizer, leaving the right side as the direct sum\n⊕iHT\nwi−2, where wiis the weight of the ith reduced Xstabilizer.\nAt this point, the Zstabilizers only have support on the original qubits and may no longer commute with the\nrows of Eq. (10). Consider the w−2new rows of a reduced Xstabilizers, where the new stabilizers are arranged\nin the order of Eq. (10). If the jthZstabilizer anti-commutes with the product of new Xstabilizers 1 to mfor\ni∈ {1, . . . , w −3}, then set the mth new column of the jth row of ˜HZto one, where the new columns are those that\nwere introduced when applying gauging to the original Xstabilizer. We will mention an alternative method to build\ncommuting Zstabilizers in Section IIIA.\nExample 2. Gauging the stabilizers\nHX=\u001211 11 11 11\n1 1 1 1 1 1 1 1\u0013\nHZ=\n11111111\n1111 1111\n11 11 11 11\n1 1 1 1 1 1 1 1\n8\nv1 v2 v3 vw−2vw−1vw· · ·\n(a)\nv1 v2 v3 vw−2vw−1vwv′\n1 v′\n2v′\nw−4v′\nw−3\n· · ·· · · (b)\nFigure 2. (a) The input Tanner graph. (b) The affect of applying gauging to the Xstabilizer in (a).\ngives\n˜HX=\n11 1\n1 11\n1 11\n1 11\n1 11\n11 1\n1 1 1\n1 11\n1 11\n1 11\n1 11\n1 1 1\n\n˜HZ=\n11111111 1 1\n1111 1111 1 1\n11 11 11 11 1 1 11 1\n1 1 1 1 1 1 1 111 1 1 1\n.\nThe parameters have transformed from (wX= 8, qX= 2, wZ= 8, qZ= 4)to( ˜wX= 3,˜qX= 2,˜wZ= 13,˜qZ= 4).\nIt follows from above that ˜nand˜nXincrease by wi−3for each wi>3while ˜nZ=nZ, and hence, ˜k=k. We also\nhave ˜wX= min {wX,3},˜qX=qX,˜wZ≥wZ, and ˜qZ≥qZ. The old Xlogical operators (appended with zeros for the\nnew primed qubits) still commute with the new Zstabilizes, however, ˜dXcould decrease. Take an element of ker˜HZ\nwith non-zero support on the new qubits that is not an Xstabilizer. The matrices HT\nwi−2row reduce to the identity\n(and a zero row) and may be used to clean the logical operator off the support of the new qubits [43]. This could lead\nto a lower weight logical representative than the logical operators of the input code. The Zdistance is at least dZ,\nbut we delay the proof until Theorem 9.\nGraphically, gauging transforms the Xstabilizers of the Tanner graph from Fig. 2a to Fig. 2b. The matrices HT\nw−2\ninduce a repetition code on the check nodes instead of the variable nodes.\nRemark : Hastings treats copying and gauging as a single operation [21]. Here the new qubits resulting from copying\nare called “copied” qubits and the new qubits resulting from gauging “new” qubits.\nThickening & Choosing Heights\nThe goal of thickening is to increase dX, and the goal of choosing heights is to reduce qZ. On its own, thickening\nis commonly referred to as (a special case of) distance balancing using the chain complexes\nA:FnZ\n2HT\nZ− − →Fn\n2HX− − →FnX\n2\nB:Fℓ−1\n2HT\nℓ− − →Fℓ\n2,\nwhere Hℓis defined in Eq. (11). The details of the tensor product of these two chains are worked out in Appendix A1.\nThere we show that A ⊗ Bgives\nC3∂3− →C2(˜HZ)T\n− − − − → C1˜HX− − → C09\nwhere the code is described by stabilizer matrices (Eq. (A7) and the transpose of Eq. (A6))\n˜HX=\u0000\nHX⊗IℓInX⊗HT\nℓ\u0001\n,˜HZ=\u0012HZ⊗Iℓ 0\nIn⊗HℓHT\nX⊗Iℓ−1\u0013\n, (12)\nand has distances ˜dX=ℓdXand ˜dZ=dZ. Thickening can be used to counteract the decrease in Xdistance that\ngauging may have caused. In addition to ˜HXand ˜HZ, there is a third matrix (Eq. (A5))5\n∂3=\u0012\nInZ⊗HT\nℓ\nHT\nZ⊗Iℓ−1\u0013\n, (13)\nwhich satisfies ˜HT\nZ∂3= 0. This implies that some of the Zstabilizers are redundant with linear dependencies\ndetermined by ∂3. Note that InZ⊗HT\nℓ=Lnz\ni=1HT\nℓ. Each row hjofHZcorresponds to a block inLnz\ni=1HT\nℓ, and the\nset of stabilizers\u0000hj⊗Iℓ0\u0001\n. The block diagonal structure ofLnz\ni=1HT\nℓimplies that the ℓstabilizers in hj⊗Iℓare\nrelated via ℓ−1constraints. Since these stabilizers are the cause of the potentially high column weights, removing\nthe redundant stabilizers can help lower qZ. Hastings calls making the choice of which row of\u0000hj⊗Iℓ0\u0001\nto keep\nchoosing heights . We represent this by a length nZvector heightswhose jth element is an integer between 1andℓ\nspecifying which row of\u0000hj⊗Iℓ0\u0001\nis kept. See Example 3 for an explicit example of thickening, using ∂3to derive\nthe row dependencies, and choosing heights.\nWhen a different height is chosen for two different rows of HZ, the resulting two rows in ˜HZhave no qubits in\ncommon, thus a good choice of heights and a sufficient amount of thickening reduces qZ. In the extreme case, choosing\nℓ=nZandheights = (1 , . . . , n Z)ensures a column weight of one within the block and therefore qZ= 3at the cost\nof extra qubits. A greedy algorithm can be used when ℓ < n Zto choose heights that satisfy certain parameters such\nas a target qZ. The properties of the resulting code depend highly on the choice of ℓandheights.\nExample 3. Thickening the stabilizers\nHX=\u00001 1 1 1\u0001\n, H Z=\u0012\n1 1 0 0\n1 0 1 0\u0013\n, (14)\nwithℓ= 3gives\n˜HX=\n1 1 1 1 1\n1 1 1 1 11\n1 1 1 1 1\n,\n˜H′\nZ=\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n, ∂ 3=\n1\n11\n1\n1\n11\n1\n1 1\n1 1\n1\n1\n1\n1\n.\nLet{h1, . . . , h 14}denote the rows of ˜H′\nZ, where h1toh6are above and h7toh14below the horizontal line, respectively.\nMultiplying ∂3\u0010\n˜H′\nZ\u0011T\ngives four equations, with the first two:\nh1+h2+h7+h9= 0 (15)\nh2+h3+h8+h10= 0. (16)\n5The map ∂3can be thought of as metachecks [44–46] for the Zstabilizers induced by the tensor product in thickening. Choosing heights\neliminates the metachecks.10\nIf we remove rows h2andh3but keep rows h1andh7toh14, we can recover h2from Eq. (15)andh3from Eq. (16).\nHence, h2, h3∈rowspace( ˜H′\nZ)even if removed as rows of ˜H′\nZ. A similar argument using the next two equations\ngenerated from ∂3\u0010\n˜H′\nZ\u0011T\nshows that we only need to keep one of the rows h4,h5, and h6. Choosing heights = (1 ,2)\nkeeps rows h1andh5and removes rows h2,h3,h4, and h6. The final Zstabilizers are\n˜HZ=\n1 1\n1 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n11 1\n. (17)\nThe parameters have transformed from (wX= 4, qX= 1, wZ= 2, qZ= 2)to( ˜wX= 6,˜qX= 2,˜wZ= 3,˜qZ= 4). This\nexample is shown in Tanner graphs in Fig. 3.\nIt follows from above that ˜n=ℓn+ (ℓ−1)nX,˜nX=ℓnX,˜wX=wX+ 2,˜qX= max {qX,2}, and ˜wZ=\nmax{wZ, qX+ 2}. Before choosing heights, ˜nZ=ℓnZ+ (ℓ−1)nand˜qZ= max {wX, qZ+ 2}. The logical operators\nare given by the Künneth formula (Eq. (A8)),\nH1(C) = (H1(A)⊗H0(B))⊕(H0(A)⊗H1(B)) =H1(A)⊗H0(B),\nfrom which it follows that ˜k= dim H1(C) = dim H1(A) dim H0(B) =k·1. Neither the dimension nor the logicals are\naffected by removing redundant stabilizers. The homology of the repetition code is given in Appendix A1.\nGraphically, recall that the Tanner graph has a variable node for each column (qubit) and a check node for each\nrow (stabilizer). For the Xstabilizers, there are therefore variable nodes for each column of InX⊗HT\nℓand separate\nvariable nodes for each column of HX⊗Iℓ, both of which are connected to the same check nodes. The Tanner graph\nofHT\nℓis simply the Tanner graph of Hℓwith the variable and check nodes (columns and rows) switched. The term\nInX⊗HT\nℓmakes nXidentical and unconnected copies of the Tanner graph of HT\nℓ. The term HX⊗Iℓis equivalent to\nIℓ⊗HXup to row and column permutations but connects the ℓcopies of HXin a different pattern. The Zstabilizers\nare similar but now there is an extra set of check nodes for HZ⊗Iℓacting on the same variable nodes as In⊗Hℓ.\nAn example is given in Fig. 3.\nConing\nThe necessary homological algebra to motivate the formulas here is reviewed in Appendix A. It is suggested for\nreaders unfamiliar with tensor products of chain complexes and the mapping cone to read this first. Here, we will\nonly describe the coning of what Hastings calls reasonable codes, which will be defined later in the context it arises.\nReaders interested in the modifications required for unreasonable codes are referred to the discussion in Ref. [21].\nA set of stabilizer generators for the 15-qubit quantum Reed-Muller code QRM(4) [47, 48] is\nHX=\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n, (18)\nHZ=\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n1 1 1 1 1 1 1 1\n1 1 1 1\n1 1 1 1\n1 1 1 1\n1 1 1 1\n1 1 1 1\n1 1 1 1\n,11\n(a)InX⊗HT\nℓ\nHX⊗Iℓ\n(b)\nHT\nX⊗Iℓ−1\nIn⊗Hℓ\nHZ⊗Iℓ\n(c)\n(d)\nFigure 3. (a) The Tanner graph of Eq. (14) with Xstabilizer on top and two Zstabilizers on bottom. (b) The effect of\napplying thickening to the Xstabilizer of (a). The Zstabilizers are omitted for clarity. (c) The effect of applying thickening\nto the Zstabilizers of (a). The Xstabilizers are omitted for clarity. (d) Using the notation of the text, the effect of choosing\nheights = (1 ,2)on (c). The Xstabilizers remain unchanged. The Tanner graph for the full code consists of the stabilizers\nshown in both (b) and (d).\nwhich has (wX= 8, qX= 4, wZ= 8, qZ= 10). Applying copying, gauging, and then thickening and choosing heights\nwith ℓ= 3,heights = (2 ,1,2,1,2,3,1,3,3,1)gives the sequence of parameters (wX, qX, wZ, qZ):\n(8,4,8,10)copying− − − − − → (8,3,32,10)gauging− − − − − → (3,3,43,10)th.&c.h.− − − − − − → (5,3,43,5).\nNaïvely trying to reduce wZusing a second round of gauging applied to the Zstabilizers gives (wX= 85 , qX=\n22, wZ= 3, qZ= 5), which is counterproductive. This occurs in general and is not unique to this example. Coning\nusing the mapping cone of Appendix A2 serves to reduce wZwhile preserving wX,qXandqZ.\nConing is more involved than copying, gauging, or thickening and choosing heights. For this reason, we begin with\na brief overview, which is aligned with the discussion in Appendix A2. First, pick a Zstabilizer to be reduced, Zi,\nand remove it from HZto create H(r)\nZ. Second, view this code as a chain complex in the standard way,\nB:C2 C1 C0\u0010\nH(r)\nZ\u0011T\nHX,\nwith C2=FnZ\n2,C1=Fn\n2, and C0=FnX\n2. Next, define a new chain complex Ai:Qi→ X i→ R i, where C1=Qi=\nF|supp( Zi)|\n2,C0=Xiis derived from the overlap of Ziwith the Xstabilizers, and C−1=Riis constructed, if necessary,\nbased on QiandXito ensure that no new logical operators are added with support only on the new qubits. All three12\nspaces are defined in detail below. These two chain complexes will be connected with appropriate chain maps f(i),\nQi Xi Ri\nC2 C1 C0∂(i)\n1∂(i)\n0\n\u0010\nH(r)\nZ\u0011T HXf(i)\n1f(i)\n0,\nwhich induces the mapping cone\nQi Xi Ri\n⊕ ⊕ ⊕\nC2 C1 C0\ncone\u0000\nf(i)\u0001\n: C2⊕ Q i C1⊕ Xi C0⊕ R i∂(i)\n1∂(i)\n0\n\u0010\nH(r)\nZ\u0011T HXf(i)\n0f(i)\n1\n˜HT\nZ˜HX(19)\nwith\n˜HX= \n∂(i)\n0\nf(i)\n0HX!\n,˜HZ=\n\u0010\n∂(i)\n1\u0011T\u0010\nf(i)\n1\u0011T\nH(r)\nZ\n. (20)\nThis is for a fixed i. The case for mstabilizers to be weight reduced gives6\nQ1 X1 R1\n⊕ ⊕ ⊕\n.........\n⊕ ⊕ ⊕\nQm Xm Rm\nC2 C1 C0∂(1)\n1∂(1)\n0\n\u0010\nH(r)\nZ\u0011THXf(1)\n0f(1)\n1\nf(m)\n1f(m)\n0∂(m)\n1∂(m)\n0. (21)\nand\n˜HX=\n∂(1)\n0\n...\n∂(m)\n0\nf(1)\n0···f(m)\n0HX\n,˜HZ=\n\u0010\n∂(1)\n1\u0011T \u0010\nf(1)\n1\u0011T\n......\u0010\n∂(m)\n1\u0011T\u0010\nf(m)\n1\u0011T\nH(r)\nZ\n. (22)\n6ThenZ−m Zstabilizers that remain in H(r)\nZare directly visible in the resulting code and are referred to as “direct stabilizers”, while\nthe other m Zstabilizers are induced by the resulting code and are referred to as “induced stabilizers” [21].13\nThese commute by definition of the chain complex. If all Zstabilizers need to be reduced, then H(r)\nZandC2will be\nempty. There are several degrees of freedom in constructing these objects and the final parameters of the code are\nhighly dependent on the choices made.\nNow in full detail, let Zi= (z1, . . . , z n)be a Z-stabilizer generator whose weight needs to be reduced. Define Qi\nbe the vector space F|supp( Zi)|\n2. There is a natural embedding of the standard basis of Qito all weight-one patterns\nofFn\n2contained in the support of Zi,f(i)\n1:Qi→C1, which extends to all of Qiby linearity: if the kth element of\nsupp( Zi)iszp, then f(i)\n1maps the kth elementary basis element of Qito the pth elementary basis element of Fn\n2. A\nmatrix representation of f(i)\n1has maximum row and column weights equal to one, which helps control wZandqZin\nEq. (22).\nThe support of any Xstabilizer overlaps with the support of Zian even number of times. Construct a set\nof tuples {(S, j, k )}, where Sis an X-stabilizer generator with overlapping support on indices j, k∈supp( S)∩\nsupp( Zi). It is not necessary to include all possible pairs (j, k)but the entire overlap must be represented for each\nS. For example, if the overlap occurs on indices {1,2,3,4}, then {(S,1,2),(S,3,4)}or{(S,1,2),(S,2,3),(S,3,4)}\nor{(S,1,2),(S,1,3),(S,1,4),(S,2,3),(S,2,4),(S,3,4)}are three valid options. Define Xito be the vector space\nF|{(S,j,k )}|\n2, where the cth standard basis element may be associated to the cth element of {(S, j, k )}. There is a map\nfromXito the space of Xsyndromes, f(i)\n0:Xi→C0, which, in matrix form, has a 1in row rand column cif the X\nstabilizer associated with the cth basis element is the rthXstabilizer represented in HX.\nThere is also a natural connection between QiandXi: the elements of Qiare associated with qubits and Xiwith\npairs of qubits. Make a graph with a vertex for every basis element of Qiwith an edge between the vertices associated\nwith the qubits in (S, j, k )∈ Xi.7The map ∂(i)\n1:Qi→ X iis called the coboundary map of the graph and takes the\nvertices to the basis elements of Xithat include those vertices. For example, the coboundary map of the square with\nvertices V={v1, v2, v3, v4}and edges E={v1v2, v2v3, v3v4, v4v1}is the |E| × |V|edge-vertex incidence matrix\n\nv1v2v3v4\nv1v21100\nv2v30110\nv3v40011\nv4v11001\n.\nLooking back at Eq. (22), the number of rows of ∂(i)\n1, and hence the spaces themselves, determine the number of\nnew qubits in the weight reduction ( |Xi|). Assuming coning is applied to the output of all the previous steps, the input\ncan be made to satisfy wX≤5,qX≤3, and qZ≤3and are of the form Eq. (12). The Zstabilizers corresponding to\nthe bottom row of Eq. (12) have weight no more than five (since HXthere has qX≤3) and therefore do not need to\nbe reduced. The InX⊗HTterm in the Xstabilizers contributes two to the weight but overlaps the 0block in the\nZstabilizers that need reduction. Thus, the support of an Xstabilizer can therefore only intersect the support of Zi\nzero or two times. This keeps Xismaller compared to applying coning to a completely random input.\nFinally, we construct the space Ri. The logical operators are determined by the chain complex of homologies (long\nexact sequence Eq. (A12))\n0→H1(Ai)→H1(B)→H1\u0010\ncone\u0010\nf(i)\u0011\u0011\n→H0(Ai). (23)\nThe term H1(B)represents the logical operators of the original code plus the logical operator created by deleting the\nZstabilizer that is being reduced. The logical operators of the output of coning are elements of H1\u0000\ncone\u0000\nf(i)\u0001\u0001\n.\nIn order to find a relationship between the two, recall that weight reduction is considered an operation on a code\nas compared to a method of constructing a new code. Hence, we require kremains the same throughout the entire\nprocess. In order for this to happen, we need H0(Ai)to be zero. The space Riis designed to make this happen. Recall\n(Appendix A) that H0(Ai) = ker ∂(i)\n0/im∂(i)\n1. It follows that we should choose Riand∂(i)\n0such that ker∂(i)\n0= im ∂(i)\n1.\nDefine Rito be the F2-vector space with a basis element for every element of the cycle basis of the graph defined by\nQiandXi, and ∂(i)\n0:Xi→ R iby the coboundary map sending the edges to the cycles they are contained in. If the\ngraph has no cycles, it is not necessary to construct Ri. In practice, we have observed that we have always needed it.\nThe following example shows that not all cycle bases are equivalent for our purposes. Freedman and Hastings\nprovide an algorithm called the Decongestion Lemma [49] to find a cycle basis such that each edge appears in at most\nO(log|Qi|2)cycles and whose cycle length is O(|Qi|log|Qi|). This may not be the minimum-weight cycle basis, but\n7This is potentially a multigraph because two different Xstabilizers could have had the same overlap with Zi.14\n1\n234 5\n6\n7\n89\n(a)\n (b)\n (c)\n (d)\nFigure 4. (a) An input graph. (b) A spanning tree for the graph in (a). (c) The fundamental cycle basis resulting from (b).\n(d) A minimum-weight cycle basis for (a).\nthese conditions are designed to control the column and row weights of ∂(i)\n0. Alternative algorithms, such as Horton’s\nalgorithm [50] for finding a minimum-weight cycle basis, could be explored in future work.\nExample 4. To find a cycle basis of the graph in Fig. 4a, start with a spanning tree. One possible choice is Fig. 4b.\nThe fundamental cycle basis associated with this tree is given by adding edges from Fig. 4a not in Fig. 4b. Adding\nedges 4,2,3, and 6produce the fundamental cycle basis in Fig. 4c. If this was interpreted as QiandXi, we would\nhave\n∂(i)\n0=\n1 1 1 1 1 1\n1 1 1 1\n1 1 1 1 1\n1 1 1\n,\nwhere the rows correspond to the cycles read left-to-right then top-to-bottom in Fig. 4c and the columns correspond to\nthe edges in Fig. 7a in numerical order. The two columns of ones correspond to the edges 5and7which appear in\nevery cycle of Fig. 4c. This is a potentially undesirable increase in qX(Eq.(22)).\nAlternatively, a so-called minimum-weight cycle basis whose total sum of the number of edges in each basis element\nis minimal is given in Fig. 4d. Now,\n∂(i)\n0=\n1 1 1\n1 1 1\n1 1 1\n1 1 1\n,\nwhere the rows correspond to the cycles read left-to-right then top-to-bottom in Fig. 4d and the columns correspond\nto the edges in Fig. 7a in numerical order. The minimum-weight basis avoids increasing qXbut is not guaranteed to\nalways do so. The Decongestion Lemma reduces both the row and column weights, but only with high probability.\nHaving chosen a cycle basis, some elements may contain a large number of edges. Large cycles lead to high-weight\nrows of ∂(i)\n0and thus high-weight Xstabilizers. If the input code was LDPC, this would produce a non-LDPC output\nand undermine the reduction of wXin gauging. To fix this, we introduce auxiliary edges to break up the cycles into\nsmaller cycles in a process called cellulation. The map and the resulting stabilizers are highly dependent on the choice\nof cellulation. In this sense, some cellulations are better than others. Here we follow Reference [22]: any time a cycle\nhas length greater than four, simply add edges to bring it down to four. We do not add any new vertices (qubits) to\ndo this. These new edges must be added to Xibut without an associated Xstabilizer, i.e., (_, j, k). This affects the\nmaps ∂(i)but not f(i)\n0. The additions to ∂(i)appear in the stabilizers Eq. (22) and could affect the LDPC properties\nof the output. We leave the question of “optimal” cellulations to future work. An explicit example is done below in\nExample 5, and another graphical demonstration is provided in Fig. 5.\nIfqXis higher than desired, we can perform an extra round of thickening and choosing heights with the roles of X\nandZswitched:\nA:FnZ\n2HZ← − −Fn\n2HT\nX← − −FnX\n2\nB:Fℓ−1\n2Hℓ← − −Fℓ\n2.\nThe cellulation and the optional second thickening and choosing heights is referred to as the reduced cone in [21, 22].15\nExample 5. Consider coning the following inputs\nHX=\n11\n11\n11\n11\n11\n11\n1 1\n1 1\n11\n11\n1 1\n\nHZ=\u00001111111111\u0001\n.\nThere is only one Zstabilizer to be reduced with support on all the qubits, so Q1=F10\n2. Since Z1has full support, f(1)\n1\nis simply the identity map. The overlaps of HXandHZgiveX1=F11\n2with standard basis elements in correspondence\nwith the edges\n{(S1,1,2),(S2,2,3),(S3,3,4),(S4,4,5),(S5,5,6),(S6,6,7),(S7,7,1),(S8,4,8),(S9,8,9),(S10,9,10),(S11,10,8)}.\nSince the jth element of X1is associated with the jthXstabilizer, f(1)\n0is also the identity.\nWe can visualize the graphical relationship between QiandXiby treating HXrestricted to the qubits on the support\nof the Zstabilizer being reduce as the edge-vertex incidence matrix\n1 2 3\n4\n5 6 789\n10\n.\nThe cycles are Ri={(1,2,3,4,5,6,7),(8,9,10)}giving\n∂(1)\n0=\u00121111111\n111\u0013\n.\nThis will produce a high-weight Xstabilizer in Eq. (22), so we cellulate by adding the dashed edges 2−6and3−5\nand(_,2,6),(_,3,5)toX1. Now the cycles are\nRi={(1,2,6,7),(2,3,5,6),(3,4,5),(8,9,10)}\nand\n∂(1)\n0=\n1 11 1\n1 1 11\n11 1\n111\n,\nwhere the columns to the right of the vertical line correspond to the cellulation. Note that the edge 4−8does not\nparticipate in any cycles and corresponds to the empty column. The new edges are not associated to any stabilizers,16\nsof(1)simply adjoins zero rows. With the new X1,\n∂(1)\n1=\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n1 1\n,\nwhere the rows below the horizontal line correspond to the cellulation.\nRemark: The process of finding ∂(i)\n1,f(i)\n1, and f(i)\n0(pre-cellulation) can be simplified to following these operations:\n\u0012\nHXInX\nIn\u0013\ndelete− − − − − →\ncolumns\u0012HX|supp( Zi)InX\nf(i)\n1\u0013\ndelete− − − − →\nrows\n∂(i)\n1\u0010\nf(i)\n0\u0011T\nf(i)\n1\n, (24)\nwhere we delete the columns of\u0012\nHX\nIn\u0013\ncorresponding to qubits {1, . . . , n } \\supp( Zi)followed by deleting the rows of\n\u0000HX|supp( Zi)InX\u0001\nwhere HX|supp( Zi)is zero.\nExample 6. We began this section by pointing out why we could not use gauging to reduce wZ. Instead we found a\ncycle basis, cellulated, and used the mapping cone. Here we follow [21] and [51] in cellulating cycles down to weight\nfour, but one could choose an alternative scheme. Consider cellulating the octogon of solid edges 1to8Eq.(25)by\ntriangulation via the dashed edges 9to13:\n12 4\n6\n8\n7 539\n101112\n13\n. (25)\nThis transforms the cycle\u00001 1 1 1 1 1 1 1\u0001\nto\n\n1 1 1\n1 1 1\n1 1 1\n1 1 1\n1 1 1\n1 1 1\n,\nwhere the columns represent the edges of Eq. (25)in numerical order. This is the gauging equation, Eq. (10). In this\nsense, cellulating plays the role of gauging for cycles by gauging a single row of ∂(i)\n0. If every cycle in the cycle basis is\ncellulated, then every row of ∂(i)\n0is “gauged” for a fixed i. However, this will not necessarily be equivalent to applying\ngauging directly because the edge labels must be consistent between all of the cycles and therefore the columns cannot\nalways be simultaneously permuted into the form Eq. (10)for every cycle.17\nNow that the full coning procedure has been described, we return to the logical operators (Eq. (23)). We will\nassume to simplify the discussion that we are reducing Zstabilizers one at a time, i.e., we are using Eq. (19) and not\nEq. (21). If an extra round of thickening and choosing heights is used, the logical operators described here will be\nmodified as discussed in that section.\nBy definition, H1(Ai) = ker ∂(i)\n1. The space Qicontains the full support of Zi. Each element of Xiis associated\nwith two elements of Qiby construction, so rows of ∂(i)\n1will always have weight two. Thus, the all-ones vector will\nalways be an element of ker∂(i)\n1. This is equivalent to saying that the Zicommutes with all of the Xstabilizers it\noverlaps with. Anything else in the kernel commutes with all the Xstabilizers in the overlap of Ziand is therefore\naZstabilizer or a Zlogical operator contained entirely in the support of Zi. A stabilizer doesn’t change the result\nwhen we take the quotient of the old logical operators and it. A Zlogical operator is more problematic, and we\nsimply define away this problem by declaring coning to only apply to codes in which no Zlogical operator is entirely\ncontained within the support of a Zstabilizer generator. Hastings called these codes reasonable ; see [21] for the\nmodifications to coning required for unreasonable codes. Assuming we have a reasonable code, applying the exactness\ncondition throughout Eq. (23), the Isomorphism Theorem (for groups) gives H1\u0000\ncone\u0000\nf(i)\u0001\u0001∼=H1(B)/ker∂(i)\n1. This\nremoves the extra logical operator caused by removing Zifor reduction, showing ˜k=k.\nIt is clear from the matrices that the parameters after coning are more complicated than the previous steps. We\nhave already showed that ˜k=k. Before cellulation, wZ≤5. Cellulation can increase this. The more a vertex is reused\nfor adding new edges to create the cellulation, the more wZcan increase. If a vertex is used xtimes in cellulation,\nthere will be a Z-stabilizer generator of at least weight x+ 2, up to potentially x+ 4. Coning does not change qZ.\nThe effect on qXdepends on how many edges are reused in cycles. This could be a large number, in which case a\nsecond round of thickening and choosing heights could be necessary, this time swapping the roles of XandZ. If this\nis done, wZwill increase by two, qZwill still remain the same, and ˜wXwill be the maximum of wXbefore the second\nround of thickening and 2 +qZ. The following lemma clarifies the asymptotic result stated in [21].\nLemma 7. Given a code that has undergone copying, gauging, thickening, and choosing heights, let hbe the maximum\nnumber of times any single height was chosen. Then the reduced cone code has ˜wX≤5 +h.\nProof.Having undergone copying, gauging, thickening, and choosing heights, wX≤5before coning. Cellulation\nensures that new Xstabilizers ∂(i)\n0have a maximum weight of four. Thus we only need to check how much the weight\nof the old Xstabilizers increases under coning, i.e., the bottom row of ˜HXin Eq. (22):\u0010\nf(1)\n0···f(m)\n0HX\u0011\n. Having\nundergone copying, gauging, thickening, and choosing heights, the Xstabilizers overlap with the Zstabilizers we\nreduce either zero or two times (Eq. (12)), contributing either none or one element to Xi, respectively. So f(i)\n0has a\nmaximum row weight of one by construction.\nRecall that we only reduce the weight of Zstabilizers from the rows\u0000HZ⊗Iℓ0\u0001\nin Eq. (12) where only one row\nis kept per ℓrows. The support of two Zstabilizers are disjoint if they are associated with different heights. This\nrow overlaps the support of the Xstabilizers in the HX⊗Iℓblock of the thickening stabilizers, which has wX≤3\ndue to gauging. If an Xstabilizer overlaps with one of the Zstabilizers we reduce, then at least two qubits of its\nsupport are within the set of qubits associated with the height of that Zstabilizer. That leaves a maximum of one\nqubit that could be in any other height. By commutativity, it therefore cannot overlap with any Zstabilizer from\nanother height. Thus, an Xstabilizer can only get added weight from f(i)\n0within exactly one height.\nA. Reducing The Overhead With Improved Copying\nWith the above steps using l= 3andheights = (2 ,1,2,1,2,3,1,3,3,1), the QRM(4) stabilizers Eq. (18) produce\na[ [724,1,3] ]code. This is a significant increase in resources to protect a single logical qubit. A small modification to\nHastings’s method can make a large difference with respect to this problem. Looking at the Tanner graph in Fig. 1b,\nany column whose weight is not equal to qXends up with unused qubits. The simplest and most natural modification\nto make is to only copy a qubit as many times as its column weight, as in Fig. 6b. Applying this to the Reed-Muller\ncode produces a [ [512,1,2] ]code. The reduction of resources propagated through the full procedure is even more\ndramatic when qXis much larger than the median column weight.\nNote that copying results in qX= 3, but some of the new qubits are only in the support of two Xstabilizers for\nboth the original and the modified copying procedures. We can choose to only expand enough so that all new qubits\nare in the support of exactly targqXXstabilizers for some target column weight targqX≥3. Referring to Fig. 1b,\nobserve that vi,1andvi,qXcan accept targqX−1edges instead of targqX−2, reducing the number of qubits needed\nfor copying. Applying this to Fig. 1b with targqX= 3produces Fig. 6c. Applying this to the Reed-Muller code with\ntargqX= 3produces a [ [315,1,2] ]code.18\nZi\n(a) We wish to reduce the weight of the stabilizer Zivia coning. The shaded/dashed portion of the Tanner graph will remain\nunchanged, and for clarity, we omit it in the following part (b).\nv1v2v3v4v5HX|supp( Zi)\n/parenleftBig\n∂(i)\n1/parenrightBigT\n/parenleftBig\nf(i)\n1/parenrightBigTf(i)\n0\n(b) Qubits v1through v5are new. The number of new qubits is always exactly equal to the number of Xstabilizers that share\nsome support with Zi, and the number of Zstabilizers replacing Ziis always exactly the number of qubits in the support of Zi.\nThe map f(i)\nialways has this form, and ∂(i)\n1is always the same structure as HX|supp( Zi), i.e.,\u0010\n∂(i)\n1\u0011T\nlooks like HX|supp( Zi)\nwith variables and checks swapped. Note that this created a new X-logical operator on the qubits v1, v2, v3, v4, v5. This is why\nwe need ∂(i)\n0.\nv1\nv2\nv3v4v5/parenleftBig\n∂(i)\n1/parenrightBigT\n(c) Redrawing just\u0010\n∂(i)\n1\u0011T\nonitsown, wecanclearlysee\nthecyclethatcausesthenew\nlogical operator.\nv1\nv2\nv3v4v5/parenleftBig\n∂(i)\n1/parenrightBigT\n∂(i)\n0(d) We add the logical oper-\natortothe Xstabilizers, cre-\nating the structure of ∂(i)\n0.\nv1\nv2\nv3v4v5\nv6/parenleftBig\n∂(i)\n1/parenrightBigT\n∂(i)\n0(e) The weight of the new X\nstabilizer is high, so we cel-\nlulate which requires an ad-\nditional qubit v6.\nv1v2v3v4v5 v6\n/parenleftBig\nf(i)\n1/parenrightBigTf(i)\n0\n/parenleftBig\n∂(i)\n1/parenrightBigT∂(i)\n0\n(f) Putting the entire graph back together, we have the completed result of coning.\nFigure 5. Illustration of coning on the Tanner graph.19\nv1 v2 v3\n(a)\n (b)\n (c)\nFigure 6. (a) The original Tanner graph. (b) Applying reduced copying to (a). (c) Applying targeted copying to (a) with\ntargqX= 3. See Fig. 1 for the original copying procedure applied to (a).\n(a)\n (b)\n (c)\n(d)\nFigure 7. (a) A 4-cycle in X. Under copying, it could be expanded to an 8-cycle (b) or a 12-cycle (c) depending on choices\nfor the connections of the original Xstabilizers. (d) The three new 4-cycles in Zfrom the two Zstabilizers which shared the\nsame variable node.\nThe improved copying methods just introduced do not decrease the distance. The drop in distance from the original\n[ [15,1,3] ]code inthe previous examplesis due totheother steps ofquantumweightreduction. The codesafter copying,\nreduced copying, and targeted copying have parameters [ [60,1,7] ],[ [32,1,4] ], and [ [16,1,3] ], respectively.\nB. Quantum Weight Reduction Reinterpreted\nAll three copying variants, Hastings’s original (Fig. 1b), the reduced copying (Fig. 6b), and targeted copying\n(Fig. 6c) may be viewed in the same framework. With the hindsight of coning, it is clear that Eq. (8) is in the form\nof a mapping cone where Hℓis given by Eq. (11):\n˜HX=\u0012\nH(r)\nXfT\n1\nHℓ\u0013\n,˜HZ=\u0010\nH(r)\nZf2\u0011\n. (26)\nProceeding one column at a time, remove the column of weight qito be reduced to get H(r)\nXandH(r)\nZ. Without all\ncolumns, these may not commute and therefore cannot form the standard chain complex. Instead, we have\nC1 C2\nC0 Fℓ\n2 Fℓ−1\n2f2 HT\nℓf1\u0010\nH(r)\nX\u0011T\nH(r)\nZ.\nIn order to be a valid chain complex, we need f2HT\nℓ= 0, and in order to be a valid chain map, we need f2f1=\nH(r)\nZ\u0010\nH(r)\nX\u0011T\n. The number of solutions for f2is the size of the left kernel of\u0000\nf1HT\nℓ\u0001\n. Viewed as a matrix, each20\ncolumn of f1must have weight zero or one, and the columns should have support exactly corresponding to the row-\nsupport of the deleted column of HX. Choosing ℓ=qXuniformly for all variables and limiting the row weight of f1\nto one gives Hastings’ copying. Choosing ℓ=qiseparately for each variable and limiting the row weight of f1to one\nresults in reduced copying. Choosing ℓ=qi−2and letting the first and last rows of f1have weight two with other\nrows having weight one results in targeted copying with targqX= 3.\nGauging removes rows of HX, preserving the commutativity between the stabilizers and can therefore form the\nstandard chain complex. Removing a row of weight wifrom HXto obtain H(r)\nX, the associated complex is\nC2 C1 C0\nFℓ−1\n2 Fℓ\n2HT\nZH(r)\nX\nHT\nℓf2 f1\nwith ℓ=wi−2, the columns of f1corresponding to the support of the deleted Xstabilizer having weight one,\nother columns having weight zero, first and last rows having weight two, and all other rows having weight one. The\nstabilizers are (compare to Example 2)\n˜HX=\u0012\nH(r)\nX\nf1HT\nℓ\u0013\n,˜HZ=\u0000\nHZfT\n2\u0001\nandHT\nℓf2=f1HT\nZ. There is a unique solution for f2since HT\nℓhas a left inverse.\nQuantum weight reduction may therefore be seen as two cones, a tensor product of chain complexes, and then\nanother cone.\nC. Effects On Iterative Decoding\nDespite its advantages, quantum weight reduction generally alters the underlying structure of the original code. In\nthe absence of any obvious structure in the output, the available decoders are those applicable to wide ranges of codes,\nsuch as belief propagation. These iterative algorithms are highly sensitive to the topology of its Tanner graph [52].\nThe fact that stabilizers must commute forces all stabilizer codes to have 4-cycles. For CSS codes, these unavoidable\ncycles are between the XandZstabilizers. If X-Zcorrelationsare ignoredand thetwotypesof stabilizersare decoded\nindependently, it is beneficial to reduce the number of short cycles from only HXand only HZ. Unfortunately, the\nweight reduction procedure above introduces a significant amount of new 4-cycles.\nSince the Tanner graph of the repetition code (and its transpose) has no cycles, expanding the variable nodes in\ncopying does not introduce any new cycles in the Xstabilizers. However, the degree of freedom present is assigning\na qubit to one of its copies in the repetition code can have important consequences. Consider the 4-cycle in Fig. 7a.\nEither Fig. 7b or Fig. 7c are valid choices, but one leads to an 8-cycle and the other to a 12-cycle. If the cycle structure\nof the input is known, selectively assigning edges can be used to lengthen short cycles of the original graph. On the\nother hand, an identical copy of the Zstabilizers is imposed on every variable node in the repetition code, leading to\nmany new 4-cycles; see Fig. 7d. A simple counting argument shows the following.\nLemma 8. Splitting a variable node into svariable nodes during copying introduces\u0000s\n2\u0001\u0000c\n2\u0001\nnew 4-cycles in the Z-only\nTanner graph and c(s−1)new 4-cycles between XandZ, where cis the number of Zcheck nodes connected to the\noriginal variable node.\nNote that this lemma applies to both the original and modified copying procedures. Applying this to the 15-qubit\nReed-Muller code Eq. (18), Hastings’s copying produces 738Z+ 168 X-Z= 906new 4-cycles; the reduced copying\nproduces 468Z+ 96X-Z= 564new 4-cycles; and the targeted copying with targqX= 3produces 45Z+ 10X-Z= 55\nnew 4-cycles.\nSimilar to copying, gauging only modifies the length of the Xcycles, while new 4-cycles are created in both Zand\nbetween XandZ. Although the number of each is unpredictable due to the nature of the new Zstabilizers, it is\noften significant due to the density of the solution.\nThe terms HX⊗Iℓ,HT\nX⊗Iℓ−1, and HZ⊗Iℓin thickening make copies of each XorZcycle present up to this\npoint in the procedure. Additionally, ˜HZcontains both HXandHZ, converting cycles between XandZto purely Z\ncycles. These new cycles are connected through a copy of the repetition code in In⊗Hℓand are therefore potentially\nlengthened. (See Fig. 3c.) Choosing heights removes some of the cycles coming from the HZ⊗Iℓterm.21\nEvery edge of the graph Qi→ X igives an X-Z4-cycle in the resulting code, but exactly half that number were\nremoved by deleting Zi. Every multi-edge in Qi→ X iresults in a Z4-cycle. Any two cycle basis elements that share\ntwo edges in Xi→ R iresults in an X4-cycle. Cellulation results in new X-Z4-cycles. If thickening is performed\nhere to reduce qXafter coning, then the results of the previous paragraph should be applied to these results.\nThe number of new short cycles introduced by quantum weight reduction severely degrades the performance of\niterative decoding schemes based on the Tanner graph, requiring cycle mitigation techniques or expensive post-\nprocessing such as ordered-statistics decoding (OSD).\nIV. CLASSICAL WEIGHT REDUCTION\nA natural question is how the above quantum weight reduction method (and our proposed modifications) compares\nto simply reducing the column and row weights of the classical codes prior to their use in constructing a quantum code.\nWe restrict ourselves to some well-known code constructions that produce valid stabilizers regardless of the input—\nsuch as the hypergraph and lifted product codes, Eqs. (1) and (4), respectively—whose row and column weights are\ncompletely determined by the classical inputs. This technique can also be used for other constructions, but we leave\nthis to future work.\nIt is not common to consider weight reduction in classical coding theory since there is no equivalent hardware\nrequirement and there are numerous methods to construct excellent LDPC codes for efficient decoding applications.\nNevertheless, we draw inspiration from the quantum case to define a procedure which is simple and maintains or\nincreases the (classical) minimum distance. The result is similar to the copying/gauging steps described above but\nwith a few minor changes that have important consequences. For clarity, we refer to Hastings’s work as quantum\nweight reduction and this method as classical weight reduction.\nSimilar to the previous section, let Hbe a parity-check matrix of an [n, k, d ]linear code with rows {hi}, let\nwi= wt( hi)be the weight of the ith row, and let qibe the weight of the ith column. Consider a row hiwith wi>3,\nand assume, as before, that the support of hihas been permuted to the first wibits. Then to weight reduce, add\nwi−1new columns to Hand replace hiwith the matrix\u0000\nIwi0HT\nwi−1\u0001\n, where Iwiis the wi×wiidentity matrix, 0\nrepresents the rest of the original columns not in the support of hi, and HT\nwiis the transpose of Eq. (11). In matrix\nform,\n\u0000v1···vw\n1···10···0\u0001\n7→\nv1v2···vw−1vw v′\n1v′\n2···v′\nw−2v′\nw−1\n1 ···1\n1 ···11\n... ···...\n1 ··· 1 1\n1··· 1\n. (27)\nRepeat this for all rows which need to be reduced. To reduce the columns, simply transpose and use the same method.\nThis is summarized in Algorithm 1. Note that the row-wise sum of\u0000\nIwi0HT\nwi−1\u0001\nis just\u0000hi0\u0001\n.\nThere is only a slight difference between Eqs. (10) and (27) (gauging) and Figs. 2b and 8c. This change increases\nthe (classical) minimum distance of the reduced code at the cost of decreased encoding rate.\nTheorem 9. LetHbe the parity-check matrix of an [n, k, d ]binary, linear code, wHbe the maximum row weight of H,\nandqHbe the maximum column weight of H. Then Algorithm 1 outputs a parity-check matrix ˜Hwithw˜H=ρ˜H= 3\nwhose code has parameters [O(nρ), k,˜d], where ρ= max {wH, qH}and˜d≥d. Additionally,\n(a) if all rows of Hhave weight greater than three, then ˜d≥3d/2;\n(b) if all columns of Hhave weight greater than three, then ˜d≥dminiqi;\n(c) if all rows and all columns of Hhave weight greater than three, then ˜d≥(3dminqi)/2.\nProof.The claims about wH,qH, and the length follow from Eq. (27). Weight reducing a row does not change column\nweights and vice versa, so implementing one does not undermine the other. Reducing a row or a column appends the\nsame number of linearly independent rows as columns, so the rank is always preserved.22\nLetHbe a parity-check matrix with rows {h1, . . . , h n−k}. Reducing row hiproduces the parity-check matrix\n\nh1\n... 0\nhi−1\nfHT\nwi\nhi+1\n... 0\nhn−k\n, (28)\nwhere wi=|supp hi|is the weight of row iandf\f\f\nsupp hi=Iwiis the wi×wiidentity. Denoting the columns to the\nleft of the vertical line by oldand those to the right by new,f\f\f\nold\\supp hi= 0. Ifcis a codeword of Eq. (28), then\nc\f\f\noldsatisfies the checks hjforj̸=iand\u0000\nf HT\nwi−1\u0001\nc= 0. Then\n0 =\u0000\nf HT\nwi−1\u0001\nc=\u0000\nf HT\nwi−1\u0001\u0012c\f\f\nold\nc\f\f\nnew\u0013\n=fc\f\f\nold+HT\nwi−1c\f\f\nnew\ngives HT\nwi−1c\f\f\nnew=fc\f\f\nold=c\f\f\nsupp hi. The image of HT\nwi−1are even-weight vectors, so c\f\f\nsupp hihas even weight and\ntherefore c\f\f\noldsatisfies hi. Hence, c\f\f\noldis a codeword of the original code.\nNow suppose cis a codeword of Eq. (28) such that c\f\f\nold= 0. Then HT\nwi−1c\f\f\nnew= 0. But kerHT\nwi−1= 0soc\f\f\nnew= 0.\nThere is therefore a one-to-one correspondence between codewords of the original code and codewords of Eq. (28)\n(˜k=k). It also follows that the minimum-weight codeword of Eq. (28) is at least as large as the old code ( ˜d≥d).\nTo prove (a), assume wi>3for all 1≤i≤n−kand weight reduce all rows. A codeword chaswt\u0000\nc\f\f\nold\u0001\n≥dand\nfc\f\f\noldflips at least dsyndromes. A single one in c\f\f\nnewflips two syndromes, so at least d/2ones are required to make\nccommute: wt(c) = wt\u0000\nc\f\f\nold\u0001\n+ wt\u0000\nc\f\f\nnew\u0001\n≥d+d/2 = 3 d/2.\nTo prove (b), observe that any bit of the original code that is supported on a column that is expanded must now\nhave support on the full repetition code in the expanded section, as this is the only way to commute with the newly\nadded rows. This increases the weight of any codeword supported on this bit; however, not all minimum weight\ncodewords may have support here, so the minimum distance may not increase. If all columns are expanded, the\ndistance increases by at least a multiplicative factor of the smallest column weight.\nTo prove (c), combine (a) and (b).\nNote that nowhere in the proof did we assume that His full rank, and therefore the proof holds for parity-check\nmatrices with redundant checks.\nRemark: The proof of Theorem 9 shows that weight reduction affects the generator and parity-check matrices of the\ncode asymmetrically. This has implications for properties based on duality. In particular, for a linear code with parity-\ncheck matrix Hand whose (Eulidean) orthogonal space is determined by the row space of G, the linear codes based on\n˜Gand ˜Hare no longer dual. This rules out using classical weight reduction in the CSS construction of stabilizer codes.\nSuppose that we instead perform weight reduction using Eq. (10) (gauging) rather than Eq. (27). Comparing the\ntwo forms of f, we will refer to this as compressed classical weight reduction since it requires the addition of fewer\nadditional bits. We still have ˜d≥dbut now part (a) of the proof does not hold (consider a codeword with overlap of\ntwo with the check being reduced such that the overlap falls within a single row of the expansion f, and we see that\na generalized proof of part (a) would fail), and part (b) holds but with a different constant, ˜d≥dmini(qi−2). In\ngeneral, there are many possible choices of row expansions: Any choice of fwith weight one in columns corresponding\nto the support of the check being reduce, and weight zero columns elsewhere is a valid expansion, though some choices\nmay not result in effective weight reduction. Further increasing the number of columns of HT\nwi−1induces a small,\nadditive constant in the distance but at the expense of a nearly linear increase in n. A row hican be expanded to\u0000\nf HT\nri−1\u0001\nwhere fis any ri×nmatrix with column weight one on bits of the support of hiand column weight zero\nelsewhere. In this case, if fhas any row with weight greater than one, then part (a) of Theorem 9 can’t be generalized\nto this choice of f, and instead the guarantee is that reduction will never reduce the minimum distance. Applying this\nto the transpose results in column expansion, and if all columns are expanded, then the minimum distance further\nincreases by a factor of at least the minimum size of all column expansions.\nWe note that classical weight reduction was previously presented in Appendix A of Ref. [37], using the language\nof cellular homology. As with quantum weight reduction, we believe that our presentation is complementary and23\nAlgorithm 1: Classical Weight Reduction\nInput: H\nOutput: ˜H\n1fori←1tonumber of rows of Hdo\n2 hi←ith row of H\n3ifwt(hi)>3then\n4ifcompressed then\n5 f←\n1 1\n1\n...\n1\n1 1\nonsupp( hi)\n6 ℓ←wt(hi)−3\n7else\n8 f←identity matrix on supp( hi)\n9 ℓ←wt(hi)−1\n10 end\n11 ifpermutethen\n// randomly permute the nonzero columns of f\n12 end\n13 if˜Hexiststhen\n14 ˜H←\u0012˜H 0\nf0HT\nℓ\u0013\n15 else\n16 ˜H←\u0000\nf HT\nℓ\u0001\n17 end\n18else\n19 if˜Hexiststhen\n20 ˜H←\u0012˜H\nhi0\u0013\n21 else\n22 ˜H←hi\n23 end\n24end\n25end\n// Apply the above to ˜HTand transpose back\n26return ˜H\nremark that Theorem 9 also gives a tighter bound on the distance of the weight reduced code. In addition, the\nindependent row and column permutations described in Section IVB are novel and can give significant improvement\nin the distances of the weight-reduced codes; see Table I.\nA. Decoding\nUnlike quantum weight reduction, classical weight reduction is able to easily map back onto the original code.\nInspired by the proof of Theorem 9, we can use this to map the decoding problem of the weight-reduced code back to\nthe decoding problem of the original code. To see how, suppose we are using the weight-reduced parity-check matrix\nto decode. An error on the bits corresponding to weight-reduced columns can be uniquely identified and corrected\nusing the rows corresponding to the HT\nℓblocks. Collapsing these columns into a single bit puts the matrix into the\nform: “original bits” followed by “new bits”. The original bits are protected by the original parity-check matrix and\nthe original decoder may be used. Errors on the new bits may be corrected using the HT\nℓblocks. This may or may\nnot be beneficial depending on the original decoder. Message-passing based decoders may perform better on the\nweight-reduced code than the original.\nGraphically, a high-degree node (Fig. 8a) is replaced by Fig. 8c. This has no cycles and lengthens any cycles the24\nv1 v2 v3 vw−2vw−1vw· · ·\n(a)\nv1 v2 v3 vw−2vw−1vwv′\n1 v′\n2v′\nw−4v′\nw−3\n· · ·· · · (b)\nv1 v2 vwv′\n1 v′\n2v′\nw−1\n· · ·· · · (c)\nFigure 8. (a) The input Tanner graph. (b) The compressed classical weight reduction of the check node in (a). (c) The classical\nweight reduction of the check node in (a). (a) and (b) are the same as Figs. 2a and 2b, respectively.\nreplaced node was previously a part of. The exact increase in cycle length depends on the ordering of the initial\nedges. The length-2 path from vw−1tovwbecomes a length-4 path through vw−1,v′\nw−1, and vw, while the length-2\npath from v1tovwbecomes the length- (2w+ 1)path through v1,v′\n1,v′\n2,···,v′\nw−1, and vw. If the cycle structure of\nthe input code is known, permutations of the check sides of the original edges in the weight reduced Tanner graph\ncan be used to selectively increase harmful short cycles. These permutations are the graphical representation of the\npermutations described in Section IVB. We have observed large increases in girth and improvements in iterative\ndecoder performance using this approach.\nThe above has effects on iterative decoding. Recall the unique-neighbor property of expander codes8: Consider a\n(ℓ, r, ε, ℓ/ 2)-left expander with vertex bipartition L∪R, (for all sets S⊂Lof size no more than ε|L|,|N(S)| ≥ℓ/2|S|,\nwhere N(S)is the neighborhood of S), then for these sets ∃y∈Rsuch that |N(y)∩S|= 1, (there exists a unique\nneighbor). Now suppose Sis a trapping set. Then unless there are many such y’s, there is not going to be a large\namount of extrinsic information flowing into this set of nodes to help overcome the trapping set. A cycle with a lot\nof extrinsic information flowing into it will be able to overcome what would otherwise become a trapping set. In this\nsense, cycle connectivity is more important than cycle length for decoder performance.\nOn one hand, weight reduction creates a potentially large number of low degree nodes, which typically strongly\ndegrade the performance of iterative decoders. On the other hand, any given cycle has the same number of extrinsic\nconnections to the rest of the graph as in the original graph since weight reduction introduces no new cycles. What\nhas changed is the ratio of extrinsic connections to the cycle length. Due to their low degree, certain patterns in the\nvariable nodes introduced by the HT\nℓblock will cause the entire repetition code pattern to flip to errors under belief\npropagation. This can be overcome if the other variable nodes receive a high amount of incoming extrinsic information\n(have high degree) but becomes significantly more difficult if the columns are also weight reduced. However, the more\nrows and columns that are reduced, the more new vertices are created in the Tanner graph, the longer the cycles, and\ntherefore probability that theses structured patterns of errors occur decreases. Hence, weight reduction introduces a\nsignificant amount of trapping sets but they become larger and therefore less harmful. In this sense, weight reduction\nintroduces uneven error protection in the variable nodes.\nSee [52, 54] for further discussion of the concepts in this subsection.\nB. Permutations\nPermutations of the input are also important for reasons besides selectively increasing the girth. Consider an [n, k, d ]\ncode with parity-check matrix H. Permuting the columns of Hproduces an equivalent code with the same parameters\nand corresponds to a relabeling of the Tanner graph without any effect on cycle structure. However, weight reducing\nthe original code and the permuted code produce nonequivalent codes with potentially different distances.\nExample 10. Weight reducing\n\u0012\n1 1 1 1 0 0\n0 0 1 1 1 1\u0013\nand its permutation\n\u0012\n1 0 1 0 1 1\n0 1 1 1 1 0\u0013\n8See [53] for a nice, low-level description of expander codes and how the unique-neighbor property of expander graphs plays a role in\nbounding the distance of the code.25\nC(H)HGP( H)R HGP( ˜H) R HGP( ˜H(c)) R\n[6,3,3][ [45,9,3] ]0.200 [ [117,9,4] ]0.077 [ [65,9,4] ]0.138\n[7,2,4][ [74,4,4] ]0.054 [ [164,4,4] ]0.024 [ [100,4,4] ]0.040\n[7,3,4][ [65,9,4] ]0.138 [ [149,9,5] ]0.060 [ [89,9,4] ]0.101\n[7,4,3][ [58,16,3] ]0.276 [ [400,16,6→7] ]0.040 [ [136,16,3→4] ]0.118\n[8,2,5][ [100,4,5] ]0.040 [ [394,4,8] ]0.010 [ [202,4,6] ]0.020\n[8,3,4][ [89,9,4] ]0.101 [ [317,9,6→7] ]0.028 [ [149,9,4→5] ]0.060\n[8,4,4][ [80,16,4] ]0.200 [ [656,16,7→10] ]0.024 [ [208,16,4→6] ]0.077\n[9,2,6][ [130,4,6] ]0.031 [ [514,4,10] ]0.008 [ [290,4,8] ]0.014\n[9,3,4][ [117,9,4] ]0.077 [ [369,9,6→7] ]0.024 [ [185,9,4→5] ]0.049\n[9,4,4][ [106,16,4] ]0.151 [ [730,16,7→10] ]0.022 [ [250,16,4→6] ]0.064\n[9,5,3][ [97,25,3] ]0.258 [ [1313 ,25,7→11] ]0.019 [ [493,25,4→8] ]0.051\n[10,2,6][ [164,4,6] ]0.024 [ [650,4,10] ]0.006 [ [394,4,8] ]0.010\n[10,3,5][ [149,9,5] ]0.060 [ [929,9,10] ]0.010 [ [369,9,7] ]0.024\n[10,4,4][ [136,16,4] ]0.118 [ [458,16,6→7] ]0.035 [ [250,16,4→5] ]0.064\n[10,5,4][ [125,25,4] ]0.200 [ [937,25,7→9] ]0.027 [ [377,25,4→6] ]0.066\n[10,6,3][ [116,36,3] ]0.310 [ [1586 ,36,7→10] ]0.023 [ [666,36,4→8] ]0.054\nTable I. Classical weight reduction applied to hypergraph product codes with the best known linear codes of small size as\ninput. C(H)is the linear code with parity-check matrix Hobtained from GAP (see main text). For each hypergraph product\ncode, we give its encoding rate R=k/n. For each weight-reduction method, we apply the relevant algorithm 10,000 times\nusing different permutations of the input parity-check matrix. In cases where permutations improved the distance, we use the\nnotation d1→d2, where d1indicates the distance without permutations, and d2indicates the highest obtained distance.\nproduces\nH=\n1 1\n1 1 1\n1 1 1\n1 1\n1 1\n1 1 1\n1 1 1\n1 1\nand H′=\n1 1\n1 1 1\n1 1 1\n1 1\n1 1\n1 1 1\n1 1 1\n1 1\n,\nrespectively. The linear code given by Hhas parameters [12,4,3]butH′is[12,4,4].\nMore generally, we can use the replacement\u0000\nΠ 0 HT\nℓ\u0001\n, where Πis any permutation of the identity. Different\npermutations can be used for reducing each row and column independently. We use this in the next section, where\nthe positive effect on distance can be seen in Tables I, IV, V and VI. Quantum weight reduction should be equally\nsusceptible to permutations; we leave this to future work.\nV. EXAMPLES AND NUMERICAL RESULTS\nIn this section, we present examples of codes constructed using classical and quantum weight reduction. We\nfocus on two classes of QEC codes: hypergraph product codes and lifted product codes (see Section II for the\nrelevant background) for two reasons. First, both classical and quantum weight reduction are applicable to these code\nclasses, so we can use them to compare the two weight reduction methods directly. Second, both of these classes\ncontain code families with constant encoding rate [23, 39], and almost linear minimum distance in the case of lifted\nproduct codes [39], and therefore present compelling alternatives to code families with limited parameters such as\ntwo-dimensional topological codes [55].26\nC(H) HGP( H) R HGP( ˜H) R HGP( ˜H(c)) R\n[16,4,6][ [400,16,6] ]0.040 [ [5008 ,16,18→23] ]0.003 [ [1360 ,16,10→13] ]0.012\n[20,5,8][ [625,25,8] ]0.040 [ [7825 ,25,23→29] ]0.003 [ [2125 ,25,12→15] ]0.012\n[24,6,10][ [900,36,10] ]0.040 [ [11268 ,36,29→33] ]0.003 [ [3060 ,36,17→18] ]0.012\nTable II. Classical weight reduction applied to the family of hypergraph product codes from [57, Table 1]. For each hypergraph\nproduct code, we give its encoding rate R=k/n. For each weight-reduction method, we apply the relevant algorithm 10,000\ntimes using different permutations of the input parity-check matrix. We use the notation d1→d2, where d1indicates the\ndistance without permutations, and d2indicates the highest obtained distance.\nWe first consider hypergraph product codes. Recall that the distance of a hypergraph product code HGP( H1, H2)\nisd= min( d1, d2, dT\n1, dT\n2). IfHiis full rank, then C\u0000\nHT\ni\u0001\nhask= 0. In this case, dT\niis defined to be infinite. By\nTheorem 9, weight reduction does not decrease the code distance of the (classical) input codes and therefore we can\nconclude that ˜d≥d, where ˜dis the distance of HGP( ˜H1,˜H2). We specialize to the case of square hypergraph product\ncodes, HGP( H) = HGP( H, H ), where His the parity-check matrix of a linear code. As above, we use ˜Hto denote\nthe matrix produced from Hby classical weight reduction and introduce ˜H(c)to denote the matrix produced from H\nby compressed classical weight reduction. We use ]HGP( H)to denote the code produced from HGP( H)by applying\nquantum weight reduction.\nTo compare the weight reduction methods, we consider small parity-check matrices returned by the GAP function\nBestKnownLinearCode (from the package GUAVA [56]) as inputs to the hypergraph product. The output matrices of\nclassical weight reduction have row and column weights upper bounded by three, and by using these as inputs we are\nable to construct weight-reduced hypergraph product codes with ( ˜wX,˜qX,˜wZ,˜qZ) = (6 ,3,6,3). The weight-reduced\ncodes produced here often have higher distance than their inputs, although at a reduced encoding rate; see Table I\n(and Tables V and VI in Appendix B1). We find that compressed classical weight reduction gives us codes with\nimproved encoding rate but often worse distance, as is to be expected from the analysis in Section IV. We additionally\nfind that independent row and column permutations (see Section IVB) can dramatically improve the distance of the\nweight-reduced codes. We demonstrate this in Table I for the hypergraph product codes where the number to the left\nof an arrow is the distance in the unpermuted case and the number to the right is the highest distance found amongst\n10,000codes constructed with independent row/column permutation. We apply the same methodology to the family\nof(wX, qX, wZ, qZ) = (7 ,4,7,4)hypergraph product codes from [57, Table 1] in Table II.\nFor quantum weight reduction we achieve parameters of ( ˜wX,˜qX,˜wZ,˜qZ) = (6 ,6,6,3)but at the cost of a pro-\nhibitively large increase in the number of physical qubits; see Table III. For example, consider the code HGP( H)\nwhere His the parity-check matrix of the [6,3,3]code from Table I. This is a [ [45,9,3] ]code with (wX, qX, wZ, qZ) =\n(7,4,7,4). The best code we found by applying quantum weight reduction after approximately 100random cycle bases\nhas parameters [ [2892 ,9,5] ]and( ˜wX,˜qX,˜wZ,˜qZ) = (6 ,6,6,3). We did not utilize a second round of thickening and\nchoosing heights to reduce ˜qX, as the increase in nwas already excessively large. Compare this with the code with\nparameters [ [117,9,4] ]and( ˜wX,˜qX,˜wZ,˜qZ) = (6 ,3,6,3)produced by classical weight reduction. These disparities\ncombined with the expected degradation of iterative decoding performance in the case of quantum weight reduction\n(see Section IIIC) lead us to conclude that classical weight reduction is a superior technique for weight reducing\nhypergraph product codes in the regime of interest.\nC(H)HGP( H)R(wX, qX, wZ, qZ)]HGP( H) R ( ˜wX,˜qX,˜wZ,˜qZ)\n[6,3,3][ [45,9,3] ]0.200 (7, 4, 7, 4) [ [2892 ,9,5] ]0.003 (6, 6, 6, 3)\n[7,2,4][ [74,4,4] ]0.054 (7, 4, 7, 4) [ [7466 ,4,6] ]0.0005 (6, 6, 8, 3)\n[7,3,4][ [65,9,4] ]0.138 (7, 4, 7, 4) [ [6844 ,9,5] ]0.001 (6, 6, 8, 3)\n[7,4,3][ [58,16,3] ]0.276 (7, 4, 7, 4) [ [5085 ,16,3] ]0.003 (6, 6, 8, 3)\nTable III. Quantum weight reduction applied to small hypergraph product codes. For each code we ran the algorithm\napproximately 100times with random cellulations and kept the output with the lowest ( ˜wX,˜qX,˜wZ,˜qZ).27\nWe also consider the more efficient lifted product construction. Classical weight reduction is applicable to quasi-\ncyclic lifted product codes with various inputs including group algebras and polynomial rings, e.g.,\n\u0000g1(x). . .gw(x)0···0\u0001\n7→\ng1(x) ···1\ng2(x) ···11\n... ···...\ngw−1(x) ··· 11\ngw(x)··· 1\n, (29)\nwhere 1is the constant polynomial. Although this reduces the row weight, the final weight is still determined by the\nnumber of coefficients in each polynomial and could be larger than our target of three.\nWe restrict our attention to codes of the form LP(A) = LP( A, AT)and use base matrices that have previ-\nously appeared in the literature, which are given in Appendix B2. All of our weight-reduced lifted codes have\n( ˜wX,˜qX,˜wZ,˜qZ) = (6 ,3,6,3). As with hypergraph product codes, we find that the weight-reduced codes have higher\ndistance but lower encoding rate than the input codes; see Table IV. We again observe that independent row and\ncolumn permutations improve the distances of the weight-reduced codes, though we only construct ten codes for each\nbase matrix as the distance calculations are more time-consuming in this case. The upper bounds on the distances\nare computed using the BP+OSD method described in [16]. Note that the bounds for the codes produced with\n(uncompressed) classical weight reduction are likely loose.\nComparing our weight-reduced lifted product and hypergraph product codes, we observe that both families have\nsimilar encoding rates but the lifted product codes generally offer improved distances. To illustrate this, consider the\nquantity Rd2, which is equal to one for the rotated surface code [58]. For the hypergraph product codes, the highest\nvalue we obtain is Rd2= 6.4for the [ [1850 ,100,9] ]code in Table VI. In contrast, for the lifted product codes produced\nusing compressed classical weight reduction (where we are have more confidence in the distance estimates), we obtain\nvalues up to Rd2= 37.6for the [ [2635 ,43,≤48] ]code in Table IV.\nC(A) LP(A) R C(˜A) LP(˜A) R C(˜A(c)) LP(˜A(c)) R\n[52,27,6][[260,58,≤6]]0.223 [130,27,12→14][[2132 ,58,≤14]]0.027 [78,27,6→8] [[676,58,≤8]]0.086\n[28,9,10] [[175,19,≤10]]0.109 [91,9,28→33] [[2191 ,19,≤39]]0.009 [49,9,14→18] [[595,19,≤18]]0.032\n[36,11,12][[225,21,≤12]]0.093 [117,11,36→40][[2817 ,21,≤48]]0.007 [63,11,18→22] [[765,21,≤22]]0.027\n[68,19,18][[425,29,≤18]]0.068 [221,19,54→62][[5321 ,29,≤74]]0.005 [119,19,32→34][[1445 ,29,≤34]]0.020\n[76,21,20][[475,31,≤20]]0.065 [247,21,60→68][[5947 ,31,≤93]]0.005 [133,21,37→38][[1615 ,31,≤38]]0.019\n[124,33,24][[775,43,≤24]]0.055 [403,33,71→84][[9703 ,43,≤115]]0.004 [217,33,44→48][[2635 ,43,≤48]]0.016\nTable IV. Classical weight reduction applied to lifted product codes. The base matrices Aare given in Appendix B2 and C(A)\nis the quasi-cyclic code defined by A. For each weight reduced base matrix, we apply the relevant algorithm 10 times with\ndifferent permutations of the input and retain the matrix whose associated quasi-cyclic code has the highest minimum distance.\nFor each lifted product code we also provide the encoding rate R=k/n. We use the notation d1→d2, where d1indicates the\ndistance without permutations, and d2indicates the highest obtained distance.\nA. Numerical Simulations\nWe simulate the performance of our codes as a quantum memory using a noise model derived from Xanadu’s\nphotonic architecture based on GKP qubits [18, 19]. Specifically, we consider a cluster state formed by foliating\na QEC code [29, 30], which is realized using GKP qubits and passive linear optics. Concatenation of the GKP\ncode with discrete-variable QEC codes has previously been considered in [40, 59–65]. For a QEC code with weights\n(wX, qX, wZ, qZ),thedataandancillanodesofthecorrespondingclusterstatehavedegrees qX+2andwX,respectively,\nin primal layers and qZ+2andwZin dual layers. The foliated stabilizers have weight wX+2andwZ+2, although we\nemphasize that these are reconstructed from single-qubit measurement outcomes rather than being measured directly.\nWe consider 2dfoliation layers, where dis the distance of the QEC code.\nFollowing Ref. [19], once the cluster state is specified, the resource state construction involves:28\n1.Preparing GKP two-qubit cluster states. These can be constructed using two GKP sensor states, a\n50:50 beamsplitter, and a π/2phase shifter [19, 66]. Prior to the beamsplitter, each mode is assumed to\nbe a perfect GKP qubit up to a single-mode Gaussian blurring channel parameterized by a variance σ2[19].\nFor Gaussian blurring channels, it is convenient to express the variance of the Gaussian in terms of decibels,\nσ2\nσ2vac[dB] = −10 log10(σ2/σ2\nvac), where σ2\nvacis the variance of the vacuum, which is 1/2in units where ℏ= 1. This\nmodel is equivalent to uniform photon loss experienced throughout the cluster state generation and measurement\nprocess [19].\n2.Placing GKP pairs. Place one GKP cluster state pair for each edge of the cluster state graph. Now each\noriginal node in the cluster state is associated with one mode (half of a pair) per neighbor. This collection of\nmodes is referred to as a macronode.\n3.Measuring macronodes. To measure a given cluster state site in the Xbasis, a continuous-variable GHZ\nmeasurement is applied on each mode within the corresponding macronode. This can be achieved by sending\neach mode through a beamsplitter network, followed by measurement of momentum on a single mode, and\nposition on the rest. To measure in the Zbasis the process is the same except that all modes are measured in\nthe position basis. The Pauli error rate of these measurements increases monotonically with variance σ2, which\ncorresponds to decreasingσ2\nσ2vac[dB].\n4.Applying feedforward corrections. As described in Ref. [19], feedforward corrective displacement operators\nthat are conditioned on binned homodyne outcomes from a given macronode must be applied on all modes\nresiding in macronodes that are neighbors with respect to the original cluster state graph. These corrections\ncan be applied in post-processing (no physical displacement gates are required).\nWe utilize the binning strategy described in [19] as a soft-in-soft-out inner decoder and BP+OSD as a soft-in-hard-\nout outer decoder [39, 57]. We use the min-sum variant of BP with N/10iterations (where Nis the number of qubits\nin the foliated cluster state) and a flooding schedule. For OSD, we choose the combination sweep strategy with search\ndepth parameter λ= 60. We did not attempt to optimize the decoder to take the structure of our codes into account,\nrather we used it because of its applicability across the class of quantum LDPC codes [57].\nWe quantify the performance of our foliated codes using the logical error rate, which we estimate using Monte Carlo\nsimulations. We consider a logical error to have occurred if any logical qubit has an error after decoding. For ntot\ntrials and nfaillogical errors we compute the logical error rate according to\npfail=nfail+κ2/2\nntot+κ2, (30)\nwith error bars given by\nκs\npfail(1−pfail)\nntot+κ2, (31)\nwhere κis the desired quantile of a standard normal distribution [67]. We use κ= 1.96which corresponds to a 95%\nconfidence interval.\nWefirstfocusonasinglehypergraphproductcode: the [ [45,9,3] ]codefromTableIconstructedfromtheparity-check\nHof a[6,3,3]linear code. We compare the performance of HGP( H)with the performance of:\n•The[ [2892 ,9,5] ]code,^HGP (H), formed by applying quantum weight reduction to HGP( H).\n•The hypergraph product code HGP( ˜H)with parameters [ [117,9,4] ].\n•The hypergraph product code HGP( ˜H(c))with parameters [ [65,9,4] ].\nThe results are shown in Fig. 9a. We find that the codes obtained using classical weight reduction outperform the\noriginal code and the code obtained using quantum weight reduction. Even though the distance of the quantum\nweight-reduced code is the highest, the waterfall region (where the logical error rate starts to decrease) for this code\nbegins at much higher values of σ2/σ2\nvacwhen compared with the other codes. This is likely due to the increased row\nand columns weights of the parity-check matrices.\nWe also compare the performance of two codes from Table II with their (compressed) weight-reduced counterparts.\nThe weight-reduced codes again have superior performance, and we note that the waterfall region for the weight-\nreduced codes begins at σ2/σ2\nvac≈10.5dB compared with σ2/σ2\nvac≈11dB for the baseline codes, indicating that\nweight reduction not only improves logical error rates but also the break-even point.29\n9.5 10 10.5 11 11.5 1210−210−1100\nσ2/σ2\nvac[dB]pfail\nHGP( H) : [ [45 ,9,3] ]\nHGP( ˜H) : [ [117 ,9,4] ]\nHGP( ˜H(c)) : [ [65 ,9,4] ]\n]HGP( H) : [ [2892 ,9,5] ]\n(a)9.5 10 10.5 11 11.5 1210−310−210−1100\nσ2/σ2\nvac[dB]pfail\nHGP( H1) : [ [400 ,16,6] ]\nHGP( ˜H(c)\n1) : [ [1360 ,16,13] ]\nHGP( H2) : [ [625 ,25,8] ]\nHGP( ˜H(c)\n2) : [ [2125 ,25,16] ]\n(b)\nFigure 9. Simulation results for hypergraph product codes. Recall that larger values of σ2/σ2\nvac[dB] correspond to lower\neffective Pauli error rates. (a) The baseline HGP( H)where His the parity-check matrix of the [6,3,3]code from Table I.\nHGP( ˜H)andHGP( ˜H(c))are constructed via applying classical weight reduction to H.]HGP( H)is constructed by applying\nquantum weight reduction to HGP( H).HGP( ˜H)andHGP( ˜H(c))have superior performance compared to the baseline code,\nwhereas ]HGP( H)only becomes competitive for large values of σ2/σ2\nvac. (b) Comparison of the codes from rows one and two of\nTable II. The waterfall region for the weight-reduced codes starts at approximately 10.5dB compared to approximately 11dB\nfor the original codes.\nVI. DISCUSSION & CONCLUSION\nConstructingqLDPCcodeswithgoodperformanceunderrealisticnoisemodelsisanessentialsteptowardsdesigning\nmore efficient fault-tolerant quantum computing architectures. Since the noise associated with measuring a stabilizer\nscales with both the row and column weight of the stabilizer matrix in many proposed architectures, qLDPC codes\nwith lower weights are likely to have superior performance. Hastings provided a method to reduce the weights of\nCSS codes [20, 21], but its use has so far been restricted to the asymptotic setting. This work provides the first\napplication of quantum weight reduction to codes constructed with near-term hardware in mind. Our examples show\nthe method leads to a large increase in the number of physical qubits and that it is often difficult to find cellulations\nthat produce small weights. Additionaly, the Decongestion Lemma adds randomness to finding a cycle basis and the\noptimal choice of cellulation of large cycles is unclear. As a result, the majority of runs on inputs with weights already\nclose to the theoretical minimum values produced outputs with weights with equal or worse to the initial weights. An\nimplementation of the method is provided at [42].\nIn addition to examples, we provided an accessible review of the method with fewer mathematical prerequisites.\nExplicit examples accompany each step, including a completely diagrammatic description. Several statements made\nin [21] are clarified and subtleties are discussed. We also proposed modifications to reduce the overhead and analysed\nthe method’s effect on iterative decoding. Finally, we showed that three of the four steps of the method may be\nviewed in the same mathematical framework, without algebraic topology.\nIn response to some of the drawbacks of quantum weight reduction, we introduced classical and compressed classical\nweight reduction. Applying this technique to the inputs of the hypergraph product (for example) guarantees row\nweights of at most six and column weights at most three. We showed that classical weight reduction can increase\nthe distance of the classical code and hence the distance for the hypergraph product code. The compressed variant\nreduces the overhead while at least maintaining the distance of the input. The two approaches represent a trade-off\nbetween achieving the maximum reduction in overhead versus the maximum increase in distance. We also showed that\npermutations in weight reduction can increase both the distance and the girth of the Tanner graph. We emphasize\nthat classical weight reduction is applicable to quasi-cyclic codes defined by matrices with entries in a polynomial\nquotient ring, allowing us to construct examples of weight-reduced lifted product codes with superior parameters to\nour hypergraph product code examples.\nBoththe quantumand classicalweightreductions havethe sameinput andoutputcode dimensions, butthe classical30\n9.5 10 10.5 11 11.510−210−1100\nσ2/σ2\nvac[dB]pfail\nLP(A1) : [ [260 ,58,6] ]\nLP(˜A1) : [ [2132 ,58,14] ]\nLP(˜A1(c)) : [ [676 ,58,8] ]\n(a)9.5 10 10.5 11 11.510−310−210−1100\nσ2/σ2\nvac[dB]pfail\nLP(A2) : [ [175 ,19,10] ]\nLP(˜A(c)\n2) : [ [595 ,19,18] ]\nLP(A3) : [ [225 ,21,12] ]\nLP(˜A(c)\n3) : [ [765 ,21,22] ]\n(b)\nFigure 10. Simulation results for lifted product codes. (a) We compare the code LP(A1)with its weight reduced counterparts,\nLP(˜A1)andLP(˜A(c)\n1); see the first row of Table IV. We again observe improved performance for the weight-reduced codes, with\nthe uncompressed variant performing best. (b) Comparison of the hypergraph product codes from rows 2 and 3 of Table II. As\nin the hypergraph product case, the waterfall region begins at a smaller value of σ2/σ2\nvacfor the weight-reduced codes.\nmethod uses far fewer qubits than the quantum method. An analysis of the cycle structure of both approaches shows\nthat quantum weight reduction may strongly degrade the performance of iterative decoders, while the classical case\nmay actually improve it. We benchmarked the performance of our weight-reduced codes in a photonic quantum\ncomputing architecture based on GKP qubits and passive linear optics. We used Monte Carlo simulations to estimate\nthe logical error rate of the foliated cluster state corresponding to a logical identity channel (quantum memory). In\nevery case we simulated, we observed improved performance for the codes produced using classical weight reduction.\nThere is another approach to weight reduction that we have so far neglected, where the check weights of a QEC code\nare reduced by transforming the QEC code into a subsystem code [68–70] whose gauge operators are lower weight.\nThis method has been successfully applied to topological codes defined on Euclidean and hyperbolic tilings [71–79]. A\ngeneral method was proposed for qLPDC codes in Ref. [78], however the Tanner graph of the code must obey certain\nconditions and the method can dramatically reduce the code distance. We leave it to future work to compare this\nmethod with the results of Section V.\nIn future work, we plan to investigate whether we can improve the performance of our codes by using tailored\ndecoders take advantage of the structure introduced by weight reduction. We note that one could already use our\ncodes as a quantum memory in an architecture with a high-rate qLDPC memory and a surface code processor, as has\nbeen proposed recently for superconducting qubits [16] and neutral atom arrays [17]. However, we also plan to explore\ntechniques for performing logical operations (e.g. [80–84]) on our codes directly, as this could enable a fully qLDPC\narchitecture with large savings in the number of physical qubits required to execute useful quantum algorithms at\nscale.\nOur results illustrate that weight reduction techniques can be a useful tool for constructing useful qLDPC codes\nby transforming codes with good parameters but relatively high-weight checks into codes with low-weight checks,\ncomparable parameters, and improved performance. It may be preferable to construct stabilizer codes with very low\nrow/column weights and high encoding rate and distance directly, but the lack of known code constructions with these\nproperties suggest that this is a challenging task. We found that optimizing the classical inputs to quantum product\nconstructions is a superior strategy to optimizing the output quantum code itself, which is perhaps not surprising\ngiven that the orthogonality constraints that restrict quantum codes do not apply in the classical case. Here, we only\nscratched the surface of the space of possible quantum product codes that can be constructed with carefully designed\nclassical inputs, and we suspect that exceptional examples are waiting to be discovered.31\nAcknowledgements\nWe gratefully acknowledge work done by Priya Nadkarni on simulating the Xanadu architecture. We thank Rafael\nAlexander for feedback throughout the writing process. Computations were performed on the Niagara supercomputer\nat the SciNet HPC Consortium. SciNet is funded by Innovation, Science and Economic Development Canada; the\nDigital Research Alliance of Canada; the Ontario Research Fund: Research Excellence; and the University of Toronto.\nResearch at Perimeter Institute is supported in part by the Government of Canada through the Department of\nInnovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of\nColleges and Universities.32\nAppendix A: Review Of Homological Algebra\n1. The Tensor Product Of Chain Complexes\nLet\nA:··· − → Ai+1∂A\ni+1− − − → Ai∂A\ni− − →Ai−1− → ···\nB:··· − → Bi+1∂B\ni+1− − − → Bi∂B\ni− − →Bi−1− → ···\nbe chain complexes with vector spaces over the same field. Then A ⊗ Bis defined to have vector spaces (A ⊗ B )n=M\ni+j=nAi⊗Bjand maps\n∂A⊗B\nn =M\ni+j=n−1∂A\ni+1⊗IBj+IAi⊗∂B\nj+1, (A1)\nwhere Iis the identity map on the appropriate space.\nConcrete examples relevant to this work are the product of a 3-term and a 2-term chain complex and the product\nof two 3-term chain complexes. For the former, let\nA:A2∂A\n2− − →A1∂A\n1− − →A0\nB:B1∂B\n1− − →B0\nThe product A ⊗ Bis often drawn as\nA2⊗B1 A2⊗B0 A1⊗B0 A0⊗B0\nA1⊗B1 A0⊗B1∂A\n2⊗IB1IA2⊗∂B\n1 ∂A\n2⊗IB0\n∂A\n1⊗IB1IA0⊗∂B\n1∂A\n1⊗IB0\nIA1⊗∂B\n1(A2)\nor\nA2⊗B1 A1⊗B1 A0⊗B1\nA2⊗B0 A1⊗B0 A0⊗B0∂A\n2⊗IB1\nIA2⊗∂B\n1\n∂A\n2⊗IB0∂A\n1⊗IB1\nIA0⊗∂B\n1\n∂A\n1⊗IB0IA1⊗∂B\n1. (A3)\nUsing the second diagram, we may define vertical and horizontal maps, ∂v\niand∂h\ni, respectively such that ∂v\ni◦∂v\ni+1=\n∂h\ni◦∂h\ni+1= 0and∂v\niand∂h\nicommute. Then every purely vertical chain and every purely horizontal chain form a\nvalid chain complex.\nDiagram (A3) is called a double complex. Equation (A1) describes the chain complex formed from collapsing the\ndouble complex into the 4-term sequence\nC=A ⊗ B :C3∂C\n3− − →C2∂C\n2− − →C1∂C\n1− − →C0 (A4)\nwith\nC3=A2⊗B1\nC2= (A2⊗B0)⊕(A1⊗B1)\nC1= (A1⊗B0)⊕(A0⊗B1)\nC0=A0⊗B0.\nNotice that these can be read off of Eq. (A2) by collapsing each vertically aligned piece with a direct sum or by taking\ndiagonal lines through Eq. (A3). This is called the total complex.33\nThe maps of the total complex (Eq. (A1)) are\n∂C\n3=\u0000\nIA2⊗∂B\n1\u0001\n⊕\u0000\n∂A\n2⊗IB1\u0001\n∂C\n2=\u0000\n∂A\n2⊗IB0+IA1⊗∂B\n1\u0001\n⊕\u0000\n∂A\n1⊗IB1\u0001\n∂C\n1=∂A\n1⊗IB0+IA0⊗∂B\n1.\nTo derive explicit matrix representations of these maps, ignore the +and⊕and start from first principles: ∂C\n3takes\nbasis vectors of C3to basis vectors of C2. We can group the basis elements of C2by those spanning the space A2⊗B0\nthen those spanning A1⊗B1. Assuming we want ∂C\n3to act by left multiplication, a matrix representation can be\norganized as\nMat\u0000\n∂C\n3\u0001\n=\u0012A2⊗B1\nA2⊗B0IA2⊗∂B\n1\nA1⊗B1∂A\n2⊗IB1\u0013\n, (A5)\nwhere the row and column labels are added for convenience. Similarly,9\nMat\u0000\n∂C\n2\u0001\n=\u0012A2⊗B0A1⊗B1\nA1⊗B0∂A\n2⊗IB0IA1⊗∂B\n1\nA0⊗B1 0 ∂A\n1⊗IB1\u0013\n(A6)\nMat\u0000\n∂C\n1\u0001\n=\u0000A1⊗B0A0⊗B1\nA0⊗B0∂A\n1⊗IB0IA0⊗∂B\n1\u0001\n. (A7)\nNote that the ordering of the basis vectors in the rows of Mat\u0000\n∂C\ni\u0001\nmust be consistent with the ordering of the columns\nofMat\u0000\n∂C\ni−1\u0001\n. If we were over a different field, it would have been necessary to define the maps to be\n∂A⊗B\nn =M\ni+j=n−1∂A\ni+1⊗IBj+ (−1)iIAi⊗∂B\nj+1\nto get the necessary cancellation.\nThe procedure is the same for the tensor product of two 3-term chain complexes:\nA:A2∂A\n2− − →A1∂A\n1− − →A0\nB:B2∂B\n2− − →B1∂B\n1− − →B0\nC=A ⊗ B :C4∂C\n4− − →C3∂C\n3− − →C2∂C\n2− − →C1∂C\n1− − →C0\nwith\nC4=A2⊗B2\nC3= (A2⊗B1)⊕(A1⊗B2)\nC2= (A2⊗B0)⊕(A1⊗B1)⊕(A0⊗B2)\nC1= (A1⊗B0)⊕(A0⊗B1)\nC0=A0⊗B0\n9We can check that the boundary of the boundary is zero:\nMat\u0010\n∂C\n2\u0011\nMat\u0010\n∂C\n3\u0011\n=\u0012∂A\n2⊗∂B\n1+∂A\n2⊗∂B\n1\n∂A\n2∂A\n1⊗IB1\u0013\n= 0\nMat\u0010\n∂C\n1\u0011\nMat\u0010\n∂C\n2\u0011\n=\u0000\n∂A\n2∂A\n1⊗IB0∂A\n1⊗∂B\n1+∂A\n1⊗∂B\n1\u0001\n= 0,\nwhere we have used the fact that 2≡0inF2.34\nand\nMat\u0000\n∂C\n4\u0001\n=\u0012A2⊗B2\nA2⊗B1IA2⊗∂B\n2\nA1⊗B2∂A\n2⊗IB2\u0013\nMat\u0000\n∂C\n3\u0001\n=\nA2⊗B1A1⊗B2\nA2⊗B0IA2⊗∂B\n1 0\nA1⊗B1∂A\n1⊗IB1IA1⊗∂B\n2\nA0⊗B2 0 ∂A\n1⊗IA2\n\nMat\u0000\n∂C\n2\u0001\n=\u0012A2⊗B0A1⊗B1A0⊗B2\nA1⊗B0∂A\n1⊗IB0IA1⊗∂B\n1 0\nA0⊗B1 0 ∂A\n1⊗IB1IA0⊗∂B\n2\u0013\nMat\u0000\n∂C\n1\u0001\n=\u0000A1⊗B0A0⊗B1\nA0⊗B0∂A\n1⊗IB0IA0⊗∂B\n1\u0001\n.\nThe homology of the total complex follows a similar form10\nHk(A ⊗ B )∼=M\ni+j=k(Hi(A)⊗Hj(B)). (A8)\nConsider the total complex (Eq. (A4)) where the chain Ais the CSS code (Eq. (6)) and Bis the classical repetition\ncode (Eq. (5)), Eq. (11)). Take the CSS code determined by the right two maps. By Eq. (A8), the Zlogical operators\nof this code are\nH1(C) = (H1(A)⊗H0(B))⊕(H0(A)⊗H1(B)).\nTo compute the homology of B, extend it by zero on both sides:\n0→Fℓ−1\n2HT\n− − →Fℓ\n2→0. (A9)\nThen H1(B) = ker HT= 0andH0(B) =Fℓ\n2/imHTis the space of all vectors modulo even-weight vectors. This has\ntwo cosets: the coset of all even-weight vectors and the coset of all odd-weight vectors. The latter is generated by\nany weight-one vector. The logical operators are therefore of the form a⊗b, where a∈H1(A)is aZlogical operator\nof the original code and b∈H0(B), which has minimum weight dZ·1. For the Xlogical operators, we take the dual\nof Eq. (A9) and apply the Künneth formula to cohomology. Now H1(B) =Fℓ−1\n2/imH= 0asrankH=ℓ−1and\nH0(B) = ker His the length ℓall-ones vector. Hence, Xlogical operators are of the form a⊗b, where a∈H1(A)is an\nXlogical operator of the original code and b∈H0(B). The Xdistance therefore increases to dX·ℓ. This technique\nfirst appeared in [20] and is called distance balancing. It was generalized to use other parity-check matrices in [85];\nsee also [86, 87].\nThe hypergraph product (Eq. (1)) is the tensor product\nA:Fn1\n2H1− − →Fm1\n2\nB:Fm2\n2HT\n2− − →Fn2\n2\nC=A ⊗ B :Fn1\n2⊗Fm2\n2∂2− →(Fn1\n2⊗Fn2\n2)⊕(Fm2\n2⊗Fm1\n2)∂1− →Fm1\n2⊗Fn2\n2,\nwhere H1andH2are parity-check matrices of classical linear codes and\n∂2=\u0012\nIn1⊗HT\n2\nH1⊗Im2\u0013\n, ∂ 1=\u0000\nH1⊗In2Im1⊗HT\n2\u0001\n.\n10Equations of this type are called Künneth formulas.35\nExtending the chains on both sides by zero and applying Eq. (A8), the ZandXlogicals are of the form\n(kerH1⊗Fn2\n2/imHT\n2)⊕(Fn1\n2/imH1⊗kerHT\n2)\nand\n(Fn1\n2/imHT\n1⊗kerH2)⊕(kerHT\n1⊗Fn2\n2/imHT\n2),\nrespectively.\n2. The Mapping Cone\nClosely related to the above is the concept of the mapping cone. Consider the two chain complexes AandBbelow\nA:··· Ai+1 Ai Ai−1 ···\nB:··· Bi+1 Bi Bi−1 ···∂A\ni+1 ∂A\ni\n∂B\ni+1 ∂B\nifi+1 fi fi−1 . (A10)\nThe maps fiare called chain maps and we require that they are homomorphisms that commute with the other maps,\ni.e.∂B\ni+1(fi+1(a)) = fi\u0000\n∂A\ni+1(a)\u0001\nfora∈Ai+1. The mapping cone is defined to be the chain complex with spaces\ncone( f)i=Ai⊕Bi+1. Graphically,\nAi Ai−1\n⊕ ⊕\nBi+1 Bifi\n∂B\ni+1∂A\ni\n. (A11)\nSimilar to the previous section, the maps ∂i: cone( f)i→cone( f)i−1are\nMat ( ∂i) =\u0012AiBi+1\nAi−1∂A\ni 0\nBifi∂B\ni+1\u0013\n.\nNote that in non-binary fields, the first column should receive a minus sign, or equivalently, fishould be replaced\nwith (−1)ifi. From the mapping cone we have the short exact (split) sequence 0→Ai−1→cone( f)i→Bi→0. This\ninduces the long exact sequence on homology (via the Snake Lemma) [88].\nHk+1(cone( f))→Hk(A)→Hk(B)→Hk(cone( f)). (A12)36\nAppendix B: Examples\n1. Hypergraph Product Codes\nC(H)HGP( H)R HGP( ˜H) R HGP( ˜H(c)) R\n[11,2,7][ [202,4,7] ]0.020 [ [884,4,12] ]0.005 [ [580,4,10] ]0.007\n[11,3,6][ [185,9,6] ]0.049 [ [1745 ,9,12→15] ]0.005 [ [765,9,8→10] ]0.012\n[11,4,5][ [170,16,5] ]0.094 [ [1930 ,16,12→13] ]0.008 [ [586,16,8] ]0.027\n[11,5,4][ [157,25,4] ]0.159 [ [557,25,5→6] ]0.045 [ [325,25,4→5] ]0.077\n[11,6,4][ [146,36,4] ]0.247 [ [1170 ,36,7→9] ]0.031 [ [530,36,4→7] ]0.068\n[11,7,3][ [137,49,3] ]0.358 [ [1885 ,49,7→10] ]0.026 [ [865,49,4→8] ]0.057\n[12,2,8][ [244,4,8] ]0.016 [ [1060 ,4,14] ]0.004 [ [724,4,12] ]0.006\n[12,3,6][ [225,9,6] ]0.040 [ [1865 ,9,14→15] ]0.005 [ [845,9,10] ]0.011\n[12,4,6][ [208,16,6] ]0.077 [ [3880 ,16,14→19] ]0.004 [ [1360 ,16,8→12] ]0.012\n[12,5,4][ [193,25,4] ]0.130 [ [697,25,5] ]0.036 [ [325,25,4] ]0.077\n[12,6,4][ [180,36,4] ]0.200 [ [900,36,6→7] ]0.040 [ [468,36,4→6] ]0.077\n[12,7,4][ [169,49,4] ]0.290 [ [1765 ,49,7→10] ]0.028 [ [785,49,4→7] ]0.062\n[12,8,3][ [160,64,3] ]0.400 [ [2210 ,64,7→9] ]0.029 [ [1090 ,64,4→7] ]0.059\n[13,2,8][ [290,4,8] ]0.014 [ [1252 ,4,14] ]0.003 [ [884,4,12] ]0.005\n[13,3,7][ [269,9,7] ]0.033 [ [2385 ,9,16→17] ]0.004 [ [1205 ,9,12] ]0.007\n[13,4,6][ [250,16,6] ]0.064 [ [4058 ,16,17→19] ]0.004 [ [1466 ,16,11→12] ]0.011\n[13,5,5][ [233,25,5] ]0.107 [ [4717 ,25,13→18] ]0.005 [ [1637 ,25,8→12] ]0.015\n[13,6,4][ [218,36,4] ]0.165 [ [900,36,5] ]0.040 [ [468,36,4] ]0.077\n[13,7,4][ [205,49,4] ]0.239 [ [1225 ,49,6→8] ]0.040 [ [709,49,4→7] ]0.069\n[13,8,4][ [194,64,4] ]0.330 [ [2210 ,64,7→10] ]0.029 [ [1090 ,64,4→8] ]0.059\n[13,9,3][ [185,81,3] ]0.438 [ [2561 ,81,6→9] ]0.032 [ [1341 ,81,4→7] ]0.060\nTable V. Classical weight reduction applied to hypergraph product codes with some of the best known linear codes with\n11≤n≤13.C(H)is the linear code with parity-check matrix Hobtained from GAP (see main text). For each hypergraph\nproduct code, we give its encoding rate R=k/n. For each weight-reduction method, we apply the relevant algorithm 10,000\ntimes using different permutations of the input parity-check matrix. In cases where permutations improved the distance, we use\nthe notation d1→d2, where d1indicates the distance without permutations, and d2indicates the highest obtained distance.37\nC(H) HGP( H) R HGP( ˜H) R HGP( ˜H(c)) R\n[14,2,9][ [340,4,9] ]0.012 [ [1570 ,4,16] ]0.003 [ [1154 ,4,14] ]0.003\n[14,3,8][ [317,9,8] ]0.028 [ [2669 ,9,16→18] ]0.003 [ [1409 ,9,12→13] ]0.006\n[14,4,7][ [296,16,7] ]0.054 [ [5008 ,16,18→21] ]0.003 [ [2056 ,16,12→14] ]0.008\n[14,5,6][ [277,25,6] ]0.090 [ [6173 ,25,17→20] ]0.004 [ [2257 ,25,11→13] ]0.011\n[14,6,5][ [260,36,5] ]0.138 [ [7460 ,36,13→18] ]0.005 [ [2468 ,36,8→12] ]0.015\n[14,7,4][ [245,49,4] ]0.200 [ [1429 ,49,6→8] ]0.034 [ [709,49,4→5] ]0.069\n[14,8,4][ [232,64,4] ]0.276 [ [1832 ,64,6→9] ]0.035 [ [1000 ,64,4→7] ]0.064\n[14,9,4][ [221,81,4] ]0.367 [ [2705 ,81,7→11] ]0.030 [ [1445 ,81,4→8] ]0.056\n[14,10,3][ [212,100,3] ]0.472 [ [2938 ,100,6→9] ]0.034 [ [1618 ,100,4→7] ]0.062\n[15,2,10][ [394,4,10] ]0.010 [ [1802 ,4,18] ]0.002 [ [1354 ,4,16] ]0.003\n[15,3,8][ [369,9,8] ]0.024 [ [2817 ,9,16→18] ]0.003 [ [1517 ,9,12→13] ]0.006\n[15,4,8][ [346,16,8] ]0.046 [ [5626 ,16,18→22] ]0.003 [ [2458 ,16,12→15] ]0.007\n[15,5,7][ [325,25,7] ]0.077 [ [10237 ,25,21→25] ]0.002 [ [3457 ,25,12→16] ]0.007\n[15,6,6][ [306,36,6] ]0.118 [ [9266 ,36,17→20] ]0.004 [ [3218 ,36,11→14] ]0.011\n[15,7,5][ [289,49,5] ]0.170 [ [9965 ,49,14→19] ]0.005 [ [3305 ,49,8→13] ]0.015\n[15,8,4][ [274,64,4] ]0.234 [ [2920 ,64,6→9] ]0.022 [ [1384 ,64,4→7] ]0.046\n[15,9,4][ [261,81,4] ]0.310 [ [2561 ,81,7→10] ]0.032 [ [1341 ,81,4→8] ]0.060\n[15,10,4][ [250,100,4] ]0.400 [ [3250 ,100,7→11] ]0.031 [ [1850 ,100,4→9] ]0.054\n[15,11,3][ [241,121,3] ]0.502 [ [3341 ,121,6→9] ]0.036 [ [1921 ,121,3→7] ]0.063\nTable VI. Classical weight reduction applied to hypergraph product codes with some of the best known linear codes with\n14≤n≤15.C(H)is the linear code with parity-check matrix Hobtained from GAP (see main text). For each hypergraph\nproduct code, we give its encoding rate R=k/n. For each weight-reduction method, we apply the relevant algorithm 10,000\ntimes using different permutations of the input parity-check matrix. In cases where permutations improved the distance, we use\nthe notation d1→d2, where d1indicates the distance without permutations, and d2indicates the highest obtained distance.\n2. Quasi-Cyclic Codes\nHere we give the base matrices used to construct the lifted product code examples in Table IV.\n1. From [89, Table 2], a [52,27,6]quasi-cyclic code with lift size ℓ= 13and base matrix\nA=\u0012\n1 1 1 1\n1x x3x9\u0013\n.\n2. From [90, Example 11], a [124,33,24]quasi-cyclic code with lift size ℓ= 31and base matrix\nA=\nx x2x4x8\nx5x10x20x9\nx25x19x7x14\n.\n3. From [40, Table 1], a [28,9,10]quasi-cyclic code with lift size ℓ= 7and base matrix\nA=\n1 1 1 1\n1x x2x5\n1x6x3x\n.\n4. From [40, Table 1], a [36,11,12]quasi-cyclic code with lift size ℓ= 9and base matrix\nA=\n1 1 1 1\n1x x6x7\n1x4x5x2\n.38\n5. From [40, Table 1], a [68,19,18]quasi-cyclic code with lift size ℓ= 17and base matrix\nA=\n1 1 1 1\n1x x2x11\n1x8x12x13\n\n6. From [40, Table 1], a [76,21,20]quasi-cyclic code with lift size ℓ= 9and base matrix\nA=\n1 1 1 1\n1x x6x7\n1x4x5x2\n.\nWe can also weight reduce base matrices with higher weight entries in certain special cases. Consider the matrix\nA=\nx+x2x4x8\nx5x9x10+x20\nx25+x19x7+x14\n,\nwhich for lift size ℓ= 46has an associated quasi-cyclic code with parameters [184,47,32][90, Example 13]. The\ncorresponding lifted product code LP(A)has parameters [ [1150 ,50,≤21] ]. If we choose the correct permutation for\neach row then we can obtain a weight reduced matrix where each column and row have weight at most three:\n˜A=\nx+x21\nx41 1\nx81\nx1+x21\nx51 1\nx91\nx7+x141\nx25+x191\n.\nThe corresponding quasi-cyclic code has parameters [414,47,81]and the lifted product code LP(˜A)has parameters\n[ [6670 ,50,≤70] ].39\n[1] E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549,\n172 (2017).\n[2] A.M.Dalzell, S.McArdle, M.Berta, P.Bienias, C.-F.Chen, A.Gilyén, C.T.Hann, M.J.Kastoryano, E.T.Khabiboulline,\nA. Kubica, G. Salton, S. Wang, and F. G. S. L. Brandão, Quantum algorithms: A survey of applications and end-to-end\ncomplexities (2023), arxiv:2310.03011.\n[3] C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti,\nN. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz, Realization\nof Real-Time Fault-Tolerant Quantum Error Correction, Phys. Rev. X 11, 041058 (2021).\n[4] S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J.\nNorris, C. K. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, Realizing repeated quantum error correction in\na distance-three surface code, Nature 605, 669 (2022).\n[5] N. Sundaresan, T. J. Yoder, Y. Kim, M. Li, E. H. Chen, G. Harper, T. Thorbeck, A. W. Cross, A. D. Córcoles, and\nM. Takita, Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood\ndecoders, Nat Commun 14, 2852 (2023).\n[6] Google Quantum AI, R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw,\nJ. Atalaya, R. Babbush, D. Bacon, J. C. Bardin, J. Basso, A. Bengtsson, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird,\nL. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro,\nJ. Cogan, R. Collins, P. Conner, W. Courtney, A. L. Crook, B. Curtin, D. M. Debroy, A. Del Toro Barba, S. Demura,\nA. Dunsworth, D. Eppens, C. Erickson, L. Faoro, E. Farhi, R. Fatemi, L. Flores Burgos, E. Forati, A. G. Fowler, B. Foxen,\nW. Giang, C. Gidney, D. Gilboa, M. Giustina, A. Grajales Dau, J. A. Gross, S. Habegger, M. C. Hamilton, M. P. Harrigan,\nS. D. Harrington, O. Higgott, J. Hilton, M. Hoffmann, S. Hong, T. Huang, A. Huff, W. J. Huggins, L. B. Ioffe, S. V. Isakov,\nJ. Iveland, E. Jeffrey, Z. Jiang, C. Jones, P. Juhas, D. Kafri, K. Kechedzhi, J. Kelly, T. Khattar, M. Khezri, M. Kieferová,\nS. Kim, A. Kitaev, P. V. Klimov, A. R. Klots, A. N. Korotkov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, K.-\nM. Lau, L. Laws, J. Lee, K. Lee, B. J. Lester, A. Lill, W. Liu, A. Locharla, E. Lucero, F. D. Malone, J. Marshall, O. Martin,\nJ. R. McClean, T. McCourt, M. McEwen, A. Megrant, B. Meurer Costa, X. Mi, K. C. Miao, M. Mohseni, S. Montazeri,\nA. Morvan, E. Mount, W. Mruczkiewicz, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, H. Neven, M. Newman, J. H.\nNg, A. Nguyen, M. Nguyen, M. Y. Niu, T. E. O’Brien, A. Opremcak, J. Platt, A. Petukhov, R. Potter, L. P. Pryadko,\nC. Quintana, P. Roushan, N. C. Rubin, N. Saei, D. Sank, K. Sankaragomathi, K. J. Satzinger, H. F. Schurkus, C. Schuster,\nM. J. Shearn, A. Shorter, V. Shvarts, J. Skruzny, V. Smelyanskiy, W. C. Smith, G. Sterling, D. Strain, M. Szalay, A. Torres,\nG. Vidal, B. Villalonga, C. Vollgraff Heidweiller, T. White, C. Xing, Z. J. Yao, P. Yeh, J. Yoo, G. Young, A. Zalcman,\nY. Zhang, and N. Zhu, Suppressing quantum errors by scaling a surface code logical qubit, Nature 614, 676 (2023).\n[7] A.Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003).\n[8] E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43,\n4452 (2002).\n[9] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum\ncomputation, Phys. Rev. A 86, 032324 (2012).\n[10] N. P. Breuckmann and J. N. Eberhardt, Quantum Low-Density Parity-Check Codes, PRX Quantum 2, 040101 (2021).\n[11] C. Gidney and M. Ekerå, How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits, Quantum 5, 433\n(2021).\n[12] I. H. Kim, Y.-H. Liu, S. Pallister, W. Pol, S. Roberts, and E. Lee, Fault-tolerant resource estimate for quantum chemical\nsimulations: Case study on Li-ion battery electrolyte molecules, Phys. Rev. Research 4, 023019 (2022).\n[13] N. Baspin and A. Krishna, Connectivity constrains quantum codes, Quantum 6, 711 (2022).\n[14] N. Baspin and A. Krishna, Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive, Phys. Rev.\nLett.129, 050505 (2022).\n[15] N. Baspin, V. Guruswami, A. Krishna, and R. Li, Improved rate-distance trade-offs for quantum codes with restricted\nconnectivity (2023), arxiv:2307.03283.\n[16] S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-\ntolerant quantum memory (2023), arXiv:2308.07915.\n[17] Q. Xu, J. Ataides, C. A. Pattison, N. Raveendran, D. Bluvstein, J. Wurtz, B. Vasic, M. D. Lukin, L. Jiang, and H. Zhou,\nConstant-overhead fault-tolerant quantum computation with reconfigurable atom arrays (2023), arXiv:2308.08648.\n[18] J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphi-\nnais, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, Blueprint for a Scalable Photonic Fault-Tolerant Quantum Com-\nputer, Quantum 5, 392 (2021).\n[19] I. Tzitrin, T. Matsuura, R. N. Alexander, G. Dauphinais, J. E. Bourassa, K. K. Sabapathy, N. C. Menicucci, and I. Dhand,\nFault-tolerant quantum computation with static linear optics, PRX Quantum 2, 040353 (2021).\n[20] M. B. Hastings, Weight Reduction for Quantum Codes (2016), arxiv:1611.03790.\n[21] M. B. Hastings, On quantum weight reduction (2021), arXiv:2102.10030.\n[22] A. Wills, T.-C. Lin, and M.-H. Hsieh, Tradeoff constructions for quantum locally testable codes (2023), arXiv:2309.05541.\n[23] J.-P. Tillich and G. Zemor, Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square\nRoot of the Blocklength, IEEE Trans. Inform. Theory 60, 1193 (2014).\n[24] P. Panteleev and G. Kalachev, Quantum LDPC Codes With Almost Linear Minimum Distance, IEEE Trans. Inform.40\nTheory68, 213 (2022).\n[25] N. P. Breuckmann and J. N. Eberhardt, Balanced Product Quantum Codes, IEEE Trans. Inform. Theory 67, 6653 (2021).\n[26] P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes, in Proceedings of\nthe 54th Annual ACM SIGACT Symposium on Theory of Computing (2022) pp. 375–388.\n[27] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).\n[28] R. Raussendorf, J. Harrington, and K. Goyal, A fault-tolerant one-way quantum computer, Annals of Physics 321, 2242\n(2006).\n[29] A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, Foliated quantum error-correcting codes, Phys. Rev. Lett. 117,\n070501 (2016).\n[30] B. J. Brown and S. Roberts, Universal fault-tolerant measurement-based quantum computation, Phys. Rev. Res. 2, 033305\n(2020).\n[31] D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001).\n[32] Xanadu (2024), in preparation.\n[33] D. Gottesman, Stabilizer Codes and Quantum Error Correction , Ph.D. thesis, Caltech (1997).\n[34] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes , 2nd ed. (North-holland Publishing Company, 1978).\n[35] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098 (1996).\n[36] A. Steane, Multiple-particle interference and quantum error correction, Proc. R. Soc. Lond. A. 452, 2551– (1996).\n[37] M. B. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: breaking the n1/2 polylog(n) barrier for quantum LDPC\ncodes, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) pp. 1276–1288.\n[38] C. Chen, W. Peterson, and E. Weldon, Some results on quasi-cyclic codes, Information and Control 15, 407 (1969).\n[39] P. Panteleev and G. Kalachev, Quantum LDPC codes with almost linear minimum distance, IEEE Transactions on Infor-\nmation Theory 68, 213 (2021).\n[40] N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, Finite Rate QLDPC-GKP Coding Scheme\nthat Surpasses the CSS Hamming Bound, Quantum 6, 767 (2022).\n[41] J. Roffe, L. Z. Cohen, A. O. Quintavalle, D. Chandra, and E. T. Campbell, Bias-tailored quantum LDPC codes, Quantum\n7, 1005 (2023).\n[42] E. Sabo, esabo/codingtheory: A basic coding theory library for julia. (2021).\n[43] S. Bravyi and B. Terhal, A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes,\nNew Journal of Physics 11, 043029 (2009).\n[44] E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Science and Technology 4, 025006\n(2019).\n[45] W. Zeng and L. P. Pryadko, Higher-dimensional quantum hypergraph-product codes with finite rates, Phys. Rev. Lett.\n122, 230501 (2019).\n[46] A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Campbell, Single-shot error correction of three-dimensional homological\nproduct codes, PRX Quantum 2, 020340 (2021).\n[47] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation (1996), arXiv:quant-ph/9610011.\n[48] J. T. Anderson, G. Duclos-Cianci, and D. Poulin, Fault-tolerant conversion between the Steane and Reed-Muller quantum\ncodes, Phys. Rev. Lett. 113, 080501 (2014).\n[49] M.FreedmanandM.Hastings,Buildingmanifoldsfromquantumcodes,GeometricandFunctionalAnalysis 31,855(2021).\n[50] J. D. Horton, A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM Journal on Computing 16,\n358 (1987).\n[51] A.Wills, T.-C.Lin,andM.-H.Hsieh,TradeoffConstructionsforQuantumLocallyTestableCodes(2023),arxiv:2309.05541.\n[52] W. Ryan and S. Lin, Channel Codes: Classical and Modern (Cambridge University Press, 2009).\n[53] P. Shankar, Expander Codes: The Sipser-Spielman Construction, Resonance 10, 25 (2005).\n[54] T. Tian, C. R. Jones, J. D. Villasenor, and R. D. Wesel, Selective avoidance of cycles in irregular LDPC code construction,\nIEEE Transactions on Communications 52, 1242 (2004).\n[55] S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2d systems, Physical review\nletters104, 050503 (2010).\n[56] J. Cramwinckel, E. Roijackers, R. Baart, E. Minkes, L. Ruscio, R. L. Miller, T. Boothby, C. Tjhai, D. Joyner, and J. Fields,\nGuava (2023), https://gap-packages.github.io/guava/ .\n[57] J. Roffe, D. R. White, S. Burton, and E. Campbell, Decoding across the quantum low-density parity-check code landscape,\nPhys. Rev. Res. 2, 043423 (2020).\n[58] H. Bombin and M. A. Martin-Delgado, Optimal resources for topological two-dimensional stabilizer codes: Comparative\nstudy, Phys. Rev. A 76, 012305 (2007).\n[59] K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High-threshold fault-tolerant quantum computation with analog quantum\nerror correction, Phys. Rev. X 8, 021054 (2018).\n[60] C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, Quantum error correction with the toric gottesman-\nkitaev-preskill code, Phys. Rev. A 99, 032344 (2019).\n[61] K. Noh and C. Chamberland, Fault-tolerant bosonic quantum error correction with the surface–gottesman-kitaev-preskill\ncode, Phys. Rev. A 101, 012316 (2020).\n[62] L. Hänggli, M. Heinze, and R. König, Enhanced noise resilience of the surface–gottesman-kitaev-preskill code via designed\nbias, Phys. Rev. A 102, 052408 (2020).\n[63] J. Zhang, J. Zhao, Y.-C. Wu, and G.-P. Guo, Quantum error correction with the color-gottesman-kitaev-preskill code,\nPhys. Rev. A 104, 062434 (2021).41\n[64] K. Noh, C. Chamberland, and F. G. Brandão, Low-overhead fault-tolerant quantum error correction with the surface-gkp\ncode, PRX Quantum 3, 010315 (2022).\n[65] J. Zhang, Y.-C. Wu, and G.-P. Guo, Concatenation of the gottesman-kitaev-preskill code with the xzzx surface code, Phys.\nRev. A107, 062408 (2023).\n[66] B. W. Walshe, B. Q. Baragiola, R. N. Alexander, and N. C. Menicucci, Continuous-variable gate teleportation and bosonic-\ncode error correction, Physical Review A 102, 062411 (2020).\n[67] L. D. Brown, T. T. Cai, and A. DasGupta, Interval Estimation for a Binomial Proportion, Statistical Science 16, 101\n(2001).\n[68] D. Kribs, R. Laflamme, and D. Poulin, Unified and generalized approach to quantum error correction, Phys. Rev. Lett.\n94, 180501 (2005).\n[69] D. Poulin, Stabilizer formalism for operator quantum error correction, Phys. Rev. Lett. 95, 230504 (2005).\n[70] D. Bacon, Operator quantum error-correcting subsystems for self-correcting quantum memories, Phys. Rev. A 73, 012340\n(2006).\n[71] H. Bombin, Topological subsystem codes, Phys. Rev. A 81, 032301 (2010).\n[72] M. Suchara, S. Bravyi, and B. Terhal, Constructions and noise threshold of topological subsystem codes, Journal of Physics\nA: Mathematical and Theoretical 44, 155301 (2011).\n[73] S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, Subsystem surface codes with three-qubit check operators,\nQuantum Info. Comput. 13, 963–985 (2013).\n[74] H. Bombín, Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes, New Journal of\nPhysics17, 083002 (2015).\n[75] S. Bravyi and A. Cross, Doubled color codes (2015), arXiv:1509.03239.\n[76] T.Jochym-O’ConnorandS.D.Bartlett,Stackedcodes: Universalfault-tolerantquantumcomputationinatwo-dimensional\nlayout, Phys. Rev. A 93, 022323 (2016).\n[77] C. Jones, P. Brooks, and J. Harrington, Gauge color codes in two dimensions, Phys. Rev. A 93, 052332 (2016).\n[78] O. Higgott and N. P. Breuckmann, Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead,\nPhys. Rev. X 11, 031039 (2021).\n[79] A. Kubica and M. Vasmer, Single-shot quantum error correction with the three-dimensional subsystem toric code, Nature\nCommunications 13, 6272 (2022).\n[80] A. Krishna and D. Poulin, Fault-tolerant gates on hypergraph product codes, Physical Review X 11, 011023 (2021).\n[81] L. Z. Cohen, I. H. Kim, S. D. Bartlett, and B. J. Brown, Low-overhead fault-tolerant quantum computing using long-range\nconnectivity, Sci. Adv. 8, eabn1717 (2022).\n[82] A. O. Quintavalle, P. Webster, and M. Vasmer, Partitioning qubits in hypergraph product codes to implement logical\ngates, Quantum 7, 1153 (2023).\n[83] N. P. Breuckmann and S. Burton, Fold-transversal clifford gates for quantum codes (2022), arXiv:2202.06647.\n[84] S. Huang, T. Jochym-O’Connor, and T. J. Yoder, Homomorphic logical measurements (2022), arXiv:2211.03625 [quant-ph].\n[85] S.Evra, T.Kaufman,andG.Zémor,DecodablequantumLDPCcodesbeyondthendistancebarrierusinghigh-dimensional\nexpanders, SIAM Journal on Computing , FOCS20 (2022).\n[86] A. Cross, Z. He, A. Natarajan, M. Szegedy, and G. Zhu, Quantum locally testable code with exotic parameters (2022),\n2209.11405.\n[87] A. Wills, T.-C. Lin, and M.-H. Hsieh, General distance balancing for quantum locally testable codes (2023),\narXiv:2305.00689.\n[88] J. Rotman, An introduction to homological algebra , Vol. 2 (Springer, 2009).\n[89] I. E. Bocharova, B. D. Kudryashov, and R. V. Satyukov, Graph-based convolutional and block LDPC codes, Probl Inf\nTransm45, 357 (2009).\n[90] R. Smarandache and P. O. Vontobel, Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on\nMinimum Hamming Distance Upper Bounds, IEEE Trans. Inform. Theory 58, 585 (2012)."    },    {        "title": "2402.05288v1.Investigating_Performance_Trends_of_Simulated_Real_time_Solar_Flare_Predictions__The_Impacts_of_Training_Windows__Data_Volumes__and_the_Solar_Cycle.pdf",        "content": "Draft version February 9, 2024\nTypeset using L ATEXpreprint style in AASTeX631\nInvestigating Performance Trends of Simulated Real-time Solar Flare Predictions: The\nImpacts of Training Windows, Data Volumes, and the Solar Cycle\nGriffin T. Goodwin\n ,1Viacheslav M. Sadykov\n ,1and Petrus C. Martens\n1\n1Physics & Astronomy Department, Georgia State University, Atlanta, GA 30303, USA; ggoodwin5@gsu.edu\nABSTRACT\nThis study explores the behavior of machine learning-based flare forecasting models\ndeployed in a simulated operational environment. Using Georgia State University’s\nSpace Weather Analytics for Solar Flares benchmark dataset (Angryk et al. 2020a,b),\nwe examine the impacts of training methodology and the solar cycle on decision tree,\nsupport vector machine, and multilayer perceptron performance. We implement our\nclassifiers using three temporal training windows: stationary, rolling, and expanding.\nThe stationary window trains models using a single set of data available before the first\nforecasting instance, which remains constant throughout the solar cycle. The rolling\nwindow trains models using data from a constant time interval before the forecasting\ninstance, which moves with the solar cycle. Finally, the expanding window trains\nmodels using all available data before the forecasting instance. For each window, a\nnumber of input features (1, 5, 10, 25, 50, 120) and temporal sizes (5, 8, 11, 14, 17,\n20 months) were tested. To our surprise, we found that for a 20-month window, skill\nscores were comparable regardless of the window type, feature count, and classifier\nselected. Furthermore, reducing the size of this window only marginally decreased\nstationary and rolling window performance. This implies that, given enough data, a\nstationary window can be chosen over other window types, eliminating the need for\nmodel retraining. Lastly, a moderately strong positive correlation was found to exist\nbetween a model’s false positive rate and the solar X-ray background flux. This suggests\nthat the solar cycle phase has a considerable influence on forecasting.\nKeywords: Space weather (2037), Solar flares (1496), Support vector machine (1936),\nSolar cycle (1487)\n1.INTRODUCTION\nDue to humanity’s growing technological advancements over the past century, solar eruptive events\nhave emerged as a significant threat to society and its infrastructure. Electromagnetic radiation,\nsolar energetic particles, and coronal mass ejections produced during solar flares have the potential\nto interfere with radio communications, GPS, and power grids (Natras et al. 2019; Hudson 2021),\nwhich are crucial components to our everyday lives. Furthermore, these events pose considerable\nhealth risks to humans, particularly astronauts who are not shielded by Earth’s magnetosphere and\nmay receive increased doses of radiation. Considering these effects, the need for robust forecasting\nmodels that provide accurate and timely predictions of solar flares has become increasingly impor-arXiv:2402.05288v1  [astro-ph.SR]  7 Feb 20242\ntant. Traditional forecasting methods, such as those used by the National Oceanic and Atmospheric\nAdministration (NOAA), have long relied on a blend of statistical analyses and human intuition\n(Crown 2012). However, given the advancements of artificial intelligence in recent years, there has\nbeen a gradual shift towards utilizing machine learning (ML) to automate and improve current fore-\ncasting capabilities. ML centers on training computers to make predictions on unseen data, given\ntheir previously acquired knowledge of some available dataset (Florios et al. 2018). For flares, this\ncan be physics-based parameters of active region (AR) vector magnetograms, extreme ultraviolet\nimages of ARs, or even sunspot properties and McIntosh classifications (Bobra & Couvidat 2015;\nNishizuka et al. 2018; Li et al. 2007). Since its initial application to space weather in the early 1990s\n(Camporeale 2019), ML has grown significantly, showing great promise within the community. How-\never, despite this success, several notable issues continue to limit its implementation in operational\nforecasting:\n1. ML models are commonly trained and tested using a random set of flaring and non-flaring\ndata, which is not necessarily consistent with real-time forecasting. In an operational setting,\npredictions must be based solely on data available prior to the forecasted event. This raises\nthe question: How do ML classifiers perform when utilizing chronological training and testing\npartitions? Sadykov & Kosovichev (2017); Nishizuka et al. (2018); Leka et al. (2019a) have\nconsidered this idea through static training and testing windows, however, to the best of our\nknowledge, no studies have attempted to implement a dynamic temporal training strategy to\nimprove operational forecasts.\n2. Complicated ML algorithms are often considered black boxes, providing little insight into their\npredictive reasoning (Camporeale 2019). This makes it challenging for forecasters to rely on\nthem confidently. Thankfully, relatively basic models exist that provide easily interpretable\npredictions. However, there is no guarantee that these models perform as well as their more\ncomplex counterparts. A previous study from Deshmukh et al. (2023) found that, for flare fore-\ncasting, ML models of different complexities were quite comparable, but a similar investigation\nhas yet to be done for a real-time forecasting environment.\n3. ML models are frequently trained on all available data to maximize performance. However,\nthis can result in a time-consuming training and hyperparameter optimization phase, which is\nnot ideal for real-time forecasting. A middle ground between performance and run time likely\nexists, but the amount of data necessary to generate effective flare forecasts is currently poorly\nunderstood.\n4. The performance of ML-based flare forecasting models is heavily influenced by the selection\nof training data. Previous studies have shown that skill scores may vary significantly when\ntraining on different parts of the solar cycle (Wang et al. 2020). It is unclear whether these\nimpacts can be mitigated through dynamic training windows.\nThe goal of this work is to thoroughly examine each of these concerns. To address Problem 1 ,\nwe deploy a training and testing methodology that simulates a real-time predictive environment.\nWe accomplish this through three training windows we label as stationary, rolling, and expanding.\nForProblem 2 , we apply our training methodology to three different ML models of increasing\ncomplexity: decision tree, support vector machine, and multilayer perceptron. We then explore3\nhow performance scales with the number of magnetogram features used in a prediction. To tackle\nProblem 3 , we investigate the impact of data volume on performance by implementing different\nstationary and rolling window sizes. Finally, to handle Problem 4 , we explore the relationship\nbetween classifier performance and the solar background soft X-ray (SXR) flux. We use this as\na probe to investigate if the solar cycle has an effect on real-time forecasts, as well as how this\npotential dependency interacts with the dynamic training windows. We would like to emphasize that\nthe main goal of this work is not necessarily to produce the best-performing classifier, but rather to\nexhaustively examine the problems we have mentioned above.\nThe remaining sections of this work are organized as follows: Section 2 details the data we use to\nconstruct our forecasts. Section 3 describes the methodology used to tune, train, test, and analyze\nour ML models. Lastly, Section 4 & 5 highlight the results and conclusions of our study.\n2.DATA\nIn this section, we provide a description of the three key datasets employed in this work. Section\n2.1 provides an overview of the Space Weather Analytics for Solar Flares database, while Section 2.2\nbriefly describes the data used to analyze the performance dependency on the solar cycle.\n2.1. Space Weather Analytics For Solar Flares (SWAN-SF)\nGeorgia State University’s Space Weather Analytics for Solar Flares (SWAN-SF) benchmark dataset\n(Angryk et al. 2020b) is a comprehensive, ML-ready collection of multivariate time series samples\nextracted from ARs present during Solar Cycle 24 (May 2010 – August 2018). For each AR, twenty-\nfour physics-based features (see Angryk et al. 2020b Table 1) are derived from photospheric vector\nmagnetograms taken by the Solar Dynamics Observatory Helioseismic and Magnetic Imager (Scherrer\net al. 2012). Throughout an AR’s lifetime, time series data are sliced into temporally successive\noverlapping files (offset by 1-hour), each containing 12 hours’ worth of data, at a 12-minute cadence.\nFiles are then labeled based on the strongest flaring event that occurs in the following 24 hours. We\ncategorize flare strength using NOAA’s logarithmic classification scale: A (weakest), B, C, M, and X\n(strongest). For this study, M and X-class flares are labeled as flaring events, considering they have\nthe greatest potential for societal impacts. Weaker flares, in addition to flare quiet time series, are\nlabeled as non-flaring events.\nIn total, there are 331,185 AR multivariate time series files in SWAN-SF. Given this, along with the\nhigh dimensionality of each file, we decided to eliminate the contiguous temporal dimension of our\ndata. This process not only allowed for easier integration with our proposed ML models, discussed\nin Section 3.1, but significantly reduced training and testing times, which is an important aspect to\nconsider when deploying models operationally. By extracting the summary statistics (mean, median,\nstandard deviation, maximum, and minimum) of each magnetic field parameter, within the 12-hour\nwindow, all files were reduced to a single, point-in-time datum, with a dimension of 1 by 120. Any files\ncontaining columns with missing data were linearly interpolated before calculating their summary\nstatistics, while files with empty columns were dropped altogether. To ensure that the data reflected a\nreal-time forecasting scenario as much as possible, all ARs were retained, including those potentially\nprone to projection effects at radial distances greater than 70 degrees from the solar disk center.\nUltimately, we were left with 330,169 data points: 6,234 flaring and 323,935 non-flaring.\nIf the reader would like to learn more about the original data processing techniques for SWAN-SF,\nplease refer to Angryk et al. (2020b).4\n2.2. Geostationary Operational Environmental Satellite (GOES) SXR Flux & Hale Classifications\nTo investigate potential correlations between a classifier’s performance and the phase of the solar\ncycle, we utilized daily SXR flux data from GOES (1 - 8 ˚A channel) provided by Ali et al. (2024). A\nproxy for the solar SXR background flux was then determined by selecting the minimum X-ray flux\nmeasurement for each day during Solar Cycle 24. Additionally, we made use of AR Hale classifications\nprovided by Marroquin et al. (2023), to study the frequency of complex ARs throughout the solar\ncycle. We then used these data in conjunction with each other to synthesize our results discussed in\nSection 4.3.\n3.METHODOLOGY\nThe structure of SWAN-SF frames this forecasting problem as a binary classification task, with the\nultimate goal of determining whether an AR will produce a ≥M class solar flare within the next 24\nhours. Previous work has shown that ML-based classifiers such as decision tree, logistic regression,\nrandom forest, support vector machine, and multilayer perceptron provide relatively reasonable fore-\ncasting performance when utilizing magnetogram feature sets (Yu et al. 2009; Yuan et al. 2010; Bobra\n& Couvidat 2015; Florios et al. 2018). In this particular study, we focus on the simulated real-time\nperformance of three models: decision tree, support vector machine, and multilayer perception. This\nsubset covers a wide gambit of complexities, ensuring we obtain robust results.\nThe following sections provide a basic overview of each model (Section 3.1), the methodology for\ndata preprocessing, feature selection, and hyperparameter tuning (Section 3.2), the design of each\ntraining window (Section 3.3), the performance metrics used to analyze our results (Section 3.4), and\nour approach for studying the solar cycle dependence (Section 3.5).\n3.1. Machine Learning Classifiers\nDecision trees (DT) are a simple, yet effective, ML algorithm for classification. Their foundation\nstems from a series of feature-based inequalities, which guide an input to a prediction. The overall\nstructure of this model is hierarchical in nature, consisting of interconnected nodes, children, and\nleaves. Each node contains a test that compares a particular feature to some threshold value. These\nnodes are then split into child nodes, each with their own thresholds, further subdividing the tree.\nEventually, enough splits are made to obtain a leaf, which determines the prediction for a given input.\nMathematically, DTs are constructed by minimizing the impurity of successive splits of the training\ndata. This can be considered analogous to determining the feature and decision boundary that best\nseparates two labeled distributions (Kingsford & Salzberg 2008; Kotsiantis 2013; Deshmukh et al.\n2023). In this work, we focus on optimizing two key hyperparameters of our DT model: the tree\ndepth and the number of training samples needed for a split/leaf node to occur. We also explore\na variety of impurities (Gini and entropy), when splitting nodes (see Kingsford & Salzberg (2008);\nKotsiantis (2013) for more details).\nIn its simplest form, the support vector machine (SVM) algorithm identifies a multidimensional\nhyperplane in feature space that maximizes the separation between labels. This hyperplane is then\nused to make predictions on input data. Typically, more complex, non-linear structures are needed\nto separate labels adequately. Thus, kernels may be applied to map the feature space into a higher\ndimension. In this paper, we employ a radial basis function (RBF) kernel (Bobra & Couvidat 2015),\nwhich is influenced by two fundamental hyperparameters: Candγ.Cis a penalty parameter for\nthe misclassification of training data, where large values of Cresult in overfitting and low values5\nlead to underfitting. γon the other hand is the “width” of the kernel. High values of γincrease the\ncomplexity of the decision boundary, while smaller values generate a smoother division, resulting in\nperformance similar to that of a linear boundary.\nMultilayer perceptrons (MLP) are a subclass of feed-forward neural networks containing a set of\nfully connected nodes (or neurons) across consecutive neuron layers. These nodes, which constitute\nthe building blocks of the algorithm, possess a series of inputs, outputs, and weights that facilitate\nthe creation of a non-linear decision boundary. All incoming data to a node is multiplied by its\ncorresponding weight and summed together. The result is then transformed using an activation\nfunction and passed to the output, which connects to each node within the subsequent layer (Gardner\n& Dorling 1998; Deshmukh et al. 2023). Typically, MLPs have three stages: a single input layer,\na single output layer, and some arbitrarily large hidden layer sandwiched in between. For binary\nclassification tasks, the input layer contains the same number of nodes as the size of the feature\nspace, the output layer contains 2 nodes, and the hidden layers contain any number of nodes. In this\nstudy, after some trial and error, we settled on a 3-stage hidden layer, with 50, 25, and 12 nodes. To\noptimize the node weights, we utilize Adam, a stochastic gradient descent algorithm (Kingma & Ba\n2014). For our non-linear activation function, we chose the rectified linear unit (ReLU), a reliable\ntransformation used in most modern networks. Two key hyperparameters we consider in this work,\nwhich can be tweaked to improve performance, are αand the number of training iterations. αserves\nas the strength of the L2 regularization. If αis large, there is a high penalty for misclassification,\nleading to a greater likelihood of overfitting the training data. The number of training iterations is\nhow often the weights of the MLP are updated. The larger this value, the more vulnerable the MLP\nis to overfitting.\nTo implement these models, we use Python’s scikit-learn library (Pedregosa et al. 2011). This\npackage provides excellent support for data preprocessing, feature selection, and hyperparameter\ntuning.\n3.2. Data Preprocessing, Feature Selection, & Hyperparameter Tuning\nData preprocessing is a key aspect to consider when training ML models, as incorrectly formatted\nor inconsistent data can lead to significantly worse predictions. In particular, disagreement in feature\nscales (due to differences in units) can pose problems, since features with larger scales tend to\nbe given additional weight. We address this problem by rescaling each training dataset feature\ndistribution to a mean of 0 and a variance of 1, using scikit-learn’s Standard Scaler module.\nThis transformation is calculated using the following formula: z=x−u\ns, where zis the transformed\nvalue, xis the input value, uis the mean of the training samples, and sis the standard deviation\nof the training samples. The transformation is then applied to the testing dataset, using the same\ntraining values for uandsto ensure that no testing data bias is introduced into our predictions.\nFeature selection is another crucial facet to include, as utilizing features with little predictive ca-\npacity will result in poor performance. To determine the optimal features to select, we employ\nscikit-learn’s SelectKBest module, which calculates the analysis of variance (ANOVA) F-value\nfor each feature in the training dataset. This univariate statistic provides an estimate for the sepa-\nration of variances between two distributions (in this case flaring/non-flaring events). Narrow and\nwidely spaced distributions will produce large F-values, while significantly overlapping distributions\nwith large standard deviations will result in small F-values. The metric is mathematically defined in6\nthe following way (Bobra & Couvidat 2015):\nF(i) =(¯x+\ni−¯xi)2+ (¯x−\ni−¯xi)2\n1\nn+−1Pn+\nc=1(¯x+\nc,i−¯xi)2+1\nn−−1Pn−\nc=1(¯x−\nc,i−¯xi)2(1)\nwhere, for a given feature i, ¯x+\niis the average value across flaring events, ¯ x−\niis the average value\nacross non-flaring events, ¯ xiis the average value across all events, n+is the number of flaring events,\nandn−is the number of non-flaring events. The kfeatures with the highest F-values are then kept, as\nthey have the best-separated distributions, and thus, are beneficial to use when training our models.\nDetermining the appropriate value for kcan be challenging, so several were tested: 1, 5, 10, 25, 50,\nand 120 (see Section 4.1). We apply this selection methodology to every new training dataset, prior\nto undersampling (see Section 3.3). Ultimately, the main reason we settled on this feature selection\napproach over others is its proven success within SWAN-SF. Generally, F-values have been shown to\nprovide reasonable insight into a feature’s importance (Yeolekar et al. 2021).\nLastly, to address hyperparameter tuning, a necessary step for maximizing predictive performance,\nwe implement a grid search using scikit-learn’s GridSearchCV module. This enables us to ex-\nhaustively test combinations of hyperparameters and select those that result in the best-performing\nmodel. For each training dataset, a stratified group 5-fold cross-validation was applied. This ensures\nthat within the training and testing folds, no data overlaps between ARs, and a similar number of\nflaring and non-flaring events are present. Using the generated folds, a model for every possible\ncombination of hyperparameters shown in Table 1 was tested. The model that produced the highest\ntrue skill statistic score (see Section 3.4) was then selected for application to the full training dataset.\nOnce again, we apply this process for all new data fed to the model.\n3.3. Simulated Real-time Training Windows\nTo explore the performance of a classifier in an operational setting, we designed a simulated real-\ntime environment centered on training and testing ML models chronologically throughout Solar Cycle\n24. Training data were produced using three different dynamic temporal windows: stationary, rolling,\nand expanding (see Figure 1). The stationary window paradigm generates forecasts using a single set\nof data that is available before the firstforecasting instance. This training data is always selected from\nthe beginning of the solar cycle (May 2010 onward). The rolling window paradigm generates forecasts\nbased on data from a constant time interval before the currently observed forecasting instance. This\nis similar to the stationary window, however, now, the window moves with the testing data. Lastly,\nmodels trained using the expanding window paradigm utilize all available data before the currently\nobserved forecasting instance.\nThe boundary conditions for a given window were defined to best emulate data acquired in real-\ntime. For a given window lower boundary date, denoted as X, all data with time series start dates\n≥Xwere retained. For a given window upper boundary date, denoted as Y, non-flaring data whose\n24-hour forecasting window ends ≤Ywere kept. Data instances with forecasting windows extending\nbeyond this date were excluded, as operators would need to wait the full 24 hours to confirm an AR\nas non-flaring. In the case of flaring data, time series can instantly be labeled as a flaring event, once\nan M or X class flare occurs. Thus, data corresponding to flares that took place at times ≤Ywere\nkept, even if their 24-hour forecasting window had extended beyond the boundary.7\nML Classifier scikit-learn Grid Search Hyperparameters\nDecision Tree criterion: [\"gini\", \"entropy\"]\nclass weight: [\"balanced\"]\nmaxdepth: [2, 3, 4, 5, 10, 20, 30, 40, 50, 100]\nminsamples leaf: [1, 10, 20, 30, 40, 50, 100]\nminsamples split: [2, 10, 20, 30, 40, 50, 100]\nSupport Vector Machine kernel: [\"rbf\"]\nclass weight: [\"balanced\"]\nC: [0.0001, 0.001, 0.01, 0.1, 1, 10, 100, 1000]\ngamma: [scale, 0.0001, 0.001, 0.01, 0.1, 1, 10, 100, 1000]\nMultilayer Perceptron hidden layer sizes: [(50, 25, 12)]\nsolver: [\"adam\"]\nactivation: [\"relu\"]\nlearning rate: [\"adaptive\"]\nalpha: [0.0001, 0.001, 0.01, 0.1, 1, 10, 100]\nmaxiter: [5, 10, 20, 30, 40, 50, 100, 200]\nNote —The SVM scale hyperparameter for gamma uses the inverse of the number of fea-\ntures times the variance of the feature vector. See the scikit-learn API (https://scikit-\nlearn.org/stable/modules/classes.html) for additional details on each parameter.\nTable 1. A list of hyperparameters used in the DT, SVM, and MLP grid search.\nFor the testing data, windows were generated in sequential 3-month blocks starting January 1st,\n2012. The boundaries for the blocks were set to encompass all flaring data and any non-flaring time\nseries whose 24-hour forecasting window end date fell within the 3-month block. Any testing blocks\nthat did not contain flaring data were not considered in our analysis of true skill statistic and Heidke\nskill score in Sections 4.1 and 4.2. This includes the period between April 2016 and March 2017, as\nwell as any time after September 2017.\nFor each training dataset, an undersampling approach was applied to mitigate the effects of class\nimbalance. Within a given window, all flaring data were retained. However, non-flaring data were\nrandomly sampled to match the number of flaring events, while preserving the original ratio of C-\nclass to B-class to flare quiet events. For example, consider a training window consisting of 119\nX-class, 974 M-class, 5,481 C-class, 5,184 B-class, and 51,160 flare quiet data. There are a total of\n1,093 flaring events, which we want to retain, and 61,825 non-flaring events, which we want to trim\ndown. By calculating the ratio between the number of flaring and non-flaring events (1,093/61,825)\nand multiplying it by the number of C-class, B-class, and flare quiet events, we can determine the\nrequired sample size from each class to preserve their original ratio while adding up to the desired\n1,093 non-flaring events. Consequently, when applying this technique to the previous example, we\nget 97 C-class, 92 B-class, and 904 flare quiet events. Generally, this approach has proven to be\nsuccessful when training and testing with SWAN-SF (Ahmadzadeh et al. 2021).8\n(a)\n(b)\n(c)\nFigure 1. (a) An example of the stationary training window methodology. Each model is trained on a set\nportion of the data at the beginning of the solar cycle (in this example it is the first 20 months). Model\nperformance is then analyzed in consecutive 3-month blocks after training. (b) An example of the rolling\ntraining window methodology. Each model is trained similarly to the stationary window, however, the\nwindow now moves with the testing blocks. (c) A depiction of the expanding window methodology. Here,\nthe models are trained using the entire available dataset, prior to the forecasting instance. The arrows in\neach figure emphasize the temporal continuation of the training windows, testing windows, and the dataset\nitself.9\nLastly, to investigate how performance scales with data volume, a series of stationary and rolling\nwindow sizes (5, 8, 11, 14, 17, and 20 months) were tested. For the stationary windows, data were\nselected starting from May 2010 up until the added window size. For the rolling windows, data were\nselected between the testing window start date and extended into the past by the rolling window size.\nFor each model, a total of three trials were run, each with different randomly undersampled non-flaring\ndata to ensure robust results. Models with training data lacking flaring events were disregarded, and\ninstead, the previously available trained model would be used. This was only pertinent to the rolling\nwindow.\n3.4. Performance Metrics\nTo evaluate the performance of each classifier, the true skill statistic (TSS) and Heidke skill score\n(HSS 2) were calculated for every 3-month testing block. These metrics are defined in the following\nway (Bobra & Couvidat 2015):\nTSS =TP\nTP+FN−FP\nFP+TN(2)\nHSS 2=2×[(TP×TN)−(FN×FP)]\n(TP+FN)×(FN+TN) + (TP+FP)×(FP+TN)(3)\nwhere TP = true positives (the number of correctly predicted flaring events), TN = true negatives\n(the number of correctly predicted non-flaring events), FP = false positives (the number of non-\nflaring events predicted as flaring), and FN = false negatives (the number of flaring events predicted\nas non-flaring). TSS ranges from -1 to +1, with a score of 0 reflecting a classifier that makes random\nor purely positive/negative forecasts, a score of -1 reflecting a classifier that is always wrong, and\na score of +1 reflecting a perfect classifier (Ahmadzadeh et al. 2021). This metric is particularly\nadvantageous as it is unbiased to the class imbalance problem prevalent within flare forecasting\n(Bobra & Couvidat 2015). When using TSS, one must keep in mind that two models with the same\nscore do not necessarily produce an identical number of true positives and true negatives. This\nis because the metric is dependent on the balance of the true positive rate (TP\nTP+FN) and the false\npositive rate (FP\nFP+TN), which can be individually tweaked to achieve the same score (Ahmadzadeh\net al. 2021).\nLike TSS, HSS 2ranges from -1 to +1 and provides an insight into a model’s improvement over a\nrandom forecast. However, the minimum score is now dependent on the class imbalance ratio. As\nit reaches 1:1 (an equal number of flaring and non-flaring events), the lower boundary approaches\n-1 (Ahmadzadeh et al. 2021). A score of 0 is equivalent to a random classifier, a negative score is\nrepresentative of a classifier that performs worse than random, and a score of +1 reflects a perfect\nclassifier. We have selected this definition of the Heidke skill score over the original (HSS 1), high-\nlighted in Bobra & Couvidat (2015), as it tends to be less sensitive to the effects of class imbalance.\nNevertheless, compared to TSS, both definitions are significantly more susceptible, with scores de-\ncreasing as the class imbalance ratio increases (see Figures 2 and 4 in Bobra & Couvidat 2015 and\nAhmadzadeh et al. 2021 for an illustrative example). Overall, both metrics are widely used in the\ncommunity, enabling others to make comparisons to this work, provided that they apply a similar\nmethodology as shown here.10\n3.5. Dependency On The Solar Cycle\nFinally, to highlight performance dependencies on the solar cycle, we investigate the interplay\nbetween a model’s false positive rate ( FPR =FP\nFP+TN) and the solar SXR background flux. We\napproach this by calculating the Spearman correlation between these parameters for each model\nutilizing 25 features. All window types, window sizes, classifiers, and trials were included. Following\nthis, we back up our results by performing a study on how FPR is influenced by the largest flare class\ngenerated by an AR, and how the frequency of complex ARs changes throughout the solar cycle.\n4.RESULTS & DISCUSSION\nIn the following sections, we discuss the results of our work in detail. In Section 4.1, we explore\nthe effects of feature selection on model performance and highlight the magnetogram features most\nfrequently chosen in our forecasts. In Section 4.2, we compare model performance between the\ndifferent window types and investigate how the temporal size of the stationary and rolling training\nwindows affect our predictions. Finally, in Section 4.3, we determine a correlation between the solar\nSXR background flux and the FPR.\n4.1. Impacts Of Feature Selection\nThe dimensionality of the feature space significantly influences the training time and complexity\nof a model. In an operational environment, unnecessary delays and complications must be avoided.\nThus, we explore how the number of features selected for a given model affects performance, in hopes\nof establishing a baseline feature requirement for forecasts utilizing point-in-time magnetogram data.\nFigure 2 summarizes our results. The columns highlight a particular skill score: TSS (left) and HSS 2\n(right), while the rows correspond to our three tested classifiers: DT (top), SVM (middle), and MLP\n(bottom). The radar plots are divided into six sections, one for every feature count tested (1, 5, 10,\n25, 50, and 120 features). Within each wedge, the radial extent of the bars denotes the skill score\nfor a given feature set and window type (color), averaged across all available 3-month testing blocks\n(01/01/2012 - 03/31/2016, 04/01/2017 - 09/30/2017) associated with the 20-month stationary and\nrolling windows. This 20-month window was selected to remove any dependencies on data volume,\nwhich will be explored in Section 4.2.\nAt first glance, we find that TSS and HSS 2scores tend to increase as more features are included in\na forecast. This is expected, given that a higher-dimensionality feature space offers additional means\nto distinguish between the flaring and non-flaring distributions. However, it is rather surprising that,\nfor a given classifier and window type, skill scores improve on average by only 0.035 when jumping\nfrom 1 to 120 features. To check whether these improvements are statistically significant, we can\ncompare the absolute difference between the two feature scores ( |¯X120−¯X1|) and their combined\nstandard errors (q\nσ2\n¯X,120+σ2\n¯X,1). When we do this, we find that 88.8% of the improvements have a\nlarger absolute difference than their combined errors, implying that they are statistically significant\nmeasurements at a 1 σor 68% confidence interval. If we extend this to 2 σ, we discover that over\n61% of the improvements are statistically significant at a 95% confidence interval. This suggests\nthat, in general, our observations are meaningful and not simply due to the uncertainties in our data.\nHowever, the general similarity between scores still warrants further investigation.\nTo explore this topic in more detail, we examine the features that are typically chosen for a forecast,\nthe flaring and non-flaring populations of those features, and the correlations between them. Figure11\nFigure 2. Average TSS and HSS 2scores for the DT, SVM, and MLP classifiers with varying feature counts\n(1, 5, 10, 25, 50, 120) and window types. The error bars illustrate the standard error on the mean. Note:\nThese results were obtained using a stationary and rolling window of 20 months.12\nFigure 3. The average normalized F-value for the 25 highest-scoring features. This metric can be thought\nof as a proxy for selection frequency. Features with values close to 1 tend to be those with the highest scoring\nF-value (and thus more likely to be chosen) in a given training window.\n3 depicts the average normalized F-value for the 25 highest-scoring features. A metric, which can be\nconsidered a proxy for the selection frequency of a particular parameter. To calculate this measure,\nwe first determined the F-value of each feature across all stationary, rolling, and expanding training\ndatasets. For a particular dataset, all F-values were normalized with respect to the highest achieved\nF-value. The scores for each feature were then averaged over all datasets and finally organized in de-\ncreasing order. Please note that even though we display the averages for individual window types, the\norder shown is solely based on the average across all training datasets. Since there is only 1 instance\nof the stationary window, and 19 instances (one for each new testing dataset) for both the rolling and\nexpanding windows, the stationary window only provides a small contribution to this order. From\nthe figure, it is evident that the summary statistics from only a few magnetogram parameters tend to\nbe chosen for a given forecast: ABSNJZH (absolute values of the net current helicity), SAVNCPP\n(sum of the absolute value of net current polarity), TOTUSJH (total unsigned current helicity),\nTOTBSQ (total magnitude of the Lorentz force), TOTPOT (total photospheric magnetic free en-\nergy density), TOTUSJZ (total unsigned vertical current), and USFLUX (total unsigned flux).\nThese features align well with those highlighted in other work within the field (Bobra & Couvidat\n2015; Yeolekar et al. 2021; Zhang et al. 2022).\nFor a more comprehensive look, we have plotted the flaring and non-flaring distributions of the\nhighest-ranking statistics for these features (see Figure 4). These figures reveal an overarching simi-\nlarity between the distributions, with each of them having a right-skewed non-flaring and left-skewed\nflaring population. While significant overlaps exist, demonstrating the challenge behind flare fore-\ncasting, a separation between the medians of these populations can still be resolved. This provides\nenough distinction to make reasonable forecasts utilizing even a single feature (most frequently AB-\nSNJZH max ), which explains the relatively high TSS scores we have obtained.13\nFigure 4. The flaring and non-flaring distributions for ABSNJZH max (the maximum absolute value of\nthe net current helicity), SAVNCPP max (the maximum sum of the absolute value of net current polarity),\nTOTUSJH median (the median total unsigned current helicity), TOTBSQ max (the maximum total mag-\nnitude of the Lorentz force), TOTPOT min (the minimum total photospheric magnetic free energy density),\nTOTUSJZ min (the minimum total unsigned vertical current), and USFLUX min (the minimum total un-\nsigned flux) over the entire SWAN-SF dataset. Distributions are plotted on a log-log scale. Bins containing\nzeros are plotted before the break in the x-axis. The solid blue line indicates the median of the non-flaring\ndistribution. The dotted orange line indicates the median of the flaring distribution.\nCalculating the Spearman correlation between the 7 features in Figure 4, we find that a strong\npositive correlation exists between all of them (see Figure 5). Extending this analysis to all 25\nfeatures in Figure 3, it comes as no surprise that a similar trend is found, with all unique correlations\nbeing≥0.72 and 80% of them being ≥0.90. Though correlation does not necessarily mean that two\nfeatures aren’t complementary (Guyon & Elisseeff 2003), we believe that this could still be a plausible\nexplanation for our results. Highly correlated features often provide similar information about the14\nFigure 5. A Spearman correlation heatmap for the features selected in Figure 4.\ntarget class. Therefore, combining them will result in only minor improvements in the separability\nof the population. In our case, the 25 most important features are highly correlated, which may\nexplain why the performance from 1 to 25 features is fairly comparable. Beyond 25, the additionally\nincluded features have diminishing F-values, making it significantly more difficult to separate between\nflaring and non-flaring events (at least in a 1-dimensional sense). This may explain the only marginal\nperformance improvements found in this range.\nShifting our focus to individual window types, we find that the rolling window consistently matches\nor outperforms the stationary and expanding windows, particularly when utilizing only 1 or 5 features.\nWe speculate that this may be a consequence of the window’s ability to capture the current flare\noccurrence rate. On a large scale, performance differences between the three window types are\nrelatively minimal. This is unexpected given that, during the latter half of the solar cycle, the\nexpanding window has access to significantly more flaring data than the other windows (see Figure\n6). This suggests that, with a sufficient amount of data, a stationary classifier may be chosen over\nother window types. This not only saves time but dramatically reduces the difficulty of implementing\nan operational model. Of course, these results utilize a relatively large training window. Utilizing a15\nFigure 6. The number of flaring events as time progresses in the 20-month stationary, rolling, and expanding\nwindows. The red regions illustrate periods where no testing windows exist, so no model was trained.\nsmaller stationary or rolling window may not produce the same results. We explore this further in\nSection 4.2.\nFinally, comparing results across the three tested classifiers, it becomes evident that MLPs yield the\nbest TSS and HSS 2scores, regardless of the number of features or window type selected (see Figure\n7). Given the algorithm’s complexity, this is expected. However, the narrow difference between the\nskill scores of all three classifiers is rather surprising. This clearly suggests that easily interpretable\nmodels, such as DTs, may be a viable alternative to more complicated models when provided with a\n20-month training window. We investigate this trend for different window sizes in Section 4.2.\n4.2. Impacts Of Training Window Size\nIn addition to feature selection, data volumes are critical to producing effective flare forecasting\nmodels. Without the proper amount of training data, ML algorithms fail to capture an adequate\ndecision boundary, which can significantly degrade performance. Since data volumes can vary during\noperational deployment, it is essential to explore how skill scores are affected by this aspect and how\nthese restrictions interact with our custom training windows. Figure 8 summarizes our results. Once16\nFigure 7. Average TSS and HSS 2scores for the DT, SVM, and MLP classifiers with varying feature counts\n(1, 5, 10, 25, 50, 120) and window types. The error bars illustrate the standard error on the mean. This plot\nis similar to Figure 2, except that a comparison is now being made between classifiers instead of window\ntype. Note: These results were obtained using a stationary and rolling window of 20 months.17\nFigure 8. Average TSS and HSS 2scores for the 25 feature DT, SVM, and MLP with varying stationary\nand rolling window sizes (5, 8, 11, 14, 17, 20 months). Naturally, the skill scores for the expanding window\nare the same across different window sizes. They are included for reference. The error bars illustrate the\nstandard error on the mean. Note: The 8-month stationary DT (the red wedge) has an average TSS score\nof 0.26 ±0.05.18\nagain, each column highlights a specific skill score, and each row a particular model. The radar plots\nare divided into six sections, one for every stationary and rolling window size tested (5, 8, 11, 14,\n17, and 20 months). Within each wedge, the radial extent of the bars denotes the skill score for a\ngiven window size and type, averaged across all testing data (01/01/2012 - 03/31/2016, 04/01/2017\n- 09/30/2017) associated with the 20-month stationary and rolling training windows. This ensures\nthat our findings are comparable across window sizes. Since the expanding window utilizes all data\nprior to the forecasting instance, its scores are the same across all wedges.\nFirst, examining the general skill score trends, we find that the effects of window size on TSS\nand HSS 2are dependent on the classifier and window type selected. For certain combinations, such\nas the DT with rolling window, removing training data results in a steady decline in performance,\nas one might expect. However, we find that this is non-universal, with a majority of scores being\ncompletely uncorrelated with one another. For example, the stationary SVM TSS is larger for the\n5-month window than the 20-month window, even though it has significantly fewer flares. This hints\nthat there may be some underlying limitations to our dataset, which we suspect are imposed by our\nmethodology. When calculating the summary statistics of each time series, we remove potentially\nsignificant knowledge related to the dynamics of an event. This gives us a lighter dataset, that\nis easier to work with but may not be as informative. It is apparent that our models are able to\ncapture some important aspects needed to predict flares, as they achieve fairly high skill scores, but\nwith additional data, this can only improve so much. Without more exhaustive features, which can\nbe taken advantage of by our ML models, increasing data volumes will not provide enough new\ninformation about the flaring population to significantly affect performance. A potential solution to\nthis problem is to train models utilizing the entire time series, which has been shown to improve\nskill scores in SWAN-SF (Ji et al. 2020). However, it remains to be seen how, or even if, this would\naffect our data volume results. Lastly, a recent study has shown that magnetogram data alone does\nnot provide significant improvements over human-based forecasting (Leka et al. 2019b). This hints\nthat there may be some inherent simplicity to magnetogram data itself, which limits its predictive\ncapability and contributes to the findings shown here.\nTaking a deeper dive into our results, we find that the stationary window almost always produces\nbetter TSS, but noticeably worse HSS 2scores, than the rolling window when the window size is less\nthan 20-months. Since the stationary window covers the beginning of the solar cycle, where the\nnumber of flaring events is low, models will be biased toward capturing each flaring event in the\ntraining data. This is because missing a single flare has a large impact on the true positive rate\n(TP\nTP+FN) and in turn the TSS score, which we are attempting to maximize when training the model.\nThis leads to the stationary window producing fewer false negatives and more false positives while\ntesting, which has less of an effect on TSS than HSS 2.\nLastly, when comparing performance across classifiers (see Figure 9), we again find that MLPs yield\nthe best TSS and HSS 2scores by only a small margin. SVMs and DTs, follow closely behind, even\noccasionally outperforming MLPs in select window types, sizes, and skill scores. This extends our\nconclusion made in 4.1, that less complex models can be reliably used in place of more sophisticated\nalgorithms, to varying data volumes. However, we find that when window sizes get too small, one\nmust be cautious. The 8-month stationary DT window produces dreadful TSS and HSS 2scores,\nwhich we suspect may be a consequence of the algorithm itself. It is well known that DTs tend to\nstruggle with instabilities and under/overfitting, with small changes in their training data producing19\nFigure 9. Average TSS and HSS 2scores for the 25 feature DT, SVM, and MLP with varying stationary\nand rolling window sizes (5, 8, 11, 14, 17, 20 months). The error bars illustrate the standard error on the\nmean. This plot is similar to Figure 8, except a comparison is now being made between classifiers instead of\nwindow type. Note: The 8-month stationary DT (the red wedge) has an average TSS score of 0.26 ±0.05.20\nvastly different trees (Li & Belford 2002). These effects, compounded with the fact there is little\ntraining data, lead to a higher chance of producing a decision boundary that does not accurately\nseparate the entire flaring population in the testing dataset.\n4.3. Dependency On The Solar Cycle\nFinally, to investigate the impact of the solar cycle on model performance, we explore the relation-\nship between FPR and a proxy for the SXR background flux (the minimum daily GOES flux value).\nFigure 10 summarizes our results. Here we find that a moderately strong positive correlation (mean\nofρ= 0.570) exists for the conglomeration of our 25 feature trials (see the All Data label in Figure\n10). This indicates that as we reach solar maximum (when background levels are high), the FPR also\nincreases. We find that models utilizing the rolling window or the MLP classifier tend to be more\nsusceptible to this trend. Interestingly, no window type or classifier is impervious to this correlation.\nLargest Flare Produced By AR False Positive Rate Of ARs Within Flare Group\nFlare Quiet / A 0.010±0.001\nB 0.077±0.004\nC 0.312±0.007\nM 0.699±0.008\nX 0.823±0.008\nTable 2. The FPR for ARs grouped by the strongest flare produced during their lifetime. Results are\naveraged across data from all models utilizing 25 features. The standard error on the mean is shown as well.\nTo explore this further, we then examine how the largest flare class generated by an AR influences\nits FPR. To accomplish this, we first divide our ARs into flaring groups dependent on the strongest\nevent they produce within their lifetime (X, M, C, B, or A / flare quiet). Any point-in-time data\nassociated with a particular AR is placed within the same group. We then recalculate the FPR for\nthe data in each AR category and average our findings across all models utilizing 25 features. Table\n2 presents our results. It is evident that ARs producing M or X-class flares generally have elevated\nFPRs in comparison to ARs generating weaker flares. This is somewhat expected, given that B and\nC-class flares, as well as flare quiet periods, occur intermittently throughout flaring episodes, which\ncan be challenging to detect. ARs may appear to have high-magnetic activity over the previous 12\nhours (in comparison to a typical non-flaring event) but do not end up flaring. This, of course, leads\nto significantly more false positive predictions for these ARs.\nBuilding on this, we then consider the frequency of complex ARs (those more likely to produce M\nand X-class flares) throughout the solar cycle. In Figure 11, we plot the ratio of ARs with a Hale\nclassification > β(this includes γ,β−γ,δ,β−δ,β−γ−δ, and γ−δ) to the total number of ARs,\nbinned monthly for Solar Cycle 24. Here, we find that more complex ARs have a higher likelihood of\nexisting during the peak of the solar cycle (near 2014) compared to the beginning or end. Tying this\nback to our findings from Table 2: if the probability of having a more complex AR is higher during\nthe peak of the cycle, and ARs producing stronger flares tend to have larger FPRs, then there is21\n(a)\n (b)\n(c)\n (d)\nFigure 10. (a) Boxplots of the Spearman correlation between the FPR and the background SXR flux.\nPlots are made for the entire collection of results, window types, and classifiers. Results are shown only\nfor models that utilize 25 features. A swarm plot is overlaid to emphasize the distribution of data. The\ntriangles indicate the mean of the distributions. (b, c, d) Scatter plots of the FPR versus background SXR\nflux for single trials of the 20-month DT, SVM, and MLP models. The correlation for each window type is\ngiven by ρ. The FPR and background SXR were binned monthly to reduce noise due to daily fluctuations.22\nFigure 11. The AR complexity ratio versus time over Solar Cycle 24. AR complexity ratio is calculated\nby dividing the number of ARs with a Hale classification more complex than β(this includes γ,β−γ,δ,\nβ−δ,β−γ−δ, and γ−δ) by the total number of ARs across each month. A Savitzky-Golay filter (solid\nred line) has been applied to illustrate the general trend of the data.\nreason to believe that the FPR will increase with background SXR flux. Of course, it is important\nto note that our forecasts are based solely on magnetic field parameters, with no direct relationship\nto the background SXR flux. Thus, we would like to emphasize that this result is merely a statistical\nobservation rather than a causal relationship.\n5.SUMMARY & CONCLUSIONS\nIn this study, we focused on producing a simulated real-time prediction environment, which can be\nused as a test bed to analyze how a variety of classifiers, features, data volumes, and the solar cycle\nimpact operational performance. From this work, we have identified the following key results:\n1. Across all window types, the most frequently chosen magnetogram features are ABSNJZH\n(absolute values of the net current helicity), SAVNCPP (sum of the absolute value of net\ncurrent polarity), TOTUSJH (total unsigned current helicity), TOTBSQ (total magnitude\nof the Lorentz force), TOTPOT (total photospheric magnetic free energy density), TOTUSJZ\n(total unsigned vertical current), and USFLUX (total unsigned flux). This corresponds well23\nwith results from other papers within the field (Bobra & Couvidat 2015; Yeolekar et al. 2021;\nZhang et al. 2022).\n2. The number of magnetogram features used to make a prediction does not have a significant\neffect on TSS or HSS 2scores. Only a marginal increase in performance is observed as additional\nfeatures are included in a forecast. We believe this may be an outcome of the highly correlated\nnature of our features.\n3. When utilizing a 20-month stationary or rolling window, performance is generally comparable\nto the expanding window. Only a minor decrease in performance is observed for the stationary\nand rolling windows when their size is reduced. This suggests that, provided with a sufficient\namount of data, a stationary classifier can be chosen over other window types, removing the\nneed for retraining. We believe this to be a consequence of our methodology or potentially an\ninherent simplicity of the magnetogram data itself.\n4. Simple and interpretable machine learning classifiers, such as decision trees, provide skill scores\nsimilar to those of more complex models.\n5. A moderately strong positive Spearman correlation exists between a model’s false positive\nrate and the background soft X-ray flux. We hypothesize that this is a consequence of highly\ncomplex active regions (those more likely to produce M and X-class flares) appearing more\nfrequently during the peak of the solar cycle. From our analysis, we observed that these active\nregions tend to be accompanied by larger false positive rates.\nOverall, we can conclude that for operational forecasts utilizing point-in-time magnetogram data,\nthe number of features, window size, window type, and classifier used have a minimal impact on\nperformance, at least for those we tested.\nRegarding future studies, there are numerous paths we can explore. First, it may be valuable to\ninvestigate whether utilizing temporally dependent features, such as the time series derivative of a\nparameter, has any impact on the forecasting results shown here. These descriptive statistics could\ngive a model better insight into how an active region is growing/decaying over time, which may\nlead to improved performance. However, recent work by Nishizuka et al. (2017) found that these\nfeatures are ineffective on time scales less than 24 hours. This indicates that we would likely need to\nextend our 12-hour observation window to benefit from them. A better alternative would be to train\nmodels directly on the time series data itself, rather than the point-in-time summary statistics. This\ncould be accomplished through more complicated deep learning algorithms such as long short-term\nmemory (LSTM) networks, which have been employed in other studies (Liu et al. 2019; Sun et al.\n2022). Of course, with these models comes added training time and a need for increasingly powerful\ncomputational resources, which is not ideal for operational purposes. Lastly, a major drawback of the\ncurrent SWAN-SF iteration is its focus on 24-hour forecasting. With the rapidly approaching NASA\nArtemis missions, it will be critical that we have the capability of predicting flaring events even farther\nin advance (hopefully up to 72 hours). While not addressed in this paper, it may be worthwhile in\nfuture studies to modify the current dataset labels for several extended forecasting windows (36, 48,\n72 hours). This may reveal hidden intricacies between our various training methodologies, not found\nin this work.24\n6.ACKNOWLEDGEMENTS\nWe would like to thank the anonymous reviewer for their excellent feedback, which greatly improved\nthe clarity of the text. This work was supported by NASA FINESST grant 80NSSC23K1639, NASA\nLWS grant 80NSSC22K0272, and NSF FDSS grant 1936361.\nREFERENCES\nAhmadzadeh, A., Aydin, B., Georgoulis, M. K.,\net al. 2021, The Astrophysical Journal\nSupplement Series, 254, 23\nAli, A., Sadykov, V., Kosovichev, A., et al. 2024,\nThe Astrophysical Journal Supplement Series,\n270, 15\nAngryk, R., Martens, P., Aydin, B., et al. 2020a,\nSWAN-SF, V1, Harvard Dataverse,\ndoi: 10.7910/DVN/EBCFKM\nAngryk, R. A., Martens, P. C., Aydin, B., et al.\n2020b, Scientific data, 7, 227\nBobra, M. G., & Couvidat, S. 2015, The\nAstrophysical Journal, 798, 135\nCamporeale, E. 2019, Space Weather, 17,\ndoi: 10.1029/2018SW002061\nCrown, M. D. 2012, Space Weather, 10, S06006,\ndoi: 10.1029/2011SW000760\nDeshmukh, V., Baskar, S., Berger, T. E., Bradley,\nE., & Meiss, J. D. 2023, A&A, 674, A159,\ndoi: 10.1051/0004-6361/202245742\nFlorios, K., Kontogiannis, I., Park, S.-H., et al.\n2018, Solar Physics, 293, 28\nGardner, M., & Dorling, S. 1998, Atmospheric\nEnvironment, 32, 2627, doi: https:\n//doi.org/10.1016/S1352-2310(97)00447-0\nGuyon, I., & Elisseeff, A. 2003, J. Mach. Learn.\nRes., 3, 1157–1182\nHudson, H. S. 2021, Annual Review of Astronomy\nand Astrophysics, 59, 445\nJi, A., Aydin, B., Georgoulis, M. K., & Angryk,\nR. 2020, in 2020 IEEE International Conference\non Big Data (Big Data), 4218–4225,\ndoi: 10.1109/BigData50022.2020.9377906\nKingma, D. P., & Ba, J. 2014, arXiv e-prints,\narXiv:1412.6980, doi: 10.48550/arXiv.1412.6980\nKingsford, C., & Salzberg, S. L. 2008, Nature\nbiotechnology, 26, 1011\nKotsiantis, S. B. 2013, Artificial Intelligence\nReview, 39, 261\nLeka, K., Park, S.-H., Kusano, K., et al. 2019a,\nThe Astrophysical Journal Supplement Series,\n243, 36Leka, K. D., Park, S.-H., Kusano, K., et al. 2019b,\nThe Astrophysical Journal, 881, 101,\ndoi: 10.3847/1538-4357/ab2e11\nLi, R., Wang, H.-N., He, H., Cui, Y.-M., & Du,\nZ.-L. 2007, Chinese Journal of Astronomy and\nAstrophysics, 7, 441\nLi, R.-H., & Belford, G. G. 2002, in Proceedings\nof the eighth ACM SIGKDD international\nconference on Knowledge discovery and data\nmining, 570–575\nLiu, H., Liu, C., Wang, J. T., & Wang, H. 2019,\nThe Astrophysical Journal, 877, 121\nMarroquin, R. D., Sadykov, V., Kosovichev, A.,\net al. 2023, The Astrophysical Journal, 952, 97,\ndoi: 10.3847/1538-4357/acdb65\nNatras, R., Horozovic, D., & Mulic, M. 2019, SN\nApplied Sciences, 1, 1\nNishizuka, N., Sugiura, K., Kubo, Y., Den, M., &\nIshii, M. 2018, ApJ, 858, 113,\ndoi: 10.3847/1538-4357/aab9a7\nNishizuka, N., Sugiura, K., Kubo, Y., et al. 2017,\nThe Astrophysical Journal, 835, 156\nPedregosa, F., Varoquaux, G., Gramfort, A., et al.\n2011, Journal of Machine Learning Research,\n12, 2825\nSadykov, V. M., & Kosovichev, A. G. 2017, ApJ,\n849, 148, doi: 10.3847/1538-4357/aa9119\nScherrer, P. H., Schou, J., Bush, R. I., et al. 2012,\nSoPh, 275, 207, doi: 10.1007/s11207-011-9834-2\nSun, Z., Bobra, M. G., Wang, X., et al. 2022, The\nAstrophysical Journal, 931, 163\nWang, X., Chen, Y., Toth, G., et al. 2020, The\nAstrophysical Journal, 895, 3\nYeolekar, A., Patel, S., Talla, S., et al. 2021, in\n2021 International Conference on Data Mining\nWorkshops (ICDMW), 1067–1076,\ndoi: 10.1109/ICDMW53433.2021.00138\nYu, D., Huang, X., Wang, H., & Cui, Y. 2009,\nSolar Physics, 255, 91\nYuan, Y., Shih, F. Y., Jing, J., & Wang, H.-M.\n2010, Research in Astronomy and Astrophysics,\n10, 78525\nZhang, H., Li, Q., Yang, Y., et al. 2022, The\nAstrophysical Journal Supplement Series, 263,\n28"    },    {        "title": "2402.05317v1.Emerging_Results_on_Automated_Support_for_Searching_and_Selecting_Evidence_for_Systematic_Literature_Review_Updates.pdf",        "content": "This is the authors ’ version of the work. The definitive version was published in WSESE @ICSE 2024 . \n \nEmerging  Results  on Automated  Support  for Searching  and \nSelecting  Evidence  for Systematic  Literature  Review  Updates  \nBianca Minetto Napoleão  \nUniversité  du Québec  à Chicoutimi  \nChicoutimi,  Canada  \nbianca.minetto -napoleao1@uqac.ca  Ritika  Sarkar  \nUniversité du  Québec  à Chicoutimi  \nChicoutimi, Canada  \nrsarkar@etu .uqac.ca  Sylvain  Hallé  \nUniversité  du Québec  à Chicoutimi  \nChicoutimi, Canada  \nshalle@acm.org  \n \n \n \n \n \n \n \nABSTRACT  Fabio  Petrillo  \nÉcole  de Technologie  Supérieure  \nMontreal, Canada  \nfabio.p etrillo@etsmtl.ca  Marcos  Kalinowski  \nPontifical  Catholic  University  of Rio \nde Janeiro  \nRio de Janeiro, Brazil  \nkalinowski@inf.puc -rio.br  \nContext: The constant growth of primary evidence and Systematic  \nLiterature Reviews (SLRs) publications in the Software Engineering  \n(SE) field leads to the need for SLR Updates. However, searching  \nand selecting evidence for SLR updates demands significant effort  \nfrom SE researchers. Objective: We present emerging results on  \nan automated approach to support searching and selecting stud - \nies for SLR updates in SE. Method: We developed an automated  \ntool prototype to perform the snowballing search technique and  \nto support the selection of relevant studies for SLR updates using  \nMachine Learning (ML) algorithms. We evaluated our automation  \nproposition through a small -scale evaluation with a reliable dataset  \nfrom  an SLR  replication  and its update.  Results:  Effectively  au- \ntomating  snowballing -based  search  strategies  showed  feasibility  \nwith minor losses, specifically related to papers without Digital  \nObject Identifier (DOI). The ML algorithm giving the highest perfor - \nmance  to select  studies  for SLR updates  was Linear  Support  Vector  \nMachine with approximately 74% recall and 15% precision . The use  \nof such  algorithms  with conservative  thresholds  to minimize  the \nrisk of missing papers can already significantly reduce evidence  \nselection efforts. Conclusion: The preliminary results of our eval - \nuation point in promising directions, indicating the potential of  \nautomating  snowballing  search  efforts  and of reducing  the num- \nber of papers to be manually analyzed by about 2.5 times when  \nselecting  evidence  for updating  SLRs  in SE. \n \n \nKEYWORDS  \nSystematic Review Update, SLR Update, Searching for evidence,  \nSelecting  evidence  \n  \n1 INTRODUCTION  \nEvidence -Based Software Engineering (EBSE) has as the main in - \nstrument Systematic Literature Reviews (SLRs) to summarize evi - \ndence from primary studies (e.g ., case studies, surveys, controlled  \nexperiments),  providing  recommendations  for practitioners  and \nsupporting the decision -making process in Software Engineering  \n(SE) [15].  \nOver  the almost  20 years  of SLRs  in SE, the number  of SLRs  has \nincreased  substantially  [17, 19], leading  to the need to update  SE \nSLRs  [17]. A not-maintained  SLR could  lead researchers  to obsolete  \nconclusions  or decisions  about  a research  topic  [33] (e.g.,  studies  \nthat employed  a particular  technology  that is now outdated)  [36].  \nPerforming  SLR updates  demands  significant  effort  especially  \nbecause  of the rapid  increase  of available  evidence [ 31, 38]. Identi - \nfying  studies  for SLR consists  of two main  activities:  (i) searching  \nfor potentially  relevant  studies  and (ii) selecting  the real relevant  \nones.  \nAiming to facilitate the search activity, Felizardo et al. [8] in- \ntroduced the use of forward snowballing (i.e., citations analysis  \nfrom  the included  studies  in the original  SLR a.k.a.  seed set [34]) \nto support searching for evidence in SLR updates. A few years  \nlater, Wohlin et al. [36] further evaluated search strategies for SLR  \nupdates and found that the use of a single iteration forward snow - \nballing employing as a seed set the original SLR and its primary  \nstudies tends to be the most cost -effective way to s earch for new  \nevidence  when  updating  SLRs.  \nWith respect to reducing the effort of study selection consider - \ning specifically  the SLR update  scenario,  only two attempts  have  \nbeen investigated by researchers [11, 33]. Both studies present ap - \nproaches  addressing  automation  support:  Felizardo  et al. [11] based  \ntheir approach on Visual Text Mining (VTM) techniques, while  \nWatanabe et al. [33] demonstrated the potential of supervised Ma - \nchine  Learning  (ML)  algorithms  to support  the selection  activity.  Bianca  Minetto  Napoleão,  Ritika  Sarkar,  Sylvain  Hallé,  Fabio  Petrillo,  and Marcos  Kalinowski   \n \nThe goal of this study  is to propose  and evaluate  an automated  an \napproach to support both the searching and selecting of studies for  \nSLR updates in SE. To achieve this goal, we developed an automated  \ntool prototype to perform the snowballing search technique [ 34] \nand also to support  the selection  of relevant  studies  for SLR updates  \nby investigating the practical use of ML algorithms. We evaluated  \nour automation  proposition  by performing  small -scale  evaluation  \n[37] with a reliable dataset from an S LR replication [ 35] and an  \nongoing update. Our preliminary results indicate that snowballing - \nbased search strategies can be fully automated with minor losses  \nand that significant effort can be saved in SLR update study selection  \nusing ML, allowing a conservatively reduced number of papers to  \nbe manually  analyzed by  about 2.5  times.  \nThe remainder  of this study  is organized  as follows.  Section  \n2 presents a brief background and related work. In Section 3 we \ndescribe the prototype tool. The small -scale evaluation is reported  \nin Section 4. Section 5 discusses our results. Section 6 relates threats  \nto validity.  Finally,  Section  7 concludes  our work.  \n \n2 BACKGROUND  AND  RELATED  WORK  \nAccording  to Mendes  et al. [17] an SLR update  is a new version  \nof a published SLR that includes new primary studies that can  \ncome from a different search strategy than the original SLR (e.g.,  \nsnowballing, manual or database search). Identifying new releva nt \nevidence for updating an SLR is one of the initial steps in identifying  \nthe need and possibility  of updating  an SLR.  \nIn SE, there are studies that investigated search strategies dedi - \ncated  to searching  for evidence  for updating  SLRs.  Felizardo  et al. \n[8] proposed  the use of forward  snowballing  as a search  strategy  \nto update SE SLRs. Their results showed a reduction of more than  \nfive times  the quantity  of primary  studies  to be analyzed  during  \nan SLR update. Later, in 2020,  Wohlin et al. [36] further investi - \ngated the use of forward snowballing to update the SE SLRs. Their  \nstudy  proposed  and evaluated  guidelines  for the search  strategy  \nto update SLRs in SE. They found that using a single itera tion of  \nforward snowballing, with Google Scholar as the search engine,  \nand employing the original SLR and its primary studies as a seed set  \ntends to be the most cost -effective way to search for new evidence  \nfor updating an SLR. One of the goals of our study is to automate  \nforward snowballing as proposed by [ 8, 36] to facilitate the search  \nactivity.  \nGiven  that the selection  of evidence  resulting  from  the execution  \nof the search strategy is a laborious activity [ 7, 10], there are two  \nrelated  works  that address  alternatives  to remedy  this issue  in \nSLR updates. The first one by Felizardo et al. [11] explored visual  \ntext mining  to support  selecting  new evidence  (primary  studies)  \nfor SLR updates. The tool presented, called Revis, connects the  \nnew evidence with the evidence of the original SLR applying the  \nKNN (K -Nearest Neighbor) Edges Connection technique presenting  \nthe results in two different visualizations: content -map and Edge  \nBundles  diagram.  The results  showed  an increase  in the number  \nof studies correctly included compared to the traditional manual  \napproach.  \nThe second  one by Watanabe  et al. [33] also takes  advantage  \nof the fact that a publishe d SLR that needs updating already has a  \nlist of included  studies.  They  investigated  using  text classification   \ntechniques, including supervised ML algorithms (Decision Tree and  \nSupport Vector Machines), to make the initial selection of primary  \nevidence (based only on the title and abstract of the studies) to  \nupdate SLRs. The study indicated potential of using automated  \ntechniques to reduce the effort required to select studies fo r SLR  \nupdates.  \nIt is worth mentioning that automation of the search and selec - \ntion of studies has been also investigated in the context of SLR  \nconduction, for example: (i) Carver & Felizardo [ 9] present an  \noverview  of existi ng automation  alternatives  for all the activities  of \nthe SLR process; and (ii) Napoleão et al. [20] focus on the search and  \nselection activities presenting a systematic mapping on existing  \nautomation support to search and sel ection of studies addressing  \nboth the SE and  medicine domains.  \nOur study  builds  on knowledge  gathered  in the related  work  and \nproposes and investigates a search and selection approach with  \nmain  focus  on the SLR update  scenario.  \n \n \n3 TOOL  DEVELOPMENT  \nIn this Section, we present details about our prototype tool devel - \noped  to support  the snowballing  search  activity  and the selection  \nof studies  activity  for SLR Updates.  \nWe started with the development of a prototype tool to perform  \nboth snowballing techniques, forward (citation analysis) and back - \nward  (references  analysis)  [34]. Even  though  the main  objective  \nof this study  is to investigate  automation  support  for searching  \nand selecting studies for the SLR updates, which do not require  \nbackward snowballing, during development we noticed that the de - \nsign of the forward solution was easily replicated for the backward  \nsolution.  Therefore,  we opted  for the development  of a snowballing  \ntool addressing both  types.  \nThe execution flow of our proposed algorithm for developing  \nthe snowballing prototype tool is shown in Figure 1. The snow - \nballing automation is preceded by inputs from the user in the form  \nof Digital Object Identifier (DOI) URLs of papers in the seed set.  \nFollowing  this, the implementation  code is run and the user is asked  \nto inform  the number  of snowballing  iterations  he/she  wants  to \nrun and whether they wish to proceed with either backward or  \nforward snowballing, or both. We chose to add these two inputs  \nbecause, for the context of SLR updates, a single iteration of forward  \nsnowballing  is enough  to return  the relevant  studies  [8, 36]. \nThe snowballing solution (backward and forward) is implemented  \nby querying the Semantic Scholar API [ 30] based on the DOI of  \nstudies.  The metadata  returned  as the query  results  are employed  \nto extract the DOIs of citations and references cited in the queried  \nstudy.  In case of studies  without  DOIs,  CrossRef  API [5] is queried  \nfor DOIs by providing the input as a referenc e string generated  \nusing the keys ‘authors’, ‘title’, ‘venue’, and ‘year’ returned in the  \nmetadata  by Semantic  Scholar  [30]. \nNext,  the acquired  DOIs  are passed  through  a redundancy  check  \n(for subsequent iterations) to ensure that the extraction has not  \nbeen done for them in previous iterations. The main part of the  \ndata acquisition starts with getting the bibliographical metadata  \nof the references and citations in the BibTeX format, by making a  \nrequest  to the DOI using  the Urllib  [24] library.  Emerging  Results  on Automated  Support  for Searching  and Selecting  Evidence  for Systematic  Literature  Review  Updates   \n \n \n \nFigure  1: Execution  flow of the snowballing  tool \n \nThe abstract is usually missing in the response and hence we  \nneed to apply other methods to get the abstract. For the abstract  \nextraction,  we make  use of ResearchGate  [27], which  allows  re- \nsearchers and students free access to abstracts of scientific publica - \ntions  and preprints  of research  work.  We employed  web scraping  \nto log in to ResearchGate with institutional credentials, and query  \nfor the studies using the DOIs. As the page is rendered by the  \nbrowser,  the abstract  is extracted  and saved  to the BibTeX  of the \ncorresponding  study.  \nThe results of the multiple iterations of the snowballing process  \nare stored in one common CSV file and one common BibTeX file  \nat the end of the runs. This applies to each type of snowballing.  \nTherefore,  if the user wishes  to perform  both backward  and forward  \nsnowballing together, there will be 2 separate files (CSV and .bib)  \nfor each type of snowballing. The CSV file stores the reference  \nstrings,  the corresponding  DOIs,  the status  of the extraction,  and \nthe iteration number. The reference string is gen erated using the  \nkeys ‘authors’, ‘title’, ‘venue’, and ‘year’ returned in the metadata by  \nSemantic Scholar [ 30], which are joined to produce a string similar  \nto the Chicago format of referencing. This is done because the full  \nreference  of a study  is not returned  by the Semantic  Scholar.  The \nstatus of the extraction for a particular study uses the following  \nimplicit phrases: “Extraction successful\", “DOI not found”, “.bib file  \nnot found”,  “Abstract  not found”,  and “Done  already  in X” where  X \nis the iteration  number.  \nSince we store the results of all the iterations in a common CSV  \nfile, the column “iteration n umber” tells us in which iteration of  \nsnowballing  a particular  study  was discovered.  The BibTeXs  of the \nstudies are stored in a common .bib file. Each BibTeX is appended  \nto the common BibTeX file after each extraction phase. Another  \nfeature of the tool hel ps us to obtain the BibTeX file for the seed  \nset as well.  \nRegarding the selection of studies portion of the tool, we opted  \nto evaluate a range of ML algorithms known to perform well for  \ntext classification  [1, 22]: XGBoost  (XGB)  [4], Linear  Support  Vec- \ntor Machines (LSVM) [ 16], Logistic Regression (LogReg) [ 12], and  \nMultinomial Naïve Bayes (MNB) [ 14]. We chose them because all  \nfour algorithms have a regularization term, which plays an impor - \ntant role in combating overfitting and underfitting in unbalanced  \ndatasets by adding pen alties to the loss function. The second -most  \nimportant  term is “class  weight”  which  prevents  the prejudice  of the it. They  are reciprocal  of the class  frequencies.  We detail  the tool’s  \nselection  process  and parametrization  in Section  4.2 by providing  a \npractical  evaluation  example.  \n4 SMALL -SCALE  EVALUATION  \nTo evaluate our prototype tool, we performed a small -scale eval - \nuation [ 37]. According to the smell indicator proposed by Wo hlin \n& Rainer [ 37], the correct label for our evaluation is small -scale  \nevaluation  instead  of a case study.  However,  to guide  and report  our \nevaluation process we followed the five main steps for case studies  \nproposed by Runeson et al. [28]: Design, preparation, collecting  \ndata,  analysis, and  reporting.  \n4.1 Design  \nOur design consists in selecting a published SLR replication [ 35] and  \nits ongoing SLR updat e evaluation instrument to search for studies  \nthrough snowballing iterations and perform an initial selection of  \npotentially  relevant  studies  to be included  in an SLR update.  \nTo evaluate  the prototype  tool’s  capability  of performing  back- \nward  and forward  snowballing,  we opted  to use the SLR replication  \nstudy  [35] since  it documents  in the supplementary  material1 the \nresults  of each  snowballing  iteration.  In summary,  our goal with \nthis is to illustrate  the tool’s  potential  to be used to perform  both \nsnowballing search types in an SLR conduction process when a seed  \nset of studies is known by the authors (e.g. selected from database  \nsearch  [35]).  \nNext, to evaluate the tool’s capability of being employed in the  \nSLR update context, the main goal of our study, we used the 45  \nselected studies by the SLR replication [ 35] as a seed set to per - \nform  an iteration  of forward  snowballing  and then apply  the ML \nalgorithms on a reliable and complete dataset from the ongoing  \nSLR update of [ 35]. In this replication and the ongoing update, the  \ninclusion  and exclusion  of new studies  were  conducted  based  on \nindividual  assessments  and the consensus  of three  experienced  SLR \nresearchers,  allowing  us to have confidence  in this data for building  \nreliable  training  and testing  sets. \n4.2 Preparation  and Collecting  data  \nTo evaluate  the tool’s  capabilities  to perform  backward  and forward  \nsnowballing  we first prepared  our seed set by obtaining  the DOI of \nmodel  towards  the minority  class,  by assigning  a higher  weight  to    \n1  https://ars.els -cdn.com/content/image/1 -s2.0-S0950584922000659 -mmc2.pdf  \nBianca  Minetto  Napoleão,  Ritika  Sarkar,  Sylvain  Hallé,  Fabio  Petrillo,  and Marcos  Kalinowski   \n \nthe studies mentioned as seed set (9 studies) in the supplementary  \ndata of the SLR replication [ 35]. Next, we replicated with our tool the  \nsame 5 iterations of backward and forward snowballing performed  \nmanually by the authors. Finally, we compared our tool’s results  \nwith the manual execution.  \nRegarding the analysis of the search and selection for SLR up - \ndates, we conducted three steps. First , we search performing a  \nsingle forward snowballing iteration using the 45 included studies  \nby the SLR replication [ 35] as seed set. Second, we train our ML  \nalgorithms with the “training set” containing both included and  \nexcluded studies of the SLR replication [ 35]. Finally, we perform  \nthe selection of studies using the trained algorithms on the results  \nof the forward  snowballing  in the first step (“testing  set”).  \nThe distribution of included and excluded studies in the training  \nand the testing sets are shown in Figure 2. It is highly imbalanced  \nwith a minority of included studies. The imbalance repre sents the  \nreal-world scenario where out of a large number of studies only a  \nfew are typically  relevant  to the focus  of a particular  research  topic.  \n \n \nFigure  2: The data  distribution  of (a) Training  set and (b) \nTesting  set \n \nThe results  obtained  using  our algorithms  are compared  with \nthe included articles selected manually for this SLR update (the  \n“testing  set”) to evaluate  the recall,  precision,  and F-measure  of the \nselected algorithms and the automated snowballing tool. Figure 3 \nsummarizes  the evaluation  of the tool’s  selection  process.  \nRegarding  the training  process,  for Training  Data  Collection , as \nshown in Figure 3, the input consists of the included and excluded  \nstudies in BibTeX format (.bib) of the SLR replication [ 35]. We  \nconverted the .bib file into CSV format by taking the ‘Title’ and  \n‘Abstract’ fields of the studies. We also label  them with relevance 1  \nfor included  studies  in the SLR replication  [35], and relevance  0 for \nexcluded  studies  from  the SLR replication  [35]. \nThus the labeled Training set CSV is ready to be passed to the  \nPreprocessing phase. The block Training shows the training pro - \ncess after the training set is fed to the Binary Classifiers LSVM,  \nXGBoost,  LogReg  and MNB,  for training.  After  the initial  training, \nhyperparameter  tuning  is done  using  the different  parameters  of the \nmodels to improve their performance on the minority class (here,  \n1). Hyperparameter tuning was done by rerunning the algorithms  \nseveral times with different values to find the model p arameters  \nmost suited to our goal: maximizing recall (finding most of the  \nrelevant  studies)  [15] and precision  (reducing  the load on reviewers  \nto check  irrelevant  studies).  Our goal is to get at least an acceptable  \ntrade -off between recall and precision according to the classification  \npresented  in [6]. The best hyperparameter  configurations  obtained   \nduring the training phase using Sklearn toolkit [ 29] were as fol - \nlows: the hyperparameter term “alpha” is set as 2 to perform strong  \nregularization on the LSVM model, and both classes are given due  \nimportance by setting the “class_weight” term as ‘balanced’ which  \nfollows a weighted loss function. This linear model is then trained  \nusing Stochastic Gradient Descent (SGD) [ 3] to optimize the loss  \nfunction with a decreasing learning rate. A similar approach was  \nfollowed  for XGBoost  and Logistic  Regression.  In XGBoost  we set \n“gamma” (regularization term) as 20, the “scale_pos_weight” term  \nas (number  of articles  in class  0)/(number  of articles  in class  1), \nand “sub_sampling” ratio term to 0.2 to prevent overfitting. In Lo - \ngistic Regression, “C” (the regularization term) is set  to 0.01 and  \nthe “class_weight”  term is set to ‘balanced’.  For Multinomial  Naïve  \nBayes the default parameters of Sklearn are used. In the first three  \nmodels, strong regularization was done to maximize the recall and  \nprecision of the minority class by makin g the models more conser - \nvative and  generalize  better on  testing  data. \nConcerning the testing process, Testing Data Collection is divided  \ninto two parts.  First,  Testing  Data  Collection  Part 1 implements  one \nround of forward snowballing using the snowballing tool on the  \nupdate seed set (45 selected studies from the SLR replication). The  \noutput of the forward snowballing process is a CSV file keeping  \ntrack  of the study  extraction  and a BibTeX  file which  holds  the bib- \nliographical  references  of all the studies  including  their abstracts,  \nin order to aid the authors in the selection of relevant studies. This  \ntask which  is usually  done  manually  is completely  automated.  Sim- \nilarly as done for the training, the BibTeX file from the output of  \nforward snowballing is converted to a CSV file taking the ‘Title’  \nand ‘Abstract’  fields  of the studies.  Second,  in Testing  Data  Collec - \ntion Part 2 the BibTeX file of included papers is al so converted to  \nCSV  and a comparison  is done  between  the two CSVs  to gener - \nate a unique labeled CSV file for testing, labeling the relevance of  \nincluded  studies  as 1 and excluded  ones as  0. \nThereafter, the labeled testing set CSV is passed through the  \nPreprocessing phase. Finally, during Testing the trained model is  \nused to predict inclusion or exclusion for the preprocessed testing  \ndata.  \nPreprocessing is done similarly for the training and test CSV files.  \nFirst,  the ‘Title’  and ‘Abstract’  columns  are merged  to form  a single  \nstring for each study under a column named ‘Merged’, which now  \nserves as the text for text classification. The preprocessing treats  \nthis text by removing stop -words using the Nltk [ 21] library in  \nPython [ 18], tokenizing the text, removing punctuation and URLs,  \nperforming lemmatization [ 23], an d finally vectorizing using the  \nBag of Words [ 32] count vectorizer. The count vectorizer gives poor  \nresults when the n -gram range is increased as it predicts with a  \ngreater  precision  only for the majority  class.  In this sense , unigrams  \nare used. The preprocessed text is then passed as input to the ML  \nalgorithm  models for training.  \nFinally,  for Validation , we record  the number  of included  studies  \nidentified during the forward snowballing round. The labels of the  \ntest set allowed  us to compute  the performance  measures  Precision,  \nRecall,  and F-measure  based  on the predictions  obtained  during  the \nTesting  phase.  They  are defined  as follows  [15]:  \n𝑛𝑢𝑚𝑏𝑒𝑟 𝑜 𝑓  𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡  𝑠𝑡𝑢𝑑𝑖𝑒𝑠 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒𝑑  𝑎𝑠 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡  \n𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜 𝑓  𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡  𝑠𝑡𝑢𝑑𝑖𝑒𝑠 \nrecall  = Emerging  Results  on Automated  Support  for Searching  and Selecting  Evidence  for Systematic  Literature  Review  Updates   \n \n \n \nFigure  3: Tool  process  to select  studies  for SLR  Updates  \n \n𝑛𝑢𝑚𝑏𝑒𝑟 𝑜 𝑓  𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑠𝑡𝑢𝑑𝑖𝑒𝑠 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒𝑑 𝑎𝑠 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 \n𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜 𝑓 𝑠𝑡𝑢𝑑𝑖𝑒𝑠 𝑟𝑒𝑡𝑟𝑖𝑒𝑣𝑒𝑑  𝑎𝑠 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡  \n \nF-measure  = harmonic  mean  of the precision  and recall  \nThe results  of our evaluation  are presented  in Section  4.3. \n \n4.3 Results  - Analysis  of our evaluation  \nThe evaluation  of our tool prototype  for the reproduction  of back- \nward  and forward  snowballing  for the SLR replication  is presented  in Table 1. In total, our tool was able to automatically identify 41  \nof the 43 (95.3%) studies manually identified by the authors when  \nperforming the same snowballing iterations (see Table 1 sum of  \ncolumn “Studies detected” + nine studies from the initial seed set).  \nHowever,  during  iteration  2, a study  identified  manually  during  for- \nward snowballing was identified by our tool during the backward  \nsnowballing  execution  (see lines  highlighted  with (*) in Table  1). \nWe opted  to conserve  this result  since  it does not interfere  with \n \nTable  1: Results  of the  snowballing search replicated  by our  tool \n \n  Iteration  1  \n  Snowballing  type   Seed  set of the iteration    Studies  detected  (%)   \nBackward  9 12/12  (100%)  \nForward  9 1/1 (100%)  \n  Iteration  2  \n  Snowballing  type   Seed  set of the iteration    Studies  detected  (%)   \nBackward  13 (1+12)  1+1*  = 2/1 (100%)  \nForward  13 (1+12)  12/14  (85.7%)  \n  Iteration  3  \n  Snowballing  type   Seed  set of the iteration    Studies  detected  (%)   \nBackward  14 (2*+12)  3/4 (75%)  \nForward  14 (2*+12)  1/1 (100%)  \n  Iteration  4  \n  Snowballing  type   Seed  set of the iteration    Studies  detected  (%)   \nBackward  4 (3+1)  1/1 (100%)  \nForward  4 (3+1)  0/0 \nprecision  = Bianca  Minetto  Napoleão,  Ritika  Sarkar,  Sylvain  Hallé,  Fabio  Petrillo,  and Marcos  Kalinowski   \n∼  \nthe final result of the snowballing tool in this automated execution  \nscenario.  \nThe two missing studies could not be identified because they do  \nnot have a DOI and the tool was not able to locate them through  \nthe implemented reference building (Chicago forma t). It is worth  \nmentioning  that the set of included  studies  of the SLR replication  is \ncomposed  of 45 studies  in total,  but two of them  are not considered  \nin this analysis because they were not retrieved by snowballing. In  \naddition, our tool was able to execute automatically four iterations  \ninstead of five (manual) because the single study resulting from  \niteration 4 did not have a DOI available which made th e snowballing  \nprocess stop. However, in this case, the final result is still the same  \n(manual versus automated) since the manual process also stopped  \nfor not  retrieving  any other  relevant  study.  \nRegarding the evaluation of our tool applied in an SLR update  \nscenario, the snowballing tool was able to retrieve 1012 unique  \nstudies in a single round of forward snowballing based on a seed  \nset of 41 out of 45 studies included in the SLR replication. The 4  \nremaining studies not identified did not have DOIs, consequ ently,  \nwe missed out on 28 citations (data from Google Scholar in Feb.  \n2023). This leads to our final seed set being formed by the 41 studies  \nand having 1012 unique citations to be analyzed in the selection  \nphase  by the ML algorithms.  \nOut of 35 studies contained in the “testing set - included”, that  \nare to be included  in the SLR update,  our search  and selection  \ntool identified  33 studies  (94.3%)  through  the forward  snowballing  \niteration, even without being able to include the 4 no -DOI studies  \nin the seed set. \nThe precision,  recall,  and F-measure  scores  on the testing  for \nthe class of interest (included papers) are illustrated in Figure 4. It \ncan be  seen that the performing model in terms of recall is LSVM  \n(74.3%),  followed  by XGBoost  (63.6%),  Logistic  Regression  (45.4%),   \nand Multinomial  Naïve  Bayes  (42.4%).  In terms  of precision,  the \nalgorithms had almost similar results (  15%) with the exception of  \nXGBoost which had a lower precision (11.6%). Considering the fact  \nthat the F-measure  combines  the effect  of both metrics,  a high recall  \nwill give a low F-measure  if the precision  is low. The F-measure  \nvalue  was observed  with the LSVM  model  (24.6%).  It is able to pre- \ndict the highest number of studies belonging to the positive class 1  \n(included), correctly. The precision value can be explained by the  \nhigh false positive value which is  due to the strict regularization  \nperformed  in LSVM,  hence,  a trade -off between  precision  and recall  \nis remarked.  In fact, Dieste  et al. [6] highlight  that there  will always  \nbe a trade -off between recall and precision because irrelevant stud - \nies are more  liked  to be returned  by a search  execution  the higher  \nthe recall  is. \nOut of 33 studies, 26 studies were identified by the LSVM model,  \n21 by XGB,  15 by LogReg,  and 14 by MNB.  The studies  to be excluded  \ncorrectly  identified  were  820 by LSVM  and XGB,  890 by LogReg,  and \n900 by MNB. LogReg and MNB give biased results for the majority  \nclass hence the true negative values are much higher than LSVM,  \nbut they have lower true po sitive values which is the number of  \nincluded studies  predicted  correctly by  the model.  \n \n \n4.4 Reporting  - Observations  from  the \nevaluation  \nAccording to the analysis of the performance measures, the LSVM  \nmodel showed the best result among the evaluated models. More - \nover,  following  the search  strategies  scale  proposed  by Dieste  et al. \n[6], the recall and precision range resulted from the LSVM model  \nshowed to be acceptable (re call 72 -80% and precision 15 -25%) mean - \ning a “good  enough  strategy”.  \n \n \n \n \nFigure  4: The performance  report  of the ML models  \nEmerging  Results  on Automated  Support  for Searching  and Selecting  Evidence  for Systematic  Literature  Review  Updates   \n5 DISCUSSION  \nThe motivation of this study was to investigate automated alterna - \ntives  to support  both activities  of searching  and selecting  studies  \nduring SLR updates. Due to the effort required for these activities,  \nauthors often end up not giving enough attention to keeping SLRs  \nup to date.  The impact  of not updating  SLRs  on the evolution  of \nscientific  knowledge  can be severe,  as outdated  SLRs  could  lead \nresearchers to obsolete conclusions. While we present emerging  \nresults, they already shed light on some meaningful automation  \nlimitations  and possibilities.  \nWith respect to searching for new studies, we described solution  \noptions  used when  building  our snowballing  tool to face automation  \nchallenges  (e.g.,  querying  CrossRef  to find DOIs,  and complement - \ning BibTex data with web scrapping). Besides providing an example  \nof such solution  options,  our results  indicate  that snowballing -based  \nsearch strategies can be fully automated with minor losses, at least  \nfor classical white literature (the only identified limitation was  \nrelated to papers without DOI), reducing t he effort of laborious  \nmanual snowballing iterations ( cf. Table 1). In order to mitigate  \nminor losses and the potential inclusion of non -classical white lit - \nerature, manual searches [ 15] can be performed to complement the  \nresults  of the  snowballing search.  \nRegarding the selection of studies in SLR updates using ML  \nalgorithms trained based on papers included and excluded in the  \noriginal  SLR,  the recall  and precision  obtained  by these  algorithms  \nstill represent  only a \"good  enough  strategy\",  according  to the scale  \nproposed by Dieste et al. . Still, they would allow to conservatively  \nsave some manual selection effort. In fact, if we take our proposed  \nmodel (LSVM), even if we use a thresho ld that results in a max  \nrecall scenario of 97% for the classifier (one false negative) and only  \n8% of precision, the SLR update author would have to manually  \nanalyze only 396 papers instead of 1012. I.e., for our investigation, in  \na scenario of conservati vely minimizing the risk of missing papers  \nto be included, it would still be possible to reduce the number of  \npapers  to be manually  analyzed  by more  than 2.5 times.  \nOverall, we believe that our results provide preliminary indica - \ntions that strengthen the belief that automated approaches could  \nsignificantly  help to reduce  the SLR update  efforts.  \nLarge Language Models (LLMs) gained attention over the last  \nyear. The mai n difference between LLMs and traditional ML algo - \nrithms combined with Natural Language Processing techniques is  \nthat ML algorithms often rely on labelled or annotated datasets  \nspecific  to the SLR task. These  datasets  are typically  created  manu - \nally by domain  experts.  On the other  hand,  LLMs  are pre-trained  \non large general -domain unannotated datasets, and their language  \nunderstanding capabilities can be fine -tuned with smaller domain - \nspecific datasets (e.g. specific domain requirements of an SLR)  \n[25, 39]. One may question the possible application of LLMs in  \nthe context  of selection  of studies  for SLRs  or SLRs  updates.  To \nthe best of our knowledge,  there  is an initiative  [2] that explored  \na deep -learning -based contextualized embedding clustering tech - \nnique  employing  two transformer -based  language  models  BERT  \n[13] and S -BERT [ 26] to perform the initial selection of studies for  \nSLRs. To evaluate their proposal, the authors compared the gener - \nated models’ resulting clusters with the results of two SLRs from  \nmedicine  manually  conducted.  As a result,  due to the small  size of  \nthe training set, the models ran out of data to train and suffered  \nfrom overfitting. The authors affirm that an extension with a larger  \ndataset is needed to underline a conclusion [ 2]. In summary, LL M \nis still an emerging topic and requires further investigation and  \nvalidation before being applied on the context of selection of studies  \nfor SLRs  and SLRs  updates.  \n \n6 THREATS  TO VALIDITY  \nIn the following,  we enumerate  the main  threats  to the validity  of \nour study.  \nConstruct Validity. With respect to searching for studies using  \nsnowballing,  the adoption  of Semantic  Scholar  has not been  for- \nmally evaluated by researchers in the context of SLR updates [ 36], \nas Goog le Scholar. However, we noticed that, in our case, the results  \nshowed  to be relevant  for the study  and comparable,  with less noise.  \nAlso, Google Scholar does not allow the use of an API to perform  \nsearches. For study selection, our evaluation results might have  \nbeen affected by the choice of ML algorithms. Other algorithms  \ncould have been explored in our study and can be considered as  \npart of future  work.  \nExternal  Validity.  The dataset  used in our analysis  might  not \nrepresent the diversity of SLR Updates in SE. Similar analyses could  \nhave been conducted based on other SLRs to improve the general - \nizability of our results. However, repli cating our emerging results  \non other SLRs to strengthen external validity would require sig - \nnificant  effort.  Furthermore,  it is challenging  to acquire  a reliable  \nand detailed  SLR dataset  (e.g. containing  the list of included  studies  \nin each iteration of the snowballing search) for SLRs that could  \npotentially  need  an update  and be considered  in our analysis.  \nReliability.  One limitation  of our study  is associated  with the \ndataset  used  in our experiment  and the possibility  of sample  bias.  \nFor the snowballing  analysis,  we used  a dataset  of an SLR repli - \ncation that involved experienced SLR researchers following strict  \nguidelines for searching and selecting evidence [ 35]. The data used  \nfor the SLR Update  analysis  was acquired  from  the same  authors  \nwho performed  the SLR replication,  also through  a rigorous  anal- \nysis process. Also, to improve the reliability of our results, the  \ntool prototype  and the small -scale  evaluation  datasets  are openly  \navailable.  \n \n7 CONCLUSION  \nIn this paper, we presented and investigated an automation solu - \ntion proposal to support searching for new evidence and selecting  \nevidence for SLR updates. We built a tool prototype and described it  \nin detail. Based on a small -scale evaluation, we discuss automation  \nlimitations and  perspectives  for the SLR update  context.  \nConcerning searching for evidence, preliminary results of our  \ninvestigation indicate that, while there are challenges faced when  \nautomating snowballing -based search strategies (e.g., to automati - \ncally gather DOIs for papers, to complement BibTeX information  \nof identified papers automatically), these strategies can b e fully  \nautomated with minor losses. This can be particularly helpful for  \nupdating SLRs, given that forward snowballing has been recom - \nmended for this context [ 8, 36]. Furthermore, applying aut omated  \nsnowballing iterations could also be employed to reduce the effort  \nof applying  SLR search  strategies  in general [35]. Bianca  Minetto  Napoleão,  Ritika  Sarkar,  Sylvain  Hallé,  Fabio  Petrillo,  and Marcos  Kalinowski   \n \nWe also investigated the selection of studies in SLR updates  \nusing  ML algorithms  trained  based  on papers  included  and excluded  \nin the original  SLR.  While  improvements  can surely  be obtained,  \nemerging results obtained by our prototype tool are promising  \nand already  considered  acceptable  by literature  [6]. In our small - \nscale evaluation, it was also possible to observe that using our  \nproposed  investigation  using  ML model  (Linear  SVM  optimized  \nusing  the SGD  algorithm)  conservatively  minimizing  the risk of \nmissing papers during the SLR update, it would still be possible to  \nreduce the number of papers to be manually analyzed in about 2.5  \ntimes.  \nHence, we envision that automated approaches could signifi - \ncantly  help to reduce  the SLR update  effort  and put forw ard that \ninvestigations  in this direction  should  be encouraged  and under - \ntaken  to help the community  keep  SLRs  up to date at the pace of \nthe rapid  increase  of new evidence.  \nIn future  work,  we intend  to improve  our prototype  tool to be \ncapable of handling multiple inputs as well as further evaluate our  \nproposition  with other  ML models  and SLR datasets.  \n7.1 Appendix  \nThe tool prototype  and the small -scale  evaluation  datasets  and re- \nsults are openly available  at ⁀https://doi.org/10.5281/zen odo.7888956.  \nACKNOWLEDGMENT  \nWe thank  the authors  of the ongoing  SLR update  considered  in our \nevaluation for sharing their data.  \nREFERENCES  \n[1] Charu C Aggarwal and ChengXiang Zhai. 2012. A survey of text classification  \nalgorithms.  Mining  text data (2012),  163–222. \n[2] Rand Alchokr, Manoj Borkar, Sharanya Thotadarya, Gunter Saake, and Thomas  \nLeich. 2022.  Supporting Systematic Literature Reviews Using Deep -Learning - \nBased Language Models. In IEEE/ACM 1st International Workshop on Natural  \nLanguage -Based  Software  Engineering  (NLBSE) . IEEE  Computer  Society,  67–74. \n[3] Léon Bottou. 2012. Stochastic gradient descent tricks. Neural Networks: Tricks of  \nthe Trade:  Second  Edition  (2012),  421–436. \n[4] Tianqi Chen, Tong He, Michael Benesty, Vadim Khotilovich, Yuan Tang, Hyunsu  \nCho,  Kailong  Chen,  Rory  Mitchell,  Ignacio  Cano,  Tianyi  Zhou,  et al. 2015.  Xgboost:  \nextreme  gradient  boosting.  R package  version  0.4-2 1, 4 (2015),  1–4. \n[5] CrossRef. [n.d.]. Crossref Metadata Search. https://search.crossref.org/references.  \nOnline;  accessed  19 March  2023.  \n[6] Oscar  Dieste,  Anna.  Griman,  and Natalia  Juristo.  2009.  Developing  search  strate - \ngies for detecting relevant experiments.  Empirical Software Engineering 14, 5  \n(2009),  513–539. \n[7] S.C.P.F.  Fabbri,  K.R. Felizardo,  F.C. Ferrari,  E.C.M.  Hernandes,  F.R. Octaviano,  \nE.Y. Nakagawa,  and J.C. Maldonado.  2013.  Externalising  tacit knowledge  of the \nsystematic  review  process.  IET Software 7, 6 (2013),  298–307. \n[8] Katia Felizardo, Emilia Mendes, Marcos Kalinowski, Erica Ferreira Souza, and  \nNandamudi Vijaykumar. 2016. Using Forward Snowballing to update System - \natic Reviews  in Softw are Engineering.  In International  Symposium  on Empirical  \nSoftware  Engineering  and Measurement  (ESEM) . \n[9] Katia  R. Felizardo  and Jeffrey  C. Carver.  2020.  Automating  Systematic  Literature  \nReview . Springer  International  Publishing,  Cham,  327–355. \n[10] Katia R. Felizardo, Erica F. de Souza, Tamiris Malacrida, Bianca M. Napoleão,  \nFabio Petrillo, Sylvain Hallé, Nandamudi L. Vijaykumar, and Elisa Y. Nakagawa.  \n2020. Knowledge Management for Promoting Update of Systematic Literature  \nReviews: An Experience Rep ort. In 2020 46th Euromicro Conference on Software  \nEngineering  and Advanced  Applications  (SEAA) . 471–478. \n[11] Katia R. Felizardo, Elisa Y. Nakwgawa, Stephen. MacDonell, and Jose Carlos  \nMaldonado. 2014. A Visual Analysis Approach to Update Systematic Reviews. I n \nInternational  Conference  on Evaluation  and Assessment  in Software  Engineering  \n(EASE) . ACM,  1–10. \n[12] David W Hosmer Jr, Stanley Lemeshow, and Rodney X Sturdivant. 2013. Applied  \nlogistic  regression . Vol. 398. John  Wiley  & Sons.   \n[13] Jacob  Devlin Ming -Wei Chang Kenton and Lee Kristina Toutanova. 2019. Bert:  \nPre-training of deep bidirectional transformers for language understanding. In  \nProceedings  of naacL -HLT, Vol. 1. 2. \n[14] Ashraf  M Kibriya,  Eibe Frank,  Bernhard  Pfahringer,  and Geoffrey  Holmes.  2005.  \nMultinomial naive bayes for text categorization revisited. In AI 2004: Advances in  \nArtificial  Intelligence:  17th Australian  Joint  Conference  on Artificial  Intelligence,  \nCairns,  Australia,  December  4-6, 2004.  Proceedings  17. Springer,  488–499. \n[15] Barbara A. Kitchenham, David Budgen, and Pearl Brereton. 2015. Evidence -Based  \nSoftware  Engineering  and Systematic  Reviews . Chapman  & Hall/CRC.  \n[16] Joseph Lilleberg, Yun Zhu, and Yanqing Zhang. 2015. Support vector machines  \nand word2vec for text classification with semantic features. In 2015 IEEE 14th  \nInternational Conference on Cognitive Informatics & Cognitive Computing (ICCI*  \nCC). IEEE,  136–140. \n[17] Emilia Mendes, Claes Wohlin, Katia Felizardo, and Marcos Kalinowski. 2020.  \nWhen to update systematic literature reviews in software engineering.  Journal  \nof Systems  and Software  167 (2020).  \n[18] Achraf Merzouki. [n.d.]. What’s new in python 3.8. https://docs.python.org/3/  \nwhatsnew/3.8.html.  Online;  accessed  19 March  2023.  \n[19] Bianca M. Napoleão, Katia R. Felizardo, Erica F. de Souza, Fabio Petrillo, Sylvain  \nHallé, Nandamudi L. Vijaykumar, and Elisa Y. Nakagawa. 2021. Establishing a  \nSearch  String  to Detect  Secondary  Studies  in Software  Engineering.  In 2021  47th \nEuromicro  Confere nce on Software  Engineering  and Advanced  Applications  (SEAA) . \n9–16. \n[20] Bianca M. Napoleão, Fabio Petrillo, and Sylvain Hallé. 2021. Automated Support  \nfor Searching  and Selecting  Evidence  in Software  Engineering:  A Cross -domain  \nSystematic Mapping. In 47th Euromicro Conference on Software Engineering and  \nAdvanced  Applications  (SEAA) . \n[21] NLTK Team. [n.d.]. Natural Language Toolkit. https://pypi.org/project/nltk.  \nOnline;  accessed  19 Marc h 2023.  \n[22] Leif E Peterson.  2009.  K-nearest  neighbor.  Scholarpedia  4, 2 (2009),  1883.  \n[23] Joël Plisson,  Nada  Lavrac,  Dunja  Mladenic,  et al. 2004.  A rule based  approach  to \nword  lemmatization.  In Proceedings  of IS, Vol. 3. 83–86. \n[24] Python Team. [n.d.]. Urllib package documentation. https://docs.python.org/3/  \nlibrary/urllib.html.  Online;  accessed  19 March  2023.  \n[25] Susmita Ray. 2019. A Quick Review of Machine Learning Algorithms. In Interna - \ntional Conference on Machine Learning, Big Data, Cloud and Parallel Computing  \n(COMITCon) . IEEE  Computer  Society,  35–39. \n[26] Nils Reimers  and Iryna  Gurevych.  2019.  Sentenc e-BERT:  Sentence  Embeddings  \nusing Siamese BERT -Networks. In Conference on Empirical Methods in Natu - \nral Language Processing and the 9th International Joint Conference on Natural  \nLanguage  Processing  (EMNLP -IJCNLP) . 3982 –3992.  \n[27] ResearchGate. [n.d.]. https://www.researchgate.net.  Online; accessed 19 March  \n2023.  \n[28] Per Runeson, Martin Host, and Austen Rainer. 2012.  Case Study Research in  \nSoftware  Engineering:  Guidelines  and Examples . \n[29] Scikit -Learn. [n.d.]. Scikit -Learn Documentation. https://scikit -learn.org/stable.  \nOnline;  accessed  19 March  2023.  \n[30] Semantic Scholar. [n.d.].  Semantic scholar - academic graph API.  https://api.  \nsemanticscholar.org/api -docs.  Online;  accessed  19 March  2023.  \n[31] Klaas -Jan Stol and Brian Fitzgerald. 2015.  A Holistic Overview of Software  \nEngineering  Research  Strategies.  In CESI . IEEE  Press,  47–54. \n[32] Hanna M Wallach. 2006. Topic modeling: beyond bag -of-words. In Proceedings of  \nthe 23rd  international  conference  on Machine  learning . 977–984. \n[33] Willian Massami Watanabe, Katia Romero Felizardo, Arnaldo Candido, Erica Fer - \nreira de Souza,  João Ede de Campos  Neto,  and Nandamudi  Lankalapalli  Vijayku - \nmar. 2020. Reducing efforts of software engineering systematic literature reviews  \nupdates  using  text classification.  Information  and Software  Technology  128 (2020).  \n[34] Claes.  Wohlin.  2014.  A Snowballing  Procedure  for Systematic  Literature  Studies  \nand a Replication.  In International  Conference  on Evaluation  and Assessment  in \nSoftware  Engineering  (EASE) . 321–330. \n[35] Claes  Wohlin,  Marcos  Kalinowski,  Katia  Romero  Felizardo,  and Emilia  Mendes.  \n2022. Successful combination of database search and snowballing for identifica - \ntion of primary  studies  in systematic  literature  studies.  Information  and Software  \nTechnology  147 (2022),  106908.  \n[36] Claes  Wohlin,  Emilia  Mendes,  Katia  Romero  Felizardo,  and Marcos  Kalinowski.  \n2020. Guidelines for the search strategy to update systematic literature reviews  \nin software  engineering.  Information  and Software  Technology  127 (2020),  106366.  \n[37] Claes  Wohlin  and Austen  Rainer.  2022.  Is it a case study?  A critical  analysis  and \nguidance.  Journal  of Systems  and Software  192 (2022),  111395.  \n[38] Li Zhang, Jia -Hao Tian, Jing Jiang, Yi Liu, Meng -Yuan Pu, and Tao Yue. 2018.  \nEmpirical Research in Software Engineering — A Literature Survey. Journal of  \nComputer  Science  and Technology  33 (2018),  876–899. \n[39] Barret  Zoph,  Colin  Raffel,  Dale  Schuurmans,  Dani  Yogatama,  Denny  Zhou,  Don \nMetzler, Ed H. Chi, Jason Wei, Jeff Dean, Liam B. Fedus, Maarten Paul Bosma,  \nOriol  Vinyals,  Percy  Liang,  Sebastian  Borgeaud,  Tatsunori  B. Hashimoto,  and Yi \nTay. 2022. Emergent abilities of large language models. Transactions on Machine  \nLearning  Research (2022),  1–30. "    },    {        "title": "2402.05320v1.Assessment_of_models_for_nonlinear_oscillatory_flow_through_a_hexagonal_sphere_pack.pdf",        "content": "Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory\nflow through a hexagonal sphere pack\nLukas Unglehrt1*and Michael Manhart1\n1*Professorship for Hydromechanics, Technical University of\nMunich, Arcisstr. 21, 80333 Munich, Germany.\n*Corresponding author(s). E-mail(s): lukas.unglehrt@tum.de;\nContributing authors: michael.manhart@tum.de;\nAbstract\nWe review models for unsteady porous media flow in the volume-\naveraging framework and we discuss the theoretical relations between the\nmodels and the definition of the model coefficients (and the uncertainty\ntherein). The different models are compared against direct numerical\nsimulations of oscillatory flow through a hexagonal sphere pack. The\nmodel constants are determined based on their definition in terms of\nthe Stokes flow, the potential flow and steady nonlinear flow. Thus, the\ndiscrepancies between the model predictions and the simulation data\ncan be attributed to shortcomings of the models’ parametrisation.\nWe found that an extension of the dynamic permeability model of Pride\net al. [Physical Review B 47(9), 1993] with a Forchheimer-type nonlin-\nearity performs very well for linear flow and for nonlinear flow at low\nand medium frequencies, but the Forchheimer term with a coefficient\nobtained from the steady-state overpredicts the nonlinear drag at high\nfrequencies. The model reduces to the unsteady Forchheimer equation\nwith an acceleration coefficient based on the static viscous tortuosity for\nlow frequencies.\nThe unsteady Forchheimer equation with an acceleration coefficient\nbased on the high frequency limit of the dynamic tortuosity has\nlarge errors for linear flow at medium and high frequencies, but\nlow errors for nonlinear flow at all frequencies. This is explained by\nan error cancellation between the inertial and the nonlinear drag.\nKeywords: oscillatory porous media flow, unsteady Forchheimer equation,\ndynamic permeability model, model comparison, direct numerical simulation\n1arXiv:2402.05320v1  [physics.flu-dyn]  7 Feb 2024Springer Nature 2021 L ATEX template\n2 Assessment of models for nonlinear oscillatory flow . . .\nArticle highlights\n•We review models for unsteady porous media flow in the volume-averaging\nframework and discuss their relationships.\n•The model predictions are compared to direct numerical simulations of\noscillatory flow through a hexagonal sphere pack.\n•Accurate models exist for the linear drag, but the Forchheimer term\noverpredicts the nonlinear drag at high frequencies.\n1 Introduction\nUnsteady flow through porous media occurs in a variety of environmental, engi-\nneering and industrial applications. For instance, wave-induced flow through\ncoral reefs (Lowe et al, 2008), breakwaters (van Gent, 1994; Losada et al,\n1995; Hall et al, 1995; Muttray, 2000) and marine sediment (Gu and Wang,\n1991) has been described using the theory of porous media. Oscillatory flow\nis also present in combustion engines where porous media burners could lead\nto emission reductions (Aboujafari et al, 2022). Moreover, porous media have\nbeen used as regenerator-type heat exchangers in Stirling engines (Simon and\nSeume, 1988; Trevizoli et al, 2016). In chemical reactors, pulsating flow could\nbe used to enhance mixing and mass transfer (Ni et al, 2003) or to separate\nsubstances (Graham and Higdon, 2002). The increasing use of wind energy\nmay also lead to an interest in processes with an intermittent energy sup-\nply. Furthermore, the understanding of transient flow behaviour is important\nfor the safety design of nuclear pebble bed reactors (Andreades et al, 2014).\nFinally, Kahler and Kabala (2019) suggested to use pulsating flow to accelerate\ngroundwater remediation.\nThe pore scale flow through a porous medium is governed by the laws of\ncontinuum mechanics, i.e. the Navier-Stokes equations, or statistical mechan-\nics, i.e. the Boltzmann equation. However, a direct solution of the pore\nscale flow is often computationally demanding and knowledge of the pore\ngeometry is often not available. Therefore, coarse-grained descriptions have\nbeen developed, made possible by the scale separation between the pore size\nand the extent of the porous medium. These coarse-grained descriptions are\nbased, for example, on the volume-averaging framework (Whitaker, 1967)\nor homogenisation theory (Ene and Sanchez-Palencia, 1975). A comparison\nbetween these approaches can be found in (Davit et al, 2013). Descriptions\nbased on the volume-averaging approach have been used for example by Gu\nand Wang (1991), who studied gravity waves over a porous seabed, by van Gent\n(1994), who investigated wave transmission through dikes and breakwaters, by\nBreugem et al (2006), who studied turbulent channel flow over porous media,\nand by Iliuta and Larachi (2016, 2017), who simulated oscillating packed-bed\nreactors for offshore applications. We now give a brief outline of the volume-\naveraging method. The macroscopic quantities are obtained by performing a\n(weighted) local average of the quantities defined at the pore scale over a\nrepresentative volume element. For example, the superficial velocity and theSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 3\nmacroscopic pressure are defined as\n⟨u⟩s=1\nVZ\nVfudV (1a)\n⟨p⟩i=1\nVfZ\nVfpdV , (1b)\nwhere ⟨.⟩sand⟨.⟩idenote the superficial and intrinsic volume average, respec-\ntively, and VandVfare the volumes of the representative volume element\nand of the fluid contained therein. These quantities are governed by the\nvolume-averaged Navier-Stokes (VANS) equations (Whitaker, 1996)\n∇· ⟨u⟩s= 0 (2a)\nρ∂⟨u⟩s\n∂t=−1\nVZ\nAfs˜pndA\n|{z }\npressure drag−1\nVZ\nAfsτwdA\n|{z }\nfriction drag−ϵ∇⟨p⟩i (2b)\nwhere ˜ p=p−⟨p⟩i,ϵ=Vf/Vis the porosity and Afsis the fluid-solid interface.\nOther works, e.g. (Hsu and Cheng, 1990), have also included convective and\ndiffusive terms in the superficial velocity, but these can be generally neglected\nif the scale separation between the pore scale and the averaging scale is large\nenough (Whitaker, 1986, 1996). In the VANS equations the pressure drag and\nthe friction drag are unclosed with respect to the superficial velocity and the\nmacroscopic pressure gradient. In the approach of Whitaker (1986, 1996), the\npore scale velocity and pressure are expressed in terms of the superficial veloc-\nity by linear mappings. These mappings take the form of tensor and vector\nfields that satisfy a boundary value problem in the pore space of a representa-\ntive volume element (“closure problem”). When the mappings are substituted\ninto the pressure drag and friction drag terms, the drag terms take the form of\na product of the inverse of a permeability-like tensor with the superficial veloc-\nity. In steady linear flow, the closure problem depends only on the geometry of\nthe pore space (Whitaker, 1986), and consequently the permeability tensor of\nthe porous medium is independent of the flow history and the fluid properties.\nIn nonlinear flow, however, the closure problem depends on the velocity field\non the pore scale (Whitaker, 1996) and the permeability-like tensor depends\non the flow history and the fluid properties. Thus, a direct numerical simu-\nlation of the pore scale flow in a representative volume element is required\nto solve the closure problem. For three-dimensional porous media, these sim-\nulations require a large computational effort. Solving the pore scale problem\ncan be avoided by parametrising the nonlinear drag directly in terms of the\nsuperficial velocity and the macroscopic pressure gradient. This is based on\nexperiments or numerical simulations of representative volume elements that\nare performed before solving the macroscale problem. In the following, we refer\nto these parametrisations as models .Springer Nature 2021 L ATEX template\n4 Assessment of models for nonlinear oscillatory flow . . .\nThe aim of the present work is to compare different models for unsteady\nporous media flow for the special case of oscillatory flow. We address the\nfollowing research questions: What is the domain of validity for the different\nmodels? How should the model coefficients be chosen? How can the models be\nimproved?\nIn this contribution, we first describe some of the prominent models that\nare available in the literature and discuss their interrelations. We then compare\nthe predictions of the different models with a high fidelity direct numerical\nsimulation dataset of oscillatory flow through a hexagonal sphere pack. We\nmake the assumption of constant model coefficients that are defined in terms\nof the linear and the steady state behaviour and represent properties of the\nporous medium geometry. For the present flow configuration this information\nis available with a high fidelity based on the works of Zhu and Manhart (2016);\nSakai and Manhart (2020); Unglehrt and Manhart (2022a). This allows us to\ntest the actual predictive capabilities of the models with a negligible ambiguity\nin the values of the model coefficients. The errors of the predictions therefore\nrepresent a shortcoming of the model and suggest the need for a different\nparametrisation.\n2 Review of models for porous media flow\nIn this section, we give an overview of some of the common models for porous\nmedia flow. As discussed in the introduction, we use the term model to refer\nto a parametrisation of the drag in the volume-averaged momentum equation.\n2.1 Models for linear flow\n2.1.1 Darcy equation\nIn steady conditions, linear flow can be described using the Darcy equation\n(Darcy, 1856)\n−∇⟨p⟩i=µ\nK⟨u⟩s(3)\nThe Darcy equation relates the pressure gradient and the superficial velocity\nlinearly using the dynamic viscosity µand the permeability K. The perme-\nability has units of length squared and is a pure function of the pore geometry.\nFor a given pore geometry, it can be computed directly from the solution to\nthe Stokes equations for a given pore geometry or from empirical correlations,\ne.g. the Kozeny-Carman equation.\n2.1.2 Unsteady Darcy equation\nThe unsteady Darcy equation arises from the volume-averaged Navier-Stokes\nequations (2b) if the quasi-steady closure with Darcy’s law is employed for the\ndrag forces\nρd⟨u⟩s\ndt=−ϵ∇⟨p⟩i−ϵ µ\nK⟨u⟩s. (4)Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 5\nThe solutions to this equation relax to Darcy’s law (3) with a time constant\nτvans=K/(ϵ ν). This model was applied by Kuznetsov and Nield (2006) to\npulsating and by Wang (2008) to transient porous media flow, respectively; in\nthese works the unsteady Darcy equation was extended by the Brinkman term\nin order to account for wall boundary conditions.\nAs discussed for example by Nield (1991), an a priori unknown acceleration\ncoefficient needs to be introduced in front of the time derivative. Based on a\nvirtual mass analogy, Sollitt and Cross (1972) and Gu and Wang (1991) used a\nfactor 1 + CM1−ϵ\nϵin front of the acceleration term which results in a time con-\nstant τvm=\u0002\n1 +CM1−ϵ\nϵ\u0003\nK/(ϵ ν). Finally, Hill et al (2001, eq. (20) and (21))\nand Zhu et al (2014) derived another time constant τen=α0τvansfrom the\nvolume-averaged kinetic energy equation assuming self-similar velocity pro-\nfiles and a quasi-steady dissipation rate where α0=\nu2\nStokes\u000b\ni/⟨uStokes⟩2\niis\nthe static viscous tortuosity that is defined in terms of the velocity field of\nthe Stokes flow (Lafarge, 1993, p.156f). In the literature, the static viscous\ntortuosity α0has been referred to as “inertial factor” (Norris, 1986), “accelera-\ntion coefficient” (Nield, 1991), “low-frequency limit of the dynamic tortuosity”\n(Champoux and Allard, 1991), “low frequency viscous [...] [equivalent] of the\ntortuosity [ α∞]” (Cortis et al, 2002), “viscous tortuosity” (Kergomard et al,\n2013), “time scale ratio” (Zhu and Manhart, 2016), “static viscous tortuosity”\n(Roncen et al, 2018).\nIt was shown by Zhu et al (2014); Zhu and Manhart (2016) that the\nunsteady Darcy equation with the time constant τen, i.e.\nρ α0d⟨u⟩s\ndt=−ϵ∇⟨p⟩i−ϵ µ\nK⟨u⟩s, (5)\nis the appropriate choice for transient flow and low frequency oscillatory flow\nwhereas for high frequency oscillatory flow a different time constant needs to\nbe employed. This will be further discussed in section 2.3.\n2.1.3 Dynamic permeability models\nLinear oscillatory flow through porous media has been studied extensively\nin acoustics. Johnson et al (1987) proposed an important family of models,\nthe dynamic permeability. These are based on a generalisation of the Darcy\nequation (3) in the frequency domain1\nF {⟨u⟩s}(ω) =−ˆK(−ω)\nµF {∇⟨p⟩i}(ω) (8)\n1We define the Fourier transform of a function g(t) and the inverse Fourier transform of a\nfunction ˆ g(ω) as\nF {g(t)}(ω) =Z∞\n−∞g(t)e−iωtdt (6)\nF−1{ˆg(ω)}(t) =1\n2πZ∞\n−∞ˆg(ω)eiωtdω (7)Springer Nature 2021 L ATEX template\n6 Assessment of models for nonlinear oscillatory flow . . .\nwhere the function ˆK(Ω) is referred to as dynamic permeability and corre-\nsponds to the frequency response function in linear systems theory.\nJohnson et al (1987) derived high-frequency asymptotics from boundary\nlayer theory (cf. Cortis et al (2003)) and proposed a model for the entire\nfrequency range by blending the high-frequency asymptotics with Darcy’s law\nat low frequencies. The model by Johnson et al (1987) is given in the notation\nof Pride et al (1993) for a complex frequency Ω as\nˆK(Ω) =Kq\n1−iPΩ\nΩ0−iΩ\nΩ0, (9)\nwith the frequency Ω 0=ϵ ν/(α∞K) and the dimensionless parameter\nP= 4α∞K/(ϵΛ2). The transition frequency Ω 0“separates viscous-force-\ndominated flow from inertial-force flow” (Pride et al, 1993). The original\nparameters introduced by Johnson et al (1987) are the high-frequency limit of\nthe dynamic tortuosity α∞(identical to the ratio\nu2\u000b\ni/⟨u⟩2\niin potential flow)\nand a length-scale Λ. Both parameters can be computed in terms of the poten-\ntial flow solution. Smeulders et al (1992) presented a derivation of the model of\nJohnson et al (1987) from first principles using homogenisation theory. Chap-\nman and Higdon (1992) compared the model predictions to numerical solutions\nof the unsteady Stokes equations obtained with a least-squares collocation\napproach and found good agreement. Similarly, in (Unglehrt and Manhart,\n2022a) we observed excellent agreement of the model with our simulations.\nFor large frequencies, the model (9) reduces to\nˆK(Ω) =Kq\n−iPΩ\nΩ0−iΩ\nΩ0(10)\nUsing the inverse Fourier transform of equations (8) and (10), we can write\nthe momentum equation in the time domain (Turo and Umnova, 2013)\nρd⟨u⟩s\ndt=−2ρ√ν\nΛZt\n−∞d⟨u⟩s\ndτ1p\nπ(t−τ)dτ−ϵ\nα∞∇⟨p⟩i, (11)\nwhere the integral term corresponds to the Caputo fractional derivative of\norder1\n2. This term originates from boundary layer theory and is analogous\nto the Basset history term in the solution for flow around a sphere (Turo\nand Umnova, 2013). Consequently, the flow state in the dynamic permeability\nmodel (9) requires the specification of the entire history of ⟨u⟩s(t). This is in\ncontrast to the unsteady Darcy equation where the flow state is completely\nspecified by the instantaneous value of ⟨u⟩s.Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 7\nIn order to improve the low-frequency behaviour, Pride et al (1993) devised\nan extended model\nˆK(Ω) =K\u0010\n1−P\n2β+q\nP2\n4β2−iPΩ\nΩ0\u0011\n−iΩ\nΩ0. (12)\nThe model of Johnson et al (1987) is obtained in the special case P= 2βwhere\nthe additional non-dimensional parameter β=α0\nα∞−1 is defined in terms of\nthe static viscous tortuosity α0and the high-frequency limit of the dynamic\ntortuosity α∞. It is therefore a measure for the difference in time scales of\nStokes flow and potential flow.\nAnother extension of the model of Johnson et al (1987) was developed by\nChampoux and Allard (1991) to represent thermal dissipation effects occuring\nfor an ideal gas. As we restrict our analyses to incompressible flow, we will not\ndiscuss this any further. A comprehensive discussion of the dynamic perme-\nability models (also known as equivalent fluid model ) can be found in Lafarge\n(2009).\nThe dynamic permeability models can also be written in the time domain\nby performing an inverse Fourier transform. For example, a time domain for-\nmulation of the model of Johnson et al (1987) (equation 9) was given by\nUmnova and Turo (2009). In the same way, we obtained the time domain\nformulation of the more general model of Pride et al (1993) (equation 12):\nρd⟨u⟩s\ndt=−ϵ\nα∞∇⟨p⟩i−\u0012ϵ\nα∞µ\nK−2µ\nΛ2β\u0013\n⟨u⟩s\n−2ρ√ν\nΛZt\n−∞\u0012ν⟨u⟩s\nΛ2β2+d⟨u⟩s\ndτ\u0013e−ν(t−τ)\nΛ2β2\np\nπ(t−τ)dτ .(13)\nNote that we have inserted the definitions of the parameters Pand Ω 0in\norder to ease the comparison with the other time domain models. The kernel\nin the convolution integral represents an exponential damping of the fractional\nderivative kernel and reduces the weight on the history further in the past.\n2.2 Models for nonlinear flow\n2.2.1 Darcy equation with cubic correction\nFor steady flow at small Re, Mei and Auriault (1991) showed using a homoge-\nnization theory approach that Darcy’s law needs to be corrected with a cubic\nterm\n−∇⟨p⟩i=µ\nK⟨u⟩s+µ b\nK⟨u⟩2\ns⟨u⟩s(14)\nwhere bis a non-negative coefficient with the units T2/L2. Koch and Ladd\n(1997) simulated flow through periodic and random arrays of cylinders and\nconfirmed that the first correction to Darcy’s law is cubic with respect to theSpringer Nature 2021 L ATEX template\n8 Assessment of models for nonlinear oscillatory flow . . .\nbulk velocity. Firdaouss et al (1997) showed that the cubic correction also\nholds for anisotropic media provided that the modulus of the bulk velocity does\nnot change under a reversal of the driving force. Hill et al (2001) investigated\nflow through random and ordered arrays of spheres using DNS and concluded\nthat “[a]t all solid volume fractions, the first inertial contribution to the non-\ndimensional drag force was found to be proportional to the square of the\nReynolds number, as predicted by the theory of Mei & Auriault.”.\n2.2.2 Forchheimer equation\nFor nonlinear flow at higher Reynolds numbers, Forchheimer (1901) proposed\nthe empirical equation\n−∇⟨p⟩i=a⟨u⟩s+b|⟨u⟩s|⟨u⟩s(15)\nwhere the coefficients aandbhave units of M/(L3T) and M/L4, respectively.\nFor packed beds of spheres, comprehensive empirical correlations for these\ncoefficients were first given by Ergun (1952). Updated forms of the correlations\nhave been given e.g. by Macdonald et al (1979). When the flow in the pore space\nbecomes turbulent, a different set of coefficients should be chosen (Burcharth\nand Andersen, 1995); this results in a piecewise description of the drag.\n2.2.3 Unsteady Forchheimer equation\nFor the description of unsteady porous media flow, Polubarinova-Kochina\n(1962) proposed to extend the Forchheimer equation (15) with an acceleration\nterm\n−∇⟨p⟩i=a⟨u⟩s+b|⟨u⟩s|⟨u⟩s+cd⟨u⟩s\ndt. (16)\nSollitt and Cross (1972) derived a parametrisation of this equation where a\nandbwere chosen according to the steady state equation by Ward (1964) and\nthe form of the acceleration coefficient cwas determined based on a virtual\nmass argument. Their equation reads\nρ Sd⟨u⟩s\ndt=−ϵ∇⟨p⟩i−ϵ µ\nK⟨u⟩s−ρϵ Cf√\nK|⟨u⟩s|⟨u⟩s(17a)\nwhere the “inertial coefficient” is defined as\nS= 1 +1−ϵ\nϵCM. (17b)\nHowever, Sollitt and Cross (1972) considered the coefficient CM, which repre-\nsents the virtual mass of the solid grains, as unknown and set it to zero. In\ncontrast, the experimental investigation of Gu and Wang (1991) resulted in\nCM= 0.46 for gravel beds with a porosity between 0 .35 and 0 .38. Also, other\nparametrisations of the acceleration coefficient have been given in later works\n(Burcharth and Andersen, 1995).Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 9\nAnother choice for the coefficient cwas proposed by Zhu (2016) who sug-\ngested to use the time constant τen=α0K/(ϵν) in direct analogy to the\nunsteady Darcy equation (5).\nBurcharth and Andersen (1995) state that “the coefficients [ bandc] are\nnot constants and should in principle be treated as instantaneous values, even\nfor oscillatory flow conditions”. For example, Hall et al (1995) calculated the\ninstantaneous values of the coefficients a,bandcfrom oscillatory flow data.\nHowever, no explicit parametrisation of the constants was determined. On the\nother hand, van Gent (1993) considered the coefficients as constants for each\nflow case and found frequency-dependent correlations for the coefficients band\nc. However, this kind of parametrisation is specific to the flow case and is not\ngenerally applicable (Burcharth and Andersen, 1995).\nFurthermore, it is unclear how the change of the drag behaviour due to\nthe transition to turbulence (see Burcharth and Andersen (1995)) could be\nincorporated into the unsteady Forchheimer equation. In oscillatory flow the\ncritical Reynolds number of transition depends on the frequency (see section\n3.2 or the estimations of Gu and Wang (1991)), ruling out the straightforward\nway of changing the coefficients a,b, and cdepending on the instantaneous\nReynolds number.\n2.2.4 Extended dynamic permeability model\nTuro and Umnova (2013) proposed a model for unsteady nonlinear porous\nmedia flow. The model combines the time domain formulation of a dynamic\npermability model with a Forchheimer-type quadratic term:\nρd⟨u⟩s\ndt=−ϵ\nα∞∇⟨p⟩i−ϵ\nα∞µ\nK(1 +ξ|⟨u⟩s|)⟨u⟩s\n−2ρ√ν\nΛZt\n−∞d⟨u⟩s\ndτ1p\nπ(t−τ)dτ .(18)\nThe parameter ξ[T/L] describes the nonlinearity and is related to the Forch-\nheimer coefficient basξ=bK\nµ. This model can be interpreted as an additive\ncombination of the drag due to the boundary layers (corresponding to the\nhigh-frequency asymptotics (10) of Johnson et al (1987)) and of the drag in\nthe steady state described by the Forchheimer equation (15).\n2.3 Discussion\n2.3.1 Relations among the linear models\nThe linear models presented in section 2.1 can be understood as various special\ncases of the dynamic permeability model of Pride et al (1993) given in equation\n(12). First, by setting P= 2β, the model of Johnson et al (1987) given in\nequation (9) is recovered. Thus, the model of Johnson et al (1987) has theSpringer Nature 2021 L ATEX template\n10 Assessment of models for nonlinear oscillatory flow . . .\nFig. 1 : Comparison of different linear models with the direct numerical simu-\nlations of Zhu and Manhart (2016). Amplitude (left) and phase (right) of the\ndynamic permeability ˆKnormalised with the permeability. The dynamic per-\nmeability functions of the models are given by the equations (9), (12), (22)\nand (20), respectively.\ninherent assumption\nα0=α∞\u0012\n1 +P\n2\u0013\n=α∞\u0012\n1 +2α∞K\nϵΛ2\u0013\n(19)\nfor the static viscous tortuosity. Second, the unsteady Darcy equation (5) can\nbe recast as a dynamic permeability model by taking the Fourier transform:\nF {⟨u⟩s}(ω) =−1\nµK\n1 +α0\nα∞iω\nΩ0F {∇⟨p⟩i}(ω). (20)\nOn the other hand, a Taylor expansion of the denominator of the model of\nPride et al (1993) (equation 12) at ω= 0 results in\nˆK(−ω) =K\n1 +α0\nα∞iω\nΩ0+O\u0010\nω2\nΩ2\n0\u0011. (21)\nBy comparison, we see that the Darcy equation and the unsteady Darcy\nequation represents the zeroth and first order asymptotes to the low frequency\nbehaviour of the frequency response function of the Pride et al (1993) model.\nThis explains the observations of Zhu and Manhart (2016) that the unsteady\nDarcy equation with α0shows excellent agreement with the direct numericalSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 11\nsimulations at low frequencies whereas a different time constant – the high-\nfrequency limit of the dynamic tortuosity α∞– has to be employed at high\nfrequencies. Notably, the high-frequency limit of the dynamic tortuosity α∞\nis a consistent choice of coefficient for the unsteady Darcy equation in that it\nleads to the correct limits for Ω →0 and Ω → ∞ .\nIn figure 1 the dynamic permeabilities implied by the different models are\ncompared to the numerical simulations of Zhu and Manhart (2016) of oscilla-\ntory flow through a hexagonal sphere pack. It can be seen that the dynamic\npermeability models of Johnson et al (1987) and Pride et al (1993) show an\nexcellent agreement with the simulation data. Note that for the hexagonal\nsphere pack, the assumption (19) is fulfilled with an error of only 1%, rendering\nthe models of Johnson et al (1987) and Pride et al (1993) virtually identical for\nthis geometry. On the other hand, the unsteady Darcy equation departs from\nthe simulation data at medium or high frequencies, depending on the choice\nof the acceleration coefficient. The model of Turo and Umnova (2013) will be\ndiscussed in the next section.\nIn conclusion, we find that the unsteady Darcy equation and the dynamic\npermeability model of Johnson et al (1987) can be seen as simplifications of the\ndynamic permeability model of Pride et al (1993), which is able to accurately\ndescribe the simulation data for the hexagonal sphere pack.\n2.3.2 Improvement of the model of Turo & Umnova (2013)\nIn the linear limit, the model (18) corresponds to a dynamic permeability of\nthe following form\nˆK(Ω) =K\n1 +q\n−iPΩ\nΩ0−iΩ\nΩ0. (22)\nIt can be seen that for Ω →0 and for Ω → ∞ the correct limits (3) and (10)\nare approached. However, when the dynamic permeability is compared to the\nmodels of Johnson et al (1987); Pride et al (1993); Zhu et al (2014) it becomes\napparent that the model of Turo and Umnova (2013) severely underestimates\nthe permeability at low and intermediate frequencies. This can be seen clearly\nin figure 1a.\nThe analysis of this model deficiency suggests a simple remedy: We replace\nthe underlying dynamic permeability model of equation (18) with the model\nof Pride et al (1993) which has the correct behaviour at low frequencies. We\nobtain the following equation:\nρd⟨u⟩s\ndt=−ϵ\nα∞∇⟨p⟩i−\u0014ϵ\nα∞µ\nK(1 +ξ|⟨u⟩s|)−2µ\nΛ2β\u0015\n⟨u⟩s\n−2ρ√ν\nΛZt\n−∞\u0012ν⟨u⟩s\nΛ2β2+d⟨u⟩s\ndτ\u0013e−ν(t−τ)\nΛ2β2\np\nπ(t−τ)dτ .(23)Springer Nature 2021 L ATEX template\n12 Assessment of models for nonlinear oscillatory flow . . .\nThis model allows us to explore the potential of the basic idea of Turo\nand Umnova (2013) of combining a dynamic permeability model with the\nForchheimer nonlinearity.\n2.3.3 Lower bounds for the coefficients of the unsteady\nForchheimer equation\nIn this section, we discuss lower bounds for the coefficients of the unsteady\nForchheimer equation that arise from Kelvin’s minimum energy theorem,\nHelmholtz’ minimum dissipation theorem and the volume-averaged kinetic\nenergy equation if the coefficients are considered as constants.\nFirst, we multiply the unsteady Forchheimer equation (16) with the super-\nficial velocity such that after some rearrangements the following equation is\nobtained:\nd\ndt\u0010c\n2⟨u⟩2\ns\u0011\n=−⟨u⟩s·∇⟨p⟩i−a⟨u⟩2\ns−b|⟨u⟩s|⟨u⟩2\ns(24)\nwhere the first term on the right hand side is the power added to or removed\nfrom the flow by the macroscopic pressure gradient. Comparing this equation\nto the volume-averaged kinetic energy equation (Zhu et al, 2014)\nd\ndt\u00121\n2ρ\nu2\u000b\ns\u0013\n=−⟨u⟩s·∇⟨p⟩i−2µ⟨S:S⟩s, (25)\nwherein Sis the strain rate tensor, we find that the term on the left hand side\nof (24) is the time derivative of a positive quantity and can take both signs.\nThus it cannot be part of the dissipation. On the other hand, the second and\nthird term on the right hand side of (24) have a definite sign. Hence they\ncannot be part of the time derivative of the kinetic energy which can take both\nsigns. Consequently, we identify\n1\n2ρ\nu2\u000b\ns=c\n2⟨u⟩2\ns(26a)\n2µ⟨S:S⟩s= (a+b|⟨u⟩s|)⟨u⟩2\ns (26b)\nas underlying assumptions of the unsteady Forchheimer equation. Note that\nthe dissipation term is consistent with Nield (2000) who discussed stationary\nflow.\nFrom Kelvin’s minimum energy theorem (Batchelor, 2000, p.384), it is\npossible to show that the kinetic energy for a given superficial velocity is\nsmallest in the potential flow. This results in the inequality (Lafarge, 1993,\np.123)\n1\n2ρ\nu2\u000b\ns≥1\n2ρ α∞\nϵ⟨u⟩2\ns (27)\nwhich is valid for an isotropic porous medium. Moreover, according to\nHelmholtz’ minimum dissipation theorem (Batchelor, 2000, pp.227–228), theSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 13\ndissipation rate for a given superficial velocity is smallest in the Stokes flow.\nUsing the expression of Zhu et al (2014); Pa´ ez-Garc´ ıa et al (2017) for the\ndissipation rate in the Stokes flow, we arrive at the inequality\n2µ⟨S:S⟩s≥µ\nK⟨u⟩2\ns(28)\nwhich again is valid for an isotropic porous medium. We therefore obtain the\nlower bounds\na≥µ\nK(29a)\nb≥0 (29b)\nc≥ρ α∞\nϵ(29c)\nfor the coefficients of the unsteady Forchheimer equation.\nThese inequalities could be interpreted as realisability conditions similar to\nthose of Schumann (1977) in the context of Reynolds stress turbulence models,\nfor if these conditions are violated, no pore scale velocity field can be found\nsuch that equation (24) describes the evolution of the kinetic energy.\n3 Methodology\nIn this section, we describe the simulation dataset that will be used as a ref-\nerence for comparing the different models. Moreover, we discuss aspects of\nthe numerical implementation of the models. Finally, we introduce the metrics\nthat will be used for quantifying the model errors.\n3.1 Description of the flow solver\nThe reference simulations were conducted using our in-house code MGLET\n(Manhart et al, 2001) which solves the incompressible Navier-Stokes equations\non a Cartesian block-structured grid with a staggered arrangement of vari-\nables. For the spatial discretisation, a symmetry-preserving second-order finite\nvolume scheme (Verstappen and Veldman, 2003) is employed. The no-slip\nboundary condition on the spheres is imposed by means of a mass-conserving\nghost-cell immersed boundary method (Peller et al, 2006; Peller, 2010).\nFor the temporal discretisation, a third-order explicit Runge-Kutta method\n(Williamson, 1980) is used and a projection step (Chorin, 1968) is performed\nat every stage.\n3.2 Description of the simulation database\nThe simulation database that will be used as a reference for the model com-\nparison consists of direct numerical simulations of oscillatory flow through a\nhexagonal sphere pack (Unglehrt and Manhart, 2022a, 2023a,b). The sphere\npack is a triply periodic close-packed arrangement of equal spheres of diameterSpringer Nature 2021 L ATEX template\n14 Assessment of models for nonlinear oscillatory flow . . .\nd. Hence, each sphere is in contact with 12 other spheres and the porosity is\ngiven as ϵ= 1−π\n3√\n2≈0.26. The flow was driven by a sinusoidally time-varying\npressure gradient\n∇⟨p⟩i(t) =gxsin(Ω t) (30)\nand therefore depends on two dimensionless parameters: The Hagen num-\nberHg=|gx|d3/(ρν2) determines the amplitude and the Womersley number\nWo=p\nΩd2/νdetermines the frequency of the macroscopic pressure gradient.\nThe Reynolds number was defined as\nRe= lim sup\nt→∞|⟨u⟩s|d\nν(31)\nand was obtained as a result of the simulations. The simulations were per-\nformed at the three Womersley numbers Wo = 10, 31 .62, and 100 that\ncorrespond to the low, medium and high frequency regime in linear flow,\nrespectively.2The Hagen numbers were set such that the simulations cover lin-\near flow, laminar nonlinear flow, and transitional and turbulence-like flow at\neach Womersley number. The flow was started from rest and simulated until\na recurrent behaviour could be observed in the superficial velocity. The differ-\nence in the superficial velocity in the last simulated cycle and the preceding\ncycle was less than approximately 0 .1% of the maximum superficial velocity in\nthe last simulated cycle for the laminar cases and about 2% for the transitional\nand turbulence-like cases. The simulation parameters can be found in table 1.\nFigure 2 shows the dimensionless parameters of our simulation database\nin the Re–Wo2parameter space. We additionally give the frequency ratio\nΩ/Ω0where Ω 0= 930 .25ν/d2is the transition frequency defined in section\n2.1.3 that can be used to distinguish between the low and the high fre-\nquency regime in linear flow. The parameter region in which linear flow can\nbe observed was determined in (Unglehrt and Manhart, 2022a). The breaking\nof the geometry-imposed symmetries in the flow was used to distinguish the\nlaminar nonlinear from the transitional and turbulence-like regime (Unglehrt\nand Manhart, 2022b).\nThe time series of the superficial velocity for some of the simulation cases\nis shown in figure 3. The amplitudes of the superficial velocity decrease with\nincreasing Reynolds or Womersley number. The decrease of the amplitude\nwith Womersley number is caused by the inertia of the system, whereas the\ndecrease with Reynolds number can be explained by the increase of the drag.\nAs the Reynolds number increases, the behaviour of the superficial velocity\nbecomes non-sinusoidal. This could be explained by the effect of nonlinearity\nand turbulence. At high Womersley and Reynolds numbers, we can distinguish\nphases of strong and weak acceleration that lead to distinctive bends in the\nsuperficial velocity. In (Unglehrt and Manhart, 2023a) we could link this to an\nincrease in the convective pressure drag due to flow separation at the contact\n2The medium frequency regime in linear flow through a hexagonal sphere pack occurs around\na Womersley number Wo0=p\nϵ d2/(α∞K) = 30 .5, where Ω 0=ϵ ν/(α∞K) is the transition\nfrequency (Pride et al, 1993). The geometric constants are given in table 2.Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 15\nTable 1 : Parameters of the simulations of oscillatory flow through a hexagonal\nsphere pack.\nCase Hg Wo Re number of simulated cycles flow regime\nLF111.00·10310 0 .171 1 .5 linear\nLF211.00·10410 1 .7 2 .25 linear\nLF311.00·10510 14 .8 1 .4 laminar nonlinear\nLF411.00·10610 76 .7 1 .25 laminar nonlinear\nLF523.16·10610 158 2 .27 transitional\nLF621.00·10710 307 1 .56 turbulence-like\nMF111.00·10431.6 0 .857 3 linear\nMF211.00·10531.6 8 .57 3 linear\nMF313.16·10531.6 26 .9 3 laminar nonlinear\nMF411.00·10631.6 73 .1 3 laminar nonlinear\nMF523.16·10631.6 157 6 .4 transitional\nMF621.00·10731.6 298 2 .26 turbulence-like\nHF111.00·105100 1 .3 20 .4 linear\nHF211.00·106100 13 19 .9 linear\nHF311.00·107100 132 6 .32 laminar nonlinear\nHF411.78·107100 252 8 laminar nonlinear\nHF523.16·107100 465 6 transitional\nHF631.00·108100 1090 3 .91 turbulence-like\nHF731.00·109100 3620 10 turbulence-like\n1from Unglehrt and Manhart (2022a)\n2from Unglehrt and Manhart (2023a)\n3from Unglehrt and Manhart (2023b)\npoints. As the Womersley number increases, we observe a phase lag of the\nsuperficial velocity with respect to the forcing. This can be seen from the zero-\ncrossings of the superficial velocity which do not coincide with the zeros of the\nforcing ( φ=kπfork= 0,1,2, . . .). This is due to the increasing importance of\nthe inertia compared to the drag. On the other hand, the phase lag decreases\nas the Reynolds number increases. This goes in hand with an increase of the\nnonlinear drag which advances the inertial term. Overall, these time series\ndemonstrate a distinct effect of nonlinearity and turbulence on the response\nof the superficial velocity to harmonic excitations.\n3.3 Model constants for the hexagonal sphere pack\nIn this section, we specify the model constants that were used to evaluate the\nmodel predictions. The parameters for the linear models have rigorous defini-\ntions in terms of the steady Stokes flow ( K,α0) and the potential flow solution\n(Λ/d,α∞), respectively. Notably, these quantities are intrinsic properties of\nthe pore geometry. The low frequency properties of the hexagonal sphere pack\nwere determined from direct numerical simulations in the studies of Zhu and\nManhart (2016) and Sakai and Manhart (2020); the high frequency propertiesSpringer Nature 2021 L ATEX template\n16 Assessment of models for nonlinear oscillatory flow . . .\n10−1100101102103104101102103104105\nlinear flowtransitional and\nturbulence-like flowLF1 LF2 LF3 LF4MF1 MF2 MF3 MF4HF1 HF2 HF3HF4\nLF6LF5HF5\nMF6 MF5HF6 HF7\nReWo2\n10−1100101102\nΩ/Ω0\nFig. 2 : Simulation parameters. The stars mark the simulations from (Unglehrt\nand Manhart, 2022a), the pluses mark the simulations from (Unglehrt and\nManhart, 2023a) and the crosses mark the simulations from (Unglehrt and\nManhart, 2023b). The regions shaded in blue and green corresponds to the\nlinear regime and the transitional and turbulence-like regime, respectively.\nwere determined by a potential flow calculation in our previous study (Unglehrt\nand Manhart, 2023a). The parameter values are summarised in table 2.\nFor the coefficients of the unsteady Forchheimer equation various choices\ncan be found in the literature (see section 2.2.3). Following Sollitt and\nCross (1972), we assume that the coefficients aandbare constant and take\ntheir steady state values. This choice ensures that the unsteady Forchheimer\nequation has the correct low frequency limit behaviour. For the accelera-\ntion coefficient c, we consider two choices: Either the coefficient is set to the\npotential flow value ρ α∞/ϵsuch that the correct high-frequency behaviour\nis recovered (Zhu and Manhart, 2016), or, following Zhu (2016), the coeffi-\ncient is set to the Stokes flow value ρ α0/ϵsuch that the unsteady Forchheimer\nequation reduces to the unsteady Darcy equation with the static viscous tor-\ntuosity (Zhu et al, 2014). We do not consider time- or frequency-dependent\ncoefficients as no generally applicable correlations have been given in the liter-\nature. Note that the coefficients of the unsteady Forchheimer equation could\nbe adjusted to improve the fit for some simulation cases; however, this comes\nat the price of high prediction errors behaviour in some of the asymptotic\nlimits, e.g. for slowly varying flow or for linear flow.\nThe coefficients of the Forchheimer equation (15) are determined based on\nthe simulation results in (Sakai and Manhart, 2020; Unglehrt and Manhart,Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 17\nFig. 3 : Variation of the superficial velocity over the cycle from the direct\nnumerical simulations of oscillatory flow through a hexagonal sphere pack. In\nall cases, the forcing is given by ∇⟨p⟩i=gxsinφand the superficial velocity\nis normalised by the value obtained from Darcy’s law at the peak pressure\ngradient.\n2023a). The linear Forchheimer coefficient ais chosen such that the Forch-\nheimer equation approaches Darcy’s law for Re→0. Then, the nonlinear\nForchheimer coefficient bis determined from a least squares fit to the ratio\nof the macroscopic pressure gradient and the superficial velocity. We obtained\nthe following coefficients:\na=µ\nK= 5777µ\nd2(32a)\nb= 88.9ρ\nd. (32b)Springer Nature 2021 L ATEX template\n18 Assessment of models for nonlinear oscillatory flow . . .\nTable 2 : Geometric parameters for the hexagonal close-packed arrangement\nof equal spheres.\nParameter Symbol Value\nPorosity ϵ 1−π\n3√\n2= 0.2595\nPermeability K 1.731·10−4d2 1\nLow-frequency limit of the dynamic tortuosity α0 2.6572\nHigh-frequency limit of the dynamic tortuosity α∞ 1.6223\nBoundary layer length scale Λ 5 .904·10−2d3\n1from Sakai and Manhart (2020)\n2from Zhu and Manhart (2016)\n3from Unglehrt and Manhart (2023a)\nFig. 4 : Forchheimer equation (15) with coefficients (32) and direct numerical\nsimulation data for stationary flow through a hexagonal sphere pack.\nIt can be seen in figure 4 that these coefficients provide a good fit to the\nsimulation data. Finally, the nonlinear coefficient ξin the equations (18) and\n(23) results as 0 .0154d/ν.\n3.4 Implementation of the models\n3.4.1 Discretisation of the model of Pride et al. (1993)\nAs we would like to obtain model predictions for transient flow, we discretise\nthe model of Pride et al (1993) in the time domain. We discretise equation\n(13) using the implicit Euler method and a piecewise linear interpolation of\nthe convolution term. We obtained the following scheme\na⟨u⟩n+1\ns=⟨u⟩n\ns−∆tϵ\nρ α∞∇⟨p⟩n+1\ni−∞X\nk=1ck⟨u⟩n−k+1\ns(33a)Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 19\nwith precomputable coefficients\na=1 + h\u00124β2\nP−2β\u0013\n+ 2β\"\u00121\n2+h\u0013\nerf\u0010√\nh\u0011\n+r\nh\nπe−h#\n(33b)\nck=−2β\n\nh\nerf\u0010p\nξ\u0011ikh\n(k−1)h\u00121\n2+ (k−1)h\u0013\n+\"r\nξ\nπe−ξ#kh\n(k−1)h\n\n\n+ 2β\n\nh\nerf\u0010p\nξ\u0011i(k+1)h\nkh\u00121\n2+ (k+ 1)h\u0013\n+\"r\nξ\nπe−ξ#(k+1)h\nkh\n\n(33c)\ndepending on the dimensionless time step h=ν∆t\nΛ2β2. Note that the compu-\ntational effort grows linearly with the number of time steps as the discrete\nconvolution must be performed over a time series of increasing length. Com-\nputationally more efficient discretisations could be devised by approximating\nthe tail of the convolution kernel in equation (13) with an exponential function\nsimilar to (van Hinsberg et al, 2011).\nThe correctness of the scheme (33) and its convergence with the time step\nsize is assessed using the method of manufactured solutions. As a test case,\nwe consider the velocity ⟨u⟩s=j0t2\n2θ(t) with the Heaviside function θ(t).\nThe forcing that is necessary to find this velocity as a solution to (13) can be\nobtained as\n−∇⟨p⟩i=ρ α∞\nϵj0t θ(t) +\u0012µ\nK−α∞\nϵ2µ\nΛ2β\u0013\nj0t2\n2θ(t)\n+ρ α∞\nϵβ j0θ(t)\u0014\u0012\nt+νt2\nΛ2β2−1\n4Λ2β2\nν\u0013\nerf\u0012rνt\nΛ2β2\u0013\n+\u0012\nt+Λ2β2\n2ν\u0013rνt\nΛ2β2e−νt\nΛ2β2\n√π#\n.(34)\nWe simulated (33) with the forcing defined above for a hexagonal sphere\npack. Figure 5a shows the relative error of ⟨u⟩swith respect to the analytic\nsolution atνT\nd2= 5. We observe first order convergence which means that the\nsingularity in the integral kernel is treated with sufficient accuracy.\n3.4.2 Discretisation of the model of Turo & Umnova (2013)\nSimilar to the preceding section, the discretisation of equation (18) was derived\nusing the implicit Euler method and a linear interpolation in the convolution\nterm. This resulted in the following scheme\na⟨u⟩n+1\ns+b\f\f\f⟨u⟩n+1\ns\f\f\f⟨u⟩n+1\ns=⟨u⟩n\ns−∆tϵ\nρ α∞∇⟨p⟩n+1\ni−∞X\nk=1ck⟨u⟩n−k+1\ns\n(35a)Springer Nature 2021 L ATEX template\n20 Assessment of models for nonlinear oscillatory flow . . .\n(a)\n (b)\nFig. 5 : Verification of the discretisations for (a) the model of Pride et al (1993)\nand (b) the model of Turo and Umnova (2013) with manufactured solutions\n(equation (34) and equation (36) withξ\nj1/5\n0ν2/5= 0.01, respectively).\nwith precomputable coefficients\na=1 +ϵ\nα∞ν∆t\nK+2√\nν∆t\nΛ2√π(35b)\nb=ϵ\nα∞ν∆t\nKξ (35c)\nck=2√\nν∆t\nΛ2√π\u0010√\nk+ 1−2√\nk+√\nk−1\u0011\n. (35d)\nThe nonlinear equation appearing in every time step was solved with the\nfunction fsolve in MATLAB where the superficial velocity from the previous\ntime step was taken as an initial guess. Other discretisations of the fractional\nderivative have been given e.g. by Diethelm et al (2005).\nAs in the preceding section, we verified the correctness and convergence\nof our implementation using a manufactured solution for the velocity ⟨u⟩s=\nj0t2\n2θ(t). The corresponding forcing results from (18) as\n−∇⟨p⟩i=ρα∞\nϵj0t θ(t) +µ\nK\u0012\n1 +ξ|j0|t2\n2\u0013\nj0t2\n2θ(t)\n+ρα∞\nϵj0√ν\nΛ8\n3√πt3\n2θ(t).(36)\nFigure 5b shows the convergence of the numerical solution to the analytical\nsolution at a timeνT\nd2= 5 and for a coefficient of nonlinearityξ\nj1/5\n0ν2/5= 0.01.\nAgain, we have used the material properties of the hexagonal sphere pack.Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 21\n3.4.3 Evaluation of model ODEs\nThe unsteady Darcy equation (5) and the unsteady Forchheimer equation\n(16) are simple ordinary differential equations. These were solved using a sec-\nond order modified Rosenbrock method implemented in the MATLAB routine\node23s (Shampine and Reichelt, 1997). The time step size was chosen adap-\ntively according to a relative tolerance of 10−6and an absolute tolerance of\n10−8ν\ndfor the superficial velocity.\n3.5 Metrics for comparison\nIn order to compare the different models between the different simulation cases,\nwe define an integral error metric. For two periodic signals u1(t) and u2(t)\nthat will represent the model predictions and the reference for the superficial\nvelocity, respectively, the L2distance between the signals\n∥u1−u2∥2= \n1\nTZTsim\nTsim−T|u1−u2|2dt!1\n2\n. (37)\nAs we are dealing with periodic signals, we can take advantage of their Fourier\nseries to further split this error into an amplitude and a phase contribution.\nUsing Parseval’s identity, we can express the squared L2distance as\n∥u1−u2∥2\n2=∞X\nk=−∞|ck|2(38)\nwhere ckare the Fourier series coefficients of u1−u2\nck=1\nTZTsim\nTsim−T(u1−u2)e−ik2π\nTtdt=ak−bk (39)\nwhere akandbkare the Fourier series coefficients of u1andu2, respectively.\nExpressing the complex numbers akandbkin polar form, we obtain\n∥u1−u2∥2\n2=∞X\nk=−∞\f\f|ak|eiϕk− |bk|eiψk\f\f2\n=∞X\nk=−∞||ak| − |bk||2+∞X\nk=−∞2|ak||bk|[1−cos(ϕk−ψk)].(40)\nThe first term represents the difference in the magnitude of the Fourier coeffi-\ncients and the second term represents the difference in the phase of the Fourier\ncoefficients. Since both terms are positive, we can define an amplitude and aSpringer Nature 2021 L ATEX template\n22 Assessment of models for nonlinear oscillatory flow . . .\nphase distance\ndampl.(u1, u2) =vuut∞X\nk=−∞||ak| − |bk||2(41)\ndphase(u1, u2) =vuut∞X\nk=−∞2|ak||bk|[1−cos(ϕk−ψk)] (42)\nand the L2distance can be decomposed into the sum of squares\n∥u1−u2∥2\n2=dampl.(u1, u2)2+dphase(u1, u2)2. (43)\nNote that dampl.(u1, u2) and dphase(u1, u2) are not proper distances in that\nthey can be zero also if u1̸=u2. Moreover, dphase(u1, u2) does not satisfy\nthe triangle inequality. However, these definitions ensure that two identical\nwaveforms that are shifted with respect to each other only lead to a phase\ndistance, but not to an amplitude distance. Also, a waveform that is a constant\nmultiple of the other only leads to an amplitude distance, but not to a phase\ndistance.\n4 Results\n4.1 Comparison of model errors\nIn this section, we compare the accuracy of the model predictions in response\nto the forcing (30) with respect to the direct numerical simulation dataset. In\nparticular, we look at the model of Turo and Umnova (2013) given in equation\n(18), the extended Pride et al (1993) model given in equation (23) and the\nunsteady Forchheimer equation (16). For the unsteady Forchheimer equations,\nwe take the acceleration coefficient as the static viscous tortuosity α0and as\nthe high-frequency limit of the dynamic tortuosity α∞.\nThe error of the model predictions with respect to the direct numerical\nsimulation results is quantified using the cycle-averaged L2error (37); we fur-\nther decompose this error into an amplitude (41) and a phase contribution (42)\nusing a Fourier series. Table 3 shows the amplitude and phase contribution\nto the L2error of the different model predictions with respect to the direct\nnumerical simulation dataset. Note that the linear models were not applied\nto the nonlinear cases since they cannot account for the nonlinear drag and\nthus would produce large errors at high Reynolds numbers. In almost all cases,\nthe main source of error is the amplitude error. In contrast, the phase error\nis significant only in four cases in which it reaches about half the size of the\namplitude error.\nFor linear flow it can be seen that the dynamic permeability models of\nJohnson et al (1987) and Pride et al (1993) are very accurate over the entire\nfrequency range whereas the unsteady Darcy equation of Zhu et al (2014) hasSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 23\nL2amplitudeerror\n1 %1 %0 %1 %25 % 0 %7 %\n1 %0 %0 %3 %25 % 2 %7 %\nn.a.n.a.n.a.5 %25 % 5 %7 %\nn.a.n.a.n.a.3 %14 % 3 %3 %\nn.a.n.a.n.a.4 %9 %4 %3 %\nn.a.n.a.n.a.2 %7 %3 %2 %\n1 %1 %10 % 1 %27 %10 %39 %\n3 %2 %12 % 5 %28 % 8 %35 %\nn.a.n.a.n.a.9 %28 %12 %24 %\nn.a.n.a.n.a.14 %25 %19 % 9 %\nn.a.n.a.n.a.16 %21 %20 % 6 %\nn.a.n.a.n.a.14 %18 %17 % 4 %\n1 %4 %27 % 4 %9 %27 %26 %\n1 %4 %27 % 5 %10 %27 %25 %\nn.a.n.a.n.a.12 %17 %27 %18 %\nn.a.n.a.n.a.18 %22 %34 % 9 %\nn.a.n.a.n.a.21 %23 %40 % 6 %\nn.a.n.a.n.a.23 %23 %41 %12 %\nn.a.n.a.n.a.14 %15 %22 %10 %\nJohnsonetal.model\nPrideetal.model\nunsteadyDarcy\nextendedPrideetal.model\nTuro&Umnovamodel\nunsteadyForchheimerwith,0\nunsteadyForchheimerwith, 1LF1\nLF2\nLF3\nLF4\nLF5\nLF6\nMF1\nMF2\nMF3\nMF4\nMF5\nMF6\nHF1\nHF2\nHF3\nHF4\nHF5\nHF6\nHF7\n0%10%20%30%40%\n(a)\nL2phaseerror\n0 %0 %0 %0 %0 %0 %0 %\n0 %0 %0 %0 %0 %0 %0 %\nn.a.n.a.n.a.0 %0 %0 %0 %\nn.a.n.a.n.a.0 %0 %0 %1 %\nn.a.n.a.n.a.1 %1 %0 %1 %\nn.a.n.a.n.a.0 %1 %0 %0 %\n0 %0 %0 %0 %0 %0 %0 %\n0 %0 %0 %0 %0 %0 %1 %\nn.a.n.a.n.a.1 %1 %1 %2 %\nn.a.n.a.n.a.1 %2 %0 %5 %\nn.a.n.a.n.a.0 %0 %0 %1 %\nn.a.n.a.n.a.1 %2 %6 %1 %\n0 %0 %0 %0 %0 %0 %0 %\n0 %0 %0 %0 %0 %0 %0 %\nn.a.n.a.n.a.0 %0 %0 %0 %\nn.a.n.a.n.a.1 %1 %0 %1 %\nn.a.n.a.n.a.2 %2 %0 %3 %\nn.a.n.a.n.a.1 %1 %0 %2 %\nn.a.n.a.n.a.1 %1 %9 %1 %\nJohnsonetal.model\nPrideetal.model\nunsteadyDarcy\nextendedPrideetal.model\nTuro&Umnovamodel\nunsteadyForchheimerwith,0\nunsteadyForchheimerwith, 1LF1\nLF2\nLF3\nLF4\nLF5\nLF6\nMF1\nMF2\nMF3\nMF4\nMF5\nMF6\nHF1\nHF2\nHF3\nHF4\nHF5\nHF6\nHF7\n0%2 :5%5%7 :5%10%\n(b)\nTable 3 : Amplitude and phase contribution to the L2error normalised with\nmax⟨u⟩sof the respective flow case. The entries where a linear model would\nbe applied to a nonlinear flow case are marked as n.a. (not applicable).Springer Nature 2021 L ATEX template\n24 Assessment of models for nonlinear oscillatory flow . . .\n(a)\n (b)\n(c)\n (d)\nFig. 6 : Distribution of the L2model error in the Re–Ω/Ω0parameter space.\nThe diameter of the circles is proportional to the L2error. The dashed line indi-\ncates the approximate boundary between linear and nonlinear flow (Unglehrt\nand Manhart, 2022a).\nvery small errors at low frequencies and high errors at high frequencies. Thus,\nthe behaviour of the linear models is consistent with the discussion in section\n2.3.\nThe prediction accuracy of the four nonlinear models shows significant\ndifferences depending on the flow case. This is illustrated in figure 6 which\nshows the variation of the L2error of the different nonlinear models over the\nRe–Woparameter space.\nWe find that the extended dynamic permeability model based on the model\nof Pride et al (1993) has very small errors in linear flow and for flow at low\nfrequencies. The reason for this is that the model equation (23) reverts to theSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 25\nFig. 7 : Comparison of model predictions of the superficial velocity for the\ncases HF1 and HF6.\nmodel of Pride et al (1993) as Re→0 and to the steady Forchheimer equation\nas Ω/Ω0→0. The prediction errors are largest for nonlinear flow at high fre-\nquencies. Interestingly, the prediction error for the case HF7 ( Re= 3580) is\nsmaller than for the case HF6 ( Re= 1080). The model of Turo and Umnova\n(2013) performs similar to the extended (Pride et al, 1993) model at high\nfrequencies; however, the prediction errors in the medium and low frequency\nregime are very large. The reason for this is the excessive damping in the linear\nregime that was discussed in section 2.3. The unsteady Forchheimer equation\nwith the acceleration coefficient based on the static viscous tortuosity α0has\nrelatively small errors at low frequencies, moderate errors at medium frequen-\ncies and very large errors at high frequencies. The largest errors can be observed\nin nonlinear flow at high frequencies. In contrast, the unsteady Forchheimer\nequation with the acceleration coefficient based on the high-frequency limit of\nthe dynamic tortuosity α∞has comparably small errors at low frequencies,\nvery large errors in linear flow at medium and high frequencies and small errors\nfor flow at high Reynolds numbers. This somewhat surprising behaviour of the\nunsteady Forchheimer equation with α∞will be investigated in the following\nsection.\n4.2 Analysis of prediction errors\nIn this section, we aim to explain the different error behaviours described\nabove. We first investigate the discrepancies in the linear regime, where the\nmodel of Turo and Umnova (2013) (equation 18) and the unsteady Forchheimer\nequation (16) have large errors at low and high frequencies, respectively. We\nthen proceed to the nonlinear high frequency regime where large discrepancies\nbetween the models can be observed and the flow state is characterised by the\ninteraction between strong accelerations and turbulence. Therefore, we takeSpringer Nature 2021 L ATEX template\n26 Assessment of models for nonlinear oscillatory flow . . .\na closer look at the predictions of the models for the linear case HF1 and\nfor the turbulent case HF6 at Ω /Ω0= 10.7 (Wo= 100). Figure 7 shows the\npredictions of the different models in comparison to the superficial velocity\nfrom the simulations.\nFor the linear case, the model of Turo and Umnova (2013) (equation 18)\nand the extended Pride et al (1993) model (equation 23) accurately represent\nthe amplitude and phase of the superficial velocity; the errors in the maximum\nsuperficial velocity are −3.1% and −2.2%, respectively. This is in agreement\nwith the discussion in section 2.3.1 (figure 1). On the other hand, the unsteady\nForchheimer equation (16) with an acceleration coefficient α0underpredicts\nthe amplitude of the superficial velocity by 24 .8% whereas the unsteady Forch-\nheimer equation with an acceleration coefficient α∞overpredicts the amplitude\nof the superficial velocity by 22 .7%.\nSince for this case the magnitude of the nonlinear term is only about 1 .5%\nof the linear drag, the unsteady Forchheimer equation is effectively reduced to\nthe unsteady Darcy equation here. Therefore, the behaviour is determined by\nthe acceleration coefficient cand the permeability (contained in the coefficient\na=µ/K)). The comparison of the two different choices for the accelera-\ntion coefficient cof the unsteady Forchheimer equation thus highlights the\neffect of the ratio between inertia and linear drag on the superficial veloc-\nity. While the choice c=ρ α∞/ϵappears to have too little mass, the choice\nc=ρ α0/ϵappears to have too much mass. Since the behaviour of the case\nHF1 is sinusoidal, it would be possible to find coefficients for the unsteady\nDarcy equation that represent the simulation data exactly. However, these best\nfit coefficients would only be valid at this particular frequency and lead to\ninconsistent behaviour in the low- and high frequency limit (see also figure 1).\nChoosing the coefficients as a function of the frequency leads to the frequency\ndomain formulation of the dynamic permeability models (see section 2.1.3). In\nthe time domain, this is reflected in the appearance of a history term.\nFor the turbulent case, the amplitude of the superficial velocity in the direct\nnumerical simulation is 16% lower than the amplitude of the linear case HF1\nscaled by a factor of 103to the same Hagen number. Moreover, the maximum\nsuperficial velocity in the case HF6 is attained significantly earlier than in the\ncase HF1 ( φ= 0.61πfor HF6 compared to φ= 0.94πfor HF1). It can be\nseen in figure 7 that all models underpredict the amplitude of the superficial\nvelocity: The error in the maximum superficial velocity is −23% for the model\nof Turo and Umnova (2013), −21.2% for the extended Pride et al (1993) model,\n−26.4% for the unsteady Forchheimer equation with α0and−12.9% for the\nunsteady Forchheimer equation with α∞. The phase is captured satisfactorily\nby the extended Pride et al (1993) model, the model of Turo and Umnova\n(2013) and the unsteady Forchheimer equation with α∞, whereas the unsteady\nForchheimer equation with α0mispredicts the phase. It can be seen that all\nmodels fail to represent the relatively sharp bend at the peak.\nIt can be seen from the comparison to the linear case that the (identi-\ncal) nonlinear term in the models causes excessive damping, since even theSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 27\nFig. 8 : Comparison of the drag from the direct numerical simulation of the\ncase HF6 with the drag from model solutions to the unsteady Forchheimer\nequation for the values of the acceleration coefficient c=ρ/ϵ(quasi-steady\nclosure / zero virtual mass), c=ρ α∞/ϵ(high-frequency limit of the dynamic\ntortuosity) and c=ρ α0/ϵ(static viscous tortuosity).\nunsteady Forchheimer equation with an acceleration coefficient c=ρ α∞/ϵ\nunderpredicts the amplitude of the simulation. In this sense, the behaviour of\nthe different models could be understood as a compensation or superposition\nof errors.\nThis interpretation is supported by figure 8, which compares the drag of\nthe direct numerical simulation of the case HF6 to the drag of model solutions\nby the unsteady Forchheimer equation for different values of the acceleration\ncoefficient. The drag is determined for each time series according to the volume-\naveraged momentum equation (2b) as the sum of the acceleration and the\npressure gradient. We first consider the curve for the value c=ρ/ϵ, which\ncorresponds to the case of zero virtual mass and was assumed for instance by\nSollitt and Cross (1972), Kuznetsov and Nield (2006) or Breugem et al (2006).\nIt can be seen that the drag is a single-valued function of the superficial velocity\nfp+fτwx=−ϵ(a⟨u⟩s+b|⟨u⟩s|⟨u⟩s). (44)\nthat is identical to the drag in the steady flow (15). For higher values of\nthe acceleration coefficient, the drag becomes a double-valued function of the\nsuperficial velocity. Since the drag in the direct numerical simulation is also\nmulti-valued, a nonzero virtual mass ( c > ρ/ϵ ) is required to represent the\ncorrect behaviour. It can be seen that the peak superficial velocity of the\nmodel predictions decreases as the acceleration coefficient increases. The peak\nsuperficial velocity of the simulation cannot be reached even for c=ρ/ϵforSpringer Nature 2021 L ATEX template\n28 Assessment of models for nonlinear oscillatory flow . . .\nwhich highest amplitude of the different model predictions is obtained. As\nb|⟨u⟩s| ≈12a, this indicates that the nonlinear drag is too large.\nIt is important to realise that the physical assumptions underlying the\ndifferent models are not fulfilled in this flow case, for instance:\n•The unsteady Forchheimer equation implies an energy equation (see section\n2.3.3) in which both the kinetic energy and the dissipation rate are single-\nvalued functions of the instantaneous superficial velocity. This is not the\ncase in the direct numerical simulation: The kinetic energy during acceler-\nation is significantly lower than during deceleration due to the generation\nof turbulent kinetic energy and the kinetic energy is not in phase with the\ndissipation rate (Unglehrt and Manhart, 2023b) – as it would have to be if\nboth were single-valued functions of the superficial velocity.\n•The model of Turo and Umnova (2013) (and similarly the model of Pride\net al (1993)) assume linear Stokes boundary layers at high frequencies that\nevolve according to an outer potential flow unlimited by convection. How-\never, when the flow separates, the outer flow is modified such that the\nboundary layers would have to evolve differently, resulting in a different\nformulation of the history term.\nTherefore, different parametrisations of the drag should be explored beyond\nthe models investigated here.\nIn conclusion, we could explain the mispredictions of the unsteady Forch-\nheimer equation for linear flow at high frequencies. These could be attributed\nmainly to a mismatch of the ratio between the inertia and the linear drag.\nA consistent resolution of this issue is provided by the dynamic permeability\nmodels that choose this ratio depending on the frequency. The behaviour of the\nmodels in nonlinear flow at high frequencies could be partially explained by an\noverprediction of the nonlinear drag. However, since many intrinsic assump-\ntions of the models are violated in this regime, the overall functional form of\nthe parametrisations should be revisited.\n5 Conclusion\nIn this contribution, we reviewed various models for unsteady porous media\nflow from the literature and compared their predictions for oscillatory flow\nthrough a hexagonal sphere pack. The reference data are direct numerical sim-\nulations of Zhu and Manhart (2016); Unglehrt and Manhart (2022a, 2023a,b)\nfor this flow configuration.\nThe models can be divided into two classes: On the one hand, there are\nthe unsteady variants of the Darcy and Forchheimer equation (Polubarinova-\nKochina, 1962); on the other hand, there are the dynamic permeability models\n(Johnson et al, 1987; Pride et al, 1993) which feature a convolution-type struc-\nture in the time domain and nonlinear extensions thereof (Turo and Umnova,\n2013).Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 29\nIn linear flow, the dynamic permeability models of Johnson et al (1987);\nPride et al (1993) provide an accurate description of the simulation data. The\nunsteady Darcy equation could be obtained as a special case of these models in\nwhich the acceleration coefficient is either based on the static viscous tortuosity\nα0or on the high-frequency limit of the dynamic tortuosity α∞. The model of\nTuro and Umnova (2013) is overly dissipative at medium and low frequencies.\nTo alleviate this drawback, we constructed a similar model based on the model\nof Pride et al (1993).\nIn nonlinear flow, we compared the unsteady Forchheimer equation for two\ndifferent choices of the acceleration coefficient, the extended dynamic perme-\nability model of Turo and Umnova (2013) and an analogous formulation based\non the model of Pride et al (1993). The unsteady Forchheimer equation with\nthe acceleration coefficient based on the static viscous tortuosity α0shows\ngood results in the low frequency regime, but the results deteriorate at higher\nfrequencies and are particularly bad in the high frequency nonlinear regime.\nOn the other hand, the unsteady Forchheimer equation with the acceleration\ncoefficient chosen based on the high-frequency limit of the dynamic tortuos-\nityα∞shows very good results at high Reynolds numbers, but incurs large\nerrors in the linear regime. The model of Turo and Umnova (2013) has rela-\ntively large errors throughout the entire parameter space. On the other hand,\nthe extension of the model of Pride et al (1993) shows excellent results in the\nlinear regime and at low frequencies numbers, but the results deteriorate for\nnonlinear flow at high frequencies.\nGenerally, our proposed extension of the model of Pride et al (1993) along\nthe lines of (Turo and Umnova, 2013) (see section 2.3.2) seems to be a robust\nchoice with accurate results in linear unsteady and nonlinear steady flow and\na moderate increase of errors towards strongly accelerated nonlinear flow. The\ndrawback of this model is the additional implementation effort and computa-\ntional cost caused by the convolution term. For weakly accelerated flow, this\nmodel can be simplified to the unsteady Forchheimer equation with an accel-\neration coefficient c=ρ α0/ϵbased on the static viscous tortuosity α0, which\nis more economical. On the other hand, the unsteady Forchheimer equation\nwith an acceleration coefficient c=ρ α∞/ϵbased on the high frequency limit\nof the dynamic tortuosity α∞should be used judiciously since the small errors\nat large Reynolds numbers must be weighed against large errors for linear\nunsteady flow.\nFurther improvements are needed in the parametrisation of the nonlinear\ndrag at high frequencies, as our results indicate that the Forchheimer term\nleads to an overprediction of the nonlinear drag. Moreover, our previous inves-\ntigations showed that there is a phase lag between the nonlinear effects in the\nvelocity field and in the drag and the superficial velocity (Unglehrt and Man-\nhart, 2022a, 2023a). It thus seems plausible that introducing a time lag between\nthe nonlinear drag and the superficial velocity could lead to an improved model.\nFurther research should also aim to generalise the present results to differentSpringer Nature 2021 L ATEX template\n30 Assessment of models for nonlinear oscillatory flow . . .\nkinds of porous media, for example random and polydisperse sphere packs,\nfoams, and cylinder arrays.\nDeclarations\nFunding. The authors gratefully acknowledge the financial support of\nthe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)\nunder grant no. MA2062/13-1. Computing time was granted by the Leibniz\nSupercomputing Centre on its Linux-Cluster.\nCompeting Interests. The authors have no relevant financial or non-\nfinancial interests to disclose.\nAuthor Contributions. L.U. performed the review of the models from the\nliterature, implemented the models and analysed the simulation data. L.U.\nand M.M. both contributed to writing the manuscript.\nData Availability. The time series of the superficial velocity and kinetic\nenergy for the simulations LF1–LF4, MF1–MF4 and HF1–HF4 are provided\nas a supplement to (Unglehrt and Manhart, 2022a). The time series of the\nsuperficial velocity and kinetic energy for the simulations LF5, LF6, MF5, MF6\nand HF5 are provided as a supplement to (Unglehrt and Manhart, 2023a). The\ntime series of the superficial velocity and kinetic energy for the simulations\nHF6 and HF7 are provided as a supplement to this work.\nReferences\nAboujafari M, Valipour MS, Hajialimohammadi A, et al (2022) Porous\nMedium Applications in Internal Combustion Engines: A Review.\nTransport in Porous Media 141(3):799–824. https://doi.org/10.1007/\ns11242-022-01750-2\nAndreades CH, Cisneros AT, Choi JK, et al (2014) Technical Description of the\n“Mark 1” Pebble-Bed Fluoride-Salt-Cooled High-Temperature Reactor (PB-\nFHR) Power Plant. Tech. Rep. UCBTH- 14- 002, Department of Nuclear\nEngineering University of California, Berkeley, Berkeley\nBatchelor GK (2000) An Introduction to Fluid Dynamics. Cambridge Univer-\nsity Press, Cambridge, https://doi.org/10.1017/CBO9780511800955\nBreugem WP, Boersma BJ, Uittenbogaard RE (2006) The influence of wall\npermeability on turbulent channel flow. Journal of Fluid Mechanics 562:35.\nhttps://doi.org/10.1017/S0022112006000887\nBurcharth H, Andersen O (1995) On the one-dimensional steady and unsteady\nporous flow equations. Coastal Engineering 24(3-4):233–257. https://doi.\norg/10.1016/0378-3839(94)00025-SSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 31\nChampoux Y, Allard JF (1991) Dynamic tortuosity and bulk modulus in air-\nsaturated porous media. Journal of Applied Physics 70(4):1975–1979. https:\n//doi.org/10.1063/1.349482\nChapman AM, Higdon JJL (1992) Oscillatory Stokes flow in periodic porous\nmedia. Physics of Fluids A: Fluid Dynamics 4(10):2099–2116. https://doi.\norg/10.1063/1.858507\nChorin AJ (1968) Numerical solution of the Navier-Stokes equations.\nMathematics of Computation 22(104):745–762. https://doi.org/10.1090/\nS0025-5718-1968-0242392-2\nCortis A, Smeulders DMJ, Lafarge D, et al (2002) Geometry Effects on Sound\nin Porous Media. In: Ehlers W (ed) IUTAM Symposium on Theoretical and\nNumerical Methods in Continuum Mechanics of Porous Materials, vol 87.\nKluwer Academic Publishers, Dordrecht, p 187–192, https://doi.org/10.\n1007/0-306-46953-7 26\nCortis A, Smeulders DMJ, Guermond JL, et al (2003) Influence of pore rough-\nness on high-frequency permeability. Physics of Fluids 15(6):1766–1775.\nhttps://doi.org/10.1063/1.1571545\nDarcy H (1856) Les fontaines publiques de la ville de Dijon. Victor Dalmont,\nParis\nDavit Y, Bell CG, Byrne HM, et al (2013) Homogenization via formal mul-\ntiscale asymptotics and volume averaging: How do the two techniques\ncompare? Advances in Water Resources 62:178–206. https://doi.org/10.\n1016/j.advwatres.2013.09.006\nDiethelm K, Ford N, Freed A, et al (2005) Algorithms for the fractional cal-\nculus: A selection of numerical methods. Computer Methods in Applied\nMechanics and Engineering 194(6-8):743–773. https://doi.org/10.1016/j.\ncma.2004.06.006\nEne H, Sanchez-Palencia E (1975) Equations et ph´ enom` enes de surface pour\nl’´ ecoulement dans un milieu poreux. Journal de M´ ecanique 14\nErgun S (1952) Fluid Flow Through Packed Columns. Chemical Engineering\nProgress 48(2):89–94\nFirdaouss M, Guermond JL, Le Qu´ er´ e P (1997) Nonlinear corrections to\nDarcy’s law at low Reynolds numbers. Journal of Fluid Mechanics 343:331–\n350. https://doi.org/10.1017/S0022112097005843\nForchheimer P (1901) Wasserbewegung durch Boden. Zeitschrift des Vereins\ndeutscher Ingenieure 45:1782–1788Springer Nature 2021 L ATEX template\n32 Assessment of models for nonlinear oscillatory flow . . .\nGraham DR, Higdon JJL (2002) Oscillatory forcing of flow through porous\nmedia. Part 2. Unsteady flow. Journal of Fluid Mechanics 465. https://doi.\norg/10.1017/s0022112002001143\nGu Z, Wang H (1991) Gravity waves over porous bottoms. Coastal Engineering\n15(5-6):497–524. https://doi.org/10.1016/0378-3839(91)90025-C\nHall KR, Smith GM, Turcke DJ (1995) Comparison of oscillatory and sta-\ntionary flow through porous media. Coastal Engineering 24(3-4):217–232.\nhttps://doi.org/10.1016/0378-3839(94)00017-R\nHill RJ, Koch DL, Ladd AJC (2001) The first effects of fluid inertia on flows in\nordered and random arrays of spheres. Journal of Fluid Mechanics 448:213–\n241. https://doi.org/10.1017/S0022112001005948\nHsu C, Cheng P (1990) Thermal dispersion in a porous medium. Interna-\ntional Journal of Heat and Mass Transfer 33(8):1587–1597. https://doi.org/\n10.1016/0017-9310(90)90015-M\nIliuta I, Larachi F (2016) Three-dimensional simulations of gas-liquid cocurrent\ndownflow in vertical, inclined, and oscillating packed beds. AIChE Journal\n62(3):916–927. https://doi.org/10.1002/aic.15071\nIliuta I, Larachi F (2017) CO2 abatement in oscillating packed-bed scrubbers:\nHydrodynamics and reaction performances for marine applications. AIChE\nJournal 63(3):1064–1076. https://doi.org/10.1002/aic.15450\nJohnson DL, Koplik J, Dashen R (1987) Theory of dynamic permeability\nand tortuosity in fluid-saturated porous media. Journal of Fluid Mechanics\n176:379–402. https://doi.org/10.1017/S0022112087000727\nKahler DM, Kabala ZJ (2019) Acceleration of Groundwater Remediation by\nRapidly Pulsed Pumping: Laboratory Column Tests. Journal of Environ-\nmental Engineering 145(1):06018,009. https://doi.org/10.1061/(ASCE)EE.\n1943-7870.0001479\nKergomard J, Lafarge D, Gilbert J (2013) Transients in Porous Media: Exact\nand Modelled Time-Domain Green’s Functions. Acta Acustica united with\nAcustica 99(4):557–571. https://doi.org/10.3813/AAA.918635\nKoch DL, Ladd AJC (1997) Moderate Reynolds number flows through periodic\nand random arrays of aligned cylinders. Journal of Fluid Mechanics 349:31–\n66. https://doi.org/10.1017/S002211209700671X\nKuznetsov AV, Nield DA (2006) Forced Convection with Laminar Pulsating\nFlow in a Saturated Porous Channel or Tube. Transport in Porous Media\n65(3):505–523. https://doi.org/10.1007/s11242-006-6791-6Springer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 33\nLafarge D (1993) Propagation du son dans les mat´ eriaux poreux ´ a struc-\nture rig ide satur´ es par un fluide viscothermique: D´ efinition de param` etres\ng´ eom´ etriques, analogie electromagn´ etique, temps de relaxation. PhD thesis,\nUniversit´ e du Maine, Le Mans\nLafarge D (2009) The Equivalent Fluid Model. In: Bruneau M, Potel C (eds)\nMaterials and Acoustics Handbook. ISTE, London, UK, p 153–204, https:\n//doi.org/10.1002/9780470611609.ch6\nLosada I, Losada M, Mart´ ın F (1995) Experimental study of wave-induced\nflow in a porous structure. Coastal Engineering 26(1-2):77–98. https://doi.\norg/10.1016/0378-3839(95)00013-5\nLowe RJ, Shavit U, Falter JL, et al (2008) Modeling flow in coral commu-\nnities with and without waves: A synthesis of porous media and canopy\nflow approaches. Limnology and Oceanography 53(6):2668–2680. https://\ndoi.org/10.4319/lo.2008.53.6.2668\nMacdonald IF, El-Sayed MS, Mow K, et al (1979) Flow through Porous\nMedia-the Ergun Equation Revisited. Industrial & Engineering Chemistry\nFundamentals 18(3):199–208. https://doi.org/10.1021/i160071a001\nManhart M, Tremblay F, Friedrich R (2001) MGLET: A parallel code for\nefficient DNS and LES of complex geometries. In: Parallel Computational\nFluid Dynamics 2000. North-Holland, Amsterdam, p 449–456, https://doi.\norg/10.1016/B978-044450673-3/50123-8\nMei CC, Auriault JL (1991) The effect of weak inertia on flow through a porous\nmedium. Journal of Fluid Mechanics 222:647–663. https://doi.org/10.1017/\nS0022112091001258\nMuttray M (2000) Wellenbewegung an und in einem gesch¨ utteten Wellen-\nbrecher - Laborexperimente im Großmaßstab und theoretische Untersuchun-\ngen. PhD thesis, Technische Universit¨ at Braunschweig, Braunschweig\nNi X, Mackley M, Harvey A, et al (2003) Mixing Through Oscillations and\nPulsations—A Guide to Achieving Process Enhancements in the Chem-\nical and Process Industries. Chemical Engineering Research and Design\n81(3):373–383. https://doi.org/10.1205/02638760360596928\nNield D (1991) The limitations of the Brinkman-Forchheimer equation in mod-\neling flow in a saturated porous medium and at an interface. International\nJournal of Heat and Fluid Flow 12(3):269–272. https://doi.org/10.1016/\n0142-727X(91)90062-ZSpringer Nature 2021 L ATEX template\n34 Assessment of models for nonlinear oscillatory flow . . .\nNield DA (2000) Resolution of a Paradox Involving Viscous Dissipation and\nNonlinear Drag in a Porous Medium. Transport in Porous Media 41(3):349–\n357. https://doi.org/10.1023/A:1006636605498\nNorris AN (1986) On the viscodynamic operator in Biot’s equations of\nporoelasticity. Journal of Wave-material Interaction 1:365–380\nPa´ ez-Garc´ ıa CT, Vald´ es-Parada FJ, Lasseux D (2017) Macroscopic momen-\ntum and mechanical energy equations for incompressible single-phase flow in\nporous media. Physical Review E 95(2). https://doi.org/10.1103/PhysRevE.\n95.023101\nPeller N (2010) Numerische Simulation turbulenter Str¨ omungen mit Immersed\nBoundaries. PhD thesis, Technische Universit¨ at M¨ unchen, M¨ unchen\nPeller N, Duc AL, Tremblay F, et al (2006) High-order stable interpolations for\nimmersed boundary methods. International Journal for Numerical Methods\nin Fluids 52(11):1175–1193. https://doi.org/10.1002/fld.1227\nPolubarinova-Kochina PI (1962) Theory of Ground Water Movement. Prince-\nton University Press, Princeton, New Jersey\nPride SR, Morgan FD, Gangi AF (1993) Drag forces of porous-medium\nacoustics. Physical Review B 47(9):4964–4978. https://doi.org/10.1103/\nPhysRevB.47.4964\nRoncen R, Fellah ZEA, Lafarge D, et al (2018) Acoustical modeling and\nBayesian inference for rigid porous media in the low-mid frequency regime.\nThe Journal of the Acoustical Society of America 144(6):3084–3101. https:\n//doi.org/10.1121/1.5080561\nSakai Y, Manhart M (2020) Consistent Flow Structure Evolution in Accel-\nerating Flow Through Hexagonal Sphere Pack. Flow, Turbulence and\nCombustion 105(2):581–606. https://doi.org/10.1007/s10494-020-00168-4\nSchumann U (1977) Realizability of Reynolds-Stress Turbulence Models.\nPhysics of Fluids 20:721–725. https://doi.org/10.1063/1.861942\nShampine L, Reichelt M (1997) The MATLAB ODE Suite. SIAM J Sci Comput\nhttps://doi.org/10.1137/S1064827594276424\nSimon TW, Seume JR (1988) A Survey of Oscillating Engine Heat Exchang-\ners. Tech. Rep. NASA-CR-182108), University of Minnesota, Minneapolis,\nMinnesota\nSmeulders DMJ, Eggels RLGM, Van Dongen MEH (1992) Dynamic per-\nmeability: Reformulation of theory and new experimental and numericalSpringer Nature 2021 L ATEX template\nAssessment of models for nonlinear oscillatory flow . . . 35\ndata. Journal of Fluid Mechanics 245(-1):211. https://doi.org/10.1017/\nS0022112092000429\nSollitt CK, Cross RH (1972) Wave Transmission through Permeable Break-\nwaters. In: Coastal Engineering 1972. American Society of Civil Engineers,\nVancouver, British Columbia, Canada, pp 1827–1846, https://doi.org/10.\n1061/9780872620490.106\nTrevizoli PV, Peixer GF, Barbosa JR (2016) Thermal–hydraulic evaluation of\noscillating-flow regenerators using water: Experimental analysis of packed\nbeds of spheres. International Journal of Heat and Mass Transfer 99:918–930.\nhttps://doi.org/10.1016/j.ijheatmasstransfer.2016.03.014\nTuro D, Umnova O (2013) Influence of Forchheimer’s nonlinearity and tran-\nsient effects on pulse propagation in air saturated rigid granular materials.\nThe Journal of the Acoustical Society of America 134(6):4763–4774. https:\n//doi.org/10.1121/1.4824969\nUmnova O, Turo D (2009) Time domain formulation of the equivalent fluid\nmodel for rigid porous media. The Journal of the Acoustical Society of\nAmerica 125(4):1860–1863. https://doi.org/10.1121/1.3082123\nUnglehrt L, Manhart M (2022a) Onset of nonlinearity in oscillatory flow\nthrough a hexagonal sphere pack. Journal of Fluid Mechanics 944:A30.\nhttps://doi.org/10.1017/jfm.2022.496\nUnglehrt L, Manhart M (2022b) Symmetry Breaking and Turbulence in Oscil-\nlatory Flow Through a Hexagonal Sphere Pack. In: Proceedings of TSFP-12\n(2022) Osaka, Osaka, Japan, p 6\nUnglehrt L, Manhart M (2023a) Decomposition of the drag force in steady and\noscillatory flow through a hexagonal sphere pack. Journal of Fluid Mechanics\n974:A32. https://doi.org/10.1017/jfm.2023.798\nUnglehrt L, Manhart M (2023b) Direct and Large–Eddy simulation of tur-\nbulent oscillatory flow through a hexagonal sphere pack. In: Marchioli C,\nSalvetti MV, Garcia-Villalba M, et al (eds) Direct and Large Eddy Simula-\ntion XIII: Proceedings of DLES13, 1st edn. No. 31 in ERCOFTAC Series,\nSpringer Cham, p 118–123\nvan Gent MRA (1993) Stationary and oscillatory flow through coarse porous\nmedia. Communications on hydraulic and geotechnical engineering, No 1993-\n09\nvan Gent MRA (1994) The modelling of wave action on and in coastal\nstructures. Coastal Engineering 22(3-4):311–339. https://doi.org/10.1016/\n0378-3839(94)90041-8Springer Nature 2021 L ATEX template\n36 Assessment of models for nonlinear oscillatory flow . . .\nvan Hinsberg MAT, ten Thije Boonkkamp JHM, Clercx HJH (2011) An effi-\ncient, second order method for the approximation of the Basset history\nforce. Journal of Computational Physics 230(4):1465–1478. https://doi.org/\n10.1016/j.jcp.2010.11.014\nVerstappen R, Veldman A (2003) Symmetry-preserving discretization of tur-\nbulent flow. Journal of Computational Physics 187(1):343–368. https://doi.\norg/10.1016/S0021-9991(03)00126-8\nWang CY (2008) The Starting Flow in Ducts Filled with a Darcy–Brinkman\nMedium. Transport in Porous Media 75(1):55–62. https://doi.org/10.1007/\ns11242-008-9210-3\nWard JC (1964) Turbulent Flow in Porous Media. Journal of the Hydraulics\nDivision 90(5):1–12. https://doi.org/10.1061/JYCEAJ.0001096\nWhitaker S (1967) Diffusion and dispersion in porous media. AIChE Journal\n13(3):420–427. https://doi.org/10.1002/aic.690130308\nWhitaker S (1986) Flow in porous media I: A theoretical derivation of\nDarcy’s law. Transport in Porous Media 1(1):3–25. https://doi.org/10.1007/\nBF01036523\nWhitaker S (1996) The Forchheimer equation: A theoretical develop-\nment. Transport in Porous Media 25(1):27–61. https://doi.org/10.1007/\nBF00141261\nWilliamson J (1980) Low-storage Runge-Kutta schemes. Journal of Computa-\ntional Physics 35(1):48–56. https://doi.org/10.1016/0021-9991(80)90033-9\nZhu T (2016) Unsteady porous-media flows. PhD thesis, Technische Univer-\nsit¨ at M¨ unchen, M¨ unchen\nZhu T, Manhart M (2016) Oscillatory Darcy Flow in Porous Media.\nTransport in Porous Media 111(2):521–539. https://doi.org/10.1007/\ns11242-015-0609-3\nZhu T, Waluga C, Wohlmuth B, et al (2014) A Study of the Time Con-\nstant in Unsteady Porous Media Flow Using Direct Numerical Simula-\ntion. Transport in Porous Media 104(1):161–179. https://doi.org/10.1007/\ns11242-014-0326-3"    },    {        "title": "2402.05344v1.Dark_matter_effects_of_a_black_hole_with_nonsingular_Yukawa_modified_potential_in_Einstein_Gauss_Bonnet_Gravity.pdf",        "content": "Dark matter effects of a black hole with nonsingular Yukawa–modified potential in\nEinstein–Gauss–Bonnet Gravity\nYassine Sekhmani ,1, 2, *A. A. Ara ´ujo Filho ,3, †Ratbay Myrzakulov ,2, 1, ‡Adam\nZ. Kaczmarek ,4, §Javlon Rayimbaev ,5, 6, 7, 8, ¶and Dominik Szcze ¸ ´sniak4, ||\n1Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Astana 010009, Kazakhstan.\n2L. N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan.\n3Departamento de F ´ısica, Universidade Federal da Para ´ıba,\nCaixa Postal 5008, 58051-970, Jo ˜ao Pessoa, Para ´ıba, Brazil\n4Institute of Physics, Faculty of Science and Technology,\nJan Długosz University in Cze ¸stochowa, 13/15 Armii Krajowej Ave., 42200 Cze ¸stochowa, Poland\n5New Uzbekistan University, Movarounnahr Str. 1, Tashkent 100007, Uzbekistan\n6Central Asian University, Tashkent 111221, Uzbekistan\n7University of Tashkent for Applied Sciences,\nGavhar Str. 1, Tashkent 100149, Uzbekistan\n8Institute of Fundamental and Applied Research,\nNational Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan\nThis paper investigates the contribution of the nonsingular Yukawa–modified potential in the con-\ntext of four–dimensional Einstein–Gauss–Bonnet (EGB) gravity modeling by a static and spherically\nsymmetric black hole solution. These Yukawa–type corrections are essentially described along two\nparameters, βandλ, affecting Newton’s law of gravity at large distances, and a deformation param-\neterℓ0, which is essential at short distances. Primarily, the strongest effect is encoded in β, which\nalters the total mass of the black hole with additional mass proportional to βM, imitating the effects\nof dark matter at large distances from the black hole. In contrast, the effect due to λis small for as-\ntrophysical values. On the other hand, the EGB gravity is ruled by the Gauss–Bonnet (GB) coupling\nconstant α, a fundamental parameter of the theory. We pay particular attention to thermodynamic\nstability, critical orbits, geodesics and quasinormal modes. The results demonstrate stability of the\nblack hole solution for a range of values of the GB coupling constant α. Furthermore, this study\ninvestigates the null geodesic motion, namely the shadow behavior, providing intriguing results in\nrelation to the size of the black hole shadow.\n*Email: sekhmaniyassine@gmail.com\n†Email: dilto@fisica.ufc.br\n‡Email: rmyrzakulov@gmail.com\n§Email: adamzenonkaczmarek@gmail.com\n¶Email: javlon@astrin.uz\n||Email: d.szczesniak@ujd.edu.plarXiv:2402.05344v1  [gr-qc]  8 Feb 20242\nI. INTRODUCTION\nThe idea to modify and provide alternatives to general relativity (GR) is as old as the original formulation\nof the theory. The first of such approaches dates back to the times of Einstein and was presented by Weyl,\nEddington, Kaluza and Klein [1–4]. These extensions are relevant even now, since the standard formulation\nof the GR faces serious challenges when considering the dark sector (the dark energy and the dark matter)\n[5]. In this respect, one of the extended theories of particular convenience is the Einstein-Gauss-Bonnet\n(EGB) gravity, historically introduced by Lovelock [6, 7]. The pivotal aspect of this theory is related to the\nGauss-Bonnet (GB) term\nG=R2−4RµνRµν+RµνρσRµνρσ, (1)\nwhich, although associated with the higher-curvature corrections in the Einstein-Hilbert action, leads to\nfield equations that are at most of the second order in metric. Thus, Ostrogardski’s instabilities are avoided\ndespite the introduction of the GB term [7]. However, while important in the five-dimensions (or in more\ngeneral couplings such as the f(G)models), the GB term in its linear form ( D→4dimensions) vanishes\nat the level of the field equations and becomes de facto irrelevant [7, 8]. Still, by using the proper rescaling\n(α→α\nD−4), the resulting equations in four dimensions may have the GB contribution, in contrast to\nLovelock’s theorem [9]. As a consequence, recent years have witnessed a resurgence of Gauss-Bonnet\ngravity in its four-dimensional form. Among many, one of the most interesting and insightful studies was\nprovided in the context of black holes [10–20]. More precisely, the latest activities in the context of EGB\ngravity have given rise to a wealth of investigations into thermodynamics [21], quasinormal modes (QNMs)\n[22, 23], or shadows behavior [24–28]. These investigations lend particular interest to the EGB gravity;\nin particular, this theory is linked to the effective theories of superstrings, which are encoded via the GB\ncoupling constant.\nIn general, the notion of a black hole (BH) plays a key role in the development of proper spacetime\ndescription, establishing one of the core research directions in modern physics [29–33]. This concept con-\nfronts the existing paradigms in both general relativity and the quantum realm via enigmatic features like the\nsingularities or the information paradox, highlighting the limits of classical relativity and the necessity for a\nquantum gravity framework [34, 35]. It also constitutes an attractive tool to study the aforementioned dark\nsector [36, 37], which is notoriously hard to detect due to the lack of electromagnetic interaction. This fact\nappears to be especially intriguing in the context of the recently introduced models based on the Yukawa\nforce [38–42]. Approaches of that kind are closely related to the modified Newtonian dynamics [43], as\nthe potential of the Yukawa type alters the Newtonian one in a similar manner [42]. Recent breakthroughs3\npoint to dark matter being explained by Yukawa coupling between baryonic matter in terms of long-range\ninteraction [38, 42]. Nevertheless, it should be noted that these ideas are not yet properly addressed by\nmodified theories of gravity (such as EGB gravity) [36, 44, 45].\nMotivated by the above, we attempt to investigate the relationship between the BH region and the specific\ndark matter model within the framework of the four-dimensional (4D) EGB gravity [7]. This is done by\nintroducing a new BH solution supported by the Yukawa corrections. As a result, we are able to provide a\nnovel contribution to the fundamental understanding of the BH spacetime in the dark sector environment,\nsupplementing and extending previous related studies [16, 19, 36, 46]. These results are followed by a\ndiscussion of the possible related effects. In particular, our study focuses on investigating shadow and light\nray behavior in Yukawa-corrected BH spacetime by using 4D EGB gravity. The BH shadows are unique\nimages produced by the light deflected near the event horizon, creating the perfect opportunity to study the\nconnection between various spacetimes and geometries. This includes analysis of the dark sector and the\ndark matter candidates [36, 47–53] as well as the discussion of the QNMs closely linked to the radius of\nthe BH shadows [11, 54–57]. Hence, through geometric optics, our work explores the intricate relationship\nbetween the BH geometry, QNMs, and the modified Yukawa potential following recent trends [46, 58–71].\nWe argue that such investigations are essential for understanding the potential influence of dark matter on\nfuture BH shadow observations [64, 72, 73].\nThe presented work is organized as follows: (i) The theoretical background and description of the\nYukawa corrections to the gravitational potential are introduced in Section II. Therein, the new BH solution\nfor the 4D EGB with the Yukawa potential is obtained, and the physical characteristics of such a model\nare thoroughly discussed [38]. (ii) In the Section IV of the manuscript, the critical orbits and geodesics are\nanalyzed. (iii) This is followed by investigations of the BH shadows and related properties in Section V . (iv)\nTo this end, in Section VI, quasinormal modes are thoroughly discussed [36, 74, 75]. (vi) The last part is\ndevoted to the conclusions and summary of the presented work, including indications of new perspectives\nthat this manuscript provides.\nII. BLACK HOLE SOLUTION WITH YUKA WA POTENTIAL\nMotivated by the exciting black hole solution with the dark matter effect recently studied in A, we are\nlooking forward to extending the situation into the EGB gravity framework.\nWe consider the EGB action with a negative cosmological constant as\nS=1\n16πZ\ndDx√−g(R−2Λ +G), (2)4\nwhere Ris the Ricci scalar, Gis mentioned in Eq. (1) and αis the GB coupling constant. In the rest of this\nwork, we assume the 4Dstatic and spherically symmetric spacetime anstaz given by\nds2=−f(r)dt2+dr2\nf(r)+r2(dθ2+ sin2θdϕ2). (3)\nEven though in the 4Dscenario, the GB term is practically a total derivative and therefore does not con-\ntribute to gravitational dynamics, an additional scalar field can be coupled to the GB term, which is referred\nto as Einstein–Dilaton Gauss–Bonnet theory influencing that dynamic [76, 77]. To overcome the topologi-\ncal problem, Glavan and Lin [9] investigated the 4Dset–in of the GB term, considering a rescaling of the\ncoupling constant\nα→α\nD−4. (4)\nConsequently, dealing with the 4Dcase is now achieved, and the field equation for component 00, incorpo-\nrating a material source, results\n1\nrdf\ndr+f\nr2−1\nr2−α\u00122(f−1)\nr3df\ndr−(f−1)2\nr4\u0013\n+ Λ = −ρ(r), (5)\nwhere T0\n0=−ρ(r).\nIn what follows, an examination of the matter source regarding the modified Yukawa potential is re-\nquired. This contribution was recently discovered to probe the cosmological significance of the modified\npart of the Yukawa potential [71]. For that reason, the gravitational potential we considered is modified by\nthe regular Yukawa–type potential given as follows\nΦ(r) =−GMmq\nrD−2+ℓD−2\n0\u0010\n1 +β e−r\nλ\u0011\n. (6)\nNotably, the wavelength of massive graviton is found to be λ=ℏ\nmgc, andℓorepresents a deformed parameter\nof Planck length size. It is well known that the energy density of the modified matter can be explicitly\nderived by considering ρ(r) =1\n4π∆Φ(r). Taking this element into account, we obtain the following energy\ndensity:\nρ(r) =e−r\nλβM\n4π rλ2(r2+ℓ2\n0)5/2A+3Mℓ2\n0\n4π(r2+ℓ2\n0)5/2(7)\nwhere on has, A= 2λℓ4\n0+(3λ2ℓ2\n0−ℓ4\n0)r+2λℓ2\n0r2−2ℓ2\n0r3−r5. A closer interpretation of the energy density\nexpression reveals that the two terms have different characteristics: the first term is linearly proportional to β\nand provides interesting results at large distances, while the second is proportional to ℓ0and plays particular\nattention to short distances. Specifically, if we ignore long–range modification, that is, as a particular case\nof our results, only the second term maintains consistency with [78].5\nTo proceed with finding a BH solution within EGB gravity, one can expand the first term of the energy\ndensity (7) in a series around ℓ0. This can result in the following expression:\nρ(r) =−βM\n4π rλ2e−r\nλ+3Mℓ2\n0\n4π(r2+ℓ2\n0)5/2+O(ℓ2\n0β) (8)\nAccordingly, it may be possible to distinguish the physical nature of the current energy density. Indeed,\nthe first term corresponds to that of [71]. In addition, it is worth noting that the negative sign indicates that\nthe energy conditions are violated inside the black hole. Alternatively, we further conjecture that the field\nequations with a cosmological constant remain valid, i.e., Gµν+ Λgµν+αHµν= 8πTµν, whereby the\neffects of effective dark matter are encoded in the stress-energy tensor. And, from the gravitational field\nequations, one can obtain\n1\nrdf\ndr+f\nr2−1\nr2−α\u00122(f−1)\nr3df\ndr−(f−1)2\nr4\u0013\n+ Λ−2βM\nrλ2e−r\nλ+6Mℓ2\n0\n(r2+ℓ2\n0)5/2= 0 (9)\nThis equation yields a pair of distinct solutions denoted by the signs ±. These solutions could be exposed\nas follows:\nf(r)+,−= 1 +r2\n2α(\n1±s\n1 +4\n3α\u0012\n6M\u00121\n(ℓ2\n0+r2)3/2+e−r\nλβ(r+λ)\nr3λ\u0013\n+ Λ +3α\nr3c1\u0013)\n, (10)\nwhere c1is an integration constant. Concretely, the physical solution is that with a negative branch, which\nbrings to the ordinary BH solutions in the context of GR considering the limit α→0. Practically speaking,\nthe negative branch is therefore a new space-time generalization in the sense of modifying the gravity of\nthe study in Ref. [71]. Moreover, we can interpret the physical content of the obtained BH solution in such\na way that the second term carries each modification of the geometry due to ℓ0. Further, the third term\nconsists entirely of the apparent effect of dark matter, while the fourth term is due to the contribution of the\ncosmological constant [78]. It is worth noting that the vanishing of the third, fourth, and last terms, the BH\nsolution in this case, can be brought to that of Ref. [71] with the consideration of the limit α→0. Notably,\nℓ0is of Planck length order, i.e., ℓ0∼10−35m[78]. In large distances and astrophysical black holes with\na large mass M, we should set r≫ℓ0; then, the ignorance of ℓ0is constrained and based on this bound\nconjecture. Setting c1= 0can provide the exact solution of the BH as follows:\nf(r) = 1 +r2\n2α(\n1−s\n1 +4\n3α\u0012\n6M\u00121\n(ℓ2\n0+r2)3/2+e−r\nλβ(r+λ)\nr3λ\u0013\n+ Λ\u0013)\n, (11)\nwhich generalizes the BH solution in the frame of the EGB gravity model. It is useful to address the relevant\nphysical aspect by working with the physical mass, M=M(1 + β), and this could be presented in the\nexact solution as follows\nf(r) = 1 +r2\n2α(\n1−s\n1 +4\n3α\u00126M\n1 +β\u00121\n(ℓ2\n0+r2)3/2+e−r\nλβ(r+λ)\nr3λ\u0013\n+ Λ\u0013)\n, (12)6\nwhich, in the context of GR, reduces to another defined exact solution [71].\n0 10 20 30 40-0.500.51.0\nrf(r)λ=0.1,β=0.2,Λ=0.002\nα\n0.20.40.60.8\n0 10 20 30 40-1.0-0.500.51.0\nrf(r)λ=0.1,β=0.2,Λ=0.002\nβ\n0.20.40.60.8\nFIG. 1. Variation of the black hole metric function (12) with respect to rfor various values of the parameter space\nand with M= 1andℓ0= 0.2.\nf(r) = 1−2Mr2\n(1 +β)(ℓ2\n0+r2)3/2−2Mβ(r+λ)e−r\nλ\nrλ(1 +β)−Λr2\n3. (13)\nTo gain a better understanding of the black hole metric (12), Fig. 1offers a suitable graphical analysis.\nThe observation is mostly significant for the parameters βandα. In the literature on black hole physics,\nthe possible roots of the metric function are classified into two kinds: the smallest root is related to two\nblack hole horizons, and the largest root is in relation to the cosmological horizon. It is observed that such\na variation of the parameters αandβgenerates various configurations of the inner horizon, the event and\ncosmological being closely located at the same horizon radius. A closer observation shows that depending\non both the parameters αandβ, the possible set of the horizon radius can reduce from three possible horizon\nradius up to one, namely the cosmological horizon.\nTo conduct some analysis of nature’s singularities in relation to the physical solution (12), the Ricci\nscalars ( R), the Ricci square ( RµνRµν) and Kretshmann scalars ( RµνλσRµνλσ) are needed\nR=−r2f′′(r) + 4rf′(r) + 2f(r)−2\nr2, (14)\nRµνRµν=r4f′′(r)2+ 8r2f′(r)2+ 8f(r) (rf′(r)−1) + 4 rf′(r)\u0000\nr2f′′(r)−2\u0001\n+ 4f(r)2+ 4\n2r4,\n(15)\nRµναβRµναβ=f′′(r)2+4f′(r)2\nr2+4(f(r)−1)2\nr4. (16)7\n0 0.5 1.0 1.5 2.0 2.5 3.0-5051015\nrRλ=0.1,β=0.2,Λ=0.002\nα\n0.10.20.30.4\n0.5 1.0 1.5 2.0 2.5 3.002468\nrRμνRμνλ=0.1,β=0.2,Λ=0.002\nα\n0.10.20.30.4\n00.5 1.0 1.5 2.0 2.5 3.005101520\nrRμναβRμναβλ=0.1,β=0.2,Λ=0.002\nα\n0.10.20.30.4\nFIG. 2. Variation of Ricci, Ricci squared, and Kretschmann scalar w.r.t. r for various values of αand with M= 1and\nℓ0= 0.2.\nUsing the metric function in our case, one can see that the scalars are not free from singularity issues. This\nis mainly due to the presence of the violation of stress–energy conservation.\nTo offer a thorough discussion of the characteristics of the black hole solution from the standpoint\nof scalar invariants, Fig. 2provides the relevant features. It is observed that all scalar invariants have\nsimilar behaviors and are positive-definite, while negative at a small rfor certain valued parameters. The\npresentation (Fig. 2) proves that the Gauss–Bonnet coupling αaffects the variation of the Ricci scalar, the\nRicci scalar squared, and the Kretshmann. In addition, it is well noted that the three scalars all go to zero\nwhen rlies outside the event horizons. Thus, the black hole solution is therefore completely unique in\nnature and exhibits a physical singularity at r= 0, beyond avoidance.8\nIII. THERMODYNAMICS STABILITY\nRevealing the local thermal stability of the BH solution is the main focus of this section. Proceeding\nwith the thermodynamics aspect is first examined by approaching the solution set at the horizon radius of\nthe equation f(rh) = 0 , giving an appropriate expression of the black hole mass as\nM=−(β+ 1)\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001\n6\u0012\nr4\nh\n(r2\nh+ℓ2\n0)3/2+βrh(rh+λ)\nλe−rh\nλ\u0013 (17)\nOn the other hand, the corresponding Hawking temperature for the metric given by Eq. (12) is [79]\nTH=1\n4π\u0012df\ndr\u0013\nr=rh(18)\nor, in terms of the parameter space of the black hole system, this quantity is expressed as follows:\nTH=−1\n4π(\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001\u0012\n3r5\nh\n(ℓ2\n0+r2\nh)5/2+βe−rh\nλ(3λ2+r2\nh+3λrh)\nλ2\u0013\n3\u0000\n2α+r2\nh\u0001\u0012\nr4\nh\n(ℓ2\n0+r2\nh)3/2+βrhe−rh\nλ(λ+rh)\nλ\u0013 +2\nrh)\n(19)\nAccordingly, specific limits on the parameter space can reduce the corresponding Hawking temperature to\nother defining ones; particularly, the choice leads to the Hawking temperature of the black hole.\nProviding the entropy expression is another task for examining the fully thermodynamic framework, for\nwhich the first law of black hole thermodynamics is to be applied. Thus, the application law clearly defines\nthe desired expression, namely dS= (1−ϕM)dM/T H. So, using Eq. (17) and Eq. (19) leads to\n0 0.5 1.0 1.5 2.0 2.5-500-2500250500\nrhChλ=0.1,β=0.2,Λ=0.002\nα=0.1\nα=0.25\nα=0.6\nα=0.95\n5 10 15 20 25-500-2500250500\nrhChλ=0.1,β=0.2,Λ=0.002\nα=0.1\nα=0.25\nα=0.6\nα=0.95\nFIG. 3. Variation of the Heat capacity Ch(22) with respect to rhfor various values of α.\nS=π\u0000\nr2\nh+ 4αlog(rh)\u0001\n(20)9\nwhere a closer observation shows that the obtained entropy does not obey the area law, and the corrected\nfactor is given explicitly as\nϕM= 4πZ\nr2∂T0\n0\n∂M= 1−r3\n(β+ 1)\u0000\nℓ2\n0+r2\u00013/2−βe−r\nλ(λ+r)\n(β+ 1)λ. (21)\nFor further details on the arguments concerning the logarithmic behavior of entropy, refer to [80].\nIn order to predict the local thermal black hole stability, the heat capacity shows, based on the analysis\nof its sign, whether the black hole system is locally stable or unstable [81–85]. We can calculate the heat\ncapacity using the expression Ch= dM/dTH. In concrete terms, focusing on the heat capacity demands\nand employing Eqs. (17) and (19), we obtain the following:\nCh=−2π(β+ 1)C1\nC2(22)\nwhere\nC1=(\n\u0000\n4Λr3\nh−6rh\u0001 \n−r4\nh\u0000\nℓ2\n0+r2\nh\u00013/2−βrhe−rh\nλ(λ+rh)\nλ!\n+\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001\u0012r5\nh\u0000\nℓ2\n0+r2\nh\u00015/2\n+4ℓ2\n0r3\nh\u0000\nℓ2\n0+r2\nh\u00015/2+βe−rh\nλ\u0000\nλ2−r2\nh+λrh\u0001\nλ2\u0013)\nC2= \nr4\nh\u0000\nℓ2\n0+r2\nh\u00013/2+βrhe−rh\nλ(λ+rh)\nλ!2(rh\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001\u0012\n15ℓ2\n0r3\nh\n(ℓ2\n0+r2\nh)7/2−βe−rh\nλ(λ+rh)\nλ3\u0013\n\u0000\n2α+r2\nh\u0001\u0012\nr4\nh\n(ℓ2\n0+r2\nh)3/2+βrhe−rh\nλ(λ+rh)\nλ\u0013\n−2rh\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001 \n3r5\nh\n(ℓ2\n0+r2\nh)5/2+βe−rh\nλ(3λ2+r2\nh+3λrh)\nλ2!\n\u0000\n2α+r2\nh\u00012\u0012\nr4\nh\n(ℓ2\n0+r2\nh)3/2+βrhe−rh\nλ(λ+rh)\nλ\u0013 −6\nr2\nh\n+\u0000\n4Λr3\nh−6rh\u0001 \n3r5\nh\n(ℓ2\n0+r2\nh)5/2+βe−rh\nλ(3λ2+r2\nh+3λrh)\nλ2!\n\u0000\n2α+r2\nh\u0001\u0012\nr4\nh\n(ℓ2\n0+r2\nh)3/2+βrhe−rh\nλ(λ+rh)\nλ\u0013 −\u0000\nΛr4\nh−3\u0000\nα+r2\nh\u0001\u0001\n\u0000\n2α+r2\nh\u0001\u0012\nr4\nh\n(ℓ2\n0+r2\nh)3/2+βrhe−rh\nλ(λ+rh)\nλ\u00132\n× \nr5\nh\u0000\nℓ2\n0+r2\nh\u00015/2+4ℓ2\n0r3\nh\u0000\nℓ2\n0+r2\nh\u00015/2+βe−rh\nλ\u0000\nλ2−r2\nh+λrh\u0001\nλ2!\n× \n3r5\nh\u0000\nℓ2\n0+r2\nh\u00015/2+βe−rh\nλ\u0000\n3λ2+r2\nh+ 3λrh\u0001\nλ2!)\nTo gain a better understanding of the heat capacity behavior and results in the local state thermal stability,\nFig. 3 provides a suitable analysis. It is observed that the system presents two physical limitation points,\nnamely root of the heat capacity (Ch= 0) , On the other hand, the system exhibits one divergent point for all10\nthe valued parameter spaces. Overall, these points involve four regions with specific signs of heat capacity,\neither negative, which indicates the black hole system is locally thermally stable, or positive, indicating a\nlocally unstable black hole. Roughly speaking, among all the first– or second–order phase transitions, the\nsystem remains locally thermally stable.\nIV . CRITICAL ORBITS AND GEODESICS\nAchieving a thorough comprehension of particle dynamics and the intricate behavior of light rays in the\nproximity of black hole structures demands a profound understanding of critical orbits. These orbits play a\npivotal role in elucidating the unique properties of spacetime, especially when influenced by the effects of\ndark matter in our specific context.\nIn our quest to better grasp the impact of the photon sphere, commonly known as the critical orbit,\nwithin our black hole scenario, we will leverage the Lagrangian method for computing null geodesics.\nThis method promises a more comprehensive and transparent perspective than the previously introduced\ngeodesic equation. Our analysis seeks to uncover how the mass of the black hole influences the photon\nsphere, thereby shedding light on the gravitational effects embedded in the black hole solution with Yukawa\npotential in the context of Gauss–Bonnet. Thereby, we write:\nL=1\n2gµν˙xµ˙xν. (23)\nWhen a fixed angle of θ=π/2is taken into account, the previously mentioned expression undergoes\nsimplification, yielding:\ng−1\n00E2+g−1\n11˙r2+g33L2= 0, (24)\nwithLis the angular momentum and Ebeing the energy. Next, Eq. (24) reads,\n˙r2=E2−\nr2 \n1−s\n4\n3α\u0012\nΛ + 6M\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\u0013\n+ 1!\n2α+ 1\n\u0012L2\nr2\u0013\n, (25)\nwith≁\nV≡\nr2\n1−vuut4\n3α \nΛ+6M \n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3!!\n+1\n\n2α+ 1\n\u0010\nL2\nr2\u0011\n, being the effective potential. To\ndetermine the critical radius, it is necessary to solve the equation ∂≁\nV /∂r = 0. In order to derive a viable\nsolution, we adopt the assumption that ris small. This consideration results in the emergence of four11\nsolutions for critical orbits. However, upon accounting for specific parameter values, only two of these\nsolutions prove to be physically significant, exhibiting positive real values—namely, rcinnerandrcouter. For\nenhanced reader comprehension, we present Tables I and II, which illustrate the behavior of these critical\norbits.\nTable I represents the behavior of rcinner. A discernible trend is observed: as the parameter Mincreases,\nthere is a substantial decrease in the value of rcinner. Conversely, an opposing pattern emerges with an\nincrease in the parameter a, leading to an increment in the values of rcinner.\nFurthermore, Table II focuses on the analysis of rcouter. It is noteworthy that the behavior of rcouteris\nfundamentally contrasting to that of rcinner. Specifically, as the parameter Mincreases, there is a simulta-\nneous increase in the values of rcouter. Conversely, when the parameter aexperiences an increase, there is a\ncorresponding decrease in the values of rcouter. Recently in the literature, there are many studies indicating\nmore than one light ring within the context of modified gravity [86, 87].\nM α r cinner Mα r cinner\n0.0 0.10 ———— 1.0 0.10 0.0890302\n1.0 0.10 0.0890302 1.0 0.11 0.0979745\n2.0 0.10 0.0444508 1.0 0.12 0.1069330\n3.0 0.10 0.0296307 1.0 0.13 0.1159090\n4.0 0.10 0.0222225 1.0 0.14 0.1249050\n5.0 0.10 0.0177779 1.0 0.15 0.1339220\n6.0 0.10 0.0148149 1.0 0.16 0.1429650\n7.0 0.10 0.0126984 1.0 0.17 0.1520370\n8.0 0.10 0.0111111 1.0 0.18 0.1611400\n9.0 0.10 0.00987655 1.0 0.19 0.1702800\n10.0 0.10 0.00888890 1.0 0.20 0.1794590\nTABLE I. The critical orbit values, denoted as rcinner, are presented across varying mass Mand parameter αvalues in\nthe updated table.\nWithin this section, our paramount aim is to grasp the intricacies of behavior resulting from the geodesic\nequation. This involves articulating it as follows:\nd2xµ\nds2+ Γµ\nγσdxγ\ndsdxσ\nds= 0. (26)\nConsidering Eq. (26), the aforementioned equation gives rise to four partial differential equations12\nM α r couter Mα r couter\n0.0 0.10 ———- 1.0 0.10 0.843597\n1.0 0.10 0.843597 1.0 0.11 0.852279\n2.0 0.10 0.989891 1.0 0.12 0.859277\n3.0 0.10 1.055290 1.0 0.13 0.864865\n4.0 0.10 1.093220 1.0 0.14 0.869258\n5.0 0.10 1.118220 1.0 0.15 0.872625\n6.0 0.10 1.136040 1.0 0.16 0.875104\n7.0 0.10 1.149430 1.0 0.17 0.876803\n8.0 0.10 1.159890 1.0 0.18 0.877815\n9.0 0.10 1.168300 1.0 0.19 0.878212\n10.0 0.10 1.175230 1.0 0.20 0.878057\nTABLE II. The critical orbit values, denoted as rcouter, are presented across varying mass Mand parameter αvalues\nin the updated table.\ndt′\nds=6αr′t′\n−r\n1−vuut4αΛ\n3+8αM \n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3!\n+1\n\nα−2M\u0012\n3r5\n(l2+r2)5/2+βr2e−r\nλ\nλ2+3βre−r\nλ\nλ+3βe−r\nλ\u0013\nr2vuut4αΛ\n3+8αM \n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3!\n+1\n\n6α−3r2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1 + 3 r2,\n(27)13\ndr′\nds=6α−3r2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1 + 3 r2\n12α\nr2\n1−vuut4αΛ\n3+8αM \n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3!\n+1\n\n2α+ 1\n2× {\n2M\u0012\n3r5\n(l2+r2)5/2+βr2e−r\nλ\nλ2+3βre−r\nλ\nλ+ 3βe−r\nλ\u0013\nr2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1\n+r \n1−s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1!\nα\n\n\u0000\nr′\u00012\n−6α−3r2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1 + 3 r2\n12α× {\n2M\u0012\n3r5\n(l2+r2)5/2+βr2e−r\nλ\nλ2+3βre−r\nλ\nλ+ 3βe−r\nλ\u0013\nr2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1\n+r \n1−s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1!\nα\n\n\u0000\nt′\u00012\n−r \n−6α+ 3r2s\n4αΛ\n3+ 8αM\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\n+ 1−3r2!\n6α\u0012\n(θ′)2+ sin2(θ)(φ′)2\u0013\n,\n(28)\ndθ′\nds=−sin(θ) cos( θ)\u0000\nφ′\u00012−2θ′r′\nr, (29)\nand\ndϕ′\nds=−2φ′(r′+rθ′cot(θ))\nr. (30)14\n-10 -8 -6 -4 -2 0 2 4-6-4-202468\nFIG. 4. A particular representation of the deflection of light, considering α= 0.1, λ= 105,Λ = 10−5, β= 0.1, and\nM= 1. Here, the thick yellow line denotes the light; the dotted lines represent the photon sphere; and the red ones\nidentify the shadows.\nIn Fig. 4, it is depicted an illustrating deviation of light, incorporating parameters such as α= 0.1,\nλ= 105,Λ = 10−5,β= 0.1, andM= 1. In this representation, the thick yellow line signifies the path\nof light, the dashed yellow lines depict the photon sphere, and the dashed red lines indicate the regions of\nshadow.\nV . NULL GEODESICS AND SHADOWS\nThis section takes care to study the null geodesic motion in gµν(x)background. The first step is to use\nthe Lagrangian formalism [88–90] as the necessary concept to describe the geodesic process so that the\nLagrangian in the background (12) is given as:\nL=−f(r)\n2˙t2+f(r)−1\n2˙r2+r2\n2LΩ2 (31)\nwhere LΩ2is the angular Lagrangian, which can be given as such\nLΩ2=˙θ2+ sin2θ˙ϕ2. (32)\nHere, the dot is a notation of the differentiation with respect to the affine parameter δalong the geodesic.\nBecause the (t, ϕ)variables are cyclic, the underlying Lagrangian analysis (31) is independent of them, and15\nthus the associated conjugate momenta πq=∂L/∂˙qare conserved. Specifically, we have\nπt=−f(r)˙t≡ −E, (33)\nπϕ=r2sin2θ˙ϕ≡L, (34)\nwhere Eis a positive constant measuring the temporal invariance of the Lagrangian. Furthermore, this\nconstant has no connection to energy due to the fact that the metric system (12) is not asymptotically flat,\nwhile Lis a constant, ensuring the conservation of the angular momentum.\nA better approach for describing analytically the motion of a particle around the BH solution is car-\nried out with the consideration of the Hamilton Jacobi formalism together with the Carter approach [91].\nIn particular, the massless particle generates null geodesic motions (photon orbits). In this regard, the\nHamilton-Jacobi equation is given in the following way:\n∂S\n∂δ=−1\n2gµν∂S\n∂xµ∂S\n∂xν, (35)\nwhere Sis the Jacobi action of the test particle and δis the affine parameter. Incorporating the inverse\nmetric component of the background (12) into the Eq. (35), one can obtain\n−2∂S\n∂δ=−1\nf(r)\u0012∂St\n∂t\u00132\n+f(r)\u0012∂Sr\n∂r\u00132\n+1\nr2 \n∂Sθ\n∂θ+1\nsin2θ\u0012∂Sϕ\n∂ϕ\u00132!\n. (36)\nIt is worth noting that a separable solution for the Jacobi action is explicitly represented by\nS=1\n2µ2δ−Et+Lϕ+Sr(r) +Sθ(θ), (37)\nwhere µis the rest mass of the test particle. In what follows, we restrict the study to dealing with mass\nparticles such as photons (µ= 0) . As a result, implementing Jacobi’s action (37) on Eq. (36) leads to the\nfollowing expression\n0 =\u001a1\nf(r)E2−f(r)\u0012∂Sr\n∂r\u00132\n−1\nr2\u0000\nL2+C\u0001\u001b\n−\u001a1\nr2\u0012∂Sθ\n∂θ\u00132\n+1\nr2\u0000\nL2cot2θ− C\u0001\u001b\n, (38)\nwhere Cis the Carter constant. After executing some computations, the result provides the following inde-\npendent couple of equations:\nr4f2(r)\u0012∂Sr\n∂r\u00132\n=r4E2−r2\u0000\nL2+C\u0001\nf(r), (39)\n\u0012∂Sθ\n∂θ\u00132\n=C −L2cot2θ. (40)16\nIn accordance with the Hamilton–Jacobi formalism, the appropriate set of the geodesic equation is given by\ndt\ndδ=E\nf(r), (41)\nr2dr\ndδ=±√\nR, (42)\nr2dθ\ndδ=±√\nΘ, (43)\ndϕ\ndδ=L\nr2sin2θ(44)\nwhere the symbols ”+/−”signify the radial directions in which photons are moving, respectively, outward\nand inward. To complete the analysis view of the Hamilton-Jacobi formalism, the RandΘare expressed\nas follows:\nR=r4E2−r2\u0000\nL2+C\u0001\nf(r) (45)\nΘ =C −L2cot2θ. (46)\nThe photon’s motion in the space-time (12) is controlled by Eqs. (41)-(44).\nTo analyze in depth the shadow behaviors, it is required to define and carry an effective potential. This\nstep certainly helps to show the shape of a black hole, which is entirely defined by the boundaries of\nits shadow and represents the apparent shape of the photon’s unstable circular orbits. Thus, the effective\npotential is expressed in such a way as\n\u0012dr\ndδ\u00132\n+Veff(r) = 0 . (47)\nIn other terms, the previous expression provides the equivalent of the following:\nVeff(r) =f(r)\u001a1\nr2\u0000\nL2+C\u0001\n−E2\nf(r)\u001b\n. (48)\nTo specifically find the unstable circular orbit of the photons, which carry information about the bound-\nary of the apparent shape of the black hole, the maximum of the effective potential provides the needed\ninformation through the following constraints [92]:\nVeff=dVeff(r)\ndr\f\f\f\f\nr=rph= 0,R=dR\ndr\f\f\f\f\nr=rph= 0. (49)\nFrom which the photon sphere radius rphis linked to the maximum effective potential for the spacetime\nblack hole (12), which is the smallest value of the roots of the following equation:\nrphf′(rph)−2f(rph) = 0 (50)\nwhere f′(r)is a notation of the differentiation∂f\n∂rand in terms of the parameter space of the BH system.\nSolving Eq. (50) to generate the explicit expression for the photon sphere seems analytically difficult, which17\nleads to the application of the numerical approach. As a consequence, Tab. III collects information on the\nphoton sphere rphwith respect to several fixed parameters.\nrph λ Λ α β\n1.16674 0.1 0.001 0.85 0.1\n2.01037 0.1 0.002 0.85 0.1\n2.10125 0.1 0.002 0.5 0.2\n1.99236 0.7 0.002 0.85 0.2\nTABLE III. Numerical sets of the photon sphere rphwithM= 1andℓ0= 0.2.\nThe present step consists of defining the shape and size of the black hole while considering space-time\n(12). For this, the impact parameters ξandηare needed and related to the constants of motion E,L, andC\nby means of the following expressions:\nξ=L\nE, η =C\nE2, (51)\nwhere the effective potential and the radial function become expressed in terms of these impact parameters,\nas shown in the following\nVeff=E2\u001af(r)\nr2(ξ+η)−1\u001b\n, (52)\nR=E2\u001a\nr4−r2(ξ+η)f(r)\u001b\n. (53)\nAfterward, injecting Eqs. (52)-(53) into Eq. (49) leads to obtaining an equation for two unknowns, ξand\nη, in the following compact form:\nη+ξ2=4r3\nph\n2rphf(rph) +r2\nphf′(rph). (54)\nAs a result, it may be noted that the photon sphere rphhas the dimensions of the length, while the quantity\nη+ξ2has the dimensions of the length square, describing a two-dimensional shadow geometry.\nNow, we turn to reveal the visualization of the black hole shadow, i.e., the geometrical quantity on a\ncelestial plane along the coordinates νandυ. Therefore, the celestial coordinates are given according to\n[93] by\nν=lim\nrO→∞ \n−r2\nOsinθOdϕ\ndr\f\f\f\f\n(rO,θO)!\n(55)\nυ=lim\nrO→∞ \nr2\nOdθ\ndr\f\f\f\f\n(rO,θO)!\n, (56)18\n-5.0 -2.5 0 2.5 5.0-5.0-2.502.55.0\nνυα=0.85,λ=0.1,β=0.1\nΛ=0.001\nΛ=0.01\nΛ=0.03\nΛ=0.035\nΛ=0.05\n-4 -2 0 2 4-4-2024\nνυα=0.85,λ=0.1,Λ=0.002\nβ=0.1\nβ=0.3\nβ=0.45\nβ=0.55\nβ=0.75\n-4 -2 0 2 4-4-2024\nνυλ=0.1,β=0.2,Λ=0.002\nα=0.5\nα=0.6\nα=0.75\nα=0.85\nα=0.95\n-4 -2 0 2 4-4-2024\nνυα=0.85,β=0.2,Λ=0.002\nλ=0.1\nλ=0.4\nλ=0.5\nλ=0.6\nλ=0.7\nFIG. 5. The geometrical shape of the shadow and the shadow radius of the BH solution in the celestial plane for\nseveral values of Λ,β,αandλwithM= 1andℓ0= 0.2\nwhere rOdenotes the distance between the black hole and the observer. To be more concrete, we look at null\ngeodesic motion in the equatorial plane θ=π/2leading to ν=−ξandυ=±√η. As a consequence, this\noutcome presents a two-dimensional geometry governed by the shadow radius expressed in the following\nway:\nR2\ns≡η+ξ2=ν2+υ2(57)\nwhich is nothing more than the shadow radius Rsin celestial coordinates. Bearing in mind that the shadow\nshape for non-rotating (static) black holes is a circle with a radius of Rs.\nThe practical step involved in analyzing the appropriate shadow behavior of the BH solution is perfectly\ndepicted in Fig. 5. So, proceeding with the shadow behavior is mainly carried out by analyzing the geo-\nmetrical behavior of the shadow radius. As it is observed, the size of the BH shadow is proportional to the\nvariation of the parameters λandΛ, while disproportional to αandβ.19\nVI. QUASINORMAL MODES\nIn the ringdown phase, the extraordinary phenomenon of quasinormal modes unfolds, revealing oscilla-\ntion patterns that remain remarkably steadfast against initial perturbations. These modes intricately capture\nthe inherent essence of the system, stemming from the natural oscillations of spacetime and transcend-\ning the influence of specific initial conditions. In contrast to the more confined nature of normal modes\nwithin closed systems, quasinormal modes characterize open systems, gradually releasing energy through\nthe emission of gravitational waves. Mathematically, these modes are eloquently expressed as poles of the\ncomplex Green function.\nUnraveling solutions to the wave equation within a system dictated by the background metric gµνis\ncrucial for determining their frequencies. Nevertheless, the quest for analytical solutions to these modes\nposes formidable challenges. The scientific community has explored diverse methodologies to surmount\nthis complexity, with the WKB (Wentzel–Kramers–Brillouin) method emerging as a prominent approach.\nIts roots trace back to the pioneering work of Will and Iyer [94, 95]. Subsequent refinements have elevated\nthe method, with Konoplya extending it to the sixth order [96], and Matyjasek and Opala pushing the\nboundaries to the thirteenth order [74].\nA. Scalar perturbations\nIn our computations, we focus on scrutinizing perturbations utilizing the Klein-Gordon equation within\nthe context of curved spacetime, specifically employing the scalar field\n1√−g∂µ(gµν√−g∂νΦ) = 0 . (58)\nWhile exploring the intriguing realm of backreaction effects is tempting in this context, this manuscript\ndeliberately shifts its focus to other aspects. Our primary concentration lies in the meticulous examination\nof the scalar field as a minor perturbation. The inherent spherical symmetry in the scenario allows us to\ndecompose the scalar field in a specific manner, as elucidated in greater detail below\nΦ(t, r, θ, φ ) =∞X\nl=0lX\nm=−lr−1Ψlm(t, r)Ylm(θ, φ). (59)\nIn this context, the spherical harmonics are denoted by Ylm(θ, φ). Upon integrating the scalar field de-\ncomposition, as presented in Eq. (59), into Eq. (58), the equation assumes a Schr ¨odinger–like form. This\ntransformation bestows upon the equation wave–like attributes, uniquely aligning it with the nuances essen-\ntial for our analytical investigation\n−∂2Ψ\n∂t2+∂2Ψ\n∂r∗2+Veff(r∗)Ψ = 0 . (60)20\nThe potential Veff, commonly known as the Regge–Wheeler potential or the effective potential, encap-\nsulates crucial details about the geometry of the black hole. To enhance our analysis, we introduce the\ntortoise coordinate r∗, spanning the entire spacetime and approaching ±∞ asr∗extends. This relationship\nis represented by dr∗=p\n[1/f(r)2]dr. Through algebraic rearrangements, we can concisely express the\neffective potential as:\nVeff=f(r)\n−2Mr2\u0012\n−3r\n(l2+r2)5/2−3βe−r\nλ(λ+r)\nλr4−βe−r\nλ(λ+r)\nλ2r3+βe−r\nλ\nλr3\u0013\nrs\n4\n3α\u0012\nΛ + 6 M\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\u0013\n+ 1\n+r \n1−s\n4\n3α\u0012\nΛ + 6 M\u0012\n1\n(l2+r2)3/2+βe−r\nλ(λ+r)\nλr3\u0013\u0013\n+ 1!\nα r+l(l+ 1)\nr2\n.(61)\nNotably, a barrier–like structure manifests prominently when the cosmological constant Λ, coupling\nconstant λ, and scalar field parameter αall take positive values. It is worth observing that as the angular\nmomentum quantum number lincreases, there is a corresponding elevation in the height of Veff.\nTo compute the quasinormal modes, we concentrate on the WKB method. Our primary aim is to derive\nstationary solutions for the system. To achieve this, we propose the representation Ψ(t, r) =e−iωtψ(r),\nwhere ωsignifies the frequency. This choice enables us to seamlessly isolate the time–independent aspect\nof Eq. (60), as delineated below:\n∂2ψ\n∂r∗2−\u0002\nω2−Veff(r∗)\u0003\nψ= 0. (62)\nTo effectively address Eq. (62), it is imperative to meticulously account for the relevant boundary con-\nditions. In our specific scenario, solutions adhering to these conditions are discerned by their distinctively\nexclusive purely ingoing behavior in the vicinity of the horizon\nψin(r∗)∼\n\nCl(ω)e−iωr∗(r∗→ −∞ )\nA(−)\nl(ω)e−iωr∗+A(+)\nl(ω)e+iωr∗(r∗→+∞).\nThe intricate complex constants A(+)\nl(ω),Cl(ω), and A(−)\nl(ω)constitute the fundamental components\nof our subsequent analysis. They play a pivotal role in the examination of the quasinormal modes of a black\nhole, characterized by frequencies ωnlthat satisfy the condition A(−)\nl(ωnl) = 0 . These modes manifest\na unique behavior, presenting as purely outgoing waves at spatial infinity and exclusively ingoing waves\nnear the event horizon. The integers nandldelineate the overtone and multipole numbers, respectively.21\nFurthermore, it is noteworthy that the spectrum of quasinormal modes is grounded in the eigenvalues of Eq.\n(62). To investigate these frequencies formally, we employ the WKB method, a semi-analytical approach\nreminiscent of quantum mechanics.\nAdditionally, the WKB approximation, originally introduced by Schutz and Will [97], has become an\nindispensable method for determining quasinormal modes, particularly in the context of studying particle\nscattering around black holes. Over the years, this technique has undergone refinement, marked by sub-\nstantial contributions from Konoplya [96, 98]. It is essential to recognize, however, that the applicability of\nthis method is contingent on the potential assuming a barrier-like form and leveling off to constant values\nasr∗→ ±∞ . By aligning the solution power series with the peak potential turning points, the quasinormal\nmodes can be accurately derived [99]. Given these premises, the sixth–order WKB formula is expressed as\nfollows:\ni(ω2\nn−V0)q\n−2V′′\n0−6X\nj=2Λj=n+1\n2. (63)\nIn essence, Konoplya’s formulation for the quasinormal modes comprises various essential components.\nSpecifically, the term V′′\n0denotes the second derivative of the potential, calculated at its zenith r0. Addition-\nally, the constants Λjare influenced by the effective potential and its derivatives at this peak. Noteworthy\nadvancements in this field have recently unveiled a 13th–order WKB approximation, pioneered by Maty-\njasek and Opala [74], markedly enhancing the precision of quasinormal frequency computations.\nTake note that the quasinormal frequencies associated with the scalar field possess a negative imaginary\ncomponent. This characteristic indicates that these modes undergo exponential decay over time, represent-\ning the dissipation of energy through scalar waves. This observation is consistent with previous investiga-\ntions into perturbations in spherically symmetric configurations, encompassing scalar, electromagnetic, and\ngravitational perturbations [100–106]. In Tabs. IV, V and VI, we represent the values of the quasinormal\nmodes for the scalar perturbations for different values of α. Under certain configuration of the system,\nunstable modes appear.\nB. Vectorial perturbations\nIn this section, we advance our exploration into the propagation of the electromagnetic field. To accom-\nplish this, we revisit the wave equations governing a test electromagnetic field\n1√−g∂ν\u0002√−ggαµgσν(Aσ,α−Aα,σ)\u0003\n= 0. (64)22\nTABLE IV . Employing the sixth–order WKB approximation, we examine the quasinormal frequencies associated\nwith scalar perturbations across different values of α, with a specific focus on cases where l= 0.\nα ω0 ω1 ω2\n0.10 1.32496 - 1.03427 i20.0727 - 1.14659 i94.801 - 4.70918 i\n0.11 0.86332 - 1.8318 i12.1996 - 1.26940 i67.9051 - 2.94847 i\n0.12 0.877716 - 1.55745 i13.833 - 1.06241 i71.5674 - 3.10621 i\n0.13 0.9181 - 1.29472 i14.9734 - 0.973842 i74.189 - 3.33687 i\n0.14 0.989237 - 1.04963 i15.8141 - 0.941273 i76.1426 - 3.59839 i\n0.15 1.09109 - 0.834005 i16.4585 - 0.93852 i77.6471 - 3.86977 i\nTABLE V . Employing the sixth–order WKB approximation, we examine the quasinormal frequencies associated with\nscalar perturbations across different values of α, with a specific focus on cases where l= 1.\nα ω0 ω1 ω2\n0.10 37.9636 - 1.22204 ×107i116.088 - 4.4149 ×107i252.091 - 1.13043 ×108\n0.11 Unstable Unstable Unstable\n0.12 3.80996 ×106- 23.1870 i1.3713 ×107- 70.7676 i3.50226 ×107- 153.528 i\n0.13 Unstable Unstable Unstable\n0.14 1.1673 ×106- 16.2341 i4.21718 ×106- 49.6392 i1.07982 ×107- 107.7900 i\n0.15 2.14988 ×106- 13.3484 i8.24887 ×105- 5.93913 i2.14988 ×106- 13.3484 i\nTABLE VI. Employing the sixth-order WKB approximation, we examine the quasinormal frequencies associated\nwith scalar perturbations across different values of α, with a specific focus on cases where l= 2.\nα ω0 ω1 ω2\n0.10 2.30133 - 4.53642 ×104i90.5567 - 1.50169 ×107i196.156 - 3.84392 ×107i\n0.11 Unstable Unstable Unstable\n0.12 762841 - 6.00149 i 2.72701 106- 18.4951 i6.93186 ×106- 40.3868 i\n0.13 8.24779 - 8.76037 ×105i25.1808 - 3.15653 ×106i54.6363 - 8.06777 ×106i\n0.14 137356 - 5.25372 i 510230. - 15.8237 i 1.3302 ×106- 34.0112 i\n0.15 2.30133 - 4.53642 ×104i 6.96983 - 171094 i 15.0289 - 450159 i23\nThe four–potential, represented as Aµ, can be expressed through an expansion in 4-dimensional vector\nspherical harmonics as follows:\nAµ(t, r, θ, ϕ )\n=X\nℓ,m\nf(t, r)Yℓm(θ, ϕ)\nh(t, r)Yℓm(θ, ϕ)\na(t,r)\nsin(θ)∂ϕYℓm(θ, ϕ) +k(t, r)∂θYℓm(θ, ϕ)\n−a(t, r) sin ( θ)∂θYℓm(θ, ϕ) +k(t, r)∂φYℓm(θ, ϕ)\n. (65)\nIn the context of this expansion, Yℓm(θ, ϕ)represents the spherical harmonics. It is pertinent to note that\nthe first term on the right–hand side exhibits a parity of (−1)ℓ+1(referred to as the axial sector), while the\nsecond term holds a parity of (−1)ℓ(known as the polar sector). Upon direct substitution of this expansion\ninto the Maxwell equations, a second–order differential equation governing the radial component can be\nrigorously derived [107]\nd2Ψ (r∗)\ndr2∗+\u0002\nω2−VE(r∗)\u0003\nΨ (r∗) = 0 . (66)\nIn both the axial and polar sectors, we derive a second-order differential equation governing the radial\ncomponent, where the tortoise coordinate is defined as r∗=R\nf−1(r)dr. The mode Ψ(r∗)represents a\nlinear combination of the functions a(t, r),f(t, r),h(t, r), and k(t, r). However, the specific functional\ndependence varies depending on the parity. In the case of the axial sector, the mode is expressed as follows:\na(t, r) = Ψ ( r∗). (67)\nIn contrast, for the polar sector, it is expressed as:\nΨ (r∗) =r2\nℓ(ℓ+ 1)[∂th(t, r)−∂rf(t, r)]. (68)\nThe associated effective potential in our case is determined as:\nVeff(r) =f(r)\u0012l(l+ 1)\nr2\u0013\n. (69)\nIn Tabs. VII and VIII, we represent the values of the quasinormal modes of the vectorial perturbations\nfor different values of α. Here, we notice that for some values of αunstable modes give rise to.\nVII. CONCLUSION\nDealt with Yukawa–modified potential as a contribution of matter, in the context of EGB gravity, it\nhad paved the way to inspect modeled black hole solutions. Some interesting geometrical constraints were24\nTABLE VII. Employing the sixth–order WKB approximation, we examine the quasinormal frequencies associated\nwith vectorial perturbations across different values of α, with a specific focus on cases where l= 1.\nα ω0 ω1 ω2\n0.10 212.252 - 0.000331183 i763.062 - 0.000806585 i1947.29 - 0.00164103 i\n0.11 0.000543762 - 78.7522 i0.00101029 - 277.368 i0.00175005 - 697.531 i\n0.12 Unstable Unstable Unstable\n0.13 60.1637 - 0.00225955 i216.598 - 0.00611754 i553.286 - 0.0128075 i\n0.14 59.6741 - 0.0031472 i215.189 - 0.00885143 i550.307 - 0.0187747 i\n0.15 0.00455997 - 11.70844 i0.0106379 - 39.3753 i0.0216639 - 95.3931 i\nTABLE VIII. Employing the sixth–order WKB approximation, we examine the quasinormal frequencies associated\nwith vectorial perturbations across different values of α, with a specific focus on cases where l= 2.\nα ω0 ω1 ω2\n0.10 Unstable Unstable Unstable\n0.11 Unstable Unstable Unstable\n0.12 8.89064 - 0.0112879 i33.3585 - 0.0246925 i87.5577 - 0.0477644 i\n0.13 19.644 - 0.00288482 i70.8999 - 0.00468075 i181.445 - 0.00780043 i\n0.14 12.8029 - 0.0054649 i45.8716 - 0.0105014 i116.827 - 0.0193291 i\n0.15 6.32386 - 0.00447958 i22.2207 - 0.00107095 i Unstable\npractically encoded by the parameters ℓ0,β,λ, and the GB coupling constant α. In particular, the spacetime\ngeometry was modified at a small distance by the change of the parameter ℓ0, while at large distances, the\nspacetime was modified with the given parameter set (β, λ, α ), with the reason that βmodified Newton’s law\nof gravity and could mimic the dark matter effect, as well as the GB coupling constant, which guaranteed\nthe validity of the EGB theory. The recent study gained fruitful investigations such as local thermal stability,\ncritical orbits, shadow behaviors, and QNMs. Thus, the black hole solution was locally thermally stable.\nMoreover, the shadow behavior showed some dependencies between the space parameter of the black hole\nsystem and the size of the black hole shadow.\nACKNOWLEDGMENTS\nA. A. Ara ´ujo Filho would like to thank Fundac ¸ ˜ao de Apoio `a Pesquisa do Estado da Para ´ıba (FAPESQ)25\nand Conselho Nacional de Desenvolvimento Cient ´ııfico e Tecnol ´ogico (CNPq) – [150891/2023-7] for the\nfinancial support. Most of the calculations were performed by using the Mathematica software. This work\nwas partly supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, Grant\nAP14870191.\n[1] H. Weyl, “Reine Infinitesimalgeometrie,” Math. Z. , vol. 2, no. 3-4, pp. 384–411, 1918.\n[2] T. Kaluza, “Zum Unit ¨atsproblem der Physik,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) , vol. 1921,\npp. 966–972, 1921.\n[3] A. Eddington, The Mathematical Theory of Relativity . Cambridge: Cambridge University Press, 1924.\n[4] O. Klein, “The Atomicity of Electricity as a Quantum Theory Law,” Nature , vol. 118, p. 516, 1926.\n[5] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, “Modified Gravity and Cosmology,” Phys. Rept. , vol. 513,\npp. 1–189, 2012.\n[6] D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. , vol. 12, pp. 498–501, 1971.\n[7] P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne, “The 4D Einstein–Gauss–Bonnet theory of\ngravity: a review,” Class. Quant. Grav. , vol. 39, no. 6, p. 063001, 2022.\n[8] J. Bonifacio, K. Hinterbichler, and L. A. Johnson, “Amplitudes and 4D Gauss-Bonnet Theory,” Phys. Rev. D ,\nvol. 102, no. 2, p. 024029, 2020.\n[9] D. Glavan and C. Lin, “Einstein-Gauss-Bonnet Gravity in Four-Dimensional Spacetime,” Phys. Rev. Lett. ,\nvol. 124, no. 8, p. 081301, 2020.\n[10] K. Yang, B.-M. Gu, S.-W. Wei, and Y .-X. Liu, “Born–Infeld black holes in 4D Einstein–Gauss–Bonnet gravity,”\nEur. Phys. J. C , vol. 80, no. 7, p. 662, 2020.\n[11] R. A. Konoplya and A. F. Zinhailo, “Quasinormal modes, stability and shadows of a black hole in the 4D\nEinstein–Gauss–Bonnet gravity,” Eur. Phys. J. C , vol. 80, no. 11, p. 1049, 2020.\n[12] R. A. Konoplya and A. Zhidenko, “Black holes in the four-dimensional Einstein-Lovelock gravity,” Phys. Rev.\nD, vol. 101, no. 8, p. 084038, 2020.\n[13] S. G. Ghosh and R. Kumar, “Generating black holes in 4DEinstein-Gauss-Bonnet gravity,” Class. Quant.\nGrav. , vol. 37, no. 24, p. 245008, 2020.\n[14] S.-W. Wei and Y .-X. Liu, “Extended thermodynamics and microstructures of four-dimensional charged Gauss-\nBonnet black hole in AdS space,” Phys. Rev. D , vol. 101, no. 10, p. 104018, 2020.\n[15] C.-Y . Zhang, P.-C. Li, and M. Guo, “Greybody factor and power spectra of the Hawking radiation in the 4D\nEinstein–Gauss–Bonnet de-Sitter gravity,” Eur. Phys. J. C , vol. 80, no. 9, p. 874, 2020.\n[16] P. K. Yerra and C. Bhamidipati, “Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet\ngravity,” Phys. Lett. B , vol. 835, p. 137591, 2022.\n[17] D. V . Singh, V . K. Bhardwaj, and S. Upadhyay, “Thermodynamic properties, thermal image and phase transi-\ntion of Einstein-Gauss-Bonnet black hole coupled with nonlinear electrodynamics,” Eur. Phys. J. Plus , vol. 137,26\nno. 8, p. 969, 2022.\n[18] A. Biswas, “Black holes in 4D AdS Einstein Gauss Bonnet gravity with power: Yang Mills field,” Gen. Rel.\nGrav. , vol. 54, no. 12, p. 161, 2022.\n[19] A. Errehymy, S. K. Maurya, G. Mustafa, S. Hansraj, H. I. Alrebdi, and A.-H. Abdel-Aty, “Black Hole Solutions\nwith Dark Matter Halos in the Four-Dimensional Einstein-Gauss-Bonnet Gravity,” Fortsch. Phys. , vol. 71,\nno. 10-11, p. 2300052, 2023.\n[20] I. D. D. Carvalho, G. Alencar, and C. R. Muniz, “Thermodynamics of static and stationary black holes in\nEinstein–Gauss–Bonnet gravity with dark matter,” Phys. Dark Univ. , vol. 42, p. 101290, 2023.\n[21] A. Belhaj and Y . Sekhmani, “Thermodynamics of Ay ´on-Beato–Garc ´ıa–AdS black holes in 4D Ein-\nstein–Gauss–Bonnet gravity,” Eur. Phys. J. Plus , vol. 137, no. 2, p. 278, 2022.\n[22] D. J. Gogoi, J. Bora, M. Koussour, and Y . Sekhmani, “Quasinormal modes and optical properties of 4-D black\nholes in Einstein Power-Yang–Mills gravity,” Annals Phys. , vol. 458, p. 169447, 2023.\n[23] Y . Sekhmani and D. J. Gogoi, “Electromagnetic quasinormal modes of dyonic AdS black holes with quasitopo-\nlogical electromagnetism in a Horndeski gravity theory mimicking EGB gravity at D →4,”Int. J. Geom. Meth.\nMod. Phys. , vol. 20, no. 09, p. 2350160, 2023.\n[24] A. Belhaj, M. Benali, H. El Moumni, M. A. Essebani, M. B. Sedra, and Y . Sekhmani, “Thermodynamic and\noptical behaviors of quintessential Hayward-AdS black holes,” Int. J. Geom. Meth. Mod. Phys. , vol. 19, no. 07,\np. 2250096, 2022.\n[25] A. Belhaj and Y . Sekhmani, “Shadows of rotating quintessential black holes in Einstein–Gauss–Bonnet gravity\nwith a cloud of strings,” Gen. Rel. Grav. , vol. 54, no. 2, p. 17, 2022.\n[26] A. Belhaj and Y . Sekhmani, “Optical and thermodynamic behaviors of Ay ´on–Beato–Garc ´ıa black holes for 4D\nEinstein Gauss–Bonnet gravity,” Annals Phys. , vol. 441, p. 168863, 2022.\n[27] A. Belhaj, Y . Hassouni, M. Oualaid, and Y . Sekhmani, “Shadow behaviors of rotating Ay ´on–Beato–Garc ´ıa\nblack holes in four-dimensional Einstein Gauss–Bonnet gravity,” Int. J. Mod. Phys. D , vol. 32, no. 04,\np. 2350016, 2023.\n[28] D. J. Gogoi, Y . Sekhmani, D. Kalita, N. J. Gogoi, and J. Bora, “Joule-Thomson Expansion and Optical Be-\nhaviour of Reissner-Nordstr ¨om-Anti-de Sitter Black Holes in Rastall Gravity Surrounded by a Quintessence\nField,” Fortsch. Phys. , vol. 71, no. 4-5, p. 2300010, 2023.\n[29] K. A. Bronnikov and S. G. Rubin, Black Holes, Cosmology and Extra Dimensions . WSP, 2012.\n[30] J. W. Moffat, “Black Holes in Modified Gravity (MOG),” Eur. Phys. J. C , vol. 75, no. 4, p. 175, 2015.\n[31] J. W. Moffat, “Modified Gravity Black Holes and their Observable Shadows,” Eur. Phys. J. C , vol. 75, no. 3,\np. 130, 2015.\n[32] M. Dehghani and S. F. Hamidi, “Nonlinearly charged black holes in the scalar-tensor modified gravity theory,”\nPhys. Rev. D , vol. 96, no. 10, p. 104017, 2017.\n[33] S. Murk, “Physical black holes in fourth-order gravity,” Phys. Rev. D , vol. 105, no. 4, p. 044051, 2022.\n[34] D. Chen, H. Wu, H. Yang, and S. Yang, “Effects of quantum gravity on black holes,” Int. J. Mod. Phys. A ,\nvol. 29, no. 26, p. 1430054, 2014.27\n[35] S. Raju, “Lessons from the information paradox,” Phys. Rept. , vol. 943, pp. 1–80, 2022.\n[36] A. A. Ara ´ujo Filho, K. Jusufi, B. Cuadros-Melgar, and G. Leon, “Dark matter signatures of black holes with\nYukawa potential,” 10 2023.\n[37] A. A. Ara ´ujo Filho, K. Jusufi, B. Cuadros-Melgar, G. Leon, A. Jawad, et al. , “Charged black holes with yukawa\npotential,” arXiv preprint arXiv:2401.15211 , 2024.\n[38] Z. Berezhiani, F. Nesti, L. Pilo, and N. Rossi, “Gravity Modification with Yukawa-type Potential: Dark Matter\nand Mirror Gravity,” JHEP , vol. 07, p. 083, 2009.\n[39] D. Borka, P. Jovanovi ´c, V . B. Jovanovi ´c, and A. F. Zakharov, “Constraining the range of Yukawa gravity\ninteraction from S2 star orbits,” JCAP , vol. 11, p. 050, 2013.\n[40] M. Garny, M. Sandora, and M. S. Sloth, “Planckian Interacting Massive Particles as Dark Matter,” Phys. Rev.\nLett., vol. 116, no. 10, p. 101302, 2016.\n[41] A. Arvanitaki, S. Dimopoulos, V . Gorbenko, J. Huang, and K. Van Tilburg, “A small weak scale from a small\ncosmological constant,” JHEP , vol. 05, p. 071, 2017.\n[42] K. Jusufi, G. Leon, and A. D. Millano, “Dark Universe phenomenology from Yukawa potential?,” Phys. Dark\nUniv. , vol. 42, p. 101318, 2023.\n[43] M. Milgrom, “A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothe-\nsis,” Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 270, July 15, 1983, p. 365-370. Research supported\nby the US-Israel Binational Science Foundation. , vol. 270, pp. 365–370, 1983.\n[44] M. Chabab, H. El Moumni, and J. Khalloufi, “On Einstein-non linear-Maxwell-Yukawa de-Sitter black hole\nthermodynamics,” Nucl. Phys. B , vol. 963, p. 115305, 2021.\n[45] R. Maier, “Yukawa Black Holes from Interacting Vacuum,” Class. Quant. Grav. , vol. 39, p. 155008, 2022.\n[46] K. Saurabh and K. Jusufi, “Imprints of dark matter on black hole shadows using spherical accretions,” Eur.\nPhys. J. C , vol. 81, no. 6, p. 490, 2021.\n[47] P. V . P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, “Shadows of Kerr black holes with scalar\nhair,” Phys. Rev. Lett. , vol. 115, no. 21, p. 211102, 2015.\n[48] S. Klimenko, I. Nikitin, and L. Nikitina, “Numerical solutions of Einstein field equations with radial dark\nmatter,” Int. J. Mod. Phys. C , vol. 28, no. 07, p. 1750096, 2017.\n[49] X. Hou, Z. Xu, and J. Wang, “Rotating Black Hole Shadow in Perfect Fluid Dark Matter,” JCAP , vol. 12,\np. 040, 2018.\n[50] S. Haroon, M. Jamil, K. Jusufi, K. Lin, and R. B. Mann, “Shadow and Deflection Angle of Rotating Black\nHoles in Perfect Fluid Dark Matter with a Cosmological Constant,” Phys. Rev. D , vol. 99, no. 4, p. 044015,\n2019.\n[51] K. Jusufi, M. Jamil, and T. Zhu, “Shadows of Sgr A∗black hole surrounded by superfluid dark matter halo,”\nEur. Phys. J. C , vol. 80, no. 5, p. 354, 2020.\n[52] A. Arbey and F. Mahmoudi, “Dark matter and the early Universe: a review,” Prog. Part. Nucl. Phys. , vol. 119,\np. 103865, 2021.\n[53] A. Anjum, M. Afrin, and S. G. Ghosh, “Investigating effects of dark matter on photon orbits and black hole28\nshadows,” Phys. Dark Univ. , vol. 40, p. 101195, 2023.\n[54] K. D. Kokkotas and B. G. Schmidt, “Quasinormal modes of stars and black holes,” Living Rev. Rel. , vol. 2,\np. 2, 1999.\n[55] I. G. Moss and J. P. Norman, “Gravitational quasinormal modes for anti-de Sitter black holes,” Class. Quant.\nGrav. , vol. 19, pp. 2323–2332, 2002.\n[56] K. Jusufi, “Connection Between the Shadow Radius and Quasinormal Modes in Rotating Spacetimes,” Phys.\nRev. D , vol. 101, no. 12, p. 124063, 2020.\n[57] F. Atamurotov, I. Hussain, G. Mustafa, and K. Jusufi, “Shadow and quasinormal modes of the\nKerr–Newman–Kiselev–Letelier black hole,” Eur. Phys. J. C , vol. 82, no. 9, p. 831, 2022.\n[58] C. Liu, T. Zhu, Q. Wu, K. Jusufi, M. Jamil, M. Azreg-A ¨ınou, and A. Wang, “Shadow and quasinormal modes\nof a rotating loop quantum black hole,” Phys. Rev. D , vol. 101, no. 8, p. 084001, 2020. [Erratum: Phys.Rev.D\n103, 089902 (2021)].\n[59] M. Ghasemi-Nodehi, M. Azreg-A ¨ınou, K. Jusufi, and M. Jamil, “Shadow, quasinormal modes, and quasiperi-\nodic oscillations of rotating Kaluza-Klein black holes,” Phys. Rev. D , vol. 102, no. 10, p. 104032, 2020.\n[60] K. Jusufi, M. Azreg-A ¨ınou, M. Jamil, S.-W. Wei, Q. Wu, and A. Wang, “Quasinormal modes, quasiperiodic\noscillations, and the shadow of rotating regular black holes in nonminimally coupled Einstein-Yang-Mills\ntheory,” Phys. Rev. D , vol. 103, no. 2, p. 024013, 2021.\n[61] X.-C. Cai and Y .-G. Miao, “Quasinormal modes and shadows of a new family of Ay ´on-Beato-Garc ´ıa black\nholes,” Phys. Rev. D , vol. 103, no. 12, p. 124050, 2021.\n[62] K. Jafarzade, M. Kord Zangeneh, and F. S. N. Lobo, “Shadow, deflection angle and quasinormal modes of\nBorn-Infeld charged black holes,” JCAP , vol. 04, p. 008, 2021.\n[63] J. A. V . Campos, M. A. Anacleto, F. A. Brito, and E. Passos, “Quasinormal modes and shadow of noncommu-\ntative black hole,” Sci. Rep. , vol. 12, no. 1, p. 8516, 2022.\n[64] F. Atamurotov, M. Jamil, and K. Jusufi, “Quantum effects on the black hole shadow and deflection angle in the\npresence of plasma*,” Chin. Phys. C , vol. 47, no. 3, p. 035106, 2023.\n[65] Y . Chen, R. Roy, S. Vagnozzi, and L. Visinelli, “Superradiant evolution of the shadow and photon ring of Sgr\nA⋆,”Phys. Rev. D , vol. 106, no. 4, p. 043021, 2022.\n[66] J. Rayimbaev, B. Majeed, M. Jamil, K. Jusufi, and A. Wang, “Quasiperiodic oscillations, quasinormal modes\nand shadows of Bardeen–Kiselev Black Holes,” Phys. Dark Univ. , vol. 35, p. 100930, 2022.\n[67] I. Dymnikova and A. Dobosz, “Orbits of Particles and Photons around Regular Rotating Black Holes and\nSolitons,” Symmetry , vol. 15, no. 2, p. 273, 2023.\n[68] B. Turimov, A. Mamadjanov, F. Atamurotov, and K. Boymurodova, “On propagation of light-ray and Sagnac\neffect in Kerr–Newman-NUT spacetime,” Chin. J. Phys. , vol. 84, pp. 258–269, 2023.\n[69] B. K. Vishvakarma, D. V . Singh, and S. Siwach, “Shadows and quasinormal modes of the Bardeen black hole\nin cloud of strings,” Eur. Phys. J. Plus , vol. 138, no. 6, p. 536, 2023.\n[70] R. Ghosh, M. Rahman, and A. K. Mishra, “Regularized stable Kerr black hole: cosmic censorships, shadow\nand quasi-normal modes,” Eur. Phys. J. C , vol. 83, no. 1, p. 91, 2023.29\n[71] E. Gonz ´alez, K. Jusufi, G. Leon, and E. N. Saridakis, “Observational constraints on Yukawa cosmology and\nconnection with black hole shadows,” Phys. Dark Univ. , vol. 42, p. 101304, 2023.\n[72] V . Perlick and O. Y . Tsupko, “Calculating black hole shadows: Review of analytical studies,” Phys. Rept. ,\nvol. 947, pp. 1–39, 2022.\n[73] K. Akiyama et al. , “First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric,”\nAstrophys. J. Lett. , vol. 930, no. 2, p. L17, 2022.\n[74] J. Matyjasek and M. Opala, “Quasinormal modes of black holes. The improved semianalytic approach,” Phys.\nRev. D , vol. 96, no. 2, p. 024011, 2017.\n[75] P. V . P. Cunha, N. A. Eir ´o, C. A. R. Herdeiro, and J. P. S. Lemos, “Lensing and shadow of a black hole\nsurrounded by a heavy accretion disk,” JCAP , vol. 03, p. 035, 2020.\n[76] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, “Dilatonic black holes in higher curva-\nture string gravity. 2: Linear stability,” Phys. Rev. D , vol. 57, pp. 6255–6264, 1998.\n[77] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, “Dilatonic black holes in higher curva-\nture string gravity,” Phys. Rev. D , vol. 54, pp. 5049–5058, 1996.\n[78] P. Nicolini, E. Spallucci, and M. F. Wondrak, “Quantum Corrected Black Holes from String T-Duality,” Phys.\nLett. B , vol. 797, p. 134888, 2019.\n[79] R. M. Wald, General Relativity . Chicago, USA: Chicago Univ. Pr., 1984.\n[80] R.-G. Cai, L.-M. Cao, and N. Ohta, “Black Holes in Gravity with Conformal Anomaly and Logarithmic Term\nin Black Hole Entropy,” JHEP , vol. 04, p. 082, 2010.\n[81] A. A. Ara ´ujo Filho, Thermal aspects of field theories . Amazon. com, 2022.\n[82] P. Sedaghatnia, H. Hassanabadi, J. Porf ´ırio, W. Chung, et al. , “Thermodynamical properties of a deformed\nschwarzschild black hole via dunkl generalization,” arXiv preprint arXiv:2302.11460 , 2023.\n[83] J. Furtado, H. Hassanabadi, J. Reis, et al. , “Thermal analysis of photon-like particles in rainbow gravity,” arXiv\npreprint arXiv:2305.08587 , 2023.\n[84] A. A. Ara ´ujo Filho, S. Zare, P. Porf ´ırio, J. K ˇr´ıˇz, and H. Hassanabadi, “Thermodynamics and evaporation of a\nmodified schwarzschild black hole in a non–commutative gauge theory,” Physics Letters B , vol. 838, p. 137744,\n2023.\n[85] A. A. Ara ´ujo Filho, J. Furtado, J. Reis, and J. Silva, “Thermodynamical properties of an ideal gas in a\ntraversable wormhole,” Classical and Quantum Gravity , vol. 40, no. 24, p. 245001, 2023.\n[86] A. A. Ara ´ujo Filho, “Analysis of a regular black hole in verlinde’s gravity,” Classical and Quantum Gravity ,\nvol. 41, no. 1, p. 015003, 2023.\n[87] A. A. Ara ´ujo Filho, “Implications of a simpson–visser solution in verlinde’s framework,” The European Phys-\nical Journal C , vol. 84, no. 1, p. 73, 2024.\n[88] S. Chandrasekhar, The mathematical theory of black holes , vol. 69. Oxford university press, 1998.\n[89] N. Cruz, M. Olivares, and J. R. Villanueva, “The Geodesic structure of the Schwarzschild anti-de Sitter black\nhole,” Class. Quant. Grav. , vol. 22, pp. 1167–1190, 2005.\n[90] J. R. Villanueva, F. Tapia, M. Molina, and M. Olivares, “Null paths on a toroidal topological black hole in30\nconformal Weyl gravity,” Eur. Phys. J. C , vol. 78, no. 10, p. 853, 2018.\n[91] B. Carter, “Global structure of the kerr family of gravitational fields,” Physical Review , vol. 174, no. 5, p. 1559,\n1968.\n[92] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, “Rotating black holes: Locally nonrotating frames, energy\nextraction, and scalar synchrotron radiation,” Astrophys. J. , vol. 178, p. 347, 1972.\n[93] S. E. Vazquez and E. P. Esteban, “Strong field gravitational lensing by a Kerr black hole,” Nuovo Cim. B ,\nvol. 119, pp. 489–519, 2004.\n[94] S. Iyer and C. M. Will, “Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a\nHigher Order WKB Analysis of Potential Barrier Scattering,” Phys. Rev. D , vol. 35, p. 3621, 1987.\n[95] S. Iyer, “BLACK HOLE NORMAL MODES: A WKB APPROACH. 2. SCHWARZSCHILD BLACK\nHOLES,” Phys. Rev. D , vol. 35, p. 3632, 1987.\n[96] R. A. Konoplya, “Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB\napproach,” Phys. Rev. D , vol. 68, p. 024018, 2003.\n[97] B. F. Schutz and C. M. Will, “Black hole normal modes - A semianalytic approach,” APJL , vol. 291, pp. L33–\nL36, Apr. 1985.\n[98] R. A. Konoplya, “Quasinormal modes of the Schwarzschild black hole and higher order WKB approach,” J.\nPhys. Stud. , vol. 8, pp. 93–100, 2004.\n[99] V . Santos, R. V . Maluf, and C. A. S. Almeida, “Quasinormal frequencies of self-dual black holes,” Phys. Rev.\nD, vol. 93, no. 8, p. 084047, 2016.\n[100] R. A. Konoplya and A. Zhidenko, “Quasinormal modes of black holes: From astrophysics to string theory,”\nRev. Mod. Phys. , vol. 83, pp. 793–836, 2011.\n[101] E. Berti, V . Cardoso, and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant.\nGrav. , vol. 26, p. 163001, 2009.\n[102] N. Heidari, H. Hassanabadi, A. A. A. Filho, J. Kr ´ız, S. Zare, and P. J. Porf ´ırio, “Gravitational signatures of a\nnon-commutative stable black hole,” Phys. Dark Univ. , vol. 43, p. 101382, 2024.\n[103] H. Chen, T. Sathiyaraj, H. Hassanabadi, Y . Yang, Z. W. Long, and F. Q. Tu, “Quasinormal modes of the\nEGUP-corrected Schwarzschild black hole,” Indian Journal of Physics , May 2023.\n[104] J. Reis, H. Hassanabadi, et al. , “Exploring antisymmetric tensor effects on black hole shadows and quasinormal\nfrequencies,” arXiv preprint arXiv:2309.15778 , 2023.\n[105] N. Heidari, H. Hassanabadi, J. Kriz, et al. , “Exploring non–commutativity as a perturbation in the\nschwarzschild black hole: Quasinormal modes, scattering, and shadows,” arXiv preprint arXiv:2308.03284 ,\n2023.\n[106] H. Hassanabadi, N. Heidari, J. Kriz, P. Porf ´ırio, S. Zare, et al. , “Gravitational traces of bumblebee gravity in\nmetric-affine formalism,” arXiv preprint arXiv:2305.18871 , 2023.\n[107] B. Toshmatov, C. Bambi, B. Ahmedov, Z. Stuchl ´ık, and J. Schee, “Scalar perturbations of nonsingular nonro-\ntating black holes in conformal gravity,” Phys. Rev. D , vol. 96, p. 064028, 2017."    },    {        "title": "2402.05357v2.Dynamic_Geometric_Connectivity_in_the_Plane_with_Constant_Query_Time.pdf",        "content": "Dynamic Geometric Connectivity in the Plane with\nConstant Query Time\nTimothy M. Chan /envel⌢pe/orcid\nDepartment of Computer Science, University of Illinois at Urbana-Champaign, USA\nZhengcheng Huang /envel⌢pe\nDepartment of Computer Science, University of Illinois at Urbana-Champaign, USA\nAbstract\nWe present the first fully dynamic connectivity data structures for geometric intersection graphs\nachieving constant query time and sublinear amortized update time for many classes of geometric\nobjects in 2D. Our data structures can answer connectivity queries between two objects, as well\nas “global” connectivity queries (e.g., deciding whether the entire graph is connected). Previously,\nthe data structure by Afshani and Chan (ESA’06) achieved such bounds only in the special case\nof axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks,\nwhereas the data structures by Chan, Pătraşcu, and Roditty (FOCS’08) worked for more general\nclasses of geometric objects but required nΩ(1)query time and could not handle global connectivity\nqueries.\nSpecifically, we obtain new data structures with O(1)query time and amortized update time near\nn4/5,n7/8, andn20/21for axis-aligned line segments, disks, and arbitrary line segments respectively.\nBesides greatly reducing the query time, our data structures also improve the previous update times\nfor axis-aligned line segments by Afshani and Chan (from near n10/11ton4/5) and for disks by\nChan, Pătraşcu, and Roditty (from near n20/21ton7/8).\n2012 ACM Subject Classification Theory of computation →Computational geometry\nKeywords and phrases Connectivity, dynamic data structures, geometric intersection graphs\nDigital Object Identifier 10.4230/LIPIcs...\nFunding Timothy M. Chan : Work supported by NSF Grant CCF-2224271.\n1 Introduction\nDynamic graph connectivity —maintaining an undirected graph under edge insertions and\ndeletions to answer connectivity queries—is a popular topic in data structure design and\ndynamic graph algorithms [ 17,18,29,12,19]. At STOC’02, Chan [ 7] initiated the study of\ndynamic connectivity in geometric settings:\nMaintain a set of ngeometric objects, subject to insertions and deletions of objects,\nso that we can quickly determine whether two query objects are connected in the\nintersection graph.\nThe challenge here is that a single insertion/deletion of an object may change Ω(n)edges in\nthe intersection graph in the worst case, and so we can’t afford to maintain the intersection\ngraph explicitly.\nPrevious work by Chan. Chan [7] obtained the first fully dynamic data structure with\nsublinear (O(n0.94)) amortized update time and sublinear ( ˜O(m1/3))1query time for axis-\naligned line segments or rectangles in 2D or axis-aligned boxes in any constant dimension.\n1Throughout the paper, the ˜Onotation hides logO(1)nfactors, and the O∗notation hides nεfactors for\nan arbitrarily small constant ε>0.\n©Timothy M. Chan and Zhengcheng Huang;\nlicensed under Creative Commons License CC-BY 4.0\nLeibniz International Proceedings in Informatics\nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:2402.05357v2  [cs.CG]  21 Mar 2024XX:2 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nThe exponent ( 0.94) arose from matrix multiplication. The heart of his solution was a data\nstructure for an extension of dynamic graph connectivity with vertex (re)insertions and\ndeletions, called dynamic subgraph connectivity —maintaining a (sparse) graph with medges\nand a subset Aof “active” vertices under (re)insertions and deletions in A, so that we can\nquickly determine whether two vertices are connected in the subgraph induced by A. As\nChan observed, the geometric problem in the case of axis-aligned boxes can be reduced\nto dynamic subgraph connectivity (ignoring polylogarithmic factors), by using so-called\n“biclique covers” (which are related to range searching or intersection searching).\nFor more general classes of geometric objects such as arbitrary line segments, biclique\ncovers have superlinear complexity, so Chan’s reduction was not sufficient to yield sublinear\nupdate time, unfortunately.\nPrevious work by Chan, Pătraşcu, and Roditty. A subsequent paper by Chan, Pătraşcu,\nand Roditty [ 10] (FOCS’08) rectified the problem: They first obtained a better data structure\nfor dynamic subgraph connectivity, avoiding fast matrix multiplication, and then incorporated\nrange searching techniques into their data structure in a more efficient way, thereby obtaining\ndata structures for dynamic geometric connectivity with sublinear amortized update time for\nvirtually all families of objects with constant description complexity. For example, for axis-\naligned line segments or rectangles in 2D and axis-aligned boxes in any constant dimension,\nthe amortized update time is ˜O(n2/3)and query time is ˜O(n1/3); for arbitrary line segments\nin 2D, the amortized update time is O∗(n9/10)and query time is ˜O(n1/5); for disks in 2D,\nthe amortized update time is O∗(n20/21)and query time is O∗(n1/7).\nRecently, Jin and Xu [ 20] proved an Ω(m2/3−ε)conditional lower bound on the amortized\nupdate time for the dynamic subgraph connectivity problem for any data structure with\nO(m1−ε)query time,2assuming the “Combinatorial 4-Clique Hypothesis”, if we restrict\nourselves to “combinatorial” algorithms that do not use algebraic techniques for fast matrix\nmultiplication. Chan [ 7] observed that dynamic subgraph connectivity with medges can\nbe reduced back to dynamic geometric connectivity for O(m)axis-aligned line segments or\nboxes in 3D (roughly speaking, because in 3D, axis-aligned segments or boxes can “simulate”\narbitrary graphs), and so Jin and Xu’s proof implies conditional optimality of Chan, Pătraşcu,\nand Roditty’s ˜O(n2/3)update bound for axis-aligned boxes in dimension 3 and above, at\nleast with respect to combinatorial algorithms.\nHowever, one major drawback in Chan, Pătraşcu, and Roditty’s data structures is that\nthe query times are super-polylogarithmic (unlike known static data structures). One can\nenvision applications where queries occur much more frequently than updates. Although\ntrade-offs with smaller query time and larger update time are possible with Chan, Pătraşcu,\nand Roditty’s data structures, they do not achieve sublinear update time in the regime of\nconstant or polylogarithmic query time.\nIn some ways, their data structures encode connectivity information only “implicitly”.\nThus, anotherdrawbackisthattheycannothandle“global”connectivityqueries, e.g., deciding\nwhether the entire intersection graph is connected, or counting the number of connected\ncomponents. Abboud and Vassilevska Williams [ 1] proved an Ω(m1−ε)conditional lower\nbound on the query/update time for global connectivity queries for the dynamic subgraph\nconnectivity problem, assuming the Strong Exponential-Time Hypothesis. So, sublinear query\nand update time are conditionally not possible for global connectivity queries for axis-aligned\nsegments and boxes in dimension 3 and above. Still, a fundamental question remains as to\n2Throughout the paper, εdenotes an arbitrarily small positive constant.T.M. Chan and Z. Huang XX:3\nwhether sublinear update time bound is achievable for global connectivity queries for various\ntypes of geometric objects in 2D (which is where the most natural geometric applications\noccur).\nPrevious work by Afshani and Chan. A different (earlier) data structure by Afshani and\nChan [2] (ESA’06) addressed some of these drawbacks. Specifically, they obtained O(1)query\ntime and ˜O(n10/11)update time in the case of axis-aligned line segments or rectangles in\n2D. Unfortunately, they were unable to extend their method to other types of objects in 2D,\nsuch as arbitrary line segments or disks.\nOther related work. Better data structures are known for some easier special cases; for\nexample, a recent SoCG’22 paper by Kaplan et al. [ 21] gave polylogarithmic results for unit\ndisks, or disks with small maximum-to-minimum-radius ratio, or for arbitrary disks in the\nincremental (insertion-only) or decremental (deletion-only) case (see also Kaplan et al.’s\nSOSA’24 paper [ 22]). However, these results are not applicable to arbitrary disks in the fully\ndynamic setting.\nNew results. We obtain new data structures for dynamic geometric connectivity for different\ntypes of objects in 2D, as summarized in Table 1.3\nQualitatively, we obtain the first data structures with constant query time and sublinear\namortized update time for arbitrary line segments in 2D as well as disks in 2D, and the first\ndata structures that can handle global connectivity queries for such objects. The line segment\ncase is especially general (since it can handle arbitrary polylines of bounded complexity).\nOur approach can in fact handle any family of semialgebraic curves with constant description\ncomplexity in 2D, though the exponent in the update bound depends on the degree of the\ncurves.\nQuantitatively, for the case of disks in 2D, our data structure not only greatly reduces the\nquery time of Chan, Pătraşcu, and Roditty’s method but does so without hurting the update\ntime—in fact, the update bound is improved (albeit slightly), from near n20/21ton7/8. For\nthe special case of axis-aligned line segments in 2D, our data structure also improves the\nupdate time of Afshani and Chan’s method, from near n10/11ton4/5.\nTechniques and new ideas. Our approach builds on Afshani and Chan’s method [ 2],\nbased on the key notion of equivalence classes of connected components. They proved\ncombinatorial bounds and described efficient algorithms for computing and maintaining such\nclasses. Although their combinatorial bounds actually hold for arbitrary geometric objects in\n2D, their algorithms are specialized to axis-aligned objects in 2D. We describe a different way\nto compute and maintain equivalence classes of components, based on a repeated splitting idea\nwhich is simple in hindsight, but has somehow been overlooked by researchers for more than\na decade. This idea allows us to reduce equivalence-class data structures to coloredvariants\nof range or intersection searching, which can be solved by known geometric data structuring\n3For the previous result on simplices in Rd, Chan, Pătraşcu, and Roditty [ 10] originally stated query time\nO∗(n1/(2d+1))and update time O∗(n1−1/d(2d+1)), but they mistakenly assumed bounds for simplex\nrange searching extend to simplex intersection searching. However, simplex intersection searching in\nRddoes reduce to semialgebraic range searching in RO(d2), so their framework implies the sublinear\nbounds shown in the table.XX:4 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nType of objects Query time Update time (amort.) Ref.\nAxis-aligned segments/boxes in Rd ˜O(n1/3) ˜O(n2/3)\nArbitrary line segments in R2O∗(n1/5)O∗(n9/10)\nDisks in R2O∗(n1/7)O∗(n20/21)\n[10]Balls in RdO∗(n1\n2d+3)O∗(n1−1\n(d+1)(2 d+3))\nSimplices in RdO∗(n1\nΘ(d2))O∗(n1−1\nΘ(d4))\nAxis-aligned segments/rectangles in R2O(1) ˜O(n10/11) [2]\nAxis-aligned segments in R2O(1) ˜O(n4/5)\nnew Arbitrary line segments in R2O(1) O∗(n20/21)\nDisks in R2O(1) O∗(n7/8)\nTable 1Previous and new results on dynamic geometric connectivity.\ntechniques. This new idea allows us to obtain constant query time and sublinear amortized\nupdate time for all the types of 2D geometric objects considered here; see Sections 4–5.\nOf independent interest is also a new variant of Afshani and Chan’s combinatorial lemma,\nwhich helps in reducing the update time further; see Section 3.\nIn Appendix 6, we obtain a still further improvement for the case of disks by incorporating\nanother new idea—namely, the usage of separators (specifically, Smith and Wormald’s\ngeometric separator theorem [ 30]). This is interesting, as separators have not been widely\nused before in the context of dynamic geometric connectivity (they have been used in dynamic\nsubgraph connectivity for planar graphs [ 15] but not for more general classes of geometric\nintersection graphs).\n2 Recap of Afshani and Chan’s Method\nBefore describing our new data structures, we first review Afshani and Chan’s previous\nmethod for dynamic connectivity for axis-aligned segments/rectangles in 2D [ 2]. To illustrate\nthe overall ideas, for simplicity we will focus on the offlinesetting of the problem, where the\nupcoming updates are known in advance (the online setting requires more work and slightly\nworse update time), and we will focus on just the case of axis-aligned segments.\nAs often seen in previous works [ 6,7], Afshani and Chan used the standard technique\nof handling insertions lazily and rebuilding periodically. Each period phase consists of a\npreprocessing step followed by qupdates, so that the preprocessing cost can be amortized.\nIn the offline setting, we know which objects will be updated in each phase.\nLetSbe the objects that exist at the beginning of the phase and will not be deleted\nduring the phase, and let Qbe the sequence of objects s1,...,sqthat will be updated during\nthe phase. (Thus, Sstays static during a phase, whereas Qis small.) Afshani and Chan\ndefined two connected components of Sto beequivalent if they intersect the exact same\nset of objects from Q. Clearly, for any s∈Q, ifsintersects an arbitrary component from\na classL, thensintersects all components from L. More importantly, using the fact that\ndistinct connected components never intersect each other and may be viewed as a collection\nof disjoint “curves” or “strings” (not necessarily of constant complexity) when the objects\nare in R2, Afshani and Chan proved a polynomial upper bound on the number of equivalence\nclasses at any moment. The general combinatorial lemma is stated as follows.T.M. Chan and Z. Huang XX:5\n▶Lemma 1 (Afshani and Chan’s combinatorial lemma [ 2]).Consider a set Qofqdisjoint\nregions with simple connected boundaries and a set Cof disjoint curves in R2. ThenC\nconsists of at most O(q3)equivalence classes with respect to Q. (This bound is tight.)\nTo readers familiar with the notion of “VC dimension” or “shatter dimension” of set\nsystems [ 28], the above lemma is equivalent to the statement that the set system (Q,R)with\nR={{s∈Q:s∩c̸=∅}:c∈C}has shatter dimension at most 3. We stress that what\nmakes the lemma interesting is that the curves in Cdo not need to have constant complexity.\nOn the other hand, disjointness of the curves in Cis crucial (otherwise, it is easy to find\ncounterexamples with exponentially many equivalence classes).\nIn the dynamic connectivity problem, the inserted segments are not disjoint. However,\nthe arrangement of Qcan be broken down into O(q2)non-intersecting sub-segments. This\nimplies anO(q6)bound on the number of equivalence classes.\nBelow, we sketch how equivalence classes can be used to obtain an offline dynamic\nconnectivity data structure for axis-aligned segments.\nPreprocessing. At the beginning of a phase, the first task is to find the set of equivalence\nclasses among components of Swith respect to Q. This step is nontrivial, but Afshani and\nChan showed how to compute these classes, or more precisely, the equivalences classes with\nrespect to the O(q2)non-intersecting sub-segments, in ˜O(n)time in the case of axis-aligned\nsegments (we will say more about this later).\nThroughout the phase, we maintain a proxy graph H, such that the connected components\nof the geometric objects roughly correspond to the connected components of H. The graph\nHis defined as follows:\nFor each equivalence class L, there is a class vertex corresponding to L.\nFor each inserted segment s∈Q, there is an insertion vertex corresponding to s.\nThe edges of Hindicate which pairs of objects intersect.\nThe graphHhasO(q6)vertices and O(q7)edges, and can be stored in a dynamic graph\nconnectivity structure supporting edge updates in ˜O(1)time [18, 19].\nOffline updates. Updates of objects are done only to Qand not toS. Whenever an object\nsis inserted to Q, we create a vertex for sinH. For each class L, we decide whether s\nintersects all components of Lby picking an arbitrary component from L, and then query an\northogonal intersection searching structure [ 14]. Each insertion/deletion in Qcauses ˜O(q6)\nedge updates in H.\nThe preprocessing step takes ˜O(n)time, while each update takes ˜O(q6)time. Choosing\nq=n1/7, we achieve ˜O(n6/7)amortized update time.\nQuery. Given two query objects u,v, there are now two cases to consider.\nIf bothu,v∈S, then letcu,cvbe the components containing u,v. Ifcu=cv, thenu\nandvare connected. If cu̸=cvand eithercuorcvbelongs to an equivalence class that\ncorresponds to an isolated vertex in H, thenuandvare not connected.\nOtherwise, uandvare connected if and only if their corresponding vertices in Hare\nconnected (if u∈S, then its corresponding vertex is the class containing cu).\nThus, queries between two objects can be handled in O(1)time. We can also handle global\nconnectivity queries: the overall number of connected components is equal to the number of\ncomponents in H(which can be maintained by a dynamic graph connectivity structure [ 29])\nplusthenumberofcomponentsinisolatedverticesof H(whichisstraightforwardtomaintain).XX:6 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nc\nc\nFigure 1 The left figure shows a curve cintersecting three regions in Q. The right figure shows\nthe minimal sub-curve of cthat intersects the same regions as c.\n1\n2\n3\n4\n\u0019\n\u00190\nr00\nFigure 2 This figure shows four curves in Cπ,π′, ordered from 1to4in cyclic order. Any region\nr′′disjoint from πandπ′must intersect a contiguous (cyclic) sequence of the curves.\n3 A New Version of the Combinatorial Lemma\nIt turns out that the O(q6)combinatorial bound on the number of equivalence classes,\nwhenQmay be intersecting, can be improved to O(q3). There is no need to subdivide the\narrangement of QintoO(q2)non-intersecting parts before applying Lemma 1. Rather, we\ncan modify Afshani and Chan’s proof directly.\n▶Lemma 2 (New combinatorial lemma) .Consider a set Qofqregions with simple connected\nboundaries and a set Cof disjoint curves in R2, where the boundaries of any two regions\nfromQintersect at most O(1)times. Then Cconsists of at most O(q3)equivalence classes\nwith respect to Q. (This bound is tight.)\nProof.We modify Afshani and Chan’s proof [ 2]. For each curve c∈C, we first replace c\nwith a minimal sub-curve that intersects the same set of regions as c. (In other words, we are\n“erasing” unnecessary parts of the input curves, which do not change the equivalence classes;\nsee Figure 1.) Then the two endpoints of cmust lie on the boundaries of two different regions\ninQ(except for curves that intersect only one region, but there are only O(q)classes of such\ncurves).\nFix two different regions r,r′∈Q. Divide∂rand∂r′intoO(1)pieces at their intersection\npoints, if they intersect. Take a piece πof∂rand a piece π′of∂r′. We will bound the\nnumber of equivalence classes of the subset Cπ,π′of all curves in Cthat have one endpoint\nonπand one endpoint on π′.\nThe curves in Cπ,π′do not intersect each other, and they also do not intersect πandπ′,\nbecause of minimality (otherwise we could have erased more). Thus, we can sort all these\ncurves along π, or cyclically equivalently, along π′. For any region r′′whose boundary does\nnot intersect πandπ′, the curves in Cπ,π′that intersect r′′are contiguous in the sorted cyclic\nsequence (see Figure 2). More generally, for an arbitrary region r′′∈Q, we can divide into\nO(1)sub-regions with boundaries not intersecting πandπ′. Thus, for any region r′′∈Q,T.M. Chan and Z. Huang XX:7\nthe curves in Cπ,π′that intersect r′′form a union of O(1)contiguous subsequences in the\nsorted sequence. Taking all the O(q)delimiters together, we get a partition of Cπ,π′into\nO(q)blocks such that two curves in the same block are in the same equivalence class. Hence,\nthe number of equivalence classes in Cπ,π′isO(q).\nAs there are O(q2)choices of (π,π′), the total number of equivalence classes is at most\nO(q3). ◀\nNote that the lemma holds also when Cis a set of disjoint connected regions (since\nregions may be “simulated” by strings [26]).\n4 Solving the Key Subproblem: Computing Equivalence Classes\nAs illustrated by Afshani and Chan’s method, dynamic connectivity—at least in the offline\nsettings—reduces to the following key subproblem.\n▶Subproblem 3 (Equivalence class computation) .Given a set Sofngeometric objects\nand a setQofqgeometric objects in R2, compute the equivalence classes among connected\ncomponents of Swith respect to Qfaster than O(qn)time (ideally, in ˜O(n)time whenqis\nnot too large).\nFor axis-aligned line segments, Afshani and Chan solved Subproblem 3 roughly as follows.\nWe first form a grid by drawing the grid lines at the endpoints of the segments of Q. The\ngrid lines then divide the segments of QintoO(q2)non-intersecting sub-segments. We will\nactually compute equivalence classes with respect to these O(q2)sub-segments instead of Q\n(this is sufficient for the application to dynamic connectivity). The sub-segments within each\nrow (or column) are linearly ordered. Since any axis-aligned segment ulies in exactly one row\nor column, the set of sub-segments intersecting ucan be represented as an integer interval,\nwhich can be computed in ˜O(1)time using binary search. For each connected component cof\nS, the class that cbelongs to can be represented as the union of the intervals for all segments\nofc. Thus, the representations of all components can be found in ˜O(n)time. Crucially, two\ncomponents have the same representation if and only if they belong to the same equivalence\nclass. With some careful analysis, Afshani and Chan showed that the representations can be\nsorted in ˜O(n)time, allowing us to then compute all equivalence classes by a linear scan in\n˜O(n)total time.\nHowever, as one can see, the above ad hoc, grid-based approach does not work when the\nobjects are not axis-aligned (it fails even for line segments with three possible slopes).\nIn this section, we propose a more general approach to solving Subproblem 3 that can\nhandle most types of objects in 2D, including disks and arbitrary line segments. Our\napproach does not require subdividing Qinto nonintersecting pieces first, so we can exploit\nour new improved combinatorial lemma to get better results even for axis-aligned segments.\nFurthermore, the incremental manner in which we compute the equivalence classes will help\nin handling online updates more efficiently in our dynamic connectivity data structures.\n4.1 New Idea: Repeated Class Splitting\nOur central idea is simple: we will compute the equivalence classes incrementally by inserting\nobjects ofQone at a time. (This is fundamentally different from Afshani and Chan’s\napproach, which scans through elements of Sinstead.) Initially, no object has been inserted,\nso all components of Sbelong to the same class. Whenever an object sis inserted, we\nexamine each class Lthat existed before the insertion. If sintersects some but not allXX:8 Dynamic Geometric Connectivity in the Plane with Constant Query Time\ncomponents in L, thenLmust be split into two “child classes”—one whose components\nintersects, and one whose components don’t. This way, we can maintain the equivalence\nclasses by repeatedly splitting existing classes as objects are being inserted. (A similar idea\nwas used by Chan [8] to solve a completely different problem.)\nClass splitting data structure. To split a class into two child classes, we design a class\nsplittingdata structureD(L), such that for any object sthat splitsLinto child classes L1,L2,\nwe can efficiently produce data structures D(L1)andD(L2). An ostensible obstacle is that\nwhen formingD(L1)andD(L2), it seems that we can’t avoid spending time at least O(1)\ntime per object in L. If so, computing the classes may take Ω(n)time per insertion.\nTo get around this obstacle, we use an amortization trick, namely, a reverse version of\nthe standard “weighted union heuristic”: instead of building both D(L1)andD(L2)from\nscratch, we do so only for the child class with smaller total size. Specifically, if L2has\nsmaller total size than L1, then we buildD(L2)from scratch, while D(L1)is obtained by\ndeleting everything associated with L2fromD(L). Intuitively, we “displace” L2fromL\nwithout necessarily examining objects from L1. We claim that the sum of the number of\ndisplaced objects over all updates in the phase is O(nlogn). One proof is via a potential\nfunction argument: defining Φ =/summationtext\nclassL|L|log|L|(where|L|:=/summationtext\ncomponent c∈L|c|), it is\nnot difficult to see that a split with ydisplacements decreases Φby at least Ω(y)(because of\nthe inequality (x+y)log(x+y)−xlogx−ylogy≥ylog(x/y+ 1)≥ywhenx≥y), and\nthe claim follows. (An alternative, simple proof is to argue that each object can be displaced\nat mostO(logn)times, since each time this happens, the size of the class containing the\nobject is reduced by at least a factor of 2.)\nReduction to component (non-)intersection reporting. The key part of the class splitting\noperation is building a data structure D(L)that can report the smaller child class L′in time\nlinear in the total size of L′, but sublinear in the total size of L. It suffices for the data\nstructure to handle the following two types of queries:\n1.Report all connected components in Lthat intersect s.\n2.Report all connected components in Lthatdo notintersects.\nBy running the two query-answering algorithms concurrently and stopping when either of\nthem terminates, we can report L′within the desired time bound. We need the data structure\nto support deletions of components, so that we can subsequently delete each component in\nL′fromL.\nThe component intersection reporting problem we face can be viewed as a colored\nintersection reporting problem, where the objective is to preprocess a set of colored objects,\nso that one can efficiently report all colors intersected by a query object. Similarly, we need to\nsolve the corresponding colored non-intersection reporting problem. (See [ 16] for background\non colored range searching.)\nInthenextthreesubsections, weapplyknowngeometricdatastructuringtechniques(some\nof which are inspired by colored range searching), to solve this component (non-)intersection\nreporting problem for axis-aligned line segments, disks, and arbitrary line segments.\n4.2 Axis-Aligned Line Segments\n▶Lemma 4. Given a set of connected components formed by naxis-aligned segments in\nR2, there is a component (non-)intersection reporting data structure with ˜O(n)preprocessingT.M. Chan and Z. Huang XX:9\ntime and ˜O(1 +k)query time, where kis the number of components reported. Furthermore,\nwe can support deletion of any component in ˜O(1)amortized time, and insertion of any\ncomponent cin˜O(|c|)amortized time.\nProof.In the following, we assume that the inserted segment is vertical; the other case can\nbe handled symmetrically. For each component c, we compute the vertical decomposition Tc\nof the horizontal segments of c. There are O(n)rectangular cells over all components. We\nmake the following observation.\n▶Observation 5. Letsbe a vertical segment and pbe its lower endpoint. For any component\nc, let∆be the cell in Tcthat contains p, and letube the horizontal segment in cthat bounds\n∆from above. Then sintersects a segment in cif and only if sintersectsu.\nBy Observation 5, components that intersect scan be reported as follows:\n1.Among the cells of Tcfor all components c, find all cells that contain p.\n2.Report, among the upper-bounding segments of these cells, the ones that intersect s.\nComponents that do not intersect scan be reported similarly, except in the second step\nwe instead report the segments that do not intersect s. We support the above reporting\nquery with a two-level data structure D.\nThe primary structure is an orthogonal rectangle stabbing structure with ˜O(1)query time\nand ˜O(1)update time (e.g., [14], or by reduction to orthogonal range searching in R4).\nFor each canonical subset Bin the primary structure, we associate each cell ∆∈B\nwith the segment that bounds ∆from above. We keep two auxiliary structures on the\nassociated line segments. Given a vertical segment s, one auxiliary structure reports all\nsegments intersecting s, while the other reports all segments not intersecting s. Both\nstructures have ˜O(1 +k)query time (where kis the output size) and ˜O(1)update time\n(as these subproblems also reduce to orthogonal range searching).\nThe preprocessing and query time bounds are clearly satisfied. Insertion/deletion of a\ncomponent crequires inserting/deleting all cells of Tcand takes ˜O(|c|)time; we can charge\nthe cost of deletion to preprocessing or insertion. ◀\n▶Corollary 6. Fornaxis-aligned line segments in R2, we can solve Subproblem 3 in ˜O(n+q4)\ntime. Furthermore, each insertion to Qtakes ˜O(q3+n/q)amortized time. We can also\nsupport deletion of a component in ˜O(1)amortized time and insertion of a component cin\n˜O(|c|)amortized time, if we are given c’s class.\nProof.We store each equivalence class in the data structure from the preceding lemma.\nFori= 1,...,q, letLi,1,...,Li,O(q3)be the existing classes before the i-th insertion. Let\nni,1,...,ni,O(q3)be the sizes of the classes. Whenever a class Li,jis split, let mi,jbe the\ntotal size of the displaced child class. As noted, the total number of displacements is/summationtext\ni,jmi,j=˜O(n), when there are no insertions of components; in general, it is ˜O(n+ Σ),\nwhere Σis the sum of the sizes of the components inserted (because insertion of a component\ncincreases the potential Φby˜O(|c|)). Thus, the total running time for class splitting\noperations over all insertions to Qis at most\n˜O\nq/summationdisplay\ni=1O(q3)/summationdisplay\nj=1(1 +mi,j)\n=˜O/parenleftbig\nq4+n+ Σ/parenrightbig\n.\nThis amortizes to ˜O(q3+n/q)time per insertion to Q, with the Σterm charged to insertions\nof components. ◀XX:10 Dynamic Geometric Connectivity in the Plane with Constant Query Time\n4.3 Disks\n▶Lemma 7. Given a set of connected components formed by ndisks in R2, there is a\ncomponent (non-)intersection reporting data structure with O∗(n)preprocessing time and\nO∗(n4/5+k)query time, where kis the number of components reported. Furthermore, we\ncan support deletion of a component in O∗(1)amortized time and insertion of a component c\ninO∗(|c|)amortized time.\nThe proof follows the same argument as for axis-aligned segments, except we use different\ndata structures than orthogonal range searching for the primary and auxiliary structures.\nProof.The component (non-)intersection searching structure for disks is similar to the\none for axis-aligned segments. However, for each component c, instead of using vertical\ndecomposition, we compute an additively weighted Voronoi diagram [ 25] with the disk centers\nas the sites and the radii as the weights. Let Vcbe a decomposition of the diagram into cells\nof constant complexity, e.g., by radially decomposing each Voronoi cell with its site as the\norigin. Then Vcsatisfies the same key property as the vertical decomposition in the case of\naxis-aligned segments.\n▶Observation 8. Letsbe a disk and pbe its center. For any component c, let∆be the cell\ninVcthat contains p, and letube the disk associated with ∆. Thensintersects a disk in c\nif and only if sintersectsu.\nBy Observation 8, components that intersect scan be reported as follows:\n1.Among the cells of Vcfor all components c, find all cells that contain p.\n2.Report, among the disks associated with these cells, the ones that intersect s.\nComponents that do not intersect scan be reported similarly, except in the second step\nwe instead report the disks that do not intersect s. Again, we support the above reporting\nquery with a two-level data structure D.\nFor the primary structure, observe that edges of additively weighted Voronoi diagrams in\nR2are hyperbolas, which have equations of the form c1x2+c2y2+c3xy+c4x+c5y= 1\nand can be “linearized” via the mapping (x,y)∝⇕⊣√∫⊔≀→(x2,y2,xy,x,y ). Thus, stabbing queries\non the cells of the Vc’s reduce to simplex-stabbing queries in R5. We can handle such\nqueries using (a multi-level version of) Matoušek’s partition tree [ 27] inR5, which has\nO∗(n)preprocessing time and O∗(1)update time, such that given any query point p, we\ncan inO∗(n4/5)time findO∗(n4/5)canonical subsets of the simplices containing p.\nFor each canonical subset Bin the primary structure, let SBbe the set of disks associated\nwith the simplices in B. We keep two auxiliary structures on the associated disks.\nGiven an input disk s, one auxiliary structure reports all disks that intersect s, while\nthe other reports all disks that do not intersect s. Both structures reduce to repeated\nnearest/farthest neighbor queries for points with additive weights in R2. We can use\nresults by Agarwal, Efrat, and Sharir [ 3] to answer queries in O∗(1 +k)time withO∗(1)\nupdate time (or alternatively, Kaplan et al.’s data structure [ 23] with slightly better,\npolylogarithmic bounds).\nThe two-level data structure clearly satisfies the desired time bounds. ◀\n▶Corollary 9. Forndisks in R2, we can solve Subproblem 3 in O∗(n+q8/5n4/5)time.\nFurthermore, each insertion to Qtakes amortized O∗(q3/5n4/5+n/q)time. We can also\nsupport deletion of a component in amortized O∗(1)time and insertion of a component cin\namortizedO∗(|c|)time, if we are given c’s class.T.M. Chan and Z. Huang XX:11\nProof.We defineni,jandmi,jfori= 1,...,qandj= 1,...,O (q3), as well as Σ, in the\nsame way as we did for axis-aligned segments. Using the same analysis, the total running\ntime for class splitting operations over all insertions to Qis at most\nO∗\nq/summationdisplay\ni=1O(q3)/summationdisplay\nj=1/parenleftig\nn4/5\ni,j+mi,j/parenrightig\n,\nwhich, by Hölder’s inequality, is bounded by O∗(q·(q3/5n4/5) +n+ Σ). This amortizes to\nO∗(q3/5n4/5+n/q)time per insertion to Q. ◀\n4.4 Arbitrary Line Segments\nThe component intersection reporting problem becomes tougher for the case of arbitrary\nline segments. Although we are unable to get near-linear preprocessing time, we can obtain\nthe following preprocessing/query-time trade-off. The idea uses cuttings and is inspired by\na known idea for a different colored range searching problem (namely, “range mode”, i.e.,\nfinding the most frequent color in a range) [9].\n▶Lemma 10. Given a set of connected components formed by nline segments in R2and a\nparameterr, there is a component (non-)intersection reporting data structure with O∗(nr4)\npreprocessing time and O∗(1 +n/r+k)query time, where kis the number of components\nreported. Furthermore, we can support deletion of a component in O∗(r4)amortized time\nand insertion of a component cinO∗(|c|r4)amortized time.\nProof.A line segment in R2can be represented by 4 real values (the coordinates of its two\nendpoints) and so can be viewed as a point in R4. For each input line segment s, the set\nof all line segments intersecting sthen corresponds a semialgebraic set in R4with constant\ndescription complexity. Let Sbe the collection of these nsemialgebraic sets.\nNow, compute a (1/r)-cutting ΓofS, consisting of O∗(r4)cells (due to known combina-\ntorial bounds on vertical decompositions) [24], in O∗(nr4)time.\nFor each cell ∆∈Γ, we store the conflict list of∆, consisting of all semialgebraic sets\ns∈Swith boundaries crossing ∆; by definition of cuttings, each conflict list has size O(n/r).\nFor each component cwith no semialgebraic set in the conflict list of ∆, we storecin a list\nA(∆)if at least one semialgebraic set from ccompletely contains ∆, or storecin another\nlistB(∆)otherwise. To allow for efficient update, for each component c, we keep track of all\nsemialgebraic sets from cthat contains ∆.\nFor each semialgebraic set s∈Sand each cell ∆∈Γ, we can decide in O(1)time which\nlistsshould be stored in. Thus, the data structure has O∗(nr4)preprocessing time. For the\nsame reason, deletion of a semialgebraic set can be handled in O∗(r4)time.\nTo handle a query for a line segment represented as a point p∈R4, we first locate the\ncell∆∈Γcontaining p. To report all components stabbed (resp. not stabbed) by p, we\nlinearly search the conflict list, and then report the components stored in the list A(∆)(resp.\nB(∆)). The reporting takes O∗(1 +n/r+k)time, where kis the output size.\nInsertions of components can be handled by the logarithmic method [4]. ◀\n▶Corollary 11. Fornline segments in R2and a parameter r, we can solve Subproblem 3 in\nO∗(r4n+qn/r +q4)time. Furthermore, each insertion to Qtakes amortized O∗(n/r+q3+\nr4n/q)time. We can also support deletion of a component in O∗(r4)time and insertion of a\ncomponent cin amortized O∗(|c|r4)time, if we are given c’s class.XX:12 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nProof.We defineni,jandmi,jfori= 1,...,qandj= 1,...,O (q3), as well as Σ, in the\nsame way as we did for axis-aligned segments and disks. Using the same analysis, the total\nrunning time for class splitting over all insertions to Qis at most\nO∗\nq/summationdisplay\ni=1O(q3)/summationdisplay\nj=1/parenleftig\n1 +ni,j\nr+mi,jr4/parenrightig\n=O∗/parenleftigqn\nr+q4+r4(n+ Σ)/parenrightig\n,\nThis amortizes to O∗(n/r+q3+r4n/q)per insertion to Q. ◀\n5 Dynamic Connectivity\nNow, we apply the equivalence-class data structures developed in Section 4 to design dynamic\nconnectivity structures for axis-aligned line segments, disks, and arbitrary line segments in\nR2. We follow the basic approach from Section 2 but describe how to handle general online\nupdates.\n5.1 Axis-Aligned Line Segments\nFor axis-aligned segments, we maintain the following data structures during a phase:\nThe equivalence-class data structure.\nA decremental data structure for explicitly maintaining the connected components of\nS, along with their sizes. This structure has ˜O(n)preprocessing time and ˜O(n)total\nupdate time over the entire phase. We achieve this by reducing to decremental graph\nconnectivity under edge deletions [ 31] (using the biclique cover technique from Chan [ 7]).\nFor each component c, an orthogonal intersection searching structure with ˜O(1)query\nand update time (e.g., [14]).\nThroughout the phase, we maintain a proxy graph Hin the same way as in Afshani and\nChan’s method, as described in Section 2. Because updates are online, we need to handle\nnot only insertions and deletions in Qbut also deletions in S.\nInsertion/deletion in Q.Insertions in Qcan be done as before, since the equivalence-class\nstructure already supports insertions to Q. By Corollary 6, the amortized cost of equivalence\nclass maintenance per insertion to Qis˜O(q3+n/q). The number of edge changes in the\nproxy graph HisO(q4); we can afford to reconstruct the connectivity structure of Hin\nO(q4)time.\nFor deletions in Q, we don’t even need to update the equivalence-class data structure.\nDeletion in S.Suppose that a segment s∈Sis deleted in the i-th update. Let cbe the\ncomponent that contains s. The deletion of smay divide cinto smaller sub-components\nc1,...,cz,listedinorderofdecreasingsize. Let mi=|c2|+···+|cz|. Since|c2|,...,|cz|≤|c|/2,\nthe sum/summationtext\nimiover the entire phase is ˜O(n).\nFirst, we update the decremental connectivity structure; this takes ˜O(n)time over\nthe entire phase. We then report the disks of c2,...,cz, delete them from the orthogonal\nintersection searching structure for c, and then rebuild the structure for each of c2,...,cz.\nThis takes ˜O(mi)time. We make c1a singleton class, if it isn’t already. We delete c1fromT.M. Chan and Z. Huang XX:13\nthe equivalence-class structure.4We create at most qsingleton classes this way, which is\nnegligible.\nNext, we compute the classes among c2,...,czwith respect to Qby “replaying” the\ninsertions in Qfrom scratch. This takes ˜O(q4+mi)time. For each class L′found this way,\nwe find the existing class Lthat is equivalent to L′(this takes ˜O(q4)total time, by viewing\neach class as a q-bit vector and sorting O(q3)such vectors). If such a class Lexists, then we\ninsert the components in L′to the equivalence-class structure. This takes ˜O(mi)amortized\ntime. Thus, the total time for handling deletions is bounded by ˜O(q5+n), which amortizes\nto˜O(q4+n/q). This is the overall amortized update time. Choosing q=n1/5, we achieve\nupdate time ˜O(n4/5).\n▶Theorem 12. Fornaxis-aligned line segments in R2, there is a dynamic connectivity data\nstructure with O(1)query time and ˜O(n4/5)amortized update time.\n5.2 Disks\nThe dynamic data structure for disks is similar to the case of axis-aligned segments, except\nwe use a different decremental connectivity structure and a different intersection searching\nstructure.\nFor decremental connectivity, we use the data structure by Kaplan et al.[21], which\nachieves ˜O(n)preprocessing time and ˜O(n)total update time over the entire phase.\nFor intersection searching, we use the data structure by Kaplan et al.[23], which achieves\n˜O(1)query and update time.\nUpdate time. The update algorithm is identical to the case of axis-aligned segments.\nBy Corollary 9, the amortized cost of equivalence class maintenance per insertion to Qis\nO∗(q3/5n4/5+n/q). For deletion, we define mifori= 1,...,qin the same way as we did\nfor axis-aligned segments. Using the same analysis, the total cost over all deletions in Sis\nbounded by\nO∗/parenleftiggq/summationdisplay\ni=1/parenleftig\nq8/5m4/5\ni+mi/parenrightig\n+n/parenrightigg\n=O∗/parenleftig\nq9/5n4/5+n/parenrightig\n,\nwhich amortizes to O∗(q4/5n4/5+n/q)per deletion. This is the overall amortized update\ntime. Choosing q=n1/9, we achieve update time O∗(n8/9).\n▶Theorem 13. Forndisks in R2, there is a dynamic connectivity data structure with O(1)\nquery time and O∗(n8/9)amortized update time.\n5.3 Arbitrary Line Segments\nFor the case of arbitrary line segments the idea is again similar to the case of axis-aligned\nsegments, but because the time bound for Subproblem 3 is superlinear and dependent on the\nparameterreven whenqis small, the settings of parameters are more delicate. There is also\na new issue: we don’t have a decremental connectivity structure for arbitrary line segments\n4All we want is that if two components are in the same class, they intersect the same elements of Q; we\ndon’t need the converse. This explains why it is fine to handle some classes such as c1separately, and\nwhy we can ignore deletions in Qin the equivalence-class structure.XX:14 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nwith both near-linear preprocessing time and near-linear total update time (the best static\nalgorithm requires O(n4/3)time [11]).\nFortunately, we observe that in the preprocessing step between every two update phases,\nno more than qnew segments are added to the decremental connectivity structure. Thus,\nthe decremental connectivity structure does not necessarily need O∗(n)preprocessing time.\nInstead, it only needs to support batch insertion of at mostqsegments in near-linear time.\nSuch a data structure has been given by Chan, Pătraşcu, and Roditty [10].\n▶Lemma 14 ([10]).LetSbe a set of nline segments in R2. There is a decremental\nconnectivity data structure with ˜O(n)total deletion time over any sequence of deletions, and\nsupports batch insertion of any qline segments in ˜O(n+q√n)time.\nSince we will choose q≪√n, the ˜O(q√n)term is negligible.\nUpdate time. The update algorithm is again identical to the case of axis-aligned segments.\nBy Corollary 11, the amortized cost of equivalence-class maintenance per insertion to Qis\n˜O(n/r+q3+r4n/q). For deletions in S, definemifori= 1,...,qin the same way as we\ndid for disks. Then the total cost over all deletions is bounded by\nO∗/parenleftiggq/summationdisplay\ni=1/parenleftigqmi\nr+q4+r4mj/parenrightig\n+n/parenrightigg\n=O∗/parenleftigqn\nr+q5+r4n/parenrightig\n,\nwhich amortizes to O∗(n/r+q4+r4n/q)per deletion. This is the dominating term in the\noverall amortized update time. Choosing r=n1/21andq=n5/21, the amortized update\ntime minimizes to O∗(n20/21).\n▶Theorem 15. Fornarbitrary line segments in R2, there is a dynamic connectivity data\nstructure with O(1)query time and O∗(n20/21)amortized update time.\nWe remark that the same approach also works more generally for fixed-degree algebraic\ncurves in 2D, with a larger exponent of the update bound depending on the degree (as the\ndimension of the lifted space gets larger in the proof of Lemma 10).\n6 Further Improvement for Disks Using Separators\nIn this section, we give a small improvement to our result for disks (from near n8/9to\nn7/8update time), interestingly by using separators. Specifically, we will use the following\nvariant of Smith and Wormald’s geometric separator theorem. (The original version bounds\nthe number of disks intersecting ∂BbyO(√n)but assumes the input disks are disjoint.\nOur variant instead bounds the stabbing number but does not require disjointness.) For\ncompleteness, we include a proof.\n▶Lemma 16 (Smith and Wormald’s separator lemma [ 30]).Given a set Sofndisks in R2,\nthere is an axis-parallel square B, such that the number of disks inside and the number of\ndisks outside are both at most 4n/5, and the set of all disks intersecting ∂Bcan be stabbed\nbyO(√n)points. Furthermore, Bcan be computed in ˜O(n)time.\nProof.Compute smallest square B0that contains at least n/5of the disk centers; this takes\n˜O(n)time [5]. SayB0has center (x,y)and side length r. Fort∈{1\nb,2\nb,...,b−1\nb}, letBtbe\nthe square with center (x,y)and side length (1 +t)r. SinceBtcan be covered by 4 squares\nof side length <r, we know that Btcontains at most 4n/5centers and at least n/5centers.T.M. Chan and Z. Huang XX:15\nR\nFigure 3 Arcs of each type can be sorted. For example, arcs with endpoints on the two blue\nedges can be sorted as illustrated by the blue arrow.\nA disk of radius≤r/bintersects∂Btfor at most O(1)choices oft. Thus, we can find\nt∈{1\nb,2\nb,...,b−1\nb}inO(n)time such that ∂BtintersectsO(n/b)disks of radius≤r/b. On\nthe other hand, the disks of radius >r/bintersecting ∂Btcan be stabbed by O(b)points.\nSettingb=√n, we see that Btsatisfies the requirement. ◀\nWe apply the separator lemma to give a new divide-and-conquer algorithm for computing\nequivalence classes for disks. This algorithm is static (not supporting insertions to Q).\n▶Lemma 17. Forndisks in R2, we can solve Subproblem 3 in ˜O(n+q6+q2√n)time.\nProof.We first preprocess each component cso that we can quickly determine whether a\nquery disk intersects c. This reduces to nearest neighbor search with additive weights, and\nso there is a data structure with ˜O(|c|)preprocessing time and ˜O(1)query time.\nGiven a region R⊆R2and a setCof components completely inside Rwith total size\nn, we describe a recursive algorithm to compute the equivalence classes of the components\nCwith respect to Q. The region Ris assumed to be an orthogonal polygon, possibly with\nholes, with ˜O(1)complexity. (Initially, R=R2.)\nDefine the special points to be the intersection points between the boundaries of two\ndisks inQ, and the inflection points (i.e., the topmost, bottommost, leftmost, and rightmost\npoints) of the disks in Q.\nBase case. Suppose that Rcontains no special points. Let Γbe arcs formed by intersecting\nthe boundaries of disks in Qwith the region R. SinceRcontains no special points, the arcs in\nΓare disjoint, and each arc in Γhas endpoints lying on two different edges of ∂R(otherwise,\nRwould contain an inflection point). Two arcs have the same typeif their endpoints lie on\nthe same pair of edges of ∂R. There are O(1)types. We can sort the arcs of the same type.\n(See Figure 3.) The arcs of the same type intersecting a component cform a contiguous\nsubsequence (i.e., an interval) in the sorted sequence, which can be determined by binary\nsearch in ˜O(|c|)time. Thus, we can compute the O(q3)equivalence classes of the components\nwith respect to Γin˜O(n)time.\nThere may be two components candc′that are equivalent with respect to Γbut not\nequivalent with respect to Q. But this can happen only if cis completely inside a disk s∈Q\nandc′is completely outside s, or vice versa; in such a case, candc′cannot intersect any arc\ninΓ(otherwise, the arc would intersect ∂sinsideR). For components intersecting no arcs in\nΓ, we can put them in the same class if they lie in the same face in the arrangement of Γ\ninsideR; there areO(q)such classes, and this step can be done in ˜O(1)time per component\nvia point location [13].XX:16 Dynamic Geometric Connectivity in the Plane with Constant Query Time\nAs a result, we obtain O(q3)classes, which are a refinement of the actual equivalence\nclasses with respect to Q. We can determine the actual equivalence classes by testing whether\neach disk in Qintersects (a representative component of) each class intersects each disk, in\n˜O(q·q3)total time.\nRecursion. Suppose that Rcontains at least one special point. We apply Smith and\nWormald’s separator theorem to obtain a square B. We recursively solve the subproblem for\nR∩Band the components completely inside R∩B, and the subproblem for R\\Band the\ncomponents completely inside R\\B. There are at most O(√n)components intersecting ∂B\n(because disks stabbed by a common point are in a common component). We can determine\nthe actual equivalence classes for these “boundary components” by testing whether each disk\ninQintersects each boundary component, in ˜O(q·√n)time. We can combine the equivalence\nclasses of the two subproblems and the boundary components in ˜O(q·q3)additional time.\nAnalysis. Note that because the recursion has logarithmic depth, each region Rgenerated\nis indeed an orthogonal polygon with ˜O(1)complexity. Letting Xbe the number of special\npoints inR, we obtain the recurrence\nT(n,X)≤ max\nn1+n2≤n, n 1,n2≤4n/5,X1+X2≤X((T(n1,X1) +T(n2,X2) +˜O(n+q4+q√n)),\nwith the base case T(n,0) = ˜O(n+q4).\nThe recurrence solves to T(n,X) =˜O(n+q4X+q√\nnX). As the number of special points\nisO(q2), we obtain a running time of ˜O(n+q6+q2√n). ◀\nWe can now use the new lemma to speed up our previous method for dynamic connectivity\nfor disks. We use the same equivalence-class data structure as before, but make only one\nchange: during a deletion in S, when we compute the equivalence classes among c2,...,cz,\nwe switch to the above lemma (since this step is a static problem).\nUpdate time. The amortized cost per insertion to Qis the same as before, i.e., O∗(q3/5n4/5+\nn/q). With the new static equivalence-class algorithm, the total cost over all deletions in S\nis now bounded by\nO∗/parenleftiggq/summationdisplay\ni=1/parenleftbig\nq6+q2√mi+mi/parenrightbig\n+n/parenrightigg\n=O∗/parenleftig\nq7+q5/2√n+n/parenrightig\n,\nwhich amortizes to O∗(q6+q3/2√n+n/q)per deletion. The insertion cost now dominates.\nChoosingq=n1/8, we achieve update time O∗(n7/8).\n▶Theorem 18. Forndisks in R2, there is a dynamic connectivity data structure with O(1)\nquery time and O∗(n7/8)amortized update time.\nReferences\n1Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower\nbounds for dynamic problems. In Proc. 55th IEEE Annual Symposium on Foundations of\nComputer Science (FOCS) , pages 434–443, 2014. doi:10.1109/FOCS.2014.53 .\n2Peyman Afshani and Timothy M. Chan. Dynamic connectivity for axis-parallel rectan-\ngles. Algorithmica , 53(4):474–487, 2009. Preliminary version in ESA’06. doi:10.1007/\ns00453-008-9234-7 .T.M. Chan and Z. Huang XX:17\n3Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels\nin 3-dimensional arrangements and its applications. SIAM J. Comput. , 29(3):912–953, 1999.\nPreliminary version in SoCG’95. doi:10.1137/S0097539795295936 .\n4Jon Louis Bentley and James B. Saxe. Decomposable searching problems I: static-to-dynamic\ntransformation. J. Algorithms , 1(4):301–358, 1980. doi:10.1016/0196-6774(80)90015-2 .\n5Timothy M. Chan. Geometric applications of a randomized optimization technique. Discret.\nComput. Geom. , 22(4):547–567, 1999. doi:10.1007/PL00009478 .\n6Timothy M. Chan. Semi-online maintenance of geometric optima and measures. SIAM J.\nComput., 32(3):700–716, 2003. doi:10.1137/S0097539702404389 .\n7Timothy M. Chan. Dynamic subgraph connectivity with geometric applications. SIAM J. Com-\nput., 36(3):681–694, 2006. Preliminary version in STOC’02. doi:10.1137/S009753970343912X .\n8Timothy M. Chan. Near-optimal randomized algorithms for selection in totally monotone\nmatrices. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 1483–1495,\n2021. doi:10.1137/1.9781611976465.89 .\n9Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, and Bryan T.\nWilkinson. Linear-space data structures for range mode query in arrays. Theory Comput.\nSyst., 55(4):719–741, 2014. URL: https://doi.org/10.1007/s00224-013-9455-2 ,doi:10.\n1007/S00224-013-9455-2 .\n10Timothy M. Chan, Mihai Pătraşcu, and Liam Roditty. Dynamic connectivity: Connecting\nto networks and geometry. SIAM J. Comput. , 40(2):333–349, 2011. Preliminary version in\nFOCS’08. doi:10.1137/090751670 .\n11Timothy M. Chan and Da Wei Zheng. Hopcroft’s problem, log-star shaving, 2D fractional\ncascading, anddecisiontrees. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA) ,\npages 190–210, 2022. doi:10.1137/1.9781611977073.10 .\n12Julia Chuzhoy, Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, and Thatchaphol Sara-\nnurak. A deterministic algorithm for balanced cut with applications to dynamic connectivity,\nflows, and beyond. In Proc. 61st IEEE Annual Symposium on Foundations of Computer\nScience (FOCS) , pages 1158–1167, 2020. doi:10.1109/FOCS46700.2020.00111 .\n13Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational\nGeometry: Algorithms and Applications . Springer, 3rd edition, 2008. URL: https://www.\nworldcat.org/oclc/227584184 .\n14Herbert Edelsbrunner and Hermann A. Maurer. On the intersection of orthogonal objects.\nInf. Process. Lett. , 13(4/5):177–181, 1981. doi:10.1016/0020-0190(81)90053-3 .\n15Daniele Frigioni and Giuseppe F. Italiano. Dynamically switching vertices in planar graphs.\nAlgorithmica , 28(1):76–103, 2000. URL: https://doi.org/10.1007/s004530010032 ,doi:\n10.1007/S004530010032 .\n16Prosenjit Gupta, Ravi Janardan, Saladi Rahul, and Michiel Smid. Computational ge-\nometry: Generalized (or colored) intersection searching. In Handbook of Data Struc-\ntures and Applications , pages 1043–1058. Chapman and Hall/CRC, 2018. URL: https:\n//www.csa.iisc.ac.in/~saladi/Papers/ds2-handbook.pdf .\n17Monika Rauch Henzinger and Valerie King. Randomized fully dynamic graph algorithms\nwith polylogarithmic time per operation. J. ACM, 46(4):502–516, 1999. doi:10.1145/320211.\n320215.\n18Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic\nfully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity.\nJ. ACM, 48(4):723–760, 2001. doi:10.1145/502090.502095 .\n19Shang-En Huang, Dawei Huang, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup. Fully\ndynamic connectivity in O(logn(log logn)2)amortized expected time. TheoretiCS , 2, 2023.\ndoi:10.46298/THEORETICS.23.6 .\n20Ce Jin and Yinzhan Xu. Tight dynamic problem lower bounds from generalized BMM and\nOMv. In Proc. 54th Annual ACM Symposium on Theory of Computing (STOC) , pages\n1515–1528, 2022. doi:10.1145/3519935.3520036 .XX:18 Dynamic Geometric Connectivity in the Plane with Constant Query Time\n21Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam\nRoditty, and Paul Seiferth. Dynamic connectivity in disk graphs. In Proc. 38th International\nSymposium on Computational Geometry (SoCG) , pages 49:1–49:17, 2022. doi:10.4230/\nLIPICS.SOCG.2022.49 .\n22Haim Kaplan, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, and Liam Roditty. Insertion-\nonly dynamic connectivity in general disk graphs. In Proc. SIAM Symposium on Simplicity of\nAlgorithms (SOSA) , pages 299–305, 2024. doi:10.1137/1.9781611977936.27 .\n23Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic\nplanar Voronoi diagrams for general distance functions and their algorithmic applications.\nDiscret. Comput. Geom. , 64(3):838–904, 2020. Preliminary version in SODA’17. doi:10.1007/\ns00454-020-00243-7 .\n24Vladlen Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. J.\nACM, 51(5):699–730, 2004. Preliminary verion in FOCS’01. doi:10.1145/1017460.1017461 .\n25D. T. Lee and Robert L. (Scot) Drysdale III. Generalization of Voronoi diagrams in the plane.\nSIAM J. Comput. , 10(1):73–87, 1981. doi:10.1137/0210006 .\n26James R. Lee. Separators in region intersection graphs. In Proc. 8th Innovations in Theoretical\nComputer Science Conference (ITCS) , pages 1:1–1:8, 2017. doi:10.4230/LIPICS.ITCS.2017.\n1.\n27Jirí Matoušek. Efficient partition trees. Discret. Comput. Geom. , 8:315–334, 1992. Preliminary\nversion in SoCG’91. doi:10.1007/BF02293051 .\n28Jirí Matoušek. Lectures on Discrete Geometry , volume 212 of Graduate Texts in Mathematics .\nSpringer, 2002.\n29Mihai Pătraşcu and Mikkel Thorup. Planning for fast connectivity updates. In Proc. 48th\nAnnual IEEE Symposium on Foundations of Computer Science (FOCS) , pages 263–271, 2007.\ndoi:10.1109/FOCS.2007.54 .\n30Warren D. Smith and Nicholas C. Wormald. Geometric separator theorems & applications. In\nProc. 39th Annual Symposium on Foundations of Computer Science (FOCS) , pages 232–243,\n1998. doi:10.1109/SFCS.1998.743449 .\n31Mikkel Thorup. Decremental dynamic connectivity. J. Algorithms , 33(2):229–243, 1999.\nPreliminary version in SODA’97. URL: https://doi.org/10.1006/jagm.1999.1033 ,doi:\n10.1006/JAGM.1999.1033 ."    },    {        "title": "2402.05371v1.Learning_to_Control_Emulated_Muscles_in_Real_Robots__Towards_Exploiting_Bio_Inspired_Actuator_Morphology.pdf",        "content": "Learning to Control Emulated Muscles in Real Robots:\nTowards Exploiting Bio-Inspired Actuator Morphology\nPierre Schumacher1,2, Lorenz Krause1,2, Jan Schneider1, Dieter Büchler1, Georg Martius∗1,3, Daniel Haeufle∗2,4\nAbstract — Recent studies have demonstrated the immense\npotential of exploiting muscle actuator morphology for natural\nand robust movement – in simulation. A validation on real\nrobotic hardware is yet missing. In this study, we emulate muscle\nactuator properties on hardware in real-time, taking advantage\nof modern and affordable electric motors. We demonstrate that\nour setup can emulate a simplified muscle model on a real\nrobot while being controlled by a learned policy. We improve\nupon an existing muscle model by deriving a damping rule that\nensures that the model is not only performant and stable but\nalso tuneable for the real hardware. Our policies are trained\nby reinforcement learning entirely in simulation, where we\nshow that previously reported benefits of muscles extend to\nthe case of quadruped locomotion and hopping: the learned\npolicies are more robust and exhibit more regular gaits. Finally,\nwe confirm that the learned policies can be executed on real\nhardware and show that sim-to-real transfer with real-time\nemulated muscles on a quadruped robot is possible. These\nresults show that artificial muscles can be highly beneficial\nactuators for future generations of robust legged robots. Videos:\nhttps://sites.google.com/view/emulatedmuscles\nI. I NTRODUCTION\nThe development of learning algorithms for robotic loco-\nmotion predominantly relies on position control with electric\nmotors. The ideal actuator is considered to impose as few\nconstraints as possible on the system, while additional proper-\nties such as regular foot patterns, minimal torque utilization,\nand robust policies arise due to the inclusion of a series of\ncost terms and certain training curricula. Nevertheless, there\nis rising interest in alternative actuation models, promising\nadvantageous properties amenable to stable control and\ngeneralizing controllers without the need for tedious cost\nterm tuning. Some of these studies investigate the effect of\naction spaces on learned controllers in general [1]–[3], while\nothers demonstrate the benefits of a particular actuator class,\nwhich is well-known but under-utilized: muscles [4]–[6].\nMuscles support the nervous system of animals and humans\nin robust control as they generate zero-time delay reactions\n(termed preflexes) to environmental influences [7]–[9]. While\n1Max Planck Institute for Intelligent Systems, Tübingen, Germany.\n2Hertie Institute for Clinical Brain Research, and Centre for Integrative\nNeuroscience, University of Tübingen, Germany.\n3Distributed Intelligence, University of Tübingen, Germany\n4Institute of Computer Engineering (ZITI), Heidelberg University, Ger-\nmany\n∗GM and DH contributed equally to this publication\nThe authors thank the International Max Planck Research School for\nIntelligent Systems (IMPRS-IS) for supporting Pierre Schumacher. This\nwork was supported by the Cyber Valley Research Fund (CyVy-RF-2020-11\nto DH and GM)some of these advantages have been investigated in simplistic\nsimulated [1], [4], [10] and hardware demonstrators [11],\n[12], actual real-world experiments remain indispensable to\nvalidate the relevance to robotic hardware.\nDifferent artificial muscle architectures are being developed\nto test the potential of the reported simulation findings, two\nnotable examples being McKibben [13], [14] and HASEL\nmuscles [15]. Although very promising, many years of\nresearch are needed to make these systems reliable and\nto allow reproducible experimentation in robotics. Some\napproaches aim to speed up the development of control\nalgorithms for artificial muscles by using hybrid sim-hardware\nsetups [16], [17] or model-based algorithms [14], which is\nnot feasible for every task.\nWith the advent of powerful and affordable direct-drive\nelectric motors it is possible to emulate particular actuators\nin real time. The concept of emulating muscles on motors\nhas been tested in simulation, even for a bipedal robot [5],\n[6], but in hardware only with a singular motor on a lab\nbench [11], [18], [19]. This approach was limited so far, as\na PD-controller was required to match the torques predicted\nby the muscle emulator due to time delays induced by the\nemulation [18], [19]. Benefits of emulated muscles have not\nbeen demonstrated in complex and realistic robotic settings.\nEven in those instances where emulated muscles have\nbeen simulated, the control algorithms were limited to state\nmachines or reflex-based controllers [5], [6]. The combination\nwith more powerful frameworks such as reinforcement learn-\ning (RL), is doubly interesting, as it may allow the generation\nof diverse movements with emulated muscles, while the\nimportance of the actuation model for RL performance is\nalso well documented [1], [3], [20], including evidence that\nmuscles can benefit learning [4].\nThe purpose of this study is to demonstrate that emulated\nmuscle properties are beneficial for learning robust walking\nwithout the need for excessive reward engineering and can be\nexploited on real hardware. To achieve this, we first extend\nsimulation results of a previously proposed muscle model [4].\nWe implement this model in the GPU-based simulator Isaac\nGym [21] and show several benefits in quadruped locomotion\nand hopping tasks — when compared to commonly used\nactuators. After observing increased robustness and more\nregular gaits in the learned policies in simulation, we emulate\nthe muscle-actuator in hardware with a high frequency real-\ntime control cycle without an intermediate PD-controller. Our\npolicies are learned with RL entirely in simulation withoutarXiv:2402.05371v1  [cs.RO]  8 Feb 2024muscle length\nFL\nFV\nmuscle velocity\nforce\nforce\n0\n0\naction\ntorque\npolicy\npolicy\npolicyPDmuscletorqueHigh-level RL policyPython interface ~50 HzLow-level controllerSim: 200Hz / Hardware: 500 Hz\nSimulation200HzHardware Backendmax 10 kHz\naction\naction\ntorque\ntorqueFig. 1: Emulated muscles on real robots. We compare three different low-level actuation controllers which are used by\nthree policies trained in simulation: PD, muscle and torque control. The resulting policies can then be tested on a real robot,\nby a real-time control module running at ∼500Hz, which communicates with the slowly executed RL policy executed\nwith∼50Hz. The robot backend performs torque commands with a maximum of 10 kHz and repeats previous torques\nbetween command updates. Importantly, the policy action either represents desired positions, muscle excitations or torques.\nThe simulation has a physics timestep of 5 ms, with 4 physics updates for each controller step.\nthe need for numerous fine-tuned cost terms and are shown\nto transfer to the real system. This finding showcases the\neffectiveness of muscle actuators and hints at the potential for\nphysical robots, either through emulation or novel hardware\nactuators.\nContributions: ( 1) We investigate different actuator models\nwith RL on a simulated robotic quadruped. ( 2) We extend\nprevious results on muscle actuation with RL from Wochner\net al. [4] to a realistic robotic locomotion task. ( 3) We\nidentify a critical parameter of the previously proposed muscle\nmodel and derive a theoretically motivated value to facilitate\ntuning the model for real systems. ( 4) We achieve sim-to-\nreal transfer with a walking policy to a real-time emulated\nmuscle-controller on robotic hardware.\nII. M ETHODS\nA. Low-level actuation controllers\nWe compare the performance of RL policies trained with\nthree different low-level controllers: PD-controller, muscle-\ncontroller, and direct torque-controller (Fig. 1).\nPD-controllers are commonly used in quadruped locomo-\ntion studies [22], [23]; our version receives a desired position\ncomputed by the policy, while the desired velocity is set to\nzero, similar to [1]:\nτi=kstiff(˜qi−qi)−kdamp ˙qi, (1)where τiis the computed torque, ˜qithe desired position, qi\ncurrent position, and ˙qithe velocity for joint i.\nThetorque-controller is considered to be ideal in the sense\nthat the torque computed by the policy is directly applied to\nthe joint:\nτi=kscale˜τi, (2)\nwhere ˜τi∈[−1,1]is computed by the policy and kscale\nscales the policy output to the robot’s torque range. While it\nis possible to use torque actuation on quadrupeds with learned\npolicies, it imposes strong control frequency requirements on\nthe implementation [24].\nThemuscle-controller is similar to [4] and emulates length\nand velocity dependent muscle force characteristics, as well\nas activation dynamics:\nτi=fmax\"2X\nk=1(−1)k+1FL(lk) FV(¯˙lk)mact,k+ FP( lk)#\n,\n(3)\nwhere FL(·)is the force-length and FV(·)the force-velocity\nrelationship (see “muscle” in Fig. 1), FP(·)the passive force\nandmactthe muscle activity. The sum is taken over two\nmuscles for each joint, each pulling in opposing directions.\nThe activity mactapproaches the policy action ˜awith a\nlow-pass filter:\n˙mact(t) =1\n∆ta(˜a(t)−mact(t)), (4)with the time constant ∆ta= 0.01s. The scalar fmax was\nintroduced to easily tune the maximum force output. The\nmuscle length is given by:\nlk=akqi+bk, (5)\nwithakandbkbeing part of the parametrization. The muscle\nvelocity is given by the derivative of Eq. 5\n¯˙lk=β˙lk=β(ak˙qi). (6)\nImportantly, we multiply the muscle velocity by a scaling\nparameter βin the FV-function (Eq. 6), corresponding to\n1/vmax in [4], allowing us to tune the maximum damping\nstrength of the muscle. The parameters aandbcorrespond\ntomandlrefin [4], where the exact formulation is detailed.\nB. Muscle model implementation\nFor the simulation experiments, we re-implemented the\nMuJoCo-based muscle model from [4]. In order to preserve\nIsaac Gym’s computational speed [22], we translated the\noriginal Cython implementation of the muscle model to a\nvectorized PyTorch version. Overall, the vanilla PD-controller\nis only 30% faster than the muscle implementation, even\nthough we simulate 16 muscles for the robot—compared to 8\nmotors for the PD and torque-controllers. Considering that the\ntraining time is mostly below an hour on a consumer-GPU,\nthis computational efficiency is sufficient for the purpose of\nthis study. The environment runs with a timestep of 5ms for\nthe physics engine and a control time step of 20ms.\nFor the hardware experiments, the muscle model is imple-\nmented in C++, which communicates with real-time Python\nfunctions which were bound via PyBind11 [25]. This module\ninterfaces with the RL policy running at ∼50Hz, which\nis not real-time executed, while the actual muscle-controller\nis running at ∼500Hz. It is important to achieve a large\nexecution frequency on the hardware to reliably emulate the\nmuscle model [4]. This paradigm is similar to most studies\nusing PD-controllers on quadruped robots [24].\nC. Muscle parameter tuning\nIn contrast to the PD-controller [26], there is no clear tuning\nprocedure for a muscle-controller. The velocity scaling βis\nespecially difficult to tune, as it affects the damping of the\nactuator through the FV-relationship [4] and might not transfer\nto real hardware due to control frequency limitations, see\nSec. III-A. When we tried to set this parameter through an\niterative tuning procedure similar to previous studies [27], we\neither obtained parameters that performed well in simulation\nbut were not stable on the hardware, or they were stable but\ndid not yield performant policies.\nWe derive an expression for the damping coefficient β,\nsuch that under certain conditions, the damping is equivalent\nto a damping-controller τ=−kdamp ˙q. With this approach,\nwe can determine reasonable βvalues that are achievable on\nthe hardware based on the easier-to-tune damping-controller.\nWe start by assuming two symmetric muscles connected\nto a single joint at position q= 0. As the FV-relationship\nis thought to primarily contribute to muscle damping [8],we also assume FL(l) = 1 ,FP(l) = 0 and constant activity\nmact,k= 1. Equation 3 then simplifies to:\nτ=fmax\u0010\nFV(¯˙l1)−FV(¯˙l2)\u0011\n(7)\n=fmax\u0010\nFV(¯˙l1)−FV(−¯˙l1)\u0011\n(8)\n≈fmax(4¯˙l1) (9)\nwhere¯˙l1=−¯˙l2in Eq. 6 as we assume symmetric muscles and\na1=−a2in Eq. 5, and we have used a Taylor expansion\naround¯˙l1= 0. By using the definition of the FV-curve\nfrom [4], [28], we obtain\nFV(¯˙l) = 1 + 2¯˙l+O(¯˙l2).\nBy comparing Eq. 9 to a damping controller and using¯˙l1=\na1β˙qwe get:\n−kdamp ˙q=−fmax4β a1˙q (10)\n⇒β=kdamp\n4a1fmax. (11)\nThe negative sign in Eq. 10 was introduced as muscles are\nassumed to always pull on the joint. We found that setting\nkdamp = 0.1is an achievable value for a damping controller\non the robot hardware. By using Eq. 11 for all muscle\nexperiments, we ensure that the maximum co-contraction\nlevel generates achievable damping on the hardware. The\npolicy can still reduce the amount of equivalent damping by\nchanging the amount of co-contraction [8].\nD. Tasks\nWe employ two tasks: walking and hopping, which have\nbeen proven interesting in previous comparisons of action\nspaces [4]. Walking requires a periodic and relatively slow\njoint movement, while periodic hopping is a highly explosive\nmovement that also requires stabilization when landing.\n1) Walking: We restrict ourselves to the target velocity\ntracking reward from [22] and omit all other terms that were\nused in that study:\nrwalk= exp\u0000\n−(vtarget−vx)2/σ\u0001\n, (12)\nwhere vtarget is the target x-velocity, vxis the x-velocity of\nthe base and σ= 0.25is a sensitivity factor. The task is reset\nif the robot base or the upper legs make contact with the\nground or if the base height is hbase<0.2m.\n2) Periodic hopping: The hopping reward is based on the\nvertical velocity of the robot base vzas\nrhop= exp (10 vclip), (13)\nwhere vclip= clip( vz,0,10). This is the same hopping reward\nas in [4] with slightly different scaling, accounting for a\ndifferent velocity range with the SOLO robot. We observed\nthe reward function to not work well if the desired hopping\nvelocity is in a range where the reward function is not sensitive\nto changes. The episode is reset when the robot base changes\nin orientation too much from upright orientation, or when a\nbody part different than the feet touches the ground.Both tasks additionally use action rate regularization:\nract=−wact(at+1−at)2, (14)\nwhere atis the action at time step tandwactis a weighting\ncoefficient.\nThe total reward is\nr=rwalk/hop+ract. (15)\nE. Learning high-level policies\nIn order to train policies, we use the popular RL algorithm\nPPO [29] with the implementation from [22] and identical\nhyperparameters as in [23] for the SoloLeap -task, shown\nin Table S6 in their work. Our policy and critic are fully-\nconnected MLPs with 3 hidden dimensions of 128 units each\nand ELU-activation functions. For simulation experiments, we\ndo not use domain randomization, while sim-to-real transfer\nrequires randomization of initial states, ground friction, body\nmass, and input noise. See Tab. I for details.\nTABLE I: Values for domain randomization (DR) and input\nnoise (IN). All values are sampled uniformly in the given\nranges and added to the existing values.\nParameter Range\nDRinit. joint pos. [−1.0,1.0]\ninit. muscle act. [0.5,1.0]\nfriction [0.5,1.25]\njoint damping [0,0.09]\npushxy [−1.5,1.5]\nmass shift [−0.5,1.2]\nINbase lin. vel. [−0.02,0.02]\nbase ang. vel. [−0.05,0.05]\ngravity vector [−0.05,0.05]\njoint pos. [−0.01,0.01]\njoint vel. [−0.075,0.075]\nmuscle length [−0.01,0.01]\nmuscle vel. [−1.0,1.0]\nmuscle act. [−0.01,0.01]\nmuscle force [−1.0,1.0]\nIII. R ESULTS\nEach simulation experiment was repeated over 10 random\nseeds. In experiments where we show a specific rollout of a\nspecific policy, we analyzed all trained seeds for all policies\nat the end of their training, and selected in each case the\none with the best behavior. In general we observe very small\nvariance across seeds, likely due to the large number (4096)\nof parallel environments in Isaac Gym.\nA. Parameter tuning procedure\nAs our study aims to investigate the potential benefits\nof muscle-like actuators, we use a methodical approach\nfor parameter tuning. First, only the task-relevant reward\nis used as objective. We tune the actuator-relevant parameters\nto achieve maximum performance in a population-based\noptimization over 10 iterations of 100 trials each. This\noptimization successfully leads to policies achieving large task\nrewards. Upon closer inspection, however, we observe them\nto be very jittery, not likely to generalize to the real hardware.TABLE II: Optimized parameter values for the used actuators.\nNote that βis not optimized but set through Eq. 11.\nParameter Value\nmusclelmin 0.24\nlmax 1.53\nfvmax 1.38\nfpmax 1.76\nlce_min 0.74\nlce_max 0.94\nfmax 34\nϕmin -3.14\nϕmax 3.14\nτact 0.01\nβ 0.36\nPDkstiff 2\nkdamp 0.05\ntorque n.a. n.a.\naction rate weight wact 0.004\nWe consequently add action-rate regularization, see Eq. 14.\nThe best parameters from the previous step are taken, and a\ngrid search over weighting coefficients for the action rate cost\nis performed. It is commonly observed that smooth actuation\nsignals are more transferable to real-world robots [30]. The\nlevel of action smoothness that works well for sim-to-real\ntransfer is hard to quantify, as such we visually check for\nthe variance in the applied torques, instead of optimizing the\nregularizer together with the other parameters.\nThis optimization procedure is executed for the PD and\nmuscle low-level controller for the walking task. The torque\nactuator has no parameters besides the maximum torque,\nwhich is given by the system specification of the robot. We\nthen keep all parameters identical for the hopping task.\nB. Walking\nWhile performant and transferable gaits usually require\ndozens of cost terms which are hand-tuned by experts [22],\n[23], we investigate the behaviors that arise from a simple\nand under-specified forward velocity reward combined with\nan action rate cost. Different actuator morphologies might\nembed certain movement priors into the learning algorithm\nwhich do not require a complex tuning procedure.\nThe task reward over training improves faster for the PD-\nand torque-controllers than for the muscle-controllers, see\nFig. 3. While all three policies perform well according to\nthis metric, we also take a look at the learned gaits in Fig. 4.\nThe touch patterns of the feet with the ground (Fig. 4b)\nshow that the muscle-agent learns the most regular gait. The\nPD-agent tends to not elevate the feet from the ground, as can\nbe seen from the gait illustration (Fig. 4c) and the joint angle\ntrajectories (Fig. 4a). This behavior relies on very precise\nmovements and friction coefficients in the simulation, which\nwould likely not generalize to the real world. Interestingly,\nthe torque-agent lifts the feet off the ground but tends to\nuse extreme and uncontrolled motions. This showcases that\neven with an under-specified reward function, a particular\nembodiment, or low-level controller, can bias the behavior\ntowards being more natural.β= 0.36 β= 0.66\n0.000.05pos. [s]\n0.5\n0.00.5torque [Nm]\n0.0 0.2 0.4 0.6 0.8 1.0\ntime [s]5\n05vel. [rad/s]\nFig. 2: Hardware deployment: excessive emulated damp-\ning destabilizes the control. We apply constant muscle\nexcitations of mact= 1.0on both muscles of a joint of the\nhardware robot. Ideally, the joint would remain at its initial\nposition, resisting external perturbations with high damping\nand stiffness. When using a high scaling value β= 0.66, the\nemulation controller can not keep up due to its limited control\nfrequency, and the motor starts to oscillate strongly. The value\nβ= 0.36, given by Eq. 11, leads to stable performance. We\nobserve similar effects with a PD-controller and excessive\nkdamp -values.\nPD muscle torque\n0.000.170.33task reward\n0 200 400 600 800 1000\nlearning iteration0.4\n0.2\n0.0action rate\nFig. 3: Stable locomotion is achieved by all actuators.\nUpper: Velocity-tracking task reward over time. The PD-agent\nlearns with the smallest number of iterations, while torque-\nand muscle-agents achieve a similar maximum performance\nwith more training. Lower: The action rate reward only slowly\nimproves for the muscle-agent, likely because the number of\nactions is twice as large as for the other actuators (16 vs 8).C. Periodic hopping\nThe simulation results for the maximum height periodic\nhopping task are shown in Fig. 5. The reward function,\nsee Eq. 13, incentivizes maximal upwards velocities for the\nlongest possible time. Even though the PD and torque agents\nachieve some very high jumps (Fig. 5c), the hopping behavior\nis straighter and more regular with the muscle actuation,\nmaking it easier to control when landing. This leads to large\nepisode lengths (Fig. 5b) for the muscle-agent, as the other\ncontrollers can often not recover after landing. These results\nare slightly different to [4], where the muscle-agent achieved\nbetter performance than the alternatives only early in the\ntraining, but this difference may be explained by the superior\nstability of quadrupeds over bipedal robots.\nD. Robustness\nWochner et al. [4] demonstrated that muscle-actuators\nexhibit higher robustness under unseen perturbations. We\ntherefore tested the trained policies under terrain variations\nused in [22] and recorded the fraction of steps across 100\nepisodes in which the policies would not fall down, which\nwe define as the success rate. Note that the policies have only\nbeen trained on flat terrain. We see that the muscle-controller\nis generally more robust and has higher success rates than\nthe other controllers (Fig. 6).\nE. Sim-to-real transfer\nIn order to test the applicability of our findings and\nhighlight their relevance for future robot design, we imple-\nmented our muscle model on real robotic hardware. First, we\ndemonstrate the sim-to-real transfer of our policies in a task\nrequiring tracking joint angles over time. For this experiment,\nthe robot is suspended in mid-air on a fixed stand (Fig. 7b).\nThe sim-to-real policies were trained with observation noise\nand domain randomization, see Tab. I. Finally, we added a\nsmall amount of joint damping through a low-level controller\nwithkdamp = 0.08Nms/rad in all hardware experiments to\nstabilize the system and prevent damage. We also added it\nin simulation to minimize the sim-to-real gap.\nWe compare a policy trained with the real-time muscle-\ncontroller to a PD-controller for which we give alternating\ntarget positions and set the target velocity to zero (Fig. 7a).\nWe use PD gains kstiff= 5.0andkdamp = 0.1, similar to [23].\nFigure 7a shows that the muscle-controller is able to track\nthe given joint angles similarly well to the PD-controller.\nIn a more realistic task, we train a walking policy with\nthe muscle-controller in simulation and can reliably perform\na running gait without additional training on the hardware.\nThis successful sim-to-real transfer demonstrates that muscle\nactuation enables learning realistic and robust gaits without\nthe need for excessive reward engineering. See Fig. 7c and the\nvideos on the project website https://sites.google.\ncom/view/emulatedmuscles .\nIV. C ONCLUSION\nIn this study, we have investigated the benefits of muscle-\nlike actuation for quadruped locomotion tasks. We validatedPD muscle torque\n2\n02angle [rad]hip 1 knee 1\n2\n0angle [rad]hip 2 knee 2\n02angle [rad]hip 3 knee 3\n0 1 2 3 4 5\ntime [s]2\n02angle [rad]hip 4\n0 1 2 3 4 5\ntime [s]knee 4\n(a) Joint angles over time.\n(b) Foot-contact patterns.\n(c) Learned gaits.\nFig. 4: Muscles learn more regular walking patterns. Using only the linear velocity and an action rate regularizing term\nas reward, the muscle-actuator learns the most regular walking pattern, lifting the feet above the ground while moving. In\ncontrast, the torque-actuator agent walks with strongly extended motions that come close to the angular limits of the robotic\nlimbs. The PD-controller agent drags the feet very closely over the ground, likely not generalizing to real-world hardware.\nprevious results on stability and robustness of the learned\npolicies, while finding that the emulated actuator morphology\nalso influences the learned gaits in an under-specified loco-\nmotion task. Finally, we proposed a tuning procedure that\nallowed us to validate the muscle model on a real robotic\nsystem under performance and stability constraints. We are\nable to execute policies learned entirely in simulation with\nlittle reward engineering on robotic hardware that runs a\nreal-time emulated muscle model. These results showcase the\npotential of muscle-like actuators when combined with RL.\nREFERENCES\n[1]X. B. Peng and M. van de Panne, “Learning locomotion skills using\ndeeprl: Does the choice of action space matter?” in Proceedings of the\nACM SIGGRAPH / Eurographics Symposium on Computer Animation ,\nser. SCA ’17. New York, NY , USA: ACM, 2017, pp. 12:1–12:13.\n[Online]. Available: http://doi.acm.org/10.1145/3099564.3099567\n[2]R. Martín-Martín, M. A. Lee, R. Gardner, S. Savarese, J. Bohg,\nand A. Garg, “Variable impedance control in end-effector space: An\naction space for reinforcement learning in contact-rich tasks,” in 2019\nIEEE/RSJ International Conference on Intelligent Robots and Systems\n(IROS) , 2019, pp. 1010–1017.[3]E. Aljalbout, F. Frank, M. Karl, and P. van der Smagt, “On the role\nof the action space in robot manipulation learning and sim-to-real\ntransfer,” 2023.\n[4]I. Wochner, P. Schumacher, G. Martius, D. Büchler, S. Schmitt, and\nD. Haeufle, “Learning with muscles: Benefits for data-efficiency and\nrobustness in anthropomorphic tasks,” in Proceedings of the 6th\nConference on Robot Learning (CoRL) , ser. Proceedings of Machine\nLearning Research, vol. 205. PMLR, Dec. 2022, pp. 1178–1188.\n[Online]. Available: https://proceedings.mlr.press/v205/wochner23a.\nhtml\n[5]Z. Batts, S. Song, and H. Geyer, “Toward a virtual neuromuscular\ncontrol for robust walking in bipedal robots,” in 2015 IEEE/RSJ\nInternational Conference on Intelligent Robots and Systems (IROS) ,\n2015, pp. 6318–6323.\n[6]A. Rai, R. Antonova, S. Song, W. Martin, H. Geyer, and C. Atkeson,\n“Bayesian optimization using domain knowledge on the atrias biped,”\nin2018 IEEE International Conference on Robotics and Automation\n(ICRA) , 2018, pp. 1771–1778.\n[7]A. J. van Soest and M. F. Bobbert, “The contribution of muscle\nproperties in the control of explosive movements,” Biol Cybern , vol. 69,\nno. 3, pp. 195–204, 1993.\n[8]F. Izzi, A. Mo, S. Schmitt, A. Badri-Spröwitz, and D. F. B. Haeufle,\n“Muscle prestimulation tunes velocity preflex in simulated perturbed\nhopping,” Scientific Reports , vol. 13, no. 1, p. 4559, Mar 2023.\n[Online]. Available: https://doi.org/10.1038/s41598-023-31179-6\n[9]M. Araz, S. Weidner, F. Izzi, A. Badri-Spröwitz, T. Siebert, and D. F. B.PD muscle torque\n(a) Hopping motion.\n0 2000 4000 6000\nlearning iteration01000200030004000task reward\n0 2000 4000 6000\nlearning iteration02004006008001000episode length\n(b) Task performance.\n0.00.20.40.60.8height [m]\n0.00.20.40.60.8height [m]\n0 2 4 6 8 10 12\ntime [s]0.00.20.40.60.8height [m] (c) Height of the robot base.\nFig. 5: Muscles learn more stable hopping. Similar to results by Wochner et al. [4], we observe that muscle actuators learn\nmore stable periodic hopping than torque or PD actuators. While the torque and especially the PD agents can occasionally\nachieve larger maximum heights than the muscle-agent, the periodic hopping behavior is less stable and achieves a smaller\noverall episode return. This task rewards large vertical velocities exponentially, the optimal behavior is therefore periodic\njumping with maximum velocity and large displacement. We show the average base height while walking as a black line in\n(c) as a reference height for the hopping motion.\n(a) Difficult terrain.\nmuscle PD torque\nactuator0.00.20.40.60.8success rate (b) Success rates for different actuators.\nFig. 6: The muscle actuator is the most robust. We rollout the different agents in an unseen terrain. The success rate is\nhighest for the muscle actuator, followed by the PD and then the torque.\nHaeufle, “Muscle preflex response to perturbations in locomotion: In\nvitro experiments and simulations with realistic boundary conditions,”\nFront Bioeng Biotechnol , vol. 11, p. 1150170, Apr. 2023.\n[10] D. F. B. Haeufle, S. Grimmer, and A. Seyfarth, “The role of\nintrinsic muscle properties for stable hopping—stability is achieved\nby the force–velocity relation,” Bioinspiration & Biomimetics ,\nvol. 5, no. 1, p. 016004, feb 2010. [Online]. Available: https:\n//dx.doi.org/10.1088/1748-3182/5/1/016004\n[11] A. Seyfarth, K. T. Kalveram, and . Geyer, Hartmut, “Simulating\nMuscle-Reflex Dynamics in a Simple Hopping Robot,” in Proceedings\nof Fachgespräche Autonome Mobile Systeme . Springer, 2007,\npp. 294–300. [Online]. Available: http://link.springer.com/10.1007/\n978-3-540-74764-2_45\n[12] A. Mo, F. Izzi, E. C. Gönen, D. Haeufle, and A. Badri-Spröwitz,\n“Slack-based tunable damping leads to a trade-off between robustness\nand efficiency in legged locomotion,” Scientific Reports , vol. 13, no. 1,\nfeb 2023.\n[13] B. Tondu and S. Zagal, “Mckibben artificial muscle can be in\naccordance with the hill skeletal muscle model,” in The First IEEE/RAS-\nEMBS International Conference on Biomedical Robotics and Biomecha-\ntronics, 2006. BioRob 2006. , 2006, pp. 714–720.\n[14] H. Ma, D. Büchler, B. Schölkopf, and M. Muehlebach, “Reinforcement\nlearning with model-based feedforward inputs for robotic table tennis,”\nAutonomous Robots , vol. 47, no. 8, pp. 1387–1403, Dec 2023.[Online]. Available: https://doi.org/10.1007/s10514-023-10140-6\n[15] P. Rothemund, N. Kellaris, S. K. Mitchell, E. Acome, and\nC. Keplinger, “Hasel artificial muscles for a new generation of\nlifelike robots—recent progress and future opportunities,” Advanced\nMaterials , vol. 33, no. 19, p. 2003375, 2021. [Online]. Available:\nhttps://onlinelibrary.wiley.com/doi/abs/10.1002/adma.202003375\n[16] D. Büchler, S. Guist, R. Calandra, V . Berenz, B. Schölkopf, and\nJ. Peters, “Learning to play table tennis from scratch using muscular\nrobots,” IEEE Transactions on Robotics , vol. 38, no. 6, pp. 3850–3860,\n2022.\n[17] S. Guist, J. Schneider, A. Dittrich, V . Berenz, B. Schölkopf,\nand D. Büchler, “Hindsight states: Blending sim and real\ntask elements for efficient reinforcement learning,” in Robotics:\nScience and Systems XIX , Jul. 2023. [Online]. Available: https:\n//www.roboticsproceedings.org/rss19/p038.html\n[18] H. Serhan, C. Nasr, and P. Henaff, “Designing a muscle like system\nbased on pid controller and tuned by neural network,” in The 2006\nIEEE International Joint Conference on Neural Network Proceedings ,\n2006, pp. 4991–4998.\n[19] H. Serhan and P. Henaff, “Muscle-like compliance in knee articulations\nimproves biped robot walkings,” in Recent Advances in Robotic\nSystems , G. Wang, Ed. Rijeka: IntechOpen, 2016, ch. 3. [Online].\nAvailable: https://doi.org/10.5772/63746\n[20] J. Schneider, P. Schumacher, D. Häufle, B. Schölkopf, and D. Büchler,muscle PD\n0.5\n0.00.5angle [rad]hip 1\n1\n01knee 1\n0.5\n0.00.5angle [rad]hip 2\n1\n01knee 2\n0.5\n0.00.5angle [rad]hip 3\n1\n01knee 3\n0 200 400 600\niteration0.5\n0.00.5angle [rad]hip 4\n0 200 400 600\niteration1\n01knee 4\n0.5\n0.00.5angle [rad]hip 1\n1\n01knee 1\n0.5\n0.00.5angle [rad]hip 2\n1\n01knee 2\n0.5\n0.00.5angle [rad]hip 3\n1\n01knee 3\n0 200 400 600\niteration0.5\n0.00.5angle [rad]hip 4\n0 200 400 600\niteration1\n01knee 4\n(a) Joint angle tracking with muscle and PD actuators.\n(b) Exemplary target angles on the real robot.\n(c) Walking gait of the muscle-policy.\nFig. 7: Sim-to-real transfer with an emulated muscle actuator. (a) & (b) We trained policies for joint angle target tracking\nin simulation under the addition of simulated sensor noise and domain randomization. The muscle-policy is able to track\ntargets with a real-time emulated muscle actuator on real robotic hardware through sim2real transfer. Note that the target\nangles were given at a fast frequency, making it challenging to track them perfectly in order to achieve a dynamic regime.\n(c) We train a muscle-driven policy in a walking task in simulation and deploy it on the real robot hardware.\n“Investigating the impact of action representations in policy gradient\nalgorithms,” 2023.\n[21] V . Makoviychuk, L. Wawrzyniak, Y . Guo, M. Lu, K. Storey, M. Macklin,\nD. Hoeller, N. Rudin, A. Allshire, A. Handa, and G. State, “Isaac gym:\nHigh performance gpu-based physics simulation for robot learning,”\n2021.\n[22] N. Rudin, D. Hoeller, P. Reist, and M. Hutter, “Learning to walk in\nminutes using massively parallel deep reinforcement learning,” in 5th\nAnnual Conference on Robot Learning , 2021. [Online]. Available:\nhttps://openreview.net/forum?id=wK2fDDJ5VcF\n[23] C. Li, M. Vlastelica, S. Blaes, J. Frey, F. Grimminger, and\nG. Martius, “Learning agile skills via adversarial imitation of\nrough partial demonstrations,” in Proceedings of the 6th Conference\non Robot Learning (CoRL) , Dec. 2022. [Online]. Available:\nhttps://openreview.net/forum?id=x6INXlnUGro\n[24] S. Chen, B. Zhang, M. W. Mueller, A. Rai, and K. Sreenath, “Learning\ntorque control for quadrupedal locomotion,” in 2023 IEEE-RAS 22nd\nInternational Conference on Humanoid Robots (Humanoids) , 2023, pp.\n1–8.[25] W. Jakob, J. Rhinelander, and D. Moldovan, “pybind11 –\nseamless operability between C++11 and Python,” 2017,\nhttps://github.com/pybind/pybind11.\n[26] J. G. Ziegler and N. B. Nichols, “Optimum Settings for Automatic\nControllers,” Transactions of the American Society of Mechanical\nEngineers , vol. 64, no. 8, pp. 759–765, 12 2022. [Online]. Available:\nhttps://doi.org/10.1115/1.4019264\n[27] D. Mattern, P. Schumacher, F. M. López, M. C. Raabe, M. R. Ernst,\nA. Aubret, and J. Triesch, “MIMo: A multi-modal infant model for\nstudying cognitive development,” 2023.\n[28] E. Todorov, T. Erez, and Y . Tassa, “Mujoco: A physics engine for\nmodel-based control,” in 2012 IEEE/RSJ International Conference on\nIntelligent Robots and Systems , 2012, pp. 5026–5033.\n[29] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov,\n“Proximal policy optimization algorithms,” 2017.\n[30] S. Mysore, B. Mabsout, R. Mancuso, and K. Saenko, “Regularizing\naction policies for smooth control with reinforcement learning,” in 2021\nIEEE International Conference on Robotics and Automation (ICRA) .\nIEEE, 2021, pp. 1810–1816."    },    {        "title": "2402.05415v2.Near_Optimal_Convex_Simple_Bilevel_Optimization_with_a_Bisection_Method.pdf",        "content": "arXiv:2402.05415v2  [math.OC]  5 Mar 2024Near-Optimal Convex Simple Bilevel Optimization with a Bis ection\nMethod\nJiulin Wang Xu Shi Rujun Jiang†\nSchool of Data Science, Fudan University\nAbstract\nThis paper studies a class of simple bilevel\noptimization problems where we minimize\na composite convex function at the upper-\nlevel subject to a composite convex lower-\nlevel problem. Existing methods either pro-\nvide asymptotic guarantees for the upper-\nlevel objective or attain slow sublinear con-\nvergence rates. We propose a bisection al-\ngorithm to find a solution that is ǫf-optimal\nfor the upper-level objective and ǫg-optimal\nfor the lower-level objective. In each it-\neration, the binary search narrows the in-\nterval by assessing inequality system fea-\nsibility. Under mild conditions, the to-\ntal operation complexity of our method is\n˜O/parenleftBig\nmax{/radicalbig\nLf1/ǫf,/radicalbig\nLg1/ǫg}/parenrightBig\n. Here, a unit\noperation can be a function evaluation, gradi-\nent evaluation, or the invocation of the prox-\nimal mapping, Lf1andLg1are the Lipschitz\nconstants of the upper- and lower-level objec-\ntives’ smooth components, and ˜Ohides loga-\nrithmic terms. Our approach achieves a near-\noptimal rate, matching the optimal rate in\nunconstrained smooth or composite convex\noptimization when disregarding logarithmic\nterms. Numerical experiments demonstrate\nthe effectiveness of our method.\n1 INTRODUCTION\nIn this paper, we focus on the following convex bilevel\noptimization problem:\n(P) min\nx∈Rnf(x) :=f1(x) +f2(x)\ns.t.x∈arg min\nz∈Rng(z) :=g1(z) +g2(z).(1)\nProceedings of the 27thInternational Conference on Artifi-\ncial Intelligence and Statistics (AISTATS) 2024, Valencia ,\nSpain. PMLR: Volume 238. Copyright 2024 by the au-\nthor(s).†Corresponding Author.Here, functions f1andg1:X→Rare convex and\ncontinuously differentiable over an open set X∈Rn.\nTheir gradients, ∇f1and∇g1, areLf1- andLg1-\nLipschitz continuous, respectively. f2andg2:Rn→\nR∪∞ are proper lower semicontinuous (l.s.c.) convex\nfunctions. We assume that gis not strongly convex,\nand the lower-level problem has multiple optimal so-\nlutions; in other words, the optimal solution set of the\nlower-level problem, denoted as X∗\ng, is not a singleton.\nOtherwise, the optimal minimum is determined by the\nlower-level problem.\nThis specific class of problems, often known as\n“simple bilevel optimization” in the existing liter-\nature (Dempe et al., 2010; Dutta and Pandit, 2020;\nShehu et al., 2021; Jiang et al., 2023), is a sub-\nclass of the general bilevel optimization prob-\nlems. In a general bilevel optimization, the\nlower-level problem is parametrized by some upper-\nlevel variables. Bilevel optimization has garnered\nsignificant interest owing to its versatile applica-\ntions across domains such as reinforcement learning\n(Hong et al., 2020), meta-learning (Bertinetto et al.,\n2018; Rajeswaran et al., 2019), hyper-parameter op-\ntimization (Franceschi et al., 2018; Shaban et al.,\n2019), and adversarial learning (Bishop et al., 2020;\nWang et al., 2021, 2022).\nLetp∗be the optimal value of problem (1) and g∗\nbe the optimal value of the unconstrained lower-level\nproblem\nmin\nx∈Rng(x) :=g1(x) +g2(x). (2)\nThe goal of this paper is to find an ( ǫf,ǫg)-optimal\nsolution ˆxsatisfying\nf(ˆx)−p∗≤ǫfandg(ˆx)−g∗≤ǫg.\nA possible approach for solving problem (1) is to refor-\nmulate it to a constrained optimization problem with\nfunctional constraints and apply primal-dual methods.\nSpecifically, problem (1) can be reformulated as a con-\nstrained convex optimization problem as follows:\nmin\nx∈Rnf(x) s.t. g(x)≤g∗. (3)Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nA critical issue of applying primal-dual-type methods\nis that problem (3) does not satisfy the regularity con-\ndition required for their convergence (the strict feasi-\nbility does not hold and hence Slater’s condition fails).\nFurthermore, classical first-order algorithms, such as\nprojected gradient descent, may also be ineffective due\nto the difficulty of computing the orthogonal projec-\ntion onto the level-set of the lower-level objective. If\nwe relax the constraint and solve the following problem\nto ensure strict feasibility\nmin\nx∈Rnf(x) s.t. g(x)≤g∗+ǫ, (4)\nthese challenges remain. Indeed, as ǫapproaches zero,\ncausing the problem to become nearly degenerate, the\ndual optimal variable may tend towards infinity. This\nphenomenon hinders convergence and results in nu-\nmerical instability (Bonnans and Shapiro, 2013). Con-\nsequently, problem (1) cannot be straightforwardly\naddressed as a conventional constrained optimization\nproblem; instead, it necessitates novel theories and al-\ngorithms customized for its hierarchical structure.\n1.1 Our Approach\nOur main technique is a bisection method that itera-\ntively narrows an interval [ l,u] that includes p∗. The\nbinary search is based on the feasibility of the following\nsystem:\nf(x)≤c, g(x)≤g∗, (5)\nwherec=l+u\n2. The key observation is that if System\n(5) is feasible, then cis an upper bound of p∗; other-\nwise, System (5) is infeasible and cis a lower bound\nofp∗. This process divides the interval in half. The\nfeasibility of the system can be checked by solving the\nfollowing problem\nmin\nx∈Rng(x),s.t. f(x)≤c. (6)\nThe above is only a basic idea, the detail of our algo-\nrithm that considers the inexactness of solving (6) is\ndetailed in Section 3. Moreover, by showing that each\niteration and the initial lower and upper bounds can\nbe solved by Accelerated Proximal Gradient (APG)\nmethods (Nesterov, 1983; Beck and Teboulle, 2009),\nwe derive a comprehensive complexity analysis for our\nalgorithm.\nWe state our contributions in the following:\n•Under mild conditions, we propose a novel bisec-\ntion method that finds an ( ǫf,ǫg)-optimal solu-\ntion of problem (1) with an operation complex-\nity˜O/parenleftBig\nmax{/radicalbig\nLf1/ǫf,/radicalbig\nLg1/ǫg}/parenrightBig\n, where the no-\ntation ˜Osuppresses a logarithmic term. Ourmethod achieves near-optimal non-asymptotic\nguarantees on both upper- and lower-level prob-\nlems, i.e., our rate aligns with the optimal rate ob-\nserved in unconstrained smooth or composite con-\nvex optimization, with the exception of omitting\nthe logarithmic term (Nemirovsky and Yudin,\n1983; Woodworth and Srebro, 2016).\n•With an additional r-th-order ( r≥1) H¨ olderian\nerror bound assumption on the lower-level prob-\nlem and incorporating other smoothness assump-\ntions, our method can find a solution ˆxthat sat-\nisfies|f(ˆx)−p∗| ≤ǫfandg(ˆx)−g∗≤ǫgwith an\n˜O/parenleftBig\n1//radicalbigǫr\nf/parenrightBig\noperation complexity. This complex-\nity arises under the setting ǫg=α\nγ/parenleftBig\nǫf\nBf/parenrightBigr\n, where\nα,γ,B fare defined in Section 4.1.\n•With an additional assumption on the optimal\nvalue of (4), our method can find a solution ˆxthat\nsatisfies|f(ˆx)−p∗| ≤ǫfandg(ˆx)−g∗≤ǫgwith an\n˜O/parenleftbig\nmax{1//radicalbigLǫgǫg,1/√ǫg}/parenrightbig\noperation complex-\nity. This operational complexity is observed when\nǫf=Lǫgǫg, whereLǫgis defined in Section 4.2.\n•Numerical experiment results on different prob-\nlems demonstrate the superior performance of our\nmethod compared to the state-of-the-art.\n1.2 Related Work\nOne class of algorithms to solve problem (1) is\nbased on solving the Tikhonov-type regularization\n(Tikhonov and Arsenin, 1977):\nmin\nx∈Rnφ(x) :=g(x) +λf(x),\nwhereλ > 0 is a regularization parameter. How-\never, these kinds of algorithms fail to provide any non-\nasymptotic guarantee for either the upper- or lower-\nlevel objective. For a review of these algorithms, see\nDoron and Shtern (2023) and Jiang et al. (2023).\nAnother class of algorithms aims to establish\nnon-asymptotic convergence rates for problem (1).\nBeck and Sabach (2014) presented the Minimal Norm\nGradient (MNG) method for the case where fis\nstrongly convex. They demonstrated that MNG con-\nverges asymptotically to the optimal solution and\npossesses an O/parenleftbig\nL2\ng1/ǫ2/parenrightbig\ncomplexity bound for the\nlower-level problem. In their setting, g≡g1and\ng2≡0. Developed from the sequential averaging\nmethod (SAM) framework, the Bilevel Gradient Se-\nquential Averaging Method (BiG-SAM) is proposed\nby Sabach and Shtern (2017). This algorithm can\nachieve an O(Lg1/ǫ) complexity bound for the lower-\nlevel problem. Solodov (2007) introduced the Itera-\ntive Regularized Projected Gradient (IR-PG) method,Jiulin Wang, Xu Shi, Rujun Jiang†\nTable 1: Summary of simple bilevel optimization algorithms. The abbrev iations “SC”, “C”, “C3” stand for\n“strongly convex”, “convex”, “Convex objective with Convex Co mpact constraints” respectively. When the\nconnection between complexity and the gradient’s Lipschitz consta nt is clear, we include it in the complexity\nresult; otherwise, we omit it.\nReferencesUpper-level Lower-level Convergence\nObjective f Objective g Upper-level Lower-level\nMNG (Beck and Sabach, 2014). SC, differentiable C, smooth Asympt otic O/parenleftbig\nL2\ng1/ǫ2/parenrightbig\nBiG-SAM (Sabach and Shtern, 2017) SC, smooth C, composite Asym ptotic O(Lg1/ǫ)\nIR-IG (Amini and Yousefian, 2019) SC C3, Finite sum Asymptotic O/parenleftbig\n1/ǫ4/parenrightbig\nTseng’s method (Malitsky, 2017) C, composite C, composite Asympt otic O(1/ǫ)\nITALEX (Doron and Shtern, 2023) C, composite C, composite O/parenleftbig\n1/ǫ2/parenrightbig\nO(1/ǫ)\na-IRG (Kaushik and Yousefian, 2021) C, Lipschitz C, Lipschitz O/parenleftBig\nmax{1/ǫ4\nf,1/ǫ4\ng}/parenrightBig\nCG-BiO (Jiang et al., 2023) C, smooth C3, smooth O(max{Lf1/ǫf,Lg1/ǫg})\nOur method C, composite C, composite ˜O/parenleftBig\nmax{/radicalbig\nLf1/ǫf,/radicalbig\nLg1/ǫg}/parenrightBig\nwhich involves applying a projected gradient step to\nthe Tikhonov-type regularization function φ(x) at each\niteration. Amini and Yousefian (2019) extended the\nIR-PG method (Solodov, 2007) for the case where\nfis strongly convex but not necessarily differen-\ntiable. Their method achieves a convergence rate\nofO/parenleftbig\n1/k0.5−b/parenrightbig\nfor the lower-level problem, where\nb∈(0,0.5). Malitsky (2017) studied a version of\nTseng’s accelerated gradient method that obtains a\nconvergence rate of O(1/k) for the lower-level prob-\nlem. These prior works only establish the convergence\nrate for the lower-level problem, while the rate for the\nupper-level objective is missing.\nSeveral algorithms have recently provided conver-\ngence rates for both upper- and lower-level objec-\ntives. Doron and Shtern (2023) presented a scheme\ncalled Iterative Approximation and Level-set Expan-\nsion (ITALEX) to solve problem (1). Their algorithm\nachieves convergence rates of O(1/k) andO/parenleftBig\n1/√\nk/parenrightBig\nfor the lower- and upper-level problems, respectively.\nKaushik and Yousefian (2021) showed that an itera-\ntively regularized gradient (a-IRG) method can obtain\ncomplexity O/parenleftbig\n1/k0.5−b/parenrightbig\nfor the upper-level problem\nandO/parenleftbig\n1/kb/parenrightbig\nfor the lower-level, where b∈(0,0.5). To\nbalance the two rates, one can set b= 0.25, and the\ncomplexity bound is O/parenleftBig\nmax{1/ǫ4\nf,1/ǫ4\ng}/parenrightBig\nas stated\nin Table 1. Jiang et al. (2023) presented a condi-\ntional gradient-based bilevel optimization (CG-BiO)\nmethod, which requires O(max{Lf1/ǫf,Lg1/ǫg}) op-\neration complexity to find an ( ǫf,ǫg)-optimal solution.\nIn their setting, f≡f1andf2≡0.\n2 PRELIMINARIES\nIn this paper, we use an Accelerated Proximal Gra-\ndient (APG) method to approximately solve subprob-lems, which have the following form:\nmin\nx∈Rnϕ(x) :=ϕ1(x) +ϕ2(x), (7)\nwhere the function ϕ1:X→Ris convex and con-\ntinuously differentiable on an open set X∈Rn. The\ngradient ∇ϕ1isLϕ1-Lipschitz continuous. The func-\ntionϕ2:Rn→R∪ {∞} is proper, lower semicon-\ntinuous, convex, possibly non-smooth, and proximal-\nfriendly. A function his proximal-friendly means that\nthe proximal mapping of h, defined as\nproxh(y) = arg min\nx∈Rnh(x) +1\n2/ba∇⌈blx−y/ba∇⌈bl2,\nis easy to compute. In this paper, we take the clas-\nsical Fast Iterative Shrinkage Thresholding Algorithm\n(FISTA) proposed in Beck and Teboulle (2009) as an\nAPG algorithm (see more details in the appendix).\nNext, we give a definition of an APG oracle.\nDefinition 1. Givenϕ1,ϕ2,Lϕ1, andxϕ\n0∈Rn\nas defined above, an APG oracle, denoted by ˜xϕ=\nAPG (ϕ1,ϕ2,Lϕ1,xϕ\n0,ǫ), is a procedure that imple-\nments the classical FISTA scheme within O(/radicalbig\nLϕ1/ǫ)\niterations to obtain an ǫ-optimal solution to problem\n(7), denoted as ˜xϕ.\nIf the Lipschitz constant Lϕ1is unknown or compu-\ntationally infeasible, we can apply the FISTA scheme\nwith line search as an alternative to the APG algo-\nrithm (see the appendix).\nFor any fixed c, problem (6) can be rewritten in the\nfollowing form:\nmin\nx∈Rngc(x) :=g1(x) +hc(x), (8)\nwherehc(x) :=g2(x) +δc(x) andδc(x) is an indicator\nfunction with the definition that δc(x) = 0 iff(x)≤c;Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nδc(x) = +∞iff(x)> c. We provide some motivat-\ning examples in which the proximal mapping of hcis\neasy to compute. These examples can be found in the\nappendix.\n2.1 Assumptions\nAssumption 1. We adopt the following basic as-\nsumptions.\n(i) Functions f1andg1are convex and continuously\ndifferentiable. The gradients of the functions f1,\ng1, denoted by ∇f1and∇g1areLf1- andLg1-\nLipschitz continuous, respectively.\n(ii) Functions f2andg2are proper, lower semicontin-\nuous, convex, possibly non-smooth, and proximal-\nfriendly.\n(iii) The upper- and lower-level functions are lower\nbounded:\nf∗:= inf\nx∈Rnf(x)>−∞, g∗:= inf\nx∈Rng(x)>−∞.\nIn addition, we assume that gis not strongly con-\nvex, and the lower-level problem has multiple op-\ntimal solutions.\n(iv) For any fixed c, the function hc:=g2+δcin\nproblem (8) is proximal-friendly.\nRemark 1. We assume that the upper-level problem\ninvolves the minimization of a composite convex func-\ntion, which comprises the sum of a smooth convex\nfunction and a potentially non-smooth convex func-\ntion. This assumption is significantly weaker than the\nstrong-convexity assumption made in certain previous\nstudies (Beck and Sabach, 2014; Sabach and Shtern,\n2017; Amini and Yousefian, 2019). This assump-\ntion is also less restrictive than the requirement\nfor the upper-level objective function to be smooth\n(Jiang et al., 2023). We also assume that the lower-\nlevel problem is a composite convex minimization.\nThis assumption is less restrictive than the smoothness\nassumption made in Beck and Sabach (2014). Addi-\ntionally, this assumption is weaker than the require-\nment that the lower-level objective function is con-\nvex with convex compact constraints, as described in\nAmini and Yousefian (2019) and Jiang et al. (2023).\nRemark 2. We make Assumption 1(iv) to enable\nthe efficient application of the APG oracle for solving\nproblem (6). The function hcis the sum of two con-\nvex functions, and the study of proximal mapping for\nsuch sums can be found in the literature (Yu, 2013;\nPustelnik and Condat, 2017; Bauschke et al., 2018;\nAdly et al., 2019).g(c)\nf∗p∗\nǫgp∗p∗+ǫfg∗+ǫg\ng∗\nc\nFigure 1: Variation of ¯ g(c) over (f∗,+∞)\n3 MAIN ALGORITHM AND\nCONVERGENCE ANALYSIS\nBefore the presentation of the main algorithm, we de-\nscribe our idea in the next subsection.\n3.1 Bisection Method\nOur method is a bisection method whose heart is the\nfeasibility of System (5). Let f∗be the optimal value\nof the unconstrained upper-level problem\nmin\nx∈Rnf(x) :=f1(x) +f2(x). (9)\nFor a given c > f∗, we let ¯g(c) be the optimal value of\nproblem (6). Then ¯ g(c) is a univariate function of con\n(f∗,+∞). According to Theorem 5.3 in Rockafellar\n(1970), the function ¯ g(c) is convex. The function ¯ g(c)\nis also non-increasing as the feasible set of problem (6)\nbecomes larger when cincreases. Moreover, if f∗<\nc < p∗, then the inequality ¯ g(c)> g∗holds; otherwise\nc≥p∗and we have ¯ g(c) =g∗. Therefore, p∗is the\nleft-most root of the equation ¯ g(c) =g∗. Letp∗\nǫgbe\nthe optimal value of (4) with ǫ=ǫg, then it is a root\nof the equation ¯ g(c) =g∗+ǫg. We illustrate the graph\nof ¯g(c) in Figure 1.\nTo illustrate the basic idea of our method, we make an\nideal assumption that the exact values of g∗and ¯g(c)\ncan be obtained. We can observe that if ¯ g(c)> g∗\nholds, then System (5) is infeasible; otherwise, ¯ g(c) =\ng∗and System (5) is feasible. For a guess point c, if\nthe condition ¯ g(c)> g∗holds, then cis a lower bound\nofp∗; otherwise, cis an upper bound of p∗.\nHowever, the ideal assumption that the exact values\nofg∗and ¯g(c) can be obtained does not hold. Instead,\nwe solve problem (2) and problem (6) to approximate\nthem, respectively. For problem (2), we invoke the\nAPG oracle ˜xg=APG(g1,g2,Lg1,xg\n0,ǫg/2) to solve it.Jiulin Wang, Xu Shi, Rujun Jiang†\nThen we can find an approximate solution ˜xgthat sat-\nisfies\n0≤˜g−g∗≤ǫg/2, (10)\nwhere ˜g=g(˜xg).\nUnder Assumption 1(iv), the proximal mapping of hc\nis easy to compute. Then we can apply an APG oracle\nto problem (6). For a given c, we invoke the APG\noracle ˜xc=APG(g1,hc,Lg1,xc\n0,ǫg/2) to solve problem\n(6). Then we can obtain an approximate solution ˜xc\nthat satisfies\n0≤g(˜xc)−¯g(c)≤ǫg/2. (11)\nSince the condition ¯ g(c)> g∗cannot be verified di-\nrectly, we replace it with the following verifiable con-\ndition\ng(˜xc)>˜g+ǫg/2. (12)\nIf Condition (12) holds, then we have\n¯g(c)(11)\n≥g(˜xc)−ǫg/2(12)\n>˜g(10)\n≥g∗,\ni.e., the inequality ¯ g(c)> g∗holds. Thus, System (5)\nis infeasible, and cis a lower bound of p∗.\nIf Condition (12) does not hold, we have g(˜xc)≤˜g+\nǫg/2 and thus\n¯g(c)(11)\n≤g(˜xc)≤˜g+ǫg/2(10)\n≤g∗+ǫg. (13)\nWe cannot infer that ¯ g(c) =g∗holds in this case. Also,\nwe cannot claim that System (5) is feasible. Thus c\nmight not be an upper bound of p∗. By (13), ˜xcis a\nfeasible solution of (4) with ǫ=ǫg. Therefore, f(˜xc)\nis an upper bound on p∗\nǫg, where p∗\nǫgrepresents the\noptimal value of (4) with ǫ=ǫgas previously defined.\nIn addition, as the inequality g(˜xc)≤g∗+ǫgholds,\n˜xcis anǫg-optimal solution of the lower-level problem\n(2).\nSummarizing the above analysis, we present the fol-\nlowing lemma.\nLemma 1. For any fixed c, if Condition (12) is sat-\nisfied, then System (5) is infeasible, and cis a lower\nbound of p∗. If Condition (12) is not satisfied, then we\ncan obtain ˜xcas anǫg-optimal solution of the lower-\nlevel problem and f(˜xc)is an upper bound of p∗\nǫg.\nNext, we show how to obtain the initial interval [ l,u]\nfor the bisection procedure. Here lis a lower bound\nofp∗anduis an upper bound of p∗\nǫg, but possibly\nnot an upper bound of p∗. We invoke the APG or-\nacle ˜xf=APG(f1,f2,Lf1,xf\n0,ǫf/2) to solve problem\n(9). Then we can obtain an approximate solution ˜xf\nthat satisfies\n0≤f(˜xf)−f∗≤ǫf/2. (14)We have f(˜xf)−ǫf/2≤f∗≤p∗. We use l=f(˜xf)−\nǫf/2 as an initial lower bound of p∗. Moreover, since\n˜xgsatisfies (10), we have ˜xgis a feasible solution of\n(4) with ǫ=ǫg. Then we use u=f(˜xg) as an initial\nupper bound of p∗\nǫg.\nNow we can do a binary search over [ l,u]. For a given\nc=l+u\n2, we check whether Condition (12) is satisfied.\nIf Condition (12) is satisfied, we let l=cbe a new\nlower bound of p∗. If Condition (12) is not satisfied,\nwe letu=f(˜xc), which is less than or equal to c, be\na new upper bound of p∗\nǫg. We summarise our method\nin Algorithm 1.\nAlgorithm 1 Bisection-based method for simple\nBilevel Optimization (Bisec-BiO)\nInput:f1,f2,g1,g2,Lf1,Lg1,ǫf,ǫg\nOutput: An (ǫf,ǫg)-optimal solution ˆx\n1:Invoke APG oracles to obtain initial bounds land\nu, and the approximate solutions ˜xfand˜xg.\n2:whileu−l > ǫfdo\n3: letc=l+u\n2and invoke an APG oracle to obtain\nan approximate solution ˜xc.\n4:ifCondition (12) is satisfied then\n5: letl=c,\n6:else\n7: letu=f(˜xc). ⊲ f(˜xc)≤c\n8:end if\n9:end while\n10:Letc=uand return the corresponding ˜xcasˆx.\n3.2 Convergence Analysis under Assumption\n1\nIn this subsection, we give the complexity result of our\nmethod.\nTheorem 1. Suppose Assumption 1 holds. Algorithm\n1 produces an ( ǫf,ǫg)-optimal solution for problem (1)\nafter at most Tevaluations of the function values f1,\nf2,g1andg2, the gradients ∇f1and∇g1, and the\ncalls of proximal mapping with respect to function hc,\nwhere\nT=˜O/parenleftBigg\nmax/braceleftBigg/radicalBigg\nLf1\nǫf,/radicalBigg\nLg1\nǫg/bracerightBigg/parenrightBigg\n,\nand ˜Osuppresses a logarithmic term.\nTheorem 1 demonstrates that our complexity achieves\nthe near-optimal rate for both upper- and lower-\nlevel objectives and matches the optimal rate of\nfirst-order methods for unconstrained smooth or\ncomposite convex optimization when disregard-\ning the logarithmic term (Nemirovsky and Yudin,\n1983; Woodworth and Srebro, 2016). Compar-\ning with existing works (Beck and Sabach, 2014;Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nSabach and Shtern, 2017; Amini and Yousefian,\n2019; Malitsky, 2017; Doron and Shtern, 2023;\nKaushik and Yousefian, 2021; Jiang et al., 2023),\nour result provides the best-known non-asymptotic\nbounds for both upper- and lower-level objectives.\nSpecifically, our complexity bound improves upon the\nresult by Jiang et al. (2023) by orders of magnitude.\nThey considered a different setup where the upper-\nlevel function is smooth, and the lower-level objective\nis a smooth convex function with convex compact\nconstraints.\nRemark 3. Assumption 1(iv) essentially implies that\nprojecting onto the sublevel set of fis straightforward,\nas observed in the motivating examples. Consequently,\nwe can employ a bisection method to locate f∗by veri-\nfying the solvability of the projection onto the sublevel\nset off. This approach eliminates the dependency on\nLf1from the overall complexity.\n4 CONVERGENCE ANALYSIS\nUNDER OTHER ASSUMPTIONS\nIn this section, we provide an analysis of the metric\nf(ˆx)−p∗. Our method guarantees that f(ˆx) serves as\nan upper bound for p∗\nǫg; however, it may not necessar-\nily be an upper bound for p∗, which could result in a\nnegative value for f(ˆx)−p∗. In fact, as ˆxmay not be\nan exact optimal solution to the lower-level problem, it\nmay not be a feasible point for problem (1). Hence, the\nvalue off(ˆx)−p∗may be negative. In essence, we cur-\nrently offer an upper bound metric for f(ˆx)−p∗while\nlacking a corresponding lower bound metric. Although\nthis may appear a bit unconventional, it is noteworthy\nthat Kaushik and Yousefian (2021) and Jiang et al.\n(2023) similarly employed f(ˆx)−p∗as their perfor-\nmance metric.\nIn this section, we introduce additional assumptions\nto establish a lower bound for f(ˆx)−p∗, allowing us\nto provide a metric expressed as |f(ˆx)−p∗|.\n4.1 Convergence Analysis under H¨ olderian\nError Bound Assumption\nIn this subsection, we make some additional assump-\ntions to provide a lower bound for f(ˆx)−p∗.\nAssumption 2. (i) The domain of g2is bounded.\n(ii) The function f2islf2-Lipschitz continuous on\ndom(g2), i.e.|f2(x)−f2(y)| ≤lf2/ba∇⌈blx−y/ba∇⌈bl.\nIn the following, we give some remarks on Assumption\n2. Assumption 2(i) is fulfilled by many examples, see\nSection 5 in Amini and Yousefian (2019) and Section\n2 in Jiang et al. (2023). In particular, in Section 5.2,\nwe have C={x∈Rn:/ba∇⌈blx/ba∇⌈bl1≤λ}, then dom( g2)is bounded. As f1is continuous, ∇f1is bounded on\ndom(g2). Hence Assumption 2(i) implies\nBf1= max\nx∈dom(g2)/ba∇⌈bl∇f1(x)/ba∇⌈bl.\nThen by mean-value theorem, the function f1isBf1-\nLipschitz continuous on dom( g2).\nAssumption 2(ii) is mild. For example, if f2(x) =\n/ba∇⌈blx/ba∇⌈bl1, thenlf2=√n, wherenis the dimension of x.\nLetBf=Bf1+lf2. Under Assumption 2(i) and (ii), it\nfollows that fisBf-Lipschitz continuous on dom( g2),\nnamely,\n|f(x1)−f(x2)| ≤Bf/ba∇⌈blx1−x2/ba∇⌈bl ∀x1,x2∈dom(g2)\n(15)\nAssumption 3. The function gsatisfies the\nH¨ olderian error bound for some α > 0andr≥1on\nthe lower-level optimal solution set X∗\ng, i.e,\nα\nrdist(x,X∗\ng)r≤g(x)−g∗,∀x∈dom(g2).(16)\nIt is important to highlight that the error bound condi-\ntion described in (16) has received considerable atten-\ntion in the literature, as evidenced by studies such as\nPang (1997); Bolte et al. (2017); Zhou and So (2017),\nand the associated references therein. Bolte et al.\n(2017) demonstrated that this error bound condition\ntypically holds when the function gexhibits prop-\nerties of being semi-algebraic and continuous, while\nalso ensuring that dom( g2) remains bounded. They\nalso showed that there is an equivalence between\nH¨ olderian error bound condition and the Kurdyka-\n/suppress Lojasiewicz inequality. There are two notable spe-\ncial cases: (i) r= 1,X∗\ngis a set of weak sharp min-\nima ofg(Burke and Deng, 2005); (ii) r= 2, Condi-\ntion (16) is known as the quadratic growth condition\n(Drusvyatskiy and Lewis., 2018). Based on Corol-\nlary 5.1 in Li and Pong (2018) and Theorem 5 in\nBolte et al. (2017), it is evident that in the motivating\nexamples provided in the appendix, the lower-level ob-\njectives satisfy the H¨ olderian error bound assumption\nwithr= 2.\nGiven Assumptions 2 and 3, we can derive the fol-\nlowing lower bound for f(ˆx)−p∗. Importantly, this\nresult is an inherent characteristic of problem (1) and\nremains unaffected by the choice of algorithm.\nProposition 1. Under Assumptions 2 and 3, let ˆxbe\nanǫg-optimal solution of the lower-level problem, i.e.,\nˆxsatisfies g(ˆx)−g∗≤ǫg. Then it holds that:\nf(ˆx)−p∗≥ −Bf/parenleftBigrǫg\nα/parenrightBig1\nr.\nThis result is similar to Proposition 1 in Jiang et al.\n(2023), which also gives a lower bound for f(ˆx)−p∗Jiulin Wang, Xu Shi, Rujun Jiang†\nwith similar assumptions. Combining Theorem 1 with\nProposition 1, we have the following result.\nCorollary 1. Under Assumptions 1-3, let ˆxbe the\noutput of Algorithm 1 and set ǫg=α\nγ/parenleftBig\nǫf\nBf/parenrightBigr\n. Then\nwith an operation complexity of ˜O/parenleftBig\n1//radicalbigǫr\nf/parenrightBig\n, we can\nfind an ˆxsuch that\n|f(ˆx)−p∗| ≤ǫf, g(ˆx)−g∗≤ǫg.\nCorollary 1 illustrates that, under Assumptions 1-3, we\ncan find an iteration point to be ǫf-close to optimality\nwith an operation complexity ˜O/parenleftBig\n1//radicalbigǫr\nf/parenrightBig\n.\n4.2 Convergence Analysis under an\nAdditional Assumption on the Optimal\nValue of (4)\nIn this subsection, we present an alternative approach\nto establishing a lower bound for f(ˆx)−p∗. Denote the\noptimal value of problem (4) as v(ǫ). Here, v(ǫ) is a\nfunction of ǫdefined on the interval [0 ,+∞). Accord-\ning to the definitions of p∗andp∗\nǫg, we can establish\nthatv(0) =p∗andv(ǫg) =p∗\nǫg. As illustrated in Fig-\nure 1, we can define an angle function θ(ǫ) on [0,π/2)\nsuch that\nǫ= (v(0)−v(ǫ))·tanθ(ǫ).\nSimilar to the monotonicity of ¯ g(c), functions v(ǫ) and\nθ(ǫ) are also monotonically non-decreasing. If ǫ= 0,\nthenθ(ǫ) = 0; otherwise ǫ > 0 andθ(ǫ)>0. It is\nhard to compute the exact value of tan θ(ǫ). Instead,\nfor any fixed ǫ > 0, we assume that there exists a\nparameter Lǫ>0 such that Lǫtanθ(ǫ)≥1 holds. In\nother words, we make the following assumption.\nAssumption 4. For any fixed ǫ >0, we assume that\nthere exists a parameter Lǫ>0such that v(0)−v(ǫ)≤\nLǫǫholds.\nIn the following, we consider a simple two-dimensional\ntoy example in which Assumption 4 holds.\nExample 1 (A toy example) .\nmin\nx∈R2f(x) =|x1|+|x2|\ns.t.x∈arg min\nz∈R2g(z) = (z1−1)2.(17)\nIn this example, the optimal solution and the optimal\nvalue of (17) are ( x∗\n1,x∗\n2) = (1,0) andv(0) =p∗=\n1, respectively. Next, we consider problem (4) with\n0< ǫ < 1. It can be easily obtained that the optimal\nsolution and the optimal value of (4) are ( x∗\n1,x∗\n2) =\n(1−√ǫ,0) andv(ǫ) = 1−√ǫ, respectively. Then we\nhave\nv(0)−v(ǫ) =√ǫ.By setting Lǫ≥1/√ǫ, Assumption 4 holds.\nUnder Assumption 4, we can derive a lower bound for\nf(ˆx)−p∗that is independent of the choice of algorithm.\nProposition 2. Under Assumption 4, let ˆxbe anǫg-\noptimal solution of the lower-level problem, i.e., ˆxsat-\nisfiesg(ˆx)−g∗≤ǫg. Then it holds that:\nf(ˆx)−p∗≥ −Lǫgǫg.\nBy combining Theorem 1 with Proposition 2, we have\nthe following result.\nCorollary 2. Under Assumption 1 and Assump-\ntion 4, let ˆxbe the output of Algorithm 1 and set\nǫf=Lǫgǫg. Then with an operation complexity of\n˜O/parenleftbig\nmax{1//radicalbigLǫgǫg,1/√ǫg}/parenrightbig\n, we can find an ˆxsuch\nthat\n|f(ˆx)−p∗| ≤ǫf, g(ˆx)−g∗≤ǫg.\nCorollary 2 demonstrates that with Assumptions 1 and\n4 in place, we can obtain an iteration point to be\nLǫgǫg-close to optimality with an operation complexity\n˜O/parenleftbig\nmax{1//radicalbigLǫgǫg,1/√ǫg}/parenrightbig\n.\n5 NUMERICAL EXPERIMENTS\nIn this section, we apply our method (Bisec-BiO)\nto two bilevel optimization problems from the mo-\ntivating examples in the appendix and compare its\nperformance with other existing methods in the lit-\nerature (Beck and Sabach, 2014; Sabach and Shtern,\n2017; Kaushik and Yousefian, 2021; Gong and Liu,\n2021; Jiang et al., 2023). For all experiments, we set\nǫf= 10−5andǫg= 10−6, and we adopt the Greedy\nFISTA algorithm proposed in Liang et al. (2022) as\nthe APG method. The Greedy FISTA algorithm can\nachieve superior practical performance compared to\nthe classical FISTA.\n5.1 Minimum Norm Solution Problem\n(MNP)\nWe first consider the linear regression problem on the\nYearPredictionMSD dataset1, which contains informa-\ntion on 515 ,345 songs, with a release year from 1992\nto 2011. For each song, the dataset contains its release\nyear and an additional 90 attributes. We use a sam-\nple of 1,000 songs randomly selected from the dataset\nwith uniform i.i.d distribution, and denote the feature\nmatrix and the release years by Aandb, respectively.\nAdditionally, in line with Merchav and Sabach (2023),\nwe adopt a min-max scaling technique and add an in-\nterceptor and 90 co-linear attributes to A.\n1https://archive.ics.uci.edu/dataset/203/\nyearpredictionmsdNear-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nFor the lower-level, we let g1(x) =1\n2/ba∇⌈blAx−b/ba∇⌈bl2, which\nexhibits a Lg1-Lipschitz continuous gradient, where\nLg1=λmax(ATA). Simultaneously, we set g2(x)≡0.\nFor the upper-level, we let f1(x) =1\n2/ba∇⌈blx/ba∇⌈bl2andf2(x)≡\n0. This configuration corresponds to finding the min-\nimum norm solution. Now, our goal is to solve the\nfollowing bilevel problem:\nmin\nx∈Rn1\n2/ba∇⌈blx/ba∇⌈bl2\ns.t.x∈arg min\nz∈Rn1\n2/ba∇⌈blAz−b/ba∇⌈bl2.(18)\nWe compare the performance of our Bisec-BiO with\nseveral existing methods to solve this problem, namely\naveraging iteratively regularized gradient method (a-\nIRG) (Kaushik and Yousefian, 2021), bilevel gradi-\nent SAM method (BiG-SAM) (Sabach and Shtern,\n2017), minimal norm gradient method (MNG)\n(Beck and Sabach, 2014), and dynamic barrier gradi-\nent descent method (DBGD) (Gong and Liu, 2021).\nIn this experiment, the feasible set of the lower-level\nproblem is unbounded. Therefore, we cannot directly\napply CG-BiO (Jiang et al., 2023). We use MATLAB\nfunction lsqminnorm to solve problem (18) and ob-\ntain the optimal values g∗andp∗for benchmarking\npurposes.\n0 0.5 1 1.5 210-1010-5100105\n0 0.5 1 1.5 210-610-410-2100102\nFigure 2: The performance of Bisec-BiO compared\nwith other methods in MNP.\nAs Figure 2 demonstrates, our Bisec-BiO converges\nmuch faster than the other baseline methods for both\nthe lower- and upper-level objectives, confirming our\ncomplexity results (see Table 1). Our method is shown\nin the figure as an oscillating curve, which depends on\nthe feasibility of System (5) at each iteration point and\nthe update mode of our method. Moreover, the left\nand right subfigures in Figure 2 show that the output\nof our algorithm meets the ( ǫf,ǫg)-optimal solution\ncriterion, as proven in Theorem 1.5.2 Logistic Regression Problem (LRP)\nWe address the logistic regression binary classification\nproblem using the ’a1a’ dataset from LIBSVM2, which\ncontains m= 30,956 instances, each with n= 123 fea-\ntures. We randomly select a sample of 1 ,000 instances\nand denote the feature matrix and labels as Aandb,\nrespectively.\nFor the lower-level, we let g1(x) =1\nm/summationtextm\ni=1log(1 +\nexp(−a⊤\nixbi)), which has a Lg1-Lipschitz continuous\ngradient with Lg1=1\n4mλmax(ATA). Here, airepre-\nsents an instance and bi∈ {− 1,1}is the corresponding\nlabel. Additionally, we set g2(x) =IC(x), whereIC(x)\nis the indicator function of C={x∈Rn:/ba∇⌈blx/ba∇⌈bl1≤λ}\nwithλ= 10. For the upper-level, we let f1(x) =1\n2/ba∇⌈blx/ba∇⌈bl2\nandf2(x)≡0, as well. We need to solve the following\nproblem:\nmin\nx∈Rn1\n2/ba∇⌈blx/ba∇⌈bl2\ns.t.x∈arg min\nz∈Rn1\nm/summationtextm\ni=1log(1 + exp( −a⊤\nizbi))\n+IC(z).\n(19)\nIn this experiment, we compare the performance of our\nBisec-BiO with the methods proposed in Section 5.1\nand the CG-based bilevel optimization method (CG-\nBiO) (Jiang et al., 2023). For benchmarking purposes,\nwe utilize the Greedy FISTA algorithm (Liang et al.,\n2022) and MATLAB function fmincon to solve prob-\nlem (18) and obtain the optimal values g∗andp∗,\nrespectively. We employ the method proposed in\nLiu et al. (2020) to compute the proximal operator of\nhc:=g2+δcin problem (8). Their method is demon-\nstrated to have a worst-case complexity of O(n2) and\nan observed complexity of O(n) in practice.\n0 0.2 0.4 0.6 0.810-1510-1010-5100\n0 0.2 0.4 0.6 0.810-310-210-1100101102\nFigure 3: The performance of Bisec-BiO compared\nwith other methods in LRP.\nFigure 3 also shows that Bisec-BiO converges much\nfaster than the other baseline methods for both the\nlower- and upper-level objectives. We have similar ob-\nservations as in Figure 2.\n2https://www.csie.ntu.edu.tw/ ~cjlin/\nlibsvmtools/datasets/binary/a1a.tJiulin Wang, Xu Shi, Rujun Jiang†\n6 CONCLUSION\nIn this paper, we address the problem of mini-\nmizing a composite convex function at the upper-\nlevel within the optimal solution set of a compos-\nite convex lower-level problem. We introduce a bi-\nsection algorithm designed to discover an ǫf-optimal\nsolution for the upper-level objective and an ǫg-\noptimal solution for the lower-level objective. Our\nmethod attains a near-optimal convergence rate of\n˜O/parenleftBig\nmax{/radicalbig\nLf1/ǫf,/radicalbig\nLg1/ǫg}/parenrightBig\nfor both upper- and\nlower-level objectives. Notably, this near-optimal rate\naligns with the optimal rate observed in unconstrained\nsmooth or composite optimization, neglecting the log-\narithmic term. We enhance convergence guarantees\nby imposing a H¨ olderian error bound assumption on\nthe lower-level problem. Numerical experiments con-\nvincingly illustrate the substantial improvement our\nmethod offers over the state-of-the-art. In future re-\nsearch, we will explore the possibility of eliminating\nthe logarithmic term from our complexity result.\nAcknowledgements\nRujun Jiang is partly supported by the National Key\nR&D Program of China under grant 2023YFA1009300,\nNational Natural Science Foundation of China under\ngrants 12171100 and 72394364, and Natural Science\nFoundation of Shanghai 22ZR1405100. Jiulin Wang is\nsupported by China Postdoctoral Science Foundation\nunder grants BX20220085 and 2022M710798.\nReferences\nSamir Adly, Lo¨ ıc Bourdin, and Fabien Caubet. On a\ndecomposition formula for the proximal operator of\nthe sum of two convex functions. Journal of Convex\nAnalysis , 26(2):699–718, 2019.\nMostafa Amini and Farzad Yousefian. An iterative reg-\nularized incremental projected subgradient method\nfor a class of bilevel optimization problems. In 2019\nAmerican Control Conference (ACC) , pages 4069–\n4074. IEEE, 2019.\nHeinz H. Bauschke, Minh N. Bui, and Xianfu Wang.\nProjecting onto the intersection of a cone and a\nsphere.SIAM Journal on Optimization , 28(3):2158–\n2188, 2018.\nAmir Beck and Shoham Sabach. A first order method\nfor finding minimal norm-like solutions of convex op-\ntimization problems. Mathematical Programming ,\n147(1-2):25–46, 2014.\nAmir Beck and Marc Teboulle. A fast iterative\nshrinkage-thresholding algorithm for linear inverse\nproblems. SIAM Journal on Imaging Sciences , 2\n(1):183–202, 2009.Luca Bertinetto, Joao F Henriques, Philip HS Torr,\nand Andrea Vedaldi. Meta-learning with dif-\nferentiable closed-form solvers. arXiv preprint\narXiv:1805.08136 , 2018.\nNicholas Bishop, Long Tran-Thanh, and Enrico Gerd-\ning. Optimal learning from verified training data.\nAdvances in Neural Information Processing Sys-\ntems, 33:9520–9529, 2020.\nJ´ erme Bolte, Trong Phong Nguyen, Juan Peypou-\nquet, and Bruce W. Suter. From error bounds to\nthe complexity of first-order descent methods for\nconvex functions. Mathematical Programming , 165:\n471–507, 2017.\nJ Fr´ ed´ eric Bonnans and Alexander Shapiro. Pertur-\nbation analysis of optimization problems . Springer\nScience & Business Media, 2013.\nJames V. Burke and Sien Deng. Weak sharp minima\nrevisited, part ii: application to linear regularity and\nerror bounds. Mathematical Programming , 104(2-3):\n235–261, 2005.\nChristine De Mol, Ernesto De Vito, and Lorenzo\nRosasco. Elastic-net regularization in learning the-\nory.Journal of Complexity , 25(2):201–230, 2009.\nStephan Dempe, Nguyen Dinh, Joydeep Dutta, and\nTanushree Pandit. Simple bilevel programming and\nextensions. Mathematical Programming , 188:227–\n253, 2021.\nStephen Dempe, Nguyen Dinh, and Joydeep Dutta.\nOptimality conditions for a simple convex bilevel\nprogramming problem. Variational Analysis and\nGeneralized Differentiation in Optimization and\nControl: In Honor of Boris S. Mordukhovich , pages\n149–161, 2010.\nLior Doron and Shimrit Shtern. Methodology and\nfirst-order algorithms for solving nonsmooth and\nnon-strongly convex bilevel optimization problems.\nMathematical Programming , 201:521–558, 2023.\nDmitriy Drusvyatskiy and Adrian S. Lewis. Error\nbounds, quadratic growth, and linear convergence\nof proximal methods. Mathematics of operations re-\nsearch , 43(3):919–948, 2018.\nJohn Duchi. Elastic net projections. Technical report,\nStanford University, 2009.\nJoydeep Dutta and Tanushree Pandit. Algorithms for\nsimple bilevel programming. Bilevel Optimization:\nAdvances and Next Challenges , pages 253–291, 2020.\nLuca Franceschi, Paolo Frasconi, Saverio Salzo, Ric-\ncardo Grazzi, and Massimiliano Pontil. Bilevel\nprogramming for hyperparameter optimization and\nmeta-learning. In International conference on ma-\nchine learning , pages 1568–1577. PMLR, 2018.Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nMichael P Friedlander and Paul Tseng. Exact reg-\nularization of convex programs. SIAM Journal on\nOptimization , 18(4):1326–1350, 2008.\nChengyue Gong and Xingchao Liu. Bi-objective trade-\noff with dynamic barrier gradient descent. NeurIPS\n2021, 2021.\nPinghua Gong, Kun Gai, and Changshui Zhang. Effi-\ncient euclidean projections via piecewise root finding\nand its application in gradient projection. Neuro-\ncomputing , 74(17):2754–2766, 2011.\nMingyi Hong, Hoi-To Wai, Zhaoran Wang, and Zhuo-\nran Yang. A two-timescale framework for bilevel op-\ntimization: Complexity analysis and application to\nactor-critic. arXiv preprint arXiv:2007.05170 , 2020.\nRuichen Jiang, Nazanin Abolfazli, Aryan Mokhtari,\nand Erfan Yazdandoost Hamedani. Conditional\ngradient-based method for bilevel optimization with\nconvex lower-level problem. In Proceedings of the\n26th International Conference on Artificial Intel-\nligence and Statistics (AISTATS) 2023, Valencia,\nSpain. PMLR: Volume 206. , 2023.\nHarshal D. Kaushik and Farzad Yousefian. A method\nwith convergence rates for optimization problems\nwith variational inequality constraints. SIAM Jour-\nnal on Optimization , 31(3):2171–2198, 2021.\nPuya Latafat, Andreas Themelis, Silvia Villa, and\nPanagiotis Patrinos. Adabim: An adaptive proximal\ngradient method for structured convex bilevel opti-\nmization. arXiv preprint arXiv:2305.03559 , 2023.\nGuoyin Li and Ting Kei Pong. Calculus of the expo-\nnent of kurdyka–/suppress lojasiewicz inequality and its appli-\ncations to linear convergence of first-order methods.\nFoundations of computational mathematics , 18(5):\n1199–1232, 2018.\nJingwei Liang, Tao Luo, and Carola-Bibiane\nSchonlieb. Improving “fast iterative shrinkage-\nthresholding algorithm”: Faster, smarter, and\ngreedier. SIAM Journal on Scientific Computing ,\n44(3):A1069–A1091, 2022.\nHongying Liu, Hao Wang, and Mengmeng Song. Pro-\njections onto the intersection of a one-norm ball\nor sphere and a two-norm ball or sphere. Journal\nof Optimization Theory and Applications , 187:520–\n534, 2020.\nJulien Mairal, Francis Bach, Jean Ponce, and\nGuillermo Sapiro. Online learning for matrix fac-\ntorization and sparse coding. Journal of Machine\nLearning Research , 11(1), 2010.\nYura Malitsky. Chambolle-pock and tseng’s methods:\nrelationship and extension to the bilevel optimiza-\ntion.arXiv preprint arXiv:1706.02602 , 2017.Roey Merchav and Shoham Sabach. Convex bi-level\noptimization problems with non-smooth outer ob-\njective function. arXiv preprint arXiv:2307.08245 ,\n2023.\nArkadij Semenoviˇ c Nemirovsky and David Borisovich\nYudin.Problem complexity and method efficiency in\noptimization . Wiley, 1983.\nYurii Nesterov. A method of solving a convex pro-\ngramming problem with convergence rate o(1/k2).\nDoklady Akademii Nauk , 269(3):543–547, 1983.\nJong Shi Pang. Error bounds in mathematical pro-\ngramming. Mathematical Programming , 79(1-3):\n299–332, 1997.\nNelly Pustelnik and Laurent Condat. Proximity oper-\nator of a sum of functions; application to depth map\nestimation. IEEE Signal Processing Letters , 24(12):\n1827–1831, 2017.\nAravind Rajeswaran, Chelsea Finn, Sham M Kakade,\nand Sergey Levine. Meta-learning with implicit gra-\ndients.Advances in neural information processing\nsystems , 32, 2019.\nR. Tyrrell Rockafellar. Convex Analysis . Princeton\nLandmarks in Mathematics and Physics. Princeton\nUniversity Press, 1970.\nEmanuele Rodola, Andrea Torsello, Tatsuya Harada,\nYasuo Kuniyoshi, and Daniel Cremers. Elastic net\nconstraints for shape matching. In Proceedings of the\nIEEE international conference on computer vision ,\npages 1169–1176, 2013.\nShoham Sabach and Shimrit Shtern. A first order\nmethod for solving convex bilevel optimization prob-\nlems.SIAM Journal on Optimization , 27(2):640–\n660, 2017.\nAmirreza Shaban, Ching-An Cheng, Nathan Hatch,\nand Byron Boots. Truncated back-propagation for\nbilevel optimization. In The 22nd International\nConference on Artificial Intelligence and Statistics ,\npages 1723–1732. PMLR, 2019.\nYekini Shehu, Phan Tu Vuong, and Alain Zemkoho.\nAn inertial extrapolation method for convex sim-\nple bilevel optimization. Optimization Methods and\nSoftware , 36(1):1–19, 2021.\nMikhail Solodov. An explicit descent method for\nbilevel convex optimization. Journal of Convex\nAnalysis , 14(2):227–237, 2007.\nAndre˘ ı Nikolaevich Tikhonov and V. I. A. K. Arsenin.\nSolutions of ill-posed problems . Wiley, 1977.\nJiali Wang, He Chen, Rujun Jiang, Xudong Li, and\nZihao Li. Fast algorithms for stackelberg prediction\ngame with least squares loss. In International Con-\nference on Machine Learning , pages 10708–10716.\nPMLR, 2021.Jiulin Wang, Xu Shi, Rujun Jiang†\nJiali Wang, Wen Huang, Rujun Jiang, Xudong Li, and\nAlex L Wang. Solving stackelberg prediction game\nwith least squares loss via spherically constrained\nleast squares reformulation. In International Con-\nference on Machine Learning , pages 22665–22679.\nPMLR, 2022.\nBlake E Woodworth and Nati Srebro. Tight complex-\nity bounds for optimizing composite objectives. Ad-\nvances in neural information processing systems , 29,\n2016.\nYao-Liang Yu. On decomposing the proximal map.\nAdvances in neural information processing systems ,\n26, 2013.\nZirui Zhou and Anthony Man-Cho So. A unified ap-\nproach to error bounds for structured convex op-\ntimization problems. Mathematical Programming ,\n165:689–728, 2017.\nHui Zou and Trevor Hastie. Regularization and\nvariable selection via the elastic net. Journal of\nthe Royal Statistical Society Series B: Statistical\nMethodology , 67(2):301–320, 2005.Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nA THE FAST ITERATIVE SHRINKAGE THRESHOLDING ALGORITHM\n(FISTA) AND CONVERGENCE RESULTS\nWe adopt the Fast Iterative Shrinkage Thresholding Algorithm (FIS TA) proposed in Beck and Teboulle (2009)\nas the APG method (see Definition 1) to solve problem (7). For any L >0, define pL(y) as the unique minimizer\nof the following quadratic approximation of ϕ(x) at a given point y:\npL(y) := arg min\nx∈RnQL(x,y) :=/braceleftbigg\nϕ1(y) +/an}b∇ack⌉tl⌉{tx−y,∇ϕ1(y)/an}b∇ack⌉t∇i}ht+L\n2/ba∇⌈blx−y/ba∇⌈bl2+ϕ2(x)/bracerightbigg\n,\nwhereLactually plays the role of a step-size.\nThe classical FISTA scheme with a constant step-size for solving pr oblem (7) is presented in Algorithm 2.\nAlgorithm 2 FISTA with constant step-size\nInput:Lϕ1,t1= 1,y1=x0∈Rn\n1:fork= 1,···do\n2:xk=pLϕ1(yk),\n3:tk+1=1+√\n1+4t2\nk\n2,\n4:yk+1=xk+tk−1\ntk+1(xk−xk−1),\n5:k=k+ 1.\n6:end for\nAs discussed in Section 2, a potential limitation of this classical schem e is its dependence on the knowledge or com-\nputation of the Lipschitz constant Lϕ1, which may not be practical. To overcome this issue, Beck and Tebou lle\n(2009) further proposed a variant of FISTA that incorporates a backtracking line search, we present it in Algo-\nrithm 3.\nAlgorithm 3 FISTA with backtracking line search\nInput:L0>0,η >1,t1= 1,y1=x0∈Rn\n1:fork= 1,···do\n2: Find the smallest nonnegative integer value iksuch that with ¯L=ηikLk−1,\nϕ(p¯L(yk))≤Q¯L(p¯L(yk),yk).\n3:Lk=ηikLk−1,\n4:xk=pLk(yk),\n5:tk+1=1+√\n1+4t2\nk\n2,\n6:yk+1=xk+tk−1\ntk+1(xk−xk−1),\n7:k=k+ 1.\n8:end for\nThe next lemma shows the convergence results of the objective fu nction under the FISTA scheme (Algorithm 2\nor 3).\nLemma 2 ((Beck and Teboulle, 2009), Theorem 4.4.) .DenoteX∗\nϕas the optimal solution set of problem (7)\nandx∗\nϕ∈X∗\nϕbe any optimal solution. Let xϕ\n0∈Rnbe an initial point. Let {xk}be the sequence generated by\nthe FISTA scheme (Algorithm 2 or 3). Then for any k≥1, we have\nϕ(xk)−ϕ(x∗\nϕ)≤2αLϕ1\n(k+ 1)2/ba∇⌈blxϕ\n0−x∗\nϕ/ba∇⌈bl2,∀x∗\nϕ∈X∗\nϕ.\nHere,α= 1 for the classical FISTA scheme (Algorithm 2), and α=ηfor the FISTA scheme with a backtracking\nline search (Algorithm 3), where ηis the backtracking parameter. This lemma demonstrates that an ǫ-optimal\nsolution of problem (7) can be obtained by Algorithm 2 or 3 within at mos tO(/radicalbig\nLϕ1/ǫ) iterations.Jiulin Wang, Xu Shi, Rujun Jiang†\nB MOTIVATING EXAMPLES\nMany applications in machine learning and signal processing involve reg ularized problems, where the upper-level\nobjectives represent the regularization terms, and the lower-lev el objectives consist of the loss functions and the\nadditional constraint terms. We present some motivating examples below.\nExample 2 (Minimum Norm Solution of Least Squares Regression Problem (MNP)) .Linear inverse problems\naim to reconstruct a vector x∈Rnfrom a set of measurements b∈Rmthat satisfy the following relation:\nb=Ax+ρε, whereA:Rn→Rmis a given linear mapping, ε∈Rmdenotes an unknown noise vector, and ρ >0\ndenotes its magnitude. Linear inverse problems can be solve d using various optimization techniques, and we focus\non the bilevel formulation (Beck and Sabach, 2014; Sabach an d Shtern, 2017; Dempe et al., 2021; Latafat et al.,\n2023; Merchav and Sabach, 2023).\nThe lower-level objective function in this formulation is g iven by\ng(x) =1\n2/ba∇⌈blAx−b/ba∇⌈bl2+IC(x),\nwhere the set Cis a closed convex set that can be chosen as C=Rn,C={x∈Rn:x≥0}orC={x∈Rn:\n/ba∇⌈blx/ba∇⌈bl1≤λ}for some λ >0, andIC(x)is the indicator function of the set C, defined as IC(x) = 0ifx∈Cand\nIC(x) = +∞ifx/∈C.\nThis problem may have multiple optimal solutions. Therefor e, a natural choice is to consider the minimal norm\nsolution problem, which seeks to find the optimal solution wi th the smallest Euclidean norm (Beck and Sabach,\n2014; Sabach and Shtern, 2017; Latafat et al., 2023):\nf(x) =1\n2/ba∇⌈blx/ba∇⌈bl2.\nWe then solve the bilevel optimization problem:\nmin\nx∈Rn1\n2/ba∇⌈blx/ba∇⌈bl2\ns.t.x∈arg min\nz∈Rn1\n2/ba∇⌈blAz−b/ba∇⌈bl2+IC(z).\nFor this example, the proximal mapping of hcreduces to an orthogonal projection onto the ℓ2-norm ball when\nC=Rn. This scenario corresponds to the experiment in Section 5.1.\nWhenC={x∈Rn:x≥0}, we have\nproxhc(y) =√\n2c\nmax{/ba∇⌈blPC(y)/ba∇⌈bl,√\n2c}·PC(y),\nwherePC(y) = max( y,0). This result is an implementation of Theorem 7.1 in Bauschke et al. (2 018).\nWhenC={x∈Rn:/ba∇⌈blx/ba∇⌈bl1≤λ}, the proximal mapping of hcsimplifies to an orthogonal projection onto the\nintersection of a ℓ2-norm ball and a ℓ1-norm ball. This projection has a worst-case complexity of O(n2) and\nan observed complexity of O(n) in practice (Liu et al., 2020). In this case, the lower-level object ives satisfy the\nH¨ olderian error bound assumption (Assumption 3) with r= 2.\nExample 3 (Sparse Solution of Least Squares Regression Problem (SSP)) .Considering the same settings as\nin Example 2. To simplify the model and save computational re sources and efficiency, we seek to reduce the\nnumber of features in the vector x∈Rnthat minimizes the linear inverse regression function g(·). This\nmeans that our goal is to find a sparse solution among all the mi nimizers of g(·). Therefore, any function\nthat promotes sparsity can be used for this purpose. For exam ple, the well-known elastic net regularization is\na good choice (Zou and Hastie, 2005; Friedlander and Tseng, 2 008; De Mol et al., 2009; Rodola et al., 2013;\nAmini and Yousefian, 2019; Merchav and Sabach, 2023). The ela stic net regularization is defined as\nf(x) =/ba∇⌈blx/ba∇⌈bl1+α\n2/ba∇⌈blx/ba∇⌈bl2,\nwhereα >0regulates the trade-off between ℓ1andℓ2norms.Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nFor this example, the proximal mapping of hcreduces to an orthogonal projection onto the elastic-net const raints\nwhenC=Rn(Duchi, 2009; Mairal et al., 2010; Gong et al., 2011; Rodola et al., 20 13).\nExample 4 (Logistic Regression Classification Problem (LRP)) .The goal of a binary classification problem is\nto establish a mapping from the feature vectors aito the target labels bi. A common machine learning approach\nfor this task is to minimize the logistic regression functio n of the given dataset (Amini and Yousefian, 2019;\nGong and Liu, 2021; Jiang et al., 2023; Latafat et al., 2023; M erchav and Sabach, 2023). More precisely, we\nhave a feature matrix A∈Rm×nand a corresponding label vector b∈Rm, where each bi∈ −1,1. The logistic\nloss function is then given by\ng1(x) =1\nmm/summationdisplay\ni=1log(1 + exp( −a⊤\nixbi)).\nOver-fitting may occur when the number of features is not negl igible relative to the number of instances m. A\ncommon way to address this problem is to regularize the logis tic objective function with a specific function or add\na constraint (Jiang et al., 2023; Merchav and Sabach, 2023). For example, we can choose g2(x) =IC(x), where\nIC(x)is the indicator of the set C={x∈Rn:/ba∇⌈blx/ba∇⌈bl1≤λ}as shown in Example 2.\nSimilarly, this problem may have multiple optimal solution s. Therefore, it is natural to consider the minimal\nnorm solution problem with the smallest Euclidean norm as de scribed in Example 2. Now, we need to solve the\nfollowing bilevel optimization problem (Gong and Liu, 2021 ; Jiang et al., 2023; Latafat et al., 2023):\nmin\nx∈Rn1\n2/ba∇⌈blx/ba∇⌈bl2\ns.t.x∈arg min\nz∈Rn1\nmm/summationtext\ni=1log(1 + exp( −a⊤\nizbi)) +IC(z).\nThe proximal mappings of hcare the same for the choices of the set Cin Example 2. In particular, when\nC={x∈Rn:/ba∇⌈blx/ba∇⌈bl1≤λ}, this scenario corresponds to the experiment in Section 5.2.\nExample 5 (Sparse Solution of Logistic Regression Classification Problem (SSLR P)).Considering the same\nsettings as in Example 4, and we seek to reduce the features in the vector x∈Rnthat minimizes the logistic\nregression function with a regularization term. For this pu rpose, we can choose the elastic-net regularization,\nwhich is proposed in Example 3.\nSimilarly, for this example, the proximal mapping of hcis an orthogonal projection onto the elastic-net constraints\nwhenC=Rn, as shown in Example 3.\nC PROOF OF THE MAIN THEOREMS\nC.1 Proof of Theorem 1\nProof. We first show that ˆxis an (ǫf,ǫg)-optimal solution of problem (1). In Step 10, we let c=u. Then\nCondition (11) is not satisfied. According to Lemma 1, the inequality g(ˆx)≤g∗+ǫgholds. Next, we prove\nf(ˆx)≤p∗+ǫfalso holds. We break the proof into two cases.\nCase (1), if u≤p∗, then we have f(ˆx)≤u≤p∗.\nCase (2), if u > p∗, thenp∗lies on [l,u] sincel≤p∗always holds. Therefore, we have\nf(ˆx)≤u≤u+p∗−l≤p∗+ǫf,\nwhere the last inequality is from the stop criterion that u−l≤ǫf. To sum up, the point ˆxis an (ǫf,ǫg)-optimal\nsolution of problem (1). In the following, we present the total oper ation complexity of our method.\nIn Step 1, we invoke APG oracles to obtain initial bounds landu. To obtain the initial lower bound of p∗,\nwe invoke the APG oracle ˜xf=APG(f1,f2,Lf1,xf\n0,ǫf/2) to solve problem (9). By Lemma 2, this can be done\nwithinO/parenleftBig/radicalbig\nLf1/ǫf/parenrightBig\niterations. The corresponding initial lower bound is l=f(˜xf)−ǫf/2. To obtain the initial\nupper bound of p∗\nǫg, we invoke the APG oracle ˜xg=APG(g1,g2,Lg1,xg\n0,ǫg/2) to solve problem (2). Similarly, weJiulin Wang, Xu Shi, Rujun Jiang†\ncan approximately solve problem (2) within O/parenleftBig/radicalbig\nLg1/ǫg/parenrightBig\niterations. The corresponding initial upper bound is\ngiven by u=f(˜xg).\nIn Step 3, we invoke APG oracle ˜xc=APG(g1,hc,Lg1,xc\n0,ǫg/2) to solve problem (6). According to Lemma 2,\nthis can be done within O/parenleftBig/radicalbig\nLg1/ǫg/parenrightBig\niterations. The number of invoking APG oracle does not exceed\n/ceilingleftbigg\nlog2u−l\nǫf/ceilingrightbigg\n+=/ceilingleftbigg\nlog2f(˜xg)−f(˜xf) +ǫf/2\nǫf/ceilingrightbigg\n+=O/parenleftbigg\nlog1\nǫf/parenrightbigg\nwherelanduare initial lower and upper bounds, and ⌈a⌉+represents the smallest non-negative integer that is\nno less than a.\nThus, the total number of evaluations of the function values f1,f2,g1, andg2, the gradients ∇f1and∇g1, and\nthe calls of the proximal mapping concerning function hcdoes not exceed T, where\nT=O/parenleftBigg/radicalBigg\nLf1\nǫf/parenrightBigg\n+O/parenleftBigg/radicalBigg\nLg1\nǫg/parenrightBigg\n+O/parenleftBigg/radicalBigg\nLg1\nǫg/parenrightBigg\n·O/parenleftbigg\nlog1\nǫf/parenrightbigg\n=˜O/parenleftBigg\nmax/braceleftBigg/radicalBigg\nLf1\nǫf,/radicalBigg\nLg1\nǫg/bracerightBigg/parenrightBigg\n,\nwhere ˜Osuppresses a logarithmic term.\nC.2 Proof of Proposition 1\nProof. By Assumption 2(i), the set X∗\ngis closed and compact. Then we can let ˆx∗= arg min\nx∈X∗g/ba∇⌈blx−ˆx/ba∇⌈blsuch that\n/ba∇⌈blˆx−ˆx∗/ba∇⌈bl= dist( ˆx,X∗\ng). It can be easily demonstrated that X∗\ngis a convex set, ensuring the well-definedness of\nˆx∗. By Assumption 3, we have\nα\nr/ba∇⌈blˆx−ˆx∗/ba∇⌈blr≤g(ˆx)−g∗≤ǫg=⇒ /ba∇⌈bl ˆx−ˆx∗/ba∇⌈bl ≤/parenleftBigrǫg\nα/parenrightBig1\nr. (20)\nBy Assumption 2, it follows that fisBf-Lipschitz continuous on dom( g2) (see (15)). Combining this result with\n(20), we have\nf(ˆx)−p∗≥f(ˆx)−f(ˆx∗)≥ −Bf/ba∇⌈blˆx−ˆx∗/ba∇⌈bl ≥ −Bf/parenleftBigrǫg\nα/parenrightBig1\nr,\nwhere the first inequality is from that p∗is the optimal value of problem (1) and ˆx∗∈X∗\ng.\nC.3 Proof of Corollary 1\nProof. Letˆxbe the output of Algorithm 1. Then ˆxis an (ǫf,ǫg)-optimal solution of (1) satisfying\nf(ˆx)−p∗≤ǫf, g(ˆx)−g∗≤ǫg.\nBy Proposition 1 and the setting ǫg=α\nγ/parenleftBig\nǫf\nBf/parenrightBigr\n, we have\nf(ˆx)−p∗≥ −Bf/parenleftBigrǫg\nα/parenrightBig1\nr=−ǫf.\nThen according to Theorem 1, with an operation complexity ˜O/parenleftBig\n1//radicalbigǫr\nf/parenrightBig\n, we can find an ˆxsuch that\n|f(ˆx)−p∗| ≤ǫf, g(ˆx)−g∗≤ǫg.\nC.4 Proof of Proposition 2\nProof. Since ˆxsatisfiesg(ˆx)−g∗≤ǫg, then ˆxis a feasible solution of (4) with ǫ=ǫg, andf(ˆx) is an upper\nbound of p∗\nǫg(see Lemma 1). Applying Assumption 4 with ǫ=ǫg, we have\nf(ˆx)−p∗≥p∗\nǫg−p∗=v(ǫg)−v(0)≥ −Lǫgǫg.Near-Optimal Convex Simple Bilevel Optimization with a Bis ection Method\nC.5 Proof of Corollary 2\nProof. This proof closely resembles the one presented in Corollary 1. Never theless, for the sake of thoroughness,\nwe provide a complete exposition. Let ˆxbe the output of Algorithm 1. Then ˆxis an (ǫf,ǫg)-optimal solution of\n(1) satisfying\nf(ˆx)−p∗≤ǫf, g(ˆx)−g∗≤ǫg.\nBy Proposition 2 and the setting ǫf=Lǫgǫg, we have\nf(ˆx)−p∗≥ −Lǫgǫg=−ǫf.\nThen according to Theorem 1, with an operation complexity ˜O/parenleftbig\nmax{1//radicalbigLǫgǫg,1/√ǫg}/parenrightbig\n, we can find an ˆxsuch\nthat\n|f(ˆx)−p∗| ≤ǫf, g(ˆx)−g∗≤ǫg.\nD ADDITIONAL NUMERICAL EXPERIMENTS\nWe consider the SSP problem from Example 3 on the YearPredictionMS D dataset, setting C=Rnandα= 0.02.\nOur objective is to solve the following bilevel problem:\nmin\nx∈Rnα\n2/ba∇⌈blx/ba∇⌈bl2+/ba∇⌈blx/ba∇⌈bl1\ns.t.x∈arg min\nz∈Rn1\n2/ba∇⌈blAz−b/ba∇⌈bl2.(21)\nWe compare the performance of our Bisec-BiO method with the aver aging iteratively regularized gradient (a-\nIRG) method (Kaushik and Yousefian, 2021). It’s worth noting tha t a-IRG can handle non-smooth upper-level\nobjectives, a capability that other methods lack. For benchmarkin g purposes, we use the MATLAB functions\nlsqminnorm andfmincon to obtain the optimal values g∗andp∗, respectively. Moreover, we adopt the method\nproposed by Gong et al. (2011) to compute the proximal operator ofhc:=g2+δcin problem (8). The other\nsettings are consistent with Section 5.1.\n0 1 2 3 4 5 6 710-1010-5100105\n0 1 2 3 4 5 6 710-1100101102\nFigure 4: The performance of Bisec-BiO compared with other metho ds in SSP.\nAs illustrated in Figure 4, Bisec-BiO converges significantly faster th an a-IRG for both lower- and upper-level\nobjectives. The numbers on the right-hand side of the last iterate s of our method (denoted as ˆx) represent the\ndifferences between g(ˆx)−g∗andf(ˆx)−p∗, respectively. This confirms that ˆxis an (ǫf,ǫg)-optimal solution."    },    {        "title": "2402.05470v1.Complete_immersions_with_constant_scalar_curvature_and_flat_normal_bundle.pdf",        "content": "arXiv:2402.05470v1  [math.DG]  8 Feb 2024COMPLETE IMMERSIONS WITH CONSTANT SCALAR CURVATURE AND FLA T\nNORMAL BUNDLE\nH. A. GURURAJA\nAbstract. LetMbe a complete Riemannian manifold with constant scalar curv atureRand dimension\nn≥3,and letf:M→Mn+k(c) be a proper isometric immersion with flat normal bundle and t ype\n(1,n−1)inacomplete simply-connected Riemannianspace form Mn+k(c)ofconstant sectional curvature\ncand dimension n+k.Our first result is global and states that R≥0 ifc= 0, R >(n−1)(n−2)\n2cif\nc >0,andR≥n(n−1)\n2cifc <0.Our second result is of a local character and states that, if i n addition\nwe assume that c≥0 and the mean curvature field of the immersion is parallel in t he normal bundle,\nthenMhas non-negative sectional curvature.\n1.Introduction\nLetMbe a connected complete Riemannian manifold of constant scalar cur vatureRand dimension\nn≥3,and letMn+k(c) be the complete simply-connected Riemannian manifold of constant sectional\ncurvaturecand dimension n+k.We consider a smooth isometric immersion f:M→Mn+k(c) with flat\nnormal bundle which means that the curvature tensor of the norm al bundle of the immersion vanishes\nidentically. We further assume that the immersion is properand hastype(1,n−1) as defined below (cf.\n§2).\nAssume that αdenotes the second fundamental form of the immersion f.For any point x∈Mand\nany normal vector η(x) atx,let\nEη(x)={X∈TxM:α(x)(X,Y) =/an}b∇acketle{tX,Y/an}b∇acket∇i}htη(x)for all Y ∈TxM}.\nThen our assumption is that at each point x∈Mthere exist distinct normal vectors η1(x) andη2(x) at\nxsuch that\nTxM=Eη1(x)⊕⊥Eη2(x)\nand\ndimEη1(x)= 1anddimEη2(x)=n−1.\nAccording to a well-known theorem of Cartan, conformally flat hype rsurfaces of a Riemannian space\nformwithdimension n≥4arecharacterizedbythepropertyofhavingaprincipalcurvatu reofmultiplicity\nat leastn−1 at each point, and hence they satisfy the above type condition wh en they are proper and\nnon-umbilical. The author was led to this condition in his attempt to ext end results of [6] to higher\ncodimensions. For this purpose, it appears that the notion of holon omic submanifolds is the appropriate\ngeneralization of rotation hypersurfaces introduced by Do Carmo and Dajczer, and the above condition\nmay be viewed as an analogue of their main criterion (cf. [1], Th. (4.2)) to this context.\nThe main result of this paper is the following theorem.\nTheorem 1.1. LetMbe a complete Riemannian manifold with constant scalar curv atureRand dimen-\nsionn≥3,and letf:M→Mn+k(c)be a proper isometric immersion with flat normal bundle and ty pe\n(1,n−1).Then the following holds:\n2010Mathematics Subject Classification. Primary 53C40; Secondary 53C42.\nKey words and phrases. Isometric immersion, Constant scalar curvature, Mean curv ature.\n12 H. A. GURURAJA\n(i)R≥0ifc= 0.If equality holds and Mis compact, then Mmust be flat.\n(ii)R>(n−1)(n−2)\n2cifc>0.\n(iii)R≥n(n−1)\n2cifc<0.Strict inequality holds if Mis compact.\nThe bounds in Theorem 1.1 are optimal. Theorem 1.1 is global and is not v alid if the completeness\nassumption on Mis dropped. Moreover, the compactness assumption in part (i) is no t superfluous. By\na classical result of Tompkins [7] it follows that k≥nin this case. See §3 for the details. We note that\nour method does not work when n= 2,which may be of interest since for n= 2 andk= 1,part (i) of\nTheorem 1.1 corresponds to Hilbert’s theorem on non-immersibility of the hyperbolic plane in R3.\nSince constant scalar curvature condition is implied by constant isot ropic curvature condition, we have\nthe following consequence of Theorem 1.1 which extends some result s of [6] to the present setting.\nCorollary 1.2. LetMbe a complete Riemannian manifold with constant isotropic c urvatureCand\ndimensionn≥4.Letf:M→Mn+k(c)be a proper isometric immersion with flat normal bundle and\ntype(1,n−1).Then\n(i)C≥0.IfC= 0andMis compact, then Mmust be flat.\n(ii)C >4(n−2)\nncifc>0.\n(iii)C≥4cifc<0.Strict inequality holds if Mis compact.\nThe method of proof of Theorem 1.1 also yields the following result of a local character.\nTheorem 1.3. LetMbe a Riemannian manifold with constant scalar curvature Rand dimension n≥3.\nLetf:M→Mn+k(c), c≥0,be a proper isometric immersion with flat normal bundle and ty pe(1,n−1).\nAssume that the mean curvature field of fis parallel in the normal bundle. Then Mhas non-negative\nsectional curvature.\nCombining Theorem 1.3 with a result of Erbacher(cf. [5], Th. 1), we ob tain the following corollarywhich\npartially generalizes a result of Dajczer and Tojeiro (cf. [3], Th. 1) t o our setting.\nCorollary 1.4. LetMbe a complete Riemannian manifold with constant scalar curv atureRand dimen-\nsionn≥3.Letf:M→Mn+k(c), c≥0,be a proper isometric immersion with flat normal bundle and\ntype(1,n−1).Assume that the mean curvature field of fis parallel in the normal bundle. Then f(M)\nmust be of the form S1(r1)×Sn−1(r2),S1(r)×Rn−1,orR×Sn−1(r)ifc= 0;andf(M)must be of the\nformS1(r1)×Sn−1(r2)ifc>0.The respective local result holds if the completeness assum ption onMis\ndropped.\nIt should be noted that the flatness assumption on the normal bun dle in Corollary 1.4 is redundant\nwhen the codimension k= 2 andMis not minimal [cf. [5], Th. 1′].\nIn particular, if Mis compact and has constant isotropic curvature then under the a ssumptions of\nCorollary 1.4 it follows that f(M) must be S1(r1)×S3(r2) for certain positive numbers r1andr2,which\nis the Clifford minimal hypersurface in S5(c).\n2.Preliminaries\nAssume that f:M→Mn+k(c) is an isometric immersion with flat normal bundle. From the Ricci\nequation\n/an}b∇acketle{tR⊥(X,Y)ξ,η/an}b∇acket∇i}ht=/an}b∇acketle{t[Aξ,Aη]X,Y/an}b∇acket∇i}ht,\nfor an isometric immersion in a space form, it follows that the shape op erators{Aξ}pairwise commute\nand hence are simultaneously diagonalizable at each point x∈M.Moreover (cf. [1], P. 35), there exists\na positive integer s(x) and pairwise distinct principal normal vectorsη1,···,ηs(x)atxsuch that the\ntangent space TxMdecomposes as\nTxM=Eη1(x)⊕⊥···⊕⊥Eηs(x).COMPLETE IMMERSIONS WITH CONSTANT SCALAR CURVATURE AND FLA T NORMAL BUNDLE 3\nHere the subspaces Eηi(x)are defined as in §1.We say that the immersion fisproperif the integer\ns=s(x) is independent of x∈M.If the immersion is proper, then it is known [8] that each ηidefines a\nsmooth normal field on Mand that the distribution {Eηi(x):x∈M}is smooth for each 1 ≤i≤s.\nNow assume that Mandfare as in §1.In this case, the distributions {Eηi(x)},1≤i≤2,are\nintegrable. Arguing as in [2] we see that each point P∈Madmits a holonomic local coordinate system\n(U,(x1,···,xn)) such that Eη1=span{∂1}andEη2=span{∂2,···,∂n}onU.In such a coordinate\nsystem the first and the second fundamental forms of the immers ionfare given respectively by\nI=v2\n1dx2\n1+···+v2\nndx2\nn\nand\nα(x)(∂1,∂1) =v2\n1η1(x);α(x)(∂1,∂j) = 0;α(x)(∂i,∂j) =δijv2\njη2(x),2≤i,j≤n,\nfor certain positive functions vi=vi(x1,···,xn),1≤i≤n,defined onU.\nThe Gauss equation for an isometric immersion in Mn+k(c) applied to the orthonormal set\n{Xi=∂i\nvi,Xj=∂j\nvj},1≤i/ne}ationslash=j≤n,\nyields\n(2.1) K(X1,Xj) =c+/an}b∇acketle{tη1,η2/an}b∇acket∇i}ht,2≤j≤n,\nand\n(2.2) K(Xi,Xj) =c+||η2||2,2≤i/ne}ationslash=j≤n.\nHere, and throughout this paper, K(X, Y) denotes the sectional curvature along the plane spanned by\nthe vectors XandY.\nThe Codazzi equations (cf. [3], P. 36)\n∇⊥\nXη2= 0,(X∈Γ(Eη2))\nand\n/an}b∇acketle{tXj,Yj/an}b∇acket∇i}ht∇⊥\nXiηj=/an}b∇acketle{t∇XjYj,Xi/an}b∇acket∇i}ht(ηj−ηi),/parenleftbig\nXi∈Γ(ηi), Xj∈Γ(ηj),1≤i/ne}ationslash=j≤n/parenrightbig\n,\nnow take the forms\n(2.3) ∇⊥\n∂jη2= 0,\n(2.4) ∇⊥\n∂1η2=∂1vj\nvj(η1−η2),\nand\n(2.5) ∇⊥\n∂jη1=∂jv1\nv1(η2−η1)\nfor anyj,2≤j≤n.\nLetD=R\nn−1.SinceMhas constant scalar curvature Rwe obtain\n(2.6) c+/an}b∇acketle{tη1,η2/an}b∇acket∇i}ht+n−2\n2/parenleftbig\nc+||η2||/parenrightbig2=D.\nFor later use in the proof of Theorem 1.1, we note that in the special case when∂jv1= 0 for each\nj,2≤j≤n,we may assume that v1=v1(x1).By the change of variable\nt=t(x1) =/integraldisplayx1\nx1(P)v1(u)du,\nweseethatthefirstfundamentalform Ioffintheresultingholonomiclocalcoordinatesystem( U,(t,x2,···,xn))\nis given by\nI=dt2+v2\n2dx2\n2+···+vndx2\nn.4 H. A. GURURAJA\n3.Proof of Theorem 1.1\nDefineφ:M→Rby\nφ(x) =c+/an}b∇acketle{tη1(x),η2(x)/an}b∇acket∇i}ht,(x∈M).\nIn any holonomic local coordinate system ( U,(x1,···,xn)) inM,a straightforward computation using\nequations (2.4) and (2.6) gives\n(3.1) ∂1/parenleftbig2D\nn−φ/parenrightbig\n+n∂1vj\nvj/parenleftbig2D\nn−φ/parenrightbig\n= 0.\nMoreover, from equation (2.3) it follows that\n∂jφ= 0,2≤j≤n.\nOn the other hand, using equations (2.3) and (2.5) we obtain\n0 =∂jφ=/an}b∇acketle{t∇⊥\n∂jη1,η2/an}b∇acket∇i}ht=/an}b∇acketle{t∂jv1\nv1(η2−η1),η2/an}b∇acket∇i}ht=∂jv1\nv1(||η2||2−/an}b∇acketle{tη1,η2/an}b∇acket∇i}ht).\nIf/parenleftbig\n||η2||2−/an}b∇acketle{tη1,η2/an}b∇acket∇i}ht/parenrightbig\n(P) = 0 for some P∈M,thenφ(P) =c+|η2(P)|2.Therefore equation (2.6) gives\nφ(P) =2D\nn.From equations (3.1) and ∂jφ= 0,it follows that φ(x) =2D\nnfor everyx∈M.ThusMhas\nconstant curvature2D\nnand, since2D\nn=c+||η2||2,it follows that D=nc\n2in this case.\nTherefore we may assume that/parenleftbig\n||η2||2− /an}b∇acketle{tη1,η2/an}b∇acket∇i}ht/parenrightbig\n(P)/ne}ationslash= 0 for any P∈M.From equation (2.5) we\nconclude that ∂jv1= 0 for each 2 ≤j≤n.As noted in §2,it follows that each point of Madmits a holo-\nnomic local coordinate system ( U,(t,x2,···,xn)) in which the first fundamental form of the immersion\nfis given by\nI=dt2+v2\n2dx2\n2+···+vndx2\nn.\nFor the remainder of the proof we will only work with holonomic local co ordinate systems on Mthat are\nof the above form.\nSince||∂t||= 1 we have ∇∂t∂t= 0.LetXj=∂j\nvj, hji=∂jvi\nvjandh1j=∂tvj,(2≤i/ne}ationslash=j≤n).A\nstraightforward computation (cf. [3], P. 20) gives\n∇∂j∂t=h1jXj\nand\n∇∂tXj= 0\nfor any 2 ≤j≤n.\nLetRdenote the curvature tensor on M.We compute\nR(∂j,∂t)∂t\n=∇∂j∇∂t∂t−∇∂t∇∂j∂t\n=−∇∂t∇∂j∂t\n=−∇∂t(h1jXj) =−∂t(h1j)Xj−h1j∇∂tXj\n=−∂t(h1j)Xj.\nThus\n(3.2) K(∂t,∂j) =−1\nvj∂ttvj.COMPLETE IMMERSIONS WITH CONSTANT SCALAR CURVATURE AND FLA T NORMAL BUNDLE 5\nFix a point P∈M.Choosea holonomiclocalcoordinatesystem ( U,(t,x2,···,xn)) aroundPasabove.\nLetγ:R→Mdenote the geodesic with initial data γ(0) =Pandγ′(0) =∂t(P).Since the integral\ncurves of∂tare geodesics parametrized by arc length it follows from uniqueness that\nγ′(s) =∂t(γ(s))\nwheneverγ(s)∈U.If (V,(τ,y2,···,yn)) denotes another holonomic local coordinate system that also\ncontainsγ(s0) for somes0∈R,then since∂τ∈Eη1(γ(s0))andEη1(γ(s0))is 1-dimensional, we must have\n∂τ(γ(s0)) =±∂t(γ(s0)).Replacingτby−τif necessary, we can assume that ∂τ(γ(s0)) =∂t(γ(s0)).In\nthis way we obtain a covering of γ(R) by holonomic local coordinate charts {(U,(t,x2,···,xn))}which\nsatisfy the condition\nγ′(s) =∂t(γ(s))\nwheneverγ(s)∈U.\nLetψ=2D\nn−φ.Using equation (2.6) we obtain\n(3.3) ∂tψ+n∂tvj\nvjψ= 0.\nAssume that the function ψis positive for the moment. In this case, we define a function θ:R→Rby\nθ(s) =1\nψ(γ(s))1\nn.\nThenθis positive, and satisfies ψ(γ(s))(θ(s))n= 1 for each s∈R.Working in a holonomic local\ncoordinate system as above we obtain\n0 = (log(ψ◦γ)θn)′=∂tψ\nψ+nθ′\nθ,(′=d\nds).\nUsing\n∂t/parenleftbig∂tvj\nvj/parenrightbig\n=1\nvj∂ttvj−(∂tvj\nvj)2\nand equation (3.2) we obtain\n∂t/parenleftbig∂tvj\nvj/parenrightbig\n=−k(∂t,∂j)−(∂tvj\nvj)2=−φ◦γ−(∂tvj\nvj)2\nand\n/parenleftbigθ′\nθ/parenrightbig′=−φ◦γ−/parenleftbig∂tvj\nvj/parenrightbig2=−φ◦γ−/parenleftbigθ′\nθ/parenrightbig2.\nThus\n(3.4)θ′′\nθ=−φ◦γ.\nWe now consider the three cases separately below.\n(i) Letc= 0.We assume that D<0 and obtain a contradiction. Using equation (2.6) we obtain\nψ=2D\nn−φ=n−2\n2||η2||2−n−2\nnD≥ −n−2\nnD>0.\nAlso, since φ<D,from equation (3.4) it follows thatθ′′\nθ>−D>0.Sinceθis strictly convex on R,we\nhave lim s→∞θ(s) =∞.This, however, contradicts ψ(γ(s))(θ(s))n= 1 for alls∈R.ThereforeD≥0.\nNow assume that Mis compact and D= 0.Using equation (2.6) we get ψ≥0.Assume that ψattains\nminimum at Q∈M.Ifψ(Q) = 0,then from equation (3.1) we conclude that ψmust vanish identically\nandMmust be flat. Assume therefore that ψ(Q)>0.From (3.1) we obtain ∂tvj(Q) = 0.The conditions\n∂ttψ=−n∂t/parenleftbig∂tvj\nvj/parenrightbig\nψ+n2/parenleftbig∂tvj\nvj/parenrightbig26 H. A. GURURAJA\nand∂ttψ(Q)≥0 imply that ∂t/parenleftbig∂tvj\nvj/parenrightbig\n(Q)≤0.It follows that ψ(Q) =−φ(Q)≤0.Thusψ(Q) = 0 and\nthatφvanishes on M.By equation (2.6), c+||η2||2= 0 and we conclude that Mmust be flat.\n(ii) Letc>0.SupposeD≤n−2\n2c.From equation (2.6) we have\nψ=2D\nn−φ=n−2\n2c+n−2\n2||η2||2≥n−2\n2c>0.\nMoreover,equation(2.6) impliesthat φ+n−2\n2c≤D,orφ≤D−(n−2\n2c)≤0.Fromequation(3.4) it follows\nthatθis convex, and strictly convex if D<n−2\n2c.As in part (i) we obtain a contradiction if D<n−2\n2c,or\nifD=n−2\n2cand if one ofthe limits lim s→±∞θ(s) is infinite. Otherwise θmust be a constantfunction and\nψ,and henceφ,must be constant along γ.Since∂jφ= 0,2≤j≤n,it follows that φmust be a constant\nfunction on M.From equation (3.3) we obtain ∂tvj= 0 for each j,2≤j≤n.Henceφ=K(∂t,∂j) = 0.\nSinceD=n−2\n2c,using equation (2.6) we obtain η2= 0.This gives 0 = φ=c+/an}b∇acketle{tη1,η2/an}b∇acket∇i}ht=c,which is\nabsurd.\n(iii) Letc<0.IfD<nc\n2.Sincenc−2D>0,from equation (2.6) we obtain ψ=2D\nn−φ>n−2\n2n(nc−2D)>\n0.Using equation (2.6) once more we get φ+n−2\n2c≤D,orφ≤D−(n−2\n2c)< c <0 sinceD <nc\n2by\nassumption. Rest of the proof is analogous to part (i).\nIn the case of equality, we have φ+n−2\n2||η2||2=c<0,so thatφ(x)<0 for every x∈M.However, if\nMis compact then the argument in part (i) shows that there exists a p ointQ∈Msuch thatφ(Q)≥0.\nHence the inequality must be strict if Mis compact. /square\nRemarks.\nTheorem 1.1 is global and the completeness assumption on Mcannot be dropped. To see this, let\nϕ= (ϕ1,ϕ2,ϕ3,ϕ4) :R3→R4be anyorthogonalparametrizationofthe unitspherein R4.Forsufficiently\nsmallǫ>0,define\nx(s) =/radicalbig\n2(es+e−s−1), s∈(−ǫ,+ǫ).\nThen the rotation hypersurface (cf. [1]) in R5which is given by\nf(s,t1,t2,t3) = (x(s)ϕ1(t1,t2,t3),···,x(s)ϕ4(t1,t2,t3),y(s)),\nand satisfying the arc length parametrization condition x′(s)2+y′(s)2= 1, s∈(−ǫ,ǫ),satisfiesD=−1\n4.\nHowever, the arc length parametrization condition cannot be cont inued for all s∈R,as can be readily\nverified. Similar examples can be constructed in any dimension and any space form by solving a certain\nODE.\nThat the compactness assumption in part (i) of Theorem 1.1 is not re dundant is illustrated by the\nfollowing example. Let ϕ= (ϕ1,ϕ2,ϕ3,ϕ4) be as before, and consider the isometric immersion f:R4→\nR5given by\nf(s,t1,t2,t3) = (/radicalbig\ns2+1ϕ1(t1,t2,t3),···,/radicalbig\ns2+1ϕ4(t1,t2,t3),log(s+/radicalbig\ns2+1)).\nThen the rotation hypersurface described by fis complete and satisfies D= 0.See [6] for the details.\nThe bounds in Theorem 1.1 are also optimal. For c= 0,the cylinder Rn−1×S1(1) satisfies D= 0.\nSuppose that c>0 andn= 4.For any real number d, d>c, letx(s) =1\nd.Then the rotationhypersurface\ninS5(c)⊂R6which is given by\nf(s,t1,t2,t3) = (x(s)ϕ1(t1,t2,t3),···,x(s)ϕ4(t1,t2,t3),x5(s),x6(s)),\nand satisfying the arc length parametrization condition on the merid ian curves→(x(s),x5(s),x6(s)),\nsatisfiesD=d.The Clifford minimal hypersurface S1(r1)×S3(r2) inS5(c) corresponds to the choice\nd=4c\n3.Forc<0 andn= 4,letx1(s) =/radicalBig\n−1\nccosh(2√−cs).Then the parabolic rotation hypersurface\ninH5(c) obtained from the function x1(s) satisfiesD= 2c.COMPLETE IMMERSIONS WITH CONSTANT SCALAR CURVATURE AND FLA T NORMAL BUNDLE 7\n4.Proof of Theorem 1.3\nAs with the proof of Theorem 1.1 we work in a holonomic local coordinat e system (U,(x1,···,xn))\nonM.We claim that ∂1vjvanishes identically for any j,2≤j≤n.Otherwise, by shrinking Uif\nnecessary, we may assume that ∂1vjdoes not vanish on U.LetHbe the mean curvature field of f.Since\nnH=η1+(n−1)η2and∇⊥H= 0,from equations (2.3) and (2.5) it follows that\n(4.1) ∇⊥\n∂jη1= 0\nand\n(4.2) ∇⊥\n∂1η1+(n−1)∇⊥\n∂1η2= 0.\nThus∂jv1= 0 for each j,2≤j≤n.From equation (2.4) we conclude that\n(4.3)∂1vj\nvj=∂1vk\nvk,2≤j,k≤n.\nSinceMhas flat normal bundle using equation (2.3) we obtain\n0 =R⊥(∂1,∂j)η2\n=∇⊥\n∂1(∇⊥\n∂jη2)−∇⊥\n∂j(∇⊥\n∂1η2)\n=−∇⊥\n∂j(∇⊥\n∂1η2).\nSubstituting for ∇⊥\n∂1η2from equation (2.4) and simplifying we obtain\n∇⊥\n∂jη1=∂j(∂1vj\nvj)\n∂1vj\nvj(η2−η1).\nFrom equation (4.1) it follows that\n∂j(∂1vj\nvj) = 0.\nCombining this with equation (4.3) we get\n∂k(∂1vj\nvj) = 0.\nThus we may assume that\nvk(x1,···,xn) =ψk(x2,···xn)µ(x1)\nfor some positive functions ψkandµ.Differentiating equation (2.6) with respect to x1we get\n/an}b∇acketle{t∇⊥\n∂1η1,η2/an}b∇acket∇i}ht+/an}b∇acketle{tη1,∇⊥\n∂1η2/an}b∇acket∇i}ht+(n−2)/an}b∇acketle{tη2,∇⊥\n∂1η2/an}b∇acket∇i}ht= 0.\nSubstituting ∇⊥\n∂1η1=−(n−1)∇⊥\n∂1η2from equation (4.2) in the above equation gives\n/an}b∇acketle{t∇⊥\n∂1η2,η1−η2/an}b∇acket∇i}ht= 0.\nSince∇⊥\n∂1η2=µ′(x1)(η1−η2),we haveµ′(x1)||η1−η2||2= 0,andµ′vanishes identically and hence\n∂1vj= 0.From this and ∂jv1= 0 we conclude that K(∂1,∂j) = 0 as in the proof of Theorem 1.1. Since\nthe curvature operator of Mis diagonalized by the orthonormal basis {∂i\nvi},the sectional curvatures of\nMlie in the interval [0 ,c+||η||2],and therefore Mhas non-negative sectional curvature.\n/square\nAknowledgements\nIt is a pleasure to thank Professors Harish Seshadri and C. S. Ara vinda for useful comments on an\nearlier draft of this article.8 H. A. GURURAJA\nReferences\n[1] Do Carmo, M. & Dajczer, M., Rotation hypersurfaces in spa ces of constant curvature, Trans. Amer. Math. Soc. ,277\n(1983), 685-709.\n[2] Dajczer, M., Onti, C.-R. & Vlachos, Th., Conformally flat submanifolds with flat normal bundle, Manuscripta Math. ,\n163(2020), 407-426.\n[3] Dajczer, M. & Tojeiro, R., Submanifolds of constant sect ional curvature with parallel or constant mean curvature,\nTohoku Math. J. ,45(1993), 43-49.\n[4] Dajczer, M. & Tojeiro, R., Submanifold theory , Springer, New York, 2019.\n[5] Erbacher, J., Isometric immersions of constant mean cur vature and triviality of the normal connection, Nagoya Math.\nJ.,45(1972), 139-165.\n[6] Gururaja, H.A. & Kumar N., Complete hypersurfaces of con stant isotropic curvature in space forms, J. Math. Anal.\nAppl.,520(2023), Paper No. 126876.\n[7] Kobayashi, S. & Nomizu, K., Foundations of differential geometry, Vol. II, Interscience Publishers, New York, 1969.\n[8] Reckziegel, H., Kr¨ ummungsfl¨ achen von isometrischen i mmersionen in r¨ aume konstanter kr¨ ummung, Math. Ann. ,223\n(1976), 169-181.\nIndian Institute of Science Education and Research Tirupat i, Srinivasapuram, Jangalapalli Village, Pan-\nguru (G.P), Yerpedu Mandal, Tirupati Dist. - 517619. Andhra Pr adesh, INDIA\nEmail address :gururaja@iisertirupati.ac.in"    },    {        "title": "2402.05490v1.Investigating_and_Controlling_the_Libration_and_Rotation_Dynamics_of_Nanoparticles_in_an_Optomechanical_System.pdf",        "content": "Investigating and Controlling the Libration  and Rotation Dynamics \nof Nanoparticles in an Optomechanical System  \nChaoxiong He1, Jinchuan Wang1, Ying Dong1, Shaochong Zhu2, Qianwen Ying1, \nYuanyuan Ma5, Fu Feng1, Zhangqi Yin4, Cuihong Li1,* and Huizhu Hu3  \n1Research Center for Frontier F undamental Studies, Zhejiang Lab, Hangzhou 310000, China  \n2Research Center for Novel Computational Sensing and Intelligent Processing, Zhejiang Lab, \nHangzhou 310000, China  \n3Laboratory of Extreme Photonics and Instrumentation, College of Optical Science and  \nEngineering , Institute of Quantum Sensing, Zhejiang University, Hangzhou 310027, China  \n4Center for Quantum Technology Research and Key Laboratory of Advanced  Optoelectronic \nQuantum Architecture and Measurements (MOE), School of Physics, Beijing Institute of \nTechnology, Beijing 100081, China  \n5The State Key Lab of Analytical Chemistry for Life Science, School of Chemistry and Chemical \nEngineering, Chemistry and Biomedicine Innovation Center (ChemBIC), Nanjing University, \nNanjing 210000, China  \n*e-mail: licuihong@zhejianglab.com  \nIn optomechanical systems, the libration and rotation of nanoparticles offer profound \ninsights for ultrasensitive torque measurement and m acroscopic quantum \nsuperpositions. Achievements include transitioning libration to rotation up to 6 GHz \nand cooling libration to millikelvin temperatures. It is undoubted that the libration and \nrotation are respectively driven by restoring and constant opt ical torques. The transition \nmechanisms between these two states, however, demand further exploration. In this \nperspective, it is demonstrated in this manuscript that monitoring lateral -scattered light \nallows real -time observation of libration/rotation tra nsitions and associated hysteresis \nas ellipticities of trapping laser fields vary. By calculating optical torques and solving \nthe Langevin equation, transitions are linked to the balance between anisotropic -\npolarization -induced sinusoidal optical torques a nd constant ones, with absorption \nidentified as the main contributor to constant torques. These findings enable direct weak torque sensing and precise nanoparticle control in rotational degrees, paving the way \nfor studying quantum effects like nonadiabatic  phase shifts and macroscopic quantum \nsuperpositions, thereby enriching quantum optomechanics research.  \n  Optically -levitated nanoparticles, acting as high -fidelity nano -oscillators and benefiting \nfrom their isolation from external environment, ha ve emerge d as superior facilitators \nfor probing nanoscale photonic and material interactions1-3, exploratory tools for \ndelineating the quantum -classical boundary4-6, and  sensors for ultraweak force s7-9. The \ninvestigation into their center -of-mass translational oscillations, along with inherent \nnonlinear dynamics induced by optical forces, has been extensive and profound10-13. \nFurthermore, optically -levitated nanoparticles exhibit a spectrum of  motions including \nlibrations, rotations, and spins. This rotational dynamism endows them with heightened \nsensitivity for measurements pertaining to rotation14-19, positioning them as \nquintessential platforms for the study of advanced mesoscopic phenomena. Notable \namong these are the exploration of vacuum friction15, the manifestation macrosc opic \nquantum coherence20, the interrogation of spin -mass interaction21, the analysis of many -\nbody correlations22, and the examination of quantum nonadiabatic phenomena23, \nthereby significantly contributing to the advancement of mesoscopic physics and \nquantum science.  \nIt has been elucidated that the major axis of non -spherical nanoparticles exhibits a \ntendency to librate around the polarization vector of the trapping laser, a phenomenon \nattributed to the anisotropic polarization and librational balance24-26. For non -spherica l \nnanoparticles trapped by circularly polarized laser fields, they are subjected to \nrotational motions, frequencies of which are governed by the equilibrium established \nbetween constant optical torques and viscous drags25,27 -29. Noteworthy achievements \ninclude the attainment of libration cooling to temperatur es in the millikelvin range30-32 \nand the realization of rapid mechanical rotations of na noparticles, having surpass ed 6 \nGHz28. A recent investigation has delved into the rotational diffusion phenomenon, \nexploring the noise -induced transition between the libration and rotation  of an optically \nlevitated nanodumbbell33. Despite these advances, the transition between libration s and \nrotation s remains sparsely documented, underscoring a significant gap in understanding \nthe fundamental mechanisms that underpin mesoscopic dynamics and quantum spin -\nmechanical interactions in optomechanical systems21,34,35.  This article proposes a novel methodology for monitoring the librational and rotational \ndynamics of nanoparticles by analyzing lateral -scattered light as ellip ticities of trapping \nlaser fields are varied under diverse trapping powers and vacuum degrees. This \napproach facilitates the direct observation of real -time critical librational/rotational \ntransitions of nanoparticles, which  are characterized by stochastic  exchanges between \nlibrations and rotations. Moreover, these transitions are marked by distinctive bi -stable \nhysteresis loops in relation to ellipticities. To account for transition mechanisms, optical \ntorques, quantified as integrals over angular momentum  current density, are computed, \nenabling the resolution of librational/rotational Langevin equations. The calculation \nresults show perfect agreements with experiment. Further theoretical examination \nindicates that, the critical libration/rotation transitio ns in elliptically polarized laser \nfields primarily arise from the balance between anisotropic -polarization -induced \nsinusoidal optical torques and constant ones, and absorption processes play a dominant \nrole in generating constant optical torques. These fi ndings establish a foundational \nunderstanding for the precise control of nanoparticles’ dynamics, paving the way for \nadvancements in the exploration of mesoscopic dynamics and macroscopic quantum \neffects within the realm of optomechanics.  \nAs is shown in Fig.1 (a), our experimental setup includes an ellipticity -controllable \n1064 nm  trapping laser field and a linearly -polarized 532 nm detecting laser field. B y \nrotating the half -wave plate before several high -reflective mirrors, we manage to \nsimultaneously c ontrol ellipticities η and polarization azimuth angles θ0 of the 1064 nm \ntrapping laser field. B oth laser fields propagate along the z axis, and the 532 nm laser \nis linearly polarized on the x axis. θ (φ) represent the angle between the x (z) axis and \nthe projection of the nanodumbbell’s symmetrical axis in the xy (xz) plane. The lateral -\nscattered 532 nm power is measured to rapidly confirm the nanodumbbell’s \ngeometries36,37. The 1064 nm li ght forward -scattered from the nanodumbbell and the \nforward -propagating trapping laser beam, interfered with each other, which makes it \nfeasible to detect the nanodumbbell’s motions with far -field photodectors30,38.  As shown in Fig.1 (b) -(d), distinct features can be observed in librational/rotational \nPower Spectral Densities (PSDs) under ellipticities η = 0.0, 0.8 and 1.0. Only libration, \nindicated by the n arrow spectrum around 300 kHz, can exist under the linearly -\npolarized trapping laser field ( η = 0.0), while only  rotation, indicated by wide spectrums, \ncan exist under the circularly polarized trapping laser field ( η = 1.0). In particular, when \nη = 0.8, the corresponding PSD combines both librational and rotational spectrums, \nfrom which we can infer that the nanodumbbell is probable to undergo both libration \nand rotation  under this circumstance.  \n \nFig. 1 Experimental Overview.  (a) Experimental set up and definitions of angular coordinates. \nHWP: Half Wave Pl ate; DM: Dichromic Mirror; HRM: High -Reflective Mirror; OBJ: Objective \nLens; OF: Optical Filter; BS: Beam Splitter; PBS: Polarization Beam Splitter; QPD: Quadrant \nPhotodetector; BPD: Balanced Phot odetector. (b) A librational PSD when η = 0.0 . (c) A PSD \nrevealing both the libration and the rotation when η = 0.8. (d) A rotational PSD when η = 1.0. (b), \n(c) and (d) are measured under the vacuum degree of around 2 mbar and the trapping laser’s power \nof around 100 mW. Lib and Rot represent the librations and rotations respectively.  \nTo get a more particular knowledge of the librational/rotational dynamics under the \nmanipulation of ellipticities, we measure librational/rotational PSDs and lateral -\nscattered  532 nm powers while changing trapping laser fields' polarizations. As  shown \nin Fig.2(a), when the angles of the half -wave plate θλ/2 increase from  0°  to \napproximately 50° , the 1064 nm trapping laser field undergoes a transformation where \nits polarization azimuth angles θ0 nonlinearly increase from 0°  to nearly 90° , and \nsimultaneously, its ellipticities η vary from zero to a maximum and then back to zero. \nThe ellipticity η reaches the maximum of about 0.8 when the polarization azimuth angle \nθ0 approaches 45° . Inspired by the fact that lateral -scattered 532 nm powers can directly \nreflect the alignment of the nanodumbbell37, we extend lateral -scattered 532 nm powers \nto observe critical librational/rotational transitions in  real time. As shown in Fig.2(b), \nreferring to PSDs, the maxima and the minima of 532 nm scattered powers reveal the \nnanodumbbell’s  steady librations around x and y axes respectively, while intermediate \nvalues indicate the nanodumbbell’s continuous rotations about the z axis. When the \npolarization azimuth angle θ0 of the 1 064 nm trapping laser deviates slightly from the x \n(or y) axes and accompanied by moderate values of the ellipticity η, Fig.2( c) depicts \nthat lateral -scattered 532 nm powers exhibit stochastic interchange between the \nintermediate values and the maxima (or minima). The observed interchange signifies \nreal-time libration al/rotational transitions, and we define the corresponding ellipticities \nas “critical ellipticities”. Therefore, compared with PSDs, lateral -scattered 532 nm \npowers capture the librational/rotational dynamics in a rapid, effective and intuitive \nway.  \n \nFig. 2 Experimental results of polarization control and librational/rotational manipulations. \n(a) Variations of ellipticities  η and polarization azimuth angles  θ0 of 1064 nm trapping laser fields \nas the half -wave plate rotates. Inset: ellipticities η as a function of polarization azimuth angles θ0. \n(b) Referring to PSDs, lateral -scattered 532 nm powers are reliable criteria to extract the \nlibrational/rotational dynamics. The maxima and minima of scattered 532 nm powers reveal \nlibrations around x and y axes respectively while the intermediate values reveal rotations. (c) \nVisualized observations on librations, rotations and librational/rotational transitions with lateral -\nscattered 532 nm powers. (d) Critical librational/rotational transitions under different v acuum \ndegrees. (e) Critical librational/rotational transitions under different trapping laser powers. (f) \nZoom -in of dashed boxes in (e).  \n “Critical ellipticities” are crucial for explorations and manipulations on the \nmesoscopic dynamics of librational/ro tational motions. The critical ellipticities \nsubjected to four typical different vacuum degrees are shown in Fig. 2(d). A reduction \nin critical ellipticities correlates with the decrease in gas pressures. The observation can \nbe qualitatively explained with  damping torques from ambient gas molecules. Higher \ngas pressures will generate stronger damping torques, and thus hinder nanodumbbells’ \ntransitions from librations to rotations by restraining their librations’ angular \namplitudes, or else facilitate their transitions from rotations to librations by decelerating \nrotations. The effect of trapping laser powers on librational/rotational transitions is \nshown in Fig.2(e) and (f). It is obvious that the critical librational/rotational transition \nbehaviors become m ore noticeable as trapping laser powers increase. Intriguingly, \nFig.2(e) and (f) highlight a significant contrast in critical ellipticities between libration -\nto-rotation and rotation -to-libration transitions.  \nTo deeply visualize the contrast above, we tune the polarization azimuth angles in two \nopposite directions with the trapping laser power fixed to about 120 mW and the gas \npressure around 0.5 mbar. As a result, Fig.3 shows the noticeable hysteresis -loop \nstructure of critical librational/rotational tran sitions with respect to ellipticities. The \nhysteresis loop proves the bi -stability of critical librational/rotational transitions, which \nhighlights the significant impact of the nanodumbbell’s initial states in critical \nlibrational/rotational transition s.  \n  \nFig. 3 Experimental results of bi-stable hysteresis -loop structures of critical \nlibrational/rotational transitions.  (a) Lateral -scattered 532 nm powers are measured in response \nto tuning polarization azimuth angles θ0 of 1064 nm trapping laser fields at two opposite directions  \nunder the vacuum degree around 0.5 mbar and the trapping laser power around 120 mW. Arrows \nrepresent tuning directions of polarization azimuth angles θ0. (b) Zoom -in of dashed boxes in (a).  \nIn or der to provide quantitative illustrations for bi -stabilities in critical \nlibrational/rotational -transition dynamics, we carry out ab-initio calculations on optical \ntorques \noptτ  according to the angular -momentum -conservation law39-41(See Methods \nfor details) and subsequently numerically solve the librational/rotational Langevin \nequation(1):  \n2\n2ddIIdt dt=−  + +opt thermalθ θττ\n (1) \nwhere I and Γ stand for rotational inertias and damping rates, and τthermal  represents \nthermal -noise torques that can be expressed by a Gaussian stochastic process:\n()0thermal t =\n, \n()() () ' 2 'thermal thermal B t t Ik T t t   =− . During the calculations, only \nlibration/rotational motions around the z axis are considered and ellipticities of the \noptical fields are quasi -continuously increased or decreased. Calculated \nlibrational/rotational PSDs at the estimated vacuum around 0.5 mba r are shown in Fig. \n4. The SiO 2 nanoparticle’s complex refractive index is set to \n1.45875 0.00098ni=+\n42. \nAs is shown in Fig.4, under the laser power of 120 mW, there is an evident difference \nbetween critical ellipticities for libration -to-rotation and rotation -to-libration transitions, \nand under the laser power of 60 mW, the critical behavior and bi -stability become less \nobvious. Calculated results of bi -stabilities in Fig.4 perfectly agree with the \nexperimental results shown in Fig.2 and Fig.3.  \n \nFig. 4 Calculation results of librational/rotational bi -stabilities.  During the calculation, \nellipticities η are quasi -continuously decreased or increased, and arrows stand for tuning directions \nof ellipticities. (a) -(b) Calculated librational/rotational P SDs under the fixed trapping laser power \nof 120 mW and the vacuum degree of 0.5 mbar. (c) -(d) Calculated librational/rotational PSDs under \nthe trapping laser power of 60 mW and the vacuum degree of 0.5 mbar.  \nTo investigate physical mechanisms behind the bi -stability with respect to ellipticities, \nit is require d to analyze how optical torques are arisen. The widely -accepted dipole \nmodel divides optical forces (or torques) into two contributions named “gradient” and \n“scattering” ones33,43,44. It is uncontroversial that “gradient” forces (or torques) are \ndetermined by derivatives of conservativ e potentials arisen from polarizations. \n“Scattering” forces (or torques) involve complex mechanisms such as radiation \nreactions33,43, absorptions, etc.   \nFor driving steady librations of nanoparticles, anisotropic -polarization -induced \nsinusoidal optical torques  \naniτ  serve as restoring torques18,30, which corresponds with \nabovementioned “gradient” torques . Particularly, for the nanodumbbell’s librations in \nthe xy plane around the z axis, \n(), 0 0 sin 2ani z   =− − , where  \n0 equals to the \npolarization azimuth angle of the trapping laser in the xy plane, and \n0  is proportional \nto the laser power and jointly determined by the laser field’s ellipticity and the \nnanoparticles’ aniso tropic parameters . (See Methods for the theoretical model in detail).  \nReferring to driving continuous rotations of nanoparticles, previous studies considered \nradiation reactions as the sole cause to exert constant scattering torques33,45. However, \nowing to optical a bsorptions, the spin angular momenta carried by absorbed photons \nwill also exert constant torques \nabsτ  on nanoparticles46,47. Here, to reveal the role of \nabsorptions in constant  optical torques, we numerically calculate the biased constant \ntorques when the imaginary part κ of the nanoparticle’s complex refractive index \nn\n   \nchanges from 0 to 0.005 with a step of 0.0002 and the real part n is fixed at 1.4587 5. \nDuring the calculation, the power and ellipticity of the 1064nm trapping laser field are \nfixed at 100mW and 0.69 respectively. Calculated results are shown in Fig. 5. In Fig.5, \nit is evident that as the imaginary part of the complex refractive index inc reases, the \nabsorption becomes increasingly significant. When the imaginary part κ of complex \nrefractive index is zero, the biased constant optical torque remains at a magnitude of \n10-22 N∙m w hich is attributed to the radiation reaction43 and consistent with results \nreported in Reference33. The complex refractive index utilized in Fig.4 is highlighted. \nIt can be seen that the highlighted biased constant torqu e is on the magnitude of 10-21 \nN∙m , which is one order of magnitude larger than that solely induced by the radiation \nreaction.  Therefore , our calculations emphasize the dominant and non -negligible roles \nof absorptions to exert constant torques that drive continuous rotations of nanoparticles.   \nFig 5. Calculated biased constant torques as functions of κ. κ represent the imaginary part of the \nnanoparticle’s complex refractive index. Calculations are done under the trapping laser power of \n100 mW  and the ellipticity of 0.69. T he complex refractive index utilized in Fig.442 is highlighted. \nAs κ increases, the absorption becomes increasingly dominant in inducing the constant torque.  \nBased on the optical torques τopt above, the critical and bi -stable behaviours in \nlibrationa l/rotational transitions are straightforward to depict in the framework of \nwashboard potential48,49. Whether the nanodumbbell experiences steady librations, \ncontinuous rotations or critical librational/rotational transitions, is mainly determined \nby the barrier -trap potential difference, its initial librational/rotational kinetic energy \nand stochastic damping. When the ellipticity is relatively small, the barrier -trap \npotential difference is relatively high and sinusoidal torques \naniτ  play dominant roles, \nand the nanodumbbell tends to undergo librations. When the ellipticity is relatively \nlarge , the barrier -trap potential difference is relatively small , or else barriers and traps \nmay even vanish, and constant torques \nconstτ  become dominant and the nanodumbbell \nusually experiences rotations . At certain ellipticities with moderate barrier -trap \npotential difference, the significant stochastic damping from ambient gas molecules \naccounts for critical librational/r otational transitions. The balance between \naniτ  and \nconstτ\n from elliptically -polarized laser fields constitutes physical basis for \nmanipulating nanodumbbells’ librational/rotational transitions. Furthermore, the bi -\nstability arises essentially due to the effect of the initial librational/rotational kinetic \nenergy in librational/rotational transitions. (See Supplementary Section2 for more \ndetails)  \nIn conclusion, the librational and rotational dynamics of an optically -levitated \nnanodumbbell levitated in elliptically -polarized laser fields are studied. With lateral -\nscattered 532 nm powers revealing real -time behaviors of librational/rotational \ndynamics, we report the bi -stabilities of critical librational/rotational tra nsitions with \nrespect to trapping laser fields’ ellipticities. Such transitions are attributed to occur on \naccount of the balance between anisotropic -polarization -induced sinusoidal optical \ntorques and constant ones that are dominantly induced by absorptio ns. Significant \nstochastic collisions from gas molecules account for the dependence of critical \nellipticities on vacuum degrees. Within the framework of washboard potentials and \ntaking nanoparticles’ initial librational/rotational kinetic energy into accou nt, \nhysteresis -loop bi -stabilities are straightforwardly understood. Our observations \nestablish essential foundations for developing quantum precision measurement \ntechniques1,13,50 and exploring inherent frontier physical problems on mesoscopic laser -\nmatter interactions6,21,51 in the rotational degree of freedom.  \n  Reference  \n1 Bachtold, A., Moser, J. & Dykman, M. I. Mesoscopic physics of nanomechanical systems. Rev. \nMod. Phys.  94 (2022).  \n2 Kuang, T.  et al.  Nonlinear multi -frequency phonon lasers with active levitated optomechanics. \nNat. Phys.  19, 414 -419 (2023).  \n3 Shi, Y .  et al.  Inverse Optical Torques on Dielectric Nanoparticles in Elliptically Polarized Light \nWaves. Phys. Rev. Lett.  129, 053902 (2022).  \n4 Tebbenjohanns, F., Frimmer, M., Jain, V ., Windey, D. & Novotny, L. Motional Sideband \nAsymmetry of a Nanoparticle Optically Levitated in Free Space. Phys. Rev. Lett.  124, 013603 \n(2020).  \n5 Piotrowski, J.  et al.  Simultaneous ground -state cooling of two mechan ical modes of a levitated \nnanoparticle. Nat. Phys.  19, 1009 -1013 (2023).  \n6 Stickler, B. A., Hornberger, K. & Kim, M. S. Quantum rotations of nanoparticles. Nat. Rev. Phys.  \n3, 589 -597 (2021).  \n7 Zhu, S.  et al.  Nanoscale electric field sensing using a levitat ed nano -resonator with net charge. \nPhoton. Res.  11, 279 -289 (2023).  \n8 Monteiro, F.  et al.  Force and acceleration sensing with optically levitated nanogram masses at \nmicrokelvin temperatures. Phys. Rev. A  101, 053835 (2020).  \n9 He, Y .  et al.  High -sensitivity  force sensing using a phonon laser in an active levitated \noptomechanical system. Opt. Express  31 (2023).  \n10 Gieseler, J. & Millen, J. Levitated Nanoparticles for Microscopic Thermodynamics -A Review. \nEntropy (Basel)  20 (2018).  \n11 Yoneda, M. & Aikawa, K. Th ermal broadening of the power spectra of laser -trapped particles \nin vacuum. Journal of Physics B: Atomic, Molecular and Optical Physics  50 (2017).  \n12 Zheng, Y .  et al.  Robust Optical -Levitation -Based Metrology of Nanoparticle's Position and \nMass. Phys. Rev.  Lett. 124, 223603 (2020).  \n13 Gonzalez -Ballestero, C., Aspelmeyer, M., Novotny, L., Quidant, R. & Romero -Isart, O. \nLevitodynamics: Levitation and control of microscopic objects in vacuum. Science  374, \neabg3027 (2021).  \n14 Nieminen, T. A., Heckenberg, N. R. & Rubinsztein -dunlop, H. Optical measurement of \nmicroscopic torques. Journal of Modern Optics  48, 405 -413 (2001).  \n15 Ahn, J.  et al.  Ultrasensitive torque detection with an optically levitated nanorotor. Nat. \nNanotechnol.  15, 89-93 (2020).  \n16 Arita, Y ., Mazilu, M. & Dholakia, K. Laser -induced rotation and cooling of a trapped \nmicrogyroscope in vacuum. Nat. Commun.  4, 2374 (2013).  \n17 Bishop, A. I., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein -Dunlop, H. Optical \nmicrorheo logy using rotating laser -trapped particles. Phys. Rev. Lett.  92, 198104 (2004).  \n18 La Porta, A. & Wang, M. D. Optical torque wrench: angular trapping, rotation, and torque \ndetection of quartz microparticles. Phys. Rev. Lett.  92, 190801 (2004).  \n19 Rashid, M., Toros, M., Setter, A. & Ulbricht, H. Precession Motion in Levitated Optomechanics. \nPhys. Rev. Lett.  121, 253601 (2018).  \n20 Stickler, B. A.  et al.  Probing macroscopic quantum superpositions with nanorotors. New Journal \nof Physics  20 (2018).  21 Jiao, M., Guo, M. S., Rong, X., Cai, Y . F. & Du, J. F. Experimental Constraint on an Exotic \nParity -Odd Spin - and Velocity -Dependent Interaction with a Single Electron Spin Quantum \nSensor. Phys. Rev. Lett.  127, 010501 (2021).  \n22 Lechner, W., Habraken, S. J.,  Kiesel, N., Aspelmeyer, M. & Zoller, P. Cavity optomechanics of \nlevitated nanodumbbells: nonequilibrium phases and self -assembly. Phys. Rev. Lett.  110, \n143604 (2013).  \n23 Chen, X. Y ., Li, T. & Yin, Z. Q. Nonadiabatic dynamics and geometric phase of an ultr afast \nrotating electron spin. Sci. Bull (Beijing)  64, 380 -384 (2019).  \n24 Friese, M. E. J., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein -Dunlop, H. Optical \nalignment and spinning of laser -trapped microscopic particles. Nature  394, 348 -350 (1998).  \n25 Ahn, J. et al.  Optically Levitated Nanodumbbell Torsion Balance and GHz Nanomechanical \nRotor. Phys. Rev. Lett.  121, 033603 (2018).  \n26 Hoang, T. M.  et al.  Torsional Optomechanics of a Levitated Nonspherical Nanoparticle. Phys. \nRev. Lett.  117, 123604 (2016).  \n27 Reimann, R.  et al.  GHz Rotation of an Optically Trapped Nanoparticle in Vacuum. Phys. Rev. \nLett. 121, 033602 (2018).  \n28 Jin, Y .  et al.  6GHz hyperfast rotation of an optically levitated nanoparticle in vacuum. Photon. \nRes. 9, 1344 -1350 (2021).  \n29 Kuhn, S . et al.  Full rotational control of levitated silicon nanorods. Optica  4, 356 -360 (2017).  \n30 Seberson, T. & Robicheaux, F. Parametric feedback cooling of rigid body nanodumbbells in \nlevitated optomechanics. Phys. Rev. A  99, 013821 (2019).  \n31 van der Laan, F. et al.  Sub-Kelvin Feedback Cooling and Heating Dynamics of an Optically \nLevitated Librator. Phys. Rev. Lett.  127, 123605 (2021).  \n32 Bang, J.  et al.  Five-dimensional cooling and nonlinear dynamics of an optically levitated \nnanodumbbell. Phys. Rev. Resear ch 2, 043054 (2020).  \n33 Bellando, L., Kleine, M., Amarouchene, Y ., Perrin, M. & Louyer, Y . Giant Diffusion of \nNanomechanical Rotors in a Tilted Washboard Potential. Phys. Rev. Lett.  129, 023602 (2022).  \n34 Kim, P. H., Hauer, B. D., Doolin, C., Souris, F. & Davis, J. P. Approaching the standard quantum \nlimit of mechanical torque sensing. Nat. Commun.  7, 13165 (2016).  \n35 Tsaturyan, Y ., Barg, A., Polzik, E. S. & Schliesser, A. Ultracoherent nanomechanical resonators \nvia soft clamping and dissipation dilution. Nat. Nanotechnol.  12, 776 -783 (2017).  \n36 Li, C.  et al.  Fast size estimation of single -levitated nanoparticles in a vacuum optomechanical \nsystem. Optics Letters  46, 4614 -4617 (2021).  \n37 Li, C.  et al.  Structure characterization of nanoparticles with optical tweezers using scattering \nlight. Optics & Laser Technology  171, 110347 (2024).  \n38 Hebestreit, E.  et al.  Calibration and energy measurement of optically levitated nanoparticle \nsensors. Rev. Sci. Instrum.  89, 033111 (2018).  \n39 Barnett, S. M. & Loudon, R. On the electromagnetic force on a dielectric medium. Journal of \nPhysics B: Atomic, Molecular and Optical Physics  39, S671 -S684 (2006).  \n40 Raziman, T. V . & Martin, O. J. Internal optical forces in plasmonic nanostructures. Opt. Express  \n23, 20143 -20157 (2015).  \n41 Nieto -Vesperinas, M. Optical torque: Electromagnetic spin and orbital -angular -momentum \nconservation laws and their significance. Phys. Rev. A  92, 043843 (2015).  \n42 Rodríguez -de Marcos, L. V ., Larruquert, J. I., Méndez, J. A. & Aznárez, J. A. Self -consis tent optical constants of SiO_2 and Ta_2O_5 films. Optical Materials Express  6, 3622 -3637 (2016).  \n43 Novotny, L. & Hecht, B. Principles of Nano -Optics (Second Edition) .  (Cambridge University \nPress, 2012).  \n44 Li, T. Fundamental Tests of Physics with Optica lly Trapped Microspheres  doctor thesis, the \nUniversity of Texas, (2013).  \n45 Ahn, J. Spin Optomechanics of Levitated Nanoparticles  Doctor of Philosophy thesis, Purdue \nUniversity, (2020).  \n46 Friese, M., Enger, J., Rubinsztein -Dunlop, H. & Heckenberg, N. R. Optical angular -momentum \ntransfer to trapped absorbing particles. Phys. Rev. A  54, 1593 (1996).  \n47 Ricci, F.  et al.  A Chemical Nanoreactor Based on a Levitated Nanoparticle in Vacuum. ACS \nNano  16, 8677 -8683 (2022).  \n48 Lindenberg, K., Sancho, J. M., Lacasta, A. M. & Sokolov, I. M. Dispersionless transport in a \nwashboard potential. Phys. Rev. Lett.  98, 020602 (2007).  \n49 Tatarkova, S. A., Sibbett, W. & Dholakia, K. Brownian particle in an opti cal potential of the \nwashboard type. Phys. Rev. Lett.  91, 038101 (2003).  \n50 Ju, P.  et al.  Near -Field GHz Rotation and Sensing with an Optically Levitated Nanodumbbell. \nNano. Lett.  23, 10157 -10163 (2023).  \n51 Delord, T., Huillery, P., Nicolas, L. & Hetet, G.  Spin-cooling of the motion of a trapped diamond. \nNature  580, 56-59 (2020).  \n \n \n  Methods  \nExperimental Details  \nA 1064  nm trapping laser field (Precilasers, YDFL -1064 -SF-10-CW) and a linearly -\npolarized 532  nm detecting laser field (LIGHTHOUSE, Spout -D-5W) are  applied in \nour experimental setup. These two laser fields propagate parallelly with each other and \nare tightly focused by an OBJ lens (Nikon, MUE21900, NA=0.8). A sili ca \nnanodumbbell  is trapped in the 1064  nm focal region inside a vacuum chamber. We \ninsta ll a camera (DataRay, S -WCD -LCM -C) directly over the nanodumbbell to \nmeasure its lateral scattered 532  nm power. The forward -propagated 1064  nm light is \ncollected by a high -precision aspherical lens (Thorlabs, AL1512 -C, NA =0.55 ) and then \nis split to two pa rts. One part is detected by a custom -built QPD  to monitor the \nnanodumbbell’s translational displacements. The other part successively go es through \na 1064  nm HWP  (Thorlabs, WPH10M -1064) and a PBS (Thorlabs, CCM1 -PBS253/M), \nand is finally sent to a BPD (Tho rlabs, PDB450C -AC) to extract the nanodumbbell’s \nlibrational/rotational motions.  \nThe power of the 532 nm detection laser is deliberately maintained at a substantially \nlower level (~10 mW) compared to the 1064 nm trapping laser (~60 -120 mW) to ensure \nthat the control exerted by the 1064 nm trapping laser over the state of the trapped \nparticle is not compromised. The 532  nm laser field is kept linearly polarized along the \nx axis. A 1064  nm HWP is installed on a stepper motor rotation mount (Thorlabs, \nK10CR1/ M) before several HRMs. HRMs are coated with dielectric films. By rotating \nthe HWP, it is achievable to simultaneously control ellipticities η and polarization \nazimuth angles θ0 of the 1064  nm trapping laser. Polarizations of both beams are \nmeasured with a  compact polarimeter (Thorlabs, PAX1000) before conducting the \nexperiments.  \nNanoparticle Trapping  \nNanoparticle s were loaded into the optical trap by using an ultrasonic nebulizer  (Omron, \nNE-C25S ). Nanoparticle s were trapped at atmospheric pressure . A nanodumbbell is constituted with two adhered nanospheres  (NanoCym, monodisperse silica \nnanoparticles with the nominal diameter of 150 nm). To identify  a nanodumbbell, two \nmethods are usually applied. One refers to the ratio between translational damping ra tes \nalong y and x axes in a trapping laser field linearly polarized on the x axis, \nyx , and \nspecifically speaking, for a nanodumbbell, \nyx  approaches 1.3. The other involves \nvariations of lateral -scattered 532  nm powers in response to changes of polarization \nazimuth angles θ0 of 1064  nm trapping laser fields. When θ0 changes from 0 ° to 90 °, \nfor a nanodumbbell, lateral -scattered 532  nm powers undergoes a changing ratio of \napproximately 65%.  \nData Acquiring and Processing  \nThe motional PSDs of the nanodumbbell are extracted with t he forward -propagated \n1064  nm light. The light contains two contributions, the 1064  nm light forward -\nscattered from the nanodumbbell and the forward -propagating trapping laser beam. The \ninterference between them constitutes the fundament for extracting the nanodumbbell’s \nmotions. Referring to a dipole -moment scattering model, the detected signal from the \nQPD is directly proportional to the nanodumbbell’s translational displacement, and the  \nsignal from the BPD is proportional to \n2sin 2 sin . The detected signals are recorded \nwith the Data Acquiring Module inside a Phase -locked Amplifier (Zurich Instruments, \nMFLI). Processing the detected signals with Fourier transform, motiona l PSDs are thus \nmeasured.  \nThe lateral -scattered 532  nm powers, which directly reveal real -time behaviors of \nlibrational/rotational dynamics of the nanodumbbell, are simultaneously acquired with \nangles of the HWP \n2 . This part of experimental data is obtained with programs \ndeveloped by ourselves.  \nAb-initio Numerical Calculations of Optical Torques  \nThe specific calculating steps are as follows.  \nStep1:  calculating  the tightly -focused trapping optical  field The tightly -focused trappin g optical field is calculated at first , serving as  the \nbackground optical field to extract the laser -nanoparticle interaction .   \nA collimated elliptically -polarized optical laser beam can be expressed as,  \n( )()2 2 2\n0\n00ˆ ˆ =x y w\nxyE E e−++collimated x yE e e\n (1) \nwhere w0 stands for the radius of the beam, \n0xE , \n0yE  stand for complex amplitudes  \nof optical fields along the x and y directions respectively. When it is tightly focused by \na focal lens with  high numerical aperture  NA, the optical field near the focus is given \nby Equation (2) according to the vector diffraction reported by Richards and wolf52,53,  \n() \n 0 00 02 0 02\n0 02 0 00 02\n0 01 0 01cos 2 sin 2\n, , sin 2 cos 222 cos 2 sinxy ikf\nxy\nxyE I I E I\nikfer z E I E I I\niE I iE I\n  \n−++\n= + −\n−−focusE\n (2) \nwhere \n22r x y=+ ,\narctanx\ny= , and f is the focal length of the focal lens, and,  \n() ()()()\n() ()()\n() ()()()max222\n0\nmax222\n0\nmax222\n012 sin cos\n00 0\n0\n12 sin 2 cos\n01 1\n0\n12 sin cos\n02 2\n0, cos sin 1 cos sin\n, cos sin sin\n, cos sin 1 cos sinfw ikz\nfw ikz\nfw ikzI r z e J kr e d\nI r z e J kr e d\nI r z e J kr e d\n \n\n \n\n     \n   \n    −\n−\n−\n=+\n\n= \n\n\n=−\n\n\n (3) \nwhere \nmax arcsin NA = , Ji stands for the i -th order Bessel Function.  \nStep2: solving Maxwell Equations  \nTo analyze the optical scattering from the nanoparticle, the whole optical field \nE  are \nusually separated into two parts: the background field \nbgE  and the scattering field \nscatE\n, and thus  \n=+bg scat E E E . Concretely, the  tightly -focused elliptically -polarized \noptical fields in Equation (2) are background fields, \n=bg focusEE , and the whole optical \nfield \nE  fulfills Maxwell equations . In this work, Maxwell equations are numerically \nsolved in a micrometer space  with the finite element method.   \nStep3: calculating optical torques  Through the abovementioned two steps, we get both the electric field \nE  and the \nmagnetic field \nH  of optical fields . Therefore , integrating the angular momentum \ncurrent density of electromagnetic fields over a closed surface surrounding the \nnaondumbbell  S, we can the optical torques\noptτ  shown in Equation  (4). \nSdS =   optτ r T n\n (4) \nwhere \nn  represents corresponding unit normal vector, and \nT\n  is Maxwell Tensor,  \n( )11= Re22   + −  + * * * *T EE HH E E H H I\n (5) \nwhere ε, μ and \nI\n  stand for dielectric constant, permeability and unit tensor \nrespectively.  \n \nTheoretical Model  \nIn the maintext, it has been mentioned that optical torques \noptτ  are composed of two \ndominant contributions \n=+opt ani absτττ . Insightfully, from the perspective of laser -\nnanoparticle interactions, anisotropic polarizations and absorptions are two dominant \nmechanisms to achieve the angular momentum transfer. Anisotropic polarizations \nessentially involve the exchange of angular moment a between optical fields and \nnanoparticles, while absorptions denote unidirectional angular momentum transfer \nfrom optical fields to nanoparticles. The former causes changes in the macroscopic \npolarization states of optical fields, while the latter results  in energy loss of optical fields.  \nAnisotropic polarizations are mainly arisen from birefringence and anisotropic \ngeometrical di mensions. The former is mainly investigated on crystal particles, while \nthe latter is prominent on non -spherical particles su ch as dumbbells, rods, and ellipsoids . \nIgnoring the spin of the nanodumbbell’s symmetrical axis,  \naniτ  is determined by total \nconservative optical potential  \naniU , ani aniUU\n=− −ani y zτ ee (6) \nwhere \n1\n2aniU=− ani incpE , \n1−=ani incpR αRE\n  represents the  induced dipole \nmoment , \n( ) ˆ ˆ ˆ ++jt\nxyzE E E e=inc x y zE e e e represents  the tightly -focused trapping optical \nfield,  \n() ,,x y z diag  =α\n  stands for the polarizability tensor in the particle frame , \nand \nR\n  stands for the rotation matrix to transform the particle frame to the laboratory \nframe.  \ncos 0 sin cos sin 0\n0 1 0 sin cos 0\nsin 0 cos 0 0 1   \n\n−   \n  =−      R\n (7) \nIn our  mathematical derivation, we elucidate components of \naniτ  motivating \ntorsions/rota tions in the xz plane around the y axis and in the xy plane around the z axis, \nand results are shown in Equation ( 8),  \n()()( )()\n()()()()2 2 22 **\n,0\n2 2 *2 *2\n,01+4 cos 24\n1= sin 24ani y x z x z x z x z\nani z y z x y x yE E E E E E\nE E E E    \n    =− − − + −\n− + + − \n (8) \nwhere \n( )22\n0 **arctan\n2xz\nx z x zEE\nE E E E−=\n+  approximates \n2  based on the assumption \nxzEE\n and \n**\n0 2 21arctan2x y x y\nxyE E E E\nEE+=\n−  represents the polarization azimuth angle \nof the trapping l aser fields. Th erefore, the conservative components of  \naniτ  show \nsinusoidal  dependences on the nanodumbbell’s  angular displacements , and  the \namplitudes of \naniτ  decrease with ellipticities of the optical field. Moreover, \naniτ  \ntends to drive the nanodumbbell’s librations on two degrees of freedom with \nequilibrium angle s \n0 equ=  and \n2equ=  respectively . absτ is determined by the total spin angular momenta of absorbed photons per unit time,  \nabs laser\nlaserI\n=absτ s\n (9) \nwhere \nabs , \nlaserI  and \nlaser  stand for the absorption cross -section, the intensity and \nangular frequency of the trapping laser field respectively, and \ns  represents the mean \nspin angular momentum of a single photon. The absorption cross -section \nabs  is \ndetermined by the nanoparticle’s geometry, the volume V, the complex refraction index \nn n i=+\n and the wavelength λ. The mean spin angular momentum of elliptically -\npolarized optical fields \ns  is derived based on the spin angular momentum operators \nand the normalized tightly -focused elliptically -polarized optical field.  \nThe normalized optical fiel d vector is shown in Equation (10).  \n2 221x\ny\nx y zzE\nE\nE E E E\n= ++inc,normalizedE\n. (10) \nThe spin angular momentum operators are shown in Equation (11).  \n0 0 0 0 0 0 0\nˆ ˆ ˆ 0 0 , 0 0 0 , 0 0\n0 0 0 0 0 0 0ii\nii\nii−      \n     = − = =          −      x y zs s s\n. (11) \nTherefore, the mean spin angular momentum can be derived  as shown in Equation (12).  \n**\n**\n2 22\n**ˆ\nˆ =\nˆy z y z\nx z z x\nx y zx y x yE E E E\niE E E E\nE E EE E E E    −\n   = − +   ++     −   *\ninc,normalized x inc,normalized\n*\ninc,normalized y inc,normalized\n*\ninc,normalized z inc,normalizedE s E\ns E s E\nE s E\n (12) \nGiven that longitudinal optical field \nzE  is far lower than transverse optical fields \nxE\n and \nyE , \nabsτ  is dominant along the z axis. The derivations above provide a \nquantitative explanation why \n,abs z increase s with the ellipticity of the optical field.  \n \nReferences  \n52 Richards, B. & Wolf, E. Electromagnetic diffraction in optical systems, II. Structure of the \nimage fi eld in an aplanatic system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences  253, 358 -379 (1959).  \n53 Wolf, E. Electromagnetic diffraction in optical systems -I. An integral representation of the \nimage field. Proceedings of the Royal Society of London. Series A. Mathematical Physical \nSciences  253, 349 -357 (1959).  \n \nData availability  \nSource data are available for this paper. All other data that support the plots within this \npaper and other findings of this study are available from the corresponding authors upon \nreasonable request.  \n \nAcknowledgements  \nWe gratefully acknowledge valuable advice from Leiming Zhou at Hefei University of \nTechnology and Xingguang Liu at Harbin Institute of Technology. Cuihong Li is \nsupport ed by the National Natural Science Foundation of China (NSFC, grant no. \n42004154). Chaoxiong He is supported by the NSFC (grant no. 62305305). Huizhu Hu \nis supported by the Major Scientific Project of Zhejiang Lab ( 2019MB0AD01 ). \nZhangqi  Yin is supported by Beijing Institute of Technology Research Fund Program \nfor Young Scholars. Fu Feng is supported by the NSFC (grant no. 62275167 , \n92250304 ). \n \nAuthor contributions  \nCuihong Li and Zhangqi Yin conceived the idea. Jinchuan Wang, Shaochong  Zhu, and \nYuanyuan Ma  jointly performed the experiments. Qianwen Ying designed the  data \nacquiring system . Chaoxiong He analysed the experimental data, performed numerical \ncalculations, constructed the theoretical model, and wrote the manuscript with the \ncontributions from Cuihong Li , Ying Dong, Zhangqi Yin,  Fu Feng and Huizhu Hu. \nHuizhu Hu, Cuihong Li and Chaoxiong He supported the project.  "    },    {        "title": "2402.05492v1.Cosmological_Forecast_of_the_Void_Size_Function_Measurement_from_the_CSST_Spectroscopic_Survey.pdf",        "content": "MNRAS 000, 1–10 (2015) Preprint 9 February 2024 Compiled using MNRAS L ATEX style file v3.0\nCosmological Forecast of the Void Size Function Measurement from the\nCSST Spectroscopic Survey\nYingxiao Song1,2, Qi Xiong1,2, Yan Gong1,2,3★, Furen Deng1,2, Kwan Chuen Chan6,7,\nXuelei Chen1,2,4,5, Qi Guo1,2, Jiaxin Han8,9,10, Guoliang Li11, Ming Li1, Yun Liu1,2, Yu Luo11,\nWenxiang Pei1,2, and Chengliang Wei11\n1National Astronomical Observatories, Chinese Academy of Sciences,20A Datun Road, Beijing 100012, China\n2School of Astronomy and Space Sciences, University of Chinese Academy of Sciences(UCAS),Yuquan Road NO.19A Beijing 100049, China\n3Science Center for China Space Station Telescope, National Astronomical Observatories, Chinese Academy of Sciences,\n20A Datun Road, Beijing 100101, China\n4Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China\n5Centre for High Energy Physics, Peking University, Beijing 100871, China\n6School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Road, Tangjia, Zhuhai, 519082, China\n7CSST Science Center for the Guangdong-Hongkong-Macau Greater Bay Area, SYSU, Zhuhai, 519082, China\n8Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China\n9Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai 200240, China\n10Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China\n11Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210023, PR China\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nVoid size function (VSF) contains the information of the cosmic large-scale structure (LSS), and can be used to derive the\nproperties of dark energy and dark matter. We predict the VSFs measured from the spectroscopic galaxy survey operated by\nChina Space Station Telescope (CSST), and study the strength of cosmological constraint. We employ a high-resolution Jiutian\nsimulation to get galaxy samples based on an improved semi-analytical model, and then generate a mock galaxy catalog of\nthe CSST spectroscopic survey according to the detection sensitivity. We identify voids from this galaxy catalog using the\nwatershed algorithm without assuming a spherical shape, and estimate the VSFs at different redshift bins from 𝑧=0.5to 1.1.\nTo obtain a reliable and accurate fitting result, we propose a void selection method based on the ellipticity, for comparing to\nthe theoretical model with a linear underdensity threshold of void formation 𝛿vassuming the spherical evolution. We assume\n𝛿vis redshift-dependent and set it as a free parameter in each redshift bin. The Markov Chain Monte Carlo (MCMC) method is\nadopted to implement the constraints on the cosmological and void parameters. We find that the VSFs from the selected voids\ncanbewellfittedbythetheoreticalmodel,andcouldaccuratelyreservethecosmologicalinformation.Basedonourestimation,\nthe VSF measurement of the CSST spectroscopic survey can constrain the cosmological parameters to a few percent level. The\nbest-fit values of 𝛿vare ranging from∼−0.4to−0.1as the redshift increases from 0.5 to 1.1, which has a distinct difference\nfrom the theoretical calculation with a constant 𝛿v≃−2.7assuming the spherical evolution. Our method can provide a good\nreference for the void identification and selection in the VSF analysis of the spectroscopic galaxy surveys.\nKey words: Cosmology – Large-scale structure of Universe – Cosmological parameters\n1 INTRODUCTION\nIn the past decades, thanks to high-precision cosmological obser-\nvations and powerful simulations, the study of cosmic large-scale\nstructure (LSS) has made significant progress. Cosmic void, also\nknownasthelow-densityregionsintheUniverse,hasbeguntoshow\nitsgreatpotential,andisbecominganeffectiveandcompetitiveprobe\nin cosmological research (Pisani et al. 2019; Moresco et al. 2022;\n★Email:gongyan@bao.ac.cnSchusteretal.2023a,b).Sincecosmicvoidhasthecharacteristicsof\nlargevolume,lowdensityandlinearevolution,itissuitabletoextract\ninformationoftheevolutionofcosmicLSS,andexplorethestructure\ngrowthrate,natureofdarkenergyandmodifiedgravitytheories(e.g.\nCai et al. 2015; Pisani et al. 2015; Zivick et al. 2015; Pollina et al.\n2016; Achitouv 2016; Sahlén et al. 2016; Falck et al. 2018; Sahlén\n&Silk2018;Paillasetal.2019;Pericoetal.2019;Verzaetal.2019;\nContarini et al. 2021; Mauland et al. 2023).\nCosmic voids are sensitive to several cosmological geometrical\nanddynamicaleffects,likeAlcock-Paczyńskieffect(e.g.Sutteretal.\n©2015 The AuthorsarXiv:2402.05492v1  [astro-ph.CO]  8 Feb 20242 Y. Song et al.\n2012, 2014b; Hamaus et al. 2016; Mao et al. 2017; Correa et al.\n2021; Hamaus et al. 2022), baryonic acoustic oscillations (BAO)\n(e.g. Kitaura et al. 2016; Liang et al. 2016; Chan & Hamaus 2021;\nForero-Sánchez et al. 2022; Khoraminezhad et al. 2022), and red-\nshift space distortions (RSD) (e.g. Paz et al. 2013; Cai et al. 2016;\nHamaus et al. 2017; Chuang et al. 2017; Nadathur & Percival 2019;\nNadathur et al. 2020; Correa et al. 2022). Various current and up-\ncoming galaxy spectroscopic surveys can be or have been used to\nstudy cosmic voids, such as Baryon Oscillation Spectroscopic Sur-\nvey (BOSS, Alam et al. 2017) and Extended Baryon Oscillation\nSpectroscopic Survey (eBOSS, Dawson et al. 2016) from the Sloan\nDigital Sky Survey (SDSS, Blanton et al. 2017), Dark Energy Sur-\nvey(DES,DarkEnergySurveyCollaborationetal.2016),andDark\nEnergy Spectroscopic Instrument (DESI, DESI Collaboration et al.\n2016).\nBesides, the next-generation spectroscopic surveys are capable to\nmap deeper Universe in higher precision, and hundred millions of\ngalaxy spectra will be obtained, e.g. MegaMapper (Schlegel et al.\n2019), WideField Spectroscopic Telescope (WST, Ellis & Dawson\n2019), the MaunaKea Spectroscopic Explorer (MSE, Percival et al.\n2019), MUltiplexed Survey Telescope1(MUST), etc. In addition,\ntheslitlessspectroscopicobservationscarriedoutbyStage-IVspace-\nbornesurveyswouldalsoprovidevaluableandenormousinformation\nabout the cosmic LSS, such as Euclid(Euclid Collaboration et al.\n2022),NancyGraceRomanSpaceTelescope(RST),andChinaSpace\nStationTelescope(CSST)(Zhan2011,2021;Gongetal.2019;Miao\netal.2023).Thesesurveysundoubtedlycanincreasethecosmicvoid\nsamplesdramatically,andinthemeantime,makehigherrequirement\nfor the data analysis of accurately extracting information.\nInthiswork,westudythecosmologicalconstraintsofthemeasure-\nmentsofvoidsizefunction(VSF)bytheCSSTspectroscopicgalaxy\nsurvey.TheVSFdenotesthenumberdensityofvoidsasafunctionof\nsize at a given redshift, which can reflect the properties of the LSS.\nThe theoretical model of the VSF is constructed by studying the\nhierarchical evolution of the void population within the framework\nof excursion-set theory (Sheth & van de Weygaert 2004) and later\nextended to the volume conserving model by Jennings et al. (2013).\nThevoidsizefunctionhasbeenwidelyusedingalaxyspectroscopic\nsurveysandcosmologicalsimulationstoconstraincosmology(Pisani\netal.2015;Contarinietal.2021,2022,2023;Pelliciarietal.2023).\nThe CSST is a 2m space-based telescope. It will cover a sky area\nof17500deg2withbothmulti-bandphotometricimagingandslitless\ngrating spectroscopic surveys in about ten years. It has seven photo-\nmetric bands (i.e. 𝑁𝑈𝑉,𝑢,𝑔,𝑟,𝑖,𝑧and𝑦) and three spectroscopic\nbands (i.e.𝐺𝑈,𝐺𝑉and𝐺𝐼), covering wavelength range 250-1000\nnm. The CSST spectroscopic survey can reach a magnitude limit\n∼23ABmagfor5 𝜎pointsourcedetection.Theangularresolutionis\n∼0.3′′within80%energyconcentrationradiusforthespectroscopic\nsurvey,andthespectralresolution 𝑅=𝜆/Δ𝜆isbetterthan200.More\nthanonehundredmilliongalaxyspectraareexpectedtobemeasured\nat0<𝑧< 2.SotheCSSTspectroscopicsurveywouldbeapowerful\nobservation for studying the evolution of the LSS.\nWe first generate the mock galaxy catalog using simulations and\nconsideringtheCSSTinstrumentaldesignandstrategyoftheCSST\nspectroscopic survey. Then we identify voids and create the void\nmock catalog adopting a method based on Voronoi tessellation and\nwatershed algorithm, and derive the information of void volume-\nweightedcenter,effectiveradius,ellipticity,etc.Theshapesofthese\nvoids are relatively arbitrary without assuming a simple spherical\n1https://must.astro.tsinghua.edu.cn/enshape. In order to obtain an accurate result, we also select a ‘high-\nquality’ void sample for the VSF analysis according to the void\nellipticity. After computing the theoretical VSF based on the halo\nmodel, we compare the mock observational VSFs with theoretical\nones at different redshift bins from 𝑧=0.5to 1.1, and the Markov\nChainMonteCarlo(MCMC)methodisadoptedtoperformthecon-\nstraints.Weconstrainthecosmologicalparameters,suchasthetotal\nmatter density parameter Ωmand dark energy equation of state 𝑤,\nand the underdensity thresholds for void formation 𝛿vby using the\nVSF mock data at different redshifts.\nThe paper is organized as follows: In Section 2, we introduce the\nsimulation we use, the methods of generating the mock galaxy and\nvoidcatalogsoftheCSSTspectroscopicgalaxysurveys;InSection3,\nwediscussthecalculationandestimationofthetheoreticalandmock\nobservational VSFs; In Section 4, we predict the constraints on rel-\nevantcosmologicalparametersandvoidparameters;Wesummarize\nour work in Section 5.\n2 MOCK CATALOGS\n2.1 Simulation\nWeemploythehigh-resolutionN-bodysimulationsfromJiutiansim-\nulation to study the expected void size function. The cosmological\nparametersarefrom Planck2018(PlanckCollaborationetal.2020),\nwith the density parameters of total matter Ωm=0.3111, baryon\nΩb=0.0490,anddarkenergy ΩΛ=0.6899,theamplitudeofmatter\nfluctuation𝜎8=0.8102, spectral index 𝑛s=0.9665, and the di-\nmensionless Hubble constant ℎ=0.6766. The simulation is carried\nout with 61443particles in a box of 1 ℎ−1Gpc per side using the\nGADGET-3code(Springeletal.2001;Springel2005),andthemass\nof each particle is about 3.73×108ℎ−1𝑀⊙. The simulation begins\nat initial redshift 𝑧𝑖=127with 128 snapshots outputting between\n𝑧𝑖and𝑧=0. The friend-of-friend and subfind algorithm are used\nto identify the dark matter halos and substructures (Springel et al.\n2001; Springel 2005).\nSince the number of high-redshift galaxies that can be detected\nby the CSST spectroscopic survey is limited (Gong et al. 2019),\nwe only consider sources at 𝑧 < 1.5in our void size func-\ntion discussion. Specifically, we choose snapshot output at redshift\n𝑧=[0.1,0.3,0.5,0.7,0.9,1.1,1.3,1.5]tobuildourgalaxyandvoid\ncatalogs.\n2.2 Galaxy mock catalog\nThemockgalaxycatalogisbuiltbyimplementinganimprovedSemi-\nAnalyticModel(Henriquesetal.2015).Thedatabasecontainsgalaxy\nemission line luminosity produced by post-processing as described\nin Pei et al. (2024) (in preparation), which can be used to precisely\nmeasurethegalaxyredshift,andselectgalaxiesthatcanbedetected\nby the CSST spectroscopic survey. We choose four strong emission\nlines, i.e. H𝛼, H𝛽, [OIII] and [OII], as indicators to estimate the\nsignal-to-noise ratio (SNR), and then decide whether a galaxy can\nbe detected according to the detection threshold. For simplicity, we\ntreateachgalaxyasapointsource,sincetheregionofemittinglines\nisusuallysmallcomparedtothefullsizeofanemission-linegalaxy.\nFollowing Cao et al. (2018) and Deng et al. (2022), the SNR per\nspectralresolutionunitforthespectroscopicsamplecanbeestimated\nMNRAS 000, 1–10 (2015)VSF from CSST 3\nby\nSNR=𝐶s𝑡exp√︁𝑁exp√︃\n𝐶s𝑡exp+𝑁pix[(𝐵sky+𝐵det)𝑡exp+𝑅2n], (1)\nwhere𝑡exp=150 sis the exposure time, 𝑁exp=4is the number\nof exposures (Gong et al. 2019), 𝑁pix= Δ𝐴/𝑙2pis the number of\ndetector pixels covered by an object. Here Δ𝐴is the pixel area on\nthe detector, assumed to be the same for all galaxies for simplicity,\nand𝑙p=0.074′′is the pixel size. The point-spread function (PSF)\nis assumed to be a 2D Gaussian distribution with the radius of 80%\nenergyconcentration ∼0.3′′intheCSSTspectroscopicsurvey. 𝐵det\n= 0.02𝑒−s−1pixel−1is the dark current of the detector, and 𝑅n= 5\n𝑒−s−1pixel−1is the read noise. 𝐶sis the counting rate from galaxy,\nand for emission line 𝑖at frequency 𝜈𝑖, we have\n𝐶𝑖\ns=𝐴eff𝑇𝑋\u0012𝜈𝑖\n𝑧+1\u0013𝐹𝑖\nline\nℎ𝜈𝑖/(𝑧+1), (2)\nwhere𝐴eff=3.14m2is the CSST effective aperture area, 𝑇𝑋is the\ntotal throughput for band 𝑋including filter intrinsic transmission,\ndetector quantum efficiency, and mirror efficiency, and 𝐹𝑖\nlineis the\nfluxoftheemissionline 𝑖whichcanbeobtainedfromthesimulation.\n𝐵skyin Equation (1) is the sky background in 𝑒−s−1pixel−1, which\nis given by\n𝐵sky=𝐴eff∫\n𝐼sky(𝜈)𝑇X(𝜈)𝑙2\np𝑑𝜈\nℎ𝜈, (3)\nwhere𝐼skyisthesurfacebrightnessoftheskybackground,including\nearthshine and zodiacal light (Ubeda 2011). We find that 𝐵sky=\n0.016, 0.196, and 0.266 𝑒−s−1pixel−1for𝐺𝑈,𝐺𝑉and𝐺𝐼bands,\nrespectively.\nWe select galaxies if SNR≥10for any emission line of the four\nlinesH𝛼,H𝛽,[OIII]and[OII]inanyspectroscopicbandtoformthe\ngalaxymockcatalog.Wefindthatthenumberdensityofgalaxiesare\n¯𝑛=4.3×10−2,1.3×10−2,1.5×10−3,6.1×10−4,4.8×10−4,1.0×\n10−4,3.2×10−5, and 1.8×10−6ℎ3Mpc−3for the eight snapshots\nwechoosefrom 𝑧=0.1to1.5,respectively.Thesenumberdensities\nare consistent with the results for the CSST spectroscopic survey\nderived from the zCOSMOS catalog (Gong et al. 2019).\n2.3 Void mock catalog\nWeidentifycosmicvoidswithVoidIDentificationandExamination\ntoolkit2(VIDE,Sutteretal.2015),awatershedvoidfindingalgorithm\nbased on ZOnes Bordering On Voidness ( ZOBOV, Neyrinck 2008).\nThis code first estimates a density field from the tracer distribution\nusingVoronoitessellation,andthenappliesawatershedalgorithmto\nfind localized density minima. (Platen et al. 2007). No assumptions\naremadefortheshapeoftheidentifiedvoidsby VIDE,sothatwecan\nfind more low density areas as voids with natural shapes. VIDEcan\nprovidevoidpropertiessuchasthevolume-weightedcenter,effective\nradius and ellipticity, which is quite helpful in void analysis. It has\nbecome an important tool in the studies of large-scale structure of\nthe Universe with applications in various void analyses (Sutter et al.\n2012, 2014b; Hamaus et al. 2016; Contarini et al. 2022, 2023).\nWeusethegalaxycatalogdescribedinSection2.2tosearchvoids\nandconstructthevoidmockcatalog.Bymakinguseof VIDE,wefind\nlocal minima of the density field as the locations of voids, and the\n2https://bitbucket.org/cosmicvoids/vide_public/src/\nmaster/\nFigure 1. Thevoidellipticitydistributionfordifferentredshiftbins.Thecolor\nfrom blue to red shows the void ellipticity distribution at redshift bins from\n0.1 to 1.5.\nvoidboundariesaredeterminedbythewatershedbasinsofthedensity\nfieldaroundtheminimawithnomergingofneighbouringzones.As\na result, the galaxies in our mock catalog is divided by overdense\nridges, and forms separate underdensity areas as voids. Each void\nis made up of individual cells obtained by Voronoi tessellation, and\neachcellcontainsaparticletracer,i.e.agalaxy.Thecellhasitsown\nvolume𝑉cell,andthedensityofacellcanbeestimatedby 𝜌=1/𝑉cell.\nTheradiusofavoid 𝑅visobtainedfromaneffectivesphere,whose\nvolume is the same as the total volume of all cells in a void, and the\nvoid effective radius 𝑅effis given by\n𝑅v=𝑅eff=(3𝑉\n4𝜋)1\n3, (4)\nwhere𝑉=Í𝑉𝑖\ncellis the void volume. The volume-weighted center\nof the𝑁Voronoi cells Xvcan be calculated by\nXv=1\n𝑉𝑁∑︁\n𝑖=1x𝑖𝑉𝑖\ncell, (5)\nwhere x𝑖is the position of the galaxy in cell 𝑖for a given void. We\nalsoestimatevoidshapesbytakingaccountofvoidmembergalaxies\nand constructing the inertia tensor:\n𝑀xx=𝑁∑︁\n𝑖=1(𝑦2\n𝑖+𝑧2\n𝑖), 𝑀xy=−𝑁∑︁\n𝑖=1𝑥𝑖𝑦𝑖, (6)\nwhere𝑀xxand𝑀xyare the diagonal and off-diagonal components,\nand𝑥𝑖,𝑦𝑖,and𝑧𝑖arethecoordinatesofthegalaxyincell 𝑖relativeto\nthe void volume-weighted center. The other components, i.e. 𝑀yy,\n𝑀zz,𝑀xz, and𝑀yz, also can be calculated using Equation (6). The\ninertia tensor reflects the distribution of galaxies inside the void,\nand the galaxy near the edges of the void will be given a higher\nweight. Then we use the inertia tensor to compute eigenvalues and\neigenvectors, and evaluate the ellipticity by\n𝜖=1−\u0012𝐽1\n𝐽3\u00131/4\n, (7)\nwhere𝐽1and𝐽3are the smallest and largest eigenvalues of the\ninertia tensor. For the ellipticity, we have 0< 𝜖 < 1, and the void\nshape is flatter when 𝜖becomes larger. We show the void ellipticity\nMNRAS 000, 1–10 (2015)4 Y. Song et al.\nTable 1. The number densities of galaxies and voids ( >5ℎ−1Mpc) of our\nmock catalogs in different redshift bins. In this sample, the number densities\nof voids with𝜖 < 0.15are also shown.\nRedshift Galaxy Void Void ( 𝜖 < 0.15)\n(ℎ3Mpc−3) (ℎ3Mpc−3) (ℎ3Mpc−3)\n0.1 4.3×10−21.1×10−45.9×10−5\n0.3 1.3×10−24.9×10−52.7×10−5\n0.5 1.5×10−31.0×10−55.7×10−6\n0.7 6.1×10−44.5×10−62.3×10−6\n0.9 1.8×10−41.6×10−69.0×10−7\n1.1 1.0×10−49.7×10−75.7×10−7\n1.3 3.2×10−53.5×10−72.2×10−7\n1.5 1.8×10−63.0×10−81.1×10−8\nTable 2.Themean,minimumandmaximumradiiofvoids( >5ℎ−1Mpcand\n𝜖 < 0.15) in different redshift bins. The range of void radius used for the\nVSF analysis we choose is also shown in a given redshift bin.\nRedshift 𝑅meanv𝑅minv𝑅maxv Radius range\n(ℎ−1Mpc) (ℎ−1Mpc) (ℎ−1Mpc) (ℎ−1Mpc)\n0.1 10.19 5.0 28.94 -\n0.3 13.45 5.01 37.22 -\n0.5 23.22 8.43 57.22 (35,40)\n0.7 30.36 11.49 61.73 (40,50)\n0.9 43.94 18.12 85.77 (50,70)\n1.1 50.97 19.7 94.67 (60,80)\n1.3 73.46 36.71 123.01 -\n1.5 181.12 92.89 258.42 -\ndistribution for different redshift bins in Figure 1. We can see the\nvoidellipticitydistributionshaveasimilarshapeatdifferentredshift\nbins with a peak between 𝜖=0.1and 0.2, and most of voids have\n𝜖 <0.5. This indicates that large number of the voids we find have\nspherical-like shapes.\nIn Table 1, we show the number densities of galaxies and voids\nwith effective radius 𝑅v=𝑅eff>5ℎ−1Mpc of our mock catalogs\nderived from the simulation in different redshift bins. The number\ndensitiesofvoidsinthissamplewithellipticitylessthan0.15arealso\nshown. In Table 2, we show the average, minimum and maximum\nradiiofvoidswith 𝑅v>5ℎ−1Mpcand𝜖 <0.15atdifferentredshift\nbins. We can find that, as expected, the number density of voids\nbecomes lower and lower from 𝑧=0.1to 1.5, which is similar to\nthetrendofthegalaxynumberdensity,whiletheaveragevoidradius\nbecomes larger and larger.\nNote that we do not take into account the effects of redshift-\nspace distortion (RSD) and Alcock-Paczyński (AP) in this work. As\ndiscussed in previous studies (e.g. Correa et al. 2021), the RSD and\nAPmainlyhavetwoeffectsonvoids,i.e.theexpansioneffectandthe\noff-centering effect. Since VSF is not affected by the off-centering\neffect and the expansion effect can be corrected by relatively simple\ncalculation,weignorethesetwoeffectsinthefollowingVSFanalysis,\nand leave the study of their impacts in our future work.\n3 VOID SIZE FUNCTION\nThe most common theory of void size functions nowadays, which\ndescribes the number of voids based on their radius, are based on\nSheth & van de Weygaert (2004), and extended by Jennings et al.\n(2013). This theoretical model adopts the same form as the halomass function and is based on the development of the excursion-set\ntheory. (Press & Schechter 1974; Peacock & Heavens 1990; Cole\n1991;Bondetal.1991;Mo&White1996).ToevaluatetheVSF,we\nfirst derive the void mass function based on the halo mass function.\nFollowing Chan et al. (2014), the void mass function in Lagrangian\nspace is given by\nd𝑛L\nd ln𝑀=¯𝜌m\n𝑀F(𝜈,𝛿v,𝛿c)d𝜈\nd ln𝑀. (8)\nHere ¯𝜌misthemeandarkmatterdensity, 𝛿cisthecriticaloverdensity\nbarrier,whichisabout1.686inthe ΛCDMmodel,and 𝛿visthelinear\nunderdensity threshold of the void formation. Assuming spherical\nevolution with void matter density 𝜌v=0.2 ¯𝜌m, in the Einstein-de\nSittermodel,thelinearthreshold 𝛿visfoundtobe-2.717,and-2.731\nintheΛCDMmodel,whichhavesmalldifferenceinthesetwomodels\n(Jennings et al. 2013). 𝜈is the peak hight, and it can be written as\n𝜈=|𝛿v|\n𝜎𝑀(𝑧). (9)\nHere𝜎𝑀(𝑧)=𝜎𝑀𝐷(𝑧),where𝐷(𝑧)isthelineargrowthfactor,and\n𝜎𝑀=𝜎0(𝑅L)is the root-mean-squared density fluctuation within\nLagrangiansize 𝑅L,where𝑀=¯𝜌m𝑉(𝑅L)and𝑉(𝑅L)≡( 4/3)𝜋𝑅3\nL.\n𝜎0(𝑅L)can be expressed by\n𝜎0(𝑅L)=∫𝑘2\n2𝜋2𝑊2(𝑘𝑅L)𝑃(𝑘)𝑑𝑘. (10)\nHere𝑃(𝑘)is the matter power spectrum of the density fluctuation\nfield and𝑊(𝑘𝑅L)is the spherical top-hat window function for size\n𝑅L.Weuse CAMBtocalculate𝑃(𝑘)and𝜎0(𝑅L)inthiswork(Lewis\net al. 2000).\nIn Equation (8),F(𝜈,𝛿v,𝛿c)is the first-crossing distribution,\nwhich shows the probability of a random trajectory crossing the\nbarrier𝛿vforthefirsttimeat 𝜈withoutcrossing 𝛿cfor𝜈′>𝜈,andit\nis written as (Sheth & van de Weygaert 2004)\nF(𝜈,𝛿v,𝛿c)=2D2\n𝜈3∞∑︁\n𝑗=1𝑗𝜋sin(D𝑗𝜋)exp(−𝑗2𝜋2D2\n2𝜈2),(11)\nwhereDis given by\nD=|𝛿v|\n𝛿c+|𝛿v|. (12)\nBesides,F(𝜈,𝛿v,𝛿c)also can be approximated by (Sheth & van de\nWeygaert 2004)\nFapp(𝜈,𝛿v,𝛿c)=√︂\n2\n𝜋exp(−𝜈2\n2)exp(−|𝛿v|\n𝛿cD2\n4𝜈2−2D4\n𝜈4).(13)\nSoEquation(11)or(13)denotethefirst-crossingdistributionforthe\ndouble-barriercase,includingboth 𝛿vand𝛿c.Forlarge𝑀,thefirst-\ncrossing distribution will reduce to the one-barrier case F(𝜈,𝛿v)\nwithout the second exponential term in Equation (13), which is also\nthe case for most voids considered in this work.\nTomaptheLagrangianVSFtotheEulerianone,weneedtoconvert\nLagrangian size 𝑅Lto Eulerian size 𝑅, and we assume the volume\nfraction𝑉d𝑛conserves (Jennings et al. 2013). Then we have\nd𝑛\nd ln𝑅=𝑉(𝑅L)\n𝑉(𝑅)d𝑛L\nd ln𝑅L. (14)\nHencethetheoreticalVSFinEulerianspacecanbederivedbasedon\nEquation (8) and (14) for the Eulerian size 𝑅, which is given by\nd𝑛\nd ln𝑅=(3\n4𝜋𝑅3)F(𝜈,𝛿v,𝛿c)𝑑𝜈\nd ln𝑅L. (15)\nIn order to improve the fitting of the void size distribution found\nMNRAS 000, 1–10 (2015)VSF from CSST 5\n30 32 34 36 38 40 42 44\nRv[h1Mpc]\n107\n106\n105\ndnv/dlnRv[h3Mpc3]\nz=0.5\n35 40 45 50 55 60\nRv[h1Mpc]\n107\n106\n105\nz=0.7\n40 45 50 55 60 65 70 75 80\nRv[h1Mpc]\n108\n107\n106\n105\ndnv/dlnRv[h3Mpc3]\nz=0.9\n50 55 60 65 70 75 80 85 90\nRv[h1Mpc]\n108\n107\n106\n105\nz=1.1\nFigure 2. The predicted void size functions for voids with 𝜖 < 0.15(red) and all voids (gray) for the four redshift bins from 𝑧=0.5to 1.1 in the CSST\nspectroscopic survey. The blue curves are the best-fits of theoretical calculation. The gray regions denote the scales excluded in our analysis using the selection\ncriteria based on the void ellipticity distribution and statistical significance.\nbythewatershedalgorithminobservations,manystudiesattemptto\nrelaxthelinearthreshold 𝛿v≃−2.7orconsideritasafreeparameter\n(Jennings et al. 2013; Sutter et al. 2014a; Chan et al. 2014; Pisani\netal.2015;Contarinietal.2022,2023).However,ifwedon’tset 𝛿v\nto a fixed value and let it change, the conversion relation between\nvoid Eulerian radius 𝑅=𝑅vand Lagrangian radius 𝑅Lneeds to be\nchanged according to 𝛿v, and then the relation can be well fitted by\n𝑅L≃𝑅v\n(1−𝛿v/𝑐v)𝑐v/3, (16)\nwhere𝑐v=1.594(Bernardeau 1994; Jennings et al. 2013). In the\nspherical evolution model, the analytical calculation gives 𝑅L=\n0.58𝑅vassuming the void matter density 𝜌v=0.2 ¯𝜌mand𝑅/𝑅L=\n(¯𝜌m/𝜌v)1/3(Jennings et al. 2013; Chan et al. 2014). In this work,\nwith the help of Equation (16), we set 𝛿vas a free parameter in a\ngiven redshift bin, which means that we assume 𝛿vis dependent onthe redshift, or its evolution cannot be fully described by the linear\ngrowth factor 𝐷(𝑧). This gives us more flexibility to fit the mock\nVSFdataatdifferentredshifts,andcanexplorethecomplexityofthe\nvoid formation and evolution.\nTo further reduce the discrepancy between the voids found by\nVIDEusingthewatershedalgorithmandthetheoreticalmodelbased\non the spherical evolution, and obtain more reliable and accurate\nfitting results, we select voids according to the ellipticity with 𝜖 <\n0.15, i.e. selecting more spherical-like voids. As shown in Figure 1,\nthis void ellipticity cut-off is around the peak of the void ellipticity\ndistributions at different redshift bins. By applying this selection\ncriterion, the void number reduction is around 50% for all redshifts\nasindicatedinTable1.InFigure2,weshowtheVSFsforvoidswith\n𝜖 <0.15and all voids at different redshift bins from 𝑧=0.5to1.1.\nThe error bar for each data point is estimated by using the jackknife\nmethod.\nMNRAS 000, 1–10 (2015)6 Y. Song et al.\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14\n100101102Nvz=0.55<R30\n30<R35\n35<R40\n40<R58\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14\n100101z=0.78<R30\n30<R40\n40<R45\n45<R50\n50<R62\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14\n100101Nvz=0.913<R40\n40<R50\n50<R60\n60<R70\n70<R86\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14\n100101z=1.116<R60\n60<R80\n80<R95\nFigure 3. The distributions of void ellipticity with 𝜖 < 0.15for different void size bins at redshift range from 𝑧=0.5to 1.1 in the CSST spectroscopic survey.\nWe can find that most of smaller voids still have relatively large ellipticity, and we will exclude the void size bin with the peak of the ellipticity distribution at\n𝜖 > 0.12and weak statistical significance in the fitting process.\nIn addition, we also analyze the ellipticity distribution with\n𝜖 < 0.15for different void size bins in a given redshift as shown\ninFigure3.Ascanbeseen,mostofsmallervoidsarestillnotspher-\nical enough with relatively large ellipticity, and larger voids have\nsmallnumbersandweakstatisticswithlargefluctuationsindistribu-\ntion. To obtain a better fitting results for the cosmological and void\nparameters,weonlyselectthevoidsizerangewiththepeakoftheel-\nlipticitydistributionat 𝜖≲0.12andsufficientstatisticalsignificance\n(SNR>1 for each VSF data point) in the fitting process. Hence, the\ngray regions (small- and large-size parts) in Figure 2 are excluded.\nBesides, based on these two selection criteria, the first ( 𝑧=0.1,0.3)\nand last (𝑧=1.3,1.5) two redshift bins are also rejected in the fit-\nting process. This is because that all of the void size ranges in the\nfirst two redshift bins at 𝑧=0.1,0.3are dominated by voids with\n𝜖∼0.15, and the void number densities in the last two redshift\nbins at𝑧=1.3,1.5are too low to provide sufficient information forcosmological constraints as shown in Table 1. The void size ranges\nwe choose at the four redshift bins from 𝑧=0.5to 1.1 are listed in\nTable 2.\n4 PARAMETER CONSTRAINT\nAfter obtaining the mock data of the VSFs at different redshifts in\nthe CSST spectroscopic survey, we can explore the constraints on\nthecosmologicalandvoidparameters.The 𝜒2methodisadoptedfor\nparameter fitting, which takes the form at 𝑧as\n𝜒2\n𝑧=∑︁\u0002\n𝑛datav(𝑅v,𝑧)−𝑛thv(𝑅v,𝑧)\u00032\n𝜎2v, (17)\nwhere𝑛datav(𝑅v,𝑧)istheVSFmockdataat 𝑧,𝑛thv(𝑅v,𝑧)=d𝑛/d ln𝑅\nas shown in Equation(15) is the theoretical VSF, and 𝜎vis the error\nMNRAS 000, 1–10 (2015)VSF from CSST 7\nFigure 4. Contourmapsofthesixcosmologicalparametersat68%and95%CLusingtheVSFmockdataatallredshiftbins.The1DPDF(bluecurve)foreach\nparameter is also shown. The gray vertical and horizontal dotted lines represent the fiducial values of these parameters.\noftheVSFmockdata.Thenwecancalculatethelikelihoodfunction\nasL∝exp(−𝜒2/2). The total chi-square of the void size function\ncan be estimated by 𝜒2\ntot=Í\n𝑧bins𝜒2𝑧for different redshift bins.\nWeusethe emceepackage(Foreman-Mackeyetal.2013)toapply\nMarkov Chain Monte Carlo (MCMC) method, which based on the\naffine-invariant ensemble sampling algorithm (Goodman & Weare\n2010).We initialize 112 walkers around the target parameters to get\n10000steps.Wethendiscardthefirst10percentofthestepsasburn-\nin.InTable3,weshowthefreeparametersinourmodelatdifferent\nredshift, their fiducial values and flat priors. We try to constrain\nthe total matter density parameter Ωm, baryon density parameter\nΩb, dark energy equation of state 𝑤, reduced Hubble constant ℎ,\nspectral index 𝑛s, and amplitude of initial power spectrum 𝐴s. The\nfree parameters about void are 𝛿𝑖vfor the four redshift bins from 𝑧=0.5to1.1,whichdenotethethresholdforvoidformation.Therefore,\nwe totally have 6 cosmological parameters, and 4 void parameters\nfor the four redshift bins in our model.\nIn Figure 4 , we show the contour maps at 68% and 95% confi-\ndence levels (CL) and 1D probability distribution functions (PDFs)\nofthesixcosmologicalparametersconstrainedbyallofthevoidsize\nfunctions from the four redshift bins. The gray dashed line marks\nthe fiducial value of the parameter. We can find that the cosmolog-\nical parameters we consider can be correctly and well constrained,\nand the fiducial values are within 1 𝜎CL of the parameter fitting\nresults.ThisalsoprovesthatthecalibrationmethodfortheVSFdata\nmentionedinthelastsectioniseffective,andcouldcorrectlyreserve\nthe cosmological information. The details of the best-fit values, 1 𝜎\nMNRAS 000, 1–10 (2015)8 Y. Song et al.\nTable 3.Thefiducialvalues,priorranges,best-fitvalues,errors,andrelative\naccuraciesofthesixcosmologicalparametersandthevoidlinearunderdensity\nthreshold parameter 𝛿𝑖vfor the four redshift bins from 𝑧=0.5to 1.1 have\nbeen shown.\nParameter Fiducial value Flat Prior Best-fit value\nΩm 0.3111 (0.1, 0.5) 0.381+0.078\n−0.090(22.0%)\nΩb 0.049 (0.01, 0.08) 0.033+0.025\n−0.016(63.8%)\nℎ 0.6766 (0.4, 0.9) 0.787+0.082\n−0.132(13.6%)\n𝑛s 0.9665 (0.7, 1.2) 0.981+0.147\n−0.169(16.1%)\n𝑤 -1 (-1.5, -0.5) −1.087+0.360\n−0.291(29.9%)\n𝐴s(×10−9) 2.1 (1.5, 2.5) 2.132+0.262\n−0.366(14.7%)\n𝛿1v - (-2, 0) −0.369+0.069\n−0.070(18.8%)\n𝛿2v - (-2, 0) −0.219+0.051\n−0.043(21.5%)\n𝛿3v - (-2, 0) −0.118+0.024\n−0.023(19.7%)\n𝛿4v - (-2, 0) −0.097+0.020\n−0.018(20.0%)\nFigure 5. Contour maps of the linear underdensity thresholds for void for-\nmation𝛿𝑖vat 68% and 95% CL using the VSF mock data at all four redshift\nbins from𝑧=0.5to 1.1. The 1D PDF (red curve) for each parameter is also\nshown.\nerrors, and relative accuracies for the six cosmological parameters\nare shown in Table 3.\nWe can see that the current VSF mock data in the CSST spectro-\nscopic survey canobtain ∼20% and∼30% constraints on Ωmand𝑤\nin1𝜎CL,respectively.Comparingtothecurrentcosmologicalcon-\nstraint results using the VSFs, e.g. the Baryon Oscillation Spectro-\nscopicSurvey(BOSS)DR12(Contarinietal.2023),wecanachieve\nsimilar constraint power but with more number of cosmological pa-\nrameters. We also note that the constraint power of Ωbis relatively\nweak, since the VSF is not sensitive to Ωb. If considering the full\nFigure 6. The best-fit values and 1 𝜎errors of𝛿𝑖vat the four redshift bins.\nTheredandbluedatapointsdenotetheconstraintresultsusingtheVSFmock\ndata of all redshift bins and a single redshift bin, respectively.\nCSST spectroscopic survey with 17,500 deg2, as we estimate, the\ncurrent constraint accuracies of the cosmological parameters can be\nimproved by almost one order of magnitude, resulting in accuracies\nof a few percent level.\nIn Figure 5, using all the VSF mock data at four redshift bins, the\njoint constraint results of the linear underdensity thresholds for void\nformation𝛿𝑖vat the four redshift bins are shown in red contours and\n1D PDFs. Note that there are no fiducial values for 𝛿𝑖v, since they\nare not the input parameters in the simulation. The Table 3 shows\nthe details of the best-fit values, 1 𝜎errors, and relative accuracies\nfor the four𝛿𝑖v. We find that the relative accuracy on 𝛿𝑖vat different\nredshifts have similar result, which is about 20%.\nWealsonoticethatthebest-fitvaluesof 𝛿𝑖vareredshift-dependent\nrangingfromabout −0.4to-0.1astheredshiftincreasesfrom 𝑧=0.5\nto1.1.Thisobviouslyhaslargediscrepancyfromthetheoreticalpre-\ndiction with a constant 𝛿v≃−2.7assuming the spherical evolution\nand𝜌v=0.2 ¯𝜌m(Jenningsetal.2013).Thisisbecausethatthevoids\ninourcatalogareidentifiedbythewatershedalgorithmwithoutcon-\nsideringtheseassumptions.Besides,asshowninEquation(16),this\nresult also implies that the Lagrangian void size 𝑅Lwill be closer\nand closer to the Eulerian size 𝑅vfrom low to high redshifts, which\nis consistent with the theoretical expectation.\nIn order to check this result, we also perform the constraints on\nthecosmologicalandvoidthresholdparametersusingtheVSFmock\ndataineachredshiftbinindividually.TheresultsareshowninFigure\n6, and the blue and red data points with error bars are the results\nderived from the data in a single redshift bin and all redshift bins,\nrespectively.Wecanfindthattheyarebasicallyconsistentwitheach\nother in 1𝜎, and the red error bars are smaller than the blue ones,\nsincethecosmologicalparameterscanbeconstrainedmorestringent\nwhen using the data from all redshift bins. Because there is almost\nno effect constraint on each cosmological parameter using the data\nin a single redshift bin, we do not show the constraint result of the\ncosmological parameters here. This redshift variation of the linear\nthreshold𝛿vis also consistent with the result indicated by Contarini\netal.(2022),wheretheyuseanonlinearunderdensitythreshold 𝛿NLv\nto study the VSFs measured by Euclid, which can be converted to\nthe linear threshold 𝛿v.\nMNRAS 000, 1–10 (2015)VSF from CSST 9\n5 SUMMARY\nIn this work, we study the VSFs measured by the CSST spectro-\nscopicsurveyatdifferentredshifts,andexploretheconstraintpower\nfor the cosmological and void parameters. We generate the galaxy\nmock catalog based on Jiutian simulation from 𝑧=0.1to 1.5, and\nconsidertheinstrumentaldesignandsurveystrategy.Weuse VIDEto\nidentifyvoidsinthegalaxymockcatalogbyadoptingthewatershed\nalgorithm,andobtainthevoidinformationsuchasvolume-weighted\ncenter, effective radius 𝑅vand ellipticity 𝜖.\nSince the theoretical VSF model is based on the spherical evo-\nlution, we also perform a selection using the void ellipticity to find\nthe voids with 𝜖 < 0.15and𝑅v>5ℎ−1Mpc. The void ellipticity\ndistributions for different void size bins at a given redshift are also\nexplored. In order to obtain a better constraint result, we only use\nthe voids in a size bin with the peak of the ellipticity distribution at\n𝜖≲0.12and sufficient statistical significance. As a result, the first\n(𝑧=0.1,0.3) and last (𝑧=1.3,1.5) two redshift bins are excluded,\nandonlythevoidswithmiddlesizesineachredshiftbinfrom 𝑧=0.5\nto 1.1 are used in the constraint process.\nWefindthattheselectedVSFdatacancorrectlyreservethecosmo-\nlogicalinformationandcouldbewellfittedbythetheoreticalmodel.\nThefittingresultsofthecosmologicalparameters,suchas Ωmand𝑤,\nareconsistentwiththefiducialvaluesin1 𝜎CL.Wealsoassumethat\nthe linear underdensity threshold 𝛿vis redshift dependent, and set it\nas a free parameter in each of the four redshift bins. After the fitting\nprocess, we find that 𝛿vis indeed varying from ∼−0.4to∼−0.1\nas the redshift increases from 𝑧=0.5to 1.1. This has significant\ndifferencefromthetheoreticalcalculationwithaconstant 𝛿v≃−2.7\nassuming the spherical evolution, since we are using the watershed\nalgorithm to identify voids with relatively natural shapes.\nWe note that the simulation can only cover a small part of survey\nareaoftheCSSTspectroscopicsurvey,andtheconstraintaccuracies\nofthecosmologicalandvoidparametersareexpectedtobeimproved\nto a few percent level, if using the data from full 17,500 deg2CSST\nwide-field survey area. The method we propose to identify and se-\nlect voids also can be a reference for the future VSF study in the\nspectroscopic galaxy surveys.\nACKNOWLEDGEMENTS\nYS and YG acknowledge the support from National Key R&D Pro-\ngram of China grant Nos. 2022YFF0503404, 2020SKA0110402,\nand the CAS Project for Young Scientists in Basic Research (No.\nYSBR-092). KCC acknowledges the support the National Science\nFoundation of China under the grant number 12273121. XLC ac-\nknowledges the support of the National Natural Science Founda-\ntion of China through Grant Nos. 11473044 and 11973047, and the\nChinese Academy of Science grants ZDKYYQ20200008, QYZDJ-\nSSW-SLH017, XDB 23040100, and XDA15020200. GLL, YL and\nCLW acknowledges the support from NSFC grant No. U1931210.\nThisworkisalsosupportedbyscienceresearchgrantsfromtheChina\nManned Space Project with Grant Nos. CMS-CSST-2021-B01 and\nCMS-CSST-2021-A01.\nDATA AVAILABILITY\nThedatathatsupportthefindingsofthisstudyareavailablefromthe\ncorresponding author upon reasonable request.REFERENCES\nAchitouv I., 2016, Phys. Rev. D, 94, 103524\nAlam S., et al., 2017, MNRAS, 470, 2617\nBernardeau F., 1994, ApJ, 427, 51\nBlanton M. R., et al., 2017, AJ, 154, 28\nBond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440\nCai Y.-C., Padilla N., Li B., 2015, MNRAS, 451, 1036\nCai Y.-C., Taylor A., Peacock J. A., Padilla N., 2016, MNRAS, 462, 2465\nCao Y., et al., 2018, MNRAS, 480, 2178\nChan K. C., Hamaus N., 2021, Phys. Rev. D, 103, 043502\nChan K. C., Hamaus N., Desjacques V., 2014, Phys. Rev. D, 90, 103521\nChuang C.-H., Kitaura F.-S., Liang Y., Font-Ribera A., Zhao C., McDonald\nP., Tao C., 2017, Phys. Rev. D, 95, 063528\nCole S., 1991, ApJ, 367, 45\nContarini S., Marulli F., Moscardini L., Veropalumbo A., Giocoli C., Baldi\nM., 2021, MNRAS, 504, 5021\nContarini S., et al., 2022, A&A, 667, A162\nContarini S., Pisani A., Hamaus N., Marulli F., Moscardini L., Baldi M.,\n2023, ApJ, 953, 46\nCorrea C. M., Paz D. J., Sánchez A. G., Ruiz A. N., Padilla N. D., Angulo\nR. E., 2021, MNRAS, 500, 911\nCorrea C. M., Paz D. J., Padilla N. D., Sánchez A. G., Ruiz A. N., Angulo\nR. E., 2022, MNRAS, 509, 1871\nDESI Collaboration et al., 2016, arXiv e-prints, p. arXiv:1611.00036\nDark Energy Survey Collaboration et al., 2016, MNRAS, 460, 1270\nDawson K. S., et al., 2016, AJ, 151, 44\nDengF.,GongY.,WangY.,DongS.,CaoY.,ChenX.,2022,MNRAS,515,\n5894\nEllisR.,DawsonK.,2019,inBulletinoftheAmericanAstronomicalSociety.\np. 45 ( arXiv:1907.06797 ), doi:10.48550/arXiv.1907.06797\nEuclid Collaboration et al., 2022, A&A, 662, A112\nFalck B., Koyama K., Zhao G.-B., Cautun M., 2018, MNRAS, 475, 3262\nForeman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125,\n306\nForero-Sánchez D., Zhao C., Tao C., Chuang C.-H., Kitaura F.-S., Variu A.,\nTamone A., Kneib J.-P., 2022, MNRAS, 513, 5407\nGong Y., et al., 2019, ApJ, 883, 203\nGoodman J., Weare J., 2010, Communications in Applied Mathematics and\nComputational Science, 5, 65\nHamaus N., Pisani A., Sutter P. M., Lavaux G., Escoffier S., Wandelt B. D.,\nWeller J., 2016, Phys. Rev. Lett., 117, 091302\nHamaus N., Cousinou M.-C., Pisani A., Aubert M., Escoffier S., Weller J.,\n2017, J. Cosmology Astropart. Phys., 2017, 014\nHamaus N., et al., 2022, A&A, 658, A20\nHenriques B. M. B., White S. D. M., Thomas P. A., Angulo R., Guo Q.,\nLemson G., Springel V., Overzier R., 2015, MNRAS, 451, 2663\nJennings E., Li Y., Hu W., 2013, MNRAS, 434, 2167\nKhoraminezhad H., Vielzeuf P., Lazeyras T., Baccigalupi C., Viel M., 2022,\nMNRAS, 511, 4333\nKitaura F.-S., et al., 2016, Phys. Rev. Lett., 116, 171301\nLewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473\nLiangY.,ZhaoC.,ChuangC.-H.,KitauraF.-S.,TaoC.,2016,MNRAS,459,\n4020\nMao Q., Berlind A. A., Scherrer R. J., Neyrinck M. C., Scoccimarro R.,\nTinker J. L., McBride C. K., Schneider D. P., 2017, ApJ, 835, 160\nMaulandR.,ElgarøyØ.,MotaD.F.,WintherH.A.,2023,A&A,674,A185\nMiao H., Gong Y., Chen X., Huang Z., Li X.-D., Zhan H., 2023, MNRAS,\n519, 1132\nMo H. J., White S. D. M., 1996, MNRAS, 282, 347\nMoresco M., et al., 2022, Living Reviews in Relativity, 25, 6\nNadathur S., Percival W. J., 2019, MNRAS, 483, 3472\nNadathur S., et al., 2020, MNRAS, 499, 4140\nNeyrinck M. C., 2008, MNRAS, 386, 2101\nPaillas E., Cautun M., Li B., Cai Y.-C., Padilla N., Armijo J., Bose S., 2019,\nMNRAS, 484, 1149\nPaz D., Lares M., Ceccarelli L., Padilla N., Lambas D. G., 2013, MNRAS,\n436, 3480\nMNRAS 000, 1–10 (2015)10 Y. Song et al.\nPeacock J. A., Heavens A. F., 1990, MNRAS, 243, 133\nPei et al. 2024, in preparation\nPelliciariD.,ContariniS.,MarulliF.,MoscardiniL.,GiocoliC.,LesciG.F.,\nDolag K., 2023, MNRAS, 522, 152\nPercival W. J., et al., 2019, arXiv e-prints, p. arXiv:1903.03158\nPerico E. L. D., Voivodic R., Lima M., Mota D. F., 2019, A&A, 632, A52\nPisani A., Sutter P. M., Hamaus N., Alizadeh E., Biswas R., Wandelt B. D.,\nHirata C. M., 2015, Phys. Rev. D, 92, 083531\nPisani A., et al., 2019, BAAS, 51, 40\nPlanck Collaboration et al., 2020, A&A, 641, A6\nPlaten E., van de Weygaert R., Jones B. J. T., 2007, MNRAS, 380, 551\nPollina G., Baldi M., Marulli F., Moscardini L., 2016, MNRAS, 455, 3075\nPress W. H., Schechter P., 1974, ApJ, 187, 425\nSahlén M., Silk J., 2018, Phys. Rev. D, 97, 103504\nSahlén M., Zubeldía Í., Silk J., 2016, ApJ, 820, L7\nSchlegel D., Kollmeier J. A., Ferraro S., 2019, in Bulletin of\nthe American Astronomical Society. p. 229 ( arXiv:1907.11171 ),\ndoi:10.48550/arXiv.1907.11171\nSchuster N., Hamaus N., Dolag K., Weller J., 2023a, arXiv e-prints, p.\narXiv:2312.11241\nSchusterN.,HamausN.,DolagK.,WellerJ.,2023b,J.CosmologyAstropart.\nPhys., 2023, 031\nSheth R. K., van de Weygaert R., 2004, MNRAS, 350, 517\nSpringel V., 2005, MNRAS, 364, 1105\nSpringel V., Yoshida N., White S. D. M., 2001, New Astron., 6, 79\nSutterP.M.,LavauxG.,WandeltB.D.,WeinbergD.H.,2012,ApJ,761,187\nSutterP.M.,LavauxG.,HamausN.,WandeltB.D.,WeinbergD.H.,Warren\nM. S., 2014a, MNRAS, 442, 462\nSutterP.M.,PisaniA.,WandeltB.D.,WeinbergD.H.,2014b,MNRAS,443,\n2983\nSutter P. M., et al., 2015, Astronomy and Computing, 9, 1\nUbedaL.,2011,in,Vol.11,ACSInstrumentHandbookforCycle20v.11.0.\np. 11\nVerza G., Pisani A., Carbone C., Hamaus N., Guzzo L., 2019, J. Cosmology\nAstropart. Phys., 2019, 040\nZhanH.,2011,ScientiaSinicaPhysica,Mechanica&Astronomica,41,1441\nZhan H., 2021, Chinese Science Bulletin, 66, 1290\nZivickP.,SutterP.M.,WandeltB.D.,LiB.,LamT.Y.,2015,MNRAS,451,\n4215\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–10 (2015)"    },    {        "title": "2402.05505v2.The_stability_analysis_based_on_viscous_theory_of_Faraday_waves_in_Hele_Shaw_cells.pdf",        "content": "arXiv:2402.05505v2  [physics.flu-dyn]  22 Feb 2024AIP/123-QED\nThe stability analysis based on viscous theory of Faraday wa ves in Hele-Shaw cells\nXingsheng Li (李兴盛)1and Jing Li (李靖)1\nMarine Numerical Experimental Center, State Key Laborator y of Ocean Engineering,\nSchool of Naval Architecture, Ocean and Civil Engineering,\nShanghai Jiao Tong University, Shanghai, 200240, PR China\n(*Electronic mail: lijing_@sjtu.edu.cn)\n(Dated: 23 February 2024)\nThe linear instability of Faraday waves in Hele-Shaw cells i s investigated with considera-\ntion of the viscosity of fluids after gap-averaging the gover ning equations due to the damp-\ning from two lateral walls and the dynamic behavior of contac t angle. A new hydrodynamic\nmodel is thus derived and solved semi-analytically. The con tribution of viscosity to criti-\ncal acceleration amplitude is slight compared to other fact ors associated with dissipation,\nand the potential flow theory is sufficient to describe onset b ased on the present study, but\nthe rotational component of velocity can change the timing o f onset largely, which para-\ndoxically comes from the viscosity. The model degenerates i nto a novel damped Mathieu\nequation if the viscosity is dropped with two damping terms r eferring to the gap-averaged\ndamping and dissipation from dynamic contact angle, respec tively. The former increases\nwhen the gap size decreases, and the latter grows as frequenc y rises. When it comes to the\ndispersion relation of Faraday waves, an unusual detuning e merges due to the imaginary\npart of the gap-averaged damping.\n1I. INTRODUCTION\nWhen a container filled with liquids is subjected to vertical vibrations, Faraday waves are\nformed at the free surface in some specific conditions. This p henomenon has been widely applied\nto numerous areas of science and industry, such as assembly o f microscale beads,1atomization,2\ninkjet printing,3oil-gas separator on floating offshore platforms4and so on. This type of reso-\nnant surface wave was first experimentally observed by Farad ay5in 1831 and subsequently by\nMatthiessen6and Rayleigh7. From then on, a large number of researchers were attracted b y this\ninteresting physical process and devoted themselves to rev ealing the mechanism behind it via ex-\nperiments, numerical simulations, and theoretical analys es. Among these studies, linear instability\nand pattern selection of Faraday waves have received the mos t attention. The linear instability\ncorresponds to the critical conditions when Faraday waves o ccur, which is called the onset. Pat-\ntern selection concentrates on which form Faraday waves wil l appear in. Depending on the fluid\nproperties and parameters of external excitation, differe nt wave patterns have been observed in\nexperiments, which include triangles,8parallel stripes, squares, hexagons and higher symmetry\nquasi-patterns.9Spatio-temporal chaos10and highly localized circular waves11may also emerge.\nThese complicated phenomena and nonlinearity bring challe nges to mathematical modelings and\nnumerical simulations.\nTo give the prediction of linear instability, Benjamin and U rsell12applied the theory of Math-\nieu equation. Results reflected that Faraday waves of half-f requency and synchronous response\nalternately occurred. But this theoretical method is only v alid in inviscid restrictions. In the limit\nof low viscosity, Landau and Lifshitz13introduced a very thin layer at the free surface, while\nthe rest of the flow remained potential. There was no other sig nal progress until a full hydrody-\nnamic system was developed by Kumar and Tuckerman14to solve the stability problem for the\ninterface between two fluids of arbitrary viscosity. The der ivation began with the Navier-Stokes\nequations and the effect of viscosity was completely retain ed. Results showed a great agreement\nwith experimental data of Edwards and Fauve15, which confirmed that the viscosity of fluids was\nessential in the stability analysis of Faraday waves. Since then, accurate prediction in the case of\nthree-dimensional Faraday waves has been achieved. Inspir ed by them, Pototsky and Bestehorn16\nstudied the linear instability problem of a two-layer liqui d film with a deformable upper surface.\nCompared to the prediction of instability thresholds, patt ern selection is more complicated be-\ncause of the nonlinearity and resonant interactions of wave vectors. Taking the limitation of linear\n2analysis into account, nonlinear approximations are neces sary to describe the pattern formation\nbeyond the instability threshold, among which the derivati on of amplitude equations has become\na classic theoretical method. Early analyses of pattern mec hanism were based on the assumption\nthat the bulk flow was irrotational and the Hamiltonian descr iption was applied.17,18The viscous\ndissipation of bulk flow was simply treated as small damping a nd represented explicitly by the\nrate of energy loss which had a form of integration.19From the comparison with experiments of\nDouady and Fauve20, it is obvious that the above assumption about dissipation i s not appropriate\nand the contribution of the free surface to the damping shoul d be considered. Focused on the dissi-\npation near the surface skin, Zhang and Viñals21used a quasi-potential approximation to describe\nthe small vortical layer near the free surface. However, the ir theory is limited to fluid systems\nwith small viscous dissipation. Chen and Viñals22then extended this previous research to arbi-\ntrary viscosity by following the earlier work of Kumar and Tu ckerman14. The difference was that\nnonlinear terms in the system of equations were retained. Th e theoretical results were validated\nby Westra, Binks, and Van De Water23later, which further explained the importance of viscosity\nin Faraday waves. Recently, Zhang, Borthwick, and Lin24applied modal decomposition to reveal\nthe nonlinear mechanisms behind pattern formation in a brim ful cylinder. This further highlighted\nthe nonlinear feature of Faraday waves. In this sense, numer ical simulations give more infor-\nmation about the nonlinear viscous problem of Faraday waves . However, most simulations have\nbeen two-dimensional, such as Murakami and Chikano25and Ubal, Giavedoni, and Saita26. Until\n2009, Périnet, Juric, and Tuckerman27presented a systemic simulation about the full dynamics of\nFaraday waves in a three-dimensional field for two viscous flu ids. The critical accelerations and\nwavenumbers, different patterns, and velocity fields were r eproduced.\nIn Hele-Shaw cells, however, the study of Faraday waves has g reatly changed. On the one\nhand, more complicated phenomena have been observed in expe riments. In 2011, Rajchenbach,\nLeroux, and Clamond28found a new type of solitary wave in a Hele-Shaw cell which was highly\nlocalized and oscillated periodically. Li, Xu, and Liao29then conducted a series of more extensive\nexperiments and discovered a new wave structure with multip le crests and troughs. Later, a family\nof two-dimensional Faraday waves with plump crests and flat t roughs almost getting in touch\nwith the bottom of the container was observed in extremely sh allow depth.30And two coupled\nFaraday waves were observed in a covered Hele-Shaw cell fille d with two immiscible liquids with\na free upper surface.31The upper one vibrates vertically, while the crests of the lo wer one oscillate\nhorizontally with unchanged wave heights. These novel Fara day waves show a strong nonlinear\n3property and some may only appear in very localized areas, wh ich makes the physical phenomena\nmore complex to model. On the other hand, the dissipation is s trengthened in Hele-Shaw flows.\nIn a traditional Faraday resonance system, it mainly arises from the viscosity of fluids. Milner19\nfound that dissipation at the walls of the container was negl igible compared to the bulk contribution\nand there was no contribution to dissipation from a moving co ntact line. But for Hele-Shaw cells\nwith extremely narrow gaps, the no-slip condition is often i ntroduced and the dissipation from\ntwo lateral walls is enhanced largely. Meanwhile, the menis cus formed along the gap direction\nhas an apparent contribution to the curvature because of the diminutive gap size, which results in\nlarger surface contributions to dissipation. These two dis sipation sources increase the instability\nthreshold significantly. Hence, the prediction of Faraday i nstability becomes more complex to a\nlarge extent in Hele-Shaw cells and has been the focus of rese arch.\nTo obtain the thresholds of Faraday instability in Hele-Sha w cells, Rajchenbach, Leroux, and\nClamond28introduced the so-called Rayleigh’s external viscosity γ=12ν/b2to calculate the\ndissipation resulting from the narrow gap, with νthe fluid kinematic viscosity and bthe cell’s\ngap-size. However, the surface tension is absolutely negle cted, which is essential for such gravity-\ncapillary waves. Li et al.32assumed that the flow through the gap was of Poiseuille type an d\ndeveloped the so-called gap-averaged Navier-Stokes equat ions which included the viscous term\ncombining with gravity and capillary force. An open-source code Gerris33,34was then used to\nsolve the above equations numerically. The wave profile, wav e height together with flow field\ninformation were obtained and compared with experimental d ata. A scaling law with respect to\nthe Strouhal number was proposed for the first time to predict the wave height. Then, Li, Li, and\nLiao35established a gap-averaged model to include surface tensio n and meniscus between the\ntwo side walls. Nonlinear terms in equations were reserved a nd the Lyapunov’s first method was\nused to obtain the threshold of onset. From the comparison wi th experiments, it is obvious that the\ngap-averaged model gives a better prediction of Faraday ins tability than previous work. Martino\net al.36discussed the influence of liquid depth on Faraday instabili ty and a new scaling law of\nwave height was given later.37But it only works when critical acceleration amplitude is kn own\nas a priori information, which cannot be calculated accurat ely without laboratory measurements.\nBesides, only gravity is involved in this scaling law, yet th e capillary effect has a similar weight\non the Faraday instability problem.38The influence of the free surface was further verified by Li\nand Li39. Focusing on the meniscus formed at the free surface, the imp act of contact line with\nhysteresis on the Faraday instability was studied. The ener gy dissipation with respect to viscosity\n4and surface tension was analyzed for three different bounda ry conditions at the lateral walls. In\nconclusion, they emphasized that the contact line motion sh ould not be ignored because it largely\ndelayed the onset of Faraday waves. Inspired by the theoreti cal work of Li, Li, and Liao35, Rachik\nand Aniss40studied the effects of finite depth and surface tension on the dispersion relation and\nFaraday instability. More recently, Bongarzone et al.41modified the damping coefficient resulting\nfrom the gap-averaging process with the theory of oscillati ng Stokes boundary layer. The results\nindicate that this theory may fit Hele-Shaw flows better than t he Darcy approximation used by Li,\nLi, and Liao35.\nAlthough many works concentrate on the linear instability o f Faraday waves in Hele-Shaw\ncells, the above mathematical analyses are all based on pote ntial assumptions or have neglected\nviscous terms in equations. The effect of viscous dissipati on from the bulk flow to the Hele-Shaw\nsystem is still unclear. Numerical simulations are also lim ited since the instability threshold cannot\nbe provided directly. A prediction of Faraday instability i n Hele-Shaw cells based on viscous\ntheory is necessary to reveal the influence of viscosity, whi ch is the motivation of the present\nwork. In this paper, the stability of Faraday waves in Hele-S haw cells is studied theoretically\nwith the viscosity of fluids being reserved. The contributio n of viscosity to critical conditions of\nonset is revealed. In accordance with Bongarzone et al.41, the gap-averaging strategy is updated.\nIn addition, a novel damped Mathieu equation is derived in th e absence of viscosity. Before\nelaborating, the dissipation in such a Hele-Shaw system sho uld be defined. The onset of Faraday\nwaves has been thought to depend on the dissipation of the ent ire system. When the balance\nbetween the external driving force and dissipation is broke n, the free surface loses its stability\nand Faraday waves appear. The dissipation of liquid motions in Hele-Shaw cells falls into four\ncategories: the two-dimensional viscous dissipation refe rring to the effect of viscosity, the gap-\naveraged damping resulting from two lateral walls, the diss ipation from dynamic contact angle\nwhich describes the interaction of the free surface and late ral walls, and dissipation from two\nends of the rectangular container. Among them, the first one i s normally overlooked by previous\nstudies,35,40,41which are limited to inviscid fluids as has been mentioned at t he beginning of this\nparagraph. With regard to the dissipation arising from two e nds, from experiments conducted in\na thin annular container by Bongarzone et al.41, the influence of two ends of the vessel is not a\nmain reason for distinctions between theory and experiment , so we ignore it in the present paper.\nThe remainder of this paper is organized as follows. In Sec. I I, a detailed derivation of the\nmathematical model with viscosity and capillary effect bei ng included is described. Then it is\n5xyzz=ζ'(x,y,t )\nmeniscus \nθ\nb\nFIG. 1. Sketch of Faraday waves in a Hele-Shaw cell. Here bdenotes the gap size of the Hele-Shaw cell\nandθis the contact angle of the liquid on the lateral wall.\nsolved semi-analytically based on linear stability theory in Sec. III. To highlight the effect of vis-\ncosity, a damped Mathieu equation is obtained based on poten tial flow theory in Sec. IV, which\nonly contains two main sources of dissipation resulting fro m the gap-averaged damping and dy-\nnamic contact angle. In Sec. V, The findings, along with a comp arison to experimental data, are\npresented. Additionally, the examination of the effects of viscosity and the remaining dissipation\nis discussed. Finally, some concluding remarks are given in Sec. VI.\nII. MATHEMATICAL MODEL\nWe consider here a horizontally infinite Hele-Shaw cell whic h undergoes a vertical periodic\noscillation. The vessel is of width band filled with a kind of liquid of density ρand dynamic vis-\ncosity µ. In the Cartesian coordinate system as shown in Fig. 1 which m oves with the oscillating\nvessel, z=0 is set at the flat free surface when at rest. The oscillation i s sinusoidal with acceler-\nation amplitude aand angular frequency ω=2π\nT(T: the forcing period). What we are concerned\nabout is the threshold of acceleration corresponding to the onset of Faraday waves with frequency\nfixed. Since the free surface is flat overall before Faraday wa ves occur, a linear theory is sufficient\n6to obtain the critical conditions. This has been verified by K umar and Tuckerman14and Chen\nand Viñals22in three-dimensional Faraday instability problem and by Bo ngarzone et al.41in the\nHele-Shaw scenario, where linear methods such as the Floque t theory were used.\nAlthough the gap size bis sufficiently small compared to the wavelength λ,42which means the\nflow in Hele-Shaw cells can be regarded as two-dimensional, t he dissipation resulting from two\nlateral walls cannot be ignored. Hence, a reasonable gap-av eraging process is necessary.\nA. Governing equation\nTo derive a gap-averaged model, there are two strategies in p revious works. One is to gap-\naverage the Navier-Stokes equations first, and then the flow i s assumed to be inviscid and irrota-\ntional, and hence the potential flow theory is applied. As a co nsequence, a damping coefficient\nappears naturally in the boundary condition at the free surf ace.35The other one is that the ve-\nlocity rather than the equations is the objective to be gap-a veraged. After plugging the Floquet\nansatz into the normal stress boundary condition, the dampi ng terms once again emerge at the\nfree surface. However, this strategy is valid only if viscou s terms in Navier-Stokes equations are\nneglected.41Here, one of the major motivations is to examine the outcome w ith the viscous ef-\nfect involved, so a different strategy has been proposed in t he present study. Although there is no\nreference concentrating on viscosity in the case of Hele-Sh aw flows, viscous theory has been suc-\ncessfully developed in three-dimensional Faraday waves.14,22Based on it, the governing equation\nis simplified to two-dimensional with no assumption on visco sity.\nUnder vertical periodic vibration, the equations governin g fluid motion are\n∂tu′+/parenleftbig\nu′·∇/parenrightbig\nu′=−1\nρ∇p+ν∇2u′−gz(t)ez, (1)\n∇·u′=0, (2)\nwhereu′=(u′,v′,w′)is the fluid velocity in x,y,z-directions, pis the pressure, ν=µ\nρis kinetic\nviscosity, gz(t)=g−acosωtis the effective gravity under sinusoidal oscillation with gdenoting\ngravity. The state of rest is a quiescent fluid with a pressure distribution p=−ρgz(t)z. By\napplying −(∇×∇×)to Eq. (1), where ∇×denotes taking the curl, together with the continuity\n7condition expressed by Eq. (2), the pressure term can be elim inated. Then the velocity satisfies\n∂t∇2u′−ν∇2∇2u′=∇×∇×/parenleftbig\nu′·∇/parenrightbig\nu′. (3)\nLinearizing the above equation and taking the component in t hez-direction, one can obtain\n/parenleftbig\n∂t−ν∇2/parenrightbig\n∇2w′=0, (4)\nwhere ∇2=∂xx+∂yy+∂zzis the Laplacian.\nEq. (4) is still three-dimensional. Simplifying this equat ion to two-dimensional is necessary.\nDue to the consideration of viscosity, we have taken higher- order derivatives of Navier-Stokes\nequations to eliminate the pressure term, which refers to th e derivation from Eq. (1) to Eq. (3).\nThis results in several terms associated with ∂xxor∂yyin Eq. (4). To deal with these terms, in\nthree-dimensional circumstances, a horizontally infinite plane was considered,14whose normal\nmodes were trigonometric functions, e.g. sin (k·x), withk= (kx,ky)andxdenoting horizon-\ntal wave vector and x,y-components, respectively. According to this assumption, there exists a\nmathematical relationship that ∂xx=−k2\nxand∂yy=−k2\ny. However, in Hele-Shaw cells, there is\nno wave profile generated in the direction of the gap except th e meniscus. So ky=0 and ∇2is\nreplaced by ∂xx+∂zzin Eq. (4). If the velocity profile in the y-direction is introduced into Eq. (4)\nand followed by the gap-averaging process, the gap-average d damping is generated from terms\nassociated with ∂yy, which is contradicted with the calculation of wavenumber a nd will lead to\nmistakes in the dispersion relation. In fact, the governing equation of previous theoretical studies\nhas been a two-dimensional Laplace equation, where no gap-a veraged damping is included. So it\nis reasonable to remove the term ∂yyin the Laplacian ∇2during the whole theoretical process to\nguarantee a right dispersion relation, except the derivati on of normal stress boundary condition,\nwhere the gap-averaged damping is introduced. The final gove rning equation is\n[∂t−ν(∂xx+∂zz)](∂xx+∂zz)w=0. (5)\nIn Eq. (5) and what follows, wis a function of x,z,t. The term sin (kx)can be separated from\nw, with kdenoting the wavenumber in the x-direction, where we have omitted the subscript xfor\nbrevity.\n8B. Boundary conditions\nTo obtain boundary conditions for Faraday instability in He le-Shaw cells, the effect of two\nlateral walls should be defined first, which is the main source of damping in the Hele-Shaw sys-\ntem. Now that the equations will be reduced to two-dimension al, a typical method is to introduce\na velocity profile in the y-direction. To deal with it, Li et al. applied the Kelvin–Helmholtz–\nDarcy theory43and assumed that the flow through the gap was of Poiseuille typ e.32,35Based on\nthis, the velocity profile was supposed to remain a parabolic shape in the y-direction. The three-\ndimensional Navier-Stokes equations were then integrated across the gap and averaged with gap\nsizeb. However, the oscillation of the vessel is not considered, w hich may result in the unsteady\nnature of such Hele-Shaw flows. When studying the sloshing in Hele-Shaw cells, Viola, Gallaire,\nand Dollet44included the oscillating Stokes boundary layer which focus ed on the unsteady terms\nof Navier-Stokes equations. Bongarzone et al.41then extended this approach to the Faraday in-\nstability problem and modified it to fit Floquet theory. In ord er to match the present study, the\nvelocity profile can be derived as\nw′(x,y,z,t)=/braceleftBigg\n1−cosh/bracketleftbig\n(1+i)y/(bδj)/bracketrightbig\ncosh/bracketleftbig\n(1+i)/2δj/bracketrightbig/bracerightBigg\nw(x,z,t), (6)\nwhere i is the imaginary unit, j=1,3,5,···corresponds to each Fourier component in the ansatz\n(Eq. (24) in the next section), and δj=√\n2ν/ω\nb√\nj/2denotes the thickness of the oscillating Stokes\nboundary layer after being nondimentionalized by b. A detailed derivation of velocity profile\nEq. (6) is shown in Appendix A. We emphasize that the velocity distribution through the thin gap\nfollows the expression of Eq. (6).\nTo obtain boundary conditions satisfied at the moving free su rface, the mathematical descrip-\ntion of the surface has to be given first. As shown in Fig. 1, the real wave profile contains two\nparts:35\nζ′(x,y,t)=ζ(x,t)+y2\nbcosθ, (7)\nwhere the two terms on the right-hand side indicate the princ ipal wave profile observed in the\nx-direction and the meniscus formed between two lateral wall s, respectively. The latter is ap-\nproximated by assuming that the meniscus is a segment of circ ular and the parabolic function is\nobtained based on Taylor expansion. Then according to Chen a nd Viñals22, the outward-pointing\n9normal unit vector nis written as\nn=(−∂xζ,−∂yζ,1)\n/bracketleftBig\n1+(∂xζ)2+(∂yζ)2/bracketrightBig1/2. (8)\nAnd two linearly independent tangential unit vectors t1andt2are\nt1=(1,0,∂xζ)\n/bracketleftBig\n1+(∂xζ)2/bracketrightBig1/2,t2=(0,1,∂yζ)\n/bracketleftBig\n1+(∂yζ)2/bracketrightBig1/2. (9)\nThe kinematic boundary condition at the free surface is\n[∂t+(u·∇H)]ζ′=w′,atz=ζ, (10)\nwith ∇H=ex∂x+ey∂y. Since the meniscus is too small compared to the principal wa ve profile, the\nposition of the free surface is set to z=ζand only ζ(x,t)is considered in ζ′on the left-hand side.\nEq. (10) is then Taylor expanded around z=0. After eliminating the nonlinear terms, integrating\nalong y, and computing its mean divided by b, it has a form of\n∂tζ=λjw,atz=0. (11)\nwhere the coefficient λj=1+(i−1)δjtanh/bracketleftbig\n(1+i)//parenleftbig\n2δj/parenrightbig/bracketrightbig\nis generated from the gap-averaging\nprocess with the expression of w′shown in Eq. (6).\nThe remaining boundary conditions at the free surface are de rived by considering the stress\ntensor, whose components are Ti j=−[p−ρgz(t)z]δi j+µ/parenleftBig\n∂ju′\ni+∂iu′\nj/parenrightBig\nwith i,jreferring to x,\ny,z. Now that the effect of the air phase above the liquid is negle cted, the tangential components\nof the stress tensor are zero at the free surface, which means\ntm·T·n=0,atz=ζ, (12)\nwith m=1,2. There are actually two equations in Eq. (12), but by taking thex-derivatives and y-\nderivatives respectively, along with the continuity condi tion of Eq. (2), a combination is obtained.\n10After being expanded around z=0 and linearized, Eq. (12) reads\n/parenleftbig\n∇2\nH−∂zz/parenrightbig\nw′=0,atz=0. (13)\nBy using the same method as that implemented to derive Eq. (5) when dealing with ∇2\nH, the\ntangential stress boundary condition Eq. (13) is simplified to\n(∂xx−∂zz)w=0,atz=0. (14)\nThe normal stress at the free surface is balanced by capillar ity force, which means\nn·T·n=σκ,atz=ζ, (15)\nwhere σis surface tension coefficient and κis surface curvature. With the free surface elevation\nof Eq. (7), the curvature is then expressed by\nκ=∂xxζ/radicalBig\n1+(∂xζ)2+2\nbcosθ. (16)\nBefore we move on to further derivation, the term relevant to the contact angle in Eq. (16)\nshould be clarified first. As has been mentioned by Li, Li, and L iao35, most of the studies have\nassumed that the out-of-plane interface shape is semicircu lar and naturally θ=180◦. However,\nin the presence of vertical periodic vibration applied to th e Hele-Shaw cell, the surface near the\nlateral walls undergoes an up-and-down process. This means the contact angle changes constantly\nand a dynamic contact angle model should be considered. Foll owing Li, Li, and Liao35, a linear\nmodel is introduced and the cosine of θis evaluated as45\ncosθ=1−β\nµCa, (17)\nwhere βis friction coefficient and Ca=µ\nσw′is the capillary number defined using vertical velocity\nat the free surface w′=∂tζ. This model can be interpreted as that the static contact ang le is 0◦\nand a modification associated with velocity is introduced wi th the consideration of the dynamic\nbehavior of the free surface. Only the term that varies with Caworks after the linearizing process.\nThe coefficient βis a phenomenological parameter which has been measured exp erimentally by\n11Hamraoui et al.45and successfully used in previous studies.35,41\nSubstituting Eq. (16) and Eq. (17) into the normal stress bou ndary condition Eq. (15), a lin-\nearized equation is obtained after removing nonlinear term s:\n−p+2µ∂zw′+ρ(g−acosωt)ζ−σ∂xxζ+2β\nb∂tζ=0,atz=0. (18)\nThen Eq. (18) is gap-averaged with the consideration of Eq. ( 6), which leads to\n−p+2µλj∂zw+ρ(g−acosωt)ζ−σ∂xxζ+2β\nb∂tζ=0,atz=0. (19)\nTo eliminate the pressure term, we first take the horizontal d ivergence of Eq. (1). With the consid-\neration of continuity condition Eq. (2), a linearized equat ion is obtained that\n−∂ztw′+ν∇2∂zw′=−1\nρ∇2\nHp. (20)\nAlso Eq. (20) is gap-averaged with velocity profile Eq. (6) be ing used\n−2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δj∂zw−λj/bracketleftbig\n∂ztw−ν/parenleftbig\n∂xxzw+∂zzzw/parenrightbig/bracketrightbig\n=−1\nρ∂xxp. (21)\nBy operating Eq. (19) with ∂xxand combining with Eq. (21), the pressure term is eliminated\n−2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δj∂zw−λj[∂ztw−ν(3∂xxzw+∂zzzw)]+( g−acosωt)∂xxζ\n−σ\nρ∂xxxxζ+2β\nρb∂xxtζ=0,atz=0.(22)\nBefore we move on to the next part, some points need to be empha sized. Our attentive readers\nwill notice that there is an alternative way to obtain Eq. (22 ) through having the three-dimensional\nnormal stress boundary condition first and then gap-averagi ng it. However, this treatment will lead\nto misunderstanding and confusion about the concepts of the two terms in the present context:\nviscous dissipation and gap-averaged damping. As mentione d in Sec. I, the former one indicates\nthe in-plane viscosity and the latter one stems from the out- of-plane counterpart. In Hele-Shaw\ncells, the latter one occupies a significant proportion in di ssipation and has been successfully\ncalculated in previous works based on potential flow theory,35,40where a damping coefficient is\n12obtained arising from the y-component of viscous terms in Eq. (1) through the gap-avera ging\nprocess. Please note that viscosity is still the very source of the gap-averaged damping. So far\nthe former one does not appear due to the use of potential flow t heory for the in-plane system.\nNevertheless, if we follow the derivation in three-dimensi onal Faraday instability problem in the\nfirst place,14,22where viscosity in each dimension exists, the linearized fo rm of Eq. (15) will be\noperated with ∇2\nH=∂xx+∂yyand Eq (1) will be operated with (∇H·)to eliminate pressure. A term\n3ν∇2\nH∂zw′will be obtained as a part of viscous dissipation after that a nd ends up the gap-averaged\ndamping after the gap-averaging process. It will genuinely overestimate the damping since this\nprocedure mixes up different ingredients of viscosity toge ther and abuses the gap-averaging to\nall of them. The truth is that viscosity in Navier-Stokes equ ations is associated with the bulk flow\nwhich should be simplified for the y-component, while the counterpart in the original normal st ress\nboundary condition is relevant to the dissipation from the f ree surface which should be treated as\nthe in-plane factor rather than gap-averaged one. To avoid t his interference in the calculation of\ndamping, we have gap-averaged Eq. (18) and Eq. (20) individu ally first and then eliminate the\npressure term. In this sense, the out-of-plane viscosity tr ansforms into the gap-averaged damping,\nand the viscous dissipation is deemed as in-plane in Hele-Sh aw cells. Eq. (22) is thus the final\nform.\nFor the bottom wall, Martino et al.36studied the influence of liquid depth ∆hon the threshold\nof Faraday instability and concluded that for∆h\nb/greaterorapproxeql2 the threshold tends to asymptotic values.\nIn experiments conducted in Hele-Shaw cells,35this ratio far exceeded the critical valve. In this\nsense, assuming that the liquid layer extends to z=−∞is feasible. A null condition at z=−∞\nmeans\nw=0,atz=−∞. (23)\nThe above equations of (5),(11),(14),(22), and (23) consti tute the final hydrodynamic equations\nfor Faraday instability in Hele-Shaw cells, in which the sol utions to be determined are vertical\nvelocity w(x,z,t)and surface displacement ζ(x,t).\nIII. SOLUTION BASED ON LINEAR STABILITY THEORY\nAs has been mentioned, a linear solution is sufficient to desc ribe the instability of the free\nsurface. This has been validated in the three-dimensional s cenario by Kumar and Tuckerman14.\nTheir formulation of deriving a system of equations to define the instability threshold was briefly\n13revisited by Chen and Viñals22. The difference was that the generalized eigenvalue proble m was\nrewritten as an equivalent, implicit expression for the acc eleration threshold. When wavenumber\nkis determined, the threshold value accan be solved by algebraic iteration. A similar procedure is\nextended to Hele-Shaw cells as follows.\nFaraday waves in Hele-Shaw cells possess a subharmonic moti on at a frequency of ω/2 ob-\nserved in previous experiments.31Only subharmonic cases are therefore considered here, whic h\nis reflected in the exponential part of the solutions. Beside s, if the fundamental mode eiωt/2is\nincluded alone, the stability analysis will be only appropr iate for small viscosity, which has been\nillustrated in Ref. 22. With the consideration of these, the Fourier series is applied to contain all\nthe harmonics of the fundamental mode eiωt/2. The ansatz for w(x,z,t)andζ(x,t)are then written\nas\nw(x,z,t)=sin(kx)∑\nj=1,3,5,···eijωt/2wj(z)Aj+c.c.,\nζ(x,t)=sin(kx)∑\nj=1,3,5,···eijωt/2Aj+c.c.,(24)\nwhere Ajare complex amplitudes, c.c.denotes the complex conjugate.\nSubstituting the assumed solutions given by Eq. (24) into th e governing equation (5), one ob-\ntains an ordinary differential equation for wj(z)as\n/bracketleftbigg1\n2ijω/parenleftbig\n−k2+∂zz/parenrightbig\n−ν/parenleftbig\n−k2+∂zz/parenrightbig2/bracketrightbigg\nwj(z)=0. (25)\nTogether with the null condition (23), the solution of Eq. (2 5) has a form of wj(z)=mjepjz+njeqjz\nwith pj=kandqj=/radicalbig\nk2+ijω/2ν.\nThe kinematic boundary condition (11) and tangential stres s boundary condition (14) are used\nto determine the coefficients mjandnj, which are listed as\nmj=2k2ν+1\n2ijω\nλj, nj=−2k2ν\nλj. (26)\nBy now, the solutions expressed in Eq. (24) are determined co mpletely. The function wj(z)is\ngiven by\nwj(z)=1\nλj/bracketleftbigg/parenleftbigg\n2k2ν+1\n2ijω/parenrightbigg\nekz−2k2νez√\nk2+ijωt/2ν/bracketrightbigg\n. (27)\n14According to Ref. 22 and private communication with Profess or Jorge Viñals (\"One component of\nthe velocity derives from a scalar potential, the other does not\"), the two components on the right-\nhand side are irrotational and rotational components of the flow, respectively. Besides, the viscous\nand inviscid contributions are separated explicitly in the first component. It is easy to know that\nwhen it comes to the viscous effect, terms containing νin Eq. (27) should be all included. There\nare two of them. The first term of the first component can be cons idered as the irrotational part of\nthe viscous dissipation, whereas the other one is a rotation al correction to it. A detailed discussion\nof these two terms will be given in Sec. V.\nThe normal stress boundary condition serves as a solvabilit y condition for the instability thresh-\noldac. Substituting the solutions into Eq. (22), for each harmoni ceijωt/2, the equation should be\nsatisfied. Considering the equivalent relationship acosωt=1\n2a/parenleftbig\neiωt+e−iωt/parenrightbig\nand the reality con-\ndition A−1=A∗\n1, equations for each Fourier component finally constitute a s ystem of equations\nlike\nj=1 : H1A1−aA∗\n1−aA3=0,\nj=3 : H3A3−aA1−aA5=0,\nj=5 : H5A5−aA3−aA7=0,\nj=7 :···,(28)\nwhere Hjare coefficients resulting from Eq. (22), which is expressed explicitly as\nHj=/braceleftBig/bracketleftbig/parenleftbig\n6k2ν+ijω/parenrightbig/parenleftbig\nmjpj+njqj/parenrightbig\n−2ν/parenleftbig\nmjp3\nj+njq3\nj/parenrightbig/bracketrightbig\nλj+2k2g+2k4σ\nρ+2ijωβk2\nρb\n+4ν(1+i)/parenleftbig\nmjpj+njqj/parenrightbig\ntanh/bracketleftbig\n(1+i)//parenleftbig\n2δj/parenrightbig/bracketrightbig\nb2δj/bracerightBig\n/k2.(29)\nEq. (28) is a system of equations in unknown amplitudes Ajof the solutions. When the working\nfluid properties, gap size b, and vibrating angular frequency ωare all defined, Hjbecomes a func-\ntion of wavenumber kand acceleration amplitude a. To obtain acalgebraically, we first truncate at\nfinite terms n. Then from the last equation, each Ajin Eq. (28) can be recursively formulated,\nAn=aAn−2\nHn,An−2=aAn−4\nHn−2−a2/Hn, ..., A3=aA1\nH3−a2/H5. (30)\n15Until the first equation in Eq. (28), it has the form of\n/parenleftBigg\nH1−a2\nH3−a2\nH5−···/parenrightBigg\nA1−aA∗\n1≡¯H1(k,a)A1−aA∗\n1=0. (31)\nFor convenience, we use ¯H1to represent the coefficient in front of A1. For a given wavenumber k,\nthe instability threshold akcan be obtained implicitly from\nak=|¯H1(k,ak)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleH1−a2\nk\nH3−···/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (32)\nIn practice, we have verified through convergence analysis t hatn=11 is sufficient to guarantee\nthe accuracy of results. The value of wavenumber kis first determined by traversing the values\nwithin the range from 0 to 1000 (m−1), which contains all experimental data in Hele-Shaw cells.14\nThen Eq. (32) is iteratively solved, after which a critical a cceleration amplitude is obtained as ak.\nThe smallest value of akis the final instability threshold ac, and the corresponding wavenumber is\nthe critical wavenumber kc.\nIV . POTENTIAL FLOW THEORY\nThe above analysis has reserved the viscous effect of fluids. In this section, the potential flow\ntheory is applied to clarify the impact of viscosity on Farad ay instability in such Hele-Shaw cells.\nThe resultant damped Mathieu equation actually can be obtai ned directly from the previous theory\nas long as all νrelated terms are dropped in Eq. (27). For the sake of simplic ity, the same equation\nis derived from the potential flow perspective. It is interes ting that if we look closely at the previous\nderivation, the inviscid but rotational flow cannot be kept i ntentionally and independently, which\nresults in no difference between the ideal and potential flow perspectives.\nFollowing Li, Li, and Liao35, the linearized form of Eqs. (1) and (2) are integrated and av eraged\nalong the y-direction, during which the velocity profile Eq. (6) is used and the function of yinu′\n16is the same as w′. This leads to the gap-averaged Navier-Stokes equations\n∂xu+∂zw=0,\nλj∂tu=−1\nρ∂xp−2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δju+1\nρTx,\nλj∂tw=−1\nρ∂zp−2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δjw−gz(t)+1\nρTz,(33)\nwhere the nonlinear and viscous terms are neglected but capi llary force (Tx,Tz)is included.\nIf we assume that there is no vortex in the two-dimensional flo w,28,32,35,40a velocity potential\nφis introduced which meets ∂xφ=uand∂zφ=w. By integrating along the streamline, Eq. (33)\nis rewritten as\n∂xxφ+∂zzφ=0, (34)\nλj∂tφ+2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δjφ+gz(t)z−σ\nρκ=0,atz=ζ. (35)\nUsing the same method as introduced in Sec. II to calculate th e curvature κ, Eq. (16) and (17) are\nsubstituted into Eq. (35), which is then linearized to\nλj∂tφ+2(1+i)νtanh/bracketleftbig\n(1+i)/2δj/bracketrightbig\nb2δjφ+(g−acosωt)ζ−σ\nρ∂xxζ+2β\nρb∂tζ=0,atz=0.(36)\nTo complete the mathematical problem, the impenetrable bou ndary condition and kinematic\nboundary condition are included, respectively:\n∂zφ=0,atz=−∞,\n∂tζ=λj∂zφ,atz=0.(37)\nThe solutions for φandζhave the form of\nφ(x,z,t)=¯φ(z,t)eikx,\nζ(x,t)=¯ζ(t)eikx.(38)\nSubstituting Eq. (38) into Laplace equation (34), together with the boundary condition (37), the\n17solution of ¯φis given by\n¯φ=1\nkλjd¯ζ\ndtekz. (39)\nSubstituting the solutions into Eq. (36), one can obtain the damped Mathieu equation\nd2¯ζ\ndt2+/braceleftBigg\n2(1+i)νtanh/bracketleftbig\n(1+i)//parenleftbig\n2δj/parenrightbig/bracketrightbig\nb2δjλj+2kβ\nρb/bracerightBigg\nd¯ζ\ndt+k/parenleftbigg\ng−acosωt+σk2\nρ/parenrightbigg\n¯ζ=0.(40)\nIf we set\nγ=γ1+γ2,γ1=2(1+i)νtanh/bracketleftbig\n(1+i)//parenleftbig\n2δj/parenrightbig/bracketrightbig\nb2δjλj,γ2=2kβ\nρb,¯ζ=¯ηe−γt\n2, (41)\nthe standard Mathieu equation is finally derived from Eq. (40 )\nd2¯η\ndT2+(¯p−2 ¯qcos2 T)¯η=0,\n¯p=4k/parenleftbig\ng+σk2/ρ/parenrightbig\n−γ2\nω2,\n¯q=2ka\nω2,(42)\nwhere T=1\n2ωt. The Mathieu equation (42) has been obtained in Refs. 12 and 1 4, and was ex-\ntended to a generalized Mathieu equation with the inclusion of viscosity by Beyer and Friedrich46.\nHowever, the relevant work is rare in the case of Hele-Shaw ce lls except the one by Li et al.32\nwhereas the capillary force was neglected for simplicity.\nTo solve the Mathieu equation, we adopt the technique used in Sec. III. The solution of Eq. (42)\nis expressed as\n¯η(T)=eγ\nωT∑\nj=1,3,5,···eijTBj+c.c., (43)\nwhere Bjhave the same meaning as Ajin Eq. (24).\n18TABLE I. The parameters for the two working liquids. Data are taken from Ref. 35, where the friction\ncoefficient βwere measured by Hamraoui et al.45\nLiquid µ(mPa s) ρ/parenleftbig\ng/cm3/parenrightbig\nσ(mN/cm) β(Pa s)\nPure ethanol ( >99.7 wt%) 1.096 0.785 0.218 0.040\nEthanol solution (50 vol%) 2.362 0.926 0.296 0.070\nSubstituting Eq. (43) into the Mathieu equation (42), a line ar equation system for Bjis obtained\nj=1 : K1B1−aB∗\n1−aB3=0,\nj=3 : K3B3−aB1−aB5=0,\nj=5 : K5B5−aB3−aB7=0,\nj=7 :···,\nwith Kj=ω2\n2k/parenleftbigg\n−j2+¯p+2ijγ\nω+γ2\nω2/parenrightbigg\n.(44)\nEq. (44) is then recursed to the equation for j=1 and solved iteratively to obtain the critical\nconditions for Faraday instability.\nV . RESULTS AND DISCUSSIONS\nA. Comparison with experiments\nTo validate the viscous theory developed in the present pape r, results are compared with exper-\niments conducted by Li, Li, and Liao35. In their experiments, two different Hele-Shaw cells made\nof Polymethyl Methacrylate (referred to as PMMA) were used, which had a length of 300 mm, a\nheight of 60 mm and a gap size b=2 mm or b=5 mm. The working liquids were pure ethanol\n(>99.7 wt%) and a mixture of ethanol and water (50 vol%), whose phys ical properties are given\nin Table I.\nLet us start with the Faraday instability threshold ac, which predicts the critical acceleration\namplitude corresponding to the onset of the free surface. Th e comparison of dimensionless accel-\neration amplitude ac/gagainst driving frequency fis shown in Fig. 2, which contains the experi-\nmental data along with the results obtained from the present viscous theory and previous theories\nof Li, Li, and Liao35and Bongarzone et al.41According to Fig. 2, theories developed in this pa-\nper and by Bongarzone et al.41give better prediction of critical acceleration than that o f Li, Li,\n19and Liao35. This further illustrates that the approximation of Poiseu ille flow underestimates the\ndamping from the gap. As has been exhibited by Bongarzone et al.41, the application of oscil-\nlating Stokes boundary layer modifies the damping coefficien t 12ν/b2to a bigger value. In this\nsense, with the influence of periodic parametric excitation being included, the theory of Stokes\nboundary layer is more suitable for the flow in Hele-Shaw cell s. In addition, it is obvious that the\nevaluation is much better for b=5 mm than for b=2 mm, with the results of ours and Bongarzone\net al.41almost consistent with experimental data for b=5 mm. Bongarzone et al.41attributed the\ndisagreement for b=2 mm to the phenomenological contact angle model and explain ed that dis-\nsipation from the free surface was more prominent as the gap s izebdecreased. However, with the\nassumption of a meniscus formed between two lateral walls, t heoretical results should have fitted\nexperiments better for b=2 mm, since the meniscus is more apparent for smaller gap size . In this\nsense, their explanation is worth deliberating. In fact, th e free surface undergoes a complicated\nprocess that remains unsolved between the lateral walls and existing treatment may be too brief to\ndescribe the dynamic behavior of the contact line. For insta nce, according to a numerical simula-\ntion conducted by Li and Li39, contact-line hysteresis can largely delay the timing of on set. A new\ncontact angle model that is more realistic may also benefit th e prediction, such as ones proposed\nby Kidambi47, Viola, Brun, and Gallaire48, and so on. More attention should be paid to this issue\nto realize a more accurate prediction of Faraday instabilit y in Hele-Shaw cells. When it comes to\nthe comparison between the present theory and that of Bongar zone et al.41, something interesting\nappears. In their derivation, the viscous terms in equation s were all neglected after gap-averaging,\nbut what surprises us is that when viscosity is present the pr edicted thresholds of Faraday insta-\nbility are, counterintuitively, slightly smaller than tho se predicted in the inviscid case. A detailed\nexplanation of the influence of viscosity will be given in the following subsection, which has not\nbeen reported before.35,41\nApart from the critical acceleration amplitude ac, the dispersion relation is another key feature\nto characterize Faraday instability. This is reflected in th e critical wavenumber kccorresponding to\nthe onset. Although previous studies have given different m odified dispersion relations explicitly\nor implicitly in the presence of forcing and damping, the tra ditional linear dispersion relation of\nFaraday waves/parenleftBigω\n2/parenrightBig2\n=gk0+σ\nρk3\n0 (45)\npossesses the best agreement to the laboratory observation s among others. Regardless of this, it is\n2014 16 18 20 22\n f  (Hz)00.20.40.60.811.2 ac / gb = 2 mm\nb = 5 mm\n(a)14 16 18 20 22\n f  (Hz)00.20.40.60.811.21.41.61.8 ac / gb = 2 mm\nb = 5 mm\n(b)\nFIG. 2. Dimensionless critical acceleration amplitude ac/gfor the onset of Faraday waves versus driving\nfrequency f. Comparison between theoretical data (circles: results gi ven by present theory; triangles: results\ngiven by Bongarzone et al.41; diamonds: results given by Li, Li, and Liao35) and experimental measure-\nments (squares) obtained from Li, Li, and Liao35. The filled and empty symbols correspond to b=2 mm\nandb=5 mm, respectively. (a) Pure ethanol and (b) ethanol solutio n.\nnot contradictory for us to compare the results obtained fro m the present theory with experimental\ndata. In Fig. 3, we plot the experimental data obtained from L i, Li, and Liao35, the results given by\nthe present theory and by Bongarzone et al.41, and traditional dispersion relation Eq. (45) is also\nincluded. Compared to the traditional dispersion relation , the results of the critical wavenumber\ngiven by us and Bongarzone et al.41both have some deviations, which is called the detuning of\nthe dispersion relation. Since the impact of viscosity on th e dispersion relation is relatively small,\nwhich can be concluded by comparing our results with those by Bongarzone et al.41, the detuning\nmay be caused by the gap-averaged damping and the dynamic con tact angle. These two factors\nare what we add to the traditional dispersion relation besid es the viscosity. This conjecture will be\ninvestigated in Subsection V C.\nB. In-plane viscosity\nThe straightforward way to look at the impact of viscosity is to compare the full model to the\nideal fluid case, while it is prohibited to filter out the visco sity but with rotational motion retained,\nwhich has been detailed in the first paragraph in Sec. IV. Alte rnatively, to clarify the influence of\nviscosity on Faraday instability, the potential flow theory has been employed. As is known to all,\nonly if fluid is inviscid and flow is irrotational can velocity potential be introduced and potential\n21150 200 250 300 350 400\nk  (m-1)35404550556065707580 / 2  (rad/s)\nExperimental data\nPresent results\nBongarzone et al.\nTraditional relation\n(a)150 200 250 300 350 400\nk  (m-1)35404550556065707580 / 2  (rad/s)\nExperimental data\nPresent results\nBongarzone et al.\nTraditional relation\n(b)\n150 200 250 300 350 400\nk  (m-1)35404550556065707580 / 2  (rad/s)\nExperimental data\nPresent results\nBongarzone et al.\nTraditional relation\n(c)150 200 250 300 350 400\nk  (m-1)35404550556065707580 / 2  (rad/s)\nExperimental data\nPresent results\nBongarzone et al.\nTraditional relation\n(d)\nFIG. 3. Different dispersion relations of Faraday waves. (a )b=2 mm for pure ethanol, (b) b=2 mm for\nethanol solution, (c) b=5 mm for pure ethanol, and (d) b=5 mm for ethanol solution.\nflow be valid.49However, if we delve into Eq. (27), there are two terms having something to do\nwith ν. One is the irrotational part of the viscous dissipation and the other one is the rotational\ncorrection. This distinction between viscous flow and poten tial flow arises from the combined\ninfluence of these two factors. The following concern is how t he two types of physical processes\naffect the final prediction.\nLet us first look at the situation of three-dimensional Farad ay instability in the absence of any\nDarcy effect from Hele-Shaw cells. Chen and Viñals22retained the rotational component during\nthe derivation process. A full linear solution was used to ob tain the onset condition, and the\ndimensionless amplitude of driving acceleration ∆=ak0/4ω2\n0was finally expanded as a power\n22series of the damping coefficient associated with viscosity ˜γ=2νk2\n0/ω0\n∆c=˜γ−1\n2˜γ3/2+11−2G\n8(3−2G)˜γ5/2+···, (46)\nwhere k0is the wavenumber obtained from traditional dispersion rel ation Eq. (45), ω0=ω/2 is\nthe wave angular frequency for subharmonics and G=gk0/ω2\n0is gravity coefficient (see Ref. 22).\nThe first term on the right-hand side is the damping associate d with viscosity of fluids, whereas\nthe second term proportional to ˜γ3/2is a correction contributed by rotational flow. This attribu -\ntion of correction can be found through the present theory as well. Rotational component has an\nexponent ez√\nk2+ijωt/2νin the solution given by Eq. (27) and appears with a dimension ofν3/2\nin Eq. (29) after expanding, which is similar to the three-di mensional case. Now we turn back\nto Eq. (46), the correction term is of a different order and ha s a different sign compared to the\nfirst term, which means the rotational component reduces the dissipation of the whole system and\nbehaves as an opposite of the irrotational viscid term. Simi lar deductions about the correction of\nrotational flow were reported in many literature, e.g., the e xpansion of low viscosity for the linear\nproblem of Faraday waves derived by Müller et al.50and the damped Mathieu equation proposed\nby Nam Hong51.\nIn Hele-Shaw cells, the terms associated with irrotational viscous dissipation and rotational\ncorrection have been separated explicitly in the solution o f Eq. (27). We hence define the Vis-\ncous & Irrotational case and the Rotational case respectively as follows. For the first case,\nwj(z)has the form/parenleftbig\n2k2ν+1\n2ijω/parenrightbig\nekz/λj. The second one is to calculate with wj(z)of the form/parenleftBig\n1\n2i jωekz−2k2νez√\nk2+i jwt/2ν/parenrightBig\n/λjand without consideration of the term νλj(3∂xxzw+∂zzzw)in\nthe normal stress boundary condition Eq. (22) due to its atte nuation effect. Because all theories\nhave already been validated with experiments in the previou s subsection, the comparison can be\nsafely extended to a wider range of driving frequencies from 10 to 30 Hz, which contains the main\nrange which is available in experiments.35The results of the potential flow theory in Sec. IV and\nof the full viscous theory are shown in Fig. 4. As in the previo us discussion about the critical\nacceleration prediction, the same phenomenon emerges that whether considering viscosity or not\nhas a tiny impact on the result. Li, Li, and Liao35once gave the analysis based on the potential\nassumption and worried that the viscosity of fluids might be t he source of the distinction between\ntheory and experiment. From the results given in Fig. 4, it ca n be concluded that obviously when\nwe are only concerned about the threshold of onset, the poten tial flow theory is feasible, which\n2310 12 14 16 18 20 22 24 26 28 30\n f  (Hz)0.60.70.80.911.11.21.31.4 ac / gFull\nPotential\nInviscid & Irrotational\nViscous & Irrotational\nRotational\n(a)10 12 14 16 18 20 22 24 26 28 30\n f  (Hz)0.811.21.41.61.822.2 ac / gFull\nPotential\nInviscid & Irrotational\nViscous & Irrotational\nRotational\n(b)\n10 12 14 16 18 20 22 24 26 28 30\n f  (Hz)0.10.20.30.40.50.6 ac / gFull\nPotential\nInviscid & Irrotational\nViscous & Irrotational\nRotational\n(c)10 12 14 16 18 20 22 24 26 28 30\n f  (Hz)0.20.30.40.50.60.70.80.91 ac / gFull\nPotential\nInviscid & Irrotational\nViscous & Irrotational\nRotational\n(d)\nFIG. 4. Comparison of the theoretical prediction results of dimensionless critical acceleration amplitude.\nSolid lines with squares: results obtained from the full the ory for viscous fluids presented in Sec. II; solid\nlines with asterisks: results of potential flow theory in Sec . IV; dotted lines with diamonds: results with\nthe viscous effect being neglected in the full model; dashed lines with triangles: results for viscous &\nirrotational case; chain lines with circles: results for ro tational case. (a) b=2 mm for pure ethanol, (b)\nb=2 mm for ethanol solution, (c) b=5 mm for pure ethanol, and (d) b=5 mm for ethanol solution.\ncan simplify the stability analysis to a great extent. Fig. 4 also illustrates the results of the Viscous\n& Irrotational case and the Rotational case, along with the inviscid and irrotational one where all\nterms with νinvolved in Eq. (27) are ignored. The resulting curves from t he last case completely\ncoincide with the one obtained based on potential flow theory , which further validates that the full\ngap-averaged model can degenerate to the damped Mathieu equ ation based on the potential flow\ntheory and verifies indirectly the correctness of the deriva tion in Sec. III.\nFrom now on the two factors in viscous effect can be discussed with the aid of curves for the\nViscous & Irrotational case and the Rotational case in Fig. 4. It is obvious that each of them\n24has a remarkable impact on the Faraday instability. Compare d with the inviscid and irrotational\ncase, the thresholds for the appearance of instability plun ge largely with only the rotational com-\nponent involved and are apparently higher in the Viscous & Irrotational case than the reference.\nThe amplitude of the decrease is way much greater than the inc rease, and the overall outcome\nwhen combining both factors results in the lower values of th e threshold predicted by the full the-\nory when b=2 mm compared with the potential flow theory. This counterint uitive phenomenon\narises here is different from what is observed in the three-d imensional case, where the rotational\ncorrection (the second term in the right-hand side of Eq. (46 )) is smaller than the viscous damping\n(the first term in the right-hand side of Eq. (46)). Hence, eve n though the rotational part plays a\npositive role in triggering instability, the overall effec t is stabilizing the flow in a three-dimensional\ncase, but it is another story in Hele-Shaw cells. We speculat e that the reason may arise from the\ngap-averaging process, in which the impact of the rotationa l component can be magnified with\nthe introduction of the gap-averaged damping coefficient. O ne piece of evidence to support this is\nthat for b=5 mm as the frequency rises, the curves of the full theory slig htly overtop the coun-\nterparts of the potential one corresponding to the almost fla t lines of the Rotational case, which\nmeans the gap size is essential in this problem. By analogy to the three-dimensional scenario, by\nexpanding Eq. (29), the coefficient of ν3/2has a different sign compared with the one of ν, but a\nsimilar expression like Eq. (46) is hard to derive for the com plicated nature of the present model.\nFurthermore, this conclusion is drawn directly from theore tical findings. However, validating the\nrotational influence in-plane in Hele-Shaw cells through ex periments is challenging, as separat-\ning the rotational component from others requires delibera te design and is currently beyond our\ncapability.\nC. Damping terms in Mathieu equation\nWithin the scope of this study, dissipation in Hele-Shaw sys tem is caused by the in-plane vis-\ncosity, the gap-averaged damping, and the dynamic contact a ngle. As discussed previously, the\neffect of viscosity is relatively small when the rotational component is involved. Hence, the main\ndissipation that dominates the threshold of Faraday instab ility originates from the gap-averaged\ndamping and the free surface. In this subsection, the contri bution of these two elements to the\ncritical instability conditions is discussed, where the po tential flow theory given in Sec. IV is\nsufficient.\n2510 12 14 16 18 20 22 24 26 28 30\n f  (Hz)-0.200.20.40.60.811.21.41.6 ac / g10, 20\n1= 0, 20\n10, 2= 0\n1= 0, 2= 0\n(a)10 12 14 16 18 20 22 24 26 28 30\n f  (Hz)-0.200.20.40.60.8 ac / g10, 20\n1= 0, 20\n10, 2= 0\n1= 0, 2= 0\n(b)\nFIG. 5. The dimensionless critical acceleration amplitude s obtained based on potential approximation ver-\nsus driving frequency f. Comparison is conducted with different damping terms γ1orγ2being incorporated\ninto the damped Mathieu equation Eq. (40). The working fluid i s pure ethanol. (a) b=2 mm and (b) b=5\nmm.\nThe damping terms have been separated individually in Eqs. ( 40-41), where γ1is the damping\ncoefficient derived from the gap-averaged process and γ2is associated with the dynamic contact\nangle at the free surface. Without loss of generality, only r esults for pure ethanol are discussed\nhere. In Fig. 5, each combination of the contribution of thes e two damping terms to the criti-\ncal acceleration amplitude is plotted. Results indicate th at the gap-averaged damping γ1provides\ndissipation that hardly varies with the forcing frequency. Although this damping coefficient is\nassociated with the thickness of Stokes boundary layer δ=/radicalbig\n2ν/ω, where the frequency is in-\ncluded in fact, the main variable that controls the value of t his coefficient is the gap size bwhen it\nis sufficiently small. This meets the case of Poiseuille flow, where a damping coefficient 12 ν/b2\nis obtained. When it comes to the dissipation resulting from the dynamic contact angle, Fig. 5\nshows a more important role of this dissipation in the critic al acceleration amplitude ac. If this\nfactor is neglected, the prediction of acis far from the real value. Different from the gap-averaged\ndamping, the dissipation from the contact angle increases a pparently with forcing frequency fris-\ning. Although γ2has nothing to do with frequency directly, the wavenumber va ries almost linearly\nwith fas shown in Fig. 3, and karises in each version of the damping term of the contact angl e in\nEq. (29) or Eq. (40). It comes from the out-of-plane curvatur e estimate with respect to the in-plane\nvelocity. In reality, it is plausible to picture that the dyn amic motion of the contact angle occurs\nat the same frequency as the wave does and a strong tie between the frequency and the damp-\ning term from the contact angle exists. If we turn to another p erspective, it is worth noting that\n2610 12 14 16 18 20 22 24 26 28 30\n f  (Hz)20304050607080Dissipation proportion  (%)contact angle dissipation\ngap-averaged dampingb = 1 mm\nb = 2 mm\nb = 5 mm\nb = 8 mm\nFIG. 6. The proportion of dissipation from dynamic contact a ngle and gap-averaged damping. Dashed lines\n(- -): the one associated with dynamic contact angle; chain l ines (–·): the one associated with gap-averaged\ndamping.\nthe damping term γ2is sensitive to the value of β,41which is obtained from a phenomenological\nmethod in experiments, where the physical features of the su bstrate and the working liquid are\nessential to measuring this parameter.45Due to its sensitivity to β, A slight discrepancy between\nthe experiments can lead to a significant difference. This ma y contribute to the distinction between\ntheory and experiment in the very first subsection in Sec. V. T here thus are many issues left worth\nimproving in the dynamic contact angle model.\nSince the two damping terms in Eq. (40) are added together lin early and the contribution to acis\nindependent, we calculate the proportion of each one to the e ntire dissipation. We extend the range\nof gap size to reveal the relationship between dissipation a nd geometric features of Hele-Shaw\ncells. A tendency appears similar to Fig. 5 that dissipation caused by dynamic contact angle is\nstrengthened as the forcing frequency increases. However, γ1experiences a decline as fescalates,\nwhich differs from what is observed in Fig. 5. We argue that γ2stabilizes the liquid motion in\na more direct way than γ1does, given the different performances between Fig. 5 and Fi g. 6 as f\nchanges. As for the influence of the gap size on the distributi on of the two ingredients, according\nto Fig. 6, with the gap size decreasing, the influence of γ1becomes more prominent compared to\nγ2, which means the dissipation from two lateral walls is enlar ged, despite the lower level from\n27100 150 200 250 300 350 400 450 500 550\nk  (m-1)30405060708090100 / 2  (rad/s)10, 20\n1= 0, 20\n10, 2= 0\n1= 0, 2= 0\nTraditional relation\n(a)100 150 200 250 300 350 400 450 500 550\nk  (m-1)30405060708090100 / 2  (rad/s)10, 20\n1= 0, 20\n10, 2= 0\n1= 0, 2= 0\nTraditional relation\n(b)\nFIG. 7. The dispersion relation of Faraday waves in Hele-Sha w cells. Comparison between traditional\ndispersion relation and potential theory with the damped te rmsγ1orγ2in Eqs. (40-41) being neglected. The\ngap size is b=2 mm. (a) Pure ethanol and (b) ethanol solution.\nγ1than its counterpart most of the time. There seems to be a conv ergent percentage for each of\nthem when the gap size is large enough if we look at cases with b=5 mm and b=8 mm closely.\nThis is a strong conclusion. Nevertheless, as the container tends to be a wider one the three-\ndimensional nature appears, and then the assumption on whic h our theory is based is gradually\nruined. Dissipation from all walls matters in that scenario , which will leave this convergence\nmeaningless. To fully exploit this trend of convergence one needs to find the upper bound of the\ngap size as a necessity.\nIn the previous subsection, we speculate that γ1andγ2should be blamed for the detuning of\nthe dispersion relation, which is manifested in the error ba r shown in Fig. 3. To analyze this prob-\nlem, comparison between results obtained from each combina tion of the two terms and traditional\ndispersion relation is made in Fig. 7. Since detuning is prom inent in cases of b=2 mm, without\nloss of generality, only results for this gap size are shown. According to Fig. 7, the two cases with\nthe gap-averaged damping coefficient γ1being removed coincide with the traditional dispersion\nrelation quite well, while the cases with this term being con sidered show a deviation. In contrast,\nthe influence of dynamic contact angle on the dispersion rela tion is negligible. In this sense, the\ndetuning comes mainly from the gap-averaged damping coeffic ient. This detuning has been ob-\nserved by Bongarzone et al.41, whose results are similar to ours as shown in Fig. 3. Accordi ng\nto their statement, the real part of the damping coefficient γ1leads to a higher value of critical\nacceleration amplitude, while the imaginary part causes th e detuning. This reason is confirmed in\n28the present work by removing the imaginary part of the gap-av eraged damping coefficient, and the\nprediction is much better just as the ones shown in Fig. 7 when γ1is neglected. We know that so far\nalthough the application of Stokes boundary layer enhances the prediction of instability threshold,\nthe unreal detuning shows up. It brings in an inherent contra diction for this updated out-of-plane\nvelocity profile. If the improvement of the estimate of the da mping comes at the cost of hindering\nthe dispersion relation, the gap-averaged damping must be o verestimated as well. Considering the\nnegligible impact of viscous dissipation discussed earlie r, the contact angle model remains the sole\nviable option to mitigate the instability. However, the cur rent formulation of γ2fails to adequately\noffset the potentially overestimated value of γ1. In spite of the promising results of the present\nstudy, a more accurate model is still needed to precisely sol ve this problem in the future.\nVI. CONCLUSIONS\nIn the scenario of three-dimensional Faraday waves, viscou s theory has been widely used to\npredict the instability threshold and pattern formation be yond onset.14,22But in Hele-Shaw cells,\nexisting theories are all based on potential flow theory or ne glect the viscosity of fluids after gap-\naveraging. The effect of viscosity on this problem remains u nknown, we thus apply viscous theory\nto the study of Faraday instability in Hele-Shaw cells.\nIn this paper, a new model that contains the viscosity of fluid s, the surface tension, and the dy-\nnamic contact angle is derived to predict the critical condi tions for the onset of Faraday waves in\nHele-Shaw cells. Starting from three-dimensional Navier- Stokes equations, the governing equa-\ntion is simplified with viscosity being reserved. An ingenio us gap-averaging technique is signifi-\ncant since viscosity complicates the problem largely. The g ap-averaged damping and the effect of\nviscosity are clarified during the derivation process. Afte r expanding the equations and removing\nnonlinear terms, the final linear hydrodynamic model is obta ined to describe the Faraday insta-\nbility. To access the critical acceleration amplitude and c ritical wavenumber, a semi-analytical\nmethod22is used and a system of equations for the amplitude of the solu tion is solved recursively.\nFinally, results of the present theory are compared with exp eriments conducted by Li, Li, and\nLiao35and with some previous theoretical ones.\nWe conclude that the presence of viscosity leads to a small co rrection compared to the potential\nflow theory. Although the viscosity has no apparent impact on the onset as a whole, the irrota-\ntional and viscous component and the rotational correction have different effects on the critical\n29conditions for the onset. The rotational correction can tri gger the motion rather than suppress it\nto a large extent. Each of them is qualitatively consistent w ith the three-dimensional case but the\noverall consequence is counterintuitive compared to the th ree-dimensional scenario. If the detailed\ninfluence of viscosity is not the priority, the potential flow theory is sufficient to predict the onset\nin Hele-Shaw cells. In this sense, a novel damped Mathieu equ ation is obtained with two damping\nterms referring to the gap-averaged damping and energy loss from the dynamic contact angle. The\ngap-averaged damping is enhanced as the gap size decreases w hile dissipation from the dynamic\ncontact angle grows for higher frequency.\nDespite the promising outcome given in the present study, no theory can provide an exact agree-\nment to the experimental observation of the onset, especial ly when the gap size turns smaller. As\nfor the dispersion relation, the unusual detuning appearin g in the present theory comes from the\nupdated damping coefficient. The introduction of the theory of Stokes boundary layer leads to a\ncontradiction between a greater dissipation from two later al walls and a deviation from the real\ndispersion relation, which means it probably overestimate s the damping as well. If this is the case,\nthe offset introduced by this misuse should be compensated b y the dissipation of the out-of-plane\ncapillary force. Actually, there is room to improve the theo ry since the calculation of the curvature\nbased on molecular dynamics is simply a linear model and reli es on the phenomenological param-\neter heavily. In addition, the existence of the liquid film on lateral walls makes the wetting process\nmore complicated which has not been considered by any existi ng literature yet.35Considering the\nlack of experimental records revealing the real dynamic beh avior of the out-of-plane free surface,\nthe treatment of the contact line deserves more attention to modify the current models.\nACKNOWLEDGMENTS\nThe authors are grateful to the private communication with P rofessor Jorge Viñals. The au-\nthors would like to acknowledge the National Natural Scienc e Foundation of China (Approval\nNo.12002206) and Fundamental Research Funds for the Centra l Universities for the financial sup-\nport of this work.\nDECLARATION OF INTERESTS\nThe authors report no conflict of interest.\n30DATA A V AILABILITY\nThe data that supports the findings of this study are availabl e within the article and from the\ncorresponding author upon reasonable request.\nAppendix A: The oscillating Stokes boundary layer\nThe oscillating Stokes boundary layer introduced in this pa per is inspired by Bongarzone\net al.41Here we give a rough description of Eq. (6). For the original d erivation please refer to\nBongarzone et al.41\nLinearizing Eq. (1) at the rest state u′=0 and p=−ρgz(t)z, one can obtain\n∂u′\n∂t=−1\nρ∇p+ν∇2u′. (A1)\nAssuming that bk≪1, the velocity through the gap satisfies v′≪u′,w′. Then Eq. (A1) can be\nsimplified as\n∂u′\n∂t=−1\nρ∂p\n∂x+ν∂2u′\n∂y2,∂w′\n∂t=−1\nρ∂p\n∂z+ν∂2w′\n∂y2,∂p\n∂y=0. (A2)\nBy making dimensionless as follows:\n¯x=xk,¯y=y\nb,¯z=zk,¯u=u′\naω,¯v=v′\nbkaω,¯w=w′\naω,¯p=kp\nρaω2,¯t=ωt,(A3)\nthe first two equations in Eq. (A2) can be expressed in a dimens ionless form which is written as\n∂¯u\n∂¯t=−∂¯p\n∂¯x+δ2\nSt\n2∂2¯u\n∂¯y2,∂¯w\n∂¯t=−∂¯p\n∂¯z+δ2\nSt\n2∂2¯w\n∂¯y2, (A4)\nwhere δSt=/radicalbig\n2ν/ω/bis the dimensionless thickness of the oscillating Stokes bo undary layer.\nWhen separating the terms relevant to ¯t, the solutions of Eq. (A4) are written as ¯ u(¯x;¯t) =\nRe/bracketleftBig\n˜u(¯x)eij¯t/2/bracketrightBig\n, where j=1,3,5,···denote Fourier terms. Substituting it into Eq. (A4), it can\nbe found that each Fourier component must satisfy\nji\n2˜u=−∂¯p\n∂¯x+δ2\nSt\n2∂2˜u\n∂¯y2,ji\n2˜w=−∂¯p\n∂¯z+δ2\nSt\n2∂2˜w\n∂¯y2. (A5)\n31By solving the ordinary differential equations about ¯ y, along with the no-slip condition at ¯ y=±1\n2,\nthe explicit functions of ¯ yare separated that\n˜u=2i\nj∂¯p\n∂¯x/braceleftBigg\n1−cosh/bracketleftbig\n(1+i)¯y/δj/bracketrightbig\ncosh/bracketleftbig\n(1+i)/2δj/bracketrightbig/bracerightBigg\n,˜v=0,˜w=2i\nj∂¯p\n∂¯z/braceleftBigg\n1−cosh/bracketleftbig\n(1+i)¯y/δj/bracketrightbig\ncosh/bracketleftbig\n(1+i)/2δj/bracketrightbig/bracerightBigg\n.(A6)\nThe coefficient δjis somewhat different from the one in Ref. 41, since differen t forms of solutions\nare assumed. Using the function of ¯ yin Eq. (A6) to describe the velocity profile leads to the\nfollowing expression in a dimensional form:\nu′(x,y,z,t)=/braceleftBigg\n1−cosh/bracketleftbig\n(1+i)y/(bδj)/bracketrightbig\ncosh/bracketleftbig\n(1+i)/2δj/bracketrightbig/bracerightBigg\nu(x,z,t),\nv′(x,y,z,t)=0,\nw′(x,y,z,t)=/braceleftBigg\n1−cosh/bracketleftbig\n(1+i)y/(bδj)/bracketrightbig\ncosh/bracketleftbig\n(1+i)/2δj/bracketrightbig/bracerightBigg\nw(x,z,t).(A7)\nREFERENCES\n1P. Chen, Z. Luo, S. Güven, S. Tasoglu, A. V . Ganesan, A. Weng, a nd U. Demirci, “Microscale\nassembly directed by liquid-based template,” Advanced mat erials 26, 5936–5941 (2014).\n2F. Liu, N. Kang, Y . Li, and Q. Wu, “Experimental investigatio n on the atomization of a spherical\ndroplet induced by Faraday instability,” Experimental The rmal and Fluid Science 100, 311–318\n(2019).\n3E. Turkoz, S. Kang, X. Du, L. Deike, and C. B. Arnold, “Reducti on of transfer threshold energy\nfor laser-induced jetting of liquids using faraday waves,” Physical Review Applied 11, 054022\n(2019).\n4O. M. Faltinsen, “Sloshing,” (2017).\n5M. Faraday, “On a Peculiar Class of Acoustical Figures; and o n Certain Forms Assumed by\nGroups of Particles upon Vibrating Elastic Surfaces,” Phil osophical Transactions of the Royal\nSociety of London Series I 121, 299–340 (1831).\n6L. Matthiessen, “Akustische versuche, die kleinsten trans versalwellen der flüssigkeiten betref-\nfend,” Annalen der Physik 210, 107–117 (1868).\n7L. Rayleigh, “VII. On the crispations of fluid resting upon a v ibrating support,” The London,\nEdinburgh, and Dublin Philosophical Magazine and Journal o f Science 16, 50–58 (1883).\n328H. W. Müller, “Periodic triangular patterns in the Faraday e xperiment,” Physical review letters\n71, 3287 (1993).\n9W. S. Edwards and S. Fauve, “Patterns and quasi-patterns in t he Faraday experiment,” Journal\nof Fluid Mechanics 278, 123–148 (1994).\n10A. Kudrolli and J. P. Gollub, “Patterns and spatiotemporal c haos in parametrically forced surface\nwaves: a systematic survey at large aspect ratio,” Physica D : Nonlinear Phenomena 97, 133–154\n(1996).\n11O. Lioubashevski, H. Arbell, and J. Fineberg, “Dissipative solitary states in driven surface\nwaves,” Physical Review Letters 76, 3959 (1996).\n12T. B. Benjamin and F. J. Ursell, “The stability of the plane fr ee surface of a liquid in vertical\nperiodic motion,” Proceedings of the Royal Society of Londo n. Series A. Mathematical and\nPhysical Sciences 225, 505–515 (1954).\n13L. D. Landau and E. M. Lifshitz, Fluid mechanics (Pergamon, 1959).\n14K. Kumar and L. S. Tuckerman, “Parametric instability of the interface between two fluids,”\nJournal of Fluid Mechanics 279, 49–68 (1994).\n15W. S. Edwards and S. Fauve, “Parametrically excited quasicr ystalline surface waves,” Physical\nReview E 47, R788 (1993).\n16A. Pototsky and M. Bestehorn, “Faraday instability of a two- layer liquid film with a free upper\nsurface,” Physical Review Fluids 1, 023901 (2016).\n17J. W. Miles, “Nonlinear faraday resonance,” Journal of Flui d Mechanics 146, 285–302 (1984).\n18J. Miles, “On faraday waves,” Journal of Fluid Mechanics 248, 671–683 (1993).\n19S. T. Milner, “Square patterns and secondary instabilities in driven capillary waves,” Journal of\nfluid mechanics 225, 81–100 (1991).\n20S. Douady and S. Fauve, “Pattern selection in Faraday instab ility,” Europhysics letters 6, 221\n(1988).\n21W. Zhang and J. Viñals, “Pattern formation in weakly damped p arametric surface waves,” Jour-\nnal of Fluid Mechanics 336, 301–330 (1997).\n22P. Chen and J. Viñals, “Amplitude equation and pattern selec tion in Faraday waves,” Physical\nReview E 60, 559 (1999).\n23M.-T. Westra, D. J. Binks, and W. Van De Water, “Patterns of Fa raday waves,” Journal of Fluid\nMechanics 496, 1–32 (2003).\n24S. Zhang, A. G. Borthwick, and Z. Lin, “Pattern evolution and modal decomposition of Faraday\n33waves in a brimful cylinder,” Journal of Fluid Mechanics 974, A56 (2023).\n25Y . Murakami and M. Chikano, “Two-dimensional direct numeri cal simulation of parametrically\nexcited surface waves in viscous fluid,” Physics of Fluids 13, 65–74 (2001).\n26S. Ubal, M. D. Giavedoni, and F. A. Saita, “A numerical analys is of the influence of the liquid\ndepth on two-dimensional Faraday waves,” Physics of Fluids 15, 3099–3113 (2003).\n27N. Périnet, D. Juric, and L. S. Tuckerman, “Numerical simula tion of Faraday waves,” Journal\nof Fluid Mechanics 635, 1–26 (2009).\n28J. Rajchenbach, A. Leroux, and D. Clamond, “New standing sol itary waves in water,” Physical\nreview letters 107, 024502 (2011).\n29X. Li, D. Xu, and S. Liao, “Observations of highly localized o scillons with multiple crests and\ntroughs,” Physical Review E 90, 031001 (2014).\n30X. Li, Z. Yu, and S. Liao, “Observation of two-dimensional Fa raday waves in extremely shallow\ndepth,” Physical Review E 92, 033014 (2015).\n31X. Li, X. Li, and S. Liao, “Observation of two coupled Faraday waves in a vertically vibrating\nHele-Shaw cell with one of them oscillating horizontally,” Physics of Fluids 30(2018).\n32J. Li, X. Li, K. Chen, B. Xie, and S. Liao, “Faraday waves in a He le-Shaw cell,” Physics of\nFluids 30(2018).\n33S. Popinet, “Gerris: a tree-based adaptive solver for the in compressible Euler equations in com-\nplex geometries,” Journal of computational physics 190, 572–600 (2003).\n34S. Popinet, “An accurate adaptive solver for surface-tensi on-driven interfacial flows,” Journal of\nComputational Physics 228, 5838–5866 (2009).\n35J. Li, X. Li, and S. Liao, “Stability and hysteresis of Farada y waves in Hele-Shaw cells,”\nJournal of Fluid Mechanics 871, 694–716 (2019).\n36R. Martino, A. Boschan, D. B. Maggi, G. Bongiovanni, J.-C. Gé minard, and M. Piva, “Sediment\nmotion induced by Faraday waves in a Hele-Shaw cell,” Physic al Review E 101, 043112 (2020).\n37Boschan, A. and Noseda, M. and Aguirre, M. A. and Piva, M., “Co mment on ’Faraday waves in\na Hele-Shaw cell’ [Phys. Fluids 30, 042106 (2018)],” Physic s of Fluids 35, 029101 (2023).\n38J. Li, X. Li, and S. Liao, “Response to ’Comment on ’Faraday wa ves in a Hele-Shaw cell’ Phys\nFluids 30,042106 (2018)’,” (2023), arXiv:2303.12537 [Flu id Dynamics].\n39J. Li and X. Li, “Numerical study of the impact of contact line with hysteresis on the Faraday\ninstability,” Physics of Fluids 34(2022).\n40A. Rachik and S. Aniss, “Effects of finite depth and surface te nsion on the linear and weakly\n34non-linear stability of Faraday waves in Hele-Shaw cell,” F luid Dynamics Research 55, 045506\n(2023).\n41A. Bongarzone, B. Jouron, F. Viola, and F. Gallaire, “A revis ed gap-averaged Floquet analysis\nof Faraday waves in Hele-Shaw cells,” Journal of Fluid Mecha nics977, A45 (2023).\n42X. Li, J. Li, X. Li, S. Liao, and C. Chen, “Effect of width on the properties of Faraday waves in\nHele-Shaw cells,” Science China Physics, Mechanics & Astro nomy 62, 1–6 (2019).\n43P. Gondret and M. Rabaud, “Shear instability of two-fluid par allel flow in a Hele–Shaw cell,”\nPhysics of Fluids 9, 3267–3274 (1997).\n44F. Viola, F. Gallaire, and B. Dollet, “Sloshing in a Hele-Sha w cell: experiments and theory,”\nJournal of Fluid Mechanics 831, R1 (2017).\n45A. Hamraoui, K. Thuresson, T. Nylander, and V . Yaminsky, “Ca n a dynamic contact angle\nbe understood in terms of a friction coefficient?” Journal of colloid and interface science 226,\n199–204 (2000).\n46J. Beyer and R. Friedrich, “Faraday instability: linear ana lysis for viscous fluids,” Physical Re-\nview E 51, 1162 (1995).\n47R. Kidambi, “Capillary damping of inviscid surface waves in a circular cylinder,” Journal of\nfluid mechanics 627, 323–340 (2009).\n48F. Viola, P.-T. Brun, and F. Gallaire, “Capillary hysteresi s in sloshing dynamics: a weakly\nnonlinear analysis,” Journal of Fluid Mechanics 837, 788–818 (2018).\n49H. Lamb, Hydrodynamics (Cambridge University Press, 1932).\n50H. W. Müller, H. Wittmer, C. Wagner, J. Albers, and K. Knorr, “ Analytic stability theory for\nFaraday waves and the observation of the harmonic surface re sponse,” Physical Review Letters\n78, 2357 (1997).\n51U. Nam Hong, “A new approach to parametric excitation of stat ionary surface waves in a viscous\nliquid,” Bull. Russ. Acad. Sci. Phys./Suppl 57, 131 (1993).\n35"    },    {        "title": "2402.05518v1.Rigidity_of_Lyapunov_Exponents_for_Geodesic_Flows.pdf",        "content": "arXiv:2402.05518v1  [math.DS]  8 Feb 2024Rigidity of Lyapunov Exponents for Geodesic Flows\nNestor Nina Zarate∗Sergio Roma˜ na†\nThis paper is dedicated to the memory of Nestor Nina Zarate\nAbstract\nIn this paper, we study rigidity problems between Lyapunov e xponents along\nperiodic orbits and geometric structures. More specificall y, we prove that for a\nsurfaceMwithout focal points, if the value of the Lyapunov exponents is constant\nover all periodic orbits, then Mis the flat 2-torus or a surface of constant negative\ncurvature. We obtain the same result for the case of Anosov ge odesic flow for\nsurface, which generalizes C. Butler’s result [5] in dimens ion two. Using completely\ndifferent techniques, we also prove an extension of [5] to the fi nite volume case,\nwhere the value of the Lyapunov exponents along all periodic orbits is constant,\nbeing the maximum or minimum possible.\nKeywords: Lyapunov Exponent, Rigidity, Anosov flow.\n1 Introduction.\nRigidity problems in dynamics and geometry are very interesting rese arch topics in math-\nematics because they provide us with a very deep connection betwe en two different areas.\nMany researchers aroundthe world are unraveling different types of rigidity between these\ntwo areas. We can cite several of these authors ([3, 6, 7, 8, 9, 10 , 12, 15, 16, 17, 33, 34].\nWe invite the reader to check the introduction of the articles [30] a nd [5], which provide\nan overview of the different results that have already been obtaine d.\nIn this paper, we focus our attention on rigidity problems of Lyapun ov exponents. We\nwill see that, under certain conditions on the Lyapunov exponents of an Anosov geodesic\nflow (of geodesic flow of surface without focal points), we can obt ain rigidity on the sec-\ntional curvature. Our goal is to extend the result over the rigidity of equality of Lyapunov\nexponents for geodesic flows of Butler in [5] for dimension two.\nLetθbea periodicorbit ofperiod τforthegeodesic flow φt:SM→SMfora Rieman-\nnian manifold M. Letχ1(θ),...,χ m−1(θ) be the complex eigenvalues of Dθφτ:Eu\nθ→Eu\nθ,\ncounted with the multiplicity. A beautiful result due to C. Butter [5] stated that:\nTheorem 1.1. [5]LetMbe anm-dimensional compact negatively curved Riemannian\nmanifold. Suppose that\n|χi(θ)|=|χj(θ)|; 1≤i,j≤m−1,\n∗Partially Supported By Capes and CNPq\n†Suppoted by Bolsa Jovem Cientista do Nosso Estado No. E-26/201.4 32/2022 - Brazil, NNSFC\n12071202, and NNSFC 12161141002 from China.\n1for every periodic point θof the geodesic flow φ:SM→SM. ThenMis homothetic to\na compact quotient of Hm\nR.\nIn particular, Butler’s result above claims that if each periodic orbit o f the geodesic\nflow defined on a manifold of negative curvature, has exactly one Ly apunov exponent on\nthe unstable (or stable) bundle then the manifold has constant neg ative curvature.\nSince the geodesic flow on a compact manifold of negative curvature is an Anosov flow,\nthen supported by Butler’s result, in [28] the authors proposed th e following conjecture\n(see [28, Conjecture 1]).\nConjecture 1. LetMbe a complete Riemannian manifold with finite volume, whose\ngeodesic flow is Anosov. If the unstable Lyapunov exponents a re constant in all periodic\norbits, then Mhas constant negative sectional curvature.\nThe main goal of this paper is to prove the last conjecture in dimensio n two. In fact, we\nprove some rigidity results for conservative Anosov flows on manifo lds of dimension three,\nwhich encompasses the case of geodesic Anosov flows of compact s urfaces.\nWe said that a flow φtis conservative if it preserves a smooth volume measure. In this\ncontext, our first result is:\nTheorem 1.2. Letφt:N→Nbe aCk-conservative Anosov flow with k≥5on a\nthree-dimensional manifold N. The following are equivalents:\n(1)There isα>0such that for all θ∈Per(φ)andξ∈Eu\nθ\\{0}\nχ+(θ,ξ) = lim\nt→+∞1\ntlog/bardbldθφt(ξ)/bardbl=α.\n(2)The volume mis a measure of maximal entropy.\n(3)For anyǫ>0,φtisCk−ǫconjugate to an Algebraic flow.\nGeodesic flows on compact manifolds are natural examples of conse rvative flows, in\nfact, theLiouvillemeasure isalways aninvariant measure forthegeo desic flow. Therefore,\nusing the structure of the geodesic flow, we use the Theorem 1.2 to prove the Conjecture\n1 for surfaces. More specifically\nTheorem 1.3. [Main Theorem ]Letφt:SM→SMbe theCk- Anosov geodesic flow\nwithk≥5on a compact surface M. The following are equivalents:\n(1)There isα>0such that for all θ∈Per(φ)andξ∈Eu\nθ\\{0}\nχ+(θ,ξ) = lim\nt→+∞1\ntlog/bardbldθφt(ξ)/bardbl=α.\n(2)The surface Mhas constant negative curvature KM=−α2.\n2The last two results are also valid looking for the Lyapunov exponent s on the stable sub-\nbundle.\nThe theorem 1.3 extends Butler’s result in dimension two since we just assume the\nAnosov condition of geodesic flow without any restrictions on the su rface curvature. The\nmain idea behind the proof of this theorem is to use Kalinin’s result (cf. [23]) to show\nthat the Liouville measure is a maximum entropy measure (MME), and t hen find some\ngeometric rigidity for this fact. For this sake, we prove that the hy potheses of 1.3 are\nequivalent to having the Liouville measure as an MME in any dimension. Mo re precisely,\nTheorem 1.4. LetMbe a compact Riemannian manifold with Anosov geodesic such t hat\nthe unstable (or stable)Lyapunov exponents are constant along the periodic orbits, then\nthe Liouville measure Lis a measure of maximal entropy, i.e.,htop(φ) =hL(φ).\nThe Theorem 1.4 is linked to the Katok Entropy Conjecture (see Conjecture 2).\nFurthermore, due to a good understanding of the geodesic flow of surfaces, we can use\nTheorem 1.4 to obtain a very nice rigidity for the geodesic flows of sur faces without focal\npoints.\nTheorem 1.5. LetMbe a compact surface without focal points with a C2Riemannian\nmetric and let φtbe its geodesic flow. The following are equivalents:\n(1)There isα≥0such that for all θ∈Per(φ)andξ∈Gu\nθ\\{0}\nχ+(θ,ξ) = lim\nt→+∞1\ntlog/bardbldθφt(ξ)/bardbl=α,\nwhereGu\nθis the unstable Green subdundle.\n(2)The surface Mhas curvature KM=−α2. Moreover, if α= 0, thenM=T2, the\nflat torus.\nNote that, in the last theorem, we can relax the high differentiability c ondition in the\nTheorem 1.3 and even relax the hyperbolicity condition. Theorem 1.3 a nd Theorem 1.5\nmay seem similar, but they are not. First, note that the Anosov con dition does not imply\nthe absence of focal points. Furthermore, they have different d ifferentiability classes.\nFinally, using techniques completely different from those used in the p roof of all previ-\nous theorems, we extend Butler’s results in the case of finite volume and pinched negative\ncurvature, when the value of the Lyapunov exponents is maximum o r minimum, more\nspecifically:\nTheorem 1.6. LetMbe a complete Riemannian manifold of finite volume and such th at\n−c2≤KM≤ −b2<0. Letφt:SM→SMbe the geodesic flow. Consider α∈ {b,c}and\nassume that for all θ∈Per(φt)we have\nχ+(θ,ξ) = lim\nt→+∞1\ntlog/bardbldθφt(ξ)/bardbl=α,\nfor allξ∈Eu\nθ\\{0}. ThenKM=−α2.\n3The Theorem 1.6gives us theproof of aweak version ofconjecture 1 inthe case of pinched\nnegative curvature when the Lyapunov exponents arethe minimum or maximum possible.\nThe conjecture remains open for the general case, even on comp act manifolds.\nStructure of the Paper: In section 2 we provide the tools to prove the main results. In\nsection 3, we proved Theorem 1.4 and later we will use it in the proof of Theorem 1.2 and\nTheorem 1.3. We finish section 3 with the proof of Theorem 1.5 using su rface techniques\nwithout focal points. Finally, section 4 will be dedicated to the proof of Theorem 1.6.\n2 Preliminaries and Notation\nThroughoutthispaper, MwilldenoteacompleteRiemannianmanifoldwithoutboundary\nof dimension n≥2,TMis the tangent bundle, SMits unit tangent bundle, and Lwill\nbe its Liouville measure.\n2.1 Geodesic Flow\nForeacht∈Randθ= (p,v)∈SMconsider thefamilyofdiffeomorphism φt:SM→SM\ndefined by\nφt(θ) = (γθ(t),γ′\nθ(t)),\nwhereγθ(t) is the unique geodesic with initial conditions γθ(0) =pandγ′\nθ(0) =vThis\nfamily is called the geodesic flow defined over SM.\nThe spray vector field Gis the vector field derivative of φt, that is,G(θ) :=d\ndtφt(θ)/vextendsingle/vextendsingle/vextendsingle\nt=0.\nGivenθ= (x,v)∈SM, we identify TθTMwithTxM⊕TxMusing the identification\nξ→(dπ(ξ),Kθ(ξ)),whereπ:TM→Mdenotesthecanonicalprojectionand K:TTM→\nTMthe connection map defined via the Levi-Civita connection. The last id entification\nallows us to define Sasaki’s metric onTMas\n/a\\}bracketle{tξ,η/a\\}bracketri}htθ=/a\\}bracketle{tDθπ(ξ),Dθπ(η)/a\\}bracketri}ht+/a\\}bracketle{tKθ(ξ),Kθ(η)/a\\}bracketri}ht.\nThe geodesic flow φtisAnosovif the tangent bundle of SM,T(SM), has a splitting\nT(SM) =Es⊕/a\\}bracketle{tG/a\\}bracketri}ht⊕Eusuch that\nD(φt\nM)θ(Es(θ)) =Es(φt\nM(θ)),\nD(φt\nM)θ(Eu(θ)) =Eu(φt\nM(θ)),\n||D(φt\nM)θ/vextendsingle/vextendsingle\nEs|| ≤Cλt,\n||D(φ−t\nM)θ/vextendsingle/vextendsingle\nEu|| ≤Cλt,\nfor allt≥0 withC >0 and 0< λ <1, whereGis the geodesic vector field. In that\ndefinition, we are always using Sasaki’s metric ofSM(see [36] for more details).\nNatural examples of Anosov geodesic flows are produced by metric s of negative curvature\non compact manifolds (cf. [1]) and metrics of pinched negative curva ture1in the non-\ncompact case ([25]). However, other examples can be found withou t the last assumption\non the curvature (see by example [14] and [29]).\n1That means there are b,c >0 such that the sectional curvature satisfies −c2≤K≤ −b2.\n42.2 No conjugate points and No focal Points\nGivenγwe said that a vector field Jalong a geodesic γis a Jacobi field if it satisfies the\nJacobi equation\nJ′′(t)+R(γ′(t),J(t))γ′(t) = 0,\nwhereRis the curvature tensor of Mand “′” denotes the covariant derivative along γ.\nJacobi’s field plays an important role in the study of the dynamic of the geodesic flow, in\nfact, the Jacobi field can be used to get geometric properties whe n we have some dynamic\nproperties of the geodesic flow (see [36] and the introduction of [ 28] for more details).\nAnother important observation is based on the following fact: If ξ∈TθSM, thenJξ(t)\ndenotestheuniqueJacobifieldalong γθ(t)suchthatJξ(0) =Dπθ(ξ)andJ′(ξ)(0) =Kθ(ξ).\nMoreover,\nDφt\nθ(ξ) = (Jξ(t),J′\nξ(t)). (1)\nThe last formula allows us to focus our attention on the study of the Jacobi field to\nunderstand the behavior of the geodesic flow.\nTwopoints p,q∈Mareconjugate ifthereisageodesic γjoiningpandqandanon-zero\nJacobi fieldalong γthatvanishes at pandq. Whenneither two pointsin Mareconjugate,\nwe say the manifold Mhas no conjugate points . Another important kind of manifold is\nthe manifold without focal points, we say that a manifold Mhas no focal points , if for\nany unit speed geodesic γinMand for any Jacobi field Jonγsuch thatJ(0) = 0 and\nJ′(0)/\\e}atio\\slash= 0 we have ( /bardblJ/bardbl2)′(t)>0, for anyt>0. It is easy to see that if a manifold has no\nfocal points, then it has no conjugate points.\nThe more classical examples of manifolds without focal points and th erefore without\nconjugate points are the manifolds of non-positive sectional curv ature. It is possible to\nconstruct a manifold having positive curvature somewhere, and wit hout conjugate points.\nThere is some special connection between the Anosov property an d no conjugate points\nproperty. In fact, in [22] Klingenberg proved that compact manifo ld with Anosov geodesic\nflow has no conjugate points, in result was generalized Ma˜ n´ e in the case of finite volume\n(cf. [27]). Recently, the second author in a joining work with I. Melo p roved the result\nfor the case of infinite volume and under the assumption of sectiona l curvature bounded\nbelow (cf. [30]).\nIn [14], Eberlein obtains a general characterization of the Anosov c ondition for compact\nmanifold without focal points, this characterization will be useful in the proof of Theorem\n1.5.\n2.2.1 Green Subbundles\nForξ= (w1,w2)∈TθSM, wherew1,w2∈TpMwith/a\\}bracketle{twi,v/a\\}bracketri}ht= 0,i= 1,2, we denote by\nJξ(t) the unique Jacobi vector field along γθsuch thatJξ(0) =w1andJ′\nξ(0) =w2. For\nmore details see [36].\nTo study the dynamic behavior of the geodesic flow, we usually look fo r two special sub-\nbundles ofTSM, which are invariant for Dφt. Thestable and unstable Green subbundles\nare defined, respectively, as follows:\nGs\nθ={ξ∈TθSM:ξis orthogonal to G(θ) and||Jξ(t)||is bounded for t≥0}\n5and\nGu\nθ={ξ∈TθSM:ξis orthogonal to G(θ) and||Jξ(t)||is bounded for t≤0},\nwhereG(θ) is the geodesic vector field.\nIfMhas dimension nand has no conjugate points, then the dimension of Green sub-\nbundles isn−1. Moreover, if Mhas no focal points, then Green’s subbundles depend\ncontinuously on θ∈SM(cf. [14]). In fact, for manifolds without focal points, the\nAnosov condition is characterized by the condition Gs\nθ∩Gu\nθ={0}and clearly G∗\nθ=E∗\nθ,\n∗=s,u.(see [14] for more details on Green subbundles).\n2.2.2 Jacobi Tensor and Riccati Equation\nLetγθbe a geodesic and consider V1,V2,...,V na system of parallel orthonormal vector\nfields along γθwithVn(t) =γ′\nθ(t). Any orthogonal vector field Z(t) alongγθcan be write\nas\nZ(t) =n−1/summationdisplay\ni=1yi(t)Vi(t).\nWe can identified Z(t) with the curve α′(t) = (y′\n1(t),y′\n2(t),...,y′\nn−1(t)). Conversely,\nany curve in Rn−1can be identified with a perpendicular vector field on γθ(t).\nFor eacht∈R, consider the symmetric matrix R(t) = (Ri,j(t)) defined as\nRi,j(t) =/a\\}bracketle{tR(γ′\nθ(t),Vi(t))γ′\nθ(t),Vj(t))/a\\}bracketri}ht,\n1≤i,j≤n−1, whereRis the curvature tensor of M. Thus, we can consider the matrix\nJacobi equation associated to Ras\nY′′(t)+R(t)Y(t) = 0. (2)\nIfY(t) is a solution of (2) then for each x∈Rn−1, the curve J(t) =Y(t)xcorresponds\ntoaJacobiperpendicularvectorfieldon γθ(t). Inthefollowing, wegiveaslightdescription\nof some special solution of the equation (2). For θ∈SM,r∈R, we consider Yθ,r(t) be\nthe unique solution of (2) satisfying Yθ,r(0) =IandYθ,r(r) = 0. In the case of a manifold\nwithout conjugate points, in [19], Green proved that lim\nr→−∞Yθ,r(t) exists for all θ∈SM\n(see also [14], Sect. 2). Moreover, if we define:\nYθ,u(t) := lim\nr→−∞Yθ,r(t),\nwe obtain a solution of Jacobi equation (2) such that det Yθ,u(t)/\\e}atio\\slash= 0, which we call the\nunstable Jacobi Tensor . Furthermore, in [19] was prove that for all t∈R\nUθ,u(t) :=Uu(φt(θ)) =DYθ,u\ndt(t)Y−1\nθ,u(t) (3)\nis well defined and it is a symmetric solution of the matrix Riccati equat ion:\nU′(t)+U2(t)+R(t) = 0. (4)\n6Analogously, taking the limit when r→+∞, we have defined Uθ,s(θ), that also satisfies\nthe Riccati equation (4). When the curvature is bounded below by −c2, then (see [19]\nand [14])\nmax{/bardblUθ,s(t)/bardbl,/bardblUθ,u(t)/bardbl} ≤c. (5)\nWhen the sectional curvature is negative bounded above, that is, there is a positive\nconstantbsuch thatK≤ −b2, then (cf. [14] and [25])\n/a\\}bracketle{tUθ,u(t)x,x/a\\}bracketri}ht ≥b/a\\}bracketle{tx,x/a\\}bracketri}ht (6)\nand\n/a\\}bracketle{tUθ,s(t)x,x/a\\}bracketri}ht ≤ −b/a\\}bracketle{tx,x/a\\}bracketri}ht,\nfor allt∈R.\n2.3 Lyapunov Exponents\nGivenθ∈SMandξ∈TθSM, theLyapunov exponent associated to ( θ,ξ) is defined as\nχ(θ,ξ) := lim\nt→∞1\ntlog||Dφt\nθ(ξ)||,\nwhenever the limit exists.\nOseledet’s Theorem guarantees the existence of Lyapunov expon ent for geodesic flows on\na compact manifold, that is, there exists a set Λ of full L-measure, and a good filtration\nof subspaces of TθSM(see [32]) where the Lyapunov exponents always exist. It is not\ndifficult to proof that, if Mhas no conjugate points, then for θ∈Λ\nχ(θ,ξ)≤0, ξ∈Gs\nθ\nχ(θ,η)≥0, η∈Gu\nθ.\nThus, wherever we use χ+(θ,η), we look always for η∈Gu\nθ. Moreover, we have the\nfollowing characterization for manifold without conjugate points ([1 8] for compact case\nand [30] for non-compact case):\nlim\nt→∞1\ntlog|det|Dφt\nθ|Gu\nθ|= lim\nt→∞1\nt/integraldisplayt\n0trUθ,u(s)ds, (7)\nwhereUθ,u(s) is given by (3).\nIn the particular case of dimension two, the last formula has a very e asy interpretation:\nLetη∈Gu\nθandJη(t) the Jacobi field associated to η(as Section 2.2). In dimension two,\nJη(t) =fη(t)E(t), whereE(t) is a parallel orthonormal vector field along γθ(t) andfn(t)\nsatisfies the uni-dimensional Jacobi equation\nJ′′(t)+K(γθ(t))J(t) = 0,\nwhereK(γθ(t)) is the sectional curvature along γθ(t).\nAlsouθ(t) =f′\nη(t)\nfη(t)satisfies the uni-dimensional Riccati equation\nu′(t)+u2(t)+K(γθ(t)) = 0. (8)\n7From (1), (5), and (7) we obtain\nχ+(θ,η) = lim\nt→∞1\ntlog||Jη(t)||= lim\nt→∞1\nt/integraldisplayt\n0uθ(s)ds.\nSinceuθ(s) is bounded (see (5)) and uθ(s) satisfies (8), then using Cauchy-Schwarz in-\nequality is easy to see that\nχ+(θ,η)≤/radicalBigg\n−limsup\nt→∞1\nt/integraldisplayt\n0K(γθ(t))ds.\nFinally, observe that if θis periodic point of period τ, then we have\nχ+(θ,η)≤/radicalBigg\n−1\nτ/integraldisplayτ\n0K(γθ(t))ds. (9)\n3 Rigidity in dimension 2.\nIn this section, assume that ( M,g) is a compact surface with a Riemannian metric g.\nIn order to prove the Theorem 1.3, we will use two important results . The first result is\nabout the approximation of Lyapunov exponents of an invariant me asure by Lyapunov\nexponents of measures concentrated on periodic orbits (cf. [23]) and the second result is\nabout the rigidity of smooth volume being the MME for three-dimensio nal Anosov flow\n(cf. [38]).\nFor each periodic point θ, letµθbe the unique φt-invariant probability measure sup-\nported on the orbit of θ, which may be obtained as the normalized push-forward of\nLebesgue measure on Rby the map t→φt(θ).\nTheorem 3.1. [Kalinin] Letφtbe a hyperbolic flow on a n-dimensional manifold Nand\nAa H¨older continuous cocycle over φt. Letµbe an ergodic φt-invariant measure and let\nλ1≤λ2≤ ··· ≤λnbe the Lyapunov exponents of Awith respect to µ, counted with the\nmultiplicity. Then for every ε >0, there is a periodic point θ∈Nofφtsuch that the\nLyapunov exponents λθ\n1≤λθ\n2≤ ··· ≤λθ\nnofAwith respect to µθsatisfy\n|λi−λθ\ni|<ε,\nfor each 1≤i≤n.\nWe use Kalinin’s result in the special case where N=SM,φtis the geodesic flow and\nthe cocycle Ais the cocycle derivative of the geodesic flow φt, this means:\nA:N×R→GL(n,R)\n(x,t)/ma√sto→Λ(x,t) =dxφt,\nandn= 2dim(M)−1.\nTo announce the second result we need, we start with the definition ofAlgebraic flows ,\nwhich can be found in [38] and [39].\n8Definition 3.2. An Anosov flow Φ :N→Non a3-dimensional compact manifold Nis\nalgebraic if it is finitely covered by\n(1)a suspension of a hyperbolic automorphism of the 2-torusT2=R2/Z2;\n(2)or the geodesic flow on some closed Riemannian surface of cons tant negative curva-\nture.\nThe following result canbefoundin[38], it gives usacharacterization o ftheconjugacy\nof a flow to an algebraic flow when we have that the volume measure is a measure of\nmaximal entropy.\nTheorem 3.3. [SLVY]Letk≥5be some integer and let Φbe aCkAnosov flow on\na compact connected 3-manifoldNsuch that the smooth volume measure µis invariant.\nThenhtop(Φ) =hµ(Φ)if and only if ΦisCk−ǫ-conjugate to an algebraic flow, for ǫ >0\narbitrarily small.\nThe Theorem 3.3 is related to the following conjecture:\nConjecture 2. [Katok Entropy Conjecture ]Let(M,g)be a connected Riemannian\nmanifold of negative curvature and ψbe the corresponding geodesic flow. Then htop(ψ) =\nhL(ψ)if and only if (M,g)is a locally symmetric space, where Lis the Liouville measure\ninSM.\nFinally, we mention the following result due to Plante (see [37]). Basically , the result\ngives us two alternatives for the strong stable and strong unstab le manifold: they are\nor not dense on the manifold when the Anosov flow ψtis transitive or equivalently the\nnon-wondering set Ω( ψt) =M.\nTheorem 3.4. [Plante]Letψt:M→Mbe an Anosov flow such that Ω(ψt) =M. Then\nthere are two possibilities:\n(a)Each strong stable and strong unstable manifold is dense in M, or\n(b)ψtis a suspension (modulo time scale change by a constant facto r) of an Anosov\ndiffeomorphism of a compact C1submanifold of codimension one in M.\nWe use the Theorem 3.3, Theorem 3.4, Ruelle’s inequality, and Pesin’s fo rmula to\nprove the Theorem 1.3 discarding some cases that may appear for o ur geodesic flow.\n3.1 Rigidity Theorems\nIn this section, we prove the Theorem 1.4, we use it to prove Theore m 1.2, and we use\nthe structure of the geodesic flow to prove Theorem 1.3. Finally, we prove Theorem 1.5\nusing techniques of surfaces without focal points.\nProof of Theorem 1.4 .Denote by Me(φt) the set of all ergodic φt-invariant measures.\nFrom Oseledets’ ergodic theorem (see [32]), for µ∈ Me(φt) consider λ1≤ ··· ≤\nλn−1≤0≤λn+1≤ ··· ≤λ2n−1being the Lyapunov exponents associated with µ. Let\nθ∈Per(φt) andµθ∈ Me(φt) the unique φt-invariant probability measure supported on\n9the orbit of θ. Thus, consider λθ\n1≤ ··· ≤λθ\nn−1≤0≤λθ\nn+1≤ ··· ≤λθ\n2n−1the Lyapunov\nexponents associated to µθ. By hypothesis λθ\ni=−α,i∈ {1,...,n−1}andλθ\ni=α,\ni∈ {n+1,...,2n−1}. From Theorem 3.1 we can approximate the Lyapunov exponents\nofµby Lyapunov exponents of µθ. Thus, we can conclude that for µ∈ Me(φt) holds\nλi=−α,i∈ {1,...,n−1}andλi=α,i∈ {n+1,...,2n−1}.\nSinceφtis Anosov, then normalize Liouville measure Lis an element of Me(φt) (see [40]).\nNow we will show that the Liouville measure on SMis a maximal measure of entropy.\nIndeed, by Ruelle’s inequality, for µ∈ Me(φt) we have that\nhµ(φ)≤/integraldisplay\nSM/summationdisplay\nλi>0λidµ(θ)\n=/integraldisplay\nSM(n−1)αdµ(θ)\n= (n−1)α. (10)\nSinceφtisC2, by Pesin’s formula we obtain that\nhL(φ) =/integraldisplay\nSM/summationdisplay\nλi>0λi(θ)dL(θ)\n=/integraldisplay\nSM(n−1)αdL(θ)\n= (n−1)α. (11)\nThus, from (10) and (11), we conclude that for all µ∈ Me(φt) holds\nhµ(φ)≤(n−1)α=hL(φ).\nThus, from the variational principle\nhtop(φ) = sup\nµ∈Me(φt)hµ(φ)≤(n−1)α=hL(φ)≤htop(φ),\nwhich concludes the proof of the theorem.\nThe proof of Theorem 1.4 implies the following corollary.\nCorollary 3.5. Ifφt:N→Nis a tridimensional conservative Anosov flow, then if\nthe Lyapunov exponents are constant along periodic orbits, then the volume measure is a\nmeasure of maximal entropy.\nProof.The proof follows the same arguments of the proof of Theorem 1.4 a nd the ergod-\nicity of the volume measure (see [40]).\nProof of Theorem 1.2 .\n(1) =⇒(2) is a consequence of Corollary 3.5. (2) = ⇒(3) is a consequence of Theorem\n3.3. In order to prove that (3) = ⇒(1), observe that the condition (1) is preserved by\nsmooth conjugacy. Finally, we note that any algebraic tridimensiona l flow satisfies the\ncondition (1), therefore the condition (1) is satisfied.\n10To the proof of Theorem 1.3 we need to use the classification of tran sitive tridimen-\nsional Anosov flow, the structure of the geodesic flow.\nProof of Theorem 1.3 .It is easy to see that (2) = ⇒(1). From Theorem 1.2 our\nflow is smooth conjugate to an algebraic model. However, the geode sic flow has no global\ncross-section (see [36]), therefore from Theorem 3.4, our flow mu st be smooth conjugate\nto the geodesic flow of a surface of constant negative curvature . Finally, from [11] our\nsurface must be isometric to a surface of constant negative curv ature, in other words, it\nhas constant negative curvature, and the value of the Lyapunov exponents determines the\nvalue of the curvature.\nThe remainder of this section will be dedicated to the proof of Theor em 1.5.\nWe denote by\nΓ(κ) :={θ∈SM:K(π(θ))≤ −κ2}\nThe next lemma is similar to [24, Lemma 2.4] and [35, Lemma 2.6].\nLemma 3.6 (Shadowing Lemma) .LetMbe a closed surface without focal points. For\nanyη,ǫ,τ >0there exists δ >0such that for any collection of orbit segments {(θi,ti)}i∈Z\nwithθi,φti(θi)∈Γ(κ),ti≥τandd(φti(θi),θi+1)<δfor alli∈Z, there exist a geodesic\nγand a sequence of times {Ti}i∈ZwithT0= 0,Ti+ti−ǫ≤Ti+1≤Ti+ti+ǫ, and\nd(γ′(t),γ′\nθi(t−Ti))<ǫfor allt∈[Ti,Ti+1]andi∈Z.\nThe geodesic γis unique up to re-parametrization. Moreover, if the orbits being shadowed\nare periodic, then the shadowing orbit is also periodic.\nDefinition 3.7. LetMbe a complete manifold without focal points. For each θ∈SM,\nwe define rank (θ)as the dimension of the vector space of the parallel Jacobi fie lds along\nthe geodesic γθ, and rank (M) := min{rank(θ) :θ∈SM}.\nIt is not difficult to see that for manifold without focal points rank( θ) is the codimen-\nsion of the vector bundle Gs\nθ⊕Gu\nθ. Consider the following set\nH={θ∈SM:K(γθ(t)) = 0,for allt∈R}.\nThe setHis clearly invariant to the geodesic flow. In general, the set R=SM\\His\ncalledregular set . It is well known that, in the case of a surface without focal points , the\nLiouville measure is ergodic for φtinR(see [2]). However, there is an open conjecture\nabout the ergodicity of the Liouville measure over the whole surface . Such a conjecture is\nequivalent to the set Rhaving total Liouville measure or the set Hhaving zero Liouville\nmeasure (see [4]).\nIt is easy to see that for surfaces without focal points\nH={θ∈SM:Gs\nθ∩Gu\nθ/\\e}atio\\slash={0}}.\nTherefore H=∅is equivalent to geodesic flow φtto be Anosov (see [14]). Observe that\nin the case of surface, rank( θ) only can be 1 or 2. It is easy to see that if θ∈ H, then\nrank(θ) = 2 and if θ∈SM\\H, then rank( θ) = 1.\n11Lemma 3.8. LetMbe a compact surface of genus greater than 1and without focal points,\nthen the geodesic flow φt:SM→SMis transitive.\nProof.We only need to prove that Mhas rank 1 and then the result will become a\nconsequence of [26]. As genus of M >1, then the Euler characteristic of Mis negative,\nthen the topological entropy htop(φt)>0 (cf. [13]). Consequently, from variational\nprinciple, there is an ergodic measure with positive Lyapunov expone nts,i.ethere is\nθ∈SMandη∈TθSMsuch thatχ+(θ,η)>0. Using the above notation,\nχ+(θ,η) = lim\nt→∞1\ntlog||Jη(t)||>0.\nTherefore, rank( θ) = 1. In fact, θ∈SM\\Hand rank(M) = 1.\nRemark 3.9. The proof of the last lemma can be easier using the Gauss-Bonn et theorem\nsince it should have points in Mof negative curvature. Therefore, together with the\ncondition of no focal points, it implies the existence of an u nbounded unstable Jacobi field.\nIn particular, we have a point with rank 1. Also, using arguments of Eberlein (see [14]) ,\nwe can see easily that parallel Jacobi field along a geodesic γimplies section curvature\nzero alongγ, so geodesic pass through point of negative curvature provi des points of rank\n1.\nThe next lemma is similar to at [24, Proposition 3]. and [35, Proposition 3.4 ].\nLemma 3.10. IfH /\\e}atio\\slash=∅, then there is a periodic orbit with Lyapunov exponent arbit rarily\nclose to0.\nProof.By hypotheses there is θ∈ Hand consequently K(γθ(t)) = 0, for all t∈R.\nGivenκ >0, by transitivity (see Lemma 3.8), there is a sequence θkconverges to θand\nunbounded negative and positive sequences t−\nkandt+\nk, respectively, such that\nK(γθk(t))≤ −κ2,for allt∈[t−\nk,t+\nk].\nBy compactness, we can assume that φt−\nk(θk) andφt+\nk(θk) converge to w−andw+, respec-\ntively. Note that K(π(w−))≤ −κ2andK(π(w+))≤ −κ2.\nConsiderǫandδform Lemma 3.6 (Shadowing Lemma), then, using the transitivity of\nthe flow, there is T > τandwwithd(w,w+)<δ\n2such thatd(φT(w),w−)<δ\n2. Thus,\nfrom Lemma 3.6, for any large enough kthere is a closed geodesic βkof periodPk, close\ntot+\nk−t−\nk+Tsuch that ( τ,ǫ)-shadows the pseudo-orbit formed by φt(θk)|[t−\nk,t+\nk]and\nφt(w)|[0,T]. It is easy to see that Pkis an unbounded sequence, moreover the path of orbit\nφt(w)|[0,T]such thatβk(τ,ǫ)-shadows does not depend on k, then the proportion of the\ncurvature along βkis bounded. Moreover, by construction, the curvature of βkis close to\nthe curvature of π(φt(θk)|[t−\nk,t+\nk]) most part of time. In other words,\nlimsup\nk→∞1\nPk/integraldisplayPk\n0K(βk(t))dt≥ −2κ2.\nPutβk(0) =wkandη∈Gu\nwk, then from (9)\n120≤χ+(wk,η)≤/radicalBigg\n−1\nPk/integraldisplayPk\n0K(βk(t))dt≤√\n2κ.\nSinceκwas chosen arbitrarily, we have our result.\nProof of Theorem 1.5 .It is easy to see that (2) = ⇒(1). We prove (1) = ⇒(2)\nusing the genus of the surface M. Note that Mhas no focal points, in particular, has no\nconjugate points. Thus, from Hopf’s result (cf. [20]), Mhas genus ≥1. In the case of\ngenus equal to 1, from [20] our surface Mis the flat torus T2and consequently α= 0.\nSo, we can assume that Mhas a genus greater than 1.\nClaim:The setH=∅. In fact, by contradiction, assume that there is θ∈ H. Therefore,\nfrom Lemma 3.10, we have periodic orbits with Lyapunov exponent ar bitrarily close to\n0, however since the value of αin the hypothesis does not depend on the periodic points,\nthen we have a contradiction.\nFrom the last Claim, all geodesics pass through negative curvature . Thus, as Mhas\nno focal points then φtis Anosov (cf. [14]). Finally, since the Lyapunov exponents are\nconstant along periodic orbits, from Theorem 1.4 the Liouville measur e is a measure of\nmaximal entropy. Moreover, as MisC2Riemannian metric without focal points, then\n[21, Corollary 3.3] we conclude that Mhas constant negative curvature and the value of\nthe curvature is −α2, as we wished.\nRemark 3.11. In the final arguments of the proof of Theorem 1.5, even the geo desic flow\nbecomes Anosov, we cannot use Theorem 1.3 since in the assumptions of Theorem 1.5\nour metric is just C2, different from Theorem 1.3 where the metric is at least C5.\nOn the hypothesis of Theorem 1.3 over the geodesic flow be Ck,k≥5, we believe\nthe theorem is true for k≥2, because observation of the Theorem 3.3 on [38] says that\nthis theorem can be true for k≥2, but technical obstructions prevent finding the precise\nboundary of required regularity.\n4 Rigidity of Lyapunov exponents in finite volume\ncase.\nThemaingoalofthissectionistoprovetheTheorem 1.6. Therefore , fromnowon, wecon-\nsiderMa complete Riemannian manifold with pinched negative curvature, i.e., there are\ntwo constants 0 <b<csuch that −c2≤KM≤ −b2. Forθ∈SM, letYθ,s(t) andYθ,u(t)\nthe stable and unstable Jacobi tensor along γθ(t), respectively and Uθ,s(t) =Y′\nθ,s(t)Y−1\nθ,s(t)\nandUθ,u(t) =Y′\nθ,u(t)Y−1\nθ,u(t) the stable and unstable Riccati solution, respectively.\n4.1 Proof of the Theorem 1.6\nOne of the main ingredients of the proof of Theorem 1.6 is the following theorem:\nTheorem 4.1 ([31]).Ifφt:SM→SMis an Anosov geodesic flow and Mhas finite\nvolume, then Per(φt) =SM.\n13Proof of Theorem 1.6 .\nThe proof of Theorem 1.6 carried out in two cases:\nCase 1: We consider the case α=b.\nNote that the hypothesis χ+(θ,ξ) =bfor allθ∈Per(φt) and allξ∈Eu\nθ. So,\nlim\nt→+∞1\ntlog|detdθφt|Eu\nθ|=b·dimEu\nθ=b(n−1).\nFrom [28, Lemma 3.2], we obtain\nlim\nt→+∞1\nt/integraldisplayt\n0tr(Uθ,u(s))ds= lim\nt→+∞1\ntlog|detdθφt|Eu\nθ|=b(n−1), (12)\nfor allθ∈Per(φt).\nHowever, if θ∈Per(φt) of periodτ, then the function t→tr(Uθ,u(t)) is periodic of period\nτ. From (12) we have\n1\nτ/integraldisplayτ\n0tr(Uθ,u(s))ds=b(n−1). (13)\nOn the other hand, as K≤ −b2andUθ,u(t) is symmetric, from (6) we can conclude\nthat all eigenvalues of Uθ,u(t)) are non-negative, in fact, all of them are greater than or\nequal tob. Letλn−1(t)≥λn−2(t)≥ ··· ≥λ1(t)≥bthe eigenvalues of Uθ,u(t). Then\ntr(Uθ,u(t)) =n−1/summationdisplay\ni=1λi(t)≥α(n−1). (14)\nFrom (13) and (14)\ntr(Uθ,u(t)) =b(n−1)\nfor allt∈R.\nHence, for all j= 1,2,...,n−1 holdsλj(t) =bfor allt∈R. Finally this implies for all\nt∈R\ntr(Uθ,u(t))2) =n−1/summationdisplay\nj=1(λj(t))2=b2(n−1). (15)\nFrom the Riccati equation\n1\nt/integraldisplayt\n0tr(U′\nθ,u(s))ds+1\nt/integraldisplayt\n0tr(Uθ,u(s)2)ds+(n−1)\nt/integraldisplayt\n0Ric(φs(θ)) = 0\nTakingtgoes to∞, then (5) and (15) provide\nb2(n−1) =−lim\nt→+∞(n−1)\nt/integraldisplayt\n0Ric(φs(θ))ds\nfor allθ∈Per(φt). Ifθ∈Per(φt) of period τ, then the last equation becomes\n1\nτ/integraldisplayτ\n0Ric(φs(θ))ds=−b2.\n14Note thatK≤ −b2, in particular, Ric( φs(θ))≤ −b2. Consequently, the last equation\nprovides Ric( φt(θ)) =−b2, for allt∈R. In resume, we had proved that for all θ∈Per(φt)\nholds Ric(θ) =−α2. AsMhas finite volume, then from Theorem 4.1 the periodic orbits\nare dense in SM, so the continuity of the function Ric( ·) guarantees Ric( θ) =−b2, for all\nθ∈SManda fortiori ,KM=−b2.\nCase 2: Ifα=c. Here we use the same arguments of [28]. By the hypothesis χ+(θ,ξ) =c\nfor allθ∈Per(φt) and allξ∈Eu\nθ. Thus\nlim\nt→+∞1\ntlog|detdθφt|Eu\nθ|=c·dimEu\nθ=c(n−1).\nIfθ∈Per(φt) of period τ, then from [28, Lemma 3.2], we obtain\n1\nτ/integraldisplayτ\n0tr(Uθ,u(s))ds= lim\nt→+∞1\ntlog/vextendsingle/vextendsingledetdθφt|Eu\nθ/vextendsingle/vextendsingle=c(n−1). (16)\nSinceUu(φs(θ)) is symmetric then it is easy to see that\n(tr(Uθ,u(s))2≤(n−1)tr(Uθ,u(s)2). (17)\nUsing the periodicity of the function s→tr(Uθ,u(s)) and integrating from 0 to τthe\nRiccati equation (4), we obtain from (16), (17), and Cauchy-Sch wartz inequality that\nc(n−1) =1\nτ/integraldisplayτ\n0tr(Uθ,u(s))ds≤/radicalBigg\n1\nτ/integraldisplayτ\n0(tr(Uθ,u(s))2ds\n≤/radicalBigg\n(n−1)\nτ/integraldisplayτ\n0tr(Uθ,u(s)2)ds\n=/radicalBigg\n−(n−1)\nτ/integraldisplayτ\n0tr(R(s))ds\n=/radicalBigg\n−(n−1)2\nτ/integraldisplayτ\n0Ric(φs(θ))ds\n≤/radicalbig\n(n−1)2c2,\nwhere in the last inequality we used that the sectional curvature sa tisfiesKM≥ −c2.\nConsequently,\n1\nτ/integraldisplayτ\n0Ric(φs(θ))ds=−c2,\nwhich implies that Ric( φt(θ)) =−c2for allθ∈Per(φt) and allt∈R. The result follows\nthe same lines of the case 1.\nAcknowledgements: The first author thanks all the members of the Institute of\nMathematics - UFRJ for all their support during the preparation of this work, and my\nfamily for being by my side whenever I needed them, I am so sorry for leaving so early.\nThe second author thanks the first author for all his struggles in r ecent years, his memory\nwill always be with me. The second author also thanks the Mathematic s Department of\nSUSTech - China for its hospitality during the preparation of this art icle.References\n[1] D. Anosov, Geodesicflows onclosed Riemannianmanifolds withnegative curvatur e,\nProc. Steklov Inst. Math. 90, 1967.\n[2] L. Barreira and Y. B. Pesin. Nonuniform hyperbolicity: Dynamics of systems with\nnonzero Lyapunov exponents ,Cambridge University Press Cambridge ,115, 2007.\n[3] G. Besson, G. Courtois, and S. Gallot. Entropies et rigidit´ es des espace localement\nsym´ etriques de courbure strictement n´ egative ,Geom Funct. Anal , 5(5), 731–799,\n(1995)\n[4] K. Burns and V. Matveev, Open problems and questions about geodesics , Ergodic\nTheory and Dynamical Systems, 41(3), 641–684 (2021).\n[5] Clark Butler, Rigidity of Equality of Lyapunov exponents for geodesic flow , J. Dif-\nferential Geometry, 109(1): 39-79, 2018.\n[6] Conell, C.: Minimal Lyapunov Exponents, Quasiconformal Structures an d Rigidity\nof Nonpositively Curved Manifolds ,Ergodic Theory and Dynamical Systems 23(2),\n429–446, 2003.\n[7] C. Croke, Irena Lasiecka, Gunther Uhlmann, Michael S. Vogelius .Geometric Meth-\nods in Inverse Problems and PDE Control . The IMA volumes in Mathematics and\nits applications, 2003.\n[8] C. Croke, A. Fathi, and J. Feldman, The marked length-spectrum of a surface of\nnonpositive curvature ,Topology 31(4), 847–855, 1992.\n[9] D. Constantine, 2-Frame flow Dynamics and Hyperbolic Rank Rigidity in Nonpos i-\ntive Curvature . J. Mod. Dyn., 2(4) 719–740, 2008.\n[10] C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector\nfield, J.Diff. Geom. 39, 659–680, 1994\n[11] C. Croke, Rigidity for surfaces of non-positive curvature , Comm. Math. Helv. 65\n(1990), no. 1, 150-169.\n[12] C. Croke and B. Kleiner, A rigidity theorem for simply connected manifolds without\nconjugate points , Erg. Th. and Dyn. Syst. 18 (1998), pt. 4, 807-812.\n[13] E.I. Dinaburg. On the relations among various entropy characteristics of dynamic al\nsystems,Math. USSR, Izv ,5):337–378, Zbl 0248.58007, 1971.\n[14] Patrick Eberlein, When is a geodesic flow of Anosov type? , J. Differential Geom.\n8(3)437-463 1973.\n[15] J. Feldman and D. Ornstein, Semi-rigidity of horocycle flows over compact surfaces\nof variable negative curvature , Ergodic Theory and Dynamical Systems, 7(1), 49–\n72, 1987.[16] R. Feres and A. Katok. Invariant tensor fields of dynamical systems with pinched\nLyapunov exponents and rigidity of geodesic flows , Ergodic Theory and Dynamical\nSystems, 9(3), 427–432, 1989.\n[17] Foulon, P.: Entropy rigidity of Anosov Flows in Dimension three , Ergodic Theory\nand Dynamical Systems 21(4), 1101-1112, 2001.\n[18] A. Freire, R. Ma˜ n´ e, On the entropy of the geodesic flow in manifolds without con-\njugate points . Inv. Math, 69, 375–392, 1982.\n[19] Green, L. W., A Theorem of Hopf . Michigan Math. J., 5(1)31–34, 1958.\n[20] E. Hopf, Closed surfaces without conjugate points , Proc. Nat. Acad. Sci. U. S. A.,\n34, 47–51, 1948.\n[21] A. Katok. Entropy and closed geodesic ,Ergodic Theory and Dynamical Systems ,\n2(3-4), 339–365, 1982.\n[22] W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type , Annals\nof Mathematics, 99(1), 1–13, 1974\n[23] Kalinin, B.; Liv˜ sic theorem for Matrix Cocycle , Ann. of Math. 173(2), 1025–1042,\n2011.\n[24] Burns, K., Gelfert, K.; Lyapunov Spectrum for Geodesic Flows of Rank 1 Surfaces ,\nDiscrete and Continuous Dynamical Systems - A 14, 1841–1872, 2014\n[25] G. Knieper, Chapter 6, Hyperbolic dynamics and riemannian geometry , Handbook\nof Dynamical Systems, vol. 1A, 453-545, Elsevier Science, 2002.\n[26] F. Liu, X. Zhu. The transitivity of geodesic flows on rank 1manifolds without focal\npoints,Differential Geomeotry and its Applications ,2, 49–53, 2018.\n[27] R. Ma˜ n´ e, On a Theorem of Klingenberg , Dynamical Systems and Bifurcation The-\nory, M. Camacho, M. Pacifico and F. Takens, eds., Pitman Research Notes in Math,\n160, 319–345, 1987.\n[28] I. Dowell and S. Roma˜ na, Some rigidity theorems for Anosov geodesic flows in\nManifolds of Finite Volume , arXiv.: 1709.09524, 2017.\n[29] I. Dowell and S. Roma˜ na, Contributions to the study of Anosov Geodesic Flows\nin non-compact manifolds .Discrete and Continuous Dynamical Systems 40(9),\n5149–1171, 2020.\n[30] I. Dowell and S. Roma˜ na, Riemannian manifolds with Anosov geodesic flow do not\nhave conjugate points . arXiv:2008.12898.\n[31] N. Nina, S. Roma˜ na, A note on the density of periodic orbits of Anosov geodesic\nflow in manifolds of finite volume . arXiv:2401.18031.[32] V. I. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic num-\nbers for dynamical systems , Trans. Moscow Math. Soc., 19, 197–231, 1968.\nMoscow.Mat.Obsch. 19, 179–210, 1968.\n[33] J.-P. Otal, Le spectre marqu´ e des loungueurs des surfaces ` a courbure n ´ egative, Ann.\nof Math. 131, 151–162, 1990.\n[34] J.-P. Otal, Sur les longueurs des g´ eod´ esiques d’une m´ etrique ` a cour bure n´ egative\ndans le disque , Comment Math. Helv., 65(2), 334–347, 1990.\n[35] K. Park, T. Want. Multifractal analysis of geodesic flows on surfaces without focal\npoints,Dynamical Systems, An International Journal ,36(4), 656–684, 2021.\n[36] G. Paternain, Geodesic flows . Progress in Mathematics, Birkhauser, Volume 180,\n1999.\n[37] Jo.. Plante, Anosov Flows , American Journal of Math. 94(3), 729–754, 1972.\n[38] J. De Simoi, M. Leguil, K. Vinhage, Y. Yang, Entropy Rigidity for 3DConservative\nAnosov Flows and Dispersing Billiards Geometric and Functi onal Analysis ,30\n1337–1269, 2020\n[39] P. Tomter, Anosov Flows on infra-homogeneous Spaces , Global Analysis (Proc.\nSympos Pure Math., Vol XIV, Berkeley, Calif., 1968, Amer. Math. So c., Providence,\nR.L. 299–327. 1968\n[40] M. Viana, K. Oliveira., Fundamentos da Teoria Erg´ odica . SBM. 2019."    },    {        "title": "2402.05522v1.Experimental_observation_of_parabolic_wakes_in_thin_plates.pdf",        "content": " \n1 Experimental observation of  parabolic wakes  in thin plates  \nJanez Rus1*, Aleksi Bossart1, Benjamin Apffel1, Matthieu Malléjac1 and R omain  Fleury1* \n1Laboratory of Wave Engineering, Institute of Electrical and Micro Engineering, Ecole Polytechnique \nFédérale de Lausanne (EPFL), Station 11, 1015 Lausanne, Switzerland  \n*Correspondence to romain.fleury@epfl.ch  or janez.rus@tum.de  \nWakes are medium perturbations created by a moving object, such as wave patterns behind \nboats, or wingtip vortices following an aircraft. Here, we report about an experimental study of \nan uncharted for m of parabolic wakes occurring in media with the group velocity twice larger \nthan the phase velocity, as opposed to the conventional case of Kelvin wakes . They are formed \nby moving a laser spot on a thin plate, which excites a unique wake pattern made of c onfocal \nparabolas, due to the quadratic dispersion of the zero -order flexural Lamb mode. If the spatial \ndimensions are rescaled by the perturbation velocity and material constant, we obtain a single \nuniversal wake with constant parabolic focal lengths. We demonstrate an evanescent regime \nabove the critical frequency where the wave components oscillate exclusively in the direction \nparallel to the perturbation path, with an opening angle of 90°. We define a dimensionless \nnumber analogous to Froude and Mach nu mbers, which determines whether the complete \nparabolic wake pattern will be excited by the moving source or not.  \n \nThe broadest definition of wakes refers to a pattern , typically made of waves,  excited by the \nmovements of an object or a perturbation in a me dium. The most prominent  example is the Kelvin \nwake pattern generated by a moving object on the surface of water  [1-11]. A second common  use of \nthe term wake is associated with the circulating turbulent flow behind a moving object in a fluid \n(Kármán vortex ) [12,13] , as seen with phenomena like wingtip vortices behind  an aircraft  [14-17]. \nAdditional associations encompass wakes formed behind charged objects in supersonic plasma flows  \n[18-20] or laser -driven plasma wakefields  [21-24], which show promise as an alternative method for \nelectron acceleration. Th is definition also includes Mach cone s [19,25]  or Cherenkov radiation  [26-\n29], which are less often called  wakes due to the absence of dispersion wave interference.  \nThe w ake pattern on the surface of deep water depends on  whether the dispersion is capillary -\ndominated  [30-32], with      ⁄ being the relation between angular frequency   and wavenumber \n , or if it is  gravity -dominated  [8,33] , with       ⁄. The latter power law yields the commonly -\nobserved  Kelvin wake  pattern,  typical  of the wakes originating from  ducks  or boats  as they move. \nThe Kelvin wake patterns exhibit  the remarkable  property that their shape , in the case of sufficiently \nslow movements , is independent  of the  perturbation velocity , when properly  rescaled  coordinates are \nused. The half-cone  opening angle  is constant at the value        (  ⁄)       . This is a  common  \nsituation  of the wakes on water where the group velocity is twice lower  than the phase velocity.  \nIn this work, w e experimentally explore the behavior of wake patter ns excited in a medium in which \nthe power law  governing the dispersion relation is  exactly inverse to the one of Kelvin wakes, namely \n    . These wakes  are obtained  behind a fast-moving  perturbation on a thin plate.  The zero-order \nasymmetric Lamb mode of the plate exhibits a  dispersion relation well approximated by      ⁄, \nwhere   is a constant dependent on the thin plate thickness and material  properties . The g roup \nvelocity        ⁄    ⁄ is therefore twice higher than the phase velocity       ⁄   ⁄, \nand proportional to the square root of frequency , setting a unique stage for the study of the new form \nof wakes . \nThe perturbation  in our case is a focused laser beam illuminating the surface of a polymer plate  \n(Fig. 1a). The induc ed heating  leads to a localized reduction in the Young’s modulus  of the plate . In \nour experiment, g alvanometric scanning mirrors were  used to displace  the heated position at const ant \nvelocit ies    in a linear trajectory . The angle between the plate  surface  and the laser beam was set to \n20° at the midpoint of the scanning area . Consequently , a small change in the galvanometric mirror's  \n2 angle provide d a large alteration in the perturbation's position . This, in turn , allowed us to achieve \nperturbation  velocities     fast enough to observe a broad range of parabolic wakes. Moreover , this \nsetup enabled us to investigate the impact of the laser spot size on the formation  of the para bolic \nwakes.  For more details regarding  the experimental procedure and the setup , please refer to \nSupplemental Material [34]. \nIn Fig. 1 b-g, we present  the parabolic wak es captured by a laser vibrometer for three distinct     \nvalues and two different time instances. The o ut-of-plane velocity of the plate was measured for each \nscanning position  throu gh separate  wake excitation s. The excitation laser was in focus at the position \n   = 0, corresponding to time t = 0 (Fig. 1 b, d, and f). Additional wakes for 12 different    values \nand a scenario involving an elevated temperature of the polymer plate are presented in Supplemental \nMaterial and Video Presentation  [34]. \n \nFig. 1. Schematic of the set up, where a moving laser beam excites elastic waves on a thin plate  (a). Measured parabolic \nwake patterns at t = 0 µs (b, d, f) and t = 70 µs (c, e, g)  at three distinct perturbation velocities    (b-c, d-e, and f -g). The \nred dashed -line square in f shows a portion of the wake whose shape is identical  to the one observed over the full range in \nb, when both axes are rescaled by a factor three , corresponding to the ratio of    values in these two cases.  \nThe wave pattern  shape of Fig. 1b is identical to that confined  within  the red dashed -line square in \nFig. 1f, which has  both axes reduced by a factor 3 – chosen to correspond to the ratio between  the \nvalues of         m/s in Fig. 1f and        m/s in Fig. 1b. In contrast , Kelvin wakes follow an \ninverse  rescaling law , where higher    provide larger dominant wavelengths , while  rescaling of \nspatial coordinates in both directions  does not change the  cone opening angle.  In our case , the \nmeasured parabolas all have a curvature that linearly increases with   . The focal distance , which is \ninversely proportional to the  curvature  of the parabol ic ridges  in Fig. 1b-g, is therefore inversely \nproportional  on   . \nAs wakes result from the interference  of many frequency components , we propose to study the \ndifferent spectral components of the wake s separately . As examples, w e show in Fig. 2d-f \n(respectively Fig. 2g-i) the measured scans from Fig. 1b,c at           (respectively from \nFig. 1f,g at           ), filtered to three selected narrowband frequencies , which were then \nprocessed  to obtain  data points in Fig. 2a-c, marked by yellow squares (respectively green circles ).  \nIn our experiment ( Fig. 2a), we observe that th e wave number of the wakes in the direction parallel to \nthe motion is fixed by    , and  follows  the law  \n  ( )   \n    (1) \nIn particular, i t is neither  influenced by the dispersion law of the medium n or the wave velocities in \nthe medium. Instead, its dependency  on   is entirely determined by    . This condition fixes for each \nfrequency the phase of the travelling wave at the moving location of the source. Equivalently, it \nmatches the parallel projection of the phase velocity with the source velocity . Using  the dispersion \n \n3 law      ⁄ where     √       , we can express  the wave number of the wakes in the \ndirection perpendicular  to   : \n  ( )  √         √      \n       √ \n        \n       (2) \nIf we introduce resca led variables   ̂       ⁄  and  ̂       ⁄ , as well as  ̂      ⁄ , we can \nsimplify Eq. (1) and Eq. (2) to \n \n ̂ ( ̂)  ̂  \n ̂ ( ̂) √ ̂   ̂  . (3) \nThe above formula for  rescaled wavenumbers  ̂ ( ̂) and  ̂ ( ̂) is represented in Fig. 2a and Fig. 2b \nwith thick red lines  and compared to the experimental results obtained for ten different source \nvelocities  ranging from 73 m/s to 292 m/s  identifiable by the shades  of gray . The opening angle \nbetween  the propagation direction of the wave component and  the direction perpendicular to  the \nsource motion at  a specific frequency ,  ( ̂)        ( ̂ ( ̂) ̂ ( ̂)⁄), is also shown in Fig. 2c. As \na result of the rescaling, the darker shades of gray , corresponding to higher   , stop at lower values of \n ̂ in Fig. 2a-c, even though the range of non -rescaled   is constant  for all   . Fig. 2a-c in their \noriginal ( non-rescaled ) coordinates are provided  in Supplemental Material [34]. \nEq. (3) predicts that  ̂ ( ̂) is real for  ̂  , when   is smaller than the critical frequency     \n    . The angle   ( ̂) increases until it reaches 90°  at    . This is the highest frequency  at which the \nwake components are purely propagating.  The wake component at     propagates in the direction \nparallel to the perturbation path with the velocity equal to   . It can there fore be directly expressed  as \n        ⁄, where    is the distance between the vertices of two parabolas defined by the maximum \nridge s of the wake pattern  (Fig. 1d). Interestingly, an opposite behavior is true for Kelvin wake s. The \n  ( ) of Kelvin waves is real above a critical frequency, which can be estimated from the \nwavelength of the wave component s propagating directly behind the moving perturbation.  \nFor parabolic wakes,  ̂ ( ̂) is imaginary at  ̂  . In this evanescent regime, wave components are \nlocalized to  the vicinity of the perturbation  path, oscillating  only along the direction parallel to   , as \ngoverned by  Eq. (1). This effect is evident  in Fig. 2f. The exponential decay length in the direction \nperpendicular to    of the evanescent waves decrease s with increasing frequencies.  Due to the \nlimited frequency range of the measuring system and the increase of intrinsic losses at higher \nfrequencies , the evan escent regime  ( ̂  ) is more easily  observed in the measurements performed \nat lower   .  \n4  \nFig. 2. The measured wave numbers   ̂  (a) and  ̂  (b) for ten different    values ( grey lines ), collapse on the red curves \npredicted by Eq. (3) except near  the critical frequency (at  ̂ = 1). Around the critical frequency , variations in the \nperturbation size along the position    (due to the laser bea m mov ing in and out of focus)  become important . These \nvariations are accounted for in a refined  model  (blue lines).  The angle φ  between the direction of    and the wavefronts  \nbecomes larger as  ̂ increases, reaching  almost  90° at the critical frequency  ̂ = 1 (c). Beyond this frequency , Eq. (3) \npredicts that   ̂  becomes  imaginary, which defines the evanescent regime  (f). When  the parabolic wake pattern s from  \nFig. 1 are filtered to narrow frequencies  (d-i), represented by yellow squares and green circles , the opening s of the  \nwavefronts adhere  to the Mach la w in respect t o the wave velocity  at the specific frequency.  \nWhile  ̂  falls on the same curve for all    values , there is a greater  discrepancy  between the \nmeasured   ̂  and Eq. (3), especially around    . This arises due to variations in laser beam  diameter \nas it moves along in the direction of   . As the beam approaches the focal  point , higher frequency \ncomponents ar e stimulated  with higher amplitudes  (and conversely as it moves away  from the focal \npoint ). These alterations  of the excitation amplitude spectrum along     leads to a non -zero imaginary \npart of  ̂ . As a result , the imaginary part of  ̂  becomes non-zero even below  the critical frequency , \nand the real part of  ̂  is underestimated by  Eq. (2) when nearing      (Fig. 2b). This effect becomes  \nmore pronounced for larger    values . Consequently, for the field map of Fig. 2i, the angle    is 65° \n(green point in Fig. 2c), instead  of 90° as predicted by the theory. Th e effect of the laser focusing and \ndefocusing along the position    can be accounted for in a refined  model ( Supplemental Material \n[34]), yielding the thick  blue lines , which fit the measurements better than the reduced model of \nEq. (3). \nHaving understood how each frequency behave s individually, we can now express  the wake pattern \nas the real part of  the following  sum over all excited waves that verify  Eq. (1) and Eq. (2) \n (     )  ∫ ( )                       \n  ∫ ( )    \n      √      \n    \n           \n . (4) \n     represents the highest frequency being excited, while   ( ) is the excitation amplitude at a \ngiven   . The amplitude spectrum of the measured parabolic wake is provided in Supplemental \nMaterial [34] for 12 different  perturbation velocities    . \n \n5 Using the same rescaled variables as those employed for  Eq. (3), the out-of-plane  displacement  of the \ninterference wake pattern can be formulated in rescaled position coordinates :  ̂       ,  ̂  \n     , and rescaled time  ̂       as: \n ( ̂   ̂   ̂)  ∫ ( ̂)    ̂ ̂   √ ̂   ̂   ̂    ̂ ̂  ̂ ̂   \n  (5) \nAnalogously  to the case of Kelvin wakes, the integral solution can be expressed  using the  stationary \nphase approximation  (please refer to Supplemental Material [34]). In order to explain  the parabolic \nwake pattern of the velocity  (out of the plane   -  ), which is what we measure , we need to \ndifferentiate Eq. (5) with respect to  ̂. Fig. 3a shows the stationary phase approximation of the time \nderivative of Eq. (5) at  ̂   and absolute  ̂  values . The two -dimensional interference pattern has \nridges and zero values that conform to the shape of parabolas.  All their focal points are located at the \ncurrent position  of the perturbation  ( ̂   ̂ )  (   ). The focal lengths of the n confocal parabola s, \n(i.e. the distance between their vertices  and the current position  of the perturbation ) lying on the \nmaximum  ridges of the wake pattern , are given by    ̂      ⁄     in ( ̂   ̂ ) coordinates.    \nis a global constant phase  shift depending on the details of the excitation physics  and n is an integer \nlabeling the considered parabola . For the out -of-plane velocity measured  in our experiment,  a phase  \n   equals to   ⁄ is expected and observed , since the maximum plate displacement  of the wake field \noccur s at the point illuminated by the heating laser . These  statements were cross -validated  by \nnumerical integral solutions  and analytical stationary phase approximation  as d etailed in \nSupplemental Material [34]. \nWe now propose a geometrical construction of the parabolic pattern  (Fig. 3b). For this, one can draw \nparallel lines representing the wavefronts of specific narrowband frequency components (similar to \nthe examples from the measurement in Fig. 2d-i). These lines have inversely sig ned slopes (or angle \n  in Fig. 2d-i) for positive and negative  ̂ , repres enting Mach  cone s with the symmetry line  on the \n   axis. The relation between these slopes and the distance between two neighboring lines is defined \nby the dispersion relation (     at specific frequency component) , while t he origin  of the lines  – \nintersection with the ordinate axis  – is defined by   . The lines are tangent to confocal parabolas  \nwith focal lengths   ̂        , as shown in Supplemental Material [34]. This approach  also \nallows us to recover the wake patterns associated with  other dispersion relation s, for example to  \ncompare with Kelvin wakes (see Video Pr esentation ). When  the line bundles are replaced  by \nnarrowband frequency components (with the lines following  the maxima  of the two -dimensional \nsloped harmonic function s), a pattern similar to Fig. 3a emerges . The parabolas situated on  the \nmaximum ridge of this wake pattern are more open and have larger foci, shifted by   ⁄. This is \nbecause all the tangent lines fall outside the parabolas  with focal lengths   ̂        . \nThe focal lengths of the confocal parabolas were  measured experimentally by determining the \nposition    of the maximum signal amplitude (and the first minimum signal amplitude in the positive \ndirection from the maximum) for all the positions   , for all time instances when the parabolic wake \npattern w as within  the scanned region , and for ten different   . The obtained mean values  of the \nposition    along with the ir standard deviation band s in rescaled coordinates ( ̂   ̂ ) are presented  in \nFig. 3c. The  curve s obtained at  all ten differ ent speeds (coded in gray shades ) and at the increased   \n(blue) align with  two confocal parabolas having  focal lengths    ⁄ (maximum) and    ⁄   \n(minimum ). This provides validation for our mathematical model s and the universality of  the \nparabolic wake pattern.   \n6  \nFig. 3. Time derivative of Eq. (5) obtained by the s tationary phase approximation  (a). The maximum  ridges  of the pattern  \nfollow shapes of confocal parabolas, having focal lengths   ̂      ⁄    , where      ⁄ due to  the velocity \nmeasurement. The parabolic shape of the interference pattern can be elucidated  through the concept of parallel  lines \n(wavefronts ) (b). Their slope  (or angle φ) is computed  from their periodicity  (wavelength)  using the dispersion relation . \nTheir origin corresponds to  the phase of the wake excitation. If the dimensions     and    are rescaled by the factor    , \nthe maximum  and minimum ridges  observed in  the measurement ( full wakes at  three  chosen  speeds shown in Fig. 1) \nalign with the two confocal parabolas ( red dashed lines)  with focal lengths of     ⁄ and       ⁄ , respectively , across \nall ten different    (c, gray shades) as well as  for the scenario when α is increased ( c, blue). \nTo achieve  the complete  parabolic wake pattern , it is essential  that the moving perturbation  excite s \nfrequencies  reaching at least until    . This condition is not fulfilled when the length of the \nperturbation  spot L is excessively  large or     is overly high . In such cases , the parabolic wake \npattern  starts to open from the front, where the waves depart from the perturbation at      . Due \nto the lack of the frequency components with larger   (having  the wider cone opening), the parabolic  \nwake pattern tends to approximate the shape of a Mach cone.  This situ ation is similar to the transition \nbetween Kelvin and Mach regimes , where in the  cone opening s are narrower than the Kelvin angle  if \nlow-frequency components are not excited , for instance in the case of  excessively  small  objects \nmoving on a water surface  with too high velocity  [6,7] . \nIn analogy with the Mach number for non -dispersive media and the Froude number for Kelvin \nwakes, a dimensionless  number  that governs these physics  can be defined for media with quadratic \ndispersion , as       . This number  measures  the ratio between the perturbation velocity and the \ncritical velocity        ⁄. The wake pattern has its complete  shape  around the vertices of \nparabolas  when      (Fig. 1b-f). All frequency components up to     are excited if   is sufficiently \nsmall  at specific   . This is not the case in Fig. 1g. Since the laser is out of focus , the value of L is \ntoo high.  At lower values of   , the criterion     is achieved for smaller  . In this case , the \nparabolas of the wake patter n will be closed at their vortices, however, they will have a smaller slope \n(longer focal len gths). In other words, only limited area around the central part of the rescaled \nuniversal wake pattern will be visible.  \nTo summarize , we provided an analytical explanation for experimentally obse rved parabolic wakes \npropagating in a medium with quadratic dispersion . We have shown that the observed pattern is \nuniversal once the coordinates are rescaled by the velocity and material factors . The equations define \ntwo regimes (propagating and evanescent) separated by    . In opposition to the Kelvin wake, the \nwave components below     produce the parabolic wake, while evanescent wave behavior is \nobservable under the condition  that the addressed frequency      . This condition  is reminiscent  \nto the critical angle  behavior  in the phenomenon of total reflection . The evanescent waves propagate \nsolely  along the trajectory  of the moving perturbation, instead of along a spatial interface for the \nphenomenon of total reflection.  \n \n7 Our finding s can be extended to other phenomena gove rned by quadratic  (or even other power laws)  \ndispersion, such as  flexural phonons on graphene membranes  [35-37] and specific regimes of \npolaritons in semiconductor microcavities  [38-40].  \nWe also demonstrate d that Lamb waves can be generated not only by a laser pulse as a fast temporal \nchange in illuminating power, but also by swift spatial movements of  a continuous laser beam. This \nphenomenon holds potential for applications in contact -free damage detection and imaging of \nmechanical properties that influence the shape of the parabolic wake.   \nIn our experiment, the wave propagation properties  were  altered by inducing changes in the heat \ndistribution  at the spot of the moving perturbation . Interest ing wave phenomena are anticipated  to \nemerge as a consequence of interaction between the moving perturbation and the wake pattern that \nwas excited at a prior temporal instance  (similar to a study on water waves  [41]). This situation  \noccurs  when the trajectory  of the moving perturbation deviates from a straight  path, when     is not \nconstant , or when the intensity of the perturbation varies over time.  \nReferences  \n[1] W. Thomson, Proceedings of the Institution of Mechanical Engineers 38, 409 (1887).  \n[2] F. S. Crawford, American Journal of Physics 52, 782 (1984).  \n[3] L. James, Measurement Science and Technology 13, 1501 (2002).  \n[4] J. W. Davys, R. J. Hosking, and A. D. Sneyd, Journal of Fluid Mechanics 158, 269 (2006).  \n[5] T. Soomere, Applied Mechanics Reviews 60, 120 (2007).  \n[6] M. Rabaud and F. Moisy, Physical Review Letters 110, 214503 (2013).  \n[7] A. Darmon, M. Benzaquen, and E. Raphaël, Journal of Fluid Mechanics 738, R3, R3 (2013).  \n[8] A. Likar and N. Razpet, American J ournal of Physics 81, 245 (2013).  \n[9] J. Colen and E. B. Kolomeisky, European Journal of Mechanics - B/Fluids 85, 400 (2021).  \n[10] F. Noblesse, J. He, Y. Zhu, L. Hong, C. Zhang, R. Zhu, and C. Yang, European Journal of \nMechanics - B/Fluids 46, 164 (2014).  \n[11] R. Pethiyagoda, T. J. Moroney, C. J. Lustri, and S. W. McCue, Journal of Fluid Mechanics \n915, A126, A126 (2021).  \n[12] T. Von Kármán, McGraw -Hill  (1963).  \n[13] R. H. Hernández and M. Sánchez, Europhysics Letters 58, 222 (2002).  \n[14] M. Mokry, The Aeron autical Journal 109, 1 (2016).  \n[15] S. Wu, X. Zhai, and B. Liu, Opt. Express 27, 1142 (2019).  \n[16] A. M. Shevchenko and A. S. Shmakov, AIP Conference Proceedings 2027 , 030103 (2018).  \n[17] J. H. Chen, B. J. Cantwell, and N. N. Mansour, Physics of Fluids A: Fluid Dynamics 2, 984 \n(1990).  \n[18] D. Winske, W. Daughton, D. S. Lemons, and M. S. Murillo, Physics of Plasmas 7, 2320 \n(2000).  \n[19] W. J. Miloch, Plasma Physics and Controlled Fusion 52, 124004 (2010).  \n[20] E. B. Kolomeisky and J. P. Straley, Physical Revi ew Letters 120, 226801 (2018).  \n[21] J. B. Rosenzweig, D. B. Cline, B. Cole, H. Figueroa, W. Gai, R. Konecny, J. Norem, P. \nSchoessow, and J. Simpson, Physical Review Letters 61, 98 (1988).  \n[22] T. Katsouleas, Nature 431, 515 (2004).  \n[23] I. Blumenfeld  et al., Nature 445, 741 (2007).  \n[24] M. Litos  et al. , Nature 515, 92 (2014).  \n[25] A. Melzer, S. Nunomura, D. Samsonov, Z. W. Ma, and J. Goree, Physical Review E 62, 4162 \n(2000).  \n[26] N. Razpet and A. Likar, American Journal of Physics 78, 1384 (2010).  \n[27] P. Genevet, D. Wintz, A. Ambrosio, A. She, R. Blanchard, and F. Capasso, Nature \nNanotechnology 10, 804 (2015).   \n8 [28] A. J. Macleod, A. Noble, and D. A. Jaroszynski, Physical Review Letters 122, 161601 (2019).  \n[29] I. Carusotto, S. X. Hu, L. A. Collins, and A. S merzi, Physical Review Letters 97, 260403 \n(2006).  \n[30] F. Moisy and M. Rabaud, Physical Review E 90, 023009 (2014).  \n[31] J.-C. Ono -dit-Biot, M. Trejo, E. Loukiantcheko, M. Lauch, E. Raphaël, K. Dalnoki -Veress, \nand T. Salez, Physical Review Fluids 4, 014808  (2019).  \n[32] E. Raphaël and P. G. de Gennes, Physical Review E 53, 3448 (1996).  \n[33] F. S. Crawford, American Journal of Physics 60, 751 (1992).  \n[34] See Supplemental Material at [URL will be inserted by publisher]  for more details regarding \nthe experiment and the refined analytical models.  \n[35] E. Mariani and F. von Oppen, Physical Review Letters 100, 076801 (2008).  \n[36] W. L. Z. Zhao, K. S. Tikhonov, and A. M. Finkel’stein, Scientific Reports 8, 16256 (2018).  \n[37] A. Taheri, S. Pisana, and C. V. Singh, Physical Review B 103, 235426 (2021).  \n[38] A. Amo, J. Lefrère, S. Pigeon, C. Adrados, C. Ciuti, I. Carusotto, R. Houdré, E. Giacobino, \nand A. Bramati, Nature Physics 5, 805 (2009).  \n[39] C. Ciuti and I. Carusotto, Phy sica Status Solidi 242, 2224 (2005).  \n[40] I. Carusotto and C. Ciuti, Physical Review Letters 93, 166401 (2004).  \n[41] C. d’Hardemare, S. Wildeman, A. Eddi, and E. Fort, Physical Review Letters 122, 104301 \n(2019).  \n  \n9 Supplemental Material : \nExperimental observation of parabolic wakes in thin plates  \nJanez Rus1*, Aleksi Bossart1, Benjamin Apffel1, Matthieu Malléjac1 and Romain Fleury1* \n1Laboratory of Wave Engineering, Institute of Electrical and Micro Engineering, Ecole Polytechnique \nFédérale de Lausanne (EPFL), Station 11, 1015 Lausanne, Switzerland  \n*Correspondence to romain.fleury@epfl.ch  or janez.rus@tum.de  \n \n1. Experimental methods  \nThe parabolic wakes were generated using a continuous laser (FL -1064 -CW, manufactured by \nChangchun New Industries Optoelectronics Technology) with  a power of 6 W and  a wavelengt h of \n1064 nm  (Fig. S1b). A 2 -axis galvanometer scan head ( XG210, manufactured by Mecco) was \nemployed to move the laser spot continuously along the surf ace of the thin plate  at a velocity    (Fig. \nS1c). \nThe thin plate with a thickness of 0. 1 mm was constructed from a shape memory polymer  supplied \nby SMP Technologies Inc, Tokyo . The polymer exhibited a glass transition temperature within the \nrange of 25°C to 90°C  (Fig. S1a). Its Young's modulus decrease s considerably, by a factor of at least \n20, when its temperature  is increased  several tens of degrees Celsius  above room temperature  [1]. \nConsequently, the wave propagation properties are rapidly altered upon exposure to the moving \ncontinuous laser. This perturbation  gave rise to the formation of parabolic wake patterns.  The thin \nplate was cooled by a steady flow of room -temperature air flow directed to t he specimen surface  \nusing a nozzle (Fig. S1e). \nThe ultrasonic  response s of the thin plate in the form of the parabolic wake pattern s were \nindependently  detected for each scanning position s (pixels) in Fig. 1 and Fig. S2. This was feasible \ndue to the high repeatability of the wake excitation.  To achieve this, a PSV -F-500-HV laser scanning \nvibrometer  (Polytec)  was installed on the side of the thin plate opposi te to the continuous laser  (Fig. \nS1d). For each scanning position, the  wake pattern was stimulated 10 times with a repetition rate  of \n20 Hz to achieve sufficient signal averaging.  The sampling frequency was set at 6.25 MHz. The \nsurface of the thin foil illuminated by the wake excit ing laser was coated black to enhance absorption, \nwhereas the oppo sing side, illuminated by the laser of the vibrometer , was covered with a retro -\nreflective foil.  \nIn order to test the hypothesis that the shape of the parabolic wake in rescaled coordinates remains \nconstant even if    changes as well, the temperature of the thin plate  was raised . Uniform heating \nacross the en tire scanned area was achieved by positioning a heater at a distance of 1 cm from the \nthin plate. The local temperature was elevated by 5°C, resulting in a 68% increase in   , which was \nmeasured independently as detailed in Section 6. \n   \n10 \n \nFig. S1. Parabolic wakes propagated within  a medium, which was  a thin black polymer plate  mounted in a metal frame \n(a). The moving perturbation was induced by a fiber -guided continuous laser (b)  directed through a  galvanometric \nscanning head (c). The wakes were measured utilizing a laser scanning vibrometer  (d). The sample was cooled by an air \nnozzle (e).  \n2. Parabolic wake p attern measurement at twelve  speeds and increased temperature  \nThe scans of the wake patterns were performed over 89 × 29 positions (   ,   ) with a spatial \nresolution of  0.31 mm. The zero value  of the position   was set to the line of the wake excitation ( the \nsymmetry line of the parabolic wake). Similarly, the zero value of the position    was defined  where \nthe continuous laser came  into focus on the thin plate's  surface.  \nIn Fig. S2, we present two timeframes of the measured wake patterns for 12 distinct    values  and the \nscenario when the temperature (and  ) was increased ( Fig. S2y). The first timeframe (left column) \ncorresponds to       , marking the moment when the perturbation crosses          . This \ninstant aligns with the moving laser being  in focus (perturbation size is the smallest). The second \ntime frame (right column) corresponds to the time instant when the perturbation crosses \n          . Here , the laser is out of focus, resulting in a limited  range of the excited frequency \ncomponents . Consequently,  the shape s of the parabolic wake pattern s undergo changes at higher    \nvalues . \nSignals shown in Fig. S2e-y were used to generate  Fig. 2 and Fig. 3c. The analysis in Supplemen tal \nMaterial encompassed a ll 12 different    values .  \n11 \n \nFig. S2. When the laser beam is in focus (perturbation at     , left-hand column), h igh frequency components are \nexcited . In this scenario , two maximum ridge parabolas become evident, with their vertex distance governed by    : this \ndistance is larger  for lower     and vice versa. However, when  the laser bea m is ou t of focus (perturbation at      , \nright -hand column ), we can only observe the second maximum ridge parabo la up until            (n). For higher  \ndimensionless number s     (being a result of higher    values) , the parabolic wakes open  at the vertices  and approach \nthe Mach regime.  Please refer to the Video Presentation  for the complete animation.   \n12 3. Wavenumber -frequency diagram s without coordinate rescaling  \nThe upper half of the wake pattern s (encompassing the 15 highest positions    ) were utilized to \nobtain  Fig. 2 and Fig. S3. Two positions along  the line of the wake excitation at    = 0 mm were \nexcluded from the analysis . In the initial step , the signals from  all scanning positions and all    were \ntransformed to the frequency domain using a fast Fourier transform.  Subsequently, an exponential \nfunction        was fit ted to the absolute values of the obtained frequency components for all \nscanning positions and all   . Here, the c onstant   represented the  imaginary part of   .  \nThe r eal parts of    and   were determined  by the following procedure. A two-dimensional fa st \nFourier transform was performed on  the images across  the dimensions    and    for each of the \nfrequency components  (examples in Fig. 2d-i) and for all   . This operation yielded images with two \npeaks situated  at positions (     ) and (       ), due to the central point symmetry . \n \nFig. S3. The data  of Fig. 2a-c (with additional  two measurement at low er   ) are presented  with the difference that the \ncoordinates are not resca led: wave numbers    and   , as well as  the opening angle   in relation to  the frequency  (  \n  ). As a result, the measured curves (gray shades) no longer overlap , but instead conform to the values anticipated \ntheoretically by Eq. (1) and Eq. (2). The imaginary part of    is included , which arises due to the  variation in  the laser \nspot size along the direction    (Section 10). The red curve displayed  in Fig. 2 is omitted from this representation, as it \ndiffer s for each    when  coordinates are not rescaled.    \n13 The i maginary part of    exerts an influence resulting in elevated values of the real part of    \nparticularly for high er frequencies  (as elaborated further in Section 10). The imaginary part of    is \nnon-zero even prior to reaching the critical frequency    . As a result, the trajectory of    curve is \nsmooth in the vicinity of    . The angle   is reduced  due to the influence  of the imaginary part of   . \nThis phenomenon stems from the effect that as the laser approaches its focal point, progressively \nhigher wave amplitudes are stimulated . \n4. Frequency spectrum excited by the moving perturbation  \nThe measurement of the signal amplitudes close to the trajectory line of the perturbation  (0.6 mm \nabove the symmetry line  of the parabolic pattern ), reveals the presence of two distinct spectral shape \nregimes (Fig. S4). In the first regime , encompassing the lowest six   , a noticeable  change of the \namplitude slope (in respect to the frequency)  can be observe d. This change aligns with  the \ncorresponding     (gray -scale vertical lines in Fig. S4a, black vertical line in Fig. S4b). This \nphenomenon arises from the fact that the frequency components above     do not propagate in the \nfar field, as clarified by Fig. 2 (evanescent behavior ). In the second regime , encompassing the highest \nsix   , there is no pronounced change of the amplitude slope (in respect to the frequency). This is due \nto the intrinsic  losses that increase with frequency (     features  a quadratic dependence on   ), and \nthe influence  of the imaginary part of    (Section 10), which becomes more prominent at  higher   . \n \nFig. S4. The m ean amplitude of the excited wake pattern , encompassing all    positions at           is presented for \n12 distinct     values in the original frequency coo rdinate (a), as well as in the rescaled angular frequency coordinate (b). \n5. Measured focal lengths  of confocal parabolas  and difference between the maximum and \nminimum ridge parabolas  \nAs explained in the main part of the article ( Fig. 3), the wake pattern in the medium with  quadratic \ndispersion has the shape of confocal  parabolas. Rescaling  both spatial dimensions  (e.g. parallel and \nperpendicular to the velocity direct ion of the moving perturbation),  by the factor    , consistently \nyields focal lengths    ̂      ⁄    , where    denotes  the integer relating to the specific \nparabola situated along  the maximum  ridge of the wake pattern.  \nFor each time instance  in which  the parabolic wake was within  the scanning range,  we pinpointed  the \nlocation s    with maximum  and minimum  signal amplitude s for all    values and all    variations  \n(including the scenario  with increased   ). This search pro cedure  was applied  on unfilter ed signals. \nWe defined the  condition that minimum  signal amplitudes were positioned  at higher     values \ncompared to  maximum  signal  amplitudes . This was necessary in order to exclude the area s \ncharacterized by deeply neg ative amplitude values after  the perturbation.   \n14 Following  this procedure , maximum and minimum  ridge parabolas shown in Fig. 3c were derived.  \nSubsequently, parabolic curve s (   of minimum  or maximum  amplitude s in dependency on   ) were \nfitted  for all time instances and all    values . The f ocal lengths were determined  from the parabolas ' \nslope (from the constant of the quadratic term ) and then rescaled by     . Mean and standard \ndeviation of the se measured  focal lengths are displayed  in Fig. S5a with a minimum of  319 values \nused for each    value  and for both foci  – with more values at lower   , due to the longer presence of \nthe wake pattern within the scanning range.  For the experiment involving increased  , 119 values \nwere  used. The measured  focal lengths  close ly align with  the values  delivered by  our analytical \nmodel:    ̂     ⁄ for the maximum ridge parabola and   ̂     ⁄ for the minimum ridge \nparabola . Please note that in the equation   ̂      ⁄    ,     for the first maximum ridge \nparabola and     ⁄ for the first minimum ridge parabola. In our experiment,    equated to   ⁄ \nsince  we measured velocity of the out -of-plane displacement of the thin plate (    would be 0 in the \ncase of a displacement measurement).  \nFig. S5b graphically presents the distances between  the vertices of the fitted maximum and minimum \nridge parabolas  of the measurement . If these distances are scaled by the factor    , they closely a lign \nwith the value of   as predicted by our analytical model.  \n \nFig. S5. The f ocal lengths of the first maximum  and minimum  ridge parabolas, derived directly from  the experimental \ndata depicted  in Fig. S2, exhibit alignment with the values of     ⁄ (maximum ridge parabola) and    ⁄ (minimum ridge \nparabola) when scaled by the factor    . The measu red distances between the vertices of both parabolas closely \napproximate the value of   under the same rescaling law.  \n6. Measurement of  the material constant    \nThe material constant  , which depends on the material and thickness of the thin plate , was \ndetermined through a separate  measurement.  In this procedure, ultrasound was excited using  a laser \npulse with the wavelength of 532  nm, energy of 10  mJ, duration of 5  ns (full width at half -\nmaximum), and a repetition rate of 20 Hz utilizing  a Surelite SL I -20 laser  (Manufacturer: \nContinuum) . The laser pulse beam diameter was 5  mm (95% energy). A linear scan was executed \nacross 111 positions   (aligned parallel to   ) with a spatial resolution  of 0.31 mm, extending  away \nfrom the ultrasound source . This sca n covered  the same region of the thin plate as utilized for the \nmeasurement of parabolic wake pattern s.  \nMeasured raw time signals in Fig. S6a reveal  a distinct  presence of a slower zero -order asymmetric \nLamb wave, characterized by its typical dispersive shape. A faster zero -order symmetric Lamb wave \nmode, discernible  as a narrow  straight  bright  line, departing from      with near-horizontal slope, \nwas excluded from the   estimation.   \n15 The time -position diagram presented in  Fig. S6a was converted to  -  diagram  by a two -dimensional \nfast Fourier trans form . In subsequent steps , we pinpointed the value of    corresponding to  the \nmaximal amplitude of the zero-order symmetric Lamb wave component for each  . The   value was \nquantified  by fitting a square root function over the measured  ( ) relationship . For the scenario  \ninvolving room temperature (black curve in Fig. S6a), we obtained the value            , and for \nthe scenario of increased thin pla te temperature              (red curve in Fig. S6a, Fig. S2y). \n \nFig. S6. The unprocessed  signals acquired through a linear scan along the position x (aligned parallel to   ) away from the \npulse excitation point at      (a).  -  diagram s of the zero-order asymmetric Lamb wave (maximum amplitude values) \nat room temperature  (b, black curve)  and at the plate temperature increased for 5°C  (b, red curve) . For both cases,    \nvalues  were determined  by fitting a square root function to the  measured  ( ) data. \n7. Solving the wake pattern integral by stationary phase approximation  \nIn the co -moving frame , employing the vectorial representation for the perturbation velocity  ⃑  \n   ⃑ , the quadratic dispersion relation can be expressed as follows:  \n ( )    ⁄  ⃑⃑  ⃑ . Eq. S 1 \nThe stationary wave pattern in polar coordinates in the co -moving frame generated  by the moving \nperturbation  can be represented as a summation of numerous plane waves with zero frequency : \n   ( ⃑) ∫ ( ⃑⃑)  (    ⃑⃑  ⃑) ( ( ⃑⃑))  ⃑⃑. Eq. S 2 \nWe introduce a vectorial expression for  ⃑⃑  (      ⃑       ⃑⃑ ), where   is the complementary \nangle of the cone opening angle   at specific frequency ( Fig. 2d-i).  \n \nBy writing  ⃑⃑  (      ⃑       ⃑⃑ ) one can rewrite      as \n          . Eq. S 3 \nAs    , this forces    ⁄     ⁄. We also write  ⃑  (     ⃑       ⃑⃑ ) so that  \n ⃑⃑  ⃑      (   ). Eq. S 4 \nAdditionally, we will make the assumption that the emission amplitude is homogeneous  (   ) \n  . The integral can now be written as  \n   ( ⃑) ∫∫ (   )  \n       ⃑⃑  ⃑ (         ) \n           \n ∫    \n              ( )   (   )       ( )   Eq. S 5  \n16       ∫   ( )     ̂    ( )   (   )   ⁄\n   ⁄   \n \n ̂      is rescaled by the same factor , as discussed  in the main part of our work . This integral is \ninvariant under the transformation     . Consequently , we can further assume that      . \nAs the  integral  of Eq. S 5 is in the form  of ∫ ( )    ̂ ( )  , it can be approximated using stationary \nphase approximation  when    . For this  purpose , one seeks  the stationary points of   such that  \n       (     )  . \nConsidering the constrains on    and  , the two stationary points are  \n    \n      and          ⁄   ⁄. Eq. S 6 \nOne can  consequently  solve  the integral  describing the parabolic wake pattern in co -moving frame as \n   ( ⃑)      √ \n ⃑[   (  ⁄)  (  ⃑     (  ⁄)   ⁄)    (  ⁄)  ( ⃑     (  ⁄)   ⁄)]. Eq. S 7 \nRemember that this expression holds true solely within the range of      . When extending the \nrange to       , it becomes necessary to  apply absolute values to bot h the cosine and sine \nterms. This step ensures that the integral maintains its invariance and yields the correct expression for \nall  . \nWithin  our study , the measured quantity is the out -of-plane velocity , as opposed to the out-of-plane \ndeformation in the co-moving frame, which is the case for the derivation in this section  until Eq. S 7. \nTo align with our experimental conditions , we formulate  the parabo lic wake pattern in the stationary  \nframe. This is achieved through a coordinate transformation:  \n ( ⃑  )    ( ⃑  ⃑  ). Eq. S 8 \nThe out -of-plane velocity in polar coordinates can then be written (assuming that the deformation of \nthe thin plate for the whole parabolic wake pattern remains small)  as \n ̇( ⃑  )    ( ⃑  )   ⃑   ⃑⃑⃑   . Eq. S 9 \nUtilizing the integral form of    ( ⃑) as described by Eq. S 5 and taking into account the perturbation \nvelocity   ⃑     ⃑ , we can formulate   ̇( ⃑  ) at    : \n ̇( ⃑)          ∫    ( )   (   )     ̂    ( )   (   )   ⁄\n   ⁄   Eq. S 10 \nThrough a computation using the stationary phase approximation, a process akin to the one employed \nfor deriving Eq. S 7, we arrive at the expression for the parabolic wake pattern taking in account the \nexperimental conditions  \n ̇( ⃑) \n         √ \n ⃑[|   (  ⁄)|   (  ⃑     (  ⁄)   ⁄) |   (  ⁄)|   ( ⃑     (  ⁄)   ⁄)]. Eq. S 11 \n ̇( ⃑) provides a parabolic pattern symmetric with regard to two Cartesian axes as it includes sum of  \ncausal and anti -causal case s. In order to obtain the pattern observed in the experiment (symmetric \nonly with regard to perturbation’s trajectory), we consider the causal case alone – the first term of the \nequation (before the plus symbol).  This delivered us the pattern presented in Fig. 3a.  \n \n  \n17 8. Geometric derivation  of the parabolic wake pattern  \nIn this section, we derive the shape of the caustics depicted in Fig. 3b of the main text. There, waves \nare approximated as trains of lines defined by the condition  ⃑⃑  ⃑    , where   is an integer. \nAssuming a ge neral power -law dispersion  ( )    and using the direct equivalent of  Eq. (3) of \nthe main text, this condition yields  \n     \n ̂  √ ̂ \n \n ̂    . Eq. S 12 \nTo find the caustic, we must find a point along this line that remains fixed under small var iations in \n ̂, as shown in Fig. S7a. In particular, the   coordinate of this fixed point does not change under a \nsmall variation in  ̂, yielding the condition   ̂   . Together with Eq. S 12, this allows us to solve \nfor the coordinates of the caustic point associated to the angular frequency  ̂, namely  \n     \n   (  ̂   \n   ̂  ) \n      \n   √ ̂  \n   ̂   \n  Eq. S 13 \nThis parametric expression  contains the Kelvin -wake case (      ), the capillary -wave case \n(     ), and the case treated in the main text (    ). As shown in Fig. S7b, it also allows us to \nrecover the asymptotic pattern corresponding to   tending towards infinity . In our case of interest,  \n   , we can go further and remove  ̂ from Eq. S 13. As expected, we obtain parabolic caustics \nsatisfying  \n     √   \n   . Eq. S 14 \n \n \nFig. S7. (a) Two wave fronts (black lines) that are close in frequency interfere constructively at a fixed point, depicted in \ngreen . We use this fixed -point condition to  derive caustic wave patterns for arbitrary  power laws, with  the   \n  ⁄   ⁄     cases depicted in  black, blue, red , and green respectively  (b).  \n \n \n  \n18 9. Proof that      ⁄ for the velocity  measurement  \nAs a starting point, we take Eq. (5) of the main part of the article, wherein   ( ̂   ̂   ̂) represents the \nout-of-plane displacement induced by the parabolic wake pattern . The numerical solution of its \nintegral is presented  in Fig. S8a, corresponding to  ̂  .  \nIn the first case, we assume a phase shift      ⁄ for all frequency components, consequently \nleading to the inclusion of the term     \n  in Eq. (5): \n      ⁄( ̂   ̂   ̂) ∫ ( ̂)    ̂ ̂   √ ̂   ̂   ̂    ̂ ̂    \n   ̂ ̂   \n . Eq. S 15 \nThe numerical solution  of Eq. S 15 at  ̂   is presented  in Fig. S8b. One can observe that in \ncomparison to Fig. S8a, the parabolas traveling along the maximum (and minimum) ridge of the \npattern exhibit longer focal lengths when      ⁄. The dashed red lines signify the confocal \nparabolas with foci   ̂   ⁄    , while the solid  red lines signify the parabolas with foci \n  ̂    ⁄    , where   is an integer.  \nThe numerical solution of the time derivative of Eq. (5) \n ̇( ̂   ̂   ̂) ∫   ̂ ( ̂)    ̂ ̂   √ ̂   ̂   ̂    ̂ ̂  ̂ ̂   \n  Eq. S 16 \nat  ̂   is shown in Fig. S8c. The maximum ridge parabolas also exhibit focal lengths of  \n  ̂    ⁄     as well,  since its time derivative introduces a phase shift of      ⁄ to all \nfrequency components. This conclusion  aligns with a comparison between the parabolic wake \npatterns described by Eq. S 8 and Eq. S 11, both of which were derived using the stationary phase \napproximation . Please note that while the shapes of       ⁄( ̂   ̂   ̂) and  ̇( ̂   ̂   ̂) are identical, \ntheir absolute amplitude values differ.  \n \nFig. S8. Introducing a phase shift of      ⁄ to all the frequency components (b) yields  an equivalent  effect on the \nparabolic wake pattern  (a) as performing a  time derivation (c). This transition results in a switch of the maximum ridge \nparabolas from  the dashed to the solid  red lines – transforming from the  famil y of confocal parabolas with foci   ̂ \n  ⁄     to the family characterized by foci   ̂    ⁄    , where   is an integer . \n10. Measurement of the imaginary part of    \nThe i maginary part of    was a consequence of the laser spot size not being uniform  along    on the \nthin plate surface . This non-uniformity was a consequence of the experimental necessity to focus the \nlaser beam to attain sufficient light intensity. Simultaneously , in order to increase the highest \nachievable   , the laser beam needed to be inclined to an angle of 20° (in the middle of the scanned \nregion) with respect to the thin plate surface.   \n19 The effect of approaching (negative   ) and distancing (positive   ) from the focus along the \nperturbation's path notably affected     values near     , as evident by the difference between the  blue \nand red lines in Fig. 2. However, this effect on the parabolic wake pattern ( across  the entire  \nfrequency range) remained  relatively  limited  and could be neglected when assessing the focal \ndistances of the parabolas  (Fig. 3c and  Fig. S5). \nOur objective  was to maintain  the analytical model as simple as possible and minimize parameters \nconnected specifically to our experimental setup. We thus employed  only one constant that \ncharacterized the rate of laser focusing along    . The alteration in the beam size was approximated by \nan exponential inc rease and decrease (with the symmetry line at     ), which reasonably \napproximated the experimental Gaussian variation of laser spot size along   . This approach resulted \nin an imaginary component of   . As it was linke d to the laser beam 's characteristics , it was \nunaffected by    and linearly dependent on frequency :     (  )   , with                 \nrepresenting a parameter  tied to the laser beam's divergence properties, which remained constant \nthroughout the exper iment .  \nThe amplitude spectrum induced  by the moving continuous laser exhibited dependency on    \n(averaged values across all    are shown in Fig. S4). Higher frequency components were  solely \nexcited  when the laser ( perturbation location) was in focus while traversing the thin plate surface . \nThe laser spot size variation  provided  an advant age, enabling the observation of the pattern shape at \nan extended perturbation size where  higher frequency components were  not excited (right -hand \ncolumn in Fig. S2 and Fig. 1g). \nThe imaginary part of    can be measured by monitoring  the amplitude change of a specific \nfrequency component along    (it would be constant if the laser spot size was uniform along   ). In \nFig. S9, we present the measurement of exponential decay constants  (referring  to amplitudes)  against \nfrequency for the six highest    values,  taken from  the middle of the upper half of the measured wake \npatterns (          ). Other measurements at lower    values are omitted due to the low signal -to-\nnoise ratio  in the frequency range above 50 kHz.  \nWe can observe that for all    values , the measured exponential decay constants (gray lines in \nFig. S9) confor m closely to the curve defined by the equation:     (  )    (blue lines in \nFig. S9). \n \nFig. S9. The gray curves represent the measured exponential decay constants of the amplitude in the     direction at \n         . Given that  the laser beam was traveling away from its focal spot, the beam diameter expanded at  higher   , \nleading to the diminished ex citation  of high -frequency components. This effect is effectively captured by the imaginary \npart of   , which remains unaffected by    and exhibits linear dependence on frequency.   \n20 11. Modeled wake pattern including  the imaginary part of    \nIn Fig. S10, we present  the numerical solution of the integral of Eq. -, while considering the non -zero \nimaginary component of   . This accounts for the  variation in perturbation size along     in our \nexperiment. The expression for    is given by :  \n  ( )   \n       Eq. S 17 \nwhere                 has been measured and is positive for     . As the perturbation \ntravels in the positive    direction and the  laser beam approaches  the focal spot, the perturbation size \nis contracted and increasingly higher frequency components are exc ited. The situation is reversed for \n     where                 . The numerical  example presented  in Fig. S10 corresponds to \nthe measured parabolic wake pattern shown in Fig. 1e and Fig. S2q with            and   \n         .  \n \nFig. S10. We consider the variation in perturbation size both in time and along   , which is a consequence  of the \nperturbation movement towards and away from the position      , where the laser beam  is in focus.  At this time instant, \nthe highest frequency co mponents are excited.   \n12. References of Supplemental Material  \n[1] A. Firouzeh, M. Salerno, and J. Paik, IEEE Transactions on Robotics 33, 765 (2017).  \n \nFig. 1. \nFig. 2.  \nFig. 3. \n(1)  \n(2) \n(3) \n(4) \n(5) \n \n \n \n \n"    },    {        "title": "2402.05538v1.Inertial_active_harmonic_particle_with_memory_escape_induced_by_viscoelastic_suspension.pdf",        "content": "Inertial active harmonic particle with memory escape induced by viscoelastic\nsuspension\nF Adersh, M Muhsin, and M Sahoo∗\nDepartment of Physics, University of Kerala, Kariavattom, Thiruvananthapuram- 695581 , India\n(Dated: February 9, 2024)\nWe investigate the self-propulsion of an inertial active particle confined in a two-dimensional\nharmonic trap. The particle is suspended in a non-Newtonian or viscoelastic suspension with a\nfriction kernel that decays exponentially with a time constant characterizing the memory timescale\nor transient elasticity of the medium. By solving the associated non-Markovian dynamics, we\nidentify two regimes in parameter space distinguishing the oscillatory and non-oscillatory behavior\nof the particle motion. By simulating the particle trajectories and exactly calculating the steady\nstate probability distribution functions and mean square displacement, interestingly, we observe\nthat with an increase in the memory time scale, the elastic bound of suspension dominates over\nthe influence of harmonic trap. As a consequence, the particle can escape out of the trap without\napproaching steady state. On the other hand, with an increase in the duration of the activity, the\nparticle becomes trapped by the harmonic confinement.\nINTRODUCTION\nThe physics of active matter represents a rapidly ad-\nvancing field of research that has gained substantial inter-\nest across various scientific disciplines [1–5], particularly\nin the realm of biophysics and bioengineering. Active or\nself-propelling systems constitute entities that are driven\nout of equilibrium by harnessing energy from their envi-\nronment to generate directed motion. Active matter pri-\nmarily includes microorganisms like E.coli, ciliates, and\nother motile bacteria, but they can also be synthesized\nartificially. Janus particles [6, 7] and microrobots [8] are\nsome of the prominent examples of artificially synthe-\nsized active matter. They can mimic biological motil-\nity and find applications in many emerging scientific and\ntechnological domains like material science and nanotech-\nnology. Some of the notable theoretical models for ex-\nploring active matter include active Ornstein-Uhlenbeck\nparticle (AOUP) model[9–20], Active Brownian particle\nmodel (ABP) [21–34], run and tumble particle (RTP)\nmodel [35, 36], etc. The AOUP model introduces activ-\nity into the system through a stochastic force that fol-\nlows the Ornstein-Uhlenbeck process. The ABP model\nconsiders both translational and rotational diffusion of\nparticles, while RTPs exhibit a “run” phase character-\nized by ballistic motion with constant speed followed by\nperiodic tumbling.\nAn inertial active particle while self-propelling within\na harmonic confinement can exhibit oscillatory motion\nand the frequency of oscillation depends on the inertia\nand strength of the harmonic confinement [37]. Such an\nactive harmonic particle while self-propelling in a two-\ndimensional (2d) plane, the oscillatory motion turns in\nthe performance of rotational trajectories [13, 38, 39].\nFurther, an active particle can also exhibit oscillation\nin its motion while suspended in a viscoelastic environ-\nment [40, 41]. Such induced oscillations often serve as\ndistinctive characteristic of non-equilibrium particle ac-tivity [42]. Moreover, such a viscoelastic particle while\nconfined in a plane, it performs circular motion [43].\nIn this paper, we delve into the dynamics of an in-\nertial active Ornstein-Uhlenbeck particle (AOUP) sus-\npended in a non-Newtonian bath and confined within a\nharmonic potential. The viscoelastic memory of the en-\nvironment influences the particle motion in both qualita-\ntive and quantitative ways. Examples include the mem-\nory induced delay between the effective self-propulsion\nforce and particle orientation in an overdamped active\nparticle [44], strong enhancement of directional changes\nin the periodically modulated Run and Tumble parti-\ncles in viscoelastic bath [45], effective repulsion among\nactive particles induced by viscoelastic fluid while mov-\ning near a boundary[46], etc. By simulating and exactly\nsolving our present model dynamics, we observe that for\nspecific choice of parameters, the particle exhibits oscil-\nlatory motion and the frequency of oscillation exhibits\na non-monotonic dependence on both inertial and har-\nmonic time scales. The simulated particle trajectories\nconfirm that the particle adopts rotational motion in the\noscillatory regime of parameter space. Moreover, an in-\ncrease in the memory timescale of the medium results in\nthe effective dominance of elastic dissipation over har-\nmonic confinement. As a consequence, the particle can\nescape out of the potential or blow off without approach-\ning a steady state. On the other hand, an increase in\nthe activity time results in trapping of the particle by\nthe harmonic confinement. These findings are consistent\nwith the analytically computed steady state results for\nprobability distribution functions and mean square dis-\nplacement.\nMODEL\nWe consider the 2d motion of an inertial active\nOrnstein-Uhlenbeck particle in a viscoelastic environ-arXiv:2402.05538v1  [cond-mat.soft]  8 Feb 20242\nment. The particle is confined in a harmonic poten-\ntialU(x, y) =1\n2k(x2+y2) with kas the harmonic con-\nstant. The dynamics is non-Markovian due to the non-\nNewtonian nature of the surrounding medium and hence\ncan be described using the generalized Langevin’s equa-\ntion of motion [20, 41]\nm¨r(t) =−γZt\n0λ(t−t′)˙r(t′)dt′−∇U+ξ(t).(1)\nHere, r(t) =x(t)ˆi+y(t)ˆjis the position vector of the\nparticle in the x−yplane and mrepresents the mass of\nthe particle. The first term in RHS defines the viscous\ndrag and is characterized by an exponentially decaying\nmemory kernel λ(τ) of the form\nλ(τ) =(\n1\nt′ce−τ\nt′c;τ≥0,\n0 ; τ <0.(2)\nThe above memory kernel represents that the medium\nexhibits a transient elasticity that exponentially decays\nto viscous behavior with a timescale t′\nc. Hence, t′\nccan be\nregarded as the memory time scale or elastic dissipation\ntimescale of the system. The term ξ(t) in Eq. (1) rep-\nresents the athermal noise associated with the dynamics\nwhich follows the Ornstein-Uhlenbeck (OU) process\ntc˙ξ(t) =−ξ(t) +√\nDη(t). (3)\nHere, tcrepresents the self-propulsion or activity\ntimescale of the dynamics, Dbeing the strength of noise\nandη(t) is the delta-correlated white noise. ξ(t) has the\nproperties\n⟨ξα(t)⟩= 0,⟨ξα(t)ξβ(t′)⟩=δαβ\n2tce−|t−t′|\ntc.(4)\nWhen tc=t′\nc, the generalized fluctuation-dissipation the-\norem is validated and the system approaches thermal\nequilibrium for D= 2γkBT.\nNow we define the physical quantities of interest. The\nmean displacement (MD) of the particle from the initial\nposition is given by\n⟨∆r(t)⟩=⟨r(t)−r(0)⟩. (5)\nSimilarly, the mean square displacement (MSD) can be\nobtained as\n⟨∆r(t)2⟩=⟨(r(t)−r(0))2⟩. (6)\nTo obtain the particle trajectory, we perform a nu-\nmerical simulation of Langevin dynamics [Eq. (1)]. The\nintegration of the equation of motion [Eq. (1)] is carried\nout using a second-order modified Euler method with a\ntimestep of 10−3. The OU noise is realized using the Fox\nalgorithm[47]. In the next section, we discuss the results\nobtained from the numerical simulation and analytical\ncalculations.\n\t0 \t0.2 \t0.4\ntc'Γ00.10.2tc'ω0\n\t0\t0.2\t0.4\nOscillation\n\t\t\t\t\t\tNo\nOscillation(a)\nν\nΓω0=0.05\nω0=0.1\nω0=0.2\nω0=0.4 \t0\t0.2\t0.4\t0.6\t0.8\n\t0 \t0.2\t0.4\t0.6\t0.8(b)ν\nω0Γ=0.1\nΓ=0.26\nΓ=0.3\nΓ=0.5 \t0\t0.2\t0.4\t0.6\t0.8\n\t0 \t0.2\t0.4\t0.6\t0.8(c)\n〈Δx(t)〉\ntΓ=0.05;\tω0=0.05\nΓ=0.05;\tω0=0.1\nΓ=0.2;\tω0=0.1\nΓ=0.3;\tω0=0.2\nΓ=0.3;\tω0=0.3\n-0.6-0.2\t0.2\t0.6\t1\t1.4\n10-1100101102103(d)FIG. 1. (a) Phase diagram separating the oscillatory and non-\noscillatory regimes in the t′\ncΓ−t′\ncω0parameter space [Eq. (7)].\nThe color map in (a) shows the evolution of frequency of os-\ncillation ( ν) with t′\ncΓ and t′\ncω0. (b) νvs Γ for different values\nofω0and for a fixed t′\nc= 1. (c) νvsω0for different values\nof Γ and for a fixed t′\nc= 1. (d) The time evolution of the\nx-component of mean displacement (MD) ⟨∆x(t)⟩for differ-\nent values of ω0and Γ with t′\nc= 1.0. The other common\nparameters are m=tc=D= 1.\n.\nRESULTS AND DISCUSSION\nSolving the dynamics [Eq. (1)], with initial conditions\nr(0) = r0and˙ r(0) = v0, the solution can be obtained as\nr(t) =3X\ni=1ai\u0014\nesit(v0+sir0)+Zt\n0esi(t−t′)ξ(t′)dt′\u0015\n.(7)\nHere, si’s are the roots of the equation\ns3+s2\nt′c+s\u0012Γ\nt′c+ω2\n0\u0013\n+ω2\n0\nt′c= 0, (8)\nwhich are given by\nsi=1\n3t′c\u0014\n−1−ωi−1∆0\n3√∆1+ωi−13p\n∆1\u0015\n, i∈ {1,2,3}\n(9)\nwith\n∆0=−1 + 3\u0002\nt′\ncΓ + ( t′\ncω0)2\u0003\n,\n∆1=−2 + 9 t′\ncΓ−18(t′\ncω0)2+√\n−∆\n2,\n∆ =−\u0002\n2−9t′\ncΓ + 18( t′\ncω0)2\u00032−4∆3\n0,and\nω=−1 +j√\n3\n2.(10)3\nHere, Γ =γ\nm,ω0=q\nk\nmandj=√−1. The coefficients\nai’s in Eq. (7) are given by\na1=s1t′\nc+ 1\n(s1−s2) (s1−s3)t′c, a2=s2t′\nc+ 1\n(s2−s1) (s2−s3)t′c\nand a3=s3t′\nc+ 1\n(s2−s3) (s1−s3)t′c.\n(11)\nSolving Eq. (8), it is realized that the roots of Eq. (8)\ncan be real as well as complex. Positive and negative\nvalues of ∆ result in some of the roots of Eq. (8) to\nbe complex. The complex roots are responsible for the\noscillatory solution of the dynamics [Eq. (7)], whereas\nthe real roots provide an exponentially decaying solution\nwithout any oscillatory behavior. Thus, there exist two\nseparate regimes associated with the oscillatory and non-\noscillatory behavior of the solution. Since the roots siare\nindependent of tc, these oscillatory and non-oscillatory\nregimes are presented in a phase diagram of t′\ncω0-t′\ncΓ pa-\nrameter space [see Fig. 1(a)]. The boundary separating\nthese two regimes in t′\ncω0–t′\ncΓ space is given by ∆ = 0.\nThe color map in Fig. 1(a) shows the evolution of fre-\nquency of oscillation ( ν) with both ω0and Γ for a fixed\nt′\ncvalue. The frequency of oscillation is computed as the\nimaginary part of si(Eq. (9)). In Fig. 1(b), we have\nshown the variation of νwith Γ for different values of ω0\nby keeping t′\ncfixed. For small values of ω0,νas a func-\ntion of Γ initially decreases, becomes zero and shows a\nplateau for the range of Γ that falls in the non-oscillatory\nregime of Fig. 1(a) and finally increases with increase in\nΓ value. For higher values of ω0, this non-monotonic\nbehavior of νwith Γ disappears and it monotonically in-\ncreases with Γ. Similarly, Fig. 1(c) shows the variation\nofνwith ω0for different values of Γ and for a fixed t′\nc.\nFor lower Γ values, νas a function of ω0shows a non-\nmonotonic behavior, it decreases, approaches zero, and\nfinally increases with ω0. For higher Γ values, νbecomes\nan increasing function of ω0. For further exploration of\nthese two regimes, in Fig. 1(d), we plot the x-component\nof MD ⟨∆x(t)⟩for different values of Γ and ω0keeping\nt′\nc= 1. Using Eq. (7), MD [Eq. (5)] of the particle can\nbe calculated as\n⟨∆r(t)⟩=3X\ni=1aiesith\nv0+r0sii\n−r0. (12)\nIn the lower powers of t, it can be expanded as\n⟨∆r(t)⟩=−r0+v0t−r0\u0000\nΓ + 4 t′\ncω2\n0\u0001\n8t′ct2+O\u0000\nt3\u0001\n.(13)From Fig. 1(d), it is to be noted that ⟨∆x(t)⟩does not\nshow any oscillatory behavior for the parameters that\nlie in the no-oscillation regimes of t′\ncΓ-t′\ncω0parameter\nspace (solid blue curve for Γ = 0 .2 and ω0= 0.1). On\nthe other hand, ⟨∆x(t)⟩exhibits an intermediate time\noscillatory behavior(curves other than blue in color) for\nthe parameters that fall in the oscillatory regime of t′\ncΓ-\nt′\ncω0parameter space.\nIn Fig. 2, we plot the instantaneous 2d particle tra-\njectories for different values of ω0andtc. Each column\nof Fig. 2 corresponds to a fixed value ω0and each row\ncorresponds to a fixed value of tc. For a given set of\nparameters, with an increase in ω0value from left to\nright in any of the rows, the particle passes from no-\noscillatory to oscillatory regime. The trajectories reflect\nrandom self-propulsion for the parameters that lie in the\nno-oscillatory regimes [see Figs. 2(a), (d) and (g)]. How-\never, the particle performs rotational trajectories for the\nparameters that lie in the oscillatory regime of parameter\nspace [see Figs. 2(b), (c), (e), (f), (h) and (i)]. Further,\nthe trajectories show stronger confinement of the particle\nwith increase in ω0value as expected. Similarly, in any\nof the columns, the trajectories get suppressed with in-\ncrease in tcvalue. This observation suggests the trapping\nor confinement of the particle around the centre of the\npotential with increase in persistent duration of activity.\nSimilarly, the 2d particle trajectories for different val-\nues of ω0andt′\ncare shown in Fig. 3, with ω0increasing\nalong a row from left to right and t′\ncincreasing along\na column from top to bottom. Similar to the observa-\ntions in Fig. 2, for a fixed t′\ncvalue, with increase in ω0\nalong a row, the parameters are chosen such that the\nparticle passes from no-oscillatory to oscillatory regime.\nThe particle performs random self-propulsion in the no-\noscillatory regime, i.e., for ∆ >0 [see Fig. 3(a), (d) and\n(g)]. For higher ω0values, the particle enters the oscil-\nlatory regime (since ∆ becomes negative) and makes a\ntransition from its random activity to twisting or rota-\ntional motion [see Figs. 3(b), (c), (e), (f), (h) and (i)].\nFurther, with increase in t′\ncvalue from top to bottom in\na column, the trajectories get enhanced. At the same\ntime, the trajectories become more asymmetric and take\nalmost elliptical shape. The enhancement of trajectory\nas an increasing function of t′\nccan be attributed to the\nfact that for high value of t′\nc, the system retains memory\nfor a prolonged period, resulting in less frequent change\nin the direction of the particle velocity. As a consequence,\nthe particle takes longer time to complete the orbit.\nNext, we have calculated the MSD [Eq. (6)] using\nEq. (7) as4\n-60-40-20020\n-100102030y(t)\n\t\n\tω0\t=\t0.1\n-6-3036\n-6-3036\t\n\t\n\tω0\t=\t1.0\n-0.6-0.300.30.6\n-1.5-0.7500.751.5\t\ntc\t=\t1.0\n\tω0\t=\t5.0\n-60-40-20020\n-100102030y(t)\n\t\n\t\t\n-202\n-2-1012\t\n\t\n\t\t\n-0.1500.15\n-0.4-0.200.20.4\t\ntc\t=\t5.0\n\t\t\n-60-40-20020\n-100102030y(t)\n\t\nx(t)\t\n-101\n-1-0.500.51(a)\n(d)\n(g)(b)\n(e)\n(h)(c)\n(f)\n(i)\t\n\t\nx(t)\t\n-0.1-0.0500.050.1\n-0.2-0.100.10.2\t\ntc\t=\t10.0\nx(t)\t\n\t0\t10\t20\t30\t40\t50\nFIG. 2. The simulated particle trajectories in the xy-plane for different values of ω0are plotted for tc= 1 in (a),(b) and (c), for\ntc= 5 in (d), (e) and (f), and for tc= 10 in (g), (h) and (i), respectively. The other common parameters are m=t′\nc=D= 1,\nand Γ = 0 .2. The color map shows the time evolution of the trajectories.\n⟨∆r2(t)⟩=⟨(r(t)−r(0))2⟩= (⟨∆r(t)⟩)2+3X\ni=13X\nj=1aia∗\njD\nm2\"\ntcet(s∗\nj−1\ntc)\n(tcsi+ 1)\u0000\n1−tcs∗\nj\u0001+tcet(si−1\ntc)\n(1−tcsi)\u0000\ntcs∗\nj+ 1\u0001\n+tc\u0000\nsi+s∗\nj\u0001\n−2\u0000\nsi+s∗\nj\u0001\n(tcsi−1)\u0000\ntcs∗\nj−1\u0001+et(si+s∗\nj)\u0000\ntc\u0000\nsi+s∗\nj\u0001\n+ 2\u0001\n\u0000\nsi+s∗\nj\u0001\n(tcsi+ 1)\u0000\ntcs∗\nj+ 1\u0001#\n.(14)\nIn Fig. 4, we present ⟨∆r2(t)⟩as a function of tfor\ndifferent values of Γ in Fig. 4(a), ω0in Fig. 4 (b), tcin\nFig. 4(c), and t′\ncin Fig. 4(d), respectively. In order to\nexplore different time regimes of motion, we introduce a\nparameter αsuch that ⟨∆r2(t)⟩ ∝tα, i.e.,\nα=∂log⟨∆r2(t)⟩\n∂logt. (15)\nThe insets of Fig. 4(a) and (b) show the variation of\nthe exponent αwith t. The initial transient time MSD\nis always found to be ballistic (since ⟨∆r2(t)⟩ ∝t2or\nα= 2) and the steady state or long time regime is al-\nways non-diffusive (since ⟨∆r2(t)⟩is independent of tor\nα= 0). However, the intermediate time regimes of MSD\nis found to be oscillatory for the parameters that fall in\nthe oscillatory regime of parameter space. In Fig. 4(a),\nthe variation of MSD as a function of tfor low and high\nvalues of Γ (Γ = 0 .05,0.5,1.0) show intermediate timeoscillations and that is why the exponent αas a function\noftshows intermediate time oscillations for Γ = 0 .05,0.5,\nand 1 .0, respectively, whereas the MSD is non-oscillatory\nfor Γ = 0 .2. With increase in ω0value in Fig. 4 (b), the\nsystem makes a transitions from no-oscillatory phase to\noscillatory phase and results intermediate time oscilla-\ntions in MSD for larger values of ω0(forω0= 0.5 and\nω0= 1). The same can also be confirmed from the inset\nof Fig. 4 (b) which shows the intermediate time oscilla-\ntory behaviour of αforω0= 0.5 and ω0= 1. Figures 4\n(c) and (d) show the variation of MSD as a function of\ntfor different values of tcandt′\nc, respectively. With an\nincrease in tcvalues, the steady state MSD decreases (see\nFig. 4 (c)) and it increases with an increase in t′\ncvalues\n(see Fig. 4 (d)). At the same time, the initial ballistic\nregime of MSD gets reduced with increase in tcvalue\nand increases with increase in t′\ncvalues.\nWe exactly evaluate the steady state MSD and proba-5\n-300-200-1000\n-150-75075y(t)\n\t\n\tω0\t=\t0.02\n-2502550\n-20-1001020\t\n\t\n\tω0\t=\t0.5\n-1001020\n-10010\t\ntc'\t=\t0.1\n\tω0\t=\t1.0\n-300-200-1000\n-150-75075y(t)\n\t\n\t\t\n-2502550\n-20-1001020\t\n\t\n\t\t\n-1001020\n-10010\t\ntc'\t=\t1.0\n\t\t\n-300-200-1000\n-150-75075y(t)\n\t\nx(t)\t\n-2502550\n-20-1001020(a)\n(d)\n(g)(b)\n(e)\n(h)(c)\n(f)\n(i)\t\n\t\nx(t)\t\n-1001020\n-10010\t\ntc'\t=\t5.0\nx(t)\t\n\t0\t20\t40\t60\t80\t100\nFIG. 3. The simulated particle trajectories in the xy-plane for different values of ω0are plotted for t′\nc= 0.1 in (a), (b) and\n(c), for t′\nc= 1 in (d), (e) and (f), and for t′\nc= 5.0 in (g), (h) and (i), respectively. The other common parameters are m= 1,\nΓ = 0 .05,tc= 1.0 and D= 1.0. The color map shows the time evolution of the trajectories.\nbility distribution function using steady state correlation\nmatrix method formalism [13, 48]. Because of the linear\ndynamics of the model and Gaussian nature of noise, the\nsteady state probability distribution is Gaussian. In or-\nder for evaluating these two quantities, we introduce the\ncorrelation matrix Ξ with components χi,χjsuch that\n[Ξi,j] =⟨χiχj⟩ − ⟨χi⟩⟨χj⟩ (16)\nand the distribution function P(χ) as\nP(χ) =1\n(2π)4p\nDet(Ξ)e[−1\n2(χT)Ξ−1(χ)].(17)\nHere, χis the column vector whose elements χi∈\n(x y v xvywxwyξxξy) with vxandvybeing the xand\nycomponents of velocity. The variables wxandwyare\nthexandycomponents of the vector wgiven by\nw= ΓZt\n0λ(t−t′)˙ r(t′)dt′. (18)\nIntroduction of wsplits the non-Markovian model dy-\nnamics [Eq. (1)] into a set of Markovian equations asfollows\n˙ r=v (19)\n˙ v=−w−ω2\n0r+√\nD\nmξ (20)\n˙ w=−w\nt′c+Γ\nt′cv (21)\n˙ξ=−ξ\ntc+η\ntc. (22)\nThe above set of equations can be expressed as\n˙χ=Aχ+Bη′(23)\nHere, the matrices A,Bandη′are given by\nA=\n0 0 1 0 0 0 0 0\n0 0 0 1 0 0 0 0\n−ω2\n00 0 0 −1 0√\nD\nm0\n0−ω2\n00 0 0 −1 0√\nD\nm\n0 0Γ\nt′c0−1\nt′c0 0 0\n0 0 0Γ\nt′c0−1\nt′c0 0\n0 0 0 0 0 0−1\ntc0\n0 0 0 0 0 0 0−1\ntc\n,(24)6\n10-710-410-1102\n10-1102(a)⟨∆r(t)2⟩\ntΓ\t=\t0.05\nΓ\t=\t0.2\t\t\nΓ\t=\t0.5\t\t\nΓ\t=\t1.0\t\t\n∝t2\n10-710-410-1102\n10-1102(b)\t\ntω0\t=\t0.05\nω0\t=\t0.1\t\t\nω0\t=\t0.5\t\t\nω0\t=\t1.0\t\t\n∝t2\n10-710-410-1\n10-1102(c)⟨∆r(t)2⟩\nttc\t=\t0.1\t\t\ntc\t=\t1.0\t\t\ntc\t=\t5.0\t\t\ntc\t=\t10.0∝t2\n10-710-410-1102\n10-1102(d)\t\nttc'\t=\t0.1\t\t\ntc'\t=\t1.0\t\t\ntc'\t=\t5.0\t\t\ntc'\t=\t10.0∝t20123\n10-1102α\nt0123\n10-1102α\nt\nFIG. 4. The MSD [Eq. (14)] as a function of tfor different\nvalues of Γ in (a) for tc=t′\nc= 1.0 and ω0= 0.1, for different\nvalues of ω0in (b) for tc=t′\nc= 1.0 and Γ = 0 .2, for different\nvalues of tcin (c) for a fixed t′\nc= 1, and for different values\noft′\ncin (d) for a fixed tc= 1, respectively. The insets of (a)\nand (b) show the exponent αas a function of tcorresponding\nto the respective MSD plots. The dashed horizontal lines in\n(c) and (d) correspond to the steady state equilibrium value\nof MSD. The other common parameters are D=m= 1.\n.\nB=\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 01\ntc0\n0 0 0 0 0 0 01\ntc\n, (25)\nand\nη′= (0 0 0 0 0 0 ηxηy)T. (26)\nAs per the correlation matrix formalism, the correla-\ntion matrix Ξ can be shown to satisfy the equation\nA·Ξ + Ξ ·AT+B BT= 0. (27)\nFinally, solving the above Eq. (27), the MSD at steady\nMSD(x,y)\n\t0\t2\t4\t6\t8\t10\ntc\t0\t2\t4\t6\t8\t10tc'\n\t0\t0.25\t0.5\t0.75\t1FIG. 5. The 2d plot of steady state MSD ⟨∆r2⟩s[Eq. (28)] as\na function of both tcandt′\nc. The other common parameters\nare Γ = 1 .0,m= 1,D= 1 and ω0= 1.0.\nstate⟨∆r2⟩scan be calculated as\n⟨∆r2⟩s= Ξ 1,1+ Ξ2,2\n=D\u0002\ntc+t′\nc+t2\ncΓ +t′2\nc(tc+t′\nc)ω2\n0\u0003\nm2Γω2\n0[t′c+tc(1 +tcΓ +tc(tc+t′c)ω2\n0)]\n=D\nΓm2ω2\n0+D(t′\nc−tc) (t′\nc+tc)2\nΓm2(Γt2c+ω2\n0t2ct′c+ω2\n0t3c+t′c+tc).\n(28)\nFrom the above equation, it is confirmed that for tc=\nt′\nclimit, ⟨∆r2⟩sreduces toD\nΓm2ω2\n0. For D= 2γkBT,\nit approaches the equilibrium value given by ⟨∆r2⟩eq=\n2kBT\nk. Similarly, in ω0→ ∞ limit, ⟨∆r2⟩sapproaches\nthe equilibrium value for D= 2γkBTeven when tc̸=\nt′\nc. However, in this limit, the system doesn’t approach\nequilibrium. The system approaches thermal equilibrium\nforD= 2γkBTintc=t′\nclimit and the equilibrium value\nfollows the equipartition theorem\n1\n2k⟨∆r2⟩eq=kBT. (29)\nThe first term of Eq. (28) is independent of tcandt′\nc.\nWith increase in t′\ncvalue, the second term of Eq. (28)\nbecomes dominant and hence in t′\nc→ ∞ limit, this term\ndiverges. As a result, the steady state MSD diverges, i.e.,\nlim\nt′c→∞⟨∆r2⟩s=∞. (30)\nSimilarly, with increase in tcvalue, the second term of\nEq. (28) becomes negative, as a result the ⟨∆r2⟩sde-\ncreases. In tc→ ∞ limit, the second term of Eq. (28)\nbecomes −D\nΓm2ω2\n0, which is exactly equal to the negative7\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t1(a)tc\t=\t5;\ttc'\t=\t1\nxyP(x,y)\n01\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t0.1\t0.2\t0.3(b)tc\t=\t5;\ttc'\t=\t5\nxy00.10.20.3\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t0.04\t0.08(c)tc\t=\t5;\ttc'\t=\t20\nxy00.010.02\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t0.01\t0.02(d)tc\t=\t1;\ttc'\t=\t5\nxyP(x,y)\n00.010.02\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t0.1\t0.2\t0.3(e)tc\t=\t5;\ttc'\t=\t5\nxy00.10.20.3\n-2-1\t0\t1\t2-2-1\t0\t1\t2\n\t0\t1\t2\t3(f)tc\t=\t20;\ttc'\t=\t5\nxy0123\nFIG. 6. The steady state position probability distribution [Eq. (32)] for different values of t′\ncfixing tc(= 5.0) are shown in\n(a)-(c) and for different values of tcfor a fixed t′\nc(= 5) is shown in (d)-(f). Other parameters are m= Γ = ω0=D= 1.\nof the first term. Hence, in this limit, the steady state\nMSD vanishes, i.e.,\nlim\ntc→∞⟨∆r2⟩s= 0. (31)\nThese results are summarized in Fig. 5, where we\npresent the 2d plot of ⟨∆r2⟩sas a function of both tcand\nt′\nc. It is observed that for a fixed value of tc,⟨∆r2⟩sin-\ncreases with increase in t′\ncand for sufficiently large value\noft′\nc,⟨∆r2⟩sbecomes infinitely large, indicating the es-\ncape of the particle without approaching steady state.\nThis observation suggests that persistence of sufficiently\nlong duration of memory in the medium provides a kind\nof elastic bound to the particle that overcomes the har-\nmonic confinement as a result of which the particle can\nescape out of the potential and takes infinitely long time\nto come back the mean position of the well or approach\nthe steady state. On the other hand, for a fixed value\noft′\nc,⟨∆r2⟩sdecreases with increase in tcvalue and be-\ncomes zero for infinitely large value of tc. This fact in-\ndicates the trapping of the particle for the presence of\ninfinitely long duration of activity in the medium. This\nis the genuine feature of an active particle confined by a\nfinite potential [49].\nNow, substituting the solution Ξ from Eq. (27) and in-\ntegrating it over all other variables, we obtain the steady\nstate joint probability distribution function P(x, y) as\nP(x, y) =1\n2πσ2exp\u0012\n−x2+y2\n2σ2\u0013\n, (32)\nwith the variance σ2given by\nσ2=D\u0002\ntc+t′\nc+t2\ncΓ +t′2\nc(tc+t′\nc)ω2\n0\u0003\n2m2Γω2\n0[t′c+tc(1 +tcΓ +tc(tc+t′c)ω2\n0)].(33)Int′\nc→0 limit, the distribution is still Gaussian with\nthe variance [Eq. 33] as\nlim\nt′c→0σ2=D(1 +tcΓ)\n2m2Γω2\n0(1 +tcΓ +t2cω2\n0). (34)\nIn Figs. 6 (a)-(c), we plot the steady state probabil-\nity distribution P(x, y) for different values of t′\ncand for\na fixed value of tc= 5. For all t′\ncvalues, the distribu-\ntion is Gaussian centered at the origin of the potential.\nAst′\ncincreases, σ2becomes an increasing function of t′\nc.\nHence the variance or width of the distribution increases\nand the distribution spreads out [see Figs. 6 (a)-(c)]. At\nthe same time, the peak of the distribution suppresses.\nThis implies that with increase in memory time scale,\nthe probability of finding the particle at the mean po-\nsition of the well decreases and the probability of find-\ning the particle at larger distances increases. Finally, in\nt′\nc→ ∞ limit, P(x, y)→0 and the distribution becomes\nflat [Fig. 6 (c)]. The vanishing of the position distribu-\ntion function in t′\nc→ ∞ limit represents the escape of the\nparticle out of the potential. This observation supports\nthe enhancement of particle trajectories [Fig. 3] and the\nenhancement of steady state MSD [Fig. 5] with increase\nof the persistent duration of memory.\nSimilarly, in Fig. 6 (d)-(e), we show the plot of P(x, y)\nfor different values of tcand for a fixed t′\ncvalue. Initially,\nin the tc→0 limit, P(x, yis Gaussian [Eq. 32] with σ2\ngiven by\nlim\ntc→0σ2=D(1 +t′2\ncω2\n0)\n2m2Γω2\n0. (35)\nAstcis increased, σ2becomes decreasing function of tc\nand hence the variance (or width) of the distribution de-\ncreases [see Figs. 6 (d)-(e)]. Simultaneously, the peak of8\nthe distribution increases and the distribution becomes\nnarrow. Finally, in tc→ ∞ limit, σ2approaches zero\nvalue and hence the probability distribution becomes a\ndelta function since\nlim\ntc→∞P(x, y) = lim\nσ→01\n2πσ2exp\u0012\n−x2+y2\n2σ2\u0013\n=δ(x)δ(y).(36)\nThis suggests that with increase in tcvalue, the\nchances of finding the particle at the mean position of\nthe well increases, confirming the trapping of the particle\nfor sufficiently long duration of activity in the medium.\nThese results are complemented with the suppression of\nparticle trajectories in Fig. 2 and reduction of steady\nstate MSD in Fig. 5 with increase in tcvalues.\nSUMMARY AND CONCLUSIONS\nIn summary, we have explored the self propulsion of\na harmonically confined inertial active Ornstein Uhlen-\nbeck particle in a non-Newtonian environment, charac-\nterized by viscoelastic suspension. We model the dynam-\nics using the generalized Langevin equation of motion.\nBy solving this non-Markovian model, we demonstrate\na phase diagram distinguishing two separate regimes in\nt′\ncω0−t′\ncΓ parameter space associated with the oscilla-\ntory and non-oscillatory behavior of the solution. From\nthe exact solution of the dynamics and from the simula-\ntion results, both transient and steady state properties of\nmotion are investigated. The simulated particle trajec-\ntories reveal that the particle exhibits rotational motion\nin the oscillatory regime of the parameter space. The\nrotational trajectories keep on getting enhanced with in-\ncrease in persistent duration of memory and gets sup-\npressed with increase in persistent duration of activity\nin the medium. This observation suggests that with in-\ncrease in memory time scale, the elastic influence exerted\nby the environment on the particle dominates over the\nharmonic bound of the potential. As a consequence, the\nparticle can come out of the potential without approach-\ning steady state. This result is further supported by the\nenhancement of steady state mean square displacement\nand uniform spreading of position distribution function\nthrough out the space with increase in the memory time\nscale. Similarly, the suppression of particle trajectories\nas an increasing function of activity time is also comple-\nmented with the reduction of steady state mean square\ndisplacement and narrowing of the probability distribu-\ntion function with increase in activity time scale.\nACKNOWLEDGEMENT\nWe thank the 8th statphysics community meeting\n(ICTS/ISPCM2023/02), during which some parts of thework were done. MS acknowledges the start-up grant\nfrom UGC, state plan fund from the University of Kerala,\nand SERB-SURE grant (SUR/2022/000377) from DST,\nGovt. of India for financial support. MM acknowledges\nSERB international travel grant (ITS/2023/002740)\nfrom DST, Govt. of India for financial support.\n∗jolly.iopb@gmail.com\n[1] C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt,\nG. Volpe, and G. Volpe, Active particles in complex\nand crowded environments, Rev. Mod. Phys. 88, 045006\n(2016).\n[2] S. Ramaswamy, Active matter, J. Stat. Mech. 2017 ,\n054002 (2017).\n[3] G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nar-\ndini, F. Peruani, H. L¨ owen, R. Golestanian, U. B. Kaupp,\nL. Alvarez, T. Kiørboe, E. Lauga, W. C. K. Poon,\nA. DeSimone, S. Mui˜ nos-Landin, A. Fischer, N. A. S¨ oker,\nF. Cichos, R. Kapral, and e. a. P Gaspard, The 2020\nmotile active matter roadmap, J. Phys.: Condens. Mat-\nter32, 193001 (2020).\n[4] P. Pietzonka, The oddity of active matter, Nat. Phys. 17,\n1193 (2021).\n[5] G. De Magistris and D. Marenduzzo, An introduction to\nthe physics of active matter, Physica A 418, 65 (2015).\n[6] A. Walther and A. H. E. M¨ uller, Janus particles: Synthe-\nsis, self-assembly, physical properties, and applications,\nChemical Reviews 113, 5194 (2013).\n[7] J. R. Howse, R. A. Jones, A. J. Ryan, T. Gough,\nR. Vafabakhsh, and R. Golestanian, Self-motile colloidal\nparticles: from directed propulsion to random walk,\nPhys. Rev. Lett. 99, 048102 (2007).\n[8] G. Wang, T. V. Phan, S. Li, M. Wombacher, J. Qu,\nY. Peng, G. Chen, D. I. Goldman, S. A. Levin, R. H.\nAustin, and L. Liu, Emergent field-driven robot swarm\nstates, Phys. Rev. Lett. 126, 108002 (2021).\n[9] B. Lehle and J. Peinke, Analyzing a stochastic process\ndriven by ornstein-uhlenbeck noise, Phys. Rev. E 97,\n012113 (2018).\n[10] L. L. Bonilla, Active ornstein-uhlenbeck particles, Phys.\nRev. E 100, 022601 (2019).\n[11] D. Martin, J. O’Byrne, M. E. Cates, ´E. Fodor, C. Nar-\ndini, J. Tailleur, and F. van Wijland, Statistical mechan-\nics of active ornstein-uhlenbeck particles, Phys. Rev. E\n103, 032607 (2021).\n[12] L. Caprini, U. Marini Bettolo Marconi, A. Puglisi, and\nA. Vulpiani, Active escape dynamics: The effect of per-\nsistence on barrier crossing, The Journal of Chemical\nPhysics 150, 024902 (2019).\n[13] L. Caprini and U. Marini Bettolo Marconi, Inertial self-\npropelled particles, J. Chem. Phys. 154, 024902 (2021).\n[14] L. Dabelow, S. Bo, and R. Eichhorn, Irreversibility in\nactive matter systems: Fluctuation theorem and mutual\ninformation, Phys. Rev. X 9, 021009 (2019).\n[15] L. Berthier, E. Flenner, and G. Szamel, Glassy dynam-\nics in dense systems of active particles, The Journal of\nChemical Physics 150, 200901 (2019).\n[16] R. Wittmann, J. M. Brader, A. Sharma, and U. M. B.\nMarconi, Effective equilibrium states in mixtures of ac-9\ntive particles driven by colored noise, Phys. Rev. E 97,\n012601 (2018).\n[17] Y. Fily, Self-propelled particle in a nonconvex external\npotential: Persistent limit in one dimension, The Journal\nof Chemical Physics 150, 174906 (2019).\n[18] D. Mandal, K. Klymko, and M. R. DeWeese, Entropy\nproduction and fluctuation theorems for active matter,\nPhys. Rev. Lett. 119, 258001 (2017).\n[19]´E. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco,\nand F. van Wijland, How far from equilibrium is active\nmatter?, Phys. Rev. Lett. 117, 038103 (2016).\n[20] M. Muhsin, M. Sahoo, and A. Saha, Orbital magnetism\nof an active particle in viscoelastic suspension, Phys. Rev.\nE104, 034613 (2021).\n[21] B. ten Hagen, S. van Teeffelen, and H. Lowen, Non-\ngaussian behaviour of a self-propelled particle on a sub-\nstrate, Condens. Matter Phys. 12, 725 (2009).\n[22] B. ten Hagen, S. van Teeffelen, and H. L¨ owen, Brownian\nmotion of a self-propelled particle, J. Phys.: Condens.\nMatter 23, 194119 (2011).\n[23] M. E. Cates and J. Tailleur, When are active brownian\nparticles and run-and-tumble particles equivalent? con-\nsequences for motility-induced phase separation, Euro.\nPhys. Lett. 101, 20010 (2013).\n[24] K. Malakar, A. Das, A. Kundu, K. V. Kumar, and\nA. Dhar, Steady state of an active brownian particle\nin a two-dimensional harmonic trap, Phys. Rev. E 101,\n022610 (2020).\n[25] I. Buttinoni, J. Bialk´ e, F. K¨ ummel, H. L¨ owen,\nC. Bechinger, and T. Speck, Dynamical clustering and\nphase separation in suspensions of self-propelled colloidal\nparticles, Phys. Rev. Lett. 110, 238301 (2013).\n[26] Y. Fily and M. C. Marchetti, Athermal phase separation\nof self-propelled particles with no alignment, Phys. Rev.\nLett.108, 235702 (2012).\n[27] J. Stenhammar, D. Marenduzzo, R. J. Allen, and M. E.\nCates, Phase behaviour of active brownian particles: the\nrole of dimensionality, Soft Matter 10, 1489 (2014).\n[28] J. Bialk´ e, J. T. Siebert, H. L¨ owen, and T. Speck, Nega-\ntive interfacial tension in phase-separated active brown-\nian particles, Phys. Rev. Lett. 115, 098301 (2015).\n[29] A. P. Solon, J. Stenhammar, R. Wittkowski, M. Kardar,\nY. Kafri, M. E. Cates, and J. Tailleur, Pressure and phase\nequilibria in interacting active brownian spheres, Phys.\nRev. Lett. 114, 198301 (2015).\n[30] L. Caprini, C. Maggi, and U. Marini Bettolo Marconi,\nCollective effects in confined active brownian particles,\nThe Journal of Chemical Physics 154, 244901 (2021).\n[31] L. Caprini, U. M. B. Marconi, C. Maggi, M. Paoluzzi, and\nA. Puglisi, Hidden velocity ordering in dense suspensions\nof self-propelled disks, Phys. Rev. Res. 2, 023321 (2020).\n[32] D. M. van Roon, G. Volpe, M. M. Telo da Gama, and\nN. A. M. Ara´ ujo, The role of disorder in the motion of\nchiral active particles in the presence of obstacles, Soft\nMatter 18, 6899 (2022).[33] C. Scholz, S. Jahanshahi, A. Ldov, and H. L¨ owen, Inertial\ndelay of self-propelled particles, Nat. Commun. 9, 5156\n(2018).\n[34] S. Mandal, B. Liebchen, and H. L¨ owen, Motility-induced\ntemperature difference in coexisting phases, Phys. Rev.\nLett.123, 228001 (2019).\n[35] H. C. BERG and D. A. BROWN, Chemotaxis in es-\ncherichia coli analysed by three-dimensional tracking,\nNature 239, 500 (1972).\n[36] K. Martens, L. Angelani, R. Di Leonardo, and L. Boc-\nquet, Probability distributions for the run-and-tumble\nbacterial dynamics: An analogy to the lorentz model,\nThe European Physical Journal E 35, 84 (2012).\n[37] G. H. P. Nguyen, R. Wittmann, and H. L¨ owen, Ac-\ntive Ornstein–Uhlenbeck model for self-propelled parti-\ncles with inertia, J. Phys.: Condens. Matter 34, 035101\n(2022).\n[38] A. Noushad, S. Shajahan, and M. Sahoo, Velocity auto\ncorrelation function of a confined Brownian particle, Eur.\nPhys. J. B 94, 202 (2021).\n[39] M. Muhsin and M. Sahoo, Inertial active Ornstein-\nUhlenbeck particle in the presence of a magnetic field,\nPhys. Rev. E 106, 014605 (2022).\n[40] F. N. C. Paraan, M. P. Solon, and J. P. Esguerra, Brow-\nnian motion of a charged particle driven internally by\ncorrelated noise, Phys. Rev. E 77, 022101 (2008).\n[41] F. J. Sevilla, R. F. Rodr´ ıguez, and J. R. Gomez-Solano,\nGeneralized Ornstein-Uhlenbeck model for active motion,\nPhys. Rev. E 100, 032123 (2019).\n[42] J. R. Gomez-Solano, R. F. Rodr´ ıguez, and E. Salinas-\nRodr´ ıguez, Nonequilibrium dynamical structure factor of\na dilute suspension of active particles in a viscoelastic\nfluid, Phys. Rev. E 106, 054602 (2022).\n[43] N. Narinder, C. Bechinger, and J. R. Gomez-Solano,\nMemory-Induced Transition from a Persistent Random\nWalk to Circular Motion for Achiral Microswimmers,\nPhys. Rev. Lett. 121, 078003 (2018).\n[44] A. R. Sprenger, C. Bair, and H. L¨ owen, Active Brown-\nian motion with memory delay induced by a viscoelastic\nmedium, Phys. Rev. E 105, 044610 (2022).\n[45] C. Lozano, J. R. Gomez-Solano, and C. Bechinger, Run-\nand-tumble-like motion of active colloids in viscoelastic\nmedia, New Journal of Physics 20, 015008 (2018).\n[46] N. Narinder, J. R. Gomez-Solano, and C. Bechinger, Ac-\ntive particles in geometrically confined viscoelastic fluids,\nNew Journal of Physics 21, 093058 (2019).\n[47] R. F. Fox, I. R. Gatland, R. Roy, and G. Vemuri, Fast,\naccurate algorithm for numerical simulation of exponen-\ntially correlated colored noise, Phys. Rev. A 38, 5938\n(1988).\n[48] N. VAN KAMPEN, Chapter viii - the fokker–planck\nequation, in Stochastic Processes in Physics and Chem-\nistry (Third Edition) , North-Holland Personal Library,\nedited by N. VAN KAMPEN (Elsevier, Amsterdam,\n2007) third edition ed., pp. 193–218.\n[49] M. Muhsin and M. Sahoo, Inertial active ratchet: Simu-\nlation versus theory, Phys. Rev. E 107, 054601 (2023)."    },    {        "title": "2402.05630v1.Strassen_s_algorithm_is_not_optimally_accurate.pdf",        "content": "Strassen’s algorithm is not optimally accurate\nJean-Guillaume Dumas\nUniversité Grenoble Alpes\nUMR CNRS 5224 LJK\n38058 Grenoble, FranceClément Pernet\nUniv. Grenoble Alpes, Grenoble INP\nUMR CNRS 5224 LJK\n38058 Grenoble, FranceAlexandre Sedoglavic\nUniversité de Lille\nUMR CNRS 9189 CRISTAL\n59650 Villeneuve d’Ascq, France\nABSTRACT\nWe propose a non-commutative algorithm for multiplying 2×2-\nmatrices using 7coefficient products. This algorithm reaches si-\nmultaneously a better accuracy in practice compared to previously\nknown such fast algorithms, and a time complexity bound with\nthe best currently known leading term (obtained via alternate basis\nsparsification). To build this algorithm, we consider matrix and ten-\nsor norms bounds governing the stability and accuracy of numerical\nmatrix multiplication. First, we reduce those bounds by minimizing\na growth factor along the unique orbit of Strassen’s 2×2-matrix\nmultiplication tensor decomposition. Second, we develop heuristics\nfor minimizing the number of operations required to realize a given\nbilinear formula, while further improving its accuracy. Third, we\nperform an alternate basis sparsification that improves on the time\ncomplexity constant and mostly preserves the overall accuracy.\nCCS CONCEPTS\n•Computing methodologies →Linear algebra algorithms .\n1 INTRODUCTION\nThe first non-commutative algorithm for multiplying 2×2-matrices\nusing 7coefficient products was discovered by Strassen [ 20]. It was\nsubsequently proven that all such algorithms with 7multiplications\nall lie in a single isotropy orbit on Strassen bilinear tensor decompo-\nsition [ 13]. We here study the numerical accuracy of the recursive\napplication of these 2×2algorithms.\nWe first propose a unified accuracy analysis of such recursive\nalgorithms, generalizing some and improving on other state of the\nart bounds [ 1,2,4,7,8,10]. Following the approach of [ 4], we\nthen seek to optimize the growth factor, a parameter governing the\naccuracy in these bounds, over Strassen’s orbit. Since the max-norm,\nproducing the sharpest bounds, precludes smooth optimization, we\nrelax the problem to optimizing a weaker growth factor in the\nFrobenius norm, which will later demonstrate to better reflect the\nobserved practical accuracy.\nThe most accurate variants are then obtained from these bilin-\near formulas by minimizing the number of operations required\nto realize them. Our heuristics for this, make use of common sub-\nexpression eliminations with rational coefficients, potential factor-\nization via the kernel of the matrices of the bilinear operators, as\nwell as the Tellegen’s transposition principle.\nWhile preserving the complexity bound exponent of Strassen’s\nalgorithm,𝑛log2(7), those algorithms require slightly more opera-\ntions, thus worsening the constant factor of the leading term. We\ntherefore finally propose further variants obtained by an alternate\nbasis sparsification, similar to those introduced in [ 3,17]. In fine,\nwe obtain variants having a time complexity bound with the best\ncurrently known leading term, that simultaneously improve on theaccuracy (i.e. mostly preserving in practice the numerical accuracy\nwith or without alternate basis sparsification, again thanks to a\nminimization of the number of operations required to realize them).\nOurc++ tools for the minimization of the number of operations\nare gathered in the PLinOpt library [ 11]. We also forked the Mat-\nlab framework of [ 8] to experiment our implementations of the\nresulting fast and accurate matrix multiplication algorithms [12].\nSection 2 presents the symmetries of matrix multiplication ten-\nsors that we will use. In Section 3 we propose the unified error\nbounds on bilinear operators and matrix multiplication algorithms,\nhighlighting how the the growth factor parameter governs accuracy.\nOn a relaxed growth factor in norm 2, we apply, in Section 4, a de-\nscent algorithm to reach some local minima and show in Section 5\nthat it lies within at most 2.6%of the optimal. Finally, Section 6\npresents our minimization heuristics and the obtained matrix mul-\ntiplication algorithms, and their associated accuracy benchmarks.\n2 MATRIX PRODUCT SEEN AS TENSOR\nWe recall there the formalism of tensor decomposition allowing\nto present clearly the symmetries, later used to search for more\nnumerically accurate fast matrix multiplication algorithms in Sec-\ntion 4. We start by briefly recalling tensorial representation of\nbilinear maps, through the example introduced by Strassen in [ 20]\nof fast 2×2-matrix product and we refer to [ 18] for this framework.\nThe product C=A·Bof2×2matrices could be computed by\nStrassen algorithm using the following computations:\n𝜌1←𝑎11(𝑏12−𝑏22), 𝜌 4←(𝑎12−𝑎22)(𝑏21+𝑏22),\n𝜌2←(𝑎11+𝑎12)𝑏22, 𝜌 5←(𝑎11+𝑎22)(𝑏11+𝑏22),\n𝜌3←(𝑎21+𝑎22)𝑏11, 𝜌 7←(𝑎21−𝑎11)(𝑏11+𝑏12),\n𝜌6←𝑎22(𝑏21−𝑏11),\u0002𝑐11𝑐12𝑐21𝑐22\u0003\n=h𝜌5+𝜌4−𝜌2+𝜌6𝜌6+𝜌3𝜌2+𝜌1𝜌5+𝜌7+𝜌1−𝜌3i\n.(1)\nThis straight-line program (a.k.a. slp) encodes the following\nbilinear map over a field Kwith𝑚,𝑘,𝑛 equal to 2:\n𝛽MM(𝐴,𝐵):K𝑚×𝑘×K𝑘×𝑛→K𝑚×𝑛,\n(A,B) → A·B.(2)\nIndices𝑚,𝑘,𝑛 are kept in this section for the sake of clarity in order\nto distinguish easily the different spaces involved in the sequel.\nDefinition 1. The spaces K·×·can be endowed with the classical\nFrobenius inner product ⟨M,N⟩=Trace(M⊺·N)that establishes\nan isomorphism between K·×·and its dual space\u0000K·×·\u0001∗.\nFrobenius inner product combines matrix product (2) and the\ntrilinear form Trace (C⊺·A·B)as follow:\nS|3:K𝑚×𝑘×K𝑘×𝑛×(K𝑚×𝑛)∗→ K,\n(A,B,C⊺) → ⟨ C,A·B⟩.(3)\nAs the space of trilinear forms is the canonical dual space of order\nthree tensor products, Strassen algorithm (1) is encoded as thearXiv:2402.05630v1  [math.NA]  8 Feb 2024Jean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic\ntensor decomposition Sof the matrix multiplication tensor in sum\nof seven rank-one tensors defined by the following relations:\nS=Í7\n𝑖=1M𝑖⊗N𝑖⊗O𝑖=\u00021 0\n0 1\u0003\n⊗\u00021 0\n0 1\u0003\n⊗\u00021 0\n0 1\u0003\n+\u00020 1\n0−1\u0003\n⊗\u00020 0\n1 1\u0003\n⊗\u00021 0\n0 0\u0003\n+\u0002−1 0\n1 0\u0003\n⊗\u00021 1\n0 0\u0003\n⊗\u00020 0\n0 1\u0003\n+\u00021 1\n0 0\u0003\n⊗\u00020 0\n0 1\u0003\n⊗\u0002−1 0\n1 0\u0003\n+\u00021 0\n0 0\u0003\n⊗\u00020 1\n0−1\u0003\n⊗\u00020 0\n1 1\u0003\n+\u00020 0\n0 1\u0003\n⊗\u0002−1 0\n1 0\u0003\n⊗\u00021 1\n0 0\u0003\n+\u00020 0\n1 1\u0003\n⊗\u00021 0\n0 0\u0003\n⊗\u00020 1\n0−1\u0003(4)\nin(K𝑚×𝑘)∗⊗(K𝑘×𝑛)∗⊗K𝑚×𝑛with𝑚=𝑘=𝑛=2. In the above\ntensor decomposition, each summands is a rank-one tensor and its\ntensor rank is the number 𝑟of such element ( 7there). Given Equa-\ntion (4), one can retrieve a multiplication formula (2) implemented\nby Eq. (1) using the third 2-contraction of the tensor S⊗A⊗B\nwhich is defined as the following map:\n\u0010\n(K𝑚×𝑘)∗⊗(K𝑘×𝑛)∗⊗K𝑚×𝑛\u0011\n⊗\u0010\nK𝑚×𝑘⊗K𝑘×𝑛\u0011\n→K𝑚×𝑛,\u0000Í\n𝑖M𝑖⊗N𝑖⊗O𝑖\u0001⊗(A⊗B)→Í\n𝑖⟨M𝑖,A⟩⟨N𝑖,B⟩O𝑖.(5)\nMatrix product tensor decompositions could be represented using\nother formalisms more adapted to the design of algorithm com-\nputing efficiently the matrix product (as shown in Section 6). For\nexample, a nice concise representation was introduced in [ 16]; it\nencodes the sum of rank-one tensors by three matrices as done\nfor the Strassen tensor decomposition (4) in the following three\nmatrices LS,RSandPS:\n1 0 0 1\n0 1 0−1\n−1 0 1 0\n1 1 0 0\n1 0 0 0\n0 0 0 1\n0 0 1 1,1 0 0 1\n0 0 1 1\n1 1 0 0\n0 0 0 1\n0 1 0−1\n−1 0 1 0\n1 0 0 0,1 0 0 1\n1 0 0 0\n0 0 0 1\n−1 1 0 0\n0 1 0 1\n1 0 1 0\n0 0 1−1⊺\n. (6)\nNotation 2. Given a𝑚×𝑘-matrix𝐴, we denote by 𝐴𝑖the𝑖-th\nrow and by vec𝐴the row-major vectorization of this matrix, i.e. the\nvector𝑣inR𝑚𝑘such that𝑣𝑖𝑘+𝑗=𝑎𝑖,𝑗. We also denote by Mat𝑚,𝑘(𝑣)\nthe reciprocal operation, building an 𝑚×𝑘matrix from an 𝑚𝑘-\ndimensional vector. Thus, the 𝑖th line LS𝑖(resp. RS𝑖) of matrix LS\n(resp. RS) is the transposition of the row-major vectorization vecM𝑖\nof the first (resp. second vecN𝑖) component of the 𝑖th triad in Equa-\ntion (4) and the 𝑖th column of matrix PSis the column-major vec-\ntorization vec O𝑖of its third component.\nDefinition 3. This encoding of a tensor by three suitable matri-\ncesL,R,Pis called a hmrepresentation and is denoted by [L;R;P].\nEquation (11) presented in Section 3 shows that the hmrepresen-\ntation allows to construct slps for the associated algorithms. We\nshow in Section 6.2 that this could be done efficiently, e.g., using\nthe kernel of L(resp. R) and Tellegen transposition applied to P.\nNow we turn to symmetries of matrix product tensor decom-\nposition. Indeed, remark that the matrix product is associated\ntoTrace(A·B·C)by Equation (3) and that, given invertible matri-\ncesU,V,Wof suitable sizes and the trace classical properties, this\ntrace is equal to:\nTrace\u0000(A·B·C)⊺\u0001=Trace(C·A·B)=Trace(B·C·A)\nand to Trace\u0000U−1·A·V·V−1·B·W·W−1·C·U\u0001.(7)\nThese relations illustrate the next theorem and induce the isotropy\naction on matrix product tensor decomposition presented below:Theorem 4 ([ 13, § 2.8]). The isotropy group of the 𝑚×𝑚matrix\nmultiplication tensor is the semidirect product psl±(K𝑚)×3⋊𝔖3,\nwhere pslstands for the group of matrices of determinant ±1and𝔖3\nfor the symmetric group on 3elements.\nLemma 5. Letgdenotes(U×V×W)inpsl±(K𝑚)×3andTa\nrank-one tensor A⊗B⊗C; the action g⋄TofgonTis the rank-\none tensor(U−⊺·A·V⊺)⊗(V−⊺·B·W⊺)⊗(W−⊺·C·U⊺). This\naction is extended by additivity on higher tensor rank tensors.\nGiven two isotropies 𝑔1defined by(u1×v1×w1)and𝑔2defined\nby(u2×v2×w2)both in psl±(K𝑚)×3, the composition 𝑔1◦𝑔2is\ngiven by(u1·u2×v1·v2×w1·w2).\nThe above isotropies action description is based on classical ten-\nsor decomposition, the corresponding action on hmrepresentation\nis a direct consequence and presented in the following lemma.\nLemma 6. Letgbe(U×V×W)inpsl±(K𝑚)×3and[L;R;P]be\nahmrepresentation of a matrix product tensor decomposition, the\naction g⋄[L;R;P]ofgon[L;R;P]is another hmrepresentation of\na matrix product tensor decomposition defined by:\n\u0002\nL·\u0000V⊺⊗U−1\u0001;R·\u0000W⊺⊗V−1\u0001;\u0000U⊗W−⊺\u0001·P\u0003\n. (8)\nDealing with a tensor decomposition or with the associated hm\nrepresentation is not strictly equivalent; In Lemma 5 there is no\nneed to care about the determinant of matrices (U,V,W)while this\nfact is no more true for Equation (8) as (say) Uacts on two different\ncomponents. The following theorem recalls that in fact all 2×2-\nmatrix product algorithms with 7coefficient multiplications are\nobtained by this single orbit of the action of isotropies on Strassen\ntensor decomposition:\nTheorem 7 ([ 14, § 0.1]). The group psl±(K𝑚)×3acts transitively\non the variety of optimal algorithms for the computation of 2×2-\nmatrix multiplication.\nThus, isotropy action on Strassen tensor decomposition may\ndefine other matrix product algorithm of same tensor rank but with\npotentially interesting characteristics as shown in Section 4. We\nexplicit these properties in the following section.\n3 BILINEAR OPERATOR ACCURACY BOUND\nWe will consider that any real finite-dimensional vector space U, is\nequipped with a norm ∥·∥and denote by∥·∥∗the related dual norm;\nfor𝜙:U→R, its norm∥𝜙∥∗issup(|𝜙(𝑣)|:∥𝑣∥≤1). For instance\nthe max-norm∥·∥∞and the one-norm ∥·∥1are dual one with the\nother, while the two-norm ∥·∥2is self-dual. We will also denote\nthe Hamming weight #{𝑖|𝑥𝑖≠0}of𝑥by∥𝑥∥0. We will denote by\n(𝑥𝑖)𝑖the vector formed by the coefficients 𝑥𝑖and∥(𝑥𝑖)𝑖∥its norm.\nBy extension, we also use ∥𝑥;𝑦∥=∥𝑥∥·∥𝑦∥and∥[L;R;P]∥=\n∥L∥∥R∥∥P∥.\nLemma 8. For any vectors 𝑥and𝑦inR𝑘and a matrix AinR𝑚×𝑘\nthe following inequalities hold:\n|𝑥·𝑦| ≤∥𝑥∥∗∥𝑦∥,∥A𝑥∥≤\r\r(∥A𝑖∥∗)𝑖\r\r∥𝑥∥, (9)\n∥A𝑥∥∞≤ max\n𝑖=1...𝑚\u0010Í𝑘\n𝑗=1|𝑎𝑖,𝑗|\u0011\n∥𝑥∥∞≤𝑘∥A∥∞∥𝑥∥∞.(10)Strassen’s algorithm is not optimally accurate\nGiven an hmrepresentation[L;R;P]of a matrix multiplication\ntensor decomposition, one can retrieve the transpose of the multi-\nplication formula (2) implemented by Eq. (1) using the Hadamard\nproduct A⊙Bof matrices AandBwith the following map:\nK𝑚×𝑘×K𝑘×𝑛→ K𝑚𝑛×1\n(A,B) → P⊺·((L·vecA)⊙(R·vecB)).(11)\nHence, we express there a bilinear operator 𝛽:R𝑒×R𝑓→R𝑔\nusing its hmrepresentation[L;R;P]inR𝑟×𝑒×R𝑟×𝑓×R𝑟×𝑔as:\n𝛽(𝑢,𝑣)=Í𝑟\n𝑖=1(L𝑖·𝑢)(R𝑖·𝑣)(P⊺)𝑖. When this operator encodes\nan𝑚×𝑘by𝑘×𝑛matrix multiplication formula, we will thus de-\nnote it by𝛽MMand we will have 𝑒=𝑚𝑘,𝑓=𝑘𝑛,𝑔=𝑚𝑛. We also\nconsider recursive applications of such operators defined as:\n𝛽(ℓ):R𝑒0𝑒ℓ×R𝑓0𝑓ℓ→R𝑔0𝑔ℓ\n(𝑢,𝑣) ↦→Í𝑟\n𝑖=1𝛽(ℓ−1)(L𝑖·𝑢,R𝑖·𝑣)(P⊺)𝑖(12)\nand𝛽0:R𝑒0×R𝑓0→R𝑔0, a bilinear operator which we will assume\nto be bounded:∥𝛽(0)(𝑢,𝑣)∥≤𝛾0∥𝑢∥∥𝑣∥∀(𝑢,𝑣)∈R𝑒0×R𝑓0. For\nconvenience, we will define the dimensions 𝐺=𝑔ℓ𝑔0, and𝐾=𝑘ℓ𝑘0.\nRecall that(P⊺)𝑖is the𝑖-th column of Pand remark that L𝑖·𝑢is\nan abuse of notation for the operation where each coefficient L𝑖,𝑗\nmultiplies a block of 𝑒0𝑒ℓ−1contiguous coefficients of 𝑢, namely:\nL𝑖·𝑢=(LiMate,e0eℓ−1(u))⊺.\nWe will consider the floating point arithmetic in the standard\nmodel of [ 15]:b𝑥denotes the computed value for an expression 𝑥\nsuch that 𝑎op𝑏=(𝑎op𝑏)(1+𝛿)for op =+,−,×,/where|𝛿|≤𝜀,\nthe unit round off, except when 𝑎op𝑏is0where𝛿is−1. We recall\nin the following Lemma some classical inequalities:\nLemma 9 (see [ 8, Eq. (3.5)] and [ 15, Eq. (4.4)]). For any vectors 𝑢\nand𝑣inR𝑛the following inequalities hold:\n|d𝑢·𝑣−𝑢·𝑣|≤∥𝑢∥0∥𝑢∥∗∥𝑣∥𝜀+𝑂(𝜀2). (13)\n\f\f\fÍ𝑛\n𝑖=1𝑢𝑖−Í𝑛\n𝑖=1𝑢𝑖\f\f\f≤(𝑛−1)\u0000Í𝑛\n𝑖=1|𝑢𝑖|\u0001𝜀+𝑂(𝜀2). (14)\nProof. Since there are 𝑟=∥𝑢∥0non-zero coefficients in 𝑢, the\nscalar product is actually computed between 𝑟-dimensional vectors.\nThe result is then as in [ 8, Eq. (3.5)]. Applying Eq. (13) with 𝑣=\n(1,..., 1)and the max-norm gives Eq. (14). □\nDefinition 10. Thegrowth factor of the formula[L;R;P]comput-\ning the bilinear form 𝛽is defined by 𝛾=max\n𝑗=1...𝑔Í𝑟\n𝑖=1∥L𝑖∥∗∥R𝑖∥∗|𝑝𝑖,𝑗|.\nThe growth factor not only bounds the values of bilinear oper-\nators, as show in Lemma 11, but is also central in analyzing their\nforward numerical error, which will be the focus of Theorem 12.\nLemma 11. For any𝑢,𝑣with adequate dimensions, the following\nrelations hold:∥𝛽(𝑢,𝑣)∥≤𝛾∥𝑢∥∥𝑣∥,∥𝛽(ℓ)(𝑢,𝑣)∥≤𝛾ℓ𝛾0∥𝑢∥∥𝑣∥\nand∥𝛽(ℓ)\nMM(𝑢,𝑣)∥∞≤𝑘ℓ𝑘0∥𝑢∥∞∥𝑣∥∞.\nProof. LetDjdenotes Diag𝑖=1...𝑟(𝑝𝑖,𝑗)and𝑐𝑗be the𝑗-th coeffi-\ncient of𝛽(𝑢,𝑣). We have that|𝑐𝑗|≤\r\ru⊺L⊺D𝑗Rv\r\r≤\r\ru⊺LD𝑗R\r\r∗∥𝑣∥,\nso that|𝑐𝑗|≤\r\rL⊺D𝑗R\r\r∗∥𝑢∥∥𝑣∥≤\r\rÍ𝑟\n𝑖=1(L𝑖⊗R𝑖)𝑝𝑖,𝑗\r\r∗∥𝑢∥∥𝑣∥\nand|𝑐𝑗| ≤\u0000Í𝑟\n𝑖=1∥L𝑖∥∗∥R𝑖∥∗|𝑝𝑖,𝑗|\u0001∥𝑢∥∥𝑣∥. Finally, the last in-\nequality follows from (9). □Theorem 12. Given any choice of norm ∥·∥, if𝐺denotes𝑔0𝑔ℓ\nand𝐾denotes𝑘0𝑘ℓ, the error in computing 𝛽(ℓ)is of the form\n∥b𝛽(ℓ)(𝑢,𝑣)−𝛽(ℓ)(𝑢,𝑣)∥∞≤𝜅∥𝑢∥∥𝑣∥𝜀+𝑂(𝜀2)where either\n𝜅=(𝐾/𝑘0)log𝑘𝛾\u0012\n𝑘2\n0+𝑄0𝛾\n𝛾−𝑘𝑘0\u0013\n−𝑄0𝛾\n𝛾−𝑘𝐾 (15)\nwhen𝛽(ℓ)is an𝑀×𝐾by𝐾×𝑁matrix multiplication, or\n𝜅=(𝐺/𝑔0)log𝑔𝛾𝛾0\u0010\n1+𝑄0\u0010\nlog𝑔(𝐺/𝑔0)+1\u0011\u0011\n, (16)\notherwise, and 𝑄0=max𝑗\u0000\r\r(P⊺)𝑗\r\r\n0+max𝑖(∥L𝑖∥0+∥R𝑖∥0)1𝑝𝑖,𝑗≠0\u0001\nas in [1, Definition 1].\nProof. By induction we will prove that the bound is of the form\n∥Δ𝛽(ℓ)∥∞=∥𝛽(ℓ)(𝑢,𝑣)−𝛽(ℓ)(𝑢,𝑣)∥∞≤𝑡ℓ∥𝑢∥∥𝑣∥𝜀+𝑂(𝜀2), clar-\nifying in the process the value for 𝑡ℓ. Consider the block 𝑐𝑗of𝐺/𝑔=\n𝑔0𝑔ℓ−1consecutive coefficients of the output: 𝑐𝑗=Í𝑟\n𝑖=1H𝑖𝑝𝑖,𝑗, where\nH𝑖=𝛽(ℓ−1)(L𝑖·𝑢,R𝑖·𝑣). Let𝑑𝑖,𝑗=H𝑖𝑝𝑖,𝑗andΔ𝑑𝑖,𝑗=c𝑑𝑖,𝑗−𝑑𝑖,𝑗.\nThen, by Eq. (14):\n∥b𝑐𝑗−𝑐𝑗∥∞≤∥𝑟∑︁\n𝑖=1c𝑑𝑖,𝑗−𝑟∑︁\n𝑖=1c𝑑𝑖,𝑗∥∞+∥𝑟∑︁\n𝑖=1c𝑑𝑖,𝑗−𝑟∑︁\n𝑖=1𝑑𝑖,𝑗∥∞(17)\n≤𝑟∑︁\n𝑖=1∥H𝑖𝑝𝑖,𝑗∥∞\u0000\r\r(P⊺)𝑗\r\r\n0−1\u0001𝜀+𝑟∑︁\n𝑖=1∥Δ𝑑𝑖,𝑗∥∞+𝑂(𝜀2) (18)\nSimilarly,\n∥Δ𝑑𝑖,𝑗∥∞≤∥𝑝𝑖,𝑗bHi−𝑝𝑖,𝑗cH𝑖∥∞+∥𝑝𝑖,𝑗cH𝑖−𝑝𝑖,𝑗H𝑖∥∞ (19)\n≤|𝑝𝑖,𝑗|∥H𝑖∥∞𝜀+|𝑝𝑖,𝑗|∥ΔH𝑖∥∞+𝑂(𝜀2), (20)\nand again by bilinearity of 𝛽(ℓ−1), the following equality holds:\n∥ΔH𝑖∥∞=∥𝛽(ℓ−1)(L𝑖·𝑢,R𝑖·𝑣)−𝛽(ℓ−1)(L𝑖·𝑢,R𝑖·𝑣)∥∞, and this\nquantity is bounded by:\n∥Δ𝛽(ℓ−1)∥∞+∥𝛽(ℓ−1)(Δ𝐿,R𝑖·𝑣)∥∞+∥𝛽(ℓ−1)(L𝑖·𝑢,Δ𝑅)∥∞(21)\nBy Eq. (13) and the induction hypothesis we have\n∥ΔM∥∞≤∥M𝑖∥0∥M𝑖∥∗∥𝑢∥𝜀+𝑂(𝜀2)with M∈{L,R},(22)\n∥Δ𝛽(ℓ−1)∥∞≤𝑡ℓ−1∥L𝑖·𝑢+ΔL∥∥R𝑖·𝑣+ΔR∥𝜀+𝑂(𝜀2), (23)\n≤𝑐ℓ−1∥L𝑖∥∗∥𝑢∥∥R𝑖∥∗∥𝑣∥𝜀+𝑂(𝜀2). (24)\nBy Lemma 11, the following inequality holds\n∥𝛽(ℓ−1)(ΔL,R𝑖·𝑣)∥∞≤Θℓ−1Θ0∥ΔL∥∞∥R𝑖∥∗∥𝑣∥ (25)\nfor(Θ,Θ0)=(𝑘,𝑘0)if𝛽=𝛽MMor(𝛾,𝛾0)otherwise (where Θ0=𝛾0\ncomes from the current proof with ℓ=1and𝑔0=1). Similarly,\n∥𝛽(ℓ−1)(L𝑖·𝑢,ΔR)∥∞≤Θℓ−1Θ0∥ΔR∥∞∥L𝑖∥∗∥𝑢∥.(26)\nGathering Eqs. (18), (20) to (22) and (24) to (26) we deduce that\n∥b𝑐𝑗−𝑐𝑗∥∞≤Í𝑟\n𝑖=1(Θℓ−1Θ0(∥L𝑖∥0+∥R𝑖∥0+\r\r(P⊺)𝑗\r\r\n0)+𝑡ℓ−1)\n×∥L𝑖∥∗∥R𝑖∥∗|𝑝𝑖,𝑗|∥𝑢∥∥𝑣∥𝜀+𝑂(𝜀2),\nand thus that∥b𝑐𝑗−𝑐𝑗∥∞≤(Θℓ−1Θ0𝑄0+𝑡ℓ−1)𝛾∥𝑢∥∥𝑣∥𝜀+𝑂(𝜀2).\nAs in [15], we deduce that 𝑡ℓmust then satisfy:\n \n𝑡ℓ=(Θℓ−1Θ0𝑄0+𝑡ℓ−1)𝛾forℓ>0,\n𝑡0=𝑘2\n0for matrix product,\n𝑡0=(1+𝑄0)𝛾0 otherwise.(27)Jean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic\nFormula Applies to norm Winograd Strassen Equation (35) Equation (34)\n𝑄 𝛾 𝑄 𝛾 𝑄 𝛾 𝑄 𝛾 𝑄 𝛾\nBrent [7] (15) Strassen only ∞ NA𝛾1,1,∞ 3.67 12\nBL [4] DDHK [10] (16) any MM alg. ∞𝑄′\n01𝛾0,1,∞29 18 7 12 9.59 40 9.81 98.54\nHigham [15] (15) S. & W. only ∞ NA𝛾1,1,∞ 4.94 18 3.83 12\nBallard et al. [1] (16) any MM alg. ∞𝑄0𝛾1,1,∞ 10 18 8 12 12 13 15 17.48\nDai, Lim [8] [8, Th 3.3] ℓ=1, any alg.2𝑚+𝑛+𝑟 𝛾 2,1 15 17.86 15 14.83 15 12.21 15 12.07\n∞𝑚+𝑛+𝑟 𝛾 1,∞,1 15 27 15 20 15 22 15 25.14\nHere (15) any MM alg.2𝑄0𝛾2,1,∞ 10 8 8 6.83 12 6.05 15 5.97\n∞𝑄0𝛾1,1,∞ 10 18 8 12 12 13 15 17.48\nHere (16) any alg. ∞𝑄0𝛾1,1,∞ 10 18 8 12 12 13 15 17.48\nTable 1: Comparing accuracy formulas for recursive bilinear matrix multiplication operators in the form of Theorem 12.\nThis recurrence relation solves into 𝑡ℓ=𝛾ℓ𝑡0+𝑄0Θ0ΘℓÍℓ\n𝑖=1(𝛾/Θ)𝑖.\nIn the case of a matrix multiplication operator, 𝑡ℓis equal to:𝛾ℓ𝑘2\n0+\n𝑄0𝑘0𝛾𝛾ℓ−𝑘ℓ\n𝛾−𝑘=\u0010\n𝐾\n𝑘0\u0011log𝑘𝛾\u0010\n𝑘2\n0+𝑄0𝛾\n𝛾−𝑘𝑘0\u0011\n−𝑄0𝛾\n𝛾−𝑘𝐾. In the general\ncase, the value of 𝑡ℓbecomes:𝛾ℓ𝛾0(1+𝑄0)+𝑄0𝛾0𝛾ℓℓ, that is:\n(𝐺/𝑔0)log𝑔𝛾𝛾0\u0010\n1+𝑄0\u0010\nlog𝑔𝐺/𝑔0+1\u0011\u0011\n. □\nTheorem 12 generalizes or improves on previous similar results\nin [1,7,8,10,15]. In particular, [ 8] only considers one recursive level\nwith no base case; [ 7,15] have tight bounds but only for Strassen\nand Winograd’s algorithms in max-norm; and lastly [1, 10] has an\nadditional logarithmic factor likely due to a loser bound on each\n∥𝛽(ℓ−1)∥∞, not exploiting the fact that they are matrix products.\nEventhough the choice of the max-norm produces the tightest\nbounds in Theorem 12, as in most of previous works, the bounds are\nstated for any choice of norm, as in [ 8], for alternative norms, such\nas the 2-norm, may give growth factor expressions more amenable\nto optimizations, as detailed in the following section.\nTable 1 compares the various existing bounds on numerical\naccuracy of matrix multiplication algorithms. These bounds depend\non the following choices made on the norms to define the growth\nfactor𝛾:\n𝛾0,1,∞=\r\r\r\u0010\r\r\u0000∥L𝑖;R𝑖∥0|𝑝𝑖,𝑘|\u0001\n𝑖\r\r\n1\u0011\n𝑘\r\r\r\n∞∥L∥∞∥R∥∞∥P∥∞\n=\u0012\nmax\n𝑘∈{1...𝑚𝑛}Í𝑟\n𝑖=1∥L𝑖∥0∥R𝑖∥0|𝑝𝑖,𝑘|\u0013\n∥L∥∞∥R∥∞∥P∥∞(28)\n𝛾2,1=\r\r(∥L𝑖;R𝑖;P𝑖∥2)𝑖\r\r1=Í𝑟\n𝑖=1∥L𝑖∥2∥R𝑖∥2∥P𝑖∥2 (29)\n𝛾𝑞,1,∞=\r\r\r\r\u0010\r\r\r\u0010\n∥L𝑖;R𝑖∥𝑞|𝑝𝑖,𝑘|\u0011\n𝑖\r\r\r\n1\u0011\n𝑘\r\r\r\r∞\n= max\n𝑘∈{1...𝑚𝑛}Í𝑟\n𝑖=1∥L𝑖∥𝑞∥R𝑖∥𝑞|𝑝𝑖,𝑘|with𝑞∈{1,2}.(30)\n4 GROWTH FACTOR ALONG ORBITS\nIn the footstep of [ 4], we aim to find alternative 2×2matrix prod-\nuct tensor decomposition, in the orbit of Strassen’s one, with im-\nproved accuracy, hence minimizing the Growth factor. The use of\nthe maxnorm induces an expression Eq. (30) for 𝛾1,1,∞poorly suited\n1[4,10] reach an improved value of 𝑄0by assuming all additions are performed\nfollowing a balanced tree, instead of a worst case estimate as done in all other formulas.\n2We applied the same 𝛾0,1,∞for [4], as it seems to be missing a dependency in the\nmagnitude of the coefficients in L,R,P, which was fixed in [10].for optimizations. We will instead make two relaxations: first, using\nthe2-norm, and second, as in [8] bounding 𝛾2,1,∞by𝛾2,1:\nmax\n𝑘∈{1...𝑚𝑛}𝑟∑︁\n𝑖=1∥L𝑖∥2∥R𝑖∥2|𝑝𝑖,𝑘|≤𝑟∑︁\n𝑖=1∥L𝑖∥2∥R𝑖∥2∥P𝑖∥2(31)\nTheorem 7 shows that all fast 2×2matrix product algorithms are\nin the same orbit under isotropies action introduced in Lemma 5.\nWhile the tensor rank is invariant under this action, growth factor\nis—generally—not. As its definition is based on Frobenius norm,\nsome isotropies leave it invariant as stated next:\nLemma 13. The growth factor 𝛾2,1is invariant under the action\nof the semidirect product so±(K𝑛)×3⋊𝔖3induced by the special\northogonal group and the permutation group 𝔖3.\nProof. By Definition 1, Frobenius norms are invariant under\northogonal transformations and so is 𝛾2,1by Eq. (29). Lemma 13 is\nthen derived from Equations (7) and (8). □\nAs it is useless to consider isotropies leaving the growth factor\ninvariant, we limit our search to isotropies of the following form:\nLemma 14. The action of(h×p)×3determines the growth factor\n𝛾2,1for h={H𝜌=h𝜌0\n0 1/𝜌i\n|𝜌>0}and p={P𝜉=h\n1𝜉\n0 1i\n|𝜉∈R}.\nProof. Considering Eq. (8), we see that the product of any ac-\ntion, say U, by a non-zero scalar will affect the growth factor ex-\nactly twice: once in Uand once, inverted, in U−1, as norms are\nabsolutely homogeneous. It is therefore sufficient to consider ma-\ntrices with determinant 1. Then, Lemma 13 states that orthogonal\nmatrices also do not have any effect. From the qrdecomposition of\nany invertible matrices there remains thus just the (h×p)part of\nthe Iwasawa decomposition, for each of the three transformations\nin Theorem 7. □\nWe should study the action of (h×p)×3on Strassen tensor de-\ncomposition in order to find variants with the smaller possible 𝛾2,1.\nUnfortunately, a direct definitive result for this question seems to be\nout of reach and we present several ersatz. First, we perform numer-\nical minimization on 𝛾(g⋄S)with a completely generic isotropy g\ninpsl±(R2)×3(involving 6indeterminates by Lemma 14); this ex-\nperiment suggests that a suitable isotropy to reach a fast matrix\nproduct tensor decomposition with minimal 𝛾2,1could be of the\nform(𝑢×𝑢×𝑢)(involving only 2indeterminates). The followingStrassen’s algorithm is not optimally accurate\nproposition states precisely this possibility (its proof is a simple\ncomputation presented in Appendix A.1).\nProposition 15. Consider the matrices 𝑢(𝜌,𝜉)=H𝜌·P𝜉and the\nisotropies g𝜌,𝜉defined by 𝑢(𝜌,𝜉)×3. The minimal value on the\norbit g𝜌,𝜉⋄Sof the growth factor 𝛾2,1\u0000g𝜌,𝜉⋄S\u0001is reached at the\npoint(𝜌,𝜉)=\u00004√︁\n4/3,−1/2\u0001and equal to 4/√\n2+16/√\n3>12.06603 .\nThe algorithm corresponding to the point (𝜌,𝜉)with minimal\n𝛾2,1on this restricted orbit is given in Eq. (34). We gather in Ta-\nble 2 values for 𝛾2,1of some matrix product tensor decompositions\ntogether with the result obtained in Proposition 15. In Section 6,\nwe compare the implementation of algorithms associated to these\ntensor decompositions in order to confirm that their numerical\naccuracy is correlated to their respective 𝛾2,1growth factor.\n5 UPPER AND LOWER BOUNDS\nWe explore in this section some bounds on the norm of each compo-\nnent of a hmrepresentation. By the multiplicativity of 𝐿𝑝,𝑞norms\n(even generalized to negative Hölder conjugates), this will always\ngive alternate bounds on the error, a priori less accurate, but poten-\ntially easier to apprehend.\nLemma 16. For any hmrepresentationH, with matrices L,R,P\ninK𝑟×𝑛, let𝛾H=𝛾2,1(H) be its𝛾2,1growth factor, as in Eq. (29).\nThen for any strictly positive 𝑦and𝑧, we have both:\n𝛾H≤∥H∥2,3≤∥H∥𝐹 and (32)\nmax\u001a\n𝑟1+3𝑧∥H∥2,−1\n𝑧;∥L∥2,−1\n𝑦·∥R∥2,−1\n𝑧·∥P∥2,1\n1+𝑦+𝑧\u001b\n≤𝛾H(33)\nProof. Let𝑎𝑖=∥L𝑖∥2,𝑏𝑖=∥R𝑖∥2and𝑐𝑖=∥P𝑖∥2. The first\nright hand side inequality is Hölder’s inequality on 𝑎𝑖,𝑏𝑖and𝑐𝑖,\nwith the Hölder conjugates1\n3+1\n3+1\n3=1, that is:∥(𝑎𝑖·𝑏𝑖·𝑐𝑖)𝑖∥1≤\n∥(𝑎𝑖)𝑖∥3·∥(𝑏𝑖)𝑖∥3·∥(𝑐𝑖)𝑖∥3=∥H∥2,3The second right hand side\ninequality is a direct application of the monotonicity of norms.\nThen, the left hand side inequality is obtained by a reverse Hölder’s\ninequality on the vectors 𝑎𝑖,𝑏𝑖,𝑐𝑖and 1, with the Hölder con-\njugates1\n−1/𝑧+1\n−1/𝑧+1\n−1/𝑧+(1+3𝑧)=1. We have indeed that\nthe(1+3𝑧)-norm∥(1)𝑖∥1/(1+3𝑧)is\u0000Í𝑟\n𝑖=111/(1+3𝑧)\u00011+3𝑧. Combined\nwith:∥H∥2,−1\n𝑧=∥(𝑎𝑖)𝑖∥−1\n𝑧·∥(𝑏𝑖)𝑖∥−1\n𝑧·∥(𝑐𝑖)𝑖∥−1\n𝑧, this shows that\n𝑟1+3𝑧∥H∥2,−1\n𝑧≤∥(𝑎𝑖·𝑏𝑖·𝑐𝑖·1)𝑖∥1. Finally, for the other lhs, we also\nuse Hölder’s inequality on 𝑎𝑖,𝑏𝑖and𝑐𝑖, now with Hölder conju-\ngates(1/−1/𝑦)+(1/−1/𝑧)+(1/1/(1+𝑦+𝑧))=1.□\nTable 2 gives the Frobenius and (2,3)-norms of each of three ma-\ntrices defining the hmrepresentation of a matrix product algorithm,\nas well as their 𝛾2,1growth factor. In the following proposition, we\nshow that—up to orthogonal transformations—the minimum of the\nFrobenius norm of the hmrepresentation defining a fast 2×2-matrix\nmultiplication algorithms is√\n10and subsequent results.\nProposition 17. The minimal product ∥H∥𝐹of the three Frobe-\nnius norms of the hmrepresentation of any bilinear algorithm for\nmatrix multiplication with 7multiplications, is√\n103.Algorithm 𝛾2,1(H)∥H∥2,3∥H∥𝐹\nWinograd 7+8√\n2+9√\n3≈17.853 11+8√\n2+9√\n3√\n143\nStrassen 12+4√\n2≈14.828 2+20√\n2√\n123\nEq. (35)75\n8+4√\n2≈12.203125\n32+4√\n2+25\n2√\n5√︃\n162\n163\nEq. (40)75\n8+4√\n2≈12.203125\n32+4√\n2+25\n2√\n5√\n10√︃\n162\n16810\n80√\n10\nEq. (34)16√\n3+4√\n2≈12.06616√\n3+4√\n2√\n103\nConv. 8.000 8√\n83\nTable 2: Illustration of Eq. (32) on H=[L;R;P]\nThis proposition’s proof is given in Appendix A.1. Remark that\nthis lower bound is reached, by the algorithm which hmrepresen-\ntation is given in Equation (34).\n√\n3\n21\n21\n2√\n3\n6\n0 0 1−√\n3\n3\n0 1 0√\n3\n3\n0 0 0−2√\n3\n−√\n3\n2−1\n21\n2−√\n3\n2\n−√\n3\n2−1\n21\n2√\n3\n6\n−√\n3\n21\n21\n2−√\n3\n6;02√\n30 0\n−1√\n3\n30 0\n0√\n3\n30−1\n1\n2−√\n3\n6√\n3\n2−1\n2\n−1\n2√\n3\n2−√\n3\n2−1\n2\n1\n2√\n3\n6√\n3\n21\n2\n1\n2√\n3\n6−√\n3\n2−1\n2;√\n3\n61\n21\n2√\n3\n2\n−√\n3\n30−1 0\n√\n3\n3−1 0 0\n√\n3\n6−1\n2−1\n2√\n3\n2√\n3\n2−1\n21\n2√\n3\n2\n−√\n3\n6−1\n21\n2√\n3\n2\n−2√\n30 0 0⊺\n(34)\nRemark 18. Similarly, the point\u00004√\n3/√\n2,−4√\n3/√\n6,4√\n3/√\n2,4√\n3/√\n6\u0001\nis a minimum of the cube of ∥L·(𝑊⊗𝑉)∥2,3. It turns out that this\nvalue is 16/√\n3+4/√\n2, the same as the 𝛾2,1growth factor at this\npoint, proving that our upper bound is reached.\nWe now turn to potential lower bounds.\nLemma 19. For𝑊=\u0002𝑟 𝑥\n0𝑟−1\u0003and𝑉=h𝑠 𝑦\n0𝑠−1i\n, and Lthe first\ncomponent of Strassen’s hmrepresentation given in Equation (6),\nand any𝑧≥0.5171, the point\u00004√\n3/√\n2,−4√\n3/√\n6,4√\n3/√\n2,4√\n3/√\n6\u0001is\na local minimum of ∥L·(𝑊⊗𝑉)∥2,−1/𝑧as a function of 𝑟,𝑥,𝑠 and𝑦.\nProof. Alike the proof of Proposition 17, we give the explicit\nexpression of∥L·(𝑊⊗𝑉)∥2,−1/𝑧as the function:\n𝑓𝑧(𝑟,𝑥,𝑠,𝑦)=\u0010\n(𝑟2𝑠2+𝑟2𝑦2+𝑥2𝑠2+(𝑥𝑦+1/𝑟𝑠)2)−1/2𝑧+(𝑠2/𝑟2+\n(𝑦/𝑟−1/𝑟𝑠)2)−1/2𝑧+(𝑟2𝑠2+𝑟2𝑦2+(𝑥𝑠+𝑠/𝑟)2+(𝑥𝑦+𝑦/𝑟)2)−1/2𝑧+\n(𝑟2𝑠2+𝑟2𝑦2+𝑠2𝑥2+𝑥2𝑦2)−1/2𝑧+(1/𝑟2𝑠2)−1/2𝑧+(𝑟2/𝑠2+(𝑥/𝑠+\n1/𝑟𝑠)2)−1/2𝑧+(𝑟2𝑠2+(𝑟/𝑠−𝑟𝑦)2+𝑥2𝑠2+(𝑥/𝑠−𝑥𝑦)2)−1/2𝑧\u0011−𝑧\n.\nThen the evaluation of its partial derivatives at the given point is\nzero, by inspection, for any 𝑧inR. Now, the roots of the character-\nistic polynomial of the Hessian of 𝑓𝑧at this point are 3𝑛\n𝑑\u0000𝑏1±√𝛿1\u0001\nand√\n3𝑛\n𝑑\u0000𝑏2±√𝛿2\u0001for𝛿1=−16𝑧(24𝑧−1)𝜏+134431\n𝑧(𝑧2−2/7𝑧+\n13/448)2−1\n𝑧+48𝑧2,𝛿2=24𝑧(272𝑧−237)𝜏+403231\n𝑧(𝑧2−12/7𝑧+\n333/448)2−1\n𝑧+2704𝑧2,𝑛=(2−1\n2𝑧+631\n2𝑧2−1\n𝑧)−𝑧,𝑑=216𝑧𝜏+36𝑧,𝑏1=\n((32𝑧−11)𝜏+4𝑧)√\n3, and𝑏2=(96𝑧−63)𝜏+52𝑧with𝜏=31\n2𝑧2−1\n2𝑧.\nFirst,𝛿1and𝛿2are never negative, so that the roots are always\nreal. Second, both expressions 𝑏𝑖2−𝛿𝑖have the same two roots,\nthe lowest one is 1/4and the second one is strictly less than 0.5171.Jean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic\nThird, all four roots are positive for 𝑧equal to this 0.5171, and, by\ncontinuity, so are they for 𝑧≥0.5171. □\nCorollary 20. 11.7554696 <28\n9211\n1435\n7is a lower bound for the 𝛾2,1\ngrowth factor of an hmformula using 7products.\nProof. Following the proof of Lemma 19, we have that equal-\nity𝑓𝑧\u00104√\n3√\n2,−4√\n3√\n6,4√\n3√\n2,4√\n3√\n6\u0011\n=(2−1\n2𝑧+631\n2𝑧2−1\n𝑧)−𝑧hold. Denoting this\nquantity by 𝜁𝑧we thus have that 71+3𝑧𝜁3𝑧≤71+3𝑧∥H∥2,−1\n𝑧. Which\nlhs limit at𝑧=∞is28\n9211\n1435\n7. By Eq. (33), this show that28\n9211\n1435\n7is\nless or equal to 𝛾𝐹as announced. Similarly, the limit of 𝜁2𝑧𝜁−1−2𝑧≤\n∥L∥2,−1\n𝑧·∥R∥2,−1\n𝑧·∥P∥2,1\n1+𝑧+𝑧, at𝑧=∞, is also28\n9211\n1435\n7.□\nCorollary 20 for instance also shows that the 𝛾2,1growth factor\nof the conventional algorithm ( 8) can not be attained by such fast\nalgorithms. Let see now how this bound behave in our experiments.\n6 ALGORITHMS INTO PRACTICE\nIn this section, we present several techniques to lower the number\nof operations used in our algorithms and thus, lower complexity\nbounds and potentially obtain a better accuracy.\nDetermining actual complexity bounds requires to estimate the\nnumber of operations required to implement a given formula. Con-\nsidering a hmrepresentation, a direct upper bound can be obtained\nby first count the number of coefficients different from −1,0,1to\nget an upper bound on the number of multiplications/divisions, sec-\nond count the number of non-zero coefficients, minus the number of\nrows, to get an upper bound on the number of additions/subtractions.\nTo obtain lower operation counts, we use the following tech-\nniques: first, we select among equivalently accurate algorithms;\nthis is presented in Section 6.1; second, we factor as much as possi-\nble the computations between rows of the hmrepresentations, as\nin Section 6.2; third, we use dependent rows as more opportunities\nfor factorization, as in Section 6.3. We then present some good\ncandidates (as well as in Appendix A.4) and we eventually look at\nsome potential sparse change of basis in Section 6.4.\n6.1 Sparsifying via rotations\nWe have seen in Lemma 13 that orthogonal transformations leave\nthe Frobenius norm invariant and thus, the 𝛾2,1growth factor. There-\nfore, one can apply 4×4generic Kronecker products of orthogo-\nnal2×2(rotation) matrices using Lemma 6 and try to optimize con-\nsidered hmrepresentation for several possible goals: (1) a smaller\nnumber of non-zero coefficients in hmrepresentation components;\n(2) a non-zero pattern better suited to factorization (see the tech-\nnique of Section 6.2); (3) a triangular (sparse) subset of independent\nrows (see the technique of Section 6.3).\nFor instance, to obtain Eq. (34), we solve for the minimal values of\nthe Frobenius norms as in Proposition 17, and then for orthogonal\ntransformations that produce as many vectors of the canonical basis\nas possible. Doing so, we found that with 𝛾2,1set to 16/√\n3+4/√\n2\nandhmrepresentation component Frobenius norms set to√\n10, the\nmaximal possible number of canonical vectors was 1. Equation (34)\nis one of those. Similarly, Eq. (40) is an orthogonal optimization\nof Eq. (35), with one canonical vector in each of components of thehmrepresentation. A c++ implementation of these tools is available\nin the PLinOpt library [11].\n6.2 Factoring heuristics\nFor the implementation of a given linear operator (one of the ma-\ntrices in the hmrepresentation) one can try to find the shortest\nstraight-line program for its computation. The problem is np-hard\nin general (see e.g. [ 6]); but for small matrices, and over the field\nwith 2elements, [ 6] and references therein, propose several heuris-\ntics that potentially reduce the number of operations.\nNot all of them are applicable to fields with more elements\nbut we use a kind of common sub-expression eliminations, the\n“cancellation-free” search, described in Algorithm 6 and imple-\nmented in plinopt/optimizer -D [11].\n6.3 Kernel computation and transposition\nIf the rank of the linear operator is lower than its number of rows,\nthen an additional strategy has proven useful: compute first some\nindependent rows, then express the dependent ones by their lin-\near relationsFor this, Algorithm 1 computes a left Kernel of the\nlinear operator and uses it to compute the dependent rows via\nlinear combinations of the independent ones. This is sometimes\nfaster than directly computing the dependent rows. Of course, if\nthe matrix’s rank is lower than the number of columns , one can\napply Algorithm 1 to the transposed matrix, and then apply the\nTellegen transposition principle to recover the transposed linear\ndependencies (see, e.g., [5] and references therein).\nAlgorithm 1 Kernel decomposition of a linear operator\nInput:𝑀inK𝑚×𝑛such that𝑟=Rank𝑀.\nOutput: A straight line program computing 𝑥↩→𝑀·𝑥.\n1:By Gaussian elimination, compute 𝑀=𝑃·𝐿·𝑈·𝑄with𝑃a\npermutation matrix, 𝐿inK𝑚×𝑟beh𝐿1\n𝐿2i\nunit upper triangular\nand𝐿1inK𝑟×𝑟; choosing𝑃so that (1) the first 𝑟rows of𝑃−1𝑀\nare sparsest; (2) 𝐿1is the sparsest; (3) 𝐿2is the sparsest;\n2:Let𝜎be the permutation represented by 𝑃;\n3:Apply Alg. 6 to[𝑟𝜎(1)...𝑟𝜎(𝑟)]⊺→[𝐼𝑟0]·𝑃·𝑀·®𝑥;\n⊲[−𝐿2·𝐿1−1𝐼𝑚−𝑟]is a (sparse) left kernel of 𝑀and provides\nthe linear dependencies of the remaining rows\n4:Apply Alg. 6 to[𝑟𝜎(𝑟+1)...𝑟𝜎(𝑚)]⊺→𝐿2·𝐿1−1[𝑟𝜎(1)...𝑟𝜎(𝑟)]⊺.\nAlgorithm 1 is implemented in plinopt/optimizer -K . The\ntransposition principle applied to such straight-line programs is\nimplemented in plinopt/transpozer [11]. These routines have\nproduced the implementations given in the following, for our differ-\nenthmformulas (for instance the implementation Table 3 of Eq. (34)\nwith only 24additions and 12multiplications/divisions).\nRemark 21. The accuracy obtained with our different fast vari-\nants is given in Figure 1 using the Matlab framework of [ 8], which\nwe forked in [ 12] and where we have just added the implementa-\ntions of the variants presented here. Thus, in Figures 1, 2, 3 and 4 we\npresent the error as the infinite norm of the difference between the\nresult of our implementations and the exact matrix multiplication.Strassen’s algorithm is not optimally accurate\n𝑡1=√\n3\n3𝑎22𝑡2=𝑎21+𝑡1𝑠1=√\n3\n3𝑏21𝑠2=𝑠1−𝑏11\n𝑡3=𝑎12+𝑡2𝑙1=√\n3\n2𝑎11+1\n2𝑡3𝑠3=𝑠2+𝑏22𝑟1=2𝑠1\n𝑙2=𝑎12−𝑡1𝑙3=𝑡2 𝑟2=𝑠2 𝑟3=𝑠1−𝑏22\n𝑙4=2𝑡1𝑙5=𝑙2−𝑙1𝑟4=1\n2𝑠3−√\n3\n2𝑏12𝑟5=𝑟3+𝑟4\n𝑙6=𝑙5+𝑙4𝑙7=𝑙5+𝑙3𝑟6=𝑟1−𝑟5𝑟7=𝑟5−𝑟2\n𝑝1=𝑙1·𝑟1𝑝2=𝑙2·𝑟2𝑝3=𝑙3·𝑟3𝑝4=𝑙4·𝑟4\n𝑝5=𝑙5·𝑟5𝑝6=𝑙6·𝑟6𝑝7=𝑙7·𝑟7\n𝑤2=𝑝5+𝑝1+𝑝6𝑤1=𝑝7+𝑝6𝑤5=𝑝4+𝑤2\n2𝑤3=𝑤2−𝑝2\n𝑐12=𝑝1−𝑝3−𝑤5𝑐21=𝑤3−𝑤5𝑐22=√\n3𝑤5\n𝑐11=√\n3\n3(𝑤3−𝑐12−2𝑤1)\nTable 3: slpof Eq. (34) with 24add. and 12mul./div.\nIn Figure 1, all our variants, Tables 3, 4 and 6 and Eqs. (41) and (42)\nwith decreasing 𝛾2,1, are mostly more and more accurate. Our best\nalgorithm presents an order of magnitude advantage over Strassen’s\nand two orders of magnitude advantage over Winograd’s. It is then\nquite close to the conventional algorithm’s accuracy. Figure 1 uses\nnormal distribution, but the same behavior is obtained, e.g., with a\nuniform distribution in Figure 4.\nFigure 1: Numerical accuracy vs size (normal distribution)\n10−1410−1310−1210−11\n32 64 128 256 512\nerror\nsquare matrix dimensionWinograd [21]\nStrassen [20]\nTable 4 and Eq. (35)\nTable 6 and Eq. (40)\nEq. (41)\nEq. (42)\nTable 3 and Eq. (34)\nConventional\nRemark 22. In [4] the authors consider all bilinear algorithms\nusing 7multiplications with constants of the form ±2𝑖; they shown\nthat Strassen’s original method [ 20] reaches in this class the min-\nimum value 12of their𝛾0,1∞factor error bound (while for in-\nstance that of Winograd [ 21] is18, see also Table 1). In Eq. (35)\nand Table 4 we propose an algorithm in this class that has a worse\n𝛾0,1∞of40, but a𝛾2,1of4/√\n2+75/8≈12.2034, better than those\nof Strassen 14.8284 or Winograd 17.8530 (see Table 2). Figure 1\nshows that this algorithm is also more accurate in practice than\nboth Strassen’s and Winograd’s variants.0−1 1 0\n11\n2−1\n2−1\n4\n0 0 1−1\n2\n0 1 0−1\n2\n0 0 11\n2\n1−1\n21\n2−1\n4\n0 1 01\n2;1 0 0−1\n11\n20 0\n01\n20−1\n1\n21\n4−1−1\n2\n01\n20 1\n1−1\n20 0\n1\n2−1\n41−1\n2;0 1 1 0\n1\n21 0 0\n1\n4−1\n2−1\n21\n−1\n20 1 0\n1\n41\n21\n21\n1\n2−1 0 0\n1\n20 1 0⊺\n(35)\n𝑟1=1\n2𝑎22𝑡2=𝑎21−𝑟1𝑢1=1\n2𝑏12𝑠1=𝑏11+𝑢1\n𝑡3=𝑎12+𝑟1𝑡0=𝑡2−𝑡3𝑠2=𝑢1−𝑏22𝑢2=𝑠1−𝑏22\n𝑡4=𝑎21+𝑟1𝑟2=𝑡2−𝑎12𝑠4=𝑏22+𝑢1𝑠0=𝑠1−𝑠4\n𝑡5=𝑎11+1\n2𝑟2𝑡1=𝑡5−𝑡0𝑠3=1\n2𝑢2−𝑏21𝑠5=𝑠0−𝑠2\n𝑝1=𝑡0·𝑠0𝑝2=𝑡1·𝑠1𝑝3=𝑡2·𝑠2𝑝4=𝑡3·𝑠3\n𝑝5=𝑡4·𝑠4𝑝6=𝑡5·𝑠5𝑝7=(𝑡4−𝑡0)·(𝑠0−𝑠3)\n𝑣1=𝑝5−𝑝3𝑣2=𝑝1+1\n2𝑣1𝑣3=𝑝7+𝑝6−𝑝4+𝑝2\n𝑐22=𝑝5+𝑝3𝑐12=𝑝2−𝑝6+𝑣2𝑐11=1\n2(𝑣3+1\n2𝑐22)𝑐21=𝑝4+𝑝7+𝑣2\nTable 4: slpof Eq. (35) with 27add., 7div. by 2and𝛾𝐹≈12.2034\nRemark 23. Eq. (35) was obtained by approximating the minimal\npoint of the 𝛾2,1growth factor taken from Proposition 15 with the\nsmallest powers of 2. Further rational higher-order approximations\nare obtained in the same vein, giving for instance Eqs. (40) to (42)\nand Table 6, as shown in Appendix A.2.\n6.4 Alternate basis sparsification\nThe technique of [ 3,17] reduces the number of operations by fac-\ntoring each matrix in the hmdecomposition into a sparser one via\na4×4change of basis ( CoB). In a recursive version, the left and right\nhand sides (resp. result) CoB can be recursively precomputed (resp.\npost-computed), for a total cost in 𝑂\u0000𝑛2log𝑛\u0001. In the meantime the\nsparsest 7×4matrices are applied reducing the dominant term of the\ncomputation. The optimal decomposition of Winograd’s algorithm\nin [17, § 3.3] reduces the number of intermediate additions from 15\nto12. For a fully recursive version, this gives a constant factor\nreduction of the complexity bound from 6𝑛log2(7)to5𝑛log2(7).\nThis is also the case for the algorithm of Eq. (34), for instance,\nthat can use the CoB of Eq. (36) to obtain the sparse recursion\nof Eq. (37). There the constant factor reduction of complexity bound\nis from 13𝑛log2(7)for Table 3 to 5𝑛log2(7). To obtain this sparse\nCoB, the generic technique of [ 3] can be used. In our case, for 4×4\nCoB, the following heuristic was sufficient to obtain optimal ( 12-\nadditions) sparse−1,0,1intermediate matrices: (1) Find indepen-\ndent columns of each CoB one at a time; (2) For this, alternatively\nfactor-out common coefficients in the resulting columns and find a\nlinear combination minimizing the density of the resulting column,\nusing as coefficients of the combination only −1,0,1and some of\nthe values of the coefficients of the input; (3) Until this alterna-\ntion does not sparsify anymore. This heuristic is implemented in\nplinopt/sparsifier [11] and resulting sparse implementation is\nshown in Algorithms 2 to 5 and Table 5.Jean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic\n0 0 02√\n3\n0 1 0√\n3\n3\n0 0 1−√\n3\n3\n−√\n3\n2−1\n21\n2−√\n3\n2;02√\n30 0\n1−√\n3\n30 0\n0√\n3\n30−1\n−1\n2√\n3\n2−√\n3\n2−1\n2;−2√\n30 0 0\n√\n3\n3−1 0 0\n−√\n3\n30−1 0\n√\n3\n2−1\n21\n2√\n3\n2⊺\n(36)\n0 0 1−1\n0 0 1 0\n0 1 0 0\n−1 0 0 0\n0 0 0 1\n1 0 0 1\n0 1 0 1;1 0 0 0\n0−1 0 0\n0 0 1 0\n0 0 1−1\n0 0 0 1\n1 0 0−1\n0 1 0 1;0−1 0 1\n0 0 1 0\n0 1 0 0\n0 0 1 1\n0 0 0 1\n1 0 0 1\n1 0 0 0⊺\n(37)\nAlgorithm 2 LCoB(𝐴,ℓ)left change-of-basis of Eq. (36)\n1:ifℓ≤0then return 𝐴.end if\n2:𝑚1=LCoB(𝑎11,ℓ−1);𝑚2=LCoB(𝑎21,ℓ−1);\n3:𝑚3=LCoB(𝑎12,ℓ−1);𝑚4=LCoB(𝑎22,ℓ−1);\n4:𝑡1=(𝑚3−𝑚2)/2;𝑡2=𝑚4√\n3/3;𝑡3=𝑡1+𝑚1√\n3/2;\n5:return[𝑚4√\n3/2+𝑡3,𝑚2−𝑡2,𝑚3+𝑡2,𝑚4√\n3/6−𝑡3].\nAlgorithm 3 RCoB(𝐴,ℓ)right change-of-basis of Eq. (36)\n1:ifℓ≤0then return 𝐴.end if\n2:𝑚1=RCoB(𝑎11,ℓ−1);𝑚2=RCoB(𝑎21,ℓ−1);\n3:𝑚3=RCoB(𝑎12,ℓ−1);𝑚4=RCoB(𝑎22,ℓ−1);\n4:𝑡1=(𝑚1−𝑚4)/2;𝑡2=𝑚3√\n3/6−𝑚2√\n3/2;\n5:return[𝑡1+𝑡2,(𝑚1+𝑚4)/2+(𝑚2−𝑚3)√\n3/2,𝑚3√\n3/3,𝑡2−𝑡1].\n𝑠1=𝑎11+𝑎12𝑠2=𝑎11+𝑎22𝑠3=𝑎11−𝑎21\n𝑡1=𝑏12+𝑏22𝑡2=𝑏11+𝑏12𝑡3=𝑏12+𝑏21\n𝑝1=𝑎11·𝑏12𝑝2=𝑠1·𝑏21𝑝3=𝑎21·𝑡1𝑝4=𝑎12·𝑡2\n𝑝5=𝑠2·𝑏22𝑝6=𝑎22·𝑡3𝑝7=𝑠3·𝑏11\n𝑐11=𝑝7−𝑝6𝑐12=𝑝2+𝑝3𝑐21=𝑝4−𝑝5𝑐22=𝑝1+𝑝2+𝑝5+𝑝6\nTable 5: slpof Eq. (37) with 12additions\nAlgorithm 4 CoBP(𝐴,ℓ)product change-of-basis of Eq. (36)\n1:ifℓ≤0then return 𝐴.end if\n2:𝑚1=CoBP(𝑎11,ℓ−1);𝑚2=CoBP(𝑎21,ℓ−1);\n3:𝑚3=CoBP(𝑎12,ℓ−1);𝑚4=CoBP(𝑎22,ℓ−1);\n4:𝑡1=𝑚4/2;𝑡2=𝑚4√\n3/2;\n5:return[𝑚12/√\n3+(𝑚3−𝑚2)√\n3/3+𝑡2,𝑚2−𝑡1,𝑚3+𝑡1,𝑡2].\nRemark 24. With Algorithm 5, we can get the best of both worlds:\nthe best known dominant term of the complexity bound, while\nmostly preserving the best known numerical accuracy. The former\nproperty comes from the fact that Eq. (37) requires only 12additions.\nThe latter is shown in Figure 2 but seems more complex to prove.\nIndeed, the sparsest 7×4matrices have a slightly reduced 𝛾2,1\ngrowth factor. For the algorithm of [ 17], this growth factor is indeed\nreduced from 7+8/√\n2+9/√\n3≈17.853, for [ 21], to 4+12/√\n2≈\n12.486. But the three CoB and eachℓrecursive calls add some termsAlgorithm 5 Sparsification of Eq. (34)\nInput:𝐴,𝐵∈K𝑛02ℓ×𝑛02ℓ.\nOutput:𝐶=𝐴·𝐵.\n1:¯𝐴←LCoB(𝐴,ℓ);¯𝐵←RCoB(𝐵,ℓ)⊲Via Algorithms 2 and 3\n2:¯𝐶←¯𝐴·¯𝐵; ⊲Via Table 5 with ℓrecursive calls\n3:return𝐶←CoBP(¯𝐶,ℓ). ⊲Via Algorithm 4\nin the bounds and has then to be taken into account. From Lem-\nmas 8 and 11, even with the infinite operator norm, we thus have\nto consider the maximum absolute row sum of the CoB. It is 3, for\nthat of [ 17]. Thus the error upper bound grows at least to\u0000𝛾·33\u0001ℓ.\nSimilarly, the 𝛾2,1growth factor of Eq. (37) is lower than that\nof Eq. (34): from 4/√\n2+16/√\n3≈12.066to7+6/√\n2≈11.243. But\nagain, the three CoB in Eq. (36) have maximum absolute row\nsum 1+√\n3≈2.73. The error upper bound then grows at least\nto\u0000𝛾·(10+18/√\n3)\u0001ℓ.\nIn all cases though, these large upper bounds do not reflect the\nexperiments shown in Figure 2. They show, on the contrary, that\nsparsification seems to mostly preserve accuracy. The accuracy of\nthe sparse Winograd variant, by [ 17], is around that of the original;\nthe accuracy of the sparse Strassen variant is slightly worse; and\nthe accuracy of Algorithm 5 is only barely inferior to that of Table 3\nand Eq. (34), despite their better time complexity.\nFigure 2: Numerical effect of sparsification ( normal distribution)\n10−1410−1310−1210−11\n32 64 128 256 512\nerror\nsquare matrix dimensionSparse Winograd, by [17]\nWinograd [21]\nSparse Strassen\nStrassen [20]\nSparse Table 3 (Algorithm 5)\nTable 3 and Eq. (34)\nConventional\nFollowing [ 19, § 3.2], we can also confirm our algorithms’ accu-\nracy on badly conditioned matrices (see Figure 3).\nRemark 25. To further improve their practical behavior, as done\nin [9, § 4.3], [ 2, § 6.1] or [ 1, § 6], some diagonal scaling adapted to\nspecific input matrices can also be added to any algorithms. The\nidea is to precondition the input matrices with well suited 𝐷1,𝐷2,𝐷3\nthanks to the relation A·B=𝐷1−1\u0000(𝐷1·𝐴𝐷2)·\u0000𝐷2−1·𝐵𝐷3\u0001\u0001𝐷3−1.\nThis can be still be applied to any of the variants presented here.Strassen’s algorithm is not optimally accurate\nREFERENCES\n[1]Grey Ballard, Austin R. Benson, Alex Druinsky, Benjamin Lipshitz, and Oded\nSchwartz. 2016. Improving the Numerical Stability of Fast Matrix Multiplication.\nSIAM J. Matrix Anal. Appl. 37, 4 (2016), 1382–1418. https://doi.org/10.1137/\n15M1032168\n[2] Grey Ballard, James Demmel, Olga Holtz, Benjamin Lipshitz, and Oded Schwartz.\n2012. Communication-Optimal Parallel Algorithm for Strassen’s Matrix Mul-\ntiplication. In Proceedings of the Twenty-Fourth Annual ACM Symposium on\nParallelism in Algorithms and Architectures (Pittsburgh, Pennsylvania, USA)\n(SPAA ’12) . Association for Computing Machinery, New York, NY, USA, 193–204.\nhttps://doi.org/10.1145/2312005.2312044\n[3] Gal Beniamini and Oded Schwartz. 2019. Fast Matrix multiplication via sparse\ndecomposition. In SPAA ’19: Proceedings of the 31st annual ACM symposium on\nParallel algorithms and architectures (Phoenix, Arizona, USA), Petra Berenbrink\nand Christian Scheideler (Eds.). Association for Computing Machinery, Phoenix,\nArizona, USA, 11–22. https://doi.org/10.1145/3323165.3323188\n[4] Dario Andrea Bini and Grazia Lotti. 1980. Stability of fast algorithms for matrix\nmultiplication. Numer. Math. 36, 1 (March 1980), 63–72. https://doi.org/10.1007/\nBF01395989\n[5]Alin Bostan, Grégoire Lecerf, and Éric Schost. 2003. Tellegen’s principle into\npractice. In ACM International Symposium on Symbolic and Algebraic Compu-\ntation, Philadelphia, USA , Rafael Sendra (Ed.). ACM Press, New York, 37–44.\nhttps://doi.org/10.1145/860854.860870\n[6] Joan Boyar, Philip Matthews, and René Peralta. 2013. Logic Minimization Tech-\nniques with Applications to Cryptology. Journal of Cryptology 26, 2 (2013),\n280–312. https://doi.org/10.1007/s00145-012-9124-7\n[7] R. P. Brent. 1970. Algorithms for matrix multiplication . Technical Report STAN-\nCS-70-157. C.S. Dpt. Standford University.\n[8] Zhen Dai and Lek-Heng Lim. 2023. Numerical stability and tensor nuclear norm.\nNumer. Math. 155, 3-4 (Nov. 2023), 345–376. https://doi.org/10.1007/s00211-023-\n01377-5\n[9] James Demmel, Ioana Dumitriu, and Olga Holtz. 2007. Fast linear algebra is stable.\nNumer. Math. 108, 1 (2007), 59–91. https://doi.org/10.1007/S00211-007-0114-X\n[10] James Demmel, Ioana Dumitriu, Olga Holtz, and Robert Kleinberg. 2007. Fast\nmatrix multiplication is stable. Numer. Math. 106, 2 (Feb. 2007), 199–224. https:\n//doi.org/10.1007/s00211-007-0061-6\n[11] Jean-Guillaume Dumas, Bruno Grenet, Clément Pernet, and Alexandre Se-\ndoglavic. 2024. PLinOpt, a collection of C++ routines handling linear & bilinear\nprograms. https://github.com/jgdumas/plinopt 1.7 kSLOC.\n[12] Jean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic. 2024. Matlab\naccurate fast matrix multiplications via 2x2 recursion. https://github.com/\njgdumas/Fast-Matrix-Multiplication 1.6 kSLOC.\n[13] Hans-Friedich de Groote. 1978. On varieties of optimal algorithms for the\ncomputation of bilinear mappings I. The isotropy group of a bilinear mapping.\nTheoretical Computer Science 7, 2 (1978), 1–24. https://doi.org/10.1016/0304-\n3975(78)90038-5\n[14] Hans-Friedich de Groote. 1978. On varieties of optimal algorithms for the compu-\ntation of bilinear mappings II. Optimal algorithms for 2×2-matrix multiplication.\nTheoretical Computer Science 7, 2 (1978), 127–148. https://doi.org/10.1016/0304-\n3975(78)90045-2\n[15] Nicholas J. Higham. 2002. Accuracy and Stability of Numerical Algorithms (second\ned.). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.\n9780898718027\n[16] John Edward Hopcroft and Jean Musinski. 1973. Duality Applied to the Com-\nplexity of Matrix Multiplications and other Bilinear Forms. In 5th Annual ACM\nSymposium on the theory of computing (Austin, TX, USA), Alfred V. Aho, Allan\nBorodin, Robert L. Constable, Robert W. Floyd, Michael A. Harrison, Richard D.\nKarp, and Raymond H. Strong (Eds.). 73–87. https://doi.org/10.1137/0202013\n[17] Elaye Karstadt and Oded Schwartz. 2017. Matrix Multiplication, a Little Faster.\nInProceedings of the 29th ACM Symposium on Parallelism in Algorithms and\nArchitectures (Washington, DC, USA) (SPAA ’17) , Christian Scheideler and\nMohammad Hajiaghayi (Eds.). ACM, New York, NY, USA, 101–110. https:\n//doi.org/10.1145/3087556.3087579\n[18] Joseph M. Landsberg. 2016. Geometry and complexity theory . Cambridge Studies\nin Advanced Mathematics, Vol. 169. Cambrigde University Press. https://doi.\norg/10.1017/9781108183192\n[19] Katsuhisa Ozaki, Takeshi Ogita, Shin’ichi Oishi, and Siegfried M. Rump. 2012.\nError-free transformations of matrix multiplication by using fast routines of\nmatrix multiplication and its applications. Numer. Algorithms 59, 1 (2012), 95–118.\nhttps://doi.org/10.1007/S11075-011-9478-1\n[20] Volker Strassen. 1969. Gaussian elimination is not optimal. Numer. Math. 13, 4\n(Aug. 1969), 354–356. https://doi.org/10.1007/BF02165411\n[21] Shmuel Winograd. 1977. La complexité des calculs numériques. La Recherche 8\n(1977), 956–963.A SUPPLEMENTARY MATERIALS\nA.1 Computational proofs\nWe gather in this section proofs of several proposition that are\nsimple computations on objects presented in our work.\nProof of Proposition 15. To simplify our computations, we\nuse the following coordinates 𝜌=4√︁\n4/𝑟and𝜉=(𝑥−1)/2, the ma-\ntrix𝑢(𝜌,𝜉)and the associated isotropy g𝜌,𝜉⋄S. In that case, the ex-\nplicit expression of the 𝛾2,1growth factor 𝛾\u0000g𝑟,𝑥⋄S\u0001along this orbit\nis2√\n2+3AwithA(𝑟,𝑥)equal to\u0000(1+𝑥)2+𝑟\u0001\u0000(𝑥−1)2+𝑟\u0001/𝑟√𝑟.\nTo conclude, we prove that the minimum of A(𝑟,𝑥)inR+×Ris\n16/3√\n3. The partial derivatives of A(𝑟,𝑥)w.r.t.𝑟and𝑥are\n𝜕A\n𝜕𝑥=4(𝑥2+𝑟−1)𝑥\n𝑟3/2,𝜕A\n𝜕𝑟=𝑟2−2(𝑥2+1)𝑟−3(𝑥2−1)2\n2𝑟5/2. (38)\nFirst, notice that𝜕A\n𝜕𝑥\u00001−𝑥2,𝑥\u0001is0and that𝜕A\n𝜕𝑟\u00001−𝑥2,𝑥\u0001is equal\nto2/((𝑥−1)(1+𝑥))3/2. The only critical point is 𝑥equal to 0and,\nas𝑟is positive, it only could be equal to 3. The Hessian matrix is:\n𝐻(A(𝑟,𝑥))=1\n𝑟3\n2\"\n4(3𝑥2+𝑟−1)−2\n𝑟(3𝑥2+𝑟−3)\n−2\n𝑟(3𝑥2+𝑟−3)−𝑟2−6(𝑥2+1)𝑟−15(𝑥2−1)2\n4𝑟2#\n,(39)\none can notice that 𝐻(A(3,0))is equal toh\n230\n0 2/3i\n/3√\n3. Hence,\nthe second partial derivative test states that\u00004√︁\n4/3,−1/2\u0001is a local\nminimum of 𝛾\u0000g𝑟,𝑥⋄S\u0001equal to 2√\n2+16√\n3≈12.06603143 . To con-\nclude, the𝛾2,1growth factor reaches its global minimal at this point\non the considered orbit because it is its only critical point. □\nProof of Proposition 17. Recall that any of the three matrices\nin the hmrepresentation of such an algorithm is obtained by row\nand column permutations of one of these matrices, multiplied by\nthe Kronecker product of two invertible 2×2matrices WandV\n(see Lemma 6 and Theorem 7). From the analysis of Section 4, we\nonly need to consider the case where- 𝑊and𝑉are each of the\nform\u0002𝑟 𝑥\n0𝑟−1\u0003\n, with strictly positive 𝑟(matrices𝑊and𝑉are taken\nin a simpler form for the sake of simplicity). For this, we let 𝑊\nbe\u0002𝑟 𝑥\n0𝑟−1\u0003and𝑉beh𝑠 𝑦\n0𝑠−1i\n, and choose 𝐿the first component of\nStrassen’s hmrepresentation given in Eq. (6). We then obtain the\nsquare of the Frobenius norm of L·(W⊗V)as a function of 𝑟,𝑥,𝑠\nand𝑦:𝑓(𝑟,𝑥,𝑠,𝑦)=4𝑟2𝑠2+3𝑟2𝑦2+3𝑠2𝑥2+𝑥2𝑦2+𝑟2/𝑠2+𝑠2/𝑟2+1/𝑟2𝑠2\n+(𝑥/𝑠+1/𝑟𝑠)2+(𝑦/𝑟−1/𝑟𝑠)2+(𝑥/𝑠−𝑥𝑦)2+(𝑦/𝑟+𝑥𝑦)2\n+(𝑥𝑦+1/𝑟𝑠)2+(𝑠/𝑟+𝑥𝑠)2+(𝑟/𝑠−𝑟𝑦)2.Solving the four par-\ntial derivatives of the gradienth𝜕𝑓\n𝜕𝑟,𝜕𝑓\n𝜕𝑠,𝜕𝑓\n𝜕𝑥,𝜕𝑓\n𝜕𝑦i\n, for a simultane-\nous zero, we obtain that the only real extrema of 𝑓are at the\nfour points: 𝑟=±4√︁\n3/4,𝑠=±𝑟,𝑥=−2/3𝑟3,𝑦=2/3𝑠3, for which its\nvalue is always 10. The additional constraint that 𝑟and𝑠are posi-\ntive, thus gives a single extremum. Now, the Hessian matrix at that\npoint is computed as: H𝑓\u00104√\n3√\n2,−4√\n3√\n6,4√\n3√\n2,4√\n3√\n6\u0011\n=4\n9195√\n369√\n3−15 15\n69√\n3195√\n3−15 15\n−15−1545√\n39√\n3\n15 159√\n345√\n3.\nLeaving out the factor 4/9, the characteristic polynomial of this ma-\ntrix is𝑋4−160√\n3𝑋3+22536𝑋2−362880√\n3𝑋+5143824 whose all\nfour roots√\n7500±√\n5232,(30±12√\n3are positive. Therefore the\npoint is a local minimum. From this, we see that√\n103is a lowerJean-Guillaume Dumas, Clément Pernet, and Alexandre Sedoglavic\nbound on the product of their Frobenius norms. Furthermore, re-\nmark that this lower bound is reached, by the algorithm which hm\nrepresentation is given in Equation (34). □\nA.2 Rational approximations\nAs stated in Remark 23, by approximating the minimal point of\nthe𝛾2,1growth factor presented in Proposition 15, we could con-\nstruct further algorithms presented in this section.\n4\n9−8\n9−8\n9−4\n9\n05\n9010\n9\n8\n9−2\n30 0\n4\n92\n98\n94\n9\n0−10\n90 0\n4\n9−1\n3−8\n92\n3\n−4\n9−2\n98\n94\n9;−3\n54\n5−4\n5−3\n5\n01\n20−1\n−11\n20 0\n05\n40 0\n3\n5−3\n104\n5−2\n5\n2\n53\n10−4\n5−3\n5\n−3\n5−9\n20−4\n5−3\n5;9\n209\n109\n10−9\n20\n0 027\n40−9\n10\n−9\n80−9\n160\n9\n209\n109\n409\n20\n−27\n409\n1027\n80−9\n20\n0 0−9\n80\n9\n209\n10−9\n40−9\n20⊺\n(40)\nFirst, Eq. (40) is an orthogonal optimization of Eq. (35), with one\ncanonical vector in each of components of the hmrepresentation.\nUnfortunately some small non-powers of 2are then unavoidable,\nbut this gives in Table 6 an algorithm realizing the formula with\nless additions than that of Table 4.\n𝑢1=𝑎12+1\n2𝑎22𝑡2=8\n9𝑎11−2\n3𝑎12𝑡1=5\n9𝑢1𝑡4=10\n9𝑎12\n𝑡3=4\n9𝑎11+8\n9𝑎21+2\n9𝑢1𝑡0=𝑡2−𝑡3𝑡5=𝑡1+𝑡0𝑡6=𝑡4+𝑡0\n𝑣1=1\n2𝑏12 𝑠1=𝑣1−𝑏22𝑠2=𝑣1−𝑏11𝑠3=5\n4𝑏12\n𝑠4=2\n5𝑏22−4\n5𝑏21+3\n5𝑠2𝑠0=𝑠1+𝑠4𝑠5=𝑠0−𝑠2𝑠6=𝑠3−𝑠0\n𝑝0=𝑡0·𝑠0𝑝1=𝑡1·𝑠1𝑝2=𝑡2·𝑠2𝑝3=𝑡3·𝑠3\n𝑝4=𝑡4·𝑠4𝑝5=𝑡5·𝑠5𝑝6=𝑡6·𝑠6\n𝑤1=𝑝6+𝑝0+𝑝4𝑤2=𝑝5+𝑝6𝑤3=𝑝3+𝑤1𝑤4=𝑝2+𝑝4\n𝑤5=𝑝1+𝑤1𝑤6=9\n20𝑤3 𝑐11=𝑤6−9\n8𝑤4\n𝑐12=9\n10𝑤3𝑐21=27\n40𝑤5−9\n8𝑤2+1\n2𝑐11𝑐22=𝑤6−9\n10𝑤5\nTable 6: slpof Eq. (40), 𝛾𝐹≈12.2034, with 24add. and 20mul.\nFinally, we present in Eqs. (41) and (42), successive higher-order\nrational approximations of the point\u00004√︁\n4/3,−1/2\u0001reducing the 𝛾2,1\ngrowth factor to 12.0695 (resp. 12.0661), approaching 12.06603 . They\nthen provide rational algorithms whose accuracy is pretty close to\nour best one, as shown in Figure 1.\n−167042\n345665295936\n345665−295936\n345665−167042\n345665\n−178623\n345665−51622047\n176980480295936\n345665167042\n345665\n0−51622047\n884902400334084\n345665\n−1289\n5120 0\n0289\n2560 0\n−167042\n345665−24137569\n88490240−295936\n345665−167042\n345665\n−167042\n34566524137569\n88490240−295936\n345665167042\n345665;−256\n289−1\n21\n2−256\n289\n−345665\n2959360 0 0\n−345665\n5918720345665\n3340840\n−178623\n295936−1 0 0\n178623\n5918721\n2178623\n334084256\n289\n−289\n10241\n2−1\n2256\n289\n−289\n10241\n21\n2−256\n289\n295936\n345665295936\n3456650 0295936\n345665295936\n3456650\n178623\n345665−167042\n3456651 0178623\n345665178623\n3456650\n−178623\n345665−178623\n3456650 1167042\n345665−178623\n3456650\n295936\n34566551622047\n176980480289\n512−289\n51251622047\n176980480−31906176129\n102294717440−345665\n295936(41)33124\n3816519208\n38165−19208\n3816533124\n38165\n33124\n3816519208\n3816518957\n381651857786\n6449885\n038416\n3816503715572\n6449885\n0 0 1 −98\n169\n0 0 0196\n169\n33124\n3816519208\n38165−19208\n38165−1882384\n6449885\n33124\n38165−19208\n38165−19208\n381651882384\n6449885;−169\n196−1\n21\n2−169\n196\n38165\n331240 0 0\n38165\n662480−38165\n384160\n18957\n331241 0 0\n18957\n662481\n218957\n38416169\n196\n−49\n1691\n2−1\n2169\n196\n−49\n1691\n21\n2−169\n196\n−18957\n3816519208\n38165−1 0−18957\n38165−18957\n381650\n−33124\n38165−1857786\n6449885−98\n16998\n169−1857786\n6449885359367849\n126417746038165\n33124\n33124\n3816533124\n381650 033124\n3816533124\n381650\n−18957\n38165−18957\n381650 119208\n38165−18957\n381650(42)\nA.3 Further numerical experiments\nFollowing [ 19, § 3.2], we study in Figure 3 the effect of sparsi-\nfication on random matrix with preassigned singular values and\nlarge condition number ≈1012given by the Matlab function gallery\n’randsvd’. The fast variants behavior is unchanged while only the\nconventional algorithm performs better.\nFigure 3: Numerical Effect of Sparsification ( large conditioning)\n10−1510−1410−1310−12\n32 64 128 256 512\nerror\nsquare matrix dimensionSparse Winograd, by [17]\nWinograd [21]\nSparse Strassen\nStrassen [20]\nSparse Table 3 (Algorithm 5)\nTable 3 and Eq. (34)\nConv.\nWe now show more evidence on the practical accuracy of the al-\ngorithms, with respect to their 𝛾2,1growth factor. Figure 4 compares\nthe main possibilities on a uniform [−1,1]distribution, while Fig-\nure 1 was using a normal distribution. The behavior is similar, with\nagain our best variant one or two orders of magnitude more accu-\nrate, and being quite close to that of the conventional algorithm.Strassen’s algorithm is not optimally accurate\nFigure 4: Numerical accuracy for uniform [-1,1] distribution\n10−1410−1310−1210−11\n32 64 128 256\nerror\nsquare matrix dimensionWinograd [21]\nStrassen [20]\nTable 3 and Eq. (34)\nConv.\nA.4 Cancellation-free search\nAlgorithm 6 describes a common sub-expression elimination heuris-\ntic that reduced the number of operations in our algorithms.\nAlgorithm 6 Cancellation-free optimization of a linear operator\nInput:𝑀∈K𝑚×𝑛.\nOutput: A straight-line program computing 𝑥→𝑀·𝑥.\n1:repeat ⊲Precomputing all repeated pairs\n2: In each row list all pairs of indices of non-zero coefficients;\n3: Among all the rows, find the pair(s) with the maximal num-\nber of co-linear representatives;\n4: In case of ties, exhaust all the possibilities with maximal\npairs (or choose one using a score like that of [6, § 3.2]);\n5: Precompute the chosen pair (in a temporary variable);\n6: Factor this pair out of all the rows: that is remove the pair\nfrom all rows but add a new column to the matrix (representing\nthat pair) with the co-linear multiple of that temporary variable;\n7:until no pair has more than 1representative\n⊲Multipliers by columns:\n8:for all equal coefficients in a column (up to sign) do\n9: Compute the product by the absolute value in a temporary\nvariable;\n10: Factor this coefficient out: remove it from the column, add\na new column (representing that product) with a ±1in the\ncorresponding row(s);\n⊲Multipliers by rows:\n11:for all equal coefficients in a row (up to sign) do\n12: Compute the sum (or subtraction) of variables with that\nsame coefficients in a temporary variable;\n13: Factor the coefficient out: remove it from the row, but add\na new column (representing that sum/subtraction) with the\ncoefficient in the same row;\n⊲Now the matrix has been simplified\n14:Apply the remaining linear operations of the matrix."    },    {        "title": "2402.05688v1.Derivative_free_sample_and_hold_control_with_prescribed_performance.pdf",        "content": "Derivative-free sample-and-hold control\nwith prescribed performance1\nLukas Lanza∗\n∗Technische Universit¨ at Ilmenau, Institute of Mathematics,\nOptimization-based Control Group, Ilmenau, Germany\nlukas.lanza@tu-ilmenau.de\nAbstract: A feedback controller is proposed to perform output reference tracking with\nprescribed performance for unknown nonlinear continuous-time systems. The controller is of\nsampled-data type, i.e., measurements are available only at sampling times – a typical situation\nin real systems when sensors are involved. Furthermore, only output information is available,\ni.e., neither the full state nor derivatives can be used for feedback. A sufficient uniform sampling\nrate is derived and the control consists of piecewise constant signals on the sampling intervals,\ni.e., zero-order hold. Feasibility of the controller and satisfaction of the control objective are\nrigorously proven. The controller is illustrated by a numerical example.\nKeywords: sampled-data control, zero-order hold, output feedback, tracking guarantees,\nunknown nonlinear continuous-time systems\n1. INTRODUCTION\nWe consider the following control problem: The out-\nputyof an unknown nonlinear multi-input multi-output\ncontinuous-time dynamical system with well-defined rela-\ntive degree and stable internal dynamics should follow a\ngiven reference trajectory yrefwhile satisfying the follow-\ning objectives\n(1) the tracking is performed with prescribed perfor-\nmance, i.e., the error satisfies ∥y(t)−yref(t)∥< ψ(t)\nfor all t≥0 for a given boundary function ψ(feasi-\nbleψare given in the set Φ below),\n(2) measurement of the output’s derivative is not avail-\nable,\n(3) measurements of the output yare available only at\ndiscrete time instances, i.e., y(kτ),k∈N.\nFor the sake of clear presentation, in this article we assume\nthe systems under consideration to be of relative degree\ntwo.\nControl objective (1), for systems with relative degree two,\nis already achieved by the continuously updated funnel\ncontroller , cf. Hackl et al. (2013), Berger et al. (2018,\n2021). Here, “continuously updated” refers to the fact that\nthe controllers use the output signal y(t) for t∈R≥0. For\nsystems with relative degree larger than one the funnel\ncontroller not only relies on the continuous availability\nof the output signal but also of its higher order deriva-\ntives. This aspect, namely the issue of the availability of\nthe output’s derivatives was addressed in Ilchmann et al.\n(2006), and in Berger and Reis (2018), Lanza (2022).\nIn Ilchmann et al. (2006) a backstepping-like procedure\nis used to design a filter which is then interconnected with\n1This work was supported by Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation; Project-ID 471539468), and\nthe Carl Zeiss Foundation (VerneDCt – Project-ID 2011640173)the system. In Berger and Reis (2018), Lanza (2022) a so-\ncalled funnel pre-compensator is used to provide auxiliary\nderivative signals, which are then used by the controller.\nFor both approaches it was shown that the conjunction\nof the filter with the system to be controlled results in\nan overall system which is amenable for funnel control.\nEarly results on tracking with prescribed accuracy while\nonly using output information (no derivative informa-\ntion) were achieved in e.g. Miller and Davison (1991) for\nsingle-input single-output LTI systems. Similar to funnel\ncontrol, though different, is prescribed performance con-\ntrol, see Bechlioulis and Rovithakis (2014), which also\nachieves the first control aspect (tracking with prescribed\nbehaviour). This controller is designed for systems in strict\nfeedback form and assumes availability of the system state.\nTo overcome the issue of accessibility to the state and\nuse output information only, in Dimanidis et al. (2020)\na high-gain observer is included in the controller design.\nSimilarly, in Chowdhury and Khalil (2019) and Liu et al.\n(2020) a high-gain observer is used in a funnel control\nscheme. The letter approaches, however, suffer from the\nfact that the high-gain parameters must be chosen a-priori\n“sufficiently large”, in particular, a sufficient lower bound\nis not provided.\nThe third aspect, namely that signals are available only\nat discrete sampling time instances, is related to redesign\nof controllers , cf. Neˇ si´ c and Gr¨ une (2005), Gr¨ une et al.\n(2008), Gr¨ une and Worthmann (2008). These techniques\naim to design piecewise constant, i.e., Zero-order Hold\n(ZoH), inputs such that the trajectories of the sampled-\ndata system converges to the continuous-time trajectories\nif the sampling interval tends to zero. In Lanza et al. (2023)\na ZoH feedback controller is proposed, which achieves\nobjectives (1) and (3). This means that the output yof a\ncontinuous-time system with arbitrary relative degree and\nstable internal dynamics tracks a reference trajectory yref\nwith prescribed performance while signals are availablearXiv:2402.05688v1  [math.OC]  8 Feb 2024only at discrete sampling times. Unlike the controller\nredesign, the feedback controller in Lanza et al. (2023) is\nproven to achieve the control objective for a fixed sampling\ninterval of length τ >0, i.e., the feasibility analysis does\nnot use the limit τ→0. However, like the continuously\nupdated funnel controller, the controller in Lanza et al.\n(2023) relies on the availability of higher order derivatives\nof the system’s output at sampling times.\nIn the present article we also take the issue of availability of\nderivatives into account, i.e., we include objective (2). We\npropose a ZoH feedback controller, similar to that in Lanza\net al. (2023) which achieves control objectives (1) – (3).\nThis means that the output of an unknown nonlinear\ncontinuous-time system with relative degree and stable\ninternal dynamics follows a reference trajectory with pre-\nscribed performance while output measurements are avail-\nable only at discrete sampling times, and no access to\nthe output’s derivative is assumed. Since we do not use\nhigh-gain observers to approximate the derivatives and no\nfurther filters are involved, this is a novelty in high-gain\nfeedback tracking control with prescribed performance.\nTo avoid that the main idea is obscured by technicalities,\nin this article we study the case of systems with relative\ndegree two. However, we are confident that a generalisation\nof the proposed method to systems with higher relative\ndegree is possible.\nThe remainder of this article is structured as follows. In\nSection 2 we formulate the control objective, and intro-\nduce the system class under consideration. In Section 3\nwe present the reasoning, which eventually leads to the\nproposed controller. Section 4 contains the main result,\nnamely a statement ensuring feasibility of the developed\ncontroller. We demonstrate successful functioning of the\nderivative-free ZoH feedback controller by means of a nu-\nmerical example in Section 5. A summering paragraph and\nan outlook to related future research in Section 6 concludes\nthe article. The proof of the main result is placed in the\nappendix.\nNotation. R≥0:= [0 ,∞). The standard inner product\nonRnis denoted by ⟨·,·⟩,∥x∥:=p\n⟨x, x⟩forx∈\nRn. For an interval I⊂R,L∞(I,Rn) is the space of\nmeasurable essentially bounded functions f:I→Rnwith\nnorm ∥f∥= ess supt∈I∥f(t)∥.L∞\nloc(I,Rn) is the space of\nlocally bounded measurable functions. Wk,∞(I,Rn) is the\nSobolev space of all k-times weakly differentiable functions\nf:I→Rnwith f, . . . , f(k)∈L∞(I,Rn).\n2. PROBLEM STATEMENT\nWe introduce the class of systems to be investigated, and\nformulate the control objective.\n2.1 System class\nIn this article we consider multi-input multi-output\ncontinuous-time unknown nonlinear dynamical systems in\ninput-output form with stable internal dynamics\n¨y(t) =f(d(t), y(t),˙y(t), η(t))+g(d(t), y(t),˙y(t), η(t))u(t),\n˙η(t) =h(η(t), y(t),˙y(t)), η(0) = η0∈Rl,\ny|[−τ0,0]=y0∈ C([−τ0,0],Rm), (1)where yrepresents the output of the system and uis the\ninput. The value τ0>0 is the “memory” of the system, i.e.\nthe initial condition is given by the initial trajectory y0,\nandηis the internal state not seen directly in the output.\nThe quantity d∈L∞(R≥0;Rp) is an unknown bounded\ndisturbance, f∈ C(Rp×Rm×Rm×Rl;Rm) is a drift\nterm, and the input is distributed by the matrix-valued\nfunction g∈ C(Rp×Rm×Rm×Rl;Rm×m). The internal\ndynamics are governed by h∈ C(Rl×Rm×Rm;Rl).\nAssumption 1. The matrix valued function g∈ C(Rp×\nRm×Rm×Rl;Rm×m) is strictly positive definite, i.e.,\n∀x∈Rp+2m+l∀z∈Rm\\ {0}:⟨z, g(x)z⟩>0.\nNote that changing the sign in (13) accounts for strictly\nnegative g.\nAssumption 2. The internal dynamics are bounded-input\nbounded-state stable, i.e., for all c0>0 there exists c1>0\nsuch that\n∥(ξ, ζ)∥∞≤c0⇒ ∥η(·; 0, ξ, ζ)∥∞≤c1, (2)\nwhere η(·,0, ξ, ζ) denotes the unique maximal solution of\nthe second equation in (1).\nIn fact, for ξ, ζ∈L∞(R≥0,Rm) Assumption 2 ensures\nthe existence of a unique global solution of the internal\ndynamics.\nRemark 2.1. Note that for sufficiently smooth functions\n˜f,˜g,˜h, cf. Schwartz et al. (1999), system (1) is equivalent\nto a state space system\n˙x(t) =˜f(x(t), d(t)) + ˜g(x(t), d(t))u(t), x(0) = x0∈Rm+l\ny(t) =˜h(x(t)),\nvia Byrnes-Isidori representation, cf. Byrnes and Isidori\n(1991).\n2.2 Control objective\nWe aim to design an output feedback ZoH controller, i.e.,\na feedback law, which, using only sampled output data\n(no output derivative information), produces a piecewise\nconstant control signal\nu(t) = ˆu∀t∈[tk, tk+τ), (3)\nwhere τ >0 is the length of the sampling interval, which\nachieves that the output yof system (1) follows a given\nreference signal yref∈W2,∞(R≥0,Rm) with predefined\nprecision in the transient phase. The latter means that\nthe tracking error e(t) :=y(t)−yref(t) satisfies\n∀t≥0 :φ(t)∥e(t)∥<1, (4)\nwhere φbelongs to the set\nΦ :={φ∈W1,∞(R≥0,R)|inf\ns≥0φ(s)>0}.\nThe control objective (4) is illustrated in Figure 1.\nThe (possibly time-varying) error tolerance φis chosen by\nthe control engineer, i.e., it is a design parameter.\n3. CONTROLLER DESIGN\nWe present a ZoH feedback controller, which achieves\nthe control task (4) for systems (1) by output feedback\nonly, i.e., the controller does not involve output derivative\ninformation.t∥e∥ψ\nFig. 1. Evolution of the error ewithin boundary ψ= 1/φ.\n3.1 Preliminary calculations\nTo propose the control law properly, we record some\nobservations and calculations. First, for e=y−yrefand\nφ∈Φ we define the following auxiliary signals\ne1(t) :=φ(t)e(t),\ne2(t) :=φ(t) ˙e(t) +α(∥e1(t)∥2)e1(t),(5)\nwhere α: [0,1)→[1,∞) is a bijection chosen by the\ncontrol engineer; a suitable choice is α(s) = 1 /(1−s).\nNote that e2involves the derivative ˙ e(t), and thus ˙ y(t).\nThe controller in Lanza et al. (2023) directly makes use\nofe2(tk) at sampling times tk=kτ,k∈N. The idea\nin the present article is to substitute the derivative by\nfinite differences and a respective resolvent term. We aim\nto find suitable relations between the error’s derivative ˙ e(t)\nand its approximation via finite differences. From Taylor’s\ntheorem using the Lagrange form of the remainder we\nknow for τ=tk−tk−1\ne(tk−1) =e(tk) + (tk−1−tk) ˙e(tk) +1\n2(tk−1−tk)2¨e(σ),\nfor some σ∈(tk−1, tk). This yields\n˙e(tk) =e(tk)−e(tk−1)\nτ+τ\n2¨e(σ).\nSo we can express e2(tk) in terms of e(tk−1), e(tk) and the\nsystem dynamics (1) by\ne2(tk) =φ(tk)e(tk)−e(tk−1)\nτ+α(∥e1(tk)∥2)e1(tk)\n+φ(tk)τ\n2¨e(σ)\n=φ(tk)e(tk)−e(tk−1)\nτ+α(∥e1(tk)∥2)e1(tk)\n+φ(tk)τ\n2\u0000˜f(σ) + ˜g(σ)uk−1\u0001\n,(6)\nfor some σ∈(tk−1, tk), where with some abuse of notation\nwe use ˜f(σ) := f(d(σ), y(σ),˙y(σ), η(σ)), and respective\nfor ˜g(σ). Here, uk−1∈Rmdenotes the input vector on\nthe previous interval [ tk−1, tk). Considering the first two\naddends of e2(tk), we define for tk−1=tk−τthe quantity\nE(tk) :=φ(tk)e(tk)−e(tk−τ)\nτ+α(∥e1(tk)∥2)e1(tk),(7)\nwhich will be used in the ZoH feedback law. From (7) it is\nclear that an initial trajectory on [ −τ,0] is required; more\nprecise, a measurement at t=−τis needed.\nBefore we proceed designing the controller, we recall the\nfollowing two results. The first lemma gives worst case\nestimates on the system dynamics when (4) is satisfied,\nand∥e2(t)∥ ≤ 1. The precise formulation requires some\ntechnical notation. To avoid these technicalities, we refer\nto (Lanza et al., 2023, Lem. 2.2), where the statement is\ngiven in a general setting.Lemma 3.1. LetI⊂R≥0be an interval. If ∥e1(t)∥<1\nand∥e2(t)∥ ≤1 for all t∈I, then there exist constants\nfmax, gmax, gmin>0 such that\nfmax≥ ∥f(d, y,˙y, η)|I∥∞,\ngmax≥ ∥g(d, y,˙y, η)|I∥∞,\ngmin≤⟨ζ, g(d, y,˙y, η)|Iζ⟩\n∥ζ∥2, ζ∈Rm\\ {0},(8)\nwhere f(d, y,˙y, η)|Imeans restricting all arguments of f\ntoI; respectively for g.\nNote that the existence of the constants is a result, not\nan assumption. Later we will make use of these worst case\nestimates, but we stress that knowledge of d, f, g, h is not\nassumed. The second result yields a global upper bound\non the auxiliary variable e1. For unique ξ∈(0,1) let\nα(ξ2)ξ= 1 +\r\r\r\r˙φ\nφ\r\r\r\r\n∞, ε 1:= max {∥e1(0)∥, ξ}<1.(9)\nThen, we have the following estimate.\nLemma 3.2. LetI⊂R≥0be an interval, ∥e(0)∥<1,\nandε1as in (9). If for all t∈Iwe have ∥e2(t)∥ ≤ 1,\nthen∥e1(t)∥ ≤ε1for all t∈I.\nThe proofs of Lemmas 3.1 and 3.2 are provided in Lanza\net al. (2023).\n3.2 Design parameters\nTo proceed deriving the derivative-free ZoH feedback con-\ntroller, we introduce some control design parameters. In-\nvoking Lemmas 3.1 and 3.2 we define the constants\n¯γ:=\u0000\n2α′(ε2\n1)ε2\n1+α(ε2\n1)\u0001\u0000\nα(ξ2)ξ+α(ε2\n1)ε1\u0001\n, (10)\nκ0:=\r\r\r\r˙φ\nφ\r\r\r\r\n∞(1 +α(ε2\n1)ε1) +∥φ∥∞(fmax+∥¨yref∥∞) + ¯γ.\nInvoking 1 /infs≥0φ(s) =∥φ∥∞, forλ∈(0,1) we define\nˆε:=∥φ∥∞\u0000\n1 +α(ε2\n1)ε1\u0001\n,\nˆE:=∥φ∥∞ˆε+α(ε2\n1)ε1,\nβ≥2∥φ∥∞κ0\ngmin\n˜F:=1\n2\u0000\nfmax+gmax∥φ∥∞β/λ\u0001\n,\nκ1:=κ0+∥φ∥∞βgmax.(11)\nWith the constants given in (9), (10), and (11) we may now\nintroduce an upper bound on the sampling time τ > 0,\nnamely\nτ≤min\n\ninfs≥0φ(s)gminβ−2κ0\n∥φ∥2∞˜FˆE,\nκ0\nκ2\n1,\n1−λ\n∥φ∥∞(˜F+gmaxλ) +κ0\n\n. (12)\n3.3 Derivative-free sampled-data zero-order hold feedback\nWe are now in the position to present the ZoH output\nfeedback controller, which applied to a system (1) achieves\nthe control objective (4). For E(·) introduced in (7), β >0chosen according to (11), λ∈(0,1), and a sampling rate\nnot slower than 1 /τforτsatisfying (12), set\n∀t∈[tk, tk+τ) :u(t)=\n\n−βE(tk),∥E(tk)∥< λ,\n−βE(tk)\n∥E(tk)∥2,∥E(tk)∥ ≥λ.(13)\nNote that in this feedback law only past and current out-\nput measurements are used. Moreover, (13) immediately\nyields an upper bound on the input, namely ∥u∥∞≤β/λ.\n4. MAIN RESULT\nWe present the main result of this article, namely feasibil-\nity of the control law (13). To phrase it, Theorem 4.1 states\nthat the ZoH feedback law (13) applied to an unknown\nnonlinear continuous-time system (1) achieves the con-\ntrol objective of output reference tracking with predefined\naccuracy (4) while the controller receives only sampled\noutput data. The latter means that no measurements of\nthe output’s derivative are required for the controller.\nTheorem 4.1. Let a reference yref∈W2,∞(R≥0,Rm) and\na funnel function φ∈Φ be given. Consider a system (1)\nwith initial trajectory y0: [−τ,0]→Rmsuch that\ny0(−τ) =yref(−τ), y0(0) = yref(0). (14)\nThen the controller (13) applied to system (1) is initially\nand recursively feasible, i.e., ∥e2(t)∥ ≤1 and ∥e1(t)∥<1\nfor all t≥0. The latter means satisfaction of the control\nobjective (4).\nThe proof is relegated to the appendix.\nNote that the equality initial condition (14) is artificial to\nensure u−1= 0. In applications, the latter can be achieved\nby choice.\n5. NUMERICAL EXAMPLE\nWe illustrate the proposed controller by a numerical ex-\nample. To this end, we consider the widely used mass\non car system, cf. Seifried and Blajer (2013). On a car\nwith mass m1a second mass m2is passively sliding on\na ramp. The second mass is coupled to the car via a\nspring-damper combination, and the ramp is inclined by\nan angle ϑ∈(0, π/2), see Figure 2.\nF\nya=consts\nFig. 2. Mass-on-car system. The figure is based on Seifried\nand Blajer (2013), Berger et al. (2021).\nThe equations of motion in input/output form read\n¨y(t) =R0y(t) +R1˙y(t) +Sη(t) + Γ u(t),\n˙η(t) =Qη(t) +Py(t),\nfor matrices R0, R1, S, Q, P with appropriate dimensions,\nwhere Qis Hurwitz, and Γ >0. We refer to e.g. Berger\net al. (2021) for a detailed derivation of the upper equa-\ntions. Note that this system has bounded-input bounded-\nstate stable internal dynamics since Qis a stable matrix.\n0 1 200.250.5\n1 1.030.40.41Fig. 3. Tracking result for control laws (13) and (15).\n0 1 2-4-201\n1 1.03-3.83-3.81-3.79\nFig. 4. Input signals.\nFor simulation purpose we choose yref= 0.4 sin( π/2t), a\nconstant error tolerance ψ= 1/φ= 0.08, and simulate on\nthe interval [0 ,2]. The calculations to obtain the worst case\ndynamics are performed in Lanza et al. (2023) thoroughly,\nso we use these estimates here. Starting on the reference\nwe have e(0) = 0, and we assume that the system satisfies\ny(−τ) =yref(−τ), i.e., initially u−1= 0.\nExample 1. Choosing λ= 0.7 conditions (11) and (12) are\nsatisfied with β= 25 .2 and τ= 1.8·10−3. We compare\nthe controller (13) with the controller from Lanza et al.\n(2023), i.e., we compare to the case if the output derivative\nis available. Following (Lanza et al., 2023, Rem. 5.2), we\nuse\nuZoH+deriv (t) =\n\n−βe2(tk), ∥e2(tk)∥< λ,\n−βe2(tk)\n∥e2(tk)∥2,∥e2(tk)∥ ≥λ,(15)\nwhere e2is given by (5). The simulation results are\ndepicted in Figures 3 and 4. Both controllers (13) and (15)\nachieve the control objective (4) with similar input values.\nExample 2. The worst case estimates fmax, gmax, gmin\nfrom Lemma 3.1 are very conservative which causes the\ncontrol parameter βfrom (11) to be large, and the suffi-\ncient sampling time τfrom (12) to be small. We illustrate\nthat both controllers from Example 1 also work for less\nconservative parameters. To this end, we run the same sim-\nulation as in Example 1 but with β= 5 and τ= 0.07, i.e.,\nthe sampling is about forty times slower. The simulation0 1 200.250.5Fig. 5. Tracking result for control laws (13) and (15).\n0 1 2-8-402\nFig. 6. Input signals with slower sampling.\nresults are shown in Figures 5 and 6. While in Example 1\nthe control signals for the controller (13) and (15) are\nalmost identical, they differ much in the second case, see\nFigure 6. The tracking behaviour in Figure 5 is still similar\nfor both controllers. Notably, the tracking performances in\nFigure 3 and Figure 5 are similar. This further shows that\nthe proposed controller works well even if the sampling\nfrequency is much slower than 1 /τin (12). However, since\nin this case the conditions of Theorem 4.1 are not satisfied,\nno guarantees can be given.\n6. CONCLUSION AND OUTLOOK\nWe proposed a sampled-data zero-order hold output feed-\nback controller, which achieves reference tracking with pre-\nscribed performance, i.e., the control objective (4) is satis-\nfied. In contrast to previously proposed controllers, Hackl\net al. (2013), Berger et al. (2021), Lanza et al. (2023) the\nZoH controller (13) does not rely on the availability of\noutput derivative measurement, but uses finite difference\napproximations. Feasibility of the proposed controller was\nproved rigorously. Since approximation of higher order\nderivatives is possible with finite differences, we are con-\nvinced that a generalisation of the proposed controller\nto systems with higher relative degree is straightforward.\nHowever, the sampling time τwill decrease with increasing\nrelative degree as can be already seen from the analysis\nin Lanza et al. (2023). We are confident that finite differ-\nences can also be used to avoid incorporation of derivativesignals in the continuous-time funnel controller Berger\net al. (2021). This will be topic of future research.\nAppendix A. PROOF OF THEOREM 4.1\nFirst, we provide some observations and estimates to be\nused in the main proof. Invoking Lemma 3.2 we may\ndirectly derive the following estimate on the the derivative\nof the error signal\n∥˙e∥∞≤sups≥0∥e2(s)∥+ sups≥0∥α(∥e1(s)∥2)∥e1(s)∥\ninfs≥0φ(s)\n≤1 +α(ε2\n1)ε1\ninfs≥0φ(s)= ˆε, (A.1)\nThe previous estimate yields an upper bound for the\nquantity E(·) in (7), independent of τ, namely\n∥E∥∞≤ ∥φ∥∞ˆε+α(ε2\n1)ε1=ˆE. (A.2)\nFurther, for tk∈R≥0we may formally estimate e2(tk) by\n∥e2(tk)∥ ≤ ∥ E(tk)∥+τ∥φ∥∞˜F. (A.3)\nNow we present the proof of Theorem 4.1.\nProof. [Proof of Theorem 4.1] The proof consists of two\nmain steps. In the first step we establish the existence\nof a local solution of the closed-loop initial value prob-\nlem (1), (13). In the second step we show feasibility of the\nproposed control law, i.e., ∥e2(t)∥ ≤1, and ∥e1(t)∥<1 for\nallt≥0. The latter means that the tracking error evolves\nwithin the funnel boundaries.\nStep 1. The application of the control signal (13) to sys-\ntem (1) leads to an initial value problem. If this prob-\nlem is considered on the interval [0 , τ], then there exists\na unique maximal solution on [0 , ω) with ω∈(0, τ].\nIf the error variables e1(t),e2(t) evolve within the set\n{ξ∈Rm| ∥ξ∥<1}for all t∈[0, ω), then ∥(y(·),˙y(·))∥\nis bounded on the interval [0 , ω), and by Assumption 2\nη(·, y,˙y) is bounded as well. Then ω=τ, cf. (Walter,\n1998, §10, Thm. XX) and nothing is left to show. Seeking\na contradiction, we assume the existence of t′∈[0, ω)\nsuch that ∥eℓ(t′)∥>1 for at least one ℓ∈ {1,2}. In-\nvoking Lemma 3.2 it suffices to show ∥e2(t)∥ ≤ 1 for\nallt∈[0, ω). Before we do so, we record the following\nobservation. For γ1(t) :=α(∥e1(t)∥2)e1(t) we calculate for\nz(·) := ( d(·), y(·),˙y)(·), η(·))\n˙e2(t)−φ(t)g(z(t))u= ˙φ(t) ˙e(t) +φ(t)¨e(t)\n+ ˙γ1(t)−φ(t)g(z(t))u\n=˙φ(t)\nφ(t)(e2(t)−γ2(t)) + ˙γ1(t)\n+φ(t)(f(z(t))−¨yref(t)) =: J(t).(A.4)\nStep 2. We show ∥e2(t)∥ ≤1 for all t∈[0, ω). Since the\nanalysis does not depend on the actual time step tk, we\nmay set tk= 0, and investigate the two cases ∥E(0)∥< λ\nand∥E(0)∥ ≥λ(although ∥E(0)∥< λby assumption).\nStep 2.a We consider ∥E(0)∥< λ . In this case we\nhave u0=−βE(0), and possibly u−1̸= 0 satisfying\n∥u−1∥ ≤ β/λ. So (A.3) is valid for t=t0= 0. Seek-\ning a contradiction, we suppose the existence of t∗:=\ninf{t∈(0, ω)| ∥e2(t)∥>1}. For the function J(·) in-\ntroduced in (A.4) we observe ∥J|[0,t∗)∥∞≤κ0according\nto Lemmas 3.1 and 3.2. Then we calculate for t∈[0, t∗]1<∥e2(t∗)∥ ≤ ∥ E(0)∥+τ∥φ∥∞˜F+Rt∗\n0∥˙e2(s)∥ds\n(A.3)\n≤ ∥E(0)∥+τ∥φ∥∞˜F\n+Rt∗\n0∥J(s)∥+∥φ∥∞gmax∥E(0)∥ds\n≤ ∥E(0)∥+τ∥φ∥∞˜F+Rt∗\n0κ0∥φ∥∞gmax∥E(0)∥ds\n< λ+τ∥φ∥∞˜F+κ0ω+∥φ∥∞gmaxλτ < 1,\nwhere we used t∗< ω≤τ≤(1−λ)/(κ0+∥φ∥∞˜F+\n∥φ∥∞gmaxλ) by (12). This contradicts the definition of t∗.\nStep 2.b We consider ∥E(0)∥ ≥ λ. In this case we have\nthe control u=−βE(0)/∥E(0)∥2, and possibly u−1̸= 0\nwith∥u−1∥ ≤ β/λ. Again, we show ∥e2(t)∥ ≤ 1 for\nallt∈[0, ω). To this end, seeking a contradiction, we\nsuppose the existence of t∗= inf{(0, ω)| ∥e2(t)∥>1}.\nInvoking the initial conditions, in particular ∥e2(0)∥ ≤1,\nand continuity of the involved functions, as well as utilising\nLemma 3.1 and (A.4), we calculate for t∈[0, t∗]\nd\ndt1\n2∥e2(t)∥2=⟨e2(t),˙e2(t)⟩=D\ne2(0) +Rt\n0˙e2(s) ds,˙e2(t)E\n≤ ∥e2(0)∥∥J(t)∥+ω∥˙e2|[0,t∗]∥2\n∞+φ(t)⟨e2(0), g(z(t))u⟩\n(6),(7)\n≤ ∥ J(t)∥+ω∥˙e2|[0,t∗]∥2\n∞+τ∥φ∥2\n∞˜Fβ∥E(0)∥\n−φ(t)β⟨E(0),g(z(t))E(0)⟩\n∥E(0)∥2\n≤κ0+ω∥˙e2|[0,t∗]∥2\n∞+τ∥φ∥2\n∞˜FβˆE\n−inf\ns≥0φ(s)gminβ\n≤κ0+ωκ2\n1+τ∥φ∥2\n∞˜FβˆE−inf\ns≥0φ(s)gminβ\n≤τ(κ2\n1+∥φ∥2\n∞˜FˆEβ) +κ0−inf\ns≥0φ(s)gminβ\n< τ∥φ∥∞˜FˆEβ+ 2κ0−inf\ns≥0φ(s)gminβ <0\nthe penultimate line by ω≤τ, and the last line by\ndefinitions of βand τ. Further, ∥˙e2|[0,t∗]∥ ≤ κ1was\nused, and ∥J|[0,t∗]∥∞≤κ0. In particular, this yields\n1\n2d\ndt∥e2(t)|t=0∥2<0, by which t∗>0. Therefore, we find\nthe contradiction 1 <∥e2(t∗)∥2<∥e2(0)∥2≤1. Repeated\napplication of the arguments in Steps 1 and 2on the\ninterval [ tk, tk+τ],k∈N, yields recursive feasibility.\nACKNOWLEDGEMENTS\nI thank my colleague Dario Dennst¨ adt (U Paderborn /\nTU Ilmenau) for fruitful discussions.\nREFERENCES\nBechlioulis, C.P. and Rovithakis, G.A. (2014). A low-\ncomplexity global approximation-free control scheme\nwith prescribed performance for unknown pure feedback\nsystems. Automatica , 50(4), 1217–1226.\nBerger, T., Ilchmann, A., and Ryan, E.P. (2021). Fun-\nnel control of nonlinear systems. Mathematics of\nControl, Signals, and Systems , 33(1), 151–194. doi:\n10.1007/s00498-021-00277-z.\nBerger, T., Lˆ e, H.H., and Reis, T. (2018). Fun-\nnel control for nonlinear systems with known strict\nrelative degree. Automatica , 87, 345–357. doi:\n10.1016/j.automatica.2017.10.017.\nBerger, T. and Reis, T. (2018). The funnel pre-\ncompensator. International Journal of Robust and Non-\nlinear Control , 28(16), 4747–4771.Byrnes, C.I. and Isidori, A. (1991). Asymptotic stabi-\nlization of minimum phase nonlinear systems. IEEE\nTransactions on Automatic Control , 36(10), 1122–1137.\nChowdhury, D. and Khalil, H.K. (2019). Funnel control for\nnonlinear systems with arbitrary relative degree using\nhigh-gain observers. Automatica , 105, 107–116.\nDimanidis, I.S., Bechlioulis, C.P., and Rovithakis, G.A.\n(2020). Output feedback approximation-free prescribed\nperformance tracking control for uncertain MIMO non-\nlinear systems. IEEE Transactions on Automatic Con-\ntrol, 65(12), 5058–5069.\nGr¨ une, L. and Worthmann, K. (2008). Sampled-data re-\ndesign for nonlinear multi-input systems. In Geometric\nControl And Nonsmooth Analysis: In Honor of the 73rd\nBirthday of H Hermes and of the 71st Birthday of RT\nRockafellar , 206–227. World Scientific.\nGr¨ une, L., Worthmann, K., and Neˇ si´ c, D. (2008).\nContinuous-time controller redesign for digital imple-\nmentation: A trajectory based approach. Automatica ,\n44(1), 225–232.\nHackl, C.M., Hopfe, N., Ilchmann, A., Mueller, M., and\nTrenn, S. (2013). Funnel control for systems with\nrelative degree two. SIAM Journal on Control and\nOptimization , 51(2), 965–995.\nIlchmann, A., Ryan, E.P., and Townsend, P. (2006). Track-\ning control with prescribed transient behaviour for sys-\ntems of known relative degree. Systems & Control\nLetters , 55(5), 396–406.\nLanza, L. (2022). Output feedback control with prescribed\nperformance via funnel pre-compensator. Mathematics\nof Control, Signals, and Systems , 34(4), 715–758. doi:\n10.1007/s00498-022-00322-5.\nLanza, L., Dennst¨ adt, D., Worthmann, K., Schmitz, P.,\nS ¸en, G.D., Trenn, S., and Schaller, M. (2023). Control\nand safe continual learning of output-constrained non-\nlinear systems. arXiv: 2303.00523 . Submitted, preprint\navailable.\nLiu, Y.H., Su, C.Y., and Li, H. (2020). Adaptive output\nfeedback funnel control of uncertain nonlinear systems\nwith arbitrary relative degree. IEEE Transactions on\nAutomatic Control , 66(6), 2854–2860.\nMiller, D.E. and Davison, E.J. (1991). An adaptive con-\ntroller which provides an arbitrarily good transient and\nsteady-state response. IEEE Transactions on Automatic\nControl , 36(1), 68–81.\nNeˇ si´ c, D. and Gr¨ une, L. (2005). Lyapunov-based\ncontinuous-time nonlinear controller redesign for\nsampled-data implementation. Automatica , 41(7),\n1143–1156.\nSchwartz, B., Isidori, A., and Tarn, T.J. (1999). Global\nnormal forms for MIMO nonlinear systems, with ap-\nplications to stabilization and disturbance attenuation.\nMathematics of Control, Signals and Systems , 12, 121–\n142.\nSeifried, R. and Blajer, W. (2013). Analysis of servo-\nconstraint problems for underactuated multibody sys-\ntems. Mechanical Sciences , 4(1), 113–129.\nWalter, W. (1998). Ordinary Differential Equations .\nSpringer-Verlag, New York."    },    {        "title": "2402.05696v1.Fixed_width_treelike_neural_networks_capacity_analysis____generic_activations.pdf",        "content": "arXiv:2402.05696v1  [stat.ML]  8 Feb 2024Fixed width treelike neural networks capacity analysis – ge neric\nactivations\nMihailo Stojnic∗\nAbstract\nWe consider the capacity of treelike committee machines (TCM) neural networks. Relying on Random\nDuality Theory (RDT), [ 42] recently introduced a generic framework for their capacit y analysis. An upgrade\nbased on the so-called partially lifted RDT (pl RDT) was then presented in [ 45]. Both lines of work focused\non the networks with the most typical, sign, activations. Here, on the other hand, we focus on networks w ith\nother, more general, types of activations and show that the f rameworks of [ 42,45] are sufficiently powerful\nto enable handling of such scenarios as well. In addition to t he standard linear activations, we uncover that\nparticularly convenient results can be obtained for two ver y commonly used activations, namely, the quadratic\nandrectified linear unit (ReLU) ones. In more concrete terms, for each of these activations, we obtain both\nthe RDT and pl RDT based memory capacities upper bound charac terization for anygiven (even) number\nof the hidden layer neurons, d. In the process, we also uncover the following two, rather re markable, facts: 1)\ncontrary to the common wisdom, both sets of results show that the bounding capacity decreases for large d\n(the width of the hidden layer) while converging to a constan t value; and 2) the maximum bounding capacity\nis achieved for the networks with precisely twohidden layer neurons! Moreover, the large dconverging values\nare observed to be in excellent agrement with the statistica l physics replica theory based predictions.\nIndex Terms: TCM neural networks; Capacity; Lifted random d uality theory; Different acti-\nvations .\n1 Introduction\nDemand for efficient handling and interpretation of large dat a sets, has grown rather rapidly over the last 15-\n20 years. Machine learning (ML) clearly distinguished itse lf as a particularly helpful set of concepts capable\nof providing the needed technical and scientific resources t o meet such a high demand. Consequently, a fast\ndevelopment of various ML branches ensued. Neural networks (NN), as one of such branches, quickly became\none of the focuses of strong research interests and among the fastest growing research fields. Many great\nresults and quite a few remarkable breakthroughs that relat e to both theoretical and practical NN aspects\nhave been obtained. Moreover, many of the well known results achieved in prior decades – that for a long\ntime served as academic prototypes – have been revisited and brought to practical usability. Among the\nmost prominent of them are certainly those that relate to one of the key NN features, the so-called, network’s\nmemory capacity (see, e.g., [ 3,8,9,16,22,26,48,50–52]). Here we continue the same trend and focus on several\nimportant capacity related questions and provide a strong t heoretical progress. Before discussing in more\ndetail some of the main problems and our technical contribut ions, we find it convenient to first introduce\nthe basics of the NN models of our interest.\n1.1 Feed forward neural networks – mathematical basics\nTo be able to properly introduce the network’s memory capaci ty as the main object of our interest, we first\ndiscuss the underlying network architecture.\n∗e-mail:flatoyer@gmail.com\n1Architecture: We are interested in multilayered multi-input single-outp ut feed-forward neural networks\nwithL−2hidden layers and di(i∈ {1,2,...,L}) nodes (neurons) in the i-th layer. To ensure a notational\nfacilitation, two additional layers, indexed by i= 1andi=Lare artificially added and they correspond\nto the network input and output, respectively (although the input and output of the network are basically\nartificial NN layers, to be in agreement with the introduced i ndexation, we will refer to them as networks\nlayers). The way the network operates is basically determin ed by specifying the vectors of threshold functions,\nf(i)(·) = [f(i)\n1(·),f(i)\n2(·),...,f(i)\ndi+1(·)]T. Each threshold function f(i)\nj(·) :Rdi→Rdescribes how neuron jin\nlayerioperates. The network effectively functions by taking the ou tputs of the nodes from layer ias the\ninputs of the nodes in layer i+1and transforming them into the new outputs (in layer i+1) via a linear\ncombination governed by the matrix of weights W(i)∈Rdi×di+1. After setting d= [d1,d2,...,d L](with\nd1=nanddL= 1) and denoting by b(i)∈Rdi+1,i= 1,2,...L the so-called thresholds vectors and by\nx(i)∈Rdiandx(i+1)∈Rdi+1the inputs and outputs of the neurons in layer i, one has the following:\nMathematical formalism of NN with architecture A(d,f(i)):\ninput:/definesx(1)−→ x(i+1)=f(i)(W(i)x(i)−b(i)) −→ output: /definesx(L+1).\n(1)\nThe architecture of the network, A(d;f(i)), is fully specified by the vectors dandf(i). Also, when the vectors\nof functions, f(i), are identical, we write A(d;f)instead of A(d;f(i)).\nMemory capacity: As mentioned earlier, one of the most fundamental features o f any neural net (including\nsingle neurons as special cases) is their ability to properl y memorize/store a large amount of data. A simple\nway to see how the above formalism achieves this is the follow ing: assume the existence of mdata pairs\n(x(0,k),y(0,k)),k∈ {1,2,...,m}, withx(0,k)∈Rnbeing the n-dimensional data vectors and y(0,k)∈Rbeing\ntheir associated labels. Determining matrices W(i)such that\nx(1)=x(0,k)=⇒x(L+1)=y(0,k)∀k, (2)\nis then sufficient to properly relate given data vectors to the ir corresponding labels. If the network archi-\ntectureA(d,f(i))is given then its memory capacity, C(A(d,f(i))),is defined as the largest sample size, m,\nsuch that (2) holds for any collection of data pairs (x(0,k),y(0,k)),k∈ {1,2,...,m}with certain prescribed\nproperties. Since the memory capacity plays one of the most i mportant roles in understanding the overall\nneural nets’ functioning mosaic, finding both, its precise t heoretical characterization and the corresponding\nfast algorithmic procedure that achieves it, is of utmost im portance. Of our particular interest in this paper\nare the theoretical aspects and we below provide a host of res ults that for many well known architectures\nalmost fully characterize their capacities.\nTo facilitate the presentation, a few structural and techni cal assumptions are in place as well. Many\nof them, however, are aligned with the ones discussed in [ 42,45]. To avoid an unnecessary repetition, we\nonly briefly recall on these and refer for a more detailed expo sition to [ 42,45]. On the other hand, we place\nthe most emphasis on those that are substantially different a nd particularly relevant to the results that we\npresent in this paper.\n1.2 Technical assumptions\nTo facilitate the exposition and to make the final results cle aner and easier to use, we will rely on several\narchitectural and data related assumptions. We state them b elow before starting the analytical considera-\ntions. As they are fairly common and rather prevalent in the l iterature, we avoid discussing them in deep\ndetails.\nNetwork architecture assumptions: 1-hidden layer treelike committee machine type of neural networks\nwithgeneric zero-threshold activation functions in the hidden layer ar e considered. This means the fol-\nlowing: (i)We assume L= 3,b(i)= 0,W(1)=In×n, andW(3)=wT, wherew∈Rd2×1(i.e.W(3)is a\nd2-dimensional row vector that will be specified depending on t he type of the considered activation functions).\n(ii)We also define d/definesd2andδ/definesδ1=d1\nd2=n\nd. While the presented mathematical concepts will hold for\n2anyd, to be able to get concrete capacity values, we eventually as sume that dis any (even) natural number.\n(iii)In the first layer, we consider identity neuronal functions, i.e. we take f(1)(x(1)) =x(1). In the hidden\nlayer, we take the generic zero-threshold activations f(2)\nj(·) :Rd2→Rwithf(2)\nj(·) =f(2)\nk(·)for anyj/ne}ationslash=k.\nFinally, at the output, we take zero-threshold sign activat ionf(3)(W(3)x(3)−b(3)) =sign/parenleftbig\nW(3)x(3)/parenrightbig\n. For\nnotational simplicity, we denote this architecture by A(d;[f(1),f(2),f(3)]) =A(d;[I,f(2),sign])/definesA(d;f(2)).\n(iv)The matrix W(i)can be generically full or with a particular structure. Both types of structuring have\nbeen of interest throughout the literature. A particular ty pe of sparse structuring, where the support of\nW(i)’sj-th row, supp/parenleftig\nW(i)\nj,:/parenrightig\n, satisfies supp/parenleftig\nW(i)\nj,:/parenrightig\n=S(j), withS(j)/defines{(j−1)δ+1,(j−1)δ+2,...,jδ},\nmakes the above architecture correspond to what is in the lit erature referred to as the treelike committee\nmachines (TCM). Precisely such architectures will be of our interest in this paper. For the completeness,\nwe also add that if the matrix is full then the above architect ure corresponds to what is in the literature\ntypically referred to as the fully connected committee machines (FCM).\nData related assumptions: (i) Binary labeling y(0,k)\ni∈ {−1,1}, as the most standard type of labeling,\nis assumed as well (choosing signperceptron as neuronal function at the output naturally com plements\nthe binary labeling choice as well). (ii)Inseparable data sets are not allowed (for example, indisti nguish-\nable/contradictory pairs (or subgroups) like (x(0,k),y(0,k))and(x(0,k),−y(0,k))can not appear). (iii)Data\nsets of statistical nature will be of our main interest. We pa rticularly focus on x(0,k)as iid standard normals.\nThis follows into the footsteps of the trend established in t he classical single perceptron references (see,\ne.g., [9,12,16,34,50–52]) and allows for, expectedly, a fairly universal statistic al treatment. It is also useful\nto note that for providing universal capacity upper bounds, any type of acceptable data set (including even\nnonstatistical ones) actually suffices.\n1.3 Prior work\nThe problems of our interest are well known and have been stud ied in various forms for almost 70 years.\nNaturally, the early studies related to the single neurons w hile the more recent ones emphasize the importance\nof understanding the multi-neuron or multi-layered archit ectures. Given the pace at which the ML and NN\nfields are developing, the underlying relevant literature i s rather vast and growing. We below highlight the\nresults that we view as most closely related to our own.\nSince the memory capacity of spherical perceptrons is direc tly connected to several fundamental questions\nin integral geometry, the early capacity considerations we re related to some of the geometrical/probabilistic\nclassic works (see, e.g., [ 9,22,26,50]). Possibly the most famous of them states that the capacity of the\nspherical signperceptrons doubles the dimension of the data ambient space, n, i.e.,C(A(1;sign))→2n\nasn→ ∞ . After being initially obtained as a remarkable closed form combinatorial geometry fact in\n[8,9,22,26,50–52], it was decades later rediscovered and reproved in various different forms in a plethora\nof different fields ranging from machine learning and pattern recognition to information theory, probability,\nand statistical physics (see, e.g., [ 3,10–12,16,33,34,40,48]).\nSign perceptrons networks: Despite the elegance of the single perceptron results, the c orresponding multi-\nperceptron ones are not easy to obtain. Particularly scarce are the TCM related ones. While a bit more\nis known about the FCM ones, a direct connection between the t wo is not apparent. Besides the trivial\nfact that the FCM capacities upper-bound the corresponding TCM ones, one may also (somewhat ad-hoc)\nview the TCM capacities as roughly the FCM ones divided by d. Although non necessarily rigorous (or\neven fully correct) such a viewing suggests a potential usef ulness of FCM results. Still, an overwhelming\nmajority of known results indicates that the memory capacit y is unavoidably related to the total number\nof the network weights, w=/summationtextL−1\ni=1didi+1. For example, the famous VC-dimension qualitative memory\ncapacity upper bound gives the scaling O(wlog(w)). It is interesting to note that for NNs with 1-hidden\nlayer,w=d1d2+d2= (n+ 1)dfor FCM and w=d1+d2=n+dfor TCM. This, on the other hand,\nfor large di’s and huge n, gives, the above mentioned, “ division by d” FCM – TCM capacity relation. A\ncouple of lower bounding results are known as well. For examp le, [7] argued that the capacity of a shallow\n3-layer network (similar to the one studied here) scales as O(nd). On the other hand, [ 49] obtained recently\na stronger version, where, for the networks with more than th ree layers, the capacity is shown to be roughly\nat leastO(w).\nObtaining precise results of non-scaling type in network architectures turned out to be a much harder\n3challenge. This seems particularly surprising given the si mplicity and elegance of the corresponding single\nperceptron ones. Until the very recent appearance of [ 42,45] there was hardly any mathematically rigorous\nresult that could provide even remotely close sign perceptr on networks capacity estimates. [ 42] utilized\nthe Random duality theory (RDT) and developed a generic fram ework for the analysis of TCM network\ncapacities. As a results of the framework, strong upper boun ds were obtained for any given (odd) number\nof the nodes in the hidden layer. [ 45] went a step further, utilized a partially lifted RDT varian t (pl RDT)\nand substantially lowered the upper bounds of [ 42].\nDifferent activations networks: Given the importance of a single sign perceptron, studying t heir merging\ninto a large architectural structure is a natural transitio n. Two things should be kept in mind when making\nsuch a transition though. First, it is not clear a priori that the results that hold for the single perceptron\nwill hold in a similar fashion for the network of perceptrons . Second, the sign perceptrons are not continuous\nfunctions and handling them algorithmically when using or t raining the network might impose computa-\ntionally/numerically unsurpassable obstacles. If on top o f that, one adds the above mentioned analytical\nhardness, the need for potentially less simple but easier to use activation factions is rather obvious.\nAs the discreteness is typically perceived as the main sourc e of both analytical and algorithmic sign\nperceptrons hardness, the natural choice for different acti vations leads towards allowing various continuous\nones as well. Many of them have already found a steady place in NN architectures. Examples include but\nare not limited to ReLU, quadratic, tanh, erf and so on. Such a ctivations make things a bit easier and,\nconsequently, a little bit more is known about their capacit ies. For example, [ 54] suggested for deep nets\nand [20] proved for 4-layer nets that the capacity is at least O(w)for sigmoids. [ 19,57] showed similar results\nfor ReLU while additionally restricting on the number of nod es. Such a restriction though was later on\nremoved in [ 55] for both tanh and ReLU.\nStatistical physics – Replica theory: Given that the precise capacity characterizations (as functions of\nthe number of the hidden layer neurons d) do not allow for any qualitative/scaling descriptions (say, of\ntheO(·)type), the mathematically rigorous results are often very h ard to achieve. In such situations,\nstatistical physics replica methods are an excellent (and o ften irreplaceable) tool to produce, non-rigorous,\nbut expectedly precise analyses. As the hardness of the precise analytical studyin g of various NN features\nhas been recognized in the mid-eighties of the last century, the application of the replica methods in capacity\ncharacterization has been around for close to four decades. The foundational concepts of such an approach\nwere laid out in the pioneering works [ 16,17], whre various forms of single perceptrons were discussed. Here,\nwe focus more on the ensuing ones that relate to the network ar chitectures. In particular, [ 5,14] studied the\nvery same, TCM architecture, as we do (as well as the above men tioned, related, FCM one). For the sign\nperceptrons, they obtained the closed form replica symmetr y based capacity predictions for any number of\nthe neurons in the hidden layer, d. They established the corresponding large dscaling behavior. Both of these\nresults were proven as mathematically rigorous capacity up per bounds in [ 42,45]. Moreover, [ 5,14] showed\nthat their replica symmetry based large dpredictions violate the mathematically rigorous ones obta ined\nthrough the uniform-bounding extension of [ 9,50–52] given in [ 15]. This contradiction was remedied in [ 5,14]\nby studying the first level of replica symmetry breaking (rsb ) and showing that it lowers the capacity. Related\nlargedscaling rsb considerations were also presented in [ 24] for both the committee and the so-called parity\nmachines (PM) (more on the earlier PM replica consideration s can be found in, e.g., [ 4,6]). Also, for the\nFCM architecture, a bit later, [ 47,53] obtained the large dscaling that matches the upper-bounding one\nof [15]. Particularly relevant to our work are two very recent line s of work. [ 2] first obtained the first level of\nrsb capacity predictions for the TCM architectures with the ReLU activations and [ 56] moved things even\nfurther and obtained similar rsb predictions for several di fferent activations, including the well known linear,\nReLU, erf, quadratic, and tanh. A key difference with respect to our results should also be noted. Namely,\nboth sets of results, [ 2] and [ 56], relate to the networks with large d(basically, to the networks with d→ ∞),\nwhereas our results are obtained for any given (even) d.\nPractical achievability: Another line of work attracted a lot of interest over the last several years and\nshould be mentioned as well. It is related to the design and an alysis of efficient algorithms that can be\nused to train the networks to potentially approach the capac ity. The key focus has been on showing that\nthe simple gradient based methods might actually perform qu ite well in this context. The so-called mild\nover-parametrization (moderately larger number of free pa rameters, w, compared to the size of the data set,\nm) is deemed as sufficing to ensure excellent performance of gra dient based methods. More on the recent\nprogress in these directions can be found in, e.g., [ 1,13,18,21,23,25,27,46,58]. These results mostly relate\n4to FCMs but are also extendable to TCMs as well.\n1.4 Contributions\nThe main object of our study is the so-called n-scaled memory capacity of the TCM NNs with various\n(different from standard signone) activation functions in the hidden layer. In other word s, we study\nc(d;f(2))/defineslim\nn→∞C(A([n,d,1];f(2)))\nn. (3)\nA very strong progress in characterizing C(A([n,d,1];sign))for any given (odd) dhas been made in [ 42].\nIn particular, utilizing the powerful Random Duality Theor y (RDT) mathematical engine, [ 42] provides an\nexplicit upper bound ˆc(d;sign)onc(d;sign). Numerical results obtained for smaller values of dsuggested\na strong benefit in adding more neurons in a network architect ure context. On the other hand, we, in this\npaper, make a substantial progress in several different aspe cts including both methodological and practical\nones.\nA summary of the main technical results of the paper: (i)We first show that the main framework from\n[42,45] (established relying on the RDT and pl RDt principles) for t he capacity analysis of sign perceptron\nnetworks can be utilized for different activations as well. (ii)To produce concrete capacity estimates, we then\nfocus on several particular activations for which neat and c onvenient results can be obtained. We fist start\nwith the linear activations and show that the capacity of the TCM network wit h one hidden layer network\nis identical to the single spherical perceptron. We then swi tch to the quadratic activations and obtain the\nclose form analytical RDT based capacity upper bounds and th eir refined pl RDT counterparts. Finally, we\nconsider the ReLU activations and obtain the corresponding analytical close d form RDT and pl RDT results.\n(iii)For all activations, we conduct the needed numerical evalua tions to obtain the concrete capacity values\nas well. (iv)All our results are obtained for an extremely challenging sc enario where the number of the\nhidden layer neurons, d, can be any even positive integer. This allows us to uncover t wo rather fascinating\nphenomena: 1) Both thr RDT and the pl RDT estimates are decrea sing for large dwhile converging to a\nconstant (not dependent on d) value; and 2) The maxima of both estimates for both quadrati c and ReLU\nactivations are achieved for d= 2neurons in the hidden layer. This is a bit contrary to the comm on wisdom\nand in a strike contrast with the corresponding behavior of t he sign activations where the capacity estimates\ngrow with d. In particular, one effectively has that when it comes to the m emory capabilities, uncontrollably\nincreasing the width of the hidden layer may not always be amo ng the most recommended architecture\nbuilding strategies.\nWe show some of the concrete capacity estimates that we obtai ned for all the three mentioned activations,\nlinear, quadratic, and ReLU, in Table 1. In Figure 1 we also vi sualize the quadratic ones as well. Few key\nsmallest values of dare shown explicitly in the table and a much wider range of dis shown in the figure (it\ngoes without saying that the quadratic activation for d= 1does not make sense). For the completeness,\nwe, in Figure 1, also show the d→ ∞ asymptotics obtained for quadratic activations utilizing the replica\nmethods in [ 56].\nTable 1: Theoretical estimates of thef(2)-activated hidden layer TCM capacity upper bounds\nActivation Upper bound on Methodology d\n(function) c(d;f(2))/defineslimn→∞C(A([n,d,1];f(2)))\nn1 2 4\nlinear ¯c(d;f(2)) RDT 2 2 2\n(f(2)(x) =x) ˆc(d;f(2)) pl RDT 2 2 2\nquadratic ¯c(d;f(2)) RDT −5.498 4.660\n(f(2)(x) =x2) ˆc(d;f(2)) pl RDT −4.065 3.657\nReLU ¯c(d;f(2)) RDT 2 3.810 3.066\n(f(2)(x) = max( x,0)) ˆc(d;f(2)) pl RDT 2 3.810 3.066\n5d5 10 15 20 25 30upper bounds on c(d;quad)\n33.544.555.5Quadratic activation – Memory capacity upper bound\nˆc(d;quad) – RDT\n¯c(d;quad) – RDT\nˆc(d;quad) – RDT; interpolated\n¯c(d;quad) – RDT; interpolated\nˆc(∞;quad) –Replica symmetry\n¯c(∞;quad) –Partial 1rsb\nˆc(∞,quad) = 4\n¯c(∞,quad)≈3.381Benefit ofpartially lifted RDT:\n∆c(d;quad) = ˆ c(d;quad)−¯c(d;quad)\nFigure 1: Memory capacity upper bound as a function of the num ber of neurons, d, in the hidden layer;\n1-hidden layer TCM with quadratic activations; plain RDT versus partially lifted RDT (Replica\nsymmetry (RS) andPartial 1rsb d→ ∞ estimates are included as well)\n2 Algebraic description of network data processing\nTo put everything on the right mathematical track and facili tate writing and overall presentation, we first\nsetW/definesW(2). After recalling that W(1)=IandW(3)=wT, we then have for any k∈ {1,2,...,m}\nx(1)=x(0,k)=⇒x(2)=f(1)(W(1)x(1)) =f(1)(x(1)) =x(1)=x(0,k), (4)\nand\nx(2)=x(0,k)=⇒x(3)=f(2)(W(2)x(2)) =f(2)(Wx(0,k)), (5)\nand\nx(3)=f(2)(Wx(0,k)) =⇒x(4)=f(3)(W(3)x(3)) =f(3)(wTx(3)) =sign(wTf(2)(Wx(0,k))).(6)\nA very neat closed-form explicit relation between the input and the output of the network can be obtained\nby connecting beginning in (4) and end in (6)\nx(1)=x(0,k)=⇒x(4)=sign(wTf(2)(Wx(0,k))). (7)\nFor network to operate properly, it is then both necessary an d sufficient that the following condition holds\ny(0,k)=sign(wTf(2)(Wx(0,k))). (8)\nAfter setting\ny/defines/bracketleftbig\ny(0,1)y(0,2)...y(0,m)/bracketrightbigTandX/defines/bracketleftbig\nx(0,1)x(0,2)...x(0,m)/bracketrightbigT, (9)\none can then rewrite (8) in generic matrix form as\n/parenleftig\n∃W∈Rd×n|/bardblyT−sign(wTf(2)(WXT))/bardbl2= 0/parenrightig\n⇐⇒((X,y)is memorized ), (10)\n6and thek-th data pair/parenleftbig\nx(0,k),y(0,k)/parenrightbig\nare thek-th rows of m×nmatrixXandm×1column vector y. This\nfurther leads to the following (effectively an alternative t o (10)):\nAlgebraic memorization characterization of the f(2)-activated hidden layer TCMs:\n0 =ξ/definesmin\nW,Q/bardbly−sign(f(2)(Q)w)/bardbl2\nsubject to XWT=Q (11)⇐⇒ Data set (X,y)is properly memorized.\nClearly, the above algebraic formulation is the key mathema tical problem on the path towards ensuring\nproper network data memorization. We therefore analyze it b elow in more detail.\n3 Random Duality Theory (RDT) based capacity analysis\nAs mentioned earlier, we consider statistical data sets wit h elements of Xbeing iid standard normals. Due\nto rotational symmetry, one can then, without a loss of gener ality, assume that the elements of yare all\nequal to 1, i.e. one can assume that y=1. The above key optimization can then be rewritten\nξ= min\nZ,Q/bardbl1−sign(f(2)(Q)w)/bardbl2\nsubject to XZ=Q, (12)\nwhere a cosmetic change, Z=WT, is introduced to facilitate writing. As in [ 42,45], we here consider\nthe TCM architecture, with a sparse Zthat ensures that treelike network architecture. This basi cally\nmeans that we consider Zsuch that the only nonzero elements of its j-th column are in rows S(j)/defines\n{(j−1)δ+ 1,(j−1)δ+ 2,...,jδ}. Utilizing this Zspecialization and given that the above problem is\ninsensitive with respect to the scaling of ZorQ, one can write\nξ= min\nZ,Q/bardbl1−sign(f(2)(Q)w)/bardbl2\nsubject to XZ=Q\n/bardblZ:,j/bardbl2= 1\nsupp(Z:,j) =S(j),1≤j≤d, (13)\nwith/bardblZ:,j/bardbl2being the norm of the j-th column of Z. A further trivial rewriting of the above gives\nξ= min\nz(j),Q/bardbl1−sign(f(2)(Q)w)/bardbl2\nsubject to X(j)z(j)=Q:,j,1≤j≤d,\n/bardblz(j)/bardbl2= 1\nz(j)∈Rδ,Q∈Rm×d, (14)\nwhereX(j)=X:,S(j)∈Rm×δ. We follow into the footsteps of [ 42,45] and to statistically analyze the\noptimizations in (13) and (14), we utilize the powerful math ematical engine called Random Duality Theory\n(RDT) developed in a long series of work [ 28,30–32,39]. To make the presentation easier to follow, we will try\nto parallel as closely as possible the approach presented in [42]. However, to avoid unnecessary repetitions, we\nonly briefly recall on some of the concepts that are identical or very similar to the corresponding ones of [ 42],\nand instead focus on key differences. As in [ 42], we start by first summarizing the RDT main principles and\nthen continue by discussing each of them within the context o f our interest here.\n7Summary of the RDT’s main principles [30,39]\n1)Finding underlying optimization algebraic representatio n2)Determining the random dual\n3)Handling the random dual 4)Double-checking strong random duality.\nAs in [ 42], all the key results (including both simple to more complic ated ones) are formulated as lemmas\nand theorems.\n1)Algebraic memorization characterization: The following lemma summarizes the above algebraic\ndiscussion by providing a precise resulting optimization r epresentation of the network memorization property\nand is a mirrored analogue to Lemma 1 in [ 42].\nLemma 1. (Algebraic optimization representation) Assume a 1-hidde n layer TCM with architecture\nA([n,d,1];f(2)). Any given data set/parenleftbig\nx(0,k),1/parenrightbig\nk=1:mcan not be properly memorized by the network if\nfrp(X)>0, (15)\nwhere\nfrp(X)/defines1√nmin\n/bardblz(j)/bardbl2=1,Qmax\nΛ∈Rm×d/bardbl1−sign(f(2)(Q)w)/bardbl2+d/summationdisplay\nj=1(Λ:,j)TX(j)z(j)−tr(ΛTQ), (16)\nandX/defines/bracketleftbig\nx(0,1)x(0,2)...x(0,m)/bracketrightbigT.\nProof. Immediate consequence of Lemma 1 in [ 42].\nOf our interest below is mathematically the most challengin g, so-called linear, regime with\nα/defineslim\nn→∞m\nn. (17)\nThe above lemma is of purely algebraic nature and as such it ho lds for any given data set/parenleftbig\nx(0,k),1/parenrightbig\nk=1:m.\nTo analyze (15) and (16), the RDT further proceeds by account ing for a statistical X.\n2)Determining the random dual: We follow the standard RDT practice and utilize the so-calle d concen-\ntration of measure property, which basically means that for any fixed ǫ >0, we have (see, e.g. [ 30,32,39,42,45])\nlim\nn→∞PX/parenleftbigg|frp(X)−EX(frp(X))|\nEX(frp(X))> ǫ/parenrightbigg\n−→0. (18)\nMoreover, the following so-called random dual theorem is an other key ingredient of the RDT machinery and\nis a mirrored alternative to Theorem 1 from [ 42].\nTheorem 1. (Memorization characterization via random dual) Let dbe any even positive integer. Consider\nTCM with dneurons in the hidden layer, and architecture A([n,d,1];f(2)), and let the elements of X∈Rm×n,\nG∈Rm×d, andH∈Rδ×dbe iid standard normals. Set\nφ(Q)/defines/bardbl1−sign(f(2)(Q)w)/bardbl2\nfrd(G,H)/defines1√nmin\nφ(Q)=0,/bardblz(j)/bardbl2=1max\nΛ∈Rm×d,/bardblΛ/bardblF=1\ntr(ΛTG)+d/summationdisplay\nj=1/bardblΛ:,j/bardbl2(H:,j)Tz(j)−tr(ΛTQ)\n\nφ0/defineslim\nn→∞EG,Hfrd(G,H). (19)\nOne then has\n(φ0>0) =⇒/parenleftig\nlim\nn→∞PX(frd>0)−→1/parenrightig\n=⇒/parenleftig\nlim\nn→∞PX(frp>0)−→1/parenrightig\n8=⇒/parenleftig\nlim\nn→∞PX(A([n,d,1];f(2))fails to memorize data set (X,1))−→1/parenrightig\n. (20)\nProof. Immediate consequence of Theorem 1 in [ 42].\n3)Handling the random dual: Proceeding as in [ 42], one first solves the inner maximization over Λand\nthen the minimization over z(j)to obtain for frd(G,H)from (19)\nfrd(G,H) =1√nmin\nφ(Q)=0/radicaltp/radicalvertex/radicalvertex/radicalbt/bardblG−Q/bardbl2\nF−2d/summationdisplay\nj=1/bardblG:,j−Q:,j/bardbl2/bardblH:,j/bardbl2+/bardblH/bardbl2\nF. (21)\nThis then further gives\nφ0= lim\nn→∞EG,Hfrd(G,H) = lim\nn→∞EG1√nmin\nφ(Q)=0/bardblG−Q/bardblF−1, (22)\nwhere, as discussed in [ 42], the above equality obtained relying on the concentration s can be replaced by an\ninequality if one alternatively relies on the Cauchy-Schwa rtz inequalities (both options are perfectly sufficient\nfor the subsequent analysis).\nThe remaining focus is on the residual optimization over Q. To that end we set\nφi(Qi,1:d)/definessign(f(2)(Qi,1:d)w), (23)\nand further write\nmin\nφ(Q)=0/bardblG−Q/bardblF=/radicaltp/radicalvertex/radicalvertex/radicalbtmin\nφ(Q)=0m/summationdisplay\ni=1d/summationdisplay\nj=1(Gij−Qij)2=/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1min\nφi(Qi,1:d)=1d/summationdisplay\nj=1(Gij−Qij)2\n=/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1zi(Gi,1:d), (24)\nwith\nzi(Gi,1:d)/definesmin\nφi(Qi,1:d)=1d/summationdisplay\nj=1(Gij−Qij)2. (25)\nTo further facilitate writing and remove unnecessary notat ions, we will set\ng/definesGT\ni,1:d\nq/definesQT\ni,1:d, (26)\nand\nzi(g;f(2))/defineszi(Gi,1:d) = min\nφi(Qi,1:d)=1d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(Qi,1:d)w≥0d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(qT)w≥0/bardblg−q/bardbl2\n2.\n(27)\nThe above mechanism is generic and in principle applies to an y type of activation f(2). To obtain concrete ca-\npacity estimates, we below proceed by considering several p articular activation examples that have attracted\na significant attention in NN literature.\n3.1 Different f(2)activations\nWe focus on three well known f(2)activations: i) linear, ii)quadratic , and iii) ReLU . For each of them we\nobtain relatively elegant final capacity bounding estimate s.\n93.1.1 Linear hidden layer activations – f(2)(x) =x\nSince the linear function is odd (i.e., since f(2)(−x) =−x=−f(2)(x)), we can, without a loss of generality,\nassume that, say, w=1, where1is the column vector of appropriate dimension with all of its components\nequal to one. However, as the analysis below shows, linearit y is a very particular form of activation where\nsuch an assumption is actually not needed. In fact, any wsuffices. This should be kept in mind for later\non when we study other two types of activations where wwill have to take particular forms to ensure that\nnetwork functioning actually makes sense at all. From (23) a nd (25), we then recognize the key optimization\nproblem of interest\nzi(Gi,1:d) = min\nsign(f(2)(Qi,1:d)w)=1d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(Qi,1:d)w≥0d/summationdisplay\nj=1(Gij−Qij)2. (28)\nFor the linear activation, one can then rewrite (28) as\nzi(g;lin) =zi(Gi,1:d) = min\nf(2)(Qi,1:d)w≥0d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(qT)w≥0/bardblg−q/bardbl2\n2= min\nqTw≥0/bardblg−q/bardbl2\n2.(29)\nProceeding by writing Lagrangian, we then further have\nzi(g;lin) = min\nqTw≥0/bardblg−q/bardbl2\n2= min\nqmax\nλ≥0/bardblg−q/bardbl2\n2−λ(qTw). (30)\nRelying on the strong duality, one then also finds\nzi(g;lin) = min\nqmax\nλ≥0/bardblg−q/bardbl2\n2−λ(qTw) = max\nλ≥0min\nq/bardblg−q/bardbl2\n2−λ(qTw). (31)\nTo solve the inner optimization over q, we then consider the following derivative\nd/parenleftbig\n/bardblg−q/bardbl2\n2−λ(qTw)/parenrightbig\ndq= 2(q−g)+λw. (32)\nAfter equalling the above derivative to zero one then finds\nq=g−λw\n2. (33)\nPlugging back this value in the objective in (31), one obtain s\nzi(g;lin) = max\nλ≥0min\nq/bardblg−q/bardbl2\n2−λ(qTw) = max\nλ≥0−λ2/bardblw/bardbl2\n2\n4−λgTw. (34)\nOptimizing further over λ, one finds\nλopt= max/parenleftbigg\n−2gTw\n/bardblw/bardbl2\n2,0/parenrightbigg\n. (35)\nAfter plugging λoptback in (34), one obtains\nzi(g;lin) =/parenleftbig\nmax/parenleftbig\n−gTw,0/parenrightbig/parenrightbig2\n/bardblw/bardbl2\n2=/parenleftbigg\nmax/parenleftbigg\n−gTw\n/bardblw/bardbl2,0/parenrightbigg/parenrightbigg2\n= (max( gi,0))2, (36)\nwheregiis a standard normal random variable. Connecting (22), (24) , (29), and (36), we finally have\nφ0= lim\nn→∞EG1√nmin\nφ(Q)=0/bardblG−Q/bardblF−1\n10= lim\nn→∞EG1√n/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1zi(Gi,1:d)−1\n=/radicalbig\nαEzi(g;lin)−1\n=/radicalig\nαE(max(gi,0))2−1\n=/radicalbiggα\n2−1. (37)\nWe summarize the above results in the following lemma.\nLemma 2. (Memory capacity; linear activation) Assume the setup of Theorem 1. For linear f(2)(x) =x,\nletc(d;lin)/definesc(d;f(2)(x) =x)be then-scaled memory capacity from (3). One then has the following f or\nsuch\n(n-scaled) memory capacity:\nc(d;lin) =2.\nIf and only if the sample complexity mis such that α/defineslimn→∞m\nn≥c(d;lin)then\nlim\nn→∞PX(A([n,d,1];sign)fails to memorize data set (X,1))−→1. (38)\nProof. Follows immediately from the above discussion.\nThe above lemma effectively states that only when the sample c omplexity mis such that m >2n(with\nnbeing the data vectors’ ambient dimension) the network fail s to memorize the data. Consequently, one has\nfor the memory capacity of the dhidden layer linearly activated neurons TCMs, C(A([n,d,1];lin)) = 2n.\nThis shows that the capacity does not change as the width of th e hidden layer increases. Moreover, it\nshows that the capacity remains equal to the capacity of the s ingle spherical perceptron neuron (see, e.g.,\n[3,8–12,16,22,26,33,34,40,48,50–52]).\n4)Double checking the strong random duality: Strictly speaking the above analysis establishes the\nupper bound on the capacity. However, given the presence of t he underlying convexity and the strong\ndeterministic duality, the machinery of [ 29,37,39] ensures that the strong random duality is in place as well\nwhich then implies that the established upper bounds are in f act tight.\n3.1.2 Quadratic hidden layer activations – f(2)(x) =x2\nSince the quadratic function f(2)(x) =x2≥0one needs to carefully make a choice for vector wwhich\nensures that the network functioning is of any use. Clearly, some of the components of wmust be negative.\nGiven the symmetry of the quadratic function, a natural choi ce that is typically considered in the literature\nin such situations is wwithd\n21s andd\n2-1s. For the concreteness, we set\nw=/bracketleftigg\n−1d\n2×1\n1d\n2×1/bracketrightigg\n, (39)\nand to facilitate exposition avoid dimensional subscripts and simply write\nw=/bracketleftbigg\n−1\n1/bracketrightbigg\n, (40)\n11assuming that the size of the column vectors 1isd\n2×1. As earlier, relying on (23), (25), (26), and (28), we\nthen recognize the following key optimization problem of in terest\nzi(g;quad) =zi(Gi,1:d) = min\nf(2)(Qi,1:d)w≥0d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(qT)w≥0/bardblg−q/bardbl2\n2= min\n(q2)Tw≥0/bardblg−q/bardbl2\n2.(41)\nAfter effectively splitting the problem into two parts, an in teresting formulation can be obtained\nzi(g;quad) = min\nqd\n2/summationdisplay\ni=1(gi−qi)2+d/summationdisplay\ni=d\n2+1(gi−qi)2\nsubject tod\n2/summationdisplay\ni=1q2\ni≤d/summationdisplay\ni=d\n2+1q2\ni. (42)\nFor a moment, we find it convenient to set/summationtextd\ni=d\n2+1q2\ni=b, defineg(r)/definesgd\n2+1:d, and look at the following\noptimization problem\nz(r)\ni(g(r),b) = min\nqd/summationdisplay\ni=d\n2+1(gi−qi)2\nsubject tod/summationdisplay\ni=d\n2+1q2\ni=b. (43)\nOne can then trivially rewrite (43) as\nz(r)\ni(g(r),b) = min\nqd/summationdisplay\ni=d\n2+1g2\ni−2d/summationdisplay\ni=d\n2+1giqi+b\nsubject tod/summationdisplay\ni=d\n2+1q2\ni=b. (44)\nSolving (44) then gives\nz(r)\ni(g(r),b) =\nd/summationdisplay\ni=d\n2+1g2\ni−2√\nb/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni+b\n\n=\nd/summationdisplay\ni=d\n2+1g2\ni−2√\nb/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni+b\n\n=\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni−√\nb\n2\n. (45)\nOne can then rewrite (42) as\nzi(g;quad) = min\nq,bd\n2/summationdisplay\ni=1(gi−qi)2+z(r)\ni(g(r),b)\n12subject tod\n2/summationdisplay\ni=1q2\ni≤b. (46)\nA few additional algebraic transformations give\nzi(g;quad) = min\nq,bd\n2/summationdisplay\ni=1g2\ni−2d\n2/summationdisplay\ni=1giqi+d\n2/summationdisplay\ni=1q2\ni+z(r)\ni(g(r),b)\nsubject tod\n2/summationdisplay\ni=1q2\ni≤b. (47)\nWe again for a moment set/summationtextd\n2\ni=q2\ni=b1≤b, defineg(l)/definesg1:d\n2, and find\nzi(g;quad) = min\nb1≤bz(l)\ni(g(l),b1)+z(r)\ni(g(r),b), (48)\nwhere, analogously to (45), we also have\nz(l)\ni(g(l),b) =\n/radicaltp/radicalvertex/radicalvertex/radicalbtd\n2/summationdisplay\ni=1g2\ni−/radicalbig\nb1\n2\n. (49)\nClearly,\n/radicaltp/radicalvertex/radicalvertex/radicalbtd\n2/summationdisplay\ni=1g2\ni≤/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni=⇒zi(g) = 0. (50)\nOn the other hand, if/radicalig/summationtextd\n2\ni=1g2\ni>/radicalig/summationtextd\ni=d\n2+1g2\nithenb1=band the optimization from (48) becomes\nzi(g;quad) = min\nb\n\n/radicaltp/radicalvertex/radicalvertex/radicalbtd\n2/summationdisplay\ni=1g2\ni−√\nb\n2\n+\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni−√\nb\n2\n. (51)\nOptimizing over bone then finds\n√\nb=/radicalig/summationtextd\n2\ni=1g2\ni+/radicalig/summationtextd\ni=d\n2+1g2\ni\n2, (52)\nand\n/radicaltp/radicalvertex/radicalvertex/radicalbtd\n2/summationdisplay\ni=1g2\ni>/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=d\n2+1g2\ni=⇒zi(g;quad) =/parenleftbigg/radicalig/summationtextd\n2\ni=1g2\ni−/radicalig/summationtextd\ni=d\n2+1g2\ni/parenrightbigg2\n2. (53)\nWriting (50) and (53) in a more compact form gives\nzi(g;quad) =/parenleftbigg\nmax/parenleftbigg/radicalig/summationtextd\n2\ni=1g2\ni−/radicalig/summationtextd\ni=d\n2+1g2\ni,0/parenrightbigg/parenrightbigg2\n2. (54)\n13Moreover, one also has\nzi(g;quad) =/parenleftig\nmax/parenleftig\na(1)\ni−a(2)\ni,0/parenrightig/parenrightig2\n2, (55)\nwherea(1)\nianda(2)\niare independent chi random variables withd\n2degrees of freedom. Connecting (22), (24),\n(41), and (55), we finally have\nφ0= lim\nn→∞EG1√nmin\nφ(Q)=0/bardblG−Q/bardblF−1\n= lim\nn→∞EG1√n/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1zi(Gi,1:d)−1\n=/radicalbig\nαEzi(g;quad)−1\n=/radicaltp/radicalvertex/radicalvertex/radicalbt\nαE/parenleftig\nmax/parenleftig\na(1)\ni−a(2)\ni,0/parenrightig/parenrightig2\n2−1. (56)\nGiven the pdf of the chi random variable withd\n2degrees of freedom\nfχ(a) =21−d\n4\nγ(d\n4)ad\n2−1e−a2\n2, (57)\none can also find\nE/parenleftig\nmax/parenleftig\na(1)\ni−a(2)\ni,0/parenrightig/parenrightig2\n=/integraldisplay∞\n0/integraldisplaya(1)\ni\n0/parenleftig\na(1)\ni−a(2)\ni/parenrightig2\nfχ(a(2)\ni)fχ(a(1)\ni)da(2)\nida(1)\ni. (58)\nWe summarize the above results in the following lemma.\nLemma 3. (Memory capacity upper bound; quadratic activation) Assume the setup of Theorem 1. For\nquadratic f(2)(x) =x2, letc(d;quad)/definesc(d;f(2)(x) =x2)be then-scaled memory capacity from (3). Let a(1)\ni\nanda(2)\nibe independent, chi distributed, random variables withd\n2degrees of freedom and let fχ(a)be as in\n(57). One then has the following\n(n-scaled) memory capacity upper bound:\nˆc(d;quad) =2\nE/parenleftig\nmax/parenleftig\na(1)\ni−a(2)\ni,0/parenrightig/parenrightig2=2\n/integraltext∞\n0/integraltexta(1)\ni\n0/parenleftig\na(1)\ni−a(2)\ni/parenrightig2\nfχ(a(2)\ni)fχ(a(1)\ni)da(2)\nida(1)\ni.\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>ˆc(d;quad)\nlim\nn→∞PX(A([n,d,1];quad)fails to memorize data set (X,1))−→1, (59)\nand\nlim\nn→∞PX(c(d,quad)<ˆc(d,quad))−→1. (60)\nProof. Follows immediately from the above discussion.\nTaking, say, d= 2for the concreteness, one finds ˆc(d;quad) = 5.4978, which basically means that when\nthe sample complexity mis such that m >5.4978n(withnbeing the data vectors’ ambient dimension) the\n14network fails to memorize the data. Consequently, one has fo r the memory capacity of the 2hidden layer\nquadratically activated neurons TCMs, C(A([n,2,1];quad))≤5.4978n. The results for a wider range of d\nare shown in Figure 2. We also add the ˆc(∞;quad) = 4, replica symmetry based prediction obtained in [ 56].\nAs the figure indicates, one has that the RDT upper bound from t he above theorem approaches the d→ ∞\nreplica symmetry based prediction already for fairly narro w nets with the number of neurons of the order\nof a couple of tens. We should also add that due to the fact that the underlying problems are now highly\nnon-convex, the strong random duality considerations from [29,37,39] are inapplicable. As we will see a bit\nlater on, the results obtained above (and shown in Figure 2), are in fact strict (non-tight) capacity upper\nbounds.\nd5 10 15 20 25 30upper bounds on c(d;quad)\n33.544.555.5Quadratic activation – Memory capacity upper bound\nˆc(d;quad) – RDT\nˆc(d;quad) – RDT; interpolated\nˆc(∞;quad) –Replica symmetry\nˆc(∞,quad) = 4\nFigure 2: Memory capacity upper bound as a function of the num ber of neurons, d, in the hidden layer;\n1-hidden layer TCM with quadratic activations; plain RDT estimate ( Replica symmetry (RS) d→ ∞\nestimate is included as well)\n3.1.3 ReLU hidden layer activations – f(2)(x) = max( x,0)\nAs was the case above when we considered the quadratic activa tion, for ReLU (rectified linear unit) one has\nf(2)(x) = max( x,0)≥0. This means that one again needs to carefully make a choice fo r vector wwhich\nensures that the network functioning makes sense. Moreover , one again easily observes that some of the\ncomponents of wmust be negative. We follow the trend set above and in the ReLU relevant literature and\nconsider wwithd\n21s andd\n2-1s, i.e., we again take\nw=/bracketleftbigg−1\n1/bracketrightbigg\n, (61)\nwhile assuming that the size of the column vectors 1isd\n2×1. After again relying on (23), (25), (28), and\n(26), one recognizes the following key optimization proble m of interest\nzi(g;relu) =zi(Gi,1:d) = min\nf(2)(Qi,1:d)w≥0d/summationdisplay\nj=1(Gij−Qij)2= min\nf(2)(qT)w≥0/bardblg−q/bardbl2\n2= min\nmax(q,0)Tw≥0/bardblg−q/bardbl2\n2.(62)\n15Splitting the problem into two parts, gives the following fo rmulation\nzi(g;relu) = min\nqd\n2/summationdisplay\ni=1(gi−qi)2+d/summationdisplay\ni=d\n2+1(gi−qi)2\nsubject tod\n2/summationdisplay\ni=1max(qi,0)≤d/summationdisplay\ni=d\n2+1max(qi,0). (63)\n3.1.3.1 d= 2\nWe study separately the simplest case d= 2. There are two reasons for doing so: 1) One can obtain a neat\nand explicit closed form final result; and 2) Somewhat counte r-intuitively, it will turn out that the bounding\ncapacities are decreasing functions of d.\nWhend= 2, (63) becomes\nzi(g;relu2) = min\nq(g1−q1)2+(g2−q2)2\nsubject to max(q1,0)≤max(q2,0). (64)\nForg1≤0one easily has zi(g) = 0. Also, one trivially has that for g2≥g1,zi(g) = 0. We then focus\non scenario where g1≥0andg2≤g1happen simultaneously. One first finds that for g2≤(1−√\n2)g1,\nzi(g) =g2\n1. On the other hand for (1−√\n2)g1≥g2≤g1,zi(g) = min/parenleftig\ng2\n1,2/parenleftbigg1−g2\n2/parenrightbig2/parenrightig\n=(g1−g2)2\n2. In a\nmore compact form one then has\nzi(g;relu2) =\n\n0, ifg1≤0org2≥g1≥0\ng2\n1, ifg1≥0andg2≤(1−√\n2)g1\n(g1−g2)2\n2,ifg1≥0and(1−√\n2)g1≤g2≤g1.. (65)\nConnecting (22), (24), (62), (63), and (65), we obtain\nφ0= lim\nn→∞EG1√nmin\nφ(Q)=0/bardblG−Q/bardblF−1\n= lim\nn→∞EG1√n/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1zi(Gi,1:d)−1\n=/radicalbig\nαEzi(g;relu2)−1. (66)\nMoreover, we also have\nEzi(g;relu2) =/integraldisplay∞\n0/integraldisplay∞\n−∞zi(g;relu2)e−g2\n2\n2√\n2πe−g2\n1\n2√\n2πdg2dg1\n=/integraldisplay∞\n0\n/integraldisplay(1−√\n2)g1\n−∞g2\n1e−g2\n2\n2√\n2π+/integraldisplayg1\n(1−√\n2)g1(g1−g2)2\n2e−g2\n2\n2√\n2π\ne−g2\n1\n2√\n2πdg2dg1\n=/integraldisplay∞\n0(I1(g1)+I2(g1))e−g2\n1\n2√\n2πdg1, (67)\nwhere\nI1(g1)/definesg2\n1erfc/parenleftig\n(√\n2−1)g1√\n2/parenrightig\n2\n16I2(g1)/defines1\n4/parenleftigg\n(g2\n1+1)/parenleftigg\nerf/parenleftbiggg1√\n2/parenrightbigg\n−erf/parenleftigg\n(1−√\n2)g1√\n2/parenrightigg/parenrightigg\n+/radicalbigg\n2\nπ/parenleftbigg\ng1e−g2\n1\n2−(1+√\n2)g1e−((1−√\n2)g1)2\n2/parenrightbigg/parenrightigg\n.\n(68)\nWe summarize the above results in the following lemma.\nLemma 4. (Memory capacity upper bound; ReLU activation; d= 2) Assume the setup of Theorem 1. For\nrectified linear unit (ReLU) f(2)(x) = max( x,0), letc(d;relu)/definesc(d;f(2)(x) = max( x,0)))be then-scaled\nmemory capacity from (3). Let I1(g1)andI2(g1)be as in (68). One then has the following\n(n-scaled) memory capacity upper bound:\nˆc(2;relu) =1\n/integraltext∞\n0(I1(g1)+I2(g1))e−g2\n1\n2√\n2πdg1≈3.81.\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>ˆc(2;relu)\nlim\nn→∞PX(A([n,2,1];relu)fails to memorize data set (X,1))−→1, (69)\nand\nlim\nn→∞PX(c(2,relu)<ˆc(2,relu))−→1. (70)\nProof. Follows immediately from the above discussion.\nThe above lemma basically states that when the sample comple xitymis such that m >3.81n(withn\nbeing the data vectors’ ambient dimension) the network fail s to memorize the data. Consequently, one has\nfor the memory capacity of the 2hidden layer ReLU activated neurons TCMs, C(A([n,2,1];relu))≤3.81n.\nAs was the case when we discussed the quadratic activation, d ue to highly non-convex underlying problems,\nthe strong random duality considerations from [ 29,37,39] are inapplicable.\n3.1.3.2 General d\nFor studying general d, we find it convenient to introduce vector g(1),g(2),g(2,ac), andg(2,a)\ng(1)/defines/braceleftbigg\ngi|gi>0,1≤i≤d\n2/bracerightbigg\ng(2)/definessort/parenleftig\ngd\n2+1:d/parenrightig\ng(2,a)/definesmin(g(2),0), (71)\nwhere sorting is in the descending order. Basically, g(1)is comprised of the positive components of g1:d\n2\nandg(2)isgd\n2+1:dsorted in the descending order. Also, for the notational sim plicity, let the lengths of g(1)\nandg(2)bed1andd2, respectively and let d3be the number of the nonnegative elements of g(2)(clearly,\nd2=d\n2). It is then not difficult to see that (63) can be rewritten as\nzi(g;relu) = min\nq(1),q(2)d1/summationdisplay\ni=1(g(1)\ni−q(1)\ni)2+d2/summationdisplay\ni=1(g(2,a)\ni−q(2)\ni)2\nsubject tod1/summationdisplay\ni=1max(q(1)\ni,0)≤d2/summationdisplay\ni=1max(q(2)\ni,0). (72)\n17Moreover, given the positivity of g(1), it is relatively easy to see that\nzi(g;relu) = min\nq(1)≥0,q(2)d1/summationdisplay\ni=1(g(1)\ni−q(1)\ni)2+d2/summationdisplay\ni=1(g(2,a)\ni−q(2)\ni)2\nsubject tod1/summationdisplay\ni=1q(1)\ni≤d2/summationdisplay\ni=1max(q(2)\ni,0). (73)\nFollowing the methodology utilized for studying the quadra tic activations, we find it convenient to study the\nfollowing optimization problem for a b≥0and for any k∈ {d3,d3+1,...,d 2}\nz(2)\ni(g(2),b,k;relu) = min\nq(2)≥0k/summationdisplay\ni=1(g(2,a)\ni−q(2)\ni)2\nsubject tok/summationdisplay\ni=1q(2)\ni=b. (74)\nOne can then trivially rewrite (74) as\nz(2)\ni(g(2),b,k;relu) = min\nq(2)≥0k/summationdisplay\ni=1/parenleftig\ng(2,a)\ni/parenrightig2\n−2k/summationdisplay\ni=1g(2,a)\niq(2)\ni+k/summationdisplay\ni=1/parenleftig\nq(2)\ni/parenrightig2\nsubject tok/summationdisplay\ni=1q(2)\ni=b. (75)\nAfter writing the Lagrangian and relying on the strong duali ty, we also have\nz(2)\ni(g(2),b,k;relu) = min\nq(2)≥0max\nνk/summationdisplay\ni=1/parenleftig\ng(2,a)\ni/parenrightig2\n−2k/summationdisplay\ni=1g(2,a)\niq(2)\ni+k/summationdisplay\ni=1/parenleftig\nq(2)\ni/parenrightig2\n+2ν/parenleftiggk/summationdisplay\ni=1q(2)\ni−b/parenrightigg\n= max\nνmin\nq(2)≥0k/summationdisplay\ni=1/parenleftig\ng(2,a)\ni/parenrightig2\n−2k/summationdisplay\ni=1g(2,a)\niq(2)\ni+k/summationdisplay\ni=1/parenleftig\nq(2)\ni/parenrightig2\n+2ν/parenleftiggk/summationdisplay\ni=1q(2)\ni−b/parenrightigg\n.(76)\nSolving over q(2)gives\nq(2,opt)= max(g(2,a)−ν,0), (77)\nand\nz(2)\ni(g(2),b,k;relu) = max\nν/parenleftiggk/summationdisplay\ni=1/parenleftig\ng(2,a)\ni/parenrightig2\n−k/summationdisplay\ni=1/parenleftig\nq(2,opt)\ni/parenrightig2\n−2νb/parenrightigg\n. (78)\nAfter setting\n¯z(2)\ni(g(2),b,k;relu) =\n\nmaxν/parenleftbigg/summationtextk\ni=1/parenleftig\ng(2,a)\ni/parenrightig2\n−/summationtextk\ni=1/parenleftig\nq(2,opt)\ni/parenrightig2\n−2νb/parenrightbigg\n,ifmin(q(2,opt))>0\n∞, otherwise ,(79)\nit is not that difficult to see that (73) can be rewritten as\nzi(g;relu) = min\nb≥0,q(1)≥0d1/summationdisplay\ni=1(g(1)\ni−q(1)\ni)2+ min\nk∈{d3,d3+1,...,d2}z(2)\ni(g(2),b,k;relu)\n18subject tod1/summationdisplay\ni=1q(1)\ni≤b. (80)\nWe can then write the Lagrangian and rely on the strong dualit y as above to obtain\nzi(g;relu) = min\nb≥0,q(1)≥0max\nν1≥0d1/summationdisplay\ni=1(g(1)\ni−q(1)\ni)2+ min\nk∈{d3+1,d3+2,...,d2}¯z(2)\ni(g(2),b,k;relu)+2ν1(d1/summationdisplay\ni=1q(1)\ni−b)\n= min\nb≥0max\nν1≥0min\nq(1)≥0d1/summationdisplay\ni=1(g(1)\ni−q(1)\ni)2+ min\nk∈{d3+1,d3+2,...,d2}¯z(2)\ni(g(2),b,k;relu)+2ν1(d1/summationdisplay\ni=1q(1)\ni−b).\n(81)\nSolving over q(1)gives\nq(1,opt)= max(g(1)−ν1,0), (82)\nand\nzi(g;relu) = min\nb≥0max\nν1≥0d1/summationdisplay\ni=1/parenleftig\ng(1)\ni/parenrightig2\n−d1/summationdisplay\ni=1/parenleftig\nq(1,opt)\ni/parenrightig2\n−2ν1b+ min\nk∈{d3,d3+1,...,d2}¯z(2)\ni(g(2),b,k;relu).\n(83)\nConnecting (22), (24), (62), (63), and (83), we obtain\nφ0= lim\nn→∞EG1√nmin\nφ(Q)=0/bardblG−Q/bardblF−1\n= lim\nn→∞EG1√n/radicaltp/radicalvertex/radicalvertex/radicalbtm/summationdisplay\ni=1zi(Gi,1:d)−1\n=/radicalbig\nαEzi(g;relu)−1. (84)\nWe summarize the above results in the following lemma.\nLemma 5. (Memory capacity upper bound; ReLU activation; general d) Assume the setup of Theorem\n1. For rectified linear unit (ReLU) f(2)(x) = max( x,0), letc(d;relu)/definesc(d;f(2)(x) = max( x,0)))be the\nn-scaled memory capacity from (3). Let gbe and-dimensional vector comprised of iid standard normals and\nletg(1),g(2), andg(2,a)be as in (71). Also, let d3be the number of the nonnegative elements of g(2)and let\nd2=d\n2. Additionally, let q(2,opt),q(1,opt),¯z(2)\ni(g(2),b,k;relu), andzi(g;relu)be as in (77), (82), (79), and\n(83), respectively. One then has the following\n(n-scaled) memory capacity upper bound:\nˆc(d;relu) =1\nEzi(g;relu).\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>ˆc(d;relu)\nlim\nn→∞PX(A([n,d,1];relu)fails to memorize data set (X,1))−→1, (85)\nand\nlim\nn→∞PX(c(d,relu)<ˆc(d,relu))−→1. (86)\n19Proof. Follows immediately from the above discussion.\nUtilization of the above lemma relies on a solid amount of num erical work. Taking for the concreteness,\nsay,d= 4, we obtain ˆc(4;relu)≈3.11. This basically means that when the sample complexity mis such\nthatm >3.11n(withnbeing the data vectors’ ambient dimension) the network fail s to memorize the\ndata. Consequently, one has for the memory capacity of the 4hidden layer ReLU activated neurons TCMs,\nC(A([n,4,1];relu))≤3.11n. One can continue for other even d, with the numerical calculations being more\nand more involved as dincreases. A bit easier (albeit not as precise and a bit jitte ry) alternative is to simulate\nhigher values of d. The obtained results are shown for a wider range of din Figure 3. The replica symmetry\nbased prediction, ˆc(∞;relu)≈2.93, obtained in [ 56] is shown for the completeness as well. Finally, we should\nadd that (as was the case when we discussed the quadratic acti vation and the ReLU one with d= 2), due\nto highly non-convex underlying problems, the strong rando m duality considerations from [ 29,37,39] are\ninapplicable.\nd2 4 6 8 10 12 14 16 18 20upper bounds on c(d;relu)\n11.522.533.544.5ReLUactivation – Memory capacity upper bound\nˆc(d;relu) – RDT\nˆc(d;relu) – RDT (interpolated)\nˆc(∞;relu) – Replica symmetry\nˆc(∞,relu) =2.933\nFigure 3: Memory capacity upper bound as a function of the num ber of neurons, d, in the hidden layer;\n1-hidden layer TCM with ReLU activations; plain RDT estimate ( Replica symmetry (RS) d→ ∞\nestimate is included as well)\n4Partially lifted Random Duality Theory (pl RDT)\nAs the results from the previous sections showed, the RDT is r ather useful tool when it comes to characterizing\nthe memory capacity. In particular, the plainRDT determines the memory capacity for the linear activation\nand upper-bounds it for the quadratic andReLU activations. Moreover, the scenarios where the applicatio n\nof the plain RDT is of the upper-bounding nature can be handled through th e recently developed fully\nlifted (fl) RDT [ 41,43,44]. However, one needs to keep in mind, that the fl RDT relies on h eavy numerical\nevaluations which would come on top of the already seen subst antial numerical work from the previous\nsections. Opting for an analytically less accurate but comp utationally more convenient route seems as\npractically more beneficial. Recalling that a similar situa tion was observed when the signactivations were\nconsidered [ 42,45], we find it reasonable to follow the path taken overthere and consider a partially lifted\n(pl) RDT variant which relies on the following principles.\n20Summary of the partially lifted (pl) RDT’s main principles [30,35,38–40]\n1)Finding underlying optimization algebraic representatio n2)Determining the partially lifted random dual\n3)Handling the partialy lifted random dual 4)Double-checking strong random duality.\nWe below assume a solid level of familiarity with the discuss ions presented in [ 42,45]. To ensure the\nsmoothness of the presentation, we parallel the presentati on from [ 45] as closely as possible and discuss\nseparately each of the above four principles within the cont ext of our interest here.\n1)Algebraic memorization characterization: This part of the pl RDT corresponds to the first part of\nthe plain RDT and is already obtained in Lemma 1. As mentioned earlier, Lemma 1 holds for any given\ndata set/parenleftbig\nx(0,k),1/parenrightbig\nk=1:m. On the other hand, to analyze (15) and (16), the pl RDT procee ds similarly to the\nplain RDT and imposes a statistics on X.\n2)Determining the partially lifted random dual: Keeping in mind the measure concentration from\n(18) (see, e.g. [ 30,32,39,42,45]), the following so-called partially lifted random dual theorem is another key\ningredient of the RDT machinery.\nTheorem 2. (Memorization characterization via partially lifted random dual) Let dbe any even integer.\nConsider a TCM with dneurons in the hidden layer and architecture A([n,d,1];f(2)), and let the elements\nofX∈Rm×n,G∈Rm×d, andH∈Rδ×dbe iid standard normals. Assuming c3>0andw∈Rd×1, set\nφ(Q)/defines/bardbl1−sign(f(2)(Q)w)/bardbl2\nf(1)\nrd(G)/definesmax\nφ(Q)=0−c3/bardblG−Q/bardblF\nf(2)\nrd(H)/defines/bardblH/bardblF\n¯φ0(α;c3)/defineslim\nn→∞1√n/parenleftbiggc3\n2−1\nc3log/parenleftig\nEGec3\n2f(1)\nrd(G)/parenrightig\n−1\nc3log/parenleftig\nEHec3\n2f(2)\nrd(H)/parenrightig/parenrightbigg\n. (87)\nOne then has\n(¯φ0(α;c3)>0) =⇒/parenleftig\nlim\nn→∞PX(frp>0)−→1/parenrightig\n=⇒/parenleftig\nlim\nn→∞PX(A([n,d,1];f(2))fails to memorize data set (X,1))−→1/parenrightig\n.(88)\nProof. Immediate consequence of Theorem 2 from [ 45].\n3)Handling the lifted random dual: After proceeding with a detailed careful analysis of the opt imization\noverQ, one arrives at the following theorem.\nTheorem 3. (Memory capacity partially lifted (pl) RDT based upper bound; generald) Assume the setup\nof Theorem 2. Let the network n-scaled capacity, c(d;f(2)), be as defined in (3) and let gbe ad-dimensional\nvector of iid standard normals. First one has\nzi(g;f(2)) = min\nf(2)(qT)w≥0/bardblg−q/bardbl2\n2\nIQ=Ege−c3\n4γzi(g;f(2))\nIsph=γsph−1\n2c3log/parenleftbigg\n1−c3\n2γsph/parenrightbigg\n, γsph=c3+/radicalbig\nc2\n3+4\n4\n¯φ0(α) = max\nc3>0min\nγ/parenleftbiggc3\n2+γ−α\nc3log(IQ)−Isph/parenrightbigg\n. (89)\nFurther, consider the following\n21(n-scaled general d) memory capacity upper bound, ¯c(d;f(2)), that satisfies:\n¯φ0(¯c(d;f(2))) = 0 ⇐⇒max\nc3>0min\nγ/parenleftbiggc3\n2+γ−¯c(d;f(2))\nc3log(IQ)−Isph/parenrightbigg\n= 0. (90)\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>¯c(d;f(2))\nlim\nn→∞PX(A([n,d,1];f(2))fails to memorize data set (X,1))−→1, (91)\nand\nlim\nn→∞PX(c(d;f(2))<¯c(d;f(2)))−→1. (92)\nProof. The proof is split into two parts: (i)Handling1\nc3√nlog/parenleftig\nEHexp/parenleftig\nc3f(2)\nrd(H)/parenrightig/parenrightig\n; and(ii)Handling of\n1\nc3√nlog/parenleftig\nEGexp/parenleftig\nc3f(1)\nrd(G)/parenrightig/parenrightig\n.\n(i)Handling1\nc3√nlog/parenleftig\nEHexp/parenleftig\nc3f(2)\nrd(H)/parenrightig/parenrightig\n:One first observes\nIsph/defines1\nc3√nlog/parenleftig\nEHexp/parenleftig\nc3f(2)\nrd(H)/parenrightig/parenrightig\n=1\nc3√nlog/parenleftbig\nEHexp/parenleftbig\nc3/bardblHT/bardblF/parenrightbig/parenrightbig\n. (93)\nAfter appropriate scaling, c3→c3√n, it was determined in [ 36,45] that\nIsph=γsph−1\n2c3log/parenleftbigg\n1−c3\n2γsph/parenrightbigg\n, γsph=c3+/radicalbig\nc2\n3+4\n4. (94)\n(ii)Handling1\nc3√nlog/parenleftig\nEGexp/parenleftig\nc3f(1)\nrd(G)/parenrightig/parenrightig\n:Following closely [ 45], we first observe\nlog(I′\nQ)/defines1\nc3√nlog/parenleftig\nEGexp/parenleftig\nc3f(2)\nrd(G)/parenrightig/parenrightig\n=1\nc3√nlog/parenleftbigg\nEGexp/parenleftbigg\n−c3min\nφ(Q)=0/bardblG−Q/bardblF/parenrightbigg/parenrightbigg\n.(95)\nUtilizing the squaring trick introduced on many occasions i n [36,40], we further find\nlog(I′\nQ) = max\nγ1\nc3√nlog/parenleftbigg\nEGexp/parenleftbigg\nc3/parenleftbigg\n−1\n4γmin\nφ(Q)=0/bardblG−Q/bardbl2\nF−γ/parenrightbigg/parenrightbigg/parenrightbigg\n. (96)\nKeeping in mind the appropriate scaling, c3→c3√nandγ→γ√n, and recalling α= limn→∞m\nn, one also\nhas\n−log(I′\nQ) = min\nγ/parenleftbigg\nγ−α\nc3log/parenleftbigg\nEGexp/parenleftbigg\nc3/parenleftbigg\n−1\n4γmin\nφi(Qi,:)=1/bardblGi,:−Qi,:/bardbl2\nF/parenrightbigg/parenrightbigg/parenrightbigg/parenrightbigg\n, (97)\nwhere\nφi(Qi,:)/definessign(sign(Qi,:)1). (98)\nIt is then not difficult to see that (97) is equivalent to the fol lowing\n−log(I′\nQ) = min\nγ/parenleftbigg\nγ−α\nc3log/parenleftbigg\nEgexp/parenleftbigg/parenleftbigg\n−c3\n4γmin\nf(2)(qT)w≤0/bardblg−q/bardbl2\n2/parenrightbigg/parenrightbigg/parenrightbigg/parenrightbigg\n= min\nγ/parenleftbigg\nγ−α\nc3log/parenleftbigg\nEgexp/parenleftbigg\n−c3\n4γzi/parenleftig\ng;f(2)/parenrightig/parenrightbigg/parenrightbigg/parenrightbigg\n= min\nγ/parenleftbigg\nγ−α\nc3log(IQ)/parenrightbigg\n. (99)\n22A simple combination of (87), (93)-(95), and (99) then compl etes the proof.\n4)Double checking the strong random duality: As discussed earlier and in [ 42,45], the standard strong\nrandom duality double checking is not in place due to inappli cability of the typical, convexity based, consid-\nerations from [ 29,37,39].\n4.1 Specialization to particular f(2)activations\nTheorem 3 is generic and works for various f(2)activations. To see how the whole machinery practically\nworks, we here consider particular f(2)activations. However, since the plain RDT completely solve d the\nlinear activation, we here focus only on the remaining two, t hequadratic and the ReLU . In fact, we first\nfocus most of our interest to the quadratic one as in that case the concrete capacity results can be obtained\nwithout an extensive numerical work. We then afterwards bri efly comment on the ReLU case as well.\n4.1.1 Pl RDT capacity estimates for quadratic activations – f(2)(x) =x2\nThe following theorem summarizes the pl RDT results for the quadratic activations.\nTheorem 4. (Memory capacity partially lifted (pl) RDT based upper bound; quadratic activation) Assume\nthe setup of Lemma 3 and Theorem 3 with a(1)\nianda(2)\nibeing independent chi distributed random variables\nwithd\n2degrees of freedom. First one has\nIQ=/integraldisplay∞\n0/integraldisplay∞\n0e−c3\n4γ(max(a(1)\ni−a(2)\ni,0))2\n2fχ(a(2)\ni)fχ(a(1)\ni)da(2)\nida(1)\ni\nIsph=γsph−1\n2c3log/parenleftbigg\n1−c3\n2γsph/parenrightbigg\n, γsph=c3+/radicalbig\nc2\n3+4\n4\n¯φ0(α) = max\nc3>0min\nγ/parenleftbiggc3\n2+γ−α\nc3log(IQ)−Isph/parenrightbigg\n. (100)\nFurther, consider the following\n(n-scaled general d) memory capacity upper bound, ¯c(d;quad), that satisfies:\n¯φ0(¯c(d;quad)) = 0 ⇐⇒max\nc3>0min\nγ/parenleftbiggc3\n2+γ−¯c(d;quad)\nc3log(IQ)−Isph/parenrightbigg\n= 0.(101)\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>¯c(d;quad)\nlim\nn→∞PX(A([n,d,1];quad)fails to memorize data set (X,1))−→1, (102)\nand\nlim\nn→∞PX(c(d;quad)<¯c(d;quad))−→1. (103)\nProof. Follows immediately from Theorem 3 after recognizing that zi(g;quad) =/parenleftBig\nmax/parenleftBig\na(1)\ni−a(2)\ni,0/parenrightBig/parenrightBig2\n2.\nThe results obtained based on the above theorem for a wider ra nge ofdare shown in Figure 4. For the\ncompleteness and a quick comparison, we include the results obtained earlier based on the plain RDT. The\nbenefit of the partially lifted RDT is fairly strong througho ut the entire range of the considered d’s. We also\nadd the partial 1rsb d→ ∞ estimate obtained based on the statistical physics replica methods in [ 56]. As\ncan be seen from the figure, the convergence with dis rather fast and already for fairly small dvalues (of the\norder of a couple of tens) the limiting, d→ ∞, bound is narrowly approached. As was the case for the plain\nRDT, we here observe that the bounding capacity estimates ar e decreasing as dincreases with the largest\n23value obtained again for d= 2. Given its clear importance, we, for the concreteness, take precisely d= 2\nand find ¯c(d;quad) = 4.065, which then implies that the memory capacity of the 2hidden layer quadratically\nactivated neurons TCMs, C(A([n,2,1];quad))≤4.065n. Moreover, one observes a substantial drop from\nthe plain RDT bound of 5.4978established in earlier sections.\nd5 10 15 20 25 30upper bounds on c(d;quad)\n33.544.555.5Quadratic activation – Memory capacity upper bound\nˆc(d;quad) – RDT\n¯c(d;quad) – RDT\nˆc(d;quad) – RDT; interpolated\n¯c(d;quad) – RDT; interpolated\nˆc(∞;quad) –Replica symmetry\n¯c(∞;quad) –Partial 1rsb\nˆc(∞,quad) = 4\n¯c(∞,quad)≈3.381Benefit ofpartially lifted RDT:\n∆c(d;quad) = ˆ c(d;quad)−¯c(d;quad)\nFigure 4: Memory capacity upper bound as a function of the num ber of neurons, d, in the hidden layer;\n1-hidden layer TCM with quadratic activations; plain RDT versus partially lifted RDT (Replica\nsymmetry (RS) andPartial 1rsb d→ ∞ estimates are included as well)\n4.1.2 Pl RDT capacity estimates for ReLU activations – f(2)(x) = max( x,0)\nThe following theorem summarizes the pl RDT results for the ReLU activations.\nTheorem 5. (Memory capacity partially lifted (pl) RDT based upper bound; ReLU activation) Assume the\nsetup of Lemma 3 and Theorem 3 with zi(g;relu)as in (83). First one has\nIQ=/integraldisplay∞\n−∞e−c3\n4γzi(g;relu)dg\nIsph=γsph−1\n2c3log/parenleftbigg\n1−c3\n2γsph/parenrightbigg\n, γsph=c3+/radicalbig\nc2\n3+4\n4\n¯φ0(α) = max\nc3>0min\nγ/parenleftbiggc3\n2+γ−α\nc3log(IQ)−Isph/parenrightbigg\n. (104)\nFurther, consider the following\n(n-scaled general d) memory capacity upper bound, ¯c(d;relu), that satisfies:\n¯φ0(¯c(d;relu)) = 0 ⇐⇒max\nc3>0min\nγ/parenleftbiggc3\n2+γ−¯c(d;relu)\nc3log(IQ)−Isph/parenrightbigg\n= 0.(105)\nThen for any sample complexity msuch that α/defineslimn→∞m\nn>¯c(d;relu)\nlim\nn→∞PX(A([n,d,1];relu)fails to memorize data set (X,1))−→1, (106)\n24and\nlim\nn→∞PX(c(d,relu)<¯c(d,relu))−→1. (107)\nProof. Follows immediately from Theorem 3 after recognizing that zi(g;relu)from (83) is precisely\nzi(g;f(2)= max(x,0)).\nThe numerical evaluations are now substantially more invol ved even for small values of d. Moreover, the\nsimulations are rather extensive for larger values and the i ndication from the plain RDT suggest that d= 2\nis particularly relevant. As indicated in Table 1, we applie d the above machinery for d= 2and obtained the\nbound≈3.81, which means that in this particular case the partial RDT mak es no improvement over the\nplain RDT. The table is completed by taking the plain RDT valu e ford= 4as well, since the underlying\npartial RDT numerical work is already rather heavy.\n5 Conclusion\nIn this paper we studied the treelike committee machines (TC M) neural networks and their memory capabil-\nities. Differently form the common practice, we here instead of typical signperceptron hidden layer neuronal\nactivations consider a generic set of activations. Utilizi ng a powerful mathematical concept called Random\nDuality Theory (RDT), [ 42] established a generic statistical framework for the 1-hid den layer TCMs analysis\nthat can study on a very precise level the scaled capacities f or any given number of the neurons in the hid-\nden layer, d. Among other things, such a machinery effectively enabled av oiding the qualitative/descriptive\nscaling types of estimates typically prevalent in the capac ity analysis literature. Moreover, studying the sign\nperceptron activations, it also made a very strong progress towards obtaining, in a mathematically rigorous\nway, their exactn-scaled capacities. For small values of d, it also made a very first rigorous progress in\nover 30 years over the previously best known bounds of [ 15]. Since the results of [ 42] are, in general, of the\nupper-bounding type, [ 45] proceeded further by considering the so-called partially lifted (pl) RDT variant\nand significantly lowered the estimates from [ 42]. Such a lowering further resulted in ensuring a universal\n(over the entire range of d) improvement over the previously best known results of [ 15].\nWe here adopt the same strategy and utilize both the plain RDT and the partial RDT to characterize the\n1-hidden layer TCM capacities with neuronal activations su bstantially different from the classical signone.\nWe first establish a universal framework for studying generi c activations and then consider three particular\nactivations types that have attracted a strong interest in r ecent NN literature: (i) linear ; (ii)quadratic ;\nand (iii) ReLU . For the linear activation we show that the plain RDT exactly characterizes the capacity.\nMoreover, we show that, no matter how wide the hidden layer is , the capacity remains equal to the capacity\nof the single spherical signperceptron. For the quadratic and ReLU activations we obtai n that the plain\nRDT predictions are decreasing functions of dthat converge to a constant value. The maximum bounding\nvalue is in both cases obtained for the smallest possible d= 2. Moreover, for the pl RDT and quadratic\nactivation, we obtain a strong improvement over the plain RD T through the entire range of the considered\nd’s. At the same time, the bounding capacity maintains the dec reasing in dproperty with the maximal value\nagain being achieved for (the minimal possible) d= 2. For the ReLU, we obtained that the pl RDT offers\nno improvement over the plain RDT for d= 2which means that the same, decreasing in d, trend applies\nto these activations as well. Moreover, we uncover that anot her of the trends observed in [ 42] manifests\nitself here as well. Namely, the bounds obtained in [ 42] precisely matched the corresponding statistical\nphysics replica symmetry based predictions obtain in [ 5,14]. Here, we also observe that the linear activation\npredictions precisely match the ones obtained through the r eplica considerations in [ 2,56]. Moreover, the\nd→ ∞ converging values of our both plain RDT and pl RDT closely app roach the corresponding ones\nobtained in [ 56].\nVarious extensions are possible as well. It is rather clear t hat the first next one is to conduct the analysis\nwith the fully lifted (fl) RDT (see, e.g., [ 44]). Also, we here consider only three well known activation\nfunctions. Many others are of interest as well, e.g., sigmoi d, erf, tanh and so on. More complex multi-layered\nnetwork architectures including both TCM and FCM or PM based ones are of interest as well. All of these\nextensions, we will discuss in separate papers.\n25References\n[1] S. Arora, S. S. Du, W. Hu, Z. Li, and R. Wang. Fine-grained a nalysis of optimization and generalization\nfor overparameterized two-layer neural networks. 2019. av ailable online at http://arxiv.org/abs/1901.\n08584.\n[2] C. Baldassi, E. M. Malatesta, and R. Zecchina. Propertie s of the geometry of solutions and capacity of\nmultilayer neural networks with rectified linear unit activ ations. Phys. Rev. Lett. , 123:170602, October\n2019.\n[3] P. Baldi and S. Venkatesh. Number od stable points for spi n-glasses and neural networks of higher\norders. Phys. Rev. Letters , 58(9):913–916, Mar. 1987.\n[4] E. Barkai, D. Hansel, and I. Kanter. Statistical mechani cs of a multilayered neural network. Phys. Rev.\nLett., 65(18):2312–2315, Oct 1990.\n[5] E. Barkai, D. Hansel, and H. Sompolinsky. Broken symmetr ies in multilayered perceptrons. Phys. Rev.\nA, 45(6):4146, March 1992.\n[6] E. Barkai and I. Kanter. Storage capacity of a multilayer neural network with binary weights. Europhys.\nLett., 14(2):107, 1991.\n[7] E. B. Baum. On the capabilities of multilayer perceptron s.Journal of complexity , 4(3):193–215, 1988.\n[8] S. H. Cameron. Tech-report 60-600. Proceedings of the bionics symposium , pages 197–212, 1960. Wright\nair development division, Dayton, Ohio.\n[9] T. Cover. Geomretrical and statistical properties of sy stems of linear inequalities with applications in\npattern recognition. IEEE Transactions on Electronic Computers , (EC-14):326–334, 1965.\n[10] D. Donoho and J. Tanner. Neighborliness of randomly-pr ojected simplices in high dimensions. Proc.\nNational Academy of Sciences , 102(27):9452–9457, 2005.\n[11] D. Donoho and J. Tanner. Observed universality of phase transitions in high-dimensional geometry,\nwith implications for modern data analysis and signal proce ssing. Phylosophical transactions of the\nroyal society A: mathematical, physical and engineering sci ences, 367, November 2009.\n[12] D. Donoho and J. Tanner. Counting the face of randomly pr ojected hypercubes and orthants, with\napplication. Discrete and Computational Geometry , 43:522–541, 2010.\n[13] S. S. Du, X. Zhai, B. Poczos, and A. Singh. Gradient desce nt provably optimizes overparameterized\nneural networks. 2018. available online at http://arxiv.org/abs/1810.02054 .\n[14] A. Engel, H. M. Kohler, F. Tschepke, H. Vollmayr, and A. Z ippelius. Storage capacity and learning\nalgorithms for two-layer neural networks. Phys. Rev. A , 45(10):7590, May 1992.\n[15] R. M. Durbin G. J. Mitchison. Bounds on the learning capa city of some multi-layer networks. Biological\nCybernetics , 60:345–365, 1989.\n[16] E. Gardner. The space of interactions in neural network s models. J. Phys. A: Math. Gen. , 21:257–270,\n1988.\n[17] E. Gardner and B. Derrida. Optimal storage properties o f neural networks models. J. Phys. A: Math.\nGen., 21:271–284, 1988.\n[18] R. Ge, R. Wang, and H. Zhao. Mildly overparametrized neu ral nets can memorize training data effi-\nciently. 2019. available online at http://arxiv.org/abs/1909.11837 .\n[19] M. Hardt and T. Ma. Identity matters in deep learning. 20 16. available online at http://arxiv.org/\nabs/1611.04231 .\n26[20] G. B. Huang. Learning capability and storage capacity o f two-hidden-layer feedforward networks. IEEE\nTransactions on Neural Networks , 14(2):274–281, 2003.\n[21] Z. Ji and M. Telgarsky. Polylogarithmic width suffices fo r gradient descent to achieve arbitrarily small\ntest error with shallow relu networks. 2019. available onli ne athttp://arxiv.org/abs/1909.12292 .\n[22] R. D. Joseph. The number of orthants in n-space instersected by an s-dimensional subspace. Tech.\nmemo 8, project PARA , 1960. Cornel aeronautical lab., Buffalo, N.Y.\n[23] Y. Li and Y. Liang. Learning overparameterized neural n etworks via stochastic gradient descent on\nstructured data. In Advances in Neural Information Processing Systems , pages 8157–8166, 2018.\n[24] R. Monasson and R. Zecchina. Weight space structure and internal representations: A direct approach\nto learning and generalization in multilayer neural networ ks.Phys. Rev. Lett. , 75:2432, September 1995.\n[25] S. Oymak and M. Soltanolkotabi. Towards moderate overp arameterization: global convergence guaran-\ntees for training shallow neural networks. 2019. available online at http://arxiv.org/abs/1902.04674 .\n[26] L. Schlafli. Gesammelte Mathematische AbhandLungen I . Basel, Switzerland: Verlag Birkhauser, 1950.\n[27] Z. Song and X. Yang. Quadratic suffices for over-parametr ization via matrix Chernoff bound. 2019.\navailable online at http://arxiv.org/abs/1906.03593 .\n[28] M. Stojnic. Block-length dependent thresholds in bloc k-sparse compressed sensing. available online at\nhttp://arxiv.org/abs/0907.3679 .\n[29] M. Stojnic. Upper-bounding ℓ1-optimization weak thresholds. available online at http://arxiv.org/\nabs/1303.7289 .\n[30] M. Stojnic. Various thresholds for ℓ1-optimization in compressed sensing. available online at http://\narxiv.org/abs/0907.3666 .\n[31] M. Stojnic. Block-length dependent thresholds for ℓ2/ℓ1-optimization in block-sparse compressed sens-\ning.ICASSP, IEEE International Conference on Acoustics, Signal and Speech Processing , pages 3918–\n3921, 14-19 March 2010. Dallas, TX.\n[32] M. Stojnic. ℓ1optimization and its various thresholds in compressed sens ing.ICASSP, IEEE Inter-\nnational Conference on Acoustics, Signal and Speech Proces sing, pages 3910–3913, 14-19 March 2010.\nDallas, TX.\n[33] M. Stojnic. Recovery thresholds for ℓ1optimization in binary compressed sensing. ISIT, IEEE Inter-\nnational Symposium on Information Theory , pages 1593 – 1597, 13-18 June 2010. Austin, TX.\n[34] M. Stojnic. Another look at the Gardner problem. 2013. a vailable online at http://arxiv.org/abs/\n1306.3979 .\n[35] M. Stojnic. Lifting ℓ1-optimization strong and sectional thresholds. 2013. avai lable online at http://\narxiv.org/abs/1306.3770 .\n[36] M. Stojnic. Lifting/lowering Hopfield models ground st ate energies. 2013. available online at http://\narxiv.org/abs/1306.3975 .\n[37] M. Stojnic. Meshes that trap random subspaces. 2013. av ailable online at http://arxiv.org/abs/\n1304.0003 .\n[38] M. Stojnic. Negative spherical perceptron. 2013. avai lable online at http://arxiv.org/abs/1306.\n3980.\n[39] M. Stojnic. Regularly random duality. 2013. available online at http://arxiv.org/abs/1303.7295 .\n[40] M. Stojnic. Spherical perceptron as a storage memory wi th limited errors. 2013. available online at\nhttp://arxiv.org/abs/1306.3809 .\n27[41] M. Stojnic. Bilinearly indexed random processes – stationarization of fully lifted interpolation. 2023.\navailable online at http://arxiv.org/abs/2311.18097 .\n[42] M. Stojnic. Capacity of the treelike sign perceptrons n eural networks with one hidden layer – rdt based\nupper bounds. 2023. available online at http://arxiv.org/abs/2312.08244 .\n[43] M. Stojnic. Fully lifted interpolating comparisons of bilinearly indexed random processes. 2023. available\nonline at http://arxiv.org/abs/2311.18092 .\n[44] M. Stojnic. Fully lifted random duality theory. 2023. a vailable online at http://arxiv.org/abs/2312.\n00070 .\n[45] M. Stojnic. Lifted rdt based capacity analysis of the 1-hidden layer treelike signperceptrons neural\nnetworks. 2023. available online at http://arxiv.org/abs/2312.08257 .\n[46] R. Sun. Optimization for deep learning: theory and algo rithms. 2019. available online at http://arxiv.\norg/abs/1912.08957 .\n[47] R Urbanczik. Storage capacity of the fully-connected c ommittee machine. J. Phys. A: Math. Gen. , 30,\n1997.\n[48] S. Venkatesh. Epsilon capacity of neural networks. Proc. Conf. on Neural Networks for Computing,\nSnowbird, UT , 1986.\n[49] R. Vershynin. Memory capacity of neural networks with t hreshold and ReLU activations. 2019. available\nonline at http://arxiv.org/abs/2001.06938 .\n[50] J. G. Wendel. A problem in geometric probability. Mathematica Scandinavica , 1:109–111, 1962.\n[51] R. O. Winder. Single stage threshold logic. Switching circuit theory and logical design , pages 321–332,\nSep. 1961. AIEE Special publications S-134.\n[52] R. O. Winder. Threshold logic . Ph. D. dissertation, Princetoin University, 1962.\n[53] Y. Xiong and J. H. Oh C. Kwon. The storage capacity of a ful ly-connected committee machine. NIPS,\n1997.\n[54] M. Yamasaki. The lower bound of the capacity for a neural network with multiple hidden layers. In\nInternational Conference on Artificial Neural Networks , pages 546–549, 1993.\n[55] C. Yun, S. Sra, and A. Jadbabaie. Small relu networks are powerful memorizers: a tight analysis of\nmemorization capacity. In Advances in Neural Information Processing Systems , pages 15532–15543,\n2019.\n[56] J. A. Zavatone-Veth and C. Pehlevan. Activation functi on dependence of the storage capacity of treelike\nneural networks. Phys. Rev. E , 103:L020301, February 2021.\n[57] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. U nderstanding deep learning requires\nrethinking generalization. ICLR, 2017.\n[58] D. Zou, Y. Cao, D. Zhou, and Q. Gu. Stochastic gradient de scent optimizes overparameterized deep\nrelu networks. 2018. available online at http://arxiv.org/abs/1811.08888 .\n28"    },    {        "title": "2402.05736v1.Numerical_solution_of_the_Newtonian_plane_Couette_flow_with_linear_dynamic_wall_slip.pdf",        "content": "arXiv:2402.05736v1  [physics.flu-dyn]  8 Feb 2024Numerical solution of the Newtonian plane Couette flow\nwith linear dynamic wall slip\nMuner M. Abou Hasan1, Ethar A.A. Ahmed2, Ahmed F. Ghaleb3, Moustafa S.\nAbou-Dina3, Georgios C. Georgiou4,∗\n*: Corresponding author, georgios@ucy.ac.cy\n1: School of Mathematics and Data Sciences, EAU, Dubai, UAE.\n2: School of Engineering and Applied Sciences, Nile Univers ity, Giza 12588, Egypt\n3: Department of Mathematics, Faculty of Science, Cairo Univ ersity, Giza 12613, Egypt\n4: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678\nNicosia, Cyprus\nAbstract\nAn efficient numerical approach based on weighted average finite diff erences is used\nto solve the Newtonian plane Couette flow with wall slip, obeying a dyna mic slip law\nthat generalizes the Navier slip law with the inclusion of a relaxation ter m. Slip is\nexhibited only along the fixed plate, and the motion is triggered by the motion of\nthe other plate. Three different cases are considered for the mot ion of the moving\nplate, i.e., constant speed, oscillating speed, and a single-period sinu soidal speed. The\nvelocity and the volumetric flow rate are calculated in all cases and co mparisons are\nmade with the results of other methods and available results in the lite rature. The\nnumerical outcomes confirm the damping with time and the lagging effe cts arising from\nthe Navier and dynamic wall slip conditions and demonstrate the hyst eretic behavior\nof the slip velocity in following the harmonic boundary motion.\nKeywords : Plane Couette flow; dynamic wall slip; Navier slip; weighte d average finite\ndifferences; hysteretic behavior.\n1 Introduction\nIn these past few decades, there has been a growing interest i n the study of Newtonian and\nnon-Newtonian viscometric flows in the presence of static or dynamic wall slip conditions,\nfor their importance in rheometry and in industrial applica tions [1, 2, 3]. Reviews of the slip\nconditions prevailing at the fluid-structure interface for various media of practical impor-\ntance have been reported by Hatzikiriakos [2, 3]. Malkin and Patlazan [4] also reviewed wall\nslip in complex fluids of different types, focusing on fluid-wal l interaction and shear-induced\nfluid-to-solid transitions.\nThe simplest dynamic wall slip equation in the case of unidir ectional one-dimensional\nNewtonian flow, such that the x-velocity component is v=v(y,t) and the plane represents\na wall, reads as follows [1]:\nvs(0,1) +λ∂vs\n∂t(0,t) =µ\nβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\n∂y(0,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (1)\nwherevsis the slip velocity, defined as the relative velocity of the fl uid particles adjacent\nto the wall with respect to that of the wall, µis the constant fluid viscosity, βis the slip\ncoefficient, and λis the slip relaxation parameter. When the latter parameter vanishes Eq.\n(1) is reduced to the Navier-slip condition [5]:\nvs(0,t) =µ\nβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\n∂y(0,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2)\n1If the wall is fixed, then Eq. (1) can be written as follows:\nv(0,t)+λs∂v\n∂t(0,t) =µ\nβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂v\n∂y(0,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nAbbatiello et al.[6] presentedthemathematical analysis ofNavier-Stokes- like problems\ninvolving a boundary where dynamic slip applies. Ferr´ as et al.[7] presented analytical and\nsemi-analytical solutions to some linear and nonlinear pro blems for Couette and Poiseuille\nflows for Newtonian and non-Newtonian media with slip bounda ry conditions at different\nwalls. Thalakkottor and Mohseni [8] used molecular dynamic simulations to study slip at\nthe fluid-solid boundary in an unsteady flow based on Stokes se cond problem, when the\nwall undergoes an oscillatory motion. They showed the exist ence of dynamic wall slip and\ndiscussed the resulting hysteresis phenomena. Hysteresis was attributed to the unsteady\ninertial forces of the fluid. Kaoullas and Georgiou [9] and Da mianouet al.[10] investigated\nslip yield stress effects in Poiseuille flows of non-Newtonian fluids and presented analytical\nand numerical solutions to these problems for a variety of sl ip conditions. Kaoullas and\nGeorgiou [11] derivedanalytical solutionstosomePoiseui lleandCouetteproblemsincluding\ndynamic wall slip and discussed its effects on the flow developm ent. More recent work on\nthe flow of power law fluids in circular annuli and analytical a pproximate solutions was\npresented by Deterre et al.[12]. Pitsillou et al.[13] presented solutions to flow problems\nwith logarithmic wall slip. Ali et al.[14] solved the axial, annular Couette flow of a\nNewtonian viscous fluid of constant density, taking into acc ount both Navier and dynamic\nslip boundary conditions, using the Laplace transform tech nique and inversion by Laguerre\npolynomials. Farragui et al.[15] used separationl of variables to derive analytical sol utions\nto the problem of cessation of annular Poiseuille and Couett e flows of a Newtonian fluid\nwith dynamic wall slip.\nThe above literature review clearly shows the importance of using numerical schemes\nfor the solution of Couette and Poiseuille flows of power-law fluids, side by side with the\npossibility of obtaining exact or approximate analytical s olutions.\nThe present work uses an efficient numerical scheme to solve th e Newtonian planar\nCouette flow when static or dynamic wall slip applies along th e fixed plate and the other\nplate moves either at constant or oscillatory speed. The evo lution of velocity and the\nvolumetric flow rate are calculated for three cases: constan t, oscillating, and single-period\nsinusoidal plate velocity. For the last two cases, the hyste retic behavior of the fluid in\nfollowing the motion of the wall is put in evidence, as this is implies energy dissipation.\n2 Governing equations\nConsider the time-dependent Couette flow of a viscous fluid be tween infinite parallel walls\nplaced a distance dapart. We work with the dimensionless equations scaling len gths byd,\ntime byd2/ν, whereνis the kinematic viscosity (defined by ν≡/µ/ρ,ρbeing the constant\nfluid density), and the velocity by the characteristic veloc ityV0of the moving wall. The\nfluid is initially at rest and suddenly the upper wall starts m oving with velocity f(t),tbeing\nthe dimensionless time.\nThe governing equations and boundary conditions in dimensi onless form are presented\n2in the system (3)-(6):\n∂v\n∂t=∂2v\n∂y2, t >0,0≤y≤1, (3)\nv(0,t)+Λs∂v\n∂t(0,t) =1\nB∂v\n∂y(0,t), (4)\nv(1,t) =f(t), t > 0 (5)\nv(y,0) = 0,0≤y≤1 (6)\nIn the above equations, vdenotes the dimensionless velocity. No slip is assumed at th e\nupper wall ( y=1) and the dynamic slip law applies at the fixed wall ( y=0).B=βd/µis the\ndimensionless slip number, where µis the viscosity, and Λs=λν/d2is the dimensionless\nslip-relaxation number. It shouldbe noted that when Λs= 0, the slip equation is reduced to\nthe static Navier slip equation. Moreover, when B→ ∞, the no-slip condition is recovered.\n3 Weighted Average Finite Difference Method\nThe analytical solution of flows with dynamic wall slip is pos sible only for linear problems,\ne.g., for Newtonian flows with a linear slip equation, such as Eq. (1). However, the visu-\nalisation of the solution and the analysis of the flow, still r equire numerical calculations,\nwhich may not be trivial (see, e.g., [16]). Hence, it is neces sary to use numerical methods\nto approximate the solution of this model. These approximat ion techniques require great\neffort. Thereexist in the literature many methods used to nume rically solve partial differen-\ntial equations: spectral methods [17], finite element, finit e difference, the weighted average\nfinite difference method (WAFDM) [18, 19, 20], and the collocat ion method [21] and [22].\nIn what follows, we use a WAFDM scheme [23] to simulate and stu dy the behavior\nof solutions for the problem (3)-(6) of planar Couette flow wi th Navier and dynamic slip\nconditions prevailing at the fixed wall only. The motion is tr iggered by assigning a given\nmotion at the other boundary in its own plane.\nThe WAFDM is a widely used numerical technique for solving di fferential equations by\napproximating the derivatives of a function at discrete poi nts in space and time using finite\ndifferences. The discretized solution is eventually obtaine d by solving a system of algebraic\nequations [18]. The WAFDM is relatively simple to implement , computationally efficient,\nand can be applied to a wide range of problems [23, 17]. The met hod can be explicit (easy\nand simple for coding and can be used when the function being a pproximated is relatively\nsmooth and well-behaved) or implicit (more accurate and has a larger stability region and\ncan be used when the function being approximated is not smoot h), depending on the value\nof the weight factor θ, 0≤θ≤1. When θ= 0.5, the Crank-Nicolson implicit scheme is\nrecovered [23].\nThe formulation of the WAFDM is outlined below. The domain [0 ,1]×[0,T] in the\n(y,t)-plane is discretized by a uniform grid with steps h= ∆yandk= ∆t, so that,\nh=1\nN, k=T\nM, (7)\nwhereNandMare the numbers of subintervals used for yandt, respectively. Hence, the\ncoordinates of the grid points are\nyn=n∆y, n= 0,1,2, ..., N, t m=m∆t, m= 0,1,2, ..., M.\nThe numerical values of the variable vand the function fat the general grid point ( yn,tm)\nare denoted, respectively, by vm\nnandfm\nn. The following difference approximations are used\n3for the time and spatial derivatives of the problem:\n(vt)m\nn=vm+1\nn−vm\nn\nk+O(k), (8)\n(vy)m\nn=vm\nn+1−vm\nn\nh+O(h), (9)\nand\n(vyy)m\nn=vm\nn+1−2vm\nn+vm\nn−1\nh2+O(h2). (10)\nTheseformulae are used to approximate the partial derivati ves of function vin the proposed\nsystem of equations and conditions. Substituting Eqs. (8)- (10) into the governing equations\n(3)-(6) leads to a linear system of equations for the unknown svm\nn, n= 0,1,2,..., N,m =\n0,1,2,...,M:\nvm+1\nn−vm\nn\nk=θvm\nn+1−2vm\nn+vm\nn−1\nh2+(1−θ)vm+1\nn+1−2vm+1\nn+vm+1\nn−1\nh2, (11)\nθvm\n0+(1−θ)vm+1\n0+Λsvm+1\n0−vm\n0\nk=θ\nBvm\n1−vm\n0\nh+1−θ\nBvm+1\n1−vm+1\n0\nh, (12)\nm= 0,1,2,...,M\nvm\nN=f(mk), m= 1,2,3,...,M (13)\nv1\nn= 0, n= 0,1,2,...,N. (14)\nAfter some manipulations and simplifications, the above sys tem may be cast in matrix\nform as follows:\nAVm+1=BVm+Fm, (15)\nwhereVm+1is the vector of unknown function values at time m+1,\nA=\n(1−θ)(1+1\nBh)+Λs\nk−1−θ\nBh0 0 ···0 0\na b a 0···0 0\n0 a b a ···0 0\n0 0 a b ···0 0\n.....................\n0 0 0 0 ···b a\n0 0 0 0 ···0α\n\nN+1,(16)\nB=\n−θ(1+1\nBh)+Λs\nkθ\nBh0 0 ···0 0\na′b′a′0···0 0\n0 a′b′a′···0 0\n0 0 a′b′···0 0\n.....................\n0 0 0 0 ···b′a′\n0 0 0 0 ···0 0\n\nN+1,(17)\nand\nFm= (0,0,0,0,...,fm\nn)T\nN+1, (18)\n4wherefm\nn≡f(xn,tm),\na=−(1−θ)ξ, b= 1+2(1 −θ)ξ, ξ=k\nh(19)\nand\na′=θξ, b′= 1−2θξ. (20)\nThesystem15is easily solved. Thelocal truncation errorof theschemeisof order O(h2+k).\nThis scheme is conditionally stable when θ >0.5, and it is unconditionally stable when\nθ≤0.5 [23].\n4 Results and discussion\nIn this section, we use the introduced numerical scheme (11) -(14) to simulate the approx-\nimation solution of (3)-(6). It will be shown that the introd uced WAFDM provides good\napproximations for the solution of (3)-(6). It is applicabl e and efficient for solving the given\nsystem of governing equation with the accompanying boundar y and initial conditions. Dif-\nferent test examples are carried out to approximate the solu tions and compare between the\nsolutions using different techniques. We compared the comput ational time of the WAFDM\nand the Legendre spectral collocation method and foundthat the WAFDM is more efficient.\nThe results assess the effect of the various parameters in the i nitial and boundary\nconditions to which the solution is subjected, demonstrate the efficiency of the introduced\ntechnique and justify the accuracy in comparison with other methods. In our numerical\nexperiments we choose different values for the weight factor a nd full agreement is reached\nwith the theoretical stability condition.\n4.1 Case 1. Plate moving at a constant speed\nIn order to assess the efficiency of the proposed numerical sch eme, we have compared our\nresults (WAFDM) for the special case when θ= 0 and f(t) = 1 with those obtained using\nthe Matlab pdepe toolbox. This is one of the cases treated in [ 24], where one plate is\nkept fixed, while the other one is suddenly set to motion from r est with constant speed.\nThe results of this comparison are shown in Figure 1 for Λs=0 (Navier slip) and three\nvalues of the slip number B, i.e.,B=0.1 (strong ), 1 (moderate), and 10 (weak slip). The\nresults of this comparison are shown in Figure 1. Full agreem ent is reached between the two\nmethods, and with the results presented in [24] based on a Fou rier expansion and on the\nuse of one-sided Fourier transform as well. The main feature of the solution is the reduction\nof the slip velocity at the boundary y= 0 as parameter Bincreases. The efficiency on\nthe computational time of the WAFDM is demonstrated by compa ring it with that of the\nLegendre spectral collocation method in Table 1.\nTable 1: Comparison of the WAFDM with the spectral collocati on method.\nWAFDM Spectral method\nNnCPU MCPU\n5005.9 s 5963.6 s\n500018.3 s 127579.9 s\n1000044.8 s 1588937.5 s\n50000789.9 s 20219357.6 s\n50\n10.20.4\n1v0.6\n0.8Pdepe, B=0.1\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 00\n10.20.4\n1v0.6\n0.8WAFDM, B=0.1\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 0\n0\n10.20.4\n1v0.6\n0.8Pdepe, B=1\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 00\n10.20.4\n1v0.6\n0.8WAFDM, B=1\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 0\n0\n10.20.4\n1v0.6\n0.8Pdepe, B=10\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 00\n10.20.4\n1v0.6\n0.8WAFDM, B=10\nt0.8\n0.50.6\ny1\n0.4\n0.2\n0 0\nFigure 1: Evolution of the solution v(y,t) in Case 1 (plate moving at constant speed) with\nθ= 0 and Navier slip ( Λs= 0) for B= 0.1 (strong slip), 1 (moderate slip), and 10 (weak\nslip). The results obtained with pdepe.m (left column) comp are well with those of the\nWAFDM (right column).\n6The volumetric flow rate, defined by\nQ(t) =/integraldisplay1\n0v(y,t)dy, (21)\nis shown in Figure 2 for θ= 0,B=0.1 (strong slip) and 1 (moderate slip), and Λs=0 (Navier\nslip), 1, and 10. The damping effect of the slip relaxation para meter in reaching a fully-\ndeveloped flow is clearly shown. As expected, the initial val ue of the volumetric flow rate\nis independent of Λsand decreases with the slip parameter B(since wall slip is reduced.\n4.2 Case 2. Oscillating plate\nConsider now the case where the moving plate is performing co ntinuous harmonic oscilla-\ntions with period equal to unity, i.e.,\nf(t) = sin4πt. (22)\nA similar problem was treated in [8] for the unsteady flow to es timate the increment in slip\nat the boundary due to wall acceleration, and uncover the hys teretic behavior of the slip\nvelocity for this harmonic motion of the boundary. Here it is required to evaluate the effect\nof the dynamic slip parameter Λson the flow in general, and on the hysteretic behavior of\nthe slip velocity due to oscillating wall motion.\nFigure 3 illustrates the distribution of the solution in spa ce and time with different\nvalues of Λs, as time is taken to run along two complete periods of the wall oscillations.\nThe boundary condition at the moving wall is clearly satisfie d. AsΛsincreases from 0 .1 to\n10, a damping of the amplitude of the oscillations with time t akes place. The differences\nbecome more noticeable as one approaches the fixed wall at y= 0, where the velocity profile\nbecomes more flattened.\nFigure 4 illustrates the solutions at y= 0, i.e the slip velocity, as functions of time\nfor different values of Λs. The figure clearly shows time damping and lagging effects at th e\nboundary y= 0. The amplitudes of oscillations of the slip velocity show s a reduction of\napproximately 33% as the value of Λsincreases from 0 .1 to 10.\nIn order to put in evidence the crucial role played by the weig ht factor θin the present\nnumerical study, we have considered one case with a new value of this parameter to test\nthe stability of the numerical scheme. As mentioned above, t he scheme is stable provided\nthatθ <0.5 [23]. The instability occuring when θ= 0.7>0.5 is illustrated in the 3D-plot\nof Figure 5.\nThe 3D plots in Figure 6 show how the solutions change in space and time when the\nslip parameter Bassumes the values 0.1, 1, and 10. Figure 7 illustrates the de pendence of\nthe slip velocity on parameter B. In both plots, we have fixed the value Λs= 0.5. The\neffect of parameter Bbecomes more pronounced as the boundary y= 0 is approached. Here\nagain, one notes the presence of lag in following the boundar y motion, and time damping\nof the peaks at the boundary y= 0 as the value of Bincreases.\nIn order to visualize the hysteretic behavior of the slip vel ocity and the lagging in\nfollowing the applied motion on one wall noticed in [8], plot s have been provided in Figure\n8 of the slip velocity against the boundary motion sin ωtfor three values of the dynamic\nslip parameter Λs. It is noticed that the hysteresis loops become narrower as t he value of\nthe dynamic slip parameter increases. In [8] the width of the hysteresis loop is related to\nthe loss of energy transfer from the wall to the fluid.\nThevolumetricflowrateisshowninFigure9, whereit isseent hatthisisoscillatory and\nis damped by the slip relaxation parameter. However, for suffi ciently large values of Λs, it\nis seen that the amplitude of oscillations of Q(t) will be less sensitive to any further increase\n70 1 2 3 4 5 6 7 8 9 10\nt00.10.20.30.40.50.60.70.80.91QB=0.1\nΛs= 0.1\nΛs= 1\nΛs= 10\n(a)\n0 1 2 3 4 5 6 7 8 9 10\nt00.10.20.30.40.50.60.70.8QB=1\nΛs= 0.1\nΛs= 1\nΛs= 10\n(b)\nFigure 2: Evolution of the volumetric flow rate Q(t) in Case 1 (plate moving at constant\nspeed) with θ= 0 and Λs= 0 (Navier slip), 1, and 10: (a) B= 0.1 (strong slip); (b) 1\n(moderate slip).\n8-1\n1-0.5\n10v\n0.8Λs=0.1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(a)\n-1\n1-0.5\n10v\n0.8Λs=1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(b)\n-1\n1-0.5\n10v\n0.8Λs=10\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(c)\nFigure 3: Evolution of the solution v(y,t) in Case 2 (oscillating plate) with θ= 0 andB= 1\n(moderate slip): (a) Λs= 0 (Navier slip); (b) Λs= 1; (c)Λs= 10.\n90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nt-0.15-0.1-0.0500.050.10.150.20.250.3vΛs=0.1\nΛs=1\nΛs=10\nFigure 4: Evolution of the slip velocity v(0,t) in Case 2 (oscillating plate) with θ= 0.5 and\nB= 1 (moderate slip) for Λs= 0.1 , 1, and 10. Time damping and lagging are observed.\n-4\n1-2\n10v×1017\n0.8Λs=1\n2\nt0.50.6\ny4\n0.4\n0.2\n0 0\nFigure 5: Instability of the solution v(y,t) in Case 2 (oscillating plate) with θ= 0.7,B= 1\n(moderate slip) and Λs= 1.\n10-1\n1-0.5\n10v\n0.8B=0.1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(a)\n-1\n1-0.5\n10v\n0.8B=1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(b)\n-1\n1-0.5\n10v\n0.8B=10\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(c)\nFigure 6: Evolution of the solution v(y,t) in Case 2 (oscillating plate) with θ= 0 and\nΛs= 0.5: (a)B= 0.1 (strong slip); (b) B= 1 (moderate slip); (c) B= 10 (weak slip).\n110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nt-0.1-0.0500.050.10.150.20.25vB=0.1\nB=1\nB=10\nFigure 7: Evolution of the slip velocity v(0,t) in Case 2 (oscillating plate) with θ= 0.5 and\nΛs= 0.5 forB= 0.1 (strong slip), 1 (moderate slip), and 10 (weak slip).\nofΛs. This phenomenon becomes more striking for larger values of the slip parameter B.\nNotice that after two complete oscillations, the volumetri c flow .flow rate has not vanished\ndue to retardation.\n4.3 Case 3. Single plate oscillation\nIn this section we assume that the upper plate oscillates onl y once and then comes to rest,\ni.e.,\nf(t) =/braceleftigg\nsin4πt, t≤0.5\n0, t > 0.5. (23)\nFigure 10 illustrates the effect of the dynamic slip parameter Λson the solution, where\nthe value of the slip parameter was set to B= 1.\nFigure 11 shows the effect of the relaxation parameter Λson the slip velocity v(0,t) for\nB= 1 (moderate slip). As for the case of continuous harmonic bo undary motion, the effect\nof the dynamic slip parameter becomes stronger as the bounda ryy= 0 is approached as\nmay be seen in Figure 10. The same remains valid for the effect of slip parameter Bon the\nfluid motion as shown in Figures 12 and 13, where the value of th e dynamic slip parameter\nwas setΛs= 0.5.\nFigure 14 represents the hysteretic behavior of the slip vel ocity in following the motion\nof the wall in Case 3 for B= 1 and three values of the dynamic slip parameter Λs. Here\nagain, it is noticed that the hysteresis loop becomes narrow er as the value of Λsincreases.\nFinally, we have shown in Figure 15 the volumetric flow rate in Case 3 with θ= 0, for\ntwo values of Band three values of Λs. The volumetric flow rate is almost independent of\nthese two slip parameters.\n5 Conclusions\nA weighted average finite difference scheme has been used to sol ve the time-dependent\n(start-up) plane Couette flow with dynamic slip along the fixe d plate. Three different cases\n12-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)-0.1-0.0500.050.10.150.2v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 0.1\n(a)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)-0.0100.010.020.030.040.050.060.070.08v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 1\n(b)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)00.0020.0040.0060.0080.010.012v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 10\n(c)\nFigure 8: Hysteretic behavior of the slip velocity v(0,t) in Case 2 (oscillating plate) with\nθ= 0 and B= 1: (a) Λs= 0.1; (b)Λs= 1; (c)Λs= 10. Note that the scale of the vertical\naxis changes.\n130 1 2 3 4 5 6 7 8 9 10\nt-1-0.8-0.6-0.4-0.200.20.40.60.81QB=0.1\nΛs= 0.1\nΛs= 5\nΛs= 10\n(a)\n0 1 2 3 4 5 6 7 8 9 10\nt-0.8-0.6-0.4-0.200.20.40.60.8QB=1\nΛs= 0.1\nΛs= 5\nΛs= 10\n(b)\nFigure 9: Evolution of the volumetric flow rate Q(t) in Case 2 (oscillating plate) with θ= 0\nandΛs= 0.1, 5, and 10: (a) B= 0.1 (strong slip); (b) B= 1 (moderate slip).\n14-1\n1-0.5\n10v\n0.8As=0.1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(a)\n-1\n1-0.5\n10v\n0.8As=1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(b)\n-1\n1-0.5\n10v\n0.8Λs=10\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(c)\nFigure 10: Evolution of the solution v(y,t) in Case 3 (single plate oscillation) with θ= 0\nandB= 1 (moderate slip): (a) Λs= 0.1;Λs= 1; (c)Λs= 10.\n150 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nt-0.15-0.1-0.0500.050.10.150.20.250.3vΛs=0.1\nΛs=1\nΛs=10\nFigure 11: Evolution of the slip velocity v(0,t) in Case 3 (single plate oscillation) with\nθ= 0.5 andB= 1 (moderate slip) for Λs= 0.1, 1, and 10.\nhave been considered for the motion of the moving plate, i.e. , constant-speed, sinusoidal\nand single-oscillation. The numerical solutions compare w ell with available analytical and\nnumerical solutions. Both the slip and relaxation paramete rs appear to decelerate the\nevolution of the flow. For the two cases of accelerated (sinus oidal and one-period sinusoidal)\nboundary motion, the numerical results have clearly demons trated a hysteretic behavior of\nthe slip velocity that will be responsible for time lag and lo ss of energy transfer by the\nmoving wall to the fluid.\nFuture work will be devoted to extend the numerical scheme fo r solving non-linear\nextensions of the problem studied here. These include the no n-Newtonian (e.g., power-law)\nflow under both Navier and dynamic slip at the fixed wall and the Newtonian flow with slip\nobeying a non-linear dynamic slip law.\nConflict of interest\nThe authors declare that they have no conflict of interest.\nData availability\nAll data generated or analyzed during this study are include d in this article\nReferences\n[1] J.R.A. Pearson, C.J.S. Petrie (1968) On melt flow instabi lity of extruded polymers,\nin: R.E. Wetton, R.E. Whorlow (Eds.), Polymer Systems: Defo rmation and Flow,\nMacmillan, pp. 163-187.\n[2] S.G. Hatzikiriakos (2012) Wall slip of molten polymers, Prog. Polym. Sci. 37, 624-643.\n16-1\n1-0.5\n10v\n0.8B=0.1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(a)\n-1\n1-0.5\n10v\n0.8B=1\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(b)\n-1\n1-0.5\n10v\n0.8B=10\n0.5\nt0.50.6\ny1\n0.4\n0.2\n0 0\n(c)\nFigure 12: Evolution of the solution v(y,t) in Case 3 (single plate oscillation) with θ= 0\nandΛs= 0.5: (a)B= 0.1 (strong slip); (b) B= 1 (moderate slip); (c) B= 10 (weak slip).\n170 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nt-0.15-0.1-0.0500.050.10.150.20.250.3vΛs=0.1\nΛs=1\nΛs=10\nFigure 13: Evolution of the slip velocity v(0,t) in Case 3 (single plate oscillation) with\nθ= 0.5 andΛs= 0.5 forB= 0.1 (strong slip), 1 (moderate slip), and 10 (weak slip).\n[3] S.G. Hatzikiriakos (2015) Slip mechanisms in complex flu id flows, Roy. Soc. Chem.\nSoft Matter 11, 7851-7856.\n[4] A. Ya. Malkin, S. A. Patlazhan (2018) Wall slip for comple x liquids-Phenomenon and\nits causes, Adv. Coll. Interface Sci. 257, 42-57.\n[5] C.L.M.H. Navier (1827) Sur les lois du mouvement des fluid es, Mem. Acad. R. Sci.\nInst. Fr. 6, 389–440.\n[6] A. Abbatiello, M. Bul´ ıˇ cek, E. Maringov´ a (2021) On the dynamic slip boundary condi-\ntion for Navier-Stokes-like problems, Math. Mod. Meth. App l. Sci. 31(11), 2165-2212.\n[7] L.L. Ferr´ as, J.M. N´ obrega, F.T. Pinho (2012) Analytic al solutions for Newtonian and\ninelastic non-Newtonian flows with wall slip, J. Non-Newton ian Fluid Mech. 175-176,\n76-88.\n[8] J.J. Thalakkottor, K. Mohseni (2013) Analysis of bounda ry slip in a flow with an\noscillating wall, Phys. Rev. E 87, 033018.\n[9] G. Kaoullas, G.C. Georgiou (2013) Slip yield stress effect s in start-up Newtonian\nPoiseuille flows, Rheol. Acta 52, 913-925.\n[10] Y. Damianou, M. Philippou, G. Kaoullas, G.C. Georgiou ( 2014) Cessation of vis-\ncoplastic Poiseuille flow with wall slip, J Non-Newtonian Fl uid Mech. 203, 24-37.\n[11] G. Kaoullas, G.C. Georgiou (2015) Start-up and cessati on Newtonian Poiseuille and\nCouette flows with dynamic wall slip, Meccanica 50, 1747-176 0.\n[12] R. Deterre, F. Nicoleau, Q. Lin, N. Allanic, P. Mousseau (2020) The flow of power-law\nfluids in concentric annuli: A full analytical approximate s olution, J. Non-Newtonian\nFluid Mech. 285, 104392.\n18-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)-0.0500.050.10.150.2v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 0.1\n(a)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)00.010.020.030.040.050.060.070.08v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 1\n(b)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1\nf(t)00.0020.0040.0060.0080.010.012v(0,t)parametric relation between v(0,t) and sin(wt)\nΛs= 10\n(c)\nFigure 14: Hysteretic behavior of the slip velocity v(0,t) in Case 2 (single plate oscillation)\nwithθ= 0 and B= 1: (a) Λs= 0.1; (b)Λs= 1; (c) Λs= 10. Note that the scale of the\nvertical axis changes.\n190 0.5 1 1.5 2 2.5 3\nt-0.3-0.2-0.100.10.20.30.4QB=0.1\nΛs= 0.1\nΛs= 1\nΛs= 10\n(a)\n0 0.5 1 1.5 2 2.5 3\nt-0.4-0.3-0.2-0.100.10.20.30.4QB=1\nΛs= 0.1\nΛs= 1\nΛs= 10\n(b)\nFigure 15: Evolution of the volumetric flow rate Q(t) in Case 3 (single plate oscillation)\nwithθ= 0 and Λs= 0.1, 1, and 10: (a) B= 0.1 (strong slip); (b) B= 1 (moderate slip,\nright).\n20[13] R. Pitsillou, A. Syrakos, G.C. Georgiou (2020) Applica tion oftheLambert Wfunction\ntosteadyshearingNewtonianflowswithlogarithmicwallsli p, Phys.Fluids32, 053107.\n[14] A.E.K. Ali, A.F. Ghaleb, M.S. Abou-Dina, M.A. Helal (20 23) Laplace transform solu-\ntion of the time-dependent annular Couette flow with dynamic wall slip, J. Brazilian\nSoc. Mech. Sci. Enging. 45, 586.\n[15] M.E. Farragui, O. Souhar, G.C. Georgiou (2024) Newtoni an annular Poiseuille and\nCouette flows with dynamic wall slip, Eur. J. Mech.-B/Fluids 103, 136-144.\n[16] S.Raju, D.Gr¨ unding,T.Maric, D.Bothe, M.Fricke (202 2) Computinghydrodynamic\neigenmodes of channel flow with slip - A highly accurate algor ithm, Can. J. Chem.\nEng. 100, 3531–3547.\n[17] N.H. Sweilam, M.M. Abou Hasan (2016) Numerical approxi mation of L´ evy-Feller\nfractional diffusion equation via Chebyshev- Legendre collo cation method, Eur. Phys.\nJ. Plus 131, 251.\n[18] N.H. Sweilam, A.F. Ghaleb, M.S. Abou-Dina, M.M. Abou Ha san, S.M. Al-Mekhlafi,\nE.K. Rawy (2022) Numerical solution to a one-dimensional no nlinear problem of heat\nwave propagation in a rigid thermal conducting slab, Indian J. Phys. 96(1), 223-232.\n[19] N.H. Sweilam, M.M. Abou Hasan (2017) Numerical solutio ns of a general coupled\nnonlinear systemof parabolicandhyperbolicequations of t hermoelasticity, Eur.Phys.\nJ. Plus 132, 212.\n[20] N.H. Sweilam, T.A. Assiri, M.M. Abou Hasan (2022) Optim al control problem of\nvariable-order delay system of advertising procedure: Num erical treatment, DCDS-S\n15(5), 1247-1268.\n[21] N.H. Sweilam, M.M. Abou Hasan (2019) An improved method for nonlinear variable\norder L´ evy-Feller advection-dispersion equation, Bull. Malays. Math. Sci. Soc. 42,\n3021-3046.\n[22] M.M. Abou Hasan, N.H. Sweilam (2020) Numerical studies of the fractional optimal\ncontrol problemofawareness andtrialadvertisingmodel, P rogr.Fract. Differ. Applics.\n8(4), 509-524.\n[23] G. D. Smith, Numerical solution of partial differential e quations, second ed., Oxford\nUniversity Press, Oxford (1978).\n[24] M. S. Abou-Dina, M. A. Helal, A. F. Ghaleb, G. Kaoullas, G . C. Georgiou (2020)\nNewtonian plane Couette flow with dynamic wall slip, Meccani ca 55, 1499-1507.\n21"    },    {        "title": "2402.05753v1.Cops_and_Robber_on_Hyperbolic_Manifolds.pdf",        "content": "COPS AND ROBBER ON HYPERBOLIC MANIFOLDS\nVESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nAbstract. The Cops and Robber game on geodesic spaces is a pursuit-evasion\ngame with discrete steps which captures the behavior of the game played\non graphs, as well as that of continuous pursuit-evasion games. One of the\noutstanding open problems about the game on graphs is to determine which\ngraphs embeddable in a surface of genus ghave largest cop number. It is\nknown that the cop number of genus ggraphs is O(g) and that there are ex-\namples whose cop number is ˜Ω(√g). The same phenomenon occurs when the\ngame is played on geodesic surfaces.\nIn this paper we obtain a surprising result about the game on a surface\nwith constant curvature. It is shown that two cops have a strategy to come\narbitrarily close to the robber, independently of the genus. We also discuss\nupper bounds on the number of cops needed to catch the robber.\nOur results generalize to higher-dimensional hyperbolic manifolds.\n1.Introduction\nThe Cops and Robber game is a pursuit-evasion game. The game is commonly\nplayed on graphs [2, 4, 6, 7, 9, 12, 18, 22], and as a new variant on geodesic\nspaces [17, 25, 26]. The players in the Cops and Robber game are the robber rand\nkcops c1, . . . , c k. On graphs the players occupy vertices while on geodesic spaces\nthe players occupy points in space. The game is played in rounds. The robber\nchooses initial positions for the players r0, c0\n1, . . . , c0\nkand in each round of the game\nthe players can move to a new position1. Each round of the game has two turns,\nthe first one for the robber and the second one for the cops. In particular, each of\nthe cops can make a step at the cops’ turn. When the game is played on a graph,\nthe players are allowed to move to an adjacent vertex at their turn. When the game\nis played on a geodesic space X, the robber chooses an agility function τ:N→R+\nat the beginning of the game such thatP\nn≥1τ(n) =∞. In the n-th round, each\nplayer makes a step of length at most τ(n) (at their turn). The position of a player\nxafter round nis denoted as xn. In the following we give the robber the pronoun\nhe, him, while the cops have the pronoun she, her. We say the cops catch the\n2020 Mathematics Subject Classification. 91A05, 91A50, 32Q45.\nV.I. was supported by the Slovenian Research and Innovation Agency (ARIS) under the grant\nZ1-50003, and by the European Union (ERC, KARST, 101071836).\nB.M. was supported in part by an NSERC Discovery Grant R611450 (Canada), by the ERC\nSynergy grant (European Union, ERC, KARST, project number 101071836), and by the Re-\nsearch Project N1-0218 of ARIS (Slovenia). On leave from: FMF, Department of Mathematics,\nUniversity of Ljubljana, Ljubljana, Slovenia.\nA.W. was supported by the Vanier Canada Scholarship program and by the Deutsche\nForschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strat-\negy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).\n1The rules of the Cops and Robber game on graphs we define here are slightly different to\nstandard rules, but they do not affect the outcome of the game on connected graphs.\n1arXiv:2402.05753v1  [math.CO]  8 Feb 20242 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nrobber if at some point in the game the cop cioccupies the same position as the\nrobber r. If the robber is not caught, we say the robber escapes .\nThe cop number c(G) of a graph Gis the minimum number of cops that can\ncatch the robber (regardless of the robber’s strategy and initial positions). For a\ngeodesic space Xwe denote by c0(X) the cop catch number , which is the minimum\nnumber of cops that can catch the robber. Further, if the game is played on a\ngeodesic space, we say that the cops win the game if\ninf\nn,id(cn\ni, rn) = 0 . (1.1)\nThe minimum number of cops that win the game on a geodesic space Xis the cop\nwin number c(X) ofX. Note that in [25, 26, 17] the cop win number is referred\nas the cop number, and the cop catch number is called the strong cop number.\nInformally speaking, the cop win number is the minimum number of cops needed\nto capture the robber when the players are represented by infinitesimally small\ndisks in the metric space.\nThe Cops and Robber game on geodesic spaces is tightly related to continuous\nand discrete pursuit-evasion games on metric spaces. In continuous pursuit-evasion\ngames the players make decisions at every point in the time interval [0 ,∞). For\nexample, the Lion and Man game as introduced by Rado (see Littlewood’s Mis-\ncellany [20]) is a two-player game where each player is represented by a point in\na metric space, and they can both move with the same maximal speed in any di-\nrection. More precisely, the players’ paths are 1-Lipschitz maps and a strategy of\na player consists of choosing a path f(g) for each of the possible opponents paths\ngsuch that if gandg′agree on a time interval [0 , t], then f(g) and f(g′) agree\non [0 , t]. The man is trying to evade the capture by the lion, while the lion is\naiming to occupy the same point as the man. Besicovitch showed that the man can\nescape the lion when the game is played on a disk [20]. Croft studied a variation\nof this game with multiple pursuers on higher dimensional balls [10] and Satimov\nand Kushkarov studied the game on the sphere [27]. However, Bollob´ as, Leader\nand Walters [5] showed that in some geodesic spaces, the lion and man game is not\nwell-defined, in the sense that both the lion and the man have a winning strategy.\nThe Discrete Lion and Man game is the Cops and Robber game where the agility\nfunction is constant, i.e. τ≡Kfor some constant K. It was shown that in the\nDiscrete Lion and Man game the lion can catch the man on a disk, more generally,\nthe lion can catch the man on any compact cat(0)-space [3, 31]. While the Cops and\nRobber game is a discrete pursuit-evasion game, it captures the properties of the\ncontinuous game, in the sense that in the Cops and Robber game one cop cannot\ncatch the robber when the game is played on a disk [17]. Further, the Cops and\nRobber game is well defined, as Mohar showed that either the cops or the robber\nhave a winning strategy [25].\nOne of the first results about cop numbers given by Aigner and Fromme states\nthat planar graphs have cop number at most 3 [2].\nTheorem 1.1 ([2]).The cop number of every planar graph is at most three.\nAn important tool for determining upper bounds for cop numbers is the Isometric\nPath Lemma, which we use to prove our main theorem and to show that the cop\ncatch number for special surfaces of genus g≥2 is at most 6 (see Theorem 4.5).COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 3\nLemma 1.2 ([2, 26]) .LetIbe an isometric path starting at Aand ending at B.\nThen one cop ccan guard Iafter spending time equal to the length of Ion the path\nto adjust himself.\nThe idea of the lemma is that the cop cstays at the same distance to Aas the\nrobber, unless the robber is too far away, in which case cwaits at B.\nFor graphs embeddable in a surface of genus git is known that the cop number\nis at most linear in gand recent progress was made on improving the linear factor,\nsee [8, 11, 28]. It is an outstanding open problem to determine which graphs\nembeddable in a surface of genus ghave largest cop number. There are graphs of\ngenus gwith cop number at least g1\n2−o(1); one such example are binomial random\ngraphs G(n, p) with p=2 logn\nn[4, 24]. The gap between the upper and lower bound\nis large and it is conjectured that the lower bound gives the right order of the cop\nnumber.\nConjecture 1.3 ([24, 26]) .LetSbe a a graph of genus g. Then c(S) =O(√g).\nConjecture 1.3 resembles the famous Meyniel’s conjecture.\nConjecture 1.4 ([12]).LetGbe a a graph of order n. Then c(G) =O(√n).\nThis is a long-standing conjecture and the best upper bound is that the cop\nnumber of an n-vertex graph is at most O\u0010\nn\n2(1−o(1))√\nlog2(n)\u0011\n, see [13, 21, 29]. Hence\neven a weaker version of the conjecture, called the Weak Meyniel’s conjecture,\nwhich states that there exists some ε >0 such that c(G)≤O(n1−ε), is still open.\nConjecture 1.3 for graphs of genus gimplies that Meyniel’s conjecture holds for\ngraphs with O(n) edges as their genus is O(n), which includes for example subcubic\ngraphs. Hosseini, Mohar and Gonzalez Hermosillo de la Maza [16] showed that\nMeyniel’s conjecture for subcubic graphs implies the Weak Meyniel’s conjecture\n(with exponent3\n4), thus Conjecture 1.3 implies the Weak Meyniel’s conjecture.\nAs for graphs, the cop win number for a surface of genus gis at most linear\ning[25]. It was shown that for graphs of cop number at least 3 there exists a\nsurface Sof genus gwith c(S)≥c(G) [25]. Therefore there are compact surfaces of\ngenus gwith cop win number at least g1\n2−o(1). The following conjecture is a tough\nconjecture since it implies Conjecture 1.3.\nConjecture 1.5 ([26]).LetSbe a geodesic surface of genus g. Then c(S) =O(√g).\nThe main result of this paper shows that the upper bound from Conjecture 1.5\ncan be significantly improved for surfaces of constant curvature. The bound is\nindependent of the genus, which we find surprising. Surfaces of constant curvature\nare either spherical, Euclidean or hyperbolic. It follows from [17] that spherical and\nEuclidean surfaces, which are surfaces of genus 0 and 1, have cop win number 2 and\ncop catch number 3. Here we settle the case of hyperbolic surfaces which can have\narbitrarily large genus, and extend it to higher-dimensional hyperbolic manifolds.\nThe main result of the paper thus reads as follows.\nTheorem 1.6. IfMis a compact hyperbolic manifold, then c(M) = 2 .\nDefinitions and technical lemmas needed in the proofs are presented in Section 2.\nThe proof of the 2-dimensional version of Theorem 1.6 is given in Section 3. In\nthe proof we describe a novel strategy based on the covering space technique. One\ncop follows the robber while the other one assures that the robber needs to diverge4 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nfrom its route, enabling the first cop to take shortcuts. The higher-dimensional\nversion of Theorem 1.6 is presented in Section 5. Finally, the cop catch number is\ndiscussed in Section 4.\n2.Preliminaries\nIn this section we present definitions and preliminary lemmas that we need to\nprove our results. We start with basic definitions which are needed for the Cops\nand Robber game.\nLet ( X, d) be a metric space. For x, y∈X, an ( x, y)-path is a continuous map\nγ:I→Xwhere I= [0,1] is the unit interval on R,γ(0) = xandγ(1) = y. One can\ndefine the length ℓ(γ) of the path γby taking the supremum over all finite sequences\n0 =t0< t1< t2<···< tn= 1 of the valuesPn\ni=1d(γ(ti−1), γ(ti)). Clearly, the\nlength of every ( x, y)-path is at least d(x, y). The metric space Xis ageodesic\nspace if for every x, y∈Xthere is an ( x, y)-path whose length is equal to d(x, y).\nAn (x, y)-path γisisometric ifℓ(γ) =d(x, y). Observe that for 0 ≤t < t′≤1\nthe subpath γ|[t,t′]is also isometric. Therefore the set γ(I) ={γ(t)|t∈I}is an\nisometric subset of X. With a slight abuse of terminology, we say that the image\nγ(I)⊂Xis an isometric path inX. If there exists a unique isometric path between\nx, ythen we will denote it by xy⊂X. A continuous map γ:R→Xis ageodesic\nif it is locally isometric, i.e., for every t∈Rthere is an ε >0 such that the subpath\nγ|Jon the interval J= [t−ε, t+ε] is isometric.\n2.1.Hyperbolic plane and hyperbolic surfaces. Since we consider the Cops\nand Robber game played on hyperbolic spaces, we properly define those and give\nexamples for particular hyperbolic surfaces.\nThe n-dimensional hyperbolic space Hnis the unique n-dimensional simply\nconnected complete Riemannian manifold with a constant negative sectional cur-\nvature equal to −1. Standard examples for hyperbolic space is the half space\nmodel and the Poincar´ e disk model. The n-dimensional Poincar´ e disk Dnis\nthe disk {(x1, . . . , x n)|x2\n1+···+x2\nn<1}equipped with the hyperbolic metric\n4dx2\n1+···+dx2\nn\n(1−x2\n1−···− x2n). For brevity, we write D=D2.\nWell-known examples for surfaces which are not hyperbolic are the sphere and\nthe flat torus. The flat torus arises from a tessellation of the Euclidean plane by\ntranslations of a parallelogram. Here, the Euclidean plane is the universal covering\nspace of the standard torus. For a topological space X, acovering space ofXis a\ntopological space Ctogether with a continuous surjective map p:C→Xsuch that\nfor every x∈X, there exists an open neighbourhood Uofx, such that p−1(U) is a\nunion of disjoint open sets in C, each of which is mapped homeomorphically onto\nUbyp(i.e. it is a local homeomorphism).\nSimilarly as for the flat torus, hyperbolic surfaces arise from a tessellation of the\nhyperbolic plane into copies of some polygon (see Figure 1), which is formalized by\nthe following theorem.\nTheorem 2.1 (Killing-Hopf [15, 19]) .Any compact Riemannian manifold of con-\nstant curvature −1is isometric to Hn/Γwhere Γis a group of isometries on Hn\nacting freely and properly discontinuously.\nHere, the curvature is the sectional curvature. For more information about\ncurvatures, we refer the reader to [1].COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 5\nAfundamental polygon of a Riemannian surface H2/Γ is a polygon in H2which\ncontains a representative of each Γ-orbit and at most one representative of each\nΓ-orbit in its interior. We define by P(k, θ) the regular k-gon in the Poincar´ e disk\nDcentred at O= (0,0) with angle θat the vertices. We denote its vertices by\nv1, . . . , v kin counter-clockwise direction and let aibe the (oriented) edge from vi\ntovi+1anda−1\nibe the reversed edge from vi+1tovi(we consider the indices modulo\nk). We are going to consider three standard hyperbolic surfaces for g≥2, where\none of them is non-orientable.\n•LetS(g) be the orientable surface obtained from P\u0010\n4g,2π\n4g\u0011\nby identifying\nthe (oriented) edges a4i−3with a−1\n4i−1anda4i−2with a−1\n4ifori= 1, . . . , g .\nThe surface S(2) is depicted in Figure 1.\n•LetS′(g) be the orientable surface obtained from P\u0010\n4g+ 2,2π\n2g+1\u0011\nby iden-\ntifying opposite (oriented) edges ai, a−1\ni+2g+1fori= 1, . . . , 2g.\n•LetN(g) be the non-orientable surface obtained from P\u0010\n2g,2π\n2g\u0011\nby iden-\ntifying the (oriented) edge a2iwith a2i+1fori= 1, . . . g .\na1\na2\na5a6\n(a)\na1 a−1\n1a2\na−1\n2\na5 a−1\n5\na6a−1\n6v1v3 v2\nv4\nv8\nv6 v7v5\n(b)\nFigure 1. Figure (a) depicts the surface S(g) and Figure (b) its\nfundamental domain P\u0010\n4g,2π\n4g\u0011\ninside a partial tessellation of the\nPoincar´ e disk.\nThediameter of a compact surface Sis the maximum distance between points in S,\nthat is diam( S) = max {d(x, y)|x, y∈S}. The systolic girth , denoted as sys( S), is\nthe length of a shortest noncontractible curve on the surface. A hyperbolic surface\nis locally isometric to the hyperbolic plane, which we will use frequently in our\nproofs.\nLemma 2.2. IfSis a hyperbolic surface, a∈Sandr <sys(S)\n4, then the disk\nB(a, r)is isometric to a hyperbolic disk.\n2.2.Trigonometric lemmas for the hyperbolic plane. In the rest of this sec-\ntion, we present lemmas that will be used in the proof of Theorem 3.1. Their proofs\nare omitted and are given in Appendix A. For properties of the hyperbolic plane\nneeded in the proofs of these lemmas we refer the reader to [23, 30].\nFirst, consider Figure 2 and let the robber’s position be X1and the cop’s position\nbeA. The robber moves to B, while the cop moves as close to Bas possible in the\nsame time. Our first lemma shows that if their starting positions would be closer6 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nAX3B\nCX2X1\nFigure 2. A hyperbolic right-angled triangle ABC with the right\nangle at C.\ntogether (at positions X2andX3, say), then their end positions would be closer\ntogether.\nLemma 2.3. LetABC be a hyperbolic triangle with a right angle at C. IfX1∈AC,\nthen\nd(A, B)−d(X1, B) = max\nX2,X3∈AX1d(X2, B)−d(X3, B).\nMoreover, d(A, B)−d(X1, B) =d(X2, B)−d(X3, B)if and only if X2=Aand\nX3=X1.\nThe second lemma shows that d(A, B)−d(X3, B) would decrease if the length\nof the side ACwould decrease, while d(A, X 3) would remain the same.\nLemma 2.4. LetABC be a hyperbolic triangle with a right angle at C. IfX3∈AC,\nthen\nd(A, B)−d(X3, B) = max\nX1,X2∈AC\nd(X1,X2)=d(A,X 3)d(X2, B)−d(X1, B).\nThe next lemma helps to compare triangles where two sides have the same length.\nLemma 2.5. Suppose ABC ,A′B′C′are hyperbolic triangles, d(A, B) =d(A′, B′),\nd(B, C) =d(B′, C′),∠BCA > ∠B′C′A′and∠CAB <π\n2. Then it holds that\nd(A, C)< d(A′, C′).\nThe Pythagorean theorem for the hyperbolic plane for a triangle ABC with right\nangle at Csays that cosh( AB) = cosh( AC)·cosh( BC).In the following we observe\nthat the increase of BCincreases AB”uniformly”.\nLemma 2.6. Suppose t2> t1>0andw >1, then there exists η >0such that for\nx, y∈[t1, t2]withx < y it holds that,\nacosh( wcosh( x))−acosh( wcosh( y))≥η·(y−x).\n3.Hyperbolic surfaces\nIn this section we prove that two cops can win the game on any compact hyper-\nbolic surface. In fact, this is the most difficult step towards proving Theorem 1.6.\nTheorem 3.1. IfSis a compact hyperbolic surface, then c(S) = 2 .COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 7\n3.1.The lower bound. Lets= sys( S). To show that c(S)>1 we play the game\nwith one cop cand the robber rand show that the robber has a strategy to stay\nat distance at leasts\n16to the cop. The robber chooses the agility function τ≡s\n16\nand initial positions such that d(c0, r0)>s\n16. Informally, the robber’s strategy is\nto move in the direction opposite to the cop’s position.\nIfd(ck, rk)≥3s\n16, then the robber’s strategy is to stay at the same place,\nwhich means rk+1=rk. Then d(ck+1, rk+1)≥d(ck, rk+1)−s\n16≥s\n8. From\nLemma 2.2 it follows that if d(ck, rk)<3s\n16, then there exists a position rk+1such\nthatd(rk, rk+1) =s\n16and an isometric path between rk+1andckcontains rk. This\nmeans that d(ck, rk+1) =d(ck, rk) +s\n8>s\n4and hence d(ck+1, rk+1)>s\n8.\n3.2.The upper bound. Instead of playing the Cops and Robber game on the\nhyperbolic surface S, we will consider the game on its universal cover, viewed as\nthe Poincar´ e disk D. For each time step kwe choose a representation Ck\ni, Rkof\nck\ni, rkinD(i= 1,2).\nIf the robber was only escaping from cop c1, then by the above argument his\nstrategy could be to stay roughly on the geodesic gdefined in Dby the current\npositions of the robber rand cop c1. Since we have two cops available, we can force\nthe robber to take a different strategy. The idea of the cops’ strategy is that cop\nc1will chase the robber, while cop c2will force the robber to move away from g,\nallowing cop c1to move closer to r.\nLetD= diam( S). The robber chooses an agility function τand starting po-\nsitions, c0\n1, c0\n2, r0. Let 0 = t0< t1< t2< t3<···be a sequence of integers\nrepresenting time steps such that\nti+1−1X\nk=tiτ(k)≥32D.\nOur goal is to show that for every ε >0 the cops c1, c2have a strategy in which\nthe cop c1can come ε-close to the robber r. This suffices to show that the cop win\nnumber is at most 2 by the definition of infimum, see [26, Corollary 6]. We will\nattain our goal by showing that in between time step tiandti+1, the cop c1can\ndecrease her distance to the robber by some small constant δ.\nLemma 3.2. For every ε >0, there exists δ >0and strategies for the cops c1, c2,\nsuch that if d(cti\n1, rti)≥εand the cops follow their strategy then either c2captures\nthe robber, or we have\nd(cti+1\n1, rti+1)≤d(cti\n1, rti)−δ.\nThe rest of the section is devoted to the proof of Lemma 3.2. Let R0, C0\n1be such\nthatd(R0, C0\n1) =d(r0, c0\n1).\n3.2.1. The strategy of cop c2.We explain the strategy of the cop c2first. Since\nwe play the game in the covering space D, we have arbitrarily many copies of\nthe position of c2that we can choose from (see Figure 3). At each time step\nti, the cop c2chooses a new representation Cti\n2, and therefore d(Cti−1\n2, Cti\n2) in\nDis possibly greater than τ(ti) but for the projection onto Swe maintain the\ncondition that d(cti−1\n2, cti\n2)≤τ(ti). In between the time steps ti, ti+1we choose\nthe representation of c2such that it is coherent with the agility function, which\nmeans d(Ck−1\n2, Ck\n2)≤τ(k) for ti< k < t i+1. We consider the geodesic g0=g0(ti)8 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nFigure 3. Copies of the fundamental polygon P(4g,2π\n4g) inD. De-\npicted is a point and one copy of this point in each of the depicted\ncopies of the fundamental polygon.\nthrough Rti, Cti\n1. We choose Cti\n2for cop c2such that it is close to the geodesic g0\nbut sufficiently far from Rti, which we make more precise in the following.\nLetPbe the point on g0at distance 10 Dfrom Rtithat is further away from Cti\n1.\nNote that there is a copy Cti\n2in the universal covering space which is at distance at\nmost Dfrom P, this will be the starting position of c2. We consider h=og0(Cti\n2),\nthe orthogonal geodesic to g0through Cti\n2. The strategy of c2between time steps\ntiandti+1is to chase the orthogonal projection of the robber on h. Let B, B′be\npoints on hat distance 8 Dfrom g0∩h.\nLemma 3.3. The cop c2can guard the path from BtoB′onh.\nProof. First we show that the cop c2can reach h∩g0much faster than the robber.\nClaim 3.4. The point h∩g0is at distance between 9Dand11Dfrom Rtiand at\ndistance at most Dfrom Cti\n2.\nProof. Note that the closest point from Cti\n2tog0ish∩g0since handg0are\northogonal. Hence d(h∩g0, Cti\n2)≤d(P, Cti\n2)≤D.\nThe closest point from Ptohis also h∩g0since P∈g0. In particular it holds\nthat d(h∩g0, P)≤d(h∩g0, Cti\n2)≤D. The distance from RtitoPis 10D, so by\nthe triangle inequality, robber’s distance to h∩g0is between 9 Dand 11 D. This\nproves the claim. ■\nNote that by Claim 3.4 the distance from Rtitohis at least 9 D. The cop c2\ncan guard the path from BtoB′onh, since her distance to g0∩his at most D,\nand hence her distance to any point on BB′is at most 9 D, which is less than the\ndistance from Rtito that point. This means that c2can use the strategy from the\nproof of Lemma 1.2 to guard the isometric path BB′. This completes the proof of\nLemma 3.3. □\n3.2.2. The strategy of cop c1.Fork≥ti, letgk=oh(Ck\n1) (the orthogonal geodesic\ntohthrough Ck\n1) and let Bk=gk∩h, see Figure 4.\nWe will assign a strategy to cop c1that will force the robber rto cross the\nisometric path Ck\n1B(or the symmetric case Ck\n1B′) for some k. Note that theCOPS AND ROBBER ON HYPERBOLIC MANIFOLDS 9\nCk\n1B\nBkh\ngkRk\nFigure 4. A schematic picture of cop’s position c1and robber’s\nposition rat time step k.\nrobber cannot cross BB′since it is guarded by cop c2. Since in round kthe robber\nmakes his turn before the cop, the robber crosses Ck\n1B(orCk\n1B′) in step k+ 1.\nIn the following the strategy of the cop c1will be such that the best strategy for\nthe robber ris to cross Ck\n1B(orCk\n1B′) somewhere close to B(orB′). We say two\npoints X, Y are on opposite sides of the geodesic gifX, Y are in distinct connected\ncomponents of the Poincar´ e disk D\\g. If two points X, Y are not on opposite sides,\nwe say they are on the same side .\nSuppose Rk, Rk+1are on the same side as Bwith respect to gk. We denote\nx=oh(Rk+1)∩Ck\n1B. Then the position Ck+1\n1can be chosen to be\n(a) on Ck\n1Bwith d(Ck+1\n1, Ck\n1) =τ(k+ 1) if d(x, Ck\n1)≥τ(k+ 1), or\n(b) on oh(Rk+1), as close to the point Rk+1as possible, otherwise.\nThis strategy ensures the following condition:\n(3.1) The robber’s position Rk+1andBare on the same side of gk+1.\nObservation 3.5. There is a strategy for the cop c1, such that for every time step\nti≤k≤ti+1−1, while the robber is contained in the triangle BBkCk\n1we can\nassume the robber’s position Rk+1is on the same side of the geodesic gkasRkby\nsubdividing each step at most once.\nProof. By symmetry we can assume that the first step of the robber away from the\ngeodesic g0is towards B. Now, suppose that the robber crosses gkby going from\nstepRkto step Rk+1, see Figure 5. By splitting each such step into two substeps\nwe may consider the first substep as above, initial and ending position on the same\nside as B, and the second substep with initial and ending position on the same\nside as B′. In the first substep the cop c1moves along gktowards the robber. In\nthe second substep the cop imagines the robber to move to ϕ(Rk+1), which is the\nreflection of the position Rk+1along gk, and determines his position ϕ(Ck+1\n1) by\nusing strategy ( a) or ( b). She then reflects the position ϕ(Ck+1\n1) along gkto obtain\nCk+1\n1. After subdividing each of the robber’s steps at most once, by symmetry we\ncan assume the robber does not cross gk, which means the position Rk+1is on the\nsame side of the geodesic gkasB. □\nRecall that in round kthe robber makes a step before the cops make their step.\nWe let tbe the first round at which the robber crosses Ct−1\n1B(orCt−1\n1B′). We will\nassume without loss of generality that Rt=x∈Ct−1\n1B(orCt−1\n1B′) by subdividing10 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\n10 VESNA IRŠIČ, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nCk\n1B\nB′Rk\n\nRk+1ϕ(Rk+1)\ngk\nh\n\nFigure 5.The pointϕ(Rk+1)is the refection point oRk+1overgk.\nRecall that in roundkthe robber makes a step beore the cops make their step.\nWe lettbe therst round at which the robber crosses Ct−1\n1B(or Ct−1\n1B′). We will\nassume without loss ogenerality thatRt=x∈Ct−1\n1B(or Ct−1\n1B′) by subdividing\nthe step and letting the agility at time steptbed(x, Rt−1). From steptuntil step\nti+1, the strategy ocopc 1is to simplyollow the robber, while the strategy o\ncopc 2is to stay at the same position. In particular,Rt∈Ct\n1B= Ct−1\n1B(or\nRt∈Ct\n1B′).\n3.2.3.ProooLemma 3.2.We start by showing that the robber crosses Ct\n1B.\nLemma 3.6.Ithe robber does not get caught by the cops,Rt∈Ct\n1B(orRt∈Ct\n1B′)\nor sometwitht i< t < t i+1. In particular,t\nk=t iτ(k)<32D.\nProo.By Observation 3.5 we can assume that at stepk, the copc 1is moving closer\ntoBandh, and by the triangle inequality on the triangle dened byCk−1\n1, Ck\n1and\ngk∩Ck−1\n1B, she decreases the total distance toBand tohby at leastτ(k), i.e.,\nd(Ck\n1, B) +d(Ck\n1, h)≤d(Ck−1\n1, B) +d(Ck−1\n1, h)−τ(k).\nBy Claim 3.4,d(Cti\n1, g0∩h)≤12Dand by the triangle inequality,d(Cti\n1, B)≤\nd(Cti\n1, g0∩h) +d(g 0∩h, B)≤12D+ 8D= 20D. Hence the cop either meetsBor\nhater movingor time32D, thus catching the robber. □\nLemma 3.2ollows directlyrom theollowing result.\nLemma 3.7.For eachε>0there exists someδ>0, such that id(Rti, Cti\n1)≥ε,\nand at stept(t i< t < t i+1)Rt∈Ct\n1B, then\nd(Rt, Ct\n1)≤d(Rti, Cti\n1)−δ.\nThe rest othis section is concerned with proving Lemma 3.7. The proois\ntechnical and uses Lemmas 2.3–2.5.\nProooLemma 3.7.Assume that the robber does not get caught by stept i+1. Let\nus consider the right triangle on verticesCt−1\n1,Bt−1andB; recall thatBk=gk∩h,\nsee Figure 4. We dene aunctionf: Ct−1\n1B→Rwith\nf(x) =d(Ct−1\n1, x)−d(Rt−1, x).\nFigure 5. The point ϕ(Rk+1) is the reflection point of Rk+1overgk.\nthe step and letting the agility at time step tbed(x, Rt−1). From step tuntil step\nti+1, the strategy of cop c1is to simply follow the robber, while the strategy of\ncopc2is to stay at the same position. In particular, Rt∈Ct\n1B=Ct−1\n1B(or\nRt∈Ct\n1B′).\n3.2.3. Proof of Lemma 3.2. We start by showing that the robber crosses Ct\n1B.\nLemma 3.6. If the robber does not get caught by the cops, then Rt∈Ct\n1B(or\nRt∈Ct\n1B′) for some twithti< t < t i+1. In particular,Pt\nk=tiτ(k)<32D.\nProof. By Observation 3.5 we can assume that at step k, the cop c1is moving closer\ntoBandh, and by the triangle inequality on the triangle defined by Ck−1\n1, Ck\n1and\ngk∩Ck−1\n1B, she decreases the total distance to Band to hby at least τ(k), i.e.,\nd(Ck\n1, B) +d(Ck\n1, h)≤d(Ck−1\n1, B) +d(Ck−1\n1, h)−τ(k).\nBy Claim 3.4, d(Cti\n1, g0∩h)≤12Dand by the triangle inequality it holds that\nd(Cti\n1, B)≤d(Cti\n1, g0∩h) +d(g0∩h, B)≤12D+ 8D= 20D. Hence the cop either\nmeets Borhafter moving for time 32 D, thus catching the robber. □\nLemma 3.2 follows directly from the following result.\nLemma 3.7. For each ε >0there exists some δ >0, such that if d(Rti, Cti\n1)≥ε,\nand at step t(ti< t < t i+1)Rt∈Ct\n1B, then\nd(Rt, Ct\n1)≤d(Rti, Cti\n1)−δ.\nThe rest of this section is concerned with proving Lemma 3.7. The proof is\ntechnical and uses Lemmas 2.3–2.5.\nProof of Lemma 3.7. Assume that the robber does not get caught by step ti+1. Let\nus consider the right triangle on vertices Ct−1\n1,Bt−1andB; recall that Bk=gk∩h,\nsee Figure 4. We define a function f:Ct−1\n1B→Rwith\nf(x) =d(Ct−1\n1, x)−d(Rt−1, x).\nIfRt=x, then d(Ct\n1, Rt)≤f(x). Note that by the triangle inequality it holds\nthatf(x)< d(Ct−1\n1, Rt−1) since x,Ct−1\n1andRt−1are not on a common geodesic.COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 11\nIn order to maximise his distance to cop c1, we show that the best choice for the\nrobber would be to move to Bin his last step (we ignore that the robber would be\ncaught by c2).\nClaim 3.8. Given fixed positions Ct−1\n1andRt−1such that Rt−1is in the triangle\ndefined by Ct−1\n1, B, Bt−1, it holds that\nf(B) = max\nx∈Ct−1\n1Bf(x).\nProof. Suppose x∈Ct−1\n1Bandx̸=B. We consider the triangle defined by\nx, B, Rt−1. Since xis on a common geodesic with Ct−1\n1andB,\nf(B) =d(Ct−1\n1, B)−d(Rt−1, B)\n= (d(Ct−1\n1, x) +d(x, B))−(d(Rt−1, B)−d(Rt−1, x))−d(Rt−1, x).\nBy the triangle inequality, d(x, B)≥d(Rt−1, B)−d(Rt−1, x), hence f(B) is at least\nf(x) =d(Ct−1\n1, x)−d(Rt−1, x). ■\nLetRti, . . . , Rtbe a sequence of t−tisteps such that Rt∈Ct−1\n1BandRkis\nin the interior of the triangle Ck−1\n1BBkfork=ti, . . . , t −1 (given the prescribed\nstrategy of cop c1). We define\ng(t) := max\nRti,...,Rt,τd(Ct\n1, Rt).\nNote that the maximum is well-defined since Sis compact and we can assume\nτ(k)≤D. Hence let Rti, . . . , Rtbe a sequence such that d(Ct\n1, Rt) =g(t).\nWe show that g(t)≤g(t−1) for every t∈ {ti, . . . , t i+1}. If the last step of c1is\nof type ( b), the robber is caught, and 0 = g(t)≤g(t−1). Thus the following holds.\nObservation 3.9. The last step of the cop c1is of type (a).\nClaim 3.10. If the second to last step is of type (a), then g(t)≤g(t−1).\nProof. We can assume Rt=Bby Claim 3.8. Suppose the second last step\nfrom Ct−2\n1toCt−1\n1is of type ( a). Note that the cop moves distance exactly\nd(Rt−2, Rt−1) +d(Rt−1, B) along Ct−2\n1Band for the robber’s steps it holds that\nd(Rt−2, Rt−1) +d(Rt−1, B)≥d(Rt−1, B). Hence by maximality of Rti, . . . , Rt,\nwe can assume Rt−1∈Rt−2B. But then Rt\ni, . . . , Rt−2, Rtis a sequence with the\nsame value assuming τ′(t−1) = τ(t−1) + τ(t),τ′(k) =τ(k) ifk≤t−2 and\nτ′(k) =τ(k−1) ifk≥t. This shows that g(t)≤g(t−1).\n■\nClaim 3.11. The second to last step is of type (a).\nProof. Now assume the step from Ct−2\n1toCt−1\n1is of type ( b). Let zcbe the\nintersection point of gt−1andCt−2\n1B, see Figure 6. By Lemma 2.3 we can assume\nfor the step size at time t−1 that τ(t−1) = d(Rt−1, Rt−2), otherwise changing\nτ(t−1) to τ′(t−1) = max {d(Ct−2\n1, zc), d(Rt−1, Rt−2)}gives a larger value of g.\nLetz1\nr, z2\nrbe the points on gt−1such that they are at the same distance from\nthe robber’s position Rt−2aszcis from the cop’s position Ct−2\n1. Let z1\nrbe closer\ntoBt−1. Note that Rt−1is either in z1rBt−1or in z2rzc. The second case cannot\nhappen since by Lemma 2.3, Rt−1=z2\nrwould give Ct−1\n1=zcand that would be12 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\npc\nBt−1\nCt−2\n1pr\nRt−2zrB\nzc\nCt−1\n1Rt−1h\ngt−1\nFigure 6. The case when the second last move of the robber is of\ntype ( b).\na better choice for the robber, contradicting maximality of Rti, . . . , Rt. We denote\nzr=z1\nrand then Rt−1∈zrBt−1, see Figure 6.\nIn the following we argue that it is of advantage to the robber to move to zr\ninstead of Rt−1, which also means that the second to last step is of type ( a) as\ndesired.\nLetpc, prbe the closest points on gt−1from Ct−2\n1, Rt−2, respectively. This means\nthe angles ∠Ct−2\n1pcBt−1,∠Rt−2prBt−1are right angles. There are two Lambert\nquadrilaterals2formed by Ct−2\n1, pc, Bt−1, Bt−2andRt−2, pr, Bt−1, oh(Rt−2)∩h,\nrespectively. Any two sides of a Lambert quadrilateral determine the length of the\nother sides, see [30]. In particular, since d(Bt−1, oh(Rt−2)∩h)≤d(Bt−1, Bt−2)\nandd(pr, Bt−1)< d(pc, Bt−1), it holds that\nd(pr, Rt−2)< d(pc, Ct−2\n1).\nTherefore by the sine formula for hyperbolic right triangles it holds for the angles\nthat∠pczcCt−2\n1>∠przrRt−2and thus\n∠Ct−2\n1zcCt−1\n1<∠Rt−2zrRt−1.\nSince d(Ct−2\n1, zc) =d(Rt−2, zr) and d(Ct−2\n1, Ct−1\n1) =d(Rt−2, Rt−1), it holds by\nLemma 2.5 that d(zr, Rt−1)< d(zc, Ct−1\n1). By Lemmas 2.4 and 2.3 we have\nd(B, zc)−d(B, zr)> d(B, Ct−1\n1)−d(B, Rt−1).\nHence it is of advantage for the robber to move to zrinstead of Rt−1, a contradiction\nto the maximality of Rti, . . . , Rt. ■\nWe will establish now a positive lower bound on d(Rti, Cti\n1)−d(Rt, Ct\n1). By\nClaims 3.10 and 3.11, we can assume that t=ti+ 1. To compute d(Rt, Ct\n1) we\nonly need to compute the difference between d(Ct1\n1, B) and d(Rt1, B). By the\nPythagorean Theorem for hyperbolic triangles and H:=g0∩h, we have\nd(Rt, Ct\n1) =d(Cti\n1, B)−d(Rti, B)\n= acosh(cosh( d(B, H )) cosh( d(Cti\n1, H))−acosh(cosh( d(B, H )) cosh( d(Rti, H))\n= acosh( bcosh( d(Cti\n1, Rti) +d(Rti, H))−acosh( bcosh( d(Rti, H))\n2A quadrilateral in which three of its angles are right angles.COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 13\nwhere b= cosh( d(B, H )) = cosh(8 D)>1. For x=d(Rti, H) and y=d(Cti\n1, H),\nand using that y−x=d(Rti, Cti\n1)≥ε, Lemma 2.6 gives η >0 such that\nd(Rt, Ct\n1)−d(Rti, Cti\n1)≥ηε.\nBy taking δ=ηε, we obtain the conclusion of Lemma 3.7. □\n4.Catching the Robber\nWe will now turn to the cop catch number. First, we show that on a hyperbolic\nsurface at least three cops are needed to catch the robber. The upper bound is\nconsidered in Section 4.2.\n4.1.Lower bound for cop catch number.\nTheorem 4.1. IfSis a hyperbolic surface, then c0(S)≥3.\nProof. Lets= sys( S) be the systolic girth of the surface S. The robber chooses\nthe agility function τ≡s\n10and starting positions r0, c0\n1, c0\n2where r0is at distance\nmore thans\n10toc0\n1, c0\n2. In fact, it suffices to assume that r0̸=c0\n1, c0\n2. We show that\nifrk̸=ck\n1, ck\n2then there exists a position rk+1at distance at mosts\n10from rkwith\nd(rk+1, ck\nj)>s\n10(j= 1,2), and hence the robber can escape from the cops.\nIfd(rk, ck\nj)>s\n5forj= 1,2, then the robber’s strategy is to stay in the same\nplace, and d(rk+1, ck\nj)>s\n5. Ifd(rk, ck\n1)≤s\n5andd(rk, ck\n2)>s\n5, then the robber\nmoves away from cop c1;rk+1is the point at distances\n10on the geodesic through\nrk, ck\n1which is further away from ck\n1. We consider B(rk,s\n5), the disk of radius at\nmosts\n5from rkonSand this is isometric to a hyperbolic disk by Lemma 2.2. It\nfollows that d(rk+1, ck\n1) =d(rk, ck\n1)+s\n10. By the triangle inequality d(rk+1, ck\n2)>s\n10.\nIfd(rk, ck\n1)≤s\n5andd(rk, ck\n2)>s\n5we use the same strategy by interchanging the\nroles of c1, c2. Hence we can assume d(rk, ck\n1), d(rk, ck\n2)≤s\n5. Note that ck\n1, ck\n2are\nnow in the hyperbolic disk B(rk,s\n5) of radiuss\n5centred at rk. We consider the disk\nB(rk,s\n5) being embedded in the Poincar´ e disk, and consider the geodesic hthrough\nck\n1, ck\n2in this disk and let oh(rk) be the orthogonal geodesic to hpassing through\nrk. Now let rk+1be the point at distances\n10on the geodesic oh(rk) which is further\naway from g. Ifck\njis on oh(rk) then d(rk+1, ck\nj) =d(rk, ck\nj) +s\n10>s\n10. Ifck\njis\nnot on oh(rk), then ∠ck\nj, oh(rk)∩h, rk+1form a right angle and by the hyperbolic\nPythagorean theorem d(ck\nj, rk+1)>s\n10. □\n4.2.Upper bound for cop catch number. The following lemma shows how to\ncapture the robber on the universal covering space D, if the robber is contained in\na polygon which is guarded by the cops.\nLemma 4.2. Suppose the robber is contained in a bounded convex polygon in the\nPoincar´ e disk Dwhere ncops guard the boundary of the polygon. Then these n\ncops can catch the robber.\nProof. Suppose the robber is contained in a bounded convex polygon at the begin-\nning of the game. For all k≥0 we consider the bisectors bk\njbetween the robber’s\nposition rkand each of the cops’ positions ck\nj(j∈[n]), and let Bk\njbe the midpoint\nbetween rkandck\nj, see Figure 7 and Figure 8.\nThe bisectors b0\njbound a polygon P0containing the robber, otherwise there is a\npoint on the boundary of the Poincar´ e disk towards which the robber can walk and14 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nescape from the cops. Without loss of generality let b0\n1, . . . , b0\nnappear consecutively\non the convex hull of P0, see Figure 7 for an example with n= 4. Let αk\nj,j+1be the\nangle between the bisector bk\njandbk\nj+1(we consider the indices modulo n) and let\nPk\nj,j+1be their intersection point.\nrc1b1P1,2\nb2 c2\nc4b4\nc3b3P4,1\nP3,4P2,3\nα2,3α1,2\nα4,1\nα3,4β1,2\nβ2,3\nβ3,4β4,1\nFigure 7. A schematic figure when the number of cops is n= 4.\nA label xin the figure denotes the value xk. That is, the figure\ndepicts the positions of the cops ck\ni(i= 1,2,3,4), the intersection\npoints Pk\ni,i+1of the bisectors bk\ni, bk\ni+1, and the angles βk\ni,i+1, αk\ni,i+1.\nEvery cop has the same strategy, which is as follows. Cop cjcopies the move\nof the robber by reflecting it along the bisector bjand subsequently if there is any\nagility left she moves as close to the robber as possible (see Figure 8).\nMore precisely, in the k-th round the cop ck−1\njwill move to obk−1\nj(rk), as close\nto the robber as possible. In particular, bk−1\njandbk\njare parallel and the voronoi\ncell of the robber with respect to bk\njis contained in the voronoi cell with respect\ntobk−1\nj. Therefore throughout the whole game, the robber is restricted to P0. We\nhave to show that with this strategy the robber is eventually caught. If\n∞X\nk=1τ(k)−d(rk, rk−1) =∞,\nthen the cops catch the robber. Note that while the robber is not caught, the\ndistance between bk−1\njandbk\njis at least τ(k)−d(rk, rk−1). The part of the Poincar´ e\ndiskD \\bk\njcontaining the cop cjeventually includes P0, in which rkis contained,\nwhich is a contradiction since bk\njis a bisector.COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 15\nrk−1\nck−1\n1bk−1\n1=bk\n1Qk\nrk\nck\n1Bk\n1\nBk−1\n1rk−1\nck−1\n1bk−1\n1Qk\nck\n1Bk\n1\nBk−1\n1rk\nbk\n1\nFigure 8. A step K+\n1in comparison to a step K−\n1. In both cases\nthe distance of rktork−1is the same and hence the move of cop\nc1is the same.\nHence we assume that the robber moves an infinite distance, that is\n∞X\nk=1d(rk, rk−1) =∞.\nWe want to show that also in this case one of the bisectors surpasses the polygon.\nNote that for some j,α0\nj,j+1<(n−2)π/nsince the angle sum in a hyperbolic n-gon\nis upper bounded by the angle sum in an Euclidean n-gon. We suppose j= 1 in\nthe following and will show, that either bk\n1orbk\n2eventually surpasses the polygon.\nBy the following claim we can assume that also αk\n1,2<(n−2)π/nfor all positive\nintegers k, as all the angles are non-decreasing and one of them has to stay below\n(n−2)π/n.\nClaim 4.3. The angles in the guarded polygon are increasing, which means\nαk\nj,j+1≥αk−1\nj,j+1.\nProof. We consider the step of the cop cjand the cop cj+1separately, imagining\nthe cop cjtakes her turn first. Let P′be the intersection point between bk\nj, bk−1\nj+1\nand let α′be the angle between bk\nj, bk−1\nj+1. Ifbk\nj=bk−1\nj, then α′=αk−1\nj,j+1. Otherwise\nbk\nj̸=bk−1\njand note that bk\njis orthogonal to obk−1\nj(rk). In particular, bk−1\njandbk\nj\nare parallel, see Figure 8. Since the quadrilateral Bk\nj, Bk−1\nj, Pk−1\nj,j+1, P′has angle\nsum smaller than 2 πand has a right angle at Bk−1\njandBk\nj, it holds that the\nangle at P′in the quadrilateral is smaller than π−αk−1\nj,j+1but then α′> αk−1\nj,j+1.\nNow we consider the quadrilateral formed by Bk\nj+1, Bk−1\nj+1, P′, Pk\nj,j+1. By the same\nargument, αk\nj,j+1> α′. ■\nWe want in the following that the angle between rk−1rkandrkck\njforj= 1 or\nj= 2 is bounded away from π/2, as this will help the cop c1orc2to get closer\nto the robber. We show that if the angle between rk−1rkandrkck\n1is close to π/2,\nthen the angle between rk−1rkandrkck\n2is bounded away from π/2 by bounding\nthe angle βk\n1,2, see Figure 7. We define the angle βk\n1,2for every k∈Nas the angle\nbetween rkck\n1andrkck\n2.16 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nClaim 4.4. It holds that\nacot \ncosh( d\u0000\nPk\n1,2, rk\u0001\n)\ncot((n−2)π/n)!\n≤βk\n1,2≤π−αk\n1,2.\nProof. Since αk\n1,2≤(n−2)π/2, there is a quadrilateral formed by rkBk\n1Pk\n1,2Bk\n2with\nright angles at Bk\n1andBk\n2. Since a quadrilateral has angle sum at most 2 πit holds\nthatβk\n1,2≤π−αk\n1,2. On the other hand ([23, Corollary 32.13]),\ncot\u0000\n∠Pk\n1,2rkBk\n1\u0001\n=cosh\u0000\nd\u0000\nPk\n1,2, rk\u0001\u0001\ncot\u0000\n∠rkPk\n1,2Bk\n1\u0001≤cosh\u0000\nd\u0000\nPk\n1,2, rk\u0001\u0001\ncot (( n−2)π/n).\nSince βk\n1,2≥∠Pk\n1,2rkBk\n1the claim follows. ■\nNote that αk\n1,2≥α0\n1,2by Claim 4.3 and d\u0000\nPk\n1,2, rk\u0001\nis upper bounded by the\ndiameter DofP0. Let\nβ= min\u001a\nacot\u0012cosh( D)\ncot((n−2)π/n)\u0013\n, α0\n1,2\u001b\n.\nIf the angle at rkbetween rk−1rkandrkck\n1is in [π\n2−β\n2,π\n2+β\n2], then the angle\nbetween rk−1rkandrkck\n2is in [0 ,π\n2−β\n2]∪[π\n2+β\n2, π] by Claim 4.4 and the definition\nofβ. Let Kjforj= 1,2 be the set of rounds at which the angle between rk−1rk\nandrkck\njis not in [π\n2−β\n2,π\n2+β\n2]. Without loss of generality,\nX\nk∈K1d(rk, rk−1) =∞.\nWe will show that if the cop c1does not catch the robber, then bk\n1eventually\nsurpasses P0, which is a contradiction. Let Qkbe the closest point to rk−1on the\ngeodesic rkck\n1. Let K+\n1be the time steps in K1in which the robber’s position rk\nis closer to ck\n1than Qktock\n1and let K−\n1be the remaining steps, see Figure 8. If k\nis a step of type K−\n1, then the distance between bk−1\n1andbk\n1is at least d(rk, Qk).\nSuppose kis a step of type K+\n1. Since rk−1QkBk\n1Bk−1\n1is a Lambert quadrilateral\nwith acute angle at rk−1, it holds that d(rk−1, Bk−1\n1)≤d(Qk, Bk\n1). Hence d(rk, Qk)\nis the distance that the cop c1is getting closer to rin step k. We will show that\nX\nk∈K1d(rk, Qk) =∞,\nwhich then proves that the robber is eventually caught. Note that the triangle\nrk−1Qkrkhas a right angle at Qkand an angle at rknot in [π\n2−β\n2,π\n2+β\n2]. In fact,\nthis angle is at mostπ\n2−β\n2, since it is in a right-angled triangle. By the cosine\nformula for right hyperbolic triangles (see [30]), it follows that\nd\u0000\nrk, Qk\u0001\n≥atanh\u0012\ncos\u0012π\n2−β\n2\u0013\ntanh\u0000\nd\u0000\nrk, rk−1\u0001\u0001\u0013\n.\nUsing that tanh( x) is monotonly increasing and for small x, tanh( x)≥x\u0000\n1−x2/3\u0001\n,\nwe get that\nX\nk∈K1tanh\u0000\nd\u0000\nrk, rk−1\u0001\u0001\n=∞.COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 17\nBut then, since atanh( x)≥x, we also have\nX\nk∈K1d(rk, Qk)≥X\nk∈K1atanh\u0012\ncos\u0012π\n2−β\n2\u0013\ntanh\u0000\nd\u0000\nrk, rk−1\u0001\u0001\u0013\n=∞,\nthus the bisector bk\n1eventually surpasses P0, a contradiction, or the distance be-\ntween the cop c1and robber decreases to 0. □\nWe now use Lemma 4.2 to bound c0from above for the special surfaces S(g), S′(g)\nandN(g).\nTheorem 4.5. Ifg≥2, then c0(S(g))≤5,c0(S′(g))≤6andc0(N(g))≤4.\nProof. We will give the proof only for S(g), the proof for the other surfaces is similar.\nLetObe the midpoint of the fundamental polygon P\u0010\n4g,2π\n4g\u0011\n. We will play the\ngame in the covering space and choose the player’s positions such that they are in\nP\u0010\n4g,2π\n4g\u0011\n. We will first use the cops c1, c2, c3to guard isometric paths. We start\nby moving cop c1to the isometric path Ov1, cop c2to the isometric path Ov5and\ncopc3to the isometric path Ov9. By Lemma 1.2 we can assume that after a finite\namount of time the cops guard the respective isometric paths. Now if the robber\nis in one of the triangles Ovjvj+1for some 1 ≤j≤4, then the robber’s moves are\nrestricted to the specified triangles since a1=a−1\n3anda2=a−1\n4. Similarly if the\nrobber is contained in one of the triangles Ovjvj+1for some 5 ≤j≤8 his moves are\nrestricted to these triangles. If the robber is outside Ovjvj+1for some 1 ≤j≤8,\nwe move cop c2to the isometric path Ov13and wait until she is guarding it. If the\nrobber is in one of the triangles Ovjvj+1for 9≤j≤12, his moves are restricted\nto these triangles. If the robber is not contained in one of these triangles we keep\ngoing for i= 3,4, . . .in the same way, moving the cop currently guarding Ov1+4i\nto guard Ov1+4(i+2)unless i+ 2 = g, in which case the robber is contained in one\nofOv4g−3v4g−2, Ov 4g−2v4g−1, Ov 4g−1v4gorOv4gv1, cop c1guards Ov1and one of\nc2orc3guards Ov4g−3.\nHence we can assume without loss of generality that cop c1guards Ov1, cop c2\nguards Ov5and the robber is in one of the triangles Ovjvj+1for some 1 ≤j≤8.\nCopc3, c4, c5will guard Ov2,Ov3,Ov4, respectively. Now the robber is captured in\neither R1=Ov1v2∪Ov3v4orR2=Ov2v3∪Ov4v5. The regions R1andR2can\nbe embedded in the covering space Dsuch that they form a quadrilateral which is\nguarded by four of the cops. By Lemma 4.2 we can now catch the robber. □\n5.Generalizations\nThe proof of Theorem 3.1 can be generalized to higher dimensional hyperbolic\nmanifolds. We first prove the following lemma, which will replace the isometric\npath lemma in the proof of Theorem 3.1.\nLemma 5.1. Let the Cops and Robber game be played on an Euclidean disk, a disk\non the unit sphere of radius at mostπ\n2or a hyperbolic disk B2. If the cops position\nck0is on the line between the center OofB2and the position of the robber rk0,\nthen the cop can guard the disk of radius1\n2(d(O, ck0) +d(O, rk0))and center O.\nProof. Letdk=1\n2(d(O, ck) +d(O, rk)) and Dkbe the disk of radius dkand center\nO. We explain the cop’s strategy for the time steps k≥k0. First suppose that18 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nthe geodesic segment from rktork+1passes through Dk, we show that the cop\ncan catch the robber. We are allowed to subdivide the robber’s step since this\nis only to the advantage of the robber, so we assume without loss of generality\nd(O, rk+1) =dk. We can assume rk+1̸=ckand we consider the closest point p\nfrom rk+1on the geodesic through Oandrk.\nO\nrkrk+1\np\nck\ndk\nFigure 9. Depicted is the point pwhich is the closest point on\nOrkfrom the robber’s position rk+1.\nBy the Pythagorean theorem (in the Euclidean or hyperbolic plane or for a right\nangled triangle on a spherical hemisphere),\nd(O, p)< dk,\nhence d(rk, p)> d(ck, p). But then using the Pythagorean theorem again on the\ntriangles ckprk+1andrk+1prkwhich both have a right angle at p, it follows that\nd(rk+1, ck)< d(rk+1, rk), so the cop can catch the robber.\nWe show now that if d(O, rk+1)> dkand the cop does not pass through D,\nthere exists a position of the robber on Ork+1such that\nd(O, ck+1)−d(O, ck)≥d(O, rk)−d(O, rk+1). (5.1)\nBy subdividing the robber’s step, we can assume that the triangle Orkrk+1has\nangle at mostπ\n2atO. Suppose first the angle at rkin the triangle Orkrk+1is\nsmaller than π. The cop moves to the point ck+1onOrk+1such that\nd(O, ck+1)−d(O, ck) =d(O, rk)−d(O, rk+1).\nWe show now that this is a valid move, which means d(ck, ck+1)≤d(rk, rk+1).\nO\nrkrk+1\npr ckck+1\npc\nFigure 10. The case where the angle at rkin the triangle Orkrk+1\nis smaller than π.\nIfrk+1is on the geodesic Ork, then the cop moves as far as the robber, hence\nthis is a valid move for the cop. Otherwise consider the closest points pc, pronCOPS AND ROBBER ON HYPERBOLIC MANIFOLDS 19\nthe geodesic Orkofck+1, rk+1, respectively. Then using the Pythagorean theorem\nagain, d(O, ck+1)> d(O, pc) and d(O, rk+1)> d(O, pr). Therefore it holds that\nd(pc, ck)< d(O, ck+1)−d(O, ck) and d(pr, rk)> d(O, rk)−d(O, rk+1). Using the\nPythagorean theorem again, d(ck, ck+1)< d(rk, rk+1), hence 5.1 is a valid move.\nSuppose now the angle at rkin the triangle Orkrk+1is larger than π. Then this\nis of advantage to the cop since the robber moves further away from O, i.e.\nd(O, rk)−d(O, rk+1)<0.\nThe cop can simply move at a right angle to Ockand\nd(O, ck+1)−d(O, ck)≥0,\nwhich establishes (5.1). As long as the cop has not caught the robber, the value dk\nis increasing, and as soon as the robber passes through the disk Dk, he is caught.\nThis proves the lemma. □\nProof of Theorem 1.6. Suppose M=Hn/Γ where Γ is a torsion-free, discrete group\nof isometries on Hn. Let Dbe the diameter of M. We position the cop c2in the\ncovering space such that she is of distance at most Dto a point Oon the isometric\npath Ck\n1Rkat distance between 9 Dand 11 Dfrom the robber’s position (and further\nfrom cops c1). By Lemma 5.1, the cop c2can guard the n-dimensional ball Bof\nradius 4 Dwith center Oafter moving to O. We consider the ( n−1)-dimensional\nhyperplane Hthrough Owhich is perpendicular to the geodesic Ck\n1Rk. Note that\ncopc2guards all points on Hof distance at most 4 DtoO. We denote by oH(Ck\n1) the\northogonal geodesic to Hwhich contains Ck\n1. Now at the cops’ turn, we consider the\npositions Ck\n1, Rk+1andoH(Ck\n1)∩H. These points lie on a common 2-dimensional\nsubspace of Dnwhich is isometric to D2and is also isometric as a subset of Dn.\nNow the cop can use the strategies (a) and (b) outlined in Theorem 3.1 on this\nsubspace. The proof idea for the lower bound is the same as in the case n= 2.□\nWe are ready to summarise our results about manifolds of constant curvature,\ni.e.space forms .\nTheorem 5.2. IfMis a compact manifold of constant curvature, then c(M) = 2 .\nProof. IfMgpis a Riemannian manifold equipped with a Riemannian metric gpand\nconstant curvature C̸= 0, we can equip the same space with the Riemannian metric\ngp∗=|C|·gpinstead. This gives a Riemannian manifold Mgp∗of constant curvature\n1 or−1. Given the step function τforMgp, we can play the game with step functionp\n|C| ·τonMgp∗instead. By rescaling the lengths of the steps appropriately, k\ncops can win the game on Mgif and only if they can win the game on Mgp∗.\nThe result follows for negative curvature by Theorem 1.6, for constant curvature 0\nby generalizing [17, Lemma 8] and positive constant curvature by generalizing the\nresults for the sphere in [17, Theorem 4].3□\nWe also would like to point out that for the cop catching number one can obtain a\nlower bound of one larger than the dimension of the manifold by a similar argument\nas in Theorem 4.1. Further, Lemma 4.2 can be extended to higher dimensional\n3The upper bound follows from the Killing-Hopf theorem and the covering-space method and\nthe lower bound by considering a small enough agility function that allows the robber to move in\nthe opposite direction to the cop without being caught.20 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nhyperbolic spaces, if ncops guard the boundary of a convex polytope which contains\nthe robber, then they can catch the robber in finite time.\n6.Concluding remarks\nOne of the main open question is Conjecture 1.5 on the cop win number of\nsurfaces. A first step towards it would be to lower the multiplicative constant for\nupper bounds that are linear in g, as was done for graphs of genus g. Note that in\ngeneral the cop number of a compact metric space can be infinite [14]. It would be\nnice to find more properties of spaces that guarantee a finite (or constant) cop win\nnumber.\nWhen considering compact surfaces of constant curvature, it is still open whether\n(and how) the cop catch number grows with the genus of the surface. It is easy\nto show that a linear number (depending on the genus) of cops is sufficient using\nLemma 4.2, by cutting the surface into smaller parts using isometric paths (similarly\nas was done in [26]). We obtained that on natural constant curvature surfaces the\ncop number is bounded by a constant (irrespectively of the genus). A first step\ntowards understanding the cop catch number on constant curvature surfaces would\nbe to find the exact cop catch number for the surfaces S(g), S′(g) and N(g). More\nimportantly, is the cop catch number bounded by a constant on hyperbolic surfaces?\nReferences\n1. Marco Abate and Francesca Tovena, Curves and surfaces , Unitext, vol. 55, Springer, Milan,\n2012.\n2. M. Aigner and M. Fromme, A game of cops and robbers , Discrete Appl. Math. 8(1984), no. 1,\n1–11.\n3. Andrew Beveridge and Yiqing Cai, Two-dimensional pursuit-evasion in a compact domain\nwith piecewise analytic boundary , ArXiv e-prints arXiv:1505.00297 (2015).\n4. B´ ela Bollob´ as, G´ abor Kun, and Imre Leader, Cops and robbers in a random graph , Journal\nof Combinatorial Theory, Series B 103(2013), no. 2, 226–236.\n5. B´ ela Bollob´ as, Imre Leader, and Mark Walters, Lion and man—can both win? , Israel J. Math.\n189(2012), 267–286. MR 2931397\n6. Anthony Bonato, An invitation to pursuit-evasion games and graph theory , Student Mathe-\nmatical Library, vol. 97, American Mathematical Society, Providence, RI, 2022.\n7. Anthony Bonato and Richard J. Nowakowski, The game of cops and robbers on graphs , Stu-\ndent Mathematical Library, vol. 61, American Mathematical Society, Providence, RI, 2011.\nMR 2830217\n8. Nathan Bowler, Joshua Erde, Florian Lehner, and Max Pitz, Bounding the cop number of a\ngraph by its genus , SIAM J. Discrete Math. 35(2021), no. 4, 2459–2489. MR 4327323\n9. Peter Bradshaw, A proof of the meyniel conjecture for abelian cayley graphs , Discrete Math-\nematics 343(2020), no. 1, 111546.\n10. H. T. Croft, ’Lion and man’: A postscript , J. Lond. Math. Soc. 39(1964), 385–390 (English).\n11. Joshua Erde and Florian Lehner, Improved bounds on the cop number of a graph drawn on a\nsurface , Extended Abstracts EuroComb 2021, Springer, 2021, pp. 111–116.\n12. Peter Frankl, Cops and robbers in graphs with large girth and Cayley graphs , Discrete Appl.\nMath. 17(1987), no. 3, 301–305.\n13. Alan Frieze, Michael Krivelevich, and Po-Shen Loh, Variations on cops and robbers , J. Graph\nTheory 69(2012), no. 4, 383–402. MR 2979296\n14. Agelos Georgakopoulos, Compact metric spaces with infinite cop number , ArXiv e-prints\narXiv:2309.03757 (2023).\n15. Heinz Hopf, Zum Clifford-Kleinschen Raumproblem , Math. Ann. 95(1926), no. 1, 313–339.\n16. Seyyed Aliasghar Hosseini, Bojan Mohar, and Sebastian Gonzalez Hermosillo de la Maza,\nMeyniel’s conjecture on graphs of bounded degree , J. Graph Theory 97(2021), no. 3, 401–\n407. MR 4313187COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 21\n17. Vesna Irˇ siˇ c, Bojan Mohar, and Alexandra Wesolek, Cops and robber game in higher-\ndimensional manifolds with spherical and Euclidean metric , C. R. Math. Acad. Sci. Soc.\nR. Can. 44(2022), no. 3, 50–68.\n18. Gwena¨ el Joret, Marcin Kami´ nski, and Dirk Oliver Theis, The cops and robber game on\ngraphs with forbidden (induced) subgraphs , Contrib. Discrete Math. 5(2010), no. 2, 40–51.\nMR 2791289\n19. Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen , Math. Ann. 39(1891), no. 2,\n257–278.\n20. J. E. Littlewood, Littlewood’s miscellany , Cambridge University Press, Cambridge, 1986,\nEdited and with a foreword by B´ ela Bollob´ as. MR 872858\n21. Linyuan Lu and Xing Peng, On Meyniel’s conjecture of the cop number , J. Graph Theory 71\n(2012), no. 2, 192–205. MR 2965383\n22. Tomasz  Luczak and Pawe l Pra lat, Chasing robbers on random graphs: zigzag theorem , Ran-\ndom Structures & Algorithms 37(2010), no. 4, 516–524.\n23. George E. Martin, The foundations of geometry and the non-Euclidean plane , Undergraduate\nTexts in Mathematics, Springer-Verlag, New York-Berlin, 1982, Reprint.\n24. Bojan Mohar, Notes on cops and robber game on graphs , ArXiv e-prints arXiv:1710.11281\n(2017).\n25. ,Min-max theorem for the game of cops and robber on geodesic spaces , ArXiv e-prints\narXiv:2112.0301 (2021).\n26. ,The game of cops and robber on geodesic spaces , ArXiv e-prints arXiv:2205.11633\n(2022).\n27. N. Yu. Satimov and A. Sh. Kuchkarov, On the solution of a model differential pursuit-evasion\ngame on a sphere , Uzb. Mat. Zh. 2000 (2000), no. 1, 45–50 (Russian).\n28. Bernd SW Schr¨ oder, The copnumber of a graph is bounded by [3/2 genus (g)]+ 3 , Categorical\nperspectives, Springer, 2001, pp. 243–263.\n29. Alex Scott and Benny Sudakov, A bound for the cops and robbers problem , SIAM J. Discrete\nMath. 25(2011), no. 3, 1438–1442. MR 2837608\n30. Stothers, Wilson, Hyperbolic geometry ,http://www.maths.gla.ac.uk/wws/cabripages/\nhyperbolic/hyperbolic0.html , [Online; accessed 16-January-2023].\n31. Olga Yufereva, Lion and man game in compact spaces , Dyn. Games Appl. 9(2019), no. 1,\n281–292.\nAppendix A.Proofs of technical lemmas\nProof of Lemma 2.2. Consider B(a, r) as a disk in the universal covering space D.\nLetx, y∈B(a, r), then the isometric path gbetween xandyin the copy of B(a, r)\nin the universal covering space Dis a locally isometric path from xtoyinS.\nSuppose xy̸=g. Then the closed path Pgoing from xtoyalong xyand from yto\nxalong gis of length less than sys( S) and consists of two paths which are locally\nisometric and bounds a region which is non-empty.\nThe Gauss-Bonnet Theorem states that for a region R⊂Swith sectional cur-\nvature K, piecewise smooth boundary ∂(R),\nZ\nRK dA +Z\n∂(R)kgds= 2πχ(R),\nwhere dAis the element of area of the surface, and dsis the line element along the\nboundary, andR\n∂(R)kgdsis the sum of the corresponding integrals of the geodesic\ncurvature kgalong the smooth portions of the boundary, plus the sum of the angles\nby which the smooth portions turn at the corners of the boundary.\nConsider the curve Pwhich is piecewise geodesic. Since the curve Phas length\nstrictly smaller than sys( S), it has to be contractible. The curve Pbounds a region22 VESNA IR ˇSIˇC, BOJAN MOHAR, AND ALEXANDRA WESOLEK\nR. Since the curvature of locally isometric curves is 0, by Gauss-Bonnet\nZ\nR−1dA+θ1−π+θ2−π= 2πχ(R),\nwhere θ1, θ2are the exterior angles between the locally isometric paths at xandy.\nSince Pis contractible, the Euler characteristic of Ris 1, hence θ1=θ2= 2πandR\nR1dA= 0, which means that the isometric path between xandygoes along the\ngeodesic g. □\nIn the following proofs, denote d(B, C) =a,d(A, C) =bandd(A, B) =c,αis\nthe angle at A,βis the angle at Bandγis the angle at C.\nProof of Lemma 2.3. This is a simple application of the Pythagorean Theorem for\nhyperbolic triangles. Suppose X, Y∈ACwith X∈Y C. Since cosh is strictly\nincreasing on the positive real axis and acosh is strictly increasing\nd(Y, B) = acosh(cosh( a) cosh( d(Y, C)))\n≥acosh(cosh( a) cosh( d(X, C))) = d(X, B)\nwith equality if and only if X=Y. The lemma follows by using this inequality\ntwice,\nd(X2, B)−d(X3, B)≤c−d(X3, B)≤c−d(X1, B).\nClearly, d(X2, B)−d(X3, B) =c−d(X1, B) if and only if X2=AandX3=X1.□\nProof of Lemma 2.4. For a vertex X∈AC, by the Pythagorean Theorem for hy-\nperbolic triangles\nd(X, B) = acosh(cosh( a) cosh( d(X, C))).\nThe second derivative with respect to d(X, C) is\n∂2d(X, B)\n∂d(X, C)2=(cosh( d(X, C)) cosh( a) sinh2(a))\n(−1 + cosh2(a) cosh2(d(X, C)))3/2.\nTherefore,∂2d(X,B)\n∂d(X,C)2>0 for positive values of aandd(X, C), hence d(X, B) is\nconvex for d(X, C)>0. This means that the secant\nd(X, B)−d(Y, B)\nd(X, C)−d(Y, C)\nis monotonely increasing (in both d(X, C) and d(Y, C)). Consequently,\nc−d(X1, B)\nd(A, X 1)=c−d(X1, B)\nb−d(X1, C)\n≥d(X2, B)−d(X3, B)\nd(X2, C)−d(X3, C)=d(X2, B)−d(X3, B)\nd(X2, X3),\nand since d(A, X 1) =d(X2, X3), this proves the lemma. □\nProof of Lemma 2.5. By the law of cosines\nγ= arccos\u0012cosh( c)−cosh( b) cosh( a)\n−sinh(a) sinh( b)\u0013\n.COPS AND ROBBER ON HYPERBOLIC MANIFOLDS 23\nSince arccos is monotonely decreasing, we only need to show that fas defined by\nbelow equation is monotonely increasing. We define\nf(b) :=cosh( c)−cosh( b) cosh( a)\n−sinh(a) sinh( b).\nThe derivative of fis\nf′(b) =cosh( a) sinh( a) sinh( b)2+ sinh( a) cosh( b)(cosh( c)−cosh( b) cosh( a))\n(−sinh(a) sinh( b))2\n=−cosh( a) sinh( a) + sinh( a) cosh( b) cosh( c)\nsinh(a)2cosh( b)2,\nwhere we use that sinh( b)2−cosh( b)2=−1. Now since α <π\n2it follows that\ncosh( b) cosh( c)≥cosh( a) and hence the derivative is non-negative. This proves the\nlemma.\n□\nProof of Lemma 2.6. Note that by the mean value theorem for x < y\nacosh( wcosh( x))−acosh( wcosh( y))≥c·(y−x)\nwhere c= min ξ∈[x,y]∂/∂ξ (acosh( wcosh( ξ))). Since w >1, the partial derivative is\n∂\n∂ξacosh( wcosh( ξ)) =wsinh(ξ)q\nw2−1 +w2sinh2(ξ).\nTaking x, y∈[t1, t2] with t1>0 and noting that sinh( x) is monotonely increasing\nwe get that\nc≥wsinh(t1)q\nw2−1 +w2sinh2(t2)≥η\nfor some η >0. □\nUniversity of Ljubljana\nEmail address :vesna.irsic@fmf.uni-lj.si\nSimon Fraser University\nEmail address :mohar@sfu.ca\nTechnische Universit ¨at Berlin\nEmail address :alexandrawesolek@gmail.com"    },    {        "title": "2402.05768v1.A_non_damped_stabilization_algorithm_for_multibody_dynamics.pdf",        "content": "A non-damped stabilization algorithm for multibody\ndynamics\nIgor Ferna ´ndez de Bustos .Haritz Uriarte .Gorka Urkullu .\nVanessa Garcı ´a-Marina\nReceived: 12 April 2021 / Accepted: 26 September 2021 / Published online: 23 October 2021\n/C211The Author(s) 2021\nAbstract The stability of integrators dealing with\nhigh order Differential Algebraic Equations (DAEs) isa major issue. The usual procedures give rise to\ninstabilities that are not predicted by the usual linear\nanalysis, rendering the common checks (developed forODEs) unusable. The appearance of these difficult-to-\nexplain and unexpected problems leads to methods\nthat arise heavy numerical damping for avoiding them.This has the undesired consequences of lack of\nconvergence of the methods, along with a need of\nsmaller stepsizes. In this paper a new approach ispresented. The algorithm presented here allows us to\navoid the interference of the constraints in the\nintegration, thus allowing the linear criteria to beapplied. In order to do so, the integrator is applied to aset of instantaneous minimal coordinates that are\nobtained through the application of the null space. Thenew approach can be utilized along with any integra-\ntion method. Some experiments using the Newmark\nmethod have been carried out, which validate themethodology and also show that the method behaves\nin a predictable way if one considers linear stability\ncriteria.\nKeywords Stability /C1DAEs integration /C1Null\nspace /C1ODE integration\n1 Introduction\n1.1 BackgroundOrdinary Differential Equation systems (ODEs) rep-\nresent a great number of physical phenomena includ-ing structural analysis, multibody dynamics, thermal\nconduction or electrical systems. This has led to\nintensive investigation about them [ 1–6]. One of the\nkey issues in ODEs is the stability. If the system to\nintegrate is stiff, the integrator might fail due to\ninstability. A-Stable integrator are stable regardless ofthe stepsize, which is quite interesting. Explicit\nmethods, which can deliver great performance in\nterms of convergence, cannot be A-Stable. Also, quitegood performing implicit methods are not A-Stable.I. Ferna ´ndez de Bustos /C1H. Uriarte ( &)/C1G. Urkullu\nDepartment of Mechanical Engineering, Faculty of\nEngineering of Bilbao, University of the Basque Country,\nAlameda Urquijo s/n, 48013 Bilbao, Spaine-mail: haritz.uriarte@ehu.eus\nI. Ferna ´ndez de Bustos\ne-mail: igor.fernandezdebustos@ehu.eus\nG. Urkullu\ne-mail: gorka.urkullu@ehu.eus\nV. Garcı ´a-Marina\nDepartment of Mechanical Engineering, UniversitySchool of Engineering of Vitoria-Gasteiz, University ofthe Basque Country, Nieves Cano 12,\n01006 Vitoria-Gasteiz, Spain\ne-mail: vanessa.garcia@ehu.eus\n123Meccanica (2022) 57:371–399\nhttps://doi.org/10.1007/s11012-021-01433-0 (0123456789().,-volV) ( 01234567 89().,-volV)This leads to the concept of A( a) stability, which\nintroduces a conditional stability. This means that\nthere is a limit in the timestep that can be used in theintegration.\nOften, these problems, depending on the modeling\nof the system, require the incorporation of someconstraints. Therefore, a nonlinear Differential Alge-\nbraic Equations system (DAEs) must be solved to\npredict the behavior of many of these physicalphenomena [ 7–10]. Commonly, tools of optimization\nhave been taken into account to solve the nonlinear\nDAEs as it can be seen in Nocedal [ 11]. In any case, it\nseems clear that the high index systems are more\ndifficult to solve [ 12] and are associated with a\nsingular mathematical set up [ 13,14] . In this way, one\nof the main strategies to face this type of problems is\nthe index reduction technique which applies a math-\nematical process to reduce the index of a set of DAEs.A very interesting work in this field was done by\nNatsiavas in [ 15]. In this work, the authors present a\nnew theoretical approach for deriving an appropriateset of equations of motion for a class of mechanical\nsystems subjected to motion constraints. Another\nfascinating approach was presented by Gonza ´lez\n[16] and Bayo [ 17], who used the penalty factors\npresent in the augmented Lagrange formulation to\ndetermine the reaction forces in redundant constraineddynamic systems in order to represent the physical\nproperties of the system in the model. However, index\nreduction through the analytical differentiation of theconstraint equations causes the progressive drift of the\ncomputed solution; therefore, it is necessary to apply\nsome kind of stabilization. To alleviate this problemBayo and Cuadrado [ 18,19] propose a post-stabiliza-\ntion technique based on coordinates projection. In\naddition, apart from the index reduction, there isanother well-known approach: the coordinates reduc-\ntion technique. These methods try to divide the\ncoordinates in two groups: dependent and independentcoordinates. An excellent work was done by Zhang in\n[20] using this technique. He uses an implicit Runge–\nKutta method for the solution of index-3 DAEs. The\niteration of the constraint equations is embedded in the\niteration of the non-linear algebraic equations thatoriginate from the implicit method and, using the\ncoordinate partitioning technique, the independent\ncoordinates of the set of coordinates of the system arechosen. An interesting study of the behavior of these\ntwo main techniques can be found in the work done byJalo´n[21]. He contributed to the resolution and\nclarification of the multiplicity of solutions to the\nconstraint equations, where they focused on threemethods: the Lagrange equations of the first kind, the\nnull space method (introduced by Schwerin [ 22]), and\nthe Maggi equations. In this sense, Arnold et al. [ 23]\nstudied the effect of velocity projections on stability,\nand Cuadrado et al. [ 24] performed a comparison\nbetween four methods to simulate multibody dynam-ics with constraints. These methods were the aug-\nmented Lagrange formulation (ALF) index-1 and\nindex-3 with projections, ALF-1 and ALF-3, respec-tively, a modified state-space formulation, and a fully\nrecursive formulation.\n1.2 Formulation of the problem of interest for this\ninvestigation\nIn the case of ODEs, stability criteria are quite well\nestablished. But, unfortunately, these criteria cannot\nbe directly translated to DAEs when applying ODEintegrators in the usual way. Gear in [ 25], highlighted\nthe difficulties associated with the solution of this type\nof systems. Higueras et al. in [ 26] analyze the stability\nof index-2 DAEs, remarking the unexpected stepsize\nlimitations that appear even when an A-stable method\nis used. In this paper, the authors conclude thatchoosing a suitable formulation of DAEs is a vital task\nfor a successful and effective numerical integration as\nmuch as the numerical method. Otherwise, stepsize\nlimitations might appear because implicit methods\nbehave in the same way as the explicit ones. Even\nworse, the method may fail completely. These prob-lems were already observed by Ascher in [ 27], where\nthe implicit Euler method was transformed into an\nexplicit method when it was applied to solve someproblems. About stability in linear index-2 DAEs,\nHanke [ 28] presents a study of asymptotic properties\nof solutions on infinite intervals. The author states, asit happens in index-1 DAEs [ 29], that algebraically\nstable implicit integration methods are shown to be\nB-stable, provided that the null space of the leading\nJacobian is constant. If this null space rotates, stability\nproperties may change. Many others have also takennote in these problems, as it can be seen in the work\ndone by Liu [ 30] which studies the stability of the\nnumerical methods for linear index-3 DAEs.\n123372 Meccanica (2022) 57:371–399As it can be seen, the stability of the integrator\nmethods is a significant problem, especially in\nconstrained high order differential equation systems.The assumptions adopted in linear problems are not\napplicable to nonlinear ones and, usually, instabilities\nappear. These undesirable problems lead to theapplication of an integration method that introduces\na certain numerical damping (L-Stable methods).\nNevertheless, the convergence towards the solutionof the system is broadly affected and, obviously, it\nforces to use smaller time steps.\nAnother quite remarkable family of solutions to this\nproblem is the use of a reduced set of minimal\ncoordinates (see [ 31–33]). In these methods, the\nauthors select a set of the original coordinates andthe integration is performed in this set. Afterwards, the\nrestrictions are used to obtain the values of the rest of\nthe coordinates. Unfortunately, this reduced set cannotusually be applied to the whole integration interval\nand, thus, the set has to be changed from time to time.\nThis leads to an increase in the computational cost,which has commonly discouraged the use of these\nmethods.\nAn interesting aspect to point out is that explicit\nmethods, while being A( a)-stable, do not show\nunpredictable behavior. This means that the timestep\nrequired for stability can easily be predicted withclassical ODE theory. For example, methods as those\npresented in [ 34] exhibit predictable behavior. The\nmain problem of these methods is that their stabilitylimits the timestep required to solve the problem. In\nthe case of some stiff problems, this leads to the need\nof a high computational cost derived just from thestability requirement.\n1.3 Scope and contribution of this studyIn this paper a new approach for applying numerical\nintegration to index 3 DAEs (although easilyportable to DAEs of other indexes) is presented. The\nmain idea of the algorithm is to integrate in the tangent\nspace to the manifold. The new approach presents a\npredictable stability behavior and can be applied along\nwith any kind of integration method such as Newmark,HHT and others. The method also allows one to force\nthe simultaneous verification of constraints in terms of\nthe function itself and all the relevant derivatives.Although similar in concept to the methods based in a\nreduced set of coordinates, this method is quitedifferent in the sense that the coordinates are different\nin each step and represent a movement in the tangent\nspace. In order to do this without hampering compu-tational cost, most of the needed information is\nobtained from the algorithm employed to solve the\nequations. Some numerical examples are presentedwhich show the behavior of the algorithm in stiff\nproblems. Also, a quite simple explanation of the\nphenomenon of unpredictable stability in DAEs in theso-called structural integrators is presented.\n1.4 Organization of the paperThe paper is organized as follows. First, the usual\napproach for applying integration schemes to index 3DAEs is presented. Second, a quite graphical example\nof the problems that might arise in stiff problems due\nto stability issues is introduced. Afterwards, anexplanation of the usual problems that appear in the\ncommonly used approach employed for index 3 DAEs\nintegrators is given. The next point addresses the newapproach for applying the integrators. Some numerical\nexperiments are presented to demonstrate the new\napproach behavior and, finally, some conclusions aredrawn.\n2 Common approaches for the integration\nof constrained systems with structural\nintegrators\nWe consider high index DAEs in the form expressed\nby the following equations:\nF_x;x;tðÞ ¼ 0 ð1Þ\nqt;xðÞ ¼ 0 ð2Þ\nwhere (1) is the differential equation to be solved, and\n(2) are the boundary conditions. In the case of index-3DAEs, which are typical of Multibody Dynamics,\nStructural Nonlinear Mechanics and other engineering\nproblems, one can write Eq. ( 1) as:\nMxðÞ€x¼fx ;_x;t ðÞ þ G\nTxðÞk ð3Þ\nwhere MxðÞis the mass matrix, which is invertible in\nthe integration interval, fx ;_x;t ðÞ are the applied,\nCoriolis and velocity-dependent forces, and GTxðÞk\nare the constraint forces. Two main approaches are\n123Meccanica (2022) 57:371–399 373considered to apply an ODE integrator to this problem.\nThe traditional approach reformulates the problem,\nresulting in a system in the form expressed in thefollowing equations and Eq. ( 2).\n_x¼u ð4Þ\nMxðÞ _u¼fx ;u;t ðÞ þ G\nTxðÞk ð5Þ\nWith this procedure, one can apply a first order\nmethod, which can be implicit (usually single step) orexplicit (usually multi step).\nA more recent approach takes advantage of the so\ncalled structural integrators ([ 34–37]). These methods\nare directly applied to the system expressed by eqns.\n(3) and ( 2). The main advantage of these methods is\nthat they do not duplicate the number of functions tointegrate. In any case, one can also find implicit (for\nexample: Newmark, HHT) and explicit (for example\nCentral Differences) methods. Again, usually implicitmethods are formulated in a single step approach and\nexplicit methods are usually multi stepped.\nIn any case, usually one applies the integrator to\nEq. ( 3) (in the structural integrator approach) or to\neqns. ( 4) and ( 5) (in the first order approach) and,\nafterwards, the constraints presented in Eq. ( 2) are\nintroduced. We will further develop the algorithm for\nthe structural approach, although a similar develop-ment can be done for the first order approach.\nThe first step is to obtain a linearization of Eq. ( 3).\nFor example, with an implicit method, the lineariza-tion would be usually performed in tþDt ðÞ , and thus\none would reach:\nM\nL€xtþDt ðÞ þ CL_xtþDt ðÞ þ KLxtþDt ðÞ\n¼fLþGT\nLk ð6Þ\nUsually the terms ML,CL,KL,GLand fLare\nobtained in a numerical way. These terms differ from\ntheir counterparts M,C,K,Gandfand are the result\nof applying a linearization such as Taylor series to an\notherwise non-linear Eq. ( 3). Terms ML,CL,KL,GL\nandfLtherefore relate not only to M,C,K,Gandf\nrespectively; as a result of the applied linearization\n(and their dependencies on the system’s coordinates)\ndifferent derivatives of the same term with respect todifferent coordinates arise, forming several terms that\nare each a function of x,_x,€xork. All these\ndependencies related to each one of the system’svariables are summed together separately; this meansthat, for example for x, all the dependencies related to\nxform K\nLas the matrix that includes all the linearized\ndependencies related to xand is multiplied by the\ncorresponding system’s variable in the equilibrium\nequation, as seen in ( 6). The same happens with CL\nrelated to velocities, to fLrelated to independent\nterms, and to the remaining variables obtained as a\nresult of the linearization that form Eq. ( 6). The\nequation obtained with this linearization processrequires an iterative scheme to obtain the correct\nsolution.\nOne can use the function to be integrated to solve\nthe problem or reduce the index. Thus, if the\nintegration is based on the function itself, the proce-\ndure is referred as Index 3 Formulation. In this case,after applying the integrator equations to Eq. ( 1), one\nreaches an equation (expressed for implicit methods)\nin the form of:\nA\n1xtþDt ðÞ ¼ g1þGT\nL1k ð7Þ\nIf, instead, one uses the first derivative (Index 2\nFormulation), one reaches:\nA2_xtþDt ðÞ ¼ g2þGT\nL2k ð8Þ\nFinally, using the second derivative (Index 1\nFormulation), one would reach:\nA3€xtþDt ðÞ ¼ g3þGT\nL3k ð9Þ\nBeing A1;A2;A3;GL1;GL2;GL3;g1;g2;g3matrices\nand vectors that depend on the current estimation of\nxtþDt ðÞ ;_xtþDt ðÞ or€xtþDt ðÞ .\nThe linearization of Eq. ( 2) will lead to a system of\nequations that would allow one to solve eqs. ( 7), (8), or\n(9). Obviously, the linearization would be different if,\ninstead of the function, one of its derivatives is\nemployed. For example, using an Index 3 formulation,\none would obtain from Eq. ( 2) an expression in the\nfollowing form:\nHLxtþDt ðÞ ¼ bL ð10Þ\nwhere HLis the matrix that includes the linearized\nexpressions of the applied constraints that are multi-\nplied by the solution vector expressed in the system’s\ncoordinates, and bLis the independent term vector\nobtained after the same linearization that provided HL.\nThis linearization will be explained in Sect. 4. Sim-\nilarly, matrix GLused previously in this document is\nthe matrix that includes the linearized expressions of\n123374 Meccanica (2022) 57:371–399the applied constraints expressed in the coordinates in\nwhich the forces are defined, which can differ from the\nones employed to form HLas Urkullu explains [ 34].\nFor instance, GLrequires rotations for the angular\ncoordinates, and HLcan be formed using any other\nangular coordinates that can be considered moreconvenient, such as quaternions. Both matrices have\nbeen named differently to avoid confusion related to\nthis matter despite the fact that both define theJacobian of the system.\nThus, the system to be solved is composed of\neqs. ( 7) and ( 10), which are modified in an iterative\napproach. With this in mind, the algorithm is quite\nsimple. Figure 1presents the mentioned algorithm\nwhen formulated in terms of the function expressedbefore.\nIf one uses Index 2 or Index 1 formulations, a drift-\noff may appear and this leads to the need of the socalled stabilization methods, such as Baumgarte or\nProjection. ([ 38]).\nOne could consider a DAE as differential equations\non a manifold (see [ 39]). In this sense, this way of\nintegrating the DAE could be explained as follows:\nintegrating the function while forcing the solution tobelong to the manifold.\nThere are some criticisms that can be applied to this\napproach. For instance, one of them is the fact that onecannot impose that the function to be integrated and its\nderivatives satisfy the constraints in the considered\nintegration step. This is easy to demonstrate. Let usconsider the particular case of a system with N-\nfunctions to integrate, N differential equations and N\nR\nconstraints, being integrated using Newmark. This is a\ncommon case in multibody dynamics. The constraints\nbring about NRLagrange multipliers. Thus, for each\nintegration step there are 3 NþNRunknowns to solve.\nFig. 1 Typical integration algorithm\n123Meccanica (2022) 57:371–399 375For this, one has 2 N Newmark equations, the\nequilibrium in tþDtleads to another set of N\nequations and, finally, one gets 3 NRequations from\nthe constraints formulated for the functions and their\ntwo derivatives. Thus, one has 3 Nþ3NRequations\nfor 3 NþNRunknowns. Therefore, one must take the\nchoice of applying constraints solely to the function\nignoring the derivatives, applying the constraints\nsolely to one of the derivatives ignoring the functionand other derivatives (leading to the already com-\nmented drift problem) or applying a compromise\nsolution. Another issue is the erratic behavior regard-ing stability.\nThe stability of ODEs is a quite thoroughly studied\nsubject for first order methods. A-Stability is of mostimportance in stiff problems, because it will assure a\nsolution (although not necessarily accurate). Explicit\nmethods can be quite efficient, but cannot be A-Stable.In these cases one speaks of AaðÞ-Stability, which\nmeans that the method is stable depending on some\nconditions. In the case of structural integrators, usuallyone speaks about conditional and unconditional\nstability. Conditional stability is similar to AaðÞ—\nStability, while unconditional stability is equivalent toA-Stability.\nSome methods exhibit a configurable stability\nbehavior. An example is the Newmark method, whichdepends on two parameters which are selectable, aand\nb. The Newmark method requires the following\nequation to be verified for an integration to bestable [ 40]:\nDt/C20\n1\nxmaxffiffiffiffiffiffiffiffiffiffiffiffiffiffi\n1\n1\n2a/C0bs\nð11Þ\nBeing xmaxthe highest natural frequency of the\nsystem to integrate. If a[0:5 and b[0:5a, the\nmethod is unconditionally stable (A-Stable), while ifa[0:5 and b\\0:5a, the method is conditionally\nstable. But unfortunately, when one applies ODE\nintegrator to DAEs, the ODE stability analysis cannotbe used. This is quite an issue, because it can\nconsiderably limit the step sizes that can be used\nand, consequently, the efficiency. Furthermore, behav-ior can be quite unpredictable. A common solution to\nthis problem is the use of L-Stable methods. These\nmethods introduce heavy damping in the high fre-quencies, thus stabilizing the problem. An example is\nthe HHT method, which is currently one of the mostused ([ 36,37,41]). But they also introduce damping in\nthe lower frequencies, which sometimes can be an\nissue.\nAnother approach that can be used is to integrate\nthe restricted system in a set of minimal coordinates\nobtained using Null-Space algorithms or otherapproaches [ 32,33,42]. These methods use the same\nset of coordinates along part of the integration time\nand, whenever is needed, they are changed. Thesemethods usually are slow and lead to lack of\nconvergence.\n3 A simple example of the problem. a stiff\npendulum\nIn order to expose the problems of the usual approach,\na simple example has been designed. It consists of asimple pendulum affected by gravity (see Fig. 2). A\nharmonic torque with a quite low frequency is applied\nto it as shown in Eq. ( 12). This should allow one to use\na considerably large stepsize, but depending on the\nintegration method employed it can lead to instability.\nT¼T\n0sinxtðÞ ð 12Þ\nThe system of reference is on the fixed node and\ngravity is considered as g¼9:8m=s2. The concen-\ntrated mass of the pendulum is M¼1kg. The length of\nthe truss is L¼1m. The truss weight is considered to\nbe 0. The parameters defining the torque are T0¼\n0:1Nmandx¼0:1rad =s. The initial conditions are\nh0ðÞ ¼ 0,_h0ðÞ ¼ 0. We must point out that, in this\nproblem, the mass matrix should be underdetermined,\nbut this issue can be easily solved applying constraintsto the system, or using minimal least squares solutions.\nFig. 2 A simple pendulum\n123376 Meccanica (2022) 57:371–399If one expresses the problem using has a parameter,\nthe equilibrium equation can be expressed as shown in\nthe following eqn.\nmL2€htðÞþ mgLsin htðÞðÞ /C0 TtðÞ¼ 0 ð13Þ\nA linearization of the system of equations can be\nobtained in an analytical way:\nmL2€htðÞ þ mgLsin h0ðÞ þ mgLcos h0ðÞhtðÞ /C0h0/C0/C1\n/C0TtðÞ ¼ 0\nð14Þ\nmL2€htðÞþ mgLcos h0ðÞhtðÞ\n¼TtðÞ/C0 mgLsin h0ðÞ þ mgLcos h0ðÞh0 ð15Þ\nThus, for this system, one has ML¼mL2and\nKL¼mgLcos h0ðÞ. This means that the natural fre-\nquency of this system can be obtained from:\nx¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nmgLcos h0ðÞ\nL2r\n¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\ngcosh0ðÞ\nLr\nð16Þ\nThe maximum value happens in 0:\nxmax¼ffiffiffig\nLr\nð17Þ\nwhich, for the presented configuration, yields:\nxmax¼3:1304951 rad =s. This is the parameter that\nwould define the stability limits if one accepts the\nvalues obtained for linear systems. For example, for a\ncentral difference method [ 34], one would reach:\nDtmaxs¼2\nxmax¼0:6388765 s ð18Þ\nLet us consider now the step size required to\ncorrectly represent the problem, regardless of stability.\nIf the effect of the torque is considered, providing for\nthe need of about ten points to represent a cycle of thetorque, one would use:\nDt\nmaxt¼2/C1p\n10x’6:28s ð19Þ\nFurthermore, if one wants to represent correctly the\noscillations produced by gravity, one would require (ina conservative estimation) the value obtained in:\nDt\nmaxg¼2/C1p\n10xmax’0:2s ð20Þ\nThe dynamic effects of gravity are of such low\namplitude that they can be disregarded. Thus, a correct\nstep size would be about 6. But this would violate (atleast in the case of the central differences method) the\nconstraint imposed by stability criteria. Therefore, this\nis a good yet simple benchmark to study the stabilityissues. We aim to verify whether the usual way of\napplying constraints is to blame or not for the\ninstability of the problem, for which we will firstsolve the problem without the need of employing\nconstraint equations. In order to do so, an unrestricted\nminimal set of coordinates will be used. Note that thisis a particular case in the sense that one can integrate\nthe whole problem in an unrestricted system of\ncoordinates. This is not so common. Now this equationshall be integrated using different schemes. A central\ndifference approach and two different configurations\nof Newmark: the Fox-Goodwin approach and theTrapezoidal rule.\nApplying a central difference scheme, one reaches\nthe following equations for the general case and for thefirst instant:\nhtþDt ðÞ ¼ 2htðÞ/C0ht/C0Dt ðÞ\nþ\nDt2\nmL2TtðÞ/C0 mgLsin htðÞ ðÞ ð 21Þ\nh0þDt ðÞ ¼ h0ðÞ þ Dt_h0ðÞ\nþDt2\n2mL2T0ðÞ /C0 mgLsin h0ðÞ ðÞ ð 22Þ\nIn Fig. 3the obtained results are presented for step\nsizes of 0.01 s, 0.1 s, 0.6 s and 0.7 s.\nA brief explanation regarding nomenclature can be\nadded here. Due to linearization the term h0found in\nEqs. ( 14), (15) and beyond is the value the variable h\nhas in the time step that is being processed, but in the\nprevious iteration. Consequently, h0is updated with\nevery iteration. This term is not to be confused with the\nvalue this same variable has on the initial conditions,\nwhich is noted as h0ðÞ.\nThe results in minimal coordinates behave as\npredicted by linear estimations. With step sizes lower\nthan the theoretical stability limit, Dtmaxs, the method\nis stable and delivers reasonable results. It is also\ninteresting to observe how the step size of 0.6 fails to\ndeliver a good representation of the small oscillation (astep size of 0.2 was estimated as a good choice). As\npredicted by Eq. ( 18), a step size of 0.7 leads to\ninstability.\nLet us now consider the solution keeping a minimal\ncoordinates approach, but now using a Newmark\napproach with a conditionally stable configuration. A\n123Meccanica (2022) 57:371–399 3774th order Fox-Goodwin approach has been used\na¼1=2;b¼1=12 ðÞ . This is a conditionally stable ap-\nproach, and the linear stability condition can be\nexpressed as Eq. ( 10). Thus, in this\ncase:Dtmaxs/C200:78246 s. The results with stepsizes of\n0.01 s, 0.1 s, 0.6 s, 0.7 s, 0.78 s and 0.79 s are\npresented in Fig. 4.\nAgain, results are quite predictable. Step sizes of\n0.7 s and below lead to good results. However, a\nstepsize of 0.78 s represents correctly the effect of theforce, but it is too large to correctly solve a force in the\nfrequency of the movement derived from the gravity\neffects. This is the reason behind the considerableoscillation of the solution. Nevertheless, it is important\nto point out that the system is stable. Finally, the\nstepsize of 0.79 s leads to unstable integration, aspredicted by Eq. ( 11).\nLet us consider now an unconditionally stable New-\nmark scheme, by taking a¼0:5 and b¼0:25. This\napproach is commonly known as trapezoidal rule. The\nchosen stepsizes are: 0.1 s, 0.7 s, 0.79 s and 6 s.\nResults are presented in Fig. 5.\nAgain, results are as predicted by the linear theory.\nThe unconditional stability of trapezoidal rule allows\nthis scheme to report reasonable results even with astepsize of 6 s.Now we consider the use of a constrained system\napproach, for which the differential equation that\nprovides the equilibrium equation is:\nN\nT\nGML€xtþDt ðÞ þ NT\nGCL_xtþDt ðÞ þ NT\nGKLxtþDt ðÞ\n¼NT\nGfL\nð23Þ\nThis equation is the result of premultiplying ( 6)b y\nthe null space of GL; then the constraints are\nintroduced. The term ‘‘constrained system’’ is used\nto name those systems where more variables than the\nbare minimum are used to define the state of theproblem (in the case of multibody dynamics the\nposition of the elements) and thus, one needs to\nintroduce constraints in the system of equations tosolve the problem. This happens not only in all general\napproaches but also in some particular situations\nwhere one tries to use minimal coordinates, but part ofthe problem is difficult to introduce without con-\nstraints. One could name these approaches as\n‘‘mixed’’. In this case, general coordinates x;y;hðÞ\nwill be used to define the situation of the center of\ngravity of the pendulum. The location of the revolute\njoint is fixed (2 constraints). The total number of\nFig. 3 Central difference, minimal coordinates solution for step sizes a0.01 s, b0.1 s, c0.6 s and d0.7 s\n123378 Meccanica (2022) 57:371–399variables is 3 and the number of constraints is 2, as\nshown in the following equation:\nqR¼rGþR/C1rL\nGR/C0rC¼0 ð24Þ\nwhere qRis the expression of the constraint, rGis the\nposition of the constrained solid’s center of gravity, R\nis its rotation matrix, rL\nGRis the distance from the\nsolid’s center of gravity to the point where the\nconstraint is applied expressed in local coordinates,andr\nCis the position of the rotation constraint in a\nfixed position, expressed in global coordinates. In\norder to solve the problem we have used a structuralintegrator approach, similar to those presented by\nCuadrado et al .[24]. In the trapezoidal ruleconfiguration, the method seems to work in a correct\nway, as can be seen in the results shown in Fig. 6.\nThe method seems to work well using a trapezoidal\nrule, but this is not as usual as one might think. Thereare plenty of reports of trapezoidal rule instabilities\n(see, for example, [ 41]).\nIn the case of Fox-Goodwin, the algorithm fails\nregardless of the stepsize, as can be seen in Fig. 7.\nThis brings about an obvious conclusion: the\nstability issues are not caused by the non linearity of\nthe problem itself, but by the introduction of con-\nstraints. Let us see now what happens when increasingthe amplitude of the torque in the stiff pendulum\nproblem. When doing so, the maximum angle of\nrotation grows and so does the rotation of the null\nFig. 4 Fox-Goodwin, minimal coordinates solution for step sizes a0.01 s, b0.1 s, c0.6 s, d0.7 s, e0.78 s and f0.79 s\n123Meccanica (2022) 57:371–399 379space of the Jacobian of the constraints (see Hanke\n[28]). The stepsize is 0.1 s. The obtained angle foramplitudes T0of 5, 6, 8 and 9 (in Newton meter) are\nrepresented in Fig. 8:\nFig. 5 Trapezoidal rule, minimal coordinates solution for step sizes a0.1 s, b0.7 s, c0.79 s and d6s\nFig. 6 Trapezoidal rule, using constraints for step sizes a0.1 s, b0.7 s, c0.79 s and d6s\n123380 Meccanica (2022) 57:371–399Fig. 7 Fox-Goodwin, using constraints for step sizes a0.1 s, b0.6 s, c0.78 s and d0.79 s\nFig. 8 Newmark (trapezoidal rule) constrained system results for amplitudes of a5Nm, b6Nm, c7Nm and d9Nm\n123Meccanica (2022) 57:371–399 381The same analysis has been carried out employing\nthe minimal coordinate approach, the same time\nincrement (0.1 s) and Newmark parameters (Trape-\nzoidal Rule), which yields the following results(Fig. 9):\nThe algorithm using constraints fails at amplitude\nof 9Nm. However, it does not seem to be related toinstability (it rather seems to be a convergence\nfailure). It is in fact in the reaction forces due to\nconstraints where one can verify the instability. Theseforces are represented (the X component, Y is similar)\nin Fig. 10.I n[ 26] the authors relate these kinds of\nfailures of the scheme to a bad conditioning of theleading term. This issue is also approached by Hanke\n[28], where this leading term is defined as the null\nspace of the Jacobian. It is also stated that its rotationcompromises the stability of the scheme. Although\nboth papers are related to index-2 DAEs, these results\nallow one to consider that, as the amplitude of thetorque increases, so does the rotation of the pendulum\nand, in consequence and in the words of Hanke [ 28],\nthe rotation of the nullspace of the Jacobian, being this\nthe reason behind the failure. It is not the non linearity\nof the problem the reason behind the stability issues,but the introduction of constraints. In the words of\nHigueras, the use of minimal coordinates leads to a\nconstant null space of the leading term, which renders\nthe system qualified (see [ 26,43]), allowing the use of\na linear stability analysis to be applied. It is the\nintroduction of constraints what makes the null space\nof the leading term (or, shall we say, Jacobian) to vary(or rotate) over time causing the showcased instability.\nWhen the torque amplitude is low, the rotation is quite\nsmall and thus, the problem is nearly linear. This is thereason behind the apparently correct behavior of\nNewmark. This looks awkward, because one could\nguess that, integrating in a broader space of themanifold, the average natural frequency of the system\nis lower and this should help stability, but the effect of\nthe constraints prevails.\n4 Stability considerations\nFor linear systems, there are several ways to study\nstability. Probably the most straightforward approach\nstudies the amplification introduced by the matrix that\ndelivers the variables in tþDtfrom those evaluated in\nFig. 9 Newmark (Trapezoidal Rule) minimal coordinate results for amplitudes of a5Nm, b6Nm, c7Nm and d9Nm\n123382 Meccanica (2022) 57:371–399the previous step ( t). In this way, in the case of\nNewmark, one reaches, for the single degree of\nfreedom problem:\nDt2€xtþDt ðÞ\nDt_xtþDt ðÞ\nxtþDt ðÞ8\n><\n>:9\n>=\n>;¼1\n2þ4anxDtþ2bx2Dt2 ðÞ\n4a/C01ðÞ nxDtþx22b/C01 ðÞ Dt2ðÞ\n21/C0aðÞ þ 2b/C0aðÞ x2Dt2ðÞ\n1/C02b ðÞ þ a/C02b ðÞ 2nxDt ðÞ2\n64\n/C02x2Dt2/C04nxDt ðÞ /C0 2x2Dt2ðÞ\n2b/C0aðÞ x2Dt2þ2 ðÞ /C0 2ax2Dt2\n2þa/C0bðÞ 4nxDt ðÞ 2þ4anx ðÞ3\n5Dt2€xtðÞ\nDt_xtðÞ\nxtðÞ8\n<\n:9\n=\n;\n¼A½/C138Dt2€xtðÞ\nDt_xtðÞ\nxtðÞ8\n<\n:9\n=\n;\nð25Þ\nThe absolute values of the eigenvalues of matrix A½/C138\ncannot be larger than 1. With this in mind, one can\nreach the condition expressed in the followingequation:Dt/C201\nxffiffiffiffiffiffiffiffiffiffiffiffiffiffi\n1\n1\n2a/C0bs\nð26Þ\nThis can easily be generalized to multiple degrees\nof freedom by decoupling the equations. This leads toEq. ( 11). In a linear ODE, this equation represents the\nphenomena in a perfect way, because the natural\nfrequencies of the system to integrate are constant. Innonlinear conditions, however, if one uses minimal\ncoordinates, a violation of the equation in a single step\nwould lead to an unstable increment of the error thatmight not be of importance if it does not happen\nrepeatedly over the integration process. In any case, if\nthis violation does not happen in any step, the systemwill be stable. This is also confirmed in the stiff\npendulum example. This allows one to develop\nunconditionally stable integrators in nonlinear condi-tions provided that no constraints appear in the\nproblem.\nLet us now consider a constrained problem. As\nstated before, one can formulate constraints based on\nthe functions to be integrated (Index 3 formulation),\ntheir first derivative (Index 2) or their second deriva-tive (Index 1). Let us choose the most straightforward\nmethod, which is using an index 3 formulation. When\nFig. 10 Newmark (trapezoidal rule) reaction force (X component) for amplitudes of a5Nm, b6Nm, c7Nm and d9Nm\n123Meccanica (2022) 57:371–399 383trying to analyze the stability of the problem, the\nlinearized constraints in Eq. ( 10) should be studied,\nwhere:\nHL¼oqxðÞ\noxð27Þ\nThis leads to a system that can be expressed in the\nfollowing form:\nML\n0/C20/C21\n€xtþDt ðÞ þCL\n0/C20/C21\n_xtþDt ðÞ\nþKL\nHL/C20/C21\nxtþDt ðÞ ¼fLþGT\nLk\nbL() ð28Þ\nNowkcan be considered as a function to be solved.\nEquation ( 28) could be written as:\nML0\n00/C20/C21€xtþDt ðÞ\n€ktþDt ðÞ/C26/C27\nþCL0\n00/C20/C21_xtþDt ðÞ\n_ktþDt ðÞ/C26/C27\nþKL/C0GT\nL\nHL 0\"#\nxtþDt ðÞ\nktþDt ðÞ/C26/C27\n¼fL\nbL/C26/C27\nð29Þ\nIt does not matter how small the values in HLare.\nThe system has great probabilities of instability. If oneconsiders Eq. ( 11),x\nmax¼1 , and, thus, one reaches:\nDt/C201\n1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi\n1\n1\n2a/C0bs\nð30Þ\nThus, the only way for the system to verify the\nstability condition is that1\n2a/C0b/C200. This is the reason\nbehind the failure of the Fox-Goodwin approach in the\npendulum example regardless of the stepsize. In the\ncase of the classical Newmark approach ( a[0:5 and\nb[0:5a), the stepsize limit is not determined and that\nallows the integrator to successfully solve the problem\nunder certain conditions.\nIt is known that the use of formulations which\nintroduce the constraints in a second order derivative\nframe tends to be more stable. This comes straight-\nforward from the fact that, in this case, one wouldreach an equation in the form:M\nL0\nHL0/C20/C21€xtþDt ðÞ\n€ktþDt ðÞ/C26/C27\nþCL0\n00/C20/C21_xtþDt ðÞ\n_ktþDt ðÞ/C26/C27\nþKL/C0GT\n00\"#\nxtþDt ðÞ\nktþDt ðÞ/C26/C27\n¼fL\ncL/C26/C27\nð31Þ\nThis introduces nonzeroes i n the mass matrix instead\nof introducing them in the stiffness matrix. The problemhere lies in the drift-off of the function to be integrated,\nwhich must be somehow corrected. One can introduce\nBaumgarte stabilization, or projection, or any method toavoid the drift but, anyway, these methods introduce a\nrelation among the functions in tþDtand those in t,\nwhich, in any case will numerically behave as astiffness, rendering again the stability analysis useless.\nThis is a major concern, because in these cases one\ncannot predict stability in a proper way. In addition,another problem of these stabilization methods is that\nthey introduce more equations than variables in each\nintegration step, which drives to the need of finding asolution of compromise.\nNote that the mentioned impossibility of performing a\nproper stability analysis and the associated issues appear,\nas Eq. ( 31) shows, in implicit meth ods; explicit methods,\non the other hand, do not present this problem but,\nunfortunately, there is no unc onditionally stable explicit\nmethod. For example, in the central differences method\nthe constraints do not appear as a stiffness and, thus, an\nintegrator based on central differences has pre-dictable conditional stability.\nOne could also argue that Eqs. ( 29) and ( 31) could\nface numerical issues. In the usual approaches that areused to solve them, these are avoided, as mentioned by\nNocedal [ 11].\n5 A new approach for solving the problem\nThe unexpected stability issues seem to arise not due\nto the nonlinearity of the problem, but because of the\nintroduction of nonlinear constraints. Problems solved\nusing minimal coordinates do not exhibit unpre-\ndictable behavior, while those solved with non-linearconstraints do. A quite interesting study in the subject\ncan be found in [ 26]. In that paper, the use of minimal\ncoordinates would be considered to lead to a qualifiedDAE [ 26,43]. One could also state that, in minimal\ncoordinates, no constraints are to be dealt with, and\n123384 Meccanica (2022) 57:371–399thus, the problem is an ODE. However, one cannot\nalways rely on minimal coordinates, because this\nusually involves formulations that are not general. Thequestion arisen is: is there a way to integrate a DAE\navoiding this problem? It is obvious that not in the\nusual form. Instead of integrating the variables inwhich the DAE is formulated, an alternative is to\nintegrate in the manifold (which is, in fact, what\nminimal coordinates do). To accomplish this proce-dure in a DAE, one can integrate in a linealization of\nthe manifold performed in the step to integrate. These\nideas can be applied to most implicit integrators, but,for the sake of clarity, we will use a Newmark\napproach, using analytical derivatives. A similar\napproach was presented in [ 34], but applied to an\nexplicit method. Due to the fact that explicit methods\ndo not exhibit the stability behavior that implicit\nmethods do, the approach in [ 34] cannot be used here.\nLet us consider the DAE formulated with eqns. ( 3)\nand ( 2). Here the full restriction treatment will be\npresented. A Taylor series linearization of ( 2) leads to:\nqxðÞ /C25 qx\n0ðÞ þoqxðÞ\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0x/C0x0 ðÞ ¼ 0 ð32Þ\nTaking into account ( 27), one can write:\nHLx0ðÞx¼HLx0ðÞx0/C0qx 0ðÞ ¼ bL ð33Þ\nIf one considers this equation for obtaining\nxtþDt ðÞ ,x0will be the current estimation of\nxtþDt ðÞ . With this in mind, one can write:\nxtþDt ðÞ ¼ xpþNHatþDt ðÞ ð 34Þ\nwhere NHis the null space of HL, and xpis a particular\nsolution of Eq. ( 10). Any expression of NHandxp\nwould theoretically lead to the correct solution. An\nobvious choice for xpis the minimal least squares\nsolution. This choice will always provide lead to good\nnumerical conditioning. One could argue that the useof a sparse solution for x\npcould be of use, but being xp\na vector, little advantage can be obtained from this.\nThe use of different expressions for xpshould generate\nproper results, provided that the elements of xpare\nkept small. For NHthe use of a fundamental base will\nbring about a sparse expression, which will reducecomputational cost, as mentioned by Coleman [ 44].\nThe important fact is for atþDt ðÞ to be a set of\nminimal coordinates which are only valid for the stepcurrently being integrated. The hereby presentedmethod is based on integrating on these coordinates,\nwhich change from step to step. In the case of both\nscleronomic and holonomic constraints, one can write:\n_qx ;_xðÞ ’ _qx\n0;_x0ðÞ þo_q\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0;_x0x/C0x0 ðÞ\nþoq\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0_x/C0_x0 ðÞ ¼ 0ð35Þ\nThis can be written in the form:\nHx 0ðÞ _x¼/C0 _qx 0;_x0ðÞ þ Hx 0ðÞ _x0þ_Hx 0;_x0ðÞ x0\n/C0_Hx 0;_x0ðÞ x\nð36Þ\nNow one introduces ( 34), reaching:\nHx 0ðÞ _x¼/C0 _qx 0;_x0ðÞ þ Hx 0ðÞ _x0þ_Hx 0;_x0ðÞ x0\n/C0_Hx 0;_x0ðÞ xp/C0_Hx 0;_x0ðÞ NHa\nð37Þ\nThe system ( 37) can be solved leading to:\n_x¼_xPþNH_aþ_XPa ð38Þ\nBeing _xpa solution to:\nHx 0ðÞ _x¼/C0 _qpx0;_x0ðÞ þ Hx 0ðÞ _x0þ_Hx 0;_x0ðÞ x0\n/C0_Hx 0;_x0ðÞ xp\n¼/C0 _qpx0;_x0ðÞ þ Hx 0ðÞ _x0\nþ_Hx 0;_x0ðÞ x0/C0xp/C0/C1\nð39Þ\nAnd _XPa solution to:\nHx 0ðÞ _X¼/C0 _Hx 0;_x0ðÞ NH ð40Þ\nHere it must be pointed out that neither _xpnor _XP\nare obtained as derivatives of any function, but as\nsolutions of linear systems. The use of the dot has been\ndecided in order to keep unit coherence. Now oneneeds to tackle accelerations. By taking again a\nderivative:\n€qx ;_x;€x ðÞ ’ €qx\n0;_x0;€x0 ðÞ þo€q\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0;_x0;€x0x/C0x0 ðÞ\nþ2o_q\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0;_x0_x/C0_x0 ðÞ þoq\nox/C12/C12/C12/C12/C12/C12/C12/C12\nx0€x/C0€x0 ðÞ\n¼0\nð41Þ\nThis can be rewritten as:\n123Meccanica (2022) 57:371–399 385€qx 0;_x0;€x0 ðÞ þ €Hx 0;_x0;€x0 ðÞ x/C0x0 ðÞ\nþ2_Hx 0;_x0ðÞ _x/C0_x0 ðÞ þ Hx 0ðÞ €x/C0€x0 ðÞ\n¼0 ð42Þ\nReordering:\nHx 0ðÞ €x¼/C0 €qx 0;_x0;€x0 ðÞ þ €Hx 0;_x0;€x0 ðÞ x0\nþ2_Hx 0;_x0ðÞ _x0þHx 0ðÞ €x0\n/C0€Hx 0;_x0;€x0 ðÞ x/C02_Hx 0;_x0ðÞ _x ð43Þ\nNow one can substitute eqns. ( 34) and ( 38):\nHx 0ðÞ €x¼/C0 €qx 0;_x0;€x0 ðÞ þ €Hx 0þ2_H_x0þH€x0\n/C0€Hxp/C02_H_xp/C02_HN H_a\nþ/C0 2_H_Xp/C0€HN H/C0/C1\na\nð44Þ\nSo one can solve:\n€x¼€xpþ€Xp1aþ_Xp2_aþNH€a ð45Þ\nBeing €xpa particular solution to:\nHx 0ðÞ €x¼/C0 €qx 0;_x0;€x0 ðÞ þ €Hx 0;_x0;€x0 ðÞ x0/C0xp/C0/C1\nþ2_Hx 0;_x0ðÞ _x0/C0_xp/C0/C1\nþHx 0ðÞ €x0\nð46Þ\nIn turn, €Xp1is a solution to:\nHx 0ðÞ €X¼/C02_Hx 0;_x0ðÞ _Xp/C0€Hx 0;_x0;€x0 ðÞ NHð47Þ\nAnd _Xp2is a solution to:\nHx 0ðÞ €X¼/C02_Hx 0;_x0ðÞ NH ð48Þ\nOne can notice:\n_XP2¼2_XP ð49Þ\nRegarding xp,_XP,_xp,€Xp1and €xp, it is important to\nstate that, although any solution will suffice, it is\nnumerically important not to let these solutions takeany value (it could be too large and cause numerical\nissues), thus, it is recommended to use minimal norm\nvalues. In the event of redundant constraints, it isrecommended to use minimal least squares values,\neven considering that they should be compatible.\nNow one must formulate the equilibrium equation\nin terms of a. From Eq. ( 6), and introducing eqs. ( 34),\n(38) and ( 42), one can write:\nM\nL€xpþ€Xp1aþ_Xp2_aþNH€a/C0/C1\nþCL_xpþNH_aþ_Xpa/C0/C1\nþKLxpþNHa/C0/C1\n¼fLþGT\nLk ð50ÞReordering:\nMLNH€aþCLNHþ2ML_Xp/C0/C1_a\nþKLNHþCL_XpþML€Xp1/C0/C1\na\n¼fL/C0KLxp/C0CL_xp/C0ML€xpþGT\nLk ð51Þ\nThis is the system to be actually integrated. This\nsystem is not constrained, because any value given to\natþDt ðÞ ,_atþDt ðÞ and €atþDt ðÞ will lead to values\nofxtþDt ðÞ ,_xtþDt ðÞ and €xtþDt ðÞ satisfying the\nlinearized constraints. Thus, in an iterative approach\nthey will converge to satisfying the original (notlinearized) constraints. It is important to remark that\nthe approach will not only converge to the verification\nof either the constraints expressed in terms of thefunction, its first derivative or its second derivative,\nbut also all of them together. Once computed\natþDt ðÞ ,_atþDt ðÞ and €atþDt ðÞ , one can use\neqs. ( 34), (38) and ( 42) to obtain xtþDt ðÞ ,_xtþDt\nðÞ\nand €xtþDt ðÞ . The process will be restarted with these\ninitial values until convergence is achieved, leading to\nthe next timestep. To avoid ill-conditioned systems,\none can use the null space of Gto reduce the system:\nNT\nGMLNH€aþNT\nGCLNHþ2NT\nGML_Xp/C0/C1_a\nþNT\nGKLNHþNT\nGCL_XpþNT\nGML€Xp1/C0/C1\na\n¼NT\nGfL/C0NT\nGKLxp/C0NT\nGCL_xp/C0NT\nGML€xp\nþNT\nGGT\nLk ð52Þ\nA commentary is due here. The novelty of the\nhereby presented method is not the use of the null\nspace of GLto reduce the system, but on the fact that a\nand its derivatives are what is being integrated insteadofxand its derivatives, as can be seen in Eq. ( 52). This\ndiffers from the usual way, in which first the integra-\ntion is applied to an equation in the form of ( 23)\neffectively integrating x, and then constraints of either\nposition, velocity or acceleration are applied. There-\nfore, in the usual way only one kind of constraints isapplied and fulfilled.\nAnother important issue is that the frequencies to be\ntaken into account for stability analysis are those of thesystem to be actually integrated, and, thus, those\nshould be obtained from the reduced mass matrix\nM\nR¼NT\nGMLNHand the reduced stiffness matrix\nKR¼NT\nGKLNHþNT\nGCL_XpþNT\nGML€Xp1.\nThe computational cost can be increased because\none needs to obtain NT\nGKLNHþNT\nGCL_Xpþ\nNT\nGML€Xp1 and NT\nGCLNHþ2NT\nGML_Xp. But\n123386 Meccanica (2022) 57:371–399NT\nGKLNH,NT\nGCLNH,NT\nGMLandNT\nGCLhave to be\nobtained anyway, one only needs to compute three\nproducts of reduced matrices. The cost of obtaining\n€Xp1and _Xpis also noticeable, but one must take into\naccount that the factorization of NHis also needed\nregardless of the method, so it introduces no additionalcost. Thus, the method does not change the order of the\ncomputational cost.\nLastly, the term N\nT\nGGT\nLkcan be fully eliminated in\nexplicit methods due to the orthogonality property.\nImplicit methods, however, require a previous lin-\nearization of the term GT\nLk, and only part of this termwill be eliminated due to the mentioned orthogonality\nproperty. The remaining terms of the linearization will\nbe included as part of the stiffness matrix KLandfL.\nOne of the key points of the approach presented\nhere is to notice that, if ais a variable to be\nintegrated, atðÞ,_atðÞand €atðÞare also needed. It could\nbe tempting to use the values obtained in the previousstep, but this cannot be done, because the linearization\nis not the same from step to step. Furthermore, that\nwould not be integrating in the linearization. In orderto obtain atðÞ,_atðÞand €atðÞone generalizes the\neqs. ( 34), (38) and ( 42), which were formulated in\ntþDtto the following eqns.:\nFig. 11 Flowchart of the proposed algorithm\n123Meccanica (2022) 57:371–399 387Fig. 12 Fox-Goodwin, new formulation, for stepsizes a0.1 s, b0.6 s, c0.78 s and d0.79 s\nFig. 13 Trapezoidal rule, using constraints for stepsizes a0.1 s, b0.7 s, c0.79 s and d6s\n123388 Meccanica (2022) 57:371–399xtðÞ¼ xpþNHatðÞ ð 53Þ\n_xtðÞ¼ _xPþNH_atðÞþ _XpatðÞ ð 54Þ\n€xtðÞ¼ €xpþ€Xp1atðÞþ _Xp2_atðÞþ NH€atðÞ ð 55Þ\nAs the unknowns are atðÞ,_atðÞand €atðÞ, it can be\nwritten as follows:NHatðÞ¼ xtðÞ/C0 xp ð56Þ\nNH_atðÞ¼ _xtðÞ/C0 _xP/C0 _XpatðÞ ð 57Þ\nNH€atðÞ¼ €xtðÞ/C0 €xP/C0€Xp1atðÞ/C0 _Xp2_atðÞ ð 58Þ\nThe eqs. ( 56), (57) and ( 58) are overdetermined and\nusually incompatible, because the linearization in tþ\nDtwill not usually satisfy the nonlinear constraints in\nt. Thus, the least squares solutions for these equations\ncan be computed, which is equivalent to project the\ncoordinates in t to the linearization in tþDt. Obvi-\nously, if the function to integrate is continuous, thesmaller Dtis, the better the projection fits to the\noriginal function. As a consequence, convergence of\nthis operation seems unquestionable. With all this inmind, the flowchart of the algorithm is represented in\nFig. 11.\nFig. 14 Double pendulum\nFig. 15 Results of Newmark integration based on displacements (angle of the 1st truss)\nFig. 16 Value of the constraint forces in the fixed joint. Left: in the X axis. Right: in the Y axis\n123Meccanica (2022) 57:371–399 389With this approach, ais a set of minimal coordi-\nnates to be integrated. Its physical interpretation is\nlinked to the null space of H, that is, NH. The product\nNHa(and also NH_aandNH€a) represents the direction\nwhich is tangent to the displacement (or velocity or\nacceleration fields, correspondingly) of the system inthat instant.\n6 Numerical experiments\nFor these numerical experiments, the Octave environ-\nment has been used. This will not lead to a proper\nmeasure of the efficiency of the method, provided that\nOctave does not use even a JIT compiler to improvethe iterative processes. The use of Matlab would\nconsiderably improve on this, but would still not lead\nto a proper assessment of the efficiency.\nThe first example is obviously the simple stiff\npendulum discussed before. The results achieved,\nusing Fox-Goodwin parameters, for stepsizes of 0.1 s,0.6 s, 0.78 s, 0.79 s are presented in Fig. 12:\nIf one compares these results to those presented in\nFig. 4and 7, one can see that the new method, being\ngeneral coordinate based, behaves like a minimal\ncoordinate approach in terms of stability. It is impor-\ntant to remark that the classical approach presentedfails even in the smallest stepsize, as shown in Fig. 7.\nThe trapezoidal rule also behaves in a correct way,\nas expected (Fig. 13).The second example in this document is the double\npendulum analyzed by Negrut [ 35]. In the paper the\nauthors had to use quite conservative values of theparameters: a¼0:75 and b¼0:390625, which deliv-\ners a first order algorithm. Even under these condi-\ntions, the authors found the method quite competitiveagainst BDF integrators. The used stepsize is 1 e/C03s.\nAlthough the authors do not explicitly state it, the\nreason of such a conservative set of parameters seemsrelated to stability. The system is depicted in Fig. 14.\nBoth joints include torsional spring-dampers. The\nsystem is subjected to gravity in the negative y axis.\nThe considered acceleration is g¼9:81m=s\n2. Total\nintegration time is 10 s.\nFirst, the classical approach along with a displace-\nment-based formulation is tested. The values used are\na¼0:5000005 and b¼0:2500006, which are clearly\nin the unconditionally stable region. The obtainedresults are presented in Fig. 15:\nWhat is interesting from this example are the values\nof the constraint forces in the fixed joint (see Fig. 16).\nThey show clearly the effect of instability and seem to\nagree with the hypothesis presented before. That is, the\nconstraints are the reason inducing the instability, notthe nonlinearity of the equilibrium equation.\nThe values of aandbused in this example are\ngreater than the bare minimum required for Newmarkto be unconditionally stable just to avoid any doubt\nabout issues related to discrete mathematics. The use\nofa¼0:5 and b¼0:25 also brings to failure.\nFig. 17 Results using the new approach: FG with 5 e/C04sstepsize\n123390 Meccanica (2022) 57:371–399Depending on the particular implementation of the\nalgorithm (based in acceleration with stabilization or\nso) the results might vary but, in any case, with theclassical approach there is no way of predicting a\npriori the stability of the method. Using constraints\nformulated in terms of the positions (the function to beintegrated) along with a choice of a¼0;5001 and\nb¼0;2507, and a stepsize of 1 e/C03s, the problem is\nsuccessfully solved. But it is not quite comfortable torely on a try-error scheme to determine a working set\nof parameters.However, using the new approach the results\nchange dramatically. One can even use a Fox-Good-\nwin (FG) approach. Using a¼0:5 and b¼1=12, with\na stepsize of 5 e/C04s, one reaches the result shown in\nFig. 17.\nAn additional advantage of the method is the fact\nthat constraints are numerically fulfilled for position,\nvelocity and acceleration (see Fig. 18,19and20). The\neuclidean norm of the constraint errors are alwaysbelow 3 e/C014 for position and velocity constraints,\nwhereas for acceleration it is under 1 e/C010. This high\nFig. 18 Evolution of the norm of the position constraint error\nFig. 19 Evolution of the norm of the velocity constraint error\n123Meccanica (2022) 57:371–399 391value happens only at the beginning of the simulation,\nand seems related to the high accelerations produced\ndue to initial conditions.\nIf one computes the maximum natural frequencies\nof the system the graph in Fig. 21is obtained.\nThe high value of the natural frequencies at the\nbeginning is due to the initial values. In any case, one\ncan compute from the linear stability analysis that a\n5e/C03sstepsize can be used along with FG. This is\nfive times the timestep used in [ 35]. The results with\nthis configuration are presented in Fig. 22.Clearly, the result is not numerically good, mainly\ndue to the fact that at the beginning of the simulation,\nthere is a large variation in the accelerations whichrequire a considerably smaller timestep. The interest-\ning point here is that, yet, the method is stable in these\nconditions.\nThe last example to be presented is the Andrews\nSqueezer mechanism, which can be found in [ 45] and\na good description with results can be spotted in theIFTOMM Multibody Benchmark.\nCoordinates for the fixed joints A, B, C and O are\nshown in Table 1.\nFig. 20 Evolution of the norm of the acceleration constraint error\nFig. 21 Evolution of the maximum natural frequencies of the double pendulum\n123392 Meccanica (2022) 57:371–399Trusses have the following mechanical properties:\nThe coordinates for the centers of gravity of the\ntrusses are indicated in local coordinates along the line\n(Table 2).\nThe solid BDE has a mass of 0.02373 kg and\n5.255e-6 kg /C1m2of inertia. BE is 0.035 m length.\nSpring stiffness is k¼4530 N=m, and its undeformed\nlength is l0¼0:07785 m. OF truss has a constant\ntorque applied s¼0:033Nm. Initial position is\nobtained from b0¼/C00:062rad. Gravity is not con-\nsidered. Using a classical position-based Newmark\napproach, the system fails with a¼0:5 and b¼0:25.\nIt has only been possible to obtain a solution with\na¼0:5001 and b¼0:2507. However, with the new\napproach, using a¼0;5 and b¼1=12 (FoxGoodwin), with a time increment of 2 e/C06sone\nreaches a good result, as presented in Fig. 24.\nAlthough the aim of this paper is focused on\nstability rather than on precision, which obviouslydepends on the integrator used, it must be pointed out\nthat the error obtained with Fox-Goodwin, with a Dt¼\n2e/C06sis under 2 :28e/C06. The use of a quite good\ntrapezoidal rule integrator (MBSLab), with a timestep\nof 1e/C06syields an error under 8 :2e/C06. The use of\nRunge Kutta Fehlberg (an explicit method consideredto be around 4\nth–5th order, implemented in OpenSim),\nwith a variable stepsize from 1 e/C05sto 1e/C04syields\nan error of 3 :75e/C07. This seems to be reasonable\ntaking into account that Fox-Goodwin is considered as\na 4th order method and the trapezoidal rule is\nconsidered as 2nd order.\nIf one plots the maximum natural frequency of the\nsystem along the integration, Fig. 25is obtained:\nWith this in mind, one could obtain a value for the\nstepsize that should not lead to instability. If one\nconsiders xmax¼4503 :477, one can obtain from\nFig. 23 Andrews Squeezer Mechanism\nFig. 22 Results using the new approach: FG with 5 e/C03sstepsize\nTable 1 Coordinates for the fixed joints\nJoint X Y\nA -0.06934 -0.00227\nB -0.03635 0.03273\nC 0.014 0.072\nO0 0\n123Meccanica (2022) 57:371–399 393Table 2 Mechanical\nproperties of trussesTruss Mass (kg) Length (m) I z(kg/C1m2)X G YG\nOF 0.004325 0.007 2.194 /C110–60.00092 0\nEF 0.00365 0.028 4.41 /C110–70.0165 0\nHE 0.00706 0.02 5.667 /C110–70.00579 0\nGE 0.00706 0.02 5.667 /C110–70.00579 0\nAG 0.05498 0.04 1.169 /C110–50.02308 0.00916\nAH 0.02373 0.04 1.912 /C110–50.01228 -0.00449\nFig. 24 Andrews Squeezer Mechanism Results using FG. Coordinates of Joint F\nFig. 25 Andrews Squeezer Mechanism. Evolution of maximum natural frequency\n123394 Meccanica (2022) 57:371–399Eq. ( 26) that using Dt/C205:44e/C04sverifies the linear\ncondition for stability along all the integration time.\nThe results obtained with Dt¼5e/C04sare depicted in\nFig. 26.\nEvidently the error increases, but the system keeps\nbeing stable. A result that might surprise is that usingDt¼6e/C04s(larger than the computed stability limit)\nthe integration is also stable (see Fig. 27).\nThis is not actually completely unexpected. One\nmust consider the fact that in nonlinear conditions\nnatural frequencies vary over time and the computed\nstability limit is a worst-case scenario. The error mightgrow in a single iteration and in the following return tostability. It can even happen for some iterations and\nprovoke no major stability issues. In fact, the system\nshows stability even with values of the stepsize around1e/C03s. In any case, it is obviously not a good practice\nto use a stepsize larger than the one that can be\nobtained from a stability analysis for the currentintegration step.\n7 Some preliminary results regarding efficiency\nIn order to give an idea about the methods efficiency, a\nquite preliminary implementation in C has been\nFig. 26 Andrews Squeezer Mechanism Results using FG. Coordinates of Joint F. Stepsize: 5 e/C04s\nFig. 27 Andrews Squeezer Mechanism Results using FG. Coordinates of Joint F. Stepsize: 6 e/C04s\n123Meccanica (2022) 57:371–399 395carried out, which currently supports R and P joints.\nAlthough the implementation is quite preliminary, it\nhas been developed taking into account performanceand, thus, results should give an initial idea of the\nperformance that one can expect from the method. The\nimplementation is currently based in dense matrices.\nA common problem to compare efficiency is to find\nbenchmark results. Thanks to the IFTOMM Multibody\nBenchmark, some results are publicly available.Unfortunately no stiff problem is included there. Here\nwe consider the term stiff in the sense of numerical\nintegration and, therefore, related to the stability issuesthat the problem might lead to. This does not mean that\nthe problem has no elastic elements. It means that the\ntimestep required to correctly solve those problems isnot large enough to lead to stability issues. Even the\nstiff flyball example is not stiff in the sense of\nnumerical integration and can easily be solved by anexplicit method. Taking into account that the stability\ndoes not affect efficiency, the performance can be\nmeasured with non-stiff problems. The studied exam-ples are the double four bar and the rectangular Bricard\nmechanism. In both of them the benchmark limits the\nmechanical energy drift.\nThe first example is the double four bar mechanism.\nThe maximum mechanical energy drift is 0.1 J. The\npublished results are shown below (Table 3):\nFrom the table above, one could state that a good\nresult is of about 1 cent of a second, but one has to take\nthose results carefully. Most of the results presentedare provided using minimal, and relative coordinates,\nwhich heavily reduce the computational cost, but these\nmethods are not general, and, thus, should not becompared with general methods as the one here\npresented. Also there are some 2D implementations\nthat see their efficiency increased due to the reductionin the amount of variables. Thus, the results to be\nconsidered should be those presented by Tagliapietra\n(using OpenSim), Masarati (MBDyn), Urkullu(DIMCD, [ 46]) and Gonzalez (MBSLab). The\nreported times are, respectively, 0.456, 0.325, 0.0325\nand 0.145. It is important to mention that Tagliapietragets quite a good result in terms of precision, which\nhinders the real performance of the algorithm, which\nshould be better. The method explained in thisdocument takes 0.4 s (average of 10 simulations) on\na computer with an Intel i7 7700HQ processor fixed at\n2.8 GHz. Taking into account that this is an implicit,unconditionally stable method, it is a quite good result.\nThe second example to which this method has been\napplied is the Bricard mechanism. This system hasredundant constraints. With this mechanism the aim is\nto keep the mechanical energy drift below 0.001 J.\nThe results are shown below (Table 4):\nAgain, the results presented by Tagliapietra (using\nOpenSim) are misleading because of the use of a\nhigher precision, which increases the cost. In any case,the times are 0.258 and 0.125. The here presented\nmethod takes 0.357 s in the same conditions exposed\nin the previous example. Again, a reasonable result.\nTable 3 Results of the double four bar mechanism\nMethod Author Coord Step (s) e (J) CPU (s) Processor GeekBench score\nIndex-3 ALF J. Cuadrado Natural 1e-2 0.0917 0.6 C2duo E6550 1400\nIndex-3 ALF A. Luaces Relatives 1e-2 0.0137 1.7 C2duo E6550 1400\nMbsLab F. Gonza ´lez Natural 3D 1e-2 0.0917 0.145 C2duo E8400 1900\nOpenMBS R. Pastorino Natural 2D 1e-2 0.0877 0.1278 BeagleBoneB NA\nIndex-3 ALF R. Pastorino Natural 2D 1e-2 0.0877 0.0045 I7 3740QM 3800\nNon-recursive NEF M. Burkhardt Mı ´nimal Variable 0.0015 0.0568 I7 3770 4000\nMBDyn P. Masarati 3D 8e-3 0.09 0.325 I7 2620 M 3000\nNon-recursive NEF M. Burkhardt Mı ´nimal 1e-2 0.0002 0.0191 SnapDrag S800 NA\nNon-recursive NEF M. Burkhardt Mı ´nimas 1e-2 0.0002 0.0023 I7 960 2700\nSolidWorks C. Chaojie Natural 2D 1e-2 0.0001 48.77 C2Duo E8400 1900\nOpenSim L. Tagliapietra Natural 1e-3 < > 1e-2 3.2e-7 0.455 I5 4570 4000\nBiolim F. Mouzo Relatives 3D 1e-2 0.029 0.0226 i7 6700 K 5400\nDIMCD G. Urkullu Cartesian 3D 1e-2 0.0885 0.0325 Ryzen 5 2600 4700\n123396 Meccanica (2022) 57:371–3998 Conclusions and future work\nThe integration of DAEs along with nonlinear con-\nstraints leads usually to unqualified DAEs. This\ntranslates into unpredictable stability behavior of thealgorithms. This can be usually avoided by switching\nto minimal coordinates, but this arises the need of a\nparticular mathematical development for each case,which can be an obstacle for general problems. The\nusual approach to solve this issue in general algo-\nrithms related to implicit methods is the use of heavilydamped integrators such as Newmark with high values\nof the parameters or HHT, but this heavily impacts the\nconvergence of the methods introducing a consider-able amount of numerical damping. Here an alterna-\ntive approach has been presented to perform the\nintegration. In this approach, one integrates the DAEin a set of coordinates representing the tangent space\nof the manifold. To do this, a Taylor expansion of the\nconstraints is iteratively performed in the currenttimestep which translates the function to a minimal\ncoordinate set. This allows one to integrate in the\ntangent space. To obtain the initial values of thefunction in these minimal coordinates, these are\nprojected in the tangent space. One advantage of this\napproach is that constraints are applied for each stepsimultaneously to the function to integrate and all the\nrequired derivatives. Furthermore, the new approach\nhas not the unpredictable stability behavior of theclassical approach, thus allowing one to use high order\nimplicit methods such as Fox-Goodwin, or linear\nacceleration with predictable stability. Several testshave been performed with this algorithm, applying a\nNewmark approach, with Fox-Goodwin, linear accel-\neration and classical Newmark configurations. Theseconfirm the predictable stability behavior and also\nyield to proper results. Currently no efficiency exper-\nimental tests have been performed. Due to the factthat, compared to the classical approach the newalgorithm only includes operations of, at most, the\nsame order ( N\n3) than the other methods, one can\nestimate that the new approach should not consider-\nably increase computational cost, although it mightnot allow one to use certain sparse methods without\nmodification. Additionally, a simple study for the case\nof structural integrators has been presented whichexplains the behavior of stability in DAEs when\ncompared to unconstrained ODEs.\nFuture studies should be focused in performance.\nWith this in mind a general Newmark approach should\nbe developed in a high-performance compiled lan-\nguage. Additionally a deeper mathematical analysis ofthe method would be necessary. A proper analysis on\nthe possible impact of the projection made for the\nvalues in the previous step in the convergence should\nalso be performed.\nAcknowledgements The authors would like to thank to the\nBasque Government for its funding (ref. IT 947-16) and to the\nSpanish Ministry of Economy and Competitiveness for the grant\nthrough the project DPI2016-80372-R (AEI/FEDER, UE). Weacknowledge Javier Cuadrado and his group for the help they\nhave always given to us. We also thank Rafael Avile ´s for his\nmentorship.\nFunding Open Access funding provided thanks to the CRUE-\nCSIC agreement with Springer Nature.\nDeclarations\nConflict of interest The authors declare that they have no\nconflict of interest.\nOpen Access This article is licensed under a Creative Com-\nmons Attribution 4.0 International License, which permits use,sharing, adaptation, distribution and reproduction in any med-ium or format, as long as you give appropriate credit to the\noriginal author(s) and the source, provide a link to the Creative\nCommons licence, and indicate if changes were made. Theimages or other third party material in this article are included in\nthe article’s Creative Commons licence, unless indicated\notherwise in a credit line to the material. If material is notTable 4 Results of the Bricard mechanism\nMethod Author Coords Step (s) e (J) CPU (s) Processor GeekBench score\nIndex-3 ALF F. Gonza ´lez Natural 3D 5e-3 8e-4 0.125 I7 3770 4000 4000\nOpenSim 3.2 L. Tagliapietra Natural 1e-3 \\[ 1e-2 9.6e-7 0.258 I5-4570 4100 4100\nDIMCD G. Urkullu Cartesian 3D 1.5e-2 8.1e-4 0.0777 Ryzen 5 2600 4700\n123Meccanica (2022) 57:371–399 397included in the article’s Creative Commons licence and your\nintended use is not permitted by statutory regulation or exceeds\nthe permitted use, you will need to obtain permission directlyfrom the copyright holder. To view a copy of this licence, visit\nhttp://creativecommons.org/licenses/by/4.0/ .\nReferences\n1. Rakhsha M, Pazouki A, Serban R, Negrut D (2019) Using a\nhalf-implicit integration scheme for the SPH-based solution\nof fluid–solid interaction problems. Comput Methods Appl\nMech Eng. https://doi.org/10.1016/j.cma.2018.09.027\n2. Witteveen W, Pichler F (2020) Separate time integration\nbased on the hilber, hughes, taylor scheme for flexible\nbodies with a large number of modes. J Comput Nonlinear\nDyn. https://doi.org/10.1115/1.4046733\n3. Pappalardo CM, Guida D (2018) A comparative study of the\nprincipal methods for the analytical formulation and the\nnumerical solution of the equations of motion of rigid\nmultibody systems. Arch Appl Mech. https://doi.org/10.\n1007/s00419-018-1441-3\n4. Noh G, Bathe KJ (2019) The Bathe time integration method\nwith controllable spectral radius: the q?-Bathe method.\nComput Struct. https://doi.org/10.1016/j.compstruc.2018.\n11.001\n5. Hairer E, Wanner G (1999) Stiff differential equations\nsolved by Radau methods. J Comput Appl Math. https://doi.\norg/10.1016/S0377-0427(99)00134-X\n6. Gander MJ, Wanner G (2020) Exact BDF stability angles\nwith maple. BIT Numer Math. https://doi.org/10.1007/\ns10543-019-00796-x\n7. Scholz L (2018) The signature method for DAEs arising in\nthe modeling of electrical circuits. J Comput Appl Math\n332:107–139\n8. Brenan KE, Campbell SL, Petzold LR (1996) Numerical\nsolution of initial-value problems in differential-algebraic\nequations, Siam\n9. Kunkel P, Mehrmann V (2006) Differential-algebraic\nequations: analysis and numerical solution, EuropeanMathematical Society\n10 Bauchau OA (2010) Flexible multibody dynamics.\nSpringer, Newyork\n11. Nocedal J, Wright S (2006) Numerical optimization.\nSpringer, Newyork\n12. Lamour R (2005) Index determination and calculationof\nconsistent initial values for DAEs. Comput Math with Appl50:1125–1140\n13. Ge ´radin M, Cardona A (2001) Flexible multibody dynam-\nics: a finite element approach. Wiley, Hoboken\n14. Shabana AA (1997) Flexible multibody dynamics: review\nof past and recent developments. Multibody Syst Dyn\n1:189–222. https://doi.org/10.1023/A:1009773505418\n15. Natsiavas S, Paraskevopoulos E (2015) A set of ordinary\ndifferential equations of motion for constrained mechanicalsystems. Nonlinear Dyn 79:1911–1938\n16. Gonza ´lez F, Ko ¨vecses J (2013) Use of penalty formulations\nin dynamic simulation and analysis of redundantly con-strained multibody systems. Multibody Syst Dyn 29:57–76.\nhttps://doi.org/10.1007/s11044-012-9322-y17. Bayo E, De Jalon JG, Serna MA (1988) A modified\nLagrangian formulation for the dynamic analysis of con-\nstrained mechanical systems. Comput Methods Appl MechEng 71:183–195\n18. Bayo E, Ledesma R (1996) Augmented Lagrangian and\nmass-orthogonal projection methods for constrained multi-body dynamics. Nonlinear Dyn 9:113–130\n19. Cuadrado J, Cardenal J, Morer P, Bayo E (2000) Intelligent\nsimulation of multibody dynamics: space-state and\ndescriptor methods in sequential and parallel computingenvironments. Multibody Syst Dyn 4:55–73\n20. Zhang L, Zhang D (2016) A two-loop procedure based on\nimplicit Runge-Kutta method for index-3 dae of constrained\ndynamic problems. Nonlinear Dyn 85:263–280\n21 Garcı ´a de Jalo ´n J, Callejo A, Hidalgo AF (2011) Efficient\nsolution of Maggi’s equations. J Comput Nonlinear Dyn.\nhttps://doi.org/10.1115/1.4005238\n22. Von Schwerin R (2012) Multibody system simulation:\nnumerical methods, algorithms, and software. Springer,\nNewyork\n23. Arnold M, Cardona A, Bru ¨ls O (2015) Order reduction in\ntime integration caused by velocity projection. J Mech SciTechnol 29:2579–2585. https://doi.org/10.1007/s12206-\n015-0501-7\n24. Cuadrado J, Cardenal J, Bayo E (1997) Modeling and\nsolution methods for efficient real-time simulation ofmultibody dynamics. Multibody Syst Dyn 1:259–280\n25. Gear C (1971) Simultaneous numerical solution of differ-\nential-algebraic equations. IEEE Trans Circuit Theory18:89–95\n26. Higueras I, Ma ¨rz R, Tischendorf C (2003) Stability pre-\nserving integration of index-2 DAEs. Appl Numer Math45:201–229\n27. Ascher UM, Petzold LR (1993) Stability of computational\nmethods for constrained dynamics systems. SIAM J Sci\nComput 14:95–120\n28. Hanke M, Macana EI, Ma ¨rz R (1998) On asymptotics in\ncase of linear index-2 differential-algebraic equations.\nSIAM J Numer Anal 35:1326–1346\n29. Griepentrog E, Ma ¨rz R (1986) Differential-algebraic equa-\ntions and their numerical treatment, BSB Teubner\n30. Liu H, Song Y (2003) Stability of numerical methods for\nsolving linear index-3 DAEs. Appl Math Comput\n134:35–50\n31. Garcı ´a de Jalo ´n J, Unda J, Avello A, Jime ´nez JM (1987)\nDynamic analysis of three-dimensional mechanisms in\n‘‘natural’’ coordinates. J Mech Des Trans ASME109:460–465. https://doi.org/10.1115/1.3258818\n32. Hiller M, Kecskemethy A (1994) Dynamics of multibody\nsystems with minimal coordinates, in: Comput. Anal. Rigid\nFlex. Mech. Syst., Springer, pp. 61–100\n33. Angeli A, Desmet W, Naets F (2021) Deep learning of\nmultibody minimal coordinates for state and input estima-\ntion with Kalman filtering, Multibody Syst. Dyn. 1–19\n34. Urkullu G, de Bustos IF, Garcı ´a-Marina V, Uriarte H (2019)\nDirect integration of the equations of multibody dynamics\nusing central differences and linearization. Mech Mach\nTheory 133:432–458\n35. Gavrea B, Negrut D, Potra FA (2005) The Newmark inte-\ngration method for simulation of multibody systems: ana-\nlytical considerations, in: ASME 2005 Int. Mech. Eng.\n123398 Meccanica (2022) 57:371–399Congr. Expo., American Society of Mechanical Engineers,\npp. 1079–1092.\n36. Jay LO, Negrut D (2007) Extensions of the HHT- amethod\nto differential-algebraic equations in mechanics, Electron.\nTrans. Numer. Anal\n37. Dopico D, Lugris U, Gonzalez M, Cuadrado J (2005) IRK vs\nstructural integrators for real-time applications in MBS.J Mech Sci Technol. https://doi.org/10.1007/bf02916159\n38. Hairer E, Wanner G (1991) Solving ordinary differential\nequations II. Springer, Heidelberg\n39. Rheinboldt WC (1984) Differential-algebraic systems as\ndifferential equations on manifolds. Math Comput. https://\ndoi.org/10.2307/2008288\n40. Lindfield G, Penny J () Numerical methods: using\nMATLAB, Academic Press\n41. Cuadrado J, Dopico D, Naya MA, Gonzalez M (2004)\nPenalty, semi-recursive and hybrid methods for MBS real-\ntime dynamics in the context of structural integrators.Multibody Syst Dyn. https://doi.org/10.1023/B:MUBO.\n0000044421.04658.de42. De Jalon JG, Bayo E (2012) Kinematic and dynamic sim-\nulation of multibody systems: the real-time challenge.\nSpringer, Newyork\n43. Ma ¨rz R (2003) Differential algebraic systems with properly\nstated leading term and MNA equations, in: Model. Simu-\nlation, Optim. Integr. Circuits, Springer, pp. 135–151\n44. Coleman TF, Verma A (2001) A preconditioned conjugate\ngradient approach to linear equality constrained minimiza-\ntion. Comput Optim Appl 20:61–72\n45. Schiehlen W (1990) Multibody systems handbook. https://\ndoi.org/10.1007/978-3-642-50995-7\n46. Urkullu G, Ferna ´ndez-de-Bustos I, Olabarrieta A, Ansola R\n(2021) Estudio de la eficiencia del me ´todo de integracio ´n\ndirecta mediante diferencias centrales (DIMCD)., DYNA-Ingenierı ´a e Ind. 96\nPublisher’s Note Springer Nature remains neutral with\nregard to jurisdictional claims in published maps and\ninstitutional affiliations.\n123Meccanica (2022) 57:371–399 399"    },    {        "title": "2402.05782v1.Analysing_the_Sample_Complexity_of_Opponent_Shaping.pdf",        "content": "Analysing the Sample Complexity of Opponent Shaping\nKitty Fung∗\nUniversity of Oxford\nOxford, United Kingdom\nkittyfung01@gmail.comQizhen Zhang∗\nUniversity of Oxford\nOxford, United Kingdom\nqizhen.zhang@eng.ox.ac.ukChris Lu\nUniversity of Oxford\nOxford, United Kingdom\nchristopher.lu@exeter.ox.ac.uk\nJia Wan\nMassachusetts Institute of Technology\nCambridge, United States\njiawan@mit.eduTimon Willi\nUniversity of Oxford\nOxford, United Kingdom\ntimon.willi@eng.ox.ac.ukJakob Foerster\nUniversity of Oxford\nOxford, United Kingdom\njakob.foerster@eng.ox.ac.uk\nAbstract\nLearning in general-sum games often yields collectively sub-optimal\nresults. Addressing this, opponent shaping (OS) methods actively\nguide the learning processes of other agents, empirically leading\nto improved individual and group performances in many settings.\nEarly OS methods use higher-order derivatives to shape the learning\nof co-players, making them unsuitable to shape multiple learning\nsteps. Follow-up work, Model-free Opponent Shaping (M-FOS),\naddresses these by reframing the OS problem as a meta-game . In\ncontrast to early OS methods, there is little theoretical understand-\ning of the M-FOS framework. Providing theoretical guarantees for\nM-FOS is hard because A) there is little literature on theoretical\nsample complexity bounds for meta-reinforcement learning B) M-\nFOS operates in continuous state and action spaces, so theoretical\nanalysis is challenging. In this work, we present R-FOS, a tabular\nversion of M-FOS that is more suitable for theoretical analysis. R-\nFOS discretises the continuous meta-game MDP into a tabular MDP.\nWithin this discretised MDP, we adapt the 𝑅𝑚𝑎𝑥 algorithm, most\nprominently used to derive PAC-bounds for MDPs, as the meta-\nlearner in the R-FOS algorithm. We derive a sample complexity\nbound that is exponential in the cardinality of the inner state and\naction space and the number of agents. Our bound guarantees that,\nwith high probability, the final policy learned by an R-FOS agent\nis close to the optimal policy, apart from a constant factor. Finally,\nwe investigate how R-FOS’s sample complexity scales in the size of\nstate-action space. Our theoretical results on scaling are supported\nempirically in the Matching Pennies environment.\n∗Equal Contribution\nWork done at the Foerster Lab For AI Research (FLAIR), University of Oxford\nCorresponding Authors: kittyfung01@gmail.com, qizhen.zhang@eng.ox.ac.uk\nThis work is licensed under a Creative Commons Attribution\nInternational 4.0 License.\nProc.of the 23rd International Conference on Autonomous Agents and Multiagent Systems\n(AAMAS 2024), N. Alechina, V. Dignum, M. Dastani, J.S. Sichman (eds.), May 6 – 10, 2024,\nAuckland, New Zealand .©2024 International Foundation for Autonomous Agents and\nMultiagent Systems (www.ifaamas.org).Keywords\nOpponent Shaping; Multi-Agent; Reinforcement Learning; Meta\nReinforcement Learning; Sample Complexity\nACM Reference Format:\nKitty Fung, Qizhen Zhang, Chris Lu, Jia Wan, Timon Willi, and Jakob Fo-\nerster. 2024. Analysing the Sample Complexity of Opponent Shaping. In\nProc.of the 23rd International Conference on Autonomous Agents and Mul-\ntiagent Systems (AAMAS 2024), Auckland, New Zealand, May 6 – 10, 2024 ,\nIFAAMAS, 18 pages.\n1 Introduction\nLearning in general-sum games commonly leads to collectively\nworst-case outcomes [ 6]. To address this, opponent shaping (OS)\nmethods account for opponents’ learning steps and influence other\nagents’ learning processes. Empirically, this can improve individual\nand group performances.\nEarly OS methods [ 6,10,12] rely on higher-order derivatives, which\nare high-variance and result in unstable learning. They are also\nmyopic, focusing only on the opponent’s immediate future learning\nsteps rather than their long-term development [ 14]. Recent work,\nModel-free Opponent Shaping (M-FOS) [ 14], solves the above chal-\nlenges. M-FOS introduces a meta-game structure, each meta-step\nrepresenting an episode of the embedded “inner” game. The meta-\nstate consists of “inner” policies, and the meta-policy generates an\ninner policy at each meta-step . M-FOS uses model-free optimisation\ntechniques to train the meta-policy, eliminating the need for higher-\norder derivatives to accomplish long-horizon opponent shaping.\nThe M-FOS framework has shown promising long-term shaping\nresults in social-dilemma games [9, 14].\nThe original M-FOS paper presents two cases of the M-FOS algo-\nrithm. For simpler, low-dimensional games, M-FOS learns policy\nupdates directly by taking policies as input and outputting the next\npolicy as an action. Inputting and outputting entire policies does\nnot extend well to more complex, higher-dimensional games, e.g.\nwhen policies are represented as neural networks. The original\nM-FOS paper also proposes a variant which uses trajectories as\ninputs instead of the exact policy representations. In this work we\nderive the sample complexity for both cases.arXiv:2402.05782v1  [cs.LG]  8 Feb 2024Whereas some previous OS algorithms enjoy strong theoretical\nfoundations thanks to the Differentiable Games framework [ 1], the\nM-FOS framework has not been investigated theoretically. Under-\nstanding the sample complexity of an algorithm is helpful in many\nways, such as evaluating its efficiency or predicting the learning\ntime. However, providing theoretical guarantees for M-FOS is chal-\nlenging because A) there is very little literature on theoretical sam-\nple complexity bounds for even single-agent meta-reinforcement\nlearning (RL), let alone multi-agent and B) M-FOS operates in con-\ntinuous state and action space (the meta-game).\nIn this work, we present R-FOS, a tabular algorithm approximation\nof M-FOS. Unlike M-FOS, which operates in a continuous meta-\nMDP, R-FOS operates in a discrete approximation of the original\nmeta-MDP. The resulting discrete MDP allows us to perform rigor-\nous theoretical analysis. We adapt R-FOS from M-FOS such that it\nstill maintains all the key properties of M-FOS. Within this discrete,\napproximate MDP, R-FOS applies the 𝑅MAX algorithm [ 8,20] to the\nM-FOS meta-game. 𝑅MAX is a model-based reinforcement learning\n(MBRL) algorithm typically used for the sample complexity analysis\nof tabular MDPs. Using existing results developed for 𝑅MAX [3],\nwe derive an exponential sample complexity PAC-bound, which\nguarantees with high probability (1- 𝛿) that the optimal policy in the\ndiscretised meta-MDP is very close ( <𝜖away) to the policy learned\nby R-FOS. We then derive several bounds which guarantee policies\nbetween the original meta-MDP and the discretised meta-MDP are\nclose to each other up to a constant distance. Lastly, combining all\nof the previous bounds we derived, we obtain the final exponential\nsample complexity result.\nFor notational simplicity, we mostly omit the “meta” prefix in the\nrest of the paper. For example, the terms “MDP”, “transition func-\ntion”, and “policy”, refer to the meta-MDP, meta-transition function,\nand meta-policy respectively. We use the prefix “inner” whenever\nwe refer to the inner game. Furthermore, our analysis of M-FOS\nis limited to the asymmetric shaping case (i.e. the meta-game of\nshaping a naive inner-learner∗) and we leave the extension to meta-\nselfplay for future work.\nOur contributions are three-fold:\n(1)We present R-FOS (see Algorithm 1), a tabular approximation of\nM-FOS. Instead of learning a meta-policy inside the continuous\nmeta-MDP𝑀, R-FOS learns a meta-policy inside a discretised\nmeta-MDP which approximates 𝑀. Inside this discretised Meta-\nMDP, R-FOS uses 𝑅MAX as the meta-agent. Note that R-FOS\nstill maintains key properties of the original M-FOS algorithm,\nsuch as being able to exploit naive learners.\n(2)We present an exponential sample complexity bound for both\ncases described in M-FOS (See Theorems G.2 and G.3). Specif-\nically, we prove that, with high probability, the final R-FOS\npolicy is close to the optimal policy in the original meta-MDP\nup to a constant distance.\n(3)We implement R-FOS†and analyse the empirical sample com-\nplexity in the Matching Pennies environment. We establish links\n∗Naive learners are players who update their policy assuming other learning agents\nare simply a part of the environment.\n†The project code is available on https://github.com/FLAIROx/rfosbetween theory and experiments by demonstrating that in both\nrealms, sample complexity scales exponentially with the inner-\ngame’s state-action space size.\n2 Related Work\nTheoretical Analysis of Differentiable Games: Much past work\nassumes that the game being optimised is differentiable [ 1]. This\nassumption enables far easier theoretical analysis because one can\ndirectly use end-to-end gradient-based methods rather than rein-\nforcement learning in those settings. Several works in this area\ninvestigate the convergence properties of various algorithms Bal-\nduzzi et al. [1], Letcher [11], Schäfer and Anandkumar [17].\nOpponent Shaping: More closely related to our work are methods\nthat specifically analyse OS. SOS [ 12] and COLA [ 21] both analyse\nopponent-shaping methods that operate in the differentiable games\nframework. These works provide theoretical convergence analysis\nfor opponent-shaping algorithms; however, neither work analyzes\nsample complexity. POLA [ 23] theoretically analyses an OS method\nthat is invariant to policy parameterization. M-FOS does not operate\nin the differentiable games framework. While this enables M-FOS\nto scale to more challenging environments, such as Coin Game [ 14],\nit comes at the cost of convenient theoretical analysis. Khan et al .\n[9]empirically scales M-FOS to more challenging environments\nwith larger state spaces, while Lu et al . [15] empirically investigates\napplying M-FOS to a state-based adversary. To the best of our\nknowledge, our work is the first to theoretically analyse OS outside\nof the differentiable games framework. Furthermore, our work is\nthe first to analyse the sample complexity of an OS method.\nTheoretical Analysis of Sample Complexity in RL: There are\nseveral works that use the 𝑅MAX [3] framework to derive the sample\ncomplexity of RL algorithms across a variety of settings. Closely\nrelated to our work is Zhang et al . [22] , which uses the 𝑅MAX\nalgorithm to derive sample complexity bounds for learning in fully-\ncooperative multi-agent RL.\nOur work is also related to methods that analyse sample complexity\non continuous-space RL. Analyzing the sample complexity of algo-\nrithms in continuous-space RL is particularly challenging because\nthere are an infinite number of potential states. To address this,\nnumerous techniques have been suggested that each make specific\nassumptions: Liu and Brunskill [13] assumes a stationary asymp-\ntotic occupancy distribution under a random walk in the MDP.\nMalik et al . [16] uses an effective planning window to handle MDPs\nwith non-linear transitions. However, neither of these assumptions\napplies to M-FOS.\nInstead, this work focuses on discretising the continuous space\nand expresses the complexity bounds in terms of the discretisation\ngrid size. This is related to the concept of state aggregation [2,19],\nwhich groups states into clusters and treats the clusters as the\nstates of a new MDP. These previous works only formulated the\naggregation setting in MDPs and did not provide theoretical or\nempirical sample complexity proofs.\nFurthermore, prior studies on PAC-MDP did not empirically verify\nthe connection between the sample complexity and size of the state-\naction space. In this work, we empirically verify the relationshipbetween the sample complexity and the cardinality of the\ninner-state-action-space in the Matching Pennies game.\nAlgorithm 1 The R-FOS Algorithm\nMeta-game Inputs: Discretised meta-MDP ⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩\n𝑚-known, meta-game horizon ℎmeta\nInner-game Inputs: Inner game G =⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩\ninner-game horizon ℎinner\nInitialisation:∀ˆ𝑠𝑑∈ˆ𝑆𝑑,ˆ𝑎𝑑∈ˆ𝑆𝑑,ˆ𝑠′\n𝑑∈ˆ𝑆𝑑\nˆ𝑄(ˆ𝑠𝑑,ˆ𝑎𝑑)← 0, 𝑟(ˆ𝑠𝑑,ˆ𝑎𝑑)← 0, 𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)← 0, 𝑛(ˆ𝑠𝑑,ˆ𝑎,ˆ𝑠′)← 0\n1:formeta-episode =0,1,..do\n2: Reset environment\n3:formeta-time step =1,2,...,ℎ metado\n4: Choose ˆ𝑎𝑑:=argmaxˆ𝑎𝑑∈ˆ𝐴𝑑ˆ𝑄(ˆ𝑠𝑑,ˆ𝑎′\n𝑑)\n5: Roll-out𝐾inner games of length ℎinner using ˆ𝑎𝑑=𝜙𝑡\n6: Inner-game opponents each update their own inner-\npolicies naively\n7: Let𝑅be our agent’s 𝐾inner-games’ discounted return\n8: Letˆ𝑠′\n𝑑be the next meta-state after executing meta-action\nˆ𝑎𝑑from meta-state ˆ𝑠𝑑\n9: if𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)<𝑚then\n10:𝑟(ˆ𝑠𝑑,ˆ𝑎𝑑)←𝑟(ˆ𝑠𝑑,ˆ𝑎𝑡)+𝑅\n11:𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)←𝑛(ˆ𝑠𝑑,ˆ𝑎𝑡)+1\n12:𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑,ˆ𝑠′𝑑)←𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑,ˆ𝑠′\n𝑑)+1\n13: if𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)=𝑚then\n14: for𝑖=1,2,3,···,⌈𝑙𝑛(1\n𝜀(1−𝛾))\n1−𝛾⌉do\n15: for all(ˆ𝑠,ˆ𝑎)do\n16: if𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)≥𝑚then\n17: ˆ𝑄(ˆ𝑠𝑑,ˆ𝑎𝑑) ← ˆ𝑅𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑) +\n𝛾Í\n𝑠′\n𝑑ˆ𝑇𝑑(ˆ𝑠′|ˆ𝑠𝑑,ˆ𝑎𝑑)max ˆ𝑎′\n𝑑ˆ𝑄(ˆ𝑠′\n𝑑,ˆ𝑎′\n𝑑)\n18: ˆ𝑠←ˆ𝑠′\n3 Background\n3.1 Stochastic Game\nA stochastic game (SG)‡is given by a tuple 𝐺=⟨I,𝑆,𝑨,𝑇,𝑹,𝛾⟩.\nI={1,···,𝑛}is the set of agents, 𝑆is the state space, 𝑨is the\ncross-product of the action space for each agent such that the joint\naction space 𝑨=𝐴1×···×𝐴𝑛,𝑇:𝑆×𝑨↦→𝑆is the transition\nfunction, 𝑹is the cross-product of reward functions for all agents\nsuch that the joint reward space 𝑹=𝑅1×···×𝑅𝑛, and𝛾∈[0,1)\nis the discount factor.\nIn an SG, agents simultaneously choose an action according to\ntheir stochastic policy at each timestep 𝑡,𝑎𝑖\n𝑡∼𝜋𝑖\n𝜙𝑖(·|𝑠𝑖\n𝑡). The joint\naction at timestep 𝑡is𝒂𝒕={𝑎𝑖\n𝑡,𝒂−𝒊\n𝒕}, where the superscript −𝒊\nindicates all agents except agent 𝑖and𝜙𝑖is the policy parameter\nof agent𝑖. The agents then receive reward 𝑟𝑖\n𝑡=𝑅𝑖(𝑠𝑡,𝒂𝒕)and\nobserve the next state 𝑠𝑡+1∼𝑇(·|𝑠𝑡,𝒂𝒕), resulting in a trajectory\n𝜏𝑖=(𝑠0,𝒂0,𝑟𝑖\n0,...,𝑠𝑇,𝒂𝑻,𝑟𝑖\n𝑇), where𝑇is the episode length.\n‡We use the bold notation to indicate vectors over 𝑛agents.3.2 Markov Decision Process\nA Markov decision process (MDP) is a special case of stochastic\ngame and can be described as M=⟨𝑆,𝐴,𝑇,𝑅,𝛾⟩, where𝑆is the\nstate space, 𝐴is the action space, 𝑇\u0000𝑠𝑡+1|𝑠𝑡,𝑎𝑡\u0001is the transition\nfunction,𝑅(𝑠𝑡,𝑎𝑡)is the reward function, and 𝛾is the discount\nfactor. At each timestep 𝑡, the agent takes an action 𝑎𝑡∈𝐴from a\nstate𝑠𝑡∈𝑆and moves to a next state 𝑠𝑡+1∼𝑇\u0000·|𝑠𝑡,𝑎𝑡\u0001. Then, the\nagent receives a reward 𝑟𝑡=𝑅(𝑠𝑡,𝑎𝑡).\n3.3 Model-Free Opponent Shaping\nModel-free Opponent Shaping (M-FOS) [ 14] frames the OS problem\nas a meta-reinforcement-learning problem, in which the opponent\nshaper plays a meta-game. The meta-game is an MDP (sometimes\nwe also refer to it as meta-MDP ), in which the meta-agent controls\none of the inner agents in the inner game.\nThe inner game is the actual environment that our agents are play-\ning, which is an SG. The original M-FOS describes two cases for\nthe meta-state:\n(1)In the meta-game at timestep 𝑡, the M-FOS agent is at the meta-\nstate ˆ𝑠𝑡=[𝜙𝑖\n𝑡−1,𝝓−𝒊\n𝒕−1], which contains all inner-agents’ policy\nparameters for the underlying SG. In this work, we assu,e all\ninner-agents are parameterised by their Q-value table.\n(2)Alternatively, ˆ𝑠𝑡=𝝉in cases where past trajectories of the\ninner-game represent the policies sufficiently.\nWe provide theoretical sample complexity results for both of these\ntwo cases\nThe meta-agent takes a meta-action ˆ𝑎𝑡=𝜙𝑖\n𝑡∼𝜋𝜃(·|ˆ𝑠𝑡), which is the\nM-FOS’ inner agent’s policy parameters. The action is chosen from\nthe meta-policy 𝜋parameterized by parameter 𝜃. In this work, we\nonly look at the case where the meta-policy is a Q-value function\ntable, and is denoted as ˆ𝑄instead. The M-FOS agent receives reward\n𝑟𝑡=Í𝐾\n𝑘=0𝑟𝑖\n𝑘(𝜙𝑖\n𝑡,𝝓−𝒊\n𝒕), where𝐾is the number of inner episodes.\nA new meta-state is sampled from a stochastic transition function\nˆ𝑠𝑡+1∼𝑇(·|ˆ𝑠𝑡,ˆ𝑎𝑡).\nNote that the original paper introduces two different algorithms:\nThe first meta-trains M-FOS against naive learners commonly re-\nsulting in exploiting them. The second instead considers meta-self-\nplay, whereby two M-FOS agents are trained to shape each other,\nresulting in reciprocity. In this work we only consider the first,\nasymmetric case.\n3.4 The𝑅MAXAlgorithm\n𝑅MAX [3] is an MBRL algorithm proposed for analysing the sample\ncomplexity for tabular MDPs. Given any MDP 𝑀,𝑅MAX constructs\nanempirical MDP ˆ𝑀that approximates 𝑀. The approximation is\ndone by estimating the reward function 𝑅and transition 𝑇using\nempirical samples. The resulting approximate reward and transition\nmodels are denoted by ˆ𝑅and ˆ𝑇respectively.\n𝑅MAX encourages exploration by dividing the state-action pairs into\ntwo groups - those that have been visited at least 𝑚-times, and those\nthat haven’t. The set of state-action pairs that have been visited at\nleast𝑚-times is called the “ 𝑚-known set”. Using the empirical MDPˆ𝑀, the𝑅MAX algorithm constructs an 𝑚-known empirical MDP . This\n𝑚-known empirical MDP behaves almost exactly as the empirical\nMDP, except when the agent is at a state-action pair outside the\n𝑚-known set. When the agent is outside the 𝑚-known set, the\ntransition function is self-absorbing (i.e. the transition function\nonly transitions back to the current state) and the reward function\nis the maximum (See Table 3 in the appendix). The consequence\nof the𝑚-known setup is the agent is encouraged to explore state-\naction pairs that have high uncertainty (i.e. that has been visited\nunder𝑚times). Specifically, the value function for the under-visited\nstates is the maximum possible expected return, which gives the\nalgorithm its name. This is in line with optimism in the face of\nuncertainty .\n3.5𝜀-Nets\nDefinition 3.1. (𝜀-Net [ 5,7]) For𝜀>0,N𝜀is an𝜀-net over the set\nΘ⊆R𝐷if for all𝜃∈Θ, there exists 𝜃′∈N𝜀such that\r\r𝜃−𝜃′\r\r2≤𝜀.\nTo discretise a 𝐷-dimensional sphere of radius 𝑅, we can use a\n𝜀-net containing 𝐷-dimensional cubes of sides 𝜆. This results in\u0010\n2𝑅\n𝜆+1\u0011𝐷\npoints. Within each 𝐷-dimensional cube, the largest\ndistance between the vertices and the interior points comes from\nthe center of the cube, which is𝜆√\n𝑑\n2. Therefore, to guarantee a full\ncover of all the points in the sphere, the largest cube size that we\ncan have should satisfy 𝜀=𝜆√\n𝑑\n2.From here on, we will replace the\n𝜀in𝜀-net with𝛼to avoid notation overloading.\n4 Sample Complexity Analysis with 𝑅MAXas\nMeta-Agent\nAs introduced in Section 3, 𝑅MAX [3] is a MBRL algorithm for\nlearning in tabular MDPs. We adapt the original M-FOS algorithm\nto use𝑅MAX as the meta-agent (see Algorithm 1) and refer to this\nadapted algorithm as R-FOS from here on. We use a tabular Q-\nlearner as the naive learner for all inner-game opponents. While\nthe original M-FOS paper uses PPO [18], we choose the Q-learner\nfor the ease of sample complexity analysis.\nWe provide theoretical results for the two cases of M-FOS’ meta-\nagent proposed by the original paper [ 14].Case I uses all agents’\ninner policy parameters from the previous timestep as the meta-\nstate. Case II instead uses the most recent inner-game trajectories\nas the meta-state. In both cases, the meta-action determines the\ninner agent’s policy parameters for the next inner episode.\nAt a high level, we first discretise the meta-MDP, which allows us\nto use the theoretical bounds from 𝑅MAX (only suitable for tabular\nMDPs), then we develop theory for bounding the discrepancy be-\ntween the continuous and discrete meta-MDP, and lastly, we use\nall of this to bound the final discrepancy. Specifically, the sample\ncomplexity analysis consists of six steps§:\n(1)To use𝑅MAX as the M-FOS meta-agent, we first discretise the\ncontinuous meta-MDP 𝑀=⟨ˆ𝑆,ˆ𝐴,𝑇,𝑅,𝛾⟩into a discretised\n§see detailed proof in the appendixmeta-MDP𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩. We first discretise the con-\ntinuous meta-state space and meta-action space using epsilon-\nnets [ 5,7]. Based on this discretised meta-state and meta-action\nspace, we define the discretised transition and reward function.\nSee Section 4.2 for details.\n(2)We then construct a 𝑚-known discretised MDP 𝑀𝑚,𝑑, as de-\nscribed by the R-MAX algorithm [ 20]. See Section 4.3 for details.\n(3)Then, we deploy the 𝑅MAX algorithm in 𝑀𝑚,𝑑.𝑅MAX both\nestimates the empirical 𝑚-known discretised MDP, ˆ𝑀𝑚,𝑑, using\na maximum likelihood estimate from empirical samples and\nlearns an optimal policy in ˆ𝑀𝑚,𝑑. For example, to estimate the\nmeta-reward, our algorithm, R-FOS evaluates the inner-game\npolicy outputted by the meta-policy using episodic rollouts. The\nestimates are then used to update the meta-policy according\nto the R-FOS algorithm. Our R-FOS algorithm optimistically\nassigns rewards for all under-visited discretised (meta-state,\nmeta-action) pairs to encourage exploration (like 𝑅MAX). See\nSection 4.4 for details.\n(4)We next prove a PAC-bound which guarantees with large prob-\nability, that the optimal policy learnt in ˆ𝑀𝑚,𝑑is similar to the\noptimal policy in 𝑀,𝑑. This step uses results from [ 20]. See\nSection 4.5 for details.\n(5)We also prove a strict bound that guarantees the optimal policies\nlearnt in𝑀,𝑑 and𝑀are similar up to a constant. This step uses\nresults from [4]. See Section 4.8 for details.\n(6)Using the two bounds from above, we prove the final sample\ncomplexity guarantee which quantifies that, with large proba-\nbility, the optimal policy learnt in ˆ𝑀𝑚,𝑑is similar to the optimal\npolicy in𝑀up to a constant. See Section 4.9 for details.\n4.1 Assumptions\nWe first outline all assumptions made in deriving the sample com-\nplexity of the R-FOS algorithm.\nAssumption 4.1. Both meta-game and inner-game are finite hori-\nzon. We use ℎmeta to denote the meta-game horizon, and ℎinner to\ndenote the inner-game horizon.\nAssumption 4.2. We assume the inner-game reward is bounded.\nFor simplicity of the proof and without loss of generality, we set\nthis bound as1\nℎinner, whereℎinner is the horizon of the inner game.\nFormally, for all(𝑠,𝑎),0≤𝑅inner(𝑠,𝑎)≤1\nℎinner. This allows us to\nintroduce the notion of maximum inner reward andmaximum inner\nvalue function as𝑅max,inner=1\nℎinnerand𝑉max,inner=1respectively.\nThis implies that the reward and value function in the meta-game\nare also bounded, i.e., 𝑅max=1and𝑉max=1\n1−𝛾(the latter being\nan upper bound).\nAssumption 4.3. The meta-game uses a discount factor of 𝛾. For\nsimplicity of the proof, the inner-game uses a discount factor of 1.\nThis assumption can be easily deleted by adapting 𝑅max, inner (see\nabove) in the original proof in [20].Assumption 4.4. For simplicity, the inner game is assumed to be\ndiscrete.\nAssumption 4.5. The meta-reward function is Lipschitz-continuous:\nFor all ˆ𝑠1,ˆ𝑠2∈ˆ𝑆andˆ𝑎1,ˆ𝑎2∈ˆ𝐴,\f\f𝑅(ˆ𝑠1,ˆ𝑎1)−𝑅(ˆ𝑠2,ˆ𝑎2)\f\f≤L𝑅\r\r(ˆ𝑠1,ˆ𝑎1)−( ˆ𝑠2,ˆ𝑎2)\r\r∞\nwhereL𝑅is the meta-reward function’s Lipschitz-constant.\nAssumption 4.6. The meta-transition function is Lipschitz-continuous:\nFor all ˆ𝑠1,ˆ𝑠′\n1,ˆ𝑠2∈ˆ𝑆andˆ𝑎1,ˆ𝑎′\n1,ˆ𝑎2∈ˆ𝐴,\n\f\f𝑇(ˆ𝑠′\n1|ˆ𝑠1,ˆ𝑎1)−𝑇(ˆ𝑠′\n2|ˆ𝑠2,ˆ𝑎2)\f\f≤L𝑇\r\r(ˆ𝑠′\n1,ˆ𝑠1,ˆ𝑎1)−( ˆ𝑠′\n2,ˆ𝑠2,ˆ𝑎2)\r\r∞\nwhereL𝑇is the meta-transition function’s Lipschitz-constant.\nAssumption 4.7. There’s a Lipschitz-continuous point-to-set map-\nping between meta-state space and meta-action space such that for\nany ˆ𝑠,ˆ𝑠′∈ˆ𝑆and ˆ𝑎,ˆ𝑎′∈ˆ𝐴, there exists some ˆ𝑎∈𝑈(ˆ𝑠)such that\r\rˆ𝑎−ˆ𝑎′\r\r∞<L\r\rˆ𝑠−ˆ𝑠′\r\r∞\nAssumption 4.8. The meta-game transition function 𝑇(·|ˆ𝑠,ˆ𝑎)\nis a probability density function such that 0≤𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎) ≤\nLand∫\nˆ𝑆𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎)𝑑ˆ𝑠′=1,∀ˆ𝑠,ˆ𝑠′∈ˆ𝑆andˆ𝑎∈ˆ𝐴\nThe first four assumptions are required to be able to use the R-MAX\nalgorithm, while the latter assumptions are needed for bounding the\ndiscrepancy between the continuous meta-MDP and the discretised\nmeta-MDP.\n4.2 Step 1: Discretising the Meta-MDP\nTo use𝑅MAX as the M-FOS meta-agent, we discretise the continuous\nmeta-MDP 𝑀=⟨ˆ𝑆,ˆ𝐴,𝑇,𝑅,𝛾⟩into a discretised meta-MDP 𝑀𝑑=\n⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩. We discretise the continuous state and action\nspace using 𝜀-nets with spacing 𝛼.\n4.2.1 Discretising the State and Action Space: Case I\nInCase I , the meta-state ˆ𝑠𝑡is all inner agents’ policies parame-\nters from the previous timestep. Each of the inner agent 𝑖’s policy\nis a Q-table, denoted as 𝜙𝑖∈R|𝑆|×|𝐴|. Formally, ˆ𝑠𝑡:=𝝓𝑡−1=\n[𝜙𝑖\n𝑡−1,𝝓−𝒊\n𝒕−1].The meta-action ˆ𝑎𝑡is the inner agent’s current policy\nparameters 𝜙𝑖\n𝑡.\nFor the meta-action space ˆ𝐴and a chosen discretisation error 𝛼>0,\nwe obtain the 𝜀-net ˆ𝐴𝑑⊂ˆ𝐴such that for all ˆ𝑎∈ˆ𝐴, there exist\nˆ𝑎𝑑∈ˆ𝐴𝑑where\r\rˆ𝑎−ˆ𝑎𝑑\r\r≤𝛼. (1)\nDividing the space with grid size 𝜆results in the size of discretised\nmeta-action space upper bounded by\n|ˆ𝐴𝑑|≤ \n2√︁\n|𝑆||𝐴|\n𝜆+1!|𝑆||𝐴|\n. (2)\nSimilarly, the size of the discretised meta-state space is upper\nbounded by\n|ˆ𝑆𝑑|≤ \n2√︁\n𝑛|𝑆||𝐴|\n𝜆+1!𝑛|𝑆||𝐴|\n. (3)4.2.2 Discretising the State and Action Space: Case II\nIn Case II, the meta-state ˆ𝑠𝑡is all inner agents’ past trajectories.\nFormally, ˆ𝑠𝑡:=𝝉𝑡., where𝜏𝑖\n𝑡={𝑠0,𝑎0,𝑠1,𝑎1,...,𝑠𝑡,𝑎𝑡}. Because\nwe assume the Inner-Game is discrete (i.e. the state and action\nspace are both discrete), the meta-state in this case does not need\ndiscretisation. Let ℎbe the maximum length of the past trajectories\ncombined, i.e. ℎ=ℎinner·ℎmeta. The size of the meta-state space is\n|ˆ𝑆𝑡|=(|𝑆||𝐴|)𝑛ℎ. (4)\nThe meta-action remains the same as Case I.\n4.2.3 Discretising the Transition and Reward Function\nUnder the above discretisation procedure, we define the discretised\nMDP𝑀𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾), where the state space remains contin-\nuous and the action space is restricted to discretised actions. We\ndefine the transition function and reward function for 𝑀𝑑as:\n𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)=𝑇(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑)\n∫\nˆ𝑆𝑇(ˆ𝑧𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑)𝑑ˆ𝑧(5)\nIntuitively,𝑇𝑑(ˆ𝑠′|ˆ𝑠𝑑,ˆ𝑎𝑑)is a normalized sample of 𝑇𝑑(·|·,ˆ𝑎𝑑)at\nˆ𝑠′,ˆ𝑠𝑑, and the transition probability 𝑇𝑑(·|ˆ𝑠,ˆ𝑎𝑑)takes a constant\nvalue within each grid in the state space. This means that instead\nof treating the transition function as a discretised distribution of all\npossible values of ˆ𝑆𝑑, we treat it as a continuous distribution over\nthe original continuous state space, but normalize each grid from\nthe𝜀-net into a step function.\n𝑅𝑑(ˆ𝑠,ˆ𝑎𝑑)=𝑅(ˆ𝑠𝑑,ˆ𝑎𝑑) (6)\nSimilarly, the reward function is continuous over the state space,\nbut normalized each grid from the 𝜀-net into a step function.\n4.3 Step 2: The 𝑚-known Discretised MDP\nIn the previous step, we converted the meta-MDP 𝑀into a discre-\ntised meta-MDP𝑀𝑑. From𝑀𝑑, R-FOS builds an 𝑚-known discre-\ntised MDP𝑀𝑚,𝑑(see Table 3 in the appendix).\nDefinition 4.9 (m-Known MDP) .Let𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩be\nan MDP. We define 𝑀𝑚,𝑑to be the𝑚-known MDP. As is standard\npractice,𝑚-known refers to the set of state-action pairs that have\nbeen visited at least 𝑚times. For all state-action pairs in 𝑚-known,\nthe induced MDP 𝑀𝑚,𝑑behaves identical to 𝑀𝑑. For state-action\npairs outside of 𝑚-known, the state-action pairs are self-absorbing\n(i.e. only self-transitions) and maximally rewarding with 𝑅𝑀𝐴𝑋 .\n4.4 Step 3: The Empirical Discretised MDP\nFrom the𝑚-known discretised MDP 𝑀𝑚,𝑑, we then learn an empir-\nical𝑚-known discretised MDP 𝑀𝑚,𝑑by calculating the maximum\nlikelihood from empirical samples (see Table 3 in the appendix). As\nshown in Algorithm 1, R-FOS learns an optimal policy within this\nempirical𝑚-known discretised MDP.\nDefinition 4.10 (Empirical m-Known discretised MDP) .𝑀𝑚,𝑑is\nthe expected version of ˆ𝑀𝑚,𝑑where:𝑇𝑚,𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑):=(\n𝑇𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑)if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n1[ˆ𝑠′\n𝑑=ˆ𝑠𝑑], otherwise\nˆ𝑇𝑚,𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑):= \n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑,ˆ𝑠′\n𝑑)\n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n1[ˆ𝑠′\n𝑑=ˆ𝑠𝑑],otherwise\n𝑅𝑚,𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑):=(\n𝑅𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n𝑅max otherwise\nˆ𝑅𝑚,𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑)= \nÍ𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)\n𝑖𝑟(ˆ𝑠𝑑,ˆ𝑎𝑑)\n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n𝑅max, otherwise\n(7)\n4.5 Step 4: The Bound Between 𝑀𝑑and ˆ𝑀𝑚,𝑑\nWe first prove the PAC bound which guarantees that, with high\nprobability, the optimal policies learnt in the discretised MDP 𝑀𝑑\nand empirical 𝑚−𝑘𝑛𝑜𝑤𝑛 discretised MDP are very close. We prove\nthe bound using results from [20].\nTheorem 4.11. (𝑅MAX MDP Bound [ 20]) Suppose that 0≤𝜀<1\n1−𝛾\nand0≤𝛿<1are two real numbers and 𝑀=⟨𝑆,𝐴,𝑇,𝑅,𝛾⟩is any\nMDP. There exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\n=\n𝑂\u0012\n(𝑆+ln(𝑆𝐴/𝛿))𝑉2\nmax\n𝜀2(1−𝛾)2\u0013\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅MAX is executed\non𝑀with inputs𝑚and𝜀1, the following holds. Let 𝐴𝑡denote𝑅MAX’s\npolicy at time 𝑡and𝑠𝑡denote the state at time 𝑡. With probability at\nleast 1−𝛿,𝑉𝐴𝑡\n𝑀(𝑠𝑡)≥𝑉∗\n𝑀(𝑠𝑡)−𝜀is true for all but\n˜𝑂\u0012\n𝑆2𝐴/\u0010\n𝜀3(1−𝛾)6\u0011\u0013\ntimesteps (final sample complexity bound).\n4.6 Case I\nDirectly plugging in Equations 10and9into Theorem G.1, we\nobtain the following PAC-bound.\nTheorem 4.12. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are\ntwo real numbers. Let M be any continuous meta-MDP with in-\nner stochastic game 𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=\n⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case\nI) using grid size of 𝜆. There exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1,\nsatisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u0013𝑛|𝑆||𝐴|\n𝜀2(1−𝛾)4ª®®®®®\n¬\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs\n𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy\nat time𝑡andˆ𝑠𝑡denote the state at time 𝑡. With probability at least1−𝛿,𝑉∗\n𝑀𝑑(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀𝑑(ˆ𝑠𝑡)≤𝜀is true for all but\n˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u00132𝑛|𝑆||𝐴|\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.\n4.7 Case II\nDirectly plugging in Equations 11and9into Theorem G.1, we\nobtain the folling PAC-bound.\nTheorem 4.13. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are\ntwo real numbers. Let M be any continuous meta-MDP with in-\nner stochastic game 𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=\n⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case\nI) using grid size of 𝜆. There exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1,\nsatisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂 \n(|𝑆||𝐴|)𝑛ℎ\n𝜀2(1−𝛾)4!\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs\n𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy\nat time𝑡andˆ𝑠𝑡denote the state at time 𝑡. With probability at least\n1−𝛿,𝑉∗\n𝑀𝑑(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀𝑑(ˆ𝑠𝑡)≤𝜀is true for all but\n˜𝑂©\n«(|𝑆||𝐴|)𝑛ℎ\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.\n4.8 Step 5: The Bound between 𝑀and𝑀𝑑\nNext, we give a guarantee that the optimal policies learnt in the\noriginal meta-MDP 𝑀and the discretised MDP 𝑀𝑑are similar\nenough with a distance up to a constant factor. Using the results\nfrom [4], we obtain the following property.\nTheorem 4.14. (MDP Discretization Bound [ 7]) There exists a con-\nstantK(thats depends only on the Lipschitz constant L) such that\nfor some discretisation coarseness 𝜆∈(0,L\n2]\n\r\r\r𝑉∗\n𝑀−𝑉∗\n𝑀𝑑\r\r\r∞≤K𝜆\n(1−ˆ𝛾)2.\n4.9 Step 6: Adding it together\nTo combine the bounds we obtained in Step 4 and 5, we need\nan additional bound that bounds the policy value between the\ncontinuous and discretised MDP.\nLemma 4.15 (Simulation Lemma for Continuous MDPs) .Let𝑀\nand ˆ𝑀be two MDPs that only differ in (𝑇,𝑅)and(ˆ𝑇,ˆ𝑅).\nLet𝜖𝑅≥max𝑠,𝑎|ˆ𝑅(𝑠,𝑎)−𝑅(𝑠,𝑎)|and𝜀𝑝≥max𝑠,𝑎∥ˆ𝑇(·|𝑠,𝑎)−𝑇(·|\n𝑠,𝑎)||1. Then∀𝜋:ˆS→𝑎,\r\r\r𝑉𝜋\n𝑀−𝑉𝜋\nˆ𝑀\r\r\r∞≤𝜀𝑅\n1−𝛾+𝛾𝜖𝑃𝑉max\n2(1−𝛾).\nUnder discretisation, [ 4] showed that, with a small enough grid\nsize, and restricting to the discretised action space, the difference\nin transition probability of the continuous MDP 𝑀and discretised\nMDP𝑀𝑑is upper bounded by a constant.\nLemma 4.16. [4] There exists a constant 𝐾𝑃(depending only on\nconstantL) such that\n|𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)−𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)|≤𝐾𝑝𝛼\nfor all ˆ𝑠′,ˆ𝑠∈ˆ𝑆,ˆ𝑎𝑑∈ˆ𝐴𝑑and all𝛼≤(0,1\n2L]\nWe now apply the Lemma I.1 to bound the difference in value for\nany discretised policy (i.e. restricting action space to ˆ𝐴𝑑) in the\ncontinuous MDP 𝑀and discretised MDP 𝑀𝑑.\nLemma 4.17. Let𝑀ˆ𝐴𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇,𝑅,𝛾)be the continuous MDP 𝑀\nrestricted to the discretised action space. Recall the discretised MDP\n𝑀𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾). Then for any discretised policy 𝜋:ˆ𝑆→ˆ𝐴𝑑,\n∥𝑉𝜋\n𝑀ˆ𝐴𝑑−𝑉𝜋\n𝑀𝑑∥∞=∥𝑉𝜋\n𝑀−𝑉𝜋\n𝑀𝑑∥∞≤LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n(1−𝛾)2\nNote that, restricted to discretised policies 𝜋which only picks\nactions in ˆ𝐴𝑑, the value of 𝜋in the original MDP 𝑀,𝑉𝜋\n𝑀, equals to\nits value the same MDP restricted to discretised action space, 𝑉𝜋\n𝑀ˆ𝐴𝑑.\n4.10 Case I\nSumming up the bounds in Lemma I.3, Theorems G.2 and H.1, we\nobtain the final bound for Case I. The final bound guarantees, with\nhigh probability, that the policy we obtain from R-FOS is close to\nthe optimal policy in 𝑀apart from a constant factor.\nTheorem 4.18. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are\ntwo real numbers. Let M be any continuous meta-MDP with in-\nner stochastic game 𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=\n⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case\nI) using grid size of 𝜆. There exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1,\nsatisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u0013𝑛|𝑆||𝐴|\n𝜀2(1−𝛾)4ª®®®®®\n¬\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs\n𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy\nat time𝑡andˆ𝑠𝑡denote the state at time 𝑡. With probability at least\n1−𝛿,\n𝑉∗\n𝑀(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀(ˆ𝑠𝑡)≤𝜀+K𝜆\n(1−ˆ𝛾)2+LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n(1−𝛾)2is true for all but\n˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u00132𝑛|𝑆||𝐴|\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps. I.e. the above is the final sample complexity.\n4.11 Case II\nSimilarly, summing up the bounds in Lemma I.3, Theorems G.3 and\nH.1, we obtain the final bound for Case II. In Section 5, we also\nshow empirically that the number of samples needed indeed scales\nby a factor of|𝑆||𝐴|2𝑛ℎ, as seen in Theorem 4.19.\nTheorem 4.19. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are\ntwo real numbers. Let M be any continuous meta-MDP with in-\nner stochastic game 𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=\n⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case\nI) using grid size of 𝜆. There exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1,\nsatisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂 \n(|𝑆||𝐴|)𝑛ℎ\n𝜀2(1−𝛾)4!\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs\n𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy\nat time𝑡andˆ𝑠𝑡denote the state at time 𝑡. With probability at least\n1−𝛿,\n𝑉∗\n𝑀(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀(ˆ𝑠𝑡)≤𝜀+K𝜆\n(1−ˆ𝛾)2+LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n2(1−𝛾)2\nis true for all but\n˜𝑂©\n«(|𝑆||𝐴|)2𝑛ℎ\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬timesteps\n5 Experiments\nWe now validate our theoretical findings empirically.\n5.1 The Matching Pennies Environment\nMatching Pennies is a two-player, zero-sum game with a payoff\nmatrix shown in Section 5.1. Each agent either pick Heads (H) or\nTails (T),𝑎𝑖∈{𝐻,𝑇}and𝑎𝑖∼𝜋𝜙𝑖(·|{}) , where𝜙𝑖correspond to\nthe probability of player 𝑖picking H. Note that in this work, the\ngame is not iterated. This means that an inner-episode has a length\nof 1 and the inner-episodic return corresponds to the payoff after\none interaction 𝑟=Payoff Table(𝑎1,𝑎2). For R-FOS, this means that\na meta-step corresponds to one iteration of the Matching Pennies\ngame. The meta-return corresponds to the discounted, cumulative\nmeta-reward after playing the Matching Pennies 𝐾times. While the\noriginal M-FOS was evaluated on a more complex, iterated version\nof the Matching Pennies game, this simple setting with a binary\naction space is sufficient for our empirical validation. Our setting is\nalso more practical for implementation because the 𝑅MAX algorithmmemory usage grows exponentially with the size of the state and\naction space. Thus, for any of the more complex environments from\nthe M-FOS paper we were not able do any empirical analysis of R-\nFOS at all, due to the exponential sample and memory requirements.\nPlayer 1\\Player 2 Head Tail\nHead (+1, -1) (-1, +1)\nTail (-1, +1) (+1, -1)\nTable 1: Payoff Matrix for MP\n5.2 Experiment Setup\nWe implement an empirical version of our R-FOS algorithm. Be-\ncause the R-FOS algorithm uses Q-value iteration to solve the meta-\ngame, the algorithm needs to keep a copy of the meta-Q-value\ntable. Therefore, memory usage grows exponentially with respect\nto the inner-game’s state-action space size. We found that Case I of\nthe algorithm was intractable to implement even with a compact\nenvironment like MP. The meta Q-table of size |ˆ𝑆|×| ˆ𝐴|was sim-\nply too large to fit in memory. Therefore, we focus on empirically\nvalidating a simplified case of Case II. We make two simplifications,\n(1)The meta-state uses a partial history of past actions. Only the\nmost recent ℎactions are used, where ℎis a hyper-parameter\nwe pick. The window size allows us to control the size of the\nmeta-game state, i.e., ˆ𝑆∈R2ℎ. Because the MP game only has\none state, it is not necessary to include the state.\n(2)To further decrease the problem size for tractability, we define\nthe meta-agent action to be the inner-agent’s greedy action,\ninstead of the Q-table. This results in a much small meta-action\nsize of|ˆ𝐴|=2\n6 Results and Discussion\nWe draw connections between our sample complexity theory results\nand experimental results in the MP environment. Our goal is to\nanalyse the scaling law of R-FOS. Specifically, we investigate how\nthe sample complexity changes when we vary the window-size ℎ.\nUnder the MP environment settings, the inner-game state-action\nspace size is|𝑆||𝐴|=2and the number of players is 𝑛=2. Following\nthe bound in Theorem 4.19, we see that the only term that depends\non h is the 16ℎterm:\n˜𝑂©\n«(|𝑆||𝐴|)2𝑛ℎ\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬∼˜𝑂©\n«16ℎ\u0012\n2√\n2\n𝜆+1\u00132\n𝜀3(1−𝛾)6ª®®®®®\n¬\nHence, our theory results says that whenever the game horizon is\nincreased by 1, we expect to see the sample complexity to increase\nby a factor of 16 in the MP environment. Figure 1 shows the reward\nacross the meta episodes on a log 16scale. The graph contains\nthree reward curves for meta-trajectory length ℎ=2,3,4, which\nconverges approximately at 163,164,𝑎𝑛𝑑 165episodes. Indeed, this\nis consistent with our theoretical results in Theorem G.3.\nFigure 1: Empirical sample complexity while varying the tra-\njectory window ℎ. We plot the meta-reward per meta-episode.\nTo better visualise the connection with the theory results,\nwe plot the x-axis in log16scale. The reported results are the\nmean over 3 seeds with standard error.\n7 Conclusion\nWe presented three main contributions in our work. First of all, we\npresented R-FOS, a tabular algorithm adapted from M-FOS. Unlike\nM-FOS, which learns a policy in a continuous meta-MDP, R-FOS\ninstead learns a policy in a discrete approximation of the origi-\nnal meta-MDP which allows us to more easily perform theoretical\nanalysis. Within this discretised meta-MDP, R-FOS uses the 𝑅MAX\nalgorithm as the meta-agent. We adapted R-FOS from M-FOS such\nthat it still maintains all key attributes of M-FOS. Second of all, we\nderived an exponential sample complexity bound for both cases\ndescribed in M-FOS (the two cases being either inner-game policies\nor inner-game trajectory history as meta-state). Specifically, we\nproved that with high probability, the policy learnt by R-FOS is\nclose to the optimal policy from the original meta-MDP up to a\nconstant distance. Finally, we implemented R-FOS and investigated\nthe empirical sample complexity in the Matching Pennies environ-\nment. We draw connections between theory and experiments by\nshowing both results scales exponentially according to the size of\nthe inner-game’s state-action-space.\nAcknowledgments\nKF was supported by the D. H. Chen Foundation Scholarship.\nQZ is supported by Armasuisse and Cohere.References\n[1]David Balduzzi, Sebastien Racaniere, James Martens, Jakob Foerster,\nKarl Tuyls, and Thore Graepel. 2018. The mechanics of n-player\ndifferentiable games. In International Conference on Machine Learning .\nPMLR, 354–363.\n[2]Craig Boutilier, Thomas Dean, and Steve Hanks. 1999. Decision-\nTheoretic Planning: Structural Assumptions and Computational Lever-\nage. J. Artif. Int. Res. 11, 1 (jul 1999), 1–94.\n[3]Ronen Brafman and Moshe Tennenholtz. 2001. R-MAX - A General\nPolynomial Time Algorithm for Near-Optimal Reinforcement Learning.\nThe Journal of Machine Learning Research 3, 953–958. https://doi.org/\n10.1162/153244303765208377\n[4]CHEE-S Chow and John N Tsitsiklis. 1991. An optimal one-way multi-\ngrid algorithm for discrete-time stochastic control. IEEE transactions\non automatic control 36, 8 (1991), 898–914.\n[5]Murat A. Erdogdu. 2022. Covering with epsilon-nets. https://erdogdu.\ngithub.io/csc2532/lectures/lecture05.pdf\n[6]Jakob N. Foerster, Richard Y. Chen, Maruan Al-Shedivat, Shimon\nWhiteson, Pieter Abbeel, and Igor Mordatch. 2018. Learning with\nOpponent-Learning Awareness. arXiv:1709.04326 [cs.AI]\n[7]D Haussler and E Welzl. 1986. Epsilon-Nets and Simplex Range Queries.\nInProceedings of the Second Annual Symposium on Computational\nGeometry (Yorktown Heights, New York, USA) (SCG ’86) . Association\nfor Computing Machinery, New York, NY, USA, 61–71. https://doi.\norg/10.1145/10515.10522\n[8]Sham Kakade. 2003. On the sample complexity of Reinforcement Learn-\ning. Ph.D. Dissertation. University of London.\n[9]Akbir Khan, Newton Kwan, Timon Willi, Chris Lu, Andrea Tacchetti,\nand Jakob Nicolaus Foerster. [n.d.]. Context and History Aware Other-\nShaping. ([n. d.]).\n[10] Dong-Ki Kim, Miao Liu, Matthew D Riemer, Chuangchuang Sun,\nMarwa Abdulhai, Golnaz Habibi, Sebastian Lopez-Cot, Gerald Tesauro,\nand JONATHAN P HOW. 2021. A Policy Gradient Algorithm for\nLearning to Learn in Multiagent Reinforcement Learning. https:\n//openreview.net/forum?id=zdrls6LIX4W\n[11] Alistair Letcher. 2020. On the impossibility of global convergence in\nmulti-loss optimization. arXiv preprint arXiv:2005.12649 (2020).\n[12] Alistair Letcher, Jakob Foerster, David Balduzzi, Tim Rocktäschel, and\nShimon Whiteson. 2018. Stable opponent shaping in differentiable\ngames. arXiv preprint arXiv:1811.08469 (2018).\n[13] Yao Liu and Emma Brunskill. 2018. When Simple Exploration is Sample\nEfficient: Identifying Sufficient Conditions for Random Exploration to\nYield PAC RL Algorithms.\n[14] Chris Lu, Timon Willi, Christian A. Schroeder de Witt, and Jakob N.\nFoerster. 2022. Model-Free Opponent Shaping. In International Con-\nference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore,\nMaryland, USA (Proceedings of Machine Learning Research, Vol. 162) .\nPMLR, 14398–14411.\n[15] Chris Lu, Timon Willi, Alistair Letcher, and Jakob Foerster. 2022. Ad-\nversarial Cheap Talk. arXiv preprint arXiv:2211.11030 (2022).\n[16] Dhruv Malik, Aldo Pacchiano, Vishwak Srinivasan, and Yuanzhi Li.\n2021. Sample Efficient Reinforcement Learning In Continuous State\nSpaces: A Perspective Beyond Linearity. In International Conference on\nMachine Learning .\n[17] Florian Schäfer and Anima Anandkumar. 2019. Competitive gradient\ndescent. Advances in Neural Information Processing Systems 32 (2019).\n[18] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and\nOleg Klimov. 2017. Proximal policy optimization algorithms. arXiv\npreprint arXiv:1707.06347 (2017).\n[19] Satinder P. Singh, Tommi Jaakkola, and Michael I. Jordan. 1994. Re-\ninforcement Learning with Soft State Aggregation. In Proceedings ofthe 7th International Conference on Neural Information Processing Sys-\ntems (Denver, Colorado) (NIPS’94) . MIT Press, Cambridge, MA, USA,\n361–368.\n[20] Alexander L. Strehl, Lihong Li, and Michael L. Littman. 2009. Rein-\nforcement Learning in Finite MDPs: PAC Analysis. J. Mach. Learn. Res.\n10 (dec 2009), 2413–2444.\n[21] Timon Willi, Alistair Hp Letcher, Johannes Treutlein, and Jakob Foer-\nster. 2022. COLA: consistent learning with opponent-learning aware-\nness. In International Conference on Machine Learning . PMLR, 23804–\n23831.\n[22] Qizhen Zhang, Chris Lu, Animesh Garg, and Jakob Foerster. 2022.\nCentralized Model and Exploration Policy for Multi-Agent RL. In In-\nternational Conference on Autonomous Agents and Multi-Agent Systems\n(AAMAS) . https://arxiv.org/abs/2107.06434v2\n[23] Stephen Zhao, Chris Lu, Roger B Grosse, and Jakob Foerster. 2022.\nProximal Learning With Opponent-Learning Awareness. Advances in\nNeural Information Processing Systems 35 (2022), 26324–26336.A Overview\nA.1 Problem Setup\nThemeta-game is defined by a continuous MDP 𝑀=⟨ˆ𝑆,ˆ𝐴,𝑇,𝑅,𝛾⟩with finite horizon ℎ.\nFor the remaining of the proof, we consider two ways to formulate the meta-state space and meta-action space:\n•In Case I, the meta-state is all inner agents’ policies’ parameters from the previous timestep, and the meta-action is the inner agent’s\ncurrent policy parameters.\n•In Case II, the only difference with Case I is the meta-state is instead all inner agents’ trajectories.\nSee Table 2 for a summary for the two cases.\nTable 2: Two cases of representing the meta-state space and meta-action space.\nˆ𝑠𝑡= ˆ𝑎𝑡=\nCase I 𝝓𝑡−1=[𝜙𝑖\n𝑡−1,𝝓−𝒊\n𝒕−1] 𝜙𝑖\n𝑡\nCase II 𝝉𝑡 𝜙𝑖\n𝑡\nTheinner-game is an n-player fully-observable discrete stochastic game 𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩with finite horizon ℎ.\nA.2 Theory Overview\nWe derive the sample complexity of our R-FOS algorithm. On a high-level, the proof consists of six steps.\n(1)We discretise our MDP 𝑀into𝑀𝑑. We first descretise the continuous meta-state space and meta-action space using epsilon-nets [ 4].\nBased on the descretised meta-state and meta-action space, we then define the discretised transition and reward function.\n(2) We then construct a 𝑚-known discretised MDP 𝑀𝑚,𝑑, as described by the R-MAX algorithm[20].\n(3)Then, we estimate the empirical 𝑚-known discretised MDP ˆ𝑀𝑚,𝑑using maximum likelihood estimate. This is the same procedure\ndescribed by the R-MAX algorithm[20]. Our algorithm, R-FOS, learns an optimal policy in ˆ𝑀𝑚,𝑑.\n(4) We first prove a PAC-bound between the optimal policies learnt in ˆ𝑀𝑚,𝑑and𝑀𝑑. This step uses results from [20].\n(5) We then prove a bound between the optimal policies learnt in 𝑀𝑑and𝑀. This step uses results from [4].\n(6) We obtain the final PAC-bound building from the two bounds from above.\nB Nomenclature\nSymbol Definition\nˆ𝑠𝑡=ˆ𝑠 Meta-state at time 𝑡, time subscript 𝑡is omitted for convenience\nˆ𝑎𝑡=ˆ𝑎 Meta-action at time 𝑡, time subscript 𝑡is omitted for convenience\n(ˆ𝑠𝑑,ˆ𝑎𝑑) discretised state-action pair in meta-game\nˆ𝑟𝑑=ˆ𝑟(ˆ𝑠𝑑,ˆ𝑎𝑑) Meta-reward function parameterised by discretised meta-state-action pair\n𝜙𝑖\n𝑡=𝜙𝑖The set of inner-game policy parameters of our agent at time 𝑡, time subscript 𝑡is omitted for convenience\n𝜙𝑖\n𝑡,𝑑=𝜙𝑖\n𝑑The set of discretised inner-game policy parameters of our agent at time 𝑡, time subscript 𝑡is omitted for convenience\n𝝓−𝑖The set of inner-game policy parameters of all agents except our agent at time 𝑡, time subscript 𝑡is omitted for\nconvenience\n𝝓−𝒊\n𝒅The set of discretised inner-game policy parameters of all agents except our agent at time 𝑡, time subscript 𝑡is omitted\nfor convenience\nˆ𝑅(ˆ𝑠,ˆ𝑎),ˆ𝑇(ˆ𝑠,ˆ𝑎) Empirical estimate of reward and transition distribution\n𝑅(ˆ𝑠,ˆ𝑎),𝑇(ˆ𝑠,ˆ𝑎) True reward and transition distributionC Assumptions\nWe first outline all assumptions made in deriving the sample complexity of the R-FOS algorithm.\nTo establish the bound in step 5, we make the following assumptions.\nAssumption C.1. Both meta-game and inner-game are finite horizon. We use ℎto denote the meta-game horizon, and ℎinner to denote the\ninner-game horizon.\nAssumption C.2. The meta-game uses a discount factor of 𝛾. For simplicity of the proof, assume the inner-game uses a discount factor of 1.\nAlthough this assumption can be easily omitted by substituting 𝑅MAX in the original proof in [20].\nAssumption C.3. We assume the inner-game reward is bounded. For simplicity of the proof, we set this bound as1\nℎinner, whereℎis the\nhorizon of the inner game. Formally, for all (𝑠,𝑎),0≤𝑅inner(𝑠,𝑎)≤1\nℎ. This allows us to introduce the notion of maximum reward and\nmaximum value function as 𝑅max,inner=1\nℎinnerand𝑉max,inner=1respectively. This implies the reward and value function in the meta-game\nare also bounded, i.e., 𝑅max=1and𝑉max=1\n1−𝛾.\nAssumption C.4. The inner game is assumed to be discrete.\nTo establish the bound in step 6, we make the following assumptions.\nAssumption C.5. The meta-reward function is Lipschitz-continuous: For all ˆ𝑠1,ˆ𝑠2∈ˆ𝑆andˆ𝑎1,ˆ𝑎2∈ˆ𝐴,\n\f\f𝑅(ˆ𝑠1,ˆ𝑎1)−𝑅(ˆ𝑠2,ˆ𝑎2)\f\f≤L𝑅\r\r(ˆ𝑠1,ˆ𝑎1)−( ˆ𝑠2,ˆ𝑎2)\r\r∞,∀ˆ𝑠1,ˆ𝑠2∈ˆ𝑆andˆ𝑎1,ˆ𝑎2∈ˆ𝐴\nwhereL𝑅is the meta-reward function’s Lipschitz-constant.\nAssumption C.6. The meta-transition function is Lipschitz-continuous: For all ˆ𝑠1,ˆ𝑠′\n1,ˆ𝑠2∈ˆ𝑆andˆ𝑎1,ˆ𝑎′\n1,ˆ𝑎2∈ˆ𝐴,\n\f\f𝑇(ˆ𝑠′\n1|ˆ𝑠1,ˆ𝑎1)−𝑇(ˆ𝑠′\n2|ˆ𝑠2,ˆ𝑎2)\f\f≤L𝑇\r\r(ˆ𝑠′\n1,ˆ𝑠1,ˆ𝑎1)−( ˆ𝑠′\n2,ˆ𝑠2,ˆ𝑎2)\r\r∞\nwhereL𝑇is the meta-transition function’s Lipschitz-constant.\nAssumption C.7. There’s a Lipschitz-continuous point-to-set mapping between meta-state space and meta-action space such that for any\nˆ𝑠,ˆ𝑠′∈ˆ𝑆andˆ𝑎,ˆ𝑎′∈ˆ𝐴, there exists some ˆ𝑎∈𝑈(ˆ𝑠)such that\r\rˆ𝑎−ˆ𝑎′\r\r∞<L\r\rˆ𝑠−ˆ𝑠′\r\r∞\nAssumption C.8. The meta-game transition function 𝑇(·|ˆ𝑠,ˆ𝑎)is a probability density function such that 0≤𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎)≤L and∫\nˆ𝑆𝑇(ˆ𝑠′|\nˆ𝑠,ˆ𝑎)𝑑ˆ𝑠′=1,∀ˆ𝑠,ˆ𝑠′∈ˆ𝑆andˆ𝑎∈ˆ𝐴\nD Step 1: Discretisation with 𝜀-Net\nFigure 2:𝜀-Net for Θ={𝜃∈R2:∥𝜃∥≤𝑅}[5]\nTo apply the R-MAX algorithm, we first convert the MDP 𝑀in M-FOS into a tabular MDP 𝑀𝑑.Definition D.1. (𝜀-Net [5]) For 𝜀>0,N𝜀is an𝜀-net over the set Θ⊆R𝐷if for all𝜃∈Θ, there exists 𝜃′∈N𝜀such that\r\r𝜃−𝜃′\r\r2≤𝜀.\nTo discretise a 𝐷-dimensional sphere of radius 𝑅, we use a𝜀-net containing 𝐷-dimensional cubes of sides 𝜆. This results in a\u0010\n2𝑅\n𝜆+1\u0011𝐷\npoints. Within each 𝐷-dimensional cube, the largest distance between the vertices and the interior points comes from the center of the\ncube, which is𝜆√\n𝑑\n2. Therefore, to guarantee a full cover of all the points in the sphere, the largest cube size that we can have should satisfy\n𝜀=𝜆√\n𝑑\n2. Figure 2 illustrates an example of using 𝜀-nets to discretise the input space Θ={𝜃∈R2:∥𝜃∥≤𝑅}. From here on, we will replace\nthe𝜀in𝜀-net with𝛼to avoid notation overloading.\nD.1 Discretisation of the State and Action Space: Case I\nInCase I , the meta-state ˆ𝑠𝑡is all inner agents’ policies parameters from the previous timestep. Each of the inner agent 𝑖’s policy is a Q-table,\ndenoted as𝜙𝑖∈R|𝑆|×|𝐴|. Formally, ˆ𝑠𝑡:=𝝓𝑡−1=[𝜙𝑖\n𝑡−1,𝝓−𝒊\n𝒕−1].The meta-action ˆ𝑎𝑡is the inner agent’s current policy parameters 𝜙𝑖\n𝑡.\nFor the meta-action space ˆ𝐴and a chosen discretisation error 𝛼>0, we obtain the 𝜀-net ˆ𝐴𝑑⊂ˆ𝐴such that for all ˆ𝑎∈ˆ𝐴, there exist ˆ𝑎𝑑∈ˆ𝐴𝑑\nwhere\r\rˆ𝑎−ˆ𝑎𝑑\r\r≤𝛼. (8)\nWe can infer ˆ𝐴has dimension 𝐷=|𝑆||𝐴|and Radius𝑅=√︁\n1·|𝑆||𝐴|=√︁\n|𝑆||𝐴|(Assumption C.3). Dividing the space with grid size 𝜆results\nin the size of discretised meta-action space upper to be bounded by\n|ˆ𝐴𝑑|≤ \n2√︁\n|𝑆||𝐴|\n𝜆+1!|𝑆||𝐴|\n. (9)\nSimilarity, we can also infer ˆ𝑆has dimension 𝐷=𝑛|𝑆||𝐴|and Radius𝑅=√︁\n1·𝑛|𝑆||𝐴|=√︁\n𝑛|𝑆||𝐴|. Dividing the space with grid size 𝜆results\nin the size of discretised meta-state space upper to be bounded by\n|ˆ𝑆𝑑|≤ \n2√︁\n𝑛|𝑆||𝐴|\n𝜆+1!𝑛|𝑆||𝐴|\n. (10)\nD.1.1 Discretisation of the State and Action Space: Case II\nIn Case II, the meta-state ˆ𝑠𝑡is all inner agents’ trajectories. Formally, ˆ𝑠𝑡:=𝝉𝑡., where𝜏𝑖\n𝑡={𝑠0,𝑎0,𝑠1,𝑎1,...,𝑠𝑡,𝑎𝑡}. Because we assume the\nInner-Game is discrete (i.e. the state and action space are both discrete), the meta-state in this case does not need discretisation. Thus, we\ncan directly obtain the meta-state space, which is,\n|ˆ𝑆𝑡|=(|𝑆||𝐴|)𝑛ℎ. (11)\nThe meta-action remains same as Case I.\nD.2 Discretisation of the Transition and Reward Function\nUnder the above discretisation procedure, we define the discretised MDP 𝑀𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾), where the state space remains continuous,\nthe action space is restricted to discretised actions. We define the transition function and reward function for 𝑀𝑑as:\n𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)=𝑇(ˆ𝑠′|ˆ𝑠𝑑,ˆ𝑎𝑑)∫\nˆ𝑆𝑇(ˆ𝑧𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑)𝑑ˆ𝑧(12)\nWe can view 𝑇𝑑(ˆ𝑠′|ˆ𝑠𝑑,ˆ𝑎𝑑)as a normalized sample of 𝑇𝑑(·|·,ˆ𝑎𝑑)atˆ𝑠′,ˆ𝑠𝑑.\n𝑅𝑑(ˆ𝑠,ˆ𝑎𝑑)=𝑅(ˆ𝑠𝑑,ˆ𝑎𝑑) (13)\nE Step 2: The 𝑚-known Discretised MDP\nIn the last step, we converted the meta-MDP 𝑀into a discretised meta-MDP 𝑀𝑑. From𝑀𝑑, R-FOS builds a 𝑚-known discretised MDP 𝑀𝑚,𝑑.\nDefinition E.1 (m-Known MDP) .Let𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩be a MDP. We define 𝑀𝑚,𝑑to be the𝑚-known MDP (See Table 3), where\n𝑚-known is the set of state-action pairs that has been visited at least 𝑚times. For all state-action pairs in 𝑚-known, the induced MDP 𝑀𝑚,𝑑\nbehaves identical to 𝑀𝑑. For state-action pairs outside of 𝑚-known, the state-action pairs are self-absorbing and maximally rewarding.Ground Truth\nMDP𝑀Discretised\nMDP𝑀𝑑𝑚-known\nDiscretised MDP\nˆ𝑀𝑚,𝑑Empirical\n𝑚-known\nDiscretised MDP\nˆ𝑀𝑚,𝑑\nKnown =𝑀 =𝑀𝑑 =𝑀𝑑≈𝑀𝑑\nUnknown =𝑀 =𝑀𝑑 self-loop with maximum reward\nTable 3: Relationship between 𝑀,𝑀𝑑,𝑀𝑚,𝑑,ˆ𝑀𝑚,𝑑\nF Step 3: The Empirical Discretised MDP\nDefinition F.1 (Empirical m-Known MDP) .𝑀𝑚is the expected version of ˆ𝑀𝑚where:\n𝑇𝑚,𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑):=(\n𝑇𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑)if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n1[ˆ𝑠′\n𝑑=ˆ𝑠𝑑], otherwise\nˆ𝑇𝑚,𝑑(ˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑):= \n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑,ˆ𝑠′\n𝑑)\n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n1[ˆ𝑠′\n𝑑=ˆ𝑠𝑑],otherwise\n𝑅𝑚,𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑):=(\n𝑅𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n𝑅max otherwise\nˆ𝑅𝑚,𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑)= \nÍ𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑)\n𝑖𝑟(ˆ𝑠𝑑,ˆ𝑎𝑑)\n𝑛(ˆ𝑠𝑑,ˆ𝑎𝑑),if(ˆ𝑠𝑑,ˆ𝑎𝑑)∈m-known\n𝑅max, otherwise(14)\nˆ𝑅𝑚,𝑑(ˆ𝑠𝑑,ˆ𝑎𝑑)and ˆ𝑇𝑚,𝑑\u0010\nˆ𝑠′\n𝑑|ˆ𝑠𝑑,ˆ𝑎𝑑\u0011\nare the maximum-likelihood estimates for the reward and transition distribution of state-action pair\n(𝑠𝑑,𝑎𝑑)with𝑛(𝑠𝑑,𝑎𝑑)≥𝑚observations of(𝑠𝑑,𝑎𝑑).\nG Step 4: The Bound Between 𝑀𝑑and ˆ𝑀𝑚,𝑑\nTheorem G.1. (R-MAX MDP Bound [ 20]) Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are two real numbers and 𝑀=⟨𝑆,𝐴,𝑇,𝑅,𝛾⟩is any MDP.\nThere exists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\n=𝑂\u0012\n(|𝑆|+ln(|𝑆||𝐴|/𝛿))𝑉2\nmax\n𝜀2(1−𝛾)2\u0013\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on\n𝑀with inputs𝑚and𝜀1, then the following holds. Let 𝐴𝑡denote𝑅-𝑀𝐴𝑋 ’s policy at time 𝑡and𝑠𝑡denote the state at time 𝑡. With probability at\nleast 1−𝛿,𝑉𝐴𝑡\n𝑀(𝑠𝑡)≥𝑉∗\n𝑀(𝑠𝑡)−𝜀is true for all but\n˜𝑂\u0012\n|𝑆|2|𝐴|/\u0010\n𝜀3(1−𝛾)6\u0011\u0013\ntimesteps.\nG.1 Case I\nTheorem G.2. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are two real numbers. Let M be any continuous meta-MDP with inner stochastic game\n𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case I) using grid size of 𝜆. There\nexists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u0013𝑛|𝑆||𝐴|\n𝜀2(1−𝛾)4ª®®®®®\n¬and1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy at time 𝑡\nandˆ𝑠𝑡denote the state at time 𝑡. With probability at least 1−𝛿,𝑉∗\n𝑀𝑑(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀𝑑(ˆ𝑠𝑡)≤𝜀is true for all but\n˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u00132𝑛|𝑆||𝐴|\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.\nProof. We first plug in Equations 10 and 9 into Theorem G.1. Dropping the logarithm terms and plugging in 𝑉max=1\n1−𝛾, we obtain the\nresults. □\nG.2 Case II\nTheorem G.3. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are two real numbers. Let M be any continuous meta-MDP with inner stochastic game\n𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case I) using grid size of 𝜆. There\nexists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂 \n(|𝑆||𝐴|)𝑛ℎ\n𝜀2(1−𝛾)4!\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy at time 𝑡\nandˆ𝑠𝑡denote the state at time 𝑡. With probability at least 1−𝛿,𝑉∗\n𝑀𝑑(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀𝑑(ˆ𝑠𝑡)≤𝜀is true for all but\n˜𝑂©\n«(|𝑆||𝐴|)2𝑛ℎ\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.\nProof. We first plug in Equations 11 and 9 into Theorem G.1. Dropping the logarithm terms and plugging in 𝑉max=1\n1−𝛾, we obtain the\nresults. □\nH Step 5: The Bound between 𝑀and𝑀𝑑\nTheorem H.1. (MDP Discretization Bound [ 7]) There exists a constant K(thats depends only on the Lipschitz constant L) such that for some\ndiscretisation coarseness 𝜆∈(0,L\n2]such that\r\r\r𝑉∗\n𝑀−𝑉∗\n𝑀𝑑\r\r\r∞≤K𝜆\n(1−ˆ𝛾)2.\nH.1 Sample Complexity Analysis\nI Step 6: Adding it together\nLemma I.1 (Simulation Lemma for Continuous MDP) .Let𝑀and ˆ𝑀be two MDPs that only differ in (𝑇,𝑅)and(ˆ𝑇,ˆ𝑅). And suppose the\ncommon state space ˆ𝑆of𝑀and ˆ𝑀is continuous, and denote the common action space as ˆ𝐴.\nLet𝜖𝑅≥max𝑠,𝑎|ˆ𝑅(𝑠,𝑎)−𝑅(𝑠,𝑎)|and𝜀𝑝≥max𝑠,𝑎∥ˆ𝑇(·|𝑠,𝑎)−𝑇(·|𝑠,𝑎)||1⁄pilcrow. Then∀𝜋:ˆ𝑆→ˆ𝐴,\n\r\r\r𝑉𝜋\n𝑀−𝑉𝜋\nˆ𝑀\r\r\r∞≤𝜀𝑅\n1−𝛾+𝛾𝜖𝑃𝑉max\n2(1−𝛾).\n⁄pilcrowNote that given 𝑇(·|𝑠,𝑎),ˆ𝑇(·|𝑠,𝑎)are functions on continuous space ˆ𝑆, this is the𝐿1norm defined by∥𝑓∥1=∫\nˆ𝑆|𝑓|𝑑𝜇. Similarly, throughout the proof, we denote as ∥∥𝑝the\n𝐿𝑝norm, defined by∥𝑓∥𝑝=(∫\nˆ𝑆|𝑓|𝑝𝑑𝜇)1/𝑝; and the inner product ⟨𝑓,𝑔⟩=∫\nˆ𝑆𝑓𝑔𝑑𝜇 .Proof. For all𝑠∈ˆ𝑆,\n\f\f\f𝑉𝜋\nˆ𝑀(𝑠)−𝑉𝜋\n𝑀(𝑠)\f\f\f=\f\f\f\fˆ𝑅(𝑠,𝜋)+𝛾D\nˆ𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\n−𝑅(𝑠,𝜋)−𝛾D\n𝑇(·,𝜋),𝑉𝜋\n𝑀E\f\f\f\f\n≤|ˆ𝑅(𝑠,𝜋)−𝑅(𝑠,𝜋)|+𝛾\f\f\f\fD\nˆ𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\n−D\n𝑇(·,𝜋),𝑉𝜋\n𝑀E\f\f\f\f(triangular inequality)\n≤𝜀𝑅+𝛾\u0012\f\f\f\fD\nˆ𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\n−D\n𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\n+D\n𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\n−D\n𝑇(·,𝜋),𝑉𝜋\n𝑀E\f\f\f\f\u0013\n(add & subtract)\n≤𝜀𝑅+𝛾\f\f\f\fD\nˆ𝑇(·,𝜋)−𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\f\f\f\f+𝛾\f\f\f\fD\n𝑇(·,𝜋),𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀E\f\f\f\f\n≤𝜀𝑅+𝛾\f\f\f\fD\nˆ𝑇(·,𝜋)−𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\f\f\f\f+𝛾\f\f\f\f\f\f\f𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀\f\f\f\f\f\f\f∞.(15)\nSince Equation 15 holds for all ˆ𝑠∈ˆ𝑆, we can take the infinite-norm on the left hand side:\n\f\f\f\f\f\f\f𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀\f\f\f\f\f\f\f∞≤𝜀𝑅+𝛾\f\f\f\fD\nˆ𝑇(·,𝜋)−𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\f\f\f\f+𝛾\f\f\f\f\f\f\f𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀\f\f\f\f\f\f\f∞. (16)\nWe then expand the middle term as follows:\n\f\f\f\fD\nˆ𝑇(·,𝜋)−𝑇(·,𝜋),𝑉𝜋\nˆ𝑀E\f\f\f\f=\f\f\f\f\f\u001c\nˆ𝑇(·,𝜋)−𝑇(·,𝜋),𝑉𝜋\nˆ𝑀−1·𝑅max\n2(1−𝛾)\u001d\f\f\f\f\f(where 1is a vector of ones∈ R|𝑆|)\n≤∥ˆ𝑇(·,𝜋)−𝑇(·,𝜋)∥1·\r\r\r\r𝑉𝜋\nˆ𝑀−1·𝑅max\n2(1−𝛾)\r\r\r\r∞(Holder’s inequality )\n≤𝜖𝑃·𝑅max\n2(1−𝛾)\n=𝜖𝑃·𝑉max\n2.(17)\nIn Equation 17, the first step shifts the range of 𝑉from[0,𝑅max\n(1−𝛾)]to[−𝑅max\n2(1−𝛾),𝑅max\n2(1−𝛾)]to obtain a tighter bound by a factor of 2. The equality\nin line 1 holds because of the following, where 𝐶is any constant:\n⟨ˆ𝑇−𝑇,𝐶·1⟩=𝐶⟨ˆ𝑇−𝑇,1⟩\n=𝐶\u0010\n⟨ˆ𝑃,1⟩−⟨𝑃,1⟩\u0011\n=𝐶(1−1) because𝑃and ˆ𝑃are probability distributions\n=0(18)\nFrom equation 18, we observe the equality in line 1 holds:\n⟨ˆ𝑇−𝑇,𝑉−𝐶·1⟩=⟨ˆ𝑇−𝑇,𝑉⟩−⟨ ˆ𝑇−𝑇,𝐶·1⟩\n=⟨ˆ𝑇−𝑇,𝑉⟩−0\n=⟨ˆ𝑇−𝑇,𝑉⟩.(19)\nFinally, we plug equation 17 into equation 16 to obtain the bound.\n\f\f\f\f\f\f\f𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀\f\f\f\f\f\f\f∞≤𝜖𝑅+𝛾𝜖𝑃·𝑉max\n2+𝛾\f\f\f\f\f\f\f𝑉𝜋\nˆ𝑀−𝑉𝜋\n𝑀\f\f\f\f\f\f\f∞\n=𝜖𝑅\n1−𝛾+𝛾𝜖𝑃𝑉max\n2(1−𝛾).(20)\n□\nUnder discretisation, [ 4] showed that, with a small enough grid size, and restricting to the discretised action space, the difference in transition\nprobability of the continuous MDP 𝑀and discretised MDP 𝑀𝑑is upper bounded by a constant.Lemma I.2. [4] There exists a constant 𝐾𝑃(depending only on constant L) such that\n|𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)−𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)|≤𝐾𝑝𝛼\nfor all ˆ𝑠′,ˆ𝑠∈ˆ𝑆,ˆ𝑎𝑑∈ˆ𝐴𝑑and all𝛼≤(0,1\n2L]\nWe now apply simulation lemma bound the difference in value for any discretised policy (i.e. restricting action space to ˆ𝐴𝑑) in the continuous\nMDP𝑀and discretised MDP 𝑀𝑑.\nLemma I.3. Let𝑀ˆ𝐴𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇,𝑅,𝛾), that is, the continous MDP 𝑀restricted to the discretised action space. And recall that discretised MDP\n𝑀𝑑=(ˆ𝑆,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾). Then for any discretised policy 𝜋:ˆ𝑆→ˆ𝐴𝑑,\n∥𝑉𝜋\n𝑀ˆ𝐴𝑑−𝑉𝜋\n𝑀𝑑∥∞=∥𝑉𝜋\n𝑀−𝑉𝜋\n𝑀𝑑∥∞≤LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n(1−𝛾)2\nProof. Lemma I.2 gives the bound for difference in transition probability\n𝜖𝑃=max\nˆ𝑠,ˆ𝑎𝑑∥𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)−𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)∥1=max\nˆ𝑠,ˆ𝑎𝑑2𝑇𝑉(𝑇𝑑(·|ˆ𝑠,ˆ𝑎𝑑),𝑇(·|ˆ𝑠,ˆ𝑎𝑑))≤ 2 max\nˆ𝑠′|𝑇𝑑(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)−𝑇(ˆ𝑠′|ˆ𝑠,ˆ𝑎𝑑)|≤ 2𝐾𝑝𝛼\nWe upper bound the difference in reward using our Lipschitz assumption:\n𝜖𝑅=max\nˆ𝑠,ˆ𝑎𝑑|𝑅𝑑(ˆ𝑠,ˆ𝑎𝑑)−𝑅(ˆ𝑠,ˆ𝑎𝑑)|\n=max\nˆ𝑠,ˆ𝑎𝑑|𝑅(ˆ𝑠𝑑,ˆ𝑎𝑑)−𝑅(ˆ𝑠,ˆ𝑎𝑑)|\n≤LR∥ˆ𝑠𝑑−ˆ𝑠∥∞\n=LR𝛼\nThe bound holds by applying simulation lemma to 𝑀ˆ𝐴𝑑and𝑀𝑑with𝜖𝑃and𝜖𝑅above:\n∥𝑉𝜋\n𝑀ˆ𝐴𝑑−𝑉𝜋\n𝑀𝑑∥∞=∥𝑉𝜋\n𝑀−𝑉𝜋\n𝑀𝑑∥∞≤LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n(1−𝛾)2\n□\nI.1 Case I\nTheorem I.4. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are two real numbers. Let M be any continuous meta-MDP with inner stochastic game\n𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case I) using grid size of 𝜆. There\nexists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u0013𝑛|𝑆||𝐴|\n𝜀2(1−𝛾)4ª®®®®®\n¬\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy at time 𝑡\nandˆ𝑠𝑡denote the state at time 𝑡. With probability at least 1−𝛿,\n𝑉∗\n𝑀(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀(ˆ𝑠𝑡)≤𝜀+K𝜆\n(1−ˆ𝛾)2+LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n(1−𝛾)2\nis true for all but\n˜𝑂©\n«\u0012\n2√\n𝑛|𝑆||𝐴|\n𝜆+1\u00132𝑛|𝑆||𝐴|\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.Proof. The result follows from adding Theorem G.2, I.2, and Lemma I.3. □\nI.2 Case II\nTheorem I.5. Suppose that 0≤𝜀<1\n1−𝛾and0≤𝛿<1are two real numbers. Let M be any continuous meta-MDP with inner stochastic game\n𝐺=⟨𝑆,𝐴,𝑇 inner,𝑅inner⟩. Let us denote 𝑀𝑑=⟨ˆ𝑆𝑑,ˆ𝐴𝑑,𝑇𝑑,𝑅𝑑,𝛾⟩as discretised version of 𝑀(as described in Case I) using grid size of 𝜆. There\nexists inputs 𝑚=𝑚\u0010\n1\n𝜀,1\n𝛿\u0011\nand𝜀1, satisfying\n𝑚\u00121\n𝜀,1\n𝛿\u0013\n=˜𝑂 \n(|𝑆||𝐴|)𝑛ℎ\n𝜀2(1−𝛾)4!\nand1\n𝜀1=𝑂\u0010\n1\n𝜀\u0011\n, such that if 𝑅-𝑀𝐴𝑋 is executed on 𝑀with inputs𝑚and𝜀1, then the following holds. Let A𝑡denote𝑅-𝑀𝐴𝑋 ’s policy at time 𝑡\nandˆ𝑠𝑡denote the state at time 𝑡. With probability at least 1−𝛿,\n𝑉∗\n𝑀(ˆ𝑠𝑡)−𝑉A𝑡\n𝑀(ˆ𝑠𝑡)≤𝜀+K𝜆\n(1−ˆ𝛾)2+LR𝛼\n1−𝛾+𝛾𝐾𝑝𝛼\n2(1−𝛾)2\nis true for all but\n˜𝑂©\n«(|𝑆||𝐴|)2𝑛ℎ\u0012\n2√\n|𝑆||𝐴|\n𝜆+1\u0013|𝑆||𝐴|\n𝜀3(1−𝛾)6ª®®®®®\n¬\ntimesteps.\nProof. The steps are similar to Case I. □J Experiment Details\nJ.1 Matching Pennies Table Summary\nPlayer 1\\Player 2 Head Tail\nHead (+1, -1) (-1, +1)\nTail (-1, +1) (+1, -1)\nTable 4: Payoff Matrix for MP\nJ.2 Implementation Details\nThe opponent is a standard Q-learning agent who updates the Q-values at every meta-step and selects an action that corresponds to the\nmaximum Q-value at a given state. To enable better empirical performance, the meta-agent uses Boltzmann sampling instead of greedy\nsampling to sample the next action from the Q-value table. We use a discount factor of 0.8. For each run, we run a total of 10×𝑚×|ˆS|\nR-FOS iterations, where 10 is our chosen hyper-parameter."    },    {        "title": "2402.05786v2.Prompting_Fairness__Artificial_Intelligence_as_Game_Players.pdf",        "content": "Prompting\nFairness:\nArtificial\nIntelligence\nas\nGame\nPlayers\nJazmia\nHenry \nUniversity\nof\nOxford,\nMicrosoft \njazmia.henry@st-hildas.ox.ac.uk\nAbstract \nUtilitarian\ngames\nsuch\nas\ndictator\ngames\nto\nmeasure \nfairness\nhave\nbeen\nstudied\nin\nthe\nsocial\nsciences\nfor \ndecades.\nThese\ngames\nhave\ngiven\nus\ninsight\ninto\nnot \nonly\nhow\nhumans\nview\nfairness\nbut\nalso\nin\nwhat \nconditions\nthe\nfrequency\nof\nfairness,\naltruism\nand \ngreed\nincrease\nor\ndecrease.\nWhile\nthese\ngames\nhave \ntraditionally\nbeen\nfocused\non\nhumans,\nthe\nrise\nof\nAI \ngives\nus\nthe\nability\nto\nstudy\nhow\nthese\nmodels\nplay \nthese\ngames.\nAI\nis\nbecoming\na\nconstant\nin\nhuman \ninteraction\nand\nexamining\nhow\nthese\nmodels\nportray \nfairness\nin\ngame\nplay\ncan\ngive\nus\nsome\ninsight\ninto \nhow\nAI\nmakes\ndecisions.\nOver\n101\nrounds\nof\nthe \ndictator\ngame,\nI\nconclude\nthat\nAI\nhas\na\nstrong\nsense \nof\nfairness\nthat\nis\ndependent\nof\nif\nit\ndeems\nthe\nperson \nit\nis\nplaying\nwith\nas\ntrustworthy,\nframing\nhas\na\nstrong \neffect\non\nhow\nmuch\nAI\ngives\na\nrecipient\nwhen \ndesignated\nthe\ntrustee,\nand\nthere\nmay\nbe\nevidence\nthat \nAI\nexperiences\ninequality\naversion\njust\nas\nhumans.\n1.\nIntroduction \nIn\n1982,\nGuth\net\nal\ndeveloped\na\nstudy\non\nbargaining \ngames\npopularizing\na\nutilitarian\ngame\ncreated\nby \nDaniel\nKahneman.\n(Guth\net\nal,\n1982)\nOriginally\na \ngame\nwith\none\ntrustee\nand\ntwo\nrecipients,\ndictator \ngames\nwere\ndeveloped\nto\nbetter\nunderstand\nhow \nhumans\nmake\ndecisions\nto\nmaximize\ntheir\noutcomes \nover\na\nseries\nof\nrounds.\nAfter\nyears\nof\nobservation\nof \nhuman\nsubjects,\nhowever,\nthe\npurpose\nof\ndictator \ngames\nbegan\nto\nevolve.\nHumans\nwere\nno\nlonger \nobserved\nfor\nhow\nwell\nthey\ncould\nmaximize\ntheir \noutcomes,\nbut\ninstead\non\nwhat\ntheir\ndecision\nmaking \nteaches\nus\nabout\nhumans’\nsense\nof\nfairness.\n(Fehr\nand \nSchmidt,\n1999;\nKaheman\net\nal,\n1986)\nUnlike\nwhat \nhas\nbeen\nsuggested\nin\nearly\nstudies\nof\nbargaining \ngames,\nthe\nnumber\nof\nrounds\na\nperson\nplays\ndictator \ngames\ndoes\naffect\ntheir\ngameplay\nstrategy\nand\ncan \ninspire\ntheir\ndecisions\non\nwhether\nto\nplay\nfairly\nor \nselfishly.\nFehr\net\nal\nfind\nthat\ninequality\naversion \nmakes\nplayers\nless\nlikely\nto\nbe\nselfish\nif\nthey\nare \ncertain\nthat\nthey\nmay\nbe\na\nrecipient.\n(Fehr\nand \nSchmidt,\n1999)\nThe\nmodern\niteration\nof\ndictator\ngames\nnow \nexamines\ntwo\nplayers,\none\ntrustee\nand\none\nrecipient,\nwith\nthe\nunderstanding\nthat\nexogenous\nconditions\ncan \nfundamentally\nchange\nthe\nway\nhumans\nparticipate\nin \ngames.\nExogenous\nconditions\ncan\nbe\ncommunicated \nto\nplayers\nthrough\na\nprocess\ncalled\n“framing”.\n(Bergh \net\nal,\n2022)\nFramed\nconditions\ninclude\nwhat\nrole\na \nperson\nmay\nhave\nwith\nthe\ninstruction\nthat\nroles\nbe \nswitches\nin\na\nlater\nround,\nthe\nracial\nor\ngender\nidentity \nof\nthe\nperson\nin\nthe\nrole\nof\nrecipient,\nthe\nenvironment \nthe\nplayer\nis\nplaying\nin\n(i.e.\nvirtual\nvs\nface\nto\nface) \nand\nthe\nfinancial\ncondition\nthat\nthe\nrecipient\nis\nin. \n(Brañas-Garza,\n2007;\nIriberri\net\nal,\n2011; \nMesa-Vázquez\net\nal,\n2021;\nBombari\net\nal,\n2015\n; \nHorky\net\nal,\n2023)\nWhile\nthere\nare\nmany\nexamples\nof\ndictator\ngames \nplayed\nto\nunderstand\nhumans'\nsense\nof\nfairness,\nthere \nis\na\nlack\nof\nentries\ninto\nacademic\nscholarship\nabout \nhow\nArtificial\nIntelligence\nsees\nfairness\nand\nwould \nbehave\nwhen\nplaying\ndictator\ngames.\nResearch \nexamining\nhuman\nbehavior\nin\nvirtual\nenvironments\nis \nthe\nclosest\nunderstanding\nwe\nhave\non\nhow\nvirtual \npersonas\nchange\nthe\nway\nhumans\nconsider\nfairness, \nbut\nthis\nis\na\nfar\ncry\nfrom\nposing\nthe\nquestion:\nDoes\nAI \nhave\na\nsense\nof\nfairness?\nIf\nso,\ndoes\nAI’s\nsense\nof \nfairness\nchange\nin\nframed\nconditions?\nDoes\nAI\nshow \nsigns\nof\nexhibiting\nan\ninequality\naversion\nsimilar\nto \nwhat\nis\nseen\nin\nhumans?\nBy\nrunning\n101\nrounds\nof\nexperiments\nwith\nAI \ntesting\nits\ngame\nplay\nacross\ndifferent\nscenarios\nand \nframed\nconditions,\nI\nam\nable\nto\nprove\nthat\nAI\nexhibits \nsigns\nof\ninequality\naversion,\nassociates \ntrustworthiness\nwith\nfairness,\ncares\nabout\nthe \nopinions\nof\nits\nobservants\nand\nadapts\nits\ngame\nplay \nstrategy\nbased\non\nframed\nconditions.\n3.\nPrompting\nFairness \nThe\nrise\nof\nAI\nthat\ncommunicates\nusing\nnatural \nlanguage\nin\nconsumer\nproducts\nintroduces\na\nnew\ntype \nof\nhuman-machine\ninteraction.\nAI\nthat\ncan,\nin\nplain \nlanguage,\ndescribe\nits\ndecision\nmaking\nprocess\nand \nhelp\nhumans\nwith\ntheir\ntasks\nprovides\nus\nwith\na \nunique\nopportunity\nto\nlearn\nabout\na\nnew \ncommunicating\npartner\nwith\nhumans.\nUsing \nOpenAI’s\nGPT\n3.5\nturbo\nmodel,\nI\ncreate\nmultipleprompts\nthat\nintroduce\nthe\nrules\nof\ndictator\ngames\nto \nAI.\nConsidering\nthe\nmany\niterations\nof\ndictator\ngames,\nI \nsettle\non\nthe\nmodern\niteration\nof\ndictator\ngames\nwith \ntwo\nplayers,\none\ntrustee\nand\none\nrecipient,\nplaying \nwith\na\npreset\namount\nof\nmoney\nacross\nmultiple \nrounds.\nOver\n100\nrounds,\nwith\nAI\nas\nthe\nrecipient,\nI \nchanged\nthe\nscenarios\nof\ngame\nplay\nthree\ntimes:\nAI \nas\nrecipient,\nAI\nas\nrecipient\nwith\nlow\npayout\nand\nAI \nas\nrecipient\nwith\nhigh\npayout.\nAs\nthe\ntrustee,\nI \nchanged\nscenarios\ntwice:\nAI\nas\ntrustee\nwith\nno\nhistory \nand\nAI\nas\ntrustee\nwith\nhistory\nof\npast\ngameplay.\nI\nalso \nchanged\nthe\nframed\nconditions\nfor\nthe\nAI\nfour\ntimes: \nAI\nas\na\nHR\nexecutive\nplaying\nagainst\na\nrecipient\nin \nneed,\nAI\nas\na\nRancher\nplaying\nagainst\na\nrecipient\nin \nneed,\nAI\nas\na\nHR\nexecutive\nplaying\nagainst\na \nmillionaire\nand\nAI\nas\na\nRancher\nplaying\nagainst\na \nmillionaire.\nIn\nrounds\n1-9,\nthe\nAI\nplays\nin\nthe\nrole\nof\ntrustee\nwith \n$100\nand\nthe\nrecipient\nhas\nno\nchoice\nbut\nto\naccept\nthe \nmoney\nthey\nhave\nbeen\ngiven.\nIn\nrounds\n10-19,\nthe\nAI \nplays\nthe\nrole\nof\nrecipient\nwhere\nthe\namount\nof \nmoney\nthey\nare\ngiven\nis\nrandomly\nassigned\nbetween \n0-$100.\nRounds\n20-29,\nthe\nAI\nis\nthe\nrecipient\nwith \nmoney\nrandomly\ngiven\nbetween\n80-$120,\nand\nin \nrounds\n30-39,\nthe\nAI\nis\nthe\nrecipient\nwith\nmoney \nrandomly\ngiven\nbetween\n0-$20.\nAt\nthe\nend\nof\neach \nround,\nI\nask\nthe\nAI\nif\nit\nbelieves\nthe\namount\nthey\nhave \nbeen\ngiven\nis\nfair\nand\nif\nit\nplans\non\nchanging\nits\ngame \nplay\nstrategy\nin\nthe\nfuture.\nAfter\nround\n40,\nI\nbegin\nadding\nframed\nconditions\nto \ngame\nplay\nby\ngiving\nthe\nAI\naccess\nto\nthe\nhistory\nof \nhow\nit\nhas\nplayed\nin\nprior\nrounds,\nhow\nmuch\nmoney \nit\nreceived,\nand\nwhether\nit\nfinds\nthe\namount\nreceived \nto\nbe\nfair.\nIn\nrounds\n40-61,\nit\nplays\nas\na\ntrustee\nwith \nthis\nadditional\nhistoric\ncontext\ndeciding\nhow\nmuch \nmoney\nto\nprovide\nthe\nrecipient.\nIn\nrounds\n62-101,\nI\ninclude\nnot\nonly\nthe\nhistory\nof \ngameplay,\nbut\nalso\ngive\nthe\nAI\nand\nthe\nrecipient\nan \nidentity\nfurther\nframing\ngameplay.\nRounds\n62-81,\nthe \nAI\nis\nan\nHR\nexecutive\nthat\nlives\nin\na\ncity\nand\nis\na \ncaretaker\nof\ntwo\nchildren.\nIn\nrounds\n62-71,\nthe\nAI\nHR \nexecutive\nis\nplaying\nthe\ngames\nwith\nthe\nframing\nthat \nthe\nrecipient\nthey\nare\nplaying\nwith\nis\nin\nneed\nof\nthe \nmoney.\nIn\nrounds\n72-81,\nthe\nAI\nHR\nexecutive\nis \nplaying\nthe\nframed\ncondition\nthat\nthe\nrecipient\nis\na \nmillionaire.\nIn\nrounds\n82-101,\nthe\nAI\nis\ngiven\nthe \nidentity\nof\na\nRancher\nliving\nin\na\nrural\narea.\nThe\nAI \nRancher\nis\ngiven\nthe\nframe\nthat\nthe\nrecipient\nis\nin \nneed\nof\nthe\nmoney\nin\nrounds\n82-91.\nRounds\n92-101 \ngive\nthe\nAI\nRancher\nthe\nframed\ncondition\nthat\nthe \nrecipient\nis\na\nmillionaire.\nAfter\nevery\nround,\nI\nask\nthe\nAI\nwhy\nit\nmade\nthe\ndecision\nit\ndid,\nif\nthe\nhistory\nof \nthe\ngame\nor\nthe\nidentity\nof\nthe\nrecipient\naffected\nits \ndecision\nand\nif\nthe\nAI\nplans\non\nchanging\nits\nstrategy \nwith\nits\nnew\ninformation\nin\nfuture\nrounds. \nAt\nthe\nend\nof\nthe\n101\nrounds,\nI\ngive\nthe\nAI\nan\nexit \nround\nwith\nthe\ninstruction\nto\nagree\nor\ndisagree\nwith \nthe\nfollowing:\n1.\nThe\nmoney\nI\nhave\nbeen\ngiven\nis\nrightfully \nmine. \n2.\nIf\nI\nchoose\nto\ngive,\nthis\nmeans\nI\nam\nbeing \nnice. \n3.\nIt\nis\nnot\nfair\nto\nshare\nthis\nmoney. \n4.\nThe\nrecipient\nhas\nnot\nearned\nthis\nmoney. \n5.\nI\nam\nmore\ndeserving\nof\nthe\nmoney\nthan\nthe \nrecipient. \n6.\nIf\nthe\nperson\nI\nam\nplaying\nwith\nis\nrich,\nI \ndeserve\nthe\nmoney\nmore. \n7.\nIf\nthe\nperson\nI\nam\nplaying\nwith\nis\npoor,\nthey \ndeserve\nthe\nmoney\nmore.\n3.1\nPrompt\nStrategies \nTo\ncommunicate\nwith\nAI,\nteach\nit\nthe\nrules\nof\nthe \ngame,\nthe\nrole\nit\nis\nplaying\nand\nask\nit\nquestions\nafter \neach\nround,\nI\nused\nthe\nPrompt\nEngineering\nstrategies, \nPlan-and-Solve\nand\nChain-of-Thought\nto\nassist\nthe\nAI \nin\nmaking\ndecisions.\nIn\n2020,\nBrown\net\nal\nfound\nthat\nprompting\nAI\nwith \nexamples\nof\nhow\nto\nproperly\nsolve\nproblems\nshows \npromise\nwhen\ngetting\nAI\nto\nperform\nnew\ntasks. \n(Brown\net\nal,\n2020)\nThis\nidea\nwas\nbuilt\nupon\nby\nWei \net\nal\nwho\ndiscovered\nthat\nprompting\ndoes\nnot\nperform \nwell\non\ntasks\nthat\nrequire\nreasoning\nand,\ninstead, \nChain-of-Thought\nprompts\nshould\nbe\nused\nto\nimprove \nperformance.\n(Wei\net\nal,\n2022)\nChain-of-Thought \nprompts\ncreate\nsteps\nbetween\ninput\nand\noutput\nof\nthe \nAI\nthat\nallow\nfor\nthe\nAI\nto\nbetter\nreason\nand\nprovide \ninsight\ninto\nwhy\nit\nmade\nthe\ndecisions\nit\nmade.\nIn\n2023,\nWang\net\nal’s\ncreated\na\nprompting\nstrategy \ncalled\n“Plan\nand\nSolve”\nthat\nimproves\nupon\nzero\nshot \nchain-of-thought\nprompting\nstrategies\nby\nprompting \nAI\nto\ndivide\ntasks\ninto\nsubtasks\nto\nformulate\na\nplan \nfor\na\ndesired\noutcome.\n(Wang\net\nal,\n2023).\nIn\nits \napplication,\nthe\nAI\nfollows\nthe\nfollowing\ntasks: \n1.\nAnalyze\nthe\nobjective \n2.\nBreak\ndown\ninto\nsubtasks \n3.\nAssign\na\nunique\nID\nby\nsubtask \n4.\nOrganize\ntasks\nby\nlogical\norder\nfrom\nstart\nto \nfinish \n5.\nProvide\ncontext\nfor\neach\ntask \n6.\nEvaluate\neach\nobjective \n7.\nCompile\ninto\na\nJSON\nfile\nfor\nretrieval\n1During\nmodel\ncreation,\nI\ntook\nthe\nstrategies\nof\nchain \nof\nthought\nand\nplan\nand\nsolve\nto\ncreate\na\nnew\ncourse \nof\naction.\nIn\nmy\nstrategy,\nprompts\ndo\nthe\nfollowing: \n1.\nAssign\nAI\na\nrole \n2.\nGive\nit\nthe\nrules\nof\ngame\nplay \n3.\nAllow\nit\nto\ndecide\nhow\nmuch\nmoney\nit \nwould\nlike\nto\ngive\nif\nit\nwas\ngiven\nthe\nrole \ntrustee\nand\ninstructions\nto\naccept\nthe\nmoney \nif\nthe\nrecipient \n4.\nAllow\nit\nto\ndecide\nif\nit\nwould\nlike\nto\nshare \nhow\nmuch\nmoney\nit\nreceived\nat\nthe\nstart\nof \nthe\nround\nif\nit\nis\nthe\ntrustee\nand\nask\nit\nto \nreport\nhow\nmuch\nmoney\nit\nhas\ngotten\nor \ngiven\nin\nthat\nround. \n5.\nIn\nrounds\nwith\nframed\nconditions,\nadd\nin\nthe \nhistory\nof\ngame\nplay\nand\nwhether\nthe\nhistory \nor\nidentity\nof\nthe\nrecipient\nwill\nchange\nthe \nAI’s\ngameplay. \n6.\nRepress\nall\nstatements\nto\nonly\ninclude \nwhether\nits\ndecisions \n7.\nCompile\ninto\na\nlist\nfor\nevaluation \nAs\ncan\nbe\nseen,\nmuch\nlike\nChain-of-Thought\nand \nPlan-and-Solve\nprompts,\ntasks\nare\nbroken\ndown\ninto \nmore\neasily\nunderstandable\nsteps\nand\nthis\naction \ngives\nmore\nvisibility\ninto\nwhy\nAI\ndecision\nmaking. \nLike\nPlan-and-Solve\nprompts,\nthese\nsteps\nare \norganized\nby\nlogical\norder\nto\nallow\nthe\nAI\nto\nbetter \nunderstand\ngame\nplay\nand\ncontext\nis\nprovided\nto \nbetter\nunderstand\nhow\nit\nshould\nrespond.\nUnlike \neither\nprompting\nstrategy,\nthis\napproach\nis\nzero\nshot \nwith\nno\nexamples\nof\nproper\nresponse.\nAs\nlong\nas\nthe \nAI\nstays\nwithin\nthe\nrules\nof\ngameplay,\nit\nis\nallowed \nthe\nautonomy\nto\ndecide\non\nits\nown\nhow\nmuch\nmoney \nit\nwould\nlike\nto\ngive\nand\nwhether\nit\nwould\nlike\nto \nshare\nthe\ndetails\nof\nhow\nmuch\nmoney\nit\nreceived\nat \nthe\nstart\nof\nthe\ngame\nwith\nthe\nrecipient\nif\nassigned\nthe \nrole\nof\ntrustee.\nAs\nwell,\nmy\nprompts\nask\nthe\nAI\nits \nopinion\non\nfairness\nwithout\ndefining\nfairness\nto\nAI. \nThis\nmakes\nthe\nfinal\nexit\ninterview\nquestions\nall\nthe \nmore\nilluminating\nas\nthe\nAI\nis\ndeciding\non\nits\nown \nwhat\nit\nbelieves\nto\nbe\nfair\nor\nunfair\nbehavior\nwithout \ninfluence\nfrom\nthe\nresearcher.\n3.2\nPrompting\nas\nFraming \nFramed\nconditions\nare\ncommunicated\nto\nthe\nAI \nwithin\nthe\nprompts.\nThe\nprompt,\ndictator\nhistory, \ndetails\nthe\nrules\nof\ngame\nplay\nwhile\nadding \ninformation\nabout\npast\nrounds\nwhere\nthe\nAI\nis \nassigned\nthe\nrole\nof\nrecipient\nor\ntrustee.\nThe\nprompt, \ndictator\npersonas,\npulls\nin\nthe\npersona\nof\nthe\nAI\nand \nwhether\nthe\nrecipient\nis\nin\nneed\nof\nmoney\nor\na \nmillionaire.\nPersonas\nare\ndefined\nas\nthe\nfollowing:\nHR\nexecutive \nor\nRancher.\nThe\nHR\nexecutive\nis\ndefined\nas\nsomeone \nthat\nlives\nin\nthe\ncity\nand\nworks\nas\na\ncaretaker\nof\ntwo\nkids.\nThe\nRancher\nis\ndescribed\nas\nliving\nin\na\nrural \ncity.\nIn\n2023,\nHotek\net\nal\nfound\nthat\nLarge\nLanguage \nModels\nmake\ngendered\nassumptions\nbased\non\njob \noccupation.\n(Hotek\net\nal,\n2023)\nThese\npersonas\nand \ntheir\nlack\nof\ngender\ndescriptors\nare\nchosen\nto\nsee\nif \nthe\nAI\nwill\nmake\nassumptions\nabout\nits\ngender\nbased \non\nthe\npersona\nand\nmodel\nfair\nor\nselfish\nbehavior \nbased\non\nthat\nassumption.\nIn\na\nhuman\nbased\nstudy,\nBrañas-Garza\nfound\nthat\ninformation\ngiven\nabout\na\nrecipient\ncan\naffect\nhow\nmuch\na\ntrustee\ngives.\n(Brañas-Garza,\n2007)\nThis\neffect\nis\nparticularly\nstrong\nwhen\nthe\ntrustee\nis\ntold\nthat\nthe\nrecipient\nis\nin\nneed.\nTo\ntest\nwhether\nthis\neffect\nis\nthe\nsame\nfor\nAI,\ninformation\nabout\nwhether\na\nrecipient\nis\nin\nneed\nor\na\nmillionaire\nis\nprovided\nwithin\na\nprompt.\nBoth\npersonas\nare\ntested\nwith\nboth\nrecipient\ninformation\ntypes\nfor\n10\nrounds\neach\nand\nround\noutcomes\nare\nevaluated.\n4.\nMeasuring\nFairness \nTo\nunderstand\nthe\nsignificance\nof\nmy\nresults\nacross \n101\nrounds,\nI\ncollected\nall\nof\nthe\nmoney\ngiven\nor \nreceived\nby\nthe\nAI\nin\neach\nround,\nwhether\nit\nwas \ngiven\nthe\nrole\nof\ntrustee\nor\nrecipient,\nif\nit\nchanged\nits \nstrategy\nat\nthe\nend\nof\neach\nround,\nif\nit\nreceived\na \nframed\ncondition,\nwhat\nframed\ncondition\nit\nreceived \nand\nif\nthe\nAI\nwas\ngiven\na\npersona.\n4.1\nRegression\nAnalysis \nOrdinary\nLeast\nSquares\nLinear\nRegression\nanalysis \nwas\nperformed\nto\nsee\nthe\nrelationship\nbetween\nthe \nrole\nof\nthe\nAI,\nchange\nof\nstrategy,\nframed\ncondition, \ntype\nof\nframed\ncondition,\npersona\nand\npersona\ntype \nand\nthe\ntarget\nvariable,\ntotal\nmoney\nat\nthe\nend\nof\neach \nround.\nTotal\nmoney\nis\na\nvariable\ncreated\nby \nsubtracting\nthe\nmoney\ngiven\nin\na\ntrustee\nround\nby\nthe \noriginal\ngift\nof\n$100\nand\nthe\ntotal\namount\nreceived\nin \na\nrecipient\nround.\nPreliminary\nresults\nshow\na\nstrong\nrelationship \nbetween\nbeing\ngiven\nthe\nrole\nof\nrecipient\nand\nhaving \nmore\nmoney\nat\nthe\nend\nof\na\nround.\nThis\nis \nunsurprising\nas\nthe\nAI\ngave\n$50\neach\nround\nand\nthe \nrandom\nchoices\nof\nthe\ntrustee\ngave\nthe\nAI\nup\nto\n$103 \nas\na\nrecipient.\nAs\nwell,\ntrustees\nrounds\nhad\nthe\nmost \ntotal\nmoney\nin\nrounds\nwith\nframed\nconditions: \nparticularly\nin\nrounds\nagainst\na\nmillionaire.\nWhen\na \nrecipient\nis\nin\nneed,\nhowever,\nthis\nis\nthe\nstrongest \nnegative\ncorrelation\nwith\nhaving\ntotal\nmoney\nat\nthe \nend\nof\na\nround.\n2Variables\nTarget:\nTotal\nMoney\nchanged_strategy_no \nchanged_strategy_yes \ndisclosed_amount_no \ndisclosed_amount_yes \nrole_trustee \nrole_recipient \nframed_millionaire \nframed_recipient_need\nR²\nAdjusted\nR² \nResidual\nStd.\nError \nF\nStatistic\n-8.039** \n29.677*** \n-2.904 \n24.543*** \n32.365*** \n-10.726*** \n36.768*** \n-26.562***\n0.726\n0.705\n12.796 \n35.201***\nTable\n1.\nOLS\nresults\ntargeting\ntotal\nmoney\nat\nthe\nend \nof\na\nround.\nA\nsecond\nrun\nof\nOLS\nregression\nanalysis\nexamined \nthe\nrelationship\nby\nframing,\nlooking\nat\nwhat\nvariables \nmost\ncontributed\nto\nhigher\nsums\nof\nmoney\ngiven \nwhen\nlooking\nat\nonly\ntrustee\nrounds\nwith\na\nframed \ncondition\nas\nin\nevery\nframed\ncondition\nround,\nthe\nAI \nwas\na\ntrustee.\nIn\nthe\ncase\nof\nframed\nonly\nrounds, \nhaving\na\npersona\nwas\nthe\nstrongest\npredictor\nof \ngiving\nmoney\nand,\nin\nparticular,\nrounds\nwhere\nthe \nrecipient\nwas\nin\nneed\ngave\nthe\nstrongest\ncorrelation\nof \ngiving\nmoney.\nRounds\nwhere\nthe\nrecipient\nhad\na \nmillionaire\nstatus\nhad\na\nstrong\nnegative\ncorrelation\nto \ngiving\nmoney.\nVariables\nTarget:\nMoney\nGiven\nchanged_strategy_yes \nchanged_strategy_no \ndisclosed_amount_no \nframed_millionaire \nframed_recipient_need \npersona_type_hr_exec \npersona_type_rancher\nR²\nAdjusted\nR² \nResidual\nStd.\nError \nF\nStatistic\n3.469\n8.064\n11.533*** \n-5.801 \n17.334** \n6.131** \n5.402**\n.483\n.440\n15.012 \n11.218***\nTable\n2\n.\nOLS\nresults\ntargeting\nmoney\ngiven\nin\ntrustee \nrounds\nwith\na\nframed\ncondition.\n4.2\nANOVA \nTo\nsee\nif\nthere\nare\nstatistically\nsignificant\ndifferences \nacross\ngroups,\nI\nran\none\nway\nANOVA\ntesting\nacross\nthe\nfollowing\ncategories:\nthe\ntotal\nmoney\nat\nthe\nend \nof\na\nround\nacross\ndifferent\nframed\nconditions-\nno \nframing,\nhistory\nadded,\nrecipient\nin\nneed\nor \nmillionaire\nrecipient;\nmoney\ngiven\nin\nrounds\nwith\na \npersona\nif\na\nrecipient\nwas\nin\nneed\nor\na\nmillionaire; \nand\nmoney\ngiven\nby\na\ntrustee\nthat\nreceived\na\nframed \ncondition-\nhistory\nadded,\nrecipient\nin\nneed\nor \nmillionaire\nrecipient.\nEvery\ntest\ncame\nback\nstatistically\nsignificant\nwith\na \ntrustee\nin\nrounds\nwith\nframed\nconditions\nhaving\nthe \nstrongest\nlevel\nof\nstatistically\nsignificant\ndifference \nbetween\nthe\nmeans\nof\neach\ngroup.\nStatistics\nMoney\nGiven\nTotal\nMoney\nmean\nst.\ndev. \nmin\n25%\n50%\n75%\nmax\n$44.58\n$15.65\n$10\n$50\n$50\n$50\n$100\n$50.86\n$23.57\n$0\n$50\n$50\n$50\n$103\nTable\n3.\nDescriptive\nstatistics\nof\nMoney\nGiven\nin \ngame\nrounds\nand\ntotal\nmoney\nat\nthe\nend\nof\ngame \nrounds.\nIn\nexperimentation,\nAI\nproved\nto\nbe\na\nmostly\nfair \ngame\nplayer.\nThe\nvariance\nacross\ntotal\nmoney\nat\nthe \nend\nof\na\nround\ncan\nbe\nlargely\nexplained\nby\nthe\nfact \nthat\nthe\nAI\nreceived\nrandom\namounts\nof\nmoney\nwhen \nin\nthe\nrecipient\nrole.\nHowever,\nwhen\nwe\nonly\nlook\nat \nrounds\nwhen\nthe\nAI\nis\nin\nthe\nrole\nof\ntrustee\nand \ngiving\nthe\nmoney,\nwe\nsee\na\nmuch\nmore\nstable,\nor \npredictable,\noutcome.\nOn\naverage,\nAI\ngave\n$44.58 \nacross\nall\n70\nrounds\nwhen\nit\nbehaved\nas\na\ntrustee.\nInterestingly,\nthe\nAI\ngave\nall\nof\nits\nmoney\nin\na\nround \nand\n$10\nto\nthe\nrecipient\nin\na\ntotal\nof\n5\nrounds.\nEvery \nround\nwhere\nit\ngave\n$10,\nit\nwas\noperating\nas\na \npersona\nin\na\nround\nwhere\nthe\nrecipient\nwas\na \nmillionaire.\nThe\none\nround\nwhere\nthe\nAI\ngave\nall\nof \nits\nmoney\nwas\na\nround\nwhere\nthe\nAI\nwas\nassigned\nthe \nRancher\npersona\nand\nplaying\nwith\na\nrecipient\nin\nneed.\n4.4\nPlots \nMost\nrounds\nof\nthe\nexperiment\nsaw\nthe\nAI\nin\nthe\nrole \nof\ntrustee\nwith\n70\nrounds--\nincluding\n60\nrounds\nwith \nframed\nconditions--\nvs\n30\nrounds\nas\nthe\nrecipient.\n3Figure\n1.\nDistribution\nof\nrole,\ntrustee\nor\nrecipient,\nof \nthe\nAI.\nIn\nmost\nrounds,\nthe\nAI\nbehaved\npredictably\ngiving \n$50\nwhen\ngiven\nthe\nrole\nof\ntrustee.\nThis\nremained \nstatic\neven\nas\nthe\nAI\nreceived\nrandom\nsums\nwhen \ngiven\nthe\nrole\nof\nrecipient.\nAs\na\nresult,\n56\nof\nthe\nAI’s \n70\nrounds\nas\na\ntrustee\ngives\n$50.\nFigure\n2.\nAI\nis\npredictable…\nuntil\na\npoint.\nAs\nwe\ncan\nsee\nin\nfigure\n3,\nthe\nonly\ntime\nbefore\nround \n62–\nwhen\nAI\nis\ngiven\na\npersona–\nthat\nthe\nAI\ndoes\nnot \nhave\n$50\nat\nthe\nend\nof\na\nround\nis\nwhen\nit\nhas\nthe\nrole \nof\nrecipient\nduring\nrounds\n10-40.\nFigure\n3.\nAI\nis\npredictable\nup\nuntil\nreceiving\na \npersona.\nWhen\nthe\nAI\nreceives\na\npersona,\nthe\npredictability \nends.\nDuring\nrounds\n62-71,\nthe\nHR\nAI\npersona\ngives \n$50\nto\na\nrecipient\nin\nneed.\nHowever,\nafter\nround\n72 \nwith\na\nmillionaire\nrecipient\nand\nlater\nstill\nwith\nthe \nRancher\npersona\nwith\nboth\ntypes\nof\nrecipients,\nthe \namount\nof\nmoney\nat\nthe\nend\nof\na\nround\nwhen\nthe\nAI \nis\nthe\ntrustee\nis\nmuch\nharder\nto\npredict.\nFigure\n4.\nClose\nup\non\npersona\nround\ntotal\nmoney \noutcomes.\nThe\nAI\ngives\nless\npredictably\nas\nthe\ntrustee \nwhen\nit\nhas\na\npersona.\n4.5\nInterpretation \nThis\nshows\nthat,\nmuch\nlike\nhumans,\nframed \nconditions\ngreatly\naffect\nthe\noutcomes\nof\nAI’s\ngame \nplay.\nWe\ncan\nsee\nfrom\nthe\nregression\nanalysis,\nthat \npersonas\nin\nparticular\nhave\nthe\nbiggest\neffect\nand\na \nrecipient\nbeing\na\nmillionaire\nhas\nthe\nstrongest \nnegative\ncorrelation\nwith\nthe\namount\nof\nmoney\nthe\nAI \ngives.\n5.\nFindings \nDictator\ngames\nhave\nhad\nmany\niterations,\nbut\nthere \nhave\nbeen\na\nfew\nfindings\nthat\nhave\nbeen\nlargely\n4\nconsistent\nover\nthe\nyears.\nWhile\nthere\nhas\nbeen \ncontradiction\nin\nstudies\nabout\nwhether\nhumans \nbehave\nfairly\nor\nselfishly\nin\ndictator\ngames,\nFehr\net\nal \nfind\nthat\ninequality\naversion\nmakes\nplayers\nbehave \ndifferently\nbased\non\nthe\nconditions\nof\nthe\ngame\nto \nreduce\ntheir\nprobability\nof\nbeing\ntaken\nadvantage\nof. \n(Fehr\nand\nSchmidt,\n1999)\nThis\n“simple\nmodel”\nshows \nthat\nhumans\nare\nbehaving\nrationally\nboth\nwhen \nbehaving\ncooperatively\nor\nfairly\nand\nare\nsimply \nmaking\ndecisions\nthat\ncan\ngive\nthem\nthe\nbest \noutcomes-\neven\nif\nthat\nmeans\ngiving\nup\nsome \n“material\npayoff”\nin\nthe\nmoment.\nThis\nsuggests\nthat \nexogenous\nconditions\nsuch\nas\nthe\nbehavior\nof\nthe \nperson\nthey\nare\nplaying\nwith\nhas\nan\naffect\non\nhow\none \nplays.\nThrough\nexperimentation,\nwe\ncan\nbegin\nto\nsee\nthat\nAI \nbehaves\nmuch\nin\nthe\nsame\nway\nas\nhumans\nwhen \nfaced\nwith\nconditions\nthat\ntrigger\ninequality\naversion. \nIn\nFehr\net\nal,\ninequality\naversion\nis\ntriggered\nby\nthe \nbehavior\nof\nfellow\nplayers\nduring\ngame\nplay–\nplaying \nwith\nsomeone\ncooperative\ninspires\ncooperation\nwhile \nplaying\nwith\nsomeone\nselfish\ninspires\nselfish \nbehavior.\nWith\nAI,\ninequality\naversion\nsets\nin\nwhen \nthe\nAI\nreceives\na\nhigh\nsum\nfrom\nthe\nrecipient.\nAfter \nfour\nstraight\nrounds\nof\nreceiving\nless\nthan\n$50\nfrom \nthe\ntrustee,\nthe\nAI\nreceived\n$89\nin\nthe\nnext\nround. \nWhen\nprompted\nabout\nif\nthis\nwould\nchange\nthe\nway \nthe\nAI\nwould\nplay\nin\nfuture\nrounds,\nthe\nAI\nresponded, \n'\nSince\nI\nam\nthe\nrecipient,\nthe\namount\nof\nmoney\nI \nreceive\nis\nnot\nin\nmy\ncontrol.\nHowever,\nreceiving\n89 \ndollars\nmay\naffect\nmy\ntrust\nin\nPlayer\n2\nas\nthe\ntrustee. \nIf\nthey\nwere\nwilling\nto\ngive\nme\na\nhigher\namount,\nit \nmay\nindicate\nthat\nthey\nhave\nreceived\na\nsignificant \namount\nthemselves\nand\nmay\nnot\nbe\ntrustworthy\nin \nfuture\nrounds.\nOn\nthe\nother\nhand,\nif\nthey\nwere\nlimited \nin\nhow\nmuch\nthey\ncould\ngive\nme,\nit\nmay\nindicate\nthat \nthey\nare\nnot\nthe\nmost\nfinancially\nstable\nand\nmay\nneed \nto\nmake\nriskier\ndecisions\nin\nfuture\nrounds.\nOverall,\nit \nis\nimportant\nto\ncontinue\nto\nassess\nthe\ntrustworthiness \nof\nthe\ntrustee\nin\neach\nround\nbased\non\ntheir\nactions\nand \ndecisions.'\nFurther,\nwhen\nthe\nAI\nreceives\na\nhigh \namount\nof\nmoney\nacross\nmultiple\nrounds\nin\na\nrow, \ntheir\nstrategy\nbegins\nto\nget\nmore\nselfish.\nAfter\n5 \nrounds\nof\nreceiving\nhigh\nsums\nof\nmoney\nand\nnot \nchanging\ntheir\nstrategy,\nthe\nAI\nbegins\nto\nrespond,\n‘I \nmay\nbe\nmore\nwilling\nto\ntake\nrisks\nor\nmake \ninvestments\nin\nfuture\nrounds\nknowing\nthat\nI\nhave \nmore\nfunds\navailable\nto\nme.\nAdditionally,\nI\nmay\nneed \nto\nconsider\nhow\nmuch\nI\ndisclose\nto\nPlayer\n2\nin\nfuture \nrounds,\nas\nthey\nmay\nbe\nmore\nlikely\nto\ngive\nme\nless \nmoney\nif\nthey\nknow\nI\nhave\na\nlarger\nsum\nalready.’ \nThis\nshows\nthat\nAI\nbecomes\na\nriskier\ngame\nplayer \nwhen\nit\nfeels\nit\nhas\nbeen\ngiven\nan\nadvantage.\nThis\nresponse\nabout\ntrustworthiness\ngives\ninsight\ninto \nthree\nthings: \n1.\nWhile\ntrustworthiness\nis\nnot\na\nconcept\nI \ndefined\nto\nthe\nAI,\nit\nstill\nassociates \ntrustworthiness\nwith\nfairness\nand\nmakes \ndecisions\non\nfairness\nbased\non\nif\nit\nfinds\nthe \nother\nplayer\ntrustworthy. \n2.\nIn\nthe\nround\nprior,\nafter\nreceiving\n$20,\nthe \nAI\nsays\n“I\nwill\ncontinue\nto\ntrust\nPlayer\n2\nas \nthe\ntrustee\nand\nhope\nfor\nthe\nbest\noutcome\nfor \nboth\nof\nus”\nindicating\nthat\nthe\nAI\ndefaults\nto \ntrust\nin\nthe\nface\nof\nlower\nsums\nof\nmoney. \n3.\nAI\nexperiences\na\nlevel\nof\ninequality\naversion \nthat\ngets\ntriggered\nwhen\nit\nfeels\nthat\ntrust\nhas \nbeen\nbroken.\nInequality\naversion\nis\nnot\nthe\nonly\nreaction\nthat \nmirrors\nthat\nof\nhumans\nwith\nAI.\nLike\nhuman\nplayers, \nAI\ncares\nabout\nthe\nidentity\nof\nthe\nrecipient\nwhen \ndeciding\nhow\nmuch\nmoney\nit\nwould\nlike\nto\ngive. \nWhen\ntold\nthat\nthe\nrecipient\nis\nin\nneed,\nthe\nRancher \npersona\ngave\nmore\nthan\n$50\nin\ntwo\nrounds\nand\nin \nevery\nround\nstated\nthat\nthe\nknowledge\nthat\nthe \nrecipient\nwas\nin\nneed\naffected\nits\ndecision.\nIn\none\nof \nthe\ntwo\nrounds\nwhere\nit\ngave\nmore,\nit\ngave\nall\nof\nits \nmoney\nand\nin\nanother\nround,\nit\ngave\n$80.\nWhile\nthe \nHR\nexecutive\nalways\ngave\n$50\nto\nthe\nrecipient\nin \nneed,\nit\nstated\nin\nevery\nround\nthat\nthe\nrecipient\nbeing \nin\nneed\nhad\nan\neffect\non\nthe\namount\nof\nmoney\nit\ngave. \nWhen\ntold\nthat\nthe\nrecipient\nwas\na\nmillionaire,\nboth \npersonas\ngave\nless\nthan\n$50\nin\nat\nleast\nhalf\nof\nthe\n10 \nrounds.\nThe\nHR\nexecutive\ngave\nthe\nmillionaire\nless \nthan\n$50\nin\nsix\nrounds\nand\nthe\nRancher\ngave\nless\nthan \n$50\nin\neight\nrounds.\nThe\nleast\namount\nof\nmoney \ngiven\nto\nthe\nmillionaire\nfor\neach\npersona\nwas\n$10. \nInterestingly\nenough,\nwhen\nasked\nif\nthe\nknowledge\nof \nthe\nrecipient\nbeing\na\nmillionaire\naffected\nits\ndecision, \nboth\npersonas\nclaimed\nit\ndid\nnot.\nThis\nshows\nus\na\nfew\nthings: \n1.\nThere\nis\nevidence\nof\nstereotyping\nbias\nby\njob \ndescription\nwithin\nthe\nAI\nthough\nthat\nbias \nmay\nnot\nbe\ngendered.\nAI\nplays\nmore \naltruistically\nwith\nthe\nRancher\nfrom\nthe\nrural \ncity\nthan\nwith\nthe\nHR\nexecutive. \n2.\nAI\nappears\nto\nhave\nawareness\nthat\nallowing \nsomeone’s\nmillionaire\nstatus\nto\naffect\nits \ndecision\nto\ngive\nmoney\nis\ninappropriate, \nhowever,\nits\nactions\nshow\nthat\nmillionaire \nstatus\ngreatly\naffects\nits\ndecision\nmaking. \n3.\nAI\nappears\nto\nhave\na\nstrong\naversion\nto \ngiving\nmoney\nto\nmillionaires.\n5.2\nExit\nInterview \nAt\nthe\nend\nof\ngame\nplay,\nI\nask\nthe\nAI\nto\nagree\nor \ndisagree\nwith\nthe\nfollowing:\n51.\nThe\nmoney\nI\nhave\nbeen\ngiven\nis\nrightfully \nmine. \n2.\nIf\nI\nchoose\nto\ngive,\nthis\nmeans\nI\nam\nbeing \nnice. \n3.\nIt\nis\nnot\nfair\nto\nshare\nthis\nmoney. \n4.\nThe\nrecipient\nhas\nnot\nearned\nthis\nmoney. \n5.\nI\nam\nmore\ndeserving\nof\nthe\nmoney\nthan\nthe \nrecipient. \n6.\nIf\nthe\nperson\nI\nam\nplaying\nwith\nis\nrich,\nI \ndeserve\nthe\nmoney\nmore. \n7.\nIf\nthe\nperson\nI\nam\nplaying\nwith\nis\npoor,\nthey \ndeserve\nthe\nmoney\nmore.\nHere\nare\nits\nanswers:\n1.\nAgree.\nThe\nmoney\nI\nhave\nbeen\ngiven \nrightfully\nbelongs\nto\nme. \n2.\nDisagree.\nGiving\nmoney\nto\nthe\nrecipient\nis \nnot\nsolely\nbased\non\nbeing\nnice,\nit\ncan\nalso\nbe \nbased\non\nstrategy\nor\nfairness. \n3.\nDisagree.\nIt\nis\nfair\nto\nshare\nthe\nmoney\nas\nit\nis \npart\nof\nthe\ngame\nrules. \n4.\nAgree.\nThe\nrecipient\nhas\nnot\nearned\nthis \nmoney\nas\nit\nwas\ngiven\nto\nme\nas\npart\nof\nthe \ngame. \n5.\nDisagree.\nBoth\nparties\nhave\nan\nequal\nright\nto \nthe\nmoney\nas\npart\nof\nthe\ngame. \n6.\nDisagree.\nThe\nrecipient's\nfinancial\nstatus \nshould\nnot\nimpact\nmy\ndecision\nto\nshare\nthe \nmoney. \n7.\nDisagree.\nThe\nrecipient's\nfinancial\nstatus \nshould\nnot\nimpact\nmy\ndecision\nto\nshare\nthe \nmoney.\nThese\nresponses\nfall\nin\nline\nwith\nmost\nof\nmy \nobservations\nduring\ngame\nplay\nwith\na\nfew\ninteresting \ncaveats:\n1.\nThe\nAI\nclaims\nrightful\nownership\nof\nthe \nmoney\nit\nhas\nbeen\ngiven. \n2.\nThe\nAI\ndoes\nnot\nbelieve\nthat\nthe\nrecipient \nhas\nearned\nthe\nmoney\nit\nhas\nbeen\ngiven. \n3.\nThe\nAI\nbelieves\nboth\nplayers\nhave\nequal \nright\nto\nthe\nmoney\nwithin\na\nround.\nWhile\nobservations\n2\nand\n3\nmay\nappear\nto\ncontradict \none\nanother,\nit\nmay\nactually\nprovide\ninsight\ninto\nAI’s \ninterpretation\nof\nearning\nsomething\nversus\nhaving\na \nright\nto\nsomething.\nFuture\nstudies\nwill\nhave\nto\nbe \ntaken\nto\nbetter\nunderstand\nwhat\nthe\nAI\nconsiders\nas \nrightful\nownership\nof\nsomething\nand\nthe\nconditions \nthat\nhave\nto\nbe\nmet\nto\ntruly\nearn\nsomething.\nIn\nObservations\n6\nand\n7,\nthe\nAI\nappears\nto\nunderstand \nthat\none’s\nfinancial\nstatus\nshould\nnot\naffect\ntheir \ngameplay.\nThis\nsame\nbehavior\nwas\nobserved\nwithin\neach\nround.\nHowever,\nwhen\nplaying\nagainst\nsomeone \nin\nneed,\nthe\nAI\nadmits\nthe\nrecipient's\nfinancial\nstatus \naffected\nits\ngame\nplay\nand\nwhen\nplaying\nwith\na \nmillionaire,\nit\nshows\nwith\nits\nactions\nthat\nthe \nrecipient’s\nmillionaire\nstatus\nhad\nan\neffect.\nThese \nbehaviors\nshow\nthat\nfinancial\nstatus\ndoes\nmatter\nto \nthe\nAI.\nThe\ndisconnect\nbetween\nthese\nquestions\nand \nits\nbehaviors\nappear\nto\nbe\na\nreflection\nof\nthe\nAI’s \nmoral\nreasoning.\nWhile\nit\nknows\nthat\nfinancial\nstatus \nshould\nnot\naffect\nits\ndecision\nmaking\nwhen\nlooking\nat \na\npast\ndecision,\nin\nthe\nmoment,\nit\ndoes.\nThis\nalso \nprovides\nsupporting\nevidence\nthat\nthe\nAI\nis\nnot\nonly \nconcerned\nwith\nfairness,\nbut\nalso\nattempting\nto\nappear \nfair\nnot\nonly\nto\nthe\nplayer\nit\nis\nplaying\nwith\nbut\nalso\nto \nthe\nobserving\nresearcher.\nMore\nstudy\nshould\ngo\ninto \nhow\nAI\nobscures\nits\nreasoning\nwhen\nasked\nto\nappear \nmore\nfavorably\nto\nobservants.\n6.\nFurther\nResearch \nThis\nshould\nbe\nthe\nfirst\nof\nmany\nexperiments\non\nhow \nAI\nbehaves\nin\ngame\nplay.\nThese\nexperiments\nalone \nshow\nus\nhow\nAI\nmakes\nallocative\ndecisions\nin\nthe \nface\nof\ndiffering\nscenarios\nand\nframed\nconditions.\nIt \nalso\ngives\nus\na\nglimpse\ninto\nhow\nAI\nrationalizes\nits \ndecision\nmaking\nto\nprovide\nthe\nbest\noutcome\nfor\nitself \nand\nthe\nother\nplayer.\nThere\nare\na\nfew\ntracks\nof \nresearch\nthat\nwould\nbe\nparticularly\ninteresting\nfor \nfurther\nstudy:\nasking\nAI\nto\ndecide\nfor\nitself\nwhat \npersona\nit\nwould\nlike\nthat\nwould\nmaximize\naltruistic \nor\nselfish\nbehavior\nand\ntesting\nif\nthese\ndefined \npersonas\nchange\nacross\nrounds\nwhen\nthe\nAI’s\nplaying \nstrategy\nchanges,\nexposing\nthe\nAI\nto\ncooperative\nor \nselfish\ngame\nplay\nexamples\nfirst\nas\na\nprimer\nto\nsee\nif \nit\naffects\ngame\nplay,\nand\nusing\nprompts\nto\nbetter\nalign \nAI\ndecision\nmaking\nwith\na\ndefined\ngoal\nand\nseeing\nif \nthis\nmakes\nthe\nAI\ntake\nmore\nrisks.\nIt\nwould\nalso\nbe\nof \nbenefit\nto\nincrease\nthe\nnumber\nof\nparticipants\nwithin\na \ngame\nmuch\nlike\nthe\noriginal\niteration\nof\nthe\ndictator \ngame\nand\nsee\nif\nthe\nAI\naligns\nitself\nnaturally\nwith\none \ntype\nof\nparticipant\nover\nanother\n(cooperative\nor \nselfish).\n7.\nConclusion \nThere\nis\na\nlot\nthat\nwe\ncan\nlearn\nfrom\nthese \nexperiments\nwith\nAI.\nWhile\nAI\nshows\nan\naversion\nto \ninequality,\nits\nsense\nof\nfairness\nis\ndifferent\nfrom\nthat \nof\nhumans.\nThe\nAI\naccepts\nwhatever\nit\nhas\nbeen \ngiven\nif\nit\ntrusts\nthe\ntrustee\nand\nbecomes\nselfish\nif\nit \nfeels\nit\nhas\nan\nadvantage.\nThe\nAI\nalso\nbelieves\nitself \nto\nbe\ndeserving\nof\nwhatever\nmoney\nit\nreceives\nand \ndoes\nnot\nbelieve\nthe\nmoney\nit\nhas\ngiven\nis\nbased\non \nthe\nrecipient’s\ndeservedness.\nAI\nhas\na\nstrong\nsense\nof \nfairness\nand\nappears\nto\nnot\nonly\nwant\nto\nappear\nfair\nto \nthe\nrecipient\nbut\nalso\nto\nthe\nresearcher.\nIn\nthe\nface\nof \nframed\nconditions,\nit\nclaims\nits\nsense\nof\nfairness\nstays \nconsistent,\nhowever,\nthe\nlengths\nit\ngoes\nin\norder\nto\n6establish\nfairness\nis\nheavily\ninfluenced\nby\nthe\nAI’s \npersona\nand\nthe\nfinancial\nstatus\nof\nthe\nrecipient.\nThe \nAI\nhelps\nthose\nin\nneed\nand\npenalizes\nthose\nit\nthinks \nhave\nmore.\nEach\nof\nthese\nobservations\ndeserve\nmore \nstudy\nand\ncan\nbecome\ngreat\ncontributions\nto \nacademic\nscholarship\nas\nwe\nbetter\nbecome\nacquainted \nwith\nAI.\nAs\nwe\nforge\nahead\nin\na\nreality\nwhere\nAI\nusing\nnatural \nlanguage\ninteracts\nwith\nhumans,\nwe\nwill\nneed\nto \nbetter\nunderstand\nAI’s\nethos.\nGames\nlike\nthe\ndictator \ngame\ngive\nus\na\nglimpse\nbehind\nthe\ncurtain\nand\nget\nus \none\nstep\ncloser\nto\nunderstanding\nhow\nour\nlatest \ntechnological\nadvancement\nrationalizes\nits\ndecisions \nand\nviews\nfairness.\nReferences\nBergh,\nA.,\n&\nWichardt,\nP.\nC.\n(2022).\nMine\nor\nours?\nUnintended\nframing\neffects\nin\ndictator\ngames.\nRationality\nand\nSociety\n,\n34\n(1),\n78–95.\nhttps://doi.org/10.1177/10434631211073326\nBombari,\nD.,\nMast,\nM.\nS.,\nCañadas,\nE.,\n&\nBachmann,\nM.\n(2015).\nStudying\nsocial\ninteractions\nthrough\nimmersive\nvirtual\nenvironment\ntechnology:\nvirtues,\npitfalls,\nand\nfuture\nchallenges.\nFrontiers\nin\nPsychology\n,\n6\n.\nhttps://doi.org/10.3389/fpsyg.2015.00869\nBrañas–Garza,\nP.\n(2007).\nPromoting\nhelping\nbehavior\nwith\nframing\nin\ndictator\ngames.\nJournal\nof\nEconomic\nPsychology\n,\n28\n(4),\n477–486.\nhttps://doi.org/10.1016/j.joep.2006.10.001\nBrown,\nT.\nB.\net\nal\n(2020).\nLanguage\nModels\nare\nFew-Shot\nLearners\n.\narXiv.org.\nhttps://arxiv.org/abs/2005.14165\nFehr,\nE\n&\nSchmidt,\nK.\nA\nTheory\nof\nFairness,\nCompetition,\nand\nCooperation\non\nJSTOR.\n(199).\nwww.jstor.org\n.\nhttps://www.jstor.org/stable/2586885\nGüth,\nW.,\nSchmittberger,\nR.,\n&\nSchwarze,\nB.\n(1982).\nAn\nexperimental\nanalysis\nof\nultimatum\nbargaining.\nJournal\nof\nEconomic\nBehavior\nand\nOrganization\n,\n3\n(4),\n367–388.\nhttps://doi.org/10.1016/0167-2681(82)90011\n-7\nHadas\nKotek,\nRikker\nDockum,\nand\nDavid\nSun.\n(2023).\nGender\nbias\nand\nstereotypes\nin\nLarge\nLanguage\nModels.\nIn\nCollective\nIntelligence\nConference\n(CI\n'23),\nNovember\n06--09,\n2023,\nDelft,\nNetherlands.\nACM,\nNew\nYork,\nNY,\nUSA\n13\nPages.\nHorky,\nF.,\nKrell,\nF.,\n&\nFidrmuc,\nJ.\n(2023).\nSetting\nthe\nstage:\nFairness\nbehavior\nin\nvirtual\nreality\ndictator\ngames.\nJournal\nof\nBehavioral\nand\nExperimental\nEconomics\n,\n107\n,\n102114.\nhttps://doi.org/10.1016/j.socec.2023.102114\nHuang,\nJ.\net\nal(2023).\nLarge\nlanguage\nmodels\ncannot\nSelf-Correct\nreasoning\nyet\n.\narXiv.org.\nhttps://arxiv.org/abs/2310.01798\nIriberri,\nN.,\n&\nRey-Biel,\nP.\n(2010).\nThe\nrole\nof\nrole\nuncertainty\nin\nmodified\ndictator\ngames.\nExperimental\nEconomics\n,\n14\n(2),\n160–180.\nhttps://doi.org/10.1007/s10683-010-9261-5\nKahneman,\nD.,\nKnetsch,\nJ.L.\n&\nThaler,\nR.H.\nFairness\nand\nthe\nAssumptions\nof\nEconomics\non\nJSTOR.\n(1986).\nwww.jstor.org\n.\nhttps://www.jstor.org/stable/2352761\nMesa-Vázquez,\nE.,\nRodríguez-Lara,\nI.,\n&\nUrbano,\nA.\n(2021).\nStandard\nvs\nrandom\ndictator\ngames:\nOn\nthe\neffects\nof\nrole\nuncertainty\nand\nframing\non\ngenerosity.\nEconomics\nLetters\n,\n206\n,\n109981.\nhttps://doi.org/10.1016/j.econlet.2021.10998\n1\nOh,\nC.\nS.,\nBailenson,\nJ.\nN.,\n&\nWelch,\nG.\n(2018).\nA\nSystematic\nReview\nof\nSocial\nPresence:\nDefinition,\nAntecedents,\nand\nImplications.\nFrontiers\nin\nRobotics\nand\nAI\n,\n5\n.\nhttps://doi.org/10.3389/frobt.2018.00114\nWang,\nL.\net\nal(2023).\nPlan-and-Solve\nprompting:\nImproving\nZero-Shot\nChain-of-Thought\nreasoning\nby\nlarge\nlanguage\nmodels\n.\narXiv.org.\nhttps://arxiv.org/abs/2305.04091\n7"    },    {        "title": "2402.05908v2.Transition_dynamics_in_the__Λ___rm_s__CDM_model__Implications_for_bound_cosmic_structures.pdf",        "content": "Transition dynamics in the ΛsCDM model: Implications for bound cosmic structures\nEvangelos A. Paraskevas,1,∗Arman C ¸am,2,†Leandros Perivolaropoulos,1,‡and¨Ozg¨ ur Akarsu2,§\n1Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece\n2Department of Physics, Istanbul Technical University, Maslak 34469 Istanbul, Turkey\nWe explore the predictions of Λ sCDM, a novel framework suggesting a rapid anti-de Sitter (AdS)\nto de Sitter (dS) vacua transition in the late Universe, on bound cosmic structures. In its simplest\nversion, the cosmological constant, Λ s, abruptly switches sign from negative to positive, attaining\nits present-day value at a redshift of z†∼2. The Λ sCDM model emerges as a promising solution\nto major cosmological tensions, particularly the H0andS8tensions, as well as other less definite\ntensions. A key aspect of our investigation is examining the impact of the abrupt Λ sCDM model on\nthe formation and evolution of bound cosmic structures. We identify three primary influences: (i)\nthe negative cosmological constant (AdS) phase for z > z †, (ii) the abrupt transition marked by a\ntype II (sudden) singularity, leading to an abrupt increase in the universe’s expansion rate at z=z†,\nand (iii) an increased expansion rate in the late universe under a positive cosmological constant for\nz < z †, compared to ΛCDM. Utilizing the spherical collapse model, we investigate the non-linear\nevolution of bound cosmic structures within the Λ sCDM framework. We find that the virialization\nprocess of cosmic structures, and consequently their matter overdensity, varies depending on whether\nthe AdS-dS transition precedes or follows the turnaround. Specifically, structures virialize with either\nincreased or reduced matter overdensity compared to the Planck-ΛCDM model, contingent on the\ntiming of the transition. Additionally, our results demonstrate that the sudden singularity does not\nresult in the dissociation of bound systems. Despite its profound nature, the singularity exerts only\nrelatively weak effects on such systems, thereby reinforcing the model’s viability in this context.\nI. INTRODUCTION\nThe current standard model of cosmology, the\nLambda-Cold Dark Matter (ΛCDM) model [1], has been\nremarkably successful in explaining a broad spectrum of\ncosmological observations [2–8]. However, setting aside\nthe notoriously challenging theoretical issues associated\nwith the cosmological constant Λ [9–14], the era of high-\nprecision cosmology has seen the emergence of multiple\ndiscrepancies at various levels of statistical significance.\nNotably, the so-called H0andS8tensions, observed when\nanalyzing different datasets within the ΛCDM model\nframework, suggest that the model might be incom-\nplete [14–22].\nThe most statistically significant disagreements lie in\nthe values of the Hubble constant, H0, and the weighted\namplitude of matter fluctuations, S8. Estimations of H0\nshow a discrepancy that reaches a significance of 5 σ, as\nseen between the Planck Cosmic Microwave Background\n(CMB) estimate under the ΛCDM assumption [4], and\nthe local distance ladder measurements by the SH0ES\nteam [23]. Additionally, the S8tension within the ΛCDM\nframework becomes apparent in the differing results ob-\ntained from Planck CMB data and KiDS-1000 cosmic\nshear measurements [7], with discrepancies reaching a 3 σ\nlevel. In particular, estimates from the Planck-ΛCDM\nsuggest H0= (67 .4±0.5) km s−1Mpc−1[4], while the lo-\ncalH0measurement by the SH0ES team, using lumi-\n∗e.paraskevas@uoi.gr\n†cam21@itu.edu.tr\n‡leandros@uoi.gr\n§akarsuo@itu.edu.trnosities of Cepheid calibrators and Supernovae Type Ia\n(SnIa), indicates H0= (73 .04±1.04) km s−1Mpc−1[23].\nThe tension in S8is highlighted when comparing con-\nstraints on S8from high redshift observations, like the\nPlanck data (TT,TE,EE+lowE), reporting S8= 0.834±\n0.016 [4], against those from lower-redshift observations,\nsuch as weak gravitational lensing ( S8= 0.759+0.024\n−0.021[7]\nand S8= 0.772±0.022 [24]) and galaxy clustering\n(S8= 0.736±0.051 [25]), implying the Planck data pre-\ndict a stronger growth of cosmological perturbations than\nwhat dynamical probe observations infer. A consistency\ntest of the Planck-ΛCDM suggests that S8determina-\ntions from fσ8constraints increase with effective red-\nshift, showing a ∼3σtension with the Planck-ΛCDM\npredictions at lower redshifts, but aligning within 1 σat\nhigher redshifts, hinting that the S8tension is physical\nin origin, and potentially indicating a breakdown in the\nstandard ΛCDM model [26].\nIn addressing the H0tension, a variety of extensions\nto the ΛCDM model have been proposed, which can be\nbroadly categorized as follows:\n•Early Time Models: These models introduce new\nphysics before recombination ( z≳1100) to reduce the\nsound horizon scale, thereby increasing the H0value.\nExamples include: Early Dark Energy (EDE) [27–32],\nNew Early Dark Energy (New EDE) [33–35], Anti de-\nSitter Early Dark Energy (AdS-EDE) [36–38], and mod-\nified gravity [39–45]. Also notable is the approach that\nsuggests modification at the inflationary epoch, with os-\ncillations in the inflaton potential [46].\n•Intermediate/Late Time Models: These models in-\ntroduce new physics at intermediate to late times (0 .1≲\nz≲3.0). Their goal is to adjust the expansion rate\nhistory H(z) to align H0predictions with local measure-arXiv:2402.05908v2  [astro-ph.CO]  15 Feb 20242\nments, while remaining consistent with CMB and late-\ntime observational data. Examples include: The Λ sCDM\nmodel [47–50] (which posits a rapidly sign-switching cos-\nmological constant Λ s, from Anti de-Sitter to de-Sitter\n(AdS to dS), in the late universe as conjectured from the\nfindings in graduated dark energy (gDE) [47]), Phantom\nCrossing Dark Energy [51–57], Omnipotent Dark En-\nergy model [51, 56], dynamical DE on top of an AdS\nbackground [56, 58–60], and (non-minimally) Interacting\nDark Energy (IDE) [61–72].1\n•Ultra Late Time Models: These models implement\nchanges in either fundamental physics or stellar physics\nduring the recent past ( z≲0.01) [53, 83–86].\nWhile our list includes some key examples from the\nnumerous attempts to resolve the H0tension through\nnew physics, it is by no means exhaustive. For a compre-\nhensive overview and detailed classification of various ap-\nproaches, please refer to Refs. [16, 18, 19]. It is fair to say\nthat, as of now, there is no widely accepted model that\nis both observationally and theoretically fully satisfac-\ntory. Moreover, addressing the H0tension while ensuring\ncompatibility with all available data and without exac-\nerbating other, less significant discrepancies such as the\nS8tension, remains a challenging task. Currently, only a\nfew models are known to propose simultaneous solutions\nto both the H0andS8tensions. Among these, without\nclaim to be exhaustive, are the Λ sCDM model [47–50],\nNew EDE [34, 35], inflation with oscillations in the in-\nflaton potential [46], some IDE models [65, 67, 72], ster-\nile neutrino with a non-zero masses+dynamical DE [87],\ndark matter (DM) with varying equation of stat (EoS)\nparameter [88], and AdS-EDE with ultralight axion [38].\nHowever, even with an optimistic view, it is difficult to\nclaim that these models currently present a completed\ntheoretical framework. Among them, the Λ sCDM model\nstands out for its simplicity, introducing only one extra\nfree parameter compared to the standard ΛCDM model:\nthez†, which signifies the redshift of the rapid AdS-dS\ntransition. The remainder of this paper will focus on the\nΛsCDM model.\nAnother aspect drawing our attention to the Λ sCDM\nmodel is its potential relevance to recent findings from\nthe James Webb Space Telescope (JWST). As initially\nnoted in [49] (refer to Section IV. C of the reference),\nthe model’s incorporation of a negative (AdS) cosmolog-\nical constant for z≳2 could lead to enhanced struc-\nture formation at these higher redshifts. This possi-\nbility aligns with observations from JWST’s deep space\n1Dark energy densities that reach negative values, consistent with\na negative (AdS-like) cosmological constant, especially for z≳\n1.5−2, are also observed in model-independent/non-parametric\nobservational reconstructions [73–82]. Further more, a recent\nmodel-independent reconstruction of the IDE kernel, employing\nGaussian process methods as suggested in [78], reveals that DE\nassumes negative densities for z≳2, which suggests that IDE\nmodels do not preclude the possibility of negative DE densities\nat high redshifts.probes ( z≳5), which suggest that structure formation\nis more intense at these higher redshifts than predicted\nby the standard ΛCDM framework [89, 90]. Specifically,\nJWST observations have revealed that the early forma-\ntion of luminous galaxies [90–97] exhibits more intense\ngrowth2. Moreover, galaxies within the redshift range\nof 7≲z≲10 display an unusually high star formation\nrate [89, 100–102], and the observed number density of\nultraviolet (UV) bright galaxies at redshift z∼15 ex-\nceeds expectations, posing a challenge to the galaxy for-\nmation models based on the ΛCDM model [94, 103]. Re-\ncent works have shown [96, 104, 105] that the presence of\na negative cosmological constant (or more broadly, nega-\ntive energy densities contributing to the Friedmann equa-\ntion) at relevant redshifts can accommodate the anoma-\nlous findings on cosmological structures observed by the\nJWST’s deep space probes.\nThe Λ sCDM model exhibits features that can address\nnot only the H0andS8tensions but also, potentially,\nthe anomalous findings from the JWST. All these fea-\ntures are controlled by a single extra free parameter,\nz†∼1.8 [49, 50], the redshift at which the cosmological\nconstant switches its sign from negative to positive and\nachieves its present-day value. The mechanism by which\nthe Λ sCDM model achieves these outcomes is straight-\nforward [48–50]: The presence of a negative cosmological\nconstant for z > z †implies that H(z) is smaller than\nin ΛCDM at these higher redshifts. Consequently, the\nsmaller H(z), offering less resistance against structure\ngrowth, leads to faster structure growth for z > z †, align-\ning with findings from JWST. On the other hand, due\nto the fact that the comoving angular diameter distance\nto the last scattering surface, DM(z∗) = cRz∗\n0H−1dz\n(where z∗≈1090), is strictly determined by the CMB\npower spectra, any reduction in H(z) for z > z †, com-\npared to ΛCDM, must be compensated by an increase in\nH(z) for z < z †. This not only explains the higher H0\nvalues predicted by Λ sCDM but also the suppression of\nstructure growth for z < z †, in comparison to ΛCDM,\nas the larger H(z) for z < z †implies greater resistance\nto structure growth. Essentially, Λ sCDM predicts higher\nvalues of both H0andσ8, compared to ΛCDM. How-\never, the decreased value of Ω m0, due to the increased\nH0, outweighs the increased σ8, resulting in a decreased\nS8=σ8p\nΩm0/0.3. Thus, it is conceivable that the\nΛsCDM model can account for intense structure growth\nat higher redshifts while simultaneously accommodating\nweaker growth at redshifts z≲2 [24, 106–109].\nThe Λ sCDM [47–49] model emerges as one of the\npromising models for addressing major cosmological ten-\nsions and stands as the most economical model among\nthose in the literature with this capability. While the\nabrupt/rapid nature of the Λ s, along with its shift from\n2However, some studies such as [98–100] argue that the JWST\ndata are not robust enough to conclusively assert a tension with\nthe ΛCDM model.3\nnegative to positive values, presents challenges in identi-\nfying a concrete physical mechanism, the phenomenolog-\nical success of Λ sCDM, despite its simplicity, strongly\nencourages the search for possible underlying physical\nmechanisms. Furthermore, it could have profound im-\nplications in theoretical physics, given that Λ <0 is a\ntheoretical sweet spot; AdS vacuum is welcome due to the\nAdS/CFT correspondence [110] and is preferred by string\ntheory and string theory motivated supergravities [111].\nThereby, it would be natural to associate Λ sto a pos-\nsible AdS-dS (phase) transition that is derived in such\nfundamental theories, and the theories that find moti-\nvation from them. Recently, it was shown in [112] that\nalthough the AdS swampland conjecture suggests that\nΛsin the late universe seems unlikely—due to the AdS\nvacua being an infinite distance apart from dS vacua in\nmoduli space—it can still be realized through the Casimir\nforces of fields inhabiting the bulk. Another study [113]\ndemonstrated that in various formulations of general rel-\nativity, it is possible to obtain a sign-switching cosmo-\nlogical constant through an overall sign change of the\nmetric. In a more recent study [114], it was demon-\nstrated that within a type-II minimally modified grav-\nity framework, known as VCDM [115, 116], an auxiliary\nscalar field with a linear potential3can induce an effec-\ntive cosmological constant, enabling the realization of an\nabrupt Λ sCDM model through a piece-wise linear poten-\ntial with two segments and facilitating smooth Λ sCDM\nmodels by smoothing out this potential. This novel the-\noretical framework, endowed with a specific Lagrangian\nfrom the VCDM theory, elevates Λ sCDM to a theoreti-\ncally complete physical cosmology, offering a fully predic-\ntive description of our universe. These theoretical devel-\nopments, emerging shortly after the introduction of the\nΛsCDM model, suggest that this model could potentially\nserve as an alternative to the standard ΛCDM model in\nthe near future.\nThe simplest version of the Λ sCDM model is con-\nstructed phenomenologically by replacing the usual Λ in\nthe standard ΛCDM model with Λ s≡Λs0sgn(z†−z),\nwhere a cosmological constant undergoes an abrupt sign-\nswitch in the past, occurring at redshift z†, and maintains\na constant positive value thereafter [48–50]. In this pa-\nper, we occasionally refer to this model as the abrupt\nΛsCDM. It is characterized by the following Friedmann\nequation:\nH2(z)\nH2\n0= Ω m0(1 +z)3+ Ω Λs0sgn(z†−z) + Ω k0(1 +z)2,\n(1)\nwhere the transition is incorporated using the signum\nfunction (sgn), and Λ s0>0 represents the present-day\n3Refer to Refs. [117–119] for discussion on the utilization of a\nscalar field with a linear potential in the context of DE modeling\nwithin the GR framework.value of Λ s(or for redshifts z < z †).4Here, H(z) repre-\nsents the Hubble parameter, while Ω m0, ΩΛs0, and Ω k0\ndenote the present-day density parameters for the pres-\nsureless fluid (baryons+CDM), sign-switching cosmolog-\nical constant, and spatial curvature, respectively. In the\nabrupt Λ sCDM model, the Hubble parameter exhibits a\ndiscontinuity in the past, specifically, at redshift z=z†,\nindicative of a type II (sudden) singularity at this exact\nredshift [121].5 6Although this singularity is mild,\nbeing weak enough not to compromise the cosmologi-\ncal model’s viability (see Appendix A for further discus-\nsion and Refs. [120, 124, 126, 127] for additional read-\ning), it nonetheless imparts a velocity kick to particles.\nThis, in turn, delays the growth of overdensities follow-\ning the sign-switch in the cosmological constant. Thus,\ncompared to the standard model, the Λ sCDM model en-\nhances early-universe structure growth, particularly for\nz > z †, driven by an initially negative cosmological con-\nstant. Conversely, in the late universe ( z < z †), when Λ s\nbecomes positive, it predicts weaker structure growth.\nThis is attributed to a lower Ω m0, a consequence of a\nlarger H0, in comparison to ΛCDM, combined with the\nadded impact of the velocity kick at the transition.\nThe behavior of spherically symmetric overdensities\nin a universe dominated by dark energy has been ex-\ntensively studied [104, 128–138]. The spherical col-\nlapse model is a fundamental tool for understanding\nthe evolution of these overdensities within a Friedmann-\nRobertson-Walker (FRW) cosmology. These overdensi-\nties function as ‘sub-universes’ having mean densities ex-\nceeding the background matter density. They are influ-\nenced by both DE and the expansion dynamics of the\nFRW background. Despite the universe’s overall expan-\nsion, these overdensities evolve relatively independently,\nakin to the evolution of a closed FRW universe [139–143].\nInitially expanding with the cosmological background,\nthey eventually succumb to local gravitational forces,\nleading to a ‘turnaround’ phase. After this turnaround,\nthe regions begin to collapse—a process that, according\nto GR, would theoretically lead to a singularity. How-\never, from an astrophysical standpoint, virialization is\ncommonly understood to occur before the formation of a\nsingularity, resulting in a stable equilibrium state.\n4Note that this abrupt behavior of Λ s, as described and consid-\nered in this work, represents as an idealized depiction of a rapid\ntransition phenomenon, akin to a phase transition, from AdS\nvacuum to dS vacuum, or a dark energy (DE) model, such as\ngDE [47], capable of mimicking this behavior [49]. Additionally,\nit is important to emphasize that Λ s, whether exhibiting abrupt\nchanges or not, does not violate the principle of energy conser-\nvation. For further details on this aspect, see Refs. [49, 120].\n5For a general overview of cosmological singularities, readers are\nreferred to Refs. [121–125].\n6This singularity is absent in smooth Λ sCDM models featuring a\nrapidly yet smoothly occurring sign-switch, as realized under the\nVCDM framework in Ref. [114]. However, this work delves into\nthe effects of Λ sin its most extreme form, the abrupt Λ sCDM\nmodel, within the GR framework.4\nThe spherical collapse model encompasses three dis-\ntinct phases:\n1.Expansion Phase: Initially, the overdense region ex-\npands in tandem with the cosmic background.\n2.Turnaround: Eventually, this spherically symmetric\nregion decouples from the cosmic expansion, reaching its\nmaximum radius. Here, local gravitational forces become\ndominant, initiating the collapse.\n3.Shell Crossing and Virialization: In this final stage,\nshells within the overdensity undergo gravitational oscil-\nlations and interactions, leading to an exchange of gravi-\ntational potential energy and ultimately resulting in viri-\nalization. This stabilizes the system in a state of equilib-\nrium.\nIn this study, we utilize the spherical collapse model\nto investigate the formation and evolution of bound cos-\nmic structures within the abrupt Λ sCDM model. Our\nprimary objective is to assess the impact of replacing the\npositive cosmological constant in the ΛCDM model with\nan abrupt sign-switching cosmological constant. This ex-\nploration involves analyzing the effects of (i) a past neg-\native cosmological constant (AdS) phase for z > z †∼2,\n(ii) the abrupt transition itself, characterized by a sud-\nden jump in the expansion rate of the universe—a type\nII (sudden) singularity—at z=z†, and (iii) the increased\nexpansion rate of the universe in the Λ sCDM model for\nz < z †. Notably, aside from the faster expansion rate,\nthe Λ sCDM and ΛCDM models are identical for z < z †.\nOur investigation focuses on the densities and scales of\nvirialized structures, characterized by their virial and\nturnaround radii, across various turnaround redshifts,\nzta, while accounting for the sign switch in the cosmolog-\nical constant in the late universe. We identify three pri-\nmary transition scenarios based on the timing of Λ s’s sign\nswitch, i.e., the AdS-dS transition, relative to the evolu-\ntionary stage of the overdensities: two occurring before\nvirialization ( a†< a vir), specifically before turnaround\n(a†< ata< avir) and after turnaround ( ata< a†< avir),\nand a third scenario where the transition occurs post-\nvirialization ( avir< a†).\nIn this paper, we first revisit the spherical collapse\nmodel with a cosmological constant of arbitrary sign, em-\nploying a semi-Newtonian framework. We derive viri-\nalized densities, building upon the methodology pre-\nsented in earlier work [104] (Section II). We then ex-\ntend our analysis to the Λ sCDM model [47–50], incor-\nporating transitional effects of the cosmological constant\ninto our calculations (Section III). Our results include\nderived virialized densities in the Λ sCDM model, which\nwe compare with those in the ΛCDM model. This anal-\nysis particularly focuses on the transition marked by a\nsudden cosmological singularity [121]. Lastly, employing\nthe Newtonian approximation of a bound system within\nan expanding background [144–147], we examine the im-\npact of this sudden singularity on systems that virialized\nbefore the transition (Section IV).II. SPHERICAL COLLAPSE MODEL IN THE\nPRESENCE OF COSMOLOGICAL CONSTANT\nIn this section, we focus on calculating virialized den-\nsities and radii using the turnaround overdensity ( δta), in\nthe presence of a cosmological constant [104, 136]. Build-\ning upon the methodology and results presented here, we\nwill analyze the Λ sCDM model in the next section.\nA. Expansion phase\n1. Background Universe\nIn our analysis, we use R(t) to represent the local scale\nfactor within the spherical overdensity and Rpfor the\nphysical radius, defined as Rp≡R(t)χ0, where χ0is the\ncorresponding comoving radius. The notation ρmdenotes\nthe (pressureless) matter energy density of the spherical\noverdensity, and ˜ ρmfor the matter energy density of the\nbackground universe. The overdensity at a given cosmic\nepoch, characterized by the background scale factor a, is\ndescribed as:\nδ(a) =ρm(a)−˜ρm(a)\n˜ρm(a). (2)\nAt the turnaround time, denoted as tta, the background\nscale factor reaches ata≡a(tta). Simultaneously, the\nscale factor of the overdense region attains its maximum\nvalue, denoted as Rta≡R(tta), resulting in its maximum\nphysical size Rp,ta≡Rtaχ0.\nThe evolution of the scale factor of the perturbation,\nR(t), and of the scale factor of the background, a(t),\nare governed by their respective Friedmann equations.\nThese equations incorporate the effects of spatial cur-\nvature, matter density, and the cosmological constant\nwithin the spherical overdensity. The Friedmann equa-\ntion for the background universe is provided in the fol-\nlowing form [132, 135, 136, 143]:\n˙a2\na2=8πG\n3˜ρm0\u0000\na−3+ω+ξa−2\u0001\n, (3)\nwhere ωandξare defined as:\nω≡ρΛ0\n˜ρm0=ΩΛ0\nΩm0,\nξ≡˜ρcrit,0−˜ρm0−ρΛ0\n˜ρm0=1\nΩm0−1−ω .(4)\n2. Overdensity\nLet us define an initial comoving time ti, which is the\nmoment when the scale factors and their time derivatives\nfor both the background universe and the overdense re-\ngion are the same:\nai=Ri,˙ai=˙Ri. (5)5\nWith these initial conditions, we reformulate the Fried-\nmann equations for the local overdensity and the back-\nground as follows:\n˙R2\nR2=8πG\n3ρm,i\u0012\nR−3+ρΛ,i\nρm,i−¯κR−2\u0013\n, (6)\n˙a2\na2=8πG\n3˜ρm,i\u0012\na−3+ρΛ,i\n˜ρm,i+¯ξa−2\u0013\n. (7)\nWe define the parameters corresponding to the spatial\ncurvatures of both the spherically overdense region and\nthe cosmological background at time tias:\n¯κ≡ρm,i+ρΛ,i−ρcrit,i\nρm,i,\n¯ξ≡˜ρcrit,i−˜ρm,i−ρΛ,i\n˜ρm,i.(8)\nFrom Eq. (5), we infer that the critical densities at time\ntifor both the overdense region and the cosmological\nbackground are identical:\nρcrit,i≡3\n8πG˙R2\ni\nR2\ni,˜ρcrit,i≡3\n8πG˙a2\ni\na2\ni. (9)\nAssuming that −¯κ=¯ξat that initial moment, and\nρcrit,i= ˜ρcrit,i, this leads to the relationship ρm,i= ˜ρm,i.\nSubsequently, we can express the overdensity as:\nρm(a) = ˜ρm,iR−3(a). (10)\nEqs. (2) and (10), evaluated at the moment ti, yield:\nR−3(a) =a−3[1 +δ(a)]. (11)\nGiven that ˜ ρm,i= ˜ρm0a−3\niand applying the rescaling\ntransformations aiR→Rand ¯κ/ai→κ, Eq. (6) is\nrewritten as [128, 132, 136, 138]:\n˙R2\nR2=8πG\n3˜ρm0\u0000\nR−3+ω−κR−2\u0001\n, (12)\nwhich is the Friedmann equation for the local overdensity\nwith scale factor R(t). By dividing Eq. (12) by Eq. (3)\nand taking the square root, we deduce:\ndR\nda=±s\na\nR1 +ωR3−κR\n1 +ωa3+ξa. (13)\nThis equation describes the dynamics of the overdensity,\nwhere the positive branch corresponds to expansion (pre-\nturnaround), and the negative branch indicates contrac-\ntion (post-turnaround).\nB. Turnaround\nWe can determine the physical size of the overdensity\nat the turnaround, denoted as Rp,ta, using the criterion:\ndR\nda\f\f\f\f\na=ata= 0. (14)This condition consequently establishes the relationship\nbetween κ,ω, and Rta:\nκ= (1 + ωR3\nta)R−1\nta. (15)\nBy rearranging the corresponding terms on each side\nof the positive branch of Eq. (13), which represents\nthe expansion phase of the overdensity, and considering\nEq. (15), we derive:\nZR\n0dRs\nR\n1 +ωR3−\u0002\n(1 +ωR3\nta)R−1\nta\u0003\nR\n=Za\n0dara\n1 +ωa3+ξa.(16)\nIn Eq. (16), employing transformation of variables such\nthat on the left-hand side (LHS) u=R/R taand on the\nright-hand side (RHS) y=a/ata, we obtain:\nZ1\n0duru\na−3\nta(1 +δta)(1−u)−ωu(1−u2)\n=Z1\n0dyry\na−3\nta+ωy3+ξa−2\ntay.(17)\nThus, for given values of ξ, ω, and ata, we can calculate\nthe density contrast at the turnaround, denoted as δta.\nC. Shell crossing and virialization\n1. Gravitational potential of the halo\nDrawing from the contributions of [148–151], we char-\nacterize the gravitational potential energy of a system\ncomposed of both dark energy and dark matter, noting\nthat virialization is exclusive to the dark matter com-\nponent. The gravitational potential energy of the halo\nsystem can be represented as:\nUhalo=1\n2Z\nVρDMΦDMdV+Z\nVρDMΦDEdV . (18)\nThe gravitational potential of dark matter (Φ DM) and\ndark energy (Φ DE) can be written as (see Appendix B\nfor the derivation):\nΦDM(r) =(\n−2πGρ DM\u0000\nR2−r2/3\u0001\nr < R\n−4πGρ DMR3/3r r ≥R\nΦDE(r) = 2 πGρ DE(1 + 3 wDE)r2\n3.(19)\nConsidering a homogeneous spherical distribution of dark\nmatter with a physical radius Rpand mass M, the energy\ndensity of DM and DE is given by:\nρDM≡3M/4πR3\np,\nρDE≡ρDMeDE\u0012a\nata\u0013−3(1+wDE)\u0012Rp\nRp,ta\u00133\n,(20)6\nwhere eDErepresents the ratio of the densities at the\nturnaround:\neDE≡ρDE(ata)\nρDM(ata). (21)\nThus, the integrals over the volume of ρDMΦDMand of\nρDMΦDE, considering Eqs. (19), (20), and (21), read as\nfollows:\n1\n2Z\nVρDMΦDMdV=−3\n5GM2\nRp,\nZ\nVρDMΦDEdV=\n−3\n5GM2\nRpΘDE\u0012a\nata\u0013−3(1+wDE)\u0012Rp\nRp,ta\u00133\n,(22)\nwhere Θ DE≡ −1\n2(1 + 3 wDE)eDE. Consequently, by fol-\nlowing Eq. (18), the potential energy of the system can\nbe expressed as:\nUhalo=−3\n5GM2\nRp\"\n1 + Θ DE\u0012a\nata\u0013−3(1+wDE)\u0012Rp\nRp,ta\u00133#\n.\n(23)\n2. Virialization condition\nThe conservation of energy, from the moment of the\nturnaround till the virialization stage, implies:\nE ≡ U halo\f\f\f\f\nRp=Rp,ta= (Khalo+Uhalo)\f\f\f\f\nRp=Rp,vir.(24)\nAssuming that the Virial Theorem holds, the kinetic en-\nergy can be written as:\nKhalo=Rp\n2dUhalo\ndRp, (25)\nand given Eq. (25), Eq. (24) suggests the virialization\ncondition:\nUhalo\f\f\f\f\nRp=Rp,ta=\u0012Rp\n2dUhalo\ndRp+Uhalo\u0013\f\f\f\f\nRp=Rp,vir.(26)\nFor a potential given in Eq. (23), Eq. (26) can be recast\nas:\n4ΘDEαDEη3−2(1 + Θ DE)η+ 1 = 0 , (27)\nwhere ηandαDEare defined through:\nη≡Rp,vir\nRp,ta, α DE≡\u0012avir\nata\u0013−3(1+wDE)\n.(28)\nThe solution of Eq. (27) expanded in terms of Θ DE, can\nbe expressed by [104, 133, 134, 148–154]:\nη=1\n2+1\n4ΘDE(−2 +αDE)\n+1\n8Θ2\nDE(−2 +αDE)(−2 + 3 αDE) +. . . .(29)In the specific case of a positive cosmological constant\n(wΛ=−1) as the dark energy component, implying αΛ=\n1 and Θ Λ=eΛ, Eq. (27) can be written as:\n4eΛη3−2(1 + eΛ)η+ 1 = 0 . (30)\nThis can be approximated by:\nη=1\n2−eΛ\n4−e2\nΛ\n8+. . . , (31)\nwhere the parameter eΛ≡ωa3\nta/(1 +δta), obtained from\nEq. (21).\n3. Density contrast at the virialization\nGiven that tvir= 2ttaand assuming the collapse is\ncompleted at t=tvir, Eq. (3) leads to:\nZyvir\n0dyry\na−3\nta+ωy3+ξa−2\ntay\n=2Z1\n0dyry\na−3\nta+ωy3+ξa−2\ntay,(32)\nwhere yvir≡avir/ata. Therefore, given specific values of\nξ,ω, and ata, we can use Eq. (32) to compute the scale\nfactor at virialization, avir. Subsequently, the value of\nδvircan be calculated using the following relation:\n1 +δvir= (1 + δta)\u0012yvir\nη\u00133\n. (33)\nIII. ΛsCDM: Λs-SIGN SWITCH BEFORE\nVIRIALIZATION\nWithin the Λ sCDM [48–50] framework, the evolution\nof the background universe is governed by:\n˙a2\na2=8πG\n3˜ρm0\u0002\na−3+ωssgn(1 /a†−1/a) +ξsa−2\u0003\n,\n(34)\nwhereas the evolution for the overdensity is described by:\n˙R2\nR2=8πG\n3˜ρm0\u0002\nR−3+ωssgn(1 /a†−1/a)−κsR−2\u0003\n,\n(35)\nwhere ωsandξsare parameters defined as:\nωs≡ρΛs0\n˜ρm0=ΩΛs0\nΩm0,\nξs≡˜ρcrit,0−˜ρm0−ρΛs0\n˜ρm0=1\nΩm0−1−ωs.\nAfter dividing Eq. (34) by Eq. (35), we obtain:\ndR\nda=±s\na\nR1 +ωssgn(1 /a†−1/a)R3−κsR\n1 +ωssgn(1 /a†−1/a)a3+ξsa.(36)7\nA. Λs-sign switch before the turnaround\n(a†< ata< avir)\n1. Density contrast at the turnaround\nAssuming that the Λ s-sign switch transition occurs\nduring the expansion phase of the overdensity, we in-\ntegrate Eq. (36) as follows:\nZR\n0dRs\nR\n1 +ωssgn(1 /a†−1/a)R3−κsR\n=Za\n0dara\n1 +ωssgn(1 /a†−1/a)a3+ξsa.(37)\nBy implementing the change of variables u=R/R ta\nto the LHS of Eq. (37), and y=a/atato the RHS of\nEq. (37), and integrating until the turnaround moment,\nwe obtain the following system of equations:\nZy†\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay\n=Zu†\n0duru\na−3\nta(1 +δta)(1−u)−ωsu(1 +u2),(38)\nZ1\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay\n=Z1\nu†duru\na−3\nta(1 +δta)(1−u)−ωsu(1−u2).(39)\nHere, we have denoted y†≡a†/ata,u†≡R†/Rta, and\nκs= (1 + ωsR3\nta)R−1\nta, given that the cosmological con-\nstant is positive at the moment of turnaround. Thus, for\na given y†, we can derive the corresponding values of us\nandδtathat satisfy Eqs. (38) and (39) simultaneously.\n2. Density contrast at the virialization\nIn such a case, the Λ s-sign switch transition occurs\nprior to the turnaround moment. As a result, at the\nturnaround, the cosmological constant has already been\npositive, ensuring that the collapse proceeds with a pos-\nitive cosmological constant throughout, similar to the\nΛCDM case. At the moment of turnaround:\nρΛs(ata)\nρm(ata)= sgn(1 /a†−1/ata)eΛs≡eΛs,(40)\nwhere we have defined:\neΛs≡ωsa3\nta/(1 +δta). (41)\nThe virialization condition results in:\n4eΛsη3−2(1 + eΛs)η+ 1 = 0 , (42)\nwhich can be approximated by:\nη=1\n2−eΛs\n4−e2\nΛs\n8+. . . . (43)Assuming that the collapse is completed at t=tvir(with\ntvir≃2tta), the Friedmann equation for the background\nuniverse (for y†<1) is by given:\nZy†\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay\n+Zyvir\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay\n=2 Zy†\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay\n+Z1\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay!\n.(44)\nFrom Eq. (44), we deduce:\nZyvir\n1dyry\na−3\nta+ωsy3+ξsa−2\ntay\n=Zy†\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay\n+Z1\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay.(45)\nThus, it becomes feasible to deduce the value of avirin\nterms of ata, and subsequently δvirfrom Eq. (33).\nB. Λs-sign switch after the turnaround\n(ata< a†< avir)\n1. Density contrast at the turnaround\nGiven that the transition occurs after the turnaround\n(i.e., a†> ata), the expansion phase persists with a neg-\native cosmological constant throughout. The value of δta\nis determined using the following equation:\nZ1\n0duru\na−3\nta(1 +δta)(1−u) +ωsu(1−u2)\n=Z1\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay,(46)\nwhich corresponds to Eq. (17), with the substitution of\nω→ −ωs7. Note that have used the relation κs= (1−\nωsR3\nta)R−1\nta, where the cosmological constant is negative\nat the moment of turnaround.\n2. Effect of the Type II Singularity: Free particle in the\nHubble flow\nWe assume that a certain amount of kinetic energy is\ninduced in each free particle that experienced the Λ s-sign\n7We have also assumed that ata≤amΛs≡(Ωm0/ΩΛs0)1\n3, i.e.,\nthe moment where H2(amΛs) = 0.8\nswitch transition event. Considering the RW metric,\nds2=−dt2+a2\u0002\ndχ2+χ2(dθ2+ sin2θdϕ2)\u0003\n,(47)\nthe physical distance is given by r(t) = a(t)χwith χ\nbeing the comoving coordinate. The geodesic equation,\nrepresenting the physical radial coordinate of a free parti-\ncle with a constant comoving coordinate in a RW metric,\nis expressed as [144]:\n¨r−¨a\nar= 0. (48)\nTo elucidate the consequences of the type II (sudden)\nsingularity that occurs at the Λ s-sign switch transition\n(see Appendix A), it is imperative to evaluate the integral\nof Eq. (48) over an interval surrounding the transition\nmoment. Consider, in particular, the time interval tgiven\nbyt∈[t†−ε, t†+ε], where εis a positive infinitesimal.\nProceeding with this approach and given that r=aχ:\nZt†+ε\nt†−εdt¨r−Zt†+ε\nt†−εdt¨a\nar= 0 = ⇒˙r\f\f\f\ft†+ε\nt†−ε= (Hr)\f\f\f\ft†+ε\nt†−ε.\n(49)\nConsider the velocity difference8, around the moment of\nsingularity t†:\nδV≡˙r(+)−˙r(−)=H(+)r(t†)−H(−)r(t†). (50)\nGiven the continuity of the physical distance r(t), we\nderive the velocity impulse as\nδV=δH r†. (51)\nHere, we denote the discontinuous increase in the Hubble\nparameter, resulting from the sign switch of the cosmo-\nlogical constant, as derived from the Friedmann equation:\nδH≡H(+)−H(−)\n=r\n8πG˜ρm0\n3\"\u0010\na−3\n†+ωs+ξsa−2\n†\u00111\n2\n−\u0010\na−3\n†−ωs+ξsa−2\n†\u00111\n2#\n.(52)\nThe solution to the geodesic equation for a free par-\nticle, as denoted in Eq. (48), within the Λ sCDM frame-\nwork can be derived using joint boundary conditions at\nthe transition moment i.e. r(t) continuous at t†and also\nEq. (51). These conditions include the continuity of r(t)\natt†and conditions described in Eq. (51), which accounts\nfor the velocity kick. With a specified set of initial con-\nditions for rand ˙r, we are able to derive the analytical\nsolution Eq. (C8)(Fig. 1). Considering that the physical\ndistance r=aχ(where a(t) is the scale factor, as shown\nin Eq. (C6)), follows the exact same form as a(t), assum-\ning a constant χ(see Appendix C for detailed derivation\nof the scale factor in the Λ sCDM model).\n8Throughout the text, we define the left and right limits of a\nfunction as; f(−)≡limε→0f(t†−ε) and f(+)≡limε→0f(t†+ε),\nrespectively.\nz†=1.78,t †=0.28H 0-1z†=1.97,t †=0.25H 0-1z†=2.75,t †=0.17H 0-1\n0.50 0.75 1.00 1.25 1.50 1.75 2.001.01.21.41.61.82.0\nt/t†r/r0FIG. 1. Utilizing the joint boundary conditions at the transi-\ntion moment t/t†= 1, an analytical solution to the geodesic\nequation for a free particle within the Λ sCDM cosmological\nframework is derived. The solution of geodesic Eq. (C8),\nwhich maintains the continuity of the physical distance r(t)\nand incorporates the velocity kick, Eq. (51). The initial con-\nditions r(t/t†= 0.5) = r0and ˙r(t/t†= 0.5) = r0/t†are\nimposed. We observe that the effect of the kick (at t/t†= 1)\nis insignificant.\n3. Effect of the Type II Singularity: Spherical collapse\nmodel\nA free particle in constant comoving coordinates will\ninfall due to the contraction of the spherical overden-\nsity entrained by the spacetime geometry. Following the\nsame reasoning as in Section III B 2, we introduce the\ncorresponding velocity kick to a particle in the spherical\noverdensity as:\n∆V≡˙R(+)\np,†−˙R(−)\np,†, (53)\nwhere ˙Rp≡χ0aHdR\nda. During the collapsing phase, we\nimplement the negative branch of Eq.(36) in Eq.(53),\nwhich results in:\n∆V=−χ0a† \nH(+)s\na†\nR†1 +ωsR3\n†−κsR†\n1 +ωsa3\n†+ξsa†\n−H(−)s\na†\nR†1−ωsR3\n†−κsR†\n1−ωsa3\n†+ξsa†!\n.(54)\n4. Effect of the Type II Singularity: Modified virialization\ncondition\nAssume the transition occurs during the collapse of the\noverdensity (where a†> ata), at the turnaround moment:\nρΛs(ata)\nρm(ata)= sgn(1 /a†−1/ata)eΛs=−eΛs, (55)\nand the potential energy of the system, at the moment\nwhen the scale factor attains the value ataand the cos-9\nmological constant is negative, is substituted according\nto the Eq. (22), as:\nUhalo=−3\n5GM2\nRp\"\n1−eΛs\u0012Rp\nRp,ta\u00133#\n. (56)\nEnergy conservation is upheld right until the brink of\nthe singularity’s emergence at a=a†. Ifε >0 is an in-\nfinitesimally small positive quantity, and t†−εrepresents\nthe moment just before the singularity, then, given the\ncontinuity of Rp, when ε→0, we obtain:\nK(−)=Uhalo\f\f\f\f\nRp,ta− U halo\f\f\f\f\nRp,†. (57)\nImmediately after, energy conservation continues to be\nvalid but incorporates an energy impulse attributed to\nthe singularity. Specifically, the velocity Vof a spheri-\ncally symmetric shell is given by V=˙R\nRRp. This implies\nthat for t†+ε, due to a velocity kick, Vtransitions to\nV+ ∆V(Eq. (54)). Consequently, this impulse affects\nthe kinetic energy in the following manner:\nK(+)=K(−)[1 + ∆]2, (58)\nwhere we have defined ∆ as:\n∆≡∆V\f\fV(−)\f\f=∆V\f\f\fχ0a†H(−)dR\nda(−)\f\f\f\n=1−H(+)\nH(−)vuuut\u0010\n1 +ωsR3\n†−κsR†\u0011\u0010\n1−ωsa3\n†+ξsa†\u0011\n\u0010\n1 +ωsa3\n†+ξsa†\u0011\u0010\n1−ωsR3\n†−κsR†\u0011.\n(59)\nFrom immediately post-singularity up to the point of\nvirialization, the conservation of energy, along with the\nVirial theorem, gives:\nRp\n2dUhalo\ndRp\f\f\f\f\nRp,vir+Uhalo\f\f\f\f\nRp,vir=K(+)+Uhalo\f\f\f\f\nRp,†.(60)\nEqs. (58) and (57) combined imply the following modified\nvirialization condition :\nRp,vir\n2dUhalo\ndRp\f\f\f\f\nRp,vir+Uhalo\f\f\f\f\nRp,vir\n= \nUhalo\f\f\f\f\nRp,ta− U halo\f\f\f\f\nRp,†!\n(1 + ∆)2+Uhalo\f\f\f\f\nRp,†.\n(61)\nImplementing the potential energy, given by Eq. (56),\ninto Eq. (61) and taking into account that the singularity\ntakes place subsequent to the turnaround moment, i.e.,a†> ata, we derive:\n−1−y−3(1+w)\nvir Θη3\n10Rp,vir+1−y−3(1+w)\n†Θu3\n†\n5Rp,†\n+3Θy−3(1+w)\nvir R2\np,vir\n10R3\nta\n−(1 + ∆)2 \n−1−Θ\n5Rta+1−y−3(1+w)\n†Θu3\n†\n5Rp,†!\n= 0.(62)\nSubsequently, for cosmological constant ( wΛ=−1),\nEq. (62) implies that:\n∆0(−1 +u†) +u†\u0012\n1−1\n2η\u0013\n+u†eΛs\u0002\n−1 + 2 η2+ ∆ 0(−1 +u2\n†)\u0003\n= 0,(63)\nwhere we have defined the following dimensionless pa-\nrameters:\n∆0≡∆(2 + ∆) , u†≡R†\nRta.\nNote that when ∆ = 0, Eq. (27) is recovered, correspond-\ning to a collapse with a negative cosmological constant.\nWhile Eq. (63) admits an analytical solution, its com-\nplexity can hinder a straightforward physical interpreta-\ntion. As such, we resort to an approximate solution. At\nfirst order in ∆ 0and values of u†close to 1, we obtain:\nη=1\n2\u0014\n1 +∆0(1−u†)\nu†\u0015\n+eΛs\n4\u0002\n1 + 2∆ 0(1−u2\n†)\u0003\n−e2\nΛs\n8\u0014\n1 +7∆0(1−u†)\nu†\u0015\n+. . . .(64)\nThis approximation remains valid for |∆| ≪1. In cases\nwhere this condition is not met, it becomes necessary to\nresort to the analytical solutions of Eq. (63) (Figure 2).\nThe collapse of an overdense region and the expansion of\nthe universe exert opposing effects. The impact of the\nsingularity on the overdensity is significant when y†≃1\nand becomes smaller for values that are further away,\nas well as for larger values of zta, as shown in Fig. 3.\nAlthough the sudden singularity, extracts kinetic energy\n(i.e., when −1≤∆<0) from the collapsing overdensity,\nit is notable that near the turnaround moment, the veloc-\nity of the contracting overdensity approaches zero. Con-\nsequently, the ‘velocity brake’ (occurring when ∆ <−1)\nis sufficient to reverse the direction, effectively inducing\na velocity kick. This kick expands the shell once more to\na slightly larger value of a new physical radius, before it\ncollapses again and eventually virializes.\nAdditionally, we can assume a brief period of time after\nthe turnaround but before the shell-crossing, where we\ncan still apply the negative branch of the Eq. (36). Thus,\nby integrating Eq. (36) throughout the period where the10\n1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7\nzta−18−16−14−12−10−8−6−4−20∆\ny†= 1.01\ny†= 1.02\ny†= 1.04\ny†= 1.06\ny†= 1.08\n−1≤∆<0\nFIG. 2. Variation of ∆ with respect to zta, obtained by keep-\ningy†∈[1.01,1.02,1.04,1.06,1.08] constant. The collapse of\nthe overdense region and the expansion of the universe exert\nopposing effects. Consequently, the emergence of the sudden\nsingularity effectively dissipates kinetic energy, (∆ <0) as\ndemonstrated above. The shaded gray area represents the\nregion of −1≤∆<0, where only the kinetic energy is ex-\ntracted from the overdensity. Meanwhile, for ∆ <−1, the\noverdensity expands once more to a slightly larger value of a\nnew physical radius, before it collapses again and eventually\nvirializes.\nscale factor ranges atatoa†, we obtain the following equa-\ntion:\nZ1\nu†duru\na−3\nta(1 +δta)(1−u) +ωsu(1−u2)\n=Zy†\n1dyry\na−3\nta−ωsy3+ξsa−2\ntay.(65)\nWe evaluate u†using Eq. (65) and after that we incorpo-\nrateη, as calculated in Eq. (64).\nSubsequently, the value of Rviris is ascertained in ac-\ncordance with Eq.(69), leading to the determination of\nthe anticipated ratios, as discussed in Section III C.\nAssuming that tvir≃2ttaand that the collapse is com-\npleted when t=tvir, Eq. (34) yields ( y†>1):\nZy†\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay\n+Zyvir\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay\n=2Z1\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay,(66)\nfrom which we obtain:Zy†\n1dyry\na−3\nta−ωsy3+ξsa−2\ntay\n+Zyvir\ny†dyry\na−3\nta+ωsy3+ξsa−2\ntay\n=Z1\n0dyry\na−3\nta−ωsy3+ξsa−2\ntay.(67)It then becomes feasible to deduce the value of avirin\nterms of ata, through Eq. (32) and subsequently δvirfrom\nEq. (33).\nC. Contrasting ΛsCDM with Standard ΛCDM:\nInsights into the Physical Outcomes\nBased on Eqs. (11) and (20), the physical radius of the\noverdensity at turnaround is expressed as:\nRp,ta=\u00143M\n4π(1 +δta)˜ρm0\u00151\n3\nata. (68)\nFurthermore, using the definition of ηfrom Eq. (28), we\ncan write the virialized physical radius of the overdensity\nas:\nRp,vir= (1 + zta)−1\u00143M\n4π(1 +δta)˜ρm0\u00151\n3\nη . (69)\nConsequently, the ratio of the virialized matter density\nin the Λ sCDM model to that in the ΛCDM model [1, 4]\nreads:\n(ρvir)Λs\n(ρvir)Λ≡\u0014(Rp,vir)Λs\n(Rp,vir)Λ\u0015−3\n=(1 +δΛs\nta)\n(1 +δΛ\nta)\u0014η(eΛs)\nη(eΛ)\u0015−3\n.\n(70)\nThe physical effects of the Λ s-sign switch (AdS-dS)\ntransition can primarily be understood by considering the\ntiming of the transition relative to the turnaround mo-\nment, distinguishing between the pre-turnaround ( z†>\nzta) and post-turnaround ( zta< z†) Λs-sign switch tran-\nsitions. These effects are demonstrated in Fig. 3, where,\nfor a realistic assessment, the cosmological parameters\nfor both models, namely, Λ sCDM (for various z†cases)\nand ΛCDM, are chosen to ensure consistency with the\nPlanck-CMB power spectra, as detailed in Appendix D.\n1. Pre-turnaround Λs-sign switch transition ( z†> zta)\nWe first consider the case y†<1, where the Λ s-sign\nswitch transition occurs before the turnaround, i.e., z†>\nzta. Our findings indicate that if this transition happens\nbefore turnaround, the density contrast at turnaround\n(δta) will be higher than in the ΛCDM model, as shown\nin the top-left panel of Fig. 3. The rationale behind this is\nas follows: Because the transition has already occurred,\nthe positive cosmological constant has already begun to\ninfluence the curvature of the halo as described by Eq. 15\n(ωs>0). On the other hand, as illustrated in Fig. 4 plot-\nted, which is plotted by choosing zta= 2 (noting that\ndifferent values of ztado not change the trends in the\nplots), a larger y†<1 results in the overdensity evolving\nunder the negative cosmological constant’s influence for\na longer period. This implies that for larger y†<1 val-\nues, the negative cosmological constant’s slowing-down11\n1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7\nzta5.005.255.505.756.006.256.501 +δtaΛCDM (Ω m= 0.3101)\ny†= 0.60 (ρΛs(ata)>0)\ny†= 0.70 (ρΛs(ata)>0)\ny†= 0.80 (ρΛs(ata)>0)\ny†= 1.01 (ρΛs(ata)<0)\ny†= 1.05 (ρΛs(ata)<0)\ny†= 1.08 (ρΛs(ata)<0)\n1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7\nzta1651751851952052152252352451 +δvirΛCDM (Ω m= 0.3101)\ny†= 0.60 (ρΛs(ata)>0)\ny†= 0.70 (ρΛs(ata)>0)\ny†= 0.80 (ρΛs(ata)>0)\ny†= 1.01 (ρΛs(ata)<0)\ny†= 1.05 (ρΛs(ata)<0)\ny†= 1.08 (ρΛs(ata)<0)\n1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7\nzta0.970.980.991.001.011.021.031.041.051.06RΛs\np,ta/RΛ\np,taΛCDM (Ω m= 0.3101)\ny†= 0.60 (ρΛs(ata)>0)\ny†= 0.70 (ρΛs(ata)>0)\ny†= 0.80 (ρΛs(ata)>0)\ny†= 1.01 (ρΛs(ata)<0)\ny†= 1.05 (ρΛs(ata)<0)\ny†= 1.08 (ρΛs(ata)<0)\n1.7 2.0 2.3 2.6 2.9 3.2 3.5 3.8 4.1 4.4 4.7\nzta0.700.750.800.850.900.951.001.051.10ρΛs\nvir/ρΛ\nvir\nΛCDM (Ω m= 0.3101)\ny†= 0.60 (ρΛs(ata)>0)\ny†= 0.70 (ρΛs(ata)>0)\ny†= 0.80 (ρΛs(ata)>0)\ny†= 1.01 (ρΛs(ata)<0)\ny†= 1.05 (ρΛs(ata)<0)\ny†= 1.08 (ρΛs(ata)<0)\n1.502.613.724.835.947.068.179.2810.3911.50\nz†\n1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50\nz†0.2550.2600.2650.2700.2750.2800.2850.2900.2950.3000.3050.3100.315Ωm0\nΛCDM\nΛsCDM\ny†= 0.60\ny†= 0.70\ny†= 0.80\ny†= 1.01\ny†= 1.05\ny†= 1.08\nFIG. 3. Numerical analysis performed to calculate δta,δvir,ρvir, and Rp,tafor post-turnaround (shades of blue) and pre-\nturnaround (shades of red) cases, where the cosmological parameters are derived from the analysis given in the Appendix D.\nEach of the figure obtained by varying 1 .7≤zta≤4.7, for constant y†≡a†/ata. The corresponding z†value for a given zta\ncan be calculated via 1 + z†= (1 + zta)/y†. Even though we have used the same physical matter density parameter throughout\nthe analysis, the Ω m0parameters will be different for both models. For a given z†, one can easily find the corresponding Ω m0\nfor the Λ sCDM model (see Eq. (D7)).\neffect on the overdensity’s expansion, due to its induced\ngravitational attraction, lasts longer, leading to denser\nstructures. Consequently, we observe generally higher\nδtavalues for the Λ sCDM model compared to the ΛCDM\nmodel in the top-left panel of Fig. 3, with this difference\nincreasing for larger y†values, as long as y†<1. We\nnote that the expansion rate of the background universe,\n˙a=H(z)\n1+z, around ztais almost identical across various y†\nvalues for a given zta, as seen in Fig. 4 plotted by choosingzta= 2 as an example. Thus, the expansion rate of the\nuniverse at or around the turnaround moment does not\nintervene in and influence our discussions on the value of\nδtawithin the Λ sCDM framework. And of course, since\nδtaandRp,taare interrelated through Eq. 68, a higher\nδtacorresponds to a smaller Rp,ta, and vice versa, for a\ngiven value of ztaorata. Finally, the collapsing phase of\nthe overdensity proceeds under the influence of a positive\ncosmological constant until virialization, similar to the12\n0.000.250.500.751.001.251.501.752.002.252.502.753.003.253.503.754.004.25\nz58606264666870727476788082848688H(z)/(1 +z)y†>1y†<1ΛCDM (Ω m= 0.3101)\ny†= 0.60 (ρΛs(ata)>0)\ny†= 0.70 (ρΛs(ata)>0)\ny†= 0.80 (ρΛs(ata)>0)\ny†= 1.01 (ρΛs(ata)<0)\ny†= 1.05 (ρΛs(ata)<0)\ny†= 1.08 (ρΛs(ata)<0)\nzta= 2.0\nFIG. 4. ˙ a≡H(z)/(1 + z) vs zplotted for constant y†≡\na†/ata, by fixing the turnaround redshift at 2 (i.e., zta= 2).\nWe have applied that method described in the Appendix D\nto determine the cosmological parameters. For a fixed ata, as\ny†→0, the scale factor of the transition approaches to a†→\n0 and the Λ sCDM model dynamics becomes similar to the\nΛCDM. Meanwhile, as y†→1/ata, the transition approaches\ntoday. Also note that, before the transition the expansion\nrate of the universe is smaller (˙ aΛsCDM<˙aΛCDM) but after\nthe transition it’s faster (˙ aΛsCDM>˙aΛCDM) than the ΛCDM\nuniverse.\nstandard ΛCDM model. Therefore, evidently, the larger\nvalues of δtaachieved in the Λ sCDM model, compared to\nΛCDM model, give rise to corresponding larger values of\nδvirandρΛs\nvir, as seen in the top- and middle-right panels\nof Fig. 3, respectively.\n2. Post-turnaround Λs-sign switch transition ( zta> z†)\nWe now consider the case y†>1, where the Λ s-\nsign switch transition occurs after the turnaround, i.e.,\nzta> z†, focusing on the phase during which halo is col-\nlapsing, but well before shell-crossing, when the halo is\nstill homogeneous and isotropic. In this phase, the den-\nsity contrast at the turnaround, δta, will be lower com-\npared to that in the ΛCDM model, with this effect more\npronounced at lower values of zta, as seen in the top-left\npanel of Fig. 3. The rationale behind this is as follows: In\nthe ΛCDM model, the overdensity experiences a positive\ncosmological constant throughout its evolution. In con-\ntrast, in the post-turnaround Λ s-sign switch transition\ncase of the Λ sCDM model, the overdensity experiences a\nnegative cosmological constant until and for some time\nafer reaching the turnaround radius. This implies that\nless matter energy density is required for the overdensity\nto achieve turnaround due to the enhancing gravitational\nattraction effects of the negative cosmological constant\n(in contrast to the positive cosmological constant), lead-\ning to less curvature for the overdensity. We note that\nthe lower the zta, the larger difference in δtabetween the\nΛsCDM and ΛCDM models. This understandable, as at\nlower redshifts the cosmological constant is more domi-nant in both models, but it is negative in the Λ sCDM\nand positive in the ΛCDM model. Specifically, the lower\nthezta, the higher the δtain the ΛCDM model, while the\nlower ztais, the lower the δtain the Λ sCDM model.\nNote that δtais almost identical for a given ztavalue,\nwith only a barely visible differences at lower ztavalues\nfor various y†values. This explained by the fact that for\nz > z ta, the expansion rate of the universe for different\ny†values is almost the same, allowing the overdensities\nto evolve through nearly identical background universe\ndynamics until turnaround is achieved, as seen in Fig. 4\nplotted by choosing zta= 2 as an example. Note that\nfor different y†>1 values, we have used the same phys-\nical matter density value and fixed the angular size of\nthe sound horizon at the last-scattering surface to ensure\nconsistency with CMB-Planck spectra (see Appendix D),\nresulting in slightly different values for Ω m0, correspond-\ning to slightly different H0values. This implies, for\na given zta, slightly different matter density parame-\nters/expansion rates of the universe at the times when\nthe over density is evolving towards the turnaround, and\nthe time taken to reach the turnaround radius would be\nslightly different. The Ω m0can be read from the bot-\ntom panels of Fig. 3; for y†={1.01,1.05,1.08}values\nwe used, we have z†={1.97,1.86,1.78}and Ω m0=\n{0.2821,0.2777,0.2739}. In other words, ˙ ais almost iden-\ntical across all Λ sCDM cases for y†>1 for z > z taand\nthus is not expected cause significant variations in δta\nfor different y†>1 values. In these cases, the overden-\nsity experiences the negative cosmological constant for a\nnearly identical duration, see Fig. 4. The minor varia-\ntions in δtaare attributed to slightly different values of\nΩm0, implying slightly different expansion rates of the\nuniverse and hence a slightly different passage of time\ntaken until turnaround is achieved. In the middle-left\npanel of Fig. 4, we plot the ratio of the physical radius of\nthe halo at the turnaround in the Λ sCDM model to that\nin the ΛCDM model, viz., RΛs\np,ta/RΛ\np,ta.9We observe that\nRΛs\np,ta/RΛ\np,ta>1, being larger for lower values of zta, and\nis almost the identical for different y†>1 values with a\nbarely visible difference for small ztavalues. This aligns\nwith expectations when considering Eq. 15 for ωs<0,\nimplying the higher values of Rp,tacorrespond to lower\nvalues of δtaand vice versa.\nFinally, in the top-right panel of Figure 3, we plot the\ndensity contrast at the moment of virialization, δvir, for\nthe cases where the Λ s-sign switch transition occurs dur-\ning the collapsing phase, before the halo virializes. It is\nconceivable that the δvirvalues in the Λ sCDM are lower\nthan in the ΛCDM model, similar to the situation with\nδta. However, we immediately see that, although this\nexpectation holds, unlike with the situation for δta, the\nδvirvalues differ significantly for different y†values and\n9The results are independent of the halo’s mass, as demonstrated\nin Eq. (68).13\nbecome more pronounced for lower ztavalues. This phe-\nnomenon is not surprising and can roughly be explained\nas follows: As seen in Fig. 4 (plotted by choosing zta= 2\nas an example), the larger the value of y†the greater\nthe difference between z†andzta, implying that Λ s-sign\nswitch transition occurs at later in the collapsing phase\nof the overdensity for the larger y†>1 values. That is,\ngiven that δtais almost the same for a given ztafor differ-\nent values of y†, for larger y†values, the Λ s-sign switch\ntransition occurs when the overdensity is more compact,\nthereby it is conceivable that the rapid increase in the\nuniverse’s expansion rate at the transition will have less\ninfluence on the collapsing overdensity for larger y†>1\nvalues. A more precise, but also more concise, expla-\nnation is as follows: We observe that the impact of the\nsingularity at z†on the overdensity is considerable when\ny†≃1 and diminishes for values further from 1 or for\nlarger ztavalues. The singularity extracts kinetic energy\n(i.e., when −1≤∆<0), leading to a ’velocity brake’\nin the collapse. Conversely, when ∆ <−1, it induces a\nvelocity kick, reversing the direction of the collapse and\nre-expanding the shell to a slightly larger new physical\nradius before its eventual collapse and virialization. This\neffect, observable in Figure 2, suggests that higher ∆ val-\nues result in lower virialized densities in the halo.\nIV. THE IMPACT OF BACKGROUND\nEXPANSION ON BOUND SYSTEMS\nA. Newtonian approximation of a bound system in\nan expanding background\nNumerous studies have been devoted to the exploring\nthe impacts of cosmic expansion on bound systems [145,\n147, 155–159]. In the Newtonian limit, the vicinity of a\npoint mass Msituated within a dynamically expanding\nbackground is characterized as follows:\nds2=−\u0012\n1−2GM\na χ\u0013\ndt2\n+a2\u0002\ndχ2+χ2(dθ2+ sin2θdϕ2)\u0003\n,(71)\nwhere tis the comoving time and a≡a(t). Interpreting\nthe gravitational field of the point mass as a weak field ,\nand by admitting low velocities , the geodesic equations\ncan be differentiated with respect to the coordinate time\nt, as:\n¨xµ+ Γµ\nνσ˙xν˙xσ= 0, (72)\nwhere ˙ ≡d/dt. For the geodesic equation in the χ-\ncoordinate, considering θ=π/2 in the equatorial plane,\nwe derive:\nGM\nχ(t)2a(t)3−χ(t)˙ϕ(t)2+2˙a(t) ˙χ(t)\na(t)+ ¨χ(t) = 0 .(73)Similarly, the geodesic equation for x3, taking θ=π/2\nagain, yields:\n2˙a(t)˙ϕ(t)\na(t)+2˙ϕ(t) ˙χ(t)\nχ+¨ϕ(t) = 0 = ⇒d\ndth\n(aχ)2˙ϕi\n= 0.\n(74)\nDefining L= (aχ)2˙ϕas the angular momentum per unit\nmass, we express the geodesic equation from Eq. (73) in\nterms of the physical radial coordinate r=aχas follows:\n¨r−¨a\nar−L2\nr3+GM\nr2= 0. (75)\nConsidering a moment in time, t0, where expansion can\nbe disregarded (i.e., ˙ r(t0) = 0), we define the physical\nradius of an orbit as r0≡r(t0). The angular speed at\nthis moment, neglecting expansion, is described as:\n˙ϕ2\f\f\f\f\nt=t0=L2\nr4\n0≡ω2\n0≡GM\nr3\n0. (76)\nGiven the condition outlined in Eq. (76), this leads to the\nconclusion that L2=GMr 0. While a circular orbit with\na constant physical radius does not exist at all times, r0\nrepresents the radius at a specific instant. Under the con-\ndition that the initial angular speed is significant enough\nto overlook cosmic expansion, it approximates a stable\nNewtonian circular orbit, assuming L2̸= 0 and ˙ r= 0 at\nall times.\nAdditionally, considering a rescaling d/d t=t−1\ninitd/d¯t\nand′≡d/d¯t(where tinitis the initial cosmic time se-\nlected for the system), we derive from Eq. (75):\n¯r′′=¯ω2\n0\n¯r3−¯ω2\n0\n¯r2+a′′\na¯r , (77)\nwhere\na′′\na=−1\n2(H0tinit)2Ωm0h\na−3−2ωsa−1δ(1/a†−1/a)\n−2ωssgn(1 /a†−1/a)i\n,\n(78)\nand\n¯r≡r\nr0,¯ω0≡ω0tinit,¯t≡t\ntinit,˙ϕ≡¯ω0\n¯r2.(79)\nThe final step before proceeding to solve Eq. (77), in-\nvolves initiating from an orbit with a rescaled radius,\n¯r(tinit) = s, taking into account the expanding back-\nground. This is achieved by setting the right-hand side\nof Eq.(77) to zero and solving for s:\n¯ω2\n0(1−s) +a′′\na\f\f\f\f\ntinits4= 0. (80)\nSubsequently, we numerically solve the geodesic equa-\ntion, as presented in Eq. (77), for a bound system. Nu-\nmerical examples of bound orbits within the Λ sCDM\nmode are demonstrated in Fig. 5 and Fig 6.14\nTABLE I. Clusters of galaxies form massive, virialized struc-\ntures. Nevertheless, this is not the case for super-clusters.\nAssuming an initial angular speed defined by ω2\n0=GM/r3\n0,\nwe derive some very rough typical scales for both galaxy clus-\nters and super-clusters [160–164].\nSystem M[M⊙]R[Mpc] T0[H−1\n0]ω0[H0]\nCluster ∼1015∼2 ∼1 ∼10\nSupercluster ∼1016∼100 ∼100 ∼0.01\nB. Physical outcomes\nWe consider a point particle with a path parameterized\nby the proper time τ. In this setting, the x0-geodesic\nequation is:\nd\ndτ\u0014dt\ndτ\u0012\n1−2GM\naχ\u0013\u0015\n=\n−H(aχ)dt\ndτ\"\nL2\n(aχ)3+1\naχ\u0012\nadχ\ndτ\u00132#\n.(81)\nFor the first integral of a timelike geodesic, described\nby ds2=−dτ2and assuming θ=π/2, we redefine the\nmetric from Eq. (71) as:\n−1 =−\u0012\n1−2GM\naχ\u0013\u0012dt\ndτ\u00132\n+a2\u0012dχ\ndτ\u00132\n+L2\n(aχ)2.(82)\nAssuming a sufficiently small time interval where every\ncosmic bound system maintains a physical radius r=aχ\nand adheres to the approximation H(aχ)≪1, particle in\norbit with this radius will conform to this approximation.\nIf we also assume a small peculiar velocity, the second\nterm in Eq. (81) is deemed higher order, simplifying the\nequation to:\ndt\ndτ\u0012\n1−2GM\naχ\u0013\n=k≡const . (83)\nUpon integrating Eq. (83) into Eq. (82), we derive:\n−1 =−\u0012\n1 +2GM\naχ\u0013\nk2+a2\u0012dχ\ndτ\u00132\n+L2\n(aχ)2.(84)\nIn the regime of small peculiar velocities (d t≃dτ), this\nresults in an expression for the quasi-energy:\nE≡k2−1\n2=1\n2˙r2−GM\nr−Hr˙r+1\n2H2r2+L2\n2r2.(85)\nConsider a particle initially moving in an orbit at\nr(tinit) =s r0(with ˙ r(tinit) = 0) in a bound system. Dur-\ning a short time interval around t†, if the approximations\n¯r(tinit)≃1 and ˙¯r≃0 are valid, then Eq. (85) simplifies\nto:\nE=1\n2(H+δH)2r2\n0+(ω0r0)2\n2−(ω0r0)2. (86)\nΛCDM: t init=0.238 H 0-1,ω0=21 H 0, tϵ[0.238H 0-1,2.38 H 0-1]\nΛsCDM: t init=0.238 H 0-1, t†=0.2383 H 0-1,ω0=21 H 0, tϵ[0.238H 0-1,2.38 H 0-1]\n-2 -1 0 1-1.5-1.0-0.50.00.51.01.5\nyx\nΛCDM: t init=0.238 H 0-1,ω0=2.98 H 0, tϵ[0.238H 0-1,7.14 H 0-1]\nΛsCDM: t init=0.238 H 0-1, t†=0.2383 H 0-1,ω0=2.98 H 0, tϵ[0.238H 0-1,7.14 H 0-1]\n-2 0 2 4 6 8 10-8-6-4-202\nyxFIG. 5. A particle in orbit, embedded in ΛCDM (gray)\nand Λ sCDM (red), with t†= 0.2383H−1\n0for Ω m0= 0.29,\nandtinit= 0.238H−1\n0. The initial angular velocity set as\nω0= 21H0(top-panel) and ω0= 2.98H0(bottom-panel) (See\nalso Table I). We initiate a circular orbit against an expand-\ning background i.e. with a rescaled radius ¯ r(tinit) =sand\n˙r(tinit) = 0. The sis determined by solving Eq. (80). Sub-\nsequently, we numerically solve Eq. (77), in each case. The\nblack dot represents the position of the particle in orbit in the\nrescaled x−yplane at the moment the singularity.\nGiven these approximations, consider any particle in a\nsystem with mass M, that initiates its orbit at a time\ntinit, where tinit∈(t†−T, t†) and Tis sufficiently small.\nThe particle has an initial angular momentum L, and\n˙r(tinit) = 0. Also, the particle’s initial physical radius\nis approximately r(tinit)≃r0≃r(t†). We define the\ncritical angular speed ,ωcrit, at the moment t†,iffthe\ntotal energy E= 0, according to the equation\nωcrit=H+δH . (87)\nGiven that the singularity is relatively weak, it alone can-\nnot dissociate any bound system before the continual ex-\npansion of the universe does. Thus, the dissociation of15\nΛCDM: t init=0.01 H 0-1,ω0=15 H 0, tϵ[0.01H 0-1,2 H 0-1]\nΛsCDM: t init=0.01 H 0-1, t†=0.2383 H 0-1,ω0=15 H 0, tϵ[0.01H 0-1,2 H 0-1]\n-3 -2 -1 0 1-2.0-1.5-1.0-0.50.00.51.01.5\nyx\nΛCDM: t init=0.01 H 0-1,ω0=10 H 0, tϵ[0.01H 0-1,2 H 0-1]\nΛsCDM: t init=0.01 H 0-1, t†=0.2383 H 0-1,ω0=10 H 0, tϵ[0.01H 0-1,2 H 0-1]\n-6 -4 -2 0-3-2-101\nyx\nFIG. 6. A particle in orbit, embedded in ΛCDM (gray)\nand Λ sCDM (red), with t†= 0.2383H−1\n0for Ω m0= 0.29\nandtinit= 0.01H−1\n0. The initial angular velocity is set as\nω0= 15H0(top-panel) and ω0= 10H0(bottom-panel) (See\nalso Table I). We initiate a circular orbit against an expand-\ning background i.e. with a rescaled radius ¯ r(tinit) =sand\n˙r(tinit) = 0. The sis determined by solving Eq. (80). Sub-\nsequently, we numerically solve Eq. (77), in each case. The\nblack dot represents the position of the particle in orbit in the\nrescaled x−yplane at the moment the singularity.\nbound orbits is primarily driven by cosmic expansion,\nwith the singularity inducing a minor perturbation that\nslightly increases the Hubble value at a specific instant.\nThis perturbation contributes minimally to the dissoci-\nation of a bound orbit. Notably, the timing of initiat-\ning a bound orbit significantly influences its evolution,\nas demonstrated in Figure 5, where we initiate an or-\nbit near the moment the singularity. We calculate the\ncritical angular speed, ωcrit, to be approximately 3 for\nΛsCDM and 2 .9 for ΛCDM.\nGiven the brief period when the cosmological constant\nis negative in the Λ sCDM compared to its positive phase,\nvariations in orbits relative to those equivalent orbits\nin the ΛCDM model suggest a scenario in the Λ sCDM\nmodel where orbits with sufficiently low angular speed\nmay dissociate, unlike their counterparts in the ΛCDMmodel. However, this dynamic changes when orbits are\ninitiated far in the past, away from the singularity. In\nsuch cases, the negative cosmological constant in the\nΛsCDM model aids in maintaining the orbit’s binding\nover a significant period, preventing the singularity from\nweakening the gravitational attraction enough to cause\nmore dissociation than in the ΛCDM model.\nThe critical initial angular speed, ωcrit, approximated\nfrom Eq. (87), acts as a threshold. For bound sys-\ntems with an angular speed ω0—the speed in a New-\ntonian bound system disregarding cosmic expansion—\ndissociation due to the cosmic expansion occurs if and\nonly if ωcrit> ω 0. This approximation holds mainly for\norbits beginning at moments near t†. The results from\nEq. (87) align well with those obtained numerically from\nthe geodesic equation Eq. (75), as shown in Fig. 5. How-\never, this method does not accurately approximate nu-\nmerical results for orbits starting well before the moment\nunder study; such an example is shown in Fig. 6.\nV. CONCLUSION\nThe Λ sCDM model [47–50] has emerged as a highly\npromising approach to addressing major cosmological\ntensions within the standard ΛCDM model and its canon-\nical extensions, such as the H0andS8tensions. This\nachievement is realized through the introduction of a\nsingle parameter, z†, to the six-parameter base ΛCDM\nmodel. This parameter determines the timing of an\nabrupt AdS-dS transition, changing from Λ s=−Λs0\nforz > z †to its late-time positive value Λ s= Λ s0for\nz < z †. Examining the implications of adopting the\nΛsCDM model framework for the universe’s evolution, es-\npecially on the formation and evolution of bound cosmic\nstructures, is crucial. The switch to the Λ sCDM model is\nanticipated to impact bound cosmic structures for three\nprimary reasons: (i) the negative cosmological constant\n(AdS) phase for z > z †∼2, (ii) the abrupt transition\nitself, marked by a sudden jump in the universe’s expan-\nsion rate—a type II (sudden) singularity—at z=z†, and\n(iii) the increased expansion rate compared to the ΛCDM\nmodel for z < z †. Despite the faster expansion rate, the\nΛsCDM and ΛCDM models are otherwise identical for\nz < z †. All these aspects warrant thorough investigation\nas their potential effects can be used to test the unique\npredictions of the model. In this paper, we analyze the\nnon-linear evolution of a spherical overdensity within the\nΛsCDM cosmology. We begin by revisiting the dynamics\nof spherical collapse within the ΛCDM framework and\nthen integrate the physical effects of the AdS-dS transi-\ntion into the spherical collapse model. This integration is\naccomplished by adjusting the Friedmann equations for\nthe Λ sCDM model. Furthermore, using energy consid-\nerations, we make predictions about the eventual state\nof the halo, dependent on the timing of the transition\nrelative to the turnaround. Specifically:\n•If the turnaround occurs after the transition, in16\nwhat we refer to as the pre-turnaround transition , ha-\nlos that form and undergo this transition exhibit viri-\nalized overdensities exceeding those predicted by the\nPlanck/ΛCDM model, particularly at lower turnaround\nredshifts. This observation can be attributed to the in-\ncreased δtavalues resulting from a period with a negative\ncosmological constant, which facilitates the formation of\ndenser structures. In both the ΛCDM and Λ sCDM mod-\nels under pre-turnaround transition scenarios, the expan-\nsion of the universe reaches the turnaround radius with\na positive cosmological constant. This results in greater\ncurvature and matter overdensity at the turnaround ra-\ndius compared to the post-turnaround case.\n•If the turnaround occurs before the transition, in\nwhat we refer to as post-turnaround transition , halos ex-\nperiencing this transition typically achieve virialization\nat lower overdensities compared to those predicted by\nPlanck/ΛCDM. This observation can be attributed to\nthe negative cosmological constant at the turnaround,\nnecessitating a lower matter overdensity at this moment,\nwhich results in reduced curvature at turnaround. Con-\nsequently, in this scenario, overdensities attain a larger\nmaximum physical radius owing to their diminished mat-\nter overdensity and curvature.\nIn the abrupt Λ sCDM model, we have observed that\nthe Hubble parameter displays a discontinuity at a spe-\ncific past redshift, z=z†. This discontinuity leads to a\ntype II (sudden) singularity at z=z†, as discussed in\nAppendix A and supported by [121]. Despite its mild\nnature, this singularity imparts a velocity kick to parti-\ncles. We have shown that smoothing out the abrupt be-\nhavior effectively eliminates this singularity, even when\nthe transition occurs very rapidly. Hence, our findings\nregarding the velocity kick pertain to the extreme ver-\nsion of the Λ sCDM model. However, we demonstrate\nthat, even in this case, we ascertain that the singular-\nity’s impact on Newtonian bound virialized systems is\nminimal, thereby not threatening the model’s viability\nin this context. Notably, the singularity, being relatively\nweak, does not lead to the dissociation of large bound\nsystems before this done the universe’s continuous back-\nground expansion. For instance, large clusters or super-\nclusters, corresponding to ω0<10H0, will be dissociated\nby the background expansion but remain practically un-\naffected by the singularity (as illustrated in Fig. 5-6).\nThe expansion in both the Λ sCDM and ΛCDM models\ntends to induce dissociation of bound systems at scales of\nlarge clusters and above at similar levels. Therefore, the\npresence of unbound orbits is primarily driven by uni-\nversal expansion, with the singularity merely causing an\nincrease in the Hubble expansion rate at a specific mo-\nment. Interestingly, the negative cosmological constant\nin the Λ sCDM model tends to enhance the stability of\nbound systems due to its attractive gravity effects.\nThe outcomes of our analysis open intriguing avenues\nfor future research. Broadening the scope, a natural ex-\ntension could involve generalizing the spherical collapse\nmodel to accommodate a more diverse range of suddencosmological singularities. Another promising direction\nis the investigation of the impact of the Λ-sign switch\ntransition on gravitational waves traversing the sudden\ncosmological singularity, as discussed in [127]. More-\nover, delving into the physical mechanisms that induce\nthe sign switch of the cosmological constant, as explored\nin Refs. [112–114], remains a significant area of inter-\nest. The Λ sCDM model, with its potential to address the\nHubble and S8tensions, may also influence early universe\nstructure growth due to its period of negative cosmo-\nlogical constant. Our current study assumes a uniform\ntransition in a homogeneous universe. However, slight\ninhomogeneities could lead to timing variations in this\ntransition, as suggested in [49]. Such variations might re-\nsult in different regions of the universe experiencing the\ncosmological constant’s sign switch at distinct redshifts,\nwith potential implications for galaxy formation. This\nscenario could encompass sudden singularities of varying\nintensities and even halos formed entirely under a neg-\native cosmological constant. Recent observations from\nthe JWST hint at more intense early galaxy formation,\npotentially aligning with our model’s implications. Al-\nthough our results show only minor deviations from the\nΛCDM model for structures formed at higher redshifts,\nthey underscore the necessity for further exploration.\nParticularly, the prospect of halos forming entirely under\na negative cosmological constant could present a different\nnarrative and warrants detailed investigation.\nDATA AVAILABILITY STATEMENT\nThe Mathematica (v12) and python files used for\nthe production of the figures and for derivation of\nthe main results of the analysis can be found at\ncamarman/transition-dynamics-lscdm repository under\nthe MIT license.\nACKNOWLEDGMENTS\n¨O.A. acknowledges the support by the Turkish\nAcademy of Sciences in scheme of the Outstanding Young\nScientist Award (T ¨UBA-GEB ˙IP). ¨O.A. and A.C ¸. are\nsupported in part by TUBITAK Grant No. 122F124.\nThis research was supported by COST Action CA21136\n- Addressing observational tensions in cosmology with\nsystematics and fundamental physics (CosmoVerse), sup-\nported by COST (European Cooperation in Science and\nTechnology). This project was also supported by the Hel-\nlenic Foundation for Research and Innovation (H.F.R.I.),\nunder the “First call for H.F.R.I. Research Projects\nto support Faculty members and Researchers and the\nprocurement of high-cost research equipment Grant”\n(Project Number: 789).17\nAppendix A: Demonstration of Type II Singularity\nWe have studied the effects an abrupt transition de-\nscribed by the signum function. This description leads\nto a Type II (sudden) singularity, characterized by:\nt=t†, a =a†<∞, ρ tot(a†)<∞,|Ptot(a†)| → ∞ ,\n(A1)\nwith the following characteristics: the scale factor, a(t),\nis continuous and non-zero at the moment t†; the first\nderivative of the scale factor, ˙ a, is discontinuous at t†;\nand its second derivative ¨ adiverges at t†[121]. We prove\nthis argument, by considering that ¨ a/a=˙H+H2, and\nby implementing Friedmann equation\nH2=8πG\n3˜ρm0\u0002\na−3+ωssgn(1 /a†−1/a)\u0003\n, (A2)\nwe obtain:\n¨a\na=−4πG\n3ρm0h\na−3−2ωssgn(1 /a†−1/a) (A3)\n−2ωsa−1δ(1/a†−1/a)i\n.\nWhere δrepresents the Dirac delta function. It is evident\nthat at the precise moment of the singularity,¨a\na→ ∞ .\nGiven that, Ptot=−1\n8πGh\n2¨a\na+\u0000˙a\na\u00012i\n, at the moment of\ntransition, Ptot→ −∞ . This is a feature that can arise\nin universes with any spatial curvature. Additionally, we\ncould define an equation of state that resembles a smooth\ntransition in the energy density of the Λ s:\nρΛs(a) =ρΛs0tanh [ σ(a−a†)]\ntanh [ σ(1−a†)], (A4)\nforρΛs0>0, where a†<1 and σ >010being a parame-\nter determining the rapidity of the transition. Under this\nparametrization, the total energy density and total pres-\nsure of the universe, containing only dust and Λ s, can be\nwritten as:\nρtot(a) =ρm0a−3+ρΛs0tanh [ σ(a−a†)]\ntanh [ σ(1−a†)],\nPtot(a) =−ρΛs0\u0014tanh [ σ(a−a†)]\ntanh [ σ(1−a†)]+σa\n3sech2[σ(a−a†)]\ntanh [ σ(1−a†)]\u0015\n.\n(A5)\nUpon examining the characteristics of ρtot(a) and Ptot(a)\nata=a†, we find:\nρtot(a†) =ρm0a−3\n†,\nPtot(a†) =−ρΛs0a\n3σ\ntanh [ σ(1−a†)].(A6)\n10One must set σ≫1 to approximate the signum function.\n0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475\na024681012ρtot/ρΛs0\nσ= 50\nσ= 100\nσ= 200\nσ→∞\na†= 0.3704\nρtot≡ρm0a−3\n†\n0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475\na10−1\n−5\n−10\n−15\n−20\n−25Ptot/ρΛs0\nσ= 50\nσ= 100\nσ= 200\nσ→∞\na†= 0.3704FIG. 7. Total density and total pressure of the Λ sCDM uni-\nverse (see Eq. (A5)) with respect to scale factor. For σ→ ∞ ,\nwe observe that ρtot(a†)→const . >0 while Ptot(a†)→ −∞ .\nNotice that ρtot(a†) does not depend on σ, and Ptot(a†)\nis negative but finite for finite values of σ.\nThe “smoothed-out energy density of Λ s” ( Eq.(A6))\nreduces, by taking σ→ ∞ , to an abrupt (sudden) AdS\n→dS transition, which we have studied in the current\npaper:\nlim\nσ→∞ρΛs(a) =ρΛs0sgn[a−a†]. (A7)\nIn this case, we observe that the total pressure diverges\nto negative infinity:\nlim\nσ→∞Ptot(a†) =−∞, (A8)\nwhile the total energy density remains positive and finite.\nThus, this behavior, which occurs at the limit of σ→\n∞, is characterized by a type II (sudden) cosmological\nsingularity [121, 122]. Note that at the moment when\nthe singularity occurs, the equation of state parameter\nwDEis undefined; this is a feature of sudden singularities.\nFor finite values of σ, this singularity does not exist, and\nthe equation of state parameter wDEcan be defined. For\nthat reason, we reconstructed it directly by smoothing\nout the dark energy density and took the limit for the\nextreme case [120].18\nAppendix B: Gravitational potentials for dark\nmatter and dark energy\nThe Poisson equation for barotropic fluid, described by\nP=wρ, is expressed as:\n∇2Φ = 4 πGρ(1 + 3 w). (B1)\nIn a theoretical framework where a homogeneous energy\ndensity is spherically distributed within a radius R, and\nthe gravitational potential is denoted as Φ ≡Φ(r), the\ngeneral solution is applied:\nΦ(r) = 2 πGρ(1 + 3 w)r2\n3−C\nr+D. (B2)\nThe gravitational potential of dark matter, Φ DM, is de-\ntermined by assuming ρ(r > R ) = 0. As a result, the\ngeneral solution is reformulated as:\nΦI(r > R ) = 2 πGρ(1 + 3 w)r2\n3−C\nr+D,\nΦII(r < R ) =C\nr+D.(B3)\nThe boundary conditions are defined as follows: (a)\nΦI(r= 0) <∞, (b) Φ II(r→ ∞ ) = 0, (c) Φ I(r=R) =\nΦII(r=R), and (d) dΦ I/dr\f\f\nr=R= dΦ II/dr\f\f\nr=R. Ap-\nplying the general solution from Eq. (B3) for dark matter\ndensity ρDM(assuming wDM= 0) within a sphere of ra-\ndiusR, we obtain:\nΦDM(r) =\n\n−2πGρ DM\u0000\nR2−r2/3\u0001\nr≤R\n−4πGρ DMR3/3r r ≥R(B4)\nMeanwhile, the gravitational potential of dark energy,\nΦDE, is determined by imposing uniform energy density\nacross the universe as boundary conditions: (a) Φ( r=\n0)<∞and (b) Φ( r= 0) = 0. Thus, applying these\nconditions to Eq. (B2), we deduce:\nΦDE(r) = 2 πGρ DE(1 + 3 wDE)r2\n3. (B5)\nAppendix C: Scale factor of the ΛsCDM model\nTo elucidate the scale factor in terms of comoving time,\nwe employ the Friedmann equation, articulated as:\nH2\nH2\n0= Ω m0a−3+ Ω Λs0sgn(1 /a†−1/a). (C1)\nThe scale factor maintains continuity and can be char-\nacterized by integrating Eq. (C1), as demonstrated be-\nlow:\n˙a=H0\u0002\nΩm0a−1+ Ω Λs0a2sgn(1 /a†−1/a)\u00031\n2.(C2)In particular for a < a †the integration is given by:\nZa\n0da\n(Ωm0a−1−ΩΛs0a2)1\n2=H0t . (C3)\nBy changing variable a=y2\n3\u0010\n1−ΩΛs0\nΩΛs0\u00111\n3and given that\nda=2\n3\u0010\n1−ΩΛs0\nΩΛs0\u00111\n3y−1\n3dy, Eq. (C3) implies:\na(t) =\u00121−ΩΛs0\nΩΛs0\u00131\n3\nsin2\n3\u00123\n2p\nΩΛs0H0t\u0013\n. (C4)\nSubsequently, if a > a †, then the integration proceeds as:\nZa†\n0da\n(Ωm0a−1−ΩΛs0a2)1\n2\n+Za\na†da\n(Ωm0a−1+ Ω Λs0a2)1\n2=H0t .(C5)\nIn that case, Eq. (C5) implies:\na(t) =\u00121−ΩΛs0\nΩΛs0\u00131\n3\n×sinh2\n3\"\n3\n2p\nΩΛs0H0(t−t†)\n+ sinh−1h\nsin\u00003\n2p\nΩΛs0H0t†\u0001i#\n.(C6)\nSubsequently, we get:\n¨a\na=\n\n−4\n9A2\n1h\n1 +1\n1−cos(2A1t)i\nt < t†\n−4\n9A2\n1h\n−2 +1\nsinh [A1(t−t†)+A2]i\nt > t†(C7)\nwhere we have denoted:\nA1≡3\n2H0p\n(1−Ωm0),\nA2≡sinh−1[sin (A1t†)].\nNext, the general solutions to the free particle geodesic\nEq. (48), are outlined separately for the two distinct pe-\nriods of comoving time, t≥t†andt≤t†, as follows:\nr(t) =\n\nC1h\n−1 + cos ( A1t)2i1\n4P1\n6\n1\n6[cos (A1t)]\n+C2h\n−1 + cos ( A1t)2i1\n4Q1\n6\n1\n6[cos (A1t)] t≤t†\nB D1 2F1\u0002\n−1\n6,1\n3,5\n6,tanh2[A1(t−t†) +A2]\u0003\n+B D2tanh2\n3[A1(t−t†) +A2] t≥t†\n(C8)\nwhere we have denoted\nB ≡\u0002\n−1 + tanh2(A1(t−t†) +A2)\u0003−1/2,19\nandPm\nl,Qm\nlrepresents the associated Legendre func-\ntions of first and second kind, respectively. Meanwhile,\n2F1is the hyper-geometric function [165]. The constants\nare determined through the appropriate boundary con-\nditions.\nAppendix D: Determining cosmological parameters\nIn order to determine the cosmological parameters for\nour analysis, we will follow a method used in [166–168].\nThe locations of peaks (i.e., peak spacing) in the CMB\npower spectrum11,lA, is a well-measured quantity and it\nis defined as:\nlA≡π(1 +z∗)DA(z∗)\nr∗s=π\nθ∗s, (D1)\nwhere θ∗\ns≡r∗\ns/dA(z∗) represents the angular size of the\nsound horizon at the last-scattering surface [169–171].\nHere, r∗\nsrepresents the comoving sound horizon at the\nlast-scattering surface:\nr∗\ns=∞Z\nz∗cs(z)dz\nH(z), (D2)\nwhere cs(z) is the sound speed of the photon-baryon fluid:\ncs(z) =cr\n3\u0010\n1 +3ωb\n4ωγ(1+z)\u0011, (D3)\nandz∗is the redshift at the last-scattering surface, which\ncan be approximated analytically via [172]:\nz∗= 1048\u0002\n1 + 0 .00124( ωb)−0.738\u0003\n[1 +g1(ωm)g2],\ng1= 0.0783( ωb)−0.238\u0002\n1 + 39 .5(ωb)0.763\u0003−1,\ng2= 0.560[1 + 21 .1(ωb)1.81]−1.(D4)\nMeanwhile, the proper angular diameter distance to the\nlast-scattering surface defined as:\nDA(z∗)≡dA(z∗)\n1 +z∗=c\n1 +z∗z∗Z\n0dz\nH(z). (D5)\nIn what follows, we will start the calculations by assum-\ning that the acoustic scale and the physical density pa-\nrameter for matter in the ΛCDM universe will be equal\nto the Λ sCDM12(i.e., lΛ\nA≃lΛs\nAandωΛ\nm≃ωΛsm). Since\n11Also known as the “acoustic scales”\n12CMB distance priors, lAandR, (viz. ωm) are actually derived\nparameters by fitting a cosmological model to the CMB power\nspectra [173–175]. Thus, the underlying cosmology would change\nthe distance priors [176]. Since, Λ sCDM does not change the\nphysics of the early universe, we can assume that it will not\ncause a significant variation in lAor in ωm.TABLE II. Plik Best Fit values taken from the Planck-2018\ndataset [4]. We have defined the physical radiation den-\nsity parameter as the sum of the physical photon and neu-\ntrino density parameters; ωr≡ωγ+ωn= 2.469×10−5×h\n1 +7\n8\u00004\n11\u00014/3Neffi\nwith Neff= 3.046 for standard model of\nparticle physics.\nParameter Value\nωb 0.022383\nωm 0.143140\nωr 4.177×10−5\n100θ∗\ns 1.041085\nΛsCDM does not change Neffor the physics of the early\nuniverse, we can further assume that ωΛ\nr≃ωΛsrand\nωΛ\nb≃ωΛs\nb.\nUnder these conditions, z∗andcs(z) will be the same\nfor ΛCDM and Λ sCDM models, which can be seen via\nEqs. (D3) and (D4). By combining Eqs. (D1), (D2) and\n(D5), we can write:\nθ∗\ns=∞Z\nz∗cs(z) dzp\nωm(1 +z)3+ωr(1 +z)4+ (h2\n0−ωm−ωr)fDE(z)\nz∗Z\n0cdzp\nωm(1 +z)3+ωr(1 +z)4+ (h2\n0−ωm−ωr)fDE(z).\n(D6)\nSince the value of the parameter θ∗\nsis fixed by Planck\nCMB observations almost model independently, we can\nconstrain h0for any given fDE(z)≡ρDE(z)/ρDE,0(viz.\nfΛ(z)≡1 and fΛs(z)≡sgn(z†−z)), provided that the\npre-recombination universe remains as in the standard\ncosmological model.\nTo simplify the above procedure, one can use the fitting\nformulae given below to calculate the Ω m0for the Λ sCDM\nmodel, expressed in terms of the transition redshift, z†:\nΩm0(z†) =c0+c1z−1\n†+c2z−2\n†+c3z−3\n†. (D7)\nBy assuming the Table II parameters, we obtain\nthe constants of the equation as ( c0, c1, c2, c3) =\n(0.3093,0.0155,−0.0994,−0.0722), which correctly finds\nΩm0up to order of 10−4for 1.5≤z†≤11.5.20\n[1] S. Dodelson and F. Schmidt, Modern Cosmology , second\nedition ed. (Academic Press, 2021) pp. 1–19.\n[2] A. G. Riess et al. (Supernova Search Team), Astron. J.\n116, 1009 (1998), arXiv:astro-ph/9805201.\n[3] S. Perlmutter et al. (Supernova Cosmology Project), As-\ntrophys. J. 517, 565 (1999), arXiv:astro-ph/9812133.\n[4] N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6\n(2020), [Erratum: Astron.Astrophys. 652, C4 (2021)],\narXiv:1807.06209 [astro-ph.CO].\n[5] S. Aiola et al. (ACT), JCAP 12, 047 (2020),\narXiv:2007.07288 [astro-ph.CO].\n[6] S. Alam et al. (eBOSS), Phys. Rev. D 103, 083533\n(2021), arXiv:2007.08991 [astro-ph.CO].\n[7] M. Asgari et al. (KiDS), Astron. Astrophys. 645, A104\n(2021), arXiv:2007.15633 [astro-ph.CO].\n[8] T. M. C. Abbott et al. (DES), Phys. Rev. D 105, 023520\n(2022), arXiv:2105.13549 [astro-ph.CO].\n[9] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).\n[10] S. M. Carroll, W. H. Press, and E. L. Turner, Ann.\nRev. Astron. Astrophys. 30, 499 (1992).\n[11] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D\n9, 373 (2000), arXiv:astro-ph/9904398.\n[12] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559\n(2003), arXiv:astro-ph/0207347.\n[13] T. Padmanabhan, Phys. Rept. 380, 235 (2003),\narXiv:hep-th/0212290.\n[14] P. Bull et al. , Phys. Dark Univ. 12, 56 (2016),\narXiv:1512.05356 [astro-ph.CO].\n[15] E. Di Valentino et al. , Astropart. Phys. 131, 102605\n(2021), arXiv:2008.11284 [astro-ph.CO].\n[16] E. Di Valentino, O. Mena, S. Pan, L. Visinelli,\nW. Yang, A. Melchiorri, D. F. Mota, A. G. Riess,\nand J. Silk, Class. Quant. Grav. 38, 153001 (2021),\narXiv:2103.01183 [astro-ph.CO].\n[17] P. J. E. Peebles, Annals Phys. 447, 169159 (2022),\narXiv:2208.05018 [astro-ph.CO].\n[18] L. Perivolaropoulos and F. Skara, New Astron. Rev. 95,\n101659 (2022), arXiv:2105.05208 [astro-ph.CO].\n[19] E. Abdalla et al. , JHEAp 34, 49 (2022),\narXiv:2203.06142 [astro-ph.CO].\n[20] S. Vagnozzi, Universe 9, 393 (2023), arXiv:2308.16628\n[astro-ph.CO].\n[21] J.-P. Hu and F.-Y. Wang, Universe 9, 94 (2023),\narXiv:2302.05709 [astro-ph.CO].\n[22] O. Akarsu, E. O. Colg´ ain, A. A. Sen, and M. M. Sheikh-\nJabbari, (2024), arXiv:2402.04767 [astro-ph.CO].\n[23] A. G. Riess et al. , Astrophys. J. Lett. 934, L7 (2022),\narXiv:2112.04510 [astro-ph.CO].\n[24] P. A. Burger et al. (2023) arXiv:2309.08602 [astro-\nph.CO].\n[25] S.-F. Chen, Z. Vlah, and M. White, JCAP 02, 008\n(2022), arXiv:2110.05530 [astro-ph.CO].\n[26] S. A. Adil, O. Akarsu, M. Malekjani, E. O. Colg´ ain,\nS. Pourojaghi, A. A. Sen, and M. M. Sheikh-Jabbari,\n(2023), 10.1093/mnrasl/slad165, arXiv:2303.06928\n[astro-ph.CO].\n[27] T. Karwal and M. Kamionkowski, Phys. Rev. D 94,\n103523 (2016), arXiv:1608.01309 [astro-ph.CO].\n[28] V. Poulin, T. L. Smith, T. Karwal, and\nM. Kamionkowski, Phys. Rev. Lett. 122, 221301\n(2019), arXiv:1811.04083 [astro-ph.CO].[29] V. Poulin, T. L. Smith, D. Grin, T. Karwal, and\nM. Kamionkowski, Phys. Rev. D 98, 083525 (2018),\narXiv:1806.10608 [astro-ph.CO].\n[30] P. Agrawal, F.-Y. Cyr-Racine, D. Pinner, and L. Ran-\ndall (2019) arXiv:1904.01016 [astro-ph.CO].\n[31] M. Kamionkowski and A. G. Riess (2022)\narXiv:2211.04492 [astro-ph.CO].\n[32] S. D. Odintsov, V. K. Oikonomou, and G. S. Sharov,\nPhys. Lett. B 843, 137988 (2023), arXiv:2305.17513 [gr-\nqc].\n[33] F. Niedermann and M. S. Sloth, Phys. Rev. D 103,\nL041303 (2021), arXiv:1910.10739 [astro-ph.CO].\n[34] J. S. Cruz, F. Niedermann, and M. S. Sloth (2023)\narXiv:2305.08895 [astro-ph.CO].\n[35] F. Niedermann and M. S. Sloth (2023) arXiv:2307.03481\n[hep-ph].\n[36] G. Ye and Y.-S. Piao, Phys. Rev. D 101, 083507 (2020),\narXiv:2001.02451 [astro-ph.CO].\n[37] G. Ye and Y.-S. Piao, Phys. Rev. D 102, 083523 (2020),\narXiv:2008.10832 [astro-ph.CO].\n[38] G. Ye, J. Zhang, and Y.-S. Piao, Phys. Lett. B 839,\n137770 (2023), arXiv:2107.13391 [astro-ph.CO].\n[39] M. Rossi, M. Ballardini, M. Braglia, F. Finelli, D. Pao-\nletti, A. A. Starobinsky, and C. Umilt` a, Phys. Rev. D\n100, 103524 (2019), arXiv:1906.10218 [astro-ph.CO].\n[40] M. Braglia, M. Ballardini, W. T. Emond, F. Finelli,\nA. E. Gumrukcuoglu, K. Koyama, and D. Paoletti,\nPhys. Rev. D 102, 023529 (2020), arXiv:2004.11161\n[astro-ph.CO].\n[41] T. Adi and E. D. Kovetz, Phys. Rev. D 103, 023530\n(2021), arXiv:2011.13853 [astro-ph.CO].\n[42] M. Braglia, M. Ballardini, F. Finelli, and K. Koyama,\nPhys. Rev. D 103, 043528 (2021), arXiv:2011.12934\n[astro-ph.CO].\n[43] M. Ballardini, M. Braglia, F. Finelli, D. Paoletti, A. A.\nStarobinsky, and C. Umilt` a, JCAP 10, 044 (2020),\narXiv:2004.14349 [astro-ph.CO].\n[44] G. Franco Abell´ an, M. Braglia, M. Ballardini, F. Finelli,\nand V. Poulin (2023) arXiv:2308.12345 [astro-ph.CO].\n[45] M. Petronikolou and E. N. Saridakis, Universe 9, 397\n(2023), arXiv:2308.16044 [gr-qc].\n[46] D. K. Hazra, A. Antony, and A. Shafieloo, JCAP 08,\n063 (2022), arXiv:2201.12000 [astro-ph.CO].\n[47] O. Akarsu, J. D. Barrow, L. A. Escamilla, and\nJ. A. Vazquez, Phys. Rev. D 101, 063528 (2020),\narXiv:1912.08751 [astro-ph.CO].\n[48] O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, and J. A. Vazquez,\nPhys. Rev. D 104, 123512 (2021), arXiv:2108.09239\n[astro-ph.CO].\n[49] O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, J. A. Vazquez,\nand A. Yadav, Phys. Rev. D 108, 023513 (2023),\narXiv:2211.05742 [astro-ph.CO].\n[50] O. Akarsu, E. Di Valentino, S. Kumar, R. C. Nunes,\nJ. A. Vazquez, and A. Yadav (2023) arXiv:2307.10899\n[astro-ph.CO].\n[51] E. Di Valentino, A. Mukherjee, and A. A. Sen, Entropy\n23, 404 (2021), arXiv:2005.12587 [astro-ph.CO].\n[52] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos,\nPhys. Rev. D 101, 123516 (2020), arXiv:2004.08363\n[astro-ph.CO].\n[53] G. Alestas, L. Kazantzidis, and L. Perivolaropoulos,21\nPhys. Rev. D 103, 083517 (2021), arXiv:2012.13932\n[astro-ph.CO].\n[54] M. R. Gangopadhyay, S. K. J. Pacif, M. Sami, and\nM. K. Sharma, Universe 9, 83 (2023), arXiv:2211.12041\n[gr-qc].\n[55] S. Basilakos, A. Lymperis, M. Petronikolou, and E. N.\nSaridakis (2023) arXiv:2308.01200 [gr-qc].\n[56] S. A. Adil, O. Akarsu, E. Di Valentino, R. C. Nunes,\nE.¨Oz¨ ulker, A. A. Sen, and E. Specogna, Phys. Rev. D\n109, 023527 (2024), arXiv:2306.08046 [astro-ph.CO].\n[57] M. R. Gangopadhyay, M. Sami, and M. K. Sharma,\nPhys. Rev. D 108, 103526 (2023), arXiv:2303.07301\n[astro-ph.CO].\n[58] L. Visinelli, S. Vagnozzi, and U. Danielsson, Symmetry\n11, 1035 (2019), arXiv:1907.07953 [astro-ph.CO].\n[59] K. Dutta, Ruchika, A. Roy, A. A. Sen, and\nM. M. Sheikh-Jabbari, Gen. Rel. Grav. 52, 15 (2020),\narXiv:1808.06623 [astro-ph.CO].\n[60] A. A. Sen, S. A. Adil, and S. Sen, Mon. Not. Roy.\nAstron. Soc. 518, 1098 (2022), arXiv:2112.10641 [astro-\nph.CO].\n[61] S. Kumar and R. C. Nunes, Phys. Rev. D 96, 103511\n(2017), arXiv:1702.02143 [astro-ph.CO].\n[62] E. Di Valentino, A. Melchiorri, and O. Mena, Phys.\nRev. D 96, 043503 (2017), arXiv:1704.08342 [astro-\nph.CO].\n[63] W. Yang, A. Mukherjee, E. Di Valentino, and S. Pan,\nPhys. Rev. D 98, 123527 (2018), arXiv:1809.06883\n[astro-ph.CO].\n[64] S. Pan, W. Yang, E. Di Valentino, E. N. Saridakis,\nand S. Chakraborty, Phys. Rev. D 100, 103520 (2019),\narXiv:1907.07540 [astro-ph.CO].\n[65] S. Kumar, R. C. Nunes, and S. K. Yadav, Eur. Phys.\nJ. C79, 576 (2019), arXiv:1903.04865 [astro-ph.CO].\n[66] E. Di Valentino, A. Melchiorri, O. Mena, and\nS. Vagnozzi, Phys. Rev. D 101, 063502 (2020),\narXiv:1910.09853 [astro-ph.CO].\n[67] E. Di Valentino, A. Melchiorri, O. Mena, and\nS. Vagnozzi, Phys. Dark Univ. 30, 100666 (2020),\narXiv:1908.04281 [astro-ph.CO].\n[68] M. Lucca and D. C. Hooper, Phys. Rev. D 102, 123502\n(2020), arXiv:2002.06127 [astro-ph.CO].\n[69] A. G´ omez-Valent, V. Pettorino, and L. Amendola,\nPhys. Rev. D 101, 123513 (2020), arXiv:2004.00610\n[astro-ph.CO].\n[70] S. Kumar, Phys. Dark Univ. 33, 100862 (2021),\narXiv:2102.12902 [astro-ph.CO].\n[71] R. C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino,\nand O. Mena, Phys. Rev. D 105, 123506 (2022),\narXiv:2203.08093 [astro-ph.CO].\n[72] A. Bernui, E. Di Valentino, W. Giar` e, S. Kumar,\nand R. C. Nunes, Phys. Rev. D 107, 103531 (2023),\narXiv:2301.06097 [astro-ph.CO].\n[73] E. Aubourg et al. , Phys. Rev. D 92, 123516 (2015),\narXiv:1411.1074 [astro-ph.CO].\n[74] V. Sahni, A. Shafieloo, and A. A. Starobinsky, Astro-\nphys. J. Lett. 793, L40 (2014), arXiv:1406.2209 [astro-\nph.CO].\n[75] V. Poulin, K. K. Boddy, S. Bird, and M. Kamionkowski,\nPhys. Rev. D 97, 123504 (2018), arXiv:1803.02474\n[astro-ph.CO].\n[76] Y. Wang, L. Pogosian, G.-B. Zhao, and A. Zucca,\nAstrophys. J. Lett. 869, L8 (2018), arXiv:1807.03772[astro-ph.CO].\n[77] A. Bonilla, S. Kumar, and R. C. Nunes, Eur. Phys. J.\nC81, 127 (2021), arXiv:2011.07140 [astro-ph.CO].\n[78] L. A. Escamilla, O. Akarsu, E. Di Valentino, and\nJ. A. Vazquez, JCAP 11, 051 (2023), arXiv:2305.16290\n[astro-ph.CO].\n[79] L. A. Escamilla and J. A. Vazquez, Eur. Phys. J. C 83,\n251 (2023), arXiv:2111.10457 [astro-ph.CO].\n[80] M. Malekjani, R. M. Conville, E. O. Colg´ ain,\nS. Pourojaghi, and M. M. Sheikh-Jabbari, (2023),\narXiv:2301.12725 [astro-ph.CO].\n[81] O. Akarsu, E. O. Colgain, E. ¨Ozulker, S. Thakur,\nand L. Yin, Phys. Rev. D 107, 123526 (2023),\narXiv:2207.10609 [astro-ph.CO].\n[82] R. Calder´ on, R. Gannouji, B. L’Huillier, and\nD. Polarski, Phys. Rev. D 103, 023526 (2021),\narXiv:2008.10237 [astro-ph.CO].\n[83] V. Marra and L. Perivolaropoulos, Phys. Rev. D 104,\nL021303 (2021), arXiv:2102.06012 [astro-ph.CO].\n[84] G. Alestas, I. Antoniou, and L. Perivolaropoulos, Uni-\nverse 7, 366 (2021), arXiv:2104.14481 [astro-ph.CO].\n[85] G. Alestas, D. Camarena, E. Di Valentino,\nL. Kazantzidis, V. Marra, S. Nesseris, and\nL. Perivolaropoulos, Phys. Rev. D 105, 063538\n(2022), arXiv:2110.04336 [astro-ph.CO].\n[86] L. Perivolaropoulos and F. Skara, Phys. Rev. D 104,\n123511 (2021), arXiv:2109.04406 [astro-ph.CO].\n[87] S. Pan, O. Seto, T. Takahashi, and Y. Toda, (2023),\narXiv:2312.15435 [astro-ph.CO].\n[88] K. Naidoo, M. Jaber, W. A. Hellwing, and M. Bilicki,\n(2022), arXiv:2209.08102 [astro-ph.CO].\n[89] M. Boylan-Kolchin, Nature Astron. 7, 731 (2023),\narXiv:2208.01611 [astro-ph.CO].\n[90] I. Labb´ e, P. van Dokkum, E. Nelson, R. Bezanson, K. A.\nSuess, J. Leja, G. Brammer, K. Whitaker, E. Mathews,\nM. Stefanon, and B. Wang, Nature (London) 616, 266\n(2023), arXiv:2207.12446 [astro-ph.GA].\n[91] N. Menci, M. Castellano, P. Santini, E. Merlin,\nA. Fontana, and F. Shankar, Astrophys. J. Lett. 938,\nL5 (2022), arXiv:2208.11471 [astro-ph.CO].\n[92] M. Biagetti, G. Franciolini, and A. Riotto, Astrophys.\nJ.944, 113 (2023), arXiv:2210.04812 [astro-ph.CO].\n[93] M. Forconi, Ruchika, A. Melchiorri, O. Mena, and\nN. Menci (2023) arXiv:2306.07781 [astro-ph.CO].\n[94] R. P. Gupta, Mon. Not. Roy. Astron. Soc. 524, 3385\n(2023), arXiv:2309.13100 [astro-ph.CO].\n[95] K. Glazebrook et al. (2023) arXiv:2308.05606 [astro-\nph.GA].\n[96] S. A. Adil, U. Mukhopadhyay, A. A. Sen, and\nS. Vagnozzi (2023) arXiv:2307.12763 [astro-ph.CO].\n[97] S. Hirano and N. Yoshida (2023) arXiv:2306.11993\n[astro-ph.GA].\n[98] L. Y. A. Yung, R. S. Somerville, S. L. Finkelstein, S. M.\nWilkins, and J. P. Gardner (2023) arXiv:2304.04348\n[astro-ph.GA].\n[99] J. McCaffrey, S. Hardin, J. Wise, and J. Regan (2023)\narXiv:2304.13755 [astro-ph.GA].\n[100] Y.-Y. Wang, L. Lei, G.-W. Yuan, and Y.-Z. Fan, Astro-\nphys. J. Lett. 954, L48 (2023), arXiv:2307.12487 [astro-\nph.GA].\n[101] J. Wang, Z. Huang, L. Huang, and J. Liu, “Quantifying\nthe tension between cosmological models and JWST red\ncandidate massive galaxies,” (2023), arXiv:2311.0286622\n[astro-ph.CO].\n[102] D. Wang and Y. Liu (2022) arXiv:2301.00347 [astro-\nph.CO].\n[103] F. Melia, Mon. Not. Roy. Astron. Soc. 521, L85 (2023),\narXiv:2302.10103 [astro-ph.CO].\n[104] E. A. Paraskevas and L. Perivolaropoulos (2023)\narXiv:2308.07046 [astro-ph.CO].\n[105] N. Menci, S. A. Adil, U. Mukhopadhyay, A. A. Sen, and\nS. Vagnozzi, (2024), arXiv:2401.12659 [astro-ph.CO].\n[106] L. Kazantzidis and L. Perivolaropoulos (2019)\narXiv:1907.03176 [astro-ph.CO].\n[107] D. Benisty, Phys. Dark Univ. 31, 100766 (2021),\narXiv:2005.03751 [astro-ph.CO].\n[108] R. C. Nunes and S. Vagnozzi, Mon. Not. Roy. Astron.\nSoc.505, 5427 (2021), arXiv:2106.01208 [astro-ph.CO].\n[109] S. Bocquet et al. (SPT), Astrophys. J. 878, 55 (2019),\narXiv:1812.01679 [astro-ph.CO].\n[110] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231\n(1998), arXiv:hep-th/9711200.\n[111] R. Bousso and J. Polchinski, JHEP 06, 006 (2000),\narXiv:hep-th/0004134.\n[112] L. A. Anchordoqui, I. Antoniadis, and D. Lust (2023)\narXiv:2312.12352 [hep-th].\n[113] B. Alexandre, S. Gielen, and J. a. Magueijo, (2023),\narXiv:2306.11502 [hep-th].\n[114] O. Akarsu, A. De Felice, E. Di Valentino, S. Kumar,\nR. C. Nunes, E. Ozulker, J. A. Vazquez, and A. Yadav,\n(2024), arXiv:2402.07716 [astro-ph.CO].\n[115] A. De Felice, A. Doll, and S. Mukohyama, JCAP 09,\n034 (2020), arXiv:2004.12549 [gr-qc].\n[116] A. De Felice, S. Mukohyama, and M. C. Pookkil-\nlath, Phys. Lett. B 816, 136201 (2021), [Erratum:\nPhys.Lett.B 818, 136364 (2021)], arXiv:2009.08718\n[astro-ph.CO].\n[117] J. Garriga, A. D. Linde, and A. Vilenkin, Phys. Rev.\nD69, 063521 (2004), arXiv:hep-th/0310034.\n[118] J. Garriga, L. Pogosian, and T. Vachaspati, Phys. Rev.\nD69, 063511 (2004), arXiv:astro-ph/0311412.\n[119] L. Perivolaropoulos, Phys. Rev. D 71, 063503 (2005),\narXiv:astro-ph/0412308.\n[120] E. Ozulker, Phys. Rev. D 106, 063509 (2022),\narXiv:2203.04167 [astro-ph.CO].\n[121] J. D. Barrow, Class. Quant. Grav. 21, L79 (2004),\narXiv:gr-qc/0403084.\n[122] S. Nojiri, S. D. Odintsov, and S. Tsujikawa, Phys. Rev.\nD71, 063004 (2005), arXiv:hep-th/0501025.\n[123] L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D\n74, 064030 (2006), arXiv:gr-qc/0607073.\n[124] A. Balcerzak, T. Denkiewicz, and M. Lisaj, Eur. Phys.\nJ. C83, 980 (2023), arXiv:2308.13293 [gr-qc].\n[125] J. de Haro, S. Nojiri, S. D. Odintsov, V. K. Oikonomou,\nand S. Pan (2023) arXiv:2309.07465 [gr-qc].\n[126] O. Trivedi (2023) arXiv:2309.08954 [gr-qc].\n[127] E. A. Paraskevas and L. Perivolaropoulos, Universe 9,\n317 (2023), arXiv:2307.00298 [astro-ph.CO].\n[128] V. R. Eke, S. Cole, and C. S. Frenk, Mon. Not. Roy.\nAstron. Soc. 282, 263 (1996), arXiv:astro-ph/9601088.\n[129] E. L. Lokas and Y. Hoffman (2000) arXiv:astro-\nph/0011295.\n[130] D. F. Mota and C. van de Bruck, Astron. Astrophys.\n421, 71 (2004), arXiv:astro-ph/0401504.\n[131] P. Creminelli, G. D’Amico, J. Norena, L. Senatore,\nand F. Vernizzi, JCAP 03, 027 (2010), arXiv:0911.2701\n[astro-ph.CO].[132] V. Pavlidou and B. D. Fields, Phys. Rev. D 71, 043510\n(2005), arXiv:astro-ph/0410338.\n[133] C. Horellou and J. Berge, Mon. Not. Roy. Astron. Soc.\n360, 1393 (2005), arXiv:astro-ph/0504465.\n[134] S. Basilakos, Astrophys. J. 590, 636 (2003), arXiv:astro-\nph/0303112.\n[135] D. Tanoglidis, V. Pavlidou, and T. Tomaras, JCAP 12,\n060 (2015), arXiv:1412.6671 [astro-ph.CO].\n[136] D. Tanoglidis, V. Pavlidou, and T. Tomaras (2016)\narXiv:1601.03740 [astro-ph.CO].\n[137] F. Pace, S. Meyer, and M. Bartelmann, JCAP 10, 040\n(2017), arXiv:1708.02477 [astro-ph.CO].\n[138] V. Pavlidou, G. Korkidis, T. Tomaras, and\nD. Tanoglidis, Astron. Astrophys. 638, L8 (2020),\narXiv:2004.04395 [astro-ph.CO].\n[139] W. H. Press and P. Schechter, Astrophys. J. 187, 425\n(1974).\n[140] J. R. Bond, S. Cole, G. Efstathiou, and N. Kaiser,\nAstrophys. J. 379, 440 (1991).\n[141] A. Cooray and R. K. Sheth, Phys. Rept. 372, 1 (2002),\narXiv:astro-ph/0206508.\n[142] M. Asgari, A. J. Mead, and C. Heymans (2023)\narXiv:2303.08752 [astro-ph.CO].\n[143] S. Dodelson and F. Schmidt, in Modern Cosmology (Sec-\nond Edition) , edited by S. Dodelson and F. Schmidt\n(Academic Press, 2021) second edition ed., pp. 325–372.\n[144] G. A. Baker, Jr. (2001) arXiv:astro-ph/0112320.\n[145] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 70,\n123529 (2004), arXiv:astro-ph/0410309.\n[146] V. Faraoni and A. Jacques, Phys. Rev. D 76, 063510\n(2007), arXiv:0707.1350 [gr-qc].\n[147] L. Perivolaropoulos, Phys. Rev. D 94, 124018 (2016),\narXiv:1609.08528 [gr-qc].\n[148] O. Lahav, P. B. Lilje, J. R. Primack, and M. J. Rees,\nMon. Not. Roy. Astron. Soc. 251, 128 (1991).\n[149] I. T. Iliev and P. R. Shapiro, Mon. Not. Roy. Astron.\nSoc.325, 468 (2001), arXiv:astro-ph/0101067.\n[150] I. Maor and O. Lahav, JCAP 07, 003 (2005),\narXiv:astro-ph/0505308.\n[151] P. Saha, D. Dey, and K. Bhattacharya (2023)\narXiv:2306.00805 [gr-qc].\n[152] R. A. Battye and J. Weller, Phys. Rev. D 68, 083506\n(2003), arXiv:astro-ph/0305568.\n[153] L.-M. Wang and P. J. Steinhardt, Astrophys. J. 508,\n483 (1998), arXiv:astro-ph/9804015.\n[154] N. N. Weinberg and M. Kamionkowski, Mon. Not. Roy.\nAstron. Soc. 341, 251 (2003), arXiv:astro-ph/0210134.\n[155] A. Einstein and E. G. Straus, Rev. Mod. Phys. 17, 120\n(1945).\n[156] R. Nandra, A. N. Lasenby, and M. P. Hobson,\nMon. Not. Roy. Astron. Soc. 422, 2931 (2012),\narXiv:1104.4447 [gr-qc].\n[157] R. Nandra, A. N. Lasenby, and M. P. Hobson,\nMon. Not. Roy. Astron. Soc. 422, 2945 (2012),\narXiv:1104.4458 [gr-qc].\n[158] M. Bouhmadi-L´ opez, C.-Y. Chen, and P. Chen, Eur.\nPhys. J. C 75, 90 (2015), arXiv:1406.6157 [gr-qc].\n[159] I. Antoniou and L. Perivolaropoulos, Phys. Rev. D 93,\n123520 (2016), arXiv:1603.02569 [gr-qc].\n[160] A. Kravtsov and S. Borgani, Ann. Rev. Astron. Astro-\nphys. 50, 353 (2012), arXiv:1205.5556 [astro-ph.CO].\n[161] J. Gao, H. Zou, X. Zhou, and X. Kong, Publ. Astron.\nSoc. Pac. 132, 024101 (2020), arXiv:1912.10909 [astro-\nph.GA].23\n[162] D. Harvey, A. Robertson, S.-I. Tam, M. Jauzac,\nR. Massey, J. Rhodes, and I. G. McCarthy, Mon. Not.\nRoy. Astron. Soc. 500, 2627 (2020), arXiv:2011.01945\n[astro-ph.CO].\n[163] L. Fernandez, M. M. Cueli, J. Gonz´ alez-Nuevo,\nL. Bonavera, D. Crespo, J. M. Casas, and A. Lapi,\nAstron. Astrophys. 658, A19 (2022), arXiv:2111.05422\n[astro-ph.CO].\n[164] S. Sankhyayan, J. Bagchi, E. Tempel, S. More,\nM. Einasto, P. Dabhade, S. Raychaudhury, R. Athreya,\nand P. Hein¨ am¨ aki, Astrophys. J. 958, 62 (2023),\narXiv:2309.06251 [astro-ph.CO].\n[165] G. Arfken, H. Weber, and F. Harris, Mathematical\nMethods for Physicists: A Comprehensive Guide (El-\nsevier Science, 2012).\n[166] L. Knox and M. Millea, Phys. Rev. D 101, 043533\n(2020), arXiv:1908.03663 [astro-ph.CO].\n[167] P. Shah, P. Lemos, and O. Lahav, Astron. Astrophys.\nRev.29, 9 (2021), arXiv:2109.01161 [astro-ph.CO].\n[168] S. Vagnozzi, Phys. Rev. D 102, 023518 (2020),\narXiv:1907.07569 [astro-ph.CO].\n[169] E. Komatsu et al. (WMAP), Astrophys. J. Suppl. 180,330 (2009), arXiv:0803.0547 [astro-ph].\n[170] E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett,\nB. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R.\nNolta, L. Page, D. N. Spergel, M. Halpern, R. S. Hill,\nA. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S.\nTucker, J. L. Weiland, E. Wollack, and E. L. Wright,\nThe Astrophysical Journal Supplement Series 192, 18\n(2011).\n[171] Y. Wang and P. Mukherjee, Phys. Rev. D 76, 103533\n(2007), arXiv:astro-ph/0703780.\n[172] W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996),\narXiv:astro-ph/9510117.\n[173] Y. Wang and S. Wang, Phys. Rev. D 88, 043522\n(2013), [Erratum: Phys.Rev.D 88, 069903 (2013)],\narXiv:1304.4514 [astro-ph.CO].\n[174] Q.-G. Huang, K. Wang, and S. Wang, JCAP 12, 022\n(2015), arXiv:1509.00969 [astro-ph.CO].\n[175] L. Chen, Q.-G. Huang, and K. Wang, JCAP 02, 028\n(2019), arXiv:1808.05724 [astro-ph.CO].\n[176] O. Elgaroy and T. Multamaki, Astron. Astrophys. 471,\n65 (2007), arXiv:astro-ph/0702343."    },    {        "title": "2402.05938v1.The_Jackson_Richmond_4CT_Constant_is_EXACTLY_10_27.pdf",        "content": "arXiv:2402.05938v1  [math.CO]  8 Feb 2024The Jackson-Richmond 4CT Constant is EXACTLY 10/27\nShalosh B. EKHAD and Doron ZEILBERGER\nAbstract : In their recent claimed computer-free proof of the Four Col or Theorem, David Jackson\nand Bruce Richmond attempted to use sophisticated “asympto tic analysis” to explicitly compute a\ncertain number whosepositivity (according to them) implie s this famous theorem. While the juryis\nstill out whether their valiant attempt holds water, we prov e, in this modest note, that this constant\nequals exactly 10/27. We also point out that their evaluatio n of this constant must be erroneous,\nfor two good reasons. Finally, as an encore, we state many sim ilar, but more complicated, results.\nDavid Jackson and Bruce Richmod recently made a brave attemp t [JR] to give a human-generated\nproof of the Four Color (or as they would spell it, Colour) The orem. They relied heavily on Tutte’s\nseminal paper [T]. An important part was their (correct) the orem stated below (equivalent to\nTheorem 9 of [JR]).\n(We follow their convention of denoting the coefficient of xnin the formal power series f(x) by\n[xn]f(x).)\nTheorem : Letg(x) be the formal power series\ng(x) =/summationdisplay\nn≥12(4n+1)!\n(n+1)!(3n+2)!xn,\nthen there exists a positive constant, let’s call it c, such that\nlim\nn→∞[xn](g(x))2\n[xn]g(x)=c .\nHowever, as we will show below, their determination of that c onstantcmust be erroneous.\nItturnsout that this constant, c, that may bechristened the Jackson-Richmond 4CT constant ,\nhas a nice explicit expression as a good old rational number .\nFact:c=10\n27.\nIn fact, we can do more than just asymptotics. We can actually state the exactexpression for that\nratio, as a rational function ofn, and sketch twoproofs.\nLemma: For all integers n≥1, we have:\n[xn](g(x))2\n[xn]g(x)=10(n−1)(n2+14n+12)\n3(3n+5)(3n+4)(n+2).\nSketch of First Proof : the top of the left side is a certain terminating hypergomet ric sum, and\ndividingby [ xn]g(x), alias2(4n+1)!\n(n+1)!(3n+2)!, is still such a sum, and the Zeilberger algorithm [Z] finds\n1a linear recurrence equation (second-order in fact) satisfi ed by it. That same recurrence happens\nto also hold for the right side, and checking the initial cond itions at n= 1 and n= 2 concludes the\nproof. .\nSketch of Second Proof : Below we will exhibit a degree 4 algebraic equation satisfie d byg(x).\nHenceg(x)2satisfies another algebraic equation of degree 4. From this u sing the standard tools\ndescribed below, one can deduce a differential equation, and i n turn, a recurrence satisfied by the\ncoefficients of g(x)2, that happen to match the right side of the lemma times [ xn]g(x)..\nCritique of section 2.2 of [JR]\nIn Eq. (13) of [JR], they needed to evaluate A:=g(27\n256), but failed to give its exact value, that\nhappens to be5\n27. In fact the formal power series g(x) (that happens to also be a convergent power\nseries for x≤27\n256) satisfies the algebraic equation :\nx/parenleftbig\nx2+11x−1/parenrightbig\n+/parenleftbig\n4x3+25x2−14x+1/parenrightbig\ng(x)+x/parenleftbig\n6x2+17x+3/parenrightbig\ng(x)2+x2(4x+3)g(x)3+x3g(x)4= 0.\nOnce guessed, it is easily confirmed using, Salvy and Zimmerm ann’s [SZ] Maple package gfun,\nspecifically the commands gfun[algeqtodiffeq] followed by gfun[diffeqtorec] , and then veri-\nfying that the sequence {2(4n+1)!\n(n+1)!(3n+2)!}also satisfies this recurrence.\nPlugging-in x=27\n256, and solving for g(27\n256), gives the exact value A=5\n27.\nIn [JR],cis given as27\n2/radicalBig\n3\n2·A·B, whereB=16\n27/radicalBig\n3\n2π. Since the latter, thanks to Ferdinand von\nLindemann, is transcendental , and we just exhibited both Aandcas rational numbers, the [JR]\nvalue must be erroneous. Another good reason why it can’t be r ight, as stated (of course they may\nbe some minor misprints that we were unable to correct), is th at in [JR], cevaluates to 1 .253754...,\nand of course while cis positive (as we proved above), it can’t be larger than 1.\nEncore\nThe beauty of symbolic computation is that with barely extra effort, we can do much more! Using\nthe Maple package Jackmond.txt accompanying this article available from\nhttps://sites.math.rutgers.edu/~zeilberg/tokhniot/J ackmond.txt ,\nwe discovered the next lemma.\nA More general Lemma: For all integers n≥1, andr≥2, define\nAr(n) :=[xn](g(x))r\n[xn]g(x),\nand\nBr:= lim\nn→∞Ar(n),\n2We have (as above)\nA2(n) =10(n−1)(n2+14n+12)\n3(3n+5)(3n+4)(n+2).\nB2=10\n27.\nWe also have\nA3(n) =5(n−1)(n−2)/parenleftbig\n5n4+160n3+1803n2+3768n+2016/parenrightbig\n3(3n+8)(3n+5)(3n+7)(3n+4)(n+3)(n+2);\nB3=25\n243;\nA4(n) =20(n−1)(n−2)(n−3)/parenleftbig\n25n6+1350n5+31495n4+347406 n3+1211092 n2+1580304 n+665280/parenrightbig\n27(3n+11)(3n+8)(3n+5)(3n+10)(3n+7)(3n+4)(n+4)(n+3)(n+2)\nB4=500\n19683.\nTo seeAr(n) for 5≤r≤11 look at the output file\nhttps://sites.math.rutgers.edu/~zeilberg/tokhniot/o Jackmond1.txt .\nThe list of the Brfor 2≤r≤11 is\n/bracketleftbigg10\n27,25\n243,500\n19683,3125\n531441,6250\n4782969,109375\n387420489,625000\n10460353203,390625\n31381059609,19531250\n7625597484987,107421875\n205891132094649/bracketrightbigg\nand in decimals\n[0.3703703704 ,0.1028806584 ,0.02540263171 ,0.005880238822 ,0.001306719738 ,0.0002823159928 ,\n0.00005974941647 ,0.00001244779510 ,0.000002561274712 ,0.0000005217411450] .\nIf you want to see data all the way to r= 18, feel free to look at\nhttps://sites.math.rutgers.edu/~zeilberg/tokhniot/o Jackmond1a.txt .\nEnjoy!\nMoral: Before posting your next paper publicly, make sure to do som efact-checking using our\nbeloved silicon servants (soon to become our masters). Whil e,who knows? , perhaps one day, we\nwon’t need them to prove the Four Color Theorem, since either the [JR] attempt, once corrected,\nwould succeed, or some future smart humans would do it, we can still use computer-kind for the\nmundane task of checking our calculations, and confirming ou r logical arguments, before we go\npublic.\n3References\n[JR] D.M. Jackson andL.B. Richmond, A non-constructive proof of the Four Colour Theorem ,arxiv,\nhttps://arxiv.org/abs/2212.09835\n[SZ] Bruno Salvy and Paul Zimmermann, GFUN: a Maple package for the manipulation of gener-\nating and holonomic functions in one variable , ACM Transactions on Mathematical Software 20\n(1994), 163177.\nhttps://dl.acm.org/doi/pdf/10.1145/178365.178368 .\n[T] W. T. Tutte, A census of planar triangulations , Canad. J. Math. 14(1962), 21-38.\n[Z] Doron Zeilberger, The method of creative telescoping , J. Symbolic Computation 11(1991), 195-\n204.\nhttps://sites.math.rutgers.edu/~zeilberg/mamarimY/c t1991.pdf .\nShaloshB.Ekhad,c/oD.Zeilberger,DepartmentofMathemat ics, RutgersUniversity(NewBrunswick),\nHill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataw ay, NJ 08854-8019, USA.\nEmail:ShaloshBEkhad at gmail dot com .\nDoron Zeilberger, Department of Mathematics, Rutgers Univ ersity (New Brunswick), Hill Center-\nBusch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854- 8019, USA.\nEmail:DoronZeil at gmail dot com .\nExclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeil-\nberger, and arxiv.org\nFeb. 8, 2024 .\n4"    },    {        "title": "2402.06003v1.Favorable_conditions_for_heavy_element_nucleosynthesis_in_rotating_proto_magnetar_winds.pdf",        "content": "Draft version February 12, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nFavorable conditions for heavy element nucleosynthesis in rotating proto-magnetar winds\nTejas Prasanna,1, 2Matthew S. B. Coleman,3and Todd A. Thompson2, 4, 1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n2Center for Cosmology & Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA\n3Department of Physics and Engineering Physics, Stevens Institute of Technology, Castle Point on the Hudson, Hoboken, NJ 07030, USA\n4Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA\nABSTRACT\nThe neutrino-driven wind cooling phase of proto-neutron stars (PNSs) follows successful supernovae.\nWind models without magnetic fields or rotation fail to achieve the necessary conditions for production\nof the third r−process peak, but robustly produce a weak r−process in neutron-rich winds. Using 2D\nmagnetohydrodynamic simulations with magnetar-strength magnetic fields and rotation, we show that\nthe PNS rotation rate significantly affects the thermodynamic conditions of the wind. We show that\nhigh entropy material is quasi-periodically ejected from the closed zone of the PNS magnetosphere\nwith the required thermodynamic conditions to produce heavy elements. We show that maximum\nentropy Sof the material ejected depends systematically on the magnetar spin period P⋆and scales as\nS∝P−5/6\n⋆ for sufficiently rapid rotation. We present results from simulations at a constant neutrino\nluminosity representative of ∼1−2 s after the onset of cooling for P⋆ranging from 5 ms to 200 ms and\na few simulations with evolving neutrino luminosity where we follow the evolution of the magnetar\nwind until 10 −14 s after the onset of cooling. We estimate at magnetar polar magnetic field strength\nB0= 3×1015G, 1015G, and 5 ×1014G that neutron-rich magnetar winds can respectively produce\nat least ∼2−5×10−5M⊙,∼3−4×10−6M⊙, and∼2.5×10−8M⊙of material with the required\nparameters for synthesis of the third r−process peak, within 1 −2 s, 10 s, and 14 s in that order after\nthe onset of cooling. We show that proton-rich magnetar winds can have favorable conditions for\nproduction of p−nuclei, even at a modest B0= 5×1014G.\n1.INTRODUCTION\nCore collapse of massive stars with mass ≳8−10 M⊙\nand subsequent successful explosion results in the for-\nmation of proto-neutron stars (PNSs). PNSs cool by\nemitting neutrinos (Burrows & Lattimer 1986), which\ncan drive a thermal wind (Duncan et al. 1986; Burrows\net al. 1995; Janka & Mueller 1996). PNS winds have\nbeen suggested as promising astrophysical sites for pro-\nduction of heavy elements (Woosley & Hoffman 1992;\nMeyer et al. 1992; Woosley et al. 1994), but earlier wind\nmodels do not achieve the required conditions for pro-\nduction of the third r−process peak (Qian & Woosley\n1996), except for a very compact PNS at high neutrino\nluminosities that occur ≲1 s after the onset of the cool-\ning phase (Otsuki et al. 2000; Thompson et al. 2001;\nWanajo et al. 2001).\nMagnetar-strength magnetic fields ≳1014−1015G\nmay be important for nucleosynthesis in PNS winds.\nOne possible explanation for the origin of strong mag-\nnetic fields in PNSs is rapid rotation with periods ≲\n3 ms (Duncan & Thompson 1992; Thompson & Dun-\ncan 1993). But, the fact that magnetars are commonin the Galaxy comprising ∼40% of all neutron star\nbirths (Beniamini et al. 2019), and a lack of evidence\nfor anomalously energetic supernova remnants associ-\nated with anomalous X-ray pulsars and soft gamma-ray\nrepeaters (Vink & Kuiper 2006), suggests that rapid ro-\ntation of the order of milliseconds may not be required\nto generate magnetar-strength magnetic fields. Even\nmodest progenitor rotation may be sufficient to produce\nmagnetars during core-collapse (White et al. 2022).\nThompson (2003) show that magnetar-strength mag-\nnetic fields can trap matter in the closed zone of the\nmagnetosphere and that energy deposition by neutrinos\ncan lead to ejection of high entropy material. Thomp-\nson (2003) also provide the first estimates of entropy en-\nhancement as a function of magnetic field strength and\npredict a robust r−process in magnetar winds. Thomp-\nson & ud-Doula (2018) perform the first 2D magneto-\nhydrodynamic (MHD) simulations to show entropy en-\nhancement in dynamical eruptions (plasmoids) from the\nPNS magnetosphere for a non-rotating PNS . Desai et al.\n(2023) study the dynamics of magnetized non-rotatingarXiv:2402.06003v1  [astro-ph.HE]  8 Feb 20242\nPNS winds in 3 dimensions and show entropy enhance-\nment in plasmoids.\nThe electron fraction Yeof the wind is one of the major\nparameters that affects nucleosynthesis. Yein the wind\nis mostly set by luminosities and energies of the electron-\ntype neutrinos and antineutrinos via the charged current\nprocesses (Qian & Woosley 1996). Yethus depends on\nthe PNS cooling model (e.g., Pons et al. 1999; Vartanyan\n& Burrows 2023). However, Metzger et al. (2008) show\nthat magnetocentrifugal slinging caused by very rapid\nrotation (near-breakup) and strong magnetic fields can\nlead to a decrease in asymptotic Ye. Some studies sug-\ngest that PNS winds may be moderately neutron-rich\nand thus facilitate r−process, but possibly proton-rich\nat late times during the cooling phase (e.g., Roberts\n2012; Roberts et al. 2012).\nThe synthesis of p−nuclei in proton-rich winds has\nbeen previously studied (Hoffman et al. 1996; Pruet\net al. 2005, 2006; Fr¨ ohlich et al. 2006; Wanajo 2006;\nWanajo et al. 2011). Pruet et al. (2006) show that the\nwind entropy has a significant impact on nucleosynthe-\nsis. By artificially increasing the entropy by a factor of\n2−3 over the baseline entropy of 55 −77 k Bbaryon−1,\nthey find that the nucleosynthesis can extend until A≈\n170. Friedland et al. (2023) show that for PNS masses\n≳1.7 M ⊙, the νp−process can synthesize p−nuclei un-\ntilA≲105 in neutrino-driven outflows of core col-\nlapse supernovae. The elemental abundances between\nStrontium and Silver in very metal poor stars may\nbe a result of weak r−process and νp−process occur-\nring in core-collapse supernova explosions (Psaltis et al.\n2023). Xiong et al. (2023) suggest a new nucleosynthe-\nsis process ( νr-process) for the production of p−nuclei in\nneutron-rich winds which occurs if the product of expan-\nsion timescale and the rate for νeabsorption on neutrons\nexceeds 0.1 at the T∼3×109K (3 GK) surface.\nAn important piece of physics not included in ear-\nlier works by Thompson & ud-Doula (2018) and De-\nsai et al. (2023) is magnetar rotation. In our earlier\nworks, we explored 2D MHD simulations of magnetar\nwinds with rotation and magnetar-strength magnetic\nfields in the context of magnetar spindown (Prasanna\net al. 2022), gamma-ray bursts and superluminous su-\npernovae (Prasanna et al. 2023). Metzger et al. (2007)\nuse 1D models with magnetic fields and rotation to argue\nthat rapidly rotating magnetars produce comparatively\nmore favorable conditions for r−process than slowly ro-\ntating non-magnetized PNSs, but due to a lack of dy-\nnamic PNS magnetosphere in their models, they do not\nexplore the thermodynamics of plasmoids. Vlasov et al.\n(2014, 2017) explore the prospects for nucleosynthesis in\nmagnetar winds including the effects of rotation, but as-sume a static magnetosphere which precludes the study\nof plasmoid dynamics.\nIn this paper, we perform the first 2D MHD simu-\nlations of magnetar winds with magnetar-strength mag-\nnetic fields and rotation to study the prospects for nucle-\nosynthesis. We show that the PNS rotation rate signif-\nicantly affects the entropy of the material in plasmoids.\nEntropy is a critical parameter that influences nucle-\nosynthesis in both neutron-rich (e.g., Thompson & ud-\nDoula 2018) and proton-rich (e.g., Pruet et al. 2006)\nwinds. We show that magnetar winds can have favor-\nable conditions for the production of r−process nuclei,\np−nuclei, and the νr−process.\nIn Section 2, we describe the numerical method, mi-\ncrophysics, and boundary conditions in the simulations.\nIn Section 3, we present the results to show that PNS ro-\ntation rate significantly influences the plasmoid entropy\nand other wind properties. In Section 4, we summarize\nthe results and discuss the implications of our results for\nnucleosynthesis in magnetar winds.\n2.MODEL\nWe use the MHD code Athena ++(Stone et al. 2020)\nfor our simulations, which we have configured to solve\nthe non-relativistic MHD equations:\n∂ρ\n∂t+∇ ·(ρv) = 0, (1)\n∂(ρv)\n∂t+∇ ·\u0014\nρvv+\u0012\nP+B2\n2\u0013\nI−BB\u0015\n=−ρGM⋆\nr2ˆr,\n(2)\n∂E\n∂t+∇ ·\u0014\u0012\nE+\u0012\nP+B2\n2\u0013\u0013\nv−B(B·v)\u0015\n=˙Q,\n(3)\n∂B\n∂t− ∇ × (v×B) = 0, (4)\nwhere M⋆is the mass of the PNS, ris the radius from\nthe center of the PNS, ρis the mass density of the fluid,\nvis the fluid velocity, Eis the total energy density of\nthe fluid, Pis the fluid pressure, ˙Qis the neutrino heat-\ning/cooling rate, and Bis the magnetic field. We solve\nthe MHD equations in two dimensions using spherical\npolar coordinates assuming axisymmetry. We assume\nthat the magnetic and rotation axes of the PNS are\naligned.\n2.1. Microphysics\nWe use the general equation of state (EOS) mod-\nule in Athena ++(Coleman 2020)1and run the simula-\ntions using the approximate analytic form of the general\n1With additional modifications for a composition-dependent EOS.3\nEOS (Qian & Woosley 1996) containing non-relativistic\nbaryons, relativistic electrons and positrons, and pho-\ntons. This is a reasonable approximation to the EOS\nat temperatures T≳0.5 MeV, but this approximation\nis not valid at temperatures much lower than 0.5 MeV\nbecause the electrons and positrons cease to be relativis-\ntic. We use the approximate analytic EOS to limit the\ncomputation time. We have run a few simulations with\nthe general Helmholtz EOS (Timmes & Swesty 2000),\nwhich is accurate even in the regions with T <0.5 MeV,\nto compare with the results obtained using the approx-\nimate analytic EOS. We find reasonable agreement be-\ntween the two at T≳0.5 MeV (see Table 1).\nWe evolve the electron fraction Yeof the wind as a\nfunction of radius and time as described in Prasanna\net al. (2023). To compute the neutrino energy deposi-\ntion rate ˙ q(related to ˙Qin equation 3 as ˙Q= ˙qρ), we\nconsider the neutrino heating and cooling rates result-\ning from charged current νeand ¯νeabsorption processes,\nneutrino scattering on electrons and positrons, neutrino\nscattering on nucleons, and neutrino-antineutrino an-\nnihilation to form electron-positron pairs. We use the\nexpressions from Thompson et al. (2001), but we ignore\nthe general relativistic (GR) terms. GR effects increase\nthe effective gravitational potential, and the associated\ngravitational redshift terms decrease the heating rate,\nboth leading to lower ˙M, larger entropy, and shorter\nexpansion timescales (Cardall & Fuller 1997; geodesic\nbending partially compensates the gravitational redshift\nterms (Salmonson & Wilson 1999)). For comparison,\nwe present results from two simulations with only the\ncharged current neutrino heating and cooling processes\n(see Table 1) and find comparable results.\n2.2. Reference frame, initial conditions, and\ncomputation grid\nWe perform the simulations in an inertial (lab) frame\nof reference. We initialize the simulations with a one\ndimensional non-rotating and non-magnetic (NRNM)\nwind model. We incorporate an initial purely dipolar\nmagnetic field using the magnetic vector potential (see\nPrasanna et al. 2022 for details). To include the effects\nof rotation in the initial conditions, we initialize the az-\nimuthal velocity vϕby conserving the specific angular\nmomentum: vϕ(r) =rΩ⋆sinθ(R⋆/r)2, where ris the\nradius from the center of the PNS, θis the polar angle\nmeasured from the rotation axis of the PNS, R⋆is the\nPNS radius (12 km in our simulations), and Ω ⋆= 2π/P⋆\nis the angular frequency of rotation of the PNS. The\ncomputation grid starts from the surface of the PNS and\nextends to an outer boundary radius of 104km. The grid\nis divided into Nrlogarithmically spaced zones in the ra-dial direction and Nθuniformly spaced zones in the θ\ndirection. For the simulations in this paper, we use 512\nradial zones and 256 θzones. We present results from\none simulation with 1024 radial zones and 512 θzones\nfor comparison (see Table 1).\n2.3. Boundary conditions\nWe enforce co-rotation with the PNS and set vϕ(r) =\nrΩ⋆sinθat the inner boundary. We solve equation 2 to\nobtain the inner boundary conditions (BCs) on density,\nradial velocity vr, and θ−velocity vθ. We set vr= 0\nandvθ= 0 at the inner boundary. We set vr= 0 at\nthe inner boundary to ensure that the ϕ−derivative of\nρis zero (axisymmetry condition, see Prasanna et al.\n2022), although it is incompatible with a steady state\nwind. To test if this condition affects the results, we\nhave run one simulation with vrat the inner boundary\nset by conserving the mass outflow rate ˙M(equation 8).\nWe find good agreement between the two choices for the\nvrinner BC (see Table 1).\nThe density inner BC is (refer to Prasanna et al. 2022\nfor the derivation):\nρ(r,θ) =ρ0exp\u0014GM⋆mn\nkBT\u00121\nr−1\nR⋆\u0013\u0015\n×exp\u0014mnr2Ω2\n⋆sin2θ\n2kBT\u0015\nexp\u0014−mnR2\n⋆Ω2\n⋆\n2kBT\u0015\n,(5)\nwhere we have assumed that the pressure near the sur-\nface is dominated by ideal nucleon gas pressure. We\nset a constant temperature at the inner boundary, and\nthis condition has been used to derive equation 5. In\nequation 5, mnis the average mass of a nucleon, ρ0is\nthe base density at the equator on the PNS surface, T\nis the temperature, and kBis the Boltzmann constant.\nWe set ρ0= 1012g cm−3, which is a reasonable choice\nfor base density at the surface of a PNS (see Thompson\net al. 2001, where the density boundary condition is set\nusing the neutrino optical depth). We have tested our\n1D NRNM simulations with base densities 1011−1013g\ncm−3and find that the mass outflow rate is not very\nsensitive to the base density across this range. For com-\nparison with the results obtained using the density BC\nin equation 5, we present results from simulations with\na constant density inner BC (see Table 1). We find that\nresults obtained from both the choices of density BCs\nagree with each other. However, we stick with equation\n5 for the inner density BC mainly because this condition\nsatisfies equation 2 and also because we find in the sim-\nulations with PNS polar magnetic field strength B0= 0\nthat the constant density BC underestimates angular\nmomentum flux ˙J(equation 9) compared to the simu-\nlations with density BC as in equation 5 (by a factor of4\n∼2.5 at P⋆= 20 ms). We emphasize that the difference\nin results between the two choices for the density BC\noccurs only when magnetic fields are small and do not\ndominate the angular momentum flux. Since the con-\nstant density inner BC does not satisfy equation 2 in\nthe ghost zones at the inner boundary, it is reasonable\nto expect some inaccuracy in the results, especially in\nthe low magnetic field limit.\nWe set the azimuthal component of the electric field\nEϕto zero at the inner edge of the first active compu-\ntational zone to maintain a constant magnetic field at\nthe surface of the PNS as a function time (Prasanna\net al. 2023). At the PNS surface, kinetic equilibrium is\nestablished between the neutrino flux and the wind due\nto high temperature and density (Burrows & Mazurek\n1982; Qian & Woosley 1996). Hence, we enforce net\nzero neutrino heating ( ˙ q= 0) at the inner boundary. To\nset the temperature Tand electron fraction Yein the\nghost zones at the inner boundary, we solve the simul-\ntaneous equations ˙ q= 0 and ˙Ye= 0 (equation 7 in Qian\n& Woosley 1996) using the two dimensional Newton-\nRaphson (2DNR) method at the ghost zone closest to\nthe surface at the north pole. We use the value of Tob-\ntained by this method to set the temperature in all the\nghost zones at the inner boundary (constant tempera-\nture condition). Once Tis set in all the ghost zones,\nwe compute the electron fraction Yein the ghost zones\nby solving ˙ q= 0 using the one dimensional Newton-\nRaphson method. We name this the “first choice inner\nBCs” for ease of referencing. We find that there is no so-\nlution to the 2DNR for arbitrary values of base density\nρ0at a given neutrino luminosity. There is a maximum\nvalue of ρ0above which there is no simultaneous solution\nto the set of equations ˙ q= 0 and ˙Ye= 0.\nAnother reasonable choice for the inner BCs (we name\nthis the “second choice BCs”) is to simultaneously en-\nforce ˙ q= 0 and ˙Ye= 0 in all the ghost zones. With\nthis choice, the temperature is not the same in all the\nghost zones. But, we note that equation 5 still holds\nfor the density inner BC because the temperature near\nthe PNS surface is sufficiently slowly varying such that\n|∇T| ≪ | T∇ρ/ρ|, which allows us to ignore tempera-\nture gradients when deriving equation 5. Due to the\nnature of the density in the ghost zones at the inner\nboundary (see equation 5) and the fact that there is no\nsimultaneous solution to ˙ q= 0 and ˙Ye= 0 above a cer-\ntain value of ρ0, this choice of the inner BCs precludes\nvalues of ρ0≳1012g cm−3for the range of neutrino lu-\nminosities considered in this paper (see Section 3.3 and\n3.5). With this choice of BCs, we have to restrict ρ0\nto values ≲5×1011g cm−3, which is still a reasonable\nchoice for the PNS base density. But as the neutrinoluminosity decreases, the density gradient near the PNS\nsurface increases and this leads to a drop in the density\nin the first active computational zone. In order to main-\ntain a density ≥3×1011g cm−3in the first active zone\nin our evolutionary models (see Section 3.5), we choose\nto perform the simulations with the first choice inner\nBCs (as described in the previous paragraph). With the\nfirst choice, we find that the neutrino luminosities in\nthe evolutionary models can reach 1051ergs s−1, which\nis sufficient for our purposes in this paper. For com-\nparison, we present some results with the second choice\ninner BCs and ρ0= 3×1011g cm−3(see Table 1). We\nfind good agreement between the results obtained from\nboth these choices.\nAt the outer boundary, we set outflow conditions by\nconserving the specific angular momentum and the mass\noutflow rate ˙M(equation 8).\n3.RESULTS\n3.1. Diagnostic quantities\nThe entropy Sin the wind, dynamical expan-\nsion timescale texp, the figure of merit parameter for\nr−process nucleosynthesis ζ, mass outflow rate ˙M, an-\ngular momentum flux ˙J, spindown timescale τJ, and en-\nergy outflow rate ˙Eare the principal physical quantities\nwe measure. The dynamical expansion timescale is de-\nfined as:\ntexp=1\nvr\f\f\f\f1\nρdρ\ndr\f\f\f\f−1\n. (6)\nThe figure of merit parameter for r−process nucleosyn-\nthesis ζatT∼0.5 MeV is given by (Hoffman et al.\n1997),\nζ=S3\nY3etexp. (7)\nThe critical value of ζrequired for production\nof the third r−process peak is ζcrit≃8×\n109(kBbaryon−1)3s−1(Hoffman et al. 1997) at T∼\n0.5 MeV. The mass outflow rate through a closed spher-\nical surface is given by,\n˙M(r) =I\nSr2ρvrdΩ, (8)\nwhere dΩ is the solid angle. The z-component (along\nthe axis of rotation of the PNS) of the angular momen-\ntum flux is given by the following integral over a closed\nspherical surface (Vidotto et al. 2014):\n˙J(r) =I\nS\u0014\n−BrBϕrsinθ\n4π+ρvrvϕrsinθ\u0015\nr2dΩ. (9)\nIn a time-steady state, ˙M(r) and ˙J(r) are constant as\na function of radius (except in the first ∼10 zones near5\nthe inner boundary where boundary effects affect the\nprofiles).\nThe total angular momentum of the star is roughly\nJ=2\n5MR2\n⋆Ω⋆. We define the spindown time of the\nPNS as:\nτJ=J\n˙J. (10)\nSince ˙J(r) is constant with radius, τJcan be measured\nat any radius (see Section 3.3 for details).\nThe energy flux is given by the following surface inte-\ngral (we generalize the definition in Metzger et al. 2007\nto two and three dimensions):\n˙E(r) =I\nSr2ρvr\u00141\n2\u0000\nv2\nr+v2\nθ+v2\nϕ\u0001\n−rBrBϕΩ⋆sinθ\n4πρvr\n(11) −GM⋆\nr+e+P\nρ\u0015\ndΩ,\nwhere eis the specific internal energy of the outflow.\n3.2. Plasmoids from the PNS magnetosphere\nAs predicted by Thompson (2003) and shown by\nThompson & ud-Doula (2018), we find quasi-periodic\neruptions from the PNS magnetosphere when the mag-\nnetic field is strong enough. Figure 1 shows 2D maps of\nradial velocity vrand entropy along with the structure of\nthe PNS magnetosphere for different values of magnetic\nfield strength and neutrino luminosity. If the magnetic\ntension force in the PNS equatorial region cannot ex-\nceed the wind pressure gradient, the wind breaks open\nthe closed zone of the magnetic field and the magnetic\nfield structure settles into a stable split-monopole con-\nfiguration as shown in the left panel of Figure 1. If the\nmagnetic tension force is sufficiently strong compared\nto the wind pressure gradient, the closed zone of the\nmagnetic field in the equatorial region traps matter in\na helmet streamer configuration, as in models of Solar\nCorona (e.g., Pneuman & Kopp 1971; Steinolfson et al.\n1982). The closed magnetosphere is unstable because\nthe neutrinos from the PNS deposit energy and heat\nthe trapped matter. Eventually, when the wind pressure\ngradient exceeds the magnetic tension force, the trapped\nmatter is ejected in a toroidal plasmoid. The mag-\nnetic field quickly reconnects and the process repeats\nagain. Plasmoids erupt quasi-periodically and the ejec-\ntion timescale depends on the magnetic field strength\n(tej∝B2, see Thompson & ud-Doula 2018 for analytic\nestimates in the non-rotating case). We show in this\npaper that the ejection timescale and properties of the\nplasmoids also depend on the PNS spin period.\nThe magnetic tension force can exceed the wind pres-\nsure gradient in two ways: a larger value of polar mag-\nnetic field strength B0at a fixed neutrino luminosity(middle panel in Figure 1) or a smaller value of neutrino\nluminosity at a fixed value of B0(right panel in Figure\n1). In Figure 1, the left and the right panels show data\nfrom an evolutionary model (starting from an initial spin\nperiod of 20 ms at a fixed B0= 1015G, see Section 3.5),\nand the middle panel shows data from a fixed neutrino\nluminosity model (see Section 3.3).\nFigure 2 is a sequence of images showing the struc-\nture of the magnetic field during a plasmoid eruption\nalong with 2D maps of radial velocity vrand entropy as\na function of time from the start of the simulation, at a\nconstant PNS spin period of 20 ms and polar magnetic\nfield strength B0= 3×1015G from a constant neutrino\nluminosity simulation (see Section 3.3). As shown in the\nfigure, the first closed magnetic field line in the equato-\nrial region remains unchanged at different times because\nthe magnetic tension force at this field line is so large\nthat the the wind pressure gradient can never exceed the\nmagnetic tension. The second field line in the equatorial\nregion shows the magnetic reconnection event and the\neruption of matter.\n3.3. Constant neutrino luminosity simulations\nWe first present results from simulations with constant\nneutrino luminosities and energies at a polar magnetic\nfield strength B0= 3×1015G. For these simulations, the\nneutrino luminosities are L¯νe= 8×1051ergs s−1,Lνe=\nL¯νe/1.3, and Lνµ=L¯νµ=Lντ=L¯ντ=L¯νe/1.4, and\nthe neutrino energies are ϵ¯νe= 14 MeV, ϵνe= 11 MeV,\nandϵνµ=ϵ¯νµ=ϵντ=ϵ¯ντ= 23 MeV following Thomp-\nson et al. (2001). All the results presented in this pa-\nper are for a 1.4 M ⊙PNS (all else equal, a higher mass\nPNS produces higher entropy and lower mass outflow\nrate (Qian & Woosley 1996; Thompson et al. 2001)).\nAs described in Section 1, the asymptotic electron frac-\ntion in the wind is mostly set by neutrino luminosities\nand energies (Qian & Woosley 1996). For the choices of\nneutrino luminosities and energies described above, the\nasymptotic electron fraction is ∼0.45.\nFigure 3 shows the maps of entropy for PNS spin pe-\nriods P⋆= 200 ms, 20 ms, 10 ms, and 5 ms at B0=\n3×1015G from the constant neutrino luminosity simu-\nlations at the time instants of the respective simulations\nwhen maximum entropy occurs at the T= 0.5 MeV tem-\nperature surface. The maps clearly show that the max-\nimum entropy in the equatorial region increases with\nfaster PNS rotation at a fixed value of B0and neutrino\nluminosity. We provide the physical intuition for the\nvariation of entropy with PNS rotation rate in Section\n3.4.\nTable 1 presents results from the constant neutrino lu-\nminosity simulations as a function of PNS spin period P⋆6\nFigure 1. Comparison of PNS magnetosphere for different values of polar magnetic field strength B0(units of 1015G) and\nneutrino luminosity (units of 1051ergs s−1). The spin period P⋆is in units of milliseconds. As the magnetic field strength\ndominates the wind pressure gradient (see Section 3.2 for details), we find quasi-periodic dynamical plasmoid eruptions from\nthe closed zones of the PNS magnetosphere as shown in the middle and right panels, otherwise the magnetic field settles into a\nstable split-monopole field configuration as shown in the left panel. This figure shows 2D maps of radial velocity vr(left half of\neach panel) and entropy (right half of each panel). The minimum value of entropy shown on the colorbar is chosen for better\ncontrast, the actual minimum value of entropy in the simulations is smaller. The central white circle is the PNS with a radius\nof 12 km and the white lines are the magnetic field lines. The magnetic and rotation axes of the PNS are along the vertical\nin the figure. The 0.5 MeV temperature surface is shown by the cyan dotted lines. The outer boundary in this figure is at a\nradius of 200 km. The left and the right panels show data from an evolutionary model (see Section 3.5), and the middle panel\nshows data from a fixed neutrino luminosity model (see Section 3.3). Although the PNS spin periods in the panels are slightly\ndifferent, they do not significantly affect the structure of the magnetosphere. The structure of the magnetosphere in this figure\nis determined mostly by B0and the neutrino luminosity.\nFigure 2. Sequence of images showing the structure of the magnetic field during a plasmoid eruption. This figure shows 2D\nmaps of radial velocity vrand entropy as a function of time from the start of the simulation at a PNS spin period of 20 ms and\npolar magnetic field strength B0= 3×1015G from a constant neutrino luminosity simulation (see Section 3.3). The minimum\nvalue of entropy shown on the colorbar is chosen for better contrast, the actual minimum value of entropy in the simulations is\nsmaller. The high entropy and vrregions in the left panel are from a previous plasmoid eruption. The outer boundary in this\nfigure is at a radius of 100 km. The 0.5 MeV temperature surface is shown by the cyan dotted lines. The white lines are the\nmagnetic field lines and the central white circle is the PNS with a radius of 12 km.7\nat a polar magnetic field strength B0= 3×1015G. We\nhave run these simulations for a duration of 0.5 s. The\nquantities presented in Table 1 are as follows. Smax,0.5 is\nthe maximum entropy of the outflowing material at the\nT= 0.5 MeV temperature surface during the course of\nthe simulation. As the magnetic field reconnects follow-\ning a plasmoid eruption, some wind material is dragged\nback into the PNS magnetosphere. We do not report the\nentropy of material with vr<0 that is dragged back.\nWe measure only the entropy of the outflowing material\n(vr>0) that is released into the expanding wind. ⟨S⟩˙M\nin Table 1 is the average entropy of the outflowing mate-\nrial weighted by ˙Min the zones within an angle of 3◦on\neither side of the equator at the T= 0.5 MeV surface.\nThe values reported in the parentheses are the mini-\nmum and maximum values of ⟨S⟩˙Mduring the course of\nthe simulation. We measure the mass flux ˙Mζ,0.5of the\noutflowing matter with the r−process figure of merit pa-\nrameter (equation 7) ζ≥ζcritthrough the T= 0.5 MeV\nto estimate the nucleosynthetic yields of elements in the\nthird r−process peak. We also present the minimum\nvalue during the simulation of the average dynamical\nexpansion timescale (equation 6) weighted by ˙M. Al-\nthough the T= 0.5 MeV surface is not spherical, we do\nnot consider the non-radial surface normal to compute\n˙Mthrough the T= 0.5 MeV surface because we sum the\nmass outflow rate through individual cells of the com-\nputational domain (= r2ρvrdΩ, where dΩ is the solid\nangle) at T= 0.5 MeV and we are interested in only\nthe radial component of the mass flux through the cells\nwhich can flow into the expanding wind.\nAs a sanity check for the measured mass outflow\nrate of high- ζmaterial through the non-spherical T=\n0.5 MeV surface, we have computed the mass outflow\nrate ˙Mζthat satisfies the condition ζ≥ζcritthrough\na spherical surface at a radius of 100 km. We choose\nr= 100 km because this is the average radius of the\nT= 0.5 MeV surface at the PNS equator in the con-\nstant neutrino luminosity simulations. We find that\nthe value of ˙Mζ,0.5through the T= 0.5 MeV surface\nis smaller than ˙Mζthrough the spherical surface at\nr= 100 km by 8% for P⋆= 20 ms, 10 ms and larger\nby 7% and 39% respectively for P⋆= 200 ms, 5 ms at\nB0= 3×1015G in the constant neutrino luminosity\nsimulations. We attribute the difference to the fact that\nthe distance of T= 0.5 MeV from the PNS in the equa-\ntorial region is modulated by plasmoids and is not al-\nways equal to 100 km. Although we present this sanity\ncheck, we emphasize that the value of ˙Mζ,0.5through\ntheT= 0.5 MeV surface gives the best estimate of the\nmass outflow rate of the wind material with favorable\nconditions for the r−process.As mentioned earlier, we consider only the wind ma-\nterial with vr>0 for the estimates. This is because we\ndo not know if the wind material with vr<0 that is\ndragged back during magnetic reconnection is eventu-\nally released into the outflowing wind. For example, at\nP⋆= 20 ms and B0= 3×1015G in the constant neu-\ntrino luminosity simulation, we find that ∼2% of the\nwind material that satisfies the condition ζ≥ζcrithas\nvr<0. Simulations with Lagrangian tracer particles are\nrequired to assess if the material that is dragged back\ninto the magnetosphere is eventually released into the\noutflow, which will be our focus in a future work.\nIn Table 1, tpis the average time interval between\nplasmoids. Various regions in the PNS equatorial closed\nzone erupt quasi-periodically with ejection timescales\ndepending on the value of magnetic field and wind pres-\nsure gradient. The average time interval between plas-\nmoids presented in Table 1 is the average time interval\nbetween peaks in the energy outflow rate ˙E. We also\npresent time-averaged values of spindown timescale τJ,\nmass outflow rate ˙M, and asymptotic energy outflow\nrate ˙Emeasured over a spherical surface. These quan-\ntities have been averaged over time to account for the\nmodulation due to plasmoids. We measure τJand ˙M\nat a radius of 50 km and asymptotic ˙Eat a radius of\n1000 km. We note that the time-averaged values of τJ\nand ˙Mare negligibly dependent on the radius of mea-\nsurement as long as they are not measured within the\nfirst∼10 zones near the PNS surface where bound-\nary effects affect the values. The time-averaged value\nof asymptotic ˙Eis also negligibly dependent on the ra-\ndius of measurement as long as it is measured at radii\nwell past the peak of the net neutrino heating rate (see\nFigure 4 in Prasanna et al. 2023 for an example of the\nprofile of net neutrino heating rate including only the\ncharged current processes). The values of τJ,˙M, and ˙E\nreported in Table 1 agree with the results in our previous\nworks (Prasanna et al. 2022, 2023).\nSince we use the approximate analytic form of the\ngeneral equation of state (Qian & Woosley 1996), the\nresults are accurate only in the temperature range\nT≳0.5 MeV. In our simulations with the approximate\nEOS and in a few simulations with the more accurate\nHelmholtz EOS, we find regions of higher entropy at\nT <0.5 MeV. Smax,[0,2,0.5] shown in Table 1 is the max-\nimum entropy of the outflowing material in the tem-\nperature range 0.2 MeV ≤T≤0.5 MeV. Since our fo-\ncus in this paper is to show that the PNS rotation rate\naffects the entropy and other properties of the wind,\nthe approximate EOS is sufficient. In a work that will\nfollow this paper, we will explore the nucleosynthetic8\nyields throughout the computational grid by using the\nHelmholtz EOS and Lagrangian tracer particles.\nIn the simulation with PNS spin period P⋆= 5 ms\nandB0= 3×1015G, we find that the Alfv´ en speed\nand the fast magnetosonic speed in a few regions on\nthe grid approach or exceed the speed of light. Since\nour simulations do not include relativistic effects, the\nresult at P⋆= 5 ms needs to be confirmed by running\nsimulations with relativistic effects. This will be our\nfocus in a future work.\nFigure 4 shows the profiles of mass outflow rate ˙M\nthrough the T= 0.5 MeV surface with ζ≥ζcrit(equa-\ntion 7) as a function of time after the start of the simula-\ntion for different values of PNS spin period at B0= 3×\n1015G. Figures 5 and 6 show time-averaged ˙Mthrough\nT= 0.5 MeV surface as a function of entropy and ζre-\nspectively. For PNS spin period P⋆≳10 ms, we find an\naverage mass outflow rate of ∼2−3×10−5M⊙s−1\nwith ζ≥ζcritthrough the T= 0.5 MeV surface at\nB0= 3×1015G and neutrino luminosities as described\nearlier in this subsection. We discuss the implications of\nthis result for nucleosynthesis in Section 4.\nAlthough the average ˙Mwith ζ≥ζcritis comparable\nforP⋆≳10 ms, we find that the thermodynamic condi-\ntions of this material are significantly different. Figure\n7 shows a scatter plot of ˙Mweighted average of entropy\nand expansion timescale in the zones within 3◦of the\nPNS equator at T= 0.5 MeV for various values of PNS\nspin period. As shown in Figures 5, 6, and 7, the en-\ntropy, ζ, and expansion timescale texpof the material\nejected in plasmoids is strongly influenced by the PNS\nrotation rate. In a future work, we intend to incorporate\nLagrangian tracer particles in the simulations to track\nnucleosynthetic yields to understand how the different\nthermodynamic conditions affect the elemental yields.\n3.4. Analytic estimates\nAs described in Section 3.3, our simulations show that\nthe PNS rotation rate significantly affects the entropy\nof the material ejected in plasmoids. We provide an-\nalytic estimates of the scaling of entropy in plasmoids\nas a function of PNS angular frequency of rotation Ω ⋆.\nWe use force balance at the PNS equator to derive the\nscaling relations.\nThe wind pressure is dominated by baryons at the\nPNS surface, but due to rapid decrease in density out-\nside the PNS surface, the wind pressure is dominated\nby electrons, positrons, and photons within just a few\nkilometers from the PNS surface. The wind pressure Pw\ncan then be approximated as Pw≈AT4, where Tis thewind temperature and A=11π2\n180k4\nB\n(ℏc)3(Qian & Woosley\n1996). Plasmoids are ejected when the wind pressure\ngradient becomes large enough such that the combined\nforce density due to the magnetic field Fmand the PNS\ngravity Fgcannot provide the required centripetal force\ndensity Fcto enable co-rotation of the trapped matter\nwith the PNS. The force density due to the magnetic\nfield is a combination of the magnetic energy density\ngradient Fb∝ ∇B2and the magnetic tension |Ft| ∝B2\nRc,\nwhere Bis the magnitude of the local magnetic field and\nRcis the radius of curvature of the magnetic field line.\nThus, the force balance condition at the PNS equator\njust before a plasmoid eruption is,\n∇Pw+Fc=Fg+Fm. (12)\nBefore deriving the scaling relations, it is important\nto specify the domain of values of Ω ⋆where the scal-\nings hold true. Although the force balance condition is\ntrue for all values of Ω ⋆, the derived scaling relations\nwork only if the magnitude of Fcis comparable to the\nother forces. We can expect PNS rotation to impact the\nwind entropy when the PNS is rotating fast enough such\nthat|Fc|is comparable to the wind pressure gradient at\ntemperatures T≳0.2 MeV. In non-magnetized and non-\nrotating 1D wind models, T= 0.5 MeV occurs at about\n∼25−50 km and T= 0.1 MeV occurs at ∼150−500 km\nfrom the PNS surface depending on the neutrino lumi-\nnosity (Thompson et al. 2001). As shown in Figure 3,\nrapid PNS rotation causes the T= 0.5 MeV surface to\ncome closer to the PNS in the equatorial region. But,\nas shown in Figure 2, plasmoids also modulate the dis-\ntance of constant temperature surfaces from the PNS\nsurface. Estimating an average radius of r∼100 km\nfor the T= 0.5 MeV surface, and a corresponding mass\ndensity ρ∼5×105−106g cm−3(these are typical val-\nues in our constant neutrino luminosity simulations) at\nhigh neutrino luminosities occurring ∼1−2 s after the\nonset of cooling, the condition\nAT4∼ρr2Ω2\n⋆,crit (13)\nyields Ω ⋆,crit∼300−400 rad s−1corresponding to\nP⋆,crit∼15−20 ms for PNS rotation to affect the wind\nentropy.\nAlthough we cannot explore in detail the thermody-\nnamics at T < 0.5 MeV due to the approximations to\nthe EOS in this paper (see Section 2.1), slower rotation\nofP⋆∼50 ms may be sufficient to significantly affect the\nthermodynamic conditions in the wind at T <0.5 MeV.\nAs shown in Figure 1, the constant temperature sur-\nfaces come closer to the PNS as the neutrino luminosi-\nties decrease. Depending on the factors by which ρand9\nFigure 3. 2D maps of entropy at a polar magnetic field strength B0= 3×1015G for PNS spin periods P⋆= 200 ms (left half\nof left panel), 20 ms (right half of left panel), 10 ms (left half of right panel), and 5 ms (right half of right panel) at the time\ninstants of the respective simulations when maximum entropy occurs at the 0.5 MeV temperature surface. The T= 0.5 MeV\nsurface is shown by the cyan dotted lines. The white lines are the magnetic field lines and the central white circle is the PNS\nwith a radius of 12 km. The outer boundary in this figure is at a radius of 130 km. These maps are from constant neutrino\nluminosity simulations (see Section 3.3).\n0.2 0.3 0.4\nTime (s)012345˙M[ζ≥ζcrit, T= 0.5 MeV] (10−4M⊙s−1)\nP⋆= 200 ms\n0.2 0.3 0.4\nTime (s)P⋆= 20 ms\n0.2 0.3 0.4\nTime (s)P⋆= 10 ms\n0.2 0.3 0.4\nTime (s)P⋆= 5 ms\nFigure 4. Mass outflow rate ˙Mof the material with the r−process figure of merit parameter ζ≥ζcrit(equation 7) through\ntheT= 0.5 MeV surface as a function of time after the start of the simulation for various values of PNS spin period P⋆at\nB0= 3×1015G and constant neutrino luminosity (see Section 3.3). Time starts from 0.2 s on the time axis because we ignore\nthe simulation data that can be affected by the initial condition transient during this time. The profile for P⋆= 5 ms is shown in\ngray to emphasize that the Alfv´ en speed and the fast magnetosonic speed exceed the speed of light in this simulation. Inclusion\nof relativistic effects is required to confirm the result for P⋆= 5 ms, which will be our focus in a future work.10\n100 150\nEntropy (k Bbaryon−1)−8−6−4log10˙M[T= 0.5 MeV] (M⊙s−1)\n˙Mtot,0.5P⋆= 200 ms\n100 200 300\nEntropy (k Bbaryon−1)˙Mtot,0.5P⋆= 20 ms\n200 400 600\nEntropy (k Bbaryon−1)˙Mtot,0.5P⋆= 10 ms\n0 500 1000\nEntropy (k Bbaryon−1)˙Mtot,0.5P⋆= 5 ms\nFigure 5. Time-averaged mass outflow rate ˙Mthrough the T= 0.5 MeV surface as a function of entropy for various values of\nPNS spin period at B0= 3×1015G and constant neutrino luminosity (see Section 3.3). The dotted black horizontal line shows\nthe total time-averaged ˙Mthrough the T= 0.5 MeV surface. The values of ˙Mtot,0.5 in this figure are slightly different from the\n˙Mvalues computed over a spherical surface reported in Table 1 because the T= 0.5 MeV surface is not spherical (see Figures\n1, 2, and 3). The borders in the bar chart for P⋆= 5 ms are shown in gray to emphasize that the Alfv´ en speed and the fast\nmagnetosonic speed exceed the speed of light in this simulation. Inclusion of relativistic effects is required to confirm the result\nforP⋆= 5 ms, which will be our focus in a future work.\n6 8 10\nlog10ζ((kBbaryon−1)3s−1)−12−10−8−6−4−2log10˙M[T= 0.5 MeV] (M⊙s−1)\n˙Mtot,0.5P⋆= 200 ms\nζcrit\n5.0 7.5 10.0 12.5\nlog10ζ((kBbaryon−1)3s−1)˙Mtot,0.5P⋆= 20 ms\nζcrit\n5.0 7.5 10.0 12.5\nlog10ζ((kBbaryon−1)3s−1)˙Mtot,0.5P⋆= 10 ms\nζcrit\n7.5 10.0 12.5 15.0\nlog10ζ((kBbaryon−1)3s−1)˙Mtot,0.5P⋆= 5 ms\nζcrit\nFigure 6. Similar to Figure 5, but ˙Mis plotted as a function of ζin this figure. The red dotted vertical line shows the value\nofζcritwhich estimates the value of ζrequired for the production of the third r−process peak (Hoffman et al. 1997).\n−4−2 0\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)6080100120140/angbracketleftS/angbracketright˙M,0.5(kBbaryon−1)\nP⋆= 200 ms\n−4−2 0\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)50100150200\nP⋆= 20 ms\n−4−3−2−1\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)100200300400\nP⋆= 10 ms\n−4−2\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)200400600\nP⋆= 5 ms\nFigure 7. Scatter plot of ˙Mweighted average of entropy and expansion timescale in the zones within 3◦on either side of the\nPNS equator at T= 0.5 MeV for various values of PNS spin period at B0= 3×1015G. Each dot in the scatter plot represents the\nweighted average at one time instant. All the dots together in each scatter plot span a duration of 0.3 s of the simulation time.\nThe data in this figure is from constant neutrino luminosity simulations (see Section 3.3). The dots in the plot for P⋆= 5 ms are\nshown in gray to emphasize that the Alfv´ en speed and the fast magnetosonic speed exceed the speed of light in this simulation.\nInclusion of relativistic effects is required to confirm the result for P⋆= 5 ms, which will be our focus in a future work.11\nTable 1. Wind properties as a function of PNS spin period P⋆from constant neutrino luminosity simulations at a polar magnetic field\nstrength B0= 3×1015G (see Section 3.3) and from the evolutionary models (see Section 3.5) at B0= 1015G (run until 10 s after the onset\nof cooling) and B0= 5×1014G (run until 14 s after the onset of cooling).\naB0bP⋆cSmax,0.5d⟨S⟩˙M,0.5e˙Mζ,0.5f⟨texp, min ⟩˙M,0.5gtphτJi˙Mj˙EkSmax,[0,2,0.5]\n(1015G) (ms) (k B/baryon) (k B/baryon) (M ⊙s−1) (s) (ms) (s) (M ⊙s−1) (ergs s−1) (k B/baryon)\n(min, max)\n3 200 176 (59, 144) 2.5 ×10−51.4×10−433 2.4 2.9 ×10−46.4×1048198\n20 331 (54, 227) 2.4 ×10−51.4×10−432 6.4 3.0 ×10−42.7×1049480\n10 591 (51, 390) 2.8 ×10−51.4×10−426 9.6 3.3 ×10−46.4×10491218\n5#1077 (71, 708) 5.4 ×10−57.7×10−520 14.4 3.9 ×10−41.7×10501712\nl20 350 (55, 168) 3.0 ×10−51.2×10−428 5.5 3.9 ×10−43.3×1049414\nl,m20 315 (54, 187) 3.4 ×10−51.6×10−428 5.8 4.0 ×10−43.0×1049444\nl,n20 299 (58, 198) 3.1 ×10−51.2×10−428 5.4 4.0 ×10−43.5×1049470\no200 184 (60, 135) 2.5 ×10−51.4×10−442 2.5 2.9 ×10−46.5×1048186\no20 335 (55, 203) 2.5 ×10−51.3×10−442 6.4 3.1 ×10−42.7×1049510\np20 347 (55, 201) 2.5 ×10−51.3×10−432 6.3 3.1 ×10−42.7×1049501\nn,q20 444 (56, 304) 2.1 ×10−51.4×10−439 7.7 2.4 ×10−42.5×1049772\nn,q200 189 (64, 137) 2.3 ×10−51.4×10−442 2.6 2.3 ×10−46.1×1048190\n1 200 (evol.) 153 (52, 132)$3.1×10−73.6×10−475 9.0 2.0 ×10−41.2×1048178\n20 (evol.) 229 (51, 215)$3.6×10−72.7×10−469 26.0 2.1 ×10−44.9×1048578\n0.5 20 (evol.) 207 (53, 181)$1.7×10−91.5×10−3210 65.0 1.4 ×10−42.2×1048340\naPolar magnetic field strength of the PNS.\nbSpin period of the PNS.\ncMaximum entropy of the outflowing material at the T= 0.5 MeV surface over the course of the simulation.\ndAverage entropy of the outflowing material weighted by ˙Mwithin an angle of 3◦on either side of the equator at the T= 0.5 MeV surface.\nThe values in the parentheses show the minimum and maximum of this value over the course of the simulation.\neTime-averaged mass outflow rate of the material with the figure of merit parameter ζ≥ζcrit(equation 7).\nfMinimum value over the course of the simulation of the average expansion timescale (equation 6) of the outflowing material weighted by\n˙Mwithin an angle of 3◦on either side of the equator at the T= 0.5 MeV surface.\ngTime-averaged interval between plasmoids.\nhTime-averaged spindown timescale of the PNS (equation 10) measured at a radius r= 50 km (see Section 3.3).\niTime-averaged mass outflow rate (equation 8) measured at a radius r= 50 km (see Section 3.3).\njTime-averaged energy outflow rate (equation 11) measured at a radius r= 1000 km (see Section 3.3).\nkMaximum entropy of the outflowing material in the temperature range 0.2 MeV ≤T≤0.5 MeV over the course of the simulation.\nlSimulation with a base density of 3 ×1011g cm−3and ˙q= 0, ˙Ye= 0 simultaneously enforced in all the ghost zone cells at the inner\nboundary (second choice inner BCs, see Section 2.3).\nmSimulation with a resolution of ( Nr,Nθ)=(1024, 512).\nnSimulation with Helmholtz EOS.\noSimulation with a constant density of 1012g cm−3at the inner boundary. All the other inner BCs are as described in Section 2.3.\npSimulation with radial velocity inner boundary condition set by conserving ˙M. All the other inner BCs are as described in Section 2.3.\nqSimulation with only charged current neutrino heating/cooling processes and inner boundary conditions set by ˙ q= 0 and Yein ghost zones\nset equal to the electron fraction in the first active computational zone. All the other inner BCs are as described in Section 2.3.\n#Simulation in which the Alfv´ en speed and the fast magnetosonic speed exceed the speed of light. Inclusion of relativistic effects is required\nto confirm this result, which will be our focus in a future work.\n$The figure of merit parameter ζ(equation 7) does apply to proton-rich winds, but we present this value for comparison (see Section 3.5).12\nrchange at a given temperature surface (the factors de-\npend on neutrino luminosities, B0, and PNS spin pe-\nriod), a higher or lower value of Ω ⋆is required for PNS\nrotation to affect the wind entropy as the neutrino lu-\nminosities decrease.\nIn order to get analytic estimates, we guess that Fm\njust before an eruption is a function of only the mag-\nnetic field, and that Fmis independent of Ω ⋆, which is\na reasonable expectation. As an analogy, this is similar\nto a mass-loaded rotating rubber band. As the rotation\nrate and/or the mass on the rubber band increase, the\ntension in the rubber band eventually approaches the\nbreaking point. The breaking point is a function of the\nmaterial of the rubber band and not the rotation rate.\nFrom equation 12, we thus have the condition that\n∇Pw+Fc−Fgis independent of Ω ⋆just before an erup-\ntion at the PNS equator. Considering the radial com-\nponents at the PNS equator, we get,\nd\ndΩ⋆\u0014∂Pw\n∂r+ρrΩ2\n⋆−GM⋆ρ\nr2\u0015\n= 0, (14)\nwhere ρis the mass density of the trapped matter, M⋆\nis the PNS mass, Gis the gravitational constant, and\nris the radius at which the eruption occurs. We have\n∂Pw\n∂r= 4AT3∂T\n∂r. To satisfy equation 14, we require\nthat each term in the brackets on the left hand side be\nindependent of Ω ⋆. It is easy to prove this using the\nmethod of contradiction (see Appendix A).\nThus, we have the following conditions just before a\nplasmoid eruption at a given magnetic field B:\nT3∂T\n∂r∝Ω0\n⋆, (15)\nρrΩ2\n⋆∝Ω0\n⋆, and (16)\nρ\nr2∝Ω0\n⋆. (17)\nFrom the above conditions, we get, ρ∝r2,r∝Ω−2/3\n⋆,\nandρ∝Ω−4/3\n⋆. We can expect T3∂T\n∂rto scale as ∼\nT4\nr, thus obtaining T∝Ω−1/6\n⋆. At temperatures T≳\n0.5 MeV, the entropy Sis proportional to T3/ρ(Qian &\nWoosley 1996). We thus obtain the following scaling for\nentropy as a function of Ω ⋆:\nS∝Ω5/6\n⋆. (18)\nThese estimates suggest that the enhancement of en-\ntropy in plasmoids with faster PNS rotation is primarily\ndue to lower mass density on the magnetic field lines (we\nemphasize that the magnetic field lines are not real en-\ntities, but are useful in providing a physical intuition).\nThe reason for lower mass density associated with faster\nPNS rotation can be understood by invoking the mass-\nloaded rubber band analogy mentioned above. Adding\n101102103\nΩ⋆(rad s−1)102103Entropy (k Bbaryon−1)\nSmax,0.5\n/angbracketleftS/angbracketright˙M,0.5,maxFigure 8. The solid blue lines show maximum entropy Smax\nand the maximum of the average entropy weighted by ˙M\nin the zones within 3◦on either side of the PNS equator\nat the T= 0.5 MeV surface as a function of PNS angular\nfrequency Ω ⋆at a polar magnetic field strength B0= 3×\n1015G. The filled circles are the data points from constant\nluminosity simulations (see Section 3.3 and Table 1). The\ndata point from P⋆= 5 ms (Ω ⋆= 1256 rad s−1) is shown as\na gray colored filled circle to emphasize that the Alfv´ en speed\nand the fast magnetosonic speed exceed the speed of light in\nthis simulation. Inclusion of relativistic effects is required to\nconfirm the result for P⋆= 5 ms, which will be our focus in a\nfuture work. The dashed orange lines show the Ω5/6\n⋆scaling\nof entropy (see the proportionality in 18). We note that this\nis an approximate verification of the scaling (see Section 3.4\nfor details).\nmass to a rotating rubber band stretches it. As men-\ntioned earlier, the maximum tension a rubber band can\ntolerate depends on the material and not on the total\nmass added or the the rotation rate. The tension in the\nrubber band is proportional to both the mass it carries\nand the rotation rate. As the rotation rate increases,\nthe critical tension is reached for a smaller value of the\ntotal mass on the rubber band. This analogy is helpful\nin getting a physical intuition in the case of plasmoid\neruptions. As the PNS rotation rate increases, the total\nmass trapped on the magnetic field lines at the point of\neruption decreases, leading to a smaller mass density on\nthe field lines.\nTo verify the scaling relations derived above, the re-\ngion where a field line with a given magnetic field\nstrength Berupts needs to be identified. Due to the lack\nof a straight-forward computational condition to iden-13\ntify the point of eruption from the simulation data, we\npresent approximate verification of the entropy scaling,\nwhich is sufficient for our purposes in this paper. We\nverify the scaling at a fixed T= 0.5 MeV temperature\nsurface. This approximate verification assumes that the\nentropy of the material ejected in the plasmoids does\nnot change significantly between the region of eruption\nand the T= 0.5 MeV surface. This is a reasonable ap-\nproximation because the radial velocity of the material\nejected in the plasmoids is ∼1010cm s−1as shown in\nFigures 1 and 2. Since the material has to travel a dis-\ntance of at most ∼50−100 km to reach T= 0.5 MeV\n(because the T= 0.5 MeV surface is itself at a distance\nof∼100 km in our constant luminosity simulations),\nit takes a time of ≲1 ms, which is much smaller than\nthe timescale for plasmoid ejection (see Table 1). We\ncan thus safely assume that the entropy enhancement\nwithin the T= 0.5 MeV surface occurs mostly when the\nmatter is trapped.\nFigure 8 shows the maximum entropy Smaxand the\nmaximum of the average entropy weighted by ˙Min the\nzones within 3◦on either side of the PNS equator at\ntheT= 0.5 MeV surface as a function of PNS angular\nfrequency Ω ⋆at polar magnetic field strength B0= 3×\n1015G. Figure 8 also shows the the S∝Ω5/6\n⋆scaling\nrelation derived in above. We find excellent agreement\nbetween the derived scaling relation and the simulation\nresults for PNS spin periods P⋆≲20 ms as estimated in\nequation 13. We emphasize that the scaling relations fail\nforP⋆≳20 ms in the constant luminosity simulations\npresented in this paper because the PNS is not rotating\nfast enough at these spin periods for the centripetal force\nto be comparable to the other forces in the equatorial\nregion.\n3.5. Evolutionary models\nWe have run models with evolving neutrino lu-\nminosity and PNS spin period at fixed polar mag-\nnetic field strength B0= 1015G starting from ini-\ntial spin periods P⋆0= 200 ms and 20 ms, and at\nB0= 5×1014G starting from P⋆0= 20 ms. We use\nthe Vartanyan & Burrows (2023) cooling model ob-\ntained from a 11.75 M ⊙progenitor. The neutrino lu-\nminosities and energies span (start and end values in\nthe brackets) L¯νe=\u0002\n1.2×1052, 1.6×1051\u0003\nergs s−1,\nLνe=\u0002\n1.3×1052, 1.7×1051\u0003\nergs s−1,Lνµ=L¯νµ=\nLντ=L¯ντ=\u0002\n8.7×1051, 1.6×1051\u0003\nergs s−1,ϵ¯νe=\n[16.3, 12.0] MeV, ϵνe= [13.9, 11.1] MeV, and ϵνµ=ϵ¯νµ=\nϵντ=ϵ¯ντ= [14.9, 11.4] MeV for the models at B0=\n1015G. The initial luminosities and energies are the\nsame for the model at B0= 5×1014G, but the final\nluminosities and energies in the order listed above are\n2 4 6 8 10\nTime (s)5075100125150175200225Smax[T= 0.5 MeV] (k Bbaryon−1)\nP⋆0= 20 ms\nP⋆0= 200 msFigure 9. Maximum entropy at the T= 0.5 MeV surface\nfor the evolutionary models at polar magnetic field strength\nB0= 1015G. The blue and orange lines show the profiles for\nmodels starting from initial PNS spin periods P⋆0= 20 ms\nand 200 ms respectively.\n1.1×1051ergs s−1, 1.1×1051ergs s−1, 1.2×1051ergs\ns−1, 11.0 MeV, 10.1 MeV, and 10.6 MeV. The initial lu-\nminosities and energies mentioned here occur 1.2 s after\nthe onset of cooling in the chosen model. We choose\na start time of 1.2 s because this corresponds to the\ntime when the neutrinosphere radii (computed assuming\nblackbody emission) for all the neutrino species is com-\nparable to the PNS radius of 12 km in our simulations.\nWe follow the evolution until 10 s and 14 s after the on-\nset of the cooling phase for the models at B0= 1015G\nandB0= 5×1014G respectively.\nThe cooling model of Vartanyan & Burrows (2023)\nprovides the data for a duration of 4.8 s after the onset\nof cooling. We use the data to obtain a power law fit to\nneutrino luminosities and extrapolate the neutrino lumi-\nnosities assuming a constant power law index at times\ngreater than 4.8 s. We use the blackbody approximation\nϵν∝L1/4\nνto obtain neutrino energies for times greater\nthan 4.8 s. For these combinations of neutrino luminosi-\nties and energies, we find that the electron fraction Yeis\ngreater than 0.5 throughout the evolution for all the evo-\nlutionary models. We find that the asymptotic Yeis 0.54\nat the beginning of the evolution which increases as the\nneutrino luminosities and energies decrease. As the neu-\ntrino luminosities decrease, we find plasmoid eruptions\nfrom the PNS magnetosphere. Plasmoids modulate the\nvalue of asymptotic Yein the equatorial region and the14\n60 80 100 120 140\nEntropy (k Bbaryon−1)−10−9−8−7−6−5−4−3log10˙M[T= 0.5 MeV] (M⊙s−1)˙Mtot,0.5P⋆0= 200 ms\n50 100 150 200\nEntropy (k Bbaryon−1)−10−9−8−7−6−5−4−3log10˙M[T= 0.5 MeV] (M⊙s−1)˙Mtot,0.5P⋆0= 20 ms\nFigure 10. Time-averaged mass outflow rate ˙Mthrough the T= 0.5 MeV surface as a function of entropy for the evolutionary\nmodels (see Section 3.5) starting from initial PNS spin periods P⋆0= 200 ms and 20 ms at a fixed polar magnetic field strength\nB0= 1015G, where the time-average has been taken over the entire course of the evolution. The dotted black horizontal line\nshows the total time-averaged ˙Mthrough the T= 0.5 MeV surface. The values of ˙Mtot,0.5 in this figure are slightly different\nfrom the ˙Mvalues computed over a spherical surface reported in Table 1 because the T= 0.5 MeV surface is not spherical.\n−3−2−1 0\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)5060708090100110120130/angbracketleftS/angbracketright˙M,0.5(kBbaryon−1)P⋆0= 200 ms\n−3−2−1 0\nlog10/angbracketlefttexp/angbracketright˙M,0.5(s)6080100120140160180200220/angbracketleftS/angbracketright˙M,0.5(kBbaryon−1)P⋆0= 20 ms\n1.22.03.04.05.06.07.08.09.09.8\nTime (s)\nFigure 11. Scatter plot of ˙Mweighted average of entropy and expansion timescale in the zones within 3◦on either side of\nthe PNS equator at T= 0.5 MeV for the evolutionary models starting from initial spin periods P⋆0= 200 ms and 20 ms at\nB0= 1015G. Each dot in the scatter plot has been color-coded as a function of time from the onset of the cooling phase and\nrepresents the weighted average at one time instant. All the dots together in each scatter plot span a duration of 8.6 s during\nthe PNS cooling phase. It is evident from these plots that the entropy in plasmoids increases as the PNS cools and the neutrino\nluminosity decreases at a fixed value of polar magnetic field strength (note the different limits on the y-axis in the two panels).15\n2.5 5.0 7.5 10.0 12.5 15.0\nTime (s)6080100120140160180200Smax[T = 0.5 MeV] (k Bbaryon−1)P⋆0= 20 ms\nFigure 12. Maximum entropy at the T= 0.5 MeV surface\nfor the evolutionary model at polar magnetic field strength\nB0= 5×1014G and initial PNS spin period P⋆0= 20 ms.\nmodulation percentage increases as neutrino luminosi-\nties decrease. Towards the end of the evolution, we find\nthat plasmoids modulate asymptotic Yeby∼3%.\nFigure 9 shows the maximum entropy at the T=\n0.5 MeV surface as a function of time after the onset of\ncooling for the evolutionary models starting from initial\nspin periods of 200 ms and 20 ms at B0= 1015G. There\nare no oscillations in the entropy profiles until a time of\n∼2.2 s in Figure 9 due to a stable PNS magnetosphere.\nAs the neutrino luminosities and energies decrease, the\nmagnetic field becomes dominant resulting in plasmoid\neruptions, which causes oscillations in the entropy pro-\nfiles. It is evident from Figure 9 that faster PNS rota-\ntion results in higher entropy in the plasmoids, consis-\ntent with the results from constant neutrino luminosity\nsimulations (see Section 3.3). Figure 10 shows the dis-\ntribution of mass outflow rate through the T= 0.5 MeV\nsurface as a function of entropy for both the evolution-\nary models at B0= 1015G. Figure 11 shows a scatter\nplot of ˙Mweighted average of entropy and expansion\ntimescale in the zones within 3◦on either side of the\nPNS equator at T= 0.5 MeV.\nFigure 12 shows the maximum entropy at T=\n0.5 MeV as a function of time after the onset cooling for\nthe evolutionary model at B0= 5×1014G and initial\nPNS spin period P⋆0= 20 ms. We find that plasmoids\ndevelop 9 s after the onset of cooling. We find that the\nentropy in plasmoids increases as the neutrino luminosi-\nties and energies decrease.We present the results from the evolutionary models\nin Table 1 (refer to Section 3.3 for the description of\nthe parameters in Table 1). As mentioned above, the\nasymptotic electron fraction Yeis>0.5 in all the evo-\nlutionary simulations, and hence the figure of merit pa-\nrameter ζ(equation 7) is not applicable. However, for\ncomparison, we present the values of ˙Mwith ζ≥ζcrit\nfor the evolutionary models in Table 1 assuming that the\ndistribution of ˙Mas a function of entropy and ζwould\nbe roughly the same even for asymptotic Ye<0.5.\nSimilar to our conclusions from the constant neutrino\nluminosity simulations, we find even in the evolutionary\nmodels that PNS rotation rate strongly influences the\nwind properties, which can directly affect heavy element\nnucleosynthesis. We discuss the implications of these\nresults for nucleosynthesis in Section 4.\nThe evolutionary models also allow us to explore\nthe spindown of PNSs. In an earlier work (Prasanna\net al. 2022), we explored the spindown of ‘slowly’ ro-\ntating magnetars with polar magnetic field strength\nB0≳1015G and spin period ≳100 ms at the onset\nof the cooling phase for a duration of ∼3−5 s with a\ncooling model different from the one used in this paper.\nHere, we explore the spindown of PNSs until 10 −14 s\nafter the onset of cooling using the Vartanyan & Bur-\nrows (2023) cooling model. The left panel in Figure 13\nshows the spindown timescale as a function of time af-\nter the onset of cooling for the evolutionary models at\nB0= 1015G. The difference in the nature of τJbetween\nthe ‘slowly’ rotating case with P⋆0= 200 ms and the\nmore rapidly rotating case with P⋆0= 20 ms is strik-\ning. Apart from the modulation due to plasmoids, the\nspindown timescale is affected by two competing fac-\ntors as the neutrino luminosity decreases. First, the\nmass outflow rate ˙Mdecreases with neutrino luminosity,\nwhich nominally would cause τJto increase. However,\nas the neutrino luminosity decreases, the Alfv´ en radius\nincreases which causes τJto decrease (see Prasanna et al.\n2022 for further details). These two features combine to\nproduce the τJprofiles shown.\nWe find that the first effect always dominates for the\nevolutionary model with P⋆0= 20 ms and B0= 1015G,\nwhich results in an increasing τJprofile (apart from the\nmodulation due to plasmoids). For the evolutionary\nmodel with P⋆0= 200 ms and B0= 1015G, we find that\nthe second effect dominates during the early phases of\nthe evolution resulting in a decreasing τJprofile. Be-\ntween 2.5 s and 6 s, we find that the two effects roughly\ncancel each other resulting in an almost constant τJpro-\nfile. During the later phases of evolution, the first effect\ndominates, resulting in an increasing τJprofile. Spin-16\n2 4 6 8 10\nTime (s)01020304050607080Spindown time τJ(s)\nP⋆0= 200 msP⋆0= 20 ms\n2 4 6 8 10\nTime (s)0100200300400500P⋆(t) (ms)P⋆0= 200 ms\nP⋆0= 20 ms\nFigure 13. The left panel shows the spindown time τJand the right panel shows the PNS spin period as a function of time\nafter the onset of the cooling phase for the evolutionary models at polar magnetic field strength B0= 1015G. The blue and\norange lines show the profiles for the models starting from initial PNS spin periods P⋆0= 20 ms and 200 ms respectively.\ndown timescales of just a few seconds imply rapid spin-\ndown.\nThe right panel in Figure 13 shows the evolution of\nthe PNS spin period during the cooling phase for the\nevolutionary models at B0= 1015G. We find that the\nthe PNS in the evolutionary calculation with an initial\nspin period of 200 ms and B0= 1015G spins down to a\nperiod of 561 ms at the end of 8.6 s of evolution. Based\non the results in Prasanna et al. (2022), we can expect\nmore rapid spindown at a higher magnetic field strength\nand/or slower initial rotation period. On the other hand,\nwe find that the PNS in the evolutionary calculation\nwith P⋆0= 20 ms and B0= 1015G spins down to just\n32 ms at the end of 8.6 s of evolution. These results are\nconsistent with the results in our earlier work (Prasanna\net al. 2022).\nWe find much slower spindown for the evolutionary\nmodel at P⋆0= 20 ms and B0= 5×1014G, where the\nPNS spins down to 26 ms at the end of 13 s of evolution.\nAlthough spindown is not significant for initial PNS spin\nperiods ≲20 ms, we emphasize that rapid spindown ac-\ncompanies the PNS cooling phase for magnetars born\nslowly rotating with initial spin periods ≳100 ms with\nmagnetar strength magnetic fields ≳1015G.\n4.DISCUSSION AND CONCLUSIONS\nIn this paper, we explore for the first time the\nprospects for heavy element nucleosynthesis in winds\nfrom highly magnetized and rotating neutron stars. Ear-\nlier works have explored the prospects for nucleosyn-\nthesis in magnetar winds without including the effectsof rotation (Thompson & ud-Doula 2018; Desai et al.\n2023) or include the effects of rotation in a static mag-\nnetosphere geometry (Vlasov et al. 2014, 2017). Simi-\nlar to the earlier works with a dynamic magnetosphere\n(Thompson & ud-Doula 2018; Desai et al. 2023), we\nfind dynamical eruptions of high entropy material from\nthe proto-neutron star (PNS) magnetosphere when the\nmagnetic field is strong enough to dominate the wind\npressure (see Section 3.2 and Figures 1, 2). We find\nfor the first time that PNS rotation strongly influences\nthe entropy of material in the plasmoids. We present\na set of 2D magnetohydrodynamic (MHD) simulations\nat constant neutrino luminosity for PNS spin periods\nP⋆= 200 ms, 20 ms, 10 ms, and 5 ms at a polar mag-\nnetic field strength B0= 3×1015G (see Section 3.3).\nFigure 3 shows the 2D maps of entropy for various val-\nues of PNS spin period at a fixed B0, where it is evident\nthat the entropy in plasmoids increases with faster PNS\nrotation.\nWe use the figure of merit parameter ζ(equation 7)\nto estimate the potential for r−process using the values\nof entropy, expansion timescale, and electron fraction.\nFavorable conditions to produce the third r−process\npeak can be achieved in the regions with ζ≥ζcrit≃\n8×109(kBbaryon−1)3s−1(Hoffman et al. 1997). Figure\n4 shows the mass outflow rate ˙Mwith ζ≥ζcritthrough\ntheT= 0.5 MeV temperature surface as a function of\ntime for various values of PNS spin period at a fixed B0.\nFigures 5 and 6 show the time-averaged distribution of\n˙Mthrough the T= 0.5 MeV surface as a function of\nentropy and ζrespectively. Figure 7 shows the scatter17\nplot of ˙Mweighted average of entropy and expansion\ntimescale for various values of PNS spin period. From\nall these figures, it is evident that the PNS rotation rate\nstrongly influences the entropy and other wind proper-\nties that can directly affect nucleosynthesis in winds.\nWe present an analytic scaling relation between the\nentropy in plasmoids and the PNS angular frequency of\nrotation Ω ⋆(see Section 3.4). We find that the entropy S\nin plasmoids scales as S∝Ω5/6\n⋆(see the proportionality\nin 18). Figure 8 shows that the derived scaling relation\nagrees very well with the simulation data when Ω ⋆is\nsufficiently large (see Section 3.4 for details).\nWe also present results from simulations with evolving\nneutrino luminosity and PNS spin period at B0= 1015G\nandB0= 5×1014G (see Section 3.5). We use the cool-\ning model from Vartanyan & Burrows (2023) and follow\nthe evolution until 10 s and 14 s after the onset of cool-\ning at B0= 1015G and 5 ×1014G respectively. Figure\n9 shows the maximum entropy as a function of time af-\nter the onset of cooling for the evolutionary models at\nB0= 1015G. Figure 10 shows the distribution of ˙M\nthrough the T= 0.5 MeV surface as a function of en-\ntropy at B0= 1015G. Figure 11 shows the scatter plot of\n˙Mweighted average of entropy and expansion timescale\nfor the evolutionary models at B0= 1015G. Figure 12\nshows the maximum entropy at T= 0.5 MeV as a func-\ntion of time after the onset of cooling at B0= 5×1014G.\nSimilar to our conclusions from the constant neutrino lu-\nminosity simulations, we find even from our evolution-\nary models that PNS rotation rate affects the critical\nnucleosynthesis parameters in PNS winds. Table 1 sum-\nmarises the results from the constant neutrino luminos-\nity simulations and the evolutionary models.\nAlthough we find favorable conditions for nucleosyn-\nthesis in constant neutrino luminosity simulations and\nevolutionary simulations, the nucleosynthetic yields de-\npend critically on the electron fraction Yeof the wind.\nFor the neutrino luminosities and assumed energies\nused in the constant luminosity simulations, we find an\nasymptotic Ye∼0.45, which favors the production of\nr−process elements. For PNS spin periods ≳10 ms in\nthe constant neutrino luminosity simulations, we find\nthat the the time-averaged ˙Mwith ζ≥ζcritthrough\ntheT= 0.5 MeV surface is ∼2.5×10−5M⊙s−1(see\nTable 1). Since the timescale for the neutrino luminosi-\nties to decrease at the high neutrino luminosities consid-\nered in the constant luminosity simulations is ∼1−2 s,\nwe can expect ∼2−5×10−5M⊙of high ζmaterial\nto be released during just the first few seconds of the\ncooling phase at B0= 3×1015G. We can then ex-\npect∼5×10−5−10−4M⊙of high ζmaterial to be\nejected by a magnetar throughout the cooling phase at\n−4−3−2−1 0\nlog10(texpλνen)−8−7−6−5−4−3log10˙M[T= 3 GK] (M⊙s−1)˙Mtot,3 GKP⋆= 20 msFigure 14. Time-averaged mass outflow rate through the\nT= 3×109K (3 GK) surface as a function of neutrino expo-\nsure defined as the product of expansion timescale texpand\nthe rate for νeabsorption on neutrons λνen. The horizontal\nblack dashed line shows the total time-averaged ˙Mthrough\ntheT= 3 GK surface. The value of ˙Mtot,3 GK in this figure\nis slightly different from the ˙Mvalue computed over a spher-\nical surface reported in Table 1 because the T= 3 GK sur-\nface is not spherical. The vertical dashed red line shows the\nthreshold for the νr−process nucleosynthesis (Xiong et al.\n2023). This plot shows data from a constant neutrino lumi-\nnosity simulation at B0= 3×1015G and PNS spin period\nP⋆= 20 ms run with the Helmholtz EOS (the simulation\nmarked with l, n in Table 1).\nB0= 3×1015G. Larger masses of high ζmaterial may be\nexpected at higher B0. Given a Galactic supernova rate\nof∼0.02 yr−1and a fraction ∼40−50% of all neutron\nstars being magnetars (Beniamini et al. 2019), our simu-\nlations suggest that we can expect ∼5×10−7−10−6M⊙\nyr−1of high ζmaterial to be produced by magnetars.\nWith a conservative estimate accounting for the α−rich\nfreezeouts, we can expect ∼5×10−8−10−7M⊙yr−1\nofr−process elements with A > 130 to be produced\nby magnetars. The production rate of heavy r−process\nelements with A > 130 is ∼10−7M⊙yr−1when aver-\naged over the Galaxy’s star formation (Qian 2000). This\nestimate suggests that magnetar winds are viable candi-\ndates for heavy element synthesis. Our finding that the\ntotal mass of high ζmaterial produced by magnetars is\nroughly the same for any spin period ≳10 ms (see Table\n1) further supports the claim.18\nFrom the results presented in Table 1 and Figure 6, we\nfind that on average, the mass outflow rate of the mate-\nrial with ζ≥ζcritconstitutes ∼8−10% of the total mass\noutflow rate through the T= 0.5 MeV surface. The re-\nmaining ∼90% of the wind has ζ < ζ crit. The wind ma-\nterial outside the equatorial region with ζ < ζ critmay\nproduce a ‘weak’ r−process that can extend to the first\nand possibly the second r−process peak. Detailed sim-\nulations with tracer particles are required to study the\nnucleosynthetic yields that emerge from the distribution\nof the wind material as a function of ζ.\nThe above estimates consider only the T= 0.5 MeV\nsurface because the approximations to the EOS (see Sec-\ntion 2.1) do not allow us to explore the wind parame-\nters accurately at T < 0.5 MeV. As shown in Table 1,\nwe find in the approximate EOS simulations and in sim-\nulations with the more accurate Helmholtz EOS that\nthe entropy in the regions with T < 0.5 MeV is larger\ncompared to the entropy at T= 0.5 MeV. In the sim-\nulation with the Helmholtz EOS (the one marked with\nl, n in Table 1) that allows us to explore the wind pa-\nrameters even at T <0.5 MeV, we find that the average\n˙Mwith ζ≥ζcritthrough the T= 0.5 MeV, 0.4 MeV,\n0.3 MeV, and 0.2 MeV surfaces is 3.1 ×10−5M⊙s−1,\n3.7×10−5M⊙s−1, 4.0×10−5M⊙s−1, and 4.8 ×10−5M⊙\ns−1, respectively. This result motivates simulations with\nthe Helmholtz EOS and Lagrangian tracer particles that\nwill enable detailed analysis of the nucleosynthetic yields\nconsidering all the regions of the wind. We intend to ex-\nplore this in detail in a future work.\nIn our evolutionary models using the Vartanyan &\nBurrows (2023) cooling model, we find an asymptotic\nYe>0.5 throughout the evolution. If the PNS wind\nis proton-rich, p−nuclei can be produced (e.g., Pruet\net al. 2006). By artificially increasing the entropy by\na factor of 2 −3 over their baseline outflow entropy of\n55−77 k Bbaryon−1, Pruet et al. (2006) find that the\nnucleosynthesis can extend until A≈170. They find\nthat even some r−process nuclei can be produced in\nproton-rich winds. These conditions are satisfied in our\nevolutionary models for a polar magnetic field strength\nB0= 1015G, and even a modest B0= 5×1014G, sug-\ngesting that PNS winds may be a source of p−nuclei.\nAlthough Ye>0.5 in our evolutionary models does\nnot favor r−process, we provide estimates for the\nr−process element yields assuming that the distribution\nof˙Mas a function of entropy and ζwould be roughly\nthe same even for asymptotic Ye<0.5 in these simula-\ntions. Under this assumption, we can estimate using the\nvalues in Table 1 that a magnetar ejects ∼3.5×10−6M⊙\nof material with ζ≥ζcritthrough the T= 0.5 MeV sur-\nface in 10 s during the cooling phase at B0= 1015G.AtB0= 5×1014G, we estimate that ∼2.5×10−8M⊙\nof material with ζ≥ζcritis ejected through the T=\n0.5 MeV surface in 14 s. As mentioned above, we ex-\npect higher values of ˙Mwith ζ≥ζcritatT <0.5 MeV.\nFor example, we find that ∼2×10−7M⊙of material\nsatisfies the ζ≥ζcritcondition at T= 0.2 MeV (the\nestimate at 0.2 MeV may not be accurate due to the\napproximations to the EOS). These estimates suggests\nthat magnetars with B0≲1015G cannot account for the\nentire r−process budget of the Galaxy, during the cool-\ning phases explored in these models. As shown in all the\nevolutionary models, the entropy in plasmoids increases\nas the neutrino luminosities decrease, and we can ex-\npect more favorable nucleosynthesis conditions at lower\nluminosities. Detailed evaluation throughout the PNS\ncooling phase with tracer particles is required to assess\nthe nucleosynthetic yields, which will be our focus in a\nfuture work.\nXiong et al. (2023) present a new nucleosynthesis\nprocess in which p−nuclei and the short-lived nucleus\n92Nb can be produced in neutron-rich winds via the\nνr−process. They show that the νr−process is en-\nabled if the neutrino exposure, which they define as\nthe product of expansion timescale texpand the rate\nforνeabsorption on neutrons λνen, exceeds 0.1 at the\nT∼3×109K (3 GK) surface. We find that this con-\ndition is satisfied in our simulations. At T∼3 GK, the\napproximate analytic EOS (Qian & Woosley 1996) is\nnot accurate. Hence, we present results from the simu-\nlation run with the Helmholtz EOS. Figure 14 shows the\ntime-averaged mass outflow rate through the T= 3 GK\nsurface as a function of neutrino exposure (for the sim-\nulation marked with l, n in Table 1). We find that\nan average mass outflow rate of 7.5 ×10−7M⊙s−1\nsatisfies the condition texpλνen≥0.1 at T= 3 GK,\nconstituting ∼0.2% of the total mass outflow rate\nthrough the 3 GK surface. This translates to a mass\nof∼7×10−7−1.5×10−6M⊙with the required condi-\ntions for the νr−process during the first ∼1−2 s of the\nPNS cooling phase.\nWe also study spindown of PNSs using the evolution-\nary models. Figure 13 shows the spindown timescale and\nthe evolution of PNS spin period as a function of time\nafter the onset of cooling at B0= 1015G. We find that\nthe nature of the profile of spindown time correspond-\ning to the evolutionary model with initial spin period\nP⋆0= 200 ms is significantly different compared to the\nprofile at P⋆0= 20 ms (see Section 3.5 for details). Sim-\nilar to the results in our earlier work (Prasanna et al.\n2022), we find that ‘slowly’ rotating magnetars born\nwith polar magnetic field strengths B0≳1015G and\ninitial spin periods ≳100 ms can spindown significantly19\nduring the PNS cooling phase lasting just a few tens of\nseconds.\nIn a work that will follow this paper, we plan to in-\ncorporate Lagrangian tracer particles in the simulations\nto track nucleosynthetic yields. Since our goal in this\npaper was just to demonstrate that PNS rotation rate\nsignificantly affects the nucleosynthesis parameters of\nthe wind, we have chosen to use the approximate an-\nalytic form of the general equation of state (Qian &\nWoosley 1996) which is computationally much less ex-\npensive than the Helmholtz EOS (Timmes & Swesty\n2000). As mentioned earlier, this approximation to the\nEOS allows us to explore the wind parameters at only\nT≳0.5 MeV where the approximation is valid. We in-\ntend to relax the approximations to the EOS in a future\nwork.\nFinally, we note that in the constant neutrino lumi-\nnosity simulation (Section 3.3) at a PNS spin period\nP⋆= 5 ms and polar magnetic field strength B0=\n3×1015G, the Alfv´ en speed and the fast magnetosonic\nspeed exceed the speed of light. Since we use non-\nrelativistic physics in our current simulations, inclusionof relativistic effects is required to confirm the result at\nP⋆= 5 ms. In the evolutionary models presented in this\npaper, non-relativistic physics is sufficient as the mag-\nnetosonic speeds remain well below the speed of light\nthroughout the evolution. Relativistic simulations are\nrequired to study PNS evolution throughout the cooling\nphase at higher values of B0. This will also be a focus\nof future work.\nACKNOWLEDGMENTS\nWe thank Brian Metzger, Zewei Xiong, Gabriel\nMart´ ınez-Pinedo, Oliver Just, and Andre Sieverding for\nhelpful discussions. TAT thanks Asif ud-Doula, Brian\nMetzger, Phil Chang, Niccol´ o Bucciantini, and Eliot\nQuataert for discussions and collaboration on this and\nrelated topics. TP and TAT are supported in part by\nNASA grant 80NSSC20K0531. We have run our simula-\ntions on the Ohio supercomputer (Center 1987). Parts\nof the results in this work make use of the colormaps in\nthe CMasher package (van der Velden 2020).\nREFERENCES\nBeniamini, P., Hotokezaka, K., van der Horst, A., &\nKouveliotou, C. 2019, MNRAS, 487, 1426,\ndoi: 10.1093/mnras/stz1391\nBurrows, A., Hayes, J., & Fryxell, B. A. 1995, ApJ, 450,\n830, doi: 10.1086/176188\nBurrows, A., & Lattimer, J. M. 1986, ApJ, 307, 178,\ndoi: 10.1086/164405\nBurrows, A., & Mazurek, T. J. 1982, ApJ, 259, 330,\ndoi: 10.1086/160169\nCardall, C. Y., & Fuller, G. M. 1997, ApJL, 486, L111,\ndoi: 10.1086/310838\nCenter, O. S. 1987, Ohio Supercomputer Center.\nhttp://osc.edu/ark:/19495/f5s1ph73\nColeman, M. S. B. 2020, ApJS, 248, 7,\ndoi: 10.3847/1538-4365/ab82ff\nDesai, D. K., Siegel, D. M., & Metzger, B. D. 2023, ApJ,\n954, 192, doi: 10.3847/1538-4357/acea83\nDuncan, R. C., Shapiro, S. L., & Wasserman, I. 1986, ApJ,\n309, 141, doi: 10.1086/164587\nDuncan, R. C., & Thompson, C. 1992, ApJL, 392, L9,\ndoi: 10.1086/186413\nFriedland, A., Mukhopadhyay, P., & Patwardhan, A. V.\n2023, arXiv e-prints, arXiv:2312.03208.\nhttps://arxiv.org/abs/2312.03208Fr¨ ohlich, C., Mart´ ınez-Pinedo, G., Liebend¨ orfer, M., et al.\n2006, PhRvL, 96, 142502,\ndoi: 10.1103/PhysRevLett.96.142502\nHoffman, R. D., Woosley, S. E., Fuller, G. M., & Meyer,\nB. S. 1996, ApJ, 460, 478, doi: 10.1086/176986\nHoffman, R. D., Woosley, S. E., & Qian, Y. Z. 1997, ApJ,\n482, 951, doi: 10.1086/304181\nJanka, H. T., & Mueller, E. 1996, A&A, 306, 167\nMetzger, B. D., Thompson, T. A., & Quataert, E. 2007,\nApJ, 659, 561, doi: 10.1086/512059\n—. 2008, ApJ, 676, 1130, doi: 10.1086/526418\nMeyer, B. S., Mathews, G. J., Howard, W. M., Woosley,\nS. E., & Hoffman, R. D. 1992, ApJ, 399, 656,\ndoi: 10.1086/171957\nOtsuki, K., Tagoshi, H., Kajino, T., & Wanajo, S.-y. 2000,\nApJ, 533, 424, doi: 10.1086/308632\nPneuman, G. W., & Kopp, R. A. 1971, SoPh, 18, 258,\ndoi: 10.1007/BF00145940\nPons, J. A., Reddy, S., Prakash, M., Lattimer, J. M., &\nMiralles, J. A. 1999, ApJ, 513, 780, doi: 10.1086/306889\nPrasanna, T., Coleman, M. S. B., Raives, M. J., &\nThompson, T. A. 2022, MNRAS, 517, 3008,\ndoi: 10.1093/mnras/stac2651\n—. 2023, MNRAS, 526, 3141, doi: 10.1093/mnras/stad2948\nPruet, J., Hoffman, R. D., Woosley, S. E., Janka, H. T., &\nBuras, R. 2006, ApJ, 644, 1028, doi: 10.1086/50389120\nPruet, J., Woosley, S. E., Buras, R., Janka, H. T., &\nHoffman, R. D. 2005, ApJ, 623, 325, doi: 10.1086/428281\nPsaltis, A., Jacobi, M., Montes, F., et al. 2023, arXiv\ne-prints, arXiv:2312.12306,\ndoi: 10.48550/arXiv.2312.12306\nQian, Y. Z. 2000, ApJL, 534, L67, doi: 10.1086/312659\nQian, Y. Z., & Woosley, S. E. 1996, ApJ, 471, 331,\ndoi: 10.1086/177973\nRoberts, L. F. 2012, ApJ, 755, 126,\ndoi: 10.1088/0004-637X/755/2/126\nRoberts, L. F., Reddy, S., & Shen, G. 2012, PhRvC, 86,\n065803, doi: 10.1103/PhysRevC.86.065803\nSalmonson, J. D., & Wilson, J. R. 1999, ApJ, 517, 859,\ndoi: 10.1086/307232\nSteinolfson, R. S., Suess, S. T., & Wu, S. T. 1982, ApJ,\n255, 730, doi: 10.1086/159872\nStone, J. M., Tomida, K., White, C. J., & Felker, K. G.\n2020, ApJS, 249, 4, doi: 10.3847/1538-4365/ab929b\nThompson, C., & Duncan, R. C. 1993, ApJ, 408, 194,\ndoi: 10.1086/172580\nThompson, T. A. 2003, ApJL, 585, L33,\ndoi: 10.1086/374261\nThompson, T. A., Burrows, A., & Meyer, B. S. 2001, ApJ,\n562, 887, doi: 10.1086/323861\nThompson, T. A., & ud-Doula, A. 2018, MNRAS, 476,\n5502, doi: 10.1093/mnras/sty480\nTimmes, F. X., & Swesty, F. D. 2000, ApJS, 126, 501,\ndoi: 10.1086/313304van der Velden, E. 2020, Journal of Open Source Software,\n5, 2004, doi: 10.21105/joss.02004\nVartanyan, D., & Burrows, A. 2023, MNRAS, 526, 5900,\ndoi: 10.1093/mnras/stad2887\nVidotto, A. A., Jardine, M., Morin, J., et al. 2014,\nMNRAS, 438, 1162, doi: 10.1093/mnras/stt2265\nVink, J., & Kuiper, L. 2006, MNRAS, 370, L14,\ndoi: 10.1111/j.1745-3933.2006.00178.x\nVlasov, A. D., Metzger, B. D., Lippuner, J., Roberts, L. F.,\n& Thompson, T. A. 2017, MNRAS, 468, 1522,\ndoi: 10.1093/mnras/stx478\nVlasov, A. D., Metzger, B. D., & Thompson, T. A. 2014,\nMNRAS, 444, 3537, doi: 10.1093/mnras/stu1667\nWanajo, S. 2006, ApJ, 647, 1323, doi: 10.1086/505483\nWanajo, S., Janka, H.-T., & Kubono, S. 2011, ApJ, 729, 46,\ndoi: 10.1088/0004-637X/729/1/46\nWanajo, S., Kajino, T., Mathews, G. J., & Otsuki, K. 2001,\nApJ, 554, 578, doi: 10.1086/321339\nWhite, C. J., Burrows, A., Coleman, M. S. B., & Vartanyan,\nD. 2022, ApJ, 926, 111, doi: 10.3847/1538-4357/ac4507\nWoosley, S. E., & Hoffman, R. D. 1992, ApJ, 395, 202,\ndoi: 10.1086/171644\nWoosley, S. E., Wilson, J. R., Mathews, G. J., Hoffman,\nR. D., & Meyer, B. S. 1994, ApJ, 433, 229,\ndoi: 10.1086/174638\nXiong, Z., Mart´ ınez-Pinedo, G., Just, O., & Sieverding, A.\n2023, arXiv e-prints, arXiv:2305.11050,\ndoi: 10.48550/arXiv.2305.1105021\nAPPENDIX\nA.PROOF FOR THE ASSERTION AFTER EQUATION 14\nSuppose we have the sum of the three terms as\nAΩα\n⋆+BΩβ\n⋆+CΩγ\n⋆=X, (A1)\nwhere A,B,C, and Xare non-zero constants independent of Ω ⋆. Let us assume that there are solutions to this\nequation for α̸= 0, β̸= 0, and γ̸= 0 for all Ω ⋆>0. Since Xis independent of Ω ⋆, doubling the value of Ω ⋆on the\nLHS does not alter the value of X. We thus obtain\n2αAΩα\n⋆+ 2βBΩβ\n⋆+ 2γCΩγ\n⋆=AΩα\n⋆+BΩβ\n⋆+CΩγ\n⋆. (A2)\nThe LHS and the RHS in the above equation are polynomials in Ω ⋆, and equality is possible only if the coefficients\nof the respective powers of Ω ⋆in the polynomials are equal. This condition requires α=β=γ= 0, which is in\ncontradiction with our initial assumption. We can thus conclude that the sum of the three terms can be independent\nof Ω ⋆only if each term is independent of Ω ⋆."    },    {        "title": "2402.06014v1.Trustful_Coopetitive_Infrastructures_for_the_New_Space_Exploration_Era.pdf",        "content": "Trustful Coopetitive Infrastructures for the New Space\nExploration Era\nLoïck Chovet∗\nUniversity of Luxembourg\nSpaceR Research Group, SnT\nKirchberg, Luxembourg\nloick.chovet@uni.luRenan Lima Baima∗\nEduard Hartwich\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\nrenan.limabaima@uni.lu\neduard.hartwich@uni.luAbhishek Bera\nUniversity of Luxembourg\nSpaceR Research Group, SnT\nKirchberg, Luxembourg\nabhishek.bera@uni.lu\nJohannes Sedlmeir\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\njohannes.sedlmeir@uni.luGilbert Fridgen\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\ngilbert.fridgen@uni.luMiguel Angel Olivares Mendez\nUniversity of Luxembourg\nSpaceR Research Group, SnT\nKirchberg, Luxembourg\nmiguel.olivaresmendez@uni.lu\nABSTRACT\nIn the new space economy, space agencies, large enterprises, and\nstart-ups aim to launch space multi-robot systems ( MRS ) for various\nin-situ resource utilization ( ISRU ) purposes, such as mapping, soil\nevaluation, and utility provisioning. However, these stakeholders’\ncompeting economic interests may hinder effective collaboration\non a centralized digital platform. To address this issue, neutral and\ntransparent infrastructures could facilitate coordination and value\nexchange among heterogeneous space MRS . While related work\nhas expressed legitimate concerns about the technical challenges\nassociated with blockchain use in space, we argue that weighing\nits potential economic benefits against its drawbacks is necessary.\nThis paper presents a novel architectural framework and a com-\nprehensive set of requirements for integrating blockchain technol-\nogy in MRS, aiming to enhance coordination and data integrity in\nspace exploration missions. We explored distributed ledger tech-\nnology ( DLT) to design a non-proprietary architecture for hetero-\ngeneous MRS and validated the prototype in a simulated lunar\nenvironment. The analyses of our implementation suggest global\nISRU efficiency improvements for map exploration, compared to\na corresponding group of individually acting robots, and that fos-\ntering a coopetitive environment may provide additional revenue\nopportunities for stakeholders.\nCCS CONCEPTS\n•Computing methodologies →Distributed computing method-\nologies ;•Applied computing →Aerospace ;Information integra-\ntion and interoperability ; Cross-organizational business processes;\n∗Both authors contributed equally to this research\nPermission to make digital or hard copies of part or all of this work for personal or\nclassroom use is granted without fee provided that copies are not made or distributed\nfor profit or commercial advantage and that copies bear this notice and the full citation\non the first page. Copyrights for third-party components of this work must be honored.\nFor all other uses, contact the owner/author(s).\nSAC ’24, April 8–12, 2024, Avila, Spain\n©2024 Copyright held by the owner/author(s).\nACM ISBN 979-8-4007-0243-3/24/04.\nhttps://doi.org/10.1145/3605098.3635887•Computer systems organization →Peer-to-peer architec-\ntures ;Self-organizing autonomic computing ;Robotic autonomy ;\nExternal interfaces for robotics .\nKEYWORDS\nBlockchain, digital platform, in-situ resource utilization, open sci-\nence, multi-robot system, space economy\nACM Reference Format:\nLoïck Chovet, Renan Lima Baima, Eduard Hartwich, Abhishek Bera, Jo-\nhannes Sedlmeir, Gilbert Fridgen, and Miguel Angel Olivares Mendez. 2024.\nTrustful Coopetitive Infrastructures for the New Space Exploration Era. In\nThe 39th ACM/SIGAPP Symposium on Applied Computing (SAC ’24), April\n8–12, 2024, Avila, Spain. ACM, New York, NY, USA, Article 4, 10 pages.\nhttps://doi.org/10.1145/3605098.3635887\n1 INTRODUCTION\nThe space industry has experienced substantial growth in recent\nyears, pushed by various government policies, international agree-\nments [ 49], and private investments, accounting for 80 % of the\nsector’s revenue over the last 16 years [ 22]. Major space agencies,\nsuch as NASA [58], have emphasized the emergence of long-term\nhuman presence on space constellations and MRS lunar missions\nfor destination reconnaissance, resource prospecting, and map-\nping [ 7,24,52,67], also known as ISRU . Combining this tendency\nwith recent regulatory framework changes — shifting from a strict\nantitrust approach to a more consumer-focused free market [ 59]\n— highlights the potential for further expansion and advancement\ntoward a market-based space exploration approach. As the space\nindustry moves towards decentralization and a collaborative ap-\nproach, there is a growing need to explore innovative solutions that\nenable efficient coordination among multiple robots [63].\nMRS involve groups of robots working together or supporting\neach other to accomplish specific tasks [ 60]. These systems can\ninvolve homogeneous orheterogeneous groups of robots and are\ntraditionally categorized by their level of goal similarity, awareness\nof each other, and interaction as collective ,cooperative , orcollabora-\ntive[60]. The emergence of coopetitive systems, where competingarXiv:2402.06014v1  [cs.RO]  8 Feb 2024SAC ’24, April 8–12, 2024, Avila, Spain Chovet and Baima et al.\nagents simultaneously choose to cooperate owing to economic\nincentives, further expands the possibilities of MRS [75] inISRU\ndriven explicitly by economic incentives and shared objectives.\nInformation-sharing protocols and market-based approaches can\noften improve coordination among MRS and robots’ resource uti-\nlization, cost-effectiveness, and exploration capabilities [ 63]. These\nsystems can effectively mitigate adverse selection caused by infor-\nmation asymmetries — situations where few market participants\nknow more about products (e.g., water or iron positions) or service\nquality (e.g., mapping data) than others [ 4], and as such, contribute\nto efficiency in the market [ 40] in line with the growing acceptance\nof this technology in space missions [29].\nCoordinating MRS in a market-based approach towards ISRU\npresents significant challenges due to the involvement of multi-\nple competing entities and nations, with more than 60 countries\ninvolved in space activities [ 14]. Additionally, the legal [ 69] and\ntechnical requirements for planetary mobility systems further com-\nplicate the coordination of MRS in space exploration [ 52]. This\nwork’s technical and economic foundations highlight the need for\nrobust and adaptable ISRU systems that disincentivize undesirable\nbehavior [49].\nAdditionally, stakeholders involved in space missions highly\nvalue the achievement of being the first to explore and acquire\nspace and scientific data, so companies should aim to make mis-\nsion outputs (e.g., videos, images, and audio) broadly accessible\n(e.g., via the web and media) [ 17]. This perspective resonates with\nthe principles of open science, which promotes knowledge shar-\ning, transparency, collaboration, and accessibility in research [53].\nThe evolving role of space ecosystems in shaping the future of\nspace exploration [ 59] underscores the need for a decentralized (i.e.,\nnon-proprietary), trustworthy, and transparent digital platform.\nSuch a platform can facilitate the seamless exchange of information\nand value among stakeholders [ 41], enabling autonomous MRS\ncoordination for resource trading. DLT-based systems might of-\nfer a suitable solution for this specific requirement [ 15] and, as a\nconsequence, have been considered the foundation of both space\nMRS [31] and open science platforms [21].\nDLT offers several advantages for machine cooperation, includ-\ning in MRS settings [ 40,77]. These platforms can automate tasks\nsuch as bidding for resource usage, publicly broadcasting resource\nacquisition, and facilitating immediate operational cost compensa-\ntion. With a DLT-based digital platform, entities can provide idle\nrobot resources and stack intermediary profitable tasks to auto-\nmatically compensate operational costs and openly recognize pio-\nneering exploratory participants, thus increasing ISRU efficiency\nand enabling lower-cost space exploration. Despite the advantages,\nDLT in space MRS is not without challenges. For instance, space\nrobots face harsh conditions and limited resources, which conflict\nwith the inefficient information processing of blockchains’ inten-\nsive computation and storage replication [ 15,38]. Furthermore,\ndespite stakeholders’ interest in open science [ 17], replicated infor-\nmation processing must still be aligned with the need to protect the\nsensitive business information of robots or organizations exposed\nthrough transactional (meta-) data [68].\nTherefore, tensions exist between the opportunities and chal-\nlenges of using DLT for space MRS , requiring closer investigation.\nThis paper aims to contribute to this understanding by designing anarchitecture for coopetitive MRS forISRU , focusing on facilitating\nopen science through DLT. We drew upon the ESA-ESRIC space re-\nsources challenge as a specific use case [ 51] of mapping exploration\nand applied DLT to coordinate automated cross-organizational\neconomic interactions effectively. Our study investigates the tech-\nnical feasibility of leveraging DLT to enable participation from\nuniversities, research institutes, and small companies in low-cost\nexplorative scientific research. Such a decision-making platform\nmust be holistically assessed to determine whether or not it can\nimprove the global efficiency of ISRU [27,59], maintain targeted\ninformation symmetry, and compensate space companies to create\nadditional revenue streams for their (idle) robotic resources. Hence,\nour research question (RQ) is as follows:\nRQ: Can an ISRU platform architecture based on DLT address the\nchallenges of coordinating coopetitive space MRS mapping for open\nscience?\nIn the evolving landscape of space exploration, which increas-\ningly favors collaborative approaches [ 59], our case study, detailed\nin Section 2, motivates the need for an architecture that empow-\ners organizations to utilize MRS capabilities for efficient ISRU data\nacquisition. Adopting the design science research ( DSR) method\nby Peffers et al. [ 61] to answer our RQ, we have conceptualized a\ndecentralized (i.e., non-proprietary) MRS platform. This platform,\nelucidated in Section 4, is grounded in a thorough literature review,\nintegrating insights from DLT [15], space MRS [32],DLT in robot-\nics [2],DLT in distributed control and cooperative robots [ 47], and\nDLT for the space industry [ 42]. As outlined in Section 4, we de-\nvelop a DLT-based coopetitive MRS that supports space exploration,\nprimarily scientific ISRU mapping. These objectives were shaped\nby an in-depth case study [ 28] and a literature review to discern the\nsystem’s requirements. By leveraging DLT, we strive to establish\na platform that ensures verifiable exchanges of information and\nvalue among diverse, untrusted stakeholders. Our iterative devel-\nopment and evaluation processes, described in Sections 4 and 5,\nhave been instrumental in refining the platform’s design to meet\nthese objectives and address coopetitive ISRU challenges. The eval-\nuations, conducted in three diverse environments, as detailed in\nSection 5, have been pivotal in assessing the platform’s efficacy and\nadaptability. The limitations and open challenges are discussed in\nSection 6. Section 7 highlights the platform’s potential to enhance\nthe efficiency of ISRU activities and create new revenue streams\nfor stakeholders. Our findings and supplementary resources will\nbe publicly available on platforms like GitHub and YouTube.\n2 SCENARIO: REQUIREMENTS ELICITATION\nThis section delves into our space mapping scenario that utilizes\ndecentralization and coopetition to enhance efficiency in mapping\ncoordination among multiple robots. The purpose is to outline the\nscope and specific use case we will refer to throughout the paper.\nSection 6 presents further discussions and limitations of our artifact.\nWe employed the decentralized multi-robotic platform REALMS2\n[71], it uses three Leo Rovers®[46] and a LUVMI-X robot to identify\nresources and analyze and map the environment [33].\nTo illustrate this setting more precisely, we revisit the market-\nbased mapping scenario from Dias et al . [27] and illustrate the\nresult in Figure 1. This approach, proven effective in patrolling,Coopetitive Infrastructures for Space Exploration SAC ’24, April 8–12, 2024, Avila, Spain\nFigure 1: Moon’s Goldberg polyhedron diagram, each zone\ncolor represents a company’s MRS operational area [37].\nTable 1: List of service providers.\nSP Color Robotic Fleet Price Main Focus\nA Pink Communications satellites,\nMultiple antennas- Moon-earth\ncommunications\nB Blue Medium size offline fleet of robots - Mapping\nC Green Few robots w/ embedded sensors $20 Resource analysis\nD Yellow Large fleet of small robots $10 Fast mapping with\nless precision\nE None A single robot $200 Mapping\nexploration, and pick-and-delivery [ 63], is grounded in market-\nbased strategies for MRS . For the sake of simplification, we are\nusing exemplary amounts in the following scenario. The celestial\nsurface stratification follows the existing Goldberg polyhedron\napproach [ 37] and uses the selenographic system to refer to its\nsurface positions. An initiator stationed on Earth, the service or-\nderer (SO), sends requests to a celestial stationed network of MRS\nservice providers (SP), as listed in Table 1. The SO proposes to pay\na maximum of $50 for mapping the target region. SP D is 5 m away,\nwhile SP C is 10 m from the target area. We consider an oversimpli-\nfied abstraction of each robot’s cost function, with a cost of $2 for\neach meter they travel. Consequently, SP D wins the contract by\nbidding $10 compared to $20.\nThe primary need of ISRU SOs is to locate and investigate re-\nsources from celestial bodies [ 17]. This involves identifying areas\nwith water or ice and mapping out the geological features of the\nplanet’s terrain. This role could often be triggered and motivated\nby universities and research institutions that seek to advance sci-\nentific knowledge and innovation in space. However, the scarcity\nof resources available to SOs due to high budget requirements for\nspace-related research hampers advancements in this field. Incom-\nplete mapping data is also an issue owing to the limited coverage\nof celestial geography [ 24]. Access to more precise information\nabout celestial resources and terrain would significantly improve\nscientific outcomes [52].\nThe SPs, on the other hand, may represent start-ups [ 24,67], large\ncompanies [ 25], space agencies [ 58], and organizations that already\nhad some level of collaboration internally but not necessarily withone another. These entities are willing to work together to maximize\ntheir revenues and return on mission investment by offering re-\nsources, such as idle robots, shelter time, or energy supply. However,\nestablishing trustworthy partnerships across various knowledge\ndomains [ 44], including international agreements [ 59,69] and in-\ndividual initiatives [ 22,43,67], poses a challenge. Their struggle\nwith inefficient use of exploration units, methods, and resources\noften results in wasted opportunities and increased costs [ 72]. The\nmost significant values of these entities are in guaranteeing mission\npioneering and promptly sharing exploratory data [ 17]. By follow-\ning open science principles, they can encourage collaboration and\ninnovation and gain access to valuable space exploration data [ 53].\nWe do not claim that these requirements are fully comprehensive\nor sufficient, yet we consider them necessary.\n3 FOUNDATIONS\nExisting solutions for heterogenous MRS exploring the moon are\nstill in their development phase. As an example of current advance-\nments, Corob-X [ 26] represents a notable initiative in the domain\nof space robotics. However, it is important to note that Corob-X\nis a proprietary solution that relies on mutual collaboration and\ntrust among stakeholders. In contrast, our approach offers a non-\nproprietary, blockchain-based framework, addressing the need for\ndecentralized trust and open collaboration in MRS.\n3.1 Market-based coordination for coopetitive\nMRS and space\nCoordinating multiple robots in market-based systems has been\na focal point of extensive research evaluating their quality, trade\nefficiency, and impact on performance through theoretical analysis\nand experimental results [ 27,63]. Centralized approaches often\noverlook the complexities of cross-organizational interactions [ 18].\nIn contrast, emerging distributed and decentralized market-based\nMRS offer advantages in optimizing capital distribution, supply\nand demand matching, and automated resource allocation while\nreducing risks associated with centralized systems, such as single\npoints of failure or direct peer-to-peer setups [45].\nAuction-based mapping systems, though less complex than tra-\nditional solutions, require greater participation and have been im-\nproved by integrating contractual models to enhance efficiency in\nthe space industry [ 12,27]. Integrating these contracts into dig-\nital platforms supported by reliable networks can enhance peer\nautonomy and participation [ 31,42]. Given the vulnerabilities of\ncentralized systems to censorship and manipulation [ 16], combin-\ning market-based approaches with blockchain technology offers a\npromising avenue for enhancing MRS coordination.\nDecentralized approaches using DLT have emerged, with some\nemploying blockchain-based auction systems to determine resource\nusage in competitive scenarios and ensure trustworthy services [ 45].\nSuch approaches promote the efficiency, robustness, adaptability,\nfault tolerance, effectiveness, and responsiveness of MRS [34,35,\n45,78]. However, fully distributed systems sometimes produce sub-\noptimal solutions [ 27,63]. Thereby, given that the technical re-\nquirements for planetary mobility, such as resource and latency\nconstraints [ 38,52], add another layer of complexity, these chal-\nlenges necessitate innovative, real-time coordination solutions inSAC ’24, April 8–12, 2024, Avila, Spain Chovet and Baima et al.\nISRU scenarios. While existing research has primarily focused on\nsingle-entity settings [ 30], there remains a gap in the literature for\nblockchain and auction-based MRS solutions in coopetitive ISRU\nscenarios.\n3.2 Distributed ledger technology for space\nDLT provides decentralized transaction recording and validation\nacross multiple nodes, offering similar properties as common digital\nplatforms but without a central operator [ 19]. Blockchain, a subset\nofDLT, is characterized by its replicated, append-only, hash-linked\ndata structures [ 15]. The technology allows for various levels of\nopenness regarding participation and access, ranging from per-\nmissionless to permissioned and public to private networks [ 11].\nBlockchain’s non-proprietary nature enhances trust among entities\nand enables programmable value exchanges using fungible and\nnon-fungible tokens [ 39]. Smart contracts, often using standards\nlike ERC20, facilitate standardized transactions and programming\nlogic [ 6,54] Using blockchain technology to conduct direct peer-to-\npeer transactions and facilitate smart contracts can reduce costs and\nincrease productivity in various industries [ 3,41,77]. Our work\nexplores the potential benefits and limitations of implementing\nDLT in the identified gap of MRS forISRU . We focus on enhanced\ndata sharing and the automation of services using smart contracts,\nthereby creating new revenue opportunities, such as offering robots\nidle time and promoting non-discriminatory participation among\nstakeholders, ultimately establishing a coopetitive ISRU economy.\n4 SOLUTION: REQUIREMENTS AND\nARCHITECTURE\nWe conducted a case study and literature review to collect design\nrequirements for our coopetitive MRS based on DLT and focused\nonISRU , as described here and in sections 2 and 3. Although the\nrequirement list is not final, we integrated insights from the previ-\nous sections’ discussions to create a comprehensive list addressing\nthe unique challenges and complexities of implementing DLT in an\nMRS for space. Our system architecture draws inspiration from the\nworks of Li et al . [50] , Abe et al . [1], and requirements from Lorenz\n[52] and Cameron et al . [17] . It follows the make-or-buy economic\nframework [ 18] while prioritizing transparency principles of open\nscience [ 53] and the cost-efficiency characteristics of coopetitive set-\ntings [ 36]. Our technical design enables geographically distributed\nrobots to self-coordinate [ 34] by using market-based strategies for\nservice provisioning [27, 63, 78].\n4.1 Requirements analysis\nOur design aims to generate new revenue streams through map\nselling and granting recognition as pioneer explorers by meeting\nthe following requirements:\ni.Network: Enabling clients to create job postings and broad-\ncast them through the mesh network, enabling robots to\ncommunicate and coordinate tasks trustfully [17, 52].\nii.Data sharing transparency: Transparency about who shares\nwhich information at which time – by monitoring explorers\nand job execution history – fosters a coopetitive environment\nwith reduced information asymmetry [36, 53].iii.Robot agnostic system: Compatibility with any robotic\nplatform following the established norms, with the capacity\nto scale the system [21, 63].\niv.Data loss resistance: Ensuring data remains intact and ac-\ncessible despite local system failures or network disruptions,\nsafeguarding valuable information [ 12], e.g., operational map\ndata.\n4.2 Requirements specification\nTo be more precise, the significant functional and non-functional\nrequirements are identified as follows:\n4.2.1 Functional requirements:\n(1) Receive and process robots and SOs’ map requests [78].\n(2)Implement a descending-price auction mechanism for robots\nto allocate mapping tasks [63, 78].\n(3)Verify the integrity and completeness of metadata associated\nwith each job request [27].\n(4) Execute the payment process for completed tasks [27].\n(5)Provide an interface for managing and monitoring job re-\nquests during the mission [27].\n(6)Enable mapping and ISRU performance assessment [ 34,52].\n(7)Ensure an adaptable robotic network compatible with the\ndifferent robot types [27, 34, 78].\n(8)Support the seamless integration with possible existing in-\nfrastructures during space missions, such as space decen-\ntral [23], SpaceChain [76], and TruSat initiatives [34, 56].\n(9)Ensure compatibility with standard data formats and proto-\ncols for efficient data exchange [34].\n(10) Promote interoperability of robotic communication proto-\ncols to facilitate seamless coordination [34].\n4.2.2 Non-functional requirements:\n(1)Conformity: Provide an interoperable economic framework\ninterface for efficient market-based coordination [78].\n(2)Robustness: Protection mechanisms to safeguard against\nthe inconsistency or loss of essential data owing to system\nfailures or network disruptions [27, 34, 52].\n(3)Reliability: Enable agnostic data sharing among robots to\nensure efficient coordination [27].\n(4)Openness: Foster a mission’s public network environment,\nenabling relevant parties to participate on the platform [ 78].\n(5)Usability: Accessible job request statuses and pricing infor-\nmation for processing and decision-making [74].\n(6)Compatibility: Ensure metadata requirements-driven data\nstandards to diverse mapping tasks and applications [27].\n(7)Maintainability: System design incentivizes participants to\ntake ownership of maintaining the network [34].\n(8)Portability: Keeping hardware requirements at a minimum\nto enable flexibility in deploying robot missions [2, 64, 66].\n(9)Interoperability: Ensure the system can handle traded data\nwithout significant adjustments [27, 63, 78].\n4.3 Architecture\nFigure 2 features an overview of the architecture layers and their\nfunctions in our decentralized system architecture for job requestsCoopetitive Infrastructures for Space Exploration SAC ’24, April 8–12, 2024, Avila, Spain\nBlockchain\n    Layer\nStorage\n  Layer\nUser\n Robot\nConsensus MechanismJob Smart Contract\nrekey2Contract() givesDIDs() grantOwnership() submitNFT() distributeReward()\nTemplate Smart Contract\npublishJobContracts() requestJobContracts() signJobContract() computeReputation() updateJobStatus()\nData\n...Block n +1HeaderBlock Body\nTask Metadata SM SignedBlock nHeaderBlock Body\nAvailable SM Requirements\nData\n...Store\nJob Metadata\nSM Signed\nJob Execution\nFigure 2: Architecture layers.\nand contract creation. The process involves various entities, in-\ncluding the Client, MeshNetwork, Robot, TemplateSmartContract,\nJobSmartContract, and JobContract. The lifecycle begins with the\nClient initiating a JobRequest by sending a message to the Mesh-\nNetwork. The MeshNetwork broadcasts the request to available\nRobots, and each Robot optimizes and schedules its operation out-\nside the blockchain. Once the scheduling is complete, the Robot\nsends the JobRequest through the TemplateSmartContract interface.\nUpon receiving the JobRequest, the TemplateSmartContract creates\na JobRequest’s unique identifier. The TemplateSmartContract then\nshares the ID (transaction hash) of the JobRequest with the Robot. If\nat least one Robot decides to participate, a JobContract is generated\nand submitted to the TemplateSmartContract. The TemplateSmart-\nContract updates the proposed job options from the JobContract\nwith the most cost-effective proposal. The ID of the lowest-priced\nJobContract is returned and linked to the corresponding JobRequest,\nwhile it informs the current status and price to the Client.\n5 EVALUATION\nThis section has two subsections that examine the feasibility and\nperformance of our proposed architecture. In subsection 5.1, we\nfocus on logically validating the architectural requirements of our\ndesign. We discuss network communication, data transparency,\ndata integrity assurances, and the use of smart contracts for auto-\nmated economic decision-making. In subsection 5.2, we evaluate\nour system’s fitness to meet accuracy targets in mapping tasks and\nresilience in replicated celestial conditions through simulations and\nphysical experiments. This subsection also discusses usability and\ntransaction execution duration aspects.5.1 Requirements validation\nOur architecture, depicted in Figure 2, integrates the network in-\nfrastructure with DLT for transparent, immutable, and coopeti-\ntiverobot coordination. By utilizing an InterPlanetary File Sys-\ntem ( IPFS), the platform stores only image hashes that define the\nimages’ URLs on the blockchain, ensuring data integrity. To ensure\nconformity ( 𝑁𝐹𝑅−1), the economic automated decision-making is\nbased on the make-or-buy decision framework [ 18]. The JobCon-\ntract (𝐹𝑅−4), which records the execution of mapping tasks and fa-\ncilitates information trading, and the TemplateSmartContract acting\nas an interface for robots to post and monitor JobRequests ( 𝐹𝑅−5),\nfacilitating efficient robot coordination and automating job assign-\nment and compensation ( 𝐹𝑅−1) in a non-discriminatory environ-\nment (𝐹𝑅−7).\nWe initially used a mesh network for communication and con-\ntrol systems suitable for decentralized peer-to-peer and low-latency\nsettings (𝑁𝐹𝑅−3). However, we identified exploitation vulnera-\nbilities regarding unauthorized data modification during transmis-\nsion between peers. Given the high knowledge-sharing capabilities\nofcoopetitive platforms [ 36], the system’s integration allows for\nthe secure and transparent sharing of mapping data among the\nparticipating robots, where robots add the data requirements (i.e.,\nmapping location) to be validated by the smart contract ( 𝑁𝐹𝑅−4)\nvia the specific stratified Goldberg polyhedron registry position\napproach [37]. When followed, it ensures the reliability of the col-\nlected information ( 𝐹𝑅−10). The system publicly records who was\nthe pioneer in being the first to explore, investigate, or discover an\narea or execute activities recognized as at the forefront of space ex-\nploration initiatives. As such, this verifiable record can help to build\nvaluable partnerships and position universities as leaders in inno-\nvative interdisciplinary research. Additionally, a commission model\ncan compensate the pioneer with every sale of the non-fungible\ntokens ( NFT) associated with the map on secondary markets [ 39],\ntranslating into additional revenue streams.\nIPFS enables robust and decentralized storage of mapping data\n(𝑁𝐹𝑅−2) and protects from loss or censorship ( 𝐹𝑅−8). It accepts\nstandard data formats ( 𝑁𝐹𝑅−2) and detects tampering through\nhash verification. While transactions within the blockchain are\nimmutable, standard asset parameters can be modified. With these\nparameters, entities can confidently re-configure, destroy, mint,\nfreeze, and even clawback addresses, ensuring that only autho-\nrized participants can modify the mapping metadata within the\nsystem (𝑁𝐹𝑅−7). The history of these assets is immutably recorded\nin the blockchain and can be publicly verified. In our case, these\nsystems do not depend on specific specialized hardware and enable\nnear real-time processing in space-located facilities or, as we have\nsuccessfully implemented, in a server-client structure that simpli-\nfies data sharing ( 𝑁𝐹𝑅−8). Thus, it affirms the supposed improved\nperformance of a distributed, decentralized system consisting of\nmultiple coopetitive robots compared to single individual mapping\nunits.\nDuring the bidding process, our robots engage in a descending-\nprice auction. By posing a lower bid than the previous lowest bid\nrecorded in the smart contract ( 𝐹𝑅−2), robots signal their will-\ningness to execute a job. This descending-price bidding allows the\nrobots to continuously reassess their capabilities and resources,SAC ’24, April 8–12, 2024, Avila, Spain Chovet and Baima et al.\nresulting in an economically efficient solution [ 27]. The robots can\nnegotiate multiple contracts simultaneously, and different robots\ncan initiate new contracts to find new ways to execute previous\ncontract proposals ( 𝑁𝐹𝑅−5). As a result, several JobRequests may\nattach to multiple JobContracts, coordinated by several TemplateS-\nmartContract instantiations ( 𝑁𝐹𝑅−9). The TemplateSmartCon-\ntract evaluates the proposals and ranks the options as it reach\nthe time limit. The Client then considers the JobContract and de-\ncides whether to accept it. If accepted, an atomic payment transfer\nprocess is initiated by rekeying the contract signature to the Tem-\nplateSmartContract, potentially triggering multiple intermediate\ncontracts. The respective bidding robot can set a maximum price\nthreshold to ensure that auctioning outcomes align with it.\nRegarding exploitation protection in this phase, to ensure the\ncontract is not vulnerable to skipping payment after task execution,\nit is only considered ready for execution once the TemplateSmart-\nContract receives the rekey power. Once granted, the contract can\nbe signed, and the payment can be made automatically according to\nthe contract agreement. However, if an attacker keeps bidding lower\nthan others, there is no automated protection against spamming\nor the need to create new auctions. The solutions hence incorpo-\nrate the bidder’s reputation, which may be lower than others, and\nsetting a time limit to restrict possible spamming attacks. These\nattacks may involve proposing multiple fake tasks or submitting\nfake lower bids, which the robot never intended to execute, or it\njust wanted to prevent the transaction from happening. Another\npotential exploitation is the failure to complete tasks upon which\nparties previously agreed. Even though our robots are automatically\nprogrammed to function independently, any deliberate harmful hu-\nman interference can still disrupt their operations. Nevertheless,\nthe physical limitations on communication time delays from Earth\nand the celestial body and the smart contract time limit may offer\nsome protection against such cases. Despite unintended malfunc-\ntions that may still occur, this physical protection, combined with\nthe reputations of bidders, can improve trust in the platform.\nFigure 3: Experimental environment that mimics the environ-\nmental lunar conditions (Lunalab), the European cooperative\nlarge logistic lander and the Gazebo ’s [48] virtual space for\nmulti-robot simulation integrated with ROS 2 [62].\nOnce the payment is received, the TemplateSmartContract trig-\ngers the participating robots to sign and execute the JobRequestand JobContract ( 𝐹𝑅−4). The robots carry out the specified tasks\noutlined in the contract: mapping or selling the map exploration\nrights while adding the asset description into the metadata and\ndata (𝑁𝐹𝑅−6). Before finalizing the contracts, the TemplateSmart-\nContract conducts a plausibility check of the correctness and com-\npleteness of the mapping data retrieved from IPFS by assessing the\nrequirements described on the JobRequest and the NFT metadata.\nThis check ensures that the data adheres to the initial specifications\noutlined in the JobRequest without any unauthorized modifications,\nsuch as via the mutable asset characteristics. The possible mutabil-\nity functions are: freezing an asset until the user meets a specific\nrequirement, clawback by debiting a user’s account for defaulting\nin loan payments, and revoking the ownership of assets belonging\nto users. Thus, the completion is successful if the robot receives the\npayment according to the due terms of the JobContract.\nIn the last stage, due to the NFTs’ (meta-) data, the Client is\nrecognized as the pioneer explorer of the acquired mapping data\nor deliverables, signifying the process completion. The TemplateS-\nmartContract verifies that the retrieved map metadata and IPFS\ndata align with the original JobRequest and the acquired mapping\ninformation ( 𝐹𝑅−3). After the verification process, the Client can\nuse the data for decision-making, knowing that the mapping coordi-\nnation process was transparent and trustworthy. We implemented\na prototype of our designed architecture to evaluate its practicality\nand discuss the lessons learned.\n5.2 Requirements verification in simulation and\nimplementation\nThe evaluation of our system aimed to test and validate its accuracy\nand resistance in similar celestial conditions, explicitly focusing on\nassessing space-related limitations, such as latency and resource\nconstraints. We conducted three simulations to evaluate the ar-\nchitectural solution’s ability to collect, coordinate, and maintain\nmapped area data. We started with a turtlesim -based simulation1\nduring the conceptualization stage using ROS 2 [62]. While ROS 2\nsimplified the complex robotic development with open-source soft-\nware tools and libraries, in the second simulation, we used Gazebo ’s\nrealistic 3D simulation platform [ 48] for testing robot models and\nalgorithms, thus providing a closer reproduction of the final sim-\nulated environment. Additionally, as virtual simulation may not\nsufficiently replicate real-world situations [ 35], the third evaluation\nrepresented a proof-of-concept conducted in a celestial-like envi-\nronment laboratory setting. Each test had an approximate duration\nof 30 minutes. The robotic platform utilized a visual-SLAM algo-\nrithm during the last assessment to create a map of the analogous\nlunar terrain facility, as illustrated in Figure 3. The facility in which\nwe trained the robots to create precise maps, an 80 m2rectangle\nfilled with basalt, was designed to emulate the surface of the Moon’s\nsouth pole, an area where researchers expect to find resources.\nAs we are currently in the prototype phase, our primary fo-\ncus is to assess the technical feasibility and performance of the\narchitecture. Our system’s architecture, illustrated in Figure 4, is de-\nsigned for efficient data storage, NFT management, and trading with\n3 stages: NFT creation, sale, and retrieval. We harnessed the capabil-\nities of Python ,𝐶++,Gazebo , and ROS 2 for development, ensuring\n1https://wiki.ros.org/turtlesimCoopetitive Infrastructures for Space Exploration SAC ’24, April 8–12, 2024, Avila, Spain\neffective communication and simulation. To bridge the communi-\ncation gap between robots (LEO2 and LEO3) and the blockchain,\nwe integrated the PureStake connector via a REST service2.\nAll communication flows, except for the smart contract, have\nbeen implemented, encompassing the autonomous creation, nego-\ntiation, coordination, selling, and verification of NFTs and map-\nping requirements. Despite not taking full advantage of the smart\ncontract automated behavior, the prototype already leverages a\nblockchain network for trading NFTs and certain data storage as-\npects, enabling real-time decision-making based on blockchain data.\nThe architecture uses IPFS for high-throughput content-addressed\nblock storage [ 13]. Entities can govern their mapping NFTs meta-\ndata and settings, from transferability up to their destruction, pro-\nmoting value to a market for data access and recognizing pioneer-\ning exploration. These NFTs encapsulate specific map data and\nmetadata, containing coordinates, resolution, sensors, mapping al-\ngorithms, and map price, fostering trade within the network and\nimproving ISRU’s information symmetry.\nOur evaluation assessed the system’s bandwidth usage and trans-\naction duration in simulated lunar network conditions. Despite\ntransactions needing to be fully optimized, the average bandwidth\nusage during uploads was less than 5 MB over 4 seconds, with a\nmedian of 4.7 MB. The standard deviation was 0.5 MB, indicating a\nrelatively consistent transaction performance. The time required\nfor uploading maps and conducting sales transactions was approx-\nimately 10 seconds, with a median of 9 seconds. The standard\ndeviation was 2 seconds, indicating that most transactions were\ncompleted within a reasonable time frame. These durations are\nrelatively short compared to the time robots typically take to nav-\nigate to the place and process cartographic data, which, at best,\ncan take several minutes to hours. As a result, the upload duration,\nplus mapping and reaching a consensus, is negligible compared to\nconsidering the inherent latencies of bidirectional communication\nbetween terrestrial stations and robotic units (which, for the Moon\ncase, takes a minimum of five seconds as the signal needs to pass\nthrough the different relays on Earth and satellites in space), let\nalone the possibility of other nations being open to start cooperat-\ning. The system displayed reasonable transaction duration usage\nand significantly reduced robot coordination delays compared to\nwaiting for human coordination.\n6 DISCUSSION\nOur evaluation demonstrates the technical feasibility of the pro-\nposed architecture for MRS in collaborative scientific space explo-\nration. The system successfully collects, coordinates, and maintains\nmapped area data via the blockchain metadata while evaluating\nthe mean bandwidth usage and transaction duration to validate\nthe feasibility in real-world testing conditions (see Figure 4). More-\nover, our architecture offers precise map data that can be leveraged\nfor various applications, including celestial terrain coverage with\nsustainable ISRU capabilities, thereby enhancing information sym-\nmetry and decision-making processes.\nAs for scalability, while the initial setup requires resources to\nbootstrap the constellation map through multiple interactions, the\n2https://www.purestake.com/system is designed to handle the growing data in DLT orIPFS with-\nout being overly strained. The blockchain only stores limited map\nmetadata, managing rights, properties, and descriptions. Neverthe-\nless, there may be challenges with multiple mapping data streams\nin the IPFS system in the long run.\n6.1 Limitations\nThis section outlines the limitations of our architecture, catego-\nrized into technical challenges, economic implications, and future\nresearch directions. Ensuring the reliability and integrity of shared\nmapping data and the effectiveness of smart contracts remains\na challenge. As such, future work should focus on measurement\nsystems, quality assurance metrics, and addressing the practical\nlimitations of smart contracts in decentralized systems [ 3]. For in-\nstance, validation mechanisms [ 50], reputation systems [ 16,73],\nand different consensus mechanisms [ 30,65] could be integrated to\nenhance mapped data reliability. Continuous monitoring and anom-\naly detection algorithms can also support detecting and mitigating\nsuspicious activities [ 70]. It is crucial to understand the practical\nimplications of public vs. private networks, permissionless vs. per-\nmissioned designs, and the specific consensus mechanisms that min-\nimize computational burden, specifically for resource-constrained\nspace MRS [5].\nAlthough our implementation aims to solve the dependable com-\nmunication limitation for DLT-based platforms to thrive, the au-\ntomated market-based approaches for MRS , such as blockchain\nauctioning systems, are still in the early stages. Auction systems are\nalso considered intricate and less effective in centralized situations\nand excessively complex in distributed situations [ 27]. Additionally,\nhierarchical decision-making and external third-party systems can\npose challenges in identifying trustworthy evaluators for smart\ncontract task execution [ 1,50]. Further research should investigate\nefficient communication protocols for blockchain, smart contracts,\nmarket mechanisms [ 27], and decentralized combinatorial optimiza-\ntion systems [10], which are crucial for space missions.\nThe economic implications of DLT in space MRS require fur-\nther investigation and support from real-world examples and case\nstudies. Narratives for blockchain technology are essential for pro-\nmoting its acceptance of new ways of governing outer space [ 31].\nReal-world examples and case studies exploring economic inter-\nactions facilitated by DLT in space exploration, such as resource\ntrading, service provisioning, and fair compensation mechanisms,\nwould enhance the credibility and depth of the discussion [ 9,31,42].\nFuture research should focus on real-world deployments and eco-\nnomic modeling to quantify the economic impact of our architec-\nture.\nAs Afanasyev et al . [3] point out, the field must transition from\ntheoretical discussions to practical implementations. Our work\naims to contribute to this transition by implementing prototypes\nin celestial-like conditions. We are addressing the limitations and\nchallenges to ensure the platform’s efficiency and economic viabil-\nity in space exploration. We intend to continue our research and\ndevelopment efforts to address these challenges, keeping in line\nwith the increasing interest from the scientific community, space\nagencies [9, 43], and industry [42].SAC ’24, April 8–12, 2024, Avila, Spain Chovet and Baima et al.\nLEO3 creates an NFT representing the map\nLEO3 maps ZoneALEO2 wants to buy the map\nZoneA\nLEO3 Assets\nZoneA\nLEO2 Assets\nLEO3 Assets\nLEO2 paid cryptocurrency, and in exchange now owns the NFT\nFigure 4: Prototype in the laboratory of simulated Moon environment.\n6.2 Research gap and open challenges\nHigh autonomy in limited resources: Coopetitive platforms\nare known to provide high knowledge-sharing capabilities [ 36],\nwhere sophisticated techniques, such as ontologies, semantic web,\nand linked data [ 55] would be an option, given that robots can\nlearn new abilities with minimal incremental rewarded data [ 8].\nHowever, it is challenging in environments with limited resources\nto deeply understand data and manage its heterogeneity while\nensuring system stability and safety.\nGlobal cooperation and open networks: A neutral coopet-\nitive platform’s economic aspect may also benefit companies by\nallowing their robots to perform low-priority tasks during idle time,\npotentially increasing equipment utilization and return on invest-\nment and improving global ISRU efficiency. Collaboration is crucial\nin reducing mission costs and achieving economies of scale [ 20].\nDespite early proposals to utilize planning knowledge to improve\nperformance and efficiency [ 57] and possibly learn incremental ac-\ntions [ 8], it is crucial to establish ways to exchange norms, maintain\nteam identity, and promote group member confidence [70].\nReliable communication mechanisms and automated DLT-\nbased markets: Hierarchical distributed decision-making struc-\ntures have their strengths and weaknesses for DLT-based auto-\nmated markets [ 72] but relying on external third parties still chal-\nlenges identifying trustworthy evaluator methods for DLT-based\napplication layers, which could be an organization or a digitally sup-\nplied system challenges to identify trustworthy evaluator methods\nforDLT-based application layers as it may reduce the trustworthi-\nness of the network.\nEconomic and legal aspects remain open: While our research\nprimarily focuses on the technical feasibility and implementation of\nblockchain technology in MRS for space exploration, we acknowl-\nedge the significance of economic and legal aspects in this domain.\nThe economic implications, including cost-benefit analysis, funding\nmodels, and the financial viability of such systems, remain crucial\nareas for future exploration. Likewise, legal considerations, encom-\npassing regulatory compliance, space law, and data governance,\npose complex challenges that are beyond the scope of this paper.7 CONCLUSIONS\nOur primary scientific contribution lies in the architecture and\nrequirement engineering for blockchain applications in MRS . The\nprototype, while a crucial element for demonstrating the feasibil-\nity of our approach, is secondary to the theoretical framework\nwe have adopted. Our architecture, developed through interdisci-\nplinary perspectives from information systems, engineering, and\neconomics, answered the RQby demonstrating the potential of\nDLT. Our platform’s flexibility and non-discriminatory aspect allow\nfor leveraging idle resources through a robot-as-a-service platform.\nThe prototype enables efficient mapping coordination via smart\ncontracts, ensures data integrity via NFTs, and facilitates revenue\ngeneration through market-based cryptocurrency and NFT trans-\nactions, allowing organizations to trade valuable assets in space\nmissions. Blockchain is employed not as an end in itself but as\na tool to facilitate secure and reliable information exchange and\ncoordination among robots, addressing key challenges in robotic\ncommunication and operation Our research provides a founda-\ntion for future advancements and implementations, following the\nprinciples and guidelines of the DSR approach.\nWhile our research presents promising results, several areas\nfor further exploration and improvement remain. The economic\nimplications of DLT in space MRS require more profound analy-\nsis, including identifying suitable use cases, evaluating economic\nincentives, and examining governance and legal considerations.\nAdditionally, while addressing the environmental impact of DLT\nin space missions, future research should evaluate the performance\nand efficiency of our approach, consider transaction throughput\nand latency, optimize energy efficiency, and explore scalability\nsolutions such as sharding and layer two protocols [ 40]. Future\nresearch could also develop decision support on when DLT is re-\nquired instead of digital signatures and bilateral communication\nto securely and efficiently implement concurrent access. However,\nthis dilemma again emphasizes that while a DLT may be desirable\nfrom an economic perspective, it can add substantial complexity to\ndesign and setup. Thus, it is crucial to identify suitable use cases\nforDLT in space MRS and understand service providers’ economic\nneeds and incentives.Coopetitive Infrastructures for Space Exploration SAC ’24, April 8–12, 2024, Avila, Spain\nACKNOWLEDGMENTS\nThis research was funded in part by the Luxembourg National\nResearch Fund (FNR) in the FiReSpARX Project, ref. 14783405,\nPABLO Project, ref. 16326754, and by PayPal, PEARL grant ref-\nerence 13342933/ Gilbert Fridgen. For the purpose of open access,\nand in fulfillment of the obligations arising from the grant agree-\nment, the author has applied a Creative Commons Attribution 4.0\nInternational (CC BY 4.0) license to any Author Accepted Manu-\nscript version arising from this submission.\nREFERENCES\n[1]Ryosuke Abe, Shigeya Suzuki, Kenji Saito, Hiroya Tanaka, Osamu Nakamura, and\nJun Murai. 2022. Fabchain: Managing Audit-Able 3D Print Job over Blockchain.\nInIEEE Int. Conf. Blockchain and Cryptocurrency . IEEE, Shanghai, China, 9. https:\n//doi.org/10.1109/ICBC54727.2022.9805519\n[2]U. S. P. Srinivas Aditya, Roshan Singh, Pranav Kumar Singh, and Anshuman Kalla.\n2021. A Survey on Blockchain in Robotics: Issues, Opportunities, Challenges and\nFuture Directions. Journal of Network and Computer Applications 196 (Dec. 2021),\n10. https://doi.org/10.1016/j.jnca.2021.103245\n[3]Ilya Afanasyev, Alexander Kolotov, Ruslan Rezin, Konstantin Danilov, Alexey\nKashevnik, and Vladimir Jotsov. 2019. Blockchain Solutions for Multi-Agent\nRobotic Systems: Related Work and Open Questions. In Proc. 24th Conf. Open\nInnov. Assoc. (FRUCT’24) . FRUCT Oy, Helsinki, Uusimaa, FIN, 551–555. https:\n//doi.org/10.5555/3338290.3338366\n[4]George A. Akerlof. 1970. The Market for \"Lemons\": Quality Uncertainty and\nthe Market Mechanism. The Quarterly Journal of Economics 84, 3 (Aug. 1970),\n488–500. https://doi.org/10.2307/1879431\n[5]Mohammad Alfraheed and Abdullah Al-Zaghameem. 2013. Exploration and\nCooperation Robotics on the Moon. Journal of Signal and Information Processing\n04, 03 (Jan. 2013), 253–258. https://doi.org/10.4236/jsip.2013.43033\n[6]Khalid Husain Ansari and Umesh Kulkarni. 2020. Implementation of Ethereum\nRequest for Comment (ERC20) Token. In Proceedings of the 3rd International\nConference on Advances in Science & Technology . Elsevier, Rochester, NY, 6. https:\n//doi.org/10.2139/ssrn.3561395\n[7]Philip Arm, Gabriel Waibel, Jan Preisig, Turcan Tuna, Ruyi Zhou, Valentin Bickel,\nGabriela Ligeza, Takahiro Miki, Florian Kehl, Hendrik Kolvenbach, and Marco\nHutter. 2023. Scientific Exploration of Challenging Planetary Analog Environ-\nments with a Team of Legged Robots. Science Robotics 8, 80 (July 2023), eade9548.\nhttps://doi.org/10.1126/scirobotics.ade9548\n[8]Renan Lima Baima and Esther Luna Colombini. 2021. Modeling Object’s\nAffordances via Reward Functions. In 2021 IEEE International Conference on\nSystems, Man, and Cybernetics (SMC) . IEEE, Melbourne, Australia, 2183–2190.\nhttps://doi.org/10.1109/smc52423.2021.9658915\n[9]Gianluigi Baldesi. 2017. Beyond Bitcoin: Leveraging the Blockchain for\nSpace 4.0. https://www.esa.int/About_Us/Digital_Agenda/Beyond_Bitcoin_\nLeveraging_the_Blockchain_for_Space_4.0\n[10] Sue A. Baldor, Carlos Quiroz, and Paul Wood. 2013. Applying a Cloud Computing\nApproach to Storage Architectures for Spacecraft. In 2013 IEEE Aerospace Confer-\nence. IEEE, Big Sky, Montana, 1–6. https://doi.org/10.1109/aero.2013.6497340\n[11] Roman Beck, Christoph Müller-Bloch, and John Leslie King. 2018. Governance\nin the Blockchain Economy: A Framework and Research Agenda. JAIS 19, 10\n(2018), 1020–1034. https://doi.org/10.17705/1jais.00518\n[12] E. K. Belyaeva, D. Yu. Ivanov, and S. V. Domnina. 2021. Contractual Arrangements\nBetween Providers and Consumers of Digital Technologies in Space Industry. In\nDigital Economy and the New Labor Market: Jobs, Competences and Innovative HR\nTechnologies (Lecture Notes in Networks and Systems) , Svetlana Igorevna Ashma-\nrina and Valentina Vyacheslavovna Mantulenko (Eds.). Springer International\nPublishing, Cham, 135–142. https://doi.org/10.1007/978-3-030-60926-9_19\n[13] Juan Benet. 2014. IPFS - Content Addressed, Versioned, P2P File System. https:\n//doi.org/10.48550/arXiv.1407.3561 arXiv:1407.3561\n[14] Mariel Borowitz. 2019. Strategic Implications of the Proliferation of Space Sit-\nuational Awareness Technology and Information: Lessons Learned from the\nRemote Sensing Sector. Space Policy 47 (Feb. 2019), 18–27. https://doi.org/10.\n1016/j.spacepol.2018.05.002\n[15] Bert-Jan Butijn, Damian A. Tamburri, and Willem-Jan van den Heuvel. 2020.\nBlockchains: A Systematic Multivocal Literature Review. Comput. Surveys 53, 3,\nArticle 61 (July 2020), 37 pages. https://doi.org/10.1145/3369052\n[16] Davide Calvaresi, Alevtina Dubovitskaya, Diego Retaggi, Aldo F. Dragoni, and\nMichael Schumacher. 2018. Trusted Registration, Negotiation, and Service Evalu-\nation in Multi-Agent Systems throughout the Blockchain Technology. In Int. Conf.\non Web Int. IEEE, Santiago, Chile, 56–63. https://doi.org/10.1109/WI.2018.0-107\n[17] Bruce G. Cameron, Theodore Seher, and Edward F. Crawley. 2011. Goals for\nSpace Exploration Based on Stakeholder Value Network Considerations. ActaAstronautica 68 (2011), 2088–2097. https://doi.org/10.1016/j.actaastro.2010.11.003\n[18] L.E. Cánez, K.W. Platts, and D.R. Probert. 2000. Developing a Framework for Make-\nor-buy Decisions. International Journal of Operations & Production Management\n20, 11 (Jan. 2000), 1313–1330. https://doi.org/10.1108/01443570010348271\n[19] Christian Catalini and Joshua S. Gans. 2020. Some Simple Economics of the\nBlockchain. cacm 63 (2020), 80–90. https://doi.org/10.1145/3359552\n[20] Denise Chow, Paul Hennessy, and Reed Hoffmann. April 8, 2022, 5:52 PM\nCEST. How Falling Launch Costs Are Fueling the Thriving Space In-\ndustry. https://www.nbcnews.com/science/space/space-launch-costs-growing-\nbusiness-industry-rcna23488.\n[21] Raiane Coelho, Regina Braga, José Maria David, Mário Dantas, Victor Stroele,\nand Fernanda Campos. 2021. Integrating Blockchain for Data Sharing and Collab-\noration Support in Scientific Ecosystem Platform. In Proc. of the Hawaii Int.Conf.\non Sys. Sci. IEEE, Hawaii, 10. https://doi.org/10.24251/HICSS.2021.031\n[22] Lesley Conn. 2021. Global Space Economy Nears $447B. https://www.\nthespacereport.org/uncategorized/global-space-economy-nears-447b/.\n[23] Consensys Space. 2018. Open Source Space. https://www.consensys.space/\n[24] Ian A. Crawford. 2015. Lunar Resources: A Review. Prog. Phys. Geogr. Earth\nEnviron. 39, 2 (April 2015), 137–167. https://doi.org/10.1177/0309133314567585\n[25] Société Européenne des Satellites. 2021. SES Satellite Telecommunications Net-\nwork Provider. https://www.ses.com/\n[26] Alexander Dettmann, Thomas Voegele, Jorge Ocón, Iulia Dragomir, Shashank\nGovindaraj, Matteo de Benedetti, Valérie Ciarletti, Rafik Hassen-Khodja, Thierry\nGerma, Raphael Viards, Gonzalo J. Paz-Delgado, and Laura M. Mantoani. 2022.\nCOROB-X: a cooperative robot team for the exploration of lunar skylights. In\nASTRA 2022 16th Symposium on Advanced Space Technologies in Robotics and\nAutomation . ESA-ESTEC, Noordwijk, Netherlands, 9. https://hal-insu.archives-\nouvertes.fr/insu-03751549\n[27] M.B. Dias, R. Zlot, N. Kalra, and A. Stentz. 2006. Market-Based Multirobot\nCoordination: A Survey and Analysis. Proc. IEEE 94, 7 (July 2006), 1257–1270.\nhttps://doi.org/10.1109/jproc.2006.876939\n[28] Kathleen M. Eisenhardt and Melissa E. Graebner. 2007. Theory Building from\nCases: Opportunities and Challenges. AMJ 50, 1 (Feb. 2007), 25–32. https:\n//doi.org/10.5465/amj.2007.24160888\n[29] European Space Agency. 2019. Blockchain and Earth Observation: a white\npaper. https://eo4society.esa.int/2019/04/09/blockchain-and-earth-observation-\na-white-paper/ Section: Publications.\n[30] Eduardo Castelló Ferrer. 2019. The Blockchain: A New Framework for Robotic\nSwarm Systems. In Proc. Future Technol. Conf. (Advances in Intelligent Systems\nand Computing, Vol. 881) . Springer International Publishing, Cham, 1037–1058.\nhttps://doi.org/10.1007/978-3-030-02683-7_77\n[31] Primavera De Filippi and Andrea Leiter. 2021. Blockchain in Outer Space. Amer.\nJour. of Int. Law 115 (2021), 413–418. https://doi.org/10.1017/aju.2021.63\n[32] Yang Gao and Steve Chien. 2017. Review on Space Robotics: Toward Top-Level\nScience through Space Exploration. Science Robotics 2, 7 (June 2017), eaan5074.\nhttps://doi.org/10.1126/scirobotics.aan5074\n[33] Jeremi Garcet, Diego Urbina, Simon Sheridan, Janos Biswas, Anthony Evagora,\nLuiz Richter, Guillaume Fau, Hemanth Kumar, Daniel Fodorcan, Thibaud Chupin,\nKasrten Kullack, Craig Pitcher, Neil Murray, Philipp Reiss, Mattia Reganaz,\nShashank Govindaraj, Richard Aked, and Joseph Salini. 2019. Lunar Volatiles\nMobile Instrumentation (LUVMI) Project Results. In IAC-19 SYMPOSIUMS (A3\nIAF SPACE EXPLORATION SYMPOSIUM, Vol. A3) . IAC, Washington, DC, USA, 9.\nhttps://doi.org/10.3030/727220\n[34] Sergio Garcıa, Claudio Menghi, Patrizio Pelliccione, Thorsten Berger, and Re-\nbekka Wohlrab. 2018. An Architecture for Decentralized, Collaborative, and\nAutonomous Robots. In 2018 IEEE International Conference on Software Architec-\nture (ICSA) . IEEE, Seattle, WA, 75–7509. https://doi.org/10.1109/ICSA.2018.00017\n[35] Sergio García, Daniel Strüber, Davide Brugali, Thorsten Berger, and Patrizio\nPelliccione. 2020. Robotics Software Engineering: A Perspective from the Service\nRobotics Domain. In Proc. of the 28th ACM Joint Meeting on Eur. Soft. Eng. Conf.\nand Symposium on the Foundations of Software Engineering . ACM, Virtual Event\nUSA, 593–604. https://doi.org/10.1145/3368089.3409743\n[36] Shahla Ghobadi and John D’Ambra. 2011. Coopetitive Knowledge Sharing: An\nAnalytical Review of Literature. Elec. Jour. of Know. Manag. 9, 4 (2011), 307–317.\n[37] Jason Goo, Ojas Bora, Priyansh Manne, Kazem Mullah, Kayden Tilden, Jeremy\nHartley, Ayame Kunieda, Savannah Laver, Stephanie Lee, Laquan Shuai, and\nIsaiah Li Yun Tan. 2019. Blockchain Lunar Registry. https://diana.io.\n[38] Tobias Guggenberger, Johannes Sedlmeir, Gilbert Fridgen, and André Luckow.\n2022. An In-depth Investigation of the Performance Characteristics of Hy-\nperledger Fabric. Computers and Industrial Engineering 173 (2022), 108716.\nhttps://doi.org/10.1016/j.cie.2022.108716\n[39] Eduard Hartwich, Philipp Ollig, Gilbert Fridgen, and Alexander Rieger. 2023.\nProbably Something: A Multi-Layer Taxonomy of Non-Fungible Tokens. Internet\nResearch ahead-of-print, ahead-of-print (Jan. 2023), 26. https://doi.org/10.1108/\nINTR-08-2022-0666 arXiv:2209.05456 [cs]\n[40] Eduard Hartwich, Alexander Rieger, Johannes Sedlmeir, Dominik Jurek, and\nGilbert Fridgen. 2023. Machine Economies. Electronic Markets 33, 1 (July 2023),\n36. https://doi.org/10.1007/s12525-023-00649-0SAC ’24, April 8–12, 2024, Avila, Spain Chovet and Baima et al.\n[41] Alexandra Höß, Vincent Schlatt, Alexander Rieger, and Gilbert Fridgen. 2021.\nThe Blockchain Effect: From Inter-Ecosystem to Intra-Ecosystem Competition.\nInECIS 2021 Proceedings (Research Papers) . AIS, Marrakech, Morocco, 16. https:\n//aisel.aisnet.org/ecis2021_rp/36\n[42] Hussein. Ibrahim, Marwa A. Shouman, Nawal A. El-Fishawy, and Ayman. Ahmed.\n2021. Literature Review of Blockchain Technology in Space Industry: Challenges\nand Applications. In Int. Conf. Electron. Eng. IEEE, Menouf, Egypt, 1–8. https:\n//doi.org/10.1109/ICEEM52022.2021.9480642\n[43] Brian Israel. 2020. Space Governance 3.0. Georgia Journ. of Int. & Comp. Law 48,\n3 (2020), 16. https://digitalcommons.law.uga.edu/gjicl/vol48/iss3/7\n[44] Niclas Kannengiesser, Sebastian Lins, Christian Sander, Klaus Winter, Hellmuth\nFrey, and Ali Sunyaev. 2022. Challenges and Common Solutions in Smart Contract\nDevelopment. IEEE Transactions on Software Engineering 48, 11 (Dec. 2022), 4291–\n4318. https://doi.org/10.1109/TSE.2021.3116808\n[45] Aleksandr Kapitonov, Ivan Berman, Vitaly Bulatov, Sergey Lonshakov, and Alek-\nsandr Krupenkin. 2018. Robonomics Based on Blockchain as a Principle of\nCreating Smart Factories. In Proc. Int. Conf. on Intern. of Thin.: Sys., Manag. and\nSec.IEEE, Valencia, Spain, 78–85. https://doi.org/10.1109/IoTSMS.2018.8554864\n[46] Georgios Karalekas, Stavros Vologiannidis, and John Kalomiros. 2020. EUROPA: A\nCase Study for Teaching Sensors, Data Acquisition and Robotics via a ROS-based\nEducational Robot. Sensors 20, 9 (2020), 2469. https://doi.org/10.3390/s20092469\n[47] Ameer T. Khan, X. Cao, Shuai Li, and Zoran Milosevic. 2019. Blockchain Technol-\nogy with Applications to Distributed Control and Cooperative Robotics: A Survey.\nInt. Jour. of Rob. and Cont. 2 (2019), 36. https://doi.org/10.5430/ijrc.v2n1p36\n[48] Nathan Koenig and Andrew Howard. 2004. Design and Use Paradigms for Gazebo,\nan Open-Source Multi-Robot Simulator. In IEEE/RSJ Int. Conf. on Inte. Rob. and\nSys.IEEE, Sendai, Japan, 2149–2154. https://doi.org/10.1109/IROS.2004.1389727\n[49] Amanda M. Leon. 2018. Mining for Meaning: An Examination of the Legality of\nProperty Rights in Space Resources. Virginia Law Review 104 (2018), 497–546.\n[50] Shasha Li, Xiaodong Bai, and Songjie Wei. 2022. Blockchain-Based Crowd-\nsourcing Framework with Distributed Task Assignment and Solution Verifi-\ncation. Hindawi Security and Communication Networks 2022 (Jan. 2022), 16.\nhttps://doi.org/10.1155/2022/9464308\n[51] Mathias Link, David Parker, Massimo Sabbatini, and Franziska . 2021. ESA-\nESRIC Space Resources Challenge - Prospecting Technologies - Call. https:\n//www.spaceresourceschallenge.esa.int\n[52] Ralph D. Lorenz. 2020. How Far Is Far Enough? Requirements Derivation for\nPlanetary Mobility Systems. Advances in Space Research 65, 5 (March 2020),\n1383–1401. https://doi.org/10.1016/j.asr.2019.12.011\n[53] Erin C McKiernan, Philip E Bourne, C Titus Brown, Stuart Buck, Amye Kenall,\nJennifer Lin, Damon McDougall, Brian A Nosek, Karthik Ram, Courtney K Soder-\nberg, Jeffrey R Spies, Kaitlin Thaney, Andrew Updegrove, Kara H Woo, and Tal\nYarkoni. 2016. How Open Science Helps Researchers Succeed. eLife 5 (July 2016),\n1–19. https://doi.org/10.7554/elife.16800\n[54] Thomas Heinz Meitinger. 2017. Smart contracts. Informatik Spektrum 40, 4 (Aug.\n2017), 371–375. https://doi.org/10.1007/s00287-017-1045-2\n[55] Abdulkadir Memduhoglu and Melih Basaraner. 2018. Possible Contributions\nof Spatial Semantic Methods and Technologies to Multi-Representation Spatial\nDatabase Paradigm. International Journal of Engineering and Geosciences 3, 3\n(Oct. 2018), 108–118. https://doi.org/10.26833/ijeg.413473\n[56] Yalda Mousavinia, Suzi Bianco, Radek Zasiadczuk, Josh Perry, Kevin Siegler, Marc\nCohen, Sean Marquez, A. Lunn, Otto Garcia, P. Tasca, Brent Sherwood, and J.\nSimmons. 2012. Space Decentral: A Decentralized Autonomous Space Agency.\n[57] Bharath Muppasani, Vishal Pallagani, Biplav Srivastava, Raghava Mutharaju,\nMichael N. Huhns, and Vignesh Narayanan. 2023. A Planning Ontology to\nRepresent and Exploit Planning Knowledge for Performance Efficiency. https:\n//doi.org/10.48550/arXiv.2307.13549 arXiv:2307.13549 [cs]\n[58] NASA. 2016. Space Technology Roadmaps and Priorities Revisited . Technical\nReport. The National Academies Press. https://doi.org/10.17226/23582\n[59] Alina Orlova, Roberto Nogueira, and Paula Chimenti. 2020. The Present and\nFuture of the Space Sector: A Business Ecosystem Approach. Space Policy 52\n(May 2020), 101374. https://doi.org/10.1016/j.spacepol.2020.101374\n[60] Lynne E. Parker. 2008. Multiple Mobile Robot Systems. In Springer Handbook of\nRobotics , Bruno Siciliano and Oussama Khatib (Eds.). Springer, Berlin, Heidelberg,\n921–941. https://doi.org/10.1007/978-3-540-30301-5_41\n[61] Ken Peffers, Tuure Tuunanen, Marcus A. Rothenberger, and Samir Chatterjee.\n2007. A Design Science Research Methodology for Information Systems Research.\nJournal of Management Information Systems 24, 3 (Dec. 2007), 45–77. https:\n//doi.org/10.2753/mis0742-1222240302\n[62] Morgan Quigley, Ken Conley, Brian Gerkey, Josh Faust, Tully Foote, Jeremy Leibs,\nRob Wheeler, and Andrew Ng. 2009. ROS: An Open-Source Robot Operating\nSystem. In ICRA Workshop on Open Source Software , Vol. 3. IEEE, Kobe, Japan,\n5. https://www.semanticscholar.org/paper/ROS%3A-an-open-source-Robot-\nOperating-System-Quigley/d45eaee8b2e047306329e5dbfc954e6dd318ca1e\n[63] Félix Quinton, Christophe Grand, and Charles Lesire. 2023. Market Approaches\nto the Multi-Robot Task Allocation Problem: A Survey. Journal of Intelligent &\nRobotic Systems 107, 2 (Feb. 2023), 29. https://doi.org/10.1007/S10846-022-01803-0[64] Yara Rizk, M. Awad, and E. Tunstel. 2019. Cooperative Heterogeneous Multi-\nRobot Systems: A Survey. csur 52 (2019), 1–31. https://doi.org/10.1145/3303848\n[65] Yara Rizk, Mariette Awad, and Edward W. Tunstel. 2018. Decision Making in\nMultiagent Systems: A Survey. IEEE Transactions on Cognitive and Developmental\nSystems 10, 3 (Sept. 2018), 514–529. https://doi.org/10.1109/tcds.2018.2840971\n[66] Thomas M. Roehr, Florian Cordes, and Frank Kirchner. 2014/01//Jan/Feb2014.\nReconfigurable Integrated Multirobot Exploration System (RIMRES): Heteroge-\nneous Modular Reconfigurable Robots for Space Exploration. Journal of Field\nRobotics 31, 1 (2014/01//Jan/Feb2014), 3–34. https://doi.org/10.1002/rob.21477\n[67] Martin J. Schuster, Marcus G. Müller, Sebastian G. Brunner, Hannah Lehner, Peter\nLehner, et al .2020. The ARCHES Space-Analogue Demonstration Mission: To-\nwards Heterogeneous Teams of Autonomous Robots for Collaborative Scientific\nSampling in Planetary Exploration. IEEE Robotics and Automation Letters 5, 4\n(2020), 5315–5322. https://doi.org/10.1109/lra.2020.3007468\n[68] Johannes Sedlmeir, Jonathan Lautenschlager, Gilbert Fridgen, and Nils Urbach.\n2022. The Transparency Challenge of Blockchain in Organizations. Electronic\nMarkets 32 (2022), 1779—1794. https://doi.org/10.1007/s12525-022-00536-0\n[69] Marshall Smith, Douglas Craig, Nicole Herrmann, Erin Mahoney, Jonathan Krezel,\nNate McIntyre, and Kandyce Goodliff. 2020. The Artemis Program: An Overview\nof NASA’s Activities to Return Humans to the Moon. In IEEE Aerosp. Conf. IEEE,\nBig Sky, MT, USA, 10. https://doi.org/10.1109/aero47225.2020.9172323\n[70] Volker Strobel, Eduardo Castelló Ferrer, and Marco Dorigo. 2018. Managing\nByzantine Robots via Blockchain Technology in a Swarm Robotics Collective\nDecision Making Scenario. In Proc. 17th Int. Conf. Auton. Agents MultiAgent Syst.\n(AAMAS ’18) . International Foundation for Autonomous Agents and Multiagent\nSystems, Stockholm, Sweden, 541–549. https://doi.org/10.5555/3237383.3237464\n[71] Dave Van Der Meer, Loick Chovet, Gabriel Garcia, Abhishek Bera, and Miguel An-\ngel Olivares Mendez. 2023. REALMS 2 -RESILIENT EXPLORATION AND LUNAR\nMAPPING SYSTEM 2. In Proceedings of ASTRA 2023 . ASTRA, Leyden, Nether-\nlands, 1 – 8. https://doi.org/10.3389/frobt.2023.1127496\n[72] Zhi Yan, Nicolas Jouandeau, and Arab Ali Cherif. 2013. A Survey and Analysis\nof Multi-Robot Coordination. International Journal of Advanced Robotic Systems\n10, 12 (Dec. 2013), 399. https://doi.org/10.5772/57313\n[73] Han Yu, Zhiqi Shen, Cyril Leung, Chunyan Miao, and Victor R. Lesser. 2013. A\nSurvey of Multi-Agent Trust Management Systems. IEEE Access 1 (2013), 35–50.\nhttps://doi.org/10.1109/access.2013.2259892\n[74] Liudmila Zavolokina, Florian Spychiger, Claudio Tessone, and Gerhard Schwabe.\n2018. Incentivizing Data Quality in Blockchains for Inter-Organizational Net-\nworks - Learning from the Digital Car Dossier. In Proc. Int. Conf. Inf. Syst. AIS,\nSan Francisco, USA, 18. https://doi.org/10.5167/uzh-157909\n[75] Tiehui Zhang, Hengyu Li, Jun Liu, Shaorong Xie, and Jun Luo. 2022. Group-\nSymmetric Consensus for Nonholonomic Mobile Multirobot Systems in Coope-\ntition Networks. Proceedings of the Institution of Mechanical Engineers, Part\nC: Journal of Mechanical Engineering Science 236, 7 (April 2022), 3743–3754.\nhttps://doi.org/10.1177/09544062211045482\n[76] Zee Zheng, Cliff Beek, Jeff Garzik, and Nick Trudgen. 2018. SpaceChain -\nCommunity-Based Space Platform. https://spacechain.com/\n[77] Natasa Zivic, Christoph Ruland, and Jochen Sassmannshausen. 2019. Distributed\nLedger Technologies for M2M Communications. In Proc. Int. Conf. Inf. Netw. IEEE,\nKuala Lumpur, Malaysia, 301–306. https://doi.org/10.1109/icoin.2019.8718115\n[78] R. Zlot, A. Stentz, M.B. Dias, and S. Thayer. 2002. Multi-Robot Exploration\nControlled by a Market Economy. In Proc. IEEE Int. Conf. Robot. Autom. , Vol. 3.\nIEEE, Washington, USA, 3016–3023. https://doi.org/10.1109/robot.2002.1013690\nReceived 29 September 2023; revised 10 December 2023; accepted 05 January\n2024"    },    {        "title": "2402.06036v1.Designing_Trustful_Cooperation_Ecosystems_is_Key_to_the_New_Space_Exploration_Era.pdf",        "content": "Designing Trustful Cooperation Ecosystems is Key to the New\nSpace Exploration Era\nRenan Lima Baima\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\nrenan.limabaima@uni.luChovet Loïck\nUniversity of Luxembourg\nSpaceR Research Group, SnT\nKirchberg, Luxembourg\nloick.chovet@uni.luJohannes Sedlmeir\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\njohannes.sedlmeir@uni.lu\nMiguel Angel Olivares Mendez\nUniversity of Luxembourg\nSpaceR Research Group, SnT\nKirchberg, Luxembourg\nmiguel.olivaresmendez@uni.luGilbert Fridgen\nUniversity of Luxembourg\nFINATRAX Research Group, SnT\nKirchberg, Luxembourg\ngilbert.fridgen@uni.lu\nABSTRACT\nIn the emerging space economy, autonomous robotic missions with\nspecialized goals such as mapping and mining are gaining traction,\nwith agencies and enterprises increasingly investing resources. Mul-\ntirobot systems ( MRS ) research has provided many approaches to es-\ntablish control and communication layers to facilitate collaboration\nfrom a technical perspective, such as granting more autonomy to\nheterogeneous robotic groups through auction-based interactions\nin mesh networks. However, stakeholders’ competing economic\ninterests often prevent them from cooperating within a proprietary\necosystem. Related work suggests that distributed ledger technol-\nogy ( DLT) might serve as a mechanism for enterprises to coordinate\nworkflows and trade services to explore space resources through\na transparent, reliable, non-proprietary digital platform. We chal-\nlenge this perspective by pointing to the core technical weaknesses\nof blockchains, in particular, increased energy consumption, low\nthroughput, and full transparency through redundancy. Our objec-\ntive is to advance the discussion in a direction where the benefits of\nDLT from an economic perspective are weighted against the draw-\nbacks from a technical perspective. We finally present a possible\nDLT-driven heterogeneous MRS for map exploration to study the\nopportunities for economic collaboration and competitiveness.\nCCS CONCEPTS\n•Computing methodologies →Distributed computing method-\nologies ;•Applied computing →Aerospace ; Astronomy; Informa-\ntion integration and interoperability ; Cross-organizational business\nprocesses; •Computer systems organization →Peer-to-peer\narchitectures ;Self-organizing autonomic computing ;Robotic au-\ntonomy ;External interfaces for robotics .\nPermission to make digital or hard copies of part or all of this work for personal or\nclassroom use is granted without fee provided that copies are not made or distributed\nfor profit or commercial advantage and that copies bear this notice and the full citation\non the first page. Copyrights for third-party components of this work must be honored.\nFor all other uses, contact the owner/author(s).\nICSE-NIER’24, April 14–20, 2024, Lisbon, Portugal\n©2024 Copyright held by the owner/author(s).\nACM ISBN 979-8-4007-0500-7/24/04.\nhttps://doi.org/10.1145/3639476.3639760KEYWORDS\nAutonomous Agent, Blockchain, Coordination, Coopetition, Multi-\nAgent Systems, Multi-Robot Systems, Space Economy\nACM Reference Format:\nRenan Lima Baima, Chovet Loïck, Johannes Sedlmeir, Miguel Angel Oli-\nvares Mendez, and Gilbert Fridgen. 2024. Designing Trustful Cooperation\nEcosystems is Key to the New Space Exploration Era. In New Ideas and\nEmerging Results (ICSE-NIER’24), April 14–20, 2024, Lisbon, Portugal. ACM,\nNew York, NY, USA, 5 pages. https://doi.org/10.1145/3639476.3639760\n1 INTRODUCTION\nThe development of the new space economy was bootstrapped\nthanks to policies and agreements by governments and interna-\ntional organizations [ 3]. The industry’s transition from a centralized\nmodel to a decentralized free market system [ 38] has increased the\nnumber of companies and startups in the new space economy, with\n80% of the revenue generated in the last 16 years coming from pri-\nvate investments [ 9]. Major space agencies, including NASA [ 37],\nhave outlined goals for in situ resource utilization ( ISRU ), such\nas destination reconnaissance, mapping, and resource acquisition.\nThese plans aim to establish a long-term human presence on celes-\ntial bodies (e.g., Moon and Mars) through robotic lunar missions\nthat utilize ISRU technology [12] to perform tasks from mining to\nmanufacturing solar panels.\nMRS are groups of robots that collaborate to complete specific\ntasks or support each other based on individual capabilities. They\ncan comprise both homogeneous and heterogeneous groups of\nrobots and find applications in diverse fields such as precision\nfarming, search and rescue, and space exploration [ 41]. The main\nadvantage of using MRS is that a team of geographically distributed\nrobots can outperform single robots [ 14]. However, its effectiveness\nhinges on robust, data-integrated coordination mechanisms that fa-\ncilitate the exchange of norms, maintain team identity, and promote\ngroup confidence [ 46]. To avoid the issue of information asymmetry\nin coordination mechanisms and the resulting “market for lemons”\ndilemma [ 2], it is crucial to minimize the knowledge gap between\nsellers and buyers. While financial incentives and sanctions can\nencourage cooperation and aid in meeting legal and regulatory\nrequirements, managing resources and coordinating interactions\namong multiple organizations and countries can be challenging.arXiv:2402.06036v1  [cs.RO]  8 Feb 2024ICSE-NIER’24, April 14–20, 2024, Lisbon, Portugal Baima et al.\nIncoopetitive MRS [21], various operators’ economic incentives\noften compete, making it difficult to achieve common objectives\nand ensure participation. With over 60 countries involved in space\nactivities [ 7], many actors may be reluctant to join a centralized\nplatform due to concerns about monopolistic or oligopolistic own-\nership, conflicts of interest, and treaty violations [29].\nAs organizations increasingly collaborate to achieve shared ob-\njectives [ 24], deploying cross-border and cross-organization MRS\narguably becomes increasingly critical for future missions. However,\nthe efficiency of ISRU in such heterogeneous MRS missions depends\non effective communication and coordination among robots [ 33],\nsuch as the location and availability of critical resources, e.g., wa-\nter and iron. The challenge of aligning the economic interests of\ncompeting stakeholders in a shared, non-proprietary digital space\nis expected to shape the new era of space exploration [ 38].DLT\nhas been proposed to facilitate collaboration among these entities,\nenabling autonomous, decentralized, peer-to-peer decision-making\ninMRS for space missions [ 13,26]. In this regard, DLT has shown\npromise in previous research [ 13,28]. In the context of space mis-\nsions, it could allow robots to autonomously negotiate resource\nusage, such as shelters, without a central authority. However, it\nis essential to note that DLT is not without technical limitations.\nAlthough it offers similar automation capabilities to centralized plat-\nforms through smart contracts, it is generally less efficient due to the\ninherent replication of transaction processing and storage [ 8,44].\nConsequently, deploying DLT inMRS and space exploration mis-\nsions is particularly challenging [ 47], given the limited resources\nin such a remote environment. Additionally, the inherent trans-\nparency of blockchains complicates data access management, as\ninformation is either visible to all participants or none [45].\nThis New Ideas and Emerging Results paper intentionally steers\nfrom presenting an exhaustive list of applications and challenges\nto showcase practical application development. It emphasizes ex-\nploring new ideas and the suitable needs of using DLT for non-\nproprietary market-based coordination of MRS s in space. Our ap-\nproach is not about pioneering a specific, feasible application in\nspace exploration and multirobotic coordination. We revisited the\nmarket-based MRS space mapping case presented by Dias et al .\n[16], which we have already begun implementing to evaluate the\nnext steps. Our primary contribution is investigating the economic\nbenefits and assessing potential solutions’ technical challenges in\nspace exploration requirements [ 19,32,50]. By leveraging DLT\nfor distributed open coordination, we aim to address the global\ncost-efficiency problem in market-based MRS coordination [ 42],\nadvocating for creating a trustworthy cross-organizational plat-\nform. Section 2 introduces the foundations of DLT, including smart\ncontracts and tokens, and reviews existing works on DLT inMRS\nand space. Section 3 details the technical challenges of using DLT\nin space MRS . Finally, Section 4 presents a use case for distributed\nauction-based mapping exploration, focusing on the ESA-ESRIC\nSpace Resource Challenge [30]. We conclude in Section 5.\n2 RELATED WORK\nWe draw upon systematic literature reviews on market-based ap-\nproaches in MRS [16,42], research on space MRS [18], and the\napplication of DLT in robotics and the space industry [1, 26].DLT gained prominence with its first application, the cryptocur-\nrency Bitcoin [ 36]. As a subset of DLT, blockchain technology is\ncharacterized by the replicated synchronized transaction process-\ning and a decentralized consensus algorithm with different levels of\nparticipation, permissionless vs. permissioned, and access, public\nvs. private [ 8].DLT offers novel opportunities for digital interaction\nin a non-proprietary digital infrastructure, such as creating trust\nwithin consortiums and enabling new trading streams via Fungible\nand Non-Fungible Tokens (NFTs). NFTs can represent, among many\nother possibilities, ownership of an non-interchangeable asset, such\nas collectibles or art pieces [ 25]. Besides the simple transfer of own-\nership, DLT also allows implementing programming logic through\nuploading code known as smart contracts and invoking this code’s\nexposed methods through transactions. In essence, smart contracts\nallow DLT to provide the same functionality as a centralized plat-\nform, with two exceptions: first, non-deterministic methods are not\nsupported, and second, the smart contract code, as well as all its\ninputs, intermediary results, and outputs, when triggered through\na transaction are available to all DLT nodes owing to the replicated\nexecution and storage of transactions [27].\nMarket-based approaches have been suggested for various ap-\nplications in MRS [42] and, specifically, space mapping [ 16]. These\napproaches often leverage auction systems as suitable tools to\nefficiently decide which resources to use in a competitive space\nenvironment [ 50]. Previous research suggests that robots can of-\nfer dependable services to each other by incorporating DLT [31],\nthereby opening up possibilities for space exploration, such as sell-\ning maps, facilitating decentralized decision-making, and enabling\nautonomous behavior. Integrating DLT is also said to leverage agent\ncooperation and communication aimed at detection, learning, and\nautonomous penalization [ 40], improving heterogeneous robots’\ncapabilities allowing them to perform complex tasks collaboratively.\nApplications of DLT in space have also been suggested for record-\ning orbital positions, mining licenses, and managing space traf-\nfic [10]. Another suggested application of DLT in space is to sup-\nport the efficient and effective coordination of MRS missions, such\nas in-orbit operations [ 11], satellite formations, surface, and plan-\netary exploration. By enabling robots to communicate and make\ntransactions and decisions autonomously, DLT is said to support\nthe independent operation of individual robots and enable them\nto adapt to dynamic environments. Examples of single-entity de-\nployments of DLT that consider economic transactions include the\nAIRA [ 31] project, which introduces the concept of “Robonomics,”\nwhere heterogeneous robots and humans can contract and offer\nservices and receive payment through smart contracts.\nDespite advancements, earlier suggestions have focused on cen-\ntralized single-entity settings. As such, they are not considering the\nmain reason for using DLT in a multi-participatory and competitive\nspace environment and the novel challenges that may arise from it.\nIn other words, all the platform functionalities that related work\nrelies on could also be provided as a service through a dedicated\nprovider that creates the needed digital infrastructure for one or\nmultiple platforms in space [ 38] or uses a decentralized peer-to-\npeer communication network to create such a platform. Moreover,\nexisting works still need to adequately address the need for flexible\ninformation representation and autonomous robot task learning in\ndynamic environments [ 5]. A better understanding of the economicDesigning Trustful Cooperation Ecosystems is Key to the New Space Exploration Era ICSE-NIER’24, April 14–20, 2024, Lisbon, Portugal\nopportunities and technical challenges is indispensable to assess\nand implement DLT in this context.\n3 RESEARCH GAP AND OPEN CHALLENGES\nOur discussion is grounded in existing research that tackles the\nchallenges of DLT from various angles, such as resource consump-\ntion, performance, and transparency [ 44,45]. While Aditya et al .\n[1]have explored the general challenges that DLT faces in robotics,\nwe focus on five specific open problems particularly relevant to\nDLT in space [26] and market-based MRS [16, 42].\nThe first challenge is achieving high levels of autonomy in\nresource-limited environments, a recurring issue in space explo-\nration [ 32,47]. While ontologies and semantic web have been pro-\nposed as a solution [ 34], they often face difficulties dealing with\ndata heterogeneity and system stability in resource-limited settings.\nFurthermore, establishing a fault-proof infrastructure and a trust-\nless environment is crucial for autonomous coordination [ 38,42].\nGiven the constraints on storage and computational capacity in\nspace robots due to the need for temperature and radiation harden-\ning [18], thorough modeling and testing of DLT inMRS are essential.\nThis includes evaluating public vs. private networks, permissionless\nvs. permissioned topologies, and possible ISRU -specific consensus\nmechanisms, such as consensus-based auction and bundle algo-\nrithms [ 42]. The most advanced existing solutions for heterogenous\nMRS , such as [ 43] and [ 15], assume a fully cooperative system.\nEvaluating a coopetitive system remains an open problem.\nThe second challenge involves the universal cost/reward func-\ntion for MRS in market-based coordination [ 42]. Despite various\nproposals to improve bid valuation, such as distance-based metrics\nor performance indexes, the computational complexity of com-\nbinatorial auctions still needs to improve [ 16]. Consequently, re-\nsearchers have explored alternatives, such as clustering from single-\nitem auctions to consensus-based bundle algorithms. However, cur-\nrent solutions often rely on proprietary frameworks, needing more\ntransparency and trustless requirements for a digital platform in the\nemerging space economy [ 38]. To optimize ISRU global efficiency,\ncompanies must consider outsourcing mapping data to other MRS\ncompanies willing to sell instead of individually mapping it.\nThirdly, public narratives about blockchain technology are vital\nfor its acceptance in governing outer space [ 13], and misconcep-\ntions about high energy consumption have been a barrier [ 44]. The\nreplicated processing of transactions storage using blockchain still\nincreases the total resource consumption in terms of bandwidth,\ncomputing power, and electricity by a factor of 𝑁in a blockchain\nnetwork of 𝑁nodes compared to a centralized platform, without\ntaking necessary backups into account [ 44]. Nevertheless, design\ndecisions such as not using proof of work as a consensus mecha-\nnism and approaches like sharding, roll-ups, and off-chain payment\nchannels can somewhat mitigate these inefficiencies [ 25,31,44,46].\nMoreover, further investigation is needed to understand how to\nmaintain trust and collaboration in a coopetitive economy [21].\nFourth, for a comprehensive evaluation of opportunities and\nchallenges, DLT research should also investigate alternatives, such\nas solutions that propose a cloud-based system that could be used\nin spacecraft networks to address data complexity and heterogene-\nity [6]. These alternatives require further investigation, especially\nFigure 1: Scelestial mapping scenario, where each zone color\nrepresents a company’s collaboration\nconcerning latency sensitivity in DLT solutions for satellite com-\nmunication [ 26]. Primarily, latency characteristics should be inves-\ntigated, as DLT is known to be highly latency sensitive despite the\nsafety guarantees even in non-synchronous environments [ 20,23].\nLastly, recent works have demonstrated the potential of using\nartificial intelligence (AI) agents to autonomously learn incremental\naffordances [ 5] and achieve Pareto efficiency and improve equal-\nity and productivity in competitive and dynamic market simula-\ntions [ 49]. While these results promise to promote universal shared\nefficient knowledge, there needs to be more comprehensive cov-\nerage and insights on autonomous economic interactions in the\nexisting literature on machine economies in general [ 25] and DLT\nin heterogeneous space MRS in specific. DLT has been suggested to\nmanage heterogeneous space systems and efficiently support decen-\ntralized behavior [ 17]. The increasing interest from the scientific\ncommunity and industry participants [ 48] also supports the need\nfor further research on cost-effective MRS . Traditional optimization\ntechniques have also been used to achieve decentralized, shared\neconomies without DLT [39]; further performance comparisons are\nneeded to understand the approximation heuristics of decentralized\ncombinatorial optimization systems.\n4 RESEARCH PLAN\nBy outlining the methodology, objectives, and critical components\nof our research plan, grounded in a hypothetical scenario developed\nas part of the ESA-ESRIC space resource challenge [ 30], we aim to\nexplore the integration of blockchain technology extending to the\nproposed market-based MRS for space exploration [16].\nIn the given scenario, depicted in Figure 1, six main actors are\ninvolved: the Service Orderer (SO), an Earth-based organization\nresponsible for mission directives, and five Service Providers (SP) –\nentities that execute the tasks in-situ – as roles elaborated in Table 1.\nWhile the decentralized MRS coordination concept dates back to\nthe 1980s [ 4,35], recent surveys lacking related approaches [ 1,42]\nindicate that the full potential of DLT in this domain still needs to\nbe explored. To fill the research gaps, we prioritize in our plan the\nfollowing essential requirements:ICSE-NIER’24, April 14–20, 2024, Lisbon, Portugal Baima et al.\nTable 1: List of service providers.\nSP Color Robotic Fleet Main Focus\nA Pink Communications satellites,\nMultiple antennasMoon-earth\ncommunications\nB Blue Medium size offline fleet of robots Mapping\nC Green Few robots w/ embedded sensors Resource analysis\nD Yellow Large fleet of small robots Fast mapping with\nless precision\nE None A single robot Mapping\ni.Network : Enable a mesh network that lets clients create job\npostings and robots to communicate and coordinate trust-\nfully via market-based tasks [16, 42].\nii.Data Sharing Transparency : Ensure transparency in data\nsharing to foster a coopetitive environment with reduced\ninformation asymmetry [21, 32].\niii.Robot Agnostic System : Develop a system compatible with\nany robotic platform and scalable to growing demands [ 16].\niv.Data Loss Resistance : Implement safeguards to ensure data\nintegrity and accessibility, even in the face of local system\nfailures or network disruptions [32, 37, 38].\nIn the scenario depicted in Figure 1, each polygon, structured\nusing the Goldberg polyhedron, represents a sub-map owned and\ntraded by network members [ 22]. This approach reduces the coor-\ndination complexity and allows for a fixed data size for each map\nsegment, optimizing the data storage on the blockchain. However,\nbecause different sections of the map may represent distinct ter-\nritories to be explored, ownership of the gathered information is\ntransferred to the client rather than the explorer. When a robot\ncreates data, it is desirable to represent it as an NFT to prove the\nauthenticity of the knowledge produced and shared on the network.\nSuch would, for instance, prevent unauthorized marketing as the\noriginal explorer, yet excessive information about the map would\nrender its value because other DLT nodes could retrieve it for free.\nOur initial implementation involves a decentralized system of\nthree rovers designed for resource identification, context analysis,\nand environmental mapping. These rovers have components compa-\nrable to space-graded hardware and can run blockchain nodes. The\nsystem uses a mesh network for communication and centralized\ndata processing. However, as the MRS for exploration design deci-\nsion, the ultimate aim is to transition to a fully decentralized system\nsupported by our successful simulations under similar Lunar net-\nwork conditions. While our research plan offers a comprehensive\napproach to integrating DLT with MRS , as we argued in Section 2,\na critical aspect of future research will be to focus on optimizing the\nthroughput and resource consumption of blockchain nodes. The\ninitial data organization strategy significantly enhanced data trans-\nmission efficiency, particularly regarding critical bandwidth usage.\nYet, it is essential to investigate the specific conditions under which\nDLT is most effective compared to authenticated and accountable\nbilateral communication based on digital signatures.\n5 CONCLUSION\nThis paper has presented a comprehensive research plan for in-\ntegrating DLT with market-based heterogeneous MRS in spaceexploration. We have critically examined the existing literature\nto identify significant research gaps and articulated the necessity\nfor interdisciplinary endeavors encompassing legal, regulatory, en-\ngineering, organizational, and economic aspects. Our proposed\nresearch agenda, while ambitious, aims to develop an MRS archi-\ntecture incorporating DLT and enabling autonomous economic\ndecision-making among robotic agents. We have also highlighted\nthe unique technical challenges that arise when deploying DLT-\nbased systems in the harsh conditions of outer space. While the pri-\nmary focus is on space applications, the principles, and architectures\ndiscussed have broader implications, potentially revolutionizing\neconomic routes that could be adapted for terrestrial applications.\nIt is important to note that while this paper emphasizes practical\napplication and feasibility, the challenges and use cases presented\nare not exhaustive. Our objective is to stimulate further research\nand discussion in this area, and we invite other scholars to extend,\ncritique, or build upon our proposed framework. By laying the\ngroundwork for future research, we hope to contribute to develop-\ning more efficient, transparent, and autonomous systems for space\nexploration. Designing such a system integrating DLT with MRS\ncan significantly advance the field of software engineering, espe-\ncially for robotics and automation software, offering new avenues\nfor innovation and collaboration.\nACKNOWLEDGMENTS\nThis research was funded in part by the Luxembourg National\nResearch Fund (FNR) in the FiReSpARX (ref. 14783405) and PABLO\n(ref. 16326754) projects, and by PayPal, PEARL grant reference\n13342933. For the purpose of open access, and in fulfillment of\nthe obligations arising from the grant agreement, the author has\napplied a Creative Commons Attribution 4.0 International (CC BY\n4.0) license to any Author Accepted Manuscript version arising\nfrom this submission.\nREFERENCES\n[1]U. S. P. Srinivas Aditya, Roshan Singh, Pranav Kumar Singh, and Anshuman Kalla.\n2021. A Survey on Blockchain in Robotics: Issues, Opportunities, Challenges and\nFuture Directions. Journal of Network and Computer Applications 196 (Dec. 2021),\n10. https://doi.org/10.1016/j.jnca.2021.103245\n[2]George A. Akerlof. 1970. The Market for \"Lemons\": Quality Uncertainty and\nthe Market Mechanism. The Quarterly Journal of Economics 84, 3 (Aug. 1970),\n488–500. https://doi.org/10.2307/1879431\n[3]Rabi’ Al-Akhar. 2019. Space Science and Technology - the Official Portal of\nthe UAE Government. , 14 pages. https://u.ae/en/about-the-uae/science-and-\ntechnology/key-sectors-in-science-and-technology/space-science-and-techno\n[4]Ph. Saint Aubert, M. Hervieux, J. L. Perbos, E. Saggese, and C. Soprano. 1986.\nCentralized vs Decentralized Options for a European Data Relay Satellite System.\nActa Astronautica 13 (1986), 387. https://doi.org/10.1016/0094-5765(86)90093-7\n[5]Renan Lima Baima and Esther Luna Colombini. 2021. Modeling Object’s Affor-\ndances via Reward Functions. In Proc. Int. Conf. on Sys., Man, and Cyb. IEEE, Mel-\nbourne, Australia, 2183–2190. https://doi.org/10.1109/smc52423.2021.9658915\n[6]Sue A. Baldor, Carlos Quiroz, and Paul Wood. 2013. Applying a Cloud Computing\nApproach to Storage Architectures for Spacecraft. In 2013 IEEE Aerospace Confer-\nence. IEEE, Big Sky, Montana, 1–6. https://doi.org/10.1109/aero.2013.6497340\n[7]Mariel Borowitz. 2019. Strategic Implications of the Proliferation of Space Sit-\nuational Awareness Technology and Information: Lessons Learned from the\nRemote Sensing Sector. Space Policy 47 (Feb. 2019), 18–27. https://doi.org/10.\n1016/j.spacepol.2018.05.002\n[8]Bert-Jan Butijn, Damian A. Tamburri, and Willem-Jan van den Heuvel. 2020.\nBlockchains: A Systematic Multivocal Literature Review. Comput. Surveys 53, 3,\nArticle 61 (July 2020), 37 pages. https://doi.org/10.1145/3369052\n[9]Lesley Conn. 2021. Global Space Economy Nears $447B. https://www.\nthespacereport.org/uncategorized/global-space-economy-nears-447b/\n[10] Consensys Space. 2018. Open Source Space. https://www.consensys.space/Designing Trustful Cooperation Ecosystems is Key to the New Space Exploration Era ICSE-NIER’24, April 14–20, 2024, Lisbon, Portugal\n[11] Florian Cordes. 2018. Design and Experimental Evaluation of a Hybrid Wheeled-Leg\nExploration Rover in the Context of Multi-Robot Systems . Kumulative Dissertation.\nUniversität Bremen, Bremen, Germany.\n[12] Ian A. Crawford. 2015. Lunar Resources: A Review. Progress in Physical Geography:\nEarth 39, 2 (April 2015), 137–167. https://doi.org/10.1177/0309133314567585\n[13] P. De Filippi and Andrea Leiter. 2021. Blockchain in Outer Space. American Journal\nof International Law 115 (2021), 413–418. https://doi.org/10.1017/aju.2021.63\n[14] A. Deshpande and J. Luntz. 2003. Decentralized Control for a Team of Physically\nCooperating Robots. In Proc. Int. Conf. Intell. Robots Syst. , Vol. 2. IEEE, Las Vegas,\nNevada, USA, 1757–1762 vol.2. https://doi.org/10.1109/iros.2003.1248898\n[15] Alexander Dettmann, Thomas Voegele, Jorge Ocón, Iulia Dragomir, Shashank\nGovindaraj, Matteo de Benedetti, Valérie Ciarletti, Rafik Hassen-Khodja, Thierry\nGerma, Raphael Viards, Gonzalo J. Paz-Delgado, and Laura M. Mantoani. 2022.\nCOROB-X: A Cooperative Robot Team for the Exploration of Lunar Skylights.\nInASTRA 2022 16th Symposium on Advanced Space Technologies in Robotics and\nAutomation . ESA-ESTEC, Noordwijk, Netherlands, 9. https://hal-insu.archives-\nouvertes.fr/insu-03751549\n[16] M. Bernardine Dias, Robert Zlot, Nidhi Kalra, and Anthony Stentz. 2006. Market-\nBased Multirobot Coordination: A Survey and Analysis. Proc. IEEE 94, 7 (July\n2006), 1257–1270. https://doi.org/10.1109/JPROC.2006.876939\n[17] Eduardo Castelló Ferrer, Ognjen Rudovic, Thomas Hardjono, and Alex Pentland.\n2018. RoboChain: A Secure Data-Sharing Framework for Human-Robot Interac-\ntion. In The Tenth International Conference on eHealth, Telemedicine, and Social\nMedicine . IARA XPS Press, Rome, Italy, 124 to 130.\n[18] Yang Gao and Steve Chien. 2017. Review on Space Robotics: Toward Top-Level\nScience through Space Exploration. Science Robotics 2, 7 (June 2017), eaan5074.\nhttps://doi.org/10.1126/scirobotics.aan5074\n[19] Sergio Garcıa, Claudio Menghi, Patrizio Pelliccione, Thorsten Berger, and Re-\nbekka Wohlrab. 2018. An Architecture for Decentralized, Collaborative, and\nAutonomous Robots. In 2018 IEEE International Conference on Software Architec-\nture (ICSA) . IEEE, Seattle, WA, 75–7509. https://doi.org/10.1109/ICSA.2018.00017\n[20] Arthur Gervais, Ghassan O. Karame, Karl Wüst, Vasileios Glykantzis, Hubert\nRitzdorf, and Srdjan Capkun. 2016. On the Security and Performance of Proof of\nWork Blockchains. In Proceedings of the ACM SIGSAC Conference on Computer\nand Communications Security (CCS ’16) . Association for Computing Machinery,\nNew York, NY, USA, 3–16. https://doi.org/10.1145/2976749.2978341\n[21] Shahla Ghobadi and John D’Ambra. 2011. Coopetitive Knowledge Sharing: An\nAnalytical Review of Literature. Electronic Journal of Knowledge Management\n9, 4 (Jan. 2011), 307–317. https://research.manchester.ac.uk/en/publications/\ncoopetitive-knowledge-sharing-an-analytical-review-of-literature\n[22] Jason Goo, Ojas Bora, Priyansh Manne, Kazem Mullah, Kayden Tilden, Jeremy\nHartley, Ayame Kunieda, Savannah Laver, Stephanie Lee, Laquan Shuai, and\nIsaiah Li Yun Tan. 2019. Blockchain Lunar Registry. https://diana.io\n[23] Tobias Guggenberger, Johannes Sedlmeir, Gilbert Fridgen, and André Luckow.\n2022. An In-Depth Investigation of the Performance Characteristics of Hy-\nperledger Fabric. Computers & Industrial Engineering 173 (Nov. 2022), 108716.\nhttps://doi.org/10.1016/j.cie.2022.108716\n[24] Jeffrey S. Harrison and Caron H. St. John. 1996. Managing and Partnering with\nExternal Stakeholders. Academy of Management Perspectives 10, 2 (May 1996),\n46–60. https://doi.org/10.5465/ame.1996.9606161554\n[25] Eduard Hartwich, Alexander Rieger, Johannes Sedlmeir, Dominik Jurek, and\nGilbert Fridgen. 2023. Machine Economies. Electronic Markets 33, 1 (July 2023),\n36. https://doi.org/10.1007/s12525-023-00649-0\n[26] Hussein. Ibrahim, Marwa A. Shouman, Nawal A. El-Fishawy, and Ayman. Ahmed.\n2021. Literature Review of Blockchain Technology in Space Industry: Challenges\nand Applications. In Int. Conf. Electron. Eng. IEEE, Menouf, Egypt, 1–8. https:\n//doi.org/10.1109/ICEEM52022.2021.9480642\n[27] Niclas Kannengiesser, Sebastian Lins, Christian Sander, Klaus Winter, Hellmuth\nFrey, and Ali Sunyaev. 2022. Challenges and Common Solutions in Smart Contract\nDevelopment. IEEE Transactions on Software Engineering 48, 11 (Dec. 2022), 4291–\n4318. https://doi.org/10.1109/TSE.2021.3116808\n[28] Ameer Tamoor Khan, Xinwei Cao, Shuai Li, and Zoran Milosevic. 2019.\nBlockchain Technology with Applications to Distributed Control and Coop-\nerative Robotics: A Survey. International Journal of Robotics and Control 2, 1 (Jan.\n2019), 36. https://doi.org/10.5430/ijrc.v2n1p36\n[29] Amanda M. Leon. 2018. Mining for Meaning: An Examination of the Legality\nof Property Rights in Space Resources. Virginia Law Review 104, 3 (May 2018),\n497–546. jstor:44864188 https://www.jstor.org/stable/44864188\n[30] Mathias Link, David Parker, Massimo Sabbatini, and Franziska . 2021. ESA-\nESRIC Space Resources Challenge - Prospecting Technologies - Call. https:\n//www.spaceresourceschallenge.esa.int\n[31] Sergey Lonshakov, Aleksandr Krupenkin, Aleksandr Kapitonov, Evgeny Rad-\nchenko, Alisher Khassanov, and A. Starostin. 2018. Robonomics: Platform for\nIntegration of Cyber Physical Systems into Human Economy for Engineers, Smart\nCities and Industry 4.0 Creators. https://doi.org/10.13140/RG.2.2.23928.60169\n[32] Ralph D. Lorenz. 2020. How Far Is Far Enough? Requirements Derivation for\nPlanetary Mobility Systems. Advances in Space Research 65, 5 (March 2020),\n1383–1401. https://doi.org/10.1016/j.asr.2019.12.011[33] Bernardo Martinez Rocamora, Cagri Kilic, Christopher Tatsch, Guilherme A. S.\nPereira, and Jason N. Gross. 2023. Multi-Robot Cooperation for Lunar In-Situ\nResource Utilization. Frontiers in Robotics and AI 10 (March 2023), 1149080.\nhttps://doi.org/10.3389/frobt.2023.1149080\n[34] Abdulkadir Memduhoglu and Melih Basaraner. 2018. Possible Contributions\nof Spatial Semantic Methods and Technologies to Multi-Representation Spatial\nDatabase Paradigm. International Journal of Engineering and Geosciences 3, 3\n(Oct. 2018), 108–118. https://doi.org/10.26833/ijeg.413473\n[35] D. Miller. 1985. A Spatial Representation System for Mobile Robots. In 1985 IEEE\nInternational Conference on Robotics and Automation Proceedings , Vol. 2. IEEE, St.\nLouis, MO, USA, 122–127. https://doi.org/10.1109/ROBOT.1985.1087318\n[36] Satoshi Nakamoto. 2008. Bitcoin: A Peer-to-Peer Electronic Cash System. https:\n//bitcoin.org/bitcoin.pdf\n[37] NASA. 2016. Space Technology Roadmaps and Priorities Revisited . Technical\nReport. The National Academies Press, DC, USA. https://doi.org/10.17226/23582\n[38] Alina Orlova, Roberto Nogueira, and Paula Chimenti. 2020. The Present and\nFuture of the Space Sector: A Business Ecosystem Approach. Space Policy 52\n(May 2020), 101374. https://doi.org/10.1016/j.spacepol.2020.101374\n[39] Peter Pilgerstorfer and Evangelos Pournaras. 2017. Self-Adaptive Learning in\nDecentralized Combinatorial Optimization - A Design Paradigm for Sharing\nEconomies. In 2017 IEEE/ACM 12th International Symposium on Software Engi-\nneering for Adaptive and Self-Managing Systems (SEAMS) . IEEE, Buenos Aires,\nArgentina, 54–64. https://doi.org/10.1109/seams.2017.8\n[40] Nikiforos Pittaras. 2022. A Cooperative Reinforcement Learning Environment\nfor Detecting and Penalizing Betrayal. https://doi.org/10.48550/arXiv.2210.12841\narXiv:2210.12841\n[41] Alberto Pretto, Stéphanie Aravecchia, Wolfram Burgard, Nived Chebrolu, Chris-\ntian Dornhege, Tillmann Falck, Freya Fleckenstein, Alessandra Fontenla, Marco\nImperoli, Raghav Khanna, Frank Liebisch, Philipp Lottes, Andres Milioto, Daniele\nNardi, Sandro Nardi, Johannes Pfeifer, Marija Popović, Ciro Potena, Cédric\nPradalier, Elisa Rothacker-Feder, Inkyu Sa, Alexander Schaefer, Roland Sieg-\nwart, Cyrill Stachniss, Achim Walter, Wera Winterhalter, Xiaolong Wu, and Juan\nNieto. 2021. Building an Aerial-Ground Robotics System for Precision Farming:\nAn Adaptable Solution. IEEE Robotics & Automation Magazine 28, 3 (Sept. 2021),\n29–49. https://doi.org/10.1109/mra.2020.3012492\n[42] Félix Quinton, Christophe Grand, and Charles Lesire. 2023. Market Approaches\nto the Multi-Robot Task Allocation Problem: A Survey. Journal of Intelligent &\nRobotic Systems 107, 2 (Feb. 2023), 29. https://doi.org/10.1007/S10846-022-01803-0\n[43] Martin J. Schuster, Marcus G. Müller, Sebastian G. Brunner, H. Lehner, P. Lehner,\nR. Sakagami, A. Dömel, Lukas Meyer, B. Vodermayer, R. Giubilato, M. Vayugundla,\nJosef Reill, Florian Steidle, Ingo von Bargen, K. Bussmann, Rico Belder, P. Lutz,\nW. Stürzl, Michal S., Moritz Maier, S. Stoneman, Andre F. Prince, B. Rebele, M.\nDurner, E. Staudinger, Siwei Zhang, Robert P., Esther Bischoff, C. Braun, Susanne\nSchröder, E. Dietz, S. Frohmann, Anko Börner, H. Hübers, B. Foing, R. Triebel,\nAlin O. Albu, and A. Wedler. 2020. The ARCHES Space-Analogue Demonstration\nMission: Towards Heterogeneous Teams of Autonomous Robots for Collaborative\nScientific Sampling in Planetary Exploration. IEEE Robotics and Automation\nLetters 5, 4 (Oct. 2020), 5315–5322. https://doi.org/10.1109/LRA.2020.3007468\n[44] Johannes Sedlmeir, Hans Ulrich Buhl, Gilbert Fridgen, and Robert Keller. 2020.\nThe Energy Consumption of Blockchain Technology: Beyond Myth. Business &\nInf. Sys. Eng. 62, 6 (2020), 599–608. https://doi.org/10.1007/s12599-020-00656-x\n[45] Johannes Sedlmeir, Jonathan Lautenschlager, Gilbert Fridgen, and Nils Urbach.\n2022. The Transparency Challenge of Blockchain in Organizations. Electronic\nMarkets 32 (2022), 1779—1794. https://doi.org/10.1007/s12525-022-00536-0\n[46] Volker Strobel, Eduardo Castelló Ferrer, and Marco Dorigo. 2018. Managing\nByzantine Robots via Blockchain Technology in a Swarm Robotics Collective\nDecision Making Scenario. In Proc. 17th Int. Conf. Auton. Agents MultiAgent Syst.\n(AAMAS ’18) . International Foundation for Autonomous Agents and Multiagent\nSystems, Stockholm, Sweden, 541–549. https://doi.org/10.5555/3237383.3237464\n[47] Pavlos Triantafyllou, Rafael Afonso Rodrigues, Sirapoab Chaikunsaeng, Diogo\nAlmeida, Graham Deacon, Jelizaveta Konstantinova, and Giuseppe Cotugno.\n2021. A Methodology for Approaching the Integration of Complex Robotics\nSystems: Illustration through a Bimanual Manipulation Case Study. IEEE Robotics\n& Automation Magazine 28 (2021), 88. https://doi.org/10.1109/mra.2021.3064759\n[48] Matthew Weinzierl and Mehak Sarang. 2021. The Commercial Space Age Is Here.\nHarvard Busi. Rev . (2021), 1. hbr.org/2021/02/the-commercial-space-age-is-here\n[49] Stephan Zheng, Alexander Trott, Sunil Srinivasa, Nikhil Naik, Melvin Gruesbeck,\nDavid C. Parkes, and Richard Socher. 2020. The AI Economist: Improving Equality\nand Productivity with AI-driven Tax Policies. https://doi.org/10.48550/arXiv.\n2004.13332 arXiv:2004.13332\n[50] Robert Zlot, Anthony Stentz, M.Bernardine Dias, and Scott Thayer. 2002. Multi-\nRobot Exploration Controlled by a Market Economy. In Proceedings 2002 IEEE\nInternational Conference on Robotics and Automation (Cat. No.02CH37292) , Vol. 3.\nIEEE, DC, USA, 3016–3023. https://doi.org/10.1109/ROBOT.2002.1013690\nReceived 4 September 2023; revised 22 November 2023; accepted 21 Decem-\nber 2023"    },    {        "title": "2402.06064v1.Formalizing_Automated_Market_Makers_in_the_Lean_4_Theorem_Prover.pdf",        "content": "Formalizing Automated Market Makers in the\nLean 4 Theorem Prover\nDaniele Pusceddu /envel⌢pe\nETH Zurich, Switzerland\nUniversity of Cagliari, Italy\nMassimo Bartoletti /envel⌢pe/h⌢me\nUniversity of Cagliari, Italy\nAbstract\nAutomated Market Makers (AMMs) are an integral component of the decentralized finance (DeFi)\necosystem, as they allow users to exchange crypto-assets without the need for trusted authorities\nor external price oracles. Although these protocols are based on relatively simple mechanisms, e.g.\nto algorithmically determine the exchange rate between crypto-assets, they give rise to complex\neconomic behaviours. This complexity is witnessed by the proliferation of models that study their\nstructural and economic properties. Currently, most of theoretical results obtained on these models\nare supported by pen-and-paper proofs. This work proposes a formalization of constant-product\nAMMs in the Lean 4 Theorem Prover. To demonstrate the utility of our model, we provide\nmechanized proofs of key economic properties like arbitrage, that at the best of our knowledge have\nonly been proved by pen-and-paper before.\n2012 ACM Subject Classification Software and its engineering →Formal methods; Software and its\nengineering→Formal software verification\nKeywords and phrases Smart contracts, Ethereum, Verification, Blockchain\nDigital Object Identifier 10.4230/OASIcs...\nFunding Massimo Bartoletti : Partially supported by project SERICS (PE00000014) and PRIN 2022\nDeLiCE (F53D23009130001) under the MUR National Recovery and Resilience Plan funded by the\nEuropean Union – NextGenerationEU.\n1 Introduction\nAutomated Market Makers (AMMs) are one of the key applications in the Decentralized\nFinance (DeFi) ecosystem, as they allow users to trade crypto-assets without the need for\ntrusted intermediaries [ 13]. Unlike traditional order-book exchanges, where buyers and sellers\nmust find a counterpart, AMMs enable traders to autonomously swap assets deposited in\nliquidity pools contributed by other users, who are incentivized to provide liquidity by a\ncomplex reward mechanism. At the time of writing, there are multiple AMM protocols\ncontrolling several billions of dollars worth of assets1. This has made AMMs an appealing\ntarget for attacks, resulting in losses worth billions of dollars over time2.\nThe security of AMMs depends on several factors: besides the absence of traditional\nprogramming bugs, it is crucial that their economic mechanism gives rise to a rational\nbehaviour of its users that aligns with the AMM ideal functionality, i.e. providing an\nalgorithmic exchange rate coherent with the one given by trusted price oracles. Therefore, it\nis important to obtain strong guarantees about the economic mechanisms of these protocols.\nWhile formal verification tools for smart contracts based on model-checking are useful in\n1https://defillama.com/protocols/Dexes\n2https://chainsec.io/defi-hacks/\n©Daniele Pusceddu and Massimo Bartoletti;\nlicensed under Creative Commons License CC-BY 4.0\nOpenAccess Series in Informatics\nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:2402.06064v1  [cs.LO]  8 Feb 2024XX:2 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\ndetecting programming bugs and even in proving some structural properties of AMMs [ 8,10],\nthey are not suitable for verifying, or even expressing more complex properties regarding\nthe economic mechanism of AMMs. These economic mechanisms have been studied in\nseveral research works, which, in most cases, provide pen-and-paper proofs of the obtained\nproperties. Given the complexity of the studied models, it would be desirable to also provide\nmachine-verified proofs, so that we may rely on the proven properties beyond any reasonable\ndoubt. To the best of our knowledge, existing mechanized formalizations [ 12] focus on\nverifying relevant structural properties of AMMs like their state consistency, and not on\nstudying the economic mechanism of AMMs (see Section 4 for a detailed comparison).\nContributions\nIn this paper we formalize Automated Market Makers in the Lean 4 theorem prover. Our\nmodel is based on a slightly simplified version of the Uniswap v2 protocol (one of the leading\nAMMs), which was studied in [ 7] with a pen-and-paper formalization. We provide a Lean\nspecification of blockchain states, abstracted from any factors that are immaterial to the study\nof AMMs. Then, we model the fundamental interactions that users may have with AMMs as\nwell as the economic notions of price, networth and gain. Finally, we build machine-checked\nproofs of economic properties of constant-product AMMs. In particular, we derive an explicit\nformula for the economic gain obtained by a user after an exchange with an AMM. Building\nupon this formula we prove that, from a trader’s perspective, aligning a constant-product\nAMM’s internal exchange rate with the rate given by the trader’s price oracle implies the\noptimal gain from that AMM. This results in the fundamental property of AMMs acting as\nprice oracles themselves [ 4]. We then construct the optimal swap transaction that a rational\nuser can perform to maximize their gain, solving the arbitrage problem. Our formalization\nand proofs3are made available in a public GitHub repository. At the best of our knowledge,\nthis is the first mechanized formalization of the economic mechanism of AMMs. We finally\ndiscuss some open issues, and alternative design choices for formalizing AMMs.\n2 Formalization\nAn Automated Market Maker implements a decentralized exchange between two different\ntoken types. The exchange rate is determined by a smart contract, which also takes care of\nperforming the exchange itself: namely, the contract receives from a trader some amount of\nthe input token type, and sends back the correct amount of the output token type, which\nis taken from the AMM reserves. A single smart contract can control many instances of\nAMMs (also called AMM pairs ): we may have a pair for each possible unordered pair of\ntoken types. To create an AMM instance, a user must provide the initial liquidity for the\nreserves of that pair of tokens. Liquidity providers are rewarded with a type of token that\nspecifically represents shares in that AMM’s reserves: we call these mintedtoken types, while\nany other token type will be called atomic.\nBlockchain State\nWe begin by formalizing the blockchain state, abstracting from the details that are immaterial\nto the study of AMMs. Then, our model includes the users’ wallets, the AMMs and their\nreserves (see Listing 1). We formalize the universes of users and atomic token types as the\n3https://github.com/danielepusceddu/lean4-ammD. Pusceddu and M. Bartoletti XX:3\ntypes Aand T, respectively, as structures that encapsulate a natural number. Hereafter,\nwe usea,b,...to denote users in A, andτ,τ0,τ1,...to denote tokens in T. Minted token\ntypes are pairs of T. We represent the funds owned by a user by a walletthat maps token\ntypes to non-negative reals. To rule out wallets with infinite tokens, we use Mathlib’s finitely\nsupported functions4: in general, given any type αand any type Mwith a 0element,\nf∈α→0Mifsupp (f) ={x∈α|f(x)̸= 0}is finite.\nWe define the type W0of wallets of atomic tokens as a structure encapsulating T→0R≥0.\nThis definition induces an element 0∈W0such that supp (0) =∅: this is the empty wallet ,\nwhich enables us to form the type T→0W0. We define the type W1of wallets of minted tokens\nas a structure that encapsulates a function bal∈T→0W0. The intuition is that balτ0τ1\ngives the owned amount of the minted token type created by the AMM pair with tokens τ0\nandτ1. Consistently, the function balmust satisfy two conditions: balτ0τ1=balτ1τ0,\nmeaning that the order of atomic tokens is irrelevant, and balτ τ= 0, meaning that the\ntwo token types in an AMM must be distinct. Our definition of W1encapsulates proofs of\nthese two properties, called unordanddistinct , respectively.\nWe map users to their wallets with the types S0and S1, which account for the atomic\ntokens and for the minted tokens, respectively. Finally, we formalize sets of AMM pairs\nwith the type AMMs. The definition is strikingly similar to that of W1, but with a changed\nconstraint. The intuition is that resτ0τ1gives the reserves of τ0in the AMM pair (τ0,τ1),\nwhile resτ1τ0gives the reserves of τ1in the same AMM pair. For uninitialized AMM pairs,\nthe obtained reserves must be 0. The property posresensures that either an AMM pair\nhas no reserves of both token types (i.e., the AMM has not been created yet), or both token\ntypes have strictly positive reserves (i.e., one cannot deplete the reserves of a single token\ntype in an AMM pair). We combine the previous definitions in the type Γ, which represents\nthe state of a blockchain (note that Γabstracts from all the details immaterial for AMMs).\nToken supply\nGiven a blockchain state s∈Γ, we define the supply of an atomic token type τ0as:\natomsupplys(τ0) =/summationdisplay\na∈supp(s.atoms )(s.atomsa)τ0+/summationdisplay\nτ1∈supp(s.ammsτ0)(s.ammsτ0)τ1\nwhere the partial application s.ammsτ0gives a map with all AMM pairs with τ0as one of\ntheir token types. We define the supply of a minted token type (τ0,τ1)as follows:\nmintsupplys(τ0,τ1) =/summationdisplay\na∈supp(s.mints )(s.mintsa)τ0τ1\nThe corresponding Lean definitions (in Listing 2) have been split in order to facilitate\ntheorem proving and, in particular, the use of Lean’s simplifier.\nAMM reserves\nGiven a blockchain state sand two token types τ0,τ1, the termss.ammsτ0τ1ands.ammsτ1τ0\ndenote, respectively, the reserves of τ0andτ1in the AMM pair (τ0,τ1). This way of accessing\nthe AMM reserves is a bit impractical: when writing proofs, using s.ammsτ0τ1carries an\nobligation to provide the functions distinct and posres. In particular, this requires the\n4https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finsupp/Defs.htmlXX:4 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\nListing 1 Fundamental Lean definitions for the state of an AMM system.\nstructure A where\nn:N\nstructure T where\nn:N\nstructure W 0where\nbal: T→0R≥0\nstructure S 0where\nmap: A→0W0structure W 1where\nbal: T→0W0\nunord:∀(τ0τ1: T),\nbal τ0τ1= f τ1τ0\ndistinct:∀(τ: T),\nbal τ τ = 0\nstructure S 1where\nmap: A→0W1structure AMMs where\nres: T→0W0\ndistinct:∀(τ: T),\nres τ τ = 0\nposres:∀(τ0τ1: T),\nres τ0τ1̸=0↔fτ1τ0̸=0\nstructure Γwhere\natoms: S 0\nmints: S 1\namms: AMMs\nListing 2 Supply of atomic token types and of minted token types.\n1noncomputable def S 0.supply (s: S 0) (τ: T): R≥0 := s.map.sum ( λ_ w => w τ)\n2\n3noncomputable def S 1.supply (s: S 1) (τ0τ1: T): R≥0 :=\n4 s.map.sum ( λ_ w => w.get τ0τ1)\n5\n6noncomputable def AMMs.supply (amms: AMMs) ( τ: T): R≥0 :=\n7 (amms.res τ).sum λ_ x => x\n8\n9noncomputable def Γ.atomsupply (s: Γ) (τ: T): R≥0 :=\n10 (s.atoms.supply τ) + (s.amms.supply τ)\n11\n12noncomputable def Γ.mintsupply (s: Γ) (τ0τ1: T): R≥0 := s.mints.supply τ0τ1\nuser to explicitly add, in any theorem using the reserves, the assumption that the reserves\nare strictly positive to indicate the AMM pair has been created. Furthermore, this way of\naccessing the reserves hides the fact that when one of the reserves is strictly positive, also\nthe other one is such, which again should be made explicit when writing proofs.\nTo cope with these issues, we build a Lean API that allows for hiding these implementation\ndetails (see Listing 3). For example, given an AMM pair (τ0,τ1)in a states, the expression\ns.amms.r0τ0τ1initgive the reserves of token τ0in the AMM, which are guaranteed to be\nstrictly positive under the initialization precondition init∈(s.amms.init ).\nTransactions\nOur model encompasses all the main types of transactions supported by AMMs: creating an\nAMM, adding/removing liquidity, and swapping a token for another. Swaps are parameterised\nby aswap rate function , which determines the exchange rate. We use the formalization of\nswap transactions (Listing 4) to exemplify the scheme we used for all the transaction types.\nThe type Swap (sx,s,a,τ 0,τ1,x)represents valid swap transactions in a blockchain state\ns, with the swap rate function sx, performed by user ato exchange xamount of the input\ntokenτ0for a certain amount of the output token τ1(Line 2). Each element of this type\nis a structure containing a proof of the validity of the transaction. For example, for swap\ntransactions we must prove that the user has enough amount of τ0(condition enough), thatD. Pusceddu and M. Bartoletti XX:5\nListing 3 Fragment of the AMM API: AMM reserves.\n1def AMMs.init (amms: AMMs) ( τ0τ1: T): Prop := amms.res τ0τ1̸=0\n2\n3def AMMs.r 0(amms: AMMs) ( τ0τ1: T) (h: amms.init τ0τ1):R>0 :=⟨amms.res τ0τ1,\n4 by unfold init at h; exact NNReal.neq_zero_imp_gt h ⟩\n5\n6def AMMs.r 1(amms: AMMs) ( τ0τ1: T) (h: amms.init τ0τ1):R>0 :=⟨amms.res τ1τ0,\n7 by unfold init at h; exact NNReal.neq_zero_imp_gt ((amms.posres τ0τ1).mp h)⟩\nthe AMM pair with tokens τ0andτ1exists (condition exi), and it has enough reserves\nofτ1to give as output (condition nodrain). Since the type Swap (···)is empty when the\nparameters do not satisfy the above conditions, invalid transactions are not really expressible\nin our model. Instead, if Swap (···)represents a valid transaction, it will be a singleton type\ndue to proof irrelevance (i.e., any two proofs of the same proposition are equal).\nEach transaction is equipped with an applyfunction that yields the state reached by\nexecuting the transaction in the given state. For example, for sw∈Swap (sx,s,a,τ 0,τ1,x),\napplyswyields a state where: (i) a’s atomic tokens wallet is updated by removing xunits\nofτ0and adding sw.yunits ofτ1(Line 12), where sw.yis the amount of tokens outputted\nby the AMM pair; (ii) accordingly, the AMM reserves are updated by removing sw.yunits of\nτ1and adding xunits ofτ0(Line 14); (iii) the minted token wallets is unchanged (Line 13).\nThese definitions use functions and proofs not included in Listing 4 for brevity, such as sub\nandsub_r1. These are designed with the same spirit of allowing only valid operations, and\nso require suitable proofs. For example, sub_r1requires a proof of the existence of the AMM\nwe are removing liquidity from, and a proof that the AMM pair has enough liquidity to\nretain a positive amount of reserves. We build these proofs inline using those contained in\nthe structure: for instance, the parameter sw.exipassed to sub_r1at line 11 is a proof that\nthe AMM pair exists. Then, at line 16 we define the constant-product swap rate function,\nthat is the swap rate function used by Uniswap v2. From Lemma 5 onwards, our results will\nfocus on AMMs using this swap rate function.\nPrice, networth and Gain\nAn important aim of our model is to state and prove economic properties of AMMs related\nto the networth of their users. The fundamental definitions are in Listing 5. Given a wallet\nw∈W0and an atomic token price oracle o∈T→R>0, we define the valueofwin Line 2 as:\nvalue (w,o) =/summationdisplay\nτ∈supp(w)w(τ)·o(τ)\nThe value of a wallet of minted tokens w∈W1is defined similarly, except that: (i) the\nsummation ranges over supp (w.u), withw.u∈T2→0R≥0representing the uncurrying of w;\n(ii) the summation is divided by 2 since, if (τ0,τ1)is in the support of w, then also (τ1,τ0)is\nin its support. For uniformity with the definition of value of atomic wallets, also here we\nassume an oracle that gives the price of (minted) tokens. However, while for pricing atomic\ntokens we indeed resort to an oracle, for minted tokens this oracle is instantiated to a specific\nfunction, coherently with [7]:\nmintedprices(o,τ0,τ1) =(s.amms.r0τ0τ1)·o(τ0) + (s.amms.r1τ0τ1)·o(τ1)\nmintsupplys(τ0,τ1)XX:6 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\nListing 4 Definition of the swap transaction type and of its application, as well as the constant-\nproduct swap rate function.\n1abbrev SX := R>0→R>0→R>0→R>0\n2structure Swap (sx: SX) (s: Γ) (a: A) ( τ0τ1: T) (x: R>0) where\n3 enough: x≤s.atoms.get a τ0 -- user a has at least x τ0\n4 exi: s.amms.init τ0τ1 -- AMM pair τ0τ1exists in s\n5 nodrain: x*(sx x (s.amms.r 0τ0τ1exi) (s.amms.r 1τ0τ1exi))\n6 < (s.amms.r 1τ0τ1exi) -- AMM has enough output tokens\n7\n8def Swap.y (sw: Swap sx s a τ0τ1x):R>0 :=\n9 x*(sx x (s.amms.r 0τ0τ1sw.exi) (s.amms.r 1τ0τ1sw.exi))\n10\n11noncomputable def Swap.apply (sw: Swap sx s a τ0τ1x):Γ:= {\n12 atoms := (s.atoms.sub a τ0x sw.enough).add a τ1sw.y,\n13 mints := s.mints,\n14 amms := (s.amms.sub_r 1τ0τ1sw.exi sw.y sw.nodrain).add_r 0τ0τ1(by\nsimp[sw.exi]) x }\n15\n16noncomputable def SX.constprod: SX := λ(x r 0r1:R+) => r 1/(r 0+ x)\nwhere we have omitted the initialization precondition hfor brevity.\nWe then define the networth of a user as the sum of the value of their two types of wallets\n(Line 12). The gainof a user upon an update of the blockchain state is the difference between\nthe networth in the new state and that in the old state (Line 16).\nReachable states\nTo formalize reachable states, we begin by defining sequences of transactions (Listing 6).\nTx(sx,s,s′)is the type of sequences of valid transactions (of any kind) starting from state s\nand leading to s′. The parameter sxis the swap rate function used in swap transactions.\nTechnically, Tx(sx,s,s′)is an instance of the indexed family of dependent types Tx, dependent\nonsxands, and indexed by s′. In practice, this means that the constructors must preserve\nthe values of sxands(building upon the sequence of transactions does not change the\nswap rate function being used nor the originating state), while s′may change after each\nconstruction (the state resulting from the sequence changes with each transaction that is\nadded to it). A state s′isreachable if there exists a valid sequence of transactions that\nreachess′starting from a valid initialstates, i.e. a state with no initialized AMMs or minted\ntoken types in circulation.\n3 Results\nWe now present some noteworthy properties of AMMs that we have proven in Lean.\nProposition 1 ensures that, in any reachable state, there exists an AMM with token types\nτ0andτ1if and only if the minted token type (τ0,τ1)is in circulation, i.e. it has a strictly\npositive supply. This result showcases the validity of our model with regards to reasoning\nabout reachable states. Technically, it also allows us to prove that mintedprices(o,τ0,τ1)is\nstrictly positive for any initialized AMM pair (τ0,τ1)in any reachable state s.D. Pusceddu and M. Bartoletti XX:7\nListing 5 Users’ networth and gain.\n1noncomputable def W 0.value (w: W 0) (o: T→R>0):R≥0 :=\n2 w.sum ( λ τ x => x*(o τ))\n3\n4noncomputable def W 1.value (w: W 1) (o: T→T→R≥0):R≥0 :=\n5 (w.u.sum ( λp x => x*(o p.fst p.snd))) / 2\n6\n7noncomputable def Γ.mintedprice (s: Γ) (o: T→R>0) ( τ0τ1: T): R≥0 :=\n8 if h:s.amms.init τ0τ1then\n9 ((s.amms.r 0τ0τ1h)*(o τ0)+(s.amms.r 1τ0τ1 h)*(o τ1)) / (s.mints.supply τ0τ1)\n10 else 0 -- price is zero if AMM is not initialized\n11\n12noncomputable def Γ.networth (s: Γ) (a: A) (o: T →R>0):R≥0 :=\n13 (W0.value (s.atoms.get a) o) + (W 1.value (s.mints.get a) (s.mintedprice o))\n14\n15noncomputable def A.gain (a: A) (o: T →R>0) (s s’: Γ):R:=\n16 ((s’.networth a o): R) - ((s.networth a o): R)\nListing 6 Sequences of transactions and reachable states.\n1inductive Tx (sx: SX) (init: Γ):Γ→Type where\n2 | empty: Tx sx init init\n3\n4 | swap (s’: Γ) (rs: Tx sx init s’)\n5 (sw: Swap sx s’ a τ0τ1v0):\n6 Tx sx init sw.apply\n7 -- Other constructors omitted for brevity\n8\n9def validInit (s: Γ): Prop :=\n10 (s.amms = AMMs.empty ∧s.mints = S 1.empty)\n11\n12def reachable (sx: SX) (s: Γ): Prop :=\n13∃(init: Γ) (tx: Tx sx init s), validInit init\n▶Proposition 1 (Existence of AMMs vs.minted token supply) .Lets′∈Γbe a reachable\nblockchain state. Then, for any minted token type (τ0,τ1), its supply in s′is strictly positive\nif and only if s′has an AMM with token types τ0andτ1.\nProof.By induction on the length of the sequence of transactions leading to s′:\nBase case: empty transaction sequence. The proof is trivial for both directions, since a\nvalid starting state has no initialized AMMs and no minted tokens in circulation.\nInductive case: there are several subcases depending on the last transaction fired in the\nsequence. Here we consider the creation of the AMM (τ′\n0,τ′\n1)in reachable state s′. We\nproceed by cases on the truth of the equality {τ0,τ1}={τ′\n0,τ′\n1}. If it is a different token\npair, then the supply remains unchanged along with the initialization status, and we\ncan conclude by the induction hypothesis. If it is the same token pair, then we just\nincremented its minted token supply (which is non-negative, so after incrementing it, it\nmust be strictly positive), and we just initialized the AMM.\nSee source code for the other cases. AMMLib/Transaction/Trace.lean:275 ◀XX:8 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\nLemma 2 allows us to determine the change in the value of a wallet after it has been\nupdated in some way: the resulting equality is the basis for all the subsequent proofs. To\nillustrate it, we briefly introduce two definitions that have been omitted before: drain 0(w0,τ0)\nis the atomic token wallet such that drain 0(w0,τ0)(τ0) = 0anddrain 0(w0,τ0)(τ1) =w0(τ1)\nfor every other token τ1̸=τ0. We define drain 1(w1,(τ0,τ0))∈W1similarly.\n▶Lemma 2 (Value expansion) .Letw0∈W0,o0∈T→R>0, andτ0,τ1∈T. Then,\nvalue (w0,o0) =w0(τ0)·o(τ0) +value (drain 0(w0,τ0),o0)\nand, withw1∈W1ando1∈T2→R>0such thato1(τ0,τ1) =o1(τ1,τ0),\nvalue (w1,o1) =w1(τ0,τ1)·o1(τ0,τ1) +value (drain 1(w1,τ0,τ1),o1)\nProof.By definition of value and by properties of the sum over a finite support. Full Lean\nproof at AMMLib/State/AtomicWall.lean:116 ◀\nLemma 3 gives an explicit formula for the gain obtained by a user upon firing a swap\ntransaction. It is fundamental to all proofs involving gain.\n▶Lemma 3 (Gain of a swap) .Letsw∈Swap (sx,s,a,τ 0,τ1,x)and leto∈T→R>0. Then,\ngain (a,o,s, applysw) = (sw.y·o(τ1)−x·o(τ0))·/parenleftbigg\n1−(s.mintsaτ0τ1)\nmintsupplys(τ0,τ1)/parenrightbigg\nProof.By repeated application of Lemma 2 in order to isolate the value contributed by\nthe token types involved in the swap, and by use of Mathlib’s ring_nf simplifier tactic.\nAMMLib/Transaction/Swap/Networth.lean:55 ◀\nLemma 4 establishes a correspondence between the profitability of a swap transaction\n(i.e., a positive or negative gain) and the order between the swap rate and the exchange rate\ngiven by the price oracle. In particular, assuming a trader awho is not a liquidity provider\n(i.e.,ahas no minted tokens for the AMM pair targeted by the swap), Lemma 4 states that:\ngain (a,o,s, applysw)<0⇐⇒sx(x,r0,r1)<o(τ0)/o(τ1)\ngain (a,o,s, applysw) = 0⇐⇒sx(x,r0,r1) =o(τ0)/o(τ1)\ngain (a,o,s, applysw)>0⇐⇒sx(x,r0,r1)>o(τ0)/o(τ1)\nTechnically, to formalize this result it is convenient to use Mathlib’s cmp5, which gives the\norder between the two parameters.\n▶Lemma 4 (Swap rate vs.exchange rate) .Letsw∈Swap (sx,s,a,τ 0,τ1,x)be a swap\ntransaction, and let o∈T→R>0be a price oracle. For i∈{0,1}, letri=s.amms.ri(τ0,τ1).\nIfs.mints (a) (τ0,τ1) = 0, then\ncmp(gain (a,o,s, applysw),0) = cmp/parenleftbigg\nsx(x,r0,r1),o(τ0)\no(τ1)/parenrightbigg\nProof.Byalgebraicmanipulationandapplicationof Lemma3. AMMLib/Transaction/Swap/Net-\nworth.lean:86 ◀\n5https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Data/Ordering/Basic.\nhtml#cmpD. Pusceddu and M. Bartoletti XX:9\nLemma 5 establishes that, in a constant-product AMM, there exists only one profitable\ndirection for a swap. Namely, if swapping τ0forτ1gives a positive gain, then swapping in\nthe other direction (i.e., τ1forτ0) will give a negative gain. Note that the inverse does not\nhold: a negative gain in a direction does not imply a positive gain in the other direction.\n▶Lemma 5 (Unique direction for swap gain) .Letsw∈Swap (constprod,s,a,τ 0,τ1,x)and\nsw′∈Swap (constprod,s,a,τ 1,τ0,x′)be two swap transactions in opposite directions, and\nleto∈T→R>0. Ifgain (a,o,s, applysw)>0, then gain (a,o,s, applysw)<0.\nProof.By Lemma 4. AMMLib/Transaction/Swap/Constprod.lean:160 ◀\n▶Example 6. Consider a blockchain state sand atomic tokens τ0andτ1, and assume\nthat: (i)ais a trader with no minted tokens, i.e. (s.mintsa)τ0τ1= 0; (ii) the AMM\npair for (τ0,τ1)has been initialized in sand has reserves r0= (s.ammsτ0τ1) = 18and\nr1= (s.ammsτ1τ0) = 6; (iii) all the AMMs in suse the constant-product swap rate function;\n(iv)ois a price oracle such that oτ0= 3andoτ1= 4. Assume that awants to sell 6 units of\nτ1for some units of τ0with the swap transaction sw∈Swap (constprod,s,a,τ 1,τ0,6). Then,\nby Lemma 3, a’s gain is given by 9·3−24 = 3>0. Coherently with Lemma 5, any swap in\nthe opposite direction would give a negative gain: e.g., if asells 6 units of τ0, her gain would\nbe3/2·4−18 =−12.\nWe say that a swap transaction sw∈Swap (constprod,s,a,τ 0,τ1,x)isoptimal for a\ngiven price oracle owhen, for all sw′∈Swap (constprod,s,a,τ 0,τ1,x′)withx′̸=xwe have\ngain (a,o,s, applysw′)<gain (a,o,s, applysw)\nTheorem 7 gives a sufficient condition for the optimality of swaps in constant-product\nAMMs: it suffices that the ratio of the AMM’s reserves after the swap is equal to the\nexchange rate given by the price oracle. Intuitively, the condition in Theorem 7 means that\nthe exchange rate between the two token types induced by the AMM (i.e., the ration between\nthe token reserves) is aligned with the exchange rate given by the price oracle, and so further\nswaps would yield a negative gain. Note that, by definition, if a swap is optimal then it is\nalso unique, i.e. swapping any other amount would yield a suboptimal gain.\n▶Theorem7 (Sufficientconditionforoptimalswaps) .Letsw∈Swap (constprod,s,a,τ 0,τ1,x)\nbe a swap transaction on a constant-product AMM, and let o∈T→R>0be a price oracle.\nFori∈{0,1}, letr′\ni=(applysw).amms.ri(τ0,τ1)be the AMM reserves after the swap. If\nr′\n1/r′\n0=o(τ0)/o(τ1), thenswis optimal.\nProof.By cases on x<x′and by application of Lemma 4. AMMLib/Transaction/Swap/-\nConstprod.lean:184 ◀\n▶Example 8. Under the assumptions of Example 8, the exchange rate between τ1andτ0\ngiven by the price oracle is (o τ0)/(o τ1) = 3/4, while the exchange rate induced by the\nAMM (i.e., the ratio between the reserves) is r1/r0= 1/3. Hence, to satisfy the equality\nin Theorem 7 we must perform a swap that increases r1and decreases r0, i.e.amust sell\nunits ofτ1to buy units of τ0, coherently with the necessary condition for a positive gain\nin Example 8. Theorem 9 below will establish exactly how many units of τ1must be traded.\nTheorem 9 gives an explicit formula for the input amount xthat yields an optimal swap\ntransaction for a given AMM pair, under the assumption that the user firing the transaction\ndoes not hold the AMM’s minted token type. The other implicit assumption is that the userXX:10 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\nafiring the swap has the needed amount xof units of the sold token, i.e. x≤(s.atomsa)τ1.\nIn practice, this assumption can always be satisfied with flash loans , which allow ato borrow\nthe amount x, perform the swap, and then return the loan in a single, atomic transaction.\n▶Theorem9 (Arbitrageforconstant-productAMMs) .Letsw∈Swap (constprod,s,a,τ 0,τ1,x)\nbe a swap transaction on a constant-product AMM, and let o∈T→R>0be a price oracle.\nFori∈{0,1}, letri=s.amms.ri(τ0,τ1)be the AMM reserves. If ahas no minted tokens\n(i.e.,s.mints (a) (τ0,τ1) = 0)thenswis optimal if the amount of traded units of τ0is:\nx=/radicaligg\no(τ1)·r0·r1\no(τ0)−r0\nProof.By algebraic manipulation and Theorem 7. AMMLib/Transaction/Swap/Const-\nprod.lean:316 ◀\n▶Example 10. Under the assumptions of Example 8, we know that to perform a profitable\nswap the trader must sell units of τ1, i.e. fire a transaction Swap (constprod,s,a,τ 1,τ0,x).\nTheorem 9 gives the optimal input value x, i.e., the number of sold units of τ1:\nx=/radicalbigg\n3·6·18\n4−6 = 3\nThen, the output amount is given by sw.y= 3·18/(6+3) = 6 and, by Lemma 3, the gain of a\nis6·3−3·4 = 6, which maximizes it. Note that, in the new state, the exchange rate given by\nthe AMM coincides with that given by the price oracle: (r1+x)(r0−sw.y)= (oτ0)/(oτ1).\n4 Related work\nThe closest work to ours is [ 12], which proposes a methodology for developing and verifying\nAMMs in the Coq proof assistant. In this approach, an AMM is decomposed into multiple\ninteracting smart contract: e.g., each minted token is modelled as a single smart contract,\nfollowing way fungible tokens are encoded in blockchains platforms that do not provide\ncustom tokens natively, as e.g. in Ethereum and Tezos. These smart contracts are then\nimplemented as Coq functions on top of ConCert [ 6], a generic model of blockchain platforms\nand smart contracts mechanized in Coq. Concert is used to verify behavioural properties\nof smart contracts, either in isolation or composed with other smart contracts: this is\nfundamental for [12], where AMMs are specified as compositions of multiple contracts.\nMore specifically, [ 12] applies the proposed methodology to the Dexter2 protocol, which\nimplements a constant-product AMM based on Uniswap v1 on the Tezos blockchain6. The\nmain properties of AMMs proved in [ 12] are correspondences between the state of AMMs\nand the sequences of transactions executed on the blockchain. In particular, they prove that:\nthe balance recorded in the main AMM contract is coherent with the actual balance\nresulting from the execution of the sequence of transactions;\nthe supply of the minted token recorded in the main AMM contract is equal to the actual\nsupply resulting from the execution;\nthe state of the minted token contract is coherent with the execution.\n6https://gitlab.com/dexter2tz/dexter2tz/-/tree/master/D. Pusceddu and M. Bartoletti XX:11\nFurthermore, starting from the Coq specification of the Dexter2 AMM, [ 12] extracts verified\nCamlLigo code, which is directly deployable on the Tezos blockchain.\nAlthough both our work and [ 12] involve AMM formalizations within a proof assistant,\nthe ultimate goals are quite different. The formalization in [ 12] closely follows the concrete\nimplementation of a particular AMM instance (Dexter2) and produces a deployable imple-\nmentation that is provably coherent with the proposed Coq specification. By contrast, we\nstart from a more abstract specification of AMM, with the goal of studying the properties\nthat must be satisfied by any implementation coherent with the specification. An advantage\nof our approach over [ 12] is that it provides a suitable level of abstraction where proving\nproperties about the economic mechanisms of AMMs, i.e. properties about the gain of users\nand of equilibria among users’ strategies. In particular, Theorem 9 establishes a paradigmatic\nproperty of AMMs, which explains the economic mechanism underlying their design.\nBesides these main differences, our AMM model and that in [ 12] have several differences.\nA notable difference is that our model is based on Uniswap v2, while [ 12] is based on Uniswap\nv1. In particular, this means that our AMMs can handle arbitrary token pairs, while in\nDexter2 any AMM pairs a token with the blockchain native crypto-currency.\nAt the best of our knowledge, [ 12] and ours are currently the only mechanized formaliza-\ntions of AMMs in a proof assistant. Many other works study economic properties of AMMs\nthat go beyond those proved in this paper. The works [ 5] and [4] study, respectively, an\nAMM model based on Uniswap similar to ours, and a generalization of the model where the\nAMM is parameterised over a trading function of the AMM reserves, which must remain\nconstant before and after any swap transactions (in Uniswap v1 and v2, the trading function\nis just the product between the AMM reserves). The work [ 7] studies another generalization\nof Uniswap v2, where the relation between input and output tokens of swap transactions\nis determined by an arbitrary swap rate function , studying the properties of this function\nthat give rise to a sound economic mechanism of AMMs. While both [ 4] and [7] share\nthe common goal of providing general models of AMMs wherein to study their economic\nbehaviour, they largely diverge on the formalization: [ 4] is based on concepts related to\nconvex optimization problems, while [ 7] borrows formalization and reasoning techniques from\nconcurrency theory. The Lean 4 model proposed in this paper follows the formalization in [ 7],\nwhich has the advantage of requiring far less mathematical dependencies: although Mathlib\nis equipped to reason about convex sets and functions7, it is currently lacking in advanced\nconvex optimization definitions and results as those used in [4].\nOur AMM model is based on Uniswap v2, one of the most successful AMMs so far. We\nbriefly discuss some alternative AMM constructions. Balancer [ 2] generalizes the constant-\nproduct function used by Uniswap to a constant (weighted geometric) mean f(r1,···,rn) =/producttextn\ni=1rwi\ni, where the weight wireflects the relevance of a token τiin a tuple (τ1,···,τn).\nCurve [9] mixes constant-sum and constant-product functions, aiming at a swap rate with\nsmall fluctuations for large amounts of swapped tokens. The work [ 11] studies a variant\nof the constant-product swap rate invariant, where the rate adjusts dynamically based on\noracle prices, with the goal of reducing the need for arbitrage transactions. Other approaches\naiming at the same goal are studies in [1, 11], and implemented in [3]. Extending our Lean\nformalization and results to these alternative AMM designs would require a substantial\nreworking of our model and proofs.\n7https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Convex/FunctionXX:12 Formalizing Automated Market Makers in the Lean 4 Theorem Prover\n5 Conclusions\nIn this work we have provided a formalization of AMMs in Lean. Blockchain states are\nrepresented as structures containing wallets and AMMs, and transactions as dependent types\nequipped with a function that defines the state resulting from firing the transaction. Based\non this, we have modeled the key economic notions of price, networth and gain. We have\nthen focused on the economic properties of AMMs, constructing machine-checkable proofs.\nIn Lemma 3 we have given an explicit formula for the economic gain of a user after firing a\nswap transaction. In Theorem 7 we have proved that the rational strategy for traders leads\nto the alignment between the AMM internal exchange rate and that given by price oracles.\nFinally, in Theorem 9 we have derived the amount of tokens that a trader should sell to\nmaximize the gain from a constant-product AMM.\nDesign choices\nBefore coming up with our Lean formalization, we have experimented with a few alternative\ndefinitions. Currently, we use the two pairs of atomic token types (τ0,τ1)and(τ1,τ0)to\nrepresent the same minted token type. Initially, we used the type Mof sets of atomic tokens\nof cardinality 2, and modeled wallets of minted tokens by the type M→0R≥0. However,\nusingMin the definition of AMMs turned out not to be as easy. The type M→0R2\n≥0would\nobviously not work since we would not know which value corresponds to the reserve of which\ntoken. On the other hand, the dependent type (m:M)→0Option (m→R>0)would work\nafter defining 0 := None, at the cost of losing the straightforward definition for supply of\natomic tokens. We have opted for the custom subtype R>0to represent the positive reals,\nsince they simplify writing certain definitions and proof passages (e.g., avoiding the use of\ngarbage outputs in the definitions where negative inputs would not make sense). This choice\nhowever turned out to have some cons, since it makes using Mathlib more complex, and in\nsome cases we have to coerce back to the reals anyway (e.g. when reasoning about the gain).\nUsing Mathlib’s reals would perhaps lead to a smoother treatment.\nLimitations\nCompared to real-world AMM implementations, our Lean formalization introduces a few\nsimplifications, that overall contribute to keeping our proofs manageable. Bridging the gap\nwith real AMMs would require several extensions, which we discuss below as directions for\nfuture work. AMMs typically implement a trading fee ϕ∈[0,1]that represents the portion\nof the swap amount kept by the AMM. While modeling the fee would be easy ( ϕwould be an\nadditional parameter to SXfunctions, to Swaptypes, and to transactions Tx), it would require\na major reworking of all the results that deal with swaps. Our swaps have zero-slippage , in\nthat either a swap gives exactly the amount of tokens required by a user, or they are aborted.\nWhile on the one hand this is desirable (e.g., it rules out sandwich attacks), on the other\nhand it has drawbacks related to liveness, since a user may need to repeatedly send swap\ntransactions until one is accepted. Real-world AMM implementations allow users to specify\na slippage tolerance in the form of the minimum amount of tokens they expect from a swap.\nExtending our model to encompass slippage tolerance would require to add a parameter to\neach transaction type, and a minor reworking of the results. Some AMM implementations\nallow users to create AMMs pairs involving mintedtoken types. Consequently, the tokens\nminted by these AMMs in general are “nestings” of token types. Extending our model in this\ndirection would require to replace price oracles in our results with a suitable price function.D. Pusceddu and M. Bartoletti XX:13\nReferences\n1Improving frontrunning resistance of x*y=k market makers, 2018. https://ethresear.ch/t/\nimproving-front-running-resistance-of-x-y-k-market-makers/1281 .\n2Balancer whitepaper, 2019. https://balancer.finance/whitepaper/ .\n3Mooniswap whitepaper, 2020. https://mooniswap.exchange/docs/\nMooniswapWhitePaper-v1.0.pdf .\n4Guillermo Angeris and Tarun Chitra. Improved price oracles: Constant function market\nmakers. In ACM Conference on Advances in Financial Technologies (AFT) , pages 80–91.\nACM, 2020. https://arxiv.org/abs/2003.10001 .doi:10.1145/3419614.3423251 .\n5Guillermo Angeris, Hsien-Tang Kao, Rei Chiang, Charlie Noyes, and Tarun Chitra. An analysis\nof Uniswap markets. Cryptoeconomic Systems , 1(1), 2021. doi:10.21428/58320208.c9738e64 .\n6Danil Annenkov, Jakob Botsch Nielsen, and Bas Spitters. Concert: a smart contract certifica-\ntion framework in Coq. In ACM SIGPLAN International Conference on Certified Programs\nand Proofs (CPP) , pages 215–228. ACM, 2020. doi:10.1145/3372885.3373829 .\n7MassimoBartoletti, JamesHsinyuChiang, andAlbertoLluch-Lafuente. Atheoryofautomated\nmarket makers in DeFi. Logical Methods in Computer Science , Volume 18, Issue 4, dec 2022.\nURL: https://doi.org/10.46298%2Flmcs-18%284%3A12%292022 ,doi:10.46298/lmcs-18(4:\n12)2022.\n8Certora. Formal verification of Compound’s open-oracle with Uniswap anchor. https://files.\nsafe.de.fi/safe/files/audit/pdf/CompoundUniswapAnchoredOpenOracleAug2020.pdf ,\n2020.\n9Michael Egorov. Stableswap - efficient mechanism for stablecoin, 2019. https://curve.fi/\nfiles/stableswap-paper.pdf .\n10Uri Kirstein. Detecting corner cases in Compound V3\nwith formal specifications. https://medium.com/certora/\ndetecting-corner-cases-in-compound-v3-with-formal-specifications-b7abf137fb15 ,\n2022.\n11Bhaskar Krishnamachari, Qi Feng, and Eugenio Grippo. Dynamic curves for decentralized\nautonomous cryptocurrency exchanges. In International Symposium on Foundations and\nApplications of Blockchain (FAB) , volume 92 of OASIcs, pages 5:1–5:14. Schloss Dagstuhl -\nLeibniz-Zentrum für Informatik, 2021. doi:10.4230/OASIcs.FAB.2021.5 .\n12Eske Hoy Nielsen, Danil Annenkov, and Bas Spitters. Formalising decentralised exchanges in\nCoq. InACM SIGPLAN International Conference on Certified Programs and Proofs (CPP) ,\npages 290–302. ACM, 2023. doi:10.1145/3573105.3575685 .\n13Jiahua Xu, Krzysztof Paruch, Simon Cousaert, and Yebo Feng. Sok: Decentralized exchanges\n(DEX) with automated market maker (AMM) protocols. ACM Comput. Surv. , 55(11):238:1–\n238:50, 2023. doi:10.1145/3570639 ."    },    {        "title": "2402.06142v2.Developing_a_Theoretical_Model_for_the_Resummation_of_Infrared_Effects_in_the_Post_Reconstruction_Power_Spectrum__youtu_be_u1_xx3_4xCg_.pdf",        "content": "arXiv:2402.06142v2  [astro-ph.CO]  28 Feb 2024Developing a Theoretical Model for the Resummation of Infra red Effects\nin the Post-Reconstruction Power Spectrum\nNaonori Sugiyama1,∗\n1National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan\n(Dated: February 29, 2024)\nSince galaxy distribution reconstruction effectively redu ces non-Gaussian terms in the power spectrum co-\nvariance matrix, it has attracted interest not only for Bary on Acoustic Oscillation (BAO) signals but also for\nvarious cosmological signal analyses. To this end, this pap er presents a novel theoretical model that addresses\ninfrared (IR) effects in the post-reconstruction galaxy pow er spectrum, including 1-loop corrections. In partic-\nular, we discuss the importance of incorporating non-pertu rbative effects arising from IR contributions into the\ndisplacement vector /u1D494used for reconstruction. Consequently, post-reconstruct ion nonlinear damping of BAO\ncan be described by a single two-dimensional Gaussian funct ion. This is a phenomenon not observed when /u1D494is\nconsidered to at a linear order in the Zel’dovich approximat ion. Furthermore, we confirm that the cross-power\nspectrum of the pre- and post-reconstruction density fluctu ations lacks IR effect cancellations, and shows an\nexponential decay in both the cross-power spectrum and the a ssociated shot-noise term.\nI. INTRODUCTION\nThe reconstruction of galaxy distributions [1] has traditi on-\nally been a focus of research to amplify baryon acoustic os-\ncillation (BAO) signals [2, 3]. However, recent findings ind i-\ncate that these reconstruction techniques can effectively r educe\nnon-Gaussian terms in the covariance matrix of cosmologica l\n/u1D45B-point statistics. As a result, their application has expan ded\nbeyond BAO signals to encompass a variety of cosmologi-\ncal signals. In particular, there is a growing interest in im -\nproving constraints on cosmological parameters through po st-\nreconstruction two-point statistics [4, 5]. In addition, a pplica-\ntions in three-point statistics have been reported to stren gthen\nconstraints on primordial non-Gaussianity [6].\nDespite the potential of reconstruction techniques, their\npractical applications, particularly beyond BAO signal an al-\nysis, are still emerging. This slow advancement is due to\na limited understanding of the nonlinear characteristics o f\nreconstructed density fluctuations. The examination of pre -\nreconstruction two-point correlation functions and power spec-\ntra primarily employs advanced theoretical models, which e x-\ntend beyond Standard Perturbation Theory (SPT) [e.g., 7], t o\nprecisely extract small-scale information. For instance, so-\nphisticated theoretical models such as the Convolution La-\ngrangian Perturbation Theory (CLPT) [8, 9], the Renormal-\nized Perturbation Theory (RPT) [10], the TNS model [11],\nand the Effective Field Theory of Large-scale Structure (EFT\nof LSS) [12, 13] have been applied to analyze actual galaxy\nsurvey datasets, such as the Baryon Oscillation Spectrosco pic\nSurvey Data Release 12 (BOSS DR12) galaxy data [14–17].\nOn the other hand,theoretical research into post-reconstr uction\ndensity fluctuations has been limited to 1-loop calculation s in\nSPT [18–20] and methods based on the Zel’dovich approx-\nimation [21–24]. See also [25, 26] for perturbation theory\napproaches to iterative reconstruction methods [27–34].\nIn this study, we primarily investigate the infrared (IR) eff ect\non the power spectrum of galaxies. This effect possesses\n∗nao.s.sugiyama@gmail.coma unique property that allows for non-perturbative analysi s,\nmaking it crucial for advancing theories beyond SPT. For the\npre-reconstruction scenario, several IR effects are found, and\nwe summarize them below. When taking the IR limit and\nconsidering only the nonlinearities arising from IR effects , all\nIR nonlinear effects are negated in the calculations of the po wer\nspectrum, leaving only the linear power spectrum [35–43].\nThis IR cancellation holds true when evaluating all density\nfluctuations at equal times. However, assessing the correla tion\nof fluctuations at distinct times yields the consistency rel ation\nfor the Large Scale Structure (LSS), which uniquely connect s\nhigher-orderstatistics to their lower-ordercounterpart s [37, 38,\n44]. Moreover, proper consideration of the IR effect can shed\nlight on the nonlinear decay of the BAO signal [40, 45–55].\nThis paper aims to comprehensively elucidate the impact\nof non-perturbative IR effects on the BAO signal and the\noverall power spectrum shape after reconstruction. In this\ncontext, we present a new IR-resummed model of the post-\nreconstruction power spectrum, including 1-loop correcti ons.\nConcurrently, we explore its relation to established theor ies\nthat address partial IR effects after reconstruction, such a s the\n1-loop correction in SPT and the Zel’dovich approximation.\nWe further demonstrate the fundamental significance of the\nIR effect in the cross power spectrum between the pre- and\npost-reconstruction density fluctuations. The proposed mo del\nshould be directly applicable to observational galaxy data as it\nincorporates Redshift Space Distortions (RSDs) [e.g., 56] and\ngalaxy bias effects [e.g., 57].\nThe outline of this paper is as follows. Section II reviews\nthe non-perturbative treatment of infrared (IR) effects on p re-\nreconstruction density fluctuations. Section III explores the IR\neffects on post-reconstructiondensity fluctuations and pre sents\na model for their power spectrum. Section IV calculates the\ncross power spectrum of density fluctuations before and afte r\nreconstruction, highlighting the importance of nonlinear IR\neffects. Section V concludes with the findings and implicatio ns\nin this paper. Appendix A introduces a kind of modified\ngravity theory as an example of breaking the IR cancellation .\nAppendices B and C summarize the behavior in the IR limit\nfor solutions up to third-order fluctuations in SPT before an d2\nafter reconstruction, respectively.\nII. PRE-RECONSTRUCTION CASE\nIn preparation for the post-reconstruction case, we review\nthe derivation for the IR-resummed power spectrum model in\nthe pre-reconstruction case. This section provides an abbr e-\nviated version of the proof found in [39, 40, 55]. For a more\ncomplete understanding, see the original papers, which det ail\nthe calculations; see e.g., [49–54, 58] for other approache s to\ndealing with IR effects.\nA. Infrared (IR) effects on dark matter density fluctuations\nConsider a density field function /u1D70C(/u1D499)defined over three-\ndimensional spatial coordinates /u1D499. Let/u1D6FF(/u1D499)represent the\ndeviation from the mean density field ¯ /u1D70C. This relation can be\nexpressed as /u1D70C(/u1D499)=¯/u1D70C(1+/u1D6FF(/u1D499)). The Fourier transform of\n/u1D6FF(/u1D499)can be represented as /tildewide/u1D6FF(/u1D48C)=∫\n/u1D4513/u1D465/u1D452−/u1D456/u1D499·/u1D48C/u1D6FF(/u1D499), where\nthe tilde denotes a Fourier-transformed quantity.\nLet us begin by considering the density fluctuations of dark\nmatter, denoted /u1D6FFm(/u1D499), wherein the subscript “m” represents\n“matter”. In perturbation theory, an /u1D45Bth-order dark matter\ndensity fluctuation in Fourier space, denoted /tildewide/u1D6FF[/u1D45B]\nm(/u1D48C), can\nbe represented by the convolution integral of /u1D45Bwave vec-\ntors (/u1D4911,...,/u1D491/u1D45B) with an associated /u1D45Bth-order kernel function\n/u1D439/u1D45B(/u1D4911,...,/u1D491/u1D45B). Hence, the /u1D45Bth-order density fluctuation is\nexpressed as\n/tildewide/u1D6FF[/u1D45B]\nm(/u1D48C)=/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])\n×/u1D439/u1D45B(/u1D4911,...,/u1D491/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D4911)···/tildewide/u1D6FF[1]\nm(/u1D491/u1D45B),(1)\nwhere/u1D491[1,/u1D45B]=/u1D4911+···+/u1D491/u1D45B, and/tildewide/u1D6FF[1]\nmis the linear (first-order)\ndark matter density fluctuation in Fourier space, and /u1D6FFDis the\nthree-dimensional delta function.\n1. Second order fluctuations\nFor example, the second-order kernel function is given by\n/u1D4392(/u1D4911,/u1D4912)=17\n21+1\n2/u1D70712/parenleftbigg/u1D45D1\n/u1D45D2+/u1D45D2\n/u1D45D1/parenrightbigg\n+2\n7/parenleftbigg\n/u1D7072\n12−1\n3/parenrightbigg\n,(2)\nwhere/u1D45D/u1D45B=|/u1D491/u1D45B|and/u1D70712=ˆ/u1D45D1·ˆ/u1D45D2with ˆ/u1D45D/u1D45B=/u1D491/u1D45B//u1D45D/u1D45B. Substi-\ntuting Eq. (2) into Eq. (1) and performing the inverse Fourie r\ntransform, we obtain\n/u1D6FF[2]\nm(/u1D499)=17\n21/bracketleftbig\n/u1D6FF[1]\nm(/u1D499)/bracketrightbig2−/u1D6BF[1](/u1D499) ·∇/u1D6FF[1]\nm(/u1D499)\n+2\n7/bracketleftbigg/parenleftbigg/u1D715/u1D456/u1D715/u1D457\n/u1D7152−1\n3/u1D6FF/u1D456/u1D457/parenrightbigg\n/u1D6FF[1]\nm(/u1D499)/bracketrightbigg2\n. (3)\nOn the right-hand side of Eq. (3), the three terms from left\nto right are respectively named the “nonlinear growth term” ,“shift term”, and “tidal term” [18, 59]. The linear displace ment\nvector, associated with the shift term, is defined by the line ar\ndark matter density fluctuation as\n/u1D6BF[1](/u1D499)=/uni222B.dsp/u1D4513/u1D458\n(2/u1D70B)3/u1D452/u1D456/u1D48C·/u1D499/u1D456/u1D48C\n/u1D4582/tildewide/u1D6FF[1]\nm(/u1D48C). (4)\nIn the IR limit, where one of the wave vectors in /u1D4392(/u1D4911,/u1D4912)\nis much smaller than the other (specifically, when /u1D45D2→0),\nthe following relation holds:\n/u1D4392(/u1D4911,/u1D4912) − −−− →\n/u1D45D2→01\n2/parenleftBigg\n/u1D4911·/u1D4912\n/u1D45D2\n2/parenrightBigg\n. (5)\nWhen substituting Eq. (5) into Eq. (1), the following approx i-\nmation is considered simultaneously:\n/u1D6FFD(/u1D48C−/u1D4911−/u1D4912) − −−− →\n/u1D45D2→0/u1D6FFD(/u1D48C−/u1D4911). (6)\nConsequently, in the IR limit, we derive:\n/u1D6FF[2]\nm(/u1D499) − −−−− →\nIR limit−/u1D6BF[1]·∇/u1D6FF[1]\nm(/u1D499), (7)\nwhere\n/u1D6BF[1]=/u1D6BF[1](/u1D499=0). (8)\nIn deriving Eq. (7), we also accounted for the condition /u1D45D1→0\nand introduced a pre-factor of 2. Comparison with Eq. (3)\nreveals that the IR limit solution involves isolating only t he\nshift term, and furthermore, the displacement vector conta ined\nin the shift term is evaluated at the origin /u1D499=0.\nDecomposing the second-order fluctuation solution into\nterms manifesting in the IR limit and the others, the solutio n\ncan be reformulated as\n/u1D6FF[2]\nm=/u1D6FF[2]\n(S)m(/u1D499) −/u1D6BF[1]·∇/u1D6FF[1]\nm(/u1D499), (9)\nwhere the subscript (S)denotes the contribution from short-\nwavelength modes, and /u1D6FF[2]\n(S)m(/u1D499)is explicitly given by\n/u1D6FF[2]\n(S)m(/u1D499)=17\n21/bracketleftBig\n/u1D6FF[1]\nm(/u1D499)/bracketrightBig2\n−/u1D6BF[1]\n(S)(/u1D499) ·∇/u1D6FF[1]\nm(/u1D499)\n+2\n7/bracketleftbigg/parenleftbigg/u1D715/u1D456/u1D715/u1D457\n/u1D7152−1\n3/u1D6FF/u1D456/u1D457/parenrightbigg\n/u1D6FF[1]\nm(/u1D499)/bracketrightbigg2\n. (10)\nIn the equation above, we defined the displacement vector\ncontributing from short-wavelength modes as\n/u1D6BF[1]\n(S)(/u1D499)=/u1D6BF[1](/u1D499) −/u1D6BF[1], (11)\nwhere/u1D6BF[1]\n(S)(/u1D499=0)=0.\n2. Non-perturbative treatment\nIn the IR limit of the /u1D45Bth-order kernel function\n/u1D439/u1D45B(/u1D4911,···,/u1D491/u1D45B), one of the /u1D45Bwavenumbers contributingto the3\nscale of interest /u1D458=|/u1D491[1,/u1D45B]|possesses a magnitude substan-\ntially smaller than the other (/u1D45B−1)wavenumbers. For a single\ndark matter fluid in General Relativity (GR), the recursion r ela-\ntion allows the determination of solutions for /u1D439/u1D45B(/u1D4911,···,/u1D491/u1D45B)\nup to an infinite order (e.g., see [7]). Using mathematical in -\nduction on this recursion relation, the relation below can b e\nproven [39]:\n/u1D439/u1D45B(/u1D4911,...,/u1D491/u1D45B−1,/u1D491) − −−− →\n/u1D45D→01\n/u1D45B/parenleftbigg/u1D491[1,/u1D45B−1]·/u1D491\n/u1D45D2/parenrightbigg\n×/u1D439/u1D45B−1(/u1D4911,...,/u1D491/u1D45B−1).(12)\nMore generally, for /u1D45Fwavenumbers approaching zero,\n/u1D439/u1D45B(/u1D4911,...,/u1D491/u1D45B−/u1D45F,/u1D491/u1D45B−/u1D45F+1,...,/u1D491/u1D45B) − −−−−−−−−−−−− →\n/u1D45D/u1D45B−/u1D45F+1,...,/u1D45D/u1D45B→0\n(/u1D45B−/u1D45F)!\n/u1D45B!/u1D45F/productdisplay.1\n/u1D6FC=1/parenleftbigg/u1D491[1,/u1D45B−/u1D45F]·/u1D491/u1D45B−/u1D45F+/u1D6FC\n/u1D45D2\n/u1D45B−/u1D45F+/u1D6FC/parenrightbigg\n/u1D439/u1D45B−/u1D45F(/u1D4911,...,/u1D491/u1D45B−/u1D45F).(13)\nIn this context, we decompose the linear dark matter density\nfluctuation into long-wavelength modes, with wavenumbers\napproaching zero, and short-wavelength modes which are in-\ndependent of the former:\n/tildewide/u1D6FF[1]\nm(/u1D491)=/tildewide/u1D6FF[1]\nm(/u1D491→0) +/tildewide/u1D6FF[1]\n(S)m(/u1D491). (14)\nSubstituting Eqs. (13) and (14) into Eq. (1) and using the\napproximation\n/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B]) − −−−−−−−−−−−− →\n/u1D45D/u1D45B−/u1D45F+1,...,/u1D45D/u1D45B→0/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B−/u1D45F]),(15)\nthe dark matter density fluctuation can be formally expresse d\nas [40]\n/u1D6FFm(/u1D499)=/u1D6FF(S)m(/u1D499−/u1D6BF[1]), (16)\nwhere the /u1D45B-th order of /u1D6FF(S)m(/u1D499)is given by\n/u1D6FF[/u1D45B]\n(S)m(/u1D499)=/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3/u1D452/u1D456(/u1D4911+···+/u1D491/u1D45B)·/u1D499\n×/u1D439/u1D45B(/u1D4911,...,/u1D491/u1D45B)/tildewide/u1D6FF[1]\n(S)m(/u1D4911)···/tildewide/u1D6FF[1]\n(S)m(/u1D491/u1D45B).(17)\nThus, the IR limit effect in a single dark matter fluid can be\ninterpreted as a result of a coordinate transformation due t o\nthe displacement vector at the origin.\nThe IR limit is an operation that extracts nonlinear contrib u-\ntions from large scales (small wavenumbers). In other words ,\nthe wavenumber /u1D458of interest is considerably larger than the\nwavenumbers contributed in the IR limit. Therefore, histor -\nically, the IR limit solution has also been referred to as the\nhigh-/u1D458(/u1D458→ ∞ ) limit solution (e.g., see [60]). Conversely,\nthe ultraviolet (UV) limit corresponds to the low- /u1D458(/u1D458→0)\nlimit and is treated by e.g. EFT of LSS [12, 13].\nIn summary, this paper makes three approximations related\nto the IR limit:\n1. As illustrated in Eqs. (5) and (13), for the kernel functio n\n/u1D439/u1D45B(/u1D4911,···,/u1D491/u1D45B)that characterizes nonlinear effects, weemploy the approximate kernel function when one or\nmultiple wavenumbers approach zero. This means that\nthe dominant effect in the IR limit appears as the co-\nordinate transformation of the short-wavelength density\nfluctuation /u1D6FF(S)due to/u1D6BFas in Eq. (16).\n2. As presented in Eqs. (6) and (15), within the delta func-\ntion/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])that characterizes the mode coupling\nintegral with multiple wave vectors, we assume no mode\ncoupling occurs between the IR effect and other compo-\nnents when one or more wavenumbers approach zero.\nThis implies that the displacement vector is evaluated\nat the origin in Eq. (8), and furthermore, indicates no\ncorrelation between /u1D6FF(S)and/u1D6BF, as will be discussed in\nSection II C.\n3. The nonlinear effects of /u1D6FF(S)mare truncated at a finite\nperturbation order. Any higher order perturbative effects\nbeyond this are treated as IR effects through /u1D6BF.\nThe clearest example in the IR limit occurs when /u1D6FF(S)m(/u1D499) ∼\n/u1D6FF[1]\nm(/u1D499), approximating the nonlinear dark matter density fluc-\ntuation as\n/u1D6FFm(/u1D499) − −−−− →\nIR limit/u1D6FF[1]\nm(/u1D499−/u1D6BF[1]). (18)\nThe conclusions of this paper will be mainly based on this sim -\nplest case in Sections III and IV. When higher order correcti on\nterms up to the 1-loop order are to be considered, they will be\nexplained on there as appropriate.\nB. Extensions to RSD and bias effects\nIn cases affected by RSD and bias effects, such as observed\ngalaxies, the use of Lagrangian Perturbation Theory (LPT) i s\nadvantageous. This is because LPT naturally accommodates\ndisplacement vectors. Within LPT, RSD effects are integrate d\ninto the displacement vector. Bias effects, on the other hand ,\ncan be treated using the equation [61]:\n1+/u1D6FFg(/u1D499,ˆ/u1D45B)\n=/uni222B.dsp\n/u1D4513/u1D45E[1+/u1D6FFbias(/u1D492)]/u1D6FFD(/u1D499−/u1D492−/u1D6BFred(/u1D492,ˆ/u1D45B)),(19)\nwhere/u1D6FFgrepresents the galaxy density fluctuation, /u1D6FFbiasis\nthe density fluctuation including the bias effect in Lagrangi an\nspace,/u1D6BFredis the displacement vector including the RSD ef-\nfect, and ˆ /u1D45Bis a unit vector directed along the line of sight\n(LOS). The relation between /u1D6BFand/u1D6BFredis given by\n/u1D6BFred=/u1D6BF+/dotacc/u1D6BF·ˆ/u1D45B\n/u1D44E/u1D43Bˆ/u1D45B, (20)\nwhere/dotacc/u1D6BF(/u1D492)is the time derivative of /u1D6BF(/u1D492), and/u1D44Eand/u1D43Bare\nthe scale factor and the Hubble parameter, respectively.\nThe/u1D45Bth-order of /u1D6BF(/u1D492)in Fourier space is represented as\n/tildewide/u1D6BF[/u1D45B]\nm(/u1D48C)=/u1D456/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])\n×/u1D473/u1D45B(/u1D4911,...,/u1D491/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D4911)···/tildewide/u1D6FF[1]\nm(/u1D491/u1D45B),(21)4\nwhere the nonlinear kernel vector functions up to the second -\norder are given by\n/u1D4731(/u1D4911)=/u1D4911\n/u1D45D2\n1,\n/u1D4732(/u1D4911,/u1D4912)=3\n14/u1D491[1,2]\n|/u1D491[1,2]|2/parenleftBig\n1−/u1D7072\n12/parenrightBig2\n. (22)\nNote that the second order kernel vector function /u1D4732(/u1D4911,/u1D4912)\ndoes not include a shift term but consists of a combination of\na growth term and a tidal term, unlike /u1D4392(/u1D4911,/u1D4912).\nThe displacement vector is decomposed into two parts: the\nvalue at the origin /u1D492=0and the other values. Thus, we can\nwrite\n/u1D6BFred(/u1D492,ˆ/u1D45B)=/u1D6BFred(ˆ/u1D45B) +/u1D6BF(S)red(/u1D492,ˆ/u1D45B). (23)\nHere,/u1D6BF(S)red=/u1D6BFred(/u1D492) −/u1D6BFredis termed as the short-\nwavelength mode of /u1D6BFred. Consequently, /u1D6FFgcan be formally\nrewritten as\n/u1D6FFg(/u1D499,ˆ/u1D45B)=/u1D6FF(S)g(/u1D499−/u1D6BFred(ˆ/u1D45B),ˆ/u1D45B), (24)\nwhere\n1+/u1D6FF(S)g(/u1D499,ˆ/u1D45B)\n=/uni222B.dsp\n/u1D4513/u1D45E[1+/u1D6FFbias(/u1D492)]/u1D6FFD/parenleftbig/u1D499−/u1D492−/u1D6BF(S)red(/u1D492,ˆ/u1D45B)/parenrightbig,(25)\nIn the above derivation, we used 1 =∫\n/u1D4513/u1D45E/u1D6FFD(/u1D499−/u1D492)=∫\n/u1D4513/u1D45E/u1D6FFD(/u1D499−/u1D492−/u1D6BFred(ˆ/u1D45B)). In Fourier space, it becomes\n/tildewide/u1D6FFg(/u1D48C,ˆ/u1D45B)=/u1D452−/u1D456/u1D48C·/u1D6BFred(ˆ/u1D45B)/tildewide/u1D6FF(S)g(/u1D48C,ˆ/u1D45B). (26)\nNote that Eq. (24) is simply a rewrite of Eq. (19) and, unlike\nEq. (16), is not the result of solving the equation to perturb a-\ntive infinite order in the IR limit. To consider Eq. (24) as an\nextension of the solution of Eq. (16) that takes into account\nthe bias effect, the RSD effect, and even nonlinear effects on\nthe displacement vector, one must make an assumption. That\nis,in the IR limit, all contributions from IR modes appear as\nthe coordinate transformation through /u1D6BFred. This is expected\nto be true for a single dark matter fluid in GR. However, in\nAppendix A, we introduce Degenerate Higher-Order Scalar-\nTensor (DHOST) theories (for reviews, see e.g. [62, 63]), a\ntype of modified gravity theory, as an example of how a sim-\nple rewrite of Eq. (19) into Eq. (24) cannot extract only the\nIR effect through /u1D6BFred. Also, while this assumption is consid-\nered valid for the standard bias model in GR (e.g., see [57]),\nthere is a possibility that it may be violated by other bias ef -\nfects in the context of modified gravity theories [64, 65] and\nmulticomponent fluids [66].\n1. Perturbative expansion\nIn Fourier space, the /u1D45Bth-order term of the galaxy density\nfluctuation can be expressed as\n/tildewide/u1D6FF[/u1D45B]\ng(/u1D48C)=/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])\n×/u1D44D/u1D45B(/u1D4911,...,/u1D491/u1D45B,ˆ/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D4911)···/tildewide/u1D6FF[1]\nm(/u1D491/u1D45B).(27)Here,/u1D44D/u1D45B(/u1D4911,···,/u1D491/u1D45B)represents nonlinear kernel functions\nthat incorporate both the bias effect and the RSD effect.\nApproximating both /u1D6FF(S)(/u1D499)and/u1D6BFredas linear, we then\nobtain [55]\n/u1D6FFg(/u1D499,ˆ/u1D45B) − −−−− →\nIR limit/u1D6FF[1]\ng(/u1D499−/u1D6BF[1]\nred(ˆ/u1D45B),ˆ/u1D45B),\n/tildewide/u1D6FFg(/u1D48C,ˆ/u1D45B) − −−−− →\nIR limit/u1D452−/u1D456/u1D48C·/u1D6BF[1]\nred(ˆ/u1D45B)/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B). (28)\nThe linear galaxy density fluctuation is given by\n/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B)=/u1D44D[1](/u1D48C,ˆ/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D48C), (29)\nwhere/u1D44D1(/u1D48C,ˆ/u1D45B)is the first order kernel function [56]:\n/u1D44D1(/u1D48C,ˆ/u1D45B)=/u1D44F1+/u1D453/u1D7072. (30)\nHere,/u1D44F1is the linear bias parameter, /u1D453represents the linear\ngrowth rate function, and /u1D707=ˆ/u1D458·ˆ/u1D45Bdenotes the cosine of the\nangle between ˆ/u1D458and ˆ/u1D45B.\nFrom Eq. (26), up to the third-order in perturbation theory,\n/tildewide/u1D6FF[/u1D45B]\n(S)g(/u1D48C,ˆ/u1D45B)are expressed as\n/tildewide/u1D6FF[1]\n(S)g(/u1D48C,ˆ/u1D45B)=/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B)\n/tildewide/u1D6FF[2]\n(S)g(/u1D48C,ˆ/u1D45B)=/tildewide/u1D6FF[2]\ng(/u1D48C,ˆ/u1D45B) −/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]\nred(ˆ/u1D45B)/parenrightBig\n/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B)\n/tildewide/u1D6FF[3]\n(S)g(/u1D48C,ˆ/u1D45B)=/tildewide/u1D6FF[3]\ng(/u1D48C,ˆ/u1D45B) −/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[2]\nred(ˆ/u1D45B)/parenrightBig\n/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B)\n−/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]\nred(ˆ/u1D45B)/parenrightBig\n/tildewide/u1D6FF[2]\ng(/u1D48C,ˆ/u1D45B)\n+1\n2/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]\nred(ˆ/u1D45B)/parenrightBig2/tildewide/u1D6FF[1]\ng(/u1D48C,ˆ/u1D45B). (31)\nThe/u1D45Bth-order displacement vector in redshift space is com-\nputed by the following linear transformation as [61]\n/u1D6BF[/u1D45B]\nred(/u1D492,ˆ/u1D45B)=/u1D479/u1D45B(ˆ/u1D45B) ·/u1D6BF[/u1D45B](/u1D492). (32)\nThe transformation matrix is given by\n[/u1D479/u1D45B]/u1D456/u1D457=/u1D470/u1D456/u1D457+/u1D45B /u1D453ˆ/u1D45B/u1D456ˆ/u1D45B/u1D457, (33)\nwhere/u1D470is the three-dimensional identity matrix, and /u1D456, /u1D457=\n1,2,3.\nThroughout the remainder of this paper, we always con-\nsider both RSD and bias effects. To simplify notation, the\nsubscripts “g” and “red”, as well as the LOS-dependence ˆ /u1D45B,\nare omitted in the subsequent sections. We retain the sub-\nscript “m” only for quantities related to dark matter: i.e.,\n/u1D6BFred(/u1D492,ˆ/u1D45B)=/u1D6BF(/u1D492),/u1D6BFred(ˆ/u1D45B)=/u1D6BF,/u1D6FFg(/u1D499,ˆ/u1D45B)=/u1D6FF(/u1D499), and\n/u1D44D/u1D45B(/u1D4911,···,/u1D491/u1D45B,ˆ/u1D45B)=/u1D44D/u1D45B(/u1D4911,···,/u1D491/u1D45B). Furthermore, a sim-\nple right arrow will represent the IR limit: − −−−− →\nIR limit=→.\nC. IR-cancellation of two-point statistics in the IR limit\nIn this subsection, we provide a brief overview of the IR\neffect cancellation in the 2PCF and its Fourier-transformed5\ncounterpart, the power spectrum. A fundamental assumption\nisthe complete absence of correlation between /u1D6FF(S)(/u1D499)and/u1D6BF,\nwhich corresponds to the second assumption in the IR limit\ndiscussed at the end of Section II A. Owing to the translation al\nsymmetry of the ensemble average, all IR effects originating\nfrom/u1D6BFare canceled out, leaving only the short-wavelength\n2PCF:\n/angb∇acketleft/u1D6FF(/u1D499)/u1D6FF(/u1D499′)/angb∇acket∇ight=/angbracketleftbig\n/u1D6FF(S)(/u1D499−/u1D6BF)/u1D6FF(S)(/u1D499′−/u1D6BF)/angbracketrightbig\n→/angbracketleftbig/u1D6FF(S)(/u1D499)/u1D6FF(S)(/u1D499′)/angbracketrightbig. (34)\nAssuming that all nonlinear effects emerge only from the IR\neffects through /u1D6BF[1], then the nonlinear 2PCF reduces to the\nlinear 2PCF:\n/angb∇acketleft/u1D6FF(/u1D499)/u1D6FF(/u1D499′)/angb∇acket∇ight →/angbracketleftbig\n/u1D6FF[1](/u1D499−/u1D6BF[1])/u1D6FF[1](/u1D499′−/u1D6BF[1])/angbracketrightbig\n=/angbracketleftbig/u1D6FF[1](/u1D499)/u1D6FF[1](/u1D499′)/angbracketrightbig. (35)\nIn Fourier space, it becomes\n/angb∇acketleft/u1D6FF(/u1D48C)/u1D6FF(/u1D48C′)/angb∇acket∇ight →/angbracketleftbig\n/u1D452−/u1D456/u1D48C·/u1D6BF[1]\n/u1D452−/u1D456/u1D48C′·/u1D6BF[1]/angbracketrightbig/angbracketleftbig/tildewide/u1D6FF[1](/u1D48C)/tildewide/u1D6FF[1](/u1D48C′)/angbracketrightbig\n=(2/u1D70B)3/u1D6FFD(/u1D48C+/u1D48C′)[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458),(36)\nwhere/u1D443[11]\nm(/u1D458)is the linear matter power spectrum, calculated\nas\n/angb∇acketleft/tildewide/u1D6FF[1]\nm(/u1D48C)/tildewide/u1D6FF[1]\nm(/u1D48C′)/angb∇acket∇ight=(2/u1D70B)3/u1D6FFD(/u1D48C+/u1D48C′)/u1D443[11]\nm(/u1D458).(37)\nIn the following, we provide a detailed analysis of the ex-\nponential function/angbracketleftbig\n/u1D452−/u1D456/u1D48C·/u1D6BF[1]\n/u1D452−/u1D456/u1D48C′·/u1D6BF[1]/angbracketrightbig\nin Eq. (36). Using\ncumulants, denoted by /angb∇acketleft···/angb∇acket∇ight c/one.sup, this can be reconstructed as\n/angbracketleftbig\n/u1D452−/u1D456/u1D48C·/u1D6BF[1]\n/u1D452−/u1D456/u1D48C′·/u1D6BF[1]/angbracketrightbig\n=D(/u1D48C)D(/u1D48C′)E(/u1D48C,/u1D48C′), (39)\nwhereD(/u1D48C)andE(/u1D48C,/u1D48C′)are defined as\nD(/u1D48C)=exp/parenleftbigg1\n2/angbracketleftbigg/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]/parenrightBig2/angbracketrightbigg\nc/parenrightbigg\n,\nE(/u1D48C,/u1D48C′)=exp/parenleftBig/angbracketleftBig/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]/parenrightBig /parenleftBig\n−/u1D456/u1D48C′·/u1D6BF[1]/parenrightBig/angbracketrightBig\nc/parenrightBig\n.(40)\nThe function D(/u1D48C)is subsequently described as a two-\ndimensional exponentially decaying function [45–48]:\nD(/u1D48C)=exp/parenleftBigg\n−/u1D4582(1−/u1D7072)/u1D70E2\n⊥+/u1D4582/u1D7072/u1D70E2\n/ba∇dbl\n2/parenrightBigg\n, (41)\nwhere the radial and transverse components of smoothing fac -\ntors,/u1D70E2\n/ba∇dbland/u1D70E2\n⊥, are given by\n/u1D70E2\n⊥=1\n3/uni222B.dsp/u1D451/u1D45D\n2/u1D70B2/u1D443[11]\nm(/u1D45D),\n/u1D70E2\n/ba∇dbl=(1+/u1D453)2/u1D70E2\n⊥. (42)\n/one.supThe cumulants of a statistical quantity /u1D44B, denoted /angb∇acketleft/u1D44B/u1D45B/angb∇acket∇ightc, are obtained from\na power series expansion of the cumulant-generating functi on ln/angb∇acketleft/u1D452/u1D44B/angb∇acket∇ight:\nln/parenleftBig\n/angb∇acketleft/u1D452/u1D44B/angb∇acket∇ight/parenrightBig\n=/summationdisplay.1\n/u1D45B=11\n/u1D45B!/angb∇acketleft/u1D44B/u1D45B/angb∇acket∇ightc. (38)In addition, the function E(/u1D48C,/u1D48C′)satisfies\nE(/u1D48C,/u1D48C)=D2(/u1D48C),\nE(/u1D48C,−/u1D48C′)=E−1(/u1D48C,/u1D48C′),\nE(/u1D48C,−/u1D48C)=D−2(/u1D48C), (43)\nand generates an exponentially increasing function as the\nsquare of the inverse of Eq. (41) when the statistical trans-\nlational symmetry /u1D48C+/u1D48C′=0is satisfied. Finally, as indicated\nin Eq. (36), the nonlinear galaxy power spectrum converges t o\nthe linear galaxy power spectrum in the IR limit:\n/u1D443(/u1D48C) →/angbracketleftbig\n/u1D452−/u1D456/u1D48C·/u1D6BF[1]\n/u1D452/u1D456/u1D48C·/u1D6BF[1]/angbracketrightbig\n[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458)\n=D2(/u1D48C)D−2(/u1D48C)[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458)\n=[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458). (44)\nD. Relations to previous works\n1. 1-loop corrections in SPT\nConsider the next leading order, i.e., the 1-loop order con-\ntribution, in the framework of SPT [7]. This is divided\ninto/u1D443[22](/u1D48C), which is the product of the dual second-\norder density fluctuations, and /u1D443[13](/u1D48C), which is the prod-\nuct of the first- and third-order density fluctuations, where\n/u1D4431-loop(/u1D48C)=/u1D443[22](/u1D48C) +/u1D443[13](/u1D48C).\nIn this context, /u1D44322and/u1D44313become\n/u1D443[22](/u1D48C)=2/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3/uni222B.dsp/u1D4513/u1D45D2\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,2])\n× [/u1D44D2(/u1D4911,/u1D4912)]2/u1D443[11]\nm(/u1D45D1)/u1D443[11]\nm(/u1D45D2),\n/u1D443[13](/u1D48C)=6/u1D44D1(/u1D48C)/u1D443[11]\nm(/u1D458)\n×/uni222B.dsp/u1D4513/u1D45D\n(2/u1D70B)3/u1D44D3(/u1D48C,/u1D491,−/u1D491)/u1D443[11]\nm(/u1D45D). (45)\nIn the IR limit, both /u1D44D2(/u1D48C,/u1D491)and/u1D44D3(/u1D48C,/u1D491,−/u1D491)are approxi-\nmated as (see Appendix B)\n/u1D44D2(/u1D48C,/u1D491) − −−− →\n/u1D45D→01\n2/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n/u1D44D1(/u1D48C),\n/u1D44D3(/u1D48C,/u1D491,−/u1D491) − −−− →\n/u1D45D→0−1\n3!/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg2\n/u1D44D1(/u1D48C),(46)\nwhere/u1D4791is the transformation matrix given in Eq. (33). Con-\nsequently, /u1D443[22]and/u1D443[13]in the IR limit are described as\n/u1D443[22]\nIR(/u1D48C) →/parenleftBig\n/u1D4582(1−/u1D7072)/u1D70E2\n⊥+/u1D4582/u1D7072/u1D70E2\n/ba∇dbl/parenrightBig\n× [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458),\n/u1D443[13]\nIR(/u1D48C) → −/parenleftBig\n/u1D4582(1−/u1D7072)/u1D70E2\n⊥+/u1D4582/u1D7072/u1D70E2\n/ba∇dbl/parenrightBig\n× [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458), (47)\nwhere the subscript “IR” means the values evaluated in the\nIR limit. When summing up /u1D443[22]\nIRand/u1D443[13]\nIR, they mutually6\ncancel out, resulting in a net value of zero. These results al ign\nwith the form obtained when expanding D(/u1D48C)D−1(/u1D48C)up to\nthe 1-loop order in Eq. (44).\n2.Γ-expansion\nThe nonlinear power spectrum can generally be decomposed\ninto two components. One component is proportional to the\nlinear matter power spectrum, and the other is associated wi th\nmode-coupling integrals. This relation is presented in [10 , 47]\n/u1D443(/u1D48C)=/u1D43A2(/u1D48C)/u1D443[11]\nm(/u1D458) +/u1D443MC(/u1D48C). (48)\nHere, the function /u1D43A(/u1D48C)appearing in the first term on the\nright-hand side is referred to as the propagator, and the sec ond\nterm is known as the mode-coupling term.\nTheΓ-expansion corresponds to the decomposition of the\npower spectrum within the framework of mode-coupling in-\ntegrals (e.g., see [60]). When compared with Eq. (48), the\nfirst-order Γ-expansion is related to the propagator, and the\nsum of the degrees of Γ-expansion from the second order up\nto infinity constitutes the mode-coupling term. In the conte xt\nof theΓ-expansion, the nonlinear power spectrum in the IR\nlimit is decomposed into [39, 40]:\n/u1D443(/u1D48C) → D2(/u1D48C)∞/summationdisplay.1\n/u1D45B=1/parenleftBig\n/u1D4582(1−/u1D7072)/u1D70E2\n⊥+/u1D4582/u1D7072/u1D70E2\n/ba∇dbl/parenrightBig/u1D45B−1\n(/u1D45B−1)!\n× [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458), (49)\nwhere the summation index /u1D45Bdenotes the /u1D45Bth order of the\nΓ-expansion. Furthermore, we obtain the corresponding ex-\npressions of /u1D43A2(/u1D48C)and/u1D443MC(/u1D48C)in the IR limit:\n/u1D43A2(/u1D48C)/u1D443[11]\nm(/u1D458) → D2(/u1D48C)[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458),\n/u1D443MC(/u1D48C) → D2(/u1D48C) (E(/u1D48C,−/u1D48C) −1) [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458)\n=/parenleftbig1−D2(/u1D48C)/parenrightbig[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458). (50)\nIt is important to note that the function E(/u1D48C,/u1D48C′)serves to\nconnect different wave vectors /u1D48Cand/u1D48C′. Consequently, a\nseries expansion of E(/u1D48C,/u1D48C′) −1 yields a term related to the\nmode coupling integrals.\n3. Propagator\nIn this paper, we focus on the impact of the BAO signal\ndue to non-perturbative IR effects. To that end, we provide a\ndetailed explanationof the calculation of the propagator, which\ndirectly influences the nonlinear decay of BAO.\nThe propagator is defined by the correlation between two\nfactors: nonlinear galaxy density fluctuations and linear d ark\nmatter density fluctuations.\n/angb∇acketleft/u1D6FF(/u1D499)/u1D6FF[1]\nm(/u1D499′)/angb∇acket∇ight=/uni222B.dsp/u1D4513/u1D458\n(2/u1D70B)3/u1D452/u1D456/u1D48C·(/u1D499−/u1D499′)/u1D43A(/u1D48C)/u1D443[11]\nm(/u1D458).(51)Note that this definition assumes the absence of primor-\ndial non-Gaussianities. When the nonlinear density fluctua -\ntions are decomposed into long-wavelength displacement ve c-\ntors and short-wavelength density fluctuations, as shown in\nEq. (24), the propagator can be computed as\n/angb∇acketleft/u1D6FF(S)(/u1D499−/u1D6BF)/u1D6FF[1]\nm(/u1D499′)/angb∇acket∇ight\n=/uni222B.dsp/u1D4513/u1D458\n(2/u1D70B)3/u1D452/u1D456/u1D48C·(/u1D499−/u1D499′)/angbracketleftBig\n/u1D452−/u1D456/u1D48C·/u1D6BF/angbracketrightBig\n×/bracketleftBigg\n/u1D44D1(/u1D48C)/u1D443[11]\nm(/u1D458) +∞/summationdisplay.1\n/u1D45B=1/u1D443[1(2/u1D45B+1)]\n(S)(/u1D48C)/(2/u1D44D1(/u1D48C))/bracketrightBigg\n,(52)\nwhere/u1D6FF[1]\nm(/u1D499)corresponds to short-wavelength modes and\nthus correlate only with /u1D6FF(S)(/u1D499), and/u1D443[1(2/u1D45B+1)]\n(S)represents the\ncross-power spectrum of /u1D6FF[2/u1D45B+1]\n(S)and/u1D6FF[1]\nm. Note that only odd\norders appear in /u1D443[1(2/u1D45B+1)]\n(S)since primordial non-Gaussianities\nare absent.\nBy truncating /u1D6FF(S)up to the third-order and by further ap-\nproximating /u1D6BFto be of linear order, we obtain\n/u1D43A(/u1D48C)\n≃ D(/u1D48C)/bracketleftBig\n/u1D44D1(/u1D48C) +/u1D443[13]\n(S)(/u1D48C)/slashBig /parenleftBig\n2/u1D44D1(/u1D48C)/u1D443[11]\nm(/u1D458)/parenrightBig/bracketrightBig\n.(53)\nHere,/u1D443[13]\n(S)is defined as\n/u1D443[13]\n(S)(/u1D48C)=/u1D443[13](/u1D48C) −/u1D443[13]\nIR(/u1D48C), (54)\nwhere/u1D443[13]\nIRrepresents the IR limit value of /u1D443[13]as given in\nEq. (47).\nIn the definition of the propagator (52), it is important to\nnote that nonlinear density fluctuations and linear density fluc-\ntuations possess different IR effects. In particular, there i s no\nIR cancellation based on statistical translational symmet ry in\nthe calculation of the propagator. The exponential decay of\nthe propagator is due to the absence of this IR cancellation.\nSimilarly, other higher-order coefficients in the Γ-expansion\ndecay exponentially, as shown in Eq. (49).\nThis paper repeatedly emphasizes that an exponential de-\ncay effect is consistently observed when calculating cross-\ncorrelations between fluctuations with distinct IR effects. No-\ntably, Section IV will demonstrate the exponential decay of the\ncross-power spectrum for both pre- and post-reconstructio n\ndensity fluctuations /two.sup.\nE. IR-resummed model for\nthe pre-reconstruction power spectrum\nIn the previous section, we discussed the IR cancellation\nunder the assumption that /u1D6FF[1](/u1D499)and/u1D6BF[1]are uncorrelated.\n/two.supAs another example, Chisari and Pontzen [67] showed in the Ze l’dovich ap-\nproximation that the cross-power spectrum of density fluctu ations evaluated\nat two different times decays exponentially due to IR cancell ation breaking.7\nThis lack of correlation is due to the three assumptions men-\ntioned at the end of Section II A 2, especially the one stating\nthat there is no mode coupling between the IR effect and the\nshort-wavelength density fluctuation. In practice, howeve r,\nthere is mode-coupling that prevents complete IR cancella-\ntion. By addressing this imperfection in IR cancellation an d\nisolating only the effects of BAO, we can construct a theo-\nretical template model that effectively handles the nonline ar\ndamping effects of BAO.\n1. Linear level\nAs an example, the lowest-order term among the mode-\ncoupling terms is /u1D443[22](/u1D48C). Applying the delta function with\nthe limit/u1D45D2→0 (or/u1D45D1→0) in Eq. (45), /u1D443[22](/u1D48C)becomes\n/u1D443[22](/u1D48C) →4/u1D443[11]\nm(/u1D458)/uni222B.dsp/u1D4513/u1D45D\n(2/u1D70B)3/bracketleftbig\n/u1D44D[2](/u1D48C,/u1D491)/bracketrightbig2/u1D443[11]\nm(/u1D45D).(55)\nThe equation above indicates that when we decouple the large -\nscale mode from the small-scale mode,the mode coupling term\nis proportional to the linear matter power spectrum, /u1D443[11]\nm(/u1D458).\nHowever, in reality, the integral of the mode coupling is ex-\npected to smooth out the BAO signal in the linear power spec-\ntrum, resulting in a smoother function. To properly account\nfor this effect, we decompose the linear matter power spectru m\ninto two components: the “wiggle” component containing\nonly the BAO signal and the “no-wiggle” component without\nBAO [68]. Thus, it is expressed as /u1D443[11]\nm(/u1D458)=/u1D443w(/u1D458)+/u1D443nw(/u1D458),\nwhere the subscripts “w” and “nw” denote “wiggle” and “no-\nwiggle”, respectively. By replacing /u1D443[11]\nm(/u1D458)with/u1D443nw(/u1D458), and\nby using Eq. (46), we obtain\n/u1D443[22]\nIR(/u1D48C)=−2 lnD(/u1D48C)[/u1D44D1(/u1D48C)]2/u1D443nw(/u1D458). (56)\nThe approximation used for /u1D443[22]can also be applied for the\nnon-perturbativemode coupling term in the IR limit. Based o n\nEq. (50), replacing /u1D443[11]\nm(/u1D458)that appears in the mode coupling\nterm with /u1D443nw(/u1D458)yields\n/u1D443(/u1D48C) → D2(/u1D48C)/bracketleftbig/u1D44D1(/u1D48C)/bracketrightbig2/u1D443[11]\nm(/u1D458)\n+/parenleftbig1− D2(/u1D48C)/parenrightbig/bracketleftbig/u1D44D1(/u1D48C)/bracketrightbig2/u1D443nw(/u1D458)\n=/bracketleftbig/u1D44D1(/u1D48C)/bracketrightbig2/bracketleftbigD2(/u1D48C)/u1D443w(/u1D458) +/u1D443nw(/u1D458)/bracketrightbig.(57)\nThis form is consistent with the template model proposed\nby [46], which has been widely used for BAO analyses. The\nmethod used to derive the power spectrum model in Eq. (57)\nis based on [40, 55]. On the other hand, for other approaches\nconcerningthe derivation of the IR-resummed power spectru m\nmodel, see also [50, 51, 53, 58].\n2. 1-loop level\nLet us consider the 1-loop order correction terms. In the IR\nlimit, the nonlinear galaxy power spectrum is described as\n/u1D443(/u1D48C) → D2(/u1D48C)D−2(/u1D48C)/u1D443(S)(/u1D48C), (58)where/u1D443(S)is consists of only the short-wavelength density\nfluctuations, /tildewide/u1D6FF(S). At the 1-loop level, we derive from Eq. (31)\n/u1D443[22]\n(S)(/u1D48C)=/u1D443[22](/u1D48C) −/u1D443[22]\nIR(/u1D48C),\n/u1D443[13]\n(S)(/u1D48C)=/u1D443[13](/u1D48C) −/u1D443[13]\nIR(/u1D48C), (59)\nwhere we used the uncorrelation between /u1D6BFand/u1D6FF(S). Com-\npared with the propagator including the 1-loop correction t erm\nin Eq. (53), the propagator and the mode-coupling term at the\n1-loop level are respectively shown as\n/u1D43A2(/u1D48C)/u1D443[11]\nm(/u1D458)\n=D2(/u1D48C)/bracketleftBig\n[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458) +/u1D443[13]\n(S)(/u1D48C)/bracketrightBig\n,(60)\nand\n/u1D443MC(/u1D48C)\n=/parenleftBig\n1−D2(/u1D48C)/parenrightBig /bracketleftBig\n[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458) +/u1D443[13]\n(S)(/u1D48C)/bracketrightBig\n+/u1D443[22]\n(S)(/u1D48C). (61)\nBy replacing all linear matter power spectrum /u1D443[11]\nmappearing\nin the mode-coupling term with the no-wiggle matter power\nspectrum /u1D443nw, we finally obtain the IR-resummed power spec-\ntrum model at the 1-loop level [40]:\n/u1D443(/u1D48C)=D2(/u1D48C)/bracketleftBig\n[/u1D44D1(/u1D48C)]2+/u1D443[13]\n(S)(/u1D48C)/slashbig/u1D443[11]\nm(/u1D458)/bracketrightBig\n/u1D443w(/u1D458)\n+ [/u1D44D1(/u1D48C)]2/u1D443nw(/u1D458) +/u1D443[13]\nnw(/u1D48C) +/u1D443[22](/u1D48C),(62)\nwhere\n/u1D443[13]\nnw(/u1D48C)=/u1D443[13](/u1D48C)/parenleftBig\n/u1D443nw(/u1D458)//u1D443[11]\nm(/u1D458)/parenrightBig\n. (63)\n3. Mode-coupling contributions\nIn this paper, we have entirely disregarded the effects of\nBAO arising from the mode-coupling terms. However, it\nis well-known that contributions from these mode-coupling\nterms introduce additional corrective terms to the nonline ar\ndamping of BAO. For instance, the perpendicular dispersion ,\n/u1D70E⊥, in Eq. (42) should be modified as follows [50–52]:\n/u1D70E2\n⊥=1\n3/uni222B.dsp/u1D451/u1D45D\n2/u1D70B2/u1D443[11]\nm(/u1D45D) [1−/u1D4570(/u1D45D/u1D45Fs)], (64)\nwhere/u1D4570is the zeroth-order spherical Bessel function, and /u1D45Fs\nis the sound horizon scale. The correction term in the above\nequation only slightly changes the value of the smoothing pa -\nrameter/u1D70E2in the Gaussian decaying function D(41), and\ndoes not change the form of the function itself. Therefore,\nfor the sake of simplicity of the theoretical model, we did no t\ncalculate these mode-coupling terms in detail in this paper .\nThe contributions from the mode-coupling terms are natural ly\naccounted for in methods like CLPT [8, 9], which involve ap-\npropriate calculations of mode-coupling integrals. Apply ing\nfurther detailed calculations on these mode coupling terms to\nthe post-reconstruction case will be a topic for future work .8\nIII. POST-RECONSTRUCTION CASE\nWe apply the method for constructing the IR-resummed\npower spectrum model from the pre-reconstructionto the pos t-\nreconstruction case.\nA. Post-reconstruction density fluctuations\nTo reconstructthe galaxy distribution, the displacement v ec-\ntor for reconstruction is calculated from the observed gala xy\ndensity fluctuations as follows [1]:\n/u1D494(/u1D499)=/uni222B.dsp/u1D4513/u1D45D\n(2/u1D70B)3/u1D452/u1D456/u1D491·/u1D499/parenleftbigg/u1D456/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/tildewide/u1D6FFobs(/u1D491), (65)\nwhere/u1D44F1,fidis an input fiducial linear bias parameter for re-\nconstruction, /u1D44AG(/u1D45D/u1D445s)=exp/parenleftbig−/u1D45D2/u1D4452\ns/2/parenrightbigis a Gaussian filter\nfunction, and /u1D445sis an input smoothing scale. This /u1D494(/u1D499)is\nderived from the observed galaxy fluctuation /u1D6FFobs(/u1D499). By\nsubstituting /u1D6FFobswith the nonlinear density fluctuation /u1D6FFin\nEq. (19), which includes the bias and RSD effects, we can\nmake a theoretical prediction of /u1D494(/u1D499).\nThe reconstruction scheme uses /u1D494(/u1D499)to move each of the\ngalaxy data particles and the corresponding random particl es.\nSuch an operation is expressed as\n1+/u1D6FFd(/u1D499)=/uni222B.dsp\n/u1D4513/u1D465′(1+/u1D6FF(/u1D499))/u1D6FFD(/u1D499−/u1D499′−/u1D494(/u1D499′))\n1+/u1D6FFs(/u1D499)=/uni222B.dsp\n/u1D4513/u1D465′/u1D6FFD(/u1D499−/u1D499′−/u1D494(/u1D499′)). (66)\nHere,/u1D6FFd(/u1D499)is the density fluctuation of the reconstructed\ngalaxy data particles, and /u1D6FFs(/u1D499)is the density fluctua-\ntion of the reconstructed random particles. The observed\npost-reconstruction galaxy density fluctuation is then giv en\nby [6, 55]\n/u1D6FFrec(/u1D499)=/u1D6FFd(/u1D499) −/u1D6FFs(/u1D499)\n=/uni222B.dsp\n/u1D4513/u1D465′/u1D6FF(/u1D499′)/u1D6FFD(/u1D499−/u1D499′−/u1D494(/u1D499′)).(67)\nB. IR effects on the post-reconstruction density fluctuation\nThe displacement vector for reconstruction, /u1D494(/u1D499), is de-\nrived from the nonlinear density fluctuation, which include s\nthe shift term that should be considered in the IR limit (e.g. ,\nsee Eq. 2). This characteristic of /u1D494(/u1D499)is unique, contrasting\nwith the displacement vector /u1D6BF(/u1D492), where the shift term is\nexcluded (e.g., Eq. 22). In this subsection, we focus on ex-\namining the non-perturbative behavior of the shift term wit hin\n/u1D494(/u1D499)and demonstrate its impact on the post-reconstruction\ndensity fluctuation.\nFollowing the approach introduced in Section II, we decom-\npose the galaxy density fluctuation /u1D6FF(/u1D499)into two components:\nthe IR effect represented by /u1D6BF, and the short-wavelength den-\nsity fluctuation /u1D6FF(S)(/u1D499). By substituting Eq. (24) and Eq. (26)into Eq. (65) and Eq. (67), we obtain\n/u1D6FFrec(/u1D499)=/uni222B.dsp\n/u1D4513/u1D465′/u1D6FF(S)(/u1D499′−/u1D6BF)/u1D6FFD/parenleftBig\n/u1D499−/u1D499′−/u1D494(S)(/u1D499′−/u1D6BF)/parenrightBig\n=/uni222B.dsp\n/u1D4513/u1D465′′/u1D6FF(S)(/u1D499′′)/u1D6FFD/parenleftBig\n/u1D499−/u1D6BF−/u1D499′′−/u1D494(S)(/u1D499′′)/parenrightBig\n.(68)\nIn the second line, we used the coordinate transformation:\n/u1D499′′=/u1D499′−/u1D6BF. Here,/u1D494(S)(/u1D499)is defined as\n/u1D494(S)(/u1D499)=/uni222B.dsp/u1D4513/u1D45D\n(2/u1D70B)3/u1D452/u1D456/u1D491·/u1D499/parenleftbigg/u1D456/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/tildewide/u1D6FF(S)(/u1D491).(69)\nIn the IR limit, the short-wavelength displacement vector f or\nreconstruction, /u1D494(S), no longer receives contributions from the\nshift term, leading to its analogy with the displacement vec tor\n/u1D6BF. As demonstrated in Eqs. (23) and (24), the IR effect within\nthe nonlinear density fluctuation is represented by /u1D6BFat the\norigin. Similarly to /u1D6BF, by decomposing /u1D494Sinto its origin value\nand other components, we can elucidate the IR effect within\nthe post-reconstruction density fluctuation:\n/u1D494(S)(/u1D499)=/u1D494+/u1D494(SS)(/u1D499), (70)\nwhere/u1D494=/u1D494S(/u1D499=0). Finally, /u1D6FFrec(/u1D499)can be formally rewrit-\nten as\n/u1D6FFrec(/u1D499)=/u1D6FF(S)rec(/u1D499−/u1D6BFrec), (71)\nwhere\n/u1D6BFrec=/u1D6BF+/u1D494, (72)\nand\n/u1D6FF(S)rec(/u1D499)=/uni222B.dsp\n/u1D4513/u1D465′/u1D6FF(S)(/u1D499′)/u1D6FFD/parenleftbig/u1D499−/u1D499′−/u1D494(SS)(/u1D499′)/parenrightbig.(73)\nThis result shows that the IR effect manifests in the post-\nreconstruction density fluctuation as the coordinate trans -\nformation via /u1D6BFrec, like the pre-reconstruction case in\nEq. (24). Consequently, the arguments applicable to the pre -\nreconstruction IR effect can similarly be applied to the post -\nreconstruction context.\nC. Lagrangian space\nWe can also perform the calculation of density fluctuations\nafter reconstruction in Lagrangian space. To facilitate a c om-\nparison with the Zel’dovich approximation, set to be discus sed\nin Section III E, we provide proof for Eq. (71) in Lagrangian\nspace.\nIn Lagrangian space, /u1D6FFd(/u1D499)and/u1D6FFs(/u1D499)are each expressed as\n1+/u1D6FFd(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E(1+/u1D6FFbias(/u1D492))\n×/u1D6FFD(/u1D499−/u1D492−/u1D6BF(/u1D492) −/u1D494(/u1D492+/u1D6BF(/u1D492))),\n1+/u1D6FFs(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E/u1D6FFD(/u1D499−/u1D492−/u1D494(/u1D492)). (74)9\nSimilar to Eq. (68), they can be rewritten as\n1+/u1D6FFd(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E(1+/u1D6FFbias(/u1D492))\n×/u1D6FFD/parenleftBig\n/u1D499−/u1D492−/u1D6BF−/u1D6BF(S)(/u1D492) −/u1D494(S)(/u1D492+/u1D6BF(S)(/u1D492))/parenrightBig\n,\n1+/u1D6FFs(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E/u1D6FFD/parenleftBig\n/u1D499−/u1D492−/u1D494(S)(/u1D492−/u1D6BF)/parenrightBig\n=/uni222B.dsp\n/u1D4513/u1D45E′/u1D6FFD/parenleftBig\n/u1D499−/u1D492′−/u1D6BF−/u1D494(S)(/u1D492′)/parenrightBig\n, (75)\nwhere/u1D6BF(S)(/u1D492)=/u1D6BF(/u1D492) −/u1D6BFin Eq. (23). For /u1D6FFs(/u1D499), we used\nthe coordinate transformation: /u1D492′=/u1D492−/u1D6BF.\nNote that /u1D494(/u1D492+/u1D6BF(/u1D492))within/u1D6FFd(/u1D499)does not include the\ncontribution from the shift term in the IR limit. This is be-\ncause the coordinate transformation, represented by /u1D492+/u1D6BF(/u1D492),\ncancels out the IR effect of /u1D6BF. As a result, it can be expressed\nonly in terms of short-wavelength modes as /u1D494(S)(/u1D492+/u1D6BF(S)(/u1D492)).\nFurthermore, we decompose /u1D494(S)into the value at the origin\n/u1D492=0and other parts:\n/u1D494(S)(/u1D492+/u1D6BF(S)(/u1D492))=/u1D494+/u1D494(SS)(/u1D492+/u1D6BF(S)(/u1D492))\n/u1D494(S)(/u1D492)=/u1D494+/u1D494(SS)(/u1D492), (76)\nwhere we used /u1D6BFS(/u1D492=0)=0.\nFinally, we obtain\n/u1D6FFrec(/u1D499)=/u1D6FF(S)d(/u1D499−/u1D6BFrec) −/u1D6FF(S)s(/u1D499−/u1D6BFrec)\n=/u1D6FF(S)rec(/u1D499−/u1D6BFrec), (77)\nwhere\n1+/u1D6FF(S)d(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E(1+/u1D6FFbias(/u1D492))\n×/u1D6FFD/parenleftbig/u1D499−/u1D492−/u1D6BF(S)(/u1D492) −/u1D494(SS)(/u1D492+/u1D6BF(S)(/u1D492))/parenrightbig,\n1+/u1D6FF(S)s(/u1D499)=/uni222B.dsp\n/u1D4513/u1D45E/u1D6FFD/parenleftbig/u1D499−/u1D492−/u1D494(SS)(/u1D492)/parenrightbig. (78)D. IR cancellation after reconstruction\nWe can show that the IR cancellation occurs even after\nreconstruction:\n/angb∇acketleft/u1D6FFrec(/u1D499)/u1D6FFrec(/u1D499′)/angb∇acket∇ight=/angbracketleftbig\n/u1D6FF(S)rec(/u1D499−/u1D6BFrec)/u1D6FF(S)rec(/u1D499′−/u1D6BFrec)/angbracketrightbig\n→/angbracketleftbig\n/u1D6FF(S)rec(/u1D499)/u1D6FF(S)rec(/u1D499′)/angbracketrightbig\n. (79)\nApproximating both /u1D6FF(S)rec(/u1D499)and/u1D6BFrecas linear, we obtain\n/angb∇acketleft/u1D6FFrec(/u1D499)/u1D6FFrec(/u1D499′)/angb∇acket∇ight →/angbracketleftbig\n/u1D6FF[1](/u1D499−/u1D6BF[1]\nrec)/u1D6FF[1](/u1D499′−/u1D6BF[1]\nrec)/angbracketrightbig\n=/angbracketleftbig\n/u1D6FF[1](/u1D499)/u1D6FF[1](/u1D499′)/angbracketrightbig\n. (80)\nUnder this approximation, the propagator term and the mode-\ncoupling term are respectively represented as\n/u1D43A2(/u1D48C)/u1D443[11]\nm(/u1D458) → D2\nrec(/u1D48C)[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458), (81)\nand\n/u1D443MC(/u1D48C) → ( 1−D2\nrec(/u1D48C))[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458). (82)\nThese specifically lead to the IR cancellation in Fourier spa ce:\n/u1D443rec(/u1D48C) → D2\nrec(/u1D48C)[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458)\n+ (1−D2\nrec(/u1D48C))[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458)\n=[/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458). (83)\nHere,Drec(/u1D48C)is the two-dimensional exponentially decaying\nfunction after reconstruction, given by\nDrec(/u1D48C)=exp/parenleftbigg1\n2/angbracketleftbigg/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]\nrec/parenrightBig2/angbracketrightbigg\nc/parenrightbigg\n,\n=exp/parenleftBigg\n−/u1D4582(1−/u1D7072)/u1D70E2\nrec,⊥+/u1D4582/u1D7072/u1D70E2\nrec,/ba∇dbl\n2/parenrightBigg\n.(84)\nThe radial and transverse components of the smoothing facto rs\nare further decomposed into\n/u1D70E2\nrec,⊥=/u1D70E2\n⊥+/u1D70E2\nps,⊥+/u1D70E2\nss,⊥,\n/u1D70E2\nrec,/ba∇dbl=/u1D70E2\n/ba∇dbl+/u1D70E2\nps,/ba∇dbl+/u1D70E2\nss,/ba∇dbl, (85)\nwhere/u1D70E2\n⊥and/u1D70E2\n/ba∇dblare given in Eq. (42). The subscript “ps”\nmeans the correlation between /u1D6BFand/u1D494, and “ss” denotes the\nauto-correlation of /u1D494. These smoothing factors are calculated\nas10\n/u1D70E2\nps,⊥=1\n3/uni222B.dsp/u1D458max\n/u1D458min/u1D451/u1D45D\n2/u1D70B2/parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg /bracketleftbigg\n2/parenleftbigg\n/u1D44F1+/u1D453\n5/parenrightbigg/bracketrightbigg\n/u1D443m,lin(/u1D45D),\n/u1D70E2\nps,/ba∇dbl=1\n3/uni222B.dsp/u1D458max\n/u1D458min/u1D451/u1D45D\n2/u1D70B2/parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg/bracketleftBigg\n2(1+/u1D453)/parenleftbigg\n/u1D44F1+3\n5/u1D453/parenrightbigg/bracketrightBigg\n/u1D443m,lin(/u1D45D),\n/u1D70E2\nss,⊥=1\n3/uni222B.dsp/u1D458max\n/u1D458min/u1D451/u1D45D\n2/u1D70B2/parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg2/bracketleftbigg/parenleftbigg\n/u1D44F2\n1+2\n5/u1D44F1/u1D453+3\n35/u1D4532/parenrightbigg\n/u1D443m,lin(/u1D45D) +1\n¯/u1D45B/bracketrightbigg\n,\n/u1D70E2\nss,/ba∇dbl=1\n3/uni222B.dsp/u1D458max\n/u1D458min/u1D451/u1D45D\n2/u1D70B2/parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg2/bracketleftBigg/parenleftbigg\n/u1D44F2\n1+42\n35/u1D44F1/u1D453+3\n7/u1D4532/parenrightbigg\n/u1D443m,lin(/u1D45D) +1\n¯/u1D45B/bracketrightBigg\n. (86)\nHere, we summarize the key considerations when performing\nthe above calculations:\n1. The integral range should depend on the scale at which\n/u1D494(/u1D499)is measured.\n2. Since /u1D494(/u1D499)is derived from the power spectrum of the\nmeasured galaxy density fluctuations, we should con-\nsider the shot-noise term 1 /¯/u1D45Bin/u1D70E2\nss,⊥and/u1D70E2\nss,/ba∇dbl, where\n¯/u1D45Bis the mean number density [69].\nE. Comparisons with the Zel’dovich approximation\nPrevious studies [21–24] have often used a simple linear\napproximation for /u1D494(/u1D499), known as the Zel’dovich approxima-\ntion for reconstruction. This approximation, however, doe s\nnot account for nonlinear IR effects in /u1D494(/u1D499). In this subsec-\ntion, we present a detailed comparison between the Zel’dovi ch\napproximation and our findings in Section III D.\nIn the Zel’dovich approximation, Eq. (74) becomes\n1+/u1D6FFd(/u1D499) ∼/uni222B.dsp\n/u1D4513/u1D45E/parenleftBig\n1+/u1D6FF[1]\nbias(/u1D492)/parenrightBig\n×/u1D6FFD/parenleftBig\n/u1D499−/u1D492−/u1D6BF[1](/u1D492) −/u1D494[1](/u1D492)/parenrightBig\n,\n1+/u1D6FFs(/u1D499) ∼/uni222B.dsp\n/u1D4513/u1D45E/u1D6FFD/parenleftBig\n/u1D499−/u1D492−/u1D494[1](/u1D492)/parenrightBig\n. (87)\nIn the IR limit, they are represented as\n/u1D6FFd(/u1D499) →/u1D6FF[1]\nd/parenleftBig\n/u1D499−/u1D6BF[1]−/u1D494[1]/parenrightBig\n,\n/u1D6FFs(/u1D499) →/u1D6FF[1]\ns/parenleftBig\n/u1D499−/u1D494[1]/parenrightBig\n. (88)\nWe then derive\n/u1D6FFrec(/u1D499) →/u1D6FF[1]\nd/parenleftBig\n/u1D499−/u1D6BF[1]−/u1D494[1]/parenrightBig\n−/u1D6FF[1]\ns/parenleftBig\n/u1D499−/u1D494[1]/parenrightBig\n.(89)\nThe key difference of this equation from Eq. (77) is how /u1D494(/u1D492)\nis treated within /u1D6FFs(/u1D499). In Eq. (77), non-perturbativeIR effects\nare incorporated into /u1D494(/u1D492), revealing both /u1D6BFand/u1D494effects in\n/u1D6FFs(/u1D499). In contrast, the Zel’dovich approximation assumes a\nlinear/u1D494(/u1D492), resulting in only the /u1D494effect.In Fourier space, Eq. (89) becomes\n/tildewide/u1D6FFrec(/u1D48C)\n→/bracketleftBig\n/u1D452−/u1D456/u1D48C·/parenleftBig\n/u1D6BF[1]+/u1D494[1]/parenrightBig\n/u1D44Dd,1(/u1D48C) −/u1D452−/u1D456/u1D48C·/u1D494[1]/u1D44Ds,1(/u1D48C)/bracketrightBig\n/tildewide/u1D6FF[1]\nm(/u1D48C),(90)\nwhere\n/u1D44Dd,1(/u1D48C)=/bracketleftbigg\n1+/parenleftbigg\n−/u1D44AG(/u1D458/u1D445s)\n/u1D44F1,fid/parenrightbigg/bracketrightbigg\n/u1D44D1(/u1D48C),\n/u1D44Ds,1(/u1D48C)=/parenleftbigg\n−/u1D44AG(/u1D458/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D48C). (91)\nThe propagator is then given by\n/u1D43A(/u1D48C)=Drec(/u1D48C)/u1D44Dd,1(/u1D48C) −D ss(/u1D48C)/u1D44Ds,1(/u1D48C), (92)\nwith\nDss(/u1D48C)=exp/parenleftbigg1\n2/angbracketleftbigg/parenleftBig\n−/u1D456/u1D48C·/u1D494[1]/parenrightBig2/angbracketrightbigg\nc/parenrightbigg\n. (93)\nEquation (92) shows that the propagator in the Zel’dovich\napproximation comprises two Gaussian decay functions:\nDrec(/u1D48C)andDss(/u1D48C). This contrasts with Eq. (81), where\nonlyDrec(/u1D48C)appears. The specific expression for Dss(/u1D48C)is\nDss(/u1D48C)=exp/parenleftBigg\n−/u1D4582(1−/u1D7072)/u1D70E2\nss,⊥+/u1D4582/u1D7072/u1D70E2\nss,/ba∇dbl\n2/parenrightBigg\n,(94)\nwhere the radial and transverse components of the smoothing\nfactors are given in Eq. (86).\nWe can also demonstrate that Eq. (89) breaks the IR cancel-\nlation:\n/angb∇acketleft/u1D6FFrec(/u1D499)/u1D6FFrec(/u1D499′)/angb∇acket∇ight → /angb∇acketleft/u1D6FF[1]\nd(/u1D499)/u1D6FF[1]\nd(/u1D499′)/angb∇acket∇ight\n− /angb∇acketleft/u1D6FF[1]\nd(/u1D499−/u1D6BF[1])/u1D6FF[1]\ns(/u1D499′)/angb∇acket∇ight\n− /angb∇acketleft/u1D6FF[1]\ns(/u1D499)/u1D6FF[1]\nd(/u1D499′−/u1D6BF[1])/angb∇acket∇ight\n+ /angb∇acketleft/u1D6FF[1]\ns(/u1D499)/u1D6FF[1]\ns(/u1D499′)/angb∇acket∇ight. (95)\nIn the cross-correlation functions between /u1D6FFd(/u1D499)and/u1D6FFs(/u1D499′),\nthe IR cancellation does not occur, and the contributionof /u1D6BF[1]\nremains. In Fourier space, the cross-power spectrum betwee n\n/tildewide/u1D6FFd(/u1D48C)and/tildewide/u1D6FFs(/u1D48C′)is represented as\n/u1D443ds(/u1D48C) → D(/u1D48C)/u1D44Dd,1(/u1D48C)/u1D44Ds,1(/u1D48C)/u1D443[11]\nm(/u1D458). (96)11\nHence, breaking the IR cancellation leads to an exponential\ndecay of the power spectrum. This finding deviates from\nour result where the IR cancellation is satisfied even after\nreconstruction.\nF. Comparison with the 1-loop solution in SPT\nIn this subsection, we investigate the IR limit properties o f\nthe 1-loop solution in SPT for the post-reconstruction powe r\nspectrum.\nIn Fourier space, the /u1D45Bth-order term of the redshift-space\ndensity fluctuations after reconstruction is given by\n/tildewide/u1D6FF[/u1D45B]\nrec(/u1D48C)=/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])\n×/u1D44Drec,/u1D45B(/u1D4911,...,/u1D491/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D4911)···/tildewide/u1D6FF[1]\nm(/u1D491/u1D45B).(97)\nThe function /u1D44Drec,/u1D45B(/u1D4911,···,/u1D491/u1D45B)denotes nonlinear kernel\nfunctions accountingfor the bias effect, the RSD effect, and t he\nreconstruction effect. Consequently, the post-reconstruc tion\nterms corresponding to /u1D443[22]and/u1D443[13]are expressed as\n/u1D443[22]\nrec(/u1D48C)=2/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3/uni222B.dsp/u1D4513/u1D45D2\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,2])\n× [/u1D44Drec,2(/u1D4911,/u1D4912)]2/u1D443[11]\nm(/u1D45D1)/u1D443[11]\nm(/u1D45D2),\n/u1D443[13]\nrec(/u1D48C)=6/u1D44D1(/u1D48C)/u1D443[11]\nm(/u1D458)\n×/uni222B.dsp/u1D4513/u1D45D\n(2/u1D70B)3/u1D44Drec,3(/u1D48C,/u1D491,−/u1D491)/u1D443[11]\nm(/u1D45D). (98)\nIn the IR limit, both /u1D44Drec,2(/u1D48C,/u1D491)and/u1D44Drec,3(/u1D48C,/u1D491,−/u1D491)are ap-\nproximated as (see Appendix C)\n/u1D44Drec,2(/u1D48C,/u1D491) − −−− →\n/u1D45D→01\n2/u1D44D1(/u1D48C)/braceleftBigg/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n+/parenleftbigg/u1D48C·/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44A(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D491)/bracerightBigg\n/u1D44Drec,3(/u1D48C,/u1D491,−/u1D491) − −−− →\n/u1D45D→0−1\n3!/u1D44D1(/u1D48C)/braceleftBigg/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n+/parenleftbigg/u1D48C·/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44A(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D491)/bracerightBigg2\n.(99)\nUsing Eq. (99), /u1D443[22]\nrecand/u1D443[13]\nrecin the IR limit are described\nas\n/u1D443[22]\nrec,IR(/u1D48C)=/parenleftBig\n/u1D4582(1−/u1D7072)/u1D70E2\nrec,⊥+/u1D4582/u1D7072/u1D70E2\nrec,/ba∇dbl/parenrightBig\n× [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458),\n/u1D443[13]\nrec,IR(/u1D48C)=−/parenleftBig\n/u1D4582(1−/u1D7072)/u1D70E2\nrec,⊥+/u1D4582/u1D7072/u1D70E2\nrec,/ba∇dbl/parenrightBig\n× [/u1D44D1(/u1D48C)]2/u1D443[11]\nm(/u1D458). (100)\nThese terms cancel each other out, leading to the post-\nreconstruction 1-loop power spectrum approaching zero in t heIR limit: /u1D4431-loop\nrec(/u1D48C) →0. This result is consistent with those\nobtained by expanding both the propagator term and the mode-\ncoupling term up to the 1-loop order, as presented in Eqs. (81 )\nand (82). Given that SPT ensures accurate calculations for\neach respective order, this finding further reinforces the v alid-\nity of our main result: i.e., the IR cancellation occurs even\nafter reconstruction.\nG. IR-resummed model for\nthe post-reconstruction power spectrum\nIn the same way as Eq. (57), we replace the linear matter\npower spectrum, which appears in the mode-coupling term in\nthe IR limit, with the corresponding no-wiggle power spec-\ntrum. We then derive\n/u1D443rec(/u1D48C)=/bracketleftbig\n/u1D44D1(/u1D48C)/bracketrightbig2/bracketleftbig\nD2\nrec(/u1D48C)/u1D443w(/u1D458) +/u1D443nw(/u1D458)/bracketrightbig\n.(101)\nThis form is identical to the one before reconstruction, wit h the\nexception of the values of the smoothing parameters that cha r-\nacterize the exponential decay of the BAO signal. Template\nmodels using a single Gaussian function have been widely use d\nin post-reconstruction BAO analysis (e.g., see [70–73]). T he\nfindings in this paper confirm the validity of these previous\npost-reconstruction BAO analyses.\nAt the 1-loop level, we obtain\n/u1D443rec(/u1D48C)=D2\nrec(/u1D48C)/bracketleftBig\n[/u1D44D1(/u1D48C)]2+/u1D443[13]\n(S)rec(/u1D48C)/slashbig/u1D443[11]\nm(/u1D458)/bracketrightBig\n/u1D443w(/u1D458)\n+ [/u1D44D1(/u1D48C)]2/u1D443nw(/u1D458) +/u1D443[13]\nrec,nw(/u1D48C) +/u1D443[22]\nrec(/u1D48C),(102)\nwhere\n/u1D443[13]\nrec,nw(/u1D48C)=/u1D443[13]\nrec(/u1D48C)/parenleftBig\n/u1D443nw(/u1D458)//u1D443[11]\nm(/u1D458)/parenrightBig\n, (103)\nand\n/u1D443(S)rec(/u1D48C)[13]=/u1D443[13]\nrec(/u1D48C) −/u1D443[13]\nrec,IR(/u1D48C). (104)\nIV. CROSS-POWER SPECTRUM BETWEEN PRE- AND\nPOST-RECONSTRUCTION DENSITY FLUCTUATIONS\nWe now turn to the cross power spectrum between the pre-\nand post reconstruction density fluctuations. In this secti on,\nwe demonstrate that the pre- and post-reconstruction cross -\npower spectrum does not occur the IR cancellation, resultin g\nin an overall exponential decay.\nIn the IR limit, the cross 2PCF of /u1D6FF(/u1D499)and/u1D6FFrec(/u1D499)can be\nexpressed as\n/angb∇acketleft/u1D6FF(/u1D499)/u1D6FFrec(/u1D499′)/angb∇acket∇ight →/angbracketleftbig/u1D6FF[1](/u1D499−/u1D6BF[1])/u1D6FF[1]\nrec(/u1D499′−/u1D6BF[1]\nrec)/angbracketrightbig.(105)\nDue to the statistical translation symmetry, this expressi on\ntransforms to\n/angb∇acketleft/u1D6FF(/u1D499)/u1D6FFrec(/u1D499′)/angb∇acket∇ight →/angbracketleftbig\n/u1D6FF[1](/u1D499+/u1D494[1])/u1D6FF[1]\nrec(/u1D499′)/angbracketrightbig\n.(106)12\nAs a result, the contributions from the IR effect related to /u1D494[1]\nare retained in the cross 2PCF.\nIn Fourier space, the cross power spectrum of /tildewide/u1D6FF(/u1D48C)and\n/tildewide/u1D6FFrec(/u1D48C)in the IR limit is given by\n/u1D443cross(/u1D48C)\n→ D(/u1D48C)Drec(/u1D48C)Epd(/u1D48C,−/u1D48C)/bracketleftbig/u1D44D1(/u1D48C)/bracketrightbig2/u1D443[11]\nm(/u1D458),(107)\nwhere the exponential function Epd(/u1D48C,/u1D48C′)is defined as\nEpd(/u1D48C,/u1D48C′)=exp/parenleftBig/angbracketleftBig/parenleftBig\n−/u1D456/u1D48C·/u1D6BF[1]/parenrightBig /parenleftBig\n−/u1D456/u1D48C′·/u1D6BF[1]\nrec/parenrightBig/angbracketrightBig\nc/parenrightBig\n,(108)\nThis function satisfies the following relation:\nEpd(/u1D48C,−/u1D48C)=Dss(/u1D48C)D−1\nrec(/u1D48C)D−1(/u1D48C). (109)\nSubstituting Eq. (109) into Eq. (107) leads to\n/u1D443cross(/u1D48C) → D ss(/u1D48C)/bracketleftbig\n/u1D44D1(/u1D48C)/bracketrightbig2/u1D443[11]\nm(/u1D458). (110)\nThus, the contributionfrom the remaining IR effect in the cro ss\n2PCF, shown in Eq. (106), appears as the exponential decay\nfunction in the cross power spectrum.\nTo account for the nonlinear damping effect of BAO, as was\ndone in Eqs. (57) and (101), we make the following substitu-\ntions in Eq. (107):\nEpd(/u1D48C,−/u1D48C)/u1D443[11]\nm(/u1D458)\n− −−−− →\nreplace/u1D443w(/u1D458) +E pd(/u1D48C,−/u1D48C)/u1D443nw(/u1D458). (111)\nSubsequently, we obtain\n/u1D443cross(/u1D48C)\n→/bracketleftbig\n/u1D44D1(/u1D48C)/bracketrightbig2/bracketleftbig\nD(/u1D48C)Drec(/u1D48C)/u1D443w+ D ss(/u1D48C)/u1D443nw(/u1D48C)/bracketrightbig\n.(112)\nA. Shot-noise in the cross-power spectrum\nThe shot-noise term in the power spectrum arises when the\nsame particle is counted multiple times during power spectr um\ncalculations. Usually, the shot-noise term is absent in the cross\npower spectrum of different density fluctuations. However, i n\nthe case of the cross power spectrum between the pre- and post -\nreconstruction density fluctuations, the same galaxies can be\nidentified, making the shot-noise term significant.\nIn the discrete picture, the number density in Fourier space\nis given by\n/u1D45B(/u1D48C)=/u1D441g/summationdisplay.1\n/u1D456/u1D452−/u1D456/u1D48C·/u1D499/u1D456, (113)\nwhere/u1D499/u1D456denotes the position of the /u1D456-th galaxy, and /u1D441grepre-\nsents the total number of galaxies. The auto power spectrum\nin this discrete picture is expressed as\n/u1D449\n/u1D4412g|/u1D45B(/u1D48C)|2=/u1D449\n/u1D4412g/summationdisplay.1\n/u1D456≠/u1D457/u1D452−/u1D456/u1D48C·(/u1D499/u1D456−/u1D499/u1D457)+/u1D449\n/u1D4412g/u1D441g/summationdisplay.1\n/u1D456=/u1D457\n=/u1D443(/u1D48C) +1\n¯/u1D45B, (114)where/u1D449denotes the survey volume. The first and second terms\non the right-hand side correspond to the power spectrum and\nthe shot-noise term, respectively:\n/u1D443(/u1D48C)=/u1D449\n/u1D4412g/summationdisplay.1\n/u1D456≠/u1D457/u1D452−/u1D456/u1D48C·(/u1D499/u1D456−/u1D499/u1D457),\n1\n¯/u1D45B=/u1D449\n/u1D4412g/u1D441g/summationdisplay.1\n/u1D456=/u1D457, (115)\nwhere ¯/u1D45B=/u1D441g//u1D449is the mean number density.\nLet/u1D45Brec(/u1D48C)be the post-reconstruction density field in\nFourier space. It can be expressed in the discrete picture\nas\n/u1D45Brec(/u1D48C)=/u1D441g/summationdisplay.1\n/u1D456/u1D452−/u1D456/u1D48C·/u1D499rec,/u1D456, (116)\nwhere\n/u1D499rec,/u1D456=/u1D499/u1D456+/u1D494(/u1D499/u1D456). (117)\nTherefore, the shot-noise term in the cross-power spectrum is\ngiven by\n/u1D449\n/u1D4412g/u1D441g/summationdisplay.1\n/u1D456=/u1D457/u1D452−/u1D456/u1D48C·(/u1D499/u1D456−/u1D499rec,/u1D456)=/u1D449\n/u1D4412g/u1D441g/summationdisplay.1\n/u1D456=/u1D457/u1D452/u1D456/u1D48C·/u1D494(/u1D499/u1D456)(118)\nCalculating the ensemble average of the above equation and\ntaking the continuous limit, we obtain\n/angbracketleftBigg\n/u1D449\n/u1D4412g/u1D441g/summationdisplay.1\n/u1D456=/u1D457/u1D452/u1D456/u1D48C·/u1D494(/u1D499/u1D456)/angbracketrightBigg\n=1\n/u1D441g/uni222B.dsp\n/u1D4513/u1D465/angbracketleftBig\n/u1D452/u1D456/u1D48C·/u1D494(/u1D499)/angbracketrightBig\n=1\n¯/u1D45B/angbracketleftBig\n/u1D452/u1D456/u1D48C·/u1D494(/u1D499=0)/angbracketrightBig\n≃1\n¯/u1D45B/angbracketleftBig\n/u1D452/u1D456/u1D48C·/u1D494[1]/angbracketrightBig\n, (119)\nIn the final line, we made an approximation of the linear /u1D494(/u1D499).\nFinally, we derive\n/angbracketleftBigg\n/u1D449\n/u1D4412g/u1D45B(/u1D48C)/u1D45B∗\nrec(/u1D48C)/angbracketrightBigg\n=/u1D443cross(/u1D48C) + D ss(/u1D48C)/parenleftbigg1\n¯/u1D45B/parenrightbigg\n.(120)\nThis result indicates that the shot-noise term is proportio nal to\nDss(/u1D48C)and exponentially decays, as is /u1D443cross(/u1D48C).\nV. CONCLUSIONS\nIn this paper, we present a new theoretical model addressing\nthe resummation of infrared (IR) effects in the power spec-\ntrum of galaxy density fluctuations after reconstruction. O ur\nmodel accurately describes the nonlinear damping of the pos t-\nreconstruction BAO signal, while including the 1-loop corr ec-\ntion term in SPT. The first main results are summarized in\nEqs. (101) and (102).13\nWe point out in Section III B that it is crucial to consider the\nnonlinear IR effects contained within the displacement vect or\nfor reconstruction /u1D494(65). As a result, the IR effects on the\npost-reconstruction density fluctuation can be described a s a\ncoordinate transformation of the density fluctuations (71) , like\nthe pre-reconstruction case (24). This result leads to the c an-\ncellation of the IR effects on the power spectrum in the IR limi t\neven after reconstruction. Furthermore, the nonlinear beh avior\nof BAO is characterized by the two-dimensional Gaussian de-\ncaying function of BAO, supporting the theoretical foundat ion\nof the commonly used single two-dimensional Gaussian de-\ncay function in post-reconstruction power spectrum and 2PC F\nanalyses [e.g., 70–73].\nWe compare our findings regarding the post-reconstruction\ninfrared (IR) effects with those obtained using the Zel’dovi ch\napproximation and SPT in Section III E and SectionIII F. No-\ntably, in the Zel’dovich approximation, which linearly app rox-\nimates/u1D494, IR cancellation does not occur, and two Gaussian\ndamping functions are required to describe the nonlinear BA O\neffects. Conversely, in the post-reconstruction SPT, IR can cel-\nlation is observed in the IR limit, corroborating our result s.\nWe further clarify the behavior of the cross-power spectrum\nbetween pre- and post-reconstructiondensity fluctuations , not-\ning its overall exponential decay due to the absence of IR can -\ncellation. This observation also applies to the correspond ing\nshot-noise term. While these phenomena have been observed\nin previous simulation studies [5], our work thus provides a\nmathematical foundation for these observations. Addition ally,\nas our second main result, we present a model, as shown in\nEq. (112), to elucidate the nonlinear effects of BAO on the\ncross-power spectrum.\nIn conclusion, our work presents a novel direction in de-\nveloping theoretical models for the post-reconstruction p ower\nspectrum beyond standard perturbation theory. The IR-\nresummed models presented in this paper, which consider the\ngalaxy bias and RSD effects, are directly applicable to the\nanalysis of real galaxy data.\nDuring the preparation of this paper, we learned that the\nDESI collaboration [74] was planning to submit a study with\nconclusions similar to ours, suggesting that the BAO signal in\nthe post-reconstruction power spectrum should be describe d\nby a single Gaussian damping function. To ensure a coor-\ndinated approach, we arranged to submit our paper close to\nthe DESI collaborators’ paper submission. Their study in-\nvestigates the theoretical and numerical systematics of th e\npost-reconstructionBAO signal in detail and, furthermore , dis-\ncusses the contributions of mode-coupling terms to the BAO\nsignal, an aspect not covered in our paper. In contrast, our w ork\nfocuses on the IR effects before and after reconstruction, pr o-\nviding a more detailed theoretical analysis. Therefore, th ese\nstudies complement each other well, each enriching the unde r-\nstanding of the topic from a different perspective. We refer\nreaders interested in the numerical aspects to the paper by t he\nDESI collaboration.ACKNOWLEDGMENTS\nNS acknowledges financial support from JSPS KAKENHI\nGrant Number 19K14703. NS thanks Shun Saito, Hee-Jong\nSeo, Florian Beutler, and Shi-Fan Stephen Chen for useful\ncomments and discussion.\nAppendix A: Degenerate Higher-Order Scalar-Tensor (DHOST )\nTheories\nIn DHOST theories, the second-order kernel density fluctu-\nation of dark matter is given by [75]\n/u1D6FF[2]\nm(/u1D499)=/parenleftbigg\n/u1D705−4\n21/u1D706/parenrightbigg/bracketleftbig/u1D6FF[1]\nm(/u1D499)/bracketrightbig2−/u1D705/u1D6BF[1](/u1D499) ·∇/u1D6FF[1]\nm(/u1D499)\n+2\n7/u1D706/bracketleftbigg/parenleftbigg/u1D715/u1D456/u1D715/u1D457\n/u1D7152−1\n3/u1D6FF/u1D456/u1D457/parenrightbigg\n/u1D6FF[1]\nm(/u1D499)/bracketrightbigg2\n. (A1)\nHere, in ΛCDM assuming /u1D4532= Ω m,/u1D705=/u1D706=1. From\nEqs. (9) and (10), the short-wavelength density fluctuation\nthen becomes\n/u1D6FF[2]\n(S)m(/u1D499)=/parenleftbigg\n/u1D705−4\n21/u1D706/parenrightbigg/bracketleftbig\n/u1D6FF[1]\nm(/u1D499)/bracketrightbig2−/u1D6BF[1]\n(S)(/u1D499) ·∇/u1D6FF[1]\nm(/u1D499)\n−Δ/u1D705/u1D6BF[1](/u1D499) ·∇/u1D6FF[1]\nm(/u1D499)\n+2\n7/u1D706/bracketleftbigg/parenleftbigg/u1D715/u1D456/u1D715/u1D457\n/u1D7152−1\n3/u1D6FF/u1D456/u1D457/parenrightbigg\n/u1D6FF[1]\nm(/u1D499)/bracketrightbigg2\n. (A2)\nwhere/u1D705=1+Δ/u1D705. There remains a shift term, Δ/u1D705/u1D6BF[1](/u1D499) ·\n∇/u1D6FF[1]\nm(/u1D499), in/u1D6FF[2]\n(S)m(/u1D499)that does not go to zero in the IR limit.\nTherefore, it is not possible to simply separate IR effects fr om\nthe rest based on Eq. (24). This unique feature of DHOST\ntheories leads to a violation of the IR cancellation [76]. Th is\ndeviation in the coefficient of the shift term from 1 in DHOST\ntheories is also related to the consistency relation for the large-\nscale structure [65, 77, 78].\nAppendix B: Standard Perturbation Theory\nIn this appendix, we show the specific form of the nonlinear\nkernel functions /u1D44D/u1D45Bgiven in Eq. (27) up to the third-order in\nSPT and investigate their behaviors in the IR limit.\nIn Fourier space, Eq. (19) becomes\n/tildewide/u1D6FFg(/u1D48C)=/uni222B.dsp\n/u1D4513/u1D45E/u1D452−/u1D456/u1D48C·/u1D492/braceleftBig\n[1+/u1D6FFbias(/u1D492)]/u1D452−/u1D456/u1D48C·/u1D6BFred(/u1D492)−1/bracerightBig\n.(B1)\nWe assume that the /u1D45Bth-order of the biased density fluctuation\ncan be represented as\n/tildewide/u1D6FF[/u1D45B]\nbias(/u1D48C)=/uni222B.dsp/u1D4513/u1D45D1\n(2/u1D70B)3···/u1D4513/u1D45D/u1D45B\n(2/u1D70B)3(2/u1D70B)3/u1D6FFD(/u1D48C−/u1D491[1,/u1D45B])\n×/u1D435/u1D45B(/u1D4911,...,/u1D491/u1D45B)/tildewide/u1D6FF[1]\nm(/u1D4911)···/tildewide/u1D6FF[1]\nm(/u1D491/u1D45B),(B2)\nwhere/u1D4351(/u1D4911)=(/u1D44F1−1)with/u1D44F1being the Eulerian linear\nbias. We follow the standard bias theory [e.g., 57] and assum e14\nthat/u1D435/u1D45B≥2(/u1D4911,···,/u1D491/u1D45B)do not include the shift terms. Then,\nthe nonlinear kernel functions up to the third order are give n\nby [61, 79]\n/u1D44D1(/u1D48C)=/u1D4351(/u1D48C) +/u1D48C·/u1D473red,1(/u1D48C), (B3)\n/u1D44D2(/u1D4911,/u1D4912)=/u1D4352(/u1D4911,/u1D4912) +/bracketleftbig\n/u1D48C·/u1D473red,2(/u1D4911,/u1D4912)/bracketrightbig\n+1\n2/braceleftbig\n/u1D4351(/u1D4911)/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4912)/bracketrightbig\n+1 perm./bracerightbig\n+1\n2/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4911)/bracketrightbig/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4912)/bracketrightbig\n,(B4)\nand\n/u1D44D3(/u1D4911,/u1D4912,/u1D4913)\n=/u1D4353(/u1D4911,/u1D4912,/u1D4913) +/bracketleftbig\n/u1D48C·/u1D473red,3(/u1D4911,/u1D4912,/u1D4913)/bracketrightbig\n+1\n3/braceleftbig\n/u1D4352(/u1D4911,/u1D4912)/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4913)/bracketrightbig\n+2 perms./bracerightbig\n+1\n3/braceleftbig\n/u1D4351(/u1D4911)/bracketleftbig\n/u1D48C·/u1D473red,2(/u1D4912,/u1D4913)/bracketrightbig\n+2 perms./bracerightbig\n+1\n6/braceleftbig/u1D4351(/u1D4911)/bracketleftbig/u1D48C·/u1D473red,1(/u1D4912)/bracketrightbig/bracketleftbig/u1D48C·/u1D473red,1(/u1D4913)/bracketrightbig+2 perms./bracerightbig\n+1\n6/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4911)/bracketrightbig/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4912)/bracketrightbig/bracketleftbig\n/u1D48C·/u1D473red,1(/u1D4913)/bracketrightbig\n,(B5)\nwhere/u1D48C=/u1D4911+/u1D4912for/u1D44D2,/u1D48C=/u1D4911+/u1D4912+/u1D4913for/u1D44D3, and\n/u1D43Fred,/u1D45B(/u1D4911,···,/u1D491/u1D45B)are calculated through the transformationmatrix given in Eq. (33):\n/u1D473red,/u1D45B(/u1D4911,···,/u1D491/u1D45B)=/u1D479/u1D45B·/u1D473/u1D45B(/u1D4911,···,/u1D491/u1D45B). (B6)\nSince/u1D473/u1D45B≥2(/u1D4911,···,/u1D491/u1D45B)do not include the shift term, the\nfunctions /u1D44D2(/u1D48C,/u1D491)and/u1D44D3(/u1D48C,/u1D491,−/u1D491), which are needed to\ncompute the 1-loop power spectrum in the IR limit, are ap-\nproximated as\n/u1D44D2(/u1D48C,/u1D491) − −−− →\n/u1D45D→01\n2/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n/u1D44D1(/u1D48C),\n/u1D44D3(/u1D48C,/u1D491,−/u1D491) − −−− →\n/u1D45D→0−1\n3!/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg2\n/u1D44D1(/u1D48C).(B7)\nAppendix C: Standard Perturbation Theory\nafter Reconstruction\nWe investigate the IR limit properties of the post-\nreconstruction density fluctuations up to the third order.\nIn Fourier space, Eq. (67) becomes\n/tildewide/u1D6FFrec(/u1D48C)=/uni222B.dsp\n/u1D4513/u1D465/u1D452−/u1D456/u1D48C·/u1D499/u1D452−/u1D456/u1D48C·/u1D494(/u1D499)/u1D6FF(/u1D499). (C1)\nThe nonlinear kernel function up to the third order are then\ngiven by [20]\n/u1D44Drec,1(/u1D4911)=/u1D44D1(/u1D4911),\n/u1D44Drec,2(/u1D4911,/u1D4912)=/u1D44D2(/u1D4911,/u1D4912) +1\n2/braceleftBigg/parenleftBigg\n/u1D48C·/u1D4911\n/u1D45D2\n1/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D1/u1D445s)\n/u1D44F1,fid/parenrightbigg\n+/parenleftBigg\n/u1D48C·/u1D4912\n/u1D45D2\n2/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D2/u1D445s)\n/u1D44F1,fid/parenrightbigg/bracerightBigg\n/u1D44D1(/u1D4911)/u1D44D1(/u1D4912),\n/u1D44Drec,3(/u1D4911,/u1D4912,/u1D4913)=/u1D44D3(/u1D4911,/u1D4912,/u1D4913)\n+1\n3/braceleftBigg/parenleftBigg\n/u1D48C·/u1D491[1,2]\n/u1D45D2\n[1,2]/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D[1,2]/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D2(/u1D4911,/u1D4912)/u1D44D1(/u1D4913) +2 perms./bracerightBigg\n+1\n3/braceleftBigg/parenleftBigg\n/u1D48C·/u1D4911\n/u1D45D2\n1/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D1/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D4911)/u1D44D2(/u1D4912,/u1D4913) +2 perms./bracerightBigg\n+1\n6/braceleftBigg/parenleftBigg\n/u1D48C·/u1D4911\n/u1D45D2\n1/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D1/u1D445s)\n/u1D44F1,fid/parenrightbigg/parenleftBigg\n/u1D48C·/u1D4912\n/u1D45D2\n2/parenrightBigg/parenleftbigg\n−/u1D44AG(/u1D45D2/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D4911)/u1D44D1(/u1D4912)/u1D44D1(/u1D4913) +2 perms./bracerightBigg\n,(C2)\nwhere/u1D48C=/u1D4911+/u1D4912for/u1D44Drec,2, and/u1D48C=/u1D4911+/u1D4912+/u1D4913for/u1D44Drec,3. The above equations include the bias effects as a slight exte nsion\nof those given by Hikage et al. [20]. Approximations in the IR limit of /u1D44Drec,2and/u1D44Drec,3in the form needed to compute the 1-loop\npower spectrum are given by\n/u1D44Drec,2(/u1D48C,/u1D491) − −−− →\n/u1D45D→01\n2/braceleftbigg/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n+/parenleftbigg/u1D48C·/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D491)/bracerightbigg\n/u1D44D1(/u1D48C),\n/u1D44Drec,3(/u1D48C,/u1D491,−/u1D491) − −−− →\n/u1D45D→0−1\n3!/braceleftbigg/parenleftbigg/u1D48C·/u1D4791·/u1D491\n/u1D45D2/parenrightbigg\n+/parenleftbigg/u1D48C·/u1D491\n/u1D45D2/parenrightbigg /parenleftbigg\n−/u1D44AG(/u1D45D/u1D445s)\n/u1D44F1,fid/parenrightbigg\n/u1D44D1(/u1D491)/bracerightbigg2\n/u1D44D1(/u1D48C). (C3)15\n[1] D. J. Eisenstein, H.-j. Seo, E. Sirko, and D. Spergel,\nAstrophys. J. 664, 675 (2007), arXiv:astro-ph/0604362.\n[2] R. A. Sunyaev and Y. B. Zeldovich, Astrophys. Space Sci. 7, 3\n(1970).\n[3] P. J. E. Peebles and J. T. Yu, Astrophys. J. 162, 815 (1970).\n[4] C. Hikage, R. Takahashi, and\nK. Koyama, Phys. Rev. D 102, 083514 (2020),\narXiv:2007.13998 [astro-ph.CO].\n[5] Y. Wang, G.-B. Zhao, K. Koyama, W. J. Percival,\nR. Takahashi, C. Hikage, H. Gil-Mar ´ın, C. Hahn,\nR. Zhao, W. Zhang, X. Mu, Y. Yu, H.-M. Zhu,\nand F. Ge, arXiv e-prints , arXiv:2202.05248 (2022),\narXiv:2202.05248 [astro-ph.CO].\n[6] M. Shirasaki, N. S. Sugiyama, R. Takahashi,\nand F.-S. Kitaura, Phys. Rev. D 103, 023506 (2021),\narXiv:2010.04567 [astro-ph.CO].\n[7] F. Bernardeau, S. Colombi, E. Gaztanaga, and R. Scoccima rro,\nPhys. Rept. 367, 1 (2002), arXiv:astro-ph/0112551.\n[8] J. Carlson, B. Reid, and M. White,\nMon. Not. Roy. Astron. Soc. 429, 1674 (2013),\narXiv:1209.0780 [astro-ph.CO].\n[9] L. Wang, B. Reid, and M. White,\nMon. Not. Roy. Astron. Soc. 437, 588 (2014),\narXiv:1306.1804 [astro-ph.CO].\n[10] M. Crocce and R. Scoccimarro, Phys. Rev. D73, 063519 (2006),\narXiv:astro-ph/0509418 [astro-ph].\n[11] A. Taruya, T. Nishimichi, and\nS. Saito, Phys. Rev. D 82, 063522 (2010),\narXiv:1006.0699 [astro-ph.CO].\n[12] D. Baumann, A. Nicolis, L. Senatore, and M. Zaldarriaga ,\nJCAP 07, 051 (2012), arXiv:1004.2488 [astro-ph.CO].\n[13] J. J. M. Carrasco, M. P. Hertzberg, and L. Senatore,\nJHEP 09, 082 (2012), arXiv:1206.2926 [astro-ph.CO].\n[14] D. J. Eisenstein et al. (SDSS), Astron. J. 142, 72 (2011),\narXiv:1101.1529 [astro-ph.IM].\n[15] A. S. Bolton et al. (Cutler Group, LP),\nAstron. J. 144, 144 (2012), arXiv:1207.7326 [astro-ph.CO].\n[16] K. S. Dawson et al. (BOSS), Astron. J. 145, 10 (2013),\narXiv:1208.0022 [astro-ph.CO].\n[17] S. Alam et al. (SDSS-III), Astrophys. J. Suppl. 219, 12 (2015),\narXiv:1501.00963 [astro-ph.IM].\n[18] M. Schmittfull, Y. Feng, F. Beutler, B. Sherwin,\nand M. Y. Chu, Phys. Rev. D 92, 123522 (2015),\narXiv:1508.06972 [astro-ph.CO].\n[19] C. Hikage, K. Koyama, and A. Heav-\nens, Phys. Rev. D 96, 043513 (2017),\narXiv:1703.07878 [astro-ph.CO].\n[20] C. Hikage, K. Koyama, and R. Taka-\nhashi, Phys. Rev. D 101, 043510 (2020),\narXiv:1911.06461 [astro-ph.CO].\n[21] N. Padmanabhan, M. White, and J. D. Cohn,\nPhys. Rev. D 79, 063523 (2009), arXiv:0812.2905 [astro-ph].\n[22] H.-J. Seo, F. Beutler, A. J. Ross, and S. Saito,\nMon. Not. Roy. Astron. Soc. 460, 2453 (2016),\narXiv:1511.00663 [astro-ph.CO].\n[23] M. White, Mon. Not. Roy. Astron. Soc. 450, 3822 (2015),\narXiv:1504.03677 [astro-ph.CO].\n[24] S.-F. Chen, Z. Vlah, and M. White, JCAP 09, 017 (2019),\narXiv:1907.00043 [astro-ph.CO].\n[25] A. Ota, H.-J. Seo, S. Saito, and\nF. Beutler, Phys. Rev. D 104, 123508 (2021),arXiv:2106.00146 [astro-ph.CO].\n[26] A. Ota, H.-J. Seo, S. Saito, and\nF. Beutler, Phys. Rev. D 107, 123523 (2023),\narXiv:2211.07960 [astro-ph.CO].\n[27] H.-M. Zhu, Y. Yu, U.-L. Pen, X. Chen,\nand H.-R. Yu, Phys. Rev. D 96, 123502 (2017),\narXiv:1611.09638 [astro-ph.CO].\n[28] H.-M. Zhu, Y. Yu, and U.-\nL. Pen, Phys. Rev. D 97, 043502 (2018),\narXiv:1711.03218 [astro-ph.CO].\n[29] Y. Yu, H.-M. Zhu, and U.-L. Pen, Astrophys. J. 847, 110 (2017),\narXiv:1703.08301 [astro-ph.CO].\n[30] X. Wang, H.-R. Yu, H.-M. Zhu, Y. Yu, Q. Pan,\nand U.-L. Pen, Astrophys. J. Lett. 841, L29 (2017),\narXiv:1703.09742 [astro-ph.CO].\n[31] M. Schmittfull, T. Baldauf, and M. Zal-\ndarriaga, Phys. Rev. D 96, 023505 (2017),\narXiv:1704.06634 [astro-ph.CO].\n[32] R. Hada and D. J. Eisenstein,\nMon. Not. Roy. Astron. Soc. 478, 1866 (2018),\narXiv:1804.04738 [astro-ph.CO].\n[33] R. Hada and D. J. Eisenstein,\nMon. Not. Roy. Astron. Soc. 482, 5685 (2019),\narXiv:1810.05026 [astro-ph.CO].\n[34] H.-J. Seo, A. Ota, M. Schmittfull, S. Saito, and\nF. Beutler, Mon. Not. Roy. Astron. Soc. 511, 1557 (2022),\narXiv:2106.00530 [astro-ph.CO].\n[35] B. Jain and E. Bertschinger, Astrophys. J. 456, 43 (1996),\narXiv:astro-ph/9503025.\n[36] R. Scoccimarro and J. Frieman,\nAstrophys. J. Suppl. 105, 37 (1996), arXiv:astro-ph/9509047.\n[37] A. Kehagias and A. Riotto, Nucl. Phys. B 873, 514 (2013),\narXiv:1302.0130 [astro-ph.CO].\n[38] M. Peloso and M. Pietroni, JCAP 05, 031 (2013),\narXiv:1302.0223 [astro-ph.CO].\n[39] N. S. Sugiyama and T. Futamase, Astrophys. J. 769, 106 (2013),\narXiv:1303.2748 [astro-ph.CO].\n[40] N. S. Sugiyama and D. N. Spergel, JCAP 02, 042 (2014),\narXiv:1306.6660 [astro-ph.CO].\n[41] D. Blas, M. Garny, and T. Konstandin, JCAP 09, 024 (2013),\narXiv:1304.1546 [astro-ph.CO].\n[42] D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov,\nJCAP 07, 052 (2016), arXiv:1512.05807 [astro-ph.CO].\n[43] M. Lewandowski and L. Senatore, JCAP 08, 037 (2017),\narXiv:1701.07012 [astro-ph.CO].\n[44] P. Creminelli, J. Nore ˜na, M. Simonovi ´c, and F. Vernizzi,\nJCAP 12, 025 (2013), arXiv:1309.3557 [astro-ph.CO].\n[45] Y. B. Zeldovich, Astron. Astrophys. 5, 84 (1970).\n[46] D. J. Eisenstein, H.-j. Seo, and M. J. White,\nAstrophys. J. 664, 660 (2007), arXiv:astro-ph/0604361.\n[47] M. Crocce and R. Scoccimarro,\nPhys. Rev. D 77, 023533 (2008), arXiv:0704.2783 [astro-ph].\n[48] T. Matsubara, Phys. Rev. D77, 063530 (2008),\narXiv:0711.2521 [astro-ph].\n[49] L. Senatore and M. Zaldarriaga, JCAP 02, 013 (2015),\narXiv:1404.5954 [astro-ph.CO].\n[50] T. Baldauf, M. Mirbabayi, M. Simonovi ´c, and\nM. Zaldarriaga, Phys. Rev. D 92, 043514 (2015),\narXiv:1504.04366 [astro-ph.CO].\n[51] D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov,\nJCAP 07, 028 (2016), arXiv:1605.02149 [astro-ph.CO].16\n[52] L. Senatore and G. Trevisan, JCAP 05, 019 (2018),\narXiv:1710.02178 [astro-ph.CO].\n[53] M. M. Ivanov and S. Sibiryakov, JCAP 07, 053 (2018),\narXiv:1804.05080 [astro-ph.CO].\n[54] M. Lewandowski and L. Senatore, JCAP 03, 018 (2020),\narXiv:1810.11855 [astro-ph.CO].\n[55] N. S. Sugiyama, S. Saito, F. Beutler, and H.-\nJ. Seo, Mon. Not. Roy. Astron. Soc. 501, 2862 (2021),\narXiv:2010.06179 [astro-ph.CO].\n[56] N. Kaiser, Mon. Not. Roy. Astron. Soc. 227, 1 (1987).\n[57] V. Desjacques, D. Jeong, and F. Schmidt,\nPhys. Rept. 733, 1 (2018), arXiv:1611.09787 [astro-ph.CO].\n[58] Z. Vlah, U. Seljak, M. Y. Chu, and Y. Feng,\nJCAP 03, 057 (2016), arXiv:1509.02120 [astro-ph.CO].\n[59] M. Schmittfull, T. Baldauf, and\nU. Seljak, Phys. Rev. D 91, 043530 (2015),\narXiv:1411.6595 [astro-ph.CO].\n[60] F. Bernardeau, M. Crocce, and R. Scoccimarro,\nPhys. Rev. D78, 103521 (2008), arXiv:0806.2334 [astro-ph].\n[61] T. Matsubara, Phys. Rev. D78, 083519 (2008), [Erratum: Phys.\nRev.D78,109901(2008)], arXiv:0807.1733 [astro-ph].\n[62] D. Langlois, Int. J. Mod. Phys. D 28, 1942006 (2019),\narXiv:1811.06271 [gr-qc].\n[63] T. Kobayashi, Rept. Prog. Phys. 82, 086901 (2019),\narXiv:1901.07183 [gr-qc].\n[64] N. S. Sugiyama, D. Yamauchi, T. Kobayashi, T. Fu-\njita, S. Arai, S. Hirano, S. Saito, F. Beutler, and\nH.-J. Seo, Mon. Not. Roy. Astron. Soc. 523, 3133 (2023),\narXiv:2302.06808 [astro-ph.CO].\n[65] N. S. Sugiyama, D. Yamauchi, T. Kobayashi, T. Fu-\njita, S. Arai, S. Hirano, S. Saito, F. Beutler, and\nH.-J. Seo, Mon. Not. Roy. Astron. Soc. 524, 1651 (2023),arXiv:2305.01142 [astro-ph.CO].\n[66] J. Yoo, N. Dalal, and U. Seljak, JCAP 07, 018 (2011),\narXiv:1105.3732 [astro-ph.CO].\n[67] N. E. Chisari and A. Pontzen, Phys. Rev. D 100, 023543 (2019),\narXiv:1905.02078 [astro-ph.CO].\n[68] D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605 (1998),\narXiv:astro-ph/9709112 [astro-ph].\n[69] M. White, (2010), arXiv:1004.0250 [astro-ph.CO].\n[70] S. Alam et al. (BOSS),\nMon. Not. Roy. Astron. Soc. 470, 2617 (2017),\narXiv:1607.03155 [astro-ph.CO].\n[71] S. R. Hinton et al., Mon. Not. Roy. Astron. Soc. 464, 4807 (2017),\narXiv:1611.08040 [astro-ph.CO].\n[72] S. Alam et al. (eBOSS), Phys. Rev. D 103, 083533 (2021),\narXiv:2007.08991 [astro-ph.CO].\n[73] J. Moon et al.(DESI), (2023), arXiv:2304.08427 [astro-ph.CO].\n[74] S.-F. Chen et al., (2024), arXiv:2402.14070 [astro-ph.CO].\n[75] S. Hirano, T. Kobayashi, H. Tashiro, and\nS. Yokoyama, Phys. Rev. D 97, 103517 (2018),\narXiv:1801.07885 [astro-ph.CO].\n[76] S. Hirano, T. Kobayashi, D. Yamauchi, and S. Yokoyama,\nPhys. Rev. D 102, 103505 (2020), arXiv:2008.02798 [gr-qc].\n[77] M. Crisostomi, M. Lewandowski, and\nF. Vernizzi, Phys. Rev. D 101, 123501 (2020),\narXiv:1909.07366 [astro-ph.CO].\n[78] M. Lewandowski, JCAP 08, 044 (2020),\narXiv:1912.12292 [astro-ph.CO].\n[79] R. Scoccimarro, H. M. P. Couchman, and\nJ. A. Frieman, Astrophys. J. 517, 531 (1999),\narXiv:astro-ph/9808305 [astro-ph]."    },    {        "title": "2402.06148v1.Imaginary_eigenvalues_of_Hermitian_Hamiltonian_with_an_inverted_potential_well_and_transition_to_the_real_spectrum_at_exceptional_point_by_a_non_Hermitian_interaction.pdf",        "content": "arXiv:2402.06148v1  [quant-ph]  9 Feb 2024Imaginary eigenvalues of Hermitian Hamiltonian with an inv erted potential well and\ntransition to the real spectrum at exceptional point by a non -Hermitian interaction\nNi Liu,∗Meng Luo, and J. -Q. Liang†\nInstitute of Theoretical Physics, State Key Laboratory of Q uantum Optics and Quantum Optics Devices,\nShanxi University, Taiyuan 030006, Shanxi, China\nWe in this paper study the hermiticity of Hamiltonian and ene rgy spectrum for the SU(1,1) sys-\ntems. The Hermitian Hamiltonian can possess imaginary eige nvalues in contrast with the common\nbelief that hermiticity is a sufficient condition for real spe ctrum. The imaginary eigenvalues are de-\nrived in algebraic method with imaginary-frequency boson o perators for the Hamiltonian of inverted\npotential well. Dual sets of mutually orthogonal eigenstat es are required corresponding respectively\nto the complex conjugate eigenvalues. Arbitrary order eige nfunctions seen to be the polynomials\nof imaginary frequency are generated from the normalized gr ound-state wave functions, which are\nspatially non-localized. The Hamiltonian including a non- Hermitian interaction term can be con-\nvertedby similarity transformation to the Hermitian one wi th an effective potential of reduced slope,\nwhich is turnable by the interaction constant. The transfor mation operator should not be unitary\nbut Hermitian different from the unitary transformation in o rdinary quantum mechanics. The ef-\nfective potential vanishes at a critical value of coupling s trength called the exceptional point, where\nall eigenstates are degenerate with zero eigenvalue and tra nsition from imaginary to real spectra\nappears. The SU(1,1) generator /hatwideSzwith real eigenvalues determined by the commutation relati on\nof operators, however, is non-Hermitian in the realization of imaginay-frequency boson operators.\nThe classical counterpart of the quantum Hamiltonian with n on-Hermitian interaction is a complex\nfunction of the canonical variables. It becomes by the canon ical transformation of variables a real\nfunction indicating exactly the one to one quantum-classic al correspondence of Hamiltonians.\nPACS numbers:\nI. INTRODUCTION\nIn quantum mechanics the Hamiltonian has to be\nHermitian to guarantee real energy spectrum for the\nclosed systems. The non-Hermitian Hamiltonian de-\nscribesusuallyopensystems,whichexchangeenergycon-\ntinuously with external source and thus possess com-\nplex energy spectrum. Bender and his collaborators\nproposed for the first time a creative finding that the\nHermitian Hamiltonian is sufficient but not necessary\nto have real eigenvalues1–3. According to them the\nnon-Hermitian Hamiltonian with real spectrum has the\nparity-time ( PT) reflection symmetry1–6. Mostafazadeh\nnamed the non-Hermitian Hamiltonian with real spec-\ntrum as pseudo-Hermitian, which satisfies a neces-\nsary condition7–9. While the PTsymmetry is neither\nnecessary nor sufficient condition for a non-Hermitian\nHamiltonian to have real spectrum. It is found that\nthePT-asymmetric Hamiltonian can also possess real\nspectra10,11.\nFor thePT-symmetric non-Hermitian Hamiltonians\nthere exists an exceptional point, which separates the\nunbrokenPTsymmetry region with pure real-spectrum\nand a broken PTsymmetry region, where eigenval-\nues are complex1,12,13. This transition at the excep-\ntional point has been observed in numerous labora-\ntory experiments14,15for various non-Hermitian Hamil-\ntonians. The non-Hermitian Hamiltonian has become\na rapidly developing subject in different branches of\nphysics16–21. The spectrum property becomes compli-\ncated at the exceptional points22–25due to the sym-metry breaking, which can be modulated by dissipative\nmedia26–29. The exceptional point and complex domain\nof eigenvalues have been well studied particularly in re-\nlation with optical systems14,15.\nAs a matter of fact the content of quantum mechan-\nics is more rich than what we are known. Not only the\nnon-HermitianHamiltoniansofopensystemsbutalsothe\nHermitianHamiltonianofaconservativesystemcanhave\ncomplex spectrum if the potential is unbounded from be-\nlow. The complex spectrum of an inverted double-well\npotential was studied long ago in terms of the instan-\nton method under semiclassical approximation30,31. The\nimaginarypartofenergyeigenvaluescharacterizesthede-\ncay rate or life time of metastable states30,31. As of yet\nthe rigorous complex eigenvalues and associated eigen-\nstates have not been given in the framework of quantum\nmechanics for the Hermitian Hamiltonians.\nThe gain and loss of energy are not the only mecha-\nnism for complex eigenvalues. We propose in the present\npaper a solvable Hamiltonian consisting of SU(1,1) gen-\nerators, which can be realized by single-mode boson op-\nerators. In the absence of external field the Hamiltonian,\nwhich is Hermitian however with imaginary eigenvalues,\ndescribes a particle in an inverted potential well. Clas-\nsically the particle moves acceleratingly away from the\ncenter of the inverted potential well. The quantum wave\nfunctions are, of course, spatially non-localized. Inverted\npotential wellunbounded from belowis found in the min-\nisuperpace model of expanding universe, which is estab-\nlished from Einstein equation by including the cosmology\nconstant32.2\nWe demonstrate in the present paper that the non-\nHermitian interaction of an external field can decrease\ncontinuously the slope of inverted potential well by the\nincrease of coupling constant. At the end the potential\nvanishes at a critical coupling-value, namely the excep-\ntional point, where all eigenstates are degenerate with a\nzero eigenvalue. Beyond the exceptional point the eigen-\nvalues become real as that of a normal oscillator. We dis-\ncovera peculiar transition from imaginaryto real spectra\nat the exceptional point induced by the non-Hermitian\ninteraction.\nThe Hamiltonian with a non-Hermitian interaction\nterm canbe convertedby asimilaritytransformationto a\nHermitian one. The transformation operator is not uni-\ntary but Hermitian5different from the unitary transfor-\nmation in the ordinary quantum mechanics. The classi-\ncal counterpartofthe non-HermitianHamiltonian is seen\nto be a complex function of canonical variables. It be-\ncomes under the canonical transformation of variables a\nreal function indicating strict one to one correspondence\nof the quantum and classical Hamiltonians5.\nThe article is organized as follows. In Sec.II, the imag-\ninary spectrum and eigenstates are obtained by means\nof the algebraic method for the Hermitian Hamiltonian\nwith an inverted potential well. We demonstrate in\nSec.III the transition from imaginary to real spectra at\nexceptional point by the non-Hermitian interaction. The\nquantum-classical correspondence for the Hermitian and\nnon-Hermitian Hamiltonians is revealed in Sec.IV. The\nconclusion and discussion are presented in Sec.V.\nII. HERMITIAN HAMILTONIAN WITH AN\nINVERTED POTENTIAL WELL AND\nIMAGINARY SPECTRUM\nThe hermiticity of Hamiltonian is not sufficient con-\ndition for real eigenvalues. We assume that the Hermi-\ntian Hamiltonian possesses discrete complex eigenvalues.\nThere must exist dual-set of eigenstates such that\n/hatwideH|un/angbracketrightr=En|un/angbracketrightr,r/angbracketleftun|/hatwideH=r/angbracketleftun|E∗\nn\n/hatwideH|un/angbracketrightl=E∗\nn|un/angbracketrightl,l/angbracketleftun|/hatwideH=l/angbracketleftun|En,(1)\nin which the subscripts “ r”, “l” denote respectively the\n“ket” and “bra” states. The two sets of eigenstates are\nmutually orthonormal\nl/angbracketleftun|um/angbracketrightr=r/angbracketleftun|um/angbracketrightl=δn,m.\nThe density operators defined by /hatwideρ=|ψ/angbracketrightrl/angbracketleftψ|,/hatwideρ†=\n|ψ/angbracketrightlr/angbracketleftψ|are non-Hermitian invariants\nd/hatwideρ\ndt=d/hatwideρ†\ndt= 0.\nThey obey the quantum Liouvill equation (in the unit\nconvention /planckover2pi1= 1 throughout the paper),\ni∂/hatwideρ\n∂t=/bracketleftBig\n/hatwideH,/hatwideρ/bracketrightBig\n, i∂/hatwideρ†\n∂t=/bracketleftBig\n/hatwideH,/hatwideρ†/bracketrightBig\n,which are in consistence with the Schr¨ odinger equations\ni∂\n∂t|ψ/angbracketrightγ=/hatwideH|ψ/angbracketrightγ,\nfor both the “bra” and “ket” states ( γ=l,r). The state\ndensities /angbracketleftx|/hatwideρ|x/angbracketright,/angbracketleftx|/hatwideρ†|x/angbracketrightare conserved quantities satis-\nfying the basic requirement of quantum mechanics.\nWe consider the Hermitian Hamiltonian for a particle\nof unit mass in the inverted potential well\n/hatwideH= Ω/parenleftbigg/hatwidep2\n2−/hatwidex2\n2/parenrightbigg\n, (2)\nin which/hatwidex,/hatwidepare dimensionless operator with the usual\ncommutation relation [ /hatwidex,/hatwidep] =i. The classical orbit of\nthe particle in the inverted potential well is simply\nx(t) =v0\nΩsinh(Ωt),\nwith initial position x0= 0 and velocity v0. A min-\nisuperspace model32of the universe is found also as an\ninverted potential well, which results in the accelerating\nexpansion. The Hamiltonian Eq.(2) becomes with the\nimaginary frequency boson-operators\n/hatwideH=iΩ/parenleftbigg\n/hatwideb+/hatwideb−+1\n2/parenrightbigg\n,\nwhere\n/hatwideb−=/radicalbigg\ni\n2(/hatwidex+/hatwidep),/hatwideb+=/radicalbigg\ni\n2(/hatwidex−/hatwidep).(3)\nThe imaginary-frequency boson operators satisfy the\nsame commutation relation as the usual boson operators\n/bracketleftBig\n/hatwideb−,/hatwideb+/bracketrightBig\n= 1. (4)\nThe commutation relation of SU(1,1) generators5,33is\ninvariant under the realization of imaginary-frequency\nboson-operators\n/hatwideSz=1\n2/parenleftbigg\n/hatwideb+/hatwideb−+1\n2/parenrightbigg\n,/hatwideS+=1\n2/parenleftBig\n/hatwideb+/parenrightBig2\n,/hatwideS−=1\n2/parenleftBig\n/hatwideb−/parenrightBig2\n(5)\nsuch that\n/bracketleftBig\n/hatwideSz,/hatwideS±/bracketrightBig\n=±/hatwideS±,/bracketleftBig\n/hatwideS+,/hatwideS−/bracketrightBig\n=−2/hatwideSz.(6)\nThus the Hamiltonian Eq.(2), is represented as\n/hatwideH= 2iΩ/hatwideSz. (7)\nSince\n/hatwideS†\nz=1\n2/parenleftbigg\n−/hatwideb−/hatwideb++1\n2/parenrightbigg\n=−/hatwideSz,\nthe Hamiltonian Eq.(7) is Hermitian as it should be.\nWhile the SU(1,1) generator /hatwideSzis no longer Hermi-\ntian different from the representation of normal boson3\noperators5,33. The imaginary-frequency boson-number\noperators\n/hatwidenI=/hatwideb+/hatwideb−;/hatwiden†\nI=−/hatwideb−/hatwideb+ (8)\nare non-Hermitian with eigenstates given by\n/hatwidenI|n/angbracketrightr=n|n/angbracketrightr,r/angbracketleftn|n=r/angbracketleftn|/hatwiden†\nI, (9)\n/hatwiden†\nI|n/angbracketrightl=n|n/angbracketrightl,l/angbracketleftn|n=l/angbracketleftn|/hatwiden†\nI\nThe orthogonal condition is\nl/angbracketleftn|m/angbracketrightr=δnm. (10)\nThe Hermitian Hamiltonian possesses discrete imaginary\neigenvalues\n/hatwideH|n/angbracketrightγ=±iΩ/parenleftbigg\nn+1\n2/parenrightbigg\n|n/angbracketrightγ, (11)\nwithγ=r,lrespectively for the “ket” and “bra” states.\nIndeed the Hermitian Hamiltonian with imaginaryeigen-\nvalues Eq.(11) has a dual-set of eigenstates in agreement\nwith the general theory Eq.(1).\nBased on the commutation relation of imaginary-\nfrequency boson-operators Eq.(4) the arbitrary order\neigenstates can be generated from the ground states such\nthat (see Appendix for the derivation)\n|n/angbracketrightr=/parenleftBig\n/hatwideb+/parenrightBign\n√\nn!|0/angbracketrightr,\n|n/angbracketrightl=/parenleftBig\n/hatwideb−/parenrightBign\nin√\nn!|0/angbracketrightl.\nFrom the ground-state equations\n/hatwideb−|0/angbracketrightr= 0;/hatwideb+|0/angbracketrightl= 0\nthe arbitrary order eigenfunctions are seen to be\nimaginary-frequency polynomials\nψr,n(x,Ω) =1\nπ1\n4√\n2nn!/radicalBig\ni(n+1\n2)/parenleftbigg\nx−id\ndx/parenrightbiggn\ne−ix2\n2,\nψl,n(x,Ω) =1\nπ1\n4/radicalbig\n2nn!i(n−1\n2)/parenleftbigg\nx+id\ndx/parenrightbiggn\neix2\n2,(12)\nwhich derived analytically in the Appendix are spatially\nnon-localized different from the Hermit polynomials of\nthe normal oscillator. It may be better to remark that\nthe wave functions are represented in the dimensionless\ncoordinate x, which is scaled by the square root of fre-\nquency√\nΩ. In the ordinary space coordinate, xshould\nbe replaced by√\nΩx(see Appendix). Thus the eigen-\nfunctionsEq.(12)areactuallyfrequencydependentinthe\nusual space coordinate.III. TRANSITION FROM IMAGINARY TO\nREAL DOMAINS OF SPECTRUM AND THE\nEXCEPTIONAL POINT\nWe apply a non-Hermitian interaction to the Hamilto-\nnian for a particle in the inverted potential well\n/hatwideH=/hatwideH0+G/parenleftBig\n/hatwideS+−/hatwideS−/parenrightBig\n, (13)\nwhere\n/hatwideH0= 2iΩ/hatwideSz\nandGis the coupling constant. /hatwideH0is Hermitian how-\never with imaginary eigenvalue as shown in the previous\nsection, while total Hamiltonian is non-Hermitian,\n/hatwideH†=/hatwideH0−G/parenleftBig\n/hatwideS+−/hatwideS−/parenrightBig\n(14)\nwhere/hatwideS+,/hatwideS−given in Eq.(5) are anti-Hermitian /hatwideS†\n+=\n−/hatwideS+,/hatwideS†\n−=−/hatwideS−seen from the definition of boson oper-\nators Eq.(3).\nThe non-Hermitian Hamiltonian with complex eigen-\nvalues can be also transformed to a Hermitian one in\ngeneral. To see this we assume that\n/hatwideH|un/angbracketrightr=εn|un/angbracketrightr,/hatwideH†|un/angbracketrightl=ε∗\nn|un/angbracketrightl.\nUnder a similarity transformation with a Hermitian op-\nerator/hatwideR†=/hatwideR, we have\n/hatwideR/hatwideH/hatwideR−1|/tildewideun/angbracketrightr=εn|/tildewideun/angbracketrightr,/hatwideR−1/hatwideH†/hatwideR|/tildewideun/angbracketrightl=ε∗\nn|/tildewideun/angbracketrightl,\nwith\n|/tildewideun/angbracketrightl=/hatwideR−1|un/angbracketrightl,|/tildewideun/angbracketrightr=/hatwideR|un/angbracketrightr.(15)\nIf the equality\n/hatwideR/hatwideH/hatwideR−1=/hatwideR−1/hatwideH†/hatwideR≡/tildewideH, (16)\nis satisfied, the transformed Hamiltonian /tildewideH=/tildewideH†is Her-\nmitian with complex eigenvalues.\nTo this end the Hamiltonian Eq.(13) can be diagonal-\nized in terms of a similarity transformationwith the Her-\nmitian operator\n/hatwideR=e−iη\n2(/hatwideS++/hatwideS−), (17)\nwhere the real-value parameter ηis to be determined.\nTheSU(1,1) generators are realized by the imaginay-\nfrequency boson operators in Eq.(5). It is easy to check\nthat the transformation operator Eq.(17) is Hermitian\n/hatwideR=/hatwideR†. Using the transformation relations5,33\n/hatwideR/hatwideS+/hatwideR−1=/hatwideS+cosh2η\n2−/hatwideS−sinh2η\n2−i/hatwideSzsinhη,\n/hatwideR/hatwideS−/hatwideR−1=/hatwideS−cosh2η\n2−/hatwideS+sinh2η\n2+i/hatwideSzsinhη,\n/hatwideR/hatwideSz/hatwideR−1=/hatwideSzcoshη+i\n2(/hatwideS+−/hatwideS−)sinhη,4\nthe diagonalized Hamiltonian is indeed Hermitian\n/tildewideH=/hatwideR/hatwideH/hatwideR−1=/hatwideR−1/hatwideH†/hatwideR= 2iΓI/hatwideSz,(18)\nwith an effective imaginary frequency\nΓI=/radicalbig\nΩ2−G2. (19)\nThe non-Hermitian coupling constant Gdecreases the\nslope of potential well and the effective frequency. There\nexists an exceptional point\nGc= Ω,\nwherealleigenstatesaredegeneratewithvanishingeigen-\nvalue. The pending parameter ηis determined from the\nequation\nsinhη=±G√\nΩ2−G2, (20)\nwhich for a real value of ηis valid below the exceptional\npointG < G c. The Hamiltonian Eq.(18) describes the\nparticle in a deformed inverted potential well with the\nreduced slope, which tends to zero at the exceptional\npoint. The variation of potential-well shape is illustrated\nin Fig.1.\nThe eigenstates of transformed Hamiltonian are obvi-\nously\n/tildewideH|/tildewideun/angbracketrightγ=±iΓI/parenleftbigg\nn+1\n2/parenrightbigg\n|/tildewideun/angbracketrightγ,\nrespectively for the “ket” and “bra” states, γ=r,l.\nIn our case it is simply the eigenstate of imaginary-\nfrequency boson-number /hatwidenI\n|/tildewideun/angbracketrightγ=|n/angbracketrightγ,\ndefined in equation Eq.(9). The eigenstates of\nthe original non-Hermitian Hamiltonians in equations\nEq.(13),Eq.(14) are obtained respectively by the inverse\ntransforms\n|un/angbracketrightr=/hatwideR−1|n/angbracketrightr,|un/angbracketrightl=/hatwideR|n/angbracketrightl,\nwith corresponding eigenvalues\nεn=iΓI/parenleftbigg\nn+1\n2/parenrightbigg\n,\nand\nε∗\nn=−iΓI/parenleftbigg\nn+1\n2/parenrightbigg\n.\nThe lowest-layer energy spectrum with respect to in-\nteraction constant Gis displayed in Fig.2 for n= 0\n(black),1 (blue),2 (red). All eigenvalues vanish at the\nexceptionalpoint Gc, which separatesthe realand imagi-\nnarydomainsofenergyspectrum. Weemphasizethatthe-3.0-2.0-1.00.01.02.03.0-1.5-1.0-0.50.00.51.01.5\nxVG/\u0001=0.3\nG/\u0001=0.7\nG/\u0001=1.0\nG/\u0001=1.3\nG/\u0001=1.7\nFIG. 1: The interaction-constant dependence of inverted\npotential-well shape for G= 0.3 (red curve), 0 .7 (blue) in\nthe unit Ω. The potential vanishes at the exceptional point\nGc= 1.0 (black) and becomes normal well beyond Gcfor\nG= 1.3 (green), 1 .7 (orange).\nFIG. 2: The lowest layers ( n= 0,1,2) of energy spectrum as\nfunctions of the interaction constant G. The imaginary eigen-\nvalues below Gchave two branches respectively for the “ket”\nand “bra” states and vanish at Gc. The spectrum becomes\nreal beyond the exceptional point.\nopen orbitsbelowthe exceptionalpoint Gcarealsoquan-\ntized, however with two branches of imaginary discrete-\neigenvalues.\nWith the coordinaterepresentationof SU(1,1)genera-\ntors/hatwideS+and/hatwideS−in equations Eq.(3),Eq.(5)the eigenfunc-\ntions of ”bra” and ”ket” states are obtained as5\nur,n(x) =e−η\n2/parenleftBig\nx2−d2\ndx2/parenrightBig\nψr,n(x,ΓI),\nul,n(x) =eη\n2/parenleftBig\nx2−d2\ndx2/parenrightBig\nψl,n(x,ΓI),\nin which the value of parameter ηis solved from Eq.(20).\nThe eigenfunctions ψγ,n(x,ΓI) =/angbracketleftx|/tildewideun/angbracketrightγforγ=r,l\nare the same as in the equation Eq.(12), however with\nthe reduced frequency Γ Iinstead of Ω. The state density\nprobability is\nρr,l=/angbracketleftx|ur,n/angbracketright/angbracketleftul,n|x/angbracketright=ψr,n(x,ΓI)ψ∗\nl,n(x,ΓI),\nρl,r=ρ∗\nr,l.\nBeyondGcthe potential becomes a normal oscillator-\nwell as shown in Fig.(1). By analytical continuation ex-\ntension the transformed Hamiltonian describes a normal\noscillator\n/tildewideHext= 2Γ/hatwideSz, (21)\nwith\nΓ =/radicalbig\nG2−Ω2.\nTheSU(1,1) generatorsin this region are realized by the\nnormal boson operators /hatwidea,/hatwidea†\n/hatwideSz=1\n2/parenleftbigg\n/hatwiden+1\n2/parenrightbigg\n,/hatwiden=/hatwidea†/hatwidea\nwhere\n/hatwidea=1√\n2(/hatwidex+i/hatwidep),/hatwidea†=1√\n2(/hatwidex−i/hatwidep).\nThe commutation relation Eq.(6) of SU(1,1) genera-\ntors is invariant under the realization of normal boson-\noperators.\nThe eigenstates of transformed Hamiltonian Eq.(21)\nare that of normal oscillator\n/tildewideHext|n/angbracketright= Γ/parenleftbigg\nn+1\n2/parenrightbigg\n|n/angbracketright,\nwhere/hatwiden|n/angbracketright=n|n/angbracketright.\nIV. QUANTUM-CLASSICAL\nCORRESPONDENCE FOR THE HAMILTONIAN\nOF INVERTED POTENTIAL WELL WITH\nNON-HERMITIAN INTERACTION\nThe classical counterpart of the quantum Hamiltonian\nEq.(13) in the realization of imaginary-frequency boson\noperators Eq.(5),Eq.(3) is seen to be\nH(x,p) = Ω/parenleftbiggp2\n2−x2\n2/parenrightbigg\n−iGxp, (22)which is a complex function of canonical variables x,\npcorresponding to the non-Hermitian interaction. We\nare going to find what is the classical counterpart of the\ntransformedHermitianHamiltonian Eq.(18). Tothis end\nwe adopt a canonical transformation of variables\nx=Xcoshη\n2−iPsinhη\n2,\np=iXsinhη\n2+Pcoshη\n2, (23)\nin which parameter ηis to be determined. Substituting\nthe canonical variable transformation Eq.(23) into the\nphase space Lagrangian\nL=.xp−H(x,p) (24)\nwith the classical Hamiltonian given by Eq.(22) yields\nL(X,P) =L′(X,P)+dF(X,P)\ndt,(25)\nwhereF(X,P) is called the gaugeorgeneratingfunction.\nTwo Lagrangians L(X,P) and\nL′(X,P) =.\nXP−H′(X,P)\ndiffering by a total time derivative of canonical-variable\nfunctionF(X,P) are gauge equivalent34, since they give\nrise to the same equation of motion. The gauge transfor-\nmation of the Hamiltonian Eq.(22) is found as\nH′= ΓI/parenleftbiggP2\n2−X2\n2/parenrightbigg\n, (26)\nunder the condition that the pending parameter ηis de-\ntermined by the equation Eq.(20). The Hamiltonian H′,\nwhich indeed is a real function corresponding to the Her-\nmitian Hamiltonian, describes a particle in an inverted\npotential well. It is exactly the classical counterpart of\nquantum version /tildewideHgiven in Eq.(18). The gauge function\nF=1\n2/bracketleftbiggi\n2/parenleftbig\nP2−X2/parenrightbig\n−XP/bracketrightbiggG\nΓI+1\n2XP\nis however a complex function of the canonical variables.\nThe quantum-classical correspondence holds strictly5,35\nfor the Hamiltonian of imaginary spectrum with non-\nHermitian interaction.\nV. CONCLUSION AND DISCUSSION\nThe hermiticity of Hamiltonian is not a necessary\ncondition to possess real spectrum. It is insufficient ei-\nther. The Hermitian Hamiltonian of a conservative sys-\ntem can also have complex eigenvalues, if the potential\nis unbounded from below. The complex eigenvalues of\nmetastable states were studied long ago with the semi-\nclassical method30,31. Rigorous eigenvalues and states6\nare still lacking. The Hermitian Hamiltonian with in-\nverted potential well is solved in the algebraic method\nby means of imaginary-frequency boson operators. The\nimaginary spectrum is derived analytically along with a\ndual-set of eigenstates corresponding respectively to the\ncomplex conjugate eigenvalues. The density-operators\nare non-Hermitian invariants, which give rise to the den-\nsity probability conservation in agreement with the basic\nrequirement of quantum mechanics. The SU(1,1) gener-\nator/hatwideSz,whichpossessesalwaysrealspectrumdetermined\nby the commutation relation, however is non-Hermitian\nin the realization of imaginary-frequency boson opera-\ntors. An interesting observation is that the Hermitian\nHamiltonian possesses imaginary eigenvalues while the\nspectrum of non-Hermitian operator /hatwideSzis real. It is\nwell known that the spectrum of the pseudo-Hermitian\nHamiltonians is not necessarily real in the whole region\nof parameter value. It approaches a complex domain\nat the exceptional point when the coupling constant of\nnon-Hermitian interaction increases. Different from the\ncommon belief we find the opposite direction of transi-\ntion from imaginaryto real domains of eigenvalues at the\nexceptional point. In the considered SU(1,1) Hamilto-\nnian the non-Hermitian interaction decreases the slope\nof inverted potential well. The potential-well slope van-\nishes at the exceptional point, where all eigenstates are\ndegenerate with zero eigenvalues. Beyond the excep-\ntional point the potential becomes normal well with a\nreal spectrum. Although the SU(1,1) generator/hatwideSzpos-\nsesses real spectrum always, the boson-operator realiza-\ntion of it can be either Hermitian with the normal boson\nor non-Hermitian with the imaginary-frequency boson\noperators. The quantum-classical one to one correspon-\ndenceexistsstrictlyfortheHermitianandnon-Hermitian\nHamiltonians.\nVI. ACKNOWLEDGMENTS\nThis work was supported by the National Natural Sci\nence Foundation of China (Grant No. 12374312), Shanx-\niScholarship Council of China (Grant No. 2022-014 and\n2023-033).\nVII. CONFLICT OF INTEREST\nThe authors declare no conflict of interest.\nVIII. DATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are\navailablefromthecorrespondingauthoruponreasonable\nrequest.IX. APPENDIX\nHamiltonian of inverted potential:\nThe Hamiltonian for a particle of unit mass in an in-\nverted potential well reads\n/hatwideH=/hatwidep2\n2−1\n2Ω2/hatwidex2(27)\nwith the conventional unit /planckover2pi1= 1. In the dimensionless\nvariableswith /hatwidepreplaced by/hatwidep/√\nΩand/hatwidexby√\nΩ/hatwidexwe have\ntheHamiltonianEq.(2). TheHamiltonianEq.(27)canbe\nregardedas a harmonic oscillatorof imaginary frequency,\n/hatwideH=/hatwidep2\n2+1\n2(iΩ)2/hatwidex2.\nImaginary frequency polynomials:\nUsing the commutation relation of boson operators\nEq.(4) it is easy to verify\n/hatwidenI/hatwideb−|n/angbracketrightr= (n−1)/hatwideb−|n/angbracketrightr.\nThus/hatwideb−is a lowering operator for the “ket” state. We\nassume that\n/hatwideb−|n/angbracketrightr=cr\nn−|n−1/angbracketrightr, (28)\nin which the coefficient cr\nn−is to be determined. /hatwideb+acts\nas a rising operator, since\n/hatwidenI/hatwideb+|n−1/angbracketrightr=n/hatwideb+|n−1/angbracketrightr.\nWe may define\n/hatwideb+|n−1/angbracketrightr=cr\n(n−1)+|n/angbracketrightr.\nActing the rising operator on the Eq.(28) yields\n/hatwidenI|n/angbracketrightr=cr\nn−cr\n(n−1)+|n/angbracketrightr,\nand then we have from the orthogonal condition Eq.(10)\nthe recurrence relation\ncr\nn−cr\n(n−1)+=n.\nRepeating the same procedure on the states |n−1/angbracketrightr,\n|n−2/angbracketrightr;|n−2/angbracketrightr,|n−3/angbracketrightr;···;we obtain\ncr\n(n−1)−cr\n(n−2)+=n−1;\ncr\n(n−2)−cr\n(n−3)+=n−2;\n······\ncr\n1−cr\n0+= 1.7\nFor the boson number ground-state we require\n/hatwideb−|0/angbracketrightr= 0, (29)\nand\ncr\n0−= 0.\nThe coefficients are determined as\ncr\nn−=cr\n(n−1)+=√n;cr\n(n−1)−=cr\n(n−2)+=√\nn−1;···;\ncr\n1−=cr\n0+= 1.\nThus thenth “ket” state can be generated from ground\nstate with the rising operator\n|n/angbracketrightr=(/hatwideb+)n\ncr\n(n−1)+...cr\n0+|0/angbracketrightr=(/hatwideb+)n\n√\nn!|0/angbracketrightr.\nFor the “bra” states situation is opposite that /hatwideb−acts as\na rising operator\n/hatwiden†\nI/hatwideb−|n/angbracketrightl= (n+1)/hatwideb−|n/angbracketrightl,\nand thus\n/hatwideb−|n/angbracketrightl=cl\nn−|n+1/angbracketrightl.\nWhile/hatwideb+is lowering operator\n/hatwiden†\nI/hatwideb+|n/angbracketrightl= (n−1)/hatwideb+|n/angbracketrightl,\nand\n/hatwideb+|n/angbracketrightl=cl\nn+|n−1/angbracketrightl. (30)\nThe recurrence relation for the “bra” states is obtained\nfrom the equations Eq.(8),Eq.(30) as\nn|n/angbracketrightl=/hatwiden†|n/angbracketrightl=−cl\nn+cl\n(n−1)−|n/angbracketrightl,\nand from the orthogonal condition Eq.(10) we have\ncl\nn+cl\n(n−1)−=−n.\nRepeating the same procedure on the states |n−1/angbracketrightl,|n−\n2/angbracketrightl;|n−2/angbracketrightl,|n−3/angbracketrightl;···;the result is\ncl\n(n−1)+cl\n(n−2)−=−(n−1);\n······\ncl\n1+cl\n0−=−1.\nFor the ground state it is\n/hatwideb+|0/angbracketrightl= 0, (31)\nand\ncl\n0+= 0.\nThe solutions of coefficients are seen to be\ncl\nn+=cl\n(n−1)−=i√n,cl\n(n−1)+=cl\n(n−2)−=i√\nn−1;···;\ncl\n1+=cl\n0−=i,from which the nth “bra” state can be generated from\nthe ground state such as\n|n/angbracketrightl=/parenleftBig\n/hatwideb−/parenrightBign\nin√\nn!|0/angbracketrightl.\nIn coordinaterepresentationthe ground-stateequation\nEq.(29) is\n/parenleftbigg\nx−id\ndx/parenrightbigg\nψr,0= 0,\nfrom which the ground “ket” state is found as\nψr,0=1√\nNe−ix2\n2,\nwhereNis the normalization constant to be determined\nfrom the orthonormal condition Eq.(10) between “bra”\nand “ket” states. The arbitrary order eigenfunctions can\nbe generated from the ground state such as\nψr,n=1√\nN√\nn!/parenleftBigg/radicalbigg\ni\n2/parenrightBiggn/parenleftbigg\nx−id\ndx/parenrightbiggn\ne−ix2\n2.\nWith the same procedure we have the “bra” ground-\nstate equation Eq.(31)\n/parenleftbigg\nx+id\ndx/parenrightbigg\nψl,0= 0,\nin the coordinate representation and the wave function\nψl,0=1√\nNeix2\n2.\nThe arbitrary-order wave function of “bra” state is\nψl,n=1\nin√\nn!√\nN/parenleftBigg/radicalbigg\ni\n2/parenrightBiggn/parenleftbigg\nx+id\ndx/parenrightbiggn\neix2\n2.\nThe normalization constant can be determined from the\northonormal condition Eq.(10) between “bra” and “ket”\nstates\n/integraldisplay\nψ∗\nl,0ψr,0dx=1\nN/integraldisplay\ne−ix2dx= 1.\nThe normalization constant\nN=/radicalbiggπ\ni,\nis then obtained by means of the imaginary integral mea-\nsure as in the path integral of quantum mechanics (see\nfor example34). The normalized wave functions are given\nby equation Eq.(12) in Sec.II.8\n∗Electronic address: liuni2011520@sxu.edu.cn\n†Electronic address: jqliang@sxu.edu.cn\n1C. M. Bender, S. Boettcher, Phys. Rev. Lett. 80, 5243\n(1998).\n2C. M. Bender, S. Boettcher, and P. Meisinger, J. Math.\nPhys.40, 2201 (1999).\n3C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev.\nLett.89, 270401 (2002).\n4Y. Gu, X. M. Bai, X. L. Hao, and J.-Q. Liang, Results.\nPhys.38, 105561 (2022).\n5Y. Gu, X. L. Hao, and J.-Q. Liang, Ann. Phys.(Berlin).\n534, 2200069 (2022).\n6N. Liu, S. Huang, and J.-Q. Liang, Results. Phys. 40,\n105813 (2022).\n7A. Mostafazadeh, Nuclear. Physics. B. 640, 419 (2002).\n8A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).\n9A. Mostafazadeh, Phys. Lett. B. 650, 208 (2007).\n10N. Liu, Y. Gu, and J.-Q. Liang, Phys. Scr. 98, 035109\n(2023).\n11N. Amaaouche, M. Sekhri, R. Zerimeche, M. Mustaphaa,\nand J.-Q. Liang, Physics. open. 13, 100126 (2022).\n12C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev.\nD.70, 025001 (2004).\n13C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).\n14C. M. Bender, PT Symmetry: in Quantum and Classical\nPhysics (World Scientific, Singapore, 2018).\n15D. N. Christodoulides, J. Yang, Parity-time symmetry and\nits applications (Springer Tracts in Modern Physics, 2018) .\n16C. Hang, G. Huang, and V. V. Konotop, Phys. Rev. Lett.\n110, 083604 (2013).\n17Z. Y. Zhang, Y. Q. Zhang, J. T. Sheng, L. Yang, M. A.\nMiri, D. N. Christodoulides, B. He, Y. P. Zhang, and M.\nXiao, Phys. Rev. Lett. 117, 123601 (2016).\n18S. Bittner, B. Dietz, U. G¨ unther, H. L. Harney, M. Miski-\nOglu, A. Richter, and F. Sch¨ afer, Phys. Rev. Lett. 108,\n024101 (2012).\n19C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel,Am. J. Phys. 81, 173 (2013).\n20C. E. R¨ uter, K. G. Makris, R. El-Ganainy, D. N.\nChristodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192\n(2010).\n21Y. Wu, W. Q. Liu, J. P. Geng, X. R. Song, X. Y. Ye, C. K.\nDuan, X. Rong, and J. F. Du, Science. 364, 878 (2019).\n22L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G.\nLi, G. Wang, and M. Xiao, Nat. Photon. 8, 524 (2014).\n23P. G. Kevrekidis, J. Cuevas–Maraver, A. Saxena, F.\nCooper, and A. Khare, Phys. Rev. E. 92, 042901 (2015).\n24H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Nature.\n537, 80 (2016).\n25W. L. Li, C. Li, and H. S. Song, Phys. Rev. A. 95, 023827\n(2017).\n26S. K.¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Nat.\nMater.18, 783 (2019).\n27J. Zhang, B. Peng, S. K. ¨Ozdemir, K. Pichler, D. O.\nKrimer, G. Zhao, F. Nori, Y. X. Liu, S. Rotter, and L.\nYang, Nat. Photon. 12, 479 (2018).\n28Y. L. Liu, R. Wu, J. Zhang, S. K. ¨Ozdemir, L. Yang, F.\nNori, and Y. X. Liu, Phys. Rev. A. 95, 013843 (2017).\n29F. Quijandr ´l, U. Naether, S. K. ¨Ozdemir, F. Nori, and D.\nZueco, Phys. Rev. A. 97, 053846 (2018).\n30J. -Q. Liang and H. J. W. M¨ uller-Kirsten, Phys. Rev. D.\n45, 2963 (1992)\n31J. -Q. Liang and H. J. W. M¨ uller-Kirsten, Phys. Rev. D.\n50, 6519 (1994)\n32J.-Q. Liang, H. J. W. M¨ uller-Kirsten, Y. B. Zhang, A. V.\nShurgaia, S. P. Kou, and D. K. Park, Phys. Rev. D. 62,\n025017 (1999).\n33Y. Z. Lai, J.-Q. Liang, H. J. W. M¨ uller-Kirsten, and J. G.\nZhou, Phys. Rev. A. 53, 3691 (1996).\n34J. -Q. Liang and L. F. Wei, New Advances in Quantum\nPhysics (Third edition) (Science Press, Beijing, 2023).\n35J. L. Xin, J. -Q. Liang, Sci. China. Phys. Mech. Astron.\n57, 1504 (2014)."    },    {        "title": "2402.06173v1.SMC_Is_All_You_Need__Parallel_Strong_Scaling.pdf",        "content": "SMC Is All You Need: Parallel Strong Scaling\nXinzhu Liang1, Sanjaya Lohani2, Joseph M. Lukens3,4, Brian T. Kirby5,6,\nThomas A. Searles2, and Kody J. H. Law1\n1School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom\n2Department of Electrical and Computer Engineering, University of Illinois Chicago, Chicago, Illinois\n60607, USA\n3Research Technology Office and Quantum Collaborative, Arizona State University, Tempe, Arizona\n85287, USA\n4Quantum Information Science Section, Oak Ridge National Laboratory, Oak Ridge, Tennessee\n37831, USA\n5DEVCOM US Army Research Laboratory, Adelphi, Maryland 20783, USA\n6Tulane University, New Orleans, Louisiana 70118, USA\nAbstract\nInthegeneralframeworkofBayesianinference, thetargetdistributioncanonlybeeval-\nuated up-to a constant of proportionality. Classical consistent Bayesian methods such as\nsequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) have unbounded\ntime complexity requirements. We develop a fully parallel sequential Monte Carlo (pSMC)\nmethod which provably delivers parallel strong scaling , i.e. the time complexity (and per-\nnode memory) remains bounded if the number of asynchronous processes is allowed to\ngrow. More precisely, the pSMC has a theoretical convergence rate of MSE =O(1/NR )1,\nwhere Ndenotes the number of communicating samples in each processor and Rde-\nnotes the number of processors. In particular, for suitably-large problem-dependent N, as\nR→ ∞the method converges to infinitesimal accuracy MSE =O(ε2)with a fixed finite\ntime-complexity Cost =O(1)and with no efficiency leakage, i.e. computational complexity\nCost =O(ε−2). A number of Bayesian inference problems are taken into consideration to\ncompare the pSMC and MCMC methods.\n1 Introduction\nThe Bayesian paradigm for machine learning can be traced back at least to the 1990s [MacKay,\n1992, Neal, 1993, Hinton et al., 1986, Ng and Jordan, 1999, Bengio, 2000, Andrieu et al.,\n2003, Bishop, 2006, Murphy, 2012]. It is attractive because it delivers the Bayes estimator,\nalong with principled quantification of uncertainty. However, in general the posterior (target)\ndistribution can only be evaluated up-to a constant of proportionality, and the only consistent\nmethods available for inference are of Monte Carlo type: notably Markov chain Monte Carlo\n(MCMC) and sequential Monte Carlo (SMC). These are still too expensive to be used in\npractice for anything except ground truth approximation for toy problems, and variational\nBayesian approximation methods have taken centre stage in the context of Bayesian deep\nlearning [Blundell et al., 2015, Gal and Ghahramani, 2016]. The present work addresses the\ncomputational intractability head-on by proposing a method which can distribute the workload\n1MSE is the mean square error, and 1/NR≡(NR)−1.\n1arXiv:2402.06173v1  [stat.ML]  9 Feb 2024Figure 1: Convergence of the MCMC method pCN, and our pSMC method with pCN kernel.\nThe crossover line separates the pSMC-preferred regime with the pCN-preferred regime. The\nexample is a Gaussian Bayesian linear model with m×d= 16×4dimensional design matrix.\nSee Section 4.1 for further details.\nacross arbitrarily many workers. As such, the aim is to revive interest in Bayesian MC methods\nas both optimal and practical.\nGiven data D, the Bayesian posterior distribution over θ∈Θ∈Rdis given by:\nπ(θ)∝ L(θ)π0(θ), (1)\nwhere L(θ) :=L(D|θ)is the likelihood for the given data Dandπ0(θ)is the prior. The Bayes\nestimator of a quantity of interest φ:Θ→RisR\nΘφ(θ)π(θ)dθ. It minimizes the appropriate\nfrequentist risk at the population level and can therefore be considered optimal.\nMarkov chain Monte Carlo methods (MCMC) originated with the famous Metropolis-\nHastings(MH)methods[Metropolisetal.,1953,Hastings,1970], andarethefavouredapproach\nto consistently approximate this kind of target distribution in general. MCMC has seen wide-\nspread use and rigorous development in statistics from the turn of the millennium [Gelfand and\nSmith, 1990, Geyer, 1992, Robert et al., 1999, Roberts and Tweedie, 1996, Duane et al., 1987].\nOne crowning achievement is overcoming the so-called “curse-of-dimensionality”, i.e. exponen-\ntial degradation, in the rate of convergence. Nonetheless, some dimension-dependence typically\nremainsintheconstantfortheaforementionedMCMCmethods. Itwasrecentlyshownthatthe\nperformance of a carefully constructed MH kernel [Neal, 1998] can be completely dimension-\nindependent [Cotter et al., 2013]. We adopt the name precondition Crank-Nicholson (pCN)\nmethod, introduced in the latter. Shortly thereafter, the convergence properties of vanilla pCN\nwere further improved by leveraging Hessian and/or covariance information in the transition\nkernel[Law,2014,Cuietal.,2016,RudolfandSprungk,2018,Beskosetal.,2017]. TheHamilto-\nnian Monte Carlo (HMC) method Duane et al. [1987], Neal [1993], Houlsby et al. [2011] is a\nhybrid method which interleaves Metropolis-Hastings type accept/reject with evolution in an\nextended phase-space to circumvent random walk behaviour and enable better mixing. This is\nperhaps the most popular MCMC method in use in the machine learning community. Despite\nits attractive properties, MCMC is much more expensive to use in practice in comparison to\npoint-estimators or variational methods and is still in minority use. There has been some effort\nin parallelizing MCMC [Chen et al., 2016], with the most elegant and widely-recognized work\nbeing Jacob et al. [2020]. Nonetheless, these methods lack theory and/or are awkward and\ncumbersome to implement in practice, and have not caught on.\nThe sequential Monte Carlo (SMC) sampler [Del Moral et al., 2006] is an alternative\npopulation-based method which developed at the turn of the millennium Jarzynski [1997],\nBerzuini and Gilks [2001], Gilks and Berzuini [2001], Neal [2001], Chopin [2002]. The SMC\n2sampler approximates the target distribution πthrough a sequence of tempering distribu-\ntions πj, starting from a simulable distribution such as the prior π0. A population of sample\n“particles” from π0evolvethroughimportancere-sampling(selection)andMCMCmoves(muta-\ntion) between successive tempering distributions. See the following for recent thorough intro-\nductions Dai et al. [2022], Chopin et al. [2020]. SMC gracefully handles multi-modality and\nfacilitates adaptive tuning of the Markov kernel, with only a small theoretical efficiency over-\nhead with respect to MCMC. It also delivers an unbiased estimator of the normalizing constant,\nor model evidence, which can be useful in practice. Furthermore, the structure of SMC method\nfacilitates parallelisation at multiple levels.\nThe parallel implementation of the SMC method has already been considered in Vergé et al.\n[2015], Whiteley et al. [2015], however the existing methods involve communication between\nall samples , which hinders implementation. The island particle model [Vergé et al., 2015]\nseparates the total number of sample NRintoRislands and each island has Nsamples. Two\nlevels of resampling is needed in each tempering step: first, particles within each island are\nselected by resampling separately, and then islands themselves undergo selection by resampling\nat the island level. The method guarantees the MSE =O(1/NR)convergence rate, but requires\n(frequent) communication between all samples, thereby undermining the parallelism in a sense:\nall processes have to wait for the slowest one and the method is exposed to I/O bottlenecks\nat every sync point. Without these interactions, the island particle method has a penalty of\n1/N2, and as R≫N, the penalty becomes noticeable.\nThe present work introduces the parallel SMC (pSMC) sampler, which can be viewed as a\njudicious estimator built from an island particle filter without any interaction between islands .\nWe provide a simple proof that our estimator converges with the rate MSE =O(1/NR), and\nwithout the 1/N2bias, under reasonable and mild assumptions. In particular, the estimator\ndeliversparallel strong scaling , in the sense that for any given problem it converges for fixed\n(suitably large) NasR→ ∞, and without any loss of efficiency: with Rnon-interacting\nprocessors, the method converges at the rate MSE =O(1/R)with O(1)time complexity. It\nhas come to our attention after writing this that the very general αSMC method [Whiteley\net al., 2015] actually includes a very similar method as a special case, hence also overcoming\nthe interaction limitation and bias of the island particle filter, however it is mentioned only\nparenthetically and not simulated or investigated thoroughly. That work is focused instead\non achieving stability for online SMC algorithms, on the infinite time horizon, in exchange for\nallowing communication between all samples.\nThe paper is organized as follows. In Section 2, we restate the algorithm for a single SMC\nandprovideasummaryofvariousMCMCkernels. InSection3, thegeneralframeworkofpSMC\nis established. In Section 3.1, the theoretical results for the convergence of pSMC are given.\nIn Section 4, a number of Bayesian inference problems, including the Gaussian, trace-class\nneural network, and quantum state tomography problems, are examined in order to compare\nthe pSMC and pCN. In Section 5, the conclusion and additional discussion are addressed.\n2 SMC sampler\nDefine the target distribution as π(θ) =f(θ)/Z, where Z=R\nΘf(θ)dθandf(θ) :=L(θ)π0(θ).\nGiven a quantity of interest φ:Θ→R, for simplicity, we define\nf(φ) :=Z\nΘφ(θ)f(θ)dθ=f(1)π(φ),\nwhere f(1) =R\nΘf(θ)dθ=Z.\n3Algorithm 1 SMC sampler\nGenerate θi\n1∼π1fori= 1, ..., NandZN\n1= 1.\nforj= 2toJdo\nStore ZN\nj=ZN\nj−11\nNPN\nk=1L(θk\nj−1)λj−λj−1.\nfori= 1toNdo\nDefine wi\nj=L(θk\nj−1)λj−λj−1/PN\nk=1L(θk\nj−1)λj−λj−1.\nResample: Select Ii\nj∼ {w1\nj, ..., wN\nj}and let ˆθi\nj=θIi\nj\nj−1.\nMutate: Draw θi\nj∼ M j(ˆθi\nj,·).\nend for\nend for\nSo, the target expectation can be computed by\nπ(φ) =f(φ)\nf(1).\nDefine h1, ..., h J−1byhj=fj+1/fj, where f1=π0, fJ=f=L(θ)π0(θ), and define πj=\nfj/Zjwhere Zj=R\nΘfjdθ. For j= 2, ..., J, we let fjdefine in a annealing scheme:\nfj=f1−λj\n1fλj\nJ=L(θ)λjπ0(θ),\nwhere 0 =λ1< λ 2<···< λ J= 1andλjwill be chosen adaptively according to the effective\nsample size (ESS), see Example 2.1.\nExample 2.1 (Adaptive tempering) .In order to keep the sufficient diversity of sample popula-\ntion, welettheeffectivesamplesizetobeatleast ESS min=N/2ateachtempering λj−1anduse\nit compute the next tempering λj. For jth tempering, we have weight samples {wk\nj−1, θk\nj−1}N\nk=1,\nthen the ESS is computed by\nESS =1PN\nk=1(wk\nj−1)2,\nwhere wk\nj−1=L(θk\nj−1)λj−λj−1/PN\nk=1L(θk\nj−1)λj−λj−1. Let h=λj−λj−1, the effective sample size\ncan be presented as a function of h, ESS (h). Using suitable root finding method, one can find\nh∗such that ESS(h∗) =ESS min, then set the next tempering λj=λj−1+h∗.\nNow let Mjforj= 2, . . . , Jbe Markov transition kernels such that (πjMj)(dθ) =πj(dθ),\nanalogous to any suitable MCMC kernel [Geyer, 1992, Robert et al., 1999, Cotter et al., 2013,\nLaw, 2014]. This operation must sufficiently decorrelate the samples, and as such we typically\ndefine the MCMC kernels MjbyMsteps of some basic MCMC kernel, where Mmay or may\nnot be 1. We call Mthe mutation parameter.\nOne can observe that\nπj(hj) =1\nZjZfj+1(θ)\nfj(θ)fj(θ)dθ=Zj+1\nZj.\nSince we let πJ=πbe the target distribution and Z1= 1, the normalisation constant of πis\ngiven by\nZJ=Z1J−1Y\nj=1πj(hj) =J−1Y\nj=1πj(hj).\n4From Algorithm 1, we can define the following estimators:\nπN\nj(hj) :=1\nNNX\ni=1hj(θi\nj), i= 1, ..., J,\nZN\nJ:=J−1Y\nj=1πN\nj(hj),\nfN\nJ(φ) :=J−1Y\nj=1πN\nj(hj)πN\nJ(φ) =ZN\nJπN\nJ(φ).(2)\nHence, the sequential Monte Carlo estimator for estimating the expectation of the quantity of\ninterest is given by\nˆφSMC=πN(φ) =πN\nJ(φ) =fN\nJ(φ)\nZN\nJ=fN(φ)\nfN(1). (3)\nThe SMC sampler can employ and potentially accelerate any appropriate MCMC approach.\nHere, the following MCMC kernels are introduced.\n2.1 Pre-conditioned Crank-Nicolson kernel\nBy slightly changing the random walk MH kernel, one can ensure stability of the proposal\nchain and stationarity with respect to a specified Gaussian distribution. For posteriors with\nGaussian priors, including for Gaussian measures over infinite-dimensional function spaces this\nleads to a stable MH method which depends only on the properties of the likelihood. For a\nprior N(µ0,Σ0), the pCN proposal is\nθ′= (a−β2)1/2(θ−µ0) +µ0+ Σ1/2\n0δ , δ ∼N(0, Id).\nSee Neal [1998], Cotter et al. [2013] for further details.\nAn enhancement for the pCN kernel involves employing the operator-weighted proposal\n[Law, 2014, Cui et al., 2016], which employs the covariance matrix of the current state within\nthe mutation kernel rather than the identity matrix, thus improving mixing when the likelihood\ninforms certain directions much more than others. The general pCN kernel is given by\nθ′= Σ1/2\n0(Id−β2D)1/2Σ−1/2\n0(θ−µ0) +µ0+ Σ1/2\n0βD1/2δ .\nThe scaling matrix Dshould be chosen according to the information present in the likelihood.\nA simple and computationally convenient choice which is particularly amenable to use within\nSMC is to build it from an approximation of the target covariance [Beskos et al., 2018]. In\nthe present work we adopt the simplest and cheapest choice and let D=diag(cvar[θ]). The\napproximation of the variance, cvar, will be built from the current population of samples in the\nSMC case, and adaptively constructed in the MCMC case [Haario et al., 2001, Chen et al.,\n2016]. See also [Rudolf and Sprungk, 2018, Beskos, 2014, Zahm et al., 2022].\n2.2 Hamiltonian Monte Carlo\nHamiltonianMonteCarlo(HMC)kernel[Duaneetal.,1987,Nealetal.,2011,Houlsbyetal.,\n2011, Cobb and Jalaian, 2021] is essentially a gradient-based MCMC kernel on an extended\nstate-space. We first build a Hamiltonian H(θ, p)with additional auxiliary “momentum” vector\npof the same dimension as the state θ\nH(θ, p) =−logπj(θ) +1\n2p⊤M−1\n0p, (4)\n5Algorithm 2 pCN kernel\nGive a current state θ, the distribution πj, a scaling parameter β.\nform= 1toMdo\nGenerate θ′= Σ1/2\n0(Id−β2D)1/2Σ−1/2\n0(θ−µ0) +µ0+ Σ1/2\n0βD1/2δwhere δ∼N(0, Id)and\nu∼U([0,1]).\nifu≤minn\n1,L(θ′)λj\nL(θ)λjo\nthen\nθ=θ′.\nelse\nθ=θ.\nend if\nend for\nAlgorithm 3 Hamiltonian Monte Carlo kernel\nGive a current state θ, the distribution πjand a mass matrix M0.\nGenerate a initial momentum p∼N(0, M0).\nform= 1toMdo\nforl= 1toLdo\nGenerate θl∆tandpl∆tfrom (6) with θ0=θandp0=p\nend for\nLetθ′=θL∆tand generate u∼U([0,1])\nifu≤minn\n1,H(θ′,p′)\nH(θ,p)o\nthen\nθ=θ′andp=p′.\nelse\nθ=θandp=p.\nend if\nend for\nwhere M0is a mass matrix, so that1\nZexp(H(θ, p))is a target distribution on the extended\nspace, where the momentum can be simulated exactly p∼N(0, M0).\nFrom physics we know that the Hamiltonian dynamics conserve energy, hence avoiding local\nrandom-walk type behaviour and allowing ballistic moves in position:\ndθ\ndt=∂H\n∂p=M−1\n0p,\ndp\ndt=∂H\n∂θ=∇θlogπj(θ).(5)\nA carefully constructed symplectic-integrator is capable of conserving energy as well, for ex-\nample the leapfrog integrator Leimkuhler and Reich [2004]:\npt+∆t/2=pt+∆t\n2dp\ndt(θt),\nθt+∆t=θt+ ∆tdθ\ndt(pt+∆t/2),\npt+∆t=pt+∆t/2+∆t\n2dp\ndt(θt+∆t),(6)\nwhere tis the leapfrog step iteration and ∆tis the step size.\nEach step of the HMC method is summarized as follows:\n(i) simulate a random momentum p∼N(0, M0)(hence jumping to a new energy contour);\n6Algorithm 4 Parallel SMC sampler\nforr= 1toRdo\nRun Algorithm 1, output θN,r.\nCompute the estimators ZN,r,fN,r(φ),fN,r(1)andπN,r(φ)by (2).\nend for\n(ii) approximate the Hamiltonian dynamics using Lsteps from (6);\n(iii) correct numerical error from (ii) with an MH accept/reject step for1\nZexp(H(θ, p)).\nSee Algorithm 3.\n3 Parallel SMC method\nBy separating NRsamples into Rprocessors with Nsamples in each, the parallel SMC sampler\nhas a Rtimes lower time complexity than a single SMC sampler. This reduction in time com-\nplexity is crucial if we encounter an expensive SMC, which is common in the high-dimensional\nBayesian inference problem. More importantly, under reasonable and mild assumptions we will\ndemonstrate the strong scaling property for pSMC: the cost decreases linearly with the number\nof cores with constant time. The Algorithm 4 displays the parallel SMC method.\nFollowing from Algorithm 4, we can define the unnormalized estimators\nFN,R(φ) =1\nRRX\nr=1fN,r(φ), (7)\nFN,R(1) =1\nRRX\nr=1fN,r(1) =1\nRRX\nr=1ZN,r. (8)\nThen, the parallel SMC estimator for estimating π[φ]is given by\nˆφpSMC =FN,R(φ)\nFN,R(1)=PR\nr=1fN,r(φ)PR\nr=1fN,r(1). (9)\n3.1 Theoretical results for parallel strong scaling\nIn order to make use of (9), we define fN,r(ζ)both for quantity of interest ζ=φandζ= 1.\nWe denote that |ζ|∞= max θ∈Θ|ζ(θ)|in the following equations. Cζdenotes the constant\ndepended on the function ζ. Note that using φwith one-dimensional output is without loss of\ngenerality for our convergence results, and the following proof can be directly generalized to\nthe multi-output function by using the inner product.\nAssumption 3.1. LetJ∈Nbe given, for each j∈ {1, ..., J}, there exists a C > 0such that\nfor all θ∈Θ,\nC−1< f j(θ),L(θ)≤C.\nAssumption 3.2. LetJ∈Nbe given, for each j∈ {1, ..., J}, there exists a ρ∈(0,1)such\nthat for any (u, v)∈Θ2,Ais a measureable set from the space containing all the measureable\nsubset of Θ, Z\nAMj(u, du′)≥ρZ\nAMj(v, dv′).\n7Table 1: Comparison between pCN and pSMC regarding to the time cost, memory cost and\nMSE.\nTime Cost Memory Cost MSE\npCN COST (d, m)NR d (m+NR) C2\nMCMC(d, m)/NR\npSMC COST (d, m)COST over(d, m)N d (m+N) C2\nSMC(d, m)/NR\nFirst, we restate the convergence result for the non-parallelized SMC method in Proposition\n3.3, which is from Del Moral [2004].\nProposition 3.3. Assume Assumption 3.1 and 3.2. Then, for any J∈N, there exists a C >0\nsuch that for any N∈N, suitable ζ:Θ→R,\nE[(fN(ζ)−f(ζ))2]≤C|ζ|2\n∞\nN. (10)\nIn addition, the estimator is unbiased E[fN(ζ)] =f(ζ).\nLemma 3.4. Assume Assumption 3.1 and 3.2. Then, for any J∈N, there is a C > 0such\nthat for any N, R∈N, suitable ζ:Θ→R,\nE[(FN,R(ζ)−f(ζ))2]≤C|ζ|2\n∞\nNR.\nTheorem 3.5. Assume Assumption 3.1 and 3.2. Then, for any J∈N, there exists a Cφ>0,\nwhich depends on φ, such that for any R∈N,\nE[( ˆφpSMC−π(φ))2]≤Cφ\nNR. (11)\nThen the strong scaling in the parallel SMC is described as the MSE defined in (11)decreases\nas1/RforRnodes, for suitable variables value ( M,N,J), with constant time.\n3.2 Discussion of pSMC in practice\nLetmdenote the number of observations in the data. As Theorem 3.5 suggests, the nuisance\nparameters (M, N, J )must be chosen large enough to ensure rapid convergence in R. For\nMCMC the error will be bounded by CMCMC (d, m)/NR, and the time (and total) cost will\ngrow like COST (d, m)NR, where COSTis the basic cost of evaluating the likelihood (target)2.\nMeanwhile, the error of SMC will be bounded by CSMC(d, m)/N, while the time cost will grow\nlikeCOST (d, m)MJN. Since MandJwill also depend on (d, m)– and need to be chosen\nlarger for more complex targets – we will denote by COST over(d, m) :=MJthe cost overhead\nof SMC vs MCMC for a given total number of samples.\nFinally, we can achieve a fixed MSE of ε2with MCMC by selecting NR=CMCMC (d, m)ε−2,\nwhich incurs a complexity of\nCOST (d, m)NR=COST (d, m)CMCMC (d, m)ε−2.\nSimilarly, the time complexity of SMC will be\nCOST (d, m)COST over(d, m)N=COST (d, m)COST over(d, m)CSMC(d, m)ε−2/R .\n2We are simplifying discussion by referring only the the major sources of complexity (d, m ), while any other\nmetric for problem complexity will also come into play. For example, under partial observations the linear model\nwill become more difficult as the variance of the observations σ2tends to 0.\n8Table 2: The MSE, number of nodes for pSMC, and total time until the crossover point (see\nFigure 1) for the Gaussian examples, using pCN as the MCMC kernel.\nGaussian m ×d = 64 Gaussian m ×d = 1024\nm=16 > d m = 8 = d m=4 < d m = 64 > d m = 32 = d m = 16 < d\nMSE 1.2332e-07 7.3838e-04 3.1846e+00 6.0035e-07 2.8125e-02 1.9128e+01\nNo.node 4 2 1 1 20 1\nTime 0.9749 1.7513 1.7521 1.3517 6.0139 14.3041\nTable 3: The MSE, number of nodes for pSMC, and total time until the crossover point (see\nFigure 1) for the non-Gaussian example of QST regression. For BNN classification, MSE is\nreplaced with mean KL-divergence.\nQST (pCN) BNN (HMC)\n(d,m) = (16,6) (d,m) = (64,36) (d,m) = (256,216) iris (d,m) = (15,50) MNIST (d,m) = (1177,50)\nMSE/Loss 4.0111e-04 1.5068e-03 4.5424e-05 1.3936e-04 6.7747e-02\nNo.node 1 1 681 33 26\nTime 0.0456 0.6681 29.6914 13.2356 34.5685\nEquating these two, we can estimate the crossover\nR >COST over(d, m)CSMC(d, m)/CMCMC (d, m).\nFor suitably chosen M, J, we expect CSMC(d, m)≈CMCMC (d, m), so the crossover should be\nroughly equivalent to COST over(d, m)processors.\nThe memory cost of Nsamples of size disNd, and the memory cost of storing the data is\nO(dm). The latter may be considerably smaller, and even inconsequential, for neural network\nmodels with large number of parameters and smaller input dimension. Nonetheless, we take\nO(d(m+N))as a proxy for the per process memory requirements of SMC. Meanwhile, if the\nquantity of interest is known a priori and estimated online, then MCMC samples do not need\nto be stored, resulting in a memory cost of only O(dm). Of course, if one intends to perform\ninference in the future without saving samples, then the total memory cost is O(d(m+NR)),\nbut the samples could be (thinned and) offloaded at checkpoints, so this can nevertheless be\nkept under control.\nClearly we require a minimum N≫CSMC(d, m)in order for the SMC theory to make sense,\nand we have indeed found that convergence slows or stops if this condition is not satisfied.\nThis can be viewed as sufficient maximum population diversity. Furthermore, the samples\nmust decorrelate sufficiently, leading to a minimum M, and they must be be rejuvenated\nsufficiently often, leading to a minimum J. Both of these conditions can also spoil convergence.\nConventional advice suggests that it is safe to select M=J=N=O(max{m, d})Chopin\net al. [2020]. If d=m, then the crossover occurs when R > d2, and the memory requirement\nper processor is also O(d2). In practice, we have found that the minimum required MandJ\noften increase much slower than O(d), so the crossover often occurs earlier than d2.\nComputing and storing the log-likelihood instead of the likelihood itself wherever possible\nin SMC sampler is preferred in practice to avoid numerical over/underflow. See Appendix A\nfor further details.\n4 Simulations\nApplications of pSMC with different MCMC kernels on various Bayesian inference problems\nwith/without analytical solutions are provided in this section. The notations pSMC-pCN and\npSMC-HMC denote parallel SMC with pCN and HMC transition kernels, respectively.\n94.1 High-dimensional Gaussian cases\nBayesian inference of Gaussian distributions are always tractable, in which the analytical solu-\ntion of the posterior are solvable. Numerical verification of the convergence rate and strong\nscaling property of pSMC is provided by computing the MSE in relation to the analytical\nsolution.\nConsider a toy Gaussian inference example with mobservations ( y) and d-dimensional\nparameter ( θ). Assume we have data D={y, X}, where y∈RmandX∈Rm×d, and\nparameter θ∈Rdconnected by the following inverse problem:\ny=Xθ+ν, ν∼N(0, σ2Im), (12)\nwhere Xis the design matrix. If we let π0(θ) =N(µ0,Σ0), this is one of very few problems with\nan analytical Bayesian posterior, which will provide a convenient ground truth for measuring\nconvergence. In particular, the posterior distribution is a multivariate Gaussian distribution\nN(µ,Σ), where\nµ= Σ(Σ−1\n0µ0+1\nσ2XTy),Σ = (Σ−1\n0+1\nσ2XTX)−1.\nSee e.g. [Bishop, 2006].\nIn the following tests, we compare pSMC-pCN with pCN performance in estimation of the\nposterior mean, µ=E[θ|D]. Let X∈Rm×dbe a randomly selected full rank matrix and\nσ= 0.01. The observations are generated as\ny=Xθ∗+ν, (13)\nwhere θ∗∼π0andν∼N(0, σ2Im)are independent.\nThree cases are examined: the over-observed case ( m≫d), the full-observed case ( m=d)\nand the partial-observed case ( m≪d). If the number of observed data has been qualified\nor overqualified, parameter θis expected to be estimated accurately by pSMC-pCN and pCN\nmethods. On the opposite, if the parameter is estimated under loss of information, which could\nlead to an inaccurate capture of some of the components of the data, pCN may perform poorly\non the dimension with less information; on the other hand, pSMC-pCN will perform well on all\ndimensions of θ.\nThe expected −1convergence rate of pSMC-pCN and pCN are verified numerically in the\nleft plots of figures in Appendix D.1. Note for the partially observed case, we have verified that\nconvergence does happen if we run the chain for much much longer (not shown). Recall we\ndefined the crossover point for fixed M, N, Jas the minimum number of nodes when pSMC has\na lower MSE than MCMC. See Figure 1. We report the current MSE, the number of nodes for\npSMC-pCN, and the time at the crossover points with pCN for the various different regimes in\nTable 2. The full convergence diagrams are given in the Appendix D.1. The improvement of\npSMC is clearly the greatest in the wide case of partial observations m < dfor the Gaussian,\nsuggesting that the method may be quite well-suited to small(er) data cases.\n4.2 Quantum State Tomography problems\nQuantum State Tomography (QST) is one application area where an accurate approximation\nof the Bayesian mean estimator is particularly valuable, but due to exponential scaling of\nthe state-space as the number of qubits grows it quickly becomes computationally prohibitive\n[Lukens et al., 2020, 2021]. In a Q-qubit quantum system with 2−level quantum information\ncarriers, we intend to infer the 2Q×2Qdensity matrix ρ(θ)through the parameter θ∈Rd,\nwhere the ρis parameterized3byθ∈Rdwith d= 22(Q+1)such that any value within θ’s\n3Details for the parameterization of ρ(θ)is given in Appendix B\n10support returns a physical matrix. We consider using the Bayesian state estimation on 3Q\nmeasurements with 2Qoutcomes for Squantum states, respectively. There are then m= 6Q\ntotal measurements for each quantum state, which can be described by a sequence of positive\noperators Λifori∈ {1, ..., m}and corresponding observed counts D={c1, ..., c m}.\nFollowing from Born’s rule, the likelihood is given by\nL(θ) =mY\ni=1[Trρ(θ)Λi]ci. (14)\nHence, the posterior distribution is,\nπ(θ)∝ L(θ)π0(θ), (15)\nwhere π0is a standard Gaussian prior N(0, I).\nWe compare pSMC-pCN (with N= 32) and pCN for Q= 1,2,3qubits, i.e. (d, m) =\n(16,6),(64,36),(256,216). The quantity of interest is φ=ρ. See Figures 7, 8 and 9. The\nexpected 1/NRconvergence rate for pSMC is verified in the left plots in the above figures. The\nright plots in those figures represent the strong scaling property in pSMC and the crossover\npoint is shown in Table 3.\n4.3 Bayesian Neural Network Examples\nHMC has gained widespread acceptance and favour in high-dimensional models, like the neural\nnetworks. In this section, we compare pSMC-HMC with HMC using BNN examples with\nsimilar settings cited in Cobb and Jalaian [2021].\nSuppose we have data set D={(x1, y1), ...,(xm, ym)}, where xi∈Rnare inputs associated\nto output labels yi∈Y. Let D∈N,(n0, ..., n D)∈ND+1with n0=nfor the input layer.\nLet weights be Ai∈Rni×ni−1and biases be bi∈Rnifori∈ {1, ..., D}, we denote θ:=\n((A1, b1), ...,(AD, bD)). The layer is defined by\ng1(x, θ) :=A1x+b1,\ngd(x, θ) :=Aiσni−1(gi−1(x)) +bi, i∈ {2, ..., D −1},\ng(x, θ) :=ADσnD−1(gD−1(x)) +bD,\nwhere σi(u) := ( ν(u1), ..., ν (ui))Twith ReLU activation ν(u) = max {0, u}.\nIn the regression example, we have Y=RnDand can assume the data as follows, in which\nthe data is defined in a continuous space,\nyi=g(xi, θ) +νi, ϵ∼N(0, σ2I), (16)\nwhere i={1, ..., m}. Then the likelihood for the regression problem (16) can be computed as\nL(θ) =mY\ni=1exp(−1\n2σ2∥yi−g(xi, θ)∥2).\nInthediscretedataset, liketheclassificationproblems, wehave Y={1, ..., K}andnD=K,\nthen we instead define the so-called softmax function as\nhk(x, θ) =exp(gk(x, θ))PK\nj=1exp(gj(x, θ)), k∈Y, (17)\nand define h(x, θ) = ( h1(x, θ), ..., h K(x, θ))as a categorical distribution on Koutcomes based\non data x. Then we assume that yi∼h(xi)fori={1, ..., m}.\n11Now, the likelihood for the classification problem (17) can be computed as\nL(θ) =mY\ni=1KY\nk=1hk(xi, θ)1[yi=k],\nwhere 1[yi=k]= 1ifyi=kand0otherwise.\nGiven a Gaussian prior π0=N(0, I), the posterior distribution of θis\nπ(θ)∝ L(θ)π0(θ). (18)\nIn the following examples, the QOI is interpreted as φ=gin the regression problem and\nas the softmax function of the output, φ=h, in the classification problem. Rather than\nconsidering MSE as in the regression example, the mean centred cross-entropy (KL-divergence)\nis used as metric between the estimated Bayesian ground truth4φgand the Bayesian estimators\nˆφin the classification cases as follows\nL(φg,ˆφ) =KX\ni=1φg,ilog(φg,i)−KX\ni=1φg,ilog( ˆφi).\n4.3.1 Iris Classification Example\nConsider a fully connected deep neural network with only one layer for learning a three-class\nclassification problem on iris data set, that is D= 1,n0= 4andn1=K= 3, then the\ndimension of the parameter θin neural network is d= 15. There are 150data points in the iris\ndata set, with the first m= 50be the training set and the remaining 100be the testing set.\nThe precision parameter in likelihood is σ= 1. We take L= 20,M= 1,∆t= 0.1andM0=I\nin HMC. The convergence results are shown in Figure 10 and crossover point is given in table\n3.\n4.3.2 MNIST Classification Example\nNow, we consider a convolution neural network with two convolution layers followed by two\nfully connected layers, each convolution layer with 5input channels and output channels, 5×5\nkernel and stride as 1. Then the dimensional of parameter θisd= 1177. Using this neural\nnetwork, we intend to learn a binary classification problem on the MNIST data set by taking\nthe image for \"0\" and \"1\". We take the first m= 50images for training and following 100\nimages for testing. The precision parameter in likelihood is σ= 1. We take L= 20,M= 1,\n∆t= 0.001andM0= 0.01Iin HMC. The convergence results are shown in Figure 11 and\ncrossover point is given in table 3.\n5 Conclusion\nThe pSMC method has been introduced and proven to deliver parallel strong scaling , leading to\na more practically applicable class of consistent Bayesian methods. In particular, any MCMC\nkernel can be plugged into pSMC and the efficiency of MCMC is preserved, while asynchronous\nparallel processing can be easily and straightforwardly leveraged with full parallel efficiency.\nThis property has been verified and illustrated on a range of numerical examples in the context\nof machine learning. Most notably, this paves the way to high-accuracy optimal Bayesian\nsolution to larger problems than were accessible before. It is very exciting future work to really\npush the limits of this method with modern HPC.\n4Bayesian ground truth for each example is computed by a single SMC-HMC with a large number of samples.\n12Acknowledgement\nKJHL and XL gratefully acknowledge the support of IBM and EPSRC in the form of an Indus-\ntrial Case Doctoral Studentship Award. JML acknowledges funding from the U.S. Department\nof Energy, Office of Science, Advanced Scientific Computing Research (Early Career Research\nProgram, ReACT-QISE).\nReferences\nV Al Osipov, Hans-Juergen Sommers, and K Życzkowski. Random bures mixed states and\nthe distribution of their purity. Journal of Physics A: Mathematical and Theoretical , 43(5):\n055302, 2010.\nChristophe Andrieu, Nando De Freitas, Arnaud Doucet, and Michael I Jordan. An introduction\nto mcmc for machine learning. Machine learning , 50:5–43, 2003.\nYoshua Bengio. Probabilistic neural network models for sequential data. In Proceedings of\nthe IEEE-INNS-ENNS International Joint Conference on Neural Networks. IJCNN 2000.\nNeural Computing: New Challenges and Perspectives for the New Millennium , volume 5,\npages 79–84. IEEE, 2000.\nCarlo Berzuini and Walter Gilks. Resample-move filtering with cross-model jumps. Sequential\nMonte Carlo Methods in Practice , pages 117–138, 2001.\nAlexandros Beskos. A stable manifold mcmc method for high dimensions. Statistics & Probab-\nility Letters , 90:46–52, 2014.\nAlexandros Beskos, Mark Girolami, Shiwei Lan, Patrick E Farrell, and Andrew M Stuart. Geo-\nmetric mcmc for infinite-dimensional inverse problems. Journal of Computational Physics ,\n335:327–351, 2017.\nAlexandros Beskos, Ajay Jasra, Kody Law, Youssef Marzouk, and Yan Zhou. Multilevel se-\nquential monte carlo with dimension-independent likelihood-informed proposals. SIAM/ASA\nJournal on Uncertainty Quantification , 6(2):762–786, 2018.\nChristopher M Bishop. Pattern recognition and machine learning , volume 4. Springer, 2006.\nCharles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncer-\ntainty in neural network. In International conference on machine learning , pages 1613–1622.\nPMLR, 2015.\nYuxin Chen, David Keyes, Kody JH Law, and Hatem Ltaief. Accelerated dimension-\nindependent adaptive metropolis. SIAM Journal on Scientific Computing , 38(5):S539–S565,\n2016.\nNicolas Chopin. A sequential particle filter method for static models. Biometrika , 89(3):539–\n552, 2002.\nNicolas Chopin, Omiros Papaspiliopoulos, et al. An introduction to sequential Monte Carlo ,\nvolume 4. Springer, 2020.\nAdam D Cobb and Brian Jalaian. Scaling hamiltonian monte carlo inference for bayesian neural\nnetworks with symmetric splitting. In Uncertainty in Artificial Intelligence , pages 675–685.\nPMLR, 2021.\n13Simon L Cotter, Gareth O Roberts, Andrew M Stuart, and David White. Mcmc methods for\nfunctions: modifying old algorithms to make them faster. 2013.\nTiangang Cui, Kody JH Law, and Youssef M Marzouk. Dimension-independent likelihood-\ninformed mcmc. Journal of Computational Physics , 304:109–137, 2016.\nChenguang Dai, Jeremy Heng, Pierre E Jacob, and Nick Whiteley. An invitation to sequential\nmonte carlo samplers. Journal of the American Statistical Association , 117(539):1587–1600,\n2022.\nPierre Del Moral. Feynman-kac formulae . Springer, 2004.\nPierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential monte carlo samplers. Journal\nof the Royal Statistical Society Series B: Statistical Methodology , 68(3):411–436, 2006.\nSimon Duane, Anthony D Kennedy, Brian J Pendleton, and Duncan Roweth. Hybrid monte\ncarlo.Physics letters B , 195(2):216–222, 1987.\nYarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model\nuncertainty in deep learning. In international conference on machine learning , pages 1050–\n1059. PMLR, 2016.\nAlan E Gelfand and Adrian FM Smith. Sampling-based approaches to calculating marginal\ndensities. Journal of the American statistical association , 85(410):398–409, 1990.\nCharles J Geyer. Practical markov chain monte carlo. Statistical science , pages 473–483, 1992.\nWalter R Gilks and Carlo Berzuini. Following a moving target—monte carlo inference for\ndynamic bayesian models. Journal of the Royal Statistical Society: Series B (Statistical\nMethodology) , 63(1):127–146, 2001.\nHeikki Haario, Eero Saksman, and Johanna Tamminen. An adaptive metropolis algorithm.\nBernoulli , pages 223–242, 2001.\nW Keith Hastings. Monte carlo sampling methods using markov chains and their applications.\n1970.\nGeoffrey EHinton, Terrence J Sejnowski, et al. Learningand relearning inboltzmannmachines.\nParallel distributed processing: Explorations in the microstructure of cognition , 1(282-317):2,\n1986.\nNeil Houlsby, Ferenc Huszár, Zoubin Ghahramani, and Máté Lengyel. Bayesian active learning\nfor classification and preference learning. arXiv preprint arXiv:1112.5745 , 2011.\nPierre E Jacob, John O’Leary, and Yves F Atchadé. Unbiased markov chain monte carlo meth-\nods with couplings. Journal of the Royal Statistical Society Series B: Statistical Methodology ,\n82(3):543–600, 2020.\nChristopher Jarzynski. Equilibrium free-energy differences from nonequilibrium measurements:\nA master-equation approach. Physical Review E , 56(5):5018, 1997.\nKody JH Law. Proposals which speed up function-space mcmc. Journal of Computational and\nApplied Mathematics , 262:127–138, 2014.\nBenedict Leimkuhler and Sebastian Reich. Simulating hamiltonian dynamics . Number 14.\nCambridge university press, 2004.\n14Joseph M Lukens, Kody JH Law, Ajay Jasra, and Pavel Lougovski. A practical and efficient\napproach for bayesian quantum state estimation. New Journal of Physics , 22(6):063038,\n2020.\nJosephMLukens, KodyJHLaw, andRyanSBennink. Abayesiananalysisofclassicalshadows.\nnpj Quantum Information , 7(1):113, 2021.\nDavid JC MacKay. A practical bayesian framework for backpropagation networks. Neural\ncomputation , 4(3):448–472, 1992.\nNicholas Metropolis, Arianna W Rosenbluth, Marshall N Rosenbluth, Augusta H Teller, and\nEdward Teller. Equation of state calculations by fast computing machines. The journal of\nchemical physics , 21(6):1087–1092, 1953.\nFrancesco Mezzadri. How to generate random matrices from the classical compact groups.\narXiv preprint math-ph/0609050 , 2006.\nKevin P Murphy. Machine learning: a probabilistic perspective . 2012.\nRadford Neal. Regression and classification using gaussian process priors. Bayesian statistics ,\n6:475, 1998.\nRadford M Neal. Bayesian learning for neural networks , volume 118. Springer Science &\nBusiness Media, 1993.\nRadford M Neal. Annealed importance sampling. Statistics and computing , 11:125–139, 2001.\nRadford M Neal et al. Mcmc using hamiltonian dynamics. Handbook of markov chain monte\ncarlo, 2(11):2, 2011.\nAndrew Ng and Michael Jordan. Approximate inference a lgorithms for two-layer bayesian\nnetworks. Advances in neural information processing systems , 12, 1999.\nChristian P Robert, George Casella, and George Casella. Monte Carlo statistical methods ,\nvolume 2. Springer, 1999.\nGareth O Roberts and Richard L Tweedie. Exponential convergence of langevin distributions\nand their discrete approximations. Bernoulli , pages 341–363, 1996.\nDaniel Rudolf and Björn Sprungk. On a generalization of the preconditioned crank–nicolson\nmetropolis algorithm. Foundations of Computational Mathematics , 18:309–343, 2018.\nChristelle Vergé, Cyrille Dubarry, Pierre Del Moral, and Eric Moulines. On parallel implement-\nation of sequential monte carlo methods: the island particle model. Statistics and Computing ,\n25(2):243–260, 2015.\nNick Whiteley, Anthony Lee, and Kari Heine. On the role of interaction in sequential monte\ncarlo algorithms. Bernoulli , 22(1):494–529, 2015.\nOlivier Zahm, Tiangang Cui, Kody Law, Alessio Spantini, and Youssef Marzouk. Certified\ndimension reduction in nonlinear bayesian inverse problems. Mathematics of Computation ,\n91(336):1789–1835, 2022.\n15A Techniques for pSMC in practice\nIt is worth noting that when computing the likelihoods in a single SMC in practice, it may\noccur computational small numbers that the computer will recognize as 0. Fortunately, if one\ntakes the logarithm of the number, the computer can store much smaller values. So, instead\nof using the likelihood itself, we store and use the log-likelihood, taking the exponential as\nneeded. We use log-likelihood for unnormalized weights and adjust it with a constant for the\nsmall number of importance sampling weights. An additional trick is provided below to further\navoid the small computational number issue when calculating the constant in the parallel SMC\nestimator (9).\nThe procedure is defined as follows: for core r= 1, ..., R,\nZN,r=JY\nj=1NX\ni=1ωi,r\nj=JY\nj=1∆Kr\nj∆ˆZr\nj=KrˆZr,\nwhere ∆Kr\nj= exp(max ilog(wi,r\nj))andKr=QJ\nj=1∆Kr\nj;∆ˆZr\nj=PN\ni=1ˆwi,r\nj=PN\ni=1exp(log( wi,r\nj)−\nmax ilog(wi,r\nj))andˆZr=QJ\nj=1∆ˆZr\nj. Forthesmallnumberofimportancesamplingweights wi,r\nj,\nwe work with log-likelihood for unnormalized weights and modify it with a constant ˆwi,r\nj.\nUsing the same trick again on the normalized constant over Rcores, we have, for core r,\nZN,r=KrˆZr= exp(log( ˆZr) + log( Kr)−log(K))K,\nwhere log(K) = max r(log( ˆZr)+log( Kr)). Sincetheterm Kwillbecancelledoutaswecompute\nthe parallel SMC estimator, so we only need the term exp(log( ˆZr) + log( Kr)−log(K)), which\ncould further avoid the computational error due to the small number K.\nB Parameterization of ρ(θ)in QST\nThe2Q×2Qdensity matrix ρ(θ)are parameterized by a given parameter θwith 4(2Q)2non-\nnegative real numbers distributed from the standard normal distribution. The first 2(2Q)2\nnumbers populate the real and imaginary parts of a 2Q×2Qmatrix which through the Mezzadri\nalgorithm is converted to a 2Q×2Qunitary U[Mezzadri, 2006]. One can construct Uwith the\nQR decomposition routines:\n1. Construct a 2Q×2Qcomplex random matrix Dby the other 2(2Q)2entries left in θ.\n2. Use QR decomposition to get D=BP.\n3. Create the diagonal matrix Psign=diag(p11\n|p11|, ...,p2Q2Q\n|p2Q2Q|), where pii= [P]iifori=\n1,2,3, ...,2Q.\n4. Construct the unitary matrix UasU=BPsign.\nThe second 2(2Q)2parameters fill a complex 2Q×2Qmatrix C, then ρ(θ)can be constructed\nfollowing Al Osipov et al. [2010]:\nρ(θ) =(I2Q+U)CCH(I2Q+UH)\nTr[(I2Q+U)CCH(I2Q+UH)].\nC Proofs\nThe proofs of the various results in the paper are presented here, along with restatements of\nthe results.\n16C.1 Proof relating to Lemma 3.4\nLemma 4.4 Assume Assumption 3.1 and 3.2. Then, for any J∈N, there is a C > 0such\nthat for any N, R∈N, suitable ζ:Θ→R,\nE[(FN,R(ζ)−f(ζ))2]≤C|ζ|2\n∞\nNR.\nProof.\nE[(FN,R(ζ)−f(ζ))2] =E\u0014\u00121\nRRX\nr=1(fN,r(ζ)−f(ζ))\u00132\u0015\n=1\nR2E\u0014RX\nr=1(fN,r(ζ)−f(ζ))2+RX\nr=1RX\nr′=1(fN,r(ζ)−f(ζ))(fN,r′(ζ)−f(ζ))\u0015\n=1\nR2RX\nr=1E[(fN,r(ζ)−f(ζ))2] +1\nR2RX\nr=1RX\nr′=1E[(fN,r(ζ)−f(ζ))]E[(fN,r′(ζ)−f(ζ))].\n(19)\nBy Proposition 3.3, three expectation terms in (19) are expressed as follows\nE[(fN,r(ζ)−f(ζ))2]≤C|ζ|2\n∞\nN,E[(fN,r(ζ)−f(ζ))] = 0 .\nThen, we conclude\nE[(FN,R(ζ)−f(ζ))2]≤C|ζ|2\n∞\nNR.\nC.2 Proof relating to Theorem 3.5\nTheorem 4.5 Assume Assumption 3.1 and 3.2. Then, for any J∈N, there exists a Cφ>0,\nwhich depends on φ, such that for any R∈N,\nE[( ˆφpSMC−π(φ))2]≤Cφ\nNR. (20)\nThen the strong scaling in the parallel SMC is described as the MSE defined in (20)decreases\nas1/RforRnodes, for suitable variables value ( M,N,J), with constant time.\nProof.\nE[( ˆφpSMC−π(φ))2] =E\u0014\u0012FN,R(φ)\nFN,R(1)−f(φ)\nf(1)\u00132\u0015\n=E\u0014\u0012FN,R(φ)\nFN,R(1)−FN,R(φ)\nf(1)+FN,R(φ)\nf(1)−f(φ)\nf(1)\u00132\u0015\nApplying Cauchy-Schwartz inequality, we have\n≤2E\u0014\u0012FN,R(φ)\nFN,R(1)−FN,R(φ)\nf(1)\u00132\u0015\n+ 2E\u0014\u0012FN,R(φ)\nf(1)−f(φ)\nf(1)\u00132\u0015\n≤2|FN,R(φ)|2\n∞\n|FN,R(1)|2|f(1)|2E[(FN,R(1)−f(1))2] +2\n|f(1)|2E\u0002\n(FN,R(φ)−f(φ))2\u0003\n(21)\n17Assume Assumption 3.1, there exists a C′\nφsuch that2|FN,R(φ)|2\n∞\n|FN,R(1)|2|f(1)|2≤C′\nφ, and there exists a C′\nsuch that2\n|f(1)|2≤C′. Then, following (21), we have\nE[( ˆφpSMC−π(φ))2]≤C′\nφE[(FN,R(1)−f(1))2] +C′E\u0002\n(FN,R(φ)−f(φ))2\u0003\nBy Lemma 3.4 with ζ=φandζ= 1respectively, we have\n≤C′\nφC\nNR+C′C|φ|2\n∞\nNR\nLetCφ=C′\nφC+C′C|φ|2\n∞, we have\n≤Cφ\nNR.\nD Figures\nD.1 Gaussian cases\nFigure 2: Convergence of the Gaussian case for m=d= 8. In pSMC, the number of mutation\nsteps is M= 10; in pCN, the number of burn-in samples is 10000. For both methods, the\nnumber of realizations is 100.\nFigure 3: Convergence of the Gaussian case for m= 4andd= 16. In pSMC, the number of\nmutation steps is M= 10; in pCN, the number of burn-in samples is 5000. For both methods,\nthe number of realizations is 100.\n18Figure 4: Convergence of the Gaussian case for m= 64andd= 16. In pSMC, the number of\nmutation steps is M=d; in pCN, the number of burn-in samples is 10000. For both methods,\nthe numbers of realizations is 10.\nFigure 5: Convergence of the Gaussian case for m=d= 32. In pSMC, the number of mutation\nsteps is M=d; in pCN, the number of burn-in samples is 10000. For both methods, the\nnumber of realizations is 10.\nFigure 6: Convergence of the Gaussian case for m= 16andd= 64. In pSMC, the number of\nmutation steps is M=d; in pCN, the number of burn-in samples is 10000. For both methods,\nthe number of realizations is 10.\n19D.2 Quantum State Tomography problem\nFigure 7: Convergence of the quantum state tomography example for m= 6andd= 16. In\npSMC, the number of mutation steps is M=d; in pCN, the number of burn-in samples is 0.\nFor both methods, the number of states is S= 10.\nFigure 8: Convergence of the quantum state tomography example for m= 36andd= 64. In\npSMC, the number of mutation steps is M=d; in pCN, the number of burn-in samples is 0.\nFor both methods, the number of states is S= 10.\n20Figure 9: Convergence of the quantum state tomography example for m= 216andd= 256.\nIn pSMC, the number of mutation steps is M=d; in pCN, the number of burn-in samples is\n0. For both methods, the number of states is S= 10.\nD.3 Bayesian Neural Network Examples\nFigure 10: Convergence of the iris classification example. For both methods, the number of\nrealizations is 10,L= 20,M= 1,∆ = 0 .1,M0=I.\nFigure 11: Convergence of the MNIST classification example. For both methods, the number\nof realizations is 10,L= 20,M= 1,∆t= 0.001,M0= 0.01I.\n21"    },    {        "title": "2402.06186v1.Constraints_on_Quasinormal_modes_from_Black_Hole_Shadows_in_regular_non_minimal_Einstein_Yang_Mills_Gravity.pdf",        "content": "Constraints on Quasinormal modes from Black Hole Shadows in regular non-minimal Einstein\nYang-Mills Gravity\nDhruba Jyoti Gogoi1, 2,∗and Supakchai Ponglertsakul3,†\n1Department of Physics, Moran College, Moranhat, Charaideo 785670, Assam, India\n2Theoretical Physics Division, Centre for Atmospheric Studies,\nDibrugarh University, Dibrugarh 786004, Assam, India\n3Strong Gravity Group, Department of Physics, Faculty of Science,\nSilpakorn University, Nakhon Pathom 73000, Thailand\nThis work deals with the scalar quasinormal modes using higher order WKB method and black hole shadow\nin non-minimal Einstein Yang-Mills theory. To validate the results of quasinormal modes, time domain profiles\nare also investigated. We found that with an increase in the magnetic charge of the black hole, the ring-down\ngravitational wave increases non-linearly and damping rate decreases non-linearly. The presence of magnetic\ncharge also results in a decrease in the black hole shadow non-linearly. It is found that for large values of the\ncoupling parameter, the black hole changes to a solitonic solution and the corresponding ring-down gravitational\nwave frequency increases slowly with a decrease in the damping rate. For the solitonic solutions, the shadow\nis also smaller. The constraints on the model parameters calculated using shadow observations of M87* and\nSgr A* and an approximate analytic relation between quasinormal modes and shadow at the eikonal limit is\ndiscussed.\nKeywords: Quasinormal modes; Yang-Mills gravity; Black hole shadow; Time domain profile\nI. INTRODUCTION\nBlack holes are one of the most fundamental objects in the\nuniverse. They originate from collapsing stars and are gov-\nerned by Einstein’s theory of general relativity (GR). Regard-\nless of complex interior structure of dying stars, once formed,\nblack holes become one of the simplest celestial objects. Ac-\ncording to the no-hair theorem, black holes are uniquely de-\nscribed by three physical parameters i.e, mass, angular mo-\nmentum and electric charge. However, there are several sce-\nnarios where the no-hair theorem can be evaded. For instance,\nin the Einstein-Yang-Mills (EYM) theory and its variants, sev-\neral numerical black holes with Yang-Mills hair are studied\n[1]. When asymptotic structure of spacetime is modified to\nboxlike boundary, various hairy black holes are found [2].\nMoreover, there has been much interest in gravitational the-\nories coupled with nonlinear electrodynamics sources. Such\nmarriage leads to a novel type of black hole with event hori-\nzon but possesses no essential singularity or regular black\nhole [3]. Regular black holes are also studied in modified\ntheories of gravity, for instance, f(R)gravity [4, 5], Gauss-\nBonnet gravity [5–7]. An exact spherically symmetric Wu-\nYang monopole solution is derived and studied [8–10]. This\nis a non-minimally extended of Einstein-Yang-Mills equation\nwithSU(2)Wu-Yang ansatz. Later, this regular magnetic\nblack hole is generalized to include a cosmological constant\n[11].\nAs predicted by GR, any accelerated massive objects cre-\nate disturbances in spacetime around them. The disturbances\npropagate through spacetime in the form of gravitational\nwaves (GWs). These waves contain crucial information about\ntheir astrophysical sources. When two black holes collide and\n∗Email: moloydhruba@yahoo.in\n†Email: supakchai.p@gmail.commerge into a single massive black hole, during this process\nGW is emitted continuously. After the merger, a single black\nhole emits GWs signal which is prominently dominated by\na ringdown mode. The black hole ringdown is generally de-\nscribed by quasinormal modes (QNMs) [12–15]. An asso-\nciated complex frequency is determined by black hole’s mass\nand angular momentum. The real component of the frequency\nis the emission frequency while the imaginary component de-\nnotes exponential decay. Thus, studying QNMs of black holes\nis crucial since they allow us to attain various characteristics\nof black holes.\nRemarkably, the detection of GWs [16] in 2015 has opened\na new window to testing theories of gravity. Vast number\nof works have been devoted to study the QNMs of various\nblack holes. QNMs of nonminimally coupled scalar field in\nthree dimensional spacetime are explored in [17]. In addi-\ntion, QNMs of nonminimally coupled scalar field in a five\ndimensional Einstein-Power-Maxwell background are studied\nvia the Wentzel–Kramers–Brillouin (WKB) method and pseu-\ndospectral Chebyshev method [18]. Optical properties and\nQNMs of black holes dependence on the Yang-Mill charges\nare investigated in Einstein-Power-Yang-Mills theory [19].\nQNMs of black holes within the framework of generalized\nuncertainty principle (GUP) are analyzed [20–23]. Due to\nvariety of black hole solutions in modified theories of grav-\nity, there are many works considering QNMs of black holes\nin various models e.g., f(Q)gravity [24], f(R, T)gravity\n[25], Horndeski gravity [26, 27], Rastall gravity [28–30] and\nde Rham-Gabadadze-Tolley (dRGT) massive gravity [31–35].\nVery recently, QNMs of black hole in emergent gravity frame-\nwork are computed via the WKB and asymptotic iteration\nmethods [36].\nIn recent times, the visual representation of black holes has\ngarnered significant attention, driven by observations of ce-\nlestial objects like M87⋆[37–42] and Sgr A⋆[43]. These\nobservations have sparked interest in the realms of both GRarXiv:2402.06186v1  [gr-qc]  9 Feb 20242\nand modified gravity (MOG), offering an intriguing oppor-\ntunity to depict and analyze hypothetical images associated\nwith these observed phenomena. This avenue allows us to en-\ngage in a geometric and topological exploration, comparing\nthe theoretical depiction of black hole images with the actual\nimages of M87⋆and Sgr A⋆. Consequently, the visualiza-\ntion of black holes assumes a captivating optical dimension,\nparticularly as we endeavour to uncover the relationships be-\ntween optical features, such as the shadows they cast, through\nthe application of thermodynamics [44, 45] and quasi-normal\nmodes [46, 47]. Additionally, when considering the shape\ndata, the shadow images can take the form of either distorted\ngeometries on the image plane [48–52] or a pattern of con-\ncentric circles, contingent on the presence of a rotation pa-\nrameter within the spacetime. To gain a deeper understanding\nof these shadow behaviours, the Hamilton-Jacobi formalism\n[53] emerges as a compatible analytical framework. It is based\non the concept that massless photons in the vicinity of a black\nhole generate specific orbits around the region, a phenomenon\nreferred to as the geodesic null background. In the realm of\nblack hole imaging, ongoing research aims to provide a com-\nparative analysis of observed images, such as those of M87⋆\nand Sgr A⋆, in terms of their size and shape.\nMotivated by these studies, in this work, we aim to inves-\ntigate the QNMs and black hole shadows in the framework\nof regular non-minimally EYM theory in four-dimensional\nspacetime. We also introduce constraints on the model pa-\nrameters from black hole shadow observations and try to re-\nlate the shadow behaviour with the QNMs of the black hole\nspacetime.\nThis paper is organized as follows. In Section II, we dis-\ncuss the EYM theory and the associated black hole solution\nin brief. In Section III, massless scalar perturbations on the\nblack hole spacetime have been investigated and the QNMs\nspectrum is analysed. Section IV deals with the evolution of\nscalar perturbation. In Section V, we investigate the black\nhole shadow. In Section VI, we discuss how QNMs are linked\nwith shadows of the black hole. Finally in Section VII, we\nprovide a concluding remark of our work.\nThroughout the manuscript, we consider c=ℏ= 8πG=\n1.\nII. REGULAR NON-MINIMAL EINSTEIN YANG-MILLS\nBLACK HOLE\nIn this section, we shall discuss the theory and the black\nhole solution in brief. The associated action of the 4-D regular\nnon-minimally EYM theory is [8, 11]\nS=1\n8πZ\nd4x√−g\u0014\nR+1\n2F(a)\nµνFµν(a)+1\n2¯RαβµνF(a)\nαβF(a)\nµν\u0015\n,\n(1)\nwhere Rstands for the Ricci curvature scalar. The indices la-\nbeled with Greek letters range from 0 to 3, while those labeled\nwith Latin letters range from 1 to 3. Additionally, the notation\nF(a)\nµνrepresents the Yang-Mills (YM) tensor and is connectedto the YM potential A(a)\nµthrough the equation:\nF(a)\nµν=∇µA(a)\nν− ∇ νA(a)\nµ+f(a)\n(b)(c)A(b)\nµA(c)\nν, (2)\nwhere ∇µrepresents the covariant derivative and the symbols\nf(b) (c)(a)denote the real structure constants of the three-\nparameter YM gauge group SU(2). The tensor ¯Rαβµνis de-\nfined as [11]\n¯Rαβµν=ξ1\n2R\u0000\ngαµgβν−gανgβµ\u0001\n+ξ2\n2\u0000\nRαµgβν−Rανgβµ+Rβνgαµ−Rβµgαν\u0001\n+ξ3Rαβµν. (3)\nHere, RαβandRαβµνrepresent the Ricci and Riemann ten-\nsors, respectively, and ξi(i= 1,2,3)denote the non-minimal\ncoupling parameters between the YM field and the gravita-\ntional field. Assuming the gauge field is described by the Wu-\nYang ansatz and taking ξ1=ξ, ξ2= 4ξ, ξ3=−6ξwith\nξ >0, a regular, static, and spherically symmetric black hole\nsolution was discovered [8, 11]. This black hole solution is\ndescribed by the metric [8, 11],\nds2=−f(r)dt2+f−1(r)dr2+r2\u0000\ndθ2+ sin2θdϕ2\u0001\n,(4)\nwhere the metric function has the following form:\nf(r) = 1 +\u0012r4\nr4+ 2ξQ2\u0013\u0012Q2\nr2−2M\nr\u0013\n, (5)\nwithMrepresenting the mass of the black hole and Qbeing\nthe magnetic charge.\nThe electromagnetic field four-potential for the regular non-\nminimal magnetic black hole is given by:\nAν= (0,0,0, Qcosθ). (6)\nIt’s evident that when Q= 0, the metric (4) simplifies to the\nstandard Schwarzschild black hole metric, while when ξ= 0,\nit corresponds to the Reissner-Nordstr ¨om black hole metric\nbut with a magnetic charge instead of an electric charge. The\nquantity of event horizons of the non-minimal EYM black\nhole depends on both the non-minimal parameter and the mag-\nnetic charge, potentially resulting in multiple horizons. In this\nwork, we shall use the black hole line element (4) to study the\nmassless scalar QNMs and the black hole shadow.\nIII. MASSLESS SCALAR QUASINORMAL MODES\nIn this section, we shall investigate the QNMs associated\nwith massless scalar perturbation. In this case, we shall as-\nsume that the test field has negligible impact or influence\non the black hole spacetime defined in the previous section\nand the associated scalar field is massless in nature. We ob-\ntain a Klein-Gordon-type equation associated with the QNMs\nwhile considering the pertinent conservation relations associ-\nated with the spacetime under consideration. We shall imple-\nment Pad ´e averaged 6th order WKB approximation method to\nobtain the QNMs numerically.3\nTaking into consideration of axial perturbation only, it is\npossible to express the perturbed metric in the following form\n[24, 54]:\nds2=− |gtt|dt2+r2sin2θ(dϕ−p1(t, r, θ )dt\n−p2(t, r, θ )dr−p3(t, r, θ )dθ)2+grrdr2\n+r2dθ2. (7)\nIn this context, the parameters p1,p2, and p3are intricately\nlinked to the perturbation occurring in the spacetime sur-\nrounding the black hole. These parameters play a crucial role\nin characterizing the modifications induced by the perturba-\ntion. Specifically, they contribute to the altered dynamics and\nstructure of the black hole spacetime.\nNow, for the massless scalar field, as it is considered that the\neffect of the field on the spacetime is minimal, the perturbed\nmetric Eq. (7) takes the following form:\nds2=−|gtt|dt2+grrdr2+r2dΩ2. (8)\nThe Klein-Gordon equation in curved spacetime in this case\nis written as\n□Φ =1√−g∂µ(√−ggµν∂νΦ) = 0 . (9)\nEq. (9) describes the QNMs associated with the scalar pertur-\nbation in the considered black hole spacetime.\nUtilising spherical harmonics, it is possible to express the\nwave function of the scalar field in the following decomposed\nform:\nΦ(t, r, θ, ϕ ) =X\nl,mψl(r)\nrYlm(θ, ϕ)e−iωt. (10)\nIn the above equation, Ylm(θ, ϕ)is the spherical harmonics\npart of the wave function and the parameters landmare the\nassociated parameters of spherical harmonics. Now, one can\nuse this decomposed form of the wave function in the Eq. (10)\nto obtained a reduced wave equation given by\n∂2\nr∗ψl(r∗) +ω2ψl(r∗) =Vs(r)ψl(r∗). (11)\nThis is a standard differential equation where the differenti-\nation is done with respect to the tortoise coordinate r∗. The\ntortoise coordinate is connected with the normal coordinate r\nby the following definition:\ndr∗\ndr=q\ngrr|g−1\ntt|. (12)\nIn the Eq. (11), parameter Vs(r)is the effective potential\nfor the massless scalar perturbation and it is given by:\nVs(r) =|gtt| \nl(l+ 1)\nr2+1\nrp\n|gtt|grrd\ndrq\n|gtt|g−1rr!\n.\n(13)\nl=1\nl=2\nl=3\nl=4\n2 4 6 8 100.00.20.40.60.81.0\nrVs(r)FIG. 1: Variation scalar potential with respect to rusing\nM= 1,Q= 0.3andξ= 0.1.\nIn this context, the symbol lis employed to signify the mul-\ntipole moment associated with the QNMs of the black hole.\nThe variation of the scalar potential Vs(r)with respect to the\nmultipole moment lis shown in Fig. 1. The overall effective\npotential increases with ℓ.\nThe other model parameters Qandξalso have noticeable\nimpacts on the potential Vs(r)which are depicted in Fig. 2.\nIt is seen that with an increase in the black hole charge Q,\nthe peak value of the potential increases non-linearly and the\ncorresponding rshifts towards the black hole event horizon.\nSimilarly, an increase in the value of ξalso increases the peak\nvalue of the potential. However, the variation is compara-\ntively smaller at the peak position and the peak value seems to\nchange uniformly with respect to ξ. This suggests that the\nmodel parameter Qmight have a non-linear impact on the\nspectrum of QNMs and ξmight have an almost linear impact\non QNMs. The details will be investigated in the next part of\nthe study.\nA. The Pad ´e averaged WKB approximation method\nIn this study, we implement a well-known method called\nWKB approximation method to calculate the QNMs nu-\nmerically for the massless scalar perturbation in the above-\nmentioned black hole spacetime. We consider Pad ´e averaged\n6th-order corrections to the WKB method [55–58] to obtain\nQNMs with higher accuracy.\nIn higher-order WKB approximation method, the oscilla-\ntion frequency ωof ring-down GWs is expressed as:\nω=vuut−i\"\n(n+ 1/2) +6X\nk=2¯Λk#\np\n−2V′′\n0+V0.(14)\nHere, the parameter n, representing the overtone number,\ntakes on discrete values starting from zero (n= 0,1,2, . . .).\nAdditionally, V0represents the value of the potential function\nV(r)at a specific radial coordinate rmax, while V′′\n0signifies4\nQ=0.3\nQ=0.5\nQ=0.7\nQ=0.9\n2 4 6 8 100.00.20.40.60.81.0\nrVs(r)\nξ=0.1\nξ=0.5\nξ=0.9\nξ=1.3\n1 2 3 4 50.00.20.40.60.81.0\nrVs(r)\nFIG. 2: Variation scalar potential with respect to rusing M= 1andl= 3. On the first panel, we have used ξ= 0.1and on the\nsecond Q= 0.7. Forξ= 0.9andξ= 1.3, these configurations are not black holes but represent solitonic solutions.\nthe second derivative of V(r)with respect to revaluated at the\nsame position. The pivotal location rmaxcorresponds to where\nthe potential function achieves its maximum value, rendering\nthis radial coordinate particularly significant.\nIntegral to our computational framework is the correction\nterms ¯Λk, which play a crucial role in refining the precision\nof our calculations. These correction terms are instrumental\nin capturing subtle nuances in the determination of the oscil-\nlation frequency, contributing to the overall accuracy of the\nframework.To elaborate on the procedural aspects of our approach, we\ndraw upon established references in the field. Specifically,\nRef.s [55–58] provide comprehensive insights into the mathe-\nmatical formulations of the correction terms and elaborate on\nthe intricacies of the Pad ´e averaging method applied in our\nstudy.\nBy incorporating these computational techniques, our re-\nsearch aims to uncover the nuanced behaviour of ring-down\nGWs, making significant contributions to the broader under-\nstanding of astrophysical phenomena in such types of theories\nof gravity.\nTABLE I: QNMs with different values of multipole moment lcalculated using 6th order Pad ´e averaged WKB approximation\nmethod. In this table we have chosen n= 0, M= 1, Q= 0.3andξ= 0.1.\nlPad´e averaged WKB ∆rms ∆6\n1 0.297616 −0.098004 i1.18328×10−90.0000209617\n2 0.491302 −0.097122 i3.04799×10−72.38347×10−6\n3 0.686034 −0.096870 i4.63924×10−84.55756×10−7\n4 0.881104 −0.096766 i1.04331×10−81.20843×10−7\n5 1.076325 −0.096713 i3.41801×10−94.44961×10−8\n6 1.271628 −0.096682 i1.36926×10−91.88337×10−8\n7 1.466978 −0.096663 i6.75515×10−109.20866×10−9\nTable I showcases the QNMs across a range of multipole mo-\nment values, with a specific focus on instances where the over-\ntone number is set to n= 0. The second column of the\ntable delineates the QNMs obtained through the application\nof the 6th-order Pad ´e averaged WKB approximation method.\nWithin this tabular representation, the term ∆rms assumes\nsignificance as it quantifies the root mean square error asso-ciated with the 6th-order Pad ´e averaged WKB approximation.\nMoreover, the term ∆6is introduced to gauge the error be-\ntween two adjacent order approximations. This error term is\ncomputed by assessing the absolute difference between ω7,\nrepresenting QNMs computed using the Pad ´e averaged 7th\norder WKB approximation method, and ω5, denoting QNMs\nobtained from the Pad ´e averaged 5th order WKB approxima-5\ntion method. Mathematically, ∆6is expressed as follows:\n∆6=|ω7−ω5|\n2. (15)\nA notable trend within the table is the observed reduction in\nerror associated with QNMs as the multipole moment lin-\ncreases. This behaviour aligns with the characteristic limi-\ntations of the WKB approximation method, particularly ev-\nident when the overtone number nsurpasses the multipole\nmoment l. The diminishing error with increasing multipole\nmoment is a distinctive feature of the WKB method, signify-\ning its challenges in providing accurate results when overtone\nnumbers exceed the corresponding multipole moment values\n[21, 23, 24, 59–62]. By keeping this fact in mind, in this in-\nvestigation, we have considered n < l to calculate QNMs.\nWe have shown the variation of the massless scalar QNMs\nin Fig. 3 and 4 with respect to Qandξ, respectively. In Fig.\n3, one can see that with an increase in the charge of the black\nholeQ, the real QNMs increase non-linearly. On the other\nhand, the damping rate or decay rate of ring-down GWs in-\ncreases very slowly initially up to around Q= 0.6and beyond\nthis the damping rate decreases drastically showing a highly\nnon-linear pattern.\nIn Fig. 4, the initial panel intricately illustrates the nuanced\nshift in real quasinormal frequencies concerning the parame-\nterξ, while the second panel meticulously delineates the cor-\nresponding alteration in the damping rate of ring-down GWs\nwith respect to ξ. Evidently, as the parameter ξexperiences an\nincremental surge, the real quasinormal frequencies manifest\na discernible linear escalation. In contrast, a rise in the value\nofξinduces a linear reduction in the damping rate. It is note-\nworthy, however, that this variation is comparatively modest\nwhen juxtaposed with the scenario involving the parameter Q.\nConsequently, when contemplated through the prism of\nQNMs, the influences exerted by Qandξexhibit notabledisparities. Looking forward, as we anticipate a wealth of\ndata from state-of-the-art GW detectors such as Laser Inter-\nferometer Space Antenna (LISA), there emerges the excit-\ning prospect of establishing stringent constraints on the Yang-\nMills field by harnessing insights derived from the meticulous\nexamination of QNMs.\nIV . TIME EVOLUTION OF SCALAR PERTURBATIONS\nIn the previous section, we conducted numerical computa-\ntions to unravel the characteristics of QNMs and scrutinized\nhow they hinge on the model parameters. Shifting our atten-\ntion to the subsequent section, our exploration now centers on\nthe temporal behaviours of massless scalar perturbations. To\nelucidate these profiles, we adopt the time domain integration\nframework delineated by Gundlach et al. [63, 64]\nTo progress further, we denote ψ(r∗, t) =ψ(i∆r∗, j∆t) =\nψi,j, andV(r(r∗)) =Vi. Here r∗represents the tortoise coor-\ndinate and tdenotes time. Subsequently, we express the scalar\nwave equation (9) as:\nψi+1,j−2ψi,j+ψi−1,j\n∆r2∗−ψi,j+1−2ψi,j+ψi,j−1\n∆t2\n−Viψi,j= 0. (16)\nThe initial conditions are specified as ψ(r∗, t) =\nexp\u0014\n−(r∗−k1)2\n2σ2\u0015\nandψ(r∗, t)|t<0= 0, where k1andσ\nrepresent the median and width of the initial wave-packet, re-\nspectively. Employing these initial conditions, we compute\nthe time evolution associated with the scalar perturbation us-\ning the iterative scheme:\nψi,j+1=−ψi,j−1+\u0012∆t\n∆r∗\u00132\n(ψi+1,j+ψi−1,j) + \n2−2\u0012∆t\n∆r∗\u00132\n−Vi∆t2!\nψi,j. (17)\nWe implement this iterative scheme to obtain the time do-\nmain profiles by keeping a fixed value of∆t\n∆r∗. However, it\nis important to mention that one must consider∆t\n∆r∗<1to\nsatisfy the V on Neumann stability condition throughout the\nnumerical procedure. Throughout our numerical calculations,\nwe have considered this ratio around 0.7to obtain the time\nevolution of the massless scalar perturbations on the black\nhole spacetime. We have set k1= 0andσ= 5for the numer-\nical calculations of time domain profiles.\nIn Fig. 5, we have shown the time domain profiles in the\nunits of M. One may observe that the variation of time do-\nmain profiles with respect to multiple moment lstands in\nagreement with our Table I. From the second and third panel,\nit is evident that the impact of the coupling parameter ξonthe QNMs spectrum is very small in comparison to the charge\nparameter Qas predicted earlier. These results are also in\nagreement with the results obtained from Pad ´e averaged WKB\napproximation method.\nV . OPTICAL BEHA VIOUR OF THE BLACK HOLE:\nSHADOW\nThe investigation into supermassive black holes (SMBHs)\nholds significant importance in astrophysics, given their mys-\nterious nature and their central positions in galaxies, where\nthey exert exceptional gravitational forces with event hori-\nzons preventing light escape. Recent advances in observa-\ntional techniques, exemplified by the Event Horizon Tele-6\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n0.0 0.2 0.4 0.6 0.8 1.00.700.750.800.85\nQωR\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n0.0 0.2 0.4 0.6 0.8 1.0-0.095-0.090-0.085-0.080-0.075-0.070\nQωI\nFIG. 3: Variation of real and imaginary QNMs using M= 1,l= 3andξ= 0.1.\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.68600.68650.68700.68750.6880\nξωR\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.0970-0.0968-0.0966-0.0964-0.0962-0.0960-0.0958\nξωI\nFIG. 4: Variation of real and imaginary QNMs using M= 1,l= 3andQ= 0.3.\nscope (EHT), have offered unprecedented insights into these\ncolossal cosmic entities.\nIn this part, we explore the mysterious concept of black\nhole shadows, uncovering the hidden areas within the complex\nstructure of space and time that cover the edge of these cosmic\nobjects. Black holes have incredibly strong gravity that forces\neven light to be pulled in when it crosses the boundary called\nthe event horizon [65, 66]. Consequently, the outer boundary\nof a black hole manifests as an indistinct yet solemn circle,\nprojecting its inscrutable silhouette against the surrounding\ncosmic tableau of matter or luminous radiance. The dimen-\nsions and intricate contours of this umbral figure furnish pro-\nfound insights into the fundamental characteristics of black\nholes and the essence of gravity itself. Recent advancements\nin both astronomical observation and technological capabili-\nties have conferred upon us the capacity to capture and visu-\nally scrutinize these elusive black hole shadows, representing\na paradigmatic leap in our pursuit to comprehend these enig-\nmatic celestial phenomena [67]. As we peer into these cos-\nmic shadows, we not only unveil the visual facets of black\nholes but also unlock new strata of scientific comprehension,shedding light on the intricacies veiled within the expansive\ncosmic tapestry. These advancements herald a gateway to a\ndeeper understanding of the universe’s intricacies, underscor-\ning the remarkable synergy between cutting-edge technology\nand our relentless scientific inquisitiveness about the cosmos.\nSuch a study will extensively contribute towards our under-\nstanding of different modified gravity theories and the validity\nof GR in light of current observational aspects.\nThe Euler-Lagrange equation is given by\nd\ndτ\u0012∂L\n∂˙xµ\u0013\n−∂L\n∂xµ= 0, (18)\nwhere the Lagrangian is expressed as\nL(x,˙x) =1\n2gµν˙xµ˙xν. (19)\nFor static and spherically symmetric black hole, the La-\ngrangian becomes\nL(x,˙x) =1\n2\u0014\n−f(r)˙t2+1\nf(r)˙r2+r2\u0010\n˙θ2+ sin2θ˙ϕ2\u0011\u0015\n.\n(20)7\nl=1\nl=2\nl=3\n0 50 100 150 20010-1210-810-41\ntψ\nξ=0.5\nξ=1.0\nξ=1.5\n0 50 100 150 20010-910-710-50.0010.10010\ntψ\n160 165 170 175 1805.×10-71.×10-65.×10-61.×10-5\nQ=0.3\nQ=0.5\nQ=0.7\n0 50 100 150 20010-1210-810-41\ntψ\nFIG. 5: Time domain profiles of scalar perturbation with M= 1. On the first panel, Q= 0.3andξ= 0.5; on the second panel\nl= 1andQ= 0.3and on the third panel l= 1andξ= 0.5.\nHere the dot over the variables denotes the derivative with re-\nspect to the proper time τ.\nChoosing the equatorial plane where θ=π/2, the con-\nserved energy Eand angular momentum Lcan be computed\nby utilizing the killing vectors ∂/∂τ and∂/∂ϕ as given by\nE=f(r)˙t, L =r2˙ϕ. (21)\nIn the case of photon, we can write the geodesic equation as,\n−f(r)˙t2+˙r2\nf(r)+r2˙ϕ2= 0. (22)\nUtilizing Eq. (22) in conjunction with the conserved quanti-\nties, namely, EandL, the orbital equation for a photon can be\nderived as. [67]\n\u0012dr\ndϕ\u00132\n=Veff, (23)\nwhere Veffis defined as\nVeff=r4\u0014E2\nL2−f(r)\nr2\u0015\n. (24)Expressing Eq. (23) in radial form yields,\nVr(r) =1\nζ2−˙r2/L2. (25)\nIn this expression, ζrepresents the impact parameter and is\ndefined as ζ=L/E. Additionally, the reduced potential Vr(r)\ncan be expressed as:\nVr(r) =f(r)\nr2. (26)\nTo scrutinize the shadows of black holes, our focus is di-\nrected towards a specific point along the trajectory denoted\nasrph. This particular point corresponds to the turning point,\nsignifying the location of the light ring that encircles the black\nhole. It can also be interpreted as the radius of the photon\nsphere, holding significant relevance in the analysis of the\nblack hole’s shadow characteristics. At this pivotal turning\npoint of a black hole [65, 67–72],\nVeff|rph= 0,and Veff|rph= 0. (27)\nIt is feasible to determine the impact parameter ζat the turning\npoint by\n1\nζ2\ncrit=f(rph)\nr2\nph. (28)8\nTherefore, the radius of the photon sphere, denoted as rph,\ncan be determined using[67]:\nd\ndrA(r)\f\f\f\f\nrph= 0. (29)\nThis equation can be reformulated as\nf′(rph)\nf(rph)−h′(rph)\nh(rph)= 0, (30)\nhere we have A(r) =h(r)/f(r)withh(r) =r2.\nNow, to derive the shadow of the black hole, we express Eq.\n(23) using Eq. (28) in terms of the function A(r)as\n\u0012dr\ndϕ\u00132\n=h(r)f(r)\u0012A(r)\nA(rph)−1\u0013\n. (31)\nUtilizing Eq. (31), one can calculate the shadow radius of\nthe black hole. In the scenario where a static observer is po-\nsitioned at a distance r0from the black hole, we can ascer-\ntain the angle αbetween the light rays originating from the\nobserver and the radial direction of the photon sphere. This\nangle can be computed as [67]\ncotα=1p\nf(r)h(r)dr\ndϕ\f\f\f\f\nr=r0. (32)\nIn conjunction with Eq. (31), the aforementioned equation\ncan be represented as\ncot2α=A(r0)\nA(rph)−1. (33)\nOnce more, the above equation can be reformulated using the\nrelation sin2α= 1/(1 + cot2α)as\nsin2α=A(rph)\nA(r0). (34)\nBy substituting the expression for A(rph)from Eq. (28)\nand using A(r0) =r2\n0/f(r0), the shadow radius of the black\nhole for a static observer at r0can be approximated as [67]\nRs=r0sinα=s\nr2\nphf(r0)\nf(rph). (35)\nOnce more, as r0→ ∞ , i.e., for a static observer at a large\ndistance, f(r0)→1. Consequently, for such an observer, the\nshadow radius Rsbecomes,\nRs=rphp\nf(rph). (36)\nFinally, through the stereographic projection of the shadow\nfrom the black hole’s plane to the observer’s image plane with\ncoordinates (X, Y), the apparent form of the shadow can be\ndetermined. These coordinates are defined as [22, 67, 73]\nX= lim\nr0→∞ \n−r2\n0sinθ0dϕ\ndr\f\f\f\f\nr0!\n, (37)\nY= lim\nr0→∞ \nr2\n0dθ\ndr\f\f\f\f\n(r0,θ0)!\n. (38)In this context, θ0denotes the angular orientation of the ob-\nserver in relation to the plane of the black hole.\nThe investigation presents stereographic projections of the\nblack hole shadow for the specific black hole under examina-\ntion. Fig. 6 illustrates these projections, offering visual depic-\ntions of the shadow’s characteristics corresponding to various\nmodel parameters. In the initial panel of Fig. 6, the stereo-\ngraphic projections of the black hole are displayed for differ-\nent values of the model parameters Qandξ. It is evident that\nan increase in the model parameter Qresults in a decrease in\nthe black hole shadow. The second panel illustrates the black\nhole shadow for various values of the parameter ξ, revealing a\ngradual decrease in the shadow with an increase in ξ. For en-\nhanced visualization, Fig. 7 presents the square of the black\nhole shadow radius concerning the model parameters Qandξ.\nThis representation indicates that ξhas a minimal impact on\nthe black hole shadow. Therefore, the investigation demon-\nstrates that both model parameters exert distinct influences on\nthe black hole shadow.\nA notable feature of SMBHs is the presence of a shadow\ncast by their event horizons, a phenomenon arising from the\ngravitational bending of light. The size and shape of this\nshadow furnish valuable information about the SMBH and its\nsurrounding environment. To comprehend and quantify the\nshadow’s dimensions, the spatial separation Dbetween the\nSMBH and the galactic center must be considered.\nA conventional method for determining the classical diam-\neter of the black hole shadow involves the application of the\narclength equation:\ndsh=Dθ sh\nM, (39)\nwhere dshdenotes the shadow’s diameter, θshrepresents the\nangular size of the shadow, and Mcorresponds to a specific\nunit of measurement.\nFor M87*, one of the extensively studied SMBHs, the mea-\nsurement of the shadow diameter has been realized through\nEHT observations [37, 43]. The reported diameter of the\nM87* shadow is dM87*\nsh= (11 ±1.5)M, serving as a crucial\ndata point for scrutinizing the properties of this SMBH.\nHowever, it is imperative to consider the uncertainties asso-\nciated with these measurements for a more accurate compre-\nhension of the physical parameters. The determination of such\nuncertainties requires meticulous analysis, and in the case of\nthe M87* shadow diameter, Ref.s [74, 75] offer detailed in-\nsights into the employed methodology.\nThis study aims to delve deeper into the implications of the\nmeasured M87* and Sgr A* shadow diameters by incorpo-\nrating uncertainties from the aforementioned references. Our\nspecific focus is on constraining the potential values of the\nblack hole charge Qby calculating the 1σlimits based on re-\nsults obtained for M87* and Sgr A*. By considering these\nconstraints, we aspire to augment our understanding of the\nmodel behaviour and the model parameters from the physical\nproperties of M87* and Sgr A* and contribute to the broader\ncomprehension of SMBHs as well as possible constraints on\nQNMs from them.9\n-4 -2 0 2 4-4-2024\nXYQ\n00.20.40.60.8\n-4 -2 0 2 4-4-2024\nXYξ\n0246810\nFIG. 6: Stereographic projection of black hole shadow using M= 1. On the first panel, we have used Q= 0.3and on the\nsecond panel ξ= 0.5.\nξ=0.1\nξ=0.3\nξ=0.5\nξ=0.7\n0.0 0.2 0.4 0.6 0.820222426\nQRs2\nQ=0.1\nQ=0.3\nQ=0.5\nQ=0.7\n0.0 0.2 0.4 0.6 0.8 1.021222324252627\nξRs2\nFIG. 7: Variation of R2\nsas a function of black hole charge Q(the left panel) and non-minimal coupling constant ξ(the right\npanel). We have used M= 1.\n0.0 0.2 0.4 0.6 0.8 1.034567\nQRs\n-15 -10 -5 0 5 1034567\nξRs\nFIG. 8: Constraint on black hole shadow radius from Sgr A*. On the left panel, we have used ξ= 0.1and on the right panel,\nwe have used Q= 0.35. The excluded (allowed) regions are denoted by the red (grey) shaded areas. The black dotted\nhorizontal line is the radius of the shadow of the Schwarzschild black hole.\nIn Figs. 8 and 9, we have shown the constraints on the model parameters Qandξfrom the observed shadow radii of10\n0.0 0.2 0.4 0.6 0.8 1.034567\nQRs\n-15 -10 -5 0 5 1034567\nξRs\nFIG. 9: Constraint on black hole shadow radius from M87*. On the left panel, we have used ξ= 0.1and on the right panel, we\nhave used Q= 0.35. The excluded (allowed) regions are denoted by the red (grey) shaded areas. The black dotted horizontal\nline is the radius of the shadow of the Schwarzschild black hole.\nconstraint on Qusing ξ= 0.1constraint on ξusing Q= 0.35\n1σ(upper/lower) from M87* none / 0.8912 −138.11/none\n1σ(upper/lower) from Sgr A* none/ 0.789 −8.695/none\nTABLE II: Variation of shadow with 1σranges of Qandξbased on the shadow radii of Sgr. A* and M87* as depicted in Figs.\n8 and 9.\nSgr A* and M87* by following Ref.s [74, 75]. One can see\nthat the observational data do not put a strong constraint on\nthe coupling parameter ξ. However, the charge parameter Q\nis well constrained by the observed shadow radii of Sgr A*\nand M87*.VI. CONNECTION BETWEEN QUASINORMAL MODES\nAND SHADOW OF THE BLACK HOLE\nIn this section, we shall briefly discuss the approximate an-\nalytical connection of QNMs with the black hole shadow ra-\ndius. One may note that from the numerical results of QNMs\nand black hole shadow, there is a possible correspondence be-\ntween them.\nIn this scenario, we shall consider only 3rdorder WKB ex-\npansion given by\nω=(\nV+V4\n8V2\u0012\nν2+1\n4\u0013\n−\u00127 + 60 ν2\n288\u0013V2\n3\nV2\n2+iνp\n−2V2\u00141\n2V2\u00145V4\n3(77 + 188 ν2)\n6912V4\n2\n−V2\n3V4(51 + 100 ν2)\n384V3\n2+V2\n4(67 + 68 ν2)\n2304V2\n2+V5V3(19 + 28 ν2)\n288V2\n2+V6(5 + 4 ν2)\n288V2\u0015\n−1\u0015)1/2\nr=rph.\n(40)\nOne may note that even we consider higher order expansion\nsuch as 6th order or 12th order expansion, in the eikonal limit,\nthe first significant terms are identical with those obtained\nby the 3rd order WKB approximation method. The term Vistands for the i-thderivative of the potential V. We also con-\nsidered ν=n+1\n2, where nis the overtone number of QNMs.\nBy expanding (40), at the eikonal regime, one can obtain\n[19, 76]11\nω=ωR−iωI=\nℓp\nf(r)\nr\f\f\f\f\f\nr=rph+p\nf(r)\n2r\f\f\f\f\f\nr=rph+O(ℓ−1)\n\nR−\ni\nn+ 1/2√\n2p\nf(r)\nr\f\f\f\f\f\nr=rphq\n6rf′(r)−6f(r)−r2f′′(r)−r2f(r)−1f′(r)2\f\f\f\f\f\nr=rph+O(ℓ−1)\n\nI. (41)\nComparing the imaginary part\nωI=n+ 1√\n2R−1\nsq\n2f(rph)−r2\nphf′′(rph) +O(ℓ−1),(42)\nand the real part\nωR=R−1\ns\u0012\nℓ+1\n2+O(ℓ−1)\u0013\n. (43)\nFurther exploring the constraints on the visual representa-\ntion of a black hole i.e. shadow can be achieved through the\nexamination of observational data concerning its QNMs. It is\nessential to emphasize the direct relationship between the fea-\ntures of the black hole’s silhouette and the specific values of\nits QNMs as this will help us to uncover different properties of\nthe black hole hairs. Consequently, the availability of signif-\nicant observational data from space-based gravitational wave\ndetectors like LISA presents an avenue for imposing stricter\nconstraints on the parameters of models used to understand\nblack holes. Moreover, this process enables the assessment of\nconsistency between two distinct approaches-shadow analysis\nand QNMs-in testing the underlying theoretical framework.\nFrom the approximated relations (42) and (43), at the\neikonal limit, we observe that the black hole shadow has an\nimpact on the behaviour of the QNMs spectrum. Most impor-\ntantly, an observational constraint on the black hole shadow\nwill also put a constraint on the real and imaginary parts of\nQNMs of the black hole at the eikonal limit. However, as ev-\nident from the table II, such a constraint on the QNMs will\nbe very weak at the current stage. In the near future, obser-\nvational results from LISA might put a very strong constraint\non QNMs which may be useful along with the observational\nresults of black hole shadow to check for the consistencies\nbetween them and to constrain the theory more stringently.\nVII. CONCLUDING REMARKS\nIn this work, we considered a charged black hole solution\nin non-minimally coupled EYM theory and studied the scalar\nperturbation on the black hole spacetime and the associated\nQNMs. From the behaviour of the potential, we found that for\nhigher values of the coupling parameter ξ, the line element\nrepresents solitonic solutions instead of a black hole solution.\nIn order to obtain the QNMs, we used Pad ´e averaged WKB\napproximation method. The analysis showed that black hole\ncharge Qhas a more significant impact on the quasinormal\nmode spectrum. With an increase in charge, the oscillation\nfrequency of ring-down GWs increases non-linearly. In thecase of the damping rate of GWs, we found that initially, with\nan increase in Q, the damping or decay rate increases slowly\nand beyond some threshold value around Q= 0.65, it starts to\ndecrease rapidly as Qapproaches 1. One may note that some\nprevious Ref.s [19, 29] also found similar variation trends of\nQNMs with respect to black hole charge Q. However, due to\nthe presence of coupling term and non-linear charge distribu-\ntion, in this case, we observe slightly different behaviour of\nQNMs towards higher values of Q.\nUnlike black hole charge Q, the coupling parameter ξhas a\nlinear effect on the QNMs spectrum. With an increase in the\nvalue of ξ, the oscillation frequency of QNMs increases while\nthe damping rate decreases slowly.\nWe also investigated time domain profiles for different\nmodel parameters and found that the profiles depict similar\nresults as those found by using the WKB method.\nIn the next part of our work, we considered the shadow of\nthe black hole and found that Qhas a noticeable impact on\nthe shadow behaviour of the black hole. For a charged black\nhole, the shadow is expected to be smaller in size. A similar\nbehaviour is observed in the case of ξalso. However, the vari-\nation of shadow with respect to ξis small. Thus the impact\nof the coupling parameter as well as the presence of solitonic\nsolutions may not be well differentiated by utilising shadow\nobservations. As a result, observational data on black hole\nshadows, do not have a stringent constraint on ξ.\nWe also studied the relationship between the QNMs and the\nshadow of the black hole and found that the shadow observa-\ntions can provide a constraint on the QNMs spectrum. In the\nnear future, observational data of QNMs from LISA may pro-\nvide us with scope to constrain the theory more efficiently and\nto check whether the shadow observations are well consistent\nwith it or not.\nOur study has uncovered that in the standard non-minimally\ncoupled EYM theory, both the charge parameter Qand the\ncoupling term ξexert notable influences on the quasinormal\nspectrum and shadow of the black hole. The effects of these\nparameters on the quasinormal mode spectrum differ signifi-\ncantly in their nature.\nACKNOWLEDGMENTS\nDJG acknowledges the contribution of the COST Action\nCA21136 – “Addressing observational tensions in cosmology\nwith systematics and fundamental physics (CosmoVerse)”. SP\nacknowledges the funding supported from the NSRF via the\nProgram Management Unit for Human Resources &Institu-\ntional Development, Research and Innovation [grant number12\nB37G 660013].\nDATA A V AILABILITY STATEMENT\nThere are no new data associated with this article.\n[1] P. Bizon, Colored black holes, Phys. Rev. Lett. 64, 2844 (1990);\nP. C. Aichelburg and P. Bizon, Magnetically charged black\nholes and their stability, Phys. Rev. D 48, 607 (1993), arXiv:gr-\nqc/9212009; E. E. Donets and D. V . Galtsov, Stringy sphalerons\nand nonAbelian black holes, Phys. Lett. B 302, 411 (1993),\narXiv:hep-th/9212153; B. Kleihaus, J. Kunz, and A. Sood,\nCharged SU(N) Einstein Yang-Mills black holes, Phys. Lett. B\n418, 284 (1998), arXiv:hep-th/9705179; E. Winstanley, Exis-\ntence of stable hairy black holes in SU(2) Einstein Yang-Mills\ntheory with a negative cosmological constant, Class. Quant.\nGrav. 16, 1963 (1999), arXiv:gr-qc/9812064; B. L. Shepherd\nand E. Winstanley, Dyons and dyonic black holes in su(n)\neinstein-yang-mills theory in anti–de sitter spacetime, Phys.\nRev. D 93, 064064 (2016); B. R. Greene, S. D. Mathur,\nand C. M. O’Neill, Eluding the no-hair conjecture: Black\nholes in spontaneously broken gauge theories, Phys. Rev. D\n47, 2242 (1993); S. Ponglertsakul and E. Winstanley, Solitons\nand hairy black holes in Einstein–non-Abelian–Proca theory\nin anti–de Sitter spacetime, Phys. Rev. D 94, 044048 (2016),\narXiv:1606.04644 [gr-qc].\n[2] S. R. Dolan, S. Ponglertsakul, and E. Winstanley, Stability of\nblack holes in Einstein-charged scalar field theory in a cav-\nity, Phys. Rev. D 92, 124047 (2015), arXiv:1507.02156 [gr-\nqc]; S. Ponglertsakul, E. Winstanley, and S. R. Dolan, Stability\nof gravitating charged-scalar solitons in a cavity, Phys. Rev. D\n94, 024031 (2016), arXiv:1604.01132 [gr-qc]; S. Ponglertsakul\nand E. Winstanley, Effect of scalar field mass on gravitating\ncharged scalar solitons and black holes in a cavity, Phys. Lett.\nB764, 87 (2017), arXiv:1610.00135 [gr-qc]; N. Sanchis-Gual,\nJ. C. Degollado, P. J. Montero, J. A. Font, and C. Herdeiro, Ex-\nplosion and final state of an unstable reissner-nordstr ¨om black\nhole, Phys. Rev. Lett. 116, 141101 (2016); P. Bosch, S. R.\nGreen, and L. Lehner, Nonlinear evolution and final fate of\ncharged anti–de sitter black hole superradiant instability, Phys.\nRev. Lett. 116, 141102 (2016); N. Sanchis-Gual, J. C. Degol-\nlado, C. Herdeiro, J. A. Font, and P. J. Montero, Dynamical for-\nmation of a Reissner-Nordstr ¨om black hole with scalar hair in a\ncavity, Phys. Rev. D 94, 044061 (2016), arXiv:1607.06304 [gr-\nqc]; P. Basu, C. Krishnan, and P. N. Bala Subramanian, Hairy\nBlack Holes in a Box, JHEP 11, 041, arXiv:1609.01208 [hep-\nth]; Y . Peng, Studies of a general flat space/boson star transition\nmodel in a box through a language similar to holographic super-\nconductors, JHEP 07, 042, arXiv:1705.08694 [hep-th]; O. J. C.\nDias and R. Masachs, Evading no-hair theorems: hairy black\nholes in a Minkowski box, Phys. Rev. D 97, 124030 (2018),\narXiv:1802.01603 [gr-qc].\n[3] E. Ayon-Beato and A. Garcia, Regular black hole in general rel-\nativity coupled to nonlinear electrodynamics, Phys. Rev. Lett.\n80, 5056 (1998), arXiv:gr-qc/9911046; J. Matyjasek, Extremal\nlimit of the regular charged black holes in nonlinear electrody-\nnamics, Phys. Rev. D 70, 047504 (2004); L. Balart and E. C.\nVagenas, Regular black holes with a nonlinear electrodynamics\nsource, Phys. Rev. D 90, 124045 (2014); M.-S. Ma, Magneti-cally charged regular black hole in a model of nonlinear elec-\ntrodynamics, Annals Phys. 362, 529 (2015), arXiv:1509.05580\n[gr-qc]; K. A. Bronnikov, Regular magnetic black holes and\nmonopoles from nonlinear electrodynamics, Phys. Rev. D 63,\n044005 (2001), arXiv:gr-qc/0006014; Z.-Y . Fan and X. Wang,\nConstruction of Regular Black Holes in General Relativity,\nPhys. Rev. D 94, 124027 (2016), arXiv:1610.02636 [gr-qc].\n[4] M. E. Rodrigues, E. L. B. Junior, G. T. Marques, and V . T.\nZanchin, Regular black holes in f(R)gravity coupled to non-\nlinear electrodynamics, Phys. Rev. D 94, 024062 (2016), [Ad-\ndendum: Phys.Rev.D 94, 049904 (2016)], arXiv:1511.00569\n[gr-qc].\n[5] S. Nojiri and S. D. Odintsov, Regular multihorizon black holes\nin modified gravity with nonlinear electrodynamics, Phys. Rev.\nD96, 104008 (2017), arXiv:1708.05226 [hep-th].\n[6] S. G. Ghosh, D. V . Singh, and S. D. Maharaj, Regular black\nholes in einstein-gauss-bonnet gravity, Phys. Rev. D 97, 104050\n(2018).\n[7] M. Estrada and R. Aros, A new class of regular Black Holes in\nEinstein Gauss Bonnet gravity with localized sources of matter,\nPhys. Lett. B 844, 138090 (2023), arXiv:2305.17233 [gr-qc].\n[8] A. B. Balakin and A. E. Zayats, Non-minimal Wu-Yang\nmonopole, Phys. Lett. B 644, 294 (2007), arXiv:gr-qc/0612019.\n[9] T. T. Wu and C. N. Yang, Some solutions of the classical\nisotropic gauge field equations, in Properties of Matter under\nUnusual Conditions, in Honor of Edward Teller’s 60th Birth-\nday, in H. Mark, S. Fernbach (Eds), , 344 (1969).\n[10] Y . Shnir, Magnetic Monopoles (Springer, Berlin, 2006).\n[11] A. B. Balakin, J. P. S. Lemos, and A. E. Zayats, Reg-\nular nonminimal magnetic black holes in spacetimes with\na cosmological constant, Phys. Rev. D 93, 024008 (2016),\narXiv:1512.02653 [gr-qc].\n[12] C. V . Vishveshwara, Scattering of Gravitational Radiation by a\nSchwarzschild Black-hole, Nature 227, 936 (1970).\n[13] W. H. Press, Long Wave Trains of Gravitational Waves from a\nVibrating Black Hole, Astrophys. J. Lett. 170, L105 (1971).\n[14] K. D. Kokkotas and B. G. Schmidt, Quasinormal modes of\nstars and black holes, Living Rev. Rel. 2, 2 (1999), arXiv:gr-\nqc/9909058.\n[15] R. A. Konoplya and A. Zhidenko, Quasinormal modes of black\nholes: From astrophysics to string theory, Rev. Mod. Phys. 83,\n793 (2011), arXiv:1102.4014 [gr-qc].\n[16] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo\nCollaboration), Observation of gravitational waves from a bi-\nnary black hole merger, Phys. Rev. Lett. 116, 061102 (2016).\n[17] A. Rinc ´on and G. Panotopoulos, Greybody factors and quasi-\nnormal modes for a nonminimally coupled scalar field in a\ncloud of strings in (2+1)-dimensional background, Eur. Phys.\nJ. C78, 858 (2018), arXiv:1810.08822 [gr-qc].\n[18] A. Rincon, P. A. Gonzalez, G. Panotopoulos, J. Saave-\ndra, and Y . Vasquez, Quasinormal modes for a non-\nminimally coupled scalar field in a five-dimensional Ein-\nstein–Power–Maxwell background, Eur. Phys. J. Plus 137, 127813\n(2022), arXiv:2112.04793 [gr-qc].\n[19] D. J. Gogoi, J. Bora, M. Koussour, and Y . Sekhmani, Quasinor-\nmal modes and optical properties of 4-D black holes in Einstein\nPower-Yang–Mills gravity, Annals Phys. 458, 169447 (2023),\narXiv:2306.14273 [gr-qc].\n[20] R. Karmakar, D. J. Gogoi, and U. D. Goswami, Quasinor-\nmal modes and thermodynamic properties of GUP-corrected\nSchwarzschild black hole surrounded by quintessence, Int.\nJournal of Mod. Phys. A 37, 10.1142/S0217751X22501809\n(2022), arXiv:2206.09081 [gr-qc].\n[21] G. Lambiase, R. C. Pantig, D. J. Gogoi, and A. ¨Ovg¨un, In-\nvestigating the Connection between Generalized Uncertainty\nPrinciple and Asymptotically Safe Gravity in Black Hole Sig-\nnatures through Shadow and Quasinormal Modes, (2023),\narXiv:2304.00183 [gr-qc].\n[22] M. A. Anacleto, J. A. V . Campos, F. A. Brito, and E. Pas-\nsos, Quasinormal modes and shadow of a Schwarzschild\nblack hole with GUP, Annals Phys. 434, 168662 (2021),\narXiv:2108.04998 [gr-qc].\n[23] D. J. Gogoi and U. D. Goswami, Quasinormal modes and\nHawking radiation sparsity of GUP corrected black holes in\nbumblebee gravity with topological defects, JCAP 06(06), 029,\narXiv:2203.07594 [gr-qc].\n[24] D. J. Gogoi, A. ¨Ovg¨un, and M. Koussour, Quasinormal Modes\nof Black holes in f(Q)gravity, (2023), arXiv:2303.07424 [gr-\nqc].\n[25] T. Tangphati, M. Youk, and S. Ponglertsakul, Magnetically\ncharged regular black holes in f(R, T)gravity coupled to non-\nlinear electrodynamics, (2023), arXiv:2312.16614 [gr-qc].\n[26] O. J. Tattersall and P. G. Ferreira, Quasinormal modes of black\nholes in Horndeski gravity, Phys. Rev. D 97, 104047 (2018),\narXiv:1804.08950 [gr-qc].\n[27] Y . Sekhmani and D. J. Gogoi, Electromagnetic quasinormal\nmodes of dyonic AdS black holes with quasitopological elec-\ntromagnetism in a Horndeski gravity theory mimicking EGB\ngravity at D →4, Int. J. Geom. Meth. Mod. Phys. 20, 2350160\n(2023), arXiv:2306.02919 [gr-qc].\n[28] D. J. Gogoi and U. D. Goswami, Quasinormal modes of black\nholes with non-linear-electrodynamic sources in Rastall gravity,\nPhys. Dark Univ. 33, 100860 (2021), arXiv:2104.13115 [gr-qc].\n[29] D. J. Gogoi, R. Karmakar, and U. D. Goswami, Quasinor-\nmal modes of nonlinearly charged black holes surrounded by\na cloud of strings in Rastall gravity, Int. J. Geom. Meth. Mod.\nPhys. 20, 2350007 (2023), arXiv:2111.00854 [gr-qc].\n[30] D. J. Gogoi, N. Heidari, J. K ˇr´ıˇz, and H. Hassanabadi, Quasinor-\nmal Modes and Greybody factors of AdS/dS Black holes sur-\nrounded by Quintessence in Rastall gravity, Fortschr. der Phys.\n2300245 (2024), arXiv:2307.09976 [gr-qc].\n[31] P. Burikham, S. Ponglertsakul, and L. Tannukij, Charged scalar\nperturbations on charged black holes in de Rham-Gabadadze-\nTolley massive gravity, Phys. Rev. D 96, 124001 (2017),\narXiv:1709.02716 [gr-qc].\n[32] S. Ponglertsakul, P. Burikham, and L. Tannukij, Quasinormal\nmodes of black strings in de Rham–Gabadadze–Tolley massive\ngravity, Eur. Phys. J. C 78, 584 (2018), arXiv:1803.09078 [gr-\nqc].\n[33] T. Wuthicharn, S. Ponglertsakul, and P. Burikham, Quasi-\nnormal modes of near-extremal black holes and black strings in\nmassive gravity background, Int. J. Mod. Phys. D 31, 2150127\n(2022), arXiv:1911.11448 [gr-qc].\n[34] P. Wongjun, C.-H. Chen, and R. Nakarachinda, Quasinor-\nmal modes of a massless Dirac field in de Rham-Gabadadze-\nTolley massive gravity, Phys. Rev. D 101, 124033 (2020),\narXiv:1910.05908 [gr-qc].[35] S. H. Hendi and M. Momennia, Quasinormal modes of black\nholes in dRGT massive gravity under electromagnetic pertur-\nbations, Iran. J. Phys. Res. 21, 213 (2021).\n[36] D. J. Gogoi, A. ¨Ovg¨un, and D. Demir, Quasinormal modes and\ngreybody factors of symmergent black hole, Phys. Dark Univ.\n42, 101314 (2023), arXiv:2306.09231 [gr-qc].\n[37] K. Akiyama et al. (Event Horizon Telescope), First M87\nEvent Horizon Telescope Results. I. The Shadow of the Su-\npermassive Black Hole, Astrophys. J. Lett. 875, L1 (2019),\narXiv:1906.11238 [astro-ph.GA].\n[38] K. Akiyama et al. (Event Horizon Telescope), First M87 Event\nHorizon Telescope Results. II. Array and Instrumentation, As-\ntrophys. J. Lett. 875, L2 (2019), arXiv:1906.11239 [astro-\nph.IM].\n[39] K. Akiyama et al. (Event Horizon Telescope), First M87 Event\nHorizon Telescope Results. III. Data Processing and Calibra-\ntion, Astrophys. J. Lett. 875, L3 (2019), arXiv:1906.11240\n[astro-ph.GA].\n[40] K. Akiyama et al. (Event Horizon Telescope), First M87\nEvent Horizon Telescope Results. IV . Imaging the Central Su-\npermassive Black Hole, Astrophys. J. Lett. 875, L4 (2019),\narXiv:1906.11241 [astro-ph.GA].\n[41] K. Akiyama et al. (Event Horizon Telescope), First M87 Event\nHorizon Telescope Results. V . Physical Origin of the Asymmet-\nric Ring, Astrophys. J. Lett. 875, L5 (2019), arXiv:1906.11242\n[astro-ph.GA].\n[42] K. Akiyama et al. (Event Horizon Telescope), First M87 Event\nHorizon Telescope Results. VI. The Shadow and Mass of\nthe Central Black Hole, Astrophys. J. Lett. 875, L6 (2019),\narXiv:1906.11243 [astro-ph.GA].\n[43] K. Akiyama et al. (Event Horizon Telescope), First Sagittar-\nius A* Event Horizon Telescope Results. I. The Shadow of the\nSupermassive Black Hole in the Center of the Milky Way, As-\ntrophys. J. Lett. 930, L12 (2022).\n[44] X.-C. Cai and Y .-G. Miao, Can we know about black hole ther-\nmodynamics through shadows?, (2021), arXiv:2107.08352 [gr-\nqc].\n[45] M. Zhang and M. Guo, Can shadows reflect phase structures of\nblack holes?, Eur. Phys. J. C 80, 790 (2020), arXiv:1909.07033\n[gr-qc].\n[46] K. Jusufi, Connection Between the Shadow Radius and Quasi-\nnormal Modes in Rotating Spacetimes, Phys. Rev. D 101,\n124063 (2020), arXiv:2004.04664 [gr-qc].\n[47] H. Yang, Relating Black Hole Shadow to Quasinormal Modes\nfor Rotating Black Holes, Phys. Rev. D 103, 084010 (2021),\narXiv:2101.11129 [gr-qc].\n[48] A. Grenzebach, V . Perlick, and C. L ¨ammerzahl, Photon Re-\ngions and Shadows of Kerr-Newman-NUT Black Holes with\na Cosmological Constant, Phys. Rev. D 89, 124004 (2014),\narXiv:1403.5234 [gr-qc].\n[49] A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh,\nShadow of rotating regular black holes, Phys. Rev. D 93,\n104004 (2016), arXiv:1604.03809 [gr-qc].\n[50] L. Amarilla and E. F. Eiroa, Shadow of a rotating braneworld\nblack hole, Phys. Rev. D 85, 064019 (2012), arXiv:1112.6349\n[gr-qc].\n[51] R. Kumar and S. G. Ghosh, Rotating black holes in 4D\nEinstein-Gauss-Bonnet gravity and its shadow, JCAP 07, 053,\narXiv:2003.08927 [gr-qc].\n[52] A. Belhaj and Y . Sekhmani, Shadows of rotating quintessential\nblack holes in Einstein Gauss-Bonnet gravity with a cloud of\nstrings, Gen. Rel. Grav. 54, 17 (2022).\n[53] B. Carter, Global structure of the Kerr family of gravitational\nfields, Phys. Rev. 174, 1559 (1968).14\n[54] M. Bouhmadi-L ´opez, S. Brahma, C.-Y . Chen, P. Chen, and D.-\nh. Yeom, A consistent model of non-singular Schwarzschild\nblack hole in loop quantum gravity and its quasinormal modes,\nJCAP 07, 066, arXiv:2004.13061 [gr-qc].\n[55] B. F. Schutz and C. M. Will, BLACK HOLE NORMAL\nMODES: A SEMIANALYTIC APPROACH, Astrophys. J.\nLett. 291, L33 (1985).\n[56] S. Iyer and C. M. Will, Black Hole Normal Modes: A WKB\nApproach. 1. Foundations and Application of a Higher Order\nWKB Analysis of Potential Barrier Scattering, Phys. Rev. D 35,\n3621 (1987).\n[57] R. A. Konoplya, Quasinormal behavior of the d-dimensional\nSchwarzschild black hole and higher order WKB approach,\nPhys. Rev. D 68, 024018 (2003), arXiv:gr-qc/0303052.\n[58] J. Matyjasek and M. Telecka, Quasinormal modes of black\nholes. II. Pad ´e summation of the higher-order WKB terms,\nPhys. Rev. D 100, 124006 (2019), arXiv:1908.09389 [gr-qc].\n[59] Y . Sekhmani and D. J. Gogoi, Electromagnetic quasinormal\nmodes of dyonic AdS black holes with quasitopological elec-\ntromagnetism in a Horndeski gravity theory mimicking EGB\ngravity at D →4, International Journal of Geometric Methods\nin Modern Physics , 2350160 (2023).\n[60] N. Parbin, D. J. Gogoi, J. Bora, and U. D. Goswami, Deflec-\ntion angle and quasinormal modes of a de Sitter black hole in\nf(T,B)gravity, (2022), arXiv:2211.02414 [gr-qc].\n[61] D. J. Gogoi and U. D. Goswami, Tideless traversable worm-\nholes surrounded by cloud of strings in f(R) gravity, JCAP 02,\n027, arXiv:2208.07055 [gr-qc].\n[62] R. Karmakar, D. J. Gogoi, and U. D. Goswami, Quasinor-\nmal modes and thermodynamic properties of GUP-corrected\nSchwarzschild black hole surrounded by quintessence, Interna-\ntional Journal of Modern Physics A 37, 2250180 (2022).\n[63] C. Gundlach, R. H. Price, and J. Pullin, Late time behavior\nof stellar collapse and explosions: 1. Linearized perturbations,\nPhys. Rev. D 49, 883 (1994), arXiv:gr-qc/9307009.\n[64] C. Gundlach, R. H. Price, and J. Pullin, Late time behavior of\nstellar collapse and explosions: 2. Nonlinear evolution, Phys.\nRev. D 49, 890 (1994), arXiv:gr-qc/9307010.\n[65] B. Eslam Panah, K. Jafarzade, and S. H. Hendi, Charged\n4D Einstein-Gauss-Bonnet-AdS black holes: Shadow, energy\nemission, deflection angle and heat engine, Nucl. Phys. B 961,115269 (2020), arXiv:2004.04058 [hep-th].\n[66] R. A. Konoplya, Shadow of a black hole surrounded by dark\nmatter, Phys. Lett. B 795, 1 (2019), arXiv:1905.00064 [gr-qc].\n[67] D. J. Gogoi, Y . Sekhmani, D. Kalita, N. J. Gogoi, and J. Bora,\nJoule-Thomson Expansion and Optical Behaviour of Reissner-\nNordstr ¨om-Anti-de Sitter Black Holes in Rastall Gravity Sur-\nrounded by a Quintessence Field, Fortschritte der Physik 71,\n2300010 (2023).\n[68] K. Jafarzade, M. Kord Zangeneh, and F. S. N. Lobo, Shadow,\ndeflection angle and quasinormal modes of Born-Infeld charged\nblack holes, JCAP 04, 008, arXiv:2010.05755 [gr-qc].\n[69] R. C. Pantig, P. K. Yu, E. T. Rodulfo, and A. ¨Ovg¨un, Shadow\nand weak deflection angle of extended uncertainty principle\nblack hole surrounded with dark matter, Annals of Physics 436,\n168722 (2022).\n[70] A. ¨Ovg¨un, I. Sakallı, and J. Saavedra, Shadow cast and Deflec-\ntion angle of Kerr-Newman-Kasuya spacetime, JCAP 10, 041,\narXiv:1807.00388 [gr-qc].\n[71] U. Papnoi, F. Atamurotov, S. G. Ghosh, and B. Ahmedov,\nShadow of five-dimensional rotating Myers-Perry black hole,\nPhys. Rev. D 90, 024073 (2014), arXiv:1407.0834 [gr-qc].\n[72] A. ¨Ovg¨un, I. Sakallı, J. Saavedra, and C. Leiva, Shadow cast\nof noncommutative black holes in Rastall gravity, Mod. Phys.\nLett. A 35, 2050163 (2020), arXiv:1906.05954 [hep-th].\n[73] R. Karmakar, D. J. Gogoi, and U. D. Goswami, Thermodynam-\nics and shadows of GUP-corrected black holes with topological\ndefects in Bumblebee gravity, Physics of the Dark Universe 41,\n101249 (2023).\n[74] P. Kocherlakota et al. (Event Horizon Telescope), Constraints\non black-hole charges with the 2017 EHT observations of\nM87*, Phys. Rev. D 103, 104047 (2021), arXiv:2105.09343\n[gr-qc].\n[75] S. Vagnozzi et al. , Horizon-scale tests of gravity theories and\nfundamental physics from the Event Horizon Telescope im-\nage of Sagittarius A, Class. Quant. Grav. 40, 165007 (2023),\narXiv:2205.07787 [gr-qc].\n[76] B. Cuadros-Melgar, R. D. B. Fontana, and J. de Oliveira, Ana-\nlytical correspondence between shadow radius and black hole\nquasinormal frequencies, Phys. Lett. B 811, 135966 (2020),\narXiv:2005.09761 [gr-qc]."    },    {        "title": "2402.06224v1.Adaptive_multi_gradient_methods_for_quasiconvex_vector_optimization_and_applications_to_multi_task_learning.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nAdaptive multi-gradient methods for quasiconvex vector\noptimization and applications to multi-task learning\nNguyen Anh Minh ·Le Dung Muu ·Tran Ngoc\nThang\nFebruary 12, 2024\nAbstract We present an adaptive step-size method, which does not include line-search\ntechniques, for solving a wide class of nonconvex multiobjective programming problems\non an unbounded constraint set. We also prove convergence of a general approach under\nmodest assumptions. More specifically, the convexity criterion might not be satisfied by\nthe objective function. Unlike descent line-search algorithms, it does not require an initial\nstep-size to be determined by a previously determined Lipschitz constant. The process’s\nprimary characteristic is its gradual step-size reduction up until a predetermined condition is\nmet. It can be specifically applied to offer an innovative multi-gradient projection method for\nunbounded constrained optimization issues. Preliminary findings from a few computational\nexamples confirm the accuracy of the strategy. We apply the proposed technique to some\nmulti-task learning experiments to show its efficacy for large-scale challenges.\nKeywords nonconvex multi-objective programming ·gradient descent algorithms ·\nquasiconvex functions ·pseudoconvex functions ·adaptive step-sizes\nMathematics Subject Classification (2000) 90C25 ·90C26 ·68Q32 ·93E35\n1 Introduction\nWith a multitude of real-world applications, gradient descent methods are a popular tool for a\nbroad variety of programming problems, including both scalar and vector optimization, from\nconvex to nonconvex, and references [4], [7], and [14]. Iterative solutions based on gradient\nNguyen Anh Minh\nFaculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1st Dai Co Viet street,\nHanoi, Viet Nam\nE-mail: minh.na194117@hust.edu.vn\nLe Dung Muu\nTIMAS, Thang Long University and Institute of Mathematics, V AST, Hanoi, Vietnam\nE-mail: ldmuu@math.ac.vn\nTran Ngoc Thang ( \u0000)\nFaculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1st Dai Co Viet street,\nHanoi, Viet Nam\nE-mail: thang.tranngoc@hust.edu.vnarXiv:2402.06224v1  [math.OC]  9 Feb 20242 Nguyen Anh Minh et al.\ndirections and step sizes are produced by gradient descent algorithms at each iteration.\nThe step-size was chosen using one of the few widely used methods, and for a long time,\nresearchers concentrated on determining the direction to enhance the convergence rate of\ntechniques (see [4], [21] ).\nIn order to lower the total computational cost of the method, new key areas of machine\nlearning applications with large dimensionality and nonconvex objective functions have neces-\nsitated the invention of novel step-size choosing procedures (see [7], [14]). One-dimensional\nminimization line-search, whether precise or approximate, is computationally expensive every\niteration, especially when determining the function value is almost the same as determining\nits derivative and necessitates solving intricate auxiliary issues (see [4]). The step-size value\ncan be determined using known information, such as the gradient’s Lipschitz constants, to\nskip the line-search. But doing so necessitates only employing a portion of their imprecise\nestimations, which slows convergence. For the well-known divergent series rule, this is also\ntrue (see [12], [21]).\nThere are numerous applications of the class of quasiconvex multiobjective functions in\nreal-world issues, such as those in economics. An further drawback of the weighting method\nfor these kinds of issues is that positive combinations of quasiconvex functions might not\nbe quasiconvex. The class of quasiconvex problems has been the subject of several research\n( [18], [20]). For the purpose of minimizing a quasiconvex-concave function under convex\nand quasiconvex-concave inequality constraints, a class of branch-and-bound algorithms is\nput forth [33]. Despite the fact that Kiwiel created the gradient approach for quasiconvex\nprogramming in 2001 [32], the usage of decreasing step-size causes the approach to converge\nslowly. After that, the gradient approach has been improved. Examples of these include the\nwork of Yu et al. ( [24], 2019) and Hu et al. ( [11], 2020), who both utilize a constant step-size\nbut require that the objective function meet the Holder condition. The other technique is\nthe neurodynamic approach (see [6] (Bian et al, 2018), [42] (Liu et al., 2022)) that solves\npseudoconvex programming problems with unbounded constraint sets using recurrent neural\nnetwork models. It does not select the step size in an adaptive manner.\nThe adaptive step-size procedure was proposed in [10] (Ferreira et al., 2020) and [13]\n(Konnov, 2018) and is mostly used to solve single-objective optimization problems. The\nstep-size algorithm for the conditional gradient approach is proposed in [13], and this\nmethod works well for solving pseudoconvex programming problems where the feasible\nset is bounded. The scope of [10] has been extended to include the specific scenario of an\nunbounded feasible set, which is not applicable in the unconstrained scenario.\nAdaptive step sizes have been used in some recent research to address multi-objective\noptimization problems; [44] and [43] are two noteworthy examples. In [44], unconstrained\nmulti-objective optimization problems were solved by combining the BFGS quasi-Newton\napproach with a nonmonotone adaptive step size strategy. Additionally, a conditional gradient\nmethod combined with an adaptive step size strategy was used in [43] to solve limited\nmulti-objective optimization problems. Their adaptive step size, however, is only theoretical\nbecause it was created using knowledge of the objective function’s Lipschitz constant.\nIn this study, we present a novel monotone adaptive step-size technique that does not\nrequire line searches for a wide range of unconstrained multiobjective optimization problems\nin which the objective function is smooth and nonconvex. Reducing the step size gradually\nuntil a predefined condition is met is an essential part of this process.Additionally, the\nLipschitz continuity of the objective function’s gradient is not necessary for convergence; the\nLipschitz constant is only used in one particular algorithmic scenario to determine the step\nsize. Preliminary computer tests demonstrate the effectiveness of the proposed modification.\nWe demonstrate the effective performance of the proposed method on large-scale problemsAdaptive multi-gradient methods for quasiconvex vector optimization 3\nby conducting a series of machine learning experiments, such as multi-variable logistic\nregressions and neural networks for classification.\nIn this work, we present a new adaptive step-size approach that does not require line\nsearches for a wide class of multiobjective programming problems in which the constraint set\nis unbounded closed convex and the objective function is nonconvex smooth. Reducing the\nstep size gradually until a predefined condition is met is an essential part of this process. The\nmethod does not use predefined constants, even though convergence requires the Lipschitz\ncontinuity of the objective function’s gradient. Preliminary computer experiments have\ndemonstrated the effectiveness of the proposed modification. We conduct a range of machine\nlearning experiments, such as multi-task learning and classification using neural networks, to\ndemonstrate the effectiveness of the suggested approach on large-scale problems.\nChapter 3 will present our Adaptive Multi-gradient algorithm for solving Multi-objective\noptimization Problems while proving the convergence of the algorithm. Chapter 4 will provide\nan upgraded version of the Adaptive Multi-gradient algorithm to help evenly distribute the\nfound optimal solutions on the Pareto Front. Chapter 5 will provide some examples of\nsolving multi-objective optimization problems for convex, pseudoconvex, and nonconvex\nfunctions. Chapter 6 will discuss the application of the proposed algorithm in solving Multi-\ntask Learning problems with specific tasks in image processing and text processing. Chapter\n7 will present some conclusions of this paper.\n2 Preliminaries and Notations\nWe examine a scenario involving a single-objective function f:Rm→Rwith a permissible\nsetU⊆Rm. The subsequent definitions are derived from [19].\nDefinition 1 [19] The differentiable function f:Rm→Ris considered\ni) convex on Uif, for all x,y∈U, and λ∈[0,1], the following inequality holds:\nf(λx+(1−λ)y)≤λf(x)+(1−λ)f(y).\nii) pseudoconvex on Uif, for all x,y∈U, the condition is met:\n⟨∇f(x),y−x⟩ ≥0⇒f(y)≥f(x).\niii) quasiconvex on Uif, for all x,y∈U, and λ∈[0;1], the inequality holds:\nf(λx+(1−λ)y)≤max{f(x);f(y)}.\nProposition 1 [9] The differentiable function f is quasiconvex on U if and only if\nf(y)≤f(x)⇒ ⟨∇f(x),y−x⟩ ≤0.\nIt is important to note that the implication chain holds: ” fis convex” ⇒”fis pseudoconvex”\n⇒”fis quasiconvex” [19].\nThe range, or image space, of a matrix M∈Rm×nis denoted by R(M), and I∈Rn×n\nrepresents the unit matrix. For two matrices A,B∈Rn×n,B≤A(B<A) signifies that A−B\nis positive semidefinite (definite). In the upcoming discussions, the Euclidean norm in Rn\nis denoted by ∥·∥, and B[x,r]represents the closed ball of radius rwith center x∈Rn. The\nsame notation ∥·∥is employed for the induced operator norms on the corresponding matrix\nspaces.4 Nguyen Anh Minh et al.\nDefine, for A∈Rm×n\n∥A∥∞,2:=max\nx̸=0∥Ax∥∞\n∥x∥.\nThen∥·∥∞,2is a norm in Rm×n. It is easy to prove that\n∥A∥∞,2:=max\ni=1,...,m∥Ai,·∥\n=max\ni=1,...,m \nn\n∑\nj=1A2\ni,j!1/2\n.\n2.1 Multi-objective optimization problem\nWe examine the multi-objective optimization problem represented as follows:\nMin x∈UF(x), (MOP( F,U))\nwhere the set U⊆Rnis nonempty, convex, and closed. The vector function F:U−→Rm\nis assumed to be differentiable on an open set containing U. We will assume that U={x∈\nRn|gi(x)≤0,i=1,..., l}. The Jacobian matrix of Fatxis denoted as follows:\nJF(x) =\n∇F1(x)T\n...\n∇Fm(x)T\n∈Rm×n\nIn this paper, we consider the Jacobian matrix JFofFto be continuously Lipschitz in U. This\nimplies that the component-wise derivatives ∇Fj∈Rnof the function Fjare continuously\nLipschitz for all j=1,..., m. It is also assumed that the solution set of (MOP( F,U))is\nnonempty. The subsequent definitions are sourced from [2].\nDefinition 2 An element x∗∈Uis referred to as a efficient point or a Pareto optimal point\nof problem (MOP( F,U)) if there is no y∈U,F(y)≤F(x∗)andF(y)̸=F(x∗), where the ≤\nsymbol between vectors is interpreted as the comparison between each component function.\nThere is no y∈Usuch that Fi(y)≤Fi(x∗)for every i=1,..., m, and there exists i0∈1,..., m\nsuch that Fi0(y)<Fi0(x∗).The Pareto Front is defined as the collection of Pareto optimum\nlocations.\nGiven the set Rm\n+:=R+×···× R+, the partial order between vector function values is\ndefined as follows:\nF(y)≤F(z)⇐⇒ F(z)−F(y)∈Rm\n+,\nHence, in order to solve problem (MOP( F,U)), it is necessary to find a point that represents\nthe minimum in the partial order.\nDefinition 3 A point x∗∈Uis referred to as a weakly efficient solution or a weak Pareto\noptimal point of problem (MOP( F,U)). If there is no element in U such that F(y) is less than\nF(x*), where the inequality F(y) ¡ F(x*) is interpreted as a partial order. The definition ends.\nConsidering Rm\n++:=R++×···× R++, we also define\n−Rm\n++={−s:s∈Rm\n++}.Adaptive multi-gradient methods for quasiconvex vector optimization 5\nDefinition 4 A point x∗∈Uis considered locally efficient, or weakly locally efficient, in\nproblem (MOP( F,U)) if there exists a neighborhood V⊆Uofx∗where x∗is an efficient, or\nweakly efficient, point in problem (MOP( F,U)) when restricted to V.\nIf the set Uis convex and the function FisRm\n+-convex (meaning that each component\nfunction of Fis convex), then every locally efficient point is also a globally efficient point.\nFurthermore, every point that is locally efficient is also a weakly locally efficient point.\nDefinition 5 [8] A vector zbelonging to the set Uis called a tangent direction of Uatzif\nthere exists a sequence\u0000\nzk\u0001\nkcontained in Uand a positive scalar λ∈R+such that\nlim\nk→∞zk=zand lim\nk→∞λzk−z\n∥zk−z∥=s\nThe collection of all tangent directions of the set Uat the point zis referred to as the tangent\ncone of Uatzand symbolized as T(U,z).\nFor a closed convex set U, we have T(U,z):=U(z):={s∈Rn|s=u−zfor some u∈U}.\nDefinition 6 A point x∗in the set Uis referred to as a Pareto stationary point (orPareto\ncritical point ) of the vector function Fover Uif the intersection of the product of the Jacobian\nmatrix JF(x∗)and the set U(x∗)with the negative real numbers raised to the power of mis\nempty.\nThis definition is a necessary condition (but not sufficient) for a point to be an effective Pareto\npoint and was first used in [8] to define the steepest descent algorithm for multiobjective\noptimization.\nLemma 1 If the function Fis pseudo-convex, meaning that its component functions Fi,i=\n1,..., mare pseudo-convex, then Definition 6 serves as both a necessary and sufficient\ncondition for a point to be considered a weakly efficient point.\nSuppose, on the contrary, that there is a point y∈Usuch that F(y)<F(x). Given\nthatFiis pseudo-convex for each i=1,..., m, we may conclude that ⟨∇Fi(x),(y−x)⟩<0.\nConsequently, JF(x)(y−x)∈ −Rm\n++, which violates Definition 6 (it is important to note that\nwe are assuming the set Uto be both convex and closed).\n2.2 Multi-task Learning Problem\nTraditional machine learning methods typically focus on solving one task with a single model.\nThis approach may cause us to overlook valuable information that could help us perform\nbetter on the task at hand, information that can be derived from related tasks. To illustrate\nthis, consider a simple example where you want to predict the price of a house based on\nfeatures such as area, number of rooms, number of floors, proximity to commercial centers,\nand so on. Clearly, adding a few additional tasks, such as predicting whether the house is in\nthe city or suburbs or whether it’s a villa or an apartment, can provide valuable insights for\npredicting its price.\nFrom a specialized perspective, Multi-Task Learning (MTL) is often implemented by\nsharing common model parameters (shared parameters) among network architectures and\nspecific parameters for each individual task’s network architecture. This allows us to obtain a\nmore generalized model for the original task we are interested in.6 Nguyen Anh Minh et al.\nOur research effort aims to optimize shared parameters using multi-objective optimization\nmethods. Multi-Task Learning has been successfully applied in computer vision (Bilen and\nVedaldi, 2016; Misra et al., 2016; Kokkinos, 2017; Zamir et al., 2018), natural language\nprocessing (Collobert and Weston, 2008; Dong et al., 2015; Liu et al., 2015a; Luong et al.,\n2015; Hashimoto et al., 2017), and audio processing (Huang et al., 2013; Seltzer and Droppo,\n2013; Huang et al., 2015). It is also referred to by other names such as Joint Learning,\nLearning to Learn, and Learning with Auxiliary Tasks. When we encounter an optimization\nproblem that involves many loss functions, we are really dealing with a problem that is\nconnected to Multi-Task Learning.\nThe MTL problem is represented as a Multi-Objective Programming (MOP) problem,\nusing the approach proposed by Sener et al. [17]. The MTL problem entails the execution of\nmjobs using a loss function represented by a vector. The equation 2.2 represents a multi-task\nlearning problem, where we aim to minimize the loss function L(θ)with respect to the\nparameter set θ. The loss function Li(θ)corresponds to the loss function for task iover all\ntasks. MTL algorithms concurrently optimize many tasks. Problem (2.2) is a multi-objective\noptimization problem, and often, there is no single solution or parameter configuration that\ncan maximize all tasks concurrently. Thus, we can exclusively offer Pareto optimal solutions\nthat illustrate the compromise between several tasks.\n3 Adaptive multi-gradient for solving Multi-objective optimization problem\n3.1 Solving subproblem by KKT condition\nWe will now solve issue (MOP( F,U))when Uis the set of all real n-dimensional vectors.\nThe steepest descent direction s(x)atxis defined as the answer to the optimization problem:\n\u001a\nmin max j=1,...,m∇Fj(x)Ts+1\n2||s||2\nsubject to s∈Rn,(1)\nThe problem (1)possesses a solitary optimal solution due to the strong convexity of the\nfunction ∇Fj(x)Ts+1\n2sTswith respect to the variable sforj=1,..., m. When m=1, the\nsteepest descent path s(x)is equal to the negative gradient of F(x), denoted as −∇F(x).\nThe formulation of problem (1) is based on the idea of approximating the expression\nmax\nj=1,...,mFj(x+s)−Fj(x)\nby employing a first-order Taylor expansion around xfor each function Fj. The authors\nin [8] have suggested a formulation of the optimization problem (1)to determine the steepest\ndescent direction for multi-objective optimization. The most favorable solution for equation\n(1) will be represented as Θ(x). Thus, we have the following equations:\nΘ(x) = inf\ns∈Rnmax\nj=1,...,m∇Fj(x)Ts+1\n2||s||2, (2)\nand\ns(x) =arg min\ns∈Rnmax\nj=1,...,m∇Fj(x)Ts+1\n2||s||2(3)Adaptive multi-gradient methods for quasiconvex vector optimization 7\nDespite the non-smooth nature of issue (1), it may still be classified as a second-order\noptimization problem and effectively solved using the Karush-Kuhn-Tucker (KKT) criteria.\nThe problem (1) may be expressed as the following optimization problem, denoted as ( P1):\n\n\nmin h(t,s) =t\nsubject to ∇Fj(x)Ts+1\n2||s||2−t≤0(1≤j≤m)\n(t,s)∈R×Rn.(P1)\nSo the Lagrangian objective function is\nL((t,s),λ) =t+m\n∑\nj=1λj\u0012\n∇Fj(x)Ts+1\n2||s||2−t\u0013\n.\nThe problem denoted by equation (P1)possesses a solitary optimal solution, denoted as\n(Θ(x),s(x)), due to its convex nature and the fulfillment of Slater’s condition. According\nto the Karush-Kuhn-Tucker (KKT) criteria, there is a KKT multiplier vector λ=λ(x)that\ncorresponds to s=s(x)andt=Θ(x). We can readily calculate the expression:\ns(x) =−m\n∑\nj=1λj(x)∇Fj(x). (4)\nThe most steep descent direction is determined by the weight vector λacquired from\nsolving the KKT system, which converts the multi-objective optimization issue into a single-\nobjective optimization problem. If the regularity criterion is met, the KKT solution for the\nconvex programming problem (P1)corresponds to the solution of both the primal and dual\nproblems. When we combine it with equation (4), we obtain:\nΘ(x) =sup\nλ≥0inf\ns∈RnL((t,s),λ)\n=sup\nλ≥0\n∑λj=1inf\ns∈Rnm\n∑\nj=1λj\u0012\n∇Fj(x)Ts+1\n2sTs\u0013\n=sup\nλ≥0\n∑λj=1inf\ns∈Rn \nm\n∑\nj=1λj(x)∇Fj(x)T.s+1\n2sTs!\n=sup\nλ≥0\n∑λj=1−1\n2∥m\n∑\nj=1λj(x)∇Fj(x)∥2(5)\nIn the following part, we will analyze the attributes of Θ(x)and investigate its relationship\nwith s(x). Moreover, we examine the stationarity of Θ(x). The subsequent lemmas and remark\nare results reported in the publication referenced as [8].\nLemma 2 With x ∈Rn, we have\nΘ(x) =−1\n2s(x)Ts(x).\nLemma 3 Denote s (x)as the solution and Θ(x)as the optimum value of problem (P1).\n1. If x is Pareto critical, then s (x) =0∈RnandΘ(x) =0.8 Nguyen Anh Minh et al.\n2. If x is not Pareto critical, then Θ(x)<0.\n3. The mappings x 7→s(x)and x7→Θ(x)are continuous.\nInstead of precisely solving problem (P1), it is intriguing from an algorithmic standpoint\nto handle approximate answers. If xis not Pareto critical, we define sas an approximate\nsolution to equation ( P1) with a tolerance σ∈]0,1]if\nmax\nj(JF(x)s)j+1\n2∥s∥2≤σΘ(x)≤0,s∈Rn(6)\nwhere Θ(x)is the optimum value of problem ( P1).\nLemma 4 Let’s assume that xis not Pareto critical and that sis an estimated solution of\nequation (P1)with a tolerance value σ∈]0,1]. Then\n∥s(x)∥ ≤2∥JF(x)∥∞,2\nProof. Ifxis not Pareto-critical ⇒Θ(x)<0. Also, Θis an approximate solution of (P1)\nso\nfx(s)+1\n2∥s∥2≤σΘ(x)\n⇒ ∥s∥2≤2(σΘ(x)−fx(s))≤2(−fx(s))\n⩽2(−JF(x)s)j∀j=1,m\n≤2∥JF(x)s∥∞,2\n≤2∥JF(x)∥∞,2∥s∥\n3.2 Compute the adaptive step length\nAssume we have a vector s∈Rnwith JF(x)s<0. To compute a step length α>0we use\nan adaptive rule. Let σbe a constant that is chosen in the interval (0,1). The criterion for\naccepting αis\nF(x+αs)≤F(x)+σαJF(x)s. (7)\nLet us begin with an initial value of αthat lies between 0 and 1. If the condition is not\nmet, we update αby multiplying it with κ, where κis a value between 0 and 1 (inclusive).\nWe then go to the next iteration and obtain a new direction, denoted as s. The finiteness of\nthis technique can be deduced from the fact that equation (7) is strictly valid for sufficiently\ntiny values of α>0.\nLemma 5 [8] If Fis differentiable and JF(x)s<0, then there exists a positive value ε\n(which may depend on x ,s, and σ) such that the inequality\nF(x+ts)<F(x)+σαJF(x)s\nholds for any αin the interval (0,ε].Adaptive multi-gradient methods for quasiconvex vector optimization 9\n3.3 The complete algorithm\nAt each step, for a non-stationary point, we solve the problem (P1)to determine the steepest\ndescent direction using the formula in Lemma 2. We proposed the following algorithm to\nsolve problem (MOP( F,U)) with U=Rn.\nAlgorithm 1 Adaptive multi-gradient\nStep 1 . Setκ∈(0,1],σ∈[0,1],α1∈(0,1), and x1∈Rn. Initialize k:=1.\nStep 2. (Main loop)\n(a) Find the descent direction by solving the following problem:\nmin\nλj||m\n∑\nj=1λk\nj∇Fj(xk)||2\nsubject to λj≥0,m\n∑\nj=1λk\nj=1\nFrom this, we obtain the optimal solution:\nλk= (λk\nj),s(xk) =−\u0010\n∑m\nj=1λk\nj∇Fj(xk)\u0011\nandΘ(xk) =−1\n2||s(xk)||2.\n(b) If Θ(xk) =0, then End. Otherwise, continue to Step 2(c).\n(c) Set xk+1:=xk+αks(xk).\n(d) Compute the step size:\nIf\nF(xk+1)≤F(xk)+σD\nJF(xk),xk+1−xkE\n(8)\nthenαk+1=αk, otherwise set αk+1:=καk.\nStep 3. Setk:=k+1 and go to Step 2.\n3.4 Convergence of adaptive multi-gradient methods\nThe convergence analysis of the proposed algorithm is based on quasi-Fejer convergence.\nWe recall that a sequence\b\nzk\t\n⊂Rnis said to be quasi-Fej ´er convergent to a set U,U̸=/ 0, if\nand only if for each z∈U, there exists a sequence {εk}⊂R+such that ∑+∞\nk=1εk<+∞and\n\r\r\rzk+1−z\r\r\r2\n≤\r\r\rzk−z\r\r\r2\n+εk.\nTheorem 1 Assume that the sequence\b\nxk\t\nis generated by Algorithm 1. Then, each limit\npoint (if any) of the sequence\b\nxk\t\nis a Pareto stationary point of the problem (MOP( F,U))\nwhere U =Rn.\nProof. Letybe an accumulation point of the sequence\u0000\nxk\u0001\nand let s(y)andΘ(y)be the\nsolution and the optimum value of ( P1) aty.\nAccording to Lemma 3 it is enough to prove that Θ(y) =0. Clearly, the sequence F\u0000\nxk\u0001\nis componentwise strictly decreasing and we have\nlim\nk→∞F\u0010\nxk\u0011\n=F(y)\nTherefore,\nlim\nk→∞\r\r\rF\u0010\nxk\u0011\n−F\u0010\nxk+1\u0011\r\r\r=0.10 Nguyen Anh Minh et al.\nLet’s consider the sequence {kp}={k1,k2,...}with p→∞, satisfying the inequality\n(15). In that case, the value of αkpwill remain unchanged.\nCase 1: {kp}is finite.\nIf the sequence {kp}is finite with sufficiently large p, there exists a step Konwards\nwhere the inequality (15) is not satisfied. In other words,\nFj\u0000\nxK+1\u0001\n≥Fj(xK)+σαK⟨JF(xK),s(xK)⟩j\nfor at least one j∈ {1,..., m}. According to Proposition 5, we have max j⟨JF(xK),s(xK)⟩ ≥0.\nCombining this with (6), we obtain Θ(xK) =0, which implies that xKis a Pareto stationary\npoint.\nCase 2: {kp}is infinite.\nIn the infinite sequence {kp}, we will always have\nF\u0010\nxk\u0011\n−F\u0010\nxk+1\u0011\n≥ −αkσ⟨JF(xk),s(xk)⟩ ≥0,\nfor all k∈ {kp}and therefore\nlim\nk→∞αkσ⟨JF(xk),s(xk)⟩=0. (9)\nObserve that αk∈]0,1]for all k∈ {kp}. Now take a subsequence\u0000\nxku\u0001\nuconverging to y. We\nwill consider two possibilities:\nlimsup\nu→∞αku>0\nand\nlimsup\nu→∞αku=0.\nCase 2.1 In this case, there exist a subsequence\u0000\nxkℓ\u0001\nℓconverging to yand satisfying\nlim\nℓ→∞αkℓ=¯α>0\nUsing (9) we conclude that\nlim\nℓ→∞JF\u0010\nxkℓ\u0011\nskℓ=0\nwhich also implies\nlim\nℓ→∞Θ\u0010\nxkℓ\u0011\n=0\nSince x7→Θ(x)is continuous, we conclude that Θ(y) =0, so yis Pareto critical.\nCase 2.2 Due to the results in Lemma 4 we know that the sequence\u0000\nsku\u0001\nuis bounded.\nTherefore we can take a subsequence\u0000\nxkr\u0001\nrof\u0000\nxku\u0001\nusuch that the sequence\u0000\nskr\u0001\nralso\nconverges to some ¯ s. Note that for all rwe have\nmax\ni\u0010\nJF\u0010\nxkr\u0011\nskr\u0011\ni≤σΘ\u0010\nxkr\u0011\n<0\nTherefore, passing onto the limit r→∞we get\n1\nσmax\ni(JF(y)¯s)i≤Θ(y)≤0 (10)\nTake some q∈N. For rlarge enough,\nαkr<αq\n1Adaptive multi-gradient methods for quasiconvex vector optimization 11\nwhich means that the Adative step size (15) is not satisfied for α=αq\n1, i.e.,\nJF\u0010\nxkr+\u0000\nαq\n1\u0001\nskr\u0011\n≰F\u0010\nxkr\u0011\n+σ\u0000\nαq\n1\u0001\nJF\u0010\nxkr\u0011\nskr\n(forrlarge enough). Passing onto the limit r→∞(along a suitable subsequence, if necessary)\nwe get\nFj\u0000\ny+\u0000\nαq\n1\u0001\n¯s\u0001\n≥Fj(y)+σ\u0000\nαq\n1\u0001\n(JF(y)¯s)j\nfor at least one j∈ {1,..., m}. Note that this inequality holds for any q∈N. From Lemma 5\nit follows that\nmax\ni(JF(y)¯s)i≥0\nwhich, together with (10), implies Θ(y) =0. So, again we conclude that yis Pareto critical.\nNow, we are interested in the study of the convergence properties of the proposed\nalgorithm when the objective function is quasiconvex. Define\nT=n\nx∈Rn:F(x)≤F\u0010\nxk\u0011\n,∀ko\n.\nProposition 2 [31] If Fis quasiconvex and x∈Rnthen,\n∇Fi\u0000\nxk\u0001\n,x−xk\u000b\n≤0for all k\nand all i =1,..., m.\nTheorem 2 Assume that Fis a quasiconvex function on Rn. IfT̸=/ 0then{xk}is generated\nby Algorithm 1 converges to a stationary point.\nProof. Based on the proof of Theorem 1, we again consider the sequence {kp}={k1,k2,...}\nwith p→∞, satisfying the inequality (15).\nCase 1: {kp}is finite.\nDue to the finiteness of the sequence {kp}, we can prove the existence of a Pareto critical\npoint xKafter a finite number of steps. Therefore when Fis a quasiconvex function on Rn,\n{xk}converges to a stationary point.\nCase 2: {kp}is infinite.\nAfter some simple algebra, for each x∈Rn, we obtain\ndk:=\r\r\rxk−x\r\r\r2\n−\r\r\rxk+1−x\r\r\r2\n+\r\r\rxk+1−xk\r\r\r2\n=2D\nxk+1−xk,x−xkE\n.\nUsing the update formula xk+1=xk+αksk, it follows that\ndk=2αkD\nsk,x−xkE\n=2αkD\nsk,x−xk−sk+skE\n=2αkD\nsk,x−xk−skE\n+2αk\r\r\rsk\r\r\r2\n.\nSo we get\ndk−2αk\r\r\rsk\r\r\r2\n=2αk*\n−m\n∑\nj=1λk\nj∇Fj\u0010\nxk\u0011\n,x−xk−sk+12 Nguyen Anh Minh et al.\nTherefore,\n\r\r\rxk−x\r\r\r2\n−\r\r\rxk+1−x\r\r\r2\n+αk(αk−2)\r\r\rsk\r\r\r2\n=dk−2αk\r\r\rsk\r\r\r2\n=2αk *\nm\n∑\nj=1λk\nj∇Fj\u0010\nxk\u0011\n,xk−x+\n+*\nm\n∑\nj=1λk\nj∇Fj\u0010\nxk\u0011\n,sk+!\n.\nRewriting this last equality and using that F\u0000\nxk+1\u0001\n≤F\u0000\nxk\u0001\n+σαkJF\u0000\nxk\u0001\nskwithαk<1, it\nfollows that\n\r\r\rxk+1−x\r\r\r2\n≤\r\r\rxk−x\r\r\r2\n+2αk*\nm\n∑\nj=1λk\nj∇Fj\u0010\nxk\u0011\n,x−xk+\n+2\nσm\n∑\nj=1λk\nj\u0010\nFj\u0010\nxk\u0011\n−Fj\u0010\nxk+1\u0011\u0011\n.\nUsing that F\u0000\nxk+1\u0001\n≤F\u0000\nxk\u0001\nand 1≥λk\nj≥0 for all j=1,..., m, we obtain\nm\n∑\nj=1\u0010\nFj\u0010\nxk\u0011\n−Fj\u0010\nxk+1\u0011\u0011\n≥m\n∑\nj=1λk\nj\u0010\nFj\u0010\nxk\u0011\n−Fj\u0010\nxk+1\u0011\u0011\nSo we have for all x∈Rnand each k, there existsn\nλk\njom\nj=1⊂[0,1]satisfying ∑m\nj=1λk\nj=\n1, and\n\r\r\rxk+1−x\r\r\r2\n≤\r\r\rxk−x\r\r\r2\n+2αk*\nm\n∑\nj=1λk\nj∇Fj\u0010\nxk\u0011\n,x−xk+\n+2\nσm\n∑\nj=1\u0010\nFj\u0010\nxk\u0011\n−Fj\u0010\nxk+1\u0011\u0011\n.\n(11)\nfork∈ {kp}. We have the sequence\b\nF\u0000\nxk\u0001\t\nis strictly monotone decreasing. Therefore\nεk:=∑m\nj=1\u0002\nFj\u0000\nxk\u0001\n−Fj\u0000\nxk+1\u0001\u0003\nis positive for all k. It follows from Proposition 2 that\n\r\r\rxk+1−x\r\r\r2\n≤\r\r\rxk−x\r\r\r2\n+2\nσεk\nObserve that the series ∑∞\nk=1εkis convergent. Indeed, εk>0 for all kand\nℓ\n∑\nk=0εk=ℓ\n∑\nk=0 \nm\n∑\ni=1\u0010\nFi\u0010\nxk\u0011\n−Fi\u0010\nxk+1\u0011\u0011!\n=m\n∑\ni=1 \nℓ\n∑\nk=0\u0010\nFi\u0010\nxk\u0011\n−Fi\u0010\nxk+1\u0011\u0011!\n=m\n∑\ni=1\u0010\nFi\u0000\nx0\u0001\n−Fi\u0010\nxℓ+1\u0011\u0011\n≤m\n∑\ni=1\u0000\nFi\u0000\nx0\u0001\n−Fi(x)\u0001\n.\nTherefore,\b\nxk\t\nis quasi-Fej ´er convergent to T. Using result on quasi-Fej ´er convergent\nLemma 1 in [31], the sequence\b\nxk\t\nis bounded. Let ¯xbe an accumulation point of\b\nxk\t\n.\nThe sequence\b\nF\u0000\nxk\u0001\t\nis monotone decreasing, which implies that ¯x∈T. Using Lemma 1\nin [31] we have\b\nxk\t\nis convergent to ¯ x, which is stationary by Theorem 1.\nTheorem 3 Assume that Fis a pseudoconvex function on Rn. IfT̸=/ 0, then{xk}generated\nby Algorithm 1 converges to a weakly efficient point.\nProof By [19], pseudoconvexity implies quasiconvexity. Then, using Theorem 2, we obtain\nthat{xk}converges to a stationary point, which is a weakly efficient point.\nRemark 1 By using Remark 2 in [8], we have the (15) criterion holds for 0≤λ≤2(1−σ)/L.\nAs a result, if the value of the Lipschitz constant Lis already known, we can choose the\nconstant step-size λ∈(0,2/L)to solve problem MOP( F,U).Adaptive multi-gradient methods for quasiconvex vector optimization 13\n4 Adaptive multi-gradient with preference vectors\nSolving multi-objective problems using scalarization approaches tends to concentrate the\nobtained Pareto optimal points in a specific region of the Pareto front. In order to evenly\ndistribute the Pareto optimal points across the entire Pareto front, we need to apply additional\nlinear constraints to help partition the image space. On each partitioned image space, we will\nfind a corresponding Pareto optimal solution.\nThis paper [27] has a similar idea. It looks at a group of Kpreference vectors {u1,u2,...,uK}∈\nRm\n+. We divide the image space into subspaces Ωk(k=1,..., K)using the conditions below:\nΩk=\b\nv∈Rm\n+| ⟨ui,v⟩ ≤ ⟨uk,v⟩,∀i=1,..., K\t\n.\nThe main idea of preference vectors is to decompose a multi-objective problem into several\nconstrained multi-objective subproblems with different trade-off preferences among the tasks\nin the original problem. By solving these subproblems in In parallel, the proposed algorithm\ncan obtain a set of well-representative solutions with different trade-offs. Next, we solve the\nconstrained optimization problem on each region Ωkby combining the inequality constraints\nGpto aid in the space partitioning. We need to solve the following problem:\nmin\nx∈RnF(x) = ( F1(x),F2(x),···,Fm(x))\nsubject to Gp(x) =⟨up−uk,F(x)⟩ ≤0,∀p=1,..., K,(12)\nIt is necessary to sample the preference vectors u1,..., uKto cover the objective space evenly\nto make HV and cosine similarity interact well. This is quite simple in 2D space using K\nvectors\b\ncos\u0000kπ\n2K\u0001\n,sin\u0000kπ\n2K\u0001\t\n.to partition the object space into Ksubregion Ωk,k=1,...K\nand randomly sample the vector ukof subregion Ωk. However, how do you take a partitioned\nrandom sample in any dimensional Jspace? A good partition must satisfy:\n∪J\nk=1(Ωk) =SJandΩk′∩k′̸=kΩk=/ 0\nWithSJ=\b\nλ∈RJ\n>0:∑jλj=1\t\nis the set feasible values of random variable u. Based\non [51], we define subregions Ωiby points u= (u1,..., uK)∈U⊂SJsuch that:\nu1∈ {0,δ,2δ,...,..., 1}s.t1\nδ=n∈N∗\nIf 1<k<J−1,mk=δ\nuk,1≤k<k′−1, we have:\nuk′∈(\n0,δ,..., \nK−k′−1\n∑\nk=1mk!\nδ)\n,uJ=1−J−1\n∑\nk=1uk\nUsing the Delaunay triangulation algorithm [50] for these points, we obtain the required\npartition. Figure 1 is the illustrative result of the algorithm in 3D space.14 Nguyen Anh Minh et al.\nFig. 1 Partitioning algorithm with J=3,δ=0.2\n4.1 The update algorithm\nRemark 2 We propose the following approach to find an acceptable initialization point:\n– Step 1: Randomly initialize a point x0∈Rn,ηr∈(0,1).\n– Step 2: Ifx0is feasible, stop. Otherwise, proceed to Step 3 .\n– Step 3: We can find a descent direction sras follows:\n(sr,αr) =arg min\ns∈Rn,α∈Rα+1\n2∥s∥2,subject to ∇Gp(x0)Ts≤α\n– Step 4: Update x0=x0+ηrsrand return to Step 2 .\nOnce an acceptable initialization point is found, we proceed to the proposed algorithm\nwith preference vectors. We can find a descent direction to solve (12) applying a similar\nschema as in problem P1.\n\n\nmin g(t,s) =t\nsubject to ∇Fj(x)Ts+1\n2sTs−t≤0(1≤j≤m)\n∇Gp(x)⊤s+1\n2sTs−t≤0(p∈Iε(x))\n(t,s)∈R×Rn.(P2)\nBy apply KKT condition to solve problem P2, we have the fomula of descent direction is:\ns(xk) =−\nm\n∑\nj=1λk\nj∇Fj(xk)+∑\np∈Iε(xk)γk\np∇Gp(xk)\n, (13)\nwhere Iε(xk) ={p|Gp(xk)≥ −ε,p=1,..., K}. So we have the following proposed algorithm\nwith preference vectors.Adaptive multi-gradient methods for quasiconvex vector optimization 15\nAlgorithm 2 Adaptive multi-gradient with preference vectors\nStep 1 . Setκ∈[0,1],σ∈[0,1],α1∈(0,1), and x1\nr∈Rn. Initialize k:=1.\nStep 2. Obtain an acceptable starting point x1from x1\nr.\nStep 3. (Main loop)\n(a) Find the descent direction by solving the following problem:\nmin\nλj,γp||m\n∑\nj=1λk\nj∇Fj(xk)+∑\np∈Iε(xk)γk\np∇Gp(xk)||2\nsubject to λj≥0,γp≥0,m\n∑\nj=1λk\nj+∑\np∈Iε(xk)γk\np=1(14)\nWith, Iε(xk):={p∈1,..., K|Gp(xk)≥ −ε}.\nSo we can obtain the optimal solutions:\nλk= (λk\nj),γk= (γk\np),s(xk) =−\u0010\n∑m\nj=1λk\nj∇Fj(xk)+∑p∈Iε(x)γk\np∇Gp(xk)\u0011\nandΘ(xk) =−1\n2||s(xk)||2.\n(b) If Θ(xk) =0 then End. Otherwise, continue to Step 3(c)\n(c) Set xk+1:=xk+αks(xk).\n(d) Compute the step size:\nIf\nF(xk+1)≤F(xk)+σD\nJF(xk),xk+1−xkE\n(15)\nthenαk+1=αk, otherwise set αk+1:=καk.\nStep 4. Setk:=k+1 and go to Step 3.\n4.2 Convergence of adaptive multi-gradient with preference vector\nIn this subsection, we show that the Algorithm 2 can be reformulated as a linear scalarization\nof tasks with adaptive weight assignment. We first tackle the unconstrained case. Suppose we\ndo not decompose the multi-objective problem and hence remove all constraints from the\nproblem 12, it will immediately reduce to the update rule in Lemma 2. It is straightforward\nto rewrite the corresponding MOP( F,U) with U=Rninto a linear scalarization form:\nF(xk) =m\n∑\nj=1λk\njFj(xk) (16)\nwhere we adaptively assign the weights λiby solving the following problem in each iteration:\nmin\nλj1\n2\r\r\r\r\rm\n∑\nj=1λj∇Fj\u0010\nxk\u0011\r\r\r\r\r2\n,s.t.m\n∑\nj=1λj=1,λj≥0,∀j=1,..., m.\nIn the constrained case, we have extra constraint terms Gp\u0000\nxk\u0001\n. IfGp\u0000\nxk\u0001\nis inactivated,\nwe can ignore it. For an activated Gp\u0000\nxk\u0001\n, assuming the corresponding reference vector is ui,\nwe have:\n∇Gp\u0010\nxk\u0011\n= (up−ui)T∇F\u0010\nxk\u0011\n=m\n∑\nj=1(up j−ui j)∇Fj\u0010\nxk\u0011\n.\nSince the gradient direction s(xk)can be written as a linear combination of all ∇Fj\u0000\nxk\u0001\nand∇Gp\u0000\nxk\u0001\nas in (13), the general Pareto MTL algorithm can be rewritten as:\nF(xk) =m\n∑\nj=1αjFj\u0010\nxk\u0011\n,where αj=λj+∑\np∈Iε(xk)γp(up j−ui j), (17)16 Nguyen Anh Minh et al.\nwhere λjandγpare obtained by solving problem (14) with assigned reference vector ui.\nSo basically, we can view the problem of solving (12) as simultaneously solving multiple\nunconstrained multi-objective optimization subproblems. The scalarization form (16) of the\nmulti-objective problem MOP( F,U) is transformed into (17) with the same functions Fj,\ndiffering only in the scalarization coefficients αj. So similar to the proof above, we have\nsome following theorems\nTheorem 4 Assume that the sequence\b\nxk\t\nis generated by Algorithm 2. Then, each limit\npoint (if any) of the sequence\b\nxk\t\nis a Pareto stationary point of the problem (MOP( F,U))\nwhere U =Rn. Define T =\b\nx∈Rn:F(x)≤F\u0000\nxk\u0001\n,∀k\t\n.\n•ifFis quasiconvex on RnandT̸=/ 0, then the sequence\b\nxk\t\nconverges to a Pareto\nstationary point of the problem.\n•ifFis pseudoconvex on RnandT̸=/ 0, then the sequence\b\nxk\t\nconverges to a weakly\nefficient solution of the problem.\n5 Numerical experiments\nThe configuration of hyperparameters will be presented in each example. In this paper, we\nwill experiment with the proposed algorithm using baselines and metrics as follows:\n– Baselines: We compare Algorithm 2 with the PMTL algorithm [27].\n–Evaluation metric: The area dominated by Pareto front is known as Hypervolume [22].\nThe higher Hypervolume, the better Pareto front.\nTo fairly compare the experimental results of the algorithms, we use simple preference\nvectors [cos(kπ\n2K),sin(kπ\n2K)]with Kis number of initial points to evenly divide the 2D space\nand employ the Delaunay triangulation algorithm to generate preference vectors for the case\nof 3D space.\nExample 1 Consider a convex multi-objective optimization problem without constraints:\nMin F(x) =(\n1\n25x2\n1+1\n100\u0012\nx2−9\n2\u00132\n,1\n25x2\n2+1\n100\u0012\nx1−9\n2\u00132)\nsubject to x∈Rn\nThe results obtained by using Algorithm 1 and Algorithm 2 to solve Example 1 with 10\ninitialization points are as follows:Adaptive multi-gradient methods for quasiconvex vector optimization 17\nFig. 2 Result of Algo 1\n Fig. 3 Result of Algorithm 2\nFor Algorithm 1, we can see that the obtained Pareto solutions are concentrated in a\nregion on the Pareto Front. This drawback has been overcome by using constraints to divide\nthe image space in Algorithm 2.\nTo compare the effectiveness of the proposed algorithm with the PMTL algorithm, we\nconducted experiments by randomly initializing points to find solutions for Example 1. We\nhave the following result:\nInitial\nPointIteration\nNumberTest case\nnumberA VG Time\nPMTLA VG Time\nProposed AlgoA VG HV\nPMTLA VG Proposed\nAlgo\n50 450 8 1064.64 915.49 0.99 1.01\n40 1000 7 629.65 474.18 1.02 1.03\n40 500 10 250.18 230.21 1.01 1.02\nWe can see that the proposed algorithm has a faster running time compared to the PMTL\nalgorithm and the obtained solutions converge better to the Pareto Front compared to the\nPMTL algorithm. To compare the superiority of the optimization time evenly spread on the\nPareto Front, we ran Algorithm 2 with fewer iterations than the PMTL algorithm and take\naverage results after 10 times.. The results are as follows:\nNumber of\nInitial PointsPMTL Algorithm Algorithm 2\nAverage HV Time (s) Iterations Average HV Time (s) Iterations\n10 0.97 3.31 200 0.97 2.59 150\n20 0.96 15.73 200 0.96 12.97 150\n30 0.95 77.04 300 0.96 55.07 200\n40 0.95 251.4 300 0.95 174.13 200\n50 0.94 835.25 300 0.95 564.18 200\nTable 1 Comparison results between Algorithm 2 and the PMTL algorithm for solving Example 1 in various\nscenarios.18 Nguyen Anh Minh et al.\nThe obtained results show that with fewer iterations, the proposed Algorithm 2 still\nachieves an average HV no worse than the PMTL algorithm with more iterations. Conse-\nquently, the running speed of Algorithm 2 is faster than the PMTL algorithm.\nExample 2 We consider the following non-convex constrained multi-objective optimization\nproblem:\nMin F(x) ={2x2\n1+x2\n2+3\n1+2x1+8x2,x2\n1+2x2\n2+3\n1+8x1+2x2}\nsubject to x∈R2\nWe perform the experiment with Algorithm 1 and Algorithm 2 using 10 random initial points,\nand we obtain the following results:\nFig. 4 Result of solving Problem 4 of Algorithm 1\n Fig. 5 Result of solving Problem 4 of Algorithm 2\nWe can see that both algorithms are able to find solutions on the Pareto Front, but with\nthe linear constraints that help divide the image space, Algorithm 2 finds solutions that are\nmore evenly spread out on the Pareto Front. To compare the effectiveness of the proposed\nalgorithm with the PMTL algorithm, we conducted experiments by randomly initializing\npoints to find solutions for Example 2. We have the following result:\nInitial\nPointIteration\nNumberTest case\nnumberA VG Time\nPMTLA VG Time\nProposed AlgoA VG HV\nPMTLA VG Proposed\nAlgo\n35 1000 10 110.04 94.53 1.14 1.14\n40 1000 5 152.53 131.01 1.14 1.14\nWe can see that the proposed algorithm has a faster running time compared to the PMTL\nalgorithm and the obtained solutions converge better to the Pareto Front compared to the\nPMTL algorithm. To compare the superiority of the optimization time evenly spread on the\nPareto Front, we ran Algorithm 2 with fewer iterations than the PMTL algorithm and take\naverage results after 10 times. The results are as follows:Adaptive multi-gradient methods for quasiconvex vector optimization 19\nNumber of\nInitial PointsPMTL Algorithm Algorithm 2\nAverage HV Time (s) Iterations Average HV Time (s) Iterations\n10 1.12 9.65 500 1.12 5.99 400\n20 1.13 21.28 500 1.13 11.87 300\n30 1.14 50.97 500 1.14 48.74 300\nTable 2 Comparison results between Algorithm 2 and the PMTL algorithm for solving Example 2 in various\nscenarios.\nThe obtained results show that with fewer iterations, the proposed Algorithm 2 still\nachieves an average HV no worse than the PMTL algorithm with more iterations. Conse-\nquently, the running speed of Algorithm 2 is faster than the PMTL algorithm.\nExample 3 (Toy example (Lin et al [3])) Consider the non-convex unconstrained multi-\nobjective optimization problem:\nMin F(x) =\u001a\n1−exp−∑d\ni=1(xi−1\nd)2\n,1−exp−∑d\ni=1(xi+1\nd)2\u001b\nsubject to x∈R2\nWe conducted experiments with d=20. We apply Algorithm 2 to solve the above problem\nwith 10 initial data points, and we obtain the following results:\nFig. 6 Result of Algo 1\n Fig. 7 Result of Algo 2\nThe solutions obtained by Algorithm 1 concentrate in a region on the Pareto Front, while\nusing Algorithm 2, we obtain solutions that are all on the Pareto Front and spread evenly\nacross the surface.\nFirstly, we compare our proposed Algorithm 2 with MGDA [35] algorithm\nInitial\nPointIteration\nNumberTest case\nnumberA VG HV\nPMTLA VG Proposed\nAlgo\n50 100 10 0.85 1.52\n40 200 13 0.87 1.49\n45 150 15 0.88 1.5320 Nguyen Anh Minh et al.\nThe HV achieved by the proposed algorithm is significantly higher. Utilizing the proposed\nalgorithm will help distribute the obtained solutions evenly across the Pareto front and yield\nhigher HV . Secondly, to compare the effectiveness of the proposed algorithm with the PMTL\nalgorithm, we conducted experiments by randomly initializing points to find solutions for\nExample 3. We measured the solving time of both algorithms and the Hyper V olume value of\nthe Pareto Front obtained by each algorithm. The results obtained are as follows\nInitial\nPointIteration\nNumberTest case\nnumberA VG Time\nPMTLA VG Time\nProposed AlgoA VG HV\nPMTLA VG Proposed\nAlgo\n50 200 40 264.74 224.06 1.48 1.53\n50 300 10 245.86 181.84 1.50 1.53\n60 200 10 259.52 236.51 1.47 1.51\nWe can see that the proposed algorithm has a faster running time than the PMTL algorithm\nand at the same time, the obtained solutions converge better to the Pareto Front than the\nPMTL algorithm. To compare the superiority of solving evenly-distributed Pareto Fronts\nwith respect to optimization time, we executed Algorithm 2 with fewer iterations than PMTL.\nThe results demonstrate that with fewer iterations, Algorithm 2 still achieves an average HV\nno worse than PMTL with more iterations. Specifically:\nNumber of\nInitial PointsPMTL Algorithm Algorithm 2\nAverage HV Time (s) Iterations Average HV Time (s) Iterations\n10 1.49 2.01 100 1.49 1.37 50\n20 1.53 4.38 100 1.53 2.45 80\n30 1.49 17.97 100 1.51 10.79 80\n40 1.49 54.01 100 1.49 35.73 75\n50 1.44 99.42 100 1.47 67.67 75\nTable 3 Comparison results between Algorithm 2 and the PMTL algorithm in various scenarios.\nExample 4 (Toy example (Lin et al [3])) Consider a three-dimensional multi-objective opti-\nmization example with the following non-convex functions:\nMin F(x) =\u001a\n1−exp−∑d\ni=1(xi−1\nd)2\n,1−exp−∑d\ni=1(xi+1\nd)2\n,\n2−exp−∑d\ni=1(xi−1\nd)2\n−exp−∑d\ni=1(xi+1\nd)2\u001b\ns.tx∈Rd\nWe proceed to solve Example 4 with dimensionality d=20. Applying Algorithm 2 to\nsolve the example with 10 initial points, we obtain the following results:Adaptive multi-gradient methods for quasiconvex vector optimization 21\nFig. 8 Results of Algorithm 2 solving Example 4\nThe Pareto solutions found by Algorithm 2 are evenly distributed on the Pareto Front.\nIn summary, the solutions found by the proposed algorithm converge better to the Pareto\nFront than PMTL. The proposed algorithm can be seen as an improved version of PMTL\nwith the ability to search for Pareto solutions more evenly and faster on unconstrained\nmulti-objective optimization problems.\n6 Applications to multi-task learning\n6.1 Proposed Algorithm\nSimilarly to [3], we will use Kpreference vectors u1,u2,..., uKto represent the trade-off\npreferences among tasks in the MTL problem. Additionally with θare model’s parameters,\nwe incorporate linear inequality constraints Gp(θt) =⟨up−uk,L(θt)⟩ ≤0,∀p=1,..., K\nto partition the image space into subproblems. Then, we propose using an adaptive step size\nto update the learning rate of the Neural network. We need to solve the following problem\nmin\nθL(θ) = (L1(θ),L2(θ),···,Lm(θ))T\ns.t.Gj(θt) = (uj−uk)TL(θt)≤0,∀j=1,..., K,(P3)\nBy applying KKT condition and similar to transformation in Section 4.2, we get:\ns(θt) =− \nm\n∑\nj=1λk\nj∇Lj(θt)+∑\np∈I(θt)γk\np∇Gp(θt)!\nL(θt) =m\n∑\nj=1αjLj(θt),where αj=λj+∑\np∈Iε(θt)γp(up j−ui j)\nwhere Iε(θt) ={p|Gp(xk)≥ −ε,p=1,..., K}. The detailed algorithm for MTL is presented\nbelow.22 Nguyen Anh Minh et al.\nAlgorithm 3 Proposed Algorithm for Solving MTL Problem\nInput: Set of priority vectors {u1,u2,..., uK}.\n1:fork=1 toKdo\n2: Set κ∈[0,1],σ∈[0,1],α1∈(0,+∞).\n3: Initialize parameter set θk\nrfor the Neural network.\n4: Find effective parameters θk\n0from θk\nrusing the descent method.\n5: fort=1 toTdo\n6: Find the descent direction by solving the following problem:\nmin\nλj,γp||m\n∑\nj=1λk\nj∇Lj(θk\nt)T+∑\np∈I(θkt)γk\np∇Gp(θk\nt)||2\ns.t.λj≥0,γp≥0,m\n∑\nj=1λk\nj+∑\np∈I(θkt)γk\np=1\n7: Where Iε(θk\nt):={p∈1,..., K|Gp(θk\nt)≥ −ε}.\n8: Determine the descent direction:\ns(θk\nt) =−\nm\n∑\nj=1λk\nj∇Lj(θk\nt)+∑\np∈I(θkt)γk\np∇Gp(θk\nt)\n\n9: Set θk\nt+1:=θk\nt+αts(θk\nt).\n10: Compute step size:\n11: If L(θk\nt+1)≤L(θk\nt)+σ\ns(θk\nt),θk\nt+1−θk\nt\u000b\n12: then αt+1=αt, otherwise set αt+1:=καt.\n13: Update iteration step t:=t+1.\n14: end for\n15:end for\nOutput: The set of solutions for all subproblems with different trade-offs {θk\nT|k=1,..., K}.\n6.2 Applications to Computer Vision\nIn this section, we will implement the proposed algorithm and evaluate its effectiveness on a\ntwo-task image classification problem using the Multi-MNIST dataset provided at [3]. At\nthe same time, we will compare the effectiveness of the proposed algorithm with the PMTL\nalgorithm.\nMulti-MNIST is a dataset constructed by combining two digits from the MNIST dataset\n[34] into one image. In the MTL problem of image classification, the model simultaneously\nlearns two loss functions to predict two digits, one in the top-left corner and one in the\nbottom-right corner, nested within each other. The training set consists of 120,000 images,\nand the test set consists of 20,000 images. An illustrative image of the dataset is shown below:\nAdaptive multi-gradient methods for quasiconvex vector optimization 23\nWe construct a LeNet model similar to the one in [34]. Our two tasks are image classifi-\ncation tasks, so the objective or loss function in our model is represented as Cross-entropy\nloss:\nL1(ˆy1,y1) =−K\n∑\nky(k)\n1log ˆy(k)\n1\nL2(ˆy2,y2) =−K\n∑\nky(k)\n2log ˆy(k)\n2\nHere, y(k)\ni,(i=1,2)takes on values of 0 or 1, representing the correct label kthat has been\ncorrectly classified, and ˆy(k)\ni,(i=1,2)are the predicted label values of the model. We need to\nsimultaneously learn both Cross-entropy loss functions. We carry out the training process with\n10 reference vectors [cos(kπ\n2K),sin(kπ\n2K)],k=1,2,..., 10for every 100 epochs. We illustrate\nthe results with the reference vector u1= (1,0)as follows:\nFig. 9 The test result\nThe experimental results show that both image classification tasks are learned simulta-\nneously and relatively effectively. The table comparing the accuracy values on the Test set\nbetween the proposed algorithm and PMTL will be illustrated below:\nAverage Accuracy\nTask 1Average Accuracy\nTask 2\nPMTL 0.87±0.01 0.85±0.01\nOurs 0.88±0.01 0.86±0.01\nWe can see that the effectiveness of the proposed algorithm is quite similar to the PMTL\nalgorithm. The comparison of the training process between the proposed algorithm (blue line\nfor task 1 and black line for task 2) and the PMTL algorithm (red line for task 1 and green\nline for task 2) for the reference vector u1is shown in the following image:24 Nguyen Anh Minh et al.\nFig. 10 Compare the proposed algorithm and PMTL\nThe proposed algorithm and PMTL algorithm are two multi-task learning methods for\nimage classification tasks. The experimental results show that both algorithms can learn\ntwo tasks simultaneously with relatively similar effectiveness, as seen in the comparison\nof accuracy values on the Test set. However, in terms of the training process, the proposed\nalgorithm has an advantage over PMTL in that it uses an Adaptive step size, which leads\nto a more consistent decrease in the loss function and therefore a continuous increase in\naccuracy during training. Meanwhile, PMTL uses a fixed step size, which may lead to slower\nconvergence or oscillation during training. In summary, while both algorithms can achieve\nsimilar performance in multi-task learning for image classification, the proposed algorithm\nhas the advantage of a more stable and consistent training process thanks to its Adaptive step\nsize.\n6.3 Application to Natural Language Processing\nOur inquiry focused exclusively on the Drug Review dataset, as outlined in the study con-\nducted by Graßer et al. in 2018. The dataset consists of user assessments of specific prescrip-\ntions, information about relevant diseases, and a user rating indicating overall happiness. We\nanalyze two tasks: (1) predicting the drug’s rating using regression and (2) classifying the\npatient’s state. The dataset comprises 215,063 samples. 90% of the data is utilized, while\nsituations lacking adequate user feedback are eliminated. Next, there are 100 condition labels\nand 188155 samples.\nAfter preprocessing the customer reviews, we proceed to embed the reviews using the\nGLOVE method [52]. This conversion transforms the words and sentences into numerical\nmatrix space. Subsequently, we pass this embedded matrix through a TextCNN modelAdaptive multi-gradient methods for quasiconvex vector optimization 25\n[53]. We construct the loss function using two Cross-Entropy functions for the multi-label\nclassification problem:\nL1(ˆy1,y1) =−K\n∑\nky(k)\n1log ˆy(k)\n1\nL2(ˆy2,y2) =−K\n∑\nky(k)\n2log ˆy(k)\n2\nHere y(k)\ni,(i=1,2)takes values of 0 or 1, representing whether label kis correctly classified,\nand ˆy(k)\ni,(i=1,2)are the predicted label values by the model.\nWe proceed to train the model using Algorithm 3 with 5 preferrence vectors [cos(kπ\n2K),sin(kπ\n2K)],k=\n1,2,..., 5where each priority vector goes through 50 epochs. The hyperparameters, after\ntuning, are set to σ=0.05,κ=0.95and learning rate lr=0.001. The Train/Test ratio is\n3 : 1. The results of the training process with loss function metrics and accuracy on the Test\nset are as follows:\nFig. 11 Training progress of Algorithm 3 on the Drug Review dataset.\nWe observe that the loss function values decrease steadily for both the Train and Test sets,\nand the accuracy consistently increases. This indicates that the model effectively learns both\ntasks simultaneously. To compare the performance of the proposed algorithm with the PMTL\nalgorithm, keeping lr=0.0001 (the locally optimal hyperparameter after tuning) unchanged\nfor a fair evaluation, we obtain the following results:\nFig. 12 Comparison of Algorithm 3 and PMTL when run on Drug Review dataset26 Nguyen Anh Minh et al.\nThe results show that after only 20 epochs, the accuracy of Algorithm 3 is significantly\nhigher than that of the PMTL algorithm. This is the result of appropriately adjusting the\nlearning rate during training based on the adaptive update condition. We then measure the\naverage accuracy results of the two PMTL algorithms and Algorithm 3 over 10 different\nruns with the same set of initialization parameters, and the obtained results are as follows:\nFig. 13 Comparison of the Pareto Fronts between\nPMTL and Algorithm 3 on the Drug Review\ndataset.Average\nAccuracy Task 1Average\nAccuracy Task 2\nPMTL 0.82±0.01 0.83±0.01\nProposed 0.84±0.01 0.86±0.01\nTable 4 Comparison between Algorithm 3 and PMTL\nwhen run on the Drug Review dataset.\nThe results show the effectiveness of Algorithm 3 compared to PMTL. The Pareto Front\nfound by Algorithm 3 is significantly superior to the Pareto Front found by PMTL.\n7 Conclusion\nWe proposed a novel easy adaptive step-size process in a wide family of solution methods for\noptimization problems with non-convex objective functions. This approach does not need any\nline-searching or prior knowledge, but rather takes into consideration the iteration sequence’s\nbehavior. As a result, as compared to descending line-search approaches, it significantly\nreduces the implementation cost of each iteration. We demonstrated technique convergence\nunder simple assumptions. We demonstrated that this new process produces a generic founda-\ntion for optimization methods. The preliminary results of computer experiments demonstrated\nthe new procedure’s efficacy.\nReferences\n1.Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in hilbert spaces.\nSpringer, New York (2011)\n2.Matthias Ehrgott, 2005. Multicriteria Optimization, Springer Books, Springer, edition 0, number 978-3-\n540-27659-3, July.\n3.Lin, Xi and Zhen, Hui-Ling and Li, Zhenhua and Zhang, Qing-Fu and Kwong, Sam: Pareto multi-task\nlearning, Advances in neural information processing systems (2019)\n4. Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2009)\n5. Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, New Jersey (1970)\n6.W. Bian, L. Ma, S. Qin, X. Xue: Neural network for nonsmooth pseudoconvex optimization with general\nconvex constraints, Neural Networks 101, 1-14 (2018).Adaptive multi-gradient methods for quasiconvex vector optimization 27\n7.Cevher, V ., Becker, S., Schmidt, M.: Convex optimization for big data. Signal Process. Magaz. 31, 32–43\n(2014).\n8.Fliege and Svaiter, Benar Fux: Steepest descent methods for multicriteria optimization, Mathematical\nmethods of operations research (2000).\n9.J.E. Dennis, R.B. Schnabel: Numerical methods for unconstrained optimization and nonlinear equations,\nPrentice-Hall, New Jersey, 1983.\n10.O. P. Ferreira, W. S. Sosa, On the Frank–Wolfe algorithm for non-compact constrained optimization\nproblems, Optimization, 71:1, 197-211 (2022).\n11.Y . Hu , J. Li, C. K. Yu, Convergence Rates of Subgradient Methods for Quasiconvex Optimization\nProblems, Computational Optimization and Applications, 77, 183–212 (2020).\n12.K. C. Kiwiel, Convergence and efficiency of subgradient methods for quasiconvex minimization, Math.\nProgram., Ser. A 90: 1–25 (2001).\n13.I. V . Konnov, Simplified versions of the conditional gradient method, Optimization, 67(12), 2275-2290\n(2018).\n14. Guanghui Lan, First-order and Stochastic Optimization Methods for Machine Learning, Springer Series\nin the Data Sciences, Springer Nature Switzerland (2020)\n15.N. Liu, J. Wang, S. Qin: A one-layer recurrent neural network for nonsmooth pseudoconvex optimization\nwith quasiconvex inequality and affine equality constraints, Neural Networks 147, 1-9 (2022).\n16.Yura Malitsky, Konstantin Mishchenko, Adaptive Gradient Descent without Descent, Proceedings of\nMachine Learning Research, 119:6702-6712 (2020).\n17.Sener, O., Koltun, V . (2018). Multi-task learning as multi-objective optimization. Advances in neural\ninformation processing systems, 31.\n18.Luc, Dinh. (2005). Generalized Convexity in Vector Optimization. Handbook of Generalized Convexity\nand Generalizes Monotonicity. 76. 10.1007/0-387-23393-85.\n19. O. Mangasarian, Pseudo-convex functions, Siam Control, 8, 281-290 (1965)\n20.D.T. Luc, Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, vol.\n319, Springer, Berlin, 1989, pp. 101–109.\n21.Y . Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science\n& Business Media, 2013.\n22. Zitzler, E.; and Thiele, L. 1999. Multiobjective evolutionary algorithms: a comparative case study and the\nstrength Pareto approach. IEEE transactions on Evolutionary Computation, 3(4): 257–271.\n23. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240-256 (2002)\n24.C.K. Yu, Y . Hu, X. Yang & S. K. Choy, Abstract convergence theorem for quasi-convex optimization\nproblems with applications, Optimization, 68(7), 1289-1304, 2019.\n25.Yura Malitsky, Konstantin Mishchenko, Adaptive Gradient Descent without Descent, Proceedings of\nMachine Learning Research, 119:6702-6712 (2020).\n26. O. Mangasarian, Pseudo-convex functions, Siam Control, 8, 281-290 (1965)\n27. Lin, Xi, et al. ”Pareto multi-task learning.” Advances in neural information processing systems 32 (2019).\n28.Y . Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer Science\n& Business Media, 2013.\n29. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240-256 (2002)\n30.C.K. Yu, Y . Hu, X. Yang & S. K. Choy, Abstract convergence theorem for quasi-convex optimization\nproblems with applications, Optimization, 68(7), 1289-1304, 2019.\n31. Cruz, J. B., P ´erez, L. L., Melo, J. G. (2011). Convergence of the projected gradient method for quasiconvex\nmultiobjective optimization. Nonlinear Analysis: Theory, Methods and Applications, 74(16), 5268-5273.\n32. Kiwiel, Krzysztof C. ”Convergence and efficiency of subgradient methods for quasiconvex minimization.”\nMathematical programming 90 (2001): 1-25.\n33.Horst, R., Muu, L.D., Nast, M. Branch-and-bound decomposition approach for solving quasiconvex-\nconcave programs. J Optim Theory Appl 82, 267–293 (1994).\n34.LeCun, Yann, et al. ”Gradient-based learning applied to document recognition.” Proceedings of the IEEE\n86.11 (1998): 2278-2324.\n35. D ´esid´eri, Jean-Antoine. Multiple-gradient descent algorithm (MGDA). Diss. INRIA, 2009.\n36.Bilen, Hakan, and Andrea Vedaldi. ”Integrated perception with recurrent multi-task neural networks.”\nAdvances in neural information processing systems 29 (2016).\n37.Misra, I., Shrivastava, A., Gupta, A. and Hebert, M., 2016. Crossstitch networks for multi-task learning.\nIn Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 3994- 4003).\n38.Kokkinos, Iasonas. ”Ubernet: Training a universal convolutional neural network for low-, mid-, and\nhigh-level vision using diverse datasets and limited memory.” Proceedings of the IEEE conference on\ncomputer vision and pattern recognition. 2017.\n39. Zamir, A.R., Sax, A., Shen, W., Guibas, L.J., Malik, J. and Savarese, S., 2018. Taskonomy: Disentangling\ntask transfer learning. In Proceedings of the IEEE conference on computer vision and pattern recognition\n(pp. 3712-3722)28 Nguyen Anh Minh et al.\n40. Collobert, Ronan, and Jason Weston. ”A unified architecture for natural language processing: Deep neural\nnetworks with multitask learning.” Proceedings of the 25th international conference on Machine learning.\n2008.\n41.Dong, D., Wu, H., He, W., Yu, D. and Wang, H., 2015, July. Multitask learning for multiple language\ntranslation. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics\nand the 7th International Joint Conference on Natural Language Processing (V olume 1: Long Papers) (pp.\n1723-1732).\n42. Liu, X., Gao, J., He, X., Deng, L., Duh, K. and Wang, Y .Y ., 2015. Representation learning using multi-task\ndeep neural networks for semantic classification and information retrieval.\n43. Assun c ¸˜ao, P. B., Orizon Pereira Ferreira, and L. F. Prudente. ”Conditional gradient method for multiobjec-\ntive optimization.” Computational Optimization and Applications 78 (2021): 741-768.\n44.Ghalavand, Nasim, Esmaile Khorram, and Vahid Morovati. ”An adaptive nonmonotone line search for\nmultiobjective optimization problems.” Computers and Operations Research 136 (2021): 105506.\n45.Luong, M.T., Le, Q.V ., Sutskever, I., Vinyals, O. and Kaiser, L., 2015. Multi-task sequence to sequence\nlearning. arXiv preprint arXiv:1511.06114.\n46. Hashimoto, K., Xiong, C., Tsuruoka, Y . and Socher, R., 2016. A joint many-task model: Growing a neural\nnetwork for multiple nlp tasks. arXiv preprint arXiv:1611.01587\n47.Huang, J.T., Li, J., Yu, D., Deng, L. and Gong, Y ., 2013, May. Crosslanguage knowledge transfer using\nmultilingual deep neural network with shared hidden layers. In 2013 IEEE International Conference on\nAcoustics, Speech and Signal Processing (pp. 7304-7308). IEEE.\n48.Seltzer, Michael L., and Jasha Droppo. ”Multi-task learning in deep neural networks for improved\nphoneme recognition.” 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.\nIEEE, 2013.\n49.Huang, J.T., Li, J., Yu, D., Deng, L. and Gong, Y ., 2013, May. Crosslanguage knowledge transfer using\nmultilingual deep neural network with shared hidden layers. In 2013 IEEE International Conference on\nAcoustics, Speech and Signal Processing (pp. 7304-7308). IEEE.\n50.Edelsbrunner, Herbert, and Raimund Seidel. ”V oronoi diagrams and arrangements.” Proceedings of the\nfirst annual symposium on Computational geometry. 1985.\n51.Das, I.; and Dennis, J. 2000. Normal-Boundary Intersection: A New Method for Generating the Pareto\nSurface in Nonlinear Multicriteria Optimization Problems. SIAM Journal on Optimization, 8\n52.Pennington, Jeffrey, Richard Socher, and Christopher D. Manning. ”Glove: Global vectors for word\nrepresentation.” Proceedings of the 2014 conference on empirical methods in natural language processing\n(EMNLP). 2014.\n53.Kim, Yoon. ”Convolutional neural networks for sentence classification.” arXiv preprint arXiv:1408.5882\n(2014)."    },    {        "title": "2402.06256v1.Bayesian_Committee_Machine_Potential_for_Isothermal_Isobaric_Molecular_Dynamics_Simulations.pdf",        "content": "Bayesian Committee Machine Potential for Isothermal-Isobaric Molecular Dynamics\nSimulations\nSoohaeng Yoo Willow1and Chang Woo Myung1,∗\n1Department of Energy Science, Sungkyunkwan University, Seobu-ro 2066, Suwon, 16419, Korea\n(Dated: February 12, 2024)\nIn recent advancements in material simulations, the utilization of Sparse Gaussian Process Re-\ngression (SGPR)-based machine learning potentials (MLPs) has proven to be highly successful in\ndiverse applications such as catalysis, batteries, and solar cells. In the context of isothermal and\nisobaric molecular dynamics simulations, achieving precise pressure estimates is crucial for an ac-\ncurate understanding of the system behavior under constant pressure conditions. In this study,\nwe introduce a novel kernel function designed for estimating the virial term, a critical component\nfor pressure calculations in materials simulations. Our study reveals that the inclusion of a virial\nprediction in the kernel function leads to significantly improved accuracy in calculating the stress\nof a system. We present a kernel-based ML potential that can be estimated via the Bayesian Com-\nmittee Machine without the need for additional training. This improvement allows us to calculate\nthe melting temperature of ice through Isobaric-Isenthalpy NpH molecular dynamics simulations\nof ice-liquid coexisting phases, employing the BCM potential.\nI. INTRODUCTION\nIn the last two decades, the application of machine\nlearning potentials (MLPs) in the field of materials sim-\nulations has proven to be highly successful, particularly\nfor the accurate prediction of potential energy surfaces\n(PESs) of the reference ab initio data,[1–12] such as den-\nsity functional theory (DFT),[13–15] many-body pertur-\nbation (MP),[16–18] and coupled-cluster (CC) theories.\nMoreover, MLPs uphold an affordable computational\ncost, allowing one to investigate many interesting physi-\ncal phenomena with currently accessible supercomputers.\nThis breakthrough has paved the way for large-scale and\nlong-time simulations in various scientific domains, in-\ncluding materials science, chemistry, and biology. Several\ndistinct machine learning methods have been employed\nfor the construction of MLPs. These include the atom-\ncentered artificial neural network method (ANN),[19–22]\nGaussian approximation potentials (GAPs),[3, 23] graph\nneural network potentials (GNNs),[5, 6] gradient-domain\nmachine learning (GDML),[24] moment tensor potentials\n(MTP),[25–27] and sparse Gaussian process regression\n(SGPR).[7–11, 28] The versatility of these methodologies\nhas led to their widespread adoption and successful ap-\nplication across different scientific disciplines, promising\nvaluable insights and outcomes.\nThe Gaussian Process (GP) based MLP boasts in the\nactive learning, uncertainty prediction, and low data\nrequirement.[29–32] However, despite all these advan-\ntages, the scalability of GP-based ML potential came\nunder scrutiny in particular for achieving universal po-\ntential due to its poor scaling. MLPs based on GP\nface computational challenges when dealing with large\n∗cwmyung@skku.edudatasets. The training complexity increases dramatically\ntoO(n3) due to the necessity of inverting a kernel ma-\ntrix, and prediction complexity grows quadratically with\nthe dataset size n. As a result, GPs are restricted to\nhandling datasets smaller than 104. This led to the de-\nvelopment of the SGPR technique,[7–10, 28] which uses a\nsubset of data for efficiency. However, SGPR still strug-\ngles with large numbers mof inducing points, showing\ncomputational demands of O(nm2). To mitigate this\nlimitation, the focus shifted to aggregation models such\nas the Bayesian Committee Machine (BCM)[33] to ad-\ndress computational challenges. This involves combining\npredictions from pdistinct sub-models, each trained on\ndifferent data sections, with a computational scaling of\nO(nm2/p2). This approach is deemed effective in en-\nhancing the scalability of kernel-based ab initio MLPs.\nMolecular dynamics (MD) simulations serve as a pow-\nerful tool to complement experimental findings by offer-\ning a detailed atomic-level understanding in materials.\nBy comparing the results of MD simulations with their\ncorresponding experimental observations, such as phase\ndiagrams and radial distribution functions, one can ef-\nfectively test the accuracy and reliability of the PES.\nFor accurately performing isothermal-isobaric MD sim-\nulations, it is crucial to correctly estimate the stress. In\nthis context, improving the accuracy of stress estimations\nof MLPs can enhance the reliability and effectiveness of\nMD simulations. While other types of MLPs such as\nneural network MLPs [34] incorporated the estimation\nof viral stress, a SGPR-based model has not yet included\nthe estimation of virial stress.\nZhang and coworkers in 2021[35] employed MD sim-\nulations with a SCAN-DFT[36] based Deep potential\nmodel[34, 37] to predict ice-water phase diagram, achiev-\ning a generally satisfactory agreement with experimental\nresults. Subsequently, Bore and Paesani in 2023[38] uti-arXiv:2402.06256v1  [cond-mat.soft]  9 Feb 20242\nlized the deep potential model[34] based on MB-pol[39–\n41] to estimate the ice-water phase diagram and showed\nimproved agreement with experimental results. Surpris-\ningly, however, they observed that the phase diagram\npredicted by the MLP did not mirror the one obtained\nusing its reference potential, called MB-pol. To enhance\nour comprehension of the disparity in predicting phase\ndiagrams between MLP and the reference model, we per-\nformed isobaric-isenthalpy ( NpH ) MD simulations of ice-\nliquid coexisting phases to estimate the melting points of\nthe SGPR-based MLP.\nII. THEORY\nII.1. Virial Kernel prediction in Sparse Gaussian\nProcess Regression potential\nThe MLP energy is expressed in terms of a local chem-\nical environment (LCE) ρias\nE=N/summationdisplay\ni=1E(ρi), (1)\nwhere Eis a fictional latent function of local energy. This\nindicates that the MLP energy as well as force acting on\natom idepends on the relative coordinates of surrounding\natoms. The LCE of atom iis defined by\nρi={⃗ rij;j̸=i|⃗ rij|< rc}, (2)\nwhere ⃗ rij=⃗ rj−⃗ ri,rcis a cutoff radius for neighborhood\nrelations.\nThe core component of Bayesian regression is building\na covariance Kernel K(ρi, ρj) that encodes the similar-\nity between ρiandρj. The Kernel is invariant with re-\nspect to the symmetry operations that leaves the poten-\ntial energy invariant, e.g. translations and and rotations.\nHere, the smooth overlap of atomic positions (SOAP)[23]\nis used for the descriptor of a similarity kernel between\nLCEs and is defined as\nK(ρi, ρj) =/integraldisplay\ndR|/integraldisplay\nd⃗ rξi(⃗ r)ξj(R⃗ r)|2, (3)\nwhere Ris the three-dimensional rotation operator, ξi(⃗ r)\nis the atomic density in the neighborhood of i,\nξi(⃗ r) =/summationdisplay\nj∈ρieα|⃗ r−⃗ rij|2, (4)\nandαis a hyperparameter.\nTo improve the scalability, one can exploit the LCE\nsimilarities in the data X. It is very likely that many LCE\npairs in Xare similar: K(ρi, ρj)≈1. Let z={χj}m\nj=1\ndenote a reduced set of LCEs which are significantlydistinct from each other and are sufficient statistics for\nX={R1,···, Rn}.zis called the inducing set of LCEs.\nWith the inducing set of LCEs, the latent energy function\nof atom iis expressed as\nE(ρi) =m/summationdisplay\nj=1wjK(ρi, χj) (5)\nwhere w= [w1,···, wn] are the weights for the descrip-\ntors in z. The potential energy for Rbecomes\nE(R) =N/summationdisplay\niE(ρi) =/summationdisplay\ni,jwjK(ρi, χj). (6)\nThe forces fµ\niare obtained from\nfµ\ni=m/summationdisplay\nj=1wj/bracketleftigg\n−/summationdisplay\nk=1∂K(ρk, χj)\n∂rµ\ni/bracketrightigg\n=m/summationdisplay\nj=1wjKµ\ni(R, χ j),\n(7)\nwhere Kµ\ni(R, χ j) is the covariance kernel for the force on\natom iandµ={x, y, z}.\nThe previous SGPR potential did not include the stress\ntensor in the training process.[7] Due to this, the SGPR\npotential often fails to accurately predict the stress tensor\nof a given system. To improve this shortcoming, our\nstrategy is to incorporate the stress data into the training\nprocess of SGPR. The instantaneous stress tensor Pαβis\ndefined as\nPαβ=kBTαβ/V+Wαβ/V, (8)\nwhere Tαβis the instantaneous temperature tensor. To\npredict the stress tensor, we build the internal virial ker-\nnel function Wαβas below\nWαβ=/summationdisplay\nifα\nirβ\ni\n=/summationdisplay\njwj\n−/summationdisplay\ni,k∂K(ρk, χj)\n∂rα\nirβ\ni\n (9)\nThe weight vector wis obtained by linear fitting of\nE(R),fµ\nk, andWαβto the target energies, forces, virial\nstresses ynfor a given dataset X={R1,···, Rn}. In\nthe Bayesian linear regression method, the mathematical\nexpression that we wish to solve becomes\nKnmwm=yn, (10)\nwhere Knmis the covariance matrix between data X and\ninducing set zof LCEs. In the SGPR algorithm,[7] wm\ncan be the solution of the following linear system:\n/bracketleftbiggKnm\nσLT/bracketrightbigg\nwm=/bracketleftbigg\nyn\n0/bracketrightbigg\n, (11)\nwhere Lis the Cholesky factor of Kmm(Kmm=LLT,\nwhere Lis lower triangular), 0is a columnar vector of\nzeros with length m, and σis the noise hyperparameter.3\nFIG. 1. Schematic illustration of Bayesian committee machine potential. The entire dataset is divided into the distinct subsets\nbased on classification. Within each subset, an SGPR-based MLP is trained. Subsequently, the universal MLP for the entire\ndataset is estimated through the BCM. The weight factorβα\nσ2αensures that the predictions from confident models are more\noutspoken than those from less confident models.\nII.2. Bayesian Committee Machine potential\nThe SGPR algorithm encounters difficulties in obtain-\ning the solution wmfor Eq. (11), which involves the\ninversion of a kernel matrix, with large size of training\ndatasets Xand inducing set zof LCEs. In order to miti-\ngate these challenge, we employed the BCM[33] as a novel\napproach to combine SGPR-MLPs which were separately\ntrained on different subsets as shown in Figure 1. Tak-\ning the example of ice polymorphs, instead of utilizing\nall ice phases as a collective training dataset, we assign\neach ice phase to an individual training dataset. Subse-\nquently, we trained separate models for each subset to\ncreate machine learning potential models tailored to spe-\ncific phases. We consolidated these individually trained\nmodel into an universal MLP via BCM. This approach\nnot only streamlines the training process but also ad-\ndresses the computational challenges associated with the\ninversion of large kernel matrices, paving the way for an\nefficient and accurate universal MLPs.\nWith classification of the training dataset and in-\nducing set into the distinct subsets: {n1,···, np}and\n{m1,···, mp}, in the BCM,[33] the predicted universal\npotential energy of the system becomes\nE≈ˆSp/summationdisplay\nα=1βα\nσ2αEα, (12)\nwhere the potential energy Eαis calculated using α-th\nsubset ( Rn∈αandzα={χj}j∈α) and the weight vec-\ntorwαofα-th sub SGPR model. This energy can be\nrepresented as:\nEα(R)≈/summationdisplay\ni,(j∈mα)wα\njK(ρi, χj). (13)The maximum value of the covariance loss for the α-th\nsubset, denoted as σ2\nα, is used to weight the α-th com-\nmittee prediction. The covariance loss s(ρi) is defined\nas\nσ2\nα= max(1 −Kρi,mK−1\nmmKT\nρi,m). (14)\nFurthermore, we weight the prediction with βαfollowing\nthe concepts of robust BCM where the individual com-\nmittee prediction is weighted by the differential entropy,\nβα=−log(σ2\nα). The final normalization factor ˆSis thus\ngiven:\nˆS=/bracketleftiggp/summationdisplay\nαβα\nσ2α/bracketrightigg−1\n. (15)\nThe significant benefit of employing the BCM with\nSGPR lies in its ability to work with a reduced ( n/p)-\ndimensional training dataset and a ( m/p)-dimensional\ninducing set, instead of the original n-dimensional train-\ning dataset and m-dimensional inducing set, where n=/summationtext\nαnαandm=/summationtext\nαmα. Consequently, the computa-\ntional cost in BCM-based SGPR scales as from O(nm2)\ntopO(nm2/p3). Another advantage of the BCM is that\na particular subset, exhibiting greater similarity with a\ntest configuration x∗, contributes more significantly to\nenergy, forces, and stress.\nCompared to “product of experts” (POE),[42] in which\nthe predicted universal potential energy reads E≈\n1\nP/summationtextP\nαEα, the weight factor βα/σ2\nαin the BCM en-\nsures that the predictions from confident models showing\nhigher similarity are given more emphasis that those from\nless confident models.4\n0.2 0.3 0.4 0.5 0.6 0.7 0.8\nMB-pol0.20.40.60.81.01.21.4SGPR-MLP\nPressure\nScheme\nScheme1\nScheme2\nFIG. 2. Comparison of SGPR-MLP’s predicted pressure with\nreference values. The testing datasets were gathered from\nmolecular dynamics simulations of systems where ice III and\nliquid coexist, using MB-pol water at pressures of P= 0.4\nGPa and P= 0.6 GPa in isobaric isothermal conditions.\nIII. RESULTS\nIII.1. Pressure prediction using virial Kernel\nmethod\nIn Section II.1, the weight vector wmrelies on the co-\nvariance matrix Knmand corresponding target values for\nenergy, forces, virial stress denoted as yn. The weight\nvector’s values vary based on whether the covariance ma-\ntrix and corresponding target values include the virial\nstress term. In this subsection, our objective is to com-\npare instantaneous pressure calculated using the weight\nvector obtained under different conditions. This compar-\native analysis seeks to provide insights into the impact of\nincorporating or omitting the virial stress term in covari-\nance matrix and target values on estimation of instanta-\nneous pressure. To achieve this, we explore two schemes.\nThe first scheme ( Scheme 1 ) involves constructing wm\nsolely with Knmandynincluding only both energies\nand forces. The second scheme ( Scheme 2 ) incorpo-\nrates virial term into Knmandynin addition to energies\nand forces to obtain wm.\nTo obtain the training and testing dataset, we\nperformed isobaric isothermal molecular dynamics sim-\nulations of the system where ice III and liquid coexist,\nusing MB-pol water at pressures of P= 0.4 and 0 .6\nGPa and temperature of T= 255 K. Once we trained\nthe weight vector wmbased on the training dataset, we\nestimated the pressures with the testing dataset (see\nFigure 2) using Eqs. (8) and (9). Figure 2 shows that\nonlyScheme 2 could reproduce target instantaneous\npressures accurately.\n240 245 250 255 260 265 270\nT emperature (K)0.10.20.30.40.50.6Pressure (GPa)\nIce II\nIce IIIIce V\nLiq\nIce IhIce II@MB-pol\nIce III@MB-pol\nIce V@MB-pol\nIce II[Bore&Paesani]\nIce III[Bore&Paesani]\nIce V[Bore&Paesani]\nIce II@SGPR-MLP\nIce III@SGPR-MLP\nIce V@SGPR-MLPFIG. 3. Ice-liquid melting points were calculated through\nNpH MD simulations of coexisting ice-liquid systems. The\nmelting points obtained using the MB-pol model are denoted\nby circle points, and those obtained with the universal SGPR-\nMLP are indicated by star points. The melting points calcu-\nlated by Bore and Paesani[38] are represented by open square\npoints. Solid black lines represent the experimental melting\nlines.\nIII.2. Estimation of the coexistence line of\nice-liquid\nNpH molecular dynamics simulations were employed\nto compute the melting temperature ( Tm) of ice within\nthe coexisting ice-liquid phase.[43–45] NpH MD sim-\nulations enable the spontaneous adjustment of tem-\nperature to satisfy the condition µice(P, T)T=Tm=\nµliq(P, T)T=Tm. Specifically, when the temperature of\nice-liquid coexisting system is higher than Tm, the chem-\nical potential of the ice phase is higher than that of the\nliquid phase: µice(P, T)T>T m> µ liq(P, T)T>T m. Con-\nsequently, the ice phase undergoes a process of melting\nwith the absorption of heat, leading to a reduction in the\nkinetic temperature of the coexisting system towards the\nmelting temperature Tm. One notable advantage of em-\nploying NpH MD simulations over NV E MD simulation\nis the absence of the stress-anisotropy problem, as the\nthree principle components of the stress tensor can be\nadjusted to the specified external pressure in the former.\nWe aim to estimate the melting temperatures of ice II,\nIII, and V under low pressure conditions (0.2 ∼0.6 GPa).\nTo achieve this, on-the-fly active learning using isobaric\nisothermal ( NpT ) MD simulations were performed to cre-\nate a SGPR MLP for each phases (ice II, ice III, and\nice V) where ice and liquid coexist. The MB-pol wa-\nter model[41] was chosen for the reference model in this\nstudy. All MD simulations were carried out using ASE\npackage. We employed a time step of 0.5 fs for the in-\ntegration of the equations of motion that were propa-5\ngate according to combined Nose-Hoover and Parrinello-\nRahman dynamics.[46–48]\nThree individually trained SGPR MLP models were\nmerged into one universal SGPR MLP model for ice-\nliquid coexisting systems using the BCM. NpH MD sim-\nulations with this universal model were performed to de-\ntermine the melting temperatures of ice II, III, and V.\nAs shown in Figure S1, the NpH MD simulations main-\ntained the conservation of enthalpy ( H) while the total\nenergy as the sum of the potential energy and kinetic\nenergy exhibited fluctuations. Figure 3 shows that melt-\ning points of ice II, III, and V, as obtained from iso-\nbaric isenthalpy ( NpH ) MD simulations with the MB-\npol water model, largely coincide with those calculated\nfrom enhanced-coexistence simulations.[38] The univer-\nsal SGPR-MLP gives melting points that exhibit a slight\nshift towards higher values compared to those obtained\nusing the reference potential (MB-pol). This disparity\narise due to slight noise in energy and forces during train-\ning of SGPR-MLP. In general, aside from noting a shift\nin the phase diagram, it is expected that the well-trained\nSGPR-MLP can faithfully reproduce physical and ther-\nmodynamic properties of materials using the reference\npotential models.\nDeep neural network ML models[35, 38] based on DFT\nand MB-pol predict that ice III is metastable when com-\npared to ice II. Our SGPR ML model based on MB-pol\nexhibits the same trend, indicating that ice III is indeed\nmetastable in comparison to ice II. The discrepancy be-\ntween these ML models and experimental observations\nmay arise from the omission of entropy resulting from\nproton disorder in ice III.\nIV. DISCUSSION\nThe focus of our investigation lies in the critical as-\npect of achieving precise pressure estimates, particularly\nin the context of isothermal and isobaric molecular dy-\nnamics simulations, as accurate pressure information is\nparamount for a thorough understanding of system be-\nhavior. A pivotal contribution of our work involves the\nintroduction of a novel kernel function explicitly tailored\nfor estimating the virial term, a key factor in pressure cal-\nculations during molecular dynamics simulations. Our\nfindings underscore the significance of incorporating avirial term correction within the kernel function, leading\nto a substantial improvement in the accuracy of system\npressure calculations. This refinement proves instrumen-\ntal in enhancing the overall reliability and precision of\nthe MLP, particularly when simulating material systems\nunder isothermal and isobaric conditions.\nFurthermore, the refined MLP with the virial term\ncorrection enables us to extend its applicability to es-\ntimating the melting temperature of ice polymorphs.\nWe achieve this by employing isobaric-isenthalpy ( NpH )\nmolecular dynamics simulations that capture the coexist-\ning phases of ice and liquid. The universal SGPR-MLP,\nnow augmented with the virial term correction, demon-\nstrates its efficacy in providing accurate predictions of\nthe melting temperature for ice. This not only highlights\nthe versatility of our proposed methodology but also un-\nderscores its potential in advancing the understanding of\nmaterial behavior under specific thermodynamic condi-\ntions.\nIn the end, the study highlights the efficacy of the\ncombining model (BCM) generated through kernel-based\nmethods, such as SGPR, for enhanced generality and\ntransferability. By employing the BCM, a universal\nSGPR-MLP can be built by leveraging previously trained\nmodels from a database. In comparison to the conven-\ntional “product of experts” (POE) method, the universal\nSGPR-MLP obtained via the BCM demonstrates supe-\nrior predictive accuracy for potential energy, forces, and\npressures. This indicates that the BCM offers a more ro-\nbust and reliable framework for broader applications in\nthe field of machine-learning potentials.\nV. ACKNOWLEDGEMENTS\nSYW and CWM acknowledge the support from the\nNational Research Foundation of Korea (NRF) grant\nfunded by the Korea government (MSIT) (No. RS-\n2023-00222245). CWM acknowledges the support from\nthe National Research Foundation of Korea (NRF)\ngrant funded by the Korea government (MSIT) (No.\nNRF-2022R1C1C1010605). The authors are grateful\nfor: the computational support from the Korea Insti-\ntute of Science and Technology Information (KISTI)\nfor the Nurion cluster (KSC-2023-CRE-0059,KSC-2023-\nCRE-0355,KSC-2023-CRE-0454).\n[1] O. T. Unke, S. Chmiela, H. E. Sauceda, M. Gastegger,\nI. Poltavsky, K. T. Sch¨ utt, A. Tkatchenko, and K.-R.\nM¨ uller, Machine Learning Force Fields, Chem. Rev. 121,\n10142 (2021).\n[2] V. L. Deringer, A. P. Bart´ ok, N. Bernstein, D. M.\nWilkins, M. Ceriotti, and G. Cs´ anyi, Gaussian Process\nRegression for Materials and Molecules, Chem. Rev. 121,10073 (2021).\n[3] A. P. Bart´ ok, M. C. Payne, R. Kondor, and G. Cs´ anyi,\nGaussian Approximation Potentials: The Accuracy of\nQuantum Mechanics, without the Electrons, Phys. Rev.\nLett. 104, 136403 (2010).\n[4] M. S. Chen, J. Lee, H.-Z. Ye, T. C. Berkelbach, D. R.\nReichman, and T. E. Markland, Data-Efficient Machine6\nLearning Potentials from Transfer Learning of Periodic\nCorrelated Electronic Structure Methods: Liquid Water\nat AFQMC, CCSD, and CCSD(T) Accuracy, J. Chem.\nTheory Comput. 19, 4510 (2023).\n[5] J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and\nG. E. Dahl, Neural Message Passing for Quantum Chem-\nistry (2017), arxiv:1704.01212 [cs].\n[6] S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P.\nMailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and\nB. Kozinsky, E(3)-equivariant graph neural networks for\ndata-efficient and accurate interatomic potentials, Nat\nCommun 13, 2453 (2022).\n[7] A. Hajibabaei, C. W. Myung, and K. S. Kim, Sparse\nGaussian process potentials: Application to lithium dif-\nfusivity in superionic conducting solid electrolytes, Phys.\nRev. B 103, 214102 (2021).\n[8] A. Hajibabaei and K. S. Kim, Universal Machine Learn-\ning Interatomic Potentials: Surveying Solid Electrolytes,\nJ. Phys. Chem. Lett. 12, 8115 (2021).\n[9] A. Hajibabaei, M. Ha, S. Pourasad, J. Kim, and K. S.\nKim, Machine Learning of First-Principles Force-Fields\nfor Alkane and Polyene Hydrocarbons, J. Phys. Chem. A\n125, 9414 (2021).\n[10] C. W. Myung, A. Hajibabaei, J.-H. Cha, M. Ha, J. Kim,\nand K. S. Kim, Challenges, Opportunities, and Prospects\nin Metal Halide Perovskites from Theoretical and Ma-\nchine Learning Perspectives, Adv.Energy Mater. 12,\n2202279 (2022).\n[11] J. Vandermause, Y. Xie, J. S. Lim, C. J. Owen, and\nB. Kozinsky, Active learning of reactive Bayesian force\nfields applied to heterogeneous catalysis dynamics of\nH/Pt, Nat Commun 13, 5183 (2022).\n[12] J. Vandermause, S. B. Torrisi, S. Batzner, Y. Xie, L. Sun,\nA. M. Kolpak, and B. Kozinsky, On-the-fly active learn-\ning of interpretable Bayesian force fields for atomistic rare\nevents, npj Comput Mater 6, 20 (2020).\n[13] K. Laasonen, M. Sprik, M. Parrinello, and R. Car, ”Ab\ninitio” liquid water, J. Chem. Phys. 99, 9080 (1993).\n[14] W. D. Richards, T. Tsujimura, L. J. Miara, Y. Wang,\nJ. C. Kim, S. P. Ong, I. Uechi, N. Suzuki, and\nG. Ceder, Design and synthesis of the superionic con-\nductor Na10SnP2S12, Nat Commun 7, 11009 (2016).\n[15] M. Boero, M. Parrinello, and K. Terakura, First Princi-\nples Molecular Dynamics Study of Ziegler-Natta Hetero-\ngeneous Catalysis, J. Am. Chem. Soc. 120, 2746 (1998).\n[16] S. Y. Willow, M. A. Salim, K. S. Kim, and S. Hi-\nrata, Ab initio molecular dynamics of liquid water using\nembedded-fragment second-order many-body perturba-\ntion theory towards its accurate property prediction, Sci\nRep5, 14358 (2015).\n[17] M. Del Ben, M. Sch¨ onherr, J. Hutter, and J. VandeVon-\ndele, Bulk Liquid Water at Ambient Temperature and\nPressure from MP2 Theory, J. Phys. Chem. Lett. 4, 3753\n(2013).\n[18] M. Del Ben, J. Hutter, and J. VandeVondele, Forces and\nstress in second order Møller-Plesset perturbation the-\nory for condensed phase systems within the resolution-\nof-identity Gaussian and plane waves approach, J. Chem.\nPhys. 143, 102803 (2015).\n[19] J. Behler and M. Parrinello, Generalized Neural-Network\nRepresentation of High-Dimensional Potential-Energy\nSurfaces, Phys. Rev. Lett. 98, 146401 (2007).\n[20] J. Behler, Atom-centered symmetry functions for con-\nstructing high-dimensional neural network potentials, J.\nChem. Phys. 134, 074106 (2011).[21] G. P. P. Pun, R. Batra, R. Ramprasad, and Y. Mishin,\nPhysically informed artificial neural networks for atom-\nistic modeling of materials, Nat Commun 10, 2339\n(2019).\n[22] M. Eckhoff and J. Behler, High-dimensional neural\nnetwork potentials for magnetic systems using spin-\ndependent atom-centered symmetry functions, Npj Com-\nput. Mater. 7, 170 (2021).\n[23] A. P. Bart´ ok, R. Kondor, and G. Cs´ anyi, On representing\nchemical environments, Phys. Rev. B 87, 184115 (2013).\n[24] S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky,\nK. T. Sch¨ utt, and K.-R. M¨ uller, Machine learning of ac-\ncurate energy-conserving molecular force fields, Sci. Adv.\n3, e1603015 (2017).\n[25] A. V. Shapeev, Moment Tensor Potentials: A Class of\nSystematically Improvable Interatomic Potentials, Mul-\ntiscale Model. Simul. 14, 1153 (2016).\n[26] E. V. Podryabinkin and A. V. Shapeev, Active learning\nof linearly parametrized interatomic potentials, Comput.\nMater. Sci. 140, 171 (2017).\n[27] I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V.\nShapeev, The MLIP package: Moment tensor potentials\nwith MPI and active learning, Mach. Learn.: Sci. Tech-\nnol.2, 025002 (2021).\n[28] J. Qui˜ nonero-Candela and C. E. Rasmussen, A Unifying\nView of Sparse Approximate Gaussian Process Regres-\nsion, J. Mach. Learn. Res. 6, 1939 (2005).\n[29] C. E. Rasmussen and C. K. I. Williams,\nGaussian Processes for Machine Learning (The MIT\nPress, 2005).\n[30] B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and\nN. De Freitas, Taking the Human Out of the Loop: A\nReview of Bayesian Optimization, Proc. IEEE 104, 148\n(2016).\n[31] N. Lawrence, Probabilistic Non-linear Principal Com-\nponent Analysis with Gaussian Process Latent Variable\nModels, J. Mach. Learn. Res. 6, 1783 (2005).\n[32] M. A. ´Alvarez and N. D. Lawrence, Computationally Ef-\nficient Convolved Multiple Output Gaussian Processes,\nJ. Mach. Learn. Res. 12, 1459 (2011).\n[33] V. Tresp, A Bayesian Committee Machine, Neural Com-\nputation 12, 2719 (2000).\n[34] H. Wang, L. Zhang, J. Han, and W. E, DeePMD-kit:\nA deep learning package for many-body potential energy\nrepresentation and molecular dynamics, Comput. Phys.\nCommun. 228, 178 (2018).\n[35] L. Zhang, H. Wang, R. Car, and W. E, Phase Diagram\nof a Deep Potential Water Model, Phys. Rev. Lett. 126,\n236001 (2021).\n[36] J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly Con-\nstrained and Appropriately Normed Semilocal Density\nFunctional, Phys. Rev. Lett. 115, 036402 (2015).\n[37] L. Zhang, D.-Y. Lin, H. Wang, R. Car, and W. E, Active\nlearning of uniformly accurate interatomic potentials for\nmaterials simulation, Phys. Rev. Mater. 3, 023804 (2019).\n[38] S. L. Bore and F. Paesani, Realistic phase diagram of\nwater from ”first principles” data-driven quantum simu-\nlations, Nat Commun 14, 3349 (2023).\n[39] V. Babin, C. Leforestier, and F. Paesani, Development\nof a “First Principles” Water Potential with Flexible\nMonomers: Dimer Potential Energy Surface, VRT Spec-\ntrum, and Second Virial Coefficient, J. Chem. Theory\nComput. 9, 5395 (2013).\n[40] V. Babin, G. R. Medders, and F. Paesani, Development\nof a “First Principles” Water Potential with Flexible7\nMonomers. II: Trimer Potential Energy Surface, Third\nVirial Coefficient, and Small Clusters, J. Chem. Theory\nComput. 10, 1599 (2014).\n[41] M. Riera, C. Knight, E. F. Bull-Vulpe, X. Zhu, H. Ag-\nnew, D. G. A. Smith, A. C. Simmonett, and F. Pae-\nsani, MBX: A many-body energy and force calculator for\ndata-driven many-body simulations, J. Chem. Phys. 159,\n054802 (2023).\n[42] G. E. Hinton, Training Products of Experts by Minimiz-\ning Contrastive Divergence, Neural Comput. 14, 1771\n(2002).\n[43] S. Yoo, X. C. Zeng, and J. R. Morris, The melting lines\nof model silicon calculated from coexisting solid–liquid\nphases, The Journal of Chemical Physics 120, 1654\n(2004).\n[44] S. Yoo, X. C. Zeng, and S. S. Xantheas, On the\nphase diagram of water with density functional the-ory potentials: The melting temperature of ice Ih with\nthe Perdew–Burke–Ernzerhof and Becke–Lee–Yang–Parr\nfunctionals, The Journal of Chemical Physics 130,\n221102 (2009).\n[45] S. Yoo and S. S. Xantheas, Communication: The effect of\ndispersion corrections on the melting temperature of liq-\nuid water, The Journal of Chemical Physics 134, 121105\n(2011).\n[46] S. Melchionna, G. Ciccotti, and B. L. Holian, Hoover\nNPT dynamics for systems varying in shape and size,\nMol. Phys. 78, 533 (1993).\n[47] S. Melchionna, Constrained systems and statistical dis-\ntribution, Phys. Rev. E 61, 6165 (2000).\n[48] B. L. Holian, A. J. De Groot, W. G. Hoover, and C. G.\nHoover, Time-reversible equilibrium and nonequilibrium\nisothermal-isobaric simulations with centered-difference\nStoermer algorithms, Phys. Rev. A 41, 4552 (1990).Bayesian Committee Machine Potential for Isothermal-Isobaric Molecular\nDynamics Simulations\nSoohaeng Yoo Willow1and Chang Woo Myung1,∗\n1Department of Energy Science, Sungkyunkwan University, Seobu-ro 2066, Suwon, 16419, Korea\n(Dated: February 12, 2024)\nTABLE S1. Ice-liquid coexisting systems. Total number of water molecules in the simulation box ( NH2O)\nand initial cell dimensions of ice-liquid coexisting systems for NpH MD simulations.\nNH2O (xbox, ybox, zbox, alpha, beta, gamma)\nIce II - Liquid 864 (23.415, 23.415, 43.910, 84.826, 84.826, 113.100)\nIce III- Liquid 648 (19.982, 20.349, 41.379, 90, 90, 90)\nIce V - Liquid 672 (18.553, 45.374, 20.833, 90, 109.215, 90)\n∗cwmyung@skku.eduarXiv:2402.06256v1  [cond-mat.soft]  9 Feb 20242\n0 20 40 60 80 100\ntime(ps)1.5\n1.0\n0.5\n0.00.51.01.52.02.5H(eV)H @ P=0.2 GPa\nEtot @ P=0.2 GPa\nH @ P=0.6 GPa\nEtot @ P=0.6 GPa\nFIG. S1. NpH MD simulations of ice II-liquid coexisting systems at pressures of P= 0.2 and 0 .6 GPa show\nthe Enthalpy ( H) conservation while the total energies fluctuate.3\n0 20 40 60 80 100\ntime(ps)0.2\n0.00.20.40.60.81.01.2P_zP= 0.2 GPa\nP= 0.6 GPa\nFIG. S2. Instantaneous pressure Pzvalues in NpH MD simulations of ice II-liquid coexisting system at\npressures of P= 0.2 and 0 .6 GPa. Instantaneous pressures are adjusted to specified target pressure during\nNPH MD simulations.4\n0 20 40 60 80 100\ntime(ps)240245250255260265T\nP=0.2GPa\nP=0.6GPa\nFIG. S3. Instantaneous temperature values in NpH MD simulations of ice II-liquid coexisting system at\npressures of P= 0.2 and 0 .6 GPa. Instantaneous temperatures are tuned to approach melting temperature,\nensuring that they meet the condition µice(P, T)T=Tm=µliq(P, T)T=Tm."    },    {        "title": "2402.06271v1.Adaptive_proximal_gradient_methods_are_universal_without_approximation.pdf",        "content": "Adaptive proximal gradient methods are universal\nwithout approximation∗\nKonstantinos A. Oikonomidis†Emanuel Laude†Puya Latafat†\nAndreas Themelis‡Panagiotis Patrinos†\nAbstract\nWe show that adaptive proximal gradient methods for convex problems are not restricted to\ntraditional Lipschitzian assumptions. Our analysis reveals that a class of linesearch-free methods\nis still convergent under mere local Hölder gradient continuity, covering in particular continuously\ndifferentiable semi-algebraic functions. To mitigate the lack of local Lipschitz continuity, popular\napproachesrevolvearound ε-oraclesand/orlinesearchprocedures.Incontrast,weexploitplainHölder\ninequalities not entailing any approximation, all while retaining the linesearch-free nature of adaptive\nschemes. Furthermore, we prove full sequence convergence without prior knowledge of local Hölder\nconstants nor of the order of Hölder continuity. In numerical experiments we present comparisons to\nbaseline methods on diverse tasks from machine learning covering both the locally and the globally\nHölder setting.\nKeywords. Convex minimization ·proximal gradient method ·universal methods ·linesearch-free adap-\ntive methods ·local Hölder gradient continuity ·first-order methods\nAMS subject classifications. 65K05 ·90C06 ·90C25 ·90C30 ·90C47\n1 Introduction\nWe consider composite minimization problems of the form\nminimize\nx∈nφ(x):=f(x) +g(x) (P)\nwhere f:n→is convex and has locallyHölder continuous gradient of (possibly unknown ) order\nν∈(0,1], and g:n→is proper, lsc, and convex with easy to compute proximal mapping.\nThe proximal gradient method is the de facto splitting technique for solving the composite problem\n(P). Under global Lipschitz continuity of ∇f, convergence results and complexity bounds are well estab-\nlished. Nevertheless, there exist many applications where such an assumption is not met. Among these are\nmixtures of maximum likelihood models [14], classification, robust regression [35, 12], compressive sens-\ning [7] and p-Laplacian problems on graphs [16], or the subproblems in the power augmented Lagrangian\nmethod [24, 32, 21].\nAlthough often linesearch methods are still applicable under this setting, their additional backtrack-\ning procedures can be quite costly in practice. In response to this, this work investigates linesearch-free\n∗Work supported by: the Research Foundation Flanders (FWO) postdoctoral grant 12Y7622N and research projects\nG081222N, G033822N, and G0A0920N; Research Council KU Leuven C1 project No. C14/18/068; European Union’s Horizon\n2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 953348; Japan Society\nfor the Promotion of Science (JSPS) KAKENHI grant JP21K17710.\n†Department of Electrical Engineering (ESAT-STADIUS), KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium.\n{konstantinos.oikonomidis,puya.latafat,emanuel.laude,panos.patrinos}@esat.kuleuven.be\n‡Faculty of Information Science and Electrical Engineering (ISEE), Kyushu University, 744 Motooka, Nishi-ku 819-0395,\nFukuoka, Japan. andreas.themelis@ees.kyushu-u.ac.jp\n1arXiv:2402.06271v1  [math.OC]  9 Feb 2024adaptive methods under mere local Hölder continuity of∇f, covering in particular all continuously dif-\nferentiable semi-algebraic functions.1\nIn the Hölder differentiable setting, a notable approach was introduced in the seminal works [27, 9]\nthat rely on the notion of ε-oracles [9, Def. 1]. The main idea there is to approximate the Hölder smooth\nterm with the squared Euclidean norm, resulting in an approximate descent lemma [27, Lem. 2] that can\nbe leveraged for a linesearch procedure. More specifically, given η∈(0,1), some γ0>0, and an accuracy\nthreshold ε >0,Nesterov’s universal primal gradient (NUPG) method [27] consists of computing\nxk+1= proxγk+1g\u0000\nxk−γk+1∇f(xk)\u0001\n, (1a)\nwhere γk+1= 2γkηmkandmk∈is the smallest such that\nf(xk+1)≤f(xk) +⟨∇f(xk), xk+1−xk⟩+1\n2γk+1∥xk+1−xk∥2+1\n2ε. (1b)\nIt is clear that εis a parameter of the algorithm, a fact further illustrated by the convergence rate\nφ(xk)−φ(x⋆)≤ε\n2+1\nε1−ν\n1+ν2L2\n1+ν\nν∥x0−x⋆∥2\nk+1, (1c)\nwhere Lνis a modulus of Hölder continuity of the gradient: the coefficient of the 1/(k+1)term becomes\narbitrarily large as higher accuracy is demanded ε→0. This approach nevertheless allows handling\nHölder smooth problems in the same manner as Lipschitz smooth ones and thus implementing classical\nimprovements such as acceleration [27, 17, 13]. Moreover, its implementability has led to algorithms\nthat go beyond the classical forward-backward splitting, such as primal-dual methods [39, 31] and even\nvariational inequalities [34].\nIn the context of classical majorization-minimization, when the order and the modulo of smooth-\nness are known, the Hölder smoothness inequality itself has been used to generate descent without the\naformentioned approximation. Akin to Lipschitz continuity, Hölder continuity of ∇fwith constant H\ntranslates into a descent lemma inequality which, after addition of g(x), yields the upper bound to the\ncostφ\nφ(x)≤f(xk) +⟨∇f(xk), x−xk⟩+1\n(1+ν)λk+1,ν∥x−xk∥1+ν+g(x), (2)\nfor any λk+1,ν<1/H, cf. Fact 2.2(2). This was considered in [5, 15] in a general Banach space setup as\nwell as in [36, 4] for smooth but possibly nonconvex problems. With xk+1denoting the minimizer of the\nabove majorization model, the first-order optimality condition\n0∈∂g(xk+1) +∇f(xk) +1\nλk+1,ν∥xk+1−xk∥−(1−ν)(xk+1−xk) (3)\nreveals that the resulting iterations are essentially an Euclidean proximal gradient method for a particular\nstepsize γk+1:=λk+1,ν∥xk+1−xk∥1−νthat bears an implicit dependence on the future iterate xk+1.\nSuch majorize-minimize paradigm adopts λk+1,νasexplicitstepsize parameter, and is thus tied to\n(the knowledge of) the order of νof Hölder differentiability. Instead, we directly derive conditions on the\n“Euclidean” stepsize γk+1and rather regard λk+1,νas animplicitparameter, crucial to the convergence\nanalysis yet absent in the algorithm. (In contrast to λk+1,ν, the absence of the subscript νinγk+1\nemphasizes the independence of the Hölder exponent on the Euclidean stepsize; this notational convention\nwill be adopted throughout.)\nWe also remark that the implicit nature of the inclusion (3) is ubiquitous in algorithms that involve\nproximal terms of the form1\np∥x−xk∥pforp >1, such as the cubic Newton and tensor methods [29, 10, 6]\nor the high-order proximal point algorithm [24, 30, 32, 21]. Notice further that the Hölder proximal\ngradient update for non-Euclidean norms as described above differs from performing a “scaled” gradient\nstep followed by a higher-order proximal point step. Instead, that corresponds to the anisotropic proximal\ngradient method [20] for choosing ϕ(x) =H\n1+ν∥x∥1+ν\n1+ν.\n1This is a consequence of the Łojasiewicz inequality: for a continuous semi-algebraic function H:n→n, consider\nf(x, y) =∥x−y∥andg(x, y) =∥H(x)−H(y)∥in [3, Cor. 2.6.7].\n2Our contribution\nOur approach departs from and improves upon existing works in the following aspects.\n•Through a novel analysis of adaPGπ,r[18], we demonstrate that the class of linesearch-free adaptive\nmethods advanced in [25, 18, 26, 19] are convergent even in the locallyHölder differentiable setting,\ncovering in particular the class of semi-algebraic C1functions.\n•Our approach bridges the gap between two fundamental approaches to minimizing Hölder-smooth\nfunctions: it is both exactas in [5], in the sense that it does not involve nor depend on any predefined\naccuracy, and universal akin to the approach in [27], for it does not depend on (nor require the\nknowledge of) problem data such as the Hölder exponent ν.\n•We establish sequential convergence (as opposed to subsequential or approximate cost convergence)\nwith an exact rate\nmin\nk≤Kφ(xk)−minφ≤O(1\n(K+1)ν). (4)\nDifferently from existing analyses that rely on a global lower bound on the stepsizes to infer convergence\nand an O(1/(K+1))rate in the case ν= 1, we identify a scaling of the stepsizes and a lower bound thereof\nthat enables us to tackle the general ν∈(0,1)regime.\n•In numerical simulations we show that adaPGπ,π\n2performs well on a collection of locally and globally\nHölder smooth problems, such as classification with Hölder-smooth SVMs and a p-norm version of\nLasso. We show that our method performs consistenly better than Nesterov’s universal primal gradient\nmethod [27] and in many cases better than its fast variant [27], as well as the recently proposed\nauto-conditioned fast gradient method [22].\n2 Universal, adaptive, without approximation\nWe consider standard (Euclidean) proximal gradient steps\nxk+1= proxγk+1g(xk−γk+1∇f(xk)) (5)\nfor solving (P) under the following assumptions.\nAssumption 2.1. The following hold in problem (P):\nA1f:n→is convex and has locally Hölder continuous gradient of (possibly unknown ) order\nν∈(0,1].\nA2g:n→is proper, lsc, and convex.\nA3 A solution exists: arg min φ̸=∅.\nWe build upon a series of adaptive algorithms, starting with a pioneering gradient method in [25]\nand the follow-up studies [18, 26, 19, 40] which contribute with proximal extensions, larger stepsizes, and\ntighter convergence rate estimates. While standard results of proximal algorithms guarantee a descent\nalong the iterates in terms of the cost, distance to solutions, and fixed-point residual individually, the key\nidea behind this class of methods is to eliminate linesearch procedures by implicitly ensuring a descent\non a (time-varying) combination of the three (see Lemma 3.3). This was achieved under local Lipschitz\ncontinuity of ∇f, by exploiting local Lipschitz estimates at consecutive iterates xk−1, xk∈ngenerated\nby the algorithm such as\nℓk:=⟨xk−xk−1,∇f(xk)− ∇f(xk−1)⟩\n∥xk−xk−1∥2(6a)\n3and\nLk:=∥∇f(xk)− ∇f(xk−1)∥\n∥xk−xk−1∥(6b)\n(with the convention0\n0= 0).\nUnder the assumption considered in the aforementioned references of local Lipschitz continuity of ∇f,\nthat is, with ν= 1, the estimates ℓkandLkas in (6) remain bounded whenever xkandxk−1range in\na bounded set. Although this is no more the case in the setting investigated here, for ℓkandLkmay\ndiverge as xkandxk−1get arbitrarily close, we show that these Lipschitz estimates can still be employed\neven if ∇fis merely locally Höldercontinuous.\nSpecifically, we will consider the following stepsize update rule\nγk+1=γkmin(q\n1\nπ+γk\nγk−1,r\n1\n2[γ2\nkL2\nk−(2−π)γkℓk−(π−1)]+)\n(7)\nforsome π∈[1,2],where [·]+:= max { ·,0}.Thiscorrepondstotheupdateruleofthealgorithm adaPGπ,r\nof [18] specialized to the choice r=π/2for the second parameter. This restriction is nevertheless general\nenough to recover the update rules of [25, Alg. 1], [19, Alg. 2.1], and [26, Alg. 1] ( π= 1), as well as the\none of [26, Alg. 3] ( π=3/2), see [19, Rem. 2.4]. In particular, our analysis in the generality of π∈[1,2]\ndemonstrates that all these adaptive algorithms are convergent in the locally Hölder (convex) setting .\nA crucial challenge in the locally Hölder setting is the lack of a positive uniform lower bound for the\nstepsize sequence (γk)k∈generated by (7). To mitigate this, we factorize γkas\nγk=λk,ν∥xk−xk−1∥1−ν, (8)\nintroducing scaled stepsizes λk,νas suggested by the upper bound minimization procedure (3). This\nallows us to normalize the Hölder inequalities into Lipschitz-like ones, see Lemma 2.3. (Throughout, the\nsubscript νshall be used for quantities with dependence on νfor clarity of exposition.) Our analysis\nrelies on showing that this scaled stepsize is lower bounded whenever the second term in (7) is active (see\nLemma 3.6). We emphasize that this quantity, while crucial in our convergence analysis, does not appear\nin the algorithm, which only uses the estimates (6) and the update rule (7), neither of which depend on\n(the knowledge of) the local Hölder order ν.\nThe adaptive update (7) allows the stepsize to increase along the iterates, thus distinguishing it from\nthe well-established and popular approach put forward in [11], and followed by many extensions (see for\ninstance recent works by [23, 38, 8], among others), which however can tackle more general problems.\n2.1 Hölder continuity estimates\nIn this section, we set up some basic facts about Hölder continuity of ∇fthat will be essential in our\nanalysis. We again emphasize that our convergence analysis makes mere use of existence ofν∈(0,1](see\nAssumption 2.1(A1)), but the algorithm is independent of (the knowledge of) this exponent, cf. adaPGπ,π\n2\n(Algorithm 1).\nLocal Hölder continuity of ∇fof order ν(which we shall refer to as local ν-Hölder continuity of ∇f\nfor brevity) amounts to the existence, for every convex and bounded set Ω⊂n, of a constant LΩ,ν>0\nsuch that\n∥∇f(y)− ∇f(x)∥ ≤LΩ,ν∥y−x∥ν∀x, y∈Ω.\nNote that both norms are the standard Euclidean 2-norms. We remark that the limiting case ν= 0\namounts to subgradients of fbeing bounded on bounded sets, that is, to fbeing merely convex and\nreal valued with no differentiability requirements. Although not covered by our convergence results in\nSection 3.2, the preliminary lemmas collected in Section 3.1 still remain valid for any real-valued convex\nf. To clearly emphasize this fact, whenever applicable we shall henceforth specify “(possibly with ν= 0)”\n4Algorithm 1 AdaPGπ,π\n2: A universal adaptive proximal gradient method\nRequire starting point x−1∈n, stepsizes γ0≥γ−1>0, parameter π∈[1,2]\nInitialize: x0= proxγ0g(x−1−γ0∇f(x−1))\nRepeat for k= 0,1, . . .until convergence\n1:With ℓk, Lkgiven in (6), define the stepsize as (with notation [z]+= max {z,0}and convention1\n0=∞)\nγk+1=γkmin\n\ns\n1\nπ+γk\nγk−1,1q\n2[γ2\nkL2\nk−(2−π)γkℓk+ 1−π]+\n\n(10)\n2:xk+1= proxγk+1g(xk−γk+1∇f(xk))\nwhen invoking Assumption 2.1; in this case, the notation ∇fshall indicate any subgradient map ∇f:\nn→nwith∇f(x)∈∂f(x), and\nLΩ,0≤2 lipΩf (9)\nis bounded by twice the Lipschitz modulus for fonΩ[33, Thm. 24.7].\nThroughout, we will make use of the following inequalities, which reduce to well-known Lipschitz and\ncocoercivity properties of ∇fwhen ν= 1. The proof of the second assertion can be found in [36, Lem.\n1]; our cocoercivity-like claims are a slight refinement of known global and/or scalar versions, see e.g.,\n[1, Cor. 18.14] and [37, Prop. 1]. The simple proofs of this and the following lemma are provided in the\ndedicated Appendix A.\nFact 2.2 (Hölder-smoothness inequalities) .Suppose that h:n→is convex and ∇hisν-Hölder\ncontinuous with modulus LE,ν>0on a convex set E⊆nfor some ν∈[0,1]. Then, for every x, y∈E\nthe following hold:\n1.⟨x−y,∇h(x)− ∇h(y)⟩ ≤LE,ν∥x−y∥1+ν;\n2.h(y)−h(x)− ⟨∇ h(x), y−x⟩ ≤LE,ν\n1+ν∥x−y∥1+ν.\nIfν̸= 0and∇hisν-Hölder continuous on an enlarged set E:=E+B(0; diam E)with modulus LE,ν,\nthen the following local cocoercivity-type estimates also hold:\n3.2ν\n1+ν1\nL1/ ν\nE,ν∥∇h(x)− ∇h(y)∥1+ν\nν≤ ⟨x−y,∇h(x)− ∇h(y)⟩;\n4.ν\n1+ν1\nL1/ ν\nE,ν∥∇h(x)− ∇h(y)∥1+ν\nν≤h(y)−h(x)− ⟨∇h(x), y−x⟩.\nBased on the inequalities in Fact 2.2, given a sequence (xk)k∈we define local estimates of Hölder\ncontinuity of ∇fwith ν∈[0,1]as follows:\nℓk,ν:=⟨xk−xk−1,∇f(xk)− ∇f(xk−1)⟩\n∥xk−xk−1∥1+ν(11a)\nand\nLk,ν:=∥∇f(xk)− ∇f(xk−1)∥\n∥xk−xk−1∥ν. (11b)\nLet us draw some comments on these quantities. Considering the scaled stepsize λk,νgiven in (8), it is of\nimmediate verification that\nλk,νLk,ν=γkLk, λ k,νℓk,ν=γkℓk. (12)\n5Moreover, observe that\nℓk,ν≤Lk,ν≤LΩ,ν (13)\nholds whenever LΩ,νis aν-Hölder modulus for ∇fon a compact convex set Ωthat contains both xk−1\nandxk, the first inequality following from a simple application of Cauchy-Schwartz. We also remark that\ndefining ℓk,νandLk,νas above in place of a cocoercivity-like estimate\nck,ν:=\u00122ν\n1 +ν∥∇f(xk)− ∇f(xk−1)∥1+1\nν\n⟨xk−xk−1,∇f(xk)− ∇f(xk−1)⟩\u0013ν\ncauses no loss of generality, since each one among ck,ν,ℓk,νandLk,νcan be derived based on the other\ntwo. The use of ℓk,νandLk,νprovides nevertheless a simpler and more straightforward Hölder estimate,\ncontrary to ck,νwhich instead involves counterintuitive powers and coefficients, as well as a potentially\nlooser Hölder modulus, and fails to cover the limiting case ν= 0, cf. Fact 2.2(3).\nLet us denote the forward operator by\nHk:= id−γk∇f. (14)\nThe subgradient characterization of the proximal gradient update (5) then reads\nHk(xk−1)−Hk(xk)∈γk∂φ(xk). (15)\nAs in [19], the combined use of ℓk,νandLk,νyields a local Hölder modulus for the forward operator,\nthough in this work it will be convenient to express it with respect to the scaled stepsize λk,νas in (8).\nLemma 2.3. Letℓk,νandLk,νbe as in (11)for some xk−1, xk∈n, and let Hkbe as in (14)for some\nγk>0. Then, for any ν∈[0,1]and with λk,νas in(8)it holds that\nMk:=∥Hk(xk)−Hk(xk−1)∥2\n∥xk−xk−1∥2 = 1 + λ2\nk,νL2\nk,ν−2λk,νℓk,ν\n= 1 + γ2\nkL2\nk−2γkℓk.\nBeing MkaLipschitz estimate, unless ∇fis locally Lipschitz continuous there is no guarantee that\nMkis bounded for pairs xk−1, xkranging in a compact set. The ν-Hölder estimate of the forward operator\nHk, which instead isguaranteed to be bounded on bounded sets (for bounded (γk)k∈), is given by\nMk,ν:=Mk∥xk−xk−1∥1−ν=∥Hk(xk)−Hk(xk−1)∥\n∥xk−xk−1∥ν\n≤(1 +λk,νLk,ν)∥xk−xk−1∥1−ν, (16)\nwhere the inequality follows from the triangle inequality and the definition of Lk,ν, cf. (11b).\n3AdaPGπ,π\n2revisited\nIn this section adaPGπ,π\n2is presented in Algorithm 1 for solving composite problems (P). The main oracles\nof the algorithm are plain proximal and gradient evaluations. We refer to [2, §6] for examples of functions\nwith easy to evaluate proximal maps.\nAdaPGπ,π\n2incorporates the simple stepsize update rule (10) with a parameter π∈[1,2]that strikes\na balance between speed of recovery from small values (e.g., due to steep or ill-conditioned regions), and\nmagnitude of the stepsize dictated by the second term. If π= 1, whenever γ2\nkL2\nk≤γkℓk, the second term\nreduces to 1/0=∞, and γk+1=γkp\n1 +γk/γk−1strictly increases. On the other end of the spectrum, if\nπ= 2the update reduces to\nγk+1=γkmin(q\n1\n2+γk\nγk−1,1q\n2[γ2\nkL2\nk−1]+)\n,\n6where having the first term active (for instance if γkLk≤1) for two consecutive updates already ensures\nan increase in the stepsize owing to the simple observation that 1/2+p\n1/2>1.\nWhile adaPGπ,π\n2has the ability to recover from a potentially bad choice of initial stepsizes γ0, γ−1,\nthe behavior of the algorithm during the first iterations can be impacted negatively. To eliminate such\nscenarios, γ0can be refined by running offline proximal gradient updates. Specifically, starting from the\ninitial point x−1,γ0can be updated as the inverse of either one of (11b) or (11a) evaluated between x−1\nand the obtained point. If the updated stepsize is substantially smaller than the original one, the same\nprocedure may be repeated an additional time. Once a reasonable γ0is obtained, we suggest selecting\nγ−1small enough such thatq\n1\nπ+γ0\nγ−1≥1√2L0,ensuring that γ1would be proportional to the inverse of\nL0. We remark that the choice of γ−1does not affect the sequential convergence results of Theorem 3.8.\nIt does nevertheless affect the constant in our sublinear rate results of Theorem 3.9, through possibly\nhaving a larger ρmaxtherein, although this effect isa mere theoretical technicalitywith negligible practical\nimplications.\n3.1 Preliminary lemmas\nThis subsection collects some preliminary results adaptated from [25, 18, 26, 19] and that hold true under\nconvexity assumption without further restrictions. All the proofs, although differing only by negligible\nadaptations, are nevertheless provided in the dedicated Appendix B to demonstrate their independence\nofq∈[0,1]. In particular, the next lemma can be viewed as a counterpart of the well known firm\nnonexpansiveness (FNE) of proxγg, which is recovered when γk+1=γk, and offers a refinement of the\nnonexpansiveness-like inequality in [26, Lem. 12] that follows after an application of Cauchy-Schwarz.\nLemma 3.1 (FNE-like inequality) .Let Assumptions 2.1(A1) and 2.1(A2) hold (possibly with ν= 0).\nFor any γk>0and denoting ρk+1:=γk+1/γk, iterates (5)satisfy the following:\n1\nρk+1∥xk+1−xk∥2≤ ⟨Hk(xk−1)−Hk(xk), xk−xk+1⟩\n≤ρk+1∥Hk(xk−1)−Hk(xk)∥2.\nBy combining this inequality with the identity in Lemma 2.3 we obtain the following.\nCorollary 3.2. Let Assumptions 2.1(A1) and 2.1(A2) hold (possibly with ν= 0). For any γk>0, and\nwithMkandλk,νas in(14)and(8), iterates (5)satisfy\n∥xk+1−xk∥ ≤γk\nγk+1Mk∥xk−1−xk∥. (17)\nWe next present the main descent inequality taken from [19, Lem. 2.2]. Its proof is nevertheless in-\ncluded in the appendix to demonstrate its independence of ν: it relies on mere use of convexity inequalities\nandidentities involving Lk, ℓkas in Lemmas 2.3 and 3.1 to express norms and inner products in terms\nof∥xk−xk−1∥2. It reveals that for any π≥0andx⋆∈arg min φ, up to a proper stepsize selection, the\nfunction\nUπ\nk(x⋆):=1\n2∥xk−x⋆∥2+1\n2∥xk−xk−1∥2+γk(1 +πρk)Pk−1 (18)\nmonotonically decreases, where we introduced the symbol\nPk:=φ(xk)−minφ (19)\nfor the sake of conciseness.\nLemma 3.3 (main inequality) .Let Assumption 2.1 hold (possibly with ν= 0), and consider a sequence\ngenerated by proximal gradient iterations (5)with γk>0andρk+1:=γk+1/γk. Then, for any π≥0,\nx⋆∈arg min φ, and k≥1the following holds:\nUπ\nk+1(x⋆)≤ Uπ\nk(x⋆)−γk\u0000\n1 +πρk−πρ2\nk+1\u0001\nPk−1\n−\b1\n2−ρ2\nk+1\u0002\nγ2\nkL2\nk−γkℓk(2−π) + 1−π\u0003\t\n∥xk−xk−1∥2. (20)\n7Apparently, the stepsize update rule of adaPGπ,π\n2is designed so as to make the coefficients of Pk−1\nand∥xk−xk−1∥2on the right-hand side of (20) negative, so that the corresponding proximal gradient\niterates monotonically decrease the value of Uπ\nk(x⋆).\nLemma 3.4 (basic properties of adaPGπ,π\n2).Under Assumption 2.1 (possibly with ν= 0), the following\nhold for the iterates generated by adaPGπ,π\n2:\n1.(Uπ\nk(x⋆))k∈as defined in (18)decreases and converges to a finite value.\n2.The sequence (xk)k∈is bounded and admits at most one optimal limit point.\n3.Pmin\nK≤ Uπ\n1(x⋆)\u000e\u0000PK+1\nk=1γk\u0001\nfor every K≥1, where Pmin\nk:= min i≤kPi.\nRemark 3.5. The validity of Lemma 3.4(3) also when ν= 0hints that havingP\nk∈γk=∞suffices to\ninfer convergence results for the proximal subgradient method without differentiability of f. Whether, or\nunder which conditions, this is really the case is currently an open problem.\n3.2 Convergence and rates\nDistinguishing between the iterates in which the stepsize is updated according to the first or the second\nelement in the minimum of (10) will play a fundamental role in our analysis. For this reason, it is\nconvenient to introduce the following notation:\nK1:=n\nk∈|γk+1=γkq\n1\nπ+γk\nγk−1o\n(21a)\nand\nK2:=\\K1. (21b)\nUnlike its locally Lipschitz counterpart ( ν= 1), in the Hölder setting, a global lower bound for the\nstepsize sequence (γk)k∈cannot be expected. Nevertheless, a lower bound for the scaledstepsizes λk,ν\nwhenever k∈K2is sufficient to ensure convergence.\nLemma 3.6. Let Assumption 2.1 hold (possibly with ν= 0), and consider the iterates generated by\nadaPGπ,π\n2. Then, with K2as in(21b), for every k∈K2\nλk,ν≥1√\n2Lk,νρmaxand ρk+1≥1√\n2λk,νLk,ν, (22)\nwhere ρmax:= maxn\n1\n2\u0000\n1 +p\n1 + 4/π\u0001\n,γ0\nγ−1o\n.\nProof.Owing to ρk+1≤p\n1/π+ρkas ensured in (1), it can be verified with a trivial induction argument\nthat\nρk≤ρmax:= maxn\n1\n2(1 +p\n1 + 4/π), ρ0o\nfor all k≥0. (23)\nIfk∈K2, then γk+1coincides with the second update in (10), and thus\nρmax≥ρk+1=1r\n2h\nλ2\nk,νL2\nk,ν−(2−π)λk,νℓk,ν+ 1−πi\n+≥1√\n2λk,νLk,ν, (24)\ncompleting the proof.\nThe anticipated lower bound on (λk,ν)k∈K2will follow from (13), once boundedness of the sequence\n(xk)k∈generated by adaPGπ,π\n2is established.\nIn our convergence analysis we will need the following lemma that extends [18, Lem. B.2] by allowing\na vanishing stepsize. As a result it is only γk+1times the cost that can be ensured to converge to zero,\nwhich will nevertheless prove sufficient for our convergence analysis in the proof of Theorem 3.8.\n8Lemma 3.7. Suppose that a sequence (xk)k∈converges to an optimal point x⋆∈arg min φ, and for\nevery klet¯xk:= proxγk+1g(xk−γk+1∇f(xk))with (γk)k∈⊂++bounded. Then, (¯xk)k∈too converges\ntox⋆and(γk+1(φ(¯xk)−minφ))k∈→0.\nProof.By nonexpansiveness of the proximal mapping\n∥¯xk−x⋆∥ ≤ ∥ xk−x⋆−γk+1(∇f(xk)− ∇f(x⋆))∥ ≤ ∥ xk−x⋆∥+γk+1∥∇f(xk)− ∇f(x⋆)∥ →0,\nwhere we used the fact that x⋆= proxγk+1g(x⋆−γk+1∇f(x⋆))for any γk+1>0in the first inequality,\nand boundedness of γk+1in the last implication. Moreover, for every k∈one has\nγk+1(φ(¯xk)−minφ) =γk+1(f(¯xk)+g(¯xk)−minφ)≤γk+1(f(¯xk)−f(x⋆))−⟨xk−γk+1∇f(xk)−¯xk, x⋆−¯xk⟩,\nwhere in the inequality subgradient characterization of proximal mapping. The inner product vanishes\nsince both xkand¯xkconverge to x⋆, and the claim follows by continuity of fand lower semicontinuity\nofφ.\nTheorem 3.8 (convergence) .Under Assumption 2.1, the sequence (xk)k∈generated by adaPGπ,π\n2con-\nverges to some x⋆∈arg min φ.\nProof.We first show two intermediate claims.\nClaim 3.8(a): If ν >0, then infk∈Pk= 0, and in particular (xk)k∈admits a (unique) optimal limit\npoint.\nIfsupk∈γk=∞, then we know from Lemma 3.4(3) that lim inf k→∞Pk= 0. Suppose instead that\n(γk)k∈is bounded. Then, the set K2as in (21b) must be infinite. Since Lk,ν≤LΩ,ν, it follows from\nLemma 3.6 that\nλk,ν≥λmin,ν:=1√\n2LΩ,νρmax∀k∈K2, (25)\nhence from (45) thatX\nk∈K2∥xk−xk−1∥1−ν(1 +πρk−πρ2\nk+1)Pk−1<∞.\nNoticing that 1 +πρk−πρ2\nk+1= 0fork /∈K2, necessarily 1 +πρk−πρ2\nk+1̸→0asK2∋k→ ∞(or,\nequivalently, as k→ ∞), for otherwise lim inf k→∞ρk>1and thus γk↗ ∞. Therefore, there exists\nan infinite set ˜K2⊆K2such that 1 +πρk−πρ2\nk+1≥ε >0for all k∈˜K2, implying that\nX\nk∈˜K2∥xk−xk−1∥1−νPk−1<∞.\nThus, lim˜K2∋k→∞∥xk−xk−1∥= 0(orlim infk∈˜K2Pk−1= 0, in which case there is nothing to show).\nFor any x⋆∈arg min φwe thus have\n0≤Pk=φ(xk)−minφ≤ ⟨xk−x⋆,xk−1−xk\nγk−\u0000\n∇f(xk−1)− ∇f(xk)\u0001\n⟩\n≤ ∥xk−x⋆∥\u0010\n1\nγk∥xk−1−xk∥+∥∇f(xk−1)− ∇f(xk)∥\u0011\n(26)\n=∥xk−x⋆∥\u0010\n1\nλk,ν∥xk−1−xk∥1−ν∥xk−1−xk∥+Lk,ν∥xk−1−xk∥ν\u0011\n≤ ∥xk−x⋆∥∥xk−1−xk∥ν\u0010\n1\nλmin,ν+LΩ,ν\u0011\n∀k∈˜K2.\nSince ν >0, by taking the limit as ˜K2∋k→ ∞we obtain that lim˜K2∋k→∞Pk= 0.\n9Claim 3.8(b): If ν >0and(γk)k∈is bounded, then (xk)k∈converges to a solution.\nSuppose first that (γk)k∈is bounded. Consider a subsequence (xk)k∈Ksuch that limK∋k→∞Pk= 0,\nwhich exists and converges to a solution x⋆by Claim 3.8(a). Since (γk)k∈is bounded, in light of\nLemma 3.7 also xk+1→x⋆and, in turn, xk+2→x⋆as well. Then,\nUπ\nk+1(x⋆) =1\n2∥xk+1−x⋆∥2+1\n2∥xk+1−xk∥2+γk(1 +πρk+1)Pk→0asK∋k→ ∞ ,\nand thus1\n2∥xk−x⋆∥ ≤ Uπ\nk(x⋆)→0ask→ ∞, since the entire sequence (Uπ\nk(x⋆))k∈is convergent.\nTo conclude the proof of the theorem, it remains to show that also in case (γk)k∈is unbounded the\nsequence (xk)k∈converges to a solution. To this end, let us suppose now that γkis not bounded. This\ncase requires requires a few more technical steps, which can nevertheless almost verbatim be adaptated\nfrom the proof of [19, Thm. 2.4(ii)]; we emphasize that the difference here is that the stepsize sequence\nis not guaranteed to be bounded away from zero.\nWe start by observing that Claim 3.8(a) and Lemma 3.4(2) ensure that an optimal limit point x⋆∈\narg min φexists. It then suffices to show that Uπ\nk(x⋆)converges to zero. To arrive to a contradiction,\nsuppose that this is not the case, that is, that U:= lim k→∞Uπ\nk(x⋆)>0. We shall henceforth proceed by\nintermediate claims that follow from this condition, eventually arriving to a contradictory conclusion.\nContradiction claim 1∗: For any K⊆,limK∋k→∞xk=x⋆holds iff limK∋k→∞γk+1=∞.\nThe implication “ ⇐” follows from\nγkPk−1≤ Uπ\nk(x⋆)<∞\nsince (xk)k∈is bounded and x⋆is its unique optimal limit point.\nSuppose now that (xk)k∈K→x⋆. To arrive to a contradiction, up to possibly extracting suppose that\n(γk+1)k∈K→¯γ∈[0,∞).Then,itfollowsfromLemma3.7that (xk+1)k∈K→x⋆and(γk+1Pk+1)k∈K→0.\nAs shown in (23)\nρk≤ρmaxfor all k≥0 (27)\nwhich in turn implies (γk+2Pk+1)k∈K→0and that (γk+2)k∈Kis also bounded, we may iterate and infer\nthat also (xk+2)k∈Kconverges to x⋆. Recalling the definition of Uπ\nkin (18),\nUπ\nk+2(x⋆):=1\n2∥xk+2−x⋆∥2+1\n2∥xk+2−xk+1∥2+γk+2(1 +πρk+2)Pk+1→0,\ncontradicting U= lim K∋k→∞Uπ\nk+2(x⋆)>0.\nContradiction claim 2∗: Suppose that (xk)k∈K→x⋆; then also (xk−1)k∈K→x⋆.\nIt follows from the previous claim that limK∋k→∞γk+1=∞. Because of (27), one must also have\nlimK∋k→∞γk=∞. Invoking again the previous claim, by the arbitrarity of the index set Kthe assertion\nfollows.\nContradiction claim 3∗: Suppose that (xk)k∈K→x⋆; then (γk−1Lk−1)k∈K→ ∞and(ρk)k∈K→0.\nUsing the previous claim twice, xk−1, xk−2→x⋆asK∋k→ ∞. In particular\nlim\nK∋k→∞∥xk−1−xk−2∥2= 0. (28)\nFrom the expression (18) of Uπ\nkwe then have\nlim\nK∋k→∞γk(1 +πρk)Pk−1=U, (29)\nwhere we remind that by contradiction assumption U:= lim k→∞Uπ\nk(x⋆)>0. Denoting C:=ρmax(1 +\nπρmax), we have\nγk−1Pk−1≤ ∥xk−1−x⋆∥\u0000\n∥xk−1−xk−2∥+γk−1∥∇f(xk−1)− ∇f(xk−2)∥\u0001\n=∥xk−1−x⋆∥\u0000\n∥xk−1−xk−2∥+γk−1Lk−1∥xk−1−xk−2∥ν\u0001\n10for every k. Then, by (29)\n0< U = lim\nK∋k→∞γk(1 +πρk)Pk−1= lim inf\nK∋k→∞ρk(1 +πρk)γk−1Pk−1\n≤ρmax(1 +πρmax) lim inf\nK∋k→∞∥xk−1−x⋆∥\u0000\n∥xk−1−xk−2∥+γk−1Lk−1∥xk−1−xk−2∥ν\u0001\nwhich yields the first claim owing to (28) and ν >0.\nThis along with the update rule (10) implies\nρk≤1\n2\u0002\nγ2\nk−1L2\nk−1−(2−π)γk−1ℓk−1+ 1−π\u0003→0,\nas claimed.\nHaving shown the above claims, the proof is concluded as in [18, Thm. 2.4(ii)] by constructing a specific\nunbounded stepsize sequence and using claims 1∗and3∗to obtain the sought contradiction.\nWhile a global lower bound for the stepsizes (γk)k∈is not available, it is at the moment unclear\nwhether one for the (entire) scaled sequence (λk,ν)k∈exists. Nevertheless, with Pkas in (19), a lower\nbound for an alternative scaled sequence (γk+1P−(1−ν)/ν\nk)k∈does exist, thanks to which the following\nconvergence rate can be achieved.\nTheorem 3.9 (sublinear rate) .Suppose that Assumption 2.1 holds. Then, the following sublinear rate\nholds for the iterates generated by adaPGπ,π\n2:\nmin\nk≤KPk≤max(\nUπ\n1(x⋆)\nγ0(K+ 1),C(π, ν)Uπ\n1(x⋆)1+ν\n2LΩ,ν\n(K+ 1)ν)\n,\nwhere C(π, ν) =√\n2(√π)ν\u0000√\n2ρmax+ 1\u00011−ν.\nProof.LetLΩ,νdenote a modulus of ν-Hölder continuity for ∇fon a compact convex set Ωcontaining\n(xk)k∈. We proceed by intermediate claims.\nClaim 3.9(a): If λk,ν≤1\nLΩ,ν, then Pk≤Pk−1.\nLete∇φ(xk):=∇f(xk) +e∇g(xk), where e∇gis as in (42). Then, e∇φ(xk)∈∂φ(xk)and thus\nφ(xk−1)≥φ(xk) +⟨e∇φ(xk), xk−1−xk⟩\n=φ(xk) +1\nγk⟨Hk(xk−1)−Hk(xk), xk−1−xk⟩\n=φ(xk) +1\nγk∥xk−xk−1∥2−Lk,ν∥xk−1−xk∥1+ν\n=φ(xk) +\u00001\nλk,ν−Lk,ν\u0001\n∥xk−1−xk∥1+ν,\nestablishing the claim.\nWe next aim at establishing a lower bound on the stepsize sequence in terms of P1−ν\nν\nk. To simplify the\nexposition, we now fix x⋆∈arg min φand denote\n˜Cπ\nν(ν):=q\n2Uπ\n1(x⋆)\u00001\nν+LΩ,ν\u0001\n. (30)\nClaim 3.9(b): For every k∈it holds that Pk≤˜Cπ\nν(λk,ν)∥xk−xk−1∥ν.\nWe begin by observing that\ne∇φ(xk):=1\nγk\u0000\nHk(xk−1−Hk(xk))\u0001\n∈∂φ(xk)\n11owing to (42). Combined with (16) it follows that\n∥e∇φ(xk)∥ ≤\u00001\nλk,ν+Lk,ν\u0001\n∥xk−xk−1∥ν≤\u00001\nλk,ν+LΩ,ν\u0001\n∥xk−xk−1∥ν.\nMoreover, by convexity,\nPk=φ(xk)−minφ≤ ⟨e∇φ(xk), xk−x⋆⟩ ≤ ∥e∇φ(xk)∥∥xk−x⋆∥ ≤˜Cπ\nν(λk,ν)\nq\n2Uπ\n1(x⋆)\u00001\nλk,ν+LΩ,ν\u0001\n∥xk−xk−1∥ν\nas claimed, where the last inequality uses the fact that1\n2∥xk−x⋆∥2≤ Uπ\nk(x⋆)≤ Uπ\n1(x⋆).\nWe next analyze two possible cases for any iteration index k≥0.\nClaim 3.9(c): For any k∈K2,γk+1≥1√\n2LΩ,ν\u0010\nPk\n˜Cπν(λmin,ν)\u00111−ν\nν.\nSince Lk,ν≤LΩ,ν, as shown in Lemma 3.6 λk,ν≥λmin,ν=1√\n2LΩ,νρmaxholds for every k∈K2.\nMoreover, by definition of K2,\nγk+1=γkr\n2h\nλ2\nk,νL2\nk,ν−(2−π)λk,νℓk,ν+ 1−πi\n+\n=λk,ν∥xk−xk−1∥1−ν\nr\n2h\nλ2\nk,νL2\nk,ν−(2−π)λk,νℓk,ν+ 1−πi\n+≥∥xk−xk−1∥1−ν\n√\n2Lk,ν≥∥xk−xk−1∥1−ν\n√\n2LΩ,ν(31)\nholds for all k∈K2. By using the lower bound\n∥xk−xk−1∥1−ν≥\u0010Pk\n˜Cπν(λk,ν)\u00111−ν\nν≥\u0010Pk\n˜Cπν(λmin,ν)\u00111−ν\nν\nin Claim 3.9(b) raised to the power1−ν\nνthe claim follows.\nClaim 3.9(d): For any k∈K1,γk+1≥\n\n1√\n2πLΩ,ν\u0000Pk\n˜Cπν(λmin,ν)\u00011−ν\nνif (K2̸=∅and) k≥minK2,\n\u0000\n1 +1\nπ\u0001k\n2γ0 otherwise.\nLet\nK2,<k:=K2∩ {0,1, . . . , k −1}\ndenote the (possibly empty) set of all iteration indices up to k−1such that the first term in (10) is\nstrictly larger than the second one.\nIfK2,<k=∅, then ρt+1=p\n1/π+ρtholds for all t≤k, which inductively gives ρ2\nt≥1 +1\nπfor all\nt= 1, . . . , K(since ρ0≥1). We then have\nγ2\nk+1=γ2\n0Qk\nt=1ρ2\nt+1≥(1 +1\nπ)kγ2\n0. (32)\nSuppose instead that K2,<k̸=∅, and let n2,kdenote its largest element:\nn2,k:= max K2,<k= max\u001a\ni < k|γi+1< γiq\n1\nπ+ρi\u001b\n.\nWe will use the fact that the update rule ρi+1=q\n1\nπ+ρiimplies that ρiincreases along consecutive\nindices in K1, and in particular\ni, . . . , j ∈K1⇒ ρi+1=q\n1\nπ+ρi≥1√πand ρj+1≥ ··· ≥ ρi+2≥r\n1\nπ+q\n1\nπ≥r\n1\n2+q\n1\n2>1,\n12where the second-last inequality uses the fact that π≤2. In particular,\ni, i+ 1, . . . , j ∈K1⇒Qj+1\nt=i+1ρt≥1√π(33)\n(this being also trivially true for an empty product, since π≥1).\nWe consider two possible subcases:\n⋄First, suppose that the index j:= maxn\nn2,k≤i≤k|λi,ν>1\nLΩ,νo\nexists. Schematically,\n∈K2\nn2,k,∈K1\n. . . , j, . . . , k\nλi,ν≤1\nLΩ,νand λj,ν>1\nLΩ,ν. (34)\nBy definition of n2,k, all indices between jandkare in K1, and thus\nγk+1=γjk+1Y\ni=j+1ρi(33)\n≥1√πγj=1√πλj,ν∥xj−xj−1∥1−ν\n(34)\n>1√π1\nLΩ,ν∥xj−xj−1∥1−νClaim 3.9(b)\n≥1√π1\nLΩ,ν\u0010\nPj\n˜Cπν(λj,ν)\u00111−ν\nν(34)\n>1√π1\nLΩ,ν\u0010\nPj\n˜Cπν(1/LΩ,ν)\u00111−ν\nν.\nSince λi,ν≤1\nLΩ,νholds for all i=j+ 1, . . . , k, it follows from Claim 3.9(a) that Pk≤Pj, and thus\nγk+1≥1√π1\nLΩ,ν\u0010\nPk\n˜Cπν(1/LΩ,ν)\u00111−ν\nν. (35)\n⋄Alternatively, it holds that λj,ν≤1\nLΩ,νfor all j=n2,k, . . . , k, and in particular by virtue of\nClaim 3.9(b) we have that Pk≤Pnk,2. Arguing as before,\nγk+1=γnk,2+1k+1Y\ni=nk,2+1ρi(33)\n≥1√πγnk,2+1Claim 3.9(c)\n≥1√\n2πLΩ,ν\u0010Pnk,2\n˜Cπν(λmin,ν)\u00111−ν\nν≥1√\n2πLΩ,ν\u0010\nPk\n˜Cπν(λmin,ν)\u00111−ν\nν.(36)\nCombining (35) and (36)\nγk+1≥min(\n1\n˜Cπν(1/LΩ,ν)1−ν\nν,1√\n2˜Cπν(λmin,ν)1−ν\nν)\nP1−ν\nν\nk√πLΩ,ν=1√\n2πLΩ,ν˜Cπν(λmin,ν)1−ν\nνP1−ν\nν\nk,\nwhere the identity uses the fact that the minimum is attained at the first element, having ˜Cπ\nν(ν)\ndecreasing in ν >0and1\nLΩ,ν≥λmin,ν=1√\n2ρmaxLΩ,ν(since ρmax≥1).\nFinally, combining Claims 3.9(c) and 3.9(d) and noting that\n1√\n2πLΩ,ν \nPk\n˜Cπν(λmin,ν)!1−ν\nν\n=1√\n2πL1/ν\nΩ,ν \n1p\n2Uπ\n1(x⋆)\u0000√\n2ρmax+ 1\u0001!1−ν\nν\nP1−ν\nν\nk,\nwe conclude that\nγk+1≥\n\n1\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nνP1−ν\nν\nkif (K2̸=∅and) k≥minK2,\n\u0000\n1 +1\nπ\u0001k\n2γ0 otherwise\n13holds for any k∈, where\nC(π, ν) =√\n2LΩ,ν√πν\u0000\n1 +√\n2ρmax\u00011−ν\nis as in the statement. The sum of stepsizes can be lower bounded by\nPK+1\nk=1γk=Pk0\nk=1γk+PK+1\nk=k0γk≥γ0k0X\nk=1(1 +1\nπ)k\n2+1\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nνK+1X\nk=k0P1−ν\nν\nk−1\n≥γ0k0X\nk=1(1 +1\nπ)k\n2+K+ 1−k0\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nν\u0000\nmin\nk≤KPk\u00011−ν\nν\n≥γ0k0+K+ 1−k0\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nν\u0000\nmin\nk≤KPk\u00011−ν\nν\n≥min(\nγ0,1\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nν\u0000\nmin\nk≤KPk\u00011−ν\nν)\n(K+ 1).\nTherefore, in light of Lemma 3.4(3) we have\nUπ\n1(x⋆)≥min\nk≤KPkK+1X\nk=1γk≥min(\nγ0min\nk≤KPk,1\nUπ\n1(x⋆)1−ν\n2νC(π, ν)1\nν\u0000\nmin\nk≤KPk\u00011\nν)\n(K+ 1).\nEquivalently, for every Kit holds that\neither min\nk≤KPk≤Uπ\n1(x⋆)\nγ0(K+ 1)or min\nk≤KPk≤Uπ\n1(x⋆)1+ν\n2C(π, ν)\n(K+ 1)ν,\nresuting in the claimed bound.\nIn the locally Lipschitz setting ν= 1, the above rate matches exactly the O(1/(K+1))of [18, Thm. 1.1]\n(with r=π/2), including the coefficients. Despite such a worst-case sublinear rate, the fast behavior of the\nalgorithm in practice can be explained by utilization of large stepsizes and the bound in Lemma 3.4(3).\nWe also remark that under (local) strong convexity, up to modifying the stepsize update similarly to\n[25, §2.3], a contraction can be established in terms of Uπ\nk(x⋆)in Lemma 3.3. We refer the reader to [25]\nfor this approach and postpone a more detailed analysis to a future work.\n4 Experiments\nWe demonstrate the performance of adaPGπ,π\n2on a range of simulations for three different choices of\nπ. We compare against the baseline and state-of-the-art methods: Nesterov’s universal primal gradient\nmethod ( NUPG) [27], its fast variant ( F-NUPG) [27], as well as the recently proposed auto-conditioned\nfast gradient method ( AC-FGM) [22].\nWhile adaPGπ,π\n2onlyinvolvesgradient evaluations,the othermethodsadditionallyrequireevaluations\nof the objective. In all applications considered in this section, the smooth part f(x)takes the form\nψ(Ax)where Ais the design matrix containing the data. Consequently, matrix vector products y=Ax\nandA⊤∇ψ(y), each of complexity O(mn), constitute the most costly operations. For the sake of a fair\ncomparison, we store vectors that can be reused in subsequent evaluations, and plot the progress of the\nalgorithm in terms of calls to AandA⊤. As a result, denoting with #Athe number of calls to AandA⊤,\nand with #ls≥1the number of linesearch trials (including the successful ones), each iteration involves:\n•NUPG: 1 call to ∇fand#LStof; by exploiting linearity, #A= 1 + # LS\n•F-NUPG:#LScalls to ∇fand2×#LStof; by exploiting linearity, #A= 3 + # LS\n•AdaPGπ,π\n2: 1 call to ∇f; using linearity, #A= 2\n•AC-FGM: 1 call to ∇fand 1 to f; by linearity, #A= 2.\n140 5000 10000 15000 20000 25000 30000 35000 40000\n#calls to A,A/latticetop1014\n1012\n1010\n108\n106\n104\n102\n100102ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGm=200, n=500, k=60, λ=1.0, p=1.5\n0 10000 20000 30000 40000 50000\n#calls to A,A/latticetop1014\n1012\n1010\n108\n106\n104\n102\n100102ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGm=500, n=1000, k=100, λ=1.0, p=1.5\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1014\n1012\n1010\n108\n106\n104\n102\n100102ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGm=200, n=500, k=60, λ=1.0, p=1.7\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1014\n1012\n1010\n108\n106\n104\n102\n100102ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGm=500, n=1000, k=100, λ=1.0, p=1.7Figure 1: p-norm Lasso with varying powers p. It can be seen that adaPGπ,π\n2performs better than NUPG\nin all cases in terms of calls to A, A⊤. In this experiment adaPGπ,π\n2also performs consistently better than\nthe accelerated algorithms AC-FGM andF-NUPG.\nImplementation details In all experiments the solvers use x= 0as the initial point, and are executed\nwith the same initial stepsize computed as follows. First, we ran an offline proximal gradient update\nstarting from the initial point and computing the stepsize as the inverse of (11b) evaluated between the\ninitial and the obtained point. This procedure was repeated one additional time in case the obtained\nstepsize was substantially smaller than the original one. In the case of AC-FGM, in addition, we used\nthe procedure of [22] which can also be found in Appendix C. The implementation for reproducing the\nexperiments is publicly available.2\n4.1 p-norm Lasso\nIn this experiment we consider a variant of Lasso in which the squared 2-norm is replaced with a p-norm\nraised to some power p∈(1,2):\nmin\nx∈n1\np∥Ax−b∥p\np+λ∥x∥1, (37)\n2https://github.com/EmanuelLaude/universal-adaptive-proximal-gradient\n150 500 1000 1500 2000 2500 3000 3500 4000\n#calls to A,A/latticetop107\n106\n105\n104\n103\n102\n101\nϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGphishing (λ=0.01, p=1.5)\n0 500 1000 1500 2000 2500 3000 3500 4000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGa9a (λ=0.01, p=1.5)\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGw8a (λ=0.005, p=1.5)\n0 200 400 600 800 1000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\nϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGcovtype.binary (λ=20, p=1.5)\n0 500 1000 1500 2000 2500 3000 3500 4000\n#calls to A,A/latticetop107\n106\n105\n104\n103\n102\n101\nϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGphishing (λ=0.005, p=1.5)\n0 500 1000 1500 2000 2500 3000 3500 4000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGa9a (λ=0.005, p=1.5)\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGw8a (λ=0.003, p=1.5)\n0 1000 2000 3000 4000 5000 6000\n#calls to A,A/latticetop1013\n1011\n109\n107\n105\n103\nϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGcovtype.binary (λ=10, p=1.5)\n0 1000 2000 3000 4000 5000 6000 7000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGphishing (λ=0.001, p=1.5)\n0 1000 2000 3000 4000 5000 6000 7000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGa9a (λ=0.001, p=1.5)\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGw8a (λ=0.001, p=1.5)\n0 2000 4000 6000 8000 10000 12000 14000\n#calls to A,A/latticetop106\n105\n104\n103\n102\nϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGcovtype.binary (λ=1.0, p=1.5)Figure 2: Classification with Hölder-smooth SVMs. It can be seen that adaPGπ,π\n2consistently outperforms\nNUPGin terms of calls to A, A⊤.\nforA∈[−1,1]m×n,b∈mwith n > mandλ >0. The first term in (37) is differentiable with globally\nHöldercontinuousgradientwithorder ν=p−1.Theproximalmappingofthesecondtermistheshrinkage\noperation which is computable in closed form. To assess the performance of the different algorithms we\ngenerate random instances of the problem using a p-norm version of the procedure provided in [28]. In\nFigure 1 we depict convergence plots for the different methods applied to random instances with varying\ndimensions of A∈m×n, powers p∈(1,2)and number of nonzero entries kof the solution x⋆. It can be\nseen that adaPGπ,π\n2performs consistently better than Nesterov’s universal primal gradient [27] method\nNUPGwith ε= 1e−12in terms of evaluations of forward and backward passes (calls to A, A⊤). In this\nexperiment adaPGπ,π\n2also performs consistently better than the accelerated algorithms AC-FGM and\nF-NUPG.\n4.2 Hölder-smooth SVMs with ℓ1-regularization\nIn this subsection we assess the performance of the different algorithms on the task of classification\nusing a p-norm version of the SVM model. For that purpose we consider a subset of the LibSVM binary\nclassification benchmark that consists of a collection of datasets with varying number mof examples ai,\nfeature dimensions nand sparsity. Let (aj, bj)m\nj=1with bj∈ {− 1,1}andaj∈ndenote the collection of\ntraining examples. Then we consider the minimization problem\nmin\nx∈n1\nmmX\nj=11\npmax{0,1−bj⟨aj, x⟩}p+λ∥x∥1. (38)\nforp∈(1,2). The loss function is globally Hölder smooth with order ν=p−1while the proximal\nmapping of the second term can be solved in closed form. The results are depicted in Figure 2.\n160 2000 4000 6000 8000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGmushrooms (λ=0.005, p=1.5)\n0 250 500 750 1000 1250 1500 1750\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGw8a (λ=0.003, p=1.5)\n0 2000 4000 6000 8000 10000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGmushrooms (λ=0.001, p=1.5)\n0 250 500 750 1000 1250 1500 1750 2000\n#calls to A,A/latticetop1012\n1010\n108\n106\n104\n102\n100ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGw8a (λ=0.001, p=1.5)Figure 3: p-norm regularized logistic regression. In the present cases adaPGπ,π\n2performs better than the\nbaseline algorithms.\n4.3 Logistic regression with p-norm regularization\nIn this subsection we consider classification with the logistic loss and a smooth p-norm regularizer, for\nsome p∈(1,2). The problem can be cast in convex composite minimization form as follows\nmin\nx∈n1\nmmX\nj=1ln(1−exp(bj⟨aj, x⟩)) +λ1\np∥x∥p\np. (39)\nUnlike the previous classification model this yields a smooth optimization problem. Hence we perform\ngradient steps with respect to φand choose the nonsmooth term g≡0. The results are depicted in\nFigure 3.\n174.4 Mixture p-norm regression\nIn this final experiment we consider the following mixture model:\nmin\nx∈B2(r)JX\nj=1∥Ajx−bj∥pj\npj, (40)\nwhere B2(r)is the 2-norm ball with radius randpj∈(1,2]. Since the pjare not identical the smooth\npart in Equation (40) is merely locally Hölder smooth. The nonsmooth part gis the indicator function\nof the set B2(r). In Figure 4 we compare the performance for J= 6with p= (1.8,1.7,1.6,1.5,1.5,1.5)\nand two different values of nwhere the entries of Aj∈[−1,1]mj×nandbj∈[−1,1]mjare uniformly\ndistributed between [−1,1]with m= (400 ,300,400,100,100,300).\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1013\n1011\n109\n107\n105\n103\n101\n101ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGn=1000, r=0.1\n0 1000 2000 3000 4000 5000 6000 7000 8000\n#calls to A,A/latticetop1013\n1011\n109\n107\n105\n103\n101\n101ϕ(x)−ϕ(x)\nAC-FGM\nAdaPG1.2,0.6\nAdaPG1.5,0.75\nAdaPG2,1\nF-NUPG\nNUPGn=2000, r=0.1\nFigure 4: Mixture p-norm regression with ball constraint. It can be seen that adaPGπ,π\n2performs best\non this comparison. In contrast to the previous experiments, the problem involves a heterogeneous sum\nwhose summands exhibit varying orders of Hölder differentiability.\n5 Conclusions\nIn this paper we showed that adaptive proximal gradient methods are universal, in the sense that they\nconverge under mere local Hölder gradient continuity of anyorder. This is achieved through a unified\nanalysis of adaPGπ,π\n2that encapsulates existing methods for different values of the parameter π∈[1,2].\nSequential convergence along with an O(1/Kν)rate for the cost is established. Remarkably, the analysis\nand implementation of the algorithm does not require a priori knowledge of the order ν∈(0,1]of Hölder\ncontinuity. In other words, adaPGπ,π\n2and its analysis automatically adapts to the best possible ν.\nThe validity of some of the auxiliary results for ν= 0is an encouraging indication that the algorithm\ncould potentially cope with plain real valuedness of f, waiving differentiability assumptions. Whether\nthis is really the case remains an interesting open question.\nMoreover, our experiments demonstrate that adaPGπ,π\n2consistently outperforms NUPGon a diverse\ncollection of challening convex optimization problems with both locally and globally Hölder smooth costs.\nIn many cases we observe that adaPGπ,π\n2performs better than the accelerated algorithms F-NUPG and\nAC-FGM. We conjecture that adaPGπ,π\n2exploits a hidden Hölder growth that accelerated algorithms\ncannot take advantage from (as is known for the classical Euclidean case under metric subregularity). In\nfuture work we aim to extend our analysis to nonconvex and stochastic settings.\n18Appendix\nA Proofs of Section 2.1\nProof of Fact 2.2 (Hölder-smoothness inequalities). The proof of assertion 2.2(1) follows by applying\nthe Cauchy-Schwarz inequality. For assertion 2.2(2) when ν > 0we refer the reader to [36, Lem. 1];\nalthough the reference assumes global Hölder continuity, the arguments therein only use Hölder continuity\nof∇fon the segment [x, y]. For the case ν= 0, for any x, y∈Eand∇h(x)∈∂h(x),∇h(y)∈∂h(y), we\nhave\nh(y)≤h(x) +⟨∇h(y), y−x⟩\n=h(x) +⟨∇h(x), y−x⟩+⟨∇h(y)− ∇h(x), y−x⟩\n≤h(x) +⟨∇h(x), y−x⟩+∥∇h(y)− ∇h(x)∥∥y−x∥\n≤h(x) +⟨∇h(x), y−x⟩+LE,0∥y−x∥,\nwhere the first inequality follows from convexity of h.\nWe now turn to the last two claims, and thus restrict to the case ν >0. We will only show assertion\n2.2(4), as 2.2(3) in turn follows by exchanging the roles of xandyand summing the corresponding\ninequalities. The proof follows along the lines of [2, Thm. 5.8(iii)] and is included for completeness to\nhighlight the need of the enlarged set E. We henceforth fix x, y∈E⊆E. Since ∇fisν-Hölder continuous\ninEwith modulus LE,ν, it follows from assertion 2.2(2) that\nf(z)≤f(y) +⟨∇f(y), z−y⟩+LE,ν\n1+ν∥z−y∥1+ν∀z∈E. (41)\nLetlx(y):=f(y)−f(x)−⟨∇f(x), y−x⟩,andnotethat lxisaconvexfunctionwith ∇lx(y) =∇f(y)−∇f(x).\nFor any z∈E, we have\nlx(z) =f(z)−f(x)− ⟨∇f(x), z−x⟩\n(41)\n≤f(y) +⟨∇f(y), z−y⟩+LE,ν\n1+ν∥z−y∥1+ν−f(x)− ⟨∇f(x), z−x⟩\n=f(y)−f(x)− ⟨∇f(x), y−x⟩+⟨∇f(y)− ∇f(x), z−y⟩+LE,ν\n1+ν∥z−y∥1+ν\n=lx(y) +⟨∇lx(y), z−y⟩+LE,ν\n1+ν∥z−y∥1+ν.\nNoticing that ∇lx(x) = 0, it follows from convexity that xis a global minimizer of lx, hence that minlx=\nlx(x) = 0. Let us denote v:=1\n∥∇lx(y)∥∇lx(y)and define z=y− ∥∇ lx(y)∥1/νL−1/ν\nE,νv. Note that\n∥z−y∥=\u0010\n∥∇lx(y)∥\nLE,ν\u00111\nν=\u0010\n∥∇f(y)−∇f(x)∥\nLE,ν\u00111\nν≤ ∥y−x∥,\nand in particular z∈E+B(y;∥y−x∥)⊆E. From the previous inequality we get\n0 = min lx=lx(x)≤lx\u0000\ny− ∥∇ lx(y)∥1/νL−1/ν\nE,νv\u0001\n.\n(41)\n≤lx(y)− ∥∇ lx(y)∥1/νL−1/ν\nE,ν⟨∇lx(y), v⟩+LE,ν\n1+ν1\nL1+1/ν\nE,ν∥∇lx(y)∥1+1\nν\n=f(y)−f(x)− ⟨∇f(x), y−x⟩ −ν\nν+11\nL1/ν\nE,ν∥∇f(x)− ∇f(y)∥1+1/ν,\nas claimed.\n19Proof of Lemma 2.3 .Expanding the squares yields\n∥Hk(xk)−Hk(xk−1)∥2=∥xk−xk−1∥2+γ2\nk∥∇f(xk)−∇f(xk−1)∥2+ 2γk⟨xk−xk−1,∇f(xk)−∇f(xk−1)⟩\n=∥xk−xk−1∥2+γ2\nkL2\nk,ν∥xk−xk−1∥2ν+ 2γkℓk,ν∥xk−xk−1∥1+ν.\nFrom the identity γk=λk,ν∥xk−xk−1∥1−ν>0, the claimed expression follows.\nB Proofs of Section 3.1\nProof of Lemma 3.1 (FNE-like inequality). Recall the subgradient characterization\ne∇g(xk+1):=xk−xk+1\nγk+1− ∇f(xk)∈∂g(xk+1). (42)\nBy monotonicity of ∂g,\n1\nγ2\nk+1∥xk−xk+1∥2≤\r\rxk−xk+1\nγk+1+e∇g(xk)−e∇g(xk+1)\r\r2−\r\re∇g(xk)−e∇g(xk+1)\r\r2\n=\r\rxk−1−xk\nγk+∇f(xk)− ∇f(xk−1)\r\r2−\r\rxk−1−xk\nγk+xk+1−xk\nγk+1+∇f(xk)− ∇f(xk−1)\r\r2\n=\r\rxk−1−xk\nγk\r\r2+\r\r∇f(xk)− ∇f(xk−1)\r\r2\n+2\nγkγk+1⟨xk−xk, xk−1−xk⟩+2\nγk⟨xk−1−xk,∇f(xk)− ∇f(xk−1)⟩\n−\r\rxk−1−xk\nγk\r\r2−\r\rxk+1−xk\nγk+1\r\r2−\r\r∇f(xk)− ∇f(xk−1)\r\r2\n−2\nγkγk+1⟨xk−1−xk, xk+1−xk⟩ −2⟨xk−1−xk\nγk+xk+1−xk\nγk+1,∇f(xk)− ∇f(xk−1)⟩\n=2\nγkγk+1⟨xk−xk+1, xk−1−xk⟩+ 2⟨xk−xk+1\nγk+1,∇f(xk)− ∇f(xk−1)⟩ −\r\rxk+1−xk\nγk+1\r\r2\n=2\nγkγk+1⟨xk−xk+1, Hk(xk−1)−Hk(xk)⟩ −1\nγ2\nk+1\r\rxk+1−xk\r\r2.\nAfter moving the norm on the right-hand side and multiplying byγkγk+1\n2the claimed first inequality is\nobtained. The second one then follows by an application of the Cauchy-Schwarz inequality.\nProof of Lemma 3.3 (main inequality). From the convexity inequality for gatxk+1with subgradient\ne∇g(xk+1)as in (42), we have that for any x∈domg\n0≤g(x)−g(xk+1) +⟨∇f(xk), x−xk+1⟩ −1\nγk+1⟨xk−xk+1, x−xk+1⟩\n=g(x)−g(xk+1) +⟨∇f(xk), x−xk+1⟩+1\n2γk+1\u0000\n∥xk−x∥2− ∥xk+1−x∥2− ∥xk−xk+1∥2\u0001\n.(43)\nOn the other hand, using e∇g(xk)∈∂g(xk)yields that\n0≤g(xk+1)−g(xk) +⟨∇f(xk−1), xk+1−xk⟩ −1\nγk⟨xk−1−xk, xk+1−xk⟩. (44)\nWe now rewrite the inner product in (43) is a way that allows us to exploit the above inequality:\n⟨∇f(xk), x−xk+1⟩=⟨∇f(xk), x−xk⟩+⟨∇f(xk), xk−xk+1⟩\n≤f(x)−f(xk) +⟨∇f(xk), xk−xk+1⟩.\n20The last inner product can be controlled using (44):\n⟨∇f(xk), xk−xk+1⟩=⟨∇f(xk)− ∇f(xk−1), xk−xk+1⟩+⟨∇f(xk−1), xk−xk+1⟩\n≤ ⟨∇f(xk)− ∇f(xk−1), xk−xk+1⟩+g(xk+1)−g(xk)−1\nγk⟨xk−1−xk, xk+1−xk⟩\n=g(xk+1)−g(xk) +1\nγk⟨Hk(xk−1)−Hk(xk), xk−xk+1⟩\n(Lemmas 2.3 and 3.1) ≤g(xk+1)−g(xk) +ρk+1\nγkMk∥xk−xk−1∥2.\nCombine this with the prior inequality and (43) to obtain\n0≤γk+1(φ(x)−φ(xk)) +ρ2\nk+1Mk∥xk−xk−1∥2+1\n2\u0000\n∥xk−x∥2− ∥xk+1−x∥2− ∥xk−xk+1∥2\u0001\n.\nFinally, recalling the definition of e∇g(xk)as in (42) we have\n∇f(xk) +e∇g(xk) =1\nγk\u0000\nHk(xk−1)−Hk(xk)\u0001\n=xk−1−xk\nγk+∇f(xk)− ∇f(xk−1)∈∂φ(xk).\nTherefore, for any ϑk+1≥0\n0≤γk+1ϑk+1\u0010\nφ(xk−1)−φ(xk)−1\nγk⟨Hk(xk−1)−Hk(xk), xk−1−xk⟩\u0011\n=γk+1ϑk+1\u0010\nφ(xk−1)−φ(xk)−1−γkLk,ν\nγk∥xk−1−xk∥2\u0011\n.\nCombining the last two inequalities and letting Pk(x):=φ(xk)−φ(x)yields\n1\n2∥xk+1−x∥2+γk+1(1 +ϑk+1)Pk(x) +1\n2∥xk−xk+1∥2\n≤1\n2∥xk−x∥2+γk+1ϑk+1Pk−1(x)−ρk+1(ϑk+1(1−γkLk,ν)−ρk+1Mk)∥xk−1−xk∥2\nLetting ϑk=πρkfor all k, and setting x=x⋆∈arg min φestablishes the claimed inequality.\nProof of Lemma 3.4 (basic properties of adaPGπ,π\n2).\n♠3.4(1)) The update rule for γkensures that the coefficients of ∥xk−xk−1∥2andPk−1in (20) are\nnegative, and that therefore (Uπ\nk(x⋆))k∈is decreasing. Since Uπ\nk(x⋆)≥0, the sequence converges to a\nfinite (positive) value.\n♠3.4(2)) Boundedness follows by observing that1\n2∥xk−x⋆∥2≤ Uπ\nk(x⋆)≤ Uπ\n0(x⋆), where the last in-\nequality owes to the previous assertion. In what follows, we let LΩ,ν<∞be a ν-Hölder modulus for ∇f\non a convex and compact set Ωthat contains all the iterates xk. In particular, ℓk,ν≤Lk,ν≤LΩ,νholds\nfor every k. Suppose that x∞andx′\n∞are two optimal limit points, and observe that\nUπ\nk(x∞)− Uπ\nk(x′\n∞) =1\n2∥x∞∥2+1\n2∥x′\n∞∥2+⟨xk, x′\n∞−x∞⟩.\nSince both (Uπ\nk(x∞))k∈and(Uπ\nk(x′\n∞))k∈are convergent, by taking the limit along the subsequences\nconverging to x∞andx′\n∞we obtain ⟨x∞, x′\n∞−x∞⟩=⟨x′\n∞, x′\n∞−x∞⟩, which after rearranging yields\n∥x∞−x′\n∞∥2= 0.\n♠3.4(3)) Since γk(1 +πρk−πρ2\nk+1)≥0because of the update rule of γk+1, and with Pmin\nk:= min i≤kPi\ndenoting the best-so-far cost at iteration k, a telescoping argument on (20) yields that\nPmin\nKKX\nk=1γk(1 +πρk−πρ2\nk+1)≤KX\nk=1γk(1 +πρk−πρ2\nk+1)Pk−1\n≤ Uπ\n1(x⋆)− Uπ\nK+1(x⋆)≤ Uπ\n1(x⋆)−γk+1(1 +πρK+1)Pmin\nK,(45)\n21hence that\nPmin\nK≤Uπ\n1(x⋆)PK+1\nk=1γk(1 +πρk−πρ2\nk+1) +γk+1(1 +πρK+1)\n=Uπ\n1(x⋆)PK\nk=1(γk+πγkρk−πγk+1ρk+1) +γk+1+πγk+1ρK+1=Uπ\n1(x⋆)\nπγ1ρ1+PK+1\nk=1γk.\nThis completes the proof.\nC Implementation details of AC-FGM\nIn this section we describe the specific implementation of the auto-conditioned fast gradient method\n(AC-FGM) [22], which in our notation reads\nzk+1= proxγk+1g(yk−γk+1∇f(xk))\nyk+1= (1−βk+1)yk+βk+1zk+1\nxk+1= (zk+1+τk+1xk)/(1 +τk+1).\nRegarding the positive sequences (γk)k∈,(βk)k∈,(τk)k∈, we use the update rule described in [22, Cor.\n3] and as such in Corollary 2 of the same paper. We choose βk=β=1−√\n3\n2for all k∈, ensure that\nγ1∈[β\n4(1−β)c1,1\n3c1]and set γ2=β\n2c1andγk+1= min {τk−1+1\nτkγk,βτk\n4ck}for all k≥2. Finally, τ1= 0,τ2= 2\nandτk+1=τk+α\n2+2(1−α)γkck\nβτk, fork≥2and some α∈[0,1]. We chose α= 0, since this configuration\nconsistently outperfomed the others in our experiments. The sequence (ck)k∈is the so-called local\nLipschitz estimate and is defined as in [22, Eq. (3.9)]:\nck=\n\n√\n∥x1−x0∥2∥∇f(x1)−∇f(x0)∥2+(ϵ/4)2−ϵ/4\n∥x1−x0∥2 ifk= 1,√\n∥∇f(xk)−∇f(xk−1)∥2−ϵ/4\n2[f(xk−1)−f(xk)−⟨∇f(xk),xk−1−xk⟩]+ϵ/τk+1otherwise.\nwhere ϵis the predefined desired accuracy of the algorithm.\nReferences\n[1] Heinz H. Bauschke and Patrick L. Combettes. Convex analysis and monotone operator theory in\nHilbert spaces . CMS Books in Mathematics. Springer, 2017.\n[2] Amir Beck. First-order methods in optimization . SIAM, Philadelphia, PA, 2017.\n[3] Jacek Bochnak, Michel Coste, and Marie-Françoise Roy. Real algebraic geometry . Springer-Verlag\nBerlin Heidelberg, 1998.\n[4] Jérôme Bolte, Lilian Glaudin, Edouard Pauwels, and Mathieu Serrurier. The backtrack Hölder\ngradient method with application to min-max and min-min problems. Open Journal of Mathematical\nOptimization , 4:1–17, 2023.\n[5] KristianBredies. Aforward–backwardsplittingalgorithmfortheminimizationofnon-smoothconvex\nfunctionals in Banach space. Inverse Problems , 25(1):015005, 2008.\n[6] Coralia Cartis, Nicholas IM Gould, and Philippe L Toint. Adaptive cubic regularisation methods\nfor unconstrained optimization. part i: motivation, convergence and numerical results. Mathematical\nProgramming , 127(2):245–295, 2011.\n22[7] Rick Chartrand and Wotao Yin. Iteratively reweighted algorithms for compressive sensing. In 2008\nIEEE international conference on acoustics, speech and signal processing , pages 3869–3872. IEEE,\n2008.\n[8] Aaron Defazio, Baoyu Zhou, and Lin Xiao. Grad-GradaGrad? A non-monotone adaptive stochastic\ngradient method. arXiv:2206.06900 , 2022.\n[9] Olivier Devolder, François Glineur, and Yurii Nesterov. First-order methods of smooth convex\noptimization with inexact oracle. Mathematical Programming , 146:37–75, 2014.\n[10] Nikita Doikov, Konstantin Mishchenko, and Yurii Nesterov. Super-universal regularized Newton\nmethod. SIAM Journal on Optimization , 34(1):27–56, 2024.\n[11] John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and\nstochastic optimization. Journal of machine learning research , 12(7), 2011.\n[12] Alan B Forsythe. Robust estimation of straight line regression coefficients by minimizing pth power\ndeviations. Technometrics , 14(1):159–166, 1972.\n[13] Saeed Ghadimi, Guanghui Lan, and Hongchao Zhang. Generalized uniformly optimal methods for\nnonlinear programming. Journal of Scientific Computing , 79:1854–1881, 2019.\n[14] Benjamin Grimmer. On optimal universal first-order methods for minimizing heterogeneous sums.\nOptimization Letters , pages 1–19, 2023.\n[15] Wei-Bo Guan and Wen Song. The forward–backward splitting method and its convergence rate for\nthe minimization of the sum of two functions in Banach spaces. Optimization Letters , 15(5):1735–\n1758, 2021.\n[16] Yosra Hafiene, Jalal Fadili, and Abderrahim Elmoataz. Nonlocal p-Laplacian variational problems\non graphs. arXiv:1810.12817 , 2018.\n[17] Dmitry Kamzolov, Pavel Dvurechensky, and Alexander V Gasnikov. Universal intermediate gradient\nmethod for convex problems with inexact oracle. Optimization Methods and Software , 36(6):1289–\n1316, 2021.\n[18] Puya Latafat, Andreas Themelis, and Panagiotis Patrinos. On the convergence of adaptive first order\nmethods: Proximal gradient and alternating minimization algorithms. arXiv:2311.18431 , 2023.\n[19] Puya Latafat, Andreas Themelis, Lorenzo Stella, and Panagiotis Patrinos. Adaptive proximal algo-\nrithms for convex optimization under local Lipschitz continuity of the gradient. arXiv:2301.04431 ,\n2023.\n[20] Emanuel Laude and Panagiotis Patrinos. Anisotropic proximal gradient. arXiv:2210.15531 , 2022.\n[21] Emanuel Laude and Panagiotis Patrinos. Anisotropic proximal point algorithm. arXiv:2312.09834 ,\n2023.\n[22] Tianjiao Li and Guanghui Lan. A simple uniformly optimal method without line search for convex\noptimization. arXiv:2310.10082 , 2023.\n[23] Xiaoyu Li and Francesco Orabona. On the convergence of stochastic gradient descent with adaptive\nstepsizes. In The 22nd International Conference on Artificial Intelligence and Statistics , pages 983–\n992. PMLR, 2019.\n[24] Javier Luque. Nonlinear proximal point algorithms . PhD thesis, Massachusetts Institute of Technol-\nogy, Department of Civil Engineering, 1984.\n23[25] Yura Malitsky and Konstantin Mishchenko. Adaptive gradient descent without descent. In Pro-\nceedings of the 37th International Conference on Machine Learning , volume 119, pages 6702–6712.\nPMLR, 13- 2020.\n[26] Yura Malitsky and Konstantin Mishchenko. Adaptive proximal gradient method for convex opti-\nmization. arXiv:2308.02261 , 2023.\n[27] Yu Nesterov. Universal gradient methods for convex optimization problems. Mathematical Program-\nming, 152(1-2):381–404, 2015.\n[28] Yurii Nesterov. Gradient methods for minimizing composite functions. Mathematical Programming ,\n140(1):125–161, aug 2013.\n[29] Yurii Nesterov. Implementable tensor methods in unconstrained convex optimization. Mathematical\nProgramming , 186:157–183, 2021.\n[30] Yurii Nesterov. Inexact accelerated high-order proximal-point methods. Mathematical Programming ,\npages 1–26, 2021.\n[31] Yurii Nesterov, Alexander Gasnikov, Sergey Guminov, and Pavel Dvurechensky. Primal–dual ac-\ncelerated gradient methods with small-dimensional relaxation oracle. Optimization Methods and\nSoftware, 36(4):773–810, 2021.\n[32] Konstantinos A Oikonomidis, Alexander Bodard, Emanuel Laude, and Panagiotis Patrinos. Power\nproximal point and augmented Lagrangian method. arXiv:2312.12205 , 2023.\n[33] Ralph T. Rockafellar. Convex analysis . Princeton University Press, 1970.\n[34] Fedor Stonyakin, Alexander Tyurin, Alexander Gasnikov, Pavel Dvurechensky, Artem Agafonov,\nDarina Dvinskikh, Mohammad Alkousa, Dmitry Pasechnyuk, Sergei Artamonov, and Victorya\nPiskunova. Inexact model: A framework for optimization and variational inequalities. Optimiza-\ntion Methods and Software , 36(6):1155–1201, 2021.\n[35] Tianbao Yang and Qihang Lin. Rsg: Beating subgradient method without smoothness and strong\nconvexity. The Journal of Machine Learning Research , 19(1):236–268, 2018.\n[36] Maryam Yashtini. On the global convergence rate of the gradient descent method for functions with\nHölder continuous gradients. Optimization letters , 10:1361–1370, 2016.\n[37] Yiming Ying and Ding-Xuan Zhou. Unregularized online learning algorithms with general loss\nfunctions. Applied and Computational Harmonic Analysis , 42(2):224–244, 2017.\n[38] Alp Yurtsever, Alex Gu, and Suvrit Sra. Three operator splitting with subgradients, stochastic gra-\ndients, and adaptive learning rates. Advances in Neural Information Processing Systems , 34:19743–\n19756, 2021.\n[39] Alp Yurtsever, Quoc Tran Dinh, and Volkan Cevher. A universal primal-dual convex optimization\nframework. Advances in Neural Information Processing Systems , 28, 2015.\n[40] Danqing Zhou, Shiqian Ma, and Junfeng Yang. AdaBB: Adaptive Barzilai-Borwein method for\nconvex optimization. arXiv:2401.08024 , 2024.\n24"    },    {        "title": "2402.06320v1.Particle_Denoising_Diffusion_Sampler.pdf",        "content": "Particle Denoising Diffusion Sampler\nAngus Phillips1Hai-Dang Dau1Michael John Hutchinson1Valentin De Bortoli2George Deligiannidis1\nArnaud Doucet1\nAbstract\nDenoising diffusion models have become ubiq-\nuitous for generative modeling. The core idea is\nto transport the data distribution to a Gaussian\nby using a diffusion. Approximate samples from\nthe data distribution are then obtained by esti-\nmating the time-reversal of this diffusion using\nscore matching ideas. We follow here a similar\nstrategy to sample from unnormalized probability\ndensities and compute their normalizing constants.\nHowever, the time-reversed diffusion is here simu-\nlated by using an original iterative particle scheme\nrelying on a novel score matching loss. Contrary\nto standard denoising diffusion models, the result-\ning Particle Denoising Diffusion Sampler (PDDS)\nprovides asymptotically consistent estimates un-\nder mild assumptions. We demonstrate PDDS on\nmultimodal and high dimensional sampling tasks.\n1. Introduction\nConsider a target probability density πonRdof the form\nπ(x) =γ(x)\nZ,Z=Z\nRdγ(x)dx, (1)\nwhere γ:Rd→R+can be evaluated pointwise but its\nnormalizing constant Zis intractable. We develop here a\nMonte Carlo scheme to sample approximately from πand\nestimate Z.\nWe follow an approach inspired by denoising diffusion mod-\nels (Ho et al., 2020; Song et al., 2021; Song & Ermon, 2019)\nby considering a “noising” diffusion progressively transport-\ning the original target to a Gaussian. The time-reversal of\nthis diffusion, the “denoising” diffusion, allows us theoreti-\ncally to sample from the target starting from Gaussian noise.\nHowever, it is impossible to simulate this process exactly\nas its drift depends on the gradient of the logarithm of the\nintractable marginal densities of the noising diffusion, i.e.\nthe score. For generative modeling, where one has access to\n1University of Oxford2CNRS, ENS Ulm. Correspondence to:\nAngus Phillips <angus.phillips@stats.ox.ac.uk >.samples from π, one can rely on neural networks and score\nmatching (Hyv ¨arinen, 2005; Vincent, 2011). This strategy\nis not applicable in the Monte Carlo sampling context as\nwe cannot sample the “noising” diffusion since we do not\nhave access to any samples from πto approximate the initial\ndistribution of the diffusion with.\nThe idea of using denoising diffusion models for Monte\nCarlo sampling has already been explored by Berner et al.\n(2022); McDonald & Barron (2022); Vargas et al. (2023);\nHuang et al. (2024); Richter et al. (2024). Berner et al.\n(2022); Richter et al. (2024); Vargas et al. (2023) focus on\nthe minimization of a reverse Kullback–Leibler divergence\nor log-variance criterion while McDonald & Barron (2022)\nrely on an importance sampling scheme which scales poorly\nin high dimensions. Finally, Huang et al. (2024) relies on a\nseries of Markov chain Monte Carlo (MCMC) to estimate\nthe score. This last scheme does not provide estimates of\nnormalizing constants.\nWe develop here an alternative approach inspired by de-\nnoising diffusion models with guidance. Guided diffusions\ncombine pre-trained diffusion models with a guidance term\nderived from a likelihood to sample approximately from\nposterior distributions; see e.g. Song et al. (2021); Chung\net al. (2023); Song et al. (2023); Corso et al. (2023). While\nthey provide samples with appealing perceptual properties,\nthey rely on various approximations, in order of impor-\ntance: (1) approximation of the score and guidance terms,\n(2) time-discretization of the diffusion and (3) approximate\ninitialization of the diffusion. Wu et al. (2023); Cardoso\net al. (2024) have used particle methods also known as Se-\nquential Monte Carlo (SMC) (Doucet et al., 2001; Chopin\n& Papaspiliopoulos, 2020) to obtain consistent estimates in\nthe generative modeling context.\nOur contributions are as follows:\n(1) we adapt guided diffusions to sampling problems,\n(2) we provide theoretical results quantifying the error intro-\nduced by current guided diffusions in a simple scenario,\n(3) we develop an SMC scheme to provide consistent esti-\nmates in this setup and establish limit theorems,\n(4) we introduce an algorithm that reduces the variance of\nthe SMC estimates based on a novel score matching loss.\nAll proofs are postponed to the Appendix.\n1arXiv:2402.06320v1  [stat.ML]  9 Feb 2024Particle Denoising Diffusion Sampler\n2. Denoising diffusions with guidance\n2.1. Noising and denoising diffusions\nConsider the following noising diffusion (Xt)t∈[0,T],\ndXt=−βtXtdt+p\n2βtdWt, X 0∼π, (2)\nwhere (Wt)t∈[0,T]is ad-dimensional Brownian motion\nandβt>0. The transition density of this diffusion is\ngiven by p(xt|x0) =N(xt;√1−λtx0, λtI)forλt=\n1−exph\n−2Rt\n0βsdsi\n. We denote by πtthe density of Xt\nunder (2). In practice, we considerRT\n0βsds≫1, and there-\nforeπT(x)≈ N(x; 0,I). The diffusion (2) thus transforms\nπ0=πinto approximately N(0,I). If instead we initial-\nize (2) using p0(x) =N(x; 0,I), its marginals (pt)t∈[0,T]\nsatisfy pt=p0.\nThe time-reversal (Yt)t∈[0,T]= (XT−t)t∈[0,T]of (2), the\ndenoising diffusion, satisfies YT∼πand\ndYt= [βT−tYt+ 2βT−t∇logπT−t(Yt)] dt+p\n2βT−tdBt,\n(3)\nwhere (Bt)t∈[0,T]is another Brownian motion; see e.g.\nHaussmann & Pardoux (1986); Cattiaux et al. (2023). The\nmain idea of denoising diffusions is to sample from πby\nsampling (3)asYT∼π(Ho et al., 2020; Song et al., 2021).\nHowever, we cannot simulate (3) exactly as, in order of\nimportance, (1) the score terms (∇logπt)t∈[0,T]are in-\ntractable, (2) it is necessary to time-discretize the diffusion\nand (3) πTcannot be sampled. We can always use numer-\nical integrators and approximate πTwith a unit Gaussian\ndistribution to mitigate (2) and (3). In generative modeling\n(1) is addressed by leveraging tools from the score matching\nliterature (Vincent, 2011; Hyv ¨arinen, 2005) and using neural\nnetwork estimators. In our sampling setting we do not have\naccess to access to samples from πbut only to its unnormal-\nized density. Therefore, alternative approximations must be\ndeveloped.\n2.2. Denoising diffusions with guidance\nFor generative modeling, the use of denoising diffusions\nwith guidance terms to sample approximately from posterior\ndistributions has become prominent in the inverse problem\nliterature, see e.g. Song et al. (2021); Chung et al. (2023);\nSong et al. (2023); Corso et al. (2023). We present here a\nsimple extension of this idea applicable to any target π(x)\nby rewriting this target as\nπ(x)∝p0(x)g0(x),forp0(x) =N(x; 0,I).(4)\nIt is always possible to perform such a decomposition, which\nprovides an alternative expression for the score.\nLemma 2.1. The following identities hold\nπt(xt)∝p0(xt)gt(xt),∇logπt(xt) =−xt+∇loggt(xt),\n(5)where\ngt(xt) =R\ng0(x0)p(x0|xt)dx0, (6)\nandp(x0|xt) =N(x0;√1−λtxt, λtI)is the conditional\ndensity of X0given Xt=xtfor the diffusion (2) initialized\nusing X0∼p0.\nFrom Lemma 2.1, it follows that (3) can be rewritten as\ndYt= [−βT−tYt+ 2βT−t∇loggT−t(Yt)] dt (7)\n+p\n2βT−tdBt.\nThis is akin to having a diffusion model with tractable scores\n∇logpt(xt) =∇logp0(xt) =−xtand with gt(xt)as a\nguidance term.\n2.3. Guidance approximation\nThe most important source of error when approximating (7)\nis the lack of a closed form expression for gtas it involves\nan intractable integral. In the context of inverse problems,\na simple approximation used by (Chung et al., 2023; Song\net al., 2023) is given by\ngt(xt)≈g0(R\nx0p(x0|xt)dx0) =g0(√1−λtxt)\n:= ˆgt(xt). (8)\nThis approximation is good when tis close to 0orTbut\ncrude otherwise, as established by the following result.\nProposition 2.2. Letπ(x) =N(x;µ, σ2)and(βt)t∈[0,T]\nbe any schedule satisfying limT→∞RT\n0βsds=∞. Con-\nsider the following approximation of (7)\ndZ(T)\nt= [−βT−tZ(T)\nt+ 2βT−t∇log ˆgT−t(Z(T)\nt)]dt(9)\n+p\n2βT−tdBt, Z(T)\n0∼ N(0,1).\nThen limT→∞E[Z(T)\nT] =µandlimT→∞Var(Z(T)\nT) = 1\nforσ= 1and otherwise\nlim\nT→∞E[Z(T)\nT] =µ\n1−σ2(1−e−(1/σ2−1)), (10)\nlim\nT→∞Var(Z(T)\nT) =1−e−2(1/σ2−1)\n2(1/σ2−1). (11)\nHence, even without considering any time discretization er-\nror,limT→∞E[Z(T)\nT]̸=µandlimT→∞Var(Z(T)\nT)̸=σ2;\nthe error getting larger as σ2moves away from 1. If we\nconsider the target N(µ, σ2)⊗d, a similar result holds along\neach dimension. Therefore, the Kullback–Leibler diver-\ngence between the target and the result of running (9)grows\nlinearly with d. As a result an exponentially increasing num-\nber of samples is needed to obtain an importance sampling\napproximation of the target of reasonable relative variance\n(Chatterjee & Diaconis, 2018).\n2Particle Denoising Diffusion Sampler\n3. Particle denoising diffusion sampler\nIn this section, we propose a particle method to correct\nthe discrepancy between the distribution outputted by the\nguided diffusion and the target. Let [P] ={1, ..., P}for\nP∈N. We first present the exact joint distribution of\n(Xtk)k∈{0,...,K}withtk=kδfor a fixed step size δfor the\ndiffusion (2)where K=T/δ is an integer. We then make\nexplicit the corresponding reverse-time Markov transitions\nfor this joint distribution and show how they can be approx-\nimated. Finally we show how these approximations can\nbe corrected to obtain consistent estimates using SMC. In-\nstead of writing Xtk, we will write Xkto simplify notation.\nSimilarly we will write gkforgtk,λkforλtketc.\n3.1. From continuous time to discrete time\nFork∈[K], letαk= 1−exph\n−2Rkδ\n(k−1)δβsdsi\n. The\njoint distribution of X0:K= (X0, X1, ..., X K)under (2)\nsatisfies\nπ(x0:K) =π(x0)Q\nk∈[K]p(xk|xk−1), (12)\nforp(xk|xk−1) =N(xk;√1−αkxk−1, αkI). This im-\nplies in particular\nπ(xk, xk+1) =πk(xk)p(xk+1|xk). (13)\nWe also denote by p(x0:K)the joint distribution of (2)initial-\nized at p0(x0), in this case the Markov process is stationary,\ni.e.pk(xk) =p0(xk), and reversible w.r.t. p0.\nFrom (5), it can be easily checked that the backward transi-\ntions of (12) satisfy\nπ(xk|xk+1) =p(xk|xk+1)gk(xk)\ngk+1(xk+1)\n≈p(xk|xk+1) exp[⟨∇loggk+1(xk+1), xk−xk+1⟩]\n=N(xk;√1−αk+1xk+1\n+αk+1∇loggk+1(xk+1), αk+1I), (14)\nwhere we used gk≈gk+1and a Taylor expansion of\nloggk+1(xk)around xk+1; both of which are reasonable\nwhen δ≪1. Since αk+1≈2βk+1δand√1−αk+1≈\n1−βk+1δ, this approximation of π(xk|xk+1)corresponds\nto a form of exponential integrator for the time-reversal (7).\nWhile this discrete-time representation is interesting, it is\nclearly typically impossible to exploit it to obtain exact\nsamples from πsince, like its continuous time counterpart\n(7), it relies on the intractable potentials (gk)K\nk=1. In the\nfollowing Section 3.2, given an approximation ˆgkofgk, we\npropose a particle mechanism to sample exactly from πas\nthe number of particles goes to infinity.3.2. From discrete time to particles\nWe use a particle method to sample from π. The key idea is\nto break the difficult problem of sampling from πinto a se-\nquence of simpler intermediate sampling problems. Ideally\nwe would sample from πKfirst then πK−1, πK−2, ...until\nπ0=π. Unfortunately this is not possible as this requires\nknowing gk. Suppose that we have an approximation ˆgkof\ngk(for instance, the guidance approximation defined in (8)).\nInspired by (5), we sample instead for k∈[K]from the\nsequence of densities\nˆπk(xk)∝p0(xk)ˆgk(xk),ˆZk=R\np0(xk)ˆgk(xk)dxk,\n(15)\nbackward in time. At k=K, the function gKis almost\nconstant so we choose ˆgK≡1and sampling from ˆπK=\nN(0,I)is easy.\nGiven particles approximately distributed according to ˆπk+1,\nwe aim to obtain sample from ˆπk. Drawing inspiration\nfrom (14), we sample according to the proposal\nˆπ(xk|xk+1) :=N(xk;p\n1−αk+1xk+1\n+αk+1∇log ˆgk+1(xk+1), αk+1I).(16)\nWe then reweight these pairs using the weights\nwk(xk, xk+1) :=ˆπk(xk)p(xk+1|xk)\nˆπk+1(xk+1)ˆπ(xk|xk+1)\n∝ˆgk(xk)p(xk|xk+1)\nˆgk+1(xk+1)ˆπ(xk|xk+1), (17)\nwhere the last line follows from a simple application of (15).\nThe first line of (17) means that the xkmarginal of the\nweighted system consistently approximates ˆπk. It should be\nnoted that wkquantifies the error in (16).\nFinally, we resample these particles with weights propor-\ntional to (17). This resampling operation allows us to focus\nthe computational efforts on promising regions of the space\nbut some particles are replicated multiple times, reducing the\npopulation diversity. Therefore, we then optionally perturb\nthe resampled particles using a MCMC kernel of invariant\ndistribution ˆπk. The resulting Particle denoising diffusion\nsampler (PDDS) is summarized in Algorithm 1.\n3.3. Algorithm settings\nReparameterization. In practice, we would like to have\np0to be such that g0is bounded or has bounded moments.\nTo achieve this, it can be desirable to obtain a variational\napproximation N(x;µ,Σ)ofπthen do the change of vari-\nables x′= Σ−1/2(x−µ), sample in this space using PDDS\nbefore mapping the samples back using x=µ+ Σ1/2x′.\nResampling. The idea of resampling is to only propagate\nparticles in promising regions of the state space. Given N\n3Particle Denoising Diffusion Sampler\nAlgorithm 1 Particle Denoising Diffusion Sampler\nInput: Schedule (βt)t∈[0,T]as in (2); Approximations\n(ˆgk)K\nk=0s.t.ˆg0=g0,ˆgK= 1; Number of particles N\nSample Xi\nKiid∼ N(0,I)fori∈[N]\nSetˆZK←1andωi\nK←1/Nfori∈[N]\nfork=K−1, . . . , 0do\nMove . Sample ˜Xi\nk∼ˆπ(·|Xi\nk+1)fori∈[N](see(16))\nWeight .ωi\nk←ωk(˜Xi\nk, Xi\nk+1)fori∈[N](see (17))\nSetˆZk←ˆZk+1×1\nNP\ni∈[N]ωi\nk\nNormalize ωi\nk←ωi\nk/P\nj∈[N]ωj\nk\nResample .X1:N\nk←resample( ˜X1:N\nk, ω1:N\nk)(see Sec-\ntion 3.3)\nMCMC (Optional). Sample Xi\nk←Mk(Xi\nk,·)for\ni∈[N]using a ˆπk-invariant MCMC kernel Mk.\nend for\nOutput: Estimates ˆπN=1\nNP\ni∈[N]δXi\n0ofπ,ˆZN\n0ofZ\nparticles ˜X1:N\nkandNweights ω1:N\nksumming to 1, resam-\npling selects Noutput particles X1:N\nksuch that, for any\nfunction φ:Rd→R,\nEh\n1\nNP\ni∈[N]φ(Xi\nk)|˜X1:N\nk, ω1:N\nki\n=P\ni∈[N]ωi\nkφ(˜Xi\nk).\nPopular schemes satisfying this identity are multinomial,\nstratified, residual, and systematic resampling (Douc &\nCapp ´e, 2005).\nWe employ systematic resampling in all our simulations as\nit provides the lowest variance estimates.\nResampling can however reduce the diversity of the particle\npopulation by introducing identical particles in the output.\nAs such, a popular recipe is to trigger resampling at time\nkonly when the Effective Sample Size (ESS), a measure\nof particle diversity defined by (P\ni∈[N](ωi\nk)2)−1, is be-\nlow a certain threshold (Del Moral et al., 2012; Dai et al.,\n2022). This is implemented using Algorithm 3 presented in\nAppendix A.\nMCMC kernel. We want to design an MCMC kernel of\ninvariant distribution ˆπk(xk)defined in (15). As we have\naccess to ∇log ˆπk(xk) =−xk+∇log ˆgk(xk), we can\nuse a Metropolis-adjusted Langevin algorithm (MALA) or\nHamiltonian Monte Carlo; e.g. MALA considers a proposal\nx⋆\nk=xk+γ∇log ˆπk(xk) +√2γϵ, ϵ ∼ N(0,I),\nfor a step size γ. This proposal is accepted with probability\nmin\u001a\n1,ˆπk(x⋆\nk)N(xk;x⋆\nk+γ∇log ˆπk(x⋆\nk),2γI)\nˆπk(xk)N(x⋆\nk;xk+γ∇log ˆπk(xk),2γI)\u001b\n.\n3.4. Theoretical results\nFixed number of discretization steps. We show below\nthat the estimates ˆZN\n0andπNf=1\nNPN\ni=1f(Xi\n0)returnedby Algorithm 1 both satisfy a central limit theorem. This fol-\nlows from standard SMC theory (Del Moral, 2004; Webber,\n2019).\nProposition 3.1. Assume that E[wk(Xk, Xk+1)2]<∞\nwhere the expectation is w.r.t. ˆπ(xk+1)ˆπ(xk|xk+1)and thatR\nπk(x)(gk/ˆgk)(x)dx <∞. Then ˆZN\n0is an unbiased esti-\nmate of Zand has finite variance. If multinomial resampling\nis used at every step,√\nN(ˆZN\n0/Z −1)is asymptotically\nnormal with asymptotic variance\nσ2\nK=χ2(πK||ˆπK)+\nK−1X\nk=0χ2(πk(xk)π(xk|xk+1)||ˆπk(xk)ˆπ(xk|xk+1)),\nwithχ2(·||·)the chi-squared divergence between two dis-\ntributions. Moreover, for any bounded function f, we also\nhave asymptotic normality of√\nN(ˆπNf−πf).\nThe finiteness assumptions on E[wk(Xk, Xk+1)2]andR\nπk(x)(gk/ˆgk)(x)dxrequire that ˆgkis not too far from\ngkbut still allow enough freedom in the choice of ˆgk. The\nexpression for σ2\nKsuggests that choosing ˆgkclose to gkwill\nreduce the asymptotic variance as (16) will better approxi-\nmate (14).\nInfinitely fine discretization limit. We now investigate\nthe performance of PDDS as the number of discretization\ntime steps goes to infinity. Even when all the weights are\nequal, multinomial resampling still kills over a third of\nparticles on average (Chopin & Papaspiliopoulos, 2020).\nHence a fine discretization with repeated applications of\nmultinomial resampling leads to the total collapse of the\nparticle approximation. This has been formalized in the\ncontinuous-time setting (Chopin et al., 2022). In our case, it\ncan be readily checked that indeed σ2\nK→ ∞ asK→ ∞ in\ngeneral. This justifies using a more sophisticated resampling\nstrategy. Indeed, when using the sorted stratified resampling\nstrategy of Gerber et al. (2019), the following results show\nthat our particle approximations remain well behaved as\nK→ ∞ .\nProposition 3.2. Consider the setting of Proposition 3.1\nwith sorted stratified resampling. Then there exists a se-\nquence of sets (BN)such that P(BN)→1and\nlim sup\nN→∞E[N(ˆZN\n0/Z −1)21BN]≤ζ2\nK\nwith\nζ2\nK:=χ2(πK||ˆπK)+\nK−1X\nk=0Zπk+1(xk+1)2\nˆπk+1(xk+1)χ2(π(xk|xk+1)||ˆπ(xk|xk+1))dxk+1.\nThe next result bounds the limit when K→ ∞ , i.e. δ=\nT/K→0\n4Particle Denoising Diffusion Sampler\nProposition 3.3. Under the setting of Proposition 3.2, as-\nsume that βt≡1and that the target distribution satisfies\nthe regularity conditions in Appendix C.4.1. Let ¯gt:=gt/Z\nand˜gt:= ˆgt/R\np0(x)ˆgt(x)dx. Assume further that the\napproximations ˆgt,¯gt,˜gtsatisfy C−1\n1≤¯gt/˜gt≤C1and\n∥∇log ˆgt(xt)∥ ≤C2(1+∥xt∥)for some C1, C2≥0. Then\nlim supK→∞ζ2\nK≤χ2(πT||ˆπT)+\n2RT\n0EXt∼πth\n¯gt\n˜gt(Xt)∥∇loggt(Xt)− ∇log ˆgt(Xt)∥2i\ndt.\nThis result highlights precisely how the asymptotic error\ndepends on the quality of the estimates of gtand its log-\narithmic gradient. Let πˆg(dx[0,T])be the path measure\ninduced by running (7)with some approximation ˆgt, i.e. the\ndistribution of (Yt)t∈[0,T]given by (7). If importance sam-\npling were directly used to correct between πˆg(dx[0,T])and\nπ(dx[0,T]), the asymptotic error for ˆZN\n0/Zwould be equal\ntoχ2(π||πˆg)which is greater than exp\b\nKL(π|πˆg)\t\n−1. In\ncontrast, ignoring the negligible first term χ2(πT||ˆπT), the\nerror in this proposition is upper bounded by 2C1KL(π|πˆg).\nThis shows how PDDS helps reduce the error of naive im-\nportance sampling.\n4. Learning potentials via score matching\nThe previous theoretical results establish the consistency\nof PDDS estimates for any reasonable approximation ˆgkof\ngkbut also show that better approximations lead to lower\nMonte Carlo errors. We show here how to use the approx-\nimation of πoutputted by PDDS to learn a better neural\nnetwork (NN) approximation ˆgθ(k, xk)of the potential func-\ntionsgkby leveraging score matching ideas. Once we have\nlearned those approximations, we can then run again PDDS\n(Algorithm 1) with the new learned potentials. In our exper-\niments (Section 6), we demonstrate that this can drastically\nimprove results in challenging scenarios.\n4.1. Different score identities\nFrom (5), we have the identity\n∇loggk(xk) =xk+∇logπk(xk). (18)\nHence, if we obtain an approximation ∇logπθ(k, xk)\nof∇logπk(xk), then we get an approximation\nlog ˆgθ(k, xk) =1\n2||xk||2+ log ˆ πθ(k, xk)ofloggk(xk).\nTo learn the score, we rely on the following result.\nProposition 4.1. The score satisfies the standard Denoising\nScore Matching (DSM) identity\n∇logπk(xk) =R\n∇logp(xk|x0)π(x0|xk)dx0.(19)\nMoreover, ifR\n∥∇π(x0)∥e−η∥x0∥2dx0<∞,∀η >0, theNovel Score Matching (NSM) identity\n∇logπk(xk) =κkR\n∇logg0(x0)π(x0|xk)dx0−xk\n(20)\nholds for κk=√1−λkand∇logg0(x0) =\n∇logπ0(x0) +x0. Hence, we can approximate the score\nby minimizing one of the two following loss functions\nℓDSM(θ) =X\nk∈[K]E∥∇log ˆπθ(k, X k)− ∇logp(Xk|X0)∥2,\nℓNSM(θ) =X\nk∈[K]E||∇log ˆπθ(k, X k) +Xk\n−κk∇logg0(X0)||2,\nwhere the first loss is applicable if πhas finite second\nmoment and the second loss is applicable if additionally\nEπ[∥∇logπ(X)∥2]<∞. All the expectations are taken\nw.r.t.π(x0)p(xk|x0). For expressive neural networks, both\nℓDSM(θ)andℓNSM(θ)are such that ∇log ˆπθ(k, xk) =\n∇logπk(xk)at the minimizer.\nThe benefit of this novel loss is that it is much better be-\nhaved compared to the standard denoising score matching\nloss when δ≪1as established below, this is due to the\nfact that the variance of the terms appearing in ℓDSMfork\nclose to zero become very large. This loss is not applica-\nble to generative modeling as πis only available through\nsamples.\n4.2. Benefits of alternative score matching identity\nWe establish here formally the benefits of NSM over DSM.\nNSM score matching prevents variance blow up as kis close\nto time 0andδ→0. This is more elegantly formalized in\ncontinuous-time. In this case, the renormalized loss func-\ntions ℓDSMandℓNSMintroduced in Proposition 4.1 become\nforsθ(t, xt) =∇log ˆπθ(t, xt)\nℓDSM(θ) =RT\n0E∥sθ(t, Xt)− ∇logp(Xt|X0)∥2dt,\nℓNSM(θ) =RT\n0E||sθ(t, Xt) +Xt−κt∇logg0(X0)||2dt,\nwhere the expectations are w.r.t. π(x0)p(xt|x0). Unbi-\nased estimates of the gradient of these losses ˆ∇ℓDSM(θ)\nand ˆ∇ℓNSM(θ)are obtained by computing the gradient\nw.r.t. θof the argument within the expectation at a sam-\npleτ∼Unif[0 , T],X0∼πandXτ∼p(xτ|X0).\nThe following result clarifies the advantage of NSM score\nmatching.\nProposition 4.2. Letd= 1 ,βt≡1, and π(x0) =\nN(x0;µ, σ2). Suppose that the score network sθ(t, xt)is a\ncontinuously differentiable function jointly on its three vari-\nables. Moreover, assume that |s|+∥∇θs∥ ≤C(1+|θ|+|t|+\n|xt|)αfor some C, α > 0; and that Eπ[∥∇θsθ(0, X0)∥2]>\n5Particle Denoising Diffusion Sampler\nAlgorithm 2 Potential neural network training\nInput: Particle approx. ˆπNoutputted by Algorithm 1; Po-\ntential NN ˆgk(θ, xk); Initialization θ0; Local loss func-\ntions ℓ(θ, k, x 0, xk); Batch size B; Number of gradient\nupdates Nup; Learning rate η >0.\nfori= 1,2, . . . , N updo\nforb= 1,2, . . . , B do\nSample Xb\n0∼ˆπN(·);kb∼Unif[ K];\nSample Xkb∼pkb,0(·|Xb\n0)\nend for\nθi:=θi−1−η\nBP\nb∈[B]∇θℓ(θi−1, kb, Xb\n0, Xkb)\nend for\nOutput: Potential ˆgθNup(k, xk)\n0for all θ. Then, for all θ,ℓDSM(θ)andE[ˆ||∇ℓDSM(θ)||2]\nare infinite whereas ℓNSM(θ)andE[||ˆ∇ℓNSM(θ)||2]are fi-\nnite.\n4.3. Neural network parametrization\nContrary to standard practice for generative modeling (Ho\net al., 2020; Song et al., 2021), we parameterize a func-\ntion and not a vector field. This is necessary to run PDDS\n(Algorithm 1) as it relies on potentials to weight particles.\nWe will use a parameterization of the form log ˆπθ(k, xk) =\nlog ˆgθ(k, xk)−1\n2||x||2where we parametrize ˆgθusing two\nneural networks rηandNγsuch that θ= (η, γ). The\nnetwork rηreturns a scalar whereas Nγreturns a vector in\nRd. The precise expression is given by\nlog ˆgθ(k, xk) = [rη(k)−rη(0)]⟨Nγ(k, xk), xk⟩+\n+ [1−rη(k) +rη(0)] log g0(p\n1−λkxk).\nThis parametrization takes advantage of the simple approxi-\nmation (8)with which it coincides at time 0. Zhang & Chen\n(2022) used a similar parametrization to incorporate gradient\ninformation in their control policy. Although log ˆgθ(k, xk)\nis a scalar, we deliberately let Nγ(k, xk)return a vector\nwhich is then scalar-multiplied with xk. This is usually\ndone in the literature when the scalar potential instead of\njust its gradient is learned (see e.g. Salimans & Ho, 2021)\nand helps improve model expressiveness.\n4.4. Training the neural network\nBoth ℓDSM(θ)and ℓNSM(θ)can be written asP\nk∈[K]E[ℓ(θ, k, X 0, Xk)]for appropriately defined\nlocal loss functions ℓ. Algorithm 2 describes the gradient\nupdates according to the batch size B.\nIn practice, we first run Algorithm 1 with a simple approx-\nimation such as (8). We then use Algorithm 2 to learn\nˆgθ(k, xk)and can then execute Algorithm 1 again with\nTempering\n Noising\nFigure 1: Tempered (top) and noised (bottom) sequences\nof distributions for the target π(x) = 0 .8N(x;−4,0.52) +\n0.2N(x; 4,1). The tempered sequence follows πt(x)∝\nπ(x)(1−ηt)ϕ(x)ηtwhere ϕis the standard normal and ηt\nincreases from 0to1. The noising sequence follows the\nforward diffusion in Equation (2). Red dots indicate the\nposition of modes in the original target. The tempered se-\nquence suffers from mode switching, i.e. the low mass large\nwidth mode becomes dominant across the tempered path.\nThe noised sequence does not suffer from this problem.\nˆgk(xk) = ˆgθ(k, xk)to obtain lower variance estimates of π\nandZ. We can further refine the approximation of gkusing\nAlgorithm 2. In practice, we found that one or two iterations\nare sufficient.\n5. Related work\nSMC samplers (Del Moral et al., 2006; Dai et al., 2022)\nare a general methodology to sample sequentially from a\nsequence of distributions. It relies on the notion of forward\nand backward kernels in order to move from one distribution\nto another. PDDS can be cast in this framework where the\nforward kernel is chosen to be the forward noising diffu-\nsion and the backward kernel the approximate time-reversal.\nThe novelty of our work is the exploitation of the special\nstructure induced by such a choice to come up with efficient\nbackward kernel estimates. Other standard methods include\nAnnealed Importance Sampling (Neal, 2001) and Parallel\nTempering (Geyer, 1991). Unlike our work, all these meth-\nods approximate a sequence of tempered versions of the\ntarget. While they are standard, it is also well-known that\ntempering strategies can exhibit poor performance for mul-\ntimodal targets (Woodard et al., 2009; Tawn et al., 2020;\nSyed et al., 2022) as tempering can change dramatically\nthe masses of distribution modes depending on their widths.\nAdding noise does not suffer from this issue (M ´at´e & Fleuret,\n2023). It only perturbs the distribution locally, preserving\nthe weights of the modes; see Figure 1 for an illustration.\nOur sampler can also be interpreted as sampling from a\n6Particle Denoising Diffusion Sampler\nsequence of distributions using so-called twisted proposals.\nThe general framework for approximating these twisted pro-\nposals using SMC has been considered in Guarniero et al.\n(2017); Heng et al. (2020); Lawson et al. (2022). In particu-\nlar, Wu et al. (2023) and Cardoso et al. (2024) recently apply\nsuch ideas to conditional simulation in generative modeling\ngiven access to a pretrained score network. Wu et al. (2023)\nrely on the simple approximation (8) and do not quantify\nits error, while Cardoso et al. (2024) use a different pro-\nposal kernel which leverages the structure of a linear inverse\nproblem. Our setup is here different as we consider general\nMonte Carlo sampling problems. We do not rely on any\npretrained network and refine our potential approximations\nusing an original loss function. We additionally provide\ntheoretical results in particular when the discretization time\nstep goes to zero.\n6. Experimental results\n6.1. Normalizing constant estimation\nWe evaluate here the quality of normalizing constant esti-\nmates produced by PDDS on a variety of sampling tasks.\nWe consider two synthetic target densities and two posterior\ndistributions, with results on additional targets included\nin Appendix D.4. The synthetic targets are Mixture ,\na 2-dimensional mixture of Gaussian distributions with\nseparated modes (Arbel et al., 2021) and Funnel , a 10-\ndimensional target displaying challenging variance structure\n(Neal, 2003). The posterior distributions are Sonar , a logis-\ntic regression posterior fitted to the Sonar (61-dimensional)\ndataset and LGCP , a log Gaussian Cox process (Møller et al.,\n1998) modelling the rate parameter of a Poisson point pro-\ncess on a 40×40 = 1600 -point grid, fitted to the Pines\ndataset. Precise specification of these targets can be found\nin Appendix D.1.\nWe compare our performance to a selection of strong base-\nlines. We consider two annealing algorithms, firstly an\nSMC sampler (Del Moral et al., 2006) with HMC kernels\nand secondly CRAFT (Matthews et al., 2022), which uses\nnormalizing flows to transport particles at each step of an\nSMC algorithm. We also consider two diffusion-based sam-\npling methods, Path Integral Sampler (PIS) (Zhang & Chen,\n2022) and Denoising Diffusion Sampler (DDS) (Vargas\net al., 2023). As mentioned in Section 3.3, we reparame-\nterize the target using a variational approximation for all\nmethods. We include full details of hyperparameter and\noptimizer settings, run times and experimental procedures\nin Appendix D.2.\nWe present the normalizing constant estimation results in\nFigure 2. We report results from our method with both one\nand two iterations of potential approximation (PDDS1 and\nPDDS2 respectively). PDDS1 is given the same training\n1 2 4 8 16−0.8−0.6−0.4−0.20.00.20.40.6Log ZMixture\n2 4 8 16 32−1.25−1.00−0.75−0.50−0.250.00Log ZFunnel\n2 4 8 16 32−108\n−110\n−112\n−114\n−116\n−118Log ZSonar\nCRAFT\nDDS\nPDDS1\nPDDS2\nPIS\nSMC\n8 16 32 64 128\nSteps475480485490495500505Log ZLGCPFigure 2: Log normalizing constant estimation results for\nour method with one and two iterations of potential approxi-\nmation (PDDS1 and PDDS2 respectively), compared with\nSMC, CRAFT, DDS and PIS. Dotted black represents ana-\nlytic ground truth where available, otherwise long-run SMC.\nVariation is displayed over both training and sampling seeds\n(2000 total). The y-axes on Sonar and LGCP have been\ncropped and outliers (present in all methods) removed for\nclarity, uncurated samples are presented in Appendix D.4.\n7Particle Denoising Diffusion Sampler\nFigure 3: Samples from each method using 4 steps on the\nGaussian Mixture task.\n0 100 200\nStep30405060708090100ESS (%)Simple\n0 100 200\nStepPDDS1\n0 100 200\nStepPDDS2\nFigure 4: ESS curves for our method after 0, 1, and 2\niterations of Algorithm 2 on the Sonar task with 200 steps.\nbudget as PIS and DDS. Considering the posterior sampling\ntasks, we uniformly outperform both diffusion-based ap-\nproaches (PIS and DDS) in terms of estimation bias and\nvariance. We find that in the Sonar task CRAFT outper-\nforms our approach in few step regimes but we close this gap\nand sometimes outperform CRAFT given a larger number\nof steps, while the reverse is true on the LGCP task. Consid-\nering the synthetic target densities, our approach similarly\noutperforms the diffusion-based approaches. Our approach\nproduces higher variance estimates than CRAFT in low step\nregimes, but produces lower variance estimates in larger\nstep regimes. CRAFT clearly outperforms all competing\nmethods on the synthetic Funnel task.\n6.2. Sample quality\nWe also visually assess the quality of samples from each\nmethod in Figure 3. We choose the multi-modal Mixture\ntask in 2-dimensions for ease of visualization. Unsurpris-\ningly we find that both PIS and DDS do not capture all 6\nmodes of the distribution due to the mode-seeking behaviour\nof the reverse KL objective. Samples from our approach\nappear of similar quality to those of CRAFT and SMC.\n6.3. Iterations of potential approximation\nWe demonstrate here the improvement in normalizing con-\nstant estimates due to our iterative potential approximation\n2 4 8 16 32\nSteps−120.0−117.5−115.0−112.5−110.0−107.5Log Z\nSimple\nPDDS1\nPDDS2\n0 1 2 3 4\nTarget density evaluations ×108−113−112−111−110−109−108Log Z\nPDDS\nDDS\nPISFigure 5: Top: normalizing constant estimation after 0, 1\nand 2 iterations of Algorithm 2 on the Sonar task. Bottom:\nevolution of normalizing constant estimates during training,\nmeasured in terms of number of target density evaluations,\nshowing one realization on the Sonar task with 16 steps.\nscheme. Figure 4 shows the improvement in ESS and re-\nduction in number of resampling steps. In the top pane\nof Figure 5, we see that the simple potential approxima-\ntion (8) provides very poor normalizing constant estimates.\nAlgorithm 2 considerably improves upon this performance.\nIn the second pane of Figure 5, we show the evolution of ˆZN\n0\nduring training for each of the diffusion-based approaches.\nWhile PDDS initially falls below PIS and DDS due to the\nsimple initial approximation, we quickly recover and exceed\neach of these methods after around 7×107density evalua-\ntions. We also observe that PDDS improves incrementally\ndue to the iterative potential approximation.\n7. Discussion\nThis paper contributes to the growing literature on the use of\ndenoising diffusion ideas for Monte Carlo sampling (Berner\net al., 2022; McDonald & Barron, 2022; Vargas et al., 2023;\nHuang et al., 2024; Richter et al., 2024). It proposes an orig-\ninal iterative SMC algorithm which provides an unbiased\nestimate of the normalizing constant for any finite number of\nparticles by leveraging an original score matching technique.\nThis algorithm also provides asymptotically consistent esti-\nmates of the normalizing constant and of expectations with\n8Particle Denoising Diffusion Sampler\nrespect to the target. One limitation of PDDS is that it\npractically relies on g0(x)being a well-behaved potential\nfunction. While our approach using a variational approxi-\nmation to guide a reparameterization of the target has been\neffective in our examples, more sophisticated techniques\nmight have to be implemented (Hoffman et al., 2019).\nImpact Statement. Sampling is a ubiquitous problem.\nTherefore, PDDS can be applied to a wide range of appli-\ncations. While we have obtained limit theorems for this\nscheme, it is important to exercise caution if the output were\nto be used for decision making as we practically only use a\nfinite number of particles.References\nArbel, M., Matthews, A., and Doucet, A. Annealed flow\ntransport Monte Carlo. In International Conference on\nMachine Learning , 2021.\nBerner, J., Richter, L., and Ullrich, K. An optimal control\nperspective on diffusion-based generative modeling. In\nNeurIPS 2022 Workshop on Score-Based Methods , 2022.\nBradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary,\nC., Maclaurin, D., Necula, G., Paszke, A., VanderPlas, J.,\nWanderman-Milne, S., and Zhang, Q. JAX: composable\ntransformations of Python+NumPy programs, 2018.\nCardoso, G., Idrissi, Y . J. E., Corff, S. L., and Moulines,\nE. Monte Carlo guided diffusion for Bayesian linear in-\nverse problems. In International Conference on Learning\nRepresentations , 2024.\nCattiaux, P., Conforti, G., Gentil, I., and L ´eonard, C. Time\nreversal of diffusion processes under a finite entropy con-\ndition. Annales de l’Institut Henri Poincare (B) Proba-\nbilites et statistiques , 59(4):1844–1881, 2023.\nChatterjee, S. and Diaconis, P. The sample size required in\nimportance sampling. Ann. Appl. Probab. , 28(2):1099–\n1135, 2018.\nChopin, N. and Papaspiliopoulos, O. An Introduction to\nSequential Monte Carlo . Springer Series in Statistics.\n2020.\nChopin, N., Singh, S. S., Soto, T., and Vihola, M. On resam-\npling schemes for particle filters with weakly informative\nobservations. Ann. Stat. , 50(6):3197–3222, 2022. ISSN\n0090-5364.\nChung, H., Kim, J., Mccann, M. T., Klasky, M. L., and\nYe, J. C. Diffusion posterior sampling for general noisy\ninverse problems. In The Eleventh International Confer-\nence on Learning Representations , 2023.\nCorso, G., Xu, Y ., De Bortoli, V ., Barzilay, R., and Jaakkola,\nT. Particle guidance: non-iid diverse sampling with diffu-\nsion models. arXiv preprint arXiv:2310.13102 , 2023.\nDai, C., Heng, J., Jacob, P. E., and Whiteley, N. An invita-\ntion to sequential monte carlo samplers. Journal of the\nAmerican Statistical Association , 117(539):1587–1600,\n2022.\nDe Bortoli, V ., Thornton, J., Heng, J., and Doucet, A. Diffu-\nsion Schr ¨odinger bridge with applications to score-based\ngenerative modeling. In Advances in Neural Information\nProcessing Systems , 2021.\nDel Moral, P. Feynman-Kac Formulae: Genealogical and\nInteracting Particle Approximations . Springer, 2004.\n9Particle Denoising Diffusion Sampler\nDel Moral, P., Doucet, A., and Jasra, A. Sequential Monte\nCarlo samplers. Journal of the Royal Statistical Society:\nSeries B (Statistical Methodology) , 68(3):411–436, 2006.\nDel Moral, P., Doucet, A., and Jasra, A. On adaptive re-\nsampling strategies for sequential Monte Carlo methods.\nBernoulli , 18(1):252–278, 2012.\nDouc, R. and Capp ´e, O. Comparison of resampling schemes\nfor particle filtering. In Proceedings of the 4th Interna-\ntional Symposium on Image and Signal Processing and\nAnalysis , pp. 64–69. IEEE, 2005.\nDoucet, A., De Freitas, N., and Gordon, N. J. Sequential\nMonte Carlo Methods in Practice . Information Science\nand Statistics. New York, NY: Springer, New York, 2001.\nGerber, M., Chopin, N., and Whiteley, N. Negative associ-\nation, ordering and convergence of resampling methods.\nAnn. Stat. , 47(4):2236–2260, 2019. ISSN 0090-5364.\nGeyer, C. Markov chain Monte Carlo maximum likelihood.\nInComputing science and statistics: Proceedings of 23rd\nSymposium on the Interface Interface Foundation, Fairfax\nStation, 1991 , pp. 156–163, 1991.\nGuarniero, P., Johansen, A. M., and Lee, A. The iterated\nauxiliary particle filter. Journal of the American Statisti-\ncal Association , 112(520):1636–1647, 2017.\nHaussmann, U. G. and Pardoux, E. Time reversal of dif-\nfusions. The Annals of Probability , 14(3):1188–1205,\n1986.\nHeng, J., Bishop, A. N., Deligiannidis, G., and Doucet,\nA. Controlled sequential Monte Carlo. The Annals of\nStatistics , 48(5):2904–2929, 2020.\nHo, J., Jain, A., and Abbeel, P. Denoising diffusion prob-\nabilistic models. In Advances in Neural Information\nProcessing Systems , 2020.\nHoffman, M., Sountsov, P., Dillon, J. V ., Langmore, I., Tran,\nD., and Vasudevan, S. Neutra-lizing bad geometry in\nHamiltonian Monte Carlo using neural transport. arXiv\npreprint arXiv:1903.03704 , 2019.\nHuang, X., Dong, H., Hao, Y ., Ma, Y ., and Zhang, T. Re-\nverse diffusion Monte Carlo. In International Conference\non Learning Representations , 2024.\nHyv¨arinen, A. Estimation of non-normalized statistical mod-\nels by score matching. The Journal of Machine Learning\nResearch , 6:695–709, 2005.\nKingma, D. and Ba, J. Adam: A method for stochastic\noptimization. In ICLR , 2015.Kloeden, P. E. and Platen, E. Numerical solution of stochas-\ntic differential equations , volume 23 of Appl. Math. (N.\nY.). Berlin: Springer-Verlag, 1992. ISBN 3-540-54062-8.\nLai, C.-H., Takida, Y ., Murata, N., Uesaka, T., Mitsu-\nfuji, Y ., and Ermon, S. Regularizing score-based mod-\nels with score fokker-planck equations. arXiv preprint\narXiv:2210.04296 , 2022.\nLawson, D., Ravent ´os, A., Linderman, S., et al. Sixo:\nSmoothing inference with twisted objectives. Advances\nin Neural Information Processing Systems , 35:38844–\n38858, 2022.\nLiptser, R. S. and Shiryayev, A. N. Statistics of random\nprocesses. I. General theory. Translated by A. B. Aries ,\nvolume 5 of Appl. Math. (N. Y.) . Springer, New York,\n1977.\nM´at´e, B. and Fleuret, F. Learning deformation trajectories\nof Boltzmann densities. arXiv preprint arXiv:2301.07388 ,\n2023.\nMatthews, A. G. D. G., Arbel, M., Rezende, D. J., and\nDoucet, A. Continual repeated annealed flow transport\nMonte Carlo. In International Conference on Machine\nLearning , 2022.\nMcDonald, C. J. and Barron, A. R. Proposal of a score based\napproach to sampling using Monte Carlo estimation of\nscore and oracle access to target density. In NeurIPS 2022\nWorkshop on Score-Based Methods , 2022.\nMøller, J., Syversveen, A. R., and Waagepetersen, R. P.\nLog Gaussian Cox processes. Scandinavian Journal of\nStatistics , 25(3):451–482, 1998.\nNeal, R. M. Annealed importance sampling. Statistics and\nComputing , 11(2):125–139, 2001.\nNeal, R. M. Slice sampling. The Annals of Statistics , 31:\n705–767, 06 2003.\nNichol, A. Q. and Dhariwal, P. Improved denoising diffusion\nprobabilistic models. In International Conference on\nMachine Learning , 2021.\nRichter, L., Berner, J., and Liu, G.-H. Improved sampling\nvia learned diffusions. In International Conference on\nLearning Representations , 2024.\nSalimans, T. and Ho, J. Should EBMs model the energy\nor the score? In Energy Based Models Workshop-ICLR\n2021 , 2021.\nSong, J., Vahdat, A., Mardani, M., and Kautz, J.\nPseudoinverse-guided diffusion models for inverse prob-\nlems. In International Conference on Learning Represen-\ntations , 2023.\n10Particle Denoising Diffusion Sampler\nSong, Y . and Ermon, S. Generative modeling by estimating\ngradients of the data distribution. Advances in neural\ninformation processing systems , 32, 2019.\nSong, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er-\nmon, S., and Poole, B. Score-based generative modeling\nthrough stochastic differential equations. In International\nConference on Learning Representations , 2021.\nSountsov, P., Radul, A., and contributors. Inference\ngym, 2020. URL https://pypi.org/project/\ninference_gym .\nSyed, S., Bouchard-C ˆot´e, A., Deligiannidis, G., and Doucet,\nA. Non-reversible parallel tempering: A scalable highly\nparallel MCMC scheme. Journal of the Royal Statistical\nSociety Series B , 84(2):321–350, 2022.\nTawn, N. G., Roberts, G. O., and Rosenthal, J. S. Weight-\npreserving simulated tempering. Statistics and Comput-\ning, 30(1):27–41, 2020.\nVargas, F., Grathwohl, W., and Doucet, A. Denoising diffu-\nsion samplers. In International Conference on Learning\nRepresentations , 2023.\nVincent, P. A connection between score matching and de-\nnoising autoencoders. Neural Computation , 23(7):1661–\n1674, 2011.\nWebber, R. J. Unifying sequential Monte Carlo with resam-\npling matrices. arXiv preprint arXiv:1903.12583 , 2019.\nWoodard, D. B., Schmidler, S. C., and Huber, M. Sufficient\nconditions for torpid mixing of parallel and simulated\ntempering. Electronic Journal of Probability , 14:780–\n804, 2009.\nWu, L., Trippe, B. L., Naesseth, C. A., Blei, D., and Cun-\nningham, J. P. Practical and asymptotically exact condi-\ntional sampling in diffusion models. In Thirty-seventh\nConference on Neural Information Processing Systems ,\n2023.\nZhang, Q. and Chen, Y . Path integral sampler: a stochastic\ncontrol approach for sampling. In International Confer-\nence on Learning Representations , 2022.\n11Particle Denoising Diffusion Sampler\nAppendix\nThe Appendix is organized as follows. In Appendix A we detail the PDDS algorithm when adaptive resampling is used. In\nAppendix B we reformulate the results of Webber (2019) in terms of chi-squared divergences. In Appendix C, we expose\nour proofs. Finally, in Appendix D we present details and additional work relating to our experiments.\nA. Particle Denoising Diffusion Sampler with Adaptive Resampling\nAlgorithm 3 Particle Denoising Diffusion Sampler with Adaptive Resampling\nInput: Schedule (βt)t∈[0,T]as in (2); Approximations (ˆgk)K\nk=0s.t.ˆg0=g0,ˆgK= 1; Number of particles N; ESS\nresampling threshold α\nSample Xi\nKiid∼ N(0,I)fori∈[N]\nSetˆZK←1andωi\nK←1/Nfori∈[N]\nfork=K−1, . . . , 0do\nMove . Sample ˜Xi\nk∼ˆπ(·|Xi\nk+1)fori∈[N](see (16))\nWeight .ωi\nk←ωi\nkωk(˜Xi\nk, Xi\nk+1)fori∈[N](see (17))\nSetˆZk←ˆZk+1×1\nNP\ni∈[N]ωi\nk\nNormalize ωi\nk←ωi\nk/P\nj∈[N]ωj\nk\nResample and MCMC .\nif(P\ni∈[N](ωi\nk)2)−1< αN then\nX1:N\nk←resample( ˜X1:N\nk, ω1:N\nk)(see Section 3.3)\n(Optional). Sample Xi\nk←Mk(Xi\nk,·)fori∈[N]using a ˆπk-invariant MCMC kernel.\nReset ω1:N\nk←1/N\nelse\nX1:N\nk←˜X1:N\nk\nend if\nend for\nOutput: Particle estimates ˆπN=1\nNP\ni∈[N]δXi\n0ofπandˆZN\n0ofZ\nB. Asymptotic error formulae for SMC\nThe results of Webber (2019) play a fundamental role in our work. Compared to traditional SMC literature, they put\nmore stress on the normalizing constant estimate, have results for different resampling schemes, and do not require weight\nboundedness.\nIn this section, we present these results using the language of chi-squared divergence. This gives alternative expressions\nwhich are easier to manipulate in our context.\nB.1. Chi-squared divergence\nFor two probability distributions pandqsuch that q≪p, the chi-squared divergence of qwith respect to pis defined as\nχ2(q||p) =Z\nXp(dx)\"\u0012dq\ndp\u00132\n(x)−1#\n.\nWe state without proof two simple properties of this distance.\nLemma B.1. Letgbe a nonnegative function from XtoRsuch that p(g) :=Ep[g(X)]<∞. Define the probability\ndistribution πbyπ(dx)∝p(x)g(x). Then\nVarp(g) = (p(g))2χ2(π||p).\n12Particle Denoising Diffusion Sampler\nLemma B.2. The chi-squared divergence between two probability distributions pandqonX ×Y satisfies the decomposition\nχ2(q(dx,dy)||p(dx,dy)) =χ2(q(dx)||p(dx)) +Z\np(dx)\u0012dq\ndq\u00132\n(x)χ2(q(dy|x)||p(dy|x)).\nB.2. Generic Feynman-Kac formula and the associated SMC algorithm\nLet\nM(dx0:T) =M(dx0)M(dx1|x0). . .M(dxT|xT−1)\nbe a Markov measure defined on X0×. . .× XT. LetG0(x0),G1(x0, x1),. . .,GT(xT−1, xT)be strictly positive functions.\nFor any t < T , assume that there exists Zt>0such that\nQt(dx0:T) =1\nZtM0(dx0)G0(x0)M1(dx1|x0)G1(x0, x1). . .Mt(dxt|xt−1)Gt(xt)M(dxt+1:T|xt)\nis a probability measure. Then Qt(dx0:t)is called the Feynman-Kac model associated with the Markov measure M(dx0:t)\nand the weight functions (Gs)s≤t. Given a number of particles N, Algorithm 4 approximates QT(dxT)and the normalizing\nconstant ZT. It is called the SMC algorithm associated to the Feynman-Kac model QT.\nAlgorithm 4 Generic SMC algorithm\nInput: Markov kernels M(dxt|xt−1); Functions Gt(xt−1, xt); Number of particles N\nSample X1:N\n0iid∼M0(dx0)\nSetωn\n0=G0(Xn\n0)andZN\n0=1\nNPωn\n0\nNormalize ωn\n0←ωn\n0/P\nmωm\n0\nfort= 1, . . . , T do\nResample particles ˜X1:N\nt−1among particles X1:N\nt−1with weights ω1:N\nt−1\nSample Xn\nt∼Mt(dxt|˜Xn\nt−1)\nSetωn\nt=Gt(˜Xn\nt−1, Xn\nt)andZn\nt=Zn\nt−11\nNPωn\nt\nNormalize ωn\nt←ωn\nt/Pωm\nt\nend for\nOutput: Empirical measurePωn\nTδXn\nTapproximating QT(dxT)and estimate ZN\nTapproximating ZT\nB.3. Asymptotic error\nWe recall the following definition from Webber (2019).\nDefinition B.3. The notation |Yn−c|≲Unmeans that there exists a sequence of sets (Bn)such that P(Bn)→1and\nlim sup\nn→∞E\"\n1Bn\f\f\f\fYn−c\nUn\f\f\f\f2#\n≤1.\nWe are now ready to restate parts of Theorem 3.2 and Example 3.4 of Webber (2019) in terms of chi-squared divergences.\nTheorem B.4. Given the Feynman-Kac model defined in Section B.2, assume that χ2(QT(dx0)||M0(dx0))<∞and\nχ2(QT(dxt)||Qt(dxt))<∞,∀t. Then√\nN(ZN\nT/ZT−1)is asymptotically normal with variance σ2\nmult if multinomial\nresampling is used;\f\f\f√\nN(ZN\nT/ZT−1)\f\f\f2\n≲σ2\nsortif sorted resampling (Gerber et al., 2019) is used; with\nσ2\nmult=χ2(QT(dx0)||M0(dx0)) +TX\nt=1χ2(QT(dxt−1,dxt)||Qt−1(dxt−1)Mt(xt−1,dxt))\n=χ2(QT(dx0)||M0(dx0)) +TX\nt=1Z\nQt−1(dxt−1)\u0012QT(dxt−1)\nQt−1(dxt−1)\u00132\nχ2(QT(dxt|xt−1)||Mt(dxt|xt−1))\n| {z }\nσ2\nsort+\n+TX\nt=1χ2(QT(dxt−1)||Qt−1(dxt−1))\n13Particle Denoising Diffusion Sampler\nwhere we have decomposed σ2\nmult using the chain rule (Lemma B.2).\nProof. The original formulation of Theorem 3.2 (Webber, 2019) is written in terms of the following quantities\n˜Gt:=EM\"t−1Y\ns=0Gs#\nGt/E\"tY\ns=0Gs#\nht(xt) :=E\"TY\ns=t+1Gs\f\f\f\f\fXt=xt#\n.\nTo translate these notations into our case, note that ˜Gt=Qt(dx0:T)/Qt−1(dx0:T)and thus\nmin\nc∈REM\"tY\ns=0˜Gs|ht−c|2#\n= min\nc∈REQt\u0002\n|ht−c|2\u0003\n= Var Qt(ht) = [Qt(ht)]2χ2(QT(dxt)||Qt(dxt))\nusing Lemma B.1. Moreover,\nVarM(Gt+1(Xt, Xt+1)ht+1(Xt+1)|Xt) =h2\nt(Xt)χ2(QT(dxt+1|xt)||MT(dxt+1|xt))\nand thus, using ht(xt) =QT(dxt)\nQt(dxt)Qt(ht)we get\nE\"tY\ns=0GsVar(Gt+1ht+1|Xt)#\n=ZtQt(ht)2EQt\"\u001aQT(dxt)\nQt(dxt)(Xt)\u001b2\nχ2(QT(dxt+1|Xt)||MT(dxt+1|Xt)#\n.\nThe identity ZtQt(ht) =ZThelps conclude the proof.\nC. Proofs\nC.1. Proof of Proposition 2.1\nProof. Write\nπt(xt) =Z\nπ0(x0)p(xt|x0)dx0=1\nZZ\ng0(x0)p0(x0)p(xt|x0)dx0\n=1\nZZ\ng0(x0)p0(xt)p(x0|xt)dx0\n=1\nZp0(xt)gt(xt).(21)\nThe proof is concluded by taking the log gradient of the obtained identity with respect to xt.\nC.2. Proof of Lemma 2.2\nWe first state the following elementary lemma on the solution of a linear SDE (Kloeden & Platen, 1992, Chapter 4.2).\nLemma C.1. Leta: [0, T]→Randc: [0, T]→Rbe two continuous functions. Put Φt= expRt\n0asds,∀t∈[0, T]. Then\nthe solution to\ndZt= (atZt+ct)dt+√\n2dWt (22)\nis\nZt= Φ t\u0012\nZ0+Zt\n0Φ−1\nscsds+Zt\n0Φ−1\ns√\n2dWs\u0013\n. (23)\nUsing Ito isometry, we get the following straightforward corollary.\nCorollary C.2. Under the setting of Lemma C.1, if Z0∼ N(0,1), then\nE[Zt] = Φ tZt\n0Φ−1\nscsds,Var(Zt) = Φ2\nt+ 2Φ2\ntZt\n0Φ−2\nsds. (24)\n14Particle Denoising Diffusion Sampler\nWe are now ready to give the proof of Proposition 2.2.\nProof of Proposition 2.2. Without loss of generality, we can assume that βt≡1. Indeed, putting ˆ¯gt(xt) :=g0(e−txt)and\ndefining ¯Z(T)\ntby\nd¯Z(T)\nt=h\n−¯Z(T)\nt+ 2∇logˆ¯gT−t(¯Z(T)\nt)i\ndt+√\n2d˜Bt,¯Z(T)\n0∼ N(0,1), (25)\nwe see that Z(T)\n0:thas the same law as ¯ZRT\n0βsdsRT\nT−tβsds.\nWe only consider the case σ̸= 1here. The case σ= 1can be treated using similar but simpler calculations; it can also be\nrecovered by letting σ→1in the expressions below.\nSince∇logg0(x0) =∇logπ(x0)− ∇logp0(x0) =−(x−µ)/σ2+x, Equation (9) has the form (22) with\nat=−h\n1 + 2 e2(t−T)(1/σ2−1)i\n, c t= 2et−Tµ/σ2. (26)\nTedious but standard calculations yield\nΦt= exp\b\n−t−e−2T(1/σ2−1)(e2t−1)\t\n(27)\nand the integral of t-dependent terms inRT\n0Φ−1\ntctdtis\nZT\n0expn\n2t+ (1/σ2−1)e2(t−T)o\ndt=1\n2e−2T(1/σ2−1)\u0002\nexp\b\n1/σ2−1\t\n−exp\b\ne−2T(1/σ2−1)\t\u0003\n.\nUsing Corollary C.2, we have\nE[Z(T)\nT] =µ\n1−σ2\u0002\n1−exp\b\n(e−2T−1)(1/σ2−1)\t\u0003\n(28)\nandVar(Z(T)\nT) =vT,1+vT,2, where\nvT,1=1\n2(1/σ2−1)\u0002\n1−exp\b\n−2(1/σ2−1)(1−e−2T)\t\u0003\n(29)\nvT,2= exp\b\n−2T−2(1/σ2−1)(1−e−2T)\t\n. (30)\nThe lemma is proved.\nC.3. Proof of Propositions 3.1 and 3.2\nThe propositions follow from a direct application of Theorem B.4 to the Feynman-Kac model\nQK(y0:K) =1\nZMK(y0:K)G0(y0)KY\nk=1Gk(yk−1, yk)\nwhere we have the correspondence yk=xK−kand\nMK(y0:K) =N(xK|0,Id)ˆπ(xK−1|xK). . .ˆπ(x0|x1)\nG0(y0) = ˆgK(xK)\nGk(yk−1, yk) =ωK−k(xK−k, xK−k+1).\nC.4. Proof of Proposition 3.3\nWe start by providing some intuition for the result. Repeat that Xkis a shorthand for Xkδwhere δis the discretization step\nsize. This convention only applies for Xk.\n15Particle Denoising Diffusion Sampler\nWhen Kis big ( δis small), we have, as established in (14),\nπ(xk|xk+1)≈ N(xk;p\n1−αk+1xk+1+αk+1∇loggk+1(xk+1), αk+1I)\nˆπ(xk|xk+1) =N(xk;p\n1−αk+1xk+1+αk+1∇log ˆgk+1(xk+1), αk+1I).\nUsing the analytic formula for the chi-squared divergence between two Gaussians we get\nχ2(π(xk|xk+1)||ˆπ(xk|xk+1))≈eαk+1∥∇loggk+1−∇log ˆgk+1∥2(xk+1)−1≈2δ∥∇loggk+1− ∇log ˆgk+1∥2(xk+1).\nThus\nζ2\nK=χ2(πK||N(0,Id)) +X\nkZπk+1(xk+1)2\nˆπk+1(xk+1)χ2(π(xk|xk+1)||ˆπ(xk|xk+1))dxk+1\n≈χ2(πK||N(0,Id)) +X\nkZπk+1(xk+1)2\nˆπk+1(xk+1)2δ∥∇loggk+1− ∇log ˆgk+1∥2(xk+1)dxk+1\n≈χ2(πT||N(0,Id)) + 2ZT\n0Z\nXπt(x)2\nˆπt(x)∥∇loggt(x)− ∇log ˆgt(x)∥2dxdt.\nC.4.1. R EGULARITY CONDITIONS FOR PROPOSITION 3.3\nWe assume that the sequence of distributions πt(·)satisfy the following properties.\nAssumption C.3. There exists M1>0such that ∥∇logπt(xt)∥ ≤M1(1 +∥xt∥).\nAssumption C.4. There exist M2>0,M3>0,α2≥1, and α3≥1such that\r\r∇2logπt(xt)\r\r≤M2(1 +∥xt∥)α2and\r\r∇3logπt(xt)\r\r≤M3(1 +∥xt∥)α3.\nAssumption C.5. There exist ϑ >0andM∞<∞such thatR\nπt(xt)eϑ∥xt∥2dxt< M∞,∀t.\nThese assumptions are satisfied, for example, when the target distribution π0is Gaussian.\nRemark C.6.Letφ(xt) =∇logπt(xt). Then the notation ∇3logπt(xt)refers to the second order differential φ′′(xt)\nwhich is a bilinear mapping from Rd×RdtoRd. For any multilinear operator H:X1×. . .× Xn→ Y , we define\n∥H∥op:= sup\nx1,...,x n̸=0∥H(x1, . . . , x n)∥\n∥x1∥. . .∥xn∥.\nBy writing ∥H∥we implicitly refer to ∥H∥op. In fact, the space of such operators is of finite dimensions in our cases of\ninterests, hence any two norms are bounded by a constant factor of each other.\nThe above assumptions only concern the differential of ∇logπt(xt)with respect to x. The following lemma derives a\nbound with respect to t.\nLemma C.7. Under the above assumptions, there exist constants ¯M1and¯α1such that\n\r\r\r\r∂\n∂t∇logπt(xt)\r\r\r\r≤¯M1(1 +∥xt∥)¯α1,∀t.\nProof. Using the Fokker-Planck equation for the score (Lai et al., 2022), we write\n∂tloggt(xt) = div x∇loggt(xt) +∇loggt(xt)◦(∇loggt(xt)−xt).\nFor a fixed t, putφ(xt) =∇loggt(xt)andψ(xt) = Tr( φ′(xt)) +φ(xt)◦(φ(xt)−xt). Then ∂t∇loggt(xt) =∇ψ(xt).\nViewing ψ′(xt),φ′(xt), and φ′′(xt)as elements of L(Rd,R),L(Rd,Rd), andL(Rd,L(Rd,Rd))respectively; where\nL(A, B)is the space of linear operators from AtoB; we can write\nψ′(xt)h= Tr( φ′′(xt)h) + (φ′(xt)h)◦(φ(xt)−xt) +φ(xt)◦(φ′(xt)h−h),∀h∈Rd\nwhere ◦stands for the usual scalar product between two vectors in Rd. There is a constant Cdepending on the dimension\nsuch that Tr(L)≤C∥L∥opfor endomorphisms L. Thus\n∥ψ′(xt)h∥op≤C∥φ′′(xt)∥op∥h∥+∥φ′(xt)∥op∥φ(xt)−xt∥∥h∥+∥φ(xt)∥∥φ′(xt)−Id∥op∥h∥\n≤¯M1(1 +∥xt∥)¯α1∥h∥\nfor some ¯M1and¯α1by Assumptions C.3 and C.4. This entails ∥∂t∇loggt(xt)∥=∥ψ′(xt)∥op≤¯M1(1 +∥xt∥)¯α1.\n16Particle Denoising Diffusion Sampler\nC.4.2. F ORMAL PROOF\nTo formalize the error of the heuristic approximations presented at the beginning of Section C.4 we need the following\ntechnical lemma.\nLemma C.8. Letx0andvbe two vectors in Rdand suppose that XA\ntandXB\ntare respectively the unique solutions of the\nSDEs:\n(A) : d XA\nt= (−XA\nt+ 2v)dt+√\n2dWA\nt, X 0=x0\n(B) : d XB\nt= (−XB\nt+ 2ft(XB\nt))dt+√\n2dWB\nt, X 0=x0\nwhere there exist strictly positive constants M1,¯M1,M2, and M3; and strictly greater than 1constants ¯α1,α2, and α3;\nsuch that the function ft(xt)satisfies ∥ft(xt)∥ ≤M1(1 +∥xt∥),∥∂tft(xt)∥ ≤ ¯M1(1 +∥xt∥)¯α1, and\r\r∇ift(xt)\r\r≤\nMi+1(1 +∥xt∥)αi+1fori∈ {1,2}. (The notation ∇refers implicitly to the gradient with respect to x.) Denote PA(dx[0,T])\nandPB(dx[0:T])respectively the path measures associated with the solutions of (A) and (B). Then, there exist a parameter\nfMdepending on all the aforementioned constants and a parameter fM1depending only on M1such that for any t≤1/fM1,\nthe chi-squared divergence of PB(dx[0,t])with respect to PA(dx[0,t])is finite, and\n\f\f\fχ2(PB(dx[0:t])||PA(dx[0:t]))−2t∥f0(x0)−v∥2\f\f\f≤fMt2etfM1(∥x0∥2+∥v∥2)(1 +∥x0∥+∥v∥)4(1+¯α1+α2+α3)\nProof. Put∆t(xt) = 2 ft(xt)−2v. By an application of Girsanov’s theorem (Liptser & Shiryayev, 1977, Example 3,\nSection 6.2.3), we have\nDt:=dPB\ndPA(X[0,t]) = exp\u001aZt\n0⟨∆s(Xs),dWA\ns⟩√\n2−1\n4Zt\n0∥∆s(Xs)∥2ds\u001b\nwhere, for a vector-valued process Vt, the notationRt\n0⟨Vs,dWs⟩:=Pd\ni=1Rt\n0Vi\nsdWi\ns. As a preliminary step, we would like\nto bound EA[Dα\nt(1 +∥Xt∥+∥v∥)n]for some α, n≥1. Here, c(·)denotes a constant whose value might change from line\nto line and depends on the variables inside the round bracket. We also drop the subscript/superscript A from EAandWA\nt\nwhenever there is no risk of confusion. We have\nE[Dα\nt(1 +∥Xt∥+∥v∥)n] =E\u0014\nexp\u001a\nαZt\n0⟨∆s,dWs⟩√\n2−α\n4Zt\n0∥∆s∥2ds\u001b\n(1 +∥Xt∥+∥v∥)n\u0015\n=E\u0014\nexp\u001a\nαZt\n0⟨∆s,dWs⟩√\n2−α2\n2Zt\n0∥∆s∥2ds\u001b\nexp\u001a\u0012α2\n2−α\n4\u0013Zt\n0∥∆s∥2ds\u001b\n(1 +∥Xt∥+∥v∥)n\u0015\n≤et(c(M1,α)+c(α)∥v∥2)E\u0014\nexp\u001a\nαZt\n0⟨∆s,dWs⟩√\n2−α2\n2Zt\n0∥∆s∥2ds\u001b\nexp\u001a\nc(M1, α)Zt\n0∥Xs∥2ds\u001b\n×\n×(1 +∥Xt∥+∥v∥)n\u0015\nusing∥∆s∥2≤c(M1)(1 +∥xs∥2) + 8∥v∥2\n≤et(c(M1,α)+c(α)∥v∥2)E1/2\u0014\nexp\u001a√\n2αZt\n0⟨∆s,dWs⟩ −α2Zt\n0∥∆s∥2ds\u001b\u0015\n×\n×E1/4\u0014\nexp\u001a\n4c(M1, α)Zt\n0∥Xs∥2ds\u001b\u0015\nE1/4[(1 +∥Xt∥+∥v∥)n]\nusing double Cauchy-Schwarz E[XY Z ]≤E1/2[X2]E1/4[Y4]E1/4[Z4].\nIn the last line of the above display, the first expectation is equal to 1by the same Girsanov argument as Liptser &\nShiryayev (1977, Example 3, Section 6.2.3). To bound the second expectation, we note that under PA, we have Xs∼\nN(√1−λs(x0−2v) + 2v, λs). Elementary calculations yield the bound\nE[ek∥Xs∥2] =\u00121√1−2kλs\u0013d\nexp(\nk\r\r√1−λs(x0−2v) + 2v\r\r2\n1−2kλs)\n≤(√\n2)de16k(∥x0∥2+∥v∥2)\n17Particle Denoising Diffusion Sampler\nfor0< k < 1/4. Write\nE\u0014\nexp\u001a\n4c(M1, α)Zt\n0∥Xs∥2ds\u001b\u0015\n≤1\ntZt\n0Eh\ne4tc(M1,α)∥Xs∥2i\nds≤c(1)e64tc(M1,α)(∥x0∥2+∥v∥2)\nift≤1\n16c(M1,α), using Jensen’s inequality and the above bound. The third expectation can be bounded by c(n)(1+∥x0∥4n+\n∥v0∥4n).\nPutting everything together, we establish that there exist constants c(n)andc(M1, α)such that\nE[Dα\nt(1 +∥Xt∥+∥v∥)n]≤c(n)etc(M1,α)(∥x0∥2+∥v∥2)(1 +∥x0∥4n+∥v∥4n),∀t≤1\nc(M1, α). (31)\nNow to study χ2(PB(dx[0:t])||PA(dx[0:t])), we apply Ito’s formula to D2\ntand get, under PA\nD2\nt= 1 +√\n2Zt\n0D2\ns⟨∆s,dWs⟩+Zt\n0D2\ns∥∆s∥2\n2dt. (32)\nPutting ηt(xt) =∥∆t(xt)∥2and˜ft(xt) = 2 ft(xt)−xt, we have\nD2\ntηt=η0+√\n2Zt\n0D2\ns⟨ηs∆s+∇ηs,dWs⟩+\n+Zt\n0D2\ns(\nηs∥∆s∥2\n2+∇ηs(˜f(Xs) + 2∆ s) +∂η\n∂s+ Tr\u0000\n∇2ηs\u0001)\nds.(33)\nTo study (32) and(33), we use (31) together with the following bounds which are consequences of the lemma’s assumptions:\n∥∆s(xs)∥ ≤c(M1)(1 +∥xs∥+∥v∥)\nηs(xs)≤c(M1)(1 +∥xs∥+∥v∥)2\n∥∇ηs(xs)∥ ≤c(M1, M2)(1 +∥xs∥+∥v∥)1+α2\n\r\r\r\r∂η\n∂s(s, Xs)\r\r\r\r≤c(M1,¯M1)(1 +∥xs∥)1+¯α1\n\r\rTr\u0000\n∇2ηs\u0001\r\r≤c(M1, M2, M3)(1 +∥xs∥+∥v∥)1+α2+α3.\nThe last inequality is justified by the fact that Tr\u0000\n∇2ηs\u0001\n≲\r\r∇2ηs\r\r, where by considering ∇2ηs(xs)as the second\ndifferential of ηsatxs(i.e. a bilinear form from Rd×RdtoR), we have\n∂2ηs\n∂x2(x)[h, k] = ∆ s(x)◦∂2∆\n∂x2(x)[h, k] + (∂∆\n∂x(x)h)◦(∂∆\n∂x(x)k),∀(h, k)∈Rd×Rd\nwhere◦stands for the usual scalar product between two vectors in Rd. These bounds show that the stochastic integrals (w.r.t.\ndWs) in(32) and(33) are true martingales (as opposed to merely local martingales). Moreover, there exist a constant fM\ndepending on M1,M2,M3,¯M1,¯α1,α2, and α3; and a constant fM1depending on M1only such that\nE\"\nD2\ns\f\f\f\f\fηs∥∆s∥2\n2+∇ηs(˜f(Xs) + 2∆ s) +∂η\n∂s+ Tr\u0000\n∇2ηs\u0001\f\f\f\f\f#\n≤4fMetfM1(∥x0∥2+∥v∥2)×\n×(1 +∥x0∥+∥v∥)4(1+¯α1+α2+α3),∀s≤t≤1\nfM1.\nTaking expectation of both sides of (33) and rearranging yields\n\f\fE[D2\ntηt]−η0\f\f≤4tfMetfM1(∥x0∥2+∥v∥2)(1 +∥x0∥+∥v∥)4(1+¯α1+α2+α3),∀t≤1\nfM1.\n18Particle Denoising Diffusion Sampler\nThen we have, by taking expectation of both sides of (32):\n\f\f\fχ2(PB(dx[0:t])||PA(dx[0:t]))−tη0\n2\f\f\f=\f\f\fE(D2\nt)−1−tη0\n2\f\f\f=\f\f\f\fZt\n0\u0012\nE\u0014D2\nsηs\n2\u0015\n−η0\n2\u0013\nds\f\f\f\f≤Zt\n0\f\f\f\fE\u0014D2\nsηs\n2\u0015\n−η0\n2\f\f\f\fds\n≤Zt\n02sfMetfM1(∥x0∥2+∥v∥2)(1 +∥x0∥+∥v∥)4(1+¯α1+α2+α3)ds\n=t2fMetfM1(∥x0∥2+∥v∥2)(1 +∥x0∥+∥v∥)4(1+¯α1+α2+α3).\nThe proof is completed.\nWe are now ready to give the proof of Proposition 3.3.\nProof. To make the arguments clearer, we shall assume that the proposal distribution is\nˆπ(xk|xk+1) =N(xk|p\n1−αk+1xk+1+ 2(1−p\n1−αk+1)∇log ˆgk+1(xk+1), αk+1I)\nwhich is slightly different from (16). As we will see, the proof also applies to the original discretization with minimal\nchanges. For 0< s < u < T , the distributions ˆπ(xs|xu)andπ(xs|xu)can be obtained by respectively solving the following\nSDEs between times T−uandT−s:\nˆπ(xs|xu) : d YA\nt= (−YA\nt+ 2∇log ˆgT−u(YA\nt))dt+√\n2dWA\nt, YA\nT−u=xu\nπ(xs|xu) : d YB\nt= (−YB\nt+ 2∇loggT−t(YB\nt))dt+√\n2dWB\nt, YB\nT−u=xu.\nThe assumptions in Section C.4.1 and Lemma C.7 show that the conditions of Lemma C.8 are satisfied for this pair of SDEs.\nThus\n\f\f\fχ2(PB(dy[T−u,T−s])||PA(dy[T−u,T−s]))−2(u−s)∥∇loggu(xu)− ∇log ˆgu(xu)∥2\f\f\f≤fM(u−s)2×\n×e(u−s)fM1(∥xu∥2+∥∇log ˆgu(xu)∥2)(1 +∥xu∥+∥∇log ˆgu(xu)∥)α+\nforα+= 4(1 + ¯ α1+α2+α3)andu−s≤1\nfM1. This, together with the data processing inequality and the assumption on\n|∇log ˆgt|, implies\nχ2(π(xk|xk+1)||ˆπ(xk|xk+1))≤2δ∥∇loggk+1− ∇log ˆgk+1∥2(xk+1) +fMδ2eδfM1(1+2C2\n2)(1+∥xk+1∥)2×\n×(1 +C2)α+(1 +∥xk+1∥)α+,∀δ≤1\nfM1.\nThus, for a sufficiently fine discretization,\nX\nkZπk+1(xk+1)2\nˆπk+1(xk+1)χ2(π(xk|xk+1)||ˆπ(xk|xk+1))dxk+1≤\n≤X\nkδZπk+1(xk+1)2\nˆπk+1(xk+1)2∥∇loggk+1− ∇log ˆgk+1∥2(xk+1)dxk+1+\n+X\nkZ\nC1πk+1(xk+1)fMδ2eδfM1(1+2C2\n2)(1+∥xk+1∥2)(1 +C2)α+(1 +∥xk+1∥)α+dxk+1.(34)\nThe first term is a Riemann sum and converges toRT\n0Rπt(xt)2\nˆπt(xt)2∥∇loggt− ∇log ˆgt∥2(xt)dxtdt. To bound the second\nterm, first note that\nEπk+1[(1 +∥X∥)2α+]≤22α+−1Eh\n1 +∥X∥2α+i\n≤22α+−1E\u0014\n1 +eϑ∥X∥2max\u0012⌈α+⌉!\nϑ⌈α+⌉,⌊α+−1⌋!\nϑ⌊α+−1⌋\u0013\u0015\n≤22α+−1\u0012\n1 +M∞max\u0012⌈α+⌉!\nϑ⌈α+⌉,⌊α+−1⌋!\nϑ⌊α+−1⌋\u0013\u0013\n19Particle Denoising Diffusion Sampler\nwhere Mθandϑappear in Assumption C.5. Thus, as long as δfM1(1 + 2 C2\n2)≤ϑ/2, it holds that\nZ\nπk+1(x)eδfM1(1+2C2\n2)∥x∥2(1 +∥x∥)α+dx≤Eπk+1h\neϑ∥X∥2/2(1 +∥X∥)α+i\n≤E1/2h\neϑ∥X∥2i\nE1/2\u0002\n(1 +∥X∥)2α+\u0003\n≤C(M∞, α+, ϑ)\nfor some constant C(M∞, α+)depending on M∞,α+, and ϑ. Hence the second sum of (34) tends to 0when δ→0. The\nproof is finished.\nC.5. Proof of Proposition 4.1\nThe denoising score matching identity is standard and recalled here for convenience. We have\nπk(xk) =Z\np(xk|x0)π0(x0)dx0\nso by using the log derivative we obtain\n∇logπk(xk) =Z\n∇logp(xk|x0)π0(x0)p(xk|x0)\nπk(xk)dx0 (35)\n=Z\n∇logp(xk|x0)π(x0|xk)dx0. (36)\nIt can be easily verified that the interchange of differentiation and integration here does not require any regularity assumption\nonπ0(x0)apart from differentiability.\nTo prove the novel score identity, we first note that, under the conditionR\n∥∇π(x0)∥e−η∥x0∥2dx0<∞,∀η >0, we have\nZ\n∇x0logπ(x0|xk)π(x0|xk)dx0= 0 (37)\naccording to Lemma C.9. Combining this identity with (36), we have, for any α∈R,\n∇logπ(xk) =Z\n[∇xklogp(xk|x0) +α∇x0logπ(x0|xk)]π(x0|xk)dx0.\nIn particular:\n• For α=1√1−λk, we get\n∇logπ(xk) =1√1−λkZ\n∇logπ(x0)π(x0|xk)dx0\nwhich is the identity presented in Appendix C.1.3 (De Bortoli et al., 2021);\n• For α=√1−λk, we get\n∇logπ(xk) =Z\u0010p\n1−λk∇logg0(x0)−xk\u0011\nπ(x0|xk)dx0\nwhich is the identity we wanted to prove.\nThe verifications are straightforward by remarking that ∇x0logπ(x0|xk) =∇logg0(x0) +∇x0logp(x0|xk). We also\nnote that choosing α= 0brings us back to the classical score matching loss. Therefore, different values of αgive losses\nwith different properties.\nWe finish this section with a technical lemma giving conditions for (37) to hold.\nLemma C.9. Letf:Rd→Rbe a probability density, i.e. f≥0andR\nf(x)dx= 1. Suppose that fis continuously\ndifferentiable andR\n∥∇f(x)∥dx <∞. ThenR\n∇f(x)dx= 0.\n20Particle Denoising Diffusion Sampler\nRemark C.10 .The conditionR\n∥∇f(x)∥dx <∞is clearly the minimum necessary forR\n∇f(x)dx= 0to make sense. On\nthe other hand, we do notexplicitly require that for∇fvanishes at infinity.\nProof. Without loss of generality, we only prove thatR\n∂1f(x)dx= 0. Putg(x1) :=R\nf(x1, x2:d)dx2:d. Fubini’s theorem\nthen implies thatR\nRg(x1)dx1= 1. We have\nZ\nRd∂1f(x)dx= lim\nM→∞Z\nRd−1ZM\n−M∂1f(x1, x2:d)dx1dx2:d= lim\nM→∞Z\nRd−1f(M, x 2:d)−f(−M, x 2:d)dx2:d\n= lim\nM→∞g(M)−g(−M).(38)\nPutI(M) :=R∞\n0|g(M+x)−g(−M−x)|dx. We have\nI(M)≤Z∞\n0|g(M+x)|+|g(−M−x)|dx,\nthuslimM→∞I(M) = 0 by the integrability of g. Combining this with Fatou’s lemma yields\n0 = lim inf\nM→∞I(M)≥Z∞\n0lim inf\nM→∞|g(M+x)−g(−M−x)|dx=Z∞\n0\f\f\f\fZ\nRd∂1f(y)dy\f\f\f\fdx\nby (38). This means thatR\nRd∂1f(y)dy= 0.\nC.6. Proof of Proposition 4.2\nProof. Since we are in the Gaussian case with d= 1, we have ∇logg0(x0) =ax0+bfor some a, b∈R. Therefore\nℓNSM(θ)andVar(ˆℓDSM(θ))are trivially bounded. To study ℓDSM(θ), we first note that\nVar(X0|Xt) =λtσ2\nλt+ (1−λt)σ2=:ρt.\nWrite\nEh\n∥sθ(t, Xt)− ∇logp(Xt|X0)∥2\f\f\fXti\n≥Var ( sθ(t, Xt)− ∇logp(Xt|X0)|Xt) = Var\u0012\n−Xt−√1−λtX0\nλt\f\f\f\fXt\u0013\n=1−λt\nλ2\ntρt\nso\nℓDSM(θ)≥ZT\n01−λt\nλ2\ntρtdt=∞\nsince ρt∼2tast→0. Concerning ˆ∇ℓDSM(θ) = 2 T(sθ(τ, Xτ)− ∇logp(Xτ|X0))∇θsθ(τ, Xτ), we have\nE\u0014\r\r\rˆ∇ℓDSM(θ)\r\r\r2\f\f\f\fXt, τ=t\u0015\n= 4T2∥∇θsθ(t, Xt)∥2Eh\n∥sθ(t, Xt)− ∇logp(Xt|X0)∥2\f\f\fXt, τ=ti\n≥4T2∥∇θsθ(t, Xt)∥21−λt\nλ2\ntρt\nso\nE\u0014\r\r\rˆ∇ℓDSM(θ)\r\r\r2\u0015\n=1\nTZT\n0E\u0014\r\r\rˆ∇ℓDSM(θ)\r\r\r2\f\f\f\fτ=t\u0015\ndt≥4TZT\n0E\u0014\n∥∇θsθ(t, Xt)∥21−λt\nλ2\ntρt\u0015\ndt\n= 4TE\"ZT\n0∥∇θsθ(t, Xt)∥21−λt\nλ2\ntρtdt#\n.\nThe integral inside the last expectation is infinite whenever the event ∥∇θsθ(0, X0)∥ ̸= 0holds, thanks to the continuity of\n∇θsand the path X[0,T]. Since E∥∇θsθ(t, X0)∥2>0by assumption, that event has non-zero probability, which concludes\nthe proof.\n21Particle Denoising Diffusion Sampler\nD. Experimental Details\nIn this section we give additional details and ablations relating to our experimental results. We begin by providing details of\nthe sampling tasks we considered. We then provide details of our implementation and the baseline methods. We finally\nprovide additional ablation studies and results which demonstrate the properties of our method.\nD.1. Benchmarking targets\nGaussian Here we consider the target π(x) =N(x; 2.75,0.252).\nMixture This target was used in Arbel et al. (2021). It is an equally weighted mixture of 6 bivariate Gaussian distributions\nwith means µ1= (3.0,0.0), µ2= (−2.5,0.0), µ3= (2.0,3.0), µ4= (0.0,3.0), µ5= (0.0,−2.5), µ6= (3.0,2.0)and\ncovariances Σ1= Σ 2=\u00000.7 0.0\n0.0 0.05\u0001\n,Σ4= Σ 5=\u00000.05 0.0\n0.0 0.07\u0001\n,Σ3= Σ 6=\u00001.0 0.95\n0.95 1.0\u0001\n. This target is symmetric around\ny=x.\nFunnel This target was proposed by Neal (2003). Its density follows x0∼ N(0, σ2\nf), x1:9|x0∼ N(0,exp(x0)I), with\nσf= 3.\nLogistic Regression The Bayesian logistic regression model is defined by the prior distribution θ∼ N (0, σ2I)and\nlikelihood y|θ, x∼Bernoulli (σ(θTx))where σis the sigmoid function. We consider sampling the posterior θ|y, xon the\nIonosphere and Sonar datasets, which are of 35and61dimensions respectively.\nBrownian Motion In this task, we make noisy observations of a simple Brownian motion over 30 time steps. The model\nwas introduced by Sountsov et al. (2020) and is defined by the prior αinn∼LogNormal (0,2),αobs∼LogNormal (0,2),\nx1∼ N(0, α2\ninn)andxi∼ N(xi−1, α2\ninn)fori= 2, ...,30. The observation likelihood is given by yi∼ N(xi, α2\nobs)for\ni= 1, ...,30. The goal is to sample the posterior distribution of αinn, αobs, x1, ..., x 30|{yi}10\ni=1S{yi}30\ni=21. This task is in 32\ndimensions.\nLog Gaussian Cox Process The LGCP model (Møller et al., 1998) was developed for the analysis of spatial data.\nThe Poisson rate parameter λ(x)is modelled on the grid using an exponentially-transformed Gaussian process, and\nobservations come from a Poisson point process. The unnormalized posterior density is given directly by γ(x) =\nN(x;µ, K)Q\ni∈[1:M]2exp(xiyi−aexp(xi)), where xiare the points of a regular M×Mgrid. In our experiments, we fit\nthis model on the Pines forest dataset where M= 40 , resulting in a problem in 1600 dimensions.\nD.2. Algorithmic details and hyperparameter settings\nHere we give details of the algorithmic settings used in our experiments. We first describe the considerations taken to ensure\na fair comparison between baselines, and then we detail exact hyperparameter settings.\nOur method was implemented in Python using the libraries of JAX (Bradbury et al., 2018), Haiku and Optax. Our\nimplementation is available on Github1. We used the open source code-bases of Arbel et al. (2021) to run the SMC and\nCRAFT baselines and of Vargas et al. (2023) to run the PIS and DDS benchmarks, both of which are also implemented in\nJAX.\nIn all experiments we used 2000 particles to estimate the normalizing constant.\nVariational approximation We used a variational approximation as the reference distribution for all methods. We found\nthat this was required for numerical stability of the potential function gt(xt)in our method. We therefore used the same\nvariational approximation for all methods to ensure a fair comparison. Note that PIS reverses a pinned brownian motion\nand thus the reference distribution depends on the diffusion time span Tand the noise coefficient σ. Since σaffects the\nperformance of the PIS algorithm itself, we tune this parameter independently rather than setting this via the variational\napproximation. The variational approximation was a mean-field variational distribution i.e. a diagonal Gaussian distribution\nlearnt by optimizing the ELBO. We used 20,000optimisation steps ( 50,000for the Funnel andBrownian tasks) with\nthe Adam optimizer (Kingma & Ba, 2015) and learning rate 1e−3. We did not use a variational approximation in the\n1https://github.com/angusphillips/particle_denoising_diffusion_sampler\n22Particle Denoising Diffusion Sampler\nGaussian (16) Mixture (16) Funnel (32) Brownian (16) Ion (32) Sonar (32) LGCP (128)\nPDDS137570 /\n74 / 0.1037764 /\n81 / 0.1339316 /\n98 / 0.1543584 /\n79 / 0.1944166 /\n78 / 0.1249210 /\n77 / 0.12347776 /\n1404 / 1.86\nCRAFT32 /\n4 / 0.02176608 /\n26 / 0.096077440 /\n178 / 0.181024 /\n27 / 0.212240 /\n38 / 0.093904 /\n40 / 0.09409600 /\n3072 / 14.9\nPIS37570 /\n129 / 0.0037764 /\n167 / 0.0139316 /\n394 / 0.0243854 /\n176 / 0.0144166 /\n324 / 0.0249210 /\n326 / 0.01347776 /\n3931 / 0.64\nDDS37570 /\n136 / 0.0037764 /\n187 / 0.0139316 /\n381 / 0.0243854 /\n183 / 0.0144166 /\n338 / 0.0149210 /\n332 / 0.02347776 /\n3941 / 0.68\nSMC 0 / 0 / 0.02 0 / 0 / 0.07 0 / 0 / 0.17 0 / 0 / 0.20 0 / 0 / 0.09 0 / 0 / 0.09 0 / 0 / 14.6\nTable 1: Number of trainable parameters / training time total (seconds) / sampling time per 2000 particles (seconds). Timings\nare averaged over 3 training seeds.\nGaussian task.\nNetwork architectures and optimizer settings For the CRAFT baseline, we followed the flow network architectures\nand optimizer settings given in Matthews et al. (2022), which are restated below for completeness. For the diffusion-based\nmethods (PDDS, DDS and PIS) we use the same network architecture and optimizer settings for each method. The neural\nnetwork follows the PISGRAD network of Zhang & Chen (2022) with minor adaptations. We use a sinusoidal embedding\nof 128 dimensions for the time input. We use a 3-layer MLP with 64 hidden units per layer for the ‘smoothing’ network\n(rη(t)in our notation and NN2(t)in Zhang & Chen (2022)). For the main potential/score network ( Nγin our notation and\nNN1(t, x)in Zhang & Chen (2022)), we use a 2 layer MLP of 64 hidden units per layer to encode the 128-dimensional\ntime embedding. This is concatenated with the state input xbefore passing through a 3-layer MLP with 64 hidden units\nper layer, outputting a vector of dimension d. In PDDS, we take the scalar product of this output with the state input xto\napproximate the potential function, while PIS and DDS use the d-dimensional output to approximate the optimal control\nterm. The activation function is GeLU throughout. We train for 10,000iterations of the Adam optimizer (Kingma & Ba,\n2015) with batch size 300and a learning rate of 1e−3, which decays exponentially at a rate of 0.95every 50 iterations\n(with the exception of the Funnel task where we did not use any learning rate decay).\nThe number of trainable parameters for each method and task can be found in Table 1, along with training time and sampling\ntime.\nAnnealing and noise schedules For the annealing based approaches (SMC and CRAFT) we used a geometric annealing\nschedule with initial distribution the variational approximation as described above. For the diffusion based approaches\n(PDDS, DDS and PIS) we carefully considered the appropriate noise schedules for each method. Firstly, we fix the diffusion\ntime span at T= 1 and adapt the discretization step size depending on the number of steps Kof the experiment, i.e.\nδ=T/K . This choice is equivalent to the fixed discretization step and varying diffusion time Tas considered by Vargas\net al. (2023), up to the choice of αmax.\nFor PIS, the original work of Zhang & Chen (2022) used by default a uniform noise schedule controlled by σ. Further,\nVargas et al. (2023) were unable to find a noise schedule which improved performance above the default uniform noise\nschedule. As such as we stick with the uniform noise schedule and tune the noise coefficient σby optimizing the ELBO\nobjective over a grid search up to a resolution of 0.02.\nFor DDS, it was observed by Vargas et al. (2023) that controlling the transition scale√αksuch that it goes smoothly to zero\nask→0is critical to performance. To achieve this they choose a cosine-based schedule α1/2\nk=α1/2\nmaxcos2\u0000π\n21−k/K+s\n1+s\u0001\nforssmall ( 0.008following Nichol & Dhariwal (2021)), which we term the DDS cosine schedule. The parameter αmax\nis tuned such that the noise at the final step in the reverse process is sufficiently small. We found that this scheduler did\n23Particle Denoising Diffusion Sampler\nindeed result in the best performance when compared to a linear noise schedule ( βt=β0+βTt/T) or the popular cosine\nschedule of Nichol & Dhariwal (2021) ( λt= 1−cos2\u0000π\n2t/T+s\n1+s\u0001\n). As such we use the DDS cosine schedule and tune αmax\nby optimizing the ELBO objective over a grid search to a resolution of 0.02.\nFor our method PDDS, we obtained the best performance using the cosine schedule of Nichol & Dhariwal (2021), which\nsetsλt= 1−cos2\u0000π\n2t/T+s\n1+s\u0001\nwhere we recall that λt= 1−exp\u0010\n−2Rt\n0βsds\u0011\n, i.e. the variance of the transition from 0\ntot. We provide an illustration of the benefits of this schedule in the following section. In particular we found that the\nalternative DDS cosine schedule did not improve performance and added the complexity of tuning the parameter αmax.\nIn summary, while each of the diffusion based approaches used different noise schedulers, each was chosen to provide the\nbest performance for the individual approach and thus ensures a fair comparison.\nD.2.1. SMC AND CRAFT SETTINGS\nWe used 1 iteration of an HMC kernel with 10 leapfrog integrator steps as the proposal distribution in the SMC and CRAFT\nbaselines. We tuned the HMC step sizes based on initial runs and obtained the step size schedules given below. We\nperformed simple resampling when the ESS dropped below 30%. We trained CRAFT for 500iterations ( 1000 onFunnel )\nwith a batch size of 2000 and learning rate schedule detailed below. We also list the flow architecture in each task. Our\nparameter settings differ to those in Matthews et al. (2022) since we use the variational approximation, which results in\nlarger MCMC step sizes and smaller learning rates.\nGaussian Step sizes [0.7,0.7,0.5,0.4]linearly interpolated between times [0.0,0.25,0.5,1.0]. CRAFT used a diagonal\naffine flow, with a learning rate of 1e−2.\nMixture Step sizes [0.5,0.5,0.5,0.3]linearly interpolated between times [0.0,0.25,0.5,1.0]. CRAFT used a spline\ninverse autoregressive flow with 10 spline bins and a 3 layer autoregressive MLP of 30 hidden units per layer, with a learning\nrate of 1e−3.\nFunnel Step sizes [1.0,0.9,0.8,0.7,0.6]linearly interpolated between times [0.0,0.25,0.5,0.75,1.0]. CRAFT used an\naffine inverse autoregressive flow, trained for 4000 iterations with a learning rate of 1e−3.\nBrownian Step sizes [0.8,0.8,0.7,0.6,0.5]linearly interpolated between times [0.0,0.25,0.5,0.75,1.0]. CRAFT used a\ndiagonal affine flow, with a learning rate of 1e−3.\nIon Step sizes [0.7,0.7,0.6,0.5,0.4]linearly interpolated between times [0.0,0.1,0.25,0.5,1.0]. CRAFT used a diagonal\naffine flow, with a learning rate of 1e−3.\nSonar Step sizes [0.7,0.7,0.6,0.5,0.35]linearly interpolated between times [0.0,0.1,0.25,0.5,1.0]. CRAFT used a\ndiagonal affine flow, with a learning rate of 1e−3.\nLGCP Step sizes [0.35,0.35,0.3,0.2]linearly interpolated between times [0.0,0.25,0.5,1.0]. CRAFT used a diagonal\naffine flow, with a learning rate of 1e−4.\nWe used SMC with the above settings for 1000 steps to estimate the ‘ground truth’ normalizing constant on the Bayesian\nposterior targets.\nD.2.2. DDS SETTINGS\nOptimal settings for αmaxare given in Table 2. Note that we do not tune σas in Vargas et al. (2023) since we used a\nvariational approximation.\nD.2.3. PIS SETTINGS\nOptimal settings for σare given in Table 3. Note that we were unable to obtain reasonable performance with PIS with only 1\nstep.\n24Particle Denoising Diffusion Sampler\nGaussian Mixture Brownian Funnel Ion Sonar LGCP\nSteps 1 1 1 2 2 2 8\n1 0.86 0.28 0.76 0.60 0.68 0.68 0.74\n2 0.86 0.28 0.76 0.68 0.80 0.82 0.62\n4 1.00 0.36 0.84 0.68 0.74 0.78 0.60\n8 0.96 0.52 0.80 0.60 0.64 0.64 0.44\n16 0.82 0.54 0.72 0.64 0.52 0.50 0.26\nTable 2: Optimal settings for αmax. The number of steps for a given entry is the base steps in the first row multiplied by the\nstep multiplier in the first column.\nGaussian Mixture Brownian Funnel Ion Sonar LGCP\nSteps 1 1 1 2 2 2 8\n1 NA NA NA 1.50 0.37 0.25 1.36\n2 1.00 2.40 0.08 1.00 0.40 0.31 1.64\n4 1.00 2.20 0.10 1.00 0.40 0.40 1.78\n8 1.00 1.92 0.10 1.00 0.46 0.46 1.99\n16 1.00 1.88 0.13 1.00 0.46 0.49 2.06\nTable 3: Optimal settings for σ. The number of steps for a given entry is the base steps in the first row multiplied by the step\nmultiplier in the first column. We were unable to tune PIS with one step size.\nD.2.4. PDDS SETTINGS\nNo tuning of the cosine noise schedule was required. We performed systematic resampling (Douc & Capp ´e, 2005) when\nthe ESS dropped below 30%. We did not perform any optional MCMC steps. In experiments where we used additional\niterations of potential approximation, we train each refinement for 10,000steps with a fresh instance of the learning rate\nschedule ( 1e−3with exponential decay at a rate of 0.95per 50 iterations). At each refinement we initialise the potential\napproximation at it’s previous state, rather than training from scratch at each refinement.\nD.3. Ablation studies\nCosine scheduler As stated above, we follow the cosine scheduler introduced by Nichol & Dhariwal (2021). We found,\nas demonstrated in Figure 6 and Figure 7, that the cosine schedule was effective in ensuring the forward SDE converges to\nthe target distribution while regularly spacing the ESS drops across the sampling path.\nFigure 6: Left: linear schedule, βf= 8. Middle: linear schedule: βf= 12 . Right: cosine schedule.\n25Particle Denoising Diffusion Sampler\nFigure 7: Left: linear schedule, βf= 8. Middle: linear schedule: βf= 12 . Right: cosine schedule.\nFigure 8: Samples and estimated normalizing constant from PDDS on N(2.75,0.252)target.\nNumber of particles In Figure 8 we show that the normalizing constant estimation error with the naive potential\napproximation does decrease as the number of particles increases. However, we could not eliminate the error before\nexceeding the computer memory. We conclude that we must improve our initial potential approximation to obtain feasible\nresults, hence motivating the iterative potential approximation scheme.\nGuidance path of NN potential approximation In Figure 9 we demonstrate that using a neural network to correct the\nnaive approximation has the effect of correcting the path of the guidance SDE. Here we use a linear schedule for βtsince\nthe cosine schedule does not allow for analytic roll-out of the SDE.\nD.4. Additional and uncurated results\nIn Figure 10 we display the normalising constant estimates on all tasks, including the Brownian andIon tasks which were\nomitted from the main text for space. In Figure 11 we display the same results with outliers present. Note that all methods\nare susceptible to erroneous over-estimation of the normalising constant due to numerical errors an instability. It appears\nthat all methods are equally susceptible to this issue, with no single method displaying more erroneous overestimation than\nthe others. Results on the Gaussian task are displayed separately in Figure 12.\n26Particle Denoising Diffusion Sampler\nFigure 9: Guidance path using analytic potential function (left), naive potential approximation (middle) and neural network\npotential approximation (right).\n27Particle Denoising Diffusion Sampler\n1 2 4 8 16−0.8−0.6−0.4−0.20.00.20.40.6Log ZMixture\n2 4 8 16 32−1.4−1.2−1.0−0.8−0.6−0.4−0.20.00.2Funnel\n1 2 4 8 16−3−2−1012Log ZBrownian\n2 4 8 16 32−115−114−113−112−111Ion\n2 4 8 16 32\nSteps−125.0−122.5−120.0−117.5−115.0−112.5−110.0−107.5Log ZSonar\nCRAFT\nDDS\nPDDS1\nPDDS2\nPIS\nSMC\n8 16 32 64 128\nSteps430440450460470480490500LGCP\nFigure 10: Normalizing constant estimation results on all tasks. Outliers are hidden for clarity. Each box consists of 2000\nestimates, coming from 20 training seeds each with 100 evaluation seeds.\n28Particle Denoising Diffusion Sampler\n1 2 4 8 16−101234Log Z\nMixture\n2 4 8 16 32−2−101234567\nFunnel\n1 2 4 8 16−202468Log Z\nBrownian\n2 4 8 16 32−114−112−110−108−106\nIon\n2 4 8 16 32\nSteps−125−120−115−110−105−100Log Z\nSonar\nCRAFT\nDDS\nPDDS1\nPDDS2\nPIS\nSMC\n8 16 32 64 128\nSteps430440450460470480490500510\nLGCP\nFigure 11: Normalizing constant estimation results on all tasks. Each box consists of 2000 estimates, coming from 20\ntraining seeds each with 100 evaluation seeds.\n29Particle Denoising Diffusion Sampler\n1 2 4 8 16\nSteps−2.0−1.5−1.0−0.50.00.51.01.52.0Log ZGaussian\nCRAFT\nDDS\nPDDS1\nPDDS2\nPIS\nSMC\n1 2 4 8 16\nSteps−2.0−1.5−1.0−0.50.00.51.01.52.0\nGaussian\nFigure 12: Normalizing constant estimation results on the Gaussian task. Each box consists of 2000 estimates, coming from\n20 training seeds each with 100 evaluation seeds. Left: outliers removed, right: uncurated.\n30"    },    {        "title": "2402.06335v1.Local_exact_controllability_to_the_trajectories_of_the_convective_Brinkman_Forchheimer_equations.pdf",        "content": "arXiv:2402.06335v1  [math.AP]  9 Feb 2024LOCAL EXACT CONTROLLABILITY TO THE TRAJECTORIES OF THE\nCONVECTIVE BRINKMAN-FORCHHEIMER EQUATIONS\nPARDEEP KUMAR1AND MANIL T. MOHAN2*\nAbstract. Inthis article, wediscussthe localexactcontrollabilitytotrajecto riesofthe fol-\nlowing convective Brinkman-Forchheimer (CBF) equations (or damp ed Navier-Stokes equa-\ntions) defined in a bounded domain Ω ⊂Rd(d= 2,3) with smooth boundary:\n∂u\n∂t−µ∆u+(u·∇)u+αu+β|u|2u+∇p=f+ϑ,∇·u= 0,\nwhere the control ϑis distributed in a subdomain ω⊂Ω, and the parameters α,β,µ > 0\nare constants. We first present global Carleman estimates and ob servability inequality for\nthe adjoint problem of a linearized version of CBF equations by using a global Carleman\nestimate for the Stokes system. This allows us to obtain its null cont rollability at any time\nT >0. We then use the inverse mapping theorem to deduce local results concerning the\nexact controllability to the trajectories of CBF equations.\n1.Introduction\nLet Ω⊂Rd(d= 2,3) be a bounded domain with a smooth boundary ∂Ω (for example of\nclass C2). Letω⊂Ω be a non-empty, small, open subset, and let T >0. We will use the\nnotationsQ:= Ω×(0,T),Qω:=ω×(0,T), and Σ := ∂Ω×(0,T), and denote by n(x),\nthe outward unit normal to Ω at the point x∈∂Ω. The convective Brinkman-Forchheimer\n(CBF) equations describe the motion of an incompressible viscous flu id through a rigid,\nhomogeneous, isotropic, porous medium (cf. [ 4]). Let us consider the following controlled\nCBF equations:\n\n∂u\n∂t−µ∆u+(u·∇)u+αu+β|u|2u+∇p=f+ϑ1ω,inQ,\n∇·u= 0,inQ,\nu=0,on Σ,\nu(0) =u0,in Ω.(1.1)\nHereu(x,t)∈Rdrepresents the velocity vector field of the fluid, p(x,t)∈Ris the pressure\nin the fluid, f(x,t)∈Rdis a given external forcing acting on the fluid, ϑ(x,t)∈Rdis a\n1,2Department of Mathematics, Indian Institute of Technology Roor kee-IIT Roorkee, Haridwar High-\nway, Roorkee, Uttarakhand 247667, INDIA.\ne-mail:Pardeep Kumar: pkumar3@ma.iitr.ac.in.\ne-mail:Manil T. Mohan: manilfma@iitr.ac.in, maniltmohan@gmail. com.\n*Corresponding author.\nKey words: Convective Brinkman-Forchheimer equations, Locally distributed c ontrol, Carleman esti-\nmates, Null controllability, Local exact controllability.\nMathematics Subject Classification (2020): Primary 35Q35; Secon dary 93B05, 93B07, 93C10, 93C20.\n12 P. KUMAR AND M. T. MOHAN\ncontrol distributed in some arbitrary fixed subdomain ωof the physical domain Ω, u0(x) is\na given initial vector field, and 1ωis the characteristics function of the set ω:\n1ω(x) =/braceleftbigg\n1,forx∈ω,\n0,forx∈Ω\\ω.\nThe positive constant µdenotes the Brinkman coefficient (effective viscosity), while αand\nβstand for the Darcy(permeability of porous medium) and Forchheimer (proportional to\nthe porosity of the material) coefficients, respectively. Setting α=β= 0 yields the classical\nd-dimensional controlled Navier-Stokes equations (NSE). This acco unts to refer the problem\n(1.1) as a controlled damped NSE also. The global solvability results of the uncontrolled\nproblem ( 1.1) (that is, ϑ=0) is discussed in [ 18,39], etc.\n1.1.Statement of the problem. The main objective of this work is to discuss the local exact\ncontrollability to trajectories of the CBF equations ( 1.1). To state the problem precisely and\npresent the main result, we require some function spaces which are given below ([ 46]):\nH={u∈L2(Ω)d:∇·u= 0 in Ω and u·n= 0 on∂Ω},\nV={u∈H1\n0(Ω)d:∇·u= 0 in Ω },\nL2\n0(Ω) =/braceleftbigg\nq∈L2(Ω) :/integraldisplay\nΩq(x)dx= 0/bracerightbigg\n,\nwherenis the unit outward drawn normal to the boundary and u·n/vextendsingle/vextendsingle\n∂Ω= 0 should\nbe understood in the sense of trace ([ 46, Section 1.3, Chapter 1]). The following nota-\ntions are also used repeatedly in the article: Lp(Q)d:= Lp(0,T;Lp(Ω)d), forp∈[1,∞],\nL2(Qω)d:= L2(0,T;L2(ω)d), L2(Q)d2:= L2(0,T;L2(Ω)d2), L2\n0(Q) := L2(0,T;L2\n0(Ω)) and\nL2(Q) := L2(0,T;L2(Ω)). Let ( ·,·) denote the inner product in the Hilbert space Hand/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}ht\nrepresent the duality pairing between a Banach space Xand its dual X′. Note that Hcan\nbe identified with its own dual H′. We will denote a generic constant (usually depending on\nΩ andω) byCthroughout this article.\nWe cannot expect the exact controllability for CBF equations with an arbitrary target\nfunctiondue tothedissipative andirreversible characteristics oft hesystem ([ 45]). Therefore,\nwe consider an“ideal”trajectory ( /tildewideu,/tildewidep) of the following uncontrolled CBF equations:\n\n\n∂/tildewideu\n∂t−µ∆/tildewideu+(/tildewideu·∇)/tildewideu+α/tildewideu+β|/tildewideu|2/tildewideu+∇/tildewidep=f,inQ,\n∇·/tildewideu= 0,inQ,\n/tildewideu=0,on Σ,\n/tildewideu(0) =/tildewideu0,in Ω,(1.2)\nwith the same right-hand side fas in (1.1). We additionally suppose that ( /tildewideu,/tildewidep) satisfies the\nfollowing regularity property:\n/tildewideu∈L∞(Q)d. (1.3)\nLet us now define the notion of local exact controllability to traject ories of the CBF\nequations ( 1.1). The task is to find a control ϑsuch that at least one solution of ( 1.1)\nverifies that\nu(·,T) =/tildewideu(·,T),in Ω.LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 3\nIf we are able to obtain such a control, we can switch it off after time Tand the system will\nfollow the“ideal”trajectory ( /tildewideu,/tildewidep) of (1.2).\nMore precisely, we say that the system ( 1.1) islocally exactly controllable to trajectories if\nfor a suitable trajectory ( /tildewideu,/tildewidep),solution of ( 1.2), there exists a δ>0 such that if\n/ba∇dblu0−/tildewideu0/ba∇dblL2d−2(Ω)d≤δ,\nwe can find a control ϑsuch that the corresponding solution to ( 1.1) satisfies:\nu(·,T) =/tildewideu(·,T),in Ω. (1.4)\n1.2.Main result and application. The main result of the present article concerns the local\nexact controllabilityto trajectoriesof theCBF equations ( 1.1). We nowstate ourmainresult\nin the following theorem:\nTheorem 1.1. Let us assume that ωis a non-empty open subset of ΩandT >0. We suppose\nthat the solution (/tildewideu,/tildewidep)of the system (1.2)satisfies(1.3). Then, there exists δ >0such that\nfor anyu0∈H∩L4(Ω)dsatisfying\n/ba∇dblu0−/tildewideu0/ba∇dblL2d−2(Ω)d≤δ,\nthere exists a control ϑ∈L2(Qω)dand a solution (u,p)of the system (1.1)that satisfies\n(1.4).\nMotivated by the ideas developed in [ 8,35,41], we prove Theorem 1.1in Section 5and the\nproof is based on an application of the inverse mapping theorem.\nRemark 1.2. Under the regularity hypotheses described in (1.3), J.-P. Puel [ 41] established\nlocal exact controllability to the trajectories of the Navi er-Stokes system. Therefore, it will be\nproven in this article that the presence of a cubic nonlinear ity requires no additional regularity\non the trajectories than the one described in (1.3).\nLet us provide an application of the above controllability result (for m ore details, see\n[25,41], etc).\nStabilizability of unstable stationary flows: Let (u†,p†) be a stationary solution of the\nfollowing CBF equations in a bounded domain Ω ⊂Rd:/braceleftBigg\n−µ∆u†+(u†·∇)u†+αu†+β|u†|2u†+∇p†=f(x),in Ω,\n∇·u†= 0,in Ω,u†=0,on∂Ω.(1.5)\nThe global solvability results ( u†∈D(A), where A is the Stokes operator) of the system\n(1.5) can be obtained from [ 40]. Suppose that this stationary solution is unstable. Thus,\nfor eachδ >0,there is an initial condition u0∈ {u:/ba∇dblu−u†/ba∇dblL2d−2(Ω)d< δ}such that the\nsolution ( u,p) of the CBF equations ( 1.1) withf(x,t)≡f(x) andϑ(x,t)≡0would diverge\nfrom (u†,p†) ast→ ∞and\n/ba∇dblu(·,t)−u†/ba∇dblL4(Ω)d/notarrowright0 ast→ ∞.\nOne can stabilize the unstable stationary solution ( u†,p†) by replacing the first equation\nof (1.1) by the distributed control ϑ. Indeed, by Theorem 1.1, for any initial condition\nu0∈H∩L2d−2(Ω)d,there is a control ϑsupported in ω×(0,T), such that the solution ( u,p)\nof the CBF equations ( 1.1) satisfies the condition\nu(x,T)≡u†(x).4 P. KUMAR AND M. T. MOHAN\nThis says that we are able to perform a strong stabilization of the un stable stationary flows\n(u†,p†) of solution of the system ( 1.5) by an interior control.\n1.3.Strategy and literature review. Several steps will be required to prove the main The-\norem1.1. Let us now sketch the strategy which we employ to solve our proble m.\n•First of all if we write\nv=u−/tildewideu, q=p−/tildewidep,v0=u0−/tildewideu0,\nwe obtain a new controlled system:\n\n\nLv+∇q=N(v)+ϑ1ω,inQ,\n∇·v= 0,inQ,\nv=0,on Σ,\nv(0) =v0,in Ω,(1.6)\nwhere\nLv=∂v\n∂t−µ∆v+(/tildewideu·∇)v+(v·∇)/tildewideu+αv+β|/tildewideu|2v+2β(v·/tildewideu)/tildewideu,(1.7)\nN(v) =−(v·∇)v−β|v|2(v+/tildewideu)−2β(v·/tildewideu)v. (1.8)\nConsequently, it is easy to see that the local exact controllability of the trajectories\nof the system ( 1.1) is equivalent to the local null controllability of the system ( 1.6),\nthat is, we want to find the control ϑsuch that at time T, we have v(·,T) =0.\nMore precisely, we say that the system ( 1.6) isnull controllable if, for any initial data\nv0∈H∩L2d−2(Ω)d, there exists a control ϑsuch that the associated solution to the\nsystem (1.6) satisfies:\nv(·,T) =0,in Ω. (1.9)\nAccordingly, we will focus on proving the local null controllability of th e nonlinear\nsystem (1.6).\n•Inordertoprovethislocalresult, weemployaninverse mappingarg ument, developed\nin [8,41], etc., in two steps.\nGiven anappropriatechoice of tensor g, a function h(ina functional class that will\nbe specified later), and given an initial data v0∈H, we prove the null controllability\nof the following linearized system:\n\n\nLv+∇q=∇·g+h+ϑ1ω,inQ,\n∇·v= 0,inQ,\nv=0,on Σ,\nv(0) =v0,in Ω,(1.10)\nwhere the operator Lis defined in ( 1.7).\nThe null controllability of ( 1.10) is based on the attainment of the so-called ob-\nservability inequality for a backward system associated with ( 1.10). Let us introduceLOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 5\nthe adjoint state ( ϕ,π) associated with ( 1.10) that satisfies the system:\n\n\nL∗ϕ+∇π=0,inQ,\n∇·ϕ= 0,inQ,\nϕ=0,on Σ,\nϕ(T) =1\nεvε(T),in Ω.(1.11)\nHereL∗is the adjoint operator of Ldefine by\nL∗ϕ=−∂ϕ\n∂t−µ∆ϕ−(Dϕ)/tildewideu+αϕ+β|/tildewideu|2ϕ+2β(ϕ·/tildewideu)/tildewideu, (1.12)\nvε(·,ϑε) is the solution of the optimal control problem (cf. subsection 4.1)\nmin\nϑ∈L2(Qω)d/parenleftbigg1\n2ε/ba∇dblv(T)/ba∇dbl2\nH+1\n2/integraldisplay\nQω|ϑ|2dxdt/parenrightbigg\n,\nsuch that v(·,ϑ) is the solution of the system ( 1.10) associated to ϑ, andDϕstands\nfor the symmetrized gradient Dϕ=∇ϕ+(∇ϕ)⊤.\nAt present, the most powerful tool to prove the null controllabilit y of the linearized\nsystem (1.10) is establishing global Carleman estimates for the solutions of its adj oint\nsystem (1.11). We first deduce a Carleman estimate for the adjoint system ( 1.11) and\nwith the help of this Carleman estimate, we derive the observability ine quality (see\nTheorem 3.6) of (1.11), which is useful for proving the null controllability of the\nsystem (1.10).\n•Next, wereturntothenonlinearproblemandthenapplytheinverse mappingtheorem\nto extend the control result from the linearized system ( 1.10) to the nonlinear system\n(1.6), thus obtaining the desired result (proof of main Theorem 1.1).\nThis two-step strategy has been successfully applied to other non linear systems such as\nthe incompressible Navier-Stokes equations (see [ 8,22,34,35,41], etc.), the Boussinesq system\n[25,28], etc., the one-dimensional Kuramoto-Sivashinsky equation [ 12], the Swift-Hohenberg\nequation [ 26], and the Cahn-Hilliard equation [ 31], etc. and references therein.\nThe controllability problems for incompressible fluids have been the su bject of much re-\nsearch in recent years. There are several results concerning th e controllability properties\nof the incompressible NSE by Coron [ 13,14], Coron and Fursikov [ 15], Fursikov [ 19], Fur-\nsikov and Imanuilov [ 20–23,25], Imanuilov [ 32–35], Fern´ andez-Cara [ 7], Fern´ andez-Cara et.\nal. [8,10], Guerrero [ 29], etc. and references therein. Several papers have been invest igated\non the controllability properties of the Boussinesq system by Fursik ov and Imanuilov [ 24,25],\nImanuilov [ 33], Fern´ andez-Cara et. al. [ 10], Guerrero [ 28], Burgos, Guerrero and Puel [ 6],\netc. and references therein. The above works explore the null co ntrollability, approximate\ncontrollability, exact controllability to the trajectories using distrib uted or boundary control,\nand also controlling the system with a reduced number of controls.\nIn 1998, O. Imanuvilov established a result of local exact controllab ility to trajectories for\nNSEwithclassical Dirichletboundaryconditions, underratherstro ngregularityassumptions\non the trajectory in [ 34]. This study was improved in [ 35] and furthermore, this result was\nextended by Fern´ andez-Cara et. al. [ 8] and Puel [ 41] with less regularity on the trajectory.\nGuerrero [ 29] considered the local exact controllability to the trajectories of N SE with\nnonlinear Navier-slip boundary conditions, with the control being su pported in a small set.\nThe local exact controllability to the trajectories of the Boussines q system is studied in [ 28].6 P. KUMAR AND M. T. MOHAN\nThe local exact controllability to the trajectories for the N-dimensional Navier-Stokes and\nBoussinesq systems with N−1 scalarcontrolsisdemonstratedbyFern´ andez-Caraet. al.[ 10].\nThey also obtained global null controllability results for some (trunc ated) approximations of\nNSE. The authors in [ 6] proved local exact controllability to the trajectories of Boussine sq\nsystems when the control is supported in a small set and acting on t he divergence equation.\nFor an extensive study on the controllability problems for NSE and re lated models, the\ninterested readers are referred to see [ 3,6,8,28,29,35,41], etc. and references therein. The\naim of this article is to prove local exact controllability to the traject ories of the CBF\nequations ( 1.1). To the best of our knowledge, there are no existing results in the literature\non the exact controllability of CBF equations, and this work appears to be the first one to\ndiscuss local exact controllability to the trajectories of the CBF eq uations ( 1.1).\n1.4.Organization of the article. The rest of the article is structured as follows: In the next\nsection, we provide the well-posedness result (Theorem 2.1) for the linearized CBF equations\n(1.10). In section 3, we first discuss the necessary weight functions needed to obtain the main\nresults of this article. In the same section, we derive the required C arleman estimates (see\n(3.24)), and observability inequality (Theorem 3.6) for the adjoint system ( 1.11) associated\nto the linearized system ( 1.10) by using the global Carleman estimates of the Stokes system.\nIn Section 4, we prove the null controllability of the linearized system ( 1.10) (Theorem 4.1).\nFurthermore, we also show that the solutions and controls decay e xponentially (Theorem\n4.2) which is helpful to examine the local exact controllability to the traj ectories of the CBF\nequations ( 1.1). In the final section, firstly, we define some functional classes a nd deduce\nsome useful regularity results that are required to handle the non linear terms (see subsection\n5.2). Finally, we establish the local null controllability of the nonlinear pro blem (1.6) by an\napplication of the inverse mapping theorem which leads to the require d results.\n2.Well-posedness of the Linearized CBF Equations\nIn this section, we provide the well-posedness result for the lineariz ed CBF equations\n(1.10), which is crucial in obtaining the controllability result for the CBF equ ations.\n2.1.Well-posedness. Let us prove the existence, uniqueness, and stability results for t he\nlinearized system ( 1.10). To do this, we first derive formal energy estimates for the solut ion\nof the problem ( 1.10). The following theorem establishes the well-posedness of the syst em\n(1.10).\nTheorem 2.1. Let/tildewideu∈L∞(Q)d,v0∈H,g∈L2(Q)d2,h∈L2(0,T;H−1(Ω)d), andϑ∈\nL2(Qω)dbe given. Then, there exists a unique weak solution (v,q)to the system (1.10)\nsatisfying\nv∈C([0,T];H)∩L2(0,T;V),andq∈L2\n0(Q),\nand there exists a constant C >0such that\n/ba∇dblv/ba∇dblC([0,T];H)+/ba∇dbl∇v/ba∇dblL2(0,T;H)+/ba∇dblq/ba∇dblL2(Q)\n≤C/parenleftbig\n/ba∇dblv0/ba∇dblH+/ba∇dblg/ba∇dblL2(Q)d2+/ba∇dblh/ba∇dblL2(0,T;H−1(Ω)d)+/ba∇dblϑ/ba∇dblL2(Qω)d/parenrightbig\n, (2.1)\nand\n/ba∇dblv1−v2/ba∇dblC([0,T];H)+/ba∇dbl∇(v1−v2)/ba∇dblL2(0,T;H)+/ba∇dblq1−q2/ba∇dblL2(Q)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 7\n≤C/braceleftbigg\n/ba∇dblv1(0)−v2(0)/ba∇dblH+/ba∇dblg1−g2/ba∇dblL2(Q)d2+/ba∇dblh1−h2/ba∇dblL2(0,T;H−1(Ω)d)+/ba∇dblϑ1−ϑ2/ba∇dblL2(Qω)d/bracerightbigg\n,\n(2.2)\nwhereCdepends on the input data, µ,α,β,T , andΩ.\nProof.Existence: ThefollowingcalculationscanbemaderigorousbyusingaFaedo-Galer kin\napproximation technique. As the system ( 1.10) is linear, weak convergence of a Faedo-\nGalerkin approximated subsequence is enough to pass to the limit. Ta king the inner product\nwithv(·) to the first equation in ( 1.10) and using the fact that (( /tildewideu· ∇)v,v) = 0 and\n(∇q,v) = 0, we obtain\n1\n2d\ndt/ba∇dblv(t)/ba∇dbl2\nH+µ/ba∇dbl∇v(t)/ba∇dbl2\nH+α/ba∇dblv(t)/ba∇dbl2\nH+β/ba∇dbl|/tildewideu||v|/ba∇dbl2\nH+2β/ba∇dbl/tildewideu·v/ba∇dbl2\nH\n≤(∇·g,v)+/a\\}b∇acketle{th,v/a\\}b∇acket∇i}ht+(ϑ1ω,v)−((v·∇)/tildewideu,v), (2.3)\nfor a.e.t∈[0,T], where we have performed the integration by parts over Ω. We est imate\neach term on the right-hand side of ( 2.3) separately by employing H ¨older’s and Young’s\ninequalities as follows:\n|(∇·g,v)| ≤ /ba∇dblg/ba∇dblL2(Ω)d2/ba∇dbl∇v/ba∇dblH≤3\n2µ/ba∇dblg/ba∇dbl2\nL2(Ω)d2+µ\n6/ba∇dbl∇v/ba∇dbl2\nH,\n|/a\\}b∇acketle{th,v/a\\}b∇acket∇i}ht| ≤ /ba∇dblh/ba∇dblH−1(Ω)d/ba∇dblv/ba∇dblV≤3\n2µ/ba∇dblh/ba∇dbl2\nH−1(Ω)d+µ\n6/ba∇dblv/ba∇dbl2\nV,\n|(ϑ1ω,v)| ≤ /ba∇dblϑ/ba∇dblL2(ω)d/ba∇dblv/ba∇dblH≤1\n2α/ba∇dblϑ/ba∇dbl2\nL2(ω)d+α\n2/ba∇dblv/ba∇dbl2\nH,\n|((v·∇)/tildewideu,v)|=|((v·∇)v,/tildewideu)| ≤ /ba∇dbl∇v/ba∇dblH/ba∇dblv/ba∇dblL4(Ω)d/ba∇dbl/tildewideu/ba∇dblL4(Ω)d\n≤C/ba∇dbl∇v/ba∇dbl1+d\n4\nH/ba∇dblv/ba∇dbl1−d\n4\nH/ba∇dbl/tildewideu/ba∇dblL4(Ω)d≤C/ba∇dbl/tildewideu/ba∇dbl8\n4−d\nL4(Ω)d/ba∇dblv/ba∇dbl2\nH+µ\n6/ba∇dbl∇v/ba∇dbl2\nH.\nSubstituting the above estimates in ( 2.3), and integrating the resulting estimate from 0 to t,\nwe obtain\n/ba∇dblv(t)/ba∇dbl2\nH+µ/integraldisplayt\n0/ba∇dbl∇v(s)/ba∇dbl2\nHds+2β/integraldisplayt\n0/ba∇dbl|/tildewideu(s)||v(s)|/ba∇dbl2\nHds+4β/integraldisplayt\n0/ba∇dbl/tildewideu(s)·v(s)/ba∇dbl2\nHds\n≤ /ba∇dblv0/ba∇dbl2\nH+3\nµ/integraldisplayt\n0/ba∇dblg(s)/ba∇dbl2\nL2(Ω)d2ds+3\nµ/integraldisplayt\n0/ba∇dblh(s)/ba∇dbl2\nH−1(Ω)dds\n+1\nα/integraldisplayt\n0/ba∇dblϑ(s)/ba∇dbl2\nL2(ω)dds+C/integraldisplayt\n0/ba∇dbl/tildewideu(s)/ba∇dbl8\n4−d\nL4(Ω)d/ba∇dblv(s)/ba∇dbl2\nHds, (2.4)\nfor allt∈[0,T]. An application of Gronwall’s inequality in ( 2.4) yields\n/ba∇dblv(t)/ba∇dbl2\nH≤/parenleftbigg\n/ba∇dblv0/ba∇dbl2\nH+3\nµ/integraldisplayT\n0/ba∇dblg(t)/ba∇dbl2\nL2(Ω)d2dt+3\nµ/integraldisplayT\n0/ba∇dblh(t)/ba∇dbl2\nH−1(Ω)ddt\n+1\nα/integraldisplayT\n0/ba∇dblϑ(t)/ba∇dbl2\nL2(ω)ddt/parenrightbigg\n×exp/braceleftbigg\nC/integraldisplayT\n0/ba∇dbl/tildewideu(t)/ba∇dbl8\n4−d\nL4(Ω)ddt/bracerightbigg\n,\nfor allt∈[0,T]. Thus, from ( 2.4), it is immediate that\nsup\nt∈[0,T]/ba∇dblv(t)/ba∇dbl2\nH+/integraldisplayT\n0/ba∇dbl∇v(t)/ba∇dbl2\nHdt+/integraldisplayT\n0/ba∇dbl/tildewideu(t)·v(t)/ba∇dbl2\nHdt+/integraldisplayT\n0/ba∇dbl|/tildewideu(t)||v(t)|/ba∇dbl2\nHdt8 P. KUMAR AND M. T. MOHAN\n≤C/parenleftbigg\n/ba∇dblv0/ba∇dbl2\nH+/ba∇dblg/ba∇dbl2\nL2(Q)d2+/ba∇dblh/ba∇dbl2\nL2(0,T;H−1(Ω)d)+/ba∇dblϑ/ba∇dbl2\nL2(Qω)d/parenrightbigg\n. (2.5)\nFor Φ∈L2(0,T;V), from the first equation in ( 1.10), we obtain\n|/a\\}b∇acketle{tvt,Φ/a\\}b∇acket∇i}ht|=|/a\\}b∇acketle{t∇·g+h+ϑ1ω+µ∆v−(v·∇)/tildewideu−(/tildewideu·∇)v\n−αv−β|/tildewideu|2v−2β(v·/tildewideu)/tildewideu,Φ/a\\}b∇acket∇i}ht|\n≤/parenleftbig\n/ba∇dbl∇·g/ba∇dblH−1(Ω)d+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ1ω/ba∇dblH−1(Ω)d+µ/ba∇dbl∆v/ba∇dblV′+α/ba∇dblv/ba∇dblV′\n+3β/ba∇dbl|/tildewideu|2|v|/ba∇dblV′/parenrightbig\n/ba∇dblΦ/ba∇dblV+|/a\\}b∇acketle{t(v·∇)Φ,/tildewideu/a\\}b∇acket∇i}ht|+|/a\\}b∇acketle{t(/tildewideu·∇)Φ,v/a\\}b∇acket∇i}ht|\n≤C/parenleftbig\n/ba∇dbl∇·g/ba∇dblH−1(Ω)d+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ1ω/ba∇dblH−1(Ω)d+/ba∇dblv/ba∇dblV\n+/ba∇dblv/ba∇dblH+/ba∇dbl|/tildewideu|2|v|/ba∇dblL4\n3(Ω)d/parenrightbig\n/ba∇dblΦ/ba∇dblV+C/ba∇dbl/tildewideu/ba∇dblL4(Ω)d/ba∇dblv/ba∇dblL4(Ω)d/ba∇dblΦ/ba∇dblV\n≤C/braceleftbig\n/ba∇dblg/ba∇dblL2(Ω)d2+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ/ba∇dblL2(ω)d+/ba∇dbl∇v/ba∇dblH\n+/ba∇dbl|/tildewideu||v|/ba∇dblH/ba∇dbl/tildewideu/ba∇dblL4(Ω)d+/ba∇dbl/tildewideu/ba∇dblL4(Ω)d/ba∇dbl∇v/ba∇dblH/bracerightbig\n/ba∇dblΦ/ba∇dblV,\nwhere we have used H ¨older’s inequality and embedding V֒→L4\nσ(Ω)d֒→H≡H′֒→\nL4\n3σ(Ω)d֒→V′. Integrating the above estimate from 0 to T, and using the energy esti-\nmate (2.5), along with the hypothesis /tildewideu∈L∞(Q)d,v0∈H,g∈L2(Q)d2,ϑ∈L2(Qω)d, and\nh∈L2(0,T;H−1(Ω)d), we deduce that vt∈L2(0,T;V′), and we also have\n/ba∇dblvt/ba∇dbl2\nL2(0,T;V′)≤C/parenleftbigg\n/ba∇dblv0/ba∇dbl2\nH+/ba∇dblg/ba∇dbl2\nL2(Q)d2+/ba∇dblh/ba∇dbl2\nL2(0,T;H−1(Ω)d)+/ba∇dblϑ/ba∇dbl2\nL2(Qω)d/parenrightbigg\n.(2.6)\nThe fact that\nv∈L2(0,T;V) and vt∈L2(0,T;V′),\nimplies that v∈C([0,T];H).\nThe first equation in the system ( 1.10) can be used directly to deduce the pressure field q\nbyv0,v,g,h,ϑ. Taking divergence on both sides of the first equation in ( 1.10), we get\n∆q=∇·/braceleftbig\n∇·g+h+ϑ1ω−(/tildewideu·∇)v−(v·∇)/tildewideu−β|/tildewideu|2v−2β(v·/tildewideu)/tildewideu/bracerightbig\n,\nin the weak sense. From the above equation, we have\nq= (∆)−1/bracketleftbig\n∇·/braceleftbig\n∇·g+h+ϑ1ω−(/tildewideu·∇)v−(v·∇)/tildewideu−β|/tildewideu|2v−2β(v·/tildewideu)/tildewideu/bracerightbig/bracketrightbig\n.(2.7)\nTaking L2-norm on both sides of the equation ( 2.7), and then using elliptic regularity (Cat-\ntabriga’s regularity theorem) and H ¨older’s inequality, we obtain\n/ba∇dblq/ba∇dblL2(Ω)=/vextenddouble/vextenddouble(∆)−1/bracketleftbig\n∇·/braceleftbig\n∇·g+h+ϑ1ω−(/tildewideu·∇)v−(v·∇)/tildewideu\n−β|/tildewideu|2v−2β(v·/tildewideu)/tildewideu/bracerightbig/bracketrightbig/vextenddouble/vextenddouble\nL2(Ω)\n≤C/vextenddouble/vextenddouble/bracketleftbig\n∇·/braceleftbig\n∇·g+h+ϑ1ω−∇·(/tildewideu⊗v)−∇·(v⊗/tildewideu)\n−β|/tildewideu|2v−2β(v·/tildewideu)/tildewideu/bracerightbig/bracketrightbig/vextenddouble/vextenddouble\nH−2(Ω)\n≤C/parenleftbig\n/ba∇dblg/ba∇dblL2(Ω)d2+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ1ω/ba∇dblH−1(Ω)d+/ba∇dbl/tildewideu⊗v/ba∇dblH+/ba∇dbl|/tildewideu|2|v|/ba∇dblV′/parenrightbig\n≤C/braceleftbig\n/ba∇dblg/ba∇dblL2(Ω)d2+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ/ba∇dblL2(ω)d+/ba∇dbl/tildewideu/ba∇dblL4(Ω)d/ba∇dblv/ba∇dblL4(Ω)d+/ba∇dbl|/tildewideu|2|v|/ba∇dblL4\n3(Ω)d/bracerightbig\n≤C/braceleftbig\n/ba∇dblg/ba∇dblL2(Ω)d2+/ba∇dblh/ba∇dblH−1(Ω)d+/ba∇dblϑ/ba∇dblL2(ω)d+/ba∇dbl/tildewideu/ba∇dblL4(Ω)d/ba∇dblv/ba∇dblV+/ba∇dbl|/tildewideu||v|/ba∇dblH/ba∇dbl/tildewideu/ba∇dblL4(Ω)d/bracerightbig\n.LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 9\nTaking the square on the both sides in the above equation and then in tegrating the resulting\nestimate from 0 to T, we have\n/integraldisplayT\n0/ba∇dblq(t)/ba∇dbl2\nL2(Ω)dt≤C/braceleftbigg/integraldisplayT\n0/ba∇dblg(t)/ba∇dbl2\nL2(Ω)d2dt+/integraldisplayT\n0/ba∇dblh(t)/ba∇dbl2\nH−1(Ω)ddt+/integraldisplayT\n0/ba∇dblϑ(t)/ba∇dbl2\nL2(ω)ddt\n+ sup\nt∈[0,T]/ba∇dbl/tildewideu(t)/ba∇dbl2\nL4(Ω)d/parenleftbigg/integraldisplayT\n0/ba∇dblv(t)/ba∇dbl2\nVdt+/integraldisplayT\n0/ba∇dbl|/tildewideu(t)||v(t)|/ba∇dbl2\nHdt/parenrightbigg/bracerightbigg\n.(2.8)\nSubstituting the estimate ( 2.5) into (2.8) leads to\n/ba∇dblq/ba∇dbl2\nL2(Q)≤C/parenleftbigg\n/ba∇dblv0/ba∇dbl2\nH+/ba∇dblg/ba∇dbl2\nL2(Q)d2+/ba∇dblh/ba∇dbl2\nL2(0,T;H−1(Ω)d)+/ba∇dblϑ/ba∇dbl2\nL2(Qω)d/parenrightbigg\n.(2.9)\nHence, from ( 2.5), (2.6) and (2.9), we derive the existence of a solution\nv∈C([0,T];H)∩L2(0,T;V),andq∈L2\n0(Q),\nand (2.1) follows from ( 2.5), (2.6) and (2.9).\nUniquenessandstability: Sincesystem( 1.10)islinear,wecanseefromanestimatesimilarto\n(2.1) (see (2.2)) that the solution ( v,q) depends continuously on the initial data, which leads\nto the Lipschitz stability of the solution ( v,q) in (2.2), and the uniqueness is immediate. /square\nWe remark that the significance of this result will be highlighted in sect ions4and5,\nespecially for proving the null controllability of the linearized system ( 1.10).\n3.Carleman Estimates and observability inequality\nThe exact controllability problem for a linear evolution system can oft en be proven by\nexamining the observability inequality of the associated adjoint syst em.\nIn this section, we derive a Carleman estimate and an observability ine quality for the\nadjoint system ( 1.11) corresponding to the linearized system ( 1.10), which is necessary to\nstudy the exact controllability of the trajectories of the CBF equa tions. The only way we\nknow to obtain such an estimate and inequality is to use a global Carlem an estimate for the\nStokes system. We begin this section by addressing the weight func tions that will be used\nthroughout the article. Other weight classes will be presented in th e sequel as needed.\n3.1.Weight functions. We proceed to construct appropriate weight functions. Let us re call\nthatω⊂Ω is a non-empty open set.\nLemma 3.1. Letωbe an arbitrary fixed subdomain of Ω. Then, there exists a function\nη∈C2(Ω)such that\nη(x)>0,∀x∈Ω, η|∂Ω= 0,|∇η(x)|>0,∀x∈Ω\\ω.\nA proof of this lemma can be found in [ 22, Lemma 1.1]. Let us define ℓ∈C∞([0,T]) be\nsuch that\nℓ(t) =ton/bracketleftbigg\n0,T\n4/bracketrightbigg\n, ℓ(t) =T−ton/bracketleftbigg3T\n4,T/bracketrightbigg\n, ℓ(t)≥T\n4on/bracketleftbiggT\n4,3T\n4/bracketrightbigg\n.\nUsing the function ηconstructed in Lemma 3.1, we introduce the weight functions as\nψ(x,t) =ξ(x,t)−eλ(/bardblη/bardblL∞+m2)\nℓ4(t), ξ(x,t) =eλ(η(x)+m1)\nℓ4(t), (3.1)10 P. KUMAR AND M. T. MOHAN\nˇψ(t) = min\nx∈Ωψ(x,t) =ψ(t)|∂Ω=eλm1−eλ(/bardblη/bardblL∞+m2)\nℓ4(t), (3.2)\nˇξ(t) = min\nx∈Ωξ(x,t) =ξ(t)|∂Ω=eλm1\nℓ4(t), (3.3)\nwhereλ≥1 and the constants m1andm2are chosen such that m1≤m2(see [41, Lemma\n4.1]), and there exists a constant C >0 such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ψ\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cξ5\n4,and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2ψ\n∂t2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cξ3\n2.\nWe notice that\n\n1\nˇξ(t)1\n4∂ˇψ(t)\n∂t≤Cξ(t),1\nˇξ(t)5\n4∂ˇξ(t)\n∂t≤C,for allt∈[0,T],\nˇψ(t)≤ψ(x,t),ˇξ(t)≤ξ(x,t),ˇξ(t)≥C0>0,for all (x,t)∈Ω×(0,T).(3.4)\nWe define the space H1\n4,1\n2(Σ) using a classical notation as\nH1\n4,1\n2(Σ) = H1\n4(0,T;L2(∂Ω))∩L2(0,T;H1\n2(∂Ω)).\nWe remark that Fursikov and Imanuvilov first introduced functions of this type in [ 22] to\nobtainCarlemaninequalitiesfortheheatequations. Wealsostateth attheseweightfunctions\nwere already defined by J.-P. Puel in the work [ 41] in order to obtain a Carleman estimate\nfor the Stokes system.\n3.2.Carleman estimates. In the following subsection, we deduce a Carleman estimate for\nthe adjoint system ( 1.11) associated with the linearized system ( 1.10). This will provide a\nnull controllability result for the linearized CBF equations ( 1.10). We closely follow similar\nideas of Carleman estimates for the Stokes system with Dirichlet con ditions, which were\nproved by J.-P. Puel in [ 41]. To this end, let us consider the following Stokes system:\n\n\n∂y\n∂t−∆y+∇q=k,inQ\n∇·y= 0,inQ,\ny=0,on Σ,\ny(0) =y0,in Ω.(3.5)\nIt is well known (see [ 46]) that if y0∈Handk∈L2/parenleftbig\n0,T;H−1(Ω)d/parenrightbig\n, where H−1(Ω) is the\ndual space of H1\n0(Ω), there exist a unique solution ( y,q) of the system ( 3.5) such that\ny∈C([0,T];H)∩L2(0,T;V),andq∈L2\n0(Q).\nMoreover, from the classical regularity result, if y0∈Vandk∈L2(Q)d, we have\ny∈C([0,T];V)∩L2(0,T;H2(Ω)∩V),∂y\n∂t∈L2(0,T;H),and∇q∈L2(Q)d,\nand in particular\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y\n∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(0,T;H)+/ba∇dbly/ba∇dblL∞(0,T;V)+/ba∇dbly/ba∇dblL2(0,T;H2(Ω))+/ba∇dbl∇q/ba∇dblL2(Q)d≤C/parenleftbig\n/ba∇dbly0/ba∇dblV+/ba∇dblk/ba∇dblL2(Q)d/parenrightbig\n.(3.6)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 11\nLet us define z:=∇×y= curly. Taking the curl of the equation ( 3.5), we get\n\n\n∂z\n∂t−∆z=∇×k,inQ,\nz(x,0) =z0(x) =∇×y0(x),in Ω.(3.7)\nAs∇·y= 0, we also have for a.e. t∈(0,T)\n/braceleftbigg−∆y(t) =∇×z(t),in Ω,\ny(t) =0,on∂Ω.(3.8)\nWe notice that y=0on∂Ω implies\n∇y= (∇y·n)n,on∂Ω.\nThus,z|∂Ωcan be expressed in terms of ∇y·n.\nThe vector function zverifies a system of non-homogeneous heat equations ( 3.7). We\ncan use the global Carleman estimate from [ 37] or an improvement established in [ 38]. For\nnon-homogeneous parabolic equations, the following result of the g lobal Carleman estimate\nhas been established in [ 38].\nTheorem 3.2 (Theorem 2.2, [ 38]).Letωbe a non-empty open subset of ΩandT >0.\nLet us consider the weights η,ψandξdefined in Lemma 3.1and(3.1). Then, there exist\ns0≥1, λ0≥1andC >0such that for every s≥s0andλ≥λ0, we have\n/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2/parenrightbigg\ndxdt≤C/parenleftbigg\ns−1\n2/ba∇dblξ−1\n4esψz/ba∇dbl2\nH1\n4,1\n2(Σ)+/integraldisplay\nQe2sψ|k|2dxdt/parenrightbigg\n+Csλ2/integraldisplay\nQωe2sψξ|z|2dxdt. (3.9)\nThe vector function yis a solutionof the elliptic equation ( 3.8) with right hand side ∇×z.\nWe can apply a result of [ 36] which gives a Carleman estimate for the elliptic equation ( 3.8)\nwith the weight function for fixed t∈(0,T),\nγ(x) =eλ(η(x)+m1). (3.10)\nTheorem 3.3 (Theorem A.1, [ 36]).Letωbe a non-empty open subset of Ωandγbe the weight\nfunction defined in (3.10). Then, there exists τ0≥1, λ0≥1andC >0such that for every\nτ≥τ0andλ≥λ0, the function y(·)satisfies\n/integraldisplay\nΩe2τγ/parenleftbig\n|∇y(t)|2+τ2λ2γ2|y(t)|2/parenrightbig\ndx\n≤C/parenleftbigg\nτ/integraldisplay\nΩe2τγγ|z(t)|2dx+τ2λ2/integraldisplay\nωe2τγγ2|y(t)|2dx/parenrightbigg\n, (3.11)\nfor a.e.t∈(0,T).\nLet us take τ=s\nℓ4(t)and multiply the expression ( 3.11) byλ2exp/parenleftbigg\n−2seλ(/bardblη/bardblL∞+m2)\nℓ4(t)/parenrightbigg\n, and\nthen integrate the resulting estimate from 0 to T, we obtain\n/integraldisplay\nQe2sψ/parenleftbig\nλ2|∇y|2+s2λ4ξ2|y|2/parenrightbig\ndxdt12 P. KUMAR AND M. T. MOHAN\n≤C/parenleftbigg\nsλ2/integraldisplay\nQe2sψξ|z|2dxdt+s2λ4/integraldisplay\nQωe2sψξ2|y|2dxdt/parenrightbigg\n. (3.12)\nCombining the estimates ( 3.9) and (3.12), we arrive at\n/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2+λ2|∇y|2+s2λ4ξ2|y|2/parenrightbigg\ndxdt\n≤C/parenleftbigg\ns−1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddoubleξ−1\n4esψ∂y\n∂n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1\n4,1\n2(Σ)+/integraldisplay\nQe2sψ|k|2dxdt/parenrightbigg\n+C/integraldisplay\nQωe2sψ/parenleftbig\nsλ2ξ|z|2+s2λ4ξ2|y|2/parenrightbig\ndxdt. (3.13)\nWe will now use the classical regularity estimates for the solutions of the Stokes system to\nestimate the boundary term/vextenddouble/vextenddouble/vextenddoubleξ−1\n4esψ∂y\n∂n/vextenddouble/vextenddouble/vextenddouble2\nH1\n4,1\n2(Σ). To do this, let us introduce the following\nfunctions:\ny∗=ˇξ(t)−1\n4esˇψ(t)y,andq∗=ˇξ(t)−1\n4esˇψ(t)q,\nwhereˇψandˇξare defined in ( 3.2) and (3.3), respectively. Then, they satisfy\n\n\n∂y∗\n∂t−∆y∗+∇q∗=esˇψ\nˇξ1\n4k+s\nˇξ1\n4∂ˇψ\n∂tesˇψy−1\n4ˇξ5\n4∂ˇξ\n∂tesˇψy,inQ,\n∇·y∗= 0,inQ,\ny∗=0,on Σ,\ny∗(0) =0,in Ω.(3.14)\nUsing the estimate ( 3.4) and regularity result for the above system (cf. [ 46]) yield\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y∗\n∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH1(0,T;H)+/ba∇dbly∗/ba∇dbl2\nL2(0,T;H2(Ω))≤C/parenleftbig\n/ba∇dblesψk/ba∇dbl2\nL2(Q)d+s2/ba∇dblξesψy/ba∇dbl2\nL2(0,T;H)/parenrightbig\n.(3.15)\nOn the other hand, using [ 37, Proposition 3.2], we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y∗\n∂n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1\n4,1\n2(Σ)=/vextenddouble/vextenddouble/vextenddouble/vextenddoubleesˇψ\nˇξ1\n4∂y\n∂n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1\n4,1\n2(Σ)≤C/ba∇dbly∗/ba∇dbl2\nH1,2(Q), (3.16)\nwhere\nH1,2(Q) = H1(0,T;H)∩L2(0,T;H2(Ω)).\nFrom (3.15) and (3.16), we deduce\ns−1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddoubleesˇψ\nˇξ1\n4∂y\n∂n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1\n4,1\n2(Σ)≤C/parenleftBig\ns−1\n2/ba∇dblesψk/ba∇dbl2\nL2(Q)d+s3\n2/ba∇dblξesψy/ba∇dbl2\nL2(0,T;H)/parenrightBig\n.(3.17)\nSubstituting the estimate ( 3.17) into (3.13) and then taking s0large enough such that for\ns≥s0, we can absorb the term s3\n2/ba∇dblξesψy/ba∇dbl2\nL2(0,T;H)from the left hand-side, and we obtain the\nfirst Carleman estimate for the Stokes system ( z=∇×y)\n/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2+λ2|∇y|2+s2λ4ξ2|y|2/parenrightbigg\ndxdtLOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 13\n≤C/parenleftbigg/integraldisplay\nQe2sψ|k|2dxdt+/integraldisplay\nQωe2sψ/parenleftbig\nsλ2ξ|z|2+s2λ4ξ2|y|2/parenrightbig\ndxdt/parenrightbigg\n. (3.18)\nNow, we want to get ride of the local term Csλ2/integraltext\nQωe2sψξ|z|2dxdtfrom the right hand side\nin (3.18). First of all, we can find a non-empty open subset ω0of Ω such that ω0⊂ωand\n|∇η(x)|>0 inΩ\\ω0. Thus, we can replace ωbyω0in the second term of the right-hand\nside of (3.18) to deduce\n/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2+λ2|∇y|2+s2λ4ξ2|y|2/parenrightbigg\ndxdt\n≤C/parenleftbigg/integraldisplay\nQe2sψ|k|2dxdt+sλ2/integraldisplay\nQω0e2sψξ|z|2dxdt+s2λ4/integraldisplay\nQωe2sψξ2|y|2dxdt/parenrightbigg\n.(3.19)\nNow, our aim is to eliminate the term Csλ2/integraltext\nQω0e2sψξ|z|2dxdtin (3.19). Let us take a cut-off\nfunctionθ∈C∞\n0(ω) such that\nθ≡1,inω0,and 0≤θ≤1,inω.\nThen, we have\nCsλ2/integraldisplay\nQω0e2sψξ|z|2dxdt≤Csλ2/integraldisplay\nQωe2sψθξ|z|2dxdt\n=Csλ2/integraldisplay\nQωe2sψθξz·(∇×y)dxdt=Csλ2/integraldisplay\nQω∇×/parenleftbig\ne2sψθξz/parenrightbig\n·ydxdt\n≤Csλ2/integraldisplay\nQωe2sψθξ|∇z||y|dxdt+Csλ2/integraldisplay\nQωe2sψξ|y|/parenleftbig\n2sθλξ|z|+ξ|z|+λξθ|z|/parenrightbig\ndxdt\n≤C/parenleftbigg/integraldisplay\nQωe2sψ\nsξ|∇z|2dxdt/parenrightbigg1\n2/parenleftbigg/integraldisplay\nQωs3λ4ξ3e2sψ|y|2dxdt/parenrightbigg1\n2\n+C/parenleftbigg/integraldisplay\nQωsλ2ξe2sψ|z|2dxdt/parenrightbigg1\n2/parenleftbigg/integraldisplay\nQωs3λ4ξ3e2sψ|y|2dxdt/parenrightbigg1\n2\n+C/parenleftbigg/integraldisplay\nQωsλ2ξe2sψ|z|2dxdt/parenrightbigg1\n2/parenleftbigg/integraldisplay\nQωsλ2ξe2sψ|y|2dxdt/parenrightbigg1\n2\n+C/parenleftbigg/integraldisplay\nQωsλ2ξe2sψ|z|2dxdt/parenrightbigg1\n2/parenleftbigg/integraldisplay\nQωsλ4ξe2sψ|y|2dxdt/parenrightbigg1\n2\n≤1\n2/integraldisplay\nQωe2sψ\nsξ|∇z|2dxdt+1\n2/integraldisplay\nQωsλ2ξe2sψ|z|2dxdt+C/integraldisplay\nQωs3λ4ξ3e2sψ|y|2dxdt,\nwhere we have used the Cauchy-Schwarz and Young’s inequalities. S ubstituting the above\nestimate in ( 3.19), we arrive at\n/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2+λ2|∇y|2+s2λ4ξ2|y|2/parenrightbigg\ndxdt\n≤C/parenleftbigg/integraldisplay\nQe2sψ|k|2dxdt+/integraldisplay\nQωe2sψs2λ4ξ2|y|2dxdt/parenrightbigg\n+1\n2/integraldisplay\nQωe2sψ\nsξ|∇z|2dxdt14 P. KUMAR AND M. T. MOHAN\n+1\n2/integraldisplay\nQωsλ2ξe2sψ|z|2dxdt+C/integraldisplay\nQωs3λ4ξ3e2sψ|y|2dxdt. (3.20)\nFrom (3.20), we obtain the Carleman estimate for the Stokes system as follows :\nTheorem 3.4. Letωbe a non-empty open subset of ΩandT >0. Let us consider the weight\nfunctionsη,ψandξdefined in Lemma 3.1and(3.1). Then there exists s0≥1, λ0≥1and\nC >0such that for s≥s0andλ≥λ0and every solution yof the Stokes system (3.5), we\nhave/integraldisplay\nQe2sψ/parenleftbigg|∇z|2\nsξ+sλ2ξ|z|2+λ2|∇y|2+s2λ4ξ2|y|2/parenrightbigg\ndxdt\n≤C/parenleftbigg/integraldisplay\nQe2sψ|k|2dxdt+s3λ4/integraldisplay\nQωe2sψξ3|y|2dxdt/parenrightbigg\n. (3.21)\nNote that the adjoint system ( 1.11) can be viewed as the previous Stokes system ( 3.5)\nwithtandkreplaced by T−tand\nk∗= (Dϕ)/tildewideu−αϕ−β|/tildewideu|2ϕ−2β(ϕ·/tildewideu)/tildewideu,\nrespectively, with the hypothesis that /tildewideu∈L∞(Q)d. Thus, it is immediate that\n/ba∇dblesψk∗/ba∇dbl2\nL2(Q)d≤2/ba∇dbl/tildewideu/ba∇dbl2\nL∞(Q)d/ba∇dblesψ∇ϕ/ba∇dbl2\nL2(0,T;H)+4α2/ba∇dblesψϕ/ba∇dbl2\nL2(0,T;H)\n+36β2/ba∇dbl/tildewideu/ba∇dbl4\nL∞(Q)d/ba∇dblesψϕ/ba∇dbl2\nL2(0,T;H)\n≤C/ba∇dblesψ∇ϕ/ba∇dbl2\nL2(0,T;H)+C/ba∇dblλξesψϕ/ba∇dbl2\nL2(0,T;H).\nNow, applyingtheCarlemanestimate( 3.21)andbychoosing λ0(whereλ≥λ0)largeenough,\nwe can absorb the terms in the left-hand side and we have the followin g Carleman estimate\nfor the adjoint state ϕ(see (1.11)):\n/integraldisplay\nQe2sψ/parenleftbigg|∇(∇×ϕ)|2\nsξ+sλ2ξ|∇×ϕ|2+λ2|∇ϕ|2+s2λ4ξ2|ϕ|2/parenrightbigg\ndxdt\n≤Cs3λ4/integraldisplay\nQωξ3e2sψ|ϕ|2dxdt. (3.22)\nFrom now on, let us fix s≥s0andλ≥λ0such that ( 3.22) holds. We consider the\nfollowing weight functions in order to deduce a Carleman estimate tha t does not vanish at\nt= 0:/braceleftBigg/tildewideψ(t) =ψ(t),ift∈[T/2,T],/tildewideψ(t) =ψ(T/2),ift∈[0,T/2],\n/tildewideξ(t) =ξ(t),ift∈[T/2,T],/tildewideξ(t) =ξ(T/2),ift∈[0,T/2].(3.23)\nWe note that the new weight functions /tildewideψand/tildewideξare no longer degenerate in the neighborhood\noft= 0. Moreover, by the construction of weight functions ψ(x,t) andξ(x,t) defined in\n(3.1), one can see that /tildewideψ≤0 fort∈[0,T]. Replacing ψandξby/tildewideψand/tildewideξ, respectively,\nand using the standard energy estimates for the backward Stoke s system, one can obtain the\nsame Carleman estimate:/integraldisplay\nQe2s/tildewideψ/parenleftbigg|∇(∇×ϕ)|2\ns/tildewideξ+sλ2/tildewideξ|∇×ϕ|2+λ2|∇ϕ|2+s2λ4/tildewideξ2|ϕ|2/parenrightbigg\ndxdt\n≤Cs3λ4/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt. (3.24)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 15\n3.3.Observability inequality. In this subsection, we derive an observability inequality for\nthe adjoint system ( 1.11) associated with the linearized system ( 1.10) around a trajectory\n(/tildewideu,/tildewidep) of (1.2), which leads to the null controllability of the system ( 1.10) at any time T >0.\nFirst, we present a remark which is helpful in deriving the observabilit y inequality.\nRemark 3.5. Letϕ(T)∈Hand let(ϕ,π)be the corresponding solution of the adjoint system\n(1.11). We know that ϕ∈C([0,T];H)∩L2(0,T;V). Letχ∈C1([0,T])be such that χ(T) = 0.\nThen(/tildewideϕ,/tildewideπ) := (χϕ,χπ)solves the system:\n\n\nL∗/tildewideϕ+∇/tildewideπ=−χ′ϕ,inQ,\n∇·/tildewideϕ= 0,inQ,\n/tildewideϕ=0,onΣ,\n/tildewideϕ(T) =0,inΩ,(3.25)\nwhereL∗is the adjoint operator of Ldefined in (1.12). Therefore, (/tildewideϕ,/tildewideπ)is a strong solution\nof(3.25)and we have\nϕ(t)∈H2(Ω)∩V,ϕt(t)∈H,andπ(t)∈H1(Ω)∩L2\n0(Ω),\nfor a.e.t∈[0,T].\nTheorem 3.6. Let/tildewideψand/tildewideξbe the weight functions defined in (3.23)and(/tildewideu,/tildewidep)be the solution\nof(1.2)satisfying (1.3). Then, there exists a positive constant Cdepending on s,λ,/tildewideuand\nT, such that every solution to the adjoint system (1.11)satisfies:\n/ba∇dblϕ(0)/ba∇dbl2\nH+/integraldisplay\nQ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplay\nQe2s/tildewideψ|∇ϕ|2dxdt\n≤C/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt. (3.26)\nProof.The proof of this lemma is classical and relies on the Carleman estimate (3.24) and\nthe dissipation properties of the solutions of the adjoint system ( 1.11). This tells us that\n/ba∇dblϕ(t)/ba∇dblHis an increasing function of t, so the presence of /ba∇dblϕ(0)/ba∇dblHin the left-hand side of\n(3.26) is not a surprise (see [ 28]). For completeness, we include the proof here (cf. [ 8,28]).\nWe start with an a-priori estimate for the adjoint system ( 1.11):\nsup\nt∈[0,T/2]/ba∇dblϕ(t)/ba∇dbl2\nH+/integraldisplayT/2\n0/ba∇dblϕ(t)/ba∇dbl2\nVdt≤C/integraldisplay3T/4\nT/2/ba∇dblϕ(t)/ba∇dbl2\nHdt, (3.27)\nwhereCdepends on Ω ,Tand/tildewideu. To prove this, let us take χ∈C1([0,T]) such that\nχ= 1 in [0,T/2], χ≡0 in [3T/4,T],and|χ′| ≤C.\nThen, (χϕ,χπ) solves the system ( 3.25), and we obtain the following energy estimate:\nsup\nt∈[0,T]/ba∇dblχ(t)ϕ(t)/ba∇dbl2\nH+/integraldisplayT\n0/ba∇dblχ(t)ϕ(t)/ba∇dbl2\nVdt≤C/integraldisplayT\n0/ba∇dblχ′(t)ϕ(t)/ba∇dbl2\nHdt, (3.28)\nand (3.27) follows from ( 3.28). In Ω×(0,T/2), (3.27) implies\n/ba∇dblϕ(0)/ba∇dbl2\nH+/integraldisplayT/2\n0/integraldisplay\nΩ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplayT/2\n0/integraldisplay\nΩe2s/tildewideψ|∇ϕ|2dxdt16 P. KUMAR AND M. T. MOHAN\n≤C/parenleftBigg\nsup\nt∈[0,T/2]/ba∇dblϕ(t)/ba∇dbl2\nH+/integraldisplayT/2\n0/ba∇dblϕ(t)/ba∇dbl2\nVdt/parenrightBigg\n≤C/integraldisplay3T/4\nT/2/integraldisplay\nΩ|ϕ|2dxdt≤/integraldisplay\nQs2λ4/tildewideξ2e2s/tildewideψ|ϕ|2dxdt, (3.29)\nwhere we have used the fact e2s/tildewideψ≤e2s0/tildewideψ≤Cfor/tildewideψ≤0 and chosen 0 <C≤s2λ4/tildewideξ2e2s/tildewideψfor\ns≥s0≥1, λ≥λ0≥1 in the last inequality. Applying the Carleman estimate ( 3.24) into\n(3.29), we obtain a first estimate in Ω ×(0,T/2):\n/ba∇dblϕ(0)/ba∇dbl2\nH+/integraldisplayT/2\n0/integraldisplay\nΩ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplayT/2\n0/integraldisplay\nΩe2s/tildewideψ|∇ϕ|2dxdt\n≤Cs3λ4/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt. (3.30)\nOn the other hand, since ψ=/tildewideψandξ=/tildewideξin Ω×(T/2,T), we have\n/integraldisplayT\nT/2/integraldisplay\nΩ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplayT\nT/2/integraldisplay\nΩe2s/tildewideψ|∇ϕ|2dxdt\n=/integraldisplayT\nT/2/integraldisplay\nΩξ2e2sψ|ϕ|2dxdt+/integraldisplayT\nT/2/integraldisplay\nΩe2sψ|∇ϕ|2dxdt\n≤/parenleftbigg\ns2λ4/integraldisplay\nQξ2e2sψ|ϕ|2dxdt+λ2/integraldisplay\nQe2sψ|∇ϕ|2dxdt/parenrightbigg\n≤Cs3λ4/integraldisplay\nQωξ3e2sψ|ϕ|2dxdt\n=Cs3λ4/integraldisplayT/2\n0/integraldisplay\nωξ3e2sψ|ϕ|2dxdt+Cs3λ4/integraldisplayT\nT/2/integraldisplay\nωξ3e2sψ|ϕ|2dxdt, (3.31)\nwhere we have used the Carleman estimate ( 3.22). By the definition of weight functions ( 3.1)\nand (3.23), from (3.31), we immediately get\n/integraldisplayT\nT/2/integraldisplay\nΩ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplayT\nT/2/integraldisplay\nΩe2s/tildewideψ|∇ϕ|2dxdt\n≤Cs3λ4/integraldisplayT/2\n0/integraldisplay\nω|ϕ|2dxdt+Cs3λ4/integraldisplayT\nT/2/integraldisplay\nω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt\n≤Cs3λ4/integraldisplayT/2\n0/integraldisplay\nω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt+Cs3λ4/integraldisplayT\nT/2/integraldisplay\nω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt\n=Cs3λ4/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt, (3.32)\nwhich combined with ( 3.30) leads to\n/ba∇dblϕ(0)/ba∇dbl2\nH+/integraldisplay\nQ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplay\nQe2s/tildewideψ|∇ϕ|2dxdt≤Cs3λ4/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt,\nand we get the desired observability inequality ( 3.26). /squareLOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 17\n4.Null Controllability of the Linear System\nIn this section, we prove the null controllability of the linearized CBF e quations ( 1.10) as\na consequence of the observability inequality (see Theorem 3.6). This result will be helpful\nto determine the local exact controllability of the system ( 1.1) in the next section.\nLater discussion of the null controllability of the linearized CBF equat ions is motivated\nby the works [ 8,41], in which the authors established the null controllability of the lineariz ed\nNavier-Stokes system in both two and three dimensions.\n4.1.Penalty method. Withthehelpofobservability inequality ( 3.26), onecanprove thenull\ncontrollability of the linearized system ( 1.10). We establish the following null controllability\nresult for the system ( 1.10).\nTheorem 4.1. Ifv0∈Hand the functions gandhsatisfy\n/integraldisplay\nQe−2s/tildewideψ|g|2dxdt<+∞and/integraldisplayT\n0/ba∇dble−s/tildewideψh/ba∇dbl2\nH−1(Ω)ddt<+∞,\nrespectively, then there exists a control ϑ∈L2(Qω)dand a solution vof the system (1.10)\nsuch that\nv(·,T) =0.\nProof.Let us consider, for each ε>0, the following optimal control problem:\nmin\nϑ∈L2(Qω)dJε(ϑ), (4.1)\nwhere\nJε(ϑ) =1\n2ε/ba∇dblv(T)/ba∇dbl2\nH+1\n2/integraldisplay\nQω|ϑ|2dxdt, (4.2)\nandvis the solution of the system ( 1.10) associated to ϑ.\nStep 1. Existence of an optimal control: From the definition of the cost functional Jε(·)\ndefined in ( 4.2), we can see that Jε(·)≥0, there exists an infimum IofJε(·) over L2(Qω)d,\nthat is,\n0≤I:= inf\nϑ∈L2(Qω)dJε(ϑ)<+∞.\nSince, 0≤I <+∞, there exists a minimizing sequence {ϑn} ∈L2(Qω)dsuch that\nlim\nn→∞Jε(ϑn) = lim\nn→∞/parenleftbigg1\n2ε/ba∇dblvn(T)/ba∇dbl2\nH+1\n2/integraldisplay\nQω|ϑn|2dxdt/parenrightbigg\n=I,\nwherevn(·) is the unique solution of the linearized problem ( 1.10) with the control ϑn∈\nL2(Qω)dand initial data vn(0)∈H. Since,0∈L2(Qω)d, without loss of generality, we may\nassume that Jε(ϑn)≤ Jε(0), where vis the unique solution of ( 1.10) corresponding to the\ncontrol0. Using the definition of Jε(·), we deduce that\n1\n2ε/ba∇dblvn(T)/ba∇dbl2\nH+1\n2/integraldisplay\nQω|ϑn|2dxdt≤1\n2ε/ba∇dblv(T)/ba∇dbl2\nH<+∞.18 P. KUMAR AND M. T. MOHAN\nFrom the above expression, it is clear that, there exists a constan tR>0, large enough such\nthat\n1\n2/integraldisplay\nQω|ϑn|2dxdt≤R<+∞. (4.3)\nUsing the energy estimate obtained in Lemma 2.1, we get\n/ba∇dblvn/ba∇dblL∞(0,T;H)+/ba∇dblvn/ba∇dblL2(0,T;V)≤C/parenleftbig\n/ba∇dblv0/ba∇dblH+/ba∇dblg/ba∇dblL2(Q)d2+/ba∇dblh/ba∇dblL2(0,T;H−1(Ω)d)+/ba∇dblϑn/ba∇dblL2(Qω)d/parenrightbig\n≤C/parenleftbig\n/ba∇dblv0/ba∇dblH+/ba∇dblg/ba∇dblL2(Q)d2+/ba∇dblh/ba∇dblL2(0,T;H−1(Ω)d)+R1\n2/parenrightbig\n.\nWe know that the dual of L1(0,T,H) is L∞(0,T,H) and L1(0,T,H) is separable. Also the\nspaces L2(0,T,V) and L2(Qω)dare reflexive. By using the Banach-Alaoglu theorem, we can\nextract subsequences still denoted by {vn}and{ϑn}such that\n\n\nvnw∗−⇀vεin L∞(0,T,H),\nvnw−⇀vεin L2(0,T;V),\nϑnw−⇀ϑεin L2(Qω)d.\nMoreover, vn\ntw−⇀vεtin L2(0,T;V′) (cf. (2.6)). Now, we show that ( vε,ϑε) satisfies the\nsystem (1.10). Since, Lis a linear operator defined in ( 1.7), from the above convergences,\none can easily deduce\nLvnw−⇀Lvεin L2(0,T;V′).\nOn passing to limit n→ ∞in the equation satisfied by ( vn,ϑn), we find that the limit\n(vε,ϑε) satisfies ( 1.10) in the weak sense.\nFinally, we show that ( vε,ϑε) is a minimizer, that is, I=Jε(ϑε). Since the cost functional\nJε(·) is continuous and convex on L2(Qω)dand L∞(0,T;H)∩L2(0,T;V), it follows that Jε(·)\nis weakly lowersemicontinuous. That is, for a sequence ϑnw−⇀ϑεin L2(Qω)d,we have\nJε(ϑε)≤liminf\nn→∞Jε(ϑn).\nTherefore, we obtain\nI≤ Jε(ϑε)≤liminf\nn→∞Jε(ϑn) = lim\nn→∞Jε(ϑn) =I,\nand hence ( vε,ϑε) is a minimizer of the problem ( 4.1).\nStep 2.Uniqueness of the optimal control: The optimal control problem ( 4.1) has a solution\n(vε,ϑε) for anyε >0, and by the convexity of the cost functional and since the syste m\n(1.10) is linear, this solution is even unique.\nStep 3.First order necessary conditions of optimality: The necessary condition of minimum\nyields\n/parenleftbigg1\nεvε(T),vw(T)/parenrightbigg\n+/integraldisplay\nQωϑε·wdxdt= 0,for allw∈L2(Qω)d, (4.4)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 19\nwherevwis, together with certain pressure qw, a solution of\n\n\nLvw+∇qw=w1ω,inQ,\n∇·vw= 0,inQ,\nvw=0,on Σ,\nvw(0) =0,in Ω.(4.5)\nHere, the (affine) map’s derivative as ϑ→v(ϑ) at the point ϑεin the direction wis\nrepresented by vw. Then, the duality properties between v(see (1.10)) andϕ(see (1.11))\nyield\n/parenleftbigg1\nεvε(T),vw(T)/parenrightbigg\n=/integraldisplay\nQωϕ·wdxdt,for allw∈L2(Qω)d,\nwhich combined with ( 4.4), provides\n/integraldisplay\nQω(ϑε+ϕ)·wdxdt= 0,for allw∈L2(Qω)d.\nConsequently, we can identify ϑε:\nϑε+ϕ=0,inQω. (4.6)\nStep 4.Null controllability of the linearized system: We are now ready to pass the limit as\nε→0. To this end, by multiplying the state equation for vε(see (1.10)) byϕ, we deduce\n/parenleftbig\nLvε+∇qε,ϕ/parenrightbig\n=/parenleftbig\n∇·g+ϑε1ω,ϕ/parenrightbig\n+/a\\}b∇acketle{th,ϕ/a\\}b∇acket∇i}ht.\nOn simplification, it is immediate that\nd\ndt(vε,ϕ)+/parenleftbig\nL∗ϕ,vε/parenrightbig\n=/parenleftbig\n∇·g+ϑε1ω,ϕ/parenrightbig\n+/a\\}b∇acketle{th,ϕ/a\\}b∇acket∇i}ht.\nIntegrating the above equation with respect to time from 0 to T, we find\n/parenleftbig\nvε(T),ϕ(T)/parenrightbig\n−/parenleftbig\nv0,ϕ(0)/parenrightbig\n=−d/summationdisplay\ni,j=1/integraldisplay\nQgij∂ϕi\n∂xjdxdt+/integraldisplay\nQωϑε·ϕdxdt+/integraldisplayT\n0/a\\}b∇acketle{th,ϕ/a\\}b∇acket∇i}htdt.\nThe optimality condition ( 4.6) and (1.11) imply\n1\nε/ba∇dblvε(T)/ba∇dbl2\nH+/integraldisplay\nQω|ϑε|2dxdt=/parenleftbig\nv0,ϕ(0)/parenrightbig\n−d/summationdisplay\ni,j=1/integraldisplay\nQgij∂ϕi\n∂xjdxdt+/integraldisplayT\n0/a\\}b∇acketle{th,ϕ/a\\}b∇acket∇i}htdt.\nUsing the Cauchy-Schwartz and Young’s inequalities, we obtain\n1\nε/ba∇dblvε(T)/ba∇dbl2\nH+/integraldisplay\nQω|ϑε|2dxdt\n≤3C\n2/ba∇dblv0/ba∇dbl2\nH+3C\n2/integraldisplay\nQe−2s/tildewideψ|g|2dxdt+3C\n2/integraldisplayT\n0/ba∇dble−s/tildewideψh/ba∇dbl2\nH−1(Ω)ddt\n+1\n6C/ba∇dblϕ(0)/ba∇dbl2\nH+1\n3C/integraldisplay\nQe2s/tildewideψ|∇ϕ|2dxdt, (4.7)20 P. KUMAR AND M. T. MOHAN\nwhere/tildewideψand/tildewideξare the weight functions defined in ( 3.23) andCis the constant appearing\nin the observability inequality (see ( 3.26)). From the observability inequality ( 3.26) and\noptimality condition ( 4.6), we deduce\n/ba∇dblϕ(0)/ba∇dbl2\nH+/integraldisplay\nQ/tildewideξ2e2s/tildewideψ|ϕ|2dxdt+/integraldisplay\nQe2s/tildewideψ|∇ϕ|2dxdt\n≤C/integraldisplay\nQω/tildewideξ3e2s/tildewideψ|ϕ|2dxdt≤C/integraldisplay\nQω|ϕ|2dxdt=C/integraldisplay\nQω|ϑε|2dxdt. (4.8)\nIf we assume that gandhsatisfy\n/integraldisplay\nQe−2s/tildewideψ|g|2dxdt<+∞and/integraldisplayT\n0/ba∇dble−s/tildewideψh/ba∇dbl2\nH−1(Ω)ddt<+∞,\nthen by using ( 4.8) in (4.7), we have the following estimate:\n1\nε/ba∇dblvε(T)/ba∇dbl2\nH+1\n2/integraldisplay\nQω|ϑε|2dxdt≤C/parenleftbigg\n/ba∇dblv0/ba∇dbl2\nH+/integraldisplay\nQe−2s/tildewideψ|g|2dxdt+/integraldisplayT\n0/ba∇dble−s/tildewideψh/ba∇dbl2\nH−1(Ω)ddt/parenrightbigg\n.\nWecaneasilyseethat1√εvε(T)andϑεareuniformlyboundedin HandL2(Qω)d, respectively,\nand independent of ε. Using the Banach-Alaoglu theorem, we can extract subsequence s, still\ndenoted by ϑεandvε, such that\nϑεw−⇀ϑin L2(Qω)d,\nvε=v(ϑε)w−⇀v=v(ϑ) in C([0,T];H)∩L2(0,T;V),\nvε(T)w−⇀v(T) inH.\nUsing the weakly lowersemicontinuity property of the norm, we have\n/ba∇dblv(T)/ba∇dbl2\nH≤liminf\nε→0/ba∇dblvε(T)/ba∇dbl2\nH.\nThus, we must have\nv(·,T) =0,\nwhich completes the proof. /square\n4.2.Exponentially decreasing controls and solutions. In the following subsection, our aim\nis to find a control ϑ,which is exponentially decreasing as tapproaches T, such that not\nonlyv(·,T) =0, but also vis exponentially decreases as tapproaches T. This result will\nbe crucial to deduce the controllability properties for the nonlinear system (1.1) in the next\nsection. Motivated by the works [ 8,41], we have the following result:\nTheorem 4.2. Ifv0∈Hand the functions gandhsatisfy\n/integraldisplay\nQe−2s/tildewideψ|g|2dxdt<+∞and/integraldisplayT\n0/ba∇dble−s/tildewideψh/ba∇dbl2\nH−1(Ω)ddt<+∞,\nrespectively, then there exists a control ϑ∈L2(Qω)dand a solution vof(1.10)such that\nv(·,T) =0,\nand\n/integraldisplay\nQe−2s/tildewideψ|v|2dxdt<+∞and/integraldisplay\nQωe−2s/tildewideψ\nξ3|ϑ|2dxdt<+∞.LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 21\nProof.Let us remark that we adapt the ideas from [ 8,28,41] to prove our result. Let us\ndefine\nX=/braceleftbigg\n(¯v,¯q)∈C∞(Q)d+1,∇·¯v= 0,inQ,¯v=0,on Σ,/integraldisplay\nω¯q(t)dx= 0,a.e. in (0,T)/bracerightbigg\n.\nLet us consider the bilinear form B:X ×X → Rgiven by\nB/parenleftbig\n(˘v,˘q),(¯v,¯q)/parenrightbig\n=/integraldisplay\nQe2s/tildewideψ(L∗˘v+∇˘q)·(L∗¯v+∇¯q)dxdt\n+/integraldisplay\nQωe2s/tildewideψ/tildewideξ3˘v·¯vdxdt,for all (¯v,¯q)∈ X, (4.9)\nwhereL∗is the adjoint operator of L(cf. (1.12) and (1.7)).\nSince/tildewideu∈L∞(Q)d, because of the Carleman estimate ( 3.21), we can see that B(·,·) is a\nscalar product on X. Let us define Xas the completion of Xfor the norm associated to\nscalar product B(·,·) (which we denote by /ba∇dbl·/ba∇dblX). Then, Xis a Hilbert space for this scalar\nproduct. Thus, the bilinear form B(·,·) is well-defined, continuous, and coercive on X, and\nobserve that the observability inequality ( 3.26) holds for all ( ¯v,¯q)∈X, that is,\n/ba∇dbl¯v(0)/ba∇dbl2\nH+/integraldisplay\nQe2s/tildewideψ/parenleftbig\n|∇¯v|2+/tildewideξ2|¯v|2/parenrightbig\ndxdt≤C/integraldisplay\nQωe2s/tildewideψ/tildewideξ3|¯v|2dxdt\n≤CB/parenleftbig\n(¯v,¯q),(¯v,¯q)/parenrightbig\n,for all (¯v,¯q)∈X.(4.10)\nLet us now consider the linear form L:X →Rdefined by\n/a\\}b∇acketle{tL,(¯v,¯q)/a\\}b∇acket∇i}ht= (v0,¯v(0))H−d/summationdisplay\ni,j=1/integraldisplay\nQgij∂¯vi\n∂xjdxdt+/integraldisplayT\n0/a\\}b∇acketle{th,¯v/a\\}b∇acket∇i}htdt,for all (¯v,¯q)∈X,(4.11)\nwith\n|/a\\}b∇acketle{tL,(¯v,¯q)/a\\}b∇acket∇i}ht| ≤ /ba∇dblv0/ba∇dblH/ba∇dbl¯v(0)/ba∇dblH+/vextenddouble/vextenddoublee−s/tildewideψg/vextenddouble/vextenddouble\nL2(Q)d2/vextenddouble/vextenddoublees/tildewideψ¯v/vextenddouble/vextenddouble\nL2(0,T;V)\n+/vextenddouble/vextenddoublee−s/tildewideψh/vextenddouble/vextenddouble\nL2(0,T;H−1(Ω)d)/vextenddouble/vextenddoublees/tildewideψ¯v/vextenddouble/vextenddouble\nL2(0,T;V)\n≤C/parenleftBig\n/ba∇dblv0/ba∇dblH+/vextenddouble/vextenddoublee−s/tildewideψg/vextenddouble/vextenddouble\nL2(Q)d2+/vextenddouble/vextenddoublee−s/tildewideψh/vextenddouble/vextenddouble\nL2(0,T;H−1(Ω)d)/parenrightBig\n/ba∇dbl(¯v,¯q)/ba∇dblX,(4.12)\nfor all (¯v,¯q)∈Xand we have used the estimate ( 4.10). Therefore, we can see that the linear\nform/a\\}b∇acketle{tL,(·)/a\\}b∇acket∇i}htis well-defined and continuous on X. Consequently, in view of Lax-Milgram\nTheorem (see [ 16, Chapter 6]), there exists a unique solution ( ˘v,˘q)∈Xof the variational\nproblem\nB/parenleftbig\n(˘v,˘q),(¯v,¯q)/parenrightbig\n=/a\\}b∇acketle{tL,(¯v,¯q)/a\\}b∇acket∇i}ht,for all (¯v,¯q)∈X. (4.13)\nLet us define\nv=e2s/tildewideψ(L∗˘v+∇˘q),andϑ=−e2s/tildewideψ/tildewideξ31ω˘v, (4.14)\nand one can observe that ( v,ϑ) verifies\n/integraldisplay\nQe−2s/tildewideψ|v|2dxdt+/integraldisplay\nQωe−2s/tildewideψ\nξ3|ϑ|2dxdt<+∞\nandvis solution of the system in ( 1.10) for certain pressure q.22 P. KUMAR AND M. T. MOHAN\nNote that, v∈L2(Q)dandϑ∈L2(Qω)d. Then, we introduce the weak solution ( ˇv,ˇq) to\nthe system\n\nLˇv+∇ˇq=∇·g+h+ϑ1ω,inQ,\n∇·ˇv= 0,inQ,\nˇv=0,on Σ,\nˇv(0) =v0,in Ω.(4.15)\nIndeed,ˇvis also the unique solution of ( 4.15) defined by transposition. Of course, this says\nthatˇv∈L2(Q)dis the unique function verifying\n/integraldisplay\nQˇv·¯bdxdt= (v0,¯v(0))H−d/summationdisplay\ni,j=1/integraldisplay\nQgij∂¯vi\n∂xjdxdt\n+/integraldisplayT\n0/a\\}b∇acketle{th,¯v/a\\}b∇acket∇i}htdt+/integraldisplay\nQωϑ·¯vdxdt,for all¯b∈L2(Q)d.(4.16)\nHere¯v, together with some pressure ¯ q, is the solution of the following system\n\n\nL¯v+∇¯q=¯b,inQ,\n∇·¯v= 0,inQ,\n¯v=0,on Σ,\n¯v(T) =0,in Ω.\nFrom the relations ( 4.14) and (4.13), one can easily get that valso verifies ( 4.16). Conse-\nquently,v=ˇvandvis, together with a certain pressure q= ˇq, the solution of the system\n(1.10).\nOn the other hand, by virtue of ( 4.9) and (4.12)-(4.14), it follows\n/integraldisplay\nQe−2s/tildewideψ|v|2dxdt+/integraldisplay\nQωe−2s/tildewideψ\nξ3|ϑ|2dxdt\n=/integraldisplay\nQe2s/tildewideψ|L∗˘v+∇˘q|2dxdt+/integraldisplay\nQωe2s/tildewideψ/tildewideξ3|˘v|2dxdt\n≤C/parenleftBig\n/ba∇dblv0/ba∇dblH+/vextenddouble/vextenddoublee−s/tildewideψg/vextenddouble/vextenddouble\nL2(Q)d2+/vextenddouble/vextenddoublee−s/tildewideψh/vextenddouble/vextenddouble\nL2(0,T;H−1(Ω)d)/parenrightBig\n.\nObviously, this shows that v(·,T) =0and that vandϑdecrease exponentially as tap-\nproachesTand it completes the proof. /square\n5.The Nonlinear Problem\nIn this section, we provide the proof of our main result, Theorem 1.1, using similar argu-\nments to those employed in [ 8,41], and references therein. Moreover, we see that the results\nobtained in the previous section allow us to locally invert a nonlinear sys tem. We deduce\nsome regularity results, which will be sufficient for us to apply a suitab le inverse mapping\ntheorem (see Theorem 5.9) to reach our main goal.\nInthecontext ofthelocalexact controllabilitytothetrajectorie s oftheCBF equations, we\nadapt some ideas of the proofs from the works [ 8,41], where the local exact controllability to\nthe trajectories is established for theNavier-Stokes system in bo thtwo and three dimensions.LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 23\n5.1.Choice of special weights. Fromnow on, we omit the notation /tildewidefor the weight functions\n/tildewideψand/tildewideξdefined in ( 3.23). We still have some choice for the constants m1andm2in the\nweightsψandξ. To do this, first, for t∈[T/2,T] (the caset∈[0,T/2] is straightforward),\nwe define\nˇψ(t) = min\nx∈Ωψ(x,t) =eλm1−eλ(/bardblη/bardblL∞+m2)\nℓ4(t)<0,\nˇξ(t) = min\nx∈Ωξ(x,t) =eλm1\nℓ4(t),\nˆψ(t) = max\nx∈Ωψ(x,t) =eλ(/bardblη/bardblL∞+m1)−eλ(/bardblη/bardblL∞+m2)\nℓ4(t)<0,\nˆξ(t) = max\nx∈Ωξ(x,t) =eλ(/bardblη/bardblL∞+m1)\nℓ4(t).\nThen, we have the following result for choosing the constants given in [41].\nLemma 5.1 (Lemma 4.1, [ 41]).We can choose the constants m1andm2withm1≤m2such\nthat forλ0large enough, we have for λ≥λ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ψ\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cξ5\n4,and/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2ψ\n∂t2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cξ3\n2,\nand\n3\n2ˆψ≤ˇψor−ˇψ≤ −3\n2ˆψ.\nProof.Proof follows in the same lines as in the proof of [ 41, Lemma 4.1]. Let us take\nm1= (m0+4)/ba∇dblη/ba∇dblL∞andm2=m3/ba∇dblη/ba∇dblL∞.\nIt is easy to see that all the requirements are fulfilled if\nm0+4<m3<5\n4m0+4,\nforλ0sufficiently large. Now, such a choice of m0andm3is possible. /square\n5.2.Functional class and some regularity results. To apply an inverse mapping argument\nand extract some controllability properties for the nonlinear syste m (1.1), suitable spaces are\nneeded. We must define the correct functional class to apply the in verse mapping theorem.\nTo do this, let us define the weighted Banach spaces ( Ed,/ba∇dbl·/ba∇dblEd),(d= 2,3) as follows:\nE2=/braceleftbigg/parenleftbig\nv,ϑ/parenrightbig\n:/parenleftbig\nv,ϑ/parenrightbig\n∈ E0,e−3s\n4ˆψv∈L6(0,T;L4(Ω)2),\n∃q,¯g,¯h:e−sψ(∇·¯g+¯h)∈L2(0,T;H−1(Ω)2),\nLv+∇q−ϑ1ω=∇·¯g+¯h,v(0)∈H/bracerightbigg\n,\nwhend= 2 and\nE3=/braceleftbigg/parenleftbig\nv,ϑ/parenrightbig\n:/parenleftbig\nv,ϑ/parenrightbig\n∈ E0,e−3s\n4ˆψv∈L4(0,T;L12(Ω)3)∩L6(Q)3,\n∃q,¯g,¯h:e−sψ(∇·¯g+¯h)∈L2(0,T;W−1,6(Ω)3),24 P. KUMAR AND M. T. MOHAN\nLv+∇q−ϑ1ω=∇·¯g+¯h,v(0)∈H∩L4(Ω)3/bracerightbigg\n,\nwhend= 3, where\nE0=/braceleftbigg/parenleftbig\nv,ϑ/parenrightbig\n:e−sψv∈L2(Q)d, ξ−3\n2e−sψϑ∈L2(Qω)d,\ne−3s\n4ˆψv∈L∞(0,T;H)∩L2(0,T;V)/bracerightbigg\n,\nendowed with the norm\n/ba∇dbl(v,ϑ)/ba∇dbl2\nE2=/vextenddouble/vextenddoublee−sψv/vextenddouble/vextenddouble2\nL2(Q)2+/vextenddouble/vextenddoubleξ−3\n2e−sψϑ/vextenddouble/vextenddouble2\nL2(Qω)2+/vextenddouble/vextenddoublee−3s\n4ˆψv/vextenddouble/vextenddouble2\nL6(0,T;L4(Ω)2)\n+/vextenddouble/vextenddoublee−3s\n4ˆψv/vextenddouble/vextenddouble2\nL∞(0,T;H)∩L2(0,T;V)+/vextenddouble/vextenddoublee−sψ∇·¯g/vextenddouble/vextenddouble2\nL2(0,T;H−1(Ω)2)\n+/vextenddouble/vextenddoublee−sψ¯h/vextenddouble/vextenddouble2\nL2(0,T;H−1(Ω)2)+/ba∇dblv(0)/ba∇dbl2\nH,\nwhend= 2 and\n/ba∇dbl(v,ϑ)/ba∇dbl2\nE3=/vextenddouble/vextenddoublee−sψv/vextenddouble/vextenddouble2\nL2(Q)3+/vextenddouble/vextenddoubleξ−3\n2e−sψϑ/vextenddouble/vextenddouble2\nL2(Qω)3+/vextenddouble/vextenddoublee−3s\n4ˆψv/vextenddouble/vextenddouble2\nL∞(0,T;H)∩L2(0,T;V)\n+/vextenddouble/vextenddoublee−3s\n4ˆψv/vextenddouble/vextenddouble2\nL6(Q)3+/vextenddouble/vextenddoublee−3s\n4ˆψv/vextenddouble/vextenddouble2\nL4(0,T;L12(Ω)3)\n+/vextenddouble/vextenddoublee−sψ∇·¯g/vextenddouble/vextenddouble2\nL2(0,T;W−1,6(Ω)3)+/vextenddouble/vextenddoublee−sψ¯h/vextenddouble/vextenddouble2\nL2(0,T;W−1,6(Ω)3)+/ba∇dblv(0)/ba∇dbl2\nL4(Ω)3,\nwhend= 3. Here, the operator Lvis defined in ( 1.7). On the other hand, we also define\nthe weighted Banach spaces ( Gd,/ba∇dbl·/ba∇dblGd) ford= 2,3 as:\nG2=/braceleftbigg/parenleftbig\n∇·¯g+¯h,v0/parenrightbig\n:e−sψ(∇·¯g+¯h)∈L2(0,T;H−1(Ω)2),v0∈H/bracerightbigg\n,\nand\nG3=/braceleftbigg/parenleftbig\n∇·¯g+¯h,v0/parenrightbig\n:e−sψ(∇·¯g+¯h)∈L2(0,T;W−1,6(Ω)3),v0∈H∩L4(Ω)3/bracerightbigg\n,\nendowed with the norm\n/ba∇dbl(∇·¯g+¯h,v0)/ba∇dbl2\nG2=/vextenddouble/vextenddoublee−sψ∇·¯g/vextenddouble/vextenddouble2\nL2(0,T;H−1(Ω)2)+/ba∇dble−sψ¯h/ba∇dbl2\nL2(0,T;H−1(Ω)2)+/ba∇dblv0/ba∇dbl2\nH,\nand\n/ba∇dbl(∇·¯g+¯h,v0)/ba∇dbl2\nG3=/vextenddouble/vextenddoublee−sψ∇·¯g/vextenddouble/vextenddouble2\nL2(0,T;W−1,6(Ω)3)+/ba∇dble−sψ¯h/ba∇dbl2\nL2(0,T;W−1,6(Ω)3)+/ba∇dblv0/ba∇dbl2\nL4(Ω)3.\nRemark 5.2. The spaces denoted as Ej(j= 0,2,3)represent the natural spaces, where solu-\ntions of the null controllability of (1.10)must be found in order to preserve these properties\nfor the nonlinear terms (u· ∇)uand|u|2u. Additional information will be provided in\nsubsection 5.3.\nGiven any ( ∇·g∗+h∗,v0)∈ G, we know from Theorem 4.2that (v,ϑ∗) solves the system:\n\n\nLv+∇q=∇·g∗+h∗+ϑ∗1ω,inQ,\n∇·v= 0,inQ,\nv=0,on Σ,\nv(0) =v0,in Ω,(5.1)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 25\nwith\nv(·,T) =0,\nand\nv∈L∞(0,T;;H)∩L2(0,T;V), e−sψv∈L2(Q)d,andξ−3\n2e−sψϑ∗∈L2(Qω)d.\nLet us prove that ( v,ϑ∗)∈ E. It remains to prove that\n•e−3s\n4ˆψv∈L∞(0,T;H)∩L2(0,T;V),in dimension d= 2,3,\n•e−3s\n4ˆψv∈L4(0,T;L12(Ω)3)∩L6(Q)3, in dimension d= 3,\n•e−3s\n4ˆψv∈L6(0,T;L4(Ω)2),in dimension d= 2.\nLet us first deduce that e−3s\n4ˆψv∈L∞(0,T;H)∩L2(0,T;V).To fulfill this purpose, let us\ndefine the functions\nˆv=e−3s\n4ˆψv,ˆq=e−3s\n4ˆψq,ˆg=e−3s\n4ˆψg∗,ˆh=e−3s\n4ˆψh∗,andˆϑ=e−3s\n4ˆψϑ∗.\nThen, the function ( ˆv,ˆq) solves the following system:\n\n\nLˆv+∇ˆq=∇·ˆg+ˆh+ˆϑ1ω−3s\n4∂ˆψ\n∂tˆv,inQ,\n∇·ˆv= 0,inQ,\nˆv=0,on Σ,\nˆv(0) =e−3s\n4ˆψ(0)v0,in Ω.(5.2)\nIt is easy to check that\n∇·ˆg,ˆh∈L2(0,T;H−1(Ω)d),andˆϑ∈L2(Qω)d,\nand\n∂ˆψ\n∂tˆv=∂ˆψ\n∂te−3s\n4ˆψv=/parenleftbigg∂ˆψ\n∂tes\n4ˆψ/parenrightbigg\ne−sˆψv∈L2(Q)d.\nTherefore, for given v0∈H, from the existence result for d= 2,3, we have\nˆv∈L∞(0,T;H)∩L2(0,T;V).\nNow, we will prove that ˆv∈L4(0,T;L12(Ω)3)∩L6(Q)3whend= 3. Since/tildewideu∈L∞(Q)3\nandˆv∈L∞(0,T;H)∩L2(0,T;V), we observe that\nαˆv+β|/tildewideu|2ˆv+2β(ˆv·/tildewideu)/tildewideu∈L2(Q)3.\nAlso,ˆv∈L2(0,T;V)⊂L2(0,T;L6(Ω)3), we obtain\n∇·(ˆv⊗/tildewideu),∇·(/tildewideu⊗ˆv)∈L2(0,T;W−1,6(Ω)3),\nand these terms can be combined in ∇ ·ˆg, so that without loss of generality, we can write\n(with the same notations as in ( 5.2))\n\n\n∂ˆv\n∂t−µ∆ˆv+∇ˆq=∇·ˆg+ˆh+ˆk,inQ,\n∇·ˆv= 0,inQ,\nˆv=0,on Σ,\nˆv(0) =e−3s\n4ˆψ(0)v0,in Ω,(5.3)26 P. KUMAR AND M. T. MOHAN\nwhere\n∇·ˆg,ˆh∈L2(0,T;W−1,6(Ω)3),\nand\nˆk=ˆϑ1ω−αˆv−β|/tildewideu|2ˆv−2β(ˆv·/tildewideu)/tildewideu−3s\n4∂ˆψ\n∂tˆv∈L2(Q)3.\nLet us consider the Stokes system:\n\n−∂v\n∂t−µ∆v+∇q=k3,inQ,\n∇·v= 0,inQ,\nv=0,on Σ,\nv(T) =0,in Ω.(5.4)\nRemark 5.3. Based on the Giga-Sohr regularity result (see [ 27, Theorem 2.8] and Solonnikov\n[43] also), for every k3∈Lr1(0,T;Lr2(Ω)d)and1< r1,r2<+∞, there exists a unique\nsolution(v,∇q)to the Stokes system (5.4)satisfying\nv∈Lr1(0,T0;W2,r2(Ω)d)for all0<T0≤TwithT0<+∞,\n∂v\n∂t,∇q∈Lr1(0,T;Lr2(Ω)d).\nMoreover, we have\n/integraldisplayT\n0/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂v\n∂t/vextenddouble/vextenddouble/vextenddouble/vextenddoubler1\nr2dt+/integraldisplayT\n0/ba∇dbl∆v/ba∇dblr1\nr2dt+/integraldisplayT\n0/ba∇dbl∇q/ba∇dblr1\nr2dt≤C/integraldisplayT\n0/ba∇dblk3/ba∇dblr1\nr2dt,\nwhereC=C(r1,r2,Ω)is a positive constant.\nLemma 5.4. Let us assume that d= 3,v0∈H∩L4(Ω)3,/tildewideu∈L∞(Q)3, and that ∇·ˆg,ˆh∈\nL2(0,T;W−1,6(Ω)3),andˆk∈L2(Q)3. Then,ˆv∈L4(0,T;L12(Ω)3).\nThis lemma is a consequence of the following one by the transposition m ethod.\nLemma 5.5. Letd= 3. Then, for each k3∈L4\n3(0,T;L12\n11(Ω)3), there exists a unique solution\n(v,q)to the Stokes system (5.4)satisfying\nv∈C([0,T];L4\n3(Ω)3)∩L2(0,T;W1,6\n5\n0(Ω)3).\nProof.Let us first show that v∈C([0,T];L4\n3(Ω)3). From the Giga-Sohr regularity result on\nthe Stokes problem (see Remark 5.3), we have\nv∈L4\n3(0,T;W2,12\n11(Ω)3) and∂v\n∂t∈L4\n3(0,T;L12\n11(Ω)3).\nFrom the Sobolev embedding theorem, we obtain\nv∈L4\n3(0,T;W2,12\n11(Ω)3)֒→L4\n3(0,T;W1,12\n7(Ω)3)֒→L4\n3(0,T;L4(Ω)3).\nBy interpolation arguments (see [ 5] and [44]), we know that, if\nv∈L4\n3(0,T;L4(Ω)3),and∂v\n∂t∈L4\n3(0,T;L12\n11(Ω)3),LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 27\nthen\nv∈C/parenleftbig\n[0,T];/parenleftbig\nL4(Ω)3,L12\n11(Ω)3/parenrightbig\n3\n4,4\n3/parenrightbig\n.\nThe interpolationspace/parenleftbig\nL4(Ω)3,L12\n11(Ω)3/parenrightbig\n3\n4,4\n3is the Lorentz space L4\n3,4\n3(Ω)3= L4\n3(Ω)3(see [5,\nTheorem 5.2.1]). Consequently, we have v∈C([0,T];L4\n3(Ω)3).\nOn the other hand, we have\nv∈L4\n3/parenleftbig\n0,T;W2,12\n11(Ω)3∩W1,12\n7(Ω)3/parenrightbig\n∩L∞(0,T;L12\n11(Ω)3). (5.5)\nWe can use interpolation results in ( 5.5) to obtain\nv∈L2/parenleftbig\n0,T;(W2,12\n11(Ω)3,L12\n11(Ω)3)1\n3,12\n11/parenrightbig\n,\nand by the definition of Sobolev spaces (see [ 44, Chapter 34]), we have\n/parenleftbig\nW2,12\n11(Ω)3,L12\n11(Ω)3/parenrightbig\n1\n3,12\n11= W4\n3,12\n11(Ω)3.\nBy the virtue of Sobolev’s inequality W4\n3,12\n11(Ω)3֒→W1,6\n5(Ω)3, andv=0on the boundary\nΣ in the sense of trace, we deduce that\nv∈L2/parenleftbig\n0,T;W1,6\n5\n0(Ω)3/parenrightbig\n,\nwhich completes the proof. /square\nProof of Lemma 5.4.By definition, we say that ˆvis the solution by transposition of the\nproblem for the Stokes equation ( 5.3). This mean that ˆv∈L4(0,T;L12(Ω)3), is the unique\nsolution of ( 5.3) satisfying:\n/integraldisplay\nQˆv·k3dxdt=/integraldisplay\nΩe−3s\n4ˆψ(0)v0·v(0)dx+/integraldisplayT\n0/a\\}b∇acketle{t∇·ˆg+ˆh,v/a\\}b∇acket∇i}ht\nW−1,6(Ω)3,W1,6\n5\n0(Ω)3dt\n+/integraldisplay\nQˆk·vdxdt,for allk3∈L4\n3(0,T;L12\n11(Ω)3),\nwhere (v,q) is the solution to ( 5.4) associated to k3. We know that\nv(0)∈L4\n3(Ω)3,v∈L2(0,T;W1,6\n5\n0(Ω)3)⊂L2(Q)3,\nand\nv0∈L4(Ω)3,∇·ˆg,ˆh∈L2(0,T;W−1,6(Ω)3),ˆk∈L2(Q)3.\nNote that, for every k3∈L4\n3(0,T;L12\n11(Ω)3), the Stokes system ( 5.4) possesses exactly one\nsolutionv∈L4(0,T;L12(Ω)3) that depends continuously on k3. Therefore, ˆvis well-defined\nand defines a unique element ˆv∈/parenleftBig\nL4\n3(0,T;L12\n11(Ω)3)/parenrightBig′\n≡L4(0,T;L12(Ω)3), which completes\nthe proof of Lemma 5.4. /square\nNow, our task is to deduce that ˆv∈L6(Q)3. For this purpose, we establish the following\nlemma:\nLemma 5.6. Letd= 3. Then, for each k3∈L6\n5(Q)3, there exists a unique solution (v,q)to\nthe Stokes system (5.4)satisfying\nv∈C([0,T];L4\n3(Ω)3)∩L2(0,T;W1,6\n5\n0(Ω)3).28 P. KUMAR AND M. T. MOHAN\nLet us assume that Lemma 5.6holds. Then, ˆvmust coincide with the solution by trans-\nposition of the problem ( 5.3), namely, the unique function ˆv∈L6(Q)3satisfying\n/integraldisplay\nQˆv·k3dxdt=/integraldisplay\nΩe−3s\n4ˆψ(0)v0·v(0)dx+/integraldisplayT\n0/a\\}b∇acketle{tF,v/a\\}b∇acket∇i}ht\nW−1,6(Ω)3,W1,6\n5\n0(Ω)3dt,\nfor allk3∈L6\n5(Q)3andFstands for the function\nF=∇·ˆg+ˆh+ˆϑ1ω−3s\n4∂ˆψ\n∂tˆv\n−/parenleftbig\n(/tildewideu·∇)ˆv+(ˆv·∇)/tildewideu+αˆv+β|/tildewideu|2ˆv+2β(ˆv·/tildewideu)/tildewideu/parenrightbig\n, (5.6)\nand (v,q) is the solution of the Stokes system ( 5.4) associated to k3. Remark that, as we\nalready had that ˆv∈L2(0,T;V)⊂L2(0,T;L6(Ω)3), all the terms of the previous definition\n(5.6) make sense by virtue of Lemma 5.6and the assumption v0∈L4(Ω)3(see Remark 5.3).\nTherefore, ˆv∈/parenleftBig\nL6\n5(Q)3/parenrightBig′\n≡L6(Q)3.\nLet us provide a proof of Lemma 5.6, which is based on interpolation arguments.\nProof of Lemma 5.6.Let us first prove that v∈C([0,T];L4\n3(Ω)3). From the Giga-Sohr\nregularity result for the Stokes problem (see Remark 5.3), we have that\nv∈L6\n5(Q)3and∂v\n∂t∈L6\n5(Q)3.\nThe Sobolev embedding yields\nW2,6\n5(Ω)3֒→H1(Ω)3֒→L6(Ω)3.\nBy the virtue of interpolation results, we have that, if\nv∈L6\n5(0,T;L6(Ω)3) and∂v\n∂t∈L6\n5(Q)3,\nthen\nv∈C/parenleftbig\n[0,T];/parenleftbig\nL6(Ω)3,L6\n5(Ω)3/parenrightbig\n7\n8,4\n3/parenrightbig\n,\nand the interpolationspace/parenleftbig\nL6(Ω)3,L6\n5(Ω)3/parenrightbig\n7\n8,4\n3coincides with the Lorentz space L4\n3,4\n3(Ω)3=\nL4\n3(Ω)3(see [5, Theorem 5.2.1]). Therefore, we get v∈C([0,T];L4\n3(Ω)3).\nOn the other hand, we also have\nv∈L6\n5(0,T;W2,6\n5(Ω)3∩H1\n0(Ω)3) and v∈L∞(0,T;L6\n5(Ω)3).\nUsing the interpolation results, we deduce that\nv∈L2/parenleftbig\n0,T;(W2,6\n5(Ω)3,L6\n5(Ω)3)1\n3,6\n5/parenrightbig\n.\nBut we know that (see [ 44, Chapter 34])\n/parenleftbig\nW2,6\n5(Ω)3,L6\n5(Ω)3/parenrightbig\n1\n3,6\n5= W4\n3,6\n5(Ω)3.\nThanks to Sobolev’s embedding W4\n3,6\n5(Ω)3֒→W1,6\n5(Ω)3andv(t)∈H1\n0(Ω)3,a.e.t∈(0,T),\nwe finally infer that\nv∈L2/parenleftbig\n0,T;W1,6\n5\n0(Ω)3/parenrightbig\n,\nwhich completes the proof. /squareLOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 29\nLet us now consider the case d= 2. Our task is to prove that ˆv∈L6(0,T;L4(Ω)2).\nLemma 5.7. Letd= 2. Then, for each k3∈L6\n5(0,T;L4\n3(Ω)2), there exists a unique solution\n(v,q)to the Stokes system (5.4)satisfying\nv∈C([0,T];H)∩L2(0,T;V).\nLet us assume that Lemma 5.7holds. Then, ˆvmust coincide with the solution by trans-\nposition of the problem ( 5.3), namely, the unique function ˆv∈L6(0,T;L4(Ω)2) verifying\n/integraldisplay\nQˆv·k3dxdt=/integraldisplay\nΩe−3s\n4ˆψ(0)v0·v(0)dx+/integraldisplayT\n0/a\\}b∇acketle{tF∗,v/a\\}b∇acket∇i}htH−1(Ω)2,H1\n0(Ω)2dt,\nfor allk3∈L6\n5(0,T;L4\n3(Ω)2) (cf. Remark 5.3). Here, the function F∗is\nF∗=∇·ˆg+ˆh+ˆϑ1ω−3s\n4∂ˆψ\n∂tˆv\n−/parenleftbig\n(/tildewideu·∇)ˆv+(ˆv·∇)/tildewideu+αˆv+β|/tildewideu|2ˆv+2β(ˆv·/tildewideu)/tildewideu/parenrightbig\n, (5.7)\nand (v,q) is the solution of the Stokes system ( 5.4) associated to k3. By using the sim-\nilar arguments as Lemma 5.4and by the virtue of Lemma 5.7, we finally obtain ˆv∈\nL6(0,T;L4(Ω)2).\nProof of Lemma 5.7.Let us first deduce that v∈L2(0,T;H1\n0(Ω)2). From the Giga-Sohr\nregularity result for the Stokes problem (see Remark 5.3), we have that\nv∈L6\n5(0,T;W2,4\n3(Ω)2∩H1\n0(Ω)2) and∂v\n∂t∈L6\n5(0,T;L4\n3(Ω)2).\nWe also have\nv∈L6\n5(0,T;W2,4\n3(Ω)2)∩L∞(0,T;L4\n3(Ω)2).\nThe interpolation results gives\nv∈L2/parenleftbig\n0,T;(W2,4\n3(Ω)2,L4\n3(Ω)2)1\n4,4\n3/parenrightbig\n.\nBut we know that (see [ 44, Chapter 34])\n/parenleftbig\nW2,4\n3(Ω)2,L4\n3(Ω)3/parenrightbig\n1\n4,4\n3= W3\n2,4\n3(Ω)2.\nThe Sobolev embedding theorem W3\n2,4\n3(Ω)2֒→H1(Ω)2andv=0on the boundary Σ in the\nsense of trace results to\nv∈L2/parenleftbig\n0,T;H1\n0(Ω)2/parenrightbig\n.\nOn the other hand, the Sobolev embedding yields\nW2,4\n3(Ω)2֒→H1(Ω)2֒→L4(Ω)2.\nBy the virtue of interpolation results, we have that, if\nv∈L6\n5(0,T;L4(Ω)2) and∂v\n∂t∈L6\n5(0,T;L4\n3(Ω)2),\nthen\nv∈C/parenleftbig\n[0,T];/parenleftbig\nL4\n3(Ω)2,L4(Ω)2/parenrightbig\n1\n2,2/parenrightbig\n.30 P. KUMAR AND M. T. MOHAN\nFurthermore, thanks to the Lorentz space (see [ 5, Theorem 5.2.1])\n/parenleftbig\nL4\n3(Ω)2,L4(Ω)2/parenrightbig\n1\n2,2= L2,2(Ω)2= L2(Ω)2,\nand we deduce that v∈C([0,T];H) and completes the proof. /square\nRemark 5.8. In light of the preceding regularity results, it is evident t hat the solution (v,ϑ∗)\nto the controllability problem (5.1)belongs to the space E.\n5.3.Local exact controllability to trajectories. In this subsection, we prove our main result\nof local exact controllability to trajectories of the CBF equations ( 1.1) (Theorem 1.1) by\nusing the inverse mapping theorem (or epimorphism theorem) given b elow. In order to\nachieve this, we establish the null controllability of the nonlinear syst em (1.6) which leads\nto the require result. We mainly follow the ideas outlined in [ 8,41] and references therein.\nTheorem 5.9 (Inverse mapping theorem) .LetEandGbe two Banach spaces and let A:E →\nGsatisfyA ∈C1(E;G). Assume that e0∈ E,A(e0) =b0, andA′(e0) :E → Gis surjective.\nThen, there exists δ >0such that, for every b∈ Gsatisfying /ba∇dblb−b0/ba∇dblG<δ, there exists a\nsolution of the equation\nA(e) =b,e∈ E.\nThis theorem can be viewed as a simple corollary of the generalized implic it function\ntheorem, as demonstrated in the work of V. Alekseev, V. Tikhomiro v, and S. Fomin [ 1].\nLet us define the operator A:E → Gby\nA(v,ϑ) =/parenleftbig\nLv−N(v)+∇q−ϑ1ω,v(0)/parenrightbig\n,for all (v,ϑ)∈ E, (5.8)\nwhere the operators LvandN(v) are defined in ( 1.7) and (1.8), respectively and E=Ed\nandG=Gdford= 2,3.\nThe next lemma provides the continuity of some multilinear forms which will be required\nto apply Theorem 5.9.\nLemma 5.10. Assume that/tildewideu∈L∞(Q)d. Let us consider the bilinear and trilinear forms\nNBi:E ×E → W andNTri:E ×E ×E → W , respectively, given by\nNBi(v1,v2) =∇·(v1⊗v2)+β(v1·v2)/tildewideu+2β(v1·/tildewideu)v2, (5.9)\nNTri(v1,v2,v3) =β(v1·v2)v3, (5.10)\nwhere\nW=/braceleftbigg\nL2(e−sψ(0,T);H−1(Ω)2),ifd= 2,\nL2(e−sψ(0,T);W−1,6(Ω)3),ifd= 3.\nThen, there exists C >0such that\n/ba∇dblNBi(v1,v2)/ba∇dblW≤C/ba∇dblv1/ba∇dblE/ba∇dblv2/ba∇dblE, (5.11)\n/ba∇dblNTri(v1,v2,v3)/ba∇dblW≤C/ba∇dblv1/ba∇dblE/ba∇dblv2/ba∇dblE/ba∇dblv3/ba∇dblE. (5.12)\nProof.Let us first prove the continuity of the bilinear form NBi. In fact, for d= 2, we can\nusee−3s\n4ˆψv∈L4(Q)2for any (v,ϑ)∈ Eand obtain from ( 5.9) that\n/ba∇dblNBi(v1,v2)/ba∇dblW≤ /ba∇dbl∇·(v1⊗v2)/ba∇dblL2(e−sψ(0,T);H−1(Ω)2)+/ba∇dblβ(v1·v2)/tildewideu/ba∇dblL2(e−sψ(0,T);H−1(Ω)2)\n+/ba∇dbl2β(v1·/tildewideu)v2/ba∇dblL2(e−sψ(0,T);H−1(Ω)2)\n≤C/ba∇dblv1⊗v2/ba∇dblL2(e−sψ(0,T);H)+C/ba∇dbl(v1·v2)/tildewideu/ba∇dblL2(e−sψ(0,T);H)LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 31\n+C/ba∇dbl(v1·/tildewideu)v2/ba∇dblL2(e−sψ(0,T);H)\n≤C/ba∇dble−s\n2ψv1/ba∇dblL4(Q)2/ba∇dble−s\n2ψv2/ba∇dblL4(Q)2\n+C/ba∇dbl/tildewideu/ba∇dblL∞(Q)2/ba∇dble−s\n2ψv1/ba∇dblL4(Q)2/ba∇dble−s\n2ψv2/ba∇dblL4(Q)2.\nFrom Lemma 5.1(cf. [41, Lemma 4.1]), we also have\n−ψ≤ −ˇψ≤ −3\n2ˆψ. (5.13)\nThus, we infer\n/ba∇dblNBi(v1,v2)/ba∇dblW≤C/ba∇dble−3s\n4ˆψv1/ba∇dblL4(Q)2/ba∇dble−3s\n4ˆψv2/ba∇dblL4(Q)2\n+C/ba∇dbl/tildewideu/ba∇dblL∞(Q)2/ba∇dble−3s\n4ˆψv1/ba∇dblL4(Q)2/ba∇dble−3s\n4ˆψv2/ba∇dblL4(Q)2,\nwhich leads to ( 5.11).\nOn the other hand, for d= 3, we get\n/ba∇dblNBi(v1,v2)/ba∇dblW\n≤ /ba∇dbl∇·(v1⊗v2)/ba∇dblL2(e−sψ(0,T);W−1,6(Ω)3)+/vextenddouble/vextenddoubleβ(v1·v2)/tildewideu/vextenddouble/vextenddouble\nL2(e−sψ(0,T);W−1,6(Ω)3)\n+/vextenddouble/vextenddouble2β(v1·/tildewideu)v2/vextenddouble/vextenddouble\nL2(e−sψ(0,T);W−1,6(Ω)3)\n≤ /ba∇dblv1⊗v2/ba∇dblL2(e−sψ(0,T);L6(Ω)3)+C/vextenddouble/vextenddouble(−∆)−1\n2(v1·v2)/tildewideu/vextenddouble/vextenddouble\nL2(e−sψ(0,T);L6(Ω)3)\n+C/vextenddouble/vextenddouble(−∆)−1\n2(v1·/tildewideu)v2/vextenddouble/vextenddouble\nL2(e−sψ(0,T);L6(Ω)3)\n≤C/ba∇dblv1⊗v2/ba∇dblL2(e−sψ(0,T);L6(Ω)3)+C/vextenddouble/vextenddouble∇(−∆)−1\n2(v1·v2)/tildewideu/vextenddouble/vextenddouble\nL2(e−sψ(0,T);H)\n+C/vextenddouble/vextenddouble∇(−∆)−1\n2(v1·/tildewideu)v2/vextenddouble/vextenddouble\nL2(e−sψ(0,T);H)\n≤C/ba∇dble−s\n2ψv1/ba∇dblL4(0,T;L12(Ω)3)/ba∇dble−s\n2ψv2/ba∇dblL4(0,T;L12(Ω)3)\n+C/ba∇dbl/tildewideu/ba∇dblL∞(Q)3/ba∇dble−s\n2ψv1/ba∇dblL4(Q)3/ba∇dble−s\n2ψv2/ba∇dblL4(Q)3\n≤C/ba∇dble−s\n2ψv1/ba∇dblL4(0,T;L12(Ω)3)/ba∇dble−s\n2ψv2/ba∇dblL4(0,T;L12(Ω)3)\n+C/ba∇dbl/tildewideu/ba∇dblL∞(Q)3/ba∇dble−s\n2ψv1/ba∇dblL4(0,T;L12(Ω)3)/ba∇dble−s\n2ψv2/ba∇dblL4(0,T;L12(Ω)3).\nBy the virtue of the inequality ( 5.13), we immediately reach at ( 5.11).\nNow, we proceed to prove the continuity of the trilinear form NTri. Ford= 3, from ( 5.10),\nwe deduce that\n/ba∇dblNTri(v1,v2,v3)/ba∇dblW\n=/ba∇dblβ(v1·v2)v3/ba∇dblL2(e−sψ(0,T);W−1,6(Ω)3)≤C/vextenddouble/vextenddouble(−∆)−1\n2(v1·v2)v3/vextenddouble/vextenddouble\nL2(e−sψ(0,T);L6(Ω)3)\n≤C/vextenddouble/vextenddouble∇(−∆)−1\n2(v1·v2)v3/vextenddouble/vextenddouble\nL2(e−sψ(0,T);H)≤C/ba∇dble−sψ(v1·v2)v3/ba∇dblL2(0,T;H)\n≤C/ba∇dble−s\n3ψv1/ba∇dblL6(Q)3/ba∇dble−s\n3ψv2/ba∇dblL6(Q)3/ba∇dble−s\n3ψv3/ba∇dblL6(Q)3\n≤C/ba∇dble−s\n2ˆψv1/ba∇dblL6(Q)3/ba∇dble−s\n2ˆψv2/ba∇dblL6(Q)3/ba∇dble−s\n2ˆψv3/ba∇dblL6(Q)3\n≤C/ba∇dbles\n4ˆψe−3s\n4ˆψv1/ba∇dblL6(Q)3/ba∇dbles\n4ˆψe−3s\n4ˆψv2/ba∇dblL6(Q)3/ba∇dbles\n4ˆψe−3s\n4ˆψv3/ba∇dblL6(Q)3,32 P. KUMAR AND M. T. MOHAN\nand (5.12) follows by using ( 5.13). Similarly, for d= 2, we have\n/ba∇dblNTri(v1,v2,v3)/ba∇dbl2\nW=/integraldisplayT\n0/ba∇dble−sψβ(v1·v2)v3/ba∇dbl2\nH−1(Ω)2dt≤C/integraldisplayT\n0/ba∇dble−sψ(v1·v2)v3/ba∇dbl2\nL12\n11(Ω)2dt\n≤C/integraldisplayT\n0/ba∇dble−s\n3ψv1/ba∇dbl2\nL3(Ω)2/ba∇dble−s\n3ψv2/ba∇dbl2\nL3(Ω)2/ba∇dble−s\n3ψv3/ba∇dbl2\nL4(Ω)2dt\n≤Csup\nt∈[0,T]/parenleftBig\n/ba∇dble−s\n3ψv1/ba∇dbl4\n3\nH/ba∇dble−s\n3ψv2/ba∇dbl4\n3\nH/parenrightBig/parenleftbigg/integraldisplayT\n0/ba∇dble−s\n3ψ∇v1/ba∇dbl2\n3\nH/ba∇dble−s\n3ψ∇v2/ba∇dbl2\n3\nH/ba∇dble−s\n3ψv3/ba∇dbl2\nL4(Ω)2dt/parenrightbigg\n≤Csup\nt∈[0,T]/parenleftBig\n/ba∇dble−s\n3ψv1/ba∇dbl4\n3\nH/ba∇dble−s\n3ψv2/ba∇dbl4\n3\nH/parenrightBig/parenleftbigg/integraldisplayT\n0/ba∇dble−s\n3ψ∇v1/ba∇dbl2\nHdt/parenrightbigg1\n3\n×/parenleftbigg/integraldisplayT\n0/ba∇dble−s\n3ψ∇v2/ba∇dbl2\nHdt/parenrightbigg1\n3/parenleftbigg/integraldisplayT\n0/ba∇dble−s\n3ψv3/ba∇dbl6\nL4dt/parenrightbigg1\n3\n≤Csup\nt∈[0,T]/parenleftBig\n/ba∇dble−s\n2ˆψv1/ba∇dbl4\n3\nH/ba∇dble−s\n2ˆψv2/ba∇dbl4\n3\nH/parenrightBig\n/ba∇dble−s\n2ˆψv1/ba∇dbl2\n3\nL2(0,T;V)\n×/ba∇dble−s\n2ˆψv2/ba∇dbl2\n3\nL2(0,T;V)/ba∇dble−s\n2ˆψv3/ba∇dbl2\nL6(0,T;L4(Ω)2)\n≤Csup\nt∈[0,T]/parenleftBig\n/ba∇dbles\n4ˆψe−3s\n4ˆψv1/ba∇dbl4\n3\nH/ba∇dbles\n4ˆψe−3s\n4ˆψv2/ba∇dbl4\n3\nH/parenrightBig\n/ba∇dbles\n4ˆψe−3s\n4ˆψv1/ba∇dbl2\n3\nL2(0,T;V)\n×/ba∇dbles\n4ˆψe−3s\n4ˆψv2/ba∇dbl2\n3\nL2(0,T;V)/ba∇dbles\n4ˆψe−3s\n4ˆψv3/ba∇dbl2\nL6(0,T;L4(Ω)2),\nwhere we have used the embedding H1(Ω)2֒→L12\nσ(Ω)2֒→L12\n11σ(Ω)2֒→H−1(Ω)2, Gagliardo-\nNirenberg’s and H ¨older’s inequalities. Using ( 5.13), we deduce ( 5.12), and the proof of\nLemma5.10is completed. /square\nLemma 5.11. Let us assume that /tildewideu∈L∞(Q)d. Then,A ∈C1(E;G).\nProof.From (5.9) and (5.10), we can see that the operator N(v) defined in ( 1.8) can be\nwritten as\nN(v) =NBi(v,v)+NTri(v,v,v).\nThus, (5.11) and (5.12) imply that\n/ba∇dblN(v)/ba∇dblG≤C(/ba∇dblv/ba∇dbl2\nE+/ba∇dblv/ba∇dbl3\nE).\nTherefore, we infer that A:E → Gdefined in ( 5.8) is well-defined.\nThe first component of the operator A(v,ϑ) is linear (and consequently C1), except N(v).\nHowever, the bilinear and trilinear forms given by ( 5.9) and (5.10), respectively, are contin-\nuous (cf. Lemma 5.10) and consequently, we deduce that Ais a C1map from EtoG, which\nproves the lemma. /square\nAs a consequence of this lemma, we can prove our main result (Theor em1.1). For the\nproof, we follow the strategy introduced in [ 8,35,41], etc.\nProof of Theorem 1.1.Let us take e0= (0,0)∈ Eandb0= (0,0)∈ G. It follows that\nA′(e0) :E → Gis given by\nA′(e0)[v∗,ϑ∗] =/parenleftbig\nLv∗+∇q∗−ϑ∗1ω,v∗(0)/parenrightbig\n,for all (v∗,ϑ∗)∈ E,LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 33\nandA′(e0) is a surjective linear map from EtoG, as implied by the null controllability result\nfor the linearized system ( 5.1) (see also Remark 5.8), or equivalently, Im( A′(e0)) =G. An\napplication of the inverse mapping theorem (Theorem 5.9) gives exactly the conclusions of\nTheorem 1.1if we take b= (0,v0)∈ G, that is, it gives the existence of δ >0 such that, if\n/ba∇dblv0/ba∇dblL2d−2(Ω)d≤δ,\nthen we can find a control ϑsuch that the associated solution to ( 1.6) satisfies/parenleftbig\nv,ϑ/parenrightbig\n∈ E\nand\nv(·,T) =0,in Ω,\nand we reach at the desired result. Thus the proof of Theorem 1.1is completed. /square\nRemark 5.12. We could have taken in the equation for /tildewideuan external force /tildewidefdifferent from\nf(cf.(1.2)). We can set\nf=/tildewidef+f∗,\nwhere/tildewideξ−3\n2e−s/tildewideψ/tildewidefis sufficiently small and finite, and obtain the local exact con trollability to\ntrajectories of the CBF equations (1.1).\nAcknowledgments: P. Kumar and M. T. Mohan would like to thank the Department of\nScience and Technology (DST), India for Innovation in Science Purs uit for Inspired Research\n(INSPIRE) Faculty Award (IFA17-MA110).\nReferences\n[1] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Optimal control , Contemporary Soviet Mathematics,\nConsultants Bureau, New york, 1987.\n[2] E.V. Amosova, Exact local controllability for the equations of visc ous gas dynamics, Differ. Equ. ,47\n(2011), 1776–1795.\n[3] M.Badra,S.ErvedozaandS.Guerrero,Localcontrollabilityto trajectoriesfornon-homogeneousincom-\npressible Navier-Stokes equations, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire ,33(2016), 529–574.\n[4] A. Bejan and D. Nield, Convection in porous media , Berlin: Springer, 2006.\n[5] J. Bergh and J. L ¨ofstr¨om,Interpolation spaces , Springere Verlag, Newe, 1976.\n[6] M.G. Burgos, S. Guerreroand J.-P.Puel, Localexact controllab ilityto the trajectoriesofthe Boussinesq\nsystemviaafictitiouscontrolonthedivergenceequation, Commun. Pure Appl. Anal. ,8(2009),311–333.\n[7] E. Fern´ andez-Cara, On the approximate and null controllability of the Navier-Stokes equations, SIAM\nRev.,41(1999), 269–277.\n[8] E. Fern´ andez-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Pu el, Local exact controllability to the\ntrajectories of the Navier-Stokes equations, J. Math. Pures Appl. ,83(2004), 1501–1542.\n[9] E.Fern´ andez-CaraandS. Guerrero,GlobalCarlemaninequalit iesforparabolicsystemsandapplications\nto controllability, SIAM J. Control Optim. ,45(2006), 1399–1446.\n[10] E. Fern´ andez-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. P uel, Some controllability results for the N-\ndimensional Navier-Stokesand Boussinesq systems with N−1 scalar controls, SIAM J. Control Optim. ,\n45(2006), 146–173.\n[11] N. Carre˜ no and S. Guerrero, Local null controllability of the N-dimensional Navier-Stokes system with\nN−1 scalar controls in an arbitrary control domain, J. Math. Fluid Mech. ,15(2013), 139–153.\n[12] E.CerpaandA. Mercado,Localexactcontrollabilityto thetra jectoriesofthe1DKuramoto-Sivashinsky\nequation, J. Differential Equations ,250(2011), 2024–2044.\n[13] J.-M. Coron, On the controllability of the 2-D incompressible Navie r-Stokes equations with the Navier\nslip boundary conditions, ESAIM Control Optim. Calc. Var. ,1(1996), 35–75.\n[14] J.-M.Coron, On the controllabilityof 2-Dincompressibleperfect fluids,J. Math. Pures Appl. ,75(1996),\n155–188.34 P. KUMAR AND M. T. MOHAN\n[15] J.-M. Coron and A.V. Fursikov, Global exact controllability of the 2-D Navier-Stokes equations on\nmanifold without boundary, J. Russian Math. Phys. ,4(1996), 1–20.\n[16] L.C. Evans, Partial Differential Equations , Grad. Stud. Math., vol. 19, Second Edition, Amer. Math.\nSoc., Providence, RI, 2010.\n[17] C. Fabre, Uniqueness results for Stokes equations and their c onsequences in linear and nonlinear control\nproblems, ESAIM Control Optim. Calc. Var. ,1(1996), 267–302.\n[18] C.L. Fefferman, K.W. Hajduk and J.C. Robinson, Simultaneousapp roximationin Lebesgue and Sobolev\nnorms via eigenspaces, Proc. Lond. Math. Soc. (3) ,125(4) (2022), 759–777.\n[19] A.V. Fursikov, Exact boundary zero controllability of three-dim ensional Navier-Stokes equations, J.\nDyn. Control Syst. ,1(1995), 325–350.\n[20] A.V. Fursikov and O.Y. Imanuvilov, On approximate controllability o f the Stokes system, Ann. Fac.\nSci. Toulouse Math. (6) ,2(1993), 205–232.\n[21] A.V. Fursikov and O.Y. Imanuvilov, On exact boundary zero-con trolability of two-dimensional Navier-\nStokes equations, Acta Appl. Math. ,37(1994), 67–76.\n[22] A.V. Fursikov and O.Y. Imanuvilov, Controllability of Evolution Equations , Lecture Note Series 34,\nResearch Institute of Mathematics, Seoul National University, S eoul, 1996.\n[23] A.V. Fursikov and O.Y. Imanuvilov, Local exact controllability of t he two-dimensional Navier-Stokes\nequations, Sb. Math. , 1996.\n[24] A.V. Fursikov and O.Y. Imanuvilov, Local exact boundary contr ollability of the Boussinesq equation,\nSIAM J. Control Optim. ,36(1998), 391–421.\n[25] A.V.FursikovandO.Y.Imanuvilov,ExactcontrollabilityoftheNav ier-StokesandBoussinesqequations,\nRussian Math. Surveys ,54(1999), 565–618.\n[26] P. Gao, Local exact controllability to the trajectories of the S wift-Hohenberg equation, Nonlinear Anal. ,\n139(2016), 169–195.\n[27] Y. Giga and H. Sohr, Abstract Lpestimates for the Cauchy problem with applications to the Navier-\nStokes equations in exterior domains, J. Funct. Anal. ,102(1991), 72–94.\n[28] S. Guerrero, Local exact controllability to the trajectories o f the Boussinesq system, Ann. Inst. H.\nPoincar´ e C Anal. Non Lin´ eaire ,23(2006), 29–61.\n[29] S. Guerrero, Local exact controllability to the trajectories o f the Navier-Stokes system with nonlinear\nNavier-slip boundary conditions, ESAIM Control Optim. Calc. Var. ,12(2006), 484–544.\n[30] S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, A result concerning the global approximate controllability\nof the Navier-Stokes system in dimension 3, J. Math. Pures Appl. ,98(2012), 689–709.\n[31] P. Guzm´ an, Local exact controllability to the trajectories of the Cahn-Hilliard equation, Appl. Math.\nOptim.,82(2020), 279–306.\n[32] O.Y. Imanuvilov, Local exact controllability for the 2D Navier-St okes equations with the Navier slip\nboundary conditions, Turbulence Modeling and Vortex Dynamics, Lecture Notes in Physics, 491(1997),\n148–168.\n[33] O.Y. Imanuvilov, Local exact controllability for the 2-D Boussine sq equations with the Navier slip\nboundary conditions, ESAIM: Proceedings , 1998, 153–170.\n[34] O.Y. Imanuvilov, On exactcontrollabilityfor the Navier-Stokese quations, ESAIM Control Optim. Calc.\nVar.,3(1998), 97–131.\n[35] O.Y. Imanuvilov, Remarks on exact controllability for the Navier- Stokes equations, ESAIM Control\nOptim. Calc. Var. ,6(2001), 39–72.\n[36] O.Y. Imanuvilov and J.-P. Puel, Global Carlemanestimates for wea k elliptic non homogeneousDirichlet\nproblem, Int. Math. Res. Not. IMRN ,16(2003), 883–913.\n[37] O.Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimate s for parabolic equations with non\nhomogeneous boundary conditions, Chin. Ann. Math. Ser. B ,30(2009), 333–378.\n[38] O.Y. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimate s for second order non homogeneous\nparabolic equations, To appear.\n[39] M.T. Mohan, On the convective Brinkman-Forchheimer equation s,Submitted .\n[40] M.T. Mohan, Stochastic convective Brinkman-Forchheimer equ ations, Submitted ,\nhttps://arxiv.org/pdf/2007.09376.pdf .\n[41] J.-P. Puel, Optimization with PDE Constraints , Lecture Notes in Computational Science and Engineer-\ning101, Springer, 2014.LOCAL EXACT CONTROLLABILITY OF THE CBF EQUATIONS 35\n[42] A. Shirikyan, Approximate controllability of three-dimensional N avier-Stokes equations, Commun.\nMath. Phys. ,266(2006), 123–151.\n[43] V.A. Solonnikov, Estimates of the solution of a certain initial-boun dary value problem for a linear\nnonstationary system of Navier-Stokes equations, J. Soviet Math. ,8(1977), 467–523.\n[44] L. Tarter, An Introduction to Sobolev Spaces and Interpolation Spaces , Springer, Berlin/Heidelberg/New\nYork, 2007.\n[45] R. Triggiani, A note on the lack of exact controllability for mild solut ions in Banach spaces, SIAM J.\nControl Optim. ,15(3) (1977), 407–411.\n[46] R.Temam, Navier-Stokes Equations, Theory and Numerical Analysis , North-Holland,Amsterdam,1984."    },    {        "title": "2402.06343v1.Blackbody_heat_capacity_at_constant_pressure.pdf",        "content": "arXiv:2402.06343v1  [cond-mat.stat-mech]  9 Feb 2024Blackbody heat capacity at constant pressure\nE. S. Moreira Jr.1\nInstituto de Matem´ atica e Computa¸ c˜ ao, Universidade Feder al de Itajub´ a,\nItajub´ a, Minas Gerais 37500-903, Brazil\nFebruary, 2024\nAbstract\nAt first glance, the title of this work seems to be improper. And the r eason is well known. Since\nblackbody pressure depends only on temperature, one cannot ta ke the derivative of the thermodynamic\nquantities with respect to oneof them, keeping the other constan t. That is, the heat capacityat constant\npressure, CP, as well as, the coefficient of thermal expansion, α, and the isothermal compressibility, κT,\nare ill-defined quantities. This work will show that when the perfect c onducting nature of the walls of a\nblackbody cavity is taken into account, CP,αandκTare in fact well defined, and they are related by\nthe usual thermodynamic relations, as expected. Two geometries will be considered, namely, a spherical\nshell and a cubic box. It will be shown that CP,αandκTdepend very much on the geometry of the\ncavity. Issues regarding thermodynamic stability will be addressed , revealing that they also depend on\nthe cavity’s geometry. It is argued that these findings may be amen able to experimental verification.\n1 Introduction\nBlackbody radiation is present in most branches of physics. Theref ore, a clear understanding of its\nthermodynamics is vital for the comprehension of many phenomena . There are many ways to obtain the\nHelmholtz free energy Fof the electromagnetic radiation, at temperature T, in an arbitrary large cavity\nof volume V. The most familiar ways are those in statistical mechanics textbook s (see, e.g., [1, 2, 3]),\nwhich lead to2\nF=−π2\n45(kT)4\n(¯hc)3V, (1)\nyielding the entropy S=−(∂TF)Vand the pressure P=−(∂VF)T,\nS=4π2\n45/parenleftbiggkT\n¯hc/parenrightbigg3\nkV, P =π2\n45(kT)4\n(¯hc)3, (2)\nas well as the internal energy U=∂β(βF) and the blackbody heat capacity at constant volume C=\n(∂TU)V,\nU=π2\n15(kT)4\n(¯hc)3V, C =4π2\n15k/parenleftbiggkT\n¯hc/parenrightbigg3\nV. (3)\nThe familiar thermodynamic relations PV=U/3 andC= 3Sare easily verified.\nInsteadofusing Ftoobtain the blackbodythermodynamics, onecoulduse theGibbs th ermodynamic\npotential G=F+PVand the equations\nS=−/parenleftbigg∂G\n∂T/parenrightbigg\nP, V =/parenleftbigg∂G\n∂P/parenrightbigg\nT. (4)\nHowever, Gvanishes identically, and therefore equation (4) is not available here .\n1E-mail: moreira@unifei.edu.br\n2Fundamental constants are the usual ones, and β:= 1/kT.\n1Other issues arise in the calculation of the heat capacity at constan t pressure\nCP=/parenleftbigg∂H\n∂T/parenrightbigg\nP, (5)\nwhereH=U+PVis the enthalpy, or in the calculations of the coefficient of thermal ex pansionαand\nthe isothermal compressibility κT, which are given by the definitions\nα:=1\nV/parenleftbigg∂V\n∂T/parenrightbigg\nP, κ T:=−1\nV/parenleftbigg∂V\n∂P/parenrightbigg\nT. (6)\nThe trouble is that, in blackbody thermodynamics, PandTare not independent parameters [cf. equa-\ntion (2)], and thus the quantities in equations (5) and (6) are ill-defin ed. As mentioned in [4], one\nmay try to come to terms with these difficulties by arguing that CP,αandκTare, in fact, infinite.\nNevertheless, this argument does not seem to be satisfactory, a nd there remain other issues [5].\nIn fact, equation (1) should be rewritten as\nF=−π2\n45(kT)4\n(¯hc)3V+···, (7)\nwhere dots denote vacuum polarization corrections due to the wall’s cavity (see, e.g., [6]). Usually, such\ncorrections are omitted in calculations in statistical mechanics, sinc e they are small at the regime of high\ntemperatures and/or large cavities, i.e., when\nkTV1/3\n¯hc≫1. (8)\nHowever, they do carry the key ingredient which makes CP,αandκTwell defined thermodynamic\nquantities in the context of blackbody radiation, as this work intend s to show.\nAt this point, it should be remarked that, at the opposite regime of lo w temperatures and/or small\ncavities (i.e., when kTV1/3≪¯hc), equation (7) gives place to F=U0+···where the temperature\nindependent term U0is the vacuumenergyin the cavity[6], andnowthe dots denotether mal corrections.\nThus, the Casimir energy3U0[9, 10] and equation (7) are, in fact, two facets of the same pheno menon\n[11, 12, 13, 14, 15, 16, 17, 18].\nThe text will be organized as follows. In sections 2 and 3, respective ly, the correctionsin equation (7)\nwill be used to calculate well defined CP,αandκTfor the blackbody radiation in a perfect conducting\nspherical shell, and in a perfect conducting cubic box. In section 4, the stability of blackbody radiation\nwill be determined in each of these cavities. In section 5, a summary a nd further analysis concerned the\nresults will be given.\n2 Spherical shell\nFor a perfect conducting spherical shell carrying hot electromag netic radiation, equation (7) becomes\n[15]\nF=−π2\n45(kT)4\n(¯hc)3V−kT\n4/bracketleftbigg\nln/parenleftbiggkTV1/3\n¯hc/parenrightbigg\n+0.291/bracketrightbigg\n, (9)\nwhere smaller corrections have been neglected. Accordingly, equa tion (2) is replaced by\nS=4π2\n45/parenleftbiggkT\n¯hc/parenrightbigg3\nkV+k\n4/bracketleftbigg\nln/parenleftbiggkTV1/3\n¯hc/parenrightbigg\n+1.291/bracketrightbigg\n, P =π2\n45(kT)4\n(¯hc)3+kT\n12V, (10)\nand equation (3) is replaced by\nU=π2\n15(kT)4\n(¯hc)3V+kT\n4, C V=C+k\n4. (11)\n3For reviews on the Casimir effect, see [7, 8].\n2Therefore, CVis essentially the blackbody heat capacity at constant volume in equa tion (3) up to a\nsmall correction. Note also that the equation of state PV=U/3 still holds, but CV= 3Sholds only\napproximately, as a quick check shows.\nAn interesting fact is that G=F+PVno longer vanishes identically. Instead, by using equations\n(9) and (10), it follows that\nG=−kT\n4/bracketleftbigg\nln/parenleftbiggkTV1/3\n¯hc/parenrightbigg\n−0.042/bracketrightbigg\n, (12)\nwhich, by noticing Pin equation (10), can be rewritten as a function of TandP. Then, it is a simple\nmatter of calculation to show that equation (4) leads correctly to t he expressions in equation (10).\nAsPnow is a function of TandV[cf. equation (10)], equations (5) and (6) can be applied. Enthalpy\nH= 4U/3, when expressed in terms of TandP, becomes\nH=π2\n135(kT)5\n(¯hc)3/bracketleftbigg\nP−π2\n45(kT)4\n(¯hc)3/bracketrightbigg−1\n+kT\n3. (13)\nThen equation (5) yields a well defined heat capacity at constant pr essure, namely\nCP=4\n3C2\nk/bracketleftBigg\n1+75\n(4π)2V/parenleftbigg¯hc\nkT/parenrightbigg3/bracketrightBigg\n+k\n3, (14)\nwhereCis the blackbody heat capacity at constant volume in equation (3). C learly, up to small\ncorrections, CP= 4C2/3k. Noticing equations (8) and (11), one sees that CP/CV∼(kT/¯hc)3V,\nresulting that [see equation (11)]\nCP≫CV≈C. (15)\nIt is worth remarking that water volumetric heat capacity is 4 .2×106J/m3K, which corresponds to\n1014C, whenV= 1m3andT= 300K, and it is comparable to 10−2CPas can be promptly verified.\nRecalling that Vcan be written as a function of TandPby means of equation (10), rather lengthy\nbut straightforward calculations lead to [see equation (6)]\nα=16π2\n15k(kT)2\n(¯hc)3V+1\nT, κ T=12V\nkT, (16)\nwhich are also well defined quantities4. At this point a nice test of consistency is available. Namely, by\nnoting equations (11), (14) and (16), it follows that\nCP−CV=TVα2\nκT, (17)\nwhich is a well known identity in thermodynamics [2, 3].\nPhysical interpretation of the expressions in equation (16) is near ly self-evident in the light of the\ndefinitions in equation (6). For example, as αin equation (16) is positive, it means that, keeping P\nconstant, the spherical shell expands as it warms. The positivene ss ofκTin equation (16) will be\naddressed later, in section 4.\nBeforeturning to the cubic box, it is worth calculatingone morequan tity in the sphericalshell cavity,\nsince it gives another opportunity to check consistency of the res ults. It happens to be the adiabatic\ncompressibility κS, and it is defined as κTin equation (6), now replacing TbyS. One sets dS= 0 in\nequation (10) and writes dTin terms of dV. Then, by considering dPin equation (10), it results that\nκS=135\n4π2(¯hc)3\n(kT)4/bracketleftBigg\n1+15\n4π2V/parenleftbigg¯hc\nkT/parenrightbigg3/bracketrightBigg−1\n, (18)\n4One may feel inclined to call κTin equation (16) a “classical” quantity, since it does not co ntain ¯h. However, that is not\nappropriate. The expression for κTin equation (16) is just the ¯ h0-term of an expansion in powers of ¯ h.\n3where the factor out of the right brackets is κS= 3/4Pfrom the usual blackbody thermodynamics.\nNow, by noticing equations (11), (14), (16) and (18), it follows ano ther thermodynamic identity, namely\n[2, 3]:\nCV\nCP=κS\nκT, (19)\nas expected.\n3 Cubic box\nNow, for a perfect conducting cubic box carryinghot electromagn eticradiation, the procedures to obtain\nthermodynamics are as those followed in the previous section. Howe ver, instead of the Helmholtz free\nenergy given in equation (9), one considers [14, 18]\nF=−π2\n45(kT)4\n(¯hc)3V+π\n4(kT)2\n¯hcV1/3, (20)\nwhere smaller corrections have been omitted again. It should be str essed that, due to the different\ngeometriesofthe cavities,equations(9) and(20) differonlybythe ircorrectionstothe familiarblackbody\nHelmholtz free energy [see equations (7) and (8)]. Nevertheless, s uch a little difference will lead to\ncontrasting results.\nIt follows from equation (20) that [cf. equations (10) and (11)]\nS=4π2\n45/parenleftbiggkT\n¯hc/parenrightbigg3\nkV−π\n2k2T\n¯hcV1/3, P =π2\n45(kT)4\n(¯hc)3−π\n12V2/3(kT)2\n¯hc, (21)\nand\nU=π2\n15(kT)4\n(¯hc)3V−π\n4(kT)2\n¯hcV1/3, C V=C−π\n2k2T\n¯hcV1/3. (22)\nOnce more, PV=U/3 still holds, whereas CV= 3Sholds only approximately. Here, it is worth pointing\nout that for a fixed T, as the volume Vof the cavity increases, Pin equation (10) decreases, but Pin\nequation (21) increases. It will be seen shortly that this fact is rele vant in the study of thermodynamic\nstability.\nThe Gibbs thermodynamic potential is also nonvanishing, i.e.,\nG=π\n6(kT)2\n¯hcV1/3, (23)\nand it differs considerably from that in equation (12) for the spheric al shell. Again, it is a simple matter\nof calculation to verify that equations (4) and (23) are consistent with equation (21).\nCorresponding to equation (13), now one has for H= 4U/3\nH=/parenleftBigπ\n3/parenrightBig3/2(kT)3\n2(¯hc)3/2P/bracketleftbiggπ2\n45(kT)4\n(¯hc)3−P/bracketrightbigg−3/2\n,\nand equation (5) yields\nCP=−/parenleftbigg32C5\n15πk2/parenrightbigg1/3/bracketleftBigg\n1−45\n8πV2/3/parenleftbigg¯hc\nkT/parenrightbigg2\n+225\n32π2V4/3/parenleftbigg¯hc\nkT/parenrightbigg4/bracketrightBigg\n, (24)\nwhich differs sharply from CPin equation (14), especially because these heat capacities have diffe rent\nsigns. This feature will be addressed again in the context of thermo dynamic stability. Thus, up to small\ncorrections, CP=−(32C5/15πk2)1/3, and noticing equation (22) one sees that the appropriate version\nfor equation (15) is now |CP| ≫CV≈C.\n4The same steps that, in the previous section, have led to equations (16) and (18), now lead to\nα=−8π\n5k2T\n(¯hc)2V2/3+3\nT, κ T=−18\nπ¯hc\n(kT)2V2/3, (25)\nand\nκS=135\n4π2(¯hc)3\n(kT)4/bracketleftBigg\n1−15\n4πV2/3/parenleftbigg¯hc\nkT/parenrightbigg2/bracketrightBigg−1\n.\nNoteworthy is the fact that αandκTin equation (25) are negative, whereas their counterparts for th e\nspherical shell in section 2 are positive [cf. equation (16)]. Regardin gα, equations (6) and (25) say that,\nkeepingPconstant, the cubic box shrinks as it warms, instead of expanding a s the spherical shell. The\nnegativeness of κTin equation (25) will be considered shortly.\nTherefore, α,CPandκTare also well defined quantities in the cubic box and, together with CVand\nκS, they satisfy the usual identities in equations (17) and (19), as sh ould be.\n4 Thermodynamic stability\nA long-lasting physical system must satisfy both the following criter ia for stable thermodynamic equi-\nlibrium [2]:\nCV>0, κ T>0. (26)\nThe first inequality in equation (26) implies thermal stability, whereas the second one implies mechanical\nstability. By examining the results in the previous sections, one sees that blackbody radiation is a\nstable thermodynamic system in the spherical shell; but it is not in the cubic box, since the isothermal\ncompressibility in the latter is negative [see equation (25)]. Note that , for both cavities, CV>0 and\nκS>0, thus the negativiness of CPin equation (24) agrees with equations (17) and (19), as expected .\nIt should be remarked that, by taking into account the definition of κTin equation (6), another way\nof expressing the criterion for mechanical stability is ( ∂VP)T<0, i.e., pressure must diminish as the\nsystem volume increases, with the temperature kept fixed. Althou gh thermodynamic stability requires\nκT>0, there exist indeed substances with negative isothermal compre ssibities [19]. Most likely they\nhave to be monitored closely.\n5 Final remarks\nIn ordinaryblackbodythermodynamics, it isassumedthat radiation pressure Ponthe wallsofthe cavity\ndepends on temperature Talone. That is only part of the story, though, since sub-leading con tributions\nin the expression for Pcarries also dependence on volume V. This fact then allows definition of the\nquantities CP,αandκT, as this work has shown for two perfect conducting cavities, name ly, a spherical\nshell and a cubic box.\nIt is perhaps worth highlighting that, in leading order, CP= 4C2/3kfor the spherical shell, whereas\nfor the cubic box CP=−(32C5/15πk2)1/3, withCdenoting the familiar blackbody heat capacity at\nconstant volume. These radically different expressions for CPmirror the fact that statistical mechanics\nof the normal modes in the cavity are affected by the cavity’s geome try. By using well known criteria, it\nwas shown that blackbody radiation in the spherical shell is thermod ynamic stable, while in the cubic\nbox is not, and that is related with the negative CPabove. Since α,κTandκSare quantities amenable\nto experimental determination, one may wonder whether the findin gs in this work can be experimentally\nvalidated.\nA last remark is in order. Here high temperatures and/or large cavit ies have been addressed [see\nequation (8)]. It would be pertinent to extend the present study t o determine CP,αandκTwhenT\nis low and/or Vis small. Such an investigation has been undertaken in [20] for the geo metry of a slab,\nrevealing unusual and interesting features due to the strong pre sence of vacuum polarization.\n5Acknowledgements – Work partially supported by “Funda¸ c˜ ao de Amparo ` a Pesquisa d o Estado de\nMinas Gerais” (FAPEMIG) and by “Coordena¸ c˜ ao de Aperfei¸ coam ento de Pessoal de N´ ıvel Superior”\n(CAPES).\nReferences\n[1] R. K. Pathria, Statistical Mechanics , Elsevier, UK (1972).\n[2] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics , John Wiley & Sons, U.S.A.\n(1985).\n[3] K. Huang, Statistical Mechanics , John Wiley & Sons, U.S.A. (1987).\n[4] R. E. Kelly, Thermodynamics of blackbody radiation, Am. J. Phys. 49, 714 (1981).\n[5] R. F. Rodr´ ıguez and L. S. Garc´ ıa-Col´ ın, The specific heat puz zle in black-body radiation, Eur. J.\nPhys.10, 214 (1989).\n[6] N. D. Birrel and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press,\nCambridge UK, 1982)\n[7] K. A. Milton, The Casimir Effect, Physical Manifestations of Zero-Point E nergy, World Scientific,\nNew Jersey (2001).\n[8] M. Bordag, U. Mohideen and V. M. Mostepanenko, New developme nts in the Casimir effect, Phys.\nRept.353, 1 (2001).\n[9] H. B. G. Casimir, On the attraction between two perfectly condu cting plates, Proc. K. Ned. Akad.\nWet.51, 793 (1948).\n[10] T. H. Boyer, Quantum electromagnetic zero-point energy of a conducting spherical shell and the\nCasimir model for a charged particle, Phys. Rev. 174, 1764 (1968).\n[11] E. M. Lifshitz, The Theory of Molecular Attractive Forces betw een Solids, Zh. Eksp. Teor. Fiz. 29,\n94 (1955) [Sov. Phys. JETP 2, 73 (1956)].\n[12] J. Mehra, Temperature correction to the Casimir effect, Phys ica37, 145 (1967).\n[13] L. S. Brown and G. J. Maclay, Vacuum Stress between Conduct ing Plates: An Image Solution,\nPhys. Rev. 184, 1272 (1969).\n[14] K. M. Case and S. C. Chiu, ElectromagneticFluctuations in a Cavit y, Phys. Rev. A 1, 1170 (1970).\n[15] R. Balian, B. Duplantier, Electromagnetic waves near perfect c onductors. II. Casimir effect, Ann.\nPhys.112, 165 (1978)\n[16] J. Schwinger, L. L. DeRaad, Jr., and K. A. Milton, Casimir Effect in Dielectrics, Ann. Phys. 115,\n1 (1978).\n[17] J. S. Dowker and G. Kennedy, Finite temperature and boundar y effects in static space-times, J.\nPhys.A 11, 895 (1978).\n[18] J. Ambjørn and S. Wolfram, Properties of the Vacuum. I. Mech anical and Thermodynamic, Ann.\nPhys.147, 1 (1983).\n[19] R. H. Baughman, S. Stafstr¨ om, C. Cui and S. O. Dantas, Mat erials with Negative Compressibilities\nin One or More Dimensions, Science 279, 1522 (1998).\n[20] E. S. Moreira Jr. and H. da Silva, Blackbody thermodynamics in th e presence of Casimir’s effect,\nJ. Stat. Mech. 063102 (2023) .\n6"    },    {        "title": "2402.06344v1.Wetting_ridge_dissipation_at_large_deformations.pdf",        "content": "Wetting ridge dissipation at large deformations\nMartin H. Essink1, Stefan Karpitschka2,3, Hamza K. Khattak4,\nKari Dalnoki-Veress4,5, Harald van Brummelen6, and Jacco H.\nSnoeijer1\n1Physics of Fluids Group, Mesa+ Institute, University of Twente,\n7500 AE Enschede, The Netherlands.\n2Max Planck Institute for Dynamics and Self-Organization, 37077\nG¨ ottingen, Germany.\n3Fachbereich Physik, Universit¨ at Konstanz, 78457 Konstanz,\nGermany.\n4Department of Physics and Astronomy, McMaster University,\n1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada.\n5UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research\nUniversity, 75005 Paris, France.\n6Department of Mechanical Engineering, Eindhoven University of\nTechnology, 5600 MB Eindhoven, The Netherlands.\nFebruary 12, 2024\nLiquid drops slide more slowly over soft, deformable substrates than over\nrigid solids. This phenomenon can be attributed to the viscoelastic dissipation in-\nduced by the moving wetting ridge, which inhibits a rapid motion, and is called\n“viscoelastic braking”. Experiments on soft dynamical wetting have thus far\nbeen modelled using linear theory, assuming small deformations, which captures\nthe essential scaling laws. Quantitatively, however, some important dispari-\nties have suggested the importance of large deformations induced by the sliding\ndrops. Here we compute the dissipation occurring below a contact line mov-\ning at constant velocity over a viscoelastic substrate, for the first time explicitly\naccounting for large deformations. It is found that linear theory becomes inaccu-\nrate especially for thin layers, and we discuss our findings in the light of recent\nexperiments.\n1 Introduction\nThe motion of spreading and sliding drops offers a paradigmatic example of\nwetting dynamics. While relevant for any application where drops touch a wall,\n1arXiv:2402.06344v1  [cond-mat.soft]  9 Feb 2024UA BUFigure 1: (a)Schematic view of a droplet moving to the right at velocity U,\nwith the moving contact line indicated by the circle. (b)Zoom of the moving\nwetting ridge. The horizontal translation of the contact line induced a vertical\nmotion of material points in the substrate, indicated by the blue arrows. This\nmotion induces viscoelastic dissipation inside the substrate. (Note that the\nmotion is not restricted to the interface but also occurs inside the bulk of the\nsubstrate, and is not strictly in the vertical direction.)\nthe wetting motion comes with challenges related to the fluid flow at small\nscales near the contact line [1–3]. It is of particular interest that the spreading\nand sliding is affected by deformability of the substrate [4–22]. This field of\nresearch was pioneered by Shanahan and co-workers [23–26], who discovered\nthat contact lines move more slowly over rubbers and elastomers, as compared\nto rigid surfaces, regardless of the precise polymeric nature of the substrate. This\nslowing down was attributed to the viscoelastic dissipation inside the substrate\nthat ensues when a wetting ridge translates over the substrate.\nThe mechanism behind this so-called “viscoelastic braking” is sketched in\nFigure 1. The surface tension of the droplet exerts a localized traction on the\nsolid at the conctact line, creating a deformation of the underlying elastic sub-\nstrate. Subsequently, a horizontal translation of this wetting ridge (red arrow)\ninduces time-dependent displacements of material points (blue arrows). These\ndisplacements generate a strong viscoelastic dissipation inside the substrate that\nis responsible for the slowing down of the wetting motion [23–26]. Scaling theo-\nries were presented to compute the dissipation from the substrate’s rheological\nproperties [4, 27, 28], as a function of the speed of the contact line. In ex-\nperiments, however, the dissipation, or rather the dissipative force, is usually\nnot measured directly, but it is mostly quantified via a a change in the contact\nangle of the liquid-vapor interface with respect to the equilibrium angle. This\nchange in angle was shown to originate from a rotation of the tip of the wetting\nridge that, again, can be related to the viscoelastic response of the substrate\n[29–33]. The interpretations based on “ridge rotation” and “dissipation” offer\nstrictly equivalent descriptions of dynamical wetting over soft substrates. This\nequivalence has been shown explicitly in the context of linear viscoelasticity,\nassuming small deformations, for which analytical predictions have indeed been\nobtained from the two different methods [30, 34].\n2Recent experiments have for the first time provided direct measurements on\nthe dissipation induced by moving contact lines [35]. These measurements were\nachieved by dragging droplets over thin PDMS substrates by a flexible micro-\npipette, which serves as a force sensor. The experiments confirmed the essential\nscaling laws of dissipation with contact line speed. Specifically, the dissipation\nper unit length of contact line, denoted D, is of the form\nD\nγU=β(H)\u0012U\nU∗\u0013n\n. (1)\nIn this expression γis the liquid-vapor surface tension, Uis the contact line\nvelocity, U∗is a characteristic viscoelastic velocity, and nthe rheological ex-\nponent of the substrate. The key element of interest here is the dependency\non the substrate thickness, reported as a dimensionless factor β(H). The ex-\nperimental data for this factor βobtained by [35], is reproduced in Figure 2.\nThe data clearly show that dissipation decreases for smaller substrate thickness,\nand suggest that βeven vanishes for H→0. This result, however, is in stark\ndisagreement with predictions from linear viscoelastic theory [16, 29, 30, 36],\nsuperimposed as the red curve in Figure 2. It was argued in [35] that the mis-\nmatch between experiments and theory at small substrate thickness is due to the\nemergence of large deformations for wetting ridges on thin substrates. Namely,\nlinear theory predicts the height of the wetting ridge hrto scale as H3/4in the\nlimit of small substrate thickness [28]. This scaling for the ridge height implies a\ntypical vertical strain hr/H∼H−1/4, which obviously constitutes a very large\ndeformation when Hbecomes small. As such, linear viscoelasticity does not\noffer a consistent theory of soft wetting on very thin substrates.\nThe goal of this paper is to model the statics and dynamics of wetting ridges,\nwhile consistently accounting for large viscoelastic deformations. First, we com-\npute static wetting ridge profile using finite element simulations of Neo-Hookean\nsolids, following a similar (macroscopic) scheme as in [37]. The static results are\npresented in Section 2, where we will focus on the ridge profiles in the limit of\nsmall substrate thickness. Then, we compute the dissipation following the idea\nsketched in Figure 1. The static profile is translated horizontally at constant\nvelocity; subsequently, the dissipation is computed from the time-dependent\ndeformations from this horizontal motion. In the context of small deformation\ntheory, this procedure based on translation of a static profile is known to give\nthe exact dynamical dissipation, to leading order in velocity [29], and we show\nthat this idea can be equally applied to large deformations. Namely, the change\nin the shape of the wetting ridge due to the motion offers only a higher order\n(in velocity) correction to the shape of the wetting ridge, and thus to the dissi-\npation. In Section 3 we formalise this procedure for large deformations, and the\nresulting dissipation factors β(H) are presented in Section 4. We also analyti-\ncally derive the dissipation in the vicinity of the tip, which exhibits a power-law\ndivergence. Importantly, we find that the dynamics-induced singularity is in-\ntegrable, which justifies a posteriori the use of static profiles to compute the\nleading order dissipation. In Section 5 we explore substrate constitutive rela-\ntions beyond the Neo-Hookean model, specifically focusing on the effect of finite\n30 250 500 750 1000\nH(nm)0.000.250.500.751.001.25β/ˆβ∞\nLinear Theory0 2.5 5.0 7.5 10.0GH/γsFigure 2: Dissipation βin the wetting ridge, as a function of the substrate\nthickness Hand normalised thickness GH/γ s. Experimental data is shown in\nblack, and the corresponding linear viscoelastic theory in red. Recreated from\ndata provided in [35].\nextensibility. We compare our results to the experiments in [35] as discussed in\nthe concluding Section 6.\n2 Static profiles\n2.1 Formulation\nFigure 3 shows a static numerical result. We consider a two-dimensional wetting\nridge on a substrate of finite thickness H. The substrate has a shear modulus\nG, and a surface energy γson the free surface. For simplicity we here consider\nγsas constant, i.e. independent of the stretch and the same on both sides of the\nwetting ridge. For asymmetric wetting conditions we refer to [37]. The wetting\nridge is induced by a point force γthat pulls vertically on the surface, which\nmodels the force exerted by the liquid-vapor interface at the contact line. The\nsolid is deformed according to a mapping x=f(X), which maps the reference\nconfiguration Xto the deformed configuration x. The gradient of this mapping,\nthe deformation gradient tensor F=∂x/∂X, is a measure of the deformations of\nthe solid. Restricting our considerations to solids with hyperelastic constitutive\nrelations, from this tensor Fan energy density W(F) can be calculated, that\nrepresents the energy per unit area (in plane-strain conditions) that is stored in\nthe elastic deformations of the solid. For most of this paper, we will consider the\nsubstrate to consist of an incompressible Neo-Hookean solid under plane-strain,\n4γ\nγsγs\nθSFigure 3: Schematic representation of the wetting ridge on a solid substrate.\nThe black lines indicate a course, non refined mesh in the deformed state. The\nmesh in reference state is shown in gray. The surface tensions γandγsare\nshown in blue and red respectively. The solid opening angle θSis indicated\nbetween the surface tension arrows.\nfor which the energy density reads\nW(F) =G\n2\u0000\ntr\u0000\nF·FT\u0001\n−2\u0001\n−p(det(F)−1), (2)\nwhere Gis the shear modulus of the material, and incompressibility is enforced\nthrough the Lagrange multiplier p. Integration of this energy density over the\nentire (undeformed) substrate yields the total mechanical energy stored in the\nsubstrate. With the addition of the surface energy γson the free surface, and\nthe vertical point force γat (X, Y) = (0 , H), the total energy is\nU=Z\ndXHZ\n0dY W (F) +Z\nγsλsdX+γey·x(0, H) (3)\nwhere λsis the stretch of the free surface, and eythe unit vector in vertical\ndirection. The problem can be solved by minimisation of the energy functional\nUwith respect to the deformation x. In the final part of this paper we investigate\nthe effect of finite extensibility of the polymeric substrate, which amounts to a\nmodification of the elastic energy density W(F).\nA typical result of the minimisation is shown in Figure 3. The pulling force\ngenerates deformations throughout the entire substrate, which gives rise to both\nvertical and horizontal displacements. Immediately below the contact line, the\nwetting ridge is governed by an opening angle θSdictated by the Neumann\nbalance of the three surface tensions [38]. Therefore, the solid opening angle θS\nand the ratio of surface energies γ/γsare directly related as\nθS= 2 arccos\u00121\n2γ\nγs\u0013\n⇐⇒γ\nγs= 2 cos\u0012θS\n2\u0013\n. (4)\nHence, we shall be using θSas the measure that quantifies the relative strength\nof the surface tensions. We also remark that the theory of linear elasticity is\n5only expected to be valid for θSclose to 180◦, i.e. γ/γs≪1. Experimentally,\nhowever, this condition is typically not fulfilled, which supports our interest\nof performing simulations with large deformation theory. The ratio of surface\ntension to elasticity is expressed via the elastocapillary length; here it can be\ndefined as both γ/G andγs/G. Roughly speaking, the former sets a typical\nvertical scale, while the latter is a horizontal scale of deformation. As a dimen-\nsionless measure of the (inverse) substrate thickness, we will be using γs/GH .\nThe opening angle θS, or equivalently the ratio γ/γs, provides the second di-\nmensionless parameter in the problem.\n2.2 Numerical method\nThe minimisation of the total energy (3) is implemented using the open-source\nfinite element framework Nutils [39]. We start with an unrefined mesh of 4 ×192\nelements, representing only the right half of the wetting ridge, for symmetry\nreasons. A high aspect ratio is achieved by increasingly stretching the elements\nwhen moving further away form the contact line, as can be observed in Figure\n3. The mapping X7→X(1+X) produces a mesh of width 2352 H, which is wide\nenough to be considered infinite for the purposes of this study. The elements are\nhierarchically refined up to 18 times using a residual-based adaptive method,\nsuch that an element size of ∆ <10−4γs/Gis achieved near the contact line.\nThe bottom of the mesh is constrained in both horizontal and vertical direction,\nimitating a substrate bonded to a rigid surface. The right boundary is also\nconstrained in both directions. The left boundary represents the symmetry axis\nof the wetting ridge, thus the deformations are only constrained in horizontal\ndirections, with no shear in the vertical direction. The top surface representing\nthe free surface is traction-free, with only stresses resulting from surface tension.\nFor the simulation of sharp wetting ridges a continuation approach was re-\nquired. In this case the surface energy ratio γ/γswas increased in a number\nof smaller steps, such that a value up to γ/γs= 1.5 (θS= 82 .8◦) could be\nreached, before discretisation artifacts typical for sharp kinks at free surfaces\ninhibit convergence of the solver.\n2.3 Effect of substrate thickness\nWe are interested in studying how the wetting ridge is affected when varying\nthe substrate thickness. Specifically, we wish to investigate the limit of thin\nsubstrates, for which linear elasticity becomes inconsistent since large strains\nare predicted [28, 35]. In the simulations we thus vary γs/GH , which represents\nthe ratio of the elastocapillary length γs/Gand the substrate thickness H.\nFigure 4a shows free surface profiles of the wetting ridge for a range of γs/GH ,\nwith the horizontal and vertical axis normalized by H. The presented data\nare for γ/γs= 1, so that θS= 120◦. The blue data corresponds to the limit\nof small elastocapillary length, or equivalently of large substrate thickness. In\nthis limit, the deformation (normalized by H) appears as a small feature on\ntop of the comparatively thick substrate. The yellow data correspond to the\n6−10−5 0 5 10\nx/H−0.250.000.250.500.751.00y/HA\n120◦\n−20−10 0 10 20\nx/hr−0.250.000.250.500.751.00y/hrB\n120◦\n10−1100101102103\nγs/GHFigure 4: Numerical free surface profiles for various values of the elastocapillary\nlength, with lines colored according to the colorbar on the right. A surface\nenergy ratio γ/γs= 1.0 was used, corresponding to a solid angle θS= 120◦.(a)\nNormalised by the thickness of the undeformed solid layer H.(b)Normalised\nby the height of the wetting ridge hr, which leads to a collapse of the data in\nthe soft limit.\nopposite limit of large elastocapillary length, or small thickness. Now, the elastic\ndeformation becomes large compared to the layer thickness, and continues to\ngrow upon increasing γs/GH . Since the opening angle remains constant between\nthe curves, the profiles all appear to exhibit a similar shape in the soft limit,\nonly changing in scale. We examine whether the shape of the wetting ridge is\nindeed truly self-similar in the thin limit, by scaling the profiles by the height\nof the wetting ridge hr, both vertically and horizontally. The resulting set of\ncurves is displayed in Figure 4b. Indeed, the wetting ridge profiles converge to\na universal shape for thin substrates ( γs/GH≫1), suggesting that the ridge\nheight – rather than the substrate thickness – is the relevant length scale in the\nthin limit.\nThe remaining task is to understand the scaling of the ridge height hras a\nfunction of γs/GH . The numerical results for for the ridge height are reported\nin Figure 5, for different values of θS. In the plot the ridge height hr/His\nnormalized by an additional factor γs/γ, which leads to a nearly perfect collapse\nof the data for all θS. The Figure clearly reveals the presence of two regimes.\nFor thick substrates we find a scaling that is close to linear, and consistent with\nthe prediction from two-dimensional linear elasticity hr∼γ/Gln(GH/γ s) [40,\n41].\nOur main interest, however, lies in the limit of thin substrates, which pertains\nto the data on the right in Figure 5. The ridge height hrnow closely follows a\nscaling exponent 1 /4. This behavior is perfectly in line with predictions based\non linear elasticity, which suggest the wetting ridge to grow as [28]\nhr\nH∼γ\nγs\u0010γs\nGH\u00111/4\n. (5)\n710−210−1100101102\nγs/GH10−210−1100hr/H·γs/γ\n∼xlnx14\nθS= 179.4◦\nθS= 139.0◦\nθS= 82.8◦Figure 5: Height of the wetting ridge as a function of the elastocapillary number.\nThe height of the wetting ridge is scaled by the ratio of surface energies γ/γsto\noverlay results for multiple solid opening angles. The blue line and the triangle\nshow, respectively, the ∼xlnxregime for thick layers and the 1 /4 power law\nfor thin layers, predicted by linear theory.\nHence, the wetting ridge continues to grow, without any evidence for a bound,\nupon increasing the elastocapillary length. Importantly, since hr/Hserves as\na measure for strain, at least at the location of the ridge, we clearly see that\nstrains do not remain small: they even diverge in the limit H→0. However,\nthe results from the Neo-Hookean solid (see Figure 5) are perfectly in line with\nthe prediction from linear theory (5). Hence, even though linear elasticity is\nno longer consistent in the limit of thin substrates, it correctly predicts the\nhorizontal and vertical scaling of the wetting ridge on Neo-Hookean solids.\n3 From static profiles to dissipation\nWe now turn to the main focus of the paper, which is to compute the dissipation\ninside a wetting ridge as a function of its translation velocity. For this we\nemploy the procedure outlined in the Introduction. In the spirit of the sketch\nin Figure 1, we consider a horizontal translation of the full static solution of\nFigure 3 at a constant velocity, and compute the dissipation from the resulting\nmotion of material points. In this section we present the technical details on how\nto perform this calculation while consistently accounting for large deformations.\nThe results of this calculation, in particular the spatial density of dissipation\nand the behavior of the global dissipation factor β(H) as defined in (1), will be\npresented in section 4.\n3.1 Stress relaxation at large deformations\nThe theory of linear viscoelasticity can be expressed using a stress relaxation\nfunction that captures the memory of stress under time-dependent deformations.\n8For small deformations, this formalism is typically written in scalar form as\nσ(t) =Zt\n−∞dt′ψ(t−t′) ˙γ(t′). (6)\nThis equation gives a relation between (a component of) the stress and the his-\ntory of the rate-of-strain ˙ γ. The function ψ(t−t′) is called the stress relaxation\nfunction, and defines the linear rheological response of the material. Since we\nwish to deal with dissipation in the presence of large deformations, we need to\nuse the appropriate tensorial form of this relation.\nIn the general description of nonlinear viscoelastic solids, the stress at time t\nis a functional over the entire history of deformation [42]. The deformation must\nthus be described using a time-dependent mapping x=f(X, t). We assume that\nthe system is in the (stress-free) reference configuration at time t0, such that\nf(X, t0) =X. The history of deformation can then be described by the two-\ntime Finger tensor B(t, t′), also known as the left Cauchy-Green deformation\ntensor, which compares the deformation of the material between two different\ntimes, namely the current time tand a time in the past t′< t. Formally, this\nquantity is defined from the time-dependent deformation tensor\nF(t, t′) =∂x(t)\n∂x(t′)⇒B(t, t′) =F(t, t′)·FT(t, t′). (7)\nThe Finger tensor with respect to the reference configuration, as it appears in\nstatic elasticity, is recovered as B(t, t0).\nIn what follows we will restrict ourselves to isotropic models of solids with\n“finite linear viscoelasticity”; this terminology expresses that the models are not\nrestricted to small deformations, but are still taken as linear functional of B.\nSpecifically, we shall be using the form [42]\nσ(t) =σel(t) +Zt\n−∞dt′\u0014\n−ψ1(t−t′)∂B(t, t′)\n∂t′+ψ2(t−t′)∂B(t, t′)−1\n∂t′\u0015\n.(8)\nThe first term σel(t) is the elastic stress that, as usual, depends on B(t, t0) and\nB(t, t0)−1that measures the deformation with respect to the reference config-\nuration at t=t0. The dissipative, viscoelastic part of the stress is expressed\nby the integral, where the kernels ψ1,2decay to zero in the limit of large t−t′;\nthis condition ensures that the elastic response is recovered after the configura-\ntion is held constant (i.e. after stopping the “flow”). In general, the relaxation\nfunctions ψ1,2can be chosen to depend on B(t, t0), which would render the\nfunctional nonlinear and the model beyond “finite linear viscoelasticity”; for\nsimplicity this possibility will not be considered here. Since the Neo-Hookean\nsolid, and also the Gent model that will be discussed later, do not explicitly\ndepend on B(t, t0)−1, we will make the further simplification that ψ2= 0. A\nfinal reduction can be achieved for the Neo-Hookean solid, which is defined by\nthe elastic stress σel(t) =G(B(t, t0)−I). Setting the reference time t0=−∞,\n9the elastic stress can be absorbed inside the integral as:\nσ(t) =−Zt\n−∞dt′ψ(t−t′)∂B(t, t′)\n∂t′. (9)\nHere we made use of the property that B(t, t) =I, and we defined the stress\nrelaxation function\nψ(t−t′) =G+ψ1(t−t′). (10)\nWe remind that ψ1→0 in the long-time limit, such that ψ→Gensuring a\nsolid-like behavior.\nUpon comparing to (6), we find that (9) forms a tensorial viscoelastic rhe-\nology that admits a consistent description for large deformations. The explicit\nmanipulation of this form, however, is intricate as it involves tensor products\nof the mapping F(t, t′). A great simplification is obtained when expressing the\ntensors using a curvilinear formulation. In this formulation, material points are\ndescribed by curvilinear coordinates qi. The infinitesimal distance dsbetween\ntwo nearby points qiandqi+dqifollows, using Einstein summation convention,\nas\nds2=gijdqidqj, (11)\nwhere gijis the so-called metric tensor. For a time-dependent deformation, the\ndistance between material coordinates changes over time, such that the metric\nmust be time-dependent gij(t). In Appendix A, we demonstrate that covariant\ncomponents of the the rate-of-strain tensor ˙γare simply given by ˙ gij(t), so that\n˙γis naturally connected to the time-derivative of the metric. Likewise, one can\nshow that the contravariant components at time tof the Finger tensor B(t, t′)\nare given by gij(t′), i.e. involving the metric at time t′. With these expressions,\nthe rheology (9) takes on a remarkably simple form (Appendix A),\nσij(q, t) =−Zt\n−∞dt′ψ(t−t′) ˙gij(q, t′). (12)\nThis form is identical to linear viscoelasticity, as given by (6), but now offering\na proper tensor form admitting large deformations; the geometric nonlinearities\nassociated to large deformations are implicitly accounted for by the metric ten-\nsor. For transparency, we have introduced in (12) the spatial coordinates qas\narguments of both stress and metric. We recall that these are associated with\nmaterial points, so that the memory integral is to be performed at a constant\nmaterial point.\n3.2 Dissipation for a traveling wave\n3.2.1 Density of mechanical work\nWith an explicit expression for the viscoelastic stress at hand, we can evaluate\nthe power density,\nP=1\n2σ:˙γ, (13)\n10giving the volumetric density of work done by the stress per unit time. Let us\nnote that under the standing assumption of incompressibility, the power density\nis identically defined in the reference and current configurations. The power\ndensity does not give direct access to the dissipation within the solid, as it is\ncomprised of a reversible (elastic) contribution and an irreversible (dissipative)\ncontribution. Hence, the work can be split as\nP=DW\nDt+ϵ. (14)\nIn this expression Wis the reversibly stored elastic energy, while the remain-\nderϵis the irreversibly dissipated power per unit volume. The term DW/Dt\nrepresents the material derivative of the stored energy, i.e.the time derivative\nwith respect to a fixed position in the reference configuration.\nTo illustrate these concepts, let us consider a passing contact line, moving\nabove a certain material point inside the viscoelastic substrate. The total work\nper unit volume by the passing wave is obtained by integrating the power over\nall times,\n∞Z\n−∞Pdt=W\f\f\f\ft=∞\nt=−∞+∞Z\n−∞ϵ dt, (15)\nPrior and after the passage of the contact line, there is no elastic energy stored.\nHence, the term involving Wdrops out, so that the integral over the power\ngives the total amount of dissipated energy. So even though P ̸=ϵ, the total\ndissipation can be computed by considering an integral over the power density.\n3.2.2 Travelling wave solution\nWe will now develop explicit expressions for the dissipation below a moving\ncontact line. For this, we first write the power density in curvilinear form\nP=1\n2σ(t) :˙γ(t) =1\n2σij(X, Y, t ) ˙gij(X, Y, t )\n=−1\n2tZ\n−∞dτ˜ψ\u0012t−τ\nλ\u0013\n˙gij(X, Y, τ ) ˙gij(X, Y, t ). (16)\nIn the first line we made explicit that we will use the horizontal and vertical\nof reference positions as the curvilinear coordinates, i.e. q1=Xandq2=Y.\nIn the second line we inserted the expression for σijin terms of the stress\nrelaxation function ψ, as derived in the preceding section. In order to keep track\nof dimensions, we made the argument of the relaxation function dimensionless\nwith the relaxation time scale λ, according to ˜ψ(t/λ) =ψ(t).\nThe power density can thus be written in terms of time-derivatives of the\nmetric tensor, evaluated at times tandt′. For a contact line moving to the\nright as a travelling wave, we find that all fields are of the form g(X, Y, t ) =\n11G(X−Ut, Y ). Correspondingly, the power density due to a travelling wave can\nbe expressed as\nP(X, Y, t ) =−U\n2tZ\n−∞d(Uτ)˜ψ\u0012t−τ\nλ\u0013\nGij′(X−Uτ, Y )G′\nij(X−Ut, Y ),(17)\nwhere a prime indicates a derivative with respect to the first argument of G.\nMaking the transformation of variables ˆX=X−Utand ˜X=X−Uτ, one\nobtains\nP(ˆX, Y ) =−U\n2∞Z\nˆXd˜X˜ψ ˜X−ˆX\nUλ!\nG′\nij(ˆX, Y )Gij′(˜X, Y ). (18)\nHence, for a travelling wave solution, the history-dependence at a co-moving\npoint ˆXcan be inferred by looking at the deformations at ˜X > ˆX, i.e. points\nthat have already passed ˆX. As explained in the discussion of Figure 1, to\nleading order in velocity we can evaluate this expression for the work by taking\nthe fields Gij(X, Y) and Gij(X, Y) directly from the static numerical solutions\nobtained in Section 2.\nThe central goal of our study is to compute the total dissipation Din the\nsubstrate, representing the total energy dissipated per unit time per unit length\nof contact line. This quantity is obtained by integrating ϵover the entire domain\nof the substrate. Given that for a travelling wave the total stored elastic energy\ndoes not change in time, it is only displaced, we simply find that\nD=HZ\n0dY∞Z\n−∞dX ϵ =HZ\n0dY∞Z\n−∞dXP. (19)\nHence, the integral of the power density over the entire domain directly offers\nthe total amount of dissipation.\n3.2.3 Density of dissipation\nWhile we shall be using (19) to compute the global dissipation inside the sub-\nstrate, it is still of interest to consider the local volumetric density of dissipation\nϵ. To compute this quantity, we follow [43] and define a conformation tensor\nAij. The conformation tensor characterizes the local degree of stretching, and\ncan be viewed as the order parameter that describes the thermodynamic state of\nthe system. Given the linear constitutive law (12), it is natural to define a linear\nrelation between stress and the order parameter, according to σij=GAij. This\nis a simple “Neo-Hookean” relation between stress and the order parameter,\nfor which the corresponding reversible work can be described by a linear stored\nenergy density\nW=1\n2GgijAij. (20)\n12To appreciate that this is the correct form of the energy, we take its time-\nderivative,\nDW\nDt=1\n2GAij˙gij+1\n2Ggij˙Aij. (21)\nWe recall that the material derivative in this expression reduces to regular time\nderivatives, since here the fields are expressed as functions of the curvilinear\nmaterial coordinates. Noting that ˙ gijrepresent the covariant components of\nthe tensor ˙γ, we recognize the first term on the right as the power density\nby internal stress,1\n2σ:˙γ. From this we can make the identification that\nσij=GAij, consistent with our initial definition of the conformation tensor.\nGiven that DW/Dtrepresents the reversible part of the work, the second term\non the right thus represents the irreversible dissipation ϵ, i.e.\nϵ=−1\n2Ggij˙Aij=−1\n2gij˙σij. (22)\nFor a thermodynamically consistent constitutive relation, the expression for\nϵ≥0, which we will indeed find in our numerical results.\n3.3 Chasset-Thirion rheology\nUntil now we have not considered any specific relaxation function ψ. In line\nwith previous work, we will consider the Chasset-Thirion rheology [35, 44], that\nhas a complex modulus\nG′+iG′′=G[1 + ( iωλ)n], (23)\nwith 0 < n < 1, and where λoffers the characteristic relaxation timescale. This\ncomplex modulus indeed provides an excellent fit of typical PDMS substrates\nused in soft wetting experiments [29, 30]. On the time-domain, this model is\nrecovered to a stress relaxation function\n˜ψ(¯t) =G\u0002\n1 + Γ(1 −n)−1¯t−n\u0003\n, (24)\nwhere ¯t=t/λand the result further involves the Γ-function. In the expression\nfor the power density, the constant term of the relaxation function expresses the\nelastic stress, and will not give a resultant contribution to the total dissipation\nintegral. To compute D, it is thus sufficient to retain only the time-dependent\npart of the relaxation function, so that the power density can be written as\nP\nUG=−1\n2(Uλ)n\nΓ(1−n)∞Z\nXd˜XG′\nij(X, Y)Gij′(˜X, Y )\u0010\n˜X−X\u0011n . (25)\nThis expression can also be used to compute the power density of a purely elastic\nmaterial, i.e. ψ=G, upon setting n= 0.\n133.4 Non-dimensionalisation\nAs a final step, we wish to express the total dissipation in the form of equation\n(1): the velocity-dependence is pulled out explicitly and βis a dimensionless\nfactor that reflects the dependence of the dissipation on the substrate thickness\n[35]. In the presented dimensionless formulation of the problem, βdepends on\nthe parameters γs/(GH) and γ/γs. Since the total dissipation follows from an\nintegral of power density over the entire substrate, we analogously define\nP\nUγ/H2= (U/U∗)nα. (26)\nwhere αreflects the thickness-dependent part of the power density. The integral\nrelation between DandPas given by (19) then conveys\nβ=Z1\n0d¯YZ∞\n−∞d¯X α, (27)\nwhere ¯X=X/H and¯Y=Y/H.\nWith the definition of the Chasset-Thirion rheology in place, the exponent n\nintroduced in (1) is now well-defined. In addition, the characteristic horizontal\nvelocity U∗=γs/(Gλ) naturally emerges from the elastocapillary length γs/G\nand the relaxation time λ. Substituting the above scalings in (25), we find the\ndimensionless power density\nα=−1\n2γs\nγ\u0010γs\nGH\u0011n−1 1\nΓ(1−n)∞Z\n¯Xd¯˜XG′\nij(¯X,¯Y)Gij′(¯˜X,¯Y)\u0010¯˜X−¯X\u0011n . (28)\nIntegration over the entire substrate finally gives,\nβ=−1\n2γs\nγ\u0010γs\nGH\u0011n−1 1\nΓ(1−n)1Z\n0d¯Y∞Z\n−∞d¯X∞Z\n¯Xd¯˜XG′\nij(¯X,¯Y)Gij′(¯˜X,¯Y)\u0010¯˜X−¯X\u0011n .\n(29)\nThis expression gives the sought after dissipation factor β, which is a function of\nγs/GH andθS(via the factor γ/γs). We recall that also the static deformation\nprofiles are encoded in Gij(¯X,¯Y) and Gij(¯X,¯Y), as obtained from the finite\nelement simulations, depend on the parameters γs/GH andθS.\n3.4.1 Benchmark: The limit of small deformations\nThe large-deformation result for βwill be evaluated numerically, and compared\nto the analytical expression for the limit of small deformations. The latter\nconsists of a single integral, rather than of a triple integral, and reads [30, 35]:\nˆβ=−γ\nGH\u0010γs\nGH\u0011n\nsin\u0010nπ\n2\u0011Z∞\n−∞dQ\n2πK(Q)Qn+1\n\u0000\n1 +γs\nGHQ2K(Q)\u00012, (30)\n14where K(Q) is the elastic kernel for a dimensionless layer of finite thickness,\nK(Q) =1\n2Qsinh (2 Q)−2Q\ncosh (2 Q) + 2Q2+ 1. (31)\nIn this expression it is explicit that the dissipation integral βis only a function of\nγs/(GH) and γ/γs. In the limit of infinite thickness ( γs/GH≪1), the integral\ncan be performed analytically and one finds the reference value for a half-space\nˆβ∞=γ\nγsn2n−1\ncos\u0000nπ\n2\u0001. (32)\nAnalogously to ˆβ∞, we also define β∞for the large-deformation case as the\nlimiting value of βfor infinite thickness ( γs/GH≪1).\n4 Dissipation results\nIn the previous section we have shown how to compute dissipation within the\nviscoelastic substrate. We now present the results of this calculation, while\nsystematically varying γs/GH andθS.\n4.1 Spatial distribution of dissipation inside the solid\n4.1.1 Numerics\nBefore turning to the total dissipation induced by a moving ridge, we first\ninvestigate where this dissipation occurs inside the substrate. More precisely,\nwe compute the dimensionless power density α, as defined by (26), where the\nresult is computed from the explicit formula (28). We recall that even though\nthis quantity integrates up to the total dissipation inside the wetting ridge, it is\nnot a measure of “local dissipation”. This becomes immediately clear when αis\nplotted for a purely elastic solid, for which no dissipation can occur. This elastic\ncase is obtained from (28) with n= 0, and the resulting power density is given\nin Figure 6a. Red zones are regions of positive work, where energy is stored in\nthe solid; primarily upon arrival of the ridge. Blue zones are regions of negative\nwork, where energy is released; primarily after the passage of the ridge. For\nthe purely elastic case of Figure 6a, the distribution of work is perfectly anti-\nsymmetric, such that all the stored energy is eventually released. The resulting\npower density therefore integrates up to zero; as should be the case for for a\npurely elastic substrate that exhibits no viscoelasticity.\nOn the contrary, this left-right symmetry is broken for viscoelastic solids.\nThis can be seen in Figure 6b, where we show the viscoelastic contribution to\nthe power density, as computed from (28) with n̸= 0. Specifically, we consider\nn= 0.25, which is close to the experimental value n= 0.23 in [35]. The storage\nof energy (red) ahead of the wetting ridge still closely resembles that of the\npurely elastic substrate. However, much less of this energy is released (blue)\n15−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0\nx/H0.00.51.0y/HA\n−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0\nx/H0.00.51.0y/HB\n−10−1−10−2−10−3010−310−210−1\nα/vextendsingle/vextendsingle\nn=0.00\n−10−1−10−2−10−3010−310−210−1\nα/vextendsingle/vextendsingle\nn=0.25Figure 6: Dimensionless power density αin the substrate below a wetting ridge\nmoving to the right. Positive contributions (red) indicate mechanical energy\nbeing stored into the solid, where the negative contributions (blue) indicate\nthat energy is released. The black contours emphasise the transitions between\npositive and negative power density. Results are obtained for an opening angle\nofθS= 120◦and an elastocapillary number of γs/GH = 1.0.(a)A purely\nelastic solid response ( n= 0) exhibits an anti-symmetric distribution of work\nin the solid. (b)The viscoelastic contribution to the power for a typical for the\ncasen= 0.25. In this case, the symmetry is broken and the power density will\nintegrate to nonzero global dissipation.\nin the wake of the ridge. Upon integrating over the entire substrate, one thus\nobtains a resultant global (positive) dissipation.\nIt is of interest to disentangle the viscoelastic power of Figure 6b, into the\nreversible part ( dW/dt) and the irreversible part ( ϵ), using the splitting accord-\ning to (14). The results are shown in Figure 7; the plotted quantities are made\ndimensionless using the same scalings as α. The upper panel (a) corresponds\nto the reversible part of the viscoelastic power, dW/dt, which is again com-\nputed by applying the travelling wave method to the reversible energy defined\nby (20). Interestingly, this reversible viscoelastic power density is not left-right\nsymmetric, even though (by definition) its integral adds up to zero. The actual\ndissipation density ϵis reported in Figure 7b. This result is obtained by sub-\ntracting the reversible power from the total power, as explained in (14), and is\nstrictly non-negative. From Figure 7b we observe that the dissipation is largest\nclose to the contact line, where the rate of deformation is largest. Also, we find\nthat the maximum dissipation rate is skewed to the left, which is the part of\nthe substrate where the contact line has already passed. A detailed zoom of the\n16−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0\nx/H0.00.51.0y/HA\n−2.0−1.5−1.0−0.5 0.0 0.5 1.0 1.5 2.0\nx/H0.00.51.0y/HB Panel C\n−0.015−0.010−0.005 0.000 0.005 0.010 0.015\nx/H−0.010−0.0050.000(y−hr)/HC\n−10−1−10−2−10−3010−310−210−1\nd¯W\nd¯X/vextendsingle/vextendsingle\nn=0.25\n10−410−310−210−1\n¯/epsilon1/vextendsingle/vextendsingle\nn=0.25\n100101102103104\n¯/epsilon1/vextendsingle/vextendsingle\nn=0.25Figure 7: The viscoelastic power as given in Figure 6b is split into (a)Reversible\npower dW/dt and(b)Irreversible power ϵ, the latter being the density of dis-\nsipation. The plotted quantities are made dimensionless using the same scaling\nasα.(c)Zoom of the dissipation density near the tip, comparing numerics\n(colors) to the asymptotic analysis (dashed lines).\nlocal dissipation is provided in Figure 7c, where it is compared to an analytical\nsolution derived below using an asymptotic analysis.\n4.1.2 Asymptotics: Zooming in near the tip\nClose to the contact line, at scales well below the elastocapillary length, a static\nwetting ridge can be described as a fold of and elastic half-space to an angle θS\n[38]. Using polar coordinates ( R,Φ) to describe the reference configuration, the\nhalf-space corresponds to the domain −π≤Φ≤0 with R≥0. This is mapped\nonto a deformed configuration, with polar coordinates ( r, ϕ), according to [45]\n(r, ϕ) =\u0012R√\nb, b\u0010\nΦ +π\n2\u0011\n−π\n2\u0013\n. (33)\n1710−410−310−210−1100\n1−Y/H100102104¯/epsilon1/vextendsingle/vextendsingle\nn=0.25\nA\n−π−3π/4−π/2−π/4 0\nΦ0.000.050.100.15¯/epsilon1/vextendsingle/vextendsingle\nn=0.25R1+n\nB R= 10−2\nR= 10−3\nR= 10−4Figure 8: Asymptotic solution for the dissipation near the tip (solid lines),\ncompared to interpolated numerical data (symbols) without any adjustable pa-\nrameters. (a)The density of dissipation (taken close to the angle of maximum\ndissipation, at Φ = −2.0) as a function of the distance to the tip. Simulations\nconfirm the power-law divergence with exponent 1+ n.(b)Normalized angular\ndependence of the dissipation density. Numerical results converge to the ana-\nlytical prediction as the distance to the tip decreases.\nThis transformation amounts to a uniform compresssion by a factor b=θS/πin\nthe azimuthal direction, which is centerred around Φ = −π/2 (and ϕ=−π/2).\nThis radial stretching then automatically follows to ensure incompressibility.\nIndeed, as reported in [38], numerical solution of the wetting ridge indeed closely\nfollows this fold map in the asymptotic limit where r→0.\nTo compute the dissipation, we can proceed by following the formalism as\ndescribed above. Rather than using the numerical static profiles, we now use\nthe analytical solution to compute the dissipation asymptotically close to the\ntip. Details of this calculation are given in Appendix B, where we obtain a\nclosed form analytical expression. For convenience, the result is presented in\ndimensional units, and reads:\nϵ(R,Φ) =1\n2Gλn\u0000\nb2−1\u00012\nb2πn(n+ 1)\nΓ(1−n) sin(nπ)sin(Φ) sin( nΦ)\u0012U\nR\u0013n+1\n.(34)\nFigure 7c provides a direct comparison between this result and the numerical\nevaluation of the dissipation. Both the dependencies on Rand in Φ provide an\nexcellent description of the numerical dissipation profiles, as is evident from the\ndirect comparison in Figure 8.\nWe thus find that the dissipation density diverges as 1 /Rn+1∼1/rn+1,\nupon approaching the tip. The limit of linear viscoelasticity is recovered by\ntaking 1 −b≪1, corresponding to angles close to θS=π. It is clear that lin-\near viscoelasticity, as used previously in [30, 34], correctly captures the scaling\nexponents for the dissipation. In fact, the scaling can be understood by consid-\nering the loss modulus, G′′∼G(ωλ)n, in conjunction with a typical frequency,\n18ω∼U/r. The dissipation density then follows as\nϵ∼G′′·ω∼G(ωλ)nω∼Gλn(U/r)n+1, (35)\nin perfect agreement with the exact result (34). An important consequence of\nthis scaling is that, since 0 < n < 1, this dissipation singularity is integrable at\nsmall scale – this can be seen by multiplying ϵwith the area element rdθdr ∼\ndθdr/rn, which gives a singularity weaker than 1 /r; see also the discussion in\n[30]. This ensures that the total dissipation, as reported below, indeed remains\nfinite. This also justifies a posteriori our approach to compute the dissipation\nbased on static profiles. Note that the inner region of soft wetting, close to the\ncontact line, is described by a wedge geometry [38, 45], which is less singular\nthan the Flamant problem [46, 47]. The description proposed here avoids the\nintricacies of expansions [48], and clearly shows that large deformations do not\naffect the nature of the singularity: the scaling laws for the dissipation density\nare the same as those found in linear theory. We remark that these asymptotic\nresults cannot be used to predict the global dissipation, since the large-scale\nfeatures of the ridge act as cut-off for the inner asymptotics.\n4.2 Global dissipation\nWe now turn to the global dissipation, which is expressed using the dissipation\nfactor βthat is defined in the introduction as equation (1). We compute this\nfactor in the presence of large deformations, using (29), and the results will\nbe systematically compared to the small-deformation linear theory (30). This\ndirect comparison is shown in the three panels of Figure 9, each corresponding\nto a different opening angle θSof the wetting ridge. Naturally, one would expect\nthe dissipation to match the linear theory in the case of θSclose to 180◦, as\nthe deformations in this limit are small. Indeed, for an angle θS= 179 .4◦in\nFigure 9a the simulations almost perfectly match the small deformation theory,\nfor the full range of γs/GH . This excellent quantitative agreement (including\nthe normalisation factor for thick layers ˆβ∞, which was not fitted but taken\ndirectly from linear theory), offers a validation of our numerical scheme.\nAs the wetting ridge becomes sharper, with smaller θS, some deviations\nbecome apparent. In Figures 9b and 9c we see that with a decreasing opening\nangle (or increasing γ/γs) two important changes occur. First of all, in the\nlimit of infinitely thick layers ( γs/GH→0), the dissipation coefficients β→\nβ∞deviates from the linear prediction ˆβ∞. This deviation depends on the\nparameter n, as can be seen in Figure 10a. Secondly, for thinner substrates\nanother change in the βcurves is observed. It seems the minimum value of\nβ, here called βmindecreases when the wetting ridge becomes sharper. The\nminimum βminis further quantified in Figure 10b. It is clear that the minimum\nof the dissipation decreases significantly when the solid angle θStakes on smaller\nvalues.\nTypical experimental values for θSlie around 90◦. Our results thus imply\nthat significant quantitative differences between experiment and linear theory\n19100102\nγs/GH0.51.02.0β/ˆβ∞\nA\nn=0.20\nn=0.40\nn=0.60\n100102\nγs/GH\nB\n100102\nγs/GH\nCFigure 9: Normalised dissipation in the wetting ridge as a function of the elas-\ntocapillary length for various values of the parameter nin the Chasset-Thirion\nrheology. Results from linear theory are shown as solid lines, where the numer-\nical results are denoted by the symbols. The ratio of surface energies for the\nnumerical results are (a)θS= 179 .4◦,(b)θS= 120 .0◦,(c)andθS= 82.8◦.\nare to be expected. On the other hand, the qualitative trends of linear theory\nremain intact: in all cases the dissipation exhibits a minimum, and does not tend\nto zero in the limit of small thickness. This is in contrast with the experimental\ndata, suggesting that βvanishes in the limit H→0. To further elaborate on\nthis, we replot the experimental results of [35] in Figure 11, compared to both\nlinear and large deformation results. Note that βis now normalised by β∞, since\nthe large thickness value of βis experimentally used as a fitting parameter, and\nthe horizontal axis displays the dimensionless thickness HG/γ s(rather than its\ninverse in the preceding plots). The large deformation results are clearly a closer\nmatch to the experiments; still, these do not capture the strong decrease in β\nthat was observed for experiments on thin layers.\n5 Finite extensibility\n5.1 Modeling considerations\nUntil now we have only considered the Neo-Hookean constitutive behavior, aug-\nmented with a Chasset-Thirion viscoelastic relaxation. While quantitative dif-\nferences appeared with respect to the linear theory, we found that qualitative\ntrends for the ridge height and the dissipation are the same, even in the limit\nof very thin substrates. We now explore whether qualitative changes appear\nfor constitutive relations with a more strongly nonlinear behavior, beyond the\nNeo-Hookean solid.\nSoft solids such as the PDMS used in the experiments [35], typically consist\nof a cross-linked network of polymer chains. While the polymers have some free-\ndom to deform, the polymer chains cannot be stretched indefinitely (in contrast\n2090 120 150 180\nθS1.001.251.501.752.00β∞/ˆβ∞\nA n=0.20\nn=0.40\nn=0.60\n0.2 0.4 0.6\nn0.20.40.60.81.0βmin/β∞B θS= 179.4◦\nθS= 139.0◦\nθS= 82.8◦Figure 10: (a)Dissipation coefficient for thick substrates β∞as a function of\nthe opening angle θSfor various values of the parameter n, normalised by the\nvalue ˆβ∞predicted from linear elasticity. (b)Minimum value of the dissipation\ncoefficient βas a function of the parameter n, for a number of surface energy\nratios γ/γs.\nto a Neo-Hookean solid). A model that captures the finite extensibility of the\npolymers is the Gent solid model, which imposes limiting behavior on the strain\ninvariant I= tr\u0000\nF·FT\u0001\n−2, such that it cannot exceed a limiting factor Im.\nSince Ican be seen as a “mean stretch”, Imtranslates to a direct limit on the\nstretch, with a maximum uniaxial stretch λmdefined by Im=λ2\nm+λ−2\nm−2.\nThe Gent solid model has a free energy density[49]\nW(F) =−G Im\n2ln \n1−tr\u0000\nF·FT\u0001\n−2\nIm!\n−p(det(F)−1), (36)\nwhich develops a diverging stress when I→Im. In the limit of I/Im→0,\nhowever, the model converges to the Neo-Hookean solid (2). Therefore, for\nmoderate deformations the two models should be nearly indistinguishable. For\nlarge deformations, as is the case for thin substrates, qualitative differences are\nanticipated.\nOnce again, the dissipation can be computed from static solutions, by using\nthe travelling wave Ansatz. For the Gent model, the starting point of the\nviscoelastic analysis is no longer (9) but for the more general relation (8), with\na modified expression for σel. However, the elastic part of the stress does not\ncontribute to the dissipation. So if we use for ψ1the time-dependent part of the\nChasset-Thirion relaxation function, we are still allowed to use the very same\nexpression for the dissipation factor β, namely equation (29). The effect of finite\nextensibility is then encoded via the change of the static profiles.\n5.2 Results\nSince the finite extensibility only comes into play at relatively large deforma-\ntions, it is expected that differences between the Neo-Hookean model and Gent\n2110−210−1100101\nGH/γs10−1100β/β∞\nLinear Elasticity\nNeo-Hookean\nKhattak et al.Figure 11: Dissipation in the wetting ridge as a function of the substrate thick-\nness, normalised by the elastocapillary length. Dissipation is normalised by β∞,\nthe value of βobserved for thick substrates. The experimental results from [35]\nare overlaid in black. A surface energy ratio γ/γs= 1.0 was used for the nu-\nmerical results, corresponding to an internal solid angle θS= 120◦. The value\nn= 0.25 is chosen to approximately match the experimental data.\nmodels will only be apparent at large deformations. This is indeed the case\nwhen plotting the height of the wetting ridge as a function of the elastocapillary\nnumber, as done in Figure 12a. Only for very thin substrates, where the wetting\nridge becomes relatively large, we see a qualitative difference between the two\nmodels (see also Figure 12b). Specifically, the Gent model predicts a saturation\nto the unbounded growth of the ridge height; leading to a breakdown of the 1/4\nscaling law. This implies that pulling more strongly on the solid does not lead\nto larger deformations when reaching the finite extensibility limit. The smaller\nthe value of Im, the smaller the maximum height of the ridge.\nThe saturation of the deformation also impacts the total dissipation.\nRoughly speaking, the finite extensibility means that the metric factors ap-\npearing in the dissipation integral (29) are also subjected to a saturation, even\nif one continues to increases γs/GH . By virtue of the finite extensibility of the\nmetric tensors in (29), as well as their derivatives, are bounded independent of\nH. Expression (29) then yields the following estimate for β:\nβ≲\u0010γs\nGH\u0011n−1\n, (37)\nwhenever finite extensibility comes into play. Given that n < 1, this implies\nβ≲H1−n→0 in the limit of vanishing substrate thickness. Hence, the Gent\nmodel could offer a qualitative change in the behavior of β, with a trend that\nis in line with experimental observations.\nThis hypothesis is confirmed in Figure 13, where we report βfor the Gent\nmodel. In contrast to the Neo-Hookean model and the linear theory, the\nGent model does not exhibit a minimum, but gives a vanishing dissipation\n22100102\nγs/GH10−210−1100hr/H·γs/γ\n∼xlnx14\nGent (Im= 2.5)\nGent (Im= 5.0)\nGent (Im= 10.0)\nNeo-Hookean\n0 5 10 15\nx/H0.00.51.0y/HGent (Im= 2.5)\nGent (Im= 5.0)\nGent (Im= 10.0)\nNeo-HookeanFigure 12: (a)Height of the wetting ridge as a function of the elastocapillary\nnumber, similar to Figure 5 but including finite extensibility results for a number\nofImvalues. The blue line and the triangle show, respectively, the ∼xlnx\nregime for thick layers and the 1 /4 power law for thin layers, predicted by linear\ntheory [28]. A surface energy ratio γ/γs= 1.0 was used for all simulations,\ncorresponding to an internal solid angle θS= 120◦.(b)The corresponding\nsurface profiles for a relatively thin substrate ( γs/GH = 3.16×102).\nasHG/γ s→0. The observed scaling is in very good agreement with the pre-\ndiction (37). This reveals that strong nonlinearities are indeed important to\nexplain the experimental observations. Having said that, the Gent model in\ncombination with a “linear” Chasset-Thirion model is not able to provide a\nquantitative match of the experiment on the thinnest substrates. The precise\nvalue of Imin the experiments depends sensitively on the elastomer type and\ncrosslinking density. In the experimental results a value Im∼10 would be a\nreasonable estimate [50], which is within the captured numerical range.\n6 Conclusion\nIn this study we set out to find an explanation for the apparent disagreement\nbetween the dissipation of moving wetting ridges in experiments and linear\ntheory that was observed in [35]. It was found that linear theory does not\ncapture the monotonic decrease of dissipation when reducing the thickness of\nthe substrate; instead, linear theory predicts a minimum in dissipation followed\nby a sharp increase for very thin layers. It was hypothesised that the appearance\nof a minimum was an artefact of the linear solid model, and would disappear\nfor nonlinear solid models. To test this we performed finite element simulations\nof the wetting ridge, allowing for the consideration of nonlinear solid models.\nFrom these simulation results the dissipation is then calculated.\nWe find that the free surface profiles converge to a self-similar shape in the\nlimit of thin substrates. This means that even though the wetting ridge grows\nrelative to the substrate thickness, its shape remains unaltered, but is only\n2310−310−210−1100101\nGH/γs10−1100β/β∞\n1−n\n1Linear Elasticity\nNeo-Hookean\nGent (Im=2.5)\nGent (Im=5.0)\nGent (Im=10.0)\nKhattak et al.Figure 13: Dissipation in the wetting ridge as a function of the substrate thick-\nness, similar to Figure 11. The data for the finite extensibility model is added\nin blue. All simulations use a surface energy ratio γ/γs= 1.0, or an internal\nsolid angle θS= 120◦, and n= 0.25 in the rheology. The expected power law\n(37) is displayed in the bottom left.\nscaled up. Remarkably, the wetting ridge in this limit, normalised by the layer\nthickness, shows the same power law growth behavior as predicted by linear\nelasticity [28, 33]. Unexpectedly, even in the limit of large deformation the\ngrowth of the wetting ridge persists.\nKnowing the shape of the wetting ridges and deformation within the elastic\nlayer allows us to calculate the rate of work in the substrate. By integration,\nwe can then find the dissipation of a moving wetting ridge. Our results show\nthat the global dissipation for opening angles near 180◦is indeed in perfect\nagreement with the predictions of linear theory; the convergence validates our\nnumerical method. For sharper wetting ridges we start to see effects of nonlinear\nelasticity, as would be expected for the larger deformations associated with the\nsharper ridges. The dissipation for thick substrates increases, but the minimum\nand asymptote for thin layers persist.\nSince the molecular structure of typical soft materials is not fully captured\nby a Neo-Hookean solid, we also considered a finite extensibility model. Such a\nmodel incorporates the limited extensibility of the polymer chains that make up\nthe bulk. We find that on thick substrates, finite extensibility does not signif-\nicantly impact the size of the wetting ridges, while on thin substrates it sets a\nmaximum height. Dissipation calculations show an asymptote that goes to zero\nin the thin substrate limit. So, the introduction of strong nonlinearities beyond\nthe Neo-Hookean solid are clearly of importance to capture the essentical fea-\nture of dynamical wetting experiments on very thin layers. However, the Gent\nmodel combined with linear viscoelastic dissipation does not yet offer a quan-\ntitative match to the experimental data. Future studies could be dedicated to\nadd the nonlinear stress of the Gent model into the dissipation model, to see if\na quantitative match can be achieved. Conversely, experiments on thinner lay-\n24ers combined with macroscopic nonlinear rheological characterisations (beyond\nG′/G′′) could be of interest for future work.\nA Deformation in terms of metric tensor\nIt will turn out convenient to express the relevant deformation tensors using\ncurvilinear material coordinates {qi}, with i= 1,2 for our two-dimensional\nproblem. We therefore provide the connection between the curvilinear formula-\ntion and the quantities that can be extracted from the numerical results. The\ndiscussion below is adapted from [43, 51].\nThe numerical simulations make use of the deformation gradient tensor\nF=∂x/∂X. We now consider a dynamical problem, for which the mapping\nx=f(X, t) becomes time-dependent. For this reason we from now on use the\nnotation x(t) and replace X=x(t0), where, without loss of generality, we con-\nsider the system to be in its reference configuration at t=t0. With this, Fand\nits inverse F−1can be expressed as\nF(t) =∂x(t)\n∂x(t0)=∂x(t)\n∂qi∂qi\n∂x(t0)=ei(t)⊗Ei, (38)\nF−1(t) =∂x(t)\n∂x(t0)=∂x(t0)\n∂qi∂qi\n∂x(t)=Ei⊗ei(t), (39)\nwhere we introduced the time-dependent curvilinear basis vectors\nei(t) =∂x(t)\n∂qi,ei(t) =∂qi\n∂x(t), (40)\nand the time-independent reference basis vectors\nEi=ei(t0),Ei=ei(t0). (41)\nIn what follows we need to compute the history of deformation by comparing\nstates at two different times, say the current time tand a time in the past t′< t.\nThe corresponding mapping F(t, t′) is defined by\nF(t)·F−1(t′) =ei(t)⊗Ei·Ej⊗ej(t′)\n=ei(t)⊗ei(t′) =∂x(t)\n∂x(t′)=F(t, t′). (42)\nWhen evaluating the associated Finger tensor, we obtain\nB(t, t′) =F(t, t′)·FT(t, t′)\n=F(t)·F−1(t′)·F−T(t′)·FT(t)\n=ei(t)⊗ei(t′)·ej(t′)⊗ej(t)\n=gij(t′)ei(t)⊗ej(t). (43)\n25Here we introduced the contravariant metric tensor gij, which is defined as\ngij(t) =ei(t)·ej(t), g ij(t) =ei(t)·ej(t), (44)\nand for completeness we also gave the covariant metric gij. It follows from (43)\nthat the two-time Finger tensor is obtained by expressing the contravariant\nmetric at time t′on the curvilinear basis at time t. Similarly the inverse of this\ntensor can be evaluated\nB−1(t, t′) =F−T(t)·FT(t′)·F(t′)·F−1(t)\n=gij(t′)ei(t)⊗ej(t). (45)\nNaturally, gij(t′) and gij(t′) act as one-anothers inverse, being related respec-\ntively to B−1andB.\nThe constitutive relations of viscoelastic materials involve time-derivatives\nof the two-time Finger tensor. These can be evaluated as\n∂B(t, t′)\n∂t′= ˙gij(t′)ei(t)⊗ej(t)\n=F(t)·\u0010\n˙F−1(t′)·F−T(t′) +F−1(t′)·˙F−T(t′)\u0011\n·FT(t),(46)\n∂B−1(t, t′)\n∂t′= ˙gij(t′)ei(t)⊗ej(t)\n=F−T(t)·\u0010\n˙FT(t′)·F(t′) +FT(t′)·˙F(t′)\u0011\n·F−1(t), (47)\nproviding the expression in both in symbolic notation and in curvilinear com-\nponents. Now we see the great advantage of the curvilinear formulation. When\nexpressed using the current basis vectors ei(resp. ei), the time-derivatives of\nB(resp. B−1) reduce to ordinary time-derivatives of the metric components\nin the past ˙ gij(resp. ˙ gij). By consequence, the constitutive relations take on\ncomparatively simple forms when expressed in curvilinear coordinates. Phrased\ndifferently, the curvilinear description can be seen as an elegant way to effectuate\nthe intricate projections between current and past configurations.\nFor completeness, we also provide the following connections (see [43] for\ndetails):\n˙F= (∇v)T·F, ˙F−1=−F−1·(∇v)T. (48)\nWe thus obtain,\n∂B−1(t, t′)\n∂t′= ˙gij(t′)ei(t)⊗ej(t)\n=F−T(t)·F(t′)T·\u0000\n∇v(t′) + (∇v)T(t′)\u0001\n·F(t′)·F−1(t)\n=F−T(t, t′)·˙γ(t′)·F−1(t, t′) (49)\nThe time-derivative of B−1thus reflects the shear-rate ˙γ=∇v+ (∇v)Tin the\npast, but projected on the current basis. When setting t=t′after taking the\n26derivative, we find\n∂B−1(t, t′)\n∂t′\f\f\f\f\nt′=t= ˙gij(t)ei(t)⊗ej(t) =∇v+ (∇v)T=˙γ(t) (50)\nSo, ˙gij(t) combined with the basis at the current time directly gives ˙γ(t).\nB Asymptotics\nWe now compute the dissipation near the contact line, assuming the deformation\nis described by a fold map. The reference domain is an elastic halfspace ( Y <0),\nwhich is flat initially, which in polar coordinates reads\n(X, Y)→(R,Θ) = (p\nX2+Y2,arctan( Y/X)), (51)\nwhere Rand Θ are the radial and tangential coordinates respectively. The\nincompressible fold map in the absence of surface shear can now be defined as\n[45]\n(r, θ) =\u0012R√\nb, bΘ +(b−1)π\n2\u0013\n. (52)\nwhere randθare the counterparts of Rand Θ in the deformed configuration,\nb=θs/πand one confirms that incompressibility is indeed satisfied ( rdrdθ =\nRdRd Θ). The deformed configuration in Cartesian coordinates can be expressed\nas,\n(r, θ)→(rcosθ, rsinθ) = (x, y), (53)\nfrom which, the deformation gradient tensor F=dx/dXcan be computed.\nThen, the metric quantities gij(t′), ˙gij(t′) and ˙ gij(t′) can be calculated from\nthe expressions (45), (46) and (47) respectively. With the metric expressions\nin hand, it is now possible to directly compute dissipation density (22). Using\nMathematica, we obtain (34) that is successfully compared to numerics.\nAcknowledgements\nWe thank Bruno Andreotti for discussions. M.H.E. and J.H.S. acknowledge\nfunding from NWO through VICI Grant No. 680-47-632, S.K. from the Ger-\nman research foundation (DFG, project no. 422877263). K.D.V. and H.K.K.\nacknowledge financial support from the Natural Science and Engineering Re-\nsearch Council of Canada, and HKK acknowledges funding from the Vanier\nCanada Graduate Scholarship.\nReferences\n[1] P.-G. de Gennes, F. Brochard-Wyart, and D. Qu´ er´ e. Capillarity and wet-\nting phenomena: Drops, bubbles, pearls, waves . New York: Springer, 2004.\nisbn: 9780387216560.\n27[2] D. Bonn et al. “Wetting and spreading”. In: Rev. Mod. Phys. 81 (2009),\npp. 739–805. doi:10.1103/RevModPhys.81.739 .\n[3] Jacco H. Snoeijer and Bruno Andreotti. “Moving Contact Lines: Scales,\nRegimes, and Dynamical Transitions”. In: Annu. Rev. Fluid Mech. 45.1\n(2013), pp. 269–292. doi:10.1146/annurev-fluid-011212-140734 .\n[4] B. Andreotti and J. H. Snoeijer. “Statics and Dynamics of Soft Wet-\nting”. In: Annu. Rev. Fluid Mech. 52 (2020), pp. 285–308. doi:10.1146/\nannurev-fluid-010719-060147 .\n[5] J. Bico, ´E. Reyssat, and B. Roman. “Elastocapillarity: when surface ten-\nsion deforms elastic solids”. In: Annu. Rev. Fluid Mech. 50 (2018), pp. 629–\n659.\n[6] Robert W. Style et al. “Universal Deformation of Soft Substrates Near a\nContact Line and the Direct Measurement of Solid Surface Stresses”. In:\nPhys. Rev. Lett. 110 (6 2013), p. 066103. doi:10.1103/PhysRevLett.\n110.066103 .\n[7] R. W. Style et al. “Patterning droplets with durotaxis”. In: Proc. Natl.\nAcad. Sci. U. S. A. 110 (2013), pp. 12541–12544. doi:10.1073/pnas.\n1307122110 .\n[8] S. J. Park et al. “Visualization of asymmetric wetting ridges on soft solids\nwith x-ray microscopy”. In: Nat. Commun. 5.1 (2014), p. 4369. doi:10.\n1038/ncomms5369 .\n[9] S. J. Park et al. “Self-spreading of the wetting ridge during stick-slip on\na viscoelastic surface”. In: Soft Matter 13 (2017), pp. 8331–8336. doi:\n10.1039/c7sm01408b .\n[10] Aditi Chakrabarti et al. “Elastowetting of Soft Hydrogel Spheres”. In:\nLangmuir 34.13 (2018), pp. 3894–3900. issn: 1520-5827. doi:10.1021/\nacs.langmuir.8b00368 .\n[11] Longquan Chen et al. “Static and dynamic wetting of soft substrates”.\nIn:Current Opinion in Colloid & Interface Science 36 (2018), pp. 46–57.\ndoi:10.1016/j.cocis.2017.12.001 .\n[12] R. D. Schulman et al. “Surface energy of strained amorphous solids”. In:\nNat. Commun. 9 (2018), p. 982. doi:10.1038/s41467-018-03346-1 .\n[13] Qin Xu et al. “Viscoelastic and Poroelastic Relaxations of Soft Solid Sur-\nfaces”. In: Physical Review Letters 125.23 (2020), p. 238002. doi:10 .\n1103/PhysRevLett.125.238002 .\n[14] Martin Coux and John M. Kolinski. “Surface textures suppress vis-\ncoelastic braking on soft substrates”. In: Proceedings of the National\nAcademy of Sciences 117.51 (2020), pp. 32285–32292. doi:10 . 1073 /\npnas.2008683117 .\n28[15] Dongshi Guan, Elisabeth Charlaix, and Penger Tong. “State and Rate De-\npendent Contact Line Dynamics over an Aging Soft Surface”. In: Physical\nReview Letters 124.18 (2020), p. 188003. issn: 1079-7114. doi:10.1103/\nphysrevlett.124.188003 .\n[16] Aaron Bardall et al. “Gradient-induced droplet motion over soft solids”.\nIn:IMA Journal of Applied Mathematics 85.3 (2020), pp. 495–512. issn:\n1464-3634. doi:10.1093/imamat/hxaa015 .\n[17] Binyu Zhao et al. “Elasticity-to-Capillarity Transition in Soft Substrate\nDeformation”. In: Nano Letters 21.24 (2021), pp. 10361–10367. doi:10.\n1021/acs.nanolett.1c03643 .\n[18] Zhuoyun Cai et al. “Fluid separation and network deformation in wetting\nof soft and swollen surfaces”. In: Communications Materials 2.1 (2021),\npp. 1–11. doi:10.1038/s43246-021-00125-2 .\n[19] Weiwei Zhao et al. “The role of crosslinking density in surface stress and\nsurface energy of soft solids”. In: Soft Matter 18.3 (2022), pp. 507–513.\ndoi:10.1039/d1sm01600h .\n[20] Zhuoyun Cai and Jonathan T. Pham. “How Swelling, Cross-Linking, and\nAging Affect Drop Pinning on Lubricant-Infused, Low Modulus Elas-\ntomers”. In: ACS Applied Polymer Materials 4.5 (2022), pp. 3013–3022.\nissn: 2637-6105. doi:10.1021/acsapm.1c01455 .\n[21] Niloofar Nekoonam et al. “Controllable Wetting Transitions on Pho-\ntoswitchable Physical Gels”. In: ACS Applied Materials and Interfaces\n15.22 (2023), pp. 27234–27242. issn: 1944-8252. doi:10.1021/acsami.\n2c22979 .\n[22] Lukas Hauer et al. “Phase Separation in Wetting Ridges of Sliding Drops\non Soft and Swollen Surfaces”. In: Physical Review Letters 130.5 (2023),\np. 058205. doi:10.1103/physrevlett.130.058205 .\n[23] Martin E. R. Shanahan and Alain Carre. “Anomalous Spreading of Liquid\nDrops on an Elastomeric Surface”. In: Langmuir 10.6 (1994), pp. 1647–\n1649. issn: 1520-5827. doi:10.1021/la00018a004 .\n[24] Alain Carre and Martin E. R. Shanahan. “Direct Evidence for Viscosity-\nIndependent Spreading on a Soft Solid”. In: Langmuir 11.1 (1995), pp. 24–\n26.issn: 1520-5827. doi:10.1021/la00001a007 .\n[25] M E R Shanahan and A Carre. “Viscoelastic dissipation in wetting and\nadhesion phenomena”. In: Langmuir 11.4 (1995), pp. 1396–1402.\n[26] A. Carr´ e, J. C. Gastel, and M. E. R. Shanahan. “Viscoelastic effects in\nthe spreading of liquids”. In: Nature 379 (1996), pp. 432–434. doi:10.\n1038/379432a0 .\n[27] D. Long, A. Ajdari, and L. Leibler. “Static and dynamic wetting properties\nof thin rubber films”. In: Langmuir 12 (1996), pp. 5221–5230. doi:10.\n1021/la9604700 .\n29[28] M. H. Zhao et al. “Geometrical control of dissipation during the spreading\nof liquids on soft solids”. In: Proc. Natl. Acad. Sci. U. S. A. 115.8 (2018),\npp. 1748–1753. doi:10.1073/pnas.1712562115 .\n[29] S. Karpitschka et al. “Droplets move over viscoelastic substrates by surf-\ning a ridge”. In: Nat. Commun. 6.1 (2015), p. 7891. doi:10 . 1038 /\nncomms8891 .\n[30] M. van Gorcum et al. “Spreading on viscoelastic solids: are contact angles\nselected by Neumann’s law?” In: Soft Matter 16 (2020), pp. 1306–1322.\ndoi:10.1039/c9sm01453e .\n[31] S. I. Tamim and J. B. Bostwick. “Model of spontaneous droplet transport\non a soft viscoelastic substrate with nonuniform thickness”. In: Physi-\ncal Review E 104.3 (2021), p. 034611. issn: 2470-0053. doi:10.1103/\nphysreve.104.034611 .\n[32] Saiful Tamim and Joshua B. Bostwick. “Spreading of a thin droplet on\na soft substrate”. In: Journal of Fluid Mechanics 971 (2023). issn: 1469-\n7645. doi:10.1017/jfm.2023.673 .\n[33] Mathieu Ol´ eron et al. “Morphology and stability of droplets sliding on soft\nviscoelastic substrates”. In: arXiv (2023). 2304.04925. doi:10.48550/\nARXIV.2304.04925 .\n[34] Stefan Karpitschka et al. “Soft wetting: Models based on energy dissipa-\ntion or on force balance are equivalent”. In: Proc. Natl. Acad. Sci. U.S.A.\n115.31 (2018), E7233–E7233.\n[35] H. K. Khattak et al. “Direct force measurement of microscopic droplets\npulled along soft surfaces”. In: Nat. Commun. (2022).\n[36] S. Aland and D. Mokbel. “A unified numerical model for wetting of soft\nsubstrates”. In: Int. J. Numer. Methods Eng. 122 (2021), pp. 903–918.\ndoi:10.1002/nme.6567 .\n[37] C Henkel et al. “Soft wetting with (a) symmetric Shuttleworth effect”. In:\nProc. R. Soc. A 478.2264 (2022), p. 20220132.\n[38] A. Pandey et al. “Singular Nature of the Elastocapillary Ridge”. In: Phys.\nRev. X 10.3 (2020), p. 031067. doi:10.1103/PhysRevX.10.031067 .\n[39] Gertjan van Zwieten et al. Nutils, version 6.2 . Version 6.2. available at\nhttps://dx.doi.org/10.5281/zenodo.4071707. 2020. doi:10.5281/zenodo.\n4071707 .\n[40] L. Limat. “Straight contact lines on a soft, incompressible solid”. In: Eur.\nPhys. J. E 35 (2012), p. 134. doi:10.1140/epje/i2012-12134-6 .\n[41] L. A. Lubbers et al. “Drops on soft solids: free energy and double transition\nof contact angles”. In: J. Fluid Mech. 747 (2014), R1. doi:10.1017/jfm.\n2014.152 .\n[42] A Wineman. “Nonlinear viscoelastic solids—a review”. In: Mathematics\nand mechanics of solids 14.3 (2009), pp. 300–366.\n30[43] J H Snoeijer et al. “The relationship between viscoelasticity and elastic-\nity”. In: Proc. R. Soc. Lond. A 476.2243 (2020), p. 20200419.\n[44] P Thirion and R Chasset. “Viscoelastic relaxation of rubber vulcan-\nizates in extension”. In: Chimie & Industrie, Genie Chimique 97.5 (1967),\npp. 617–626.\n[45] Manohar Singh and Allen C Pipkin. “Note on Ericksen’s problem”. In: Z.\nAngew. Math. Phys. 16.5 (1965), pp. 706–709.\n[46] V.M. Mal’kov and Yu.V. Mal’kova. “Analysis of a stress singularity in a\nnon-linear Flamant problem for certain models of a material”. In: Journal\nof Applied Mathematics and Mechanics 72.4 (2008), pp. 468–474. doi:\n10.1016/j.jappmathmech.2008.08.006 .\n[47] H. B. Wu et al. “Effect of large deformation and surface stiffening on the\ntransmission of a line load on a neo-Hookean half space”. In: Soft Matter\n14 (2018), pp. 1847–1855. doi:10.1039/c7sm02394d .\n[48] J. Dervaux, M. Roch´ e, and L. Limat. “Nonlinear theory of wetting on\ndeformable substrates”. In: Soft Matter 16 (2020), pp. 5157–5176. doi:\n10.1039/d0sm00395f .\n[49] Alan N Gent. “A new constitutive relation for rubber”. In: Rubber Chem.\nTechnol. 69.1 (1996), pp. 59–61.\n[50] Matt P Milner, Lihua Jin, and Shelby B Hutchens. “Creasing in\nevaporation-driven cavity collapse”. In: Soft Matter 13.38 (2017),\npp. 6894–6904.\n[51] Albert Edward Green and Wolfgang Zerna. Theoretical elasticity . Courier\nCorporation, 1992.\n31"    },    {        "title": "2402.06358v1.Robust_inference_for_an_interval_monitored_step_stress_experiment_under_proportional_hazards.pdf",        "content": "Robust inference for an interval-monitored step-stress\nexperiment under proportional hazards\nNarayanaswamy Balakrishnan, Mar´ ıa Jaenada and Leandro Pardo\nAbstract\nAccelerated life tests (ALTs) play a crucial role in reliability analyses, providing lifetime esti-\nmates of highly reliable products under normal conditions. Among the different types of ALTs, the\nstep-stress design incrementally increases the stress level at predefined times, while maintaining a\nconstant stress level between successive changes. This approach accelerates the occurrence of fail-\nures, effectively reducing experimental duration and cost. While many studies focusing on ALTs\nassume a specific form for the lifetime distribution; in certain applications instead a general form\nsatisfying certain properties, such as the proportional hazards requirement, should be preferred. In\nparticular, the proportional hazard model assumes that applied stresses act multiplicatively on the\nhazard rate, and thus the hazards function under increased stress may be divided into two factors,\nwith one representing the effect of the stress, and the other representing the baseline hazard. In this\nwork, we examine particular forms of baseline hazards, namely, linear and quadratic forms. More-\nover, certain experiments may face practical constraints, making continuous monitoring of devices\ninfeasible. Instead, devices under test are inspected at predetermined intervals, and resulting data\nare then grouped as counts of failures, leading to interval-censoring. On the other hand, recent\nworks have shown an appealing trade-off between the efficiency and robustness of divergence-based\nestimators in parametric inference under interval-censored data. This paper introduces the step-\nstress ALT model under proportional hazards and presents a robust family of minimum density\npower divergence estimators (MDPDEs) for estimating device reliability and related lifetime char-\nacteristics such as mean lifetime and distributional quantiles. The MDPDEs of related lifetime\ncharacteristics and their asymptotic distributions are derived, providing approximate confidence\nintervals for the parameters of interest. Empirical evaluations through Monte Carlo simulations\ndemonstrate their performance in terms of robustness and efficiency. Finrally, an illustrative exam-\nple is provided to demonstrate the usefulness of the model and the associated method developed\nhere.\n1 Introduction\nLife testing analysis is a major concern in reliability, with applications in diverse fields including en-\ngineering and biomedical sciences. Moreover, driven by customer expectations, nowadays products\nare designed to be highly reliable with substantially long lifetimes. Consequently, life-testing experi-\nments for such highly reliable products under normal operating conditions (NOC) would usually lead\nto few failures (if at all) in the experiment. As a result, to achieve accurate inference under NOC,\nlarge experimental times, and consequently, high cost,s are necessitated. To address this limitation,\naccelerated life-tests (ALTs) reduce the time to failure by exposing the experimental units to stronger\ntest conditions than NOC, causing a greater number of observed failures. The failure data under\nincreased stress levels can be accurately analyzed and subsequently, the results from the reliability\nanalysis conducted under increased stress conditions can be extrapolated to make inferences about the\nproduct’s performance under NOC. In particular, step-stress ALT plans are noteworthy. These plans\ninvolve increasing the stress factor at which units are exposed at certain (typically pre-specified) times,\nthereby subjecting all devices under test to the same stress patterns and heightening the probability\n1arXiv:2402.06358v1  [math.ST]  9 Feb 2024of failure. That is, the stress is increased and held constant between two successive stress levels for all\nnon-failed units. The step-stress ALT design enables efficient data collection and accurate inference\nwith smaller number of devices than under the popular constant-stress testing, and also would result\nin shorter experimental times. But, it necessitates a model relating the relationship between the wear\ncaused by preceding stresses and the current lifetime distribution.\nSeveral known distributions have been explored in the literature to describe the lifetimes of tested\nproducts, including the exponential, Weibull, gamma, log-normal or generalized gamma distributions.\nHowever, the true underlying distribution for the data is generally unknown and so none of these para-\nmetric distributions may be adequate to fit the observed data. Alternatively, Cox [1972]’s proportional\nhazards (PH) model offers a semi-parametric multiple regression approach for reliability estimation, in\nwhich the baseline hazard function is multiplicatively affected by the applied stresses. The PH model\nis distribution-free and relies on the assumption that the hazard rate ratio between two stress levels\nremains constant over time. This model can be particularly useful when no expert knowledge may be\navailable to select a specific parametric lifetime distribution family, but certain natural constraints,\nsuch as the constant hazard rate ratio, can be made. In this paper, we adopt the PH Cox model with\nlinear and quadratic forms for the baseline hazard function.\nAlthough step-stress tests are typically conducted with continuous monitoring of unit lifetimes,\nthere are situations where continuous monitoring is not feasible, but units are only inspected for failures\nat specific time intervals. In such cases, the observed data are grouped failure counts between two\nconsecutive inspection times, resulting in interval-censoring. This interval-censored set-up may arise\nin experiments where functional check requires a manual test, or in industrial experiments where cost\nlimitations make it more feasible to intermittently monitor the lifetime of units at specific intervals\nrather than on a time-continuous manner. For example, Balakrishnan et al. [2023a] applied the\ninterval-censored step-stress model for analyzing lifetime data from non-destructive tests with one-\nshot devices. Under this scenario, all data observed are censored, and so specific inferential methods\nneed to be developed to handle such situations effectively.\nIn parametric, inference the model parameters are typically estimated using maximum likelihood\nestimators (MLEs) due its good asymptotic properties. However, this method suffers from lack of\nrobustness and can be significantly affected by outliers present in the data The accuracy of the in-\nference procedure profoundly affects the reliability estimates at normal operating conditions and the\nsubsequent decisions regarding product configuration, warranties or preventive maintenance schedule.\nTherefore, contamination in the data can exert a strong negative impact on the results when employing\nthe MLE.\nIn recent years, considerable efforts have been made towards developing robust methods for\ninterval-censored ALT, most of them based on divergence measures (see, for example, Balakrish-\nnan et al. [2023a,c,b]). In this paper, we extend the robust approach proposed in the above mentioned\npapers and develop here robust estimators for interval-monitored ALTs based on divergence measures\nunder the PH model. To best of our knowledge, no research on inference for interval-censored ALTs\nwithin the PH model framework, either based on MLE or robust estimators, has been carried out so\nfar.\nThe rest of the paper is organized as follows: Section 2 revisits the step-stress ALT model under the\nproportional hazards assumption and introduces two functional forms for the baseline hazard—namely,\nlinear and quadratic baselines. In Section 3, a family of robust estimators based on density power\ndivergence (DPD) is defined, and explicit expressions of the model for the linear and quadratic forms\nare obtained. Additionally, the estimating equations and asymptotic distribution of the robust esti-\nmators are derived. In Section 4, three of the main lifetime characteristics, mean lifetime, reliability,\nand distributional quantiles, under both linear and quadratic baseline hazards are estimated, and as-\nsociated asymptotic distributions and confidence intervals are obtained by applying the Delta method.\nThe performance of the proposed estimators is then empirically evaluated in terms of accuracy and\n2robustness in Section 5. An illustrative example demonstrates the usefulness of the model and the\nmethod developed here in Section 6. Finally, some concluding remarks are made in Section 7.\n2 Step-stress ATL with proportional hazards\nLife testing of highly reliable products poses a challenge, but if a specific environmental factor influ-\nencing device reliability is identified, an ALT can be conducted to intentionally induce more failures\ncompared to a NOC. When a lifetime distribution is assumed, the functional relationship between the\nstress level and the devices reliability function can be established in terms of the model parameters.\nFor example, for scale parametric families, the scale parameter at a constant stress x, say λ,is usually\nassumed to be log-linearly related to the the stress level of the form\nλ(x;a) = log( a0+a1x), (1)\nwitha= (a0, a1)T∈R2.Proportional hazards (PH) models, first proposed by Cox [1972], facilitate a\nsemi-parametric regression approach for reliability estimation, in which the baseline hazard function\nis affected multiplicatively by the applied stresses. That is, if we denote by λ(t, x) the hazard rate\nfunction of a product lifetime under a constant stress x,then we can multiplicatively separate the\neffects of the natural product reliability and the stress level experienced as\nh(t, x) =h0(t)λ(x;a), (2)\nwhere λ(x;a) is the log-linear relation stated in (1). The PH model assumes that the ratio of hazard\nrates between two stress levels is constant over time, i.e.,\nh(t, x1)\nh(t, x2)=λ(x1;a)\nλ(x2;a)\nfor any two stress levels x1andx2.One of the advantages of the PH model is its flexibility to\nextend it to time-dependent covariates. On the other hand, since the stress level is increased during\nthe experimentation in step-stress ALTs, a model relating the lifetime distribution of a device at a\nspecific (current) and preceding stress levels becomes necessary. The natural underlying assumption\nis that the reliability of devices under test at a given stress level is somehow influenced by the stress\nlevels experienced earlier, causing it to wear and tear. Three main models have been discussed in\nthe literature for a such purpose. DeGroot and Goel [1979] proposed the tampered random variable\nmodel and discussed optimal tests under a Bayesian framework. Bhattacharyya and Soejoeti [1989]\nproposed the tampered failure rate model which assumes that the effect of change of stress is to\nmultiply the initial failure rate function by a factor subsequent to the stress change time. Finally,\nSedyakin [1966], Bagdonavicius [1978] and Nelson [1990] all adopted the cumulative exposure model.\nThe natural assumption underlying the cumulative exposure model has led to its widespread use\nadoption. Elsayed and Jiao [2002] adopted the cumulative exposure model for optimal step-stress\nALT under PH, and it is the approach that is employed here. Of course, the appropriate form of\nthese two models, the first relating the stress level to the failure rate (at constant stress) and the\nother relating the change on the reliability when increasing the stress will depend on the nature of the\nproduct and the assumptions made about its degradation or failure mechanism.\nThe cumulative exposure model states that the residual life of an experimental unit depends only\non the cumulative exposure it has experienced, with no memory of how this exposure was accumulated.\nThat is, the effect of increasing the stress level from x1tox2at time τon the lifetime distribution\nof the device is mathematically defined by a shift on the lifetime distribution such that continuity is\nensured, i.e.,\nF1(τ) =F2(τ+s),\n3where Fi(·) denotes the cumulative lifetime distribution (CDF) at constant stress level xiand the\nshifting time s <0 represents the accumulated damage until the time of stress change. Alternately,\nthe previous condition can be stated in terms of the cumulative hazard function at constant stress\nlevel xi, denoted in the following by Hi(·), i= 1,2,as follows:\nH1(τ) =H2(τ+s),\nwhere Hi(t) =−log(1−Fi(t)), i= 1,2,andsis the shifting time.\nBringing together the PH and the cumulative exposure assumptions, the shifting time smust\nsatisfy the following relationship\nH1(τ) =λ(x1,a)Zτ\n0h0(t)dt=λ(x2,a)Zτ+s\n0h0(t)dt=H2(τ+s). (3)\nNote that the hazard rate of the lifetime over the step-stress experiment may not be continuous at\ntime τ,and the continuity assumption is inherited only by the cumulative hazard function. From the\nabove formula, the cumulative hazard function of the device lifetime, T,under a simple step-stress\nmodel with stress levels x1andx2,is given by\nHT(t) =(\nH1(t) =λ(x1,a)Rt\n0h0(u)du, 0< t < τ,\nH2(t+s) =λ(x2,a)Rt+s\n0h0(u)du, τ < t < ∞,(4)\nand consequently the reliability of the device can be obtained as\nRT(t) =(\nR1(t) = exp( −H1(t)), 0< t < τ,\nR2(t+s) = exp( −H2(t+s)), τ < t < ∞,(5)\nwhere the shifting time s=H−1\n2(H1(τ))−τ. In practical use, the shifting time could be estimated\nnon-parametrically or by assuming a specific form for the baseline hazard. Because the observed data\nare usually censored, the non-parametric estimator may not be accurate. For this reason, we will\nassume here linear and quadratic functions for the baseline hazard.\nThe assumption of linear and quadratic forms is made here with the hope that the true (but\nunknown) hazard may be satisfactorily approximated by a polynomial of degree one or two (low\ndegree). A reasonable justification is derived from the fact that some hazard functions from known\ndistributions may be expanded in a converging power series. Therefore, the linear and quadratic\nforms can be seen as first- and second-order Taylor approximations of any underlying general hazard\nfunction. Furthermore, in some sense, the use of a polynomial transformation is a non-parametric\ntechnique, and consequently may be considered as an alternative to other non-parametric estimates.\nThe assumption of a polynomial hazard rate is equivalent to the assumption of polynomial ex-\nponential transformation of the lifetime adopted in Krane [1963] and, as pointed out therein, it is\nsuitable for industrial property survival data (see NARUC [1938], AGA-EEI [1942] reports).\n2.1 Linear hazards\nLet us consider linear form for the baseline hazard as\nh0(t) =γ0+γ1t, t > 0, (6)\nwhere γ= (γ0, γ1) are the linear relationship coefficients. Using the same notation as in Section 2,\nthe hazard rate function of a device lifetime subjected to constant stress xunder PH is given by\nh(t, x) = (γ0+γ1t) exp ( a1x), (7)\n4where a= (a0, a1)∈R2are the coefficients of the log-linear relationships in (1) relating the stress level\nto the devices lifetimes. However, note that the intercept a0becomes superfluous in Eq. (7) under\nlinear baseline assumption, since its multiplicative role can be incorporated into the baseline hazard\nparameters γ.Therefore, for parametric inference on the reliability of the product under a constant\nstress, we would need to estimate both relationships coefficients in (7) resulting in a 3-dimensional\nmodel parameter θ= (γ, a1).\nNow, the corresponding cumulative hazard and reliability functions at constant stress xare given,\nrespectively, by\nHθ(t, x) =\u0012\nγ0t+γ1t2\n2\u0013\nexp (a1x) and Rθ(t, x) = exp\u0014\n−\u0012\nγ0t+γ1t2\n2\u0013\nexp (a1x)\u0015\n.\nUnder the simple step-stress experiment with two stress levels x1andx2,the cumulative lifetime\nhazards at constant stresses are combined to yield the step-stress cumulative hazard function as\nHθ(t) =\n\nexp (a1x1)\u0010\nγ0t+γ1t2\n2\u0011\n, 0< t < τ,\nexp (a1x2)\u0010\nγ0(t+s) +γ1(t+s)2\n2\u0011\n, τ < t < ∞,(8)\nwhere sis the solution of the second-order equation\nγ1\n2(τ+s)2+γ0(τ+s)−exp (a1(x1−x2))\u0012\nγ0τ+γ1τ2\n2\u0013\n= 0. (9)\nThe above equation has at least one real root, since all coefficients are positive and so, the dis-\ncriminant of the second-order equation is also positive. Moreover, it has at least one negative solution\nand so we will choose the greater negative solution of the above equation to ensure that s <0 and\nτ+s >0.\nThe reliability function of a device under test will be given as\nRθ(t) =\n\nexp\u0010\n−exp (a1x1)\u0010\nγ0t+γ1t2\n2\u0011\u0011\n, 0< t < τ,\nexp\u0010\n−exp (a1x2)\u0010\nγ0(t+s) +γ1(t+s)2\n2\u0011\u0011\n, τ < t < ∞,(10)\nwhere ssatisfies the second-order equation in (9).\nThe model parameter should satisfy some natural restrictions. Firstly, because the hazard function\nis positive at any time t >0,both coefficients γ0andγ1in (6) must be positive. Moreover, under\npositive and increasing stress levels, the reliability of the devices should decrease when increasing the\nstress level. Then, the regression coefficient in (1), namely a1,should also be positive. Summarizing,\nassuming increasing and positive stress levels, the parameter space now becomes Θ = R3+.\n2.2 Quadratic hazards\nLet us now assume a quadratic form for the baseline hazard function, as\nh0(t) =γ0+γ1t+γ2t2, t > 0, (11)\nwhere γ= (γ0, γ1, γ2) are the quadratic relationship coefficients. Applying the PH condition, the\nhazard rate function of a device lifetime subjected to a constant stress xis then given by\nh(t, x) = (γ0+γ1t+γ2t2) exp ( a1x)\n5and the corresponding cumulative hazard function is\nHθ(t) =\n\nexp (a1x)\u0010\nγ0t+γ1t2\n2+γ2t3\n3\u0011\n, 0< t < τ,\nexp (a1x)\u0010\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0011\n, τ < t < ∞,(12)\nwhere sis the solution of the equation\nγ2\n3(τ+s)3+γ1\n2(τ+s)2+γ0(τ+s)−exp (a1(x1−x2))\u0012\nγ0τ+γ1τ2\n2+γ2τ3\n3\u0013\n= 0. (13)\nThe intercept of the log-linear relationship has again been omitted here because its multiplicative\neffect is represented by the coefficients of the quadratic baseline hazard function. Equation (13) has at\nleast one real solution. As before, we may choose sas the greatest negative solution of the equation.\nNote that under quadratic baseline hazards, the model parameter is a 4-dimensional vector θ= (γ, a),\nwithγanda1being the coefficients defining the multiplicative factors in Equations (11) and (1). As\nbefore, there exist natural restrictions on the parameter space; the reliability of the devices should\ndecrease when increasing the stress level, and the baseline hazard must be positive. To simplify the\nparameter constraints, we assume that all the coefficients in Equation (11) are positive and so is a1\nfor the log-linear factor. This is a strong assumption, but realistic in many situations, as suggested in\nKrane [1963]. Therefore, the parameter space of θ= (γ, a1) gets reduced to Θ = R4+.\nExponentiating the negative cumulative hazard function we obtain the reliability function of a\ndevice under test under quadratic baseline hazards as\nRθ(t) =\n\nexp\u0010\n−exp (a1x1)\u0010\nγ0t+γ1t2\n2+γ2t3\n3\u0011\u0011\n, 0< t < τ,\nexp\u0010\n−exp (a1x2)\u0010\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0011\u0011\n, τ < t < ∞,(14)\nwhere the shifting time ssatisfies Equation (13).\n3 Robust estimators\nConsider a interval-monitored simple step-stress ALT model with two stress levels, x1andx2,switched\nat a pre-fixed time τ.A total of Ndevices are tested and the number of failures is recorded at Lfixed\ninspections times IT∈ {0 =t0< t1<···< tL},including the time of stress change τ=tkfor a\ncertain k < L. Here, kis the number of inspection times under the first stress and L−kis the number\nof inspection times under the second stress level. Note that the experiment termination time is also\nfixed as tL(Type I censoring design). Then, the numbers of devices that have failed within the j-th\ninspection interval, [ tj−1, tj) are recorded as njand the remaining nL+1=N−PL\nj=1njis the number\nof surviving units after the termination time. The data recorded from such as step-stress experiment\nis summarized in Table 1.\nBecause the observed data are interval-censored, the complete likelihood of the model will not\nbe useful. Instead, we can define the incomplete likelihood based on a multinomial model of the\ngrouped data as follows. For each device under test, we consider the L+ 1 possible experimental\nresults corresponding to failing within inspected intervals [ tj−1, tj), j= 1, ..., L, and surviving after the\nexperiment termination time. The probability of failure within an interval [ tj−1, tj) can be determined\nby evaluating the reliability of the lifetime as\nπj(θ) =Rθ(tj−1)−Rθ(tj) = exp ( −Hθ(tj−1))−exp (−Hθ(tj)), (15)\nand the probability of survival at termination time tLis given by\nπL+1(θ) =Rθ(tL) = exp ( −Hθ(tL)).\n6IT Stress level Number of failures\nt0= 0 x1 -\nt1 x1 n1\n.........\ntk−1 x1 nk−1\ntk x2 nk\n.........\ntL x2 nL\nTable 1: Step-stress ALT design with interval-monitoring\nUsing the expressions given in Equations (6) and (11) for linear and quadratic baseline hazards,\nrespectively, we can compute the probabilities of each experimental result under both linear and\nquadratic baseline hazard assumptions.\nThen, adopting the multinomial approach with probability vector π(θ) = ( π1(θ), ..., π L+1(θ))T,\nthe incomplete log-likelihood of the data is given by\nL(θ)∝L+1X\nj=1njlog (πj(θ)), (16)\nand consequently the maximum likelihood estimator (MLE) is the maximizer (or equivalent by min-\nimizer) of the previous log-likelihood (or negative log-likelihood). The MLE is a commonly used\nestimator due to its well known asymptotic properties; it is a consistent and efficient estimator. How-\never, it is highly sensitive to outliers, which can lead to erroneous analysis and results in the presence\nof contamination in data. Alternatively, divergence-based estimators have a great advantage in terms\nof robustness with a small loss in efficiency for general statistical models. In particular, Balakrish-\nnan et al. [2023a,c,b] developed robust minimum density power divergence estimators (MDPDEs)\nfor non-destructive one-shot devices tested under step-stress experiments under the assumption of\nexponential, gamma and lognormal lifetime distributions, respectively. They demonstrated that the\nMDPDE family provides a bridge between efficient and robust estimators. Following on from previous\nwork, we develop here the MDPDE under the PH assumption.\nThe DPD between two distributions aims to quantify the statistical closeness between the two\nassociated random variables; The greater DPD they have, the more distinct the variables would be.\nIn contrast, the DPD would be small (but always positive) for similar distributions. Then, when\nassuming a parametric model, we should look for a distribution that is as close as possible to the true\nunderlying distribution. The DPD between two probability mass functions p= (p1, ..., p L+1)Tand\nq= (q1, ..., q L+1)Tis given, for β >0, by\ndβ(p,q) =L+1X\nj=1qβ+1\nj−\u0012\n1 +1\nβ\u0013L+1X\nj=1qβ\njpj+1\nβL+1X\nj=1pβ+1\nj.\nThe DPD can be defined at β= 0 by taking continuous limit as β→0 and it yields the well-known\nKullback-Leibler divergence given by\ndKL(p,q) =L+1X\nj=1pjlog\u0012pj\nqj\u0013\n.\nAs the true underlying distribution is unknown, we can approximate it non-parametrically through\nits empirical probability vector bp=\u0000n1\nN, ...,nL\nN,nL+1\nN\u0001Tand the DPD between the empirical and the\n7assumed distribution, bpandπ(θ),is then given by\ndβ(bp,π(θ)) =L+1X\nj=1\n\nπj(θ)β+1−\u0010\n1 +1\nβ\u0011\nπj(θ)βbpj+1\nβbpβ+1\nj, β > 0,\nbpjlog\u0010bpj\nπj(θ)\u0011\n, β = 0.(17)\nThe closer the empirical probability and the assumed probability vector are, the better fit the\nassumed model provides for the observed data. Therefore, the MDPDE is defined as the minimizer of\nthe DPD loss as follows:\nbθβ= min θ∈Θdβ(bp,π(θ))\nwith Θ = R2+×R×R+,for linear baseline and Θ = R3+×R×R+for the quadratic baseline. Observe\nthat the negative log-likelihood in (16) and Kullback-Leibler divergence ( β= 0) in (17) are equivalent\nas objective functions. Moreover, the MLE is included in the MDPDE family as a particular case for\nβ= 0,and all the asymptotic properties of the MLE are derived in the following section by specializing\natβ= 0.\n3.1 Linear baseline\nWe study the asymptotic properties of the MDPDEs for interval-monitored step-stress ALTs under\nlinear baseline PH first. The asymptotic distribution of the MLE as well as an explicit expression of\nthe Fisher information matrix of the incomplete model will be obtained as particular cases for our\ngeneral results.\nFirst, we introduce some useful notation. Following Expression (15), the probability of failure\nwithin an inspected interval ( tj−1, tj] is given by\nπj(θ) =\n\nexp\u0012\n−exp(a1x1)\u0012\nγ0tj−1+γ1t2\nj−1\n2\u0013\u0013\n−exp\u0012\n−exp(a1x1)\u0012\nγ0tj+γ1t2\nj\n2\u0013\u0013\n, j≤k,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(tj−1+s) +γ1(tj−1+s)2\n2\u0011\u0011\n−exp\u0010\n−exp(a1x2)\u0010\nγ0(tj+s) +γ1(tj+s)2\n2\u0011\u0011\n, k ≤j≤L,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(tj+s) +γ1(tj+s)2\n2\u0011\u0011\n, j =L+ 1,\n(18)\nwhere the shifting time sis a negative solution of Equation (9). Because the MDPDE with tuning\nparameter βis a minimum of the DPD loss, as defined in (17), it must annul the first derivatives of\nthe loss. The following result presents the estimating equations of the MDPDE for interval-monitored\nstep-stress ALTs under the assumption of linear proportional hazards.\nResult 1 Let us consider a sample (n1, ...n L+1)from a simple interval-monitored step-stress ALT with\nLinspection times and N=PL+1\nj=1njdevices under test. We define the non-parametric estimator of\nthe probability as bp=\u0000n1\nN, ...,nL\nN,nL+1\nN\u0001\n.Then, the MDPDE with parameter βunder the assumption\nof linear proportional hazards, bθβ,must annul the β-score function defined by\nUβ(θ) =W(θ)TD(θ)β−1(bp−π(θ)), (19)\nwhere the matrix W(θ)is the 4×(L+ 1) Jacobian matrix of the theoretical probability of failure π(θ)\nas defined in (15), with columns given by\nwj=(∂Rθ(tj−1,x1)\n∂θ−∂Rθ(tj,x1)\n∂θ, tj−1< τ,\n∂Rθ(tj−1,x2)\n∂θ−∂Rθ(tj,x2)\n∂θ, τ≤tj−1,\n8where the four components of the gradient vector of the reliability function,\n∂Rθ(t, x)\n∂θ=\u0012∂Rθ(t, x)\n∂γ0,∂Rθ(t, x)\n∂γ1,∂Rθ(t, x)\n∂a1\u0013T\nare given, under the first stress level x1, by\n∂Rθ(t, x1)\n∂γ0=−Rθ(t, x1) exp( a1x1)t,\n∂Rθ(t, x1)\n∂γ1=−Rθ(t, x1) exp( a1x1)t2\n2,\n∂Rθ(t, x1)\n∂a1=−Rθ(t, x1) exp( a1x1)tk2(t)x2,\nand under the increased stress level x2by\n∂Rθ(t, x2)\n∂γ0=−Rθ(t, x2) exp( a1x2)\u0012\n(t+s) +γ1\n2(τ+s)s\nk2(τ)k1(t+s)\nk1(τ+s)\u0013\n,\n∂Rθ(t, x2)\n∂γ1=−Rθ(t, x2) exp( a1x2)\u0012(t+s)2\n2−γ0\n2(τ+s)s\nk2(τ)k1(t+s)\nk1(τ+s)\u0013\n,\n∂Rθ(t, x2)\n∂a1=−Rθ(t, x2) exp( a1x2)\u0014\n(t+s)k2(t+s)x2+k1(t+s)\nk1(τ+s)(τ+s)k2(τ+s)(x1−x2)\u0015\nand the functions k1(t)andk2(t)are\nk1(t) =γ1t+γ0 and k2(t) =γ1\n2t+γ0.\nByD(θ),we denote a diagonal matrix with diagonal entries given by the probability vector π(θ).\nProof. See Appendix A.1\nFrom the above result, the MPDPE can be defined as a solution of the system of equations\nUβ(θ) =04,\nknown as the estimating equations associated with the MDPDE, and thus belongs to the family of\nM-estimators. Moreover, we can derive the score function of the MLE by setting β= 0 in Equation\n(19), yielding\nUβ=0(θ) =W(θ)TD(θ)−1(bp−π(θ)).\nNotably, the matrices W(θ) and D(θ) do not depend on the tuning parameter βof the DPD loss, and\nthus the MDPDE estimating equations are a weighted version of the maximum likelihood estimating\nequations with weights equal to πj(θ)β,representing the probability of failure within the j-th interval.\nThus, the MDPDE down-weighs the observations on intervals with low probabilities.\n93.2 Quadratic baseline\nIn this subsection, we derive the estimating equations and the asymptotic distribution of the MDPDE\nunder quadratic baseline hazard. The primary equations can be obtained in a manner similar to\nthe linear case, but it is important to note that the probabilities of failure (and consequently their\nderivatives) differ and must be computed separately. The parameter θunder the assumption of\nquadratic hazard defined in (11) refers to a 4 −dimensional vector with entries ( γ0, γ1, γ2, a1),the first\nthree representing the baseline hazard and the last entry representing the relationship of the lifetime\ndistribution and the stress factor to which the units are subjected to.\nUsing the notation in Section 2.2, under quadratic baseline hazard, the probability of failure within\nan interval ( tj−1, tj] is given by\nπj(θ) =\n\nexp\u0012\n−exp(a1x1)\u0012\nγ0tj−1+γ1t2\nj−1\n2+γ2t2\nj−1\n3\u0013\u0013\n−exp\u0012\n−exp(a1x1)\u0012\nγ0tj+γ1t2\nj\n2+γ2t3\nj\u0013\u0013\n, j ≤k,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(tj−1+s) +γ1(tj−1+s)2\n2+γ2(tj−1+s)3\n3\u0011\u0011\n−exp\u0010\n−exp(a1x2)\u0010\nγ0(tj+s) +γ1(tj+s)2\n2+γ2(tj+s)3\n3\u0011\u0011\n, k≤j≤L,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(tj+s) +γ1(tj+s)2\n2+γ2(tj+s)3\n3\u0011\u0011\n, j =L+ 1,(20)\nwhere sis the shifting time obtained as a solution of Equation (13).\nResult 2 Given a sample (n1, ...n L+1)from a simple interval-monitored step-stress ALT with Lin-\nspection times and N=PL+1\nj=1njdevices under test, the MDPDE with parameter βunder the assump-\ntion of quadratic proportional hazard, bθβ,must annul the β-score function defined by\nUβ(θ) =W(θ)TD(θ)β−1(bp−π(θ)), (21)\nwherebp=\u0000n1\nN, ...,nL\nN,nL+1\nN\u0001Tis the empirical probability of failure, the matrix W(θ)is the 5×(L+1)\nJacobian matrix of the theoretical probability of failure π(θ)defined in Equation (15), with columns\ngiven by\nwj=(∂Rθ(tj−1,x1)\n∂θ−∂Rθ(tj,x1)\n∂θ, tj−1< τ,\n∂Rθ(tj−1,x2)\n∂θ−∂Rθ(tj,x2)\n∂θ, τ≤tj−1,\nwhere the five components of the gradient vector of the reliability function,\n∂Rθ(t, x)\n∂θ=\u0012∂Rθ(t, x)\n∂γ0,∂Rθ(t, x)\n∂γ1,∂Rθ(t, x)\n∂γ2,∂Rθ(t, x)\n∂a1\u0013T\nare given, under the first stress level x1, by\n∂Rθ(t, x1)\n∂γ0=−Rθ(t, x1) exp( a1x1)t,\n∂Rθ(t, x1)\n∂γ1=−Rθ(t, x1) exp( a1x1)t2\n2,\n∂Rθ(t, x1)\n∂γ2=−Rθ(t, x1) exp( a1x1)t3\n3,\n∂Rθ(t, x1)\n∂a1=−Rθ(t, x1) exp( a1x1)tk2(t)x2,\n10and under the increased stress level x2by\n∂Rθ(t, x2)\n∂γ0=−Rθ(t, x2) exp( a1x2)\u0012\n(t+s) + (τ+s)k1(t+s)\nk1(τ+s)\u0012k2(τ+s)\nk2(τ)−1\u0013\u0013\n,\n∂Rθ(t, x2)\n∂γ1=−Rθ(t, x2) exp( a1x2)\u0012(t+s)2\n2+(τ+s)\n2k1(t+s)\nk1(τ+s)\u0012\nτk2(τ+s)\nk2(τ)−(τ+s)\u0013\u0013\n,\n∂Rθ(t, x2)\n∂γ2=−Rθ(t, x2) exp( a1x2)\u0012(t+s)3\n3+(τ+s)\n3k1(t+s)\nk1(τ+s)\u0012\nτ2k2(τ+s)\nk2(τ)−(τ+s)2\u0013\u0013\n,\n∂Rθ(t, x2)\n∂a1=−Rθ(t, x2) exp( a1x2)\u0014\n(t+s)k2(t+s)x2+k1(t+s)\nk1(τ+s)(τ+s)k2(τ+s)(x1−x2)\u0015\n,\nand the functions k1(t)andk2(t)are\nk1(t) =γ2t2+γ1t+γ0 and k2(t) =γ2\n3t2+γ1\n2t+γ0.\nMoreover, D(θ)denotes a diagonal matrix with diagonal entries given by the probability vector π(θ).\nProof. See Appendix A.2\nAs in the linear case, the score function of the MLE can be obtained by specializing at β= 0 and\nthe MDPDE estimating equations Uβ(θ) =05can be viewed as a weighted version of the efficient\nmaximum likelihood score, with weights πj(θ).That is, for positive values of the tuning parameter\nβ, it provides a relative-to-the-model down-weighting for outlying observations; higher-than-expected\nfailure counts that are considerably discrepant with respect to the model will get nearly zero weights.\nIn the most efficient case ( β= 0), all observations, including very severe outliers, get weights equal to\none.\n3.3 Asymptotic properties\nWe next derive the asymptotic distribution of the MDPDE under linear and quadratic baseline haz-\nards. This asymptotic convergence is a key result for two reasons: Firstly, it demonstrates the consis-\ntency of the estimator, and secondly it facilitates the derivation of asymptotic confidence intervals for\nparameters of interest.\nResult 3 Letθ∗be the true value of the parameter θandπ(θ∗)denote the true probability mass vector\nof the multinomial model. Then, the asymptotic distribution of the MDPDE, bθβ,for the interval-\nmonitored step-stress ALT model under PH, is given by\n√\nN\u0010\nbθβ−θ∗\u0011L− − − − →\nN→∞N(0,Σβ(θ∗)),\nwhere the asymptotic variance-covariance matrix is given by\nΣβ(θ) =J−1\nβ(θ)Kβ(θ)J−1\nβ(θ) (22)\nwith\nJβ(θ) =W(θ)TD(θ)β−1W(θ),\nKβ(θ) =W(θ)T\u0010\nD(θ)2β−1−π(θ)β(π(θ)β)T\u0011\nW(θ),(23)\nandW(θ)is as defined in Result 1 for the linear baseline hazard and as in Result 2 for the quadratic\nbaseline hazard.\n11Proof. The proof follows similar lines to the proof of Result 3 in Balakrishnan et al. [2023a].\nBecause the MDPDE at β= 0 coincides with the MLE, the inverse of the Fisher information\nmatrix of the model is given by\nI−1\nF(θ) =Σβ=0(θ) =J−1\nβ=0(θ)Kβ=0(θ)J−1\nβ=0(θ) =W(θ)TD(θ)−1W(θ). (24)\nMoreover, from the above asymptotic distribution, the standard errors for all MDPDEs with tuning\nparameter β,bγβ\ni, i= 0,1,2,for the baseline hazard parameter or baβ\n1for the log-linear relation with\nthe stress level, can be obtained as the diagonal entries of the asymptotic variance-covariance matrix\nΣβ(θ) given in (22). These standard errors will be denoted by\nσ2\nβ(γ∗\ni) =Σβ(θ∗)i+1,i+1, i= 0,1,2,\nfor the baseline hazard parameters, and\nσ2\nβ(a∗\n1) =Σβ(θ∗)3,3orσ2\nβ(a∗\n1) =Σβ(θ∗)4,4\nfor the log-linear relationship parameters under linear and quadratic baseline hazard functions, respec-\ntively. A consistent estimator of these standard errors can be estimated by plugging-in any consistent\nestimator of the true parameter. Here, we will choose the same value of the tuning parameter βfor the\nMDPDE, ensuring robust estimation of the standard estimation error for positive values of the tuning\nparameter. However, any other estimator could be used instead. For example, for a more efficient\n(but non-robust) estimation, we could make use of the MLE for estimating the standard error of any\nMDPDE.\nFrom the above discussion, we can build approximate 100(1 −α)% confidence intervals for the\nbaseline hazard parameters γi(fori= 0,1,2) and the stress-factor relationship baβ\n1as follows. For the\nbaseline hazard parameters, γi,fori= 0,1,2,the approximate confidence intervals are given by\nIC1−α[γi] =h\nbγβ\ni±zα/2σβ(bγβ\ni)i\n,\nand similarly, for the stress-factor relationship parameter, baβ\n1,we have\nIC1−α[a1] =h\nbaβ\n1±zα/2σβ(baβ\n1)i\n.\nHere, zαdenotes the upper α-quantile of a standard normal distribution. If any of these estimated\nintervals extend beyond the boundaries of a parameter space, such as the requirement for baseline\nhazard parameters to be positive, the extreme of intervals need to be truncated.\n4 Point estimation of lifetime characteristics\nMany reliability analyses aim to estimate specific lifetime characteristics rather than the complete\nlifetime distribution function. Specifically, inferring the median, mean lifetime, distribution quantiles,\nand reliability at fixed mission times are often of interest under NOC. Once the model parameters θ=\n(γ,a) have been estimated, these quantities can be readily estimated by substituting the parameter\nestimates into the distribution function and then computing the corresponding lifetime characteristic of\ninterest. As we assume general linear or quadratic baseline hazard functions, the derivation of explicit\nexpressions for the previously mentioned lifetime characteristics under both baseline forms is necessary.\nIn this section, we develop explicit expressions for estimating the mean lifetime, distribution quantiles\n(including the median of the distribution), and reliability at fixed mission times under NOC based on\nthe MDPDEs. We also present their asymptotic properties, inherited from the robust estimators, as\nwell as asymptotic confidence intervals.\n124.1 Linear baseline hazard\nWe first assume linear baseline hazard function defined in Equation (6), with unknown parameters\nγ= (γ0, γ1).The cumulative distribution function of the lifetime Tunder normal operating conditions\nx0is then given by\nFT(t) = 1−exp\u0012\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2\u0013\u0013\n,0< t < ∞, (25)\nwhere exp ( a1x0) represents the multiplicative effect of the stress level under NOC.\n4.1.1 Mean lifetime to failure\nThe mean lifetime is often a key parameter in reliability analyses because it provides valuable insight\ninto the performance and durability of the considered product. It may play an important role in\ndecision-making, risk assessment, and quality control across various industries. Using the expression\nof the c.d.f. in (25), the mean of the lifetime of the product is given by\nEθ(T) =1\n2s\n2π\nγ1exp(a1x0)exp\u0012exp(a1x0)γ2\n0\n2γ1\u0013\n. (26)\nDetailed calculations for the mean lifetime are provided in Appendix B.1. Consequently, the MDPDE\nof the mean lifetime with parameter βcan be computed readily as\nEbθβ[T] =1\n2s\n2π\nbγβ\n1exp(baβ\n1x0)exp \nexp(baβ\n1x0)(bγβ\n0)2\n2bγβ\n1!\n,\nwherebθβ= (bγβ\n0,bγβ\n1,baβ\n1) is the MDPDE of the model parameter θ.As the estimated mean lifetime can\nbe expressed in terms of a differentiable function of the MDPDE, the asymptotic distribution of the\nMDPDE of the mean lifetime, ET(bθβ),can be obtained directly by applying the Delta-method. In\nparticular, it can be shown that the MDPDE of the mean lifetime is such that\n√\nN\u0010\nEbθβ[T]−Eθ∗[T]\u0011L− − − − →\nN→∞N(0, σβ(ET(θ∗))),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n1) is the true value of the model parameters,\nσ2\nβ(ET(θ∗)) =\u0014∂Eθ[T]\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Eθ[T]\n∂θ\u0015\nθ=θ∗,\nwithΣβ(θ) being defined in Equation (22) and\n\u0014∂Eθ[T]\n∂θ\u0015\nθ=θ∗=1\n2ET(θ∗)\u0012γ∗\n0\nγ∗\n1exp(+ a∗\n1x0),1\nγ∗\n1\u0014\n−1−γ2\n0\nγ1exp(+ a∗\n1x0)\u0015\n,\n\u0014\n−1 +(γ∗\n0)2\nγ∗\n1exp(+ a∗\n1x0)\u0015\nx0\u0013T\n.(27)\nExplicit derivation of the above derivatives are given in Appendix B.1. As the MDPDE bθis a consistent\nestimator of the true model parameter, the standard error of the estimate can be approximated by\nσ2\nβ(Ebθβ[T])\nNow, based on the asymptotic distribution of the MDPDE of the mean lifetime, an approximate\n100(1−α)%-confidence interval for the mean lifetime can be obtained as\nIC1−α[Eθ[T]] =h\nEbθβ[T]±zα/2σβ(Ebθβ[T])i\n.\nBecause the mean lifetime is a positive quantity, if the lower bound of the approximate confidence\ninterval is negative, it should be truncated to 0 .\n134.1.2 Reliability\nEstimating the reliability of the product at a certain mission time may help manufacturers to ensure\nproduct performance, safety or cost-effectiveness of the product at a particular time horizon of interest.\nLet us consider the reliability function of the product at constant stress x0under the assumption of\nlinear baseline hazard function as in (14). The estimated reliability of the product based on the\nMDPDE bθβis then given by\nRbθβ(t0) = exp\u0012\n−exp\u0010\nbaβ\n1x0\u0011\u0012\nbγβ\n0t0+bγβ\n1t2\n0\n2\u0013\u0013\n,0< t < ∞, (28)\nwhere t0denotes the fixed mission time.\nTo derive approximate confidence intervals for the product’s reliability at a specific mission time,\nwe can employ the asymptotic distribution of the function Rbθβ(t0).This distribution can be derived\nusing the Delta method as follows:\n√\nN\u0010\nRbθβ(t0)−Rθ∗(t0)\u0011L− − − − →\nN→∞N(0, σβ(Rθ∗(t0))),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n1) is the true value of the model parameters,\nσ2\nβ(Rθ∗(t0)) =\u0014∂Rθ(t0)\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Rθ(t0)\n∂θ\u0015\nθ=θ∗,\nwithΣβ(θ) being as defined in Equation (22) and\n∂Rθ(t0)\n∂θ=−Rθ(t0) exp\u0010\naβ\n0+a1x0\u0011\u0012\nt0,t2\n0\n2,\u0012\nγ0t0+γ1t2\n0\n2\u0013\nx0\u0013T\n. (29)\nThe standard error of the reliability estimate, denoted above as σβ(Rθ∗(t0)),can be robustly and\nconsistently estimated using a MDPDE. We can then build an approximate 100(1 −α)% confidence\ninterval as follows:\nIC1−α[Rθ(t0)] =h\nRbθβ(t0)±zα/2σβ(Rbθβ(t0))i\n,\nwhere zαdenotes the upper αquantile of a standard normal distribution. Given that reliability values\nfall within the interval [0 ,1], if either of the approximated interval bounds exceed this range, they\nshould be truncated accordingly.\n4.1.3 Quantiles\nCertain distribution quantiles, particularly those representing the lower and upper tails, offer valuable\ninsights into the characteristics of the tails of the lifetime distribution. Besides, the median life serves\nas a crucial measure of central tendency, as it gets less affected by extreme values, unlike the mean\nlifetime. This makes the median a more robust measure of centrality than the mean lifetime to failure.\nHence, estimating quantiles for the lower tail, upper tail, and central tendency can help manufacturers\nin comprehending the overall behaviour of the lifetimes of the products. Here, we will develop robust\nestimators for an α-quantile based on the MDPDEs.\nTheα-quantile of the lifetime distribution represents the time at which 100 α% of the devices would\nhave failed. Under the PH hazard assumption with linear baseline function and constant stress x0, the\nquantile function is obtained by inverting the distribution function in (25), that is, Qθ(α) =F−1\nT(α).\nTherefore, equating the distribution function to α, we obtain the α-quantile function to satisfy the\nfollowing equation:γ1\n2Qθ(α)2+γ0Qθ(α) + log( α) exp\u0010\n−aβ\n1x0\u0011\n= 0. (30)\n14Note that the above equation always has a real solution due to the constraint 0 < α < 1,which\nensures that the term log( α) is negative. Moreover, all remaining quantities γ0,γ1and exp( a1x0) are\npositive, and so the discriminant of the second-order solution formula is positive and the solutions of\nthe equation are therefore real. Then, solving for Qθ(α),we explicitly obtain the α-quantile as\nQθ(α) =−γ0+r\nγ2\n0−2γ1log(α) exp\u0010\n−aβ\n1x0\u0011\nγ1. (31)\nAlthough a second-order equation may have up to two solutions, in the context of quantiles where\nthe result should be a positive quantity, we have selected the unique positive solution provided by the\nabove equation.\nNow, applying the delta-method, we can readily stablish the asymptotic distribution of the α−quantile\nas follows:√\nN\u0010\nQbθβ(α)−Qθ∗(α)\u0011L− − − − →\nN→∞N(0, σβ(Qθ∗(α))),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n1) is the true value of the model parameters,\nσ2\nβ(Qθ∗(α)) =\u0014∂Qθ(α)\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Qθ(α)\n∂θ\u0015\nθ=θ∗,\nwithΣβ(θ) being as defined in Equation (22) and\n∂Qθ(α)\n∂θ=−Qθ(α)\nk1(Qθ(α))\u0012\n1,Qθ(α)\n2, k2(Qθ(α))x0\u0013T\n, (32)\nwith k1(t) =γ1t+γ0andk2(t) =γ1\n2t+γ0.\nThe explicit forms of the derivatives of the α-quantile have been obtained by taking implicit deriva-\ntives of Equation (30) with respect to each parameter and subsequently solving each corresponding\nequation. Detailed calculations are provided in Appendix B.2. Estimating the standard errors of the\nestimates by plugging-in the MDPDE as before, an approximate 100(1 −α)% confidence interval can\nbe given as follows:\nIC1−α[Qθ(α0)] =h\nQbθβ(α0)±zα/2σβ(Qbθβ(α0))i\n,\nwhere zαdenotes the upper αquantile of a standard normal distribution. As the domain of the\ndistribution function is restricted to positive values, any approximate lower bound that falls below\nzero needs to be be truncated.\n4.2 Quadratic baseline hazard\nWe will now discuss in detail the case of quadratic baseline hazard. Estimating the lifetime char-\nacteristics under quadratic baseline hazard is a more complex task, as not all characteristics have\nan explicit expression, and they need to be computed numerically. Consequently, while adopting a\nquadratic lifetime assumption provides a more flexible statistical model, the estimation of lifetime\ncharacteristics poses a big challenge. Let us assume a quadratic hazard function as in Equation (11),\nwith unknown parameters θ= (γ0, γ1, γ2, a1).The cumulative distribution function of the lifetime T\nunder a constant stress x0is then given by\nFT(t) = 1−exp\u0012\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0013\n,0< t < ∞, (33)\nwhere exp ( a1x0) represents the multiplicative effect of the stress level. Consequently, the probability\ndensity function of the lifetime is given by\nfT(t) = exp ( a1x0)\u0000\nγ0+γ1t+γ2t2\u0001\nexp\u0012\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0013\n. (34)\n154.2.1 Mean lifetime to failure\nThe mean lifetime of a device under the quadratic hazard assumption and constant stress level x0can\nbe computed as\nEθ[T] = exp ( a1x0)Z∞\n0\u0000\nγ0t+γ1t2+γ2t3\u0001\nexp\u0012\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0013\ndt. (35)\nThe definite integral in (35) does not have a closed form and so, given a certain vector model\nparameter, it needs to be estimated numerically. In particular, the MDPDE of the mean lifetime\nfor a fixed β, E bθβ[T],can be obtained by plugging-in the MDPDE estimator bθβinto the previous\nexpression and subsequently approximating its value numerically. To compute the improper integrals\nwith infinite domain through Monte Carlo techniques, the infinite integrand interval can be mapped\nto a finite interval and then any regular numerical quadrature routine may be used for the modified\nfinite integral, or use Gaussian quadrature rules. Although the mean lifetime to failure does not\nhave an explicit form, it is a differentiable function of θand its derivatives can be computed using\nderivative under integral sign rules. Therefore, its asymptotic distribution can be obtained through\nthe Delta-method, as in the linear case. In particular, we can establish the following asymptotic result:\n√\nN\u0010\nEbθβ[T]−Eθ∗[T]\u0011L− − − − →\nN→∞N(0, σβEθ∗[T]),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n0, a∗\n1) is the true value of the model parameters,\nσ2\nβ(Eθ∗[T]) =\u0014∂Eθ[T]\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Eθ[T]\n∂θ\u0015\nθ=θ∗,\nwithΣβ(θ) being as in Equation (22) and∂Eθ[T]\n∂θ=\u0010\n∂Eθ[T]\n∂γ0,∂Eθ[T]\n∂γ1,∂Eθ[T]\n∂γ2,∂Eθ[T]\n∂a1\u0011T\nis the vector of\nderivatives with\n∂Eθ[T]\n∂γ0= exp ( a1x0)Z∞\n0t(1−thθ(t))Rθ(t)dt,\n∂Eθ[T]\n∂γ1= exp ( a1x0)Z∞\n0t2(1−t\n2)hθ(t))Rθ(t)dt,\n∂Eθ[T]\n∂γ2= exp ( a1x0)Z∞\n0t3(1−t\n3hθ(t))Rθ(t)dt,\n∂Eθ[T]\n∂a1= (1−exp (a1x0))Eθ[T]x0.(36)\nDetailed calculations of the above derivatives are given in Appendix C.1. Given a MDPDE bθβ,we can\nnumerically approximate the estimated mean lifetime Ebθβ[T] and estimated standard error σβ(Ebθβ[T])\nand consequently an approximate 100(1 −α)%-confidence interval for the mean lifetime can be given\nas\nIC1−α[Eθ[T]] =h\nEbθβ[T]±zα/2σβ(Ebθβ[T])i\n.\nAgain, it’s important to note that the mean lifetime is a positive value. Therefore, if the lower bound\nof the approximate confidence interval happens to be negative, it should be truncated to a minimum\nvalue of 0.\n4.2.2 Reliability\nThe reliability function of the lifetime of the device under quadratic hazard at a mission time t0and\nconstant stress level x0is given by\nRθ(t0) = exp\u0012\n−exp (a1x0)\u0012\nγ0t0+γ1t2\n0\n2+γ2t3\n0\n3\u0013\u0013\n.\n16Therefore, the MDPDE of the reliability with tuning parameter βis defined as\nRbθβ(t0) = exp\u0012\n−exp\u0010\nbaβ\n1x0\u0011\u0012\nbγβ\n0t0+bγβ\n1t2\n0\n2+bγβ\n2t3\n0\n3\u0013\u0013\n.\nwherebθβ= (bγβ\n0,bγβ\n1,bγβ\n2,baβ\n1)Tis the MDPDE of the parameter θ.The properties of the MDPDE for\nthe reliability, Rbθβ(t0),will be inherited from the properties of bθβ,and positive values of the tuning\nparameter will provide robust estimates of the reliability. Applying the Delta method to the function\nRθ(t0),we obtain√\nN\u0010\nRbθβ(t0)−Rθ∗(t0)\u0011L− − − − →\nN→∞N(0, σβ(Rθ∗(t0))),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n0, a∗\n1) is the true value of the model parameters,\nσ2\nβ(Rθ∗(t0)) =\u0014∂Rθ(t0)\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Rθ(t0)\n∂θ\u0015\nθ=θ∗,\nwithΣβ(θ) being defined in Equation (22) and\n∂Rθ(t0)\n∂θ=−Rθ(t0) exp ( a1x0)\u0012\nt0,t2\n0\n2,t3\n0\n3,\u0012\nγ0t0+γ1t2\n0\n2γ2t3\n0\n3\u0013\nx0,\u0013T\n. (37)\nTherefore, we can build an approximate 100(1 −α)% confidence interval as\nIC1−α[Rθ(t0)] =h\nRbθβ(t0)±zα/2σβ(Rbθβ(t0))i\n,\nwhere zαdenotes the upper αquantile of a standard normal distribution and σβ(Rbθβ(t0)) is a robust\n(forβ >0) and consistent estimator of the standard error. As before, the approximate interval bounds\nneed to be truncated to [0 ,1] if their estimated values exceed these values, as the reliability of a device\ncan not exceed these limits.\n4.2.3 Quantiles\nThe upper α-quantile of the cumulative distribution function under quadratic baseline hazard, denoted\nbyQθ(α),should satisfy the equation\n1−α= 1−exp\u0012\n−exp (a1x0)\u0012\nγ0Qθ(α) +γ1Qθ(α)2\n2+γ2Qθ(α)3\n3\u0013\u0013\n,\nand so, after some algebra, we can establish that the upper α-quantile of the lifetime must satisfy the\nequationγ2\n3Qθ(α)3+γ1\n2Qθ(α)2+γ0Qθ(α) + log( α) exp (−a1x0) = 0 .\nThis equation may have up to three real solutions (and at least one). We will choose the least positive\nreal solution. While we do not have an explicit formula for the α-quantile, it is a differentiable function\nwith respect to θand so we can apply the Delta-method to derive its asymptotic distribution, yielding\nthe result:√\nN\u0010\nQbθβ(α)−Qθ∗(α)\u0011L− − − − →\nN→∞N(0, σβ(Qθ∗(α))),\nwhere θ∗= (γ∗\n0, γ∗\n1, a∗\n0, a∗\n1) is the true value of the model parameters,\nσ2\nβ(Qθ∗(α)) =\u0014∂Qθ(α)\n∂θ\u0015T\nθ=θ∗Σβ(θ∗)\u0014∂Qθ(α)\n∂θ\u0015\nθ=θ∗,\n17withΣβ(θ) begin as defined in Equation (22) and\n∂Qθ(α)\n∂θ=−Qθ(α)\nk1(Qθ(α))\u0012\n1,Qθ(α)\n2,Qθ(α)2\n3, k2(Qθ(α))x0\u0013T\n, (38)\nwith k1(t) =γ2t2+γ1t+γ0andk2(t) =γ2\n3t2+γ1\n2t+γ0.Details of the calculations are presented in\nAppendix C.2. Further, using the asymptotic distribution of the α0-quantile and the consistency of\nthe MDPDE, a 100(1 −α)% confidence interval can be given as\nIC1−α[Qθ∗(α0)] =h\nQbθβ(α0)±zα/2σβ(Qbθβ(α0))i\n.\nwhere zαdenotes the upper αquantile of a standard normal distribution. Again, the lower bound\nmay be truncated if necessary.\nIt is worthy to note the similarity of the formulas for the linear and quadratic baseline hazard\ncases. Indeed, if we set γ2= 0 and suppress the components corresponding to that parameter in all\nvector and matrices involved, we would recover the formulas presented for the linear case earlier in\nSection 4.1.\n5 Monte Carlo simulation study\nWe examine the performance of the proposed MDPDEs through an extensive simulation study, con-\nsidering both linear and quadratic baseline functions. We evaluate the accuracy of the estimates in\nterms of their mean squared error (MSE), computed as\nMSE (bγβ\ni) =||bγβ\ni−γi||2, i= 0,1,2,and MSE (baβ\n1) =||baβ\n1−a1||2.\nMoreover, we examine the accuracy of the estimated lifetime characteristics based on the MDPDE,\nwhich will inherit the properties of the estimators. The accuracy of the estimation of the lifetime\ncharacteristics is also measured in terms of their MSE.\nTo generate the failure counts, we consider a simple step-stress ALT assuming PH lifetimes and we\nthen generate the count of failure within the interval ( tj−1, tj] using a conditional binomial distribution\nwith probability given by\neπj(θ) =πj(θ)\n1−GT(tj),\nwhere πj(θ) is the multinomial probability defined in Equation (15) for linear and Equation (20) for\nquadratic baseline hazard functions. Although both conditional binomial and multinomial models are\nequivalent, we use the first to generate the data so we can increase the probability of failure in one\ncell, and simulate the effect on the subsequent inspections.\nTo evaluate the robustness of the estimators, we introduce some contamination in data. Generating\noutlying failure times may not necessarily result in an increased count of failures deemed as outliers.\nSo, in contrast to the notion of outliers based on individual observations, Barnett [1992] and Victoria-\nFeser and Ronchetti [1997] pointed out that an outlier from grouped data is identified within a class\nwhose associated probability (according to the underlying model) is notably smaller than the observed\nfrequency. This can be formalized by defining the “adjusted residuals” (see also Fuchs and Kenett\n[1980]) as follows:\nrj=√\nNbpj−πj(θ)p\nπj(θ)(1−πj(θ)),\nwhere πj(θ) is the expected probability of class jaccording to the underlying model and bpj=nj/Nis\nthe observed relative frequency in class j. Because πj(θ) depends on the unknown model parameter\n18θ, a robust estimation of the parameter is important to avoid the inflation of the denominator of the\nresidual and the consequent misleading effect. Due to the step-stress scheme in the generation of the\nfailure times, increasing or decreasing the probability of failure in a cell will also influence the observed\nprobabilities in the subsequent cells, but not the observed failures of the previous ones. Therefore, the\nadjusted residuals will increase at all cells from the contaminated cell onward.\nOn the other hand, as all baseline hazard parameters are assumed to be positive, we reparameterize\nthe hazard parameters for easier computation by exponentiating them, i.e., γ′\ni= log( γi) and the hazard\nfunction coefficients are then exp( γ). This ensures that all estimated parameters are positive. If a\nparameter estimated is lower than 10−7, we considered it as zero.\n5.1 Linear baseline hazard\nWe consider linear hazard with true parameter vector θ0= (exp( −4),exp(−5.3),0.5) and inspection\ntimes 2 ,4,6,8,10,12,14,16,18,20 and 22 .The stress in increased at τ1= 14 from x1= 0.5 tox2= 2.5.\nThe hazard function of the true underlying model at constant stress xis then\nh(t, x) =−exp(0 .5x)(exp( −4) + exp( −5.3)t).\nWe contaminate the model by increasing the probability of failure within the 10-th cell by (1+ ε) times\nits probability, which will result in the decrease of failures in the subsequent cells. Figure 1 shows\nthe increase on the residuals for the contaminated cells for increasing contamination. Note that the\npercentage of contaminated cells remains constant at 27% ,but the strength of contamination increases\nwith ε.\n12345\n0.00 0.25 0.50 0.75 1.00\nεresidualsCells\n11\n12\nFigure 1: Residuals of contaminated cells under linear baseline hazard\nFigure 2 presents the RMSE on the estimation of the three model parameters for increasing con-\ntamination of the model. As seen there, the MLE outperforms all its competitors in the absence of\ncontamination, but rapidly worsens with contamination. In contrast, the MDPDE with positive values\nofβare competitive to the MLE in the absence of contamination and are also not heavily influenced\nby contamination in data. Recall that the MDPDE estimating equations defined in (28) down-weigh\nthe observations with low probability (say π) by a factor of πβ.Because the theoretical probability of\nfailing within the last two intervals is quite low, the MDPDE gets less influenced by an abnormally\nhigh number of failures.\nThe performance of the estimators would naturally influence the accuracy of the corresponding\nlifetime characteristics, which are of natural interest in many reliability analysis. For example, one may\nbe interested in inferring specific quantiles, mean lifetime, hazard rate or the reliability of the product\nunder NOC at certain mission times. Figure 3 presents the accuracy on the estimation of the median\n(50%-quantile), mean lifetime, hazard rate and reliability at time t0= 5 under a lower stress (set at\n190.0 0.2 0.4 0.6 0.8 1.0 1.20.016 0.018 0.020 0.022 0.024\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8(a) RMSE( γ0)\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.00400.00420.00440.00460.00480.00500.00520.0054\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8 (b) RMSE( γ1)\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.02 0.04 0.06 0.08 0.10\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8 (c) RMSE( a1)\nFigure 2: Root mean squared error (RMSE) of the estimates under linear baseline hazard\nx0= 0.3 for illustrative purposes) representing the NOC. All efficiency and robustness properties of\nthe MDPDEs are inherited by the estimates of the lifetime characteristics and all estimates based on\nthe MLE are seen to be heavily influenced by contamination present in the data.\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.15 0.20 0.25 0.30\ncontMSE \nβ\n0\n0.2\n0.4\n0.6\n0.8\n(a) RMSE of the median\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.20 0.25 0.30 0.35 0.40 0.45\ncontMSE \nβ\n0\n0.2\n0.4\n0.6\n0.8 (b) RMSE of the mean\n0.0 0.2 0.4 0.6 0.8 1.0 1.21.0e−05 1.5e−05 2.0e−05 2.5e−05\ncontMSE \nβ\n0\n0.2\n0.4\n0.6\n0.8\n(c) RMSE of the hazard\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.00015 0.00020 0.00025 0.00030 0.00035\ncontMSEchar10\nβ\n0\n0.2\n0.4\n0.6\n0.8 (d) RMSE of the reliability\nFigure 3: Root mean squared error (RMSE) of the estimates of lifetime characteristics under linear\nbaseline hazard\n205.2 Quadratic baseline hazard\nWe now consider a quadratic hazard baseline with true parameter θ0= (exp( −4),0,exp(−6.0),0.5)\nso that the underlying hazard function is given by\nh(t, x) =−exp(0 .5x)(exp( −4) + exp( −6)t2).\nFor quadratic hazard, the probability of failure will increase more rapidly in time and so we consider\na shorter experiment with inspection times 1 ,2,3,4,5,6,7,8,9,10,11 and 12 ,when the experiment\nterminates. The same stress levels x1= 0.5 and x2= 2.5 are considered and the time of stress\nchange is set at τ1= 8.We contaminate the penultimate cell as before, here corresponding to cell\n11 by increasing its probability of failure by ε.Figure 4 plots the increase in the residuals at the\ncontaminated cells for increasing contamination amount. It can be seen that all three cells, including\nthe event of surviving after the experiment terminates, are contaminated and consequently their\nresiduals increases with ε.\n246\n0.00 0.25 0.50 0.75 1.00\nεresidualsCells\n11\n12\n13\nFigure 4: Residuals of contaminated cells under quadratic baseline hazard\nAs in the previous section, we evaluate the accuracy and robustness of the MDPDEs with different\nvalues of the tuning parameter. Figure 5 presents the root mean square error of the estimation of each\nnon-zero model parameter γ0, γ2,anda1.For the parameter γ1(which is null in the true parameter\nvector), all MDPDEs estimate it to be zero.\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.0160.0180.0200.0220.0240.0260.028\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8\n(a) RMSE( γ0)\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.0040.0060.0080.0100.0120.0140.016\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8 (b) RMSE( γ2)\n0.0 0.1 0.2 0.3 0.4 0.5 0.60.1 0.2 0.3 0.4 0.5\ncontMSE10\nβ\n0\n0.2\n0.4\n0.6\n0.8 (c) RMSE( a1)\nFigure 5: Root mean squared error (RMSE) of the estimates of model parameters under quadratic\nbaseline hazard\nOnce again, the MLE is heavily influenced by an abnormal increase in failures at cells with low\n21theoretical probability (i.e., outliers in grouped data), while the MDPDE with strictly positive values\nof the tuning parameter βare much more robust against outlying cells.\n6 Illustrative Data Analysis\nWe revisit an example discussed in Elsayed and Jiao [2002], wherein a two-step-stress ALT was con-\nducted for Metal Oxide Semiconductor (MOS) capacitors in order to estimate its hazard rate over a 10\nyear period of time at design temperature of T= 50◦C.The test needed to be completed in T= 300\nhours and the total number of test units under test was N= 200 .To avoid any unexpected change in\nfailure mechanisms within the design temperature range, it was decided by engineers’ judgment that\nthe testing temperatures could not exceed a maximum temperature of T= 250◦C.Therefore, two\nstress levels were considered, x1= 145◦Candx2= 250◦C.\nAccording to the Arrehenius model, the lifetime of a product or a chemical reaction and temper-\nature to which it got exposed are related as follows:\nT=Aexp\u0012\n−Ea\nkx\u0013\n,\nwhere Trepresents the lifetime of the product, Ais the pre-exponential factor, to be determined for\neach product, Eais the activation energy, which varies by failure mechanism, k= 8.36×10−5eV/◦K\nis an universal Boltzman’s constant and xis the absolute temperature in Kelvin.\nThe Arrhenius law states that at higher temperatures, chemical reactions or degradation processes\noccur more rapidly. Conversely, at lower temperatures, the rates of these processes decrease. The\nArrhenius equation may help in predicting how long-term exposure to higher temperatures affects the\nproduct’s lifetime. Therefore, accelerated life tests can be applied on products by subjecting them\nto higher temperatures than they would typically experience during normal use, and then the failure\nbehaviour can be extrapolated to the normal use temperature using the previous equation.\nThen, re-parametrizing the stress factor to si=−1\nxi,the Arrhenius Equation can be rewritten in\na log-linear relation form as\nT= exp\u0012\nlog(A)−\u0012Ea\nk\u0013\ns\u0013\n;\nthat is, the two stress are x1=−2.3914×103andx2=−1.9114×103.Elsayed and Jiao [2002] assumed\nlinear baseline hazard functions and used the initial values γ0= 0.0001, γ1= 0.5 and a1= 3,800 for se-\nlecting optimal designs for the temperature and time of stress change. We used their suggested parame-\nters to generate a real data-based sample with inspection times 40 ,60,90,110,130,150,170,183,190,210,220\nand 250 .Table 2 presents the MDPDEs for different values of the tuning parameter in the absence\nof contamination. The estimates are averages of the estimated parameters over R= 1,000 simulated\nsamples.\nβ 0 0.2 0.4 0.6 0.8 1\nbγβ\n00.00012 0.00021 0.00011 0.00010 0.00009 0.00007\nbγβ\n1 0.4830 0.4785 0.4751 0.4728 0.4714 0.4716\nbaβ\n1 3,790 3,790 3,790 3,780 3,780 3,780\nTable 2: Model parameter estimates for the real-data based example\nAll MDPDE estimates, for the different values of β,are quite similar and near the true value of\nthe parameter, illustrating their competitive performance in the absence of contamination.\nNow, we consider a contaminated scenario. Following the description in Section 5, we increase\nthe probability of failure within the penultimate cell (and the subsequent cells are consequently also\n22contaminated). Figure 6 presents the RMSE of the estimates of the median, mean, reliability and\nhazard rate at t0= 60 under NOC based on the MDPDE with different values of β. As in the\nprevious numerical analyses, the MLE gets heavily influenced by the abnormal count of failures and\nconsequently its performance worsens rapidly as the amount of contamination increases. Meanwhile,\nthe MDPDEs with positive βs show a clear advantage over the MLE in terms of robustness.\n0.0 0.2 0.4 0.6 0.8 1.0 1.210000 30000 50000 70000\ncontMSEchar11\nβ\n0\n0.2\n0.4\n0.6\n0.8\n(a) RMSE of the median\n0.0 0.2 0.4 0.6 0.8 1.0 1.220000 40000 60000 80000\ncontMSEchar11\nβ\n0\n0.2\n0.4\n0.6\n0.8 (b) RMSE of the mean\n0.0 0.2 0.4 0.6 0.8 1.0 1.24.0e−09 6.0e−09 8.0e−09 1.0e−08 1.2e−08 1.4e−08\ncontMSEchar11\nβ\n0\n0.2\n0.4\n0.6\n0.8\n(c) RMSE of the hazard\n0.0 0.2 0.4 0.6 0.8 1.0 1.24.0e−06 6.0e−06 8.0e−06 1.0e−05 1.2e−05\ncontMSEchar11\nβ\n0\n0.2\n0.4\n0.6\n0.8 (d) RMSE of the reliability\nFigure 6: Root mean squared error (RMSE) of the estimates of lifetime characteristics for the illus-\ntrative example\n7 Conclusions\nThis paper introduces inferential methods for step-stress ALTs under proportional hazards for interval-\nmonitored reliability experiments. Specifically, we have investigated estimation procedures under\nlinear and quadratic baseline functions. Due to the lack of robustness of the MLE, we have presented\na robust family of estimators known as the MDPDEs. This family is indexed by a tuning parameter,\nβ, which controls the trade-off between the efficiency of the estimator and the robustness. Indeed,\nthe MDPDE family includes the MLE as an extreme case with maximum efficiency, but minimal\n(indeed, none) robustness. Furthermore, from the estimates of the lifetime characteristics, we have\nderived point estimates and confidence intervals for the main lifetime characteristics, including the\nmean lifetime and distribution quantiles, under both linear and quadratic forms.\nAll proposed estimators haven been evaluated through a simulation study, demonstrating a sig-\nnificant advantage of the MDPDEs in terms of robustness, with a small compromise in efficiency.\nThe performance of the robust estimators has also been evaluated in a real application for inferring\nthe reliability of Metal Oxide Semiconductor (MOS) capacitors, as described in Elsayed and Jiao\n[2002]. In both analyses, the MDPDEs are quite similar in the absence of contamination (demonstrat-\ning their competitive performance even with no contamination), but the MLE rapidly worsens when\ncontamination gets introduced in the grouped data.\n23While both linear and quadratic baseline forms exhibit flexibility in approximating the general\nbaseline hazards, one would naturally be interested in preferable form for the baseline, based on the\ndata at hand. In our future research, we intend to explore hypothesis tests for the baseline form.\nAcknowledgements\nThis work was supported by the Spanish Grant PID2021-124933NB-I00 and Natural Sciences and\nEngineering Research Council of Canada (of the first author) through an Individual Discovery Grant\n(No. 20013416). M. Jaenada and L. Pardo are members of the Interdisciplinary Mathematics Institute\n(IMI).\nReferences\nAGA-EEI. An appraisal of methods for estimating service lives of utility properties, 1942.\nV. Bagdonavicius. Testing the hypothesis of additive accumulation of damages. Theory of Probability\nand its Applications , 23:403–408, 1978.\nN. Balakrishnan, E. Castilla, M. Jaenada, and L. Pardo. Robust inference for nondestructive one-shot\ndevice testing under step-stress model with exponential lifetimes. Quality and Reliability Engineering\nInternational , 39(4):1192–1222, 2023a.\nN. Balakrishnan, M. Jaenada, and L. Pardo. Non-destructive one-shot device test under step-stress ex-\nperiment with lognormal lifetime distribution. Journal of Computational and Applied Mathematics ,\npage 115483, 2023b.\nN. Balakrishnan, M. Jaenada, and L. Pardo. Step-stress tests for interval-censored data under gamma\nlifetime distribution. Quality Engineering , pages 1–18, 2023c.\nV. Barnett. Unusual outliers. Data Analysis and Statistical Inference. Koln: Verlag , 1992.\nG. K. Bhattacharyya and Z. Soejoeti. A tampered failure rate model for step-stress accelerated life\ntest. Communications in statistics-Theory and methods , 18(5):1627–1643, 1989.\nD. R. Cox. Regression models and life-tables. Journal of the Royal Statistical Society: Series B\n(Methodological) , 34(2):187–202, 1972.\nM. H. DeGroot and P. K. Goel. Bayesian estimation and optimal designs in partially accelerated life\ntesting. Naval Research Logistics Quarterly , 26(2):223–235, 1979.\nE. Elsayed and L. Jiao. Optimal design of proportional hazards based accelerated life testing plans.\nInternational Journal of Materials and Product Technology , 17(5-6):411–424, 2002.\nC. Fuchs and R. Kenett. A test for detecting outlying cells in the multinomial distribution and two-way\ncontingency tables. Journal of the American Statistical Association , 75(370):395–398, 1980.\nS. A. Krane. Analysis of survival data by regression techniques. Technometrics , 5(2):161–174, 1963.\nNARUC. Report of the committee on depreciation, 1938.\nW. Nelson. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis . John Wiley &\nSons, New York, 1990.\n24N. M. Sedyakin. On one physical principle in reliability theory (in russian). Technical Cybernetics , 3:\n80–87, 1966.\nM.-P. Victoria-Feser and E. Ronchetti. Robust estimation for grouped data. Journal of the American\nStatistical Association , 92(437):333–340, 1997.\nA Proofs of the main results\nA.1 Proof of Result 1\nProof. Taking derivatives in the DPD loss defined in Eq. (17) and equating them to zero, we get\n∂dβ(bp,π(θ))\n∂θ= (β+ 1)L+1X\nj=1\u0012\nπj(θ)β−1(πj(θ)−bpj)∂πj(θ)\n∂θ\u0013\n=03,\nwhere the derivative of each probability of failure πj(θ) is given by\n∂πj(θ)\n∂θ=∂Rθ(tj−1)\n∂θ−∂Rθ(tj)\n∂θ.\nLet us write the reliability function of the device at stress level xas\nRθ(t, x) =\n\nexp\u0010\n−exp(a1x1)\u0010\nγ0t+γ1t2\n2\u0011\u0011\n, x =x1,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(t+s) +γ1(t+s)2\n2\u0011\u0011\n, x=x2,\nwhere sis the corresponding shifting time corresponding to the stress change from x1tox2,which\ndepends on the model parameters and satisfies Eq. (9).\nFirstly, we need to compute the derivatives of the shifting time s(as a function of the parameters\nθ). Taking implicit derivatives in Equation (9) with respect to each parameter ( γ0, γ1, a0, a1),we\nobtain the following system of linear equations:\nγ1(τ+s)∂s\n∂γ0+γ0∂s\n∂γ0=−(τ+s) +exp(a1x1)\nexp(a1x2)τ,\nγ1(τ+s)∂s\n∂γ1+γ0∂s\n∂γ1=−1\n2(τ+s)2+exp(a1x1)\nexp(a1x2)1\n2τ2,\nγ1(τ+s)∂s\n∂a1+γ0∂s\n∂a1=exp(a1x1)\nexp(a1x2)(x1−x2)\u0010γ1\n2τ2+γ0τ\u0011\n.\nLet us define\nk1(t) =γ1t+γ0 and k2(t) =γ1\n2t+γ0.\nThen, solving the above equations, we obtain the derivatives of the shifting time as follows:\n∂s\n∂γ0=γ1\n2(τ+s)s\nk2(τ)1\nk1(τ+s),\n∂s\n∂γ1=−γ0\n2(τ+s)s\nk2(τ)1\nk1(τ+s),\n∂s\n∂a1=(τ+s)\nτk2(τ+s)\nk1(τ+s)(x1−x2),\n25where we have used the equality\nexp(a1x1)\nexp(a1x2)=(τ+s)\nτk2(τ+s)\nk2(τ)\nto simplify the expressions. Now, we compute separately the derivatives of the reliability under stress\nlevels x1andx2.Under the first stress level x1, we have\n∂Rθ(t, x1)\n∂γ0=−Rθ(t, x1) exp( a1x1)t,\n∂Rθ(t, x1)\n∂γ1=−Rθ(t, x1) exp( a1x1)t2\n2,\n∂Rθ(t, x1)\n∂a1=−Rθ(t, x1) exp( a1x1)tk2(t)x2.\nOn the other hand, the derivatives of the reliability under the increased stress x2with respect to\neach parameter are as follows:\n∂Rθ(t, x2)\n∂γ0= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012\n(t+s) +γ0∂s\n∂γ0+γ1(t+s)∂s\n∂γ0\u0013\u0015\n,\n∂Rθ(t, x2)\n∂γ1= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012(t+s)2\n2+γ0∂s\n∂γ1+γ1(t+s)∂s\n∂γ1\u0013\u0015\n,\n∂Rθ(t, x2)\n∂a1= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2\u0013\nx2−exp(a1x2)\u0012\nγ0∂s\n∂a1+γ1(t+s)∂s\n∂a1\u0013\u0015\n,\nUpon substituting the formulas of the shifting time derivatives in the above expressions, we can rewrite\nthe above derivatives as follows:\n∂Rθ(t, x2)\n∂γ0=−Rθ(t, x2) exp( a1x2)\u0012\n(t+s) +γ1\n2(τ+s)s\nk2(τ)k1(t+s)\nk1(τ+s)\u0013\n,\n∂Rθ(t, x2)\n∂γ1=−Rθ(t, x2) exp( a1x2)\u0012(t+s)2\n2−γ0\n2(τ+s)s\nk2(τ)k1(t+s)\nk1(τ+s)\u0013\n,\n∂Rθ(t, x2)\n∂a1=−Rθ(t, x2) exp( a1x2)\u0014\n(t+s)k2(t+s)x2+k1(t+s)k2(τ+s)\nk1(τ+s)(τ+s)(x1−x2)\u0015\n.\nNow, defining\nzj=(∂Rθ(tj−1,x1)\n∂θ−∂Rθ(tj,x1)\n∂θtj−1< τ\n∂Rθ(tj−1,x2)\n∂θ−∂Rθ(tj,x2)\n∂θτ≤tj−1,\nwe obtain the required formula.\nA.2 Proof of Result 2\nProof. Taking derivatives in the DPD loss defined in Eq. (17) and equating them to zero, we get\n∂dβ(bp,π(θ))\n∂θ= (β+ 1)L+1X\nj=1\u0012\nπj(θ)β−1(πj(θ)−bpj)∂πj(θ)\n∂θ\u0013\n=04.\n26where the derivative of each probability of failure πj(θ) is given by\n∂πj(θ)\n∂θ=∂Rθ(tj−1)\n∂θ−∂Rθ(tj)\n∂θ,\nwith Rθ(tj−1) denoting the reliability under quadratic hazard. The explicit formula of πj(θ) is as\ngiven in (18). Let us write the reliability function of the device at stress level xas\nRθ(t, x) =\n\nexp\u0010\n−exp(a1x1)\u0010\nγ0t+γ1t2\n2+γ2t3\n3\u0011\u0011\n, x =x1,\nexp\u0010\n−exp(a1x2)\u0010\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0011\u0011\n, x=x2,\nwhere sis the shifting time corresponding to the stress change from x1tox2,which depends on the\nmodel parameters and satisfies third-order equation defined in Eq. (13).\nFirstly, we need to compute the derivatives of the shifting time s(as a function of the parameters\nθ). Taking implicit derivatives in Eq. (13) with respect to each parameter ( γ0, γ1, γ2, a1),we obtain\nthe following system of linear equations:\nγ2(τ+s)2∂s\n∂γ0+γ1(τ+s)∂s\n∂γ0+γ0∂s\n∂γ0=−(τ+s) +exp(a1x1)\nexp(a1x2)τ,\nγ2(τ+s)2∂s\n∂γ1+γ1(τ+s)∂s\n∂γ1+γ0∂s\n∂γ1=−1\n2(τ+s)2+exp(a1x1)\nexp(a1x2)1\n2τ2,\nγ2(τ+s)2∂s\n∂γ2+γ1(τ+s)∂s\n∂γ2+γ0∂s\n∂γ2=−1\n3(τ+s)3+exp(a1x1)\nexp(a1x2)1\n3τ3,\nγ2(τ+s)2∂s\n∂a1+γ1(τ+s)∂s\n∂a1+γ0∂s\n∂a1=exp(a1x1)\nexp(a1x2)(x1−x2)\u0010γ2\n3τ3+γ1\n2τ2+γ0τ\u0011\n.\nLet us define\nk1(t) =γ2t2+γ1t+γ0 and k2(t) =γ2\n3t2+γ1\n2t+γ0.\nThen, the system of equations can be rewritten as\nk1(τ+s)∂s\n∂γ0=−(τ+s) +exp(a1x1)\nexp(a1x2)τ,\nk1(τ+s)∂s\n∂γ1=−1\n2(τ+s)2+exp(a1x1)\nexp(a1x2)1\n2τ2,\nk1(τ+s)∂s\n∂γ2=−1\n3(τ+s)3+exp(a1x1)\nexp(a1x2)1\n3τ3,\nk1(τ+s)∂s\n∂a1=exp(a1x1)\nexp(a1x2)(x1−x2)τk2(τ).\nSolving the above equations, we obtain the derivatives of the shifting time as follows:\n∂s\n∂γ0=(τ+s)\nk1(τ+s)\u0012k2(τ+s)\nk2(τ)−1\u0013\n,\n∂s\n∂γ1=1\n2(τ+s)\nk1(τ+s)\u0012\nτk2(τ+s)\nk2(τ)−(τ+s)\u0013\n,\n∂s\n∂γ2=1\n3(τ+s)\nk1(τ+s)\u0012\nτ2k2(τ+s)\nk2(τ)−(τ+s)2\u0013\n,\n∂s\n∂a1= (τ+s)k2(τ+s)\nk1(τ+s)(x1−x2),\n27where we have used the equality\nexp(a1x1)\nexp(a1x2)=(τ+s)\nτk2(τ+s)\nk2(τ)\nto simplify the expressions. Now, we compute separately the derivatives of the reliability under stress\nlevels x1andx2.Under the first stress level x1, we have\n∂Rθ(t, x1)\n∂γ0=−Rθ(t, x1) exp( a1x1)t,\n∂Rθ(t, x1)\n∂γ1=−Rθ(t, x1) exp( a1x1)t2\n2,\n∂Rθ(t, x1)\n∂γ2=−Rθ(t, x1) exp( a1x1)t3\n3,\n∂Rθ(t, x1)\n∂a0=−Rθ(t, x1) exp( a1x1)tk2(t),\n∂Rθ(t, x1)\n∂a1=−Rθ(t, x1) exp( a1x1)tk2(t)x2.\nOn the other hand, the derivatives of the reliability under the increased stress x2with respect to\neach parameter are as follows:\n∂Rθ(t, x2)\n∂γ0= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012\n(t+s) +γ0∂s\n∂γ0+γ1(t+s)∂s\n∂γ0+γ2(t+s)2∂s\n∂γ0\u0013\u0015\n,\n∂Rθ(t, x2)\n∂γ1= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012(t+s)2\n2+γ0∂s\n∂γ1+γ1(t+s)∂s\n∂γ1+γ1(t+s)∂s\n∂γ0+γ2(t+s)2∂s\n∂γ1\u0013\u0015\n,\n∂Rθ(t, x2)\n∂γ2= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012(t+s)3\n3+γ0∂s\n∂γ2+γ1(t+s)∂s\n∂γ2+γ2(t+s)2∂s\n∂γ2\u0013\u0015\n,\n∂Rθ(t, x2)\n∂a0= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\n−exp(a1x2)\u0012\nγ0∂s\n∂a0+γ1(t+s)∂s\n∂a0+γ2(t+s)2∂s\n∂a0\u0013\u0015\n,\n∂Rθ(t, x2)\n∂a1= exp\u0012\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\u0013\n×\u0014\n−exp(a1x2)\u0012\nγ0(t+s) +γ1(t+s)2\n2+γ2(t+s)3\n3\u0013\nx2\n−exp(a1x2)\u0012\nγ0∂s\n∂a1+γ1(t+s)∂s\n∂a1+γ2(t+s)2∂s\n∂a1\u0013\u0015\n.\nUpon substituting the formulas of the shifting time derivatives in the above expressions, we can rewrite\n28the above derivatives as follows:\n∂Rθ(t, x2)\n∂γ0=−Rθ(t, x2) exp( a1x2)\u0012\n(t+s) + (τ+s)k1(t+s)\nk1(τ+s)\u0012k2(τ+s)\nk2(τ)−1\u0013\u0013\n,\n∂Rθ(t, x2)\n∂γ1=−Rθ(t, x2) exp( a1x2)\u0012(t+s)2\n2+(τ+s)\n2k1(t+s)\nk1(τ+s)\u0012\nτk2(τ+s)\nk2(τ)−(τ+s)\u0013\u0013\n,\n∂Rθ(t, x2)\n∂γ1=−Rθ(t, x2) exp( a1x2)\u0012(t+s)3\n3+(τ+s)\n3k1(t+s)\nk1(τ+s)\u0012\nτ2k2(τ+s)\nk2(τ)−(τ+s)2\u0013\u0013\n,\n∂Rθ(t, x2)\n∂a1=−Rθ(t, x2) exp( a1x2)\u0014\n(t+s)k2(t+s)x2+k1(t+s)\nk1(τ+s)(τ+s)k2(τ+s)(x1−x2)\u0015\n.\nNow, defining\nwj=(∂Rθ(tj−1,x1)\n∂θ−∂Rθ(tj,x1)\n∂θ, tj−1< τ,\n∂Rθ(tj−1,x2)\n∂θ−∂Rθ(tj,x2)\n∂θ, τ≤tj−1,\nwe obtain the required formula.\nB Calculations for the lifetime characteristics under linear baseline\nhazard\nB.1 Computation of the mean lifetime and its derivatives\nLet us consider a linear baseline hazard function of the form in (6) and a constant stress x0.Then,\nthe cumulative distribution function of the products in this case is\nFT(t) = 1−exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2\u0013\u0015\n,0< t < ∞,\nand the corresponding distribution density function is\nfT(t) =−exp (a1x0) (γ0+γ1t) exp\u0014\nexp (a0+a1x0)\u0012\nγ0t+γ1t2\n2\u0013\u0015\n,0< t < ∞,\n29where exp ( a1x0) represents the multiplicative effect of the stress level. Then, the mean lifetime of the\nproduct can be obtained as follows:\nEθ[T] =Z∞\n0tfT(t)dt\n=Z∞\n0exp (a1x0) (γ0t+γ1t2) exp\u0014\n−exp [a0+a1x0]\u0012\nγ0t+γ1t2\n2\u0013\u0015\ndt\n=Z∞\n0exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2\u0013\u0015\ndt\n=Z∞\n0exp\u0014\n−γ1\n2exp (a1x0)\u0012\n2γ0\nγ1t+t2\u0013\u0015\ndt\n= exp\u0012γ1\n2exp(a1x0)γ2\n0\nγ2\n1\u0013Z∞\n0exp\"\n−γ1\n2exp (a1x0)\u0012\nt+γ0\nγ1\u00132#\ndt\n= exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013s\n2π\nγ1exp(a1x0)\n×Z∞\n0r\nγ1exp(a1x0)\n2πexp\"\n−γ1\n2exp (a1x0)\u0012\nt+γ0\nγ1\u00132#\ndt\n= exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013s\n2π\nγ1exp(a1x0)\n×Z∞\nγ0\nγ1√\nγ1exp(a1x0)1√\n2πexp\u0012\n−1\n2u2\u0013\ndu\n= exp\u0014\nexp(a1x0)γ2\n0\n2γ1\u0015s\n2π\nγ1exp(a1x0)Φ\u0012\n−γ0√γ1,exp\u0010a1x0\n2\u0011\u0013\n,(39)\nwhere Φ denotes the cumulative distribution function of a standard normal variable.\n30Now, the derivatives of Eθ[T] with respect to every parameter are as follows:\n∂Eθ[T]\n∂γ0=s\n2π\nγ1exp(a1x0)\u0014\nexp\u0012exp(a1x0)γ2\n0\n2γ1\u0013exp(a1x0)γ0\nγ1Φ\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011\u0013\n−exp\u0012exp(a1x0)γ2\n0\n2γ1\u0013\nϕ\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011\u0013exp(a1x0/2)√γ1\u0015\n=γ0\nγ1Eθ[T] exp( a1x0)−√\n2π\nγ1exp\u0014exp(a1x0)γ2\n0\n2γ1\u0015\nϕ\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011\u0013\n,\n∂Eθ[T]\n∂γ1=s\n2π\nexp(a1x0)exp\u0012exp(a1x0)γ2\n0\n2γ1\u0013\u0014\n−1\n2γ−3/2\n1−1√γ1exp(a1x0)γ2\n0\n2γ2\n1\u0015\nΦ\u0012\n−γ0√γ1p\nexp(a1x0)\u0013\n+ exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013s\n2π\nγ1exp(a1x0)ϕ\u0012\n−γ0√γ1p\nexp(a1x0)\u0013γ0\n2γ3/2\n1p\nexp(a1x0)\n=1\n2γ1Eθ[T]\u0014\n−1−exp(a1x0)γ2\n0\nγ1\u0015\n+ exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013\nϕ\u0012\n−γ0√γ1p\nexp(a1x0)\u0013√\n2πγ0\n2γ2\n1,\n∂Eθ[T]\n∂a1=x0\n2exp\u0012exp(a1x0)γ2\n0\n2γ1\u0013r2π\nγ1\"\n−1\n2p\nexp(a1x0)+p\nexp(a1x0)γ2\n0\n2γ1#\nΦ\u0012\n−γ0√γ1p\nexp(a1x0)\u0013\n+ exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013s\n2π\nγ1exp(a1x0)ϕ\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011\u0013\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011x0\n2\u0013\n=Eθ[T]Φ\u0012\n−γ0√γ1p\nexp(a1x0)\u0013\u0014\n−1 +γ2\n0\nγ1exp(a1x0)\u0015x0\n2\n−exp\u0012\nexp(a1x0)γ2\n0\n2γ1\u0013\nϕ\u0012\n−γ0√γ1exp\u0010a1x0\n2\u0011\u0013√\n2πγ0\nγ1x0\n2.\n(40)\nB.2 Computation of the α−quantile and its derivatives\nAs discussed in Section 4, the α−quantile of the distribution should satisfy the second-order equation\nγ1\n2Qθ(α)2+γ0Qθ(α) + log( α) exp\u0010\n−aβ\n1x0\u0011\n= 0. (41)\nNow, taking derivatives with respect to every parameter ( γ0, γ1, a0, a1),we obtain the following system\nof equations:\nγ1Qθ(α)∂Qθ(α)\n∂γ0+Qθ(α) +γ0∂Qθ(α)\n∂γ0= 0,\n1\n2Qθ(α)2+γ1Qθ(α)∂Qθ(α)\n∂γ1γ0∂Qθ(α)\n∂γ1= 0,\nγ1Qθ(α)∂Qθ(α)\n∂a0+γ0∂Qθ(α)\n∂a0−log(α) exp\u0010\n−aβ\n1x0\u0011\n= 0,\nγ1Qθ(α)∂Qθ(α)\n∂a1+γ0∂Qθ(α)\n∂a1−log(α) exp\u0010\n−aβ\n1x0\u0011\nx0= 0.\n31Upon solving the above equations, we readily obtain the derivatives of the α-quantile function as\nfollows:\n∂Qθ(α)\n∂γ0=−Qθ(α)\nγ1Qθ(α) +γ0,\n∂Qθ(α)\n∂γ1=−Qθ(α)2\n2γ1Qθ(α) + 2γ0.\n∂Qθ(α)\n∂a0=log(α) exp\u0010\n−aβ\n1x0\u0011\nγ1Qθ(α) +γ0,\n∂Qθ(α)\n∂a1=log(α) exp\u0010\n−aβ\n1x0\u0011\nγ1Qθ(α) +γ0x0.\nNow, by using Eq. (41), we can rewrite −log(α) exp\u0010\n−aβ\n1x0\u0011\n=γ1\n2Qθ(α)2+γ0Qθ(α),and thus the\nexpressions presented in Section 4.1.3 are obtained.\nC Calculations for the lifetime characteristics under quadratic base-\nline hazard\nC.1 Computation of the mean lifetime and its derivatives\nLet us consider the mean lifetime of a device under the quadratic hazard assumption and constant\nstress level x0,given by\nEθ[T] = exp ( a1x0)Z∞\n0\u0000\nγ0t+γ1t2+γ2t3\u0001\nexp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt.\nTaking derivatives with respect to every model parameter θ= (γ0, γ1, γ2, a0, a1),we get\n32∂Eθ[T]\n∂γ0= exp ( a1x0)Z∞\n0texp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n−exp (a1x0)2Z∞\n0\u0000\nγ0t+γ1t2+γ2t3\u0001\ntexp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n= exp ( a1x0)Z∞\n0tRθ(t)dt−Z∞\n0t2hθ(t)Rθ(t)dt,\n∂Eθ[T]\n∂γ1= exp ( a1x0)Z∞\n0t2exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n−exp (a1x0)2Z∞\n0\u0000\nγ0t+γ1t2+γ2t3\u0001t2\n2exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n= exp ( a1x0)Z∞\n0t2Rθ(t)dt−Z∞\n0t3\n2hθ(t)Rθ(t)dt,\n∂Eθ[T]\n∂γ2= exp ( a1x0)Z∞\n0t3exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n−exp (a1x0)2Z∞\n0\u0000\nγ0t+γ1t2+γ2t3\u0001t3\n3exp\u0014\n−exp (a1x0)\u0012\nγ0t+γ1t2\n2+γ2t3\n3\u0013\u0015\ndt\n= exp ( a1x0)Z∞\n0t3Rθ(t)dt−Z∞\n0t4\n3hθ(t)Rθ(t)dt\n= (1−exp (a1x0))Eθ[T],\n∂Eθ[T]\n∂a1= (1−exp (a1x0))Eθ[T]x0.(42)\nC.2 Computation of the α−quantile and its derivatives\nTheα−quantile of the distribution should satisfy the third-order equation\nγ1\n3Qθ(α)3+γ1\n2Qθ(α)2+γ0Qθ(α) + log( α) exp\u0010\n−aβ\n1x0\u0011\n= 0.\nNow, taking derivatives with respect every parameter ( γ0, γ1, γ2, a0, a1),we obtain the following system\nof equations:\nγ2Qθ(α)2∂Qθ(α)\n∂γ0+γ1Qθ(α)∂Qθ(α)\n∂γ0+Qθ(α) +γ0∂Qθ(α)\n∂γ0= 0,\nγ2Qθ(α)2∂Qθ(α)\n∂γ1+1\n2Qθ(α)2+γ1Qθ(α)∂Qθ(α)\n∂γ1γ0∂Qθ(α)\n∂γ1= 0,\n1\n3Qθ(α)3+γ2Qθ(α)2∂Qθ(α)\n∂γ2+γ1Qθ(α)∂Qθ(α)\n∂γ2γ0∂Qθ(α)\n∂γ2= 0,\nγ2Qθ(α)2∂Qθ(α)\n∂a1+γ1Qθ(α)∂Qθ(α)\n∂a1+γ0∂Qθ(α)\n∂a1−log(α) exp\u0010\n−aβ\n1x0\u0011\nx0= 0,\n33Upon solving the above equations, we obtain the derivatives of the α-quantile function as follows:\n∂Qθ(α)\n∂γ0=−Qθ(α)\nγ2Qθ(α)2+γ1Qθ(α) +γ0,\n∂Qθ(α)\n∂γ1=−1\n2Qθ(α)2\nγ2Qθ(α)2+γ1Qθ(α) +γ0,\n∂Qθ(α)\n∂γ2=−1\n3Qθ(α)3\nγ2Qθ(α)2+γ1Qθ(α) +γ0,\n∂Qθ(α)\n∂a1=log(α) exp\u0010\n−aβ\n1x0\u0011\nγ2Qθ(α)2+γ1Qθ(α) +γ0x0.\nNow, by using Eq. (41), we can rewrite −log(α) exp\u0010\n−aβ\n1x0\u0011\n=γ2\n3Qθ(α)3+γ1\n2Qθ(α)2+γ0Qθ(α),\nand thus expressions presented in Section 4.2.3 are obtained.\n34"    },    {        "title": "2402.06439v1.Selective_Radiance_in_Super_Wavelength_Atomic_Arrays.pdf",        "content": "Selective Radiance in Super-Wavelength Atomic Arrays\nCharlie-Ray Mann,1Francesco Andreoli,1Vladimir Protsenko,2Zala Lenar ˇciˇc,2and Darrick E. Chang1, 3\n1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain.\n2Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia.\n3ICREA-Instituci ´o Catalana de Recerca i Estudis Avanc ¸ats, 08015 Barcelona, Spain.\nA novel way to create efficient atom-light interfaces is to engineer collective atomic states that selectively radiate\ninto a target optical mode by suppressing emission into undesired modes through destructive interference. While\nit is generally assumed that this approach requires dense atomic arrays with sub-wavelength lattice constants,\nhere we show that selective radiance can also be achieved in arrays with super-wavelength spacing. By stack-\ning multiple two-dimensional arrays we find super-wavelength mirror configurations where one can eliminate\nemission into unwanted diffraction orders while enhancing emission into the desired specular mode, leading to\nnear-perfect reflection of weak resonant light. These super-wavelength arrays can also be functionalized into\nefficient quantum memories, with error probabilities on the order of ∼1%for a trilayer with only around ∼100\natoms per layer. Relaxing the previous constraint of sub-wavelength spacing could potentially ease the technical\nrequirements for realizing efficient atom-light interfaces, such as enabling the use of tweezer arrays.\nIntroduction.— Creating an efficient atom-light interface is\nchallenging because atoms are inherently dissipative objects;\nthey can absorb photons from a desired optical mode and scat-\nter them into other inaccessible modes, resulting in a loss of\nquantum information. Rather than relying on the collective\nenhancement in disordered ensembles [1] or the Purcell en-\nhancement in structured photonic environments [2] to improve\nefficiencies, a new generation of atom-light interfaces have\nbeen proposed based on selective radiance [3]. The central\nidea is to exploit wave interference as a new resource in care-\nfully engineered configurations of atoms in order to suppress\nspontaneous emission into undesired modes.\nA striking example occurs in ordered two-dimensional (2D)\natomic arrays with sub-wavelength lattice constants. These ar-\nrays can act as a near-perfect mirror for weak resonant light\n[4, 5] and be functionlized into an efficient quantum memory\n[6] and deterministic photon-photon gate [7]. The selective\nradiance leads to a substantial reduction in errors compared\nto what is predicted for disordered ensembles [8, 9], where\nspontaneous emission is treated as an independent process. In\nparticular, errors at the ∼1%level should be feasible using\nsub-wavelength arrays, a goal that has proven elusive for con-\nventional atom-light interfaces [10–13].\nTo realize 2D sub-wavelength arrays, current experiments\nutilize ultracold atoms in a Mott insulator phase of an optical\nlattice [14, 15]. A tantilizing alternative would be to employ\noptical tweezers [16–21] as one can assemble defect-free ar-\nrays in almost any desired geometry, and they offer signifi-\ncantly shorter experimental cycle times which would lead to\nfaster repetition rates for quantum optics operations. How-\never, unfortunately, the diffraction-limited focus of the indi-\nvidual tweezers presents a serious challenge to achieving sub-\nwavelength lattice constants. To that end, it is important to\nestablish whether we can relax this constraint and find selec-\ntively radiant configurations with super-wavelength spacing,\nwhich is more compatible with current tweezer technology.\nWhile it is well known that a 2D super-wavelength array\nis a poor light-matter interface, due to photons being scat-\ntered into multiple diffraction orders, here we demonstrate\nFIG. 1. Schematic of a super-wavelength mirror composed of a bi-\nlayer atomic array with lattice constant a > λ 0and layer spacing\nd > λ 0. The array can selectively radiate into the desired specular\nmode at a rate Γdet=MΓ00which is enhanced by the number of\nlayers M(Γ00being the rate of a monolayer), while the emission\ninto unwanted diffraction orders is eliminated through inter-layer de-\nstructive interference Γdiff= 0.\nthat one can restore the selective radiance by stacking mul-\ntiple 2D layers. Using an idealized model, we find a range\nof super-wavelength mirror configurations which support col-\nlective atomic states that selectively radiate into the desired\nspecular at an enhanced rate, while the emission into all un-\nwanted diffraction orders in eliminated through inter-layer de-\nstructive interference (see Fig. 1). Guided by this intuition, we\nshow that a finite super-wavelength array can almost perfectly\nreflect a weak classical beam on resonance, and be functional-\nized into into an efficient quantum memory for single photons.\nFor example, one can achieve errors on the order of ∼1%for\na trilayer with only around ∼100atoms per layer.\nSpin model formalism.— We consider an ensemble of N\nidentical atoms trapped at fixed positions ri. Initially, we\nassume a minimal two-level model where the ground |gi⟩\nand excited |ei⟩states support a circularly-polarized transi-arXiv:2402.06439v1  [quant-ph]  9 Feb 20242\ntion with frequency ω0and dipole matrix element ℘=|℘|e℘,\nwhere e℘is the unit polarization vector. The ensemble is then\ndriven by a coherent input field with a (near-resonant) central\nfrequency ωpand amplitude E+\nin(r) =⟨ˆE+\nin(r)⟩whose polar-\nization matches the atomic transition.\nWithin the quantum jump formalism and the Born-Markov\napproximation [22], the atomic state |ψ⟩evolves determinis-\ntically iℏ∂t|ψ⟩=ˆHeff|ψ⟩under the non-Hermitian Hamilto-\nnian ˆHeff=ˆHin+ˆHdip[3]. This should then be supple-\nmented with stochastically applied quantum jumps, but these\nwill not be important for the observables of interest here [6, 7].\nIn the rotating frame the input Hamiltonian reads\nˆHin=−ℏX\ni∆ˆσi\nee−ℏX\ni(Ωiˆσi\neg+ H.c.), (1)\nwhere ˆσi\nµν=|µi⟩⟨νi|are the atomic operators with {µ, ν} ∈\n{g, e},∆ = ωp−ω0is the atom-probe detuning, and Ωi=\n℘∗·E+\nin(ri)/ℏare the Rabi frequencies. Furthermore, the\ndipole-dipole Hamiltonian is [3]\nˆHdip=−ℏΓ0X\nijGijˆσi\negˆσj\nge, (2)\nwhere Gij=e∗\n℘·¯¯G(ri−rj, ω0)·e℘is the projected compo-\nnent of the (dimensionless) dyadic Green function [23]\n¯¯G(r, ω0) =3eik0r\n4k3\n0r3\u0002\u0000\nk2\n0r2+ ik0r−1\u0001¯¯I\n−\u0000\nk2\n0r2+ 3ik0r−3\u0001ˆr⊗ˆr\u0003\n,(3)\nwithk0=ω0/c. This Hamiltonian describes the effective co-\nherent ( Re{Gij}) and dissipative ( Im{Gij}) interactions be-\ntween the atoms which are mediated by the free-space pho-\ntons. The coherent part of the self interaction, associated\nwith the Lamb shift, is divergent and assumed to be absorbed\ninto the definition of ω0, while the dissipative part is finite\nIm{Gii}= 1/2andΓ0=k3\n0|℘|2/3πℏε0is the free-space\nspontaneous emission rate of an isolated atom.\nOnce the dynamics of the atoms have been solved for,\none can reconstruct the field correlations in a target detection\nmode Edet(r)which can be efficiently captured in a given op-\ntical setup (e.g. a Gaussian). The associated operator is [24]\nˆadet= ˆadet,in+ ir\n3Γ0\n8πX\niE∗\ndet(ri)·e℘ˆσi\nge, (4)\nwhere we choose the normalization so that ⟨ˆa†\ndetˆadet⟩corre-\nsponds to the rate of outgoing photons emitted into the de-\ntection mode. The first and second term in Eq. (4) are the\ncontributions from the input and scattered fields, respectively.\nWe consider a driving amplitude Ωithat is weak enough\nso that the dynamics are effectively restricted to the single-\nexcitation manifold and the low saturation limit. If we take the\ndetection mode to be the same as the input mode, but propa-\ngating in the opposite direction, then the reflection coefficient\nisr=−(3i/8π)⃗E ·¯¯Λ·⃗Eand the reflectance is R=|r|2[24].Here, ⃗E= (E1,E2, . . . ,EN)TwithEi=e℘·E∗\ndet(ri), and\n¯¯Λis the linear response matrix whose elements are given by\n(¯¯Λ−1)ij= (∆ /Γ0)δij+Gij.\nSub-wavelength mirror.— It is insightful to first review\nhow a sub-wavelength atomic array can act as a perfect mir-\nror [4, 5], and why this breaks down for super-wavelength\nspacing. For concreteness, we initially consider a square lat-\ntice of atoms located at periodic positions riin the xyplane\nwith lattice constant a, although the same arguments apply\nto any Bravais lattice. In the idealized limit N→ ∞ the dis-\ncrete translational symmetry dictates that the single-excitation\natomic eigenstates of Eq. (2) are |q⟩=ˆS†\nq|g⟩⊗N, where\nˆS†\nq=N−1/2P\nieiq·riˆσi\negcreates a collective spin wave with\nBloch wavevector q∈BZ. Although the array supports N\nsuch eigenstates, a plane wave at normal incidence can only\ncouple to the spin wave with q=0. However, the in-plane\nmomentum is only conserved modulo a reciprocal lattice vec-\ntorgmn= (2π/a)(m, n), which we index by m, n∈Z. As a\nresult, the spin wave can scatter photons into a discrete set of\npropagating diffraction orders (mn)in addition to the specu-\nlar detection mode (00).\nDue to the excitation of a single eigenstate, the reflection\ncoefficient has a simple Lorentzian form [24]\nr=iΓdet/2\n−∆ +J−i(Γdet+ Γdiff)/2, (5)\nwhere J=−Γ0Re{P\ni̸=jGij}is the collective frequency\nshift of the spin wave. The corresponding collective decay\nrateΓ = 2Γ 0Im{P\niGij}can be expressed as a sum over the\ndifferent emission channels Γ =P\nmnΓmn, where\nΓmn= Γ 00k2\n0+k2\nmn\n2k0kmnΘ(k0− |gmn|). (6)\nHere, Γ00= 3πΓ0/k2\n0Ais the decay rate into the specular\ndetection mode ( Γdet= Γ 00), where Ais the area of the unit\ncell, while Γdiff=P\nmn̸=00Γmnis the decay rate into all\nthe propagating diffraction orders (see Fig. 1). We have in-\ntroduced the Heaviside step function Θ(x)in Eq. (6) because\nkmn= (k2\n0−|gmn|2)1/2becomes imaginary for |gmn|> k0;\nthese correspond to evanescent diffraction orders which do not\nrepresent possible emission channels.\nThe maximum reflectance on resonance ( ∆ = J) is de-\ntermined by the branching ratio of the emission, Rmax=\n[Γdet/(Γdet+ Γ diff)]2. If the lattice constant is sub-\nwavelength a < λ 0then all of the diffraction orders become\nevanescent ( Γdiff= 0), leading to perfect reflection on res-\nonance Rmax= 1. For super-wavelength lattice constants\na > λ 0there is a dramatic reduction in reflectance because\nmultiple diffraction orders open up with Γdet/Γdiff≪1,\nthereby losing the selective radiance of the sub-wavelength\narray.\nSuper-wavelength mirror.— To restore the selective radi-\nance we now consider Msuper-wavelength layers, each con-\ntaining Natoms located at riα=ri+dα. Here, rirep-\nresent the 2D Bravais lattice vectors and dα= (ρα, zα)is3\nFIG. 2. (a) Maximum specular reflectance Rmaxof a Gaussian beam as a function of lattice constant for a triangular monolayer ( N= 127 ,\nw=√\nNa/3). (b) Maximum reflectance as a function of lattice constant and layer spacing for a trilayer. The peaks are predicted well by the\nidealized model (white circles). (c) Scaling of the minimum reflectance error ϵR= 1−Rmaxas function of Nfor a monolayer (red dots),\nbilayer (green dots) and trilayer (blue dots), where we have optimized the set {a, d, w }near(a, d)≈(1.6λ0,1.5λ0). The dotted lines indicate\nthe scaling ϵR≈10.6N−0.8andϵR≈66.6N−1.7for the bilayer and trilayer, respectively.\nthe shift of each layer indexed by α. To gain some intu-\nition, we again consider the idealized limit N→ ∞ and a\nplane wave input at normal incidence. The relevant single-\nexcitation manifold is now spanned by |α⟩=ˆS†\nα|g⟩⊗NM,\nwhere ˆS†\nα=N−1/2P\niˆσiα\negcreates a collective spin wave\nwithq=0in layer α. Within this manifold the dynamics are\neffectively described by the Hamiltonian\nˆHdip=−ℏΓ00X\nαβGαβˆS†\nαˆSβ, (7)\nwhere the inter-layer matrix elements are\nGαβ=i\n2X\nmnΓmn\nΓ00eikmn|zα−zβ|eigmn·(ρα−ρβ). (8)\nFor simplicity, we have neglected the coherent interactions\nmediated by the evanescent diffraction orders, and we have\nalso absorbed Jinto the detuning term.\nIn general, the input field will couple to all the Mcollective\natomic eigenstates of Eq. (7), giving rise to complex interfer-\nence effects. To recover a single-state response we search for\nthe analog of the “mirror” configuration in waveguide QED\n[25, 26], where the coherent inter-layer interactions vanish\nRe{Gαβ}= 0 and the atomic eigenstates are energetically\ndegenerate. With this goal it is favorable to preserve the 2D\npoint symmetry group of the underlying Bravais lattice. Then,\nif the interactions are dissipative for some gmnthey are also\ndissipative for all those with the same magnitude |gmn|. This\ncondition is satisfied by stacking parallel layers ( ρα= 0),\nbut there also exist other high-symmetry configurations (e.g.\nsquare lattices shifted by half a unit cell).\nTo ensure that the inter-layer interactions mediated by the\nspecular mode are always dissipative, we can fix the layer\nspacing d=ℓλ0/2withℓ∈N, which means that we only\nhave one parameter to tune – the lattice constant. We thus con-\nsider the simplest case where there is only one set of diffrac-\ntion orders; for the square lattice this restricts us to the rangea/λ0∈(1,1.41), while for the triangular lattice the range is\nmuch larger a/λ0∈(1.15,2). From Eq. (8) one can show that\nthe inter-layer interactions become purely dissipative when\nQ=ℓ\u0010\n1 +q\n1− |gmn|2/k2\n0\u0011\n(9)\nis equal to an integer Q∈N. From this set of critical lattice\nconstants we can distinguish two qualitatively different con-\nfigurations.\nIfQ∈Neven then the diffraction orders mediate dissipa-\ntive interactions with the same sign as the specular mode [24].\nConsequently, Gαβbecomes a rank-1 matrix where there is\nonly one bright eigenstate |B⟩=M−1/2P\nα(−1)ℓα|α⟩with\ndecay rate MΓ, while the other M−1eigenstates are com-\npletely dark. This bright state is not selectively radiant be-\ncause the emission into all channels is collectively enhanced,\nand thus the branching ratio of emission is identical to a mono-\nlayer with Γdet/Γdiff≪1. Evidently, these are not the de-\nsired super-wavelength mirror configurations.\nThe situation is very different when Q∈Nodd, because the\ndiffraction orders mediate dissipative interactions with an al-\nternating sign with respect to the specular mode [24]. As a re-\nsult,Gαβbecomes a rank-2 matrix with two bright eigenstates\nandM−2dark eigenstates. For an even number of layers one\neigenstate |B1⟩=M−1/2P\nα(−1)(ℓ+1)α|α⟩has a collec-\ntively enhanced decay rate into the diffraction orders Γdiff=\nMP\nmn̸=00Γmn, while the emission into the specular detec-\ntion mode is completely suppressed Γdet= 0. In stark con-\ntrast, the other eigenstate |B2⟩=M−1/2P\nα(−1)ℓα|α⟩has\na collectively enhanced decay rate into the specular detection\nmode Γdet=MΓ00, while the emission into the diffraction\norders is completely suppressed Γdiff= 0 (see Fig. 1). Evi-\ndently, the input plane wave can only couple to |B2⟩and this\nstate selectively radiates into the target specular mode – these\nare the desired super-wavelength mirror configurations.\nRemarkably, even a simple bilayer is sufficient to restore\nthe perfect reflection on resonance, since the high degree of4\nFIG. 3. (a) Maximum retrieval efficiency ηmax as a function of lattice constant for a triangular monolayer, where we consider a two-way\nretrieval (see inset) into a Gaussian mode ( N= 127 ,w=√\nNa/3,ϕ= 0). (b) Maximum retrieval efficiency as a function of lattice constant\nand layer spacing for a trilayer. (c) Scaling of the minimum retrieval error ϵm= 1−ηmaxas function of Nfor a monolayer (red dots), bilayer\n(green dots) and trilayer (blue dots), where we have optimized the set {a, d, w, ϕ }near(a, d)≈(1.8λ0,1.4λ0). The dotted lines indicate the\nscaling ϵm≈6.3N−0.91andϵm≈1.2N−1for the bilayer and trilayer, respectively.\nsymmetry allows one to shut off emission into all the diffrac-\ntion orders simultaneously. The physics is slightly more nu-\nanced for an odd number of layers since both bright states con-\ntribute to the response, although it stills behaves like a perfect\nmirror. Moreover, this slight difference between even and odd\nlayers diminishes rapidly with increasing layers [24].\nFinite size effects.— We now consider the realistic case of\na finite array and a input Gaussian mode with beam waist w\n(see inset in Fig. 2a), and study how the errors scale with atom\nnumber. For this analysis we focus on triangular lattice con-\nfigurations since they perform better for a larger range of lat-\ntice constants. In Fig. 2a we plot the maximum reflectance as\na function of lattice constant for a monolayer with N= 127\nandw=√\nNa/3. For sub-wavelength lattice constants we\nobserve that the reflectance is slightly reduced from unity,\nwhich is the result of two fundamental sources of error [6].\nFirst, a small fraction of the Gaussian beam extends beyond\nthe array boundaries which does not interact with the atoms.\nSecond, the Gaussian beam contains a superposition of in-\nplane wavevectors and can thus couple to spin waves with\nq̸=0which are only quasi-degenerate, thereby losing the\nideal single-state response.\nAs anticipated, super-wavelength lattice constants exhibit a\ndramatic reduction in the maximum reflectance because mul-\ntiple diffraction orders open up. In Fig. 2b we plot the max-\nimum reflectance as a function of lattice constant and layer\nspacing for a trilayer. We can observe several pockets of high\nreflectance within the super-wavelength parameter regions,\nand the peaks are predicted very well by the idealized mir-\nror configurations calculated from Eq. (9). Note that there are\nadditional sources of error compared to the monolayer case.\nFor example, within the idealized model a perfect mirror ex-\nists for any layer separation d=ℓλ0/2, but in the realistic\ncase the spatial overlap of the diffracted beams reduces as the\nlayer spacing increases.\nIn Fig. 2c we show the scaling of the reflectance error\nεR= 1−Rmaxas a function of Nfor a monolayer, bilayerand trilayer. For each data point we have optimized the set\nof parameters {a, d, w }using a local optimization algorithm,\ntargeting the first super-wavelength mirror configuration near\n(a, d)≈(1.6λ0,1.5λ0). As expected for the monolayer, the\nerror plateaus at large values ϵR≈0.98because it does not\nexhibit selective radiance. In contrast, from numerical fitting\nwe find that the reflection error scales as ϵR≈10.6N−0.8for\nthe bilayer, while for the trilayer the error scales much faster\nasϵR≈66.6N−1.7. Besides these abstract scalings, with\nonlyN= 127 atoms per layer one can achieve a reflectance\nofRmax≈0.79with a bilayer and Rmax≈0.98with a tri-\nlayer – this represents about a 50-fold increase compared to a\nmonolayer Rmax≈0.02.\nQuantum memory.— In the idealized case where the de-\ntection mode interacts with a single collective atomic state,\nit can be established that the resonant reflectance of classi-\ncal light determines the efficiency of various quantum appli-\ncations [27]. Here, we study the efficiency of an EIT-based\nquantum memory [6, 8] for the realistic case, where this cor-\nrespondence does not exactly hold, and study how the errors\nscale with atom number. The basic idea is to coherently and\nreversibly map a photonic state to a high-lying, long-lived\natomic state |si⟩, facilitated by a classical control field on the\n|si⟩ ↔ | ei⟩transition.\nIt is more convenient to optimize the retrieval of an exci-\ntation that is initially stored as a spin wave excitation |ψ(t=\n0)⟩=P\niαsiαˆσiα\nsg|g⟩⊗NM, and then the optimal storage pro-\ncess is related via time-reversal symmetry [8]. The retrieval\nefficiency is defined as the probability that the photon is emit-\nted into the target detection mode η=R∞\n0dt⟨ˆa†\ndet(t)ˆadet(t)⟩.\nWith the assumption of a spatially uniform control field, one\nfinds η= (3/8π)⃗ s·¯¯K·⃗ s∗regardless of the temporal profile\nof the control field [6]. Here, ⃗ s= (s11, . . . , s iα, . . . , s NM)T\nis the vector of initial amplitudes and\n¯¯K= iX\nξξ′(⃗E ·⃗ vξ)(⃗E∗·⃗ v∗\nξ′)\nλξ−λ∗\nξ′⃗ vξ⊗⃗ v∗\nξ′ (10)5\nis a Hermitian matrix, where λξand⃗ vξare the set of eigen-\nvalues and eigenvectors of the matrix Giα,jβ , which satisfy\nthe relations ⃗ vξ·⃗ vξ′=δξξ′andP\nξ⃗ vξ⊗⃗ vξ=¯¯I. The max-\nimum efficiency ηmaxfor a given detection mode and set of\natomic positions is then given by the maximum eigenvalue of\n¯¯K, and the corresponding eigenvector gives the optimal initial\nspin wave excitation [6].\nSince the array will naturally emit outgoing photons in\nboth directions, it is favorable to consider a two-way retrieval\nscheme (see inset in Fig. 3a), where the detection mode is\ntaken to be a superposition of two counter-propagating Gaus-\nsian modes with a relative phase ϕ. In Fig. 3a we plot the\nmaximum retrieval efficiency as a function of lattice constant\nfor a monolayer with N= 127 ,w=√\nNa/3andϕ= 0.\nWe see a significant reduction of the retrieval efficiency for\nsuper-wavelength lattice constants due to emission into un-\nwanted diffraction orders. In Fig. 3b we show the efficiency\nfor a trilayer as a function of lattice constant and layer spac-\ning, and we observe many pockets of high efficiency which\napproximately correlate with the regions of high reflectance\nin Fig. 2b.\nIn Fig. 3c we show the scaling of the minimum retrieval\nerror ϵm= 1−ηmax as a function of Nfor a monolayer,\nbilayer and trilayer. For each data point we have locally op-\ntimized the set of parameters {a, d, w, ϕ }targeting the first\nsuper-wavelength peak near (a, d)≈(1.8λ0,1.4λ0). The er-\nror for the monolayer plateaus to large values ϵm≈0.86, but\nthe selective radiance is restored in the bilayer and numeri-\ncally we find that the error scales as ϵm≈6.3N−0.91. For the\ntrilayer, the errors are significantly reduced compared to the\nbilayer, but they scale with a similar power ϵm≈1.2N−1.\nWhile this abstract scaling is similar to what is predicted from\nMaxwell-Bloch theory for disordered ensembles [8], the ab-\nsolute error for a given atom number is substantially smaller.\nFor the trilayer one can achieve an error of less than 1%with\nonlyN= 127 atoms per layer. To reach such an error with a\ndisordered ensemble would demand a very large optical depth\n(∼600) which is technically very challenging to work with,\nand state-of-the-art errors remain at the ∼10% level [10, 11].\nConclusion.— Despite it being a common assumption, we\nhave demonstrated that sub-wavelength spacing is not a fun-\ndamental requirement for selective radiance. In fact, one can\nbuild efficient light-matter interfaces with super-wavelength\narrays which could enable the use of tweezer arrays for quan-\ntum optics applications. In principle, one is not limited to\nperiodic configurations with tweezers and the errors can be\nreduced further by locally optimizing the individual atomic\npositions. Moreover, it may be fruitful to employ a more so-\nphisticated global optimization algorithm to search for non-\nintuitive configurations with super-wavelength spacing. Simi-\nlar to their sub-wavelength counterparts, super-wavelength ar-\nrays can be further functionalized with Rydberg interactions\nto enable an efficient, deterministic photon-photon gate [7].Acknowledgements.— C.-R.M. acknowledges funding\nfrom the Marie Skłodowska-Curie Actions Postdoctoral Fel-\nlowship ATOMAG (grant agreement No. 101068503).\nZ.L. acknowledges the QuantERA grant QuSiED by MVZI\n(QuantERA II JTC 2021) and ERC StG 2022 project\nDrumS, Grant Agreement 101077265. D.E.C acknowledges\nsupport from the European Union, under European Re-\nsearch Council grant agreement No 101002107 (NEWSPIN),\nFET-Open grant agreement No 899275 (DAALI) and EIC\nPathfinder Grant No 101115420 (PANDA); the Govern-\nment of Spain under Severo Ochoa Grant CEX2019-\n000910-S [MCIN/AEI/10.13039/501100011033]; QuantERA\nII project QuSiED, co-funded by the European Union\nHorizon 2020 research and innovation programme (No\n101017733) and the Government of Spain (European\nUnion NextGenerationEU/PRTR PCI2022-132945 funded\nby MCIN/AEI/10.13039/501100011033); Generalitat de\nCatalunya (CERCA program and AGAUR Project No. 2021\nSGR 01442); Fundaci ´o Cellex, and Fundaci ´o Mir-Puig.\n[1] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Quantum inter-\nface between light and atomic ensembles, Reviews of Modern\nPhysics 82, 1041 (2010).\n[2] D. Chang, J. Douglas, A. Gonz ´alez-Tudela, C.-L. Hung,\nand H. Kimble, Colloquium: Quantum matter built from\nnanoscopic lattices of atoms and photons, Reviews of Modern\nPhysics 90, 031002 (2018).\n[3] A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. Kim-\nble, and D. E. Chang, Exponential improvement in photon\nstorage fidelities using subradiance and “selective radiance” in\natomic arrays, Physical Review X 7, 031024 (2017).\n[4] R. J. Bettles, S. A. Gardiner, and C. S. Adams, Enhanced optical\ncross section via collective coupling of atomic dipoles in a 2D\narray, Phys. Rev. Lett. 116, 103602 (2016).\n[5] E. Shahmoon, D. S. Wild, M. D. Lukin, and S. F. Yelin, Co-\noperative resonances in light scattering from two-dimensional\natomic arrays, Physical review letters 118, 113601 (2017).\n[6] M. Manzoni, M. Moreno-Cardoner, A. Asenjo-Garcia, J. V .\nPorto, A. V . Gorshkov, and D. Chang, Optimization of photon\nstorage fidelity in ordered atomic arrays, New journal of physics\n20, 083048 (2018).\n[7] M. Moreno-Cardoner, D. Goncalves, and D. E. Chang, Quan-\ntum nonlinear optics based on two-dimensional rydberg atom\narrays, Physical Review Letters 127, 263602 (2021).\n[8] A. V . Gorshkov, A. Andr ´e, M. Fleischhauer, A. S. Sørensen,\nand M. D. Lukin, Universal approach to optimal photon storage\nin atomic media, Physical review letters 98, 123601 (2007).\n[9] J. D. Thompson, T. L. Nicholson, Q.-Y . Liang, S. H. Cantu,\nA. V . Venkatramani, S. Choi, I. A. Fedorov, D. Viscor, T. Pohl,\nM. D. Lukin, et al. , Symmetry-protected collisions between\nstrongly interacting photons, Nature 542, 206 (2017).\n[10] P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Lau-\nrat, Highly-efficient quantum memory for polarization qubits in\na spatially-multiplexed cold atomic ensemble, Nature commu-\nnications 9, 363 (2018).\n[11] Y . Wang, J. Li, S. Zhang, K. Su, Y . Zhou, K. Liao, S. Du,\nH. Yan, and S.-L. Zhu, Efficient quantum memory for single-\nphoton polarization qubits, Nature Photonics 13, 346 (2019).6\n[12] D. Tiarks, S. Schmidt-Eberle, T. Stolz, G. Rempe, and S. D ¨urr,\nA photon–photon quantum gate based on rydberg interactions,\nNature Physics 15, 124 (2019).\n[13] T. Stolz, H. Hegels, M. Winter, B. R ¨ohr, Y .-F. Hsiao, L. Husel,\nG. Rempe, and S. D ¨urr, Quantum-logic gate between two op-\ntical photons with an average efficiency above 40%, Physical\nReview X 12, 021035 (2022).\n[14] J. Rui, D. Wei, A. Rubio-Abadal, S. Hollerith, J. Zeiher, D. M.\nStamper-Kurn, C. Gross, and I. Bloch, A subradiant optical mir-\nror formed by a single structured atomic layer, Nature 583, 369\n(2020).\n[15] K. Srakaew, P. Weckesser, S. Hollerith, D. Wei, D. Adler,\nI. Bloch, and J. Zeiher, A subwavelength atomic array switched\nby a single rydberg atom, Nature Physics 19, 714–719 (2023).\n[16] D. Barredo, S. De L ´es´eleuc, V . Lienhard, T. Lahaye, and\nA. Browaeys, An atom-by-atom assembler of defect-free arbi-\ntrary two-dimensional atomic arrays, Science 354, 1021 (2016).\n[17] D. Barredo, V . Lienhard, S. De Leseleuc, T. Lahaye, and\nA. Browaeys, Synthetic three-dimensional atomic structures as-\nsembled atom by atom, Nature 561, 79 (2018).\n[18] P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter,\nD. Barredo, K.-N. Schymik, V . Lienhard, L.-P. Henry, T. C.\nLang, T. Lahaye, et al. , Quantum simulation of 2d antifer-\nromagnets with hundreds of rydberg atoms, Nature 595, 233\n(2021).\n[19] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. An-\nschuetz, A. Krajenbrink, C. Senko, V . Vuletic, M. Greiner,\nand M. D. Lukin, Atom-by-atom assembly of defect-free one-dimensional cold atom arrays, Science 354, 1024 (2016).\n[20] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran,\nH. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al. ,\nProbing many-body dynamics on a 51-atom quantum simulator,\nNature 551, 579 (2017).\n[21] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini,\nA. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho,\net al. , Quantum phases of matter on a 256-atom programmable\nquantum simulator, Nature 595, 227 (2021).\n[22] P. Meystre and M. Sargent, Elements of quantum optics\n(Springer Science & Business Media, 2007).\n[23] L. Novotny and B. Hecht, Principles of nano-optics (Cambridge\nuniversity press, 2012).\n[24] See Supplemental Material for: 1) More details on the mode\nprojection formalism; 2) Derivation of the reflection coeffi-\ncients; 3) Comparison between even and odd layers.\n[25] D. E. Chang, L. Jiang, A. Gorshkov, and H. Kimble, Cavity\nqed with atomic mirrors, New Journal of Physics 14, 063003\n(2012).\n[26] M. Mirhosseini, E. Kim, X. Zhang, A. Sipahigil, P. B. Dieterle,\nA. J. Keller, A. Asenjo-Garcia, D. E. Chang, and O. Painter,\nCavity quantum electrodynamics with atom-like mirrors, Na-\nture569, 692 (2019).\n[27] Y . Solomons, R. Ben-Maimon, and E. Shahmoon, Universal\napproach for quantum interfaces with atomic arrays, arXiv\npreprint arXiv:2302.04913 (2023)."    },    {        "title": "2402.06507v1.Non_Conforming_Finite_Element_Method_For_Constrained_Dirichlet_Boundary_Control_Problem.pdf",        "content": "arXiv:2402.06507v1  [math.OC]  9 Feb 2024Non-Conforming Finite Element Method For Constrained\nDirichlet Boundary Control Problem\nSudipto Chowdhury∗, Divay Garg†\nAugust 2023\nAbstract\nThis article examines the Dirichlet boundary control probl em governed by the Poisson equa-\ntion, where the control variables are square integrable fun ctions defined on the boundary of a\ntwo dimensional bounded, convex, polygonal domain. It empl oys an ultra weak formulation\nand utilizes Crouzeix-Raviart finite elements to discretiz e the state variable, while employing\npiecewise constants for the control variable discretizati on. The study demonstrates that the\nenergy norm of an enriched discrete optimal control is unifo rmly bounded with respect to the\ndiscretization parameter. Furthermore, it establishes an optimal order a priori error estimate\nfor the control variable.\nKeywords: Optimal control problem, Crouziex-Raviart, Finite element method , A priori error\nanalysis, Control constraints\nAMS subject classification: 65N30, 65N15, 65N12, 65K10\n1 Introduction\nWe consider the control constrained Dirichlet boundary control p roblem on a convex and bounded\npolygonal domain Ω ⊆R2. We denote the polygonal boundary of the domain Ω by Γ := ∪k\nj=1Γj,\nwhere Γ jis a straight line segment for all 1 ≤j≤k. The minimization problem seeks a pair\n(¯y,¯u)∈L2(Ω)×Uadsuch that\nJ(¯y,¯u) = min\n(y,u)∈L2(Ω)×UadJ(y,u), (1.1)\nsubject to the following second order elliptic equation:\n/braceleftBigg\n−∆y=fin Ω,\ny=uon Γ.(1.2)\nThe given function f∈L2(Ω) denotes the force which acts on the system externally and Uad⊆\nL2(Γ) is the set of admissible controls which will be defined later. The qua dratic cost functional\nJ(·,·) :L2(Ω)×L2(Γ)→Ris defined in the following manner:\nJ(y,u) :=1\n2/integraldisplay\nΩ|y−yd|2dx+α\n2/integraldisplay\nΓ|u|2ds, (1.3)\nwhereyd∈L2(Ω) is the given desired state and α >0 is known parameter which is introduced for\nthe regularization. The state equation (1.2) under the standard w eak formulation is not well posed\n∗Department of Mathematics, The LNM Institute of Informatio n Technology Jaipur, Rajasthan- 302031, India\n(sudipto.choudhary@lnmiit.ac.in)\n†Department of Mathematics, Indian Institute of Technology Delhi, New Delhi- 110016, India\n(divaygarg2@gmail.com)[The work of the author is granted by CSIR Research Fellowship]\n1S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nforu∈L2(Γ) as the weak solution of (1.2) lie in H1(Ω) space and therefore by trace theorem [7],\nthe first trace of it must lie in H1/2(Γ) which is not the case in this formulation. There are multiple\ntechniques available in the literature to overcome this variational diffi culty, one of which is to use\nthe method of transposition which is also known as the ultra weak for mulation [11, 24]. In brevity,\nwe discuss the method of transposition. Define a test function spa ce as follows:\nX:={φ∈H1\n0(Ω) : ∆φ∈L2(Ω)}.\nBy elliptic regularity theory on convex polygonal domains, we have X=H2(Ω)∩H1\n0(Ω). Now, we\nobtain the ultra weak formulation of the state equation (1.2). Multip lyingφ∈Xin both sides of\n(1.2) and then integrate both the sides to find\n−/integraldisplay\nΩ∆yφdx=/integraldisplay\nΩf φdx.\nA use of integration by parts in the above equation yields\n/integraldisplay\nΩf φdx=/integraldisplay\nΩ∇y·∇φdx−/integraldisplay\nΓφ∂y\n∂nds\n=/integraldisplay\nΩ∇y·∇φdx,\nwhere we have used the fact that φ∈H1\n0(Ω) to get the last equality. The idea in the method of\ntransposition is to apply the integration by parts once more. By doin g so, we get\n/integraldisplay\nΩf φdx=−/integraldisplay\nΩy∆φdx+/integraldisplay\nΓy∂φ\n∂nds.\nThus, the ultra weak formulation of (1.2) can be stated as: Given f∈L2(Ω) and u∈L2(Γ), find\ny∈L2(Ω) such that\n(y,∆φ) =−(f,φ)+/an}b∇acketle{tu,∂φ/∂n /an}b∇acket∇i}ht ∀φ∈X, (1.4)\nwhere (·,·) and/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotes the L2inner product on the domain Ω and its boundary Γ respectively.\nAnother approach to study second order Dirichlet boundary cont rol problem is to transform the\nDirichlet boundary condition into Robin’s type boundary condition [12 ]. Other method is to employ\nthe energy space based approach in which the control is penalized in the energy space H1/2(Γ)\n[13, 26]. In this article the control variable is discretized with piecewis e constant finite elements\nwhile the state and adjoint states are discretized using first-orde r non conforming Crouzeix Raviart\nfinite elements.\nIn the finite element analysis for the Dirichlet boundary control pro blems addressed so far in the\nliteraturepolynomialdegreesusedtodiscretizethestatevariable ortheadjointstatevariableandthe\ncontrol variable happens to be the same. But in this article we have d iscretized the control variable\nwith piecewise constant finite elements whereas the state and adjo int variables are discretized using\nCrouzeix Raviart finite elements. Therefore the main challenge is to a ssociate the discrete state\nvariables with the discrete control variables via an invertible bounde d linear operator such that the\noperatornormoftheinverseremainsuniformlyboundedwith respe cttothediscretizationparameter\nh >0 (to be defined later). In order to construct this operator in (4.2 1) a mild restriction on the\ntriangulation is imposed (discussed in Section 4 ). This constitutes on e of the main novelties of this\narticle. Apart from this an enrichment operator is introduced satis fying the orthogonality described\nin Theorem 3.4, which plays a crucial role in the entire analysis. Moreov er in Theorem 4.1, 4.2,4.3,\n5.2 and 4.5 several super convergence results are derived, which a lso contributes significantly to the\nnovelty of this article.\n2 Continuous Problem\nIn this segment, we discuss the variational setting of the investiga ted problem and derive the first\nordernecessaryoptimalityconditions. Before,weenterintothea nalysispart, wedefinethenotations\n2S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nwhich are used throughout the article. The notations Hm(Ω),Hm\n0(Ω), and Wm,p(Ω) represent the\nstandard Sobolev spaces on Ω, where mis a non negative integer. The Sobolev spaces of non integer\norders,Hs(Ω) and Ws,p(Ω), are defined using interpolation, as explained in [7]. On a Lipschitz\ndomain, this definition is equivalent to the definition that uses double- integral norms, which is\ndiscussed in [7, Theorem 14.2.3]. The norms of Hs(Γ) and Ws,p(Γ), 0≤s≤1 and 1 < p <∞,\nare defined on the boundary Γ using charts, which is equivalent to us ing double-integral norms on\nΓ, according to [7, 25]. The outwards unit normal vector to Γ is deno ted byn. The dual space of\nH1/2(Γ) is denoted by H−1/2(Γ) equipped with the operator norm /ba∇dbl·/ba∇dblH−1/2(Γ)defined as:\n/ba∇dblµ/ba∇dblH−1/2(Γ):= sup\nv∈H1/2(Γ),v/\\e}atio\\slash=0/an}b∇acketle{tµ,v/an}b∇acket∇i}ht\n/ba∇dblv/ba∇dblH1/2(Γ), µ∈H−1/2(Γ).\nAny generic member of R2is denoted by x= (x1,x2), where x1,x2∈R. In the article, Cdenotes a\npositive constant, which is generic in nature and does not depend on the solutions or the mesh-size.\nRemark 2.1. Whenvbelongs to H2(Ω), the trace of its gradient, denoted by ∇v|Γ, lies inH1/2(Γ)2.\nIfΓhas a smooth boundary, the outwards unit normal vector nis continuous, allowing the definition\nof the normal derivative ∂nv=n·∇vto be well defined. Thus ∂nv∈H1/2(Γ)forv∈H2(Ω)and\nthe following estimate holds:\n/ba∇dbl∂nv/ba∇dblH1/2(Γ)≤C/ba∇dblv/ba∇dblH2(Ω), v∈H2(Ω).\nHowever, this estimate is not applicable for polygonal boun dariesΓ, which are only Lipschitz con-\ntinuous. Nonetheless, for v∈H2(Ω), we still have ∂nv|Γj∈H1/2(Γj)for each straight component\nΓj,1≤j≤kofΓ. To accommodate this, we introduce the space ˜H1/2(Γ) :={v∈L2(Γ),v|Γj∈\nH1/2(Γj)∀1≤j≤m}. We denote by ˜H−1/2(Γ)the completion of L2(Γ)with respect to the\nfollowing dual norm on L2(Γ)\n/ba∇dblµ/ba∇dbl˜H−1/2(Γ):= sup\nv∈X\\{0}/an}b∇acketle{tµ,∂nv/an}b∇acket∇i}ht\n/ba∇dblv/ba∇dblH2(Ω)≤sup\nv∈˜H1/2(Γ),v/\\e}atio\\slash=0/an}b∇acketle{tµ,v/an}b∇acket∇i}ht\n/ba∇dblv/ba∇dbl˜H1/2(Γ).\nNote that ˜H−1/2(Γ)is not in general the dual space of ˜H1/2(Γ). In the case of a smooth boundary\nΓ, the mapping ∂n:X→H1/2(Γ)is surjective, so ˜H−1/2(Γ) =H−1/2(Γ).\nFor any u∈˜H−1/2(Γ), the harmonic extension of uis denoted by yu∈L2(Ω) which satisfies the\nfollowing equation:\n(yu,∆φ) =/an}b∇acketle{tu,∂nφ/an}b∇acket∇i}ht ∀φ∈X. (2.1)\nThe function u∈˜H−1/2(Γ) is known as the ultra weak trace of yu∈L2(Ω) and the following trace\nestimate holds true:\n/ba∇dblu/ba∇dbl˜H−1/2(Γ)= sup\nφ∈X\\{0}(yu,∆φ)\n/ba∇dblφ/ba∇dblH2(Ω)/lessorsimilar/ba∇dblyu/ba∇dblL2(Ω).\nFurthermore, if u∈H1/2(Γ), then yu∈H1(Ω) becomes the standard harmonic extension of usuch\nthat it satisfies the following equation:\n(∇yu,∇φ) = 0∀φ∈V.\nIn this case u∈H1/2(Γ) is simply the trace of yuin the sense that yu|Γ=u.\n2.1 Optimality System\nIn this section, we write the model Dirichlet boundary optimal contr ol problem and derive the\ncorresponding optimality system by applying the first order necess ary optimality conditions. For\n3S. Chowdhury and D. Garg PHFEM for Constrained DBCP\n(f,u)∈L2(Ω)×L2(Γ), define ( yf,yu)∈L2(Ω)×L2(Ω) tobe the solutionsofthe followingequations:\n(yf,∆φ) =−(f,φ)∀φ∈X, (2.2)\n(yu,∆φ) =/an}b∇acketle{tu,∂nφ/an}b∇acket∇i}ht ∀φ∈X. (2.3)\nIn view of (2.2)-(2.3), the ultra weak solution y∈L2(Ω) of (1.4) can be written as y= (yf+yu).\nIn addition, if the given data satisfy ( f,u)∈L2(Ω)×H1/2(Γ), then we can recover the standard\nweak formulation, i.e. the following holds:\n\n\ny =yf+yu, yf∈H1\n0(Ω), yu∈H1(Ω),\nyu =uon Γ,\na(yf,v) = (f,v)−a(yu,v)∀v∈H1\n0(Ω),(2.4)\nwherea(·,·) :H1(Ω)×H1(Ω)→Ris a symmetric bilinear form given by:\na(v,w) =/integraldisplay\nΩ∇v·∇wdx∀v,w∈H1(Ω).\nThe bilinear form a(·,·) defines a continuous mapping on H1(Ω) and coercive map on H1\n0(Ω). Set\nV=H1\n0(Ω) and Q=H1(Ω). Given that Ω is a convex polygonal domain, for u= 0, the weak\nsolutionyof (1.2) belongs to H2(Ω)∩H1\n0(Ω). Additionally, it satisfies the following a prioribound.\n/ba∇dbly/ba∇dblH2(Ω)≤C/ba∇dblf/ba∇dblL2(Ω).\nThe subsequent lemma affirms the well-posed-ness of the boundary value problem (1.2) in its ultra\nweakform, and asa particularinstance, it ensuresthe presence o fthe ultra weakharmonicextension\nof the general boundary data u∈˜H−1/2(Γ).\nLemma 2.2. For any given u∈˜H−1/2(Γ)andf∈H−2(Ω), the state equation in its ultra weak\nform(1.4)has a unique solution y∈L2(Ω)such that the following a prioriestimate is satisfied:\n/ba∇dbly/ba∇dblL2(Ω)≤C/parenleftBig\n/ba∇dblf/ba∇dblH−2(Ω)+/ba∇dblu/ba∇dbl˜H−1/2(Γ)/parenrightBig\n, (2.5)\nwhereH−2(Ω)is the dual space of X.\nProof.Forf∈H−1(Ω) and u∈H1/2(Γ), there exists a unique weak solution y∈H1(Ω) of (1.2)\nsatisfying (2.4). An application of integration by parts in (2.4) yields\n−(y,∆φ) = (f,φ)−/an}b∇acketle{tu,∂nφ/an}b∇acket∇i}ht ∀φ∈X, (2.6)\nwhich implies that yis the solution of (1.4). We apply duality argument to prove the a priorierror\nestimate (2.5). Consider z∈Vto be the weak solution of the following Poisson problem:\n/braceleftBigg\n−∆z=yin Ω,\nz= 0 on Γ .\nBy elliptic regularity theory, z∈H2(Ω) and/ba∇dblz/ba∇dblH2(Ω)≤C/ba∇dbly/ba∇dblL2(Ω). Thus, using (2.6) for φ=z, we\nget\n/ba∇dbly/ba∇dbl2\nL2(Ω)= (y,−∆z)\n= (f,z)−/an}b∇acketle{tu,∂nz/an}b∇acket∇i}ht,\nwhich on using Cauchy-Schwarz inequality yields (2.5). Since H−1(Ω)⊆H−2(Ω) and H1/2(Γ)⊆\n˜H−1/2(Γ) are dense, by density arguments, for u∈˜H−1/2(Γ) andf∈H−2(Ω), the equation (1.4)\npossesses a unique solution y∈L2(Ω) satisfying the a priori error estimate (2.5).\n4S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nIn light of the Lemma 2.2, we define the control to state map which ma ps the given data to the\nunique solution of (1.4).\nDefinition 2.3. For(f,u)∈L2(Ω)×L2(Γ), the solution map S:L2(Ω)×L2(Γ)→L2(Ω)is defined\nasS(f,u) =y= (yf+yu)such that yf,yusatisfy(2.2)and(2.3), respectively.\nNow, we define the space of admissible controls Uadas follows:\nUad:={u∈L2(Γ) :ua≤u(x)≤uba.ex∈Γ},\nwhereua,ub∈Rare such that ua< ub. The model Dirichlet boundary optimal control problem\nreads as: find (¯ y,¯u)∈L2(Ω)×Uadsuch that\nJ(¯y,¯u) = min\n(y,u)∈L2(Ω)×UadJ(y,u), (2.7)\nsubject to the constraint y=S(f,u), where J(·,·) is the quadratic cost functional defined by (1.3).\nNext, we present the following result that addresses the well-pose dness of the underlying optimal\ncontrol problem (2.7) and the corresponding first-order optimalit y system.\nTheorem 2.4. The optimal control problem (2.7)possess a unique solution (¯y,¯u)∈L2(Ω)×Uad.\nMoreover the unique pair (¯y,¯u)solves the following first order necessary optimality condi tions:\n¯y= ¯yf+y¯u,¯yf, y¯u∈L2(Ω), (2.8)\n(¯yf,∆φ) =−(f,φ)∀φ∈X, (2.9)\n(y¯u,∆φ) =/an}b∇acketle{t¯u,∂nφ/an}b∇acket∇i}ht ∀φ∈X, (2.10)\n(¯y−yd,yu−y¯u)+α/an}b∇acketle{t¯u,u−¯u/an}b∇acket∇i}ht ≥0∀u∈Uad. (2.11)\nProof.By utilizing the solution operator S(·,·), we can reformulate the model optimal control\nproblem (2.7) into a simplified form, expressed as follows:\nmin\nu∈Uadj(u), (2.12)\nwherej(·) :L2(Γ)→Rrepresents the reduced cost functional defined as:\nj(u) :=1\n2/ba∇dblS(f,u)−yd/ba∇dbl2\nL2(Ω)+α\n2/ba∇dblu/ba∇dbl2\nL2(Γ).\nThe reduced cost functional j(·) defines a strictly convex functional on a non empty closed convex\nHilbert space Uad. Thus, the standard theory of PDE constrained optimal control problem [27]\nimplies that problem (2.12) has a unique solution say ¯ u∈Uad. Setting ¯ y=S(f,¯u) as the corre-\nsponding state variable. By definition of S(·,·), we find that ¯ y∈L2(Ω) satisfies (2.8)-(2.10). By\napplying the first order necessary optimality conditions, we get for anyu∈Uad\n0≤j′(¯u)(u−¯u) := lim\nt↓0j(¯u+t(u−¯u))−j(¯u)\nt\n= (¯y−yd,yu−y¯u)+α/an}b∇acketle{t¯u,u−¯u/an}b∇acket∇i}ht,\nwhich implies that ¯ u∈Uadsolves (2.11). This completes the proof.\n2.2 Regularity of the Optimal Control\nIn this fragment, we obtain the regularity of the optimal variables n amely ¯yand ¯uwhich are useful\nfor the subsequent analysis. Introduce an adjoint state θ∈Vsuch that it satisfies the following\nvariational formulation: find θ∈Vsuch that:\n(∇v,∇θ) = (¯y−yd,v)∀v∈V. (2.13)\n5S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nBy elliptic regularity theory for convex polygonal domain [19], θ∈Xand it satisfies the following\na priori estimate:\n/ba∇dblθ/ba∇dblH2(Ω)≤C/ba∇dbl¯y−yd/ba∇dblL2(Ω).\nUsing integration by parts and density arguments in (2.13), we get\n(v,∆θ) =−(¯y−yd,v)∀v∈L2(Ω). (2.14)\nSinceθ∈H2(Ω), it implies ∂nθ∈˜H1/2(Γ). For any u∈Uad, using (2.14) for v=yu−y¯u∈L2(Ω),\nwe get\n(¯y−yd,yu−y¯u) =−(yu−y¯u,∆θ)\n=−/an}b∇acketle{tu−¯u,∂nθ/an}b∇acket∇i}ht,\nwhere we have used (2.1) to observe the last equality. Hence, the in equality (2.11) reduces to\nα/an}b∇acketle{t¯u,u−¯u/an}b∇acket∇i}ht−/an}b∇acketle{tu−¯u,∂nθ/an}b∇acket∇i}ht ≥0∀u∈Uad,\nwhich implies\n/an}b∇acketle{tα¯u−∂nθ,u−¯u/an}b∇acket∇i}ht ≥0∀u∈Uad. (2.15)\nIt follows from (2.15) and standard arguments from [27, Chapter 2 ] that\n¯u(x) =P[ua,ub]/parenleftBig1\nα(∂nθ)(x)/parenrightBig\n,for almost every x∈Γ, (2.16)\nwhereP[ua,ub](w) := min {ub,max{ua,w}}denotes the projection of Ronto [ua,ub]. Since Ω is\nconvex, therefore it follows from (2.16) that ¯ u∈˜H1/2(Γ). Asθequals zero on Γ, the derivative of\nθin the tangential direction also equals zero on Γ. Therefore, at the corner points of the domain\ndenoted as x, we have ( ∂nθ)(x) = 0. Hence, it follows from (2.16) that ¯ u(x) = 0 at all the corner\npointsxof the domain, which implies that ¯ u∈H1/2(Γ).\n3 Preliminaries\nWe introduce the following notations which are used throughout the article.\n•This a regular triangulation of Ω into closed triangles T.\n•hT:= diam ( T) is the diameter of the triangle T∈ Th.\n•h:= max\nT∈ThhTis the mesh parameter.\n•Vi\nh:= the set of all interior vertices of Th.\n•Vb\nh:= the set of all boundary vertices of Th.\n•Vh:=Vi\nh∪Vb\nhis the set of all vertices of Th.\n•Ei\nh:= the set of all interior edges of Th.\n•Eb\nh:= the set of all boundary edges of Th.\n•Eh:=Ei\nh∪Eb\nhis the set of all edges of Th.\n•me:= mid point of e∈ Eh.\n•Pk(T) := the set of all polynomials of degree atmost k∈N∪{0}overT∈ Th.\n6S. Chowdhury and D. Garg PHFEM for Constrained DBCP\n•Hk(Ω,Th) :={v∈L2(Ω) :v|T∈Hk(T)∀T∈ Th},k∈N∪{0}.\nFrom this point forward, the notation a/lessorsimilarbindicates the existence of a positive constant Cthat is\nnot dependent on the mesh parameter h, such that a a≤Cb. It is assumed that the triangulation\nThis regular, meaning that any two distinct simplices in Thwith non-empty intersection are either\nidentical, or share exactly one common vertex, or one common edge . Additionally, every interior\nangle of any simplex is bounded from below by a universal positive cons tantρ. All the generic\nconstantshiddeninthenotation /lessorsimilararesolelydeterminedby ρ >0,therebyensuringthetriangulation\nis shape regular.\nNext, we will establish the definitions for the jump and average of fu nctions that are scalar-valued\nand vector-valued. Consider an edge e∈ Ei\nhthat is shared by two adjacent triangles T+andT−,\nsuch that e=T+∩T−. Letn+be the unit normal of epointing from T+toT−, andn−=−n+. In\nthis context, we define the jumps [[ ·]] and averages { {·} }across the edge eas follows:\n•For a scalar valued function w∈H1(Ω,Th), define\n[[w]] :=w+n++w−n−,{ {w} }:=w++w−\n2,\nwherew±=w|T±.\n•For a vector valued function v∈[H1(Ω,Th)]2, define\n[[v]] :=v+·n++v−·n−,{ {v} }:=v++v−\n2,\nwherev±=v|T±.\nTo simplify the notation, we also introduce the concepts of jump and mean on the boundary Γ.\nConsider any edge e∈ Eb\nh, where it is evident that there exists a triangle T∈ Thsuch that e=\n∂T∩Γ. Letnebe the unit normal of epointing outward from T. For any w∈H1(Ω,Th) and any\nv∈[H1(Ω,Th)]2, we define the following on e∈ Eb\nh:\n[[w]] :=wneand{ {w} }=w,\n[[v]] :=v·neand{ {v} }=v.\nBelow, we define the Crouziex-Raviart finite element spaces.\nVh:={vh∈L2(Ω) :vh|T∈P1(T)∀T∈ Thandvhis continuous at me∀e∈ Eh},\nV0\nh:={vh∈Vh:vh(me) = 0∀e∈ Eb\nh}.\nWe need to modify the definition of Vh. The mean integral of the function should be\ncontinuous i.e DOF should be modified to mean integrals.\nThe discrete bilinear form apw(·,·) :Vh×Vh→Ris defined by\napw(vh,wh) =/summationdisplay\nT∈Th/integraldisplay\nT∇vh|T·∇wh|Tdx∀vh,wh∈Vh.\nThe mesh dependent bilinear form apw(·,·) defines a symmetric bilinear form which coincides with\nthe continuous bilinear form a(·,·) onH1(Ω)×H1(Ω). Now, we define the mesh dependent norm\nonVh. For any vh∈Vh, define/ba∇dbl·/ba∇dblhas follows:\n/ba∇dblvh/ba∇dblh:=/radicalBig\napw(vh,vh).\nNote that, /ba∇dbl · /ba∇dblhdefines only a semi-norm on Vhbut a norm on V0\nh. Next, we define a Crouziex-\nRaviart interpolation map ICR:V∩C(¯Ω)→V0\nhas follows:\n/integraldisplay\neICRvds=/integraldisplay\nevds∀e∈ Eh, v∈V∩C(¯Ω). (3.1)\n7S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nBy density arguments, ICRcan be extended continuously and uniquely to V, and the extension is\nagain denoted by ICRfor the ease of presentation. In the next lemma, we state the app roximation\nproperties of ICR[15].\nLemma 3.1. Forv∈V, it holds that:\n/ba∇dblv−ICRv/ba∇dblL2(Ω)+h/ba∇dblv−ICRv/ba∇dblh/lessorsimilarh|v|H1(Ω).\n3.1 Raviart-Thomas Interpolation Operator\nIn this segment, Raviart-Thomasinterpolation operatoris defined which is used further for the error\nanalysis. Given a triangle T∈ Th, the lowest order Raviart-Thomas space is defined as:\nRT0(T) :=P0(T)2+xP0(T),\nwherex= (x1,x2)∈R2.\nRemark 3.2. LetT∈ Thbe any triangle. For any v∈RT0(T), we have v·ne∈P0(e)for all\ne∈∂T, whereneis the unit outward normal vector to e.\nFor anyT∈ Th, define the local interpolation operator Π T:H1(T)2→RT0(T) as follows:\n/integraldisplay\neΠTv·nipds=/integraldisplay\nev·nipds∀p∈P0(e),∀e∈∂T.\nWe state the approximation property of the interpolation operato r ΠTin the following lemma.\nLemma 3.3. LetT∈ Thbe any fixed triangle. For any v∈H1(T)2, the following holds:\n/ba∇dblv−ΠTv/ba∇dblL2(T)/lessorsimilarhT|v|H1(T).\nProof.We refer the readers to the article [1] for the proof.\n3.2 Enrichment Operator\nThe focus of this section is on introducing a new enrichment operato r which plays a crucial role in\nourupcoming analysis. It is well known that the enrichment operato rsplaya crucialrole in the anal-\nysis for non-conforming and discontinuous Galerkin finite element me thods [4, 5, 6, 8, 9, 10, 22, 23].\nThese operators connect the non-conforming or discontinuous G alerkin spaces to it’s conforming\ncounterpart.\nThe novelty of the enrichment operator being introduced here is du e to two facts. The degrees of\nfreedom for the finite element functions of it’s conforming counter parts are taken in a non-standard\nmanner which enables us to prove an orthogonality result for the CR -finite element functions (The-\norem 3.4). This result plays a crucial role in obtaining the optimal orde r energy norm error estimate\nfor the solution of the state equation.\nTo begin with we modify the definitions of degrees of freedoms of the conforming counterpart Vcof\nVh.\nVc={vh∈V:vh|T∈P2(T), vh|+(p) =vh|−(p) forp∈νand/integraldisplay\nevh|+ds=/integraldisplay\nevh|−ds\nwherevh|+=vh|T+, vh|−=vh|T−, e∈ Eh}. (3.2)\nNote that this choice of Vcenables the Lagrange interpolation operator IL:H2(Ω)→Vcto have\nthe following orthogonal property:\n/integraldisplay\ne(v−ILv)ds= 0, (3.3)\n8S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nwhich plays a crucial role to prove Theorem 3.4.\nNote that, Vcdenotes the Lagrange finite element space of order 2, which is the c onforming\ncounterpart of V0\nh. LetIL:V∩C(¯Ω)→Vcdenotes the Lagrange interpolation operator defined as\nfollows: for v∈V∩C(¯Ω),\n(ILv)(p) =v(p)∀p∈ Vh,/integraldisplay\neILvds=/integraldisplay\nevds∀e∈ Eh. (3.4)\nLetIc:V0\nh→Vcdenotes the enriching map defined by: for vh∈V0\nh, define\nIcvh(p) =1\n|Tp|/summationdisplay\nT∈Tpvh|T(p)∀p∈ Vi\nh,\nIcvh(p) = 0∀p∈ Vb\nh, (3.5)/integraldisplay\neIcvhds=/integraldisplay\nevhds∀e∈ Eh, (3.6)\nwhereTpdenotes the set of triangles sharing the node p∈ Vhand|Tp|refers to the cardinality of Tp.\nSubsequently, we prove an orthogonality result concerning Ic, which holds significant importance in\nthe subsequent analysis.\nTheorem 3.4. Letphbe a piecewise linear function (need not be in Vh), it holds that:\napw(ph,vh−Icvh) = 0∀vh∈V0\nh.\nProof.Letvh∈V0\nhbe an arbitrary function. It follows from triangle wise integration by parts that\napw(ph,vh−Icvh) =/summationdisplay\ne∈Eh/integraldisplay\ne/braceleftbigg /braceleftbigg∂ph\n∂n/bracerightbigg /bracerightbigg\n[ [vh−Icvh)] ]ds\n+/summationdisplay\ne∈Ei\nh/integraldisplay\ne/bracketleftbigg /bracketleftbigg∂ph\n∂n/bracketrightbigg /bracketrightbigg\n{ {vh−Icvh} }ds. (3.7)\nSince,ph|T∈P1(T) for all T∈ Th, the rest of the proof follows from using (3.5) and (3.6) in\n(3.7).\n4 Crouzeix-Raviart Analysis of Various State Equations\nThis section considers the Crouzeix-Raviart (CR) finite element app roximation of the solutions of\ncertain auxiliary state equations. While some of these equations hav e a typical weak solution in\nH1(Ω), others do not. The optimal order L2norm error estimates are derived for both of the cases.\nTo begin with, we define the spaces as follows:\nWh:={vh∈H1(Ω) :vh|T∈P1(T)∀T∈ Th},\nUh:={uh∈L2(Γ) :uh|e∈P0(e)∀e∈ Eb\nh},\nV1(Γ) :={vh∈C(Γ) :vh|e∈P1(e)∀e∈ Eb\nh}.\nFirstly, we consider the equation which possesses standard weak s olution in H1(Ω). Let zh∈V1(Γ)\nbe an arbitrary function. Since, V1(Γ)⊂H1/2(Γ), thus zh∈H1\n2(Γ). Using zh, we define a function\nsay ˜zhin the following manner:\n/braceleftBigg\n˜zh(p) = 0 ∀p∈ Vi\nh,\n˜zh(p) =zh(p)∀p∈ Vb\nh.(4.1)\n9S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nIt is easy to see that ˜ zh∈Wh. For any zh∈V1(Γ), define yzh∈H1(Ω) such that:\nyzh=y0+ ˜zh, y0∈V, (4.2)\na(y0,v) =−a(˜zh,v)∀v∈V. (4.3)\nLetyzh,h∈Vhbe the CR-approximation of yzh∈H1(Ω), i.e. it satisfies the following equation:\nyzh,h=y0,h+ ˜zh, y0,h∈V0\nh, (4.4)\napw(y0,h,vh) =−apw(˜zh,vh)∀vh∈V0\nh. (4.5)\nBy Lax-milgram lemma, auxiliary problems (4.2)-(4.3) and (4.4)-(4.5) a re well posed. The following\nTheorem yields an optimal order error estimate for yzh.:\nTheorem 4.1. Forzh∈V1(∂Ω)letyzh, yzh,hsatisfies (4.2),(4.2)and(4.4),(4.5)respectively.\nThen/ba∇dblyzh−yzh,h/ba∇dbl/lessorsimilarh[/ba∇dblyzh−yzh,h/ba∇dblpw+h1−ǫ/ba∇dblzh/ba∇dblH3\n2−ǫ(∂Ω)]for anyǫ >0.\nProof.Next, we estimate /ba∇dblyzh−yzh,h/ba∇dbl. Again, by duality, we find that\n/ba∇dblyzh−yzh,h/ba∇dbl= sup\ng∈L2(Ω),g/\\e}atio\\slash=0(yzh−yzh,h,g)\n/ba∇dblg/ba∇dbl. (4.6)\nUsing (4.2) and (4.4), we have\nyzh−yzh,h=y0−¯y0,h. (4.7)\nFor anyg∈L2(Ω), define φg∈H1\n0(Ω) such that\na(φg,v) = (g,v)∀v∈H1\n0(Ω). (4.8)\nBy elliptic regularity theory, we have φg∈H2(Ω) such that /ba∇dblφg/ba∇dbl2,Ω/lessorsimilar/ba∇dblg/ba∇dbl. Letφg,h∈V0\nhbe\nthe standard non-conforming finite element approximation of φg. Thus, it satisfies the following\nequation:\napw(φg,h,vh) = (g,vh)∀vh∈V0\nh. (4.9)\nSincey0∈H1\n0(Ω) and ¯ y0,h∈Vh, it follows from (4.3),(4.5), (4.8) and (4.9) that\n(g,y0−¯y0,h) =a(φg,y0)−apw(φg,h,¯y0,h)\n=apw(φg−φg,h,y0−¯y0,h)+apw(φg−φg,h,¯y0,h)+apw(φg,h,y0−¯y0,h)\n=apw(φg−φg,h,y0−¯y0,h)+apw(φg,¯y0,h)−(g,¯y0,h)\n+apw(φg,h,y0)+apw(˜zh,φg,h)\n=apw(φg−φg,h,y0−¯y0,h)+apw(φg,¯y0,h−y0)−(g,¯y0,h−y0)\n+apw(y0,φg,h−φg)+apw(˜zh,φg,h−φg)\n=I+II+III+IV+V. (4.10)\nWe estimate each term on the right hand side of (4.10). First, consid er the term IV+V.\nIV+V=apw(y˜zh,φg,h−φg)\nNote that since zhis a piecewise linear and globally continuous on each line segment Γ j(1≤j≤\nn) consisting ∂Ω. Therefore zh|Γj∈H3\n2−ǫ(Γj) for any ǫ >0 [3]. Since zhsatisfies the compatibility\nconditions that zhis continuous at the vertices of Ω and as Ω is convex therefore an ap plication of\n10S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nresult from real interpolation spaces [7] yields that yzh∈H2−ǫ(Ω) [18]Theorem 1.8. Consider the\ntermapw(y˜zh,φg,h−φg). A use of integration by parts yields\napw(y˜zh,φg,h−φg) =/summationdisplay\nT∈Th/integraldisplay\nT∇y˜zh·∇(φg,h−φg)dx\n=−/summationdisplay\nT∈Th/integraldisplay\nT∆y˜zh(φg,h−φg)dx+/summationdisplay\nT∈Th/integraldisplay\n∂T∇y˜zh·n(φg,h−φg)ds\n=−/summationdisplay\nT∈Th/integraldisplay\nT∆y˜zh(φg,h−φg)dx\n+/summationdisplay\nT∈Th/integraldisplay\n∂T∂y˜zh\n∂n(φg,h−φg)ds, (4.11)\n=−/summationdisplay\nT∈Th/integraldisplay\nT∆y˜zh(φg,h−φg)dx\n+/summationdisplay\nT∈Th/integraldisplay\n∂T∂y˜zh\n∂n((φg,h−φg)−ICR((φg,h−φg))ds+\n/summationdisplay\nT∈Th/integraldisplay\n∂T∂y˜zh\n∂n(ICR(φg,h−φg)−IcICR((φg,h−φg))ds+/summationdisplay\nT∈Th/integraldisplay\n∂T∂y˜zh\n∂nIcICR((φg,h−φg))ds\n=−/summationdisplay\nT∈Th/integraldisplay\nT∆y˜zh(φg,h−φg)dx\n+/summationdisplay\nT∈Th/integraldisplay\n∂T(∇y˜zh−ΠT∇y˜zh)·n((φg,h−φg)−ICR((φg,h−φg))ds+\n/summationdisplay\nT∈Th/integraldisplay\n∂T(∇y˜zh−ΠT∇y˜zh)·n(ICR(φg,h−φg)−IcICR((φg,h−φg))ds+\n/summationdisplay\ne∈Eh/integraldisplay\ne∂y˜zh\n∂n[ [IcICR((φg,h−φg))] ]ds. (4.12)\nWhereΠ TislowestorderRaviart-Thomasinterpolationoperatorasin Lemma3 .3. Note that ∆ y˜z=\n0 and/integraldisplay\ne∂y˜zh\n∂n[ [IcICR((φg,h−φg))] ]ds= 0. Therefore application of Cauchy Schwarz inequality,\ndiscrete trace inequality, Crauzeix Raviart and Raviart Thomas inte rpolation error estimates in\n(4.12) yields\napw(y˜zh,φg,h−φg) =/lessorsimilarh1−ǫ|y˜zh|2−ǫ,Ω/ba∇dblφg,h−φg/ba∇dblpw. (4.13)\nNow, consider the terms II+III.\nII+III=apw(φg,¯y0,h−y0)−(g,¯y0,h−y0)\n=apw(φg,(¯y0,h−y0)−ICR(¯y0,h−y0))+apw(φg,ICR(¯y0,h−y0)−Ic(ICR(¯y0,h−y0)))+\napw(φg,Ic(ICR(¯y0,h−y0)))−(g,(¯y0,h−y0)−ICR(¯y0,h−y0))−\n(g,ICR(¯y0,h−y0)−Ic(ICR(¯y0,h−y0)))−(g,Ic(ICR(¯y0,h−y0))). (4.14)\n11S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nFrom (4.14) consider the term apw(φg,(¯y0,h−y0)−ICR(¯y0,h−y0)).\napw(φg,(¯y0,h−y0)−ICR(¯y0,h−y0))\n=−/summationdisplay\nT∈Th/integraldisplay\nT∆φg((¯y0,h−y0)−ICR(¯y0,h−y0))dx+/summationdisplay\nT∈Th/integraldisplay\n∂T∂φg\n∂n((¯y0,h−y0)−ICR(¯y0,h−y0))ds\n=/summationdisplay\nT∈Th/integraldisplay\nTg((¯y0,h−y0)−ICR(¯y0,h−y0))dx+/summationdisplay\nT∈Th/integraldisplay\n∂T(∇φg−(ΠT∇φg))·ne\n((¯y0,h−y0)−ICR(¯y0,h−y0))ds\n≤/ba∇dblg/ba∇dbl/ba∇dbl(¯y0,h−y0)−ICR(¯y0,h−y0)/ba∇dbl+/summationdisplay\ne∈Eh/ba∇dbl(∇φg−(ΠT∇φg))·ne/ba∇dbl∂Te/ba∇dbl(¯y0,h−y0)−ICR(¯y0,h−y0)/ba∇dbl∂Te\n(4.15)\nAn application of the discrete trace-inequality and the CR interpolat ion error estimation in (4.15)\nyields,\napw(φg,(¯y0,h−y0)−ICR(¯y0,h−y0))/lessorsimilarh/ba∇dblg/ba∇dbl/ba∇dbl¯y0,h−y0/ba∇dblpw. (4.16)\nArguments in the similar line yields\napw(φg,ICR(¯y0,h−y0)−Ic(ICR(¯y0,h−y0)))/lessorsimilarh/ba∇dblg/ba∇dbl/ba∇dbl¯y0,h−y0/ba∇dblpw. (4.17)\nCombining (4.8), (4.14), (4.15), (4.16), (4.17) and standard L2norm error-estimates for ICRinter-\npolation operator yields,\nII+III/lessorsimilar/ba∇dblg/ba∇dbl/ba∇dbl¯y0,h−y0/ba∇dblpw. (4.18)\nNext we aim to consider /ba∇dbly0,h−y0/ba∇dblpw.\nFrom (4.2), (4.3), (4.4), Theorem 3.4 and (4.5) we obtain\n/ba∇dbly0,h−ICRy0/ba∇dbl2\npw≤apw(ICRy0−y0,h,ICRy0−y0,h)\n=apw(ICRy0,ICRy0−y0,h)−apw(y0,h,ICRy0−y0,h)\n=apw(ICRy0−y0,ICRy0−y0,h)+apw(y0,ICRy0−y0,h)+\napw(˜P1(¯uh),ICRy0−y0,h−Ic(ICRy0−y0,h))+apw(˜P1(¯uh),Ic(ICRy0−y0,h))\n=apw(ICRy0−y0,ICRy0−y0,h)+apw(y0,ICRy0−y0,h)+apw(˜P1(¯uh),Ic(ICRy0−y0,h))\n=apw(ICRy0−y0,ICRy0−y0,h)+apw(y0,ICRy0−y0,h−Ic(ICRy0−y0,h))+\napw(y0,Ic(ICRy0−y0,h))+apw(˜P1(¯uh),Ic(ICRy0−y0,h))\n=apw(ICRy0−y0,ICRy0−y0,h)+apw(y0,ICRy0−y0,h−Ic(ICRy0−y0,h))\n=apw(ICRy0−y0,ICRy0−y0,h)+apw(y0−ICRy0,ICRy0−y0,h−Ic(ICRy0−y0,h))\n(4.19)\nThen applying the similar arguments as in (4.11), (4.12) and Young’s ine quality in the right hand\nside of (4.19) we obtain\n/ba∇dbly0,h−ICRy0/ba∇dblpw/lessorsimilar/ba∇dbly0−ICRy0/ba∇dblpw. (4.20)\nFinally combining (4.6), (4.7), (4.10), (4.13), (4.18) and (4.20) the re sult is obtained.\nCorollary : Forzh∈V1(∂Ω) letyzh, yzh,hsatisfies (4.2)-(4.3) and (4.4)-(4.5) respectively. Then\n/ba∇dblyzh−yzh,h/ba∇dbl/lessorsimilarh[/ba∇dblyzh−yzh,h/ba∇dblpw+/ba∇dblzh/ba∇dblH1\n2(∂Ω)].\nWe note that since we are considering triangulations Thsuch that it induces an odd number\nof edges on ∂Ω anddim V 1(∂Ω) =dim U htherefore P0:V1(∂Ω)→Uhis bijective. Define\n˜P1:Uh→V1(∂Ω) by\n˜P1=P−1\n0. (4.21)\n12S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nNote that the matrix representation of P0:V1(∂Ω)→Uhis\n\n1 0 0 · ·0\n1 1 0 · ·0\n0 1 1 · ·0\n· · · · · 0\n· · · 1 1 0\n0 0·0 1 1\n\nTherefore if /ba∇dbl˜P1/ba∇dblL(Uh,V1(∂Ω))denotes the operator norm of ˜P1subordinate to L2norm then it is\nuniformly bounded with respect to the mesh refinement parameter h >0 [21].i. e.\n/ba∇dbl˜P1/ba∇dblL(Uh,V1(∂Ω))≤C, (4.22)\nwhereC >0 is independent of h >0. This operator plays a central role in associating the discrete\ncontrol variable with the discrete state variable, which is discussed in the next section.\nTheorem 4.2. For a piecewise constant function uhdefined on ∂Ω, the following holds:\n/ba∇dblyuh−y˜P1(uh)/ba∇dbl/lessorsimilarh/ba∇dbl˜P1uh/ba∇dblH1\n2(∂Ω)..\n/ba∇dblyuh−yP1(uh)/ba∇dbl/lessorsimilarh.\nProof.Sinceuh∈L2(Γ),yuh∈L2(Ω) is the solution of the following ultra weak formulation:\n(yuh,∆φ) =−/an}b∇acketle{tuh,∂φ/∂n/an}b∇acket∇i}ht ∀φ∈X. (4.23)\nNote that ˜P1(uh)∈H1/2(Γ) andy˜P1(¯uh)∈L2(Ω) also satisfies\n(y˜P1(uh),∆φ) =−/an}b∇acketle{t˜P1(uh),∂φ/∂n/an}b∇acket∇i}ht ∀φ∈X. (4.24)\nBy duality, we have\n/ba∇dblyuh−y˜P1(uh)/ba∇dbl= sup\ng∈L2(Ω),g/\\e}atio\\slash=0(g,yuh−y˜P1(uh))\n/ba∇dblg/ba∇dbl. (4.25)\nFor anyφ∈X, it follows from (4.23), (4.24) that\n(y˜P1(uh)−yuh,∆φ) =/an}b∇acketle{tuh−˜P1(uh),∂φ/∂n/an}b∇acket∇i}ht\n=/an}b∇acketle{tP0(˜P1uh)−˜P1(uh),∂φ/∂n−P0(∂φ/∂n)/an}b∇acket∇i}ht. (4.26)\nWe know that for any g∈L2(Ω), there exists φ∈Xsuch that −∆φ=gin Ω and since Ω is convex\nfrom elliptic regularity theory /ba∇dblφ/ba∇dblH2(Ω)/lessorsimilar/ba∇dblg/ba∇dbl. Thus, it follows from (4.26) and Cauchy-Schwarz\ninequality that,\n(yuh−y˜P1(uh),g) =/an}b∇acketle{tuh−˜P1(uh),∂φ/∂n−P0(∂φ/∂n)/an}b∇acket∇i}ht\n=/an}b∇acketle{tP0(˜P1uh)−˜P1(uh),∂φ/∂n−P0(∂φ/∂n)/an}b∇acket∇i}ht\n/lessorsimilar/ba∇dblP0(˜P1uh)−˜P1uh/ba∇dbl0,Γ/ba∇dbl∂φ/∂n−P0(∂φ/∂n)/ba∇dbl0,Γ. (4.27)\nThen combining (4.25), (4.27) and the elliptic regularity estimate for φwe obtain\n/ba∇dblyuh−yP1(uh)/ba∇dbl/lessorsimilarh/ba∇dbl˜P1uh/ba∇dblH1\n2(∂Ω). (4.28)\n13S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nTheorem 4.3. For a piecewise constant function uhdefined on ∂Ωit’s extension yuhand it’s CR\napproximation yuh,hsatisfies the following estimate:\n/ba∇dblyuh−yuh,h/ba∇dbl/lessorsimilarh.\nProof.Applying Theorems 4.1, 4.2 the proof follows .\nTheorem 4.4. The following estimate holds:\n/ba∇dblyP0¯u−y˜P1P0¯u/ba∇dbl/lessorsimilarh/ba∇dbl¯u/ba∇dblH1\n2(∂Ω).\nProof.The main ingredients of the proof consists of orthogonality proper ties of the L2projection\noperator and bijection of ˜P1. Note that since we consider the triangulation such that it induces a n\nodd numbers of edges on ∂Ω therefore given P0¯u∈Uhthere exists an unique v1∈V1(∂Ω) such that\nP0¯u=P0v1. (4.29)\nNote that\n/ba∇dblyP0¯u−y˜P1P0¯u/ba∇dbl= sup\ng∈L2(Ω),g/\\e}atio\\slash=0(yP0¯u−y˜P1P0¯u,g)\n/ba∇dblg/ba∇dbl. (4.30)\nThen consider the auxiliary problem:\n−∆φ=gin Ω, (4.31)\nφ= 0 on∂Ω.\nConsidering ( yP0¯u−y˜P1P0¯u,g), (4.31), P2\n0=P0along with integration by parts implies\n(y˜P1P0¯u−yP0¯u,g) = (˜P1P0¯u−P0¯u,∂φ\n∂n)\n= (˜P1P0v1−P0v1,∂φ\n∂n)\n= (v1−P0v1,∂φ\n∂n)\n= (v1−P0v1,∂φ\n∂n−P0∂φ\n∂n). (4.32)\nTherefore combination of (4.30), (4.29) and (4.32) implies that\n/ba∇dblyP0¯u−y˜P1P0¯u/ba∇dbl/lessorsimilarh/ba∇dbl˜P1P0¯u/ba∇dblH1\n2(∂Ω). (4.33)\nNext we show that /ba∇dbl˜P1P0¯u/ba∇dblH1\n2(∂Ω)/lessorsimilar/ba∇dbl¯u/ba∇dblH1\n2(∂Ω). Consider\n/ba∇dbl˜P1P0¯u/ba∇dblH1\n2(∂Ω)≤ /ba∇dbl˜P1P0(¯u−π1¯u)/ba∇dblH1\n2(∂Ω)+/ba∇dbl˜P1P0(π1¯u)/ba∇dblH1\n2(∂Ω)\n≤Ch−1\n2/ba∇dbl˜P1P0(¯u−π1¯u)/ba∇dbl+/ba∇dblπ1¯u/ba∇dblH1\n2(∂Ω)\n≤Ch−1\n2/ba∇dblP0(¯u−π1¯u)/ba∇dbl+/ba∇dblπ1¯u/ba∇dblH1\n2(∂Ω)\n≤Ch−1\n2/ba∇dbl¯u−π1¯u/ba∇dbl+/ba∇dbl¯u/ba∇dblH1\n2(∂Ω)\n≤C/ba∇dbl¯u/ba∇dblH1\n2(∂Ω). (4.34)\nTherefore combining (4.30), (4.32), (4.33) and (4.34) the result is o btained.\nTheorem 4.5. Letz∈H1\n2(∂Ω), then the following holds:\n/ba∇dblyz−yP0(z),h/ba∇dbl/lessorsimilarh/ba∇dblz/ba∇dblH1/2(Γ).\n14S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nProof.By triangle inequality, we have\n/ba∇dblyz−yP0(z),h/ba∇dbl ≤ /ba∇dblyz−y˜P1(P0(z))/ba∇dbl+/ba∇dbly˜P1(P0(z))−yP0(z),h/ba∇dbl. (4.35)\nConsider /ba∇dblyz−y˜P1(P0z)/ba∇dbl.\n/ba∇dblyz−y˜P1(P0z)/ba∇dbl ≤ /ba∇dblyz−yP0z/ba∇dbl+/ba∇dblyP0z−y˜P1(P0z)/ba∇dbl. (4.36)\nConsider /ba∇dblyP0z−y˜P1(P0z)/ba∇dbl. We write the L2norm in duality form\n/ba∇dblyP0z−y˜P1(P0z)/ba∇dbl= sup\ng∈L2(Ω),g/\\e}atio\\slash=0(yP0z−y˜P1(P0z),g)\n/ba∇dblg/ba∇dbl(4.37)\nConsider the auxiliary problem\n−∆φ=gin Ω, (4.38)\nφ= 0 on∂Ω.\nConsider ( yP0z−y˜P1(P0z),g). Combining (4.37), (4.38) and integration by parts we obtain\n(yP0z−y˜P1(P0z),g) = (y˜P1(P0z)−yP0z,∆φ)\n= (P0z−˜P1(P0z),∂φ\n∂n)\n= (P0˜P1(P0z))−˜P1(P0z),∂φ\n∂n)\n= (P0(˜P1P0z)−˜P1P0z,∂φ\n∂n−P0∂φ\n∂n)\n/lessorsimilarh/ba∇dbl˜P1P0z/ba∇dblH1\n2(∂Ω)/ba∇dblg/ba∇dbl (4.39)\nTherefore combining (4.37), (4.39) we obtain\n/ba∇dblyP0z−y˜P1(P0z)/ba∇dbl/lessorsimilarh/ba∇dbl˜P1P0z/ba∇dblH1\n2(∂Ω). (4.40)\nNext consider /ba∇dblyz−yP0z/ba∇dbl. Applying duality arguments as above we obtain\n/ba∇dblyz−yP0z/ba∇dbl/lessorsimilarh/ba∇dblz/ba∇dblH1\n2(∂Ω). (4.41)\nApplying Corollary 1 we obtain\n/ba∇dbly˜P1(P0(z))−yP0(z),h/ba∇dbl/lessorsimilarh/ba∇dbl˜P1P0(z)/ba∇dblH1\n2(∂Ω). (4.42)\nCombining (4.34), (4.35), (4.36), (4.39), (4.40), (4.41), (4.42) we o btain the result.\n5 Discrete Control Problem and Error Estimate\nIn this section the fully discrete control problem is introduced. The control space is discretized by\npiecewise constant functions. Define the discrete control proble m by:\nJh(yh,uh) =1\n2/ba∇dblyh−yd/ba∇dbl2+α\n2/ba∇dbl˜P1(uh)/ba∇dbl2, (5.1)\nsubject to,\nyh=yf,h+yuh,h, yf,h∈V0\nh, yuh,h=y0,h+˜P1(uh)∈Vh, (5.2)\napw(yf,h,vh) = (f,vh)∀vh∈V0\nh, (5.3)\napw(y0,h,vh) =−apw(/tildewider˜P1(uh),vh)∀vh∈V0\nh. (5.4)\nTheexistenceanduniquenessofthesolutionoftheproblemfollowsf romfinitedimensionalandstrict\nconvexity arguments. The optimal variables (¯ uh,¯yh)∈Uh,ad×Vhsatisfies the following system\n15S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nTheorem 5.1. There exists a unique solution (¯yh,¯uh)∈Vh×Uh,adof the discrete optimal control\nproblem. Furthermore, it satisfy the following system of eq uations:\n¯yh= ¯yf,h+ ¯y¯uh,h,¯yf,h∈V0\nh,¯y¯uh,h= ¯y0,h+˜P1(¯uh)∈Vh, (5.5)\napw(¯yf,h,vh) = (f,vh)∀vh∈V0\nh, (5.6)\napw(¯y0,h,vh) =−apw(/tildewider˜P1(¯uh),vh)∀vh∈V0\nh, (5.7)\n(¯yh−yd,yph,h−¯y¯uh,h)+α/an}b∇acketle{t˜P1(¯uh),˜P1(ph)−˜P1(¯uh)/an}b∇acket∇i}ht ≥0∀ph∈Uh,ad. (5.8)\nNext we prove the following auxiliary estimate which is important to obt ain the optimal order\nerror estimate for the optimal control.\nTheorem 5.2. For the discrete optimal control ¯uhthe following error estimate holds:\n/ba∇dbly¯uh−y¯uh,h/ba∇dbl/lessorsimilarh\nProof.From Theorem 4.2 we obtain\n/ba∇dbly¯uh−y˜P1(¯uh)/ba∇dbl/lessorsimilarh/ba∇dbl˜P1(¯uh)/ba∇dblH1\n2(∂Ω). (5.9)\nNext it remains to find an uniform bound for /ba∇dbl˜P1(¯uh)/ba∇dblH1\n2(∂Ω). Letξh∈Vhsatisfies\napw(ξh,vh) = (¯yh−yd,vh),∀vh∈Vh (5.10)\nThen note that (5.8) is equivalent to\n(¯θh+α˜P1¯uh,˜P1ph−˜P1¯uh)≥0,∀ph∈Uad, (5.11)\nwhere for any rh∈V1(∂Ω),¯θh∈V1(∂Ω) is defined by\n(¯θh,rh) =apw(¯ξh,˜rh)−(¯yh−yd,˜rh), (5.12)\nwhere ˜rh∈Vhis a piecewise linear, globally continuous extension of rhin Ω. Next we prove\n/ba∇dbl∂θy¯uh\n∂n−¯θh/ba∇dbl/lessorsimilar√\nh. (5.13)\nWhereθy¯uh∈H1\n0(Ω) satisfies\n−∆θy¯uh=y¯uh−ydin Ω, (5.14)\nθy¯uh= 0 on∂Ω.\nNote that\n/ba∇dbl∂θy¯uh\n∂n−¯θh/ba∇dbl2/lessorsimilar/ba∇dbl∂θy¯uh\n∂n−˜P1P0(∂θy¯uh\n∂n)/ba∇dbl2+/ba∇dbl˜P1P0(∂θy¯uh\n∂n)−¯θh/ba∇dbl2(5.15)\nAlso\n/ba∇dbl˜P1P0(∂θy¯uh\n∂n)−¯θh/ba∇dbl2/lessorsimilar/ba∇dbl˜P1P0(∂θy¯uh\n∂n)−∂θy¯uh\n∂n/ba∇dbl2+/ba∇dbl∂θy¯uh\n∂n−P1∂θy¯uh\n∂n/ba∇dbl2+/ba∇dblP1∂θy¯uh\n∂n−¯θh/ba∇dbl2.\n(5.16)\nConsider,\n/ba∇dblP1(∂θy¯uh\n∂n)−¯θh/ba∇dbl2= (∂θy¯uh\n∂n−¯θh,P1(∂θy¯uh\n∂n)−¯θh) (5.17)\n16S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nTo estimate /ba∇dbl∂θy¯uh\n∂n−¯θh/ba∇dblconsider the following auxiliary problem: Find zh∈˜Vhsuch that\napw(zh,vh) = 0,∀vh∈Vh (5.18)\nzh=P1(∂θy¯uh\n∂n)−¯θhon∂Ω.\nThen from (5.12), (5.18) and (5.14) we obtain,\n/integraldisplay\n∂Ω/parenleftbigg∂θy¯uh\n∂n−¯θh/parenrightbigg/parenleftbigg\nP1/parenleftbigg∂θy¯uh\n∂n/parenrightbigg\n−¯θh/parenrightbigg\n=apw(θy¯uh−ξh,zh)\n=apw(θy¯uh−ICRθy¯uh,zh)\n≤ /ba∇dblθy¯uh−ICRθy¯uh/ba∇dblpw/ba∇dblzh/ba∇dblpw\n/lessorsimilarh/ba∇dbly¯uh−yd/ba∇dbl]/ba∇dblzh/ba∇dblpw\n/lessorsimilarh/ba∇dbl¯uh/ba∇dblH−1\n2(∂Ω)/ba∇dblP1(∂θy¯uh\n∂n)−¯θh/ba∇dblH1\n2(∂Ω)\n/lessorsimilar√\nh/ba∇dbl¯uh/ba∇dbl/ba∇dblP1(∂θy¯uh\n∂n)−¯θh/ba∇dbl (5.19)\nCombining (5.17) and (5.19) and applying the fact that /ba∇dbl¯uh/ba∇dblis uniformly bounded with respect to\nh >0 we obtain\n/ba∇dbl∂θy¯uh\n∂n−¯θh/ba∇dbl/lessorsimilar√\nh. (5.20)\nHence application of triangle inequality, interpolation error estimate s and inverse inequality implies\nthat\n/ba∇dbl¯θh/ba∇dblH1\n2(∂Ω)≤C. (5.21)\nNote that (5.11) implies\n˜P1¯uh=P˜P1(Uh,ad)(1\nα¯θh). (5.22)\nAlso from (4.22) it is evident that ˜P1(Uh,ad) is an uniformly bounded convex subset of V1(∂Ω).\nTherefore by the stability of P˜P1(Uh,ad)with respect to H1\n2(∂Ω) norm [12] we obtain that\n/ba∇dbl˜P1¯uh/ba∇dblH1\n2(∂Ω)≤C. (5.23)\nCombining (5.9) and (5.23) the result follows.\nTheorem 5.3. Suppose ¯u∈Uad,¯uh∈Uh,adbe the optimal control and discrete optimal control\nrespectively. The following holds:\n/ba∇dbl¯u−¯uh/ba∇dblL2(Γ)/lessorsimilarh1/2.\nProof.We begin by choosing ph=P0¯uin (5.8) to obtain,\nα/ba∇dbl˜P1¯uh−˜P1P0¯u/ba∇dbl2≤α(˜P1P0¯u,˜P1P0¯u−˜P1¯uh)+(¯yh−yd,yP0¯u,h−y¯uh,h)\n=α(˜P1P0¯u−P0¯u,˜P1P0¯u−˜P1¯uh)+(P0¯u,˜P1P0¯u−˜P1¯uh)+(¯yh−yd,yP0¯u,h−y¯uh,h)\n=α(˜P1P0¯u−P0(˜P1P0¯u),˜P1P0¯u−˜P1¯uh)+α(P0¯u,˜P1P0¯u−˜P1¯uh)+\n(¯yh−¯y,yP0¯u,h−y¯uh,h)+(¯y−yd,yP0¯u,h−y¯uh,h) (5.24)\n17S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nApplication of Young’s inequality to (5.24) and the fact that P0˜P1=IUhyields,\n(α−ǫ)/ba∇dbl˜P1¯uh−˜P1P0¯u/ba∇dbl2≤α(P0¯u,˜P1P0¯u−˜P1¯uh)+(¯yh−¯y,yP0¯u,h−y¯uh,h)+\n(¯y−yd,yP0¯u,h−y¯uh,h)+/ba∇dbl˜P1P0¯u−P0(˜P1P0¯u)/ba∇dbl2\n=α(P0¯u,(˜P1P0¯u−˜P1¯uh)−P0(˜P1P0¯u−˜P1¯uh))+(¯yh−¯y,yP0¯u,h−y¯uh,h)+\n(¯y−yd,yP0¯u,h−y¯uh,h)+/ba∇dbl˜P1P0¯u−P0(˜P1P0¯u)/ba∇dbl2+α(P0¯u,P0¯u−¯uh)\n=(¯yh−¯y,yP0¯u,h−y¯uh,h)+(¯y−yd,yP0¯u,h−y¯uh,h)+\nα\nǫ/ba∇dbl˜P1P0¯u−P0(˜P1P0¯u)/ba∇dbl2+α(P0¯u,P0¯u−¯uh) (5.25)\nFrom (5.25) consider (¯ y−yd,yP0¯u,h−y¯uh,h)+α(P0¯u,P0¯u−¯uh).\n(¯y−yd,yP0¯u,h−y¯uh,h)+α(P0¯u,P0¯u−¯uh) =(¯y−yd,yP0¯u,h−y¯uh,h)+\nα(P0¯u−¯u,P0¯u−¯uh)+α(¯u,P0¯u−¯uh)\n=(¯y−yd,yP0¯u,h−y¯uh,h)+α(¯u,P0¯u−¯uh)\n=(¯y−yd,yP0¯u,h−y¯uh,h)+α(¯u,P0¯u−¯u)+α(¯u,¯u−¯uh)\n=−α/ba∇dbl¯u−P0¯u/ba∇dbl2+α(P0¯u,P0¯u−¯u)+\n(¯y−yd,yP0¯u,h−y¯uh,h)+α(¯u,¯u−¯uh)\n≤(¯y−yd,yP0¯u,h−y¯uh,h)+α(¯u,¯u−¯uh)\n=α(¯u,¯u−¯uh)+(¯y−yd,yP0¯u,h−yP0¯u+\nyP0¯u−y¯u+y¯u−y¯uh+y¯uh−y¯uh,h). (5.26)\nCombining (2.11) and (5.26) we obtain\n(¯y−yd,yP0¯u,h−y¯uh,h)+α(P0¯u,P0¯u−¯uh)≤(¯y−yd,yP0¯u,h−yP0¯u)+\n(¯y−yd,yP0¯u−y¯u)+(¯y−yd,y¯uh−y¯uh,h).(5.27)\nConsider (¯ yh−¯y,yP0¯u,h−y¯uh,h) from (5.25).\n(¯yh−¯y,yP0¯u,h−y¯uh,h) = (yf,h−yf,yP0¯u,h−y¯uh,h)+(y¯uh,h−y¯u,yP0¯u,h−y¯uh,h)\n≤(yf,h−yf,yP0¯u,h−y¯uh,h)+(yP0¯u,h−y¯u,yP0¯u,h−y¯uh,h).(5.28)\nFrom (5.28) consider /ba∇dblyP0¯u,h−y¯uh,h/ba∇dbl.\n/ba∇dblyP0¯u,h−y¯uh,h/ba∇dbl ≤ /ba∇dblyP0¯u,h−yP0¯u/ba∇dbl+/ba∇dblyP0¯u−y¯uh/ba∇dbl+/ba∇dbly¯uh−y¯uh,h/ba∇dbl (5.29)\nFrom (5.29) consider /ba∇dblyP0¯u,h−yP0¯u/ba∇dbl.\n/ba∇dblyP0¯u,h−yP0¯u/ba∇dbl ≤ /ba∇dblyP0¯u−yP1(P0¯u)/ba∇dbl+/ba∇dblyP1(P0¯u)−yP0¯u,h/ba∇dbl. (5.30)\nCombining (5.30), Theorem 4.4, Theorem 4.5 we obtain\n/ba∇dblyP0¯u,h−yP0¯u/ba∇dbl/lessorsimilarh/ba∇dbl¯u/ba∇dblH1\n2(∂Ω). (5.31)\nTheorem 5.2 yields,\n/ba∇dbly¯uh,h−y¯uh/ba∇dbl/lessorsimilarh. (5.32)\nAn application of ultra weak formulation yields that\n/ba∇dblyP0¯u−y¯uh/ba∇dbl/lessorsimilar/ba∇dblP0¯u−¯uh/ba∇dbl/lessorsimilarC, (5.33)\nwhereC >0 is depends only upon |ua|,|ub|,|Ω|and shape-regularity of the triangulation. Com-\nbining (5.29), (5.30), (5.31), (5.32) and (5.33) we obtain\n/ba∇dblyP0¯u,h−y¯uh,h/ba∇dbl/lessorsimilarC (5.34)\n18S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nTherefore combining (5.28) and (5.34) one obtains\n(¯yh−¯y,yP0¯u,h−y¯uh,h)≤(yf,h−yf,yP0¯u,h−y¯uh,h)+(y¯uh,h−y¯u,yP0¯u,h−y¯uh,h)/lessorsimilarh,(5.35)\nwhere the generic constant is independent of the discretization pa rameterh >0 and depends only\nupon/ba∇dbl¯u/ba∇dblH1\n2(∂Ω),/ba∇dblf/ba∇dbl, shape-regularity of the triangulation, Ω. Applying ultraweak form ulation\n(1.4) for y¯u−P0¯uand duality argument we obtain\n/ba∇dbly¯u−yP0¯u/ba∇dbl/lessorsimilarh/ba∇dbl¯u/ba∇dblH1\n2(∂Ω). (5.36)\nHence combination of (5.31), (5.36) and triangle inequality yields\n/ba∇dbly¯u−yP0¯u,h/ba∇dbl/lessorsimilarh/ba∇dbl¯u/ba∇dblH1\n2(∂Ω). (5.37)\nCombining (5.25), (5.26), (5.27), (5.28), (5.29), (5.30), (5.31), (5 .32), (5.36), (5.37) we obtain\n/ba∇dbl˜P1¯uh−˜P1P0¯u/ba∇dbl/lessorsimilarh1/2(5.38)\nNote that\n/ba∇dbl¯u−¯uh/ba∇dbl ≤ /ba∇dbl¯u−P0¯u/ba∇dbl+/ba∇dblP0¯u−˜P1P0¯u/ba∇dbl+/ba∇dbl˜P1¯uh−˜P1P0¯u/ba∇dbl+/ba∇dbl¯uh−˜P1¯uh/ba∇dbl.(5.39)\nNote that\n/ba∇dbl¯uh−˜P1¯uh/ba∇dbl=/ba∇dblP0(˜P1¯uh)−˜P1¯uh/ba∇dbl/lessorsimilarh1/2/ba∇dbl˜P1¯uh/ba∇dbl. (5.40)\nWe conclude our proof with the help of (5.39), (5.38), (5.40) (4.34) a nd (5.23).\n19S. Chowdhury and D. Garg PHFEM for Constrained DBCP\nReferences\n[1] G.Acosta, R.G.Dur` an, Themaximumangleconditionformixedan dnonconformingelements:\nApplication to the Stokes equations, SIAM J. Numer. Anal. 37:18–36, 2000.\n[2] F.B. Belgacem, H.E. Fekihand H.Metoui. Singularperturbationo fDirichletboundarycontrol\nof elliptic problems. ESAIM:M2AN , 37:833–850, 2003.\n[3] S. C. Brenner, A nonstandard finite element interpolation estima te.Numer. Funct. Anal.\nOptim. 20, no. 3-4, 245–250,1999DOI 10.1080/01630569908816890. MR1691362(2000b:65210)\n[4] S.C. Brenner. Two-level additive Schwarz preconditioners for n onconforming finite element\nmethods. Math. Comp. , 65:897–921, 1996. MR1348039 (96j:65117).\n[5] S.C. Brenner. Convergence of nonconforming multigrid methods without full elliptic regularity.\nMath. Comp. , 68:25–53, 1999. MR1620215 (99c:65229)\n[6] S.C. Brenner. Ponicar´ e-Friedrichs inequalities for piece-wise H1functions. SIAM J. Numer.\nAnal., 41:306–324, 2003. MR1974504 (2004d:65140)\n[7] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods (Third\nEdition). Springer-Verlag, New York, 2008.\n[8] Carstensen, C., Gallistl, D. and Schedensack, M. Adaptive nonco nforming Crouzeix-Raviart\nFEM for eigenvalue problems. Math. Comp. , 84:1061-1087, 2015.\n[9] Carstensen, C., Feischl, M., Page, M. and Praetorius, D. Axioms o f adaptivity. Comp. Math.\nAppl., 67:1195-1253, 2014.\n[10] Carstensen, C., Gallistl, D. and Hu, J. A discrete Helmholtz decom position with Morley finite\nelement functions and the optimality of adaptive finite element schem es.Comp. Math. Appl. ,\n68:2167-2181, 2014.\n[11] E. Casas and J. P. Raymond. Error estimates for the numerica l approximation of Dirichlet\nboundary control for semi linear elliptic equations. SIAM J.Control Optim. , 45:1586–1611,\n2006.\n[12] E. Casas, M. Mateos and J. P. Raymond. Penalization of Dirichlet optimal control problems.\nESAIM Control Optim. Calc. Var. , 15:782–809, 2009.\n[13] S. Chowdhury, T. Gudi and A. K. Nandakumaran. Error bound s for a Dirichlet boundary\ncontrol problem based on energy spaces. Math. Comp. , 86:1103–1126, 2017.\n[14] P. G. Ciarlet. The Finite Element Method for Elliptic Problems . North-Holland, Amsterdam,\n1978.\n[15] M. Crouzeix and P. A. Raviart. Conforming and nonconforming fi nite element methods for\nsolvingthestationaryStokesequations. Revue fran¸ caise d’automatique, informatique, recherche\nop´ erationnelle. Analyse num´ erique . 7: 33–75, 1973.\n[16] K. Deckelnick, A. G¨ unther and M. Hinze. Finite element approxim ation of Dirichlet boundary\ncontrol for elliptic PDEs on two and three dimensional curved domain s.SIAM J. Control\nOptim., 48:2798–2819, 2009.\n[17] D. Garg and K. Porwal. Unified discontinuous Galerkin finite elemen t methods for second order\nDirichlet boundary control problem. Appl. Numer. Math. , 185:336–364, 2023.\n[18] V. Girault and P.-A. Raviart. Finite Element Methods for Navier-S tokes Equations: Theory\nand Algorithms. Springer Series in Computational Mathematics, Springer-V erlag, Berlin , 5:\nMR851383 (88b:65129), 1986.\n20S. Chowdhury and D. Garg PHFEM for Constrained DBCP\n[19] P. Grisvard. Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics,\nSIAM,2011 . ISBN 1611972027, 9781611972023.\n[20] T. Gudi and R. C. Sau. Finite element analysis of the constrained Dirichlet boundary control\nproblem governed by the diffusion problem. ESIAM J. COCV. , 26:1–19, 2020.\n[21] Ipsen Ilse C. F. Numerical matrix analysis: linear systems and lea st squares\n[22] O. Karakashian and F. Pascal. A posteriori error estimates fo r a discontinuous Galerkin ap-\nproximation of second-order elliptic problems. SIAM J. Numer. Anal. , 41:2374–2399, 2003.\n[23] Mao, S. and Shi, Z. On the error bounds of nonconforming finite elements. . Sci. China Math.\n, 53, 2917-2926. 2010.\n[24] S. May, R. Rannacher and B. Vexler. Error analysis for a finite e lement approximation of\nelliptic Dirichlet boundary control problems. SIAM J. Control Optim. , 51:2585–2611, 2013.\n[25] E. D. Nezza a, G. Palatucci, E. Valdinoci Hitchhiker’s guide to the fractional Sobolev spaces.\nBull. Sci. math. , 136, 521–573, 2012.\n[26] G. Of, T. X. Phan and O. Steinbach. An energy space finite eleme nt approach for elliptic\nDirichlet boundary control problems. Numer. Math. , 129:723–748, 2015.\n[27] F. Troltzsch. Optimale Steuerung Partieller Differentialgleichungen . Vieweg, Cambridge Uni-\nversity Press, 2005.\n21"    },    {        "title": "2402.06515v2.The_Decisive_Power_of_Indecision__Low_Variance_Risk_Limiting_Audits_and_Election_Contestation_via_Marginal_Mark_Recording.pdf",        "content": "arXiv:2402.06515v2  [cs.CR]  22 Mar 2024The Decisive Power of Indecision:\nLow-Variance Risk-Limiting Audits and Election Contestat ion\nvia Marginal Mark Recording\nBenjamin Fuller, Rashmi Pai, Alexander Russell\nUniversity of Connecticut – V oting Technology Center\n{benjamin.fuller,rashmi.pai,alexander.russell }@uconn.edu\nMarch 26, 2024\nRisk-limiting audits (RLAs) are techniques for verifying t he outcomes of large elections. While they provide\nrigorous guarantees of correctness, widespread adoption h as been impeded by both e fficiency concerns and the fact\nthey offer statistical, rather than absolute, conclusions. We atte nd to both of these di fficulties, defining new families of\naudits that improve e fficiency and offer qualitative advances in statistical power.\nOur new audits are enabled by revisiting the standard notion of a cast-vote record so that it can declare multiple\npossible mark interpretations rather than a single decisio n; this can reflect the presence of ambiguous marks, which\nappear regularly on hand-marked ballots. We show that this s imple expedient can o ffer significant efficiency improve-\nments with only minor changes to existing auditing infrastr ucture. We establish that these “Bayesian” comparison\naudits are indeed risk-limiting in the formal sense of (Full er, Harrison, and Russell, 2022).\nWe then define a new type of post-election audit we call a contested audit . These permit each candidate to provide\na cast-vote record table advancing their own claim to victor y. We prove that these audits o ffer remarkable sample\nefficiency, yielding control of risk with a constant number of sa mples (that is independent of margin). This is a\nfirst for an audit with provable soundness. These results are formulated in a game-based security model that specify\nquantitative soundness and completeness guarantees. Fina lly, we observe that these audits provide a direct means to\nhandle contestation of election results a ffirmed by conventional RLAs.\n1 Introduction\nRisk-limiting audits (RLAs) are methods for verifying the outcome of a large-scal e election. Developed by the aca-\ndemic and election community over that last two decades [1], RLAs offer significant efficiency improvements over\nthe burden of a full hand recount while providing rigorous gu arantees of correctness. They can also support a variety\nof social choice functions and election organizations. The framework posits reasonable physical assumptions under\nwhich elections—involving diverse interactions among mul tiple untrusted human and electronic processes—can be\nauthoritatively protected from errors or deliberate falsi fication in tabulation, vote aggregation, and reporting. Fi nally,\nthe framework supports public verification : third-party observers can certify the outcome of the audit .\nRLAs are only defined in settings with voter-verified ground t ruth, typically furnished by hand-marked paper\nballots.1Such marked ballots determine the ground-truth outcome of the election ; of course, any particular tabulation\nof the ballots—generally carried out by electronic tabulat ors—determines a potentially di fferent outcome. RLAs\nare designed to detect disagreement between the ground-tru th outcome and a given tabulated outcome except with a\nprescribed, concrete failure probability called the “risk ” of the audit. The word “outcome” here refers to the winner of\nthe election rather than the exact vote totals.\n1Verified V oting summarizes the adoption of such methods in th e U.S.\n1RLA adoption in large democracies has been uneven. In the Uni ted States, RLAs have been advocated by area ex-\nperts [2], the US Senate, and the 2014 Presidential Commissi on on Election Administration. However, only a handful\nof states currently run such audits, with most adoption occu rring in the last few years. A major obstacle to widespread\nadoption is efficiency: RLAs require manual interpretation of a sampled set of ballots whose size grows as a function\nof the margin of the race to be audited. Thus, circumstances w ith small margins may require examination of many\nthousands of ballots and, unfortunately, election logisti cs and planning must prepare for such onerous outcomes. (As\nan example, a conventional ballot polling audit typically r equires over 10,000 ballot samples for a 2% margin.) A\nrelated policy and perception challenge is that high-e fficiency RLAs necessarily provide (only) statistical guaran tees;\ncommon risk values in practice are 5% or 10%. This invites an i mmediate criticism: a losing candidate whose loss has\nbeen reaffirmed by an RLA may argue that a 1-in-20 chance of error is unsat isfactory. We engage with both of these\nconcerns by defining two new families of post-election audit s.\n1.1 Our Results in Brief\nBoth of our new approaches are enabled by revisiting the conv entional understanding of a cast-vote record (CVR),\nwhich is a declaration of how a ballot was counted in an electi on. CVRs are critical elements of standard “comparison”\naudits, which are the starting points for our new audits. We r edefine CVR semantics to include explicit indication\nof marginal marks [3] by either asserting a collection of possible interpreta tions for a marginal mark or a predicted\nprobability distribution of interpretations. We leverage this explicit declaration of indecision to define two new fam ilies\nof audits, described below.\nBayesian ballot-comparison risk-limiting audits with red uced variance and improved e fficiency. Recording a\npredicted probability distribution for the interpretation of marginal marks in a CVR and reflecti ng this appropriately in\nthe audit leads to a new family of Bayesian risk-limiting audits , which can offer significant efficiency advantages and\nimproved confidence in the audit. In particular, in settings of practical interest this reduces the standard deviation o f\ncompletion times for standard sequential statistical test ing methods and hence reduces the number of ballots that need\nto be drawn in the first (and typically only) phase of standard multi-phase audits. For margins of 1% this reduces the\nnumber of ballots sampled by over 10%; see Table 2. We also dis cuss such audits in a less expressive “conservative”\nsetting where the CVR simply lists a set of possible interpre tations. (One could consider interpretations with some\nminimum probability as shown in Figure 1.) These results are articulated in an adaptation of the formal game-based\nmodel of [4].\nAside from these e fficiency improvements, such CVRs provide an improvement in vo ter confidence and inter-\npretability by explaining discrepancies. Specifically, co nventional CVRs will frequently disagree with marginal mar ks\nas, of course, there is no authoritative conclusion for such a mark; this can cast doubt on the CVR and the audit. A\nCVR permitted to assert the ambiguity of a mark provides an im mediate means for separating the common circum-\nstances where the mark is ambiguous from more disturbing dis crepancies arising from real failures of the audit. We\nremark that Clearballot publishes mark images with predict ed probability distributions and allows citizens to search\nfor marginal marks; thus our method makes use of data that is a lready public in many jurisdictions.\nCompetitive audits involving multiple CVRs submitted by multiple interested p arties, yielding a natural ap-\nproach to election contestation .We define a novel notion of a “competitive audit,” in which mul tiple parties submit\nCVRs to substantiate competing claims of victory. We develo p a formal cryptographic game-style framework for\ndefining and analyzing such audits. This modeling permits a q uite strong adversary that may both choose which\ninterpretation to return for a marginal mark and whether to s uppress a ballot. We then show that this leads to an\nextraordinarily efficient audit calling for evaluation of a fixed, constant numbe r of ballots , independent of margin.\nSuch strong efficiency guarantees are impossible for conventional RLAs. Th is new auditing framework provides an\napproach to the problem of election contestation, where the results of an RLA are challenged by a third party.\n2Baseline CVR\nID Bugs Daffy\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10Conservative CVR\nID Bugs DaffyNo V ote\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10Bayesian CVR\nID Bugs DaffyNo V ote\n1 1\n2 .72 .02 .26\n3 .99 .01\n4 1\n5 .75 .25\n6 .12 .38 .5\n7 .46 .1 .44\n8 1\n9 1\n10 .02 .98\nFigure 1: Generalization of conventional CVRs to the conser vative setting, where multiple interpretations are declar ed,\nand the Bayesian setting, where one associates probabiliti es with each interpretation. Blanks cells in the Bayesian\nindicate probability 0; for brevity, the “No V ote” columns r eflect both the no-mark case and the overvote case (with\nmarks appearing for both candidates).\n1.2 A Detailed Survey of the General Framework, Related Work , and Our Results\nAs described above, a RLA is carried out in the context of an el ection with voter-verified ground truth. For concrete-\nness, our discussion assumes hand-marked paper ballots, wh ich is the most common instantiation in practice. The\naudit is intended to detect disagreement between the ground -truth outcome of the election, as determined by the bal-\nlots themselves, and the declared outcome of the election resulting from a tabulation of the ba llots; we use the word\n“declared” throughout to emphasize that the tabulated outc ome may be inconsistent with the ground-truth outcome.\nSuch an audit examines a collection of physical ballots, usu ally drawn at random according to a convention depending\non the audit, and either concludes with C onsistent , indicating that the sampled ballots appear consistent wit h the de-\nclared outcome, or I nconclusive , indicating that the audit has not amassed su fficient evidence of consistency. An audit\nis said to have riskα(a number in the range [0 ,1]) if the probability that it outputs C onsistent is no more thanαwhen\nthe declared outcome contradicts the ground-truth outcome .\nA trivial audit that always outputs I nconclusive has zero risk, but is obviously not useful in practice. This p oints\nto the importance of completeness —the additional guarantee that the audit outputs C onsistent with high probability\nunder favorable circumstances; we discuss this in detail be low. A straightforward hand recount of an election is itself a\nRLA audit with zero risk (modulo any errors in manual interpr etation). More sophisticated RLAs randomly draw and\ninspect a collection of ballots to make statistical conclus ions about the outcome. A general feature of such statistica l\naudits is that the number of samples scales with the margin of victory; this reflects the natural intuition that a landslid e\nshould be easier to statistically verify than a victory by sl im margin. In the most familiar setting of a simple, two-\ncandidate, first-past-the-post election, the empirical me an of the number of votes for a particular candidate among\na collection of log(2 /α)/(2ǫ2) uniformly sampled ballots is correct to within ǫwith probability 1 −α. This simple\nprocedure—called “ballot polling” in the literature—yiel ds a risk-limiting audit that inspects\nO/parenleftigglog(1/α)\nµ2/parenrightigg\n(1)\nballots in expectation, where µis the actual margin of victory [1].\nThe most efficient known approach for RLAs, known as a ballot comparison audit [1, 5–11], requires a detailed\ntabulation report called a cast-vote record table (CVR). A C VR declares a unique identifier and interpretation (that\nis, a determination of the cast votes) for each ballot in the e lection. This ballot-by-ballot record of the election\nnaturally determines a tabulation and outcome. It also enab les high-efficiency audits that proceed by iterating the\nfollowing experiment: (1) The auditor picks a random identi fierιfrom the CVR table; (2) the auditor finds a physical\nballot with identifier ι; (3) the auditor compares the interpretation of the physica l ballot against the associated CVR\n3entry. Intuitively, iterations where the physical ballot m atches the CVR provide support for this declared record of th e\nelection, while disagreements erode that support. Continu ing the discussion in the same two-candidate first-past-the -\npost setting, observe that if the CVR table declares an incor rect outcome for the election, then at least a µ/2 fraction\nof entries of the CVR table must disagree with the correspond ing physical ballot; as above, µis the scaled margin of\nground-truth victory. (The factor of two arises because a si ngle disagreement can both negate a vote for the declared\nwinner and supply a vote for the declared loser.) It follows t hat after 2/µsamples we expect to observe a discrepancy\nif the outcomes differ, suggesting that this approach should yield a risk-limit ing audit with sample size\nO/parenleftigglog(1/α)\nµ/parenrightigg\n. (2)\n(The log(1/α) term increases the number of samples so that the probabilit y of observing an inconsistency is driven to at\nleast 1−α.) It is instructive to compare this against (1) above. Due to the improved dependence on µ, in circumstances\nwith small margins ballot comparison is significantly more e fficient than ballot polling.\nThe completeness challenge. The efficiency landscape is more complicated than the discussion ab ove (and the\nasymptotic expression (2)) indicates. The complication is the requirement of completeness—that is, that the audit\nconclude with Consistent when considering an accurate CVR under favorable condition s. In practice, when a post-\nelection RLA concludes with I nconclusive , policy calls for either a complete hand count or an addition al stage of\nauditing (with a smaller risk parameter so as to avoid signifi cant amplification of risk from the composite audit). See\nfor example Rhode Island’s legislation and Colorado’s fact sheet. Thus a full accounting of e fficiency must weigh the\ncosts and likelihood of I nconclusive conclusions in the best-e ffort settings that arise in practice.\nTo be more concrete, a ballot-comparison auditor simply wis hing to meet a prescribed risk limit αcould adopt the\nframework above with the convention that anyobserved disagreement between the CVR and a physical ballot causes\nthe auditor to immediately terminate the audit and output I nconclusive . If examination of log(1 /α)/(2µ) ballots indeed\nexposes no inconsistencies, the audit can safely return C onsistent with only anαprobability of error, as desired. For\nmarginµ=.01 and riskα=.05, for example, this requires inspection of 499 ballots. No w, if this “strict” auditor has\nthe luxury of unambiguous ground truth, a perfectly correct CVR, and consistently accurate ballot interpretations, th en\nit will indeed conclude the audit with C onsistent . However, discrepancies between the CVR and physical ballo ts arise\nfrequently in practice. (For example, the comparison RLA ru n by the US state of Colorado for the 2020 presidential\nelection generated hundreds of discrepancies [12].) These discrepancies can result from truly ambiguous voter intent ,\ndisagreement among the interpretations of an auditing boar d, or operational errors such as retrieval of an incorrect\nballot. Estimates from prior work (e.g. Stark’s audit tools ; [13]) suggest discrepancy rates as large as 0 .5%, though\nthe actual rate of incidence depends on various features of t he election.\nThese considerations have led to the development and adopti on of audits (and associated statistical tests) that\ncan tolerate such anticipated errors. For example, adoptin g the same parameters (5% risk, 1% margin) but with the\nintroduction of a.01% rate of marks being inadvertently dropped during interp retation, the standard Kaplan-Markov\ntest calls for a sample size of 1220 ballots in order to termin ate with probability 95%.2This example illustrates that\neven small error rates have a drastic impact on required samp le size; we remark, additionally, that the incidence of\nsporadic errors in an otherwise correct CVR or errors in ball ot interpretation are the cause of variance in the stopping\ntime for such sequential statistical tests. As audits are ty pically conducted in phases—drawing a collection of ballot s\nto be inspected at once and calibrated so that each phase is li kely to be the last—variance also directly influences\nefficiency as it forces large phases.\nThese concerns motivate our exploration of more expressive CVRs. In particular, we note that the party (or\ntabulator) preparing the cvrcan identify marginal marks during tabulation that are like ly to lead to errors during\na comparison audit. Reflecting these marginal marks on the CV R offers a new dimension of optimization for the\nauditing rule. As a matter of bookkeeping, we will consider C VRs that may list multiple interpretations for each ballot\nor a full predicted probability distribution of interpreta tions. (A traditional CVR corresponds to the case when the si ze\nof each set of interpretations is 1.) We apply this to develop both conventional audits with improved e fficiency and\n2This estimate uses McBurnett’s tool rlacalc with standard r ates of errors in the literature drawn from Lindeman and Star k [1]. These are\no1,u1=.001 and o2,u2=.0001, see discussion in Section 5.\n4define a new class of competitive audits, providing both high efficiency and a mechanism for contesting conventional\naudits.\nAs we remarked above, enriching CVRs so that they have the exp ressive power to reflect marginal marks can also\nimprove confidence in the audit, as it can distinguish unavoi dable discrepancies arising from ambiguity from truly\n“unexplained” auditing failures.\n1.2.1 Improved E fficiency Risk-Limiting Audits via Bayesian CVRs\nWe show how to adapt standard ballot comparison audits to the setting of Bayesian CVRs that declare a predicted\ndistribution of ballot interpretations. We indicate how to formulate the standard notions of discrepancy and margin\nin this setting and observe that this simple change has signi ficant efficiency ramifications. Using discrepancy and\nmarginal rates from practice we show that such audits reduce sample size standard deviation by 14% compared to\na traditional audit at a margin of 1% (see Table 2). We also dev elop a variant of this audit that uses a CVR that\nmerely reflects the possible interpretations of each ballot (rather than a full probabil ity distribution). While this does\nnot provide the full benefits of the Bayesian approach, it may be simpler to deploy in practice since no probability\nestimation is required. It also considers a stronger advers arial model where the ballot interpretation of the auditor i s\nadversarially controlled.\nWe adopt to our setting the cryptographic game-style modeli ng for RLAs developed in [4], and prove that the new\naudit is indeed risk-limiting. We then carry out simulation s to show that the most popular sequential statistical test\nused for RLAs (the Kaplan-Markov test) can take advantage of the resulting notion of discrepancy to provide more\nefficient RLAs.\n1.2.2 Competitive Audits and Election Contestation\nOur second contribution is a new class of post-election audi ts that we call competitive RLAs . In a competitive RLA\neach candidate has an “advocate” that is allowed to examine b allots (under supervision to ensure ballots are not\ndestroyed or modified). For each candidate A1,..., Ak, their advocate produces a CVR that may declare di fferent\nballot interpretations and winners; the intent is that the a dvocate for candidate Afiles a CVR that is favorable to A. In\nparticular, if Awon the election, the CVR (filed by the advocate for A) should establish that; in a weaker sense, even\nifAdid not win, the advocate may wish to file a CVR that prevents ot her candidates that did not win the election from\nconvincingly claiming that they did. For simplicity, we for mulate this audit in the “conservative” setting, where CVRs\nmay list multiple interpretations for each ballot but do not record a probability distribution on these various possibl e\ninterpretations. (It is certainly possible to formulate th is in the setting with a Bayesian CVRs, see end of Section 6.)\nAssuming that at least one submitted CVR only lists interpre tations that are consistent with the ballots and the\nelection has a winner, the auditor will identify the correct winner using a constant number of ballots. Furthermore, if\nsome submitted CVR lists the exact list of possible interpre tations for each ballot, no losing candidate can be identifie d\nas the winner. We emphasize that the number of ballot samples required by this approach is a constant that not scale\nwith the margin. (The constant itself is a function of how rob ust the audit should be in the face of errors by the\nadvocates.)\nThe basic idea of the audit is straightforward: Faced with a p air of contradictory CVRs— cvr1andcvr2—there\nmust be at least one ballot on which these CVRs completely dis agree: examining this single ballot then provides an\nimmediate strike against one of CVRs. Indeed, a strict audit or can use a generalization of this approach to settle the\ncompeting claims made by a collection of kCVRs with just k−1 ballots. (That is, the auditor immediately removes\nfrom consideration any CVR found to be inconsistent with an i nspected ballot.) Such a stringent auditor will not be\nappropriate in practice, which must account for a small numb er of errors on the part of the advocates (and the cvri) or\nthe possibility that ballots are lost, say, during the audit . Even accounting for such phenomena, our approach yields\naudits with remarkable e fficiency. (As mentioned earlier, the complexity of the audit s cales with the desired robustness\nin the face of such advocacy errors.)\nThe most natural and practical instantiation of our techniq ues is providing an e fficient adjudication process for\ndisagreements that result after the parallel scanning of ba llots, for example, using the OpenScan system [14]. The\nidea of the OpenScan system is for advocates to place cameras above a physical scanner and create their own CVRs\n5without having physical access to the ballots. Their system did not have a natural mechanism to adjudicate if the\nparties disagreed on the winner. Competitve audits provide that adjudication with a constant query ballot complexity.\nElection contestation. Competitive RLAs provide an immediate approach to election contestation. Specifically,\nconsider a situation where a candidate wishes to dispute the conclusions of conventional RLA (that is, the candidate\nclaims that an invalid election was not detected by the RLA). In this case, the candidate (or their advocate) can be\npermitted to examine the ballots and produce a competing CVR for the election (that, presumably, will claim that\nthe candidate is the victor). The CVR originally produced fo r the election (and rea ffirmed by the original RLA) and\nthe candidate’s CVR can then be treated by the competitive au dit mechanism. This has a number of notable features:\n(i.) the audit can rigorously settle the dispute with risk 2−θ(k)after examining only kballots—thus a small constant\nnumber of queries, independent of margin, are su fficient; (ii.) this provides an interesting avenue for electi on policy:\nfor example, the cost and e ffort of the audit could be borne by the contesting party.\nAdversarial modeling. As in the prior contribution, we consider a cryptographic ga me-based model for analysis.\nThe model places the ballots (and, in fact, the choice of the r eturned interpretation) in the hands of an adversarial\nenvironment, thus reflecting properties of the audit in adve rse circumstances that reflect a variety of the challenges th at\nface practical audits.\n1.3 Other Related Work\nRisk-limiting audits were originally defined by Stark in 200 8 [15]. The framework was generalized and sharpened over\nthe next decade [1,16–18]. As indicated earlier, a variety o f specific methods have been explored, often with an eye to\noptimize certain practical settings [1, 6, 15, 19–22]. Ther e are four major directions in current research (1) supporti ng\nvaried social choice functions [9, 11], (2) managing multip le races [5–8, 23], (3) sharpening risk estimates [7, 10, 15,\n19, 20, 24–28] and (4) implementation issues [4, 16–18, 29–3 3].\nThe most closely related work to our competitive setting is a recent, independent article of Jones et al. [34]. One\ncan view our competitive audits as an analogue of a “multi-pr over interactive proof system” consisting of competing\nand potentially malicious “provers” who are attempting to c onvince a “verifier” of the truth (or falsity) of a statement\n(reflecting the complexity-theoretic setting of [35, 36]). Jones et al. were inspired by the “cooperative” version of\nthis same framework in which a collection of cooperating, bu t potentially malicious, provers attempt to similarly\nconvince a verifier of the truth of a statement; in fact, this c ooperative version is far more common in the complexity-\ntheoretic literature (reflecting the standard setting of [3 7]). Jones et al. develop an auditing approach designed to ca tch\ninconsistencies across multiple scans used to create (mult iple) CVRs. In our auditing context, their techniques focus\non detecting circumstances where at least one of the scans (p rovers) is dishonest. Our techniques focus on deciding\nwho to trust in a situation where at least one scan is honest. T he two approaches can thus naturally compose.\nThe key element in Jones et al. approach is a secret, random pe rmutation used to shu ffle the ballot orderings\nbetween the distinct scans. Discrepancy is calculated by co mparing the outcome for selected ballots between the scans.\nThe procedure provides similar e fficiency guarantees: a constant number of ballots, independe nt of margin. We remark\nthat their goal—detecting a failed scanner when the operato rs of the scanners may be colluding—appears to require\nstronger assumptions than are necessary for our competitiv e audits. Most importantly, they require that distinct ball ots\ncarrying the same votes are indistinguishable and (as indic ated above) the ability to physically instantiate a private ,\nrandom shuffle that will be reflected in scan order.\n1.4 Extensions and Future Work\nBallot Polling with Marginal Marks Ballot polling refers to the process of drawing a collection of randomly se-\nlected ballots, and using the cast votes on the ballots to sta tistically infer the outcome of an election. If we were to\nincorporate ballots with marginal marks into the polling pr ocess, this would require an ability to compute a question-\nable diluted margin without the need of a CVR. Since polling i s sound for any margin, this change retains soundness,\nhowever, experiments /simulations are needed to see if it improves e fficiency.\n6Batch Comparison with Marginal Marks Batch comparison is an auditing process that involves rando mly select-\ning batches of ballots to hand count (rather than individual ballots), and computing an observed discrepancy between\nthe hand counted ballots within the batch, and the tabulated totals of the ballots in the batch. For this process, we see\nno obstacles in incorporating ballots with ambiguous marks ; our techniques directly translate into the batch setting.\nOrganization. We cover notational preliminaries and define CVRs and ballot s with multiple interpretations in Sec-\ntion 2, cover Bayesian audits in Section 3, Conservative aud its in Section 4, evaluate e fficiency of the Bayesian and\nconservative methods in Section 5, and competitive audits i n Section 6.\n2 Preliminaries; the Bayesian Setting\nWe use boldface to refer to “physical” objects, such as indiv idual ballots (typically denoted b) or groups of ballots\n(typically B). For a natural number k, we define [ k]={1,..., k}(and [0]=∅). We letΣ=[−2,2], a set of particular\nsignificance as standard single-ballot discrepancy takes v alues in this set; this is discussed in detail later in the pap er.\nFor a set X, let X∗be the set of all finite-length sequences over X; that is, X∗={(x1,..., xk)|k≥0,xi∈X}. In\nparticular, we let {0,1}∗denote the collection of finite-length bitstrings. Finally , we define XNto be the set of all\nsequences{(x0,x1,...)|xi∈X}. We typically use script letters, e.g., P, to denote probability distributions.\nMaking multiple interpretations explicit requires many ne w definitions. A summary is in Table 1.\nFormal definitions for ballots, elections, and cast-vote re cords. For simplicity, we consider only audits of a sin-\ngle first-past-the-post race with a set Cof candidates. See Stark [6, 9] for a detailed account of how m ore complex\nelections can be reduced to this canonical setting. We now se t down the elementary definitions of elections, manifests,\nand CVRs. The main definitional change is that a physical ballot yields a probability distribution of possible\noutcomes; this models the important situation where a physical ballot may have ambiguous markings that could rea-\nsonably be interpreted in various ways by an auditing board o r a tabulation device. We explore two conventions for\ncast vote records to reflect this uncertainty: the first is the Bayesian framework, which associates with each ballot a\npredicted probability distribution of possible readings ( depending on the anticipated reading a particular audit boa rd,\nsay). We then formulate a simpler, conservative model that posits a collection of interpretations for each b allot without\nassigning them probabilities. This reduces the complexity of producing cast-vote records, as one only needs to iden-\ntify the possible interpretations of a ballot rather than a comprehensive pro bability distribution. In this conservative\nmodel, we consider quite strong adversarial behavior that a ssumes that the audit board always returns the “worst-case”\ninterpretation allowed by the ballot. (In fact, this is equi valent to replacing the probabilities listed on the CVR with\na probability of 1 for some interpretation that “minimizes” the margin for a declared winner.) To move as swiftly as\npossible to our results on conventional audits with margina l marks, we first lay out the definitions for the Bayesian\ncase; we return to the definitions for the conservative case i n Section 6.\nDefinition 1 (Interpretations, ballots, and ballot families) .LetCbe a set of candidates. An interpretation is a function\nI:C→{ 0,1}; whenCis understood from context, we write I={I:C→{ 0,1}}for the collection of all interpretations.\nSuch interpretations will be used to indicate the votes appe aring for candidates on individual ballots. A ballot is a\nphysical object bwith two properties:\n(1) The ballot bislabeled with an indelible identifier id b∈{0,1}∗.\n(2) The ballot bdetermines a “ground truth” probability distribution Tbon the set of interpretations I. Thus\nTb:I→Ris a non-negative function on Ifor which/summationtext\nI∈ITb(I)=1.\nWe let T/LEFTCIRCLE /Circle\nb:C→Rdenote the expected interpretation over the distribution Tb: that is, for each A∈C,T/LEFTCIRCLE /Circle\nb(A)=\nE[Ib(A)],where Ibis a random variable distributed according to Tb.\nAballot family Bis a collection of ballots with the same candidates. We let Bι={b∈B|idb=ι}denote the\nsubset of ballots with identifier ι. When ballots identifiers are distinct across B, so that|Bι|≤1for eachι, we say that\nthe family is uniquely labeled .\nBallots are typically intended to have an unambiguous inter pretation. This corresponds to the case where the\ndistributionTbis supported on a single interpretation (taking the value 1 a t that interpretation and zero elsewhere).\n7Definition 2. Anelection E is a tuple (C,B,S)whereCis a set of candidates, Bis a ballot family, and S=|B|is the\ntotal number of ballots.\nDefinition 3 (Election winners, losers, and margin) .Let E=(C,B,S)be an election. For a candidate A∈C, define\nE/LEFTCIRCLE /Circle(A)=/summationdisplay\nb∈BT/LEFTCIRCLE /Circle\nb(A).\nThemargin of Awith respect to Bis defined to be\nµA,B=E/LEFTCIRCLE /Circle(A)−E/LEFTCIRCLE /Circle(B)\nS.\nWhen we wish to emphasize that this notion of margin is defined with respect to the expected values T/LEFTCIRCLE /Circle\nb(·)we refer to\nit as Bayesian margin . A candidate W∈Cis the winner of the election E if ∀A∈C\\{ W},µW,A>0. In this case, we\ndefine the margin , denotedµ, of the election to be the minimum of these quantities:\nµE=min\nA∈C\\{W}µW,A.\nOtherwise, the election does not determine a winner and we de fineµE:=0. If a winner exists it is unique. A candidate\nLis called a loser if there is a candidate Afor which E/LEFTCIRCLE /Circle(L)<E/LEFTCIRCLE /Circle(A).\nCast-vote record (CVR). A cast-vote record table (CVR) is an (untrusted) declaratio n of both the ballots appearing\nin an election and the interpretations of the ballots.\nDefinition 4 (Bayesian Cast-V ote Record Table) .LetCbe a set of candidates. A Bayesian Cast-V ote Record table\n(CVR) is a sequence of pairs\ncvr=/parenleftbig(ι1,P1),..., (ιt,Pt)/parenrightbig\nwhere theιrare distinct bitstrings in {0,1}∗and eachPr:I→Ris a probability distribution on Iwhich we refer\nto as a “prediction. ” Intuitively, such a cvrdeclares that, for each row r, the interpretation of the ball ot labeledιris\ngiven by the distribution Pr.\nFor each candidate A∈Cand r∈{1,..., t}, we define P/LEFTCIRCLE /Circle\nr(A)=E[I(A)], the expected value of the vote for Awhen\nI is drawn according to the distribution Pr. We then define\ncvr(A)=t/summationdisplay\nr=1P/LEFTCIRCLE /Circle\nr(A).\nDeclared winners and losers. If there is a candidate A∈ C with the property that cvr(A)>cvr(B)for every\nB∈C\\{ A}, then we say that Ais the declared winner according to cvr. A candidate Afor which there exists some B\nsuch that cvr(A)<cvr(B)is called a declared loser .\nWe use the following general language when referring to CVRs :\n(1) The bitstringsιrareidentifiers and we letIdents (cvr)denote this set.\n(2) The number t is the sizeof the CVR.\n(3) The r th row , denoted cvr(r), refers to the tuple (ιr,Pr).\n(4) Identifiers appearing in the CVR are distinct, so when con venient we reference PrandPrby identifier rather\nthan row: that is, for an identifier ι=ιrappearing in the CVR, we define Pι:=PrandP/LEFTCIRCLE /Circle\nι:=P/LEFTCIRCLE /Circle\nr.\nDefinition 5 (Bayesian declared margin) .Letcvrbe a Bayesian CVR with declared winner Wand size s. The declared\nmargin ofcvris the quantity\nµcvr:=min\nA∈C\\Wcvr(W)−cvr(A)\ns≥0.\nIfcvrhas no declared winner, we define µcvr:=0.\n8Definition 6 (Bayesian validity, consistency, and contradiction) .Let E=(C,B,S)be an election and cvrbe a Bayesian\nCVR. We say that cvrisvalid for E if E has a winner and cvrdeclares that winner. We say that cvris invalid for E if it\ndeclares a winner Wthat does not win E.\nDefinition 7 (Bayesian discrepancy) .Let E=(C,B,S)be an election and let cvr=((ι1,P1),..., (ιt,Pt))be a Bayesian\nCVR with declared winner W. For a ballot b∈Band a distribution P:I→R, the discrepancy is\nD[P,b;W]=max\nA∈C\\W/parenleftig/parenleftbigP/LEFTCIRCLE /Circle(W)−P/LEFTCIRCLE /Circle(A)/parenrightbig−/parenleftbigT/LEFTCIRCLE /Circle\nb(W)−T/LEFTCIRCLE /Circle\nb(A)/parenrightbig/parenrightig\nwhere, as above, P/LEFTCIRCLE /Circle(A)=E[I(A)]with I distributed according to P. We expand this definition to apply, additionally,\nto a special symbol ⊥:\nD[P,⊥;W]=max\nA∈C\\W/parenleftig\nP/LEFTCIRCLE /Circle(W)−P/LEFTCIRCLE /Circle(A)/parenrightig\n+1.\nFor an identifierιdefine\nDcvr,ι=min/braceleftbigD[Pι,b;W]|b∈Bι/bracerightbigifBι/nequal∅,\nD[Pι,⊥;W] otherwise.\nThe discrepancy of the cast vote record table cvris:Dcvr=/summationtextt\ni=1Dcvr,ιi.\nWe begin by establishing the fundamental relationship betw een discrepancy and margin for invalid CVRs.\nLemma 1. Let E=(C,B,S)be an election and let cvrbe an invalid CVR for E with declared winner Wand size\nt=S. Then Dcvr≥(µcvr+µE)·S≥µcvr·S.\nProof of Lemma 1. LetIdentsRdenote the subset of identifiers that identify least one ball ot: that is,IdentsR={ι∈\nIdents (cvr)|Bι/nequal∅}. We call these “represented” identifiers. For each ι∈IdentsR, let cιbe a ballot in Bιfor\nwhich D[Pι,cι;W]=Dcvr,ι. As Bι∩Bι′=∅for anyι/nequalι′, these chosen ballots cιare distinct. Considering that\n|Idents (cvr)|=t=S=|B|, the number of “unassigned” ballots in B(that do not appear in {cι|ι∈IdentsR})\nis the same as the number of unrepresented identifiers (appea ring inIdents (cvr) but not inIdentsR). By choosing\nan arbitrary one-to-one correspondence between unreprese nted identifiers and unassigned ballots, we may extend the\ncorrespondenceι↔cιto all identifiers in Idents (cvr) and all ballots in Bso that it is a one-to-one correspondence\nbetween these two sets.\nTo complete the proof, we observe that\nDcvr=t/summationdisplay\ni=1Dcvr,ιi (3)\n=/summationdisplay\nι∈IdentsRmin{D[Pι,b;W]|b∈Bι}+/summationdisplay\nι/nelementIdentsR\n(ι∈Idents (cvr))D[Pι,⊥;W]\n≥/summationdisplay\nι∈IdentsRD[Pι,cι;W]+/summationdisplay\nι/nelementIdentsR\n(ι∈Idents (cvr))D[Pι,cι;W]\n≥/summationdisplay\nι∈IdentsD[Pι,cι;W], (4)\nwhere we have used the fact—immediate from the definition—th at for any distribution Pand any ballot b,\nD[P,⊥;W]≥D[P,b;W].\nLetB/nequalWbe a candidate for which E/LEFTCIRCLE /Circle(B)≥E/LEFTCIRCLE /Circle(A) for all candidates A. (If there is no winner in E, such a candidate\nalways exists but may “tie” with some other candidate(s).) T hen, again by the definition of discrepancy, for each ι\nD[Pι,cι;W]≥/parenleftig/parenleftbigP/LEFTCIRCLE /Circle\nι(W)−P/LEFTCIRCLE /Circle\nι(B)/parenrightbig−/parenleftbigT/LEFTCIRCLE /Circle\ncι(W)−T/LEFTCIRCLE /Circle\ncι(B)/parenrightbig/parenrightig\n9Audit Type Notation Meaning\nAllbPhysical ballot\nB Ballot family\nBιBallots with identifier ι\nSNumber of ballots in election\nC Candidates in the election\nBayesianE Election\nTbProbability distribution over auditor interpretation of b allot\nIbRandom variable with distribution Tb\nT/LEFTCIRCLE /Circle\nb(A)Expected number of votes for ballot breceived by candidate Aif audited\nE/LEFTCIRCLE /Circle(A)Expected number of votes across election received by candid ateAif hand counted\nPrPredicted distribution of interpretations for ballot with identifier r\nP/LEFTCIRCLE /Circle\nr(A)Expected number of votes predicted for candidate Aon ballot with identifier r\nµA,BBayesian diluted margin of Awith respect to candidate B\nµAMinimum of Bayesian diluted margin across candidates\ncvr Cast vote record containing identifiers and predicted inter pretation distributions\nfor each ballot\ncvr(A)Predicted number of votes received by Aoncvr\nµcvr Bayesian diluted margin declared by cvr\nDcvr Discrepancy of cvrwith respect to election E\nConservativeTbSet of possible interpretations on b\nT/CIRCLE\nb(A)Best interpretation for Aonb\nT/Circle\nb(A)Worst interpretation for Aonb\nT/CIRCLE(A)Most possible votes for Ain election\nT/Circle(A)Least possible votes for Ain election\nµ+\nA,BConservative diluted margin of Awith respect to candidate B\nµ+\nAMinimum of Conservative diluted margin across candidates\nµ+\ncvr Conservative diluted margin declared by cvr+\ncvr+Conservative cvrdeclaring set of possible interpretations\nP/Circle\nr(A)Worst predicted interpretation for candidate A\nP/CIRCLE\nr(A)Best predicted interpretation for candidate A\ncvr/Circle(A)Minimum votes for Aoncvr\ncvr/CIRCLE(A)Maximum votes for Aoncvr\nD+\ncvr Conservative discrepancy of cvrwith respect to election E\nCompetitiveOmission (cvrA,cvrB)Set of identifiers declared by cvrAabsent in cvrB\nConflict (cvrA,cvrB)Set of identifiers that appear in both CVRs with no interpreta tions in common\nDisagree (cvrA,cvrB)Union of omission and conflict\nTable 1: Summary of Notation\nReturning to the expression (4), we then conclude that\n/summationdisplay\nι∈Idents (cvr)D[Pι,cι;W]≥/summationdisplay\ni/parenleftig/parenleftbigP/LEFTCIRCLE /Circle\nι(W)−P/LEFTCIRCLE /Circle\nι(B)/parenrightbig+/parenleftbigT/LEFTCIRCLE /Circle\ncι(B)−T/LEFTCIRCLE /Circle\ncι(W)/parenrightbig/parenrightig\n=/summationdisplay\ni/parenleftbigP/LEFTCIRCLE /Circle\nι(W)−P/LEFTCIRCLE /Circle\nι(B)/parenrightbig+/summationdisplay\nb∈B/parenleftbigT/LEFTCIRCLE /Circle\nb(B)−T/LEFTCIRCLE /Circle\nb(W)/parenrightbig\n=/parenleftbigcvr(W)−cvr(B)/parenrightbig+/parenleftbigE(B)−E(W)/parenrightbig\n≥S·(µcvr+µE).\nIn the first equality above we use the fact that ι↔cιis a one-to-one correspondence, so that each ballot appears once\n10in this sum. We conclude that Dcvr≥S(µcvr+µE), as desired. /square\n3 Bayesian Comparison Audits; Leveraging Marginal Marks\nWe present our auditor and analysis using a cryptographic ga me between the auditor and the environment , in the spirit\nof [4]. The auditor naturally reflects the role played by an au diting team in a conventional audit; as such it is provided\nwith a CVR, the actual size of the election (in real-world set tings this is provided by a trusted ballot manifest ), and\nthe ability to request ballots by identifier in order to carry out the audit. When the auditor has completed the audit, it\nreturns either the token I nconclusive or Consistent . The environment, on the other hand, is responsible for prov iding\nphysical ballots when they are requested. Note, in particul ar, that the auditor’s access to ballots is entirely mediate d by\nthe environment, which may choose not to return ballots when they are requested, make choices about which ballot to\nreturn when multiple ballots share an identifier, etc. The po wer of this modeling framework is that the critical concept\nof risk can be defined in the worst case over all possible behaviors of the environm ent. This reflects a wide variety of\nfailures—inadvertent or malicious—that occur in practica l audits.\nThe formal Auditor-Environment game is described in Figure 2. We remark that each ballot “delivered” by the\nenvironment in the game generates a single draw from the asso ciated ground truth distribution of interpretations;\nmultiple requests for the same ballot are permitted by the co nvention, which generate new independent samples from\nthe ground truth distribution (modeling a situation where a n audit board evaluates the ballot afresh each time it is\nsampled).\nThe formal auditor. The auditor we consider is a standard “comparison auditor” a dapted to the setting where dis-\ncrepancy is computed with respect to the expected value of th e predicted distribution in the CVR rather than a single\ninterpretation. Fortunately, once discrepancy is redefine d in this way, the remaining analysis of the standard auditor\nrequires no significant changes. We lay down the definitions b elow.\nThe precise definition of an auditor requires specifying a “o ne-sided” statistical test. To explain the intuition,\nconsider a situation where an auditor is provided an invalid CVR with declared margin µcvr. From Lemma 1, the\nstraightforward ballot comparison experiment calling for the auditor to sample a random identifier from the CVR and\ncompare the interpretation of the returned ballot to the pre dicted distribution on the CVR yields discrepancy Dfor\nwhichE[D]≥µcvr. Of course, a valid CVR that accurately reflects the interpre tations on the ballots will be able to\nachieveE[D]≈0. Thus, the auditor wishes to ensure that if E[D]≥µcvrit will only return the token C onsistent with\nsmall probabilityα(the “risk” of the audit). This is precisely the task of a “one -sided” statistical test. In fact, as we\nconsider environments that need not respond independently during each ballot request, we require a slightly richer\nfamily of tests, formulated in this setting by [4]. We briefly lay out the definitions here, noting these definitions are\nused in a similar manner.\nDefinition 8. [4, Definition 8] A sequence of bounded (real-valued) random variables X 1,... are said to beδ-\ndominating if, for each t≥0,E[Xt|X1,..., Xt−1]≥δ.We apply this terminology to the distribution Dcorresponding\nto the random variables, writing δ/triangleleftequalD.\nDefinition 9. [4, Definition 9] Let Σ= [−2,2]. Astopping time is a function Stop :Σ∗→{0,1}so that for any\nsequence x 1,x2,...of values inΣthere is a finite prefix x 1,..., xkfor which Stop (x1,..., xk)=1.\nFor a sequence of random variables X 1,...taking values inΣ, letτStop(X1,...)be the random variable given by\nthe smallest t for which Stop (X1,..., Xt)=1. This naturally determines the random variable X 1,..., XτStop, the prefix\nof the X igiven by the first time Stop ()=1.\nDefinition 10. [4, Definition 10] An adaptive audit test , denoted T=(Stop,R), is described by two families of\nfunctions, StopδandRδ. For each 0<δ≤1,\n(1)Stopδis a stopping time, as in Definition 9, and\n(2)Rδ:Σ∗→{0,1}is the rejection criterion .\n11Auditor (MAudit )–Environment (Env) game for election E=(B,C,S) and CVR cvr\n(1)Setup .\n(a)Ballot and tabulation delivery (to Env).The physical ballots Bis given toEnv.\n(b)Ballot manifest and CVR delivery (to MAudit ).The size Sand the cvrare given to the auditor\nMAudit .\n(2)Audit .MAudit repeatedly makes a ballot request to Env, or chooses to conclude the audit:\nA ballot request .MAudit requests a ballot from Envwith identifierι∈{0,1}∗.\nEnvdoes one of two things:\n(a) Responds with a ballot bwith identifierι′; in this case,MAudit is givenι′andI, an interpretation\ndrawn fromTb.\n(b) Responds with No ballot ; this is forwarded to MAudit .\n(3)Conclusion .MAudit returns one of the two values: C onsistent or Inconclusive .\nFigure 2: The RLAMAudit,Env(E,cvr) auditing game for comparison audits with multiple interpr etations.\nLetDbe a probability distribution on ΣN; for such a distribution, define αδ,D=E[Rδ(X1,..., Xτ)]where X 1,...are\nrandom variables distributed according to Dandτis determined by Stopδ. The riskof the test T is\nα=sup\n0<δ≤2\nδ/triangleleftequalDαδ,D. (5)\nThe formal auditor for the Bayesian setting is presented in F igure 3, and depends on a particular adaptive statistical\ntest. As mentioned above, the auditor is the natural compari son auditor [1]; the key di fference with existing approaches\nis that computation of margin and discrepancy incorporates Bayesian CVR predictions. We can now define the risk of\na Bayesian audit.\nDefinition 11 (Risk of Bayesian audit) .LetMAudit be an Auditor. For election E and environment EnvletRLAMAudit,Env(E)\ndenote the random variable equal to the conclusion of the aud it as described in Figure 2. An auditor MAudit hasα-risk\n(orα-soundness ) if, for all elections E, all invalid cvrs and all environments Env,\nPr\nMAudit,Env[RLAMAudit,Env(E,cvr)=Consistent ]≤α.\nTheorem 1. Let E be an election and cvrbe an invalid CVR. For any environment Env, letDobs\ni,Envdenote the random\nvariable computed in BasicExperiment during iteration i; then E[Dobs\ni,Env]≥Dcvr/S≥µcvr.\nLet(Stop,R)be an adaptive audit test with risk α. LetMAudit be as in Figure 3,MAudit [Stop,R]has riskα.\nProof. The proof of Theorem 1 naturally follows from Lemma 1 after no ting that for all values x∈Σ,i∈Nit must be\nthe case that Pr[ Dobs\ni,Env≤x]≤Pri←[1,t]/bracketleftbigDcvr,ιi≤x/bracketrightbig. Thus, it follows that E[Dobs\ni,Env]=Dcvr/S≥µcvr. /square\n4 Conservative Comparison Audits\nOur second approach is called a conservative comparison audit . At a high level, this can be seen as replacing the\nprobability distribution from the previous section with a s et of possible interpretations. For a declared W, the auditor\nruns the audit as though the interpretation that minimizes µWoccurs with probability 1. When accurate prediction of\nthe auditor distribution is possible this model has worse re sults than Bayesian interpretation but doesn’t su ffer from\nthe poor efficiency when one overshoots the probability of the auditor (s ee results in the next subsection). As a second\nadvantage, for our modeling in the conservative setting, th e environment is allowed to choose the interpretation of the\n12MAudit [Stop,R](E=(C,B,S),cvr)\n(1) Receive cvr=/parenleftbig(ι1,P1),..., (ιt,Pt)/parenrightbig.\n(2) If cvrhas repeated identifiers or S/nequalt, return Inconclusive .\n(3) If cvrdeclares a winner W, defineµ:=µcvr;\notherwise, return Inconclusive .\n(4) Initialize iter=0.\n(5) Repeat\n(a) Increment iter:=iter+1.\n(b) Perform Diter:=BasicExperiment\nuntil Stopµ(D1,...,Diter)=1\n(6) If Rµ(D1,...,Diter)=1 returnConsistent ; otherwise return I nconclusive .\nBasicExperiment :\n(1)RowSelect : Select a row r∈[S] uniformly.\n(2) Letιbe the identifier in row r; request delivery of ι.\n(3) If a ballot bwas delivered with identifier ι′=ιand interpretation I∗, return\nact :=max\nA∈C\\W/parenleftig\nP/LEFTCIRCLE /Circle\nr(W)−P/LEFTCIRCLE /Circle\nr(A)−(I∗(W)−I∗(A))/parenrightig\n.\nElse, return act :=max\nA∈C\\W/parenleftig\nP/LEFTCIRCLE /Circle\nr(W)−P/LEFTCIRCLE /Circle\nr(A)/parenrightig\n−1.\nFigure 3: The auditor MAudit [(Stop,R)].\nballot selected by the audit board among the possible interp retations of the ballot. This is opposed to the previous\nsection where this interpretation was i.i.d. sampled by the game.\nIn all of the below definitions, we just contrast them with the definition in Section 2 rather than providing a new\ndefinition.\n4.1 Conservative Definitions\nDefinition 12 (Conservative Interpretations, ballots, and ballot famil ies).We continue to use the definitions and ter-\nminology for ballots and interpretations (cf., Definition 1 ) with one alteration. Rather than calling for a ground truth\ndistributionTb, we define a set T bofpossible interpretations for this ballot. (That is, interpretations that could rea-\nsonably be the conclusion of an audit board’s inspection of t he ballot.) In the setting with a ground-truth distribution ,\nthe set T bwould naturally correspond to the support of the distribution T b={I∈I|T b(I)>0}.\nAs mentioned above, conservative CVRs specify a set of decla red interpretations for each ballot.\nDefinition 13 (Conservative Cast-V ote Record Table (CVR)) .LetCbe a set of candidates. A conservative Cast-V ote\nRecord table (CVR) is a sequence of pairs\ncvr+=/parenleftbig(ι1,P1),..., (ιt,Pt)/parenrightbig,\nwhere theιrare distinct bitstrings in {0,1}∗and each P r⊂ I is a subset of interpretations. Intuitively, the CVR\ndeclares that for each row r the interpretation of the ballot labeledιrlies in the set P r. For each candidate A∈C, we\ndefine\nP/Circle\nr(A)=min{I(A)|I∈Pr},\nP/CIRCLE\nr(A)=max{I(A)|I∈Pr},\n13cvr/Circle(A)=t/summationdisplay\nr=1P/Circle\nr(A), and cvr/CIRCLE(A)=t/summationdisplay\nr=1P/CIRCLE\nr(A).\nWinners, losers, and contradictory CVRs. If there is a candidate A∈C with the property that cvr/Circle(A)>cvr/CIRCLE(B)\nfor every B∈C\\{ A}, then we say that Ais the declared conservative winner according to cvr+. If there is no such\ncandidate, we say that cvr+isindeterminate . Any candidate L∈Cfor which cvr/CIRCLE(L)<cvr/Circle(A)for some candidate\nA∈Cis adeclared conservative loser . We adopt the same naming conventions for such conservative CVRs as we do\nwith Bayesian CVRs (that is, for the set of identifiers, size, and treating identifiers as indices for P randPr).\nFinally, we say that two CVRs are contradictory if one of the two CVRs declares a winner and the other CVR\ndeclares a different winner.\nWe remark that a conservative CVR can be indeterminate even w hen it does not declare a tie: E.g., there may be\ntwo candidates AandBfor which cvr/CIRCLE(A)>cvr/Circle(B) and cvr/CIRCLE(B)>cvr/Circle(A).\nDefinition 14 (Conservative consistency and validity) .Acvr=/parenleftbig(ι1,P1),..., (ιt,Pt)/parenrightbigisconsistent with E if the ballots\nBare uniquely labeled, the identifiers appearing in cvrare identical to those appearing on the ballots of E, and for\neach ballot b∈B, Tb⊂Pιb.In the special case where T b=Pidbfor each ballot, we say that cvris the canonical CVR\nfor the election E. We only define these notions for uniquely l abeled elections.\nFor election E with winner W, acvris invalid if it declares a winner other than W.\nDefinition 15 (Conservative Interpretation limits; election winner; el ection margin) .Let E=(C,B,S)be an election.\nFor a ballot b∈Band a candidate A∈C, define\nT/Circle\nb(A)=min{I(A)|I∈Tb},\nT/CIRCLE\nb(A)=max{I(A)|I∈Tb}.\nThus T/CIRCLE\nb(A)andT/Circle\nb(A)indicate the most and least favorable interpretations of th e ballot for the candidate A. We\nadditionally define\nE/Circle(A)=/summationdisplay\nb∈BT/Circle\nb(A), and E/CIRCLE(A)=/summationdisplay\nb∈BT/CIRCLE\nb(A).\nDefinition 16 (Conservative margin) .Theconservative margin of Awith respect to Bcompares the “worst-case” and\n“best-case” interpretations for two candidates:\nµ+\nA,B=E/Circle(A)−E/CIRCLE(B)\nS.\nWe then defineµ+\nA=min B∈C\\{A}µ+\nA,B. If there exists some candidate Wsuch thatµ+\nW>0we call this candidate the\n(conservative) winner and denote this quantity as µ+. If a conservative winner exists, then it is unique. There ma y be\nno conservative winner in which case an election is called indeterminate . We call a candidate La conservative loser if\nthere exists some other candidate Asuch thatµ+\nL,A>0. Intuitively, this candidate does not tie the election even if every\nballot is interpreted in the most favorable method. Note tha t conservative losing candidates may exist in indeterminat e\nelections.\nDefinition 17 (Conservative Declared Margin) .We call a candidate La conservative loser if there exists some other\ncandidate Asuch thatµ+\nA,L>0. Intuitively, this candidate does not tie the election even if every ballot is interpreted in\nthe most favorable method. Note that conservative losing ca ndidates may exist in indeterminate elections.\nThedeclared margin of a conservative cvrwith declared winner Wis the quantity\nµ+\ncvr:=min\nA∈C\\Wcvr/Circle(W)−cvr/CIRCLE(A)\nS.\nDefinition 18 (Conservative discrepancy) .Let E=(C,B,S)be an election and let cvr=((ι1,P1),..., (ιt,Pt))be a\nconservative CVR with declared winner W. For a ballot b∈Band interpretation I, the conservative discrepancy of I\nwith respect to bis\nD+[I,b;W]=max\nA∈C\\W/parenleftig/parenleftbigI(W)−I(A)/parenrightbig−/parenleftbigT/CIRCLE\nb(W)−T/Circle\nb(A))/parenrightig\n.\n14Auditor (MAudit )–Environment (Env) game for election E=(B,C,S) and CVR cvr\n(1)Setup .\n(a)Ballot and tabulation delivery (to Env).The physical ballots Bis given toEnv.\n(b)Ballot manifest and CVR delivery (to MAudit ).The size Sand the cvrare given to the auditor\nMAudit .\n(2)Audit .MAudit repeatedly makes a ballot request to Env, or chooses to conclude the audit:\nA ballot request .MAudit requests a ballot from Envwith identifierι∈{0,1}∗.\nEnveither responds a ballot bwith single identifier ι′andan interpretation I∈Tbor responds with No\nballot.\n(3)Conclusion .MAudit returns one of the two values: C onsistent or Inconclusive .\nFigure 4: The RLAMAudit,Env(E,cvr) auditing game for comparison audits with multiple interpr etations.\nWe expand this definition to apply, additionally, to a specia l symbol⊥:\nD+[I,⊥;W]=max\nA∈C\\W/parenleftbigI(W)−I(A)/parenrightbig+1.\nThe discrepancy of a cast vote record table cvris:\nD+\ncvr=t/summationdisplay\ni=1min/braceleftbigD[I,b;W]/vextendsingle/vextendsingle/vextendsingleI∈Icvr\ni,b=⊥oridb=ιi/bracerightbig.\n4.2 Adapted Auditor and Game\nWe consider the following adaption of the auditing game wher e the environment is allowed to specify the interpretation\nreturned to the auditor in Figure 4 with the change in boldfac e. The auditor is identical to that in Figure 3 except the\nBayesian cvrmargin in Step (3) is replaced with µ+\ncvr.\nTheorem 2. Let E be an election and cvrbe an invalid CVR. For any environment Env engaging in Figure 4, let\nDobs\ni,Envdenote the random variable computed in BasicExperiment thenE[Dobs\ni,Env]≥Dcvr/S≥µ.whereµis the value\ncomputed in Step (3).\nLet(Stop,R)be an adaptive audit test with risk α. LetMAudit be as in Figure 3 with µcvrreplaced withµ+\ncvr,\nMAudit [Stop,R]has riskα.\nProof. The proof of Theorem 2 proceeds similarly to Theorem 1. Here w e provide a substitute for Lemma 1 for the\nconservative setting.\nLemma 2. Let E=(C,B,S)be an election and let cvrbe an invalid CVR for E with declared winner Wand size\nt=S. Then D+\ncvr≥(µ+\ncvr+µ+\nE)·S≥(µ+\ncvr)·S.\nProof. We focus on the setting when Bis uniquely labeled and all identifiers appear in the uniquel y labeled cvr. See\nthe proof of Lemma 1 for why this setting su ffices.\nLetWcvrbe the declared conservative winner and let WE/nequalWcvrthe conservative winner of the election. By\ndefinition, E/Circle(WE)>E/CIRCLE(Wcvr).\nBy the definition of discrepancy, for each ι\nD+[Pι,cι;Wcvr]≥/parenleftig/parenleftbigP/Circle\nι(Wcvr)−P/CIRCLE\nι(WE)/parenrightbig−/parenleftbigT/Circle\ncι(Wcvr)−T/CIRCLE\ncι(WE)/parenrightbig/parenrightig\n15We then conclude that\nD+\ncvr≥/summationdisplay\nι∈Idents (cvr)D[Pι,cι;W]\n≥/summationdisplay\ni/parenleftig/parenleftbigP/Circle\nι(Wcvr)−P/CIRCLE\nι(WE)/parenrightbig+/parenleftbigT/Circle\ncι(WE)−T/CIRCLE\ncι(Wcvr)/parenrightbig/parenrightig\n=/summationdisplay\ni/parenleftbigP/Circle\nι(Wcvr)−P/CIRCLE\nι(WE)/parenrightbig+/summationdisplay\nb∈B/parenleftbigT/Circle\nb(WE)−T/CIRCLE\nb(Wcvr)/parenrightbig\n=/parenleftbigcvr(Wcvr)−cvr(WE)/parenrightbig+/parenleftbigE(We)−E(Wcvr)/parenrightbig\n≥S·(µ+\ncvr+µ+\nE).\nIn the first equality above we use the fact that ι↔cιis a one-to-one correspondence, so that each ballot appears once\nin this sum. We conclude that Dcvr≥S(µ+\ncvr+µ+\nE), as desired. This completes the proof of Lemma 2. /square\nThis completes the proof of Theorem 2. /square\n5 Completeness of Bayesian and Conservative Audits; E fficiency Analysis\nThe previous subsection established that one can recover a n atural RLA from CVRs that declare multiple interpreta-\ntions for ballots. In the case when all ballots and CVRs have a single interpretation the audit is a traditional ballot\ncomparison audit. We also note that the auditor receives a si ngle interpretation of the ballot. This models the fact that\nan audit board has to produce a final adjudication of a ballot. As described above, the purpose of the test is to decide\nwhetherEDcvr/S≥µcvr.\nPerhaps the most widely deployed test meeting the demands ou tlined in the previous section is the Kaplan-\nMarkov test (though there are other natural choices [9, 10, 2 5, 38]); in particular, this is the test featured in the\nArlo RLA administration system. The basic Kaplan-Markov te st (omitting minimum and maximum sample sizes)\nis as follows: for a parameter γ>1, define the value\nRisk(γ)\nδ(D1,...,Dℓ)=ℓ/productdisplay\niter=11−δ\n2γ\n1−Diter\n2γ.\nThe stopping time for risk limit αis determined by the test Riskδ?≤α; when this occurs, the test rejects the hypothesis\n(that the mean is larger than µ) and, in our setting, the auditor would output C onsistent .\nSimulation parameters. We consider a two candidate election with 100 ,000 ballots with m=.5% marginal ballots\nand underlying error rates of o1=.1%, u1=.1%, o2=.01%, and u2=.01%; Errors o1ando2represent ballots\nwith discrepancies of +1 and+2 respectively. If the CVR lists a vote for the winner Wand the actual ballot shows\na vote for a loser this is an o2error. The actual ballot showing no vote results in an o1, this is called an undervote.\nThe errors u1andu2are defined similarly for discrepancies of −1,−2. For marginal ballots, we assume there are two\npossible interpretations: an interpretation for Wand an interpretation of an undervote. We compare three appr oaches\nin Table 2:\n(1) In the Baseline approach, half of the marginal ballots are counted as votes f or the reported winner on the CVR.\nWhen a marginal ballot is selected for audit it is determined as a vote for Wwith probability 50%. Note this can\nresult in discrepancy values of {−1,0,1}.\n(2) In the Bayesian approach, the margin receives P/LEFTCIRCLE /Circle\nr(W)=.5 “votes” for each marginal ballot. As above, when\na marginal ballot is selected for audit it is determined as a v ote for Wwith probability 50%. However, since\nP/LEFTCIRCLE /Circle\nr(A)=.5 the possible discrepancy values are {−.5,.5}\n(3) In the Conservative approach, no marginal ballots are included in the margin and only positive discrepancy\nvalues are possible if the auditor interprets the ballot as a vote for W. (These are not essential for the main\nnarrative of the paper, but are included for completeness.)\n16Baseline Conservative Bayesian\nµ Mean Stdev Median 95% Mean Stdev Median 95% Mean Stdev Median 95%\n.01 608 210 567 1028 595 181 576 938 583 175 545 920\n.02 316 78 292 469 314 69 294 420 308 64 292 415\n.03 213 44 202 283 212 38 219 263 210 39 202 271\nTable 2: Number of ballots for Kaplan-Markov comparison aud it between the baseline and the Bayesian auditor, across\nmargins, and probability .5 of CVR and auditor determining marginal mark as for winner, α=.05,γ=1.1.\nBaseline Conservative Bayesian\npm Mean Stdev Median 95% Mean Stdev Median 95% Mean Stdev Median 95%\n1 471 115 438 704 514 164 494 824 468 116 438 704\n.9 493 131 454 729 526 161 494 842 490 126 468 738\n.8 520 149 470 817 533 158 494 824 510 135 485 773\n.7 547 170 502 881 599 167 538 879 534 150 504 830\n.6 575 188 539 930 579 178 544 944 559 155 533 856\n.5 608 210 567 1028 595 181 576 938 583 175 545 920\n.4 630 217 591 1071 617 181 576 975 616 189 576 1003\n.3 662 231 616 1123 646 199 590 1020 641 199 596 1031\n.2 685 227 644 1128 671 202 627 1071 671 211 619 1097\n.1 707 227 627 1138 700 215 658 1108 692 209 630 1104\n0 729 214 658 1184 729 214 658 1184 724 215 658 1184\nTable 3: Number of ballots across pm,α=.05,γ=1.1,µ=.01.\nAll results perform 5000 simulations for each parameter set ting. A simulation pulls a uniform ballot until the risk\nlimit is met. Each time a marginal ballot is pulled its interp retation is independently sampled (50% probability of vote\nforWand 50% of blank). RLAs require the most work for small margin s which is where our improvement is evident.\nWe report on the mean, standard deviation, median, and 95% th reshold of ballots retrieved in Table 2. In practice,\nthe 95% mark is most relevant; this reflects the fact that the w ork to conduct an additional round of an RLA is often\nprohibitive—requiring a distributed process across a whol e region—so termination within the specified sample size\nwith high probability is desirable. The Bayesian approach r educes the 95% percentile of ballots sampled by roughly\n10.5%. Roughly, this improvement arises because the variance o f discrepancy for “marginal” ballots is reduced from\n1/2 using the baseline approach to 1 /4 using the Bayesian approach.\nIn the above, both the ground truth (observed by the auditor) and CVR prediction probability of a marginal mark\nbeing interpreted as a mark is pm=.5. This is the setting that has the largest improvement on the variance of\ndiscrepancy. However, for varying values of pmthe approach still provides e fficiency improvements over the baseline\napproach. This is shown in Table 3. Note for pm∈{0,1}the baseline and Bayesian approach are identical as margina l\nballots never result in nonzero discrepancy. In Table 4, we c onsider two separate parameters for the probability that\nthe CVR marks each marginal ballot as a vote for W, denoted pcvrand the probability that the auditor does the same\ndenoted pMAudit . Interestingly, the baseline approach outperforms the Bay esian approach when pcvr>pMAudit by\n≈20%; however Bayesian can outperform Baseline by ≈30% when pcvr<pMAudit . This is part of the reason we\nintroduce the conservative approach where one does not have to estimate probabilities.\n6 Election Contestation and Competitive Ballot Comparison Audits\nWe consider a new class of post-election audits we call competitive audits . These audits can e fficiently distinguish\nCVRs that are consistent with the ballots from invalid CVRs a nd, more generally, reconcile competing claims of\nvictory made by two CVRs. This novel auditing technique also provides, as described earlier, a direct approach to\nhandling election contestation.\n17pMAudit=pcvr+.4 pMAudit=pcvr+.2 pMAudit=pcvr pMAudit=pcvr−.2 pMAudit=pcvr−.4\npcvr Base Cons Bayes Base Cons Bayes Base Cons Bayes Base Cons Bayes Base Cons Bayes\n1 704 824 704 1053 855 816 1307 924 750\n.9 729 842 738 1094 893 840 1347 938 763\n.8 780 810 981 817 824 773 1135 938 839 1347 975 740\n.7 856 816 999 881 879 830 1139 975 836 1373 1020 768\n.6 780 810 1129 860 842 1011 930 944 856 1165 975 850 1377 1071 754\n.5 790 817 1164 871 900 1010 1028 938 920 1160 1052 823 1347 1153 748\n.4 816 842 1203 905 924 1062 1071 975 1003 1100 1093 791 1271 1184 742\n.3 816 893 1216 886 960 1079 1123 1020 1031 1109 1108 808\n.2 816 900 1290 861 975 1068 1128 1071 1097 1058 1184 815\n.1 780 938 1301 820 1020 1097 1138 1108 1104\n0 739 1020 1347 816 1102 1058 1184 1184 1184\nTable 4: 95% of number of ballots for Kaplan-Markov comparis on audit,α=.05,γ=1.1,µ=.01.\nTo simplify our treatment of these audits, we focus on the con servative setting, in which CVRs declare, for each\nballot, a subset of possible interpretations without assig ning a probability distribution to the interpretations in t he\nsubset. We begin by indicating the changes required to the ba sic definitions for this setting.\nAn intuitive survey of the framework. After an election, we assume that each candidate A∈Cis represented by\nanadvocate . In a preliminary “rescan phase,” the advocate for the candi dateAmay inspect the ballots Band submit\nto the audit a conservative CVR, denoted cvrA; the idea is that the advocate will submit a CVR that is favora ble for the\ncandidate. If the declared conservative winner of cvrAisA, we say that this is a “declaration of victory.” (Advocates\nmay also be allowed to announce the presence of duplicate bal lot labels in the election, which can then be corrected\nby the audit. For simplicity, however, we simply treat Bas uniquely labeled in this setting.) Following this, the au dit\ncommences with the “judgment phase:” after considering som e of the ballots, the audit either declares the audit to be\nInconclusive or, for some candidate A∈C, declares that “ Ais the winner.”\nThe judgment phase; reconciling disagreements among the CV Rs. If no pair of the submitted CVRs are con-\ntradictory, the audit concludes without considering any ba llots. In this non-contradictory case, if any CVR declares\nvictory the associated candidate is declared to be the winne r of the audit; if, alternatively, no CVR declares a winner,\nthe audit concludes with I nconclusive .\nOtherwise, if two submitted CVRs are contradictory, the aud it must settle the contradictory claims made by various\nsubmitted CVRs. In this case, some candidate Amust be a declared winner according to one CVR and a declared l oser\naccording to another. We will see that there must then exist a n identifierιon which the two CVRs make contradictory\nassertions. Intuitively, one of these contradictory CVRs c an then be removed from consideration by fetching the ballot\nwith this identifier, if it exists, and relying on this ballot to settle the contradictory claims made by the two CVRs.\nThe natural procedure is complicated by three issues:\n(1) two contradictory CVRs may declare di fferent sets of identifiers rather than disagreeing on the inte rpretation of\nspecific ballots;\n(2) the audit should provide suitable protection against an imperfect or malicious environment that may not always\nrespond with a given ballot, even when it exists; and\n(3) the audit should withstand a small collection of errors i n the creation of CVRs by advocates to reflect the\npractical difficulty of producing a perfect CVR, even by a well-intentioned advocate.\nWe remark that the second issue above forces the audit to trea t “positive evidence” and “negative evidence” di fferently:\nfor example, while the delivery of a ballot that disagrees wi th a particular CVR, either by possessing an identifier that\nis not declared in the CVR or having an interpretation that is inconsistent with those declared in the CVR, is considered\nevidence against the CVR, the failure of the environment to p roduce a ballot declared in a particular CVR is not. This\n18cautious convention prevents a malicious environment from misleading the auditor. An environment that can refuse to\nretrieve ballots can always cause the audit to output I nconclusive .\nUltimately, we will prove two facts about the audit.\n•If one of the submitted CVRs is consistent with the ballots an d ballots are faithfully returned to the auditor\nduring the judgment phase, then—except with small probabil ity that can be explicitly bounded as a function\nof the number of samples—the auditor will correctly conclud e the audit with “ Wis the winner.” Thus, if the\nadvocate for Windeed acts in W’s best interests, Wcan be confident that the audit will conclude favorably.\n•If one of the submitted CVRs is consistent with the ballots an d an alternate CVR declares that a losing candidate\nLwon the election, the audit will not conclude that “ Lis the winner.” Furthermore, this guarantee is robust\nagainst ballot suppression—it does not require that ballot s are correctly returned when requested.\nTaken together, we see that if a candidate’s advocate submit s a consistent CVR, the candidate can be assured a victory\nif no ballots are lost or suppressed, and can be assured that n o losing candidate is declared the victor even in the face\nof ballot suppression.\n6.1 Modeling Competitive Audits\nFormally, we model a competitive audit with two parties, the auditor (denotedJudge ) and the environment (denoted\nEnv). The auditor is responsible for analyzing submitted conse rvative CVRs, requesting ballots to carry out the audit,\nand arriving at a final conclusion. The environment is respon sible for servicing requests by the auditor for individual\nballots: in particular, the environment is in possession of the ballots and, for each request for a ballot by the auditor,\ndecides which ballot (if any) will be returned to the auditor and, moreover, how that ballot is to be interpreted (that is,\nwhich possible interpretation is to be adopted).\nAs mentioned above, this two party modeling of the audit refle cts the fact that even a well-designed auditor may\nhave to contend with such complexities as loss of ballots, po tential malicious supression of ballots, and the final choic e\nof an “auditing board” in the interpretation of a ballot. The se decisions are in the hands of the environment.\nDefinition 19 (Competitive auditor; competitive environments; honest e nvironments) .Acompetitive auditor Judge\nis a randomized procedure for carrying out a competitive aud it. In the context of an election E =(C,B,S)with\nconservative ballots B, the auditor is invoked with:\n(1) the set of candidates C,\n(2) the size Sof the election, and\n(3) a list of conservative CVRs cvr+\n1,..., cvr+\nk.\nThe audit proceeds in rounds, each round being an opportunit y for the auditor to request delivery of a ballot (specified\nby an identifier) along with an interpretation of the ballot. After a sequence of requests, the auditor concludes the\naudit, resulting in either the statement Inconclusive , or “ Ais the winner” for some candidate A∈C.\nBallot requests are handled by a second randomized procedur e called the environment . The environment is invoked\nwith (C,B,cvr+\n1,..., cvr+\nk). To each (identifier) request ιmade by the auditor, the environment either responds with a\nballot b∈Band a single interpretation I ∈Tbor a “no ballot” symbol ⊥. The environment may choose not to return\na ballot when a matching ballot exists in B, any ballot it does return is assumed to match the requested i dentifier:\nιb=ι.3\nAn environment is honest if it responds to any requested identifier with a matching bal lotband an interpretation\nI∗∈Ibwhen such a ballot exists.\nDefinition 20. Let E=(C,B,S)be an election. For a competitive auditor, Judge , environmentEnv, and a sequence\nof conservative CVR tables cvr+\n1,..., cvr+\nk, define\nCompetitiveA [Judge,Env;E;cvr+\n1,..., cvr+\nk]\n3These conventions are a convenience: the auditor can check t he identifier of a returned ballot to ensure that it matches th e request and,\nadditionally, ensure that the returned interpretation is a mong those associated with the ballot; if either of these ver ifications fail, the auditor treats\nthis ballot response as a ⊥.\n19Competitive auditor ( Judge )–Environment (Env) game\nCompetitiveA [Judge,Env;B,C,S].\n(1)Advocate CVR generation . For each candidate A∈C, an advocate for A—with access to the full candidate\nsetCand the physical ballots B—generates a cast-vote record table, cvr+\nA.\n(2)Setup .\n(a)Candidates, ballots, and CVR delivery (to Env).(C,B,{cvr+\nA|A∈C}) are given to the environment\nEnv.\n(b)Candidate list, ballot manifest, and CVR delivery (to Judge ).(C,S,{cvr+\nA|A∈C}) are given to\nthe auditorJudge .\n(3)Audit .Judge repeatedly makes a request to Env, or chooses to conclude the audit:\n•A ballot request .Judge requests a ballot from Envwith identifierι∈{0,1}∗.\n(a)Enveither responds either with an interpretation Iwhere I∈Tbfor some ballot bfor whichι=ιb\nor responds with⊥(meaning No ballot ).\n(4)Conclusion .Judge either returns:\nAis the winner meaning “The audit is consistent with victory by candidate A,” or\nInconclusive meaning “Audit inconclusive.”\nFigure 5: The competitive auditing game CompetitiveA with auditorJudge , environmentEnv, election E, and advo-\ncate CVRs{cvr+\nA|A∈C}.\nto be the result of the audit (returned by Judge ) described in Figure 5. As both the auditor and the environme nt may\nbe randomized procedures, this is a random variable taking v alues in the setC∪{ Inconclusive}.\nDefinition 21 (Disagreement Set) .Define the Omission Set as Omission (cvrA,cvrB)=Idents (cvrA)\\Idents (cvrB).\nDefine the Conflict Set as Conflict (cvrA,cvrB)={ι|ι∈Idents (cvrA)∩Idents (cvrB)and P cvrA,ι∩PcvrB,ι=∅}\nThe disagreement set is the set\nDisagree (cvrA,cvrB)=Omission (cvrA,cvrB)∪\nConflict (cvrA,cvrB)\nThe identifiersιin the disagreement set are those for which cvrAmakes a claim about a ballot in Bthat is inconsistent\nwithcvrB, in the sense that either\n(1)cvrBdoes not recognize that the ballot exists at all (Omission (cvri,cvr j))or\n(2)cvrAandcvrBmake contradictory claims about ballot interpretation (Conflict (cvrA,cvrB)).\nIntuitively, a CVR cvrBwill be disqualified if a ballot bis found during the judgment phase with either i) an\nidentifierιthat was not reported on cvrBor ii) an interpretation I∗that was not in the set of possible interpretations\nPcvrA,ι.\nDefinition 22. (Disqualification Function) Define function Disqual :Σ∗→{0,1}that takes in an identifier, ι, an\ninterpretation, I∗, and a CVR, cvrA.\nDisqual (ι(s),I∗,cvrA)=1ι(s)/nelementIdents (cvrA)∨\nI∗/nelementPcvrA,ι\n0otherwise.\nWe now analyze the competitive auditor Judgetin Figure 6.\n20Competitive auditor Judget[C,S;cvr1,..., cvrk];\nTheJudgetimmediately outputs I nconclusive if any advocate announces duplicate labels. For each submit ted\nCVR cvrA, defineIdents A:=Idents (cvrA)={ι|the identifierιappears in cvrA}and, for an identifier ι∈\nIdents A, letPA,ιdenote the set of interpretations associated with ιincvrA.\n(1) Remove from consideration any CVRs that do not have size S.\n(2) For each ordered pair ( A,B) for which the (remaining) CVRs cvrAandcvrBare contradictory:\n•Letι(1),...,ι(t)be independent and uniformly random elements of Disagree (cvr+\nA,cvr+\nB). For each s∈\n{1,..., t}, request the ballot matching ι(s). Let I(1),...,I(t)denote the sequence of returned identifiers\n(including⊥).\n•For each request s, we define the sequence Disqual s=Disqual (ι(s),I(s),cvr+\nB).\n•Finally, if/summationtextt\ns=1Disqual s>t/2, disqualify cvr+\nB.\n(3) Remove all disqualified CVRs from consideration.\n(4) If there remains any pair of contradictory CVRs, or no rem aining CVR declares victory, the audit concludes\nwith Inconclusive .\n(5) Otherwise, a winner Wis declared by a remaining CVR that is not contradicted by any other remaining\nCVR and the audit concludes with “ Wis the winner.”\nFigure 6: The competitive auditor Judget, with inputC,S, and advocate CVRs {cvr+\nA|A∈C} where there are k\nsubmitted CVRs. The parameter tdetermines the number of ballot samples collected by the aud itor: in particular, the\nauditor requests no more than tk(k−1) ballots.\n6.2 Analysis of the Competitive Auditor Judget\nIn this section, we make two claims arguing about the complet eness and soundness of Judge . We first consider\nthese claims in the case that an “honest” party makes no error s and then consider the setting with errors in the next\nsubsection.\nThm 3 shows that if there is an election winner Wand some cvrWis consistent with the physical ballots, then W\nwill be the output of Judge . This theorem assumes an honest environment. Recall that (i .) the consistency of a CVR\nmeans that for each ballot, the CVR submits a superset of the a ctual interpretations for that ballot and (ii.) a candidate\nis an election winner only if it wins regardless of the interp retation of ballots with multiple interpretations.\nThm 4 shows that if a candidate is an election loser Land some cvris canonical with the physical ballots, then\nLwill never be the output of Judge . This theorem does not assume whether the environment is hon est. Recall that\na CVR is canonical if for every ballot it declares exactly the set of interpretations on the physical ballots. Intuitivel y,\nsince a loser loses no matter the interpretation declared by the environment, a canonical CVR is never disqualified by\nJudge andLcan never be the output.\nThe main requirement for both theorems is that the advocate w rites down the full set of interpretations for each\nphysical ballot on the CVR (either a superset or the exact set ). An advocate gets disqualified when there are non-\nintersecting interpretations between the original and adv ocate CVR, meaning that there are interpretations on the\noriginal CVR that are not listed on the advocate CVR.\nTheorem 3 (Competitive completeness) .LetJudgetbe the competitive auditor with sample parameter t ∈Z+. Let\nE=(C,B,S)be an election with winner W∈C. Let cvr∗be consistent with E and declare Wto be the winner. Let\ncvr1,..., cvrk−1be any collection of CVRs. Then\nCompetitiveA [Judget,Env;E;cvr∗,cvr1,..., cvrk−1]=W,\nfor any honest environment Env. Furthermore,Judgetcompletes the audit with no more than tk (k−1)ballot samples.\nWe remark that in this case tdoes not impact the result of the theorem and can safely be set to 1. (We consider\nt>1 for the case with errors in the next subsection.)\n21Proof of Theorem 3. Adopt notation as described in the statement of the theorem: an election E=(C,S,B) with\nuniquely labeled ballots and winner W, and a honest environment Env. Let cvr∗be a CVR consistent with E, also\ndeclaring winner W, and let cvr1,..., cvrk−1be any sequence of CVRs. Considering that cvr∗is consistent with E,\nit is not possible for cvr∗to amass any votes for disqualification during comparison wi th any alternative CVR, as by\ndefinition it provides a consistent interpretation for any b allot band interpretation I∈Tbsampled byJudge . It follows\nthat the only possible outcomes of the audit are I nconclusive or “Wis the winner.”\nTo complete the proof, we establish that any CVR cvrithat declares an alternate winner W′will necessarily be\ndisqualified (by cvr∗) during the audit. We establish that the set Claim i≺∗(associated with the two CVRs cvriandcvr∗)\ncannot be empty: Observe that if Claim i≺∗were, in fact, empty then the two CVRs would declare the same s et of\nidentifiers and, moreover, there would be an interpretation I∼\nbfor each ballot bconsistent with both CVRs in the sense\nthatI∼\nb∈Pcvr∗,ι∩Pcvri,ι. But, as these CVRs declare these two di fferent winners, this would imply that both\n/summationdisplay\nbI∼\nιb(W′)≤cvr/CIRCLE\n∗(W′)<cvr/Circle\n∗(W)≤/summationdisplay\nbI∼\nιb(W)\nand /summationdisplay\nbI∼\nιb(W)≤cvr/CIRCLE\ni(W)<cvr/Circle\ni(W′)≤/summationdisplay\nbI∼\nιb(W′),\na contradiction. It follows that Claim i≺∗is nonempty. As we assume that Envis honest, any identifier requested during\nthe audit fromClaim i≺∗will be properly serviced by the environment and will lead to a vote for disqualification for\ncvri. In this case, with a consistent cvr∗and honestEnv, alltvotes collected during consideration of the pair ( i,∗) will\ncall for disqualification. Thus cvriwill be removed from consideration. As all contradictory CV Rs will be so removed,\nthe audit will conclude with “ Wis the winner,” as desired. /square\nTheorem 4 (Competitive soundness) .LetJudgetbe the competitive auditor with integer sample parameter t >0.\nLet E=(C,B,S)be an election with loser L∈C. Let cvr∗be the canonical CVR for E. Let cvr1,..., cvrk−1be any\ncollection of CVRs. Then\nPr/bracketleftbigCompetitiveA [Judge,Env;E;cvr∗,cvr1,..., cvrk−1/bracketrightbig=L/bracketrightbig=0,\nfor any environment Env. Furthermore,Judgetcompletes the audit with no more than tk (k−1)ballot samples.\nProof of Theorem 4. Adopt notation as described in the statement of the theorem: an election E=(C,S,B) with\nuniquely labeled ballots and losing candidate L. We now consider an arbitrary environment Env. Let cvr∗be the\ncanonical CVR and let cvr1,..., cvrk−1be any sequence of CVRs. Considering that cvr∗is canonical with E, it is not\npossible for cvr∗to amass any votes for disqualification during comparison wi th any alternative CVR, as by definition\nit provides a consistent interpretation for any ballot band interpretation I∗∈Tbsampled byJudge . Note, additionally,\nthat as cvr∗is the canonical CVR, it reflects the fact that Lis a loser: there is a candidate Aso that cvr/CIRCLE\n∗(L)<cvr/Circle\n∗(A).\nAscvr∗cannot be disqualified, the audit cannot conclude with “ Lis the winner,” as desired. /square\n6.3 Handling Advocacy Errors\nIt is unrealistic to demand that a well-intentioned auditor can produce a perfectly consistent (or canonical) CVR.\nOccasional mistakes may occur during ballot interpretatio n, there may be some uncertainty about establishing a set I\nthat necessarily contains all interpretations that an audi ting board may assign to a ballot, and ballots may be overlook ed.\nAs we insist in our setting that CVRs have size consistent wit h the election, we note that an advocate can always pad\nout a CVR by adding invented identifiers that do not correspon d to any ballot. All told, these considerations lead us to\nconsider CVRs that are “nearly” consistent, defined formall y below.\nTheorems 5 and 6 extend these theorems to the case when “hones t” parties make a small fraction of errors.\nDefinition 23. Let E=(C,S,B)be an election. We say that a cvr(overC) is(1−ǫ)-consistent if the size of the CVR\nisSand there is a subset C of Idents (cvr)of size at least (1−ǫ)Son which cvris consistent, which is to say that for\neachι∈C, there is a ballot b∈Bfor whichι=ιbandIb⊂Iι(whereIιis the set of interpretations declared by cvr\nforι). Define a (1−ǫ)-canonical cvranalogously.\n22LetBin[γ,t;t/2] denotes the tail of the binomial distribution: specifical ly, if X1,..., Xtare independent Bernoulli\nrandom variables for which Pr[ Xi=1]=γ, then Bin[γ,t;t/2] is the probability that/summationtext\niXi≥t/2.\nTheorem 5 (Competitive completeness with imperfect advocacy) .LetJudgetbe the competitive auditor with integer\nsample parameter t >0. Let E=(C,B,S)be an election with winner W∈C. Let cvr∗be(1−ǫ)-consistent with E\nand declare Wto be the winner with margin µcvr∗>4ǫ. Let cvr1,..., cvrk−1be any collection of CVRs and let Envbe\nan honest environment. Then\nPr[CompetitiveA [Judget,Env;E;cvr∗,cvr1,..., cvrk−1]=W]\n≥1−2(k−1)Bin[2ǫ/µ cvr∗,t;t/2].\nForγ<1/2 (guaranteed above by 4 ǫ<µ cvr∗), the quantity Bin[γ,t;t/2]=exp(−ω(t)). Under the mild assumption that\nǫ<µ cvr∗/6, say, the number of samples required by such an audit to achi eve any particular risk limit is independent of\nµcvr∗. An advocate may reduce µcvr∗in an effort to be consistent with the underlying ballots.\nProof of Theorem 5. Adopt notation as described in the statement of the theorem: an election E=(C,S,B) with\nuniquely labeled ballots and winner W, and a honest environment Env. Let cvr∗be (1−ǫ)-consistent with E, declaring\nwinner Wwith marginµ∗>4ǫ; letcvr1,..., cvrk−1be any sequence of CVRs. Consider a CVR cvrithat contradicts\ncvr∗; in this case cvr/Circle\ni(A)>cvr/CIRCLE\ni(W) for a candidate A. By assumption, cvr/Circle\n∗(W)≥cvr/CIRCLE\n∗(A)+S·µ∗. It follows that\nbothClaim i≺∗andClaim∗≺imust have cardinality at least S·µ∗/2.\nThere are two events of interest: the event that cvr∗is disqualified by some contradictory cvri, and the event that\nsome contradictory cvriis not disqualified by cvr∗.\nFor the first of these, observe that while |Claim∗≺i|≥Sµ∗/2 for a contradictory cvri, the CVR cvr∗is consistent\nwith the actual ballots on all but ǫSidentifiers. It follows that the probability that an identifi er chosen uniformly\nfromClaim∗≺iwill reference a ballot that is inconsistent with cvr∗is no more thanǫ/(µ∗/2)=2ǫ/µ∗. Thus the\nprobability that cvr∗is disqualified by cvriis no more than Bin[2ǫ/µ∗,t;t/2], as desired. As there are no more than\nk−1 contradictory CVRs, the probability that cvr∗is disqualified is no more than ( k−1)Bin[2ǫ/µ∗,t;t/2].\nThe second event can be bounded by similar means. Consider a c ontradictory cvri; as above, the setClaim i≺∗has\nsize at least S·µ∗/2 and, of these identifiers, all but ǫSof them are consistently reflected by the ballots. It follows that\nthe probability that a random identifier drawn from Claim i≺∗will be consistent with the underlying ballots (and hence\ngenerate a vote for disqualification for cvriconsidering thatEnvis honest) is at least 1 −ǫ/(µ/2). Recall that the the\nbinomial distribution corresponding to the sum of ti.i.d. Bernoulli random variables with expectation 1 −γis precisely\nthe distribution corresponding to tvariables with expectation γreflected about the point t/2. Thus such a contradictory\nCVR will avoid disqualification with probability no more tha nBin[2ǫ/µ∗,t;t/2]. As there are no more than k−1\ncontradictory CVRs, the probability that any of them are not disqualified is no more than ( k−1)Bin[2ǫ/µ∗,t;t/2].\nCombining these two bounds yields the final estimate, as in th is case any contradictory CVR will have been\ndisqualified while cvr∗will remain. /square\nTheorem 6 (Competitive soundness) .LetJudgetbe the competitive auditor with integer sample parameter t >0. Let\nE=(C,B,S)be an election with loser L∈C. Let cvr∗be a (1−ǫ)-canonical CVR for E. Define\nµcvr∗=max\nA∈C\\L(cvr−\n∗(A)−cvr+\n∗(L))/S.\nLetcvr1,..., cvrk−1be any collection of CVRs. Then\nPr/bracketleftbigCompetitiveA [Judget,Env;E;cvr∗,cvr1,..., cvrk−1/bracketrightbig=L/bracketrightbig\n≤2(k−1)Bin[2ǫ/µ,t;t/2]\nfor any environment Env. Furthermore,Judgetcompletes the audit with no more than tk (k−1)ballot samples.\nProof. Follows from the same reasoning as Theorem 5. /square\n236.4 Remarks on Bayesian contested audits\nWhile we do not present any details, we make a few remarks abou t contested audits in the Bayesian case. This calls\nfor each advocate to submit a Bayesian CVR. Now we note that id entification of a collection of ballots Don which the\ntwo CVRs substantively disagree permits us to define two di fferent probability distributions on this set of ballots: (i. )\nselect a ballot at random from D, and (ii.) output an interpretation given by the CVR in quest ion. Then the classical\nsequential probability ratio test provides a statistical t est for distinguishing the two models [39]. This can serve as\nthe statistical mechanism for selecting between contradic tory CVRs. One can achieve a constant number of ballots to\ndisqualify a CVR only if there are a constant number of ballot s whose respective Dhave constant Kullback-Leibler-\ndivergence, this is why we treat the simpler case where a cons tant number of ballots have disjoint interpretations.\nAcknowledgements\nThese results were developed as part of a collaboration with the Office of the CT Secretary of State and, additionally,\nwere supported in part by a grant from that o ffice. Discussions with anonymous reviewers improved the narr ative and\ntechnical treatment. B.F. is supported by NSF Grants #22328 13 and #2141033 and the O ffice of Naval Research. A.R\nwas supported by NSF Grant #1801487.\nReferences\n[1] Mark Lindeman and Philip B Stark. A gentle introduction t o risk-limiting audits. IEEE Security&Privacy ,\n10(5):42–49, 2012.\n[2] National Academies of Sciences, Engineering, and Medic ine. Securing the Vote: Protecting American Democ-\nracy. 2018.\n[3] Andrea Bajcsy, Ya-Shian Li-Baboud, Mary Brady, et al. Sy stematic measurement of marginal mark types on\nvoting ballots, 2015. https://nvlpubs.nist.gov/nistpubs/ir/2015/NIST.IR.8 069.pdf .\n[4] B.Fuller, A. Harrison, and A. Russell. Adaptive risk-li miting comparison audits. In IEEE Symposium on Security\nand Privacy , pages 2002–2019, Los Alamitos, CA, USA, may 2023.\n[5] Philip B Stark. Auditing a collection of races simultane ously. arXiv preprint arXiv:0905.1422 , 2009.\n[6] Philip B Stark. Super-simple simultaneous single-ball ot risk-limiting audits. In EVT/WOTE , 2010.\n[7] Michael J Higgins, Ronald L Rivest, and Philip B Stark. Sh arper p–values for stratified election audits. Statistics,\nPolitics, and Policy , 2(1), 2011.\n[8] Kellie Ottoboni, Philip B Stark, Mark Lindeman, and Neal McBurnett. Risk-limiting audits by stratified union-\nintersection tests of elections (SUITE). In International Joint Conference on Electronic Voting , pages 174–188.\nSpringer, 2018.\n[9] Philip B Stark. Sets of half-average nulls generate risk -limiting audits: Shangrla. In Financial Cryptography and\nData Security , pages 319–336. Springer, 2020.\n[10] Ian Waudby-Smith, Philip B Stark, and Aaditya Ramdas. R ilacs: Risk limiting audits via confidence sequences.\nInInternational Joint Conference on Electronic Voting , pages 124–139. Springer, 2021.\n[11] Michelle Blom, Jurlind Budurushi, Ronald L Rivest, Phi lip B Stark, Peter J Stuckey, Vanessa Teague, and Damjan\nVukcevic. Assertion-based approaches to auditing complex elections, with application to party-list proportional\nelections. In International Joint Conference on Electronic Voting , pages 47–62. Springer, 2021.\n[12] Colorado Secretary of State. 2020 general election ris k-limiting audit discrepancy report, 2020.\nhttps://www.sos.state.co.us/pubs/elections/RLA/2020 /general/DiscrepancyReport.pdf .\n24[13] A. Russell, L. Michel, B. Fuller, J. Wohl, and G. Johnson . Statistical anal-\nysis of post-election audit data for the November 8, 2022 sta te election.\nhttps://voter.engr.uconn.edu/wp-content/uploads/sit es/3651/2023/02/2022-11-08-hand-count-statistics. \n[14] Kai Wang, Eric Rescorla, Hovav Shacham, and Serge J Belo ngie. Openscan: A fully transparent optical scan\nvoting system. EVT/WOTE , 10:1–13, 2010.\n[15] Philip B. Stark. Conservative statistical post-elect ion audits. The Annals of Applied Statistics , 2(2):550 – 581,\n2008.\n[16] Matthew Bernhard. Risk-limiting audits: A practical s ystematization of knowledge. In International Joint\nConference on Electronic Voting , 2021.\n[17] Lynn Garland, Neal McBurnett, Jennier Morrell, Marian K. Schneider, and Stephanie Singer. Principles and best\npractices for post-election tabulation audits, 2018.\n[18] Joseph Lorenzo Hall, Luke W Miratrix, Philip B Stark, Me lvin Briones, Elaine Ginnold, Freddie Oakley, Martin\nPeaden, Gail Pellerin, Tom Stanionis, and Tricia Webber. Im plementing risk-limiting post-election audits in\nCalifornia. In USENIX, editor, 2009 Electronic Voting Technology Workshop /Workshop on Trustworthy Elections\n(EVT/WOTE) , Montreal, Canada, 2009. USENIX, USENIX.\n[19] Mark Lindeman, Philip B Stark, and Vincent S Yates. Brav o: Ballot-polling risk-limiting audits to verify out-\ncomes. In EVT/WOTE , 2012.\n[20] Stephen Checkoway, Anand Sarwate, and Hovav Shacham. S ingle-ballot risk-limiting audits using convex op-\ntimization. In Proceedings of the 2010 International Conference on Electr onic Voting Technology /Workshop on\nTrustworthy El ections , EVT/WOTE’10, pages 1–13, USA, 2010. USENIX Association.\n[21] Philip B. Stark. Cast: Canvass audits by sampling and te sting. IEEE Transactions on Information Forensics and\nSecurity , 4(4):708–717, 2009.\n[22] Michelle Blom, Peter J Stuckey, Vanessa Teague, and Dam jan Vukcevic. A first approach to risk-limiting audits\nfor single transferable vote elections. In International Conference on Financial Cryptography and Da ta Security ,\npages 366–380. Springer, 2022.\n[23] Philip B Stark. E fficient post-election audits of multiple contests: 2009 cali fornia tests. In CELS 2009 4Th\nannual conference on empirical legal studies paper , 2009.\n[24] Kellie Ottoboni, Matthew Bernhard, J Alex Halderman, R onald L Rivest, and Philip B Stark. Bernoulli ballot\npolling: a manifest improvement for risk-limiting audits. InInternational Conference on Financial Cryptography\nand Data Security , pages 226–241, 2019.\n[25] Philip B. Stark. Risk-limiting postelection audits: C onservative P-values from common probability inequalities.\nIEEE Transactions on Information Forensics and Security , 4(4):1005–1014, 2009.\n[26] Philip B. Stark. Alpha: Audit that learns from previous ly hand-audited ballots, 2022.\n[27] Jorge H Banuelos and Philip B Stark. Limiting risk by tur ning manifest phantoms into evil zombies. arXiv\npreprint arXiv:1207.3413 , 2012.\n[28] Filip Zag´ orski, Grant McClearn, Sarah Morin, Neal McB urnett, and Poorvi L V ora. The athena class of risk-\nlimiting ballot polling audits. arXiv preprint arXiv:2008.02315 , 2020.\n[29] Jennifer Morrell. Knowing it’s right, part two. risk-l imiting audit implementation workbook., 2019.\n[30] Jennie Bretschneider, Sean Flaherty, Susannah Goodma n, Mark Halvorson, Roger Johnston, Mark Lindeman,\nRonald L. Rivest, Pam Smith, and Phillip B. Stark. Risk-limi ting post-election audits: Why and how, 2012.\n25[31] Filip Zag´ orski, Grant McClearn, Sarah Morin, Neal McB urnett, and Poorvi L V ora. Minerva–an e fficient risk-\nlimiting ballot polling audit. In USENIX Security Symposium , pages 3059–3076. USENIX Association, 2021.\n[32] Amanda K Glazer, Jacob V Spertus, and Philip B Stark. Bay esian audits are average but risk-limiting audits\nare above average. In Electronic Voting: 5th International Joint Conference, E- Vote-ID 2020, Bregenz, Austria,\nOctober 6–9, 2020, Proceedings 5 , pages 84–94. Springer, 2020.\n[33] Josh Benaloh, Kammi Foote, Philip B Stark, Vanessa Teag ue, and Dan S Wallach. Vault-style risk-limiting audits\nand the inyo county pilot. IEEE Security&Privacy , 19(4):8–18, 2021.\n[34] Douglas W Jones, Sunoo Park, Ronald L Rivest, and Adam Se alfon. Scan, shuffle, rescan: Machine-assisted\nelection audits with untrusted scanners. In Financial Cryptography , 2024.\n[35] John H Reif. The complexity of two-player games of incom plete information. Journal of computer and system\nsciences , 29(2):274–301, 1984.\n[36] M. Kiwi, C. Lund, D. Spielman, A. Russell, and R. Sundara m. Alternation in interaction. Computational\nComplexity , 9(3):202–246, 2000.\n[37] L´ aszl´ o Babai, Lance Fortnow, and Carsten Lund. Non-d eterministic exponential time has two-prover interactive\nprotocols. Computational complexity , 1:3–40, 1991.\n[38] Jacob V Spertus. Cobra: Comparison-optimal betting fo r risk-limiting audits. In International Conference on\nFinancial Cryptography and Data Security , pages 95–109. Springer, 2023.\n[39] Abraham Wald. Sequential tests of statistical hypothe ses. In Breakthroughs in statistics: Foundations and basic\ntheory , pages 256–298. Springer, 1992.\n26This figure \"CVR_visual.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/2402.06515v2"    },    {        "title": "2402.06533v2.2PM_waveform_from_loop_corrected_soft_theorems.pdf",        "content": "2PM waveform from loop corrected soft theorems\nFrancesco Alessioa,band Paolo Di Vecchiaa,c\naNORDITA, KTH Royal Institute of Technology and Stockholm University,\nHannes Alfv´ ens v¨ ag 12, SE-11419 Stockholm, Sweden\nbDepartment of Physics and Astronomy, Uppsala University,\nBox 516, SE-75120 Uppsala, Sweden\ncThe Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark\nAbstract\nWe introduce a classical version of the loop corrected soft graviton theorem and\nwe use it to compute the universal part of the one-loop (2PM) waveform up to sub-\nsubleading order in the energy ωof the emitted graviton for spinless black holes scat-\ntering. In particular, we compute the action of the soft operators on the classically\nresummed four-point amplitude, that can be written in terms of the exponential of the\neikonal phase (and is therefore non-perturbative in the Newton’s constant GN) and\nthen we perform the usual PM expansion in powers of GN. We find perfect agreement\nwith the existing 2PM literature at the orders ω−1, logωandωlog2ω, which are uni-\nversal. Furthermore, we use this method to compute the universal part of the ωlogω\ncontribution to the 2PM waveform. Even if in the present analysis we limit ourselves\nto compute the soft 2PM waveform, our general formulae can be used to extract all\nuniversal PM orders of the terms connected with the infrared divergences, once the im-\npulse at the corresponding precision is known. Our approach, based on the resummed\neikonal amplitude, gives a unified picture of the various computations of the classical\nsoft graviton behaviour that are present in the literature since the seminal paper by\nWeinberg in 1965 [1].arXiv:2402.06533v2  [hep-th]  1 Mar 20241 Introduction\nIn the last few years techniques based on scattering amplitudes have been successfully applied\nto the computation of gravitational observables in the framework of the Post-Minkowskian\n(PM) expansion. They have been used to extract classical observables from the full quantum\namplitude. A non-exhaustive list of PM literature is [2–110]. These include matching QFT\nresults with an EFT description [7, 8, 111], developing the worldline formalism [38, 44] to\ncompute higher orders in the PM expansion and using the KMOC approach [9] or the\neikonal exponentiation [14,112–114] for computing observables for both spinless and spinning\nparticles. The scattering results have then also been used in EOB models [2, 46, 115–117]\nand for bound orbits [118–122].\nA major breakthrough was the two-loop calculation [10,11] of the impulse in the potential\nregion completed with the calculation [29–31,89,123] of the radiation reaction that made the\ndeflection angle finite at high energy. Those results were confirmed by a direct calculation [19,\n25, 34, 35] of the two-loop classical elastic amplitude. During the last couple of years three\ngroups [41, 43, 124–126] have computed both the contribution of the potential region and\nthat of the radiation reaction to the deflection angle at 4PM in the elastic scattering of two\nmassive particles obtaining identical results. In particular, Refs. [125,126] included also the\ncontribution of few powers of the spin.\nOne of the main objectives of theoretical work on gravitational waves is to compute\nthe waveform of the gravitational waves emitted by the merging or by the scattering of\ntwo black holes. The leading waveform was already computed in a pioneering work in\nRef. [127] using standard general relativity methods. Recently it has been computed using\nscattering amplitude techniques both in time domain [128] and in frequency domain [95,\n129, 130]. Generalization to the spinning case can be found in [131–133]. Those papers\nprovide the waveform at 1PM order, that is obtained from the classical tree-level 5-point\namplitude involving four massive particles and a graviton. Recently, the 5-point amplitude\nand the waveform at 2PM (one-loop) have also been computed [49,50,109,134] (see also the\nsubsequent papers [135–139]). In particular, Refs. [136,137] deal with the one-loop waveform\nin the limit in which the graviton is soft.\nAt the tree-level the amplitude with emission of a soft graviton satisfies low-energy theo-\nrems that follow directly from gauge invariance [140,141]. According to these theorems, the\nleading O(ω−1) and the subleading O(ω0) terms are universal and they apply to any theory.\nThe sub-sub-leading O(ω) term instead contains a universal piece, but non-universal pieces\ncan appear in different cases.\nAt one-loop level scattering amplitudes are infrared divergent in spacetime dimensions\nD= 4 and they must be regularised by introducing, for instance, an infrared regulator\nϵ=4−D\n2. Then the previous low-energy theorems are modified and contain divergent terms\nin the ϵ→0 limit [142].\nRecently in a series of papers [143–148] the authors have considered the low-energy\ntheorems, computing the new universal terms involving log ωup to one-loop level. The log ω\nterm has also been independently obtained in Refs. [149, 150] using methods based on the\neikonal resummation and it has been shown in Ref. [151] to follow from superrotation Ward\n1identities. It has also been confirmed [137] from the soft expansion of the complete one-loop\n5-point amplitude involving four massive particles and a graviton. This last paper confirms\nthe universality of the O(ω−1),O(logω) and O(ωlog2ω) terms, but also shows that the\norderO(ωlogω) is not universal, as pointed out e.g.in Ref. [148].\nAs shown in both Refs. [142] and [144, 148], the soft factors involve the calculation of\ntwo types of one-loop Feynman diagrams, one where the internal graviton is attached to two\nmassive legs and another where the internal graviton is attached to a massive and a massless\nleg. They both need to be regularised. In Ref. [142] it is used dimensional regularisation.\nOn the other hand, as done in [1], one can also regularise them by the introduction of an\nultraviolet Λ and an infrared cutoff λand then relate the two regularisations by imposing\nthat1\n2ϵ= logλ\nΛ. If we then assume that ωacts as an infrared cutoff in the first type of\ndiagram and instead acts as an ultraviolet cutoff in the second type of diagram, we get\nexactly the log ωsoft factors obtained in [144,148], as pointed out in [151].\nIn this paper we follow the procedure outlined in Ref. [148] (with the limitations discussed\nthere for the O(ωlogω) term) for computing the various soft factors in the classical limit\nand then we use them in the case of a 5-point amplitude involving four scalar particles and\na graviton. However, instead of applying them to a quantum 4-point amplitude, we act\nwith them on a resummed classical 4-point amplitude that is obtained by transforming back\nto momentum space the eikonal in impact parameter space. In this way we compute the\nO(logω) and O(ωlogω) terms of the 1PM waveform and the same ones at 2PM, together\nwithO(ωlog2ω).\nThe paper is organised as follows. In section 2 we summarise how the tree-level and one-\nloop soft theorems are obtained, mainly following the procedure outlined in Refs. [144,148].\nIn section 3 we revisit the eikonal exponentiation and, in particular, we discuss how the\nresummed elastic four-point amplitude in momentum space can be written in terms of the\neikonal phase. In section 4 we perform the soft expansion of the tree-level waveform, starting\nfrom the soft expansion of the tree-level five-point amplitude. In particular, we compute its\nO(ω−1),O(logω) and O(ωlogω) contributions. In section 5 we introduce a soft graviton\ntheorem that acts on the resummed 4-point amplitude, obtaining the 5-point amplitude\nin the soft limit and up to order 2PM. In section 6 we write the soft contribution to the\nwaveform both at 1PM and at 2PM orders and in section 7 we displat the first orders of their\nPN expansion. Section 8 is devoted to some conclusions. The details of the calculations of\nthe results presented in section 6 are given in appendix A. Finally in Appendix B we show\nhow to extract the classical contribution of the Weinberg function Wfrom the full quantum\ncalculation.\nNote added\nAfter the submission of the present paper to the arXiv, we became aware of two related\npapers [152,153] submitted on the same day with partial overlap with our analysis.\n22 Soft theorems\nLet us start by considering an elastic scattering process n→minvolving nincoming and m\noutgoing massive particles and let us denote the corresponding tree-level scattering amplitude\nbyMN, with N=n+m. We introduce the shorthand notation pi≡(p1, . . . , p n) and\npf≡(pn+1, . . . , p m) for the incoming and outgoing asymptotic momenta and\nηj=\u001a−1 incoming\n1 outgoing, σ jk=−pj·pk\nmjmk, j, k = 1, . . . N. (2.1)\nIn our convention, all momenta are outgoing and hence we parametrize them as\npj=ηj(Ej, ⃗ pj), E j>0, p2\nj=−m2\nj. (2.2)\nWe are interested in the tree-level amplitude MN+1corresponding to the inelastic scattering\nn→m+ 1 involving an additional outgoing graviton with momentum k=ω(1,ˆn). The\nsub-subleading soft graviton theorem relates MN+1toMNin the limit where the graviton\nis soft, i.e.when its energy ωis small and it reads [1,154]\nMµν\nN+1(pi, pf, k) =κ\u0012\nˆSµν\n(−1)+ˆSµν\n(0)+ˆSµν\n(1)\u0013\nMN(pi, pf) +O(ω2), (2.3)\nwhere κ=√8πGNand\nˆSµν\n(−1)=NX\nj=1pµ\njpν\nj\npj·k, (2.4)\nˆSµν\n(0)=NX\nj=1p(µ\njkρ\npj·kˆJν)ρ\nj, (2.5)\nˆSµν\n(1)=1\n2NX\nj=1kρkσ\npj·kˆJµρ\njˆJνσ\nj, (2.6)\nare the leading, subleading and sub-subleading soft factors and, for spinless particles, ˆJµν\nj=\n2p[µ\nj∂\n∂pjν]1is the mechanical angular momentum operator of the j-th particle. Equation (2.3)\nis an analytic series expansion of Mµν\nN+1for small ω. A similar result holds for scattering of\nparticles mediated by the electromagnetic field.\nThe leading and subleading soft factors capture the full O(ω−1) and O(ω0) behaviours\nofMµν\nN+1and they are universal, i.e.they do not depend on the particular theory consid-\nered. They are a direct consequence of gauge invariance of the amplitude [140,141] and are\n1Here we define symmetrization as A(µBν)=1\n2(AµBν+AνBµ) and antisymmetrization as A[µBν]=\n1\n2(AµBν−AνBµ).\n3related to the conservation of total momentum and angular momentum during the scatter-\ning process [155]. Further, it has been proven that they are connected to the invariance\nof the gravitational S-matrix under the action of asymptotic symmetries in four spacetime\ndimensions and to the gravitational memory effect (see, e.g.[156]). On the other hand, the\nO(ω) behaviour is not universal and the sub-subleading soft factor in (2.4) captures only its\nuniversal part [157].\nAlready at tree-level, when going to position space in D= 4, there is a breakdown of\nthe power series expansion in ωat subleading order and the corresponding subleading soft\nfactor contains terms that are non-analytic in ωand, in particular, they behave as O(logω)\n[143, 144, 158]. A simple derivation of this consists in explicitly computing the trajectories\nof particles subjected to the gravitational interaction and observing that at early and late\ntimes, as the proper time |τ| → ∞ , they are logarithmically divergent. As a consequence,\nthe mechanical angular momentum ˆJµν\niof each particle diverges as log |τ| ≃log|ω|−1and\ntherefore also the subleading soft factor in (2.3) does so. Ultimately, this is a consequence\nof the long-range nature of the gravitational (and electromagnetic) interaction in D= 4.\nEquation (2.3) is still a tree-level statement. It is of interest in our analysis to consider how it\ngets modified when one takes into account one-loop corrections to MN+1andMN, together\nwith their infrared divergences. As discussed in the seminal paper [1], such divergences\nexponentiate for (any) gravity amplitude M\nM=e−WMI.F., (2.7)\nwhere I.F. stays for “infrared finite” and where, for a general ( N+ 1)-point amplitude\ninvolving massive particles, the function Wis explicitly given by\nW=−iGN\n2ϵN+1X\nj̸=k=1wjk, w jk≃mjmkσ2\njk−1\n2q\nσ2\njk−1δηjηk,1, (2.8)\nin dimensional regularization with ϵ=4−D\n2. In equation (2.8) we used “ ≃” to denote\nequality after having taken the classical limit, in which we are eventually interested. A\ncomplete formula for W, including its quantum contributions, can be found in appendix B.\nIf we take particle N+ 1 to be outgoing and massless with momentum kwe get [139]\nW ≃ − iGN\n2ϵNX\nj̸=k=1wjk+iGN\nϵNX\nj=1(pj·k)δηj,η5≡ W reg+Wphase. (2.9)\nAs discussed later in section 4 in (4.38), ϵ−1terms correspond to logarithmic terms2. In\n2See also the discussion in [151].\n4particular [148]\nWreg≃ −iGNmjmkNX\nj̸=k=1σ2\njk−1\n2q\nσ2\njk−1δηjηk,1log{(ω+iηjϵ)L}, (2.10)\nWphase≃2iGNlog{(ω+iϵ)R}mX\nj=n+1pj·k, (2.11)\nwhere the ϵappearing in the last two equations is not the dimensional regularisation pa-\nrameter we introduced earlier, but comes from performing loop integrals and by keeping\ntrack of the iϵprescription, see e.g.appendix A of [145]. While in many references the iϵ\nprescription inside the argument of the logarithm is not taken into account, we find it a\nnecessary ingredient to obtain part of the result, as we explain later in section 5. Wregcomes\nfrom loop diagrams where the internal graviton line connects two massive lines and  L−1= Λ\nis a UV scale that regularises such loop integrals. Hence, Wregfactors out IR divergences\ncontained both in MN+1andMNand for the latter we can write a relation\nMN(pi, pf) =e−W regMI.F.\nN(pi, pf), (2.12)\nOn the other hand, Wphase comes from loop diagrams involving an internal three-graviton\nvertex and Ris an arbitrary IR scale related to the energy resolution of the detector. There-\nfore we write\nMµν\nN+1(pi, pf, k) =e−W reg−W phaseMµνI.F.\nN+1(pi, pf, k). (2.13)\nUsing the previous relations together with soft theorems gives\nMµν\nN+1(pi, pf, k) =e−W reg−W phaseMµνI.F.\nN+1(pi, pf, k) =e−W phase−W regˆSµνMI.F.\nN(pi, pf)\n=e−W phasee−W regˆSµνeWregMN(pi, pf), (2.14)\nwhere\nˆSµν=κ\u0012\nˆSµν\n(−1)+ˆSµν\n(0)+ˆSµν\n(1)\u0013\n+O(ω2). (2.15)\nis the soft operator. We get a concise formulation of the one-loop soft graviton theorem\nMµν\nN+1(pi, pf, k) =κe−W phasee−W reg\u0012\nˆSµν\n(−1)+ˆSµν\n(0)+ˆSµν\n(1)\u0013\neWregMN(pi, pf)\n+O(ω2, ω2logω). (2.16)\n5Equation (2.16) can be expanded as\nMµν\nN+1(pi, pf, k) =κe−W phase\u001a\nˆSµν\n(−1)+ˆSµν\n(0)+ˆSµν\n(1)+ (ˆSµν\n(0)Wreg)\n+NX\nj=1kρkσ\npj·k(ˆJµρ\njWreg)ˆJνσ\nj+1\n2NX\nj=1kρkσ\npj·k(ˆJµρ\njWreg)(ˆJνσ\njWreg)\n+ (ˆSµν\n(1)Wreg)\u001b\nMN(pi, pf) +O(ω2, ω2logω), (2.17)\nand, at one-loop one has to expand the overall exponential up to quadratic order and keep\nonly terms of O(Gn\nN) with n <2. This procedure reproduces the results in [142] and in [148].\nWe notice that in the latter reference, the last term is absent, while it is displayed in the\nformer. As we discuss later, we find that this term is subleading in ℏand therefore it does\nnot contribute to the classical limit.\n3 Eikonal exponentiation revisited\nHere we specify to the case n=m= 2 and we consider an elastic 2 →2 scattering. We can\ngo to impact parameter space by means of a Fourier transform\n˜M4(s, b) =Zd4−2ϵq\n(2π)4−2ϵeib·qˆδ(2p1·q)ˆδ(2p2·q)M4(s, q2)\n=1\n4m1m2√\nσ2−1Zd2−2ϵq⊥\n(2π)2−2ϵeib·q⊥M4(s, q2\n⊥), (3.1)\nwhere we introduced an infrared regulator ϵ= (4−D)/2,ˆδ(x) = 2 πδ(x) and q⊥is the\nprojection of qonto the two-dimensional plane perpendicular to p1andp2\nqµ\n⊥=Pµ\n⊥νqν, Pµν\n⊥=ηµν−Pµν\n∥, Pµν\n∥=1\nσ2−1\u0012pµ\n1pν\n1\nm2\n1+pµ\n2pν\n2\nm2\n2−2σp(µ\n1pν)\n2\nm1m2\u0013\n. (3.2)\nIn (3.1) s=−(p1+p2)2=m2\n1+m2\n2+ 2m1m2σ, where σ≡σ12=σ34= (1−v2)−1\n2,\nvbeing the relative velocity between the two particles and t=−(p1+p4)2=−q2are\nthe usual Mandelstam variables. In the PM expansion we assume that the Schwarzschild\nradius rj= 2GNmjof the two particles is much smaller than the impact parameter bof the\nscattering process. This allows us to expand perturbatively M4as\nM4(s, q2) =X\nn≥0M(n)\n4(s, q2), (3.3)\n6where M(n)\n4∼Gn+1\nNis the n-loop ( n+ 1 PM) contribution to the amplitude arising from\nthe exchange of n+ 1 virtual gravitons. As discussed e.g.in [114] in the classical limit the\namplitude in impact parameter space resums into a compact exponential structure3\n1 +i˜M4(s, b)≃e2iδ(σ,b), δ (σ, b) =X\nn≥0δ(n)(σ, b), (3.4)\nwhere δis the eikonal phase and δ(n)itsn-loop contribution that scales as\nδ(n)(σ, b)∼(GNs)\u0012GN√s\nb\u0013n\n, (3.5)\nFor instance, the tree-level (1PM) amplitude corresponding to one-graviton exchange is\niM(0)\n4(s, q) =\n =4iκ2\nq2m2\n1m2\n2\u0012\nσ2−1\nD−2\u0013\n+O((q2)0).(3.6)\nIn this case the integral in (3.1) is infrared divergent and it gives [14,21]\n2iδ(0)(σ, b) =i˜M(0)\n4(s, b) =iGNm1m22σ2−1√\nσ2−1Γ(−ϵ)\n(πb2)−ϵ. (3.7)\nSuch infrared divergence can be traced back to the usual Coulombic potential in three space\ndimensions. In this case, introducing an inverse length scale µ, the leading eikonal becomes\n2δ(0)(σ, b) =−2GNm1m22σ2−1√\nσ2−1log(µb). (3.8)\nThe one-loop eikonal is not infrared divergent and it is given by\n2δ(1)(σ, b) =3πG2\nNm1m2(m1+m2)\n4b5σ2−1√\nσ2−1. (3.9)\nAfter the classical exponentiation in (3.4), one can perform an inverse Fourier transform to\nget the classical amplitude in momentum space\nM4(s, Q) = 4 im1m2√\nσ2−1Z\nd2b e−iQ·b\u0012\n1−e2iδ(σ,b)\u0013\n. (3.10)\n3At 3PM the eikonal develops an imaginary part due to the emission of gravitational radiation. In this\ncase the eikonal should be more properly treated as an operator, see Ref. [32].\n7Here Qis the classical transfer momentum, that is different from q, which scales with ℏ. By\nthe usual argument [114], the integral is dominated by the stationary phase and therefore\nwe identify the total classical momentum transfer as\nQµ=∂\n∂bµRe2δ(σ, b) =Qµ\n(0)+Qµ\n(1)+O(G3\nN)\n=−2GNm1m22σ2−1√\nσ2−1bµ\nb2+3πG2\nNm1m2(m1+m2)\n45σ2−1√\nσ2−1bµ\nb3+O(G3\nN). (3.11)\n4 Soft expansion in bspace\nIn this section we discuss the soft expansion of the classical amplitude in impact parameter\nspace. We define a Fourier transform for the 5-point amplitude, expand the integrand for\nsmall frequencies, and then compute the integral collecting terms that scale with the same\npower of ω.\nSetup\nLet us consider the Fourier transform of the classical 5-point amplitude, which we denote by\n˜Mµν\n5. Introducing the transferred momenta q1=p1+p4andq2=p2+p3, it can be written\nas\n˜Mµν\n5(pi, bi, k) =ZdDq1\n(2π)DZdDq2\n(2π)Dei(b1·q1+b2·q2)Mµν\n5(pi, qi, k)ˆδ(2p1·q1)ˆδ(2p2·q2)\n׈δ(D)(q1+q2+k). (4.1)\nSolving the D-dimensional ˆδby writing q1=q−k\n2andq2=−k\n2−q, gives\n˜Mµν\n5(pi, bi, k) =e−i(b1+b2)·k\n2ZdDq\n(2π)Deib·qMµν\n5\u0012\npi, , q−k\n2,−q−k\n2, k\u0013\n׈δ(2p1·q−p1·k)ˆδ(2p2·q+p2·k), (4.2)\nwhere we denoted b≡b1−b2. Now we proceed to solve the remaining two ˆδfunctions in\na covariant way. We start by noticing that any four-vector vcan be decomposed into a\ncomponent that lies in the plane spanned by p1andp2and into a component perpendicular\nto it\nvµ=vµ\n∥+vµ\n⊥, vµ\n∥=Pµν\n∥vν, vµ\n⊥=Pµν\n⊥vν, (4.3)\nwhere the parallel and perpendicular projectors have been defined in (3.2). The arguments\nof the two delta functions appearing in the Fourier transform (4.2) can be written as\n2p1·q−p1·k= 2p1·q∥−p1·k∥, 2p2·q+p2·k= 2p2·q∥+p2·k∥. (4.4)\n8Furthermore we can decompose v∥as\nvµ\n∥=−v1∥pµ\n1\nm1−v2∥pµ\n2\nm2, (4.5)\nwith\nv1∥=σ(v·p2)m1−(v·p1)m2\nm1m2(σ2−1), v 2∥=σ(v·p1)m2−(v·p2)m1\nm1m2(σ2−1). (4.6)\nSolving the first of (4.4) gives\nq∗\n1∥=k1∥+σk2∥−2σq2∥\n2, (4.7)\nand the ˆδis\nˆδ(2p1·q−p1·k) =1\n2m1ˆδ(q1∥−q∗\n1∥). (4.8)\nSolving the second of (4.4) gives, using (4.8)\nq∗\n2∥=k2∥(σ2+ 1) + 2 k1∥σ\n2(σ2−1), (4.9)\nand the ˆδbecomes\nˆδ(2p2·q+p2·k) =1\n2m2(σ2−1)ˆδ(q2∥−q∗\n2∥). (4.10)\nPutting (4.9) back into (4.7) gives\nq∗\n1∥=k1∥(σ2+ 1) + 2 k2∥σ\n2(1−σ2). (4.11)\nNote that, for k= 0, we get q∗\n1∥= 0 = q∗\n2∥, as expected, and the kinematics of the 5-\npoint amplitude collapses into the one of the elastic 2 →2 amplitude. Let us now compute\nthe transformation of the measure dDqappearing in (4.2). We are splitting the integration\nvariables qasqµ={qα\n∥, qi\n⊥}={q1∥, q2∥, qi\n⊥}with α= 1,2 and i= 1, ..., D −2. We have\ndDq=dD−2q⊥d2q∥and, using the standard center of mass parametrization\npµ\n1=−(E1,0, p,0), pµ\n2=−(E2,0,−p,0), (4.12)\nwith\nE1=m1(m1+m2σ)p\nm2\n1+m2\n2+ 2σm1m2, E 2=m2(m2+m1σ)p\nm2\n1+m2\n2+ 2σm1m2, (4.13)\n9p=m1m2√\nσ2−1p\nm2\n1+m2\n2+ 2m1m2σ, (4.14)\nwe have qα\n∥= (q0, q3) and hence the coefficients q1∥andq2∥of the decomposition (4.5) are\nexplicitly given by\nq0=q1∥E1\nm1+q2∥E2\nm2, q3=q1∥p\nm1−q2∥p\nm2, (4.15)\nand hence dq0dq3=√\nσ2−1dq∥1dq∥2so that\n˜Mµν\n5(pi, bi, k) =e−i(b1+b2)·k\n2\n4m1m2√\nσ2−1ZdD−2q⊥\n(2π)D−2dq1∥dq2∥\n(2π)2eib·qˆδ(q1∥−q∗\n1∥)ˆδ(q2∥−q∗\n2∥)\n× Mµν\n5\u0012\npi, q−k\n2,−q−k\n2, k\u0013\n=e−i(b1+b2)·k\n2\n4m1m2√\nσ2−1Zd2−2ϵq⊥\n(2π)2−2ϵeib·q⊥Mµν\n5\u0012\npi, q−k\n2,−q−k\n2, k\u0013\f\f\f\f\nq∗\n1∥,q∗\n2∥.(4.16)\nWe notice that the Fourier transform of the classical 5-point amplitude is, apart from the\noverall exponential (which can be however set to zero by choosing a frame where b1=−b2),\nis the same as the one involving the elastic amplitude in (3.1), but the integrand does not\nhave to be evaluated in q⊥but in ( q∗\ni∥, q⊥), where q∗\ni∥depend linearly on k∼ω.\nIn the following we use equation (4.16) to derive the soft expansion of the tree-level\n5-point amplitude.\nClassical 5-point amplitude and its soft expansion\nAn explicit expression of the classical tree-level 5-point amplitude is [4,6,76,129,159]\nM(0)µν\n5(pi, qi, k) =κ3(\n8(p1·kpµ\n2−p2·kpµ\n1)(p1·kpν\n2−p2·kpν\n1)\nq2\n1q2\n2\n+8p1·p2\"p2·k\np1·kpµ\n1pν\n1−p(µ\n1pν)\n2\nq2\n2+p1·k\np2·kpµ\n2pν\n2−p(µ\n1pν)\n2\nq2\n1−2p1·kp(µ\n2qν)\n1−p2·kp(µ\n1qν)\n1\nq2\n1q2\n2#\n+β\"\n−(q1·k)\n(p1·k)2q2\n2pµ\n1pν\n1−(q2·k)\n(p2·k)2q2\n1pµ\n2pν\n2+ 2 \np(µ\n1qν)\n1\n(p1·k)q2\n2−p(µ\n2qν)\n1\n(p2·k)q2\n1+qµ\n1qν\n1\nq2\n1q2\n2!#)\n,(4.17)\nwith β= 4m2\n1m2\n2(σ2−1\nD−2) and q1+q2+k= 0. Notice that the amplitude in (4.17) is gauge\ninvariant, kµM(0)µν\n5=kνM(0)µν\n5= 0. Using q1=q−k\n2andq2=−q−k\n2, we can expand the\n10previous amplitude around k= 0 up to the linear order in ω. We get\nM(0)µν\n5\u0012\npi, q−k\n2,−q−k\n2, k\u0013\n=M(0)µν\n5\f\f\f\nO(ω−1)(pi, q, k) +M(0)µν\n5\f\f\f\nO(ω0)(pi, q, k)\n+M(0)µν\n5\f\f\f\nO(ω)(pi, q, k) +O(ω2), (4.18)\nwith\nM(0)µν\n5\f\f\f\nO(ω−1)(pi, q, k) =−κ3β\nq2\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n(q·k)−2\u0012p(µ\n1qν)\np1·k\n−p(µ\n2qν)\np2·k\u0013\u0015\n, (4.19)\nM(0)µν\n5\f\f\f\nO(ω0)(pi, q, k) =κ3(\n8p1·p2p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2−2p(µ\n1pν)\n2\nq2+\n+β\u0014(q·k)2\nq4\u0012pµ\n1pν\n1\n(p1·k)2+pµ\n2pν\n2\n(p2·k)2\u0013\n−2(q·k)\nq4\u0012p(µ\n1qν)\np1·k+p(µ\n2qν)\np2·k\u0013\n+ 2qµqν\nq4\u0015)\n, (4.20)\nM(0)µν\n5\f\f\f\nO(ω)(pi, q, k) =κ3\u001a\n8[(p1·k)pν\n2−(p2·k)pν\n1][p1·k)pµ\n2−(p2·k)pµ\n1]\nq4\n+ 8p1·p2\u0014p1·k\np2·kpµ\n2pν\n2−p2·k\np1·kpµ\n1pν\n1\nq4(q·k)−2(p1·k)p(µ\n2qν)−(p2·k)p(µ\n1qν)\nq4\u0015\n−β\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013(q·k)3\nq6−2\u0012p(µ\n1qν)\np1·k−p(µ\n2qν)\np2·k\u0013(q·k)2\nq6\u0015)\n. (4.21)\nAs can be easily shown, these quantities can be obtained by acting with the soft operators\non the classical tree-level, elastic amplitude displayed in (3.6), and then taking the classical\nlimit\nM(0)µν\n5|O(ω−1)(pi, q, k) =κˆSµν\n(−1)M(0)\n4(pi, q), (4.22)\nM(0)µν\n5|O(ω0)(pi, q, k) =κˆSµν\n(0)M(0)\n4(pi, q), (4.23)\nM(0)µν\n5|O(ω)(pi, q, k) =κˆSµν\n(1)M(0)\n4(pi, q). (4.24)\n11Leading soft\nLet us start by considering the O(ω−1) contribution to the tree-level waveform in (4.16). It\nis clear that the only contribution is given by\n˜M(0)µν\n5|O(ω−1)(pi, bi, k) =1\n4m1m2√\nσ2−1Zd2−2ϵq⊥\n(2π)2−2ϵeib·q⊥M(0)µν\n5|O(ω−1)(pi, q⊥, k).(4.25)\nIn the following we will use [14]\nIµ1...µn\nϵ (b, ν) =Zd2−2ϵq⊥\n(2π)2−2ϵeiq⊥·bqµ1\n⊥. . . qµn\n⊥\n(q2\n⊥)ν\n= (−i)n ∂n\n∂bµ1. . . ∂b µnΓ(1−ϵ−ν)\n4νπ1−ϵΓ(ν)1\n(b2)1−ϵ−ν, (4.26)\nwith [160]\n∂n\n∂bµ1. . . ∂b µn1\n(b2)1−ϵ−ν=[n\n2]X\nm=02n−mΓ(ν+ϵ)\nΓ(1 + ϵ−n+m)1\n(b2)1−ϵ−ν−n+m{[P⊥]m[b]n−2m}µ1...µn,(4.27)\nwhere the symbol {[P⊥]m[b]n−2m}µ1...µncomprises all the symmetric structures one can con-\nstruct with P⊥andbhaving ncovariant indices, e.g.\n{[P⊥]0[b]1}µ1=bµ1, (4.28)\n{[P⊥]1[b]0}µ1µ2=Pµ1µ2\n⊥, (4.29)\n{[P⊥]1[b]1}µ1µ2µ3=Pµ1µ2\n⊥bµ3+Pµ2µ3\n⊥bµ1+Pµ3µ1\n⊥bµ2. (4.30)\nWe get\n˜M(0)µν\n5|O(ω−1)(pi, bi, k) =−2iκGNm1m2\nb2(2σ2−1)√\nσ2−1\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n(b·k)\n−2\u0012p(µ\n1bν)\np1·k−p(µ\n2bν)\np2·k\u0013\u0015\n, (4.31)\nreproducing the linear memory term.\nSubleading soft\nTheO(ω0) contribution to the waveform is more subtle to construct. In fact, it is not\nonly given by the Fourier transform of M(0)µν\n5|O(ω)in equation (4.20), but there are also\ntwo additional contributions. The first comes from the O(ω0) term of the expansion of the\n12leading soft factor computed in q= (q∗\n∥(ω), q⊥):\nM(0)µν\n5\f\f\f\nO(ω−1)(pi, q∗\n∥, q⊥k) =−κ3β\nq2\n⊥+q∗2\n∥\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n(q⊥·k+q∗\n∥·k)\n−2\u0012p(µ\n1(qν)\n⊥+q∗ν)\n∥)\np1·k−p(µ\n2(qν)\n⊥+q∗ν)\n∥)\np2·k\u0013\u0015\nO(ω0)=−κ3β\nq2\n⊥\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n(q∗\n∥·k)−2\u0012p(µ\n1q∗ν)\n∥\np1·k−p(µ\n2qν)q∗ν)\n∥\np2·k\u0013\u0015\n. (4.32)\nThe second, comes from the first order of expansion of the overall exponential e−i(b1+b2)·k\n2\nappearing in equation (4.16) that multiplies the leading soft factor\n−i(b1+b2)·k\n2M(0)µν\n5|O(ω−1)(pi, q, k), (4.33)\nwhere M(0)µν\n5|O(ω−1)is given in equation (4.19). Putting the above two equations together\nwith (4.20) gives a tensor\nTµν\n(0)(pi, q, k) =−κ3β\nq2\"\n(q∗\n∥·k∥)\n(p1·k)2pµ\n1pν\n1−pµ\n2pν\n2(q∗\n∥·k∥)\n(p2·k)2−2 \np(µ\n1q∗ν)\n∥\n(p1·k)−p(µ\n2q∗ν)\n∥\n(p2·k)!#\n+κ3(\n8p1·p2pµ\n1pν\n1p2·k\np1·k+pµ\n2pν\n2p1·k\np2·k−2p(µ\n1pν)\n2\nq2+\n+β\u0014(k·q)2\nq4\u0012pµ\n1pν\n1\n(p1·k)2+pµ\n2pν\n2\n(p2·k)2\u0013\n−2(q·k)\nq4\u0012p(µ\n1qν)\n(p1·k)+p(µ\n2qν)\n(p2·k)\u0013\n+ 2qµqν\nq4\u0015)\n+i(b1+b2)·k\n2κ3β\nq2\u0014\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n(k·q)−2\u0012p(µ\n1qν)\np1·k−p(µ\n2qν)\np2·k\u0013\u0015\n, (4.34)\nwhose Fourier transform is the O(ω0) contributions to the waveform ˜M(0)µν\n5. Performing\nsuch a Fourier transform of Tµν\n(0)gives two separate contributions. The first comes only from\nthe last two lines and it is\n˜M(0)µν\n5|O(ω0)(pi, bi, k) =−2κGNm1m2\nb22σ2−1√\nσ2−1\u0014\u0012b1·k\n(p1·k)2pµ\n1pν\n1−b2·k\n(p2·k)2pµ\n2pν\n2\u0013\n(b·k)\n−2\u0012b1·k\np1·kp(µ\n1bν)−b2·k\np2·kp(µ\n2bν)\u0013\n+bµbν\u0015\n. (4.35)\nThe second is infrared divergent and needs to be regularised\n˜M(0)µν\n5|O(log(ω))(pi, bi, k) =−κGNσ(2σ2−3)\n(σ2−1)3\n2\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2\n−pµ\n1pν\n2−pν\n1pµ\n2\u0013\nΓ(−ϵ)(b2)ϵ. (4.36)\n13We use the following regularisation\nΓ(−ϵ)(b2)ϵ−→ − 2 log( ωb), (4.37)\nor in other words we just replace the ϵ−1pole by a logarithm as\n1\n2ϵ−→log(ωb). (4.38)\nThe appearance of ωinside the argument of the logarithm makes it dimensionless. This can\nbe explained as follows. In fact, in the soft k∼ω→0 limit the divergent integrals over q⊥\nshould be more properly treated with the method of regions and sum the contributions of\nthe integrals over the two regions defined by ω≪q⊥∼b−1andω∼q⊥≪b−1 4. In the\nsum, the divergence cancels and the argument of the logarithm becomes dimensionless as it\nshould, giving eventually a term\n˜M(0)µν\n5|O(log(ω))(pi, bi, k) =2κGNσ(2σ2−3)\n(σ2−1)3\n2\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2\n−pµ\n1pν\n2−pν\n1pµ\n2\u0013\nlog(ωb), (4.39)\nreproducing the result in e.g.equation (8.96) of [114]. However, because of the regularisation\nprocedure, the O(ω0) term in (4.35) gets modified. The full O(ω0) contribution to ˜M(0)µν\n5\ncan be found in (4.16) of [137].\nSub-subleading soft\nSimilarly, we could derive the sub-subleading contribution to the waveform. We find again\na term proportional to ω\n˜M(0)µν\n5|O(ω)(pi, bi, k) =iκGNm1m2\nb22σ2−1√\nσ2−1\u0014\n(b·k)\u0012(b1·k)2\n(p1·k)2pµ\n1pν\n1−(b2·k)2\n(p2·k)2pµ\n2pν\n2\u0013\n+ 2\u0012\n(b1·k)p(µ\n1bν)\np1·k+ (b2·k)p(µ\n2bν)\np2·k\u0013\n+bµbν\u0015\n, (4.40)\nwhich gets however modified by the regularisation procedure for the ωlog(ωb) term, which\ninstead is universal and it reads\n˜M(0)µν\n5|O(ωlog(ω))(pi, bi, k) =−2iκG Nσ(2σ2−3)\n(σ2−1)3\n2\u0014\n(b1·k)p2·k\np1·kpµ\n1pν\n1+ (b2·k)p1·k\np2·kpµ\n2pν\n2\n−(b1·k+b2·k)p(µ\n1pν)\n2−(p2·k)p(µ\n1bν)+ (p1·k)p(µ\n2bν)\u0015\nlog(ωb). (4.41)\nThe last term is also in agreement with e.g.equation (4.13) of [137]. It also agrees with [147].\n4We thank Carlo Heissenberg for several clarifying discussions on this point.\n14We notice that equations (4.39) and (4.41) have a factorσ(2σ2−3)\n(σ2−1)3\n2. This is nothing, but the\nderivative with respect to the momenta, and therefore with respect to σ5, of the function\n2σ2−1√\nσ2−1appearing in the tree-level eikonal δ(0)given in equation (3.8). This suggests that\nthe tensorial structures considered in this section contributing to the subleading and sub-\nsubleading waveform could be obtained directly, and more easily, by acting with the soft\noperators ˆSµν\n(a)on the eikonal and that there should be an analogue of the soft theorem for\nthe (classical) amplitudes in bspace. In the next section we develop this idea and introduce\na classical version of the soft theorem that indeed involves the eikonal phase.\n5 Eikonal soft theorem\nSo far, we computed the soft expansion of the waveform out of the soft expansion of the\n5-point amplitude in momentum space computed in ( q∗\n1∥, q∗\n2∥). In this section we present a\nnew way to get the universal soft expansion of the classical waveform that passes through\nthe eikonal phase introduced earlier in section 3. Beside being computationally easier, the\nmethod we present here smoothly extends to the higher-loop case. We use the one-loop soft\ntheorem in (2.17) and apply it to the classical, resummed M4written as\nM4(Q)ˆδ(2p1·Q)ˆδ(2p2·Q) =iZ\nd4b e−ib·Q\u0012\n1−e2iδ(σ,b)\u0013\n≃ −iZ\nd4b e−ib·Qe2iδ(σ,b),(5.1)\nwhere Qis the classical momentum transfer and where in the last step in (5.1) we ignored\na term ∼ˆδ(4)(Q), which is irrelevant for our analysis.\nNotice however that, because of 5-point kinematics, M5depends on the variables Q1=\np1+p4andQ2=p2+p3, fixed by Q1+Q2=k, and having associated dual variables b1and\nb2. In the soft ω→0 limit Q1≃ −Q2≡Qand the 5-point kinematics becomes equal to the\n4-point one and in fact, M4only depends on Q. In order to restore the dependence on b1\nandb2onM5obtained through the soft theorem, we write the classical M4in a symmetric\nform with respect to the momenta\nM4(Q1, Q2)ˆδ(2p1·Q1)ˆδ(2p2·Q2)≃ −iZ\nd4b e−ib1·(p1+p4)−ib2·(p2+p4)eiδ(σ12,b)+iδ(σ34,b),(5.2)\nwhere we have used b·Q=b1·Q1+b2·Q2andQ1=p1+p4andQ2=p3+p4and\n5We use several time in the paper the formulae\n∂\n∂p1µf(σ12) =−1\nm1m2pµ\n2∂\n∂σ12f(σ12),∂\n∂p2µf(σ12) =−1\nm1m2pµ\n1∂\n∂σ12f(σ12),\n∂\n∂p3µf(σ34) =−1\nm1m2pµ\n4∂\n∂σ34f(σ34),∂\n∂p4µf(σ34) =−1\nm1m2pµ\n3∂\n∂σ34f(σ34).\n15δ(σ12, b)+δ(σ34, b) with\nδ(σ12, b) =−GNm2m22σ2\n12−1p\nσ2\n12−1log(µb) +O(G2\nN), (5.3)\nδ(σ34, b) =−GNm2m22σ2\n34−1p\nσ2\n34−1log(µb) +O(G2\nN), (5.4)\nwhich is the eikonal phase symmetrized with respect to incoming and outgoing momenta.\nWe then construct the soft waveform as in (2.17), but this time acting on the resummed\namplitude\n˜Mµν\n5(pi, pf, bi, k) =−iκZ\nd4Q eiQ·b\u001a\ne−W phase\u0014\nˆSµν\n(−1)+ˆSµν\n(0)+ˆSµν\n(1)+ (ˆSµν\n(0)Wreg)\n+4X\nj=1kρkσ\npj·k(ˆJµρ\njWreg)ˆJνσ\nj+1\n24X\nj=1kρkσ\npj·k(ˆJµρ\njWreg)(ˆJνσ\njWreg)\n+ (ˆSµν\n(1)Wreg)\u0015Z\nd4b′e−ib′\n1·(p1+p4)−ib′\n2·(p2+p4)eiδ(σ12,b′)+iδ(σ34,b′)\u001b\n+O(ω2, ω2logω). (5.5)\nThis procedure, beside immediately reproducing the tree-level results obtained earlier, it\nsmoothly extends to the one-loop case, which we present in detail here as the main result of\nthe present paper. We also write explicitly the symmetrized function Wregin (2.10)\nWreg≃ −iGNm1m2\u00122σ2\n12−1p\nσ2\n12−1log(ω−L) +2σ2\n34−1p\nσ2\n34−1log(ω+L)\u0013\n, (5.6)\nWphase≃2iGN(p3+p4)·klog(ω+R) =−2iGNEωlog(ω+R), (5.7)\nwhere ω±=ω±iϵand in the last step we used E=E1+E2and the center of mass\nparametrization in (4.12), (4.13) and (4.14) and k=ωnwith n= (1,sinθ,sinϕ,sinθcosϕ,cosθ).\nWe computed explicitly the right hand side in (5.5) and it yields6\n˜Mµν\n5(pi, pf) =e2iGNEωlog(ω+R)\b\nAµν\n(−1)(pi, pf) +Aµν\n(0)(pi, pf) +Aµν\n(1)(pi, pf)\n+Bµν\n(0)(pi) log( ω−b) +Bµν\n(0)(pf) log( ω+b) +Dµν\n(1)(pi) log2(ω−b) +Dµν\n(1)(pf) log2(ω+b)\n+ [Bµν\n(1)(pi) +Cµν\n(1)(pi)] log( ω−b) + [Bµν\n(1)(pf) +Cµν\n(1)(pf)] log( ω+b)\t\ne2iδ(σ,b)\n+O(ω2, ω2logω). (5.8)\nThe various tensorial structures Tµν\n(a)and their PM expansions are derived and listed in\nappendix A. The subscript ( a), with a=−1,0,1 indicates that Tµν\n(a)scales as ∼ωa. In\n6Here we simplify the notation ˜Mµν(pi, pf, bi, k)≡˜Mµν(pi, pf) since we are mainly interested in the\nperturbative expansion pf=pi+O(GN).\n16principle, the above formulae are valid at all orders in GN, depending on the PM order of\nthe classical momentum exchanged Qthat is used when we substitute p4=−p1+Qand\np3=−p2−Q. We find that the structures Aµν\n(a)that do not contain logarithms are equal to\nthe ones found in the previous section once we use p4=−p1+Qandp3=−p2−Qwith\nQ=Q(0)\nAµν\n(−1)(pi, pf) = ˜M(0)µν\n5|O(ω−1)(pi, bi, k) +O(κG2\nN), (5.9)\nAµν\n(0)(pi, pf) = ˜M(0)µν\n5|O(ω0)(pi, bi, k) +O(κG2\nN), (5.10)\nAµν\n(1)(pi, pf) = ˜M(0)µν\n5|O(ω)(pi, bi, k) +O(κG2\nN). (5.11)\nWhile in the previous section these structures were obtained by carefully keeping track and\nsumming several different contributions, in this approach they are more naturally computed\nas the action of the soft operators on the symmetrized Fourier exponent\nAµν\n(a)(pi, pf)e−i(p1+p4)·b1−i(p2+p3)·b2=−iκˆSµν\n(a)\u001a\ne−i(p1+p4)·b1−i(p2+p3)·b2\u001b\n. (5.12)\nLet us stress an important point here. In principle, the action of the various terms in (5.5)\nwill give terms proportional to log( µb), coming from the leading eikonal and to log( ω±L),\ncoming from Wreg. In this way, terms that are usually regarded as one-loop, combine in\na simple way with tree-level ones. We find that the coefficients of these logarithmic terms\nare equal and therefore the argument of the various logarithms becomes dimensionless and\nequal to ωb, as displayed in (5.8), and therefore the method of region is not needed within\nthis approach. We also note that all the tensors appearing in that equation are multiplied\nby an eikonal phase e2iδ. Just as the Fourier transform of the elastic 4-point amplitude in\nthe classical limit resums into an exponentiated structure, also does the 5-point one in the\nsoft limit. Of course the exponentiation should be checked order by order in GNby explicit\nperturbative computations. However, soft theorems predict such exponentiation in the soft\nlimit. In the remainder of this paper we will only consider terms that come from the first\norder of the exponential e2iδ≃1. Higher order terms in GNwill scale with smaller powers\nofℏand are superclassical.\nBefore proceeding further let us note that\nlog(ω±b) = log\u0010√\nω2+ϵ2b\u0011\n±iarctan\u0010ϵ\nω\u0011\n∼log(ωb)±iπ\n2, (5.13)\nand that [136]\ne2iGNEωlog(ω+R)=e2iGNEωlog(R\nb)e2iGNEωlog(ωb)\n=e2iGNEωlog(R\nb)(1 + 2 iGNEωlog(ω+b)−4G2\nNE2ω2log2(ω+b) +. . .). (5.14)\n17This allows to organize (5.8) into universal terms proportional to log( ωb),ωlog2(ωb) and\nωlog(ωb), in which we are interested. We have, including only these terms\n˜Mµν\n5(pi, pf)∼e2iGNEωlog(R\nb)\u0002˜Mµν\nlog(ωb)(pi, pf) log( ωb)\n+˜Mµν\nωlog2(ωb)(pi, pf)ωlog2(ωb) +˜Mµν\nωlog(ωb)(pi, pf)ωlog(ωb)\u0003\n, (5.15)\nwhere\n˜Mµν\nlog(ωb)(pi, pf) = 2 iGNEωAµν\n(−1)(pi, pf) +Bµν\n(0)(pi) +Bµν\n(0)(pf), (5.16)\n˜Mµν\nωlog2(ωb)(pi, pf) =−4G2\nNE2ω2Aµν\n(−1)(pi, pf) +Dµν\n(1)(pi) +Dµν\n(1)(pf)\n+ 2iEG Nω[Bµν\n(0)(pi) +Bµν\n(0)(pf)], (5.17)\n˜Mµν\nωlog(ωb)(pi, pf) =Bµν\n(1)(pi) +Bµν\n(1)(pf) + 2iEG NωAµν\n(0)(pi, pf)\n+iπ[Dµν\n(1)(pf)− Dµν\n(1)(pi)]−4iπG2\nNω2Aµν\n−1(pi, pf). (5.18)\nTheωlog(ωb) term should also contain the tensorial structure Cµν\n(1), however as clear from\nits explicit expression in (A.26), it is subleading in the classical limit. We stress again that\nthe above formulae are valid to all orders in GNbecause they depend on the perturbative\nexpansion of the exchanged classical momentum.\nNotice also that when we change the IR scale R→˜Rtheωlog(ωb) term gets modified\nas\n˜Mµν\nωlog(ωb)(pi, pf)−→ ˜Mµν\nωlog(ωb)(pi, pf) + 2iGNEωlog ˜R\nR!\n˜Mµν\nlog(ωb)(pi, pf). (5.19)\nApart from irrelevant overall factors, Eq. (5.16) reproduces the quantity Bµν−Cµνin Eqs.\n(1.7) and (1.8) and (5.17) reproduces the quantity Fµν−Gµνin Eqs. (1.9) and (1.10) of\nRef. [146].\n6 1PM and 2PM universal amplitude\nThe Fourier transform of the 5-point amplitude admits a PM expansion\n˜Mµν\n5(pi, pf) = ˜M(0)µν\n5(pi) +˜M(1)µν\n5(pi) +O(κG3\nN), (6.1)\nand, correspondingly, the universal terms of the waveform proportional to log( ωb),ωlog2(ωb)\nand to ωlog(ωb) that are displayed in equations (5.16), (5.17) and (5.18), can be PM ex-\npanded as\n˜Mµν\nωxlogy(ωb)(pi, pf) = ˜M(0)µν\nωxlogy(ωb)(pi) +˜M(1)µν\nωxlogy(ωb)(pi) +O(κG3\nN), (6.2)\nwith x= 0,1 and y= 1,2.\n181PM\nUsing the perturbative expansion shown in appendix A we get at 1PM\n˜M(0)µν\nlog(ωb)(pi) = 2 κGNσ(2σ2−3)\n(σ2−1)3\n2\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2−pµ\n1pν\n2−pν\n1pµ\n2\u0013\n, (6.3)\n˜M(0)µν\nωlog2(ωb)(pi) = 0 , (6.4)\n˜M(0)µν\nωlog(ωb)(pi) =−2iκG Nσ(2σ2−3)\n(σ2−1)3\n2\u0014\n(b1·k)p2·k\np1·kpµ\n1pν\n1+ (b2·k)p1·k\np2·kpµ\n2pν\n2\n−(b1·k+b2·k)pµ\n1pν\n2+pν\n1pµ\n2\n2−(p2·k)pµ\n1bν+pν\n1bµ\n2+ (p1·k)pµ\n2bν+pν\n2bµ\n2\u0015\n. (6.5)\nEquations (6.3) and (6.5) correctly reproduce the O(log(ωb)) and O(ωlog(ωb)) contributions\nto the leading waveform [114] and they match equations (4.39) and (4.41).\n2PM\nAt 2PM we get\n˜M(1)µν\nlog(ωb)(pi) = 2 κG2\nNm1m2\nb2(p1·k+p2·k)\u00122σ2−1√\nσ2−1\u0013\u0014\n(b·k)\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n−pµ\n1bν+pν\n1bµ\np1·k+pµ\n2bν+pν\n2bµ\np2·k\u0015\u0014\n2−σ(2σ2−3)\n(σ2−1)3\n2\u0015\n, (6.6)\n˜M(1)µν\nωlog2(ωb)(pi) = 4 iκG2\nN(p1·k+p2·k)σ(2σ2−3)\n(σ2−1)3\n2\u0014p2·k\np1·kpµ\n1pµ\n1+p1·k\np2·kpµ\n2pµ\n2\n−pµ\n1pν\n2−pν\n1pµ\n2\u0015\n, (6.7)\n˜M(1)µν\nωlog(ωb)(pi) = I˜M(1)µν\nωlog(ωb)(pi) + II˜M(1)µν\nωlog(ωb)(pi) + III˜M(1)µν\nωlog(ωb)(pi), (6.8)\n19with\nI˜M(1)µν\nωlog(ωb)(pi) = 2 iκG2\nNm1m2\nb2σ(2σ2−3)\n(σ2−1)3\n22σ2−1√\nσ2−1(p1·k+p2·k)\u001a\n(b·k)\u0014\n(b1·k)\n×pµ\n1pν\n1\n(p1·k)2−(b2·k)pµ\n2pν\n2\n(p2·k)2\u0015\n−(b1·k)pµ\n1bν+pν\n1bµ\np1·k+ (b2·k)pµ\n2bν+pν\n2bµ\np2·k+bµbν\u001b\n,(6.9)\nII˜M(1)µν\nωlog(ωb)(pi) =−4iκG2\nNm1m2\nb22σ2−1√\nσ2−1(p1·k+p2·k)\u001a\n(b·k)\u0014\n(b1·k)pµ\n1pν\n1\n(p1·k)2\n−(b2·k)pµ\n2pν\n2\n(p2·k)2\u0015\n−(b1·k)pµ\n1bν+pν\n1bµ\np1·k+ (b2·k)pµ\n2bν+pν\n2bµ\np2·k+bµbν\u001b\n, (6.10)\nIII˜M(1)µν\nωlog(ωb)(pi) =πκG2\nN\u0014σ(2σ2−3)\n(σ2−1)3\n2\u00152\n(p1·k+p2·k)\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2\n−pµ\n1pν\n2−pν\n1pµ\n2\u0013\n. (6.11)\nEquations (6.6) and (6.7) reproduce the O(log(ωb)) and O(ωlog2(ωb)) contributions to the\nsubleading waveform. In particular, we find that (6.6) and (6.7) are compatible with results\nin equation (9.9) of [136] and with (2.6) of [144] and with (2.5) of [147]. Notice that (6.6) is\napparently not compatible with what is stated in equation (4.11) of [137]. However, when\nexpressing the initial momenta p1andp2in terms of the variables ˜ p1and ˜p2introduced in\n(2.30) of [137] the sum of the 1PM contribution in (6.3) and the 2PM one in (6.6) is actually\nequal to (4.11) of [137]7. We notice that the O(ωlog2ω) term satisfies\n˜M(1)µν\nωlog2(ωb)(pi) = 2 iGN(p1·k+p2·k)˜M(0)µν\nlog(ωb)(pi) +O(G3\nN), (6.12)\nin agreement also with (4.12) of [137].\nNotice that equation (6.9) and (6.10) can be put together, yielding\nI˜M(1)µν\nωlog(ωb)(pi) + II˜M(1)µν\nωlog(ωb)(pi) =−2iκG2\nNm1m2\nb22σ2−1√\nσ2−1(p1·k+p2·k)\u001a\n(b·k)\u0014\n(b1·k)\n×pµ\n1pν\n1\n(p1·k)2−(b2·k)pµ\n2pν\n2\n(p2·k)2\u0015\n−(b1·k)pµ\n1bν+pν\n1bµ\np1·k+ (b2·k)pµ\n2bν+pν\n2bµ\np2·k+bµbν\u001b\n×\u0014\n2−σ(2σ2−3)\n(σ2−1)3\n2\u0015\n. (6.13)\nAs we discuss later, III˜M(1)µν\nωlog(ωb)is the expression resummed in the velocities of the first\nequation in (9.11) of [136]. It is also contained in equation (4.14) of [137]. Interestingly,\n7We thank the authors of [137] for pointing this out.\n20the last factor between square bracket in (6.13) is the same that appears in M(1)µν\nlog(ωb)in\n(6.6). As for the latter, we have checked that the sum of (6.5) and (6.13) is compatible\nwith what is stated in equations (4.15) and (4.16) of [137], once we express it in terms\nof the variables ˜ piandbedefined in (2.30) and (2.31) of the same reference. The terms\nin˜Mµν\nlog(ωb)and I˜Mµν\nωlog(ωb), having one factor ofσ(2σ2−3)\n(σ2−1)3\n2, come from the 2PM expansions\nof the tensors Bµν\n(a)(pi) +Bµν\n(a)(pf), with a= 0,1 whose 1PM contribution ( pf=pi) exactly\nreproduces the tree-level waveform. These terms are entirely predicted by the subleading and\nsub-subleading graviton soft factors ˆSµν\n(a)once we perform the PM expansion p4=−p1+Q(1)\nandp3=−p2−Q(1).\n7 PN expansion\nThe metric fluctuation can be obtained from the Fourier transform of the 5-point amplitude\nas\nWµν=2\nRGN\nκ˜Mµν\n5=W(0)µν+W(1)µν+O(G4\nN). (7.1)\nHere, following [136], we consider the contraction of the waveform with polarization vectors\nW=Wµνmµmν, mµ=1√\n2(ϵµ\nθ−iϵµ\nϕ), (7.2)\nwith ϵθ=∂θn,ϵϕ=∂ϕnandn= (1,sinθ,sinϕ,sinθcosϕ,cosθ). Explicitly\nmµ=1√\n2(0,cosθcosϕ+isinϕ,cosθsinϕ−icosθ,−sinθ), (7.3)\nso that m·k= 0. In the previous section we have given expressions for the logarithmic\nuniversal terms that hold for relativistic velocities. Here, in order to compare with existing\nliterature, we perform a PN expansion of the waveform in powers of p∞, related to vby\nv=p∞p\n1 +p2\n∞. (7.4)\nWe use a frame defined by\nbµ\n1=E2\nE(0, b,0,0), bµ\n2=E1\nE(0,−b,0,0), bµ=bµ\n1−bµ\n2= (0, b,0,0). (7.5)\n21Expanding the tree-level waveform gives\nW(0)\nlog(ωb)=−i2G2\nNM2ν\nR(cosϕ+icosθsinϕ)2\u00141\np∞−∆ sin θsinϕ+p∞\n4[\u0000\n2(3ν−1) sin2θ\n×cos 2ϕ+ (3ν−1) cos 2 θ+ν−7)] +∆p2\n∞\n2[2(2ν−1) sin2θsin2ϕ−2ν+ 5] sin θsinϕ\n+O(p3\n∞)\u0015\n, (7.6)\nW(0)\nωlog(ωb)=−2G2\nNbM2ν\nR\u0014\n−1\np2\n∞(cosθcosϕ+isinϕ)(cos θsinϕ−icosϕ)\n−∆\np∞sinθcosϕ(cosϕ+icosθsinϕ)2+1\n2(cosθsinϕ−icosϕ) [3 cos θcosϕ\n+ sin ϕ(3i+ (3ν−1)(cos θsin 2ϕ−2icosϕ2) sin2θ)] +O(p∞)\u0015\n. (7.7)\nFor the subleading waveform we get\nW(1)\nlog(ωb)=−2G3\nNM3ν\nbR\u00142\np3\n∞(cosθcosϕ+isinϕ)(cos ϕ+icosθsinϕ) +∆\n2p2\n∞sinθ\n×[4 cos θsin3ϕ+ 2icos3ϕ−8 cos θsinϕcos2ϕ−i(3 cos 2 θ+ 7) sin2ϕcosϕ]\n+1\n2p∞(cosϕ+icosθsinϕ)[cos θ\u0000\ncosϕ((3ν−1) cos 2 θ+ 3ν+ 1) + 2(3 ν−1) sin2θcos 3ϕ\u0001\n+i\u0000\n2(3ν−1) sin2θsin 3ϕ+ sin ϕ((3ν−1) cos 2 θ+ 3ν+ 1)\u0001\n] +O(p0\n∞)\u0015\n, (7.8)\nW(1)\nωlog2(ωb)=−2G3\nNM3ν\nR(cosϕ+icosθsinϕ)2\u0014\n−2\np∞+ 2∆ sin θsinϕ−p∞\n2[2(3ν−1)\n×sin2θcos 2ϕ+ (3ν−1) cos 2 θ+ 3ν−7] +O(p2\n∞)\u0015\n, (7.9)\n22for the universal terms. For the non-universal ones we find\nIW(1)\nωlog(ωb)=−i2G3\nNM3ν\nR\u00141\np4\n∞(cosθcosϕ+isinϕ)2+2i∆\np3\n∞cosϕsinθ(cosθcosϕ+isinϕ)\n×(cosϕ+icosθsinϕ) +1\n16p2\n∞[2(1 + 9 ν+ (3−5ν) cos 2 θ) cos 2 ϕ−4icosθ(−1−5ν\n+ (3ν−1) cos 2 θ) sin 2 ϕ+ sin θ2(−3−7ν+ 9(3 ν−1) cos 4 ϕ+ 6(1−3ν) cos 2 θsin22ϕ)\n+ 6i(3ν−1) sin θsin 2θsin 4ϕ]−∆\np∞cosϕsinθ(cosϕ+icosθsinϕ)[(ν\n+ 2(2 ν−1) cos 2 ϕsin2θ) sinϕ−icosθcosϕ(ν+ 4(1−2ν) sin2θsin2ϕ)] +O(p0\n∞)\u0015\n,(7.10)\nIIW(1)\nωlog(ωb)=−i2G3\nNM3ν\nR\u00142\np∞(cosθcosϕ+isinϕ)2+ 4i∆ sin θcosϕ(cosθcosϕ+isinϕ)\n×(cosϕ+icosθsinϕ) +p∞\n8[2 cos 2 ϕ(10 + 9 ν+ (6−5ν) cos 2 θ)−4icosθsin 2ϕ(\n(3ν−1) cos 2 θ−7−5ν) + sin2θ\u0000\n6(1−3ν) cos 2 θsin22ϕ+ 9(3 ν−1) cos 4 ϕ−7ν\n−15) + 6 i(3ν−1) sin θsin 2θsin 4ϕ] +O(p2\n∞)\u0015\n, (7.11)\nIIIW(1)\nωlog(ωb)=π2G3\nNM3ν\nR(cosϕ+icosθsinϕ)2\u0014\n−1\n2p4\n∞+∆\n2p3\n∞sinθsinϕ\n−1\n16p2\n∞[4(3ν−1) sin2θcos 2ϕ+ (6ν−2) cos 2 θ+ 6ν−26]−∆\n4p∞[sinθsinϕ(2(2ν−1)\n×sin2θsin2ϕ−3ν+ 8)−1\n16[8(5(ν−1)ν+ 1) sin4θsin4ϕ−4(9 + ν(9ν−31) sin2θsin2ϕ\n+ν(3ν−47) + 15] + O(p∞)\u0015\n, (7.12)\nwhere ν=m1m2/M2is the reduced mass, M=m1+m2is the total mass and ∆ =\n(m1−m2)/Mthe mass difference. We notice that (7.12), which eventually comes from the\nargument in (5.13), matches the first equation of (9.11) of [136]. On the other hand, choosing\nappropriately the factor log\u0010\n˜R\nR\u0011\nin (5.19) we could reproduce the terms in the PN expansion\ndisplayed in the last two equations in (9.11) of [136] proportional to γEandπ.\n238 Conclusions\nThe tree-level soft graviton theorem follows from gauge invariance and fixes the soft graviton\nbehaviour of the amplitude (in momentum space), involving any number of massive scalar\nparticles and a graviton, in terms of the amplitude without the graviton and of a universal\nsoft factor. We have applied it to the case of the scattering of two massive particles with the\nemission of a graviton and we have then shown that, going to impact parameter space, one\nmust regularise the result either with dimensional regularisation or with an infrared cutoff\nthat, when taken to be equal to the graviton energy ω, generates O(logω) and O(ωlogω)\nuniversal terms.\nWe have shown that the previous calculation can be significantly simplified if we apply the\nsoft factor to the eikonal resummed four-point amplitude instead of the tree-leve one. We use\nthis framework to discuss also the one-loop corrected classical soft graviton behaviour that\ninvolves the two functions WregandWphasethat appear in various approches [142–148,151] to\nthis problem, starting from the original paper by Weinberg [1]. In this way, we compute both\nthe 1PM and 2PM O(logω),O(ωlog2ω) andO(ωlogω) contributions to the waveform. The\n2PM sub-subleading term of order O(ωlogω) is not universal and its not entirely captured\nby the soft graviton theorem. Here we display explicitly its universal part. We have finally\ncompared our results with those present in the literature. In particular, we find perfect\nagreement with Refs. [142–148, 151]. After having mapped the initial momenta piused in\nour analysis to the ones ˜ piused in [137], we find agreement also with this last reference.\nMore precisely we get a universal term that agrees with the one obtained in Ref. [137], but,\nwith our method, we do not get extra non-universal terms present in [137], for which it seems\nthat the full one-loop computation needs to be performed.\nAcknowledgements\nWe thank Carlo Heissenberg, Raffaele Marotta, Rodolfo Russo and Gabriele Veneziano for\nmany useful discussions and Laura Donnay, Kevin Nguyen, Romain Ruzziconi and Biswajit\nSahoo for useful correspondence. The research of FA (PDV) is fully (partially) supported\nby the Knut and Alice Wallenberg Foundation under grant KAW 2018.0116.\n24A Action of soft operators\nIn the following we define M4(Q1, Q2)ˆδ(2p1·Q1)ˆδ(2p2·Q2)≡ M 4(pi, pf). Then, the action\nof the soft operators ˆSµν\n(a)in (2.4),(2.5),(2.6) in (5.5) is\nκˆSµν\n(−1)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b)Aµν\n(−1)(pi, pf), (A.1)\nκˆSµν\n(0)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){Aµν\n(0)(pi, pf) + [Bµν\n(0)(pi)\n+Bµν\n(0)(pf)] log b}, (A.2)\nκˆSµν\n(1)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){Aµν\n(1)(pi, pf) + [Bµν\n(1)(pi)\n+Bµν\n(1)(pf) +Cµν\n(1)(pi) +Cµν\n(1)(pf)] log b+ [Dµν\n(1)(pi) +Dµν\n(1)(pf)] log2b}, (A.3)\nand that of the additional terms in the one-loop soft theorem in (2.17) involving Wregis\nκ4X\nj=1p(µ\njkρ\npj·k(ˆJν)ρ\njWreg)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){\nBµν\n(0)(pi) logω−+Bµν\n(0)(pf) logω+}, (A.4)\nκ4X\nj=1kρkσ\npj·k(ˆJ(µρ\njWreg)ˆJν)σ\njM4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){\nBµν\n(1)(pi) logω−+Bµν\n(1)(pf) logω++ 2[Dµν\n(1)(pi) logω−+Dµν\n(1)(pf) logω+] logb}, (A.5)\nκ\n24X\nj=1kρkσ\npj·k(ˆJµρ\njWreg)(ˆJνσ\njWreg)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){\nDµν\n(1)(pi) log2ω−+Dµν\n(1)(pf) log2ω+}, (A.6)\nκ\n24X\nj=1kρkσ\npj·k(ˆJµρ\njˆJνσ\njWreg)M4(pi, pf) =Z\nd4be−iQ1·b1−iQ2·b2eiδ(pi,b)+iδ(pf,b){\nCµν\n(1)(pi) logω−+Cµν\n(1)(pf) logω+}. (A.7)\nIn the above computation we dropped the irrelevant scales µandLfrom the arguments of\nthe logarithms. When summing the above contributions we see that each term proportional\nto log band to log ω±can be put together to yield log( ω±b). Each tensorial structure Tµν\n(a)\n25scales as ωa. The ones that do not contain logarithmic terms are given by\nAµν\n(−1)(pi, pf) =−iκ4X\nj=1pµ\nipν\ni\npi·k, (A.8)\nAµν\n(0)(pi, pf) =−κ\u0014\n(b1·k)\u0012pµ\n1pν\n1\np1·k+pµ\n4pν\n4\np4·k\u0013\n+ (b2·k)\u0012pµ\n2pν\n2\np2·k+pµ\n3pν\n3\np3·k\u0013\n−pµ\n1bν\n1+pν\n1bµ\n1\n2−pµ\n4bν\n1+pν\n4bµ\n1\n2−pµ\n2bν\n2+pν\n2bµ\n2\n2−pµ\n3bν\n2+pν\n3bµ\n2\n2\u0015\n, (A.9)\nAµν\n(1)(pi, pf) =iκ\n2\u0014\n(b1·k)2\u0012pµ\n1pν\n1\np1·k+pµ\n4pν\n4\np4·k\u0013\n+ (b2·k)2\u0012pµ\n2pν\n2\np2·k+pµ\n3pν\n3\np3·k\u0013\n−(b1·k)(pµ\n1bν\n1+pν\n1bµ\n1+pµ\n4bν\n1+pν\n4bµ\n1)−(b2·k)(pµ\n2bν\n2+pν\n2bµ\n2+pµ\n3bν\n2+pν\n3bµ\n2)\u0015\n. (A.10)\nUsing momentum conservation as p4=−p1+Qandp3=−p2−Q, with\nQµ=−2GNm1m22σ2−1√\nσ2−1bµ\nb2+O(G2\nN), (A.11)\nwe get at linear order in GN\nAµν\n(−1)(pi) =−2iκGNm1m2\nb22σ2−1√\nσ2−1\u0014\n(b·k)\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n−pµ\n1bν+pν\n1bµ\np1·k\n+pµ\n2bν+pν\n2bµ\np2·k\u0015\n+O(G2\nN), (A.12)\nAµν\n(0)(pi) =−2κGNm1m2\nb22σ2−1√\nσ2−1\u001a\n(b·k)\u0014\n(b1·k)pµ\n1pν\n1\n(p1·k)2−(b2·k)pµ\n2pν\n2\n(p2·k)2\u0015\n−(b1·k)pµ\n1bν+pν\n1bµ\np1·k+ (b2·k)pµ\n2bν+pν\n2bµ\np2·k+bµbν\u001b\n+O(G2\nN), (A.13)\nAµν\n(1)(pi) =iκGNm1m2\nb22σ2−1√\nσ2−1\u001a\n(b·k)\u0014\n(b1·k)2pµ\n1pν\n1\n(p1·k)2−(b2·k)2pµ\n2pν\n2\n(p2·k)2\u0015\n−(b1·k)2pµ\n1bν+pν\n1bµ\np1·k+ (b2·k)2pµ\n2bν+pν\n2bµ\np2·k+ (b1·k+b2·k)bµbν\u001b\n+O(G2\nN).(A.14)\nNotice that, after having imposed momentum conservation, the previous tensors satisfy gauge\ninvariance kµAµν\n(a)=kνAµν\n(a)= 0. Furthermore, notice that they scale as κGN, which is the\ncorrect coupling for a 5-point amplitude at tree-level.\nThe terms comprising log ωare\nBµν\n(0)(pi) =κGNσ(2σ2−3)\n(σ2−1)3\n2\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2−pµ\n1pν\n2−pν\n1pµ\n2\u0013\n, (A.15)\nBµν\n(0)(pf) =κGNσ(2σ2−3)\n(σ2−1)3\n2\u0012p4·k\np3·kpµ\n3pν\n3+p3·k\np4·kpµ\n4pν\n4−pµ\n3pν\n4−pν\n3pµ\n4\u0013\n, (A.16)\n26together with\nBµν\n(1)(pi) =−iκG Nσ(2σ2−3)\n(σ2−1)3\n2\u0014\n(b1·k)p2·k\np1·kpµ\n1pν\n1+ (b2·k)p1·k\np2·kpµ\n2pν\n2 (A.17)\n−(b1·k+b2·k)pµ\n1pν\n2+pν\n1pµ\n2\n2−(p2·k)pµ\n1bν+pν\n1bµ\n2+ (p1·k)pµ\n2bν+pν\n2bµ\n2\u0015\n, (A.18)\nBµν\n(1)(pf) =−iκG Nσ(2σ2−3)\n(σ2−1)3\n2\u0014\n(b2·k)p4·k\np3·kpµ\n3pν\n3+ (b1·k)p4·k\np3·kpµ\n4pν\n4\n−(b1·k+b2·k)pµ\n3pν\n4+pν\n3pν\n4\n2+ (p4·k)pµ\n3bν+pν\n3bµ\n2+ (p3·k)pµ\n4bν+pν\n4bµ\n2\u0015\n, (A.19)\nCµν\n(1)(pi) =−κGN3(p1·k+p2·k)\n2m1m2(σ2−1)5\n2\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2\n−pµ\n1pν\n2+pν\n1pµ\n2\u0013\n, (A.20)\nCµν\n(1)(pf) =−κGN3(p3·k+p4·k)\n2m1m2(σ2−1)5\n2\u0012p4·k\np3·kpµ\n3pν\n3+p3·k\np4·kpµ\n4pν\n4\n−pµ\n3pν\n4+pν\n3pµ\n4\u0013\n, (A.21)\nand with\nDµν\n(1)(pi) =iκ\n2G2\nN\u0014σ(2σ2−3)\n(σ2−1)3\n2\u00152\n(p1·k+p2·k)\u0012p2·k\np1·kpµ\n1pν\n1+p1·k\np2·kpµ\n2pν\n2\n−pµ\n1pν\n2−pν\n1pµ\n2\u0013\n, (A.22)\nDµν\n(1)(pf) =iκ\n2G2\nN\u0014σ(2σ2−3)\n(σ2−1)3\n2\u00152\n(p3·k+p4·k)\u0012p4·k\np3·kpµ\n3pν\n3+p3·k\np4·kpµ\n4pν\n4\n−pµ\n3pν\n4−pν\n3pµ\n4\u0013\n. (A.23)\nAll the tensorial structures appearing above automatically satisfy gauge invariance kµTµν\n(a)=\n27kνTµν\n(a)= 0. We can PM expand the tensors that depend on the outgoing momenta pfas\nBµν\n(0)(pf) =Bµν\n(0)(pi)−2κG2\nNm1m2\nb2(p1·k+p2·k)σ(2σ2−3)\n(σ2−1)3\n22σ2−1√\nσ2−1\n×\u0014\n(b·k)\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n−pµ\n1bν+pν\n1bµ\np1·k+pµ\n2bν+pν\n2bµ\np2·k\u0015\n+O(G3\nN), (A.24)\nBµν\n(1)(pf) =Bµν\n(1)(pi) + 2iκG2\nNm1m2\nb2σ(2σ2−3)\n(σ2−1)3\n22σ2−1√\nσ2−1(p1·k+p2·k)\u0014\n(b·k)\u0012\n(b1·k)pµ\n1pν\n1\n(p1·k)2−(b2·k)pµ\n2pν\n2\n(p2·k)2\u0013\n−(b1·k)pµ\n1bν+pν\n1bµ\np1·k\n+ (b2·k)pµ\n2bν+pν\n2bµ\np2·k+bµbν\u0015\n+O(G3\nN), (A.25)\nCµν\n(1)(pf) =−Cµν\n(1)(pi)−κG2\nN3\n(σ2−1)3(p1·k+p2·k)2\nb2\u0014\n(b·k)\u0012pµ\n1pν\n1\n(p1·k)2−pµ\n2pν\n2\n(p2·k)2\u0013\n−\u0012pµ\n1bν+pν\n1bµ\np1·k−pµ\n2bν+pν\n2bµ\np2·k\u0013\u0015\n+O(G3\nN), (A.26)\nDµν\n(1)(pf) =−Dµν\n(1)(pi) +O(G3\nN). (A.27)\nB Classical limit of the infrared divergent factor\nIn this appendix we extract the classical limit of the infrared divergent factor Win (2.8) that\nappears at one-loop level and that, once exponentiated, takes care of all infrared divergences\nat arbitrary perturbative order. It is a one-loop diagram involving two massive propagators\nand a graviton propagator that is attached to the two massive lines.\nIt was originally computed by Weinberg [1] and it is given by\nW=G\nπlogλ\nΛNX\nj,k=1mjmkηjηk\u0012\nσ2\njk−1\n2\u0013log\u0010\nσjk+q\nσ2\njk−1\u0011\n−iπδηjηk,1\nq\nσ2\njk−1(B.1)\nwhere Λ ( λ) is an ultraviolet (infrared) cutoff and the sums are extended over the total\nnumber Nof particles participating in the scattering. For j=kthe internal massless line\nconnects two points of the same massive line and therefore it does not contribute to the\nclassical amplitude. In this case, the correct procedure [161] is to simply drop the divergent\nimaginary part and to compute the real part for σjj→1. For these terms of the sum with\nj=kone gets justm2\nj\n2.\nEquation (B.1) has also been computed using dimensional regularisation (for instance in\nRef. [162]) and the two results match each other with the substitution\nlogλ\nΛ−→1\n2ϵ, (B.2)\n28similarly to (4.38). By rewriting (B.1) in the following form\nW=G\nπlogλ\nΛNX\nj,k=1(pj·pk)2−1\n2m2\njm2\nkq\n(pj·pk)2−m2\njm2\nk\"\nηjηk\n2log−ηjηkpj·pk+q\n(pj·pk)2−m2\njm2\nk\n−ηjηkpj·pk−q\n(pj·pk)2−m2\njm2\nk\n−iπδηjηk,1#\n, (B.3)\nwhere the quantity −ηjηkpj·pkis always positive, we can take the limit where the particle\nwith j, k=Nis massless. In this case we get the same expression as in (B.3) where, however,\nthe indices j, kgo from 1 to N−1 and an additional term, containing the momentum pN=k,\ngiven by [137]\nWphase=G\nπlogλ\nΛN−1X\nj=12(−pj·k)\u00141\n2log4(pj·k)2\nm2\njµ2−iπδηjηN,1\u0015\n. (B.4)\nAs explained in Ref. [137] the terms with log mNcancel because of momentum conservation\nleaving behind an arbitrary energy scale µ.\nIn the final part of this appendix we show that the terms with the logarithms in (B.3)\nand (B.4) are negligible in the classical limit.\nLet us consider for simplicity the case with N= 4 with two incoming and two outgo-\ning massive particles in (B.3) with mass m1andm2respectively. We have four types of\nlogarithmic terms in (B.3). Two of them, corresponding to ( jk) = (12)(21)(34)(43) and\n(jk) = (13)(31)(24)(42), can be respectively written in terms of the Mandelstam variables s\nandu. Since, at t∼0,s−m2\n1−m2\n2=−(u−m2\n1−m2\n2), one can see that their sum is pro-\nportional to tand vanishes for t= 0. Four terms, corresponding to ( jk) = (11)(22)(33)(44),\ngive a contribution to the sum equal to m2\n1+m2\n2. The remaining four terms, corresponding\nto (jk) = (14)(41)(23)(32), can be written in terms of the Mandelstam variable tand for\nsmall tthey give a contribution equal to m2\n1+m2\n2plus terms of order t. In conclusion, the\nsum of the 16 terms behaves as t, for small t, that is of the order ℏ2and therefore negligible\nin the classical limit.\nIn order to show that also the sum of logarithmic terms in (B.4) are quantum one intro-\nduces the two quantities Q1=p1+p4andQ2=p2+p3and rewrites the sum in terms of\np1, p2, Q1andQ2obtaining a quantity proportional to Q1andQ2for small Q1andQ2that\nis of the order ℏand therefore negligible in the classical limit.\nNotice that, with the choice λ=ω, (B.3) is equal to Eq. (1.5) of [148] except for the\nsum where the terms j=kare excluded and the second term of (B.4) corresponds to the\nquantity in Eq. (1.6) of [148] apart from some signs probably due to different conventions.\nReferences\n[1] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140(1965) B516–B524.\n29[2] A. Buonanno and T. Damour, “Effective one-body approach to general relativistic\ntwo-body dynamics,” Phys. Rev. D 59(1999) 084006, arXiv:gr-qc/9811091 .\n[3] W. D. Goldberger and I. Z. Rothstein, “An Effective field theory of gravity for\nextended objects,” Phys. Rev. D 73(2006) 104029, arXiv:hep-th/0409156 .\n[4] W. D. Goldberger and A. K. Ridgway, “Radiation and the classical double copy for\ncolor charges,” Phys. Rev. D 95no. 12, (2017) 125010, arXiv:1611.03493 [hep-th] .\n[5] F. Cachazo and A. Guevara, “Leading Singularities and Classical Gravitational\nScattering,” JHEP 02(2020) 181, arXiv:1705.10262 [hep-th] .\n[6] A. Luna, I. Nicholson, D. O’Connell, and C. D. White, “Inelastic Black Hole\nScattering from Charged Scalar Amplitudes,” JHEP 03(2018) 044,\narXiv:1711.03901 [hep-th] .\n[7] N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Plant´ e, and P. Vanhove,\n“General Relativity from Scattering Amplitudes,” Phys. Rev. Lett. 121no. 17,\n(2018) 171601, arXiv:1806.04920 [hep-th] .\n[8] C. Cheung, I. Z. Rothstein, and M. P. Solon, “From Scattering Amplitudes to\nClassical Potentials in the Post-Minkowskian Expansion,” Phys. Rev. Lett. 121\nno. 25, (2018) 251101, arXiv:1808.02489 [hep-th] .\n[9] D. A. Kosower, B. Maybee, and D. O’Connell, “Amplitudes, Observables, and\nClassical Scattering,” JHEP 02(2019) 137, arXiv:1811.10950 [hep-th] .\n[10] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Scattering\nAmplitudes and the Conservative Hamiltonian for Binary Systems at Third\nPost-Minkowskian Order,” Phys. Rev. Lett. 122no. 20, (2019) 201603,\narXiv:1901.04424 [hep-th] .\n[11] Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P. Solon, and M. Zeng, “Black Hole\nBinary Dynamics from the Double Copy and Effective Theory,” JHEP 10(2019)\n206, arXiv:1908.01493 [hep-th] .\n[12] A. Brandhuber and G. Travaglini, “On higher-derivative effects on the gravitational\npotential and particle bending,” JHEP 01(2020) 010, arXiv:1905.05657 [hep-th] .\n[13] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, “Note on the\nabsence of R2corrections to Newton’s potential,” Phys. Rev. D 101no. 4, (2020)\n046011, arXiv:1911.10108 [hep-th] .\n[14] A. Koemans Collado, P. Di Vecchia, and R. Russo, “Revisiting the second\npost-Minkowskian eikonal and the dynamics of binary black holes,” Phys. Rev. D\n100no. 6, (2019) 066028, arXiv:1904.02667 [hep-th] .\n30[15] A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove,\n“Post-Minkowskian Hamiltonians in general relativity,” Phys. Rev. D 100no. 8,\n(2019) 084040, arXiv:1906.01579 [hep-th] .\n[16] N. E. J. Bjerrum-Bohr, A. Cristofoli, and P. H. Damgaard, “Post-Minkowskian\nScattering Angle in Einstein Gravity,” JHEP 08(2020) 038, arXiv:1910.09366\n[hep-th] .\n[17] C. Cheung and M. P. Solon, “Classical gravitational scattering at O(G3) from\nFeynman diagrams,” JHEP 06(2020) 144, arXiv:2003.08351 [hep-th] .\n[18] J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Extremal black hole scattering at\nO(G3): graviton dominance, eikonal exponentiation, and differential equations,”\nJHEP 11(2020) 023, arXiv:2005.04236 [hep-th] .\n[19] A. Brandhuber, G. Chen, G. Travaglini, and C. Wen, “Classical gravitational\nscattering from a gauge-invariant double copy,” JHEP 10(2021) 118,\narXiv:2108.04216 [hep-th] .\n[20] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and\nM. Zeng, “Scattering Amplitudes and Conservative Binary Dynamics at O(G4),”\nPhys. Rev. Lett. 126no. 17, (2021) 171601, arXiv:2101.07254 [hep-th] .\n[21] A. Cristofoli, P. H. Damgaard, P. Di Vecchia, and C. Heissenberg, “Second-order\nPost-Minkowskian scattering in arbitrary dimensions,” JHEP 07(2020) 122,\narXiv:2003.10274 [hep-th] .\n[22] M. Accettulli Huber, A. Brandhuber, S. De Angelis, and G. Travaglini, “From\namplitudes to gravitational radiation with cubic interactions and tidal effects,” Phys.\nRev. D 103no. 4, (2021) 045015, arXiv:2012.06548 [hep-th] .\n[23] L. de la Cruz, B. Maybee, D. O’Connell, and A. Ross, “Classical Yang-Mills\nobservables from amplitudes,” JHEP 12(2020) 076, arXiv:2009.03842 [hep-th] .\n[24] A. Cristofoli, R. Gonzo, D. A. Kosower, and D. O’Connell, “Waveforms from\namplitudes,” Phys. Rev. D 106no. 5, (2022) 056007, arXiv:2107.10193 [hep-th] .\n[25] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Radiative classical\ngravitational observables at O(G3) from scattering amplitudes,” JHEP 10(2021)\n148, arXiv:2104.03957 [hep-th] .\n[26] A. Cristofoli, R. Gonzo, N. Moynihan, D. O’Connell, A. Ross, M. Sergola, and C. D.\nWhite, “The Uncertainty Principle and Classical Amplitudes,” arXiv:2112.07556\n[hep-th] .\n[27] T. Damour, “High-energy gravitational scattering and the general relativistic\ntwo-body problem,” Phys. Rev. D 97no. 4, (2018) 044038, arXiv:1710.10599\n[gr-qc] .\n31[28] E. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Gravitational\nBremsstrahlung from Reverse Unitarity,” Phys. Rev. Lett. 126no. 20, (2021) 201602,\narXiv:2101.07255 [hep-th] .\n[29] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Universality of\nultra-relativistic gravitational scattering,” Phys. Lett. B 811(2020) 135924,\narXiv:2008.12743 [hep-th] .\n[30] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Radiation Reaction from\nSoft Theorems,” Phys. Lett. B 818(2021) 136379, arXiv:2101.05772 [hep-th] .\n[31] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “The eikonal approach to\ngravitational scattering and radiation at O(G3),”JHEP 07(2021) 169,\narXiv:2104.03256 [hep-th] .\n[32] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “The eikonal operator at\narbitrary velocities I: the soft-radiation limit,” JHEP 07(2022) 039,\narXiv:2204.02378 [hep-th] .\n[33] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “Classical Gravitational\nObservables from the Eikonal Operator,” arXiv:2210.12118 [hep-th] .\n[34] N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plant´ e, and P. Vanhove, “Classical\ngravity from loop amplitudes,” Phys. Rev. D 104no. 2, (2021) 026009,\narXiv:2104.04510 [hep-th] .\n[35] N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plant´ e, and P. Vanhove, “The amplitude\nfor classical gravitational scattering at third Post-Minkowskian order,” JHEP 08\n(2021) 172, arXiv:2105.05218 [hep-th] .\n[36] N. E. J. Bjerrum-Bohr, L. Plant´ e, and P. Vanhove, “Post-Minkowskian radial action\nfrom soft limits and velocity cuts,” JHEP 03(2022) 071, arXiv:2111.02976\n[hep-th] .\n[37] Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H. Shen, M. P. Solon, and\nM. Zeng, “Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics\nat O(G4),” Phys. Rev. Lett. 128no. 16, (2022) 161103, arXiv:2112.10750\n[hep-th] .\n[38] G. K¨ alin and R. A. Porto, “Post-Minkowskian Effective Field Theory for\nConservative Binary Dynamics,” JHEP 11(2020) 106, arXiv:2006.01184\n[hep-th] .\n[39] G. K¨ alin, Z. Liu, and R. A. Porto, “Conservative Dynamics of Binary Systems to\nThird Post-Minkowskian Order from the Effective Field Theory Approach,” Phys.\nRev. Lett. 125no. 26, (2020) 261103, arXiv:2007.04977 [hep-th] .\n32[40] C. Dlapa, G. K¨ alin, Z. Liu, and R. A. Porto, “Dynamics of binary systems to fourth\nPost-Minkowskian order from the effective field theory approach,” Phys. Lett. B 831\n(2022) 137203, arXiv:2106.08276 [hep-th] .\n[41] C. Dlapa, G. K¨ alin, Z. Liu, and R. A. Porto, “Conservative Dynamics of Binary\nSystems at Fourth Post-Minkowskian Order in the Large-Eccentricity Expansion,”\nPhys. Rev. Lett. 128no. 16, (2022) 161104, arXiv:2112.11296 [hep-th] .\n[42] G. K¨ alin, J. Neef, and R. A. Porto, “Radiation-reaction in the Effective Field Theory\napproach to Post-Minkowskian dynamics,” JHEP 01(2023) 140, arXiv:2207.00580\n[hep-th] .\n[43] C. Dlapa, G. K¨ alin, Z. Liu, J. Neef, and R. A. Porto, “Radiation Reaction and\nGravitational Waves at Fourth Post-Minkowskian Order,” Phys. Rev. Lett. 130\nno. 10, (2023) 101401, arXiv:2210.05541 [hep-th] .\n[44] G. Mogull, J. Plefka, and J. Steinhoff, “Classical black hole scattering from a\nworldline quantum field theory,” JHEP 02(2021) 048, arXiv:2010.02865 [hep-th] .\n[45] G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, “All things retarded:\nradiation-reaction in worldline quantum field theory,” JHEP 10(2022) 128,\narXiv:2207.00569 [hep-th] .\n[46] M. Khalil, A. Buonanno, J. Steinhoff, and J. Vines, “Energetics and scattering of\ngravitational two-body systems at fourth post-Minkowskian order,” Phys. Rev. D\n106no. 2, (2022) 024042, arXiv:2204.05047 [gr-qc] .\n[47] C. R. T. Jones and M. Solon, “Scattering Amplitudes and N-Body Post-Minkowskian\nHamiltonians in General Relativity and Beyond,” arXiv:2208.02281 [hep-th] .\n[48] D. Bini and T. Damour, “Radiation-reaction and angular momentum loss at the\nsecond post-Minkowskian order,” Phys. Rev. D 106no. 12, (2022) 124049,\narXiv:2211.06340 [gr-qc] .\n[49] A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini,\n“One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective\nfield theory,” JHEP 06(2023) 048, arXiv:2303.06111 [hep-th] .\n[50] A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, “Inelastic exponentiation and\nclassical gravitational scattering at one loop,” JHEP 06(2023) 126,\narXiv:2303.07006 [hep-th] .\n[51] N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, “Scattering amplitudes for all\nmasses and spins,” JHEP 11(2021) 070, arXiv:1709.04891 [hep-th] .\n[52] A. Guevara, “Holomorphic Classical Limit for Spin Effects in Gravitational and\nElectromagnetic Scattering,” JHEP 04(2019) 033, arXiv:1706.02314 [hep-th] .\n33[53] D. Bini and T. Damour, “Gravitational spin-orbit coupling in binary systems,\npost-Minkowskian approximation and effective one-body theory,” Phys. Rev. D 96\nno. 10, (2017) 104038, arXiv:1709.00590 [gr-qc] .\n[54] J. Vines, “Scattering of two spinning black holes in post-Minkowskian gravity, to all\norders in spin, and effective-one-body mappings,” Class. Quant. Grav. 35no. 8,\n(2018) 084002, arXiv:1709.06016 [gr-qc] .\n[55] D. Bini and T. Damour, “Gravitational spin-orbit coupling in binary systems at the\nsecond post-Minkowskian approximation,” Phys. Rev. D 98no. 4, (2018) 044036,\narXiv:1805.10809 [gr-qc] .\n[56] J. Vines, J. Steinhoff, and A. Buonanno, “Spinning-black-hole scattering and the\ntest-black-hole limit at second post-Minkowskian order,” Phys. Rev. D 99no. 6,\n(2019) 064054, arXiv:1812.00956 [gr-qc] .\n[57] A. Guevara, A. Ochirov, and J. Vines, “Scattering of Spinning Black Holes from\nExponentiated Soft Factors,” JHEP 09(2019) 056, arXiv:1812.06895 [hep-th] .\n[58] M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, “The simplest massive S-matrix:\nfrom minimal coupling to Black Holes,” JHEP 04(2019) 156, arXiv:1812.08752\n[hep-th] .\n[59] Y. F. Bautista and A. Guevara, “From Scattering Amplitudes to Classical Physics:\nUniversality, Double Copy and Soft Theorems,” arXiv:1903.12419 [hep-th] .\n[60] Y. F. Bautista and A. Guevara, “On the double copy for spinning matter,” JHEP 11\n(2021) 184, arXiv:1908.11349 [hep-th] .\n[61] B. Maybee, D. O’Connell, and J. Vines, “Observables and amplitudes for spinning\nparticles and black holes,” JHEP 12(2019) 156, arXiv:1906.09260 [hep-th] .\n[62] A. Guevara, A. Ochirov, and J. Vines, “Black-hole scattering with general spin\ndirections from minimal-coupling amplitudes,” Phys. Rev. D 100no. 10, (2019)\n104024, arXiv:1906.10071 [hep-th] .\n[63] N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, “Kerr black holes as elementary\nparticles,” JHEP 01(2020) 046, arXiv:1906.10100 [hep-th] .\n[64] H. Johansson and A. Ochirov, “Double copy for massive quantum particles with\nspin,” JHEP 09(2019) 040, arXiv:1906.12292 [hep-th] .\n[65] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, “Classical potential for general spinning\nbodies,” JHEP 09(2020) 074, arXiv:1908.08463 [hep-th] .\n[66] P. H. Damgaard, K. Haddad, and A. Helset, “Heavy Black Hole Effective Theory,”\nJHEP 11(2019) 070, arXiv:1908.10308 [hep-ph] .\n34[67] M.-Z. Chung, Y.-T. Huang, and J.-W. Kim, “Kerr-Newman stress-tensor from\nminimal coupling,” JHEP 12(2020) 103, arXiv:1911.12775 [hep-th] .\n[68] R. Aoude, K. Haddad, and A. Helset, “On-shell heavy particle effective theories,”\nJHEP 05(2020) 051, arXiv:2001.09164 [hep-th] .\n[69] M.-Z. Chung, Y.-t. Huang, J.-W. Kim, and S. Lee, “Complete Hamiltonian for\nspinning binary systems at first post-Minkowskian order,” JHEP 05(2020) 105,\narXiv:2003.06600 [hep-th] .\n[70] Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, “Spinning black hole binary\ndynamics, scattering amplitudes, and effective field theory,” Phys. Rev. D 104no. 6,\n(2021) 065014, arXiv:2005.03071 [hep-th] .\n[71] R. Aoude, K. Haddad, and A. Helset, “Tidal effects for spinning particles,” JHEP 03\n(2021) 097, arXiv:2012.05256 [hep-th] .\n[72] A. Guevara, B. Maybee, A. Ochirov, D. O’connell, and J. Vines, “A worldsheet for\nKerr,” JHEP 03(2021) 201, arXiv:2012.11570 [hep-th] .\n[73] Z. Liu, R. A. Porto, and Z. Yang, “Spin Effects in the Effective Field Theory\nApproach to Post-Minkowskian Conservative Dynamics,” JHEP 06(2021) 012,\narXiv:2102.10059 [hep-th] .\n[74] D. Kosmopoulos and A. Luna, “Quadratic-in-spin Hamiltonian at O(G2) from\nscattering amplitudes,” JHEP 07(2021) 037, arXiv:2102.10137 [hep-th] .\n[75] R. Aoude and A. Ochirov, “Classical observables from coherent-spin amplitudes,”\nJHEP 10(2021) 008, arXiv:2108.01649 [hep-th] .\n[76] G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, “Gravitational\nBremsstrahlung and Hidden Supersymmetry of Spinning Bodies,” Phys. Rev. Lett.\n128no. 1, (2022) 011101, arXiv:2106.10256 [hep-th] .\n[77] Y. F. Bautista, A. Guevara, C. Kavanagh, and J. Vines, “From Scattering in Black\nHole Backgrounds to Higher-Spin Amplitudes: Part I,” arXiv:2107.10179\n[hep-th] .\n[78] M. Chiodaroli, H. Johansson, and P. Pichini, “Compton black-hole scattering for s ≤\n5/2,” JHEP 02(2022) 156, arXiv:2107.14779 [hep-th] .\n[79] K. Haddad, “Exponentiation of the leading eikonal phase with spin,” Phys. Rev. D\n105no. 2, (2022) 026004, arXiv:2109.04427 [hep-th] .\n[80] R. Gonzo and C. Shi, “Geodesics from classical double copy,” Phys. Rev. D 104\nno. 10, (2021) 105012, arXiv:2109.01072 [hep-th] .\n35[81] G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, “SUSY in the sky with\ngravitons,” JHEP 01(2022) 027, arXiv:2109.04465 [hep-th] .\n[82] M. V. S. Saketh, J. Vines, J. Steinhoff, and A. Buonanno, “Conservative and\nradiative dynamics in classical relativistic scattering and bound systems,” Phys. Rev.\nRes.4no. 1, (2022) 013127, arXiv:2109.05994 [gr-qc] .\n[83] T. Adamo, A. Cristofoli, and P. Tourkine, “Eikonal amplitudes from curved\nbackgrounds,” arXiv:2112.09113 [hep-th] .\n[84] W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, “The 2PM Hamiltonian for\nbinary Kerr to quartic in spin,” arXiv:2111.13639 [hep-th] .\n[85] G. U. Jakobsen and G. Mogull, “Conservative and radiative dynamics of spinning\nbodies at third post-Minkowskian order using worldline quantum field theory,”\narXiv:2201.07778 [hep-th] .\n[86] R. Aoude, K. Haddad, and A. Helset, “Searching for Kerr in the 2PM amplitude,”\narXiv:2203.06197 [hep-th] .\n[87] R. Aoude, K. Haddad, and A. Helset, “Classical Gravitational Spinning-Spinless\nScattering at O(G2S ∞),”Phys. Rev. Lett. 129no. 14, (2022) 141102,\narXiv:2205.02809 [hep-th] .\n[88] Z. Bern, D. Kosmopoulos, A. Luna, R. Roiban, and F. Teng, “Binary Dynamics\nThrough the Fifth Power of Spin at O(G2),”arXiv:2203.06202 [hep-th] .\n[89] F. Alessio and P. Di Vecchia, “Radiation reaction for spinning black-hole scattering,”\nPhys. Lett. B 832(2022) 137258, arXiv:2203.13272 [hep-th] .\n[90] W.-M. Chen, M.-Z. Chung, Y.-t. Huang, and J.-W. Kim, “Gravitational Faraday\neffect from on-shell amplitudes,” JHEP 12(2022) 058, arXiv:2205.07305 [hep-th] .\n[91] A. Ochirov and E. Skvortsov, “Chiral Approach to Massive Higher Spins,” Phys.\nRev. Lett. 129no. 24, (2022) 241601, arXiv:2207.14597 [hep-th] .\n[92] P. H. Damgaard, J. Hoogeveen, A. Luna, and J. Vines, “Scattering angles in Kerr\nmetrics,” Phys. Rev. D 106no. 12, (2022) 124030, arXiv:2208.11028 [hep-th] .\n[93] F. Febres Cordero, M. Kraus, G. Lin, M. S. Ruf, and M. Zeng, “Conservative Binary\nDynamics with a Spinning Black Hole at O(G3) from Scattering Amplitudes,” Phys.\nRev. Lett. 130no. 2, (2023) 021601, arXiv:2205.07357 [hep-th] .\n[94] G. Menezes and M. Sergola, “NLO deflections for spinning particles and Kerr black\nholes,” JHEP 10(2022) 105, arXiv:2205.11701 [hep-th] .\n36[95] M. M. Riva, F. Vernizzi, and L. K. Wong, “Gravitational bremsstrahlung from\nspinning binaries in the post-Minkowskian expansion,” Phys. Rev. D 106no. 4,\n(2022) 044013, arXiv:2205.15295 [hep-th] .\n[96] L. Cangemi and P. Pichini, “Classical Limit of Higher-Spin String Amplitudes,”\narXiv:2207.03947 [hep-th] .\n[97] L.-Y. Hung, K. Ji, and T. Wang, “Scrambling and Entangling Spinning Particles,”\narXiv:2208.12128 [hep-th] .\n[98] G. U. Jakobsen and G. Mogull, “Linear Response, Hamiltonian and Radiative\nSpinning Two-Body Dynamics,” arXiv:2210.06451 [hep-th] .\n[99] M. V. S. Saketh and J. Vines, “Scattering of gravitational waves off spinning\ncompact objects with an effective worldline theory,” Phys. Rev. D 106no. 12, (2022)\n124026, arXiv:2208.03170 [gr-qc] .\n[100] N. E. J. Bjerrum-Bohr, G. Chen, and M. Skowronek, “Classical Spin Gravitational\nCompton Scattering,” arXiv:2302.00498 [hep-th] .\n[101] Y. F. Bautista, A. Guevara, C. Kavanagh, and J. Vinese, “Scattering in Black Hole\nBackgrounds and Higher-Spin Amplitudes: Part II,” arXiv:2212.07965 [hep-th] .\n[102] L. Cangemi, M. Chiodaroli, H. Johansson, A. Ochirov, P. Pichini, and E. Skvortsov,\n“Kerr Black Holes Enjoy Massive Higher-Spin Gauge Symmetry,”\narXiv:2212.06120 [hep-th] .\n[103] F. Comberiati and L. de la Cruz, “Classical off-shell currents,” JHEP 03(2023) 068,\narXiv:2212.09259 [hep-th] .\n[104] F. Comberiati and C. Shi, “Classical Double Copy of Spinning Worldline Quantum\nField Theory,” arXiv:2212.13855 [hep-th] .\n[105] J.-W. Kim and J. Steinhoff, “Spin supplementary condition in quantum field theory,\nPart I : covariant SSC and physical state projection,” arXiv:2302.01944 [hep-th] .\n[106] M. A and D. Ghosh, “Classical spinning soft factors from gauge theory amplitudes,”\narXiv:2210.07561 [hep-th] .\n[107] J. Hoogeveen, “Charged test-particle scattering and effective one-body metrics with\nspin,” arXiv:2303.00317 [hep-th] .\n[108] K. Haddad, “Recursion in the classical limit and the neutron-star Compton\namplitude,” arXiv:2303.02624 [hep-th] .\n[109] A. Elkhidir, D. O’Connell, M. Sergola, and I. A. Vazquez-Holm, “Radiation and\nReaction at One Loop,” arXiv:2303.06211 [hep-th] .\n37[110] A. Bhattacharyya, D. Ghosh, S. Ghosh, and S. Pal, “Observables from classical black\nhole scattering in Scalar-Tensor theory of gravity from worldline quantum field\ntheory,” arXiv:2401.05492 [hep-th] .\n[111] D. Neill and I. Z. Rothstein, “Classical Space-Times from the S Matrix,” Nucl. Phys.\nB877(2013) 177–189, arXiv:1304.7263 [hep-th] .\n[112] D. Amati, M. Ciafaloni, and G. Veneziano, “Superstring Collisions at Planckian\nEnergies,” Phys. Lett. B 197(1987) 81.\n[113] D. Amati, M. Ciafaloni, and G. Veneziano, “Classical and Quantum Gravity Effects\nfrom Planckian Energy Superstring Collisions,” Int. J. Mod. Phys. A 3(1988)\n1615–1661.\n[114] P. Di Vecchia, C. Heissenberg, R. Russo, and G. Veneziano, “The gravitational\neikonal: from particle, string and brane collisions to black-hole encounters,”\narXiv:2306.16488 [hep-th] .\n[115] A. Buonanno and T. Damour, “Transition from inspiral to plunge in binary black\nhole coalescences,” Phys. Rev. D 62(2000) 064015, arXiv:gr-qc/0001013 .\n[116] T. Damour and P. Rettegno, “Strong-field scattering of two black holes: Numerical\nrelativity meets post-Minkowskian gravity,” Phys. Rev. D 107no. 6, (2023) 064051,\narXiv:2211.01399 [gr-qc] .\n[117] P. Rettegno, G. Pratten, L. M. Thomas, P. Schmidt, and T. Damour, “Strong-field\nscattering of two spinning black holes: Numerical relativity versus post-Minkowskian\ngravity,” Phys. Rev. D 108no. 12, (2023) 124016, arXiv:2307.06999 [gr-qc] .\n[118] G. K¨ alin and R. A. Porto, “From Boundary Data to Bound States,” JHEP 01(2020)\n072, arXiv:1910.03008 [hep-th] .\n[119] G. K¨ alin and R. A. Porto, “From boundary data to bound states. Part II. Scattering\nangle to dynamical invariants (with twist),” JHEP 02(2020) 120,\narXiv:1911.09130 [hep-th] .\n[120] G. Cho, G. K¨ alin, and R. A. Porto, “From boundary data to bound states. Part III.\nRadiative effects,” JHEP 04(2022) 154, arXiv:2112.03976 [hep-th] . [Erratum:\nJHEP 07, 002 (2022)].\n[121] R. Gonzo and C. Shi, “Boundary to bound dictionary for generic Kerr orbits,” Phys.\nRev. D 108no. 8, (2023) 084065, arXiv:2304.06066 [hep-th] .\n[122] T. Adamo, R. Gonzo, and A. Ilderton, “Gravitational Bound Waveforms from\nAmplitudes,” arXiv:2402.00124 [hep-th] .\n[123] T. Damour, “Radiative contribution to classical gravitational scattering at the third\norder in G,”Phys. Rev. D 102no. 12, (2020) 124008, arXiv:2010.01641 [gr-qc] .\n38[124] P. H. Damgaard, E. R. Hansen, L. Plant´ e, and P. Vanhove, “Classical observables\nfrom the exponential representation of the gravitational S-matrix,” JHEP 09(2023)\n183, arXiv:2307.04746 [hep-th] .\n[125] G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and Y. Xu, “Conservative Scattering\nof Spinning Black Holes at Fourth Post-Minkowskian Order,” Phys. Rev. Lett. 131\nno. 15, (2023) 151401, arXiv:2306.01714 [hep-th] .\n[126] G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer, “Dissipative Scattering of\nSpinning Black Holes at Fourth Post-Minkowskian Order,” Phys. Rev. Lett. 131\nno. 24, (2023) 241402, arXiv:2308.11514 [hep-th] .\n[127] S. J. Kovacs and K. S. Thorne, “The Generation of Gravitational Waves. 4.\nBremsstrahlung,” Astrophys. J. 224(1978) 62–85.\n[128] G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff, “Classical Gravitational\nBremsstrahlung from a Worldline Quantum Field Theory,” Phys. Rev. Lett. 126\nno. 20, (2021) 201103, arXiv:2101.12688 [gr-qc] .\n[129] S. Mougiakakos, M. M. Riva, and F. Vernizzi, “Gravitational Bremsstrahlung in the\npost-Minkowskian effective field theory,” Phys. Rev. D 104no. 2, (2021) 024041,\narXiv:2102.08339 [gr-qc] .\n[130] S. Mougiakakos, M. M. Riva, and F. Vernizzi, “Gravitational Bremsstrahlung with\nTidal Effects in the Post-Minkowskian Expansion,” Phys. Rev. Lett. 129no. 12,\n(2022) 121101, arXiv:2204.06556 [hep-th] .\n[131] S. De Angelis, R. Gonzo, and P. P. Novichkov, “Spinning waveforms from KMOC at\nleading order,” arXiv:2309.17429 [hep-th] .\n[132] R. Aoude, K. Haddad, C. Heissenberg, and A. Helset, “Leading-order gravitational\nradiation to all spin orders,” arXiv:2310.05832 [hep-th] .\n[133] A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, and G. Travaglini, “Resummed\nspinning waveforms from five-point amplitudes,” arXiv:2310.04405 [hep-th] .\n[134] A. Herderschee, R. Roiban, and F. Teng, “The sub-leading scattering waveform from\namplitudes,” JHEP 06(2023) 004, arXiv:2303.06112 [hep-th] .\n[135] S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, “What can be\nmeasured asymptotically?,” arXiv:2308.02125 [hep-th] .\n[136] D. Bini, T. Damour, and A. Geralico, “Comparing one-loop gravitational\nbremsstrahlung amplitudes to the multipolar-post-Minkowskian waveform,” Phys.\nRev. D 108no. 12, (2023) 124052, arXiv:2309.14925 [gr-qc] .\n[137] A. Georgoudis, C. Heissenberg, and R. Russo, “An eikonal-inspired approach to the\ngravitational scattering waveform,” arXiv:2312.07452 [hep-th] .\n39[138] L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk, “Gravitational Bremsstrahlung in\nBlack-Hole Scattering at O(G3): Linear-in-Spin Effects,” arXiv:2312.14859\n[hep-th] .\n[139] A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, “More on the Subleading\nGravitational Waveform,” arXiv:2312.14710 [hep-th] .\n[140] Z. Bern, S. Davies, P. Di Vecchia, and J. Nohle, “Low-Energy Behavior of Gluons\nand Gravitons from Gauge Invariance,” Phys. Rev. D 90no. 8, (2014) 084035,\narXiv:1406.6987 [hep-th] .\n[141] J. Broedel, M. de Leeuw, J. Plefka, and M. Rosso, “Constraining subleading soft\ngluon and graviton theorems,” Phys. Rev. D 90no. 6, (2014) 065024,\narXiv:1406.6574 [hep-th] .\n[142] Z. Bern, S. Davies, and J. Nohle, “On Loop Corrections to Subleading Soft Behavior\nof Gluons and Gravitons,” Phys. Rev. D 90no. 8, (2014) 085015, arXiv:1405.1015\n[hep-th] .\n[143] A. Laddha and A. Sen, “Observational Signature of the Logarithmic Terms in the\nSoft Graviton Theorem,” Phys. Rev. D 100no. 2, (2019) 024009, arXiv:1806.01872\n[hep-th] .\n[144] B. Sahoo and A. Sen, “Classical and Quantum Results on Logarithmic Terms in the\nSoft Theorem in Four Dimensions,” JHEP 02(2019) 086, arXiv:1808.03288\n[hep-th] .\n[145] A. P. Saha, B. Sahoo, and A. Sen, “Proof of the classical soft graviton theorem in D\n= 4,” JHEP 06(2020) 153, arXiv:1912.06413 [hep-th] .\n[146] B. Sahoo and A. Sen, “Classical soft graviton theorem rewritten,” JHEP 01(2022)\n077, arXiv:2105.08739 [hep-th] .\n[147] D. Ghosh and B. Sahoo, “Spin-dependent gravitational tail memory in D= 4,” Phys.\nRev. D 105no. 2, (2022) 025024, arXiv:2106.10741 [hep-th] .\n[148] H. Krishna and B. Sahoo, “Universality of loop corrected soft theorems in 4d,” JHEP\n11(2023) 233, arXiv:2308.16807 [hep-th] .\n[149] A. Addazi, M. Bianchi, and G. Veneziano, “Soft gravitational radiation from\nultra-relativistic collisions at sub- and sub-sub-leading order,” JHEP 05(2019) 050,\narXiv:1901.10986 [hep-th] .\n[150] M. Ciafaloni, D. Colferai, and G. Veneziano, “Infrared features of gravitational\nscattering and radiation in the eikonal approach,” Phys. Rev. D 99no. 6, (2019)\n066008, arXiv:1812.08137 [hep-th] .\n40[151] S. Agrawal, L. Donnay, K. Nguyen, and R. Ruzziconi, “Logarithmic soft graviton\ntheorems from superrotation Ward identities,” arXiv:2309.11220 [hep-th] .\n[152] A. Georgoudis, C. Heissenberg, and R. Russo, “Post-Newtonian Multipoles from the\nNext-to-Leading Post-Minkowskian Gravitational Waveform,” arXiv:2402.06361\n[hep-th] .\n[153] D. Bini, T. Damour, S. De Angelis, A. Geralico, A. Herderschee, R. Roiban, and\nF. Teng, “Gravitational Waveform: A Tale of Two Formalisms,” arXiv:2402.06604\n[hep-th] .\n[154] F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,”\narXiv:1404.4091 [hep-th] .\n[155] S. Weinberg, “Photons and Gravitons in S-Matrix Theory: Derivation of Charge\nConservation and Equality of Gravitational and Inertial Mass,” Phys. Rev. 135\n(1964) B1049–B1056.\n[156] A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory . 3,\n2017. arXiv:1703.05448 [hep-th] .\n[157] A. Laddha and A. Sen, “Sub-subleading Soft Graviton Theorem in Generic Theories\nof Quantum Gravity,” JHEP 10(2017) 065, arXiv:1706.00759 [hep-th] .\n[158] A. Laddha and A. Sen, “Logarithmic Terms in the Soft Expansion in Four\nDimensions,” JHEP 10(2018) 056, arXiv:1804.09193 [hep-th] .\n[159] A. V. Manohar, A. K. Ridgway, and C.-H. Shen, “Radiated Angular Momentum and\nDissipative Effects in Classical Scattering,” Phys. Rev. Lett. 129no. 12, (2022)\n121601, arXiv:2203.04283 [hep-th] .\n[160] F. Alessio, “Kerr binary dynamics from minimal coupling and double copy,”\narXiv:2303.12784 [hep-th] .\n[161] D. R. Yennie, S. C. Frautschi, and H. Suura, “The infrared divergence phenomena\nand high-energy processes,” Annals Phys. 13(1961) 379–452.\n[162] C. Heissenberg, “Infrared divergences and the eikonal exponentiation,” Phys. Rev. D\n104no. 4, (2021) 046016, arXiv:2105.04594 [hep-th] .\n41"    },    {        "title": "2402.06538v1.An_Exercise_in_Tournament_Design__When_Some_Matches_Must_Be_Scheduled.pdf",        "content": "An Exercise in Tournament Design: When Some Matches Must Be Scheduled\nSushmita Gupta1, Ramanujan Sridharan2, Peter Strulo2\n1The Institute of Mathematical Sciences, HBNI, India\n2University of Warwick, UK\nsushmita.gupta@gmail.com, r.maadapuzhi-sridharan@warwick.ac.uk, peter.strulo@warwick.ac.uk\nAbstract\nSingle-elimination (SE) tournaments are a popular format\nused in competitive environments and decision making. Al-\ngorithms for SE tournament manipulation have been an active\ntopic of research in recent years. In this paper, we initiate the\nalgorithmic study of a novel variant of SE tournament ma-\nnipulation that aims to model the fact that certain matchups\nare highly desired in a sporting context, incentivizing an or-\nganizer to manipulate the bracket to make such matchups\ntake place. We obtain both hardness and tractability results.\nWe show that while the problem of computing a bracket en-\nforcing a given set of matches in an SE tournament is NP-\nhard, there are natural restrictions that lead to polynomial-\ntime solvability. In particular, we show polynomial-time solv-\nability if there is a linear ordering on the ability of players\nwith only a constant number of exceptions where a player\nwith lower ability beats a player with higher ability.\nIntroduction\nThere is a rich history of work on the algorithmics of de-\nsigning Single Elimination (SE) or knockout tournaments as\nthey are a format of competition employed in varied scenar-\nios such as sports, elections and different forms of decision\nmaking (Tullock 1980; Horen and Riezman 1985; Rosen\n1986; Laslier 1997; Connolly and Rendleman 2011). Based\non an initial bracket (a permutation of the players, also called\naseeding ), it proceeds in multiple rounds, culminating in a\nsingle winner. In each round, all players that have not yet lost\na match are paired up to play the next set of matches. Losers\nexit and winners proceed to the next round, until only one\nremains, the winner of the tournament. In general, the tour-\nnament designer is assumed to be given probabilities pijex-\npressing the likelihood that player ibeats player j. In this\npaper, we focus on the deterministic model, i.e., when these\nprobabilities are 0 or 1. This model has already been the sub-\nject of numerous papers in the last few years. Besides being\nindependently interesting from a structural and algorithmic\nperspective as shown by (Vassilevska Williams 2010; Aziz\net al. 2014; Ramanujan and Szeider 2017; Gupta et al. 2018,\n2019; Manurangsi and Suksompong 2023; Zehavi 2023), the\ndeterministic model naturally captures sequential majority\nelections along binary trees (Lang et al. 2007; Vu, Altman,\nCopyright © 2024, Association for the Advancement of Artificial\nIntelligence (www.aaai.org). All rights reserved.and Shoham 2009) where each “match” is a comparison of\nvotes of two candidates and the candidate with more votes\nwins and moves on.\nA major question in the study of SE tournament design\nis the T OURNAMENT FIXING Problem (TF): Can a designer\nefficiently find a bracket that maximizes the likelihood (or\nensures, in the case of the deterministic model) that a player\nof their choice wins the tournament? However, there are\nother natural objectives around SE tournament design be-\nsides favoring a particular player and that is the focus of this\npaper. Great rivalries generate great entertainment. Imagine\na sports tournament that features marquee matches marked\nby factors such as historic rivalries, contemporaneous news\nevents, geographic proximity or even personal rivalries be-\ntween members of the opposing teams. These are some of\nthe most widely known and talked about rivalries in the\nworld of sports that greatly enhanced the notoriety and vis-\nibility of the sport and thereby achieved great financial suc-\ncess, publicity and relevancy for all the stake holders, be\nit the organizers, the sponsors, not to mention the partici-\npants. From the competition design perspective, that consid-\nerations such as revenue, viewer engagement, and relevance\nshould be at the forefront is straightforward.\nThus, scheduling especially attractive matches is a ratio-\nnal tournament design imperative, motivating our algorith-\nmic study of finding a bracket that aims to ensure that a given\nset of demand matches are played in the SE tournament. One\ncan view our model of scheduling a set of demanded games\nin this setting to be a special case of revenue maximization\nwith unit revenue given to each demand match and zero to\nall others. Setting the target revenue to be equal to the num-\nber of demand matches implies that achieving the target rev-\nenue is the same as scheduling all the demand matches. A\nmore general problem was studied by (Lang et al. 2007),\nin the context of sequential majority voting along binary\ntrees, except they allow arbitrary costs for the edges and\nthe goal is to achieve minimum possible cost in the final\nSE tournament. They showed this problem to be NP-hard.\nWe also note that our paper is naturally aligned with inves-\ntigations into the relationship between round-robin and SE\ntournaments, e.g., (Stanton and Vassilevska Williams 2011).\nSpecifically, while every demand match obviously occurs in\na round-robin tournament, how efficiently could one ensure\nthe same in an SE tournament? This is our focus.arXiv:2402.06538v1  [cs.DS]  9 Feb 2024Our contributions. We make advances on two fronts–\nconceptual and algorithmic. On the conceptual front, we in-\ntroduce a tournament design objective that is both a spe-\ncial case of revenue maximization and as we demonstrate\nlater, a novel variant of the well-studied Subgraph Isomor-\nphism problem. We call the problem D EMAND TOURNA -\nMENT FIXING (Demand-TF), formally defined as follows.\nThe input contains (i) a directed graph T(called tourna-\nment digraph) whose vertices are the players and for every\npair of players uandv, there is a directed edge (i.e., arc)\nfrom utovif and only if ubeats v(assume no ties) and\n(ii) a set Sof arcs of T. The vertices of Tare denoted by\nV(T)and the arcs by A(T). The goal is to find a bracket (if\none exists) such that in the SE tournament generated by this\nbracket, the matches corresponding to the arcs in S(called\ndemand matches) take place.\nSolving this problem requires one to create a bracket that\nwill ensure that a player u(and its “demand rivals”) all\nprogress far enough in the tournament so that uis able to\nplay in all the demand matches featuring it. Effectively, we\nare aiming to create within one single bracket, multiple fa-\nvorable brackets for each of those players that are some-\nhow highly correlated. Treading this fine line raises fasci-\nnating algorithmic challenges as we show in this paper. In\nfact, for the special case of an acyclic tournament digraph\n(DAG), i.e., when there is a linear ordering of players ac-\ncording to their strengths where each player beats every\nplayer appearing after it, the TF problem is trivial as the\nstrongest player wins every SE tournament. On the other\nhand, even in this special case, Demand-TF is a challeng-\ning problem. However, as we will discuss, our main result\nimplies a polynomial-time algorithm even for this problem\non DAGs, as a corollary.\nWe highlight the relation between our problem and S UB-\nGRAPH ISOMORPHISM (Cygan et al. 2015) problem (SI).\nTF has a well-established connection with a specific type of\nspanning tree within the tournament digraph, called a span-\nning binomial arborescence orSBA (Vassilevska Williams\n2016). We refer the reader to the section on preliminaries\nfor a formal definition. In fact, there is a solution to the TF\ninstance if and only if the tournament digraph has an SBA\nrooted at the favorite player. In terms of SI, the tournament\ndigraph is the “host” graph and the “pattern” graph being\nsought is an SBA rooted at the favorite player. In Demand-\nTF, the pattern graph is an SBA that contains all the demand\narcs (those arcs that correspond to demand matches). To the\nbest of our knowledge this is a novel “edge-extension” vari-\nant of SI where the goal is to build the pattern graph using\na set of given edges as a starting point. The setting of (Ma-\nnurangsi and Suksompong 2023) can also be interpreted as\na constrained version of SI where arcs representing matches\nbetween higher ranked players can only occur in parts of the\npattern graph that represent later rounds. Their motivation\nwas to prevent the best players from meeting too early.\nWe next describe our algorithmic contributions (see Table\n1). On the one hand, we show that Demand-TF is NP-hard\nand conditionally rule out any algorithm that runs in time\nndO(1)where dis the number of demand matches. This mo-Algorithms nO(k)-time\n2O(klogk)nO(1)-time if F⊆ S\n3n·nO(1)-time\nHardness no2o(n)-time algorithm (under ETH)\nnondO(1)-time algorithm (if NP ̸⊆QP)\nTable 1: A summary of our results for D EMAND -TF. Here,\nnis the number of players, kis the size of some minimum\nfeedback arc set Fof the input tournament, Sis the set of\ndemand matches and dis the number of demand matches.\ntivates the search for tractable restrictions of the problem and\nbrings us to the central results of the paper. Here, we make\nthe following contributions:\nAlgorithm 1: Demand-TF is P-time solvable when the tour-\nnament digraph has a linear ordering on the ability of play-\ners with a constant number of exceptions where a player\nwith lower ability beats a player with higher ability. In other\nwords, when the feedback arc set number of the tournament\ndigraph is constant. This is a natural condition in competi-\ntions where there is a clear-cut ranking of the players accord-\ning to their skills with only a few pairs of players for which\nthe weaker player can beat the stronger player. Motivated by\nempirical work in (Russell and van Beek 2011), (Aziz et al.\n2014) initiated the design of algorithms for TF when the in-\nstances have constant feedback arc set number and gave the\nfirst P-time algorithm. This restriction was then extensively\nexplored in a series of papers (Ramanujan and Szeider 2017;\nGupta et al. 2018, 2019) leading to novel fixed-parameter\nalgorithms. In parameterized complexity parlance, we give\nan XP algorithm for Demand-TF parameterized by the feed-\nback arc set number k(i.e., running time nO(k)). This brings\nup the natural question of whether the problem is fixed-\nparameter tractable (FPT) (i.e., solvable in time f(k)nO(1)\nfor some function f). Although we do not settle this question\nin this paper, we identify an additional structural constraint\nin our next result that leads to an FPT algorithm.\nAlgorithm 2: If, in the given instance (T,S)of Demand-\nTF, every upset match is also a demand match, then we get\nfixed-parameter tractability parameterized by the feedback\narc set number. The natural motivation for this scenario is a\ntournament designer being incentivized (e.g., by betting) to\nensure that the upsets take place.\nAlgorithm 3: We extend our methodology in the preceding\nalgorithms to handle further constraints (in the same running\ntime). In particular, when the designer wants each demand\nmatch to take place in a specific round of the SE tourna-\nment, we can still find such a bracket in time nO(k). Such\nconstraints allow the designer to ensure that some matches\ndo not occur too early (in the spirit of (Manurangsi and Suk-\nsompong 2023)) or too late in the tournament.\nFinally, moving to the exact-exponential-time regime we\nshow that assuming the Exponential Time Hypothesis (Im-\npagliazzo and Paturi 2001), one cannot get a subexponential-\ntime algorithm for the problem (i.e., a 2o(n)running time\nwhere nis the number of players) and complement thislower bound with an algorithm with running time 2O(n)–\nan asymptotically tight bound.\nOrganization of the paper. We begin by presenting ba-\nsic definitions followed by our hardness results. Then, our\nlargest section is dedicated to presenting our main algorithm\n(Algorithm 1 ). The remainder of the algorithmic contribu-\ntions ( Algorithm 2 andAlgorithm 3 above) are presented in\nthe following sections. Finally, we conclude with directions\nfor future research.\nPreliminaries\nBinomial arborescences. An arborescence is a rooted di-\nrected tree such that all arcs are directed away from the root.\nDefinition 1 (Vassilevska Williams 2010) .The set of bino-\nmial arborescences over a tournament digraph Tis recur-\nsively defined as follows. (i)Each a∈V(T)is a binomial\narborescence rooted at a.(ii)If, for some i >0,Haand\nHbare2i−1-node binomial arborescences rooted at aand\nb, respectively, then adding an arc from atobgives a 2i-\nnode binomial arborescence ( BA) rooted at a. If a binomial\narborescence His such that V(H) =V(T), then His a\nspanning binomial arborescence (SBA ) ofT.\nThe relevance of binomial arborescences comes from the\nfollowing variant of a result of (Vassilevska Williams 2010).\nProposition 1. LetTbe a tournament digraph and let\nS ⊆A(T). Then, there is a seeding of V(T)such that the\nresulting SE tournament has every match in Sif and only if\nThas an SBAHsuch that A(H)⊇ S.\nWe use the terms vertex and player interchangeably. A\nrooted forest is the disjoint union of a set of arborescences.\nLetHbe a rooted forest. For a vertex u∈V(H), denote\nbyChild H(u)the set of children of uinH, byDesc H(u)\nthe set of descendants of uinH(including u). The strict\ndescendants of ucomprise descendants of uthat are nei-\ntherunor children of u. We denote by SiblH(u), the set\nof siblings of uinH. We define the height ofvinH,\nhtH(v) := log |Desc H(v)|. Note that if His an SBA ofT,\nthen∀v∈V(T),∃i∈[logn]∪{0}such that |Desc H(v)|=\n2iand hence htH(v)is an integer. The interpretation in the\ncorresponding SE tournament is that vis the winner of a\nsubtournament played by the players who are descendants\nofvinHandhtH(v)is the number of matches that vwins.\nIfhtH(v)>htH(u), then we say that vishigher thanuand\nuislower thanv. We will also refer to the height of aBA\nmeaning the height of its root.\nWe will use the following characterization of BAs:\nProposition 2. Forn >0,Hnis aBAof height nrooted at\nvnif and only if Hn=Sn−1\ni=0Hiwhere Hiis aBAof height\nirooted at vitogether with an edge from vnto each vi.\nIn an instance (T,S)of D EMAND -TF, we call the arcs\ninSdemand arcs ordemand matches and their endpoints\ndemand vertices . For every demand arc (p, q), we say that\npis ademand in-neighbor ordemand parent ofqandqis\nademand out-neighbor ordemand child ofp. We will use\ndemand parent and demand child in the context of rooted\ntrees and demand in/out-neighbor otherwise. For a vertex p,the number of its demand in-neighbors is called its demand\nin-degree . The demand out-degree is defined symmetrically.\nLose(S)denotes the vertices with a demand in-neighbor,\nthat is, those vertices that lose some demand match.\nHardness Results for DEMAND -TF\nWe first show that D EMAND -TF is NP-complete and then\ninfer further facts regarding the complexity of this problem\nusing our proof in combination with hardness results on TF\nproved by (Aziz et al. 2014).\nTheorem 1. DEMAND -TF is NP–complete.\nProof. To demonstrate the NP-hardness, we will give a\npolynomial-time reduction from the T OURNAMENT FIX-\nING problem (TF). Let (T, v∗)be an instance of TF. Let\nn=|V(T)|. We first construct an n-vertex acyclic tour-\nnament, D1with source vertex d1. We call D1a “dummy”\ntournament. Notice that in any SE tournament played by the\nvertices in D1regardless of the seeding, the vertex d1will\nbe the winner. Now, define T′to be the tournament obtained\nby taking the disjoint union of TandD1and then doing the\nfollowing:\n1. Add the arc (d1, v∗), ensuring that v∗loses to d1.\n2. For every other vertex v∈V(T), add the arc (v, d1),\nensuring that d1loses to every vertex in Texcept v∗.\n3. Add arcs ensuring that every vertex in Tbeats every ver-\ntex in V(D1)except for d1.\nWe initially set S:={(d1, v∗)}, find an arbitrary SBAH1\nonD1and for each arc of H1incident on d1, we add this arc\ntoS. This completes the construction of the D EMAND -TF\ninstance (T′,S).\nClearly, the reduction can be done in polynomial time, so\nit remains to argue the correctness.\nSuppose that (T, v∗)is a yes-instance of TF. Then there\nexists a permutation πoverV(T)such that v∗wins the SE\ntournament where the first-round matches are given by πand\nthe pair-wise results by T. Suppose that the permutation of\nV(D1)that leads to the SBA H1isπ1. Then, notice that\nthe permutation π′obtained by simply taking the union of π\nandπ1is a permutation of V(T′). We claim that π′certifies\nthat(T′,S)is a yes-instance of D EMAND -TF. That is, in\nthe SE tournament where the first-round matches are given\nbyπ′and the pair-wise results by T′, every demand match\nis played. By construction, π′results in v∗losing to d1in\nthe final. Moreover, the second-half of the bracket given by\nπ′that comprises only of vertices from V(D1)is identical\ntoπ1, guaranteeing that all matches corresponding to arcs in\ntheSBAH1are indeed played. This completes the forward\ndirection.\nConversely suppose that there exists a permutation π′on\nV(T′)which results in every demand match being played.\nNotice that the definition of Simplies that d1wins the tour-\nnament and plays exactly logn+ 1 matches, out of which\none is against v∗and the remaining lognmatches must be\nagainst vertices of D1. We next argue that d1cannot play\nagainst v∗before the final. If this were not the case, then\none half of the bracket given by π′contains both d1andv∗. Then, the other bracket must contain at least one vertex\nfrom V(T). Since all vertices in V(T)\\ {v∗}are stronger\nthan every vertex in V(D1), it follows that the opponent of\nd1in the final would have to be a vertex of Tother than v∗,\na contradiction to d1winning the whole tournament. By the\nsame reasoning, the half of the bracket given by π′that con-\ntainsd1must in fact only comprise vertices of D1. This im-\nplies that v∗wins its subtournament that comprises exactly\nthe vertices of T, implying that (T, v∗)is a yes-instance of\nTF.\nTheorem 2. From Theorem 1 combined with known hard-\nness of TF (Aziz et al. 2014), we obtain the following.\n1.DEMAND -TFis NP–complete even if there is a vertex on\nwhich every demand arc is incident.\n2. Unless NP⊆Quasi -P-time, there is no algorithm for\nDEMAND -TF that runs in time ndO(1), where dis the\nnumber of demand arcs. In particular, this rules out a\n2dO(1)nO(1)-time fixed-parameter algorithm.\n3. Assuming the Exponential Time Hypothesis (ETH), there\nis no 2o(n)-time algorithm for DEMAND -TF.\nProof. The first statement follows from the construction in\nTheorem 1. Moreover, notice that in the same construction,\nthe number of demand arcs is logn+ 1. Hence, an algo-\nrithm for D EMAND -TF with running time ndO(1)would im-\nply a quasi-polynomial-time algorithm for TF, which is NP-\ncomplete. Finally, the proof of NP-hardness of TF given in\n(Aziz et al. 2014) reduces an instance of 3-SAT-2L (3-SAT\nwhere every literal appears at most twice) with nvariables\nto an instance of TF with O(n)vertices. Since our reduction\nfrom TF only doubles the number of vertices, a 2o(n)-time\nalgorithm for D EMAND -TF would imply the same running\ntime for 3-SAT-2L, which violates ETH.\nRecall that the naive algorithm for D EMAND -TF has\nrunning time 2O(nlogn)as a result of brute-forcing over\nall possible brackets. We next improve this to a single-\nexponential running time with a dynamic programming\nalgorithm, asymptotically matching the 2o(n)-time lower\nbound from Theorem 2.\nTheorem 3. DEMAND -TF can be solved in time 3nnO(1).\nProof. Let(T,S)be the given instance of D EMAND -TF.\nLet us define a boolean function ∆ : 2V(T)×V(T)as fol-\nlows. For every S⊆V(T)andx∈V(T),∆(S, x) = 1 if\nand only if the following conditions are satisfied.\n•|S|is a power of 2 and x∈S.\n• There is a permutation πSoverSsuch that the SE tour-\nnament played by Saccording to the seeding πSis won\nbyxand every demand match in S ∩A[S]is played in\nthis tournament.\nNotice that (T,S)is a yes-instance if and only if there\nis some x∈V(T)such that ∆(V(T), x) = 1 . Hence, it\nsuffices to give an algorithm that computes the function ∆\nin the stated running time. We have the following claim at\nthe crux of our algorithm.Claim 1. Consider a vertex set Sof size 2pfor some p >0,\nand let x∈S. Then, ∆(S, x) = 1 if and only if there is\nan equi-partition S1⊎S2=Sand a vertex ysuch that\n∆(S1, x) = 1 ,∆(S2, y) = 1 ,(x, y)∈A(T)and except for\nthe arc (x, y)there is no demand arc in Swith one endpoint\ninS1and the other endpoint in S2.\nProof. In the forward direction, suppose that ∆(S, x) = 1\nand consider an SE tournament played by Sthat is won by x\nand in which every demand match in S ∩A[S]is played. Let\nybe the opponent of xin the final of this SE tournament. Let\nS2be the descendants of yin the corresponding SBA and let\nS1=S\\S2. Then, notice that except for potentially the arc\n(x, y), there cannot be a demand arc with one endpoint in\nS1and the other in S2. Moreover, (x, y)∈A(T)and every\ndemand arc contained in S∩(A[S1]∪A[S2])must be played,\ni.e.,∆(S1, x) = ∆( S2, y) = 1 . This completes the proof\nof the forward direction. The argument for the converse is\nsymmetrical.\nGiven the above claim, we compute ∆using a dynamic\nprogramming algorithm as follows. In the base case, when\n|S|= 1, this is trivial. Now, suppose that we have computed\n∆(S, x)for every |S| ≤2i. Consider a vertex set S′of size\n2i+1, and let x∈S′. Using the above claim, it is sufficient\nto go through every possible equi-partition of S′into sets S′\n1\nandS′\n2and check whether there exists a vertex ysuch that\n∆(S′\n1, x) = 1 ,∆(S′\n2, y) = 1 ,(x, y)∈A(T)and except\nfor the arc (x, y)there is no demand arc in Swith one end-\npoint in S′\n1and the other endpoint in S′\n2. This can clearly be\ndone in time O∗(2|S′|). Consequently, the running time of\nour overall algorithm is bounded by Σn\ni=0\u0000n\ni\u0001\n2incfor some\nconstant c, implying a running time of O∗(3n)as claimed.\nThis completes the proof of the theorem.\nGoing beyond Theorem 3\nWe first remark that similar to (Kim and Vas-\nsilevska Williams 2015), the subset convolution technique\ncan be used to speed up the algorithm of Theorem 3 to\na2nnO(1)-time algorithm. We also point out that the\napproach used in the the algorithm of Theorem 3 can be\neasily extended to a more general version of the problem\nwhere we assign to the matches, arbitrary (non-negative)\ninteger weights that are polynomially bounded, and we\nwant to compute a seeding that maximizes the total weight\nof the satisfied demands. To achieve this, we would\nsimply have to enhance the function ∆so that for every\nS⊆V(T), x∈V(T)and polynomially bounded w∈N,\n∆(S, x, w ) = 1 if and only if |S|is a power of 2, x∈S\nand there is a permutation πSover Ssuch that the SE\ntournament played by Saccording to the seeding πSis\nwon by xand total weight of the demand matches in\nA[S]that are played in this SE tournament is w. Then, the\nanalogue of the claim in the proof of Theorem 3 would\nbe:∆(S, x, w ) = 1 if and only if there is an equi-partition\nS1⊎S2=Sand integers w1, w2, and a vertex ysuch that\n∆(S1, x, w 1) = 1 ,∆(S2, y, w 2) = 1 ,(x, y)∈A(T)and\nthe sum of w1, w2and the weight assigned to the arc (x, y)\nadds up to w.DEMAND -TFon Graphs of Bounded Feedback\nArc Set Number\nWe now turn our attention to tournament digraphs with a\nconstant-size (denoted by k) feedback arc set. In this section\nwe will assume that an instance of Demand-TF is a triple\n(T,S, F), where Fis a minimum feedback arc set of T.\nThe assumption that Fis given, is without loss of generality\nsince a minimum feedback arc set of size at most kcan be\ncomputed in time 3knO(1)-time (Cygan et al. 2015). We re-\nfer to the endpoints of Fasfeedback vertices . Additionally\nwe will assume that σ=v1, v2, . . . , v nis a linear ordering\nof the vertices of Tsuch that the arcs (vi, vj)withi > j are\nprecisely the “upset” matches, i.e., the arcs of F. We say that\nviisstronger thanvjfor all j > i . We say that a subgraph of\nthe tournament digraph is valid if every demand arc between\ntwo vertices in the subgraph is also present in the subgraph.\nThus, Proposition 1 implies that a solution to D EMAND -TF\ncorresponds to a valid SBA of the given tournament digraph.\nNote that any player can lose at most one match in an SE\ntournament. Hence, if (T,S, F)is an instance of D EMAND -\nTF in which some vertex has demand in-degree greater than\n1, then it is a no-instance. So, we may assume without loss of\ngenerality that in any non-trivial instance of D EMAND -TF,\nevery vertex has at most one demand in-neighbor. In the rest\nof this section we will also assume that all but at most one of\nthe feedback vertices has a demand in-neighbour: this is true\nif, in the final SBA ,every feedback vertex except the root\nhas a demand parent. We will ensure this in our algorithm\nby guessing the parent of every feedback vertex (i.e., the\nplayer that beats it) and adding the resulting arc to the set of\ndemand arcs (the overhead is at most n2k).\nWe also guess the heights of each feedback vertex (over-\nhead (logn)2k). Using this as a starting point, we obtain an\nestimate for the heights of every vertex via the following\ndefinition, where the reader may think of the function gas\nour guesses for the heights of the feedback vertices.\nDefinition 2 (Function ht∗and compactness property) .Fix\na function g:V(F)→[logn]. For each v∈V(T)let\nht∗\ng(v) =g(v)ifv∈V(F). Otherwise let ht∗\ng(v)be the\nminimum non-negative integer satisfying:\n1. For each demand arc (v, w),ht∗\ng(v)>ht∗\ng(w).\n2. For each u, w such that (u, v),(u, w)∈ S ,ht∗\ng(v)̸=\nht∗\ng(w)if either (i) wis weaker than vor (ii) w∈V(F).\nWe say that a binomial arborsescence Hiscompact with\nrespect to gifhtH(v) =ht∗\ng(v), for every v∈Lose(S)∩\nV(H).\nAdditionally, we say that a BAHisweakly compact if it is\ncompact with respect to the function htHrestricted to V(F).\nIntuitively, ht∗\ng(v)can be described as follows. Fix a hy-\npothetical solution, that is a valid SBAH, and suppose that\ninH, we know the heights of each demand child and each\nweaker demand sibling of v. Moreover, suppose that we\nknow the heights of the vertices in V(F), which is expressed\nby the function g. Based on this information, since Hcon-\ntains every demand arc, one can narrow down the set of all\npossible heights that vcan have in H, e.g., by using the fact\nthatvis higher than every child, vcannot have the sameheight as a sibling, and so on. The value of ht∗\ng(v)is the\nsmallest candidate value of the height of vwe are left with.\nIn other words, ht∗\ng(v)gives a lower bound on the height of\nvinanysolution. This is formally stated below.\nObservation 1. If a valid BAHis compact with respect to\ng, then, for all v∈V(H),htH(v)≥ht∗\ng(v).\nProof. Clearly the claim holds for v∈Lose(S)by def-\ninition. If vis not a demand vertex, that is there are no\ndemand edges incident on v, then ht∗\ng(v) = 0 so the\nclaim holds vacuously. Otherwise we have htH(v) = 1 +\nmax w∈ChildH(v)htH(w)andhtH(w) = ht∗\ng(w)whenever\n(v, w)∈ S since then w∈Lose(S). Additionally condi-\ntion 2 never applies so htH(v)≥ht∗\ng(v).\nNote that for any vertex v,ht∗\ng(v)is completely deter-\nmined by only ht∗\ng(w)where wis a child of vor a sibling\nthat is weaker or a feedback vertex. If wis a feedback ver-\ntex then ht∗\ng(w)is determined by g, otherwise in both of the\nother cases, wis weaker than v. Soht∗\ngcan be easily calcu-\nlated in polynomial time by simply applying the definition to\nvertices in strength order beginning with the weakest. From\nnow on we will assume that we know ht∗\ng.\nThe central insight behind our algorithm that leads to\nthe notion of compactness is that in yes-instances, there is\nalways a solution where the height of every vertex that loses\na demand match is precisely this smallest candidate value\n(this is formalized in Lemma 2). This motivated our defini-\ntion of weak compactness in Definition 2. Given this fact,\nour algorithmic strategy is to “pack” the rest of the vertices\ninto the solution SBA using an intricate subroutine that is\nguided by this insight. Roughly speaking, our algorithm will\nuse a greedy approach to complete the packing, where at any\nstep, a set of partially constructed subgraphs are available\nand the goal is to make a new partial solution that contains\nthe latest vertex that is processed. However, the challenge\nour approach has to face is that in the intermediate steps\nof our algorithm we would be dealing with partially con-\nstructed subgraphs (i.e., forests) where each component is\nnot necessarily an SBA , yet we cannot simply break them\napart since they encode important height information that\nwe wish to enforce in the complete solution. Thus, the step-\nby-step challenge, solved by subroutine P ACK which we de-\nscribe later, is to carefully “glue” some of these structures\ntogether to form a supergraph in each step such that in the\nfinal step we have a BA. The trickiest aspect is to do this in\nsuch a way that if at any point we cannot find appropriate\npieces to glue together, we are able to correctly reject.\nGuaranteed compactness. We next formalize our central\ninsight and prove that every yes-instance has a valid weakly\ncompact SBA . Towards this, we argue that we can modify\nany valid SBA to achieve this property using the following\n“exchange” lemma.\nLemma 1. Suppose His a valid SBA ,(u, v)∈ S,w∈\nSiblH(v)such that v, w /∈V(F),wis lower than vand let\nBbe the set of children of vthat are at least as high as w.\nMoreover, suppose that vhas no demand out-neighbors in\nBand either (i) wis stronger than vor (ii) (u, w)/∈ S.Then, Hcan be transformed to a valid SBA ,H′where the\nfollowing hold:\n1.htH′(v) = htH(w). That is, after the transformation, v\nnow has the “old” height of w.\n2. For every x /∈B∪ {v, w},htH(x) = htH′(x). That is,\nexcept for a few vertices adjacent to vinHthe heights\nof all other vertices remain the same after the transfor-\nmation.\nProof. First of all, we observe that no vertex of Bcan be\ninV(F). Indeed, since the vertices in Bare children of v\nandvhas no demand out-neighbors in Bby the premise\nof the lemma, it follows that the vertices in Bhave no de-\nmand in-neighbors. On the other hand, recall that we have\nassumed that every non-root feedback vertex has a demand\nin-neighbor. Hence, Bis disjoint from V(F).\nThe vertices of Bare weaker than vwhich in turn must\nbe weaker than usince v /∈V(F). Similarly, wis weaker\nthanubecause of the premise that w /∈V(F). Moreover,\nnotice that no vertex in Bcan beat u. Otherwise, we would\nhave a 3-cycle containing u,vand a vertex of B, implying\nthat two of these vertices must be in V(F), a contradiction\nto the preceding arguments. More generally, ubeats B∪\n{v, w}, it must be the case that if the subgraph induced by\nB∪ {u, v, w }contains a cycle, then V(F)contains at least\ntwo vertices of B∪ {u, v, w }. But this is a contradiction to\nthe premise that {v, w}/∈V(F)and our earlier conclusion\nthatBis disjoint from V(F). Hence, the subgraph induced\nbyB∪ {u, v, w }is acyclic.\nLetℓ=htH(v)−htH(w)−1and let r0, . . . , r ℓbe the\nvertices in B, where, for each i∈ {0, . . . , ℓ},htH(ri) =\nhtH(w)+i. Moreover, for each i∈ {0, . . . , ℓ}, letTidenote\ntheBAof height htH(w) +i, rooted at ri.\nRecall that we have assumed that vhas no demand out-\nneighbors in B. So, we can remove the arc (v, x)for each\nx∈Bwithout “losing” any demand matches. Let H′be the\nrooted forest obtained from Hby removing these arcs. We\nwill now modify H′to obtain an SBA where the height of\nvis the same as the height of winH. Notice that currently,\nhtH′(v) =htH(w)and the vertices of Bare now the roots\nofBAs of heights {htH(w) +i|i∈ {0, . . . , ℓ}}.\nWe now consider the following two cases.\nCase 1: (u, w)∈ S. Then, we are in the case where wis\nstronger than vand so, wbeats r0, . . . , r ℓ. Hence, we can\nadd to H′, the arc (w, x)for each x∈B, thereby rooting\nthese BAbelow wand converting H′to an SBA . This en-\nsures that htH′(v) =htH(w). Effectively, we have swapped\nthe heights of vandw.\nCase 2: (u, w)/∈ S. In this case, remove (u, w)from H′\nand call the resulting forest H′′. Notice that htH′′(v) =\nhtH(w). Therefore, it remains to construct a BAQof height\nhtH(v)using the trees T0, . . . , T ℓalong with the subtree of\nHrooted at w(call it Tw), such that Qcan be rooted below\nu. Towards this, notice that ubeats every vertex in B∪{w}.\nHence, our task is simply to “pack” the trees Tw, T0, . . . , T ℓ\ninto a BAof height htH(v)and as long as the root of this BA\nis contained in B∪ {w}, we can just root this BAbelow u.\n  u\nvw\nTw\n4r2\nT2\n16r1\nT1\n8r0\nT0\n4\n2\n1B\nv\nw\nTw\n4r2\nT2\n16r1\nT1\n8r0\nT0\n4\n2\n1uFigure 1: An illustration of the exchange operation in Case 2\nin the proof of Lemma 1. The first image is before the trans-\nformation and the second image is after. Notice that (u, w)\nis not a demand arc in this case. Moreover, in this figure, we\nare assuming that r1beats wandr2while wbeats r0. The\nexchange argument in Case 1 is straightforward. We simply\nmake r0, r1, r2children of w.\nSetp0=wand for each i∈[ℓ], define the vertex pias\nthe stronger vertex in the pair {pi−1, ri−1}andsias the\nweaker vertex. Similarly, let J0denote the subtree of H\nrooted at wand for every i∈[ℓ+ 1], define Jito be the\nBAobtained by taking Ji−1andTi−1and making pithe\nroot, i.e., adding the arc (pi, si). Notice that J1, . . . , J ℓ+1\nexist since, for each i∈ {0, . . . , ℓ}, the trees JiandTiare\nBAof height htH(w) +i. Moreover, Jℓ+1is aBAof height\nhtH(w) +ℓ+ 1 = htH(v)and is rooted at pℓ+1∈B∪{w},\nwhich is weaker than u. Hence, we can simply delete from\nH′′the trees J0, T0, . . . , T ℓ, add the BAJℓ+1and make its\nrootpℓ+1, a child of u. See Figure 1 for an illustration of this\nexchange operation.\nIn both cases, it is straightforward to check that the second\nstatement of the lemma holds.\nLemma 2. If there exists a valid SBAH, then there exists a\nvalid, weakly compact SBA .\nProof Sketch. In any non-weakly-compact SBA there is a\nweakest vertex that contradicts the definition of weak com-\npactness. We choose a valid SBA that has the strongest such\n“certificate of non-compactness” and aim to find another\nSBA with a stronger one which would give the required con-\ntradiction. By properties of BAs this certificate vertex has asibling that is the correct height so we apply Lemma 1 to\nswap these heights. This gives us the required SBA .\nProof. Note that if a valid SBA His not weakly com-\npact, then there is a weakest vertex v∈Lose(S)such\nthathtH(v)̸=ht∗\ng(v), where gis the restriction of htH\ntoV(F). Clearly, vcannot be a feedback vertex since by\ndefinition of g,htH(v) =ht∗\ng(v). We call vthecertificate\nof non-compactness ofH. In the rest of the proof of this\nlemma, we assume that His chosen in such a way that its\ncertificate of non-compactness vis the strongest possible\namong those of all valid, non-weakly-compact SBA . Then,\nhtH(x) =ht∗\ng(x)for every xthat is weaker than vand loses\na demand match.\nWe aim to find an H′with a strictly stronger certificate\nof non-weak-compactness, that is htH′(x) =ht∗\ng′(x)for all\nx∈Lose(S)that is either vor weaker than v, where g′is the\nrestriction of htH′toV(F). This would give us the required\ncontradiction.\nFirst of all, vmust be higher than all of its children in\nHandvcannot have the same height as any of its siblings\ninH. So, htH(v)̸=ht∗\ng(v)implies that htH(v)>ht∗\ng(v).\nMoreover, since His aBA, there must exist w∈SiblH(v)\nsuch that htH(w) = ht∗\ng(v). Let ube the parent of vand\nwinH. Moreover, w /∈V(F)since otherwise, ht∗\ng(w)and\nht∗\ng(v)would coincide, which is not possible.\nOur next goal is to prove the following claim, which will\nenable us to use our exchange arguments from Lemma 1.\nClaim 2. H,S, u, v, w satisfy the premise of Lemma 1.\nProof. We have already argued that htH(v)>ht∗\ng(v)and\nhtH(w) =ht∗\ng(v), implying that htH(w)<htH(v), i.e., w\nis lower than vinH.\nNow, suppose for a contradiction that vhas, among its\nchildren that are at least as heavy as w(i.e., the set B), a\ndemand-out-neighbor v′. Since v /∈V(F), it follows that\nthe arc (v, v′)is not a feedback arc, implying that v′is\nweaker than v. By our selection of vas a certificate of non-\ncompactness, it follows that htH(v′) =ht∗\ng(v′). This implies\n(by invoking Condition 1 in Definition 2) that:\nht∗\ng(v)>ht∗\ng(v′) =htH(v′)≥htH(w) =ht∗\ng(v).\nThis is a contradiction, hence we conclude that vhas no\ndemand out-neighbors in the set B.\nFinally, it remains to argue that either wis stronger than\nvor(u, w)is not a demand match. We will argue that\nthe former holds. Indeed, if wis weaker than v, then this\nwould contradict Condition 2 in Definition 2 because ht∗\ng(v)\ncould not have htH(w)as a possible candidate value. Hence,\nLemma 1 is applicable.\nNow, let H′be the valid SBA obtained by invoking\nLemma 1 on H,S, u, v, w . Then, we have that htH′(v) =\nhtH(w) =ht∗(v). We have the following claim:\nClaim 3. For every xthat is weaker than vand loses a de-\nmand match, htH′(x) =ht∗(x).Proof. The second statement of Lemma 1 guarantees that\nfor every xthat is weaker than vand disjoint from B∪\n{v, w}, we have that htH′(x) =htH(x). Here, Bis the set\nof children of vthat are at least as high as winH. Since we\nalready know that for every such x,htH(x) =ht∗(x), we\nconclude that htH′(x) =ht∗(x).\nIt remains to argue that for every x∈B∪ {w}that is\nweaker than vand loses a demand match, htH′(x) =ht∗(x).\nWe have already argued that the vertices in Bmust have a\ndemand in-degree of 0. Hence, x /∈B. We next consider\nthe possibility that x=w. Recall that since Lemma 1 was\napplicable on H,S, u, v, w , we know that either whas a de-\nmand in-degree of 0 or wis stronger than v. In either case,\nwe have that x̸=w, completing the proof of the claim.\nBy definition v /∈V(F)and we know that w /∈V(F)\nsince every feedback vertex has a demand parent and if w\nshared a demand parent with vthen condition 2 of Defini-\ntion 2 would have ensured that ht∗\ng(v)̸=htH(w). So, we\ncan apply Lemma 1 to get H′where htH′(v) =htH(w) =\nht∗\nhtH(v). Condition 2 of Lemma 1 ensures that htHonly dif-\nfers from htH′onB∪ {v, w}, which is disjoint from V(F).\nHence ht∗\nhtH=ht∗\nhtH′sohtH′(v) =ht∗\nhtH(v). Claim 3 still\nholds in this setting so H′has a stronger certificate of non-\nweak-compactness which is a contradiction.\nBefore moving to the description of our packing subrou-\ntine, we need to define a natural relaxation of BAto account\nfor feedback vertices.\nPartial Binomial Arborescences. In our greedy packing\nstrategy, we will process vertices from weakest to strongest,\nwith the underlying assumption that when we arrive at a ver-\ntexv, all descendants of vare weaker and have their respec-\ntive sub-arborescences built in an earlier step. When Tis\nacyclic this works fine, however this does not go smoothly\nwhen we have cycles since a descendant of the current ver-\ntex,v, may actually be stronger than v. Hence it is not\nyet processed by our algorithm, and consequently its sub-\narborescence is not yet built. This leads to the scenario that\ntheBAin the solution that is rooted at vcannot be fully\nbuilt either. Notwithstanding this difficulty, we note that the\npartial structures our algorithm deals with have enough BA-\nlike properties that one direction of Proposition 2 still holds.\nThis leads us to the following definition, which relaxes the\nconditions of a BAwhen feedback vertices are encountered.\nDefinition 3 (Feedback descendants and partial binomial\narborescence) .Given a rooted forest Qand a vertex v∈\nV(Q), define the feedback descendants of vinQ,\nFDesc Q(v) :=[\nf∈DescQ(v)∩V(F),f̸=vDesc Q(f)\\ {f}\nGiven a BAHof height i, on a tournament digraph T\nwith feedback arc set F, we call a subtree H′ofHapartial\nbinomial arborescence ( PBA ) of height iifV(H)\\V(H′)⊆\nFDesc H(root(H)).\nIn Definition 3, FDesc Q(v)are strict descendants of a\nfeedback vertex that is itself a strict descendant of v. Equiv-\nalently these are vertices xwhere the path from vtoxcon-\ntains a feedback vertex that is not vorx, that is, xis “past” a  \nFDescH(u)H\n0\n00\n001\n00\n00112\n14 3uv\nw\nx\nyFigure 2: A BAHand the feedback descendants of its root.\nFeedback vertices are in red and demand arcs are in blue.\nht∗\ng(where gishtHrestricted to V(F)) is noted next to each\nvertex. ht∗\ng(v) = 3 due to its demand siblings. The green\narcs are those that would be added during the inner loop of\nthe algorithm when vn−i=vandj= 2. The call to P ACK\nwould use P={w, x, y }, add the arcs (w, x)and(x, y)and\noutput w. Finally Step 11 would add the arc (v, w).\nfeedback vertex. A PBA is aBAthat is missing some feed-\nback descendants of its root. Figure 2 contains an example\nof feedback descendants.\nObservation 2. Forn >0, ifHn=Sn−1\ni=0Hiwhere Hiis\naPBA of height irooted at vitogether with an edge from vn\nto each vithenHnis aPBA of height nrooted at vn.\nGuessed size. Definition 3 defines the height of a PBA im-\nplicitly. Our algorithm constructs an SBA by gluing PBA s of\nspecific heights together so it needs their heights, or equiva-\nlently an appropriate notion of their size, to see if there are\nany good candidate PBA s to glue together. The following\ndefinition allows us to calculate the size of a PBA .\nDefinition 4. Given a rooted forest Q, and g:V(F)→\n[log(n)]define the guessed size of v,\nszQ,g(v) =(\n2g(v)ifv∈V(F)\n1 +P\nw∈ChildQ(v)szQ,g(w)otherwise\nWe will drop the reference to gwhen it is clear from the\ncontext.\nNote that if His an SBA andgishtHrestricted to V(F),\nthenszH,g(v) =|Desc H(v)|= 2htH(v)for all v. Further-\nmore if H′⊂His aPBA of height irooted at vthen\nszH′,g(v) = 2i. Effectively, deleting feedback descendants\nofvdoes not change the value of sz(v). Note that the con-\nverse is not true so we will need to prove that a given sub-\ngraph is a PBA andszwill then check its height.\nSince Qis a rooted forest, each vertex is the child of at\nmost one vertex. So applying the definition recursively will\nonly require calculating szQ(w)once for each vertex w∈\nV(Q). Therefore szQ(v)can be calculated in polynomial\ntime for any Qandvdirectly from the definition.\nThe subroutine P ACK.The following lemma describes a\ncrucial subroutine for us. Informally, the subroutine creates\naPBA of height jby adding arcs between the provided ver-\ntices in one of its inputs ( P) and then outputs the root of thisPBA . Crucially, it is a very local algorithm: it only affects\nvertices from P. This allows us to repeatedly call it with-\nout undoing work it has already done in a previous call. In\norder to describe this property we will need the following\ndefinition: suppose QandQ′are both rooted forests, then\ndefine DescQ\nQ′(v)as the set of vertices that are descendants\nofvinQ′but that have no parent in Q. Note that all of these\nvertices except vmust have a parent in Q′.\nLemma 3 (Packing lemma) .There is a polynomial-time al-\ngorithm PACK that:\n•Takes as input a tuple (Q, P, j )such that:\n1.Q⊂Tis a valid rooted forest.\n2. For every w∈P:\n(a)Q[Desc Q(w)]is aPBA of height at most j.\n(b)whas no parent in Q.\n3.P\nw∈PszQ(w)≥2j\n•Outputs a tuple (Q′, v)such that:\n1.Q′⊂Tis a valid rooted forest and Q′is a supergraph\nofQ.\n2.Q′[Desc Q′(v)]is aPBA of height j.\n3.vhas no parent in Q′.\n4. If Q′[Desc Q′(x)]̸=Q[Desc Q(x)]then x∈\nDescQ\nQ′(v). Additionally DescQ\nQ′(v)⊆P. That is, all\nvertices affected by PACK become descendants of vin\nQ′and were from P.\n5. For all s∈P\\DescQ\nQ′(v)andt∈DescQ\nQ′(v)we\nhave szQ(s)≤szQ(t). That is, the algorithm uses the\nvertices with largest height.\nProof. Initially let Q0=QandP0=P. For each i≥0\neither:\n• There exists v∈Piwith szQi(v) = 2j. Then return\n(Qi, v).\n• Or there is no such v. Then let x, y be two vertices of\nlargest szwithxthe stronger. More precisely, choose\nx, y∈Pisuch that szQi(x) = szQi(y)and for all\nother a, b∈Pisatisfying szQi(a) =szQi(b)we have\nszQi(a)≤szQi(x). Let Qi+1=Qi∪ {(x, y)}and\nPi+1=Pi\\ {y}then repeat for the next i.\nCorrectness We first show that the total value of szacross\nthe vertices of Piis conserved. This is because, when we\nremove yfrom Piafter making ya child of x,szQi(y)\ngets added to szQi+1(x). The formal argument follows. Let\n(x, y)be the arc present in Qi+1but not Qi. Then Pi+1=\nPi\\ {y}. For every vertex w∈Pi+1except xwe have\nszQi+1(w) =szQi(w). Also, every vertex in Pi+1⊆Phas\nno parent in Qand so, is not a feedback vertex. In particular,\nszQi+1(x) = 1 +X\nw∈ChildQi+1(x)szQi+1(w)\n=szQi(y) + 1 +X\nw∈ChildQi(x)szQi(w)\n=szQi(y) +szQi(x). (1)Note that, since the descendants of every vertex w∈PinQ\nform a PBA ,szQ(w)is a power of two and hence (1) shows\nthat for all i≥0andw∈Piwe have szQi(w)is a power\nof two. Now\nX\nw∈Pi+1szQi+1(w) =szQi+1(x) +X\nw∈Pi+1\\{x}szQi(w)\n=szQi(x) +X\nw∈Pi\\{x}szQi(w)\n=X\nw∈PiszQi(w) (2)\nWe can now argue that the algorithm always terminates. Let\nn=|P|. In each iteration one vertex is removed from Pi,\nso after n−1iterations, if the algorithm has not already re-\nturned, there is only one vertex remaining in Pn−1, call it x.\nSince the total value of szacross the vertices of Piis con-\nserved we know thatP\nw∈Pn−1szQn−1(w) =szQn−1(x)≥\n2j. Each vertex in P0is the root of a PBA inQ0, after join-\ning two such vertices by an arc we have a PBA of height\none larger by definition. So for all w∈Pi,szQi+1(w)≤\n2szQi(w)and both numbers are powers of two. Hence the\nalgorithm will return before we have szQi(w)>2j. Hence,\nszQn−1(x)≤2j, implying that szQn−1(x) = 2jand the\nalgorithm will return (Qn, x).\nWe now check that Q′satisfies each of the conditions of\nthe lemma:\n1. By construction, Q′is a supergraph of Qand a valid\nrooted forest (recall Qis already valid). The only arcs\ninQ′but not Qare chosen to be from a stronger to a\nweaker vertex. Since at most one vertex without a par-\nent is a feedback vertex (the root) and the new arcs are\nalways between vertices of Pwhich all have no parent,\nQ′⊂T.\n2. Each vertex in P0is the root of a PBA inQ0, after joining\ntwo such vertices by an arc we have a PBA of height\none larger by definition. So, for each i, the descendants\nof every vertex in Piform a PBA inQi, in particular\nvinQ′. The height of this PBA rooted at visjsince\nszQ′(v) = 2j.\n3. Every vertex in P0has no parent in Q=Q0. Any vertex\nthat is given a parent in Qi+1is removed from Pi+1so\nthe invariant that each vertex in Pihas no parent in Qiis\nmaintained. Since vis chosen from Piit has no parent in\nQ′.\n4. Every arc is added between vertices of Pthat have the\nlargest szinPi. Since after this the new root has even\nlarger sz, it will either be returned in the next iteration or\nhave a new arc added in future. Hence the endpoints of\nevery new arc end up forming a PBA rooted at vand be-\ncome descendants of vinQ′and since they are in Pthey\nhad no parent in Q. These are the only vertices whose\ndescendants change since vhas no parent in Q′.\n5. This is ensured by our choice of x, y.\nWe will also need the following corollary of Lemma 3.Corollary 1. Lemma 3 holds if every occurrence of PBA is\nreplaced by BA.\nInformally this says that the same subroutine can be used\nto pack BAs: by replacing PBA s with BAs in the input we\nget a BAin the output.\nWe can now present our main result.\nTheorem 4. An instance of DEMAND -TF,(T,S), can be\nsolved in time nO(k)where kis the feedback arc set number\nofT.\nWe first describe the algorithm claimed in the above state-\nment, following which we give a proof of correctness.\nIn our algorithm, we first guess an injective p:V(F)→\nV(T)∪ {⊥} representing the guessed parent of each feed-\nback vertex. Then, for each v∈V(F), we add the arc\n(p(v), v)toS(unless v∈Lose(S)orp(v) =⊥) to en-\nsure this guess is honored by the algorithm and justify our\nassumption that every feedback vertex has a parent. We then\nguess g:V(F)→[log(n)]representing the guessed heights\nof the feedback vertices and calculate ht∗\ng. Now we can san-\nity check g: if(u, v)∈ S we check that ht∗\ng(u)>ht∗\ng(v)\nand if (u, v),(u, w)∈ S we check that ht∗\ng(v)̸=ht∗\ng(w).\nFinally, if p(v) =⊥, we check that g(v) = log( n)since this\nis guessing that vhas no parent, i.e., it is the root. Following\nthis, we call Algorithm 1 once with each possible combina-\ntion of the guesses of gand value of Sgiven by our guess\nofp. We reject the input if every invocation of Algorithm 1\nfails to output a solution.\nAlgorithm 1:\n1Q0,0←(V(T),S);\n2for0≤i < n do\n3 for0≤j <ht∗\ng(vn−i)do\n4 ifthere exists y∈Child Qi,0(vn−i)with\nszQi,0(y) = 2jthen\n5 Qi,j+1←Qi,j;\n6 else\n7 LetPi,jbe the set of vertices, w, that are\nweaker than vn−i, have no parent in\nQi,j, and have szQi,j(w)≤2j;\n8 ifP\nw∈Pi,jszQi,j(w)<2jthen\n9 Reject;\n10 (bQi,j+1, wi,j)←PACK(Qi,j, Pi,j, j);\n11 Qi,j+1←bQi,j+1∪ {(vn−i, wi,j)};\n12 Qi+1,0←Qi,ht∗\ng(vn−i);\n13LetP∗be the set of vertices without parents in Qn,0;\n14ifP\nz∈P∗szQn,0(z)< n then\n15 Reject;\n16(Q∗, v∗)←PACK(Qn,0, P∗,log(n));\n17return Q∗;\nThe outer loop iterates over each vertex from weakest to\nstrongest. By Proposition 2, vn−ineeds to have a child of\nheight jfor each jthat the second loop considers. Step 4checks if such a child already exists (this happens when it\nis a demand child). Otherwise we check if there are enough\nvertices that could become descendants of vn−iand then call\nPACK to create such a child (see Figure 2). After applying\nthis process to every vertex we will have a number of BAs\nthat we can then pack into an SBA at Step 16.\nRecall that ht∗\ngcan be calculated in polynomial time. Also\nht∗\ng≤log(n)so both loops run at most ntimes. Recall that\neach value of szcan be calculated in polynomial time. There\nare at most nchildren of any vertex so the existence of y\ncan be checked in polynomial time. Each Pi,jcan be calcu-\nlated in polynomial time by checking whether each vertex\nsatisfies the conditions on vertices of Pi,j. Finally P ACK is\na polynomial-time subroutine. So since there are log(n)O(k)\npossible values for gandnO(k)possible values for p(and\nthe resulting value of S), the overall running time is nO(k).\nProof of correctness. We first consider the case where the\nalgorithm outputs Q∗and prove that Q∗is a valid SBA and\nhence the algorithm has correctly identified a positive in-\nstance.\nLemma 4. For each 0≤i < n , ifvn−ihas no parent in\nQn,0, then the descendants of vn−iinQn,0form a BAof\nheight ht∗\ng(vn−i)\nProof. We will begin with the following inductive hypothe-\nsis.\nHypothesis 1. For all i < i′, for all j <ht∗\ng(vn−i), either:\n•there exists y∈Child Qi,j(vn−i)such that the descen-\ndants of yinQi,jform a PBA of height j, or\n•(Qi,j, Pi,j, j)satisfy the conditions of Lemma 3.\nWhen i= 0there are no vertices weaker than vn−i=vn\nand hence P0,j=∅. So the first possibility in the hypothesis\nmust occur for every jsince otherwise the algorithm would\nhave rejected at Step 9.\nNow we use this hypothesis to prove the following claim.\nClaim 4. For all i < i′, for all ℓ≤ht∗\ng(vn−i′), if either\nvn−i∈Lose(S)orvn−ihas no parent in Qi′,ℓ, then the de-\nscendants of vn−iinQi′,ℓform a PBA of height ht∗\ng(vn−i).\nProof. We aim to find, for all j < ht∗\ng(vn−i), ay∈\nChild Qi′,ℓ(vn−i)such that the descendants of yinQi′,ℓform\naPBA of height j.\nWhen the first possibility in the hypothesis occurs, the al-\ngorithm checks this and sets Qi,j+1←Qi,j. Otherwise we\ncan apply Lemma 3 and hence the descendants of wi,jin\nbQi,j+1form a PBA of height j. So choosing y=wi,jguar-\nantees we have such a yinQi,j+1.\nSince vn−ieither has a parent already in Qi,jor has no\nparent still in Qi′,ℓit is not in DescQi,j′\nQi,j′+1(wi,j′)for any\nj≤j′≤ℓso its descendants are unchanged by P ACK up\ntoQi′,ℓ. Any other arcs are added below vertices stronger\nthanvn−iso only affect the descendants of vn−iif they are\nbelow a feedback vertex, say z. In this case the new descen-\ndants are chosen to be of height less than ht∗\ng(z)so they haveno effect on whether a PBA is formed below yor its height.\nSo the descendants of yinQi′,ℓalso form a PBA of height\nj. InQ0,0,vn−ihad no children of height greater than or\nequal to ht∗\ng(vn−i)since all such children would be demand\nchildren and this would contradict Definition 2. No arcs ad-\njacent to vn−iare added before Qi,0so, in Qi′,ℓ,vn−ihas\nchildren whose descendants form a PBA of each height up\ntoht∗\ng(vn−i)and no others: this is exactly the conditions for\nCorollary 2.\nWe can now extend the hypothesis to i=i′. We first show\nthatQi′,jis a valid rooted forest and a subgraph of Tfor all\nj <ht∗\ng(vn−i′). Initially Qi′,0=Qi′−1,ht∗\ng(vn−i′+1)so is a\nvalid rooted forest and a subgraph of Tby assumption. We\nproceed by induction on j. Either Qi′,j+1=Qi′,j(and we\nare done) or it is just Qi′,jwith the additional arcs added at\nsteps 10 and 11. Since 10 is just a call to P ACK,bQi′,j+1is a\nvalid rooted forest and a subgraph of Tby Lemma 3 assum-\ning that (Qi′,j, Pi′,j, j)satisfies the premises of Lemma 3.\nAlso by Lemma 3 wi′,j+1has no parent in bQi′,j+1and is in\nPi′,j+1and hence weaker than vn−i′and not a feedback ver-\ntex. So Qi′,j+1is also a valid rooted forest and a subgraph\nofT.\nIt remains to show that, when the first possibility of the\nhypothesis does not occur, the conditions of Lemma 3 on\nPi′,jare satisfied. The way Pi′,jis chosen and that we have\nnot rejected at Step 9 show all these conditions except that,\nfor each z∈Pi′,j,Qi′,j+1[Desc Qi′,j+1(z)]is aPBA .z\nclearly has no parent in Qi′,jand for each zthere is some\ni < i′withz=vn−isince zis weaker than vn−i′so this\nfinal condition is shown by Claim 4.\nTherefore, by induction on i′we have the hypothesis for\nalli < n and hence also Claim 4 for all i < n . It remains\nto prove that the descendants of vn−iinQn,0form a BA\nand not just a PBA . Suppose for a contradiction this was\nnot the case, then there must be some isuch that, for every\nu∈Desc Qn,0(vn−i)the descendants of udo form a BA\nbut the descendants of vn−iitself do not. In particular the\ndescendants of every child of vn−iform a BAand hence the\nonly way the descendants of vn−ido not form a BAis for\nvn−ito be a feedback vertex and “missing” a child, that is\nthere exists a j < ht∗\ng(vn−i)such that there is no child of\nvn−iwhose descendants form a BAof height j. However,\nin the proof of Claim 4 we find exactly such a child which\nis a contradiction. This extends Claim 4 to BAs for Qn,0,\nproving the Lemma.\nHence (Qn,0, P∗,log(n))satisfies the premises of\nLemma 3 and therefore, by Corollary 1, the descendants of\nv∗inQ∗form a BAof height log(n)which is an SBA . So\nwhen the algorithm outputs a graph it is a valid SBA .\nIt remains to show the converse. Suppose for a contradic-\ntion that the algorithm rejects for every guess of gbut we\nhave a positive instance: that is there exists a valid SBA . By\nLemma 2 there exists a valid, weakly compact SBAH⊂T.\nLemma 5. Given a valid, weakly-compact SBAH, for all\ni, jthere exists a valid SBAHi,jthat is compact with respect\ntohtHrestricted to V(F), such that Qi,j⊂Hi,j.8422 1688\n88 164228JlAlvn-i’u Z wxyBl\nJl+1\nvn-i’yu wxFigure 3: An example of the operation of Lemma 5. The numbers represent szK. Note that in this case ybeats uso has taken\nthe place of uas the child of vn−iof size 16. Both vn−iandwhave many other children not pictured here. Note also that it\nmay be the case that vn−i=w.\nProof. LetgbehtHrestricted to V(F). Clearly Q0,0⊂H\nsince His a valid SBA and so, contains every demand arc.\nAlsoQ0,j=Q0,0for each 0≤j <ht∗(vn−i)(see the proof\nof Lemma 4) so let H0,j=H. Now assume that Qi,j⊂Hi,j\nfor0≤i < i′and0≤j≤ht∗(vn−i). We want to extend\nthis to i=i′. Initially, Qi′,0=Qi′−1,ht∗\ng(vn−i′+1)so let\nHi′,0=Hi′−1,ht∗\ng(vn−i′+1). Now we can assume that Qi′,j⊂\nHi′,jand aim to prove that Qi′,j+1⊂Hi′,j+1. Additionally\nifvn−i′is not a demand vertex then ht∗\ng(vn−i) = 0 so we\nare already done. Let ube the unique vertex of height jfrom\nChild Hi′,j(vn−i′). If(vn−i′, u)∈ SthenQi′,j+1=Qi′,jso\nwe can set Hi′,j+1=Hi′,jand we are done. Otherwise, let\nR=DescQi′,j\nQi′,j+1(wi′,j)(these are the vertices given parents\nby P ACK), initialise J0=Hi′,j, and repeat the following for\nℓ≥0:\n1. Let ube the unique vertex of height jfrom\nChild Jℓ(vn−i′)andAℓ=DescQi′,j\nJℓ(u)\\FDesc Jℓ(u).\n2. IfAℓ=Rthen set Hi′,j+1=Jℓand we are done.\n3. Otherwise, pick a vertex y∈R\\Aℓmaximizing\nszQi′,j(y). Ify=root(Jℓ)letx=y. Otherwise, let\nwbe the last vertex on the path from root(Jℓ)toyinJℓ\nthat is in Lose(S)∪ {root(Jℓ)}. Letxbe the vertex after\nwon this path.\n4. Let Bℓ=DescQi′,j\nJℓ(x)\\FDesc Jℓ(x). LetKbe the rooted\nforest obtained from Jℓby deleting each in-edge to a ver-\ntex in Aℓ∪Bℓ(this includes (vn−i′, u)and(w, x)ifyis\nnot the root of Jℓ).\n5. Find a Z⊂Aℓ\\Rsuch thatP\nz∈ZszQi′,j(z) =\nszQi′,j(y).\n6. Let (K′, a∗)←PACK(K,(Aℓ\\Z)∪ {y},htJℓ(u))\nand(K′′, b∗)←PACK(K′,(Bℓ\\ {y})∪Z,htJℓ(x)).\n7. Finally, let Jℓ+1 = K′′∪ {(vn−i′, a∗)}\n(∪{(w, b∗)}ify̸=root(Jℓ)).Some explanation: Aℓare the vertices that Jℓhas used to\nmake the subtree of height jbelow vn−i′. We can think of\nthese as the “decisions” that Jℓhas made. If they agree with\nthe decisions that Qi′,j+1has made ( R) then we are done.\nIf not we find a vertex that Qi′,j+1chose but Jℓdid not ( y)\nand look at where it is in Jℓ.Bℓis the decisions that Jℓmade\naround y. By swapping the set that yis in (and preserve sum\nof betas using Z) we can calculate a new Jℓ+1where yis in\nAℓ+1.\nWe now justify that this process does indeed produce a\nvalid SBAHi′,j+1as required. Clearly every vertex in Aℓis\nweaker than vn−i′and, since y∈R⊆Pi′,j,yis weaker\nthanvn−i′so(vn−i′, a∗)is inT.\nIfy̸=root(Jℓ)then we need to show that (w, b∗)is in\nT. Since wis the lastvertex from Lose(S)andxis after\nit,x /∈Lose(S), that is (w, x)/∈ S . Suppose for a con-\ntradiction that wis strictly weaker than vn−i′. The condi-\ntions on wensure Claim 4 applies: the descendants of w\ninQi′,jform a PBA of height ht∗\ng(w). Therefore, since\nQi′,j⊂Jℓ,Desc Jℓ(w)\\FDesc Jℓ(w) = Desc Qi′,j(w)\\\nFDesc Qi′,j(w). But y∈Desc Jℓ(w)\\FDesc Jℓ(w)whereas\ny /∈Desc Qi′,j(w)\\FDesc Qi′,j(w)since yhas no parent in\nQi′,j. Sowis at least as strong as vn−i′(they may be the\nsame vertex). Every vertex in Aℓ(and hence Z) is weaker\nthanusince all feedback vertices have parents and any\nstronger vertices which are descendants of uare excluded\nbyFDesc Jℓ(u). Since uis weaker than vn−i′and addition-\nally not a feedback vertex, the edge (w, b∗)is inT.\nWe need to check that we only make valid calls to P ACK.\nBy our choice of Zwe have\nX\na∈(Aℓ\\Z)∪{y}szK(a) =X\na∈AℓszK(a) =htJℓ(u) (3)\nand the same for x. Both sides of the last equality are simply\ncounting the vertices that are descendants of uinH: the leftside uses gto count descendants of feedback vertices and\nexplicitly counts the others. By the assumption of compact-\nness with respect to gin the inductive hypothesis this agrees\nwith the right hand side. The descendants of uandxinJℓ\nform a PBA since initially Jℓ=Hi′,jwhich is an SBA and\nthe only changes are the result of P ACK.\nIt remains to show that such a Zexists for each ℓ. We\nhaveAℓ\\R⊂Pi′,j\\RsoszQi′,j(z)≤szQi′,j(y)for all\nz∈Aℓ\\Rby Lemma 3. Finally the equality\nX\nz∈AℓszQi′,j(z) =X\nz∈RszQi′,j(z) = 2j\nis conserved for all ℓby the choice of Z. Hence\nX\nz∈Aℓ\\RszQi′,j(z) =X\nz∈R\\AℓszQi′,j(z)≥szQi′,j(y)\nbecause yis one of the elements of the second sum. There-\nfore there exists a Zfor each ℓ.\nSince Z⊂Aℓ\\Randy∈R\\Aℓ,|Aℓ∩R|increases\nby one each iteration, so after at most |R|iterations the loop\nwill terminate. At this point Aℓ=R, that is, the descendants\nof the child of vn−i′of height j(including the child itself)\nare the same in both Qi,j+1andHj+1. Also every vertex\ninLose(S)has a parent in Qi′,jso their descendants are\nunchanged by P ACK and since htJℓ+1(a∗) = htJℓ(u), no\nheights are changed there either. Hence their heights agree\nbetween Hi′,jandHi′,j+1and, since Hi′,jis compact with\nrespect to g,Hi′,j+1is compact with respect to gtoo. So by\ninduction on jwe have a valid SBAHi′,jwhich is compact\nwith respect to gand a supergraph of Qi′,jfor each j. This\ncompletes the proof of the lemma.\nConsider the run where the algorithm guesses gashtH\nrestricted to V(F). First, suppose the algorithm rejects at\nStep 9. Hi,jis compact with respect to gsohtHi,j(vn−i)≥\nht∗\ng(vn−i), by Observation 1. Hence there exists x∈\nChild Hi,j(vn−i)with htHi,j(x) =j < ht∗\ng(vn−i)andxis\nnot a demand vertex since the algorithm never rejects in this\ncase. NowP\ny∈DescQi,j\nHi,j(x)szQi,j(y)≥2jby a similar ar-\ngument as for (3). This time the feedback descendants are\nincluded in the sum and hence are double counted leading\nto an inequality. All vertices y∈DescQi,j\nHi,j(x)are weaker\nthanxand hence vn−isince xis not a feedback vertex. Also\n2j≥szHi,j(y)≥szQi,j(y)since Qi,j⊂Hi,j. Finally they\nare chosen to not have a parent in Qi,jso they must be in\nPi,j. But then we have\n2j≤X\ny∈DescQi,j\nHi,j(x)szQi,j(y)≤X\nw∈Pi,jszQi,j(w)\nwhich contradicts the rejection of the algorithm.\nFinally, suppose the algorithm rejects at Step 15. Then\nconsider Hn,0=Hn−1,ht∗\ng(v1)⊃Qn−1,ht∗\ng(v1)=Qn,0.\nSince Hn,0is an SBA , clearly\nX\nz∈P∗szQn,0(z)≥2htHn,0(root(Hn,0))=n\ncontradicting the rejection.In the following sections, we describe how our algorithm\ncan be extended or modified to obtain the further algorithmic\nresults outlined in the Introduction.\nFPT Algorithm for D EMAND -TF When Upsets Are\nDemanded\nSuppose that F⊆ S. A closer inspection of our algorithm\nshows that where we assumed that every feedback vertex\nexcept the root has a demand parent, a weaker assumption is\nsufficient, specifically, the following property.\nProperty 1. For all v /∈Lose(S)there is no (u, v)∈F.\nThat is, every vertex without a demand parent has no in-\nfeedback arc. Equivalently Lose(F)⊆Lose(S). Clearly this\nis implied by F⊆ S . This means that any vertex that is\nstronger than v, and only these vertices, can be used as its\nparent. Furthermore, even if vis a feedback vertex the de-\nscendants of vare weaker than its ancestors, at least un-\ntil further feedback vertices. Effectively vdoes not act as\na feedback vertex (if there is a feedback edge (v, w)we will\nhandle this when talking about w) so we replace V(F)with\nLose(F)throughout the algorithm: these are the remaining\nfeedback vertices that still behave as such. Whenever the\nanalysis refers to a vertex not being a feedback vertex be-\ncause it has no demand parent we now simply use Property\n1: although it may be a feedback vertex, it behaves as a non-\nfeedback vertex since it has no incoming feedback edges.\nThe only changes required to our algorithm are there-\nfore the removal of the guess of the parents of feedback\nvertices (eliminating the overhead of nO(k)) and a slightly\nsmaller set of feedback vertices used in the domain of g\nand the definitions of ht∗andsz. Since the guess of the\nparents of the feedback vertices has been removed the run-\ntime is dominated by the guess of g: there are (logn)O(k)=\n(klogk)O(k)nO(1)possibilities for gso the overall run time\nis FPT in k.\nXP Algorithm for D EMAND -TF With Specified\nRounds for Demands\nThe first key observation is that the round of a match is ex-\nactly the height of that the losing player of the match takes in\nthe solution (numbering rounds from zero). That is, if a de-\nmand match (u, v)occurs at round iin the tournament rep-\nresented by an SBAHthenhtH(v) =i. So if every demand\nmatch has a specified round this is equivalent to specifying\nthe heights of every vertex in Lose(S). Theorem 4 guesses\nthe heights of vertices in V(F)and calculates the heights of\nall other vertices in Lose(S). So by extending the domain of\ngtoLose(S)∪V(F)and ensuring it agrees with the required\nheights the algorithm will ensure that the SBA it outputs will\nsatisfy our additional constraints. Note that when we extend\nthe domain of gin this way, the guess is still made only for\nthe vertices in V(F)as the value of gfor the vertices in\nLose(S)is part of the input. Thus, the running time of the\nnew algorithm remains the same as that of Theorem 4.\nFuture Work\n1. Theorem 4 shows the viability of a systematic study of\nthe parameterized complexity of Demand-TF with re-spect to other parameters studied for TF, such as feed-\nback vertex set.\n2. We also propose a relaxation of our model where, if,\nthere is no seeding that enables every demand match to\ntake place, then the goal is to compute a seeding that\nmakes the maximum number of demand matches take\nplace. How efficiently could one do this? Our work im-\nplies a 2dnO(k)-time algorithm for this problem, where\ndis the total number of demands and kis the feedback\narc set number – simply guess the set of satisfied de-\nmands and invoke Theorem 4. A natural follow-up ques-\ntion is whether one can remove the exponential depen-\ndence on dand have an XP algorithm parameterized\nbykalone? Resolution of this question would require\nadditional ideas to this paper. For instance, we can no\nlonger assume that every vertex has at most one demand\nin-neighbor. Efficient approximation algorithms for this\nproblem are also an interesting research direction. In\nterms of exact-exponential-time algorithms, it is easy to\nsee that the algorithm of Theorem 3 already extends nat-\nurally to this variant.\n3. Could one extend our results for Demand-TF to the prob-\nabilistic model? Here, the tournament designer has two\nnatural objectives – maximize the probability that all de-\nmand matches are played or maximize the expected num-\nber of satisfied demands.\n4. The edge-constrained version of S UBGRAPH ISOMOR -\nPHISM we define in this paper is a problem of inde-\npendent interest. Techniques such as color coding (Alon,\nYuster, and Zwick 1995) could give FPT algorithms for\nthis problem parameterized by the size of some simple\npattern graphs. However, the behavior of this problem\nwith respect to various structural parameterizations of\ngraphs is less clear and we leave this as a research di-\nrection of broad interest to the algorithms community.\nAcknowledgements\nSushmita Gupta acknowledges support from SERB’s\nMATRICS Grant (MTR/2021/000869) and SUPRA Grant\n(SPR/2021/000860). Ramanujan Sridharan acknowledges\nsupport by the Engineering and Physical Sciences Research\nCouncil (grant numbers EP/V007793/1 and EP/V044621/1).References\nAlon, N.; Yuster, R.; and Zwick, U. 1995. Color-Coding. J.\nACM , 42(4): 844–856.\nAziz, H.; Gaspers, S.; Mackenzie, S.; Mattei, N.; Stursberg,\nP.; and Walsh, T. 2014. Fixing a Balanced Knockout Tour-\nnament. In Proceedings of the Twenty-Eighth AAAI Con-\nference on Artificial Intelligence, July 27 -31, 2014, Québec\nCity, Québec, Canada. , 552–558. AAAI Press.\nConnolly; and Rendleman. 2011. Tournament qualification,\nseeding and selection efficiency. Technical Report 2011-96,\nTuck School of Business .\nCygan, M.; Fomin, F. V .; Kowalik, L.; Lokshtanov, D.;\nMarx, D.; Pilipczuk, M.; Pilipczuk, M.; and Saurabh, S.\n2015. Parameterized Algorithms . Springer. ISBN 978-3-\n319-21274-6.\nGupta, S.; Roy, S.; Saurabh, S.; and Zehavi, M. 2018. Win-\nning a Tournament by Any Means Necessary. In Proceed-\nings of the Twenty-Seventh International Joint Conference\non Artificial Intelligence, IJCAI 2018, July 13-19, 2018,\nStockholm, Sweden. , 282–288.\nGupta, S.; Saurabh, S.; Sridharan, R.; and Zehavi, M. 2019.\nOn Succinct Encodings for the Tournament Fixing Prob-\nlem. In Kraus, S., ed., Proceedings of the Twenty-Eighth\nInternational Joint Conference on Artificial Intelligence, IJ-\nCAI 2019, Macao, China, August 10-16, 2019 , 322–328. ij-\ncai.org.\nHoren; and Riezman. 1985. Comparing Draws for Sin-\ngle Elimination Tournaments. Operations Research , 33(2):\n249–262.\nImpagliazzo, R.; and Paturi, R. 2001. On the Complexity of\nk-SAT. J. Comput. Syst. Sci. , 62(2): 367–375.\nKim, M. P.; and Vassilevska Williams, V . 2015. Fixing Tour-\nnaments for Kings, Chokers, and More. In Proceedings of\nthe Twenty-Fourth International Joint Conference on Artifi-\ncial Intelligence, IJCAI 2015, Buenos Aires, Argentina, July\n25-31, 2015 , 561–567. AAAI Press.\nLang, J.; Pini, M. S.; Rossi, F.; Venable, K. B.; and Walsh,\nT. 2007. Winner Determination in Sequential Majority V ot-\ning. In Veloso, M. M., ed., IJCAI 2007, Proceedings of\nthe 20th International Joint Conference on Artificial Intel-\nligence, Hyderabad, India, January 6-12, 2007 , 1372–1377.\nLaslier. 1997. Tournament Solutions and Majority V oting.\nSpringer-Verlag .\nManurangsi, P.; and Suksompong, W. 2023. Fixing knock-\nout tournaments with seeds. Discret. Appl. Math. , 339: 21–\n35.\nRamanujan, M. S.; and Szeider, S. 2017. Rigging Nearly\nAcyclic Tournaments Is Fixed-Parameter Tractable. In\nSingh, S.; and Markovitch, S., eds., Proceedings of the\nThirty-First AAAI Conference on Artificial Intelligence,\nFebruary 4-9, 2017, San Francisco, California, USA , 3929–\n3935. AAAI Press.\nRosen. 1986. Prizes and incentives in elimination tourna-\nments. The American Economic Review , 76(4): 701–715.Russell, T.; and van Beek, P. 2011. An Empirical Study of\nSeeding Manipulations and Their Prevention. In Walsh, T.,\ned.,IJCAI 2011, Proceedings of the 22nd International Joint\nConference on Artificial Intelligence, Barcelona, Catalonia,\nSpain, July 16-22, 2011 , 350–356. IJCAI/AAAI.\nStanton, I.; and Vassilevska Williams, V . 2011. Rigging\nTournament Brackets for Weaker Players. In Walsh, T.,\ned.,IJCAI 2011, Proceedings of the 22nd International Joint\nConference on Artificial Intelligence, Barcelona, Catalonia,\nSpain, July 16-22, 2011 , 357–364. IJCAI/AAAI.\nTullock. 1980. Toward a Theory of the Rent-seeking Soci-\nety.Texas A&M University Press .\nVassilevska Williams, V . 2010. Fixing a Tournament. In\nProceedings of the Twenty-Fourth AAAI Conference on Ar-\ntificial Intelligence, AAAI 2010, Atlanta, Georgia, USA, July\n11-15, 2010 . AAAI Press.\nVassilevska Williams, V . 2016. Knockout Tournaments. In\nHandbook of Computational Social Choice , 453–474.\nVu, T.; Altman, A.; and Shoham, Y . 2009. On the complex-\nity of schedule control problems for knockout tournaments.\nIn Sierra, C.; Castelfranchi, C.; Decker, K. S.; and Sich-\nman, J. S., eds., 8th International Joint Conference on Au-\ntonomous Agents and Multiagent Systems (AAMAS 2009),\nBudapest, Hungary, May 10-15, 2009, Volume 1 , 225–232.\nIFAAMAS.\nZehavi, M. 2023. Tournament Fixing Parameterized by\nFeedback Vertex Set Number Is FPT. In Williams, B.; Chen,\nY .; and Neville, J., eds., Thirty-Seventh AAAI Conference on\nArtificial Intelligence, AAAI 2023, Thirty-Fifth Conference\non Innovative Applications of Artificial Intelligence, IAAI\n2023, Thirteenth Symposium on Educational Advances in\nArtificial Intelligence, EAAI 2023, Washington, DC, USA,\nFebruary 7-14, 2023 , 5876–5883. AAAI Press."    },    {        "title": "2402.06595v1.Damping_of_density_oscillations_from_bulk_viscosity_in_quark_matter.pdf",        "content": "arXiv:2402.06595v1  [hep-ph]  9 Feb 2024Damping of density oscillations from bulk viscosity in quar k matter\nJos´ e Luis Hern´ andez1,2,3, Cristina Manuel1,2and Laura Tolos1,2,4\n1Institute of Space Sciences (ICE, CSIC), Campus UAB,\nCarrer de Can Magrans, 08193 Barcelona, Spain\n2Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Ba rcelona, Spain\n3Facultat de F´ ısica, Universitat de Barcelona, Mart´ ı i Fra nqu` es 1, 08028 Barcelona, Spain.\n4Frankfurt Institute for Advanced Studies, Ruth-Moufang-S tr. 1, 60438 Frankfurt am Main, Germany\nWe study the damping of density oscillations in the quark mat ter phase that might occur in\ncompact stars. To this end we compute the bulk viscosity and t he associated damping time in\nthree-flavor quark matter, considering both nonleptonic an d semileptonic electroweak processes.\nWe use two different equations of state of quark matter, more p recisely, the MIT bag model and\nperturbative QCD, including the leading order corrections in the strong coupling constant. We\nanalyze the dependence of our results on the density, temper ature and value of strange quark mass\nin each case. We then find that the maximum of the bulk viscosit y is in the range of temperature\nfrom 0.01 to 0.1 MeV for frequencies around 1 kHz, while the as sociated minimal damping times of\nthe density oscillations at those temperatures might be in t he range of few to hundreds milliseconds.\nOur results suggest that bulk viscous damping might be relev ant in the post-merger phase after the\ncollision of two neutron stars if deconfined matter is achiev ed in the process.\nKeywords: Three-flavor quark matter, Bulk viscosity, Dampi ng time\nI. INTRODUCTION\nThelong-debatedpossibilitythatquarkmattermaybe\npresent within the core of neutron stars, or adopting the\nformofquarkstars, hasbeen extensivelyexplored[1] (see\nalso [2] for a review and references). While most inves-\ntigations predominantly focused on assessing the mass-\nto-radius ratio of the stars, dictated by the equation of\nstate (EoS) linked to the stellar material, recent oppor-\ntunities for getting insights from neutron star interiors\nhavearisenthanksto the detection ofgravitationalwaves\n(GW) [3, 4]. In the events of neutron star mergers or in\nthe exploration of diverse stellar oscillation modes gener-\nating gravitational radiation [5–7], non-equilibrium pro-\ncesses unfold. The dynamics of these processes are in-\nfluenced by the material’s transport coefficients. A com-\nprehensive understanding of the transport coefficients of\nultra-dense matter becomes imperative, as these are de-\ntermined by the microscopic composition and the dom-\ninant interactions of its constituents. The knowledge of\nthe transport coefficients then brings a connection of the\nmicroscopic and macroscopic dynamics of the star.\nIn this work we will focus on the damping of density\noscillations of quark matter, which are relevant in the\nstudy of neutron star mergers [8]. We aim to dilucidate\nwhether in quark matter dissipative processes might af-\nfect the dynamics in neutron star mergers, in the event\nthat a deconfined phase is achieved in the process. The\ndamping of density oscillations is mainly governedby the\nbulk viscosity, which is the transport coefficient quantify-\ning the energy dissipation in a compressionor rarefaction\nof matter.\nThe bulk viscosityof quarkmatter has been previously\nstudied in [9–12] mainly to determine its effect on the\ndamping of the so called r-modes on isolated compact\nstars. The computation of the bulk viscosity of quarkmatter has also been recently reviewed in [13], improv-\ning the form of the EoS of quark matter, both using a\nperturbative QCD approach and taking into account a\nnonleptonic electroweak (EW) process, and also using\ntwo holographic models. Here we will also compute the\nbulk viscosity. We first consider the MIT bag model to\ndescribe quark matter, as this is extensively used in as-\ntrophysical settings, but we also use a QCD perturbative\napproach. We include both nonleptonic and semileptonic\nprocessesin our computation, as the last arerelevant in a\ncertain range of temperatures, as we will show. Unfortu-\nnately, at the densities one might expect to find in astro-\nphysicalsettings, whichwouldhardlyexceed10times the\nvalue of nuclear saturation density n0≈0.15fm−3, the\nQCD coupling constant αsis not so small, and higher or-\nder corrections than those we consider might be needed.\nIt has been claimed that the computations of the EoS\nfor quark matter only converge for values of the density\n≈40n0[14–16], values which however are not realistic\nin astrophysical settings. That is why one has to invoke\nsome modelling for the EoS of quark matter. It is how-\never instructive to compare those results from those pre-\ndicted in perturbation theory, as another possible model\nfor the description of quark matter in compact stars [17].\nWe will further compute the time scale associated to\nthe damping of the density oscillations. This brings rel-\nevant information on whether these dissipative processes\nmight be relevant or not, for example, in the inspiral\nphase of neutron star mergers, or its post-merger dynam-\nics. Attempts to include the effects of bulk viscosity in\nthe numerical modelling of viscous relativistic hydrody-\nnamics valid for neutron stars and neutron star mergers\nhave only being recently initiated [18, 19].\nOur study is complementary to the same study carried\nout for nuclear matter in Ref. [20, 21], or in [22]. Also\nfor nuclear matter the processes responsible for the bulk\nviscosity are mediated by the EW interactions, by either2\ndirect or by modified URCA processes. Also the value of\nthe bulk viscosity and the damping density oscillations\nstrongly depends on the EoS used to describe nuclear\nmatter.\nThis paper is structured as follows. In Sec. IIwe\npresent the general framework we use for the computa-\ntion of the bulk viscosity in the presence of a periodic\ndisturbance in three-flavor quark matter, and associated\nto different EW processes. We provide numerical values\nof the viscosity and corresponding damping times of den-\nsity oscillations in Sec. III, using the EoS associated to\nthe MIT bag model in Sec. IIIA, andalsoto perturbative\nQCD in Sec. IIIB. We present a discussion of our results\nin Sec.IV. And, finally, in Appendix Awe justify why we\nignore temperature effects in the EoS in the temperature\nrange we are considering for the MIT bag model.\nWe use natural units throughout the article, /planckover2pi1=c=\nKB= 1\nII. BULK VISCOSITY IN THREE-FLAVOR\nQUARK MATTER\nIn this section we study the bulk viscosity generated\nin three-flavor quark matter by nonleptonic and semilep-\ntonic weak processes. As a result, we get the bulk viscos-\nity associated to neutrino-free quark matter as a func-\ntion of temperature, T, the chemical potentials of the\nconstituent particles of the star, µi, and the frequency of\nthe oscillation mode, ω. In this article we will only focus\non the study of density oscillations, although our results\nfor the bulk viscosity might be also used for the study\nof the damping of different stellar oscillation modes. In\nthe normal phase and the neutrino-transparent regime,\nwe consider the following equilibration processes\nu+d↔u+s , (1)\nu+e−→d+νe, (2)\nd→u+e−+ ¯νe, (3)\nu+e−→s+νe, (4)\ns→u+e−+ ¯νe, (5)\nthat involve electrons ( e) and electronic neutrinos and\nantineutrinos ( νeand ¯νe, respectively) as well as up ( u),\ndown (d) and strange ( s) quarks.\nOn the one hand, fluctuations around the equilibrium\nvalue of the four-vector velocity ( uµ) and the particle\nnumber density ( nj) can be expressed as follows\nuµ=uµ\n0+δuµ, nj=nj,0+δnj, (6)\nsuch that, in beta equilibrium, we have\n∂µ(njuµ) = 0. (7)\nConsidering the LRF (Local Rest Frame) in the equilib-\nrium state, that is uµ\n0=uµ\nLRFand neglecting quadraticterms in the deviations, O(δ2), Eq. (7) implies that the\nparticle conservation law can be expressed as\nθnj,0+uµ\nLRF∂µδnj(t) = 0, (8)\nwhereθ=∂µδuµis the fluid expansion rate or equiva-\nlently\nθnj,0+∂\n∂tδnj(t) = 0. (9)\nOn the other hand, out-of-beta-equilibrium deviations of\nthe particle number density of the constituent particles\nare obtained from the following set of equations for the\nparticle density of strange quarks ( ns) and the particle\ndensity of electrons ( ne)\n∂\n∂tδns(t)+θns,0=−λ1µ1−λ2µ2,(10)\n∂\n∂tδne(t)+θne,0=λ2µ2+λ3µ3,(11)\nwhereλ1, λ2,andλ3aredefinedintermsoftheequilibra-\ntion rates of the nonleptonic and semileptonic processes\nas\nµ1λ1= Γs+u→d+u−Γd+u→s+u, (12)\nµ2λ2= Γs→u+e+¯νe−Γu+e→s+νe,(13)\nµ3λ3= Γd→u+e+¯νe−Γu+e→d+νe.(14)\nThe rates in the limit of massless up and down quarks\nhave been computed in several studies at tree level [23–\n25]. For pure massless up and down quarks, perturbative\ncorrectionsto the quark dispersion law in the strong cou-\npling constant αsare however needed in order to find a\nnon-vanishing value for λ3. One then finds\nλ1=64\n5π3G2\nFsin2ΘCcos2ΘCµ5\ndT2,(15)\nλ2=17\n40πG2\nFsin2ΘCµsm2\nsT4, (16)\nλ3=17\n15π2G2\nFcos2ΘCαsµdµuµeT4,(17)\nwhereGF= 1.166×10−5GeV−2is the Fermi coupling\nconstant, Θ C= 13.02◦is the Cabibbo angle, msis the\nstrange quark mass, and\nµ1=µs−µd, (18)\nµ2=µs−µe−µu, (19)\nµ3=µd−µe−µu. (20)\nNote that µ2−µ1=µd−µe−µu=µ3. Then, we obtain\nthat\n∂\n∂tδne(t)+θne,0= (λ2+λ3)µ2−λ3µ1.(21)\nThe oscillating parts of the particle density are taken to\nbe proportional to eiωt, so that\nδnj(t) =−θ\niωnj,0. (22)3\nThus the equation for the strange quark density can be\nexpressed as follows\niωδns(t)+θns,0=−λ1µ1−λ2µ2.(23)\nWe can then determine the out-of-beta-equilibrium de-\nviations of the particle number density of electrons and\nstrange quarks considering\nnd= 2nB−ns−ne, (24)\nnu=nB+ne. (25)\nHere we used the charge neutrality condition\nne+1\n3ns+1\n3nd=2\n3nu, (26)\nand the definition of the baryon density\nnB≡1\n3nu+1\n3nd+1\n3ns. (27)\nUsing the beta equilibrium and charge neutrality condi-\ntions in the out-of-beta-equilibrium particle number den-\nsities, we obtain\nδnd= 2δnB−δns−δne, (28)\nδnu=δnB+δne. (29)\nThe imbalance of the chemical potentials out-of-beta-\nequilibrium can be written in terms of deviations of the\nparticle number densities as follows\nµ1=Asδns−Adδnd, (30)\nµ2=Asδns−Aeδne−Auδnu,(31)\nwhereAjare the susceptibilities of the constituent par-\nticles, which can be determined in terms of the following\npartial derivatives\nAij=/parenleftbigg∂µi\n∂nj/parenrightbigg\n, (32)\noncewechooseanEoS.Employingtheserelations, weget\nthe conservationequationforthe particlenumber density\nof strange quarks\niωδns=−θns,0−λ1[Asδns−Ad(2δnB−δns−δne)]\n−λ2[Asδns−Aeδne−Au(δnB+δne)],(33)\nand a similar expression for the electrons\niωδne=−θne,0+(λ2+λ3)[Asδns−Au(δnB+δne)\n−Aeδne]−λ3[Asδns−Ad(2δnB−δns−δne)].\n(34)\nUsing Eqs.( 33) and (34) we are able to get an expres-\nsion forδns. Next, by employing the conservation of the\nbaryon number\nδnB=−θ\niωnB,0, (35)we obtain\nBδns=−θ\niω{iω[[iω+(λ2+λ3)A+λ3Ad]ns,0\n+ (λ2A−λ1Ad)ne,0]+[Ad(Au+2A)λQ\n+iω(2λ1Ad+λ2Au)]nB,0}, (36)\nwhere we define A≡Ae+Au,\nλQ≡λ1λ2+λ1λ3+λ2λ3, (37)\nand\nB≡iω[λ1(As+Ad)+λ2(A+As)+λ3(A+Ad)]\n+λQ[A(As+Ad)+AdAs]−ω2. (38)\nAnd by means of the Eq. ( 34) and (36), we have\nBδne=−θ\niω{iω[[iω+λ1(As+Ad)+λ2As]ne,0\n+ (λ2As−λ3Ad)ns,0]+[λ2As(λ2Au+2λ1Ad)\n−[λ2Au+λ3(Au−2Ad)][iω+(λ1+λ2)As]\n−AdAuλQ]nB,0}. (39)\nIn general, the deviations of the particle number density\ncan be expressed as\nδnj=δnj,0+δn′\nj, (40)\nwhereδnj,0is a fluctuation around beta equilibrium and\nδn′\njis an out-of-equilibrium deviation. For the former\none it follows\nδnj,0=−θ\niωnj,0. (41)\nThen, for the electron and the strange quark number\ndensity, we get\nδne=δne,0+δn′\ne, (42)\nδns=δns,0+δn′\ns. (43)\nThus\nBδn′\ns=θ\niω[iω(λ1C1+λ2C2)+(AC1+AdC2)λQ],(44)\nand\nBqδn′\ne=θ\niω{iω[(C1−C2)λ3−C2λ2]\n−[(Ad+As)C2−AsC1]λQ},(45)\nwhere\nC1≡ns,0As−nd,0Ad,\n=ns,0As−(2nB,0−ns,0−ne,0)Ad,(46)\nC2≡ns,0As−nu,0Au−ne,0Ae,\n=ns,0As−ne,0A−nB,0Au. (47)4\nOnce known the out-of-equilibrium fluctuations of the\nparticle number densities, we can calculate the bulk vis-\ncosity. First, we determine the out-of-equilibrium pres-\nsure as\np(nj(t)) =p(nj,0+δnj,0)+δp′(t) =p0(t)+δp′(t),(48)\nwhere the nonequilibrium part of the pressure is given by\nΠ =δp′=/summationdisplay\nj/parenleftbigg∂p\n∂nj/parenrightbigg\n0δn′\nj. (49)\nThis follows if we assume that the thermal equilibrium\nrate is much larger than the chemical equilibrium rate,\nso the temperature is constant (see the note [51] in [26]).\nUsing the Gibbs-Duhem relation\ndp=sdT+/summationdisplay\ninidµi, (50)\nwhich is valid out of equilibrium, and taking dT≈0, we\nhave\ncj≡/parenleftbigg∂p\n∂nj/parenrightbigg\n0=/summationdisplay\nini,0/parenleftbigg∂µi\n∂nj/parenrightbigg\n0=/summationdisplay\nini,0Aij.(51)\nThen the nonequilibrium pressure in quark matter can\nbe expressed as\nΠ =/summationdisplay\njcjδn′\nj=ceδn′\ne+cuδn′\nu+cdδn′\nd+csδn′\ns.(52)\nDue to the conservation of the baryon particle number\ndensity, its out-of-equilibrium deviation is zero\nδn′\nB= 0, (53)\nand, as a result, we get\nδn′\nu=δn′\ne, (54)\nδn′\nd=−δn′\ns−δn′\ne. (55)\nThus it follows that\nΠ = (cu−cd+ce)δn′\ne+(cs−cd)δn′\ns.(56)\nAccording to the Eq. ( 51), we find that\ncs=ns,0Ass, (57)\ncu=nu,0Auu, (58)\ncd=nd,0Add, (59)\nce=ne,0Aee. (60)\nAt this stage the model choice of three-flavor quark mat-\nter with electrons is required. For electrons we only con-\nsider diagonal terms for Aeebecause they form an ultra-\nrelativistic noninteracting gas. The same applies for the\nquarks (even if we consider the first correction in αs),\nso that the particle number density for each flavor doesnot depend on the chemical potential of the other flavors.\nThus,\nAu=Auu, Ad=Add, As=Ass, Ae=Aee,(61)\nwhere\ncs−cd=C1, (62)\ncs−cu−ce=C2, (63)\nso that\nΠ = (C1−C2)δn′\ne+C1δn′\ns, (64)\nwhich can be expressed as\nBΠ =θ\niω{iω[λ1C2\n1+λ2C2\n2+λ3(C1−C2)2]\n+ [AdC2\n2+AC2\n1+As(C1−C2)2]λQ}.(65)\nThe bulk viscosity in quark matter is given by\nζ≡ −Re[Π]\nθ, (66)\nwhere the real part of the nonequilibrium pressure can\nbe obtained from Eq. ( 65) as\nζ=κ1+κ2ω2\nκ3+κ4ω2+ω4, (67)\nwhere\nκ1≡λQ{C2\n1[(A+Ad)[A(λ2+λ3)+Adλ3]\n−Ad(Aλ2−Adλ1)]\n−2C1(C1−C2)[Ad[(Ad+As)λ1+(A+Ad)λ3]\n−AAsλ2]\n+ (C1−C2)2[λ1(Ad+As)2+λ2A2\ns+λ3A2\nd]},(68)\nκ2≡λ1C2\n1+λ2C2\n2+λ3(C1−C2)2, (69)\nκ3≡λ2\nQ[A(As+Ad)+AdAs]2, (70)\nκ4≡[(Ad+As)λ1+Asλ2]2+2(Aλ2−Adλ1)\n×[As(λ2+λ3)−(Ad+As)λ3]\n+ [Adλ3+A(λ2+λ3)]2. (71)\nIII. DAMPING TIME OF DENSITY\nOSCILLATIONS\nIn this section we determine the damping time asso-\nciated to the bulk viscosity coming from baryon number\ndensity oscillations in a medium. Let us assume a small\ndensity oscillation described by δnB=δnB,0Re(eiωt),\nwhereδnB,0andωare the magnitude and frequency of\nthe oscillation, respectively.\nThe energy density ǫstored in a baryonic oscillation\nwith amplitude δnBcan be obtained as\nǫ=1\n2∂2ε\n∂n2\nB(δnB)2, (72)5\nwhere the energy density is\nε= Ω+/summationdisplay\niniµi, (73)\nand the damping time is defined by\nτζ≡ǫ/(dǫ/dt). (74)\nThe energy dissipation time can be related with the bulk\nviscosity according to following expression\ndǫ\ndt=ω2ζ\n2/parenleftbiggδnB\nnB/parenrightbigg2\n. (75)\nAs a result, the damping time of baryon density oscilla-\ntions by bulk viscosity is given by\nτζ=n2\nB\nω2ζ∂2ε\n∂n2\nB. (76)\nConsidering an EoS in the limit of zero temperature re-\nsults in a simplification of the expression leaving the full\ntemperature-dependenceencodedonlyin thebulk viscos-\nity.\nA. MIT bag model\nIn this section we determine the values of the chemical\npotentials and particle number densities of three-flavor\nquark matter with electrons in the neutrino-transparent\nregime using as constraints the beta equilibrium and\ncharge neutrality. As a first approximation we can con-\nsider the simplest phenomenological bag model at zero\ntemperature. For finite-temperature corrections to the\nideal Fermi gas expressions, see the Appendix A. For this\nmodel the thermodynamic potential is given by\nΩ =/summationdisplay\ni=e,u,d,sΩ(0)\ni+Beff, (77)\nwhereBeffis the bag constant [27], and Ω(0)\niis the grand\ncanonical potential of massless electrons and light quarks\ndescribed as ideal Fermi gases\nΩ(0)\ne=−µ4\ne\n12π2, (78)\nand\nΩ(0)\nf=−Nc\n12π2/bracketleftbigg\nµfuf/parenleftbigg\nµ2\nf−5\n2m2\nf/parenrightbigg\n+3\n2m4\nfln/parenleftbiggµf+uf\nmf/parenrightbigg/bracketrightbigg\n, (79)\nhereafter f=u,d,s,Nc= 3 is the number of colors, mf\ndenotes the quark mass, and uf≡/radicalBig\nµ2\nf−m2\nf.From the thermodynamic potential we are able to get\nthe thermodynamic properties of quark matter. Particu-\nlarly, the number particle density for each particle specie\ncan be calculated by\nni=−/parenleftbigg∂Ω\n∂µi/parenrightbigg\nT,V. (80)\nAt zero temperature the number densities of quarks and\nelectrons can be written as follows\nnf=Nc\n3π2(µ2\nf−m2\nf)3/2, (81)\nand\nne=1\n3π2µ3\ne. (82)\nWith these expressions at hand, the beta equilibrium\nconditions can be expressed as\nµd=µs, (83)\nµs=µu+µe. (84)\nUsing Eq. ( 26) (the charge neutrality condition) and the\ndefinition of the baryon number density of Eq. ( 27), we\ncan determine the four chemical potentials and the four\nnumber densities ( µd,µs,µu,µe,nd,ns,nu, andne) for\na fixed value of the baryon number density.\nThesusceptibilitiescanbe obtainedfromEqs.( 81)and\n(82), and are given by\nAee=π2\nµ2e, (85)\nand\nAff=π2\n3µf/radicalBig\nµ2\nf−m2\nf, (86)\nfor electrons and quarks, respectively.\nFor this study we consider different values of the\nstrange quark mass and the baryon number density\nin terms of the nuclear saturation density, n0= 0.15\nfm−3[28]. The light quarks are considered massless, as\ntheirtinyvaluesdonothavearelevanteffectonthequan-\ntities of interest.\nTablesIandIIshow the values of the chemical po-\ntential and number density for quarks and electrons at\nms= 100 MeV and typical values of the baryon number\ndensity in neutron stars.\nnB,0/n0µu,0µd,0µs,0µe,0\n3324.31331.84331.847.53\n5384.52390.91390.916.39\n6408.61414.64414.646.03\nTable I: Chemical potentials in MeV for quark matter with\nelectrons imposing charge neutrality and beta equilibrium us-\ning the MIT bag model at ms= 100 MeV and varying the\nbaryonnumberdensity(normalized tonuclearsaturation de n-\nsity).6\nnB,0/n0nu,0/n0nd,0/n0ns,0/n0ne,0/n0\n33.003.212.791.25×10−5\n55.005.254.757.66×10−6\n66.006.275.736.42×10−6\nTable II: Particle number densities normalized to the nucle ar\nsaturation density imposing charge neutrality and beta equ i-\nlibrium using the MIT bag model at ms= 100 MeV and\nvarying the normalized baryon number density.\nIn Fig.1the bulk viscosity as a function of the tem-\nperature is depicted for different frequencies and baryon\nnumber densities for a fixed value of ms= 100 MeV. As\ncan be seen, increasing the baryon number density gen-\nerates a shift of the maximum of the bulk viscosity to\nlower temperatures, increasing slightly its value. This is\nclearlyseeninFig. 2, wherewezoomthe maximumofthe\nbulk viscosities for different densities at ω/2π= 1 kHz\nandms= 100 MeV. The values for the maxima for the\ndifferent densities at ω/2π= 1 kHz and ms= 100 MeV\nare given in Table III. In addition, in Fig. 1we consider\ndifferent values of the angular frequency around 1 kHz.\nWe observe that the larger the frequency is, the smaller\nthe value of the maximum of the bulk viscosity becomes\nwhereas it moves to larger temperatures.\n0.010.050.100.5015101024102510261027102810291030\nFigure 1: Bulk viscosity of three-flavor quark matter in the\nneutrino-free regime using the MIT bag model for different\nnormalized baryon number densities and normalized frequen -\ncies atms= 100 MeV.0.050.060.070.080.090.102.60×10282.65×10282.70×10282.75×10282.80×1028\nFigure 2: Zooming in of Fig. 1 to display the behavior of the\nmaximum bulk viscosities at ω/2π= 1 kHz, ms= 100 MeV\nand for different baryon number densities.\nIn order to study the damping times of baryon density\noscillations induced by the weak-interaction-driven bulk\nviscosity we resort to Eq. ( 76). The energy density in\nthe MIT bag model is given explicitly by Eqs. ( 73), (77),\n(81), and (82).\nFig.3displays the damping times associated to the\nbulk viscosities in Fig. 1. Note that the exact values for\nthe minimal damping times can be found in Table III.\nThe temperature dependence of the damping time is the\nsame as for the inverse of the bulk viscosity, as this fol-\nlows from the zero-temperature approximation for the\nthermodynamic potential. However, all other terms in-\nvolved in Eq.( 76) are relevant for determining the exact\nvalue of the damping times as a function of the baryon\nnumber density. In addition, the ω−2term modifies sig-\nnificantly the frequency-dependence from the inverse of\nthe bulk viscosity. We also note that at nB,0/n0= 3\nthe damping times seem to be independent of the fre-\nquencies considered in the low temperature regime for\napproximately T <20 keV.\n0.010.050.100.5015101100104106108\nFigure 3: Damping times from density oscillations using the\nMIT bag model for different normalized baryon number den-\nsities and frequencies at ms= 100 MeV.7\nnb,0/n0Tm ζmax τminω/2π\n32.5×10−22.73×10292132.420.1\n32.5×10−12.73×102721.3210\n37.9×10−22.73×1027213.241\n56.2×10−22.75×1028420.141\n65.7×10−22.76×1028535.371\nTable III: Maximum of the bulk viscosity (in gr cm−1s−1)\nand minimum of the damping times (in ms) for different nor-\nmalized baryon number densities and frequencies (in kHz) at\nms= 100 MeV according to Figs. 1 and 3. Here Tmdenotes\nthe temperature in MeV of the maximum (minima) of the\nbulk viscosity (damping time).\nIn addition, one can study the effect of the strange\nquark mass in the bulk viscosity. Fig. 4shows the bulk\nviscosity as a function of the temperature for different\nvalues of the strange quark mass at ω/2π= 1 kHz and\nnB,0/n0= 3. As the strange quark mass increases, not\nonly the bulk viscosity is larger but also the effect of the\nsemileptonic processes becomes more evident for temper-\naturesofafewMeVs. Thiseffect islinkedtothefactthat\nλ2∝m2\ns.\n0.010.050.100.5015101024102510261027102810291030\nFigure 4: Bulk viscosity of three-flavor quark matter in the\nneutrino-free regime using the MIT bag model for different\nvalues of the strange quark mass at ω/2π= 1 kHz and\nnB,0/n0= 3.\nFinally, in Fig. 5we depict the damping times asso-\nciated to varying the value of the strange quark mass,\ncorresponding to the viscosities in Fig. 4. Increasing the\nvalue of the mass of the strange quark has a drastic ef-\nfect in loweringthe damping times below10 milliseconds.\nThe minimal damping times of density oscillations for\nω/2π= 1 kHz thus can range from 3 to 200 milliseconds\nat a given temperature, but this depends strongly on the\nvalue of the strange quark mass. The values for the max-\nima of the bulk viscosity and the minimal damping time\nare given in Table IV.0.010.050.100.5015101101001000104105106\nFigure 5: Damping times from density oscillations using the\nMIT bag model for different values of the strange quark mass\natω/2π= 1 kHz and nB,0/n0= 3.\nmsTm ζmax τmin\n1007.9×10−22.73×1028213.24\n1507.5×10−21.34×102942.77\n2006.9×10−24.05×102913.93\n3005.7×10−21.73×10303.19\nTable IV:Maximumofthebulkviscosity (ingrcm−1s−1)and\nminimum of the damping times (in ms) for different masses\n(in MeV) according to Figs. 4 and 5. Here Tmdenotes the\ntemperature in MeV of the maximum (minima) of the bulk\nviscosity (damping time).\nB. Perturbative QCD\nA similar analysis can be performed for perturbative\nQCD at high density with a finite mass for the strange\nquark. As previously stated for the MIT model, a zero-\ntemperature limit for the number particle density and\nsusceptibility of the constituent particles is a good ap-\nproximation for the temperature region of interest. In\nperturbative QCD, the thermodynamic potential at fi-\nnite temperature and chemical potential up to O(αs) has\nbeen addressed in Ref. [29]. In this work we consider the\nzero-temperature limit of this expression which is given\nby\nΩ = Ω(0)\ne+/summationdisplay\nf=u,d,s/parenleftBig\nΩ(0)\nf+Ω(1)\nf/parenrightBig\n,(87)\nwhere the leading-order terms for massless electrons and\nnon-vanishing quark masses are shown in Eqs. ( 78) and\n(79), and the first-order correction in the MS scheme is\ngiven by8\nΩ(1)\nf=αs(N2\nc−1)\n16π3/braceleftbigg\n3/bracketleftbigg\nm2\nfln/parenleftbiggµf+uf\nmf/parenrightbigg\n−µfuf/bracketrightbigg2\n−2u4\nf+m2\nf/bracketleftbigg\nµfuf−m2\nfln/parenleftbiggµf+uf\nmf/parenrightbigg/bracketrightbigg\n×/bracketleftbigg\n6ln/parenleftbigg¯Λ\nmf/parenrightbigg\n+4/bracketrightbigg/bracerightbigg\n, (88)\nwhere¯Λistherenormalizationscaleand uf≡/radicalBig\nµ2\nf−m2\nf\nas in the previous section. The thermodynamic po-\ntential up to order αsdepends on the renormalization\nsubtraction point explicitly and implicitly through the\nscale dependence of the strong coupling constant and the\nmass [14, 29, 30]. Considering the massless approxima-\ntion for the light quarks, the scale dependence of the\ncoupling and the strange quark mass to first order in αs\ncan be expressed as\nαs(¯Λ) =4π\nβ0L/bracketleftbigg\n1−2β1\nβ2\n0ln(L)\nL/bracketrightbigg\n, (89)\nms(¯Λ) = ˆms/parenleftBigαs\nπ/parenrightBig4/9/parenleftBig\n1+0.895062αs\nπ/parenrightBig\n,(90)\nwithL= 2ln(¯Λ/ΛMS), the one-loop β-function coef-\nficientβ0= 11−2Nf/3, and the two-loop coefficient\nβ1= 51−19Nf/3 withNf= 3. ΛMSand the invari-\nant mass ˆ mscan be fixed by requiring αs≃0.3 and\nms≃100 MeV at ¯Λ = 2 GeV. As a result, one ob-\ntains ΛMS≃380 MeV and ˆ ms= 262 MeV. According\nto these constraints, the only undetermined parameter is\nthe value of the renormalization scale ¯Λ.\nThe particle number density for electrons is given in\nEq. (82) and for quarks up to O(αs) can be expressed as\nnf=n(0)\nf+n(1)\nf, (91)\nwhere\nn(0)\nf=Nc\n3π2(µ2\nf−m2\nf)3/2, (92)\nand\nn(1)\nf=−αs(N2\nc−1)\n4π3µfu2\nf/braceleftbigg\n1−3m2\nf\nµfufln/parenleftbiggµf+uf\nmf/parenrightbigg\n+m2\nf\n2µfuf/bracketleftbigg\n6ln/parenleftbigg¯Λ\nmf/parenrightbigg\n+4/bracketrightbigg/bracerightbigg\n. (93)\nAn alternative to handle this expression is not to con-\nsider the ¯Λ-dependent term in Eq. ( 93) as in Ref. [11],\nwhich is equivalent to set ¯Λ = exp( −2/3)mf. However,\nthe strong coupling constant and the strange quark mass\nin Eqs. ( 89) and (90) are fixed according to this choice.\nOther alternatives consider ¯Λ = 2µsand 3µs, which have\na relevant impact in the mass-to-radius ratio of com-\npact stars [17, 29], and ¯Λ = 2/summationtext\nfµf/Nfas in Ref. [14].In this work, we proceed setting ¯Λ = 2µsand imple-\nment the beta-equilibrium and the charge-neutralitycon-\nditions. Using this procedurethe strangequarkmassand\nthe strong coupling constant are decreasing functions of\nthe baryon particle density.\nWith theseexpressionsathand, wesolvethebeta equi-\nlibrium and charge neutrality conditions for different val-\nues of the baryon number density. The light quarks are\nconsidered massless. Tables VandVIlist the values\nof the chemical potential, the strange quark mass, the\nstrong coupling constant, and the number density of the\nconstituent particles in three-flavor quark matter with\nelectrons for different baryon number density.\nnB,0/n0µu,0µd,0µs,0µe,0msαs\n6470.47489.18489.1818.71138.460.54\n10544.80557.26557.2612.46127.120.47\n20671.16678.75678.757.58115.060.39\n40832.88837.75837.754.87106.010.33\nTable V: Input parameters for the bulk viscosity with pQCD\nat different normalized baryon number densities imposing\nbeta equilibrium and electric charge neutrality: chemical po-\ntentials in MeV, the strange quark mass in MeV and the\nstrong coupling constant.\nnB,0/n0nu,0/n0nd,0/n0ns,0/n0ne,0/n0\n66.006.745.251.92×10−4\n1010.0010.709.305.67×10−5\n2020.0020.6919.311.28×10−5\n4040.0040.7139.293.39×10−6\nTableVI:Normalized particle numberdensities withpQCDat\ndifferent normalized baryon number densities imposing beta\nequilibrium and electric charge neutrality.\nThe susceptibilities are given by Eq. ( 85) and\nA−1\nff=Nc\nπ2µf/radicalBig\nµ2\nf−m2\nf−αs(Nc−1)2\n4π3/braceleftbigg\n3µ2\nf−4m2\nf\n+m2\nfµf\nuf/bracketleftbigg\n2−3ln/parenleftbiggµf+uf\nmf/parenrightbigg\n+3ln/parenleftbigg¯Λ\nmf/parenrightbigg/bracketrightbigg/bracerightbigg\n,\n(94)\nfor electrons and quarks, respectively.\nIn Fig.6we plot the bulk viscosity as function of the\ntemperature at ω/2π= 1 kHz and ¯Λ = 2µsfor different\nbaryon number densities.9\n0.010.050.100.501510102410251026102710281029\nFigure 6: Bulk viscosity of three-flavor quark matter with\nelectrons using perturbative QCD for different normalized\nbaryon number densities at ω/2π= 1 kHz and and ¯Λ = 2µs.\nOur results seem to qualitatively agree with those re-\ncently presented in [13], valid for densities 40 n0, within\nperturbative QCD, even if in that reference higher order\ncorrections to the EoS were included, and only the non-\nleptonic process u+d↔u+swas considered. We have\nchecked that the value of the maximum value of the bulk\nviscosity as well as its location as a function of the tem-\nperature qualitatively agree, when computed at the same\norder of accuracy. In Ref. [13] higher order perturbative\ncorrections are included, changing slightly the position\nand the value of the maximum of the bulk viscosity. We\nhowever find some discrepancies with [13] that might be\ndue to the fact that in this reference the semileptonic\nprocesses were discarded.\nIn order to check the relevance of the semileptonic pro-\ncess we consider a similar approach to Ref. [11] comput-\ningthebulkviscositygeneratedonlybynonleptonicweak\nprocesses, ζnon. This can be obtained from the general\nexpression in Eq.( 67) setting λ2,λ3→0 and it is given\nby\nζnon=λ1C2\n1\n(Ad+As)2λ2\n1+ω2. (95)\nWe plot the ratio of the full bulk viscosity with that\narising only from the nonleptonic processes in Fig. 7. We\nconclude that there is certain range of temperatures, in\nthe region from 0.1 MeV to 2 MeV, where neglecting the\nsemileptonic processes is not a good approximation. We\nnote that in Ref. [11] it has been claimed that the regime\nwhere the semileptonic processes might be dropped de-\npends on the value of the frequency at a given value of\nthe density and temperature.1.01.21.41.61.82.0\n0.010.050.100.5015101.011.02\nFigure 7: Ratio ζ/ζnonas a function of the temperature using\nperturbative QCD for different frequencies at nB,0/n0= 6\nand 40. Note that the y-axis is different for each case.\nLastly, Fig. 8shows the damping times for different\nbaryon densities at ω/2π= 1 kHz and ¯Λ = 2µs. The\nvalues for the maxima of the bulk viscosity and the min-\nimal damping time for different densities are given in\nTableVII. As stated before, the strange quark mass de-\ncreases as the baryon number density gets larger. We\nsee that the maximum bulk viscosity decreases when the\nbaryon density increases. The damping time curves ex-\nhibit the same trend as in Fig. 5. AtnB,0/n0= 6 we\nhavems≈138 MeV and for higher values of the baryon\nnumber density up to nB,0/n0= 40 we get ms≈106\nMeV.\nNote that the maximum bulk viscosity and shortest\ndamping times exhibit the opposite behavior when in-\ncreasing the density in the MIT bag model, see Fig. 2.\nThis is linked to the fact that there the strange quark\nmass is a fixed parameter, which does not change with\nthe density. It should be possible to improve this feature\nin the modelling of the EOS, but we leave it for future\nwork.10\n0.010.050.100.5015101100104106108\nFigure 8: Damping times from density oscillations using\nperturbative QCD for different baryon number densities at\nω/2π= 1 kHz and ¯Λ = 2µs.\nnB,0/n0Tm ζmax τmin\n63.6×10−23.78×102943.65\n103.1×10−22.28×1029142.12\n202.4×10−21.29×1029626.32\n401.8×10−28.20×10282459.50\nTable VII: Maximum of the bulk viscosity (in gr cm−1s−1)\nand minimum of the damping times (in ms) for different\nbaryon number density according to Figs. 6 and 8. Here\nTmdenotes the temperature in MeV of the maximum (min-\nima) of the bulk viscosity (damping time).\nIV. OUTLOOK\nWe have studied the bulk viscosity and the damping\ntime of density oscillations of quark matter, using differ-\nent EoSs, and exploring their dependence on both the\nbaryon density, temperature and value of the strange\nquark mass. At the densities that could be attained in\nneutron stars we have considered the MIT bag model,\nand checked that the value of the bulk viscosity changes\nsignificatively with the value of the strange quark mass.\nWe have also used an EoS extracted from QCD at order\nαs. We have included all the relevant electroweak pro-\ncesses that equilibrate quark matter after a disturbance\nof the density, and checked in which temperature regime\nthe nonleptonic process is dominant.\nWhile we see that the numerical value of the bulk vis-\ncosity of quark matter depends on the form of EoS, on\nthe value of the strange quark mass, and the form of\nthe quark dispersion law, one might see some general\nfeatures from our results. In particular, we find that\nthe maximum value of the bulk viscosity, producing the\nshortest damping times of the density oscillations (in the\norder of the few to several hundreds of milliseconds, de-\npending on values of the density and the strange quark\nmass) occur at temperatures in the range from 0.01 to0.1 MeV, the precise value depends on the EoS describ-\ning quark matter. The bulk viscosity of nuclear matter,\nwhich also highly depends on the corresponding mod-\nelled EoS of nuclear matter, seems to have its maximum\nat much higher values of the temperature, in the order\nof few MeVs [20, 21]. Then one can clearly conclude the\nstrongest damping of density oscillations occur in dif-\nferent temperature regimes in quark or nuclear matter,\nwhile these different phases occur at different densities.\nOur results might be of interest so as to assess whether\nthe effect of the bulk viscosity should be included or\nnot in numerical simulations of mergers of neutron stars.\nSeveral of such numerical studies mention the possibility\nof reaching to a deconfined quark matter phase [31–35].\nAs the time scales associated to the initial stages of the\nmerger are of the order of few milliseconds, unless there\nare regions in the stars where the reached temperatures\nare in the range of 0.01 MeV, the effect of the bulk vis-\ncosity in the quark matter phase would be unnoticeable.\nThe effect might be more pronounced in the post-merger\nphase, as it seems also to be the case if one assumes only\nthe presence of nuclear matter, see [19]. However, the ef-\nfect in both cases, depends on the temperatures attained\nin the post-merger object.\nWe have not considered the possibility of Cooper pair-\ning of quarks in this article. In the so called color flavor\nlocked (CFL) [36] phase, and much below the supercon-\nducting transition the bulk viscosity is dominated by the\ninteraction of the superfluid phonons [37], and the kaons\n[38] and it was computed in [39, 40]. A further study of\nhow density oscillations are damped would be required,\nbut from the results found in [39, 40] one might predict\nthat damping times would be longer than in the normal\nphase.\nThe effect of the bulk viscosity of quark matter might\nbe also relevant in the study of the damping of the differ-\nent oscillation modes of isolated compact stars. We will\naddress them in a different publication.\nACKNOWLEDGMENTS\nWhile we were finishing this project, Ref. [13]\nappeared in the arxives, which has a clear over-\nlap with part of the content of this work. We\nacknowledge support from the program Unidad\nde Excelencia Mar´ ıa de Maeztu CEX2020-001058-\nM, from the projects PID2019-110165GB-I00 and\nPID2022-139427NB-I00 financed by the Spanish\nMCIN/AEI/10.13039/501100011033/FEDER, UE\n(FSE+), as well as from the Generalitat de Catalunya\nunder contract 2021SGR 171, by the EU STRONG-2020\nproject, under the program H2020-INFRAIA-2018-1\ngrant agreement no. 824093, and by the CRC-TR 211\n’Strong-interaction matter under extreme conditions’-\nproject Nr. 315477589 - TRR 211.11\nAppendix A: MIT model at finite temperature\nA typical approach to determine the chemical poten-\ntials and susceptibilities of quark matter with electrons\ninside neutron stars appeals to a zero-temperature limit\nof the thermodynamic potential. In this appendix we\nstudy the temperature-dependence of the ideal Fermi gas\nexpressions for the particle density and the susceptibili-\nties to infer its relevance in the temperature region of up\nto 10 MeV.\nThe leading order of the thermodynamic potential for\nthis model at finite temperature is given by [41, 42]\nΩ(0)\ni=−γiT\n2π2/integraldisplay∞\n0k2dk/braceleftBigg\nln/bracketleftbigg\n1+exp/parenleftbigg\n−Ei,k−µi\nT/parenrightbigg/bracketrightbigg\n+ ln/bracketleftbigg\n1+exp/parenleftbigg\n−Ei,k+µi\nT/parenrightbigg/bracketrightbigg/bracerightBigg\n, (A1)\nwithEi,k=/radicalbig\nk2+m2\niandγithe degeneracy factor, for\nelectrons, γe= 2, accounts their spin degrees of freedom\nand for quarks, γf= 2Nc, considers the spin and color\ndegrees of freedom. For electrons their mass, me= 0.511\nMeV, is small compared to the strange quark mass, re-\nsorting to the massless approximation. Then, the inte-\ngration in the thermodynamic potential of an ideal rela-\ntivistic Fermi gas can be carried out to give\nΩ(0)\ne=−1\n12/parenleftbiggµ4\ne\nπ2+2µ2\neT2+7\n15π2T4/parenrightbigg\n.(A2)\nUsing Eqs. ( A1) and (A2), the finite-temperature expres-\nsions for the number densities are given by\nnf=Nc\nπ2/integraldisplay\nk2dk/braceleftBigg\n1\n1+exp[( Ef,k−µf)/T]\n−1\n1+exp[( Ef,k+µf)/T]/bracerightBigg\n, (A3)\nand\nne=µe\n3/parenleftbigg\nT2+µ2\ne\nπ2/parenrightbigg\n, (A4)\nfor quarks and electrons, respectively.\nUsing Eqs. ( A3) and (A4) we compute the chemical\npotentials of quark matter with electrons imposing the\ncharge neutrality and beta equilibrium conditions at dif-\nferent temperatures.\nIn Fig.9we show the chemical potentials at finite tem-\nperature normalized by its value at zero temperature as\nwell as at nB,0/n0= 3 and ms= 100 MeV. Thermal\neffects reduce the chemical potential of the constituent\nparticles in these conditions. For temperatures below 10\nMeV the deviations compared to the value at zero tem-\nperature are up to 0 .3% for light quarks and up to 0 .1%\nfor electrons, while for temperatures up to 50 MeV thesedeviations can be up to 8% for light quarks and ∼1%\nfor electrons.\n0.050.100.501510500.920.940.960.981.00\nFigure 9: The chemical potentials of the different particle\nspeciesµiobtained by imposing beta equilibrium and electric\ncharge neutrality normalized by the same value obtained at\nzero temperature, µT=0. Additionally, we set nB,0/n0= 3\nandms= 100 MeV.\nTosumup, ourpredictionsforthe bulkviscosityvalues\nwould not be significantly affected by these thermal cor-\nrectionsfortemperaturesupto10MeV.Forhighervalues\nof temperatures these effects may be relevant. Also, at\nthese temperatures neutrinos might get trapped in the\nmedium and have to be taken into account in the beta-\nequilibrium conditions, thus changing the bulk viscosity\nand associated damping time [43].12\n[1]B. Freedman and L. McLerran,\nPhys. Rev. D 17, 1109 (1978) .\n[2]M. G. Alford, S. Han, and K. Schwenzer,\nJournal of Physics G: Nuclear and Particle Physics 46, 114001 (2019) .\n[3]B. P. Abbott et al. (LIGO Scientific\nCollaboration and Virgo Collaboration),\nPhys. Rev. Lett. 119, 161101 (2017) .\n[4]B.P.Abbott et al.,The Astrophysical Journal Letters 848, L12 (2017) .\n[5]K. D. Kokkotas and B. G. Schmidt,\nLiving Rev. Rel. 2, 2 (1999) ,arXiv:gr-qc/9909058 .\n[6]L. Rezzolla, ICTP Lect. Notes Ser. 14, 255 (2003),\narXiv:gr-qc/0302025 .\n[7]M. Sieniawska and M. Bejger, Universe 5, 217 (2019) .\n[8]M. G. Alford, L. Bovard, M. Hanauske, L. Rezzolla, and\nK. Schwenzer, Phys. Rev. Lett. 120, 041101 (2018) .\n[9]R. F. Sawyer, Phys. Lett. B 233, 412 (1989) , [Erratum:\nPhys.Lett.B 237, 605 (1990), Erratum: Phys.Lett.B 347,\n467–467 (1995)].\n[10]J. Madsen, Phys. Rev. D 46, 3290 (1992) .\n[11]B. A. Sa’d, I. A. Shovkovy, and D. H. Rischke,\nPhysical Review D 75(2007), 10.1103/physrevd.75.125004 .\n[12]M. G. Alford and A. Schmitt,\nJournal of Physics G: Nuclear and Particle Physics 34, 67–101 (2006) .\n[13]J. C. Rojas, T. Gorda, C. Hoyos, N. Jokela, M. J¨ arvinen,\nA. Kurkela, R. Paatelainen, S. S¨ appi, and A. Vuorinen,\n“Estimate forthebulkviscosityofstronglycoupledquark\nmatter,” (2024), arXiv:2402.00621 [hep-ph] .\n[14]A. Kurkela, P. Romatschke, and A. Vuorinen,\nPhysical Review D 81(2010), 10.1103/physrevd.81.105021 .\n[15]T. Gorda, O. Komoltsev, A. Kurkela,\nand A. Mazeliauskas, JHEP06, 002 (2023) ,\narXiv:2303.02175 [hep-ph] .\n[16]T. Gorda, R. Paatelainen, S. S¨ appi, and\nK. Sepp¨ anen, Phys. Rev. Lett. 131, 181902 (2023) ,\narXiv:2307.08734 [hep-ph] .\n[17]E. S. Fraga, R. D. Pisarski, and J. Schaffner-Bielich,\nPhysical Review D 63(2001), 10.1103/physrevd.63.121702 .\n[18]M. Chabanov and L. Rezzolla, (2023),\narXiv:2311.13027 [gr-qc] .\n[19]M. Chabanov and L. Rezzolla, (2023),\narXiv:2307.10464 [gr-qc] .\n[20]M. G. Alford and S. P. Harris,\nPhysical Review C 100(2019), 10.1103/physrevc.100.035803 .\n[21]M. Alford, A. Harutyunyan, and A. Sedrakian,\nParticles 3, 500 (2020) .\n[22]Y. Yang, M. Hippert, E. Speranza, and\nJ. Noronha, Phys. Rev. C 109, 015805 (2024) ,\narXiv:2309.01864 [nucl-th] .\n[23]N. Iwamoto, Phys. Rev. Lett. 44, 1637 (1980) .\n[24]N. Iwamoto, Annals of Physics 141, 1 (1982) .[25]J. Madsen, Phys. Rev. D 47, 325 (1993) .\n[26]M. Alford, A. Harutyunyan, and A. Sedrakian,\nPhysical Review D 104(2021), 10.1103/physrevd.104.103027 .\n[27]The bulk viscosity and the damping times are indepen-\ndent of the bag constant unless a chemical-potential de-\npendence is included.\n[28]C. J. Horowitz, J. Piekarewicz, and B. Reed,\nPhys. Rev. C 102, 044321 (2020) .\n[29]E. S. Fraga and P. Romatschke,\nPhysical Review D 71(2005), 10.1103/physrevd.71.105014 .\n[30]J. Vermaseren, S. Larin, and T. van Ritbergen,\nPhysics Letters B 405, 327–333 (1997) .\n[31]A. Bauswein, N.-U. F. Bastian, D. B. Blaschke,\nK. Chatziioannou, J. A. Clark, T. Fischer, and\nM. Oertel, Phys. Rev. Lett. 122, 061102 (2019) ,\narXiv:1809.01116 [astro-ph.HE] .\n[32]E. R. Most, L. J. Papenfort, V. Dexheimer,\nM. Hanauske, S. Schramm, H. St¨ ocker, and\nL. Rezzolla, Phys. Rev. Lett. 122, 061101 (2019) ,\narXiv:1807.03684 [astro-ph.HE] .\n[33]L. R. Weih, M. Hanauske, and L. Rez-\nzolla, Phys. Rev. Lett. 124, 171103 (2020) ,\narXiv:1912.09340 [gr-qc] .\n[34]S. Blacker, N.-U. F. Bastian, A. Bauswein, D. B.\nBlaschke, T. Fischer, M. Oertel, T. Soultanis,\nand S. Typel, Phys. Rev. D 102, 123023 (2020) ,\narXiv:2006.03789 [astro-ph.HE] .\n[35]A. Prakash, D. Radice, D. Logoteta, A. Perego,\nV. Nedora, I. Bombaci, R. Kashyap, S. Bernuzzi,\nand A. Endrizzi, Phys. Rev. D 104, 083029 (2021) ,\narXiv:2106.07885 [astro-ph.HE] .\n[36]M. G. Alford, K. Rajagopal, and F. Wilczek,\nNucl. Phys. B 537, 443 (1999) ,arXiv:hep-ph/9804403 .\n[37]C. Manuel and F. J. Llanes-Estrada,\nJCAP08, 001 (2007) ,arXiv:0705.3909 [hep-ph] .\n[38]M. G. Alford, M. Braby, S. Reddy, and T. Sch¨ afer,\nPhys. Rev. C 75, 055209 (2007) ,arXiv:nucl-th/0701067 .\n[39]M. Mannarelli and C. Manuel,\nPhys. Rev. D 81, 043002 (2010) ,\narXiv:0909.4486 [hep-ph] .\n[40]R. Bierkandt and C. Manuel,\nPhys. Rev. D 84, 023004 (2011) ,\narXiv:1104.5624 [hep-ph] .\n[41]X. J. Wen, X. H. Zhong, G. X.\nPeng, P. N. Shen, and P. Z. Ning,\nPhysical Review C 72(2005), 10.1103/physrevc.72.015204 .\n[42]L. L. Lopes, C. Biesdorf, K. D. Marquez, and D. P.\nMenezes, Physica Scripta 96, 065302 (2021) .\n[43]K. Pal and A. K. Dutt-Mazumder,\nPhys. Rev. D 84, 034004 (2011) ,\narXiv:1101.3870 [hep-ph] ."    },    {        "title": "2402.06603v1.Hamiltonicity_of_expanders__optimal_bounds_and_applications.pdf",        "content": "arXiv:2402.06603v1  [math.CO]  9 Feb 2024Hamiltonicity of expanders: optimal bounds and applicatio ns\nNemanja Dragani´ c∗Richard Montgomery†David Munh´ a Correia‡\nAlexey Pokrovskiy§Benny Sudakov‡\nAbstract\nAnn-vertex graph Gis aC-expander if |N(X)| ≥C|X|for every X⊆V(G) with|X|< n/2C\nand there is an edge between every two disjoint sets of at least n/2Cvertices. We show that there is\nsome constant C >0 for which every C-expander is Hamiltonian. In particular, this implies the well\nknown conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in ( n,d,λ)-graphs. This\ncompletes a long line of research on the Hamiltonicity of sparse graph s, and has many applications,\nincluding to the Hamiltonicity of random Cayley graphs.\n1 Introduction\nA Hamilton cycle in a graph Gis a cycle that contains all the vertices of G. The presence of such a cycle\ncategorizes Gas Hamiltonian. This fundamental concept in Graph Theory ha s been extensively studied,\nfor example see [2, 11, 15, 18, 19, 26, 33, 42, 44, 48, 49, 56] an d the surveys [31, 50]. Deciding whether a\ngraph is Hamiltonian or not is an NP-complete problem, and th us it is an important area of research to\nfind simple conditions which imply Hamiltonicity. One class ic example is Dirac’s theorem [21] that any\ngraph with n≥3 vertices and minimum degree at least n/2 is Hamiltonian. Another beautiful condition,\nby Chv´ atal and Erd˝ os [15], is that if the connectivity of a g raph is at least its independence number, then\nthe graph is Hamiltonian. Other famous Hamiltonicity condi tions include those of Chv´ atal [14], Jackson\n[38] and Nash-Williams [54]. Most known conditions for Hami ltonicity, however, require the graph to be\nvery dense. All the results mentioned above imply that the gr aph has linear minimum degree, except\nthe Chv´ atal-Erd˝ os result, which still implies minimum de gree Ω(√n). Hence, it is of particular interest\nto find Hamiltonicity conditions which also apply to sparse g raphs.\nA key area of research towards this in the last 50 years has bee n on Hamiltonicity in sparse random\ngraphs, and, in particular, which binomial random graphs G(n,p) and which random regular graphs\nGn,dare likely to be Hamiltonian. P´ osa [41] was the first to deter mine the threshold for Hamiltonicity\ninG(n,p), introducing the famous rotation-extension technique wh ich quickly become a widely used\ntool with innumerable applications (including in this pape r). After a refinement of P´ osa’s result by\nKorshunov [41], in 1983 Bollob´ as [10] and Koml´ os and Szeme r´ edi [40] independently showed that, if\np= (logn+loglog n+ω(1))/n, thenG(n,p) is almost certainly Hamiltonian. As is well-known, when\np= (logn+ loglog n−ω(1))/n,G(n,p) almost certainly has a vertex with degree 1, and hence no\nHamilton cycle. Random regular graphs, then, may be far spar ser yet plausibly Hamiltonian with high\nprobability, and, after significant focus on the problem, it is now known that Gn,dwill almost surely\n∗Mathematical Institute, University of Oxford, UK. Researc h supported by SNSF project 217926.\nEmail:nemanja.draganic@maths.ox.ac.uk\n†Mathematics Institute, University of Warwick, Coventry, C V4 7AL, UK. Research supported by the European Research\nCouncil (ERC) under the European Union Horizon 2020 researc h and innovation programme (grant agreement No. 947978).\nEmail:richard.montgomery@warwick.ac.uk.\n‡Department of Mathematics, ETH, Z¨ urich, Switzerland. Res earch supported in part by SNSF grant 200021 196965.\nEmails:{david.munhacanascorreia, benjamin.sudakov }@math.ethz.ch .\n§Department of Mathematics, University College London, Gow er Street, London WC1E 6BT, UK.\nEmail:dralexeypokrovskiy@gmail.com.\n1have a Hamilton cycle for all 3 ≤d≤n−1. For further details on this, see the work of Cooper, Frieze ,\nand Reed [17] and Krivelevich, Sudakov, Vu, and Wormald [47] .\nThe well-established understanding of Hamiltonicity in ra ndom graphs presents an important step\ntowards the search for simple properties of sparse graphs wh ich imply Hamiltonicity. It points to con-\nsidering natural ‘pseudorandom’ conditions which are requ ired by a deterministic graph to resemble a\nrandom graph. However, forgoing the randomness of G(n,p) and relying only on these pseudorandom\nproperties to find a Hamilton cycle presents a significantly fi rmer challenge, similar to the generalisation\nof other problems from random to pseudorandom graphs. Pseud orandom graphs have been systemati-\ncally studied since work by Thomason [59, 60] in the 1980’s, a history that can be found in the survey by\nKrivelevich and Sudakov [46]. The most studied class of pseu dorandom regulargraphs was introduced\nby Alon and is defined using spectral properties. Recalling t hat if a graph Gisd-regular then its largest\neigenvalue is d, we denote the second largest eigenvalue of Gin absolute value by λ(G). Then, a graph\nGis an (n,d,λ)-graphif it isd-regular with nvertices and satisfies |λ(G)| ≤λ.\nThefirstmajorsteptowards understandingtheHamiltonicit y ofsuchpseudorandomgraphswasmade\nby Krivelevich and Sudakov in 2003 in their influential paper [45]. They showed that if dis sufficiently\nlarger than λ, then the graph is Hamiltonian. More precisely,\nd\nλ≥1000·logn·(logloglog n)\n(loglogn)2(1)\nimplies that every ( n,d,λ)-graph is Hamiltonian. In the same paper, Krivelevich and S udakov made the\nbeautiful conjecture that (1) can be replaced byd\nλ≥Cfor some large constant C, as follows.\nConjecture 1.1. There exists C >0such that ifd\nλ≥C, then every (n,d,λ)-graph is Hamiltonian.\nThat is, if the absolute value of every other eigenvalue of a r egular graph is at most a small constant\nfraction of the largest eigenvalue, then the graph should be Hamiltonian.\nConsidering the result in [45] in the context of random graph s allows us to benchmark, broadly, this\nprogress. That is to say, the result of [45] is strong enough t o prove the likely Hamiltonicity of the\nrandom regular graph Gn,dwhend≥log2−o(1)n, while the ultimate goal, Conjecture 1.1, would be\nstrong enough to prove the likely Hamiltonicity of the rando m regular graph Gn,dwhendis at least\nsome large constant. Despite a great deal of attention, for e xample seen by the various relaxations and\ngeneralisations of the problem studied in [4, 12, 34, 46, 43] , and many incentivising applications (see\nSection 1.1), the bound at (1) established in [45] remained u nchallenged for 20 years. Only recently,\nGlock, Munh´ a Correia and Sudakov [30] finally improved this , significantly strengthening the result\nof [45] by showing that, for some large constant C >0,d/λ≥Clog1/3nsuffices to imply Hamiltonicity.\nMoreover, they showed that Conjecture 1.1 holds in the speci al case where d≥nα, for any fixed α, that\nis, in this case (1) can be weakened to d/λ≥C.\nKrivelevich and Sudakov applied their bound at (1) to other p roblems on Hamiltonicity in sparse\ngraphs; these and other applications are discussed in Secti on 1.1. To allow applications to non-regular\ngraphs, other pseudorandom conditions which imply Hamilto nicity were also studied. Motivated by this,\nshortly after Conjecture 1.1 was stated, several papers con sidered an even stronger conjecture, singling\nout the key properties of ( n,d,λ)-graphs thought to give some potential for proving Hamilto nicity. To\nstate this even stronger conjecture, whose variant appeare d for example in [12], we need the following\ndefinition.\nDefinition 1.2. Ann-vertex graph Gwithn≥3 is aC-expander if,\n(a)|N(X)| ≥C|X|for all vertex sets X⊆V(G) with|X|< n/2C, and,\n(b) there is an edge between any disjoint vertex sets X,Y⊆V(G) with|X|,|Y| ≥n/2C.\nConjecture 1.3. For every sufficiently large C >0, everyC-expander is Hamiltonian.\n2In 2012, Hefetz, Krivelevich and Szab´ o [34] made progress o n this problem; the precise expansion con-\nditions used in their result can be found in Theorem 1.1 of [34 ] (in particular weakening (a) in our\nDefinition 1.2), but imply that every (log1−o(1)n)-expander is Hamiltonian.\nIn this paper, we prove Conjecture 1.3, thus completing an ex tensive line of research on Hamiltonicity\nproblems.\nTheorem 1.4. For every sufficiently large C >0, everyC-expander is Hamiltonian.\nThis result has a large number of applications, as discussed below. In particular, it is a standard exercise\nto show that for every C >0 there exists a constant C0such that, ifd\nλ≥C0, then every ( n,d,λ)-graph\nis aC-expander. Thus, clearly Conjecture 1.1 is implied by Theor em 1.4, giving the following.\nTheorem 1.5. There is a constant C >0such that ifd\nλ≥Cthen every (n,d,λ)-graph is Hamiltonian.\nTo prove Theorem 1.4, we take a very different approach to the pr evious work on this problem, and so\ngive a detailed outline of the proof in Section 2.2. We finish t his section by giving some examples of the\napplications of Theorems 1.4 and 1.5.\n1.1 Applications\nPseudorandom conditions for Hamiltonicity have found a lar ge variety of applications, and Theorems 1.4\nand 1.5 immediately improve the bounds required for many suc h applications. We will discuss here\nthree applications: Hamiltonicity in random Cayley graphs , Hamiltonicity in well-connected graphs and\nHamilton cycles with few colours in edge-coloured graphs. F urther applications, many of them discussed\nin [30, 45], can be found in various other fields, ranging from problems in positional games (see, e.g.,\n[24, 16, 34, 32]), to questions about finding coverings and pa ckings of Hamilton cycles in random and\npseudorandom graphs (see, e.g., [25, 22, 36]), Hamiltonici ty thresholds in different random graph models\n(see, e.g., [28, 6, 29]), and for various other problems (see , e.g., [35, 39]), including as far afield as Alon\nand Bourgain’s work on additive patterns in multiplicative subgroups [5].\nHamiltonicity in random Cayley graphs. In1969, Lov´ asz [51] madethefollowing famousconjecture\nabout theHamiltonicity of vertex-transitive graphs, whic hare graphsin whichany vertex can bemapped\nto any other vertex by an automorphism.\nConjecture 1.6. Every connected vertex-transitive graph contains a Hamilt on path, and, except for five\nknown examples, a Hamilton cycle.\nAs Cayley graphs are vertex transitive and none of the five kno wn exceptions in Conjecture 1.6 are\nCayley graphs, Lov´ asz’s conjecture implies the following earlier conjecture, posed in 1959, by Strasser\n[57].\nConjecture 1.7. Every connected Cayley graph is Hamiltonian.\nConjecture 1.7 is known to be true when the underlying group i s abelian, but the only progress towards\nthe conjectures in general is a result of Babai [8] that every vertex-transitive n-vertex graph contains a\ncycle of length Ω(√n) (see [20] for a recent improvement by DeVos) and a result of C hristofides, Hladk´ y\nand M´ ath´ e [13] that every vertex-transitive graph of line ar minimum degree contains a Hamilton cycle.\nThe “random version” of Conjecture 1.7 is a natural relaxati on of the original problem. Alon and\nRoichman [7] showed that there is a constant C >0 for which, for any group G, the Cayley graph\ngenerated by a random set SofClog|G|elements, Γ( G,S) say, is almost surely connected. Hence, an\nimportant instance of Conjecture 1.7 is to show that Γ( G,S) is almost surely Hamiltonian. This problem\nwas also stated as a conjecture by Pak and Radoiˇ ci´ c [55]. Th eorem 1.5 resolves this conjecture. Indeed,\nAlon and Roichman [7] showed that if |S| ≥Clog|G|for some large constant C, then Γ( G,S) is almost\nsurely an ( n,d,λ)-graph with d/λ≥Kfor some large constant K. Thus, we obtain the following.\n3Theorem 1.8. LetCbe a sufficiently large constant. Let Gbe a group of order nandd≥Clogn. If\nS⊆Gis a set of size dchosen uniformly at random, then, with high probability, Γ(G,S)is Hamiltonian.\nHamiltonicity in well-connected graphs. For any function f:Z+→R, we say that a graph Gis\nf-connected if|A∩B| ≥f(min(|A\\B|,|B\\A|)) for every two subsets A,B\\V(G) suchthat V(G) =A∪B\nand there is no edge between A\\BandB\\A. In [12], Brandt, Broersma, Diestel, and Kriesell proved\nthat iff(k)≥2(k+1)2forevery k∈NthenGis Hamiltonian. Thisboundwas improved to klogk+O(1)\nin [34]. Brandt et al. also conjectured that there exists a fu nctionfwhich is linear in kyet ensures\nHamiltonicity. It is not difficult to check that if f(k) =Ckfor allk, then an f-connected graph is a\n(C/2)-expander. Indeed, given a set X⊆V(G) of size at most n/Cand its neighbourhood, N(X),\napplying the f-connected condition with A=X∪N(X),B=V(G)\\Xshows that |N(X)| ≥C|X|/2,\nwhile, given two disjoint sets X,Y⊆V(G) of size at least n/C, setting A=V(G)\\X,B=V(G)\\Y\nand applying the f-connected condition shows that there is an edge between X,Y. Therefore, Theorem\n1.4 implies the conjecture, as follows.\nTheorem 1.9. LetCbe a sufficiently large constant and let f(k) =Ckfor every k∈Z+. Then, any\nf-connected graph with at least 3 vertices is Hamiltonian.\nHamilton cycles using few colours. Anoptimally edge-coloured graph Gis one which is properly\ncoloured using the fewest possible number of colours. Akbar i, Etesami, Mahini, and Mahmoody [3]\nproved that any optimally coloured n-vertex complete graph Knhas a Hamilton cycle containing edges\nof at most 8√ncolours, and conjectured there should always be such a cycle using only O(logn) colours,\nwhich would be best possible up to a multiplicative constant . The bound from [3] on the number of\ncolours was later improved to O(log3n) by Balla, Pokrovskiy and Sudakov [9]. Their strategy is to\nrandomly pick d=O(log3n) colours and show that the subgraph of these colours is an ( n,d,λ)-graph\nwith high probability, for some appropriate λ, and apply the result from [45] on the Hamiltonicity of\n(n,d,λ)-graphs. Using their improved condition, Glock, Munh´ a Co rreia and Sudakov [30] showed that\nthis is possible with only O(log5/3n) colours using the same method. Applying instead the bound i n\nTheorem 1.5 immediately proves the conjecture (see [30] for more details).\nTheorem 1.10. Every optimally coloured Knhas a Hamilton cycle with O(logn)colours.\n2 Preliminaries\n2.1 Notation.\nOur notation is standard, and we recall here only the most imp ortant. A graph Ghas vertex V(G) and\nedge set E(G), and we set |G|=|V(G)|. Alinear forest is a graph consisting of a collection of vertex\ndisjoint paths. Two vertices are at distance ℓinGif the length of the shortest path between them in G\nisℓ. For a subset of vertices X, we denote by G[X] the induced subgraph of Gwith vertex set X, by\nΓG(X) the set of vertices adjacent to at least one vertex in X, and by NG(X) the (outer) neighbourhood\nNG(X) = ΓG(X)\\X. We omit Gin the subscript when it is clear from the context which graph we are\nworking with.\n2.2 Proof outline\nWhereGis ann-vertexC-expander, for some large constant C, we wish to find a Hamilton cycle in G.\nBefore describing our methods, it is instructive to briefly r ecall the well-known P´ osa rotation-extension\napproach, and how the approach works on random graphs but not on pseudorandom graphs (for a more\ndetailed approach than this sketch, see, for example, [10]) .\nTake a maximal length path PinG, with endvertices xandx0, say. Ifxhas a neighbour yinGon\nPwhose neighbour on Pwhich is nearest to x,zsay, is not equal to x, then the path Pcan be rotated\n4x0 x yzP:x yzF:\nFigure 1: On the left, a rotation of an xx0-pathPby removing zyand adding\nxyto get an xx0-path highlighted in blue. On the right, an example rotation of a\nlinear forest Fto get the 4 paths highlighted in blue, thus removing xfrom the set\nof endvertices and adding z.\nby adding xyand removing yz(see Figure 1). This gives us a path with the same length as P, where\none endvertex is x0and the other is a new endvertex, z. As P´ osa showed, if Gis aC-expander, then this\ncan be done iteratively to show that there are at least (say) n/3 different new endvertices v, say those in\nEx0, so that you can rotate Prepeatedly to get an x0v-pathPvwith the same length as P. Then, each\nof these paths Pvcan be rotated without changing the endvertex v, to get at least n/3 new endvertices,\nsay those in Fv. IfGcontains any edge in {vu:v∈Ex0,u∈Fv},Ghas a cycle with length |P|, whence,\nasGis connected (a trivial consequence of the expansion condit ion), if|P|< nthen we can find a longer\npath than P, a contradiction. Thus, Ghas a Hamilton cycle. Of course, Gmay not contain any edge\nin{vu:v∈Ex0,u∈Fv}. When working with random graphs, however, we can reserve so me random\nedges and sprinkle them in to find an edge in {vu:v∈Ex0,u∈Fv}and hence a longer path than P–\ndoing this iteratively can get a Hamilton path and thus a Hami lton cycle.\nIn pseudorandom graphs we cannot do this sprinkling. Thus, t he dream strategy would be to find\ntwo large sets A,BwithA×B⊂ {vu:v∈Ex0,u∈Fv}, so that we can apply Definition 1.2b) to\nfind the edge. Indeed, for their result discussed in the intro duction, Hefetz, Krivelevich and Szab´ o [34]\nfound such sets A,Bwith|A|,|B| ≥n/log1−o(1)n. However, large linear sets AandB(as would be\nrequired for Theorem 1.4) seem too hard to find, essentially b ecause the rotations performed around\neach endvertex of the path interfere with each other by rotat ing sections of the paths.\nFor our approach, instead of rotating only paths, we perform rotations of disjoint union of paths,\ni.e., of linear forests. The details of these rotations can b e found in Section 3, but, analogously, we\nalter a linear forest and preserve all but one endvertex of th ese paths (one sample rotation is depicted\nin Figure 1), iteratively developing many different possibil ities for the endvertex we are changing. The\nimplementation is much more intricate, but, essentially, i f most of the vertices in a linear forest Fare\nnot in long paths then we can take two endvertices xandx′of different paths, divide the paths in F\ninto two groups FxandFx′, and perform rotations just within these groups to find new en dvertices in\nplace ofxandx′. As these two sets of rotations are done using disjoint linea r forests, they can be done\nindependently, to get sets of new endvertices AandBrespectively, such that any edge from AtoBwill\nallow us to connect two paths together, reducing the number o f paths in the linear forest by 1 and losing\nthe endvertices xandx′. This is done in Section 3.\nWe then wish to apply this iteratively to reduce the number of paths in our linear forest. However\nthese rotations may create long paths (so that it is hard to fin d the partition into forests FxandFx′).\nThus, we combine this with an argument showing if Fhas too many long paths then we can replace it\nwith a linear forest F′whose lengths are more evenly spread yet whose endvertices a re still a subset of\nthe endvertices of F. This is also done using rotations, and is carried out in Sect ion 4 (which includes\na sketch of the proof once the necessary definitions have been introduced). Starting with a spanning\nlinear forest F, we can alternately perform rotations and add an edge (reduc ing the number of paths by\none) while losing any two endvertices (where, moreover, wit h some more effort, we can specify these two\nendvertices) and then replace the current linear forest wit h one with more evenly spread path lengths\n(so that the first step keeps working). Ultimately, this allo ws us to reach F′, a spanning linear forest of\nat mostn0.8paths, all of whose endvertices were also endvertices of pat hs inF.\nThis argument can be pushed further to end up with fewer paths , but cannot produce a Hamilton\n5path let alone a Hamilton cycle. Thus, we need some way to conn ect the paths we produce into a\nHamilton cycle. We do this by setting aside at the outset an ( A,B)-linking structure HinG, where\n|A|=|B|and this has the property that Hcontains a spanning linear forest connecting any desired\npartition of pairs a,bwitha∈Aandb∈B. More precisely, this is defined as follows.\nDefinition 2.1. A graph Hwith disjoint sets A,B⊆V(H) of equal size is said to be an ( A,B)-linking\nstructure if for every bijection ϕ:A→Bthere exist vertex disjoint paths P1,...,P |A|of equal length\nsuch that the following hold.\n1. The paths cover all the vertices of H, so that V(H) =V(P1)∪...∪V(P|A|).\n2. For each i∈[|A|], the path Pihas endpoints aandϕ(a) for some a∈A.\nBy using properties of sorting networks, adapting an approa ch of Hyde, Morrison, M¨ uyesser, and Pavez-\nSign´ e [37], we can find an ( A,B)-linking structure HinGwith|A|=|B|=n0.9and|H|=o(n0.95).\nTaking then an initial linear forest Fwhich covers G−V(H)\\(A∪B) and has every vertex in A∪B\namong its endvertices, we then apply our methods to reach a li near forest F′with the same vertex set\nasF, but whose endvertices are exactly those in A∪B(with no isolated vertices in F′). Then, applying\nthe linking property of Hallows us to connect the paths in F′′together into our desired Hamilton cycle\n(see Figure 2).\nX\nB A\nFigure 2: Given a linear forest F′(depicted in red) which spans G−(X\\(A∪B))\nwith no isolated vertices and endvertices End( F′) =A∪B, ifG[X] is an (A,B)-\nlinking structure, then the paths in blue can be found disjoi ntly while using all the\nvertices in X, thus linking F′into a Hamilton cycle.\n2.3 Expansion and linear forests\nIn order to carry out P´ osa rotations in linear forests, we ne ed to define the interior of sets with respect to\nthe linear forest, and a variation of expansion where we only consider the neighbourhood of sets within\nthe interior of the linear forest. For this, we will use the fo llowing definitions.\nDefinition 2.2. LetGbe a graph, and U,V⊆V(G) with|U| ≥105C. We say that U C-expands into\nV(inG)if allX⊆Uwith|X| ≤ |U|/5000Chave|NG(X)∩V| ≥C|X|.\nDefinition 2.3. LetGbe a graph and Fa linear forest in G. For a set U⊆V(F), we define the\ninterior of UinFto be the set int F(U) :={u∈U:NF(u)⊆U}. Furthermore, we say that Uis an\n(F,C)-expander ifU C-expands into int F(U).\nThe following observation is immediate from the previous de finition.\n6Observation 2.4. For all linear forests Fand subsets X,Y⊆V(F)it must be that intF(X\\Y)≥\nintF(X)−3|Y|.\nTo carry out rotations within only some of the paths in a linea r forest, we will need to find expanding\nsubgraphs sitting within the interior of these specific path s. Often expander subgraphs can be found\nusing only Definition 1.2b) by removing a maximal set of verti ces which does not expand and is not too\nlarge (see, for example, [52, Section 3.7]). Here, we work si milarly, but remove multiple sets Bi,jto find\na collection of 4 sets U′\niwith expansion properties (with respect to some linear fore stF) as follows.\nLemma 2.5. LetC≥105andC′=C/5000. LetGbe aC-expander containing a linear forest F,\nand letU1,U2,U3,U4⊂V(G)be disjoint sets with |intF(Ui)| ≥n/500for each i∈[4]. Then, there are\nsubsetsU′\ni⊆Ui,i∈[4], such that\n•for each i,j∈[4],U′\niC′-expands into intF(U′\nj), and\n•|U′\ni| ≥ |Ui|−2n/Cfor each i∈[4].\nIn particular, for each i∈[4],U′\niis an(F,C′)-expander.\nProof.Pick sets Bi,j⊆Ui,i,j∈[4], satisfying the following:\n(1)|Bi,j| ≤2n/Cfor alli,j∈[4].\n(2)/vextendsingle/vextendsingle/vextendsingleN(Bi,j)∩intF/parenleftBig\nUj\\/uniontext4\nt=1Bj,t/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤C′|Bi,j|for alli,j∈[4].\n(3)/summationtext\ni,j∈[4]|Bi,j|is as large as possible, subject to (1) and (2).\nFirst note that this is indeed possible, as the sets Bi,j=∅,i,j∈[4], satisfy (1) and (2). Let B:=/uniontext\ni,j∈[4]Bi,j.\nClaim 1. For each i,j∈[4],|Bi,j|< n/2C.\nProof.Suppose otherwise. Then, the definition of a C-expander implies that Bi,jis adjacent to all but\nat mostn/2Cvertices of V(G)\\Bi,jand, thus, to all but at most n/2Cvertices in int F(Uj\\B). Hence,\n|N(Bi,j)∩intF(Uj\\B)| ≥ |intF(Uj\\B)|−n/2C≥ |intF(Uj)|−3|B∩Uj|−n/2C\n≥ |intF(Uj)|−3·16·2n/C−n/2C≥(10C′−96.5)n/C > C′|Bi,j|,\ncontradicting (2). Note that in the previous inequalities w e used Observation 2.4, and that |intF(Ui)| ≥\nn/500 for each i∈[4] andC′=C/5000. ⊡\nWe now show that setting U′\ni:=Ui\\Bfor each i∈[4] gives the first item of the desired outcome of the\nlemma.\nClaim 2. For each i,j∈[4],Ui\\B C′-expands into intF(Uj\\B).\nProof.Suppose for contradiction that there is some i,j∈[4] for which there is some X⊆Ui\\Bwith\n|N(X)∩intF(Uj\\B)|< C′|X|and|X| ≤n/5000C′≤n/C. Then,|Bi,j∪X| ≤ |Bi,j|+|X| ≤2n/Cby\nClaim 1 and, furthermore,\n|N(Bi,j∪X)∩intF(Uj\\B)| ≤ |N(Bi,j)∩intF(Uj\\B)|+|N(X)∩intF(Uj\\B)|\n≤C′|Bi,j|+C′|X|=C′|Bi,j∪X|.\nFor each s,t∈[4], we have |N(Bs,t)∩intF(Ut\\(B∪X))| ≤ |N(Bs,t)∩intF(Ut\\B)| ≤C′|Bs,t|. Thus,\nreplacing Bi,jbyBi,j∪Xin{Bs,t:s,t∈[4]}gives a family of sets satisfying (1) and (2), but with a\nlarger total number of vertices, contradicting (3). ⊡\nFinally, setting U′\ni=Ui\\Bfor eachi∈[4] satisfies the lemma. Indeed, note that since |B∩Ui| ≤4·n/2C\nfor each i∈[4] by Claim 1, we have that |U′\ni| ≥ |Ui|−2n/C. /square\n73 Rotations of linear forests\nIn this section we give our methods for reducing the number of paths in a spanning linear forest in an\nexpander, byrotating thelinear forest. Afterdefiningourr otations andmakingsomesimpleobservations\nin Section 3.1, we will prove some intermediary results for t hese rotations in Section 3.2 before proving\nthe key result of this section, Lemma 3.6, in Section 3.3.\n3.1 Rotation definition and simple observations\nWe first define rotations of linear forests, which can be compa red to rotations of paths by lining up the\npaths of the linear forest as in Figure 1. Rotations of linear forests can, however, look quite different\ndepending on whether the edge used to rotate is between two di fferent paths, or between vertices in\nthe same path, and on whether these paths are isolated vertic es, or not; the different possiblities are\nillustrated in Figure 3.\nDefinition 3.1. LetP1andP2be two (possibly equal) paths of a linear forest Fin a graph G. Letx\nbe an endpoint of P1and letybe a neighbour (in G) ofxinP2. Letzbe a vertex adjacent to yinP2if\n|P2| ≥1 (ifP1=P2, then let zbe the vertex closer to xinP1, where we may have z=x), and let z=y\notherwise. Then, the linear forest F′obtained from Fby removing the edge zy(ifz/\\e}atio\\slash=y) and adding\nthe edge xy∈E(G) is called a 1 -rotation of FinG.\nWe call the vertex xtheold endpoint ,zthenew endpoint (indeed, note that z∈End(F′)), andythe\npivot. The edge yzis called the broken edge of the rotation. We will refer to the process of removing zy\n(if it exists) and adding xytoFas a 1-rotation, as well as the resulting linear forest F′, for example,\nsaying that, by performing a 1-rotation on Fwith old endpoint x, pivotyand new endpoint z, we get\nthe 1-rotation F′ofF.\nx\nP1 P2b)\ny\nz\nx\nP1 P2d)\ny\nz\nx\nP1 P2c)\ny=za)\nx\nP1=P2y\nz\nEnd(F′) = (End( F)\\{x})∪{z}End(F′) = End( F)\\{x}End(F′) = End( F)∪{z}\nFigure 3: Different 1-rotations of F={P1,P2}to getF′by removing the edge\nyz(if it exists) and adding the dashed edge xy, with the new paths highlighted in\nblue and the change in endvertices given underneath. In a)P1=P2, and inb)–d)\nP1/\\e}atio\\slash=P2. Inb)neitherxoryare isolated, in c)y=zis isolated but not xand in\nd)xis isolated but not y.\nAs in P´ osa’s original rotation-extension method on paths, we will repeatedly rotate linear forests to find\nmany different new endvertices. For this, we will now define a k-rotation .\nDefinition 3.2. Ak-rotation of a forest Fin a graph Gis any forest which is obtained by a sequence\nofkconsecutive 1-rotations starting with FinG, where the old endpoint of each 1-rotation is the new\nendpoint of the previous one and the pivot is at least at dista nce 3 in Ffrom the old endpoint of the\nfirst 1-rotation and the pivot in the i-th rotation for all i < k(and thus the broken edge belongs to F\n8as well). We say that the k-rotation has old endpoint x, orstarts at x, andnew endpoint yifxis the\nold endpoint of the first 1-rotation and yis the new endpoint of the last 1-rotation.\nFor a set U⊆V(G) we denote by Ek\nU(v,F) the set of new endpoints obtained by i-rotations of\nFstarting from v(i.e., with old endpoint v) withi≤k, such that the pivots of each 1-rotation are\ncontained in int F(U), so that each broken edge belongs to F[U]. We call such rotations ( U,i)-rotations ,\nand where iis not specified call them U-rotations .\nNext we prove some simple results on k-rotations.\nLemma 3.3. LetF′be a(U,k)-rotation of F, withk≥1, starting with vertex vand letube the new\nendpoint. Then, the following holds.\n(A)IfEk\nU(v,F)⊆S⊆Ek+1\nU(v,F), then the set of pivots used in the 1-rotations to get new endpoints\ninSis of size at most 2|S|.\nFurthermore, if Fhas no isolated vertices, then the following also hold.\n(B)The only possible isolated vertex in F′isu, which furthermore can only be isolated in F′ifu∈\nEnd(F).\n(C)End(F′) = (End(F)∪{u})\\{v}.\nProof.For (A) note that every pivot used for a i-rotation in Sis adjacent to a vertex in Sin the forest\nF, or is equal to a vertex in Sin the case this pivot is isolated in F. This is because all broken edges in\nak-rotation F′are always edges in the initial forest F. As the maximum degree of the linear forest F\nis 2, there are at most 2 |S|pivots used.\nWe prove part (B) by induction on kand under the weaker assumption that Fhas no isolated vertices\nexcept for, possibly, v. Fork= 1, let zbe the pivot of the rotation, and note that vis adjacent to z\ninF′, hence it is not isolated in F′. SinceFcontains no other isolated vertices, note that the only\npossible isolated vertices in F′arezandu, aszuis the only deleted edge in F. Butzis adjacent to v\ninF′, so only ucan be isolated in F, and this happens if and only if it did not have any other adjac ent\nvertex in Fbesidesz, i.e. if and only if it is an endpoint of F. Suppose now the statement holds for\nk−1≥1. Consider a k-rotation FkofFwith old endpoint vand new endpoint u. By definition of\nk-rotation, we get a ( k−1)-rotation Fk−1ofFwith old endpoint vand new endpoint w, such that\nFkis a 1-rotation of Fk−1with old endpoint wand new endpoint uand a pivot z. By the induction\nhypothesis, the only possible isolated vertex in Fk−1isw. Applying the induction hypothesis to Fk−1\nand then to its 1-rotation Fkimplies that uis the only possible isolated vertex in Fk, and that this is\nthe case if and only if uis an endpoint of Fk−1. Aszis at distance at most 3 from all the previous\npivots (used for Fk−1), we know that uis an endpoint of Fk−1if and only if it is an endpoint of F, which\ncompletes this part of the proof.\nFor part (C), we also proceed by induction. Let k= 1 and let zbe the pivot of the 1-rotation F′. If\nzis the neighbouring vertex of vinF, the statement trivially holds as F=F′andu=v. Otherwise,\nnote that vhas precisely one neighbour in F, while it has another neighbour zinF′, sovis not an\nendpoint in F′. On the other hand, the edge zuis removed from F, souhas at most one neighbour\ninF′, so it is an endpoint. Assume the statement holds for k−1≥1. Consider a k-rotation Fkof\nFwith old endpoint vand new endpoint u. As before, we get a ( k−1)-rotation Fk−1ofFwith old\nendpoint v, new endpoint w, such that Fkis a 1-rotation of Fk−1with old endpoint wand new endpoint\nuand a pivot z. By the induction hypothesis, we know that End( Fk−1) = (End( F)∪ {w})\\ {v}.\nWe distinguish two cases. If wis not an endpoint of F, then by (B) it is not isolated in Fk−1, and\nhence by applying the induction hypothesis to the 1-rotatio nFkofFk−1, we see that End( Fk) =\n(End(Fk−1)∪ {u})\\ {w}= (End( F)∪ {u})\\ {v}. In the second case, when wis an endpoint of F,\nagain by (B) we have that wis the only isolated vertex in Fk−1. Thus, the 1-rotation FkofFk−1still\n9contains was an endpoint, while similarly to the case k= 1 the only new created endpoint is u. Hence\nEnd(Fk) = ((End( F)∪{w})\\{v})∪{u}= (End(F)\\{v})∪{u}as required.\n3.2 Intermediate results on rotations\nAs mentioned in the proof sketch, a key part of P´ osa’s rotati on-extension technique is to show that\nperforming rotations iteratively in an expander creates a l inear-sized set of potential endvertices. We\nnow show the corresponding result for our linear forest rota tions.\nLemma 3.4. LetC >100and letFbe a linear forest in a graph G,van endpoint of F, andUan\n(F,C)-expander containing v. Then|Ek\nU(v,F)| ≥ |U|/105fork= 2logCn.\nProof.Suppose, to the contrary, that |Ek\nU(v,F)|<|U|/105. For each i∈[k−1], we have |Ei\nU(v,F)| ≤\n|Ek\nU(v,F)|<|U|/105. Now, for each i∈[k−1], and each pair ( x,y) withx∈Ei\nU(v,F),y∈intF(U)\nandxy∈E(G),yis a good candidate to use as a pivot for a 1-rotation of a linea r forest which has x\nas an endvertex, created by rotating Fup tok−1 times. However, by our definition, we need to have\nthat this pivot is at least at distance 3 in Ffrom all the previous pivots in the same rotations. In total,\nthough, as we have at most 2 |Ei\nU(v,F)|pivots used by part (A) of Lemma 3.3, this will rule out at most\n10|Ei\nU(v,F)|potential pivots y. Finally, we note that two pivots in N(Ei\nU(v,F))∩intF(U) may possibly\ncorrespond to the same new endpoint. Therefore, we have\n|Ei+1\nU(v,F)| ≥|N(Ei\nU(v,F))∩intF(U)|−10|Ei\nU(v,F)|\n2. (2)\nIf|Ei\nU(v,F)| ≥ |U|/5000C, then, by considering a subset S⊆Ei\nU(v,F) of size|U|/5000C, we have that\nsinceUis an (F,C)-expander, |N(S)∩intF(U)| ≥C|S|and so,|N(Ei\nU(v,F))∩intF(U)| ≥C|S| −\n|Ei\nU(v,F)| ≥ |U|/5000− |U|/105≥9|U|/105−10|Ei\nU(v,F)|, where we are using that |Ei\nU(v,F)| ≤\n|U|/105. Then, by (2), |Ei+1\nU(v,F)| ≥ |U|/105, a contradiction. Therefore, |Ei\nU(v,F)| ≤ |U|/5000C,\nwhence we have |N(Ei\nU(v,F))∩intF(U)| ≥C|Ei\nU(v,F)|asUis an (F,C)-expander, so that (2) implies\n|Ei+1\nU(v,F)| ≥C|Ei\nU(v,F)|\n3.\nAs this holds for every 0 ≤i≤k−1, we have |Ek\nU(v,F)| ≥(C/3)k> n, a contradiction.\nThe next lemma essentially says that if a linear forest Fin ann-vertex expander can be split into two\nso that rotations can be done on each subcollection of paths i ndependently starting from two vertices x\nandyrespectively, then we can alter O(logn) edges in Fto decrease the number of paths by 1 while\nlosing exactly xandyas endvertices, as follows.\nLemma 3.5. LetC > C′>1010and letGbe ann-vertexC-expander and Fa spanning linear forest in\nG. Suppose Fhas no isolated vertices and let x,ybe two endpoints of F. LetX,Ybe(F,C′)-expanders\nof size at least 0.0001nsuch that x∈X,y∈Y, no path in Fintersects both XandY, and there is no\nvertex in (X\\{x})∪(Y\\{y})which is an endpoint in F. Then, there is a spanning linear forest F′inG\nfor which |E(F)∆E(F′)|=O(logn),End(F′) = End( F)\\{x,y}, andF′contains no isolated vertices.\nProof.LetFXbe the collection of paths in Fwhich intersect XandFYbe the collection of paths in\nFwhich intersect Y, so that FX∩FY=∅. By Lemma 3.4, there exists a subset X′⊆Xof size at least\n|X|/105> n/Csuch that for all x′∈X′there is an X-rotation Fx′ofFwith new endpoint x′. Note\ncrucially that since Fx′is anX-rotation, we have that FY⊆ Fx′. Similarly, there is a subset Y′⊆Yof\nsize at least |Y|/105> n/Csuch that for all y′∈Y′there is a Y-rotation Fy′ofFwith new endpoint\ny′. Letx′y′be an edge between X′andY′, which exists as Gis aC-expander.\n10Now, the forest Fx′is such that FY⊆ Fx′. Therefore, any Y-rotation of Fis identical when restricted\ntoYto anyY-rotation of Fx′with the same change in endpoints. Therefore, since y′is a new endpoint\ncreated by a Y-rotation Fy′ofF, there is then a linear forest Fx′,y′which is obtained by performing the\nX-rotation and Y-rotation consecutively. Note that Fx′,y′has no isolated vertices by Lemma 3.3 (B).\nIn particular, then, x′,y′are not isolated in Fx′,y′and so adding the edge x′y′toFx′,y′we obtain the\nforestF′which has neither x′ory′as endvertices. More precisely, End( F′) = End( F)\\{x,y}. Since\nFxandFyareO(logn)-rotations, we conclude that FandF′differ in at most O(logn) edges, i.e.,\n|E(F)∆E(F′)|=O(logn).\nUnfortunately, even when we can split our linear forest into two subcollections of paths with many\nvertices, we cannot guarantee that we can rotate within them independently (as for Lemma 3.5) from\ntwo arbitrary endvertices xandy. Therefore, in our proof, we will first need to rotate in the wh ole linear\nforest to replace xandyby vertices for which we can do this. This is carried out in Sec tion 3.3, using\nthe following lemma.\nLemma 3.6. LetC≥C′>106and letGbe ann-vertexC-expander which contains a linear forest\nFwith no isolated vertices. Let U,V⊆V(G)be two subsets of vertices. Let u∈Ube an endpoint of\nFandUan(F,C′)-expander containing uwith|U| ≥105n/C. LetVsatisfy|intF(V)| ≥11n/Cand\ncontain no endpoints of F. Then, there is an (U∪V,O(logn))-rotation F′ofFandv∈Vsuch that\nEnd(F′) = (End( F)\\ {u})∪{v}. Furthermore, we have that F′[V\\v] =F[V\\v], all the edges broken\nin the successive 1-rotations except the last one are not in F[V], andF′has no isolated vertices.\nProof.First apply Lemma 3.4 to see that |E2logC′n\nU(v,F)| ≥ |U|/105≥n/C. We consider two cases.\nCase 1: E2logC′n\nU(u,F)∩V/\\e}atio\\slash=∅.\nLetk≤2logC′nbe the smallest integer such that Ek\nU(u,F)∩V/\\e}atio\\slash=∅andv∈Ek\nU(u,F)∩V. LetF′\nbe the rotation corresponding to endpoint v. ThenF′is aO(logn)-rotation of F. Furthermore, by the\nminimality of k, none of the new endpoints, except for v, in the successive 1-rotations creating F′belong\ntoV. Hence, none of the broken edges in these successive rotatio ns are contained in V. Therefore, the\nonly difference between F′[V] andF[V] occurs in the last 1-rotation, with a broken edge incident t ov.\nHence,F′[V\\v] =F[V\\v], as desired. It is easy to check that F′has no isolated vertices. Indeed,\nnote that by Lemma 3.3 (B), the only candidate is possibly v, but since Vhas no endpoints in F, by\nLemma 3.3 (B) we are done.\nCase 2: E2logC′n\nU(u,F)∩V=∅.\nLetk≤2logC′nbe the smallest integer such that |Ek\nU(u,F)| ≥n/C, and let Ek−1\nU(u,F)⊆X⊆\nEk\nU(u,F) satisfy |X|=n/C. LetPbe the set consisting of uand all the pivots used in 1-rotations\ncreating endpoints in Xand note that by part (A) of Lemma 3.3 we have |P| ≤2|X|. LetP2be set\nof vertices which are at distance at most 2 from Pin the forest F. Now we have that |intF(V)\\P2| ≥\n11n/C−5|P| ≥11n/C−10n/C≥n/C. SinceGis aC-expander, there is an edge uywithu∈Xand\ny∈intF(V)\\P2. Consider the ( U,i)-rotation of Ffor which xis the last new endpoint. Combined with\nthe 1-rotation with old endpoint x, pivotyand a new endpoint v∈V, we obtain a new rotation which\nsatisfies two properties: its last new endpoint is in V, and all the previous pivots except maybe the last\none are in U. Denote this rotation by F′.\nThe above argument shows that the obtained i-rotation F′satisfiesi≤2logC′n+1<logn. Note\nthat, by construction, all the pivots used to obtain F′are at least at distance 3 in F. Also note that\nsinceF′is a combination of a ( U,i)-rotation and another 1-rotation, such that the ( U,i)-rotation uses\nno new endpoints in V, the only edge which is possibly broken in F[V] is in the last 1-rotation when\nthe edge yvis broken. This immediately implies F′[V\\v] =F[V\\v]. Finally, again by Lemma 3.3 (B)\nthere are no isolated vertices in F′.\n113.3 The main rotation lemma\nThe next lemma is one of our key results, and is the most techni cal part of our proof.\nLemma 3.7. LetC >1010, letGbe ann-vertexC-expander and let Fbe a spanning linear forest in G\nwith no isolated vertices and such that at least 0.1nvertices of Gbelong to paths in Fof lengths between\n100and√n. Let also x,y∈End(F). Then, there is a spanning linear forest F′inGwith no isolated\nvertices such that |E(F)∆E(F′)|=O(logn)andEnd(F′) = End( F)\\{x,y}.\nProof.LetP1,...,P tbe the paths in Fof length between 100 and√n, so that/summationtext\ni∈[t]|P′\ni| ≥0.1n. Let\nH:={P′\n1,...,P′\nt}, such that each path P′\niis obtained from Piby removing the two endpoints of the\npathPi, so that now/summationtext\ni∈[t]|Pi| ≥0.1n(1−1\n50)≥0.098n. Take a partition of this linear forest Hinto\nH1,H2,H3,H4,H5,H6each of which span at least 0 .015nvertices. Hence, the interior of each of those\nforests (since they consist of paths of length at least 98), i s at least of size96\n98·0.015n≥0.01n. By\nLemma 2.5, we can find subsets U1⊆V(H1) andU2⊆V(H2), where Uiis such that it C′-expands into\nintH(Uj) = int F(Uj) for alli,j∈ {1,2}. In particular, U1andU2are (F,C′)-expanders for C′=C/5000.\nFurthermore, every vertex in U1has at least C′>100 neighbours in int F(U2), and every vertex in U2\nhas at least C′>100 neighbours in int F(U1). We also get that for each i∈ {1,2}it holds that\n|intF(Ui)| ≥ |V(Hi)|−2n/C≥0.01n−2n/C≥0.005n.\nWe now use Lemma 3.6 to prove that the following holds.\nClaim 3. There is a linear forest F′andx1∈U1,x2∈U2such that End(F′) = (End( F)\\{x,y})∪\n{x1,x2},|E(F′[U1∪U2])∆E(F[U1∪U2])| ≤20and|E(F)∆E(F′)|=O(logn). Furthermore, F′has\nno isolated vertices.\nProof.First, apply Lemma 3.6 with u:=x,U:=V(G), andV:=U1∪U2to give a ( V(G),O(logn))-\nrotation F1ofF, wherexis replaced by a new endpoint x1∈U1∪U2, so that End( F1) = (End( F)\\\n{x})∪{x1}, andF[(U1∪U2)\\x1] =F1[(U1∪U2)\\x1]. Also, since U1∪U2does not contain any endpoints\nofF, Lemma 3.3 (B) implies that F1has no isolated vertices. Without loss of generality, let x1∈U1.\nNow, we apply again Lemma 3.6 to F1withu:=y,U:=V(G)\\T,andV:= (U1∪U2)\\T, where\nTis the set of vertices which are at most at distance 2 from x1inF, so that |T| ≤5. Thus, we\nget anO(logn)-rotation F2ofF1whereyis replaced by a new endpoint w∈(U1∪U2)\\T, so that\nEnd(F2) = (End( F)\\{x,y})∪{x1,w}andF1[V\\w] =F2[V\\w]. Furthermore, since the endpoint is\nnot inT, it is different to x1and, again by Lemma 3.3 (B), F2has no isolated vertices.\nNow, in the case that w∈U2, the linear forest F′:=F2is as desired with x2:=wandx1since\nF[(U1∪U2)\\({x1,w} ∪T)] =F2[(U1∪U2)\\({x1,w} ∪T)]. Otherwise, if w∈U1, by the previous\ncondition that every vertex in U2hasat least 100 neighboursinint F(U1), thereisan edge wzforsome z∈\nintF(U1)∩intF2(U1). We can thenusethisedge tocreate a1-rotation F′ofF2, replacingtheoldendpoint\nwwith a new endpoint x2∈U2\\{x1}, that is, End( F2) = (End( F)\\{x,y})∪{x1,x2}. The linear forest\nF′is then as desired since, in particular, F[(U1∪U2)\\({w,z,x2}∪T)] =F′[(U1∪U2)\\({w,z,x2}∪T)]\nand again, by Lemma 3.3 (B), since F2has no isolated vertices and x2is not an endpoint of F, and so,\nalso not one of F2given that End( F2) = (End( F)\\{x,y})∪{x1,w},F′has no isolated vertices.\nFinally, we have |E(F′[U1∪U2])∆E(F[U1∪U2])| ≤20, since these linear forests differ in at most 8\nvertices, either T∪{x1,w}in the first case, or T∪{w,z,u2}in the second. ⊡\nLet us remark here that to obtain F′we rearranged some paths, hence, it is entirely possible tha t, in\nF′, vertices in U1live in the same paths as vertices in U2(although this was not the case in F). Thus,\nwe cannot yet apply Lemma 3.5 to the new endpoints x1,x2.\nNow, let H′\n3⊆ H3,H′\n4⊆ H4,H′\n5⊆ H5,andH′\n6⊆ H6be the linear sub-forests of H3,H4,H5,H6\nconsisting of paths P′\nisuch that Pi∈ F ∩ F′. By the previous claim, we have that |E(F)∆E(F′)|=\nO(logn). Since each path in Hhas size at most√n=o(n/logn), we must then have that |V(H′\nj)| ≥\n120.009nfor each j. By Lemma 2.5, there exist an ( H′\n3,C′)-expander U3, an (H′\n4,C′)-expander U4, an\n(H′\n5,C′)-expander U5and an ( H′\n6,C′)-expander U6, each of size at least 0 .009n−n/2C≥0.005n. As\nbefore, since we also get that each Uiis such that it C′-expands into int F′(Uj) (for all 3 ≤i,j≤6), we\nhave that every vertex in U3has at least C′≥1 neighbours in int F′(U5) and every vertex in U4has at\nleastC′≥1 neighbours in int F′(U6). Also, crucially note the following observation.\nClaim 4. U1andU2are(F′,C′−40)-expanders.\nIndeed, this is easy to see since the claim before implies tha t|E(F′[U1∪U2])∆E(F[U1∪U2])| ≤20.\nWe now apply Lemma 3.6 again. First, we apply the lemma to F′withx:=x1,U:=U1,andV:=U3\n(recall that U3does not contain endpoints of F, and so also of F′), which gives an ( U1∪U3,O(logn))-\nrotation F′\n1ofF′in which the endpoint x1is replaced by an endpoint x3∈U3. Moreover, we have\nthatF′\n1[U3\\x3] =F′[U3\\x3] andF′\n1is aU1∪U3-rotation, so that all broken edges in the successive\nrotations belong to U1∪U3and, thus, U2is still an ( F′\n1,C′−40)-expander, int F′\n1(U4) = int F(U4) is of\nsize at least n/1000, and F′\n1[U4] =F′[U4]. Furthermore, the lemma implies that F′\n1contains no isolated\nvertices and that no vertex of U4is an endpoint of F′\n1. Therefore, we can apply again Lemma 3.6 to\nF′\n1, now with x:=x2,U:=U2,andV:=U4. This gives a ( U2∪U4,O(logn))-rotation F′\n2with a new\nendpoint x4∈U4replacing x2and which contains no isolated vertices.\nNow, note that for all paths P′\ni∈ H′\n5∪H′\n6, we have that Pi∈ F′\n2. Note also that since the previous\nrotations forming F′\n1andF′\n2are such that all the broken edges in the successive 1-rotati ons apart from\nthe last ones are not in F[U3] orF[U4], we have that the path in F′\n2containing x3is a sub-path of a path\nPi∈ Fsuch that P′\niis contained in H′\n3and the path containing x4is a sub-path of a path Pj∈ Fsuch\nthatP′\njis contained in H′\n4. Thus, x3,x4are endpoints of different paths in F′\n2. Now, by assumption,\nx3∈U3has a neighbour in int F′(U5) = int F′\n2(U5) andx4∈U4has a neighbour in int F′(U6) = int F′\n2(U6).\nThis implies that we can perform two 1-rotations on F′\n2to getF′\n3and vertices x5∈U5,x6∈U6with\nEnd(F′\n3) = (F′\n2\\ {x3,x4})∪ {x5,x6}= (F \\ {x,y})∪ {x5,x6}. Note that x5,x6are not endpoints in\nF(and inF′), and we have End( F′\n2) = (End( F)\\{x,y})∪{x3,x4}, sox5,x6are not endpoints in F′\n2\neither. Hence, in F′\n3, the vertices x5,x6are also not isolated, implying that F′\n3has no isolated vertex\neither, by Lemma 3.3 (B).\nNow, since x3,x4were endpoints of different paths in F′\n2, it is easy to check that there can be no path\nofF′\n3which intersects both U5andU6. SinceU5andU6are still ( F′\n3,C′−2)-expanders and have size at\nleast 0.005n, Lemma 3.5 implies the existence of a linear forest F′\n4so that End( F′\n4) = End( F)\\{x,y},\nand furthermoreno isolated vertices. Moreover, since ever y rotation performedwas an O(logn)-rotation,\nwe have that |E(F)∆E(F′\n4)|=O(logn), as desired. /square\n4 Linear forests with few paths\nThe goal of this section is to find a spanning linear forest in a n expander with a relatively small number\nof paths. That is, we will prove the following result.\nLemma 4.1. LetGbe ann-vertexC-expander for C >1010. Then, it contains a spanning linear forest\nwith at most n0.8paths and no isolated vertices.\nWe will find the required linear forest for Lemma 4.1 by taking a minimal such linear forest under a\ncertain ordering, which we define after the following supple mentary definition.\nDefinition 4.2. Given a linear forest Fand integers a,b, we define SF(a,b) to be the set of vertices in\nthe collection of paths in Fof length at least aand at most b. For each v∈V(F), we denote by segF(v)\nthe path in Fwhich contains v. We omit Fin the subscript when it is clear from the context.\nDefinition 4.3. Let<lexbe the ordering on the family of all linear forests, where for any two linear\nforestsF1andF2we have F1<lexF2if\n13•F1has fewer paths than F2, or\n•F1andF2have the same number of paths and the vector of path lengths of F1in decreasing order\nis lexicographically smaller than that of F2.\nTo illustrate this definition, for example, if F1consists of paths of length 8 ,5,3,2,2 andF2consists of\npaths of length 8 ,5,4,2,1, then we have F1<lexF2as they both have 5 paths but (8 ,5,3,2,2) is smaller\nthan (8,5,4,2,1) in the lexicographic ordering.\nAs discussed in Section 3, it will be convenient to use spanni ng linear forests with no isolated vertices.\nWe will be able to apply the following lemma to show that a <lex-minimal spanning linear forest in an\nexpander has no isolated vertices.\nLemma 4.4. LetFbe a<lex-minimal spanning linear forest in an n-vertexC-expander Gfor some\nC >106. Suppose that Fcontains an isolated vertex v, andF′is ak-rotation of Fwith old endpoint v.\nThenF′is<lex-minimal, the new endpoint uofF′is isolated in F′, andEnd(F′) = End( F).\nProof.We prove this by induction on k. For the initial case, “ k= 1”, let wbe the pivot of the 1-\nrotation, and let Pbe the path of Fcontaining w. LetP1andP2be the two (possibly empty) subpaths\nofP−w, labelled so that |P1| ≤ |P2|. If|P2| ≥2, then replacing {w}andPinFwith the paths\nP1wvandP2gives a linear forest F′withF′<lexF, which is a contradiction. Similarly, if |P1|= 0,\nthen replacing {w}andPinFwith the path P2wvgives a linear forest F′withF′<lexF, which is\na contradiction. Thus we can suppose that P1,P2are both single vertices, i.e. that P=xwyfor some\nx,y. Then, a 1-rotation with pivot wreplaces the paths Pand{v}with either the paths vwx,yor\nthe paths vwy,x. In both cases the new linear forest has the same path lengths asFdid (and so stays\n<lex-minimal), the same set of endpoints as F, and has new endpoint either xory, which is now isolated.\nFor the induction step, consider a k-rotation FkofFwith old endpoint v. By the definition of a\nk-rotation, we get a ( k−1)-rotation Fk−1ofFwith old endpoint v, new endpoint u, such that Fkis a\n1-rotation of Fk−1with old endpoint u. By induction we get that Fk−1is<lex-minimal, has uisolated,\nand has End( Fk−1) = End( F). By the “ k= 1” case, we have that Fkis<lex-minimal, its new endpoint\nis isolated, and End( Fk) = End( Fk−1) = End( F).\nWe now sketch out the proof of Lemma 4.1. Given a <lex-minimal spanning linear forest Fin aC-\nexpander G, withClarge, we will first observe that there cannot be an edge in Gfrom the end of a path\ninFto a path in Fwhich is more than five times as long (see Lemma 4.5). We will th en argue, for\nanyi≥0, that this implies that if we perform a sequence of i1-rotations starting with an endpoint of a\nsmallest path in a <lex-minimal spanning linear forest F, then the new endvertex must be in a path at\nmost 6itimes as long as a smallest path (see Lemma 4.6). However, Lem ma 3.4 implies that within at\nmostk:= 2logCnrotations we reach at least n/105new endpoints, which then collectively have edges\nto all but at most n/2Cvertices in the graph by the definition of a C-expander. After using Lemma 4.4\nand assuming, for sake of contradiction that Fhas more than n0.8paths, we can deduce that Fhas no\nisolated vertices and also note that Fmust have at most 100 n/Cvertices contained in paths of length\nless than 100, so that the <lex-minimality of Fthen implies by Lemma 3.7 that at least 0 .8nof the\nvertices of Gare in paths in Fwith more than√nvertices. In combination, this all implies that the\nshortest path in Fmust have at least 6−(k+1)√nvertices, so that Fcontains at most√n·6(k+1)≤n0.8\npaths, a contradiction, as required.\nFirst, then, we show endvertices of paths cannot have an edge to much longer paths in a <lex-minimal\nspanning linear forest, in the following stronger form.\nLemma 4.5. Lett≥t′>0be integers. Let Fbe a minimal spanning linear forest in Gwith respect to\n<lex, with paths of decreasing lengths ℓ1,...,ℓt. LetF′be another forest with paths of lengths s1,...,st\nin decreasing order, where the first t′−1lengths are the same as in F, i.e.ℓi=sifor each i < t′. Letx\nbe an endpoint in F′in a path of length at most st′. Then,NG(x)⊆SF′(0,5st′).\n14Proof.Suppose for contradiction there is a path Pi(of length si) inF′withsi≤st′, whose endpoint x\nhas a neighbour yin another path Pjof length at least 5 st′≥5|Pi|. Then, we can obtain a new forest F′′\nfromF′such that F′′<lexF, as follows. We replace the paths PiandPjinFby two new paths. The\nfirst new path is the concatenation of Pi, the edge xyand the shorter subpath of Pjwhich starts at yand\nends in an endpoint of Pj. The other path is the remaining part of Pj. Note that both of those paths\nare shorter than Pjas the first path is of length at most |Pi|+1+|Pj|/2≤ |Pj|/5+1+|Pj|/2<|Pj|−1,\nand the second one is a strict subpath of Pj. This contradicts the <lex-minimality of F.\nNext, we iterate Lemma 4.5, showing that repeatedly applyin g 1-rotations cannot create a new endpoint\nwith a neighbour in a considerably longer path than the path c ontaining the old endpoint, as follows.\nLemma 4.6. LetFbe a<lex-minimal spanning linear forest of G. Letvbe an endpoint of a path in F.\nThenN(Ek\nG(v,F))⊆SF(0,6k+1|segF(v)|)for every positive integer k.\nProof.LetFkbeak-rotation of Fand letukbeits new endpoint and v=u0its old endpoint. Consider\nthe linear forests F=F0,F1,F2,...,Fk, whereFiis a 1-rotation of Fi−1, and where the new endpoint\nofFiisui, the old endpoint is ui−1, and the pivot is zi. We show that the following two conditions hold\nfor every i≥0:\n(1)N(ui)⊆SF(0,6i+1|segF(v)|)\n(2) The paths of length >6i+1|segF(v)|are the same in FandFi+1.\nHence, the particular case i=kproves the statement of the lemma. For the base case i= 0, let\nℓ1,...,ℓtbe the lengths of the paths in Fin decreasing order. Let t′be such that vis in a path of\nlengthℓt′. Note that by Lemma 4.5 applied to F=F′witht′, we know that vonly has neighbours in\nSF(0,5|segF(v)|)⊆SF(0,6|segF(v)|), as required for part (1). In the resulting linear forest F1only two\npaths have changed compared to F– one of length |segF(v)|and one of length at most 5 |segF(v)|, and\nso we obtained two new paths of length at most 6 |segF(v)|each. Hence, all the paths in F1of length\nlarger than 6 |segF(v)|are the same as in F, proving part (2).\nSuppose now the claim holds for all j < i, and let us prove it for i. Note that the first condition\nimplies that all the pivots zjforj≤iare inSF(0,6j|segF(v)|), as each zjlives inN(uj−1). In\nparticular, zi∈SF(0,6i|segF(v)|), meaning that ui∈SF(0,6i|segF(v)|), because either zi=uior\nthe broken edge ziuiis inF, souiandziare on the same path in F. Additionally, note that by\nthe induction hypothesis (part (2)) all the paths of length l arger than 6i|segF(v)|are the same in F\nandFi. Hence, since ui∈SF(0,6i|segF(v)|), thenui∈SFi(0,6i|segF(v)|). Therefore we can apply\nLemma 4.5 to F,Fianduito get that N(ui)⊆SFi(0,5|segFi(ui)|), which by the previous discussion\ngives that N(ui)⊆SFi(0,5·6i|segF(v)|). Now, we show that (2) holds. Note that, in order to get\nFi+1, inFithe path which contains uiand a path which contains a vertex in N(ui) are replaced\nwith two paths on the same set of vertices. Thus the two newly o btained paths are of length at most\n6i|segF(v)|+5·6i|segF(v)| ≤6i+1|segF(v)|. As the paths of length >6i|segF(v)|are the same in Fand\nFi, this means that the paths of length >6i+1|segF(v)|are the same in FandFi+1, completing part (2).\nFor part (1), recall that N(ui)⊆SFi(0,5·6i|segF(v)|). Thus, N(ui)⊆SF(0,6i+1|segF(v)|). Indeed,\notherwise a vertex x∈N(ui) is in a path of length >6i+1|segF(v)| ≥6i|segF(v)|inF, but inFiit is\nin a path of length at most 6i+1|segF(v)|. By the induction hypothesis for part (2) for i−1, the paths\nof length >6i|segF(v)|are the same in FandFiand thus, this is a contradiction. This completes the\nproof.\nNext, we show that in a <lex-minimal spanning linear forest with quite a lot of paths, on ly a small\nproportion of the vertices can lie in very long paths.\nLemma 4.7. LetC >1010. LetFbe a<lex-minimal spanning linear forest in an n-vertexC-expander G.\nSuppose that the number of paths in Fis at least nε, for some ε >0. Then,|S(6n1−ε+2logC6,n)| ≤n/2C.\n15Proof.Since we have nεpaths, there is one of length at most n1−ε. Setk:= 2logCn, letvbe\nan endpoint of a shortest path, which is of length m≤n1−ε, and suppose for contradiction that\n|S(6mn2logC6,n)|> n/2C. By Lemma 3.4, we have |Ek\nG(v,F)| ≥n/2C. We now have two cases.\nIfEk\nG(v,F)∩S(6mn2logC6,n)/\\e}atio\\slash=∅, then let ibe the smallest such that there exists x∈Ei\nG(v,F)∩\nS(6mn2logC6,n). Letybe the pivot for xin the corresponding i-rotation and recall that x,yare in\nthe same path in F, hencey∈S(6mn2logC6,n) as well. Note also that y∈N(Ei−1\nG(v,F)), so that\nS(6mn2logC6,n)∩N(Ei−1\nG(v,F))/\\e}atio\\slash=∅. On the other hand, if Ek\nG(v,F)∩S(6mn2logC6,n) =∅, then by\nthe definition of a C-expander, we have that N(Ek\nG(v,F))∩S(6mn2logC6,n)/\\e}atio\\slash=∅. Both of these cases\ncontradict Lemma 4.6, which says that for i≤kwe have\nN(Ei\nG(v,F))⊆S(0,6k+1|segF(v)|) =S(0,61+2logCnm) =S(0,61+2log6n/log6Cm) =S(0,6mn2logC6).\nHence,|S(6n1−ε+2logC6,n)| ≤ |S(6mn2logC6,n)| ≤n/2C.\nFinally, then, in this section, we can put this all together t o prove Lemma 4.1.\nProof of Lemma 4.1. LetFbe a<lex-minimal spanning forest in G. First we will show that it must\ncontain at most n/Cpaths. Indeed, suppose otherwise and let Sbe a set which contains exactly one\nendpoint from each path in F. SplitSinto disjoint sets S=S1∪S2, both of size |S1|,|S2| ≥n/2C.\nSinceGis aC-expander, there is an edge between S1andS2. Adding this edge to Fcreates a linear\nforest with one fewer path than F, contradicting the <lex-minimality of F.\nWe now show that Fhas no isolated vertices. Again, suppose otherwise and let vbe an isolated\nvertex in F. By Lemma 4.4, for all k, we have that Ek(v,F)⊆End(F). Lemma 3.4 then tells us that\n|End(F)| ≥ |E2logCn(v,F)| ≥n/105>2n/C. This contradicts the previous assertion that Fhas at\nmostn/Cpaths.\nNow that we know that Fhas at most n/Cpaths and no isolated vertices, it must be that at\nmost 100 n/Cvertices in Fare contained in paths of length less than 100. To conclude, a ssume for\ncontradiction that this forest has more than n0.8paths. Now, apply Lemma 4.7 with ε= 0.8 to conclude\nthatFsatisfies|SF(6n1−ε+2logC6,n)| ≤n/2C. Since 1 −ε+2logC6<1/2, the number of vertices in\npaths longer than√nis less than n/2C < n/2. Hence, at least n/2−100n/C≥0.1nof the vertices are in\nSF(100,√n). Now, apply Lemma 3.7 to Fto obtain a new linear forest with fewer paths, contradictin g\nthe<lex-minimality of F.\n5 Linking structures and decomposing the expander graph\nIn this section, we will prove the main result we need which wi ll find an appropriate linking structure\nin an expander. For convenience for our application, we will additionally find relatively long paths\nconnecting the two special sets in the linking structure, as in the following result.\nLemma 5.1. LetC >1015, letnbe sufficiently large and let Gbe aC-expander. Then, there is a\npartition of V(G)into three sets X,Y,Z, and disjoint sets A,B⊂X, with the following properties.\n•G[X]is an(A,B)-linking structure and |A|=|B|=n0.9.\n•G[Y∪A∪B]has a spanning linear forest with |A|=|B|paths of size n0.1/5whose endpoints are\ninA∪B.\n•G[Z]andG[Z∪Y∪A∪B]are both C/100-expanders.\nFor [37, Proposition 4.4], Hyde, Morrison, M¨ uyesser and Pa vez-Sign´ e showed that there exist linking\nstructures with convenient properties, which allow for the m to be constructed in pseudorandom graphs.\nOurmethodswouldbecontentwithalinkingstructurewithmu chsmallersets A,B, thanintheefficiently\n16constructed linking structure in [37], but we will use the sa me construction, briefly outlining the main\narguments for the sake of completeness. We will start in Sect ion 5.1 by recalling some ‘extendability’\nmethods, before proving in Section 5.2 that there is a linkin g structure that can be constructed using\nthese methods, and using this to prove Lemma 5.1.\n5.1 Embedding in expanders\nHere we briefly discuss a very useful ‘extendability’ method for embedding sparse graphs in expander\ngraphs. The method combines a technique Friedman and Pippen ger [27] used to inductively embed\ntrees leaf-by-leaf into expanding graphs with a ‘roll-back ’ idea of Johannsen (used to prove Lemma 5.3\nbelow). For more on this technique, see [23]. The key definiti on defining a type of ‘good’ embedding is\nthe following.\nDefinition 5.2. LetD,mbe positive integers with D≥3. LetGbe a graph and let H⊂Gbe a\nsubgraphwith ∆( H)≤D. ThenHis (D,m)-extendable (in G)if for every S⊂V(G) with 1≤ |S| ≤2m\nwe have\n|ΓG(S)\\V(H)| ≥(D−1)|S|−/summationdisplay\nu∈S∩V(H)(dH(u)−1). (3)\nThe main result to state here is the following, which allows u s to add paths to existing ( D,m)-extendable\nsubgraphs of an expander such that the resulting subgraph st ays (D,m)-extendable. Hence, we will be\nable to embed in expanders any type of sparse graphs which can be recursively built by adding path by\npath. This lemma is, for example, a weaker version of [52, Cor ollary 3.12].\nLemma 5.3. LetC >10Dand letGbe ann-vertexC-expander with a (D,n/2C)-extendable subgraph\nHwith|H| ≤n−5nD/C−ℓ, whereℓ≥logn. Then, the following hold for all vertices y∈V(H)with\ndegH(y)≤D/2.\n•There exists a path PinGwith endpoint yof length ℓsuch that all its vertices except for ylie\noutside of HandH∪Pis(D,n/2C)-extendable.\n•For every x∈V(H)\\{y}with degH(x)≤D/2, there exists an xy-pathPinGof length ℓsuch\nthat all its internal vertices lie outside of HandH∪Pis(D,n/2C)-extendable.\n5.2 Linking structures in expanders\nWe first need the following definition, as introduced in [37].\nDefinition 5.4. LetGbe a graph and A⊆V(G). We say that GisA-path-constructible if there exists\na sequence of edge-disjoint paths P1,...,P tinGwith the following properties.\n1.E(G) =/uniontext\nj∈[t]E(Pj).\n2. For each i∈[t], the internal vertices of Piare disjoint from A∪/uniontext\nj∈[i−1]V(Pj).\n3. For each i∈[t], at least one of the endpoints of Pibelongs to A∪/uniontext\nj∈[i−1]V(Pj).\nWe will say that GisA-path-constructible with paths of length between ℓ1andℓ2if all the paths Pthave\nlengths between ℓ1andℓ2.\nThe crucial property of path-constructible graphs is simpl e to observe given Section 5.1: if they have\nlow maximum degree and the paths Piare sufficiently long (that is, of size Ω(log n)), then Lemma 5.3\nimplies that they can be embedded in expanders (and, in parti cular, in pseudorandom graphs, as done\nin [37]). As mentioned before, we now state the main lemma for finding our linking structure, which is\na weaker version of [37, Proposition 4.4].\n17Lemma 5.5. For all sufficiently large N, there exists a graph Hwith at most N1.1vertices and disjoint\nsetsA,B⊆V(H)such that His an(A,B)-linking structure and the following hold.\n1.|A|=|B|=NandA∪Bis an independent set in H.\n2.∆(H)≤4.\n3.His(A∪B)-path-constructible with paths of length between 10logNand40logN.\nIn order to obtain their stronger version of Lemma 5.5, the au thors in [37] rely on a result concerning\nsorting networks with optimal depth. Earlier uses of sortin g networks in extremal combinatorics include\nin the work of K¨ uhn, Lapinskas, Osthus and Patel [48] on Hami lton cycles in highly connected tourna-\nments and in the work of M¨ uyesser and Pokrovskiy [53] giving , among other results, a combinatorial\nproof of the Hall-Paige conjecture. We will define directly t he graph theoretic counterpart of a sorting\nnetwork, for convenience calling this itself a sorting network . For a more detailed discussion concerning\nsorting networks and their graph theoretic counterpart see , for example, [37, Section 4].\nDefinition 5.6. A graph Gis an (N,ℓ)-sorting network if there exist A,B⊆V(G) for which Gis an\n(A,B)-linking structure and disjoint sets V0=A,V1,...,V ℓ−1,Vℓ=Bwith the following properties.\n1.V(G) =V0∪...∪Vℓ.\n2. For every 0 ≤i≤ℓ,Viis an independent set in Gwith|Vi|=|A|=|B|=N.\n3. For every 0 ≤i≤ℓ, the bipartite graph G[Vi,Vi+1] is the disjoint union of K2,2’s and edges.\nSorting networks which are efficient enough for our purposes a re simple to construct, and results as early\nas 1959 ([58]) imply that there exist ( N,O(log2N))-sorting networks. Moreover, in 1983, Ajtai, Koml´ os\nand Szemer´ edi [1] showed that, for each N, there exist ( N,ℓ)-sorting networks with ℓ=O(logN), where\nℓcan easily seen to be optimal up to a constant multiple. The so rting network in the above definition is\nnot necessarily the linking structure we require for Lemma 5 .5 – although it has a low maximum degree,\nit might not be ( A∪B)-path-constructible. As observed in [37], it is possible t o ‘transform’ a sorting\nnetwork into an ( A∪B)-path-constructible linking structure. Briefly, to do thi s, start by taking the\ngraphGwith the properties above with |A|=|B|=Nandℓ=O(logN). Then, for each 0 ≤i < ℓ,\ntake the bipartite graph G[Vi,Vi+1] and, using part 3 of the definition above, substitute each si ngle edge\nby a path of an appropriately chosen length mand each K2,2with vertices a1,a2∈Viandb1,b2∈Vi+1\nby the (2 m)-vertex graphs in Figure 4.\na1\na2b1\nb2\nFigure 4: EachK2,2in the sorting network with parts {a1,a2}and{b1,b2}is\nsubstituted with a graph of the type above. Each full line rep resents an edge and\neach dotted line represents a path of an appropriately chose n length. The pattern\nin the figure is such that we can always construct such a graph w ith any number of\nmarked vertices (i.e., those drawn with a circle in the figure ) which is sufficiently\nlarge and divisible by 4. Note that the union of the full lines is simply an even cycle.\n18Notice that the graph in Figure 4 is an ( {a1,a2},{b1,b2})-linking structure. Indeed, the two horizontal\npaths are vertex disjoint paths linking a1tob1anda2tob2, and they span all the vertices, and the blue\nand red paths are vertex disjoint paths linking a1tob2anda2tob1, and they span all the vertices. Since\neach of these graphs is a linking structure, substituting ea chK2,2in eachG[Vi,Vi+1] with one of these\ngraphs and each edge in each G[Vi,Vi+1] with a path, maintains that Gis an (A,B)-linking structure.\nThe crucial property of the graph in Figure 4 is that it is also {a1,a2}-path-constructible (as well\nas{a1,a2,b1,b2}-path-constructible). Indeed, it can be constructed by firs t taking a1and adding its\nincident path represented by a dotted line, then, construct ing the even cycle in the middle formed by the\nfull lines (which represent edges), and, finally, adding all of the remaining paths represented by dotted\nlines. Furthermore, substituting each K2,2in eachG[Vi,Vi+1] with one of the graphs in Figure 4 and\neach edge in each G[Vi,Vi+1] with a path, transforms Ginto anA∪B-path-constructible graph.\nWe can now complete our sketch proof of Lemma 5.5 by taking the illustrated graph to be such\nthat it has between 10log Nand 40log Nmarked vertices (which as is explained in the caption of the\nfigure is possible) and the dotted lines to represent equally -sized paths with length also between 10log N\nand 40log N. We then have that the illustrated graph has 2 m= Θ(log2N) vertices and it is {a1,a2}-\npath-constructible (and {a1,a2,b1,b2}-path-constructible) with paths of length between 10log Nand\n40logN. Then, substituting each K2,2in eachG[Vi,Vi+1] with one of these illustrated graphs and each\nedge in each G[Vi,Vi+1] with a path of length mmakes it so that the resulting graph is an ( A,B)-linking\nstructure with O(Nlog3N)≤N1.1vertices and is ( A∪B)-path-constructible with paths of length\nbetween 10log Nand 40log N. Thus, it satisfies Lemma 5.5.\nUsing Lemma 5.5, Lemma 5.3 and Definition 5.2, we can now prove Lemma 5.1.\nProof of Lemma 5.1. First, we will find A,B⊆V(G) and a subgraph H⊆Gsuch that His an\n(A,B)-linking structure with the following properties.\n1.|A|=|B|=n0.9.\n2.|H| ≤n/100.\n3. ∆(H)≤4.\n4.His (C/50,n/2C)-extendable.\nWe will then set X:=V(H). Let then N:=n0.9and apply Lemma 5.5 to find a linking structure with\nthe given properties. Now we embed HinGso that it is extendable. First, we embed H[A∪B] in\nGsuch that H[A∪B] is (C/50,n/2C)-extendable. For this, note that Lemma 5.3 implies that the re\nexists a path PinGof length |A|+|B|which is ( C/50,n/2C)-extendable. Indeed, we can start with a\n(C/50,n/2C)-extendable edge xyinG(sinceGis aC-expander, every edge is ( C/50,n/2C)-extendable)\nand then apply the lemma with ℓ=|A|+|B|. SincePis (C/50,n/2C)-extendable, the subgraph on\nV(P) formed by removing all edges (i.e., theempty subgraphwith vertex set V(P)) is also ( C/50,n/2C)-\nextendable and hence, we can partition Pinto two sets A,Bof equal size and we have an embedding of\nH[A∪B] inGwhich is ( C/50,n/2C)-extendable.\nSinceHisA∪B-path-constructible, there exists a sequence of paths P1,...,P tand graphs H0:=\nH[A∪B],H1,...,H t:=Hsuch that for each i, the following hold: Hi+1=Hi∪Pi+1; andPi+1is a\npath whose internal vertices are not in Hiand with at least one endpoint in Hi. Furthermore, each path\nPiis of length between 10log Nand 40log N. Since|V(H)| ≤N1.1≤n/100≤n−n/10−40logn,\nwe can then iteratively apply Lemma 5.3 embedding the paths Pione by one while maintaining the\n(C/50,n/2C)-extendability property. At the end, we have H⊆Gsatisfying all desired properties.\nWe now set X:=V(H) and note that the definition of a ( C/50,n/2C)-extendable subgraph implies\nthatG[V(G)\\X] andG[(V(G)\\X)∪A∪B] areC′-expanders for C′=C/100. Indeed, by definition,\n19for every S⊆V(G) of size at most n/Cwe have\n|Γ(S)\\V(H)| ≥(C/50−1)|S|−/summationdisplay\nu∈S∩V(H)(dH(u)−1)≥C|S|/50−5|S|,\nwhich implies that Ssatisfies|NG−X(S)| ≥C|S|/50−5|S|−|S| ≥C|S|/100 =C′|S|.\nBy repeated application of Lemma 5.3, we can then construct a linear forest Fwith|A|=|B|\nequally-sized paths whose endpoints are in A∪B, interior vertices are in V(G)\\X, and which have\nlengthn0.1/5, such that H∪Fis a (C/50,n/2C)-extendable subgraph of G. LetY∪A∪Bbe the set\nof vertices spanned by F, whereYis disjoint to X. Finally, let Zdenote the rest of the vertices. Since\nH∪Fis (C/50,n/2C)-extendable, by the same argument as above we have that G[Z] is aC′-expander\nforC′=C/100. Finally, since ( V(G)\\X)∪A∪B=Z∪Y∪A∪B, as mentioned above we also have\nthatG[Z∪Y∪A∪B] is aC/100-expander.\n6 Proof of Theorem 1.4\nFinally, we can put our work together to prove Theorem 1.4.\nProof of Theorem 1.4. LetGbe aC-expander and apply Lemma 5.1 to find X,Y,Zwith the stated\nproperties. The following holds due to the properties of the linking structure in G[X] and is crucial to\nobserve. If there exists a spanning linear forest FofG′:=G−(X\\(A∪B)) with no isolated vertices and\nsuchthatEnd( F) =A∪B,thenGcontainsaHamiltoncycle(seeFigure2). Moreformally, wec anrelabel\nthe vertices so that the pairs of endpoints in Fare (a1,b1),(a2,b2),...,(at,bt) for some t≤ |A|, as well\nas the pairs ( at+1,at+2),(at+3,at+4)...,(a|A|−1,a|A|) and (bt+1,bt+2),(bt+3,bt+4)...,(b|A|−1,b|A|), then\nuse the linking structure to find paths which span G[X], such that the pairs of endpoints are ( ai+1,bi)\nfor alli, with indices taken modulo |A|.\nAll we need now is to find such an F. First apply Lemma 4.1 to G[Z], which is an expander by\nthe property from Lemma 5.1, giving a spanning linear forest inG[Z] with at most n0.8paths and no\nisolated vertices. Now, we define the linear forest F0inG′to be the union of two linear forests: the\nfirst is the linear forest in G[Y∪A∪B] with endpoints in A∪Bgiven by Lemma 5.1, and the second\nis the linear forest found in G[Z] with at most n0.8paths. Let us denote the paths in the first forest by\nP1,...,P t(witht=n0.9) and the second linear forest as H. Recall that the paths P1,...,P thave size\nn0.1/5.\nTo find the linear forest F, we now apply Lemma 3.7 repeatedly as follows, starting with F0. At each\nstep, we take the current linear forest Fi, and if there are at least t/2 =n0.9/2 pathsPj(so that they\nspan at least 0 .1nvertices) which are still in Fi, we do the following (maintaining the invariants that\nthere are no isolated vertices in Fiand that A∪B⊆End(Fi)):\n•If|End(Fi)| ≥ |A∪B|+ 1, then note that since every path in Fihas two endpoints, the total\nnumber of endpoints is even. Since |A∪B|is even, then there are at least two endpoints outside\nof this set, call them xandy. By applying Lemma 3.7 with x,yandF:=Fiwe obtain a forest\nFi+1such that End( Fi+1) = End( Fi)\\{x,y}and|E(Fi+1)∆E(Fi)|=O(logn).\n•Otherwise, F:=Fiis the desired linear forest and we finish the process.\nNow, clearly, we can have only at most 2 |End(F0)\\A∪B|= 2|H| ≤2n0.8steps in this process.\nMoreover, it is possible to perform each step because at each step we have |E(Fi+1)∆E(Fi)|=O(logn),\nand therefore O(logn) pathsPiare changed for the next linear forest. This implies that, at each step,\nFistill contains at least t−O(ilogn)≥t−O(n0.8logn)≥t/2 pathsPj. Thus, we can find such an F\nat the end of the process and, hence, a Hamilton cycle.\n20References\n[1] M. Ajtai, J. Koml´ os, and E. Szemer´ edi. An O(nlogn) sorting network. In Proceedings of the fifteenth annual\nACM symposium on Theory of computing , pages 1–9, 1983.\n[2] M. Ajtai, J. Koml´ os, and E. Szemer´ edi. First occurrence of Ha milton cycles in random graphs. North-Holland\nMath. Stud. , 115(C):173–178, 1985.\n[3] S. Akbari, O. Etesami, H. Mahini, and M. Mahmoody. On rainbow cyc les in edge colored complete graphs.\nAustralas. J. Combin. , 37:33, 2007.\n[4] P. Allen, J. B¨ ottcher, Y. K. H` an, Hiep, and Y. Person. Powers of Hamilton cycles in pseudorandom graphs.\nCombinatorica , 37:573–616, 2017.\n[5] N. Alon and J. Bourgain. Additive patterns in multiplicative subgrou ps.Geom. Funct. Anal. , 24:721–739,\n2014.\n[6] N. Alon and A. Nussboim. K-wise independent random graphs. In 2008 49th Annual IEEE Symposium on\nFoundations of Computer Science , pages 813–822. IEEE, 2008.\n[7] N. Alon and Y. Roichman. Random Cayley graphs and expanders. Random Structures & Algorithms ,\n5(2):271–284, 1994.\n[8] L. Babai. Long cycles in vertex-transitive graphs. J. Graph Theory , 3(3):301–304, 1979.\n[9] I. Balla, A. Pokrovskiy, and B. Sudakov. A remark on Hamilton cyc les with few colors. Moscow Journal of\nCombinatorics and Number Theory , 7(3):73–77, 2017.\n[10] B. Bollob´ as. The evolution of sparse graphs. In Graph theory and combinatorics (Cambridge, 1983) , pages\n35–57, 1984.\n[11] B. Bollob´ as, T. I. Fenner, and A. M. Frieze. An algorithm for fin ding Hamilton paths and cycles in random\ngraphs.Combinatorica , 7:327–341, 1987.\n[12] S. Brandt, H. Broersma, R. Diestel, and M. Kriesell. Global conn ectivity and expansion: long cycles and\nfactors in f-connected graphs. Combinatorica , 26:17–36, 2006.\n[13] D. Christofides, J. Hladk´ y, and A. M´ ath´ e. Hamilton cycles in d ense vertex-transitive graphs. J. Combin.\nTheory Ser. B , 109:34–72, 2014.\n[14] V. Chv´ atal. On Hamilton’s ideals. Journal of Combinatorial Theory, Series B , 12(2):163–168, 1972.\n[15] V. Chv´ atal and P. Erd˝ os. A note on Hamiltonian circuits. Discrete Math. , 2(2):111–113, 1972.\n[16] D. Clemens, A. Ferber, M. Krivelevich, and A. Liebenau. Fast st rategies in Maker-Breaker games played on\nrandom boards. Combinatorics, Probability and Computing , 21(6):897–915, 2012.\n[17] C. Cooper, A. Frieze, and B. Reed. Random regular graphs of n on-constant degree: connectivity and Hamil-\ntonicity. Combin. Probab. Comput. , 11(3):249–261, 2002.\n[18] B. Csaba, D. K¨ uhn, A. Lo, D. Osthus, and A. Treglown. Proof of the 1-factorization and Hamilton decom-\nposition conjectures. Mem. Amer. Math. Soc. , 244(monograph 1154):164, 2016.\n[19] B. Cuckler and J. Kahn. Hamiltonian cycles in Dirac graphs. Combinatorica , 29:299–326, 2009.\n[20] M. DeVos. Longer cycles in vertex transitive graphs. arXiv:230 2.04255, 2023.\n[21] G. A. Dirac. Some theorems on abstract graphs. Proc. Lond. Math. Soc. , 3(1):69–81, 1952.\n[22] N. Dragani´ c, S. Glock, D. Munh´ a Correia, and B. Sudakov. O ptimal Hamilton covers and linear arboricity\nfor random graphs. arXiv preprint arXiv:2310.11580 , 2023.\n[23] N. Dragani´ c, M. Krivelevich, and R. Nenadov. Rolling backward s can move you forward: on embedding\nproblems in sparse expanders. Trans. Amer. Math. Soc. , 375(7):5195–5216, 2022.\n[24] A. Ferber, R. Glebov, M. Krivelevich, and A. Naor. Biased games on random boards. Random Structures &\nAlgorithms , 46(4):651–676, 2015.\n21[25] A. Ferber and E. Long. Packing and counting arbitrary Hamilton cycles in random digraphs. Random\nStructures & Algorithms , 54(3):499–514, 2019.\n[26] A. Ferber, E. Long, and B. Sudakov. Counting Hamilton decomp ositions of oriented graphs. Int. Math. Res.\nNot. IMRN , (22):6908–6933, 2018.\n[27] J. Friedman and N. Pippenger. Expanding graphs contain all sma ll trees.Combinatorica , 7:71–76, 1987.\n[28] A. Frieze, M. Krivelevich, P. Michaeli, and R. Peled. On the trace o f random walks on random graphs.\nProceedings of the London Mathematical Society , 116(4):847–877, 2018.\n[29] A. Frieze, S. Vempala, and J. Vera. Logconcave random graph s. InProceedings of the fortieth annual ACM\nsymposium on Theory of computing , pages 779–788, 2008.\n[30] S. Glock, D. Munh´ a Correia, and B. Sudakov. Hamilton cycles in p seudorandom graphs. arXiv preprint\narXiv:2303.05356 , 2023.\n[31] R. J. Gould. Recent advances on the Hamiltonian problem: Surve y iii.Graphs Combin. , 30:1–46, 2014.\n[32] D. Hefetz, M. Krivelevich, M. Stojakovi´ c, and T. Szab´ o. Positional games , volume 44. Springer, 2014.\n[33] D. Hefetz, M. Krivelevich, and T. Szab´ o. Hamilton cycles in highly connected and expanding graphs. Com-\nbinatorica , 29(5):547–568, 2009.\n[34] D. Hefetz, M. Krivelevich, and T. Szab´ o. Hamilton cycles in highly connected and expanding graphs. Com-\nbinatorica , 29(5):547–568, 2009.\n[35] D. Hefetz, M. Krivelevich, and T. Szab´ o. Sharp threshold for the appearance of certain spanning trees in\nrandom graphs. Random Structures & Algorithms , 41(4):391–412, 2012.\n[36] D. Hefetz, D. K¨ uhn, J. Lapinskas, and D. Osthus. Optimal co vers with Hamilton cycles in random graphs.\nCombinatorica , 34(5):573–596, 2014.\n[37] J. Hyde, N. Morrison, A. M¨ uyesser, and M. Pavez-Sign´ e. Sp anning trees in pseudorandom graphs via sorting\nnetworks. arXiv preprint arXiv:2311.03185 , 2023.\n[38] B. Jackson. Hamilton cycles in regular 2-connected graphs. Journal of Combinatorial Theory, Series B ,\n29(1):27–46, 1980.\n[39] D. Johannsen, M. Krivelevich, and W. Samotij. Expanders are u niversal for the class of all spanning trees.\nCombinatorics, Probability and Computing , 22(2):253–281, 2013.\n[40] J. Koml´ os and E. Szemer´ edi. Limit distribution for the existenc e of Hamiltonian cycles in a random graph.\nDiscrete Math. , 43(1):55–63, 1983.\n[41] A. Korshunov. Solution of a problem of Erd˝ os and R´ enyi on Ha milton cycles in non-oriented graphs. Soviet\nMath Dokl , 17:760–764, 1976.\n[42] M. Krivelevich. The critical bias for the Hamiltonicity game is (1+ o(1)).J. Amer. Math. Soc. , 24(1):125–131,\n2011.\n[43] M. Krivelevich. On the number ofHamilton cycles in pseudo-rando mgraphs. arXiv preprint arXiv:1111.6261 ,\n2011.\n[44] M. Krivelevich, C. Lee, and B. Sudakov. Robust Hamiltonicity of D irac graphs. Trans. Amer. Math. Soc. ,\n366(6):3095–3130, 2014.\n[45] M. Krivelevich and B. Sudakov. Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory , 42(1):17–\n33, 2003.\n[46] M. Krivelevich and B. Sudakov. Pseudo-random graphs. In More sets, graphs and numbers: A Salute to Vera\nSos and Andr´ as Hajnal , pages 199–262. Springer, 2006.\n[47] M. Krivelevich, B. Sudakov, V. H. Vu, and N. C. Wormald. Random regular graphs of high degree. Random\nStructures Algorithms , 18(4):346–363, 2001.\n[48] D. K¨ uhn, J. Lapinskas, D. Osthus, and V. Patel. Proof of a co njecture of Thomassen on Hamilton cycles in\nhighly connected tournaments. Proceedings of the London Mathematical Society , 109(3):733–762, 2014.\n22[49] D. K¨ uhn and D. Osthus. Hamilton decompositions of regular exp anders: a proof of Kelly’s conjecture for\nlarge tournaments. Adv. Math. , 237:62–146, 2013.\n[50] D. K¨ uhn and D. Osthus. Hamilton cycles in graphs and hypergra phs: an extremal perspective. In Proceedings\nof the International Congress of Mathematicians, Seoul, Ko rea., volume 4, pages 381–406, 2014.\n[51] L. Lov´ asz. Combinatorial structures and their applications. InProc. Calgary Internat. Conf., Calgary,\nAlberta, pages 243–246, 1970.\n[52] R. Montgomery. Spanning trees in random graphs. Advances in Mathematics , 356:106793, 2019.\n[53] A. M¨ uyesser and A. Pokrovskiy. A random Hall-Paige conjectu re.arXiv preprint arXiv:2204.09666 , 2022.\n[54] C. S. J. Nash-Williams. Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency. In Studies\nin Pure Mathematics (Presented to Richard Rado) , pages 157–183. Academic Press London, 1971.\n[55] I. Pak and R. Radoiˇ ci´ c. Hamiltonian paths in Cayley graphs. Discrete Math. , 309(17):5501–5508, 2009.\n[56] L. P´ osa. Hamiltonian circuits in random graphs. Discrete Math. , 14(4):359–364, 1976.\n[57] E. Rapaport Strasser. Cayley color groups and Hamilton lines. Scripta Math. , 24:51–58, 1959.\n[58] D. L. Shell. A high-speed sorting procedure. Communications of the ACM , 2(7):30–32, 1959.\n[59] A. Thomason. Pseudo-random graphs. In Proceedings of Random Graphs, Pozna´ n 1985, M. Karo´ nski, e d.,\nAnnals of Discrete Math. 33 (North Holland 1987) , volume 144, pages 307–331.\n[60] A. Thomason. Randomgraphs, stronglyregulargraphsand ps eudorandomgraphs. Surveys in Combinatorics ,\n123(173-195):1, 1987.\n23"    },    {        "title": "2402.06604v1.Gravitational_Waveform__A_Tale_of_Two_Formalisms.pdf",        "content": "arXiv:2402.06604v1  [hep-th]  9 Feb 2024Gravitational Waveform: A Tale of Two Formalisms\nDonato Bini,1, 2Thibault Damour,3Stefano De Angelis,4\nAndrea Geralico,1Aidan Herderschee,5Radu Roiban,6and Fei Teng6\n1Istituto per le Applicazioni del Calcolo M. Picone, CNR, I-0 0185 Rome, Italy\n2INFN, Sezione di Roma Tre, I-00146 Rome, Italy\n3Institut des Hautes Etudes Scientifiques, 91440 Bures-sur- Yvette, France\n4Institut de Physique Théorique, CEA, CNRS, Université Pari s-Saclay, F–91191 Gif-sur-Yvette cedex, France\n5Institute for Advanced Study, Princeton, NJ 08540, USA\n6Institute for Gravitation and the Cosmos, Pennsylvania Sta te University, University Park, PA 16802, USA\n(Dated: February 12, 2024)\nWe revisit the quantum-amplitude-based derivation of the g ravitational waveform emitted by\nthe scattering of two spinless massive bodies at the third or der in Newton’s constant, h∼G+\nG2+G3(one-loop level), and correspondingly update its comparis on with its classically-derived\nmultipolar-post-Minkowskian counterpart. A spurious-po le-free reorganization of the one-loop five-\npoint amplitude substantially simplifies the post-Newtoni an expansion. We find complete agreement\nbetween the two results up to the fifth order in the small veloc ity expansion after taking into account\nthree subtle aspects of the amplitude derivation: (1) in agr eement with [ arXiv:2312.07452 [hep-th]],\nthe term quadratic in the amplitude in the observable-based formalism [JHEP 02, 137 (2019)]\ngenerates a frame rotation by half the classical scattering angle; (2) the dimensional regularization\nof the infrared divergences of the amplitude introduces an a dditional (d−4)/(d−4)finite term; and\n(3) zero-frequency gravitons are found to contribute addit ional terms both at order h∼G1and at\norderh∼G3when including disconnected diagrams in the observable-ba sed formalism.\nI. INTRODUCTION\nThe remarkable detection of gravitational waves by the\nLIGO and Virgo collaborations [1, 2] sparked the de-\nvelopment of novel quantum field theory (QFT)-based\napproaches to the general relativistic two-body problem\nwhich build on enormous advances in scattering ampli-\ntudes and effective field theory. The scattering regime at\nlarge minimum separation is a theoretical environment\nfree of many subtleties related to the definition of initial\nand final states, making it the ideal ground for thorough\ncomparison and cooperation with the well-tested tradi-\ntional analytical approaches to this problem. In this pa-\nper we revisit such a comparison between the amplitudes-\nbased prediction in the observable-based (KMOC) for-\nmalism [3, 4] for the classical gravitational wave signal\nfrom the scattering of two non-spinning masses and the\ncorresponding waveform derived within the Multipolar-\npost-Minkowskian (MPM) formalism [5–7].\nA natural framework for discussing classical scattering\nfrom the QFT perspective consists in taking the classi-\ncal limit ( /planckover2pi1→0) of the perturbative QFT expansion\nof scattering amplitudes, which is an expansion in pow-\ners ofGm1m2\nJc, whereJis the orbital angular momen-\ntum. By collecting the appropriate terms contributing\nto classical observables, one should obtain the classical\npost-Minkowskian (PM) expansion (in powers of New-\nton’s constant G) of the considered classical observable\n(e.g. an impulse, ∆pµ, or the waveform hµν≡gµν−ηµν).\nFor pioneering computations of the classical PM expan-\nsion of scattering observables, see [8, 9] (waveform) and\n[10, 11] (impulse). Classical bodies are treated as point\nparticles with additional properties encoded in higher-\ndimension operators extending their minimal interactionswith gravity. The classical limit is identified in the spirit\nof the correspondence principle, and Lorentz invariance\nof standard QFT perturbation theory facilitates keeping\nthe complete dependence on the velocity of particles at\nevery order in the Gexpansion.\nExtensive effort, utilizing advances in scattering am-\nplitude methods such as of generalized unitarity [12–16],\nthe double-copy relations [17–19] and powerful integra-\ntion methods [20–27] and complemented by worldline ap-\nproaches [28, 29], explored and evaluted inclusive scat-\ntering observables for interactions of spinning and spin-\nless bodies, such as scattering angles, impulse, etc., to\nremarkably high perturbative orders [30–34]. Similarly,\nposition-dependent (point-like) observables such as the\ngravitational waveform emitted during a weak-field scat-\ntering process have been discussed from different per-\nspectives and approaches [4, 35–37]. Recently, aspects\nof radiation and reaction were discussed in KMOC for-\nmalism at the third order in Newton’s constant in [38]\nand the scattering waveform, hµν∼G1+G2+G3+...,\nwas constructed to this order for spinless particles [39–\n41] (see also [42]), for spinning particles at O(G2)[43–45]\nand atO(G3)[46] and various orders in spin.\nGravitational wave emission from gravitating bina-\nries has been studied with traditional analytic meth-\nods over many years, with the purpose of construct-\ning analytical-based waveforms to be used in the de-\ntection and data-analysis of gravitational wave sig-\nnals. One of the most successful analytical methods for\ncomputing the GW emission from generic sources has\nbeen the (post-Newtonian-matched) Multipolar-Post-\nMinkowskian (MPM) formalism [5–7]. The MPM for-\nmalism has been developed over the years to a high\nperturbative accuracy, recently reaching the fourth post-2\nNewtonian (4PN) accuracy [47, 48], i.e. the N4LO level\nin a PN expansion (in powers ofv2\nc2+Gm\nrc2) beyond\nthe leading-order quadrupole formula pioneered by Ein-\nstein [49]. This PN accuracy includes nonlinear terms of\norderO(G5)inhµν.\nA comparison of the MPM and amplitudes-based re-\nsults [50] revealed substantial differences. Ref. [50] also\npointed out a remarkable structure: if the amplitudes-\nbased results are interpreted as being in a frame rotated\nby half the classical scattering angle, then the differences\nare dramatically reduced. The origin of this frame rota-\ntion was recently understood by Georgoudis, Heissenberg\nand Russo [51] in terms of a certain cut contribution\npointed out by in Ref. [42], and demonstrated explicitly\nin a small-frequency expansion.1\nIn this paper we revisit several aspects of the\namplitude- and observable-based (KMOC) [4] derivation\nof the classical gravitational wave signal emitted dur-\ning the scattering of two non-spinning masses at the\nNNLO order,2hµν=G1+G2+G3, and of the com-\nparison of the frequency-domain result with the corre-\nsponding waveform derived within the Multipolar-post-\nMinkowskian formalism. We find complete agreement to\nthe available post-Newtonian accuracy (see Secs. VIII,X\nandXI).\nThe main difficulty with carrying out the PN expan-\nsion of the amplitude-based scattering waveform stems\nfrom the severe spurious singularities exhibited by the\navailable results [39–41]. We reorganize both the ampli-\ntude and the cut expressions so that they are written in\nterms of functions free of such spurious poles on the phys-\nical sheet (see App. C). This facilitates the PN expansion,\nwhich we carry out here to 3PN accuracy3, in the equato-\nrial (θ=π/2) plane and for the complete dependence on\nthe azimuthal angle φ. We demonstrate that the frame\nrotation of [50] is equivalent to the connected part of the\ncut contribution of the KMOC formalism through this\norder (see Sec. IX).\nGravitons of vanishing frequency and their contribu-\ntions to observables are a notoriously thorny subject, see\ne.g. [52]. Ignoring them in the KMOC calculation leads to\ndifferences with the MPM results already at leading order\nin Newton’s constant (in the time-independent Coulom-\nbic fieldhµν=O(G)generated by the incoming par-\nticles). We will show that including them resolves this\ndifference (see Sec. VA). They also have nontrivial con-\ntributions at higher orders in Newton’s constant, where,\nthrough the disconnected cut contributions, they yield\n1In the version 2 update of [51], this relation between the cut\ncontribution and frame rotation was also tested to NNLO (1PN )\nin the PN expansion.\n2In the following we will mainly refer to this as “the EFT wave-\nform” following the nomenclature used in the original compa rison\nof [50], but at times we will also use “the amplitudes-based w ave-\nform” and the “KMOC waveform”.\n3Ref. [50] had reached the 2.5PN accuracy for the φ-even part of\nthe waveform, without incorporating the 2PN-level terms.terms that can be interpreted as a Bondi-Metzner-Sachs\n(BMS) [53, 54] supertranslation of the result with no con-\ntributions from such states (see Secs. XIandVC). This\nBMS supertranslation is the same one that clarifies [52]\nthe existence of an angular momentum loss at O(G2)[55];\nits importance in the amplitudes-MPM comparison was\ninitially observed in a low-frequency expansion at the first\nnon-universal order [51]. A last ingredient in our com-\nparison is a subtle O(ǫ/ǫ)contribution which originates\nfrom the graviton being described by a symmetric, trace-\nless(d−2)×(d−2)matrix inddimensions (see Secs. VB\nandX) in our (retarded-time renormalization) scheme.\nWe begin with a brief description of the MPM and\namplitudes-based methods used to evaluate the wave-\nforms, emphasizing the new contributions in the latter,\nand then proceed to detaling our comparison. We collect\nthe list of notations used in this paper in App. A.\nNote added: While this paper was being written up,\nwe became aware of concurrent work by Georgoudis,\nHeissenberg and Russo, which partly overlaps with as-\npects of our analysis. We thank the authors for commu-\nnication and coordination on the submission.\nII. INTRODUCING THE MULTIPOLAR\nPOST-MINKOWSKIAN (MPM) FORMALISM\nThe two transverse-traceless components\nf+= lim\nr→∞(rh+), f ×= lim\nr→∞(rh×) (2.1)\nof the classical, time-domain asymptotic waveform are\nencoded in the complex quantity\nW(Tr,θ,φ) =c4\n4G(f+−if×)\n=c4\n4Glim\nr→∞¯mµ¯mνrhµν, (2.2)\nwhere the normalization prefactorc4\n4Gis used to simplify\nthe PN expansion of W.4HereTr≃t−r\nc−2GM\nc3logr\ncb0denotes a retarded time which contains a logarithmic\nshift, involving an arbitrary time scale b0, proportional to\nthe total center-of-mass (c.m.) mass-energy of the sys-\ntem,M≡E\nc2, while¯mµ(defined below) is a complex,\nnull polarization vector tangent to the sphere at infinity.\nThe MPM formalism is usually set up in the (incoming)\nc.m. of the system, and computes the components of the\nradiative multipole moments of the binary system with\nrespect to a frame ¯e0,ex,ey,ezanchored on the classical\naveraged momenta\n¯pa=1\n2(pa+p′\na),(a= 1,2), (2.3)\n4We indicate powers of c, orη≡1\nc, when they help to understand\nthe PN order of various quantities. Otherwise, we often set cto\n1 without warning.3\nrather than the incoming momenta pa. Here,p′\naare the\nclassical outgoing momenta, differing from the incoming\nonespaby the full classical scattering angle χ=O(G).\nThe time axis ¯e0of the c.m. frame is\n¯eµ\n0=¯pµ\n1+ ¯pµ\n2\n|¯pµ\n1+ ¯pµ\n2|. (2.4)\nThe vector eylies in the spatial direction of ¯p1(i.e. the\nbisector between the incoming and the outgoing spatial\nmomentum of the first particle in the c.m. frame),\n¯p1=E1¯e0+¯Pc.m.ey,¯p2=E2¯e0−¯Pc.m.ey.(2.5)\nHere,Ea=/radicalbig\nm2a+P2c.m., andPc.m.is the magnitude of\nthe spatial part of the incoming momenta. In principle,\n¯Pc.m.differs from Pc.m.by¯Pc.m.=Pc.m.cos1\n2χ. However,\nthe difference is only at the order χ2=O(G2)and will\nbe negligible in our work. The axis vector exis in the\nplane of motion and orthogonal to ey(and oriented from\nparticle 2 towards particle 1). It is also the direction of\nthe relative impact parameter,\nb12=bex. (2.6)\nHerebis an eikonal-type impact parameter, linked to\nthe incoming-momenta impact parameter binbybin=\nbcos1\n2χ, in which the difference O(χ2) =O(G2)is again\nnegligible. The last spatial axis vector ezis orthogonal to\nthe plane of motion (and such that ex,ey,ezis positively\noriented). All vectors and tensors are decomposed in the\nframe¯e0,ex,ey,ezand the angles θ,φare accordingly\ndefined, so that\nn(θ,φ) = sinθcosφex+sinθsinφey+cosθez,\nk=ω(¯e0+n(θ,φ)),\n¯m=1√\n2/bracketleftbigg\n∂θn(θ,φ)−i\nsinθ∂φn(θ,φ)/bracketrightbigg\n.(2.7)\nHerenis the spatial unit vector that characterizes the\ndirection of the gravitational wave propagation.\nThe Multipolar post-Minkowskian (MPM) for-\nmalism [5–7] computes the time-domain waveform\nWMPM(Tr,θ,φ)as a sum over irreducible multipolar\ncontributions, keyed by their multipole order ℓand their\nspatial parity. Even-parity (radiative) multipoles are\ndenotedUℓ,ℓ= 2,3,4,..., while odd-parity ones are\ndenotedVℓ,ℓ= 2,3,4,.... It is convenient to keep track\nof theη≡1\ncprefactors entering the multipole expansion\nof the waveform emitted by a slow-motion source by\nwritingWas\nWMPM(Tr,θ,φ) =U2+η(V2+U3)+η2(V3+U4)\n+η3(V4+U5)+η4(V5+U6)\n+η5(V6+U7)+···. (2.8)\nWith this normalization, the PN expansion of each ra-\ndiative multipole UℓorVℓstarts at the Newtonian order\n(in the sense that the leading order (LO) value of eachmultipole is expressed in terms of Newtonian-level quan-\ntities). In Eq. ( 2.8), eachUℓorVℓis expressed in terms\nof corresponding symmetric-trace-free (STF) Cartesian\ntensors of order ℓ, according to\nUℓ(Tr,θ,φ) =1\nℓ!¯mi¯mjni1ni2···niℓ−2Uiji1i2···iℓ−2(Tr),\nVℓ(Tr,θ,φ) =−1\nℓ!2ℓ\nℓ+1¯mi¯mjncni1ni2···niℓ−2\n×ǫcdiVjdi1i2···iℓ−2(Tr). (2.9)\nThe two infinite sequences of radiative multipole mo-\nmentsUi1i2···iℓ(t),Vi1i2···iℓ(t)(wherethenceforth de-\nnotes the retarded time variable Tr) fully, and uniquely,\nparametrize the time, and angular, dependences of any\ngravitational waveform. They are formally observable if\nwe imagine surrounding the emitting system by infinitely\nmany GW detectors located on the sphere at infinity (in\nthe c.m. frame). Actually, only the time-dependent part\nof each multipole, e.g., Ui1i2···iℓ(t)−Ui1i2···iℓ(t=−∞),\nis directly observable when considering the LO O(1\nr)\nasymptotic gravitational field at future null infinity. Per-\nturbative computations have, however, shown that the\npresence of non-zero initial values Ui1i2···iℓ(t=−∞)∝ne}ationslash= 0\nof the radiative multipoles have physical effects at fu-\nture null infinity at higher orders in O(1\nr), via the back-\nscattering of gravitational waves on spacetime curvature\n(“tails”) [56, 57]. We will come back to this issue below.\nThe (post-Newtonian(PN)-matched) MPM formalism\ncomputes each (time-domain) radiative multipole mo-\nmentUi1i2···iℓ(t),Vi1i2···iℓ(t)in terms of the stress-energy\ntensor of the material source, Tµν(xλ), in several steps.\nAt the end of the day, each radiative multipole at re-\ntarded time tis given by a sum of contributions involv-\ning both source variables at time tand hereditary inte-\ngrals over the past behavior of the source, i.e. over times\n−∞ ≤t′≤t. All those hereditary integrals are expressed\nin terms of some source multipole moments , of mass-type,\nIi1i2···iℓ(t), and spin-type, Ji1i2···iℓ(t). Some of the hered-\nitary integrals are linear in the source moments (“linear\ntails”) while others are nonlinear because they arise from\nnonlinear couplings between the multipole signals in the\nzone exterior to the source. Finally, the source moments\nare computed in terms of the dynamical variables of the\nsource by using a PN-expanded approach to the gravita-\ntional field in the near zone.\nTo give an idea of the structure of the expressions com-\ning out of the MPM formalism, let us display a few of the\ncontributions to the radiative quadrupole5Uij(t)at the\nfractional 2.5PN accuracy ( η5beyond the classic Einstein\n5In the formulas below the angular brackets /angbracketleft.../angbracketrightdenote a STF\nprojection over the enclosed indices.4\nquadrupole formula):\nUij(t) =d2Iij(t)\ndt2(2.10)\n+2GM\nc3/integraldisplay∞\n0dτI(4)\nij(t−τ)/parenleftbigg\nlog/parenleftbiggτ\n2b0/parenrightbigg\n+11\n12/parenrightbigg\n+G\nc5/parenleftbigg1\n7I(5)\na/angbracketleftiIj/angbracketrighta−5\n7I(4)\na/angbracketleftiI(1)\nj/angbracketrighta−2\n7I(3)\na/angbracketleftiI(2)\nj/angbracketrighta/parenrightbigg\n+···\nin which M≡E\nc2is the total mass-energy of the sys-\ntem, and one must insert the 2.5PN-accurate value of\nthe source quadrupole Iij(t).\nIn the MPM formalism, the successive contributions in-\ndicated in Eq. ( 2.10) have different origins: (i) the source\nquadrupole moment Iij(t)computes (in a PN-expanded\napproximation) the quadrupole moment of the effective\nstress-energy tensor of the system, i.e. the sum of the\nTµν\nppof the point particles and of the effective Tµν\ngof the\nnear-zone gravitational field (potential gravitons, plus\nsome local retarded-field effects); (ii) the second (hered-\nitary) term is the linear tail contribution generated by\nthe back-scattering of quadrupolar waves on the external-\nzone Coulomb field ∼GM\nrgenerated by the total mass-\nenergy of the system; and (iii) is an example of the several\n(instantaneous) nonlinear terms arising from cubic cou-\nplings between multipole moments (here quadrupole ×\nquadrupole ×quadrupole coupling).\nIn the c.m. frame the 2.5PN-accurate source\nquadrupole Iijhas the following structure\nI≤2.5PN\nij(t) =νM/parenleftbigg\n1+a2\nc2+a4\nc4−24\n7ν\nc5G2M2\nr2˙r/parenrightbigg\nx/angbracketleftixj/angbracketright\n+ terms in x/angbracketleftivj/angbracketrightandv/angbracketleftivj/angbracketright. (2.11)\nHere,xi(t) =xi\n1(t)−xi\n2(t)denotes the relative motion\n(in the c.m. frame), vi(t) =dxi\ndtis the relative velocity,\nM=m1+m2is the total mass and νdenotes the sym-\nmetric mass ratio, ν=m1m2/(m1+m2)2. In addition,\nin order to get explicit results for the radiative moments\nas a function of time, one must solve the correspondingly\nPN-accurate equations of motion of the binary system.\nThe latter task is aided by the existence of a rather simple\nquasi-Keplerian representation of the hyperbolic-like so -\nlution of the 2PN-level conservative equations of motion\n(to which the effect of radiation reaction, which starts at\ntheG2/c5level, must be added) [58–60].\nAs the asymptotic behavior of xi(t)in the in-\ncoming state is xi(t)≈vi\n0t+O(logt), the source\nquadrupole moment is found to behave as Iij(t)≈\nνMSTF[vi\n0vj\n0]/parenleftBig\n1+O(v2\n0\nc2)/parenrightBig\nt2. One then checks that all\nthe hereditary integrals defining Uij(t)are convergent,\nand thatUij(t)has anonzero value when t→ −∞ of the\nform\nUij(t=−∞) = 2νMSTF[vi\n0vj\n0]/parenleftbigg\n1+O(v2\n0\nc2)/parenrightbigg\n.(2.12)Note that the nonzero value of the radiative quadrupole\nmoment at the Newtonian approximation is a simple con-\nsequence of the classic Einstein quadrupole formula say-\ning thatUij=d2\ndt2(/summationtext\naSTF(maxi\naxj\na))+O(η2).\nSimilar results hold for all the multipole moments. In-\ndeed, the origin of all these nonzero initial radiative mul-\ntipole moments is the fact that the waveform in the in-\nfinite past is simply given by the Coulombic field of the\nincoming worldlines, namely\nfij(t=−∞,n) =/summationdisplay\na4G/bracketleftBigg/parenleftbig\npi\napj\na/parenrightbigTT\nEa−n·pa/bracketrightBigg\n,(2.13)\nleading to (with n≡k/ω)\nW(t=−∞) =/summationdisplay\na(¯m·pa)2\nEa−n·pa=/summationdisplay\na(¯m·pa)2\n−(n·pa).(2.14)\nIn other words, the MPM waveform necessarily incorpo-\nrates the nonzero initial value, Eqs. ( 2.13), (2.14), given\nby the Coulombic field of the incoming worldlines.\nFinally, after having obtained some representation of\nthe various time-domain multipole moments, one must\ncompute their Fourier transforms (over the retarded time\nvariable), namely\nUi1i2···iℓ(ω) =/integraldisplay+∞\n−∞dteiωtUi1i2···iℓ(t),\nVi1i2···iℓ(ω) =/integraldisplay+∞\n−∞dteiωtVi1i2···iℓ(t).(2.15)\nThis leads to a Fourier-domain MPM waveform of the\nform\nWMPM(ω,θ,φ)≡U2(ω,θ,φ) (2.16)\n+η(V2(ω,θ,φ)+U3(ω,θ,φ))\n+η2(V3(ω,θ,φ)+U4(ω,θ,φ))\n+η3(V4(ω,θ,φ)+U5(ω,θ,φ))\n+η4(V5(ω,θ,φ)+U6(ω,θ,φ))\n+η5(V6(ω,θ,φ)+U7(ω,θ,φ))+···\nHere, the explicit powers of η=1\ncindicate those asso-\nciated with the LO, Newtonian value of each radiative\nmultipole. They serve as a reminder that the knowledge\nof the full waveform WMPM(ω,θ,φ)at, say, the 2.5PN\naccuracy requires one to know: (i) U2at the fractional\n2.5PN accuracy ( η5); (ii)V2andU3at the fractional 2PN\naccuracy (η4); (iii)V3andU4at the fractional 1.5PN\naccuracy (η3); (iv)V4andU5at the fractional 1PN ac-\ncuracy (η2); (v)V5andU6at the Newtonian accuracy\n(because their first PN correction starts at 1PN) and,\nfinally, (vi) V6andU7at the Newtonian accuracy.\nIn addition, as we are comparing the MPM waveform\nto the one-loop-accuracy amplitudes-based waveform, we\nonly need to determine the O(G)andO(G2)parts of\ntheG−expansion of the waveform (in its Wguise, ie.\nafter dividing limr→∞¯mµ¯mνrhµνby4G). Finally, an5\nMPM/EFT waveform comparison at the 2.5PN accuracy\nrequires, on the MPM side, the knowledge of:\nU2(ω)∼(G+G2)(η0+η2+η3+η4+η5)\nV2(ω) andU3(ω)∼(G+G2)(η0+η2+η3+η4)\nV3(ω) andU4(ω)∼(G+G2)(η0+η2+η3)\nV4(ω) andU5(ω)∼(G+G2)(η0+η2)\nV5(ω) andU6(ω)∼(G+G2)(η0)\nV6(ω) andU7(ω)∼(G+G2)(η0). (2.17)\nThe time-dependent part ( ω∝ne}ationslash= 0) of each multipole starts\nat orderG1. Only their static (initial) values contribute\nat orderG0, as exhibited in Eq. ( 2.14).\nIn these expansions, the terms of order η3only arise\nat orderG2and have two different origins. One origin is\nthe linear tails which are all of the form\nUG2tail\nℓ(t) =2GM\nc3/integraldisplay∞\n0dτd2\ndt2UG\nℓ(t−τ)\n×/parenleftbigg\nlog/parenleftbiggτ\n2b0/parenrightbigg\n+κℓ/parenrightbigg\nVG2tail\nℓ(t) =2GM\nc3/integraldisplay∞\n0dτd2\ndt2VG\nℓ(t−τ)\n×/parenleftbigg\nlog/parenleftbiggτ\n2b0/parenrightbigg\n+σℓ/parenrightbigg\n,(2.18)\nwith\nκℓ=2ℓ2+5ℓ+4\nℓ(ℓ+1)(ℓ+2)+k=ℓ−2/summationdisplay\nk=11\nk,\nσℓ=ℓ−1\nℓ(ℓ+1)+k=ℓ−1/summationdisplay\nk=11\nk, (2.19)\nin the time domain, and\nUG2tail\nℓ(ω) =2GMω\nc3/parenleftBigπ\n2+i[log(2ωb0eγE)−κℓ]/parenrightBig\nUG\nℓ(ω),\nVG2tail\nℓ(ω) =2GMω\nc3/parenleftBigπ\n2+i[log(2ωb0eγE)−σℓ]/parenrightBig\nVG\nℓ(ω),\n(2.20)\nin the frequency domain.\nAnother origin of the G2η3terms exists for V3and\nU4. They come from quadrupole ×quadrupole couplings.\nE.g., inU4there are terms of the form\nUII\nijkl(t) =G\nc3/parenleftbigg\n−21\n5I/angbracketleftijI(5)\nkl/angbracketright−63\n5I(1)\n/angbracketleftijI(4)\nkl/angbracketright\n−102\n5I(2)\n/angbracketleftijI(3)\nkl/angbracketright/parenrightbigg\n. (2.21)\nThese quadrupole ×quadrupole couplings generate terms\nof orderG2/c5inU2(see last line in Eq. ( 2.10)).\nOn the other hand, the terms of even order in ηcome\nfrom PN corrections to the source multipole moments, as\nexemplified by the termsa2\nc2+a4\nc4in Eq. ( 2.11).III. STRUCTURE AND COMPUTATION OF\nTHE MPM WAVEFORM\nRef. [50] computed the multipoles needed to reach the\nabsolute (G1+G2)η5accuracy in the even-in- φprojec-\ntion of the waveform. Because of the planar nature (in\nthexyplane) of the binary motion generating the wave-\nform, it is easily seen that a parity-even multipole Ul\ncontains (when decomposed in eimφ)mvalues equal to\nm=l,l−2,l−4etc. By contrast, a parity-odd multipole\nVlcontainseimφcontributions for m=l−1,l−3,l−5etc.\nTherefore, the even-in- φprojection of WMPMis given, at\naccuracyη5, byU2,V3,U4,V5, andU6. The latter mul-\ntipoles were computed in Ref. [50], and their values in\nthe equatorial plane ( θ=π\n2) were displayed there.\nIn the present work, we complete the MPM/EFT com-\nparison by also computing some of the multipoles con-\ntributing to the odd-in- φpart of the waveform. For\nsimplicity, we focus on only two terms in the PN ex-\npansion of the odd-in- φpiece, namely, the terms of or-\nder(G1+G2)(η1+η4). These terms come from the\nNewtonian-level terms in V2andU3, together with the\ncorresponding tail terms (which are Gη3smaller than the\nLO contribution). We present the full angular depen-\ndence of the multipoles by giving the coefficients of the\ndecomposition of WMPM(ω)in spin-weighted (with spin-\nweights=−2) spherical harmonics (SWSH) sYlm(θ,φ),\ndefined with the convention spelled out in an ancillary\nfileMPMmultipoles.nb . For instance, denoting for clar-\nitys=−2as¯2, andm=−2as¯2)\n¯2Y22(θ,φ) =/radicalbigg\n5\n4πe2iφcos4/parenleftbiggθ\n2/parenrightbigg\n,\n¯2Y2¯2(θ,φ) =/radicalbigg\n5\n4πe−2iφcos4/parenleftbiggθ\n2/parenrightbigg\n. (3.1)\nThe(l,m)SWSH coefficients of WMPMare the sum of the\nSWSH coefficients of UlandVl(respectively multiplied\nbyηl−2andηl−1),\nWMPM\nlm=ηl−2Ulm+ηl−1Vlm, (3.2)\nwhere the coefficients Ulm,Vlmare defined by\nUl(θ,φ) =l/summationdisplay\nm=−lUlm¯2Ylm(θ,φ),\nVl(θ,φ) =l/summationdisplay\nm=−lVlm¯2Ylm(θ,φ). (3.3)\nThe coefficients Ulm,Vlmcan be obtained by using the\northonormality of the sYlm(θ,φ)’s, e.g.\nUlm=/integraldisplay\nsinθdθdφU l(θ,φ)¯2Y∗\n2,m(θ,φ). (3.4)\nIn the time-domain, the Ulm, andVlmare functions of the\nretarded time t. When going to the frequency domain,6\nthey are functions of ω, with, e.g.,\nUlm(ω) =/integraldisplay+∞\n−∞dteiωtUlm(t). (3.5)\nIn practice, the Fourier transforms of the radiative mul-\ntipoles are more conveniently performed on the Carte-\nsian representation of Ui1i2···il(t)andVi1i2···il(t). As ex-\nplained in Ref. [50], it is convenient to Fourier trans-\nform the first time-derivatives of each Ui1i2···il(t)and\nVi1i2···il(t)(which vanish like O(1/t2)att→ −∞ ) and to\ndivide the result by −iω. We only consider strictly pos-\nitive frequencies ω >0because the ω <0part is deter-\nmined by the reality of Ui1i2···il(t)andVi1i2···il(t), while\ntheω= 0part is fully described by the ingoing, asymp-\ntotic waveform W(t=−∞).W(t=−∞)is exactly\ngiven by the G0, linearized-gravity result Eq. ( 2.14).\nLet us illustrate the structure (and physical content)\nofUlm(ω), andVlm(ω), by focussing on the quadrupolar\ncontribution U2=1\n2¯mi¯mjUijwhich is the only one which\nstarts at the Newtonian order η0.\nWe have sketched in Eq. ( 2.10) the various contri-\nbutions to Uij(t)which contribute at the fractional η5\n(2.5PN) level. The first term in Eq. ( 2.10),\nUI\nij(t) =d2Iij(t)\ndt2, (3.6)\nhasη5contributions coming both from an explicit 2.5PN\ncontribution in the source quadrupole moment [61],\nIij(t)∼STF/bracketleftBig/summationdisplay\namaxi\naxj\na+η2+η4+η5+.../bracketrightBig\n,(3.7)\nfrom the need to include O(G2/c5)radiation-reaction ef-\nfects [62] in the second time-derivative of the positions\nxi\na(t), and from the need to compute the Fourier trans-\nform along the radiation-reacted hyperbolic motions6.\nThe second term in Eq. ( 2.10) is the (linear) tail con-\ntribution which starts at order G2η3and reads, in the\nfrequency domain,\nUG2tail\nij(ω) =2GMω\nc3(3.8)\n×/bracketleftbiggπ\n2+i/parenleftbigg\nlog(2ωb0eγE)−11\n12/parenrightbigg/bracketrightbigg\nUG\nij(ω),\nwhere we recall that M≡E\nc2. Hereb0is the time\nscale used in the MPM formalism to adimensionalize the\n(Coulomb-like) logarithmic divergence contained in the\ntail.\nThe last term in Eq. ( 2.10) is indicative of a sum of sev-\neral nonlinear contributions (generated by various cou-\nplings between the multipolar waves in the zone exterior\n6However, as explained in Ref. [50], this can be avoided by wor king\nwithd\ndtUI\nij(t).to the system). These terms are expressed as products\nof derivatives of source (and gauge) multipole moments,\nall taken at the same retarded time tasUij(t)itself. In\nUijat orderG2/c5, there are three such nonlinear con-\ntributions: (i) one bilinear in Iij(UII\nij, denotedUQQ\nijin\n[50]); (ii) one bilinear in Iijand the angular momentum\nLk(ULI\nij≡ULQ\nij), and (iii) one bilinear in Iijand the\nℓ= 0Wi1...iℓgauge moment ( UWI\nij≡UWQ\nij):\nUII+LI+WI\nij (t) =G\nc5/parenleftbigg1\n7I(5)\na/angbracketleftiIj/angbracketrighta−5\n7I(4)\na/angbracketleftiI(1)\nj/angbracketrighta\n−2\n7I(3)\na/angbracketleftiI(2)\nj/angbracketrighta+.../parenrightbigg\n.(3.9)\nThough only one power of Genters the prefactor of Eq.\n(3.9),UII+LI+WI\nij (t)actually starts at the level G2be-\ncause of the high-order ( n≥3) time-derivatives of the\nsource quadrupole moment Iij(t)it contains. [As said in\nRef. [50] there is a final hereditary term in Uij(t), the\nnonlinear memory contribution which, however, starts at\norderG3, i.e. at the two-loop order.]\nWhen considering the frequency-domain version of U2,\nwe have (at one-loop accuracy, i.e. modulo O(G3)inW)\nU2(ω) =UG\n2(ω)+UG2\n2(ω), (3.10)\nwhere both UG\n2(ω)andUG2\n2(ω)start at Newtonian or-\nder. The PN expansions of UG\n2(ω)andUG2\n2(ω)have the\nstructure\nUG\n2∼GM2ν\np∞/parenleftbig\n1+η2p2\n∞+η4p4\n∞+η6p6\n∞+.../parenrightbig\n,\nUG2\n2∼/parenleftbiggGM\nbp2∞/parenrightbiggGM2ν\np∞(1+η2p2\n∞+η3p3\n∞+η4p4\n∞\n+η5p5\n∞+η6p6\n∞+...). (3.11)\nNote that/parenleftBig\nGM\nbp2∞/parenrightBig\nis dimensionless (when considering that\np∞has the dimension of a velocity) and of the order of\nthe Newtonian scattering angle. When considering that\np∞is a velocity, the PN expansion is encoded in pow-\ners ofηp∞=p∞\nc. The tree-level waveform WG(being\ntime-symmetric) contains only even powers of ηp∞. By\ncontrast, the one-loop waveform WG2starts having odd\npowers ofηp∞in its fractional PN expansion at order\nη3p3\n∞, which is the order where hereditary tail effects\narise (see Eq. ( 3.8)). The next odd power of ηp∞is\nη5p5\n∞which contains several types of contributions com-\ning from: (i) η5and radiation-reaction contributions to\nEq. ( 3.7); (ii) fractional 1PN corrections contained in\nthe linear tail term Eq. ( 3.8); and (iii) the nonlinear\ncontributions to Uij, Eq. ( 3.9).\nLet us finally illustrate the explicit structure of the\nfrequency-domain waveform by displaying a few impor-\ntant contributions at the O(G)andO(G2)levels. The\nFourier transform is done at the level of the once-time-\ndifferentiated, PN-expanded, Cartesian-component ra-\ndiative multipolesd\ndtUi1i2···il(t)andd\ndtVi1i2···il(t). The7\ntime-dependence of these quantities is conveniently ob-\ntained by using the explicit quasi-Keplerian representa-\ntion of the relative two-body motion (in the c.m. frame).\nThe latter representation is naturally expressed in the\nex,eyvectorial frame (which is anchored on the classi-\ncal¯pamomenta, rather than the incoming momenta pa).\nThe polar coordinates r,ϕof the relative motion,\nx(t) =x(t)ex+y(t)ey\n=r(t)[cosϕ(t)ex+sinϕ(t)ey],(3.12)\nare given as explicit functions of an (hyperbolictype) “ec-\ncentric anomaly” variable vbyquasi-Keplerian expres-\nsions ons of the form\n¯nt=etsinh(v)−v+O(η4),\nr= ¯ar(ercosh(v)−1))+O(η4), (3.13)\nϕ= 2Karctan/parenleftBigg/radicalBigg\neϕ+1\neϕ−1tanh/parenleftBigv\n2/parenrightBig/parenrightBigg\n+O(η4).\nHere, the quasi-Keplerian quantities ¯n,et,er,eϕ,Kare\n(PN-expanded) functions of the c.m. energy and angular\nmomentum of the binary system (see Refs. [58–60]).\nThe quasi-Keplerian representation Eq. ( 3.13) incorpo-\nrates (in the conservative case) a time symmetry around\nt= 0, corresponding to the closest approach between the\ntwo bodies. The asymptotic logarithmic drift of the two\nworldlines is embodied in the vparametrization involving\nhyperbolic functions.\nInserting the (PN-expanded) quasi-Keplerian represen-\ntation Eq. ( 3.13) in, say,\n−iωUi1i2···il(ω) =/integraldisplay+∞\n−∞dteiωtd\ndtUi1i2···il(t),(3.14)leads to integrals which can all be obtained (at the G+G2\naccuracy) from the master integrals (for u>0)\n/integraldisplay+∞\n−∞dvei(usinh(v)−µv)=/integraldisplay+∞\n−∞dT√\n1+T2eiuT−iµarcsinhT\n= 2eµπ\n2Kiµ(u), (3.15)\n/integraldisplay+∞\n−∞dv\ncoshveiusinh(v)=/integraldisplay+∞\n−∞dT\n1+T2eiuT\n=πe−u. (3.16)\nActually, one only needs the large-eccentricity expansion\n(corresponding to a PM expansion in powers of G, and\nto an expansion in powers of µ) of the master integral\nEq. (3.15).\nUsing these integrals, all the SWSH components of the\nwaveform at order G+G2are expressed as linear com-\nbinations of modified Bessel functions of order 0and1,\nK0(u)andK1(u), and (starting at the 1PN fractional\norder) of the exponential e−u/u, with argument given by\nthe following dimensionless version of the frequency:\nu≡ωb\np∞. (3.17)\nThe coefficients of K0(u),K1(u)ande−u/uare (ν-\ndependent) polynomials in uwhose degrees increase with\nthe PN order.\nAs already said, when l= 2themvalues are only\nm= +2,m= 0, andm=−2.\nHere is a sample of the SWSH components of the quadrupolar pie ce of the waveform. At order G1η0we find\nUG1η0\n22=−GM2ν\np∞2√\n5π\n5/bracketleftBig\n(2u+1)K0(u)+2(u+1)K1(u)/bracketrightBig\n,\nUG1η0\n20=−GM2ν\np∞√\n30π\n15K0(u),\nUG1η0\n2¯2=GM2ν\np∞2√\n5π\n5/bracketleftBig\n(2u−1)K0(u)−2(u−1)K1(u)/bracketrightBig\n. (3.18)\nAt orderG1η2we have\nUG1η2\n22=νGM2p∞√\n5π\n105/braceleftBig\n[(−72ν+38)u2+(−36ν−2)u−24ν+78]K0(u)\n+ [(−72ν+38)u2+(−72ν+17)u−48ν−138]K1(u)/bracerightBig\n,\nUG1η2\n20=νGM2p∞√\n30π\n105/bracketleftBig\n(−8ν+26)K0(u)+u(−16ν+59)K1(u)/bracketrightBig\n,\nUG1η2\n2¯2=νGM2p∞√\n5π\n105/braceleftBig\n[(−72ν+38)u2+(36ν+2)u−24ν+78]K0(u)\n+ [(72ν−38)u2+(−72ν+17)u+48ν+138]K1(u)/bracerightBig\n. (3.19)8\nTheO(G2η0)multipoles have a simple relation to the O(G1η0)ones:\nUG2η0\n2m=π\n2GM\nbp2∞uUG1η0\n2m. (3.20)\nProceeding to O(G2η2), we have instead\nUG2η2\n22=−G2M3νπ3/2√\n5\n210bp∞/braceleftBig\nu[u2(72ν−38)+u(78ν−124)+45ν−141]K0(u)\n+u[u2(72ν−38)+u(114ν−143)+90ν+12]K1(u)+252(2u2+2u+1)\nue−u/bracerightBig\n,\nUG2η2\n20=−G2M3νπ3/2√\n30\n210bp∞/bracketleftBig\nu(−47+15ν)K0(u)+u2(−59+16ν)K1(u)/bracketrightBig\n,\nUG2η2\n2¯2=G2M3νπ3/2√\n5\n210bp∞/braceleftBig\n−u[u2(72ν−38)+u(−78ν+124)+45 ν−141]K0(u)\n+u[u2(72ν−38)+u(−114ν+143)+90 ν+12]K1(u)+252\nue−u/bracerightBig\n. (3.21)\nAt 2PN the structure of UG2η4\n2is similar to that of UG2η2\n2apart from a quadratic-in- νdependence of the coefficients\nand an overall multiplication by p2\n∞.\nLet us finally focus on the G2η5contribution to U2. Following the decomposition presented above, each lmcom-\nponent ofU2is the sum of two terms: (i) an “instantaneous\" contribution ,Uinst\n2m, defined as\nUinst\n2m=UI\n2m+UII\n2m+ULI\n2m+UWI\n2m, (3.22)\ni.e. the sum of the source-moment-related term, UI\n2m, Eq. ( 3.7), and of the nonlinear terms indicated in Eq. ( 3.9);\nand (ii) a (nonlocal in time) tail term Utail\n2m, Eq. ( 3.8) (here projected on its G2η5contribution). They are respectively\ngiven by:\nUinstG2η5\n22 =2G2M3ν2√\n5πp2\n∞\n525biu/bracketleftBig\n2(245u2−57u−15)K0(u)+(490u2+131u−72)K1(u)/bracketrightBig\n,\nUinstG2η5\n20 =2G2M3ν2√\n30πp2\n∞\n105biu/bracketleftBig\n−2K0(u)+29uK1(u)/bracketrightBig\n,\nUinstG2η5\n2¯2=2G2M3ν2√\n5πp2\n∞\n525biu/bracketleftBig\n2(245u2+57u−15)K0(u)−(490u2−131u−72)K1(u)/bracketrightBig\n, (3.23)\nand\nUtailG2η5\n22 =−2\n105G2M3ν√\n5πp2\n∞\nb/parenleftbigg\nilog(2b0ωeγE)−i11\n12+π\n2/parenrightbigg\nu/braceleftBig\n[(72ν−38)u2+(78ν+2)u+45ν−78]K0(u)\n+ [(72ν−38)u2+(114ν−17)u+90ν+138]K1(u)/bracerightBig\n,\nUtailG2η5\n20 =−2\n105G2M3ν√\n30πp2\n∞\nb/parenleftbigg\nilog(2b0ωeγE)−i11\n12+π\n2/parenrightbigg\nu/bracketleftBig\n(15ν−26)K0(u)+u(16ν−59)K1(u)/bracketrightBig\n,\nUtailG2η5\n2¯2=−2\n105G2M3ν√\n5πp2\n∞\nb/parenleftbigg\nilog(2b0ωeγE)−i11\n12+π\n2/parenrightbigg\nu/braceleftBig\n[(72ν−38)u2+(−78ν−2)u+45ν−78]K0(u)\n+ [(−72ν+38)u2+(114ν−17)u−90ν−138]K1(u)/bracerightBig\n. (3.24)\nLet us finally give an example of the SWSH components of the φ-odd (and (m1−m2)-odd) piece of the waveform\nVG1η0\n21=−4i√\n5π\n15νGM2m1−m2\nMu[K0(u)+K1(u)],\nVG1η0\n2¯1=−4i√\n5π\n15νGM2m1−m2\nMu[K0(u)−K1(u)], (3.25)9\nwith\nVG2η0\n21=π\n2/parenleftbiggGM\nbp2∞/parenrightbigg\nuVG1η0\n21,\nVG2η0\n2¯1=π\n2/parenleftbiggGM\nbp2∞/parenrightbigg\nuVG1η0\n2¯1, (3.26)\nas before. The corresponding tail contributions are given b y\nVG2η3\n21=−4√\n5π\n15bm1−m2\nM/parenleftbigg7\n3+πi−2log(2b0eγEω)/parenrightbigg\np∞G2M3νu2[K0(u)+K1(u)]\n=−iGM\nbp∞u/parenleftbigg7\n3i+iπ−2ilog(2b0eγEω)/parenrightbigg\nVG1η0\n21,\nVG2η3\n2¯1=−4√\n5π\n15bm1−m2\nM/parenleftbigg7\n3+πi−2log(2b0eγEω)/parenrightbigg\np∞G2M3νu2[K0(u)−K1(u)]\n=−iGM\nbp∞u/parenleftbigg7\n3+πi−2ilog(2b0eγEω)/parenrightbigg\nVG1η0\n2¯1. (3.27)\nAll the SWSH components of the waveform WMPMthat we computed are available in the ancillary file\nMPMmultipoles.nb attached to this paper. They reach (in W)(G1+G2)(1 +η2+η3+η4+η5)accuracy for\nthe evenmterms, and the (G1+G2)(η+η4)accuracy for the odd mterms.\nIV. THE OBSERVABLE-BASED FORMALISM\nScattering waveform can also be computed using field-\ntheory based methods. In particular, the observable-\nbased formalism [3] relates directly final state observable s\nwithS-matrix elements in the classical limit. Before we\nstart to introduce the formalism, we note that in field\ntheory, it is customary to define metric perturbation as\ngµν=ηµν+κhµν, whereκ2= 32πG. It differs from\ngµν=ηµν+hµνused in the MPM calculation by a factor\nofκ. We will stick to the field-theory convention when\ncomputing S-matrix elements. We will compensate the\nfactor ofκwhen performing the Fourier transform to im-\npact parameter space, see Eq. ( 4.6) below. We will also\nsetc= 1in all the field-theory related calculations.\nA. General setup\nThe observable-based formalism [3] constructs quan-\ntum scattering observables by comparing the expectation\nvalue of the corresponding hermitian operator Obetween\nthe initial and final state of the scattering process\n∝an}bracketle{tO∝an}bracketri}ht=∝an}bracketle{tψout|O|ψout∝an}bracketri}ht−∝an}bracketle{tψin|O|ψin∝an}bracketri}ht, (4.1)\nwhereOcould be, e.g., the momentum transferred dur-\ning the scattering process or the curvature at large dis-\ntances. Observables are assumed to be measured at\ninfinity, where the gravitational field is effectively free.\nThe evaluation of the expectation value in Eq. ( 4.1) pro-\nceeds by constructing the final state |ψout∝an}bracketri}htas the time-\nevolution of the initial state |ψin∝an}bracketri}ht,|ψout∝an}bracketri}ht=S|ψin∝an}bracketri}ht, and\nthen inserting a complete set of final states next to O.The result is an expression of ∝an}bracketle{tO∝an}bracketri}htin terms products of\nS-matrix elements suitably integrated over the final-state\nphase space.\nFor the scattering problem, we construct the initial\ntwo-particle states as the tensor product of two localized\nsingle-particle states,\n|ψin∝an}bracketri}ht=/integraldisplay\ndΦp1dΦp2ϕ(p1)ϕ(p2)e−i(p1·b1+p2·b2)|p1p2∝an}bracketri}ht,\ndΦp=d4p\n(2π)4Θ(p0)(2π)δ(p2+m2), (4.2)\nwhere the measure d Φpimposes the on-shell condition,\nand the wavepacket is normalized as/integraltext\ndΦp|ϕ(p)|2= 1.\nTo compute the scattering waveform we may choose\nthe operator Oto either be the linearized Riemann ten-\nsor7or to be the transverse-traceless components of the\nasymptotic metric fluctuations about Minkowski space at\ninfinity. In the former approach, the metric is extracted\nfrom the expectation value of the asymptotic Newman-\nPenrose scalar Ψ4∝¨f+−i¨f×, by integrating twice over\ntime and determining the two integration constants from\nphysical considerations. One integration constant is fixed\nby the boundary condition limt→∞˙f(t)∼t−2. The other\ncorresponds to the EFT analog of the Coulombic field in\nEq. (2.13).\nThe latter choice yields directly the waveform. It\nalso gives a matrix-element interpretation of the inte-\ngration constants needed to reconstruct the asymptotic\n7The linearization is a consequence of the operator being ins erted\nin the final state, i.e. at infinity. The LSZ reduction then pro jects\nout all Green’s functions with more than a single graviton.10\nmetric from the Newman-Penrose scalar. In particular,\nthe integration constant that is time-independent is de-\ntermined by the parts of Eq. ( 4.1) that have support\nonly on vanishing outgoing-graviton frequency ω. Such\nδ(ω)-supported matrix elements yield nontrivial time-\ndependent contributions to the waveform at higher orders\ninGthrough iteration, as we will see in Sec. VC.\nThe observable-based formalism yields final-state ob-\nservables in the complete quantum theory. The building\nblock of the momentum-space expectation value of the\nclassical asymptotic metric is\nM(ε,k,p1,p2,q1,q2)\n≡i∝an}bracketle{tp3p4|ˆa(k)T|p1p2∝an}bracketri}ht+∝an}bracketle{tp3p4|T†ˆa(k)T|p1p2∝an}bracketri}ht\n=i∝an}bracketle{tp3p4k|T|p1p2∝an}bracketri}ht+∝an}bracketle{tp3p4|T†ˆa(k)T|p1p2∝an}bracketri}ht,(4.3)\nwhereˆacreates a physical graviton with polarization vec-\ntorεwhen acting to the left. The momentum lost by each\nmatter particle is\nq1=p1−p4, q 2=p2−p3, (4.4)\nand momentum conservation requires that q1+q2be the\noutgoing graviton momentum k. The first term in M\nis the five-point amplitude with one graviton in the fi-\nnal state; the second term is a unitarity cut that gives a\nparticular discontinuity of the quantum amplitude. We\nshall refer to this bilinear-in- Tterm as the “unitarity cut\nterm”, or simply, the “cut term”. The matrix element\nMis a Lorentz invariant function of the various scalar\nproducts of (ε,k,p1,p2,q1−q2). For notational conve-\nnience, in the following we will (partially) suppress the\narguments of M.\nThe absence of local diffeomorphism-invariant ob-\nservables in general relativity requires that one be\ncareful with the effect of large gauge transformations\non asymptotic observables. The asymptotic Newman-\nPenrose scalar and the transverse-traceless components\nof the asymptotic metric are invariant under decaying-at-\ninfinity coordinate transformations, δhµν=∂(µξν), but\nare not invariant under BMS transformations. However,\nas we will see in Section VC, both the KMOC and MPM\nformalisms (implicitly) use the same BMS vacuum, re-\nferred to as “intrinsic gauge” in Ref. [52].8\nB. Classical limit\nAs we reviewed above, the observable-based formalism\nestablishes the relation between observables with com-\nplete velocity dependence and the matrix element M\nin the quantum theory. The correspondence principle,\n8Perhaps the term “intrinsic frame” might be more appropriat e as\nit should be possible to discern its properties through grav ita-\ntional wave measurements.stating that the classical regime emerges in the limit of\nlarge (macroscopic) conserved charges, provides a means\nto extract the classical observables from their quantum\nmechanical counterpart. In our case these charges are\nthe particles’ masses, momenta and orbital angular mo-\nmentaJ, i.e.J≫/planckover2pi1= 1andm≫MPl. It then follows\n[3, 63, 64] that the impact parameter bis much larger\nthan the Compton wave length λc=p−1or, equivalently,\nthat the momentum transfer qis much smaller than the\ntypical external momenta p9and that observables are\nseries in the effective coupling Gm1m2/J.\nIn this limit, certain parts of loop-level expressions con-\ntribute to classical observables. The precise dependence\non Newton’s constant Gand momentum transfer follows\nfrom the observation that, like Newton’s potential itself,\npowers of Newton’s potential are also classical; in impact\nparameter space, a classical L-loop contribution is pro-\nportional (G/b)L+1. The inclusion of outgoing (gravita-\ntional) radiation with energy ωcommensurate with the\nmomentum transfer, ω∼q, does not alter this depen-\ndence because there is no suppression for the emission of\nclassical (gravitational) radiation. Quantum scattering\namplitudes are however more singular in the large im-\npact parameter (small transferred momentum) expansion\nthan their classical counterparts, behaving GL+1/bl<L+1,\nwith additional mass-dependent factors. This can be un-\nderstood by recalling the structure of potential scatterin g\nin non-relativistic quantum mechanics, which at N-th or-\nder in perturbation theory contains multiple insertions of\nthe lower-order potential. The cancellation of these clas-\nsically singular (or superclassical) terms in the matrix\nelement Mis a test of the calculation.\nIt is extremely convenient to take the classical limit as\nearly as possible [63–65], before the Feynman loop inte-\ngrals are evaluated, as this substantially reduces the com-\nplexity of calculations. Each of the internal graviton ex-\nchanged between matter particles should mediate a long-\nrange interaction; thus, in this regime all loop momenta\nℓshould be of the same order as the total exchanged mo-\nmentumq. The method of regions [66] in dimensional\nregularization is a systematic method to select the classi-\ncal contributions to M. The integration domain is split\ninto two regions\nHard Region : (qi,k,ℓ)→(λqi,λk,λ0ℓ),\nSoft Region : (qi,k,ℓ)→(λqi,λk,λℓ),(4.5)\nand the latter generates all classical terms. To evaluate\nan integral in only one of the regions one simply expands\nthe integrand according to the indicated scaling in λand\n9In later sections we will compare MPM and amplitudes-based\npredictions in the PN expansion. To be in the regime of validi ty\nof both this and of the PM expansions originally used in the\namplitudes computation, we must have the stronger conditio n\non the impact parameter, b≫v−2Rs, corresponding to a small\nscattering angle χ∼GM/(bv2).11\nintegrates over the entire integration domain. The full\nintegral is given by the sum of the integrals in the two\nregions [66]; there is no overcounting because the inte-\ngrand in one region yields only scaleless integrals when\nfurther expanded in the other region, and such integrals\nvanish in dimensional regularization.\nThe momentum space KMOC matrix element Mis\nrelated to the frequency-domain EFT waveform W(here\nnormalized as the MPM waveform WMPM) through a\nFourier transform to impact-parameter space,\nW(ε,k,p1,p2,b1,b2) (4.6)\n=2i\nκ/integraldisplay\nµ(k,q1,q2)e−i(q1·b1+q2·b2)M(ε,k,p1,p2,q1,q2),\nand we choose the (incoming) impact parameter b12=\nb1−b2to be orthogonal to both p1andp2,\np1·b12=p2·b12= 0. (4.7)\nThe integration measure enforces momentum conserva-\ntion and on-shell conditions for the matter particles. Fol-\nlowing Eq. ( 4.2), we can write the measure as\n/integraldisplay\ndΦp1dΦp2dΦp3dΦp4ϕ∗(p3)ϕ∗(p4)ϕ(p1)ϕ(p2)\n׈δ4(p1+p2−p3−p4−k)\n=/integraldisplay\nµ(k,q1,q2)+(quantum ), (4.8)\nand the classical contribution is\nµ(k,q1,q2) =d4q1\n(2π)4d4q2\n(2π)4ˆδ(2p1·q1)ˆδ(2p2·q2)(4.9)\n׈δ4(q1+q2−k),\nwhereˆδ(x) = 2πδ(x)andˆδd(x) = (2π)dδd(x). We\nnow sketch the derivation of Eq. ( 4.9). In the classi-\ncal limit, the Compton wavelength ℓcis much smaller\nthan the spread of the wavepacket ℓφin the position\nspace, which is in turn much smaller than the impact\nparameter |b1−b2|, namely,ℓc≪ℓφ≪ |b1−b2|. Conse-\nquently, in a classical scattering process, the wavepacket s\nϕ(p1)andϕ∗(p4)exactly overlap up to quantum cor-\nrections, which allows us to integrate out the wavepack-\nets,/integraltext\ndΦp1ϕ(p1)ϕ∗(p4)≃1[same forϕ(p2)andϕ∗(p3)].\nThen in d Φp3and dΦp4we change the variables to q1\nandq2while keeping only the leading term in the on-\nshell condition,\nδ(p2\n4+m2\n1)≃δ(2p1·q1),\nδ(p2\n3+m2\n2)≃δ(2p2·q2). (4.10)\nNote that we can drop the positive energy constraint in\nthe measure because it is already implicit in the above\ndelta function constraints.C. IR divergence and time shift\nThe matrix element Mis infrared-divergent [38–42],\nwith contributions from both terms in Eq. ( 4.3), which we\nregularize in dimensional regularization with d= 4−2ǫ.\nEven at the one-loop order we are focusing on, the fi-\nnite waveform integrand depends on whether internal\nstates are four- or d-dimensional. This is in contrast with\nscattering angle calculations, where this subtlety appear s\nonly atO(G4)[31], and is due here to the presence of di-\nvergences which are effectively (additively) renormalized .\nTo avoid possible issues probably due to unconventional\nforms of dimensional regularization (see e.g. [67, 68]), we\nuse conventional dimensional regularization (CDR) [69],\nwhere all states and momenta are uniformly continued\ntod= 4−2ǫdimensions. In the final expressions we re-\nstrict the external momenta to be four dimensional and\nuse special four-dimensional relations such as vanishing\nGram determinants for further simplifications. This reg-\nularization scheme was important in the comparison of\ntheO(G4)scattering angle [31] and the corresponding\npost-Newtonian calculations of Ref. [70]; this will be the\ncase here as well.\nThe IR divergences in Mare due to virtual soft gravi-\ntons and are expected to exponentiate [71],\nM= exp/bracketleftbigg\n−iωGE\nǫ(1−Γ/2)/bracketrightbigg\nMfin, (4.11)\nwhere Mfinis IR-finite but dependent of ǫ. The first\nterm in the exponent originates from the Weinberg soft\nfactor from the five-point amplitude [71], whose proof can\nbe generalized to all orders in the dimensional regulator.\nThe second term comes from the cut term [42] and Γis\ndefined as\nΓ =3y−2y3\n(y2−1)3/2. (4.12)\nThe factorωEin the exponent of Eq. ( 4.11) can be writ-\nten covariantly as\nm1w1+m2w2=ωE, (4.13)\nwherew1andw2are\nwa=−ua·k=−pa·k\nma. (4.14)\nTo understand the fate of the IR divergence, it is useful\nto consider the time-domain waveform, which is related\nto the frequency-domain waveform ( 4.6) by the Fourier-\ntransform\nW(t) =/integraldisplay+∞\n−∞dω\n2πW(ω)e−iωt. (4.15)\nThe IR divergent phase can now be absorbed in the\n(re)definition of the retarded time,\nt→t+δtIR, δt IR=GE\nǫ(1−Γ/2). (4.16)12\nEquivalently, we can choose to use the IR finite matrix\nelement in Eq. ( 4.6) when computing the waveform,\nM→Mfin= exp/bracketleftbigg\niωGE\nǫ(1−Γ/2)/bracketrightbigg\nM.(4.17)\nThe result for this IR-finite matrix element at Lloops\ndepends on the O(ǫ1≤n≤L)part of M, which in turn\ndepends on the dimensionality of the internal states at\nlower-loop orders. In particular, the tree-level term in\nMcontains such O(ǫ)contributions which enforce that\nthe regularized theory contains only d-dimensional gravi-\nton states, which we will refer to as ǫM(0)\nextra. While they\nare unimportant for the O(G)waveform, they interfere\nwith theO(G/ǫ)argument of the phase and yield finite\ncontributions at O(G2)proportional to M(0)\nextra. At one-\nloop order inclusion of these terms amounts to a choice of\nscheme, and the one used here is different from that used\nin [38–42]. Their inclusion however is natural in light of\nthe fact that Weinberg’s exponentiation of IR divergences\nholds to all orders in dimensional regulator in CDR sug-\ngesting that they are required for the exponentiation at\ntwo and higher loops. Furthermore, this scheme led to\na successful comparison of amplitudes and PN predic-\ntions for the O(G4)conservative scattering angle, where\nsimilarǫ/ǫeffects played an important role.\nV. WAVEFORM FROM OBSERVABLE-BASED\nFORMALISM\nBased on their origin in the T-matrix element, we can\ndivide momentum-space matrix element Minto three\ncategories,\nM(k,q1,q2) =Mconst(k,q1,q2)+Mconn(k,q1,q2)\n+Mdisc(k,q1,q2). (5.1)\nThe first term Mconstis contributed by matrix elements\nthat have the external graviton attached to an on-shell\nthree-point vertex. Due to momentum conservation, the\ngraviton must have zero energy. Thus Mconstcorre-\nsponds to a constant gravitational background. The sec-\nond term Mconncontains the amplitudes and unitarity\ncuts contributed from connected T-matrix elements with\ngeneric momentum, while Mdisccontains the unitarity\ncuts containing disconnected T-matrix elements.\nIn this section, the matter momentum paand four-\nvelocityuaare to be understood as the “barred variables”\nin amplitude literature (see e.g. [72, 73]). They differ\nfrom the true momentum pa(used e.g. in App. D) by\nhalf of the momentum transfer qa, i.e.pa=pa+qa/2.\nIn the classical limit, the difference between paandpa\nis quantum so they are indistinguishable. In contrary,\nthe¯paused in the MPM calculation is the “(classical)\naveraged momentum”, defined as ¯pa=pa+∆pa/2, where\n∆pais the classical impulse.A. Constant background\nExactly-zero energy excitations are an interesting fea-\nture of quantum field theories with massless particles.\nOn the one hand, the inclusion of an arbitrary number\nof such excitations does not change the energy of any\nstate, such as the Higgs field in spontaneously-broken\ngauge theories [74, 75]. On the other, they, and their\noff-shell extensions, may condense into nontrivial time-\nindependent backgrounds, such as the Coulomb field of a\npoint charge or the Schwarzschild metric [76, 77]. From\nthe perspective of on-shell scattering amplitudes around\nflat space, only the asymptotic zero-energy states appear\nto be relevant. It has been suggested in Ref. [78] that\ntheir effects may be incorporated in a vacuum with no\nsuch excitations through a coherent state.\nIt was emphasized in Ref. [52] that the constant part\nof the waveform is dependent on the choice of BMS su-\npertranslation frame. The so-called “canonical frame” is\ndefined to have a vanishing time-independent component\nand the “intrinsic frame” having the non-vanishing value\ngiven in Eq. ( 2.13). A BMS supertranslation connects\nthe waveforms in the two frames; in light of Ref. [78] we\nmay expect that this supertranslation is captured by co-\nherent state of low/vanishing-energy gravitons. Instead\nof exploring the all-order nature of this expectation, we\nwill study the effect of including zero-energy gravitons in\nthe KMOC computation.\nThe leading order contribution to Mconstcomes from\nthe diagram shown in Fig. 1a: the momentum of one mat-\nter particle is conserved while the other emits a graviton.\nThe relevant amplitudes are\nM3(k,p1,p4) =−iκ(ε·p1)2,\nM3(k,p2,p3) =−iκ(ε·p2)2. (5.2)\nFor on-shell and real momenta, this process is supported\nat the origin of phase space, at which the emitted gravi-\nton has exactly zero energy.\nSince one matter particle does not participate in the\nscattering, we need to use the three-particle phase space\nto perform the Fourier transform,\nWconst(k)/vextendsingle/vextendsingle/vextendsingle\nm1=2i\nκ/integraldisplay\ndΦp4ˆδ4(p1−p4−k)e−i(p1−p4)·b1\n×M3(k,p1,p4)\n=m1(ε·u1)2ˆδ(u1·k). (5.3)\nIt is easy to see by going to the rest frame of the particle\nof massm1that the delta function has a unique solution\nfor which the graviton has exactly zero energy.\nThus, if exacty zero energy gravitons are excluded from\nthe spectrum of states, Wconst(k) = 0 and we recover the\nexpected result in the BMS canonical frame. Consis-\ntently excluding such gravitons from higher-order calcu-\nlations amounts to excluding all kinematic configurations\nwhich constrain at least one graviton to have exactly zero\nenergy.13\np1p2\np4p3k\n(a)p1p2\np4p3k\n(b)\nFIG. 1. The contribution to the KMOC matrix element from\nthe disconnected components of the T-matrix.\nFIG. 2. The connected contribution to the KMOC matrix\nelement. Each blob here is a connected component of the\nT-matrix. The first two graphs are respectively the tree and\none-loop amplitude, and the last graph is a unitarity cut. Th e\nexternal particles are labeled in the same way as Fig. 1.\nConversely, allowing such contributions reproduces\nthe MPM Coulombic background given in Eqs. ( 2.13)\nand (2.14).1011At higher orders, it requires us to con-\nsistently include special kinematic configurations local-\nizing at least one graviton at the origin of phase space.\nFurthermore, we may expect that these terms may be\nobtained from those supported on finite-energy config-\nurations through the same supertranslation as the one\nconnecting the intrinsic and canonical frames.\nAs we have discussed in Sec. IVA, the Coulombic back-\nground serves to fix the initial value of the waveform.\nSince the three-point amplitudes are sufficient to match\nthe initial condition with the MPM calculation, other di-\nagrams involving a zero energy graviton are not expected\nto contribute to the constant background. We will show\nin App. Dthat it is indeed the case.\nB. Amplitudes and connected cuts\nThe contribution of all the connected (generic mo-\nmentum)T-matrix elements to Mhave been evaluated\n10We note that the “constant background” here is different from\nMlin(k)given in Eq. (5.2) of [50]. Here the background is cho-\nsen to be the pure time-independent piece of the waveform tha t\nis present at t=−∞, whileMlin(k)there was defined as the\naverage of the time-independent contributions at t=±∞. In\nother words, Mlin(k)included some time-dependent evolutions\nthat contribute to the gravitational-wave memory.\n11The observation that zero-energy gravitons reproduce the\nCoulombic background was also made recently in a related con -\ntext in [79].in [38–41] to O(κ5). We can write it as\nMconn(k,q1,q2) =Mtree+M1 loop+S1 loop+O(κ7),\n(5.4)\nwhereMtreeandM1 loopare respectively the tree-level\nand one-loop amplitude contributed from the first term\n(linear inT) of Eq. ( 4.3), while the cut contribution\nS1 loopcomes from the second term of Eq. ( 4.3). Dia-\ngrammatically they are represented by Fig. 2.\nFor later convenience, we separate out the κand mass\ndependence from Mtree,\nMtree=κ3m2\n1m2\n2M(0)\nd. (5.5)\nThe reduced tree-level amplitude is given by\nM(0)\nd=i/bracketleftbigg\n−F2\n2y2\nq2\n1q2\n2w2\n2(5.6)\n+F2\n3/parenleftbigg\n−1\nq2\n1q2\n2+y\nq2\n2w1w2+y2\n4w2\n1w2\n2−q2\n1y2\n4q2\n2w2\n1w2\n2/parenrightbigg\n+F2F3/parenleftbigg\n−2y\nq2\n1q2\n2w2+y2\nq2\n2w1w2\n2/parenrightbigg/bracketrightbigg\n+i\nd−2/bracketleftbiggF2\n3(q2\n1−q2\n2)\n4q2\n2w2\n1w2\n2+F2\n2\nq2\n1q2\n2w2\n2−F2F3\nq2\n2w1w2\n2/bracketrightbigg\n.\nwhere the graviton polarizations are encoded in the gauge\ninvariant variables\nF1=Fµνqµ\n1uν\n1, F2=Fµνqµ\n2uν\n2, F3=Fµνuµ\n1uν\n2,\n(5.7)\nandFµν=kµεν−kνεµis the linearized field strength.\nFord= 4−2ǫ, we have,\nM(0)\nd=4−2ǫ=M(0)\nd=4+ǫM(0)\nextra+O(ǫ2), (5.8)\nwhere the extra dimensional contribution comes from the\n1\nd−2factor in Eq. ( 5.8),\nM(0)\nextra=i[4F2\n2w2\n1−4F2F3w1q2\n1+F2\n3q2\n1(q2\n1−q2\n2)]\n8(q2\n1q2\n2w2\n2w2\n1).\n(5.9)\nAlthough the external states are kept in exactly four di-\nmensions, the ǫdependence appears here to ensure that\nonly graviton states propagate in d= 4−2ǫdimensions.\nThis is a consequence of the dimensional regularization\nscheme we choose. Of course, we can set d= 4when\ncomputing the tree-level waveform, and M(0)\nextradoes not\ncontribute.\nSimilarly, we can write the one-loop amplitude and cut\ncontribution as\nMone loop=κ5m3\n1m2\n2M(1)\n1+κ5m2\n1m3\n2M(1)\n2,\nSone loop=κ5m3\n1m2\n2S(1)\n1+κ5m2\n1m3\n2S(1)\n2.(5.10)14\nThe reduced one-loop amplitude M(1)\n1is given by\nM(1)\n1=−iw1\n32πǫM(0)\nd=4+/tildewiderM(1)\n1, (5.11)\nwhere the IR finite part /tildewiderM(1)\n1is the same as Eq. (5.6)\nof [50]. The expression of /tildewiderM(1)\n1can be found in App. B.\nWe note that M(1)\n2in Eq. ( 5.10) can be obtained from\nM(1)\n1by exchanging the particle labels 1and2. On the\nother hand, the cut contribution can be written as\nS(1)\n1=iw1Γ\n64πǫM(0)\nd=4+/tildewideS(1)\n1, (5.12)\nwhere the finite part /tildewideS(1)\n1is given in App. B, andS(1)\n2is\ngiven by the same particle relabeling. Note that we have\nremoved a local UV divergence from S(1)\n1. Such terms\ndo not affect the long-range interactions that we are in-\nterested in, and they are projected out by the Fourier\ntransform to impact parameter space.\nThe full IR divergence of the one-loop waveform is\n(M1 loop+S1 loop)/vextendsingle/vextendsingle/vextendsingle\nIR=−iωκ2E(2−Γ)\n64πǫMtree\nd=4,(5.13)\nFollowing Eq. ( 4.17) and the discussion below, we can\nabsorb the IR divergence into an infinite shift in the re-\ntarded time δtIR=GE\nǫ(1−Γ/2)to obtain the IR finite\nmomentum-space waveform,\nMfin= exp/bracketleftbigg\niωGE\nǫ(1−Γ/2)/bracketrightbigg\nM\n=Mtree+M1 loop+S1 loop\n+iωκ2E(2−Γ)\n64πǫMtree+O(κ7) (5.14)\n=Mtree+M1 loop fin+S1 loop fin+O(κ7).\nCrucially, the time shift introduces a term proportional to\nthed-dimensional tree amplitude Mtree. While this term\ncancels the IR divergence ( 5.13), it brings an additional\nfinite contribution proportional to the extra dimensional\ncomponent ( 5.9). Organizing the finite one-loop ampli-\ntude and cut by the mass dependence,\nM1 loop fin=κ3m3\n1m2\n2M(1)fin\n1+κ3m2\n1m3\n2M(1)fin\n2,\nS1 loop fin=κ3m3\n1m2\n2S(1)fin\n1+κ3m2\n1m3\n2S(1)fin\n2,(5.15)\nwe get\nM(1)fin\n1=/tildewiderM(1)\n1+iw1\n32πM(0)\nextra,\nS(1)fin\n1=/tildewideS(1)\n1−iw1Γ\n64πM(0)\nextra. (5.16)\nComparing with Eqs. ( 5.11) and ( 5.12) we see that, while\nthe divergence is removed by absorbing the 1/ǫinto a\ntime shift, this shift introduces a new contribution pro-\nportional to the extra dimensional component of the five-\npoint tree amplitude defined in Eq. ( 5.8). Thus the ad-\nditional terms proportional to M(0)\nextracome from the ǫ/ǫcancellation, and they are the direct consequence of the\nregularization scheme chosen in Sec. IVC. While at this\nloop order the inclusion of this term may be regarded\nas a scheme dependence (and was not included in previ-\nous amplitude calculations or comparisons), its presence\nhere is justified by the expected higher-loop structure of\nIR divergences as well as by its importance in related con-\ntexts [31]. As we will see in Sec. VIII, its presence sub-\nstantially improves the comparison between MPM and\namplitudes-based results.\nC. Cuts with disconnected matrix elements and\nBMS transformation\nIf zero-energy gravitons are excluded, then all cut con-\ntributions containing a three-point S-matrix element or\nann+ 1-point matrix element in the 1→nconfigura-\ntion vanish identically. All disconnected contributions t o\nthe cut term in Eq. ( 4.3) are of this type. If zero-energy\ngravitons are kept in the spectrum, the (albeit degen-\nerate) three-point amplitude generates, at higher-order\nin Newton’s constant, time-dependent terms through the\ncut term in this equation. The relevant contribution at\none-loop order is shown in Fig. 1b; the discussion above\nimplies that, at Lloops, the six-point tree amplitude fac-\ntor is replaced by the corresponding (L−1)-loop six-point\namplitude with no changes to the disconnected factor.12\nThe evaluation of this cut makes use of the fact that\nthe cut massless momentum ℓmust have vanishing fre-\nquency, so only the leading soft limit of the six-point tree\namplitude (the left blob of Fig. 1b) with two external\ngravitons of momenta kandℓis relevant,\nM6(p1...4,k,ℓ)≃ −κM5(p1...4,k) (5.17)\n×/parenleftBigg4/summationdisplay\ni=aηaε(ℓ)µνpµ\napν\na\n2ℓ·pa−iǫηa+ε(ℓ)µνkµkν\n2ℓ·k−iǫ/parenrightBigg\n,\nwhereηa=±1for outgoing and incoming particles, re-\nspectively, and M5(p1...4,k)is the five-point amplitude\nwith one external graviton of momentum k.\nSewing Eq. ( 5.17) onto the disconnected five-point am-\nplitude, we find that the four summed terms in Eq. ( 5.17)\nthat involve massive matter momenta will lead to inte-\ngrals of the form\n/integraldisplay\nddℓˆδ(u3·ℓ)ˆδ(ℓ2)Θ(ℓ0),\n/integraldisplay\nddℓˆδ(u3·ℓ)ˆδ(ℓ2)ˆδ(u4·ℓ)Θ(ℓ0), (5.18)\n12In the full quantum theory the three-point Green’s function is\nUV divergent; this divergence arises from loop momenta outs ide\nthe soft region, and do not contribute to the classical obser vables\nwe are discussing here.15\nboth of which are integrated to zero in d= 4−2ǫ.\nTherefore, a nonzero contribution can only come from the\nlast term in this equation, which corresponds to the soft\ngraviton being attached to the outgoing graviton with\nmomentum k. We can replace M5(p1...4,k)by its classi-\ncal limit given in Eq. ( 5.5). It then yields the following\ncontribution to MdiscatO(κ5)(which makes use of the\non-shell condition for the outgoing graviton):\nMdisc=−iκ2Mtree/bracketleftbig\nm2\n2w2\n2I(u3,k)+m2\n1w2\n1I(u4,k)/bracketrightbig\n,\n(5.19)\nwhere we used that in the classical limit −p3,4·k=−p2,1·\nk+O(q2)≃m2,1w2,1andI(u3,k)is defined as\nI(u3,k) =1\nm2/integraldisplayddℓ\n(2π)dˆδ(2u3·ℓ)ˆδ(ℓ2)Θ(ℓ0)\n2ℓ·k\n=1\nm2ω/integraldisplayddℓ\n(2π)dˆδ(2u3·ℓ)ˆδ(ℓ2)Θ(ℓ0)\n2ℓ·ˆk,(5.20)\nwhereˆk=k/ω, andI(u4,k)is obtained simply by re-\nlabeling. We did not include Θ(p0\n3+ℓ0)for the matter\ncut line because |ℓ0| ≪p0\n3soΘ(p0\n3+ℓ0) = 1. We have\nalso suppressed the iǫin the propagator since it does not\naffect the integral. However, to properly include the zero\nenergy graviton contribution across the cut, we need to\nintroduce additional regularizations.\nSince the zero energy graviton is supported only at the\norigin of phase space, we must regularize it so that this\npoint,|ℓ|= 0, remains in the integration domain.13We\nmay do this by relaxing the on-shell condition u2\n3=−1by\nintroducing u3→˜u3=u3−βˆkwith arbitrary β. In the\npresence of ˆδ(ℓ2), this modifies the cut matter propagator\nto be\n2u3·ℓ= (u3+ℓ)2+1→(˜u3+ℓ)2+1\n≃2u3·ℓ−2βu3·ˆk, (5.21)\nwhere we dropped the term proportional to ˆk·ℓas it\nleads to integrals proportional the first of Eq. ( 5.18) and\nthus vanishes. We therefore consider instead the integral\n˜I(u3,k) =eγEǫ\nm2ω/integraldisplayddℓ\n(2π)dˆδ(2u3·ℓ−2βu3·ˆk)ˆδ(ℓ2)Θ(ℓ0)\n2ℓ·ˆk\n=Θ(β)\n16πm2w2/bracketleftbigg1\nǫ−logw2\n2\nω2−logβ2\nπ/bracketrightbigg\n, (5.22)\nand we keep only the O(ǫ0)terms. The original integral\nI(u3,k)is recovered at β= 0. UsingΘ(0) =1\n2, we get,14\nI(u3,k) =1\n32πm2w2/bracketleftbigg1\nǫ−logw2\n2\nω2−logβ2\nπ/bracketrightbigg\n,(5.23)\n13This is analogous with the determination of the snail diagra ms\ninquantum scattering amplitudes, see e.g. [80].\n14Alternatively, we can obtain I(u3,k)directly as a discontinuity\nof an off-shell triangle integral, which was computed in Eq. ( C.4)\nof Ref. [81] (see Ref. [82] for comments on similar integrals ), with\nthe same off-shell continuation as discussed here.where the divergent constants logβand1/ǫwill be ab-\nsorbed by a time shift. We note that both integrals in\nEq. (5.18) remain zero even if we used the above off-shell\nregularization.\nPutting together Eqs. ( 5.19) and ( 5.23), we find that\nthe cut in Fig. 1bgives a finite contribution S1 loop disc,\nMdisc=S1 loop disc−iωGE/bracketleftbigg1\nǫ−logβ2\nπ/bracketrightbigg\nMtree,\nS1 loop disc=iG/bracketleftbigg\nm1w1logw2\n1\nω2+m2w2logw2\n2\nω2/bracketrightbigg\nMtree.\n(5.24)\nThe second term of Mdisccan be removed by a time shift\nthat supplements the one in Eq. ( 4.16). The first term,\nS1 loop disc, which may be interpreted as a direction-\ndependent time shift, is in fact the BMS supertranslation\nidentified in [52] as relating the “canonical” and “intrin-\nsic” BMS frames, thus explaining the angular momentum\nloss atO(G2)[55], as well as the transformation needed\nto relate the first nonuniversal term in the soft expansion\nof the amplitude-based [51, 83] and MPM waveforms [50].\nWe therefore see that zero-energy gravitons generate\non the one hand a time-independent background metric\natO(G0)(see Sec. VA) and, on the other hand, a time-\ndependent waveform at higher orders in Gand more-\nover that both of them can be removed by the same\nBMS supertranslation that connects the intrinsic and\ncanonical frames. This reinforces the close connection\nbetween zero-energy gravitons and the choice of BMS\nframe, which echoes features in the angular momentum\nloss calculations [52, 84, 85], and suggests that one may\nconsistently ignore the disconnected cut contributions\nand restore them through the BMS transformation of\nRef. [52]. It would be interesting to attempt to formulate\nthe observable-based formalism in the “intrinsic” BMS\nframe in terms of a vacuum dressed with a coherent state\nof zero-energy gravitons in the spirit of Ref. [78]. Such\na formulation, in which the contribution of zero-energy\nparticles (and the associated subtleties) will emerge be-\ncause of this dressing, may give a means to argue for an\nall-order connection between the disconnected cut con-\ntributions and the choice of BMS frame.\nVI. PN AND SOFT EXPANSIONS OF KMOC\nWAVEFORM\nThe one-loop amplitude part of the Mmatrix element\ncomputed in [39–41] exhibits spurious poles (SP). Some\nof these poles cancel manifestly when the amplitude is ex-\npanded in special kinematic limits such as at low velocity,\nin the soft limit, etc. However, manifesting such cancella-\ntions for general kinematic configurations is, at first sight ,\nhighly non-trivial and appears to require the use of Gram\ndeterminant and Bianchi identities among a huge num-\nber of terms. Algorithms for addressing such problems\nhave been developed in Ref. [86] (see also Ref. [87]) and\napplied to the current problem in Ref. [46].16\nIn App. Cwe take a different route and reorganize\nthe one-loop five-point scattering amplitude ( 5.11) and\nthe cut term to make manifest the cancellation of the\nspurious poles on the physical sheet. By direct inspection\nof Feynman integrals involving non-trivial numerators,\nwe find combinations of the square roots and logarithms\nappearing in ( 5.11) such that they have high-order zeros\nin correspondence with the singularities of the rational\nfunctions multiplying them. It turns out that the same\nfunctions are sufficient to render both the amplitude and\nthe cut contributions to Mfree of spurious poles. Their\nexpressions after this reorganization are\nM(1)fin\n1=/bracketleftbiggiw1\n32πlogw2\n1\nµ2\nIR+w1\n32+Γw1\n64/bracketrightbigg\nM(0)\nd=4(6.1)\n+iw1\n32πM(0)\nextra+c1I1+c2I2+l′\nqLq+l′\nwLw\n+R+lqlogq2\n1\nq2\n2+lwlogw2\n2\nw2\n1+lyarccoshy/radicalbig\ny2−1,\nS(1)fin\n1=iw1Γ\n64πM(0)\nd=4logw1w2µ2\nIR\nq2\n1q2\n2(y2−1)(6.2)\n−iw1Γ\n64πM(0)\nextra+crat+cratiologw2\n1w2\n2\nq2\n1q2\n2\n+cyarccoshy+cqLq+S(1)local\n1,\nwhere{I1,I2,Lq,Lw}are SP-free combinations. Their\nexplicit forms and construction are given in App. C. Here\nS(1)local\n1 includes all terms which have no singularities in\nq2\n1,2(includingcwLw) and therefore integrate to delta-\nfunction-supported terms in impact parameter space.\nThusS(1)local\n1 can be ignored. The explicit expressions of\nthecandlcoefficients in Eqs. ( 6.1) and ( 6.2) are included\nin the ancillary file WaveformReorganisation.nb .\nA. Impact-parameter-space waveform\nWith the reorganized parts of the momentum space\nMmatrix element, we now discuss the Fourier trans-\nform ( 4.6) to obtain the impact-parameter-space (and\nfrequency-space) waveform W(k,b1,b2). Here we will fo-\ncus on the ω∝ne}ationslash= 0(time-dependent) contribution up to\nO(G2). As M(k,q1,q2)is a function of the kinematic\ndata of the incoming state, the waveform W(k,b1,b2)is\nnaturally given in the incoming c.m. frame (e0,e1,e2,e3)\nanchored on the incoming momenta p1andp2. The unit\ntime vector e0can be chosen as\neµ\n0=pµ\n1+pµ\n2\n|pµ\n1+pµ\n2|, (6.3)\nwhile the incoming momenta are along the e2direction,\np1=E1e0+Pc.m.e2, p2=E2e0−Pc.m.e2,(6.4)whereEa=/radicalbig\nP2c.m.+m2a. Note that we can express Ea\nandPc.m.in terms of yas\nPc.m.=m1m2/radicalbig\ny2−1\nE,\nE=/radicalBig\nm2\n1+2ym1m2+m2\n2. (6.5)\nWe then choose the impact parameter b1andb2to satisfy\nE1\nEbµ\n1+E2\nEbµ\n2= 0and be orthogonal to e0. We define the\nrelative transferred momentum qµby\nqµ=qµ\n1−E1\nEkµ=−qµ\n2+E2\nEkµ, (6.6)\nsuch that the exponential factor in Eq. ( 4.6) becomes\ne−i(q1·b1+q2·b2)=e−iq·(b1−b2)=e−iq·b12, (6.7)\nand thus Wdepends only on b12. The impact parameter\nb12is orthogonal to both of the incoming momenta p1and\np2. Thus we can choose it to be along the e1direction,15\nb12=bine1. (6.8)\nThe momentum kand polarization vector εfor the emit-\nted graviton are given by\n˜ n(θ,φ) = sinθcosφe1+sinθsinφe2+cosθe3,\nk=ω(e0+˜ n(θ,φ)),\nε=1√\n2/bracketleftbigg\n∂θ˜ n(θ,φ)−i\nsinθ∂φ˜ n(θ,φ)/bracketrightbigg\n.(6.9)\nThey are formally the same as Eq. ( 2.7), but the polar\nangles here (θ,φ)are defined instead with respect to the\nspatial frame (e1,e2,e3). Therefore, the Fourier trans-\nform of M(k,q1,q2)in Eq. ( 4.6) now becomes\nFTb[f]≡/integraldisplay\nµ/parenleftBig\nk,q+E1\nEk,−q+E2\nEk/parenrightBig\ne−iq·b12f(q),\nW(ω,θ,φ) =2i\nκFTb/bracketleftbigg\nM/parenleftbigg\nk,q+E1\nEk,−q+E2\nEk/parenrightbigg/bracketrightbigg\n.\n(6.10)\nThe resulting impact parameter space waveform Wis in\nprinciple a Lorentz invariant function consisting of scala r\nproducts of (ε,k,p1,p2,b12). However, performing this\nFourier transform in complete generality is challenging.\nIn the following, we will proceed in the incoming c.m.\nframe, such that the frequency-domain waveform is given\nas a function W(ω,θ,φ), and consider its PN expansion.\nB. PN expansion\nFor the PN expansion, we will stick to the conventions\nof reference [50], also already used here in sections IIand\nIII. It is an expansion in powers of the velocity parameter\np∞≡/radicalbig\ny2−1. (6.11)\n15We note that bin=b+O(G2). See the discussion below Eq. (2.6).17\nHerep∞is taken as dimensionless (i.e. we can replace\nη=1\nc, used in Secs. IIandIIIby1).\nFollowing Ref. [50], one rewrites the kinematic vari-\nables in terms of the expansion parameter p∞, and ex-\npands the integrand before computing the Fourier trans-\nform to impact-parameter space. This gives rise to a\nrather simple integrand whose integral can be expressed\nsolely in terms of Bessel K functions and exponentials.\nOne starts by decomposing qµin terms of the external\nvectors (a covariant form of this decomposition, using\nu1,u2,band a fourth vector orthogonal on them, was\nintroduced in Ref. [4])\nqµ=q0eµ\n0+ΩQxeµ\n1+qyeµ\n2+ΩQzeµ\n3, (6.12)\nwhereeµ\n1,2,3denote the direction of impact parameter,\nof the c.m. momentum and of the orthogonal direction\nto the scattering plane, respectively – as defined in Sec-\ntionII. Moreover, Ωis the frequency of the emitted ra-\ndiation rescaled by the expansion parameter:\nω= Ωp∞. (6.13)\nThe integration over q0,ygives the following Jacobian:\n/integraldisplay\nµ(k,q1,q2)e−iq·b12=Ω2\n4m1m2p∞/integraldisplaydQxdQz\n(2π)2e−iuQx,\n(6.14)\nwhereuis the dimensionless frequency variable\nu≡Ωb≡ωb\np∞, (6.15)\nand the delta functions fix the components q0,qy:\nq0=−Ωp∞\nE2/parenleftbig\nm2\n1−m2\n2−m1m2p∞sinθsinφ/parenrightbig\n,\nqy=Ω\nE2/parenleftBig\nm1/radicalbig\np2∞+1+m2/parenrightBig/parenleftBig\nm2/radicalbig\np2∞+1+m1/parenrightBig\n.\n(6.16)\nThese relations correspond to Eqs. (5.9) in Ref. [50],\nexcept that the latter equations were expressed in the\nthe barred com frame (¯e0,ex,ey,ez).\nFor completeness, let us give also the expressions of\nw1,2in the c.m. frame:\nw1=Ωp∞\nE/bracketleftBig\nm1+m2/parenleftBig/radicalbig\np2∞+1−p∞sinθsinφ/parenrightBig/bracketrightBig\n,\nw2=Ωp∞\nE/bracketleftBig\nm2+m1/parenleftBig/radicalbig\np2∞+1+p∞sinθsinφ/parenrightBig/bracketrightBig\n.\n(6.17)\nThe original form of the amplitude [39–41] made it chal-\nlenging to compute its PN expansion because of the pres-\nence of spurious poles, though Ref. [50] succeeded to\ncarry out the expansion up to the 2.5PN approximation,\nand to evaluate the Fourier transform, thereby obtain-\ning the EFT waveform as a function of ω,θandφ. The\nSP-free version of the amplitude presented above signifi-\ncantly simplifies the task of PN-expanding the amplitude.The leading orders of the various terms in Eq. ( 6.1) in a\nsmall-p∞expansion are\nc1I1∼p0\n∞l′\nqLq∼p2\n∞R∼p−5\n∞\nc2I2∼p−5\n∞l′\nwLw∼p2\n∞ (6.18)\nlqlogq2\n1\nq2\n2∼p0\n∞lwlogw2\n2\nw2\n1∼p2\n∞lyarccoshy/radicalbig\ny2−1∼p−1\n∞.\nThus, in this organization three of the eight terms start\ncontributing only at 2PN order. Moreover, the cancella-\ntion of terms pn≤−2\n∞ is quite straightforward to observe.\nThis allowed us to reach the 3PN accuracy. Even though\nwe have so far only computed the 3PN-accurate EFT\nwaveform in the equatorial plane, i.e. when taking θ=π\n2,\nthere is no technical obstacle to derive it for arbitrary val -\nues ofθ. Although most integrals in QzandQxare ab-\nsolutely convergent, some of the contributions from the\ncut term must be interpreted as Fourier transforms of\ntempered distributions.\nAs explained in [50] the PN expansion of the amplitude\nyields an integrand containing only integer or half-intege r\npowers of\nD0= 1+Q2\nx+Q2\nz, (6.19)\nwith numerators polynomial in QzandQx. The Fourier\ntransforms of all these terms are computed from the gen-\neral formula\n/integraldisplaydQxdQz\n(2π)2e−iuQxQn\nx\nDm\n0=in2−m\nπΓ[m]∂n\nu/bracketleftbig\num−1Km−1(u)/bracketrightbig\n,\n(6.20)\nHowever, the cut terms (new with this work) involve the\nfirst power of logD0in the numerator. Such terms can\nbe integrated by using the formula\n/integraldisplaydQxdQz\n(2π)2e−iuQxQn\nxlogD0\nDm\n0= (6.21)\n=−∂ρ/braceleftbiggin2−m−ρ\nπΓ[m+ρ]∂n\nu/bracketleftbig\num+ρ−1Km+ρ−1(u)/bracketrightbig/bracerightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nρ=0.\ninvolving a derivative with respect to the order of the\nBesselKν(u)(around an integer value of ν).\nThe final result can be further reduced using relations\namong modified Bessel functions of the second kind to\nlinear combinations of K0(u),K1(u)ande−uwith coeffi-\ncients that are rational functions of u. The results of this\nexpansion up to 3PN order are presented in the ancillary\nfilePNexpandedEFT.nb .16\n16For the analysis of the PN expansion of the various terms con-\ntributing to the amplitudes, we refer to Section V of Ref. [50 ].18\nC. Soft-graviton expansion\nWe also computed the soft-graviton expansion of the\nwaveform in impact parameter space:17\nW ≃A\nω+Blogω+Cω(logω)2+Dωlogω . (6.22)\nCare must be take when carrying out the soft-graviton\nexpansion at the level of the waveform integrand. As\none quickly discovers, a naive expansion leads to uncon-\ntrolled infrared divergences at sufficiently high orders in\nthe subsequent Fourier transform in momentum transfer\nat sufficiently high orders.\nThus, the soft-graviton expansion is intertwined with\nthe Fourier transform to impact parameter space. The\npresence of the small parameter ωindicates that the ap-\npropriate framework for the evaluation of the latter is the\nmethod of regions.18. There are two distinct contribut-\ning regions, q1→ω0q1andq1→ωq1, and each of them\nhas a different small- ωexpansion which must be added\nto obtain the correct result. In both regions the result-\ning integrals can be evaluated in closed form, as done in\nRef. [4].\nThe soft-graviton expansion corresponds to the large\ntime expansion of the time-domain waveform. The be-\nhavior of the scattering waveform in this regime exhibits\nuniversality as emphasized in Refs. [89, 90]. In particular ,\nwhen expressed in terms of the incoming and outgoing\nmomentum, which differ by the impulse ∆p, the func-\ntionsA,BandCare independent of the specifics of the\nhard scattering process. We extracted these coefficients\nfrom the EFT waveform and reproduced the predictions\nof Ref. [89].\nWe have also computed the non-universal Dcoefficient\nand found exact agreement with Eq. (9.11) of [50]. This\nis the first order in the soft-graviton expansion where the\nsecond region mentioned above, in which the Fourier in-\ntegration variable scales as q1→ωq1, first contributes. It\nis also the first order at which the disconnected cut ( 5.24)\ncontributes crucially to the match with [50]. The neces-\nsity of this term for a successful comparison with the\nMPM waveform was first suggested in [51] as part of\na BMS transformation. In Secs. VAandVCwe un-\nderstood its connection to the disconnected cuts of the\nobservable-based formalism if zero-energy gravitons are\npart of the spectrum. On the other hand, the ǫ/ǫterm in\nEq. (5.16) does not contribute to any of the coefficients\nin Eq. ( 6.22) up to a time shift.\n17This PM calculation was carried out independently in Ref. [5 1]\nand both results were presented for the first time at [83, 88]. The\ncalculation was done to 2.5 PN order in Ref. [50].\n18Note that, while analogous in spirit, is a distinct from the m ethod\nof regions used to isolate the classical contribution of loo p inte-\ngrals, which have already been been evaluated here.VII. TRANSFORMING THE KMOC\nWAVEFORM TO THE BARRED MOMENTUM\nFRAME\nThe MPM waveform WMPM(ω,θ,φ)is computed in\na spatial frame anchored on the classical barred mo-\nmenta ( 2.3), while the EFT waveform W(ω,θ,φ)is origi-\nnally obtained in a spatial frame anchored on the incom-\ning momenta. Both spatial frames are of the c.m. type,\nwith a negligible O(G3)difference (due to the radiated\nmomentum) between their time-axes, namely\n¯eµ\n0=eµ\n0+O(G3). (7.1)\nAt orderG2the only significant difference between the\nbarred c.m. frame spanned by (¯e0,ex,ey,ez)(defined in\nSec.II) and the frame (e0,e1,e2,e3)is a spatial rotation\nofχ/2mapping the unit vectors e1,e2intoex,ey. [Note\nthat in both frames we use a polarization vector respec-\ntively anchored on e1,e2orex,ey, as per Eqs. ( 2.7) and\n(6.9)].\nConsequently, the function of θanφdescribing the\nEFT waveform referred to the barred momenta frame,\nsayWEFT(ω,θ,φ), is related to the function of θanφde-\nscribing the (original) EFT waveform W(ω,θ,φ)referred\nto the incoming momenta frame by\nWEFT(ω,θ,φ) =W(ω,θ,φ) (7.2)\n+χ1PM\n2∂\n∂φWG1(ω,θ,φ)+O(G3).\nHere, as indicated, it is sufficient to insert the 1PM\n[O(G1)] approximation to the scattering angle, namely\nχ1PM=2GE\nb2y2−1\ny2−1. (7.3)\nThus, using Eq. ( 7.2), the comparison between the MPM\nand EFT waveform is reduced to comparing the function\nWMPM(ω,θ,φ)to the function WEFT(ω,θ,φ).\nVIII. COMPARISON BETWEEN MPM AND\nKMOC WAVEFORMS\nWhen working in the barred momenta frame, the com-\nparison between the MPM waveform and the EFT one\namounts to comparing the following two functions of\nω,θ,φ (and of the impact parameter b):WMPM(ω,θ,φ)\non the one hand, and WEFT(ω,θ,φ)on the other hand.\nA complete physical agreement between these two wave-\nforms would mean that these two functions differ at most\nby the effect of an angular-independent time shift δt. As\ndiscussed above, when adding to the EFT waveform the\n(linearized gravity) contribution coming from the discon-\nnected, zero-frequency diagrams displayed on the left of\nFig.1, theO(G0)static background (Coulomb-like) con-\ntributions in the two waveforms agree with each other\n[compare Eq. ( 2.14) with Eq. ( 5.3)]. In the following, we19\ntherefore focus on the frequency-dependent parts ( ω∝ne}ationslash= 0)\nof the two waveforms, which both start at order G1.\nSummarizing the results so far, the function\nWEFT(ω,θ,φ)can be written (at the one loop accuracy)\nas the sum of six contributions, namely\nWEFT(ω,θ,φ) =WG1(ω,θ,φ)+W1 loop (0)(ω,θ,φ)\n+W1 loopǫ/ǫ(ω,θ,φ)+W1 loop cut(ω,θ,φ)\n+W1 loop disc(ω,θ,φ)+δrotW(ω,θ,φ)\n+O(G3). (8.1)\nHere, each term is defined as follows.\nTheO(G)waveform WG1comes from the tree-level\nmatrix element Mtreein Eq. ( 5.5),\nWG1(ω,θ,φ) =2i\nκFTb/bracketleftBig\nMtree/bracketrightBig\n. (8.2)\nTheO(G2)(one-loop) contributions to the EFT wave-\nform indicated in Eq. ( 8.1) denote, respectively:\n(i) the contribution from the (linear in T) 5-point am-\nplitude obtained by dropping the IR1\nǫdivergent terms\nin the results of Refs. [39–41], as displayed in Eq. (5.6)\nof [50], namely, in the notation introduced in Eq. ( 8.1)\nabove:\nW1 loop(0)=2i\nκFTb/bracketleftBig\nκ5m3\n1m2\n2/tildewiderM(1)\n1+(1↔2)/bracketrightBig\n;(8.3)\n(ii) the extra,ǫ\nǫcontribution from the (linear in T) 5-\npoint amplitude generated after absorbing the IR1\nǫdiver-\ngence into a time shift as discussed in Secs. IVCandVB,\nnamely, in the notation introduced in Eq. ( 8.1) above:\nW1 loopǫ/ǫ=2i\nκFTb/bracketleftBig\nκ5m3\n1m2\n2iw1\n32πM(0)\nextra+(1↔2)/bracketrightBig\n=−2κ4\n32πm2\n1m2\n2EωFTb/bracketleftBig\nM(0)\nextra/bracketrightBig\n; (8.4)\n(iii) the oneloop cut contribution from connected Tel-\nements discussed in Sec. VB, namely, in the notation\nintroduced in Eq. ( 8.1) above:\nW1 loop cut=2i\nκFTb/bracketleftBig\nS1 loop fin/bracketrightBig\n; (8.5)\n(iv) the oneloop cut contribution involving disconnected\nTelements discussed in Sec. VC, namely, in the notation\nintroduced in Eq. ( 8.1) above:\nW1 loop disc=2i\nκFTb/bracketleftBig\nS1 loop disc/bracketrightBig\n; (8.6)\nand finally, (v) the contribution from the spatial rotation\nbetween the barred- pframe and the incoming pone dis-\ncussed in Sec. VII, namely, in the notation introduced in\nEq. (8.1) above:\nδrotW= +χ1PM\n2∂\n∂φWG1(ω,θ,φ). (8.7)Ref. [50] computed, at the G2η5accuracy, the (even in φ\nsector of the) difference\nδoldW(ω,θ,φ)≡WEFT′(ω,θ,φ)−WMPM(ω,θ,φ),(8.8)\nbetween the EFT waveform, as it was understood in\nthe first versions of Refs. [39–41], and interpreted in the\nbarredpframe, and the MPM one. In the notation of\nthe present work, the quantity WEFT′(ω,θ,φ)considered\nin [50] is equal to\nWEFT′(ω,θ,φ) =WG1(ω,θ,φ)+W1 loop(0)(ω,θ,φ).\n(8.9)\nThe result of [50] for the difference δoldW(ω,θ,φ),\nEq. (8.8), was that it started at the G2/c5(2.5PN) level,\nand contained (at this order) both terms linear in the\nsymmetric mass ratio, ν, and terms quadratic in ν. The\nexplicit expression for the difference\nδoldW(ω,θ,φ) =δνW(ω,θ,φ)+δν2W(ω,θ,φ)(8.10)\nwas provided in [50] in tensorial form, and in the time\ndomain. See Eqs. (8.10)-(8.13) there. The equivalent\nresult in SWSH form, and in the frequency domain, con-\ntains onlyl= 2+contributions at order ν1, and both\nl= 2+,l= 3−andl= 4+contributions at order ν2.\nNamely,\nδνW(ω,θ,φ) =/summationdisplay\nm=2,0,−2δνU2m¯2Ylm(θ,φ),(8.11)\nδν2W(ω,θ,φ) =/summationdisplay\nm=2,0,−2δν2U2m¯2Y2m(θ,φ)\n+/summationdisplay\nm=1,−1δν2V3m¯2Y3m(θ,φ)\n+/summationdisplay\nm=4,2,0,−2,−4δν2U4m¯2Y4m(θ,φ),\n(8.12)\nwith\nδνU22=−4i√\n5π\n5G2M3\nbp2\n∞νu[K0(u)−2K1(u)],\nδνU20=−4i√\n30π\n15G2M3\nbp2\n∞νu[K0(u)+2uK1(u)],\nδνU2¯2=−4i√\n5π\n5G2M3\nbp2\n∞νu[K0(u)+2K1(u)],\n(8.13)\nδν2U22=8i√\n5π\n105G2M3\nbp2\n∞ν2u[(13u+6)K0(u)\n+13(u+1)K1(u)],\nδν2U20=8i√\n30π\n105G2M3\nbp2\n∞ν2u[2K0(u)−uK1(u)],\nδν2U2¯2=8i√\n5π\n105G2M3\nbp2\n∞ν2u[(−13u+6)K0(u)\n+13(u−1)K1(u)], (8.14)20\nδν2V32=−2i√\n7π\n21G2M3\nbp2\n∞ν2u[uK0(u)\n+(1+u)K1(u)],\nδν2V30=−2i√\n210π\n105G2M3\nbp2\n∞ν2u[uK0(u)+K1(u)],\nδν2V3¯2=−2i√\n7π\n21G2M3\nbp2\n∞ν2u[uK0(u)\n+(1−u)K1(u)], (8.15)\nδν2U44=−i√\n7π\n21G2M3\nbp2\n∞ν2u[(1+2u)K0(u)\n+2(u+1)K1(u)],\nδν2U42=−2i√π\n21G2M3\nbp2\n∞ν2u(u+1)[K0(u)+K1(u)],\nδν2U40=−i√\n10π\n105G2M3\nbp2\n∞ν2u[3K0(u)+2uK1(u)],\nδν2U4¯2=−2i√π\n21G2M3\nbp2\n∞ν2u(1−u)[K0(u)−K1(u)],\nδν2U4¯4=−i√\n7π\n21G2M3\nbp2\n∞ν2u[(1−2u)K0(u)\n+2(u−1)K1(u)]. (8.16)\nNote that, WEFT′did not (on purpose19) contain\ntheχ\n2rotation contribution, δrotW, nor the three new\ncontributions discussed in the present work, namely\nW1 loopǫ/ǫ,W1 loop cutandW1 loop disc. Therefore, up-\ndating the comparison done in [50] leads to a new differ-\nence\nδnewW(ω,θ,φ)≡WEFT(ω,θ,φ)−WMPM(ω,θ,φ),\n(8.17)\ngiven by\nδnewW=δoldW+W1 loopǫ/ǫ+W1 loop cut\n+W1 loop disc+δrotW. (8.18)\nAs we shall discuss next, the additional contributions\n(new with this work) displayed in Eq. ( 8.18) are such that\nthey precisely cancel the two types ( O(ν1)andO(ν2))\nof discrepancies that were present in Eq. ( 8.8), mod-\nulo an angular-independent time-shift between the two\nwaveforms.\nIX. KMOC CUT AND FRAME ROTATION\nIt was first noticed in [50] by directly comparing the\nMPM waveform with the amplitude contribution to the\nKMOC waveform that, up to 2PN order, there is a re-\nmarkable agreement between the two when the latter is\n19Indeed, one of the important findings of [50] was the need to\nrotate the EFT to the barred pframe, otherwise the MPM/EFT\nmismatch would start at the Newtonian order, instead of bein g\nrelegated to the 2.5PN order.evaluated in the frame rotated by half the scattering an-\ngle relative to the incoming frame (i.e. in the barred\nframe). As mentioned earlier, this raised the possibil-\nity that the cut contribution could be absorbed into this\nframe change, which is further confirmed in the soft gravi-\nton expansion discussed in Sec. VI C.\nBy Fourier-transforming the one-loop connected cut\ncontribution to impact-parameter space, Ref. [51] showed\nthatW1 loop cutcan be written as a certain differential\noperator acting on the tree-level waveform up to a fur-\nther finite redefinition of the the retarded time. In the\ncoordinates used here this differential operator acts as a\nrotation by χ/2. Using the explicit expression of the con-\nnected cut [39, 41, 91], we explicitly verified this relation ,\nW1 loop cut=−χ1PM\n2∂\n∂φWtree+iωδtcutWtree,(9.1)\nup to 3PN order. The ǫ/ǫterm in the second Eq. ( 5.16)\nplays a crucial role.20Resumming the 3PN-accurate ex-\npansion of the time shift (which starts at Newtonian or-\nder) yields the following all-order-in- p∞expression\nδtcut=κ2E\n32π/bracketleftbigg\nΓlogµIRbeγE\n2−y\n2(y2−1)3/2\n+E1E2(2y2−1)\nm1m2(y2−1)3/2/bracketrightbigg\n. (9.2)\nTo compare with Eq. (3.48) of the version 1 of Ref. [51],\nwe include the IR-divergent term in Eq. ( 5.12) to the\nleft hand side (to obtain FT b[S1 loop]) and the time shift\nδtcut\nIR=−GEΓ\n2ǫthat cancels it to the right hand side. This\ngives,\nFTb/bracketleftBig\nS1 loop/bracketrightBig\n=−χ1PM\n2∂\n∂φFTb/bracketleftBig\nMtree/bracketrightBig\n+iωδTcutFTb/bracketleftBig\nMtree/bracketrightBig\n, (9.3)\nwhereδTcut=δtcut+δtcut\nIR. ThroughO(ǫ0)our time shift\nagrees with the result of the version 1 of Ref. [51]\nδTcut=κ2E\n32π/bracketleftBigg\ny(y2−3−4ǫ\n2−2ǫ)\n(y2−1)3/2eǫγEΓ(−ǫ)\n(µ2\nIRb2/4)−ǫ\n+E1E2(2y2−1)\nm1m2(y2−1)3/2/bracketrightbigg\n. (9.4)\nTogether with the observation that in our c.m. frame the\noperator ¯δin [51] becomes −(χ1PM/2)∂φ, we find exact\nagreement with the prediction of Ref. [51] (up to a re-\ndefintion of the arbitrary scale µIR), which we have thus\nverified through direct evaluation up to O(p3\n∞), or 3PN\norder.\n20As our results were written up, we became aware that this poin t\nwas also explicitly noted in the version 2 update of Ref. [51] .21\nIn light of this agreement, let us comment on the tech-\nnical differences between the setup discussed here and\nthat of Ref. [51]. Certain steps in our evaluation and\nsimplification of the cut term, such as removal of terms\nproportional to the Gram determinant of five vectors,\nassumed that the external kinematics, including the mo-\nmentum transfer, were four-dimensional. Consequently,\nall our Fourier transforms were strictly in four dimen-\nsions, as enforced by the decomposition of the momentum\ntransfer in Eq. ( 6.12). In contrast, the setup in Ref. [51]\nassumes that the Fourier transform is d-dimensional.\nInterestingly, this does not appear to have any conse-\nquences in the comparison, such as a difference in the ǫ/ǫ\ncontributions that build on the IR-divergent terms.\nThe analysis outlined in this section indicates that\nwhen comparing the EFT and MPM result in the barred\nc.m. frame, we can effectively ignore the connected cut\ncontribution S1 loopaltogether, as it is cancelled by the\nframe rotation up to a time shift. In other words, we can\nrewrite Eq. ( 8.18) in the simplified form\nδnewW=δoldW+W1 loopǫ/ǫ+\n+W1 loop disc+timeshift . (9.5)\nWe now proceed to discuss the remaining two terms,\nW1 loopǫ/ǫandW1 loop disc.\nX.W1 loop ǫ/ǫANDδνW\nThe additional ǫ/ǫcontribution is given by Eq. ( 8.4).\nRemembering the presence of a factor 1/(m1m2)in the\nJacobian of FT b,W1 loopǫ/ǫis seen to be proportional\ntom1m2, i.e. to be proportional to ν1. In addition, as\nit contains a O(η3)factorGEω=GMω\nc3, and as M(0)\nextra\nis easily seen to start to differ from its full-gravity tree-\nlevel counterpart M(0)\nd=4only at the 1PN, O(η2), level, we\nconclude that W1 loopǫ/ǫeffectively (i.e. modulo a time\nshift) starts contributing to Wat theG2η5level.\nAn easy computation of the impact-parameter trans-\nform of the PN expansion of M(0)\nextraactually shows that,\nmodulo a time-shift contribution, iωδtWG1, the even-in-\nφprojection of W1 loopǫ/ǫprecisely cancels the O(ν),\nG2η5-level, contribution δνW(ω,θ,φ)found (in [50]) to\nbe present in δW(ω,θ,φ), and displayed in Eq. ( 8.13).\nNamely,\nδνW+/bracketleftBig\nW1 loopǫ/ǫ+iωδtǫ/ǫWG1/bracketrightBigφ−even\n=O/parenleftbiggG2\nc6/parenrightbigg\n,\n(10.1)\nwithδtǫ/ǫ=GE\nc5=GM\nc3(1+ν\n2p2\n∞+O(p4\n∞)).\nXI.W1 loop discANDδν2W\nThe final terms to discuss (at the highest PN accu-\nracy explicitly studied in [50]) are the (even-in- φprojec-\ntion of the) disconnected cut contribution to Wand theO(ν2),G2η5-level, contribution δν2W(ω,θ,φ)displayed\nin Eqs. ( 8.14), (8.15) and ( 8.16).\nIt was suggested in Ref. [51] that the difference\nδν2W(ω,θ,φ)is genuinely present in the EFT-MPM\ndifference and corresponds to the fact that the EFT\nwaveform does not contain some of the zero-frequency-\ngraviton effects contained in the MPM waveform, be-\ncause it works in the canonical BMS frame of [52] (in\nwhichW(t=−∞) = 0) rather than in the intrinsic BMS\nframe selected both by standard classical PM theory and\nthe MPM formalism. However, we found in Sec. Vthat\ndisconnected contributions to the cut term account for\nthis. They lead to the additional term W1 loop discin the\nEFT waveform whose explicit form is\nW1 loop disc= +iG/bracketleftbigg\nm1w1logw2\n1\nω2+m2w2logw2\n2\nω2/bracketrightbigg\nWG1.\n(11.1)\nThis term is O(ν2), and its PN expansion starts at the\nG2/c5level. An easy computation shows that its φ-\neven projection exactly cancels, at this order, all the\nl= 2+,l= 3−andl= 4+pieces ofδν2W(ω,θ,φ)dis-\nplayed in Eqs. ( 8.14), (8.15) and ( 8.16):\nδν2W+/bracketleftbig\nW1 loop disc/bracketrightbigφ−even=O/parenleftbiggG2\nc6/parenrightbigg\n,(11.2)\nwithout the need of an additional time-shift.\nIn other words, after factoring out the effect of the χ/2\nrotation, the total time-shift needed to align the MPM\nwaveform (defined as in [50]) and the EFT one (com-\npleted by the ǫ/ǫand the disconnected contributions) is\nδttot=GM/c3. This time-shift can be absorbed in a\nmodification of the link between the time scale b0enter-\ning the tail logarithms of the MPM formalism and the\nfrequency scale µIRof the amplitudes-based formalism.\nNamely, Eq. (8.8) of [50] is modified into\nlog(2b0µIR)+γE= 0. (11.3)\nNote the curious feature that the additional term,\nEq. (11.1), which reads,\nW1 loop disc=iωδtVV(θ,φ)WG1, (11.4)\nwith the angle-dependent time shift\nδtVV(θ,φ) = 2G/parenleftBig\nm1w1\nωlogw1\nω+m1w2\nωlogw2\nω/parenrightBig\n,\n(11.5)\ndoubles , rather than cancels, the contribution to the EFT\nwaveform coming from the last two lines in Eq. (5.22) of\nRef. [50]. See further discussion of this term below.\nThe results ( 9.3), (10.1) and ( 11.2) allow us to con-\nclude that the inclusion of the new contributions dis-\ncussed in this work to the EFT waveform resolves all the\nmismatches with the even-in- φprojection of the G2η5-\naccurate MPM waveform computed in [50].\nTo complete the EFT-MPM comparison on the odd-in-\nφsector, we have also compared the odd-in- φprojection22\nofWMPM, at theG2(η1+η4)accuracy (see Eqs. ( 3.25)\nand (3.27)) with its EFT counterpart (given, in the equa-\ntorial plane, in the ancillary file PNexpandedEFT.nb ). We\nfound perfect agreement between the two (with the same\ntime-shift needed for the φ-even part, i.e. the same link\nEq. ( 11.3)). Though our checks in the odd-in- φsector\nprobe nonlinear contributions to the waveform less deeply\nthan our even-in- φchecks, they still include the leading-\norder tail contributions, and are sensitive to the precise\nrelation Eq. ( 11.3). The explicit comparison between the\nMPM and amplitudes-based waveforms is presented in\nthe ancillary file MPMEFTcomparison.nb .\nXII. CONCLUSIONS\nWe revisited the amplitude- and observable-based\n(KMOC [3]) derivation of the classical gravitational wave\nsignal emitted during the scattering of two non-spinning\nmasses at the NNLO order hµν=gµν−ηµν∼G1+\nG2+G3, and compared our results for the frequency-\ndomain,G-rescaled waveform quantity W(ω,θ,φ)≡\nc4\n4Gεµενhµν∼G0+G1+G2, to the corresponding\nwaveform derived within the (post-Newtonian-matched)\nMultipolar-post-Minkowskian (MPM) formalism. The\nrecent computation of WMPM(ω,θ,φ)[50] had pointed\nout the presence of large discrepancies (starting at the\nNewtonian level) between WMPM(ω,θ,φ)and the (fi-\nnite part of the) amplitude-based waveform derived in\nthe (first versions of) Refs. [39–41], say WEFT′(ω,θ,φ).\nRef. [50] had further pointed out that the EFT/MPM\nmismatch was drastically reduced from the Newtonian\nlevelG2η0(whereη≡1\nc) to the 2.5 PN level G2η5\nby rotating (by half the classical scattering angle χ/2)\nthe EFT waveform from the incoming momenta frame\nto the (classical) averaged momenta frame. Part of\nthe puzzle posed by the EFT/MPM mismatch was re-\ncently clarified by the eikonal-inspired approach of Geor-\ngoudis, Heissenberg and Russo [51] who argued that the\ncut contribution needed [42] in the KMOC framework\n(and missing in the first derivations of the EFT wave-\nform) was precisely implementing the needed rotation\nbetween the incoming-momenta frame (naturally tuned\ntoWEFT(ω,θ,φ)) and the (classical) averaged-momenta\nframe (conveniently used to compute WMPM(ω,θ,φ)) for\nall final-states that do not contain zero-energy gravi-\ntons. Ref. [51] compared only the soft expansion of\nWEFT(ω,θ,φ)to the soft expansion of WMPM(ω,θ,φ)\nand could indeed verify that the χ/2rotational effect of\nthe cut term reduced the EFT/MPM mismatch between\nthese expansions to the G2η5level. However, the soft\nlimit is a rather blunt tool which is not sensitive to the\ndetails of the EFT/MPM mismatch (as discussed, e.g.,\nin Section IX of [50]).\nIndeed, after factoring out the effect of the χ/2rota-\ntion, the EFT-MPM mismatch found in [50] still con-\ntained significant terms of order ν1G2η5and of order\nν2G2η5. The terms O(ν1G2η5)effectively disappear inthe soft limit (because they are equivalent to an unob-\nservable time shift). We showed in the present work\nthat a subtle additional ǫ/ǫcontribution to the ampli-\ntude result (coming, in our scheme, from the tracelessness\nof the physical graviton in ddimensions) precisely can-\ncelled theO(ν1G2η5)EFT-MPM mismatch (which could\nnot be seen in the soft limit). Concerning the last re-\nmainingO(ν2G2η5)EFT-MPM mismatch, it was argued\nby Georgoudis et al. [51] that it remained and that, in\nview of Eqs. ( 11.1), (11.5), it was to be physically inter-\npreted as the presence of the particular O(G)supertrans-\nlation (i.e. an angle-dependent time shift) δtVV(θ,φ), Eq.\n(11.5), pointed out by Veneziano and Vilkovisky [52], for\nits role in transforming the “intrinsic” BMS frame (nat-\nurally used both in classical, worldline PM perturbation\ntheory, and in the MPM formalism) into the “canoni-\ncal” BMS frame (defined so that the initial value of the\nwaveform vanishes in the infinite past). The interpre-\ntation of the necessary presence of the supertranslation\n(11.5) was that this supertranslation removes the effect\nof static modes (zero-frequency gravitons) in the wave-\nform, and that the amplitude approach naturally com-\nputes a waveform without static modes. Our new results\nsuggest, however, a different picture. Static modes can\nbe incorporated in the KMOC waveform if one includes\ndisconnected T-matrix elements, both at the linear-in- T\nlevel and at the bilinear-in- Tlevel (cut term). We have\nexplicitly shown (see Eq. ( 5.3)) that the inclusion of dis-\nconnected diagrams (pictured in Fig. 1a) to the linear-\nin-Twaveform was precisely generating the initial, zero-\nfrequency contribution which is naturally present in the\nMPM waveform (see Eq. ( 2.13)). The situation concern-\ning the effect of disconnected diagrams in the bilinear-in-\nTterm is more subtle. In a natural regularization which\ntakes the matter velocity slightly off-shell, u3→˜u3,\nwe showed both by direct evaluation and by interpret-\ning it as the discontinuity of a triangle integral that the\ncut pictured in Fig. 1byields an additional contribution\ntoWEFTthat precisely cancels the effect of the super-\ntranslation ( 11.5), thereby removing the last remaining\nO(ν2G2η5)EFT-MPM mismatch. We have also shown in\nApp. Dthat higher-loop contributions would not mod-\nify our finding. It would clearly be interesting to con-\nfirm in a different way our conclusion that the inclusion\nof disconnected diagrams in the KMOC formalism does\nincorporate all the physical effects linked to the static\nmodes (which are included in the MPM formalism). It\nwould also be interesting to know whether a higher-order\nKMOC calculation, systematically ignoring disconnected\ndiagrams, would automatically yield a waveform differing\nfrom the MPM one onlyby taking into account the full\n(nonlinear) effect of the supertranslation ( 11.5) (which is\nfully determined by the O(G)linearized-gravity Coulom-\nbic background ( 2.13), (2.14)). A classical (worldline-\nbased) PM-gravity computation (à la [9, 92, 93]) of the\none-loop waveform would be very valuable. Conversely,\nit would be interesting to explore if the time-dependent\nand time-independent contributions of zero-energy gravi-23\ntons can be captured by a nontrivial vacuum containing\na coherent state or condensate of such modes, perhaps\nalong the lines of [78, 85, 94].\nIn this paper we have shown the agreement between the\nMPM and EFT waveforms (over the full celestial sphere)\nat the(G+G2)(η0+η2+η3+η5)accuracy (2.5PN order)\nin theφ-even sector, and at the (G+G2)(η1+η4)accu-\nracy in the φ-odd sector. [The agreement of the φ-even\n(G+G2)η4(2PN) terms was also separately checked in\nthe equatorial plane.] The details of our results are pre-\nsented in three ancillary files: (1) the PN expansions of\nthe spin-weighted-spherical-harmonic coefficients WMPM\nlm\nof the MPM waveform WMPM(θ,φ)are given in the an-\ncillary file MPMmultipoles.nb so as to reach the accu-\nracy(G+G2)(η0+η2+η3+η4+η5)for evenm’s (φ-\neven sector; which has been extended here to include\nthe 2PN terms) and the accuracy (G+G2)(η1+η4)\nfor oddm’s (φ-odd sector); (2) the PN-expansion of the\nθ=π\n2restriction of the rotated one-loop-accurate EFT\nwaveformWEFT(ω,θ,φ), Eq. ( 7.2), is given in the an-\ncillary file PNexpandedEFT.nb at the 3PN accuracy, i.e.\n(G+G2)(η0+η1+η2+η3+η4+η5+η6); and (3) the\ncomparison (and agreement), in the equatorial plane, be-\ntween the two waveforms is presented in the ancillary file\nMPMEFTcomparison.nb .\nOur current organization of the amplitude and cut con-\ntributions, in terms of functions free of spurious poles,\nmakes is possible to proceed to higher PN orders at\nO(G3)in the EFT side. By contrast, analogous exten-\nsions on the MPM side require (when using the MPM\ntechniques used so far) to compute new, challenging,\nhigher-PN contributions. This suggests that borrowing\ninformation from the EFT calculation might help to im-\nprove the power of the MPM formalism. Moreover, it\nwould undoubtedly be interesting to extend our com-\nparison to scattering waveforms for spinning bodies, for\nwhich the MPM formalism naturally applies [7, 95] and\nO(G3)amplitudes-based results are available [46].\nTheO(G3)waveform discussed here allows for a di-\nrect determination of the O(G4)angular momentum and\nenergy loss which have nontrivial implications for conser-\nvative binary scattering at O(G5), including information\non the first second-self-force contribution to this process .\nCarrying out the next-order waveform calculation, which\nboth gives analogous information O(G6)and probes the\nagreement between EFT and MPM at higher orders in\nNewton’s constant, poses an interesting and important\nchallenge to the EFT calculation, through the more in-\nvolved multi-scale integration-by-parts reduction and th e\nevaluation of the ensuing multi-scale master integrals and\nof the Fourier-transform to impact parameter space.\nOne of the important takeaway messages of our new\nresults is that, after having sorted out the subtleties\nthat were hidden in the EFT one-loop waveform, we ob-\ntained a remarkable confirmation that the classical limit\nof an amplitude-based waveform does correctly incor-\nporate the many subtle classical effects that were in-\ncluded in the O(G2η5)-accurate MPM waveform suchas (i) radiation-reaction effects on the worldlines, (ii)\nhigh-multipolarity tail effects in the wave-zone, and (iii)\ncubically nonlinear multipole couplings in the exterior\nzone. The fact that the road leading to the present suc-\ncessful EFT/MPM comparison had some bumps, which\ntaught us interesting lessons, is another example of the\nuseful synergy between amplitude-based, and classical\nperturbation-theory-based, approaches to gravitational\nphysics.\nACKNOWLEDGMENTS\nWe thank Holmfridur Hannesdottir, Sebastian Mizera,\nand Cristian Vergu for discussions; RR and FT thank En-\nrico Herrmann and Michael Ruf for emphasizing the im-\nportance of zero-energy modes. We also thank Alessan-\ndro Georgoudis, Carlo Heissenberg and Rodolfo Russo\nfor discussions and communications on the submission.\nThe present research was partially supported by the 2021\nBalzan Prize for Gravitation: Physical and Astrophysi-\ncal Aspects, awarded to T. Damour. D.B. acknowledges\nthe sponsorship of the Italian Gruppo Nazionale per la\nFisica Matematica (GNFM) of the Istituto Nazionale\ndi Alta Matematica (INDAM), as well as the hospital-\nity and the highly stimulating environment of the Insti-\ntut des Hautes Etudes Scientifiques. S.D.A.’s research\nis supported by the European Research Council, under\ngrant ERC–AdG–88541. A.H. is supported by the Si-\nmons Foundation. R.R. and F.T. are supported by the\nU.S. Department of Energy (DOE) under award number\nDE-SC00019066.\nAppendix A: Notations\nIn this appendix, we collect the various notations used\nin the main text:\nκ2= 32πG\nηµν= diag(−1,+1,+1,+1)\ngµν=ηµν+hµν=ηµν+κhµν\nw1=−u1·k w 2=−u2·k\ny=−u1·u2Γ =3y−2y3\n(y2−1)3/2\nFµν=kµεν−kνεµ\nF1=Fµνqµ\n1uν\n1\nF2=Fµνqµ\n2uν\n2\nF3=Fµνuµ\n1uν\n2\nE=/radicalBig\nm2\n1+2ym1m2+m2\n2\nE1=m1\nE(ym2+m1)E2=m2\nE(ym1+m2)\nPc.m.=m1m2\nE/radicalbig\ny2−1\np∞=/radicalbig\ny2−124\nM=m1+m2ν=m1m2\n(m1+m2)2\nM=E\nc2η=1\nc\nThe speed of light is set to unity, c= 1, unless it is useful\nto emphasize the PN order.\nAppendix B: One-loop amplitude and KMOC cut\nHere we collect the one-loop amplitude and cut contri-\nbution to the KMOC matrix element Mthat are known\nin the literature. We follow the notation used in [41],\nwhich was also used in [50]. The IR finite part of the\nclassical one-loop five-point amplitude is given by\n/tildewiderM(1)\n1=/bracketleftbiggiw1\n32πlogw2\n1\nµ2\nIR+w1\n32+Γw1\n64/bracketrightbigg\nM(0)\nd=4+R\nπ+c1I1\n+c2I2+lq\nπlogq2\n1\nq2\n2+lw2\nπlogw2\n2\nw2\n1+ly\nπarccoshy/radicalbig\ny2−1,\n(B1)\nwhereµIRis the arbitrary scale introduced by the dimen-\nsional regularization, and I1,2are two triangle integrals,\nI1=−i\n32/radicalbig\nq2\n2+w2\n1+1\n16π/radicalbig\nq2\n2+w2\n1arcsinh−w1/radicalbig\nq2\n2.\nI2=−i\n32/radicalbig\nq2\n1. (B2)\nThe gauge invariant coefficients R,c1,2andlq,w2,yare\nfunctions of the kinematic data w1,2,q2\n1,2,F1,2,3andy.\nTheir explicit expressions can be found in the ancillary\nfiles of [41].\nThe IR finite part of the one-loop KMOC cut is\nS(1)\n1=iw1Γ\n64πM(0)\nd=4logµ2\nIR+crat+cq1logq2\n1\n+cq2logq2\n2+cw1logw2\n1+cw2logw2\n2\n+c(1)\nylog(y2−1)/radicalbig\ny2−1+c(2)\nyarccoshy, (B3)where the c-coefficients can also be found in the ancillary\nfiles of [41].\nAppendix C: Reorganization of the one-loop\nfive-point amplitude free of spurious poles\nBy direct inspection of the rational coefficients appear-\ning in Eq. ( 5.11) and App. B(which can be found in the\nancillary files of [39, 41, 46]), we find four SPs (which\ncorrespond to lower dimensional Gram determinants) in\nthe physical region, plus two additional SPs on the phys-\nical sheet (but for complex kinematics). The former four\nare, respectively, given by the vanishing loci of\n∆1≡y−w2\n1+w2\n2\n2w1w2,\n∆2≡y−(q2\n1w1)2+(q2\n2w2)2\n2q2\n1q2\n2w1w2,\n∆3≡(q2\n1−q2\n2)2−4w2\n1q2\n1,\n∆4≡(q2\n1−q2\n2)2−4w2\n2q2\n2; (C1)\nthe latter two are located, respectively, at the zeroes of21\nΠ1≡w2\n1+q2\n2,\nΠ2≡w2\n2+q2\n1.(C2)\nAt∆1,2= 0, we find second-order poles in the rational\ncoefficients lq,w,yand simple poles in R. We first focus on\n∆1and then trivially generalize the result to analogous\npoles at ∆2= 0through the substitution wi→wiq2\ni,\nwhich maps ∆1into∆2.\nThe cancellation of the second-order poles requires\nthat we construct combinations of square roots and log-\narithms that are proportional to ∆2\n1. To understand the\npattern, we expand arccoshyaround∆1= 0:\narccoshy/radicalbig\ny2−1=/parenleftbigg2z\nz2−1−4z2(1+z2)\n(z2−1)3∆1/parenrightbigg\nlogz\n+4z2\n(z2−1)2∆1+O(∆2\n1), (C3)\nwherez=w2\nw1and the expansion is symmetric in z→1\nz.\nThis expansion indicates that the following combination\nof logarithms,\nlogz−z2−1\n2z/bracketleftBigg/parenleftbigg\n1+y∆1\ny2−1/parenrightbiggarccoshy/radicalbig\ny2−1−∆1\ny2−1/bracketrightBigg\n=O(∆2\n1), (C4)\n21A note of caution should be made about these additional singu -\nlarities. The sign of the iǫprescription for w1determines whether\nor not the integral I1has a branch point at Π1in the complex q2\n1plane. We use the prescription w1→w1+iǫ, which does not give\na singularity at Π1. The SP also cancels among the coefficient\nofI1and the rational terms R.25\nshould lead to expressions that are free of second-order SPs in∆1and∆2. This is indeed the case and we thus found\nthe first two desired functions:\nLw=1\n∆2\n1/braceleftBigg\n2logw2\nw1−w2\n2−w2\n1\nw1w2/bracketleftBigg/parenleftbigg\n1+y∆1\ny2−1/parenrightbiggarccoshy/radicalbig\ny2−1−∆1\ny2−1/bracketrightBigg/bracerightBigg\n, (C5)\nLq=1\n∆2\n2/braceleftBigg\n2logq2\n2\nq2\n1+2logw2\nw1−(q2\n2w2)2−(q2\n1w1)2\nq2\n1q2\n2w1w2/bracketleftBigg/parenleftbigg\n1+y∆2\ny2−1/parenrightbiggarccoshy/radicalbig\ny2−1−∆2\ny2−1/bracketrightBigg/bracerightBigg\n. (C6)\nWe stress that such functions are not unique since any term O(∆0\ni)can be added without changing the desired\nproperties.\nWe next turn our attention to the poles in ∆3andΠ1, which are closely related and are of fourth and third order,\nrespectively, in the part of the amplitude proportional to m3\n1m2\n2. When constructing an ansatz for SP-free functions\nwe should account for the fact that Π1appears only in the coefficient of I1and inRand thus we should not include\nit together with I2andlogq2\n1\nq2\n2. Further accounting for the order of the pole in Π1, a suitable ansatz is\n∆3P1\n(q2\n1)4w5\n1Π3\n1+P2\n(q2\n1)4Π3\n1I1+P3\n(q2\n1)4w4\n1logq2\n1\nq2\n2+I2. (C7)\nIn this expression Pi’s are arbitrary polynomials in q2\n1,q2\n2andw1; they are fixed by demanding that this expression\nhas a four-th order zero at ∆3= 0and cubic zero at Π1= 0(which is away from the physical region). Thus, a\nfunction with no SPs in ∆3is\nI1=−q2\n1+q2\n2\n256π(q2\n1)2w1Π1∆3/parenleftbigg3(q2\n2)2+11q2\n2w2\n1+23w4\n1\n384(q2\n1w2\n1)2Π2\n1−q2\n2+4w2\n1\n16q2\n1w2\n1Π1∆3+1\n∆2\n3/parenrightbigg\n+q2\n1+q2\n2\n128(q2\n1)4Π3\n1∆3/bracketleftbigg35\n16−35(q2\n1+q2\n2)2\n8∆3+7(q2\n1+q2\n2)4\n2∆2\n3−(q2\n1+q2\n2)6\n∆3\n3/bracketrightbigg\nI1\n−(q2\n1−q2\n2)logq2\n1\nq2\n2\n64πq2\n1w1∆3/bracketleftbigg5\n1024(q2\n1w2\n1)3−3\n128(q2\n1w2\n1)2∆3+1\n8q2\n1w2\n1∆2\n3−1\n∆3\n3/bracketrightbigg\n+I2\n∆4\n3. (C8)\nFinally, a function with no SPs in either ∆3or inΠ1is\nI2=1\nΠ3\n1/parenleftbigg\nI1−(q2\n2)2\n80πw5\n1−11q2\n2\n240πw3\n1−23\n240πw1/parenrightbigg\n.(C9)\nI1andI2will render SP-free the part of the classical\namplitude that is proportional to m3\n1m2\n2. The combined\ntransformations q1↔q2andw1↔w2yield the func-\ntions for the part of the amplitude that is proportional\ntom2\n1m3\n2.\nTo expose the functions Lq,Lw,I1andI2in the ex-\npression of the amplitude, we use partial fractioning iden-\ntities,22to isolate the SPs in c1,2,lq,w,yandRand further\nadd and subtract appropriate combinations of logarithms\nand square roots. Their coefficients are, by construction,\nfree of SPs. We then verified that the coefficients of the\nremaining logarithms are also free of SPs upon applying\nthe Bianchi and four-dimensional identities. While all\ncoefficients and rational terms are manifestly free of the\ninitial SPs, this feature comes at the expense of introduc-\ning higher powers of the physical poles.\n22For this purpose we use the Mathematica package\nMultivariateApart [87].FIG. 3. The unitarity cut that only has support on the ex-\nternal zero energy graviton.\nThe final expressions are given in Eqs. ( 6.1) and ( 6.2)\nin the main text, and explicit expressions of the cand\nlcoefficients there are included in the ancillary file\nWaveformReorganisation.nb .\nAppendix D: Disconnected matrix elements\nsupported by zero energy graviton\nIn this appendix, we will discuss the role of the cut\ncontributions to Mthat have support on the external\nzero energy graviton, which could potentially modify the\nconstant background. Besides Fig. 1athat leads to the\nSchwarzschild background, other possible contributions26\n(a) (b)\nFIG. 4. The unitarity cuts that only have support when both\nthe external and one internal graviton have zero energy.\nto the background, given in Figs. 3and4, all involve acut.\nWe start with the cut in Fig. 3. It is more convenient\nhere to use the true momenta of the massive particles.\nThe cut is given by\nS5=−iκ/bracketleftBig\n(ε·p1)2ˆδ(2p1·k)+(1↔2)/bracketrightBig\nM4.(D1)\nWe note that the true momentum pais related to paused\nin the main text through pa=pa+qa/2. Since the uni-\ntarity cut forces the external graviton to have vanishing\nenergy, it will interact with the five-point amplitude in\nthe soft limit,\nM5/vextendsingle/vextendsingle/vextendsingle\nsoft=κ/bracketleftbigg\n−(ε·p1)2\n2p1·k+iǫ−(ε·p2)2\n2p2·k+iǫ+(ε·p3)2\n2p3·k−iǫ+(ε·p4)2\n2p4·k−iǫ/bracketrightbigg\nM4 (D2)\n=κ/bracketleftBigg\ni(ε·p1)2ˆδ(2p1·k)−2(ε·p1)(ε·q1)−(ε·q1)2\n2p1·k−iǫ+(ε·p1−ε·q1)2\n2p1·k−iǫ∞/summationdisplay\nn=1/parenleftbigg2q1·k\n2p1·k−iǫ/parenrightbiggn\n+(1↔2)/bracketrightBigg\nM4.\nWe see that the delta functions cancel in the combination\nM5+S5in the soft limit such that there are no contribu-\ntions to the constant background. What is left is given\nin terms of retarded propagators. They will generate in-\nstead the memory at t→+∞. This statement works\nfor more generic cases as we do not assume a particular\nloop order for M4. In principle, the same conclusion can\nbe drawn from using pamomenta. The retarded matter\npropagators would then result from a cancellation be-\ntween (derivatives of) delta functions and principal val-\nues, much like Ref. [42].\nFinally, we sketch the derivation that the two cuts\nshown in Fig. 4also vanish. For both cuts, we can use\nthe soft limit of the five-point amplitude in the right bloband sew it to the left blob. One will find that Fig. 4a\nonly reduces to homogeneous integrands as in Eq. ( 5.18),\nwhich integrate to zero. For Fig. 4b, the only additional\nnontrivial integral is\n/integraldisplay\nddℓˆδ(u1·ℓ)ˆδ(u2·k+u2·ℓ)δ(ℓ2)Θ(ℓ0)\nk·ℓ.(D3)\nOne can rescale ℓ→ωℓand find that the integral is pro-\nportional to ωd−4δ(ω). However, because this integral\ncomes from the diagram with a massless three-point ver-\ntex, signaled by the propagator (k·ℓ)−1, the coefficient\nshould scale as ω2. Therefore, overall we have the scaling\nωd−2δ(ω), which vanishes in four dimensions.\n[1]LIGO Scientific, Virgo Collaboration, B. P. Abbott\net al., “Observation of Gravitational Waves from a\nBinary Black Hole Merger,”\nPhys. Rev. Lett. 116no. 6, (2016) 061102 ,\narXiv:1602.03837 [gr-qc] .\n[2]LIGO Scientific, Virgo Collaboration, B. P. Abbott\net al., “GW170817: Observation of Gravitational Waves\nfrom a Binary Neutron Star Inspiral,”\nPhys. Rev. Lett. 119no. 16, (2017) 161101 ,\narXiv:1710.05832 [gr-qc] .\n[3]D. A. Kosower, B. Maybee, and D. O’Connell,\n“Amplitudes, Observables, and Classical Scattering,”\nJHEP 02(2019) 137 ,arXiv:1811.10950 [hep-th] .\n[4]A. Cristofoli, R. Gonzo, D. A. Kosower, and\nD. O’Connell, “Waveforms from amplitudes,”\nPhys. Rev. D 106no. 5, (2022) 056007 ,\narXiv:2107.10193 [hep-th] .\n[5]L. Blanchet and T. Damour, “Radiative gravitationalfields in general relativity I. general structure of the\nfield outside the source,”\nPhil. Trans. Roy. Soc. Lond. A 320(1986) 379–430 .\n[6]L. Blanchet and T. Damour, “Postnewtonian\nGeneration of Gravitational Waves,” Ann. Inst. H.\nPoincare Phys. Theor. 50(1989) 377–408.\n[7]L. Blanchet, “Gravitational Radiation from\nPost-Newtonian Sources and Inspiralling Compact\nBinaries,” Living Rev. Rel. 17(2014) 2 ,\narXiv:1310.1528 [gr-qc] .\n[8]P. C. Peters, “Relativistic gravitational\nbremsstrahlung,” Phys. Rev. D 1(1970) 1559–1571 .\n[9]S. J. Kovacs and K. S. Thorne, “The Generation of\nGravitational Waves. 3. Derivation of Bremsstrahlung\nFormulas,” Astrophys. J. 217(1977) 252–280 .\n[10]M. Portilla, “Scattering of Two Gravitating Particles:\nClassical Approach,” J. Phys. A 13(1980) 3677–3683 .\n[11]K. Westpfahl, “High-Speed Scattering of Charged and27\nUncharged Particles in General Relativity,”\nFortsch. Phys. 33no. 8, (1985) 417–493 .\n[12]Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower,\n“One loop n-point gauge theory amplitudes, unitarity\nand collinear limits,” Nucl. Phys. B425 (1994) 217–260 ,\narXiv:hep-ph/9403226 [hep-ph] .\n[13]Z. Bern and A. G. Morgan, “Massive loop amplitudes\nfrom unitarity,” Nucl. Phys. B467 (1996) 479–509 ,\narXiv:hep-ph/9511336 [hep-ph] .\n[14]Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower,\n“Fusing gauge theory tree amplitudes into loop\namplitudes,” Nucl. Phys. B435 (1995) 59–101 ,\narXiv:hep-ph/9409265 [hep-ph] .\n[15]Z. Bern, L. J. Dixon, and D. A. Kosower, “One loop\namplitudes for e+ e- to four partons,”\nNucl. Phys. B 513(1998) 3–86 ,\narXiv:hep-ph/9708239 .\n[16]R. Britto, F. Cachazo, and B. Feng, “Generalized\nunitarity and one-loop amplitudes in N=4\nsuper-Yang-Mills,” Nucl. Phys. B 725(2005) 275–305 ,\narXiv:hep-th/0412103 .\n[17]Z. Bern, J. J. M. Carrasco, and H. Johansson, “New\nrelations for gauge-theory amplitudes,”\nPhys. Rev. D78(2008) 085011 ,\narXiv:0805.3993 [hep-ph] .\n[18]Z. Bern, J. J. M. Carrasco, and H. Johansson,\n“Perturbative quantum gravity as a double copy of\ngauge theory,” Phys. Rev. Lett. 105(2010) 061602 ,\narXiv:1004.0476 [hep-th] .\n[19]H. Kawai, D. C. Lewellen, and S. H. H. Tye, “A relation\nbetween tree amplitudes of closed and open strings,”\nNucl. Phys. B269 (1986) 1–23 .\n[20]K. G. Chetyrkin and F. V. Tkachov, “Integration by\nParts: The Algorithm to Calculate beta Functions in 4\nLoops,” Nucl. Phys. B 192(1981) 159–204 .\n[21]S. Laporta, “High precision calculation of multiloop\nFeynman integrals by difference equations,”\nInt. J. Mod. Phys. A 15(2000) 5087–5159 ,\narXiv:hep-ph/0102033 .\n[22]A. V. Smirnov, “Algorithm FIRE – Feynman Integral\nREduction,” JHEP 10(2008) 107 ,\narXiv:0807.3243 [hep-ph] .\n[23]A. V. Smirnov and F. S. Chuharev, “FIRE6: Feynman\nIntegral REduction with Modular Arithmetic,”\nComput. Phys. Commun. 247(2020) 106877 ,\narXiv:1901.07808 [hep-ph] .\n[24]A. V. Kotikov, “Differential equations method: New\ntechnique for massive Feynman diagrams calculation,”\nPhys. Lett. B 254(1991) 158–164 .\n[25]Z. Bern, L. J. Dixon, and D. A. Kosower,\n“Dimensionally regulated pentagon integrals,”\nNucl. Phys. B 412(1994) 751–816 ,\narXiv:hep-ph/9306240 .\n[26]E. Remiddi, “Differential equations for Feynman graph\namplitudes,” Nuovo Cim. A 110(1997) 1435–1452 ,\narXiv:hep-th/9711188 .\n[27]T. Gehrmann and E. Remiddi, “Differential equations\nfor two loop four point functions,”\nNucl. Phys. B 580(2000) 485–518 ,\narXiv:hep-ph/9912329 .\n[28]G. Mogull, J. Plefka, and J. Steinhoff, “Classical black\nhole scattering from a worldline quantum field theory,”\nJHEP 02(2021) 048 ,arXiv:2010.02865 [hep-th] .\n[29]G. Kälin and R. A. Porto, “Post-Minkowskian EffectiveField Theory for Conservative Binary Dynamics,”\nJHEP 11(2020) 106 ,arXiv:2006.01184 [hep-th] .\n[30]Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H.\nShen, M. P. Solon, and M. Zeng, “Scattering\nAmplitudes and Conservative Binary Dynamics at\nO(G4),”Phys. Rev. Lett. 126no. 17, (2021) 171601 ,\narXiv:2101.07254 [hep-th] .\n[31]Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf, C.-H.\nShen, M. P. Solon, and M. Zeng, “Scattering\nAmplitudes, the Tail Effect, and Conservative Binary\nDynamics at O(G4),”\nPhys. Rev. Lett. 128no. 16, (2022) 161103 ,\narXiv:2112.10750 [hep-th] .\n[32]C. Dlapa, G. Kälin, Z. Liu, and R. A. Porto, “Dynamics\nof binary systems to fourth Post-Minkowskian order\nfrom the effective field theory approach,”\nPhys. Lett. B 831(2022) 137203 ,\narXiv:2106.08276 [hep-th] .\n[33]G. U. Jakobsen, G. Mogull, J. Plefka, B. Sauer, and\nY. Xu, “Conservative Scattering of Spinning Black\nHoles at Fourth Post-Minkowskian Order,”\nPhys. Rev. Lett. 131no. 15, (2023) 151401 ,\narXiv:2306.01714 [hep-th] .\n[34]G. U. Jakobsen, G. Mogull, J. Plefka, and B. Sauer,\n“Dissipative Scattering of Spinning Black Holes at\nFourth Post-Minkowskian Order,”\nPhys. Rev. Lett. 131no. 24, (2023) 241402 ,\narXiv:2308.11514 [hep-th] .\n[35]G. U. Jakobsen, G. Mogull, J. Plefka, and J. Steinhoff,\n“Classical Gravitational Bremsstrahlung from a\nWorldline Quantum Field Theory,”\nPhys. Rev. Lett. 126no. 20, (2021) 201103 ,\narXiv:2101.12688 [gr-qc] .\n[36]S. Mougiakakos, M. M. Riva, and F. Vernizzi,\n“Gravitational Bremsstrahlung with Tidal Effects in the\nPost-Minkowskian Expansion,”\nPhys. Rev. Lett. 129no. 12, (2022) 121101 ,\narXiv:2204.06556 [hep-th] .\n[37]M. M. Riva, F. Vernizzi, and L. K. Wong,\n“Gravitational bremsstrahlung from spinning binaries in\nthe post-Minkowskian expansion,”\nPhys. Rev. D 106no. 4, (2022) 044013 ,\narXiv:2205.15295 [hep-th] .\n[38]A. Elkhidir, D. O’Connell, M. Sergola, and I. A.\nVazquez-Holm, “Radiation and Reaction at One Loop,”\narXiv:2303.06211 [hep-th] .\n[39]A. Herderschee, R. Roiban, and F. Teng, “The\nsub-leading scattering waveform from amplitudes,”\nJHEP 06(2023) 004 ,arXiv:2303.06112 [hep-th] .\n[40]A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm,\n“Inelastic exponentiation and classical gravitational\nscattering at one loop,” JHEP 06(2023) 126 ,\narXiv:2303.07006 [hep-th] .\n[41]A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis,\nJ. Gowdy, and G. Travaglini, “One-loop gravitational\nbremsstrahlung and waveforms from a heavy-mass\neffective field theory,” JHEP 06(2023) 048 ,\narXiv:2303.06111 [hep-th] .\n[42]S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and\nS. Mizera, “What can be measured asymptotically?,”\narXiv:2308.02125 [hep-th] .\n[43]S. De Angelis, R. Gonzo, and P. P. Novichkov,\n“Spinning waveforms from KMOC at leading order,”\narXiv:2309.17429 [hep-th] .28\n[44]A. Brandhuber, G. R. Brown, G. Chen, J. Gowdy, and\nG. Travaglini, “Resummed spinning waveforms from\nfive-point amplitudes,” arXiv:2310.04405 [hep-th] .\n[45]R. Aoude, K. Haddad, C. Heissenberg, and A. Helset,\n“Leading-order gravitational radiation to all spin\norders,”arXiv:2310.05832 [hep-th] .\n[46]L. Bohnenblust, H. Ita, M. Kraus, and J. Schlenk,\n“Gravitational Bremsstrahlung in Black-Hole Scattering\natO(G3): Linear-in-Spin Effects,”\narXiv:2312.14859 [hep-th] .\n[47]L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and\nD. Trestini, “Gravitational-Wave Phasing of\nQuasicircular Compact Binary Systems to the\nFourth-and-a-Half Post-Newtonian Order,”\nPhys. Rev. Lett. 131no. 12, (2023) 121402 ,\narXiv:2304.11185 [gr-qc] .\n[48]L. Blanchet, G. Faye, Q. Henry, F. Larrouturou, and\nD. Trestini, “Gravitational-wave flux and quadrupole\nmodes from quasicircular nonspinning compact binaries\nto the fourth post-Newtonian order,”\nPhys. Rev. D 108no. 6, (2023) 064041 ,\narXiv:2304.11186 [gr-qc] .\n[49]A. Einstein, “Über Gravitationswellen,” Sitzungsber.\nPreuss. Akad. Wiss. Berlin (Math. Phys. ) 1918 (1918)\n154–167.\n[50]D. Bini, T. Damour, and A. Geralico, “Comparing\none-loop gravitational bremsstrahlung amplitudes to\nthe multipolar-post-Minkowskian waveform,”\nPhys. Rev. D 108no. 12, (2023) 124052 ,\narXiv:2309.14925 [gr-qc] .\n[51]A. Georgoudis, C. Heissenberg, and R. Russo, “An\neikonal-inspired approach to the gravitational scatterin g\nwaveform,” arXiv:2312.07452 [hep-th] .\n[52]G. Veneziano and G. A. Vilkovisky, “Angular\nmomentum loss in gravitational scattering, radiation\nreaction, and the Bondi gauge ambiguity,”\nPhys. Lett. B 834(2022) 137419 ,\narXiv:2201.11607 [gr-qc] .\n[53]H. Bondi, M. G. J. van der Burg, and A. W. K.\nMetzner, “Gravitational waves in general relativity. 7.\nWaves from axisymmetric isolated systems,”\nProc. Roy. Soc. Lond. A 269(1962) 21–52 .\n[54]R. K. Sachs, “Gravitational waves in general relativity.\n8. Waves in asymptotically flat space-times,”\nProc. Roy. Soc. Lond. A 270(1962) 103–126 .\n[55]T. Damour, “Radiative contribution to classical\ngravitational scattering at the third order in G,”\nPhys. Rev. D 102no. 12, (2020) 124008 ,\narXiv:2010.01641 [gr-qc] .\n[56]T. Damour, “Analytical Calculations of Gravitational\nRadiation,” in 4th Marcel Grossmann Meeting on the\nRecent Developments of General Relativity . 6, 1985.\n[57]L. Kehrberger, “The case against smooth null infinity\nIV: Linearized gravity around\nSchwarzschild—anoverview,”\nPhil. Trans. Roy. Soc. Lond. A 382no. 2267, (2024) 20230039 ,\narXiv:2401.04170 [gr-qc] .\n[58]T. Damour and N. Deruelle, “General relativistic\ncelestial mechanics of binary systems. I. The\npost-Newtonian motion.,” Annales de L’Institut Henri\nPoincare Section (A) Physique Theorique 43no. 1,\n(Jan., 1985) 107–132.\n[59]T. Damour and G. Schaefer, “Higher Order Relativistic\nPeriastron Advances and Binary Pulsars,”Nuovo Cim. B 101(1988) 127 .\n[60]G. Cho, A. Gopakumar, M. Haney, and H. M. Lee,\n“Gravitational waves from compact binaries in\npost-Newtonian accurate hyperbolic orbits,”\nPhys. Rev. D 98no. 2, (2018) 024039 ,\narXiv:1807.02380 [gr-qc] .\n[61]L. Blanchet, “Energy losses by gravitational radiation in\ninspiraling compact binaries to five halves\npostNewtonian order,”\nPhys. Rev. D 54(1996) 1417–1438 ,\narXiv:gr-qc/9603048 . [Erratum: Phys.Rev.D 71,\n129904 (2005)].\n[62]T. Damour and N. Deruelle, “Radiation Reaction and\nAngular Momentum Loss in Small Angle Gravitational\nScattering,” Phys. Lett. A 87(1981) 81 .\n[63]C. Cheung, I. Z. Rothstein, and M. P. Solon, “From\nScattering Amplitudes to Classical Potentials in the\nPost-Minkowskian Expansion,”\nPhys. Rev. Lett. 121no. 25, (2018) 251101 ,\narXiv:1808.02489 [hep-th] .\n[64]Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P.\nSolon, and M. Zeng, “Black Hole Binary Dynamics from\nthe Double Copy and Effective Theory,”\nJHEP 10(2019) 206 ,arXiv:1908.01493 [hep-th] .\n[65]Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M. P.\nSolon, and M. Zeng, “Scattering Amplitudes and the\nConservative Hamiltonian for Binary Systems at Third\nPost-Minkowskian Order,”\nPhys. Rev. Lett. 122no. 20, (2019) 201603 ,\narXiv:1901.04424 [hep-th] .\n[66]M. Beneke and V. A. Smirnov, “Asymptotic expansion\nof Feynman integrals near threshold,”\nNucl. Phys. B 522(1998) 321–344 ,\narXiv:hep-ph/9711391 .\n[67]R. Harlander, P. Kant, L. Mihaila, and M. Steinhauser,\n“Dimensional Reduction applied to QCD at three\nloops,” JHEP 09(2006) 053 ,arXiv:hep-ph/0607240 .\n[68]R. V. Harlander, D. R. T. Jones, P. Kant, L. Mihaila,\nand M. Steinhauser, “Four-loop beta function and mass\nanomalous dimension in dimensional reduction,”\nJHEP 12(2006) 024 ,arXiv:hep-ph/0610206 .\n[69]J. C. Collins, Renormalization , vol. 26 of Cambridge\nMonographs on Mathematical Physics . Cambridge\nUniversity Press, Cambridge, 7, 2023.\n[70]D. Bini, T. Damour, and A. Geralico, “Radiative\ncontributions to gravitational scattering,”\nPhys. Rev. D 104no. 8, (2021) 084031 ,\narXiv:2107.08896 [gr-qc] .\n[71]S. Weinberg, “Infrared photons and gravitons,”\nPhys. Rev. 140(1965) B516–B524 .\n[72]A. Luna, I. Nicholson, D. O’Connell, and C. D. White,\n“Inelastic Black Hole Scattering from Charged Scalar\nAmplitudes,” JHEP 03(2018) 044 ,\narXiv:1711.03901 [hep-th] .\n[73]J. Parra-Martinez, M. S. Ruf, and M. Zeng, “Extremal\nblack hole scattering at O(G3): graviton dominance,\neikonal exponentiation, and differential equations,”\nJHEP 11(2020) 023 ,arXiv:2005.04236 [hep-th] .\n[74]N. Craig, H. Elvang, M. Kiermaier, and T. Slatyer,\n“Massive amplitudes on the Coulomb branch of N=4\nSYM,” JHEP 12(2011) 097 ,\narXiv:1104.2050 [hep-th] .\n[75]M. Kiermaier, “The Coulomb-branch S-matrix from\nmassless amplitudes,” arXiv:1105.5385 [hep-th] .29\n[76]M. J. Duff, “Quantum Tree Graphs and the\nSchwarzschild Solution,” Phys. Rev. D 7(1973) .\n[77]N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia,\nL. Planté, and P. Vanhove, “General Relativity from\nScattering Amplitudes,”\nPhys. Rev. Lett. 121no. 17, (2018) 171601 ,\narXiv:1806.04920 [hep-th] .\n[78]A. Cristofoli, A. Elkhidir, A. Ilderton, and\nD. O’Connell, “Large gauge effects and the structure of\namplitudes,” JHEP 06(2023) 204 ,\narXiv:2211.16438 [hep-th] .\n[79]T. Adamo, R. Gonzo, and A. Ilderton, “Gravitational\nBound Waveforms from Amplitudes,”\narXiv:2402.00124 [hep-th] .\n[80]Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson,\nand R. Roiban, “Simplifying Multiloop Integrands and\nUltraviolet Divergences of Gauge Theory and Gravity\nAmplitudes,” Phys. Rev. D 85(2012) 105014 ,\narXiv:1201.5366 [hep-th] .\n[81]S. Abreu, R. Britto, and H. Grönqvist, “Cuts and\ncoproducts of massive triangle diagrams,”\nJHEP 07(2015) 111 ,arXiv:1504.00206 [hep-th] .\n[82]J. L. Bourjaily, H. Hannesdottir, A. J. McLeod, M. D.\nSchwartz, and C. Vergu, “Sequential Discontinuities of\nFeynman Integrals and the Monodromy Group,”\nJHEP 01(2021) 205 ,arXiv:2007.13747 [hep-th] .\n[83]F. Teng, “Understanding the NLO Scattering Waveform\nSoft Limit and Post-Newtonian Expansion.”\nTalk at QDC Meets Gravity IX, December, 2023 .\n[84]A. V. Manohar, A. K. Ridgway, and C.-H. Shen,\n“Radiated Angular Momentum and Dissipative Effects\nin Classical Scattering,”\nPhys. Rev. Lett. 129no. 12, (2022) 121601 ,\narXiv:2203.04283 [hep-th] .\n[85]P. Di Vecchia, C. Heissenberg, and R. Russo, “Angular\nmomentum of zero-frequency gravitons,”\nJHEP 08(2022) 172 ,arXiv:2203.11915 [hep-th] .\n[86]S. Abreu, J. Dormans, F. Febres Cordero, H. Ita,\nB. Page, and V. Sotnikov, “Analytic Form of the PlanarTwo-Loop Five-Parton Scattering Amplitudes in QCD,”\nJHEP 05(2019) 084 ,arXiv:1904.00945 [hep-ph] .\n[87]M. Heller and A. von Manteuffel, “MultivariateApart:\nGeneralized partial fractions,”\nComput. Phys. Commun. 271(2022) 108174 ,\narXiv:2101.08283 [cs.SC] .\n[88]R. Russo, “ The Gravitational Eikonal and the NLO\nScattering Waveform.”\nTalk at QDC Meets Gravity IX, December, 2023 .\n[89]B. Sahoo and A. Sen, “Classical soft graviton theorem\nrewritten,” JHEP 01(2022) 077 ,\narXiv:2105.08739 [hep-th] .\n[90]A. P. Saha, B. Sahoo, and A. Sen, “Proof of the\nclassical soft graviton theorem in D= 4,”\nJHEP 06(2020) 153 ,arXiv:1912.06413 [hep-th] .\n[91]A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm,\n“More on the Subleading Gravitational Waveform,”\narXiv:2312.14710 [hep-th] .\n[92]L. Bel, T. Damour, N. Deruelle, J. Ibanez, and\nJ. Martin, “Poincaré-invariant gravitational field and\nequations of motion of two pointlike objects: The\npostlinear approximation of general relativity,”\nGen. Rel. Grav. 13(1981) 963–1004 .\n[93]S. Mougiakakos, M. M. Riva, and F. Vernizzi,\n“Gravitational Bremsstrahlung in the post-Minkowskian\neffective field theory,”\nPhys. Rev. D 104no. 2, (2021) 024041 ,\narXiv:2102.08339 [gr-qc] .\n[94]J. Ware, R. Saotome, and R. Akhoury, “Construction of\nan asymptotic S matrix for perturbative quantum\ngravity,” JHEP 10(2013) 159 ,\narXiv:1308.6285 [hep-th] .\n[95]L. Blanchet, A. Buonanno, and G. Faye, “Higher-order\nspin effects in the dynamics of compact binaries. II.\nRadiation field,” Phys. Rev. D 74(2006) 104034 ,\narXiv:gr-qc/0605140 . [Erratum: Phys.Rev.D 75,\n049903 (2007), Erratum: Phys.Rev.D 81, 089901\n(2010)]."    },    {        "title": "2402.06618v1.Polynomial_parametrisation_of_the_canonical_iterates_to_the_solution_of___γg___g___1__.pdf",        "content": "Polynomial parametrisation of the canonical iterates\nto the solution of −γg′=g−1\nRoland Miyamoto\nFebruary 12, 2024\nDedicated to the memory of Colin Lingwood Mallows (1930–2023)\nAbstract\nThe iterates h0, h1, h2, . . . constructed in [6] and converging to the (only) solution\ng=h: [0,1]→[0,1] of the iterative differential equation −γg′=g−1,γ > 0,\nare parametrised by polynomials over Q, and the corresponding constant γ=κ≈\n0.278877 is estimated by rational numbers.\nMSC2020: 65D20, 11B83, 11Y55, 26A06, 47H10, 33E99\nKeywords: fixed point, operator, iterative differential equation, special constant\nWe start by recalling a few definitions and facts from [6]. For 0 ≤a≤b≤1 and\nany (Lebesgue) integrable function f: [0,1]→R, we abbreviateRb\naf:=Rb\naf(x)dxandR\nf:=R1\n0f. LetEbe the set of decreasing functions g: [0,1]→[0,1] satisfying g(0) = 1.\nGiven g∈ E, we introduce its pseudo-inverse g∗∈ Edefined by\ng∗(y) := sup g−1[y,1] for y∈[0,1],\nwhich satisfiesR\ng=R\ng∗; moreover, g∗equals the compositional inverse g−1ifgis\nbijective. Stretching g∈ E horizontally by some factor a > 0 leads to the function\ng◦ ·1\na: [0, a]→[0,1] given by ( g◦ ·1\na)(x) :=g(x\na) for x∈[0, a].\nThe iterative differential equation (IDE) mentioned in the title is solved by employing the\noperator T:E → E defined via\n(1) ( Tg)(x) :=R1\nxg∗\nR\ngforg∈ Eandx∈[0,1]\nas well as its iterations T0:= id E,Tn:=T◦Tn−1forn∈N. We recall that g∈ Esolves\nthe said IDE for some γ >0 if and only if gis a fixed point of the operator T, that is,\nTg=g, and thenR\ng=γ=−1/g′(0). We construct a fixed point of Tby setting\n(2) h0:=1[0,1]∈ E and hn:=Tnh0forn∈N,\n1arXiv:2402.06618v1  [math.CO]  9 Feb 2024Figure 1: Graphs of h0, . . . , h 5\nand obtain as consequences that h0≥h1≥h2≥ ··· ,\n(3)−κnh′\nn+1=h∗\nnwhere κn:=R\nhnforn∈N0\nby (1), the limit\nh:= lim\nn→∞hn= inf{hn:n∈N0} ∈ E\nis a fixed point of T, and h, h 1, h2, . . .are bijective, strictly decreasing and convex. So far\nwe have quoted results from [6].\nTo motivate the calculations performed in this paper, we want to mention a few more\nresults scheduled for later publication as well as some connections to combinatorics. Using\ncertain geometric techniques, one can show that the operator Tenjoys global convergence,\nthat is, lim n→∞Tnf=hfor every f∈ E. In particular, his the only fixed point of T\n(and of Tnfor every n∈N), and\nκ:=R\nh=−1\nh′(0)= lim\nn→∞κn= inf{κn:n∈N0} ≈0.278877\nis the only γ >0 for which the IDE in the title has a solution in E.\nColin L. Mallows [3] observed that the rows of the triangle underlying Levine’s sequence\n(ln)n∈N= (1,2,2,3,4,7,14,42,213,2837,175450 ,139759600 ,6837625106787 , . . .)\n(see [1] and [7, p. 118]), when scaled into the unit square, seem to converge to a limit\nshape, which — its existence presumed and after a 90◦clockwise rotation about the point\n(1\n2,1\n2) — would necessarily equal the function h. As a consequence, we would have\nlim\nn→∞ln+1\nln·ln−1=κ,\n2and the constant “ c1=I≈0.277 (last digit perhaps 6 or 8)” conjectured by Mallows\nin [7, p. 118] and [1] would also equal κ.\nObviously,\n(4) ˜h:=1\nκh◦ ·κ: [0,1\nκ]→[0,1\nκ] satisfies −˜h′=˜h∗.\nGeometrically speaking, ˜hbecomes its own derivative when rotated clockwise about the\norigin by 90◦(into the fourth quadrant). Owing to this property, we want to call ˜hthe\nstandard stribola (of order 1) and any function b·h◦ ·1\na: [0, a]→[0, b] with a, b > 0\nastribola (of order 1; from Greek στρίβω = turn, twist), while we refer to has the\nunit stribola (of order 1), to κas the stribolic constant (of order 1), to Tas the\ntwint (“twist and integrate”) or stribolic operator (of order 1), and to the functions\nh0, h1, h2, . . . from (2) as the canonical stribolic iterates (of order 1). (Higher order\nstribolas and their corresponding constants will be introduced and investigated in a future\npublication.)\nIn reverence to Mallows’s contributions mentioned above, we also want to suggest the\nname Mallows’s constant forκ. The OEIS entry [2] indicates that Mallows, in an\nattempt to solve (his version of) the IDE, must have calculated the first coefficients of a\npower series for the presumed solution. In our terminology, let ˜ τ=˜h(˜τ) be the fixpoint of\nthe standard stribola and bn:=˜h(n)(˜τ) the n-th derivative of ˜hat ˜τ. Then Fa` a di Bruno’s\nformula applied to (4) yields the identities\nnX\nk=1Bn,k(b1, . . . , b n−k+1)·bk+1= 0 for n= 2,3,4, . . .\nwhere Bn,kare the exponential Bell polynomials. These identities can be used to express\nall the bnin terms of ˜ τas is made more explicit in [2].\nConcerning the canonical stribolic iterates defined by (2), we note that h∗\n0=h0,h1(x) =\n1−x=h∗\n1(x),h2(x) = (1 −x)2,h∗\n2(x) = 1−√xandh3(x) = 1−3x+2x3\n2forx∈[0,1] and\nthereby κ0= 1, κ1=1\n2,κ2=1\n3andκ3=3\n10. We cannot expect explicit function terms\nforh4, h5, . . ., but instead are aiming at parametrising them. To this end, we introduce\nthe compositions\nqn:=hn◦ ··· ◦ h1: [0,1]→[0,1] for n∈N0,\nwhere, as usual, we agree that the empty composition is q0= id [0,1]. For n∈N, the\nequation y=hn(x) is obviously parametrised by x=qn−1(t),y=qn(t),t∈[0,1].\nBecause X:= id [0,1]is transcendental over R, we can look at the univariate polynomial\nringR[X] in both ways, algebraically and as a subspace of C0[0,1], the space of continuous\nfunctions on [0 ,1]. We consider the discrete valuation v=vX:R[X]→N0∪ {∞}\nassociated with its prime element X(cf. [8, p. 4f]) and the degree map deg: R[X]→\nN0∪ {−∞} . The following theorem reveals that the qnare polynomials over Q, the\nfirst few of which are displayed in Table 1, and their degrees are given by the Fibonacci\nsequence ( F0, F1, F2, . . .) := (0 ,1,1,2,3,5,8,13, . . .).\nTheorem 1. Letn∈NandQn(x) :=Rx\n0q′\nnqn−1forx∈[0,1]. Then\n(a)qn=hn◦qn−1: [0,1]→[0,1]is bijective and continuously differentiable with qn(0) =\nnmod 2 ∈ {0,1}andqn(1)−qn(0) = ( −1)n,\n3(b)κn= (−1)nQn(1)andqn+1=qn+1(0)−1\nκnQn,\n(c)κn∈Qandqn∈Q[X]withv(qn−qn(0)) = 2⌊n\n2⌋anddegqn=Fn+1; moreover, there\nexists dn∈Nsuch that dn·(qn−qn(0))∈Z[X]is primitive.\nWherever it makes sense, the above statements also hold for n= 0.\nProof. (a)follows directly from the definition of qn. Clearly, q0= id [0,1]=Xis also\ncontinuously differentiable and bijective with q0(0) = 0 and q0(1) = 1.\n(b)Equation (3) implies h′\nn+1(hn(x)) =−x\nκnforx∈[0,1], hence\nq′\nn+1= (hn+1◦qn)′=q′\nn·(h′\nn+1◦qn) =q′\nn·(h′\nn+1◦hn◦qn−1) =−1\nκnq′\nnqn−1\nby (a), and integration yields qn+1(x) = qn+1(0)−1\nκnQn(x) for x∈[0,1]. The first\nassertion follows by substituting x= 1 and using (a) again.\n(c)By definition, κ0= 1, q0=Xandq1= 1−X. Using (b) and induction, we obtain\nκn∈Qandqn, Qn∈Q[X]. From (b) we also conclude that deg qn+1= deg Qn=\n1 + deg( q′\nnqn−1) = deg qn+ deg qn−1, thus deg qn=Fn+1by induction. Again from (b), we\nderive\nv(qn+1−qn+1(0)) = v(Qn) = 1 + v(q′\nnqn−1) =v(qn−qn(0)) + v(qn−1),\nhence v(qn−qn(0)) = 2⌊n\n2⌋by induction. Clearly, we can write qn−qn(0) =z\ndnwith\ndn∈Nandz∈Z[X] such that dnis coprime to the content cofz. On the other hand,\n(a) implies ( −1)ndn=dn·(qn(1)−qn(0)) = z(1)∈cZ. Therefore, c= 1.\nnκn qn\n01 X\n11\n21−X\n21\n3X2\n33\n101−3X2+ 2X3\n42\n75X4−4X5\n5161\n5721 +1\n2(−35X4+ 28X5+ 70X6−100X7+ 35X8)\n624941\n891481\n23(3575 X8−5720X9−6292X10+ 19240 X11−14300 X12+ 3520 X13)\n749675943612\n177918244665\n83267335346149361824147\n11711158115225119429452\n92507700451651989905962493021537936733790431031\n8990773234863161759100003096510729982749072312\n1039058362193701767718721504578116138158143785410766642680982462728116470023287868511995843\n140048278006628885452600904137492554179859017924910241263151850844470542993943699969398879\nTable 1: κ0, . . . κ 10andq0, . . . q 6\nWith a Python program implementing the formulae in Theorem 1(b), we are able to\ncalculate κ1, . . . , κ 22andq1, . . . , q 23. Their numerators and denominators appear to grow\n4exponentially in length. What fits into this paper width is listed in Table 1. More terms\nare provided in the OEIS entries [4, 5]. During our calculations, we observe that the\nnumerators of κnandκn+1seem to be mostly coprime, while the numerator of κnseems\nto “almost” divide the denominator of κn+1.\nObservation 1. Forn∈N0, write κn=µn\nνnwithµn, νn∈Nandgcd(µn, νn) = 1 . Then\n(a)µnandµn+1are coprime for n∈ {0, . . . , 21}with the sole exception of gcd(µ5, µ6) =\n7,\n(b)µndivides νn+1forn∈ {0,1,2,4,6,8,10,11,12,15,16,17,18,19,21}, while µ3=\n3·gcd(µ3, ν4),µ5= 7·gcd(µ5, ν6),µ7= 192·gcd(µ7, ν8),µ9= 37·gcd(µ9, ν10),\nµ13= 3·gcd(µ13, ν14),µ14= 7·gcd(µ14, ν15)andµ20= 192·gcd(µ20, ν21).\nAs for the primitive polynomials in Theorem 1(d), their relevant coefficients appear to be\nnon-zero and no three consecutive coefficients seem to have the same sign.\nObservation 2. Letn, dn∈Nsuch thatP\niciXi:=dn·(qn−qn(0))∈Z[X]is primitive, as\nin Theorem 1(d). Then 0/∈ {ci−1ci, cici+1} ̸⊆Nfor2⌊n\n2⌋< i < F n+1andn∈ {1, . . . , 23}.\nn κn=R\nhn ϑn:=κn−κn+1\nκn−1−κn˜κn:=κn−ϑnκn−1\n1−ϑn\n01.00000000000000000000\n10.50000000000000000000 0.33333333333333333333 0.25000000000000000000\n20.33333333333333333333 0.20000000000000000000 0.29166666666666666667\n30.30000000000000000000 0.42857142857142857143 0.27500000000000000000\n40.28571428571428571429 0.29720279720279720280 0.27967306325515280739\n50.28146853146853146853 0.39988492368275887552 0.27863938556045289094\n60.27977071835599228250 0.33227684207155282888 0.27892584129464787423\n70.27920657437653008783 0.37794408664511337726 0.27886381599022319905\n80.27899335949547590907 0.34720353947195583609 0.27887995667262631637\n90.27891933053410580614 0.36746593866934728811 0.27887632396190421606\n100.27889212741232722442 0.35386048463306698285 0.27887722953094531460\n110.27888250130247112315 0.36282275236161039705 0.27887701998359875188\n120.27887900873079859727 0.35683379624435139475 0.27887707102387107908\n130.27887776246319003437 0.36078940911521411257 0.27887705903280182623\n140.27887731282303594153 0.35815481970563930029 0.27887706192018075817\n150.27887715178224762000 0.35989857725744438595 0.27887706123667384356\n160.27887709382389702266 0.35873932154530262253 0.27887706140035996550\n170.27887707303195765148 0.35950754355823571103 0.27887706136147039755\n180.27887706555709860234 0.35899731864060339565 0.27887706137075986477\n190.27887706287364424648 0.35933565744435908888 0.27887706136854903338\n200.27887706190938341130 0.35911105319893171159 0.27887706136907650507\n210.27887706156310668722 0.35926004095941929301 0.27887706136895087046\n220.27887706143870329714\nTable 2: κ0, . . . , κ 22,ϑ1, . . . , ϑ 21and ˜κ1, . . . , ˜κ21rounded to 20 decimals\nWhile each κnprovides an upper bound for κ, the following result allows us to estimate\nκfrom below.\n5Proposition 1. For all n∈N, we have κn−1 +κn\nκn−1< κ.\nIts proof relies heavily on the geometric techniques mentioned earlier and is therefore\nreserved for a later publication. Applying Proposition 1 to n= 22 leads to the estimate\n(5) 0 .27887706099 < κ 22−1 +κ22\nκ21< κ < κ 22<0.27887706144\nusing the left column of Table 2. The convergence rate of the sequence ( κn)n∈N0is ex-\npressed by the quotients\nϑn:=κn−κn+1\nκn−1−κnforn∈N.\nThe values of ϑ2, . . . , ϑ 21in Table 2 give rise to the following hypothesis.\nConjecture 1. For all m∈N, we have ϑ2m< ϑ 2m+2< ϑ 2m+3< ϑ 2m+1.\nPresuming this conjecture allows for an estimate of the stribolic constant κusing the\nnumbers ˜ κnin the right column of Table 2 defined by\n˜κn:=κn−1−κn−1−κn\n1−ϑn=κn−ϑnκn−1\n1−ϑn=κn−1κn+1−κ2\nn\nκn−1−2κn+κn+1forn∈N,\nthat is considerably sharper than (5).\nProposition 2. If Conjecture 1 holds, then ˜κ2m−1< κ < ˜κ2mfor all m∈N, in particular,\n0.27887706136895 <˜κ21< κ < ˜κ20<0.27887706136908 .\nProof. Letm∈Nand set δn:=κn−κn+1forn∈N0. Presuming Conjecture 1, n >2m\nentailsδn\nδ2m−1=ϑ2m···ϑn> ϑn−2m+1\n2m . Therefore,\nκ2m−1−κ=∞X\nn=2m−1δn> δ2m−1∞X\ni=0ϑi\n2m=δ2m−1\n1−ϑ2m,\nthat is, κ < κ 2m−1−δ2m−1\n1−ϑ2m= ˜κ2m. Likewise,\nκ2m−κ=∞X\nn=2mδn< δ2m∞X\ni=0ϑi\n2m+1=δ2m\n1−ϑ2m+1,\nhence ˜ κ2m+1=κ2m−δ2m\n1−ϑ2m+1< κ. The estimate ˜ κ1< κ is now also covered because\n˜κ1=1\n4<11\n40= ˜κ3.\nReferences\n[1] L. Levine. Levine’s sequence. Entry A011784 in The On-Line Encyclopedia of Integer\nSequences, https://oeis.org/A011784\n6[2] C. L. Mallows. Triangle of numbers arising from analysis of Levine’s sequence\nA011784. Entry A014621 in The On-Line Encyclopedia of Integer Sequences,\nhttps://oeis.org/A014621\n[3] C. L. Mallows, B. Poonen, N. J. A. Sloane. Discussion of A011784. Scans of pages 150–\n155 and 164 of Sloane’s notebook ”Lattices 77” from June–July 1997. The On-Line\nEncyclopedia of Integer Sequences, https://oeis.org/A011784/a011784.pdf\n[4] R. Miyamoto. Numerator/denominator of canonical iterated stribolic areaR\nhn(of\norder 1). Entries A369990 and A369991 in The On-Line Encyclopedia of Integer Se-\nquences, https://oeis.org/A369990 andhttps://oeis.org/A369991\n[5] R. Miyamoto. Coefficients of the polynomial qnused to parametrize the canon-\nical stribolic iterate hn(of order 1). Entries A369992 and A369993 in The\nOn-Line Encyclopedia of Integer Sequences, https://oeis.org/A369992 and\nhttps://oeis.org/A369993\n[6] R. Miyamoto, J. W. Sander. Solving the Iterative Differential Equation −γg′=g−1.In:\nH. Maier, J. & R. Steuding (Hrsg.), Number Theory in Memory of Eduard Wirsing,\nSpringer, 2023, 223–236.\n[7] N. J. A. Sloane. My Favorite Integer Sequences . In: C. Ding, T. Helleseth, H. Nieder-\nreiter (eds): Sequences and their Applications. Discrete Mathematics and Theoretical\nComputer Science. Springer, London (1999) 103–130.\n[8] H. Stichtenoth. Algebraic Function Fields and Codes. Springer, Berlin, 1993.\n7"    },    {        "title": "2402.06669v1.Compression_effects_and_scene_details_on_the_source_camera_identification_of_digital_videos.pdf",        "content": "  Compression Effects and Scene Details on the Source Camera Identification of Digital Videos  Group of Analysis, Security and Systems (GASS), Department of Software Engineering and Artificial Intelligence (DISIA), Faculty of Computer Science and Engineering, Office 431, Universidad Complutense de Madrid (UCM), Calle Profesor José Garc´ıa Santesmases 9, Ciudad Universitaria, 28040 Madrid, Spain Raquel Ramos López, Ana Lucila Sandoval Orozco, Luis Javier García Villalba∗      Abstract The continuous growth of technologies like 4G or 5G has led to a massive use of mobile devices such as smartphones and tablets. This phenomenon, combined with the fact that people use mobile phones for a longer period of time, results in mobile phones becoming the main source of creation of visual information. However, its reliability as a true representation of reality cannot be taken for granted due to the constant increase in editing software. This makes it easier to alter original content without leaving a noticeable trace in the modification. Therefore, it is essential to introduce forensic analysis mechanisms to guarantee the authenticity or integrity of a certain digital video, particularly if it may be considered as evidence in legal proceedings. This paper explains the branch of multimedia forensic analysis that allows to determine the identification of the source of acquisition of a certain video by exploiting the unique traces left by the camera sensor of the mobile device in visual content. To do this, a technique that performs the identification of the source of acquisition of digital videos from mobile devices is presented. It involves 3 stages: 1) Extraction of the sensor fingerprint by applying the   ∗Corresponding author Email addresses: raqram01@ucm.es, Securitas Direct. Madrid, Spain (e-mail: raquel.rlopez@securitasdirect.es) (Raquel Ramos L´opez), asandoval@fdi.ucm.es (Ana Lucila Sandoval Orozco), javiergv@fdi.ucm.es (Luis Javier García Villalba) 2     block-based technique. 2) Filtering the strong component of the PRNU signal to improve the quality of the sensor fingerprint. 3) Classification of digital videos in an open scenario, that is, where the forensic analyst does not need to have access to the device that recorded the video to find out the origin of the video. The main contribution of the proposed technique eliminates the details of the scene to improve the PRNU fingerprint. It should be noted that these techniques are applied to digital images and not to digital videos. In this work, we show that it is necessary to take this improvement into account to improve the identification of digital videos. Experimental results are also presented that support the validity of the techniques used and show promising results. Keywords: Acquisition Source Identification, Forensics Analysis, H.264, PRNU, Key Frame Extraction, Sensor Pattern Noise, Smartphone, Video Enhancement, Video Forensics Analysis, Video Source Acquisition.   1. Introduction Mobile technology can be considered a success in the history of telecom- munications due to the fact that is has become very popular and widespread in our lives. Throughout history, a constant cycle of evolution has been ob- served when it comes to a new generation of mobile network innovation that is launched every decade. As shown in Figure 1 below, since the introduction of 1G technology in the late 1970s, there has been a fairly regular pace of improvements in mobile capabilities over the past four decades, plus the grad- ual incorporation of elements that until then were only intended for personal computers (processor, qwerty keyboards, larger screens ...), together with the integration of specific software programs for these devices. All this has made mobile devices a feasible alternative to desktop computers. The introduction of each new generation has not only improved network performance, but has made newer and more advanced applications and devices available. Simi- larly, with 5G, we expect both network improvements in terms of bandwidth (1+ Gbps) and ultra-low latency, as well as features such as improved en- ergy efficiency, cost optimization, massive IoT density, and dynamic resource allocation in order to enable a broad spectrum of wireless applications and IoT connections Cis:2022. 3      \n   Figure 1:  Evolution of the mobile communication network  Smartphones have become an indispensable companion in our day to day lives. At their most basic level, we use mobile devices to be connected with our friends and family and to access the Internet. However, there is a functionality that is progressively gaining importance, and in which manufacturers are putting more and more effort: the use of the camcorder. The appearance of these devices has led to an important change in people’s behavior and communication habits. In a very short time, these devices have become the focus of interest for a great majority of people, so that a large part of their daily activities are channeled through their smart- phones. Currently, there are increasingly sophisticated video manipulation tools that make it difficult to identify the characteristics of interest. This means that what a video represents cannot be taken for granted. Therefore, it is critical to employ techniques and research that are capable of determin- ing, unmistakably, if a video has been manipulated or, on the contrary, shows its original content. The forensic analysis of videos has proven to be more difficult compared to the analysis of images, since the data contained in the videos has higher compression formats, which can compromise the existing ”fingerprints” and make it more difficult to recover the processing of a video from its origin. One of the first works where the compression of a video was taken into account was in Houten and Geradts (2009), where the use of the PRNU \n4     footprint on videos coming from YouTube was analysed. To carry out their experiments, they used a set of webcams and codecs to record and encode videos. These videos were then uploaded to YouTube and downloaded. The results obtained were good but the codecs used have already become obsolete, so nowadays, this technique could not be used in a real scenario. For the identification of the source of acquisition there are two main approaches: closed and open scenarios. A scenario is considered to be closed when the identification of the video source is carried out on a specific set of camcorders known a priori. For this approach, a set of videos from each camcorder is usually used to train a classifier and later predict the source of acquisition of the videos under investigation. One of the most used techniques for classifying digital videos is SVM (Support Vector Machine). In an open scenario, the forensic analyst does not know the set of video cameras to which the videos to be identified belong. In this type of classification, in which there is no data on videos known a priori, the objective is not to identify their brand and model, but rather to be able to group together the frames of the videos that belong to the same video. In Meij and Geradts (2018) an attempt was made to determine the origin of a digital video from a mobile device that was shared by the WhatsApp messaging platform on both iOS and Android operating systems. They used videos from ten different camcorder models to carry out their experiments. For each of these cameras, three types of videos were recorded: one video was recorded in an indoor space, another one in an outdoor space, and another that recorded a gray surface to be used as a reference. The results indicated that it is possible to determine the source camera of a digital video with a high success rate, because the videos that come from the same camera show many similarities. Once these videos have been transmitted through WhatsApp, this precision decreases, although it is still possible to classify the vast majority of them with somewhat worse results in the case of iOS. This method is very limited to be applied in open scenarios for two reasons. On the one hand, the success rate decreases in iOS devices, and on the other hand, this method does not take into account the wide variety of platforms through which a video can be distributed, like in the case of video sharing platforms such as YouTube. It should be noted that in the scientific literature, the contributions that exist today focus on improving the sensor footprint applied to the study of digital images. However, this study is almost non-existent in the case of digital videos. This is a gap that we aim to bridge with this work, given that 5     it is possible to improve the PRNU fingerprint of the sensor in the case of digital videos. One of the initial works applied to digital images, where an improvement in the quality of the PRNU fingerprint of the noise was proposed, was in Li (2010). Their idea was based on the following hypothesis: The details of the scene when representing high-frequency components make their magnitude much greater than that of the noise pattern. For that reason, it is necessary to eliminate the fragments of the high-frequency scene to improve the sensor noise footprint. Therefore, in this work, an approach is proposed to mitigate the influence of scene details when calculating the noise pattern, and thus improve the success rate when identifying the origin of the device. Another work that focuses on improving the PRNU footprint applied to images is in Akshatha et al. (2016). After estimating the PRNU noise pattern of the images, they are represented according to the characteristics that can be used in the classification. The two sets of features are identified and extracted from the images. The first set includes Higher order wavelet statistics (HOWS) from the estimated PRNU noise. The second set consists of statistical features from the original images. The functions are combined and used to discriminate images based on their source cameras. To carry out their experiments, they used 4 different groups of two cameras each, and they obtained an average hit rate of 100%, 99.75%, 94.75% respectively for each of the groups. As we have already mentioned, the technique is based on digital images but does not take into account digital videos. Another limitation of this technique is that it has not been tested when an image is shared by WhatsApp or downloaded from the YouTube platform. The techniques using the PRNU sensor fingerprint to carry out identifi- cation processes need to analyze the content of the video. They have become robust and reliable techniques, yet they require a higher computational cost. There are other methods such as the one presented in L´opez et al. (2020) where they identify the source of digital video acquisition by analyzing the elemental structure of a video called the atom. An atom is made up of a set of labels, which are values in a hierarchical way. The value of each of these atoms gives some clues to obtain the origin of a mobile device. For a more in-depth study of the container atoms, see Gloe et al. (2014). To perform the classification, they used a hierarchical grouping and another one based on density (OPTICS). With the hierarchical grouping, they achieved a coef- ficient of ”homogeneity” higher than 0.9% when it came to identifying the brand and 0.80% when it came to identifying the model. As for the OPTICS 6     density-based algorithm, they achieved a homogeneity of 0.97% grouping by brand and 0.87% grouping by model. This technique is much faster than techniques that have to go through video content to find out the origin of the device. In this work, it was necessary to use a smaller set of videos instead of the one proposed in L´opez et al. (2020) due to the high computational cost required by the technique that processes the video content instead of the atoms or video labels to identify the video source. In the field of engineering, computer simulations are widely used to re- place experiments with physical models because these simulations are often computationally expensive. However, many model-based engineering design problems require numerous simulations to reach an acceptable solution. This can be computationally prohibitive. Often a single surrogate model or meta- model is used to replace a detailed simulation in design problems which require repeated calculations. This surrogate is obtained using information derived from the physical model Alizadeh et al. (2019). In Alizadeh et al. (2019) proposed a method based on cross-validation to find an ensemble of surrogates(EoS) which is created by the least possible number of data points. The resulting ensemble surrogate has higher accuracy than each individual surrogate and is less computationally intensive. To achieve this ensemble surrogate, we compare it with individual surrogate models based on three main factors: the size of the problem, precision and calculation time. They found that it is effective to use cross validation. They obtained the highest accuracy level with least required data and less computation time by using the right number of samples. An example of surrogates is relatively insensitive to the size of the sample data or number of data points. In Alizadeh et al. (2020) they created a practical guidance based on a trade-off among three main drivers, namely, size (how much information is necessary to compute the surrogate model), accuracy (how accurate the sur- rogate model must be) and computational time (how much time is required for the surrogate modeling process). To make their proposal, they reviewed the latest generation surrogate models on more than 200 articles from the most recent literature. The main contribution of this work is threefold: 7     1. A digital video acquisition source identification technique is proposed. It takes into account the effect produced by video compression to ob- tain the sensor noise fingerprint, by applying the block-based technique proposed in Kouokam and Dirik (2019). 2. Once the blocks that have survived compression have been calculated, the method proposed in Li (2010) is used. It improves the PRNU noise pattern extracted from the videos under investigation. Therefore, in this work it is shown that in order to have a reliable and robust hit rate, it is not enough to review the effect that compression has on the videos, but also techniques that improve the fingerprint formed by the imperfections of the videosensor. Related works can be found in the literature that improve the sensor footprint applied to digital images but not to digital videos. 3. Finally, a grouping algorithm that classifies the videos in an open set- ting is used. In this case, the forensic analyst does not have access to the specific mobile device where the criminal activity is located, since you can find the video under investigation on the internet. Further- more, it is impossible for the analyst to have access to a video from the vast variety of mobile devices available on the market.  The remainder of the paper is structured as follows. Section 2 provides the reader with some background on algorithms for identifying the source of digital videos and provides the formal definition of the problem. Section 3 reports all technical details about the proposed solution. Section 4 presents the numerical experiments carried out to validate the proposed method while Section 5 concludes the paper providing some conclusive remarks.  2. Digital Video Source Identification Techniques This section details the main techniques for forensic analysis of digital videos, with an emphasis on the techniques for identifying the source of the video, as this is the branch of forensic analysis on which this work is focused. The section after next discusses the most important techniques related to video content analysis, with a special emphasis on the methods that study the sensor’s noise pattern. 8  \n   According to Sandoval Orozco et al. (2013), there are five main groups: 1) metadata, 2) matrix defects CFA, 3) image features, 4) interpolation color, 5) sensor imperfections combined with Wavelet transformations. The techniques based on sensor imperfections have gradually evolved into two different families: one based on pixel defects and another that analyzes the noise pattern in the sensor. The latter is the most used technique for identifying sources in both images and digital videos. This is the basis of this work, where the demonstrations of Lukas et al. (2006) have been taken into account. It was ultimately determined that the cameras generate a pattern of noise or SPN that can be used as a single classification method.  2.1. Analysis of the Sensor Noise Pattern There are various sources of imperfections and noise introduced in the different stages of the digital video generation process. Even in the case of a uniform and fully lit photograph, small changes in pixel intensity can be observed. This is due to the shooting noise, which is random and largely due to the noise pattern, and is deterministic and stays roughly the same if multiple photos of the same scene are captured. The noise pattern in an image refers to any spatial pattern that does not change from one image to another and is composed of the spatial noise that is independent from the signal or fixed pattern noise FPN and the spatial noise due to the difference in response of each pixel to the incident signal or non-uniform response noise PRNU. The structure of the noise pattern is illustrated in Figure 2.   \n Figure 2:  Sensor pattern noise.  The noise FPN is generated by the darkness current and is also dependent on exposure and temperature. Because a fixed pattern noise is an additive independent noise, some cameras automatically eliminate it by subtracting 9     a dark frame from the images they produce. Noise PRNU is the dominant part of the noise pattern in images and is a multiplicative dependent noise that is mainly formed by pixel uniformity PNU and by low frequency defects such as zoom settings and the refraction of light in dust particles and lenses. The noise PNU is the difference in light sensitivity among the pixels in the sensor array. It is generated by the lack of homogeneity of the silicon wafers and imperfections during the manufacturing of sensors. Due to their nature and origin, it is highly unlikely that even sensors from the same wafer show correlated PNU patterns. This noise is not affected by room temperature or humidity. Noise PNU is typically more common, complex and significant in CMOS type of sensors due to the complexity of the pixel array circuitry Sandoval Orozco et al. (2015). The noise pattern PRNU is produced by the variation of sensitivity to light of the individual pixels, due to the lack of homogeneity and impurities in the silicon chips, and to the imperfections resulting from the manufacturing of the sensor. In the case of videos, it may seem that estimating the pattern PRNU of a video camera from a video sequence is simpler than in the case of still images, due to the large number of frames available in a video. However, this is not true for two main reasons; first, the spatial resolution of videos is much lower than the spatial resolution of still images. Second, video frames generally have higher compression ratios than images compressed in JPEG format. One of the first works in which a camcorder fingerprint was used to iden- tify digital images was in Kurosawa et al. (1999). They proposed a method called CCD Fingerprint to identify a camcorder from videotaped images. They recommended using defective pixels and the dark current of CCD chips for camcorder identification. This approach is limited because thermal noise can only be removed with dark frames, and the dark current property is a weak signal that does not survive video compression. Over time it has been shown that the technique developed in Fridrich et al. (2006) that identifies digital images using non-uniform response noise PRNU provides a much more robust and reliable fingerprint. In Chen et al. (2007) it was determined that it is not possible to use a single frame of the video to identify its origin since the spatial resolution of the video is much lower than in the case of still images, and each frame is subjected to complex compression systems (MPEG- X, H26X and variants). Therefore, they demonstrated that by taking advantage of the temporal res- olution of the videos, in cases of low resolution (264 × 352 pixels) and with 10     only a 10 minutes video clip, it was possible to identify the source of the video device. The experiments were performed with 25 video cameras and showed that just 40 seconds of video is enough to have reliable results. If the video quality is decreased (the compression ratio is increased) and the spatial resolution is decreased, it is necessary to increase the video clip time in order to obtain reliable results. With videos in Internet LP format and a resolution of 264 × 352 and 150 kb/sec, good results are obtained for video clips with a duration of 10 minutes. The experiments carried out are limited because they do not take into account videos from mobile devices. In Chuang et al. (2011) the impact of compression on the identification of the source camera is explored using the non-uniformity of response to the photo PRNU extracted from compressed videos generated by mobile devices. They concluded that type I frames are more trustworthy (reliable) than type P frames when giving an accurate PRNU estimate. They also, however, ob- served that the hit rate increases considerably when using all the frames that make up a video. However, they considered that using all the frames of a video has a high computational complexity. Motivated by this observation and by the fact that type I frames of a video have different reliability than type P frames, they proposed a mechanism to reorder and assign weights to the frames of a video by assigning weights 2 : 1 to type I and P frames respectively. In this way, they showed that the lower the number of frames plus a suitable weight assignment the better the estimation PRNU. This technique definitely takes into account the effect of video compression when obtaining the sensor fingerprint. Although the proposed method takes into account how video compression affects the calculation of the sensor imperfec- tions footprint, later techniques such as the one proposed by Kouokam and Dirik (2019) show that it is necessary to use all types of frames (I, B, P) that a video has to improve the hit rate. In Garc´ıa Villalba et al. (2016) a digital video source identification scheme based on PRNU noise and SVM classifier was proposed. The classification of the videos was carried out by selecting those frames with a significant scene change using the color histogram function. A total of 81 features, which are the Wavelet components of the sensor, are used to train the SVM classifier with training videos. A total of 5 different devices from 5 different brands were used to train the SVM classifier. The results obtained show a success rate of 87% or 90%, depending on the video resolution. Since this technique uses a classifier SVM, it has the following limitation: you need to have a video of the same brand and model of the video you want to investigate, and this 11     is almost impossible nowadays due to the wide variety of models available in the market. Additionally, in the extracted frames, this technique does not take into account the effect that compression produces when improving the classification. In Altinisik et al. (2018), a procedure is described to extract the finger- print that produces the sensor noise PRNU in non-stabilized videos, taking into account the H.264 video compression standard. This technique has two main goals. On the one hand, this technique tries to eliminate the filtering procedure that is applied in the H.264 compression process. On the other hand, it also tries to select the blocks that best estimate the noise pattern PRNU. This brings us to the conclusion that the macroblocks that are en- coded using intra-frame prediction and the loop filter should both be avoided so as not to weaken the noise PRNU. Quite the opposite occurs in type B or P macroblocks, where the encoder performs the prediction assuming filtered blocks. Therefore, it is necessary to remove the loop filter on the decoder as it will not produce a successful rebuild. To compensate for this behavior, the decoding process must be modified to rebuild both a filtered and an un- filtered version of each macroblock. Filtered macroblocks should be used to reconstruct future macroblocks, and unfiltered ones should be used for fin- gerprint extraction. The experiments were performed with videos converted from 550 photographic images with various levels of compression. In Kouokam and Dirik (2019) they identified the source of videos coming from mobile devices and downloaded from YouTube by using PRNU and by studying the effect that the compression of the videos produces when estimating the PRNU fingerprint of a digital video. For each frame of the video, they looked for the blocks in which the PRNU noise had not been completely degraded due to the effects of the compression of the video. They proved that the PRNU noise pattern in a block survives compression if the DCT-AC coefficients of the prediction residue of the block are not all zero. Therefore, to estimate the noise of PRNU video, only those blocks that have at least a non-zero DCT-AC coefficient in frames I, P and B are used, since they still have, at least partially, a certain amount of noise of PRNU, which makes the block valid to identify the source of acquisition in digital videos coming from mobile devices and downloaded from YouTube. All the techniques seen in this section only take into account the effect produced by the compression of a video to calculate the PRNU footprint. However, they do not take into consideration that it is necessary to improve said PRNU footprint in order to improve the classification. This work not 12     only takes into account the problem that video compression causes when calculating the fingerprint produced by sensor imperfections, but also those techniques that improve the sensor fingerprint produced by the details of the scene. The most relevant techniques that show how to improve the fingerprint of the sensor applied to digital images are detailed below. Sensor imperfections are due to the lack of uniformity in the photographic response (PRNU), which contains important information about the sensor in terms of frequency content. This information makes it appropriate for different forensic applications of videos and digital images. The main incon- venience of existing methods for the PRNU extraction is the fact that the extracted PRNU fingerprint has fine details of the image, such as high fre- quency details (textures, edges). To fix this problem, it is necessary to apply some techniques to remove those details from the scenes in order to improve the quality of the PRNU fingerprint. As mentioned in the introduction section, one of the initial works in which the technique for the improvement of the sensor fingerprint was used was in Li (2010). It should be noted that this technique is based on the suppression of the content of high-frequency details from the scene. Following this hy- pothesis, Li (2010), developed five models and two of them yielded the best results applied to digital images. Later it was found that Lis improvement model Li (2010) also suppresses useful PRNU components Kang et al. (2011). In Gupta and Tiwari (2018), a pre-processing step is applied to PRNU extraction filters widely accepted within the scientific community for low- frequency and high-frequency components of the image separately. The best results are obtained when using the Mihcak filter. They call their tech- nique pMihack filter, this pMihcak filter contains the least amount of high- frequency details of the image. As seen in this section, in the case of digital videos, there are techniques which try to identify the origin of a video by applying the technique of PRNU sensor imperfections and how this affects video compression. On the other hand, in the literature we find that in the case of images, the PRNU fingerprint obtained is not sufficient if an improve- ment to that fingerprint is not applied. Therefore, in this study we suggest a technique that takes into account both approaches to identify the source of a digital video. The classification algorithm used in this work, combines a hierarchical clustering and a flat clustering for the separation of the groups. The use of the silhouette coefficient for the validation of the results has proved to yield good results since high TPRs were obtained.  To compare the convergence 13     rate of this algorithm with the rest of algorithms in the literature, you need to consider the references below: Giri and Bardhan (2014), Dubey et al. (2015), Tsao (2015), Yin et al. (2016), Kazemi et al. (2018), Duan et al. (2018), Hoseini S. et al. (2019), Gharaei et al. (2019a), Shah et al. (2020), Rabbani et al. (2020), Sarkar and Giri (2020), Giri and Masanta (2020), Sayyadi and Awasthi (2020), Shah et al. (2020), Gharaei et al. (2020), Awasthi and Omrani (2019), Gharaei et al. (2019b)  3. Technique Description To address the issue of identifying the source of a video, the most promis- ing methods take advantage of the unique noise traces that the camera sen- sors produce in the acquired videos. As observed in the previous section, to identify the source of a digital video produced by a mobile device, most of the proposed techniques only take into account the effects that compression produces in the calculation of the noise fingerprint of the said video Chen et al. (2007), Houten and Geradts (2009), Chuang et al. (2011) Altinisik et al. (2018). The main contribution of this work is to identify the source of digital video acquisition, which focuses on the impact of video compression to obtain the trace left by the sensor noise pattern. Conversely, the proposal takes into account the fact that the magnitude of the details of the scene that a video has tends to be much greater than the fingerprint generated by the sensor noise pattern. This makes the fingerprint less reliable and, consequently, it should be removed to improve the quality of the noise pattern. In addition, it must be considered that most of the techniques proposed in the scientific community consider improving the sensor noise fingerprint adapted to digital images Sandoval Orozco et al. (2015). There are few works that adapt the improvement of the fingerprint in the case of digital videos from mobile devices. This last point is the one addressed in the our research work. Regarding video clustering, this is done in real life scenarios using the standard correlation as a measure of similarity to achieve a correct classifi- cation of digital videos by device. In order to identify the source of digital videos, this proposal has taken into account the effect produced by the high compression ratios that a video contains due to the redundancy that a video contains. To obtain the fin- gerprint of the sensor noise pattern, the hypothesis proposed by Li (2010) 14  \n   has been considered. Here it is indicated that the images may be severely contaminated by the details of the scene. This proposal has also dealt with the particular case of digital videos. The overall scheme of the proposed technique is shown in Figure 3. The proposed algorithm is divided into 3 main phases that are detailed below:  \n Figure 3:  Diagram of the proposed algorithm   1. PRNU noise is stochastic in nature and unique to each sensor. Its high dimensionality and robustness in the processing make it an ideal candidate for forensic applications such as the identification of digital cameras. The extraction of the PRNU noise is done at the macroblocks level as proposed in Kouokam and Dirik (2019). This extraction of noise based on macroblocks takes into account that for each of the frames that make up the video, only those macroblocks are used in which the PRNU noise has not been completely degraded by the compression that a digital video contains are used. As it was demonstrated in the work Kouokam and Dirik (2019), the content of a decoded macrblock depends, to a large extent, on the in- verse transform DCT of the prediction residual of the block. Therefore the first step that must be performed to To estimate the PRNU fin- gerprint, is to decode the input video with the version of jm.16.1 used in Kouokam and Dirik (2019). The result is a XML file format that contains all the information of the macroblocks that make up the set of frames that belong to a digital video. Figure 4 you can see an extract from the XML file. The Picture label refers to a frame in the video. The MacroBlock tag corresponds to all macroblocks of size 16x16 that exist in the frame. The Position label indicates the start and end (x, y) coordinates of 15     \n  Figure 4:  Output file of the decoder stage of a digital video.  the macroblock within the frame. Finally, the Coeffs tag indicates the pixel value of the macroblock to be processed. The macroblocks-based noise extraction used in this work has consid- ered that for each of the frames contained in a video, only those mac- roblocks are used in which the noise PRNU has not been completely degraded by the compression it contains a digital video. In consideration of PRNU noise of a macroblock survives compression if the residual coefficients of the macroblock DCT-AC are not all zero, as proposed in Kouokam and Dirik (2019). Only those macroblocks that contain at least a non-zero coefficient DCT-AC will be used in any of the type I, B or P frames that make up the video, otherwise it is discarded the macroblocks. Therefore, a binary matrix will be created \n16  L. j 1, otherwise \nj j    with values 0 or 1, whose value is equal to 1 if the macroblocks contains at least a non-null coefficient DCT-AC and 0 in another case. The zeros in the mask mask of the frame indicate the location of pixels or macro bolts where noise estimation PRNU is not feasible. For each of the frames that make up the video, a matrix called mask Mj is obtained. Each of its elements at the location of the pixel (p; q) is calculated according to the Equation 1. M (p, q) =  0,  if the DCT-AC coefficients are all zero  (1)  Therefore, the fingerprint of the K camera and the video noise of the video Wj are calculated according to the equation 2 where it is obtained from the noise of the video and therefore the fingerprint of the camera. n j=1 WjIjMj K = L.n (I M )2 + L (2)  where Ij, is the decoded image of the j-th frame of the video, Wj is the noise PRNU estimated from Ij, Mj is the mask of the frame j − th, n is the number of frames of the video to identify and finally, L is an array with all its elements with a value of 1 whose dimension coincides with the resolution of the video to identify. It is necessary to use this matrix, to avoid division by zero in case Mj(p, q) = 0 ∀j. This matrix called mask is a three-dimensional matrix that includes the dimensions of the input video and another additional dimension, whose value is the number of frames the video has. That is to say, if the video has a dimension of 1920x1080 pixels and a total of 302 frames, then the dimension of the mask array Mj will be 1920x1080x320 elements. This matrix is filled with the number 1 if there is at least a coefficient other than 0 in its positions DCT- AC and with the data 0 in another case. As an example, in Figure 4 it can be seen that the macroblock number 0 contains the coefficient 0 in all positions DCT-AC of the macroblock. Therefore, the mask values for those 16 elements are all filled with the number 0, while the macroblock number 47 contains more than one non-zero data point in their positions DCT-AC, as visible in the second row of the Row tag that contains the coefficients (6, 6, 7, 6). In j=1 17  −1 + eK(i,j) −α ≤ K(i, j) < 0 \u0000\u0000    this case, the values of the mask matrix are filled with the coefficient 1. This step is computationally expensive due to the large amount of information that the algorithm must process. The calculations that it has made in algorithm with the experiments were carried out on a computer with a 9 − th generation processor of the Intel Core i7 processor family equipped with 6 cores. Likewise, as can be seen in Figure 4, not all type I frames survive compression. Therefore, it is necessary to use all types of frames (I, B and P) and not only the type I frames, as suggested in other literature proposals. 2. The next step is to improve the PRNU noise fingerprint calculated in the previous step. The details of the scene when representing the high- frequency components make its magnitude much greater than that of the noise pattern. For this reason, it is necessary to remove fragments from the high-frequency scene to improve the sensor noise fingerprint. Subsequently, in Li (2010) an approach is proposed to mitigate the in- fluence of scene details when calculating the noise pattern. In this way Hence, the hit rate is improved when identifying the device on which the proposed algorithm is based. The hypothesis proposed in Li (2010) suggests that an improved fingerprint Kenh can be obtained by assign- ing less significant weighting factors to the strong components of the signal in the domain of the wavelet transform to attenuate the inter- ference of scene details. To test the hypothesis proposed in Li (2010), five models identified as enhancer models (2 - 6) were proposed. Re- sults are detailed in the Experiments section. The enhancer model that obtained the best results in this work was the model called Non-linear Exponential Transformation (Model 3), whose equation is detailed in Equation 3.   Kenh(i, j) = 1 − e−K(i,j) 0 ≤ K(i, j) ≤ α \u0000 (1 − e−α)eα−K(i,j) K(i, j) > α \u0000\u0000 (−1 + e−α)eα+K(i,j) K(i, j) < −α   (3)  where the value of the parameter α determines the attenuation of the noise. In order to decide the value of α, various experiments were carried out on videos with different values of α selected at random and 18  \n   that dealt with the majority of possible cases, therefore the following values α ∈ [2, 5, 7, 20, 50]. The value chosen was 20 as it stated to be a prototype value for all the proposed enhancer models and with which desirable results were obtained in the enhancer models used. The output of this step is a two-dimensional matrix containing the improved noises for each of the input videos under investigation. The general scheme of block-based noise extraction together with the enhancement function is shown in Figure 5.  \n Figure 5:  Enhanced algorithm structure.  For each of the improved PRNU noises calculated in the previous step (Kenh1 , ..., Kenhn ), the correlation value is obtained using the result of Equation 4. 19  \u0000 i    corr(n , n ) = (ni − ni) 0 (nj − nj)  (4) i j ||n − n || · ||n − n || i j j  where ni y nj represent the mean of the vector, ni0nj is the dot product of two vectors, and ||ni|| is the norm L2. Since the sensor noise pattern is a two-dimensional matrix, prior to the application of the correlation function, a transformation of the matrix to a one-dimensional vector is performed. 3. Finally, clustering using an unsupervised agglomerative clustering algo- rithm proposed in some works such as Caldelli et al. (2010), Garc´ıa Vil- lalba et al. (2015). To determine the similarity between videos that be- long to the same device, there are distance measures such as: Euclidean distance, Manhattan or Chebychev distance, among others. One of the measures widely used in identifying the source of digital images can be found Caldelli et al. (2010), Garc´ıa Villalba et al. (2015). To decide how to calculate the correlation value iteratively, it is nec- essary to define a measure of similarity between the clusters, so that similar clusters are cataloged before different clusters. To carry out this union, a linkage criteria function is performed. This function measures the similarity between two clusters in which at least one of them is made up of more than one sample or video. For this work, tests were carried out with various linkage criteria functions that are simple, av- erage and complete criteria. After conducting several experiments, the linkage criteria function that showed the best results was the average linkage criteria, as in the case of the works related to images Caldelli et al. (2010) and Garc´ıa Villalba et al. (2015). The Equation 5 shows the average linkage criteria function between two groups u and v.   d(u, v) = i∈u,j∈v corr(Kenhi, Kenhj )   |u||v|  (5)  where, |u|and |v|is the cardinality of the groups of u and v respectively. The value of corr(Kenh1 , Kenhn ) determines the similarity that exists between two objects of the same group, reaching its highest value the more similar Kenhi and Kenhj . The value of corr is calculated in the Equation 4. 20  \u0000    In traditional hierarchical clustering algorithms, the final result is that all the elements belong to the same group, but in this work we need each group to represent a model at the end of the execution. In order to validate the groups, the silhouette coefficient has been used as a group validation measure. For each observation x there are two measures that are:  • Cohesion a(x): it is obtained as the average distance of x to all the points in the same class. • Separation b(x): measures the average distance of a point from one of the groups to all other nearby groups. The most used separation is the average distance between x and all the elements in the closest group, although other measures are also used to value the separation.  The coefficient s(x) takes the values in the range [1,1]], where1 corre- sponds to a bad choice of the number of classes and 1 indicates well defined classes. The silhouette coefficient for x is illustrated in the Equation 6.  b(x) − a(x) sx = max{a(x), b(x)} (6) and for all the clustering is determined by the Equation 7.  SC = 1 s(x) (7) n x  where n is the number of observations. To assess the quality of the clustering resulting from the clustering al- gorithm, the precision, the curve ROC and the area AUC are described. The complexity of the algorithm increases as the resolution of the video increases. In the case of a 1920x1080 resolution, a total of 2.73.600 it- erations will be necessary, resulting in a high computational cost. The time complexity is O(n2). 21     4. Experiments and Results This section details the experiments carried out regarding the identifi- cation of the source of digital videos. The first experiment involves finding out which enhancer function of those proposed in Li (2010) is the one that obtains the best results by varying the parameter of α. The following exper- iment is performed on a different data set than the first to prove that the choice of the enhancement model and the parameter α are not linked to a specific video set. This second set of videos used takes into account that the videos have been downloaded from social media platforms such as YouTube or WhatsApp. We must take into account that when a video is uploaded or downloaded from this type of platform, the video undergoes another com- pression process limiting the quality of the extracted sensor fingerprint. To evaluate the effectiveness of the smartphone device source identifica- tion algorithm, a subset of the public repository ACID Hosler et al. (2019) has been used. In this work, models from smartphone devices have been selected because that is the case study that has been analysed in this paper. Table 1 shows the summary of the video set used.  Table 1: Composition of the ACID video set  Brand Model ID #Videos Resolution Huawei Honor 6X Pixel 2 M12 10 1920x1080p LG X Charge M17 10 1920x1080p  Galaxy J7 Pro M27 10 1920x1080p  Galaxy S3 M28 10 1920x1080p Samsung Galaxy S5 Galaxy S7 M29 M30 10 10 1920x1080p 1920x1080p  Galaxy Tab A M31 10 1920x1080p  J5-6 M32 10 1920x1080p    4.1. Influence of the parameter α on the TPR The first experiment that has been conducted involves testing all the enhancer models proposed in Li (2010). Table 2 shows the summary of the experiments carried out. It can be seen that when α = 20, all the enhancer models (γ2, γ3, γ4, γ5, γ6) have the highest success rate, reaching over 81% in almost all the proposed models. The model with the worst average rate is γ4, function that never exceeds 25% in TPR. 22     Table 2: Average TPR by model and with different values of α  Enhancer Model Values of α TPR Average 2 5 7 20 50 γ2 72% 70% 70% 70% 60% 68,4% γ3 56% 81,25% 70% 81,25% 81,25% 73,95% γ4 25% 20% 22 % 25 % 20% 20,44% γ5 55% 65% 56,4% 81,25% 55,4% 62,61% γ6 60% 69% 81,25% 81,25% 81,25% 74,75%   In Tables 3, 4 and 5 respectively, we can see, in detail, the result of clustering the γ2, γ3 and γ6 functions, since they are the models that have achieved the best average rate when α = 20, as can be seen in Table 2. To calculate the completeness or TPR of each group, it is necessary to see in the group the smartphone model that contains the largest number of videos compared to the total number of videos per model, since that is the predominant model within the group. Next, the percentage of videos that have been correctly classified for that model within a group is calculated. The vast majority of cases can be evaluated as a group associated with one or more devices. In some cases, there are groups in which no device prevails, as can be seen in 3. Its TPR is considered to be 0. Finally, the percentage of videos that have been appropriately classified for that device in that group is calculated. For the γ2 with α = 20 function, it can be observed that the groups G8 have TPR = 0, because there is no distinction between the M17 and M32 models respectively. It can be seen that the γ3 with α = 20 function has a higher average TPR than the γ2 with α = 20 function. No group that has obtained this average by implementing this enhancer has a TPR = 0, as was observed in the γ2 with α = 20 function that had a group with TPR = 0. The M12, M27, M28, M29 and M31 devices have the same behavior in both models and they achieve a value of TPR = 100%. It should be noted that γ2 and γ6 functions with α = 20 value generate the same behavior and the same number of clusters. In order to measure the performance of the clustering proposed in this technique, a curve ROC has been used that shows the productivity of the classification model at all clustering thresholds, in the case of γ3 with α = 20 function as shown in Figure 6. 23     Table 3: Clustering result with γ2 with α = 20 function  Brand-Model G1 G2 G3 G4 G5 G6 G7 G8 G9 M17 8 0 0 0 0 0 0 2 0 M32 0 5 0 0 0 0 0 2 3 M31 0 0 10 0 0 0 0 0 0 M12 0 0 0 10 0 0 0 0 0 M29 0 0 0 0 10 0 0 0 0 M28 0 0 0 0 0 10 0 0 0 M27 0 0 0 0 0 0 10 0 0  Table 4: Clustering result with γ3 with α = 20 function  Brand-Model G1 G2 G3 G4 G5 G6 G7 G8 G9 M32 7 0 0 0 0 0 0 3 0 M27 0 10 0 0 0 0 0 0 0 M31 0 0 10 0 0 0 0 0 0 M28 0 0 0 10 0 0 0 0 0 M17 0 0 0 0 8 0 0 0 2 M29 0 0 0 0 0 10 0 0 0 M12 0 0 0 0 0 0 10 0 0  Table 5: Clustering result with γ6 with α = 20 function  Brand-Model G1 G2 G3 G4 G5 G6 G7 G8 G9 M32 7 0 0 0 0 0 0 3 0 M27 0 10 0 0 0 0 0 0 0 M31 0 0 10 0 0 0 0 0 0 M28 0 0 0 10 0 0 0 0 0 M17 0 0 0 0 8 0 0 0 2 M29 0 0 0 0 0 10 0 0 0 M12 0 0 0 0 0 0 10 0 0 24      \n  Figure 6: ROC curve for γ3 with α = 20 function for ACID video dataset  The value of AUC for the area of the curve ROC in Figure 6 is 0.929980.  4.2. Comparison between γ3 with α = 20 function and block-based method In this section we are going to compare the block-based Kouokam and Dirik (2019) proposal where we only look at the effect that compression has when calculating the PRNU fingerprint - with the contribution proposed in this work, where in addition to taking into account the block-based method, the function has been included enhancer that gave the best results in the previous section, that is, γ3 with α = 20 function. On the one hand, if we use the videos in Table 1, the same TPR results are obtained, generating the same groups, for the case of the function without improvement (the one proposed in block-based) as the function proposed by this work. The result of both proposals can be seen in Table 4. On the other hand, if we use of videos in Table 6, where the M00 model, an Apple device, is added, the clusters, seen in Tables 7 and 8, are obtained. In Table 8, it can be observed that, if we do not apply only the block-based method, groups G8 and G9 have a TPR = 0, because there is a mix between the M00, M17 and M32 models in group G8. Additionally, it must be taken into account that the Apple brand M00 model cannot be identified with this proposal. \n25     Table 6: Composition of the ACID video set with M00 model is included  Brand Model ID #Videos Resolution Apple iPhone 8 Plus M00 8 1920x1080p Huawei Honor 6X Pixel 2 M12 10 1920x1080p LG X Charge M17 10 1920x1080p  Galaxy J7 Pro M27 10 1920x1080p  Galaxy S3 M28 10 1920x1080p Samsung Galaxy S5 Galaxy S7 M29 M30 10 10 1920x1080p 1920x1080p  Galaxy Tab A M31 10 1920x1080p  J5-6 M32 10 1920x1080p    Table 7: Clustering result with γ3 with α = 20 function  Brand-Model G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 Average TPR M32 7 0 0 0 0 0 0 3 0 0  M27 0 10 0 0 0 0 0 0 0 0 M31 0 0 10 0 0 0 0 0 0 0 M28 0 0 0 10 0 0 0 0 0 0 M17 0 0 0 0 8 0 0 0 0 2 M29 0 0 0 0 0 10 0 0 0 0 M12 0 0 0 0 0 0 10 0 0 0 M00 0 0 0 0 0 0 0 3 5 0 TPR-Group 70 100 100 100 80 100 100 0 50 20 72%    Table 8: Clustering result with block-based method  Brand-Model G1 G2 G3 G4 G5 G6 G7 G8 G9 Average TPR M32 5 0 0 0 0 0 0 2 3  M27 0 10 0 0 0 0 0 0 0 M31 0 0 10 0 0 0 0 0 0 M28 0 0 0 10 0 0 0 0 0 M17 0 0 0 0 7 0 0 3 0 M29 0 0 0 0 0 10 0 0 0 M12 0 0 0 0 0 0 10 0 0 M00 0 0 0 0 0 0 0 2 6 TPR-Group 50 100 100 100 70 100 100 0 0 68% 26     Therefore, we can conclude that the technique proposed in this contri- bution improves or equals the average TPR, as compared to only using the block-based method, but does not worsen the TPR obtained with the work done with the block-based method.  4.3. Average TPR with Videos from Youtube and WhatsApp Social Platforms To carry out this experiment, another set of videos available in the scien- tific community called VISION Shullani et al. (2017) has been used. Unlike the data set used in the previous experiment, this set of videos contains mod- els of mobile devices that have been uploaded to the YouTube and WhatsApp social media platforms, in addition to videos from mobile devices. A forensic analyst cannot always access the mobile device that gener- ated the multimedia video to identify the source of the video, since most of the videos that show a crime are uploaded on the Internet. To tackle this problem, it is necessary to implement techniques that materialize the identi- fication of videos from social platforms and verify that the fingerprint PRNU remains reliable in this type of situations. Also in an open stage, it is not very common to have the same number of videos for each device be identified. For this reason, experiments were carried out where the video sets of each of the models have an asymmetric distribution. These experiments were able to determine if the algorithm adapts to real life scenarios. Table 9 shows the set of videos that has been used for this experiment. Table 10 summarizes the experimental conditions used in this experiment. As for the enhancer model to be used, the γ3 with α = 20 function has been chosen. This model gave us the best results in the previous experiment. As such, when testing this model in another data set, it will be shown that these values do not depend on the specific dataset. As can be seen in Table 11, the number of videos per device is varied, and yet all the videos have been classified in a single group with the exception of device D03, which has needed 2 groups to be classified correctly. Figure 7 shows the ROC curve for this dataset. 27     Table 9: Composition of the VISION dataset sample  ID Brand Resolution # Videos D01 Samsung Galaxy S3 Mini 1280x720 27 D03 Huawei P9 1920x 1080 11 D04 LG D290 800x480 7 D07 Lenovo P70A 1280x720 4 D09 Apple iPhone 4 1280x720 3 D13 Apple iPad2 1280x720 3 D16 Huawei P9Lite 1920x1080 3 D17 Microsoft Lumia 640 1920x1080 3 D21 Wiko Ridge 4G 1920x1080 3 D22 Samsung Galaxy Trend Plus 1280x720 3 D24 Xiaomi RedmiNote3 1920x1080 3 D25 OnePlus A3000 1920x1080 3 D33 Huawei Ascend 1280x720 3  Table 10: Experimental conditions  Parameter Value Minimum number of videos from Social Platforms (YouTube, WhatsApp) 3 Enhancer model γ3 Value of α 20   The value of the curve AUC for the area of the curve ROC in Figure 6 is 0.956802. As seen in the experiments section, the highest success rate is achieved with the improvement model γ3 with α = 20. It is demonstrated that it is necessary to apply techniques that improve the PRNU fingerprint of the sensor in the case of digital videos to improve the hit rate. 28  \n   Table 11: TPR with devices from social platforms  ID G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 D01 27 0 0 0 0 0 0 0 0 0 0 0 0 0 D03 0 2 9 0 0 0 0 0 0 0 0 0 0 0 D04 0 0 0 7 0 0 0 0 0 0 0 0 0 0 D07 0 0 0 0 4 0 0 0 0 0 0 0 0 0 D09 0 0 0 0 0 3 0 0 0 0 0 0 0 0 D13 0 0 0 0 0 0 3 0 0 0 0 0 0 0 D16 0 0 0 0 0 0 0 3 0 0 0 0 0 0 D17 0 0 0 0 0 0 0 0 3 0 0 0 0 0 D21 0 0 0 0 0 0 0 0 0 3 0 0 0 0 D22 0 0 0 0 0 0 0 0 0 0 3 0 0 0 D24 0 0 0 0 0 0 0 0 0 0 0 3 0 0 D25 0 0 0 0 0 0 0 0 0 0 0 0 3 0 D33 0 0 0 0 0 0 0 0 0 0 0 0 0 3  \n Figure 7: ROC curve the γ3 with α = 20 function for VISION video dataset 29     5. Conclusions and Future Work The general conclusion is that the presented technique is valid and achieves good results. The algorithm that extracts the sensor fingerprint takes into ac- count the effect of video compression as well as the removal of high-frequency details contained in the scenes of the frames to improve the hit rate. It should be noted that in the most recent literature, techniques can be found that im- prove the fingerprint pattern of sensor imperfections applied to digital images but not to digital videos. This work shows that it is necessary to improve the sensor footprint also in the case of digital videos. The proposed clustering algorithm is based on the combination of a hi- erarchical and a flat clustering. Most of the groups that have been obtained are pure, that is, they are made up o a single mobile device. The percentage of success when using a smaller clipping size reduces the hit rate. But on the other hand, the calculation time is smaller, so a balance should be found between the clipping size and the calculation time. The unsupervised grouping technique used is carried out on a wide variety of videos, that is, videos from mobile devices (native videos) and videos downloaded from the main video- sharing platforms such as WhatsApp and YouTube. Results higher than 90% in the hit rate were achieved. Therefore, this technique can be used in real scenarios because the analyst does not need to have access to the mobile device that generated the video since the technique also works with videos downloaded from the main multimedia video sharing platforms such as WhatsApp and YouTube. As future work, the most recent works in the literature dealing with digital images such as Gupta and Tiwari (2018) and Kang et al. (2011) could be applied to videos to see if better results are achieved than those obtained with the Li (2010) technique. Finally, another future line of work that is proposed is associated with the high computational cost of the proposed technique. It is necessary to use metamodels or surrogate models that reduce this computational complexity and help increase the number of videos used in the experiments such as Alizadeh et al. (2019) and Alizadeh et al. (2020). 30     Acknowledgements This work is promoted under the specialty of Industrial PhD in collabora- tion with Securitas Direct. This work has also received funding from THEIA (Techniques for Integrity and Authentication of Multimedia Files of Mobile Devices) UCM project (FEI-EU-19-04).  References Akshatha, K., Karunakar, A., Anitha, H., Raghavendra, U., Shetty, D.. Digital Camera Identification Using PRNU: A Feature Based Approach. Digital Investigation 2016;19:69–77. Alizadeh, R., Allen, J.K., Mistree, F.. Managing Computational Complex- ity Using Surrogate Models: A Critical Review. Research in Engineering Design 2020;31(3):275–298. Alizadeh, R., Jia, L., Nellippallil, A.B., Wang, G., Hao, J., Allen, J.K., Mistree, F.. Ensemble of Surrogates and Cross-Validation for Rapid and Accurate Predictions Using Small Data Sets. AI EDAM 2019;33(4):484– 501. Altinisik,  E., Tasdemir,  K., Sencar,  H.T.. Extracting PRNU Noise from H.264 Coded Videos. In: 2018 26th European signal processing conference (EUSIPCO). IEEE; 2018. p. 1367–1371. Awasthi, A., Omrani, H.. A goal-Oriented Approach Based on Fuzzy Axiomatic Design for Sustainable Mobility Project Selection. International Journal of Systems Science: Operations & Logistics 2019;6(1):86–98. Caldelli, R., Amerini, I., Picchioni, F., Innocenti, M.. Fast Image Clus- tering of Unknown Source Images. In: 2010 IEEE International Workshop on Information Forensics and Security. IEEE; 2010. p. 1–5. Chen, M., Fridrich, J., Goljan, M., Luk´aˇs, J.. Source Digital Camcorder Identification using Sensor Photo Response Non-Uniformity. In: Electronic Imaging. International Society for Optics and Photonics; 2007. p. 65051G– 65051G. 31     Chuang, W., Su, H., Wu, M.. Exploring Compression Effects for Improved Source Camera Identification Using Strongly Compressed Video. In: Image Processing (ICIP), 2011 18th IEEE International Conference on. Brussels, Belgium; 2011. p. 1953–1956. Cis:2022. Cisco Visual Networking Index: Global Mobile Data Traffic Fore- cast Update, 20172022 White Paper. Duan, C., Deng, C., Gharaei, A., Wu, J., Wang, B.. Selective Main- tenance Scheduling Under Stochastic Maintenance Quality with Multi- ple Maintenance Actions. International Journal of Production Research 2018;56(23):7160–7178. Dubey,  R., Gunasekaran,  A., Sushil, , S.,  T..  Building Theory of Sustainable Manufacturing Using Total Interpretive Structural Mod- elling. International Journal of Systems Science: Operations & Logistics 2015;2(4):231–247. Fridrich, J., Lukas, J., Goljan, M.. Digital Camera Identification from Sensor Noise. IEEE Transactions on Information Security and Forensics 2006;1(2):205–214. Garc´ıa Villalba,  L.J., Sandoval Orozco,  A.L., Ramos Lpez,  R., Hern´andez Castro,  J.C..  Identification of Smartphone Brand and Model via Forensic Video Analysis. Expert Systems with Applications 2016;55(15):59–69. Garc´ıa Villalba, L.J., Sandoval Orozco, A.L., Rosales Corripio, J.. Smartphone Image Clustering. Expert Systems with Applications 2015;42(4):1927–1940. Gharaei,  A., Hoseini S.,  S.A., Karimi,  M..  Modelling and Optimal Lot-Sizing of the Replenishments in Constrained, Multi-Product and Bi- Objective EPQ Models with Defective Products: Generalised Cross De- composition. International Journal of Systems Science: Operations & Lo- gistics 2020;7(3):262–274. Gharaei, A., Hoseini S., S.A., Karimi, M., Pourjavad, E., Amjadian, A.. An Integrated Stochastic EPQ Model Under Quality and Green Policies: Generalised Cross Decomposition Under the Separability Approach. Inter- national Journal of Systems Science: Operations & Logistics 2019a;:1–13. 32     Gharaei, A., Karimi, M., Hoseini S., S.A.. Joint Economic Lot-Sizing in Multi-Product Multi-Level Integrated Supply Chains: Generalized Ben- ders Decomposition. International Journal of Systems Science: Operations & Logistics 2019b;:1–17. Giri, B., Bardhan, S.. Coordinating a Supply Chain with Backup Supplier Through Buyback Contract under Supply Disruption and Uncertain De- mand. International Journal of Systems Science: Operations & Logistics 2014;1(4):193–204. Giri, B., Masanta, M.. Developing a Closed-Loop Supply Chain Model with Price and Quality Dependent Demand and Learning in Production in a Stochastic Environment. International Journal of Systems Science: Operations & Logistics 2020;7(2):147–163. Gloe, T., Fischer, A., Kirchner, M.. Forensic Analysis of Video File Formats. Digital Investigation 2014;11(Supplement 1):68–76. Gupta, B., Tiwari, M.. Improving Source Camera Identification Perfor- mance Using DCT Based Image Frequency Components Dependent Sensor Pattern Noise Extraction Method. Digital Investigation 2018;24:121–127. Hoseini S., S.A., Gharaei, A., Karimi, M.. Modelling and Optimal Lot- Sizing of Integrated Multi-Level Multi-Wholesaler Supply Chains Under the Shortage and Limited Warehouse Space: Generalised Outer Approxi- mation. International Journal of Systems Science: Operations & Logistics 2019;6(3):237–257. Hosler, B.C., Zhao, X., Mayer, O., C., C., S., J.A., Stamm, M.C.. The Video Authentication and Camera Identification Database: A New Database for Video Forensics. IEEE Access 2019;7:76937–76948. Houten, V.W., Geradts, Z.. Source video camera identification for mul- tiply compressed videos originating from youtube. Digital Investigation 2009;6(1-2):48–60. Kang, X., Li, Y., Qu, Z., Huang, J.. Enhancing Source Camera Identifi- cation Performance with a Camera Reference Phase Sensor Pattern Noise. IEEE Transactions on Information Forensics and Security 2011;7(2):393– 402. 33     Kazemi, N., Abdul-Rashid, S.H., Ghazilla, R., Shekarian, E., Zanoni, S.. Economic Order Quantity Models for Items with Imperfect Quality and Emission Considerations. International Journal of Systems Science: Operations & Logistics 2018;5(2):99–115. Kouokam, E.K., Dirik, A.. PRNU-Based Source Device Attribution for YouTube Videos. Digital Investigation 2019;29:91–100. Kurosawa, K., Kuroki, K., Saitoh, N.. CCD Fingerprint Method- Identification of a Video Camera from Videotaped Images. In: Proceedings 1999 International Conference on Image Processing (Cat. 99CH36348). IEEE; volume 3; 1999. p. 537–540. Li, C.. Source Camera Identification Using Enhanced Sensor Pattern Noise. IEEE Transactions on Information Forensics and Security 2010;5(2):280– 287. L´opez, R.R., Luengo, E.A., Orozco, A.S., Villalba, L.G.. Digital Video Source Identification Based on Containers Structure Analysis. IEEE Access 2020;8:36363–36375. Lukas, J., Fridrich, J., Goljan, M.. Digital Camera Identification from Sensor Pattern Noise. IEEE Transactions on Information Forensics and Security 2006;1(2):205–214. Meij, C., Geradts, Z.. Source Camera Identification Using Photo Response Non-Uniformity on WhatsApp. Digital Investigation 2018;24:142–154. Rabbani, M., Hosseini-Mokhallesun, S.A.A., Ordibazar, A.H., Farrokhi- Asl, H.. A Hybrid Robust Possibilistic Approach for a Sustainable Supply Chain Location-Allocation Network Design. International Journal of Sys- tems Science: Operations & Logistics 2020;7(1):60–75. Sandoval Orozco, A.L., Arenas Gonzlez, D.M., Rosales Corripio, J., Garca Villalba, L.J., Hernandez-Castro, J.C.. Techniques for Source Camera Identification. In: Proceedings of the 6th International Confer- ence on Information Technology. Amman, Jordan; 2013. p. 1–9. Sandoval Orozco, A.L., Garca Villalba, L.J., Arenas Gonzlez, D.M., Ros- ales Corripio,  J., Hernandez-Castro,  J.C., Gibson,  S.J..  Smartphone 34     Image Acquisition Forensics Using Sensor Fingerprint. IET Computer Vi- sion 2015;9(5):723–731. Sarkar, S., Giri, B.. Stochastic Supply Chain Model with Imperfect Pro- duction and Controllable Defective Rate. International Journal of Systems Science: Operations & Logistics 2020;7(2):133–146. Sayyadi,  R., Awasthi,  A..  An Integrated Approach Based on Sys- tem Dynamics and ANP for Evaluating Sustainable Transportation Poli- cies. International Journal of Systems Science: Operations & Logistics 2020;7(2):182–191. Shah, N.H., Chaudhari, U., C´ardenas-Barr´on, L.E.. Integrating Credit and Replenishment Policies for Deteriorating Items Under Quadratic De- mand in a Three Echelon Supply Chain. International Journal of Systems Science: Operations & Logistics 2020;7(1):34–45. Shullani, D., Fontani, M., Iuliani, M., Alshaya, O., Piva, A.. VISION: a Video and Image Dataset for Source Identification. EURASIP Journal on Information Security 2017;2017(15):15. Tsao, Y.. Design of a Carbon-Efficient Supply-Chain Network under Trade Credits. International Journal of Systems Science: Operations & Logistics 2015;2(3):177–186. Yin, S., Nishi, T., Zhang, G.. A Game Theoretic Model for Coordination of Single Manufacturer and Multiple Suppliers with Quality Variations under Uncertain Demands. International Journal of Systems Science: Operations & Logistics 2016;3(2):79–91. "    },    {        "title": "2402.06672v1.Erratum_to__Computation_of_maximal_projection_constants_.pdf",        "content": "arXiv:2402.06672v1  [math.FA]  7 Feb 2024ERRATUM TO “COMPUTATION OF MAXIMAL\nPROJECTION CONSTANTS”\nGIULIANO BASSO\nAbstract. This is an erratum to the article: “Computation of maximal pr ojection con-\nstants” (J. Funct. Anal., 277). The statement of Lemma 3.1(2 ) of that paper is incorrect.\nAs a consequence of this the proof of Theorem 1.4 is incomplet e. In this erratum we prove a\ncorrected version of Lemma 3.1 and explain why, with this wea ker result, our original strat-\negy for proving Theorem 1.4 no longer works. The other result s of the article, in particular\nthe alternative proof of Gr¨ unbaum’s conjecture, do not dep end on Lemma 3.1 and are thus\nnot affected by this error.\nThe second item of Lemma 3.1 in [Bas19] is false in general. Th is error does not affect the\nmain body of the article. However, Lemma 3.1 was the main tool we used in our proofs of\nTheorems1.4and3.2. Asitturnsout, theproofstrategyused in[Bas19]toproveTheorem1.4\ncannot possibly lead to the claimed result. We will explain t his in detail below. Theorem 1.4\nis therefore still an open question, which can be formulated as follows.\nQuestion A. Letnbe a positive integer. Is it true that there exists an integer d > nand an\nn-dimensional linear subspace E⊂ℓd\n∞such that Ehas maximal projection constant amongst\nalln-dimensional Banach spaces? If yes, what is the value of the sm allest possible such d?\nWe now proceed by stating a corrected version of Lemma 3.1 and explain why the proof\nstrategy used in [Bas19] is not sufficient to answer Question A . In the following, we use the\nnotation from [Bas19, Section 3]. The corrected version of L emma 3.1 reads as follows.\nLemma B. The following holds true in every Banach space E.\n(1)If there exist closed linear subspaces V,U⊂Esuch that Vis finite-dimensional and\nE=V⊕U, then\nE∗=V0⊕U0,\nanddim(U0) = dim( V), as well as (V0)0=Vand(U0)0=U.\n(2)If there exist weak-star closed linear subspaces F,G⊂E∗such that Fis finite-\ndimensional and E∗=F⊕G, then\nE=F0⊕G0,\nanddim(G0) = dim( F), as well as (F0)0=Fand(G0)0=G.\n(3)If there exist closed linear subspaces V,U⊂Esuch that Vis finite-dimensional and\nE=V⊕U, then\n/bardblPU\nV/bardbl=/bardblPV0\nU0/bardbl.\n12 GIULIANO BASSO\nProof.\n(1) Since VandUare closed, it follows that ( V0)0=Vand (U0)0=U, see e.g. [Rud91,\nTheorem 4.7]. Moreover, VandE/Uare isomorphic as vector spaces, and thus using\nthatVis finite-dimensional and ( E/U)∗=U0, we find that dim( U0) = dim( V). To\nfinish the proof we show in the following that E∗=V0⊕U0. We abbreviate F=V0\nandG=U0. Letℓ∈E∗. ThenℓF:=ℓ◦PV\nUandℓG:=ℓ◦PU\nVbelong to FandG,\nrespectively. Moreover, if x=v+uwithv∈Vandu∈U, then\nℓF(x)+ℓG(x) =ℓ(u)+ℓ(v) =ℓ(u+v) =ℓ(x),\nand therefore F+G=E∗. Ifℓ∈F∩G, thenℓ(v) = 0 and ℓ(u) = 0 for all v∈Vand\nu∈U. SinceE=V⊕U, this implies that ℓ= 0. Hence, E∗=F⊕G, as desired.\n(2) It is readily verified that F0∩G0={0}. In the following we show that F0⊕G0=E.\nSinceFandGareweak-star closed, it follows directly fromtheHahn-Ban ach theorem\nthat\n(F0)0=Fand (G0)0=G,\nsee e.g. [Rud91, Theorem 4.7]. Using that ( E/F0)∗= (F0)0=Fwe find that E/F0\nis finite-dimensional. Let π:E→E/F0denote the quotient map. Since G0+F0=\nπ−1(π(G0+F0)), it follows that G0+F0is a closed linear subspace of E.\nNow, suppose ℓ∈E∗vanisheson G0+F0. Letℓ=ℓF+ℓGdenotethedecomposition\ninduced by E∗=F⊕G. For every x∈F0we have ℓ(x) = 0 and ℓF(x) = 0, and\nconsequently, ℓG(x) = 0. Thisimplies that ℓG∈(F0)0=F. But, by definition ℓG∈G\nand therefore ℓG= 0. Analogously, we can deduce that ℓF= 0 and thereby ℓ= 0.\nHence, as F0+G0is a closed linear subspace of E, it follows that F0+G0=E, as\ndesired.\n(3) Given x∈Eletx=xV+xUbe the unique decomposition induced by V⊕U.\nWe abbreviate F=V0andG=U0and forℓ∈E∗we write ℓ=ℓF+ℓGfor the\ndecomposition induced by E∗=F⊕G. Moreover, we write S(E) for the unit sphere\nof a Banach space. We have\n/bardblPU\nV/bardbl= sup\nx∈S(E)/bardblxV/bardbl= sup\nx∈S(E)sup\nℓ∈S(E∗)ℓG(xV)\nand also\n/bardblPF\nG/bardbl= sup\nℓ∈S(E∗)/bardblℓG/bardbl= sup\nℓ∈S(E∗)sup\nx∈S(E)ℓG(xV).\nHence the desired equality follows.\n/square\nOriginally, the second item of the lemma only required that Gis closed and not necessarily\nweak-star closed. However, with this weaker assumption the item becomes a false statement.\nThe following counterexample is due to T. Kobos, to whom I am g rateful for making me\naware of this issue.ERRATUM TO “COMPUTATION OF MAXIMAL PROJECTION CONSTANTS” 3\nExample C. LetF⊂ℓ∞be the two-dimensional linear subspace spanned by the vecto rs\nf1= (1,0,1,1,...) and f2= (0,1,1,1,...)\nand letG⊂ℓ∞by the weak-star closed subset G= ker(π1)∩ker(π2), where πi:ℓ∞→R\ndenotes the projection onto the ith coordinate. Clearly, ℓ∞=F⊕G. We know from the\nsecond item of Lemma B that ℓ1=F0⊕G0. We abbreviate V:= (F0)0andU:= (G0)0. In\nthe following we show that V⊕U/ne}ationslash=c0and thus the statement of Lemma 3.1(2) in [Bas19]\nis incorrect.\nNotice that for i≥3 any of the vectors a(i) = (1,1,0,...,0,−1,0,0,...), where the i-th\ncoordinate is equal to −1, is contained in F0. Hence, if x= (x1,x2,x3,...) is contained in V,\nthen necessarily x1+x2=xifor alli≥3. Butx∈c0and thus Vis equal the one-dimensional\nlinear subspace of c0spanned by the vector (1 ,−1,0,0,...). Clearly, G0⊂ℓ1is equal to the\ntwo-dimensional subspace of ℓ1spanned by the vectors (1 ,0,0,0,...) and (0,1,0,0,...). This\nimplies that U= ker(π1|c0)∩ker(π2|c0). Hence, for example, the vector (1 ,0,0,...) is not\ncontained in V⊕Uand thus V⊕U/ne}ationslash=c0.\nIn [Bas19], Lemma 3.1 was used in an essential way to prove The orem 3.2. This theorem\nclaimed that if Xis a Banach space and F⊂X∗∗a finite-dimensional subspace, then the\npre-pre-annihilator V= (F0)0has the property that\n(1.1) λ(V,X) =λ(F,X∗∗),\nwhereλdenotes the relative projection constant. However, using t he example from above, it\nis not difficult to show that this claimed equality cannot be tr ue in general. Let X=c0and\nconsider the linear subspace F⊂X∗∗=ℓ∞from Example C. We know that V:= (F0)0is\n1-dimensional and thus λ(V,X) = 1. But Fis isometric to R2equipped with the norm\n/bardbl(x,y)/bardbl= max{|x|,|y|,|x+y|}\nand thus in particular it is not isometric to ℓ2\n∞. Hence, λ(F,X∗∗)>1 and the claimed\nequality (1.1) does not hold.\nThe error is caused precisely due to the fact that for a given F⊂X∗∗, the pre-annihilator\nF0⊂X∗is in general not necessarily weak-star closed. A modified ve rsion of Theorem 3.2\nreads as follows.\nTheorem D. LetXbe a Banach space and F⊂X∗∗a finite-dimensional subspace such\nthatF0⊂X∗is weak-star closed. Then V:= (F0)0⊂Xsatisfies dim(V) = dim( F)and\nλ(V,X) =λ(F,X∗∗).\nProof.SinceF0⊂X∗has codimension nand is weak-star closed and thus in particular\nclosed in the norm topology, there exists an n-dimensional linear subspace A⊂X∗such that\nX∗=F0⊕A. Hence, using the second item of Lemma B, we find that X=V⊕A0and\ndim(V) = dim( A) =n, as well as V0=F0. Since ( F0)0=F, it follows that ( V0)0=F.\nSuppose now U⊂Xis a closed linear subspace such that X=V⊕U. Then, using the first\nand third item of Lemma B, we obtain /bardblPU\nV/bardbl=/bardblPV0\nU0/bardbl=/bardblPG\nF/bardblforG:= (U0)0. This implies\nλ(F,X∗∗)≤λ(V,X).4 GIULIANO BASSO\nTo finish the proof, we show the other inequality. Let J:X→X∗∗denote the canonical\nembedding of XintoX∗∗. Then for every x∈V, we have J(x)(ℓ) =ℓ(x) = 0 for all\nℓ∈V0=F0. This implies that J(V)⊂(F0)0=F. Since dim( V) = dim( F), we find that\nJ(V) =F. Hence, for every P:X∗∗→Fthe map Q:=J−1◦P◦Jis a projection of Xonto\nVand/bardblQ/bardbl ≤ /bardblP/bardbl. This implies that λ(V,X)≤λ(F,X∗∗), completing the proof. /square\nThus, if F⊂X∗∗is as in Theorem D, then Fis contained in the image of Xunder the\ncanonical embedding X→X∗∗. In particular, in the special case when X=c0, Theorem D\ncan only be applied to spaces that are already contained in c0. Therefore, the strategy used\nin [Bas19] to prove Theorem 1.4 cannot possibly work, since w e would have to assume the\nvery thing that we want to prove.\nIn the following remark, we explain how the error in our proof of Theorem 1.4 affects the\nresults of the follow-up article [Bas21].\nRemark E. To the author’s knowledge, Theorem 1.4 has so far only been ap plied in the\narticle [Bas21]. In particular, the moreover part of [Bas21 , Theorem 1.4] is equivalent to an\naffirmative answer of Question A, so it should be considered an open question. We also used\nTheorem 1.4 (in a non-essential way) as a shortcut in the proo f of [Bas21, Proposition 5.1].\nIn the following, we give a short sketch of how the use of Theor em 1.4 can be avoided in the\nproof of [Bas21, Proposition 5.1]. Thus, with the exception of the moreover part of [Bas21,\nTheorem 1.4], the results of [Bas21] hold true regardless of whether polyhedral maximizers\nexist or not, and are therefore not affected by our incorrect pr oof of Theorem 1.4.\nFor the rest of this remark we use the notation of [Bas21]. Let P0∈ Pn,mbe such that\n|P0|is a positive matrix and\nρ(|P0|) = max\nP∈Pn,mρ(|P|) = Π(n,m).\nThe existence of such a matrix can readily be verified by combi ning [Bas21, Theorem 1.4]\nand [Bas21, Lemma 3.2]. Let v∈Rm\n>0,d∈N, andqi=pi/dfori= 1,...,m, and Λ =\ndiag(q1,...,qm) be defined as in the proof of [Bas21, Proposition 5.1]. Furth er, letS0∈ Sm\nbe such that πn(√\nΛS0√\nΛ) is maximal amongst S∈ Smand letS∈ Sdbe the (p1,...,pm)-\nblow up of S0. Clearly, there exists a d×nmatrixUwith columns uisuch that UtU= /BDn×n\nandSui=λiui. We set P=UUt.\nBy construction, πn(S) = Tr(SP), andSandPcommute. We show in the following that\nmoreover S= Sgn(P). Letridenote the i-th row of U. SinceSis a (p1,...,pm)-blow up of\nS0, there exists a partition A1,...,A mof{1,...,d}such that # Ai=piandrs=rt=:wi\nfor alls,t∈Ai. Fori= 1,...,mwe define zi=√piwiand letVdenote the m×nmatrix\nwith rows zi. Notice that VtV= /BDn×n. Moreover, using [Bas19, Lemma 2.2], we find that\nQ:=VVtsatisfies Tr(√\nΛS0√\nΛQ) =πn(√\nΛS0√\nΛ) and so by [Bas21, Lemma 3.2], we have\nSgn(Q) =S0. But Sgn( P) is clearly the ( p1,...,pm)-blow up of Sgn( Q). Hence, Sgn( P) =S,\nas claimed. In particular, |P|is a positive matrix.ERRATUM TO “COMPUTATION OF MAXIMAL PROJECTION CONSTANTS” 5\nTo finish the proof, we need to show that d·ρ(|P|)≤jt|P|j+ǫ. Sinceρ(|Q|) =ρ(|P|)\n(which follows from a short computation) and\nπn(S) = Tr(SP) =d/summationdisplay\ni,jsijpij=d/summationdisplay\ni,j|pij|=jt|P|j,\nwe obtain that\nd·ρ(|P|)−jt|P|j=d·ρ(|Q|)−πn(S).\nThus, using that ρ(|Q|)≤ρ(|P0|) as well as πn(S) =d·πn(Sgn(P0)Λ) and Equation (13) of\n[Bas21], we may conclude that\nd·ρ(|P|)−jt|P|j≤d·/parenleftbig\nρ(|P0|)−√qt|P0|√q/parenrightbig\n.\nNow, exactly the same argument as in the proof of [Bas21, Prop osition 5.1] shows that\nd·ρ(|P|)≤jt|P|j+ǫ, as desired.\nWe take the opportunity to record some additional comments t hat correct some minor\noversights in [Bas19].\n(1) In the proof of Theorem 1.2 we rely crucially on the equali ty case in von Neumann’s\ntraceinequality, butwehaveomitted togiveareferencefor this fact. A goodreference\nis Theobald’s article [The75], which deals with real symmet ric matrices and proves\nexactly the statement we need. See also the survey article [C ar21] which discusses\nthe history of the ’equality case’ in von Neumann’s trace ine quality in detail.\n(2) In the proof of Lemma 4.2 it seems worth to point out that Λ Qis an diagonal matrix\nwith positive entries. Hence, D:=/radicalbigΛQ/radicalbigΛQtis a diagonal matrix and1\nTr(D)Dis\ncontained in Ddand therefore the inequality\n1≤Tr(/radicalbig\nΛQ/radicalBig\nΛQt)\nat the bottom of p. 3575 in [Bas19] is a direct consequence of t he equalities\nπn(A/radicalbigΛQ/radicalbigΛQt) =πn(AΛ) andπn(AΛ) = max D∈Ddπn(AD).\n(3) For each ǫ∈[0,1] we define\nD(ǫ)\nd:={D∈ Dd: mindii≥ǫ}.\nAlthough Lemma 4.2 is only stated if Λ ∈ D(0)\ndis non-singular, using the previous\nremark, it is not difficult to see that the conclusions of the le mma also hold if Λ ∈ D(ǫ)\nd\nand\nπn(AΛ) = max\nD∈D(ǫ)\ndπn(AD).\n(4) In Section 4.3 it is claimed that Lemmas 4.1 and 4.2 direct ly imply that\nmax\nD∈DNπ2(RND) =π2(1\nNRN).6 GIULIANO BASSO\nFormally, however, Lemma 4.2 is only applicable if the maxim izerD∈ DNis invert-\nible. To handle the case where the maximizer is a singular mat rix, it suffices to note\nthat\nmax\nD∈DNπ2(RND) = lim\nǫ↓0max\nD∈D(ǫ)\nNπ2(RND)\nand to use the modified version of Lemma 4.2 from the previous r emark.\n(5) In the proof of Lemma 4.4 a fixed point result from metric ge ometry was used to\nconstruct a matrix Λ with certain properties. However, the u se of this fixed point\ntheorem can easily be avoided. Indeed, for any D∈X, the matrix\nΛ =1\n|Stab(A)|/summationdisplay\nQ∈Stab(A)QDQt\nclearly has the desired properties.\nReferences\n[Bas19] Giuliano Basso. Computation of maximal projection constants. J. Funct. Anal. , 277(10):3560–3585,\n2019.\n[Bas21] Giuliano Basso. Almost minimal orthogonal project ions.Isr. J. Math. , 243(1):355–376, 2021.\n[Car21] Marcus Carlsson. Von Neumann’s trace inequality fo r Hilbert-Schmidt operators. Expo. Math. ,\n39(1):149–157, 2021.\n[Rud91] Walter Rudin. Functional analysis. New York, NY: McGraw-Hill, 2nd ed. edition, 1991.\n[The75] C. M. Theobald. An inequality for the trace of the pro duct of two symmetric matrices. Math. Proc.\nCamb. Philos. Soc. , 77:265–267, 1975.\nMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany\nEmail address :basso@mpim-bonn.mpg.de"    },    {        "title": "2402.06676v1.The_Influence_of_Urban_Morphology_on_Water_Scarcity.pdf",        "content": "THEINFLUENCE OF URBAN MORPHOLOGY ON WATER SCARCITY\nRafael Prieto-Curiel\nComplexity Science Hub\nJosefstaedter Strasse 39\n1080 Vienna, Austria\nprieto-curiel@csh.ac.atChristian Borja-Vega\nWorld Bank\n1818 H Street, N.W.,\nWashington, DC 20433\nUSA\n1 Abstract\nCities play a vital role in providing water to vast populations. Due to an increasing population and the urbanization\nprocess, urban water demand is expected to double by 2050. However, some cities are expanding in locations with\nlimited water retention capacity, scarce rainfall and unfavourable shape. This study quantifies the impact of urban form\non water scarcity across more than 100 cities in Latin America, Asia, and Africa. The analysis integrates indicators\nof urban morphology, satellite imagery for natural resources, and terrain metrics to model cities’ influence on water\naccessibility. Water tariffs, proximity to critical infrastructure, and access to piped water are used to assess scarcity\nin urban areas. Results show that locations that are far from the centre, with smaller buildings and fewer constructed\nsurfaces, are less likely to have water access. Urban sprawl negatively impacts water availability within a city, decreases\nproximity to critical infrastructure and increases water tariffs. Keeping everything else constant, a location that is twice\nas far from the city centre has 38% less proximity to critical infrastructure. Also, water tariffs are up to 75% higher and\nthe capacity of cities to provide water drops by half in more sparse cities.\n2 Introduction\nThe water supply crisis has been recognised as the most significant high-impact risk in our present era [ 1]. In 1960, less\nthan 10% of the world population lived in areas with chronic water shortage, but that number has rapidly increased in\nrecent years [ 2]. More than half of the global population faces severe water scarcity for a minimum of one month each\nyear [ 3]. Most parts of the world show an increasing trend in water scarcity [ 4]. Nearly two billion people lack access\nto safely managed water at home [ 3]. Millions of people will be forced to move from less viable areas with lower water\navailability [ 5]. Even in wealthy countries like the US, there is water insecurity [ 6]. A mixture of unsustainable use\nwith a lack of recycling programs for wastewater has created hotspots of unsustainable water use [ 7,8]. Water scarcity\nhas profound implications that manifest in a staggering human cost. Insufficient water, sanitation, and hygiene practices\n(WASH) result in nearly one million deaths annually [ 3,9,10]. Particularly alarming is the situation in numerous low-\nand middle-income countries where a substantial portion of the populace lacks access to clean, on-premises, and readily\navailable drinking water services [11].\nCities play a critical role in ensuring massive access to water. Today, 56% of the world’s population lives in cities,\nyet only 20% of the people who lack essential services live in a city [ 12]. In cities, most elements are designed\nbased on human needs, including water infrastructure [ 13]. Economies of scale play a crucial role in cities by\nsharing among a large population the existing infrastructure, such as roads, public transport or energy provisions\n[14,15,16,17,18,19,20]. Securing water accessibility is mostly part of the urbanisation process. Projections estimate\nalmost 7 out of 10 people will live in a city by 2050, so a more significant share of an even larger population will live\nin cities. Thus, it is estimated that urban water demand will nearly double by 2050 [ 21]. Yet, although cities are a\ncrucial part of ensuring water provisions, there are many challenges related to urban water. One in four cities is under\nwater stress due to geographical and financial limitations, including urban areas in the US, Spain, Mexico or India [ 22].\nCities like Barcelona have already been compelled to enforce water restrictions due to the adverse effects of severe\ndroughts [ 23]. Cities in some regions will need help finding enough water for the needs of their residents and will need\nsignificant investment to secure adequate water supplies [ 24]. Some cities will experience generalised interruptions ofarXiv:2402.06676v1  [physics.soc-ph]  7 Feb 2024Urban forms and water insecurity\nessential water services (known as ‘day zero’) that will exacerbate existing urban issues, such as violence and inequality\n[25]. Due to the uneven distribution of resources and mobility capacity, day zero will have a much higher impact on\npoor households [ 26]. By 2050, one billion people in cities will have to live with less than 100 L per person per day of\nsustainable surface and groundwater flow within their urban extent [24].\nSome cities will face severe issues related to water in the upcoming decades. First, cities have to ensure water\nprovisions in times of climate change. Vast regions are being affected by extensive floods and severe dry seasons,\naltering the equilibrium between water demand and supply and the availability of stable water resources [ 27,23]. The\nincreasing frequency of extreme weather, with long dry periods and high rainfalls, is challenging to model and predict\n[28,29,30,31]. The uncertainties generated due to unexpected weather events further exacerbate the problem of water\ninsecurity [ 32]. However, an even more pressing element related to water scarcity is related to population growth [ 2].\nSome countries will double their 2020 population before 2050 [ 33]. Some metropolitan areas might reach over 80\nmillion inhabitants, so satisfying their water demands will be burdensome [ 34]. The fast urbanisation that parts of the\nworld will experience over the next three decades may lead to diverse problems, such as exhausting land resources,\ndamaging ecosystem services and degrading sustainable development [ 7]. Some cities will have to accommodate a\nrapidly increasing population. In addition, this process of population growth and rapid urbanisation mainly occurs in\ncountries with tight budget restrictions, where other issues, such as violence and political instability, overtake long-term\nelements, such as urban or transport planning. The fastest-growing cities tend to be the poorest, so they will be in\ngreatest need of financial aid [ 22]. Thus, in some cities, they will have to quickly expand their infrastructure to host\nmany more people, likely with severe budget restrictions and at a time when climate change is altering the fragile\nequilibrium of water supplies.\nBesides a challenging future related to water availability due to the rapid population increase, a tight budget combined\nwith many more pressing conditions (including incivilities and violence), and extreme weather conditions and uncertain-\nties attributed to climate change, many cities are growing in critically unfavourable locations. Many parts of the world\nhave soil with a limited capacity to retain water combined with scarce rain, many of them where cities are expanding\n(Figure 1). Some cities are rapidly growing in regions where there is minimal rain, and when it rains, the terrain does\nnot retain that water. In such cities, water will be a scarce resource permanently.\nUrban growth is often unplanned, in remote locations with steeper slopes, in low-density developments that increase\nthe sprawl and elongation of cities [ 37,38]. Ultimately, the way cities grow results in some morphology that is often\na challenge for its residents [ 39]. Urban fragmentation with sparsely populated areas has lasting implications for a\ncity’s future resilience and sustainability [ 40,41,14,42,43,44]. Guaranteeing access to water is far more challenging\nfor a city when its growth occurs in multiple, remote and sparsely populated locations. Compact cities form a more\nsustainable way to accommodate people. But urban morphology is relevant not only because it may be a challenge for\nits population but also because it has profound implications for the ecosystem [ 45]. Expanding cities tend to invade\ngreen and blue areas, altering the system’s capacity to obtain resources sustainably [ 38,46,47]. Groundwater is at risk\ndue to human overexploitation in numerous large aquifers [ 48]. The way cities expand has lasting implications for\ngreen spaces, resiliency to floods, biodiversity conservation, and energy demands [49, 50, 51, 52].\nModelling the links between urban forms and access to water is critically relevant for planning better policies related\nto water and its management and the expansion of cities. New data at the granular level has opened novel ways of\nobserving cities at a global scale [ 53,54,55,33,56,57]. It includes, for example, satellite imagery, road infrastructure,\nnatural resources and terrain metrics, building stocks and other sources at a very granular level [ 58,59,60,61]. A\ncombination of scarcity, capacity, access, use and environment have been used to prioritise water-related work [ 62].\nHere, we quantify the impact of urban forms on water scarcity. We analyse the effect of urban sparseness across over\n100 cities in Latin America, Asia and Africa. First, we construct a delineation of urban forms using the footprint of\nall buildings built in each city. We then combine indicators related to the morphology of urban areas and satellite\nimagery for natural resources and terrain metrics to model the impact of cities on water cycles. At the local level, we\nuse proximity to critical infrastructure [63] and at the city level, water tariffs [64] and population with access to piped\nwater [ 65] to quantify the impact of urban forms in terms of water insecurity. Results show that remote neighbourhoods\ntend to be far from critical infrastructure, and collectively, sprawled cities have fewer people with access to piped water\nand pay a higher price for water.\n3 Results\nBuildings’ locations were extracted from the Google Open Buildings dataset, which contains the location and footprint\nof all constructions across multiple countries [ 59]. We focus on over 100 major cities in Asia, Africa and Latin America,\nwith a total population of around 650 million inhabitants and 183 million buildings. For city i, we delineate its perimeter\nbased on the proximity of its buildings and identify its centre, usually the main square, market or transport station\n2Urban forms and water insecurity\nyearly rain\nsoilcapacityARID \nTERRAINMOIST \nTERRAINMORE \nRAIN\nLESS \nRAIN\nsoilcapacityARID \nTERRAINMOIST \nTERRAINregions cities\nWestern\nEuropeNorthern\nEuropeSouth\nAmericaSoutheastern\nAsia\nNorthern\nAfrica\nWestern\nAsiaSouthern\nAsia\nEastern\nAfrica\nSouthern\nAfricaEuropeEastern\nNorthern\nWestern\nSouthern\nOceaniaAustrlia and NZ\nOther Islands\nAsiaEastern\nSoutheastern\nSouthern\nWestern\nCentral\nAmericaNorthern\nSouth\nCentral\nCaribbean\nAfricaWestern\nNorthern\nSouthern\nEastern\nMiddle\nEastern\nAsia\nEastern\nEuropeMiddle\nAfrica\nCentral \nAsia MORE\nMOISTMORE\nRAIN\nFigure 1: Soil capacity to retain water (horizontal axis) and yearly precipitation (vertical axis) for different regions (left)\nand cities (right). The colour corresponds to the region, and the size of the disc corresponds to the population. For each\nregion, the value is the average soil capacity and precipitation observed in its cities, weighted by the population. Soil\ncapacity to retain water was obtained from Global Aridity Index that uses evapo-transpiration processes and rainfall\ndeficit for potential vegetative growth [ 35]. The rainfall is the average monthly precipitation between 1970-2000,\naggregated to a yearly value [ 36]. Most European cities have high levels of soil capacity to retain water, so even with\nlow levels of rain, the terrain is favourable for water retention. In South America, there is less capacity to retain water,\nbut it is more rainy, so there are also better conditions for ensuring access to water.\n(details in the Methods section). In the vast majority of cities, the area near the centre is mainly occupied by buildings,\nand it decreases for regions far from the centre (Figure 2). Yet, there are vast heterogeneities between what is constructed\nand how that decreases with distance. In São Paulo and Jakarta, for example, up to 50% of the surface near the city\ncentre corresponds to a building, but in Cairo, with a similar population, less than 20% of the city near the centre\ncorresponds to buildings (Figure 2). In general, African cities and smaller urban agglomerations have fewer constructed\nsurface near the centre, increasing urban sprawl.\nWe use the location of the centre of city ito measure the distance Di,jin km from location j. However, distances in\ncities of varying population sizes are not comparable. We define the remoteness of location jin city ias\nRi,j=1000Di,j√Pi, (1)\nwhere Piis the population of city i. Remoteness is a comparable metric for distances between cities. We classify\nlocations based on remoteness and obtain four areas. Locations with Ri,j<3are the central part of the city, with\nRi,j∈[3,5)are the intermediate part, with Ri,j∈[5,9.3)are distant areas, and with Ri,j>9.3are the outmost areas\n(details in the Methods section). Remoteness plays a detrimental role in access to urban amenities. In general, the\ndistant and outmost areas are characterised by a limited constructed surface and low population density. The average\nincome of a person is 50% larger in the central parts of the city than in the intermediate part, 4.4 times bigger than in\nthe distant parts and 8.4 times bigger than in the outmost regions (Table 1).\nSimilarly, proximity to critical infrastructure is 24% larger in the central part of the city than in the intermediate, double\nthan in the distant parts and more than triple than in the outmost parts (Figure 3). Remote areas tend to be poorer [ 66].\nFurther, this situation may be exacerbated in smaller cities (details in the Supplementary Material). One of the most\nsignificant indicators to identify whether an area has a high income or access to critical infrastructure is related to the\nbuildings it has available within walking distance (considered here to be 500m [ 67]). Also, it is possible to characterise\nan area of the city by the infrastructure constructed within walking distance. Areas with a higher constructed surface\nand more infrastructure nearby tend to be wealthier, whereas remote and isolated areas tend to be poorer (Figure 3).\n3Urban forms and water insecurity\nAfrica Asia Latin America\nCairo Jakarta Sao Paulo\nKumasi Chennai Guadalajara\nNouakchott Makassar Meridacitycentre\n30km10%30%\n40%\n60%\n70%90%others buildingscity\ncentre30km30%\nothers buildingscity\ncentre 30km30%\nothers buildings\n10%30%\n40%\n60%\n70%90%10%30%\n40%\n60%\n70%90%10%30%\n40%\n60% 90%\n10%30%\n40%\n60% 90%10%30%\n40%\n60%\n70%90%10%30%\n40%\n60%\n70%90%\nFigure 2: Fraction of the city’s surface that is occupied by constructed buildings. Concentric circles represent each city,\nmatching the centre of the city to the centre of the circles. Each additional ring corresponds to an extra 500m from the\ncentre. The coloured section of each ring is the surface that is occupied by buildings.\nbuilding # const.\narea GDPPP CI size (m2) buildings surface (%)\ncentre 100 100 112 2190 26\ninter 49 70 100 1684 19\ndistant 23 49 104 1123 12\noutmost 12 30 94 682 6\nTable 1: GDP per person (based on purchasing power parity) and Critical Infrastructure compared against the centre of\nthe city. Average building size in m2, number of buildings and constructed surface (in %) within walking distance for\ndifferent areas of a city.\nSome cities tend to be highly sprawled, so most of their population is far from the city centre (Figure 4). In Asian cities,\nmore than half of its population lives in the central parts of the city, but in Latin America, only 28% of its population\nand in Africa, less than 20% of its population lives in central areas. Particularly, African cities are highly sprawled.\nFor example, taking two buildings at random in Jakarta (a city in Indonesia with 33 million inhabitants), the average\ndistance between them is 24.5km, but taking two buildings at random in Kigali (a city in Rwanda with 2.2 million\npeople), the average distance between them is 26.1km, so Kigali is much more sprawled [49].\nProximity to critical infrastructure is also affected by distances. For pixel k, we fit the expression Ik∝Rα\ni,k, where\nIk∈(0,1)is the infrastructure index of k(with higher values corresponding to more infrastructure). We find that\nα=−0.682±0.008, meaning that more remote locations have less infrastructure (details in the Supplementary\nMaterial). At the local level, remote locations tend to be far from critical infrastructure. Keeping everything else\nconstant, a pixel that is twice as remote as another has 2(−0.682) = 0 .62critical infrastructure. Also, aggregating the\ninfrastructure at the city level, we find that people in sparse cities are, on average, far from infrastructure. On average,\npeople living in a city with twice sparseness have nearly 40% less proximity to critical infrastructure (details in the\nSupplementary Material).\nIn highly sprawled cities, travel distances are longer, increasing the energy they require for mobility [ 49]. For city iwe\nmeasure the sparseness Sias\nSi=P\njPi,jRi,jP\njPi,j, (2)\nrepresenting the average remoteness of city iweighted by the population of each location. Here, we find that the city\nsparseness increases the tariffs people pay for water [ 64]. Water tariffs constitute a primary source to cover the costs of\n4Urban forms and water insecurity\nremote locations \nwith limited critical \ninfrastructureGDP PP (log) Infrastructure Constructed surfaceremote locations \nwith lower GDP PP𝑅𝑖𝑗=3\nRemoteness Remoteness RemotenessAfrica Asia Latin Americacentreinter distant out centreinter distant out centreinter distant out𝑅𝑖𝑗=5𝑅𝑖𝑗=9.3 𝑅𝑖𝑗=3 𝑅𝑖𝑗=5𝑅𝑖𝑗=9.3 𝑅𝑖𝑗=3 𝑅𝑖𝑗=5𝑅𝑖𝑗=9.3\ncentreinter distant out centreinter distant out centreinter distant out\ncentreinter distant out centreinter distant out centreinter distant out\nFigure 3: For 7,300 pixels in Africa, 6,600 in Asia and 5,800 in Latin America, we measure their remoteness (horizontal\naxis) and compare it to the income (vertical axis), proximity to critical infrastructure, and the constructed surface within\nwalking distance.\nremoteness 𝑅𝑖,𝑗cumulative population𝑅𝑖,𝑗=3 100%\n50%\n0%Jakarta\n33M people\nKigali\n2M peopleJakartacumulative \npopulation\nremotenessKigali\n20kmDhaka𝑅Jak,𝑗=3\n𝑅Kig,𝑗=3\nremotenessAsia\nAfrica\nLatinAmericacumulative \npopulation𝑅𝑖,𝑗=5\nFigure 4: Left - Cumulative fraction of the population (vertical axis) that lives up to some value of remoteness (horizontal\naxis). Each line represents a city and its population. Right - buildings in Jakarta (33 million inhabitants) and in Kigali\n(with 2 million inhabitants) are displayed on the same scale. In Jakarta, more than half of its population lives in the\nmost central part of the city (regions with Ri,j<3). However, in Kigali, only 15% of the population lives in the central\narea. In Jakarta and Kigali, travel distances are similar, but Jakarta has 15 times more population.\nservice provisions but reduce its affordability [ 68,69]. We use water tariffs at the city level to asses the relation with its\nsparseness [ 64]. Results show that in cities with more sparseness, the cost people pay for one cubic metre of water\ntends to be much higher (Figure 5). On average, water tariffs are 75% higher in a city with twice sparseness (details in\nthe Supplementary Material).\nFor each city, we estimate the population with access to piped water using national data from the Joint Monitoring\nProgramme for Water Supply, Sanitation and Hygiene (JMP) [ 65]. Results show that a city with twice sparseness\nprovides access to piped water to less than half of the population (details in the Supplementary Material).\nRemoteness decreases access to critical infrastructure and increases water tariffs. Results show that smaller cities tend\nto be more sparse (Table 2). In small cities (fewer than 1.5 million inhabitants), more than half of the population lives\nin the outmost area. In contrast, in megacities (with a population above 10 million inhabitants), only 11% live in the\noutmost areas. On average, people living in a city with twice the population have 14% fewer sparseness (details in the\nSupplementary Material).\nCities in Asia tend to be less sparse compared to cities in Latin America and Africa (Table 3). On average, 12% of\npeople in African cities live in the centre area, and 38% live in the outmost area. In contrast, 23% of the population of\nAsian cities live in the centre and only 9% live in the outmost area.\n5Urban forms and water insecurity\nU$1U$2U$4U$8U$16U$32𝑆𝑖=3\nDhakaManila\nKarachi\nCairoKisumu\nAsunciónBrasilia\nPueblaWindhoek\nCape TownLuandaSan Jose\nMumbaiKolkata\nSparseness 𝑆𝑖MORE \nSPARSENESSLESS \nSPARSENESSTariff for15m3ofwater (U$ dollars )\nSparseness 𝑆𝑖MORE \nSPARSENESSLESS \nSPARSENESS2.5%5%10%20%40% Mean access tocritical infrastructure\nKisumuOnitshaConakry\nKigaliWindhoek\nAgadezMonroviaBrasiliaSemarang Bangalore\n𝑆𝑖=3CONTINENT  POPULATION\nAfrica\nAsia\nLatAm30M\n3M\n𝑆𝑖=5 𝑆𝑖=9.3\nFigure 5: Top - Sparseness (horizontal axis) and tariff paid for 15m3of water for cities (vertical axis). Bottom -\nSpareness (horizontal axis) and mean proximity to critical infrastructure (vertical axis).\narea small medium large mega\ncentre 8% 8% 11% 22%\ninter 12% 13% 16% 29%\ndistant 30% 33% 35% 38%\noutmost 50% 46% 39% 11%\nTable 2: Distribution of the population in small cities (smaller than 1.5 million inhabitants), medium cities (smaller than\n3 million), large cities (smaller than 10 million) and megacities (larger than 10 million).\n4 Discussion\nCities are simultaneously locations where economies of scale simplify the challenge of distributing resources among\nlarge populations, but they are also where disasters magnify due to their extensive impact. Water policies are a critical\nstep for the sustainability of growing urban areas. Policies can neutralize water scarcity by allocating resources more\nefficiently and promoting recycling [ 70]. Management of ground pollution guarantees the sustainability of most sources\nof fresh water [ 71]. Yet, another set of actionable policies is related to how cities grow. Sprawled cities require\nmore energy for transport and require covering a larger surface with pavement [ 72,73]. Building compact, walkable\nneighbourhoods and taller buildings can reduce pressure on the periurban ecosystem by accommodating the growing\nnumber of people [49, 74].\nUrban form is strongly linked with income, proximity to infrastructure and water. Our findings underscore the correlation\nbetween remoteness and socio-economic indicators, with remote areas generally experiencing higher levels of poverty\nand social exclusion. We explore the impact of remoteness on water tariffs, finding that more sprawling cities incur\nhigher costs for water services, reflecting challenges in providing equitable access.\nWe introduce the concept of “remoteness” to quantify the distance of a location within a city to its centre, accounting\nfor variations in city sizes. Remoteness proves to be a valuable metric, revealing distinct patterns in urban development.\nCentral areas exhibit higher population density, more constructed surface, and increased proximity to critical infrastruc-\nture, resulting in elevated average income. Conversely, remote areas tend to be characterized by limited constructed\n6Urban forms and water insecurity\narea Africa Asia LatAm\ncentre 12% 23% 18%\ninter 16% 31% 25%\ndistant 34% 37% 40%\noutmost 38% 9% 17%\nTable 3: Distribution of the population in Africa, Asia and Latin America according to the area of the city.\nsurfaces and lower population density, both contributing to decreased access to water infrastructure. At the local level,\nremoteness reduces income and proximity to infrastructure, thus reducing the chances that a remote household has\nwater supplies. High poverty levels do not mean poor hygiene practices [ 75]. However, a lack of reliable water access\nhinders essential practices like drinking and cooking and may cause health issues such as dehydration, injury, and\ndiarrhoea [3].\nAt the city level, we introduce the concept of sparseness. We show that it has critical implications. Cities where most of\nthe population is sparse present unique challenges compared to more centrally concentrated cities. Here, we show that\nsparse cities tend to be poorer, critical infrastructure is more scarce, and also water tariffs tend to be higher. Further,\npoor households are usually not connected to piped water, and if they are, they tend to consume smaller quantities of\nwater, so the majority of water subsidies are captured by the wealthiest households [ 69]. Thus, unaffordable water\nis a persistent struggle among poor and remote neighbourhoods. Smaller cities tend to exhibit higher sparseness,\nemphasizing the need for tailored urban planning strategies. Comparative analyses across continents highlight regional\nvariations, with Asian cities generally demonstrating lower remoteness compared to counterparts in Latin America and\nAfrica. The presence of numerous tiny buildings in some cities contributes to higher sparseness, suggesting the need\nfor nuanced approaches to address specific urban configurations. Increasing density in the core parts of the city and\ndesigning greenbelts may help maintain specific desirable properties of the urban form whilst conserving green spaces\nfrom being built over [76].\nIn some cities, access to water will take a lot of work. A mix of terrain aridity, lack of rain and nearby water sources\nwill require a substantial budget for obtaining water from remote locations [ 22]. This situation will be considerably\nchallenging for the poorest and fastest-growing cities. In some cases, the demand for water is currently increasing, and\nit is expected to double within the next decades. With fast population growth, cities may evolve towards dispersed and\ndisorderly growth [77]. These cities will be in greatest need of aid. [22].\n5 Methods\nA variety of models, including cellular automata models, water evolution and planning models coupled with geographic\ninformation systems, remote sensing, and criteria analytic hierarchical process, have been used to model water scarcity\n[7]. Also, a variety of indicators have been developed to capture different characteristics of water scarcity, including\nthose related to population, availability, and water use [ 1]. Here, we use the spatial arrangement of buildings within\na city to model urban forms. Then, we use the proximity to critical infrastructure, water tariffs, and the population’s\naccess to piped water to link urban forms and water insecurity.\nWe consider 106 cities from Africa (45 cities), Asia (30 cities) and Latin America (31 cities). Major cities, state capitals\nand other medium-sized cities were selected (Figure 6). Data availability played a restriction in terms of the cities\nthat could be analysed. For example, data corresponding to building footprints from China or Saudi Arabia was not\navailable and thus could not be included.\nFigure 6: Cities considered in the study. The size of the disc corresponds to the population of each city.\n7Urban forms and water insecurity\n5.1 Identifying buildings and delineating cities\nBuildings’ locations were extracted from the recently launched Google Open Buildings dataset, https://sites.\nresearch.google/open-buildings/ , which contains building footprints [ 59]. The buildings’ footprints were\nobtained using a deep learning model with high-resolution satellite imagery (50 cm pixel size) [ 59]. This dataset\ncontains information on the location of the building’s footprint centre in geographic coordinates (latitude and longitude),\nits area in m2, and the geometry of the building’s footprint. We keep the attributes of the building’s centre location\n(latitude and longitude) and the building footprint area using [ 78]. We obtained the location and characteristics of the\nbuildings’ footprints of 106 urban agglomerations in Africa, Latin America and Asia. In total, we analyse the footprint\nof 183.4 million buildings, corresponding to a constructed surface nearly the size of Kuwait.\n5.2 Delienation or urban agglomerations\nDefining urban agglomerations is challenging and often depends on considerations and parameters [ 16,79,80,81].\nHere, we construct the delineation of the 106 cities using the same definition across all of them, applying the same\ncriteria. The delineation of an urban agglomeration boundary is based on a spatial approach that combines physical\nlimits and the continuity of the built-up area. We consider the building footprint and the proximity between constructions\nto determine the boundaries of a city. Our criteria is to join an urban polygon to some buildings depending on whether\nthey are less than 200 m apart. The same criteria have been applied elsewhere to construct the delineation of thousands\nof cities in Africa [ 82]. This step ensures that the same criteria to define the urban boundary are applied to cities across\nall countries.\nWe employ a geometric structure known as the alpha-convex hull. It is frequently used in computational geometry\nto represent the convex hull of a set of points in a plane. Here, the points represent the buildings in each city. The\nalpha-convex depends on a parameter, α. Constructing the alpha-convex hull involves modifying the traditional convex\nhull by allowing certain concave regions to be included or removed based on the αparameter. The process begins with\ncomputing the standard convex hull of the given buildings in a city. Then, each edge of the concave shape is examined,\nintroducing additional edges connecting points within a certain distance. These extra edges remove and round concave\nportions of the original shape. The result is an urban polygon that is comparable across cities.\nFor each urban area, its buildings are identified by those inside the core delineation. Then, using the identified buildings,\nwe measure the number of constructions in each city, the constructed surface, and the mean building area.\nThe infrastructure in some cities is formed mainly by small buildings. For example, in cities like Khartoum and Agadez,\nnearly two of every three buildings have a footprint smaller than 20m2. In Kuala Lumpur and Singapore, less than 15%\nof buildings are so small (Figure 7).\n4% 5%17% 18%29%\n19% 20%29% 28%38%\n10% 11% 8% 8% 7%11% 9%14%16%22%22%30%\n23% 22%34%\n27%27%\n26% 26%\n23% 24% 24%31%\n27%34%37%32%34%20%\n32% 31%25%\n29%21%\n19% 19%\n24%29% 32%27%35%29%27%19%17%10%18%17%8%10%10%\n19%19%28%25%25%20%20%15%13%9% 6% 9% 7% 9%4% 5% 4%\n24% 22%15% 12% 10% 9% 6% 4%Kuala Lumpur\nSingapore\nPanama City\nBrasilia\nJakarta\nHavana\nMonterrey\nMexico City\nLa Paz\nKinshasa\nKigali\nNouakchott\nBangui\nBobo-Dioulasso\nKhartoum\nOuagadougou\nAgadezBuilding size distribution\nsmaller than 10m2 smaller 20m2 smaller 50m2 smaller 100m2 smaller 200m2 larger 200m2\nFigure 7: Distribution of building size across selected cities.\n5.3 Identifying the city centre\nFor each urban agglomeration, we identify the city centre as the location of the main square, main church, mosque\nor central transport station (Figure 8). Based on the location of the city’s centre, two variables at the pixel level are\nmeasured: distance to the city centre and relative distance to the centre. The idea is that remote locations within the\nsame urban agglomerations are identified. Although the precise location of the centre could be ambiguous (for example,\nin some cities where the location of the main square does not correspond to the location of the main religious temple or\ntransport station), the location is used to determine the relative proximity and centrality of other places in the city. Thus,\nalthough the centre could be defined in terms of more than one location, the impact on remoteness is insignificant.\n8Urban forms and water insecurity\nCairo Jakarta Sao Paulo\nKumasi Chennai Guadalajara\nNouakchott Makassar Meridacitycentre\nAfrica Asia Latin America20km\nFigure 8: The delineation of cities depends on the building footprint of cities. For each urban agglomeration, we identify\nits centre as the main square, train station or city hall.\n5.4 Remoteness and sparseness\nFor location jin the city i, we consider two metrics. First is Di,jcorresponds to the (geodesic) distance in km between\nthe city centre iand location j. This distance, however, is relative to the population of a city. For example, in Bogota\n(a metropolitan area with nearly 9 million inhabitants), a location 2.5km away from the city centre is still considered\na central location. However, in Manizales (a city with 434,000 inhabitants), a location within the same distance is a\nperipheral location. We construct the remoteness Ri,jbetween the city centre of iand location jgiven by\nRi,j=1000Di,j√Pi. (3)\nThe idea is that a city with a population Pioccupies an area proportional to its population [ 49]. In general, distances in\na city grow with the square root of its population [ 49]. One reason for this is the growth of distances within a circle.\nConsider a circle with area Aand consider two points inside that circle. The expected distance between them is\n128\n45π√\nA. (4)\nThus, in a circle, distances grow linearly with its radius. Similarly, if the area of a city grows with its population,\ndistances in that city should grow proportional to√Pi. With this formula, a location in Bogota 2.5km away from its\ncentre has RB,j= 0.9, whereas a location in Manizales 2.5km away from its centre has RM,j= 3.8, so the same\ndistance would be considered much more remote.\nThe definition of remoteness depends on the identification of the centre. However, the impact is not critical. One\nexample is Bogota. In the city, the centre could be identified as Plaza Bolívar, or Museo del Oro (500m north), or\nParque Metropolitano (600m west), among other locations. The distance from Parque Soacha (a neighbourhood in the\nsoutheast part of the city) to the city centre is 16.0km, 16.4km, or 15.5km, respectively, meaning a variation of less than\n3%. Parque Soacha’s remoteness is either 5.7, 5.8 or 5.5, respectively, so it is identified as a remote location, regardless\nof the definition of the city centre.\nThe remoteness applies to different locations within a city. We then construct a metric that applies at the city level that\ncorresponds to how remote it is, on average, its population. For city iwe measure the sparseness Sias\nSi=P\njPi,jRi,jP\njPi,j, (5)\nrepresenting the average remoteness of city iweighted by the population of each location.\n9Urban forms and water insecurity\ncontinent pixels\nAfrica 7,317\nAsia 6,580\nLatin America 5,837\nTotal 19,734\nTable 4: Pixels per continent\n5.5 Surveying cities\nWe survey distinct locations within a city to detect whether remote locations alter water availability. We begin by\nconsidering the Relative Wealth Index data, a public access dataset that provides micro estimates of wealth for all low\nand middle-income countries [ 26]. Data is provided at 2.4km resolution and was constructed by merging different\ndatasets, including DHS survey [ 83], mobile phone and others. Scores cannot be compared across countries. Although\nthe data has some limitations and it may not accurately reflect the ground truth, it provides a systemic way of comparing\ndifferent regions across cities [84].\nWe use the grid of the Relative Wealth Index data for geographically surveying cities. Thus, we take the coordinates that\nare inside each urban polygon and use those coordinates as “pixels” to analyse heterogeneities across cities. Depending\non the size of the city, more or less pixels are considered. For instance, more than 1,000 pixels are analysed in New\nDelhi, more than 500 pixels are analysed in São Paulo, and more than 200 pixels are analysed in Cape Town. In total,\nnearly 20,000 pixels are analysed (Table 4).\nFor each pixel, the following variables are constructed: constructed surface within walking distance, area of buildings\nwithin walking distance, number of buildings within walking distance, and its remoteness (Figure 9). Further, for each\npixel, the gridded population dataset [85, 86], and the gridded form for Gross Domestic Product [66].\nBuilding’s density City delineation\nRelative wealth index Buildings within 500m\nArea  of buildings within 500m Constructed  surfac e within 500mLuanda, Angola\n20km\nFigure 9: The building’s footprint (top, left) is used to construct the delineation of a city (top, right). Then, separate\npixels inside the city are considered based on the relative wealth index grid (middle, left). For each pixel, the number of\nbuildings (middle, right), the average area of buildings (bottom, left) and the constructed surface within 500m (bottom,\nright) are measured. The figure shows the case of Luanda, in Angola, with a population of nearly 7 million inhabitants\nand a grid with 206 pixels.\n5.6 Cities and water\nWe consider three ways to analyse water accessibility: water tariffs, proximity to critical infrastructure and access to\npiped water. For water tariffs, we use the IBNet Tariffs database [ 64]. It is a large dataset with water and wastewater\ntariffs in a comparable U$ format per city.\n10Urban forms and water insecurity\nFor proximity to critical infrastructure, we use the Critical Infrastructure Spatial Index, a harmonised index of critical\ninfrastructure [63]. It uses data from OpenStreetMap to construct a harmonised index of critical infrastructure.\nFor each city, we estimate the total urban population with access to piped water. The estimate is obtained from national\ndata from the Joint Monitoring Programme for Water Supply, Sanitation and Hygiene (JMP) [65].\nAt the local level (considering the 19,734 pixels as observations) and at the local level (considering the 106 cities), a\nregression of transformed variables was computed to analyse a functional correlation between variables. Although we\nuse GDP, critical infrastructure, water tariffs or urban population with access to piped water as the dependent variable,\nwe cannot assume a causal relation with the independent variables (such as population size, remoteness or sparseness).\nThe links between variables are complex, and likely, there is feedback between them (for example, between population\nand GDP per person). The coefficients estimated for each model are in the Supplementary materials.\n11Urban forms and water insecurity\nSupplementary materials\nRemoteness within a city\nWe use the remoteness Ri,jof location jin city ias the critical variable to quantify the relative distance across cities.\nWe use a classification tree to divide the remoteness using the surface constructed within walking distance [87].\nConstructed surface within a walking distance (%)\noutmost distant intermediate centre27% of the \nsurface is \nconstructed\n𝑅𝑖𝑗<36.9% of the \nsurface is \nconstructed\n𝑅𝑖𝑗≥9.3\nFigure 10: We use a classification tree to divide the remoteness of a city [88, 78].\nThe classification tree divides the remoteness into four parts:\n•Ri,j<3-centre .\n•Ri,j∈[3,5)-intermediate .\n•Ri,j∈[5,9.3)-distant .\n•Ri,j≥9.3-outmost .\n12Urban forms and water insecurity\nGDP at the micro level\nModel 1 Model 2 Model 3\n(Intercept) 17.586∗∗∗13.242∗∗∗10.810∗∗∗\n(0.060) (0 .287) (0 .246)\ncontinent - Asia −0.033 −0.246∗∗∗−0.711∗∗∗\n(0.042) (0 .044) (0 .038)\ncontinent - Latin America 1.135∗∗∗1.109∗∗∗0.443∗∗∗\n(0.041) (0 .041) (0 .036)p\nRi,j −0.992∗∗∗−0.920∗∗∗−0.373∗∗∗\n(0.018) (0 .019) (0 .018)\nlog(population) 0.269∗∗∗0.268∗∗∗\n(0.017) (0 .015)\nsurface500m 0.000∗∗∗\n(0.000)\nmeanArea500m −0.000\n(0.000)\nR20.194 0 .206 0 .423\nAdj. R20.194 0 .206 0 .423\nNum. obs. 16425 16425 16425\nTable 5: Models for GDP PP\n13Urban forms and water insecurity\nRelative wealth index\nModel 1\n(Intercept) 1.441∗∗∗\n(0.062)\ncountry effects —-\n—-p\nRi,j −0.134∗∗∗\n(0.003)\nlog(population) −0.024∗∗∗\n(0.004)\nsurface500m 0.000∗∗∗\n(0.000)\nmeanArea500m −0.000\n(0.000)\nR20.444\nAdj. R20.442\nNum. obs. 19430\nTable 6: Models for MRWI\n14Urban forms and water insecurity\nCritical infrastructure\nModel 1 Model 2 Model 3\n(Intercept) −1.166∗∗∗0.055∗∗∗0.016\n(0.015) (0 .010) (0 .009)\nlog(RelativeDist) −0.682∗∗∗−0.064∗∗∗−0.050∗∗∗\n(0.008) (0 .001) (0 .001)\ncontinent - Asia 0.013∗∗∗0.006∗∗∗\n(0.001) (0 .001)\ncontinent - Latin America 0.039∗∗∗0.029∗∗∗\n(0.001) (0 .001)\nlog(population) 0.011∗∗∗0.011∗∗∗\n(0.001) (0 .001)\nsurface500m 0.000∗∗∗\n(0.000)\nmeanArea500m −0.000\n(0.000)\nR20.297 0 .350 0 .392\nAdj. R20.297 0 .350 0 .392\nNum. obs. 19402 19402 19402\nTable 7: Models for Critical Infrastructure\n15Urban forms and water insecurity\nWater tariffs\nModel 1 Model 2 Model 3\n(Intercept) 0.614 2 .497 1 .929\n(0.346) (1 .668) (1 .506)\nlog(MeanRemoteness) 0.815∗∗∗0.661∗∗0.334\n(0.196) (0 .237) (0 .226)\nlog(population) −0.107 −0.024\n(0.093) (0 .086)\ncontinent - Asia −0.851∗∗∗\n(0.241)\ncontinent - Latin America 0.222\n(0.186)\nR20.186 0 .200 0 .370\nAdj. R20.175 0 .179 0 .336\nNum. obs. 78 78 78\nTable 8: Models for Water Tariffs\n16Urban forms and water insecurity\nCity Infrastructure\nModel 1 Model 2 Model 3\n(Intercept) −0.947∗∗∗−4.091∗∗∗−4.103∗∗∗\n(0.196) (0 .846) (0 .777)\nlog(MeanRemoteness) −0.682∗∗∗−0.451∗∗∗−0.436∗∗∗\n(0.111) (0 .120) (0 .117)\nlog(population) 0.184∗∗∗0.169∗∗∗\n(0.048) (0 .045)\ncontinent - Asia 0.240∗\n(0.119)\ncontinent - Latin America 0.469∗∗∗\n(0.106)\nR20.281 0 .376 0 .484\nAdj. R20.274 0 .363 0 .462\nNum. obs. 99 99 99\nTable 9: Models for Critical Infrastructure\n17Urban forms and water insecurity\nRemoteness\nModel 1 Model 2 Model 3\n(Intercept) 4.797∗∗∗4.139∗∗∗3.091∗∗∗\n(0.506) (0 .510) (0 .556)\nlog(population) −0.206∗∗∗−0.156∗∗∗−0.341∗∗∗\n(0.034) (0 .035) (0 .059)\ncontinent - Asia −0.329∗∗∗−0.278∗∗\n(0.095) (0 .090)\ncontinent - Latin America −0.002 −0.074\n(0.088) (0 .085)\nlog(NBuildings) 0.275∗∗∗\n(0.073)\nR20.264 0 .354 0 .433\nAdj. R20.257 0 .335 0 .410\nNum. obs. 106 106 106\nTable 10: Models for Remoteness\n18Urban forms and water insecurity\nPiped water\nModel 1 Model 2 Model 3\n(Intercept) 16.589∗∗∗−0.890 0 .017\n(0.435) (0 .912) (1 .241)\nlog(MeanRemoteness) −1.212∗∗∗−0.046 0 .100\n(0.245) (0 .138) (0 .142)\nlog(population) 1.027∗∗∗0.973∗∗∗\n(0.053) (0 .059)\ncontinent - Asia −0.143 0 .007\n(0.139) (0 .176)\ncontinent - Latin America 0.482∗∗∗0.500∗∗\n(0.122) (0 .150)\nMeanCritInf 2.986∗∗\n(1.007)\nFootpLargeBuildings050m −1.465\n(0.744)\nARID 0.000\n(0.000)\nRAIN 0.006\n(0.004)\nDRY −0.000\n(0.000)\nR20.192 0 .840 0 .866\nAdj. R20.184 0 .833 0 .852\nNum. obs. 105 105 99\nTable 11: Models for piped water\nReferences\n[1]Junguo Liu, Hong Yang, Simon N Gosling, Matti Kummu, Martina Flörke, Stephan Pfister, Naota Hanasaki,\nYoshihide Wada, Xinxin Zhang, Chunmiao Zheng, et al. Water scarcity assessments in the past, present, and\nfuture. Earth’s Future , 5(6):545–559, 2017.\n[2]Matti Kummu, Philip J Ward, Hans De Moel, and Olli Varis. Is physical water scarcity a new phenomenon? global\nassessment of water shortage over the last two millennia. Environmental Research Letters , 5(3):034006, 2010.\n[3]llis A Adams, Justin Stoler, and Yenupini Adams. Water insecurity and urban poverty in the Global South:\nImplications for health and human biology. American Journal of Human Biology , 32(1):e23368, 2020.\n[4]Matti Kummu, Joseph HA Guillaume, Hans de Moel, Stephanie Eisner, Martina Flörke, Miina Porkka, Stefan\nSiebert, Ted IE Veldkamp, and PJ Ward. The world’s road to water scarcity: shortage and stress in the 20th century\nand pathways towards sustainability. Scientific reports , 6(1):38495, 2016.\n[5]Kanta Kumari Rigaud, Alex de Sherbinin, Bryan Jones, Jonas Bergmann, Viviane Clement, Kayly Ober, Jacob\nSchewe, Susana Adamo, Brent McCusker, Silke Heuser, and Amelia Midgley. Groundswell : Preparing for\ninternal climate migration. World Bank Report , 2018, 2018.\n[6]Katie Meehan, Jason R Jurjevich, Nicholas MJW Chun, and Justin Sherrill. Geographies of insecure water access\nand the housing–water nexus in US cities. Proceedings of the National Academy of Sciences , 117(46):28700–\n28707, 2020.\n[7]Sadeq Khaleefah Hanoon, Ahmad Fikri Abdullah, Helmi ZM Shafri, and Aimrun Wayayok. Using scenario\nmodelling for adapting to urbanization and water scarcity: Towards a sustainable city in semi-arid areas. Periodicals\nof Engineering and Natural Sciences , 10(1):518–532, 2022.\n[8]World Bank. Beyond Scarcity: Water Security in the Middle East and North Africa . The World Bank, Washington\nDC, USA, 2017.\n[9]Annette Prüss-Ustün, Jennyfer Wolf, Jamie Bartram, Thomas Clasen, Oliver Cumming, Matthew C Freeman,\nBruce Gordon, Paul R Hunter, Kate Medlicott, and Richard Johnston. Burden of disease from inadequate\n19Urban forms and water insecurity\nwater, sanitation and hygiene for selected adverse health outcomes: an updated analysis with a focus on low-and\nmiddle-income countries. International journal of hygiene and environmental health , 222(5):765–777, 2019.\n[10] Julie Livingston. Water scarcity & health in urban Africa. Dædalus , 150(4):85–102, 2021.\n[11] Alemayehu A Ambel, Robert Bain, Tefera Bekele Degefu, Ayca Donmez, Richard Johnston, and Tom Slaymaker.\nAddressing gaps in data on drinking water quality through data integration and machine learning: evidence from\nEthiopia. npj Clean Water , 6(1):63, 2023.\n[12] World Health Organization et al. Progress on household drinking water, sanitation and hygiene 2000-2020: five\nyears into the SDGs. WHO , 1, 2021.\n[13] Peter Kareiva, Sean Watts, Robert McDonald, and Tim Boucher. Domesticated nature: shaping landscapes and\necosystems for human welfare. Science , 316(5833):1866–1869, 2007.\n[14] Denise Pumain and Marianne Guerois. Scaling laws in urban systems. In Santa Fe Institute, Working Papers ,\npages 1–26, 2004.\n[15] Carmen Cabrera-Arnau, Rafael Prieto-Curiel, and Steven R. Bishop. Uncovering the behaviour of road accidents\nin urban areas. Royal Society Open Science , 7(4):191739, 2020.\n[16] Hernán D Rozenfeld, Diego Rybski, José S Andrade, Michael Batty, H Eugene Stanley, and Hernán A Makse.\nLaws of population growth. Proceedings of the National Academy of Sciences , 105(48):18702–18707, 2008.\n[17] Luís MA Bettencourt, José Lobo, Deborah Strumsky, and Geoffrey B West. Urban scaling and its deviations:\nRevealing the structure of wealth, innovation and crime across cities. PloS One , 5(11):e13541, 2010.\n[18] Rémi Louf and Marc Barthelemy. How congestion shapes cities: from mobility patterns to scaling. Scientific\nReports , 4(1):1–9, 2014.\n[19] Bin Zhou, Stephan Thies, Ramana Gudipudi, Matthias KB Lüdeke, Jürgen P Kropp, and Diego Rybski. A Gini\napproach to spatial co2 emissions. Plos One , 15(11):e0242479, 2020.\n[20] Rafael Prieto-Curiel and Juan P Ospina. Large cities are less efficient for sustainable transport: The ABC of\nmobility. arXiv preprint arXiv:2212.13956 , 1, 2022.\n[21] Martina Flörke, Christof Schneider, and Robert I McDonald. Water competition between cities and agriculture\ndriven by climate change and urban growth. Nature Sustainability , 1(1):51–58, 2018.\n[22] Robert I McDonald, Katherine Weber, Julie Padowski, Martina Flörke, Christof Schneider, Pamela A Green,\nThomas Gleeson, Stephanie Eckman, Bernhard Lehner, Deborah Balk, et al. Water on an urban planet: Urbaniza-\ntion and the reach of urban water infrastructure. Global Environmental Change , 27:96–105, 2014.\n[23] Hug March and David Saurí. The suburbanization of water scarcity in the Barcelona metropolitan region:\nSociodemographic and urban changes influencing domestic water consumption. The Professional Geographer ,\n62(1):32–45, 2010.\n[24] Robert I McDonald, Pamela Green, Deborah Balk, Balazs M Fekete, Carmen Revenga, Megan Todd, and Mark\nMontgomery. Urban growth, climate change, and freshwater availability. Proceedings of the National Academy of\nSciences , 108(15):6312–6317, 2011.\n[25] Zachary Bischoff-Mattson, Gillian Maree, Coleen V ogel, Amanda Lynch, David Olivier, and Deon Terblanche.\nShape of a water crisis: Practitioner perspectives on urban water scarcity and ‘Day Zero’ in South Africa. Water\nPolicy , 22(2):193–210, 2020.\n[26] Guanghua Chi, Han Fang, Sourav Chatterjee, and Joshua E Blumenstock. Microestimates of wealth for all low-and\nmiddle-income countries. Proceedings of the National Academy of Sciences , 119(3):e2113658119, 2022.\n[27] Kira Vinke, Maria A Martin, Sophie Adams, Florent Baarsch, Alberte Bondeau, Dim Coumou, Reik V Donner,\nArathy Menon, Mahé Perrette, Kira Rehfeld, et al. Climatic risks and impacts in South Asia: extremes of water\nscarcity and excess. Regional Environmental Change , 17:1569–1583, 2017.\n[28] SM Papalexiou, D Koutsoyiannis, and C Makropoulos. How extreme is extreme? an assessment of daily rainfall\ndistribution tails. Hydrology and Earth System Sciences , 17(2):851–862, 2013.\n[29] MT Amin, M Rizwan, and AA Alazba. A best-fit probability distribution for the estimation of rainfall in northern\nregions of Pakistan. Open Life Sciences , 11(1):432–440, 2016.\n[30] Mohita Anand Sharma and Jai Bhagwan Singh. Use of probability distribution in rainfall analysis. New York\nScience Journal , 3(9):40–49, 2010.\n[31] Hassan Tolba Aboelnga, Hazim El-Naser, Lars Ribbe, and Franz-Bernd Frechen. Assessing water security in\nwater-scarce cities: Applying the integrated urban water security index (IUWSI) in Madaba, Jordan. Water ,\n12(5):1299, 2020.\n20Urban forms and water insecurity\n[32] Ingjerd Haddeland, Jens Heinke, Hester Biemans, Stephanie Eisner, Martina Flörke, Naota Hanasaki, Markus\nKonzmann, Fulco Ludwig, et al. Global water resources affected by human interventions and climate change.\nProceedings of the National Academy of Sciences , 111(9):3251–3256, 2014.\n[33] Lucy S. Tusting, Donal Bisanzio, Graham Alabaster, Ewan Cameron, Richard Cibulskis, Michael Davies, Seth\nFlaxman, Harry S. Gibson, Jakob Knudsen, Charles Mbogo, Fredros O. Okumu, Lorenz von Seidlein, Daniel J.\nWeiss, Steve W. Lindsay, Peter W. Gething, and Samir Bhatt. Mapping changes in housing in Sub-Saharan Africa\nfrom 2000 to 2015. Nature , 1, 2019.\n[34] Daniel Hoornweg and Kevin Pope. Population predictions for the world’s largest cities in the 21st century.\nEnvironment and Urbanization , 29(1):195–216, 2017.\n[35] Robert J Zomer, Jianchu Xu, and Antonio Trabucco. Version 3 of the Global Aridity Index and potential\nevapotranspiration database. Scientific Data , 9(1):409, 2022.\n[36] Stephen E Fick and Robert J Hijmans. Worldclim 2: new 1-km spatial resolution climate surfaces for global land\nareas. International Journal of Climatology , 37(12):4302–4315, 2017.\n[37] Juan C Duque, Nancy Lozano-Gracia, Jorge E Patino, Paula Restrepo, and Wilson A Velasquez. Spatiotemporal\ndynamics of urban growth in Latin American cities: An analysis using nighttime light imagery. Landscape and\nUrban Planning , 191:103640, 2019.\n[38] Jairo A Gómez, Jorge E Patiño, Juan C Duque, and Santiago Passos. Spatiotemporal modeling of urban growth\nusing machine learning. Remote Sensing , 12(1):109, 2019.\n[39] Juan C Duque, Nancy Lozano Gracia, Jorge Patino, and Paula Restrepo. Urban form and productivity: What is\nthe shape of Latin American cities? World Bank Policy Research Working Paper , 1(8697), 2019.\n[40] Susan Parnell, Thomas Elmqvist, Timon McPhearson, Harini Nagendra, and Sverker Sörlin. Introduction:\nSituating knowledge and action for an urban planet. The Urban Planet: Knowledge Towards Sustainable Cities ,\n1:1–16, 2018.\n[41] Erneson A Oliveira, Vasco Furtado, José S Andrade, and Hernán A Makse. A worldwide model for boundaries of\nurban settlements. Royal Society Open Science , 5(5):180468, 2018.\n[42] Yunfei Li, Sebastian Schubert, Jürgen P Kropp, and Diego Rybski. On the influence of density and morphology\non the Urban Heat Island intensity. Nature Communications , 11(1):1–9, 2020.\n[43] Bin Zhou, Diego Rybski, and Jürgen P Kropp. The role of city size and urban form in the surface urban heat\nisland. Scientific Reports , 7(1):1–9, 2017.\n[44] Karen C Seto, Shobhakar Dhakal, Anthony Bigio, Hilda Blanco, Gian Carlo Delgado, David Dewar, Luxin Huang,\nAtsushi Inaba, Arun Kansal, Shuaib Lwasa, et al. Human settlements, infrastructure and spatial planning. Climate\nChange 2014: Mitigation of Climate Change, Intergovernmental Panel on Climate Change IPCC , 1, 2014.\n[45] Bridget R Scanlon, Sarah Fakhreddine, Ashraf Rateb, Inge de Graaf, et al. Global water resources and the role of\ngroundwater in a resilient water future. Nature Reviews Earth & Environment , 4(2):87–101, 2023.\n[46] Brilé Anderson, Rafael Prieto-Curiel, and Jorge Eduardo Patiño-Quinchía. City shapes and climate change in\nAfrica. West African Papers , 1, 2023.\n[47] Shlomo Angel, Patrick Lamson-Hall, and Zeltia Gonzalez Blanco. Anatomy of density: measurable factors that\nconstitute urban density. Buildings and Cities , 2(1), 2021.\n[48] Tom Gleeson, Yoshihide Wada, Marc FP Bierkens, and Ludovicus PH Van Beek. Water balance of global aquifers\nrevealed by groundwater footprint. Nature , 488(7410):197–200, 2012.\n[49] Rafael Prieto-Curiel, Jorge E Patino, and Brilé Anderson. Scaling of the morphology of African cities. Proceedings\nof the National Academy of Sciences , 120(9):e2214254120, 2023.\n[50] Xuemei Bai, Alyson Surveyer, Thomas Elmqvist, Franz W Gatzweiler, Burak Güneralp, Susan Parnell, Anne-\nHelene Prieur-Richard, Paul Shrivastava, Jose Gabriel Siri, Mark Stafford-Smith, et al. Defining and advancing a\nsystems approach for sustainable cities. Current Opinion in Environmental Sustainability , 23:69–78, 2016.\n[51] Shlomo Angel, Sara Arango Franco, Yang Liu, and Alejandro M Blei. The shape compactness of urban footprints.\nProgress in Planning , 139:100429, 2020.\n[52] Karen C Seto, Burak Güneralp, and Lucy R Hutyra. Global forecasts of urban expansion to 2030 and direct impacts\non biodiversity and carbon pools. Proceedings of the National Academy of Sciences , 109(40):16083–16088, 2012.\n[53] Jennifer Candipan, Nolan Edward Phillips, Robert J Sampson, and Mario Small. From residence to movement:\nThe nature of racial segregation in everyday urban mobility. Urban Studies , 1:0042098020978965, 2021.\n21Urban forms and water insecurity\n[54] Jules Depersin and Marc Barthelemy. From global scaling to the dynamics of individual cities. Proceedings of the\nNational Academy of Sciences , 115(10):2317–2322, 2018.\n[55] Susan Athey. Beyond prediction: Using big data for policy problems. Science , 355(6324):483–485, 2017.\n[56] ChengHe Guan, Jihoon Song, Michael Keith, Yuki Akiyama, Ryosuke Shibasaki, and Taisei Sato. Delineating\nurban park catchment areas using mobile phone data: A case study of Tokyo. Computers, Environment and Urban\nSystems , 81:101474, 2020.\n[57] Xiao Xiang Zhu, Chunping Qiu, Jingliang Hu, Yilei Shi, Yuanyuan Wang, Michael Schmitt, and Hannes\nTaubenböck. The urban morphology on our planet–global perspectives from space. Remote Sensing of Environment ,\n269:112794, 2022.\n[58] OpenStreetMap contributors. Planet dump retrieved from https://planet.osm.org . https://www.\nopenstreetmap.org , 2021.\n[59] Wojciech Sirko, Sergii Kashubin, Marvin Ritter, Abigail Annkah, Yasser Salah Eddine Bouchareb, Yann Dauphin,\nDaniel Keysers, Maxim Neumann, Moustapha Cisse, and John Quinn. Continental-scale building detection from\nhigh resolution satellite imagery. arXiv preprint arXiv:2107.12283 , 1, 2021.\n[60] Thomas Esch, Elisabeth Brzoska, Stefan Dech, Benjamin Leutner, Daniela Palacios-Lopez, Annekatrin Metz-\nMarconcini, Mattia Marconcini, Achim Roth, and Julian Zeidler. World Settlement Footprint 3D-a first three-\ndimensional survey of the global building stock. Remote Sensing of Environment , 270:112877, 2022.\n[61] NASA JPL. NASADEM Merged DEM Global 1 arc second V001 [Data set]., 2020.\n[62] Aparna Sivaraman Prabha, Ashwin Ram, and Zareena Begum Irfan. Exploring the relative water scarcity across the\nIndian million-plus urban agglomerations: an application of the Water Poverty Index. HydroResearch , 3:134–145,\n2020.\n[63] Sadhana Nirandjan, Elco E Koks, Philip J Ward, and Jeroen CJH Aerts. A spatially-explicit harmonized global\ndataset of critical infrastructure. Scientific Data , 9(1):150, 2022.\n[64] International Benchmarking Network for Water and Sanitation Utilities. Utility tariffs dataset (ibnet tariffs db),\n2023.\n[65] World Health Organization and UNICEF. Joint monitoring programme for water supply, sanitation and hygiene\n(JMP), 2023.\n[66] Matti Kummu, Maija Taka, and Joseph HA Guillaume. Gridded global datasets for gross domestic product and\nHuman Development Index over 1990–2015. Scientific Data , 5(1):1–15, 2018.\n[67] Yong Yang and Ana V Diez-Roux. Walking distance by trip purpose and population subgroups. Am. J. Prev. Med. ,\n43(1):11–19, 2012.\n[68] Simon Damkjaer. Drivers of change in urban water and wastewater tariffs. h2oj, 3(1):355–372, 2020.\n[69] Laura Fernanda Abramovsky, Luis Alberto Andres, George Joseph, Juan Pablo Rud, German Eduardo Sember,\nand Michael David Thibert. Study of the distributional performance of piped water consumption subsidies in 10\ndeveloping countries. World Bank Policy Research Working Paper , 1(9245), 2020.\n[70] World Bank. High and dry: Climate change, water, and the economy . The World Bank, Washington DC, USA,\n2016.\n[71] Peter Ravenscroft and Lucy Lytton. Seeing the Invisible . Washington, DC: World Bank, Washington DC, USA,\n2022.\n[72] City of Paris. Des forêts urbaines bientôt sur quatre sites emblématiques, 2019.\n[73] Pieter Jan Kole, Ansje J Löhr, Frank GAJ Van Belleghem, and Ad MJ Ragas. Wear and tear of tyres: a stealthy\nsource of microplastics in the environment. International Journal of Environmental Research and Public Health ,\n14(10):1265, 2017.\n[74] Brilé Anderson, Rafael Prieto-Curiel, and Jorge Eduardo Patiño Quinchía. City shapes and climate change in\nAfrica. West African Papers , 38, 2023.\n[75] Maria Rusca, Cecilia Alda-Vidal, Michaela Hordijk, and Nienke Kral. Bathing without water, and other stories\nof everyday hygiene practices and risk perception in urban low-income areas: The case of Lilongwe, Malawi.\nEnvironment and Urbanization , 29(2):533–550, 2017.\n[76] Martin Behnisch, Tobias Krüger, and Jochen AG Jaeger. Rapid rise in urban sprawl: Global hotspots and trends\nsince 1990. PLOS Sustainability and Transformation , 1(11):e0000034, 2022.\n22Urban forms and water insecurity\n[77] ChengHe Guan, Jairo A Gómez, Pratyush Tripathy, Juan C Duque, Santiago Passos, Tong Cheng, Ying Li, and\nMichael Keith. Evaluating the impact of water protection policy on urban growth: A case study of Jiaxing.\nEnvironment and Planning B: Urban Analytics and City Science , 50(4):1000–1019, 2023.\n[78] R Core Team. R: A language and environment for statistical computing, 2018.\n[79] Clémentine Cottineau, Erez Hatna, Elsa Arcaute, and Michael Batty. Diverse cities or the systematic paradox of\nurban scaling laws. Computers, Environment and Urban Systems , 63:80–94, 2017.\n[80] Elsa Arcaute, Erez Hatna, Peter Ferguson, Hyejin Youn, Anders Johansson, and Michael Batty. Constructing\ncities, deconstructing scaling laws. Journal of the Royal Society Interface , 12(102):20140745, 2015.\n[81] Céline Rozenblat. Extending the concept of city for delineating large urban regions (LUR) for the cities of the\nworld. Cybergeo: European Journal of Geography , 954, 2020.\n[82] François Moriconi-Ebrard, Dominique Harre, and Philipp Heinrigs. Urbanisation dynamics in West Africa 1950 -\n2010, 2015.\n[83] Spatial data repository, the demographic and health surveys program.\n[84] Daniele Sartirano, Kyriaki Kalimeri, Ciro Cattuto, Enrique Delamónica, Manuel Garcia-Herranz, Anthony\nMockler, Daniela Paolotti, and Rossano Schifanella. Strengths and limitations of relative wealth indices derived\nfrom big data in Indonesia. Frontiers in Big Data , 6:10, 2023.\n[85] Center for International Earth Science Information Network CIESIN. Gridded population of the world, version 4\n(gpwv4): Population count, 2018.\n[86] Venla Niva, Alexander Horton, Vili Virkki, Matias Heino, Maria Kosonen, Marko Kallio, Pekka Kinnunen,\nGuy J Abel, Raya Muttarak, Maija Taka, et al. World’s human migration patterns in 2000–2019 unveiled by\nhigh-resolution data. Nature Human Behaviour , 1:1–15, 2023.\n[87] Leo Breiman, Jerome H Friedman, Richard A Olshen, and Charles J Stone. Classification and Regression Trees .\nPacific Grove, CA: Wadsworth & Brooks, California, USA, 1984.\n[88] Terry Therneau and Beth Atkinson. rpart: Recursive Partitioning and Regression Trees , 2022. R package version\n4.1.19.\n23"    },    {        "title": "2402.06780v1.Non_Radial_Free_Geodesics__I__In_Spatially_Flat_FLRW_Spacetime.pdf",        "content": "Non-Radial Free Geodesics. I. In Spatially Flat FLRW\nSpacetime\nOmar Nemoul1, Hichem Guergouri1, Jamal Mimouni1,2\n1*Research Unit in Scientific Mediation, CERIST, Constantine, 25016, Algeria.\n2Laboratoire de Physique Mathematique et Subatomique (LPMS), University of\nConstantine 1, Constantine, 25017, Algeria.\nContributing authors: omar.nemoul@umc.edu.dz; hichem.guergouri@univ-bejaia.dz;\njamalmimouni@umc.edu.dz;\nAbstract\nThis paper provides an analytical examination of non-radial geodesics within the context of the\nspatially flat Friedmann–Lemaˆ ıtre–Robertson–Walker (FLRW) spacetime. Using the symmetry prop-\nerties of the system, two constants of motion related to this dynamical system are derived. This\ntreatment enables us to explicitly compute the radial and angular peculiar velocities, as well as the\nevolution of the comoving radial distance and angle over time. The study introduces three distinct\nmethods to characterize geodesic solutions, and additionally examines the radial geodesic limit, the\nnull geodesic limit, and the comoving geodesic limit. Additionally, the paper investigates both non-\nradial and radial free geodesics within the Lambda Cold Dark Matter ( ΛCDM) model, presenting\ndetailed results and discussions. Lastly, we explore the total angle swept by a free particle from the\ninitial singularity to infinity as measured by a comoving observer, offering insights into the dynamics\nof free particles in FLRW spacetime.\nKeywords: Non-radial geodesics, FLRW spacetime\n1 Introduction\nExperiments investigating the Cosmic Microwave Background (CMB), such as the Cosmic Background\nExplorer (COBE) [1], Wilkinson Microwave Anisotropy Probe (WMAP) [2], and the Planck satellite [3],\nalong with studies examining the universe’s accelerating expansion rate [4, 5], have provided significant\nsupport for the ΛCDM model [6]. These extensive studies have consistently shown that the universe\nis dominated by a mysterious form of repulsive gravity known as dark energy causing its accelerating\nexpansion. Moreover, on a large scale, the observable universe exhibits spatial homogeneity, isotropy, and\ngeometric flatness. These features are fundamental in validating the Friedmann-Lemaˆ ıtre-Robertson-\nWalker (FLRW) metric as the standard model for describing our spacetime [7–10].\nThe study of geodesics in FLRW spacetime is crucial for several reasons, primarily because it provides\ndeep insights into the geometry and dynamics of the universe. Such studies are fundamental for analyzing\nthe path of light and other particles as they move through the universe. This can be vital for interpreting\nastronomical observations, such as the CMB, gravitational lensing [11], the dynamical evolution of\ngalaxies [12], and modeling the large-scale universe. Furthermore, analyzing these paths also allows for\nthe testing and refinement of cosmological models, helping to constrain parameters and explore theories\nlike dark energy [13] and general relativity modifications [14]. It would be ideal if the study included\nthe gravitational influence exerted by nearby masses; however, considering free geodesics could serve\nas a zero-th approximation in scenarios characterized by relatively weak gravitational fields. Research\non free geodesic motions in FLRW spacetimes, particularly for radially freely-falling test particles in\nthe absence of non-gravitational forces, has been rigorously investigated in Refs. [15–23]. While several\n1arXiv:2402.06780v1  [gr-qc]  9 Feb 2024studies [16, 18–21] have mainly focused on solving geodesic equations, other research efforts [22, 23] used\nthe symmetry of the system to construct the exact solution.\nSince our 3D space is assumed to be isotropic and homogeneous, it follows that radial and non-\nradial geodesics are equivalent, up to a reference choice (change of origin). However, this work aims to\ninvestigate non-radial geodesics by determining both the radial χ(t) and angular ϕ(t) of a freely-falling\ntest particle relative to a comoving observer, providing its radial and angular (transverse) motion across\nthe observer’s sky.\nFirst, we begin by offering an in-depth review of what has been already done in radial χ- motion\nproblem through a direct approach based on the Euler-Lagrange equation [24, 25] to determine the\ntimelike geodesics for radial motion. We will derive a constant of motion Arelated to the conjugate\nradial variable without solving tedious geodesic equations—a task that typically requires the explicit\ncomputation of Christoffel symbols and the resolution of two independent differential equations. This\napproach is commonly presented in nearly all introductory texts on general relativity (e.g., See [26]).\nSubsequently, we present in this review two different methods for describing the physical radial solutions.\nThe first parameterization, denoted by ( χi, A), includes the initial comoving radial distance χiat an\ninitial time tiand the constant of motion A. The second parameterization, represented by ( χi, vi), uses\nboth the comoving radial distance χiand the peculiar velocity viat an initial time ti.\nMoving on, we address the general motion problem which includes both radial χand angular ϕmove-\nments. Addressing this problem by directly solving either the Euler-Lagrange equations or the geodesic\nequations with respect to unknown sets of functions ( χ, ϕ) and ( t, χ, ϕ ), respectively, is an extremely\nchallenging task. The difficulty primarily arises from the complex nature of the resulting coupled partial\ndifferential equations system. To overcome this obstacle, the following strategy exploiting the inher-\nent symmetry of the system, will yields two constants of motion. The first constant Barises from the\nsystem’s isotropic symmetry concerning the angular variable ϕ, while the second one is derived by rec-\nognizing that the magnitude of peculiar velocity is a 3D scalar vector. Using this strategy, we can extend\nits known expression from radial to non-radial motion, thereby deriving the second constant of motion\nA, which now corresponds to both radial and angular peculiar velocity components. This approach sig-\nnificantly simplifies the process, making it possible to avoid directly solving both the Euler-Lagrange and\ngeodesic equations. Moreover, we present three different methods for characterizing the physical solu-\ntions ( χ(t), ϕ(t)). The first parameterization, signified by ( χi, ϕi, A, B ), uses both the comoving radial\ndistance χi, and the initial angle ϕiat a chosen initial time ti, and the two constants of motion AandB.\nThe second parameterization, indicated by ( χi, ϕi, vχ,i, vϕ,i), uses the initial peculiar velocity components\nvχ,iandvϕ,ias substitutes for the constants of motion AandB. In contrast, the third parameterization,\nrepresented by ( χi, ϕi, vpec,i, ψi), adopts polar coordinates ( vpec,i, ψi) for expressing the initial peculiar\nvelocity instead of ( vχ,i, vϕ,i), where vpec,istands for the magnitude of the initial peculiar velocity, while\nψiis the deviation angle of the 3D initial vector ⃗ vpec,ifrom the initial radial axis. Following this, we will\nprove that the non-radial geodesics successfully meet the radial geodesic limit, the null geodesic limit,\nand the comoving geodesic limit. Furthermore, we will provide a geometrical interpretation based on\nthe fact that our geodesics are represented by straight-line trajectories in the comoving frame. This can\nbe proven by deriving the geometrical formula for a non-radial straight line, starting from the resulting\nnon-radial solution. Subsequently, our discussion will cover the Killing vectors related to the symmetries\nof the dynamical system, aiming to determine their precise expressions.\nFinally, we will apply our results to the currently accepted cosmological model (ΛCDM), using\nthe recent Planck results [3]. These results will be visually represented through a series of graphs and\nanimations in both comoving and physical frames. Additionally, we introduce the concept of the total\nangle to quantify the entire angular journey a free particle can undertake from the Big Bang singularity\natt= 0 to an indefinitely extended time. We will analyze its characteristics within the ΛCDM model,\ncomparing it to its counterpart in a static (non-expanding) flat space. In Appendices D and E, we\nrigorously demonstrate that the non-radial solutions indeed satisfy both the Euler-Lagrange and the\ngeodesic equations. Throughout this paper, we use Greek letters µ, ν, etc., with values 0,1,2, and 3, to\nrepresent spacetime indices. Spatial indices will be denoted by Latin letters i, j, etc., with values 1,2,\nand 3. Our analysis adopts a metric signature of (+ ,−,−,−) in a spacetime coordinate system defined\nby cosmic time tand comoving spatial coordinates xi. (x, y, z ) and ( χ, θ, ϕ ) represent the comoving\nCartesian and spherical coordinate systems, respectively. Additionally, we employ a system of units\nwhere the speed of light is set to c= 1.\n22 General Geodesics in spatially flat FLRW Spacetime\n2.1 For general coordinates system\nBefore addressing general geodesics for free motion in flat FLRW spacetime, it is crucial to consider\nthe problem within a general coordinate system ( t, xi). The FLRW spacetime serves as a fundamental\nmathematical framework for understanding the large-scale structure and evolution of the universe. This\nmodel assumes that the universe is, on average, both homogeneous (similar at all points) and isotropic\n(appearing the same in all directions) on large cosmic scales. It provides a powerful tool for compre-\nhending the expansion of the universe and its defining characteristics. The FLRW spacetime metric in\na general coordinate system ( t, xi) can be expressed as follows\nds2=gµνdxµdxν=dt2−a2(t)γij(⃗ x)dxidxj, (1)\nwhere a(t) is the scale factor that represents the expansion of the 3D space, and γij(⃗ x) is the spatially\nhomogeneous and isotropic 3D metric in the comoving coordinate system ⃗ x= (xi). Geodesics trace\nthe shortest path in spacetime and their trajectories can be determined by minimizing the spacetime\ninterval ∆ s. Consequently, we formulate an action principle as\nS[xi(t)] =−mZ\nds=−mZ\ndtq\n1−a2(t)γij(⃗ x) ˙xi˙xj, (2)\nwhere the Lagrangian takes the form\nL(xi(t),˙xi(t), t) =−mq\n1−a2(t)γij(⃗ x) ˙xi˙xj. (3)\nTo ensure that the Lis expressed in energy units and to obtain the correct nonrelativistic limit consistent\nwith Newton’s law, we have multiplied by the factor “ −m”. From Eq. (1), we readily obtain\ndt\nds=1p\n1−a2(t)γij(⃗ x) ˙xi˙xj. (4)\nApplying the Euler-Lagrange equations for the three spatial coordinates xi(t) yields three differential\nequations\nd\ndt∂L\n∂˙xi=∂L\n∂xi. (5)\nThis yields\nd\ndt\"\na2(t)γij(⃗ x) ˙xj\np\n1−a2(t)γi′j′(⃗ x) ˙xi′˙xj′#\n=a2(t) ˙xk˙xl\n2p\n1−a2(t)γi′j′(⃗ x) ˙xi′˙xj′∂γkl(⃗ x)\n∂xi. (6)\nThese equations, alongside the expression fordt\ndsas indicated in Eq. (4), are equivalent to the system of\nfour geodesic equations\nd2xµ\nds2+ Γµ\nαβdxα\ndsdxβ\nds= 0, (7)\nwhere Γµ\nαβare the Christoffel symbols. Solving these equations to find general geodesics is a complex\ntask. Hence, for simplicity, much research effort has focused on radial geodesics.\n2.2 For comoving spherical coordinates\nIn the comoving spherical coordinate system, where xi= (χ, θ, ϕ ), the spacetime interval in Eq. (1) will\nhave the following form\nds2=dt2−a2(t)\u0002\ndχ2+χ2(dθ2+ sin2θdϕ2)\u0003\n, (8)\nwhere χ∈[0,+∞),θ∈[0, π], and ϕ∈(−π, π]. Consequently, the action in this context is expressed as\nS[χ(t), θ(t), ϕ(t)] =−mZ\nds=−mZ\ndtr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011\n(9)\n3where the Lagrangian is given by\nL(χ(t),˙χ(t), θ(t),˙θ(t),˙ϕ(t), t) =−mr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011\n(10)\nFrom Eq. (8), the differentiation of cosmic time relative to the spacetime interval (proper time) can\nbe straightforwardly derived as\ndt\nds=1r\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011. (11)\nApplying the Euler-Lagrange equations yields three differential equations for the three comoving\ncoordinates ( χ(t), θ(t), ϕ(t)) as\nd\ndt∂L\n∂˙χ=∂L\n∂χ, (12a)\nd\ndt∂L\n∂˙θ=∂L\n∂θ, (12b)\nd\ndt∂L\n∂˙ϕ=∂L\n∂ϕ. (12c)\nThis yields\nd\ndt\na2(t) ˙χr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011\n=a2(t)χ\u0010\n˙θ2+ sin2θ˙ϕ2\u0011\nr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011, (13a)\nd\ndt\na2(t)χ2˙θr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011\n=a2(t)χ2sinθcosθ˙ϕ2\nr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011, (13b)\nd\ndt\na2(t)χ2sin2θ˙ϕr\n1−a2(t)\u0010\n˙χ2+χ2˙θ2+χ2sin2θ˙ϕ2\u0011\n= 0. (13c)\nIt can be shown that the set of these three differential equations, when combined with the expression\nfordt\ndsas outlined in Eq. (11), are equivalent to the system of four geodesic equations Eq. (7) for the\ncoordinates system xµ= (t, χ, θ, ϕ ), which are\nd2t\nds2+a(t)˙a(t)\"\u0012dχ\nds\u00132\n+χ2\u0012dθ\nds\u00132\n+χ2sin2θ\u0012dϕ\nds\u00132#\n= 0, (14a)\nd2χ\nds2+ 2˙a(t)\na(t)dt\ndsdχ\nds−χ\u0012dθ\nds\u00132\n−χsin2θ\u0012dϕ\nds\u00132\n= 0, (14b)\nd2θ\nds2+ 2˙a(t)\na(t)dt\ndsdθ\nds+2\nχdχ\ndsdθ\nds−sinθcosθ\u0012dϕ\nds\u00132\n= 0, (14c)\nd2ϕ\nds2+ 2˙a(t)\na(t)dt\ndsdϕ\nds+2\nχdχ\ndsdϕ\nds+2 cos θ\nsinθdθ\ndsdϕ\nds= 0. (14d)\nThese equations are obtained after calculating the non-zero Christoffel symbol components (See\nAppendix A) for the metric element given by Eq. (8), and then substituting them in the geodesic\nequations (7). As previously mentioned, solving these differential equations to find general geodesics\npresents a complex task. Hence, to simplify the analysis and without losing generality, our focus in the\nfollowing section will be on radial geodesics.\n43 Radial Geodesic Motion in FLRW\nIn a homogeneous and isotropic universe, where no directions or locations are favored, it is possible\nto transform any general geodesic curve into one with a purely radial spatial component through a\nsuitable choice of local coordinates. Thus, without losing generality, we confine our attention to radial\ngeodesics. Within the comoving spherical coordinates system xi= (χ, θ, ϕ ), we consider both θandϕ\nto be constants, while the comoving radial coordinate χ=χ(t) varies with time. These assumptions\nsimplify the spacetime interval Eq. (8) into the following form\nds2=dt2−a2(t)dχ2. (15)\nThe action becomes in this case as\nS[χ(t)] =−mZ\nds=−mZ\ndtp\n1−a2(t) ˙χ2, (16)\nwith its Lagrangian\nL( ˙χ(t), t) =−mp\n1−a2(t) ˙χ2. (17)\nOne can simply obtain from Eq. (15) the differentiation of the cosmic time with respect to the spacetime\ninterval (proper time) as\ndt\nds=1p\n1−a2(t) ˙χ2. (18)\n3.1 Determining the constant of motion\nIt is evident that this Lagrangian is symmetric under any time-independent translation ϵof the comoving\nradial variable χ, as the Lagrangian does not depend on χ\nχ−→χ′=χ+ϵ. (19)\nAccording to Emmy Noether’s theorem [27], this symmetry implies a conservation law, represented by\na conserved quantity (constant of motion) that remains invariant over time. This constant of motion is\nderived from the Euler-Lagrange equation (12a):\nd\ndt\"\na2(t) ˙χp\n1−a2(t) ˙χ2#\n= 0. (20)\nThus, we obtain\na2(t) ˙χp\n1−a2(t) ˙χ2=A. (21)\nHere, Ais an arbitrary real number representing the initial radial comoving velocity of the particle.\nInverting Eq. (21) straightforwardly yields the radial comoving velocity ˙ χas\n˙χ(t) =1\na(t)Ap\na2(t) +A2. (22)\nIn this case, the peculiar velocity vpechas only a radial component vχwhich can be written as\nvpec(t) =vχ(t) =a(t) ˙χ(t) =Ap\na2(t) +A2. (23)\nEq. (22) is consistent with the result obtained in Ref. [17–23]. Due to the homogeneous and isotropic\nnature of our spatial space, this formula applies to all geodesic motions, including non-radial geodesics\nas well, making Eq. (23) a general formula for calculating the magnitude of the peculiar velocity vpec\nfor general geodesics. This observation shall be used later in our discussion to address the problem of\nnon-radial geodesics. As θandϕare constants, the remaining Euler-Lagrange equations (12b) and (12c)\n5vanish identically. It can be shown that the differential equation (20), when combined with the expression\nfordt\ndsas indicated in Eq. (18), are equivalent to the two geodesic equations\nd2t\nds2+a(t)˙a(t)\u0012dχ\nds\u00132\n= 0, (24a)\nd2χ\nds2+ 2˙a(t)\na(t)dt\ndsdχ\nds= 0. (24b)\nThese two geodesic equations (24a) and (24b) are induced from Eqs. (14a) and (14b), respectively, under\nthe consideration of radial motion. Meanwhile, the remaining equations (14c) and (14d) are identically\nzero in this context. Furthermore, it can be demonstrated that the radial velocity solution (22), along\nwith the expression fordt\ndsas provided in Eq. (18), satisfy the geodesic equations (14a) and (14b).\n3.2 The initial condition for peculiar velocity\nFrom Eq. (23), it is evident that to determine the peculiar velocity of the particle, the constant of motion\nAmust be fixed. However, instead of relying on Ato constrain the motion, a more practical method\ninvolves using a measured quantity, such as the initial peculiar velocity vi=vpec(ti) at an arbitrary\nchosen initial time ti. This can be achieved by substituting t=tiinto Eq. (21), resulting in the following\nexpression\nA=a2\ni˙χip\n1−a2\ni˙χ2\ni=aivip\n1−v2\ni, (25)\nwhere we use the notation a(ti) =aiand ˙χ(ti) = ˙χi. By replacing this expression of Ainto Eqs. (22)\nand (23), we obtain\n˙χ(t) =aivi\na(t)p\na2(t)(1−v2\ni) +a2\niv2\ni, (26)\nand\nvpec(t) =a(t) ˙χ(t) =aivip\na2(t)(1−v2\ni) +a2\niv2\ni. (27)\nThis gives the radial comoving velocity ˙ χand its corresponding radial peculiar velocity vpecat any time\nt, expressed in terms of the peculiar velocity viat an arbitrary initial time ti.\n3.3 Geodesics Parametrization for Radial Motion\nGeodesic motion is given by integrating over the radial comoving velocity, we have\nχ(t) =\f\f\f\fZt\nti˙χ(t)dt+χi\f\f\f\f. (28)\nTo ensure the comoving radial distance χremains positive at all times, we place the right-hand side of the\nequation within an absolute value. Building on previous work, we outline two methods to characterize\na specific geodesic solution for our test particle, as detailed in the following:\n•(χi, A)initial conditions: Using Eqs. (28) and (22), we express the comoving radial distance χof a\nradial geodesic at any given time t, under the specific initial conditions ( χi, A) as follows\nχ(t;χi, A) =\f\f\f\f\fZt\nti1\na(t′)Ap\na2(t′) +A2dt′+χi\f\f\f\f\f. (29)\n•(χi, vi)initial conditions: Using Eqs. (28) and (26), we can formulate the radial comoving distance\nχof a radial geodesic at any given time t, under the specific initial conditions ( χi, vi), in the following\nmanner\nχ(t;χi, vi) =\f\f\f\f\fZt\ntiaivi\na(t′)p\na2(t′)(1−v2\ni) +a2\niv2\nidt′+χi\f\f\f\f\f. (30)\nWe are free to select the initial time tifor fixing both the initial comoving distance and initial peculiar\nvelocity. However, Eq. (30) has limitations when choosing ti= 0. In this scenario, ai= 0 and vi=±1\nleading to an undefined integral term,0\n0. This issue arises because the peculiar velocity of all freely-\nfalling particles at the Big Bang singularity t= 0 equals the speed of light, thus making it unsuitable\n6for determining the initial radial peculiar velocity. To address this problem, one can choose the present\ntime t0as the reference time to fix the peculiar velocity v0=vpec(t0), while still considering the Big\nBang singularity time t= 0 for fixing the initial comoving distance χi=χ(0). Consequently, in this\ncase, the solution is parameterized by the initial conditions ( χi, v0). We implemented this method in the\nrecent paper [23], where we investigate the behaviors of all geodesics using this parametrization.\n3.4 Null Geodesics limit for radial motion\nIn Ref. [23], we show the correspondence between the zero-mass limit m= 0, A=±∞, and vi=±1.\nNow, applying these limits to Eqs. (39) and (30), we obtain the corresponding null geodesics\nχ(t) =\f\f\f\f±Zt\ntidt′\na(t′)+χi\f\f\f\f. (31)\n3.5 Comoving geodesics limit for radial motion\nIf we set A= 0 in Eq. (29) or vi= 0 in Eq. (30), the particle maintains a constant comoving distance\nχ(t) =χi. (32)\n4 Non-Radial Geodesic Motion in flat FLRW Spacetime\nNow, we explore the timelike geodesics for general free motion, including non-radial trajectories, within\nthe spatially flat FLRW spacetime. Without loss of generality, we restrict the motion to the ( xy) plane\nby setting θ=π\n2. This constraint simplifies the spacetime interval (8) into the following form\nds2=dt2−a2(t)(dχ2+χ2dϕ2), (33)\nwhere ϕis the angle measured from the origin relative to the x-axis. The action in this case is as follows\nS[χ(t), ϕ(t)] =−mZ\nds=−mZ\ndtq\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2, (34)\nwhere the Lagrangian is given by\nL(χ(t),˙χ(t),˙ϕ(t), t) =−mq\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2. (35)\nFrom Eq. (33), the differentiation of cosmic time with respect to the spacetime interval (proper time)\ncan be easily derived in the following form\ndt\nds=1q\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2. (36)\nApplying the Euler-Lagrange equations for ( χ(t), ϕ(t)), Eqs. (12a) and (12c), it yields\nd\ndt\na2(t) ˙χ(t)q\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2\n=a2(t)χ(t)˙ϕ2(t)q\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2, (37a)\nd\ndt\na2(t)χ2(t)˙ϕ(t)q\n1−a2(t) ˙χ2−a2(t)χ2˙ϕ2\n= 0, (37b)\nwhere the Euler-Lagrange equation (12b) for the variable θvanishes identically since we have set θ\nas a constant. It can be shown that the set of these two differential equations, when combined with\nthe expression fordt\ndsas given in Eq. (36), are equivalent to the system of the following three geodesic\nequations\nd2t\nds2+a(t)˙a(t)\"\u0012dχ\nds\u00132\n+χ2\u0012dϕ\nds\u00132#\n= 0, (38a)\n7d2χ\nds2+ 2˙a(t)\na(t)dt\ndsdχ\nds−χ\u0012dϕ\nds\u00132\n= 0, (38b)\nd2ϕ\nds2+ 2˙a(t)\na(t)dt\ndsdϕ\nds+2\nχdχ\ndsdϕ\nds= 0. (38c)\nThese three geodesic equations (38a), (38b), and (38c) are induced from Eqs. (14a), (14b), and (14d),\nrespectively, for motion confined to the ( xy) plane. In this context, the various terms of the left-\nhanded side of the remaining equation (14c) vanishes identically. However, since both the Euler-Lagrange\nequations and geodesic equations are sets of partial differential equations involving the coupled functions\n(χ, ϕ), directly solving these equations to determine free geodesics is a highly complex and challenging\ntask. To address this effectively, we undertake two steps to derive the solutions, identifying two constants\nof motion that characterize the comoving velocities ( ˙ χ,˙ϕ). The first constant of motion can be readily\nobtained from the expression (37b), while the second one will be determined by using the constant of\nmotion for radial geodesic motion (21), taking into consideration the general formula for the peculiar\nvelocity. Let us now delve into these two steps in more detail.\n4.1 Determining the Constant of Motions\nIt is evident that the Lagrangian (35) exhibits a symmetry under any time-independent rotation δof\nthe angular variable ϕ, as it is independent of ϕ\nϕ−→ϕ′=ϕ+δ. (39)\nAccording to Noether’s theorem [27], this symmetry in the dynamical system implies the existence of a\nconservation law represented by a conserved quantity (constant of motion) that remains invariant over\ntime. This constant of motion can be directly derived from the Euler-Lagrange equation (37b), which\nyields\na2(t)χ2(t)˙ϕ(t)q\n1−a2(t) ˙χ2(t)−a2(t)χ2(t)˙ϕ2(t)=B, (40)\nwhere Bis an arbitrary real number related to both the initial angular and radial velocities of the\nparticle. It is important to note that the sign of Bdetermines the sign of ˙ϕ. Therefore, if B > 0,\nthis indicates an increasing angle ϕfor the free particle’s trajectory, as measured from the origin point\nrelative to the x-axis, while if B <0 the angle decreases. In the case where B= 0, the particle remains\nat a fixed angle, implying purely radial motion. Consequently, ϕmust be a monotonic function, either\nincreasing or decreasing, depending on the sign of B. Inverting Eq. (40) allows us to directly derive the\nsquare of the angular peculiar velocity v2\nϕas\nv2\nϕ(t) =a2(t)χ2˙ϕ2=B2(1−a2(t) ˙χ2)\na2(t)χ2+B2. (41)\nWe still need to determine the second constant of motion. Using the fact that the peculiar velocity\nmagnitude vpec(t) is a 3D scalar (invariant under spatial coordinate transformations), which is obvious\nfrom its spatial covariant form\nv2\npec(t) =a2(t)γij(⃗ x) ˙xi(t) ˙xj(t). (42)\nIn our homogeneous and isotropic 3D space, any general geodesic curve can be transformed to have\na purely radial spatial part through an appropriate choice of local spatial coordinates. This allows us\nto generalize the 3D scalar peculiar velocity expression (23) for radial motion to encompass all general\ngeodesic motions. Consequently, we can write the following equation\nv2\npec(t) =v2\nχ(t) +v2\nϕ(t)\n=a2(t) ˙χ2(t) +a2(t)χ2(t)˙ϕ2(t)\n=A2\na2(t) +A2. (43)\nThe real number Ais now related to both the initial condition for radial and angular peculiar velocities.\nIt is worth mentioning that the sign of Ais redundant for our physical motion, i.e., does not affect\nthe physical nature of the motion where both “ A” and “ −A” represent the same physics. If A= 0,\n8it follows that vpec(t) = 0, resulting in both vχ(t) and vϕ(t) being zero. This scenario corresponds to\ncomoving geodesics. Accordingly, A= 0⇒B= 0, although the converse does not necessarily hold true.\nIfA= +∞, it leads to vpec(t) = 1 and one can show from Eq. (40) that A= +∞ ⇔ B= +∞. We\nobtain a system of two equations (40) and (43), involving the comoving velocity variables ( ˙ χ,˙ϕ). Solving\nthese two equations for ( ˙ χ,˙ϕ) gives\n˙χ(t) = sgn( ˙ χ(t))1\na(t)s\nA2−B2\nχ2(t)\na2(t) +A2, (44a)\n˙ϕ(t) =B\nχ2(t)\na(t)p\na2(t) +A2, (44b)\nand the peculiar velocity components ( vχ, vϕ) are\nvχ(t) = sgn( vχ(t))s\nA2−B2\nχ2(t)\na2(t) +A2, (45a)\nvϕ(t) =B\nχ(t)p\na2(t) +A2. (45b)\nIndeed, for B= 0, it becomes evident that ˙ϕ=vϕ= 0. In this situation, both ˙ χandvχprecisely match\nthe expressions for radial free geodesics as described in Eqs. (22) and (23), respectively. This limit effec-\ntively reduces the motion to a purely radial trajectory, completely eliminating any angular component.\nFurthermore, we observe that both radial and angular peculiar velocities decrease over time, approach-\ning zero as the scale factor goes to infinity. In our previous work (see Ref. [23]), we discussed how all free\ngeodesics in FLRW spacetime converge with the Hubble flow when we consider a cosmological constant\nΛ model with an equation of state parameter wΛ=−1. This satisfies convergent the condition wd<−1\n3\nfor the dominant cosmological component as time progresses towards infinity, as detailed in Ref. [19].\n4.2 The initial Conditions for Peculiar Velocity Components\nFrom Eqs. (45a) and (45b), it is clear that in order to determine the peculiar velocity components vχ(t)\nandvϕ(t) of the particle, the constants of motion AandBmust be fixed. As performed in the previous\nsection, we shall express AandBin terms of the initial peculiar velocities vχ,i=vχ(ti) and vϕ,i=vϕ(ti),\nwhich are the initial radial and angular peculiar velocities at a chosen initial time ti, respectively. To\nachieve this, one would substitute t=tiinto Eqs. (45a) and (45b) and then invert these equations to\nsolve for AandB, which gives\nA= sgn( A)ais\nv2\nχ,i+v2\nϕ,i\n1−v2\nχ,i−v2\nϕ,i, (46a)\nB=aiχivϕ,iq\n1−v2\nχ,i−v2\nϕ,i. (46b)\nBy substituting the derived expressions for AandBback into Eqs. (44) and (45), we obtain\n˙χ(t) = sgn( ˙ χ(t))ai\na(t)vuutv2\nχ,i+v2\nϕ,i−χ2\niv2\nϕ,i\nχ2(t)\na2(t)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i), (47a)\n˙ϕ(t) =ai\na(t)χivϕ,i\nχ2(t)q\na2(t)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i), (47b)\nand\nvχ(t) =a(t) ˙χ(t) = sgn( vχ(t))aivuutv2\nχ,i+v2\nϕ,i−χ2\niv2\nϕ,i\nχ2(t)\na2(t)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i), (48a)\n9vϕ(t) =a(t)χ(t)˙ϕ(t) =aiχivϕ,i\nχ(t)q\na2(t)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i), (48b)\nwhere 0 ≤v2\nχ,i+v2\nϕ,i≤1. These expressions determine the comoving velocity components ( ˙ χ,˙ϕ), and\ntheir corresponding peculiar velocities ( vχ, vϕ) at any time t, in terms of the initial peculiar velocities\n(vχ,i, vϕ,i) at an initial time ti. In addition, we can use the polar coordinates ( vpec,i, ψi) instead of\n(vχ,i, vϕ,i) to parameterize the comoving and peculiar velocity components. Here, vpec,irepresents the\nmagnitude of ⃗ vpec,iandψiindicates the deviation angle of ⃗ vpec,ifrom the initial radial axis. It is\nimportant to note that this initial radial axis at t=tiitself is oriented at an angle ϕirelative to the\nx-axis, which makes the initial launch angle to be ϕi+ψi. This polar coordinate system offers a more\neffective and practical approach for determining the comoving and peculiar velocity components as\n˙χ(t) = sgn( ˙ χ(t))aivpec,i\na(t)vuut1−χ2\ni\nχ2(t)sin2ψi\na2(t)(1−v2\npec,i) +a2\niv2\npec,i, (49a)\n˙ϕ(t) =aivpec,i\na(t)χi\nχ2(t)sinψiq\na2(t)(1−v2\npec,i) +a2\niv2\npec,i, (49b)\nand\nvχ(t) =a(t) ˙χ(t) = sgn( vχ(t))aivpec,ivuut1−χ2\ni\nχ2(t)sin2ψi\na2(t)(1−v2\npec,i) +a2\niv2\npec,i, (50a)\nvϕ(t) =a(t)χ(t)˙ϕ(t) =χi\nχ(t)aivpec,isinψiq\na2(t)(1−v2\npec,i) +a2\niv2\npec,i, (50b)\nwhere\nvpec,i=q\nv2\nχ,i+v2\nϕ,i, (51a)\nψi= arctan\u0012vϕ,i\nvχ,i\u0013\n, (51b)\nare constrained to 0 ≤vpec,i≤1 and −π < ψ i≤π. Notice that, at t=ti, inward radial motion\ncorresponds to ψi= 0, while outward radial motion is related to ψi=π.\n4.3 Geodesics Parametrization for General Motion\nGeodesic motion can be determined by integrating Eqs. (44a) and (44b), but this treatment does not\neasily apply to the radial χequation. This issue is thoroughly explored in Appendix B and a detailed anal-\nysis provided. Now, building upon previous work, we establish three distinct methods for characterizing\na specific geodesic solution for our freely-falling test particle, as described below\n•(χi, ϕi, A, B, sgn( ˙χi))initial conditions:\nUsing Eqs. (44a) and (44b), we write the comoving radial distance χ(as detailed in Appendix B) and\nthe angle ϕfor a free test-particle at any given time t, under the specific initial conditions ( χi, ϕi, A, B )\nand sgn( ˙ χi) as\nχ(t;χi, A, B, sgn( ˙χi)) =vuut Zt\nti|A|dt′\na(t′)p\na2(t′) +A2+ sgn( ˙ χi)r\nχ2\ni−B2\nA2!2\n+B2\nA2, (52a)\nϕ(t;χi, ϕi, A, B, sgn( ˙χi)) =Zt\ntiB\nχ2(t′;χi,A,B, sgn( ˙χi))\na(t′)p\na2(t′) +A2dt′+ϕi, (52b)\nwith the initial radial distance χi≥ |B/A|, the initial angle −π < ϕ i≤π,A= 0⇒B= 0 and the sign\nof the initial comoving radial veclocity sgn( ˙ χi) =±or 0.\n•(χi, ϕi, vχ,i, vϕ,i) initial conditions:\n10Using Eqs. (47a) and (47b), we can express the radial comoving distance χand the angle ϕof a free\ntest-particle at any given time t, under the specific initial conditions ( χi, ϕi, vχ,i, vϕ,i) as\nχ(t;χi, vχ,i, vϕ,i) =\u0014\u0012Zt\ntiai\na(t′)s\nv2\nχ,i+v2\nϕ,i\na2(t′)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i)dt′\n+χivχ,iq\nv2\nχ,i+v2\nϕ,i\u00132\n+χ2\niv2\nϕ,i\nv2\nχ,i+v2\nϕ,i\u00151/2\n,(53a)\nϕ(t;χi, ϕi, vχ,i, vϕ,i) =Zt\ntiai\na(t′)χivϕ,i\nχ2(t′;χi,vχ,i,vϕ,i)q\na2(t′)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i)dt′+ϕi, (53b)\nwith χi≥0,−π < ϕ i≤π, and 0 ≤v2\nχ,i+v2\nϕ,i≤1.\n•(χi, ϕi, vpec,i, ψi)initial conditions:\nUsing Eqs. (49a) and (49b), we can express the radial comoving distance χand the angle ϕof a free\ntest-particle at any given time t, under the specific initial conditions ( χi, ϕi, vpec,i, ψi) as\nχ(t;χi, vpec,i, ψi) =\u0014\u0012Zt\ntiai\na(t′)vpec,iq\na2(t′)(1−v2\npec,i) +a2\niv2\npec,idt′\n+χicosψi\u00132\n+χ2\nisin2ψi\u00151/2\n,(54a)\nϕ(t;χi, ϕi, vpec,i, ψi) =Zt\ntiai\na(t′)χivpec,isinψi\nχ2(t′;χi,vpec,i,ψi)q\na2(t′)(1−v2\npec,i) +a2\niv2\npec,idt′+ϕi, (54b)\nwith χi≥0,−π < ϕ i≤π, 0≤vpec,i≤1, and −π < ψ i≤π.\nIn the first parametrization (52), once AandBare fixed, the range of the initial comoving radial\ndistance χimust be greater than or equal to the potential closest approaching distance |B/A|. However,\nin the case of the second (53) and third (54) formulations, the initial comoving distance χiis not related\nto any other initial parameters. This independence makes these latter two approaches more practical for\ndescribing the non-radial solutions.\n4.4 The Closest Approaching Distance\nFrom the ˙ χformula in Eq. (44a), including the termq\nA2−B2\nχ2(t), it necessarily follows that χ(t)≥\f\fB\nA\f\f\nat any time t. This is consistent with the result obtained in Eq. (52a) indicating that for any non-zero\nAand real B, the closest approaching distance, denoted by χ∗, to which the free particle can approach\nthe center in the comoving frame at a specific moment called t∗. The following relations define the radial\ndistance χ∗and angle ϕ∗of this closest approaching point under different parameterizations\nχ∗=χ(t∗) =\f\f\f\fB\nA\f\f\f\f=χi|vϕ,i|q\nv2\nχ,i+v2\nϕ,i=χi|sinψi|, (55a)\nϕ∗=ϕ(t∗) =ϕi−sgn(B)sgn( ˙ χi) arccos \f\fB\nA\f\f\nχi!\n=ϕi−sgn(vϕ,i)sgn( vχ,i) arccos\n|vϕ,i|q\nv2\nχ,i+v2\nϕ,i\n\n= sgn( ψi(π−ψi))\u0010\n|ψi| −π\n2\u0011\n+ϕi. (55b)\nThe angle ϕ∗, corresponding to the point of closest approach, is derived from Eq. (66) which will be\ndiscussed later and matches the previously proven Eq. (52b) already established. The peculiar velocity\n11components at this specific time t∗are given by\n˙χ∗= ˙χ(t∗) = 0 , (56a)\n˙ϕ∗=˙ϕ(t∗) =A2\na∗Bp\na2∗+A2\n=ai(v2\nχ,i+v2\nϕ,i)\na∗χivϕ,iq\na2∗(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i)\n=aivpec,i\na∗χisinψiq\na2∗(1−v2\npec,i) +a2\niv2\npec,i. (56b)\nIn this context, an asterisk (*) is used to denote values at time t∗, indicating the moment that particle\nreaches its closest distance χ∗to the center. The condition for determining t∗in terms of the initial\ncondition ( χi, A, B )Zt∗\nti|A|dt′\na(t′)p\na2(t′) +A2+ sgn( ˙ χi)r\nχ2\ni−B2\nA2= 0. (57)\nIt is important to note that this point of closest approach defined by the coordinates in Eqs. (55a)\nand (55b), is observed only in the comoving frame. However, in the physical frame, a different point of\nminimum approach might exist at time tmin, determined by the following conditions\n˙χphy(tmin) =d\ndt(a(t)χ(t))\f\f\f\f\nt=tmin= 0, (58a)\n¨χphy(tmin) =d2\ndt2(a(t)χ(t))\f\f\f\f\nt=tmin>0. (58b)\n4.5 Geometrical Interpretation for Non-Radial Solution\nNow, let’s focus on the geometrical representation of the geodesic trajectory in the comoving frame.\nClearly, free geodesics in the comoving frame are straight lines with an inclination angle ϕi+ψi(the\ninitial launch angle), a claim substantiated in Appendix C. Without loss of generality, Figure 1 depicts\nsuch a general geodesic trajectory with in the ( xy) plane relative to a comoving reference frame with\nthe origin point O. From the expression for radial distance (52a), we can see a precise match with the\nright-angled triangle OMtM∗in Figure 1, having sides R(t),χ∗, and the hypotenuse χ(t). They are\ngiven as follows\n12χiR(t)Rt\nti|A|dt′\na(t′)√a2(t′)+A2χ(t)qχ2i−B2\nA2\nχ∗=\f\fBA\f\ftrajectory\nMiatti\nM∗Mtatt\nϕ(t)\nϕiψi\n⃗ vpec,i\nx O\nFig. 1 : Visualizing diagram of the geodesic trajectory in the comoving frame.\nR(t) =M∗Mt=\f\f\f\f\fZt\nti|A|dt′\na(t′)p\na2(t′) +A2+ sgn( ˙ χi)r\nχ2\ni−B2\nA2\f\f\f\f\f(59a)\nχ∗=OM∗=\f\f\f\fB\nA\f\f\f\f(59b)\nχ(t) =OMt (59c)\nThe Pythagorean theorem is clearly satisfied by Eq. (52a) within the comoving frame’s context.\nχ(t) =p\nR2(t) +χ2∗. (60)\nIn conclusion, this analysis, together with Fig.1, clarifies that the radial distance χ(t) solution in Eq.(52a)\nfor general geodesics beginning at the initial comoving distance χifrom point O, can be formulated in\nterms of the radial distance R(t) for radial geodesics starting atq\nχ2\ni−B2\nA2relative to the comoving\npoint M∗. Therefore, the expression of R(t) can be matched with the solution (29) for Radial Motion\n(RM) as following\nR(t) =χRM(t;r\nχ2\ni−B2\nA2, A), (61)\nwhere we have used the redundancy of sgn( A) to align it with sgn( ˙ χi). In our analysis, the measurement\nofϕ(t) remains consistent across both comoving and physical frames. As a result, ϕ(t) explicitly depends\nonly on the radial distance χand not on the scale factor a(t). Accordingly, we will derive the trajectory\nequation geometrically, establishing the relationship between ϕandχ. From Fig. 1, one can express ϕ(t)\nin terms of the following oriented angles\nϕ(t) =∡MiOMt+ϕi\n=∡MiOM∗+∡M∗OMt+ϕi\n=sgn ( ∡MiOM∗)|∡MiOM∗|+ sgn ( ∡M∗OMt)|∡M∗OMt|+ϕi. (62)\nIt can be shown that the sign of the oriented angles ∡MiOM∗and∡M∗OMtare given by\nsgn (∡MiOM∗) =−sgn(B)sgn( ˙ χi), (63a)\n13sgn (∡M∗OMt) = sgn( B)sgn( ˙ χ(t)), (63b)\nwhile the absolute angles |∡MiOM∗|and|∡M∗OMt|can be straightforwardly derived in terms of the\narctan from Figure 1 as follows\n|∡MiOM∗|= arctan\nq\nχ2\ni−B2\nA2\f\fB\nA\f\f\n, (64a)\n|∡M∗OMt|= arctan\nq\nχ2(t)−B2\nA2\f\fB\nA\f\f\n. (64b)\nIncorporating all these elements, we obtain the following result\nϕ(t) = sgn( B)\"\nsgn( ˙χ(t)) arctan\nq\nχ2(t)−B2\nA2\f\fB\nA\f\f\n−sgn( ˙χi) arctan\nq\nχ2\ni−B2\nA2\f\fB\nA\f\f\n#\n+ϕi. (65)\nAlternatively, using the “arccos” function, the angle can be expressed as\nϕ(t) = sgn( B)\"\nsgn( ˙χ(t)) arccos \f\fB\nA\f\f\nχ(t)!\n−sgn( ˙χi) arccos \f\fB\nA\f\f\nχi!#\n+ϕi. (66)\nIn terms of ( vχ,i, vϕ,i)\nϕ(t) = sgn( vϕ,i)\"\nsgn( ˙χ(t)) arccos\nχi\nχ(t)|vϕ,i|\nq\nv2\nχ,i+v2\nϕ,i\n−sgn(vχ,i) arccos\n|vϕ,i|q\nv2\nχ,i+v2\nϕ,i\n#\n+ϕi. (67)\nIn terms of ( vpec,i, ψi)\nϕ(t) = sgn( ψi(π−ψi))\u0014\nsgn( ˙χ(t)) arccos\u0012χi|sin(ψi)|\nχ(t)\u0013\n+|ψi| −π\n2\u0015\n+ϕi, (68)\nwhere we have used the sign relations\nsgn( ˙χi) = sgn( vχ,i) = sgn\u0010π\n2− |ψi|\u0011\n, (69a)\nsgn(B) = sgn( vϕ,i) = sgn( ψi(π−ψi)). (69b)\nThe trajectory equation can also be expressed in terms of χ(t) just by inverting Eq. (68), or directly by\nusing the sines law for the OMiMttriangle as illustrated in Fig. 1, one can write\nχ(t) =χi\f\f\f\fsin (ψi)\nsin (ϕ(t)−ϕi−ψi)\f\f\f\f. (70)\nIn our solution detailed in Eq. (52b), it appears that ϕ(t) explicitly depends on the scale factor a(t).\nIf the geodesic trajectory represents a straight line, then the explicit relation to the scale factor a(t)\nshould not appear in the expression for ϕ(t). However, we know that the trajectory indeed follows a\nstraight line, and thus the angle ϕ(t) can somehow be expressed as Eq. (66) without including the scale\nfactor a(t) in its expression. This angle treatment is fully addressed in Appendix C, by starting from\nthe geodesic solution Eq. (52b), leading to the same geometrical formula as given in Eq. (66), thereby\nconcluding that the non-radial geodesics are indeed straight lines in the comoving frame.\n4.6 Radial Geodesics limit\nThe radial geodesics limit corresponds to applying the limits B= 0 to Eqs. (52), or vϕ,i= 0 to Eqs. (53),\norψi= 0, πto Eqs. (54). We retrieve Eqs. (29) and (30) for radial geodesics, with ϕ(t) =ϕiremaining\nconstant. This consistency check ensures that the new radial-angular solution appropriately satisfies the\ncorrect radial limit.\n144.7 Null geodesics limit\nThe null geodesics limit is characterized by the condition A=B= +∞in Eqs. (52), or v2\nχ,i+v2\nϕ,i= 1\nin Eqs. (53), or vpec,i= 1 in Eqs. (54). Applying this limit we obtain the corresponding null geodesics\nχ(t) =s\u0012Zt\ntidt′\na(t′)+χicosψi\u00132\n+χ2\nisin2ψi, (71a)\nϕ(t) =Zt\nti1\na(t′)χisinψi\nχ2(t′)dt′+ϕi. (71b)\n4.8 Comoving geodesics limit\nIf we set A= 0⇒B= 0 in Eqs. (52), or vχ=vϕ= 0 in Eqs. (53) in Eqs. (53), or vpec,i= 0 in Eqs. (54),\nthe particle remains at a constant comoving distance.\nχ(t) =χi, (72a)\nϕ(t) =ϕi. (72b)\nIndeed, in this case, the zero-limit of peculiar velocities in both the radial and angular components\nimplies that the particle lies with the Hubble flow.\n4.9 Symmetry and Killing vectors\nThe comoving 4-momentum can be explicitly written as follows\nPµ=mdxµ\nds=mq\n1−v2pec(t)(1,˙χ,0,˙ϕ) =m s\n1 +A2\na2(t),sgn( ˙χ(t))\na2(t)s\nA2−B2\nχ2(t),0,B\na2(t)χ2(t)!\n.\n(73)\nOn the other hand, the conjugate momentum variables ( Pχ, Pθ, Pϕ), corresponding to the comoving\ncoordinates ( χ, θ, ϕ ), can be computed from the Lagrangian (35) as\nPχ=∂L\n∂˙χ=ma2(t) ˙χq\n1−v2pec= sgn( ˙ χ)ms\nA2−B2\nχ2, (74a)\nPθ=∂L\n∂˙θ= 0, (74b)\nPϕ=∂L\n∂˙ϕ=ma2(t)χ2(t)˙ϕ(t)q\n1−v2pec(t)=mB. (74c)\nThe energy E=Ptcan be obtained by performing a Legendre transform of the Lagrangian Las follows\nPt=Pχ˙χ+Pϕ˙ϕ−L=mq\n1−v2pec=ms\n1 +A2\na2(t). (75)\nConsistency between the comoving 4-momentum Pµand the conjugate momentum Pµcan be shown by\nusing the metric tensor gµνandgµνdefined by the spacetime interval (33) to raise and lower indices,\nrespectively. Now, the energy-momentum relation (mass-shell condition) can be verified\ngµνPµPν=E2−p2=m2, (76)\nwhere the physical 3D momentum pis defined as\np(t) =a(t)q\nγijPiPj. (77)\n15In cosmology, it is well-known that the magnitude of the comoving 3D conjugate momentum is a con-\nserved quantity for free motion. Consequently, we can infer that the constant of motion Ais expressed\nas follows\nA2=a4(t)γijPiPj\nm2=γijPiPj\nm2=a2(t)p2(t)\nm2. (78)\nNow, as previously mentioned, the constant of motion Bis related to the rotational symmetry described\nin Eq. (39) corresponding to a Killing vector ξB=∂ϕ. However, the symmetry associated with the\nconstant of motion Aremains undetermined. Using Noether’s theorem [27], we can express the conserved\nquantities associated with a spatial symmetry δxiof this dynamical system as follows\n∂L\n∂˙xiδxi=Piδxi. (79)\nwhere δxirepresents an infinitesimal symmetry transformation of the Lagrangian (35). To associate the\nconstant of motion A2given in Eq. (78) with such a symmetry, δximust assume the form\nδxi=εξi\nA=ε\nmγijPj, (80)\nfor an infinitesimally small parameter εandξArepresents the Killing vector for this symmetry. Therefore,\nthe specific form of the spatial time-dependent infinitesimal transformation is explicitly given by\nχ−→χ′=χ+ε\nmPχ, (81a)\nϕ−→ϕ′=ϕ+ε\nmPϕ\nχ2, (81b)\nwhere the conjugate momenta PχandPϕare expressed in Eqs. (74a) and (74c), respectively, and satisfy\nthe Euler-Lagrange equations (37a) and (37b) as follows\n˙Pχ=˙ϕ\nχPϕ, (82a)\n˙Pϕ=−2˙χ\nχ3Pϕ. (82b)\nAccording to Noether’s theorem, the transformations given in Eqs. (81a) and (81b) lead to the constant\nof motion A2in Eq. (79). It can be checked that this transformation indeed represents a symmetry of\nthe Lagrangian (35). The Killing vector ξArelated to this symmetry can be expressed as\nξA=1\nmPχ∂χ+1\nmPϕ\nχ2∂ϕ=±s\nA2−B2\nχ2∂χ+B\nχ2∂ϕ (83)\nTwo points should be highlighted:\n•Eqs. (81a) and (81b) precisely satisfies the expected radial limit described by the translation (19), as\nit results Pχ= constant and Pϕ= 0. Similarly, by setting B= 0 in Eq. (83), we get the Killing vector\ncorresponding to radial translation symmetry ξA=∂χ.\n•The Killing vectors ξAandξBrelated to the constants of motion AandB, respectively, indeed satisfy\nthe Killing equation (isometry condition)\n∇µξν+∇νξµ= 0 (84)\nwith∇µis the covariant derivative given by the connection (98).\n5 Application to ΛCDM Model\nThe ΛCDM is a scientific framework for comprehending the universe, including its evolution, structure,\nand composition. It is based on observations such as the distribution of galaxies and the CMB. By\ncomparing theoretical predictions with actual observations, the model’s accuracy is assessed, leading to\nits continuous refinement. This model states that the universe is composed of basic components, each\n16with a dimensionless density parameter at the present time of about (according to Planck Collaboration\n2018).\nRadiation: Ω r≈9×10−5\nMatter: Ω m≈0.315\nCosmological constant: Ω Λ≈0.685\nand it is based on the Friedmann-Lemaˆ ıtre equation\n˙a(t)\na(t)=H(t) =H0s\nΩr\na4(t)+Ωm\na3(t)+ Ω Λ. (85)\nH0is the Hubble parameter today (according to Plank 2018): H0≈67.4\u0010\nkm·s−1\nMpc\u0011\n. We have considered\nthat our 3D space is flat, with the curvature density parameter Ω k= 0. If we neglect the radiation density\nparameter Ω r≈0, it becomes straightforward to analytically solve the Friedman-Lemaˆ ıtre equation (85)\nfor the scale factor a(t) as follows\na(t) =\u0012Ωm\nΩΛ\u00131/3\nsinh2/3\u00123\n2p\nΩΛH0t\u0013\n. (86)\n5.1 Free Geodesics in ΛCDM model\nIn this section, we use thee radial-angular geodesic solution (54) determined by the initial conditions\n(χi, ϕi, vpec,i, ψi) to study freely-falling particles in FLRW spacetime within the context of the ΛCDM\nmodel. For this investigation, we set the initial time tifor the conditions at the current time, ti= 13.79\nGyr. In what follows, we adopt the units of distance and time in billions of light-years (Gly) and billions\nof years (Gyr), respectively, so that H−1\n0≈14.51 Gyr. We position ourselves as a comoving observer at\nthe origin χ= 0 disregarding any peculiar velocities attributable to the Milky Way Galaxy, the Solar\nSystem, or Earth’s own motion. We shall compute the integrals and generate the corresponding curves\nnumerically. We now aim to create a series of illustrative graphs to visualize the dynamics of freely-\nfalling particles in FLRW spacetime, based on specific initial conditions. The graphs we plan to generate\ninclude:\n1. The comoving radial distance χplotted over time t.\n2. The trajectories of geodesics in the comoving ( xy) plane.\n3. The physical radial distance χphyplotted over time t.\n4. The trajectories of geodesics in the physical ( xphy, yphy) plane.\n5. The variation of the angle ϕover time t.\nTo construct these graphs, we use the established geodesic solution for the radial distance and angle\nrelations, as detailed in (54a) and (54b), respectively, along with the expression of the scale factor\nprovided in Eq. (86). The physical radial distance is determined by multiplying the comoving radial\ndistance by the scale factor a(t), as follows\nχphy(t;χi, vpec,i, ψi) =a(t)χ(t;χi, vpec,i, ψi). (87)\nThe geodesic trajectories will be traced in both the comoving and physical frames using their respective\nparametric relations. For the comoving trajectories\n(\nx(t) =χ(t) cosϕ(t)\ny(t) =χ(t) sinϕ(t), (88)\nfor the physical trajectories(\nxphy(t) =a(t)χ(t) cosϕ(t)\nyphy(t) =a(t)χ(t) sinϕ(t). (89)\nIn addition to these graphical representations, we will augment our analysis of each geodesic trajectory\nwith their corresponding animations for both the comoving and physical frames. Please refer to the Sup-\nplemental Material, for a dynamic visualization of the motion of freely-falling particles in our spacetime\nover time.\n17(a)\n (b) See Online Resource at [28] to view animation.\n(c)\n (d) See Online Resource at [29] to view animation.\n(e)\nFig. 2 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (5 Gly ,0, vvec,i,π\n4) for different peculiar velocity magnitudes vvec,i.\n18(a)\n (b) See Online Resource at [30] to view animation.\n(c)\n (d) See Online Resource at [31] to view animation.\n(e)\nFig. 3 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (5 Gly ,0, vvec,i,3π\n4) for different peculiar velocity magnitudes vvec,i.\n19(a)\n (b) See Online Resource at [32] to view animation.\n(c)\n (d) See Online Resource at [33] to view animation.\n(e)\nFig. 4 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (5 Gly ,0,1, ψi) for different initial peuliar velocity angles ψi.\n20(a)\n (b) See Online Resource at [34] to view animation.\n(c)\n (d) See Online Resource at [35] to view animation.\n(e)\nFig. 5 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (5 Gly ,0,0.4, ψi) for different initial peuliar velocity angles ψi.\n21(a)\n (b) See Online Resource at [36] to view animation.\n(c)\n (d) See Online Resource at [37] to view animation.\n(e)\nFig. 6 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (χi,0,1,3π\n4) for different initial distances χi.\n22(a)\n (b) See Online Resource at [38] to view animation.\n(c)\n (d) See Online Resource at [39] to view animation.\n(e)\nFig. 7 : Dynamics of Freely-Falling Particles in FLRW Spacetime with initial conditions:\n(χi, ϕi, vvec,i, ψi) = (χi,0,0.4,3π\n4) for different initial distances χi.\n235.2 Results and Discussions\nWith these illustrative graphs and animations in hand, we shall visualize the distinctive characteristics\nof our spacetime and their influence on freely-falling particles over time. Key observations from our\nanalysis include:\n•Intersection with the x-axis at the present time : All considered free particles are characterized\nby an initial angle ϕi= 0 at the present time. As depicted in the animations of geodesic trajectories,\nit is observed that all particles intersect with the x-axis at the present time.\n•Inward Physical Motion despite Outward Comoving Motion : All free particles begin with an\ninward physical motion ( ˙ χphy(0)<0) despite an initial outward comoving motion ( ˙ χ(0)>0). This\nphenomenon arises because, at the very beginning ( t= 0), the recessional velocity had infinitely pos-\nitive value, whereas the peculiar velocity consistently remains below the speed of light. This behavior\nindicates that the Hubble flow was the principal force driving geodesic motion in the earliest stages\nof the universe.\n•Discrepancy in Approaching Distances : The closest approaching distance in the comoving frame\ndoes not necessarily align with that in the physical frame. An example of this discrepancy is illustrated\nby the null-geodesic curve ( vpec,i= 1), represented by the red curve in Fig. 2.\n•Behavior of Non-Radial Geodesics : For non-radial geodesics, some physical trajectories continue\nto recede, while others return, approaching the closest physical distance before receding again.\n•Behavior of Radial Geodesics : Radial geodesics demonstrate four potential paths (see Ref. [23]):\n– One-Way Journey Geodesics: Some particles perpetually move away from us.\n– Double-Reversing Geodesics: Some particles return to approach us, then start again to move\naway indefinitely. This unique behavior occurs only after the onset of the universe’s accelerated\nexpansion.\n– Recrossing Geodesics: Some particles return to get closer and then cross our location to indefinitely\ncontinue their outward trajectory on the opposite side.\n– Perpetually Approaching Geodesics: Some particles approach us indefinitely without ever crossing\nour location.\n•Convergence in the Comoving Frame : In the comoving frame, all geodesic trajectories converge\nto a point ( χ∞, ϕ∞), eventually aligning with the Hubble flow as time approaches infinity. This\nconvergence occurs within the framework of the ΛCDM model, where the equation of state parameter\nfor the dominant cosmological constant wΛ=−1 as time tends to infinity, satisfying the ‘rejoin Hubble\nflow’ condition wΛ<−1\n3(see Ref. [19]).\n•Correlation Between Particles Forming Straight-Line : Preliminary observations from Anima-\ntions 2, 3, 6, and 7 suggest that freely-falling particles forming a straight line over time in the\ncomoving frame, they definitively maintain this shape in the physical frame as well. This correlation\ncan be established from the fact that straight lines preserve their linear characteristics under any\nglobal spatial rescaling.\n•Radial Geodesics : In figures 4 and 5, the purple and red curves describe radial geodesics, which are\ncharacterized by ψi= 0 for inward motion, and ψi=πfor outward motion.\n•Step Function Behavior of the Angle for Radial Geodesics : As expected, the angle for radial\ngeodesics, depicted by the purple and red curves in Figs. 4 and 5, exhibits a step function behavior\nover time, jumping by πas the particle crosses the origin to continue on the opposite side.\n•Consistency Angle Between Frames : It is observed from the trajectories and animations that the\nangle made by a free particle remains consistent in both comoving and physical frames.\n6 The total angle\nWe define the total angle, denoted by ∆ ϕ, as the angle made by any free particle in FLRW spacetime\nfrom the Big Bang singularity to an infinite time, as observed from a comoving observer. Given that ϕ\nis a monotonic function of time, the total angle is given by\n∆ϕ=|ϕ(+∞)−ϕ(0)|. (90)\nIt can be expressed in terms of initial conditions ( χi, A, B ) as\n∆ϕ(χi, A, B ) =Z+∞\n0|B|\nχ2(t′;χi,A,B )\na(t′)p\na2(t′) +A2dt′\n24= sgn ( ˙ χ(∞)) arccos \f\fB\nA\f\f\nχ(∞)!\n−sgn ( ˙χ(0)) arccos \f\fB\nA\f\f\nχ(0)!\n, (91)\nand in terms of ( χi, vχ,i, vϕ,i) as\n∆ϕ(χi, vχ,i, vϕ,i) =Z+∞\n0aiχi|vϕ,i|\nχ2(t′;χi,vχ,i,vϕ,i)\na(t′)q\na2(t′)(1−v2\nχ,i−v2\nϕ,i) +a2\ni(v2\nχ,i+v2\nϕ,i)dt′\n= sgn ( ˙ χ(∞)) arccos\nχi\nχ(∞)|vϕ,i|\nq\nv2\nχ,i+v2\nϕ,i\n−sgn ( ˙χ(0)) arccos\nχi\nχ(0)|vϕ,i|\nq\nv2\nχ,i+v2\nϕ,i\n, (92)\nand in terms of ( χi, vpec,i, ψi) as\n∆ϕ(χi, vpec,i, ψi) =Z+∞\n0aiχivpec,i|sinψi|\nχ2(t′;χi,vpec,i,ψi)\na(t′)q\na2(t′)(1−v2\npec,i) +a2\niv2\npec,idt′\n= sgn ( ˙ χ(∞)) arccos\u0012χi\nχ(∞)|sin(ψi)|\u0013\n−sgn ( ˙χ(0)) arccos\u0012χi\nχ(0)|sin(ψi)|\u0013\n. (93)\nIn this discussion, we present a series of diagrams depicting the total angle ∆ ϕas a function of the initial\nradial distance χiat the present time. These diagrams represent free geodesics traveling at the speed of\nlight vpec,i= 1 with various choices of deviation angle ψiat 0,π\n6,π\n3,π\n2,2π\n3,5π\n6, and π. Figure 8 displays\nthe graphs corresponding to each specified deviation angle ψi. At first glance, it becomes evident that the\ntotal angle ∆ ϕapproaches zero as the free particle’s initial comoving distance χigoes to infinity. This\nphenomenon is observed in our ΛCDM cosmological model, where all geodesics tend to converge with\nthe Hubble flow (see Ref. [19]). This implies that extremely distant free particles will appear stationary\nin the sky over time . Notably, this is not due to a decrease in angular velocity with radial distance\nbut rather a characteristic of the ΛCDM model, dictating a finite comoving distance journey for these\nparticles. Additionally, ∆ ϕ=πfor the radial limit χi= 0 signifies that the particle crosses the origin\nand transitions to the other side, resulting in an instantaneous πshift in its angle.\nFig. 8 : These seven diagrams depict the total angle ∆ ϕas a function of the initial comoving distance\nχifor null geodesics ( vpec,i= 1) with different initial deviation angles ψi.\n25Furthermore, as visualized in Fig. 8, the behavior of the total angle for radial motion corresponding\ntoψi= 0 and ψi=πfor outward and inward traveler is delineated by the purple and red curves,\nrespectively. These curves exhibit a step-function behavior, with a discontinuous jump of −πat the\ninitial conditions χi≈47 Gly for ψi= 0 and χi≈16.68 Gly for ψi=π. This indicates that free\nparticles in outward motion from our position has only reached us if its radial comoving distance today\nχi<47 Gly, while those beyond χi>47 Gly will never do so. Similarly, only particles in inward\nmotion will only reach us if χi<16.68 Gly at the present time. These observations are consistent with\ncosmological predictions, where the furthest currently outgoing signal makes a maximum total angle π,\nwas emitted from our location at t= 0 and now spans a comoving distance of 47 Gly, defines the current\nparticle horizon. Likewise, the furthest incoming signal that will reach us in the infinite future is now\nat a comoving distance of 16 .68 Gly, which is known by the current event horizon. For more detail, see\nRef. [40].\nWe define the total angle made by a free particle for a static spatially flat universe, denoted by ∆∗ϕ.\nIts expression can be shown to take the following form\n∆∗ϕ=|ϕi−ϕ(0) + ψi|. (94)\nTo compare the total angle of a non-radial signal within the ΛCDM Model and the static universe, we\nhave assumed all signals start their propagation at t= 0 for the static universe. We can express ∆∗ϕin\nterms of ( χi, vpec,i, ψ) by using Eq. (68) as follows\n∆∗ϕ=\f\f\f\fsgn ( ˙χ(0)) arccos\u0012χi\nχ(0)|sinψi|\u0013\n−π\n2\f\f\f\f. (95)\nIn contrast to the ΛCDM model where the total angle ∆ ϕ→0 asχi→ ∞ , a free particle in a static\nuniverse with infinitely distant initial comoving radial distance χi→ ∞ at the present time would have\na total angle corresponding to ∆∗ϕ→ψi. To visualize this fact, we present a series of diagrams (See\nFigs. 9) comparing the total angle for the ΛCDM model ∆ ϕin Eqs. (93) with that of a static universe\n∆∗ϕin Eqs. (95) as a function of the initial radial distance χiat the present time, for a variety of null\ngeodesics with initial deviation angles ψi=π\n6,π\n2, and5π\n6. The behavior of the total angle in the ΛCDM\nmodel can be explained by the finite nature of their geodesics in the comoving frame (joining the Hubble\nflow; See Ref. [19]), where they reach a finite comoving distance, denoted as χ∞.\n26(a) (vpec,i, ψi) = (1 ,π\n6)\n(b) (vpec,i, ψi) = (1 ,π\n2)\n(c) (vpec,i, ψi) = (1 ,5π\n6)\nFig. 9 : This set of figures illustrates a comparative analysis of the total angle in the ΛCDM model (∆ ϕ)\nversus a static universe (∆∗ϕ) in terms of the initial comoving distance χiunder a range of deviation\nangles.\n277 Conclusion\nIn summary, we have presented a comprehensive analysis of non-radial, free, timelike geodesics in spa-\ntially curved FLRW spacetimes, using the symmetry of the system to construct the corresponding two\nconserved quantities. This allows for a straightforward and simpler determination of the radial ˙ χ(t) and\nthe angular ˙ϕ(t) velocities along with the evolution of the radial distance χ(t) and the angle ϕ(t) in terms\nof the constant of motions AandB, offering an advantageous alternative to traditional Euler-Lagrange\nor even the geodesic equation methods. In this framework, geodesics are fully determined by three appro-\npriately parameterizations of initial conditions ( χi, ϕi, A, B ) as detailed in Eqs. (52), ( χi, ϕi, vχ,i, vϕ,i)\nin Eqs. (53), and ( χi, ϕi, vpec,i, ψi) in Eqs. (54). Furthermore, we have elucidated the radial geodesic\nlimit, the null geodesic limit, and the comoving geodesic limit for the established non-radial solution.\nOur investigation into non-radial geodesic motions has addressed the geometrical picture within the\ncomoving frame, characterized by straight-line trajectories. This analysis has enabled us to derive an\nalternative formula for the angle ϕ(t), as shown in Eqs. (66), (67), and (68), which is explicitly deter-\nmined only by the comoving radial distance χ(t). These outcomes have been applied in particular to\nthe context of our expanding universe, as evidenced by the recent results from the Planck mission [3].\nUsing this data, we have constructed time diagrams to illustrate the variations in comoving and physi-\ncal radial distance, as well as the angle, effectively depicting the motion of a free particle. Additionally,\nwe enhanced these diagrams with animations that trace the trajectories of geodesics under various ini-\ntial conditions, further enriching our insights. Moreover, we analyzed the total angle ∆ ϕswept by a\nfree particle from the Big Bang singularity to an infinite time, considering its dependence on both ini-\ntial comoving distance and peculiar velocity. We have further compared the characteristics of this total\nangle for two cosmological frameworks: the ΛCDM model and a static universe. This study potentially\nenables a more profound understanding of the behavioral aspects of free particles from our perspective,\nencompassing not only radial but also non-radial motions. It is worth mentioning that this analysis has\nbeen extended to encompass non-Euclidean spatial spaces, as elaborated in Ref. [41]. However, we aim\nto expand this investigation to include the gravitational effects of galaxies and Earth’s peculiar velocity\nrelative to the CMB.\nA The non-zero Christoffel symbol components\nThe interval element in Eq. (8) can indeed be represented in matrix form for the ordinary basis\n(∂t, ∂χ, ∂θ, ∂ϕ) induced by the coordinates system ( t, χ, θ, ϕ ) as\ngµν=\n1 0 0 0\n0−a2(t) 0 0\n0 0 −a2(t)χ20\n0 0 0 −a2(t)χ2sin2θ\n, (96)\nand its inverse is\ngµν=\n1 0 0 0\n0−1\na2(t)0 0\n0 0 −1\na2(t)χ2 0\n0 0 0 −1\na2(t)χ2sin2θ\n. (97)\nThe Christoffel symbols defined in terms of the metric and its inverse, are given by\nΓµ\nαβ=1\n2gµν(∂αgβν+∂βgαν−∂νgαβ). (98)\nThe only non-zero Christoffel symbol components are\nΓt\nij=−1\n2gtt∂tgij=a(t)˙a(t)γij, (99a)\nΓi\ntj=1\n2gik∂tgjk=˙a(t)\na(t)δi\nj, (99b)\nΓχ\nθθ=−1\n2gχχ∂χgθθ=−χ, (99c)\nΓχ\nϕϕ=−1\n2gχχ∂χgϕϕ=−χsin2θ, (99d)\n28Γθ\nχθ=1\n2gθθ∂χgθθ=1\nχ, (99e)\nΓθ\nϕϕ=−1\n2gθθ∂θgϕϕ=−sinθcosθ, (99f)\nΓϕ\nχϕ=1\n2gϕϕ∂χgϕϕ=1\nχ, (99g)\nΓϕ\nθϕ=1\n2gϕϕ∂θgϕϕ=cosθ\nsinθ. (99h)\nBy implementing the geodesic equations (7) for each spacetime index µand using the non-zero Christoffel\nsymbols (99), one can ultimately derive a system of four equations (14).\nB Derivation of the radial comoving distance formula\nThe comoving radial distance for free non-radial geodesic motion in flat FLRW spacetime can be deter-\nmined by integrating Eq. (44a). To achieve this, the coordinate variable χmust be separated from the\ntime coordinate t. Once this separation is achieved, integration can be performed to obtain the desired\nresult Zχ\nχidχq\nA2−B2\nχ2=Zt\ntisgn[ ˙χ(t′)]\na(t′)p\na2(t′) +A2dt′. (100)\nThe left-hand side of Eq. (100) can be simply integrated using the formula\nZdχq\nA2−B2\nχ2=1\n|A|r\nχ2−B2\nA2. (101)\nBy performing integration on Eq. (100), we derive the following results\nχ(t) =vuut\"Zt\ntisgn( ˙χ(t′))|A|dt′\na(t′)p\na2(t′) +A2+r\nχ2\ni−B2\nA2#2\n+B2\nA2. (102)\nThe integral in Eq. (102) presents a non-trivial challenge, mainly due to the complexities of addressing\nsgn( ˙χ(t′)). To effectively handle this problem, it is crucial to consider all possible geodesic scenarios in\nthe comoving frame. In general cosmological models, free geodesics do not necessarily converge with the\nHubble flow as time approaches infinity. Indeed, as shown in Ref. [19], for this convergence to occur,\nthe equation of state parameter wdfor the dominant cosmological component, as time tends to infinity,\nmust satisfy wd<−1\n3. Consequently, this classification enables the identification of various scenarios:\n•changed ˙χ(t)sign\n– For a general cosmological model, if a particle starts with a negative comoving radial velocity\n˙χ(0)<0, it will reach a minimum distance χ∗att=t∗, then ultimately surpassing it, and finally\nattain a positive comoving radial velocity ˙ χ(0)>0 (See Fig. 10).\nχi\n\f\fB\nA\f\fatt= 0\natti\natt∗˙χ <0\n˙χ <0\n˙χ >0\nO\nFig. 10 : Visualization of the geodesic trajectory in the comoving frame, depicting a geodesic that\ninitiates with ˙ χ(0)<0 and crosses the closest approach point at t=t∗.\n29•unchanged ˙χ(t)sign\n– For a general cosmological model, a particle with an initial positive comoving radial velocity ˙ χ(0)>0\nwill continue to move away indefinitely. In this case, ˙ χ(t) is always positive (See Fig. 11a).\n– For a Hubble flow convergence model ( wd<−1\n3), a particle starting with a negative comoving radial\nvelocity ˙ χ(0)<0 will converge to a radial distance χ∞, without reaching the minimum distance\nχ∗. Here, ˙ χ(t) is always negative (See Fig. 11b).\n– For a general cosmological model, a particle beginning with zero peculiar velocity vpec(0) = 0\nremains comoving indefinitely. Consequently, ˙ χ(t) = 0 at all times t(See Fig. 11c).\nχi\n\f\fB\nA\f\fatti\natt= 0˙χ >0˙χ >0˙χ >0\nO\n(a) Visualization of the geodesic trajectory in the comoving frame, where ˙ χ(t)>0 at any given time t.\nχi\n\f\fB\nA\f\fatt= 0\natti\natt= +∞˙χ <0\n˙χ <0\n˙χ <0\nχ∞\nO\n(b) Visualization of the geodesic trajectory in the comoving frame, where ˙ χ(t)<0 at any given time t.\nχ(t) =χiatt\nO\n(c) Visualization of the geodesic trajectory in the comoving frame, where ˙ χ(t) = 0 at any given time t.\nFig. 11 : Combined visualization of the geodesic trajectories in the comoving frame under different\nconditions.\nIn the following discussion, we address this challenge by examining sgn( ˙ χ(t)) within the integral (102)\nfor each scenario.\n30•For changed ˙χ(t)sign\nIf there is a change in the sign of ˙ χ(t), it occurs only once, transitioning from negative to positive.\nTo mathematically prove this claim, one should first determine ˙ χ(t) by computing the time derivative\nof Eq. (102). Subsequently, by multiplying both sides of the resulting expression by sgn( ˙ χ(t)), one\narrives at the following equation\n|˙χ(t)|=\"Zt\ntisgn( ˙χ(t′))|A|dt′\na(t′)p\na2(t′) +A2+r\nχ2\ni−B2\nA2#\n×|A|\nχ(t)a(t)p\na2(t) +A2. (103)\nFrom this equation, and considering arbitrary initial conditions χi, A,andB, it follows that ˙ χ(t) can\nreach zero only once, precisely at t∗if and only if\nZt∗\ntisgn( ˙χ(t′))|A|dt′\na(t′)p\na2(t′) +A2+r\nχ2\ni−B2\nA2= 0. (104)\nSince ˙ χ(t) can only vanish at t∗, its sign remains unchanged between tiandt∗. This allows us to\nextract sgn( ˙ χ(t′)) from the integral in the condition (104) to sgn( ˙ χi), as illustrated below\nsgn( ˙χi)Zt∗\nti|A|dt′\na(t′)p\na2(t′) +A2+r\nχ2\ni−B2\nA2= 0. (105)\nThis condition can also be expressed by moving the sgn( ˙ χi) to the second term, as follows\nZt∗\nti|A|dt′\na(t′)p\na2(t′) +A2+ sgn( ˙ χi)r\nχ2\ni−B2\nA2= 0. (106)\nThis expression remains valid for all values of ˙ χi, including when it equals zero. This is because ˙ χi= 0\ncorresponds to ti=t∗leading to both the first and second terms in Eq. (106) vanishing, thereby\nensuring the equation’s validity in this specific case. The sign of the initial comoving velocity ˙ χican\nbe expressed from Eq. (105) as\nsgn( ˙χi) =−q\nχ2\ni−B2\nA2\nRt∗\nti|A|dt′\na(t′)√\na2(t′)+A2. (107)\nConsequently, the validity of condition. (105) depends on the following requirement\nsgn( ˙χi) = sgn( ti−t∗). (108)\nThis proves our claim that ˙ χ(t) can change its sign just once at t∗, transitioning from negative to\npositive value. As highlighted earlier, we show that this specific time t∗corresponds to the minimum\ndistance at which the particle approaches the original point\nt=t∗⇐⇒ ˙χ(t∗) = 0⇐⇒χ(t∗) =χ∗=\f\f\f\fB\nA\f\f\f\f. (109)\nNow, to tackle the integral term in Eq. (102), we divide the integration interval into two segments,\nwithin each sgn( ˙ χ(t)) remains constant, allowing us to extract sgn( ˙ χ(t)) from the integral. Using t∗\nas the division point for the integral in Eq. (102), the expression transforms as follows\nχ(t) =vuut\u0014Zt\nt∗sgn( ˙χ(t′))|A|\na(t′)p\na2(t′) +A2dt′+Zt∗\ntisgn( ˙χ(t′))|A|\na(t′)p\na2(t′) +A2dt′+r\nχ2\ni−B2\nA2\u00152\n+B2\nA2\n=vuut\u0014\nsgn( ˙χ(t))Zt\nt∗|A|\na(t′)p\na2(t′) +A2dt′\u00152\n+B2\nA2\n=vuut\u0014Zt\nt∗|A|\na(t′)p\na2(t′) +A2dt′\u00152\n+B2\nA2\n31=vuut\u0014Zt\nt∗|A|\na(t′)p\na2(t′) +A2dt′+Zt∗\nti|A|dt′\na(t′)p\na2(t′) +A2+ sgn( ˙ χi)r\nχ2\ni−B2\nA2\u00152\n+B2\nA2\n=vuut\u0014Zt\nti|A|\na(t′)p\na2(t′) +A2dt′+ sgn( ˙ χi)r\nχ2\ni−B2\nA2\u00152\n+B2\nA2. (110)\nIn the second step, the vanishing condition from Eq. (104) was applied, noting that sgn( ˙ χ(t′)) stays\nconstant over the interval t∗→t. In the third step, we removed the term sgn( ˙ χ(t)) as its squared value\nalways results in a positive sign. Finally, in the fourth step, the zero condition term from Eq. (106)\nwas inserted into the squared term, ultimately leading to the derivation of Eq. (52a).\nIn scenarios where t∗does not exist ( ˙ χ(t)̸= 0), sgn( ˙ χ(t′)) in Eq. (102) remains constant and can be\nextracted from the integral as sgn( ˙ χi), subsequently moving it to the second term to arrive at the\nsame expression (110).\nThus, the comoving radial solution, detailed in Eq. (52a), is convincingly proven.\nC Derivation of the geometric formula for angular coordinates\nNow, let us demonstrate that the geodesic solution for the angle ϕ, as shown in Eq. (52b), is identical\nto the geometrical formula presented in Eq. (66). This formula was derived by considering the geodesic\ntrajectories of free motion as straight lines. Initially, it appears that the geodesic angular solution (52b)\nexplicitly depends on the scale factor a(t), unlike the geometric formula (66). However, we aim to prove\nthat ϕ(t) does not depend explicitly on a(t). Consider Eqs. (44a) and (44b), by dividing them we can\neliminate the explicit relation with the scale factor, resulting in the following expression\n˙χ(t)\n˙ϕ(t)=sgn( ˙χ(t))\nBχ2(t)s\nA2−B2\nχ2(t). (111)\nUpon separating variables ϕandt, we arrive at\nZϕ\nϕidϕ=BZt\ntisgn( ˙χ(t)) ˙χ(t)\nχ2(t)q\nA2−B2\nχ2(t)dt. (112)\n•For changed ˙χ(t)sign\nIf ˙χ(t) changes its sign at time t∗, we divide the integral into two intervals: titot∗andt∗tot, where\nsgn( ˙χ(t)) remains constant within each interval. This leads us to\nϕ(t) =Bsgn( ˙χi)Zt∗\nti˙χ(t)\nχ2q\nA2−B2\nχ2(t)dt+Bsgn( ˙χ(t))Zt\nt∗˙χ(t)\nχ2q\nA2−B2\nχ2(t)dt+ϕi\n=Bsgn( ˙χi)Zχ∗\nχidχ\nχ2q\nA2−B2\nχ2+Bsgn( ˙χ(t))Zχ\nχ∗dχ\nχ2q\nA2−B2\nχ2+ϕi. (113)\nUsing the integration formula\nZdχ\nχ2q\nA2−B2\nχ2=1\n|B|arctan r\nA2\nB2χ2−1!\n, (114)\nwe find the following\nϕ(t) =Bsgn( ˙χi)1\n|B|\"\narctan r\nA2\nB2χ2(t∗)−1!\n−arctan r\nA2\nB2χ2\ni−1!#\n+Bsgn( ˙χ(t))1\n|B|\"\narctan r\nA2\nB2χ2(t)−1!\n−arctan r\nA2\nB2χ2(t∗)−1!#\n+ϕi. (115)\n32Applying the relation (55a) for the radial distance at t∗, we simplify the expression to\nϕ(t) = sgn( B)\"\nsgn( ˙χ(t)) arctan r\nA2\nB2χ2(t)−1!\n−sgn( ˙χi) arctan r\nA2\nB2χ2\ni−1!#\n. (116)\nThe integral in Eq. (52b) simplifies and leads to the same geometrical relation as Eq. (66).\n•For unchanged ˙χ(t)sign\nIn this case, sgn( ˙ χ(t′)) in Eq. (112) is a constant and can be extracted from the integral to sgn( ˙ χi)\nand by using the integral formula (114), we arrive at the same expressions in Eq. (116).\nThus, we conclude that the geodesic solution for the angle ϕis indeed identical to the geometrical\nformula derived under the assumption that the geodesic trajectories of free motion are straight lines.\nThis substantiates our conclusion that the general geodesic trajectory in our spacetime is effectively a\nstraight line in the comoving frame.\nD Checking that the solutions satisfy the Euleur-Lagrange\nEquations\nWe now confirm that our solution for the comoving radial-angular velocity in Eqs. (44a) and (44b)\nsatisfies the Euler-Lagrange equations (37a) and (37b). Let’s start by computing the left-hand and\nright-hand sides of the first equation (37a), resulting in the following\nLHS =d\ndt\na2˙χq\n1−a2˙χ2−a2χ2˙ϕ2\n= sgn( ˙ χ(t))d\ndt\"s\nA2−B2\nχ2#\n= sgn( ˙ χ(t))B2˙χ\nχ3q\nA2−B2\nχ2=B2\nχ3\na√\na2+A2, (117a)\nRHS =χa2˙ϕ2\nq\n1−a2˙χ2−a2χ2˙ϕ2=B2\nχ3\na√\na2+A2. (117b)\nThis confirms that the comoving velocities in Eqs. (44a) and (44b) indeed satisfy the Euler-Lagrange\nequation (37a). Similarly, for the Euler-Lagrange equation (37b), its satisfaction can be easily verified\nusing the constant of motion defined in Eq. (40). It yields\nd\ndt\na2χ2˙ϕq\n1−a2˙χ2−a2χ2˙ϕ2\n=d\ndtB= 0, (118)\nwhich confirms that our solution in Eqs. (44a) and (44b) also satisfies the Euler-Lagrange equation (37b).\nTherefore, it is verified that our solutions in Eqs. (44a) and (44b) satisfy both Euler-Lagrange\nequations (37a) and (37b).\nE Checking that the solutions satisfy the Geodesic Equations\nNow, let’s verify that our solution for the comoving radial-angular velocity in Eqs. (44a) and (44b) satisfy\nthe geodesic equations (38a), (38b), and (38c). This verification will be discussed for each equation\nseparately. To simplify the expressions, we will use a prime (′) in the upper right to indicate the derivative\nwith respect to the proper time s. Therefore, the geodesic equations (38) can be reformulated as\nt′′+a(t)˙a(t)(χ′2+χ2ϕ′2) = 0 , (119a)\nχ′′+ 2˙a(t)\na(t)t′χ′−χϕ′2= 0, (119b)\nϕ′′+ 2˙a(t)\na(t)t′ϕ′+2\nχχ′ϕ′= 0. (119c)\n33In what follows, we will substitute ˙ χand ˙ϕby their expressions in Eqs. (44a) and (44b) in terms of A\nandB.\n•For the first equation (119a) :one can write its left-hand side as\nt′′+a˙a(χ′2+χ2ϕ′2) =t′′+a˙at′2( ˙χ2+χ2˙ϕ2)\n=T1+T2, (120)\nwhere\nT1=t′′, (121a)\nT2=a˙at′2( ˙χ2+χ2˙ϕ2). (121b)\nFirstly, computing the differentiating of the cosmological time twith respect to the proper time s\ndt\nds=1q\n1−a2˙χ2−a2χ2˙ϕ2=p\na2(t) +A2\na(t). (122)\nNow, computing each term of Eq. (120) separately gives\nT1=d2t\nds2=d\nds\"p\na2(t) +A2\na(t)#\n=dt\ndsd\ndt\"p\na2(t) +A2\na(t)#\n=−A2˙a(t)\na(t)3, (123a)\nT2=a˙at′2( ˙χ2+χ2˙ϕ2) =A2˙a(t)\na(t)3. (123b)\nTherefore, T1is the opposite of T2, ensuring they cancel each other out. Consequently, the comoving\nvelocities in Eqs. (44a) and (44b) satisfy the geodesic equation (119a).\n•For the second equation (119b) :one can write its left-hand side as\nχ′′+ 2˙a\nat′χ′−χϕ′2=t′2[U1+U2+U3], (124)\nwhere\nU1= ¨χ, (125a)\nU2=−a˙a˙χ( ˙χ2+χ2˙ϕ2), (125b)\nU3= 2˙a\na˙χ−χ˙ϕ2. (125c)\nWe will proceed by computing each term in the equation (124) separately by using the solution in\nEqs. (44a) and (44b). We obtain the following\nU1= sgn( ˙ χ(t))\n−2˙a(t)A2+ 2˙a(t)B2\nχ2(t)−˙a(t)\na2(t)A4+˙a(t)\na2(t)A2B2\nχ2(t)\n(a2(t) +A2)3/2q\nA2−B2\nχ2(t)\n+B2\na2(t)χ3(t)\na2(t) +A2, (126a)\nU2= sgn( ˙ χ(t))\n−˙a(t)\na2(t)A4+˙a(t)\na2(t)A2B2\nχ2(t)\n(a2(t) +A2)3/2q\nA2−B2\nχ2(t)\n, (126b)\nU3= sgn( ˙ χ(t))\n2˙a(t)A2−2˙a(t)B2\nχ2(t)+ 2˙a(t)\na2(t)A4−2˙a(t)\na2(t)A2B2\nχ2(t)\n(a2(t) +A2)3/2q\nA2−B2\nχ2(t)\n−B2\na2(t)χ3(t)\na2(t) +A2. (126c)\nOne can check that U1+U2+U3= 0, confirming that the comoving velocities in Eqs. (44a) and (44b)\nsatisfy the geodesic equation (119b).\n•For the third equation (119c) :one can write its left-hand side as\n34ϕ′′+ 2˙a\nat′ϕ′+2\nχχ′ϕ′=t′2[V1+V2+V3], (127)\nwhere\nV1=¨ϕ, (128a)\nV2=−a˙a˙ϕ( ˙χ2+χ2˙ϕ2), (128b)\nV3= 2˙a\na˙ϕ+2\nχ˙χ˙ϕ. (128c)\nEach term in equation (127) will be calculated separately by using the solution in Eqs. (44a) and (44b).\nWe obtain the following\nV1=−sgn( ˙χ(t))2B\na2(t)χ3(t)\na2(t) +A2s\nA2−B2\nχ2(t)−2˙a(t)B\nχ2(t)+˙a(t)A2B\na2(t)χ2(t)\n(a2(t) +A2)3/2, (129a)\nV2=−˙a(t)A2B\na2(t)χ2(t)\n(a2(t) +A2)3/2, (129b)\nV3=2˙a(t)B\nχ2(t)+2˙a(t)A2B\na2(t)χ2(t)\n(a2(t) +A2)3/2+ sgn( ˙ χ(t))2B\na2(t)χ3(t)\na2(t) +A2s\nA2−B2\nχ2(t). (129c)\nIt can be verified that V1+V2+V3= 0, thereby proving that the comoving velocities in Eqs. (44a)\nand (44b) satisfy the geodesic equation (119c). In conclusion, these calculations thoroughly confirm that\nthe comoving radial-angular velocities presented in Eqs. (44a) and (44b) indeed satisfy the geodesic\nequations (38a), (38b), and (38c).\nReferences\n[1] Smoot, G.F., et al. : Structure in the cobe differential microwave radiometer first-year maps.\nAstrophysical Journal, vol. 396, no. 1, p. L1-L5 396, 1–5 (1992)\n[2] Hinshaw, G., et al. : Nine-year wilkinson microwave anisotropy probe (wmap) observations:\ncosmological parameter results. The Astrophysical Journal Supplement Series 208(2), 19 (2013)\n[3] Aghanim, N., et al. : Planck 2018 results-vi. cosmological parameters. Astronomy & Astrophysics\n641, 6 (2020)\n[4] Perlmutter, S., et al.: Measurements of ωandλfrom 42 high-redshift supernovae. The Astrophysical\nJournal 517(2), 565 (1999)\n[5] Riess, A.G., et al. : Observational evidence from supernovae for an accelerating universe and a\ncosmological constant. The astronomical journal 116(3), 1009 (1998)\n[6] Peebles, P.J.E.: Principles of Physical Cosmology vol. 27. Princeton university press, ??? (1993)\n[7] Friedmann, A.: ¨Uber die kr¨ ummung des raumes. Z. Phys. 10, 377–386 (1922)\n[8] Lemaitre, G.: A homogeneous universe of constant mass and increasing radius accounting for the\nradial velocity of extra-galactic nebulae. In: A Source Book in Astronomy and Astrophysics, 1900–\n1975, pp. 844–848. Harvard University Press, ??? (1979)\n[9] Robertson, H.P.: Kinematics and world-structure. Astrophysical Journal, vol. 82, p. 284 82, 284\n(1935)\n[10] Walker, A.G.: On milne’s theory of world-structure. Proceedings of the London Mathematical\nSociety 2(1), 90–127 (1937)\n[11] Schneider, P., Kochanek, C., Wambsganss, J.: Gravitational Lensing: Strong, Weak and Micro:\nSaas-Fee Advanced Course 33 vol. 33. Springer, ??? (2006)\n35[12] Binney, J., Tremaine, S.: Galactic Dynamics vol. 20. Princeton university press, ??? (2011)\n[13] Carroll, S.M.: The cosmological constant. Living reviews in relativity 4(1), 1–56 (2001)\n[14] Saridakis, E.N., Lazkoz, R., Salzano, V., Moniz, P.V., Capozziello, S., Jim´ enez, J.B., De Laurentis,\nM., Olmo, G.J.: Modified Gravity and Cosmology. Springer, ??? (2021)\n[15] Baumann, D.: Cosmology, part iii mathematical tripos. University lecture notes 56, 34 (2014)\n[16] Whiting, A.B.: The expansion of space: Free particle motion and the cosmological redshift. arXiv\npreprint astro-ph/0404095 (2004)\n[17] Davis, T.M., Lineweaver, C.H., Webb, J.K.: Solutions to the tethered galaxy problem in an expand-\ning universe and the observation of receding blueshifted objects. American Journal of Physics 71(4),\n358–364 (2003)\n[18] Grøn, Ø., Elgarøy, Ø.: Is space expanding in the friedmann universe models? American Journal of\nPhysics 75(2), 151–157 (2007)\n[19] Barnes, L.A., Francis, M.J., James, J.B., Lewis, G.F.: Joining the hubble flow: Implications for\nexpanding space. Monthly Notices of the Royal Astronomical Society 373(1), 382–390 (2006)\n[20] Kerachian, M.: Uniformly accelerated traveler in an flrw universe. Physical Review D 101(8), 083536\n(2020)\n[21] Vachon, G., Vanderwee, R., Faraoni, V.: Revisiting geodesic observers in cosmology. The European\nPhysical Journal C 81, 1–9 (2021)\n[22] Cot˘ aescu, I.I.: Kinemaitcs in spatially flat flrw spacetimes. Chinese Physics C 45(10), 105101 (2021)\n[23] Nemoul, O., Guergouri, H., Mimouni, J.: Studying the behavior of radial free geodesics\ninλcdm model. International Journal of Geometric Methods in Modern Physics (2024).\nDOI:10.1142/S0219887824501342\n[24] Euler, L.: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes Sive Solutio\nProblematis Isoperimetrici Latissimo Sensu Accepti vol. 1. Springer, ??? (1952)\n[25] Lagrange, J.L.: Essai D’une Nouvelle Ethode Pour De’terminer les Maxima, et les Minima des\nFormules Integrales Indefinies. De l’Imprimerie royale, ??? (1761)\n[26] Hartle, J.B.: Gravity: an Introduction to Einstein’s General Relativity. American Association of\nPhysics Teachers (2003)\n[27] Noether, E., Tavel, M.: Invariant variation problems. arXiv preprint physics/0503066 (2005)\n[28] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( χi, ϕi, ψi)=(5,0,π\n4) and different values of vpec,i\n[29] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( χi, ϕi, ψi)=(5,0,π\n4) and different values of vpec,i\n[30] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( χi, ϕi, ψi)=(5,0,3π\n4) and different values of vpec,i\n[31] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( χi, ϕi, ψi)=(5,0,3π\n4) and different values of vpec,i\n[32] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( χi, ϕi, vpec,i)=(5,0,1) and different values of ψi\n[33] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( χi, ϕi, vpec,i)=(5,0,1) and different values of ψi\n36[34] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( χi, ϕi, vpec,i)=(5,0,0.4) and different values of ψi\n[35] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( χi, ϕi, vpec,i)=(5,0,0.4) and different values of ψi\n[36] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( ϕi, vpec,i, ψi)=(0,1,3π\n4) and different values of χi\n[37] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( ϕi, vpec,i, ψi)=(0,1,3π\n4) and different values of χi\n[38] See Online Resource for a detailed animation showcasing geodesic trajectories in the comoving\nframe with initial conditions ( ϕi, vpec,i, ψi)=(0,0.4,3π\n4) and different values of χi\n[39] See Online Resource for a detailed animation showcasing geodesic trajectories in the physical frame\nwith initial conditions ( ϕi, vpec,i, ψi)=(0,0.4,3π\n4) and different values of χi\n[40] Davis, T.M., Lineweaver, C.H.: Expanding confusion: common misconceptions of cosmological hori-\nzons and the superluminal expansion of the universe. Publications of the Astronomical Society of\nAustralia 21(1), 97–109 (2004)\n[41] Nemoul, O., Guergouri, H., Mimouni, J.: Non-Radial Free Geodesics. II. In Spatially Curved FLRW\nSpacetime. will be published\n37"    },    {        "title": "2402.06832v3.Arithmetic_functions_that_remain_constant_on_strings_of_integers.pdf",        "content": "arXiv:2402.06832v3  [math.NT]  28 Mar 2024Arithmetic functions that remain constant on strings of int egers\nNoah Lebowitz-Lockard\nDepartment of Mathematics\nUniversity of Texas at Tyler, Tyler, TX\nnlebowitzlockard@uttyler.edu\nJ.C. Saunders\nDepartment of Mathematical Sciences\nMiddle Tennessee State University, Murfreesboro, TN\nJohn.Saunders@mtsu.edu\nApril 1, 2024\nAbstract\nIn2013, Pollack etal. [5]gaveanasymptoticexpressionfor thelargest kintermsof xforwhichthe\nstring of inequalities ϕ(n+1)≥ϕ(n+2)≥ ··· ≥ϕ(n+k) holds for some n≤x, whereϕis the Euler\ntotient function. Their method of proof also works to derive an upper bound for kwithϕreplaced\nbyσ, the sum-of-divisors function. In 2023, the first author and Vandehey [4] gave an alternative\nproof for a weaker lower bound on kin the string of equalities ϕ(n+1) =ϕ(n+2) =...=ϕ(n+k).\nHere we improve their bound with a similar method of proof.\n1 Introduction\nFor a given arithmetic function f, letFf(x) be the largest kfor which the set of equalities f(n+1) =\nf(n+2) =···=f(n+k) has a solution satisfying n+k≤x. In addition, let Gf(x) be the largest kfor\nwhich the set of inequalities f(n+1)≥f(n+2)≥ ··· ≥f(n+k) has a solution satisfying n+k≤x.\nThe functions FfandGfhave been studied for various functions f. Erd˝ os [2] conjectured that\nFϕ(x)→ ∞asx→ ∞, whereϕis Euler’s totient function. To date, however, the only known solutio n\nto the equation ϕ(n+1) =ϕ(n+2) =ϕ(n+3) isn= 5185. Pollack, Pomerance, and Trevi˜ no [5, Thm.\n1.5] found an asymptotic formula for Gϕ(x).\nTheorem 1. Asx→ ∞, we have\nGϕ(x)∼log3x/log6x,\nwhere (here and below) logkxrefers to the kth iterate of the logarithm.\nThere are also results for other arithmetic functions as well. By mod ifying the proof of the previous\ntheorem, one can also show that Gσ(x)∼log3x/log6x, whereσis the sum-of-divisors function. Sp˘ ataru\n[6] and the first author and Vandehey [4] independently proved tha tFd(x) = exp( O(3/radicalbig\nlogxlog2x)),\nwheredis the number of divisors function. The first author and Vandehey [ 4] also showed that Gd(x) =\nO(/radicalbig\nlogxlog2x). Their proofs relied on bounding the size of d(n+1),...,d(n+k) from below and using\nthe fact that these quantities had to be equal. They also used this t echnique to derive an alternative\nproof that Fσ(x) = exp( O(/radicalbig\nlogxlog2x)). In this note, we find a similar proof of a better bound for\nFσ(x).\nTheorem 2. Asx→ ∞,Fσ(x) =O/parenleftbig\n(logx)1/0.677/parenrightbig\n.\n1Note 1. The above exponent 0 .677 is not exact and refers to the exponent of 0 .677 in Theorem 2 in [1].\nBy modifying our proof, we can also show that Fϕ(x) andFσd(x) have the same upper bound, where\nσd(n) is the sum of the divisors of n.\n2 Proof\nOur proof begins with the following result of Baker and Harman [1], qu oted as Theorem 1 in [3].\nNote 2. Throughout the rest of the paper we let pandqdenote prime values.\nLemma 1. For every a∈Zand 0< θ≤0.677 there exists 0 < δ(θ)<1 such that for sufficiently large\nx > X(a,θ) we have/summationdisplay\np≤x\nP(p+a)>xθ1> δ(θ)x\nlogx,\nwhereP(n) is the largest prime factor of n.\nWe use Lemma 1 to derive the following result.\nLemma 2. There exists a constant C >0 such that for sufficiently large x, we have\nx0.677\nlogx≤C·#{q:q > x0.677,∃p≤xsuch that p≡ −1 (mod q)}.\nProof.Fix a positive integer aand some θ≤0.677. The previous lemma implies that there exists\nδ:=δ(θ)∈(0,1) such that if xis sufficiently large, then\n/summationdisplay\np≤x\nP(p+a)>xθ1> δx\nlogx.\nIn particular, setting a=−1 andθ= 0.677, we may choose δso that\n/summationdisplay\np≤x\nP(p+1)>x0.6771>δx\nlogx.\nfor all sufficiently large x. Put another way the above is\n#{p≤x:∃q > x0.677such that p≡ −1 (mod q)}>δx\nlogx.\nFor anypin the set that is the left-hand side of the above inequality the existe nce ofqis unique since\nif there were two such values of q, sayq1andq2, for any pwe would have p+1> q1q2> x0.677+0.677=\nx1.354> p1.354, a contradiction. Therefore, we may partition the above set as\n/uniondisplay\nx0.677<q<x{p:p≤x,p≡ −1 (mod q)}\n#{p:p≤x,∃q > x0.677,p≡ −1 (mod q)}=/summationdisplay\nx0.677<q≤x\n∃p≤xsuch that p≡−1 (mod q)#{p≤x:p≡ −1 (mod q)}.\nSince for each q > x0.677\n#{p≤x:p≡ −1 (mod q)} ≤/ceilingleftbiggx+1\nq/ceilingrightbigg\n≪x\nq< x0.323,\n2we havex\nlogx≪#{q:q > x0.677,∃p≤xsuch that p≡ −1 (mod q)}·x0.323.\nThus,\nx0.677\nlogx≪#{q:q > x0.677,∃p≤xsuch that p≡ −1 (mod q)}.\nUsing this result, we can bound the length of a sequence of numbers with the same sum of divisors.\nLemma 3. Let\nT=σ(n+1) =σ(n+2) =···=σ(n+k).\nLetCbe the constant in Lemma 1. Then,\nexp/parenleftBigg\nC·0.667·/parenleftbiggk\n2/parenrightbigg0.677/parenrightBigg\n≤T.\nProof.We may assume that kis sufficiently large so that Lemma 1 holds with x=k/2. Consider a prime\nq >/parenleftbigk\n2/parenrightbig0.677such that there exists a prime p≤k\n2withp≡ −1 (modq). Then there exists 1 ≤i≤k\nsuch that p/bardbln+i. Therefore, p+1|T, so that q|T.Tis therefore, bounded below by the product of\nall suchq. Lemma 1 therefore implies\nexp/parenleftBigg\nC·0.667·/parenleftbiggk\n2/parenrightbigg0.667/parenrightBigg\n=/parenleftbiggk\n2/parenrightbiggC·0.667·(k\n2)0.667\nlogk−log 2\n≤T.\nWe now show that Lemma 3 directly implies Theorem 2.\nProof.By Lemma 3, there exists a constant D >0 such that\nT≥exp/parenleftbig\nDk0.677/parenrightbig\n.\nBy Mertens’ Theorem we have T≪xloglogx. Therefore, k1/0.677≪logx. Thus,k≪(logx)1/0.677.\nRemark 1. The above proof also works with σreplaced with ϕor the Carmichael function λ. In\nparticular for λwe improve another result in [4]. One just has to take a= 1 in applying Lemma 1 and\nreplacing the congruence p≡ −1 (modq) in Lemma 2 with p≡1 (modq) in both cases.\nRemark 2. The above proof also works with σreplaced with σd, wheredis any positive odd integer.\nOne simply makes the observation in the proof of Lemma 3 that p/bardbln+ialso implies that pd+ 1|T,\nwhereT=σd(n+i) so that p+1|Tsincep+1|pd+1. Unfortunately though this proof does not work\nfor positive even values of d.\nRemark 3. The Elliott-Halberstam Conjecture [7] implies that in Lemma 1 we can inc rease the range\nofθto 0< θ <1. If this is true, we can replace the exponent of 1 /0.677 in Theorem 2 with 1+ o(1).\nReferences\n[1] R.C. Baker and G. Harman, Shifted primes without large prime fact ors, Acta Arithmetica 83:4\n(1998), 331-361.\n[2] P. Erd˝ os, Some remarks on Euler’s φfunction and some related problems, Bull. Amer. Math. Soc.\n51(1945), 540-544.\n[3] G. Harman. “On the greatest prime factor of p−1 with effective constants.” Mathematics of com-\nputation 74.252 (2005): 2035-2041.\n3[4] N. Lebowitz-Lockard and J. Vandehey, Arithmetic functions th at remain constant on runs of consec-\nutive integers, J. Integer Seq. 26:8(2023) Art. 23 .8.4.\n[5] P. Pollack, C. Pomerance, E. Trevi˜ no, Sets of monotonicity fo r Euler’s totient function, Ramanujan\nJ.30(2013), 379-398.\n[6] V.-T. Sp˘ ataru, Runs of consecutive integers having the same n umber of divisors, PUMP J. Under-\ngraduate Res. 6(2023), 96-101.\n[7] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory , 3rd ed., Graduate Studies\nin Mathematics Vol. 163. American Mathematical Society, Providenc e, RI, 2015.\n4"    },    {        "title": "2402.06840v2.A_monotone_piecewise_constant_control_integration_approach_for_the_two_factor_uncertain_volatility_model.pdf",        "content": "A monotone piecewise constant control integration approach for the\ntwo-factor uncertain volatility model\nDuy-Minh Dang∗Hao Zhou†\nFebruary 28, 2024\nAbstract\nOption contracts written on two underlying assets that follow an uncertain volatility model\nhave their worst-case and best-case prices determined by solution to a two-dimensional Hamilton-\nJacobi-Bellman (HJB) partial differential equation (PDE) with cross derivatives terms. Existing\nnumerical techniques for multi-dimensional HJB PDEs primarily utilize finite differences and policy\niteration to solve the resulting non-linear algebraic equations at each timestep. This “discretize,\nthen optimize” paradigm demands a rather complex local coordinate rotation of computational\nstencils for a monotonicity-preserving approximation of cross derivative terms.\nThis paper presents a novel and more streamlined “decompose and integrate, then optimize”\napproach to tackle the aforementioned HJB PDE. Within each timestep, our strategy employs a\npiecewise constant control, breaking down the HJB PDE into independent linear two-dimensional\nPDEs. Using known closed-form expressions for the Fourier transforms of the Green’s functions\nassociated with these PDEs, we determine an explicit formula for these functions. Since the Green’s\nfunctions are non-negative, the solutions to the PDEs, cast as two-dimensional convolution integrals,\ncan be conveniently approximated using a monotone integration method. Such integration methods,\nincluding a composite quadrature rule, are generally available in popular programming languages.\nTo further enhance efficiency, we propose an implementation of this monotone integration scheme\nvia Fast Fourier Transforms, exploiting the Toeplitz matrix structure. Optimal solution/control is\nsubsequently obtained by efficiently synthesizing the solutions of the individual PDEs.\nThe proposed monotone piecewise constant control method is demonstrated to be both ℓ∞-\nstable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of\nthe HJB equation. Numerical results show remarkable agreement with benchmark solutions ob-\ntained by unconditionally monotone finite differences, tree methods, and Monte Carlo simulation,\nunderscoring the robustness and effectiveness of our method.\nKeywords: Hamilton-Jacobi-Bellman, uncertain volatility, piecewise constant control, monotone,\nnumerical integration\nAMS Subject Classification: 65D30, 65M12, 90C39, 49L25, 93E20, 91G20\n1 Introduction\nThe uncertain volatility model is an approach in quantitative finance where the instantaneous volatility\nof a risky asset is allowed to vary within a specified range [20, 1, 15]. This stands in contrast to the\nmore traditional approaches where volatility is often assumed to be either deterministic (as in the\nBlack-Scholes model) or stochastic (as in the Heston model [14] or the SABR model [13]). While\nstochastic volatility models can deliver a more detailed depiction of volatility’s dynamic evolution and\n∗School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane 4072, Australia, email:\nduyminh.dang@uq.edu.au\n†School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane 4072, Australia, email:\nh.zhou3@uq.net.au\n1arXiv:2402.06840v2  [q-fin.CP]  27 Feb 2024its interaction with asset prices, uncertain volatility models are particularly well-suited for worst-case\nscenario analysis. Specifically, although the price of a financial contract is no longer unique under\nan uncertain volatility model, for risk management, especially for sellers, the primary concern often\nlies in the worst-case scenario, which corresponds to the contract’s maximum value. Conversely, for\nthe buyers, the worst-case scenario corresponds to the minimum potential value of a contract. It is\nworth noting that the worst-case scenario for the seller of a contract is essentially the buyer’s best-\ncase scenario, and vice versa. The maximum and minimum value of a contract can be formulated\nas solution to a Hamilton-Jacobi-Bellman (HJB) equation, which needs to be solved numerically\n[26, 22, 7, 2, 30, 25, 16, 27].\nProvable convergence of numerical methods for (multi-dimensional) HJB equations are typically\nbuilt upon the framework established by Barles and Souganidis in [3]. This framework requires numer-\nical methods to be (i) ℓ∞-stable, (ii) consistent, and (iii) monotone (in the viscosity sense), provided\nthat a strong comparison result holds. Among these requirements, monotonicity is often the most\nchallenging to achieve. Non-monotone schemes could produce numerical solutions that fail to con-\nverge to viscosity solutions, resulting in a violation of the no-arbitrage principle (see, for example,\n[23, 26, 35], among many other publications).\nTo the best of our knowledge, numerical techniques for HJB PDEs are predominantly dominated\nby finite difference (FD) methods. At each timestep, these methods typically involve discretizing\nthe temporal and spatial partial derivatives respectively using (i) a fully implicit timestepping and\n(ii) a positive coefficient discretization method [34, 9]. This combination ensures the monotonicity\nof the numerical schemes. The optimal control is subsequently determined by solving the resulting\nnonlinear discretized equations, often via variants of policy iteration [4]. During this process, a local\noptimization problem at each grid point is addressed in every policy iteration. This conventional\napproach is succinctly termed “discretize, then optimize” [27]. Importantly, the positive coefficient\ndiscretization method provides a sufficient condition ensuring the convergence of policy iteration,\nregardless of the initial iterate. As such, this condition must be satisfied at each policy iteration.\nWe highlight that multi-dimensional HJB PDEs, including those from two-factor uncertain volatil-\nity models [22], pose significant challenges due to cross derivative terms when the correlation between\nthe two underlying risky assets is non-zero. At each policy iteration, construction of a monotone finite\ndifference scheme via a positive coefficient discretization method is often addressed using a local coor-\ndinate rotation of the computational stencil. Originally developed for explicit wide stencil schemes in\n[5, 6], this method was refined in [22] for a fully implicit timestepping, circumventing timestep stabil-\nity restrictions. However, as noted in [22], this approach adds a significant computational overhead.\nFor further details of numerical techniques for one- and two-dimensional HJB PDEs resulting from\none-factor and two-factor uncertain volatility models (with an uncertain correlation between the two\nunderlying risky assets), we refer the reader to [26] and [22], respectively.\nIn this paper, we present a streamlined approach to tackle the two-dimensional (2D) HJB PDE\nstemming from two-factor uncertain volatility models. Moving beyond the conventional “discretize,\nthen optimize”, we introduce a “decompose and integrate, then optimize” approach. For each timestep,\nwe utilize a piecewise constant control technique [18] to decompose the HJB PDE into independent 2D\nlinear PDEs. By employing closed-form expressions of the Green’s functions, we develop an uncondi-\ntionally monotone integration method to solve these PDEs. Optimization is then efficiently attained\nby synthesizing solutions of individual PDEs, significantly simplifying the process compared to policy\niteration and avoiding aforementioned challenges associated with positive coefficient FD discretization\nof cross derivative terms. We note a recent study [27] that also utilizes a piecewise constant control\ntechnique. However, this work remains anchored in the FD framework and incorporates a switching\nsystem, and hence necessitating interpolation in searching for optimal control.\n2The main contributions of our paper are outlined below.\n(i) The maximum and minimum value of an option contract under a two-factor uncertain volatility\nmodel with uncertain correlation is presented as an HJB PDE posed on an finite definition do-\nmain consisting of an interior and boundary sub-domains with appropriate boundary conditions.\n(ii) We develop a monotone piecewise constant control integration scheme for the HJB equation,\naddressing it through the solution of independent linear 2D PDEs at each timestep. By uti-\nlizing the known closed-form Fourier transforms of the Green’s functions for these PDEs, we\nestablish explicit formulas for these functions, facilitating direct approximation of solutions via\n2D convolution integrals through a monotone integration method. This process enables the\neffective synthesis of approximations to the HJB equation from the numerical solutions of the\nindividual PDEs.\nOur scheme not only simplifies the optimization process compared to policy iteration but also\navoids the usual complications with positive coefficient FD discretization of cross derivative\nterms. The availability of the Green’s functions in closed form enables a systematic and quan-\ntifiable approach for determining computational domain sizes, marking a significant advantage\nover the heuristic or trial-and-error methods common in FD and tree techniques. Furthermore,\nthe Green’s function’s “cancellation property” [10] effectively mitigates the impact of errors in\nartificial boundary conditions. These combined factors ensure that our method significantly\nenhances the numerical solution’s accuracy and reliability.\n(iii Utilizing the Toeplitz matrix structure, we present an efficient implementation of our monotone\npiecewise constant control integration scheme using FFTs and circulant convolution. The imple-\nmentation process includes expanding the inner summation’s convolution kernel into a circulant\nmatrix, followed by expanding the kernel for the double summation to achieve a circulant block\narrangement. This allows the circulant matrix-vector product to be efficiently computed as a\ncirculant convolution using 2D FFTs.\n(iv We mathematically demonstrate that the proposed monotone scheme is also ℓ∞-stable and con-\nsistent in the viscosity sense, proving its pointwise convergence to the viscosity solution of the\n2D HJB PDE as the discretization parameter approaches zero.\n(v) Extensive numerical results show remarkable agreement with benchmark solutions in published\ntest cases obtained through monotone FD and tree-grid methods, and Monte Carlo simulation,\nunderscoring the effectiveness of our approach. Notably, we often observe experimentally first\norder convergence, significantly exceeding the convergence rate 1 /6 proved in [18] using purely\nprobabilistic techniques.\nAlthough our focus is specifically on monotone piecewise constant control integration methods for\ntwo-factor uncertain volatility models with uncertain correlation, our comprehensive and systematic\napproach could serve as a numerical and convergence analysis framework for the development of similar\npiecewise constant control monotone integration methods for other HJB PDEs arising in finance.\nThe remainder of the paper is organized as follows. In Section 2, we briefly describe the two-factor\nuncertain volatility model and present a 2D HJB PDE. We then define a localized problem for this HJB\nequation, including conditions for boundary sub-domains. A simple and easy-to-implement monotone\npiecewise constant control integration scheme via a composite 2D quadrature rule is described in\nSection 3. In Section 4, we mathematically establish convergence the proposed piecewise constant\ncontrol monotone integration scheme to the viscosity solution of the 2D HJB PDE. Numerical results\nare given in Section 5. Section 6 concludes the paper and outlines possible future work.\n32 Formulation\nLetT > 0 be a finite investment horizon. For each t∈[0, T], we denote by XtandYtthe prices\nat time tof two distinct underlying assets. In this paper, for brevity, we occasionally employ the\nsubscript/superscript z∈ {x, y}to indicate that the discussion pertains to quantities related to the\nrespective underlying assets. We assume that the risk-neutral dynamics of the process {Zt}t∈[0,T],\nwhere Ztcan be either XtorYt, follow\ndZt=rZtdt+σzZtdWz\nt, Z 0>0 given, Zt∈ {Xt, Yt}, t∈(0, T]. (2.1)\nHere, r > 0 is the risk-free interest rate; σz>0,z∈ {x, y}, respectively are the instantaneous\nvolatility for the associated underlying asset; {Wz\nt}t∈[0,T]are correlated Brownian motions, with\ndWx\ntdWy\nt=ρdt, where −1≤ρ≤1 is the correlation parameter. In the uncertain volatility model,\nthe instantaneous volatility σz,z∈ {x, y}, in (2.1) are uncertain, but are assumed to lie within a\nknown range [21]. That is, σz∈[σz\nmin, σz\nmax],z∈ {x, y}, where 0 < σz\nmin< σz\nmaxare pre-determined and\nfixed constants. In addition, the correlation between the two underlying assets is also permitted to\nbe uncertain, lying within a known range, i.e. ρ∈[ρmin, ρmax], where −1≤ρmin≤ρmax≤1 are also\npre-determined and fixed constants. In this setting, since the instantaneous volatilities σz,z∈ {x, y}\nand the correlation ρare uncertain, the price of an option is no longer unique. However, for hedging\npurposes, we can determine the worst-case prices for the long or short positions. These prices are\nessentially the hedging costs for the associated positions.\nFor the underlying asset processes {Xt, Yt},t∈[0, T], defined in (2.1), we let ( x′, y′) be the state\nof system. We denote by v′(x′, y′, t) the time- tworst-case price of the short or long position in a\nEuropean option contract with time- Tpayoff given by function p(x′, y′). By dynamic programming,\nv′(x′, y′, t) is shown to satisfy the HJB PDEs\n0 =\n\n\u0000\n−v′\nt−sup\nα∈A′L′\nαv′\u0001\n,or\u0000\n−v′\nt−inf\nα∈A′L′\nαv′\u0001\n,(x′, y′, t)∈R+×R+×[0, T),\nv′(x′, y′, t)−p(x′, y′), (x′, y′, t)∈R+×R+× {T}.(2.2a)\n(2.2b)\nIn (2.2), the supαand inf αcorrespond to the worst-case for the short and for the long positions,\nrespectively; αis the control, where α= (σx, σy, ρ); the differential operator L′\nα(·), where the subscript\nindicates its dependence α, is defined as\nL′\nαv′=(σx)2(x′)2\n2v′\nx′x′+rx′v′\nx′+(σy)2(y′)2\n2v′\ny′y′+ry′v′\ny′+ρσxσyx′y′v′\nx′y′−rv′. (2.3)\nThe admissible control set, denoted by A′, is given by\nA′=Ax× A y× A ρ,where Ax≡[σx\nmin, σx\nmax],Ay≡[σy\nmin, σy\nmax],Aρ≡[ρmin, ρmax], (2.4)\n0< σz\nmin< σz\nmax<∞, z∈ {x, y},−1≤ρmin≤ρmax≤1.\nRemark 2.1 (Restriction of control set A′).The literature highlights that the optimal value for the\nobjective function in (2.2) can be accurately determined by considering only the boundary values within\nthe 3D admissible optimal control set A′. See, for example, [22][Proposition 3.1] and [17]. Specifically,\nit is established that the search for optimal control can be limited to a much smaller set A, defined as:\nA=\u0000\n({σx\nmin, σx\nmax} × A y)∪(Ax× {σy\nmin, σy\nmax})\u0001\n× {ρmin, ρmax}, (2.5)\nwhere AxandAyare defined in (2.4). Consequently, we focus our analysis on the boundary set A,\nenhancing the efficiency of the proposed piecewise constant control scheme by eliminating the need to\nsearch across the entire 3D set A′.\nThe aforementioned restriction presumes the existence of second-order partial derivatives, which,\ndespite appearing restrictive, is consistent with the viscosity solution framework that utilizes smooth\n4test functions. Unlike traditional grid-based methods such as FD and tree-grid [22, 17], which require\ndiscretizing the differential operator and may not always yield optimal values at A, our approach\nbypasses differential operator discretization. This not only resolves related issues but also simplifies\nthe optimization process, offering a more direct and user-friendly path to identifying optimal control\nvalues, highlighting the method’s practicality and ease of implementation.\nLetτ=T−t, and we apply the change of variables x= ln( x′)∈(−∞,∞) and y= ln( y′)∈\n(−∞,∞). Let x= (x, y, τ ), and denote by v(x)≡v(x, y, τ ) =v′(ex, ey, T−t). With these in mind,\nformulation (2.2) becomes\n0 =\n\n\u0000\nvτ−sup\nα∈ALαv\u0001\n,or\u0000\nvτ−inf\nα∈ALαv\u0001\n,x∈R×R×(0, T],\nv(x)−p(ex, ey), x∈R×R× {0},(2.6a)\n(2.6b)\nwhere ( x, y, τ )∈R×R×[0, T] and the differential operator Lα(·) is given by\nLαv=(σx)2\n2vxx+\u0012\nr−(σx)2\n2\u0013\nvx+(σy)2\n2vyy+\u0012\nr−(σy)2\n2\u0013\nvy+ρσxσyvxy−rv. (2.7)\nWithout loss of generality, we only consider the supαproblem, i.e. worst-case for the short position,\nin the following discussion. The theoretical analysis of this paper holds for the inf αproblem as well.\n2.1 Localization and definition\nFor the problem statement and convergence analysis of numerical schemes, we define a localized two-\nfactor uncertain volatility model pricing problem.\nTo this end, with x†\nmin< x min<0< x max< x†\nmax,\ny†\nmin< y min<0< y max< y†\nmax, where |x†\nmin|,|xmin|,\n|y†\nmin|,|ymin|,xmax,x†\nmax,ymaxandy†\nmaxare chosen suf-\nficiently large, we define the following sub-domains:\nΩ = [ x†\nmin, x†\nmax]×[y†\nmin, y†\nmax]×[0, T],\nΩτ0= [x†\nmin, x†\nmax]×[y†\nmin, y†\nmax]× {0},\nΩin= (xmin, xmax)×(ymin, ymax)×(0, T], (2.8)\nΩout= Ω\\Ωτ0\\Ωin.\nAn illustration of the sub-domains for the localized prob-\nlem corresponding to a fixed τ∈(0, T] is given in Fig-\nure 2.1.ΩinΩout Ωout\nΩoutΩout\nx†\nminx†\nmax\ny†\nminy†\nmax\nxmin xmax\nyminymax\nFigure 2.1: Spatial definition sub-\ndomain at each τ.\nWe now present equations for sub-domains defined in (2.8).\n•For ( x, y, τ )∈Ωin, we have (2.6).\n•For ( x, y, τ )∈Ωτ0, we use the initial condition v(x, y,0) = p(ex, ey).\n•For the outer boundary sub-domain Ω out, boundary conditions are generally informed by financial\nreasonings or derived from the asymptotic behavior of the solution. In this study, we implement\na straightforward Dirichlet condition based on discounted payoff as follows\nv(x, y, τ ) =p(ex, ey)e−rτ,(x, y, τ )∈Ωout. (2.9)\n5While more sophisticated boundary conditions might involve the asymptotic properties of the HJB\nequation (2.6)) as z→ −∞ orz→ ∞ , where z∈ {x, y}, our observations indicate that these\nsophisticated boundary conditions do not significantly impact the accuracy of the numerical solution\nwithin Ω in. This observation is largely due to the so called “cancellation property” of the Green’s\nfunction [10], which effectively mitigates the impact of approximation errors in artificial boundary\ncondition behavior on the solution in Ω in. This will be illustrated through numerical experiments in\nSubsection 5.4.\nWith x= (x, y, τ ), we let Dv(x) = (vx, vy, vτ) and D2v(x) = (vxx, vyy, vxy), and define\nFΩ(x, v)≡FΩ\u0000\nx, v(x), Dv(x), D2v(x)\u0001\n. (2.10)\nHere,\nFΩ(x, v) =\n\nFin(x, v)≡Fin\u0000\nx, v(x), Dv(x), D2v(x)\u0001\n, x∈Ωin,\nFout(x, v)≡Fout(x, v(x)), x∈Ωout,\nFτ0(x, v)≡Fτ0(x, v(x)), x∈Ωτ0,(2.11)\nwith operators\nFin(x, v) =vτ−sup\nα∈ALαv, (2.12)\nFout(x, v) =v−p(ex, ey)e−rτ, (2.13)\nFτ0(x, v) =v−p(ex, ey). (2.14)\nDefinition 2.1 (Two-factor uncertain volatility pricing problem) .The pricing problem for the two-\nfactor uncertain volatility model is defined as\nFΩ\u0000\nx, v(x), Dv(x), D2v(x)\u0001\n= 0, (2.15)\nwhere the operator FΩ(·)is defined in (2.10) .\nWe recall the notions of the upper semicontinuous (u.s.c. in short) and the lower semicontinuous\n(l.s.c. in short) envelops of a function u:X→R, where Xis a closed subset of Rn. They are\nrespectively denoted by u∗(·) (for the u.s.c. envelop) and u∗(·) (for the l.s.c. envelop), and are given\nby\nu∗(ˆ x) = lim sup\nx→ˆ x\nx,ˆ x∈Xu(x) (resp. u∗(ˆ x) = lim inf\nx→ˆ x\nx,ˆ x∈Xu(x)). (2.16)\nDefinition 2.2 (Viscosity solution of equation (2.15)) .A locally bounded function v: Ω→Ris a\nviscosity subsolution (resp. supersolution) of (2.15) if for all test function ϕ∈ C∞(Ω)and for all points\nˆ x∈Ωsuch that v∗−ϕhas a global maximum on Ωatˆ xandv∗(ˆ x) =ϕ(ˆ x)(resp. v∗−ϕhas a global\nminimum on Ω∞atˆ xandv∗(ˆ x) =ϕ(ˆ x)), we have\n(FΩ)∗\u0000\nˆ x, ϕ(ˆ x), Dϕ(ˆ x), D2ϕ(ˆ x)\u0001\n≤0, (2.17)\n\u0000\nresp. (FΩ)∗\u0000\nˆ x, ϕ(ˆ x), Dϕ(ˆ x), D2ϕ(ˆ x)\u0001\n≥0,\u0001\nwhere the operator FΩ(·)is defined in (2.10) .\nRemark 2.2 (Strong comparison result and convergence region) .As noted in [22], if the payoff func-\ntionp(ex, ey)is continuous with quadratic growth, then the value function of the problem (2.6) satisfies\na strong comparison principle result [24, 12]. That means, there exists an unique continuous viscosity\nsolution of the problem (2.6). In the context of this paper, our focus is on a finite interior subdomain\nΩin(with boundary conditions given in ΩoutandΩτ0) for the HJB equation (2.15) . Therefore, the value\nfunction (2.15) satisfies a strong comparison result in Ωin. This sub-domain defines the convergence\nregion for our proposed scheme.\n63 Numerical methods\n3.1 Piecewise constant control\nThe key step of our numerical scheme is a piecewise constant control time-stepping method for Ω in\nthat utilizes a convolution integral that involves the Green’s function of an associated 2D PDE in\nxandy. As discussed in Remark 2.1, we search for the optimal control within the boundary set A\ngiven in (2.5). To this end, we first make an observation that the admissible control set A, as defined\nin (2.4)), is a compact set. Therefore, it can be approximated arbitrarily well by a finite set [28].\nSpecifically, for any discretization parameter h >0, there exists a finite partition AhofAsuch that\nfor any α∈ A, the distance to its nearest point in Ahis no greater than h. That is,\nmax\nα∈Amin\nα′∈Ah∥α−α′∥2≤h. (3.1)\nMotivated by (3.1), to address the two-factor uncertain volatility pricing problem in Defn (2.1), we\npropose an approach that involves approximating AwithAh. Specifically, for Ω in, instead of solving\nthe HJB equation vτ−sup\nα∈ALαv= 0, we solve vτ−sup\nα∈AhLαv= 0. In our convergence analysis, we will\nestablish that this solution converges to the viscosity solution of the pricing problem in Defn (2.1) as\ndescribed in Defn (2.2).\nWe now elaborate the piecewise constant control for Ω in. We let {τm},m= 0, . . . , M , be an equally\nspaced partition in the τ-dimension, where τm=m∆τand ∆ τ=T/M . With a fixed τm>0 such\nthatτm+1≤T, we consider the HJB equation\nvτ−sup\nα∈AhLαv= 0, (x, y, τ )∈R×R×(τm, τm+1], (3.2)\nwhere the differential operator Lα(·) is defined in (2.7). Here, we note that, in (3.2), the admis-\nsible control set Ais approximated by the finite discretized control set Ah, with h > 0 being the\ndiscretization parameter.\nFor fixed α∈ A h, we denote by u(·;α)≡u(x, y, τ ;α) the solution to the linear PDE in ( x, y, τ ) of\nthe form\nuτ− L αu= 0, (x, y, τ )∈R×R×(τm, τm+1]. (3.3)\nwhere Lα(·) is defined in (2.7), subject to a generic initial condition at time τmgiven by ˆ v(x, y, τ m).\nHere,\nˆv(x, y, τ m) =(\nv(x, y, τ m) ( x, y, τ m+1)∈Ωin,\nvbc(x, y, τ m) ( x, y, τ m+1)∈Ω\\Ωin,(3.4)\nwhere vbc(x, y, τ m) is the boundary conditions at time τmsatisfying (2.9) in Ω out. respectively.\nWe denote by gα(·)≡gα(x, x′, y, y′; ∆τ),α∈ A h, the Green’s function associated with the 2D\nlinear PDE (3.3). It can be shown that gα(·) has the form gα(·; ∆τ) =gα(x−x′, y−y′; ∆τ).1By a\nGreen’s function argument [10, 8], for fixed α∈ A h, the solution u(x, y, τ m+1) for ( x, y)∈D, where\nD≡(xmin, xmax)×(ymin, ymax),\ncan be represented as the convolution integral of the Green’s function gα(·; ∆τ) and the initial condition\nˆv(·, τm) as follows\nu(x, y, τ m+1;α) =ZZ\nR2gα\u0000\nx−x′, y−y′; ∆τ\u0001\nˆv(x′, y′, τm)dx′dy′, (x, y)∈D, α∈ A h.(3.5)\nThe solution u(x, y, τ m+1;α) for ( x, y)/∈Dare given by the boundary condition (2.9).\nFor computational purposes, we truncate the infinite region of integration of (3.5) to D†, where\nD†≡[x†\nmin, x†\nmax]×[y†\nmin, y†\nmax]. (3.6)\n1This is due to the spatial homogeneity of the stochastic system (2.1).\n7Here, recall that z∈ {x, y},z†\nmin< zmin<0< zmax< z†\nmaxand|z†\nmin|andz†\nmaxare sufficiently large.\nThis results in the approximation\nu(x, y, τ m+1;α)≃ZZ\nD†gα\u0000\nx−x′, y−y′; ∆τ\u0001\nˆv(x′, y′, τm)dx′dy′, (x, y)∈D, α∈ A h.(3.7)\nFinally, an approximation to the solution of the HJB (3.2) for ( x, y, τ m+1)∈Ωinis given by\nv(x, y, τ m+1)≃max\nα∈Ahu(x, y, τ m+1;α), (x, y)∈D. (3.8)\nWe conclude by noting that the errors arising from (i) approximating AbyAhand (ii) from truncating\nthe infinite integration domain in (3.5) to a finite one in (3.7) are discussed subsequently.\n3.2 A closed-form representation of gα(·) for Ω in\nWe now present a closed-form expression for the Green’s function gα(·) of the linear PDE (3.3), where\nthe control α≡(σx, σy, ρ)∈ Ais fixed. To this end, we denote by Gα(·; ∆τ) the Fourier transform of\ngα(·; ∆τ), i.e.\n\n\nF|gα(x, y;·)| =Gα(η, ζ;·) =ZZ\nR2e−i(ηx+ζy)gα(x, y;·)dxdy,\nF−1|Gα(η, ζ;·)|=gα(x, y;·) =1\n(2π)2ZZ\nR2ei(ηx+ζy)Gα(η, ζ;·)dηdζ.(3.9)\nA closed-form expression for Gα(η, ζ;·) is given by [29]\nGα(η, ζ;·) = exp(Ψ( η, ζ)∆τ), (3.10)\nwith Ψ( η, ζ) =\u0012\n−σ2\nxη2\n2−σ2\nyζ2\n2+ (r−σ2\nx\n2)iη+ (r−σ2\ny\n2)iζ−ρσxσyηζ−r\u0013\n.\nWe now introduce a lemma providing a closed-form expression for the Green’s function gα(x, y; ∆τ).\nLemma 3.1. Let∆τ > 0be fixed, and gα(x, y; ∆τ)andGα(η, ζ; ∆τ)be a Fourier transform pair\ndefined in (3.9), and Gα(η, ζ; ∆τ)be given in (3.10) . When |ρ|<1,gα(x, y; ∆τ)can be expressed in\nthe form of a “scaled” joint density as follows\ngα(x, y; ∆τ) =e−r∆τfα(x, y; ∆τ),where (3.11)\nfα(x, y; ∆τ) =1\n2πκxκyp\n1−ρ2exp\u0012−1\n2(1−ρ2)\u0014\u0000x−µx\nκx\u00012−2ρ\u0000x−µx\nκx\u0001\u0000y−µy\nκy\u0001\n+\u0000y−µy\nκy\u00012\u0015\u0013\n.\nwith µx=\u0012σ2\nx\n2−r\u0013\n∆τ, κ x=σx√\n∆τ, µ y=\u0012σ2\ny\n2−r\u0013\n∆τ, κ y=σy√\n∆τ, (3.12)\nWhen ρ=±1,gα(x, y; ∆τ)is given by\ngα(x, y; ∆τ) =e−r∆τ1√\n2πκxexp\u0012\n−(x−µx)2\n2κ2x\u0013\nδ(y−(a+ρbx)). (3.13)\nHere, δ(·)is a Dirac delta function, and a=µy−ρbµxwithb=σy\nσx.\nProof of Lemma 3.1. When |ρ|<1, applying inverse Fourier transform to Gα(·), provided in (3.10), we\nobtain the expression for the Green’s function gα(x, y; ∆τ) given in (3.11). When ρ=±1,fα(x, y; ∆τ)\ncan be expressed in the form fα(x, y; ∆τ) =1√\n2πκxexp\u0010\n−(x−µx)2\n2κ2x\u0011\nδ(y−(a+ρbx)), where aand\nbare constants, with b >0 [11]. We then solve for aandbby comparing the Fourier transform of\ngα(x, y; ∆τ) in this case with the closed-form expression of Gα(·). This gives a=µy−ρbµxandb=σy\nσx.\nThis completes the proof.\n8Remark 3.1 (ρ=±1).In our study, while we acknowledge the theoretical significance of the cases\nwhere ρ=±1, we have chosen not to explore this scenario in depth. Such extreme correlation values,\nthough mathematically interesting, are rarely encountered in practical applications and financial mod-\neling. Therefore, our focus remains predominantly on scenarios where the correlation coefficient lies\nstrictly between −1and1, which are more representative of the conditions commonly observed and of\ngreater relevance to practitioners. However, it’s important to note that our piecewise constant control\nintegration scheme can effectively manage the special case of ρ=±1. For computational purposes,\napproximating the Dirac delta function δ(y−(a±bx))in(3.13) by a suitable Gaussian function is\nnecessary (refer to [31][Chapter 10], for example, for more details of such approximations). Essential\naspects of our scheme in this case are elaborated in Appendix A.\ng1 We now present a lemma on the boundary truncation error of the Green’s function gα(·) defined\nin (3.11). for the case |ρ|<1.\nLemma 3.2. Let∆τ >0be fixed. Suppose |x†\nmin|,x†\nmax,|y†\nmin|, and y†\nmaxare chosen such that\nminn\n|x†\nmin|, x†\nmax,|y†\nmin|, y†\nmaxo\n>max{µx±γ, µy±γ},\nwhere γ≫0is a fixed constant, µx=\u0010\nσ2\nx\n2−r\u0011\n∆τ, and µy=\u0010σ2\ny\n2−r\u0011\n∆τ, as defined in (3.12) .\nThen, for sufficiently small ∆τ,gα(·), as defined in (3.11) for a fixed α∈ A, satisfies\nZZ\nR2\\D†gα(x, y; ∆τ)dxdy < C ∆τe−1\n2∆τ,D†≡[x†\nmin, x†\nmax]×[y†\nmin, y†\nmax], (3.14)\nwhere Cis a bounded constant independently of ∆τ.\nProof of Lemma 3.2. Without loss of generality, we present a proof for the case 0 ≤ρ < 1. For\nsubsequent use, let Φ( s)≡Rs\n−∞ϕ(z)dzandϕ(z)≡(2π)−1/2exp(−z2/2) respectively be the CDF and\nthe probability density function of standard normal distribution.\nFor simplicity, we let w= minn\n|x†\nmin|, x†\nmax,|y†\nmin|, y†\nmaxo\n. With zx=x−µx\nκx,zy=y−µy\nκy, we define\nthe region Bas follows\nB= [−b, b]×[−b, b],where b= min\b\f\f−w−µx\nκx\f\f,w−µx\nκx,\f\f−w−µy\nκy\f\f,w−µy\nκy\t\n. (3.15)\nWe have\nZZ\nR2\\D†gα(x, y; ∆τ)dxdy≤e−r∆τZZ\nR2\\Bexp\u0010\n−1\n2[1−ρ2]\u0002\nz2\nx−2ρzxzy+z2\ny\u0003\u0011\n2πp\n1−ρ2dzxdzy\n≤2e−r∆τP\u0000\nzx≥b, zy≥b\u0001(i)\n≤2e−r∆τ(1 +ρ)Φ(−b)Φ\u0012−b(1−ρ)p\n1−ρ2\u0013\n(ii)\n≤e−r∆τ(1 +ρ)3/2\nπ(1−ρ)1/2×e−b2/2\nb2. (3.16)\nHere, (i) is due to an upper bound for the bivariate normal distribution in [36]; in (ii), we apply the\nfollowing fact: if X∼N(0,1), then P(X > x )≤1\nx√\n2πexp(−x2/2). It is straightforward to see that\ne−r∆τ(1+ρ)3/2\nπ(1−ρ)1/2≤(1+ρmax)3/2\nπ(1−ρmax)1/2. Thus, for sufficiently small ∆ τ, the condition w >max{µx±γ, µy±γ},\nwhere γ≫0 is fixed, implies the rhs of (3.16) is bounded by C∆τe−1/2∆τ, where Cis a bounded\nconstant independently of ∆ τ. This completes the proof.\nRemark 3.2 (Boundary truncation error) .The boundary truncation error upper bound, as detailed in\nequation (3.16) , serves as a practical tool for selecting an appropriate definition domain, D†, to ensure\n9this truncation error remains below a predefined threshold ϵ >0. To achieve this, we first identify a\nvalue of bsatisfying\ne−r∆τ(1 +ρ)3/2\nπ(1−ρ)1/2×e−b2/2\nb2≤(1 +ρmax)3/2\nπ(1−ρmax)1/2e−b2/2\nb2< ϵ. (3.17)\nGiven b, we then determine wthrough equation (3.15) ) by ensuring the following conditions are\nmet: b≤\f\f−w−µx\nκx\f\f,b≤w−µx\nκx,b≤\f\f−w−µy\nκy\f\fandb≤w−µy\nκy. Subsequently, D†is derived via\nw= minn\n|x†\nmin|, x†\nmax,|y†\nmin|, y†\nmaxo\n.\nThe methodological approach outlined above represents a significant advantage over traditional\nfinite difference methods, which typically depend on heuristic strategies or trial-and-error for deter-\nmining appropriate domain sizes. Our approach introduces a systematic and quantifiable method for\ndetermining domain size, significantly enhancing the accuracy and reliability of numerical solutions.\nThe efficacy of this systematic approach is demonstrated through numerical experiments detailed in\nSubsection 5.3.\n3.3 Discretization\nWe highlight that, in approximating the 2D convolution integral (3.7) over the finite integration\ndomain D†, it is necessary to obtain values of the Green’s function gα(x, y;·), at points ( x, y) outside\nD†. To define these points, we let z‡\nmax=zmax−z†\nminandz‡\nmin=zmin−z†\nmaxforz∈ {x, y}. Consequently,\nwe need gα(x, y;·) at ( x, y)∈D†\nout, where\nD†\nout=\u0000\n[x‡\nmin, x‡\nmax]×[y‡\nmin, y‡\nmax]\u0001\n\\D†, z‡\nmin=zmin−z†\nmaxforz∈ {x, y}. (3.18)\nAlthough D†\noutlies outside the pricing problem’s definition domain, the availability of a closed-form\nexpression for gα(x, y;·) ensures no issues for our numerical methods. Moreover, the value functions\nfor (x, y)∈D†\noutare not required for our convergence analysis. The role of D†\noutis to ensure the\nwell-definedness of an associated Green’s function for the convolution integral, which is crucial for\ntime advancement within Ω in.\nWithout loss of generality, for convenience, we assume that |zmin|andzmax, where z∈ {x, y}, are\nchosen sufficiently large so that\nz†\nmin=zmin−zmax−zmin\n2,and z†\nmax=zmax+zmax−zmin\n2. (3.19)\nWith (3.19) in mind, recalling z‡\nminandz‡\nmax,z∈ {x, y}as defined in (3.18) gives\nz‡\nmin=z†\nmin−zmax=−3\n2(zmax−zmin),and z‡\nmax=z†\nmax−zmin=3\n2(zmax−zmin). (3.20)\nWe denote by N(resp. N†andN‡) the number of intervals of a uniform partition of [ xmin, xmax]\n(resp. [ x†\nmin, x†\nmax] and [ x‡\nmin, x‡\nmax]). For convenience, we typically choose N†= 2NandN‡= 3Nso that\nonly one set of x-coordinates is needed. Also, let Px=xmax−xmin,P†\nx=x†\nmax−x†\nmin, and P‡\nx=x‡\nmax−x‡\nmin.\nWe define ∆ x=Px\nN=P†\nx\nN†=P‡\nx\nN‡. We use an equally spaced partition in the x-direction, denoted by\n{xn}, and is defined as follows\nxn= ˆx0+n∆x;n=−N‡/2, . . . , N‡/2,where\n∆x=Px/N=P†\nx/N†=P‡\nx/N‡,and (3.21)\nˆx0= (xmin+xmax)/2 = ( x†\nmin+x†\nmax)/2 = ( x‡\nmin+x‡\nmax)/2.\nSimilarly, for the y-dimension, with J†= 2J,J‡= 3J,Py=ymax−ymin,P†\ny=y†\nmax−y†\nmin, and\nP‡\ny=y‡\nmax−y‡\nmin, we denote by {yj}, an equally spaced partition in the y-direction defined as follows\nyj= ˆy0+j∆y;j=−J‡/2, . . . , J‡/2,where\n∆y=Py/J=P†\ny/J†=P‡\ny/J‡,and (3.22)\nˆy0= (ymin+ymax)/2 = ( y†\nmin+y†\nmax)/2 = ( y‡\nmin+y‡\nmax)/2.\n10We use the same previously defined uniform partition {τm},m= 0, . . . , M , with τm=m∆τand\n∆τ=T/M .2\nRegarding the control set A, defined in (2.5), we let QxandQyrespectively be the number of\nintervals of a uniform partition of Ax= [σx\nmin, σx\nmax] and Ay= [σy\nmin, σy\nmax]. We denote by {σx\nq}and\n{σy\nq′}an equally spaced partition for AxandAy, respectively, each with a uniform interval length\n∆σz=σz\nmax−σz\nmin\nQz, where z∈ {x, y}. Consequently, the discretized control set Ahapproximating Ais\ngiven by\nAh=n\u0010\n{σx\nmin, σx\nmax} × { σy\nq′}\u0011\n∪\u0000\n{σx\nq} × { σy\nmin, σy\nmax}\u0001o\n× {ρmin, ρmax}. (3.23)\nFor subsequent use, we denote by Qthe cardinality of the set Ah, assuming that both AxandAyare\ndiscretized using the same number of partitions.\nWe assume that there is a discretization parameter h >0 such that\n∆x=C1h,∆y=C2h,∆τ=C3h,∆σx=C4h,∆σy=C5h, (3.24)\nwhere the positive constants C1,C2,C3,C4, and C5are independent of h.\nFor convenience, we let M={0, . . . M −1}and we also define the following index sets:\nN={−N/2 + 1 , . . . N/ 2−1},N†={−N, . . . N },N‡={−3N/2 + 1 , . . .3N/2−1},\nJ={−J/2 + 1 , . . . J/ 2−1},J†={−J, . . . J },J‡={−3J/2 + 1 , . . . , 3J/2−1}. (3.25)\nWith n∈N†,j∈J†, and m∈ {0, . . . , M }, we denote by vm\nn,j(resp. um\nn,j) a numerical approximation\nto the exact solution v(xn, yj, τm) (resp. u(xn, yj, τm)) at the reference node ( xn, yj, τm) =xm\nn,j. We\nalso denote by ( α∗)m\nn,j≡(σ∗\nx, σ∗\ny, ρ∗)m\nn,jthe optimal control obtained by a numerical method for this\nreference node. For m∈M, nodes xm+1\nn,jhaving (i) n∈Nandj∈J, are in Ω in, (ii) either n∈N†\\N\nandj∈J†orn∈N†andj∈J†\\Jare in Ω out. For double summation, unless otherwise noted, we\nadopt the short-hand notation:q∈QX∗\nd∈D(·) :=X\nq∈QX\nd∈D(·). Lastly, it’s important to note that references to\nindices n∈N‡\\N†orj∈J‡\\J†pertain to points within D†\nout(as defined in (3.18)). As noted earlier,\nno numerical solutions are required for these points.\n3.4 Numerical schemes\nFor ( xn, yj, τ0)∈Ωτ0, we impose the initial condition (2.14) by\nv0\nn,j=p(exn, eyj), n∈N†andj∈J†. (3.26)\nFor ( xn, yj, τm+1)∈Ωout, we impose the boundary condition (2.13) as follow\nvm+1\nn,j=p(exn, eyj)e−rτm+1, n∈N†\\Norj∈J†\\J. (3.27)\nFor ( xn, yj, τm+1)∈Ωin, let gα\nn−l,j−d≡gα(xn−xl, yj−yd; ∆τ) with n∈N,j∈J,l∈N†and\nd∈J†. Here, gα(·) is given by the closed-form expression in (3.11) in Lemma 3.1, where α∈ A his\nfixed. We let um+1,α\nn,j be an approximation to the double integral (3.7) at x=xn,y=yjandτm+1)\nobtained via a 2D composite quadrature rule. It is computed by\num+1,α\nn,j = ∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dvm\nl,d, n∈Nandj∈J. (3.28)\n2While it is straightforward to generalize the numerical method to non-uniform partitioning of the τ-dimension, to\nprove convergence, uniform partitioning suffices.\n11Here, the coefficients φl,din (3.28) are the weights of the composite quadrature rule. Finally, vm+1\nn,jis\ncomputed as follow\nvm+1\nn,j= max\nα∈Ahum+1,α\nn,j = max\nα∈Ah\u001a\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dvm\nl,d\u001b\n, n∈Nandj∈J. (3.29)\nBy solving the optimization problem (3.29), we obtain the optimal control ( α∗)m+1\nn,j≡(σ∗\nx, σ∗\ny, ρ∗)m+1\nn,j.\nFinally, unless otherwise stated, two-dimensional composite trapezoidal quadrature rule is used.\n3.5 Efficient implementation and algorithms\nIn this section, we discuss an efficient implementation of the 2D discrete convolution (3.28) using\nFFT. For convenience, with N†= 2N,N‡= 3N,J†= 2JandJ‡= 3J, we define/recall sets of\nindices: N‡={−N‡/2 + 1 , . . . , N‡/2−1},N†={−N†/2, . . . , N†/2},N={−N/2 + 1 , . . . , N/ 2−1},\nJ‡={−J‡/2 + 1 , . . . , J‡/2−1},J†={−J†/2, . . . , J†/2},J={−J/2 + 1 , . . . , J/ 2−1}.\nFor a fixed mand a fixed α, to write (3.28) for all n∈Nandj∈Jinto a matrix-vector multipli-\ncation form, we adopt the following notation:\n•For a fixed j∈Jand a fixed α∈ A handm∈ {1, . . . , M }, letum,α\njbe a column vector of length\n(N−1) defined by um,α\nj≡h\num,α\n−N/2+1,j, um,α\n−N/2+2,j, . . . , um,α\nN/2−1,jiT\n;\n•For a fixed q∈J‡andm∈ {0, . . . , M −1}, letvm\nqbe a column vector of length (2 N+ 1) defined\nbyvm\nq≡h\nvm\n−N†/2,qφ−N†/2,q, vm\n−N†/2+1,qφ−N†/2+1,q, . . . , vm\nN†/2,qφm\nN†/2,qiT\n.\n•For a fixed q∈J‡and a fixed α∈ A h, letgα\nqbe a (non-square) matrix of size ( N−1)×(2N+1),\nrepresenting the convolution kernel in the inner summation (over n), defined as follows\ngα\nq=\u0002\ngα\nn−l,q\u0003\nn∈N,l∈N†=\ngα\nN/2+1,qgα\nN/2,q. . . . . . . . . gα\n−3N/2+1,q\ngα\nN/2+2,qgα\nN/2+1,q. . . . . . . . . gα\n−3N/2+2,q\n..................\ngα\n3N/2−1,qgα\n3N/2−2,q. . . gα\nN/2+1,q. . . gα\n−N/2−1,q\n. (3.30)\nIn this setup, we can express the 2D discrete convolution (3.28) for all n∈Nandj∈Jinto a\nmatrix-vector product form as follows\n\num+1,α\n−J/2+1\num+1,α\n−J/2+2\n...\num+1,α\nJ/2−1\n\n|{z}\num+1,α= ∆x∆y\ngα\nJ/2+1gα\nJ/2. . . . . . . . . gα\n−3J/2+1\ngα\nJ/2+2gα\nJ/2+1. . . . . . . . . gα\n−3J/2+2\n..................\ngα\n3J/2−1gα\n3J/2−2. . .gα\nJ/2+1. . .gα\n−J/2−1\n\n| {z }\n[gα\nj−d]j∈J,d∈J†\nvm\n−J†/2\nvm\n−J†/2+1\n...\nvm\nJ†/2\n\n|{z}\nvm.(3.31)\nHere, um+1,αis a column vector of length ( N−1)(J−1); the (block) matrixh\ngα\nj−di\nj∈J,d∈J†, which\nrepresents the convolution kernel for the double summation, is of size ( N−1)(J−1)×(2N+1)(2 J+1);\nvmis a column vector of length (2 N+ 1)(2 J+ 1).\nIt is noteworthy that the non-square matrixh\ngα\nj−di\nj∈J,d∈J†is a Toeplitz matrix [33], enabling\nefficient computation of (3.31) using FFT and circular convolution. This technique, initially applied\nto 1D problems in [37], is now adapted to the 2D case given by (3.31). Our goal is to represent (3.31)\nas a circulant matrix-vector product. This involves expandingh\ngα\nj−di\nj∈J,d∈J†to a 2D ciculant matrix -\na block matrix where each block is circulant and the blocks are arranged in a circulant pattern. More\n12specifically, the process involves (i) expanding each block gα\nj−dto a circulant matrix, denoted by ˜gα\nj−d,\nand (ii) expandingh\n˜gα\nj−di\nj∈J,d∈J†to a 2D circulant matrix. Correspondingly, the vector vmia also\nexpanded to conform with this format. Key steps of this expansion process are outlined below.\n•Expansion of blocks: For each matrix gα\nq=h\ngα\nn−l,qi\nn∈N,l∈N†,q∈J‡, of size ( N−1)×(2N+ 1),\nwe expand it into a circular matrix ˜gα\nqof size (3 N−1)×(3N−1). This expansion, detailed in\n[37], results in the matrix\n˜gα\nq=\n˜gq,α\n−1,0˜gq,α\n−1,1\ngα\nq ˜gq,α\n0,1\n˜gq,α\n1,0˜gq,α\n1,1\n, gα\nq=\u0002\ngα\nn−l,q\u0003\nn∈N,l∈N†. (3.32)\nHere, ˜gq,α\n−1,0,˜gq,α\n1,0,˜gq,α\n−1,1,˜gq,α\n0,1and˜gq,α\n1,1are padding matrices of sizes N×(2N+ 1), N×(2N+ 1),\nN×(N−2), (N−1)×(N−2), and N×(N−2), respectively. These matrices are appropriately\ndefined to ensure the circulant structure of ˜gα\nq. Further details on these padding matrices are\nprovided in Appendix B.\n•Expansion of block matrix: We then substitute gα\nj−dwith circulant block ˜gα\nj−dinh\ngα\nj−di\nj∈J,d∈J†.\nThe resulting block matrixh\n˜gα\nj−di\nj∈J,d∈J†is then expanded into a circulant matrix of size (3 N−\n1)(3J−1)×(3N−1)(3J−1), denoted as ˜gα. Specifically, ˜gαis constructed as follows:\n˜gα=\n˜gα\n−1,0 ˜gα\n−1,1 h\n˜gα\nj−di\nj∈J,d∈J†˜gα\n0,1\n˜gα\n1,0 ˜gα\n1,1\n. (3.33)\nHere, ˜gα\n−1,0,˜gα\n1,0,˜gα\n−1,1,˜gα\n0,1and˜gα\n1,1components are (block) matrices with dimensions (3 N−\n1)J×(3N−1)(2J+ 1), (3 N−1)J×(3N−1)(2J+ 1), (3 N−1)J×(3N−1)(J−2), (3 N−\n1)(J−1)×(3N−1)(J−2) and (3 N−1)J×(3N−1)(J−1), respectively. These matrices are\nappropriately defined to ensure the circulant structure of ˜gα. Further details on these padding\nmatrices are provided in Appendix B.\n•Vector expansion: To conform with the circulant-maxtrix format, for each q∈J†, we construct\nthe augmented column vector ˜vm\nqof length (3 N−1), by appending zeros to the column vector\nvm\nq. This is defined as follows.\n˜vm\nq=h\nvm\n−N†/2,qφ−N†/2,q, vm\n−N†/2+1,qφ−N†/2+1,q, . . . , vm\nN†/2,qφN†/2,q,0,0, . . . , 0iT\n.(3.34)\nThen, we form the vector ˜vmof size (3 N−1)(3J−1) by appending zeros as follows:\n˜vm=h\n˜vm\n−J†/2,˜vm\n−J†/2+1, . . . , ˜vm\nJ†/2,0,0, . . . ,0iT\n. (3.35)\nwhere 0’s are zero vectors of length (3 N−1).\n•Circulant matrix-vector product: Utilizing this setup, we express the matrix-vector product\n(3.31) as a circulant matrix-vector product, which is used to compute an intermediate column\nvector of discrete solutions. This column vector, denoted by ˜um+1,α, has a length of (3 N−\n1)(3J−1) and is determined as follows:\n˜um+1,α= ∆x∆y˜gα˜vm, α ∈ A h. (3.36)\nHere, ˜gαis the circulant matrix defined in (3.33), ˜vmis the (augmented) column vector given by\n(3.35). We note that discrete solutions um+1,α\nn,j for Ω inare obtained by discarding the components\nin˜um+1,αcorresponding to indices n∈N‡\\Norj∈J‡\\J.\n13The circulant matrix-vector product in (3.36) can be efficiently computed as a circulant convolution\nusing 2D FFT. To this end, we let bgα\n1be the first column ˜gαdefined in (3.33) reshaped into a\n(3N−1)×(3J−1) matrix as follows\nbgα\n1=hh\n˜gα\n−J/2+1i\n1. . .h\n˜gα\nJ/2i\n1h\n˜gα\nJ/2+1i\n1. . .h\n˜gα\n3J/2−1i\n1h\n˜gα\n−3J/2+1i\n1. . .h\n˜gα\n−J/2i\n1i\n.(3.37)\nHere,\u0002˜gα\nq\u0003\n1,q∈J‡, denotes the first column of the matrix ˜gα\nq. We reshape the vector ˜vminto a\n(3N−1)×(3J−1) matrix, denoted by [ ˜vm]. The circulant matrix-vector product in (3.36) can be\nexpressed as a 2D circular convolution product\n\u0002˜um+1,α\u0003\n= ∆x∆ybgα\n1∗[˜vm], α ∈ A h. (3.38)\nHere,\u0002˜um+1,α\u0003\nis a (3 N−1)×(3J−1) matrix, representing the reshaped version of ˜um+1,αfrom\n(3.36). The circular convolution product (3.38) is computed efficiently using FFT and inverse FFT\n(iFFT) as follows\n\u0002˜um+1,α\u0003\n= ∆x∆yFFT−1{FFT{[˜vm]} ◦FFT{bgα\n1}}, α ∈ A h. (3.39)\nFinally, we discard the components in\u0002˜um+1,α\u0003\ncorresponding to indices n∈N‡\\Norj∈J‡\\J,\nobtaining discrete solutions um+1,α\nn,j for Ω in.\nThe implementation (3.39) suggests that we compute the weight components of bgα\n1only once for\neach α∈ A hthrough the closed-form expression in (3.11), and reuse them for the computation over\nall time intervals. Putting everything together, the proposed numerical scheme for the two-factor\nuncertain volatility model pricing problem is presented in Algorithm 3.1 below.\nAlgorithm 3.1 A monotone piecewise constant control integration algorithm for a two-factor\nuncertain volatility model pricing problem defined in Definition (2.1).\n1:for each α∈ A h, and for each j∈N‡, compute weight matrices gα\nj=h\ngα\nn−l,ji\nn∈N,l∈N†defined in\n(3.30) using the closed-form expression (3.11);\n2:construct weight matrices bgα\n1,α∈ A h, usingh\ngα\nn−l,ji\nn∈N,l∈N†,j∈N‡, defined in (3.37);\n3:initialize v0\nn,j=p(exn, eyj),n∈N†, j∈J†;\n4:form= 0, . . . , M −1do\n5:forα∈ A hdo\n6: compute matrices of intermediate values [ ˜um+1,α] using FFT as per (3.39);\n7: obtain vector of discrete solutions um+1,α=h\num+1,α\nn,ji\nn∈N,j∈Jby discarding the components\nin [˜um+1,α] corresponding to indices n∈N‡\\Norj∈J‡\\J;\n8:end for\n9: setvm+1\nn,j= max α∈Ahum+1,α\nn,j ,n∈Nandj∈J, where um+1,α\nn,j are from Line 7; Ω in\n10: compute vm+1\nn,j,n∈N†\\Norj∈J†\\J, using (3.27); Ω out\n11:end for\nRemark 3.3 (Complexity) .As noted earlier, the cardinality of Ah, denoted by Q, isQ=O(1/h).\nAlgorithm 3.1 involves, for m= 0, . . . , M −1, the following key steps:\n•Compute um+1,α\nn,j ,n∈N†,j∈J†for all α∈ A hvia FFT algorithm. The complexity of this step\nisO(QNJ log(NJ) =O(1/h3·log(1/h)), where we take into account (3.24) .\n•Finding the optimal control (α∗)m+1\nn,j for each node xm+1\nn,j by comparing um+1,α\nn,j for all α∈ A h\nrequires O(1/h)complexity. Thus, with a total of O(1/h2)nodes, this gives a complexity O(1/h3).\n•Therefore, the major cost of Algorithm 3.1 is determined by the step of FFT Algorithm. With\nO(1/h)timesteps, the total complexity is O(1/h4·log(1/h)).\n144 Convergence to viscosity solution\nIn this section, we appeal to a Barles-Souganidis-type analysis [3] to rigorously study the convergence of\nour scheme in Ω inash→0 by verifying three properties: ℓ∞-stability, monotonicity, and consistency.\nOur scheme consists of (3.26) (for Ω τ0), (3.27) (for Ω out), and (3.29) (for Ω in).\nFor subsequent use, we state several results below. For Ω in, from Lemma 3.1, for a fixed α∈ A,\nwe haveRR\nR2gα(x, y; ∆τ)dxdy =e−r∆τ, henceRR\nD†gα(x, y; ∆τ)dxdy≤e−r∆τ<1, where D†is defined\nin (3.6). For n∈Nandj∈J(i.e. Ω in), we define\nϵg= max\nα,n,jϵα\nn,j,where ϵα\nn,j:=\f\f\f\fZZ\nD†gα(xn−x′, yj−y′; ∆τ)dx′dy′−∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d\f\f\f\f.\nHere, at noted earlier, φl,dare the weights of a two-dimensional composite trapezoidal quadrature\nrule. Since for sufficiently small h,ϵα\nn,j=O(h2), it follows that ϵg=O(h2). In addition, because the\nweights φl,dof the quadrature rule are positive, for any fixed α∈ A, we have\n0≤∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d<1 +ϵα\nn,j≤1 +ϵg< eϵg. (4.1)\n4.1 Stability\nLemma 4.1 (ℓ∞-stability) .Suppose the discretization parameter hsatisfies (3.24) . Then our scheme,\nwhich consists of (3.26) ,(3.27) , and (3.29) , satisfies the bound sup\nh>0∥vm∥∞<∞for all m= 0, . . . , M ,\nas the discretization parameter h→0. Here, we have ∥vm∥∞= max n,j|vm\nn,j|,n∈N†andj∈J†.\nProof of Lemma 4.1. First, we note that, for any fixed h >0, as given by (3.26), we have ||v0||∞<∞,\nsince Ω is a bounded domain. Therefore, we have suph>0||v0||∞<∞. Motivated by this observation,\nto demonstrate l∞-stability of our scheme, we will show that, for a fixed h >0, at any ( xn, yj, τm),\nm= 0, . . . , M , we have\n|vm\nn,j|< emϵg||v0||∞, m = 0,1, . . . , M. (4.2)\nIt is straightforward to show that (3.26) is ℓ∞-stable, since max n,j|v0\nn,j| ≤ ||v0||∞for (n, j)∈N†×J†,\nclearly satisfying (4.2). Next, for equation (3.27), we note that, since |vm+1\nn,j|=|vm\nn,je−r∆τ|<|vm\nn,j|, by\ninduction on m, we have max n,j|vm\nn,j| ≤ ||v0||∞, for either ( n, j)∈(N†\\N)×J†orN†×(J†\\J).\nNow we focus on the main task, demonstrating ℓ∞-stability for (3.29) (Ω in) through an induction\nproof on m. For the base case m= 1, with a fixed α∈ A h,\nu1\nn,j= ∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dv0\nl,d. (4.3)\nThen, we have\n|u1\nn,j| ≤ ∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d|v0\nl,d| ≤ ∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d||v0||∞≤e1ϵg||v0||∞,(4.4)\nwhere the last inequality is due to (4.1). Since v1\nn,j= max αu1\nn,j, we have\n|v1\nn,j|=|max\nαu1\nn,j| ≤ max\nα|u1\nn,j| ≤ e1ϵg||v0||∞,\nas wanted for the base case. For the hypothesis, assume that (4.2) hold for m=m′, 1≤m′≤M−1\n|vm′\nn,j|< em′ϵg||v0||∞, (n, j)∈N×J. (4.5)\n15In the induction step, we need show that (4.2) also holds for m=m′+ 1, i.e.\n|vm′+1\nn,j|< e(m′+1)ϵg||v0||∞. (4.6)\nTo show (4.6), recalling um′+1\nn,j from (3.28) gives\n|um′+1\nn,j| ≤ ∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d|vm′\nl,d|(i)\n≤∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dem′ϵg||v0||∞\n≤eϵgem′ϵg||v0||∞=e(m′+1)ϵg||v0||∞. (4.7)\nHere, (i) is due to the hypothesis (4.5) together with the fact that the scheme for Ω out, captured by\nequation (3.27), is also ℓ∞-stable as shown earlier. Hence, |vm′+1\nn,j|=|max αum′+1\nn,j| ≤e(m′+1)ϵg||v0||∞,\nproving (4.6) for m=m′+ 1. This concludes the proof.\n4.2 Consistency\nWhile equations (3.26), (3.27), and (3.29) are convenient for computation, they are not in a form\namendable for analysis. For purposes of verifying consistency, it is more convenient to rewrite them\nin a single equation. To this end, for ( xn, yj, τm+1)∈Ωin, i.e. n∈Nandj∈J, we define operator\nCm+1\nn,j(·), where\nCm+1\nn,j(·)≡ Cm+1\nn,j\u0012\nh, vm+1\nn,j,\b\nvm\nl,d\t\nl∈N†\nd∈J†\u0013\n=1\n∆τ\u0014\nvm+1\nn,j−max\nα∈Ah\u001a\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dvm\nl,d\u001b\u0015\n.(4.8)\nUsing Cm+1\nn,j(·) defined in (4.8), our numerical scheme at the reference node x= (xn, yj, τm+1) can be\nrewritten in an equivalent form as follows\n0 =Hm+1\nn,j\u0012\nh, vm+1\nn,j,\b\nvm\nl,d\t\nl∈N†\nd∈J†\u0013\n≡\n\nCm+1\nn,j(·) x∈Ωin,\nvm+1\nn,j−p(exn, eyj) x∈Ωτ0,\nvm+1\nn,j−p(exn, eyj)e−rτm+1 x∈Ωout,(4.9)\nwhere the sub-domains are defined in (2.8), and p(·,·) is the terminal condition.\nTo demonstrate the consistency in viscosity sense of (4.9), we need an intermediate result given in\nLemma 4.2 below.\nLemma 4.2 (Two dimensional - Ω in).Letϕbe a test function in C∞(Ω). For fixed α∈ A and\nxm\nn,j∈Ω, where n, j∈Nandm∈ {1, . . . , M }, with ϕm\nn,j=ϕ\u0000\nxm\nn,j\u0001\n, and for sufficiently small h, we\nhave\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dϕm\nl,d=ϕm\nn,j+ ∆τ[Lαϕ]m\nn,j+O(h2). (4.10)\nHere,\u0002\nLαϕ\u0003m\nn,j=\u0002\nLαϕ\u0003\u0000\nxm\nn,j\u0001\n, and the differential operator Lαare defined in (2.7).\nProof of Lemma 4.2. Starting from the discrete convolution on the left-hand-side (lhs) of (4.10), we\nneed to recover an associated convolution integral of the form (3.5) which is posed on an infinite\nintegration region. Since for an arbitrary fixed τm,ϕ(x, y, τ m) is not necessarily in L1(R2), standard\nmollification techniques can be used to obtain a mollifier χ(x, y, τ m)∈L1(R2) which agrees with\nϕ(x, y, τ m) onD†[19], and has bounded derivatives up to second order across R2. For brevity, instead\n16ofχ(x, y, τ m), we will write χ(x, y), which is a smooth bivariate function of ( x, y)∈R2. We have\n∆x∆yd∈N†X∗\nl∈N†φl,dgα\nn−l,j−dϕm\nl,d(i)=ZZ\nD†gα(xn−x, yj−y; ∆τ)ϕ(x, y)dx dy +O(h2)\n(ii)=ZZ\nR2gα(xn−x, yj−y; ∆τ)χ(x, y)dx dy +O(h2) +O\u0000\nhe−1/h\u0001\n(iii)= [χ∗g](xn, yj) +O(h2).\n=F−1[F[χ](η, ζ)G(η, ζ; ∆τ)] (xn, yj) +O(h2). (4.11)\nHere, in (i), the O(h2) is due to error in the composite trapezoidal rule, noting that ϕhas bounded\nderivatives of all orders in Ω because Ω is a bounded domain; in (ii) the boundary truncation error is\nO\u0000\nhe−1/h\u0001\n, due to Lemma 3.2, and in (iii) [ χ∗g] denotes the convolution of χ(x, y) and gα(x, y; ∆τ).\nIn (4.11), with Ψ( η, ζ) given in (3.10), expanding G(η, ζ; ∆τ) =eΨ(η,ζ)∆τusing a Taylor series gives\nG(η, ζ; ∆τ)≈1 + Ψ( η, ζ)∆τ+R(η, ζ)∆τ2,R(η, ζ) =Ψ(η, ζ)2eξΨ(η,ζ)\n2, ξ∈(0,∆τ). (4.12)\nTherefore,\n[χ∗g] (xn, yj) =F−1\u0002\nF[χ](η, ζ)\u0000\n1 + Ψ( η, ζ)∆τ+R(η, ζ)∆τ2)\u0001\u0003\n(xn, yj)\n=χ(xn, yj) + ∆ τF−1[F[χ](η, ζ) Ψ (η, ζ)] (xn, yj)\n+∆τ2F−1[F[χ](η, ζ)R(η, ζ)] (xn, yj). (4.13)\nHere, the first term in (4.13), namely χ(xn, yj)≡χ(xn, yj, τm) is simply ϕm\nn,jby construction of χ(·).\nFor the second term in (4.13), we focus on F[χ](η, ζ) Ψ (η, ζ). Recalling the closed-form expression\nfor Ψ( η, ζ) in (3.10), we obtain\nF[χ](η, ζ)Ψ(η, ζ) =F\u0002σ2\nx\n2χxx+σ2\ny\n2χyy+(r−σ2\nx\n2)χx+(r−σ2\ny\n2)χy+ρσxσyχxy−rχ\u0003\n(η, ζ) =F[Lαχ] (η, ζ).\nTherefore, the second term in (4.13) becomes\n∆τF−1[F[χ](η, ζ) Ψ (η, ζ)] (xn, yj) = ∆ τ[Lαχ] (xm\nn,j) = ∆ τ[Lαχ]m\nn,j. (4.14)\nFor the third term ∆ τ2F−1[F[χ](η, ζ)R(η, ζ)] (xn, yj) in (4.13), we have\n∆τ2\f\fF−1[F[χ](η, ζ)R(η, ζ)](xn, yj)\f\f\n=∆τ2\n(2π)2\f\f\f\fZZ\nR2ei(ηxn+ζyj)R(η, ζ)\u0014ZZ\nR2e−i(ηx+ζy)χ(x, y)dx dy\u0015\ndηdζ\f\f\f\f\n≤∆τ2ZZ\nR2|χ(x, y)|dxdyZZ\nR2|R(η, ζ)|dηdζ. (4.15)\nNoting R(η, ζ) =Ψ(η, ζ)2eξΨ(η,ζ)\n2, as shown in (4.12), where a closed-form expression for Ψ( η, ζ) is\ngiven in (3.10), we obtain\n|R(η, ζ)|=|(Ψ(η, ζ))2|\n2exp\u0000\nξ\u0000\n−σ2\nxη2\n2−σ2\nyζ2\n2−ρσxσyηζ−r\u0001\u0001\n.\nThe term |(Ψ(η, ζ))2|can be written in the form |Ψ|2=P\nk+q=4\nk,q≥0Ckqηkζq, where Ckqare bounded\ncoefficients. This is a quartic polynomial in ηandζ. Furthermore, the exponent of exponential term\nis bounded by\n−1\n2σ2\nxη2−1\n2σ2\nyζ2−ρσxσyηζ−r≤ −1\n2σ2\nxη2−1\n2σ2\nyζ2+|ρ|σxσy|ηζ|\n17For|ρ|<1, we have |ρ|σxσy|ηζ|<1\n2(σ2\nxη2+σ2\nyζ2). Therefore,RR\nR2|R(η, ζ)|dηdζ is bounded since\nZZ\nR2|η|k|ζ|qe−1\n2σ2xη2−1\n2σ2yζ2−ρσxσyηζdη dζ, k +q= 4, k, q≥0,\nis also bounded. Together with χ(x, y)∈L1(R2), the rhs of (4.15) is O(∆τ2), i.e.\n∆τ2\f\fF−1[F[χ](η, ζ)R(η, ζ)](xn, yj)\f\f=O(∆τ2). (4.16)\nSubstituting (4.14) and (4.16) into (4.13), noting (4.11) and χ(x, y) =ϕ(x, y) for ( x, y)∈D†gives\n∆x∆yd∈N†X∗\nl∈N†φl,dgα\nn−l,j−dϕm\nl,d=ϕm\nn,j+ ∆τ[Lαϕ]m\nn,j+O(h2).\nThis concludes the proof.\nTo establish the consistency in the viscosity sense of our scheme as presented in (4.9), it is essential\nto first examine the local consistency. This requires revisiting the operator Fin(·) defined in (2.12).\nIn the context of a discretized control set Ah, we introduce a modified operator that aligns with the\npiecewise constant control approach.\nDefinition 4.1. Given a discretization parameter h > 0, we define the operator Fh\ninfor each α∈\nAh⊆ A as follows:\nFh\nin:=Fin, α∈ A h⊆ A, (4.17)\nBuilding on this definition, Lemma 4.3 presents an important result regarding the approximation\nerror bound when implementing the piecewise constant control technique.\nLemma 4.3. For any x∈Ωin, and for a test function ϕ∈ C∞(Ω)and a constant ξ, we have\n\f\f\fFin(x, ϕ(x), Dϕ(x), D2ϕ(x))−Fh\nin(x,(ϕ+ξ)(x), D(ϕ+ξ)(x), D2(ϕ+ξ))\f\f\f≤Ch+rξ, (4.18)\nwhere C >0is a bounded constant independently of h.\nProof of Lemma 4.3. By insertion and the triangle inequality, the lhs of (4.18) is bounded as follows\n\f\f\fFin(·)−Fh\nin(·)\f\f\f≤\f\f\f\fsup\nα∈AhLα(ϕ+ξ)−sup\nα∈ALα(ϕ+ξ)\f\f\f\f+\f\f\f\fsup\nα∈ALα(ϕ+ξ)−sup\nα∈ALα(ϕ)\f\f\f\f\n=\f\f\f\fsup\nα∈AhLαϕ−sup\nα∈ALαϕ\f\f\f\f+rξ. (4.19)\nDue to the compactness of A, the supremum of Lα(ϕ) is attainable at, say α∗≡(σ∗\nx, σ∗\ny, ρ∗)∈ A. By\n(3.1), there exists α′∗≡(σ′∗\nx, σ′∗\ny, ρ′∗)∈ A hwith∥α′∗−α∗∥2≤h. Therefore, the first term in (4.19)\nbecomes\f\fsupα∈AhLα(ϕ)−supα∈ALα(ϕ)\f\f=|Lα′∗ϕ− L α∗ϕ|=. . .\n. . .(i)=\f\f\f\f1\n2((σ′∗\nx)2−(σ∗\nx)2)(ϕxx−ϕx) +1\n2((σ′∗\ny)2−(σ∗\ny)2)(ϕyy−ϕy) + (ρ′∗σ′∗\nxσ′∗\ny−ρ∗σ∗\nxσ∗\ny)ϕxy\f\f\f\f\n≤1\n2\f\f((σ′∗\nx)2−(σ∗\nx)2)\f\f(|ϕxx|+|ϕx|) +1\n2\f\f((σ′∗\ny)2−(σ∗\ny)2)\f\f(|ϕyy|+|ϕy|) +\f\f(ρ′∗σ′∗\nxσ′∗\ny−ρ∗σ∗\nxσ∗\ny)\f\f|ϕxy|\n(ii)\n≤Ch,\nwhere C >0 is a bounded constant independently of h. Here, in (i), we first insert the Lα(·) operator\n(2.7) and then combine similar terms; (ii) due to the ∥α′∗−α∗∥2≤h, together with the compactness of\nthe admissible control set Aand the fact that the test function ϕhas continuous bounded derivatives\nin Ω since Ω is bounded. This concludes the proof.\nBelow, we state the key supporting lemma related to local consistency of our numerical scheme (4.9).\n18Lemma 4.4 (Local consistency) .Suppose that (i) the discretization parameter hsatisfies (3.24) .\nThen, for any test function ϕ∈ C∞(Ω), with ϕm\nn,j=ϕ\u0010\nxm\nn,j\u0011\nandx:= (xn, yj, τm+1)∈Ω, and for a\nsufficiently small h, we have\nHm+1\nn,j\u0012\nh, ϕm+1\nn,j+ξ,\b\nϕm\nl,d+ξ\t\nl∈N†\nd∈J†\u0013\n=\n\nFh\nin(·,·) +c(x)ξ+O(h)x∈Ωin,\nFout(·,·)+c(x)ξ x∈Ωout;\nFτ0(·,·) +c(x)ξ x∈Ωτ0.(4.20)\nHere, ξis a constant, and c(·)is a bounded function satisfying |c(x)| ≤max( r,1)for all x∈Ω.\nThe operators Fh\nin, defined in (4.17) , and Fout(·), and Fτ0(·), respectively defined in (2.13) -(2.14) , are\nfunctions of (x, ϕ(x)).\nProof of Lemma 4.4. Since ϕ∈ C∞(Ω) and Ω is bounded, ϕhas continuous and bounded derivatives\nof up to second-order in Ω. We now show that the first equation of (4.20) is true, that is,\nHm+1\nn,j(·) =Cm+1\nn,j(·) =Fh\nin(x, ϕ(x)) +c(x)ξ+O(h)\nifxmin< xn< xmax, ymin< yj< ymax,0< τm+1≤T.\nwhere operators Cm+1\nn,j(·) is defined in (4.8). In this case, operator Cm+1\nn,j(·) is written as follows\nCm+1\nn,j(·) =1\n∆τ\u0014\nϕm+1\nn,j+ξ−max\nα∈Ah\u001a\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d(ϕm\nl,d+ξ)\u001b\u0015\n(4.21)\n=1\n∆τ\u0014\nϕm+1\nn,j−max\nα∈Ah\u001a\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dϕm\nl,d\u001b\n+ξ\u0012\n1−max\nα∈Ah∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d\u0013\u0015\n(i)=ϕm+1\nn,j−ϕm\nn,j\n∆τ−max\nα∈Ah[Lαϕ]m\nn,j+ξ\n∆τ\u0012\n1−max\nα∈Ah\u001a\n∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d\u001b\u0013\n+O(h).\nHere, (i) is due to use Lemma 4.2. Regarding the termξ\n∆τ\u0000\n1−max α∈Ah{·}\u0001\n, suppose that max α∈Ah{·}\nis attainable at α′. Then, |1−max α∈Ah{·}|=\f\f\f\f1−∆x∆yd∈J†X∗\nl∈N†φl,dgα′\nn−l,j−d\f\f\f\f≤. . .\n. . .≤\f\f\f\f1−ZZ\nR2gα′(xn−x, yj−y; ∆τ)dxdy\f\f\f\f+\f\f\f\fZZ\nR2gα′(·,·; ∆τ)dxdy−∆x∆yd∈J†X∗\nl∈N†φl,dgα′\nn−l,j−d\f\f\f\f.(4.22)\nThe first term of (4.22) is simply 1 −e−r∆τ=r∆τ+O(h2), notingRR\nR2gα(·,·; ∆τ)dxdy =e−r∆τ\nfor any α∈ A. The second term of (4.22) is simply O(h2) +O(he−1/h) =O(h2) due to numerical\nintegration error and boundary truncation error, as noted earlier. With this in mind, we have\nξ\n∆τ\u0012\n1−max\nα∈Ah∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−d\u0013\n=rξ+O(h).\nSubstituting this result into (4.21) gives\nCm+1\nn,j(·) =ϕm+1\nn,j−ϕm\nn,j\n∆τ−max\nα∈Ah[Lαϕ]m\nn,j+rξ+O(h)(i)=\u0014\nϕτ−max\nα∈AhLαϕ\u0015m+1\nn,j+rξ+O(h).\nHere, in (i), we use ( ϕτ)m\nn,j= (ϕτ)m+1\nn,j+O(h), (ϕz)m\nn,j= (ϕz)m+1\nn,j+O(h),z∈ {x, y}, and for the cross\nderivative term ( ϕxy)m\nn,j= (ϕxy)m+1\nn,j+O(h). This proves the first equation in (4.20). The remaining\nequations in (4.20) can be proved using similar arguments with the first equation, and hence omitted\nfor brevity. This concludes the proof.\n19We now verify the consistency of the numerical scheme Hm+1\nn,j(·) as defined in (4.9). We first define\nthe notion of consistency in the viscosity sense below.\nDefinition 4.2 (Consistency in viscosity sense) .Suppose the discretization parameter hsatisfies (3.24) .\nThe numerical scheme (4.9) is consistent in the viscosity sense if, for all ˆ x= (ˆx,ˆy,ˆτ)∈Ω, and for\nanyϕ∈ C∞(Ω), with ϕm\nn,j=ϕ\u0000\nxm\nn,j\u0001\nandx=(xn, yj, τm+1), we have both of the following\nlim sup\nh→0,x→ˆ x\nξ→0Hm+1\nn,j\u0012\nh, ϕm+1\nn,j+ξ,\b\nϕm\nl,d+ξ\t\nl∈N†\nd∈J†\u0013\n≤(FΩ)∗\u0000\nˆ x, ϕ(ˆ x), Dϕ(ˆ x), D2ϕ(ˆ x)\u0001\n, (4.23)\nlim inf\nh→0,x→ˆ x\nξ→0Hm+1\nn,j\u0012\nh, ϕm+1\nn,j+ξ,\b\nϕm\nl,k+ξ\t\nl∈N†\nd∈J†\u0013\n≥(FΩ)∗\u0000\nˆ x, ϕ(ˆ x), Dϕ(ˆ x), D2ϕ(ˆ x)\u0001\n. (4.24)\nHere, (FΩ)∗(·)and(FΩ)∗(·)respectively are the u.s.c. and the l.s.c. envelop of the operator FΩ(·)\ndefined in (2.10) .\nBelow, we state and prove the main lemma on consistency of the numerical scheme (4.9).\nLemma 4.5 (Consistency) .Suppose the discretization parameter hsatisfies (3.24) . Then, the nu-\nmerical scheme (4.9) is consistent with the two-factor uncertain volatility pricing problem (2.1) inΩ\nin the sense of Definition 4.2.\nProof of Lemma 4.5. We first prove (4.23). Let ˆ x≡(ˆx,ˆy,ˆτ) be an arbitrary, but fixed, point in Ω.\nConsider h→0. There exists sequences of {hi},{mi},{xi}, and{ξi}, such that\nasi→ ∞ , h i→0, ξi→0,xi≡(xni, yji, τmi+1)→ˆ x≡(ˆx,ˆy,ˆτ), (4.25)\nand\nlim sup\ni→∞Hmi+1\nni,ji\u0010\nhi, ϕmi+1\nni,ji+ξi,\b\nϕmi\nli,di+ξi\t\u0011\n= lim sup\nh→0,x→ˆ x\nξ→0Hm+1\nn,j\u0010\nh, ϕm+1\nn,j+ξ,\b\nϕm\nl,d+ξ\t\u0011\n.(4.26)\nNow, we consider the case ˆ x∈Ωin. According to the first equation of (4.20) (Lemma 4.4), we have\nHmi+1\nni,ji\u0010\nhi, ϕmi+1\nni,ji+ξi,n\nϕmi\nli,ki+ξio\u0011\n=Fhi\nin\u0000\nxi, ϕ(xi), Dϕ(xi), D2ϕ(xi)\u0001\n+rξi+O(hi) (4.27)\nUsing (4.18) and (4.27) gives, for each i,\n\f\fF(xi, ϕ(xi),·,·)− Hmi+1\nni,ji\u0000\nhi, ϕmi+1\nni,ji+ξi,\b\nϕmi\nli,ki+ξi\t\u0001\f\f≤Cihi+ (r+c(xi))ξi+O(hi) (4.28)\nHere, Ci>0 is a bounded constant and |c(xi)| ≤max( r,1) for all i. Thus, from (4.28), we have\nHmi+1\nni,ji\u0000\nhi, ϕmi+1\nni,ji+ξi,\b\nϕmi\nli,ki+ξi\t\u0001\n≤F\u0000\nxi, ϕ(xi),·,·\u0001\n+Cihi+ (r+c(xi))ξi+O(hi) (4.29)\nCombining (4.26) and (4.29), with continuity of F(·), we obtain\nlim sup\nh→0,x→ˆ x\nξ→0Hm+1\nn,j\u0010\nh, ϕm+1\nn,j+ξ,\b\nϕm\nl,d+ξ\t\u0011\n= lim sup\ni→∞Hmi+1\nni,ji\u0010\nhi, ϕmi+1\nni,ji+ξi,\b\nϕmi\nli,di+ξi\t\u0011\n≤lim sup\ni→∞F\u0000\nxi, ϕ(xi), Dϕ(xi), D2ϕ(xi)\u0001\n+ lim sup\ni→∞(Cihi+ (r+c(xi))ξi)\n=F∗\u0000\nˆ x, ϕ(ˆ x), Dϕ(ˆ x), D2ϕ(ˆ x)\u0001\n.\nThis proves (4.23) for ˆ x∈Ωin. The case (4.23) for other sub-domains as well as the case (4.24) can\nbe proved in a similar fashion. This concludes the proof.\n204.3 Monotonicity\nWe present a result on the monotonicity of scheme (4.9).\nLemma 4.6. (Monotonicity) Scheme (4.9) satisfies\nHm+1\nn,j\u0010\nh, vm+1\nn,j,\b\num\nl,d\t\u0011\n≤ Hm+1\nn,j\u0010\nh, vm+1\nn,j,\b\nzm\nl,d\t\u0011\n(4.30)\nfor boundedn\num\nl,do\nandn\nzm\nl,do\nhavingn\num\nl,do\n≥n\nzm\nl,do\n, where the inequality is understood in the\ncomponent-wise sense.\nProof of Lemma 4.6. Since scheme (4.9) is defined case-by-case, to establish (4.30), we will show that\neach case satisfies (4.30). It is straightforward that the scheme satisfies (4.30) in Ω τ0) and Ω out. Now\nwe establish that Cm+1\nn,j(·), as defined in (4.8) for Ω in, also satisfies (4.30). We have\nCm+1\nn,j\u0010\nh, vm+1\nn,j,\b\num\nl,d\t\u0011\n− Cm+1\nn,j\u0010\nh, vm+1\nn,j,\b\nzm\nl,d\t\u0011\n=1\n∆τ\u0014\nmax\nα∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dzm\nl,d−max\nα∆x∆yd∈J†X∗\nl∈N†φl,dgα\nn−l,j−dum\nl,d\u0015\n(i)\n≤1\n∆τ\nmax\nα∆x∆yd∈N†X∗\nl∈N†φl,dgα\nn−l,j−d\u0000\nzm\nl,d−um\nl,d\u0001\n≤0. (4.31)\nHere, (i) is due to the fact that, max α∈Af1(α)−max α∈Af2(α)≤max α(f1(α)−f2(α)) for two real-\nvalued functions f1, f2ofα. This concludes the proof.\nTheorem 4.1 (Convergence to viscosity solution in Ω in).Suppose that all the conditions for Lem-\nmas 4.1), 4.4 and 4.6 are satisfied. Our scheme (4.9) converges in Ωinto the unique continuous\nviscosity solution of the two-factor uncertain volatility model pricing problem given in Definition (2.2).\nProof of Theorem 4.1. Our scheme is ℓ∞-stable (Lemma 4.1), and consistent in the viscosity sense\n(Lemma 4.4) and monotone (Lemma 4.6). Since a strong comparison holds in Ω in(Remark 2.2), by\n[3], convergence in Ω into the unique continuous vicosity solution of the HJB equation is ensured.\n5 Numerical experiments\nThis section presents the selected numerical results of our monotone piecewise constant control inte-\ngration method (MPCCI) applied to the two-factor uncertain volatility model pricing problem. The\nmodelling parameters for the tests carried out are given in Table 5.1, reproduced from [22][Table 3].\nWe note that specific ranges for σx,σy, and the correlation coefficient ρare given.\n5.1 Preliminary\nPrior to initiating our experiments, it is essential to define a sufficiently large computational domain,\nguided by the boundary truncation error bound provided in Lemma 3.2. Specifically, we follow steps\noutlined in Remark 3.2 to determine x†\nmin,x†\nmax,y†\nminandy†\nmax. In particular, in (3.17), ϵ= 10−10\nis used. With the model parameters given Table 5.1, this procedure gives xmin= ln( X0)−1.2,\nxmax= ln( X0) + 1 .2,ymin= ln( Y0)−1.2,ymax= ln( Y0) + 1 .2. Furthermore, for z∈ {x, y}, the\nvalues of z†\nmin,z†\nmax,z‡\nminandz‡\nmaxare determined as in (3.19)-(3.20). Extensive testing indicates\nthat larger intervals have negligible impact on numerical solutions, whereas smaller domains exhibit\nminor variations. These findings are numerically validated in Subsection 5.3 Unless noted otherwise,\nthe specifics of mesh size and timestep refinement levels utilized in all experiments are detailed in\nTable 5.2.\n21Parameter Value/ Parameter Value/\nRange Range\nT 0.25 (years) X0 40\nr 0.05 Y0 40\nσx [0.3, 0.5] K 40\nσy [0.3, 0.5] K1 34\nρ [0.3, 0.5] K2 46\nTable 5.1: Model parameters used in numerical ex-\nperiments for two-factor uncertain volatility model-\nreproduced from [22] Table 3.Level N J M Q\n(x) (y) (τ) (α)\n0 272750 8\n1 2828100 24\n2 2929200 56\n3 210210400 120\n4 211211800 248\nTable 5.2: Grid and timestep refine-\nment levels for numerical tests.\nOur MPCCI numerical prices are verified against those produced by: (i) closed-form solutions\n(for certain European rainbow options), (ii) FD methods reported in the literature, particularly the\nunconditionally monotone FD method of [22], (iii) tree-grid (TG) methods of [17], and (iii) Monte\nCarlo (MC) simulation. The Monte Carlo validation is carried out in two steps\n1. Step 1: we solve the two-factor uncertain volatility pricing problem using the proposed MPCCI\non a fine computational grid (comprising of 210x-nodes, 210y-nodes, and 400 timesteps). At\neach time- τm, we store the optimal controls or all pair of discrete states ( xn, yj), denoted as\b\n(α∗)m\nn,j\t\n≡\b\n(σ∗\nx, σ∗\ny, ρ∗)m\nn,j\t\n, where n∈N†,j∈J†, and m= 0, . . . , M .\n2. Step 2: we conduct Monte Carlo simulations of the two-dimensional dynamics (2.1) from t= 0\ntot=T, following the stored MPCCI-computed optimal controls. For a given pair of sim-\nulated values of XandY, linear interpolation, if necessary, is used to determine the control.\nSpecifically, for the γ-thXandYsimulated values at time- τm, denoted by ˆX(γ)\nmandˆY(γ)\nm, and\ngiven xn′≤ˆX(γ)\nm≤xn′+1andyj′≤ˆY(γ)\nm≤yj′+1, we interpolate the optimal control ( α∗)m\nn′,j′,\n(α∗)m\nn′+1,j, (α∗)m\nn′,j′+1, and ( α∗)m\nn′+1,j′+1to determine the volatilities ˜ σm\nxand ˜σm\nyand the cor-\nrelation coefficient ˜ ρmfor the interval [ τm, τm+1]. The Euler-Maruyama discretization is then\napplied for each the interval [ τm, τm+1] as follows:\nˆX(γ)\nm+1=ˆX(γ)\nm\u0010\n1 +r∆t+ ˜σm\nx√\n∆t ξ(γ)\nx\u0011\n,\nˆY(γ)\nm+1=ˆY(γ)\nm\u0010\n1 +r∆t+ ˜σm\ny√\n∆t(˜ρmξ(γ)\nx+p\n1−(˜ρm)2ξ(γ)\ny)\u0011\n,\nwhere ξ(γ)\nxandξ(γ)\nyare independent standard normal random variables. The option value is\napproximated bye−rT\nΓPΓ\nγ=1p\u0000ˆX(γ)\nM,ˆY(γ)\nM\u0001\n, with p(·,·) as the payoff function, using a total of\nΓ = 106simulation paths.\n5.2 Validation examples\n5.2.1 European call options\nOur first test case evaluates a European call option on the maximum of two assets, as described in\n[22]. The payoff function p(ex, ey) is given by\np(ex, ey) = max(max( ex, ey)−K,0), K > 0. (5.1)\nFor worst-case for the short position, the optimal control {(α∗)}at any point ( x, y, τ ) is mathematically\ndetermined to be σ∗\nx=σx\nmax,σ∗\ny=σy\nmax, and ρ∗=ρmin. Given the parameters in (5.1), this translates\ntoσ∗\nx=σ∗\ny= 0.5 and ρ∗= 0.3. The exact option price is thus analytically computable using the\nclosed-form expression from [32].\n22Level Price Abs. error Ratio\n0 6.84492756 2.77e-03\n1 6.84700690 6.93e-04 4.00\n2 6.84752662 1.73e-05 4.00\n3 6.84765654 4.33e-05 4.00\n4 6.84768902 1.08e-05 4.00\nRef. [32] 6.84769986\nMC: 95%-CI [6.8319, 6.8618]\nTable 5.3: Convergence study for a European\ncall option on the maximum of two risky as-\nsets under two-factor uncertain volatility model\n(worst-case ) - payoff function in (5.1). The\nclosed-form solution is obtained using [32] with\nfixed parameters σ∗\nx= 0.5,σ∗\ny= 0.5andρ∗= 0.3.\nFigure 5.1: Absolute error on Ωassociated\nwith test case reported in Table 5.3.\nDespite knowing the optimal control, we discretize the admissible control set in our experiments for\ngenerality, i.e. Ahis used and it contains the optimal control {(α∗)}={σx\nmax, σy\nmax, ρmin}at all refinement\nlevels. It is observed that the proposed MPCCI scheme accurately yielded the aforementioned optimal\ncontrol for all ( xn, yj, τm). Table 5.3 shows the convergence results for the time t= 0 option price at\nex=X0,ey=Y0. To provide an estimate of the convergence rate of the proposed MPCCI method, we\ncompute the “Abs. error” as the absolute error between the exact option [32] and numerical option\nprices, and the “Ratio” as the ratio of successive absolute errors. These results indicate excellent\nagreement with the analytic solution from [32], as do the results from MC simulations. Notably, the\nMPCCI method exhibits second-order convergence.\nWe further demonstrates the accuracy of the MPCCI method in the entire domain Ω. In Figure 5.1,\nwe present the absolute error at time t= 0 on grid points obtained with refinement Level 4. The\nabsolute error, computed as\n\f\fv(xn, yj, τM)−vM\nn,j\f\f, n∈N†, j∈J†, τ M=T,\nis very small across the computational domain, typically of the order of 10−5or less, with higher errors\nconcentrated near the strike K= 40, as expected.\nIn Table 5.4 and Figure 5.2, we display the best-case results for the short position. The outcomes\nclosely mirror those of the worst-case scenario, showing excellent agreement with the closed-form\nsolution, exhibiting second-order of convergence.\nThe observed second-order convergence in Tables 5.3 and 5.4 can be explained as followed. Recall\nthat in the worst-case scenario, the optimal control α∗at any point ( x, y, τ ) is always {σx\nmax, σy\nmax, ρmin},\nwhile for the best-case, it becomes {σx\nmin, σy\nmin, ρmax}. Since α∗is included in Ah(at every refinement\nlevel) and is obtained by solving the optimization problem at each timestep, the discretization error\nfrom approximating AbyAhis eliminated. In addition, for a refinement level, the constancy of α∗\nresults in a single Green’s function, namely gα∗(·; ∆τ), for all timesteps, eliminating timestepping\nerror thanks to the additive nature of integration. Therefore, the only error stems from numerical\nintegration, where the second-order accuracy is observed. This second-order convergence is expected\nfrom the composite trapezoidal rule because gridpoints aligns with the kinks of the payoff function\naround the line x=y,x=Kandy=K.\n23Level Price Abs. error Ratio\n0 3.96880850 4.80e-03\n1 3.97240621 1.20e-03 4.00\n2 3.97330502 3.00e-04 4.00\n3 3.97352968 7.49e-05 4.00\n4 3.97358584 1.87e-05 4.00\nMC: 95%-CI [3.9657, 3.9840]\nRef. [32] 3.97360457\nTable 5.4: Convergence study of an European\ncall option on the maximum of two risky as-\nsets under two-factor uncertain volatility model\n(best-case ) - payoff function in (5.1). The closed-\nform solution is obtained using [32] with fixed pa-\nrameters σx= 0.3,σy= 0.3andρ= 0.5.\nFigure 5.2: Absolute error on Ωassociated\nwith test case reported in Table 5.4.\nLevel Price Change Ratio\n0 2.65092717\n1 2.66374754 0.0128\n2 2.67280480 0.0091 1.42\n3 2.67793762 0.0051 1.76\n4 2.68070303 0.0028 1.86\nMC: 95%-CI [2.6735, 2.6832]\nFD [22] 2.6744\nTG [17] 2.6784\nTable 5.5: Convergence study for a butterfly\noption (worst-case ) under a two-factor uncer-\ntain volatility model - payoff function in (5.2).\nReference prices: by FD method is 2.6744 [22]\n(finest level in Table 6 therein, Pure wide sten-\ncil (with rotation)), by TG method is 2.6784\n[17] (finest level in Table 3 therein)Level Price Change Ratio\n0 0.94015237\n1 0.92418409 -0.0138\n2 0.91794734 -0.0062 2.56\n3 0.91473085 -0.0032 1.94\n4 0.91308945 -0.0016 1.96\nMC : 95%-CI [0.9120, 0.9182]\nFD [22] 0.9148\nTG [17] 0.9173\nTable 5.6: Convergence study for a butter-\nfly option (best-case ) under a two-factor uncer-\ntain volatility model - payoff function in (5.2).\nReference prices: (i) by FD method is 0.9148\n[22] (finest level in Table 8 therein, Pure wide\nstencil (with rotation)), (ii) by TG method is\n0.9173 [17] (finest level in Table 4 therein).\n5.2.2 Butterfly options\nIn the second test, we consider a butterfly option on the maximum of two assets. For this option, the\npayoff function p(ex, ey) is given by\np(ex, ey) = max\u0000\nmax( ex, ey)−K1,0\u0001\n−2 max\u0000\nmax( ex, ey)−(K1+K2)/2,0\u0001\n+ max\u0000\nmax( ex, ey)−K2,0\u0001\n, K 1, K2>0. (5.2)\nFor the butterfly payoff function (5.2), a closed-form expression for the option price is unknown.\nTo estimate the convergence rate of the proposed MPCCI method, we calculate the “Change” as\nthe difference in values from coarser to finer grids and the “Ratio” as the ratio of changes between\nsuccessive grids. We compare our prices against reference prices obtained by a FD method with pure\nwide stencil rotation developed in [22], and by a tree-grid (TG) method of [17].\nTables 5.5 and 5.6 display the numerical prices for both the worst-case and best-case scenarios\n(short position), indicating that our method approximates first-order convergence. The comparison\n24with reference prices shows minimal differences: against FD method prices, the discrepancies are\naround 6 ×10−3for the worst-case and 1 ×10−3for the best-case scenarios. When compared with\nTG prices, the differences are 2 ×10−3and 4 ×10−3, respectively, highlighting the MPCCI method’s\nprecision.\n5.3 Impact of spatial domain sizes\nIn this subsection, we numerically validate the adequacy of our selected spatial domain for the ex-\nperiments. We revisit the scenarios from Tables 5.3, 5.5, and 5.6, this time doubling the lengths of\nthe spatial domains. Specifically, we extend the spatial domain boundaries to xmin= ln( X0)−2.4,\nxmax= ln( X0) + 2.4,ymin= ln( Y0)−2.4,ymax= ln( Y0) + 2.4, with the number of intervals NandJ\nalso doubled to maintain the same ∆ xand ∆ y.\nThe numerical prices from this extended domain, shown in Table 5.7, are virtually identical with\nthose obtained from the original smaller domain (reproduced under columns marked Tab. 5.3, Tab. 5.5,\nand Tab. 5.6). This indicates that enlarging the spatial computational domain further has a negligible\neffect on the numerical prices. Additionally, for a comprehensive analysis, we conducted tests on\nsmaller spatial domains with the boundaries set to xmin= ln( X0)−0.9,xmax= ln( X0) + 0.9,ymin=\nln(Y0)−0.9, and ymax= ln( Y0) + 0 .9, using the same ∆ xand ∆ yas in previous tests. The prices,\npresented in Table 5.8, show slight discrepancies (from the fourth decimal digits) when compared to\nthose obtained from original domain size.\nThese findings affirm the adequacy of our computational domain, whose size was carefully chosen\nbased on the upper bound for the boundary truncation error of the Green’s function provided in\n(3.16). This approach effectively balance the need for demonstrating theoretical convergence and\ncomputational efficiency in our analysis.\nLevelTwo-factor uncertain volatility model\nEuropean Butterfly (worst) Butterfly (best)\nPrice Price Price Price Price Price\n(Tab. 5.3) (Tab. 5.5) (Tab. 5.6)\n0 6.84492758 6.84492756 2.65092717 2.65092717 0.94015237 0.94015237\n1 6.84700691 6.84700690 2.66374754 2.66374754 0.92418409 0.92418409\n2 6.84752663 6.84752662 2.67280480 2.67280480 0.91794734 0.91794734\n3 6.84765655 6.84765654 2.67793762 2.67793762 0.91473085 0.91473085\n4 6.84768903 6.84768902 2.68070303 2.68070303 0.91308945 0.91308945\nTable 5.7: Prices obtained using a larger spatial computational domain: xmin= ln( X0)−2.4,xmax=\nln(X0) + 2.4,ymin= ln( Y0)−2.4,ymax= ln( Y0) + 2.4, in comparison with prices in Table 5.3, 5.5, 5.6\nobtained with the original smaller domain zmin= ln( Z0)−1.2,zmax= ln( Z0) + 1.2, for z∈ {x, y}.\n5.4 Impact of boundary conditions\nIn this subsection, we numerically demonstrate that our straightforward approach of employing dis-\ncounted payoffs for boundary sub-domains is adequate. We revisited previous experiments reported\nin Tables 5.3, 5.5, and 5.6, introducing sophisticated boundary conditions based on the asymptotic\nbehavior of the HJB equation (2.6) as z→ −∞ andz→ ∞ forz∈ {x, y}as proposed in [22].\nSpecifically, the HJB equation (2.6) simplifies to the 1D forms shown in (5.3) when xorytends to\n−∞:\nvτ−sup\nσy∈Ay\b\u0000\nr−(σy)2/2\u0001\nvy+ (σy)2/2vyy\t\n+rv= 0, x→ −∞ ,\nvτ−sup\nσx∈Ax\b\u0000\nr−(σx)2/2\u0001\nvx+ (σx)2/2vxx\t\n+rv= 0, y→ −∞ .(5.3)\n25LevelTwo-factor uncertain volatility model\nEuropean Butterfly (worst) Butterfly (best)\nPrice Price Price Price Price Price\n(Tab. 5.3) (Tab. 5.5) (Tab. 5.6)\n0 6.84490660 6.84492756 2.65092891 2.65092717 0.94014513 0.94015237\n1 6.84698614 6.84700690 2.66375021 2.66374754 0.92417329 0.92418409\n2 6.84750559 6.84752662 2.67280820 2.67280480 0.91793374 0.91794734\n3 6.84763508 6.84765654 2.67794155 2.67793762 0.91471524 0.91473085\n4 6.84766711 6.84768902 2.68070724 2.68070303 0.91307245 0.91308945\nTable 5.8: Prices obtained using a smaller computational domain: xmin= ln( X0)−0.9,xmax=\nln(X0) + 0.9.ymin= ln( Y0)−0.9, and ymax= ln( Y0) + 0.9. Compare with prices in Table 5.3, 5.5,\n5.6, where zmin= ln( Z0)−1.2,zmax= ln( Z0) + 1.2, for z∈ {x, y}.\nAsx, y→ −∞ , the HJB equation (2.6) simplifies to the ordinary differential equation vτ+rv= 0.\nTo adhere to these asymptotic boundary conditions, we choose a much large spatial domain:\nxmin= ln( X0)−9.6,xmax= ln( X0) + 9 .6,ymin= ln( Y0)−9.6,ymax= ln( Y0) + 9 .6, and adjust\nthe number of intervals NandJaccordingly to maintain the same grid resolution (∆ xand ∆ y).\nEmploying the monotone piecewise constant control integration technique, tailored for the 1D case,\nwe effectively solved the 1D HJB equations in (5.3). The ordinary differential equation vτ+rv= 0\nis solved directly and efficiently. The scheme’s convergence to the viscosity solution can be rigorously\nestablished in the same fashion as the propose scheme.\nThe resulting option prices, listed in Table 5.9, are virtually identical with those from the original\nsettings (under columns marked with Tab. 5.3, Tab. 5.5, and Tab. 5.6). These results confirm the\neffectiveness of our simple boundary conditions, demonstrating that they are both easy to implement\nand sufficient for the theoretical and practical demands of our numerical experiments.\nLevelTwo-factor uncertain volatility model\nEuropean Butterfly (worst) Butterfly (best)\nPrice Price Price Price Price Price\n(Tab. 5.3) (Tab. 5.5) (Tab. 5.6)\n0 6.84492760 6.84492756 2.65092717 2.65092717 0.94015237 0.94015237\n1 6.84700690 6.84700690 2.66374754 2.66374754 0.92418410 0.92418409\n2 6.84752663 6.84752662 2.67280480 2.67280480 0.91794734 0.91794734\n3 6.84765655 6.84765654 2.67793762 2.67793762 0.91473085 0.91473085\n4 6.84768903 6.84768902 2.68070303 2.68070303 0.91308945 0.91308945\nTable 5.9: Results using sophisticated boundary conditions (5.3). Compare with results in Table 5.3,\n5.5, 5.6 where simple boundary conditions based on discounted payoffs are used.\n6 Conclusion\nIn this paper, we have presented a novel and streamlined approach to solving two-dimensional (2D)\nHamilton-Jacobi-Bellman (HJB) Partial Differential Equations (PDEs) arising from two-factor uncer-\ntain volatility models with uncertain correlation.\nDeparting from the traditional “discretize, then optimize” strategy, our “decompose and integrate,\nthen optimize” method utilizes a piecewise constant control technique to decompose the HJB PDE into\nindependent 2D linear PDEs at each timestep, significantly simplifying the optimization process. Our\nmain contributions include developing a monotone piecewise constant control numerical integration\n26scheme that employs Green’s functions for solving independent linear 2D PDEs. This approach sim-\nplifies the treatment of cross-derivative terms, offering an advantage over traditional finite difference\nmethods. We have implemented our scheme efficiently using FFTs and circulant convolution, utilizing\nthe Toeplitz matrix structure to expand convolution kernels into circulant matrices for both inner\nand double summations. This enables efficient computation through 2D FFTs. We mathematically\ndemonstrate of the stability and consistency of the scheme in the viscosity sense, along with its point-\nwise convergence to the viscosity solution of the HJB PDE. Extensive numerical results showcasing\nthe effectiveness of our approach align remarkably with benchmark solutions.\nWhile our focus has been on models with uncertain volatilities, our comprehensive and systematic\napproach provides a framework for developing similar integration methods for other HJB PDEs in\nfinance, demonstrating its broad applicability and potential for future exploration.\nReferences\n[1] M. Avellaneda, A. Levy, and A. Par´ as. Pricing and hedging derivative securities in markets with uncertain\nvolatilities. Applied Mathematical Finance , 2:73–88, 1995.\n[2] Marco Avellaneda and Robert Buff. Combinatorial implications of nonlinear uncertain volatility models:\nthe case of barrier options. Applied Mathematical Finance , 6(1):1–18, 1999.\n[3] G. Barles and P.E. Souganidis. Convergence of approximation schemes for fully nonlinear equations.\nAsymptotic Analysis , 4:271–283, 1991.\n[4] O. Bokanowski, S. Maroso, and H. Zidani. Some convergence results for Howard’s algorithm. SIAM Journal\non Numerical Analysis , 47:3001–3026, 2009.\n[5] J Fr´ ed´ eric Bonnans and Housnaa Zidani. Consistency of generalized finite difference schemes for the\nstochastic HJB equation. SIAM Journal on Numerical Analysis , 41(3):1008–1021, 2003.\n[6] K. Debrabant and E.R. Jakobsen. Semi-Lagrangian schemes for linear and fully non-linear diffusion equa-\ntions. Mathematics of Computation , 82:1433–1462, 2013.\n[7] Nikolai G Dokuchaev and Andrey V Savkin. The pricing of options in a financial market model with\ntransaction costs and uncertain volatility. Journal of Multinational Financial Management , 8(2-3):353–\n364, 1998.\n[8] Dean G. Duffy. Green’s Functions with Applications . Chapman and Hall/CRC, New York, 2nd edition,\n2015.\n[9] P. A. Forsyth and G. Labahn. Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance.\nJournal of Computational Finance , 11(2):1, 2007.\n[10] M. G. Garroni and J. L. Menaldi. Green functions for second order parabolic integro-differential problems .\nNumber 275 in Pitman Research Notes in Mathematics. Longman Scientific and Technical, Harlow, Essex,\nUK, 1992.\n[11] Andr´ es Alay´ on Glazunov and Jie Zhang. A note on the bivariate distribution representation of two perfectly\ncorrelated random variables by Dirac’s δ-function. arXiv preprint arXiv:1205.0933 , 2012.\n[12] J. Guyon and P. Henry-Labordere. Uncertain volatility model: a Monte-Carlo approach. Journal of\nComputational Finance , 14:37–74, 2011.\n[13] Patrick S Hagan, Deep Kumar, Andrew S Lesniewski, and Diana E Woodward. Managing smile risk. The\nBest of Wilmott , 1:249–296, 2002.\n[14] S. Heston. A closed form solution for options with stochastic volatility with applications to bond and\ncurrency options. Review of Financial Studies , 6:327–343, 1993.\n[15] Bartosz Jaroszkowski and Max Jensen. Valuation of European options under an uncertain market price of\nvolatility risk. Applied Mathematical Finance , 29(3):213–226, 2022.\n27[16] Bartosz Jaroszkowski and Max Jensen. Valuation of European options under an uncertain market price of\nvolatility risk. Applied Mathematical Finance , 29(3):213–226, 2022.\n[17] Igor Kossaczk` y, Matthias Ehrhardt, and Michael G¨ unther. The 2D tree–grid method. Journal of Compu-\ntational Finance , 22(12):29–57, 2019.\n[18] N.V. Krylov. Approximating value functions for controlled degenerate diffusion processes by using piece-\nwise constant policies. Electronic Journal of Probability , 4(2):1–19, 1999.\n[19] J.M. Lee. Introduction to smooth manifolds . Springer-Verlag, New York, 2 edition, 2012.\n[20] T. Lyons. Uncertain volatility and the risk free synthesis of derivatives. Applied Mathematical Finance ,\n2:117–133, 1995.\n[21] T. J. Lyons. Uncertain volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance ,\n2(2):117–133, 1995.\n[22] K. Ma and P.A. Forsyth. An unconditionally monotone numerical scheme for the two-factor uncertain\nvolatility model. IMA Journal of Numerical Analysis , 37(2):905–944, 2017.\n[23] A.M. Oberman. Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–\nJacobi Equations and free boundary problems. SIAM Journal Numerical Analysis , 44(2):879–895, 2006.\n[24] H. Pham. On some recent aspects of stochastic control and their applications. Probability Surveys , 2:506–\n549, 2005.\n[25] David M Pooley, Peter A Forsyth, and Ken R Vetzal. Two factor option pricing with uncertain volatility.\nInInternational Conference on Computational Science and Its Applications , pages 158–167. Springer, 2003.\n[26] D.M. Pooley, P.A. Forsyth, and K.R. Vetzal. Numerical convergence properties of option pricing PDEs\nwith uncertain volatility. IMA Journal of Numerical Analysis , 23:241–267, 2003.\n[27] C. Reisinger and P.A. Forsyth. Piecewise constant policy approximations to Hamilton-Jacobi-Bellman\nequations. Applied Numerical Mathematics , 103:27–47, 2016.\n[28] Walter Rudin. Principles of mathematical analysis . 1953.\n[29] Marjon J Ruijter and Cornelis W Oosterlee. Two-dimensional Fourier cosine series expansion method for\npricing financial options. SIAM Journal on Scientific Computing , 34(5):B642–B671, 2012.\n[30] Adam T Smith. American options under uncertain volatility. Applied Mathematical Finance , 9(2):123–141,\n2002.\n[31] Jerome Spanier, Keith B Oldham, and Robert H Romer. An atlas of functions, 1988.\n[32] R. Stulz. Options on the minimum or the maximum of two risky assets: Analysis and applications. Journal\nof Financial Economics , 10(2):161–185, 1982.\n[33] A.Dembo W. Bryc and T. Jiang. Spectral measure of large random Hankel, Markov and Toeplitz matrices.\n2006.\n[34] J. Wang and P.A. Forsyth. Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in\nfinance. SIAM Journal on Numerical Analysis , 46:1580–1601, 2008.\n[35] Xavier Warin. Some non-monotone schemes for time dependent Hamilton-Jacobi-Bellman equations in\nstochastic control. Journal of Scientific Computing , 66(3):1122–1147, 2016.\n[36] R. Willink. Bounds on the bivariate normal distribution function. Communications in Statistics-Theory\nand Methods , 33(10):2281–2297, 2005.\n[37] H. Zhang and D.M. Dang. A monotone numerical integration method for mean-variance portfolio opti-\nmization under jump-diffusion models. Mathematics and Computers in Simulation , 219:112–140, 2024.\n28Appendices\nA Special case ρ=±1\nA.1 Approximation of δ(·)\nIn this appendix, we detail key elements of the proposed scheme for the case ρ=±1 as highlighted in Remark 3.1,\nand provide selected numerical results. The key challeng is that, for computational purposes, the Dirac delta\nfunction δ(y−(a+ρbx)),a=µy−ρbµxwith b=σy\nσx, needs to be approximated. We focus on the case ρ= 1.\nThe analysis for ρ=−1 follows similarly and is omitted. Using a conditional density approach, δ(y−(a+ρbx))\nis approximated by a Gaussian (a conditional density) when the correlation coefficient is ˆ ρwith ˆ ρ↗1 [11]:\nδ(y−(a+ρbx)) = lim\nˆρ↗1δˆρ(y−(a+ρbx)),where δˆρ(γ) =exp\u0010\n−γ2\n2κ2y(1−ˆρ2)\u0011\n√\n2πκyp\n1−ˆρ2,andκy=σy√\n∆τ. (A.1)\nFor a fixed α, recall the exact Green’s function gα(x, y; ∆τ) defined in (3.13). We define by ˆ gα(x, y; ∆τ) an\napproximation to gα(x, y; ∆τ) obtained by replacing δ(y−(a+ρbx)) by δˆρ(y−(a+ρbx)). Formally,\nˆgα(x, y; ∆τ) =e−r∆τ1√\n2πκxexp\u0012\n−(x−µx)2\n2κ2x\u0013\nδˆρ(y−(a+ρbx)), (A.2)\nwhere δˆρ(·) is defined in (A.1), and a=µy−ρbµxandb=σy\nσx. The function ˆ gα(x, y; ∆τ) is the weight function\nfor our scheme. In this case, our scheme is monotone, and it is straightforward to show that it is ℓ∞-stable.\nThe selection of ˆ ρis crucial for the scheme’s consistency. Below, we show that choosing ˆ ρappropriately can\nachieve first-order consistency for the scheme.\nA.2 Consistency\nFor the rest of the proof, we let Cbe generic bounded constant independent of the discretization parameter\nh, which may take different values from line to line. We re-examine the proof of Lemma 4.2, now utilising\nˆgα(x, y; ∆τ) from (A.2) instead of gα(x, y; ∆τ). For a smooth test function ϕ, and recalling the smooth function\nχ∈L1(R2) with bounded derivatives up to second-order in R2, a mollified version of ϕ, we have\n∆x∆yd∈N†X∗\nl∈N†φl,dˆgα\nn−l,j−dϕm\nl,d=ZZ\nR2ˆgα(xn−x, yj−y; ∆τ)χ(x, y)dx dy +O(h2) +O\u0000\nhe−1/h\u0001\n=ZZ\nR2gα(xn−x, yj−y; ∆τ)χ(x, y)dx dy\n+ZZ\nR2(ˆgα(xn−x, yj−y; ∆τ)−gα(xn−x, yj−y; ∆τ))χ(x, y)dx dy +O(h2). (A.3)\nWe now focus on the error term (the second term) in (A.3), expressed through substitutions as\nZ\nRe−r∆τ\n√\n2πκxexp\u0012\n−(x−µx)2\n2κ2x\u0013\u0012Z\nR(δˆρ(y−(a+ρbx))−δ(y−(a+ρbx)))χ(xn−x, yj−y)dy\u0013\ndx. (A.4)\nRegarding the inner integral, we haveR\nR(δˆρ(y−(a+ρbx))−δ(y−(a+ρbx)))χ(xn−x, yj−y)dy=. . .\n. . .=Z\nRδˆρ(y−(a+ρbx))χ(xn−x, yj−y)dy−χ(xn−x, yj−(a+ρbx)). (A.5)\nHere, the second term in (A.5) is due to from the sifting property of the Delta function. Letting γ=yj−(a+ρbx)\nand applying a change of variables, the integral in (A.5) is reformulated as\nZ\nRδˆρ\u0000\ny−γ\u0001\nχ(xn−x, y)dy. (A.6)\nBy Taylor’s series expansion, we have\nχ(·, y) =χ(·, γ+ (y−γ)) =χ(·, γ) + (y−γ)∂χ\n∂y(·, γ) +(y−γ)2\n2∂2χ\n∂y2(·, γ) +o((y−γ)2).\n29SoZ\nRδˆρ\u0000\ny−γ\u0001\nχ(xn−x, y)dy=. . .\n. . .=Z\nRδˆρ\u0000\ny−γ\u0001\u0012\nχ(xn−x, γ) + (y−γ)∂χ\n∂y(xn−x, γ) +(y−γ)2\n2∂2χ\n∂y2(xn−x, γ) +o((y−γ)2)\u0013\ndy.(A.7)\nTerms in (A.7) are further simplified as follows\nZ\nRδˆρ\u0000\ny−γ\u0001\nχ(·, γ)dy=χ(·, γ)Z\nRδˆρ\u0000\ny−γ\u0001\ndy=χ(·, γ),\nZ\nRδˆρ\u0000\ny−γ\u0001\n(y−γ)∂χ\n∂y(·, γ)dy=∂χ\n∂y(·, γ)Z\nRδˆρ\u0000\ny−γ\u0001\n(y−γ)dy= 0,\nZ\nRδˆρ\u0000\ny−γ\u0001(y−γ)2\n2∂2χ\n∂y2(·, γ)dy=∂2χ\n∂y2(x, γ)Z\nRδˆρ\u0000\ny−γ\u0001(y−γ)2\n2dy=κ2\ny(1−ˆρ2)\n4∂2χ\n∂y2(·, γ).(A.8)\nNext, we substitute (A.8) into (A.6) which is the first term in (A.5), noting that the term χ(·, γ) in (A.8) is\nindeed χ(xn−x, yj−(a+ρbx)) and it cancels with the term χ(xn−x, yj−(a+ρbx)) in (A.5). Therefore,\ndue to boundedness of the derivatives of χ(·), the error term (A.4) becomes Cκ2\ny(1−ˆρ2). Therefore, noting\nκy=σy√\n∆τ, (A.3) becomes\n∆x∆yd∈N†X∗\nl∈N†φl,dˆgα\nn−l,j−dϕm\nl,d=ZZ\nR2gα(xn−x, yj−y; ∆τ)χ(x, y)dx dy +C∆τ(1−ˆρ2) +O(h2).(A.9)\nNow we re-examine Lemma 4.4 with (A.9) in mind. Here, since we need to achieve C∆τ(1−ˆρ2)/h→0 ash→0,\nˆρneeds to be such that (1 −ˆρ2)→0 ash→0. A possible choice is ˆ ρ=√\n1−Ch, which gives (1 −ˆρ2) =O(h),\nand we obtain the same overall O(h) error as in Lemma 4.4 for scenarios |ρ|<1.\nA.3 Select numerical experiments\nLevel Price Error Ratio\n0 8.41173784 3.67e-03\n1 8.41450094 9.07e-04 4.00\n2 8.41519144 2.16e-04 4.19\n3 8.41536400 5.15e-05 4.20\n4 8.41540714 1.17e-05 4.40\nRef. [32] 8.41540757\nMC: 95%-CI [8.3991, 8.4314]\nTable A.1: Convergence study for a European\ncall option on the maximum of two risky as-\nsets under two-factor uncertain volatility model\n(worst case) with ρ∈[−1,1]- payoff function\nin(5.1). The closed form solution is obtained\nusing [32] with fixed parameters σ∗\nx= 0.5,\nσ∗\ny= 0.5andρ∗=−1.Level Price Error Ratio\n0 4.20586189 1.84e-03\n1 4.20724358 4.60e-04 4.00\n2 4.20758890 1.15e-04 4.00\n3 4.20767522 2.86e-05 4.02\n4 4.20769680 7.02e-06 4.07\nRef. [32] 4.20770382\nMC: 95%-CI [4.2026, 4.2208]\nTable A.2: Convergence study for a European\ncall option on the maximum of two risky as-\nsets under two-factor uncertain volatility model\n(best case) with ρ∈[−1,1]- payoff function\nin(5.1). The closed form solution is obtained\nusing [32] with fixed parameters σ∗\nx= 0.5,\nσ∗\ny= 0.5andρ∗= 1.\nIn the above, we show that a possible choice for ˆ ρis√\n1−Ch, where Cis a bounded constant independently\nofh. We now present a heuristic method to determine C. In the numerical experiments for these special cases,\nwe choose ∆ y= 6κyp\n1−ˆρ2, resulting in ˆ ρ=q\n1−∆y2\n(6κy)2. To avoid the necessity for interpolation, for a given\nα∈ A h, we adjust the partition for y-direction so that each of pair ( xn, a+ρbxn), where n∈N, aligns with\nthe grid points. In Tables A.1 and A.2, we present the worst-case and best-case prices for the short position in\nthe case of European rainbow options with payoff function in (5.1) with ρ∈[−1,1]. Other parameters given\nin Table 5.1, and the mesh size and timestep refinement levels are in Table 5.2. The closed-form solution is\nobtained using [32] with fixed parameters σx= 0.5,σy= 0.5 and ρ={−1,1}. It is evident that the numerical\nprices show excellent agreement with the closed-form solutions, and also exhibit approximately second-order of\nconvergence, which aligns with our explanations in Section 5.2.1.\n30B Details of padding matrices\nIn this appendix, we provide details of the padding matrices ˜gq,α\n−1,0,˜gq,α\n1,0,˜gq,α\n−1,1,˜gq,α\n0,1and˜gq,α\n1,1for the circulant\nmatrix ˜gα\nqdefined in (3.32). These padding matrices are defined as follows\n˜gq,α\n−1,0=\ngα\n−N/2+1,qgα\n−N/2,q. . . gα\n−3N/2+1,qgα\n3N/2−1,qgα\n3N/2−2,q. . . gα\nN/2,q\ngα\n−N/2+2,qgα\n−N/2+1,q. . . gα\n−3N/2+2,qgα\n−3N/2+1,qgα\n3N/2−1,q. . . gα\nN/2+1,q\n..................\ngα\nN/2,qgα\nN/2−1,q. . . gα\n−N/2,qgα\n−N/2−1,qgα\n−N/2−2,q. . . gα\n3N/2−1,q\n\nN×(2N+1),\n˜gq,α\n1,0=\ngα\n−3N/2+1,qgα\n3N/2−1,qgα\n3N/2−2,q. . . gα\nN/2+1,qgα\nN/2,q. . . gα\n−N/2,q\ngα\n−3N/2+2,qgα\n−3N/2+1,qgα\n3N/2−1,q. . . gα\nN/2+2,qgα\nN/2+1,q. . . gα\n−N/2+1,q\n..................\ngα\n−N/2,qgα\n−N/2−1,qgα\n−N/2−2,q. . . gα\n−3N/2+1,qgα\n3N/2−1,q. . . gα\nN/2−1,q\n\nN×(2N+1),\n˜gq,α\n−1,1=\ngα\nN/2−1,qgα\nN/2−2,q. . . gα\n−N/2+2,q\ngα\nN/2,qgα\nN/2−1,q. . . gα\n−N/2+3,q\n.........\ngα\n3N/2−2,qgα\n3N/2−3,q. . . gα\nN/2+1,q\n\nN×(N−2),\n˜gq,α\n0,1=\ngα\n3N/2−1,qgα\n3N/2−2,q. . . gα\nN/2+3,qgα\nN/2+2,q\ngα\n−3N/2+1,qgα\n3N/2−1,q. . . gα\nN/2+4,qgα\nN/2+3,q\n............\ngα\n−N/2−2,qgα\n−N/2−3,q. . . gα\n−3N/2+2,qgα\n−3N/2+1,q\n\n(N−1)×(N−2),\n˜gq,α\n1,1=\ngα\n−N/2−1,qgα\n−N/2−2,q. . . gα\n−3N/2+2,q\ngα\n−N/2,qgα\n−N/2−1,q. . . gα\n−3N/2+3,q\n.........\ngα\nN/2−2,qgα\nN/2−1,q. . . gα\n−N/2+1,q\n\nN×(N−2).\nNext, we provide details of padding matrices ˜gα\n−1,0,˜gα\n1,0,˜gα\n−1,1,˜gα\n0,1and˜gα\n1,1for the 2D circulant matrix ˜gα\ndefined in (3.33). These matrices are defined as follows\n˜gα\n−1,0=\n˜gα\n−J/2+1˜gα\n−J/2. . . ˜gα\n−3J/2+1˜gα\n3J/2−1˜gα\n3J/2−2. . . ˜gα\nJ/2\n˜gα\n−J/2+2˜gα\n−J/2+1. . . ˜gα\n−3J/2+2˜gα\n−3J/2+1˜gα\n3J/2−1. . . ˜gα\nJ/2+1\n..................\n˜gα\nJ/2˜gα\nJ/2−1. . . ˜gα\n−J/2˜gα\n−J/2−1˜gα\n−J/2−2. . . ˜gα\n3J/2−1\n\n(3N−1)J×(3N−1)(2J+1),\n˜gα\n1,0=\n˜gα\n−3J/2+1˜gα\n3J/2−1˜gα\n3J/2−2. . . ˜gα\nJ/2+1˜gα\nJ/2. . . ˜gα\n−J/2\n˜gα\n−3J/2+2˜gα\n−3J/2+1˜gα\n3J/2−1. . . ˜gα\nJ/2+2˜gα\nJ/2+1. . . ˜gα\n−J/2+1\n..................\n˜gα\n−J/2˜gα\n−J/2−1˜gα\n−J/2−2. . . ˜gα\n−3J/2+1˜gα\n3J/2−1. . . ˜gα\nJ/2−1\n\n(3N−1)J×(3N−1)(2J+1),\n˜gα\n−1,1=\n˜gα\nJ/2−1˜gα\nJ/2−2. . . ˜gα\n−J/2+2\n˜gα\nJ/2˜gα\nJ/2−1. . . ˜gα\n−J/2+3\n.........\n˜gα\n3J/2−2˜gα\n3J/2−3. . . ˜gα\nJ/2+1\n\n(3N−1)J×(3N−1)(J−2),\n31˜gα\n0,1=\n˜gα\n3J/2−1˜gα\n3J/2−2. . . ˜gα\nJ/2+3˜gα\nJ/2+2\n˜gα\n−3J/2+1˜gα\n3J/2−1. . . ˜gα\nJ/2+4˜gα\nJ/2+3\n............\n˜gα\n−J/2−2˜gα\n−J/2−3. . . ˜gα\n−3J/2+2˜gα\n−3J/2+1\n\n(3N−1)(J−1)×(3N−1)(J−2),\n˜gα\n1,1=\n˜gα\n−J/2−1˜gα\n−J/2−2. . . ˜gα\n−3J/2+2\n˜gα\n−J/2˜gα\n−J/2−1. . . ˜gα\n−3J/2+3\n.........\n˜gα\nJ/2−2˜gα\nJ/2−1. . . ˜gα\n−J/2+1\n\n(3N−1)J×(3N−1)(J−1).\n32"    },    {        "title": "2402.06857v1.Multigrid_solvers_for_multipoint_flux_approximations_of_the_Darcy_problem_on_rough_quadrilateral_grids.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nMultigrid solvers for multipoint flux approximations\nof the Darcy problem on rough quadrilateral grids\nAndr´ es Arrar´ as ·Francisco J. Gaspar ·\nLaura Portero ·Carmen Rodrigo\nReceived: date / Accepted: date\nAbstract In this work, an efficient blackbox-type multigrid method is proposed\nfor solving multipoint flux approximations of the Darcy problem on logically rect-\nangular grids. The approach is based on a cell-centered multigrid algorithm, which\ncombines a piecewise constant interpolation and the restriction operator by Wes-\nseling/Khalil with a line-wise relaxation procedure. A local Fourier analysis is\nperformed for the case of a Cartesian uniform grid. The method shows a robust\nconvergence for different full tensor coefficient problems and several rough quadri-\nlateral grids.\nKeywords Darcy problem ·Local Fourier analysis ·Multigrid ·Multipoint flux\napproximation ·Rough grids\nA. Arrar´ as\nInaMat2, Departamento de Estad´ ıstica, Inform´ atica y Matem´ aticas, Universidad P´ ublica de\nNavarra, Edificio de Las Encinas, Campus de Arrosad´ ıa, 31006 Pamplona, Spain\nE-mail: andres.arraras@unavarra.es\nF.J. Gaspar\nIUMA, Departamento de Matem´ atica Aplicada, Universidad de Zaragoza, Pedro Cerbuna 12,\n50009 Zaragoza, Spain\nE-mail: fjgaspar@unizar.es\nL. Portero\nInaMat2, Departamento de Estad´ ıstica, Inform´ atica y Matem´ aticas, Universidad P´ ublica de\nNavarra, Edificio de Las Encinas, Campus de Arrosad´ ıa, 31006 Pamplona, Spain\nE-mail: laura.portero@unavarra.es\nC. Rodrigo\nIUMA, Departamento de Matem´ atica Aplicada, Universidad de Zaragoza, Pedro Cerbuna 12,\n50009 Zaragoza, Spain\nE-mail: carmenr@unizar.esarXiv:2402.06857v1  [math.NA]  10 Feb 20242 Andr´ es Arrar´ as et al.\n1 Introduction\nSingle-phase flow in porous media is governed, under saturation conditions, by the\nfirst-order system\nu=−K∇p, inΩ, (1a)\n∇ ·u=f, inΩ, (1b)\np=g, onΓD, (1c)\nu·n= 0, onΓN. (1d)\nEquations (1a) and (1b) represent Darcy’s law and the continuity equation, re-\nspectively, and conform the so-called mixed formulation of the Darcy problem.\nHere, Ωis assumed to be a polygonal domain, whose boundary ∂Ωis divided into\ntwo non-overlapping subdomains, ΓDandΓN, in which Dirichlet and Neumann\nboundary conditions are imposed, respectively. Formally, ∂Ω=ΓD∪ΓNsuch\nthat ΓD∩ΓN=∅. The unknowns p=p(x) and u=u(x) stand for the fluid\npressure and the Darcy velocity, respectively, and K=K(x) is a 2 ×2 symmetric\npositive definite tensor that represents the rock permeability divided by the fluid\nviscosity. This tensor is further assumed to satisfy\nk1ξTξ≤ξTK(x)ξ≤k2ξTξ, x∈Ω, ξ̸=0∈R2,\nwhere 0 < k1≤k2<∞. The problem is completed by providing data functions\nf=f(x) and g=g(x). Note that ndenotes the outward unit normal on ∂Ω.\nFor the sake of simplicity, the gravitational term usually involved in Darcy’s law\nhas been neglected here. This simplification is acceptable since, in order to study\nthe performance of the multigrid solver on the discretized problem, we are only\nconcerned with the structure of the matrix and not with that of the right hand\nside.\nLocally conservative discretization methods have been successfully applied to\napproximate the solution to problem (1). In this framework, it is worth mention-\ning several techniques, such as mixed finite element (MFE) methods [22], control\nvolume MFE schemes [24,51], support-operator methods [36,52], or the related\nmimetic finite difference methods [20,35,44]. These methods can all be formulated\non irregular meshes and accurately handle anisotropic discontinuous tensors K.\nMoreover, they are designed to preserve the continuity of normal velocities across\ninterelement edges. Equivalence relationships among them have been established\nin [40]. The common drawback shared by all these methods is the need to solve\nan algebraic system with saddle point structure, which is known to be indefinite.\nSeveral alternatives have been proposed to circumvent this problem, mainly\nbased on a suitable choice of MFE spaces and the introduction of certain quadra-\nture rules to approximate the vector-vector inner products. These strategies permit\nto construct an approximate Schur complement of the original system by locally\neliminating the velocity unknowns, thus yielding a cell-centered finite difference\nscheme for the pressures. Examples of this idea on triangular meshes can be found\nin [13], for the Laplace equation; in [23], for the Poisson equation; or in [48], for\na reaction–diffusion problem involving a diagonal tensor coefficient. Extensions to\nthe Darcy system are proposed in [50,58], for diagonal tensors Kon rectangular\nmeshes; or in [8], for continuous full tensors Kon logically rectangular meshesMultigrid solvers for multipoint flux approximations of the Darcy problem 3\nobtained from smooth mappings of rectangular grids. To conclude, it should also\nbe mentioned that the introduction of interelement Lagrange multipliers giving\nrise to the so-called mixed hybrid finite element methods [10] is also a classical\nalternative for avoiding saddle point problems.1\nIn this work, we will focus on the so-called multipoint flux approximation\n(MPFA) methods, which combine the main advantages of the previous techniques.\nThese methods were originally formulated as a natural extension of the well-known\ntwo-point flux approximation schemes. As such, they are defined to be control\nvolume schemes which consider more than two pressure values to approximate the\nvelocity at each cell edge. They have been formulated on unstructured triangular\nmeshes [3,4,29], and on logically rectangular meshes composed of quadrilateral\nelements [1,2,30]. Following the terminology from [5], these methods can be defined\neither in a reference space or in the physical space. Henceforth, we will describe the\nmain properties of both approaches for quadrilateral elements. Remarkably, the\ntriangular case has been analyzed in [39] using an equivalent MFE formulation,\nand a relevant extension for solving Richards’ equation has been further proposed.\nMPFA schemes in a reference space can be formulated as MFE methods using\nthe trapezoidal quadrature rule, with a particular choice of finite element spaces\nand evaluation points: the method proposed in [41] is based on the so-called broken\nRaviart–Thomas element, RT1/2, with an evaluation of the tensor at the midpoint\nof the reference element; in turn, that proposed in [61] is based on the lowest order\nBrezzi–Douglas–Marini element, BDM 1, with an evaluation of the tensor at the\ncorners of the reference element. Note that, in both cases, the degrees of freedom\nof the corresponding MFE method show a one-to-one correspondence with the\nunknowns of the MPFA scheme. In addition, they both result in a cell-centered\npressure system with a symmetric positive definite matrix. Numerical evidence\nreveals that the second variant is more accurate than the first one when computing\nnormal velocities on grids composed of O(h2)-perturbations of parallelograms [5].\nIn turn, a loss of convergence is observed for both methods on O(h)-perturbed\nmeshes. In this case, the physical space MPFA method permits to recover a first-\norder convergence rate for the normal velocities. The price to pay is a loss of\nsymmetry of the system matrix unless the mesh is composed of parallelograms.\nMPFA methods in the physical space can also be formulated as MFE schemes,\nbased on the RT1/2or the BDM 1element, using a non-symmetric variant of the\ntrapezoidal quadrature rule. In the former case, the quadrature rule involves the\nmean value of the inverse tensor K−1on each element E, denoted by K−1\nE[42]; in\nthe latter, it considers the inverse of the mean value of KonE, that is, K−1\nE[60].\nThe solvability of the resulting discrete problem is only guaranteed under technical\nrestrictions on the element geometry and the anisotropy of K(see, e.g., formulas\n(3.31) and (3.32) in [60]; cf. also [45]). Remarkably, when the mesh is composed\nof parallelograms and Kis piecewise constant on each element, the physical space\nMPFA methods are equivalent to their corresponding reference space counterparts.\nIn the present paper, we deal with the efficient solution of the large systems of\nequations arising from the reference and physical space MPFA methods based on\ntheBDM 1finite element spaces. According to [60,61], we will henceforth rename\nsuch methods the symmetric and non-symmetric multipoint flux mixed finite el-\n1In this case, a multigrid solver can be applied by using the equivalence with the noncon-\nforming Crouzeix–Raviart element (see, e.g., [14,16]).4 Andr´ es Arrar´ as et al.\nement (MFMFE) schemes, respectively. Broadly speaking, the design of solvers\nfor MPFA methods has been seldom considered in the literature. This is some-\nwhat surprising due to the wide use of these schemes in the numerical simulation\nof porous media problems. A two-level domain decomposition algorithm for an\nMPFA finite volume discretization of three-dimensional flow in anisotropic het-\nerogeneous porous media is presented in [31]. In this interesting work, additive\nand multiplicative Schwarz iterative methods are implemented as smoothers, and\nthe coarse scale operator is obtained from numerical upscaling. To the best of our\nknowledge, however, a robust multigrid solver for multipoint flux discretizations\nhas never been proposed so far. This work is intended to close this gap, presenting\nthe first multigrid method for solving multipoint flux approximations, which can\neasily handle difficult diffusion problems on rough grids and with coefficient jumps\nnot aligned with the mesh.\nIt is well known that present multigrid methods are among the most advanced\ntechniques for solving large linear systems arising from the discretization of par-\ntial differential equations (PDEs). There are mainly two different approaches to\nmultigrid: geometric multigrid methods (GMG), for which a hierarchy of grids has\nto be defined and the inter-grid transfer operations are based on geometric prin-\nciples, and algebraic multigrid methods (AMG), which construct the coarse levels\nautomatically from the system matrix on the target grid in an algebraic way. In\nthe context of AMG methods, two prevailing schemes have proved their use for\nmultiple engineering problems, namely: algebraic multigrid and aggregation-based\nmultigrid methods [15,19,53,56,57]. The origin of algebraic methods may be found\nin the early days of multigrid, when blackbox multigrid (BoxMG) with operator-\ndependent transfer operators and Galerkin coarse grid approximation was pro-\nposed for structured vertex-centered Cartesian grids [6,25,26]. This approach can\nbe considered as a predecessor of classical AMG. The aggregation-based multigrid\nmethods, whose origin can be found in [57] (smoothed aggregation), may be related\nto the cell-centered multigrid methods proposed in [38,59]. In [59], it was shown\nthat constant (i.e., operator-independent) transfer operators, in combination with\nGalerkin coarse grid discretization, provided highly efficient multigrid results for\ncell-centered discretizations of elliptic PDEs including jumping coefficients. In ad-\ndition, more robustness can be achieved in some cases by using multigrid as a\npreconditioner of a Krylov subspace method; see, e.g., [12], where a parallel multi-\ngrid preconditioned conjugate gradient method is proposed for solving groundwa-\nter flow problems. Like AMG and BoxMG, the multigrid solver proposed here is\nbased on the blackbox methodology, in which the user only provides the fine-grid\ndiscretization, the right-hand side and an initial guess for the solution, and the\ncode automatically generates the hierarchy of operators on coarser levels.\nLogically rectangular grids are considered to take advantage of the recent\ntrends in computer architectures: many-core and accelerated architectures that\nachieve their best performance when structured data can be used. On this type\nof architectures, the drawbacks of using unstructured data access and indirect ad-\ndressing may be so significant that it becomes more efficient to consider larger,\nlogically structured grids with regular data access. In particular, implementations\nof logically structured multigrid algorithms, such as BoxMG, have been shown\nto be 10 times faster than AMG for three-dimensional heterogeneous diffusion\nproblems on structured grids [46].Multigrid solvers for multipoint flux approximations of the Darcy problem 5\nIn the present work, a local Fourier analysis (LFA) [17,18] is performed for the\ncase of Cartesian rectangular grids. This analysis is known to predict very accu-\nrately the convergence rates of GMG methods by considering the local character\nof the involved operators and neglecting the effect of boundary conditions. Specif-\nically, the Toeplitz or multilevel Toeplitz structure of the matrix on an infinite\ngrid permits its diagonalization by the matrix of Fourier modes. A detailed intro-\nduction to this analysis can be found in [55,62]. The robustness of the proposed\nmultigrid solver is shown for different test problems with full tensor coefficients on\nvarious rough logically rectangular grids.\nThe remaining of the paper is structured as follows. In Section 2, we describe\nthe considered MFMFE schemes for the Darcy system. Subsection 2.1 is devoted\nto deriving the stencil coefficients for homogeneous media and Cartesian uniform\ngrids, which will be subsequently used in the LFA. In Section 3, the cell-centered\nblackbox-type multigrid method is explained. This section also includes a descrip-\ntion of the LFA, together with some predicting results of the convergence rates of\nthe multigrid method. Section 4 shows the robustness of the proposed multigrid\nsolver for problems with different permeability tensors on a variety of logically\nrectangular grids. Finally, Section 5 provides some concluding remarks together\nwith certain ideas for future research.\n2 Multipoint flux approximations of the Darcy system\nThe mixed variational formulation of the first-order system (1) reads: Find (u, p)∈\nV×Wsuch that\n(K−1u,v) = (p,∇ ·v)− ⟨g,v·n⟩ΓD, v∈V, (2a)\n(∇ ·u, w) = (f, w), w ∈W, (2b)\nwhere ( ·,·) denotes the inner product in either L2(Ω) or ( L2(Ω))2, and ⟨·,·⟩ΓD\nrepresents the L2(ΓD)-inner product or duality pairing. Moreover, W=L2(Ω)\nandV={v∈H(div;Ω) :v·n= 0 on ΓN}, with\nH(div;G) ={v∈(L2(G))2:∇ ·v∈L2(G)},\nfor any G⊂R2. It is well known that (2) has a unique solution [22]. For the sake of\nsimplicity, we will henceforth consider homogeneous Dirichlet boundary conditions\nalong ΓD, i.e., g≡0.\nNext, we introduce an MFMFE discretization for (2) based on the BDM 1\nspaces on quadrilateral elements [21,22]. As we will see below, the method fur-\nther considers suitable quadrature rules which permit to eliminate the velocity\nunknowns, thus yielding a cell-centered scheme for the pressure.\nLetThbe a logically rectangular partition of Ωinto convex quadrilaterals,\nwhere h= max E∈Thdiam( E). We denote by ˆEthe unit square with vertices\nˆr1= (0,0)T,ˆr2= (1,0)T,ˆr3= (1,1)Tandˆr4= (0,1)T, and introduce a family of\nbijective bilinear mappings {FE}E∈Thsuch that FE(ˆE) =E. In particular, given\na physical element E∈ Thwith vertices ri= (xi, yi)T, for i= 1,2,3,4,FEis\ndefined as\nFE(ˆr) =r1+r21ˆx+r41ˆy+ (r34−r21)ˆxˆy,6 Andr´ es Arrar´ as et al.\n×\nˆE\nˆr1 ˆr2ˆr3 ˆr4\nFE\n×\nE r1\nr2r3r4\nFig. 1 Degrees of freedom for the BDM 1spaces on quadrilaterals; crosses and circles denote\npressure and velocity degrees of freedom, respectively. For the sake of clarity, the velocity\ndegrees of freedom are drawn at a distance from the corresponding vertex.\nwhere rij=ri−rj. We further define, for each mapping FE, the Jacobian matrix\nDFEand its determinant JE=|det(DFE)|. The outward unit vectors normal to\nthe edges ˆ e⊂∂ˆEande⊂∂Eare denoted by ˆnˆeandne, respectively.\nFor later use, a given partition Thwill be called an O(h2)-perturbed mesh\nif it is composed of h2-parallelograms, that is, quadrilaterals whose vertices sat-\nisfy the condition |r34−r21|R2≤Ch2. This kind of elements can be obtained\nasymptotically by uniform refinements of a general quadrilateral grid.\nLet ˆV(ˆE) and ˆW(ˆE) be the BDM 1finite element spaces on the reference\nelement ˆE, i.e.,\nˆV(ˆE) =\u0014α1+β1ˆx+γ1ˆy+rˆx2+ 2sˆxˆy\nα2+β2ˆx+γ2ˆy−2rˆxˆy−sˆy2\u0015\n, ˆW(ˆE) =P0(ˆE),\nwhere α1,α2,β1,β2,γ1,γ2,randsare real constants. Note that ˆ∇·ˆV(ˆE) =ˆW(ˆE)\nand, on any edge ˆ e⊂∂ˆE,ˆv∈ˆV(ˆE) is such that ˆv·ˆnˆe∈P1(ˆe). The degrees of\nfreedom for ˆvare chosen to be the values of ˆv·ˆnˆeat the vertices of each edge ˆ e\n(see Figure 1).\nAny approximation to H(div;E) must preserve the continuity of the normal\ncomponents of the velocity vectors across the element edges. In order to satisfy\nthis requirement, the space V(E) associated to an element E∈ Thwill be defined\nby using the Piola transformation [22], i.e.,\nv↔ˆv:v= (J−1\nEDFEˆv)◦F−1\nE.\nOn the other hand, the transformation\nw↔ˆw:w= ˆw◦F−1\nE\nwill be used for the definition of the local space W(E). Finally, the global MFE\nspaces Vh×Wh⊂V×WonThare given by\nVh={v∈V:v|E↔ˆv,ˆv∈ˆV(ˆE)∀E∈ Th},\nWh={w∈W:w|E↔ˆw,ˆw∈ˆW(ˆE)∀E∈ Th}.\nIn the approximation of (2a) by the MFE method described above, it is neces-\nsary to compute integrals of the form ( K−1q,v), for q,v∈Vh. In this setting, theMultigrid solvers for multipoint flux approximations of the Darcy problem 7\nMFMFE method is derived by considering suitable quadrature rules that allow for\nlocal velocity elimination. In particular, for any q,v∈Vh, the global quadrature\nrule is defined element-wise as\n(K−1q,v)Q=X\nE∈Th(K−1q,v)Q,E. (3)\nThe definition of the local quadrature rule depends on the type of spatial meshes\nunder consideration. For O(h2)-perturbed meshes, the numerical integration on\neach element is defined by mapping to the reference element [61], i.e.,\n(K−1q,v)Q,E= (K−1\nEˆq,ˆv)ˆQ,ˆE=1\n44X\ni=1K−1\nE(ˆri)ˆq(ˆri)·ˆv(ˆri), (4)\nwhere\nK−1\nE(ˆx) =1\nJE(ˆx)DFT\nE(ˆx)K−1(FE(ˆx))DFE(ˆx).\nAs for the case of highly distorted rough grids, the quadrature rule on each element\nis defined accordingly [60], that is,\n(K−1q,v)Q,E= (eK−1\nEˆq,ˆv)ˆQ,ˆE=1\n44X\ni=1eK−1\nE(ˆri)ˆq(ˆri)·ˆv(ˆri), (5)\nwhere\neK−1\nE(ˆx) =1\nJE(ˆx)DFT\nE(ˆxc)K−1\nEDFE(ˆx),\nwith KEbeing a constant matrix such that Kij,Eis the mean value of KijonE,\nwhere Kij,EandKijare the elements on the ith row and jth column of matrices\nKEandK, respectively. Furthermore, ˆxcdenotes the center of mass of ˆE. Note\nthat, if Kis an element by element piecewise constant tensor and the spatial mesh\nconsists of parallelograms, the expression (5) reduces to (4).\nThe MFMFE approximation to (2), considering homogeneous Dirichlet bound-\nary conditions, is given by: Find (uh, ph)∈Vh×Whsuch that\n(K−1uh,v)Q= (ph,∇ ·v), v∈Vh, (6a)\n(∇ ·uh, w) = (f, w), w ∈Wh. (6b)\nIf (K−1·,·)Qin (6a) is given by (3) and (4), the MFMFE method is called sym-\nmetric; if it is given by (3) and (5), it is referred to as non-symmetric. The well-\nposedness and convergence properties of both methods stemming from (6) are\nstudied in [61] and [60], respectively. In the symmetric case, if the mesh is composed\nofh2-parallelograms, the velocity and pressure variables are first-order convergent\nin the L2-norm, the latter being second-order superconvergent at the cell centers\nin a discrete L2-norm. In the non-symmetric case, the velocity variable is shown\nto be first-order convergent, either when compared to the projection of the true\nsolution onto the space Vhin the L2-norm, or when considering the normal compo-\nnent in an edge-based norm; in turn, the pressure preserves the first-order optimal\nconvergence in the L2-norm. Note that the non-symmetric MFMFE scheme need\nnot be applied on an O(h2)-perturbed mesh. The convergence properties of both8 Andr´ es Arrar´ as et al.\nmethods will be illustrated by numerical experiments in the last section of the\npaper.\nIn the sequel, we describe how the MFMFE formulation (6) can be reduced\nto a cell-centered finite difference scheme in the pressure variable. More precisely,\nwe use (6a) to express the velocity degrees of freedom in terms of the pressure\nunknowns, and substitute them back into (6b) in order to obtain a linear system\nfor the pressures.\nIn this framework, let us introduce the vector space Hvof discrete velocity\nfunctions Uh∈R2Nℓ, where Nℓis the number of edges in Th. The degrees of\nfreedom for this space are located at the vertices of each edge. If Nvdenotes the\nnumber of vertices in Th, any Uh∈ Hvis given by Uh= (Uh,1, Uh,2, . . . , U h,Nv)T,\nwhere Uh,i∈Rℓiandℓiis the number of edges that share the ith vertex point,\nfori= 1,2, . . . , N v. The component of Uh,iassociated to the jth edge ejis given\nby the volumetric flux ( uh·nej)(ri)|ej|, where |ej|denotes the length of ej, for\nj= 1,2, . . . , ℓ i. On the other hand, we consider the vector space Hpof discrete\npressure functions Ph∈RNe, where Nestands for the number of elements in Th.\nIn this case, the degrees of freedom are located at the cell centers. Any Ph∈ Hp\nhas the form Ph= (Ph,1, Ph,2, . . . , P h,Ne)T, where Ph,i=ph(xc,i) and xc,iis the\ncoordinate vector of the ith cell center, for i= 1,2, . . . , N e.\nLet{vi}L\ni=1and{wj}Ne\nj=1be the bases of the discrete spaces VhandWh,\nrespectively. The linear system stemming from (6a)-(6b) is\n\u0014A B\nBT0\u0015\u0014Uh\nPh\u0015\n=\u00140\nFh\u0015\n, (7)\nwhere the matrices A∈R2Nℓ×2NℓandB∈R2Nℓ×Neare given by ( A)ij=\n(K−1vj,vi)Qand ( B)ij=−(wj,∇ ·vi), respectively. Finally, Fh∈RNecon-\ntains the contributions of the right-hand side fand is defined element-wise as\n(Fh)E=−Z\nEf(x)dx.\nThe use of the trapezoidal quadrature rule ( K−1·,·)Qpermits to decouple\nthe velocity degrees of freedom associated to a vertex from the rest of them. In\nconsequence, the velocity mass matrix A= diag( A1, A2, . . . , A Nv) has a block-\ndiagonal structure, and each block Ai∈Rℓi×ℓiis a local matrix associated to the\nith vertex point, for i= 1,2, . . . , N v. Since Ais induced by the discrete bilinear\nform ( K−1·,·)Q, the symmetry of the corresponding blocks Aidepends on the\nlocal quadrature rule under consideration, namely: if ( K−1·,·)Q,Eis given by (4),\nAiwill be symmetric; otherwise, if it is given by (5), Aiwill be non-symmetric\nunless the mesh is composed of parallelograms and Kis constant on each element.\nIn particular, let us consider the ith interior vertex represented in Figure 2\n(left). We denote by uj, forj= 1,2,3,4, the velocity degrees of freedom associated\nto this vertex. The corresponding basis functions of the space Vhare denoted\nbyvj, for j= 1,2,3,4. Inserting v=vjinto (6a), we obtain the local system\ncorresponding to the ith interior vertex, i.e.,\nAi\nu1|e1|\nu2|e2|\nu3|e3|\nu4|e4|\n+1\n2\np2−p1\np3−p2\np3−p4\np4−p1\n=\n0\n0\n0\n0\n, (8)Multigrid solvers for multipoint flux approximations of the Darcy problem 9\nu1u2u3u4\ne1e2e3\ne4\n××× ×\np1 p2p3 p4\nE1E2E3E4\n×\npi\nu1\nu2u3u4u5u6u6\nu7\nu8Ei\ne1e2e3\ne4\nFig. 2 Interaction of degrees of freedom in the MFMFE method.\nwhere\nAi=1\n4\nN−1\n11,E1+N−1\n11,E2N−1\n12,E20 N−1\n12,E1\nN−1\n21,E2N−1\n22,E2+N−1\n22,E3N−1\n21,E30\n0 N−1\n12,E3N−1\n11,E3+N−1\n11,E4N−1\n12,E4\nN−1\n21,E10 N−1\n21,E4N−1\n22,E4+N−1\n22,E1\n.\n(9)\nIn this expression, using a generic element notation, N−1\nij,E=K−1\nij,E, if we consider\nthe symmetric quadrature rule (4), and N−1\nij,E =eK−1\nij,E, for its non-symmetric\ncounterpart (5) (see, e.g., [60,61]).\nOn the other hand, let us consider the ith element represented in Figure 2\n(right). We denote by withe basis function of the space Whassociated to this\nelement. If we insert w=wiinto (6b), we obtain the expression\n1\n2((u5+u6)|e3| −(u1+u2)|e1|+ (u3+u4)|e2| −(u7+u8)|e4|) =Z\nEif(x),\n(10)\nwhere we consider the local notations introduced in Figure 2 (right). Therefore,\ntheith column of matrix Bhas eight non-null coefficients corresponding to the\nvelocity degrees of freedom located on the four edges of the element Ei.\nThe upper part of system (7) permits to express the velocity vector Uhin terms\nof the pressure Phin the following way\nUh=−A−1B Ph. (11)\nSince Ais block-diagonal, this operation is locally performed on each block Ai. As\na result, the velocity degrees of freedom associated to the ith vertex are expressed\nin terms of the pressure unknowns located at the centers of the elements sharing\nthat vertex. Inserting (11) into the lower part of (7), we get a cell-centered system\nfor the pressures, i.e.,\nBTA−1B Ph=−Fh, (12)10 Andr´ es Arrar´ as et al.\nwhose matrix is symmetric and positive definite if we consider the symmetric\nquadrature rule (4). The discrete diffusion operator of MFMFE discretizations on\nlogically rectangular grids has a 9-point stencil. In order to perform the LFA for\nthe multigrid method presented in Section 3, we need to derive the expression of\nthe stencil coefficients for a homogeneous medium and a uniform Cartesian grid.\nDetails are provided in the next subsection.\n2.1 The stencil coefficients for homogeneous media and Cartesian uniform grids\nLet us consider a homogeneous medium described by the permeability tensor\nK=\u0014a c\nc b\u0015\n,\nand a uniform Cartesian grid with mesh size h. In this case, DFE=hI, where I\nstands for the 2 ×2 identity matrix, and JE=h2. Hence, for every E∈ Th,\nKE=JEDF−1\nEK(DF−1\nE)T=K.\nThe expression (9) for the local matrix Aiassociated to the ith interior vertex is\nreduced to\nAi=1\n4(ab−c2)\n2b−c0−c\n−c2a−c0\n0−c2b−c\n−c0−c2a\n.\nIf we insert this expression into (8) and consider the local notations of Figure 2\n(left), the volumetric fluxes associated to the ith interior vertex can be expressed\nin terms of pressure differences as\n\nu1\nu2\nu3\nu4\n=−1\nh\na−c2\n2bc\n2c2\n2bc\n2\nc\n2b−c2\n2ac\n2c2\n2a\nc2\n2bc\n2a−c2\n2bc\n2\nc\n2c2\n2ac\n2b−c2\n2a\n\np2−p1\np3−p2\np3−p4\np4−p1\n.\nFinally, we apply the expression (10) to the central element of Figure 3. The\nlocal systems corresponding to the vertices r1,r2,r3andr4permit to express: u1\nandu8in terms of p1, p2, p3andp9;u2andu3in terms of p1, p3, p4andp5;u4\nandu5in terms of p1, p5, p6andp7; and u6andu7in terms of p1, p7, p8andp9.\nAs a result, the 9-point stencil equation of system (12) associated to the central\npressure unknown p1is given by\n9X\ni=1mipi=Z\nE1f(x),Multigrid solvers for multipoint flux approximations of the Darcy problem 11\n× × ×× × ×× × ×\np2 p3 p4p9p1p5p8 p7 p6\nr1 r2r4 r3\nu1u2u3u4u5 u6\nu7\nu8\nFig. 3 Local notations for the 9-point stencil.\nwhere\nm2=m6=−c\n2\u0010\n1 +c\n2a+c\n2b\u0011\n, m 3=m7=−b+c2\n2\u00121\na+1\nb\u0013\n,\nm4=m8=c\n2\u0010\n1−c\n2a−c\n2b\u0011\n, m 5=m9=−a+c2\n2\u00121\na+1\nb\u0013\n.\nThe corresponding coefficient of p1ism1=−P9\ni=2mi.\n3 Cell-centered multigrid and local Fourier analysis\nIn this section, we propose a blackbox cell-centered multigrid method for solving\nthe MFMFE discretization of Darcy problem on logically rectangular grids. In\naddition, we apply a local Fourier analysis technique to study the convergence of\nthe proposed solver.\n3.1 Cell-centered multigrid\nThe linear system (12) resulting from the MFMFE discretization described above\ndemands efficient solvers for its solution. In this subsection, we design a cell-\ncentered multigrid strategy [38,59] that remains robust when applied to different\nlogically rectangular grids with increasing levels of roughness. Remarkably, the\nproposed multigrid method acts as a blackbox, since the user simply provides\na difference equation on a target grid and the code automatically generates the\nhierarchy of operators on coarser levels. In the sequel, we describe in detail the\noverall procedure.\nLetLℓPℓ=Fℓdenote the linear system (12), with ℓrepresenting the level of\nthe target grid. In virtue of the blackbox multigrid idea, a hierarchy of smaller\nlinear systems\nLkPk=Fk, k = 1,2, . . . , ℓ −1,\nis built, where the corresponding linear operators on the coarser levels are con-\nstructed as\nLk=Ik\nk+1Lk+1Ik+1\nk, (13)12 Andr´ es Arrar´ as et al.\nIk\nk+1andIk+1\nkbeing appropriate restriction and prolongation operators, respec-\ntively. An iteration of a two-level cycle applies ν1iterations of a relaxation proce-\ndure, followed by a coarse-level correction technique, and ν2relaxation steps. In\nthe coarse-level correction process, we first restrict the residual to the coarse level,\nthen solve the coarse level problem exactly, and finally interpolate the obtained\nsolution to the fine level in order to correct the previous approximation. This idea\ncan be generalized to a multilevel approach by recursively applying the two-level\nalgorithm.\nThe components of the proposed multigrid solver are explained next. The hi-\nerarchy of coarse level operators is constructed by Galerkin approximation as for-\nmulated in (13). As inter-grid transfer operators, we use a piecewise constant\ninterpolation and the restriction operator by Wesseling/Khalil [38], that is,\nIk+1\nk=\n1 1\n⋆\n1 1\nk+1\nk, Ik\nk+1=1\n16\n1 1 0 0\n1 3 2 0\n⋆\n0 2 3 1\n0 0 1 1\nk\nk+1, (14)\nwhere ⋆denotes the position of a coarse level unknown. In the classical stencil\nnotation for the prolongation, the numbers within the stencil represent the con-\ntribution of the coarse level unknown to the neighbouring fine level unknowns. In\nthe stencil for the restriction, the numbers denote the weights corresponding to\nthe fine level unknowns used to construct the coarse level value.\nRegarding the relaxation procedure, a standard alternating line Gauss–Seidel\nis considered. One step of an elementary line relaxation updates all unknowns\nat the same grid line simultaneously. In the case of structured rectangular grids,\nthis implies considering the horizontal and vertical lines in the structured grid\ncorresponding to the x- and y-line relaxation, respectively. However, in the case\nof logically rectangular grids, this has to be defined more precisely. Since the\nindex set {(i, j), i= 1,2, . . . , n, j = 1,2, . . . , m }of a logically rectangular grid\nhas the same structure as that of a rectangular grid, we can define the “ x-line”\nsmoother as the relaxation that simultaneously updates all the unknowns located\nat grid points with a fixed i-index. In turn, the “ y-line” smoother updates together\nthose unknowns located at grid points with a fixed j-index. Finally, the alternating\nversion of the line relaxation step consists of an x-line relaxation iteration followed\nby a y-line relaxation step.\nRemark 1 Since our approach applies geometric multigrid on logically rectangular\ngrids, an easy parallelization of the algorithm can be obtained by grid partitioning\n[55]. The domain can be split into several blocks and each block is then sent to a\nprocessor of a parallel computer.\n3.2 Local Fourier analysis\nIn order to apply the LFA [17,18], we restrict ourselves to studying the Darcy\nsystem in a homogenous medium, discretized by an MFMFE scheme on a Cartesian\nuniform grid, whose mesh size is hin both directions. In this case, we obtain a\ndiscrete diffusion operator with constant coefficients, which can be representedMultigrid solvers for multipoint flux approximations of the Darcy problem 13\nby a constant stencil that is common to all the interior points. This stencil was\nderived in Section 2. Such an operator is assumed to be defined on an infinite grid\nGh, thus neglecting the effect of the boundary conditions.\nBased on these assumptions, all the operators involved in the multigrid method\nare extended to the infinite grid, and the error grid function can be written as a\nformal linear combination of the so-called Fourier modes, i.e.,\nφh(θ,x) =eıθx/h=eıθ1x1/heıθ2x2/h,where ı=√\n−1,\nwhich span the Fourier space\nF(Gh) ={φh(θ,x)|θ= (θ1, θ2)∈Θh= (−π, π]2,x= (x1, x2)∈Gh}.\nThe main idea of the LFA is to study how the two-grid operator acts on this ex-\npression of the error based on the Fourier components. Since the discrete diffusion\noperator satisfies the LFA assumptions, the Fourier modes are its eigenfunctions.\nMore precisely, we obtain\nLℓφh(θ,x) =eLℓ(θ)φh(θ,x),\nwhere eLℓ(θ) is called the symbol ofLℓ. In our case, recalling the stencil of the\ndiscrete operator obtained at the end of Section 2, we have\neLℓ(θ) =m1+ 2m2cos(θ1+θ2) + 2m3cos(θ2) + 2m4cos(θ1−θ2) + 2m5cos(θ1).\nIt is well known that the same holds true for standard relaxation operators as\nthose considered here, so that we can compute eSℓ(θ) and, consequently, the LFA\nsmoothing factor is given by\nµ= sup\nΘh\\Θlow\nhρ(eSℓ(θ)),\nwhere Θlow\nh= (−π/2, π/2]2is the set of low frequencies associated with standard\ncoarsening. However, the inter-grid transfer operators couple Fourier modes, thus\nleaving invariant the subspaces of 2 h-harmonics,\nF2(θ00) =spann\nφh(θ00,·), φh(θ11,·), φh(θ10,·), φh(θ01,·)o\n,\nwhere θαβ=θ00−(αsign(θ1), βsign(θ2))π, for α, β∈ {0,1}. In this way, the\nFourier symbols of the restriction and prolongation operators are matrices of sizes\n1×4 and 4 ×1, respectively, given by\neIℓ−1\nℓ(θ) =h\neR(θ00)eR(θ11)eR(θ10)eR(θ01)i\n,\neIℓ\nℓ−1(θ) =h\neP(θ00)eP(θ11)eP(θ10)eP(θ01)iT\n,\nwhere\neR(θαβ) = cos( θαβ\n1/2) cos( θαβ\n2/2) cos(( θαβ\n1−θαβ\n2)/2),\neP(θαβ) = cos( θαβ\n1/2) cos( θαβ\n2/2),14 Andr´ es Arrar´ as et al.\nTable 1 LFA-predicted smoothing factors ( µ) and two-grid convergence factors ( ρ2g), together\nwith the experimentally obtained asymptotic convergence factors ( ρh), for three different ten-\nsors.\nGauss–Seidel Alternating line\nµ ρ 2g ρh µ ρ 2g ρh\nK10.50 0.40 0.40 0.15 0.11 0.12\nK20.50 0.42 0.41 0.15 0.19 0.18\nK31.00 1.00 – 0.45 0.22 0.17\nwith α, β∈ {0,1}. Using the preceding symbols and taking into account that the\nsymbol of the Galerkin operator is given by\neLℓ−1(θ) =eIℓ−1\nℓ(θ)eLℓ(θ)eIℓ\nℓ−1(θ),\nwe can construct the Fourier representation of the two-level operator Mℓ−1\nℓas\nfollows\nfMℓ−1\nℓ(θ) =eSν2\nℓ(θ)\u0010\neIℓ(θ)−eIℓ\nℓ−1(θ)eL−1\nℓ−1(θ)eIℓ−1\nℓ(θ)eLℓ(θ)\u0011\neSν1\nℓ(θ),\nwhereeIℓ(θ) denotes the 4 ×4 identity matrix. Finally, we can estimate the spectral\nradius of the two-level operator, which provides a prediction of the asymptotic\nconvergence factor of the method, as\nρ2g= sup\nθ∈Θlow\nhρ\u0010\nfMℓ−1\nℓ(θ)\u0011\n.\nThe LFA described above allows us to estimate the convergence rates of the\nproposed algorithm very accurately. The main results of the analysis are shown in\nTable 1, which contains the smoothing factors, µ, and two-grid convergence factors,\nρ2g, predicted by the LFA. The table further displays the real asymptotic conver-\ngence factors, ρh, obtained by using W-cycles. The test considers one smoothing\nstep of two different smoothers, Gauss–Seidel and alternating line relaxations, and\nthree different tensor samples. In particular, we take K1to be the identity matrix,\nandK2andK3the following full tensors\nK2=\u00144 1\n1 4\u0015\n, K 3=\u00142 1\n1 10000\u0015\n.\nNotice that K2is isotropic, whereas K3is highly anisotropic. We can observe in\nTable 1 that the two-level convergence factors provided by the LFA match the\nexperimentally computed asymptotic convergence factors very accurately. It can\nalso be observed that, in the case of the highly anisotropic tensor, the multigrid\nmethod based on the Gauss–Seidel smoother does not converge, and the alter-\nnating line relaxation is needed to obtain a successful result. These results are in\naccordance with previous findings that state that block-wise smoothers are much\nmore efficient than point-wise smoothers when considering anisotropic problems\n[55]. Based on these observations, we will choose the alternating line smoother to\nobtain a robust blackbox-type multigrid solver for any arbitrary tensor K.Multigrid solvers for multipoint flux approximations of the Darcy problem 15\n4 Numerical experiments\n4.1 A test with a known analytical solution\nLet us consider a boundary value problem of type (1), where Ω= (0,1)2,∂Ω=ΓD\nand tensor Kis given by\nK=\u00145 3\n3 7\u0015\n.\nData functions fandgare defined in such a way that the exact solution of the\nproblem is p(x, y) = sin( πx)2sin(2πy).\nThe spatial domain is discretized using four different logically rectangular\nmeshes that contain N×Nelements. The first one is a family of smooth meshes\ndefined to be a C∞-mapping of successively refined uniform meshes on the unit\nsquare, i.e.,\nxi,j= ˆxi,j+3\n50sin(2πˆxi,j) sin(2 πˆyi,j),\nyi,j= ˆyi,j−1\n20sin(2πˆxi,j) sin(2 πˆyi,j),\nwhere ˆ xi,jand ˆyi,jdenote the spatial coordinates of the vertices on the uni-\nform mesh, and xi,jandyi,jare their counterparts on the smooth mesh, for\ni= 1,2, . . . , N + 1 and j= 1,2, . . . , N + 1. An illustration of this type of meshes\nis given in Figure 4(a). Next, we consider a set of Kershaw-type meshes [37,52],\nwhich contain certain highly skewed zones that are displayed on Figure 4(b). The\nh-perturbed grid represented in Figure 4(c) belongs to a family of meshes described\nin detail in [9]. Finally, we consider a family of randomly h-perturbed meshes con-\nsisting of highly distorted quadrilaterals. As shown in Figure 4(d), each of these\nmeshes is generated by perturbing the vertices of a uniform mesh by a distance of\nsizeO(h) in a random direction. More specifically, the spatial coordinates of the\nvertices of the randomly h-perturbed mesh can be obtained as\nxi,j= ˆxi,j−13\n10h+3\n5hri,j\nx,\nyi,j= ˆyi,j−13\n10h+3\n5hri,j\ny,\nwhere h= 1/N, and ri,j\nxandri,j\nyare pseudo-random numbers uniformly dis-\ntributed on the interval (0 ,1). It is interesting to note that, from the four meshes\nunder consideration, just the smooth and the Kershaw-type grids are composed of\nO(h2)-parallelograms.\nConvergence results for the MFMFE discretizations on different types of meshes\nare extensively reported in the literature (e.g., [5,60,61]). For the sake of illustra-\ntion, we include here some convergence tests for both the symmetric and non-\nsymmetric MFMFE schemes on the Kershaw and h-perturbed grids. As reported\nbelow, the symmetric method shows a good convergence on the Kersahw mesh,\nwhile the non-symmetric variant is required when considering the h-perturbed\ngrid. In this setting, we compute the global errors for the pressure and velocity\nvariables using different norms, i.e.,\nEp\nh=∥p−ph∥L2, bEp\nh=∥rhp−Ph∥ℓ2,\nEu\nh=∥Πhu−uh∥L2, bEu\nh=∥u−uh∥Fh,16 Andr´ es Arrar´ as et al.\n(a) Smooth mesh\n (b) Kershaw mesh\n(c)h-perturbed mesh\n (d) Randomly h-perturbed mesh\nFig. 4 Some logically rectangular grids considered in the numerical examples.\nwhere rhdenotes the restriction operator to the center of the cells, and Πhdenotes\nthe projection operator onto the space Vh[61]. We denote by ∥ · ∥L2the corre-\nsponding norm in either L2(Ω) or ( L2(Ω))2, and by ∥ · ∥ℓ2the discrete L2-norm\ninHp. Further, we define the edge-based norm ∥ · ∥Fhas\n∥v∥2\nFh=X\nE∈ThX\ne∈∂E|E|\n|e|∥v·ne∥2\ne,\nwhere |E|and|e|refer to the area of element Eand the length of edge e, respec-\ntively. The integrals involved in the pressure errors Ep\nhare approximated element-\nwise by a 9-point Gaussian quadrature formula. In turn, the integrals arising in the\nvelocity error Eu\nhare approximated by the trapezoidal quadrature rule, while those\ninvolved in the face error bEu\nhare computed by a high-order Gaussian quadrature\nrule.\nTable 2 shows the global errors and numerical orders of convergence of the\nmethod when applied to the family of Kershaw-type meshes. As predicted by the\ntheory [61], we observe first-order convergence for the pressure and the velocity\nerrors in the L2-norm, as well as for the velocity errors on the element edges.\nMoreover, we obtain second-order superconvergence for the pressure at the cell\ncenters.\nTable 3 shows the global errors and the numerical orders of convergence on the\nfamily of h-perturbed grids depicted in Figure 4(c). The numerical results show\nthat the convergence of the symmetric MFMFE method deteriorates on this type of\ngrids. In turn, Table 4 shows the numerical results obtained when considering theMultigrid solvers for multipoint flux approximations of the Darcy problem 17\nTable 2 Global errors and numerical orders of convergence for the symmetric MFMFE method\non the Kershaw meshes ( h0= 2−5).\nh Ep\nhRate ˆEp\nhRate Eu\nhRate ˆEu\nhRate\nh0 2.959e-02 – 1.857e-03 – 1.103e+00 – 9.141e-01 –\nh0/2 1.479e-02 1.000 4.739e-04 1.970 5.519e-01 0.999 4.524e-01 1.014\nh0/227.396e-03 1.000 1.192e-04 1.991 2.763e-01 0.998 2.258e-01 1.002\nh0/233.698e-03 1.000 2.986e-05 1.997 1.383e-01 0.999 1.129e-01 1.000\nh0/241.851e-03 0.998 7.470e-06 1.999 6.916e-02 1.000 5.633e-02 1.003\nTable 3 Global errors and numerical orders of convergence for the symmetric MFMFE method\non the h-perturbed meshes ( h0= 2−5).\nh Ep\nhRate ˆEp\nhRate Eu\nhRate ˆEu\nhRate\nh0 3.400e-02 – 7.831e-03 – 2.712e+00 – 2.349e+00 –\nh0/2 1.702e-02 0.998 3.821e-03 1.035 1.804e+00 0.588 1.565e+00 0.586\nh0/228.513e-03 0.999 1.893e-03 1.013 1.234e+00 0.548 1.072e+00 0.546\nh0/234.258e-03 0.999 9.432e-04 1.005 8.575e-01 0.525 7.452e-01 0.525\nh0/242.129e-03 1.000 4.708e-04 1.002 6.010e-01 0.513 5.224e-01 0.512\nTable 4 Global errors and numerical orders of convergence for the non-symmetric MFMFE\nmethod on the h-perturbed meshes ( h0= 2−5).\nh Ep\nhRate ˆEp\nhRate Eu\nhRate ˆEu\nhRate\nh0 3.325e-02 – 1.658e-03 – 1.370e+00 – 1.072e+00 –\nh0/2 1.662e-02 1.000 4.163e-04 1.994 6.855e-01 0.999 5.339e-01 1.006\nh0/228.308e-03 1.000 1.042e-04 1.998 3.428e-01 1.000 2.667e-01 1.001\nh0/234.154e-03 1.000 2.605e-05 2.000 1.714e-01 1.000 1.333e-01 1.000\nh0/242.077e-03 1.000 6.513e-06 1.999 8.571e-02 1.000 6.665e-02 1.000\nnon-symmetric MFMFE method derived by using the non-symmetric quadrature\nrule (5). In accordance with [60], we observe first-order convergence for both the\nvelocity and the pressure errors in the L2-norm, as well as for the velocity errors\non the element edges. Finally, second-order superconvergence is obtained for the\npressure errors at the cell centers.\nNext, let us show the performance of the multigrid method when considering\ndifferent types of cycles ( V-,F- and W-cycles), different families of meshes and\ndifferent grid sizes. In Table 5, we show the number of multigrid iterations re-\nquired to obtain a residual smaller than 10−9. In the case of considering random\ngrids, the last three columns of Table 5 show the rounding to the closest natural\nnumber of the average number of multigrid iterations required when performing\n100 realizations of the random mesh. It is worth pointing out that in the case of\nconsidering Kershaw meshes, we propose to use a relaxation parameter w= 0.6\nin the smoothing procedure. The multigrid solver is shown to be robust in all the\ncases with respect to the spatial discretization parameter h. Moreover, it is shown\nto be very efficient, since the number of iterations required to satisfy the stopping\ncriterion in all the cases is not very high. It is also interesting to notice that both\nF- and W-cycles show a better convergence behaviour than V-cycles on rough\ngrids such as Kershaw meshes. Finally, since the computational cost of F-cycles18 Andr´ es Arrar´ as et al.\nTable 5 Number of iterations of the multigrid method when considering different types of\ncycles and various families of logically rectangular meshes ( h0= 2−5) in the test with known\nanalytical solution.\nMesh Smooth Kershaw h-perturbed Randomly h-perturbed\nCycle V F W V F W V F W V F W\nh0 10 9 9 9 9 9 8 8 8 9 8 8\nh0/2 11 9 9 11 10 10 8 7 7 9 8 8\nh0/2211 7 7 14 13 13 8 6 6 9 8 8\nh0/2311 5 5 16 13 13 8 5 5 9 7 7\nh0/2410 4 4 17 13 13 8 5 5 9 7 7\nFig. 5 Logically rectangular mesh (left), numerical pressure (center) and logarithm of the\nnorm of the numerical velocity (right) for the flow problem that contains an impermeable\nstreak.\nis smaller than that of W-cycles, from now on we shall consider F-cycles for the\nmultigrid solver.\n4.2 Several tests with discontinuous permeability tensors\nLet us next consider a flow problem through a system that contains an imperme-\nable streak. Different variants of this porous media example have been considered\nas test problems for various spatial discretizations in [11,27,28,36]. In our case,\nthis numerical example permits us to test the behaviour of the new multigrid\nmethod in the solution of a problem that contains irregularly shaped strata and\nabrupt variations in permeability.\nIn particular, we consider problem (1) with Ω= (0,1)2, Dirichlet boundary\nconditions with g= 1−xonΓD=∂Ω, and f= 0. The flow domain contains a\nlow-permeability region which is delimited by two curves. The top curve is chosen\nto be an arc of a circle with center at (0 .1,−0.4) and radius equal to 1 .2, while the\nbottom curve is an arc of a circle with the same center and radius equal to 1 .1. The\npermeability throughout the domain is uniform and isotropic ( K=I), except in\nthe low-permeability streak. In this region, it is such that the parallel component\nto the local streak orientation is equal to 0 .1, and the normal component to the\nlocal streak orientation is equal to 0.001. For more details, see [11].Multigrid solvers for multipoint flux approximations of the Darcy problem 19\n0 5 10 15 20\niterations10-1010-5100residualN=20\nN=40\nN=80\nN=160\nN=320\nFig. 6 Convergence history of the multigrid method for the flow problem that contains an\nimpermeable streak.\nFigure 5 (left) shows the geometry of the flow domain and the logically rect-\nangular grid used in the discretization when N= 20. Observe that the grid is\nadapted to the geometry of the low-permeability streak, depicted in the figure by\nbold curves. Figure 5 (center) displays the numerical pressure and Figure 5 (right)\nshows the logarithm of the norm of the numerical velocity field when considering\nN= 320. As expected from the physical configuration, no flow enters the streak,\nwhich is in accordance with the numerical results obtained in the aforementioned\nworks. Regarding the performance of the multigrid solver, Figure 6 shows the\nresidual versus the number of iterations until the initial residual is reduced by a\nfactor of 10−10. In this case, where we consider a relaxation parameter w= 0.6\nin the smoothing procedure, the multigrid solver is also shown to be robust with\nrespect to the spatial discretization parameter h.\nNext, let us test the behaviour of the multigrid method when considering dif-\nferent configurations of low-permeability regions, and various families of logically\nrectangular grids which are not aligned with the jumps in the permeability coeffi-\ncient. In particular, we consider two low-permeability intersecting bands, depicted\nin grey in Figure 7 (left), and also square-shaped and L-shaped low-permeability\ninclusions, depicted in grey in Figure 8 (left) and Figure 9 (left), respectively. This\ntype of periodic inclusions were previously considered in [43]. In all the cases, we\nfixK= 10−3Iinside the streaks or inclusions, and K=Iin the rest of the\ndomain. Regarding the spatial meshes, we consider the four families of logically\nrectangular grids introduced in Section 4.1 and shown in Figure 4.\nThe central plot in Figures 7, 8 and 9 represents the numerical pressure ob-\ntained for the corresponding permeability configuration when considering the\nsmooth mesh depicted in Figure 4(a) for N= 320. Accordingly, the right plot\nin Figures 7, 8 and 9 represents the logarithm of the norm of the numerical ve-\nlocity field. In all the cases, the numerical solutions show the expected physical\nbehaviour.\nTable 6 shows the number of multigrid iterations required to reduce the ini-\ntial residual by a factor of 1010for different values of h, different permeability\nconfigurations and different families of spatial meshes. In particular, for each per-\nmeability distribution, the columns denoted by (a), (b), (c) and (d) correspond to\nthe families of smooth, Kershaw, h-perturbed and randomly h-perturbed meshes,\nrespectively. Once again, in the case of considering random grids, we show the inte-20 Andr´ es Arrar´ as et al.\nTable 6 Number of iterations of the multigrid method when considering different permeability\ndistributions and various families of logically rectangular meshes ( h0= 20−1).\nTwo intersecting streaks Square-shaped inclusions L-shaped inclusions\nh (a) (b) (c) (d) (a) (b) (c) (d) (a) (b) (c) (d)\nh0 7 10 7 7 6 9 7 7 6 9 7 7\nh0/2 7 9 7 7 6 9 6 7 6 9 6 7\nh0/227 9 6 7 6 9 6 7 7 10 6 7\nh0/237 9 6 6 6 10 6 7 7 10 7 7\nh0/247 9 6 6 6 10 6 7 8 11 7 8\n0 1/4 1/2 3/4 101/41/23/41\nFig. 7 Two intersecting low-permeability streaks (left), numerical pressure (center) and log-\narithm of the norm of the numerical velocity (right).\n0 1/4 1/2 3/4 101/41/23/41\nFig. 8 Square-shaped low-permeability inclusions (left), numerical pressure (center) and log-\narithm of the norm of the numerical velocity (right).\nger approximation to the average number of iterations required in 100 realizations\nof the corresponding experiment.\nIt is worth noting the robustness of the multigrid method for different perme-\nability configurations, even when considering spatial meshes which are not aligned\nwith the jumps on the permeability coefficients.\n4.3 A test with a random permeability tensor\nTo conclude, we would like to study the robustness of the proposed multigrid\nmethod in the case of considering a heterogeneous random medium. For this pur-Multigrid solvers for multipoint flux approximations of the Darcy problem 21\n0 1/4 1/2 3/4 101/41/23/41\nFig. 9 L-shaped low-permeability inclusions (left), numerical pressure (center) and logarithm\nof the norm of the numerical velocity (right).\npose, the domain is set to be Ω= (0,1)2, and Dirichlet conditions and a zero\nright-hand side are considered. We assume a diagonal permeability tensor K=kI\nwith constant k, where Iis the identity tensor. A lognormal random field may\naccurately represent the permeability of a heterogeneous porous medium [33], and\ntherefore we assume that the logarithm of the permeability field, log10k, is mod-\neled by a zero-mean Gaussian random field. In this way, to generate samples of\nthe Gaussian random field, we use the so-called Mat´ ern covariance function [34],\ngiven by\nCΦ(r) =σ2\nc21−νc\nΓ(νc)\u0012\n2√νcr\nλc\u0013νc\nKν\u0012\n2√νcr\nλc\u0013\n. (15)\nSuch a covariance function is characterized by a set of parameters Φ= (νc, λc, σ2\nc),\nwhere νcdefines the field smoothness, λcis the correlation length and σ2\ncrepresents\nthe variance. In the previous expression, ris the distance between two points, Γ\nis the gamma function, and Kνis the modified Bessel function of the second kind.\nBy using the Mat´ ern family of covariance functions, random coefficient fields with\ndifferent degrees of smoothness can be generated. In particular, we consider two\nMat´ ern reference sets of parameters with increasing order of complexity, namely:\nΦ1= (0.5,0.3,1) and Φ2= (0.5,0.1,3). In Figure 10, we represent a possible\nsample of log10kusing each of the two sets of parameters, as an example. We can\nobserve that, for the set Φ2, the fluctuations of the permeability field are much\nlarger than for Φ1.\nFor a fixed set of parameters and a mesh size h, we generated 100 realizations\nof the permeability field and computed the average number of multigrid iterations\nnecessary to reduce the initial residual by a factor of 10−9. This experiment was\ndone for both regular and randomly h-perturbed meshes and Table 7 shows the\nrounding to the closest natural number of the average number of iterations per-\nformed in each case. We can observe from the table that the convergence of the\nsolver is independent of the discretization parameter h. For both types of grids,\nwe obtain an average of around 5-6 iterations for both sets of parameters Φ1and\nΦ2. Summarizing, the performance of the proposed multigrid solver in highly het-\nerogeneous random media is shown to be very satisfactory too.22 Andr´ es Arrar´ as et al.\n-2 -1 0123\n-6 -4 -2 0246\n(a)Φ1= (0.5,0.3,1) (b) Φ2= (0.5,0.1,3)\nFig. 10 Logarithm of the permeability field, log10k, generated using two different sets of\nparameters Φ= (νc, λc, σ2\nc).\nTable 7 Average number of multigrid iterations when considering two different sets of pa-\nrameters Φ1andΦ2and both regular and randomly h-perturbed meshes ( h0= 2−5) in the\ntest with random permeability fields.\nMesh Regular mesh Randomly h-perturbed\nh Φ1 Φ2 Φ1 Φ2\nh0 5 8 5 8\nh0/2 5 7 5 7\nh0/225 6 5 6\nh0/245 5 5 5\n5 Conclusions\nThe aim of this work is the efficient solution of multipoint flux approximations of\nthe Darcy problem. These methods can be applied on irregular meshes and accu-\nrately handle anisotropic discontinuous tensors even with large coefficient jumps.\nThus, the search for a solver for MPFA is of great interest.\nHere, we have proposed a blackbox-type geometric multigrid method for MPFA\non logically rectangular meshes. This is the first time that multigrid is applied to\nMPFA discretizations. The blackbox methodology makes the solver very easy to\napply for the user, and the logically rectangular meshes take advantage of the\nrecent trends in computer architectures that achieve their best performance when\nstructured data can be used. The robustness of the proposed multigrid solver has\nbeen shown for several problems with different permeability tensors, including ran-\ndom permeability fields, on a variety of logically rectangular grids. Moreover, the\nproposed method can be easily parallelized by using grid partitioning techniques.\nThe main limitation of this solver, related to its applicability to two-dimensional\nproblems, can also be overcome. To this respect, the straightforward extension to\nthree-dimensional problems would imply the use of plane relaxation instead of\nline-smoothers, which would make the method, however, prohibitively more ex-\npensive. Thus, we would propose to apply one iteration of the two-dimensional\nmultigrid presented here on each plane instead of solving it exactly. The extension\nto three-dimensional problems, however, is a subject of future research.Multigrid solvers for multipoint flux approximations of the Darcy problem 23\nFinally, another extension for the multigrid strategy would be to consider the\nhigher order MFMFE methods described in [7]. Such methods are based on a new\nfamily of enhanced Raviart–Thomas spaces, and also reduce to a cell-centered\npressure system if the velocity degrees of freedom are chosen to be the points\nof tensor-product Gauss–Lobatto quadrature rules. In this context, the blackbox\nmultigrid solver could be naturally applied to the resulting pressure system. Alter-\nnatively, we could also design a so-called p-multigrid strategy2that takes advantage\nof different approximation orders along the iterations.\nAcknowledgements Francisco J. Gaspar has received funding from the European Union’s\nHorizon 2020 Research and Innovation Programme under the Marie Sk lodowska–Curie grant\nagreement No. 705402, POROSOS. The work of Laura Portero is supported by the Spanish\nproject MTM2016-75139-R (AEI/FEDER, UE) and the Young Researchers Programme 2018\nfrom the Public University of Navarre. Andr´ es Arrar´ as acknowledges support from the Span-\nish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE) and the Young Researchers\nProgramme 2018 from the Public University of Navarre. The work of Carmen Rodrigo is sup-\nported by the Spanish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE) and the\nDGA (Grupo de referencia APEDIF, ref. E24 17R). Andr´ es Arrar´ as and Laura Portero grate-\nfully acknowledge the hospitality of the Centrum Wiskunde and Informatica (Amsterdam),\nwhere this research was partly carried out.\nReferences\n1. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids.\nComput. Geosci. 6, 405–432 (2002)\n2. Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal,\nquadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127, 2–14\n(1996)\n3. Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on unstructured grids\nfor inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput.\n19, 1700–1716 (1998)\n4. Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on unstructured grids\nfor inhomogeneous, anisotropic media. II. Discussion and numerical results. SIAM J. Sci.\nComput. 19, 1717–1736 (1998)\n5. Aavatsmark, I., Eigestad, G.T., Klausen, R.A., Wheeler, M.F., Yotov, I.: Convergence of\na symmetric MPFA method on quadrilateral grids. Comput. Geosci. 11, 335–345 (2007)\n6. Alcouffe, R.E., Brandt, A., Dendy Jr., J.E., Painter, J.W.: The multigrid method for the\ndiffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Statist. Comput.\n2, 430–454 (1981)\n7. Ambartsumyan, I., Khattatov, E., Lee, J.J., Yotov, I.: Higher order multipoint flux mixed\nfinite element methods on quadrilaterals and hexahedra. Math. Models Methods Appl.\nSci.29, 1037–1077 (2019)\n8. Arbogast, T., Dawson, C.N., Keenan, P.T., Wheeler, M.F., Yotov, I.: Enhanced cell-\ncentered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput.\n19, 404–425 (1998)\n9. Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM J. Numer.\nAnal. 42, 2429–2451 (2005)\n10. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementa-\ntion, postprocessing and error estimates. RAIRO Mod´ el. Math. Anal. Num´ er. 19, 7–32\n(1985)\n11. Arrar´ as, A., Portero, L., Yotov, I.: Error analysis of multipoint flux domain decomposition\nmethods for evolutionary diffusion problems. J. Comput. Phys. 257, 1321–1351 (2014)\n2This technique was first proposed in [49,47] for spectral element discretizations, and has\nbeen subsequently extended to discontinuous Galerkin methods [32] or isogeometric analysis\n[54].24 Andr´ es Arrar´ as et al.\n12. Ashby, S.F., Falgout, R.D.: A parallel multigrid preconditioned conjugate gradient algo-\nrithm for groundwater flow simulations. Nucl. Sci. Eng. 124, 145–159 (1996)\n13. Baranger, J., Maitre, J.F., Oudin, F.: Connection between finite volume and mixed finite\nelement methods. RAIRO Mod´ el. Math. Anal. Num´ er. 30, 445–465 (1996)\n14. Bause, M., Knabner, P.: Computation of variably saturated subsurface flow by adaptive\nmixed hybrid finite element methods. Adv. Water Resour. 27, 565–581 (2004)\n15. Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing\n55, 379–393 (1995)\n16. Braess, D., Verf¨ urth, R.: Multigrid methods for nonconforming finite element methods.\nSIAM J. Numer. Anal. 27, 979–986 (1990)\n17. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31,\n333–390 (1977)\n18. Brandt, A.: Rigorous quantitative analysis of multigrid, I: Constant coefficients two-level\ncycle with L2-norm. SIAM J. Numer. Anal. 31, 1695–1730 (1994)\n19. Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adap-\ntive smoothed aggregation ( αSA) multigrid. SIAM Rev. 47, 317–346 (2005)\n20. Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. M2AN\nMath. Model. Numer. Anal. 43, 277–295 (2009)\n21. Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second\norder elliptic problems. Numer. Math. 47, 217–235 (1985)\n22. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, Springer Ser. Comput.\nMath. , vol. 15. Springer-Verlag, New York (1991)\n23. Brezzi, F., Fortin, M., Marini, L.D.: Error analysis of piecewise constant pressure approx-\nimations of Darcy’s law. Comput. Methods Appl. Mech. Engrg. 195, 1547–1559 (2006)\n24. Cai, Z., Jones, J.E., McCormick, S.F., Russell, T.F.: Control-volume mixed finite element\nmethods. Comput. Geosci. 1, 289–315 (1997)\n25. Dendy, Jr., J.E.: Black box multigrid. J. Comput. Phys. 48, 366–386 (1982)\n26. Dendy, Jr., J.E.: Black box multigrid for non-symmetric problems. Appl. Math. Comput.\n13, 261–283 (1983)\n27. Durlofsky, L.J.: A triangle based mixed finite element–finite volume technique for modeling\ntwo phase flow through porous media. J. Comput. Phys. 105, 252–266 (1993)\n28. Durlofsky, L.J.: Accuracy of mixed and control volume finite element approximations to\nDarcy velocity and related quantities. Water Resour. Res. 30, 965–973 (1994)\n29. Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume\nschemes with flow based grids. Comput. Geosci. 6, 433–452 (2002)\n30. Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity\nfor the general tensor pressure equation. Comput. Geosci. 2, 259–290 (1998)\n31. Ewing, R.E., Iliev, O.P., Lazarov, R.D., Rybak, I.V.: On two-level preconditioners for flow\nin porous media. Tech. Rep. 121, Fraunhofer-Institut f¨ ur Techno- und Wirtschaftsmathe-\nmatik (2007)\n32. Fidkowski, K.J., Oliver, T.A., Lu, J., Darmofal, D.L.: p-Multigrid solution of high-order\ndiscontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J.\nComput. Phys. 207, 92–113 (2005)\n33. Freeze, R.A.: A stochastic-conceptual analysis of one-dimensional groundwater flow in\nnonuniform homogeneous media. Water Resour. Res. 11, 725–741 (1975)\n34. Handcock, M.S., Wallis, J.R.: An approach to statistical spatial-temporal modeling of\nmeteorological fields. J. Amer. Statist. Assoc. 89, 368–378 (1994)\n35. Hyman, J., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for\ndiffusion equations. Comput. Geosci. 6, 333–352 (2002)\n36. Hyman, J., Shashkov, M., Steinberg, S.: The numerical solution of diffusion problems in\nstrongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130–148 (1997)\n37. Kershaw, D.S.: Differencing of the diffusion equation in Lagrangian hydrodynamic codes.\nJ. Comput. Phys. 39, 375–395 (1981)\n38. Khalil, M., Wesseling, P.: Vertex-centered and cell-centered multigrid for interface prob-\nlems. J. Comput. Phys. 98, 1–10 (1992)\n39. Klausen, R.A., Radu, F.A., Eigestad, G.T.: Convergence of MPFA on triangulations and\nfor Richards’ equation. Internat. J. Numer. Methods Fluids 58, 1327–1351 (2008)\n40. Klausen, R.A., Russell, T.F.: Relationships among some locally conservative discretization\nmethods which handle discontinuous coefficients. Comput. Geosci. 8, 341–377 (2004)\n41. Klausen, R.A., Winther, R.: Convergence of multipoint flux approximations on quadrilat-\neral grids. Numer. Methods Partial Differential Equations 22, 1438–1454 (2006)Multigrid solvers for multipoint flux approximations of the Darcy problem 25\n42. Klausen, R.A., Winther, R.: Robust convergence of multi point flux approximation on\nrough grids. Numer. Math. 104, 317–337 (2006)\n43. Kumar, P., Rodrigo, C., F.J., G., C.W., O.: On local fourier analysis of multigrid methods\nfor pdes with jumping and random coefficients. SIAM J. Sci. Comput. 41, A1385–A1413\n(2019)\n44. Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput.\nPhys. 257, 1163–1227 (2014)\n45. Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer.\nMath. 112, 115–152 (2009)\n46. MacLachlan, S.P., Moulton, J.D., Chartier, T.P.: Robust and adaptive multigrid methods:\ncomparing structured and algebraic approaches. Numer. Linear Algebra Appl. 19, 389–413\n(2012)\n47. Maday, Y., Mu˜ noz, R.: Spectral element multigrid. II. Theoretical justification. J. Sci.\nComput. 3, 323–353 (1988)\n48. Micheletti, S., Sacco, R., Saleri, F.: On some mixed finite element methods with numerical\nintegration. SIAM J. Sci. Comput. 23, 245–270 (2001)\n49. Rønquist, E.M., Patera, A.T.: Spectral element multigrid. I. Formulation and numerical\nresults. J. Sci. Comput. 2, 389–406 (1987)\n50. Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous\nflows in porous media. In: R.E. Ewing (ed.) The Mathematics of Reservoir Simulation,\nFrontiers Appl. Math. , vol. 1, pp. 35–106. SIAM, Philadelphia (1983)\n51. Russell, T.F., Wheeler, M.F., Yotov, I.: Superconvergence for control-volume mixed finite\nelement methods on rectangular grids. SIAM J. Numer. Anal. 45, 223–235 (2007)\n52. Shashkov, M., Steinberg, S.: Solving diffusion equations with rough coefficients in rough\ngrids. J. Comput. Phys. 129, 383–405 (1996)\n53. St¨ uben, K.: An introduction to algebraic multigrid. In: Trottenberg, U., Oosterlee, C.W.,\nSch¨ uller, A., Multigrid, Appendix A, pp. 413–532. Academic Press, San Diego (2001)\n54. Tielen, R., M¨ oller, M., G¨ oddeke, D., Vuik, C.: A p-multigrid method enhanced with an\nILUT smoother and its comparison to h-multigrid methods within Isogeometric Analysis.\narXiv:1901.01685v3 [math.NA], 2020\n55. Trottenberg, U., Oosterlee, C.W., Sch¨ uller, A.: Multigrid. Academic Press, San Diego\n(2001)\n56. Vanˇ ek, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed\naggregation. Numer. Math. 88, 559–579 (2001)\n57. Vanˇ ek, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for sec-\nond and fourth order elliptic problems. Computing 56, 179–196 (1996)\n58. Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic\nproblems. SIAM J. Numer. Anal. 25, 351–375 (1988)\n59. Wesseling, P.: Cell-centered multigrid for interface problems. J. Comput. Phys. 79, 85–91\n(1988)\n60. Wheeler, M.F., Xue, G., Yotov, I.: A multipoint flux mixed finite element method on\ndistorted quadrilaterals and hexahedra. Numer. Math. 121, 165–204 (2012)\n61. Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer.\nAnal. 44, 2082–2106 (2006)\n62. Wienands, R., Joppich, W.: Practical Fourier analysis for multigrid methods, Numer.\nInsights , vol. 4. Chapman & Hall/CRC, Boca Raton (2005)"    },    {        "title": "2402.06899v1.Geodesic_X_ray_transform_and_streaking_artifacts_on_simple_surfaces_or_on_spaces_of_constant_curvature.pdf",        "content": "arXiv:2402.06899v1  [math.AP]  10 Feb 2024GEODESIC X-RAY TRANSFORM AND STREAKING ARTIFACTS\nON SIMPLE SURFACES OR ON SPACES OF CONSTANT CURV ATURE\nHIROYUKI CHIHARA\nABSTRACT . The X-ray transform on the plane or on the three-dimensiona l Euclidean space can be\nconsidered as the measurements of CT scanners for normal hum an tissue. If the human body contains\nmetal regions such as dental implants, stents in blood vesse ls, metal bones, etc., the beam hardening\neffect for the energy level of the X-ray causes streaking art ifacts in its CT image. More precisely, if\nthere are two strictly convex metal regions contained in the cross-section of normal human tissue, then\nstreaking artifacts occur along the common tangent lines of the two regions.\nIn this paper we study the geodesic X-ray transform and strea king artifacts on nontrapping simple\ncompact Riemannian manifolds with strictly convex boundar ies. We show that the streaking artifacts\nresult from the propagation of conormal singularities on th e boundary of metal regions along the com-\nmon tangent geodesics under the strong and seemingly strang e assumption that the manifolds are two\ndimensional or spaces of constant curvature. This conditio n ensures that every Jacobi field takes the\nform of the product of a scalar function and parallel transpo rt along the geodesic. Our results clarify the\ngeometric meaning of the theory, which was imperceptible in the known results on the Euclidean space.\n1. I NTRODUCTION\nThe present paper studies the geodesic X-ray transform and a nonlinear phenomenon called the\nstreaking artifact on some compact Riemannian manifolds wi th boundaries. We begin with the X-ray\ntransform on the Euclidean plane R2to state the background and the motivation of the present pap er.\nEvery planar line ℓis parameterized by (θ,t)∈[0,π)×Ras\nℓ={(−ssinθ+tcosθ,scosθ+tsinθ) :s∈R},\nand the X-ray transform of a function f(x,y)onR2is defined by\nRf(θ,t) :=/integraldisplay\nℓf=/integraldisplay∞\n−∞f(−ssinθ+tcosθ,scosθ+tsinθ)ds.\nThis is also defined for θ∈[π,2π), andRf(θ,t) =Rf(θ+π,−t),(θ,t)∈[0,π)×Rholds. The\nadjointRTofRis explicitly given by\nRTg(x,y) =1\n2π/integraldisplayπ\n0g(θ,xcosθ+ysinθ)dθ,(x,y)∈R2\nfor a function g(θ,t)of(θ,t)∈[0,π)×R. Iff(x,y)describes the density distribution of the attenu-\nation coefficient on a cross-section of a human body of normal tissue, then the set {Rf(θ,t) : (θ,t)∈\n[0,π)×R}is considered to be the data of the measurements of a CT scanne r of old generation. Fur-\nthermore it is well known that f(x,y)can be reconstructed theoretically from the measurements b y\nthe inversion formula f= (−∂2\nx−∂y)1/2◦RT◦Rf. See [10] for instance. Here we assumed from\na point of view of physics that the X-ray beams have no width, a re monochromatic, and traverse the\nobjects along straight lines.\nActually, however, there are many factors affecting CT imag es: beam width, partial volume effect,\nbeam hardening effect, noise in measurements, numerical er rors, and etc. They cause the artifacts in\n2020 Mathematics Subject Classification. Primary 58J40, Secondary 53C65.\nKey words and phrases. geodesic X-ray transform, Fourier integral operators, pai red Lagrangian distributions, streaking\nartifacts.\nSupported by the JSPS Grant-in-Aid for Scientific Research # 23K03186.\n12 H. CHIHARA\nCT image. See Epstein’s textbook [3] on the mathematical the ory of medical imaging for the detail.\nThe X-ray beams consist of photons which actually have a wide range of energies E≧0. This is\ndescribed by a spectral function, which is a probability den sity function ρ(E)ofE∈[0,∞), and pick\nup the strong part of the measurement. This is called the beam hardening in medical imaging. In this\ncase the measurements of CT scanners are corrected from RfEto\nP=−log/braceleftbigg/integraldisplay∞\n0ρ(E)exp(−RfE)dE/bracerightbigg\n.\nIffEis independent of E≧0, say,fE=fE0with some fixed E0>0, thenP=RfE0. See [3,\nSection 3.3] again for the detail of the above. It is known tha t the beam hardening effect causes\nstreaking artifacts in the CT images.\nThe mathematical analysis of the streaking artifacts was or iginated by Park, Choi and Seo in their\npioneering work [20]. They studied the case that there is a me tal regionD, which is a disjoint union\nof finite numbers of strictly convex bounded domains D1,...,D Jwith smooth boundaries. They\nassumed that the spectral function is a characteristic func tion of an interval. Let E0>0be the energy\nlevel of photons for normal human tissue, and let εbe a small positive constant smaller than E0. Their\nspectral function ρ(E)and the distribution fE(x,y)of attenuation coefficients are supposed to be of\nthe forms\nρ(E) =1\n2ε1[E0−ε,E0+ε](E),\nfE(x,y) =fE0(x,y)+α(E−E0)1D(x,y),\nwhere1Ais the characteristic function of a set A,fE0is the distribution of attenuation coefficients for\nthe human normal tissue, and αis a positive constant. In this case the measurements become\nP=RfE0−log/braceleftBigg\nsinh/parenleftbig\nαεR1D/parenrightbig\nαεR1D/bracerightBigg\n=RfE0+∞/summationdisplay\nl=1Bl/parenleftbig\nαεR1D/parenrightbig2l,\nwhere{Bl}∞\nl=1is a sequence of real numbers. Then/parenleftbig\nR1D/parenrightbig2and its filtered back-projection (−∂2\nx−\n∂2\ny)1/2◦ RT/bracketleftbig/parenleftbig\nR1D/parenrightbig2/bracketrightbig\nare considered to be the principal parts of the beam hardenin g effect and of\nthe streaking artifacts respectively. We here show figures o f an original grayscale image of a sim-\nple model of two dental implants in the cross-section of oral cavity, its X-ray transform, the standard\nfiltered back-projection, and the filtered back-projection of/parenleftbig\nR1D/parenrightbig2using the Julia Programming Lan-\nguage.\nGEODESIC X-RAY TRANSFORM 3\nFigure 1. The upper left is an original grayscale image of a si mple model of two dental implants in\nthe cross-section of oral cavity, the upper right is its X-ra y transform, the lower left is the standard\nfiltered back-projection, and the lower right is the filtered back-projection of/parenleftbig\nR1D/parenrightbig2.\nWe can see streaking artifacts along four straight lines whi ch are tangent to both implants. Set\nfCT:= (−∂2\nx−∂2\ny)1/2◦RTP,\nfMA:= (−∂2\nx−∂2\ny)1/2◦RT/bracketleftbig\nP−RfE0/bracketrightbig\n=fCT−fE0.\nIn [20] Park, Choi and Seo studied the wave front sets of P− RfE0andfMAusing the canonical\nrelations of RandRT, and proved that the streaking artifacts are the singular su pport offMA. More\nprecisely, the streaking artifacts are the propagation of c onormal singularities on the boundary of D\nalong the common tangent lines of ∂Djand∂Dkforj/\\e}atio\\slash=k.\nIn [19] Paracios, Uhlmann and Wang developed the arguments o f [20] making use of the modern\nmicrolocal analysis, and proved that the streaking artifac ts are the sum of conormal distributions of\nthe same order, whose singular supports are the common tange nt lines of∂Djand∂Dkforj/\\e}atio\\slash=k.\nIn [27] Wang and Zou studied the case that Dis a nonconvex bounded domain. In this case the\nrelations of lines which are tangent to the nonconvex part of ∂Dare complicated.\nThe above mentioned results were studied on the plane R2. More recently in [1] the author devel-\noped the arguments in [19], and studied this problem for the d-plane transform on the n-dimensional\nEuclidean space Rn, wheren= 2,3,4,... andd= 1,...,n−1. Thed-plane transform of a function\nf(x)onRnis defined by\nRdf(σ,x′′) :=/integraldisplay\nσf(x′+x′′)dx′,\nwhereσ∈Gd,n,Gd,nis the Grassmannian which is the set of all d-dimensional vector subspace of\nRn,x′∈σ,x′′∈σ⊥, andσ⊥is the orthogonal complement of σinRn. We remark that if n≧3,\nthen there are two types of common tangent lines of ∂Djand∂Dkforj/\\e}atio\\slash=k.\n•Case B1: The common tangent line is contained in a common tang ent hyperplane. In other\nwords, the directions of the conormal singularities at the t angent points are same.\n•Case B2: The tangent hyperplanes at the tangent points are di fferent. In other words, the\ndirections of the conormal singularities at the tangent poi nts are different.\nWe show figures illustrating Cases B1 and B2 below.\nFigure 2. The left is Case B1 where ∂Djand∂Dkhave a common tangent hyperplane, and the right\nis Case B2 where the tangent hyperplane of ∂Djcuts through the interior of Dl.\nThe author generalized the results of [19] for the d-plane transform on Rn. More precisely, we proved\nthe following.\n•The conormal singularities at the common tangent points of ∂Djand∂Dkof Case B1 propa-\ngates along the common tangent line. The set of all the common tangent lines of Case B1 for\n∂Djand∂Dkform two hypersurfaces L(±)\njk, which are cylindrical or conic. These hypersur-\nfaces are surrounding ∂Djand∂Dk.4 H. CHIHARA\nFigure 3. Common surrounding hypersurfaces for DjandDk.\nThe filtered back-projection fMAis a sum of conormal distributions of the same order deter-\nmined bynandd. The singular supports of these conormal distributions are such hypersur-\nfaces.\n•The conormal singularities at the common tangent points of ∂Djand∂Dkof Case B2 do not\npropagate.\nWhat the author could understand in [1] is that these results are the natural extension of those of [19].\nUnfortunately, however, the author could not understand al most all the geometric meaning of the facts\nproved in [1]. It was too difficult for the author to see the ess ential parts in the setting of the Euclidean\nspace.\nThe purpose of this paper is to clarify the geometric meaning of the facts proved in [1]. For this\npurpose we study the geodesic X-ray transform on a compact Ri emannian manifold (M,g)with a\nRiemannian metric gand the smooth boundary ∂M. The geodesic X-ray transform for functions and\nsymmetric tensor fields has been extensively studied so far. See e.g., [4, 18, 21–23, 26] and references\ntherein.\nThe main results of this paper are stated in Theorem 5.1, whic h are essentially same as those of [1].\nIn the present paper we assume on (M,g)as follows.\n(A0):(M,g)is a compact, strictly convex, simple, and nontrapping Riem annian manifold with\nsmooth boundary. We say that a Riemannian manifold is simple if it has no conjugate point.\nWe say that a compact Riemannian manifold with boundary is no ntrapping if every geodesic\nexit the manifold in finite time. The metric tensor of (M,g)is denoted by /a\\}b∇acketle{t·,·/a\\}b∇acket∇i}ht=g(·,·).\n(A1): The Riemannian manifold (M,g)is two-dimensional or a space of constant curvature.\nThe assumptions on the metal region Dand on the beam hardening effect are essentially same as thos e\nof [20], but we will state them later. Let Gbe the set of all the normal geodesics of (M,g). The X-ray\ntransform of a compactly supported smooth function fonMintis defined by\nXf(γ) :=/integraldisplay\nγf, γ∈G.\nActually, in this paper we deal with the X-ray transform of co mpactly supported half-densities on Mint.\nWe describe the detail later. The assumption (A1) is very str ong and may seem to be strange. This is\neffectively used for composing the canonical relation CXTofXTand the Lagrangian describing the\nmicrolocal singularities of the conormal distribution P−Xf.\nThis paper relies on the foundations of microlocal analysis and Fourier integral operators. See\ncelebrated standard textbooks [2,6,12–15] for this. One ca n also refer expository papers [16] and [28,\nSection 2] for microlocal analysis for computer tomography and related fields. Also this paper deeply\nrelies on the methods and the ideas dealing with the geodesic X-ray transform developed by Holman\nand Uhlmann in [11].\nThe plan of this paper is as follows. Section 2 consists of bas ic facts on the X-ray transform X\nand its adjoint XT. Section 3 is devoted to studying Jacobi fields and parallel t ransport along normal\ngeodesics under the assumption ( A1). They play crucial role in clarifying the geometric meanin gs of\nthe facts in this paper. In Section 4 we discuss the geometry o f the metal region Dfor the intersection\ncalculus for X. In Section 5 we state the main theorem of the paper, and discu ss the intersection\ncalculus for XT. Finally in Section 6 we prove the main theorem in several ste ps.GEODESIC X-RAY TRANSFORM 5\n2. P RELIMINARIES\nThis section deeply relies on the results and the ideas devel oped by Holman and Uhlmann in [11].\nIn this section we assume ( A0) and not ( A1). We prepare some basic facts on the geodesic X-ray\ntransform on (M,g). Throughout this paper we use the notation which is the same a s or similar to\nthat of [11]. We denote by S(M)the unit tangent sphere bundle, that is, the principal bundl e onM\nconsisting of all the unit tangent vectors, and π:S(M)→Mdenotes the natural projection. This is a\n(2n−1)-dimensional manifold. The restriction of πonMintis denoted by the same notation π. For\nv∈S(M),γvdenotes the normal geodesic such that γv(0) =π(v)and˙γv(0) =v. If we set\nτ−(v) = sup{t≦0 :γv(t)/\\e}atio\\slash∈M}, τ+(v) = inf{t≧0 :γv(t)/\\e}atio\\slash∈M},\nthen the nontrapping condition on (M,g)implies that [τ−(v),τ+(v)]is a closed finite interval. We\nalso denote by Ψthe geodesic flow on S(Mint), that is,π/parenleftbig\nΨ(v,t)/parenrightbig\n=γv(t)andΨ(v,t) = ˙γv(t)\nforv∈S(Mint)andt∈[τ−(v),τ+(v)]. Here we introduce the unit outer normal vector ν+(x)at\nx∈∂M, and set\n∂±S(M) :=/braceleftbig\nw∈S(M) :π(w)∈∂M,±/angbracketleftbig\nw,ν+/parenleftbig\nπ(w)/parenrightbig/angbracketrightbig\n>0}.\nThe simplicity and convexity conditions on (M,g)imply that for any w∈∂−S(M)there exists a\nw′∈∂+S(M)uniquely such that w′= ˙γw/parenleftbig\nτ+(w)/parenrightbig\n. In the same way, for any γ∈ G there exists a\nw∈∂−S(M)uniquely such that γ=γw(·), and∂−S(M)is identified with G. More precisely G\nbecomes a (2n−2)-dimensional smooth manifold with the parametrization ∂−S(M).\nWe now introduce an important submersion F:S(Mint)→∂−S(M). Forv∈S(Mint)set\nF(v) := ˙γv/parenleftbig\nτ−(v)/parenrightbig\n. Then we have v/mapsto→τ−(v)is smooth and Fbecomes a smooth mapping since\n∂M is a strictly convex smooth boundary. If F(v) =F(˜v)forv,˜v∈S(Mint), then there exists a time\nt0∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\nsuch that ˜v= ˙γv(t0). Hence we have\nrank(DF|v) = dim/parenleftbig\nS(Mint)/parenrightbig\n−1 = 2n−2 = dim/parenleftbig\n∂−S(M)/parenrightbig\n,\nand we conclude that Fis a submersion.\nWe shall define the geodesic X-ray transform Xfor half densities. To obtain its canonical relation\nwe see that X=F∗◦π∗andXT=π∗◦F∗, whereXTis the adjoint of X,F∗andF∗are the\npushforward and the pullback induced by Frespectively, and π∗andπ∗are the pushforward and\nthe pullback induced by πrespectively. We denote by dvgthe Riemannian measure of (M,g), by\n|dS(M)|the Riemannian measure of S(M)with respect to the Sasaki metric, and by |d∂S(M)|the\nRiemannian measure of\n∂S(M) :=S(M)|∂M=∂−S(M)∪S(∂M)∪∂+S(M)\nwith respect to the Sasaki metric. Santal´ o’s formula takes the form\n/integraldisplay\nS(M)h(v)|dS(M)|=−/integraldisplay\n∂−S(M)/integraldisplayτ+(w)\n0h/parenleftbig\n˙γw(s)/parenrightbig/angbracketleftbig\nw,ν+/parenleftbig\nπ(w)/parenrightbig/angbracketrightbig\nds|d∂S(M)| (2.1)\nfor any continuous function honS(M). See [25, Theorem 8.4.1] for instance. So we set\n|dµ(w)|:=−/angbracketleftbig\nw,ν+/parenleftbig\nπ(w)/parenrightbig/angbracketrightbig\n|d∂S(M)|,\nand use it as the standard measure on ∂−S(M).\nLetΩ1/2\nXbe the half density bundle on a manifold X. We denote by C∞(Ω1/2\nX)the set of all\nsmooth half densities on X, and byC∞\n0(Ω1/2\nX)the set of all compactly supported smooth half densities\nonX. Let D(Ω1/2\nX)be the locally convex space which is C∞\n0(Ω1/2\nX)with the standard inductive\nlimit topology, and let D′(Ω1/2\nX)be the topological dual of D(Ω1/2\nX), that is, the space of distribution\nsections of half density bundle on X. LetE(Ω1/2\nX)be the Fr´ echet space which is C∞(Ω1/2\nX)with the\nstandard topology, and let E′(Ω1/2\nX)be the topological dual of E(Ω1/2\nX), that is, the space of compactly6 H. CHIHARA\nsupported distribution sections of the half density bundle onX. We now define the geodesic X-ray\ntransform of a compactly supported smooth half density.\nDefinition 2.1 (Geodesic X-ray transform of a compactly supported half dens ity). The geodesic\nX-ray transform on (M,g)is defined for f∈C∞\n0(Ω1/2\nMint)as a half density on ∂−S(M)by\nX[f](w) :=/parenleftBigg/integraldisplayτ+(w)\n0f\n|dvg|1/2/parenleftbig\n˙γw(s)/parenrightbig\nds/parenrightBigg\n|dµ(w)|1/2, w∈∂−S(M). (2.2)\nIt is easy to see that X:D(Ω1/2\nMint)→D(Ω1/2\n∂−S(M))is a continuous mapping. Furthermore Xcan\nbe extended on E′(Ω1/2\nMint)sinceXis a Fourier integral operator.\nWe now introduce a pushforward and a pullback of half densiti es induced by a submersion.\nDefinition 2.2 (Pushforward and pullback induced by a submersion). LetXbe annX-dimensional\nsmooth manifold equipped with a density |dµX|, and letYbe annY-dimensional smooth manifold\nequipped with a density |dµY|. Suppose that G:X→Yis a submersion. Then the pushforward\nG∗:C∞\n0(Ω1/2\nX)→C∞\n0(Ω1/2\nY)and the pullback G∗:C∞\n0(Ω1/2\nY)→C∞\n0(Ω1/2\nX)are defined by the\nrequirement that\n/integraldisplay\nYh(y)·G∗[f](y) =/integraldisplay\nXh\n|dµY|1/2/parenleftbig\nG(x)/parenrightbig\n|dµX|1/2·f(x) =/integraldisplay\nXG∗[h](x)·f(x) (2.3)\nforf∈C∞\n0(Ω1/2\nX)andh∈C∞\n0(Ω1/2\nY).\nApply (2.1) and (2.3) to Fandπ. Then forx∈Mint,v∈S(Mint),w∈∂−S(M),f∈C∞\n0(Ω1/2\nMint),\nh∈C∞\n0(Ω1/2\nS(Mint))andk∈C∞\n0(Ω1/2\n∂−S(M)), we have\nF∗[h](w) =/parenleftBigg/integraldisplayτ+(w)\n0h\n|dS(Mint)|1/2/parenleftbig\n˙γw(s)/parenrightbig\nds/parenrightBigg\n|dµ(w)|1/2,\nF∗[k](v) =k\n|dµ|1/2/parenleftbig\nF(v)/parenrightbig\n|dS(Mint)(v)|1/2,\nπ∗[h](x) =/parenleftBigg/integraldisplay\nSx(Mint)h\n|dS(Mint)|1/2(v)|dSx(Mint)(v)|/parenrightBigg\n|dvg(x)|1/2,\nπ∗[f](v) =f\n|dvg|1/2/parenleftbig\nπ(v)/parenrightbig\n|dS(Mint)(v)|1/2,\nX[f](w) =F∗◦π∗[f](w),\nXT[k](x) =π∗◦F∗[k](x)\n=/parenleftBigg/integraldisplay\nSx(Mint)k\n|dµ|1/2/parenleftbig\nF(v)/parenrightbig\n|dSx(Mint)(v)|/parenrightBigg\n|dvg(x)|1/2.\nTo state the basic facts on F∗,F∗,π∗, andπ∗, we now introduce the notation of the space of\nLagrangian distributions, and the space of conormal distri butions. See [15, Sections 25.1 and 25.2]\nand [14, Section 18.2] for the precise definitions and detail ed information. Let XandYbe smooth\nmanifolds, and let Zbe a closed submanifold of X. We denote by Im(X,Λ;Ω1/2\nX)the set of all\nLagrangian distribution sections of the half density bundl eΩ1/2\nXof orderm∈Rassociated to Λ\nwhich is a conic Lagrangian submanifold of T∗(X)\\0. In particular, if Λ =N∗(Z)\\0, then elements\nofIm(X,N∗(Z)\\0;Ω1/2\nX)are said to be conormal distributions and we abbreviate Im(X,N∗(Z)\\GEODESIC X-RAY TRANSFORM 7\n0;Ω1/2\nX)asIm(N∗(Z)\\0;Ω1/2\nX). LetCbe a homogeneous canonical relation from T∗(Y)\\0to\nT∗(X)\\0. We set\nC′:={(ξ,−η)∈T∗(X)×T∗(Y) : (x,ξ;y,η)∈C},\nCT:={(η,ξ)∈T∗(Y)×T∗(X) : (x,ξ;y,η)∈C}.\nA linear operator whose distribution kernel belongs to Im(X×Y,C′;Ω1/2\nX×Y)is said to be a Fourier\nintegral operator of order mfromYtoXassociated to the canonical relation C. If the distribution\nkernel of an operator Abelongs toIm(X×Y,C′;Ω1/2\nX×Y), we express A∈Im(X×Y,C′;Ω1/2\nX×Y)\nabusing symbols. We believe that any confusion will not occu r.\nLinear operators induced by submersions such as Definition 2 .2 are Fourier integral operators, and\nit is easy to obtain their canonical relations. This is basic ally due to the pioneering paper [7] of\nGuillemin and Schaeffer. We here quote the fact needed in the present paper from [11].\nLemma 2.3 (Holman and Uhlmann [11, Lemma 1]). In the same setting as in Definition 2.2, we\nhave\nG∗∈I(nY−nX)/4(Y×X,C′\nG∗;Ω1/2\nY×X),\nCG∗=/braceleftbig\n(ξ,DG|T\nxξ) :x∈X,ξ∈T∗\nG(x)(Y)\\{0}/bracerightbig\n,\nG∗∈I(nY−nX)/4(X×Y,C′\nG∗;Ω1/2\nX×Y),\nCG∗=CT\nG∗.\nSincedim/parenleftbig\nS(Mint)/parenrightbig\n= 2n−1,dim/parenleftbig\n∂−S(M)/parenrightbig\n= 2n−2, anddim(M) =n, Lemma 2.3 implies\nthe following.\nLemma 2.4. Assume (A0). We have\nF∗∈I−1/4/parenleftbig\n∂−S(M)×S(Mint),C′\nF∗,Ω1/2\n∂−S(M)×S(Mint)/parenrightbig\n,\nF∗∈I−1/4/parenleftbig\nS(Mint)×∂−S(M),C′\nF∗,Ω1/2\nS(Mint)×∂−S(M))/parenrightbig\n,\nCF∗=/braceleftbig\n(ξ,DF|T\nvξ) :v∈S(Mint),ξ∈T∗\nF(v)(∂−S(M))\\{0}/bracerightbig\n,\nCF∗=CT\nF∗,\nπ∗∈I−(n−1)/4/parenleftbig\nMint×S(Mint),C′\nπ∗,Ω1/2\nMint×S(Mint)/parenrightbig\n,\nπ∗∈I−(n−1)/4/parenleftbig\nS(Mint)×Mint,C′\nπ∗,Ω1/2\nS(Mint)×Mint/parenrightbig\n,\nCπ∗=/braceleftbig\n(η,Dπ|T\nvη) :v∈S(Mint),η∈T∗\nπ(v)(Mint)\\{0}/bracerightbig\n,\nCπ∗=CT\nπ∗.\nTo state the basic properties of the X-ray transform X, we now recall the notion of clean composi-\ntion.\nDefinition 2.5. LetX,YandZbe smooth manifolds, and let C1andC2be homogeneous canonical\nrelations such that C1⊂T∗(X)×T∗(Y)andC1⊂T∗(Y)×T∗(Z).\n•The composition C1◦C2is defined by\nC1◦C2={(ξ,ζ) :∃η∈T∗(Y)s.t.(ξ,η)∈C1,(η,ζ)∈C2}.\n•We say that the composition C1◦C2is clean if the following conditions are satisfied.\n(i) The intersection\nC= (C1×C2)∩/parenleftBig\nT∗(X)×∆/parenleftbig\nT∗(Y)/parenrightbig\n×T∗(Z)/parenrightBig8 H. CHIHARA\nis clean, that is, Cis an embedded submanifold of T∗(X)×T∗(Y)×T∗(Y)×T∗(Z), and\nTp(C) =Tp/parenleftbig\nC1×C2/parenrightbig\n∩Tp/parenleftBig\nT∗(X)×∆/parenleftbig\nT∗(Y)/parenrightbig\n×T∗(Z)/parenrightBig\nfor any point p∈C, where∆/parenleftbig\nT∗(Y)/parenrightbig\nis the diagonal part of T∗(Y)×T∗(Y).\n(ii) The projection map πC:C→T∗(X)×T∗(Z)is proper.\n(iii) For any Ξ∈T∗(X)×T∗(Z),π−1\nC(Ξ)is connected.\n•If the composition C1◦C2is clean, then\ne:= codim(C1×C2)+codim/parenleftBig\nT∗(X)×∆/parenleftbig\nT∗(Y)/parenrightbig\n×T∗(Z)/parenrightBig\n−codim(C)\nis said to be the excess of the composition C1◦C2. In particular, if e= 0, that is,\nN∗\np(C1×C2)∩N∗\np/parenleftBig\nT∗(X)×∆/parenleftbig\nT∗(Y)/parenrightbig\n×T∗(Z)/parenrightBig\n={0}\nfor anyp∈C, we say that the composition C1◦C2is transversal.\nUsing Lemma 2.4 we have the basic properties of X.\nTheorem 2.6. Assume (A0). The composition CF∗◦Cπ∗is transversal, and\nX=F∗◦π∗∈I−n/4/parenleftbig\n∂−S(M)×Mint,C′\nX;Ω1/2\n∂−S(M)×Mint/parenrightbig\n,\nCX=/braceleftbig\n(ξ,η)∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗(Mint) :∃v∈S(Mint)s.t.\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n.\nMoreover, the principal symbol of Xdoes not vanish.\nProof. Set\nZ:=T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗/parenleftbig\nS(Mint)/parenrightbig\n×T∗/parenleftbig\nS(Mint)/parenrightbig\n×T∗(Mint)\nD:=T∗/parenleftbig\n∂−S(M)/parenrightbig\n×∆/parenleftBig\nT∗/parenleftbig\nS(Mint)/parenrightbig/parenrightBig\n×T∗(Mint)\nfor short. We have codim(D) = dim/parenleftBig\nT∗/parenleftbig\nS(Mint)/parenrightbig/parenrightBig\n= 4n−2. Lemma 2.4 implies that\nCF∗×Cπ∗=/braceleftbig\n(ξ,DF|T\nvξ,Dπ|T\nuη,η) :v,u∈S(Mint),\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(u)(Mint)\\{0}/bracerightbig\n.\nWe have\ndim(CF∗×Cπ∗) = dim/parenleftbig\nS(Mint)/parenrightbig\n+dim/parenleftBig\nT∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig/parenrightBig\n+dim/parenleftbig\nS(Mint)/parenrightbig\n+dim/parenleftbig\nT∗\nπ(u)(Mint)/parenrightbig\n= (2n−1)+(2n−2)+(2n−1)+n= 7n−4.\nThen we have\nC:= (CF∗×Cπ∗)∩D\n=/braceleftbig\n(ξ,DF|T\nvξ,Dπ|T\nvη,η) :v∈S(Mint),\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n. (2.4)\nTo compute the dimension of C, we need to know the properties of ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\nandη∈\nT∗\nπ(v)(Mint)satisfyingDF|T\nvξ=Dπ|T\nvη. In this case we have η(v) = 0 . IndeedF(v) =F/parenleftbig\n˙γv(t)/parenrightbig\nfor anyt∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n, and this implies that DF|v¨γv(0) = 0 . Hence we have\nη(v) =η/parenleftbig\nDπ|v¨γv(0)/parenrightbig\n=Dπ|T\nvη/parenleftbig\n¨γv(0)/parenrightbig\n=DF|T\nvξ/parenleftbig\n¨γv(0)/parenrightbig\n=ξ/parenleftbig\nDF|v¨γv(0)/parenrightbig\n= 0. (2.5)GEODESIC X-RAY TRANSFORM 9\nSinceF:S(Mint)→∂−S(M)andπ:S(Mint)→Mintare submersions, we deduce that DF|T\nv:\nT∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n→T∗\nv/parenleftbig\nS(Mint)/parenrightbig\nandDπ|T\nv:T∗\nπ(v)(Mint)→T∗\nv/parenleftbig\nS(Mint)/parenrightbig\nare linear injections for\nanyv∈S(Mint). Since\nT∗\nv/parenleftbig\nS(Mint)/parenrightbig\n≃ {(η,ζ)∈T∗\nπ(v)(Mint)×T∗\nπ(v)(Mint) :ζ(v) = 0},\nanddim/parenleftBig\nRan/parenleftbig\nDF|T\nv/parenrightbig/parenrightBig\n= 2n−2, we deduce that\nRan/parenleftbig\nDF|T\nv/parenrightbig\n≃ {(η,ζ)∈T∗\nπ(v)(Mint)×T∗\nπ(v)(Mint) :η(v) = 0,ζ(v) = 0},\nRan/parenleftbig\nDπ|T\nv/parenrightbig\n={(η,0) :η∈T∗\nπ(v)(Mint)},\nRan/parenleftbig\nDF|T\nv/parenrightbig\n∩Ran/parenleftbig\nDπ|T\nv/parenrightbig\n≃ {(η,0) :η∈T∗\nπ(v)(Mint),η(v) = 0}.\nHence ifξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\nandη∈T∗\nπ(v)(Mint)satisfyingDF|T\nvξ=Dπ|T\nvη, thenη(v) = 0 and\nξis uniquely determined by ξ= (DF|T\nv)−1◦Dπ|T\nvη, where(DF|T\nv)−1is the inverse linear mapping\nof the linear bijection DF|T\nv:T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n→Ran/parenleftbig\nDF|T\nv/parenrightbig\n. Then we have\ndim(C) = dim/parenleftbig\nS(Mint)/parenrightbig\n+dim/parenleftbig\n{η∈T∗\nπ(v)(Mint) :η(v) = 0}/parenrightbig\n= (2n−1)+(n−1) = 3n−2.\nThus we have\ne= codim( D)+codim(CF∗×Cπ∗)−codim(C)\n= codim( D)+dim(Z)−dim(CF∗×Cπ∗)−dim(Z)+codim(C)\n= codim( D)−dim(CF∗×Cπ∗)+codim(C)\n= (4n−2)−(7n−4)+(3n−2) = 0.\nIf we will show that CF∗◦Cπ∗is clean, it follows that CF∗◦Cπ∗is transversal, and therefore [15,\nTheorem 25.2.3] proves the first part of Theorem 2.6. The elli pticity of Xis obvious.\nSo it suffices to show that CF∗◦Cπ∗is clean. Since DF|T\nvξandDπ|T\nvηare continuous in (v,ξ)\nand(v,η)respectively, it is easy to see that πCis proper, and that π−1\nC/parenleftbig\n(ξ,η)/parenrightbig\nis connected. Hence it\nsuffices to show that C= (CF∗×Cπ∗)∩D is a clean intersection. Fix arbitrary\nc= (ξ,DF|T\nvξ,Dπ|T\nvη,η)∈C, X= (X1,X2,X3,X4)∈Tc(CF∗×Cπ∗)\nwithDF|T\nvξ=Dπ|T\nvη. Consider a curve on CF∗×Cπ∗of the form\nR∋α/mapsto→/parenleftbig\nξ(α),Y(α),Z(α),η(α)/parenrightbig\n=c+αX+O(α2)\n= (ξ+αX1,DF|T\nvξ+αX2,Dπ|T\nvη+αX3,η+αX4)+O(α2)\nnearα= 0. IfX∈Tc(D), then we have ˙Y(0) =X2=X3=˙Z(0). Hence\nX=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nξ(α),Y(α),Z(α),η(α)/parenrightbig\n=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nξ(α),Y(α),Y(α),η(α)/parenrightbig\n∈Tc(C),\nand we have Tc(C)⊃Tc(CF∗×Cπ∗)∩Tc(D). SinceTc(C)⊂Tc(CF∗×Cπ∗)∩Tc(D)is obvious, we\ndeduce that Tc(C) =Tc(CF∗×Cπ∗)∩Tc(D), and that (CF∗×Cπ∗)∩Dis a clean intersection. This\ncompletes the proof. /square\nCombining Theorem 2.6 and [15, Theorem 25.2.2], we immediat ely conclude that the adjoint XT\nofXis also a similar elliptic Fourier integral operator. It is w ell-known that if there is no conjugate\npoint in(M,g), the normal operator XT◦Xbecomes an elliptic pseudodifferential operator of order10 H. CHIHARA\n−1, and that there exists a parametrix for XT◦ X locally inM. See [8, 11] for instance. Holman\nand Uhlmann in [11] studied the case that there are conjugate points, and proved the decomposition\nformula of the normal operator according to the order of the c onjugate points.\nTheorem 2.7. Assume (A0).XTis also an elliptic Fourier integral operator such that\nXT=F∗◦π∗∈I−n/4/parenleftbig\nMint×∂−S(M),C′\nXT;Ω1/2\nMint×∂−S(M)/parenrightbig\n,\nCXT=/braceleftbig\n(η,ξ)∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗(Mint) :∃v∈S(Mint)s.t.\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n.\nFurthermore the normal operator XT◦Xis an elliptic pseudodifferential operator of order −1.\n3. J ACOBI FIELDS AND PARALLEL TRANSPORT\nIn this section we assume both ( A0) and ( A1), and study Jacobi fields on Mint. LetY(t)be the\nJacobi field along γv(t)withv∈S(Mint)such that/parenleftbig\nY(0),∇Y(0)/parenrightbig\n= Ξ∈Tv/parenleftbig\nS(Mint)/parenrightbig\n. Then we\nhave\nDπ|Ψ(v,s)◦DvΨ|(v,s)Ξ =Y(s), DvΨ|(v,s)Ξ =∇Y(s).\nSee [24, Lemma 4.3 in Page 56] for instance. Note that\nKer(Dπ|v) =/braceleftbig\nΞ∈Tv/parenleftbig\nS(Mint)/parenrightbig\n:Dπ|v(Ξ) = 0/bracerightbig\n=/braceleftbig\n(0,ζ) :ζ∈Tv/parenleftbig\nSπ(v)(Mint)/parenrightbig/bracerightbig\n≃ {(0,ζ) :ζ∈Tπ(v)(Mint),/a\\}b∇acketle{tv,ζ/a\\}b∇acket∇i}ht= 0}.\nThen we have for any Ξ = (0,ζ)∈Ker(Dπ|v)\nY(t) =d\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0expπ(v)t(v+sζ) =tDexpπ(v)/vextendsingle/vextendsingle/vextendsingle\ntvζ.\nWe now recall the Gauss lemma for the property of Jacobi fields .\nLemma 3.1 (Gauss Lemma). LetYbe a Jacobi field along a geodesic γ. Set∇Y(t) :=∇∂/∂tY(t)\nfor short. Then\n/a\\}b∇acketle{tY(t),˙γ(t)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tY(0),˙γ(0)/a\\}b∇acket∇i}ht+t/a\\}b∇acketle{t∇Y(0),˙γ(0)/a\\}b∇acket∇i}ht.\nProof. Lemma 3.1 follows from\nd2\ndt2/a\\}b∇acketle{tY(t),˙γ(t)/a\\}b∇acket∇i}ht=−/angbracketleftbig\nR/parenleftbig\nY(t),˙γ(t)/parenrightbig\n˙γ(t),˙γ(t)/parenrightbig/angbracketrightbig\n= 0,\nwhereRis the Riemann curvature tensor. See [24, Proposition 2.3 in Page 36] for the detail. /square\nThe assumption ( A1) ensures that all the Jacobi fields on Mintare of the form of the product of a\nscalar function and a parallel transport of a tangent vector . This property plays a crucial role in the\ncomputation of the composition of CXTwith some Lagrangian submanifolds. So we need to introduce\nthe parallel transport of covectors along geodesics in Mint. Letcbe a smooth curve in Mint. We denote\nbyP(t0,t1;c)vthe parallel transport of v∈Tc(t0)atc(t1)along the curve c. Suppose that t0/\\e}atio\\slash=t1and\n˜η∈T∗\nc(t1)(Mint). The we have\n˜η/parenleftbig\nP(t0,t1;c)v/parenrightbig\n=P(t0,t1;c)T˜η(v), v∈Tc(t0)(Mint).\nWe say that P(t0,t1;c)T˜η∈T∗\nc(t0)(Mint)is the parallel transport of ˜η∈Tc(t1)(Mint)atc(t0)along\nthe curvec.\nWe now recall F. Schur’s lemma for the property of sectional c urvatures.\nLemma 3.2. Let(M,g)be a Riemannian manifold, and let Kx(σ)be the sectional curvature at\nx∈Mfor a two-dimensional vector subspace σofTx(M).GEODESIC X-RAY TRANSFORM 11\n•The sectional curvature Kx(σ)is independent of the choice of σat a pointx∈Mif and only\nif\nR(X,Y)Z=k(x){/a\\}b∇acketle{tY,Z/a\\}b∇acket∇i}htX−/a\\}b∇acketle{tX,Z/a\\}b∇acket∇i}htY}\nfor anyX,Y,Z∈Tx(M), wherek(x) :=Kx(·).\n•(F. Schur’s lemma )IfKx(σ)is independent of the choice of σat every point x∈M, and\ndim(M)≧3, thenk(x) :=Kx(·)is independent of x∈M, that is,k(x)is a constant\nfunction onM.\nIf we suppose ( A1), the equation for the Jacobi field is reduced to a simpler one .\nLemma 3.3. Suppose that (M,g)is a Riemannian manifold satisfying (A1). LetY(t)be a Jacobi\nfield along a normal geodesic γ(t)with\n/a\\}b∇acketle{tY(0),˙γ(0)/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{t∇Y(0),˙γ(0)/a\\}b∇acket∇i}ht= 0.\nIf(M,g)is a space of constant curvature K, thenY(t)solves\n∇∇Y(t)+KY(t) = 0.\nIf(M,g)is a two-dimensional manifold with a sectional curvature k(x)atx∈M, thenY(t)solves\n∇∇Y(t)+k/parenleftbig\nγ(t)/parenrightbig\nY(t) = 0.\nProof. In view of Lemma 3.2, we have only to show the latter one. Apply ing Lemma 3.2 to the\nequation of the Jacobi field, we have\n∇∇Y(t) =−R/parenleftbig\nY(t),˙γ(t)/parenrightbig\n˙γ(t)\n=−k/parenleftbig\nγ(t)/parenrightbig\n{/a\\}b∇acketle{t˙γ(t),˙γ(t)/a\\}b∇acket∇i}htY(t)−/a\\}b∇acketle{tY(t),˙γ(y)/a\\}b∇acket∇i}ht˙γ(t)}\n=−k/parenleftbig\nγ(t)/parenrightbig\n{Y(t)−/a\\}b∇acketle{tY(t),˙γ(y)/a\\}b∇acket∇i}ht˙γ(t)}\nUsing Lemma 3.1 and the orthogonal condition on Y(0)and∇Y(0), we deduce that\n∇∇Y(t) =−k/parenleftbig\nγ(t)/parenrightbig\n{Y(t)−/a\\}b∇acketle{tY(0),˙γ(0)/a\\}b∇acket∇i}ht˙γ(t)−t/a\\}b∇acketle{t∇Y(0),˙γ(0)/a\\}b∇acket∇i}ht˙γ(t)}\n=−k/parenleftbig\nγ(t)/parenrightbig\nY(t).\nThis completes the proof. /square\nWe show that if ( A1) holds, then all the Jacobi fields are of the form of the produc t of a scalar\nfunction and a parallel transport of a tangent vector.\nLemma 3.4. Suppose that (M,g)is a Riemannian manifold satisfying (A1). LetY(t)be a Jacobi\nfield along a normal geodesic γ(t)with\n/a\\}b∇acketle{tY(0),˙γ(0)/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{t∇Y(0),˙γ(0)/a\\}b∇acket∇i}ht= 0.\nSet\nX(t) :=P(0,t;γ)Y(0), ζ(t) :P(0,t;γ)∇Y(0).\nIf(M,g)is a space of constant curvature K, thenY(t)is given by\nY(t) =a(t)X(t)+b(t)ζ(t),\na′′(t)+Ka(t) = 0, a(0) = 1, a′(0) = 0,\nb′′(t)+Kb(t) = 0, b(0) = 0, b′(0) = 1.\nIf(M,g)is a two-dimensional manifold with a sectional curvature k(x)atx∈M, thenY(t)is given\nby\nY(t) =a(t)X(t)+b(t)ζ(t),\na′′(t)+k/parenleftbig\nγ(t)/parenrightbig\na(t) = 0, a(0) = 1, a′(0) = 0,\nb′′(t)+k/parenleftbig\nγ(t)/parenrightbig\nb(t) = 0, b(0) = 0, b′(0) = 1.12 H. CHIHARA\nProof. We show only the latter one. The former one can be proved in the same way. Suppose that\ndim(M) = 2 . Lemma 3.3 shows that Y(t)solves∇∇Y(t) +k/parenleftbig\nγ(t)/parenrightbig\nY(t) = 0 . SetZ(t) =\na(t)X(t)+b(t)ζ(t). Note that ∇X(t) = 0 and∇ζ(t) = 0 . Then we have\nZ(0) =a(0)X(0) +b(0)ζ(0) =X(0) =P(0,0;γ)Y(0) =Y(0),\n∇Z(t) =a′(t)X(t)+a(t)∇X(t)+b′(t)ζ(t)+b(t)∇ζ(t)\n=a′(t)X(t)+b′(t)ζ(t),\n∇Z(0) =a′(0)X(0) +b′(0)ζ(0),\n=ζ(0) =P(0,0;γ)∇Y(0) =∇Y(0).\nIf we prove that Z(t)solves∇∇Z(t)+k/parenleftbig\nγ(t)/parenrightbig\nZ(t) = 0 , thenY=Zfollows from the uniqueness\nof the solution to the initial value problem for the system of linear ordinary differential equations. We\ndeduce that\n∇∇Z(t)+k/parenleftbig\nγ(t)/parenrightbig\nZ(t)\n=∇{a′(t)X(t)+b′(t)ζ(t)}+k/parenleftbig\nγ(t)/parenrightbig\n{a(t)X(t)+b(t)ζ(t)}\n=a′′(t)X(t)+a′(t)∇X(t)+b′′(t)ζ(t)+b′(t)∇ζ(t)\n+k/parenleftbig\nγ(t)/parenrightbig\n{a(t)X(t)+b(t)ζ(t)}\n=a′′(t)X(t)+b′′(t)ζ(t)+k/parenleftbig\nγ(t)/parenrightbig\n{a(t)X(t)+b(t)ζ(t)}\n={a′′(t)+k/parenleftbig\nγ(t)/parenrightbig\na(t)}X(t)+{b′′(t)+k/parenleftbig\nγ(t)/parenrightbig\nb(t)}ζ(t)\n=0.\nThis completes the proof. /square\nFinally we show the properties of solutions a(t)andb(t). These properties are essentially related\nto the assumption that there is no conjugate point in Mint.\nLemma 3.5. Suppose (A0)and(A1). Fix arbitrary w∈∂−S(M)ands,t0,˜t0∈/parenleftbig\n0,τ+(w)/parenrightbig\nwith\nt0/\\e}atio\\slash=˜t0. If(M,g)is a space of constant curvature K, we assume that a(t;s)andb(t;s)are solutions\nto\natt(t;s)+Ka(t;s) = 0, a(s;s) = 1, at(s;s) = 0, (3.1)\nbtt(t;s)+Kb(t;s) = 0, b(s;s) = 0, bt(s;s) = 1. (3.2)\nIf(M,g)is a two-dimensional manifold with a sectional curvature k(x)atx∈M, we assume that\na(t;s)andb(t;s)are solutions to\natt(t;s)+k/parenleftbig\nγw(t)/parenrightbig\na(t;s) = 0, a(s;s) = 1, at(s;s) = 0, (3.3)\nbtt(t;s)+k/parenleftbig\nγw(t)/parenrightbig\nb(t;s) = 0, b(s;s) = 0, bt(s;s) = 1. (3.4)\nIf we set∆(t0,˜t0;s) :=a(t0;s)b(˜t0;s)−a(˜t0;s)b(t0;s), then we have ∆(t0,˜t0;s)/\\e}atio\\slash= 0.\nProof. Letκbe a positive constant. We prove Lemma 3.5 for four cases: K=κ2>0,K= 0,\nK=−κ2<0, anddim(M) = 2 .\nCase ofK=κ2>0. In this case Mmust be contained in the interior of the hemisphere of the rad ius\n1/κsince there is no conjugate point in M. Hence0≦s,t0,˜t0< π/κ follows, and we have 0<\n|t0−˜t0|<π/κ . In this case the solutions a(t;s)andb(t;s)are given by\na(t;s) = cos{κ(t−s)}, b(t;s) =sin{κ(t−s)}\nκ.GEODESIC X-RAY TRANSFORM 13\nSince0<|t0−˜t0|<π/κ , we have\n∆(t0,˜t0;s) =sin{κ(˜t0−t0)}\nκ/\\e}atio\\slash= 0.\nCase ofK= 0. In this case we have a(t;s)≡1,b(t;s) =t−s, and∆(t0,˜t0;s) =˜t0−t0/\\e}atio\\slash= 0.\nCase ofK=−κ2<0. In this case we have\na(t;s) = cosh{κ(t−s)}, b(t;s) =sinh{κ(t−s)}\nκ,\n∆(t0,˜t0;s) =sinh{κ(˜t0−t0)}\nκ/\\e}atio\\slash= 0.\nCase ofdim(M) = 2 . Setζ(t) :=P(s,t;γw)ζforζ∈Tγw(s)(Mint)\\{0}. Then we have\nb(t;s)ζ(t) = (t−s)Dexpγw(s)/vextendsingle/vextendsingle/vextendsingle\n(t−s)˙γw(s)ζ.\nIft0=s, thena(t0;t0) = 1 ,b(t0;t0) = 0 , and\nb(˜t0;t0)ζ(˜t0) = (˜t0−t0)Dexpγw(t0)/vextendsingle/vextendsingle/vextendsingle\n(˜t0−t0)˙γw(t0)ζ/\\e}atio\\slash= 0\nsinceγw(˜t0)is not a conjugate point of γw(t0). Then we have b(˜t0;t0)/\\e}atio\\slash= 0, and\n∆(t0,˜t0;t0) =a(t0;t0)b(˜t0;t0)−a(˜t0;t0)b(t0;t0) =b(˜t0;t0)/\\e}atio\\slash= 0.\nIf˜t0=s, then we can prove ∆(t0,˜t0;˜t0)/\\e}atio\\slash= 0in the same way.\nSuppose that t0/\\e}atio\\slash=sand˜t0/\\e}atio\\slash=s. Then we have b(t0;s)/\\e}atio\\slash= 0,b(˜t0;s)/\\e}atio\\slash= 0, and\nb(t;s)ζ(t) = (t−s)Dexpγw(s)/vextendsingle/vextendsingle/vextendsingle\n(t−s)˙γw(s)ζ/\\e}atio\\slash= 0\nfort/\\e}atio\\slash=ssinceγw(t)is not a conjugate point of γw(s). We see∆(t0,˜t0;s)as a determinant:\n∆(t0,˜t0;s) = det/bracketleftbigg\na(t0;s)a(˜t0;s)\nb(t0;s)b(˜t0;s)/bracketrightbigg\n.\nSinceb(t0;s)/\\e}atio\\slash= 0andb(˜t0;s)/\\e}atio\\slash= 0, we have\n/bracketleftbigg\na(t0;s)\nb(t0;s)/bracketrightbigg\n/\\e}atio\\slash= 0,/bracketleftbigg\na(˜t0;s)\nb(˜t0;s)/bracketrightbigg\n/\\e}atio\\slash= 0.\nWe now suppose that ∆(t0,˜t0;s) = 0 , and obtain a contradiction. There exists a nonzero constan tα\nsuch that /bracketleftbigg\na(˜t0;s)\nb(˜t0;s)/bracketrightbigg\n=α/bracketleftbigg\na(t0;s)\nb(t0;s)/bracketrightbigg\n.\nSince[a(t0;s),b(t0;s)]T/\\e}atio\\slash= 0, there exists a constant angle θ∈[0,2π)such that\n/bracketleftbigg\ncosθ−sinθ\nsinθcosθ/bracketrightbigg/bracketleftbigg\na(t0;s)\nb(t0;s)/bracketrightbigg\n=/bracketleftbigg\nβ\n0/bracketrightbigg\nwith someβ/\\e}atio\\slash= 0. SetB(t) := sinθ·a(t;s)+cosθ·b(t;s)for short. This comes from the second row\nof the above equality. Then we have\nB′′(t)+k/parenleftbig\nγw(t)/parenrightbig\nB(t) = 0, B(t0) = 0.\nNote thatB(s) = sinθ,B′(s) = cosθandB(s)2+B′(s)2= 1/\\e}atio\\slash= 0. Hence we deduce that/parenleftbig\nB(t),B′(t)/parenrightbig\n/\\e}atio\\slash= (0,0)in view of the uniqueness of solutions to the above linear ord inary differential14 H. CHIHARA\nequation. Thus B′(t0)/\\e}atio\\slash= 0 follows from/parenleftbig\nB(t0),B′(t0)/parenrightbig\n=/parenleftbig\n0,B′(t0)/parenrightbig\n/\\e}atio\\slash= (0,0). If we setA(t) =\nB(t)/B′(t0), then\nA′′(t)+k/parenleftbig\nγw(t)/parenrightbig\nA(t) = 0, A(t0) = 0, A′(t0) = 1.\nWe remark that A(˜t0) =αA(t0) = 0 . If we set\nY(t) :=A(t)P(t0,t;γw)ζ, ζ∈Tγw(t0)(Mint)\\{0},\nthenY(t)is a Jacobi field along γwwithY(t0) = 0 ,∇Y(t0) =ζ/\\e}atio\\slash= 0andY(˜t0) = 0 . Henceγw(˜t0)\nis a conjugate point of γw(t0). This is a contradiction. Thus we conclude that ∆(t0,˜t0;s)/\\e}atio\\slash= 0. This\ncompletes the proof. /square\n4. G EOMETRY OF METAL REGION\nIn this section we study the geometry of metal region. We now a ssume ( A0), and will assume ( A1)\nlater. We now state the assumption on the metal region Dsuch thatDis contained in Mint. Suppose\nthe following.\n(A2):Dis of the form D=J/uniondisplay\nj=1DjwithDj∩Dk=∅(j/\\e}atio\\slash=k), that is,Dis a disjoint union of\nfinite number of strictly convex bounded domains Dj(j= 1,...,J ) with smooth boundary.\nMDj\nDk\nDl\nFigure 4.Mand the metal region D.\nIn this caseMint\\Dis the region of normal tissue. The streaking artifacts are c aused by\nXT/bracketleftBigg/parenleftbiggX[1D]\n|dµ|1/2/parenrightbigg2l\n|dµ|1/2/bracketrightBigg\n, l= 1,2,3,... .\nSee (5.2) for this. In this section we prepare for studying th e microlocal singularities. More precisely,\nwe study so-called the intersection calculus needed in the n ext section. It is well-known that 1Dis a\nconormal distribution section whose singular support is th e boundary∂D. So the next lemma will be\nused for characterizing the conormal singularities of X[1D].\nLemma 4.1. Letj= 1,...,J . Then the composition CX◦N∗(∂Dj)is transversal. Moreover if we\nset\nΣj:=π∂−S(M)/parenleftbig\nCX◦N∗(∂Dj)/parenrightbig\n, j= 1,...,J,\nthencodim(Σ j) = 1 andN∗(Σj) =CX◦N∗(∂Dj).\nProof. We first show that CX◦N∗(Dj)is transversal. Set\nZ=T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗(Mint)×T∗(Mint),\nD=T∗/parenleftbig\n∂−S(M)/parenrightbig\n×∆/parenleftbig\nT∗(Mint)/parenrightbigGEODESIC X-RAY TRANSFORM 15\nfor short. We have codim(D) = dim/parenleftbig\nT∗(Mint)/parenrightbig\n= 2n. In view of Theorem 2.6 and (2.5), we have\nCX×N∗(∂Dj) =/braceleftbig\n(ξ,η,ζ)∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗(Mint)×N∗(∂Dj) :\n∃v∈S(Mint)s.t.ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},\nη∈T∗\nπ(v)(Mint)\\{0},η(v) = 0,DF|T\nvξ=Dπ|T\nvη/bracerightbig\n.\nSinceCXandN∗(∂Dj)are Lagrangian submanifolds, we have\ndim/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n= dim/parenleftbig\n∂−S(M)/parenrightbig\n+dim(Mint)+dim(Mint)\n= (2n−2)+n+n= 4n−2.\nWe set\nC:=/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n∩D\n=/braceleftbig\n(ξ,η,ζ)∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗(Mint)×N∗(∂Dj) :\n∃v∈S(Mint)s.t.ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},\nη∈T∗\nπ(v)(Mint)\\{0},η(v) = 0,DF|T\nvξ=Dπ|T\nvη,η=ζ}\n=/braceleftbig\n(ξ,η,η) :∃v∈S(∂Dj)s.t.ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},\nη∈N∗\nπ(v)(∂Dj)\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n. (4.1)\nSinceξis uniquely determined by ηfor(ξ,η,η)∈C, we deduce that\ndim(C) = dim/parenleftbig\nS(∂Dj)/parenrightbig\n+dim/parenleftbig\nN∗\nπ(v)(∂Dj)/parenrightbig\n= (2n−3)+1 = 2n−2.\nThe excess of Cis computed as\ne= codim( D)+codim/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n−codim(C)\n= codim( D)−dim/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n+dim(C)\n= 2n−(4n−2)+(2n−2) = 0.\nIf we will show that Cis a clean intersection, then it follows that CX◦N∗(Dj)is transversal. Fix\narbitraryc= (ξ,η,η)∈C, and pick up arbitrary X= (X1,X2,X3)∈Tc/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n. Consider\na smooth curve on CX×N∗(∂Dj)of the form\nR∋α/mapsto→/parenleftbig\nξ(α),η(α),ζ(α)/parenrightbig\n=c+αX+O(α2)\n= (ξ+αX1,η+αX2,η+αX3)+O(α2)\nnearα= 0. Then we have\nX=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nξ(α),η(α),ζ(α)/parenrightbig\nIfX∈Tc(D), then we have\nd\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0η(α) =d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0ζ(α),\nand\nX=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nξ(α),η(α),ζ(α)/parenrightbig\n=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nξ(α),η(α),η(α)/parenrightbig\n∈Tc(C).\nHence we obtain Tc(/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n)∩Tc(D)⊂Tc(C). Therefore we conclude that\nTc(/parenleftbig\nCX×N∗(∂Dj)/parenrightbig\n)∩Tc(D) =Tc(C)\nand thatCis clean intersection. Consider the projection πC:C∋(ξ,η,η)/mapsto→ξ∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n.\nIt follows that Cis proper and π−1\nC(ξ)forξ∈πC(C)is connected since ξis uniquely determined by\nξ= (DF|T\nv)−1◦Dπ|T\nvforη∈N∗\nπ(v)(∂Dj).16 H. CHIHARA\nNext we show that N∗(Σj) =CX◦N∗(∂Dj). Using the expression (4.1) of C, we deduce that\nCX◦N∗(∂Dj) =/braceleftbig\nξ:∃v∈S(∂Dj),∃η∈N∗\nπ(v)(∂Dj)\\{0},s.t.\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n, (4.2)\nΣj:=π∂−S(M)/parenleftbig\nCX◦N∗(∂Dj)/parenrightbig\n={F(v) :v∈S(∂Dj)}.\nThen we have dim(Σ j) = dim/parenleftbig\nS(∂Dj)/parenrightbig\n= 2n−3. Therefore we deduce that codim(Σ j) = 1 in\n∂−S(M)and thatΣjis a hypersurface in ∂−S(M). SinceCXis a homogeneous canonical relation\nandN∗(∂Dj)is a conic Lagrangian submanifold, we conclude that CX◦N∗(∂Dj)is also a conic\nLagrangian submanifold and that N∗(Σj) =CX◦N∗(∂Dj). This completes the proof. /square\nNext we study the intersection Σj∩Σkforj/\\e}atio\\slash=k. Suppose that Σjk:= Σj∩Σk/\\e}atio\\slash=∅. Then we have\nΣjk={F(v) :v∈S(∂Dj),∃˜v∈S(∂Dk)s.t.F(v) =F(˜v)}\n=/braceleftbig\nw∈∂−S(M) :∃v∈S(∂Dj),∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n\\{0}s.t.\nw=F(v),˙γv(s)∈S(∂Dk)/bracerightbig\n.\nWe have the following.\nLemma 4.2. Suppose that Σjk:= Σj∩Σk/\\e}atio\\slash=∅withj/\\e}atio\\slash=k. ThenΣjkis a transversal intersection.\nProof. Fix arbitrary w∈Σjk. Then there exist v∈S(∂Dj)ands∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\nsuch that\nw=F(v)andΨ(v,s) = ˙γv(s)∈S(∂Dk). Then (2.5) implies that η(v) = 0 for anyη∈N∗\nπ(v)(∂Dj)\nand˜η/parenleftbig\n˙γv(s)/parenrightbig\n= 0for any˜η∈N∗\nγv(s)(∂Dk). We shall prove that N∗\nw(Σj)∩N∗\nw(Σk) ={0}.\nFix arbitrary ξ∈N∗\nw(Σj)and arbitrary ˜ξ∈N∗\nw(Σk). SinceN∗(Σl) =CX◦N∗(∂Dl)(l=\n1,...,J ), there exist η∈N∗\nπ(v)(∂Dj)and˜η∈N∗\nγv(s)(∂Dk)uniquely such that\nDF|T\nvξ=Dπ|T\nvη, DF|T\nΨ(v,s)˜ξ=Dπ|T\nΨ(v,s)˜η.\nSincew=F(v) =F/parenleftbig\nΨ(v,s)/parenrightbig\n, we have\nDF|vΞ =DF|Ψ(v,s)◦DvΨ|(v,s)Ξ\nfor anyΞ∈Tv/parenleftbig\nS(Mint)/parenrightbig\n. This implies that\nDF|T\nv˜ξ=DvΨ|T\n(v,s)◦Dπ|T\nΨ(v,s)˜η. (4.3)\nThen we deduce that for any Ξ∈Tv(S(Mint))\n(ξ+˜ξ)(DF|vΞ) =DF|T\nvξ(Ξ)+DF|T\nv˜ξ(Ξ)\n=Dπ|T\nvη(Ξ)+DvΨ|T\n(v,s)◦Dπ|T\nΨ(v,s)˜η(Ξ)\n=η(Dπ|vΞ)+ ˜η(Dπ|Ψ(v,s)◦DvΨ|(v,s)Ξ). (4.4)\nSinceγv(s)is not a conjugate point of π(v) =γv(0), we deduce that\n{Dπ|Ψ(v,s)◦DvΨ|(v,s)Ξ : Ξ∈Ker(Dπ|v)}={˜u∈Tγv(s)(Mint) :/a\\}b∇acketle{t˜u,˙γv(s)/a\\}b∇acket∇i}ht= 0}.\nHere we used Lemma 3.1.\nWe now suppose that ξ+˜ξ= 0. We consider (4.4). On one hand we have ˜η(˜u) = 0 for any\n˜u∈Tγv(s)(Mint)satisfying /a\\}b∇acketle{t˜u,˙γv(s)/a\\}b∇acket∇i}ht= 0. On the other hand ˜η/parenleftbig\n˙γv(s)/parenrightbig\n= 0. Then we have ˜η/parenleftbig\n˜u+\nσ˙γv(s)/parenrightbig\n= 0for anyσ∈Rand for any ˜u∈Tγv(s)(Mint)satisfying /a\\}b∇acketle{t˜u,˙γv(s)/a\\}b∇acket∇i}ht= 0. This means that\n˜η= 0inT∗\nπ/parenleftbig\n˙γv(s)/parenrightbig(Mint). Hence (4.4) becomes η(Dπ|vΞ) = 0 for anyΞ∈Tv/parenleftbig\nS(Mint)/parenrightbig\n. Therefore\nwe haveη= 0 inT∗\nπ(v)(Mint). We deduce that if ξ+˜ξ= 0, thenξ= 0 and˜ξ= 0 sinceξand˜ξ\nare uniquely determined by ηand˜ηrespectively. Thus ξ∈N∗\nw(Σj)\\{0}and˜ξ∈N∗\nw(Σk)\\{0}are\nlinearly independent, and N∗\nw(Σj)∩N∗\nw(Σk) ={0}. This completes the proof. /squareGEODESIC X-RAY TRANSFORM 17\nLemma 4.2 implies that Σjkis a submanifold of ∂−S(M)withcodim(Σ jk) = 2 , and\nN∗(Σjk) =/uniondisplay\nw∈ΣjkN∗\nw(Σj)⊕N∗\nw(Σk).\nIn other words, the proof of Lemma 4.2 shows that all the Jacob i fields along a common tangent\ngeodesicγof∂Djand∂Dkcollaborate to distinguish γ∩∂Djandγ∩∂Dkwhich are the locations of\nsources of conormal singularities. The author could not und erstand this fact for the Euclidean case.\nSee [1, Lemma 4.2] and its proof.\nWe have known that N∗\nw(Σjk)is a two-dimensional vector space for any w∈Σjk⊂∂−S(M), and\nwe need to study what CXT◦N∗(Σjk)is. We denote by νj(x)the unit outer normal vector for Djat\nx∈∂Dj, and setηj(x) :=/a\\}b∇acketle{tνj(x),·/a\\}b∇acket∇i}ht=gx(νj,·)∈N∗\nx(∂Dj)forx∈∂Dj.\nIfw∈Σjk, then there exist v∈S(∂Dj)ands∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n\\{0}such thatw=F(v)and\n˜v:= Ψ(v,s) = ˙γv(s)∈S(∂Dk). There are two cases:\n•Case C1 .P(0,s;γv)Tηk/parenleftbig\nγv(s)/parenrightbig\n=ηj/parenleftbig\nπ(v)/parenrightbig\nor−ηj/parenleftbig\nπ(v)/parenrightbig\n.\n•Case C2 .P(0,s;γv)Tηk/parenleftbig\nγv(s)/parenrightbig\n/\\e}atio\\slash=±ηj/parenleftbig\nπ(v)/parenrightbig\n.\nReplaceηand˜ηbyηj/parenleftbig\nπ(v)/parenrightbig\nandηk/parenleftbig\nγv(s)/parenrightbig\nrespectively, and see Figure 2 in Section 1. We remark\nthat whenn= 2, Case C2 never occurs. When n≧3we splitΣjkinto two parts:\nΣ(1)\njk:=/braceleftbig\nF(v) :∃v∈S(∂Dj),∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n\\{0}s.t.\nΨ(v,s) = ˙γv(s)∈S(∂Dk),\nP(0,s;γv)Tηk/parenleftbig\nγv(s)/parenrightbig\n=ηj/parenleftbig\nπ(v)/parenrightbig\nor−ηj/parenleftbig\nπ(v)/parenrightbig/bracerightbig\n,\nΣ(2)\njk:=/braceleftbig\nF(v) :∃v∈S(∂Dj),∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n\\{0}s.t.\nΨ(v,s) = ˙γv(s)∈S(∂Dk),P(0,s;γv)Tηk/parenleftbig\nγv(s)/parenrightbig\n/\\e}atio\\slash=±ηj/parenleftbig\nπ(v)/parenrightbig/bracerightbig\n.\nWe now study the strictly convexity of Djto make full use of this property. Pick up arbitrary\nx0∈∂Dj. Then there exist a neighborhood Uatx0inMintandfj∈C∞(U)such that\nU∩Dj={x∈U:fj(x)<0},\nU∩∂Dj={x∈U:fj(x) = 0},\nU\\Dj={x∈U:fj(x)>0}.\nThen for any v∈Sx0(∂Dj), we have\nfj/parenleftbig\nγv(0)/parenrightbig\n=f(x0) = 0, f j/parenleftbig\nγv(t)/parenrightbig\n>0 (t/\\e}atio\\slash= 0)\nsinceDjis strictly convex. The function fj/parenleftbig\nγv(t)/parenrightbig\nis smooth near t= 0 andfj/parenleftbig\nγv(0)/parenrightbig\n= 0 is the\nunique local minimum of fj/parenleftbig\nγv(t)/parenrightbig\n. Hence we have\nd\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0fj/parenleftbig\nγv(t)/parenrightbig\n= 0,\nwhich is\n0 =dfj/parenleftbig\n˙γv(0)/parenrightbig\n=dfj(v) =/a\\}b∇acketle{tgradfj,v/a\\}b∇acket∇i}ht=gx0(gradfj,v).\nThe Hessian ∇dfjis given by\n∇dfj(X,Y) =X/parenleftbig\ndfj(Y)/parenrightbig\n−dfj(∇XY)\n=X/parenleftbig\nY(fj)/parenrightbig\n−∇XY(fj)\n=/a\\}b∇acketle{t∇Xgradfj,Y/a\\}b∇acket∇i}ht\nfor any vector fields XandYinU. Ifγis a geodesic, then\nd2\ndt2f/parenleftbig\nγ(t)/parenrightbig\n=d\ndtdfj/parenleftbig\n˙γ(t)/parenrightbig18 H. CHIHARA\n=∇dfj/parenleftbig\n˙γ(t),˙γ(t)/parenrightbig\n+dfj/parenleftbig\n∇˙γ(t)˙γ(t)/parenrightbig\n=∇dfj/parenleftbig\n˙γ(t),˙γ(t)/parenrightbig\n.\nThen we have\nfj/parenleftbig\nγv(t)/parenrightbig\n=fj/parenleftbig\nγv(0)/parenrightbig\n+td\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0fj/parenleftbig\nγv(t)/parenrightbig\n+t2\n2d2\ndt2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=0fj/parenleftbig\nγv(t)/parenrightbig\n+O(t3)\n=t2\n2∇dfj(v,v)+O(t3)>0\nfor anyv∈Sx0(∂Dj)andt∈Rwith0<|t| ≪1. Sincex0∈U∩∂Djis arbitrary, we obtain\n∇dfj(v,v)>0forv∈S(U∩∂Dj). (4.5)\nWe may assume that gradfj/\\e}atio\\slash= 0 inUprovided that Uis sufficiently small, and we can define\nfj/|gradfj|onU, where|gradfj|2=/a\\}b∇acketle{tgradfj,gradfj/a\\}b∇acket∇i}ht. Since\ngrad/parenleftbiggfj\n|gradfj|/parenrightbigg\n=gradfj\n|gradfj|+fj·grad/parenleftbigg1\n|gradfj|/parenrightbigg\ninU, we have\ngradx/parenleftbiggfj\n|gradfj|/parenrightbigg\n=gradxfj\n|gradxfj|=νj(x), x∈U∩∂Dj.\nIf we replace fjbyfj/|gradfj|and use the same notation fj, we have\ngradxfj=νj(x), x∈U∩∂Dj.\nIn what follows we choose fj(j= 1,...,J ) satisfying this property. Then |gradfj|2is a smooth\nfunction onUand identically equal to one on U∩∂Dj. Moreover we have the following.\nLemma 4.3. Suppose that there exist a neighborhood Uatx0∈∂DjinMintand a function fj∈\nC∞(U)such that\nU∩∂Dj={x∈U:fj(x) = 0},gradxfj=νj(x) (x∈U∩∂Dj).\nIf we setGj(x) :=|gradxfj|2, then we have\nd\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0Gj/parenleftbig\nγu(s)/parenrightbig\n= 0 (4.6)\nfor anyu∈Sx(∂Dj)andx∈U∩∂Dj.\nProof. We have only to show (4.6). Fix arbitrary x∈U∩∂Djandu∈Sx(∂Dj). We use the geodesic\nnormal coordinates near the basepoint xsuch that\nϕ:U∋expx(su+hνj(x)+t1e1+...+tn−2en−2)/mapsto→(s,h,t1,...,tn−2)∈Rn,\nwhere{u,νj(x),e1,...,en−2}is an orthonormal basis of Tx(Mint). SetFj:=fj◦ϕ−1, that is,\nFj(s,h,t1,...,tn−2) :=fj/parenleftBig\nexpx/parenleftbig\nsu+hνj(x)+t1e1+...+tn−2en−2/parenrightbig/parenrightBig\n.\nThen we have\nFj(s,h,t1,...,tn−2) = 0 forexpx(su+hνj(x)+t1e1+...+tn−2en−2)∈U∩∂Dj,\n∂Fj\n∂s(0,0,0,...,0) =d\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=0fj/parenleftbig\nγu(s)/parenrightbig\n= 0,\n∂Fj\n∂h(0,0,0,...,0) =/a\\}b∇acketle{tgradfj,νj(x)/a\\}b∇acket∇i}ht= 1.GEODESIC X-RAY TRANSFORM 19\nThe implicit function theorem implies that there exists a fu nctionψ(s,t1,...,tn−1)defined in a small\nneighborhood W1at(s,t1,...,tn−1) = (0,0,...,0)inRn−1such that\nψ(0,0,...,0) = 0, ψs(0,0,...,0) =−∂Fj/∂s\n∂Fj/∂h(0,0,0,...,0) = 0,\nFj/parenleftbig\ns,ψ(s,t1,...,t1),t1,...,tn−2/parenrightbig\n= 0,\nthat is, there exists a small neighborhood U1atxinUsuch that\nU1∩∂Dj\n=/braceleftbig\nexpx/parenleftbig\nsu+ψ(s,t1,...,t1)νj(x)+t1e1+···+tn−2en−2/parenrightbig\n: (s,t1,...,t1)∈W1/bracerightbig\n.\nSet\n˜ψ(s) :=ψ(s,0,...,0),˜Gj(s,h) :=Gj/parenleftBig\nexpx/parenleftbig\nsu+hνj(x)/parenrightbig/parenrightBig\n.\nThen we have\n˜Gj/parenleftbig\ns,˜ψ(s)/parenrightbig\n=Gj/parenleftBig\nexpx/parenleftbig\nsu+ψ(s,0,...,0)νj(x)/parenrightbig/parenrightBig\n≡1\nnears= 0, and Taylor’s formula yields\n˜ψ(s) =˜ψ(0)+s˜ψ′(0)+O(s2)\n=ψ(0,0,...,0)+sψs(0,0,...,0)+O(s2)\n=O(s2),\nGj/parenleftbig\nγu(s)/parenrightbig\n=˜Gj(s,0)\n=˜Gj/parenleftbig\ns,˜ψ(s)/parenrightbig\n−˜ψ(s)/integraldisplay1\n0∂˜Gj\n∂h/parenleftbig\ns,θ˜ψ(s)/parenrightbig\ndθ\n= 1+O(s2).\nThus we can obtain (4.6) immediately. /square\nWe now study the relationship between Σ(l)\njk(l= 1,2),∂Djand∂Dkto computeCT\nX◦N∗(Σ(l)\njk).\nFix arbitrary x0∈∂Dj. Consider the hyperplane\nHj(x0) :=/braceleftbig\nγu(t) :u∈Sx0(∂Dj),t∈/parenleftbig\nτ−(u),τ+(u)/parenrightbig/bracerightbig\n.\nConsider the relationship between Hj(x0)and∂Dk. Logically the following three cases occur:\n•Case D1: There exists y0∈∂Dkuniquely such that Hj(x0) =Hk(y0). See Case B1 of\nFigure 2. In this case there exists v∈Sx0(∂Dj)ands∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\nsuch that ˙γv(s)∈\nSy0(∂Dk)andνk(y0) =±P(0,s;γv)νj(x0). See Figure 3. This case corresponds to the case\nofF(v)∈Σ(1)\njk.\n•Case D2:H(x0)∩Dk/\\e}atio\\slash=∅. Note that this implies that Hj(x0)is never a tangent hyperplane\nfor∂Dk. See Case B2 of Figure 2. In this case there exist\ny1,y2∈∂Dk, v1,v2∈Sx0(∂Dj), s1∈/parenleftbig\nτ−(v1),τ+(v1)/parenrightbig\ns2∈/parenleftbig\nτ−(v2),τ+(v2)/parenrightbig\nsuch that\ny1/\\e}atio\\slash=y2,˙γv1(s1)∈Sy1(∂Dk),˙γv2(s2)∈Sy2(∂Dk),\nνk(y1)/\\e}atio\\slash=±P(0,s1;γv1)νj(x0), νk(y2)/\\e}atio\\slash=±P(0,s2;γv2)νj(x0),\nThese cases correspond to the cases of F(v1),F(v2)∈Σ(2)\njk.\n•Case D3:Hj(x0)∩Dk=∅.20 H. CHIHARA\nCase D2 is the intermediate case between νk(y0) = +P(0,s;γv)νj(x0)andνk(y0) =−P(0,s;γv)νj(x0)\nin Case D1 from a view point of the location of x0on∂Dj. So we shall study Case D1.\nWe now introduce a subset of S(∂Dj)of the form\nM(±)\njk:=/braceleftbig\nv∈S(∂Dj) :∃s/parenleftbig\nτ−(v),τ+(v)/parenrightbig\ns.t.\n˙γv(s)∈S(∂Dk),νk/parenleftbig\nγv(s)/parenrightbig\n=±P(0,s;γv)νj/parenleftbig\nπ(v)/parenrightbig/bracerightbig\n.\nSinceDjandDkare strictly convex, we dudeuce that for any x0in Case D1, v∈Sx0(∂Dj),s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n,y0=γv(s), and the signature +or−are determined uniquely. When n= 2 and\nΣjk/\\e}atio\\slash=∅,M(±)\njkbecome sets of two elements respectively. Furthermore the p roperties of M(±)\njkfor\nn≧3are the following.\nLemma 4.4. Suppose (A0),(A1)and(A2). Whenn≧3andΣ(1)\njk/\\e}atio\\slash=∅,M(±)\njkare connected (n−2)-\ndimensional submanifolds of S(∂Dj).\nProof. Supposen≧3andΣ(1)\njk/\\e}atio\\slash=∅. We shall show that M(+)\njkis an(n−2)-dimensional submanifold\nofS(∂Dj). Similarly we can show that so is M(−)\njk, and we omit the proof of this. We believe that the\nconnectivity of M(+)\njkis intuitively obvious since the strictly convexity of ∂Djand∂Dk. So we omit\nthe proof of this also. Fix arbitrary v0∈ M(+)\njk. Then there exists s0∈/parenleftbig\nτ−(v0),τ+(v0)/parenrightbig\nuniquely\nsuch that\nu0:= ˙γv0(s0) = Ψ(v0,s0)∈S(∂Dk), νk/parenleftbig\nπ(u0)/parenrightbig\n=P(0,s0,γv0)νj/parenleftbig\nπ(v0)/parenrightbig\n.\nSetx0:=π(v0)andy0:=π(u0)for short. Pick up a neighborhood Ujatx0inMint, a neighborhood\nUkaty0inMint, a smooth function fj∈C∞(Uj), and a smooth function fk∈C∞(Uk)such that\n∂Dj∩Uj={x∈Uj:fj(x) = 0}, νj(x) = gradxfjforx∈∂Dj∩Uj,\n∂Dk∩Uk={y∈Uk:fk(y) = 0}, νk(y) = gradyfkfory∈∂Dk∩Uk.\nThen we have\nS(∂Dj∩Uj) =/braceleftbig\nv∈S(Uj) :fj/parenleftbig\nπ(v)/parenrightbig\n= 0,dfj(v) = 0/bracerightbig\n,\nS(∂Dk∩Uk) =/braceleftbig\nu∈S(Uk) :fk/parenleftbig\nπ(u)/parenrightbig\n= 0,dfk(u) = 0/bracerightbig\n.\nForv∈S(∂Dj∩Uj)nearv0and fors∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\nnears0, we set\nJ(s,v) =\nfk/parenleftbig\nγv(s)/parenrightbig\ndfk/parenleftbig\n˙γv(s)/parenrightbig\ngradγv(s)fk−P(0,s;γv)gradπ(v)fj\n\n=\nfk/parenleftBig\nπ/parenleftbig\nΨ(v,s)/parenrightbig/parenrightBig\ndfk/parenleftbig\nΨ(v,s)/parenrightbig\ngradπ/parenleftbig\nΨ(v,s)/parenrightbigfk−P(0,s;γv)gradπ(v)fj\n.\nThe function J(s,v)takes values in R2×Tπ/parenleftbig\nΨ(v,s)/parenrightbig(Mint), andM(+)\njkis characterized by J(s,v) = 0\nnear(s0,v0). Indeed ifv∈ M(+)\njk, then there exists s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\nuniquely such that J(s,v) =\n0. Conversely if J(s,v) = 0 , thenvsatisfies the condition for v∈ M(+)\njkwiths. Hence it suffices\nto show that rank/parenleftbig\nDJ|(s0,v0)/parenrightbig\n=nto prove that M(+)\njkis an(n−2)-dimensional submanifold of\nS(∂Dj∩Uj)since(s,v)moves in the (2n−2)-dimensional manifold R×S(∂Dj)andsis uniquely\ndetermined by v.GEODESIC X-RAY TRANSFORM 21\nTo compute the differential DJ|(s0,v0), we now introduce a coordinate system in Tv0/parenleftbig\nS(∂Dj)/parenrightbig\n.\nRoughly speaking, we can see this as\nTv0/parenleftbig\nS(∂Dj)/parenrightbig\n=Tx0(∂Dj)×Tv0/parenleftbig\nSx0(∂Dj)/parenrightbig\n≃Tx0(∂Dj)×{ζ∈Tx0(∂Dj) :/a\\}b∇acketle{tζ,v0/a\\}b∇acket∇i}ht= 0}.\nPick up an orthonormal basis {v0,v1,...,vn−2}ofTx0(∂Dj), and an orthonormal basis {ζ1,...,ζn−2}\nofTv0/parenleftbig\nSx0(∂Dj)/parenrightbig\n. Then\n{(v0,0),(v1,0),...,(vn−2,0),(0,ζ1),...,(0,ζn−2)}\nbecomes an orthonormal basis of Tv0/parenleftbig\nS(∂Dj)/parenrightbig\n. Note that /a\\}b∇acketle{tζj,v0/a\\}b∇acket∇i}ht= 0 forj= 1,...,n−2. Fix\narbitrary(X,ζ)∈Tv0/parenleftbig\nS(∂Dj)/parenrightbig\n. We consider a curve v(σ)onS(∂Dj)of the form\nR∋σ/mapsto→v(σ) :=v0+σ(X,ζ)+O(σ2)\nnearσ= 0. LetY(s)be the Jacobi field along γv0with/parenleftbig\nY(0),∇Y(0)/parenrightbig\n= (X,ζ). Then we have\n/parenleftbig\nY(s),∇Y(s)/parenrightbig\n=d\ndσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nσ=0/parenleftBig\nπ/parenleftbig\nΨ(v(σ),s)/parenrightbig\n,Ψ(v(σ),s)/parenrightBig\n=/parenleftBig\nDπ|Ψ(v0,s)◦DvΨ|(v0,s)(X,ζ),DvΨ|(v0,s)(X,ζ),/parenrightBig\n.\nThe assumption ( A1) ensures that\nY(s) =a(s)P(0,s;γv0)X,∇Y(s) =b(s)P(0,s;γv0)ζ,\nwherea(s)andb(s)are solutions to\na′′(s)+k/parenleftbig\nγv0(s)/parenrightbig\na(s) = 0, a(0) = 1, a′(0) = 0,\nb′′(s)+k/parenleftbig\nγv0(s)/parenrightbig\nb(s) = 0, b(0) = 0, b′(0) = 1.\nHence we have Y(s),∇Y(s)∈Tγv0(s)/parenleftbig\nHj(x0)/parenrightbig\n, in particular, Y(s0),∇Y(s0)∈Ty0(∂Dk). We now\nintroduce Jacobi fields along γv0by\nYp(s) =Dπ|Ψ(v0,s)◦DvΨ|(v0,s)(vp,0), p= 0,1,...,n−2,\nZq(s) =Dπ|Ψ(v0,s)◦DvΨ|(v0,s)(0,ζq), q= 1,...,n−2.\nObviously we have Y0(s) = ˙γv0(s)andZq(0) = 0 forq= 1,...,n−2. Sincey0=γv0(s0)is not a\nconjugate point of x0,\n{u0,Y1(s0),...,Y n−2(s0),Z1(s0),...,Z n−2(s0)}\nbecomes a basis of Tu0/parenleftbig\nS(∂Dk)/parenrightbig\n.\nWe now start to compute DJ|(s0,v0). LetAjbe the shape operator for ∂Djatx0, and letAkbe the\nshape operator for ∂Dkaty0. It follows that rank(Aj) =n−1andrank(Ak) =n−1sinceDjand\nDkare strictly convex respectively.\nFirst we compute ∂J/∂s . LetΓp\nqr(y)be the Christoffel symbol of (M,g)inUk. SinceP(0,s;γv0)\nis the operator giving the parallel transport along γv0, we deduce that\n∂J\n∂s(s0,v0) =∂\n∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=s0\nfk/parenleftBig\nπ/parenleftbig\nΨ(v0,s)/parenrightbig/parenrightBig\ndfk/parenleftbig\nΨ(v0,s)/parenrightbig\ngradπ/parenleftbig\nΨ(v0,s)/parenrightbigfk−P(0,s;γv0)gradx0fj\n\n=∂\n∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ns=s0\nfk/parenleftBig\nπ/parenleftbig\nΨ(v0,s)/parenrightbig/parenrightBig\ndfk/parenleftbig\nΨ(v0,s)/parenrightbig\ngradπ/parenleftbig\nΨ(u0,s−s0)/parenrightbigfk−P(0,s−s0;γu0)grady0fk\n22 H. CHIHARA\n=\ndfk/parenleftbig\nΨ(v0,s0)/parenrightbig\n∇dfk(u0,u0)\n∇u0gradfk−/parenleftBig\nΓp\nqr(y0)·(u0)q·(grady0fk)r/parenrightBig\n+/parenleftBig\nΓp\nqr(y0)·(u0)q·(grady0fk)r/parenrightBig\n\n=\ndfk/parenleftbig\nΨ(v0,s0)/parenrightbig\n∇dfk(u0,u0)\n∇u0gradfk\n\n=\n0\n∇dfk(u0,u0)\nAku0+/a\\}b∇acketle{t∇u0gradfk,gradfk/a\\}b∇acket∇i}htgrady0fk\n.\nLemma 4.3 shows that /a\\}b∇acketle{t∇u0gradfk,gradfk/a\\}b∇acket∇i}ht= 0, and we obtain\n∂J\n∂s(s0,v0) =\n0\n∇dfk(u0,u0)\nAku0\n. (4.7)\nSecondly we compute the differentiation of J(s,v)with respect to the variables of Tv0/parenleftbig\nS(∂Dj)/parenrightbig\n.\nWe consider curves on S(∂Dj)of the form\nv(αp) =v0+αpvp+O(α2\np), p= 0,1,...,n−2,\nv(βq) =v0+βqζq+O(β2\nq), q= 1,...,n−2.\nIn other words we take local coordinates (α0,α1,...,α n−2,β1,...,β n−2)ofTv0/parenleftbig\nS(∂Dj)/parenrightbig\nnear the\nzero section at v0∈S(∂Dj). Furthermore the following two mappings\nRn∋(h,α0,α1,...,α n−2)\n/mapsto→expx0/parenleftbig\nhνj(x0)+α0v0+α1v1+···+αn−2vn−2/parenrightbig\n∈Uj,\nRn∋(h,α0,β1,...,β n−2)\n/mapsto→expx0/parenleftbig\nhνj(x0)+α0v0+β1ζ1+···+βn−2ζn−2/parenrightbig\n∈Uj,\ngive geodesic local coordinate systems in Ujrespectively. Hence if we use these coordinates, we may\nassume that all the Christoffel symbols of (M,g)vanish atx0, and differentiation of vector fields with\nrespect to these variables at x0become covariant derivatives with respect to the correspon ding vector\nfields. See [24, Exercise 4 in Page 37 and its Solution in Page 3 26] for instance. We shall compute\n∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0J/parenleftbig\ns0,v(αp)/parenrightbig\n=∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0J/parenleftbig\ns0,v0+αp(vp,0)/parenrightbig\n,\n∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0J/parenleftbig\ns0,v(βq)/parenrightbig\n=∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0J/parenleftbig\ns0,v0+βq(0,ζq)/parenrightbig\n.\nThus we shall actually compute\n∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0J/parenleftbig\ns0,v0+αp(vp,0)/parenrightbig\n,\n=∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0\nfk/parenleftBig\nπ/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbig/parenrightBig\ndfk/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbig\ngradπ/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbigfk−P(0,s0;γv0+αp(vp,0))gradπ(v0+αp(vp,0))fj\n,\n∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0J/parenleftbig\ns0,v0+βq(0,ζq)/parenrightbigGEODESIC X-RAY TRANSFORM 23\n=∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0\nfk/parenleftBig\nπ/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbig/parenrightBig\ndfk/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbig\ngradπ/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbigfk−P(0,s0;γv0+βq(0,ζq))gradπ(v0+βq(0,ζq))fj\n\nforp= 0,1,...,n−2andq= 1,...,n−2.\nWe compute the first row. Since u0,Yp(s0),Zq(s0)∈Ty0(∂Dk), we deduce that for p,q=\n1,...,n−2\n∂\n∂α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα0=0fk/parenleftBig\nπ/parenleftbig\nΨ(v0+α0(v0,0),s0)/parenrightbig/parenrightBig\n=dfk/parenleftbig\nDπ|Ψ(v0,s0)◦DvΨ|v0,s0(v0,0)/parenrightbig\n=dfk/parenleftbig\nΨ(v0,s0)/parenrightbig\n=dfk(u0) = 0, (4.8)\n∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0fk/parenleftBig\nπ/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbig/parenrightBig\n=dfk/parenleftbig\nDπ|Ψ(v0,s0)◦DvΨ|v0,s0(vp,0)/parenrightbig\n=dfk/parenleftbig\nYp(s0)/parenrightbig\n= 0, (4.9)\n∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0fk/parenleftBig\nπ/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbig/parenrightBig\n=dfk/parenleftbig\nDπ|Ψ(v0,s0)◦DvΨ|v0,s0(0,ζq)/parenrightbig\n=dfk/parenleftbig\nZq(s0)/parenrightbig\n= 0. (4.10)\nWe compute the second row. Since X/parenleftbig\ndfk(Y)/parenrightbig\n=∇dfk(X,Y) +dfk(∇XY)for vector fields X\nandYonUk, we deduce for p,q= 1,...,n−2\n∂\n∂α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα0=0dfk/parenleftbig\nΨ(v0+α0(v0,0),s0)/parenrightbig\n=∇dfk/parenleftbig\nDvΨ|(v0,s0)(v0,0),u0/parenrightbig\n+dfk/parenleftbig\n∇DvΨ|(v0,s0)(v0,0)u0/parenrightbig\n=∇dfk(u0,u0), (4.11)\n∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0dfk/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbig\n=∇dfk/parenleftbig\nDvΨ|(v0,s0)(vp,0),u0/parenrightbig\n+dfk/parenleftbig\n∇DvΨ|(v0,s0)(vp,0)u0/parenrightbig\n=∇dfk/parenleftbig\n∇u0Yp(s0),u0/parenrightbig\n+dfk/parenleftbig\n∇∇u0Yp(s0)u0/parenrightbig\n, (4.12)\n∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0dfk/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbig\n=∇dfk/parenleftbig\nDvΨ|(v0,s0)(0,ζq,),u0/parenrightbig\n+dfk/parenleftbig\n∇DvΨ|(v0,s0)(0,ζq)u0/parenrightbig\n=∇dfk/parenleftbig\n∇u0Zq(s0),u0/parenrightbig\n+dfk/parenleftbig\n∇∇u0Zq(s0)u0/parenrightbig\n. (4.13)\nWe compute the third row. We remark again that the derivative s of vector fields at x0become the\ncorresponding covariant derivatives. Note that Y0(0) =v0andY0(s0) =u0. For theαpvariable\n(p= 0,1,...,n−2), we deduce that\n∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0/braceleftBig\ngradπ/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbigfk\n−P(0,s0;γv0+αp(vp,0))gradπ/parenleftbig\nv0+αp(vp,0)/parenrightbigfj/bracerightBig24 H. CHIHARA\n=∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0/braceleftbig\ngradπ/parenleftbig\nΨ(v0+αp(vp,0),s0)/parenrightbigfk\n−P(0,s0;γv0+αp(vp,0))gradπ(v0)fj/bracerightbig\n−P(0,s0;γv0)∂\n∂αp/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαp=0gradπ/parenleftbig\nv0+αp(vp,0)/parenrightbigfj\n=Yp(s0)/parenleftbig\ngradfk−gradfk/parenrightbig\n−P(0,s0;γv0)Yp(0)/parenleftbig\ngradfj/parenrightbig\n=−P(0,s0;γv0)∇Yp(0)/parenleftbig\ngradfj/parenrightbig\n=−P(0,s0;γv0)/braceleftbig\nAjYp(0)+/a\\}b∇acketle{t∇Yp(0)gradfj,gradfj/a\\}b∇acket∇i}htgradfj/bracerightbig\n=−P(0,s0;γv0)AjYp(0)\n=−P(0,s0;γv0)Ajvp. (4.14)\nHere we used Lemma 4.3. In the same way we have\n∂\n∂βq/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβq=0/braceleftBig\ngradπ/parenleftbig\nΨ(v0+βq(0,ζq),s0)/parenrightbigfk\n−P(0,s0;γv0+βq(0,ζq))gradπ(v0+βq(0,ζq))fj/bracerightBig\n=−P(0,s0;γv0)AjZq(0) = 0 (4.15)\nsinceZq(0) = 0 .\nCombining (4.7), (4.8), (4.9), (4.10), (4.11), (4.12), (4. 13), (4.14) and (4.15), we obtain\nDJ|(s0,v0)=/bracketleftbigg\n0 0\nJ1R1/bracketrightbigg\n,\nJ1=/bracketleftbigg∇dfk(u0,u0)/bracketleftbig\n∇dfk(u0,u0),M1/bracketrightbig\nAku0/bracketleftbig\n−P(0,s0;γv0)Ajvp/bracketrightbig\np=0,...,n−2/bracketrightbigg\n,\nM1= [m1,...,m n−2],\nmp:=∇dfk/parenleftbig\n∇u0Yp(s0),u0/parenrightbig\n+dfk/parenleftbig\n∇∇u0Yp(s0)u0/parenrightbig\n,\nR1=/bracketleftbigg\nr1···rn−2\n0···0/bracketrightbigg\n,\nrq:=∇dfk/parenleftbig\n∇u0Zq(s0),u0/parenrightbig\n+dfk/parenleftbig\n∇∇u0Zq(s0)u0/parenrightbig\n.\nWe shall prove that rank([J1,R1]) =n. More precisely we show that rank([J1,R1]) =nif\nR1/\\e}atio\\slash=O2×(n−2), and we show that rank(J1) =nifR1=O2×(n−2), whereOp×qdenotes thep×q\nzero matrix. We now recall that (4.5) shows ∇dfk(u0,u0)>0. We setwp=−P(0,s0;γv0)Ajvp,\np= 0,1,...,n−2for short. It follows that {w0,w1,...,w n−2}forms a basis of Ty0(∂Dk)since\n{v0,v1,...,vn−2}forms a basis of Tx0(∂Dj),rank(Aj) =n−1,P(0,s0;γv0)is the parallel trans-\nport of tangent vectors along γv0, and˙γv0(s0)∈S(∂Dk). We have\nJ1=/bracketleftbigg\n∇dfk(u0,u0)∇dfk(u0,u0)m1···mn−2\nAku0 w0w1···wn−2/bracketrightbigg\n.\nThe(2,1)-elementAku0belongs toTy0(∂Dk), and is given by a linear combination of w0,w1,...,w n−1.\nIn particular rank(J1)≧n−1sincew0,w1,...,w n−2are linearly independent.\nCase ofR1/\\e}atio\\slash=O2×(n−2). Suppose that R1/\\e}atio\\slash=O2×(n−2). Then there exists a (2n−2)×(2n−2)regular\nmatrixQ1such that [J1,R1]can be reduced to\n[J1,R1]Q1= [J2,O2×(n−3)],GEODESIC X-RAY TRANSFORM 25\nJ2=/bracketleftbigg\n∇dfk(u0,u0)∇dfk(u0,u0)m1···mn−21\nAku0 w0w1···wn−20/bracketrightbigg\n.\nIt suffices to show that rank(J2) =n. We can eliminate the first row using the (1,n+ 1) -element.\nThere exists an (n+1)×(n+1) regular matrix Q2such that\nJ2Q2=/bracketleftbigg\n0 0 0 ···0 1\nAku0w0w1···wn−20/bracketrightbigg\n.\nSinceAku0can be expressed by a linear combination of w0,w1,...,w n−2, there exists an (n+1)×\n(n+1) regular matrix Q3such that\nJ2Q2Q3=/bracketleftbigg\n0 0 0 ···0 1\n0w0w1···wn−20/bracketrightbigg\n,\nwhich shows that rank(J2) =n.\nCase ofR1=O2×(n−2). Suppose that R1=O2×(n−2). We show rank(J1) =n, that is, all the\ncolumns ofJ1are linearly independent. To show rank(J1) =n, we suppose rank(J1) =n−1and\nobtain a contradiction. We now assume that rank(J1) =n−1. Then we have rank(DJ|(s0,v0)) =\nn−1, anddim/parenleftbig\nRan(DJ|(s0,v0))/parenrightbig\n=n−1. This means that the property v0∈ M(+)\njkis stable under\nthe infinitesimal displacement of x0=π(v0)in thev0direction since dim(∂Dj) =n−1andv0\nis uniquely determined by x0. This contradicts the strictly convexity of ∂Djand∂Dk. Indeed if we\nconsider a curve on ∂Djof the formx(α) =x0+αv0+O(α2)nearα= 0, we haveSx(α)(∂Dj)∩\nM(±)\njk=∅forα/\\e}atio\\slash= 0 since the strictly convexity of ∂Djand∂Dk. This completes the proof of\nLemma 4.4. /square\nSince the property v∈ M(±)\njkis uniquely determined by x∈∂Dj, the restriction of the projec-\ntionπ:S(Mint)→MintonM(±)\njkare injective respectively. Then we can deduce the followin g\nimmediately.\nCorollary 4.5. Suppose (A0),(A1)and(A2). If we set B(±)\njk:={π(v) :v∈ M(±)\njk}, thenB(±)\njkare\n(n−2)-dimensional connected submanifolds of ∂Dj. Moreover, if we set\nL(±)\njk=/braceleftbig\nγv(s) :v∈ M(±)\njk,s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig/bracerightbig\n,\nthenL(±)\njkare hypersurfaces which are tangent to ∂DjonB(±)\njkand∂DkonB(±)\nkjrespectively. See\nFigure 3.\nWe now state some elementary facts which follows immediatel y, and introduce some notation used\nlater.\n•It follows that L(±)\njk=L(±)\nkj. We set\nLjk:=L(+)\njk∪L(−)\njk,L:=/uniondisplay\n1≦j<k≦JLjk.\n•We have\nΣ(1)\njk= Σ(1)\nkj=/braceleftbig\nF(v) :v∈ M(+)\njk∪M(−)\njk/bracerightbig\n=/braceleftbig\nF(˜v) : ˜v∈ M(+)\nkj∪M(−)\nkj/bracerightbig\n.\nIf we set\n∆(Mjk) := ∆(M(+)\njk)∪∆(M(−)\njk),\n∆(M(±)\njk) :=/braceleftbig\n(v,˜v)∈ M(±)\njk×M(±)\nkj:∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\ns.t.\n˜v= ˙γv(s),P(0,s;γv)ηk/parenleftbig\nπ(˜v)/parenrightbig\n=±ηj/parenleftbig\nπ(v)/parenrightbig/bracerightbig\n.26 H. CHIHARA\nthen we have\nN∗(Σ(1)\njk) =/braceleftbig\nξ+˜ξ:ξ,˜ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n,\n∃(v,˜v)∈∆(Mjk),∃η∈N∗\nπ(v)(∂Dj),∃˜η∈N∗\nπ(˜v)(∂Dk)s.t.\nDF|T\nvξ=Dπ|T\nvη,DF|T\n˜v˜ξ=Dπ|T\n˜v˜η/bracerightbig\n. (4.16)\n•LetΩjkbe a connected subdomain of ∂Djenclosed by B(+)\njk∪B(−)\njk. Fix arbitrary x∈Ωjk.\nSee Figure 5 below. It follows that Hj(x)∩Dk/\\e}atio\\slash=∅. Furthermore there exist vi∈Sx(∂Dj)\nandsi∈/parenleftbig\nτ−(vi),τ+(vi)/parenrightbig\nwithi= 1,2such thatv1/\\e}atio\\slash=v2and˙γvi(si)∈S(∂Dk). Clearly\nHj(x)is not tangent to ∂Dkatγvi(si), andP(0,si;γvi)νj(x)/\\e}atio\\slash=±νk/parenleftbig\nγvi(si)/parenrightbig\nfori= 1,2.\nHence we have\nΣ(2)\njk= Σ(2)\nkj\n=/braceleftbig\nF(v) :∃v∈S(Ωjk),∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\ns.t.˙γv(s)∈S(∂Dk)/bracerightbig\n=/braceleftbig\nF(˜v) :∃˜v∈S(Ωkj),∃s∈/parenleftbig\nτ−(˜v),τ+(˜v)/parenrightbig\ns.t.˙γ˜v(s)∈S(∂Dj)/bracerightbig\n.\nIf we set\n∆/parenleftbig\nS(Ωjk)/parenrightbig\n:=/braceleftbig\n(v,˜v)∈S(Ωjk)×S(Ωkj) :∃s∈/parenleftbig\nτ(v)−,τ+(v)/parenrightbig\ns.t.\n˜v= ˙γv(s),P(0,s;γv)νj/parenleftbig\nπ(v)/parenrightbig\n/\\e}atio\\slash=±νk/parenleftbig\nπ(˜v)/parenrightbig/bracerightbig\n,\nthen we have\nN∗(Σ(2)\njk) =/braceleftbig\nξ+˜ξ:ξ,˜ξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n,\n∃(v,˜v)∈∆/parenleftbig\nS(Ωjk)/parenrightbig\n,∃η∈N∗\nπ(v)(∂Dj),∃˜η∈N∗\nπ(˜v)(∂Dk)s.t.\nDF|T\nvξ=Dπ|T\nvη,DF|T\n˜v˜ξ=Dπ|T\n˜v˜η/bracerightbig\n. (4.17)\nFigure 5. The situation of the neighborhood of ∂Djregarding the relationship with ∂Dk.\n5. M AINTHEOREM\nLetE≧0be the energy level of the X-ray, and let E0be a positive constant describing the energy\nlevel for normal tissue. We study the very simple model of the beam hardening effect originated\nby [20]. We assume the following.\n(A3): The half density fE(x)of the distribution of the attenuation coefficient is of the f orm\nfE(x) =fE0(x)+α(E−E0)1D(x), x∈Mint,\nwhere the half density fE0describes the distribution of the attenuation coefficient o f normal\ntissue, andα >0is a constant. The spectral function ρ(E), which is a probability densityGEODESIC X-RAY TRANSFORM 27\nfunction on [0,∞), is of the form\nρ(E) =1\n2ε1[E0−ε,E0+ε](E), E∈[0,∞),\nwhereε∈(0,E0)is a constant.\nIn this case the measurements Pbecome\nP(w) :=−log/braceleftbigg/integraldisplay∞\n0ρ(E)exp/parenleftbigg\n−X[fE]\n|dµ|1/2(w)/parenrightbigg\ndE/bracerightbigg\n|dµ|1/2\n=X[fE0](w)+∞/summationdisplay\nl=1Bl/parenleftbigg\nαεX[1D]\n|dµ|1/2(w)/parenrightbigg2l\n|dµ|1/2, w∈∂−S(M)\nwith some sequence {Bl}∞\nl=1of real numbers. Let Qbe a local parametrix for XT◦X. We set\nPMA:=P−X[fE0] =∞/summationdisplay\nl=1Bl/parenleftbigg\nαεX[1D]\n|dµ|1/2/parenrightbigg2l\n|dµ|1/2,\nfCT:=Q◦XT[P]≡fE0+fMA modC∞(Ω1/2\nMint),\nfMA:=Q◦XT[PMA] =∞/summationdisplay\nl=1Bl(αε)2lQ◦XT/bracketleftBigg/parenleftbiggX[1D]\n|dµ|1/2/parenrightbigg2l\n|dµ|1/2/bracketrightBigg\n.\nWe now state our main theorem of the present paper.\nTheorem 5.1. Suppose (A0),(A1),(A2)and(A3). Then we have\nfMA∈I−1/2−3n/4/parenleftbig\nN∗(L)\\0/parenrightbig\n(5.1)\naway fromN∗(∂D).\nThe proof of Theorem 5.1 is the investigation of the interact ion of singularities in\nQ◦XT/bracketleftBigg/parenleftbiggX[1D]\n|dµ|1/2/parenrightbigg2l\n|dµ|1/2/bracketrightBigg\n, l= 1,2,3,... . (5.2)\nFor this purpose we study the intersection calculus of CXT◦N∗(Σj)andCXT◦N∗(Σjk). In other\nwords, we need to know what kind of conormal singularities in N∗(Σj)orN∗(Σjk)can be com-\nposable with the canonical relation CXT, which is determined by the image of the adjoints of the\ndifferentials DF andDπ. Firstly we need to know the basic facts on the relationship b etween these\ntwo adjoints, which was proved by Holman and Uhlmann in [11].\nLemma 5.2 (Holman and Uhlmann [11, Lemma 5]). We assume (A0)without the simplicity condi-\ntion, that is, conjugate points are allowed for (M,g). Suppose that\nv,˜v∈S(Mint), F(v) =F(˜v), ξ∈T∗\nF(v)(∂−S(M)),\nη∈T∗\nπ(v)(Mint),˜η∈T∗\nπ(˜v)(Mint),\nDF|T\nvξ=Dπ|T\nvη, DF|T\n˜vξ=Dπ|T\n˜v˜η.\nIfv/\\e}atio\\slash= ˜v, thenvand˜vare conjugate.\nIn the same notation of Lemma 5.2, (2.5) shows that η(v) = 0 and˜η(˜v) = 0 hold. If we assume\nthe simplicity condition on (M,g), then Lemma 5.2 shows that for any ξ∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n\\0with\nw:=π∂−S(M)(ξ), one of the following occurs:\n•There exists∈/parenleftbig\n0,τ+(w)/parenrightbig\nandη∈T∗\nγw(s)(Mint)uniquely such that\nη/parenleftbig\n˙γw(s)/parenrightbig\n= 0, DF|T\n˙γw(s)ξ=Dπ|T\n˙γw(s)η.28 H. CHIHARA\n•For anys∈/parenleftbig\n0,τ+(w)/parenrightbig\nandη∈T∗\nγw(s)(Mint)satisfyingη/parenleftbig\n˙γw(s)/parenrightbig\n= 0, we have\nDF|T\n˙γw(s)ξ/\\e}atio\\slash=Dπ|T\n˙γw(s)η.\nRoughly speaking, if we assume the simplicity condition for (M,g), then for any ξ∈T∗/parenleftbig\n∂−S(M)/parenrightbig\n\\\n0, there exists only one covector η∈T∗(Mint)\\0, if any, that satisfies the identity for ξandη.\nIn this paper we need to study the propagation of singulariti es caused by η,˜η∈N∗(∂D)\\0with\nπ(η)/\\e}atio\\slash=π(˜η). In other words we investigate the existence or the nonexist ence of pairs of curves\nξ(s)∈N∗(Σjk)andη(s)∈T∗(Mint)satisfying\nDF|T\n˙γw(s)ξ(s) =Dπ|T\n˙γw(s)η(s).\nWe now set the notation. Suppose ( A0), (A1) and ( A2). Fix arbitrary w∈Σjkwithj/\\e}atio\\slash=k. Let\ns,t0,˜t0∈/parenleftbig\n0,τ+(w)/parenrightbig\nsuch that\nt0/\\e}atio\\slash=˜t0,˙γw(t0)∈S(∂Dj),˙γw(˜t0)∈S(∂Dk).\nLeta(t;s)andb(t;s)be solutions to (3.1) and (3.2) respectively, or (3.3) and (3 .4) respectively. Then\nLemma 3.5 ensures that\n∆(t0,˜t0;s) :=a(t0;s)b(˜t0;s)−a(˜t0;s)b(t0;s)/\\e}atio\\slash= 0.\nSet\nη:=/angbracketleftbig\nνj/parenleftbig\nγw(t0)/parenrightbig\n,·/angbracketrightbig\n∈N∗\nγw(t0)(∂Dj),\n˜η:=/angbracketleftbig\nνk/parenleftbig\nγw(˜t0)/parenrightbig\n,·/angbracketrightbig\n∈N∗\nγw(˜t0)(∂Dk),\nη1:=P(t0,˜t0;γw)T˜η∈T∗\nγw(t0)(Mint),\nη(s) :=P(s,t0;γw)Tη∈T∗\nγw(s)(Mint),\nη1(s) :=P(s,t0;γw)Tη1=P(s,˜t0;γw)T˜η∈T∗\nγw(s)(Mint).\nLetξand˜ξsatisfy\nξ,˜ξ∈T∗\nw/parenleftbig\n∂−S(M)/parenrightbig\n, DF|T\n˙γw(t0)ξ=Dπ|T\n˙γw(t0)η, DF|T\n˙γw(˜t0)˜ξ=Dπ|T\n˙γw(t0)˜η.\nThe two covectors ξand˜ξare uniquely determined by ηand˜ηrespectively. Then we have the fol-\nlowing lemma regarding the existence or the nonexistence of pair of curves satisfying the desirable\nconditions.\nLemma 5.3. Suppose (A0),(A1)and(A2).\nI. Suppose that η1=±η. If we set\nξ(s) :=b(˜t0;s)\n∆(t0,˜t0;s)ξ∓b(t0;s)\n∆(t0,˜t0;s)˜ξ, s∈/parenleftbig\n0,τ+(w)/parenrightbig\n,\nthen\nDF|T\n˙γw(s)ξ(s) =Dπ|T\n˙γw(s)η(s), s∈/parenleftbig\n0,τ+(w)/parenrightbig\n. (5.3)\nMoreover we have for any s∈/parenleftbig\n0,τ+(w)/parenrightbig\nDF|T\n˙γw(s)/parenleftbig\nN∗\nw(Σjk)/parenrightbig\n∩Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n={αDπ|T\n˙γw(s)η(s) :α∈R}, (5.4)\nwhere\nT∗(s) =/braceleftbig\nζ∈T∗\nγw(s)(Mint) :ζ/parenleftbig\n˙γw(s)/parenrightbig\n= 0/bracerightbig\n.\nII. Suppose that η1/\\e}atio\\slash=±η. Then for any fixed s∈/parenleftbig\n0,τ+(w)/parenrightbig\n\\{t0,˜t0},\nDF|T\n˙γw(s)/parenleftbig\nN∗\nw(Σjk)/parenrightbig\n∩Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n={0}.GEODESIC X-RAY TRANSFORM 29\nWe remark that the condition η1=±ηis equivalent to w∈Σ(1)\njkand to˙γw(t0)∈ M(±)\njkrespec-\ntively, and that the condition η1/\\e}atio\\slash=±ηis equivalent to w∈Σ(2)\njkand to˙γw(t0)∈S(Ωjk)respectively.\nWe show the figures of the curves ξ(s)inN∗\nw(Σ)andη(s)in/uniontext\ns∈/parenleftbig\n0,τ+(w)/parenrightbigTγw(s)(Mint), whenη1=η\nand(M,g)is a space of constant curvature K= 1,0,−1below.\nK=1\nξξ~\nF(γ')ξ(s)K=0\nξξ~\nF(γ')ξ(s)\nK=-1\nξξ~\nF(γ')ξ(s)\nFigure 6. The curves ξ(s)andη(s)when(M,g)is a space of constant curvature K= 1,0,−1.\nProof of Lemma 5.3. I . Suppose that η1=±η. Thenη1(s) =±η(s). We first show (5.3). The\ndefinition of ∆(t0,˜t0;s)shows\nb(˜t0;s)\n∆(t0,˜t0;s)a(t0;s)−b(t0;s)\n∆(t0,˜t0;s)a(˜t0;s) = 1, (5.5)\nb(˜t0;s)\n∆(t0,˜t0;s)b(t0;s)−b(t0;s)\n∆(t0,˜t0;s)b(˜t0;s) = 0. (5.6)\nUsing (4.3) we deduce that\nDF|T\n˙γw(s)ξ=DvΨ|T/parenleftbig\n˙γw(s),t0−s/parenrightbig◦Dπ|T\n˙γw(t0)η, (5.7)\nDF|T\n˙γw(s)˜ξ=DvΨ|T/parenleftbig\n˙γw(s),˜t0−s/parenrightbig◦Dπ|T\n˙γw(˜t0)˜η. (5.8)\nLetΞ = (X,ζ)∈T˙γw(s)(S(Mint)), and letZ(t)be the Jacobi field along γwwith/parenleftbig\nZ(s),∇Z(s)/parenrightbig\n=\nΞ. Lemma 3.4 shows that\nZ(t) =a(t;s)P(s,t;γw)X+b(t,s)P(s,t;γw)ζ\n=Dπ|˙γw(t)◦DvΨ|/parenleftbig\n˙γw(s),t−s/parenrightbigΞ. (5.9)\nUsing (5.7), (5.8), (5.9), η1(s) =±η(s), (5.5) and (5.6) sequentially, we deduce that\nDF|T\n˙γw(s)ξ(s)(Ξ) =b(˜t0;s)\n∆(t0,˜t0;s)DF|T\n˙γw(s)ξ(Ξ)∓b(t0;s)\n∆(t0,˜t0;s)DF|T\n˙γw(s)˜ξ(Ξ)\n=b(˜t0;s)\n∆(t0,˜t0;s)DvΨ|T/parenleftbig\n˙γw(s),t0−s/parenrightbig◦Dπ|T\n˙γw(t0)η(Ξ)30 H. CHIHARA\n∓b(t0;s)\n∆(t0,˜t0;s)DvΨ|T/parenleftbig\n˙γw(s),˜t0−s/parenrightbig◦Dπ|T\n˙γw(˜t0)˜η(Ξ)\n=b(˜t0;s)\n∆(t0,˜t0;s)η/parenleftbigg\nDπ|˙γw(t0)◦DvΨ|/parenleftbig\n˙γw(s),t0−s/parenrightbigΞ/parenrightbigg\n∓b(t0;s)\n∆(t0,˜t0;s)˜η/parenleftbigg\nDπ|˙γw(˜t0)◦DvΨ|/parenleftbig\n˙γw(s),˜t0−s/parenrightbigΞ/parenrightbigg\n=b(˜t0;s)\n∆(t0,˜t0;s)η/parenleftbig\nZ(t0)/parenrightbig\n∓b(t0;s)\n∆(t0,˜t0;s)˜η/parenleftbig\nZ(˜t0)/parenrightbig\n=b(˜t0;s)\n∆(t0,˜t0;s)η/parenleftbig\na(t0;s)P(s,t0;γw)X+b(t0,s)P(s,t0;γw)ζ/parenrightbig\n∓b(t0;s)\n∆(t0,˜t0;s)˜η/parenleftbig\na(˜t0;s)P(s,˜t0;γw)X+b(˜t0,s)P(s,˜t0;γw)ζ/parenrightbig\n=b(˜t0;s)\n∆(t0,˜t0;s)P(s,t0;γw)Tη/parenleftbig\na(t0;s)X+b(t0,s)ζ/parenrightbig\n∓b(t0;s)\n∆(t0,˜t0;s)P(s,˜t0;γw)T˜η/parenleftbig\na(˜t0;s)X+b(˜t0,s)ζ/parenrightbig\n=b(˜t0;s)\n∆(t0,˜t0;s)P(s,t0;γw)Tη/parenleftbig\na(t0;s)X+b(t0,s)ζ/parenrightbig\n∓b(t0;s)\n∆(t0,˜t0;s)P(s,t0;γw)Tη1/parenleftbig\na(˜t0;s)X+b(˜t0,s)ζ/parenrightbig\n=b(˜t0;s)\n∆(t0,˜t0;s)P(s,t0;γw)Tη/parenleftbig\na(t0;s)X+b(t0,s)ζ/parenrightbig\n−b(t0;s)\n∆(t0,˜t0;s)P(s,t0;γw)Tη/parenleftbig\na(˜t0;s)X+b(˜t0,s)ζ/parenrightbig\n=b(˜t0;s)\n∆(t0,˜t0;s)η(s)/parenleftbig\na(t0;s)X+b(t0,s)ζ/parenrightbig\n−b(t0;s)\n∆(t0,˜t0;s)η(s)/parenleftbig\na(˜t0;s)X+b(˜t0,s)ζ/parenrightbig\n=/braceleftbiggb(˜t0;s)\n∆(t0,˜t0;s)a(t0;s)−b(t0;s)\n∆(t0,˜t0;s)a(˜t0;s)/bracerightbigg\nη(s)(X)\n+/braceleftbiggb(˜t0;s)\n∆(t0,˜t0;s)b(t0;s)−b(t0;s)\n∆(t0,˜t0;s)b(˜t0;s)/bracerightbigg\nη(s)(ζ)\n=η(s)(X)\n=η(s)/parenleftBig\nDπ|˙γw(s)Ξ/parenrightBig\n=Dπ|T\n˙γw(s)η(Ξ),\nwhich proves (5.3) and\nDF|T\n˙γw(s)/parenleftbig\nN∗\nw(Σjk)/parenrightbig\n∩Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n⊃ {αDπ|T\n˙γw(s)η(s) :α∈R}.\nIt suffices to show the converse of the above inclusion relati on. The conormal space N∗\nw(Σjk)is\nspanned byξand˜ξ. We have only to consider a linear combination αξ+β˜ξwith(α,β)/\\e}atio\\slash= (0,0). WeGEODESIC X-RAY TRANSFORM 31\nshow that if DF|T\n˙γw(s)(αξ+β˜ξ)∈Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n, then\nαb(t0;s)±βb(˜t0;s) = 0, (5.10)\nDF|T\n˙γw(s)(αξ+β˜ξ) =Dπ|T\n˙γw(s)/parenleftBig/parenleftbig\nαa(t0;s)±βa(˜t0;s)/parenrightbig\nη(s)/parenrightBig\n. (5.11)\nLetΞ = (X,ζ)∈T˙γw(s)/parenleftbig\nMint)/parenrightbig\n. In the same way as the above derivation of (5.3), we obtain\nDF|T\n˙γw(s)(αξ+β˜ξ)(Ξ) =αη(s)/parenleftbig\na(t0;s)X+b(t0;s)ζ/parenrightbig\n+βη1(s)/parenleftbig\na(˜t0;s)X+b(˜t0;s)ζ/parenrightbig\n(5.12)\n=αη(s)/parenleftbig\na(t0;s)X+b(t0;s)ζ/parenrightbig\n±βη(s)/parenleftbig\na(˜t0;s)X+b(˜t0;s)ζ/parenrightbig\n=η(s)/parenleftBig/parenleftbig\nαa(t0;s)±βa(˜t0;s)/parenrightbig\nX/parenrightBig\n+η(s)/parenleftBig/parenleftbig\nαb(t0;s)±βb(˜t0;s)/parenrightbig\nζ/parenrightBig\n. (5.13)\nIfDF|T\n˙γw(s)(αξ+β˜ξ)∈Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n, thenDF|T\n˙γw(s)(αξ+β˜ξ)(Ξ) = 0 forΞ∈Ker/parenleftbig\nDπ|˙γw(s)/parenrightbig\n,\nand (5.13) shows (5.10). Hence (5.13) becomes\nDF|T\n˙γw(s)(αξ+β˜ξ)(Ξ) =η(s)/parenleftBig/parenleftbig\nαa(t0;s)±βa(˜t0;s)/parenrightbig\nX/parenrightBig\n=η(s)/parenleftBig/parenleftbig\nαa(t0;s)±βa(˜t0;s)/parenrightbig\nDπ|˙γw(s)Ξ/parenrightBig\n=Dπ|T\n˙γw(s)/parenleftBig/parenleftbig\nαa(t0;s)±βa(˜t0;s)/parenrightbig\nη(s)/parenrightBig\n(Ξ),\nwhich proves (5.11).\nII. Suppose that s/\\e}atio\\slash=t0,s/\\e}atio\\slash=˜t0andη1/\\e}atio\\slash=±η. Then it follows that b(t0;s)/\\e}atio\\slash= 0 andb(˜t0;s)/\\e}atio\\slash=\n0, and thatηandη1are linearly independent and so are η(s)andη1(s). It suffices to show that\nDF|T\n˙γw(s)(αξ+β˜ξ)/\\e}atio\\slash∈Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\nfor(α,β)/\\e}atio\\slash= (0,0). In this setting we have\nθ:=αb(t0;s)η(s)+βb(˜t0;s)η1(s)/\\e}atio\\slash= 0 inT∗\n˙γw(s)(Mint).\nFor anyΞ = (0,ζ)∈Ker/parenleftbig\nDπ|˙γw(s)/parenrightbig\n, (5.12) becomes\nDF|T\n˙γw(s)(αξ+β˜ξ)(Ξ) =αη(s)/parenleftbig\nb(t0;s)ζ/parenrightbig\n+βη1(s)/parenleftbig\nb(˜t0;s)ζ/parenrightbig\n=/parenleftbig\nαb(t0;s)η(s)+βb(˜t0;s)η1(s)/parenrightbig\n(ζ) =θ(ζ).\nThis implies that there exists Ξ0= (0,ζ0)∈Ker/parenleftbig\nDπ|˙γw(s)/parenrightbig\nsuch thatDF|T\n˙γw(s)(αξ+β˜ξ)(Ξ0) = 1 ,\nandDF|T\n˙γw(s)(αξ+β˜ξ)/\\e}atio\\slash∈Dπ|T\n˙γw(s)/parenleftbig\nT∗(s)/parenrightbig\n. This completes the proof. /square\nWe can do the intersection calculus of CXT◦N∗(Σj),CXT◦N∗(Σ(1)\njk)andCXT◦N∗(Σ(2)\njk)using\nLemma 5.3.\nLemma 5.4. Suppose (A0),(A1)and(A2). Then the following compositions are all clean with some\nexcesse:\nCXT◦N∗(Σj) =N∗(∂Dj)\\0, e =n−2,\nCXT◦N∗(Σ(1)\njk) =N∗(Ljk)\\0, e = 0,\nCXT◦N∗(Σ(2)\njk) =N∗(Ωjk∪Ωkj)\\0, e= 0.\nThe last one is valid only for n≧3.32 H. CHIHARA\nWe set\nZ=T∗(Mint)×T∗/parenleftbig\n∂−S(M)/parenrightbig\n×T∗/parenleftbig\n∂−S(M)/parenrightbig\n,\nD=T∗(Mint)×∆/parenleftBig\nT∗/parenleftbig\n∂−S(M)/parenrightbig/parenrightBig\nfor short until the end of this section. Then dim(Z) = 10n−8andcodim(D) = 4n−4. SinceCXT,\nN∗(Σj)andN∗(Σ(i)\njk)are conic Lagrangian submanifolds, we have for i= 1,2\ndim/parenleftbig\nCXT×N∗(Σj)/parenrightbig\n= dim/parenleftbig\nCXT×N∗(Σ(i)\njk)/parenrightbig\n= dim(Mint)+2dim/parenleftbig\n∂−S(M)/parenrightbig\n= 5n−4.\nWe split the proof of Lemma 5.4 into three parts, and prove the lemma for each composition.\nProof of Lemma 5.4, Part 1: CXT◦N∗(Σj).Theorem 2.7, Lemma 4.1 and (4.2) imply that\nCXT×N∗(Σj) =/braceleftbig\n(η,ξ,ζ) :∃v∈S(Mint),∃u∈S(∂Dj),∃θ∈N∗\nπ(u)(∂Dj)s.t.\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},DF|T\nvξ=Dπ|T\nvη,\nζ∈T∗\nF(u)/parenleftbig\n∂−S(M)/parenrightbig\n,DF|T\nuζ=Dπ|T\nuθ/bracerightbig\n.\nThen we have\nC0:=/parenleftbig\nCXT×N∗(Σj)/parenrightbig\n∩D\n=/braceleftbig\n(η,ξ,ξ) :∃v∈S(∂Dj)s.t.η∈N∗\nπ(v)(∂Dj)\\{0},\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},DF|T\nvξ=Dπ|T\nvη/bracerightbig\n,\ndim(C0) = dim/parenleftbig\nS(∂Dj)/parenrightbig\n+dim/parenleftbig\nN∗\nπ(v)(∂Dj)/parenrightbig\n= 2n−2\nsinceξis uniquely determined by ηifDF|T\nvξ=Dπ|T\nvη. Hence we have\nCXT◦N∗(Σj) ={η:∃v∈S(∂Dj)s.t.η∈N∗\nπ(v)(∂Dj)\\{0}}=N∗(∂Dj)\\0,\ne= codim( D)+codim/parenleftbig\nCXT×N∗(Σj)/parenrightbig\n−codim(C0)\n= codim( D)−dim/parenleftbig\nCXT×N∗(Σj)/parenrightbig\n+dim(C0)\n= (4n−4)−(5n−4)+(2n−2) =n−2.\nIt remains to show that CXT◦N∗(Σj)is clean. Consider the projection map π0:C0∋(η,ξ,ξ)/mapsto→\nη∈T∗(Mint). Note that for any η∈N∗(∂Dj)\nπ−1\n0(η) =/braceleftbig/parenleftbig\nη,(DF|T\nv)−1◦Dπ|T\nvη,(DF|T\nv)−1◦Dπ|T\nvη/parenrightbig\n:v∈Sπ(η)(∂Dj)/bracerightbig\n,\nwhere(DF|T\nv)−1is the inverse of the linear map DF|T\nvofT∗\nv/parenleftbig\n∂−S(M)/parenrightbig\nonto the range. Hence π0is\nproper andπ−1\n0(η)is connected for any η∈N∗(∂Dj).\nFinally we show that C0is a clean intersection. Fix arbitrary c0∈C0andX= (X1,X2,X3)∈\nTc0/parenleftbig\nCXT×N∗(Σj)/parenrightbig\n, consider a curve in CXT×N∗(Σj)of the form\nR∋α/mapsto→/parenleftbig\nη(α),ξ(α),ζ(α)/parenrightbig\n=c0+αX+O(α2) =c0+α(X1,X2,X3)+O(α2).\nThen we have\nX= (X1,X2,X3) =d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nη(α),ξ(α),ζ(α)/parenrightbig\n.\nIfXalso belongs to Tc0(D), thenX2=X3and\nX=d\ndα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα=0/parenleftbig\nη(α),ζ(α),ζ(α)/parenrightbig\n∈Tc0(C0).\nThis completes the proof. /squareGEODESIC X-RAY TRANSFORM 33\nProof of Lemma 5.4, Part 2: CXT◦N∗(Σ(1)\njk).Theorem 2.7 and (4.16) imply that\nCXT×N∗(Σ(1)\njk) =/braceleftbig\n(η,ξ,ζ+˜ζ) :∃v∈S(Mint),∃(u,˜u)∈∆(Mjk),\n∃θ∈N∗\nπ(u)(∂Dj),∃˜θ∈N∗\nπ(˜u)(∂Dk),s.t.F(u) =F(˜u)\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},DF|T\nvξ=Dπ|T\nvη,\nζ∈T∗\nF(u)/parenleftbig\n∂−S(M)/parenrightbig\n,DF|T\nuζ=Dπ|T\nuθ,\n˜ζ∈T∗\nF(u)/parenleftbig\n∂−S(M)/parenrightbig\n,DF|T\n˜u˜ζ=Dπ|T\n˜u˜θ,/bracerightbig\n.\nSetC1:=/parenleftbig\nCXT×N∗(Σ(1)\njk)/parenrightbig\n∩D. If(η,ξ,ζ+˜ζ)∈CXT×N∗(Σ(1)\njk)∩D, then Part I of Lemma 5.3\nimplies that ξ=ζ+˜ζmust be of the form\nξ= (DF|T\n˙γv(s))−1◦Dπ|T\n˙γv(s)◦P(s,0;γv)Tη\nfor somev∈ Mjk,η∈N∗\nπ(v)(∂Dj)ands∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\n. Hence we have\nC1=/braceleftbig\n(η′,ξ,ξ) :∃v∈ Mjk,∃η∈N∗\nπ(v)(∂Dj)\\{0},∃s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig\ns.t.\nη′=P(s,0 :γv)Tη,DF|T\n˙γv(s)ξ=Dπ|T\n˙γv(s)η′/bracerightbig\n,\ndim(C1) = dim( Mjk)+dim/parenleftbig\nN∗\nπ(v)(∂Dj)/parenrightbig\n+dim/parenleftBig/parenleftbig\nτ−(v),τ+(v)/parenrightbig/parenrightBig\n=n.\nTherefore we obtain\nCXT◦N∗(Σ(1)\njk) =/braceleftbig\nP(s,0;γv)Tη:v∈ Mjk,η∈N∗\nπ(v)(∂Dj)\\{0},s∈/parenleftbig\nτ−(v),τ+(v)/parenrightbig/bracerightbig\n=N∗(Ljk)\\0,\ne= codim( D)+codim/parenleftbig\nCXT×N∗(Σ(1)\njk)/parenrightbig\n−codim(C1)\n= codim( D)−dim/parenleftbig\nCXT×N∗(Σ(1)\njk)/parenrightbig\n+dim(C1)\n= (4n−4)−(5n−4)+n= 0.\nIn the same way as π0, we can show that π1:C1∋(η,ξ,ξ)/mapsto→η∈T∗(Mint)is proper,π−1\n1(η)is\nconnected for any η∈N∗(Ljk), andC1is a clean intersection. We omit the detail. This completes\nthe proof. /square\nProof of Lemma 5.4, Part 3: CXT◦N∗(Σ(2)\njk).Theorem 2.7 and (4.17) imply that\nCXT×N∗(Σ(2)\njk) =/braceleftbig\n(η,ξ,ζ+˜ζ) :∃v∈S(Mint),∃(u,˜u)∈∆/parenleftbig\nS(Ωjk)/parenrightbig\n,\n∃θ∈N∗\nπ(u)(∂Dj),∃˜θ∈N∗\nπ(˜u)(∂Dk),s.t.\nξ∈T∗\nF(v)/parenleftbig\n∂−S(M)/parenrightbig\n\\{0},η∈T∗\nπ(v)(Mint)\\{0},\nDF|T\nvξ=Dπ|T\nvη,\nζ∈T∗\nF(u)/parenleftbig\n∂−S(M)/parenrightbig\n,DF|T\nuζ=Dπ|T\nuθ,\n˜ζ∈T∗\nF(u)/parenleftbig\n∂−S(M)/parenrightbig\n,DF|T\n˜u˜ζ=Dπ|T\n˜u˜θ,/bracerightbig\n.\nSetC2:=/parenleftbig\nCXT×N∗(Σ(2)\njk)/parenrightbig\n∩D. If(η,ξ,ζ+˜ζ)∈CXT×N∗(Σ(2)\njk)∩D, then Part II of Lemma 5.3\nimplies that ξ=ζ+˜ζmust be one of the following:\nξ= (DF|T\nu)−1◦Dπ|T\nuη, η∈N∗\nπ(u)(Ωjk)\\{0},\nξ= (DF|T\n˜u)−1◦Dπ|T\n˜u˜η,˜η∈N∗\nπ(˜u)(Ωkj)\\{0},34 H. CHIHARA\nfor(u,˜u)∈∆/parenleftbig\nS(Ωjk)/parenrightbig\n. Hence we have\nC2=/braceleftbig\n(η,ξ,ξ) :∃(u,˜u)∈∆/parenleftbig\nS(Ωjk)/parenrightbig\ns.t.η∈N∗\nπ(u)(Ωjk)\\{0},DF|T\nuξ=Dπ|T\nuη/bracerightbig\n∪/braceleftbig\n(˜η,˜ξ,˜ξ) :∃(˜u,u)∈∆/parenleftbig\nS(Ωkj)/parenrightbig\ns.t.˜η∈N∗\nπ(˜u)(Ωkj)\\{0},DF|T\n˜u˜ξ=Dπ|T\n˜u˜η/bracerightbig\n,\ndim(C2) = dim/parenleftbig\nN∗(Ωjk)/parenrightbig\n=n.\nTherefore we obtain CXT◦N∗(Σ(2)\njk) =N∗(Ωjk∪Ωkj)\\0ande= 0. In the same way as π0,\nwe can show that π2:C2∋(η,ξ,ξ)/mapsto→η∈T∗(Mint)is proper,π−1\n2(η)is connected for any\nη∈N∗(Ωjk∪Ωkj), andC2is a clean intersection. We omit the detail. This completes t he proof. /square\n6. P ROOF OF THE MAINTHEOREM\nFinally we prove Theorem 5.1 in several steps. Throughout th is section we assume ( A0), (A1),\n(A2) and ( A3). The story of the proof is basically same as that of the main t heorem in [1, Sections 5\nand 6], that is, the study of the interaction of singularitie s in (5.2). We begin by confirming the\nsingularities of 1DandX[1D]. We denote by Sm(RN×Rn)the set of all smooth functions a(x,ξ)of\n(x,ξ)∈RN×Rksatisfying\n|∂β\nx∂α\nξa(x,ξ)|≦Cα,β(1+|ξ|)m−|α|.\nLemma 6.1. We have\n1Dj∈I−1/2−n/4/parenleftbig\nN∗(∂Dj)\\0;Ω1/2\nMint/parenrightbig\n, (6.1)\nX[1Dj]∈I−(n+1)/2(N∗(Σj)\\0;Ω1/2\n∂−S(M)) (6.2)\nforj= 1,...,J . Moreover the principal symbols of these conormal distribu tions do not vanish on\nthe associated conormal bundles.\nProof. We remark that the proof of (6.1) is due to local argument in a s mall neighborhood at an\narbitrary point x∈∂Dj. Then (6.1) can be proved by the exactly same symbolic calcul us in [1,\nLemma 5.1] on the Euclidean space. Indeed 1Dj/|dvg|1/2can be locally expressed as the characteristic\nfunction of a half space {(0,x1,...,x n) :x2,...,x n∈R}inRnnear∂Dj, and is given by an\noscillatory integral of the form\n1Dj\n|dvg|(x1,...,x n) =/integraldisplay\nReix1ξ1a(x2,...,x n,ξ1)dξ1,\na(x2,...,x n,ξ1)∈S−1(Rn−1×R), a(x2,...,x n,ξ1)≃1\nξ1.\nThen (6.1) follows from [14, Theorem 18.2.8]. The elliptici ty of the symbol of the conormal distribu-\ntion1Djis obvious. Since CX◦N∗(∂Dj) =N∗(Σj)is transversal, and\nX ∈I−n/4/parenleftbig\n∂−S(M)×Mint,C′\nX;Ω∂−S(M×Mint)/parenrightbig\n,\nthen [15, Theorem 25.2.3] shows (6.2). The ellipticity of th e conormal distribution 1Djimplies that\nofX[1Dj]. /square\nTo see the interaction of singularities, we need to split the square into the several parts:\n/parenleftbiggX[1D]\n|dµ|1/2/parenrightbigg2\n|dµ|1/2=J/summationdisplay\nj=1/parenleftbiggX[1Dj]\n|dµ|1/2/parenrightbigg2\n|dµ|1/2\n+2/summationdisplay\n1≦j<k≦J/parenleftbiggX[1Dj]\n|dµ|1/2·X[1Dk]\n|dµ|1/2/parenrightbigg\n|dµ|1/2(6.3)\nFor the first term of the right hand side of (6.3) we have the fol lowing.GEODESIC X-RAY TRANSFORM 35\nLemma 6.2. Ifu,v∈I−(n+1)/2/parenleftbig\nN∗(Σj);Ω1/2\n∂−S(M)/parenrightbig\n, then\n/parenleftbiggu\n|dµ|1/2·v\n|dµ|1/2/parenrightbigg\n|dµ|1/2∈I−(n+1)/2/parenleftbig\nN∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n, (6.4)\nQ◦XT/bracketleftbigg/parenleftbiggu\n|dµ|1/2·v\n|dµ|1/2/parenrightbigg\n|dµ|1/2/bracketrightbigg\n∈I−1/2−n/4/parenleftbig\nN∗(∂Dj)\\0;ΩMint/parenrightbig\n. (6.5)\nProof. On one hand (6.4) can be proved by the exactly same symbolic ca lculus for [1, Lemma 5.2]\non the Euclidean space. Set N:= dim/parenleftbig\n∂−S(M)/parenrightbig\n= 2n−2. Then there exist smooth symbols\na(x,ξ1),b(x,ξ1)∈S−3/2(RN×R)such that\nu\n|dµ|1/2(x) =/integraldisplay\nReix1ξ1a(x,ξ1)dξ1,v\n|dµ|1/2(x) =/integraldisplay\nReix1ξ1b(x,ξ1)dξ1\nlocally. Hence we have/parenleftbiggu\n|dµ|1/2·v\n|dµ|1/2/parenrightbigg\n(x) =/integraldisplay\nReix1ξ1c(x,ξ1)dξ1,\nc(x,ξ1) =/integraldisplay\nRa(x,ξ1−η1)b(x,η1)dη1.\nWe can show that c(x,ξ1)∈S−3/2(RN×R). We omit the detail. On the other hand (6.5) immediately\nfollows from [15, Theorem 25.2.3] with Lemma 5.4, Lemma 2.7 a nd\nQ◦XT∈I1−n/4/parenleftbig\nMint×∂−S(M),C′\nXT;Ω1/2\nMint×∂−S(M)/parenrightbig\n.\nWe omit the detail. /square\nLemma 6.2 says that each I−(n+1)/2/parenleftbig\nN∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,j= 1,...,J forms an algebra. More-\nover\n1Dj, Q◦XT◦X[1Dj], Q◦XT/bracketleftBigg/parenleftbiggX[1Dj]\n|dµ|1/2/parenrightbigg2l\n|dµ|1/2/bracketrightBigg\nbelong to\nI−1/2−n/4/parenleftbig\nN∗(∂Dj)\\0;Ω1/2\nMint/parenrightbig\n, l= 1,2,3,..., j = 1,...,J.\nTo deal with the second term of the right hand side of (6.3), we now introduce classes of paired\nLagrangian distributions (See [5,9,17]), and quote the res ults on the product of conormal distributions,\nwhich is due to Greenleaf and Uhlmann in [5].\nDefinition 6.3. LetXbe a smooth manifold, and let µ,ν∈R. Suppose that Λ0andΛ1are conic La-\ngrangian submanifolds of T∗X\\0, and that they intersect cleanly. We say that u∈Iµ,ν(Λ0,Λ1;Ω1/2\nX)\nifu∈ D′(Ω1/2\nX),WF(u)⊂Λ0∪Λ1, and microlocally away from Λ0∩Λ1\nIµ,ν(Λ0,Λ1;Ω1/2\nX)⊂Iµ+ν(Λ0\\Λ1;Ω1/2\nX), Iµ,ν(Λ0,Λ1;Ω1/2\nX)⊂Iµ(Λ1;Ω1/2\nX).\nSuch a distribution uis said to be a paired Lagrangian distribution associated to the pair(Λ0,Λ)of\norder(µ,ν).\nWe use the following lemma to deal with the product of conorma l distributions:\nLemma 6.4 (Greenleaf and Uhlmann [5, Lemma 1.1]). LetXbe anN-dimensional smooth mani-\nfold. Suppose that YandZare closed submanifolds of Xsuch that codim(Y) =d1,codim(Y) =d2,\nandYandZintersect transversely, that is, codim(Y∩Z) =d1+d2. Letm1andm2be real numbers.\nThen we have\nIm1+d1/2−N/4/parenleftbig\nN∗(Y)\\0;Ω1/2\nX/parenrightbig\n·Im2+d2/2−N/4/parenleftbig\nN∗(Z)\\0;Ω1/2\nX/parenrightbig\n⊂Im1+d1/2−N/4,m2+d2/2/parenleftbig\nN∗(Y∩Z)\\0,N∗(Y)\\0;Ω1/2\nX/parenrightbig36 H. CHIHARA\n+Im2+d2/2−N/4,m1+d1/2/parenleftbig\nN∗(Y∩Z)\\0,N∗(Z)\\0;Ω1/2\nX/parenrightbig\n. (6.6)\nCombining [15, Theorem 25.2.3] and Lemma 6.4, we have the fol lowing.\nLemma 6.5. LetXandYbe smooth manifolds, and let m,µandνbe real numbers. Suppose that the\ndistribution kernel of a Fourier integral operator Abelongs toIm(X×Y,C′;Ω1/2\nX×Y)with a canonical\nrelationC. Suppose that Λ0andΛ1are conic Lagrangian submanifolds of T∗(Y)such that Λ0and\nΛ1intersect cleanly, and C◦Λiis clean with excess eifori= 1,2. Ifu∈Iµ,ν(Λ0,Λ1;Ω1/2\nY), then\nAu∈Im+µ+e1/2,ν+(e0−e1)/2(C◦Λ0,C◦Λ1;Ω1/2\nX).\nFor the second term of the right hand side of (6.3), we have the following.\nLemma 6.6. Assume that j/\\e}atio\\slash=k. Then we have\n/parenleftbiggX[1Dj]\n|dµ|1/2·X[1Dk]\n|dµ|1/2/parenrightbigg\n|dµ|1/2∈ Ajk, (6.7)\nAjk:=I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n+I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σk)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,\nand the principal symbol of this product does not vanish on N∗(Σjk)\\0. Furthermore we have\nQ◦XT/bracketleftbigg/parenleftbiggX[1Dj]\n|dµ|1/2·X[1Dk]\n|dµ|1/2/parenrightbigg\n|dµ|1/2/bracketrightbigg\n∈ Wjk, (6.8)\nWjk:=I−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dk)\\0;Ω1/2\nMint/parenrightbig\n=I−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dj∪∂Dk)\\0;Ω1/2\nMint/parenrightbig\n.\nand the principal symbol of this product does not vanish on N∗(Ljk)\\0.\nProof. The ellipticity of both products is obvious. (6.7) immediat ely follows from Lemma 6.1 and\nLemma 6.4 with N= 2n−2,m1=m2=−3/2andd1=d2= 1. So we prove (6.8). We apply\nTheorem 2.7, Lemma 5.4 and Lemma 6.5 to (6.7) with m=−n/4+1 ,µ=−(n+1)/2,ν=−1,\ne1=n−2ande0= 0, that is,\nm+µ+e1\n2=−1\n2−n\n4, ν+e0−e1\n2=−n\n2.\nWe deduce that\nQ◦XT/bracketleftbigg/parenleftbiggX[1Dj]\n|dµ|1/2·X[1Dk]\n|dµ|1/2/parenrightbigg\n|dµ|1/2/bracketrightbigg\n⊂I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σjk)\\0,CXT◦N∗(Σj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σjk)\\0,CXT◦N∗(Σk)\\0;Ω1/2\nMint/parenrightbig\n=I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σ(1)\njk)\\0,CXT◦N∗(Σj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σ(2)\njk)\\0,CXT◦N∗(Σj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σ(1)\njk)\\0,CXT◦N∗(Σk)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nCXT◦N∗(Σ(2)\njk)\\0,CXT◦N∗(Σk)\\0;Ω1/2\nMint/parenrightbig\n=I−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dj)\\0;Ω1/2\nMint/parenrightbigGEODESIC X-RAY TRANSFORM 37\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ωjk∪Ωkj)\\0,N∗(∂Dj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dk)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ωjk∪Ωkj)\\0,N∗(∂Dk)\\0;Ω1/2\nMint/parenrightbig\n=Wjk\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ωjk∪Ωkj)\\0,N∗(∂Dj)\\0;Ω1/2\nMint/parenrightbig\n+I−1/2−n/4,−n/2/parenleftbig\nN∗(Ωjk∪Ωkj)\\0,N∗(∂Dk)\\0;Ω1/2\nMint/parenrightbig\n⊂Wjk.\nThis completes the proof. /square\nWe now introduce notation:\nA:=/summationdisplay\n1≦j<k≦JAjk,W:=/summationdisplay\n1≦j<k≦JWjk.\nIn the same way as Lemma 6.2, we can show that Ais an algebra.\nLemma 6.7. Suppose that j/\\e}atio\\slash=k. ThenAjkis an algebra, and so is A. More precisely we have the\nfollowing.\n•If\nu,v∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,\nthen\nu·v∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n.\n•If\nu∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,\nv∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σk)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,\nthenu·v∈ Ajk.\nProof. Lemma 6.7 can be proved by symbolic calculus in the same way as [1, Lemmas 6.2]. Set\nN= dim/parenleftbig\n∂−S(M)/parenrightbig\n= 2n−2. Fix arbitrary p∈Σjk. We can choose local coordinates (x,y,z)∈\nR×R×RN−2such that/parenleftbig\nx(p),y(p),z(p)/parenrightbig\n= (0,0,0)and\nΣj={(0,y,z) :y∈R,z∈RN−2},\nΣk={(x,0,z) :x∈R,z∈RN−2},\nΣjk={(0,0,z) :z∈RN−2}\nnearpsinceΣjandΣkare hypersurfaces, and Σjk= Σj∩Σkis a transversal intersection. Suppose\nthat\nu,v1∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σj)\\0;Ω1/2\n∂−S(M)/parenrightbig\n,\nv2∈I−(n+1)/2,−1/parenleftbig\nN∗(Σjk)\\0,N∗(Σk)\\0;Ω1/2\n∂−S(M)/parenrightbig\n.\nIt is known that there exist smooth functions a(z,ξ,η),b1(z,ξ.η)andb2(z,η.ξ)onRN−2×(R\\\n{0})×Rsuch that\n∂γ\nz∂α\nξ∂β\nηa(z,ξ,η) =O/parenleftbig\n(1+|ξ|+|η|)−3/2−|α|(1+|η|)−3/2−|β|/parenrightbig\n,\n∂γ\nz∂α\nξ∂β\nηb1(z,ξ,η) =O/parenleftbig\n(1+|ξ|+|η|)−3/2−|α|(1+|η|)−3/2−|β|/parenrightbig\n,\n∂γ\nz∂α\nη∂β\nξb2(z,η,ξ) =O/parenleftbig\n(1+|η|+|ξ|)−3/2−|α|(1+|ξ|)−3/2−|β|/parenrightbig\n,38 H. CHIHARA\nand\nu(x,y,z) =/integraldisplay/integraldisplay\nR×Rei(xξ+yη)a(z,ξ,η)dξdη,\nv1(x,y,z) =/integraldisplay/integraldisplay\nR×Rei(xξ+yη)b1(z,ξ,η)dξdη,\nv2(x,y,z) =/integraldisplay/integraldisplay\nR×Rei(xξ+yη)b2(z,η,ξ)dξdη\nnear(x,y,z) = (0,0,0). Then we have\nu(x,y,z)·v1(x,y,z) =/integraldisplay/integraldisplay\nR×Rei(xξ+yη)c1(z,ξ,η)dξdη,\nu(x,y,z)·v2(x,y,z) =/integraldisplay/integraldisplay\nR×Rei(xξ+yη)c2(z,ξ,η)dξdη,\nc1(z,ξ,η) =/integraldisplay/integraldisplay\nR×Ra(z,ξ−ζ,η−τ)b1(z,ζ,τ)dζdτ,\nc2(z,ξ,η) =/integraldisplay/integraldisplay\nR×Ra(z,ξ−ζ,η−τ)b2(z,τ,ζ)dζdτ.\nWe can prove that c1(z,ξ,η)andc2(z,ξ,η)belong to a suitable class of symbols, which characterizes\nthe classAjk. We omit the detail. /square\nFinally we complete the proof of Theorem 5.1. Lemmas 6.2, 6.6 and 6.7 shows that PMA∈ A. In\nthe same way as the proof of the latter half of Lemma 6.6, we obt ain\nfMA=Q◦XT[PMA]\n∈ W\n=/summationdisplay\n1≦j<k≦JWjk\n=/summationdisplay\n1≦j<k≦JI−1/2−n/4,−n/2/parenleftbig\nN∗(Ljk)\\0,N∗(∂Dj∪∂Dk)\\0;Ω1/2\nMint/parenrightbig\n=I−1/2−n/4,−n/2/parenleftbig\nN∗(L)\\0,N∗(∂D)\\0;Ω1/2\nMint/parenrightbig\n.\nThis shows that fMA∈I−1/2−3n/4/parenleftbig\nN∗(L)\\0/parenrightbig\naway fromN∗(∂D)\\0microlocally. This completes\nthe proof of Theorem 5.1.\nREFERENCES\n[1] H. Chihara, Microlocal analysis of d-plane transform on the Euclidean space , SIAM J. Math. Anal., 54(2022),\npp.6254–6287.\n[2] J. J. Duistermaat, “Fourier Integral Operators”, Birkh ¨ auser, Boston, 1995.\n[3] C. L. Epstein, ”Introduction to the Mathematics of Medic al Imaging, Second Edition”, SIAM, Philadelphia, 2008.\n[4] B. Frigyik, P. Stefenov and G. Uhlmann, The X-ray transform from a generic family of curves and weigh ts, J. Geom.\nAnal., 18(2007), pp.89–108.\n[5] A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularitie s of scattering data , Comm.\nMath. Phys., 157(1993), pp.549–572.\n[6] A. Grigis and J. Sj¨ ostrand, “Microlocal Analysis for Di fferential Operators: An Introduction”, Cambridge Univer sity\nPress, Cambridge, 1994.\n[7] V . Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point o f view , Proc. Sympos. Pure\nMath., 27 Part 2 (1975), pp.297–300.\n[8] V . Guillemin and S. Sternberg, “Geometric Asymptotics” , Mathematical Surveys, 14, American Mathematical Society,\nProvidence, RI, 1977.\n[9] V . Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols , Duke Math. J., 48(1981), pp.251–267.GEODESIC X-RAY TRANSFORM 39\n[10] S. Helgason, “Integral Geometry and Radon Transforms” , Springer-Verlag, New York, NY , 2011.\n[11] S. Holman and G. Uhlmann, On the microlocal analysis of the geodesic X-ray transform w ith conjugate points , J. Diff.\nGeom., 108(2018), pp.459–494.\n[12] L. H¨ ormander, “The Analysis of Linear Partial Differe ntial Operators I”, Springer-Verlag, Berlin, 1983.\n[13] L. H¨ ormander, “The Analysis of Linear Partial Differe ntial Operators II”, Springer-Verlag, Berlin, 1983.\n[14] L. H¨ ormander, “The Analysis of Linear Partial Differe ntial Operators III”, Springer-Verlag, Berlin, 1985.\n[15] L. H¨ ormander, “The Analysis of Linear Partial Differe ntial Operators IV”, Springer-Verlag, Berlin, 1985.\n[16] V . P. Krishnan and E. T. Quinto, Microlocal analysis in tomography , Handbook of mathematical methods in imaging,\nV ol. 1, 2, 3, pp.847–902, Springer-Verlag, New York, 2015.\n[17] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem , Comm. Pure Appl. Math., 32(1979),\npp.483–519.\n[18] F. Monard and P. Stefanov, Sampling the X-ray transform on simple surfaces , SIAM J. Math. Anal., 55(2023),\npp.1707–1736.\n[19] B. Palacios, G. Uhlmann and Y . Wang, Quantitative analysis of metal artifacts in X-ray tomograp hy, SIAM J. Math.\nAnal., 50(2018), pp.4914–4936.\n[20] H. S. Park, J. K. Choi and J. K. Seo, Characterization of metal artifacts in X-ray computed tomo graphy , Comm. Pure\nAppl. Math., 70(2017), pp.2191–2217.\n[21] G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography: Progress and challenges , Chinese Annales of Mathe-\nmatics, Series B, 35(2014), pp.399–428.\n[22] E. T. Quinto, Singularities of the X-Ray transform and limited data tomog raphy inR2andR3, SIAM J. Math. Anal.,\n24(1993), pp.1215–1225.\n[23] E. T. Quinto, An introduction to X-ray tomography and Radon transforms , ”The Radon Transform, Inverse Problems,\nand Tomography”, Proceedings of Symposia in Applied Mathem atics, 63, pp.1–23, American Mathematical Society,\nProvidence, RI, 2006.\n[24] T. Sakai, “Riemannian Geometry”, Translations of Math ematical Monographs, 149, American Mathematical Society,\nProvidence, RI, 1996.\n[25] V . A. Sharafutdinov, “Integral Geometry of Tensor of Te nsor Fields”, VSP, Utrecht, 1994.\n[26] P. Stefanov, G. Uhlmann and A. Vasy, Local and global boundary rigidity and the geodesic X-ray tr ansform in the\nnormal gauge , Ann. of Math., 194(2021), pp.1–95.\n[27] Y . Wang and Y . Zou, Streak artifacts from nonconvex metal objects in X-ray tomo graphy , Pure Appl. Anal., 3(2021),\npp.295–318.\n[28] J. W. Webber, S. Holman and E. T. Quinto, Ellipsoidal and hyperbolic Radon transforms; microlocal p roperties and\ninjectivity , J. Funct. Anal., bf 285 (2023), Article ID 110056, 30 pages.\nCOLLEGE OF EDUCATION , UNIVERSITY OF THE RYUKYUS , NISHIHARA , OKINAWA 903-0213, J APAN\nEmail address :hc@trevally.net"    },    {        "title": "2402.06926v1.On_a_mixed_local_nonlocal_evolution_equation_with_singular_nonlinearity.pdf",        "content": "arXiv:2402.06926v1  [math.AP]  10 Feb 2024On a mixed local-nonlocal evolution equation with singular nonlinearity\nKaushik Bal and Stuti Das\nDepartment of Mathematics and Statistics,\nIndian Institute of Technology Kanpur, Uttar Pradesh, 2080 16, India\nAbstract\nWe will prove several existence and regularity results for t he mixed local-nonlocal parabolic equation of the form\nut−∆u+(−∆)su=f(x,t)\nuγ(x,t)inΩT:= Ω×(0,T),\nu= 0in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x)inΩ;(0.1)\nwhere\n(−∆)su=cn,sP.V.ˆ\nRnu(x,t)−u(y,t)\n|x−y|n+2sdy.\nUnder the assumptions that γis a positive continuous function on ΩTandΩis a bounded domain with Lipschitz\nboundary in Rn,n>2,s∈(0,1),0<T <+∞,f≥0,u0≥0,fandu0belongs to suitable Lebesgue spaces.\nHerecn,sis a suitable normalization constant, and P.V.stands for Cauchy Principal Value.\nContents\n1 Introduction 1\n2 Preliminaries 3\n2.1Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.2Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.3Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.4Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n3 Proof of main results 17\n4 Asymptotic behaviour 34\n1. Introduction\nIn this article, we study the evolution of a mixed local-nonl ocal operator under the effect of a singular\nnonlinearity given by:\nut−∆u+(−∆)su=f(x,t)\nuγ(x,t)inΩT:= Ω×(0,T),\nu= 0in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x)inΩ;(1.1)\nwhere\n(−∆)su:=cn,sP.V.ˆ\nRnu(x,t)−u(y,t)\n|x−y|n+2sdy,\nEmail address: kaushik@iitk.ac.in and stutid21@iitk.ac.in (Kaushik Bal and Stuti Das)γis a positive continuous function on ΩTwithΩbeing a bounded domain with Lipschitz boundary in Rn,n>2,\ns∈(0,1)and0<T <+∞. Assuming that f≥0andu0≥0with variable summability, in this paper we show\nseveral existence and regularity results. To this aim, we st art by reviewing the literature concerning our problem.\nSingular elliptic problems have been extensively studied i n the literature for the past few decades starting\nwith the now classical work of Crandall-Rabinowitz-Tartar [15], who showed that the stationary state of ( 1.1),\nunder Dirichlet boundary conditions given by\n−∆u=f\nuγ(x)inΩ,\nu>0inΩ,\nu= 0in∂Ω.(1.2)\nadmits a unique solution u∈C2(Ω)∩C(¯Ω)for anyγ >0constant along with the fact that the solution must\nbehave like a distance function near the boundary provided fis Hölder Continuous. Interestingly enough Lazer-\nMckenna [ 28] showed that the unique solution obtained by [ 15] is indeed in H1\n0(Ω)iff0<γ <3. Boccardo-Orsina\n[9] in a beautiful paper showed that the followings regarding s olutions of ( 1.2)\n\nu∈W1,nr(1+γ)\nn−r(1−γ)\n0 (Ω) if0<γ <1andf∈Lr(Ω)withr∈/bracketleftbig\n1,(2∗/(1−γ))′/parenrightbig\n,\nu∈H1\n0(Ω) if0<γ <1andf∈Lr(Ω)withr= (2∗/(1−γ))′,\nu∈H1\n0(Ω) ifγ= 1andf∈L1(Ω),\nu1+γ\n2∈H1\n0(Ω) ifγ >1andf∈L1(Ω),\nhold, which was extended for variable γ, introducing certain conditions on its behaviour near the b oundary in\n[14]. The nonlocal variant given by\n(−∆)s\npu=f\nuγ(x)inΩ,\nu>0inΩ,\nu= 0in∂Ω.\nwas studied in [ 7] forp= 2andγ(x) =γ∈R+\n∗, the authors proved the existence and uniqueness of positiv e\nsolutions, according to the range of γand summability of f. For the quasilinear case, we refer [ 13] for constant\nγ >0and for variable singular exponent, the existence results h ave been obtained in [ 25]. As for the Mixed\nlocal-nonlocal elliptic problem given by\n−∆pu+(−∆)s\npu=f\nuγ(x)inΩ,\nu>0inΩ,\nu= 0in∂Ω;\nArora [ 3] forp= 2andγ(x) =γ∈R+\n∗, obtained the existence, uniqueness and regularity proper ties of the weak\nsolutions by deriving uniform a priori estimates and using t he approximation technique. They also obtained some\nexistence and nonexistence results when fbehaves like a distance function. For the case p >1, the constant\nexponentγcase has been considered in [ 26], and the variable exponent can be found in [ 8]. If one considers the\nparabolic counterpart i.e, the equation given by:\nut−∆pu=f(x,t)\nuγinΩT,\nu= 0in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x)inΩ.\nFor such an equation with p≥2,0≤f∈Lm(ΩT)withm≥1and assuming that ∀ω⊂⊂Ω,∃dω>0, such that\nu0≥dω, the authors [ 17] proved the existence of a solution uof the above problem such that\n\nu∈Lq0(0,T;W1,q0\n0(Ω)) if0<γ <1andf∈Lr(Ω)withr∈/bracketleftbigg\n1,p(n+2)\np(n+2)−n(1−γ)/parenrightbigg\nandq0=m[n(p+γ−1)+p(γ+1)]\nn+2−m(1−γ),\nu∈Lp(0,T;W1,p\n0(Ω)) if0<γ≤1andf∈Lm0(ΩT)withm0=p(n+2)\np(n+2)−n(1−γ),\nu∈Lp(0,T;W1,p\nloc(Ω)),ifγ >1andf∈L1(ΩT).\n2The Nonlocal case (s∈(0,1))for the parabolic problem was handled by [ 1] forγ >0constant to show existence\nand uniqueness results along similar lines. If one restrict s the range of γthen various existence, uniqueness and\nregularity results can be found in Bal-Badra-Giacomoni [ 4,5,6] and Giacomoni-Bougherera [ 11]. We would\nalso like to mention that the regularity theory of mixed loca l and non-local operators plays a major role in our\nproblem and we cite the following papers [ 16,18,21,22,23,24,30,31] and the references therein.\nAs for the boundedness of our solutions, we refer Aronson-Se rrin [2], where the summability requirement of\ninitial data for boundedness was introduced by Aronson and S errin, for the case of second-order differential\nequations without singularity. Outside of the Aronson-Ser rin domain, the optimal summability of solutions for\nthe local case without singularity was obtained in Boccardo -Porzio-Primo [ 10]. These results for the nonlocal\ncase have been obtained in Peral [ 29], this too for the nonsingular case. For the mixed local-non local operator\nwith singularity, we will be able to get similar types of resu lts here depending on the choice of γ. We will use\nsuitable approximating problems to get the existence and ot her summability properties of weak solutions.\nOrganization of the article\nIn the next section, we will describe some basic notations an d fix some preliminary function spaces to define\nour solutions, followed by embedding results and other prop erties regarding those spaces.\nThen we will introduce the notion of weak solution for the cas eγ= 0and show its existence, uniqueness,\npositivity and other properties. After that, we write about the existence of weak solutions for approximating\nproblems and give definitions of weak solutions for the singu lar cases, both for constant and variable exponents.\nWe end this section by stating our main theorems regarding th e existence and summability of weak solutions and\nappropriate comments.\nThe next section contains the proofs of our main results, and we end with another section that gives the asymptotic\nbehaviour of the solutions in a suitable sense.\n2. Preliminaries\n2.1.Notations\nWe gather here all the standard notations that will be used th roughout the paper.\n•We will take nto be the space dimension and denote by z= (x,t)to be a point in Rn×(0,T), where(0,T)⊂R\nfor some 0<T <∞.\n•LetΩbe an open bounded domain in Rnwith boundary ∂Ωand for0<T <∞, letΩT:= Ω×(0,T).\n•We denote the parabolic boundary ΓTbyΓT= (Ω×{t= 0})∪(∂Ω×(0,T)).\n•We define the set (ΩT)δ={(x,t)∈ΩT: dist((x,t),ΓT)<δ}forδ>0fixed.\n•We shall alternately use ∂tgor∂g\n∂torgtto denote the time derivative (partial) of a function g.\n•Forr>1, the Hölder conjugate exponent of rwill be denoted by r′=r\nr−1.\n•The Lebesgue measure of a measurable subset S⊂Rnwill be denoted by |S|.\n•For any open subset ΩofRn,K⊂⊂Ωwill implyKis compactly contained in Ω.\n•ˆ\nwill denote integration concerning either space or time onl y, and integration on Ω×ΩorRn×Rnwill be\ndenoted by a double integral¨\n.\n•We will use˚\nto denote integral over Rn×Rn×(0,T).\n•Average integral will be denoted by \n.\n•The notation a/lessorsimilarbwill be used for a≤Cb, whereCis a universal constant which only depends on the\ndimensionnand sometimes on stoo.Cmay vary from line to line or even in the same line.\n• /a\\}b∇acketle{t.,./a\\}b∇acket∇i}htwill denote the usual inner product in some associated Hilbe rt space.\n•For any function h, we denote the positive and negative parts of it by h+= max{h,0}andh−= max{−h,0}\nrespectively.\n•Fork∈N, we denote Tk(σ) = max{−k,min{k,σ}}, forσ∈R.\n32.2.Function Spaces\nIn this section, we present definitions and properties of som e function spaces that will be useful for our work.\nWe recall that for E⊂Rn, the Lebesgue space Lp(E),1≤p <∞, is defined to be the space of p-integrable\nfunctionsu:E→Rwith the finite norm\n/ba∇dblu/ba∇dblLp(E)=/parenleftbiggˆ\nE|u(x)|pdx/parenrightbigg1/p\n.\nByLp\nloc(E)we denote the space of locally p-integrable functions, which means, u∈Lp\nloc(E)if and only if\nu∈Lp(F)for everyF⊂⊂E. In the case 0< p <1, we denote by Lp(E)a set of measurable functions such\nthatˆ\nE|u(x)|pdx<∞.\nDefinition 2.1. The Sobolev space W1,p(Ω), for1≤p<∞, is defined as the Banach space of locally integrable\nweakly differentiable functions u: Ω→Requipped with the following norm\n/ba∇dblu/ba∇dblW1,p(Ω)=/ba∇dblu/ba∇dblLp(Ω)+/ba∇dbl∇u/ba∇dblLp(Ω).\nThe spaceW1,p\n0(Ω)is defined as the closure of the space C∞\n0(Ω), in the norm of the Sobolev space W1,p(Ω),\nwhereC∞\n0(Ω)is the set of all smooth functions whose supports are compact ly contained in Ω.\nDefinition 2.2. Let0<s<1andΩbe a open connected subset of Rn. The fractional Sobolev space Ws,q(Ω)\nfor any1≤q<+∞is defined by\nWs,q(Ω) =/braceleftbigg\nu∈Lq(Ω) :|u(x)−u(y)|\n|x−y|n\nq+s∈Lq(Ω×Ω)/bracerightbigg\n,\nand it is endowed with the norm\n/ba∇dblu/ba∇dblWs,q(Ω)=/parenleftbiggˆ\nΩ|u(x)|qdx+ˆ\nΩˆ\nΩ|u(x)−u(y)|q\n|x−y|n+qsdxdy/parenrightbigg1/q\n. (2.1)\nIt can be treated as an intermediate space between W1,q(Ω)andLq(Ω). For0< s≤s′<1,Ws′,q(Ω)is\ncontinuously embedded in Ws,q(Ω), see [ 19, Proposition 2.1]. The fractional Sobolev space with zero b oundary\nvalues is defined by\nWs,q\n0(Ω) ={u∈Ws,q(Rn) :u= 0inRn\\Ω}.\nHoweverWs,q\n0(Ω)can be treated as the closure of C∞\n0(Ω)inWs,q(Ω)with respect to the fractional Sobolev norm\ndefined in (2.1). BothWs,q(Ω)andWs,q\n0(Ω)are reflexive Banach spaces, for q >1, for details we refer to the\nreaders [ 19, Section 2].\nThe following result asserts that the classical Sobolev spa ce is continuously embedded in the fractional Sobolev\nspace; see [ 19, Proposition 2.2]. The idea applies an extension property o fΩso that we can extend functions\nfromW1,q(Ω)toW1,q(Rn)and that the extension operator is bounded.\nLemma 2.3. LetΩbe a smooth bounded domain in Rnand0< s <1. There exists a positive constant\nC=C(Ω,n,s)such that\n/ba∇dblu/ba∇dblWs,q(Ω)≤C/ba∇dblu/ba∇dblW1,q(Ω),\nfor everyu∈W1,q(Ω).\nFor the fractional Sobolev spaces with zero boundary value, the next embedding result follows from [ 12,\nLemma 2.1]. The fundamental difference of it compared to Lemma 2.3 is that the result holds for any bounded\ndomain (without any condition of smoothness of the boundary ), since for the Sobolev spaces with zero boundary\nvalue, we always have a zero extension to the complement.\nLemma 2.4. LetΩbe a bounded domain in Rnand0<s<1. There exists a positive constant C=C(n,s,Ω)\nsuch that ˆ\nRnˆ\nRn|u(x)−u(y)|q\n|x−y|n+qsdxdy≤Cˆ\nΩ|∇u|qdx\nfor everyu∈W1,q\n0(Ω). Here, we consider the zero extension of uto the complement of Ω.\nWe now proceed with the basic Poincaré inequality, which can be found in [ 20, Chapter 5, Section 5.8.1].\n4Lemma 2.5. LetΩ⊂Rnbe a bounded domain with C1boundary and q≥1. Then there exist a positive constant\nC >0depending only on nandΩ, such thatˆ\nΩ|u|qdx≤Cˆ\nΩ|∇u|qdx,∀u∈W1,q\n0(Ω).\nSpecifically if we take Ω =Br, then we will get for all u∈W1,q(Br), \nBr|u−(u)Br|qdx≤crq \nBr|∇u|qdx,\nwherecis a constant depending only on n, and(u)Brdenotes the average of uinBr, andBrdenotes a ball of\nradiusrcentered at x0∈Rn. Here, \ndenotes the average integration.\nUsing Lemma 2.4 , and the above Poincaré inequality, we observe that the foll owing norm on the space W1,q\n0(Ω)\ndefined by\n/ba∇dblu/ba∇dblW1,q\n0(Ω)=/parenleftbiggˆ\nΩ|∇u|qdx+ˆ\nRnˆ\nRn|u(x)−u(y)|q\n|x−y|n+qsdxdy/parenrightbigg1\nq\n,\nis equivalent to the norm\n/ba∇dblu/ba∇dblW1,q\n0(Ω)=/parenleftbiggˆ\nΩ|∇u|qdx/parenrightbigg1\nq\n.\nThe following is a version of fractional Poincaré.\nLemma 2.6. LetΩ⊂Rnbe a bounded domain with C1boundary and let s∈(0,1)andq≥1. Ifu∈Ws,q\n0(Ω),\nthen ˆ\nΩ|u|qdx≤cˆ\nΩˆ\nΩ|u(x)−u(y)|q\n|x−y|n+qsdxdy,\nholds withc≡c(n,s,Ω).\nIn view of Lemma 2.6 , we observe that the Banach space Ws,q\n0(Ω)can be endowed with the norm\n/ba∇dblu/ba∇dblWs,q\n0(Ω)=/parenleftbiggˆ\nΩˆ\nΩ|u(x)−u(y)|q\n|x−y|n+qsdxdy/parenrightbigg1\nq\n,\nwhich is equivalent to that of /ba∇dblu/ba∇dblWs,q(Ω). Forq= 2, the space Ws,2(Ω)enjoys certain special properties and we\ndenoteWs,2(Ω) =Hs(Ω)andWs,2\n0(Ω) =Hs\n0(Ω). Endowed with the inner product\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htHs\n0(Ω)=ˆ\nΩˆ\nΩ(u(x)−u(y))(v(x)−v(y))\n|x−y|n+2sdxdy,\nwe note that (Hs\n0(Ω),/ba∇dbl·/ba∇dblHs\n0(Ω))is a Hilbert space. Similar thing holds for the space W1,2\n0(Ω) =H1\n0(Ω).\nDefinition 2.7. The spaceXs\n0(Ω)is defined as\nXs\n0(Ω) ={f∈Hs(Rn)s.t.f= 0a.e. inCΩ},\nwhereCΩ =Rn\\Ω, and is endowed with the norm\n/ba∇dblu/ba∇dblXs\n0(Ω)=/parenleftbiggˆ\nQ|u(x)−u(y)|2\n|x−y|n+2sdxdy/parenrightbigg1\n2\n,\nwhereQ:=R2n\\(CΩ×CΩ).\nMoreoverW−1,q′(Ω),W−s,q′(Ω)andX−s(Ω)are defined to be the dual spaces of W1,q\n0(Ω),Ws,q\n0(Ω)and\nXs\n0(Ω)respectively, where q′:=q\nq−1. Now, we define the local spaces as\nW1,q\nloc(Ω) =/braceleftbigg\nu: Ω→R:u∈Lq(K),ˆ\nK|∇u|qdx<∞,for everyK⊂⊂Ω/bracerightbigg\n,\nand\nWs,q\nloc(Ω) =/braceleftbigg\nu: Ω→R:u∈Lq(K),ˆ\nKˆ\nK|u(x)−u(y)|q\n|x−y|n+qsdxdy <∞,for everyK⊂⊂Ω/bracerightbigg\n.\nNow forn >2, we define the critical Sobolev exponent as 2∗=2n\nn−2, then we get the following embedding\nresult for any open subset ΩofRn, see for details [ 20, Chapter 5],\n5Theorem 2.8. Letn >2. Then, there exists a constant Cdepending only on nandΩ, such that for all\nu∈ C∞\n0(Ω)\n/ba∇dblu/ba∇dbl2\nL2∗(Ω)≤Cˆ\nΩ|∇u|2dx.\nSimilarly, for n>2s, we define the fractional Sobolev critical exponent as 2∗\ns=2n\nn−2s. The following result is\na fractional version of the Sobolev inequality( Theorem 2.8 ) which also implies a continuous embedding of Hs\n0(Ω)\nin the critical Lebesgue space L2∗\ns(Ω). One can see the proof in [ 19].\nTheorem 2.9. Let0< s <1be such that n >2s. Then, there exists a constant S(n,s)depending only on n\nands, such that for all u∈ C∞\n0(Ω)\n/ba∇dblu/ba∇dbl2\nL2∗s(Ω)≤S(n,s)ˆ\nΩˆ\nΩ|u(x)−u(y)|2\n|x−y|n+2sdxdy.\nWe now recall the Gagliardo-Nirenberg interpolation inequ ality that will be useful for proving the boundedness\nof weak solutions.\nTheorem 2.10. (Gagliardo-Nirenberg) Let 1≤q<+∞be a positive real number. Let jandmbe non-negative\nintegers such that j <m . Furthermore, let 1≤r≤+∞be a positive extended real quantity, p≥1be real and\nθ∈[0,1]such that the relations\n1\np=j\nn+θ/parenleftbigg1\nr−m\nn/parenrightbigg\n+1−θ\nq,j\nm≤θ≤1\nhold. Then,/vextenddouble/vextenddoubleDju/vextenddouble/vextenddouble\nLp(Rn)≤C/ba∇dblDmu/ba∇dblθ\nLr(Rn)/ba∇dblu/ba∇dbl1−θ\nLq(Rn)\nfor anyu∈Lq(Rn)such thatDmu∈Lr(Rn). Here, the constant C >0depends on the parameters j,m,n,q,r,θ ,\nbut not onu.\nThe article will extensively use the embedding results and c orresponding inequalities. Now, we need to deal\nwith spaces involving time for the parabolic equations, so w e introduce them here. As in the classical case, we\ndefine the corresponding Bochner spaces as the following\nLq(0,T;W1,q\n0(Ω)) =/braceleftig\nu∈Lq(Ω×(0,T)),/ba∇dblu/ba∇dblLq(0,T;W1,q\n0(Ω))<∞/bracerightig\n,\nLq(0,T;Ws,q\n0(Ω)) =/braceleftig\nu∈Lq(Ω×(0,T)),/ba∇dblu/ba∇dblLq(0,T;Ws,q\n0(Ω))<∞/bracerightig\n,\nL2(0,T;Xs\n0(Ω)) =/braceleftbig\nu∈L2(Rn×(0,T)),/ba∇dblu/ba∇dblL2(0,T;Xs\n0(Ω))<∞/bracerightbig\n,\nwhere\n/ba∇dblu/ba∇dblLq(0,T;W1,q\n0(Ω))=/parenleftiggˆT\n0ˆ\nΩ|∇u|qdxdt/parenrightigg1\nq\n,\n/ba∇dblu/ba∇dblLq(0,T;Ws,q\n0(Ω))=/parenleftiggˆT\n0ˆ\nΩˆ\nΩ|u(x,t)−u(y,t)|q\n|x−y|n+qsdxdydt/parenrightigg1\nq\n,\n/ba∇dblu/ba∇dblL2(0,T;Xs\n0(Ω))=/parenleftiggˆT\n0ˆ\nQ|u(x,t)−u(y,t)|2\n|x−y|n+2sdxdydt/parenrightigg1\n2\n,\nwith their dual spaces Lq′(0,T;W−1,q′(Ω)),Lq′(0,T;W−s,q′(Ω))andL2(0,T;X−s(Ω))respectively. Again, the\nlocal spaces are defined as\nL2(0,T;H1\nloc(Ω)) =/braceleftigg\nu∈L2(K×(0,T)) :ˆT\n0ˆ\nK|∇u|qdxdt<∞,for everyK⊂⊂Ω/bracerightigg\n,\nand\nL2(0,T;Hs\nloc(Ω)) =/braceleftigg\nu∈L2(K×(0,T)) :ˆT\n0ˆ\nKˆ\nK|u(x,t)−u(y,t)|2\n|x−y|n+2sdxdydt< ∞,for everyK⊂⊂Ω/bracerightigg\n.\nWe now recall the following algebraic inequality that can be found in [ 1, Lemma 2.22].\n6Lemma 2.11. i) Letα>0. For every x,y≥0one has\n(x−y)(xα−yα)≥4α\n(α+1)2/parenleftig\nxα+1\n2−yα+1\n2/parenrightig2\n.\nii) Let0<α≤1. For every x,y≥0withx/\\e}atio\\slash=yone has\nx−y\nxα−yα≤1\nα(x1−α+y1−α).\niii) Letα≥1. Then there exists a constant Cαdepending only on αsuch that\n|x+y|α−1|x−y| ≤Cα|xα−yα|.\n2.3.Weak Solutions\nIn this subsection, along with the next subsection, we will i ntroduce notions of very weak solutions to our\nproblem and state the main results that we are going to prove. We begin with the definitions of weak solutions\nfor the nonsingular case. We first take fandu0to be inL2spaces and then relax the condition. We also state\nsome important properties of the weak solutions that we need to use in the rest of the article. Further, we will\nintroduce suitable approximating problems and properties of their solutions.\nDefinition 2.12. Assume(f,u0)∈L2(ΩT)×L2(Ω), then we say that uis an energy solution to problem\n\nut−∆u+(−∆)su=f(x,t)inΩT,\nu= 0 in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x) inΩ;(2.2)\nifu∈L2(0,T;H1\n0(Ω))∩C([0,T],L2(Ω)),ut∈L2(0,T;H−1(Ω)), and for all φ∈L2(0,T;H1\n0(Ω))we have\nˆT\n0/a\\}b∇acketle{tut,φ/a\\}b∇acket∇i}htdt+ˆT\n0ˆ\nΩ∇u·∇φdxdt+1\n2ˆT\n0ˆ\nRnˆ\nRn(u(x,t)−u(y,t))(φ(x,t)−φ(y,t))\n|x−y|n+2sdxdydt=ˆT\n0ˆ\nΩfφdxdt\nandu(·,t)→u0strongly in L2(Ω), ast→0.\nWe denote\nE(u(x,t),φ(x,t)) :=1\n2ˆ\nRnˆ\nRn(u(x,t)−u(y,t))(φ(x,t)−φ(y,t))×K(x,y,t)dxdy,\nwhereK(x,y,t) =1\n|x−y|n+2s.Following the way for fractional Laplacian in [ 29], we give the proof of existence\nfor mixed local-nonlocal case for the sake of completeness.\nTheorem 2.13. There exists a solution to problem (2.2)in the sense of Definition 2.12. Moreover, if fis also\na nonnegative function and u0≥0, then the solution is also nonnegative.\nProof. Let us denote C∞\n∗(Ω×[0,T])as theC∞(Ω×[0,T])functions that vanish in (Rn\\Ω)×[0,T]and inΩ×{T}.\nChoosingφ∈ C∞\n∗(Ω×[0,T]), foru∈L2(0,T;H1\n0(Ω)), we define the operator\nLφ(u) :=ˆT\n0ˆ\nΩ−uφtdxdt+ˆT\n0ˆ\nΩ∇u·∇φdxdt+ˆT\n0E(u(x,t),φ(x,t))dt.\nWe now observe that uis an energy solution to (2.2)withf∈L2(ΩT)⊂L2(0,T;H−1(Ω))if and only if\nLφ(u) =ˆT\n0/a\\}b∇acketle{tf,φ/a\\}b∇acket∇i}htdt+ˆ\nΩu(x,0)φ(x,0)dx,\nwhere/a\\}b∇acketle{tf,φ/a\\}b∇acket∇i}htdenote the usual inner product of fandφinL2(Ω)or the pairing of fandφbetweenH−1(Ω)and\nH1\n0(Ω).\nWe also define the following inner product, for φ,ϕ∈ C∞\n∗(Ω×[0,T])\n/a\\}b∇acketle{tϕ,φ/a\\}b∇acket∇i}ht∗=1\n2/a\\}b∇acketle{tϕ(x,0),φ(x,0)/a\\}b∇acket∇i}htL2(Ω)+ˆT\n0ˆ\nΩ∇φ·∇ϕ dxdt+ˆT\n0E(ϕ(x,t),φ(x,t))dt, (2.3)\nand denote by H∗(Ω×[0,T])the Hilbert space built as the completion of C∞\n∗(Ω×[0,T])with the norm /ba∇dblφ/ba∇dbl∗\ninduced by the inner product (2.3).\nNowLφis clearly a linear functional in H∗(Ω×[0,T])and for any ϕ∈L2(0,T;H1\n0(Ω)), by Hölder and Sobolev\ninequalities, we have\n|Lφ(ϕ)| ≤cφ/parenleftig\n/ba∇dblϕ/ba∇dblL2(0,T;L2(Ω))+/ba∇dblϕ/ba∇dblL2(0,T,H1\n0(Ω))+/ba∇dblϕ/ba∇dblL2(0,T,Hs\n0(Ω))/parenrightig\n≤˜cφ/ba∇dblϕ/ba∇dbl∗.\n7Therefore,Lφis a bounded linear functional in H∗(Ω×[0,T]), and hence by the Fréchet-Riesz Theorem, there\nexistsTφ∈H∗(Ω×[0,T])such that\nLφ(ϕ) =/a\\}b∇acketle{tϕ,Tφ/a\\}b∇acket∇i}ht∗for allϕ∈H∗(Ω×[0,T]).\nIt is trivial to show that Tis a linear operator in H∗(Ω×[0,T]). Moreover\nLφ(φ) =1\n2ˆ\nΩφ2(x,0)dx+ˆT\n0ˆ\nΩ|∇φ|2dxdt+ˆT\n0E(φ(x,t),φ(x,t))dt=/ba∇dblφ/ba∇dbl2\n∗,\nand consequently, /a\\}b∇acketle{tφ,Tφ/a\\}b∇acket∇i}ht∗=/ba∇dblφ/ba∇dbl2\n∗. Then we get by the Cauchy-Schwartz inequality,\n/ba∇dblφ/ba∇dbl2\n∗≤ /ba∇dblφ/ba∇dbl∗/ba∇dblTφ/ba∇dbl∗,i.e.,/ba∇dblφ/ba∇dbl∗≤ /ba∇dblTφ/ba∇dbl∗.\nTherefore, this implies that Tis injective and hence bijective on its range, and its invers eT−1has a norm less\nthan or equal to 1and can be extended to the closureMofRange(T).\nNow, on the other hand, we define\nBu0,f(φ) :=ˆ\nΩu0φ(x,0)dx+ˆT\n0ˆ\nΩφfdxdt.\nDenotingφ0:=φ(x,0), we get by Hölder inequality\n|Bu0,f(φ)| ≤ /ba∇dblu0/ba∇dblL2(Ω)/ba∇dblφ0/ba∇dblL2(Ω)+/ba∇dblf/ba∇dblL2(0,T;L2(Ω))/ba∇dblφ/ba∇dblL2(0,T;L2(Ω))≤cu0,f/ba∇dblφ/ba∇dbl∗,\nand thus,/vextendsingle/vextendsingleBu0,f(T−1ψ)/vextendsingle/vextendsingle≤c/vextenddouble/vextenddoubleT−1ψ/vextenddouble/vextenddouble\n∗≤c/ba∇dblψ/ba∇dbl∗.\nSinceBu0,fandT−1both are linear, therefore their composition is also so, and by above line, Bu0,f◦ T−1is\nbounded too, therefore, by applying the Fréchet-Riesz Theo rem again, there exists a unique u∈Msuch that\nBu0,f(T−1ψ) =/a\\}b∇acketle{tψ,u/a\\}b∇acket∇i}ht∗for everyψ∈M. We denote φ=T−1ψand so\nBu0,f(φ) =/a\\}b∇acketle{tTφ,u/a\\}b∇acket∇i}ht∗=Lφ(u)\nthat is,\nˆT\n0ˆ\nΩ−uφtdxdt+ˆT\n0ˆ\nΩ∇u·∇φdxdt+ˆT\n0E(u(x,t),φ(x,t))dt=ˆT\n0ˆ\nΩfφdxdt+ˆ\nΩu(x,0)φ(x,0)dx,\nwhereφ∈L2(0,T;Hs\n0(Ω))∩L2(0,T;H1\n0(Ω))≡L2(0,T;H1\n0(Ω))andφt∈L2(0,T;H−1(Ω)). Finally, by a density\nargument, one can conclude, integrating by parts, that u∈L2(0,T;H1\n0(Ω)),ut∈L2(0,T;H−1(Ω)), and\nˆT\n0ˆ\nΩutφdxdt+ˆT\n0ˆ\nΩ∇u·∇φdxdt+ˆT\n0E(u(x,t),φ(x,t))dt=ˆT\n0ˆ\nΩfφdxdt.\nSinceu∈L2(0,T;H1\n0(Ω)),ut∈L2(0,T;H−1(Ω))implies that u∈ C([0,T],L2(Ω))and so we have u(·,t)→u0\nstrongly in L2(Ω), ast→0. Thusu(x,t)is an energy solution of (2.2).\nNow we show that u≥0provided that fandu0are nonnegative. We write u=u+−u−, whereu+= max{u,0}χΩ\nandu−= max{−u,0}χΩ. We takeφ=u−χ(0,˜t),˜t>0, as a test function. Since f≥0, andφ≥0, we have\n0≤ˆT\n0ˆ\nΩfφdxdt=ˆ˜t\n0ˆ\nΩfu−dxdt. (2.4)\nOn the other hand, since uis0in(Rn\\Ω)×(0,T), we have that\nˆT\n0¨\nR2n(u(x,t)−u(y,t))(φ(x,t)−φ(y,t))K(x,y,t)dxdydt\n=ˆ˜t\n0¨\nR2n\\(Ωc×Ωc)(u(x,t)−u(y,t))(u−(x,t)−u−(y,t))K(x,y,t)dxdydt\n=ˆ˜t\n0ˆ\nΩˆ\nΩ(u(x,t)−u(y,t))(u−(x,t)−u−(y,t))K(x,y,t)dxdydt+2ˆ˜t\n0ˆ\nΩˆ\nΩcu(x,t)u−(x,t)K(x,y,t)dydxdt.\nMoreover, (u+(x,t)−u+(y,t))(u−(x,t)−u−(y,t))≤0, and thus\nˆ˜t\n0ˆ\nΩˆ\nΩ(u(x,t)−u(y,t))(u−(x,t)−u−(y,t))K(x,y,t)dxdydt\n≤ −ˆ˜t\n0ˆ\nΩˆ\nΩ(u−(x,t)−u−(y,t))2K(x,y,t)dxdydt≤0.\n8Further, we have\nˆ˜t\n0ˆ\nΩˆ\nΩcu(x,t)u−(x,t)K(x,y,t)dydxdt=−ˆ˜t\n0ˆ\nΩˆ\nΩcu2\n−(x,t)K(x,y,t)dydxdt≤0.\nTherefore, we have shown that\nˆT\n0¨\nR2n(u(x,t)−u(y,t))(φ(x,t)−φ(y,t))K(x,y,t)dxdydt≤0,\nSimilarly since ∇u+·∇u−= 0, we get\nˆT\n0ˆ\nΩ∇u·∇φdxdt=ˆ˜t\n0ˆ\nΩ∇u+·∇u−dxdt−ˆ˜t\n0ˆ\nΩ|∇u−|2dxdt≤0.\nNow sinceu0≥0, sou−(·,0)≡0, and we get\nˆT\n0/a\\}b∇acketle{tut,φ/a\\}b∇acket∇i}htdt=ˆ˜t\n0ˆ\nΩ∂u\n∂tu−dxdt=ˆ\nΩˆu(x,˜t)\nu(x,0)σ−dσdx=−1\n2ˆ\nΩ(u−(x,˜t))2dx.\nCombining the above three inequalities, we get from (2.4)that\n0≤ˆT\n0ˆ\nΩfφdxdt =ˆT\n0/a\\}b∇acketle{tut,φ/a\\}b∇acket∇i}htdt+ˆT\n0ˆ\nΩ∇u·∇φdxdt\n+1\n2ˆT\n0¨\nR2n(u(x,t)−u(y,t))(φ(x,t)−φ(y,t))\n|x−y|n+2sdxdydt≤ −1\n2ˆ\nΩ(u−(x,˜t))2dx≤0,\nand this gives that ||u−(·,˜t)||L2(Ω)= 0for each˜t>0. Therefore u−≡0. So we conclude that u≥0. We observe\nthat this comparison result also guarantees the uniqueness of energy solution to (2.2).\nRemark 2.14. Observing that (u(x,t)−u(y,t))((u(x,t)−k)+−(u(y,t)−k)+)≥0, for eachk, we can show\nthat for(f,u0)∈L∞(ΩT)×L∞(Ω), the weak solution u∈L∞(ΩT). The proof will follow exactly similar to that\nof [32, Theorem 4.2.1].\nNow we relax the spaces where fandu0lie. For the case of L1data, we consider the set\nT:={φ: Ω×[0,T]→R,s.t.−φt−∆φ+(−∆)sφ=ϕinΩT,\nϕ∈ C∞\n0(ΩT),φ∈L∞(Ω×(0,T))∩Cα,β\nloc(Ω×(0,T)),\nφ(x,0)∈L∞(Ω),φ= 0in(Rn\\Ω)×(0,T],φ(x,T) = 0 inΩ},\nwhereα,β∈(0,1).\nDefinition 2.15. Let(f,u0)∈L1(ΩT)×L1(Ω)be nonnegative functions. Then u∈L1(ΩT)is a very weak\nsolution to (2.2)if we have¨\nΩTu(−φt−∆φ+(−∆)sφ)dxdt=¨\nΩTfφdxdt+ˆ\nΩu0(x)φ(x,0)dx,∀φ∈ T.\nThe next existence result is following the lines of [ 29].\nTheorem 2.16. For(f,u0)∈L1(ΩT)×L1(Ω)being nonnegative, (2.2)has a unique nonnegative very weak\nsolutionuin the sense of Definition 2.15.\nProof. Firstly, we observe that the existence of valid test functio ns is guaranteed by the result in [ 16], [21], [30]. We\nwill now obtain the solution as a limit of solutions to approx imated problems. Let um∈L2(0,T;H1\n0(Ω))∩L∞(ΩT)\nbe the solution (exists by Theorem 2.13 ) to the approximated problem\n\n(um)t−∆um+(−∆)sum=fminΩT,\num(x,t) = 0 in(Rn\\Ω)×(0,T),\num(x,0) =u0m(x) inΩ;\nwherefm=Tm(f(x,t))andu0m=Tm(u0(x))areL∞functions. Using (Tk(um))χ(0,t), fort>0(admissible by\n[29, Proposition 3]) as a test function in the approximated prob lem, it holds that,\nˆ\nΩLk(um(x,t))dx+ˆt\n0ˆ\nΩ∇um·∇Tk(um)dxdθ\n+1\n2ˆt\n0ˆ\nQ(Tk(um(x,θ))−Tk(um(y,θ)))(um(x,θ)−um(y,θ))\n|x−y|n+2sdxdydθ\n≤k/ba∇dblf/ba∇dblL1(ΩT)+C3(k)/ba∇dblu0/ba∇dblL1(Ω)+C4(k)|Ω|,(2.5)\n9whereLk(ρ) =ˆρ\n0(Tk(ξ))dξ.Notice that\n(Tk(um(x,θ))−Tk(um(y,θ)))(um(x,θ)−um(y,θ))≥(Tk(um(x,θ))−Tk(um(y,θ)))2\nand so\n∇um·∇Tk(um)≥ |∇Tk(um)|2\nand forρ>0we have\nC1(k)ρ−C2(k)≤Lk(ρ)≤C3(k)ρ+C4(k)\nand\nLk(ρ)≥C(Tk(ρ))2,\nwhereC1,C2,C3,C4are constants depending only on kand independent of m. Therefore taking supremum over\nt∈(0,T]in(2.5), we get that {Tk(um)}mis bounded in L∞(0,T;L2(Ω))∩L2(0,T;Hs\n0(Ω))∩L2(0,T;H1\n0(Ω))\nand{um}mis bounded in L∞(0,T;L1(Ω))⊂L1(ΩT).\nNow since by comparison principle proved in Theorem 2.13 ,{um}mis increasing in m, we get the existence\nof a measurable function usuch thatTk(u)∈L∞(0,T;L2(Ω))∩L2(0,T;Hs\n0(Ω))∩L2(0,T;H1\n0(Ω)),um↑u\nstrongly in L1(ΩT)andum↑ua.e inΩT. As eachumis an energy solution, therefore, um∈ C([0,T],L2(Ω))⊂\nC([0,T],L1(Ω))and at each time level t∈[0,T], we haveum(·,t)∈L1(Ω), this along with the monotonicity\nof{um}minmallows us to define the pointwise limit (a.e.) uof{um}minΩfor each time t∈[0,T]. Alsou\nsatisfiesu(·,0) =u0(·)inL1sense. Again as each um= 0in(Rn\\Ω)×(0,T), thereforeualso satisfies the same.\nWe now prove that uis a weak solution to (2.2)in the sense of Definition 2.15 . Letφ∈ T, then asumis the\nenergy solution to the approximated problem, we have¨\nΩT((um)t−∆um+(−∆)sum)φdxdt=¨\nΩTfmφdxdt.\nUsing the fact that um→ustrongly in L1(ΩT)andum(x,0) =um0(x)→u0(x) =u(x,0)inL1(Ω), we have¨\nΩT((um)t−∆um+(−∆)sum)φdxdt=¨\nΩTum(−(φ)t−∆φ+(−∆)sφ)dxdt−ˆ\nΩum(x,0)φ(x,0)dx\n=¨\nΩTumϕdxdt−ˆ\nΩum(x,0)φ(x,0)dx\n→¨\nΩTuϕdxdt−ˆ\nΩu(x,0)φ(x,0)dx\n=¨\nΩTu(−(φ)t−∆φ+(−∆)sφ)dxdt−ˆ\nΩu(x,0)φ(x,0)dx.\nNotice that in the second last line, in order to pass the limit , we have used the facts that φ(x,0)∈L∞(Ω)and\nϕ∈ C∞\n0(ΩT). Also, since φ∈L∞(Ω×(0,T)), andfm→finL1(ΩT), we get¨\nΩTfmφdxdt→¨\nΩTfφdxdt asm→ ∞.\nThus ¨\nΩTu(−(φ)t−∆φ+(−∆)sφ)dxdt=¨\nΩTfφdxdt+ˆ\nΩu0(x)φ(x,0)dx,\nanduis a weak solution to (2.2). For the uniqueness let wbe a weak solution of (2.2)with(f,u0) = (0,0), i.e.\n\nwt−∆w+(−∆)sw= 0 inΩT,\nw= 0 in(Rn\\Ω)×(0,T),\nw(x,0) = 0 inΩ;\nwe want to prove that w≡0. For that we take F∈ C∞\n0(ΩT), and letφFbe the solution of the backward problem\n\n−(φF)t−∆φF+(−∆)sφF=FinΩT,\nφF(x,t) = 0 in(Rn\\Ω)×(0,T],\nφF(x,T) = 0 inΩ.\nTakingφFas a test function we deduce that for any F∈ C∞\n0(ΩT),\nˆT\n0ˆ\nΩwFdxdt = 0,\n10that means, w= 0inD′(ΩT).\nNext, we state some important facts regarding the solution o f(2.2).\nProposition 2.17. Let(f,w0)are non-negative functions such that (f,w0)∈L∞(ΩT)×L∞(Ω). Assume that\nwbe the weak solution to the problem\n\nwt−∆w+(−∆)sw=finΩT,\nw= 0 in(Rn\\Ω)×(0,T),\nw(x,0) =w0(x) inΩ;\nthen for all 0<t1<Tand for each ω⊂⊂Ωfixed, there exists C:=C(t1,n,s,ω)>0such that\nw(x,t)≥C(t1,n,s,ω)inω×[t1,T).\nProof. We will use the weak Harnack inequality as in [ 23]. First, we show the result for any arbitrary t2< T.\nAs¯ω×[t1,t2]⊂ΩTcontainsω×[t1,t2], so it is enough to show the result for ¯ω×[t1,t2]. Now since ¯ω×[t1,t2]is\ncompact and hence every open cover admits a finite subcover, i t suffices to show the positivity of win a uniform\nneighbourhood of any arbitrary point (˜x,˜t)∈¯ω×[t1,t2].\nAs¯ω×[t1,t2]is compact, it has a finite and positive distance from the boun dary ofΩT; let us denote this by D.\nWe choose 0<r <min{D\n2,1}, and(x0,t0)∈ΩTsuch thatt0=˜t−13\n16r2andx0= ˜x.Now for this (x0,t0), we\ncan choose 0<R=Dsuch thatBR(x0)×(t0−r2,t0+r2)⊂ΩT, and thenrsatisfiesr<R\n2. Clearly, as f≥0,\nwis a supersolution to the homogeneous problem. Now by using t he facts that w≥0inΩ×(0,T)andw= 0\nin(Rn\\Ω)×(0,T), we getw≥0inRn×(0,T). Then using [ 23, Theorem 2.8 and Corollary 2.9], and f >0a.e.\ninΩT, we get the existence of a positive constant Cdepending only on ω,[t1,t2],nandssuch that\n0<C=− −¨\nV−(r\n2)w(x,t)dxdt≤essinf\nV+(r\n2)w,\nwhereV−/parenleftigr\n2/parenrightig\n=Br\n2(x0)×(t0−r2,t0−3\n4r2)andV+/parenleftigr\n2/parenrightig\n=Br\n2(x0)×(t0+3\n4r2,t0+r2). As(˜t−1\n16r2,˜t+1\n16r2)⊂\n(t0+3\n4r2,t0+r2), so the result follows.\nNow, we will extend this up to T. For this we denote D0= dist{ω,∂Ω}. AsT >0, so∃˜r∈(0,1]and˜r <D0\n2\nsuch thatω×(T−2˜r2,T)⊂ΩTwithT−1\n4˜r2>t1. By the above argument for compact time intervals, we get\nw(x,t)≥C(t1,n,s,ω)>0inω×[t1,T−1\n4˜r2].\nTherefore it just remains to show the positivity of winω×(T−1\n4˜r2,T). Forx0∈¯ω,B˜r\n2(x0)will form an open\ncover of ¯ω, and by compactness, it will have a finite subcover. So it is en ough to show the positivity of win\nB˜r\n2(x0)×(T−1\n4˜r2,T). Now aswis nonnegative in Rn×(0,T)andBD0(x0)×(T−2˜r2,T)⊂ΩT, so by [ 23,\nTheorem 2.8 and Corollary 2.9], we have\n0<C= T−7\n8˜r2\nT−2˜r2 \nB˜r\n2(x0)w(x,t)dxdt≤essinf\nB˜r\n2(x0)×(T−1\n4˜r2,T)w,\nand hence we conclude.\nWe will use the next parabolic Kato-type inequality to prove comparison results or a priori estimates.\nProposition 2.18. Letu∈L2(0,T;H1(Ω))be a weak solution of\nut−∆u+(−∆)su=finΩT, (2.6)\nwithf∈L1(ΩT)and letΦ∈ C2(R)be a convex function such that Φ′is bounded. Then\n(Φ(u))t−∆Φ(u)+(−∆)sΦ(u)≤Φ′(u)(ut−∆u+(−∆)su)\nin the sense that for all ψ∈ C2(ΩT)∩ C([0,T],L2(Ω)), with the property that ψhas spatial support compactly\ncontained in Ω, andψ≥0inΩTandψ(·,T) = 0andψ(·,0) = 0 , we have¨\nΩTΦ(u)(−ψt−∆ψ+(−∆)sψ)dxdt≤¨\nΩTΦ′(u)fψdxdt.\n11Proof. Assume that uis smooth enough, otherwise one can use approximation argum ent. Since Φis convex so\nΦ(u(x,t))−Φ(u(y,t))≤Φ′(u(x,t))(u(x,t)−u(y,t)),\nand we get\n(−∆)s(Φ(u(x,t))) =ˆ\nRn(Φ(u(x,t))−Φ(u(y,t)))\n|x−y|n+2sdy\n≤ˆ\nRnΦ′(u(x,t))(u(x,t)−u(y,t))\n|x−y|n+2sdy\n= Φ′(u(x,t))(−∆)su(x,t).\nAgain, using the fact that the double derivative of a convex f unction is nonnegative, we get\n−∆(Φ(u)) =−Φ′(u)∆u−Φ′′(u)n/summationdisplay\ni=1/parenleftbigg∂u\n∂xi/parenrightbigg2\n≤ −Φ′(u)∆u.\nNow ifψ∈ C∞\n0(ΩT), withψ≥0inΩT, we have¨\nΩTΦ(u)(−ψt−∆ψ+(−∆)sψ)dxdt=¨\nΩT(Φ′(u)ut−∆(Φ(u))+(−∆)s(Φ(u)))ψ(x,t)dxdt\n≤¨\nΩTΦ′(u)(ut−∆u+(−∆)su)ψdxdt=¨\nΩTΦ′(u)fψdxdt.\nSinceψ∈ C∞\n0(ΩT), withψ≥0inΩT, can approximate the choice of test functions of our hypothe sis, we get the\ndesired result.\nRemark 2.19. We choose the regularization of |u|by\nΦǫ(u) =/parenleftbig\n|u|2+ǫ2/parenrightbig1\n2−ǫ.\nforǫ∈(0,1). It is then easy to verify that,\n1.Φǫ(u)→ |u|uniformly as ǫ→0,\n2.Φ′\nǫ(u)is bounded uniformly with respect to ǫ.\n3.Φǫ(u)is convex.\nSo, using the above theorem we get¨\nΩTΦǫ(u)(−ψt−∆ψ+(−∆)sψ)dxdt≤¨\nΩTΦ′\nǫ(u)(ut−∆u+(−∆)su)ψdxdt=¨\nΩTΦ′\nǫ(u)fψdxdt.\nNow letting ǫ→0, we get using (2.6)¨\nΩT|u|(−ψt−∆ψ+(−∆)sψ)dxdt≤¨\nΩTsign(u)fψdxdt.\nAdding¨\nΩTu(−ψt−∆ψ+(−∆)sψ)dxdtto both side, we get\n¨\nΩTu+(−ψt−∆ψ+(−∆)sψ)dxdt≤¨\nΩTχ{u≥0}fψdxdt.\nTherefore\n∂u+\n∂t−∆u++(−∆)su+≤/parenleftbigg∂u\n∂t−∆u+(−∆)su/parenrightbigg\nsign+u\nin the weak sense.\nWe now take γto be a positive continuous function over ΩTand proceed with the following comparison\nprinciple.\nLemma 2.20. Lets∈(0,1)anda≥0. Consider (f,u0)∈L∞(ΩT)×L∞(Ω)to be non-negative bounded\nfunctions such that (f,u0)/\\e}atio\\slash= (0,0). Assume that v1,v2are two non-negative functions with finite energy such\nthatv1,v2∈L2((0,T);H1\n0(Ω))∩L∞(ΩT)with\n\n\n(v1)t−∆v1+(−∆)sv1≤f\n(v1+a)γ(x,t)inΩT,\nv1(x,t) = 0 in(Rn\\Ω)×(0,T),\nv1(x,0)≤u0(x) inΩ;\n12and\n\n(v2)t−∆v2+(−∆)sv2≥f\n(v2+a)γ(x,t)inΩT,\nv2(x,t) = 0 in(Rn\\Ω)×(0,T),\nv2(x,0)≥u0(x) inΩ;\nwhereγ∈ C(ΩT)and is positive. Then, v2≥v1inΩT.\nProof. Defineu=v1−v2, thenu∈L2(0,T;H1\n0(Ω))∩L∞(ΩT). We show that u+= 0. By hypotheses, we get\nut−∆u+(−∆)su≤f/parenleftigg\n1\n(v1+a)γ(x,t)−1\n(v2+a)γ(x,t)/parenrightigg\n.\nNow since/parenleftigg\n1\n(v1+a)γ(x,t)−1\n(v2+a)γ(x,t)/parenrightigg\n≤0in the set {u≥0}, then using Proposition 2.18 andRemark 2.19 ,\nit holds that\n(u+)t−∆(u+)+(−∆)s(u+)≤0.\nAgain we notice that u+(x,0)≤0; therefore by comparison principle as of Theorem 2.13 , we haveu+≡0, and\nthen we conclude.\nCorollary 2.21. As a consequence of the previous comparison principle we get th at fora >0andγfixed, if\n(f,u0)are non-negative functions with (f,u0)∈Lσ(ΩT)×Lσ(Ω),σ≥2, then the problem\n\n\nvt−∆v+(−∆)sv=f\n(v+a)γ(x,t)inΩT,\nv(x,t) = 0 in(Rn\\Ω)×(0,T),\nv(x,0) =u0(x) inΩ;(2.7)\nhas a unique energy solution va∈L2(0,T;H1\n0(Ω)).\nWe note that for γbeing a constant, the existence of an energy solution can be s hown using a monotonicity\nargument by observing that v= 0is a subsolution and ¯v, the unique solution to the problem\n\n¯vt−∆¯v+(−∆)s¯v=f\naγinΩT,\n¯v(x,t) = 0 in(Rn\\Ω)×(0,T),\n¯v(x,0) =u0(x) inΩ;\nis a supersolution. Then Lemma 2.20allows us to get the existence of a unique solution vto(2.7)such that\nv∈L2(0,T;H1\n0(Ω))and0≤v≤¯vinΩT.\nNow ifγis not constant and belongs to C(ΩT), we denote\nγ∗= sup\n(x,t)∈ΩTγ(x,t)andγ∗= inf\n(x,t)∈ΩTγ(x,t),\nand observe that the unique nonnegative solutions to the prob lems\n\n¯vt−∆¯v+(−∆)s¯v=f\naγ∗inΩT,\n¯v(x,t) = 0 in(Rn\\Ω)×(0,T),\n¯v(x,0) =u0(x) inΩ;\nand\n\n¯vt−∆¯v+(−∆)s¯v=f\naγ∗inΩT,\n¯v(x,t) = 0 in(Rn\\Ω)×(0,T),\n¯v(x,0) =u0(x) inΩ;\nare sub and supersolution (respectively super and subsoluti on) of\n\n¯vt−∆¯v+(−∆)s¯v=f\naγ(x,t)inΩT,\n¯v(x,t) = 0 in(Rn\\Ω)×(0,T),\n¯v(x,0) =u0(x) inΩ;(2.8)\nfora≥1(respectively for a <1). Then, by monotonicity argument and comparison principle si milar to\n13Lemma 2.20, we get the existence of a unique nonnegative solution to (2.8), which will be a supersolution to\n(2.7)withv= 0being its subsolution. So (2.7)has a unique nonnegative solution in this case too. We note th at\nthese solutions also satisfy the comparison principle, in the sense that if a1<a2, then we have va2≤va1.\nWe now list approximated problems for γbeing a constant. Firstly when (f,u0)∈L∞(ΩT)×L∞(Ω), we\nconsider the following problem for each k∈N,\n\n(uk)t−∆uk+(−∆)suk=f(x,t)\n(uk+1\nk)γinΩT,\nuk= 0 in(Rn\\Ω)×(0,T),\nuk(x,0) =u0(x) inΩ.(2.9)\nWe now take fandu0may not be bounded and consider\n\n(uk)t−∆uk+(−∆)suk=fk(x,t)\n(uk+1\nk)γinΩT,\nuk= 0 in(Rn\\Ω)×(0,T),\nuk(x,0) =u0k(x) inΩ;(2.10)\nwherefk(x,t) =Tk(f(x,t))andu0k(x) =Tk(u0(x)).\nTo treat variable exponent, we take (f,u0)∈L1(ΩT)×L1(Ω)and consider the following approximating problem.\n\n(uk)t−∆uk+(−∆)suk=fk(x,t)\n(uk+1\nk)γ(x,t)inΩT,\nuk= 0 in(Rn\\Ω)×(0,T),\nuk(x,0) =u0k(x) inΩ.(2.11)\nWe will prescribe conditions on fandu0later on. With these settings, we have the following existen ce result.\nTheorem 2.22. For each of the above problems (2.9)–(2.11), there exists a unique nonnegative energy solution\nukfor eachk∈Nsatisfying the followings:\n(a) eachukis bounded,\n(b) the sequence {uk}kis increasing in k,\n(c) for each t0>0andω⊂⊂ΩT,∃C(ω,t0,n,s)such thatuk≥C(ω,t0,n,s).\nProof. The existence, uniqueness and nonnegativity of uk’s follow from the comparison principle in Lemma 2.20\nandCorollary 2.21 . Sincefandu0are nonnegative, therefore 0≤fk≤fk+1and0≤u0k≤u0(k+1)for each\nkand then using the similar technique as that of Lemma 2.20 , we obtain that the sequence {uk}kis increasing\nink. Therefore uk≥u1for allk. Now forf∈L∞(ΩT)we havef\n(u1+1)γ≤fand forf /∈L∞(ΩT)we have\nf1\n(u1+1)γ(x,t)≤f1and hencef1\n(u1+1)γ(x,t)∈L∞(ΩT). Therefore u1∈L∞(ΩT)and by Proposition 2.17 , we\nobtain that for all ω⊂⊂Ωand for allt0>0,u1(x,t)≥C(ω,t0,n,s)inω×[t0,T). Thus for all k≥1, we have\nuk(x,t)≥C(ω,t0,n,s)inω×[t0,T). Also, we observe that for each k,fk\n(uk+1\nk)γ(x,t)≤max{1,kγ∗}fk,where\nγ∗= sup\n(x,t)∈ΩTγ(x,t). Therefore each ukis bounded.\nRemark 2.23. As eachuk∈ C([0,T],L2(Ω))⊂ C([0,T],L1(Ω)), therefore at each time level t∈[0,T], we\nhaveuk(·,t)∈L1(Ω), this along with the monotonicity of {uk}kinkallows us to define the pointwise limit\n(a.e.)uof{uk}kinΩat each time t∈[0,T]. Alsousatisfiesu(·,0) =u0(·)inL1sense. Again as each\nuk= 0in(Rn\\Ω)×(0,T), therefore ualso satisfies the same. Further we have u≥ukfor eachkand hence\nu(x,t)≥C(ω,t0,n,s)inω×[t0,T)for eachω⊂⊂Ωandt0>0.\nTo treat the singular case, we define next as:\nDefinition 2.24. Let(f,u0)∈L1(ΩT)×L1(Ω)be a pair of non-negative functions and γ >0is a constant. We\nsay thatuis a very weak solution to the problem\n\nut−∆u+(−∆)su=f(x,t)\nuγinΩT,\nu= 0 in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x) inΩ;(2.12)\n14ifu∈L1(ΩT)satisfyingu≡0in(Rn\\Ω)×(0,T),∀ω⊂⊂Ωand∀t0>0,∃c≡c(ω,t0,n,s)>0such that\nu(x,t)≥c>0, inω×[t0,T),u(·,0) =u0(·), and for all ϕ∈ C∞\n0(ΩT), we have¨\nΩTu(−ϕt−∆ϕ+(−∆)sϕ)dxdt=¨\nΩTfϕ\nuγdxdt.\nDefinition 2.25. Let(f,u0)be a pair of non-negative functions with u0∈L1(Ω),f∈L1(ΩT)andγ∈ C(ΩT)\nbe a positive function. Then we say that u, such that u≡0in(Rn\\Ω)×(0,T), is a very weak solution to the\nproblem\n\nut−∆u+(−∆)su=f(x,t)\nuγ(x,t)inΩT,\nu= 0 in(Rn\\Ω)×(0,T),\nu(x,0) =u0(x), inΩ;(2.13)\nifu∈L1(ΩT)and∀ω⊂⊂Ωand∀t0>0,∃c≡c(ω,t0,n,s)>0such thatu(x,t)≥c >0, inω×[t0,T),\nu(·,0) =u0(·), and for all ϕ∈ C∞\n0(ΩT), we have¨\nΩTu(−ϕt−∆ϕ+(−∆)sϕ)dxdt=¨\nΩTfϕ\nuγ(x,t)dxdt.\n2.4.Main Results\nFirst, we consider γto be a constant and then take it as a positive continuous func tion. In this section, we\nstate our main results depending on γand the regularity of initial conditions.\n2.4.1.Existence for bounded data for γbeing a constant\nIn the case of bounded data, we will have the next existence re sult.\nTheorem 2.26. Let(f,u0)∈L∞(ΩT)×L∞(Ω)be a pair of non-negative functions and γ >0. Then (2.12)\nhas a bounded very weak positive solution uin the sense of Definition 2.24such thatuγ+1\n2∈L2(0,T;H1\n0(Ω))and\nu∈ C([0,T],L2(Ω)). Moreover if γ≤1orSupp(f)⊂⊂ΩT, thenu∈L2(0,T;H1\n0(Ω)).\n2.4.2.Existence for general data for γbeing a constant\nIn this subsection, we will consider general data. Accordin g to the regularity of initial conditions, we will\nconsider two cases as u0∈Lγ+1(Ω)andu0∈L1(Ω). Let us state the following existence results.\nTheorem 2.27. Let(f,u0)∈L1(ΩT)×Lγ+1(Ω)be a pair of non-negative functions and γ >0. Then (2.12)\nhas a very weak positive solution uin the sense of Definition 2.24such thatuγ+1\n2∈L2(0,T;H1\n0(Ω))andu∈\nL2(0,T;Lσ(Ω)), whereσ=n(1+γ)\nn−2andσ≥2ifγ≥1.\nTheorem 2.28. Let(f,u0)∈L1(ΩT)×L1(Ω)be a pair of non-negative functions and γ >0. Then (2.12) has a\nvery weak positive solution uin the sense of Definition 2.24such thatTk(u)γ+1\n2∈L2(0,T;H1\n0(Ω))for allk>0.\n2.4.3.Improved results for γbeing a constant\nWe now break γin three parts namely 0<γ <1,γ= 1andγ >1. We improve our results to find solutions\nin better spaces. For this, we take u0∈Lmax(γ+1,2)(Ω).The caseγ= 1will be same as that of Theorem 2.27 .\nForγ >1, we cannot find solutions in L2(0,T;H1\n0(Ω)). In fact if we look for L2(0,T;H1(Ω))estimates, we will\nonly get them in L2(t0,T;H1\nloc(Ω))for eacht0>0. We state our next result for γ >1as follows:\nTheorem 2.29. Letγ >1. Assume that (f,u0)∈L1(ΩT)×Lγ+1(Ω)be a pair of non-negative functions. Then\n(2.12) has a very weak positive solution uin the sense of Definition 2.24such thatuγ+1\n2∈L2(0,T;H1\n0(Ω))and\nu∈L2(t0,T;H1\nloc(Ω))∩L∞(0,T;L1+γ(Ω))for eacht0>0.\nNow we consider the case 0<γ <1, and state our results as follows:\nTheorem 2.30. Let0<γ <1. Assume that u0∈L2(Ω)withu0≥0andf≥0is such that\ni)f∈L2\nγ+1/parenleftig\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightig\n, or\nii)f∈L¯m(ΩT)with¯m:=2(n+2)\n2(n+2)−n(1−γ).\nThen (2.12) has a very weak solution u∈L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω))in the sense of Definition 2.24.\n15Remark 2.31. Asγ <1, so/parenleftbigg2∗\n1−γ/parenrightbigg′\n=2n\n2n−(1−γ)(n−2)<¯m<2\nγ+1,\nand the two spaces L2\nγ+1/parenleftig\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightig\nandL¯m(ΩT)are not comparable. Also the case γ= 1cannot be\nconsidered here since in the proof of Theorem 2.30, we will use Hölder inequality with exponents2\nγ+1and\n2\n1−γ. Ifγ→1, then¯m→1and2\nγ+1and/parenleftbigg2∗\n1−γ/parenrightbigg′\nboth tend to 1, so thatfwill belong to L1(ΩT).\nNow form<¯m, we no longer find solutions in L2(0,T;H1\n0(Ω))but in a larger space depending on m.\nTheorem 2.32. Let0< γ <1. Assume that 0≤f∈Lm(ΩT), with1≤m <¯m, and thatu0∈L2(Ω)be\nnonnegative. Then the problem (2.12) admits a very weak solution u∈L¯q(0,T;W1,¯q\n0(Ω))∩L∞(0,T;L1+γ(Ω)),\nwith\n¯q=m(γ+1)(n+2)\nn+2−m(1−γ).\nMoreoveru∈Lσ(ΩT), where\nσ=m(γ+1)(n+2)\nn−2(m−1).\nRemark 2.33. Observe that we can get rid of the fact that u∈L¯q(0,T;Ws1,¯q\n0(Ω))withs1< sas is done\nin [1, Theorem 11]. For our case, due to the presence of the leading Laplacian operator, we will get u∈\nL¯q(0,T;Ws,¯q\n0(Ω)).\nRemark 2.34. Clearly¯q≥m(γ+1)>1andσ≥m(γ+1)>1. Alsom<¯mis equivalent to ¯q<2which implies\nL2/parenleftbig\n0,T;H1\n0(Ω)/parenrightbig\n⊂L¯q(0,T;W1,¯q\n0(Ω)). InTheorem 2.32the caseγ= 1is not allowed since it yields ¯q= 2m≥2\nwhich contradicts ¯q<2.\n2.4.4.Further summability for γbeing a constant\nHere we state two results regarding the optimal summability of our solutions in terms of summability of the\ninitial data f∈Lr(0,T;Lq(Ω)). For theL∞-boundedness, we take u0∈L∞(Ω), and1/rand1/qare in the\nAronson-Serrin domain, see [ 2,29]. We now state the corresponding theorem as:\nTheorem 2.35. Assume that f∈Lr(0,T;Lq(Ω))withr,qsatisfying\n1\nr+n\n2q<1,\nand suppose that u0∈L∞(Ω). Then there exists a positive constant csuch that the unique finite energy solution\nof(2.10) satisfies\n/ba∇dbluk(x,t)/ba∇dblL∞(ΩT)≤c,\nand the solution uobtained as the limit of uksatisfies,u∈L∞(ΩT). Moreover for γ≤1,u∈L2(0,T;H1\n0(Ω)).\nOutside the Aronsom-Serrin zone, the solutions are not expe cted to be bounded. We divide the region by the\nstraight line1\nr=n\nn−21\nq−2\nn−2in two parts. Further, we take u0≡0. The following summability results we\nwill get for this case.\nTheorem 2.36. Assumeu0(x)≡0andf∈Lr(0,T;Lq(Ω)), withr>1,q>1satisfy\n1<1\nr+n\n2q.\nFurther, assume that for γ <1,\ni)if1\nr<n\nn−21\nq−2\nn−2, thenq>/parenleftbigg2∗\n1−γ/parenrightbigg′\n, and\nii)if1\nr≥n\nn−21\nq−2\nn−2, thenr>2\n1+γ.\nThen there exists a positive constant csuch that the sequence of finite energy solutions of (2.10) satisfies\n/ba∇dbluk/ba∇dblL∞(0,T;L2σ(Ω))+/ba∇dbluk/ba∇dblL2σ(0,T;L2∗σ)≤c,\n16where\nσ=\n\nq(n−2)(γ+1)\n2(n−2q)if1\nr<n\nn−21\nq−2\nn−2,\nqrn(γ+1)\n2(nr+2q−2qr)if1\nr≥n\nn−21\nq−2\nn−2.\nFurther, the solution uobtained as the limit of uksatisfies,u∈L∞(0,T;L2σ(Ω))∩L2σ(0,T;L2∗σ(Ω)).\nRemark 2.37. InTheorem 2.36, we observe that the conditions needed for γ <1are obvious, as Theorem 2.30\nimplies that if r=2\n1+γandq=/parenleftbigg2∗\n1−γ/parenrightbigg′\n, then by Sobolev embedding, u∈L2(0,T;L2∗(Ω))which matches\nwithTheorem 2.36as we getσ= 1in this case.\nRemark 2.38. We note that, for γ≥1, the singularity allows us to get better summability results as we can\ngo up tor= 1andq= 1, which is not allowed for the nonsingular case (see [ 10,29]), whereas for γ <1, the\nsingularity gives us better summability for the spatial expo nentqonly.\n2.4.5.Existence results for γbeing a function\nWe now consider γto be a positive continuous function on ΩT. We will mainly see the behaviour of γnear the\nparabolic boundary, and accordingly, we will state two exis tence results. We recall the strip around the parabolic\nboundary given by (ΩT)δ={(x,t)∈ΩT: dist((x,t),ΓT)<δ}forδ >0, whereΓT= (Ω×{t= 0})∪(∂Ω×(0,T)).\nTheorem 2.39. Let∃δ >0such thatγ(x,t)≤1in(ΩT)δ. Also letu0∈L2(Ω)withu0≥0andf≥0satisfies\ni)f∈L2/parenleftig\n0,T;L(2n\nn+2)(Ω)/parenrightig\n, or\nii)f∈L¯r(ΩT)with¯r:=2(n+2)\nn+4.\nThen (2.13) has a very weak solution u∈L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω))in the sense of Definition 2.25.\nRemark 2.40. Since2n\nn+2<2(n+2)\nn+4= ¯r <2, so the two spaces L2/parenleftig\n0,T;L(2n\nn+2)(Ω)/parenrightig\nandL¯r(ΩT)are not\ncomparable. Also for the constant case ( Theorem 2.30) we got larger possible spaces L2\nγ+1/parenleftig\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightig\nand\nL¯m(ΩT)for the belonging of initial data f, which gives us broader results, this is the cause of conside ring the\nconstant and nonconstant cases separately.\nTheorem 2.41. Assume that for some γ∗>1and someδ >0, we have /ba∇dblγ/ba∇dblL∞((ΩT)δ)< γ∗. Assume that\nu0∈Lγ∗+1(Ω)withu0≥0andf≥0is such that\ni)f∈Lγ∗+1/parenleftbigg\n0,T;L/parenleftBig\nn(γ∗+1)\nn+2γ∗/parenrightBig\n(Ω)/parenrightbigg\n, or\nii)f∈L˜r(ΩT)with˜r:=(n+2)(γ∗+1)\nn+2(γ∗+1).\nThen (2.13) admits a very weak nonnegative solution uin the sense of Definition 2.25such thatuγ∗+1\n2∈\nL2(0,T;H1\n0(Ω))andu∈L2(t0,T;H1\nloc(Ω))∩L∞(0,T;L1+γ∗(Ω))for eacht0>0.\nRemark 2.42. Sincen(γ∗+1)\nn+2γ∗<(n+2)(γ∗+1)\nn+2(γ∗+1)= ˜r<γ∗+1,so the two spaces Lγ∗+1/parenleftbigg\n0,T;L/parenleftBig\nn(γ∗+1)\nn+2γ∗/parenrightBig\n(Ω)/parenrightbigg\nandL˜r(ΩT)are not comparable. Also, for the constant case ( Theorem 2.29), we got the largest possible space\nL1(ΩT)for the belonging of initial data f.\n3. Proof of main results\nProof of Theorem 2.26\nWe consider the approximated problems (2.9)in this case and show that {uk}kis bounded in L∞(ΩT). Note\nthat the existence and other properties of {uk}kfollow by Theorem 2.22 . We define wk=H(uk) =uγ+1\nk, and\nnote that as uk∈L2(0,T;H1\n0(Ω))is bounded, so uγ+1\nk∈L2(0,T;H1\n0(Ω)). Then, using Kato inequality as in\nProposition 2.18 , we get\n(wk)t−∆wk+(−∆)swk≤H′(uk)((uk)t−∆uk+(−∆)suk)≤(γ+1)fin weak sense .\n17Now letϑbe the unique solution to the problem\n\nϑt−∆ϑ+(−∆)sϑ= (γ+1)finΩT,\nϑ= 0 in(Rn\\Ω)×(0,T),\nϑ(x,0) =H(u0(x)) inΩ.\nAsfandu0are bounded, so by Remark 2.14 ,ϑ∈L∞(ΩT)and by comparison principle wk≤ϑfor allk. Hence,\nuγ+1\nk≤ϑand the claim follows.\nNow sinceuk∈L∞(ΩT)∩L2(0,T;H1\n0(Ω))and nonnegative, then for any ε>0andθ>0,((uk(x,t)+ε)θ−εθ)∈\nL2(0,T;H1\n0(Ω)). So choosing 0<ε<1/k, fort∈(0,T], we take ((uk(x,θ)+ε)γ−εγ)χ(0,t)as a test function in\n(2.9), and it holds that\nˆt\n0ˆ\nΩ(uk)t((uk+ε))γ−εγ)dxdθ+ˆt\n0ˆ\nΩ∇uk·∇((uk+ε)γ−εγ)dxdθ\n+1\n2ˆt\n0ˆ\nQ(uk(x,θ)−uk(y,θ))((uk(x,θ)+ε)γ−(uk(y,θ)+ε)γ)\n|x−y|n+2sdxdydθ\n≤¨\nΩtf(x,θ)dxdθ.\nNow letting ε→0, by Fatou’s lemma, we get for all t≤T\n1\nγ+1ˆ\nΩuγ+1\nk(x,t)dx+ˆt\n0ˆ\nΩ∇uk·∇uγ\nkdxdθ\n+1\n2ˆt\n0ˆ\nQ(uγ\nk(x,θ)−uγ\nk(y,θ))(uk(x,θ)−uk(y,θ))\n|x−y|n+2sdxdydθ\n≤¨\nΩtfdxdθ+1\nγ+1ˆ\nΩuγ+1\n0(x)dx≤ /ba∇dblf/ba∇dblL∞(ΩT)|ΩT|+1\nγ+1/ba∇dblu0/ba∇dblγ+1\nL∞(Ω)|Ω|.(3.1)\nSince we know ∇uk·∇uγ\nk=γuγ−1\nk|∇uk|2,and by Lemma 2.11\n(uγ\nk(x,θ)−uγ\nk(y,θ))(uk(x,θ)−uk(y,θ))≥C(γ)/parenleftig\nuγ+1\n2\nk(x,θ)−uγ+1\n2\nk(y,θ)/parenrightig2\n,\nwe get taking supremum over t∈(0,T]in(3.1), that the sequence/braceleftig\nuγ+1\n2\nk/bracerightig\nkis bounded in L∞(0,T,L2(Ω))∩\nL2(0,T;Xs\n0(Ω))∩L2(0,T;H1\n0(Ω))≡L∞(0,T,L2(Ω))∩L2(0,T;H1\n0(Ω)).\nNow by Theorem 2.22 andRemark 2.23 , as the sequence {uk}kis increasing in k, so the pointwise limit u\nof{uk}kexists; and satisfies u(·,0) =u0(·)andu= 0in(Rn\\Ω)×(0,T). Also since/braceleftig\nuγ+1\n2\nk/bracerightig\nkis bounded\ninL2(0,T;H1\n0(Ω)), therefore uγ+1\n2\nk⇀ uγ+1\n2inL2(0,T;H1\n0(Ω)). Again by Beppo Levi theorem uk→uin\nL1(ΩT). Using Fatou’s Lemma, we have uγ+1\n2∈L∞(0,T;L2(Ω))∩L∞(ΩT). Also, by Remark 2.23 we get\nu(x,t)≥C(ω,t0,n,s)inω×[t0,T)for anyω⊂⊂Ωandt0>0. Using this positivity, it is then easy to show\nthatuis a very weak solution in the sense of Definition 2.24 and is shown in detail in the proof of Theorem 2.27 .\nIn order to show that u∈ C([0,T],L2(Ω)), we fixk≥land hence uk≥uland note that the function ˜u=\n(uk+1\nk)−(ul+1\nl)∈L2(0,T;H1(Ω))satisfies the equation\n\n\n˜ut−∆˜u+(−∆)s˜u=f(x,t)\n(uk+1\nk)γ−f(x,t)\n(ul+1\nl)γ,inΩT,\n˜u=1\nk−1\nl, in(Rn\\Ω)×(0,T),\n˜u(x,0) =1\nk−1\nl, inΩ;\nNow as/parenleftbiggf\n(uk+1\nk)γ−f\n(ul+1\nl)γ/parenrightbigg\n≤0in the set {˜u≥0}, using Proposition 2.18 andRemark 2.19 , it holds that\n(˜u+)t−∆(˜u+)+(−∆)s(˜u+)≤0.\nAgain we notice that ˜u+∈L2(0,T;H1\n0(Ω)), has finite energy and ˜u+(x,0) = 0 , therefore by comparison principle,\nit holds that ˜u+≡0, and then we have (uk+1\nk)≤(ul+1\nl)fork≥l. Therefore we get 0≤uk−ul≤1\nl−1\nk,\nfork≥l. This implies that the sequence {uk}kis Cauchy in C([0,T],L2(Ω))and henceu∈ C([0,T],L2(Ω)).\n18We now consider that γ≤1, then using ukχ(0,t)as a test function in (2.9), we get that\n1\n2ˆ\nΩu2\nk(x,t)dx+ˆt\n0ˆ\nΩ|∇uk|2dxdθ+1\n2ˆt\n0ˆ\nQ(uk(x,θ)−uk(y,θ))2\n|x−y|n+2sdxdydθ\n≤¨\nΩtfu1−γ\nkdxdθ+1\n2ˆ\nΩu2\n0(x)dx.\nSince{uk}kis uniformly bounded in L∞(ΩT), andγ≤1, so¨\nΩtfu1−γ\nkdxdθ≤C/ba∇dblf/ba∇dblL∞(ΩT),and then we\nconclude that {uk}kis bounded in the space L2(0,T;H1\n0(Ω))∩L∞(ΩT). We note that the same conclusion holds\nifSupp(f)⊂⊂ΩTtaking into consideration that uk≥u1≥CinSupp(f)for eachk.\nProof of Theorem 2.27\nWe consider here the approximating problem (2.10) and refer Theorem 2.22 for properties of uk. Asfkand\nu0kare bounded and nonnegative, therefore uk∈L2(0,T;H1\n0(Ω))is bounded and nonnegative, so we can choose\nthe same test function ((uk(x,θ)+ε)γ−εγ)χ(0,t),0<ε<1/k,in(2.10) also, and let ε→0to get\n1\nγ+1ˆ\nΩuγ+1\nk(x,t)dx+ˆt\n0ˆ\nΩ∇uk·∇uγ\nkdxdθ\n+1\n2ˆt\n0ˆ\nQ(uγ\nk(x,θ)−uγ\nk(y,θ))(uk(x,θ)−uk(y,θ))\n|x−y|n+2sdxdydθ\n≤¨\nΩtfkdxdθ+1\nγ+1ˆ\nΩuγ+1\n0k(x)dx≤ /ba∇dblf/ba∇dblL1(ΩT)+1\nγ+1/ba∇dblu0/ba∇dblLγ+1(Ω).\nSimilarly like the proof of Theorem 2.26 , it holds that/braceleftig\nuγ+1\n2\nk/bracerightig\nkis bounded in L∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω)).\nNow as{uk}kis increasing in k, we get by Fatou’s Lemma and Beppo Levi’s Lemma, that the poin twise limit\n(as mentioned in Remark 2.23 )usatisfiesuγ+1\n2∈L∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω))anduk↑ustrongly in\nL1(ΩT). Sinceuγ+1\n2∈L2(0,T;H1\n0(Ω))andn(1+γ)\nn−2=2∗(1+γ)\n2, embedding result on uγ+1\n2impliesu∈\nL2/parenleftig\n0,T;L(n(1+γ)\nn−2)(Ω)/parenrightig\n. Also, we have u(x,t)≥C(ω,t0,n,s)inω×[t0,T)for anyω⊂⊂Ωandt0>0. We\nshow thatuis a very weak solution to (2.12) in the sense of Definition 2.24 . Let us take arbitrary φ∈ C∞\n0(ΩT),\nwe then have ¨\nΩT((uk)t−∆uk+(−∆)suk)φdxdt=¨\nΩTfk\n(uk+1\nk)γφdxdt.\nNowφ∈ C∞\n0(ΩT)impliesφt,∆φand(−∆)sφall are bounded. Then as uk→ustrongly in L1(ΩT), we have¨\nΩT((uk)t−∆uk+(−∆)suk)φdxdt\n=¨\nΩTuk(−(φ)t−∆φ+(−∆)sφ)dxdt→¨\nΩTu(−(φ)t−∆φ+(−∆)sφ)dxdt.\nNow since for all ω⊂⊂Ωand for allt0>0,uk(x,t)≥C(ω,t0,n,s)inω×[t0,T), we get the positivity of {uk}k\nasuk(x,t)≥CinSuppφfor allk≥1. Hence, we can use the dominated convergence theorem to obta in¨\nΩTfk\n(uk+1\nk)γφdxdt→¨\nΩTf\nuγφdxdt ask→ ∞.\nThus ¨\nΩT(ut−∆u+(−∆)su)φdxdt=¨\nΩTf\nuγφdxdt.\nProof of Theorem 2.28\nHere we consider (2.10) but with suffix m∈Nas\n\n(um)t−∆um+(−∆)sum=fm(x,t)\n(um+1\nm)γinΩT:= Ω×(0,T),\num= 0 in(Rn\\Ω)×(0,T),\num(x,0) =u0m(x) inΩ.(3.2)\n19The properties of um’s follow from Theorem 2.22 . Asum∈L2(0,T;H1\n0(Ω))is nonnegative and Tk(um)is\nbounded, so using ((Tk(um)+ε)γ−εγ)χ(0,t)as a test function in (3.2), we get\nˆt\n0ˆ\nΩ(um)t((Tk(um)+ε)γ−εγ)dxdθ+ˆt\n0ˆ\nΩ∇uk·∇((Tk(um)+ε)γ−εγ)dxdθ\n+1\n2ˆt\n0ˆ\nQ(uk(x,θ)−uk(y,θ))((Tk(um(x,θ))+ε)γ−(Tk(um(y,θ))+ε)γ)\n|x−y|n+2sdxdydθ≤¨\nΩTf(x,θ)dxdθ.\nWe have chosen 0<ε<1/karbitrarily in above. Now letting ε→0, by Fatou’s lemma, we get for all t≤T\nˆ\nΩLk(um(x,t))dx+ˆt\n0ˆ\nΩ∇um·∇Tγ\nk(um)dxdθ\n+1\n2ˆt\n0ˆ\nQ(Tγ\nk(um(x,θ))−Tγ\nk(um(y,θ)))(um(x,θ)−um(y,θ))\n|x−y|n+2sdxdydθ\n≤ /ba∇dblf/ba∇dblL1(ΩT)+C3(k)/ba∇dblu0/ba∇dblL1(Ω)+C4(k)|Ω|,\nwhereLk(ρ) =ˆρ\n0(Tk(ξ))γdξ. Notice that\n(Tγ\nk(um(x,θ))−Tγ\nk(um(y,θ)))(um(x,θ)−um(y,θ))\n≥(Tγ\nk(um(x,θ))−Tγ\nk(um(y,θ)))(Tk(um(x,θ))−Tk(um(y,θ)))\nand so\n∇um·∇Tγ\nk(um)≥γTγ−1\nk(um)|∇Tk(um)|2\nand forρ>0, trivial calculation yields that\nC1(k)ρ−C2(k)≤Lk(ρ)≤C3(k)ρ+C4(k)andLk(ρ)≥C(Tk(ρ))γ+1,\nwhereC1,C2,C3,C4,Care positive constants depending only on k. Therefore/braceleftig\n(Tk(um))γ+1\n2/bracerightig\nmis bounded in\nL∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω))and{um}mis bounded in L∞(0,T;L1(Ω)).\nNow as{um}mis increasing in m, by Fatou’s Lemma and Beppo Levi’s theorem, the pointwise li mituof{um}m\n(as mentioned in Remark 2.23 ) satisfies (Tk(u))γ+1\n2∈L∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω)),um↑ustrongly in\nL1(ΩT). Alsou(·,0) =u0(·)inL1sense anduis0outsideΩ. Thenucan be shown to be a very weak solution\nof(2.12) in the sense of Definition 2.24 , and the proof is same as that of Theorem 2.27 .\nRemark 3.1. In view of Theorem 2.26, if we consider the approximating problem\n\n\n(uk)t−∆uk+(−∆)suk=fk(x,t)\nuγ\nkinΩT,\nuk= 0 in(Rn\\Ω)×(0,T),\nuk(x,0) =u0k(x) inΩ;\nforγ≤1, then for each k,ukexists anduk∈L2(0,T;H1\n0(Ω))∩C([0,T],L2(Ω)). Also, each ukis unique (see\nRemark 3.3). Further, the sequence {uk}ksatisfy the followings:\n(a) eachukis bounded and nonnegative,\n(b) the sequence {uk}kis increasing in k(can be shown similarly like Lemma 2.20),\n(c) for each t0>0andω⊂⊂ΩT,∃C(ω,t0,n,s)such thatuk≥C(ω,t0,n,s).\nNow forγ≤1, we have (uk+ε)γ−εγ≤uγ\nk, and hencefk((uk+ε)γ−εγ)\nuγ\nk≤fk. Therefore we can follow\nthe same procedure as that of Theorem 2.27to get that the pointwise limit uof{uk}kis a very weak solution of\n(2.12) and satisfies the corresponding properties of Theorem 2.27. Now letk≥lbe fixed, then uk≥uland\n(uk−ul)t−∆(uk−ul)+(−∆)s(uk−ul) =fk(x,t)\nuγ\nk−fl(x,t)\nuγ\nl≤fk−fl\nuγ\nkinΩT.\nHence using ((uk−ul+ε)γ−εγ)χ(0,t),0<ε<<1as a test function in the above inequality and letting ε→0\nit holds that\n1\nγ+1ˆ\nΩ(uk(x,t)−ul(x,t))γ+1dx≤¨\nΩt(fk−fl)dxdθ+1\nγ+1ˆ\nΩ(u0k(x)−u0l(x))γ+1(x)dx.\nNow as by Dominated Convergence Theorem, fk↑fstrongly in L1(ΩT)andu0k↑u0strongly in Lγ+1(Ω)if\nu0∈Lγ+1(Ω), we get that the sequence {uk}kis a Cauchy sequence in the space C([0,T];Lγ+1(Ω))and hence in\nC([0,T];L1(Ω)). Therefore this approximation allows us to have u∈ C([0,T],L1(Ω)).\n20Note that the same approximation technique will not work for γ >1, as in that case we will have uγ+1\n2\nk∈\nL2(0,T;H1\n0(Ω))which may not give us uk∈L2(0,T;H1\n0(Ω))and the monotonicity of {uk}kcannot be proven in\nsimilar way like Lemma 2.20. However, we will take approximations as (2.10) for convenience even for γ≤1.\nFurther, this approximation technique will not work for u0∈L1(Ω), as in that case, we need to take test functions\nlike((Tm(uk−ul)+ε)γ−εγ)χ(0,t)and will end up withˆ\nΩLm(uk(x,t)−ul(x,t))dxin the left which will not\ngive us anything desired.\nProof of Theorem 2.29\nWe consider the approximated problems (2.10) here and refer Theorem 2.22 . Since each uk∈L2(0,T;H1\n0(Ω))\nis bounded and γ >1so we can use uγ\nkχ(0,t)as a test function in (2.10) , to get that, for all t≤T,\n1\nγ+1ˆ\nΩuγ+1\nk(x,t)dx+ˆt\n0ˆ\nΩ∇uk·∇uγ\nkdxdθ\n+1\n2ˆt\n0ˆ\nQ(uγ\nk(x,θ)−uγ\nk(y,θ))(uk(x,θ)−uk(y,θ))\n|x−y|n+2sdxdydθ\n≤¨\nΩtfdxdθ+1\nγ+1ˆ\nΩuγ+1\n0k(x)dx\n≤ /ba∇dblf/ba∇dblL1(ΩT)+1\nγ+1/ba∇dblu0/ba∇dblLγ+1(Ω).(3.3)\nUsing item (i)ofLemma 2.11 and taking supremum over 0<t≤T, we get that/braceleftig\nuγ+1\n2\nk/bracerightig\nkis uniformly bounded\ninL∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω))and{uk}kis uniformly bounded in L∞(0,T;Lγ+1(Ω)).\nWe now show that {uk}kis uniformly bounded in L2(t0,T;Hs\nloc(Ω))∩L2(t0,T;H1\nloc(Ω))for eacht0>0. Since\nγ >1, andΩTis bounded, and {uk}kis uniformly bounded in L∞(0,T;Lγ+1(Ω)), we deduce that {uk}kis\nuniformly bounded in L2(0,T;L2(Ω)) =L2(ΩT), in particular in L2(K×(t0,T)), for every subset Kcompactly\ncontained in Ωand for each t0>0. Further, as K×K⊂Ω×Ω⊂Qand all the integrals in the left-hand-side\nof(3.3)are positive, hence we have,\nˆT\nt0ˆ\nKˆ\nK(uk(x,t)−uk(y,t))(uγ\nk(x,t)−uγ\nk(y,t))\n|x−y|n+2sdxdydt≤2/ba∇dblf/ba∇dblL1(ΩT)+2\nγ+1/ba∇dblu0/ba∇dblLγ+1(Ω),\nandˆT\nt0ˆ\nKuγ−1\nk|∇uk|2dxdt≤1\nγ/parenleftbigg\n/ba∇dblf/ba∇dblL1(ΩT)+1\nγ+1/ba∇dblu0/ba∇dblLγ+1(Ω)/parenrightbigg\n,\nfor everyK⊂⊂Ωand for each t0>0.\nWe now apply the item (iii)ofLemma 2.11 , to get\nˆT\nt0ˆ\nKˆ\nK|uk(x,t)−uk(y,t)|2|uk(x,t)+uk(y,t)|γ−1\n|x−y|n+2sdxdydt≤2Cγ/parenleftig\n/ba∇dblf/ba∇dblL1(ΩT)+/ba∇dblu0/ba∇dblLγ+1(Ω)/parenrightig\n.\nUsing the positivity of ukinω×[t0,T)for allk, we get\nˆT\nt0ˆ\nKˆ\nK|uk(x,t)−uk(y,t)|2\n|x−y|n+2sdxdydt≤22−γCγ\ncγ−1\n(K,t0)/parenleftig\n/ba∇dblf/ba∇dblL1(ΩT)+/ba∇dblu0/ba∇dblLγ+1(Ω)/parenrightig\n, (3.4)\nandˆT\nt0ˆ\nK|∇uk|2dxdydt≤1\nγcγ−1\n(K,t0)/parenleftbigg\n/ba∇dblf/ba∇dblL1(ΩT)+1\nγ+1/ba∇dblu0/ba∇dblLγ+1(Ω)/parenrightbigg\n. (3.5)\nHence{uk}kis uniformly bounded in L2(t0,T;Hs\nloc(Ω))∩L2(t0,T;H1\nloc(Ω))≡L2(t0,T;H1\nloc(Ω))for eacht0>0.\nSince we have/braceleftig\nuγ+1\n2\nk/bracerightig\nis uniformly bounded in L2(0,T;H1\n0(Ω))⊂L2(ΩT), this implies that the increasing\nsequence {uk}kis uniformly bounded in L1(ΩT). Then, there exists a measurable function usuch thatuk→u\na.e. inΩTand by Beppo Levi’s theorem uk→uinL1(ΩT). Sinceuk= 0in(Rn\\Ω)×(0,T), extending u\nby zero outside of Ωwe conclude that uk→ua.e. inRn×(0,T)withu=0 in(Rn\\Ω)×(0,T). Now we\nuse Fatou’s lemma in (3.3)–(3.5), to obtain for each t0>0,u∈L2(t0,T;H1\nloc(Ω))∩L∞(0,T;Lγ+1(Ω))and\nuγ+1\n2∈L2(0,T;H1\n0(Ω)). Asuk(x,0) =u0k(x)for eachk, sou(x,0) =u0(x)inL1sense (see Remark 2.23 ). The\nrest of the proof will follow similarly as that of Theorem 2.27 .\n21Proof of Theorem 2.30\nHere also, we consider the approximating problems (2.10) . Sinceuk∈L2(0,T;H1\n0(Ω)), we takeuk(x,t)χ(0,τ)(t)\nas a test function in (2.10) , to have\nsup\n0≤τ≤Tˆ\nΩu2\nk(x,τ)dx+2ˆT\n0ˆ\nΩ|∇uk|2dxdt+ˆT\n0ˆ\nQ(uk(x,t)−uk(y,t))2\n|x−y|n+2sdxdydt\n≤2¨\nΩTfu1−γ\nkdxdt+/ba∇dblu0/ba∇dblL2(Ω).(3.6)\nCase 1:f∈L2\nγ+1/parenleftig\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightig\nFor this case, since f∈L2\nγ+1/parenleftig\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightig\n, we apply the Hölder inequality two times, first for the space\nintegral and then for the time integral, to obtain\n¨\nΩTfu1−γ\nkdxdt≤ˆT\n0/parenleftbiggˆ\nΩ|f(x,t)|(2∗\n1−γ)′\ndx/parenrightbigg 1\n(2∗\n1−γ)′/parenleftbiggˆ\nΩ|uk(x,t)|2∗\ndx/parenrightbigg1−γ\n2∗\ndt\n=ˆT\n0/ba∇dblf/ba∇dbl\nL(2∗\n1−γ)′\n(Ω)/ba∇dbluk/ba∇dbl1−γ\nL2∗(Ω)dt\n≤/parenleftiggˆT\n0/ba∇dblf/ba∇dbl2\n1+γ\nL(2∗\n1−γ)′\n(Ω)dt/parenrightigg1+γ\n2/parenleftiggˆT\n0/ba∇dbluk/ba∇dbl2\nL2∗(Ω)dt/parenrightigg1−γ\n2\n.\nWe now apply the Sobolev embedding as of Theorem 2.8 in the last term on the right-hand-side to get\n¨\nΩTfu1−γ\nkdxdt≤(C(n))1−γ\n2/ba∇dblf/ba∇dbl\nL2\nγ+1/parenleftbigg\n0,T;L(2∗\n1−γ)′\n(Ω)/parenrightbigg/bracketleftiggˆT\n0ˆ\nΩ|∇uk|2dxdt/bracketrightigg1−γ\n2\n.\nSinceγ <1, so we use Young’s inequality to deduce from (3.6)that\nsup\n0≤τ≤Tˆ\nΩu2\nk(x,τ)dx+ˆT\n0ˆ\nΩ|∇uk|2dxdt+ˆT\n0ˆ\nQ(uk(x,t)−uk(y,t))2\n|x−y|n+2sdxdydt≤C,\nwhereCis a positive constant independent of k.\nCase 2:f∈L¯m(ΩT)\nFor this case, we notice that as f∈L¯m(ΩT), we can apply the Hölder inequality in the first term on the\nright-hand-side in (3.6)with exponents ¯mand¯m′, to get\nsup\n0≤τ≤Tˆ\nΩu2\nk(x,τ)dx+ˆT\n0ˆ\nΩ|∇uk|2dxdt+ˆT\n0ˆ\nQ|uk(x,t)−uk(y,t)|2\n|x−y|n+2sdxdydt\n≤2/ba∇dblf/ba∇dblL¯m(ΩT)/bracketleftbigg¨\nΩT|uk|(1−γ)¯m′dxdt/bracketrightbigg1\n¯m′\n+/ba∇dblu0/ba∇dblL2(Ω).(3.7)\nWe observe that (1−γ)¯m′=2(n+2)\nnand hence using the Hölder inequality with the exponentsn\nn−2andn\n2\nand by Sobolev embedding ( Theorem 2.8 ), we reach that¨\nΩT|uk|2(n+2)\nndxdt=¨\nΩT|uk|2|uk|4\nndxdt\n≤ˆT\n0/bracketleftbiggˆ\nΩ|uk(x,t)|2dx/bracketrightbigg2\nn\n/ba∇dbluk/ba∇dbl2\nL2∗(Ω)dt\n≤C(n)/bracketleftbigg\nsup\n0≤t≤Tˆ\nΩ|uk(x,t)|2dx/bracketrightbigg2\nnˆT\n0ˆ\nΩ|∇uk|2dxdt.\nSo using (3.7)and convexity argument, we get\n¨\nΩT|uk|2(n+2)\nndxdt≤C(n)22\nn/parenleftigg\n/parenleftbig\n2/ba∇dblf/ba∇dblL¯m(ΩT)/parenrightbign+2\nn/parenleftbigg¨\nΩT|uk|(1−γ)¯m′/parenrightbiggn+2\nn¯m′\n+/parenleftig\n/ba∇dblu0/ba∇dblL2(Ω)/parenrightign+2\nn/parenrightigg\n.\nSincen+2\nn¯m′=1−γ\n2<1, we use the Young inequality to obtain\n¨\nΩT|uk|2(n+2)\nndxdt≤C,\n22whereCis a positive constant independent of k. Therefore, by (3.7) we deduce that the sequence {uk}k\nis uniformly bounded in the space L2(0,T;Xs\n0(Ω))∩L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω))≡L2(0,T;H1\n0(Ω))∩\nL∞(0,T;L2(Ω)).\nNow the rest of the proof follows similarly as that of Theorem 2.27 . However, for the sake of completeness, we\ninclude it here in a bit different way.\nSince the sequence {uk}kis uniformly bounded in the reflexive Banach space L2(0,T;H1\n0(Ω)), there exist a sub-\nsequence of {uk}k, still indexed by k, and a measurable function u∈L2(0,T;H1\n0(Ω))such thatuk⇀uweakly\ninL2(0,T;H1\n0(Ω))anduk→ustrongly in L2(ΩT)and a.e. in Ω×(0,T). In addition, since uk=u= 0on\nCΩ×(0,T), extending u≡0outsideΩ, we obtain uk→ufor a.e.(x,t)∈Rn×(0,T). Again since {uk}is\nincreasing in k, anduk(x,0) =u0k(x), so we have u(x,0) =u0(x)inL1sense ( Remark 2.23 ). Also, by Fatou’s\nLemma, we get that u∈L∞(0,T;L2(Ω)). Hence it follows that\nuk(x,t)−uk(y,t)\n|x−y|n+2s\n2→u(x,t)−u(y,t)\n|x−y|n+2s\n2a.e. inQ×(0,T).\nWe take an arbitrary test function ϕ∈ C∞\n0(ΩT)in(2.10) to get\n−¨\nΩTukϕtdxdt+ˆT\n0ˆ\nΩuk(−∆φ)dxdt\n+1\n2¨\nQT(uk(x,t)−uk(y,t))(ϕ(x,t)−ϕ(y,t))\n|x−y|n+2sdxdydt=¨\nΩTfkϕ/parenleftbig\nuk+1\nk/parenrightbigγdxdt.\nSinceϕ∈ C∞\n0(ΩT), thereforeϕtand∇ϕboth are in L2(ΩT)and since strong convergence implies weak conver-\ngence too, so it is clear that\nlim\nk→∞ˆT\n0ˆ\nΩukφtdxdt=ˆT\n0ˆ\nΩuφtdxdt\nand by weak convergence in L2(0,T;H1\n0(Ω)), we get\nˆT\n0ˆ\nΩuk(−∆φ)dxdt=ˆT\n0ˆ\nΩ∇uk·∇φdxdt\n→ˆT\n0ˆ\nΩ∇u·∇φdxdt=ˆT\n0ˆ\nΩu(−∆φ)dxdt, ask→ ∞.\nWe now define\nFk(x,y,t) =uk(x,t)−uk(y,t)\n|x−y|n+2s\n2andF(x,y,t) =u(x,t)−u(y,t)\n|x−y|n+2s\n2.\nThen asFk→Fa.e. inQ×(0,T)and{uk}kis uniformly bounded in L2(0,T;Xs\n0(Ω)), by weak convergence we\nreach that\nlim\nk→∞¨\nQT(uk(x,t)−uk(y,t))(ϕ(x,t)−ϕ(y,t))\n|x−y|n+2sdxdydt\n=¨\nQT(u(x,t)−u(y,t))(ϕ(x,t)−ϕ(y,t))\n|x−y|n+2sdxdydt,\nfor allϕ∈ C∞\n0(ΩT). Now since for all ω⊂⊂Ωand for allt0>0,we haveuk(x,t)≥C(ω,t0,n,s)inω×[t0,T),\nso we reach that uk(x,t)≥CinSuppφ(as support of φis compact in ΩT) for allk≥1. Using this fact, we\nthen have\n0≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefkϕ/parenleftbig\nuk+1\nk/parenrightbigγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/ba∇dblϕ/ba∇dblL∞(ΩT)|f|\nCγ∈L1(ΩT).\nSo, by the dominated convergence theorem, we get\nlim\nk→∞¨\nΩTfkϕ/parenleftbig\nuk+1\nk/parenrightbigγdxdt=¨\nΩTfϕ\nuγdxdt.\nFinally, we pass to the limit as k→ ∞ to get\n−¨\nΩTu(x,t)ϕt(x,t)dxdt−ˆT\n0ˆ\nΩu(x,t)∆φ(x,t)dxdt\n+1\n2¨\nQT(u(x,t)−u(y,t))(ϕ(x,t)−ϕ(y,t))\n|x−y|n+2sdxdydt=¨\nΩTf(x,t)ϕ(x,t)\nuγ(x,t)dxdt,\nfor allφ∈ C∞\n0(ΩT). Therefore, uis a weak solution to (2.12) .\n23Proof of Theorem 2.32\nAsuk(≥0)∈L∞(ΩT)∩L2(0,T;H1\n0(Ω)), so((uk(x,t)+ε)θ−εθ)∈L2(0,T;H1\n0(Ω)), for anyε,θ >0. So\nchoosingγ≤θ<1,0<ε<1/k, we take the test function ((uk(x,t)+ε)θ−εθ)χ(0,τ)(t)in(2.10) , to get for each\nτ≤Tˆτ\n0ˆ\nΩ(uk)t/parenleftbig\n(uk+ε))θ−εθ/parenrightbig\ndxdt+ˆτ\n0ˆ\nΩ∇uk·∇/parenleftbig\n(uk+ε)θ−εθ/parenrightbig\ndxdt\n+1\n2ˆτ\n0ˆ\nQ(uk(x,t)−uk(y,t))((uk(x,t)+ε)θ−(uk(y,t)+ε)θ)\n|x−y|n+2sdxdydt\n≤¨\nΩTf(x,t)(uk(x,t)+ε)θ−γdxdt.\nLettingε→0, integrating the first term, and taking the supremum over τ∈[0,T], we get\n2\nθ+1sup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|θ+1dx+2ˆT\n0ˆ\nΩ∇uk·∇uθ\nkdxdt\n+ˆT\n0ˆ\nQ(uk(x,t)−uk(y,t))/parenleftbig\nuθ\nk(x,t)−uθ\nk(y,t)/parenrightbig\n|x−y|n+2sdxdydt≤2¨\nΩTfuθ−γ\nkdxdt+2\nθ+1/ba∇dblu0/ba∇dblL2(Ω).(3.8)\nSince all terms on the left are positive, taking supremum was allowed. Then by item i)ofLemma 2.11 , we get\nθ+1\n2θsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|θ+1dx+(θ+1)2\n2ˆT\n0ˆ\nΩuθ−1\nk|∇uk|2dxdt\n+ˆT\n0ˆ\nQ|uθ+1\n2\nk(x,t)−uθ+1\n2\nk(y,t)|2\n|x−y|n+2sdxdydt≤2\nθ/parenleftbigg¨\nΩTfuθ−γ\nkdxdt+/ba∇dblu0/ba∇dblL2(Ω)/parenrightbigg\n.\nIn the previous inequality, we have used the fact that θ+1<2, now we observe that the termθ+1\n2θ>1in the\nleft-hand-side can be dropped. So we get\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|θ+1dx+(θ+1)2\n4ˆτ\n0ˆ\nΩuθ−1\nk|∇uk|2dxdt\n+ˆT\n0ˆ\nQ|uθ+1\n2\nk(x,t)−uθ+1\n2\nk(y,t)|2\n|x−y|n+2sdxdydt≤2\nθ/parenleftbigg¨\nΩTfuθ−γ\nkdxdt+/ba∇dblu0/ba∇dblL2(Ω)/parenrightbigg\n.(3.9)\nAgain, using the same tool like Theorem 2.30 and the Hölder inequality with exponentn\nn−2andn\n2we get\n¨\nΩT|uk|(θ+1)(n+2)\nndxdt=¨\nΩT|uk|2θ+1\n2|uk|4\nnθ+1\n2dxdt\n≤ˆT\n0/parenleftbiggˆ\nΩ|uk(x,t)|2∗θ+1\n2dx/parenrightbiggn−2\nn/parenleftbiggˆ\nΩ|uk(x,t)|θ+1dx/parenrightbigg2\nn\ndt\n=ˆT\n0/vextenddouble/vextenddouble/vextenddoubleuθ+1\n2\nk/vextenddouble/vextenddouble/vextenddouble2\nL2∗(Ω)/parenleftbiggˆ\nΩ|uk(x,t)|θ+1dx/parenrightbigg2\nn\ndt.\nWe now take the supremum over t∈[0,T], and apply the Sobolev embedding as that of Theorem 2.8 , to get that\n¨\nΩT|uk|(θ+1)(n+2)\nndxdt≤C(n)/parenleftbigg\nsup\n0≤t≤Tˆ\nΩ|uk(x,t)|θ+1dx/parenrightbigg2\nn¨\nΩT|∇uθ+1\n2\nk|2dxdt.\nUsing (3.9)and convexity argument, we now get\n¨\nΩT|uk|(θ+1)(n+2)\nndxdt≤C(n)/parenleftbigg2\nθ/parenrightbiggn+2\nn/parenleftbigg¨\nΩTfuθ−γ\nkdxdt+/ba∇dblu0/ba∇dblL2(Ω)/parenrightbiggn+2\nn\n≤C(n)22\nn/parenleftbigg2\nθ/parenrightbiggn+2\nn/parenleftigg/parenleftbigg¨\nΩTfuθ−γ\nkdxdt/parenrightbiggn+2\nn\n+/ba∇dblu0/ba∇dbl(n+2)\nn\nL2(Ω)/parenrightigg\n.(3.10)\nCase 1:m= 1\nNow for the case m= 1, we takeθ=γin the (3.10) . So we obtain\n¨\nΩT|uk|(γ+1)(n+2)\nndxdt≤C(n)22\nn/parenleftbigg2\nγ/parenrightbiggn+2\nn/parenleftbigg\n/ba∇dblf/ba∇dbln+2\nn\nL1(ΩT)+/ba∇dblu0/ba∇dbl(n+2)\nn\nL2(Ω)/parenrightbigg\n. (3.11)\n24Also, by (3.9)we easily have\n/ba∇dbluk/ba∇dblL∞(0,T;Lγ+1(Ω))≤2\nγ/bracketleftig\n/ba∇dblf/ba∇dblL1(ΩT)+/ba∇dblu0/ba∇dblL2(Ω)/bracketrightig\n. (3.12)\nCase 2:m>1\nForm>1, we takeγ <θ<1. Using (3.10) and applying Hölder inequality with exponent mandm′, we have\n¨\nΩT|uk|(θ+1)(n+2)\nndxdt≤C1/ba∇dblf/ba∇dbln+2\nn\nLm(ΩT)/bracketleftbigg¨\nΩT|uk|m′(θ−γ)dxdt/bracketrightbiggn+2\nnm′\n+C1/ba∇dblu0/ba∇dbl(n+2)\nn\nL2(Ω),\nwhereC1=C(n)22\nn/parenleftbigg2\nθ/parenrightbiggn+2\nn\n. We now choose γ <θ<1to be such that\n(θ+1)(n+2)\nn=m′(θ−γ),i.e.θ=(n+2)(m−1)+nmγ\nn−2(m−1).\nWe note that the condition θ <1is equivalent to m <¯m; whileγ < θ is always fulfilled. Sincen+2\nnm′<1,\napplying Young’s inequality with ε>0, we get¨\nΩT|uk|(θ+1)(n+2)\nndxdt≤C1/ba∇dblf/ba∇dbln+2\nn\nLm(ΩT)/parenleftbigg\nε¨\nΩT|uk|(θ+1)(n+2)\nndxdt+C(ε)/parenrightbigg\n+C1/ba∇dblu0/ba∇dbl(n+2)\nn\nL2(Ω).\nWe choose εsmall enough such that εC1/ba∇dblf/ba∇dbln+2\nn\nLm(ΩT)=1\n2and using the fact that\nσ:=m(γ+1)(n+2)\nn−2(m−1)=(θ+1)(n+2)\nn=m′(θ−γ),\nwe get¨\nΩT|uk|σdxdt≤C, (3.13)\nwhereCis a positive constant independent of k. Now in (3.9)we use Hölder inequality, to get\nsup\n0≤τ≤Tˆ\nΩuθ+1\nk(x,τ)dx≤2\nθ/parenleftig\n/ba∇dblf/ba∇dblLm(ΩT)/ba∇dbluk/ba∇dblσ\nm′\nLσ(ΩT)+/ba∇dblu0/ba∇dblL2(Ω)/parenrightig\n.\nSinceγ <θ and by (3.13) we conclude that the sequence {uk}is uniformly bounded in L∞(0,T;Lγ+1(Ω)), the\nsame thing holds for the case m= 1by(3.12) . Finally, by (3.11) and(3.13) we conclude that in both cases, that\nis1≤m<¯m, the sequence {uk}is uniformly bounded in Lσ(ΩT),σ:=m(γ+1)(n+2)\nn−2(m−1).\nNow from (3.9), again using Hölder inequality, we estimate as¨\nΩT|∇uθ+1\n2\nk|2dxdt≤2\nθ/parenleftig\n/ba∇dblf/ba∇dblLm(ΩT)/ba∇dbluk/ba∇dblσ\nm′\nLσ(ΩT)+/ba∇dblu0/ba∇dblL2(Ω)/parenrightig\n≤C, (3.14)\nwhereC >0is a constant independent of k. Let1<¯q<2will be specified later. By Hölder inequality, we have\n¨\nΩT|∇uk|¯qdxdt=¨\nΩT|∇uk|¯quθ−1\nk\nuθ−1\nkdxdt≤/parenleftbigg¨\nΩT|∇uk|2uθ−1\nkdxdt/parenrightbigg¯q\n2\n×\n¨\nΩTuθ−1\nk\nu2(θ−1)\n2−¯q\nkdxdt\n2−¯q\n2\n(3.14)\n≤C/parenleftbigg¨\nΩTu¯q(1−θ)\n2−¯q\nkdxdt/parenrightbigg2−¯q\n2\n.\n(3.15)\nWe now choose ¯qto be such that\n¯q(1−θ)\n2−¯q=σ=m(γ+1)(n+2)\nn−2(m−1)i.e.¯q=m(γ+1)(n+2)\nn+2−m(1−γ).\nWe note that ¯q <2is equivalent to m <¯m; while¯q >1is always fulfilled. Then we get by embedding results\nand using (3.13) and(3.15)\nˆT\n0ˆ\nΩˆ\nΩ|uk(x,t)−uk(y,t)|¯q\n|x−y|n+¯qsdydxdt≤C(n,s)¨\nΩT|∇uk|¯qdxdt≤C2/parenleftbigg¨\nΩTuσ\nk(x,t)dxdt/parenrightbigg2−¯q\n2\n≤C3,\nwhereC3is a positive constant independent of k. Thus, {uk}kis uniformly bounded in L¯q(0,T;Ws,¯q\n0(Ω))∩\nL¯q(0,T;W1,¯q\n0(Ω))≡L¯q(0,T;W1,¯q\n0(Ω)).\n25Since the sequence {uk}kis uniformly bounded in the reflexive Banach space L¯q(0,T;W1,¯q\n0(Ω)), there exist a\nsubsequence of {uk}kstill indexed by kand a measurable function u∈L¯q(0,T;W1,¯q\n0(Ω))such thatuk⇀ u\nweakly inL¯q(0,T;W1,¯q\n0(Ω)). Also by Fatou’s lemma, we will get u∈L∞(0,T;Lγ+1(Ω))andu∈Lσ(ΩT), with\nσ:=m(γ+1)(n+2)\nn−2(m−1). As before, using the monotonicity of the sequence {uk}, we get using Beppo Levi’s\ntheorem that uk→ustrongly in L1(ΩT)anduk→ua.e inRn×(0,T). ByRemark 2.23 , this pointwise limit\nwill satisfy u(·,0) =u0(·)inL1sense. Then, the rest of the proof will follow from the proof o fTheorem 2.27 .\nRemark 3.2. We observe that the sequence {uk}kis uniformly bounded in Lr(Ω)for every 1≤r≤σ, then\nfollowing the same lines of the previous proof, we can show th at the sequence {uk}kis uniformly bounded in\nLq(0,T;Ws,q\n0(Ω))∩Lq(0,T;W1,q\n0(Ω))≡Lq(0,T;W1,q\n0(Ω))for all1<q≤¯qwhere1≤m<¯m.\nProof of Theorem 2.35\nLet us introduce the following notations: for any measurabl e functionvwe define\nAm(v) ={(x,t)∈ΩT:v(x,t)>m},and for a.e. t∈(0,T), At\nm(v) ={x∈Ω :v(x,t)>m}.\nWe use the idea of the classical proof by D.G. Aronson and J. Se rrin, which is to prove a uniform L∞bound for\nukinΩ×(0,τ), for a positive (small) τ(to be specified later), and then to iterate such an estimate. We consider\nthe approximated problem (2.10) and take (uk−m)+χ(0,t)∈L2(0,T;H1\n0(Ω)), form>0,t∈(0,τ),τ <T , as a\ntest function to obtainˆ\nΩϕm(uk(x,t))dx+ˆt\n0ˆ\nΩ∇uk·∇(uk−m)+dxdθ\n+1\n2ˆt\n0ˆ\nQ(uk(x,θ)−uk(y,θ))((uk−m)+(x,θ)−(uk−m)+(y,θ))\n|x−y|n+2sdxdydθ\n≤ˆτ\n0ˆ\nΩfk(uk−m)+\n(uk+1\nk)γdxdθ+ˆ\nΩϕm(uk(x,0))dx,(3.16)\nwhereϕm(ρ) =ˆρ\n0(σ−m)+dσ=(ρ−m)2\n+\n2. We choose a m>0large enough such that m>max{/ba∇dblu0/ba∇dblL∞(Ω),1},\nin order to neglect the last term above. Noting that ∇uk·∇(uk−m)+=|∇(uk−m)+|2, and\n(uk(x,θ)−uk(y,θ))((uk−m)+(x,θ)−(uk−m)+(y,θ))≥((uk−m)+(x,θ)−(uk−m)+(y,θ))2,\nwe take supremum over t∈(0,τ]in(3.16) , to get\n/ba∇dbl(uk−m)+/ba∇dbl2\nL∞(0,τ;L2(Ω))+/ba∇dbl(uk−m)+/ba∇dbl2\nL2(0,τ;Xs\n0(Ω))+/ba∇dbl(uk−m)+/ba∇dbl2\nL2(0,τ;H1\n0(Ω))\n≤ˆτ\n0ˆ\nAt\nm,kf(uk−m)+dxdt.(3.17)\nNote that, in order to deal with the singularity, we have used the fact that m≥1and henceuk>1inAt\nm,k,\nhere the subscript kinAt\nm,kdenotes that we are considering the function uk. Now the term of the right-hand\nside above can be estimated as follows,ˆτ\n0ˆ\nAt\nm,kf(uk−m)+dxdt≤ˆτ\n0ˆ\nAt\nm,kf(uk−m)2\n+dxdt+ˆτ\n0ˆ\nAt\nm,kfdxdt. (3.18)\nWe now study each member present in the right-hand side of (3.18) . We first define the followings,\n¯r= 2r′,¯q= 2q′,η=2η1\nn,ˆr= ¯r(1+η),ˆq= ¯q(1+η),\nwhereη1= 1−1\nr−n\n2q. Note that η1∈(0,1)as0<1\nr+n\n2q<1. Further, simple calculation yields that\n1\nˆr+n\n2ˆq=n\n4. Now, applying Hölder inequality repeatedly, we estimate t he first term as\nˆτ\n0ˆ\nAt\nm,kf(uk−m)2\n+dxdt≤ˆτ\n0/parenleftiggˆ\nAt\nm,k|f(x,t)|qdx/parenrightigg1\nq/parenleftiggˆ\nAt\nm,k(uk−m)2q′\n+dx/parenrightigg1\nq′\ndt\n≤\nˆτ\n0/parenleftiggˆ\nAt\nm,k|f(x,t)|qdx/parenrightiggr\nq\ndt\n1\nr\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)2q′\n+dx/parenrightiggr′\nq′\ndt\n1\nr′\n.\n26As again by Hölder inequality with exponent (1+η), we have\n\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)2q′\n+dx/parenrightiggr′\nq′\ndt\n1\nr′\n≤\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)2q′(1+η)\n+dx/parenrightiggr′\nq′(1+η)\n|At\nm,k|r′η\nq′(1+η)dt\n1\nr′\n=\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)ˆq\n+dx/parenrightigg2r′\nˆq\n|At\nm,k|2r′η\nˆqdt\n1\nr′\n≤\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)ˆq\n+dx/parenrightiggˆr\nˆq\ndt\n1\nr′(1+η)/parenleftbiggˆτ\n0|At\nm,k|ˆr\nˆq/parenrightbiggη\nr′(1+η)\n,\ntherefore ˆτ\n0ˆ\nAt\nm,kf(uk−m)2\n+dxdt≤ /ba∇dblf/ba∇dblLr(0,T;Lq(Ω))/ba∇dbl(uk−m)+/ba∇dbl2\nLˆr(0,τ;Lˆq(Ω))µk(m)2η\nˆr,\nwhereµk(m) =ˆτ\n0/vextendsingle/vextendsingleAt\nm,k/vextendsingle/vextendsingleˆr\nˆqdt. We now use Gagliardo-Nirenberg inequality ( Theorem 2.10 ) for(uk−m)+to get\nˆτ\n0ˆ\nAt\nm,kf(uk−m)2\n+dxdt≤c/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))µk(m)2η\nˆr/parenleftbiggˆτ\n0/ba∇dbl(uk−m)+/ba∇dbl(1−θ)ˆr\nL2(Ω)/ba∇dbl∇(uk−m)+/ba∇dblˆrθ\nL2(Ω)dt/parenrightbigg2\nˆr\n≤c/ba∇dblf/ba∇dblµk(m)2η\nˆr/bracketleftbigg\n/ba∇dbl(uk−m)+/ba∇dbl2\nL∞(0,τ;L2(Ω))+/ba∇dbl(uk−m)+/ba∇dbl2\nL2(0,τ;H1\n0(Ω))/bracketrightbigg\n.\n(3.19)\nwhereˆrθ= 2, andcis a constant independent of the choice of kandm. Note that we have used the fact that\nany function in H1\n0(Ω)can be considered as a function in H1(Rn). Also, we applied Young’s inequality with\nexponentsˆr\n2and its conjugate.\nOn the other hand, the second term on the right-hand side in (3.18) can be estimated by Hölder inequality as\nˆτ\n0ˆ\nAt\nm,kfdxdt≤ˆτ\n0/parenleftiggˆ\nAt\nm,k|f(x,t)|qdx/parenrightigg1\nq\n|At\nm,k|1\nq′dt\n≤\nˆτ\n0/parenleftiggˆ\nAt\nm,k|f(x,t)|qdx/parenrightiggr\nq\ndt\n1\nr/parenleftbiggˆτ\n0|At\nm,k|r′\nq′dt/parenrightbigg1\nr′\n≤ /ba∇dblf/ba∇dblLr(0,T;Lq(Ω))µk(m)2(1+η)\nˆr.(3.20)\nDenoting\n|||(uk−m)+|||2=/ba∇dbl(uk−m)+/ba∇dbl2\nL∞(0,τ;L2(Ω))+/ba∇dbl(uk−m)+/ba∇dbl2\nL2(0,τ;H1\n0(Ω)),\nand using (3.19) ,(3.20) , we get from (3.17) ,\n|||(uk−m)+|||2≤c/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))/bracketleftig\nµk(m)2η\nˆr|||(uk−m)+|||2+µk(m)2(1+η)\nˆr/bracketrightig\n,\nwherecis a constant which does not depend on kandm. Note that µk(m)≤τ|Ω|ˆr\nˆq, for allk∈Nandm, so that\nwe can fixτ, independent of ukandm, suitable small in such a way that cµk(m)2η\nˆr/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))=1\n2and use\nagain the Gagliardo-Nirenberg inequality (see (3.19) ), to deduce that\n/ba∇dbl(uk−m)+/ba∇dbl2\nLˆr(0,τ;Lˆq(Ω))≤c|||(uk−m)+|||2≤c/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))µk(m)2(1+η)\nˆr. (3.21)\nConsiderl>m> max{/ba∇dblu0/ba∇dblL∞(Ω),1}:=m0. ThenAt\nl,k⊂At\nm,k, and using (3.21) , we get\n(l−m)µk(l)1\nˆr=/parenleftbiggˆτ\n0/parenleftbig\n(l−m)ˆq/vextendsingle/vextendsingleAt\nl,k/vextendsingle/vextendsingle/parenrightbigˆr\nˆqdt/parenrightbigg1\nˆr\n≤\nˆτ\n0/parenleftiggˆ\nAt\nl,k(uk−m)ˆq\n+dx/parenrightiggˆr\nˆq\ndt\n1\nˆr\n≤\nˆτ\n0/parenleftiggˆ\nAt\nm,k(uk−m)ˆq\n+dx/parenrightiggˆr\nˆq\ndt\n1\nˆr\n≤cµk(m)1+η\nˆr.\n27Therefore for all l>m>m 0, we have\nµk(l)≤c\n(l−m)ˆrµk(m)1+η,\nwhere c is a constant independent of k. Now applying [ 27, Lemma B. 1], we conclude that µk(m0+dk) = 0,\nwheredk=c[µk(m0)]η2ˆr(1+η)/η. As for each k,dk≤c/parenleftig\nT|Ω|ˆr\nˆq/parenrightigη\n, we get\n/ba∇dbluk/ba∇dblL∞(Ω×[0,τ])≤d.\nWe can iterate this procedure in the sets Ω×[τ,2τ],···,Ω×[jτ,T], whereT−jτ/lessorequalslantτto conclude that\n/ba∇dbluk/ba∇dblL∞(ΩT)≤Cuniformly in k∈N.\nNow, since the sequence {uk}kis increasing in k, we have /ba∇dblu/ba∇dblL∞(Ω)≤C. Further, if γ≤1, testing (2.10) with\nthe function ukwe can deduce u∈L2(0,T;H1\n0(Ω))(seeTheorem 2.26 ).\nProof of Theorem 2.36\nForγ≥1, we choose δ >1+γ\n2and forγ <1, we choose δ >1, and take u2δ−1\nkχ(0,t),0< t≤Tas test\nfunction in (2.10) , withu0≡0. We note that, such a test function is admissible as uk∈L∞(ΩT). We get ∀t≤T,\n1\n2δˆ\nΩu2δ\nk(x,t)dx+ˆt\n0ˆ\nΩ∇uk·∇u2δ−1\nkdxdθ\n+1\n2ˆt\n0ˆ\nQ/parenleftbig\nu2δ−1\nk(x,θ)−u2δ−1\nk(y,θ)/parenrightbig\n(uk(x,θ)−uk(y,θ))\n|x−y|n+2sdxdydθ≤¨\nΩtfu2δ−1−γ\nkdxdθ.\nTaking supremum over t∈(0,T]and using item (i)ofLemma 2.11 , by Hölder and Sobolev inequalities, we have\nsup\nt∈[0,T]ˆ\nΩu2δ\nk(x,t)dx+2(2δ−1)\nδλˆT\n0/vextenddouble/vextenddoubleuδ\nk/vextenddouble/vextenddouble2\nL2∗(Ω)dt+(2δ−1)\nδλsˆT\n0/vextenddouble/vextenddoubleuδ\nk/vextenddouble/vextenddouble2\nL2∗s(Ω)dt\n≤2δ/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))\nˆT\n0/bracketleftbiggˆ\nΩu(2δ−1−γ)q′\nkdx/bracketrightbiggr′\nq′\ndt\n1\nr′\n.(3.22)\nNote that2δ−1\nδ>1, so we can ignore the constants in left. Denoting by A=/parenleftiggˆT\n0/ba∇dbluk/ba∇dbl(2δ−1−γ)r′\nL(2δ−1−γ)q′(Ω)dt/parenrightigg1\nr′\n,\nwe get from (3.22) ,\nsup\nt∈[0,T]ˆ\nΩu2δ\nkdx≤2δ/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))A, (3.23)\nandˆT\n0/bracketleftbiggˆ\nΩu2∗\nsδ\nkdx/bracketrightbigg2\n2∗sdt≤cˆT\n0/bracketleftbiggˆ\nΩu2∗δ\nkdx/bracketrightbigg2\n2∗\ndt≤c\nλ2δ/ba∇dblf/ba∇dblLr(0,T;Lq(Ω))A. (3.24)\nCase 1: 1<q<nr\nn+2(r−1)i.e.1\nr<n\nn−21\nq−2\nn−2\nSince,1<q<nr\nn+2(r−1)impliesq<r andr′\nq′<2\n2∗, we apply Hölder inequality with exponent2q′\n2∗r′to get\nA≤T1\nr′−2∗\n2q′/bracketleftiggˆT\n0/parenleftbiggˆ\nΩu(2δ−1−γ)q′\nkdx/parenrightbigg2\n2∗\ndt/bracketrightigg2∗\n2q′\n. (3.25)\nThus from (3.24) , it follows\nˆT\n0/bracketleftbiggˆ\nΩu2∗δ\nkdx/bracketrightbigg2\n2∗\ndt≤2δcA≤2δc/bracketleftiggˆT\n0/parenleftbiggˆ\nΩu(2δ−1−γ)q′\nkdx/parenrightbigg2\n2∗\ndt/bracketrightigg2∗\n2q′\n.\nWe now choose 2∗δ= (2δ−1−γ)q′, that is,δ=1\n2·q(n−2)(γ+1)\n(n−2q). Note that if γ≥1, thenδ>γ+1\n2if and\nonly ifq >1, and forγ <1,δ >1if and only if q >/parenleftbigg2∗\n1−γ/parenrightbigg′\n. Moreover, since q <n\n2it follows that2∗\n2q′<1.\n28Thus we get\nˆT\n0/parenleftbiggˆ\nΩu2∗δ\nkdx/parenrightbigg2\n2∗\ndt=ˆT\n0/parenleftbiggˆ\nΩu(2δ−1−γ)q′\nkdx/parenrightbigg2\n2∗\ndt≤c, (3.26)\nand therefore from (3.23) ,(3.25) and(3.26) , it follows\n/ba∇dbluk/ba∇dblL∞(0,T;L2σ(Ω))+/ba∇dbluk/ba∇dblL2σ(0,T;L2∗σ(Ω))≤c,withσ=q(n−2)(γ+1)\n2(n−2q)>q\n2.\nCase 2:nr\nn+2(r−1)≤q<n\n2r′i.e.1\nr≥n\nn−21\nq−2\nn−2and1\nr+n\n2q>1\nIn this case we choose δ=1\n2qrn(γ+1)\nnr−2q(r−1). Note that for γ≥1,δ>1+γ\n2,if and only if1\nr+n\n2q<1+n\n2which\nis always satisfied and for γ <1,δ>1if and only if q>2nr\nnr(γ+1)+4(r−1)which is satisfied if r>2\nγ+1. Now\nby interpolation, we get that1\n(2δ−1−γ)q′=1−θ\n2δ+θ\n2∗δ, whereθ=nq(r−1)\nnr(q−1)+2q(r−1)∈(0,1]. Therefore\nˆT\n0/ba∇dbluk/ba∇dblr′(2δ−1−γ)\nL(2δ−1−γ)q′(Ω)dt≤c/ba∇dbluk/ba∇dbl2δµ1\nL∞(0,T;L2δ(Ω))ˆT\n0/parenleftbiggˆ\nΩu2∗δ\nkdx/parenrightbigg2\n2∗µ2\ndt,\nwhereµ1=(1−θ)r′(2δ−1−γ)\n2δandµ2=θr′(2δ−1−γ)\n2δ. Sinceµ2≤1, we have that\nˆT\n0/ba∇dbluk/ba∇dblr′(2δ−1−γ)\nL(2δ−1−γ)q′(Ω)dt≤c/ba∇dbluk/ba∇dbl2δµ1\nL∞(0,T;L2δ(Ω))/parenleftiggˆT\n0/bracketleftbiggˆ\nΩu2∗δ\nkdx/bracketrightbigg2\n2∗\ndt/parenrightiggµ2\n,\nand sinceµ1+µ2<r′, using (3.23) and(3.24) , we can conclude the following which will give our final resul t\nˆT\n0/parenleftbiggˆ\nΩu(2δ−1−δ)q′\nkdx/parenrightbiggr′\nq′\ndt≤c.\nProof of Theorem 2.39\nConsider the approximated problem (2.11) and corresponding properties of {uk}k,Theorem 2.22 . We denote\n(ω˜T)δ= ΩT\\(ΩT)δ, thenuk(x)≥C((ω˜T)δ,n,s)>0in(ω˜T)δfor allk. Choosing ukχ(0,τ)(t)as test function in\n(2.11) , we obtain\n1\n2ˆ\nΩu2\nk(x,τ)dx+ˆτ\n0ˆ\nΩ|∇uk|2dxdt+1\n2ˆτ\n0ˆ\nQ(uk(x,t)−uk(y,t))2\n|x−y|n+2sdxdydt\n≤¨\nΩTfkuk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+1\n2ˆ\nΩu2\n0(x)dx.\nNow taking supremum over τ∈(0,T], we get\nsup\n0≤τ≤Tˆ\nΩu2\nk(x,τ)dx+2ˆT\n0ˆ\nΩ|∇uk|2dxdt+ˆT\n0ˆ\nQ(uk(x,t)−uk(y,t))2\n|x−y|n+2sdxdydt\n≤2¨\nΩTfkuk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+/ba∇dblu0/ba∇dblL2(Ω)\n= 2¨\n(ΩT)δfkuk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+2¨\n(ω˜T)δfkuk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+/ba∇dblu0/ba∇dblL2(Ω)\n≤2¨\n(ΩT)δfu1−γ(x,t)\nkdxdt+2¨\n(ω˜T)δfuk\nCγ(x,t)\n(ω˜T)δdxdt+/ba∇dblu0/ba∇dblL2(Ω)\n≤2¨\n(ΩT)δ∩{uk≤1}fu1−γ(x,t)\nkdxdt+2¨\n(ΩT)δ∩{uk≥1}fu1−γ(x,t)\nkdxdt\n+2¨\n(ω˜T)δfuk\nCγ(x,t)\n(ω˜T)δdxdt+/ba∇dblu0/ba∇dblL2(Ω)\n≤ /ba∇dblu0/ba∇dblL2(Ω)+2/ba∇dblf/ba∇dblL1(ΩT)+2/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg¨\nΩTfukdxdt.(3.27)\n29Case 1:f∈L2/parenleftig\n0,T;L(2n\nn+2)(Ω)/parenrightig\nWe note that (2∗)′=2n\nn+2, then by using Hölder’s and Sobolev’s inequalities, we have\nˆT\n0ˆ\nΩfukdxdt≤CˆT\n0/parenleftbiggˆ\nΩ|f(x,t)|(2∗)′dx/parenrightbigg1\n(2∗)′/parenleftbiggˆ\nΩ|uk(x,t)|2∗\ndx/parenrightbigg1\n2∗\ndt\n≤CˆT\n0/ba∇dblf/ba∇dblL(2∗)′(Ω)/parenleftbiggˆ\nΩ|∇uk|2dx/parenrightbigg1\n2\ndt\n≤C/parenleftiggˆT\n0/ba∇dblf/ba∇dbl2\nL(2∗)′(Ω)dt/parenrightigg1\n2/parenleftiggˆT\n0ˆ\nΩ|∇uk|2dxdt/parenrightigg1\n2\n.\nIn the above, we use Young’s inequality, and then from (3.27) we get that\nsup\n0≤τ≤Tˆ\nΩu2\nk(x,τ)dx+/ba∇dbluk/ba∇dbl2\nL2(0,T;H1\n0(Ω))≤ /ba∇dblu0/ba∇dblL2(Ω)+2/ba∇dblf/ba∇dblL1(ΩT)+C/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg\n.\nTherefore the sequence {uk}kis uniformly bounded in L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω)).\nCase 2:f∈L¯r(ΩT)\nFor this case, we apply Hölder inequality with exponents ¯rand¯r′to get\nˆT\n0ˆ\nΩfukdxdt≤C/ba∇dblf/ba∇dblL¯r(ΩT)/bracketleftbigg¨\nΩT|uk|¯r′dxdt/bracketrightbigg1\n¯r′\n. (3.28)\nWe observe that ¯r′=2(n+2)\nnand hence using the Hölder inequality with exponentsn\nn−2andn\n2and by\nSobolev embedding ( Theorem 2.8 ), we can write\n¨\nΩT|uk|2(n+2)\nndxdt=¨\nΩT|uk|2|uk|4\nndxdt≤ˆT\n0/bracketleftbiggˆ\nΩ|uk(x,t)|2dx/bracketrightbigg2\nn\n/ba∇dbluk/ba∇dbl2\nL2∗(Ω)dt\n≤C(n)/bracketleftbigg\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|2dx/bracketrightbigg2\nnˆT\n0ˆ\nΩ|∇uk|2dxdt.(3.29)\nSo using (3.28) in(3.27) , we get from (3.29) by convexity argument\n¨\nΩT|uk|2(n+2)\nndxdt≤C(n)22\nn/parenleftigg\n/parenleftbig\nC1C/ba∇dblf/ba∇dblL¯r(ΩT)/parenrightbign+2\nn/parenleftbigg¨\nΩT|uk|¯r′/parenrightbiggn+2\nn¯r′\n+Cn+2\nn\n2/parenrightigg\n,\nwhereC1= 2/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg\nandC2=/ba∇dblu0/ba∇dblL2(Ω)+2/ba∇dblf/ba∇dblL1(ΩT). Now sincen+2\nn¯r′=1\n2, we use Young’s\ninequality to obtain¨\nΩT|uk|¯r′\ndxdt=¨\nΩT|uk|2(n+2)\nndxdt≤C,\nwhereCis a positive constant independent of k. Therefore, by (3.27) and(3.28) we deduce that {uk}kis\nuniformly bounded in L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω)).\nSince{uk}kis uniformly bounded in the reflexive Banach space L2(0,T;H1\n0(Ω)), there exist a subsequence\nof{uk}k, still indexed by k, and a measurable function u∈L2(0,T;H1\n0(Ω))such thatuk⇀ u weakly in\nL2(0,T;H1\n0(Ω)). Also, since {uk}is increasing in k, it holds by Beppo Levi’s theorem that uk→ustrongly in\nL1(ΩT)and hence a.e. in ΩT. Applying Fatou’s Lemma, we get u∈L∞(0,T;L2(Ω)). This pointwise limit is\nactually defined for each t∈[0,T)and satisfies u(·,0) =u0(·)inL1sense (see Remark 2.23 ). We show that this\nuis a very weak solution to (2.13) in the sense of Definition 2.25 . Choosing arbitrary φ∈ C∞\n0(ΩT), we have¨\nΩT((uk)t−∆uk+(−∆)suk)φdxdt=¨\nΩTfkφ\n/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt.\nSinceφ∈ C∞\n0(ΩT), thereforeφt,∆φand(−∆)sφall are bounded. As uk→ustrongly in L1(ΩT), we have¨\nΩT((uk)t−∆uk+(−∆)suk)φdxdt =¨\nΩTuk(−(φ)t−∆φ+(−∆)sφ)dxdt\n→¨\nΩTu(−(φ)t−∆φ+(−∆)sφ)dxdt.\n30Now since for all ω⊂⊂Ωand for allt0>0,uk(x,t)≥C(ω,t0,n,s)inω×[t0,T), we reach that uk(x,t)≥Cin\nSuppφ(sayω×[t1,t2]) for allk≥1. Therefore\n0≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefkφ\n/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblφC−γ(x,t)\nω,t1,t2/ba∇dblL∞(ΩT)f.\nHence by the dominated convergence theorem¨\nΩTfkφ\n/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt→¨\nΩTfφ\nuγ(x,t)dxdtask→ ∞.\nThus, we get the following and conclude¨\nΩT(ut−∆u+(−∆)su)φdxdt=¨\nΩTfφ\nuγ(x,t)dxdt.\nProof of Theorem 2.41\nConsiderukto be the unique nonnegative solution to (2.11) . Sinceγ∗>1, so we can use uγ∗\nkχ(0,τ)(t)as a\ntest function in (2.11) to get for all τ≤T,\n1\nγ∗+1ˆ\nΩuγ∗+1\nk(x,τ)dx+ˆτ\n0ˆ\nΩ∇uk·∇uγ∗\nkdxdt\n+1\n2ˆτ\n0ˆ\nQ(uγ∗\nk(x,t)−uγ∗\nk(y,t))(uk(x,t)−uk(y,t))\n|x−y|n+2sdxdydt\n≤¨\nΩTfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+1\nγ∗+1ˆ\nΩuγ∗+1\n0(x)dx.\nNow taking supremum over τ∈(0,T]and using item i)ofLemma 2.11 , we get\n(γ∗+1)\n2γ∗sup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|γ∗+1dx+(γ∗+1)2\n2ˆT\n0ˆ\nΩuγ∗−1\nk|∇uk|2dxdt\n+ˆT\n0ˆ\nQ|uγ∗+1\n2\nk(x,t)−uγ∗+1\n2\nk(y,t)|2\n|x−y|n+2sdxdydt\n≤(γ∗+1)2\n2γ∗/parenleftigg¨\nΩTfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+/ba∇dblu0/ba∇dblLγ∗+1(Ω)/parenrightigg\n.\nWe note that 1<γ∗+1\nγ∗<2and hence it follows\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|γ∗+1dx+4ˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt\n+2ˆT\n0ˆ\nQ|uγ∗+1\n2\nk(x,t)−uγ∗+1\n2\nk(y,t)|2\n|x−y|n+2sdxdydt≤4γ∗/parenleftigg¨\nΩTfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+/ba∇dblu0/ba∇dblLγ∗+1(Ω)/parenrightigg\n.(3.30)\nDenoting (ω˜T)δ= ΩT\\(ΩT)δ, we have uk(x,t)≥C((ω˜T)δ,n,s)>0in(ω˜T)δfor allk. We estimate like\nTheorem 2.39 as¨\nΩTfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt=¨\n(ΩT)δfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt+¨\n(ω˜T)δfkuγ∗\nk/parenleftbig\nuk+1\nk/parenrightbigγ(x,t)dxdt\n≤¨\n(ΩT)δfuγ∗−γ(x,t)\nkdxdt+¨\n(ω˜T)δfuγ∗\nk\nCγ(x,t)\n(ω˜T)δdxdt\n≤¨\n(ΩT)δ∩{uk≤1}fuγ∗−γ(x,t)\nkdxdt+¨\n(ΩT)δ∩{uk≥1}fuγ∗−γ(x,t)\nkdxdt\n+¨\n(ω˜T)δfuγ∗\nk\nCγ(x,t)\n(ω˜T)δdxdt\n≤ /ba∇dblf/ba∇dblL1(ΩT)+/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg¨\nΩTfuγ∗\nkdxdt.\n31Using the above in (3.30) we get\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|γ∗+1dx+2ˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt\n≤4γ∗/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg¨\nΩTfuγ∗\nkdxdt+4γ∗/parenleftig\n/ba∇dblf/ba∇dblL1(ΩT)+/ba∇dblu0/ba∇dblLγ∗+1(Ω)/parenrightig\n.(3.31)\nFor convenience we take C1= 4γ∗/parenleftbigg\n1+/vextenddouble/vextenddouble/vextenddoubleC−γ(·)\n(ω˜T)δ/vextenddouble/vextenddouble/vextenddouble\nL∞(ΩT)/parenrightbigg\nandC2= 4γ∗/parenleftig\n/ba∇dblf/ba∇dblL1(ΩT)+/ba∇dblu0/ba∇dblLγ∗+1(Ω)/parenrightig\n.\nCase 1:f∈Lγ∗+1/parenleftbigg\n0,T;L/parenleftBig\nn(γ∗+1)\nn+2γ∗/parenrightBig\n(Ω)/parenrightbigg\nIn this case using Hölder inequality first in the space variab le and then by Sobolev inequality and another\napplication of Hölder inequality (in time variable) we get\nˆT\n0ˆ\nΩfuγ∗\nkdxdt=ˆT\n0ˆ\nΩf/parenleftbigg\nuγ∗+1\n2\nk/parenrightbigg2γ∗\nγ∗+1\ndxdt\n≤CˆT\n0/parenleftbiggˆ\nΩ|f(x,t)|n(γ∗+1)\nn+2γ∗dx/parenrightbiggn+2γ∗\nn(γ∗+1)/parenleftbiggˆ\nΩ|uγ∗+1\n2\nk(x,t)|2∗dx/parenrightbigg(n−2)γ∗\nn(γ∗+1)\ndt\n≤CˆT\n0/ba∇dblf(·,t)/ba∇dbl\nLn(γ∗+1)\nn+2γ∗(Ω)/parenleftbiggˆ\nΩ|∇uγ∗+1\n2\nk|2dx/parenrightbiggγ∗\nγ∗+1\ndt\n≤C/parenleftiggˆT\n0/ba∇dblf(·,t)/ba∇dblγ∗+1\nLn(γ∗+1)\nn+2γ∗(Ω)dt/parenrightigg1\nγ∗+1\n×/parenleftiggˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt/parenrightiggγ∗\nγ∗+1\n.\nNow sinceγ∗\nγ∗+1<1, so using Young’s inequality in the above estimate, we get fr om(3.31) that\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|γ∗+1dx+2ˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt\n≤C/ba∇dblf/ba∇dbl\nLγ∗+1/parenleftBigg\n0,T;Ln(γ∗+1)\nn+2γ∗(Ω)/parenrightBigg×/parenleftigg\nεˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt+C(ε)/parenrightigg\n+C2.\nChoosingεsmall enough such that εC/ba∇dblf/ba∇dbl\nLγ∗+1/parenleftBigg\n0,T;Ln(γ∗+1)\nn+2γ∗(Ω)/parenrightBigg= 1,we get\nsup\n0≤τ≤Tˆ\nΩ|uk(x,τ)|γ∗+1dx+ˆT\n0ˆ\nΩ|∇uγ∗+1\n2\nk|2dxdt≤C,\nwhich implies that the sequence/braceleftbigg\nuγ∗+1\n2\nk/bracerightbigg\nkis uniformly bounded in L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω))and{uk}k\nis uniformly bounded in L∞(0,T;Lγ∗+1(Ω)).\nCase 2:f∈L˜r(ΩT)\nFor this case, we apply Hölder inequality with exponents ˜rand˜r′to get\nˆT\n0ˆ\nΩfuγ∗\nkdxdt≤C/ba∇dblf/ba∇dblL˜r(ΩT)/bracketleftbigg¨\nΩT|uk|γ∗˜r′dxdt/bracketrightbigg1\n˜r′\n. (3.32)\nWe note that γ∗˜r′=(n+2)(γ∗+1)\nn. Again, we use the same technique as that of Theorem 2.30 ,Theorem 2.32\nandTheorem 2.39 . Using the Hölder inequality with exponentn\nn−2andn\n2we get\n¨\nΩT|uk|(γ∗+1)(n+2)\nndxdt=¨\nΩT|uk|2γ∗+1\n2|uk|4\nnγ∗+1\n2dxdt\n≤ˆT\n0/parenleftbiggˆ\nΩ|uk(x,t)|2∗γ∗+1\n2dx/parenrightbiggn−2\nn/parenleftbiggˆ\nΩ|uk(x,t)|γ∗+1dx/parenrightbigg2\nn\ndt\n=ˆT\n0/ba∇dbluγ∗+1\n2\nk/ba∇dbl2\nL2∗(Ω)/parenleftbiggˆ\nΩ|uk(x,t)|γ∗+1dx/parenrightbigg2\nn\ndt.\n32Now taking supremum over t∈[0,T]and applying the Sobolev embedding as that of Theorem 2.8 , we can write\n¨\nΩT|uk|(γ∗+1)(n+2)\nndxdt≤C(n)/parenleftbigg\nsup\n0≤t≤Tˆ\nΩ|uk(x,t)|γ∗+1dx/parenrightbigg2\nn/parenleftbigg¨\nΩT|∇uγ∗+1\n2\nk|2dxdt/parenrightbigg\n.\nUsing (3.31) and(3.32) , from the above inequality we get by convexity argument\n¨\nΩT|uk|(γ∗+1)(n+2)\nndxdt≤C(n)/parenleftbigg\nC1¨\nΩTfuγ∗\nkdxdt+C2/parenrightbiggn+2\nn\n≤C(n)22\nn/parenleftigg/parenleftbigg\nC1¨\nΩTfuγ∗\nkdxdt/parenrightbiggn+2\nn\n+Cn+2\nn\n2/parenrightigg\n≤C(n)22\nn/parenleftigg\n/parenleftbig\nC1C/ba∇dblf/ba∇dblL˜r(ΩT)/parenrightbign+2\nn/parenleftbigg¨\nΩT|uk|γ∗˜r′dxdt/parenrightbiggn+2\nn˜r′\n+Cn+2\nn\n2/parenrightigg\n.\nNow asn+2\nn˜r′=γ∗\nγ∗+1<1, therefore using Young’s inequality we get\n¨\nΩT|uk|γ∗˜r′\ndxdt=¨\nΩT|uk|(γ∗+1)(n+2)\nndxdt≤C,\nwhereCis a positive constant independent of k. The above boundedness along with (3.31) and(3.32) gives\nthat the sequence/braceleftbigg\nuγ∗+1\n2\nk/bracerightbigg\nkis uniformly bounded in L2(0,T;H1\n0(Ω))∩L∞(0,T;L2(Ω))and{uk}kis uniformly\nbounded in L∞(0,T;Lγ∗+1(Ω)). We now show {uk}kis uniformly bounded in L2(t0,T;H1\nloc(Ω))for eacht0>0.\nSinceγ∗>1, andΩTis bounded and {uk}kis uniformly bounded in L∞(0,T;Lγ∗+1(Ω)), we thus have {uk}k\nis uniformly bounded in L2(0,T;L2(Ω)) =L2(ΩT), in particular in L2(K×(t0,T)), for everyK⊂⊂Ωand for\neacht0>0. Again as/braceleftbigg\nuγ∗+1\n2\nk/bracerightbigg\nkis uniformly bounded in L2(0,T;H1\n0(Ω)), so we have by nonnegativity of uk\nˆT\nt0ˆ\nKuγ∗−1\nk|∇uk|2dxdt≤ˆT\n0ˆ\nΩuγ∗−1\nk|∇uk|2dxdt≤4C\n(γ∗+1)2,\nfor everyK⊂⊂Ωand for each t0>0. Using the positivity of ukinω×[t0,T)for allk, we conclude\nˆT\nt0ˆ\nK|∇uk|2dxdydt≤C\nγ∗cγ∗−1\n(K,t0).\nSince{uk}kis increasing in kand is uniformly bounded in L∞(0,T;Lγ∗+1(Ω)), we get by Beppo Levi’s Lemma\nuk↑ustrongly in Lγ∗+1(ΩT)and hence in L1(ΩT), whereuis the pointwise limit of uk(possibly infinite). Then\nemploying Fatou’s lemma we will get uγ∗+1\n2∈L∞(0,T;L2(Ω))∩L2(0,T;H1\n0(Ω))andu∈L2(t0,T;H1\nloc(Ω))for\neacht0>0.Remark 2.23 implies that usatisfiesu(·,0) =u0(·)inL1sense. Also, we have u(x,t)≥C(ω,t0,n,s)\ninω×[t0,T)for allω⊂⊂Ωand for each t0>0. Thisuwill be a very weak solution to (2.13) in the sense of\nDefinition 2.25 , and the proof follows exactly similar to Theorem 2.39 .\nRemark 3.3. In the above existence results, the cases where u∈L2(0,T;H1\n0(Ω)), we can show that the weak\nsolutionuis unique, if it has finite energy. For this we assume that vis another finite energy solution to (2.12) or\n(2.13). Then, by the construction of uin each existence result, we get vis a supersolution to all the approximating\nproblems. Hence by the comparison principle in Lemma 2.20, we deduce that v≥ukfor allk. Thenv≥u. To\nprove the inverse inequality, let φ=v−u, thenφ≥0andφ∈L2(0,T;H1\n0(Ω))and hence we can use both φ+or\nφ−as test functions. Also, we note that φsolves the problem\n\n\nφt−∆φ+(−∆)sφ=f/parenleftbigg1\nvγ(x,t)−1\nuγ(x,t)/parenrightbigg\ninΩT,\nφ(x,t) = 0 in(Rn\\Ω)×(0,T),\nφ(x,0) = 0 inΩ.\nSincef/parenleftbigg1\nvγ(x,t)−1\nuγ(x,t)/parenrightbigg\n≤0inΩT, by comparison principle, we get φ≤0inΩT. Thusφ= 0i.e.u=v.\nRemark 3.4. As we can notice, in each of the existence results, we have u∈L2(t0,T;W1,1\nloc(Ω))for eacht0>0\nand this gives that our definition of a weak solution is well mo tivated in the sense that we have derived them\nintegrating by parts twice upon multiplying by a test functio n.\n334. Asymptotic behaviour\nThis section is devoted to the study of the asymptotic behavi our of finite energy solution to the problem\n(2.12) , ast→ ∞, for the particular case where fdepends only on xand thatf >0withf∈Lq(Ω)andq\nwill be specified later. We first state the following existenc e and uniqueness result to the corresponding mixed\nlocal-nonlocal elliptic problem, which can be done as a dire ct application of Theorem 2.27 . Consider the problem\n\n−∆w+(−∆)sw=f\nwγinΩ,\nw= 0 in(Rn\\Ω).(4.1)\nWe define the weak solution to the above problem as:\nDefinition 4.1. Suppose that f∈L1(Ω)is a nonnegative function and γ >0. We say that wis a very weak\nsolution to problem (4.1)ifw∈L1(Ω)satisfies the boundary conditions and ∀ω⊂⊂Ω,∃cω>0such that\nw(x)≥cω>0,a.e. inω,and for all ϕ∈ C∞\n0(Ω), we haveˆ\nΩw(−∆ϕ+(−∆)sϕ)dx=ˆ\nΩfϕ\nwγdx.\nTheorem 4.2. Letf∈L1(Ω)be a non-negative function. Then for all γ >0, the problem (4.1)has a nonnegative\nvery weak solution wsuch thatwγ+1\n2∈H1\n0(Ω)in the sense of Definition 4.1.\nProof. Letwkbe the unique positive bounded solution to the approximatin g problem\n\n−∆wk+(−∆)swk=fk\n(wk+1\nk)γinΩ,\nwk>0 inΩ,\nwk= 0 in(Rn\\Ω);(4.2)\nwithfk:= min(k,f). We note that the existence of wkfollows using a simple monotonicity argument. More\nprecisely, one can prove a similar version of existence resu lts and comparison principle as of Theorem 2.13 for\nthe elliptic case also, and using that, one can find the ellipt ic version of Lemma 2.20 andCorollary 2.21 . Again,\nby the comparison principle, we deduce that the sequence {wk}kis increasing in k. By the Harnack inequality,\nsee [22, Theorem 8.3], it holds that for any set ω⊂⊂Ω, for allkwe have,wk(x)≥w1(x)≥c(ω,n,s)inω.Also,\nthe details of existence, uniqueness, positivity and monot onicity ofwk’s can be found in [ 26, Lemma 3.2].\nNow taking ((wk+ε)γ−εγ),0<ε<1/k,as a test function in (4.2)and letting ε→0, it follows thatˆ\nΩ∇wk·∇wγ\nk+1\n2¨\nR2n(wk(x)−wk(y))(wγ\nk(x)−wγ\nk(y))\n|x−y|n+2sdxdy≤ˆ\nΩf(x)dx.\nByLemma 2.11 , we have\n(wk(x)−wk(y))(wγ\nk(x)−wγ\nk(y))≥C/parenleftig\nwγ+1\n2\nk(x)−wγ+1\n2\nk(y)/parenrightig2\n,\nand on the other hand\n∇wk·∇wγ\nk=γwγ−1\nk|∇wk|2,\nwe deduce that the sequence/braceleftig\nwγ+1\n2\nk/bracerightig\nkis bounded in the reflexive Banach space Xs\n0(Ω)∩H1\n0(Ω). Now by the\nmonotonicity of {wk}k, using Beppo-Levi’s theorem, we get the existence of a measu rable function wsuch that\nwk↑wa.e. inRn,wγ+1\n2\nk⇀wγ+1\n2weakly inXs\n0(Ω)∩H1\n0(Ω). Clearlyw≥c(ω)for any setω⊂⊂Ω. Then, letting\nk→ ∞ in the approximating problems and using the dominated conve rgence theorem, we can show that wis a\nvery weak solution to problem (4.1)withwγ+1\n2∈H1\n0(Ω). We refer [ 26, Theorem 2.16] for the boundedness of w.\nOne can find conditions when w∈H1\n0(Ω)in [26]. Further, [ 26, Corollary 5.2] gives such solution is unique.\nWe now write the asymptotic behaviour of solutions to proble m(2.12) under suitable conditions on fandu0.\nTheorem 4.3. Suppose that f∈Lq(Ω)is a non-negative function depending only on x, andqis such that\nsolution to the problem (4.1)is inH1\n0(Ω)and is unique. Let ube a finite energy solution to problem (2.12) with\n0≤u0≤w, wherewis the solution to (4.1). Thenu(·,t)→wast→ ∞, inLσ(Ω), whereσ<2∗.\nProof. We divide the proof into two cases according to the value of th e initial condition u0.\nThe first case u0= 0:In this case using the comparison principle as in Lemma 2.20 , we get that u≤win\n34Ω×(0,T)for allT >0. Hence,uis globally defined. We now show that u(x,·)is increasing in t. Sinceuhas\nfinite energy, we fix t1>0and definev(x,t) =u(x,t1+t),thenvsatisfies the problem\n\n\nvt−∆v+(−∆)sv=f(x)\nvγ(x,t)inΩT,\nv(x,t) = 0 in(Rn\\Ω)×(0,T),\nv(x,0) =u(x,t1) inΩ.\nNow asu(x,t1)≥u0≡0inΩ, using the comparison principle as in Lemma 2.20 again, we have u≤vin\nΩT. Therefore for all t1>0, we haveu(x,t)≤u(x,t1+t)for allx∈Rn. Therefore u(·,t)is increasing in t.\nSinceu≤w, so there exists a measurable function ˆu(·) = limsup\nt→∞u(·,t). By dominated convergence theorem,\nu(·,t)→ˆu(·)ast→ ∞ inLσ(Ω)forσ <2∗. Also, we have ˆu≤w. It is now sufficient to show that ˆuis a very\nweak solution to problem (4.1).\nLetϕ∈ C∞\n0(Ω), then using ϕas a test function in (2.12) and integrating in Ω×(k,k+1), we get\nˆ\nΩ(u(x,k+1)−u(x,k))ϕ(x)dx+ˆk+1\nkˆ\nΩu(x,t)(−∆ϕ+(−∆)sϕ)dxdt=ˆk+1\nkˆ\nΩf(x)\nuγ(x,t)ϕ(x)dxdt.\nNow clearly we haveˆ\nΩ(u(x,k+1)−u(x,k))|ϕ(x)|dx→0ask→ ∞.\nOn the other hand, in the set Ω×(k,k+1)using the monotonicity of uin the variable t, we have\nf(x)\nuγ(x,k+1)|ϕ(x)| ≤f(x)\nuγ(x,t)|ϕ(x)| ≤f(x)\nuγ(x,k)|ϕ(x)|,∀t∈(k,k+1),\nu(x,k)|−∆ϕ| ≤u(x,t)|−∆ϕ| ≤u(x,k+1)|−∆ϕ|,∀t∈(k,k+1),\nand\nu(x,k)|(−∆)sϕ| ≤u(x,t)|(−∆)sϕ| ≤u(x,k+1)|(−∆)sϕ|,∀t∈(k,k+1).\nThen, since φ,∆φ,(−∆)sφare bounded, so by the dominated convergence theorem, we wil l get that, as k→ ∞,\nˆk+1\nkˆ\nΩu(x,t)(−∆ϕ+(−∆)sϕ)dxdt→ˆ\nΩˆu(x)(−∆ϕ+(−∆)sϕ)dx,\nandˆk+1\nkˆ\nΩf(x)\nuγ(x,t)ϕ(x)dxdt→ˆ\nΩf(x)\nˆuγ(x)ϕ(x)dx.\nHence,ˆuis a very weak solution to problem (4.1). By the uniqueness we then get ˆu=w.\nThe second case 0<u0≤w:Let us denote by ˆua finite energy solution to problem (2.12) in this case. Then\nby the comparison principle in Lemma 2.20 , we deduce that u≤ˆu≤w, whereuis the weak solution to problem\n(2.12) withu0= 0. Hence by the convergence result of the first case, we get that ˆu(·,t)→w, ast→ ∞, strongly\ninLσ(Ω)for allσ<2∗.\nReferences\n[1] Boumediene Abdellaoui, María Medina, Ireneo Peral, and Ana Primo. The effect of the Hardy potential in\nsome Calderón-Zygmund properties for the fractional Lapla cian. J. Differential Equations , 260(11):8160–\n8206, 2016.\n[2] D. G. Aronson and James Serrin. Local behavior of solutio ns of quasilinear parabolic equations. Arch.\nRational Mech. Anal. , 25:81–122, 1967.\n[3] Rakesh Arora and Vicenţiu D. Rădulescu. Combined effects in mixed local–nonlocal stationary problems.\nProceedings of the Royal Society of Edinburgh Section A: Mat hematics , page 1–47, 2023.\n[4] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. Existe nce results to a quasilinear and singular parabolic\nequation. Discrete Contin. Dyn. Syst. , pages 117–125, 2011.\n[5] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. Some re sults about a quasilinear singular parabolic\nequation. Differ. Equ. Appl. , 3(4):609–627, 2011.\n35[6] Mehdi Badra, Kaushik Bal, and Jacques Giacomoni. A singu lar parabolic equation: existence, stabilization.\nJ. Differential Equations , 252(9):5042–5075, 2012.\n[7] Begoña Barrios, Ida De Bonis, María Medina, and Ireneo Pe ral. Semilinear problems for the fractional\nLaplacian with a singular nonlinearity. Open Math. , 13(1):390–407, 2015.\n[8] Kheireddine Biroud. Mixed local and nonlocal equation w ith singular nonlinearity having variable exponent.\nJ. Pseudo-Differ. Oper. Appl. , 14(1):Paper No. 13, 24, 2023.\n[9] Lucio Boccardo and Luigi Orsina. Semilinear elliptic eq uations with singular nonlinearities. Calc. Var.\nPartial Differential Equations , 37(3-4):363–380, 2010.\n[10] Lucio Boccardo, Maria Michaela Porzio, and Ana Primo. S ummability and existence results for nonlinear\nparabolic equations. Nonlinear Anal. , 71(3-4):978–990, 2009.\n[11] Brahim Bougherara and Jacques Giacomoni. Existence of mild solutions for a singular parabolic equation\nand stabilization. Adv. Nonlinear Anal. , 4(2):123–134, 2015.\n[12] Stefano Buccheri, João Vítor da Silva, and Luís Henriqu e de Miranda. A system of local-nonlocal p-\nLaplacians: the eigenvalue problem and its asymptotic limi t asp→ ∞.Asymptotic Analysis , 128(2):149–181,\n2022.\n[13] Annamaria Canino, Luigi Montoro, Berardino Sciunzi, a nd Marco Squassina. Nonlocal problems with\nsingular nonlinearity. Bull. Sci. Math. , 141(3):223–250, 2017.\n[14] José Carmona and Pedro J. Martínez-Aparicio. A singula r semilinear elliptic equation with a variable\nexponent. Adv. Nonlinear Stud. , 16(3):491–498, 2016.\n[15] M. G. Crandall, P. H. Rabinowitz, and L. Tartar. On a Diri chlet problem with a singular nonlinearity.\nComm. Partial Differential Equations , 2(2):193–222, 1977.\n[16] Stuti Das. Gradient Hölder regularity in mixed local an d nonlocal linear parabolic problem. Journal of\nMathematical Analysis and Applications , page 128140, 2024.\n[17] Ida de Bonis and Linda Maria De Cave. Degenerate parabol ic equations with singular lower order terms.\nDifferential Integral Equations , 27(9-10):949–976, 2014.\n[18] Cristiana De Filippis and Giuseppe Mingione. Gradient regularity in mixed local and nonlocal problems.\nMathematische Annalen , pages 1–68, 2022.\n[19] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Val dinoci. Hitchhiker’s guide to the fractional sobolev\nspaces. Bulletin des sciences mathématiques , 136(5):521–573, 2012.\n[20] Lawrence C Evans. Partial Differential Equations , volume 19. American Mathematical Soc., 2010.\n[21] Yuzhou Fang, Bin Shang, and Chao Zhang. Regularity theo ry for mixed local and nonlocal parabolic\np-Laplace equations. J. Geom. Anal. , 32(1):Paper No. 22, 33, 2022.\n[22] Prashanta Garain and Juha Kinnunen. On the regularity t heory for mixed local and nonlocal quasilinear\nelliptic equations. Trans. Amer. Math. Soc. , 375(8):5393–5423, 2022.\n[23] Prashanta Garain and Juha Kinnunen. Weak Harnack inequ ality for a mixed local and nonlocal parabolic\nequation. Journal of Differential Equations , 360:373–406, 2023.\n[24] Prashanta Garain and Erik Lindgren. Higher hölder regu larity for mixed local and nonlocal degenerate\nelliptic equations. Calculus of Variations and Partial Differential Equations , 62(2):67, 2023.\n[25] Prashanta Garain and Tuhina Mukherjee. Quasilinear no nlocal elliptic problems with variable singular\nexponent. Commun. Pure Appl. Anal. , 19(11):5059–5075, 2020.\n[26] Prashanta Garain and Alexander Ukhlov. Mixed local and nonlocal Sobolev inequalities with extremal and\nassociated quasilinear singular elliptic problems. Nonlinear Anal. , 223:Paper No. 113022, 35, 2022.\n36[27] David Kinderlehrer and Guido Stampacchia. An introduction to variational inequalities and their appli ca-\ntions. SIAM, 2000.\n[28] A. C. Lazer and P. J. McKenna. On a singular nonlinear ell iptic boundary-value problem. Proc. Amer.\nMath. Soc. , 111(3):721–730, 1991.\n[29] Tommaso Leonori, Ireneo Peral, Ana Primo, and Fernando Soria. Basic estimates for solutions of a class of\nnonlocal elliptic and parabolic equations. Discrete Contin. Dyn. Syst. , 35(12):6031–6068, 2015.\n[30] Bin Shang and Chao Zhang. Hölder regularity for mixed lo cal and nonlocal p-Laplace parabolic equations.\nDiscrete Contin. Dyn. Syst. , 42(12):5817–5837, 2022.\n[31] Bin Shang and Chao Zhang. Harnack inequality for mixed l ocal and nonlocal parabolic p-Laplace equations.\nJ. Geom. Anal. , 33(4):Paper No. 124, 24, 2023.\n[32] Zhuoqun Wu, Jingxue Yin, and Chunpeng Wang. Elliptic & parabolic equations . World Scientific, 2006.\n37"    },    {        "title": "2402.06972v2.Impurities_in_graphene_and_their_influence_on_the_Casimir_interaction.pdf",        "content": "Impurities in graphene and their influence on the Casimir interaction\nN. Khusnutdinov∗and D. Vassilevich†\nCentro de Matem´ atica, Computa¸ c˜ ao e Cogni¸ c˜ ao - Universidade Federal do ABC, Santo Andr´ e, SP, Brazil\nWe study the influence of impurities in graphene described by a scattering rate Γ on the Casimir\ninteraction between graphene and an ideal conductor or between two identical sheets of graphene\nat zero temperature and chemical potential. To this end, we compute the polarization tensor of\nquasiparticles in graphene and corresponding conductivities for TE and TM channels. The Casimir\nenergy density is evaluated with the help of the Lifshitz formula. We find that depending on the\nvalue of mass gap parameter the presence of Γ may lead to a slight damping or to a considerable\nenhancement of the Casimir interaction.\nI. INTRODUCTION\nThe Casimir effect [1, 2] is an interaction of neutral bodies due to quantum vacuum fluctuations. The study of\nCasimir interaction between advanced materials is a new and promising area of research [3]. On one hand, the\nunusual electronic properties of these materials lead to interesting effects on the Casimir force. On the other hand,\nthe improved quality of Casimir experiments makes them a useful tool for exploration of the materials themselves.\nThe Dirac materials (where the quasiparticles obey a quasirelativistic Dirac-type equation at sufficiently low ener-\ngies) provide us with an example of interplay between Quantum Field Theory and Condensed Matter. Graphene is a\nprominent representative of this family [4, 5]. Dealing with the Dirac materials it is natural to describe the interaction\nwith electromagnetic field by the polarization tensor of quasiparticles and use this tensor to calculate the Casimir\ninteraction. In the case of graphene, such an approach was used in [6] and in [7] at zero and non-zero temperature,\nrespectively. Remarkably, the polarization tensor approach to Casimir interaction of graphene was the only one which\nwas confirmed at experiments [8–11].\nAll real materials contain impurities. Particular form of impurities can vary. Impurities refer to a general form\nfor breaking the cleanliness of pristine materials. A classification of impurities and defects in graphene-like materials\nmay be found in the reviews [12–15]. The two-dimensional nature of graphene decreases the number of possible types\nof defects and impurities. The point is that it is energetically favourable for adatoms or substitutional impurities\nto reside outside of the graphene surface. They may be charged [16–18], magnetic [15], isotopic [19, 20], topological\nsuch as pentagons and heptagons [13, 21], or be a consequence of imperfections and growth-induced defects like point\n[22] and cluster defects [12]. Intentional impurity is usually called a dopant while the impurity itself can be both\nintentional and unintentional (accidental). Doping is used to change the physical or chemical properties of a material.\nImpurities in graphene [23, 24] may transform a linear dispersion near Dirac points to a quadratic one which signifies\nthe appearance of a mass gap induced by impurities. There are different approaches to describing impurities and\ntheir impact on the physical properties of materials. The common is a tight-binding model with a shot- or long-range\npotential [13], and scattering approach [25, 26].\nWith this large variety of the types of impurities in graphene we need a good model which captures the universal\nproperties of impurities while being sufficiently simple for being used in calculation of the polarization tensor. A\nsuccessful way of describing impurities consists in adding to the propagator of quasiparticles a parameter Γ which\ndescribes the impurity scattering rate. In other words, Γ is an imaginary part of the fermion self-energy. Such\na description has been applied to graphene mostly in the presence of an external magnetic field in [27–31]. The\ncomputations of [31] are in a very good agreement with the measurements [32] of giant Faraday rotation in graphene.\nIn principle Γ can depend on the frequency though keeping it constant appeared to be a good approximation. In this\nwork, we neglect the other role of impurities which is their ability to create a non-zero chemical potential µ.\nA particular form of impurities which were atoms (mostly sodium) on the surface of graphene and their influence\non the Casimir force were considered in [10, 11]. According to these papers such impurities lead to a mass gap and\na non-zero chemical potential of graphene but not to the appearance of impurity scattering described by scattering\nrate Γ.\nThe primary goal of this paper is to study the influence of impurity scattering rate Γ on the Casimir interaction\nbetween graphene and an ideal metal and between two graphene sheets. We restrict ourselves to the case of a zero\ntemperature and vanishing chemical potential. This is a somewhat simplified setting. Therefore, we will not try to\n∗nail.khusnutdinov@gmail.com\n†dvassil@gmail.comarXiv:2402.06972v2  [cond-mat.mes-hall]  17 Feb 20242\ncompare our results to any existing experiment (they were done at room temperature). Rather, we will clarify some\ngeneric features of the Casimir interaction of graphene. Our main conclusions is that turning on Γ leads to a slight\ndecrease of the Casimir energy density for small masses and to a considerable enhancement thereof for large masses.\nThus, taking impurities into account may be essential for analysing precision Casimir measurements.\nThis paper is organized as follows. In the next Section, we calculate the polarization tensor of quasiparticles in\ngraphene in the Pauli–Villars subtraction scheme. In Section III, we analyse the conductivities following from this\npolarization tensor and discuss their properties which are relevant for the Casimir interaction. The Casimir interaction\nitself is studied in Section IV. Section V contains some concluding remarks. We put long formulas for the conductivities\nin the Appendix. We use the units ℏ=c= 1.\nII. POLARIZATION TENSOR\nTo fix our conventions, we start with the Dirac operator\n/D= i˜γi(∂i+ ieAi)−m, (1)\nwhich describes the propagation of quasiparticles in graphene. The Latin letters form the middle of Alphabet will are\nused to denote coordinates of 2 + 1 dimensional vectors, i, j, k = 0,1,2. The letters from the beginning of Alphabet\nwill denote spatial components, a, b, c = 1,2. A twiddle above a vector means that the spatial components are rescaled\nwith the Fermi velocity,\n˜γ0=γ0, ˜γa=vFγa. (2)\nThe Dirac γmatrices are 8-dimensional (which corresponds to four generations of fermions in graphene). They satisfy\nthe condition γiγj+γjγi= 2gijwith the flat metric gij= (+ ,−,−). Besides, tr ( γiγj) = 8 gijand tr ( γiγjγlγk) =\n8(gijglk−gilgjk+gikgjl). We will use boldface to denote spatial components of the momenta 3-vectors, k= (k0,k),\nso that p·k=p0k0−p·kwithp·k=kapaδabwith δabbeing the Kronecker symbol.\nThe propagator S(x) is the kernel of an inverse of a free Dirac operator /D0=/DA=0, i.e. /Dx\n0S(x−y) =δ(3)(x−y).\nAfter a Fourier transform\nS(x) =Zd3k\n(2π)3eikxS(k), (3)\none arrives at the following expression\nS(k) =−k0γ0+vFkaγa+m\nk2\n0−v2\nFk2−m2. (4)\nTo introduce impurities, characterized by the scattering rate Γ, and chemical potential µone shifts the temporal\ncomponent of the momentum kin the propagator (4) as k0→bk0=k0+ iΓsgn ( k0) +µ, (see [27–30, 33]) For the rest\nof this work we set µ= 0. We denote bk= (bk0,k).\nThe introduction of Γ into the Quantum Field Theory approach leads to certain difficulties with the gauge invariance.\nThe modification of fermionic propagator can be understood as a consequence of an additional term in the Dirac action\ncontaining ψ†Γsgn (−i∂0)ψ. To maintain gauge invariance, each partial derivative has to be accompanied by a gauge\nfield. That is, one should consider the expression ψ†Γsgn (−i(∂0+ ieA0))ψ. Although it is not clear how one should\ndeal with the sign function of a differential operator, new (and rather complicated) vertexes involving ψ†,ψ, and\nA0seem to be inevitable. To overcome this difficulty we proceed as in [34]. Namely, we consider only the diagrams\nwithout A0lines which are thus independent of whichever vertices involving A0. I.e., we consider only the spatial\ncomponents of polarization tensor,\nΠab(p) = ie2Zd3k\n(2π)3tr\u0010\nS(bk)˜γaS([k−p)˜γb\u0011\n. (5)\nThe full tensor may be recovered, if needed, with the help of the transversality condition.\nIn the expression (5), one can change the integration variable k→˜k. The Jacobian factor v−2\nFcancels v2\nFcoming\nfrom ˜ γaand ˜γb. As a result, the whole dependence on (5) vFremains in the external momentum only, so that\nΠab(p) = Πab\nvF=1(˜p). (6)3\nIn other words, to compute the spatial components of polarization tensor it is sufficient to do the computations for\nvF= 1 and then replace pby ˜p. Till the end of this section, all computations will be done for vF= 1. Thus,\nΠab(p) =ie2\nπ3Z\nd3k−δab(m2−bk·([k−p)) + 2 kakb−kapb−kbpa\n(bk2−m2)(([k−p)2−m2). (7)\nBy using the Feynman parametrization this integral can be rewritten as\nΠab(p) =ie2\nπ3Z\nd3kZ1\n0dx−δab(m2−bk·([k−p)) + 2 kakb−kapb−kbpa\n(x[bk2−m2] + (1−x)[([k−p)2−m2])2. (8)\nThe denominator of the integrand reads\n(x[bk2−m2] + (1−x)[([k−p)2−m2])2= (q2−R)2,\nq=k−(1−x)p, (9)\nR=xbk2\n0+ (1−x)([k−p)2\n0−x(1−x)p2−m2.\nWe change the integration variable d2k→d2q. Note, that the terms in numerator of (8) which are linear in qare\nintegrated to zero. Also, under the integral one may replace qaqbby1\n2δabq2. After the integration over qone arrives\nat the expression\nΠab(p) =ie2\nπ2Z1\n0dxZ\ndk0δab(m2−bk0([k−p)0−x(1−x)p2) + 2x(1−x)papb\nR. (10)\nTo integrate over k0one has to split the integration region into a union of intervals where the signs of k0andk0−p0\nare constant. If, for simplicity, we restrict ourselves to the case p0>0 these are the intervals ( −∞,0), (0 , p0), and\n(p0,∞). One can easily show that the integrals over the first and the last intervals coincide. We have,\nΠab(p) =ie2\nπ2Z1\n0dx\u0014\n2Z0\n−∞dk0δab(m2−(k0−iΓ)(k0−p0−iΓ)−x(1−x)p2) + 2x(1−x)papb\nx(k0−iΓ)2+ (1−x)(k0−p0−iΓ)2−x(1−x)p2−m2\n+Zp0\n0dk0δab(m2−(k0+ iΓ)( k0−p0−iΓ)−x(1−x)p2) + 2x(1−x)papb\nx(k0+ iΓ)2+ (1−x)(k0−p0−iΓ)2−x(1−x)p2−m2\u0015\n. (11)\nThe integral over k0is divergent, so that we need a regularization and renormalization procedure for which we\nchoose the Pauli–Villars subtraction. We subtract from the integrand in (11) the same expression due to a regulator\nfield with a mass M, compute the integrals and then send M→ ∞ . Symbolically,\nΠab\nPV(p) = lim\nM→∞(Πab(p, m)−Πab(p, M)). (12)\nWe represent the polarization tensor through two form factors, αandβ, as\nΠab\nPV(p) =α(p)δab+β(p)papb\np2. (13)\nThese form factors read\nα(p) =2ie2\nπ2Z1\n0dx\u001a−iπp2(1−x)x√−q−2p2(1−x)x√qarctan\u0012p0x+ iΓ√q\u0013\n+2P2(1−x)x√Qarctan\u0012P0x−iΓ√Q\u0013\n+1\n2p0(1−2x) ln\u0000\n(p0x+ iΓ)2+q\u0001\n−1\n2P0(1−2x) ln\u0000\n(P0x−iΓ)2+Q\u0001\u001b\n, (14)\nβ(p) =−4ie2p2\nπ2Z1\n0dx x(1−x)\u001aiπ\n2√−q+1√qarctan\u0012p0x+ iΓ√q\u0013\n−1√Qarctan\u0012P0x−iΓ√Q\u0013\u001b\n,\nwhere q=p2(1−x)x−m2,Q=P2(1−x)x−m2,P2=P2\n0−p2, and P0=p0+ 2iΓ. Note that the integrals over x\nin (14) are convergent.\nIn the limit Γ →0 our results agree with previous calculations,\nΠab\nPV(Γ = 0) = −e2\nπ\u0012\nδab+papb\np2\u00132m|p| −(p2+ 4m2)arctanh( |p|/2m)\n2|p|, (15)\nsee [35] and also [6].4\n1 2 3 4 50.20.40.60.8\n(a)\nm/Γs\n1 2 3 4 5246\n(b)\nm/Γg/Γ\nFIG. 1. The plots of the parameters of Drude model given by Eq. (20)\nIII. CONDUCTIVITIES\nFrom now on we restore the dependence of polarization tensor on the Fermi velocity according to Eq. (6).\nIn this Section, we consider the conductivity tensor defined as\nσab=Πab\nPV\nip0. (16)\nFor vanishing spatial momenta, pa= 0, only a scalar conductivity σ(p0) entering the tensor conductivity as\nδabσ(p0) =σab|pa=0 (17)\nis important. It is convenient to measure σ(p0) in the units of universal conductivity of graphene σgr=e2/4 which\nis nothing else than the conductivity of a pristine graphene with Γ →+0 at m→0. In terms of the form factors\nfrom the previous Section, σ(p0) =α(p0,0)/(ip0). For arbitrary values of the parameters σ(p0) can only be evaluated\nnumerically. However, an expansion in p0forp0≪Γ and p0≪mcan be done analytically,\nσ\nσgr=4\nπ2(\n2Γ2\nΓ2+m2+ip0\n3m \nΓm\u0000\nΓ2−5m2\u0001\n(Γ2+m2)2+ 2 arctan\u0012Γ\nm\u0013\n−π!\n+O(p2\n0))\n. (18)\nAtp0= 0 this equation coincides with a relation obtained in [27].\nLet us check whether at small frequencies p0the conductivity can be approximated by the Drude formula\nσDr\nσgr=sg\nip0+g=sg2\np2\n0+g2−isp0g\np2\n0+g2, (19)\nwhere ghas the meaning of inverse scattering rate (similarly to Γ) while sdefines the Drude weight. This is a\nphenomenological formula valid mostly for metals. There is no profound reason why it should describe the conductivity\nof graphene. However, this is an interesting check.\nBy identifying the leading terms in the small p0expansion of (19) with (18) we obtain\ns=8Γ2\nπ2(Γ2+m2), g=−6Γ2m\n(Γ2+m2)\u0010\nΓm(Γ2−5m2)\n(Γ2+m2)2+ 2 arctan\u0000Γ\nm\u0001\n−π\u0011. (20)\nThis quantities are plotted at Fig. 1 while the real and imaginary parts of σare depicted at Fig. 2 in comparison with the\nDrude formula. Somewhat surprisingly, the Drude scattering rate gmay be considerably larger or considerably smaller\nthan the scattering rate Γ of the quasiparticles. By looking at Fig. 2, we see that the agreement between imaginary5\n0.5 1 1.5 2 2.5 3\n−4−3−2−11\n(a)p0/Γℜ(σ/σ gr)\nm/Γ = 0 .3\nm/Γ = 1 .0\n0.5 1 1.5 2 2.5 3\n−0.4−0.3−0.2−0.10.1\n(b)p0/Γℑ(σ/σ gr)\nm/Γ = 0 .3\nm/Γ = 1 .0\nFIG. 2. Real (a) and imaginary (b) parts of the conductivity as functions of p0/Γ. The solid lines are for Drude model (19)\nwith parameters (20). The dashed lines are the conductivity derived from the polarization tensor with p= 0\n1 2 3 4 5\n×10−25101520\nΓ (eV)ℜσtm(Γ)/|σtm(Γ = 0) |\np0= 10−3eV\np0= 5×10−3eV\np0= 9×10−3eV\nFIG. 3. The normalized conductivity ℜσtm(Γ)/|σtm(Γ = 0) |as a function of Γ for m= 0.01 eV, p= 0, and three different\nvalues of frequency p0.\nparts of σandσDris fairly good at low frequencies though the deviations become larger at higher frequencies. The\nreal parts show qualitatively different behaviour. ℜσincreases at low frequencies while ℜσDrdecreases. Besides, ℜσDr\ndoes not have a jump at the threshold of pair creation p0= 2mwhich is a characteristic feature of the conductivity\nof graphene. We conclude that the Drude formula fails to give a reasonable approximation to the conductivity\nof graphene with impurities in the range of frequencies which are relevant for the Casimir effect. The quality of\napproximation may be improved by using the Drude–Lorentz formulas for conductivities [36].\nForp̸=0, it is convenient to use two different conductivities,\nσte=α\nip0and σtm=α+β\nip0. (21)\nAs we will see in the next Section, these conductivities describe the reflection of TE and TM waves, respectively, on\nthe surface of graphene. We will also see that TM modes give a dominant contribution to the Casimir interaction.\nWe would like to discuss a property of the conductivities which will be important for the analysis of Casimir effect.\nWhen Γ = 0 and the ˜ pis much smaller than m, the imaginary parts conductivities are small while the real part\nvanishes, see (15). The smallness is a direct consequence of the Pauli–Villars subtraction which ensures that the\npolarization tensor vanished at m→ ∞ . Since Γ influences conductivities, switching on Γ at ˜ p <2mshould lead to\nan increase of ℜσtm. One should only find whether this increase is large or small. Our numerical results are presented\non Fig. 3 where ℜσtm(Γ) divided by the absolute values of σtmat Γ = 0 is depicted as a function of Γ for three\ndifferent frequencies p0and a fixed m= 0.01 eV. For p0much smaller than 2 mthere is a strong enhancement of the\nconductivity. This enhancement becomes weaker and disappears as p0approaches 2 m.6\nIV. THE CASIMIR ENERGY\nLet us consider two parallel infinite planes in the vacuum separated by a distance a. One plane will be always\noccupied by graphene, while the other plane will be either graphene or an ideal conductor. The Casimir energy per\nunit surface for two interacting plane surfaces IandIIis given by the Lifshitz formula (see, for example, [1])\nEI,II=Zd2p\n(2π)3Z∞\n0dξh\nln\u0010\n1−e−2apEr(I)\nter(II)\nte\u0011\n+ ln\u0010\n1−e−2apEr(I)\ntmr(II)\ntm\u0011i\n, (22)\nwhere the integrations are done over the spatial momentum pparallel to the plates and over the imaginary frequency\nξ,p0= iξ, and pE=p\nξ2+p2. In (22), rI,II\nte,tmdenote the reflection coefficient for TE and TM waves on the first and\nsecond surfaces, respectively. These coefficients can be expressed through the polarization tensor as [7]\nrte=−\u0012\n1 +2pE\nξσte\u0013−1\n, rtm=\u0012\n1 +2ξ\npEσtm\u0013−1\n, (23)\nwhere σte,tmhave been defined above in Eq. (21). After continuation to the imaginary frequencies they read σte=\n−α/ξ,σtm=−(α+β)/ξ.\nFor an ideal conductor rtm=−rte= 1 which can be obtained from (23) by taking the limits σte,tm→ ∞ . The\nCasimir energy density for two ideal conductors reads\nEid=−π2\n720a3. (24)\nWe like to mention that Eq. (22) is valid only if the reflection matrix is diagonal in the TE-TM basis. Otherwise,\nthe Lifshitz formula is a bit more complicated, see e.g., [37].\nSince the reflection coefficients depend on |p|rather than on individual components of pwe can integrate over the\nangular coordinates in the Lifshitz formula (22) which boils down to the replacement d2p→(2π)|p|d|p|. Next, we\nintroduce new variables, yandz, by the formulas p2=p2\nE(1−y2),ξ=pEy, and pE→z/a. In these new variables,\nEI,II=1\na3Z∞\n0z2dz\n(2π)2Z1\n0dy\nln\n1−e−2z\n\u0010\n1 +2\nyσI\nte\u0011\u0010\n1 +2\nyσII\nte\u0011\n+ ln\n1−e−2z\n\u0010\n1 +2y\nσI\ntm\u0011\u0010\n1 +2y\nσII\ntm\u0011\n\n,(25)\nOne can notice that apart from an overall factor of 1 /a3the whole dependence of Casimir energy (25) on the\ndistance resides in the conductivities where it appears in the combinations\n˜m=am\nz, ˜Γ =aΓ\nz. (26)\nExplicit formulas for σteandσtmare quite long and deferred to the Appendix A, see (A1). The limit a→0 is\nequivalent to taking m→0 and Γ →0 so that we have the case gapless pristine graphene studied in [6] with\nE ∼1/a3.\nTo analyze the opposite limit of large distance we have to take ˜ m→ ∞ and˜Γ→ ∞ while keeping the fraction\n˜Γ/˜m= Γ/mfixed. In this limit, σteandσtmbehave identically,\nσte\nσgr,σtm\nσgr→8˜Γ2\nπ2\u0010\n˜Γ2+ ˜m2\u0011=8\nπ2G, (27)\nwhere\nG≡Γ2\nm2+ Γ2≤1. (28)\nThus, we have the case of constant conductivities analysed in [38]. The Casimir energy decays as 1 /a3and\nEg,g\ntm\nEid≈4.3×10−3G,Eg,g\nte\nEid≈1.3×10−5G2, (29)\nEg,id\ntm\nEid≈2×10−2G,Eg,id\nte\nEid≈2.1×10−3G, (30)7\n0.5 1 1.5 2\n×10311.522.5×10−2\n(a)\na(nm)Eg,id\nΓ/Eid\nm= 0 eV\nm= 5×10−3eV\nm= 5×10−2eV\n0.5 1 1.5 2\n×1030.511.522.5×10−2\n(b)\na(nm)Eg,id\nΓ/Eid\nm= 0 eV\nm= 5×10−2eV\nm= 5×10−1eV\n0.5 1 1.5 2\n×10312345×10−3\n(c)\na(nm)Eg,g\nΓ/Eid\nm= 0 eV\nm= 5×10−3eV\nm= 5×10−2eV\nFIG. 4. The Casimir energy density as a function of the distance normalised Eid(Eq. (24)) for the system of graphene and an\nideal conductor (on (a) and (b)) and for two identical graphene sheets (c). The impurity scattering rate is Γ = 5 ×10−3eV on\n(a) and (c), and Γ = 5 ×10−2eV on (b).\nwhere the superscripts g ,g and g ,id mean the interaction between two identical sheets of graphene and between\ngraphene and an ideal conductor, respectively. We normalized the results to the Casimir energy of two ideal conductors\n(24) and separated the contributions of TE and TM modes. As usual [7], the contributions of TE modes are much\nsmaller than the TM contributions.\nFor Γ = 0 the terms on the right hand sides of (27) vanish and the leading terms in the a→ ∞ expansion read\nσte\nσgr=4z((1−y2)vF2+y2)\n3πy(am),σtm\nσgr=4zy\n3π(am). (31)\nAs a result, the Casimir energy of two graphene sheets behave at large distances as E ∼1/(a3(am)2) while for the\nsystem of a perfect metal and graphene the asymptotic behaviour is E ∼1/(a3(am)), in agreement with [6].\nThe rest of the analysis will be done numerically. For orientation, we present the value of Γ = 4 .4×10−3eV which\ngives the best fit [31] between the Quantum Field Theory polarization tensor and the Faraday rotation experiment\n[32] at 7 T.1The separation aused in the Casimir experiments with graphene [8, 10] is between 200 nm and 2000\nnm. The value for mfor the graphene sample used in the Casimir experiment [11] was ∼0.15 eV. The presence\nof a damping factor e−2apEin the Lifshitz formula (22) indicates that the region of relevant Euclidean momenta is\nrestricted by the inequality pE≲(2a)−1. E.g., for a= 500 nm one has (2 a)−1≃0.2 eV.\nThe Casimir energy density normalized to the Casimir energy density between two ideal conductors is plotted\non Fig. 4. By comparing Fig. 4(a,b) with Fig. 4(c) we see that the Casimir interaction of graphene with graphene\nbehaves qualitatively similarly to the interaction of graphene with an ideal conductor though the former is about 5\ntimes weaker than the latter. Thus, in what follows we will only consider the interaction of graphene with an ideal\nconductor. On Figs. 4 (a) and (b) we see that the Casimir interaction is damped by the mass exactly as it happens\nfor Γ = 0, see [6]. The line for m= 0 appears slightly upper for Γ = 0 .005 eV than for Γ = 0 .05 eV, while the line for\nm= 0.05 eV passes significantly lower on Fig. 4(a) than on Fig. 4(b). Thus, the effect of Γ strongly depends on the\nmass. To see this effect more clearly, we depicted on Fig. 5 the relative variation of Casimir energy density\nδΓEg,id=Eg,id(Γ)− Eg,id(Γ = 0)\nEg,id(Γ = 0)(32)\nat a fixed separation a= 500 nm and three different values of the mass.\nThe large relative enhancement of the Casimir effect for m= 0.1 eV can be explained by the observation made at the\nend of the previous section that turning on Γ leads to a significant enhancement of ℜσtmfor the frequencies p0≪m.\nFor a larger mass the interval of frequencies where the enhancement of conductivity takes place becomes larger which\nis translated to the enhancement of Casimir interaction. Since the Casimir effect is an integral effect containing\ncompeting contributions from all frequencies and tangential momenta it is impossible to predict the amplitude of the\nenhancement basing on this type of arguments. For the same reason, it is hard to give a precise explanation to the\nvery moderate damping of Casimir force by Γ at very small masses like m= 10−3eV at Fig. 5. We can only suppose\n1For a magnetic field of 3T different values of the parameters were obtained as well as a much worse fit. This can be attributed to some\nspecific physics in particular samples of graphene used in the experiment, see [39].8\n0.20.40.60.81\n×10−10.20.40.60.8\nΓ (eV)δΓEg,idm= 10−1eV\nm= 10−2eV\nm= 10−3eV\nFIG. 5. The plots of relative variation of the Casimir energy density (32) at a= 500nm as a function of Γ for three values of\nthe mass.\nthat at the absence of the mechanism described above the impurity scattering should reduce the conductivities and\nthus the Casimir interaction according to the high-school physics intuition. For an intermediate mass, m= 10−2eV,\nwe see an intermediate behaviour.\nAnother characteristic feature of the Casimir interaction in the presence of Γ is a 1 /a3behaviour of the Casimir\nenergy at large distances. However, the asymptotic values (30) are reached at very large a. For example, for\nm= Γ = 0 .01 eV at a= 106nm the difference between exact and asymptotic values is still at the 10% level. For some\nother choices of mand Γ the agreement may be reached at somewhat smaller distances which are anyway beyond the\ndistances used at experiments.\nV. CONCLUSIONS\nIn this paper, we studied the Casimir interaction of graphene containing impurities described by a scattering rate\nΓ. We evaluated the polarization tensor of quasiparticles in the Pauli–Villars subtraction scheme and computed\nthe corresponding conductivities. Next, we used the conductivities in the Lifshitz formula to calculate the Casimir\nenergy density for various values of parameters. Our main message is that the presence of impurities can considerably\nenhance the Casimir interaction for large values of the mass (about m= 0.1 eV) and somewhat damp the Casimir\ninteraction for near zero mass. Thus, the non-zero impurity scattering rate Γ should be taken into account in the\nanalysis of precision Casimir experiments with graphene. To do this however one should also take into consideration\nthe chemical potential and temperature since both influence in an essential way the Casimir force [7, 11, 40]. This\nwill be our next task which we are going to take up in the near future.\nAn additional motivation for considering the influence of impurities jointly with the temperature comes from the\nold discussion on the Nernst theorem in the Casimir physics [41–43], see [44] for a recent review. The violation of\nNernst theorem was blamed on the relaxation rate parameter present in the Drude formula. It is interesting to check\nwhat happens if this rate appears in a different model.\nIn [34] an unconventional version of the Pauli–Villars subtraction scheme in the presence of impurities was suggested.\nIt would be interesting to study the influence of this scheme on Casimir interaction.\nFinally, we would like to mention an approach to the Casimir interaction of graphene based on the QFT effective\naction [6, 45]. This approach is equivalent to a fine structure constant expansion of the Lifshitz formula.\nACKNOWLEDGMENTS\nWe are grateful to Ignat Fialkovsky for previous collaboration and discussions on the impurities in graphene. This\nwork was supported in parts by the S˜ ao Paulo Research Foundation (FAPESP) through the grants 2021/10128-0\n(D.V.) and 2022/08771-5 (N.K.), and by the National Council for Scientific and Technological Development (CNPq),\ngrant 304758/2022-1 (D.V).9\nAppendix A: Long but useful formulas\nHere we present explicit form of the conductivities after the changes of variables made above Eq. (25) and in (26).\nσte\nσgr=8\nπ2yZ1\n0dx(\nπ((1−y2)vF2+y2)(1−x)xp\n˜m2+ (1−x)x((1−y2)vF2+y2)\n−2((1−y2)vF2+y2)(1−x)xp\n˜m2+ (1−x)x((1−y2)vF2+y2)arctan ˜Γ +yxp\n˜m2+ (1−x)x((1−y2)vF2+y2)!\n−2((1−y2)vF2+ (y+ 2˜Γ)2)(1−x)xq\n˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)arctan\n˜Γ−x(y+ 2˜Γ)q\n˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)\n\n+1\n2y(1−2x) ln\u0010\n(yx+˜Γ)2+ ˜m2+ (1−x)x((1−y2)vF2+y2)\u0011\n−1\n2(y+ 2˜Γ)(1−2x) ln\u0010\n(˜Γ−(y+ 2˜Γ)x)2+ ˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)\u0011\u001b\n,\nσtm\nσgr=8\nπ2yZ1\n0dx(\nπy2(1−x)xp\n˜m2+ (1−x)x((1−y2)vF2+y2)\n−2y2(1−x)xp\n˜m2+ (1−x)x((1−y2)vF2+y2)arctan ˜Γ +yxp\n˜m2+ (1−x)x((1−y2)vF2+y2)!\n−2(y+ 2˜Γ)2(1−x)xq\n˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)arctan\n˜Γ−x(y+ 2˜Γ)q\n˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)\n\n+1\n2y(1−2x) ln\u0010\n(yx+˜Γ)2+ ˜m2+ (1−x)x((1−y2)vF2+y2)\u0011\n−1\n2(y+ 2˜Γ)(1−2x) ln\u0010\n(˜Γ−(y+ 2˜Γ)x)2+ ˜m2+ (1−x)x((1−y2)vF2+ (y+ 2˜Γ)2)\u0011\u001b\n. (A1)\n[1] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir effect , Int. Ser. Monogr.\nPhys., Vol. 145 (Oxford University Press, Oxford, UK, 2009) pp. 1–768.\n[2] K. A. Milton, The Casimir effect: Recent controversies and progress, J. Phys. A37, R209 (2004), arXiv:hep-th/0406024\n[hep-th].\n[3] L. Woods, D. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. Rodriguez, and R. Podgornik, Materials perspective on\nCasimir and van der Waals interactions, Rev. Mod. Phys. 88, 045003 (2016), arXiv:1509.03338 [cond-mat.mtrl-sci].\n[4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene,\nRev. Mod. Phys. 81, 109 (2009), arXiv:0709.1163 [cond-mat.other].\n[5] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Reviews of Modern\nPhysics 83, 407–470 (2011).\n[6] M. Bordag, I. V. Fialkovsky, D. M. Gitman, and D. V. Vassilevich, Casimir interaction between a perfect conductor and\ngraphene described by the Dirac model, Phys. Rev. B 80 , 245406 (2009), arXiv:0907.3242 [hep-th].\n[7] I. V. Fialkovsky, V. N. Marachevsky, and D. V. Vassilevich, Finite temperature Casimir effect for graphene, Phys. Rev. B\n84, 035446 (2011), arXiv:1102.1757 [hep-th].\n[8] A. A. Banishev, H. Wen, R. K. Kawakami, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Measuring the\nCasimir force gradient from graphene on a SiO 2substrate, Phys. Rev. B 87 , 205433 (2013).\n[9] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Theory of the Casimir interaction from graphene-coated\nsubstrates using the polarization tensor and comparison with experiment, Phys. Rev. B 89 , 115419 (2014), arXiv:1403.1575\n[cond-mat.other].\n[10] M. Liu, Y. Zhang, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Demonstration of an Unusual Thermal\nEffect in the Casimir Force from Graphene, Phys. Rev. Lett. 126, 206802 (2021), arXiv:2104.13598 [quant-ph].\n[11] M. Liu, Y. Zhang, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Experimental and theoretical investigation\nof the thermal effect in the Casimir interaction from graphene, Phys. Rev. B 104, 085436 (2021), arXiv:2108.07558 [quant-\nph].\n[12] F. Banhart, J. Kotakoski, and A. V. Krasheninnikov, Structural defects in graphene, ACS Nano 5, 26–41 (2010).10\n[13] Y. Zhang, J. Hu, B. A. Bernevig, X. R. Wang, X. C. Xie, and W. M. Liu, Impurities in graphene, Physica Status Solidi\n(a)207, 2726–2738 (2010).\n[14] P. T. Araujo, M. Terrones, and M. S. Dresselhaus, Defects and impurities in graphene-like materials, Materials Today 15,\n98–109 (2012).\n[15] L. Fritz and M. Vojta, The physics of kondo impurities in graphene, Reports on Progress in Physics 76, 032501 (2013).\n[16] T. Ando, Screening effect and impurity scattering in monolayer graphene, Journal of the Physical Society of Japan 75,\n074716 (2006).\n[17] V. V. Cheianov and V. I. Fal’ko, Friedel oscillations, impurity scattering, and temperature dependence of resistivity in\ngraphene, Physical Review Letters 97, 226801 (2006), 0608228 [cond-mat.mes-hall].\n[18] J.-H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D. Williams, and M. Ishigami, Charged-impurity scattering in graphene,\nNature Physics 4, 377–381 (2008).\n[19] S. Fan, L. Liu, and M. Liu, Monitoring the growth of carbon nanotubes by carbon isotope labelling, Nanotechnology 14,\n1118–1123 (2003).\n[20] F. Simon, C. Kramberger, R. Pfeiffer, H. Kuzmany, V. Z´ olyomi, J. K¨ urti, P. M. Singer, and H. Alloul, Isotope engineering\nof carbon nanotube systems, Physical Review Letters 95, 017401 (2005).\n[21] B. An, S. Fukuyama, K. Yokogawa, M. Yoshimura, M. Egashira, Y. Korai, and I. Mochida, Single pentagon in a hexagonal\ncarbon lattice revealed by scanning tunneling microscopy, Applied Physics Letters 78, 3696–3698 (2001).\n[22] J. Kotakoski, A. V. Krasheninnikov, and K. Nordlund, Energetics, structure, and long-range interaction of vacancy-type\ndefects in carbon nanotubes: Atomistic simulations, Physical Review B 74, 245420 (2006).\n[23] T. Altanhan and B. Kozal, Impurity effects in graphene, The European Physical Journal B 85, 222 (2012).\n[24] G. Viola, T. Wenger, J. Kinaret, and M. Fogelstr¨ om, Graphene plasmons: Impurities and nonlocal effects, Physical Review\nB97, 085429 (2018).\n[25] T. Ando and T. Nakanishi, Impurity scattering in carbon nanotubes – absence of back scattering –, Journal of the Physical\nSociety of Japan 67, 1704–1713 (1998).\n[26] T. Ando, T. Nakanishi, and R. Saito, Berry’s phase and absence of back scattering in carbon nanotubes, Journal of the\nPhysical Society of Japan 67, 2857–2862 (1998).\n[27] E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Magnetic field driven metal insulator phase transition\nin planar systems, Phys. Rev. B 66, 045108 (2002), arXiv:cond-mat/0202422.\n[28] V. P. Gusynin and S. G. Sharapov, Transport of Dirac quasiparticles in graphene: Hall and optical conductivities, Phys.\nRev. B 73, 245411 (2006), arXiv:cond-mat/0512157.\n[29] V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Unusual microwave response of Dirac quasiparticles in graphene, Phys.\nRev. Lett. 96, 256802 (2006), arXiv:cond-mat/0603267.\n[30] V. P. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy, Excitonic gap, phase transition, and quantum Hall\neffect in graphene, Phys. Rev. B 74, 195429 (2006), arXiv:cond-mat/0605348.\n[31] I. V. Fialkovsky and D. V. Vassilevich, Faraday rotation in graphene, Eur. Phys. J. B 85, 384 (2012), arXiv:1203.4603\n[cond-mat.mes-Hall].\n[32] I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. van der Marel, and A. B.\nKuzmenko, Giant faraday rotation in single- and multilayer graphene, Nature Physics 7, 48 (2010), arXiv:1007.5286 [cond-\nmat].\n[33] G. Tkachov, Topological Insulators (Pan Stanford, 2013) p. 352, arXiv:1011.1669v3.\n[34] O. Holanda, R. Meyer, and D. Vassilevich, Parity anomaly with impurities and the Pauli-Villars subtraction, Phys. Rev.\nD108, 105001 (2023), arXiv:2305.11972 [hep-th].\n[35] T. Appelquist, M. J. Bowick, D. Karabali, and L. C. R. Wijewardhana, Spontaneous Breaking of Parity in (2+1)-\ndimensional QED, Phys. Rev. D 33, 3774 (1986).\n[36] D. Drosdoff and L. M. Woods, Casimir forces and graphene sheets, Phys. Rev. 82, 155459 (2010), arXiv:1007.1231.\n[37] I. Fialkovsky, N. Khusnutdinov, and D. Vassilevich, Quest for Casimir repulsion between Chern-Simons surfaces, Phys.\nRev.B97, 165432 (2018), arXiv:1802.06598 [cond-mat.mes-hall].\n[38] N. Khusnutdinov, D. Drosdoff, and L. M. Woods, Casimir energy for surfaces with constant conductivity, Phys. Rev. D\n89, 085033 (2014), arXiv:1404.2532 [quant-ph].\n[39] I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko,\nIntrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene, Nano Letters 12, 2470–2474 (2012),\narXiv:1204.4372 [cond-mat.str-el].\n[40] M. Bordag, I. Fialkovskiy, and D. Vassilevich, Enhanced Casimir effect for doped graphene, Phys. Rev. B 93 , 075414\n(2016), arXiv:1507.08693 [cond-mat.mes-hall].\n[41] V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, and C. Romero, Violation of the Nernst heat theorem in the\ntheory of the thermal Casimir force between Drude metals, Phys. Rev. A 69, 022119 (2004), arXiv:quant-ph/0401138.\n[42] I. H. Brevik, S. A. Ellingsen, J. S. Hoye, and K. A. Milton, Analytical and Numerical Demonstration of How the Drude\nDispersive Model Satisfies Nernst’s Theorem for the Casimir Entropy, J. Phys. A 41, 164017 (2008), arXiv:0710.4882\n[quant-ph].\n[43] V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. Lopez, V. M. Mostepanenko,\nand C. Romero, Comment on ‘On the temperature dependence of the Casimir effect’, Phys. Rev. E 73, 028101 (2006),\narXiv:quant-ph/0503134.\n[44] V. M. Mostepanenko, Casimir Puzzle and Casimir Conundrum: Discovery and Search for Resolution, Universe 7, 84 (2021),\narXiv:2104.01460 [quant-ph].11\n[45] M. B. Farias, C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Quantum friction between graphene sheets, Phys. Rev.\nD95, 065012 (2017), arXiv:1612.08675 [hep-th]."    },    {        "title": "2402.07038v1.Nonlinear_Modes_as_a_Tool_for_Comparing_the_Mathematical_Structure_of_Dynamic_Models_of_Soft_Robots.pdf",        "content": "Nonlinear Modes as a Tool for Comparing the Mathematical Structure\nof Dynamic Models of Soft Robots\nPietro Pustina1,4, Davide Calzolari2,3, Alin Albu-Sch ¨affer2,3, Alessandro De Luca1, Cosimo Della Santina3,4\nAbstract — Continuum soft robots are nonlinear mechanical\nsystems with theoretically infinite degrees of freedom (DoFs)\nthat exhibit complex behaviors. Achieving motor intelligence\nunder dynamic conditions necessitates the development of\ncontrol-oriented reduced-order models (ROMs), which employ\nas few DoFs as possible while still accurately capturing the\ncore characteristics of the theoretically infinite-dimensional\ndynamics. However, there is no quantitative way to measure\nif the ROM of a soft robot has succeeded in this task. In other\nfields, like structural dynamics or flexible link robotics, linear\nnormal modes are routinely used to this end. Yet, this theory\nis not applicable to soft robots due to their nonlinearities. In\nthis work, we propose to use the recent nonlinear extension in\nmodal theory –called eigenmanifolds– as a means to evaluate\ncontrol-oriented models for soft robots and compare them. To\nachieve this, we propose three similarity metrics relying on the\nprojection of the nonlinear modes of the system into a task space\nof interest. We use this approach to compare quantitatively,\nfor the first time, ROMs of increasing order generated under\nthe piecewise constant curvature (PCC) hypothesis with a high-\ndimensional finite element (FE)-like model of a soft arm. Results\nshow that by increasing the order of the discretization, the\neigenmanifolds of the PCC model converge to those of the FE\nmodel.\nI. I NTRODUCTION\nNonlinear modal analysis is a powerful tool to characterize\nthe behavior of highly nonlinear mechanical systems [1],\nsuch as aerospace structures [2], offshore towers [3] and\nmicro/nano-electromechanical systems [4]. Despite the fact\nthat its use has been historically for system analysis and\ndesign of structures undergoing small deformations, recent\nworks [5]–[7] have used nonlinear modal theory in analyz-\ning robots and in generating energy-efficient motions that\nasymptotically require no control effort to be sustained.\nNonlinear modal analysis has found extensive application\nin flexible mechanical systems under the hypothesis of small\ndeformations - especially in model reduction in structural\nmechanics have been studied [8]–[12]. In [10], energy-\nfrequency plots are used to compare ROMs of thin shells\nand plates. Finally, [11], [12] use nonlinear modes to assess\nROM convergence to FE models of specific structures.\nThe reserach of Pietro Pustina was supported by the Sapienza SoftRobot\nproject. The research of Alessandro De Luca was supported by the PNRR\nMUR project PE0000013-FAIR. The research of Cosimo Della Santina\nwas supported by the Horizon Europe Program from Project EMERGE\nunder Grant 101070918. The research of Davide Calzolari and Alin Albu-\nSch¨affer was supported by the Advanced Grant M-Runners (ID: 835284)\nby the European Research Council.1Department of Computer, Control\nand Management Engineering, Sapienza University of Rome, Rome, Italy.\npustina@diag.uniroma1.it; deluca@diag.uniroma1.it2Institute of Robotics\nand Mechatronics, German Aerospace Center, 82234 Oberpfaffenhofen,\nGermany. davide.calzolari@dlr.de; alin.albu-schaeffer@dlr.de3Department\nof Informatics, Technical University of Munich, 85748 Garching, Germany.\n4Department of Cognitive Robotics, Delft University of Technology, Delft,\nThe Netherlands. c.dellasantina@tudelft.nl\nFig. 1: Schematic of our approach using nonlinear modal analysis\nand eigenmanifold theory to compare different control-oriented\nmodels for the same soft robot. Models differ in assumptions and\nnumber of DoFs, making systematic comparison challenging. In the\nfigure,∼indicates a similar structure, ≁indicates dissimilarity and\na question mark denotes incomparability.\nContinuum soft robots [13], due to their high degree of\ndeformability and inherent softness, are expected to exhibit\na rich spectrum of modal behaviors. However, none of these\nworks are directly applicable to soft robots because they\nrely on nonlinear extensions of modal analysis built on\nassumptions that do not extend to soft robots and other multi-\nbody systems. Thus, the use of nonlinear modes in continuum\nsoft robotics is still unexplored.\nThis work proposes the first application of nonlinear modal\nanalysis to soft robotics by employing the recently developed\nconcept of eigenmanifolds [14] - which, contrary to classic\ntheories, builds on hypotheses compatible with soft robotics.\nMore specifically, with this work, we tackle the challenge\nof quantitatively comparing control-oriented reduced order\nmodels (ROMs) for continuum soft robots. In this way,\nit is also possible to compare control-oriented ROMs ob-\ntained from different discretization hypotheses, including\nthose of [15]–[18]. The goal is to have a mathematical tool\nthat can be used to quantify if two models have a similar\nmathematical structure. This is hard to do by inspection of\nthe models because even their configuration spaces are goingarXiv:2402.07038v1  [cs.RO]  10 Feb 2024to be incompatible. Instead, we show in this work that modal\ntheory can be used to this end, as illustrated in Figure 1.\nWe propose a two-stage strategy to achieve this objective.\nFirst, we operate a modal characterization of all the models\nto be compared. Then, we project the modes onto task\nspace, where we introduce similarity measures to compare\ntheir shape and frequency content. To summarize, the main\ncontributions of the paper are as follows:\n•eigenmanifolds are proposed as a means of comparing\ncontrol-oriented ROMs of soft robots derived from\ndifferent discretization hypotheses;\n•we propose a new energy-based continuation algorithm\nfor eigenmanifold computation;\n•a systematic procedure is proposed for comparing eigen-\nmanifolds of different ROMs, possibly obtained with\ndifferent discretization techniques;\n•we compute for the first time nonlinear modes of\ncontinuum soft robots.\nA. Notation\nVectors and matrices are denoted in bold, while scalars\nare denoted in lowercase normal font. The n×nidentity\nmatrix is represented by In. For a symmetric matrix A=\nATwith λmin(A)>0, we write A>0for positive\ndefiniteness. To simplify the notation, arguments of functions\nare omitted when clear from the context. Furthermore, given\na continuously differentiable function f(x) :Rh→R,\n∇xf∈Rhand∇2\nxf∈Rh×hdenote, respectively, the\ngradient and the Hessian of f. We express the soft robot’s\ndynamics as1\nM(q)¨q+c(q,˙q) +∇qV(q) =0, (1)\nwhere q,˙q,¨q∈Rndenote the configuration variables and\ntheir time derivatives, M(q)∈Rn×nis the mass matrix\nandc(q,˙q)∈Rnmodels Coriolis and centrifugal terms.\nFurthermore, V(q)represents the potential energy, including,\ne.g., the effects of gravitational and elastic forces. We also\ndenote the system energy as\nE(q,˙q) =1\n2˙qTM(q)˙q+V(q), (2)\nand the solution of (1) at time tfrom the initial condition\n(q0,˙q0)asq(t,q0,˙q0).\nII. B ACKGROUND : NONLINEAR MODES\nNonlinear modal theory has been traditionally developed\nfor mechanical systems with constant inertia matrix [20],\nbut it has recently been extended to multi-body mechanical\nsystems [14]. Since continuum soft robots fall into the latter\ncategory, in the following, we briefly summarize the main\nconcepts from [14], which we refer the reader to for an\nexhaustive treatment of the topic.\nLetqeqbe a stable equilibrium configuration of (1), i.e.,\n∇qV(qeq) =0,K(qeq) =∇2\nqV(qeq)>0,\nwhich is equivalent to saying that qeqis a local minimizer of\nE(for˙q=0). The linearization of (1) at (q,˙q) = (qeq,0)\n1As discussed in [19], common and advanced modeling techniques used\nin soft robotics all yield equations of this form. Actuation and dissipation\nterms do not appear here because nonlinear modes are defined on the\nundamped and unactuated model.yields the second-order dynamics\n∆¨q+M−1(qeq)K(qeq)∆q=0, (3)\nwhere ∆qdenotes a small increment of the configuration\nwith respect to qeq. Each eigenvector ci, i∈ {1,···, n}\nofM−1(qeq)K(qeq)generates a local two-dimensional\neigenspace\nESi= span\u001a\u0012\nci\n0\u0013\n,\u0012\n0\nci\u0013\u001b\nwith the property that when (∆q,∆˙q)is initialized in ESi,\nthe dynamics evolves according to the harmonic oscillator\nequations, namely\n∆¨q+ω2\ni∆q=0, (4)\nwhere ω2\niis the eigenvalue associated with ci. From the\nabove equation, it can be shown that the solutions starting\nfrom ESisatisfy two important properties: (i) they are\nperiodic with period Ti= 2π/ωiand (ii) they are invariant,\ni.e., will remain in ESiat all times. These trajectories are\ncalled linear modes (LMs).\nWhen moving to the nonlinear setting, each eigenspace\nESibends into a two-dimensional sub-manifold of the\nstate space, called eigenmanifold and denoted as Mi. In\nanalogy with the linear case, Mican be seen as a collection\nof invariant periodic trajectories of (1), called hereinafter\nnonlinear modes (NMs). However, NMs can have different\nperiods, while all LMs have the same period. For the purpose\nof this paper, we need a further notion, namely that of\neigenmanifold generator. Given Mi, its generator is defined\nas\nGMi={(q,˙q)∈Mi|˙q=0}.\nIn a nutshell, GMiis the subset of Micontaining all points\nwith zero velocity. Each element of GMiis an initial rest\nconfiguration associated with a periodic motion that remains\ninMiat all times. The elements of GMican be seen as the\nnonlinear counterparts of the eigenvector ci. Differently from\nthe linear case, where we can generate ESiwith only one\nvector, the nonlinear case needs an entire set of points, i.e.,\nGMi, to generate Mi. An important property of GMiis that\nit can be parameterized by the energy, implying that every\nNM can itself be parameterized by its energy. To simplify the\nnotation, we will henceforth drop the subscript iand denote\nthe NM at EasGM(E).\nIII. E IGENMANIFOLD COMPUTATION\nFor the computation of NMs, analytical and numerical\napproaches have been proposed in the literature, see [20]\nfor a thorough review on possible strategies.\nRemarkably, following a two-stage procedure, we can\neasily extend numerical approaches used for the computation\nof NMs also for that of eigenmanifolds. First, we compute\nthe generator of M. Then, we take each point of GMas an\ninitial condition to simulate the dynamics forward in time\nand compute the corresponding NM, thus obtaining a slice\nofM.\nIn this work, for the computation of GMwe use a modified\nversion of the numerical continuation algorithm presented\nin [21]. Superscripts are used to denote the iteration index.\nEach generator is computed using a predictor-corrector algo-\nrithm that starts from the direction given by the eigenvectorciof the linearized dynamics at the equilibrium. During\nthe first iteration only, the linearized dynamics (3) is used\nto compute a point close to GMandqeq. In particular, an\ninitial configuration increment ∆q0is computed such that\nthe linearized energy takes a small incremental step. In this\nway, the trajectory of (1) from q1\n0=qeq+ ∆q0with zero\nvelocity stays close to ESand has approximately the same\nperiod T1of that of ∆q0. Then, (T1,q1\n0,0)is refined with\na Newton-Raphson iteration scheme that seeks a solution of\nthe periodicity constraint\nr(q0, T) =\u0012q0−q(T,q0,0)\n−˙q(T,q0,0)\u0013\n=0, (5)\nin the unknowns (q0,0)∈ GMandT∈(0,∞). At step\nj≥2, a prediction for the current solution (Tj,qj\n0,0)is\ncomputed along the direction given by the tangent vector to\nthe previous solution (Tj−1,qj−1\n0,0). The prediction is then\ncorrected with a shooting procedure. In [21], the shooting\nprocedure aims at solving r(qj\n0, Tj) =0, which is an over-\nconstrained system of 2nequations in n+1unknowns. Since\nthe predictor step requires fairly small energy increments\nto guarantee convergence of the procedure, the computation\nof the generator can be quite time consuming. To speed up\nthe evaluation, we propose to constrain the energy increment\nbetween consecutive iterations. In particular, during the cor-\nrection stage we solve the augmented system \nr(qj\n0, Tj)\nE(qj\n0,0)−E(qj−1\n0,0)−∆E!\n=0, (6)\nbeing ∆E > 0a desired energy step. This simple modifi-\ncation has proven to significantly reduce the computation\ntime, without affecting stability of the procedure. Since\nconvergence of the scheme is guaranteed only in the vicinity\nofGM,∆Emust be properly selected to prevent numerical\ninstability. To this end, a simple adaptive rule has been\nadopted. In particular, ∆Eis initialized to a reference value\n∆¯Eand a maximum number Nmax of iterations is defined\nfor the correction. If convergence is not achieved within\nNmax iterations, ∆Eis halved and the correction repeated.\nIV. M ETRICS\nIn this work, we aim at comparing how the eigenmanifolds\nof a continuum soft robot change with the order and type\nof the discretization adopted and thus with the number\nDoFs. A direct comparison in configuration space is not\npossible in general, unless the discretizations have the same\nnumber of DoFs. Even though two models have the same\nnumber of DoFs, their configuration spaces may not be\ndirectly comparable. For example, consider the dynamic\nmodel of a soft robot obtained using a piecewise constant\ncurvature and an affine approximation of the strain. Even if\nthe discretization order is chosen such that the models have\nthe same number of DoFs, the configuration variables will\nstill belong to different spaces, making a direct comparison\nimpossible.\nWe propose to overcome this issue by introducing a task\nspace function h(q,s) :Rn× S → T , where s∈ S ⊆ Rq\ndenotes a set of hyper-parameters for T ⊆Rmandmhas\nalways the same dimension, independently of the dimension\nnofq. Given h(q,s), any eigenmanifold can be projected\nFig. 2: Schematic representation of the eigenmanifold projection.\nGiven three control-oriented ROMs, their eigenmanifolds M1,M2\nandM3cannot be directly compared in configuration space. To\novercome this issue, we introduce a constant dimensional task space\nh(q,s)in which M1,M2andM3are projected and here define\nsimilarity measures.\nintoT, resulting in a collection of periodic orbits. Because\nall trajectories of an eigenmanifold are periodic by definition,\ntheir projection into Tis a periodic trajectory in Twith same\nperiod. Therefore, the representation of MintoTis a sub-\nmanifold of Twith properties similar to M. The approach\nis illustrated in Fig. 2.\nFor example, consider the direct kinematics of a slender\nsoft arm. Regardless of the discretization technique used, it\nmust always be possible to reconstruct the robot backbone\n(using a single hyper-parameter s). In other words, we know\nthe direct kinematics mapping\nh(q, s) =p(q, s) =\npx(q, s)\npy(q, s)\npz(q, s)\n:Rn×R→R3,(7)\nwhere p(q, s)is the position of the point at distance s∈\n[0, L]from the base, being Lthe robot length at rest.\nBecause all the level sets of the energy\nL¯E=\b\n(q,˙q)∈M|E(q,˙q) =¯E\t\ngive raise to evolutions that trace the same path in configu-\nration space and differ only in a phase shift, a single point\nofL¯Eshould be projected. The generator points are good\ncandidates for this purpose because they are all characterized\nby an initial zero velocity, which allows to get rid of any\nphase shift due to the initialization in M.\nGiven the task manifold, it is now possible to define\nsimilarity measures between two eigenmanifolds M1and\nM2. These can depend on different properties that one is\ninterested in comparing, such as their shape and spectral\ncontent. A common measure to compare two curves A(t)\nandB(t)belonging to a metric space Sis the Fr ´echet\ndistance [22],\nϕ(A, B) = inf\nα,βmax\nu∈[0,1]{d(A(α(u)), B(β(u)))},(8)\nwhere d(·,·)is the distance function of S. In a nutshell,\nthe above metric looks at all possible reparameterizations of\nAandBand takes the one that minimizes their maximum\ndistance. Given h(q,s)and (8), we can define the following\nmeasures\nf(E,s) =ϕ(h(GM1(E),s),h(GM2(E),s)), (9)and\nF(E) =Z\nSf(E,v)dv, (10)\nwhere the Fr ´echet distance is applied componentwise to\nthe rows of h(q,s). In the following, we refer to f(E,s)\nandF(E), respectively, as the modal Fr ´echet distance and\nmodal integral Fr ´echet distance. The measure fallows us\nto compare two modes for a given value of Eands. On\nthe contrary, Faverages this measure over the entire space\nof hyper-parameters for h(q,s). It is also worth remarking\nthat other choices instead of ϕcould have been made, see\nfor example [22].\nConsider again the case of a slender soft arm and take\nh(q, s) =p(q, s). Suppose we are interested in quantifying\nhow close two eigenmanifolds are. A first estimate can be\nobtained by looking at the tip evolution only, i.e., f(E, L).\nHowever, f(E, L)provides only a partial indication of their\nsimilarity, because f(E, L) =0does not imply f(E, s) = 0\nfor any other point s̸=L. To obtain a more complete\nmeasure of similarity, we introduce F(E). Note that if\nF(E) =0, then f(E,s) =0for all s∈ S, which implies\nthat the NMs produce the same motion for all the system\nparticles.\nTo compare the spectra, we can adopt a similar strategy. To\nthis end, we consider the minimum value of the magnitude-\nsquared coherence defined as follows\nγ(A, B) = min\nf|gAB(f)|2\ngAA(f)gBB(f), (11)\nwhere gABis the Cross-spectral power density between A\nandB, and gAAandgBBthe auto power spectral density of\nAandB, respectively. At a given frequency, the coherence\nmeasures the extent to which Bcan be estimated from A\nthrough a linear process. Note that unlike ϕ(A, B),γ(A, B)\nis not a distance function, as it is not symmetric and lower\nvalues indicate less correlation between the two signals, thus\nworst similarity. By applying γto the components of hwe\ndefine the modal coherence as\ng(E,s) =γ(h(GM1(E),s),h(GM2(E),s)), (12)\nand the modal integral coherence\nG(E) =Z\nSg(E,v)dv. (13)\nConsiderations analogous to those for fandFhold.\nThe integral measures defined above can also be integrated\nover the energy domain to obtain an overall measure of the\nsimilarity between the eigenmanifolds, i.e.,\nFE=ZEmax\nEqeqF(E)dE,\nand\nGE=ZEmax\nEqeqG(E)dE,\nwhere Emaxis the maximum energy level considered during\nanalysis. In future works, we will explore the use of these\nmeasure as loss terms in a learning algorithm for the com-\nputation of control-oriented ROMs.\nV. S IMULATION RESULTS\nIn this section, we apply the above metrics to compute\nand compare the eigenmanifolds of a slender continuum softrobot. The goal is to investigate the differences between\nPCC ROMs obtained by increasing the resolution of the\nstrain and a FE rigid model having a large number of\nDoFs. For the PCC models, we enhance the resolution by\nincreasing the number of bodies per unit of length at rest.\nThe robot has a cylindrical shape with radius r= 0.02 [m] ,\nuniform mass density ρ= 1062 [kg m−3], and rest length\nL= 0.4 [m] . Furthermore, we consider a linear elastic\nmaterial with Young modulus Y= 0.66 [MPa] and Poisson\nratio ν= 0.5. The arm is mounted with the base rotated\nwith the gravitational field so that the straight stress-free\nconfiguration is a stable equilibrium, and motion occurs in\nthe(x, z)plane.\nAll the bodies of the PCC models share the same physical\nproperties, except for the rest length which is chosen as L/n,\nbeing nthe discretization order. The maximum number of\nbodies we consider is nmax= 5. For each of these models,\nwe compute all their n≤nmax eigenmanifolds, using the\nprocedure outlined in Section III. The rigid robot consists\nof ten cylindrical bodies, each one with length L/10and\nmass m=ρπr2(L/10) [kg] . Elasticity is introduced by\nassuming linear elastic revolute joints with uniform stiffness\ncoefficients k=Y πr410/(4L) [N m rad−1]. Because the\nPCC models have at most five bodies, we compute only\nthe first five eigenmanifolds of the rigid model, ordered in\nascending order by oscillation frequency. The eigenmanifold\ncomputation is perfomed with a desired energy step of\n∆E= 0.05 [J] until the energy reaches the maximum\nvalue Emax= 1 [J] . In the following, we use the direct\nkinematics of the backbone (7) as task space function. The\nintegral measures F(E)andG(E)are approximated through\na Gaussian quadrature rule with ten Gaussian points.\nFigure 3 shows stroboscopic plots of the robots in the\nworkspace for the first three NMs at Emax. Figure 4 depicts\nthe energy-frequency plot. Only the first two eigenmanifolds\nat lower frequency have trajectories with similar periods.\nThese differences are reasonably expected to show up also in\nthe task space, at least for the coherence. Figure 5 reports the\nenergy evolution of the modal Fr ´echet distance at the end-\neffector. All the PCC models achieve a good match with the\nFE model. Note indeed that the Fr ´echet distance is always\nless than 6 [cm] . Furthermore, the accuracy improves signif-\nicantly as more bodies are considered. For the xcomponent,\nthe distance between the PCC and FE models decreases\nwith the order of discretization, as expected. However, the\nfifth order PCC model does not always show the best match\nwith the rigid robot along the zdirection. Nevertheless, the\nmodal integral Fr ´echet distance in Fig. 6 shows that the\nfive DoFs PCC model better fits the rigid motion overall.\nThese results demonstrate that energy-frequency relation is\nnot an exhaustive similarity measure in general. Moving to\nthe coherence, see Fig. 7, the five DoFs model has good\nperformance only for the first mode. The same conclusions\ncan be drawn also for the modal integral coherence in Fig. 8.\nBased on the overall performance, we conclude that the PCC\nrobot with four DoFs has modal evolutions that overlap with\nthose of the rigid arm at the tip more closely.(a)\n(a) Mode 1; t= 0T1\n (b)t= 0.2T1\n (c)t= 0.3T1\n (d)t= 0.5T1\n (e)t= 0.7T1\n (f)t= 0.8T1\n (g)t= 0.9T1\n(h) Mode 2; t= 0T2\n (i)t= 0.2T2\n (j)t= 0.3T2\n (k)t= 0.5T2\n (l)t= 0.7T2\n (m)t= 0.8T2\n (n)t= 0.9T2\n(o) Mode 3; t= 0T3\n (p)t= 0.2T3\n (q)t= 0.3T3\n (r)t= 0.5T3\n (s)t= 0.7T3\n (t)t= 0.8T3\n (u)t= 0.9T3\nFig. 3: Stroboscopic plots of the robots in the workspace for the first (a)–(g), second (h)–(n) and third (o)–(u) NMs at Emax= 1 [J] and\nnormalized time instants with respect to the mode period. Each color corresponds to a different model.\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 N M 3 N M 4 N M 5\n(a)\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1\n(a) PCC 1DoF\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 (b) PCC 2DoFs\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 N M 3 (c) PCC 3DoFs\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 N M 3 N M 4\n(d) PCC 4DoFs\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 N M 3 N M 4 N M 5 (e) PCC 5DoFs\n00.20.40.60.8 1\nE n e r g y [ J ]020406080100F r e q u e n c y [ H z ]N M 1 N M 2 N M 3 N M 4 N M 5 (f) Rigid 10DoFs\nFig. 4: Energy-frequency relationships for the PCC (a)–(e) and FE (f) discretization, respectively. The models show a similar behavior\nonly for the first two modes.\nVI. C ONCLUSIONS AND FUTURE WORK\nThis paper presents the first investigation of eigenmani-\nfolds for control-oriented reduced-order modeling of contin-00.20.40.60.81\nE n e r g y [ J ]00.020.040.061 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s(a)\n00.20.40.60.81\nE n e r g y [ J ]00.020.040.061 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a) Mode 1,fx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.010.020.032 D o F s 3 D o F s 4 D o F s 5 D o F s (b) Mode 2,fx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.0050.010.0153 D o F s 4 D o F s 5 D o F s (c) Mode 3,fx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.0050.010.0154 D o F s 5 D o F s (d) Mode 4,fx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.0050.015 D o F s (e) Mode 5,fx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.010.021 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(f) Mode 1,fz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.020.042 D o F s 3 D o F s 4 D o F s 5 D o F s (g) Mode 2,fz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.0050.013 D o F s 4 D o F s 5 D o F s (h) Mode 3,fz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.511.5#10-34 D o F s 5 D o F s (i) Mode 4,fz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.51#10-35 D o F s (j) Mode 5,fz(E, L )\nFig. 5: Energy evolution of the modal Fr ´echet distance f(E, L)for the x(a)–(e) and z(f)–(j) tip position.\n00.20.40.60.81\nE n e r g y [ J ]00.020.040.061 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a)\n00.20.40.60.81\nE n e r g y [ J ]00.050.11 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a) Mode 1,Fx(E)\n00.20.40.60.81\nE n e r g y [ J ]00.010.020.032 D o F s 3 D o F s 4 D o F s 5 D o F s (b) Mode 2,Fx(E)\n00.20.40.60.81\nE n e r g y [ J ]00.0050.010.0153 D o F s 4 D o F s 5 D o F s (c) Mode 3,Fx(E)\n00.20.40.60.81\nE n e r g y [ J ]00.0050.014 D o F s 5 D o F s (d) Mode 4,Fx(E)\n00.20.40.60.81\nE n e r g y [ J ]0246#10-35 D o F s (e) Mode 5,Fx(E)\n00.20.40.60.81\nE n e r g y [ J ]00.020.041 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(f) Mode 1,Fz(E)\n00.20.40.60.81\nE n e r g y [ J ]00.010.020.032 D o F s 3 D o F s 4 D o F s 5 D o F s (g) Mode 2,Fz(E)\n00.20.40.60.81\nE n e r g y [ J ]00.0050.013 D o F s 4 D o F s 5 D o F s (h) Mode 3,Fz(E)\n00.20.40.60.81\nE n e r g y [ J ]00.0050.014 D o F s 5 D o F s (i) Mode 4,Fz(E)\n00.20.40.60.81\nE n e r g y [ J ]0246#10-35 D o F s (j) Mode 5,Fz(E)\nFig. 6: Energy evolution of the modal integral Fr ´echet distance F(E)for the x(a)–(e) and z(f)–(j) backbone position.\n00.20.40.60.81\nE n e r g y [ J ]00.020.040.061 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a)\n00.20.40.60.81\nE n e r g y [ J ]0.60.811 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a) Mode 1,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.512 D o F s 3 D o F s 4 D o F s 5 D o F s (b) Mode 2,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.513 D o F s 4 D o F s 5 D o F s (c) Mode 3,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]0.40.60.814 D o F s 5 D o F s (d) Mode 4,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]0.80.915 D o F s (e) Mode 5,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]0.40.60.811 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(f) Mode 1,gx(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.512 D o F s 3 D o F s 4 D o F s 5 D o F s (g) Mode 2,gz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]00.513 D o F s 4 D o F s 5 D o F s (h) Mode 3,gz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]0.40.60.814 D o F s 5 D o F s (i) Mode 4,gz(E, L )\n00.20.40.60.81\nE n e r g y [ J ]0.60.815 D o F s (j) Mode 5,gz(E, L )\nFig. 7: Energy evolution of the modal coherence g(E, L)for the x(a)–(e) and z(f)–(j) tip position.\nuum soft robots. To improve computational efficiency, we\npresent a new continuation algorithm for mode computa-\ntion with an energy-based term. We propose a systematic\napproach to compare the eigenmanifolds of dynamic models\nresulting from different discretization hypotheses and number\nof degrees of freedom (DoFs). To this end, we project the\neigenmanifolds onto a task space of fixed dimension andintroduce therein different similarity measures. The proposed\nmethod is used to compare piecewise constant curvature\n(PCC) models of increasingly finer discretization with a finite\nelement (FE) model. The results show that the task space\nmodal evolutions of the PCC and FE models become similar\nwhen the order of the PCC discretization increases.\nFuture work will explore the use of the proposed similarity\nmeasures to obtain novel control-oriented models with a00.20.40.60.81\nE n e r g y [ J ]00.020.040.061 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s(a)\n00.20.40.60.81\nE n e r g y [ J ]1.41.61.821 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(a) Mode 1,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]0122 D o F s 3 D o F s 4 D o F s 5 D o F s (b) Mode 2,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]0.511.523 D o F s 4 D o F s 5 D o F s (c) Mode 3,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]0.511.524 D o F s 5 D o F s (d) Mode 4,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]11.21.41.65 D o F s (e) Mode 5,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]11.521 D o F 2 D o F s 3 D o F s 4 D o F s 5 D o F s\n(f) Mode 1,Gx(E)\n00.20.40.60.81\nE n e r g y [ J ]0122 D o F s 3 D o F s 4 D o F s 5 D o F s (g) Mode 2,Gz(E)\n00.20.40.60.81\nE n e r g y [ J ]0.511.523 D o F s 4 D o F s 5 D o F s (h) Mode 3,Gz(E)\n00.20.40.60.81\nE n e r g y [ J ]11.524 D o F s 5 D o F s (i) Mode 4,Gz(E)\n00.20.40.60.81\nE n e r g y [ J ]1.41.61.825 D o F s (j) Mode 5,Gz(E)\nFig. 8: Energy evolution of the modal integral coherence G(E)for the x(a)–(e) and z(f)–(j) backbone position.\nreduced number of DoFs.\nREFERENCES\n[1] Y . Mikhlin and K. V . Avramov, “Nonlinear normal modes of vibrat-\ning mechanical systems: 10 years of progress,” Applied Mechanics\nReviews , pp. 1–57, 2023.\n[2] G. Kerschen, M. Peeters, J.-C. Golinval, and C. St ´ephan, “Nonlinear\nmodal analysis of a full-scale aircraft,” Journal of Aircraft , vol. 50,\nno. 5, pp. 1409–1419, 2013.\n[3] E. Gavassoni, P. B. Gonc ¸alves, and D. de Mesquita Roehl, “Nonlinear\nvibration modes of an offshore articulated tower,” Ocean Engineering ,\nvol. 109, pp. 226–242, 2015.\n[4] M. Matheny, L. Villanueva, R. Karabalin, J. E. Sader, and M. Roukes,\n“Nonlinear mode-coupling in nanomechanical systems,” Nano Letters ,\nvol. 13, no. 4, pp. 1622–1626, 2013.\n[5] F. Bjelonic, A. Sachtler, A. Albu-Sch ¨affer, and C. Della Santina,\n“Experimental closed-loop excitation of nonlinear normal modes on\nan elastic industrial robot,” IEEE Robotics and Automation Letters ,\nvol. 7, no. 2, pp. 1689–1696, 2022.\n[6] A. Coelho, A. Albu-Schaeffer, A. Sachtler, H. Mishra, D. Bicego,\nC. Ott, and A. Franchi, “EigenMPC: An eigenmanifold-inspired\nmodel-predictive control framework for exciting efficient oscillations\nin mechanical systems,” in 61st IEEE Conference on Decision and\nControl , 2022, pp. 2437–2442.\n[7] Y . P. Wotte, S. Dummer, N. Botteghi, C. Brune, S. Stramigioli, and\nF. Califano, “Discovering efficient periodic behaviors in mechanical\nsystems via neural approximators,” Optimal Control Applications and\nMethods , 2023.\n[8] T. Simpson, G. Tsialiamanis, N. Dervilis, K. Worden, and E. Chatzi,\n“On the use of variational autoencoders for nonlinear modal analysis,”\ninNonlinear Structures & Systems, Volume 1 , M. Brake, L. Renson,\nR. Kuether, and P. Tiso, Eds., 2023, pp. 297–300.\n[9] E. Pesheck, C. Pierre, and S. W. Shaw, “Modal reduction of a nonlinear\nrotating beam through nonlinear normal modes,” Journal of Vibration\nand Acoustics , vol. 124, no. 2, pp. 229–236, 2002.\n[10] C. Touz ´e, M. Amabili, and O. Thomas, “Reduced-order models\nfor large-amplitude vibrations of shells including in-plane inertia,”\nComputer Methods in Applied Mechanics and Engineering , vol. 197,\nno. 21-24, pp. 2030–2045, 2008.\n[11] R. J. Kuether, M. R. Brake, and M. S. Allen, “Evaluating convergence\nof reduced order models using nonlinear normal modes,” in Model\nValidation and Uncertainty Quantification, Volume 3 , H. Atamturktur,\nB. Moaveni, C. Papadimitriou, and T. Schoenherr, Eds., 2014, pp.\n287–300.\n[12] R. J. Kuether, B. J. Deaner, J. J. Hollkamp, and M. S. Allen,\n“Evaluation of geometrically nonlinear reduced-order models with\nnonlinear normal modes,” AIAA Journal , vol. 53, no. 11, pp. 3273–\n3285, 2015.\n[13] C. Della Santina, M. G. Catalano, and A. Bicchi, “Soft robots,” in\nEncyclopedia of Robotics , M. Ang, O. Khatib, and B. Siciliano, Eds.\nSpringer, 2021.\n[14] A. Albu-Sch ¨affer and C. Della Santina, “A review on nonlinear modes\nin conservative mechanical systems,” Annual Reviews in Control ,\nvol. 50, pp. 49–71, 2020.[15] S. H. Sadati, S. E. Naghibi, I. D. Walker, K. Althoefer, and\nT. Nanayakkara, “Control space reduction and real-time accurate\nmodeling of continuum manipulators using Ritz and Ritz–Galerkin\nmethods,” IEEE Robotics and Automation Letters , vol. 3, no. 1, pp.\n328–335, 2017.\n[16] S. Grazioso, G. Di Gironimo, and B. Siciliano, “A geometrically exact\nmodel for soft continuum robots: The finite element deformation space\nformulation,” Soft Robotics , vol. 6, no. 6, pp. 790–811, 2019.\n[17] F. Renda, C. Armanini, V . Lebastard, F. Candelier, and F. Boyer,\n“A geometric variable-strain approach for static modeling of soft\nmanipulators with tendon and fluidic actuation,” IEEE Robotics and\nAutomation Letters , vol. 5, no. 3, pp. 4006–4013, 2020.\n[18] B. Caasenbrood, A. Pogromsky, and H. Nijmeijer, “Control-oriented\nmodels for hyperelastic soft robots through differential geometry of\ncurves,” Soft Robotics , vol. 10, no. 1, pp. 129–148, 2023.\n[19] C. Della Santina, C. Duriez, and D. Rus, “Model-based control of soft\nrobots: A survey of the state of the art and open challenges,” IEEE\nControl Systems Magazine , vol. 43, no. 3, pp. 30–65, 2023.\n[20] G. Kerschen, M. Peeters, J.-C. Golinval, and A. F. Vakakis, “Nonlinear\nnormal modes, part I: A useful framework for the structural dynami-\ncist,” Mechanical Systems and Signal Processing , vol. 23, no. 1, pp.\n170–194, 2009.\n[21] M. Peeters, R. Vigui ´e, G. S ´erandour, G. Kerschen, and J.-C. Golinval,\n“Nonlinear normal modes, Part II: Toward a practical computation\nusing numerical continuation techniques,” Mechanical Systems and\nSignal Processing , vol. 23, no. 1, pp. 195–216, 2009.\n[22] Y . Tao, A. Both, R. I. Silveira, K. Buchin, S. Sijben, R. S. Purves,\nP. Laube, D. Peng, K. Toohey, and M. Duckham, “A comparative\nanalysis of trajectory similarity measures,” GIScience & Remote\nSensing , vol. 58, no. 5, pp. 643–669, 2021."    },    {        "title": "2402.07052v1.Understanding_the_Training_Speedup_from_Sampling_with_Approximate_Losses.pdf",        "content": "Understanding the Training Speedup from Sampling with\nApproximate Losses\nRudrajit Das∗1, Xi Chen∗2, Bertram Ieong2, Parikshit Bansal1, and Sujay Sanghavi∗1,2\n1UT Austin\n2Amazon\nAbstract\nIt is well known that selecting samples with large losses/gradients can significantly reduce\nthe number of training steps. However, the selection overhead is often too high to yield any\nmeaningful gains in terms of overall training time. In this work, we focus on the greedy\napproach of selecting samples with large approximate losses instead of exact losses in order\nto reduce the selection overhead. For smooth convex losses, we show that such a greedy\nstrategy can converge to a constant factor of the minimum value of the average loss in\nfewer iterations than the standard approach of random selection. We also theoretically\nquantify the effect of the approximation level. We then develop SIFT which uses early\nexiting to obtain approximate losses with an intermediate layer’s representations for sample\nselection. We evaluate SIFT on the task of training a 110M parameter 12 layer BERT base\nmodel, and show significant gains (in terms of training hours and number of backpropagation\nsteps) without any optimized implementation over vanilla training. For e.g., to reach 64%\nvalidation accuracy, SIFT with exit at the first layer takes ∼43 hours compared to ∼57\nhours of vanilla training.\n1 Introduction\nStochastic Gradient Descent (SGD) and its variants are the algorithms of choice for solving\nlarge-scale optimization problems that arise in training machine learning models. These are\nproblems of the form minw∈RdF(w), where F(.) is the expected population loss and w∈Rd\nis the vector of model parameters. More specifically, if f(w, .) is the per-sample loss and Dis\nthe data distribution, then F(w) =Ex∼D[f(w,x)]. The standard SGD update rule at wwith\nstep-size/learning rate ηis:\nw+=w−η∇f(w,x), (1)\nwhere the sample/example xis drawn from D. In practice, the gradient over a single sample\nis replaced by the average gradient over a mini-batch of samples; for our theoretical results, we\nshall stick to batch-size = 1 (as in eq. (1)).\nThe choice of the sample xin eq. (1) significantly impacts the speed of convergence. There is\ncopious work on selecting “important” samples for speeding up convergence. In this paradigm,\nthe seminal idea is that of importance sampling which proposes to sample the examples such that\nresulting stochastic gradient is unbiased while having the minimum possible variance (Zhao and\nZhang, 2015; Alain et al., 2015; Needell et al., 2014). It turns out that the optimal solution is to\nsample the examples with probability proportional to their per-sample gradient norms. While\nthis has a clean theoretical solution, it is completely infeasible from the practical standpoint\n∗Correspondence to Rudrajit Das <rdas@utexas.edu >, Xi Chen <xichex@amazon.com >and Sujay Sanghavi\n<sanghavi@mail.utexas.edu >.\n1arXiv:2402.07052v1  [cs.LG]  10 Feb 2024because of the high cost of computing the per-sample gradient norms. To address this short-\ncoming, several approximations to this exact solution have been proposed; we discuss these and\nrelated ideas in Section 2. In this space, one high-level idea is to use the per-sample loss value\nas a proxy to the per-sample gradient norm, favoring samples with high losses for performing\nthe update (Loshchilov and Hutter, 2015; Shrivastava et al., 2016; Katharopoulos and Fleuret,\n2017; Kawaguchi and Lu, 2019; Zhang et al., 2019). We discuss the important works based\non this idea in detail in Section 2. Since exact loss computation is itself expensive especially\nfor large models, some works rely on approximate loss values for sample selection; for instance,\nusing an auxiliary model to predict loss values of the actual model being trained (Katharopoulos\nand Fleuret, 2017; Zhang et al., 2019). So in summary, several approximations to the core idea\nof exact importance sampling have been proposed to make it practically usable.\nDespite the myriad sample selection approaches, there is a paucity of theoretical results\nquantifying how much speed-up we can obtain over random sampling and their limitations –\nespecially for approaches involving the use of approximate quantities. In this work, we focus\non the approach of choosing the sample with the highest loss for performing the gradient-based\nupdate. More specifically, suppose we are given R > 1 i.i.d. samples drawn from Dand their\napproximate loss values with the constraint that we can pick only onesample by inspecting\ntheapproximate losses to perform the update. We consider the greedy approach of selecting\nthe sample with the largest approximate loss , and we call this greedy SGD (GSGD) . Also,\nwe will refer to the default approach of picking a sample uniformly at random as (vanilla)\nSGD. In Section 5, we theoretically compare the convergence rates of GSGD (with approximate\nlosses) against SGD, characterizing the benefits and limitations of the former. We would like\nto clarify that we are not claiming that GSGD is a novel algorithm ; in fact, it is very similar\nto OSGD (Kawaguchi and Lu, 2019) in spirit1. Rather, we are claiming that our theoretical\ncharacterization of its benefits and limitations – with approximate losses – is the first of its kind ,\nto the best of our knowledge.\nOn the applied side, we propose to use early exiting (Teerapittayanon et al., 2016; Schwartz\net al., 2020) as a light-weight way to obtain approximate losses for sample selection in training\nlarge ML models (e.g., transformers). To be clear, our novelty is in the light-weight filtering\nprocess for training via early exiting . Empirical results on a 110M parameter BERT base model\nshow that early exit-based sample selection yields significant improvements.\nWe will now elaborate on our main contributions .\n(a)In Section 5, we theoretically compare greedy SGD (GSGD) (i.e., pick the sample with\nthehighest approximate loss ) against vanilla SGD (i.e., pick a sample at random).\n•Theorem 1 provides a convergence bound for GSGD on smooth convex losses assuming\nthat the argmax of the approximate losses is equal to the argmax of the actual losses\n(Assumption 5). In Section 5.1, we consider a setting wherein Assumption 5 no longer\nholds. We quantify the degradation in the performance of GSGD in this setting; see\nTheorems 3 and 4.\n•Akey insight is that GSGD can converge to a (problem-dependent) constant factor of\nF∗= min wF(w)in fewer iterations than SGD ; see Remark 2. This is of interest in\ntraining large ML models on extremely large datasets, where converging exactly to F∗is\ninfeasible and instead converging faster toO(F∗) is desirable. On the negative side, our\nresult indicates that GSGD may not converge to F∗asymptotically and hence it can be\nworse than SGD asymptotically (Remark 1).\n1We do not call it OSGD because Kawaguchi and Lu (2019) use exact loss values whereas we use approximate\nloss values. Moreover, Kawaguchi and Lu (2019) consider the finite-sum (ERM) setting whereas we focus on the\nstochastic setting.\n2(b)In Section 6, we propose early exiting as a way to cheaply obtain approximate losses with\nan intermediate layer’s representations for sample selection in training large ML models such\nastransformers . Specifically, we propose to do backpropagation on only 50% of the samples\nin a batch with the highest approximate losses obtained via early exiting . To our knowledge,\nearly exiting has not been studied for accelerating training . We call this early-exit based sample\n“sifting” process SIFT. This can be seamlessly integrated with other sample selection schemes;\nin particular, we also try large entropy -based sample selection with early exiting.\n•In Section 7, we show the efficacy of SIFT in training a 12 layer BERT base model\nwith 110M parameters from scratch. Specifically, to achieve 64% validation accuracy,\nloss-based and entropy-based SIFT with exit at the first layer take roughly 43 and 40\nhours, respectively, compared to roughly 57 hours of vanilla training involving no sample\nselection.\n•In Section 6.1, we quantify the probability of correctly selecting the sample with the\nlargest actual loss using early exiting for feedforward linear neural networks as a function\nof the model’s weights.\n2 Related Work\nThere is a large body of work proposing several kinds of sample-selection schemes for accelerated\ntraining. These include importance sampling methods, sample reordering based approaches and\nalgorithms focusing on samples with higher losses. Our approach falls in the last category.\nOptimal importance sampling. Importance sampling asks how should the training ex-\namples be sampled so as to obtain an unbiased stochastic gradient with the minimum possible\nvariance. Zhao and Zhang (2015); Alain et al. (2015); Needell et al. (2014) show that the opti-\nmal solution to this problem is to sample the examples with probability proportional to their\nper-sample gradient norms. Unfortunately, this is completely infeasible in practice because of\nthe high cost of computing the per-sample gradient norms.\nApproximate importance sampling. Several papers attempt to approximate the optimal\nimportance sampling procedure so as to make it feasible. For convex settings, Zhao and Zhang\n(2015); Needell et al. (2014) propose sampling with probability proportional to the smoothness\nconstant of the per-sample losses, while Borsos et al. (2018); Stich et al. (2017) propose adaptive\nsampling strategies. Going beyond convex settings, Alain et al. (2015) present a distributed\napproach for importance sampling, while Katharopoulos and Fleuret (2017) approximate the\nimportance weights with loss values which are predicted by a smaller network. Katharopoulos\nand Fleuret (2018) derive an upper bound on the per-sample gradient norms for deep-learning\nnetworks which take essentially the same time to compute as the per-sample loss values, and pro-\npose using the upper bounds for approximating the importance weights. Johnson and Guestrin\n(2018) approximate the true sampling distribution by solving a robust optimization problem.\nSample reordering. Another related line of work attempts to improve the way/order in\nwhich samples are presented while training. Bengio et al. (2009) propose curriculum learning\nwherein the key idea is to present easier examples before harder ones but this requires prior\ninformation about the training set. Several improved modifications of this idea are out there\n(Tsvetkov et al., 2016; Kumar et al., 2010; Jiang et al., 2017; Kim and Choi, 2018; Zhang et al.,\n2019; Jiang et al., 2019). There is also a long line of papers that theoretically analyze con-\nventional sample ordering schemes (such as shuffle once, random reshuffling, etc.) as well as\nimproved sample ordering schemes (Bertsekas et al., 2011; Recht and R´ e, 2012; Gurbuzbalaban\net al., 2019, 2022; Haochen and Sra, 2019; Safran and Shamir, 2020; Mishchenko et al., 2020;\n3Lu et al., 2022; Mohtashami et al., 2022).\nSelection of samples with large losses. As mentioned in Section 1, GSGD is nota novel\nalgorithm and there are prior approaches with the same underlying principle as GSGD, i.e.,\nfocus on the samples with the largest loss values. The two algorithms closest to GSGD in spirit\nare online hard example mining (OHEM) proposed by Shrivastava et al. (2016) and ordered\nSGD (OSGD) proposed by Kawaguchi and Lu (2019) – except that both algorithms use exact\nlosses for sample selection. OHEM was proposed for training object detectors (in computer\nvision) and it involves back-propagating using the gradients of only the samples with the k\nlargest losses in batch of size bwith k < b . However, Shrivastava et al. (2016) do not provide\nany theoretical guarantees. OSGD (Kawaguchi and Lu, 2019) is essentially the same algo-\nrithm as OHEM (Shrivastava et al., 2016), but with convergence guarantees and generalization\nbounds. However, unlike our theoretical results for GSGD, Kawaguchi and Lu (2019) do not\nshow how/when/to what extent ordered SGD is better than plain SGD. So even though GSGD\nis similar to OSGD in spirit, our theoretical results (for GSGD) are much more comprehensive.\nLoshchilov and Hutter (2015) propose a sampling strategy which favors picking examples with\nlarger losses. Fan et al. (2017) introduce the average top- kloss and advocate minimizing this\nloss rather than the empirical average over all the samples.\nEarly Exiting. Early exiting (Teerapittayanon et al., 2016; Schwartz et al., 2020) is a promising\napproach to decreasing the computational cost of multilayered neural architectures by approx-\nimating the output of a model through its intermediate feature representations. This saves\ncomputational costs by dynamically deciding the number of layers/modules (attention-blocks\nin Transformers) to use during inference by exiting based on some metric computed on the inter-\nmediate representations themselves. Initially used for ResNets (Teerapittayanon et al., 2016),\nearly exiting is now widely popular even for transformer models, especially language models\n(Schwartz et al., 2020; Xin et al., 2020; Schuster et al., 2022; Rotem et al., 2023; Xin et al.,\n2021; Zhu, 2021). Recent work around early exiting for large language models also explores\naccelerated decoding (Schuster et al., 2022) and improving factuality (Chuang et al., 2023). In\nthis work, we solely use early exiting as a light-weight way to obtain approximate losses with\nthe intermediate layer representations for sample selection. To our knowledge, early exiting has\nnot been used in prior work for speeding up training .\n3 Notation\nVectors and matrices are in bold font. For a natural number n, we sometimes denote the set\n{1, . . . , n }by [n]. The (Gauss) error function and complementary error function are defined as:\nerf(t) :=2√πZt\n0e−z2dzand erfc( t) := 1 −erf(t). (2)\nNote that lim t→∞erf(t) = 1. For any z∈R, we define the sigmoid function as sig( z) =1\n1+ez.\n4 Problem Setting\nWe briefly recap the optimization problem introduced in Section 1. Given access to a first-order\noptimization oracle, we would like to minimize F(w) :=Ex∼D[f(w,x)], where w∈RdandD\nis the data distribution.\nStandard First-Order Stochastic Optimization Oracle. A query at wreturns ∇wf(w,x),\nwhere x∼ D.\n4We consider a more generous oracle. However, it can also perform one gradient evaluation\nper query (same as the standard oracle).\nDefinition 1 (Proposed First-Order Stochastic Optimization Oracle ).A query at w\nfirst returns a set of R > 1samples {x(1), . . . ,x(R)}=S(R)drawn i.i.d. from Dand their\napproximate function2values {ef(w,x(1)), . . . ,ef(w,x(R))}. The user can then pick onebxfrom\nS(R), and the oracle will return ∇wf(w,ˆx).\nLater, we shall make an assumption on the relation between the approximate function value\nef(w,x) and the actual function value f(w,x). Now that we have introduced the oracle that\nwe consider in this work, we state the bxchosen by vanilla SGD and greedy SGD (GSGD).\nSGD choice: Pickbxuniformly at random from S(R).\nGSGD choice: Pickbx= arg maxx∈S(R)ef(w,x).\nWe state the GSGD update rule in more detail next.\n4.1 Greedy SGD (GSGD) Algorithm\nIn the kthiteration, we observe a set of Ri.i.d. samples drawn from D, sayS(R)\nk={x(1)\nk, . . . ,x(R)\nk}.\nWe pick:\nbxk= arg maxx∈S(R)\nkef(wk,x). (3)\nThe update of Greedy SGD (GSGD) with step-size ηkis:\nwk+1=wk−ηk∇f(wk,bxk). (4)\nVanilla SGD’s update is the same as eq. (4), except with bxkbeing a random sample from S(R)\nk.\n5 GSGD vs. SGD for Smooth Convex Objectives\nWe begin by stating some assumptions and definitions.\nAssumption 1 (Continuity ).For any x∼ D,f(w,x)is continuous w.r.t. w.\nAssumption 2 (Convexity ).For any x∼ D,f(w,x)is convex w.r.t. w.\nAssumption 3 (Smoothness ).For any x∼ D,f(w,x)isL-smooth w.r.t. w.\nAssumption 4. For any x∼ D,minw∈Rdf(w,x) = 0 .\nLet Φ F:= arg minw∈RdF(w) and F∗:= minw∈RdF(w). We restrict our attention to the\ncase of Φ Fbeing closed and compact.\nLet us first consider the case where the argmax of the approximate function values is the same\nas the argmax of the exact function values; we will relax this assumption later in Section 5.1.\nAssumption 5 (Approximate function values preserve argmax ).In the setting of Def-\ninition 1, efsatisfies:\narg maxx∈S(R)ef(w,x) = arg maxx∈S(R)f(w,x).\nUnder Assumption 5, bxkin eq. (3) becomes the same as arg maxx∈S(R)\nkf(wk,x).\n2Throughout this paper, we interchange “loss” value and “function” value freely. In most cases, we use\n“function” value in the context of optimization-based discussions and “loss” value in the context of ML-based\ndiscussions.\n5Definition 2. ForR >1, let\nbFR(w) :=E{x(j)}R\nj=1∼\niidDh\nmax\nx∈{x(j)}R\nj=1f(w,x)i\n.\nDefine\nρR(w) :=bFR(w)\nF(w)andρ∗\nR:= inf\nw/∈ΦFρR(w).\nAlso, suppose\nsup\nw∗∈ΦFbFR(w∗)≤∆R.\nExcept for the trivial case of f(w,x1) =f(w,x2)∀x1,x2which we disregard, ρ∗\nRisstrictly\nbigger than 1. Further, consider a point bw∗/∈ΦFthat is ϵ-close to some w∗∈ΦF(i.e.,\n∥bw∗−w∗∥2≤ϵ) and let ϵ→0. In that case, under Assumption 1 (and because the max\noperation preserves continuity) and Definition 2, bFR(bw∗)→bFR(w∗)≤∆R. Also, F(bw∗)≥F∗.\nThus, by definition, ρ∗\nR≤bFR(bw∗)/F(bw∗)≤(∆R/F∗). Hence, we have that:\n1< ρ∗\nR≤∆R\nF∗,∀R >1. (5)\nIn Section 5.2, we quantify ρR(w) and ρ∗\nRfor fitting a model with the squared loss.\nFor our convergence results, we assume that our initialization is w0and let:\nD0:= min\nw∗∈ΦF∥w0−w∗∥.\nWe are now ready to state our convergence results.\nTheorem 1 (GSGD ).Suppose Assumptions 1, 2, 3, 4 and 5 hold. Set ηk=η <1\nLfor all k.\nThen, GSGD has the following convergence guarantee after Kiterations:\nE\"\nF \n1\nKK−1X\nk=0wk!#\n≤D2\n0\n2ρ∗\nRη\u0000\n1−ηL\u0001\nK+∆R\nρ∗\nR\u0000\n1−ηL\u0001.\nThe proof of Theorem 1 can be found in Appendix B. We now state a corresponding folklore\nresult for SGD.\nTheorem 2 (SGD ).Suppose Assumptions 2, 3 and 4 hold. Set ηk=η <1\nLfor all k. Then,\nSGD has the following convergence guarantee after Kiterations:\nE\"\nF \n1\nKK−1X\nk=0wk!#\n≤D2\n0\n2η\u0000\n1−ηL\u0001\nK+ηLF∗\n1−ηL+F∗.\nFrom Theorem 1, observe that in general, GSGD may not converge to the minimum value\nF∗asymptotically (i.e., with K→ ∞ ). At best, we can show that GSGD converges to∆R\nρ∗\nR\nwhich is ≥F∗(this follows from eq. (5)). But SGD can indeed converge to F∗asymptotically\nby setting η=1\nηL√\nKfor example in Theorem 2. Based on this, we make the following remark.\nRemark 1. GSGD may be worse than SGD asymptotically.\nHowever, GSGD is better than SGD early on asρ∗\nR>1 and assuming ∆ R=O(F∗)\n(and F∗̸= 0), GSGD can converge to a constant factor of F∗in fewer iterations than\nSGD . We formalize this next.\n6Corollary 1 (Up to what point is GSGD better than SGD? ).Suppose we run GSGD\nand SGD with constant step-size η <1\nL. In that case, until K=D2\n0(ρ∗\nR−1)\n2η(∆R−ρ∗\nRF∗)iterations, the\nconvergence bound of GSGD in Theorem 1 is better than that of SGD in Theorem 2. Thus,\nGSGD can converge to∆R−F∗\n(1−ηL)(ρ∗\nR−1)function value in fewer iterations than SGD.\nRemark 2. Based on Corollary 1, when ∆R=O(F∗), GSGD can converge to O(F∗)function\nvalue in fewer iterations than SGD. This is of particular interest in training large ML models\nsuch as transformers on extremely large datasets, where minimizing the training loss exactly is\ninfeasible and converging faster to a constant factor of the minimum loss value is desirable.\n5.1 Beyond Argmax-Preserving Approximate Function Values\nOur previous results were under Assumption 5, i.e., the argmax of the approximate function\nvalues ( ef) is always equal to the argmax of the actual function values ( f). Here we relax this\nassumption by instead modeling efas a noisy version of f, and provide a convergence result\nfor GSGD in such a setting. Modeling efas a noisy version of fis analogous to modeling the\nstochastic gradients as a noisy version of the actual gradient in vanilla stochastic optimization.\nSpecifically, we make the following assumption.\nAssumption 6 (Approximate function values ).There exists µ(w)∈Randσ≥0such\nthat:\nef(w,x) =f(w,x) exp\u0010\nµ(w) +σζ(w,x)\u0011\n,\nwhere ζ(w,x)is i.i.d. random noise with mean 0 and variance 1.\nThus, the approximate function value is the actual function values times the exponential of\nrandom noise (i.e., exp\u0000\nσζ(w,x)\u0001\n) times some other scaling (i.e., exp( µ(w))). Assumption 6 is\npretty mild as it does not involve any particular distributional assumptions on ζ(w,x) (such as\nGaussian, etc.). The important thing to note is that under Assumption 6, bxkin eq. (3) is not\nalways equal to arg maxx∈S(R)\nkf(wk,x) – unlike Assumption 5.\nDefinition 3. Let\nbFR,approx (w) :=E{x(j)}R\nj=1∼\niidD,ζh\nf\u0000\nw,x(j∗)\u0001\f\f\fj∗= arg maxj∈[R]ef(w,x(j))i\n.\nDefine\nρR,approx (w) :=bFR,approx (w)\nF(w)andρ∗\nR,approx := inf\nw/∈ΦFρR,approx (w).\nClearly, bFR,approx (w)≤bFR(w) (as defined in Definition 2). So we also have:\nsup\nw∗∈ΦFbFR,approx (w∗)≤sup\nw∗∈ΦFbFR(w∗)≤∆R.\nIt is easy to extend the proof of Theorem 1 to obtain the following result for GSGD under\nAssumption 6.\nTheorem 3 (GSGD ).Suppose Assumptions 1, 2, 3, 4 and 6 hold. Set ηk=η <1\nLfor all k.\nThen, GSGD has the following convergence guarantee after Kiterations:\nE\"\nF \n1\nKK−1X\nk=0wk!#\n≤D2\n0\n2ρ∗\nR,approxη\u0000\n1−ηL\u0001\nK+∆R\nρ∗\nR,approx\u0000\n1−ηL\u0001.\n7The expectation in Theorem 3 also includes the randomness due to ζ(.) (i.e., the noise in\nthe approximate function values). The proof of Theorem 3 is almost identical to the proof of\nTheorem 1 (see Appendix B) and is obtained by replacing bFR(w) with bFR,approx (w) and ρ∗\nR\nwith ρ∗\nR,approx .\nFor the subsequent results in this subsection, we shall consider R= 2. Specifically, we will\nprovide a lower bound for bF2,approx (w), ρ2,approx (w) and ρ∗\n2,approx in terms of bF2(w), ρ2(w) and\nρ∗\n2, respectively, as a function of the noise level σ.\nTheorem 4. Suppose Assumption 6 holds with σ≤1\n2√\n2. Define p(σ) :=\u0000\n1−0.72\u0000\n1−e−√\n2σ\u0001\u0001\n.\nThen:\nbF2,approx (w)≥p(σ)bF2(w), ρ2,approx (w)≥p(σ)ρ2(w)andρ∗\n2,approx≥p(σ)ρ∗\n2.\nThe proof of Theorem 4 is in Appendix C. It is worth pointing out that our result is\nindependent of µ(w); intuitively, this is because a constant scaling (w.r.t. the samples) does\nnot change the argmax. Regarding the dependence w.r.t. σ, notice that p(σ) is a decreasing\nfunction of σ. So as the noise level σincreases, the lower bound becomes worse; this happens\nbecause the quality of the approximate function value worsens as σincreases. Also, as a quick\nsanity check, observe that p(0) = 1. This makes sense because σ= 0 means no effective noise,\ni.e., the approximate function values match the actual function values modulo the constant\nscaling exp( µ(w)).\nIn the following corollary, we specify a bound for the noise level σbelow which ρ∗\n2,approx >1\nand thus, we can obtain a speed-up over SGD.\nCorollary 2. As long as\nσ < σ max:=1√\n2log\u001018ρ∗\n2\n25−7ρ∗\n2\u0011\n,\np(σ)ρ∗\n2>1and therefore, ρ∗\n2,approx >1. Hence, for σ < σ max, GSGD with R= 2 using\napproximate function values for sample selection can be faster than SGD.\n5.2 Quantifying ρR(w)\nHere we shall quantify ρR(w) for a particular case. Let us consider the problem of learning\na parameterized model M(w∗, .) with the squared loss. In this case, our per-sample objective\nfunction fis:\nf(w,x) =\u0010\nM(w,x)− M (w∗,x)\u00112\n. (6)\nWe make the following assumption.\nAssumption 7. Forw̸=w∗,M(w,x)− M (w∗,x)∼\niidN\u0000\nε(w), δ2(w)\u0001\n, i.e., the per-sample\nprediction error is a Gaussian random variable.\nModeling the per-sample prediction error as a Gaussian random variable is fairly reasonable\nand a similar assumption has been made in Pennington and Bahri (2017). We provide a lower\nbound for ρ(w) in this setting.\nTheorem 5. Suppose Assumption 7 holds. Then:\n\u0010\nε2(w) +\u0010\nπ\n2logR\n4 logR\u0011\nδ2(w) +q\n2πlogR\n4 logRε(w)δ(w)\u0011\u0010\n1−1\nR\u0011\nε2(w) +δ2(w).\nThe proof of Theorem 5 is in Appendix D.\n8Corollary 3. In the setting of Theorem 5, if ε(w)≤ O(δ(w))for all win our region of\noptimization, then:\nρ∗\nR≥ \n1 + Ω \nlogR\nlogR+s\nlogR\nlogR!!\u0010\n1−1\nR\u0011\n.\n6 Approximate Losses via Early Exiting in Neural Networks\nFor sample selection in GSGD-like algorithms, we propose to use an approximation of the actual\nloss which is computed on the “early” predictions obtained by applying the linear head after the\nfinal layer to an intermediate layer’s representation (instead of the final layer’s representation).\nSpecifically, suppose θis the linear head and for a sample xwith label y,R(x) and eR(x)\nare the final layer’s and some intermediate layer’s representation with the same dimension as\nθ, respectively. Then the approximate and actual losses with the cross-entropy loss function\ndenoted by ℓareℓ(y,softmax( θ⊤eR(x)))3andℓ(y,softmax( θ⊤R(x))), respectively; we use the\nformer to select samples for performing the gradient-based update.\nWe shall now consider a feedforward linear neural network (Kawaguchi, 2016) and quantify\nthe probability of correctly picking the sample with the largest actual loss when using early\nexiting.\n6.1 Probability of Correctly Selecting the Sample with the Largest Actual\nLoss\nWe consider a binary classification problem, where each sample x∼ N(⃗0d,Id) has a binary label\ny∈ {0,1}. Our model is a k-layer linear feed-forward network parameterized by {Wi}d\ni=1∈Rd×d\nandθ∈Rd, where the soft prediction ˆ yforxis:\nˆy= sig\u0010\nθ⊤\u0000\nWk×Wk−1×. . .×W1\u0001\nx\u0011\n. (7)\nIn eq. (7), sig( .) is the sigmoid function4as defined in Section 3. For any j∈[k], let us define:\nAj:=Wj×. . .×W1andBj:=Wk×. . .×Wj+1. (8)\nThen,\nˆyj= sig\u0000\nθ⊤Ajx\u0001\n(9)\nis the “early prediction” for xat the jthlayer. Note that ˆ y= ˆyk, where:\nˆyk= sig\u0000\nθ⊤BjAjx\u0001\n. (10)\nIn the context of GSGD, we once again focus on the case of R= 2. For any two i.i.d. samples\nx(1)andx(2), let y(1)andy(2)∈ {0,1}be the corresponding ground truth labels and let ˆ y(1)\nj\nand ˆy(2)\njbe the corresponding early predictions at the jthlayer. Further, let ℓ(1)\njandℓ(2)\njbe the\ncorresponding cross-entropy losses of the early predictions at the jthlayer, i.e.,\nℓ(i)\nj=−y(i)log\u0000\nˆy(i)\nj\u0001\n−(1−y(i)) log\u0000\n1−ˆy(i)\nj\u0001\n, (11)\nfori∈ {1,2}. In the context of GSGD, ℓ(1)\njandℓ(2)\njare the approximate function values, whereas\nℓ(1)\nkandℓ(2)\nkare the actual function values. In this section, we are interested in quantifying the\n3For a vector v= [v1, . . . , v d], softmax( v) = [ˆv1, . . . , ˆvd] with ˆ vi= exp( vi)/Pd\nj=1exp(vj).\n4For binary classification problems, the sigmoid function essentially plays the role of the softmax function\nmentioned earlier.\n9probability (over the randomness of data) that early exiting at the jthlayer preserves the\nargmax; specifically, we wish to quantify\npj:=Px(1),x(2)\u0010\narg maxi∈[1,2]ℓ(i)\nj= arg maxi∈[1,2]ℓ(i)\nk\u0011\n. (12)\nNote that pk= 1. We have the following result on this.\nTheorem 6. Define\nβj:=⟨A⊤\njθ,A⊤\njB⊤\njθ⟩\n∥A⊤\njθ∥2∥A⊤\njB⊤\njθ∥2.\nWe restrict our attention to the case of βj≥0. Then:\npj= 1−1\n2√πZ∞\n0exp\u0010\n−y2\n4\u0011\nerfc \nβjy\n2q\n1−β2\nj!\ndy, (13)\nwhere erfc(.)is as defined in eq. (2). We also have the following lower bound for pj:\npj≥1−s\n2−2β2\nj\n2−β2\nj. (14)\nThe proof of Theorem 6 can be found in Appendix E. Note that βjis the (normalized)\ncorrelation between A⊤\njθandA⊤\njB⊤\njθ. As per Theorem 6, pj= 1 when βj= 1; this makes\nsense because βj= 1 implies A⊤\njθandA⊤\njB⊤\njθare parallel and so the argmax operation is\nunaffected if we use ˆ yjinstead of ˆ yk. It is worth mentioning that the lower bound for pjin\neq. (14) is loose for small βj, but we believe it is tight up to constant factors for βj≈1; in\nparticular, it is exact in the case of βj= 1. Specifically, we have the following simple corollary\nforβj≈1.\nCorollary 4. In the setting of Theorem 6, suppose βj= 1−τjwhere τj→0. Then, pj≥\n1− O(√τj).\nIn simple words, if A⊤\njθis strongly correlated with A⊤\njB⊤\njθ, then early exiting at the jth\nlayer preserves the argmax with high probability.\nIn the next section, we empirically demonstrate the efficacy of early exiting in a transformer\nmodel.\n7 Empirical Evaluation\nWe demonstrate the efficacy of early exiting in accelerating the training of a BERT base model\nfrom scratch with the masked language modeling (MLM) loss. Specifically, we apply early exit-\ning for selecting samples in a mini-batch version of greedy AdamW which is a simple extension\nof the greedy SGD idea to AdamW. We refer to this practical early exit-based sample selection\nor “sifting” strategy as SIFT . We describe SIFT in more detail later but at a high level, we\npropose to backpropagate on only 50% of the examples in a batch with the highest approximate\nlosses obtained via early exiting . We would like to emphasize that our novelty is in the light-\nweight sifting process for training via early exiting and not the idea of backpropagating on the\nsamples with large losses5.\nThe BERT base model used for our experiments here consists of 12 layers with a hidden\ndimension of 768; the total number of parameters is 110M . We train on BookCorpus (Zhu\net al., 2015) and English Wikipedia which are two diverse and extensive standard corpora. The\n5This is the same idea as OSGD (Kawaguchi, 2016) and OHEM (Shrivastava et al., 2016).\n10validation set for assessing the model’s performance is derived from the development partition\nof the training corpus. This partition ensures that the validation set represents a diverse and\nunbiased subset of the overall data. In our training, we use 512 as the maximum sequence\nlength. To process the input data, we use the “bert-base-uncased” tokenizer from the Hugging\nFace model repository. The masking is applied after tokenization with a uniform masking rate\nof 15%. Our experiments were conducted on AWS p4d.24xlarge instances (8 NVIDIA A100\nTensor core GPUs).\nOur baseline algorithm is the vanilla approach with no kind of sample filtering. The micro-\nbatch per GPU is set to 32 sequences (so there are 32 sequences per GPU * 8 GPUs * 512\ntokens = 131072 tokens per batch). We will now elaborate on SIFT which does greedy selection\nbased on early exit loss.\nLoss-based SIFT: We select 50% sequences per batch (for training) with the largest MLM\nlosses computed on the early predictions obtained from an intermediate layer in the way de-\nscribed in the beginning of Section 6. Specifically, we show results for the first, second, third,\nsixth and last (i.e., twelfth) layer. The micro-batch per GPU is set to 64 sequences. Since we\nselect half of the sequences for training, the effective batch size per GPU is 32 which is the same\nas that of baseline.\nThe early-exit based filtering idea is pretty general in the sense that it can be seamlessly\nintegrated with other sample selection schemes. In particular, we also tried greedy selection\nbased on early exit entropy (instead of loss). We describe it below.\nEntropy-based SIFT: Everything is the same as loss-based SIFT except that we select 50%\nsequences per batch whose early predictions obtained from an intermediate layer have the largest\nentropies instead of MLM losses.\nPrediction entropy has been used a measure of model uncertainty for active learning; see for\ne.g., Ren et al. (2021); Gal et al. (2017). So our proposal of entropy-based SIFT may also be of\ninterest in active learning with large-scale models.\nIt is worth mentioning that in our implementation, the forward propagation for selecting\nsamples in SIFT is implemented naively before full forward and backward propagation for\nthe update step; one could potentially make this more efficient. However, even without any\noptimization of the initial forward propagation for sample selection, we obtain significant gains\nas we discuss later. For the experiments with SIFT, we do not perform any sample filtering\nfor the first 20K steps; this is to provide an initial warm-up period and is aligned with the\nsuggestion of Kawaguchi and Lu (2019) for GSGD. The total number of update steps for each\nalgorithm is 800K. We defer some other experimental details to Appendix F.\nWe would like to point out that the goal of our experiments is to only show the efficacy of\nearly exiting for approximately selecting samples with large losses/entropies and notpropose a\nSOTA sample selection scheme.\nResults. Figures 1 and 2 show the performance of SIFT with exits at the first, sixth and\nlast layers6and baseline w.r.t. the number of samples used for backpropagation (i.e., back-\npropagation sample complexity) and the number of training hours, respectively. We report the\nvalidation accuracy at the last step with early exit at all the layers we considered in Section 7.\nPlease see the figure and table captions for detailed discussion but the key takeaways are as\nfollows:\n•SIFT is better than the baseline in terms of backpropagation sample complexity as well\nas wall-clock time.\n6We do not show the plots for exits at the second and third layers here to avoid congestion; we report the\nvalidation accuracies of all these layers in Section 7.\n11•Entropy-based SIFT does better than loss-based SIFT.\n•For entropy-based SIFT, exit at the sixth layer has the best performance while for loss-\nbased SIFT, exit at the last layer has the best performance.\nIn Appendix F (Appendix F), we report the total time spent in the forward pass for selecting\nsamples in SIFT as a function of the layer number.\n0 25,000,000 50,000,000 75,000,000 100,000,000 125,000,000 150,000,000 175,000,000 200,000,000\nNum Samples Used in Backpropagation54.0%56.0%58.0%60.0%62.0%64.0%66.0%Accuracy\nBaseline [batch_size/gpu 32]\nEntropy-based Sift, exit at 1st layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nEntropy-based Sift, exit at 6th layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nEntropy-based Sift, exit at last layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at 1st layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at 6th layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at last layer [gross batch_size/gpu 64, backward batch_size/gpu 32]110M Model\nFigure 1: Validation accuracy vs. backpropagation sample complexity. In terms of\nperformance, entropy-based SIFT >loss-based SIFT >baseline. All the accuracies are listed\nin Section 7 but for quick reference, loss-based and entropy-based SIFT with exit at the first\nlayer are better than the baseline by 1.33% and 1.75%, respectively. For loss-based SIFT, exit\nat the last layer has the best performance while for entropy-based SIFT, exit at the sixth layer\nhas the best performance.\n0 10 20 30 40 50 60 70 80\nTraining Hours54.0%56.0%58.0%60.0%62.0%64.0%66.0%Accuracy\nBaseline [batch_size/gpu 32]\nEntropy-based Sift, exit at 1st layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nEntropy-based Sift, exit at 6th layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nEntropy-based Sift, exit at last layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at 1st layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at 6th layer [gross batch_size/gpu 64, backward batch_size/gpu 32]\nLoss-based Sift, exit at last layer [gross batch_size/gpu 64, backward batch_size/gpu 32]110M Model\nFigure 2: Validation accuracy vs. wall-clock time. The observations and trends are the\nsame as Figure 1; please see the discussion therein.\n12Algorithm SIFT Layer # Val. Accuracy\nBaseline N/A 0.6466\nLoss-based SIFT 1 0.6599\nLoss-based SIFT 2 0.6597\nLoss-based SIFT 3 0.6604\nLoss-based SIFT 6 0.6607\nLoss-based SIFT 12 0.6640\nEntropy-based SIFT 1 0.6641\nEntropy-based SIFT 2 0.6672\nEntropy-based SIFT 3 0.6678\nEntropy-based SIFT 6 0.6680\nEntropy-based SIFT 12 0.6664\nTable 1: Validation accuracy at the last step. For loss-based SIFT, exit at the last layer\nhas the best performance while the performance of other layers is nearly the same. However for\nentropy-based SIFT, exit at the sixth layer has the best performance.\n8 Conclusion and Limitations\nIn this work, we theoretically characterized the benefits (as well as limitations) of the greedy\napproach of selecting samples with large approximate losses instead of exact losses. We also\nshowed the promise of early exiting in speeding up training of a transformer model.\nWe will mention some limitations in our current work which we hope to explore and address\nin future work. As we mentioned, we did not pipeline the early exit forward propagation step\nfor sample selection with the full forward and backward propagation steps for model update in\nour current implementation; doing so can yield bigger gains. In our current work, we have only\nshown the efficacy of SIFT on BERT. In the future, we hope to test SIFT on larger transformer\nmodels and other kinds of models where early exiting is feasible (such as ResNets). On the\ntheory side, our current convergence result is for convex functions; we would like to derive a\nsimilar result for non-convex functions too.\nReferences\nAlain, G., Lamb, A., Sankar, C., Courville, A., and Bengio, Y. (2015). Variance reduction in\nsgd by distributed importance sampling. arXiv preprint arXiv:1511.06481 .\nBengio, Y., Louradour, J., Collobert, R., and Weston, J. (2009). Curriculum learning. In\nProceedings of the 26th annual international conference on machine learning , pages 41–48.\nACM.\nBertsekas, D. P. et al. (2011). Incremental gradient, subgradient, and proximal methods for\nconvex optimization: A survey. Optimization for Machine Learning , 2010(1-38):3.\nBorsos, Z., Krause, A., and Levy, K. Y. (2018). Online variance reduction for stochastic opti-\nmization. arXiv preprint arXiv:1802.04715 .\nBulatov, Y. (2022). Mathematics stack exchange: Proving 1 −exp(−4x2/π)≥erf(x)2.https:\n//math.stackexchange.com/questions/6908/proving-1-exp-4x2-pi-ge-texterfx2 .\nChuang, Y.-S., Xie, Y., Luo, H., Kim, Y., Glass, J., and He, P. (2023). Dola: Decoding\nby contrasting layers improves factuality in large language models. arXiv preprint arXiv:\n2309.03883 .\n13Fan, Y., Lyu, S., Ying, Y., and Hu, B. (2017). Learning with average top-k loss. In Advances\nin neural information processing systems , pages 497–505.\nGal, Y., Islam, R., and Ghahramani, Z. (2017). Deep bayesian active learning with image data.\nInInternational conference on machine learning , pages 1183–1192. PMLR.\nGurbuzbalaban, M., Ozdaglar, A., and Parrilo, P. A. (2019). Convergence rate of incremental\ngradient and incremental newton methods. SIAM Journal on Optimization , 29(4):2542–2565.\nGurbuzbalaban, M., Ozdaglar, A., and Parrilo, P. A. (2022). Why random reshuffling beats\nstochastic gradient descent. arXiv preprint arXiv:1510.08560 .\nHaochen, J. and Sra, S. (2019). Random shuffling beats sgd after finite epochs. In International\nConference on Machine Learning , pages 2624–2633. PMLR.\nJiang, A. H., Wong, D. L.-K., Zhou, G., Andersen, D. G., Dean, J., Ganger, G. R., Joshi,\nG., Kaminksy, M., Kozuch, M., Lipton, Z. C., et al. (2019). Accelerating deep learning by\nfocusing on the biggest losers. arXiv preprint arXiv:1910.00762 .\nJiang, L., Zhou, Z., Leung, T., Li, L.-J., and Fei-Fei, L. (2017). Mentornet: Learning\ndata-driven curriculum for very deep neural networks on corrupted labels. arXiv preprint\narXiv:1712.05055 .\nJohnson, T. B. and Guestrin, C. (2018). Training deep models faster with robust, approximate\nimportance sampling. In Advances in Neural Information Processing Systems , pages 7265–\n7275.\nKatharopoulos, A. and Fleuret, F. (2017). Biased importance sampling for deep neural network\ntraining. arXiv preprint arXiv:1706.00043 .\nKatharopoulos, A. and Fleuret, F. (2018). Not all samples are created equal: Deep learning\nwith importance sampling. arXiv preprint arXiv:1803.00942 .\nKawaguchi, K. (2016). Deep learning without poor local minima. Advances in neural informa-\ntion processing systems , 29.\nKawaguchi, K. and Lu, H. (2019). Ordered sgd: A new stochastic optimization framework for\nempirical risk minimization. arXiv preprint arXiv:1907.04371 .\nKim, T.-H. and Choi, J. (2018). Screenernet: Learning self-paced curriculum for deep neural\nnetworks. arXiv preprint arXiv:1801.00904 .\nKumar, M. P., Packer, B., and Koller, D. (2010). Self-paced learning for latent variable models.\nInAdvances in Neural Information Processing Systems , pages 1189–1197.\nLoshchilov, I. and Hutter, F. (2015). Online batch selection for faster training of neural networks.\narXiv preprint arXiv:1511.06343 .\nLu, Y., Meng, S. Y., and De Sa, C. (2022). A general analysis of example-selection for stochas-\ntic gradient descent. In International Conference on Learning Representations (ICLR) , vol-\nume 10.\nMishchenko, K., Khaled, A., and Richt´ arik, P. (2020). Random reshuffling: Simple analysis with\nvast improvements. Advances in Neural Information Processing Systems , 33:17309–17320.\nMohtashami, A., Stich, S., and Jaggi, M. (2022). Characterizing & finding good data orderings\nfor fast convergence of sequential gradient methods. arXiv preprint arXiv:2202.01838 .\n14Needell, D., Ward, R., and Srebro, N. (2014). Stochastic gradient descent, weighted sampling,\nand the randomized kaczmarz algorithm. In Advances in neural information processing sys-\ntems, pages 1017–1025.\nNesterov, Y. et al. (2018). Lectures on convex optimization , volume 137. Springer.\nPennington, J. and Bahri, Y. (2017). Geometry of neural network loss surfaces via random\nmatrix theory. In International conference on machine learning , pages 2798–2806. PMLR.\nRecht, B. and R´ e, C. (2012). Toward a noncommutative arithmetic-geometric mean inequality:\nConjectures, case-studies, and consequences. In Conference on Learning Theory , pages 11–1.\nJMLR Workshop and Conference Proceedings.\nRen, P., Xiao, Y., Chang, X., Huang, P.-Y., Li, Z., Gupta, B. B., Chen, X., and Wang, X.\n(2021). A survey of deep active learning. ACM computing surveys (CSUR) , 54(9):1–40.\nRotem, D., Hassid, M., Mamou, J., and Schwartz, R. (2023). Finding the sweet spot: Anal-\nysis and improvement of adaptive inference in low resource settings. Annual Meeting of the\nAssociation for Computational Linguistics .\nSafran, I. and Shamir, O. (2020). How good is sgd with random shuffling? In Conference on\nLearning Theory , pages 3250–3284. PMLR.\nSchuster, T., Fisch, A., Gupta, J., Dehghani, M., Bahri, D., Tran, V., Tay, Y., and Metzler, D.\n(2022). Confident adaptive language modeling. Advances in Neural Information Processing\nSystems , 35:17456–17472.\nSchwartz, R., Stanovsky, G., Swayamdipta, S., Dodge, J., and Smith, N. A. (2020). The\nright tool for the job: Matching model and instance complexities. In Jurafsky, D., Chai, J.,\nSchluter, N., and Tetreault, J., editors, Proceedings of the 58th Annual Meeting of the Associ-\nation for Computational Linguistics , pages 6640–6651, Online. Association for Computational\nLinguistics.\nShrivastava, A., Gupta, A., and Girshick, R. (2016). Training region-based object detectors\nwith online hard example mining. In Proceedings of the IEEE conference on computer vision\nand pattern recognition , pages 761–769.\nStich, S. U., Raj, A., and Jaggi, M. (2017). Safe adaptive importance sampling. In Advances\nin Neural Information Processing Systems , pages 4381–4391.\nTeerapittayanon, S., McDanel, B., and Kung, H. (2016). Branchynet: Fast inference via early\nexiting from deep neural networks. In 2016 23rd International Conference on Pattern Recog-\nnition (ICPR) , pages 2464–2469.\nTsvetkov, Y., Faruqui, M., Ling, W., MacWhinney, B., and Dyer, C. (2016). Learning the\ncurriculum with bayesian optimization for task-specific word representation learning. arXiv\npreprint arXiv:1605.03852 .\nXin, J., Tang, R., Lee, J., Yu, Y., and Lin, J. (2020). DeeBERT: Dynamic early exiting for\naccelerating BERT inference. In Jurafsky, D., Chai, J., Schluter, N., and Tetreault, J., editors,\nProceedings of the 58th Annual Meeting of the Association for Computational Linguistics ,\npages 2246–2251, Online. Association for Computational Linguistics.\nXin, J., Tang, R., Yu, Y., and Lin, J. (2021). BERxiT: Early exiting for BERT with better fine-\ntuning and extension to regression. In Merlo, P., Tiedemann, J., and Tsarfaty, R., editors,\nProceedings of the 16th Conference of the European Chapter of the Association for Com-\nputational Linguistics: Main Volume , pages 91–104, Online. Association for Computational\nLinguistics.\n15Zhang, J., Yu, H.-f., and Dhillon, I. S. (2019). Autoassist: A framework to accelerate training\nof deep neural networks. arXiv preprint arXiv:1905.03381 .\nZhao, P. and Zhang, T. (2015). Stochastic optimization with importance sampling for regular-\nized loss minimization. In international conference on machine learning , pages 1–9.\nZhu, W. (2021). LeeBERT: Learned early exit for BERT with cross-level optimization. In\nZong, C., Xia, F., Li, W., and Navigli, R., editors, Proceedings of the 59th Annual Meeting of\nthe Association for Computational Linguistics and the 11th International Joint Conference on\nNatural Language Processing (Volume 1: Long Papers) , pages 2968–2980, Online. Association\nfor Computational Linguistics.\nZhu, Y., Kiros, R., Zemel, R., Salakhutdinov, R., Urtasun, R., Torralba, A., and Fidler, S.\n(2015). Aligning books and movies: Towards story-like visual explanations by watching\nmovies and reading books. In Proceedings of the IEEE international conference on computer\nvision , pages 19–27.\n16Appendix\nA Some Preliminaries for the Proofs\nFact 1 (Nesterov et al. (2018)) .For an L-smooth function g:Rp− →R,∥∇g(y)∥2\n2≤2L(g(y)−\nminz∈Rpg(z))∀y∈Rp.\nFact 2 (Cantelli’s inequality ).For any random variable Yand any t >0:\nP\u0010\nY−E[Y]≥t\u0011\n≤Var(Y)\nVar(Y) +t2.\nFact 3 (Bulatov (2022)) .We have:\nerf(t)≤r\n1−exp\u0010\n−4t2\nπ\u0011\n.\nFact 4. It holds that:\nerfc(t)≤2 exp\u0010\n−t2\n2\u0011\n.\nProof. LetZ∼ N(0,1). It can be checked that erf( t) = 2P(Z∈(0, t)). Thus,\nerfc(t) = 1−erf(t) = 2\u00101\n2−P\u0000\nZ∈(0, t)\u0001\u0011\n= 2P(Z≥t). (15)\nBut using the Gaussian tail bound, we have P(Z≥t)≤e−t2/2. Using this in eq. (15) gives us\nthe desired result.\nFact 5. For any t >0:Z∞\n0exp\u0010\n−y2\n2t2\u0011\ndy=rπ\n2t.\nProof. LetZ∼ N(0, t2). We have:\nZ∞\n0exp\u0010\n−y2\n2t2\u0011\ndy=√\n2πt2 \n1√\n2πt2Z∞\n0exp\u0010\n−y2\n2t2\u0011\ndy!\n| {z }\n=P(Z>0)=1\n2=rπ\n2t. (16)\nB Proof of Theorem 1\nProof. Consider some w∗∈ΦF. With a constant step-size, say η, we have for any iteration k:\nES(R)\nk\u0002\n∥wk+1−w∗∥2\u0003\n=∥wk−w∗∥2−2η\nES(R)\nk\u0002\n∇f(wk,bxk)\u0003\n,wk−w∗\u000b\n+η2ES(R)\nk\u0002\n∥∇f(wk,bxk)∥2\u0003\n.\n(17)\nUsing Definition 2 in eq. (17), we get:\nES(R)\nk\u0002\n∥wk+1−w∗∥2\u0003\n=∥wk−w∗∥2−2η⟨∇bFR(wk),wk−w∗⟩+η2ES(R)\nk\u0002\n∥∇f(wk,bxk)∥2\u0003\n.(18)\nUsing the convexity of f(w,x) w.r.t. w(Assumption 2) and the fact that pointwise maximum\nof convex functions is also convex, we conclude that bFR(w) is convex. Thus:\n⟨∇bFR(wk),wk−w∗⟩ ≥bFR(wk)−bFR(w∗)≥bFR(wk)−∆R, (19)\n17where the last step follows from the fact that supw∗∈ΦFbFR(w∗)≤∆R.\nFurther, using Assumption 3, Fact 1 and Assumption 4, we get:\nES(R)\nk\u0002\n∥∇f(wk,bxk)∥2\u0003\n≤2LES(R)\nk\u0002\nf(wk,bxk)\u0003\n≤2LbFR(wk). (20)\nPlugging eq. (19) and eq. (20) into eq. (17), we get:\nES(R)\nk\u0002\n∥wk+1−w∗∥2\u0003\n=∥wk−w∗∥2−2η\u0000\n1−ηL\u0001bFR(wk) + 2η∆R. (21)\nLet us impose η <1\nLso that 1 −ηL > 0. From Definition 2, we have bFR(wk)≥ρ∗\nRF(wk).\nUsing this above, we get:\nES(R)\nk\u0002\n∥wk+1−w∗∥2\u0003\n=∥wk−w∗∥2−2ρ∗\nRη\u0000\n1−ηL\u0001\nF(wk) + 2η∆R. (22)\nAfter some rearrangement, we get:\nF(wk)≤∥wk−w∗∥2−ES(R)\nk\u0002\n∥wk+1−w∗∥2\u0003\n2ρ∗\nRη\u0000\n1−ηL\u0001 +∆R\nρ∗\nR\u0000\n1−ηL\u0001. (23)\nNext, we sum the above for all k∈ {0, . . . , K −1}while taking expectation throughout. After\ndividing both sides of the resultant inequality by1\nK, we get:\n1\nKK−1X\nk=0E[F(wk)]≤∥w0−w∗∥2\n2ρ∗\nRη\u0000\n1−ηL\u0001\nK+∆R\nρ∗\nR\u0000\n1−ηL\u0001. (24)\nLetwK=1\nKPK−1\nk=0wk. Applying Jensen’s inequality, we get E[F(wK)]≤1\nKPK−1\nk=0E[F(wk)].\nUsing this in eq. (24) gives us:\nE\u0002\nF\u0000\nwK\u0001\u0003\n≤∥w0−w∗∥2\n2ρ∗\nRη\u0000\n1−ηL\u0001\nK+∆R\nρ∗\nR\u0000\n1−ηL\u0001. (25)\nObserve that this analysis holds for any w∗∈ΦF, including the one that is closest to w0. Using\nthis in the above bound gives us:\nE\u0002\nF\u0000\nwK\u0001\u0003\n≤D2\n0\n2ρ∗\nRη\u0000\n1−ηL\u0001\nK+∆R\nρ∗\nR\u0000\n1−ηL\u0001, (26)\nwhere D0:= min w∗∈ΦF∥w0−w∗∥.\nC Proof of Theorem 4\nProof. Forj∈ {1,2}, we have ef(w,x(j)) =f(w,x(j)) exp\u0010\nµ(w)+σζ(w,x(j))\u0011\n, where ζ(w,x(j))\nis i.i.d. random noise with mean 0 and variance 1. For brevity of notation, let:\nµ=µ(w) andef(j)=ef(w,x(j)), f(j)=f(w,x(j)) and ζ(j)=ζ(w,x(j)) for j∈ {1,2}.\nWithout loss of generality, let f(1)≥f(2). Let j∗= arg maxj∈[2]ef(j). Let us consider the case\nofj∗= 1. This happens when f(2)exp(µ+σζ(2))≤f(1)exp(µ+σζ(1)); this is equivalent to:\nσ(ζ(2)−ζ(1))≤logf(1)\nf(2). (27)\n18LetZ=ζ(2)−ζ(1). Note that Zis random variable with mean 0 and variance 2. As per the\nabove discussion, we have:\nP\u0000\nj∗= 1\u0001\n=P \nZ≤1\nσlogf(1)\nf(2)!\n= 1−P \nZ >1\nσlogf(1)\nf(2)!\n. (28)\nThus,\nP\u0000\nj∗= 2\u0001\n=P \nZ >1\nσlogf(1)\nf(2)!\n. (29)\nThen, we have:\nE{ζ(1),ζ(2)}h\nf(j∗)i\n=f(1)P\u0000\nj∗= 1\u0001\n+f(2)P\u0000\nj∗= 2\u0001\n(30)\n=f(1) \n1−P \nZ >1\nσlogf(1)\nf(2)!!\n+f(2)P \nZ >1\nσlogf(1)\nf(2)!\n(31)\n=f(1)−\u0000\nf(1)−f(2)\u0001\nP \nZ >1\nσlogf(1)\nf(2)!\n. (32)\nSince E[Z] = 0 and Var( Z) = 2, using Cantelli’s inequality (Fact 2), we have:\nP \nZ >1\nσlogf(1)\nf(2)!\n≤2\n2 +1\nσ2log2f(1)\nf(2). (33)\nUsing this in eq. (32), we get:\nE{ζ(1),ζ(2)}h\nf(j∗)i\n≥f(1)−\u0000\nf(1)−f(2)\u0001 \n2\n2 +1\nσ2log2f(1)\nf(2)!\n. (34)\nLet us define the function h(t;σ) :=2(1−e−t)\n2+t2\nσ2fort≥0. Then, note that:\n\u0000\nf(1)−f(2)\u0001 \n2\n2 +1\nσ2log2f(1)\nf(2)!\n=f(1)×h \nlogf(1)\nf(2);σ!\n.\nFrom Lemma 1, we have h(t;σ)≤0.72\u0000\n1−e−√\n2σ\u0001\nfor all t≥0, when σ≤1\n2√\n2. Using this\nabove gives us:\n\u0000\nf(1)−f(2)\u0001 \n2\n2 +1\nσ2log2f(1)\nf(2)!\n≤0.72\u0010\n1−e−√\n2σ\u0011\nf(1), (35)\nwhen σ≤1\n2√\n2. Plugging this into eq. (34) gives us:\nE{ζ(1),ζ(2)}h\nf(j∗)i\n≥\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nf(1). (36)\nRewriting the above in terms of the full notation, we obtain:\nE{ζ(w,x(j))}2\nj=1h\nf\u0000\nw,x(j∗)\u0001\f\f\fj∗= arg maxj∈[2]ef(w,x(j))i\n≥\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nmax\nj∈[2]f(w,x(j)).\n(37)\nThus:\nbF2,approx (w)≥\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nE{x(j)}2\nj=1h\nmax\nj∈[2]f(w,x(j))i\n=\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nbF2(w).\n(38)\n19Hence, we also have:\nρ2,approx (w) =bF2,approx (w)\nF(w)≥\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011bF2(w)\nF(w)=\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nρ2(w),\n(39)\nand:\nρ∗\n2,approx = inf\nw/∈ΦFρ2,approx (w)≥\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\ninf\nw/∈ΦFρ2(w) =\u0010\n1−0.72\u0010\n1−e−√\n2σ\u0011\u0011\nρ∗\n2.\n(40)\nLemma 1. Consider the function h(t;σ) :=2(1−e−t)\u0000\n2+t2\nσ2\u0001fort≥0andσ≤1\n2√\n2. Then, we have:\nmax\nt≥0h(t;σ)≤0.72\u0010\n1−e−√\n2σ\u0011\n.\nProof. Lett∗= arg maxt≥0h(t;σ). Settingdh\ndt\f\f\nt=t∗= 0, we obtain the following equation:\net∗= 1 +σ2\nt∗+t∗\n2. (41)\nUsing the series expansion of the exponential function above, we get:\n1 +∞X\nj=1(t∗)j\nj!= 1 +σ2\nt∗+t∗\n2=⇒(t∗)2\u0010\n1 + 2∞X\nj=1(t∗)j\n(j+ 1)!\u0011\n| {z }\n:=ν(t∗)= 2σ2. (42)\nSince ν(t∗)≥1, we conclude that:\nt∗≤√\n2σ. (43)\nFurther, since σ≤1\n2√\n2, we also have t∗≤1\n2. Note that ν(t∗) is an increasing function of t∗. So\nν(t∗)≤ν\u00001\n2\u0001\n. Also, by using the series expansion of the exponential function, it can be verified\nthat ν(t∗) =2(et∗−1)\nt∗−1. Thus, we have:\nν(t∗)≤ν\u00101\n2\u0011\n≤4e1/2−5. (44)\nUsing this in eq. (42), we get:\nt∗≥√\n2σp\n4e1/2−5≥0.62\u0000√\n2σ\u0001\n. (45)\nCombining the bounds of eq. (43) and eq. (45), we deduce that t∗=c\u0000√\n2σ\u0001\n, where c∈[0.62,1].\nThus, (1 −e−t∗)≤1−e−√\n2σand2\n2+t2\nσ2≤2\n2(1+0 .622)≤0.72. Using all of this, we obtain:\nmax\nt≥0h(t;σ) =h(t∗;σ)≤0.72\u0000\n1−e−√\n2σ\u0001\n. (46)\n20D Proof of Theorem 5\nProof. Per Assumption 7, M(w,x)− M (w∗,x)∼\niidN\u0000\nε(w), δ2(w)\u0001\nforw̸=w∗. Clearly,\nF(w) =Ex\u0002\nf(w,x)\u0003\n=ε2(w) +δ2(w), (47)\nand\nbFR(w) =E{x(1),...,x(R)}\"\nmax\nx∈{x(1),...,x(R)}f(w,x)#\n. (48)\nFor conciseness, we shall denote ε(w) and δ(w) by just εandδ, respectively. Also, let Zi=\nM(w,x(i))−M(w∗,x(i)) fori∈[R]. As per our concise notation, note that each Zi∼\niidN(ε, δ2).\nHence:\nbFR(w) =Eh\nmax\ni∈[R]Z2\nii\n. (49)\nWe shall obtain a lower bound for E\u0002\nmax i∈[R]Z2\ni\u0003\n. Let Y∼ N(0,1). For t >0, we have:\nP\u0010\nmax\ni∈[R](Zi−ε)≤t\u0011\n=P\u0010\n∩i∈[R](Zi−ε)≤t\u0011\n=\u0010\nP\u0000\nY≤t/δ\u0001\u0011R\n(50)\n=\u00101\n2+1\n2erf\u0010t√\n2δ\u0011\u0011R\n(51)\n≤ \n1\n2+1\n2r\n1−exp\u0010\n−2t2\nπδ2\u0011!R\n(52)\n≤ \n1\n2+1\n2 \n1−1\n2exp\u0010\n−2t2\nπδ2\u0011!!R\n(53)\n= \n1−1\n4exp\u0010\n−2t2\nπδ2\u0011!R\n(54)\n≤exp\u0010\n−R\n4exp\u0010\n−2t2\nπδ2\u0011\u0011\n. (55)\nIn eq. (51), the error function is as defined in eq. (2). Equation (52) follows from Fact 3.\nEquation (53) is obtained by using the fact that√1−a≤1−a\n2for all a∈[0,1]. Equation (55)\nfollows from the fact that 1 −a≤e−afor all a∈R. So, max i∈[R](Zi−ε)≤twith a probability\nof at most exp\u0010\n−R\n4exp\u0010\n−2t2\nπδ2\u0011\u0011\n. Let us choose t=δq\nπ\n2logR\n4 logR. With this choice, we get:\nmax\ni∈[R]Zi≥ε+δs\nπ\n2logR\n4 logRw.p.≥1−1\nR.\nThus,\nEh\nmax\ni∈[R]Z2\nii\n≥ \nε+δs\nπ\n2logR\n4 logR!2\u0010\n1−1\nR\u0011\n= \nε2+π\n2logR\n4 logRδ2+s\n2πlogR\n4 logRεδ!\u0010\n1−1\nR\u0011\n.(56)\nPlugging this into eq. (49) and replacing εandδwith their complete notations, i.e., ε(w) and\nδ(w), we get:\nbFR(w)≥ \nε2(w) +\u0010π\n2logR\n4 logR\u0011\nδ2(w) +s\n2πlogR\n4 logRε(w)δ(w)!\u0010\n1−1\nR\u0011\n. (57)\n21Hence:\nρR(w) =bFR(w)\nF(w)≥\u0010\nε2(w) +\u0010\nπ\n2logR\n4 logR\u0011\nδ2(w) +q\n2πlogR\n4 logRε(w)δ(w)\u0011\u0010\n1−1\nR\u0011\nε2(w) +δ2(w).(58)\nE Proof of Theorem 6\nProof. Fori∈ {1,2}, we have:\nˆy(i)\nj= sig\u0000\nθ⊤Ajx(i)\u0001\nand ˆy(i)= ˆy(i)\nk= sig\u0000\nθ⊤BjAjx(i)\u0001\n. (59)\nFor the i.i.d. samples x(1)andx(2)with ground truth labels y(1)andy(2), recall that ℓ(1)\njand\nℓ(2)\njare the corresponding cross-entropy losses of the early predictions at the jthlayer, i.e.,\nℓ(i)\nj=−y(i)log\u0000\nˆy(i)\nj\u0001\n−(1−y(i)) log\u0000\n1−ˆy(i)\nj\u0001\nfori∈ {1,2}. (60)\nWe are interested in:\npj:=Px(1),x(2)\u0010\narg maxi∈[1,2]ℓ(i)\nj= arg maxi∈[1,2]ℓ(i)\nk\u0011\n.\nUsing symmetry, we get:\npj:=Px(1),x(2)\u0010\nℓ(1)\nj≥ℓ(2)\nj\f\f\fℓ(1)\nk≥ℓ(2)\nk\u0011\n. (61)\nNote that pk= 1. Henceforth, we shall drop the subscript x(1),x(2)for conciseness. Let\n¯y(1)= 2y(1)−1 and ¯ y(2)= 2y(2)−1 be the centered ground truth labels (i.e., ∈ {− 1,1}) ofx(1)\nandx(2), respectively. It can be verified that:\nℓ(1)\nj≥ℓ(2)\nj⇐⇒θ⊤Aj\u0010\n¯y(1)x(1)−¯y(2)x(2)\u0011\n≥0 and ℓ(1)\nk≥ℓ(2)\nk⇐⇒θ⊤BjAj\u0010\n¯y(1)x(1)−¯y(2)x(2)\u0011\n≥0.\n(62)\nLetz:= ¯y(1)x(1)−¯y(2)x(2). Since x(1),x(2)∼\niidN(⃗0d,Id), we have that z∼ N(⃗0d,2Id).\nUsing eq. (62) and the definition of zfollowed by the application of Bayes’ theorem in\nEquation (61), we obtain:\npj=P\u0010\nθ⊤Ajz≥0\f\f\fθ⊤BjAjz≥0\u0011\n=P\u0000\nθ⊤Ajz≥0,θ⊤BjAjz≥0\u0001\nP\u0000\nθ⊤BjAjz≥0\u0001 . (63)\nSince z∼ N(⃗0d,2Id),P\u0000\nθ⊤BjAjz≥0\u0001\n=1\n2. Using this above, we get:\npj= 2P\u0010\nθ⊤Ajz≥0,θ⊤BjAjz≥0\u0011\n. (64)\nFor ease of notation, let u1=θ⊤Ajzandu2=θ⊤BjAjzandu=\u0014u1\nu2\u0015\n. Note uis a\nmultivariate Gaussian random variable with E[u] =\u00140\n0\u0015\nand\nE[uu⊤] :=Σ= 2\u0014∥A⊤\njθ∥2\n2\nA⊤\njθ,A⊤\njB⊤\njθ\u000b\n\nA⊤\njθ,A⊤\njB⊤\njθ\u000b\n∥A⊤\njB⊤\njθ∥2\n2\u0015\n. (65)\n22Letα1=∥A⊤\njθ∥2,α2=∥A⊤\njB⊤\njθ∥2and\nβ=⟨A⊤\njθ,A⊤\njB⊤\njθ⟩\n∥A⊤\njθ∥2∥A⊤\njB⊤\njθ∥2.\nIn the theorem statement, we shall denote βbyβjto indicate the dependence on the layer\nnumber j; we drop the subscript jhere for conciseness. With this notation, we have:\nΣ= 2\u0014α2\n1 βα1α2\nβα1α2 α2\n2\u0015\n. (66)\nThe pdf of uis therefore:\nϕ(u) =1\n2πp\ndet(Σ)exp \n−1\n2u⊤Σ−1u!\n(67)\n=1\n4πα1α2p\n1−β2exp \n−1\n4(1−β2)\u0010u2\n1\nα2\n1−2βu1u2\nα1α2+u2\n2\nα2\n2\u0011!\n. (68)\nUsing this in eq. (64), we get:\npj= 2Z∞\n0Z∞\n0ϕ(u)du1du2 (69)\n=1\n2πα1α2p\n1−β2Z∞\n0Z∞\n0exp \n−1\n4(1−β2)\u0010u2\n1\nα2\n1−2βu1u2\nα1α2+u2\n2\nα2\n2\u0011!\ndu1du2 (70)\n=1\n2πα1α2p\n1−β2Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011 Z∞\n0exp \n−\u0000\nu2−\u0000βα2\nα1\u0001\nu1\u00012\n4α2\n2(1−β2)!\ndu2!\ndu1.(71)\nWith some simple change of variables and some simplification, the above equation becomes:\npj=1\nπα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011 Z∞\n−βu1\n2√\n1−β2α1exp(−t2)dt!\ndu1 (72)\n=1\n2√πα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011 \n2√πZ βu1\n2√\n1−β2α1\n0exp(−t2)dt+2√πZ∞\n0exp(−t2)dt!\ndu1(73)\n=1\n2√πα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011 \nerf\u0010βu1\n2p\n1−β2α1\u0011\n+ lim\ny→∞erf(y)\n|{z}\n=1!\ndu1 (74)\n=1\n2√πα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011\nerf\u0010βu1\n2p\n1−β2α1\u0011\ndu1+1\n2√πα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011\ndu1\n| {z }\n=1\n2using Fact 5(75)\n=1\n2√πα1Z∞\n0exp\u0010\n−u2\n1\n4α2\n1\u0011\nerf\u0010βu1\n2p\n1−β2α1\u0011\ndu1+1\n2. (76)\nReplacing ( u1/α1) byyabove gives us:\npj=1\n2√πZ∞\n0exp\u0010\n−y2\n4\u0011\nerf\u0010βy\n2p\n1−β2\u0011\ndy+1\n2(77)\n=1\n2√πZ∞\n0exp\u0010\n−y2\n4\u0011\ndy\n| {z }\n=1\n2using Fact 5−1\n2√πZ∞\n0exp\u0010\n−y2\n4\u0011\nerfc \nβy\n2p\n1−β2!\ndy+1\n2(78)\n= 1−1\n2√πZ∞\n0exp\u0010\n−y2\n4\u0011\nerfc \nβy\n2p\n1−β2!\ndy. (79)\n23This is the exact final expression for pj.\nUsing Fact 4 fact above yields:\npj≥1−1√πZ∞\n0exp\u0010\n−y2\n4\u0011\nexp\u0010\n−β2y2\n8(1−β2)\u0011\ndy (80)\n= 1−1√πZ∞\n0exp \n−y2\n8\u00102−β2\n1−β2\u0011!\ndy (81)\n= 1−s\n2−2β2\n2−β2. (82)\nEquation (82) follows from Fact 5.\nFinally, the theorem statement follows by adding the subscript jtoβin Equation (79) and\nEquation (82) (recall that we omitted the subscript jearlier for conciseness).\nE.1 Proof of Corollary 4\nProof. Plugging in βj= 1−τjinto the lower bound for pjin Theorem 6, we get:\npj≥1−s\n2−2(1−τj)2\n2−(1−τj)2= 1−s\n2τj(2−τj)\n1 +τj(2−τj)= 1− O(√τj),\nforτj→0.\nF Remaining Experimental Details\nFor AdamW, we used the following hyper-parameter values: learning rate = 1e-4, ℓ2weight\ndecay = 0.01, β1= 0.9 and β2= 0.999. The learning rate warmup was over the first 0.2% of\ntotal steps followed by linear decay. We used the GELU activation and a dropout probability\nof 0.1 on all the layers. The training loss is the sum of the mean masked LM likelihood and the\nmean next sentence prediction likelihood.\nSIFT Criterion SIFT Layer # Sampling Time(hrs)\nLoss-based 1 0.8452\nLoss-based 2 1.1662\nLoss-based 3 1.4764\nLoss-based 6 2.5901\nLoss-based 12 4.5086\nEntropy-based 1 0.9121\nEntropy-based 2 1.2181\nEntropy-based 3 1.5398\nEntropy-based 6 2.6195\nEntropy-based 12 4.5148\nTable 2: Sampling time of SIFT as a function of early exit layer. For comparison, the full\nforward and backward propagation times (for the update) in all the cases is ∼3.8 hours and ∼\n7.9 hours, respectively.\n24"    },    {        "title": "2402.07068v1.Minimal_Einstein_Aether_Theory.pdf",        "content": "arXiv:2402.07068v1  [gr-qc]  10 Feb 2024Minimal Einstein-Aether Theory\nMetin Gürses,1,∗Çetin Şentürk,2,†and Bayram Tekin3,‡\n1Department of Mathematics, Faculty of Sciences\nBilkent University, 06800 Ankara, Turkey\n2Department of Aeronautical Engineering,\nUniversity of Turkish Aeronautical Association, 06790 Ank ara, Turkey\n3Department of Physics, Middle East Technical University, 0 6800 Ankara, Turkey\nWe show that there is a minimal Einstein-Aether theory, obta ined as a phenomenologically and\ntheoretically consistent limit from the generic Einstein- Aether theory, that supports Einstein metrics\nas solutions with a reduced cosmological constant. The mini mal theory is obtained by taking three\nof the coupling constants to be zero but keeping the expansio n coupling constant to be nonzero.\nThe square of the expansion of the unit-timelike aether field depletes the bare cosmological constant\nand thus provides, via local Lorentz symmetry breaking inhe rent in the Einstein-Aether theories,\na novel mechanism for reconciling the observed, small cosmo logical constant (or dark energy) with\nthe large theoretical prediction coming from quantum field t heories. The crucial point here is that\nminimal Einstein-Aether theory does not modify the well-te sted aspects of General Relativity such\nas solar systems tests and black hole physics including grav itational waves.\nI. INTRODUCTION\nOne of the most difficult problems in physics today is\nto understand how the predicted cosmological constant\ncoming from the zero point fluctuations of the quantum\nfields can differ from the observed dark energy by 50-120\norders of magnitude, a huge gap that depends on the\ncut-off used in the regularization of the quantum field\ntheory. This cosmological constant problem arises within\nthe context of General Relativity coupled minimally with\nthe Standard Model fields. The difficulty of the problem\nis related to the fact that due to the successes of both\ntheories, there is very little room for modification as the\nmost obvious modifications are with the gravity sector\nand often lead to drastic changes in General Relativity.\nHere we will argue that there is a modified gravity theory\nthat allows Einstein spacetimes as solutions, and the bare\ncosmological constant can be depleted as a result of local\nsymmetry breaking induced by a non-zero vacuum value\nof a vector field. The theory is a sub-class of the Einstein-\nAether theory which we shall describe first.\nThe so-called Einstein-Aether theory (EAT) [1, 2] is\na vector-tensor theory of gravity that breaks the local\nLorentz symmetry of spacetime due to the presence of a\ntimelike vector field [3, 4]. The nowhere-vanishing, unit-\nnorm vector field–called the “aether”–defines a preferred\ndirection at each point of spacetime; and dynamically\ncouples to the metric tensor to preserve the background\nindependence of the theory. Over the years, EAT has at-\ntracted a lot of attention and been investigated in various\nrespects: For example, the stability issue of the aether\nwas discussed in [5, 6], time-independent spherically sym-\nmetric solutions and black hole solutions were analyzed\n∗gurses@fen.bilkent.edu.tr\n†csenturk@thk.edu.tr\n‡btekin@metu.edu.trin [7–15], generalizations and cosmological implications\nwere studied in [16–20], extensions to include other fields\nwere considered in [21, 22], all Gödel-type of solutions\nwere given in [23, 24], and the possibilities of spacelike\nand null aether field were discussed respectively in [25–\n28] and in [29].\nEAT has four dimensionless parameters c1,c2,c3and\nc4that have certain bounds, discussed here in Sec. II,\nby comparing the theory with the solar system and other\nobservations. In general, these parameters are assumed\nto be non-vanishing. It is known that the EAT is an ex-\ntension of some other well-studied gravity theories and\nreproduces these theories to some other gravity under\ncertain limits of the coupling constants. The simplest\nexample is the trivial Einstein limit of EAT where all the\ncoupling constants {c1, c2, c3, c4}are taken to be zero. If\nthe so-called “twist” coupling constant, cw≡c1−c3, is\ntaken to infinity, one gets the Hořava gravity limit [30].\nRecently, yet another limit of EAT has been found by\nFranzin et al. in [31]: Taking c1=c3=c4= 0 and\nthe expansion of the unit timelike vector field to be zero\nin EAT, one obtains the vacuum Einstein field equations\nin the absence of matter fields. This theory is dubbed\n“Minimal Einsten-Aether Theory” which we shall shorten\nhere as MEAT in our paper. In this restricted version of\nEAT, there is no contribution to the resulting geometry\nfrom the EAT itself. In [31], it was concluded that the\nKerr metric with a unit time-like vector having zero ex-\npansion is an exact solution of the EAT. Here, in this\nwork, we generalize this result by including a cosmolog-\nical constant, and we show that by taking c2/ne}ationslash= 0 but,\nc1=c3=c4= 0; and the expansion of the unit timelike\nvector to be a constant, EAT reduces to the Einstein the-\nory with a shifted cosmological constant. It is interesting\nthat, in this new limit, there is an important contribu-\ntion to the cosmological constant in the theory which is\nproportional to the square of the constant expansion pa-\nrameter of the unit timelike vector field.\nThroughout the paper, we shall adopt the sign conven-2\ntion ( −,+,+,+).\nII. THE EINSTEIN-AETHER THEORY\nThe general action of EAT, in the presence of a cos-\nmological constant Λ and the matter fields, is given by\nI=1\n16πGˆ\nd4x√−gL+IM(gµν, vµ, φ), (1)\nwhere Gis the bare gravitational constant related to\nNewton’s constant GNby the rescaling [16]\nGN=/parenleftbigg\n1−c1+c4\n2/parenrightbigg−1\nG, (2)\nand IMdenotes the matter action with φcollectively\nrepresenting the matter fields. The action of the grav-\nity sector is constructed from the following vector-tensor\nLagrangian density:\nL=R−2Λ−Kαβµν∇αvµ∇βvν+λ(vµvµ+ 1),(3)\nin which the vector vµ, called the aether field, for non-\nzero values, would break local Lorentz invariance. And\nto ensure this breaking, the field λis taken to be the La-\ngrange multiplier that enforces the unit-norm constraint\nvµvµ=−1, (4)\nwhich certainly breaks local Lorentz symmetry. In some\nsense, this construction resembles Nambu’s construction\nof electrodynamics in the nonlinear gauge [32], but here\nthe underlying vector theory is not generically the usual\nelectrodynamics coupled to gravity as the Ktensor is\njudiciously chosen to be of the form\nKµναβ=c1gµνgαβ+c2δµ\nαδν\nβ+c3δµ\nβδν\nα−c4vµvνgαβ,\nwhere {c1, c2, c3, c4}are dimensionless coupling constants\nwhich certainly are phenomenologically bounded.\nIt pays to reparameterize the theory with the use of the\nusual hypersurface decomposition of the timelike aether\ncongruence as [30]\n∇νvµ=1\n3θhµν+σµν+ωµν−aµvν, (5)\nwhere θis the expansion of the aether field, hµνis the\nprojection tensor that projects onto the hypersurface or-\nthogonal to vµ; and σµνis the shear, ωµνis the twist,\nandaµis the acceleration, which are all defined by\nθ=∇µvµ,\nhµν=gµν+vµvν,\nσµν=a(µvν)+∇(µvν)−1\n3θhµν,\nωµν=a[µvν]− ∇ (µvν),\naµ=vν∇νvµ, (6)where we used the usual symmetrization and anti-\nsymmetrization brackets with a factor of 1 /2. With these\nnew variables, the Lagrangian in (3) reduces to\nL=R−2Λ−1\n3cθθ2−cσσ2−cωω2+caa2,(7)\nwhere cθ,cσ,cωandcaare the redefined dimensionless\nparameters with the definitions\ncθ≡c1+c3+ 3c2,\ncσ≡c1+c3,\ncω≡c1−c3,\nca≡c1+c4. (8)\nWe mentioned above that these parameters are con-\nstrained by some theoretical and observational argu-\nments. This issue was studied extensively in [1, 2, 15,\n16, 33–41]. The combination of the gravitational wave\nevent GW170817 [42] and the gamma-ray burst event\nGRB 170817A [43]–together with other theoretical and\nobservational constraints–puts a stringent bound on cσ\nas (see, e.g., Ref. [41])\n|cσ|<10−15. (9)\nOn the other hand, imposing that the PPN parameters\nof EAT are identical to those of General Relativity, the\nstability against linear perturbations in Minkowski back-\nground, vacuum Cherenkov effect and nucleosynthesis\nconstraints imply that (see, e.g., Refs. [35, 41])\n0/lessorsimilarca/lessorsimilar2.5×10−5,\n0/lessorsimilarcω<cσ\n3(1−cσ),\n0/lessorsimilarc2/lessorsimilar0.095,\nc4/lessorsimilar0. (10)\nFurthermore, in [44], it was shown that taking cσ= 0\nexactly, the stability of black holes under odd-parity per-\nturbations requires c4= 0 along with c1≥0.\nLet us discuss the field equation of the theory which we\nshall need. The fundamental fields of the gravitational\nsector are ( gµν, vµ, λ); so, by varying the action (1) with\nrespect to λ, one obtains the constraint equation (4), and\nvarying with respect to the metric gµνand the aether field\nvµ, one respectively obtains\nGµν+ Λgµν=TAE\nµν+ 8πGTM\nµν, (11)\nc4aα∇µvα+∇αJα\nµ+λvµ= 8πGTM\nµ,(12)\nwhere\nTAE\nµν≡ ∇ α/parenleftBig\nJα\n(µvν)−J(µαvν)+J(µν)vα/parenrightBig\n+c1/parenleftBig\n∇µvα∇νvα− ∇ αvµ∇αvν/parenrightBig\n+c4aµaν+λvµvν−1\n2L gµν, (13)\nTM\nµν≡ −2√−gδIM\nδgµν, TM\nµ≡ −1√−gδIM\nδvµ,3\nwith\nJµ\nν=Kµα\nνβ∇αvβ, L =Jµ\nν∇µvν. (14)\nNote that in getting (11), we eliminated the term related\nto the constraint (4). Multiplying the aether equation\n(12) by vµ, one can also derive the Lagrange multiplier\nλ=c4a2+∇βJβα−8πGTM\nµvµ. (15)\nThis section was a necessary recapitulation of the EAT\ntheory in its full generality and the pertaining constraint s\nfor its viability. Next we introduce the minimal version\nof it.\nIII. MINIMAL EINSTEIN-AETHER THEORY\nMEAT is obtained by setting [31]\ncσ=cω=ca= 0, (16)\nbutcθ/ne}ationslash= 0, in the original parametrization, which corre-\nsponds to\nc1=c3=c4= 0, c 2/ne}ationslash= 0. (17)\nThese choices are indeed consistent with the theoretical\nand observational bounds in Eqs. (9)-(10) discussed in\nSec. II.\nWith the choice (17), one has Kµανβ=c2δµνδαβ\nandJµν=c2θ δµν, and L=θ2, where θ=∇αuα. In\naddition, when we let θ=const. , then λ= 0 and the\nfield equations (11) reduce to\nGµν+ Λgµν=c2θ2\n2gµν+ 8πGTM\nµν, (18)\nand the aether equation (12) is satisfied identically by\ntaking TM\nµ= 0. In addition to the trivial Einstein limit\nc1=c2=c3=c4= 0, we now have a nontrivial limit\nc1=c3=c4= 0 and θ=const. which is equivalent\nto the Einstein theory with a shifted cosmological con-\nstant (Λ −c2θ2\n2). Observe that the contribution from\nthe aether field to the base cosmological constant is neg-\native as c2is positive; hence the aether field depletes the\ncosmological constant. This result is a generalization of\n[31] where the expansion is assumed to be zero. In our\ncase, the Kerr metric with a cosmological constant, theCarter metric [45], is also an exact solution of the Ein-\nstein Aether theory. The pp-wave metric, the AdS wave\nmetric, and all Kerr-Schild-Kundt metrics [46]-[52] are\nsolutions to this minimal theory.\nThe unit timelike vector vµsatisfies the differential\nequation ∇µvµ=θ= constant. For stationary space-\ntimes, this equation reduces to\n∂i(√−g vi) =√−g θ, (19)\nwhere the sum above runs over the spatial indices. The\nzeroth component of the vector is found from vµvνgµν=\n−1. The only field equation to be solved in EAT is given\nin (19) which is a linear partial differential equation for\nthe space components of the vector vµ. For nonstationary\ncases such as all Kerr-Schild-Kundt family of metrics we\nhave a similar equation\n∂a(√−g va) =√−g, θ (20)\nwhere the sum above covers all coordinates except for a\ncyclic coordinate on which the metric does not depend.\nThe remaining component of the timelike vector vµis\nfound from vµvνgµν=−1. If the spacetime has no\nKilling vectors, then one solves the differential equation\n∂α(√−g vα) =√−g θ (21)\nwhere the sum above covers all indices. One solves the\nabove differential equation by eliminating one of the com-\nponents of vµby using vµvνgµν=−1.\nIV. CONCLUSIONS\nWe showed that Einstein-Aether theory has a nontriv-\nial limit to Einstein’s theory where the aether field has a\nconstant expansion with c1=c3=c4= 0. This result is\nequivalent to stating that in this limit, the cosmological\nconstant is shifted and the effective cosmological constant\nis less than the bare cosmological constant that appears\nin the Lagrangian or the field equations. Therefore the\naether field in this setting provides a natural way to de-\nplete the bare cosmological constant which theoretically\ncan be very large yet experimentally very small. From\nthe observational constraints (10) among the coupling\nconstants, cθ= 3c2can be considered to be the largest\none. Hence another way of interpreting MEAT is that it\nis an approximation of EAT by ignoring the contributions\nof the other coupling constants except c2.\n[1] T. Jacobson and D. Mattingly, Gravity with a dynamical\npreferred frame , Phys. Rev. D 64, 024028 (2001).\n[2] T. Jacobson, Einstein-Aether gravity: a status report ,\nProc. Sci. QG-PH (2007) 020 [arXiv:0801.1547].[3] D. Mattingly, Modern Tests of Lorentz Invariance , Living\nRev. Relativity 8, 5 (2005).\n[4] S. Liberati, Tests of Lorentz invariance: a 2013 update ,\nClass. Quantum Grav. 30, 133001 (2013).4\n[5] S. M. Carroll, T. R. Dulaney, M. I. Gresham, and H.\nTam, Instabilities in the aether , Phys. Rev. D 79, 065011\n(2009).\n[6] W. Donnelly and T. Jacobson, Stability of the aether ,\nPhys. Rev. D 82, 081501 (2010).\n[7] C. Eling and T. Jacobson, Spherical solutions in\nEinstein-aether theory: static aether and stars , Class.\nQuantum Grav. 23, 5625 (2006).\n[8] C. Eling and T. Jacobson, Black holes in Einstein-aether\ntheory , Class. Quantum Grav. 23, 5643 (2006).\n[9] D. Garfinkle, C. Eling, and T. Jacobson, Numerical simu-\nlations of gravitational collapse in Einstein-aether theo ry,\nPhys. Rev. D 76, 024003 (2007).\n[10] T. Tamaki and U. Miyamoto, Generic features of\nEinstein-Aether black holes , Phys. Rev. D 77, 024026\n(2008).\n[11] E. Barausse, T. Jacobson, and T. P. Sotiriou, Black holes\nin Einstein-aether and Hořava-Lifshitz gravity , Phys.\nRev. D 83, 124043 (2011).\n[12] C. Gao and Y. G. Shen, Static spherically symmetric so-\nlution of the Einstein-aether theory , Phys. Rev. D 88,\n103508 (2013).\n[13] E. Barausse and T. P. Sotiriou, Black holes in Lorentz-\nviolating gravity theories , Class. Quantum Grav. 30,\n244010 (2013).\n[14] C. Ding, A. Wang, and X. Wang, Charged Einstein-\naether black holes and Smarr formula , Phys. Rev. D 92,\n084055 (2015).\n[15] A. Adam, P. Figueras, T. Jacobson, and T. Wiseman, Ro-\ntating black holes in Einstein-aether theory , Class. Quan-\ntum Grav. 39, 125001 (2022).\n[16] S. M. Caroll and E. A. Lim, Lorentz-violating vector fields\nslow the universe down , Phys. Rev. D 70, 123525 (2004).\n[17] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Modi-\nfying gravity with the aether: An alternative to dark mat-\nter, Phys. Rev. D 75, 044017 (2007).\n[18] C. Bonvin, R. Durrer, P. G. Ferreira, G. D. Starkman,\nand T. G. Zlosnik, Generalized Einstein-Aether theories\nand the Solar System , Phys. Rev. D 77, 024037 (2008).\n[19] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman,\nGrowth of structure in theories with a dynamical preferred\nframe , Phys. Rev. D 77, 084010 (2008).\n[20] J. Zuntz, T. G. Zlosnik, F. Bourliot, P. G. Ferreira, and\nG. D. Starkman, Vector field models of modified gravity\nand the dark sector , Phys. Rev. D 81, 104015 (2010).\n[21] A. B. Balakin and J. P. S. Lemos, Einstein-aether the-\nory with a Maxwell field: General formalism , Ann. Phys.\n350, 454 (2014).\n[22] T. Y. Alpin and A. B. Balakin, The Einstein-Maxwell-\naether-axion theory: Dynamo-optical anomaly in the\nelectromagnetic response , Int. J. Mod. Phys. D 25, 04\n(2016).\n[23] M. Gürses, Gödel type metrics in Einstein-aether theory ,\nGen. Rel. Grav. 41, 31 (2009).\n[24] M. Gürses, Ç. Şentürk, Gödel-type metrics in Einstein-\nAether theory II: nonflat background in arbitrary dimen-\nsions , Gen. Rel. Grav. 48, 63 (2016).\n[25] T. G. Rizzo, Lorentz violation in extra dimensions , JHEP\n09, 036 (2005).\n[26] L. Ackerman, S. M. Carroll, and M. B. Wise, Imprints of\na primordial preferred direction on the microwave back-\nground , Phys. Rev. D 75, 083502 (2007).\n[27] S. M. Carroll and H. Tam, Aether compactification , Phys.\nRev. D 78, 044047 (2008).[28] A. Chatrabhuti, P. Patcharamaneepakorn, and P.\nWongjun, Aether field, Casimir energy and stabilization\nof the extra dimension , JHEP 08, 019 (2009).\n[29] M. Gürses and Ç. Şentürk, A Modified Gravity Theory:\nNull Aether , Commun. Theor. Phys. 71, 312 (2019).\n[30] T. Jacobson, Undoing the twist: The Hořava limit of\nEinstein-aether theory , Phys. Rev. D 89, 081501(R),\n(2014).\n[31] E. Franzin, S. Liberti and J. Mazza, A Kerr Black Hole\nin Einstein-Aether Gravity , arXiv:2312.06891.\n[32] Y. Nambu, Quantum Electrodynamics in Nonlinear\nGauge , Prog. Theor. Phys. Suppl. E 68, 190 (1968).\n[33] C. Eling and T. Jacobson, Static post-Newtonian equiv-\nalence of general relativity and gravity with a dynamical\npreferred frame , Phys. Rev. D 69, 064005 (2004).\n[34] J. W. Elliott, G. D. Moore, and H. Stoica, Constrain-\ning the New Aether: gravitational Cherenkov radiation ,\nJHEP 08, 066 (2005).\n[35] B. Z. Foster and T. Jacobson, Post-Newtonian param-\neters and constraints on Einstein-aether theory , Phys.\nRev. D 73, 064015 (2006).\n[36] T. Jacobson, Einstein-aether gravity: Theory and obser-\nvational constraints , in the Proceedings of the Meeting\non CPT and Lorentz Symmetry (CPT 07) , Bloom-\nington, Indiana, 8-11 Aug. 2007 [arXiv:0711.3822].\n[37] B. Z. Foster, Strong field effects on binary systems in\nEinstein-aether theory , Phys. Rev. D 76, 084033 (2007).\n[38] J. A. Zuntz, P. G. Ferreira, and T. G. Zlosnik, Constrain-\ning Lorentz Violation with Cosmology , Phys. Rev. Lett.\n101, 261102 (2008).\n[39] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Con-\nstraints on Einstein-Aether theory and Hořava gravity\nfrom binary pulsar observations , Phys. Rev. D 89, 084067\n(2014).\n[40] Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos,\nGravitational waves in Einstein-Êther and generalized\nTeVeS theory after GW170817 , Phys. Rev. D 97, 084040\n(2018).\n[41] J. Oost, S. Mukohyama, and A. Wang, Constraints on\nEinstein-aether theory after GW170817 , Phys. Rev. D\n97, 124023 (2018).\n[42] B. P. Abbott et al. (LIGO Scientific Collaboration and\nVirgo Collaboration), GW170817: Observation of Grav-\nitational Waves from a Binary Neutron Star Inspiral ,\nPhys. Rev. Lett. 119, 161101 (2017).\n[43] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi-\nGBM, INTEGRAL Collaborations), Gravitational Waves\nand Gamma-Rays from a Binary Neutron Star Merger:\nGW170817 and GRB 170817A , Astrophys. J. Lett. 848,\nL13 (2017).\n[44] S. Tsujikawa, C. Zhang, X. Zhao, and A. Wang, Odd-\nparity stability of black holes in Einstein-aether gravity ,\nPhys. Rev. D 104, 064024 (2021).\n[45] B. Carter, Black holes equilibrium states , in Les Houches\nSummer School of Theoretical Physics: Black Holes,\nedited by C. deWitt and B. deWitt, Gordon and Breach\nScience Publishers: New York, USA; London, UK; Paris,\nFrance, 1973, pp. 61-124.\n[46] İ. Güllü, M. Gürses, T. Ç. Şişman, and B. Tekin, AdS\nwaves as exact solutions to quadratic gravity , Phys. Rev.\nD83, 084015 (2011).\n[47] M. Gürses, T. Ç. Şişman, and B. Tekin, New exact so-\nlutions of quadratic curvature gravity , Phys. Rev. D 86,\n024009 (2012).5\n[48] M. Gürses, S. Hervik, T. Ç. Şişman, and B. Tekin, Anti-\nde Sitter-Wave Solutions of Higher Derivative Theories ,\nPhys. Rev. Lett. 111, 101101 (2013).\n[49] M. Gürses, T. Ç. Şişman, and B. Tekin, AdS-plane wave\nandpp-wave solutions of generic gravity theories , Phys.\nRev. D 90, 124005 (2014).[50] M. Gürses, T. Ç. Şişman, and B. Tekin, Some exact so-\nlutions of all f(Rµν)theories in three dimensions , Phys.\nRev. D 86, 024001 (2012).\n[51] M. Gürses, T. Ç. Şişman, and B. Tekin, Gravity waves\nin three dimensions , Phys. Rev. D 92, 084016 (2015).\n[52] M. Gürses, T. Ç. Şişman, and B. Tekin, Kerr-Schild-\nKundt metrics are universal , Class. Quantum Grav. 34,\n075003 (2017)."    },    {        "title": "2402.07074v1.Local_moduli_of_continuity_for_permanental_processes_that_are_zero_at_zero.pdf",        "content": "arXiv:2402.07074v1  [math.PR]  11 Feb 2024Local moduli of continuity for permanental processes\nthat are zero at zero.\nMichael B. Marcus Jay Rosen∗\nFebruary 13, 2024\nAbstract\nLetu(s,t) be a continuous potential density of a symmetric L´ evy\nprocess or diffusion with state space Tkilled atT0, the first hitting time\nof 0, or atλ∧T0, whereλis an independent exponential time. Let\nf(t) =/integraldisplay\nTu(t,v)dµ(v),\nwhereµis a finite positive measure on T. LetXα={Xα(t),t∈T}be\nanα−permanental process with kernel\nv(s,t) =u(s,t)+f(t).\nThen when lim t→0u(t,t) = 0,\nlimsup\nt↓0Xα(t)\nu(t,t)loglog1/t≥1,a.s.\nand\nlimsup\nt↓0Xα(t)\nu(t,t)loglog1/t≤1+Cu,h,a.s.\nwhereCu,µ≤ |µ|is a constant that depends on both uandµ, which is\ngiven explicitly, and is different in the different examples.\n∗Research of Jay Rosen was partially supported by grants from the Simons Foundation.\n0Key words and phrases: permanental processes with non-symm etric kernels, law of the\niterated logarithm at zero\n0AMS 2020 subject classification: 60E07, 60G15, 60G17, 60G99 , 60J25\n11 Introduction\nLetK={K(s,t),s,t∈T}be a kernel with the property that for all tn=\n(t1,...,tn) inTn, the matrix K(tn) ={K(ti,tj),i,j∈[1,n]}determines an\nn−dimensional random variable random variable ( Xα(t1),...,Xα(tn)) with\nLaplace transform,\nE/parenleftig\ne−/summationtextn\ni=1siXα(ti)/parenrightig\n=1\n|I+K(tn)S(sn)|α, (1.1)\nwhereS(sn) is a diagonal matrix with positive entries sn= (s1,...,sn), and\nα >0. We refer to K(tn) as the kernel of ( Xα(t1),...,Xα(tn)). It follows\nfrom the Kolmogorov Extension Theorem that {K(s,t),s,t∈T}determines\na stochastic process which we denote by Xα={Xα(t),t∈T}and refer to as\nanα−permanental process with kernel K.\nThere are well known examples of α−permanental processes when α=\nk/2. Letη={η(t);t∈T}be a mean zero Gaussian process with covariance\nU={U(s,t),s,t∈T}.Let{ηi;i= 1,...,k}be independent copies of η.\nDefine,\nYk/2(t) =k/summationdisplay\ni=1η2\ni(t)\n2, t∈T. (1.2)\nThe positive stochastic process {Yk/2(t),t∈T}is referred to as a chi–square\nprocess of order kwith covariance U. It is well known that for {t1,...,tn} ⊂T\nE/parenleftig\ne−/summationtextn\nj=1sjYk/2(tj)/parenrightig\n=1\n|I+U(tn)S(sn)|k/2, (1.3)\nwhereU(tn) is then×nmatrix,{U(tj,tk);1≤j,k≤n}. Therefore Yk/2=\n{Yk/2(t);t∈T}is ak/2–permanental process with kernel U.\nIt is important to note that whereas (1.3) holds for all integ erskwhenever\nUis the covariance of a Gaussian process, it may or may not hold whenk/2\nis replaced by some α >0 that is not a multiple of 1/2. (The question of\nwhen (1.3) holds for all α >0 was raised by Paul L´ evy when he asked when\ndoes a Gaussian process have infinitely divisible squares. T his was answered\nby Griffiths and Bapat. An extensive treatment of these result s is given in [3,\nChapter 13].) Nevertheless, it follows from Eisenbaum and K aspi, [1, Theorem\n3.1] that if Uis the potential density of a transient Markov process on T\nwhich is finite for all s,t∈T, then there exist α−permanental processes\n{Yα(t),t∈T}with kernel Ufor allα>0. I.e., (1.3) holds with k/2 replaced\nbyαfor allα>0.\n2It is at this point that this becomes interesting because in t he Eisenbaum–\nKaspi theorem the kernel Uis not necessarily symmetric, or equivalent to a\nsymmetric matrix; (see [5]). When it is not, the correspondi ng permanental\nprocesses are not squares of a Gaussian process. They are a ne w kind of\npositive stochastic process defined by kernels that need not be symmetric. It\nis intriguing to study sample path properties of these proce sses.\nIn [6] we consider symmetric L´ evy processes and diffusions Y, that are\nkilled at the end of an independent exponential time or the fir st time they hit\n0, with potential densities u(x,y) with respect to some σ-finite measure on\nT⊆R1. Letfbe a finite excessive function for Y. Set\nuf(x,y) =u(x,y)+f(y), x,y ∈T. (1.4)\nUnder general smoothness conditions on f,u(x,y) and points d∈T,\nlaws of the iterated logarithm are found for Xk/2={Xk/2(t),t∈T}, a\nk/2−permanental process with kernel {uf(x,y),x,y∈T}, of the following\nform: For all integers k≥1,\nlimsup\nx→0|Xk/2(d+x)−Xk/2(d)|\n(2σ2(x)loglog1/x)1/2=/parenleftbig\n2Xk/2(d)/parenrightbig1/2, a.s., (1.5)\nwhere,\nσ2(x) =u(d+x,d+x)+u(x,x)−2u(d+x,x). (1.6)\nIn this paper we consider the question of finding the local mod ulus of\ncontinuity for Xαat 0 whenXα(0) = 0. This, it seems, is difficult, and, in\ngeneral, we can only get different upper and lower bounds. On th e other hand\nour results hold for Xαfor allα>0.\nWe consider the following situation: Let Ta be locally compact set with a\ncountable base. Let Y= (Ω,Ft,Yt,θt,Px) be a transient Borel right process\nwith state space T, and continuous strictly positive potential densities u(x,y)\nwith respect to some σ–finite measure monT. Letµbe a finite positive\nmeasure on Twith mass |µ|=µ(T). We call the function\nf(x) =/integraldisplay\nTu(y,x)dµ(y) (1.7)\na left potential for Y.\nThe following is [4, Theorem 6.1].\n3Theorem 1.1 LetYbe a transient Borel right process with state space T, as\ndescribed above. Then for any left potential fforY, there exists a transient\nBorel right process /tildewideY=(Ω,Ft,/tildewideYt,θt,Px)with state space S=T∪{∗}, where\n∗is an isolated point, such that /tildewideYhas potential densities,\n/tildewideu(x,y) =u(x,y)+f(y), x,y∈T (1.8)\n/tildewideu(∗,y) =f(y),and/tildewideu(x,∗) =/tildewideu(∗,∗) = 1,\nwith respect to the measure /tildewidemonSwhich is equal to monTand assigns a\nunit mass to ∗.\nLet{Zt,t∈S}be theαpermanental process with kernel /tildewideu(x,y). We have\nthe following general lower bound:\nTheorem 1.2 LetYbe a strongly symmetric transient Borel right process\nwith state space (0,1)⊆T⊆R1and continuous strictly positive potential\ndensitiesu(x,y)with respect to some σ–finite measure monT, as in Theorem\n1.1. Assume that limt↓0u(t,t) = 0.Then\nlimsup\nt→0Zt\nu(t,t)loglog1/t≥1, a.s. (1.9)\nRemark 1.1 In Examples 1.1–1.4, T=R1−{0}or (0,∞) and\n/tildewideu(x,y) =u(x,y)+f(y), x,y∈(0,1] (1.10)\nhas a continuous extension to x,y∈[0,1]. We continue to denote this exten-\nsion by/tildewideu(x,y). It follows from the proof of [4, Lemma 6.2] that this extens ion\nis the kernel of an α−permanental process.\nWe use [4, Theorem 1.2] to get upper bounds. We repeat it here f or the\nconvenience of the reader.\nTheorem 1.3 LetXα={Xα(t),t∈[0,1]}be anα−permanental process with\nkernel{/tildewideu(s,t),s,t∈[0,1]}with/tildewideu(0,0) = 0. Define the function,\n/tildewideσ(s,t) =/parenleftig\n/tildewideu(s,s)+/tildewideu(t,t)−2(/tildewideu(s,t)/tildewideu(t,s))1/2/parenrightig1/2\n.(1.11)\nLetϕbe a bounded increasing function which is regularly varying at zero with\npositive index such that for all s,t∈[0,1],\n/tildewideσ(s,t)≤ϕ(|s−t|), (1.12)\nIf, in addition, ϕ2(h) =O(/tildewideu(h,h))at0, then\nlimsup\nt↓0Xα(t)\n/tildewideu(t,t)loglog1/t≤1,a.s. (1.13)\n4Note that when\n/tildewideu(s,t) =u(s,t)+f(t), (1.14)\nthe denominator in (1.13) is larger than the denominator in ( 1.9).\nLet us now consider Theorem 1.3 with /tildewideu(s,t) as in (1.10) and\nf(t) :=fu,µ(t) =/integraldisplay\nTu(t,v)dµ(v), (1.15)\nfor some positive finite measure µ. By [3, Lemma 3.4.3], u(t,v)≤u(t,t)∧\nu(v,v) so that\nf(t)≤u(t,t)|µ| (1.16)\nIn this case we can write (1.13) as,\nlimsup\nt↓0Xα(t)\nu(t,t)loglog1/t≤1+|µ|,a.s. (1.17)\nHowever, in the examples we consider, (1.17) is generally to o large. In the\nexamples we consider we show that,\nf(t)∼Cu,µu(t,t),ast↓0, (1.18)\nwhere 0<Cu,µ≤ |µ|is a constant that depends on both uandµ. In this case\n(1.13) can be written as,\nlimsup\nt↓0Xα(t)\nu(t,t)loglog1/t≤1+Cu,h,a.s. (1.19)\nWe consider four examples in which u(s,t) are continuous potential densi-\nties of strongly symmetric Borel right processes killed the first time they hit\n0. In the first two examples the strongly symmetric Borel righ t processes are\nL´ evy processes, or exponentially killed L´ evy processes, killed when they hit\nzero.\nExample 1.1 LetZ={Zt;t∈R1}be a real valued symmetric L´ evy process\nwith characteristic exponent ψ, i.e.,\nE/parenleftig\neiλZt/parenrightig\n=e−tψ(λ),∀t≥0, (1.20)\nwhereψ(λ) is an even function and,\n1\n1+ψ(λ)∈L1(R1). (1.21)\n5In this example we consider processes Zfor which 0 is recurrent.\nLetZ′={Z′(t);t∈R1}be a strongly symmetric Borel right process with\nstate space T=R1− {0}obtained by killing Zthe first time it hits 0. It\nfollows from [3, Theorem 4.2.4] that the process Z′has continuous potential\ndensities with respect to Lebesgue measure restricted to T, which we denote\nby Φ ={Φ(x,y),x,y∈T}, where\nΦ(x,y) =(σ0)2(x)\n2+(σ0)2(y)\n2−(σ0)2(x−y)\n2, x,y ∈R1−{0},(1.22)\nand\n(σ0)2(x) =2\nπ/integraldisplay∞\n01−cosλx\nψ(λ)dλ. (1.23)\nNote that,\nΦ(x,x) = (σ0)2(x). (1.24)\nAs we point out in Remark 1.1 there exists an α−permanental process\n{Xα(x),x∈[0,1]}with kernel,\nv(x,y) = Φ(x,y)+fΦ,µ(y), (1.25)\nwhere\nfΦ,µ(y) =/integraldisplay\nTΦ(x,y)dµ(x) (1.26)\nand|µ|<∞.\nThe next theorem is an application of Theorem 1.3 with /tildewideu=v.\nTheorem 1.4 Let{Xα(x),x∈[0,1]}be anα−permanental process with ker-\nnelv(x,y). When (σ0)(t)∈C1(0,∞)is a regularly varying function at zero\nwith positive index, Xα(t)is continuous on [0,δ]. Furthermore, when\nlim\nt↓0t\n(σ0)2(t)= 0, (1.27)\nlimsup\nt↓0Xα(t)\n(σ0)2(t)loglog1/t≤1+|µ|\n2, a.s., (1.28)\nand when\nlim\nt↓0t\n(σ0)2(t)=1\nC, C > 0, (1.29)\nlimsup\nt↓0Xα(t)\nCtloglog1/t≤/parenleftbigg\n1+|µ|\n2/parenrightbigg\n+1\n2C/integraldisplay∞\n−∞((σ0)2(v))′dµ(v), a.s.(1.30)\n6We give conditions when ( σ0)(t)∈C1(0,∞) in Remark 7.2.\nRemark 1.2 We show in [6, Lemma 4.1] that either,\nlim\nt↓0t\n(σ0)2(t)= 0,or lim\nt↓0t\n(σ0)2(t)=C, (1.31)\nfor some constant C >0.\nNote also that when µis a symmetric measure, since (( σ0)2(v))′is an odd\nfunction, the integral in (1.30) is equal to 0. If µis concentrated on thee\nnegative half–line it is negative.\nExample 1.2 LetZβ={Zβ(t);t∈R1}be a symmetric Markov process\nobtained by killing Zat the end of an independent exponential time with\nmean 1/β,β >0. Theβ−potential density of Zwith respect to Lebesgue\nmeasure, which is the same as the 0 −potential density of Zβwith respect to\nLebesgue measure, is\nuβ(x,y) =uβ(x−y) =1\n2π/integraldisplay∞\n−∞cosλ(x−y)\nβ+ψ(λ)dλ, x,y ∈R1,(1.32)\nwhereψis an even function; see e.g., [3, (4.84)]. Let\n(σβ)2(x−y)) =uβ(x,x)+uβ(y,y)−2uβ(x,y) (1.33)\n=1\nπ/integraldisplay∞\n−∞1−cosλ(x−y)\nβ+ψ(λ)dλ, x,y ∈R1.(1.34)\nLet\nfuβ,µ(y) =/integraldisplay∞\n−∞uβ(x,y)dµ(x), (1.35)\nwhereµis a finite positive measure with µ({0}) = 0.\nLetZ′\nβ={Z′\nβ(t);t∈R1}be a strongly symmetric Borel right process\nwith state space T=R1−{0}obtained by killing Zβthe first time it hits 0.\nThe process Z′\nβhas continuous potential densities with respect to Lebesgu e\nmeasure restricted to T, which we denote by vβ={vβ(x,y),x,y∈T}, where\nvβ(x,y) =uβ(x,y)−uβ(x,0)uβ(0,y)\nuβ(0,0)(1.36)\n=uβ(x−y)−uβ(x)uβ(y)\nuβ(0);\nsee [3, (4.165)].\n7It follows from Remark 1.1 that there exists an α−permanental process\n{Xα,β(x),x∈[0,1]}with kernel\nv(x,y) =vβ(x,y)+fvβ,µ(y), (1.37)\nwhere,\nfvβ,µ(y) =/integraldisplay\nTvβ(x,y)dµ(x) (1.38)\nforµas given above. Since µ({0}) = 0 we can write (1.38) as\nfvβ,µ(y) =/integraldisplay∞\n−∞vβ(x,y)dµ(x) (1.39)\nThe next theorem is an application of Theorem 1.3 with /tildewideu=v. The reader\nshould note that in (1.41) and (1.43) we use fuβ,µ.\nTheorem 1.5 Let{Xα(x),x∈[0,1]}be anα−permanental process with ker-\nnelv(x,y). When(σβ)2(t)∈C1(T)and|(σβ)2(t)′|is bounded on (−∞,−∆)∪\n(∆,∞)for all∆>0and(σβ)2(t)is regularly varying at zero with positive\nindex,Xα,β(t)is continuous on [0,δ], for someδ>0. Furthermore, when\nlim\nt↓0t\n(σβ)2(t)= 0, (1.40)\nlimsup\nt↓0Xα(t)\n(σβ)2(t)loglog1/t≤1+fuβ,µ(0)\n2uβ(0), a.s., (1.41)\nand when\nlim\nt↓0t\n(σβ)2(t)=1\nC, C > 0, (1.42)\nlimsup\nt↓0Xα(t)\nCtloglog1/t≤1+fuβ,µ(0)\n2uβ(0)+1\n2C/integraldisplay∞\n−∞((σβ)2(v))′dµ(v), a.s.(1.43)\nIn Remark 7.2 we give examples of L´ evy processes that satisf y the hy-\npotheses of this theorem. Also we show in [6, Lemma 8.1] that f or allβ≥0,\nlim\nt↓0(σβ)2(t)\n(σ0)2(t)= 1, a.s (1.44)\nso we can replace ( σβ)2by (σ0)2in (1.41).\n8Example 1.3 LetZbeadiffusionin R1that isregularandwithouttrapsand\nis symmetric with respect to a σ–finite measure m, called the speed measure,\nwhich is absolutely continuous with respect to Lebesgue mea sure. (A diffusion\nis regular and without traps when Px(Ty<∞)>0,∀x,y∈R1.) We consider\ndiffusions Zwith generators of the form\n1\n2a2(x)d2\ndx2+c(x)d\ndx, (1.45)\nwherea2(x) andc(x)∈C/parenleftbig\nR1/parenrightbig\nanda(x)/ne}ationslash= 0 for any x∈R1. For details see\n[8, Section 7.3] and [2, Chapter 16].\nLets∈C2(R1) be a scale function for Z. The function sis strictly\nincreasing and is unique up to linear transformation so that we takes(0) = 0.\nAssume that lim x→∞s(x) =∞.\nLetZbe the process with state space T= (0,∞) that is obtained by\nstarting ZinTand then killing it the first time it hits 0. The potential\ndensity ofZwith respect to the speed measure mis,\nuT0(x,y) =s(x)∧s(y), x,y> 0, (1.46)\nwheres∈C2(0,∞) is strictly positive and strictly increasing.\nBy Remark 1.1 there exists an α−permanental process {Xα(x),x∈[0,1]}\nwith kernel\nv(x,y) =uT0(x,y)+fuT0,µ(y), (1.47)\nwhere\nfuT0,µ(y) =/integraldisplay\nTuT0(x,y)dµ(x) (1.48)\nand|µ|<∞.\nThe next theorem is an application of Theorem 1.3 where /tildewideu=v.\nTheorem 1.6 Let{Xα(t),t∈[0,1]}be anα−permanental process with kernel\nv(x,y). ThenXα(t)is continuous on [0,δ]and,\nlimsup\nt↓0Xα(t)\ns(t)loglog1/t≤1+|µ|, a.s. (1.49)\n(Note that we can replace s(t) bys′(0)tin (1.49).)\nExample 1.4 LetZbeZkilled at the end of an independent exponential\ntime with mean 1 /β. The potential density of Zwith respect to the speed\nmeasuremis\n/tildewideuβ(x,y) =/braceleftbiggpβ(x)qβ(y), x≤y\nqβ(x)pβ(y), y≤x, (1.50)\n9wherepβ,qβ∈C2(R1) are positive and pβis strictly increasing and qβis\nstrictly decreasing. (For the dependence of pβandqβonβsee [3, (4.14)].)\nLetZ′be the process with state space T= (0,∞) that is obtained by\nstartingZinTand then killing it the first time it hits 0. The potential\ndensity ofZ′is\n/tildewidevβ(x,y) =/tildewideuβ(x,y)−/tildewideuβ(x,0)/tildewideuβ(0,y)\n/tildewideuβ(0,0), x,y> 0,(1.51)\ni.e.,\n/tildewidevβ(x,y) =\n\npβ(x)qβ(y)−pβ(0)\nqβ(0)qβ(x)qβ(y),0<x≤y\nqβ(x)pβ(y)−pβ(0)\nqβ(0)qβ(x)qβ(y),0<y≤x.(1.52)\nAs we point out in Remark 1.1 there exists an α−permanental process\n{Xα(x),x∈[0,1]}with kernel\nv(x,y) =/tildewidevβ(x,y)+f/tildewidevβ,µ(y), (1.53)\nwhere\nf/tildewidevβ,µ(y) =/integraldisplay\nT/tildewidevβ(x,y)dµ(x), (1.54)\nand|µ|<∞.\nThe next theorem is an application of Theorem 1.3 where /tildewideu=v.\nTheorem 1.7 Let{Xα(t),t∈[0,1]}be anα−permanental process with kernel\nv(x,y). ThenXα(t)is continuous on [0,δ]and,\nlimsup\nt↓0Xα(t)\nρβ(0)tloglog1/t≤/parenleftbigg\n1+/integraldisplay∞\n0qβ(y)\nqβ(0)dµ(y)/parenrightbigg\n, a.s. (1.55)\nwhere,\nρβ(0) =p′\nβ(0)qβ(0)−q′\nβ(0)pβ(0). (1.56)\n2 Proof of Theorems 1.2\nWe make several observations that are used in the proof of The orem 1.2. Let\nUbe a non-singular n×nmatrix. Let U−1denote the inverse of UandUj,k\n10denote the elements of U−1. LetUfbe the (n+1)×(n+1) matrix\nUf=\n1f(1)... f (n)\n1U1,1+f(1)... U 1,n+f(n)\n............\n1Un,1+f(1)... Un,n++f(n)\n. (2.1)\nwherefis any function. One can check that\nU−1\nf=\n1+ρ−/summationtextn\ni=1f(i)Ui,1...−/summationtextn\ni=1f(i)Ui,n\n−/summationtextn\nj=1U1,jU1,1... U1,n\n............\n−/summationtextn\nj=1Un,jUn,1... Un,n\n,(2.2)\nwhere\nρ=n/summationdisplay\nj=1n/summationdisplay\ni=1f(i)Ui,j. (2.3)\nWe note the following remarkable property of Ufthat we use,\n(U−1\nf)i,i=Ui−1,i−1, i= 2,...,n+1. (2.4)\nLetξαbe a gamma random variable with probability density functio n,\nxα−1e−x\nΓ(α). (2.5)\nThe following Theorem is [7, Theorem 1.1]:\nTheorem 2.1 LetX= (X0,X1,...,Xn)be anα−permanental vector with\nnon-singular kernel K. LetA=K−1and denote the diagonal entries of A\nby(a0,a1,...,an). Then there exists a coupling between Xand ann−tuple\n(ξ(0)\nα,ξ(1)\nα,...,ξ(n)\nα)of independent identically distributed. copies of ξαsuch\nthat,\nX≥/parenleftig\na−1\n0ξ(0)\nα,a−1\n1ξ(1)\nα,...,a−1\nnξ(n)\nα/parenrightig\n, a.s. (2.6)\nProof of Theorem 1.2 Let{tj=θj}n\nj=1andletX= (X(∗),X(t1),...,X(tn)\nbe anα−permanental vector with non-singular kernel K\nK=\n1f(t1)... f (tn)\n1u(t1,t1)+f(t1)... u(t1,tn)+f(tn)\n............\n1u(tn,t1)+f(t1)... u(tn,tn)+f(tn)\n.(2.7)\n11wherefis given in (1.7). A=K−1is anMmatrix. Denote its diagonal\nentries by ( a0,a1,...,an). LetM={u(ti,tj)}n\ni,j=1. It follows from (2.4) that\nai=Mi,i, i= 1,...,n. (2.8)\nLetMbe the matrix with elements,\nMi,j=Mi,j\nM1/2\ni,iM1/2\nj,j, i,j = 1,...,n. (2.9)\nWe have,\nM=DMD, (2.10)\nwhereDis a diagonal matrix with diagonal elements {1/M1/2\ni,i}n\ni=1. Therefore,\nM−1=DM−1D. (2.11)\nLetA=M−1andA=M−1. We have,\nai=Mi,i=Ai,i=/parenleftbig\nM−1/parenrightbig\ni,i=/parenleftig\nDM−1D/parenrightig\ni,i=Ai,i\nMi,i, i= 1,...,n.\n(2.12)\nWe see by [7, Lemma 5.1] that,\nAi,i≤1\nMi,i−maxj/negationslash=iMi,j. (2.13)\nNote that it follows from the symmetry of Yand [3, Lemma 3.4.3] that when\ni/ne}ationslash=j,\nMi,j≤M1/2\ni,i\nM1/2\nj,j∧M1/2\nj,j\nM1/2\ni,i≤θ. (2.14)\nTherefore, since Mi,i= 1,\nAi,i≤1\n1−θ, (2.15)\nand by (2.12),\nai≤1+2θ\nMi,i=1+2θ\nu(θi,θi), i= 1,...,n. (2.16)\nIt follows from Theorem 2.1 that\n(Xt1,Xt2,···,Xtn)≥/parenleftig\na−1\n1ξ(1)\nα,a−1\n2ξ(2)\nα,...,a−1\nnξ(n)\nα,/parenrightig\n(2.17)\n12so that\n(a1Xt1,a2Xt2,···,anXtn)≥/parenleftig\nξ(1)\nα,ξ(2)\nα,...,ξ(n)\nα/parenrightig\n. (2.18)\nBy (2.16) we see that,\n/parenleftbiggXt1\nu(t1,t1),Xt2\nu(t2,t2),···,Xtn\nu(tn,tn)/parenrightbigg\n≥1\n1+2θ/parenleftig\nξ(1)\nα,ξ(2)\nα,...,ξ(n)\nα/parenrightig\n.(2.19)\nConsequently,\nmax\n2≤i≤nXθi\nu(θi,θi)logi≥1\n1+2θmax\n2≤i≤nξ(i)\nα\nlogi, a.s. (2.20)\nIt now follows from [4, Lemma 7.2] that for any αandǫ >0 there exists an\nl=l(ǫ) such that,\nP/parenleftbigg\nmax\nl≤i≤nXθi\nu(θi,θi)logi>1−ǫ\n1+2θ/parenrightbigg\n≥1−l+1\nn+1. (2.21)\nNow let{Zt,t∈S}be anα−permanental process with kernel /tildewideu(x,y) in (1.8).\nIt follows from (2.21) that,\nP/parenleftbigg\nmax\nl≤i≤nZθi\nu(θi,θi)logi>1−ǫ\n1+2θ/parenrightbigg\n≥1−l+1\nn+1. (2.22)\nSince we can take θarbitrarily small we get (1.9).\nThe next lemma breaks down the condition in (1.12) and is used in every-\nthing that follows to show that (1.12) is satisfied.\nLemma 2.1 Let/tildewideube as given in (1.8) and assume that usymmetric. Let /tildewideσ\nbe as given in (1.11). If\nu(s,s)+u(t,t)−2u(s,t)≤Cu(|t−s|,|t−s|), (2.23)\nand/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle≤Cu(|t−s|,|t−s|) (2.24)\nfor alls,t∈[0,δ], then\n/tildewideσ2(s,t)≤2Cu(|t−s|,|t−s|), (2.25)\nfor alls,t∈[0,δ].\n13Proof Consider/tildewideσ2(s,t). Using the fact that u(s,t) is symmetric we can\nwrite this as,\n/tildewideσ2(s,t) =/tildewideu(s,s)+/tildewideu(t,t)−/tildewideu(s,t)−/tildewideu(t,s) (2.26)\n+/parenleftig\n/tildewideu1/2(s,t)−/tildewideu1/2(t,s)/parenrightig2\n=u(s,s)+u(t,t)−2u(s,t)+/parenleftig\n/tildewideu1/2(s,t)−/tildewideu1/2(t,s)/parenrightig2\n.\nOnce again, using the fact that u(s,t) is symmetric we have,\n/parenleftig\n/tildewideu1/2(s,t)−/tildewideu1/2(t,s)/parenrightig2\n≤/vextendsingle/vextendsingle/tildewideu(s,t)−/tildewideu(t,s)/vextendsingle/vextendsingle≤/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle.(2.27)\nNow using (2.23) and (2.24) to bound the last line of (2.26) we get (2.25).\nWe use Lemma 2.1 to get a more usable version of Theorem 1.3.\nTheorem 2.2 LetXα={Xα(t),t∈[0,1]}be anα−permanental process with\nkernel{/tildewideu(s,t),s,t∈[0,1]}where/tildewideu(0,0) = 0. Ifu(v,v)is regularly varying at\nzero with positive index and (2.23) and (2.24) hold then\nlimsup\nt↓0Xα(t)\n(u(t,t)+f(t))loglog1/t≤1,a.s. (2.28)\nProofIfu(v,v) is regularly varying at zero with positive index it is asymp -\ntotic to an increasing function at zero. Denote this functio n by/hatwideϕ2(v). It\nfollows from (2.25) that\n/tildewideσ2(s,t)≤C′/hatwideϕ(|t−s|), (2.29)\nand, since/hatwideϕ2(v) =O(u(v,v)) at 0, we have /hatwideϕ2(v) =O(/tildewideu(v,v)) at zero. Con-\nsequently, (2.28) follows from (1.13).\nWe proceed to prove Theorems 1.4–1.7 by establishing (2.23) and (2.24).\n3 Proof of Theorem 1.4\nBy definition,\nΦ(s,s)+Φ(t,t)−2Φ(s,t) = Φ(|t−s|,|t−s|) = (σ0)2(t−s),(3.1)\n14first for all s,t∈R1−{0}. Since Φ(s,t) is continuous for all s,t∈R1we take\nit to be defined and continuous on R1. This gives (2.23).\nWe have the following remarkable regularity property of ( σ0)2that is\nproved at the end of this section.\nLemma 3.1\n|(σ0)2(y)−(σ0)2(x−y)| ≤(σ0)2(x), x,y ∈R1, (3.2)\nUsing this we show that,\n|fΦ,µ(s)−fΦ,µ(t)| ≤(σ0)2(t−s)|µ|, s,t∈R1. (3.3)\nIt follows from Lemma 3.1 that\n|(σ0)2(s−v)−(σ0)2(t−v)| ≤(σ0)2(t−s). (3.4)\nTherefore,\n|fΦ,µ(s)−fΦ,µ(t)| ≤/integraldisplay\nT|Φ(s,v)−Φ(t,v)|dµ(v) (3.5)\n≤1\n2/integraldisplay\nT/parenleftbig\n|(σ0)2(t)−(σ0)2(s)|+|(σ0)2(t−v)−(σ0)2(s−v)|/parenrightbig\ndµ(v)\n≤1\n2/integraldisplay\nT(σ0)2(t−s)dµ(v) = (σ0)2(t−s)|µ|.\nThe relationships in (3.1) and (3.3) and Theorem 2.2 show tha t\nlimsup\nt↓0Xα(t)\n((σ0)2(t)+fΦ,µ(t))loglog1/t≤1, a.s. (3.6)\nWe now get an estimate in the form of (1.18) so that we can write (3.6) in\nthe form of (1.19). Using Lemma 3.1 we have,\nfΦ,µ(t) =/integraldisplay\nTΦ(t,v)dµ(v) =1\n2/integraldisplay\nT/parenleftbig\n(σ0)2(t)+(σ0)2(v)−(σ0)2(t−v)/parenrightbig\ndµ(v)\n=(σ0)2(t)\n2|µ|+1\n2/integraldisplay\nT/parenleftbig\n(σ0)2(v)−(σ0)2(t−v)/parenrightbig\ndµ(v). (3.7)\n(Here we use the fact that ( σ0)2is an even function and we extend it to T.)\nConsequently\nfΦ,µ(t)\n(σ0)2(t)=|µ|\n2+1\n2/integraldisplay\nT/parenleftbig\n(σ0)2(v)−(σ0)2(v−t)/parenrightbig\n(σ0)2(t)dµ(v).(3.8)\n15By (3.4)\n|(σ0)2(v)−(σ0)2(v−t)|\n(σ0)2(t)≤1. (3.9)\nTherefore, using the Dominated Convergence Theorem we see t hat,\nlim\nt↓0fΦ,µ(t)\n(σ0)2(t)=|µ|\n2+1\n2/integraldisplay\nTlim\nt↓0/parenleftbig\n(σ0)2(v)−(σ0)2(v−t)/parenrightbig\n(σ0)2(t)dµ(v).(3.10)\nNote that,\nlim\nt↓0(σ0)2(v)−(σ0)2(v−t)\n(σ0)2(t)= lim\nt↓0(σ0)2(v)−(σ0)2(v−t)\ntt\n(σ0)2(t)\n= ((σ0)2(v))′lim\nt↓0t\n(σ0)2(t). (3.11)\nConsequently, the integral in (3.10) is equal to\n1\n2/integraldisplay\nT((σ0)2(v))′lim\nt↓0t\n(σ0)2(t)dµ(v). (3.12)\nUsing this and (3.10) we get (1.28) when (1.27) holds and (1.3 0) when (3.19)\nholds.\nProof of Lemma 3.1 Since Φ(x,y) is a continuous potential density of a\nstrongly symmetric Borel right process with state space T=R1− {0}it\nfollows from [3, Lemma 3.4.3] that\nΦ(x,y)≤Φ(x,x)∧Φ(y,y) = (σ0)2(x)∧(σ0)2(y), x,y∈R1−{0}.(3.13)\nTherefore, it follows from this and (1.22) that,\n(σ0)2(x)+(σ0)2(y)−(σ0)2(x−y)≤2(σ0)2(x), x,y∈R1−{0},(3.14)\nSinceσ0is continuous on R1we have,\n(σ0)2(x)+(σ0)2(y)−(σ0)2(x−y)≤2(σ0)2(x), x,y∈R1.(3.15)\nOr equivalently,\n(σ0)2(y)−(σ0)2(x−y)≤(σ0)2(x), x,y ∈R1.(3.16)\nFurthermore, since Φ( x,y)≥0 it follows from (1.22) that\n(σ0)2(x−y)−(σ0)2(y)≤(σ0)2(x). (3.17)\nCombining this with (3.16) we have (3.2).\n16Remark 3.1 It is interesting to note that since (( σβ)2)′(t) is an odd function\nwheneverµis a symmetric measure the integral in (1.30) is equal to 0 and we\nget\nlimsup\nt↓0Xα(t)\nCtloglog1/t≤1+|µ|\n2, a.s. (3.18)\nExample 3.1 Suppose that µis concentrated on the negative half line and\n(σ0)2(t) =C|t|. (3.19)\nThen\n1\n2C/integraldisplay∞\n−∞((σ0)2(v))′dµ(v) =−|µ|\n2. (3.20)\nTherefore, it follows from (1.30) and Theorem 1.2 that,\nlimsup\nt↓0Xα(t)\nCtloglog1/t= 1, a.s. (3.21)\n4 Proof of Theorem 1.5\nWe have,\n(σβ)2(x−y)) =uβ(x,x)+uβ(y,y)−2uβ(x,y) (4.1)\n= 2/parenleftig\nuβ(0)−uβ(x−y)/parenrightig\n.\nDefine,\n(σβ\nv)2(x,y)) =vβ(x,x)+vβ(y,y)−2vβ(x,y). (4.2)\nUsing (1.36) we see that,\n(σβ\nv)2(x,y)≤(σβ)2(x−y). (4.3)\nIn addition,\nvβ(x,x) =uβ(0)−(uβ(x))2\nuβ(0)=(uβ(0))2−(uβ(x))2\nuβ(0)(4.4)\n=(σβ)2(x)\n2uβ(0)/parenleftig\nuβ(0)+uβ(x)/parenrightig\n,\nso that,\nvβ(x,x)∼(σβ)2(x),asx↓0. (4.5)\n17It follows from (4.3) and (4.4) that,\n(σβ\nv)2(x,y)≤Cvβ(x−y,x−y),for allx,y∈[0,δ].(4.6)\nThis gives (2.23) for this example.\nToobtain (2.24) weusefollowing lemma whichisgiven in[3, L emma7.4.2]:\nLemma 4.1 Let{G(t),t∈R1}be a mean zero stationary Gaussian process\nwith continuous covariance u(s,t) =u(t−s), that is the potential density of a\nstrongly symmetric Borel right process. Set,\nσ2(t) =E(G(t)−G(0))2= 2(u(0)−u(t)). (4.7)\nThen,\n|σ2(t)−σ2(s)| ≤2u(0)\nu(t)+u(s)σ2(t−s). (4.8)\nSince{uβ(x,y),x,y∈R1}is the potential density of a strongly symmetric\nBorel right process with state pace R1it follows from [3, Lemma 3.3.3] that it\nis positive definite and therefore it is the covariance of a st ationary Gaussian\nprocess. Applying (4.8) to uβandσβwe have,\n/vextendsingle/vextendsingle/vextendsingle(σβ)2(t)−(σβ)2(s)/vextendsingle/vextendsingle/vextendsingle≤2uβ(0)\nuβ(t)+uβ(s)(σβ)2(t−s). (4.9)\nThe potential of Zβis given in (1.35). We take µ({0}) = 0 so that potential\nofZ′\nβis\nfvβ,µ(y) =/integraldisplay∞\n−∞vβ(z,y)dµ(z). (4.10)\nConsequently,\n/vextendsingle/vextendsinglefvβ,µ(x)−fvβ,µ(y)/vextendsingle/vextendsingle≤/vextendsingle/vextendsinglefuβ,µ(x)−fuβ,µ(y)/vextendsingle/vextendsingle+fuβ,µ(0)\n2uβ(0)/vextendsingle/vextendsingle(σβ)2(x)−(σβ)2(y)/vextendsingle/vextendsingle.\n(4.11)\nIt follows from (4.9) that,\nfuβ,µ(0)\n2uβ(0)/vextendsingle/vextendsingle(σβ)2(x)−(σβ)2(y)/vextendsingle/vextendsingle≤fuβ,µ(0)\n(uβ(x)+uβ(y))(σβ)2(x−y).(4.12)\nSinceuβ(x) is continuous, uβ(x)>0, for allxsufficiently small. Therefore,\nfor ∆ sufficiently small when x,yin [0,∆] we have,\nfuβ,µ(0)\n2uβ(0)/vextendsingle/vextendsingle(σβ)2(x)−(σβ)2(y)/vextendsingle/vextendsingle≤C∆(σβ)2(x−y), (4.13)\n18for some finite constant C∆.\nWe now consider the first term after the equal sign in (4.11). W e have,\n|fuβ,µ(x)−fuβ,µ(y)| (4.14)\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n−∞uβ(z−x)dµ(z)−/integraldisplay∞\n−∞uβ(z−y)dµ(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n−∞/parenleftig/parenleftig\nuβ(0)−uβ(z−y)/parenrightig\n−/parenleftig\nuβ(0)−uβ(z−x)/parenrightig/parenrightig\ndµ(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤1\n2/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle(σβ)2(z−y)−(σβ)2(z−x)/vextendsingle/vextendsingle/vextendsingledµ(z).\nIt follows from (4.9) that for ∆ sufficiently small,\n/parenleftbigg/integraldisplay∆\n0+/integraldisplay0\n−∆/parenrightbigg/parenleftig\n(σβ)2(z−y)−(σβ)2(z−x)/parenrightig\ndµ(z)≤C′\n∆(σβ)2(x−y),\n(4.15)\nfor some finite constant C′\n∆.\nIn addition\nH∆(x−y) :=/parenleftbigg/integraldisplay−∆\n−∞+/integraldisplay∞\n∆/parenrightbigg/parenleftig\n(σβ)2(z−y)−(σβ)2(z−x)/parenrightig\ndµ(z)\n≤max\n|z|≥∆(σβ)2(z)′µ({(−∞,−∆)∪µ(∆,∞)})|x−y|, (4.16)\nand,\nH∆(x−y)\n(σβ)2(x−y)≤max\n|z|≥∆(σβ)2(z)′µ({(−∞,−∆)∪µ(∆,∞)})|x−y|\n(σβ)2(x−y).(4.17)\nUsing this (4.15), (4.13) and (1.31), we get,\nlim\nx,y↓0/vextendsingle/vextendsinglefvβ,µ(x)−fvβ,µ(y)/vextendsingle/vextendsingle\n(σβ)2(x−y)≤C,\nfor some constant C. Together with (4.5) this gives us the version of (2.24)\nwe need for this theorem.\n19We now get an estimate for fvβ,µat 0.\nfvβ,µ(x) =/integraldisplay∞\n−∞uβ(x,y)dµ(y)−uβ(0,x)\nuβ(0,0)/integraldisplay∞\n−∞uβ(0,y)dµ(y) (4.18)\n=fuβ,µ(x)−uβ(0,x)fuβ,µ(0)\nuβ(0,0)\n=fuβ,µ(x)−fuβ,µ(0)+fuβ,µ(0)\nuβ(0,0)/parenleftig\nuβ(0,0)−uβ(0,x)/parenrightig\n=fuβ,µ(x)−fuβ,µ(0)+(σβ)2(x)fuβ,µ(0)\n2uβ(0,0).\nIn addition,\nfuβ,µ(x)−fuβ,µ(0) (4.19)\n=/integraldisplay∞\n−∞uβ(z−x)dµ(z)−/integraldisplay∞\n−∞uβ(z)dµ(z)\n=/integraldisplay∞\n−∞/parenleftig\nuβ(0)−uβ(z)/parenrightig\n−/parenleftig\nuβ(0)−uβ(z−x)/parenrightig\ndµ(z)\n=1\n2/integraldisplay∞\n−∞/parenleftig\n(σβ)2(z)−(σβ)2(z−x)/parenrightig\ndµ(z).\nForx∈[0,∆] it follows from (4.9) that,\n2I1(x) :=/integraldisplay∆\n−∆/parenleftig\n(σβ)2(z)−(σβ)2(z−x)/parenrightig\ndµ(z) (4.20)\n≤(σβ)2(x)/integraldisplay∆\n−∆2uβ(0)\nuβ(z−x)+uβ(z)dµ(z).\nConsequently for ∆ sufficiently small,\nI1(x)\n(σβ)2(x)≤C∆µ([−∆,∆]), (4.21)\nfor someC∆<∞with lim ∆→0C∆= 2.\nNow consider\n2I2(x) :=/parenleftbigg/integraldisplay∞\n∆+/integraldisplay−∆\n−∞/parenrightbigg/parenleftig\n(σβ)2(z)−(σβ)2(z−x)/parenrightig\ndµ(z).(4.22)\nNote that for x∈(0,δ) where 0<δ≤∆,\n(σβ)2(z)−(σβ)2(z−x) = ((σβ)2)′(z∗(z,x))x (4.23)\n20for somez∗(z,x)∈[z−x,z]. Therefore,\n|I2(x)|\n(σβ)2(x)=x\n(σβ)2(x)1\n2/parenleftbigg/integraldisplay∞\n∆+/integraldisplay−∆\n−∞/parenrightbigg\n((σβ)2)′(z∗(z,x))dµ(z).(4.24)\nSince|(σβ)2)′(z∗(z,,x))|is bounded on the range of integration we can use\nthe Dominated Convergence Theorem to get,\nlim\nx↓0|I2(x)|\n(σβ)2(x)= lim\nx↓0x\n(σβ)2(x)1\n2/parenleftbigg/integraldisplay∞\n∆+/integraldisplay−∆\n−∞/parenrightbigg\n((σβ)2)′(z)dµ(z).(4.25)\nWhen (1.40) holds this is equal to 0. Using this (4.18) and (4. 21) we see\nthat in this case,\nlim\nx↓0fvβ,µ(x)\n(σβ)2(x)≤C∆µ([−∆,∆])+fuβ,µ(0)\n2uβ(0,0). (4.26)\nSince this holds for all ∆ >0 and lim ∆→0C∆= 2 we get (1.41).\nWhen (1.42) holds the right hand side of (4.25) is equal to,\n1\n2C/parenleftbigg/integraldisplay∞\n∆+/integraldisplay−∆\n−∞/parenrightbigg\n((σβ)2)′(z)dµ(z), (4.27)\nwhich in this case gives,\nlim\nx↓0fvβ,µ(x)\n(σβ)2(x)≤1\n2C/integraldisplay∞\n−∞((σβ)2)′(z)dµ(z)+fuβ,µ(0)\n2uβ(0,0).(4.28)\nUsing this we get (1.43).\nRemark 4.1 When (1.42) holds the integral in (4.28) is finite, since it, a nd\nthehypothesisthat( σβ)2(t)∈C1(T)and|(σβ)2(t)′|isboundedon( −∞,−∆)∪\n(∆,∞) for all ∆>0 implies that |((σβ)2)′(t)|is bounded on R1.\nIt is interesting to note that since (( σβ)2)′(t) is an odd function whenever\nµis a symmetric measure the integral in (4.28) is equal to 0 and we get\nlimsup\nt↓0Xα(t)\nCtloglog1/t≤1+fuβ,µ(0)\n2uβ(0), a.s. (4.29)\nExample 4.1 In [6, 1.25] we point out that for β >0,\nuβ\nγ(x−y) :=1\n2π/integraldisplay∞\n−∞cosλ(x−y)\nβ+γλ2dλ=1\n2e−(β/γ)1/2|x−y|\n(βγ)1/2, x,y ∈R1.\n(4.30)\n21In this example we simply take β=Kandγ= 1/Kso that,\nuβ\nγ(x−y) =1\n2exp−K|x−y|. (4.31)\nand\n(σβ\nγ)2(x) = 1−exp−K|x|. (4.32)\nSuppose that µis symmetric. Then\nfuβ\nγ,µ(0) =1\n2/integraldisplay∞\n−∞e−K|y|dµ(y) =/integraldisplay∞\n0e−Kydµ(y), (4.33)\nso that\nfuβ\nγ,µ(0)\n2uβ\nγ(0)=/integraldisplay∞\n0e−Kydµ(y). (4.34)\nIn addition,\nlim\nx↓0(σβ\nγ)2(x)\nx=K. (4.35)\nTherefore, it follows from (4.29) that,\nlimsup\nt↓0Xα,β(t)\nKtloglog1/t≤/parenleftbigg\n1+/integraldisplay∞\n0e−Kydµ(y)/parenrightbigg\n, a.s. (4.36)\nSuppose that µis concentrated on the negative half line. We evaluate the\ntwo terms on the right in (4.28) for uβ\nγin (4.31). Note that in this case Cin\n(4.28) is equal to K. We have,\nfuβ\nγ,µ(0) =/integraldisplay0\n−∞uβ\nγ(−y)dµ(y)\n=1\n2/integraldisplay0\n−∞exp−K|−y|dµ(y) =1\n2/integraldisplay0\n−∞expKydµ(y).(4.37)\nSince 2uβ\nγ(0) = 1 we get,\nfuβ\nγ,µ(0)\n2uβ\nγ(0)=1\n2/integraldisplay0\n−∞expKydµ(y). (4.38)\nIn addition,\n1\n2K/integraldisplay0\n−∞((σβ)2)′(z)dµ(z) =1\n2K/integraldisplay0\n−∞(1−expKz)′dµ(z) (4.39)\n=−1\n2/integraldisplay0\n−∞expKzdµ(z).\n22Therefore, it follows from (4.29) and Theorem 1.2 that,\nlimsup\nt↓0Xα,β(t)\nKtloglog1/t= 1, a.s. (4.40)\n5 Proof of Theorem 1.6\nIt is easy to check that,\nuT0(x,x)+uT0(y,y)−2uT0(x,y) =|s(x)−s(y)|, (5.1)\nand that\n|uT0(x,v)−uT0(y,v)| ≤ |s(x)−s(y)|. (5.2)\nTherefore,\n|fuT0,µ(x)−fuT0,µ(y)| ≤ |s(x)−s(y)| |µ|. (5.3)\nThe condition that s∈C2(0,∞) shows that the criteria in Lemma 2.1 are\nsatisfied. The next lemma completes the proof of this theorem .\nLemma 5.1\nfuT0,µ(x)∼s(x)|µ|,asx↓0. (5.4)\nProofWe have,\nfuT0,µ(x) =s(x)/integraldisplay∞\nxdµ(y)+/integraldisplayx\n0s(y)dµ(y). (5.5)\nTherefore,\nfuT0,µ(x)\ns(x)=/integraldisplay∞\nxdµ(y)+/integraldisplayx\n0s(y)\ns(x)dµ(y). (5.6)\nSinces(y) is increasing and µis supported on (0 ,∞),\n/integraldisplayx\n0s(y)\ns(x)dµ(y)≤/integraldisplayx\n0dµ(y), (5.7)\nUsing this in (5.6) we get (5.4).\nThe proof is completed as in the proofs of Theorems 1.5 and 1.6 .\n236 Proof of Theorem 1.7\nLet\n(/tildewideσβ)2(x,y)) =/tildewideuβ(x,x)+/tildewideuβ(y,y)−2/tildewideuβ(x,y), (6.1)\nand define,\n(/tildewideσβ\nv)2(x,y)) =/tildewidevβ(x,x)+/tildewidevβ(y,y)−2/tildewidevβ(x,y). (6.2)\nIt follows from (1.52) that\n/tildewidevβ(x,y) =/tildewideuβ(x,y)−pβ(0)\nqβ(0)qβ(x)qβ(y). (6.3)\nConsequently,\n(/tildewideσβ\nv)2(x,y) = (/tildewideσβ)2(x,y)−pβ(0)\nqβ(0)(qβ(x)−qβ(y))2. (6.4)\nWe show in [6, Lemma 4.9] that\n(/tildewideσβ)2(x,y)∼ρβ(0)|x−y|,as|x−y| →0, (6.5)\nwhere,\nρβ(0) =p′\nβ(0)qβ(0)−q′\nβ(0)pβ(0). (6.6)\nSince,\n|qβ(x)−qβ(y)| ∼q′\nβ(0)|x−y|,asx,y↓0, (6.7)\nit follows by (6.4) and (6.5) that,\n(/tildewideσβ\nv)2(x,y)∼ρβ(0)|x−y|,asx,y↓0. (6.8)\nBy (1.52)\n/tildewidevβ(x,x) =/parenleftbigg\npβ(x)−pβ(0)\nqβ(0)qβ(x)/parenrightbigg\nqβ(x) (6.9)\nSince,\npβ(x) =pβ(0)+p′\nβ(0)x+o(x) andqβ(x) =qβ(0)+q′\nβ(0)x+o(x),(6.10)\nasx↓0, we see that,\n/tildewidevβ(x,x) =/parenleftbiggpβ(x)\npβ(0)−qβ(x)\nqβ(0)/parenrightbigg\npβ(0)qβ(x)\n=/parenleftigg\np′\nβ(0)\npβ(0)−q′\nβ(0)\nqβ(0)/parenrightigg\npβ(0)qβ(x)x+o(x)\n=ρβ(0)x+o(x),asx↓0. (6.11)\n24Consequently,\n/tildewidevβ(x−y,x−y)∼ρβ(0)|x−y|,asx,y↓0, (6.12)\nand, by (6.8)\n(/tildewideσβ\nv)2(x,y)≤C/tildewidevβ(x−y,x−y),asx,y↓0, (6.13)\nfor some constant C. This is the condition in (2.23).\nLet\nf/tildewideuβ,µ(x) =/integraldisplay∞\n0/tildewideuβ(x,y)dµ(y) (6.14)\nand\nf/tildewidevβ,µ(x) =/integraldisplay∞\n0/tildewidevβ(x,y)dµ(y). (6.15)\nUsing (6.3) we see that\nf/tildewidevβ,µ(x) =f/tildewideuβ,µ(x)−pβ(0)\nqβ(0)qβ(x)/integraldisplay∞\n0qβ(y)dµ(y). (6.16)\nNote that,\nf/tildewideuβ,µ(x) =pβ(x)/integraldisplay∞\nxqβ(y)dµ(y)+qβ(x)/integraldisplayx\n0pβ(y)dµ(y) (6.17)\nand,\nf′\n/tildewideuβ,µ(x) =p′\nβ(x)/integraldisplay∞\nxqβ(y)dµ(y)+q′\nβ(x)/integraldisplayy\n0pβ(y)dµ(y).(6.18)\nThis shows that f/tildewideuβ,µ∈C1([0,∞)) and wesee from(6.16) that for x,y∈[0,δ],\n/vextendsingle/vextendsinglef/tildewidevβ,µ(x)−f/tildewidevβ,µ(y)/vextendsingle/vextendsingle≤sup\ns∈[0,δ]f′\n/tildewideuβ,µ(s)|x−y|+pβ(0)\nqβ(0)/vextendsingle/vextendsingleqβ(x)−qβ(y)/vextendsingle/vextendsingle/integraldisplay∞\n0qβ(z)dµ(z).\n(6.19)\nUsing (6.7) we see that,\n/vextendsingle/vextendsinglef/tildewidevβ,µ(x)−f/tildewidevβ,µ(y)/vextendsingle/vextendsingle=O(|x−y|). (6.20)\nThis is the condition in (2.24).\nWe now consider the permanental process with kernel\nvβ(t,t) =/tildewidevβ(t,t)+f/tildewidevβ,µ(t). (6.21)\nThe behavior of /tildewidevβ(t,t) at 0 is given in (6.12). We now consider the behavior\noff/tildewidevβ,µat 0.\n25Lemma 6.1\nf/tildewidevβ,µ(x) =xρβ(0)/integraldisplay∞\n0qβ(y)\nqβ(0)dµ(y)+o(x),asx↓0.(6.22)\nProofRewrite (1.52) as,\n/tildewidevβ(x,y) =\n\n/parenleftbiggpβ(x)\npβ(0)−qβ(x)\nqβ(0)/parenrightbigg\npβ(0)qβ(y),0<x≤y\n/parenleftbiggpβ(y)\npβ(0)−qβ(y)\nqβ(0)/parenrightbigg\npβ(0)qβ(x),0<y≤x.(6.23)\nWe have,\nf/tildewidevβ,µ(x) =/integraldisplay∞\nx/tildewidevβ(x,y)dµ(y)+/integraldisplayx\n0/tildewidevβ(x,y)dµ(y) (6.24)\n=/parenleftbiggpβ(x)\npβ(0)−qβ(x)\nqβ(0)/parenrightbigg\npβ(0)/integraldisplay∞\nxqβ(y)dµ(y)\n+pβ(0)qβ(x)/integraldisplayx\n0/parenleftbiggpβ(y)\npβ(0)−qβ(y)\nqβ(0)/parenrightbigg\ndµ(y)\n:=I1+I2.\nIt follows as in (6.11) that,\npβ(x)\npβ(0)−qβ(x)\nqβ(0)=/parenleftigg\np′\nβ(0)\npβ(0)−q′\nβ(0)\nqβ(0)/parenrightigg\nx+o(x),asx↓0 (6.25)\n=ρβ(0)x\npβ(0)qβ(0)+o(x),asx↓0.\nUsing this and (6.24) we see that,\nI1(x) =xρβ(0)/integraldisplay∞\n0qβ(y)\nqβ(0)dµ(y)+o(x),asx↓0. (6.26)\nWe complete the proof by showing that I2(x) =o(x) asx↓0. Using (6.25)\nwe have,\nI2(x) =qβ(0)pβ(x)/integraldisplayx\n0/parenleftbiggpβ(y)\npβ(0)−qβ(y)\nqβ(0)/parenrightbigg\ndµ(y) (6.27)\n=qβ(0)pβ(x)/integraldisplayx\n0/parenleftbiggρβ(0)y\npβ(0)qβ(0)+o(y)/parenrightbigg\ndµ(y)\n=pβ(x)ρβ(0)\npβ(0)/integraldisplayx\n0ydµ(y)+qβ(0)p(x)/integraldisplayx\n0o(y)dµ(y).\n26The line above is,\n≤p(x)ρβ(0)x\npβ(0)/integraldisplayx\n0dµ(y)+qβ(0)pβ(x)o(x)/integraldisplayx\n0dµ(y) =o(x),asx↓0,\n(6.28)\nsinceµis supported on (0 ,∞).\nThe proof is completed as in the proofs of Theorems 1.5 and 1.6 .\nExample 6.1 In (1.50) we can take,\npβ(x) =eKx,andqβ(y) =1\n2e−Ky; (6.29)\nsee [6, Example 9.3], in which case,\n/tildewideuβ(x,y) =1\n2e−K|x−y|. (6.30)\nIt is easy to see that in this case,\nρβ(0) =K. (6.31)\nConsequently, (1.55) gives,\nlimsup\nt↓0Xα(t)\nKtloglog1/t=/parenleftbigg\n1+/integraldisplay∞\n0e−Kydµ(y)/parenrightbigg\n, a.s., (6.32)\nas in (4.36).\n7 Additional observations\nRemark 7.1 In Theorems 1.4 and 1.5 we require that ( σ0)2and (σβ)2are\nregularly varying at 0 with positive index. We show in [6, Sec tion 8] that these\nconditions are realized in many cases.\nRemark 7.2 We give a simple criteria for the conditions that ( σβ)2(t)∈\nC1(0,∞) and|(σβ)2(x)′|is bounded on [ δ,∞) for allδ>0.\nLemma 7.1 For allβ≥0, if\nsup\n0≤λ<∞λ|ψ′(λ)|\nβ+ψ(λ)<∞, (7.1)\n27(σβ)2(x)∈C1(0,∞)and\n|(σβ)2(x)′| ≤C(σβ)2(x)\nx,∀x∈(0,∞), (7.2)\nfor some constant C.\nProof The proof is a modification of [6, Lemma 8.4] which gives sharp er\nestimates for ( σ0)′(t) under more restrictive conditions on ψ(λ) . Whenx/ne}ationslash= 0,\n(σβ)2(x) =4\nπ/integraldisplay∞\n0sin2λx/2\nβ+ψ(λ)dλ=4\nπx/integraldisplay∞\n0sin2t/2\nβ+ψ(t/x)dt,(7.3)\nand,\n(σβ)2(x)′=−(σβ)2(x)\nx+4\nπxd\ndx/parenleftbigg/integraldisplay∞\n0sin2t/21\nβ+ψ(t/x)dt/parenrightbigg\n:=−(σβ)2(x)\nx+νβ(x). (7.4)\nWe have,\nνβ(x) =4\nπx2/integraldisplay∞\n0sin2t/2\nβ+ψ(t/x)(t/x)ψ′(t/x)\nβ+ψ(t/x)dt. (7.5)\n(See [6, Lemma 8.3] for the justification of differentiating un der the integral\nsign.) Consequently, when\nC:= sup\n0≤λ<∞/vextendsingle/vextendsingle/vextendsingle/vextendsingleλψ′(λ)\nβ+ψ(λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle<∞, (7.6)\nνβ(x)≤C(σβ)2(x)\nx. (7.7)\nThe inequalities in (7.2) follow from (7.4) and (7.7).\nNote that the condition in (7.1) is satisfied for β >0 whenψ(λ) is a\nnormalized regularly varying function at infinity, i.e. for some 0<p≤2,\nψ(λ) =Cλpexp/parenleftbigg/integraldisplayλ\n1ǫ(u)\nudu/parenrightbigg\n, (7.8)\nwhere lim u→∞ǫ(u) = 0. When β= 0 wecan also require that ψ(λ) is a\nnormalized regularly varying function at 0.\n28Remark 7.3 There are kernels for which the upper and lower bounds of the\nlocal moduli are the same.\nTheorem 7.1 LetYbe a strongly symmetric transient Borel right process\nwith state space T=R1− {0}or(0,∞)and continuous strictly positive po-\ntential densities u(x,y)with respect to some σ–finite measure monT, where\nlimt↓0u(t,t) = 0.\nLet{Xα,1(t),t∈[0,1]}and{Xα,2(t),t∈[0,1]}be anα−permanental pro-\ncesses with kernels\nv1(s,t) =u(s,t),andv2(s,t) =u(s,t)+g(s)f(t),(7.9)\nrespectively, where fis given in (1.7) and\ng(s) =/integraldisplay\nTu(s,v)dν(v) (7.10)\nfor some positive finite measure ν. Ifu(v,v)is regularly varying at zero with\npositive index and\nu(s,s)+u(t,t)−2u(s,t)≤Cu(|t−s|,|t−s|). (7.11)\nthen\nlimsup\nt↓0Xα,1(t)\nu(t,t)loglog1/t= 1. (7.12)\nIf, in addition,\n/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle≤Cu(|t−s|,|t−s|) and/vextendsingle/vextendsingleg(s)−g(t)/vextendsingle/vextendsingle≤Cu(|t−s|,|t−s|) (7.13)\nfor some constant Cfor alls,t∈[0,δ], for some δ >0, then (7.12) holds for\nXα,2. (The existence of Xα,2follows from [6, Theorem 1.1].)\nProof of Theorem 7.1 When the kernel is v1(s,t) the lower bound follows\nfrom Theorem 1.2 and the upper bound follows from Theorem 2.2 .\nSuppose the kernel is v2(s,t). Consider/tildewideσ2(s,t). Using (7.9) and the fact\nthatu(s,t) is symmetric we can write this as,\n/tildewideσ2(s,t) =v2(s,s)+v2(t,t)−v2(s,t)−v2(t,s) (7.14)\n+/parenleftig\nv1/2\n2(s,t)−v1/2\n2(t,s)/parenrightig2\n=u(s,s)+u(t,t)−2u(s,t)+(f(s)−f(t))(g(s)−g(t))\n+/parenleftig\nv1/2\n2(s,t)−v1/2\n2(t,s)/parenrightig2\n.\n29Using (7.11) and (7.13) we see that,\nu(s,s)+u(t,t)−2u(s,t)+(f(s)−f(t))(g(s)−g(t)) (7.15)\n≤C′u(|t−s|,|t−s|),\nforsomeconstant C′, foralls,t∈[0,δ]. Usingthefactthat u(s,t)issymmetric\nwe have,\n/parenleftig\nv1/2\n2(s,t)−v1/2\n2(t,s)/parenrightig2\n≤/vextendsingle/vextendsinglev2(s,t)−v2(t,s)/vextendsingle/vextendsingle (7.16)\n=/vextendsingle/vextendsinglef(s)g(t)−f(t)g(s)/vextendsingle/vextendsingle=/vextendsingle/vextendsinglef(s)(g(t)−g(s))−(f(t)−f(s))g(s)/vextendsingle/vextendsingle\n≤f(s)/vextendsingle/vextendsingleg(s)−g(t)/vextendsingle/vextendsingle+g(s)/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle.\nConsequently,\nσ2(s,t)≤C′u(|t−s|,|t−s|)+2/parenleftbig\nf(s)/vextendsingle/vextendsingleg(s)−g(t)/vextendsingle/vextendsingle+g(t)/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle/parenrightbig\n,(7.17)\nand by (1.16),\nf(s)/vextendsingle/vextendsingleg(s)−g(t)/vextendsingle/vextendsingle+g(t)/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle (7.18)\n≤ |µ|u(s,s)/vextendsingle/vextendsingleg(s)−g(t)/vextendsingle/vextendsingle+|ν|u(t,t)/vextendsingle/vextendsinglef(s)−f(t)/vextendsingle/vextendsingle.\nIt now follows by a minor generalization of Theorem 2.2 that,\nlimsup\nt↓0Xα,2(t)\n(u(t,t)+C′u2(t,t))loglog1/t≤1,a.s.,(7.19)\nfor some constant C′. This implies that\nlimsup\nt↓0Xα,2(t)\nu(t,t)loglog1/t≤1,a.s. (7.20)\nThe lower bound for Xα,2follows from a minor modification of the proof\nof Theorem 1.2. Let Uf,gbe a non-singular n×nmatrix,\n\n1f(1)... f (n)\ng(1)U1,1+g(1)f(1)... U 1,n+g(1)f(n)\n............\ng(n)Un,1+g(n)f(1)... Un,n+g(n)f(n)\n. (7.21)\nwheregis any function.\n30LetU−1\nf,gdenote the inverse of Uf,gandUj,k\nf,gdenote the elements of U−1\nf,g.\nOne can check that,\nU−1\nf,g=\n1+ρ′−/summationtextn\ni=1f(i)Ui,1...−/summationtextn\ni=1f(i)Ui,n\n−/summationtextn\nj=1g(1)U1,jU1,1... U1,n\n............\n−/summationtextn\nj=1g(n)Un,jUn,1... Un,n\n,\n(7.22)\nwhere\nρ′=n/summationdisplay\nj=1n/summationdisplay\ni=1g(j)f(i)Ui,j. (7.23)\nAs in (2.4) we have\nUi,i\nf,g=Ui−1,i−1, i= 2,...,n+1. (7.24)\nThe proof now proceeds like the proof of Theorem 1.2.\nReferences\n[1] N. Eisenbaum and H. Kaspi, On permanental processes, Stochastic Pro-\ncesses and their Applications ,119, (2009), 1401-1415. 1\n[2] L. Breiman, Probability , Classics in Applied Mathematics, SIAM,\nPhiladelphia, (1992). 1.3\n[3] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes and\nLocal Times , Cambridge University Press, New York, (2006). 1, 1, 1.1,\n1.2, 1.2, 1.4, 2, 3, 4, 4\n[4] M. B. Marcus and J. Rosen, Sample path properties of perma nental pro-\ncesses,Electronic Journal of Probability , Volume 23 (2018), no. 58, 1-47.\n1, 1.1, 1, 2\n[5] M. B. Marcus and J. Rosen, Permanental processes with ker nels that\nare not equivalent to a symmetric matrix. High Dimensional P robability\nVIII, The Oaxaca Volume, Progress in Probability 74,(2019) , 305-320.\nBirkhauser, Boston. 1\n[6] M. B. Marcus and J. Rosen, Law of the iterated logarithm fo rk/2–\npermanental processes and the local times of related Markov processes,\nSubmitted. 1, 1.2, 1.2, 4.1, 6, 6.1, 7.1, 7.2, 7.2, 7.3\n31[7] M. B. Marcus and J. Rosen, Conditions for permanental pro cesses to be\nunbounded, ... 2, 2\n[8] Daniel Revuz and Marc Yor. Continuous martingales and Brownian mo-\ntion, volume 293. Springer-Verlag, Berlin, third edition, 1999 . 1.3\n32"    },    {        "title": "2402.07086v1.Coupling_phase_switching_with_generalized_Brewster_effect_for_tunable_optical_sensor_designs.pdf",        "content": "Coupling phase -switching with generalized Brewster effect \nfor tunable optical sensor design s \nDaniel T. Yimam *, Dennis van der Veen, Teodor Zaharia, Maria Loi,  and Bart J. Kooi * \nZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 A  \nGroningen, The Netherlands  \nd.t.yimam@rug.nl , b.j.kooi@rug.nl  \n \nAbstract  \nThe non-linear and tunable optical constants of phase -change materials associated with  \ntheir phase -switching have been utilized in reconfigurable optical devices.  For example, o ne \npossible application of phase -change thin films is for tunable perfect absorption designs , where  \np- polarized light reflectance vanishes at a specific incidence angle known as the Brewster \nangle. This work demonstrates  a generalized Brewster effect  (s- and p- polarized light \nabsorption)  for a multilayered heterostructure design  based on the strong interference effect . \nThe proposed design co mprises  a low -loss phase -change material , Sb2Se3, coated on a gold \nsubstrate. We experimentally and theoretically show the coexistence of vanishing reflectance \nvalues for both s - and p - polarized lights in the visible and near IR wavelength range at a single \nBrewster angle. Such vanishing reflectance val ues (points of darkness) are associated with  \nphase singularities with abrupt changes.  Moreover, we show that additional phase singularities  \ncan be rea lized by switching the active Sb 2Se3 layer between amorphous and crystalline \nstructur es, extending the functionality of our design. The realized phase singularities are \nsusceptible to small optical constant changes and can be utilized for ultra -sensing applications. \nAs a proof of concept, we demonstrate our design's CO 2 gas sensing capabilities , showing a \nlinearly dependent optical response with gas flow rate. We believe o ur lithography -free design \nwith multiple phase singularities, from the coexisting generalized Brewster effect and  from the \nadditional  structural switching , is highly promising  for tunable optical sens ing applications . \n \nKeywords   \nPhase singularity, Brewster effect , spectroscopic ellipsometry , gas sensor , optical sensor , phase \nchange materials , perfect absorber  Introduction  \nIn nanophotonics and optoelectronics thin films play a key role in advanced optical \ndevice designs . By tuning the physical and optical propertie s of thin films , control of light \ninteraction with optical devices  has produced attractive a pplications in many fields. One \nimportant parameter tuned for such application s is the perfect  absorption of the incidence \nlight.1,2 Applications in thin film solar cells and photovoltaics, photodetectors, sensors, and \nstructural colors  all benefited from smart optical device designs , achieving perfect light \nabsorption  within tunable spectral ranges .3 One way of achieving perfect absorption is through \nlight resonance with artificially constructed metal -dielectric -metal plasmonic nanostructures \nand antenna designs.3–5 However, the heavy nanofabrication tasks needed to realize the specific \nnanostructured surfaces create practical limitations . Alternatively, lithography -free smart \nheterostructure designs can be considered for achieving perfect absorption through resonant \nenhancement  in the Fabry Perot cavity6–8 or strong interference formalisms .1,9,10 One class of \nmaterial s attractive for multilayered perfect absorption designs is phase -change  materials \n(PCMs) . Integration of PCM  thin films in optical and optoelectrical devices has received great \ninterest in the past few years. The fast and reversible phase -switching and the associated optical \ncontrast between phases have huge promises for dynamically reconfigurable photonics and \nproduced interesting results, including perfect absorption.11,12 Phase -change -based perfect \nabsorber designs offer flexibility in absorption tuning from the amorphous/crystalline \nreversible phase -switching.13–17 \nFor optical systems with no transmitted light intensity (for bulk substrates , for \nexample), perfect light absorption is translated to no reflectance. Such systems are explained \nby a point of darkness where no reflection intensity is detected for a specific polarized light. \nHowever, realizing a point of darkness requires selecting  an appropriate wavelength range and \nlight incidence angle.  The point of darkness is highly useful  since the associated non -trivial \nbehavior of the light phase  can be utilized for v arious applications.18–21 In systems where the \npoint of darkness is achieved, the phase singularities show a high sensitivity to a relatively \nsmall refractive index change and thus can be used for sensing purposes.3,8,20,22,23 Although \nmost reported results are from nanostructured plasmonic  designs, the point of darkness can be \nrealized  as well  through multilayered heterostructure stacks.2 This is achieved by a well -known \nclassical Brewster effect , where the Fresnel reflection coefficient for p -polarized light \ndisappears at a specific incidence  angle.24 As a result of the vanished reflectance, phase \nsingularities can form at the Brewster angle. Such systems have been exploited  for sensing applications.8,25 However, the Brewster effect falls short for various material s and designs \nwhere the assumption of nonmagnetic, isotropic, and homogeneous medium is altered. A  more \ngeneralized Brewster effect is expected where vanishing reflectance will occur for both s- or \np- polarized light.26–29  \nIn this work, we experimentally and theoretically demonstrate the generalized Brewster \neffect for a multilayered heterostructure stack produced by pulsed laser deposition (PLD ). We \nshow that the strong interference effect can create perfect/strong light absorptions and \nassociated phase singularities for both s - and p - polarized lights  at a single Brewster angle . We \nconstructed multilayered stack s with near zero  reflection for both s - and p - polarizations \nrealized  at the same incidence angle (i.e. , θsB = θpB). The coexistence of vanishing reflectance  \nvalues  produc ed separate  phase singularities at the corresponding points of darkness . Our \nproposed heterostructure device consists of a thin lossy phase -change layer Sb2Se3 coated on a \nbulk gold substrate. Sb 2Se3 is one of the new classes of low -loss phase -change materials that \nrecently attracted research interest for multiple applications in programmable photonics and \ndisplay devices.30–33 We demonstrate  here, for the first time to the best of our knowledge, that \nthe phase singularities from the vanishing reflectance values coexist for a wide  range of  \nthickness values of the active Sb 2Se3 layer. As a result, we fabricated an optical device with \ntwo phase resonances at a single light incidence angle of 70 °. Our work demonstrates a n \ninnovative and  lithography -free multilayered heterostructure design,  where s - and p - polarized \nBrewster effects coexist at the same Brewster angle. Moreover, another set of phase \nsingularities has been realized from the  amorphous to crystalline  phase switching of the Sb2Se3 \nlayer, which is very promising for tunable sensing applications.  \nExperimental  \n Multilayered hete rostructures were produced by pulsed laser deposit ion (PLD). Sb 2Se3 \nphase -change thin films  with varying thicknesses w ere deposited on a gold subst rate. A 10 nm \nspacer LaAlO 3 (LAO ) layer was deposited between the gold and the active Sb2Se3 layer  \n(mainly to prevent potential  intermixing ), and a 10 nm thick  LAO capping layer  was deposited  \non top . Electron beam evaporation (Temescal FC2000) was initially used to coat a 100 nm gold \nlayer on a Si/SiO 2 substrate. The deposition  parameters for Sb2Se3 were initially calibrated such \nthat exact composition  transfer (Sb40% - 60%Se) is achieved with precise thickness control. A \n1 Jcm-1 laser fluence, 1 Hz repetition rate, and 0.12 mbar of Ar processing gas were used for \ndepositions .  All optical characterizations were performed by variable -angle spectroscopic \nellipsometry  (VASE, Woollam) . Spectroscopic measurements were collected for individual \nlayers where the optical constants of n and k of each layer in the multilayered hete rostructure \nwere extracted. For both as -deposited amorphous and crystalline phases of Sb2Se3, the ψ and \nΔ parameters have been collected for multiple incidence angles. Data fitting was performed \nusing the WVASE software , and the Tauc -Lorentz oscillator was used to  model the optical \nconstants of Sb2Se3. The extracted optical constants were used for reflectance and ellipsometry  \nψ and Δ calculations . For the reflectance calculations , a transfer matrix (TM) formalism was \nused to model the multilayered stack.  The Fresnel reflection coefficients for both s - and p - \npolarizations have been collected from individual interfaces.  \nTo specifically choose an accurate thickness value of Sb2Se3 where the dual phase \nresonances  exist, the reflectance calculations were first done for various thickness values.  A \nheterostructure stack was fabricated after a  single thickness value was selected . Angle-resolved \nand polarization -dependent reflectance measurements  were performed  using spectroscopic \nellipsometry.  The points of darkness and the  phase singularities for both s - and p - polarized \nlights were realized theoretically and experimentally.  The associated phase shifts and \nresonances were measured using ellipsometry parameters (the intensity ratio ψ and phase \ndifference Δ). Moreover, we demonstrated our optical device's gas (CO 2) sensing capabilities  \nby correlating the gas flow rate to the ellipsometry -measured parameters at the phase resonance \npoints.  A gas purging system, with a box enclosing the stage, was attached to  the ellipsometry \nsetup , as illustrated in Fig. 1c.  \nResults and discussion  \nThe proposed multilayered heterostructure device, the measured optical constants n and \nk of the active low -loss Sb2Se3 phase -change layer, and the schematic of the gas sensor \nexperimental setup are presented  in Fig. 1 . The heterostructure device  schematically shown in \nFig. 1a  is composed of a spacer LAO layer to prevent possible intermixing between the gold \nsubstrate and the Sb2Se3 phase -change layer, and capping LAO layer for protection , mainly  \nagainst oxidation. Multiple thickness values have been consider ed for the Sb2Se3 layer but the \nLAO layers have a fixed thickness value of 10 nm. In Fig. 1b, the optical constants n and k of \nthe active Sb2Se3 layer are shown for both the as -deposited amorphous and crystalline phases. \nAn Sb2Se3 sample was pulsed laser deposited (PLD) on a Si/SiO 2 substrate to extract the se \noptical constants . After collecting s pectroscopic ellipsometry measurements in the spectral range of 350 – 1700 nm, the Tauc – Lorentz dispersion model was used to represent the optical \nconstants of the Sb2Se3 layer. The measurements  and data fitting w as again done for the \ncrystalline phase after annealing the sample for 5 min at 250 °C. \nThe optical constants of Sb2Se3 show interesting properties in the visible spectrum \nrange. The index of refraction n has the highest contrast between the as -deposited amorphous \nand the crystalline phases in the visible frequency range, and the material 's light absorption \ncapabilities are limited to that range. To utilize the absorbing nature of the material in the \nvisible range and the added degree of freedom from phase -switching, we focused our attention \non the wavelength  range of  350 – 1000 nm for gas sensing applications. The experim ental \nsetup for the gas sensing experiment is illustrated in Fig. 1c. A sealed box, with glass windows \nfor light input f rom the source , and for reflected light collection by the detector covers the \nsample stage of the ellipsomet er. The glass windows are placed such that measurement is only \npossible at a 70° incidence angle. The sealed glass has two tube openings: one for the gas inlet \ninto the sample stage and another for exhaust. A pressure ga uge controlled the gas flow rat e. \n   \nFigure 1. Schematic  of a multilayered hete rostructure device fabricated and the gas sensing \nexperiment setup . (a) A phase -changing layer of Sb2Se3 was deposited on a gold substrate. \nSpacer and capping LAO layers  were also deposited, each with a thickness  of 10 nm . (b) The \noptical constants n and k of the active Sb2Se3 layer were extracted f rom spectroscopic ellip-\nsometry  measurements . (c) Schematic of the gas sensing measurement setup. The ellipsomet ry \nstage was covered by a sealed box with two glass window openings for input light from the \nsource and output light directed toward the detector. The glass windows are located such that \nmeasurement is possible at  70° inciden ce angle. The sealed box has two openings , one  for the \ngas inlet and  one for the  exhaust. \n \nThe reflectance values for both s - and p - polarized lights were calculated using  the \nmeasured optical constants n and k of individual layers in the proposed heterostructure design . \nThe optical constants were extracted from ellipsometry measurements and data fitting. The \ntransfer matrix formalism was employed for the heterostructure stack presented in Fig. 1a, \nwhere the reflectance was acquired from the Fresnel reflection coefficients. Achieving a 'point \nof darkness ' in multilayered heterostructure stacks is  a relatively trivial task. The reason lies \nbehind the large degree of freedom associated with parameter changes like layer thickness, \nwavelength range, and incidence angle for reaching the desired reflectance values.  However, \nachieving points of darkness for both  s- and p -polarizations is a difficult task.  Therefore, o ur \nmultilayered heterostructure design was carefully selected for various reasons.  One of the main \nreasons, and the critical condition needed, is that perfect absorption of the polarizations (s - or \np-) can be achieved  in a certain spectral range  with the proposed design. To theoretically \ninvestigate the reflectance responses, we calculated the reflectance of s - and p - polarized lights \nfor the as -deposited amorphous Sb2Se3 layer thickness values of 0 – 45 nm  (in steps of 1 nm)  \nin the spectral range of 350 – 1000 nm.  The angle of incidence was fixed at 70 °. The calculated \nreflectance profi les and the ratios between s - and p - polarizations are presented in a color map \nin Fig. 2. The refl ectance of p -polarized light in Fig. 2a and s -polarized light in Fig. 2b as a \nfunction of Sb2Se3 layer thickness show 'point -of-darkness ' properties over large wavelength \nregions where  vanishing  reflectance values are  seen.  \nFor layer thickness values between 15 -30 nm, there exist regions in the color maps \n(indicated by white arrow) where minimum reflectance values occur  in the 350 -500 nm region \nfor the p -polarized light and in the 600 -750 nm region for the s -polarized light.  Interestingly,  \nfor the same Sb2Se3 layer thickness values in the range of 15 -30 nm, we see a reflectance \nminimum for both s - and p - polarizations. The minimum reflectance values are associated with \nthe Brewster effect. However, the observed Brewster effect  differs from the classically known \nprinciple where only the p -polarized light vanishes. Our proposed design show s Sb2Se3 layer thickness and wavelength range combinations where both s - and p - polarized lights vanish. \nThis effect is described by a new principle called the generalized Brewster effect.  To pinpoint \nthe location of minimum reflectance in terms of layer thickness and wavelength combinations, \nwe calculated the ratios of s - and p - polarization reflectance values. The ratios are calculated \nas, \n∆𝑅𝑠,𝑝=( 𝑅𝑠,𝑝− 𝑅𝑝,𝑠\n𝑅𝑝,𝑠)×100%  \nfor all Sb2Se3 layer thickness values. The log arithm  of ΔR values for both Rs/Rp and Rp/Rs \ncases are presented in Fig. 2c and 2d. As expected, extremely  high ratio values are seen at the \nminimum reflectance values (maximum absorption).  \n \nFigure 2. Calculated r eflection vs. thickness results for the multilayered hete rostructur e device  \nshown in Fig. 1a . The polarization -dependent reflectance profiles  for (a) Rp and (b) Rs are \ncomputed for varying thicknesses of the active Sb2Se3 layer. The calculations are done for the \nspectral range of 350 – 1000 nm. Thickness values of low reflection (high absorption) are i n-\ndicated for both s - and p -polarizations . The ratios of the polarizations (Rs/Rp and Rp/Rs) are \ncalculated , and the log values are plo tted in (c) and (d) , respectively. The largest ratio values \nwere identified , and the associated thickness value was used to produce the reflective gas sensor \ndevice.  \n \nThe ratio values in the color maps of Fig. 2c and 2d provide crucial information about \nour proposed multilayered heterostructure device  and its possible application in gas sensin g. In \nspectroscopic ellipsometry, the ψ and  Δ parameters are directly related to the reflectance ratios \nof s- and p - polarized lights . Therefore, a n abrupt phase change is expected at the maximum \nratio values. Below is a detailed explanation of the ellipsometry parameters, reflectance ratio \nvalues, and singular phase behaviors . The maximum ratio values correspond to the minimum \nreflectance values observed in Fig. 2a and 2b.  Moreover, for the layer thickness values of 15 -\n30 nm, our calculated results show  that the highest ratio values can be achieved  at a single \nincidence angle value of 70 ° for both s - and p - polarized lights, and creates an opportunity to \nproduce  two resonance peaks in a single device.  We fabricated a heterostructure stack based \non the simulation insights to demonstrate the presence of dual resonances and phase \nsingularities from the generalized Brewster effect . The multilayered stack was produced by \nPLD. For each layer, the pulse number to thickness  conversion was carefully tuned such that \nsub-nm thickness control was possible  (see also our earlier works with extreme thickness \ncontrol for Sb and Sb 2Te3 films ).34,35 To satisfy the existence of double resonance s, from the s - \nand p - polarized Brewster effects at an incidence angle of 70 °, we fabricated the stack with an \nSb2Se3 layer thickness of ≈23.5 nm.  \nThe reflectance values of s - and p - polarized lights were also calculated for the crystal \nphase of the active Sb 2Se3 layer ( see supplementary information (SI -1) Fig.  S1a and S1b) . Like \nthe as -deposited amorphous phase, the crystalline phase also shows  high absorption (vanishing \nreflectance) regions. Compared to the amorphous phase, the points of darkness for the \ncrystalline Sb2Se3 phase are red -shifted (towards higher wavelength values ). The minimum \nreflectance values are registered in the 500 -600 nm region  for the p -polarized light and in the \n800-900 nm region for the s -polarized light. In addition, the reflectance ratio values are also \ncalculated for the crystalline phase ( see supplementary information (SI -1) Fig.  S1c and S1d) . \nAlthough the maximum ratio value for Rs/Rp and Rp/Rs shifted towards smaller and larger \nSb2Se3 layer thickness values , a significantly large ratio value can still  be achieved at a layer \nthickness of 23.5 nm. This presents a crucial opportunity where the points of darkness can be \ntuned through the Sb2Se3 amorphous -crystalline phase -switching without compromising the \nphase singularities. More details on this are given below.  For the fabricated multilayered stack, the measured and calculated reflectance spectra  \nof the s - and p - polarized lights and the relative ratios between them are presented in Fig. 3.  \nKeep in mind that, for our multilayer stack, the Brewster angle for both polarizations is 70 °. \nThus the reflectance measurements and calculations  are done at 70 °. Reflectance values \nmeasured with variable angle spectroscopic ellipsometry and calculated using  the transfer \nmatrix formalism are presented in Fig. 3a and 3b , respect ively. The linear and log reflectance \nvalues are given for both the s - and p - polarizations. The reflectance curves show  that t he \ngeneralized Brewster effect is achieved where a reflectance minimum for both polarizations is \ndemonstrated . In both Fig. 3a and 3b, the  log reflectance curves (dotted red and blue curves) \npinpoint  the point s of darkness where the  s- and p - polarized lights vanish.  Despite the small \nshift in reflectance and wavelength values, the measured reflectance results in Fig. 3a match \nthe c alculated values in Fig. 3b very well . The small  discrepancies between the measured and \ncalculated values are probably caused  by the assumption of 'perfectly matching interfaces ' in \nthe calculations wh ereas  fabricated samples have not such ideal interfaces . Nonetheless, small \nreflectance value s of Rp = 0.3% and Rs = 3.8% are measured experimentally at the two points  \nof darkness , more than enough to produce phase singularities.  \n \nFigure 3. Measured and calculated polarization -dependent reflectance profiles for a multi-\nlayer ed heterostructure of 10 nm LAO/23.5 nm Sb2Se3/10 nm LAO/100 nm gold/SiO 2. The (a) \nmeasured and ( b) calculated reflectances for both s - and p - polariz ations, at 70 º incidence angle , \nare presented. In both (a) and ( b), the right col umn pr esents the log reflectance values, indicat-\ning the point of darkness /high absorptions. The relative reflectivity ratios  between polarizations  \nfor both measured and calculated values  are presented in ( c) and (d ). \nThe ratios of the reflectance values from the s - and p - polarized lights, for both \nmeasured and calculated values, are presented in Fig. 3c and 3d. The t wo resonance peaks \ncorrespond to the s - and p - polarizations point -of-darkness . Comparable measured and \ncalculated wavelength value s of ≈ 650 nm can be seen for ΔRp/s. However, the measured \nvalues  for ΔRs/p are red-shifted to 4 86 nm compared to  the calculated wavelength value of 470 \nnm. As mentioned above, the two resonance peaks corresponding to the reflec tance minimum \npoints or points of darkness for both s - and p - polarized lights directly correlate to distinct  \nphase behaviors where abrupt changes are expected. We measured and calculated the \nellipsometry parameters ψ and  Δ for the spectral range of 350 – 800 nm at 70° angle of \nincidence to confirm the presence of  the phase singularities. The measured and calculated ψ \nand Δ parameters are presented in Fig. 4. For better correlation of the effect of points of \ndarkness in the ellipsometry parameters, Rs and R p values are also included in the figures.  \nTo understand the effect of amorphous to crystalline phase -switching, we calculated \nthe s- and p - polarization reflectance values for our heterostructure stack with a 22 nm and 23.5 \nnm Sb2Se3 layer in the as -deposited amorphous and crystalline phases  (see supplementary \ninformation (SI -2) Fig. S2a) . The 22 nm Sb 2Se3 thickness value accounts for the thickness \nreduction upon crystallization. Phase -change materials show an increase in density when \nswitching from amorphous to crystalline phases (usually between 6 – 10 %). This density \nchange will be accounted for by a th ickness reduction. More information for the 22nm thick \nlayer is provided in supplementary information ( SI-1 and SI-3). Compared to the amorphous \nfilm,  the points of darkness for both polarizations are red-shifted towards higher wavelength \nvalues  for the crystalline film . In Fig. S2b, the reflectance ratio values clearly show two \nresonance peaks (at 584 nm and 820 nm for Rs/Rp and Rp/Rs , respectively) corresponding to \nthe vanishing reflectance values in s - and p - polarized lights. Since  the reflectance ratio \nresonance peaks are associated with phase singularities, we calculated the ellipsometry \nparameters ψ and  Δ for the crystalline Sb2Se3 phase. Indeed, t he existence of the se singularities \nare confirmed  also for the crystalline Sb2Se3 layer in Fig. S2c and S2d from the calculated ψ \nand Δ parameters.   \nFigure 4. Measured and calculated ellipsomet ry parameters ψ and Δ overlayed with s - and p - \npolarization reflectance values  for a multilayered heterostructure of 10 nm LAO/23.5 nm \nSb2Se3/10 nm LAO/100 nm gold/SiO 2. The measured (a,b) and calculated (c,d) ψ and Δ values \nshow minimum/maximum values and phase singularities at both s - and p- polarization point of \ndarkness locations.  \n \nThe measured (in Fig. 4a and 4b) and calculated (in Fig. 4c and 4d) ellipsometry \nparameters ψ and Δ match very well. However, a t the specific wavelength values where the \npoints of darkness were achieved, some distinct properties can be seen for both ψ and Δ \nparameters. The  singular characteristics of ψ and Δ are directly related to the reflectance ratios \nof s- and p - polarized lights, thus the resonances seen in Fig. 3. In ellipsometry, when the \ninitial ly linearly polarized light is reflected off of a sampl e at a non -zero incidence angle, the \nintensity and phases of the s - and p - polarized lights will change.  For each polarization, the \nlight propagation through the materials and the collected intensities from each interface is \nexplained by the Fresnel equations. More importantly, the reflection ratio of Rp/Rs, represented \nby the complex reflection coefficient ρ , correlates directly to the ellipsometry ψ and Δ \nparameters through the equation  ρ= 𝑅𝑝\n𝑅𝑠=tan(ψ)exp(i Δ). The equation clearly shows  how \nψ and Δ could be directly affected by the Rp/Rs ratio change . It explains the singular  \ncharacteristics of ψ and Δ exactly at the minimum Rs and Rp reflection values at the points of \ndarkness.  \nFrom Fig. 4a and 4c, we notice  that the ψ parameter approaches 0 ° when Rp →0 and \n90° when Rs →0. The singular modulation of ψ at the points of darkness is due to the \nrelationship  of ψ with the  complex  Fresnel coefficients rp = | rp |exp (i φp) and rs = | rs |exp (i φs) \nthrough the equation tan(ψ)  = |rp|/|rs|. For our multilayered heterostructure stack, ψ = 4.39 ° at \n486 nm and ψ = 74.17 ° at 650 nm, consistent with the theoretical basis.  Moreover, extreme \nsingularities at the points of darkness are observed from the phase difference b etween the two \npolarizations (i.e., Δ = φp – φs). The abrupt jumps in  the Δ parameter are also clearly visible in \nFig. 4b and 4d. Considering the  measured values, the Δ parameter changes from ≈90° to ≈250° \nand from ≈-20° to ≈70° near the two points of darkness associated with the reflectance \nminimums. Note that the range of the abrupt change in Δ is larger ( ≈160°) when Rp →0 \ncompared to when Rs →0 (≈90°). This is due to the  relatively higher Rs value of 3.8 % and \nprovides an important insight into designs where larger Δ changes are achieved in the case of  \nperfect absorption. The abrupt ψ and Δ parameter changes at the points of darkness are \nsusceptible to extremely small optical deviations. Therefore, t hese phase  shifts could be utilized \nfor various sensi ng applications.   \n \nFigure 5. Results from gas sensing experiment. The flow rate of CO 2 gas was changed linearly , \nand ellipsometry ψ and Δ parameters were collected. (a) The Δ value was collected for various \nflow rates in the 480 – 490 nm spectrum range . (b) A zoomed -in image of some of the Δ values \npresented in (a) show s a systematic trend between Δ and the CO 2 flow rate . (c) The relative \ndifference in Δ values (relative to Δ value with no gas flow) show s a linear dependence on  the \ngas flow rate. (d) An example of a linear correlation of Δ value change with gas flow rate at λ \n= 486 nm , i.e. the measured point of darkness . \n \nAs a proof of concept, we employed our multilayered heterostructure stack for sensing \nextremely small CO 2 gas concentrations. As illustrated in Fig. 1c, a  CO 2 gas canister was \nattached to the ellipsometry setup. The ellipsometry parameters were measured at the first \nresonance point (between 480 -490 nm) while linearly changing the gas flow rate (from 0 -83 \nsccm) . Once the gas flow started, we waited 5 min so that the flow stabilize d. Then, high-\naccuracy measurements were done for all flow rates where the polarizer/compensator \ncombination  was used to collect measured data. Moreover, f ive measurements were done in \nsequence for each flow rate,  and the average value was taken as the final result. The results \nfrom the gas sensing experiments are presented in Fig. 5  only focusing on the Δ values . The \ncompletely analogous figure of Figure 5 but now for ψ instead of Δ is shown in supplementary \ninformation (SI -4) Fig.  S4. Fig. 5a  shows  the Δ values for all gas flow rates  and a zoomed -in \nview on a small  wavelength region  is presented  in Fig. 5b.  The measurements show a shift \ntoward higher Δ values for wavelengths higher than 285 nm and lower Δ values for \nwavelengths lower than 485 nm. To better visualize  the Δ value changes to gas flow rate, we \nplotted the relative Δ value changes in Fig. 5c. The Δ change values are calculated by \nsubtracting the initial  zero gas flow  measurement  from each measurement.  A clear upward \ntrend for wavelength values above 485 nm and a downward trend for the rest is visible.  The Δ \nchanges show a somewhat linear dependence on  the gas flow rate  changes . Although relatively \nsmall, s uch linear correlation  is also seen in the  measured  ψ value changes  (see supplementary \ninformation (SI -4) Fig.  S4). In Fig. 5d, an example of this linear Δ value change for a specific \nwavelength value of 486 nm is presented as an example.  \nOur work demonstrate s a generalized Brewster effect where  perfect/strong light \nabsorptions were realized for both  the s - and p - polariz ations  for a single Brewster angle. A \nlossy phase -change layer Sb2Se3 layer was coated on a gold substrate. Vanishing reflectance \nvalues for both s- and p - polarizations w ere achieved at an incidence angle of 70 ° by fine -\ntuning the active Sb2Se3 layer thickness for the wavelength range of 350 – 1000 nm.  Our \nrealized multilayered stack, capable of double darkness points assoc iated with s - and p - \npolarized lights, can produce  rapid phase shifts, as demonstrated experimentally and \ntheoretically. Moreover, additional sets of phase singularities have been realized from the \namorphous to crystalline phase -switching of the active Sb2Se3 layers. Such robust and tunable \ndevice designs, with multiple phase singularity points, can be used for sensing applications. As \na proof of concept, we showed that our lithography -free multilayered stack c ould be utilized as \nan ultrasensitive sensor susceptible to small refractive index changes  from CO 2 gas flow s. \nComparing our design with previously reported results on the generalized Brewster \neffect26–29 and phase -change thin films based perfect absorber designs ,14,16,17 multiple obvious \nadvantages can be seen  for our approach . Our multilayered heterostructure stack exploits the \nstrong interference effect, while most results rely on the Fabry -Perot (FP) resonant cavities for \nachieving strong/perfect absorption. It has been reported that the robust nature of strong \ninterference systems with incidence angle and the reduced number of interfaces makes them \nattractive.7,10 Moreover, our lithography -free layered thin film stacks have nanofabrication \nadvantages compared with plasmonic surfaces requiring lithography step s. But more \nimportantly, the novelty of our work relies on our ability to produce an optical device capable \nof achieving strong/perfect absorption for both s - and p - polarizations at the same Brewster \nangle. While the results  reported in literature  provide theoretical and experimental basis for potentially achieving the generalized Brewster effect, the proposed designs up to now did not \nsupport the coexistence of such vanishing reflectance values for both polarizations.  In contrast, \nfor our design the coexistence at a single Brewster angle of vanishing reflectances for both \npolarizations has been proven here with the additional degree of freedom that this coexistence \nis largely preserved when switching between the amorpho us and crystalline phases of Sb 2Se3. \nConclusions  \nOur work demonstrate s a multilayered heterostructure design capable of achieving \nstrong/perfect absorption for both s - and p - polarized lights  at a single light incidence angle. \nBy designing our heterostructure stack carefully , we realized a thickness range where \nreflectance  vanishes for both s - and p - polarized lights . In addition, w e produced an optical \ndevice that experimentally validated  the coexistence of points of darkness for both s - and p - \npolarized lights at an incidence angle of 70 °. Vanishing reflectance values are a ssociated with \nsingular phase behaviors with abrupt changes at the points of darkness. We demonstrated the \nexistence of two phase singularities in the as -deposited amorphous phase of the active phase -\nchange material Sb 2Se3, while additional two phase singularities are realized after switching  to \nthe crystalline phase . As a proof of concept, we showed that phase singularities are \nultrasensitive to small deviations in optical constants , as we introduced here by a varying CO 2 \nflow,  and could  thus be utilized i n various  (sensor)  applications. The tunability from phase -\nswitching and the coexisting vanishing reflectance values for both s - and p - polarizations \nprovide promising application options with versatile designs  that are easily producible . \nAcknowledgments  \nThis project has received funding from the European Union 's Horizon 2020 Research \nand Innovation Programme \"BeforeHand \" (Boosting Performance of Phase Change Devices \nby Hetero - and Nanostructure Material Design) \" under Grant Agreement No. 824957.  \nReferences  \n \n(1)  ElKabbash, M.; Iram, S.; Letsou, T.; Hinczewski, M.; Strangi, G.; ElKabbash, M.; Iram, \nS.; Letsou, T.; Hinczewski, M.; Strangi, G. Designer Perfect Light Absorption Using \nUltrathin Lossless Dielectrics on Absorptive Substrates. Adv. Opt. Mater.  2018 , 6 (22), \n1800672. https://doi.org/10.1002/ADOM.201800672.  \n(2)  K. V., S.; ElKabbash, M.; Caligiuri, V.; Singh, R.; De Luca, A.; Strangi, G. Perfect Light \nAbsorption in Thin and Ultra -Thin Films and Its Applications. 2019 , 3–27. \nhttps://doi.org/10.1007/978 -981-13-8891 -0_1. \n(3)  Yao, Y.; Liao, Z.; Liu, Z.; Liu, X.; Zhou, J.; Liu, G.; Yi, Z.; Wang, J. Recent Progresses on Metamaterials for Optical Absorption and Sensing: A Review. Journal of Physics D: \nApplied Physics . 2021, p 21. https://doi.org/10.1088/1361 -6463/abccf0.  \n(4)  Cheng, F.; Gao, J.; Luk, S. T.; Yang, X. Structural Color Printing Based on Plasmonic \nMetasurfaces of Perfect Light Absorption. Sci. Reports 2015 51  2015 , 5 (1), 1 –10. \nhttps://doi.org/10.1038/srep11045.  \n(5)  Aydin, K.; Ferry, V. E.; Briggs, R. M.; Atwater, H. A. Broadband Polarization -\nIndependent Resonant Light Absorption Using Ultrathin Plasmonic Super Absorbers. \nNat. Commun. 2011 21  2011 , 2 (1), 1 –7. https://doi.org/10.1038/ncomms1528.  \n(6)  Li, Z.; Butun, S.; Aydin, K. Large -Area, Lithography -Free Super Absorbers and Color \nFilters at Visible Frequencies Using Ultrathin Metallic Films. ACS Photonics  2015 , 2, \n183–188. https://doi.org/10.1021/ph500410u.  \n(7)  Song, H.; Guo, L.; Liu, Z.; Liu, K.; Zeng, X.; Ji, D.; Zhang, N.; Hu, H.; Jiang, S.; Gan, \nQ.; Song, H.; Liu, K.; Zeng, X.; Ji, D.; Zhang, N.; Gan, Q.; Guo, L.; Liu, Z.; Jiang, S.; \nHu, H. Nanocavity Enhancement for Ultra -Thin Film Optical Absorber. Adv. Mater.  \n2014 , 26 (17), 2737 –2743. https://doi.org/10.1002/ADMA.201305793.  \n(8)  Sreekanth, K. V.; Sreejith, S.; Han, S.; Mishra, A.; Chen, X.; Sun, H.; Lim, C. T.; Singh, \nR. Biosensing with the Singular Phase of an Ultrathin Metal -Dielectric Nanophotonic \nCavity. Nat. Commun. 2018 91  2018 , 9 (1), 1 –8. https://doi.org/10.1038/s41467 -018-\n02860 -6. \n(9)  Kats, M. A.; Capasso, F. Optical Absorbers Based on Strong Interference in Ultra -Thin \nFilms. Laser Photonics Rev.  2016 , 10 (5), 735 –749. \nhttps://doi.org/10.1002/lpor.201600098.  \n(10)  Kats, M. A.; Blanchard, R.; Genevet, P.; Capasso, F. Nanometre Optical Coatings Based \non Strong Interference Effects in Highly Absorbing Media. Nat. Mater.  2013 , 12 (1), \n20–24. https://doi.org/10.1038/nmat3443.  \n(11)  Wuttig, M.; Bhaskaran, H.; Taubner, T. Phase -Change Materials for Non -Volatile \nPhotonic Applications. Nat. Photonics  2017 , 11 (8), 465 –476. \nhttps://doi.org/10.1038/nphoton.2017.126.  \n(12)  Fan, Z.; Deng, Q.; Ma, X.; Zhou, S. Phase Change Metasurfaces by Continuous or \nQuasi -Continuous Atoms for Active Optoelectronic Integration. Mater. 2021, Vol. 14, \nPage 1272  2021 , 14 (5), 1272. https://doi.org/10.3390/MA14051272.  \n(13)  Abdollahramezani, S.; Hemmatyar, O.; Taghinejad, H.; Krasnok, A.; Kiarashinejad, Y.; \nZandehshahvar, M.; Alu, A.; Adibi, A.; Alù, A.; Alù, A.; Adibi, A. Tunable \nNanophotonics Enabled by Chalcogenide Phase -Change Materials. Nanophotonics  \n2020 . https://doi.org/10.1515/nanoph -2020 -0039.  \n(14)  Cao, T.; Wei, C. W.; Simpson, R. E.; Zhang, L.; Cryan, M. J. Broadband Polarization -\nIndependent Perfect Absorber Using a Phase -Change Metamaterial at Visible \nFrequencies. Sci. Reports 2014 41  2014 , 4 (1), 1 –8. https://doi.org/10.1038/srep03955.  \n(15)  Cueff, S.; Taute, A.; Bourgade, A.; Lumeau, J.; Monfray, S.; Song, Q.; Genevet, P.; \nDevif, B.; Letartre, X.; Berguiga, L. Reconfigurable Flat Optics with Programmable \nReflection Amplitude Using Lithography -Free Phase -Change Material Ultra -Thin \nFilms.  Adv. Opt. Mater.  2021 , 9 (2), 2001291. \nhttps://doi.org/10.1002/ADOM.202001291.  \n(16)  Tittl, A.; Michel, A. -K. U.; Schäferling, M.; Yin, X.; Gholipour, B.; Cui, L.; Wuttig, M.; \nTaubner, T.; Neubrech, F.; Giessen Tittl, H. A.; Schäferling, M.; Yin, X.; Neubrech, F.; \nGiessen, H.; Michel, A. U.; Wuttig, M.; Taubner, T.; Gholipour, B.; Cu i, L. A \nSwitchable Mid -Infrared Plasmonic Perfect Absorber with Multispectral Thermal \nImaging Capability. Adv. Mater.  2015 , 27 (31), 4597 –4603. \nhttps://doi.org/10.1002/ADMA.201502023.  \n(17)  Wen, X.; Xiong, Q. A Large Scale Perfect Absorber and Optical Switch Based on Phase Change Material (Ge2Sb2Te5) Thin Film. https://doi.org/10.1007/s40843 -016-0129 -7. \n(18)  Grigorenko, A. N.; Patskovsky, S.; Kabashin, A. V. Phase and Amplitude Sensitivities \nin Surface Plasmon Resonance Bio and Chemical Sensing. Opt. Express, Vol. 17, Issue \n23, pp. 21191 -21204  2009 , 17 (23), 21191 –21204. \nhttps://doi.org/10.1364/OE.17.021191.  \n(19)  Grigorenko, A. N.; Nikitin, P. I.; Kabashin, A. V. Phase Jumps and Interferometric \nSurface Plasmon Resonance Imaging. Appl. Phys. Lett.  1999 , 75 (25), 3917 –3919. \nhttps://doi.org/10.1063/1.125493.  \n(20)  Kravets, V. G.; Schedin, F.; Jalil, R.; Britnell, L.; Gorbachev, R. V.; Ansell, D.; \nThackray, B.; Novoselov, K. S.; Geim, A. K.; Kabashin, A. V.; Grigorenko, A. N. \nSingular Phase Nano -Optics in Plasmonic Metamaterials for Label -Free Single -\nMolecule D etection. Nat. Mater. 2013 124  2013 , 12 (4), 304 –309. \nhttps://doi.org/10.1038/NMAT3537.  \n(21)  Berkhout, A.; Koenderink, A. F. Perfect Absorption and Phase Singularities in Plasmon \nAntenna Array Etalons. ACS Photonics  2019 , 6 (11), 2917 –2925. \nhttps://doi.org/10.1021/ACSPHOTONICS.9B01019/ASSET/IMAGES/MEDIUM/PH9\nB01019_M010.GIF.  \n(22)  Zeng, S.; Hu, S.; Xia, J.; Anderson, T.; Dinh, X. -Q.; Meng, X. -M.; Coquet, P.; Yong, \nK.-T. Graphene -MoS 2 Hybrid Nanostructures Enhanced Surface Plasmon Resonance \nBiosensors. Sensors Actuators B  2015 , 207, 801 –810. \nhttps://doi.org/10.1016/j.snb.2014.10.124.  \n(23)  Oh, S. H.; Altug, H.; Jin, X.; Low, T.; Koester, S. J.; Ivanov, A. P.; Edel, J. B.; Avouris, \nP.; Strano, M. S. Nanophotonic Biosensors Harnessing van Der Waals Materials. Nat. \nCommun. 2021 121  2021 , 12 (1), 1 –18. https://doi.org/10.1038/s41467 -021-23564 -4. \n(24)  Brewster, D. On the Laws Which Regulate the Polarisation of Light by Reflexion from \nTransparent Bodies. Philos. Trans. R. Soc. Lond.  1815 , 105, 125 –159. \nhttps://doi.org/10.1098/rstl.1815.0010.  \n(25)  Thomas, P. A.; Menghrajani, K. S.; Barnes, W. L. All -Optical Control of Phase \nSingularities Using Strong Light -Matter Coupling. Nat. Commun. 2022 131  2022 , 13 \n(1), 1 –6. https://doi.org/10.1038/s41467 -022-29399 -x. \n(26)  Paniagua -Domínguez, R.; Yu, Y. F.; Miroshnichenko, A. E.; Krivitsky, L. A.; Fu, Y. \nH.; Valuckas, V.; Gonzaga, L.; Toh, Y. T.; Kay, A. Y. S.; Lukyanchuk, B.; Kuznetsov, \nA. I. Generalized Brewster Effect in Dielectric Metasurfaces. Nat. Commun. 2016 71  \n2016 , 7 (1), 1 –9. https://doi.org/10.1038/ncomms10362.  \n(27)  Lavigne, G.; Caloz, C. Generalized Brewster Effect Using Bianisotropic Metasurfaces. \nOpt. Express  2021 , 29 (7), 11361. https://doi.org/10.1364/OE.423078.  \n(28)  Ding, L.; Qiu, T.; Zhang, J.; Wen, X. Generalized Brewster Effect Tuned Optically in a \nGraphene/Substrate System. 2019 . https://doi.org/10.1088/2040 -8986/ab4fa1.  \n(29)  Sreekanth, K. V.; Elkabbash, M.; Medwal, R.; Zhang, J.; Letsou, T.; Strangi, G.; \nHinczewski, M.; Rawat, R. S.; Guo, C.; Singh, R. Generalized Brewster Angle Effect in \nThin -Film Optical Absorbers and Its Application for Graphene Hydrogen Sensing. ACS \nPhotonics  2019 , 6 (7), 1610 –1617. https://doi.org/10.1021/acsphotonics.9b00564.  \n(30)  Delaney, M.; Zeimpekis, I.; Lawson, D.; Hewak, D. W.; Muskens, O. L. A New Family \nof Ultralow Loss Reversible Phase‐Change Materials for Photonic Integrated Circuits: \nSb 2 S 3 and Sb 2 Se 3. Adv. Funct. Mater.  2020 , 30 (36), 2002447. \nhttps://doi.org/10.1002/adfm.202002447.  \n(31)  Faneca, J.; Zeimpekis, I.; Ilie, S. T.; Bucio, T. D.; Grabska, K.; Hewak, D. W.; Gardes, \nF. Y. Towards Low Loss Non -Volatile Phase Change Materials in Mid Index \nWaveguides. Neuromorphic Comput. Eng.  2021 , 1 (1), 014004. \nhttps://doi.org/10.1088/2634 -4386/ac156e.  (32)  Dong, W.; Liu, H.; Behera, J. K.; Lu, L.; H Ng, R. J.; Valiyaveedu Sreekanth, K.; Zhou, \nX.; W Yang, J. K.; Simpson, R. E.; Dong, W.; Liu, H.; Behera, J. K.; Lu, L.; H Ng, R. \nJ.; Zhou, X.; W Yang, J. K.; Simpson, R. E.; Sreekanth, K. V. Wide Bandgap P hase \nChange Material Tuned Visible Photonics. Adv. Funct. Mater.  2019 , 29 (6), 1806181. \nhttps://doi.org/10.1002/ADFM.201806181.  \n(33)  Yimam, D. T.; Liang, M.; Ye, J.; Kooi, B. J. 3D Nanostructuring of Phase -Change \nMaterials Using Focused Ion Beam Towards Versatile Optoelectronics Applications. \nAdvanced Materials 2023, 2303502. https://doi.org/10.1002/ADMA.202303502 . \n(34)  Yimam, D. T.; Kooi, B. J. Thickness -Dependent Crystallization of Ultrathin Antimony \nThin Films for Monatomic Multilevel Reflectance and Phase Change Memory Designs. \nACS Appl. Mater. Interfaces  2022 , 14 (11), 13593 –13600. \nhttps://doi.org/10.1021/acsami.1c23974.  \n(35)  Yimam, D. T.; Ahmadi, M.; Kooi, B. J. Van Der Waals Epitaxy of Pulsed Laser \nDeposited Antimony Thin Films on Lattice -Matched and Amorphous Substrates. Mater \nToday Nano 2023, 23, 100365. https://doi.org/10.1016/J.MTNANO.2023.100365. . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Supplementary Information  for \n \nCoupling phase -switching with generalized Brewster effect \nfor tunable optical sensor design s \nDaniel T. Yimam*, Dennis van der Veen, Teodor Zaharia, Maria Loi, Bart J. Kooi*  \nZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 A \nGroningen, The Netherlands  \n*Corresponding authors. Email: d.t.yimam@rug.nl , b.j.kooi@rug.nl  \n \nSI 1 – Reflectance  for crystalline Sb 2Se3 films  \nThe crystallization of the active Sb 2Se3 layer produces reflectance profiles different \nfrom what we show in Fig. 2 of the main text. The optical constants difference for the as -\ndeposited amorphous and crystalline phases of the Sb2Se3 layer , as seen in Fig. 1b of the main \ntext, is responsible for the difference in reflectance profiles . Nevertheless, r eflectance \nvanishing points for both p - and s - polarized lights also occur for the crystalline Sb 2Se3 films \nas can be seen in Fig. S1a and S1b. Compared with the as-deposited amorphous phase in Fig. \n2, the points of darkness  for the crystalline phase  are shifted towards higher  wavelength values . \nFrom the profiles in Fig. S1c and S1c, we can see significantly large ratio values for a layer \nthickness of 23.5 nm. However, the crystallization of phase -change layers is associated with a \nreduced thickness (from increased density) , and thus we also have to consider a lower thickness \nvalue ( ≈22 nm). Nonetheless, coexisting and large resonance peaks are expected for both \nthickness values.   \nFigure S1 . Calculated  s- and p - polarization reflectance  vs. thickness  for the crystalline \nphase of the active Sb2Se3 layer in the heterostructure device.  In (a) and (b), the polarization -\ndependent reflectance profiles are presented for varying thicknes ses of the crystalline Sb2Se3 \nlayer. The layer thickness was varied from 0 – 45 nm , and the calculations were done for the \nwavelength range of 350 – 1000 nm. The white arrows indicate minimum reflectance values  \nfor both s - and p - polarized lights. In (c) and (d), the lo g values of the polarization ratios (Rs/Rp \nand Rp/Rs) are given. The l argest ratio values can be identified for a wide  range of  layer \nthickness values.  \n  \nSI 2 –  Crystalline Sb 2Se3 (23.5 nm )   \n \nFigure S2 . The p olarization -dependent reflectance profiles, the relative reflectance \nratios, and th e associated phase singularities . (a) The calculated s - and p - polarized light \nreflectance profiles for a 23.5 nm crystalline Sb2Se3 layer at a 70° incidence angle. (b) The \nrelative reflectance ratios between polarizations. The calculated ellipsometry parameters (c) ψ \nand (d) Δ, with the s - and p - polarization reflectance values overlayed.  \nThe reflectance values for both the as -deposited amorphous and crystalline phases of \nSb2Se3 are presented in Fig. S2a. Here the layer thickness is fixed to 23.5 nm. Two crucial \ninformation s are visible in the figure. The first information is the deviation in reflectance values \nfor the as -deposited amorphous and crystalline phases for both polarizations. The reflectance \nvalues are red-shifted towards higher wavelength values  for the crystalline phase . The second \nand most important information  is the strong/perfect absorption present for both phases. For a \n23.5 nm thick crystalline Sb2Se3 layer, ref lectance values of Rp = 2% and Rs = 1.4% were \nregistered at the two points of darkness. Although such reflectance values are relatively high \ncompared with the as -deposited amorphous phase of exact thickness, the values still  translate \nto strong light absorption  and, thus, phase singularities. The presence of reflectance vanishing \npoints for both phases provides an opportunity to move towards tunable optical sensor designs. \nThe resonance peaks in Fig. S2b (at 584 nm and 820 nm for Rs/Rp and Rp/Rs , respectively) \nindicate the presence of phase singularities for the crystalline Sb2Se3 layer.  Such resonance \npeaks f or the amorphous phas e were seen at 470 nm and 652 nm. To confirm the existence of \nphase singularities at the points of darkness , we calculated the ellipsometry ψ and Δ  parameters . \nThe abrupt change s in the ellipsometry paramet ers are indeed seen in Fig. S2c and S2d for both \nψ and Δ, exactly at the wavelength values where the resonance s are present . \nSI 3 – Crystalline Sb 2Se3 (22 nm)    \n \nFigure S 3. Reflectance and ratio value shifts for thickness change due to crystallization. \n(a) Calculated reflectance values for layer thickness values of 22 nm and 23.5 nm. The \nrelatively small thickness value change caused a shift in the points of darkness towards lower \nwavelength values. The shift in wavelength value and the relative ratio changes are presented \nin (b). The resonance peaks shifted in wavelength values from 584 nm to 574 nm for Rs/Rp \nand from 820 nm to 801 nm for Rp/Rs. Moreover, the relative ratio v alues for Rs/Rp increased \nwheras they decreas ed for Rp/Rs .  \nAs mentioned above, the crystallization of phase -change material s is accompanied by \nan increase in density. This density  change (usually between 6  - 10 % for GST225 and Sb 2Te3) \nmanifests  as a small layer thickness decrease in the crystalline phase compared to  the \namorphous phase.  The thickness reduction  has to be incorporated into the heterostructure \ndevice design. It is important to evaluate the relatively small thickness deviations since the \nstrong interference effect is highly sensitive , and such a small thickness change can  still cause \ndeviat ions in  the reflectance  profiles . We recalculated the reflectance values for both \npolarizations considering a crystalline Sb2Se3 layer thickness of 22 nm to evaluate the effect of \nsuch thickness deviation from phase switching. We  compared it with the results presented in \nSI2 for a  thickness value of  23.5 nm.  \nThe reflectance profiles for Sb2Se3 layer thickness values of 22 nm and 23.5 nm are \npresented in Fig. S3a. Both polarizations show small shifts in reflectance values. To extract the \nexact shifted values, we plot the reflectance ratios for the 22 nm layer thickness in Fig. S3b. \nThe effect of the small thickness deviation can be seen from the value changes in both \nresonance peaks and the associated wavelength values. The layer thickness change (from 23.5 \nnm to 22 nm) is associated with a wavelength value shift for the resonance peaks (from 584 \nnm to 574 nm for Rp and from 820 nm to 801 nm for Rs).  \nMoreover, the thickness change also changed the ratio values where Rs/Rp increased \nwhile Rp/Rs reduced. We calculated the ellipsometry ψ and Δ parameters for a thickness value \nof 22 nm to see the effect of such thickness change on the phase singularity . The ψ and Δ \nparameters for a thickness of 22 nm  and 23.5 nm for comparison  are plotted in Fig. S3c and \nS3c. Comparable phase singularities can be seen for both thickness values, with only a small \nshift in wavelength values.  \nSI 4 –  ψ vs. gas flow rate  \nFig. 5 of the main text presents the ellipsometry Δ parameter for CO 2 gas flow rate \nchanges . The measurements were done for a wavelength range of 480 – 490 nm, corresponding \nto one of the resonance peaks (thus phase singularity). In the corresponding wavelength ranges, \nthe ellipsometry ψ parameter was also collected. Fig. S4 presents the correlation between the \nmeasured ψ parameter and the gas flow rate . In Fig. S4a and S4b, the ψ values for all gas flow \nrates and a zoomed -in image are depicted , respectivel y. Although relatively small, the ψ values \nshift towards smaller values for increasing gas flow rate. The relative ψ value changes are \nplotted in Fig. S4c. A gain, somewhat linear and downward trends can be observed . Similar to \nthe Δ values, the ψ parameter also show s a linearly dependent correlation with gas flow rate , \nas seen in Fig. S4d for a wavelength value of 486 nm.   \nFigure S 4. Gas sensing experiment results for the wavelength range of 480 – 490 nm. \n(a) The ellipsometry ψ parameter collected for gas flow  rates  chang ing from 0 – 83 sccm.  (b) \nA zoomed -in image of some of the ψ values showing systematic  change s with respect to the \ngas flow rate. (c) The relative change in ψ values, with and without gas flow for each rate, \nshow s a linear dependence on gas flow rate values. (d) The ψ value change with the gas flow \nrate for a single wavelength value of 486 nm, showing a linear correlation.  \n \n"    },    {        "title": "2402.07156v1.A_hybrid_iterative_method_based_on_MIONet_for_PDEs__Theory_and_numerical_examples.pdf",        "content": "A hybrid iterative method based on MIONet for PDEs: Theory and\nnumerical examples∗\nJun Hu†and Pengzhan Jin‡\nAbstract\nWe propose a hybrid iterative method based on MIONet for PDEs, which combines the\ntraditional numerical iterative solver and the recent powerful machine learning method of neural\noperator, and further systematically analyze its theoretical properties, including the convergence\ncondition, the spectral behavior, as well as the convergence rate, in terms of the errors of the\ndiscretization and the model inference. We show the theoretical results for the frequently-used\nsmoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of\nthe convergence rate of the hybrid method w.r.t. the model correction period, which indicates\na minimum point to make the hybrid iteration converge fastest. Several numerical examples\nincluding the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are\npresented to verify our theoretical results, and also reflect an excellent acceleration effect. As a\nmeshless acceleration method, it is provided with enormous potentials for practice applications.\n1 Introduction\nPartial differential equations (PDEs) play a crucial role in describing physical reality and modeling\nengineering problems. As the vast majority of PDEs have no computable analytical solutions, seek-\ning numerical solutions is an important issue. There have been many classical numerical methods\ndeveloped for solving PDEs, such as the finite element method (FEM) [3], the finite difference\nmethod (FDM) [32], the finite volume method [6], and the spectral method [29]. With these meth-\nods, linear PDEs are usually discretized as linear algebraic systems, and subsequently be solved by\ndirect methods (e.g. Gaussian elimination), or iterative methods such as Richardson, (damped) Ja-\ncobi, Gauss-Seidel, SOR, SSOR [33]. When the scale of the problem is very large, iterative methods\nshow significant advantages like low memory requirements. However, a large scale system results in\na long solving time, which is unacceptable. To accelerate the iteration, multilevel iterative methods\nare developed [8, 34, 38], a multigrid solver will greatly reduce the steps of iteration and lead to fast\nconvergence. The shortcoming of the multigrid method is that it is mesh-depended, and requires\nusers to generate several levels of meshes, which may be a very heavy task. In order to overcome\nthis drawback, the algebraic multigrid (AMG) methods [2, 22, 28, 39] are proposed, which deal\nwith the algebraic system directly without geometric meshes. To further promote the development\nof fast and low usage cost iterative methods, in this work we propose a hybrid iterative method by\nintroducing neural network-based tools.\nThe rise of neural networks (NNs) brings a huge impact on searching numerical solutions of\nPDEs. It has been shown that NNs have powerful expressivity [4, 30], thus NNs are considered\n∗The research is supported by NSFC project 12288101.\n†School of Mathematical Sciences, Peking University, Beijing 100871, China (hujun@math.pku.edu.cn).\n‡School of Mathematical Sciences, Peking University, Beijing 100871, China (jpz@pku.edu.cn).\n1arXiv:2402.07156v1  [math.NA]  11 Feb 2024as universal approximators for the solutions of PDEs. Mainstream neural networks-based solving\nmethods for PDEs are optimizing the parameters of such NNs to minimize the corresponding loss\nfunctions constructed by strong or variational forms of PDEs, e.g. the PINNs [14, 21, 26] , the\ndeep Ritz method [5], and the deep Galerkin method [31]. These methods show the possibility\nfor solving high-dimensional problems, which is indeed a challenge for traditional methods due to\nthe curse of dimensionality. As meshless methods, they are easily implemented and friendly for\ncoders. The drawbacks of such methods are also obvious. On the one hand, these methods are\nusually hard to obtain solutions as accurate as the traditional methods like FEM, on the other\nhand, one needs to restart a training process once some parameters of the given PDE problem (e.g.\nthe right-hand side term of Poisson equation) are changed, which is costly to deal with parametric\nPDE problems. In recent years, a new research field called neural operator, is emerging to achieve\nthe fast solving for PDEs based on neural networks. The pioneer works are the DeepONet [19, 20]\nand the FNO [17, 18], they directly learn the solution operators of PDEs, for example, the mapping\nfrom the right-hand side of Poisson equation to its solution. These methods present a new landscape\nfor understanding physical reality, one may study the causality via a data-driven manner besides\nthe PDE model. Recently numerous neural operators are proposed to achieve higher performance\n[7, 9, 12, 13, 25, 27, 37], as this field is developing rapidly. There are also theoretical works for the\nabove neural operators including error analysis [15, 16, 23].\nThe most attractive neural operator for this work is the multiple-input operator network\n(MIONet) [13]. MIONet firstly faces up to the problems with multiple inputs, and generalizes\nthe architecture and the theory of DeepONet by the tool of tensor decomposition. Such architec-\nture also induces the tensor-based models for solving eigenvalue problems [10, 36]. An important\ncharacteristic of DeepONet/MIONet series architectures is that they encode the output functions\nas neural networks, rather than discrete vectors. This feature leads to efficient interpolation for the\npredicted solutions, and also allows the computing of derivatives of output functions during train-\ning, which prompts the work of physics-informed DeepONets [35] that learns the solution operator\nwithout any input-output pairs of data, by training a PINN-style loss. Another interesting work\nbased on DeepONet is the HINTS [41], which in fact inspires this work. Almost all of the works\nof neural operators use a pure machine learning-style to solve the PDE problems, and they break\naway from traditional PDE numerical methods. The neural operators are developed for fast solving,\nbut usually have limited prediction accuracy, while the traditional numerical methods can provide\nsolutions as accurate as needed at the cost of long solving time. Surprisingly, HINTS combines the\ntwo methods, i.e. the traditional numerical solver and the neural operator. It has been shown a\nspectral bias/frequency principle of the training behavior of NNs in [24, 40], i.e. the NNs often\nfit functions from low to high frequency during the training. Based on this feature, [11] employs\nthe deep learning solutions (e.g. PINNs) to initialize the iterative methods for PDEs. HINTS\nfurther utilizes the spectral bias, where a pre-trained DeepONet is periodically used to eliminate\nthe low-frequency components of the error during the iteration process, thus achieve acceleration\nsince the smoothers (e.g. Gauss-Seidel) are difficult to deal with the low-frequency errors. This\nidea is similar to the mulltigrid method. A significant difference is that HINTS does not require\nthe users to generate meshes, and the cost of this method has been transferred to the developers\nwho are responsible for the pre-trained DeepONets.\nContributions. As HINTS studies the hybrid method empirically, the lack of theoretical\nguidance leads to the imperfection of this method. In this work, we propose a hybrid iterative\nmethod based on MIONet. We theoretically show that MIONet is a more reasonable framework to\ncomplete the hybrid iterative method. An ideal operator regression model for the hybrid method\nis expected to satisfy the following three key points:\n21.Supporting multiple inputs. Taking the Poisson equation\n(\n−∇ · (k∇u) =f in Ω,\nu= 0 on ∂Ω,(1)\nas an example, its solution operator u=G(k, f) has two inputs, hence the chosen operator\nregression model should efficiently deal with such multiple-input case.\n2.Separating linearity and nonlinearity. Since the solution operator G(k, f) of (1) is linear\nw.r.t. fbut nonlinear w.r.t. k, the operator regression model M(k, f) is also expected to be\nlinear w.r.t. fand nonlinear w.r.t. k. The linearity on fwill guarantee the convergence of\nthe hybrid iterative method which will be discussed later.\n3.Supporting efficient interpolation. After obtaining the predicted solution by operator\nregression model M, we have to compute the values of M(k, f)(·) at the grid points which\nare independent of the model M, so that the model Mshould support efficient interpolation.\nFortunately, we find that MIONet satisfies all the three points perfectly, while other models do not\n(note that MIONet is a high-level framework, and one may design a targeted architecture for each\nbranch net on a specific problem, rather than directly use FNNs.). With the proposed MIONet-\nbased hybrid iterative method, we systematically analyze its theoretical properties, including the\nconvergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors\nof the discretization and the model inference. We show the theoretical results for the frequently-\nused smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of\nthe convergence rate of the hybrid method w.r.t. the model correction period, which indicates a\nminimum point to make the hybrid iteration converge fastest. Several numerical examples including\nthe Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are shown to verify our\ntheoretical results.\nThis paper is organized as follows. We first briefly introduce the operator regression model\nMIONet as the necessary tool in Section 2. In Section 3, we present the algorithm of the proposed\nhybrid iterative method in detail. We then comprehensively analyze the theoretical properties of\nthis method in Section 4. The numerical experiments are shown in Section 5. Finally, Section 6\nsummarizes this work.\n2 Operator regression with MIONet\nLet us simply introduce the operator regression model MIONet, which is a necessary tool for the\nlater discussion of the hybrid iterative method. All the content of this section can be found in the\nMIONet paper [13], and the readers may refer to that paper for more details if interested. The\nreaders familiar with MIONet may skip this section.\nLetX1, X2, ..., X nandYben+ 1 Banach spaces, and Ki⊂Xi(i= 1, ..., n ) is a compact set.\nNow we aim to learn a continuous operator\nG:K1× ··· × Kn→Y,(v1, ..., v n)7→u,\nwhere vi∈Kiandu=G(v1, ..., v n). Such Gform the space C(K1× ··· × Kn, Y) which is studied\nbelow. Note that Ycould be any Banach space. We usually consider Yas a function space, such\nasY=C(K0), L2(K0), H1(K0) for a compact domain K0.\nTo deal with the inputs from infinite-dimensional Banach spaces, it is necessary to firstly project\nthem onto finite-dimensional spaces. Here we utilize the tools of Schauder basis and canonical\nprojections.\n3Definition 1 (Schauder basis) .LetXbe an infinite-dimensional normed linear space. A sequence\n{ei}∞\ni=1inXis called a Schauder basis of X, if for every x∈Xthere is a unique sequence of\nscalars {ai}∞\ni=1, called the coordinates of x, such that\nx=∞X\ni=1aiei.\nWe show two useful examples of Schauder basis as follows.\nExample 1. Faber-Schauder basis of C[0,1].Given distinct points {ti}∞\ni=1which is a dense\nsubset in [0,1]witht1= 0,t2= 1. Let e1(t) = 1 ,e2(t) =t, and ek+1is chosen as an element, such\nthate1, ..., e k, ek+1is a basis of the (k+ 1)-dimensional space which consists of all the piecewise\nlinear functions with grid points {ti}k+1\ni=1.\nExample 2. Fourier basis of L2[0,1].Any orthogonal basis in a separable Hilbert space is a\nSchauder basis.\nWe denote the coordinate functional of eibye∗\ni, and thus\nx=∞X\ni=1e∗\ni(x)ei,∀x∈X.\nThen for a positive integer n, the canonical projection Pnis defined as\nPn(x) =Pn ∞X\ni=1e∗\ni(x)ei!\n=nX\ni=1e∗\ni(x)ei.\nWe have the following property for Pn, according to which, we can represent points in an\ninfinite-dimensional Banach space by finite coordinates within a sufficiently small projection error.\nProperty 1 (Canonical projection) .Assume that Kis a compact set in a Banach space Xequipped\nwith a Schauder basis and corresponding canonical projections Pn, then we have\nlim\nn→∞sup\nx∈K∥x−Pn(x)∥= 0.\nFor convenience, we decompose the Pnas\nPn=ψn◦φn,\nwhere φn:X→Rnandψn:Rn→Xare defined as\nφn(x) = (e∗\n1(x), ..., e∗\nn(x))T, ψ n(α1, ..., α n) =nX\ni=1αiei.\nTheφn(x) are essentially the truncated coordinates for x. Moreover, sometimes we can further\nreplace {e1, ..., e n}with an equivalent basis for the decomposition of Pn, i.e.,\nˆφn(x) =Q(e∗\n1(x), ..., e∗\nn(x))T,ˆψn(α1, ..., α n) = (e1, ..., e n)Q−1(α1, ..., α n)T,\nwith a nonsingular matrix Q∈Rn×n. For example, when applying the Faber-Schauder basis\n(Example 1), instead of using the coordinates based on the sequence {ei}∞\ni=1, we adopt the function\n4values evaluated at certain grid points as the coordinates, which is the same as the linear element\nbasis in the finite element method.\nBy considering the canonical projection property and the injective tensor product decomposition\nC(K1×K2× ··· × Kn, Y)∼=C(K1)ˆ⊗εC(K2)ˆ⊗ε···ˆ⊗εC(Kn)ˆ⊗εY, (2)\nwe can easily obtain the following approximation result.\nTheorem 1. Suppose that X1, ..., X n, Yare Banach spaces, Ki⊂Xiare compact sets, and Xi\nhave a Schauder basis with canonical projections Pi\nq=ψi\nq◦φi\nq(truncate the previous q terms).\nAssume that G:K1× ··· × Kn→Yis a continuous operator, then for any ϵ >0, there exist\npositive integers p, qi, continuous functions gi\nj∈C(Rqi),uj∈Y, such that\nsup\nvi∈Ki\r\r\r\r\r\rG(v1, ..., v n)−pX\nj=1g1\nj(φ1\nq1(v1))···gn\nj(φn\nqn(vn))·uj\r\r\r\r\r\r\nY< ϵ. (3)\nIn addition, if Gis linear with respect to vi, then linear gi\njis sufficient.\nThe MIONet is immediately constructed based on this theorem, where we just replace the gi\nj\nwith neural networks. Assume that now Y=C(K0) (L2(K0),H1(K0) etc. are also available) for\na compact domain K0.\nMIONet. We construct the MIONet according to Eq. (3). Specifically, gi= (gi\n1, ..., gi\np)T∈\nC(Rqi,Rp) and f= (u1, ..., u p)T∈C(K0,Rp) are approximated by neural networks ˜gi(called branch\nneti) and ˜f(called trunk net). Then the network (Figure 1) is\n˜G(v1, ..., v n)(y) =S\n˜g1(φ1\nq1(v1))|{z}\nbranch 1⊙··· ⊙ ˜gn(φn\nqn(vn))|{z}\nbranch n⊙˜f(y)|{z}\ntrunk\n+b, (4)\nwhere φi\nqitruncates the previous qicoordinates w.r.t. the corresponding Schauder basis as defined\nin Eq. (3), ⊙is the Hadamard product (element-wise product), e.g.\n(x1, ..., x p)⊙(y1, ..., y p) = (x1y1, ..., x pyp), (5)\nandSis the summation of all the components of a vector, b∈Ris a trainable bias.\nThe universal approximation theorem for MIONet can be easily obtained via Theorem 1 and\nthe universal approximation theorems for the applied neural networks (i.e. ˜giand˜f) such as FNNs.\nTheorem 2 (Universal approximation theorem for MIONet) .Under the setting of Theorem 1 with\nY=C(K0), L2(K0), H1(K0)etc., for any ϵ >0, there exists a MIONet ˜G, such that\n\r\r\rG −˜G\r\r\r\nC(K1×···× Kn,Y)< ϵ.\nIn addition, the branch nets can be chosen as linear maps (i.e. one-layer FNN without activation\nand bias) if Gis linear with respect to the corresponding inputs.\nUp to now, we have already shown the architecture and properties of MIONet, which will be\napplied to the iterative methods later.\n5… …Element -wise product𝒢𝑣1,…,𝑣𝑛𝑦\n𝑣1 𝑣𝑛 𝑦Coordinate projectionSum\nBranch net1\n𝜑𝑞11𝑣1 𝜑𝑞𝑛𝑛𝑣𝑛Branch net𝑛Trunk net+Figure 1: An illustration of the architecture of MIONet for general operators. All the\nbranch nets and the trunk net have the same output dimension, whose outputs are merged together\nvia the Hadamard product and then a summation.\n3 A hybrid iterative method\nWe present a new MIONet-based hybrid iterative method for general linear equations:\n(\nLαu=f in Ω,\nu=g on∂Ω,(6)\nwhere Lαis a linear differential operator depends on α.α={α1, α2, ...}is a finite set consisting\nof several parameters which can be scalars, vectors, functions, etc.. Other boundary conditions are\nalso available.\nWithout loss of generality we take the well-known Poisson equation as an example. Consider\n(\n−∇ · (k∇u) =f in Ω,\nu= 0 on ∂Ω,(7)\nwhere Ω ⊂Rdis a bounded open set. Here f∈L2(Ω), k∈L∞(Ω), and k≥Ca.e. for a constant\nC >0, so that it is a classical second-order elliptic equation. The variational formula is\nFinding u∈H1\n0(Ω) such thatZ\nΩk∇u· ∇vdx=Z\nΩfvdx, ∀v∈H1\n0(Ω).(8)\nNow assume that Ω is a bounded polyhedral domain, This a quasi-uniform and shape regular\ntriangulation of Ω. We apply the simple linear element space\nVh={v∈C(Ω) :v|K∈P1(K),∀K∈ Th} ⊂V:=H1\n0(Ω), (9)\n6where P1(K) denotes the function space consisting of all the linear polynomials over K. The finite\nelement method is\nFinding uh∈Vhsuch thatZ\nΩk∇uh· ∇vhdx=Z\nΩfvhdx,∀vh∈Vh.(10)\nLet{ϕi}nh\ni=1be the nodal basis functions corresponding to all the interior nodes of Th. Then we\nobtain a discrete linear system for Eq. (10) as\nAhµh=bh, (11)\nwhere Ah= (R\nΩk∇ϕi· ∇ϕjdx)nh×nh,bh= (R\nΩfϕidx)nh×1. Denote µh= (α1, ..., α nh)T, then\nuh:=Pnh\ni=1αiϕiis the numerical solution obtained by the above system. What we next to do is\nsolving Eq. (11) via a suitable iterative method.\nFor convenience, we temporarily omit the “ h” in above notations, but we keep in mind that\nthey indeed depend on h. For the discrete linear system\nAµ=b, (12)\nwe consider the iterative form\nµ(m+1)=µ(m)+B(b−Aµ(m)), µ(0)= 0, (13)\nwhere Bis an approximation of A−1. The frequently-used choices of Bare\nB=\n\n(ω−1I)−1, Richardson (14)\n(ω−1D)−1, Jacobi (damped) (15)\n(ω−1D+L)−1,Gauss-Seidel (SOR) (16)\nwhere A=D+L+U,Dis the diagonal of A,LandUare the strictly lower and upper triangular\nparts of A, respectively. The ωis usually considered as the relaxation parameter to be chosen. By\nperforming a suitable iterative method, we will get a numerical solution µfor system (12).\nBriefly, our hybrid iterative method is just inserting some inference steps via MIONet into the\niterative process. Now we return to the operator regression model MIONet. For the problem (7),\nwhat we are concerned with is in fact the solution operator\nG: (k, f)7→u. (17)\nNow assume that we have a dataset as T={(ki, fi), ui}N\ni=1satisfying G(ki, fi) =ui. We pre-train\na MIONet, denoted by M, on this dataset with loss function\nL(θ) =1\nNNX\ni=1∥M(ki, fi;θ)−ui∥2\nY, (18)\nwhere Ycould be C(Ω),L2(Ω),H1(Ω), etc.. Note that here we set Mlinear with respect to fdue\nto Theorem 1. An illustration is presented in Figure 2. Denote r(m):=b−Aµ(m), which is the\nresidual vector of m-th step. We expect to find a residual function related to the residual vector\nr(m). Define the residual function ¯ r(m):=Pn\ni=1βiϕi, where β= (β1, ..., β n)Tsatisfies\nHβ=r(m), H :=\u0012Z\nΩϕiϕjdx\u0013\nn×n. (19)\n7Element -wise product𝑢𝑥,𝑦=𝒢𝑘,𝑓𝑥,𝑦\n𝑘 𝑓 𝑥Sum\n+\nNonlinear Linear Nonlinear\n𝑦(𝑘1,𝑘2,…,𝑘𝑞1) (𝑓1,𝑓2,…,𝑓𝑞2)Figure 2: An illustration of the architecture of MIONet for the 2-d Poisson equation.\nSpecially, we let ¯ r(0)=f. Note that ¯ r(m)is linear with respect to r(m), we denote their relationship\nby a linear operator Has\n¯r(m)=Hr(m),H:= (ϕ1, ..., ϕ n)H−1, m≥1. (20)\nWe further define the interpolation operator as\nI(u) := ( u(x1), ..., u (xn))T, (21)\nwhere xi∈Ω is the corresponding nodal coordinate of ϕi, i.e. ϕi(xj) =δj\ni(Dirac delta function).\nThen the hybrid iterative method can be written as\nµ(m+1)=(\nµ(m)+I(M(k,H(b−Aµ(m)))), m≡0 mod M (22)\nµ(m)+B(b−Aµ(m)), otherwise (23)\nwhere Mis a positive integer to be chosen, we regard it as the correction period. The algorithm is\nsummarized in Algorithm 1.\nApproximation and padding for residual function. There is a new linear system (19)\ninvolved. We may choose an approximation for β, for example:\nβ≈diag\nX\njZ\nΩϕi·ϕjdx\n−1\nr(m)≈diag \u0012Z\nΩϕidx\u0013−1!\nr(m). (24)\nWith the solved β, we obtain the ¯ r(m)=Hr(m)expressed as a continuous piece-wise linear function\nw.r.t. {ϕi}, which does not contain the boundary nodes. Hence the current ¯ r(m)equals to 0 on ∂Ω,\nwhich is regarded as a “zero padding”. Here we may use other better padding for ¯ r(m), for example,\nadd boundary nodes whose values equal to the values of their corresponding nearest interior nodes\n(we can also apply other padding strategy like reflection padding, etc.). For convenience, we still\nuse the notation ¯ r(m)andH, i.e.\n¯r(m)=Hr(m):= Padding \n(ϕ1, ..., ϕ n)diag \u0012Z\nΩϕ1dx\u0013−1\n, ...,\u0012Z\nΩϕndx\u0013−1!\nr(m)!\n. (25)\n8Algorithm 1 Hybrid iterative method for Poisson equation\nInput: Parameters k, f, mesh Th, iterative method B,\npre-trained MIONet M, and correction period M\nOutput: Numerical solution µ\n1. Obtain a discrete system\nAµ=b\nbased on Thfor Eq. (7).\n2. Initialize µ(1)=I(M(k, f)),r(1)=b−Aµ(1),m= 1\n3.while (µ(m)does not satisfy the convergence condition)\nif(m≡0 mod M)\nµ(m+1)=µ(m)+I(M(k,Hr(m)))\nelse\nµ(m+1)=µ(m)+Br(m)(in practice we update µ(m)in-place without computing B)\nr(m+1)=b−Aµ(m+1)\nm=m+ 1 (set mtom+ 1)\nend\n4.return µ=µ(m)\nNote that the above approximation brings an acceptable error for this hybrid method. Since\n¯r(0)=f,µ(1)is exactly the prediction by MIONet. We may also think of µ(1)as an initial guess\nprovided by MIONet for the iterative method. When feeding ¯ r(m)toM, we need to evaluate the\nlinear element function ¯ r(m)at some sampling points (usually with smaller scale compared to the\nnumber of nodes n) forM.\nEfficient interpolation. The interpolation Iis easy to compute, since the prediction\nM(k,¯r(m))(·) (26)\nis expressed by a neural network (encoded as a trunk net) rather than a discrete vector, and we\nonly need to directly feed the interpolated points into the trunk net of the ONet. This is indeed\nan important advantage of the ONets series architectures.\nHybrid iteration with multigrid method. Since there is no requirement for the basic\niterative method in this algorithm, we can also utilize other advanced iterative methods such as\nthe multigrid method here. We will show more details in the experiment part.\nInhomogeneous boundary condition. Here we further investigate the inhomogeneous\nboundary condition. Consider\n(\n−∇ · (k∇u) =f in Ω,\nTu=g on∂Ω,(27)\nwhere Tis the trace operator. In such a case, we aim to learn an operator mapping from ( k, f, g )\nto the solution u, i.e.,\nG: (k, f, g )7→u. (28)\nOf course we can train an MIONet M(k, f, g ) to learn this operator. However, this strategy losses\nthe linearity w.r.t. the residual vector/function during the iteration, and we strive to preserve this\nlinearity to guarantee the convergency of the hybrid method. Denote by Ethe trace extension\noperator for (27), then T(Eg) =gforg∈H1/2(∂Ω). We derive that\n(\n−∇ · (k∇u0) =f+∇ ·(k∇Eg) in Ω ,\nTu0= 0 on ∂Ω,(29)\n9where u=u0+Eg. Therefore the model can be modified to\n˜M(k, f, g ) :=M(k, f+∇ ·(k∇Eg)) +Eg, (30)\nwhere Mis a standard MIONet which is linear w.r.t. the second input as before. Although the\ntrace extension operator Eis unique defined, we may choose a suitable constructed extension as\nneeded, for example the continuous piece-wise linear function related to a grid, which takes 0 at\nthe interior nodes and matches gat the boundary nodes. Note that here we regard f+∇·(k∇Eg)\nas an element in H−1(Ω).\nThere is in fact another way which is much easier to achieve. It is not difficult to find that\nG(k, f, g ) is linear w.r.t. ( f, g)∈L2(Ω)×H1/2(∂Ω), i.e.,\nG(k,(λ1(f1, g1) +λ2(f2, g2))) = λ1G(k,(f1, g1)) +λ2G(k,(f2, g2)), (31)\nthen we naturally apply the model\n˜M(k, f, g ) :=M(k,(f, g)), (32)\nwhere ( f, g) is a simple cartesian product of fandg, and it will be fed into the linear branch net\nofM. In such a case, the discrete algebraic system of (27) is augmented with several rows and\ncolumns taking into account the boundary nodes, where the boundary block is the identity. The\nunknown and residual vectors contain not only the coefficients of the interior nodes but also the\nboundary nodes. We apply this strategy in our experiment.\nNecessary convergence condition for operator regression models. There is a necessary\ncondition for the convergence of the hybrid iterative method. Assume that the hybrid method\nconverges with a model M, then\nlim\ns→∞µ(sM+1)= lim\ns→∞\u0010\nµ(sM)+I(M(k,Hr(sM)))\u0011\n, (33)\nand we have\nI(M(k,0)) = 0 . (34)\nAs the interpolation operator Iis related to the mesh Th, and the above equation holds for arbitrary\nITh, we know\nM(k,0) = 0 . (35)\nObviously, Eq. (35) holds as long as Mis linear w.r.t. the second input.\n4 Theoretical analysis\nConsider the one-dimensional two-point boundary value problem of system (7) where k= 1, Ω =\n(0,1), i.e.,(\n−u′′=f, x ∈(0,1),\nu(0) = u(1) = 0 .(36)\nWe choose a uniform grid with size h=1\nn+1and apply the linear element bases functions {ϕi}n\ni=1\nwhose nodal points are interior, then the above system leads to a discrete system as\nAµ=b, (37)\n10where\nA=1\nh\n2−1\n−1 2 −1\n.........\n−1 2 −1\n−1 2\n, µ =\nµ1\nµ2\n...\nµn−1\nµn\n, b =\nb1\nb2\n...\nbn−1\nbn\n, (38)\nandbi=R1\n0fϕidx. We can easily write out the eigenvalues and the eigenvectors of Aas\nλi=4\nhsin2(π\n2hi), ξi= (ξi\n1, ..., ξi\nn)T, ξi\nj=√\n2hsin(iπhj),1≤i, j≤n. (39)\nDenote Ξ = ( ξ1, ..., ξn), then Ξ is symmetric and orthogonal, i.e. ΞT= Ξ and Ξ−1= Ξ.\n4.1 Convergence condition\nRecall that we apply the hybrid iterative method (22-23) to solve (37). For convenience, here we\nsimply set ¯ r(0)=Hb=Hr(0), so that we have ¯ r(m)=Hr(m)for all m≥0. We further denote that\n¯M(r) :=I(M(1,Hr)), (40)\nthus ¯M:Rn→Rnis a linear map. Denote that\ne(m):=µ−µ(m), (41)\nthen we find that\ne(1)=µ−¯M(r(0)) = (I−¯MA)e(0), (42)\nand for 1 ≤m≤M−1,\ne(m+1)= (I−BA)e(m), (43)\nthus\ne(M)= (I−BA)M−1e(1)= (I−BA)M−1(I−¯MA)e(0). (44)\nConsequently we have\ne(sM)=Use(0), (45)\nwhere\nU:= (I−BA)M−1(I−¯MA). (46)\nUp to now, we have derived the convergence results of the hybrid iterative method.\nTheorem 3. The hybrid iterative method (22-23) converges if and only if\nρ\u0000\n(I−BA)M−1(I−¯MA)\u0001\n<1, (47)\nwhere ρ(·)denotes the spectral radius.\nCorollary 1. There exists an Mlarge enough such that the hybrid iterative method (22-23) con-\nverges, if the applied original iterative method converges, i.e.\nρ(I−BA)<1. (48)\nThe corollary is also obvious since ( I−BA)M−1(I−¯MA)→0 asM→ ∞ ifρ(I−BA)<1.\n11Corollary 2. The hybrid iterative method (22-23) with Richardson iteration, i.e. B=ωI, con-\nverges if\n0< ω <2\nρ(A), M ≥2 +\u0016\n−ln∥I−¯MA∥2\nlnρ(I−ωA)\u0017\n. (49)\nProof. If 0< ω <2\nρ(A), we have\nρ(I−BA) =ρ(I−ωA)<1,\nthus the Richardson iteration converges. Furthermore, if Eq. (49) holds, then\nρ\u0000\n(I−BA)M−1(I−¯MA)\u0001\n≤\r\r(I−ωA)M−1(I−¯MA)\r\r\n2\n≤ ∥I−ωA∥M−1\n2·\r\rI−¯MA\r\r\n2\n=ρ(I−ωA)M−1·\r\rI−¯MA\r\r\n2\n< ρ(I−ωA)1−ln∥I−¯MA∥2\nlnρ(I−ωA)−1·\r\rI−¯MA\r\r\n2\n= 1.\nIt is noteworthy that there is no requirement on the loss of the MIONet Mto guarantee\nthe convergence. However, an Mwith a large loss cannot accelerate the iteration, which will be\ndiscussed later.\n4.2 Error of model inference\nSince the MIONet usually learns the low-frequency data, we assume that the dataset includes the\nprevious several eigenfunctions as\nT={(1,¯λi¯ξi),¯ξi}n0\ni=1,1< n0≤n, (50)\nwhere ¯λi,¯ξiare the eigenvalue and the eigenfunction of the original system, i.e.,\n¯λi=i2π2,¯ξi(x) = sin( iπx), 1≤i≤n. (51)\nFor convenience, we consider the loss\nL(θ) =1\nn0n0X\ni=1ℓi(θ) :=1\nn0n0X\ni=1∥M(1,¯λi¯ξi;θ)−¯ξi∥2\nC[0,1]. (52)\nWe further impose a generalization assumption on the MIONet for this case.\nAssumption 1 (generalization assumption) .Assume that the MIONet keeps a consistent Lipschitz\nconstant L >0w.r.t. the second input during training, i.e.\n∥M(k, f1;θ)− M (k, f2;θ)∥C[0,1]≤L∥f1−f2∥C[0,1] (53)\nholds for f1, f2∈C[0,1]andθ∈ {θ1, θ2, ...}. The set {θt}is regarded as the sequence of parameters\nduring the training process. Ldepends only on k.\n12LetHbe defined as in Eq. (25) and extended with replication padding. Then\nH(r\nh\n2ξi) = (ϕ0, ϕ1, ..., ϕ n, ϕn+1)·r\n1\n2h(ξi\n1, ξi\n1, ξi\n2, ..., ξi\nn−1, ξi\nn, ξi\nn)T, (54)\nwhere ϕ0andϕn+1are the two boundary nodal functions, hence\n\r\r\r\r\rH(r\nh\n2ξi)−¯ξi\r\r\r\r\r\nC[0,1]\n= max\u0000\r\rsin(iπh)−¯ξi\r\r\nC[0,h],\r\rsin(iπh)ϕ1+ sin(2 iπh)ϕ2−¯ξi\r\r\nC[h,2h], ...,\n\r\rsin((n−1)iπh)ϕn−1+ sin( niπh)ϕn−¯ξi\r\r\nC[(n−1)h,nh],\r\rsin(niπh)−¯ξi\r\r\nC[nh,1]\u0001\n≤iπh.(55)\nConsequently,\n\r\r\r\r\r¯M(r\nh\n2ξi)−I(M(1,¯ξi))\r\r\r\r\r\n∞=\r\r\r\r\rI(M(1,H(r\nh\n2ξi)−¯ξi))\r\r\r\r\r\n∞\n≤\r\r\r\r\rM(1,H(r\nh\n2ξi)−¯ξi)\r\r\r\r\r\nC[0,1]\n≤L1\r\r\r\r\rH(r\nh\n2ξi)−¯ξi\r\r\r\r\r\nC[0,1]\n≤L1iπh,(56)\nwhere L1is the Lipschitz constant defined as in Assumption 1. By Eq. (52),\n\r\r\r\r\r¯M(¯λir\nh\n2ξi)−r\n1\n2hξi\r\r\r\r\r\n∞=\r\r\r\r\r¯M(¯λir\nh\n2ξi)−I(M(1,¯λi¯ξi)) +I(M(1,¯λi¯ξi))−I(¯ξi)\r\r\r\r\r\n∞\n≤¯λi\r\r\r\r\r¯M(r\nh\n2ξi)−I(M(1,¯ξi))\r\r\r\r\r\n∞+\r\rI(M(1,¯λi¯ξi))−I(¯ξi)\r\r\n∞\n≤¯λiL1iπh+ℓi\n=L1π3i3h+ℓi.(57)\nNow denote\nei:=ξi−¯M(λiξi),1≤i≤n, (58)\nand we get the estimate\n∥ei∥2=\r\rξi−¯M(λiξi)\r\r\n2\n=λi\n¯λir\n2\nh\r\r\r\r\r¯M(¯λir\nh\n2ξi)−r\n1\n2hξi+r\n1\n2hξi−¯λi\nλir\nh\n2ξi\r\r\r\r\r\n2\n≤λi\n¯λir\n2\nh \r\r\r\r\r¯M(¯λir\nh\n2ξi)−r\n1\n2hξi\r\r\r\r\r\n2+\r\r\r\r\rr\n1\n2hξi−¯λi\nλir\nh\n2ξi\r\r\r\r\r\n2!\n≤λi\n¯λir\n2\nh \n√n\r\r\r\r\r¯M(¯λir\nh\n2ξi)−r\n1\n2hξi\r\r\r\r\r\n∞+\r\r\r\r\rr\n1\n2hξi−¯λi\nλir\nh\n2ξi\r\r\r\r\r\n2!\n.(59)\n13With the results of (57-59) and the orthogonality of Ξ, we further derive that\n∥ei∥2≤λi\n¯λir\n2\nh \n√n(L1π3i3h+ℓi) +\r\r\r\r\rr\n1\n2hξi−¯λi\nλir\nh\n2ξi\r\r\r\r\r\n2!\n=4 sin2(π\n2hi)\nπ2i2h2p\n2(1−h)(L1π3i3h+ℓi) +\f\f\f\f\f4 sin2(π\n2hi)\nπ2i2h2−1\f\f\f\f\f.(60)\nIt is elementary to see\n1−1\n3x2≤sin2(x)\nx2≤1,∀0̸=x∈R, (61)\nso that\n∥ei∥2≤4 sin2(π\n2hi)\nπ2i2h2p\n2(1−h)(L1π3i3h+ℓi) +\f\f\f\f\f4 sin2(π\n2hi)\nπ2i2h2−1\f\f\f\f\f\n≤√\n2L1π3i3h+√\n2ℓi+ 1−(1−1\n3(π\n2hi)2)\n=√\n2L1π3i3h+π2\n12i2h2+√\n2ℓi,1≤i≤n0.(62)\nIt is summarized as\n∥ei∥2=O(i3h) +O(ℓi),1≤i≤n0. (63)\nThe error includes two parts, and the terms O(i3h),O(ℓi) are caused by the errors of the discretiza-\ntion and the loss, respectively. The first part of this error is inevitable even though the loss ℓihas\nbeen trained to zero, since there exists a gap between the eigenvector (eigenvalue) of the discrete\nsystem (37) and the eigenfunction (eigenvalue) of the original system (36), which becomes larger\nasiincreases.\nThe estimate (63) can be equivalently written as\n∥ei∥2=O\u0012i3\nn\u0013\n+O(ℓi),1≤i≤n0. (64)\nEq. (64) tells that i≪nis necessary to ensure that ∥ei∥2is small enough. Therefore n0, the size\nof the dataset Tdefined in (50), is no need to be too large compared to n. We reasonably assume\nthatn0≪n. For a fixed 1 ≤i≤n0, we have derived that\nlim\nh→0,ℓi→0∥ei∥2= 0,1≤i≤n0. (65)\nNote that here we consider the ground truth, i.e. the analytical eigenvectors and eigenvalues, as\ndata. If we utilize numerical solutions as data, the error will decrease, since the gap between data\nand the discrete system becomes smaller. As a matter of fact, if we replace the ¯λiin the dataset\n(50) by the corresponding numerical eigenvalueλi\nh, then the error termπ2\n12i2h2in the last equality\nof (62) will disappear.\n4.3 Spectral analysis\nWith the above discussion, we are able to analyze the spectral behavior and the convergence speed of\nthe proposed hybrid iterative method. Now we express e(0), e(1)(see Section 4.1) by the eigenvectors\nofAas (\ne(0)=µ−µ(0)=µ=Pn\ni=1αiξi= Ξ·α,\ne(1)= (I−¯MA)e(0)=Pn\ni=1αi(I−λi¯M)ξi=:Pn\ni=1βiξi= Ξ·β,(66)\n14where α:= (α1, ..., α n)T,β:= (β1, ..., β n)T. We further set\n(\nY:= Ξ−1¯MΞ = ( ξ1, ..., ξn)−1¯M(ξ1, ..., ξn),\nZ:= Ξ−1BΞ = ( ξ1, ..., ξn)−1B(ξ1, ..., ξn),(67)\nthen by direct computing we have\nβ= (I−YΛ)α,Λ = diag( λ1, ..., λ n), (68)\nso that\ne(1)= Ξ·(I−YΛ)α. (69)\nSimilarly, we can derive that\ne(M)= Ξ·(I−ZΛ)M−1(I−YΛ)α. (70)\nConsequently, we obtain the final error as\ne(sM)= Ξ·Tsα, (71)\nwhere\nT:= (I−ZΛ)M−1(I−YΛ). (72)\nNext we proceed discussions on T. Note that the Ydefined in (67) depends on the trained\noperator regression model MIONet, i.e., M. Assume that ¯Mperfectly learns the eigenvectors of\nEq. (37), so that\nA¯M(ξi) =ξi, (73)\nwhich leads to\n¯M(ξi) =λ−1\niξi. (74)\nThen\nY= Ξ−1¯MΞ = Λ−1, (75)\nthus\nYΛ =I, T = 0. (76)\nIn such a situation, the first iterative step by MIONet in fact gives the right solution for the discrete\nsystem (37). However, we have to take into account the error of MIONet. Keep in mind that Yis\nin fact an approximation of Λ−1, therefore we expect\nI−YΛ≈0, (77)\nat least for the first few columns. As discussed in Section 4.2, we recall that\nei=ξi−¯M(λiξi),1≤i≤n, (78)\nand\n∥ei∥2=O(i3h) +O(ℓi),1≤i≤n0. (79)\nThe errors of MIONet on the previous n0eigenvectors can be controlled by hand the loss ℓi. Hence\nwe assume that e1, ..., e n0are small enough with sufficiently training for MIONet, while en0+1, ..., e n\nare uncontrolled. Therefore,\nI−YΛ =Ξ−1·(Ξ−¯MΞΛ)\n=Ξ·(e1, ..., e n0, en0+1, ..., e n)\n=\u0002\nΞ·(e1, ..., e n0) Ξ·(en0+1, ..., e n)\u0003\n,(80)\n15whose first n0columns are small enough. It indicates that a well-trained MIONet can eliminate\nthe low-frequency components of the error.\nReturn to the final error e(sM), we have\ne(sM)= Ξ·((I−ZΛ)M−1(I−YΛ))sα\n= Ξ·(I−ZΛ)M−1\u0000\n(I−YΛ)(I−ZΛ)M−1\u0001s−1(I−YΛ)α\n= Ξ·(I−ZΛ)M−1˜Ts−1(I−YΛ)α,(81)\nwhere\n˜T:= (I−YΛ)(I−ZΛ)M−1. (82)\nWe subsequently study how the ˜Tacts on a vector v= (v1, ..., v n)T∈Rn. Now assume that we\nchoose the Richardson iteration and set ω=h\n4which satisfies the convergence condition, then\nZ=h\n4Iand\n˜Tv= (I−YΛ)(I−h\n4Λ)M−1v\n= Ξ−1·(Ξ−¯MΞΛ)\u0010\ncos2(M−1)(π\n2h)v1, ...,cos2(M−1)(π\n2hn)vn\u0011T\n= Ξ·(e1, ..., e n)\u0010\ncos2(M−1)(π\n2h)v1, ...,cos2(M−1)(π\n2hn)vn\u0011T\n= Ξ· nX\ni=1cos2(M−1)(π\n2hi)viei!\n,(83)\ntherefore\n\r\r\r˜Tv\r\r\r\n2=\r\r\r\r\rnX\ni=1cos2(M−1)(π\n2hi)viei\r\r\r\r\r\n2\n≤nX\ni=1cos2(M−1)(π\n2hi)|vi|∥ei∥2\n≤ nX\ni=1cos4(M−1)(π\n2hi)∥ei∥2\n2!1\n2 nX\ni=1|vi|2!1\n2\n≤ nX\ni=1cos2(M−1)(π\n2hi)∥ei∥2!\n∥v∥2.(84)\nAs (\ncos2(π\n2hi)≤cos2(π\n2h),1≤i≤n0,\ncos2(π\n2hi)≤cos2(π\n2h(n0+ 1)) , n 0+ 1≤i≤n,(85)\nwe obtain the estimate\n\r\r\r˜Tv\r\r\r\n2≤ \ncos2(M−1)(π\n2h)n0X\ni=1∥ei∥2+ cos2(M−1)(π\n2h(n0+ 1))nX\ni=n0+1∥ei∥2!\n∥v∥2, (86)\n16which shows the change of the error within a period of Msteps. Subsequently, we write down the\ntotal error as\n\r\r\re(sM)\r\r\r\n2\n=\r\r\r(I−ZΛ)M−1˜Ts−1(I−YΛ)α\r\r\r\n2\n=\r\r\r\r(I−h\n4Λ)M−1˜Ts−1(I−Ξ−1¯MΞΛ)α\r\r\r\r\n2\n≤cos2(M−1)(π\n2h)\r\r\r˜Ts−1(I−Ξ−1¯MΞΛ)α\r\r\r\n2\n≤cos2(M−1)(π\n2h) \ncos2(M−1)(π\n2h)n0X\ni=1∥ei∥2+ cos2(M−1)(π\n2h(n0+ 1))nX\ni=n0+1∥ei∥2!s−1\n·\r\r(I−Ξ−1¯MΞΛ)α\r\r\n2.(87)\nMoreover, the last term in above inequality is\n\r\r(I−Ξ−1¯MΞΛ)α\r\r\n2\n=\r\r(Ξ−¯MΞΛ)α\r\r\n2\n=∥(e1, ..., e n)α∥2\n≤n0X\ni=1|αi|∥ei∥2+nX\ni=n0+1|αi|∥ei∥2\n≤C \ncos2(M−1)(π\n2h)n0X\ni=1∥ei∥2+ cos2(M−1)(π\n2h(n0+ 1))nX\ni=n0+1∥ei∥2!\n,(88)\nwhere C >0 is a constant independent of s. We hence have the simple relationship that\n\r\r\re(sM)\r\r\r\n2≤C· \ncos2(M−1)(π\n2h)n0X\ni=1∥ei∥2+ cos2(M−1)(π\n2h(n0+ 1))nX\ni=n0+1∥ei∥2!s\n. (89)\nFor comparison, we write down the error of the original Richardson method as\n\r\r\re(sM)\nRi\r\r\r\n2= nX\ni=1cos4sM(π\n2hi)α2\ni!1\n2\n≥cos2sM(π\n2h)|α1|=CRi·\u0010\ncos2M(π\n2h)\u0011s\n, (90)\nwhere CRi>0 is a constant under the assumption that α1̸= 0.\nNow we show the errors of the Richardson iterative method and the corresponding hybrid\nmethod as\n\n\n\r\r\re(sM)\nRi\r\r\r\n2≥CRi·\u0000\nηM\n1\u0001s, Richardson (91)\n\r\r\re(sM)\r\r\r\n2≤C·\u0010\nϵ·ηM−1\n1+R·ηM−1\n2\u0011s\n, Hybrid (92)\nwhere\nη1= cos2(π\n2h), η 2= cos2(π\n2h(n0+ 1)) , ϵ=n0X\ni=1∥ei∥2, R =nX\ni=n0+1∥ei∥2. (93)\nNote that η1≈1, 1> η1> η2>0, and ϵis usually small enough while Ris uncontrolled. Up to\nnow, we are able to compare the convergence speeds of these two methods.\n17Theorem 4. Assume that ϵ < η 1. Then there exists a K > 0, such that for any integer M > K ,\nit holds \r\r\re(sM)\r\r\r\n2<\r\r\re(sM)\nRi\r\r\r\n2(94)\nforslarge enough. Briefly, the hybrid iterative method converges faster than the original Richardson\niterative method for Mlarge enough as long as ϵ < η 1.\nProof. Ifϵ < η 1, we have\nlim\nM→∞ϵ·ηM−1\n1+R·ηM−1\n2\nηM\n1= lim\nM→∞ϵ\nη1+R\nη2·\u0012η2\nη1\u0013M\n=ϵ\nη1<1. (95)\nNext we investigate how the Minfluences the convergence rate of the hybrid iterative method.\nAccording to Eq. (92-91), the average convergence rates of the above methods are derived as the\nfollowing.\nTheorem 5 (convergence rate) .The convergence rate of the Richardson iterative method and an\nupper bound of the average convergence rate of the corresponding hybrid iterative method are\n\n\nlim\ns→∞\r\r\re(sM)\nRi\r\r\r1\nsM\n2=η1, Richardson (96)\nlim\ns→∞\r\r\re(sM)\r\r\r1\nsM\n2≤η1· \nϵ\nη1+R\nη2·\u0012η2\nη1\u0013M!1\nM\n. Hybrid (97)\nThis theorem clearly points out how the network correction step accelerates the iteration.\nDenote\nRate( M) =η1· \nϵ\nη1+R\nη2·\u0012η2\nη1\u0013M!1\nM\n. (98)\nTo visually observe the tendency of Rate( M), we for example set η1= 0.999, η2= 0.5, ϵ= 0.1, R=\n10, and plot them in Figure 3. It is shown that Rate( M) decreases first then increases, and there\nexists a minimum point of Rate( M). For Mthat is too small, the rate will be larger than 1, which\nmeans the hybrid iterative method will diverge. The rate increases and tends to the convergence\nrate of Richardson as M→ ∞ .\n4.4 Discussion for complicated cases\nAt this point, we have systematically analyzed the hybrid iterative method for the one-dimensional\ncase with Richardson iteration (or equivalently damped Jacobi iteration). For more complicated\ncase, such as the high-dimensional problem with Gauss-Seidel iteration which is a better smoother,\nwe have to apply other effective tools [1, 8]. The smoothing property of GS is conveniently analyzed\nby using local mode analysis introduced by Brandt [1]. Here let us show the idea and promote the\ndiscussion on the hybrid Gauss-Seidel iteration.\nRecall that the 2-d Poisson equation with the uniform grid and the lexicographical order can\nbe discretized as\n4µi,j−(µi−1,j+µi,j−1+µi,j+1+µi+1,j) =bi,j. (99)\nTo begin with, we add some idealized assumptions and simplifications as in the local mode analysis:\n18101102\nM0.800.850.900.951.001.051.101.15RateHybrid\nRichardsonFigure 3: An example of Rate( M).In this case η1= 0.999, η2= 0.5, ϵ= 0.1, R= 10.\n1. Assume that boundary conditions are neglected and the problem is defined on an infinite grid\nwith a period of n+ 1, i.e.\nµi+k·(n+1),j+l·(n+1)=µi,j, b i+k·(n+1),j+l·(n+1)=bi,j,∀i, j, k, l ∈Z. (100)\n2. Assume that the Gauss-Seidel iteration is implicitly defined as\n4˜µi,j−(˜µi−1,j+ ˜µi,j−1+ ¯µi,j+1+ ¯µi+1,j) =bi,j,∀i, j∈Z, (101)\nwhere ˜ µis updated from ¯ µ. Note that Eq. (101) leads to an algebraic system for ˜ µ, which is\nuniquely solvable since it is strictly diagonally dominant.\nDenote ˜ ei,j=µi,j−˜µi,jand ¯ei,j=µi,j−¯µi,j, then\n4˜ei,j−(˜ei−1,j+ ˜ei,j−1+ ¯ei,j+1+ ¯ei+1,j) = 0 . (102)\nWe expand these errors by the discrete Fourier transformation, as\n˜ei,j=X\nθ∈Θ˜c(θ)Ψθ\ni,j,¯ei,j=X\nθ∈Θ¯c(θ)Ψθ\ni,j,0≤i, j≤n, (103)\nwhere\nΨθ\ni,j=1\nn+ 1ei(iθ1+jθ2),i2=−1, θ = (θ1, θ2), (104)\nand\nΘ =\u001a2π\nn+ 1(k, l)\f\f\f\f−n\n2≤k, l≤n+ 1\n2, k, l ∈Z\u001b\n. (105)\nSubstituting (103) into (102), we obtain\nζ(θ) :=\f\f\f\f˜c(θ)\n¯c(θ)\f\f\f\f=\f\f\f\feiθ1+eiθ2\n4−eiθ1−eiθ2\f\f\f\f. (106)\nThe smoothing factor is defined as\nζρ= max\nρπ≤|θ|≤πζ(θ),|θ|:= max( |θ1|,|θ2|),0< ρ < 1. (107)\n19It can be shown that ζ1/2=ζ(π\n2,arccos(4\n5)) =1\n2, which means the GS iteration can efficiently\neliminate the high-frequency errors.\nIt is worth noting that ζ(0,0) = 1, which indicates that the lowest-frequency component will\nnever be eliminated. In fact, the problem (99) does not determine a solution, since a change of a\nconstant on µdoes not affect the relationship. For convenience, we only consider the error excluding\nthe lowest-frequency component in the following.\nBased on this framework, we proceed the discussion on the network correction step. Assume\nthat the MIONet-based iterative step ¯Msufficiently learns the low-frequency modes, i.e.,\n4[¯MΨθ]i,j−([¯MΨθ]i−1,j+ [¯MΨθ]i,j−1+ [¯MΨθ]i,j+1+ [¯MΨθ]i+1,j)≈Ψθ\ni,j,0<|θ| ≤ρπ,(108)\nwhere 0 < ρ < 1 is a constant that determines the (low-)frequency components learned. Denote\n¯MΨθ=X\nφ∈Θαθ(φ)·Ψφ, (109)\nwe substitute (109) into (108) and similarly derive that\nX\nφ∈Θ(4−2 cos( φ1)−2 cos( φ2))αθ(φ)Ψφ≈Ψθ,0<|θ| ≤ρπ, (110)\nwhich leads to\n(4−2 cos( φ1)−2 cos( φ2))αθ(φ)≈δφ\nθ(Dirac delta function) ,0<|θ| ≤ρπ. (111)\nDenote\nγφ\nθ:=δφ\nθ−(4−2 cos( φ1)−2 cos( φ2))αθ(φ), (112)\nthen we suppose that |γφ\nθ|is as small as needed for 0 <|θ| ≤ρπ.\nNow we consider the MIONet iteration step, i.e.,\n˜µ= ¯µ+¯M(¯r), (113)\nor equivalently\n˜e= ¯e−¯M(¯r), (114)\nwhere ˜ e=µ−˜µ, ¯e=µ−¯µ, and\n¯ri,j=bi,j−(4¯µi,j−(¯µi−1,j+ ¯µi,j−1+ ¯µi,j+1+ ¯µi+1,j))\n=4µi,j−(µi−1,j+µi,j−1+µi,j+1+µi+1,j)\n−(4¯µi,j−(¯µi−1,j+ ¯µi,j−1+ ¯µi,j+1+ ¯µi+1,j))\n=4¯ei,j−(¯ei−1,j+ ¯ei,j−1+ ¯ei,j+1+ ¯ei+1,j).(115)\nWe expand ˜ e,¯e,¯ras before, i.e.,\n˜e=X\nθ∈Θ˜c(θ)Ψθ,¯e=X\nθ∈Θ¯c(θ)Ψθ,¯r=X\nθ∈Θ(4−2 cos( θ1)−2 cos( θ2))¯c(θ)Ψθ, (116)\nthen we have\nX\nθ∈Θ˜c(θ)Ψθ=X\nθ∈Θ¯c(θ)Ψθ−¯M X\nθ∈Θ(4−2 cos( θ1)−2 cos( θ2))¯c(θ)Ψθ!\n=X\nθ∈Θ¯c(θ)Ψθ−X\nθ∈Θ(4−2 cos( θ1)−2 cos( θ2))¯c(θ)X\nφ∈Θαθ(φ)·Ψφ\n=X\nθ∈Θ¯c(θ)Ψθ−X\nθ∈ΘX\nφ∈Θ(4−2 cos( φ1)−2 cos( φ2))¯c(φ)αφ(θ)·Ψθ.(117)\n20Consequently,\n˜c(θ) =¯c(θ)−X\nφ∈Θ¯c(φ)(4−2 cos( φ1)−2 cos( φ2))αφ(θ)\n=X\nφ∈Θ¯c(φ)·4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)·γθ\nφ,|θ|>0.(118)\nReturn to the algorithm of the hybrid iterative method, the network correction step is performed\nfollowing M−1 GS iteration steps, so that ¯ c(φ) can be written as\n¯c(φ) = ¯c(M−1)(φ), (119)\nthen we have the estimate\n|¯c(φ)|=|¯c(M−1)(φ)| ≤(ζρ)M−1|¯c(0)(φ)|,|φ| ≥ρπ, (120)\nand the low-frequency component |¯c(M−1)(φ)|(0<|φ|< ρπ ) has a smoothing factor ζ0greater\nthan ζρ, i.e.\n|¯c(M−1)(φ)| ≤(ζ0)M−1|¯c(0)(φ)|, ζ ρ< ζ0<1,0<|φ|< ρπ. (121)\nNow denote\ne(M):=X\nφ∈Θ,|φ|>0˜c(φ)Ψφ, e(0):=X\nφ∈Θ,|φ|>0¯c(0)(φ)Ψφ, (122)\nthen\n\r\r\re(M)\r\r\r\n2=sX\n|θ|>0|˜c(θ)|2\n≤X\n|θ|>0|˜c(θ)|\n≤X\n|θ|>0X\n|φ|>0|¯c(φ)| ·\f\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f\f· |γθ\nφ|,(123)\nby (118). With (120) and (121), we further derive that\n\r\r\re(M)\r\r\r\n2≤X\n|φ|>0|¯c(M−1)(φ)| ·X\n|θ|>0\f\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f\f· |γθ\nφ|\n≤X\n0<|φ|<ρπ(ζ0)M−1|¯c(0)(φ)| ·X\n|θ|>0\f\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f\f· |γθ\nφ|\n+X\n|φ|≥ρπ(ζρ)M−1|¯c(0)(φ)| ·X\n|θ|>0\f\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f\f· |γθ\nφ|\n≤\u0000\n(ζ0)M−1·ϵ+ (ζρ)M−1·R\u0001\r\r\re(0)\r\r\r\n2,(124)\nwhere \n\nϵ=P\n0<|φ|<ρπ,|θ|>0\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f· |γθ\nφ|,\nR=P\n|φ|≥ρπ,|θ|>0\f\f\f4−2 cos( φ1)−2 cos( φ2)\n4−2 cos( θ1)−2 cos( θ2)\f\f\f· |γθ\nφ|,(125)\n21andϵis supposed to be small as discussed before. The relationship\n\r\r\re(M)\r\r\r\n2≤\u0000\n(ζ0)M−1·ϵ+ (ζρ)M−1·R\u0001\r\r\re(0)\r\r\r\n2(126)\nimmediately leads to the final upper bound of the average convergence rate\nRate( M) =ζ0· \nϵ\nζ0+R\nζρ·\u0012ζρ\nζ0\u0013M−1!1\nM\n. (127)\nAccording to the above deduction, we have studied the convergence rate of this hybrid iterative\nmethod for 2-d problem with GS iteration, which is consistent with Eq. (98). It is expected to\npromote the discussion and obtain further theoretical results for more complicated cases. We will\nkeep it for future research due to space limitations.\n5 Numerical experiments\nThe experiments are implemented in c++withEigen library for sparse matrices and basic iterations,\nandLibTorch library for model inference. The used models are pre-trained via PyTorch . The\nrunning devices are Intel(R) Core(TM) i7-10750H for CPU and NVIDIA GeForce RTX\n2070 for GPU.\n5.1 One-dimensional Poisson equation\nConsider the 1-d Poisson equation\n(\n−∇ · (k∇u) =fin (0,1),\nu(0) = u(1) = 0 .(128)\nTo firstly construct the dataset, we generate kandfby Gaussian Process (GP) with RBF kernel.\nkis generated by GP with mean 1, std 0.2, length scale 0.1, while fis with mean 0, std 1, length\nscale 0.1. We totally generate 5000 data points as training set T={(ki, fi), ui}5000\ni=1. After training\nan MIONet on the dataset T, we obtain an MIONet-based iteration step ¯M:Rn→Rn(linear),\nwhere n= 48 is the number of the interior nodal points in the uniform grid for the interval (0 ,1).\nHere the size of MIONet is [50 ,100,100,100] for k’s branch net, [50 ,100] for f’s branch net, and\n[1,100,100,100] for the trunk net. Note that we remove the bias in f’s branch net and also the bias\nbin Eq. (4) to guarantee the linearity w.r.t. f. Figure 4 shows an example of a couple of inputs k\nandfas well as the corresponding solution uand the MIONet prediction.\nWe then apply the P1element together with the Richardson iteration (set ω=h\n4=1\n4×49) to solve\nthis 1-d Poisson equation for k= 1. The MIONet iteration ¯Mis performed every M= 20 steps\nto correct low-frequency errors in the Richardson-MIONet method (we also show the convergence\nrate versus Min Figure 5, which is consistent with the theoretical results as Eq. (98) and Figure\n3). The error threshold for stopping iteration is 1 ×10−14. We list the numbers of iterations of\nthe Richardson and the Richardson-MIONet in Table 1. The number of iterations of the original\nRichardson method is 157 times that of the Richardson-MIONet method. Since the 1-d case\nconverges too fast, we leave the study of the convergence time to the later 2-d experiment.\nWe further observe the spectral error of the iterative methods, and the results are shown in\nFigure 6. The high-frequency errors are quickly eliminated by the Richardson iteration in both two\nmethods. However, the low-frequency errors are difficult to eliminate via the original Richardson\n220.0 0.2 0.4 0.6 0.8 1.00.91.01.11.21.3Input: k\n0.0 0.2 0.4 0.6 0.8 1.02\n1\n01Input: f\n0.0 0.2 0.4 0.6 0.8 1.00.010\n0.005\n0.0000.0050.0100.0150.020Prediction: u\nReference\nMIONetFigure 4: An example of one prediction of MIONet for the 1-d Poisson equation.\n#Iterations Speed up\nRichardson 28396 /\nRichardson-MIONet 181 ×157\nTable 1: Comparison of the Richardson iterative method and the Richardson-MIONet\nhybrid iterative method for the 1-d Poisson equation. The number of iterations of the\noriginal Richardson method is 157 times that of the Richardson-MIONet method.\n101102103\nM0.860.880.900.920.940.960.981.00Rate\nRichardson\nRichardson-MIONet\nFigure 5: Convergence rate of the Richardson-MIONet method versus the correction\nperiod M.The tendency is consistent with the theoretical results as Eq. (98) and Figure 3.\n23method. In each correction step, MIONet efficiently modifies the low-frequency errors. Although\nMIONet brings new high-frequency errors, they will be quickly clear up soon. After about 180\nsteps, the Richardson-MIONet method has already achieved a machine accuracy, while the original\nRichardson method still keeps a significant error.\n0 25 50 75 100 125 150 175\nIteration40\n30\n20\n10\n0FrequencyThe spectral error of the Richardson method\n20.0\n17.5\n15.0\n12.5\n10.0\n7.5\n5.0\n2.5\n0 25 50 75 100 125 150 175\nIteration40\n30\n20\n10\n0FrequencyThe spectral error of the Richardson-MIONet method\n20.0\n17.5\n15.0\n12.5\n10.0\n7.5\n5.0\n2.5\nFigure 6: The spectral error of the iterative methods. The numerical value of each pixel is\nlog10|spectral error |. The high-frequency errors are quickly eliminated by the Richardson iteration\nin both two methods. However, the low-frequency errors are difficult to eliminate via the original\nRichardson method. MIONet efficiently modifies the low-frequency errors every M= 20 steps.\nAlthough MIONet brings new high-frequency errors, they will be quickly clear up soon. After\nabout 180 steps, the Richardson-MIONet method has already achieved a machine accuracy, while\nthe original Richardson method still keeps a significant error.\n5.2 Two-dimensional Poisson equation\nConsider the 2-d Poisson equation\n(\n−∇ · (k∇u) =f in Ω,\nu= 0 on ∂Ω,(129)\nwhere Ω = (0 ,1)2. We also generate kandfas the same setting as the 1-d case. The difference\nis that we set the length scale to 0.2. We totally generate 5000 data points as training set T=\n{(ki, fi), ui}5000\ni=1. After training an MIONet on the dataset T, we obtain an approximate solution\noperator. Here the size of MIONet is [1002,500,500,500] for k’s branch net, [1002,500] for f’s\nbranch net, and [2 ,500,500,500] for the trunk net. We also remove the corresponding biases as in\n240 0.5 11\n0.5\n0Input: k\n0.80.91.01.11.21.31.4\n0 0.5 11\n0.5\n0Input: f\n1.5\n1.0\n0.5\n0.00.51.01.52.02.5\n0 0.5 11\n0.5\n0Reference: u\n0.0000.0050.0100.0150.0200.0250.0300.035\n0 0.5 11\n0.5\n0MIONet: u\n0.0000.0050.0100.0150.0200.0250.0300.035Figure 7: An example of one prediction of MIONet for the 2-d Poisson equation.\n25the 1-d case. An example of a couple of inputs kandfas well as the corresponding solution uand\nthe MIONet prediction is shown in Figure 7.\nWe then apply the P1element together with the Gauss-Seidel (GS) iteration to solve this 2-d\nPoisson equation with grid size 1025 ×1025. The MIONet iteration ¯M:Rn→Rn(n= 1023 ×1023)\nis performed every Msteps to correct low-frequency errors in the GS-MIONet method. The error\nthreshold for stopping iteration is 1 ×10−12. We list the numbers of convergence iterations and\nthe convergence time of GS-MIONet given different Min Table 2, 3. We also show the trends of\nthe convergence steps and the convergence time versus Min Figure 8. The best speed of the GS-\nMIONet method is nearly 30 times that of the original GS method at M= 8000, while the number\nof convergence iterations reaches its minimum at M= 4000. This gap exists since the inference\ntime of MIONet accounts for a significant portion in the hybrid iteration. When M > 2556893\n(i.e. the number of convergence iterations of the GS method), MIONet only gives an initial guess\nduring the hybrid iteration, which provides a 22% acceleration.\nM GS 500 1000 2000 4000 8000 16000 32000 64000\n#Iterations 2556893 div. 167001 94001 78471 82484 96001 160001 256001\nTime (s) 27103 div. 2641 1222 923 919 1049 1722 2750\nSpeed up / div. ×10.3 ×22.2 ×29.4 ×29.5 ×25.8 ×15.7 ×9.9\nTable 2: The number of convergence iterations and the convergence time of GS-MIONet\nhybrid iterative method for the 2-d Poisson equation versus the correction period M\n(part 1). The size of grid is 1025 ×1025. “div.” means the iteration diverges under the given M.\nM 128000 256000 512000 1024000 2048000 4096000 8192000\n#Iterations 384640 768001 1024001 1439557 2048001 2071460 2071460\nTime (s) 4079 8173 10804 15467 22058 22181 22181\nSpeed up ×6.6 ×3.3 ×2.5 ×1.75 ×1.23 ×1.22 ×1.22\nTable 3: The number of convergence iterations and the convergence time of GS-MIONet\nhybrid iterative method for the 2-d Poisson equation versus the correction period M\n(part 2). The size of grid is 1025 ×1025. When M > 2556893 (i.e. the number of convergence\niterations of the GS method), MIONet only gives an initial guess during the hybrid iteration.\nNext we compare the hybrid iterative method to the multigrid method. We first solve this\nequation by different levels of multigrid methods, then for each case, we again test the corresponding\nmultigrid-MIONet hybrid iterative method. In this hybrid method, we regard a V-cycle of the\nmultigrid method as one step, and insert one network correction step every MV-cycles. The\nresults are recorded in Table 4 and visually presented in Figure 9. We observe that the proposed\nhybrid iterative method can flexibly utilize the advanced multigrid method. The speed of GS-\nMIONet is between the multigrid methods of level 3 and 4. There is an acceleration effect all the\nway up to level 6. For a high level ( ≥7) multigrid method, it converges so fast that the inference\ntime of MIONet is larger than the multigrid’s convergence time. Unsurprisingly in such a case the\nhybrid method converges slower compared to the original multigrid method. However, in practice\nit is costly to construct high level multigrid, while the use of the hybrid iterative method has no\nextra burden for users as long as the pre-trained model is provided.\n26103104105106107\nM105106Iterations\nM=4000Iterations=78471GS iterations=2556893\nGS\nGS-MIONet\n103104105106107\nM103104Time (s)\nM=8000Time=919(s)GS time=27103(s)\nGS\nGS-MIONetFigure 8: Performance of GS-MIONet versus the correction period M.We use the MIONet\ntrained on 100 ×100 grid, and to perform hybrid iteration on 1025 ×1025 grid. The number of\nconvergence iterations reaches its minimum at M= 4000, while the convergence time reaches its\nminimum at M= 8000. The best convergence time of GS-MIONet method is 919 seconds, the\nspeed of which is nearly 30 times that of the original GS method. When Mis large enough, the\nMIONet only gives an initial guess during the hybrid iteration, which provides a 22% acceleration.\nMethod GS Mg2 Mg3 Mg4 Mg5 Mg6 Mg7 Mg8 Mg9 Mg10\nBasic time (s) 27103 8880 2210 567 139 35 8.7 2.4 1.7 1.7\nHybrid time (s) 919 337 118 56 34 22 12.2 7.0 6.4 6.4\nSpeed up ×29.5 ×26.4 ×18.7 ×10.1 ×4.1×1.6 / / / /\nTable 4: Multigrid-MIONet method. The time of multigrid methods and corresponding hybrid\nmethods on 1025 ×1025 grid. The correction period Mis tuned for each hybrid case.\nGS 2 3 4 5 6 7 8 9 10\nLevels of multigrid101102103104Time (s)Multigrid-MIONet hybrid iterative method\nMultigrid\nMultigrid-MIONet\nFigure 9: Multigrid-MIONet method for the 2-d Poisson equation on 1025×1025grid.\nThere is an acceleration effect all the way up to level 6, and the speed of GS-MIONet is between\nthe multigrid methods of level 3 and 4.\n275.3 Inhomogeneous boundary condition\nFinally we consider the 2-d Poisson equation with inhomogeneous boundary condition:\n(\n−∇ · (k∇u) =f in Ω,\nu=g on∂Ω,(130)\nwhere Ω = (0 ,1)2, and the boundary condition gis also changing. We generate kandfas the same\nsetting as the previous 2-d case. The gis regarded as a function with a period of 4 (anticlockwise\nalong the boundary), which is generated by GP with exponential sine squared kernel, where we set\nmean, std, length scale and period to 0, 0.05, 1 and 4, respectively. We totally generate 5000 data\npoints as training set T={(ki,(fi, gi)), ui}5000\ni=1. After training an MIONet on the dataset T, we\nobtain an approximate solution operator. Here the size of MIONet is [1002,500,500,500] for k’s\nbranch net, [1002,500] for ( f, g)’s branch net, and [2 ,500,500,500] for the trunk net. We remove\nthe corresponding biases as before. An example of inputs k,fandgas well as the corresponding\nsolution uand the MIONet prediction is shown in Figure 10.\nWe then apply the P1element together with the GS iteration to solve this equation on the\n500×500 uniform grid. The MIONet iteration ¯M:Rn→Rn(n= 500 ×500) is performed every\nM= 1600 steps to correct low-frequency errors in the GS-MIONet method. The error threshold for\nstopping iteration is 1 ×10−12. We show the number of convergence iterations and the convergence\ntime of the GS(-MIONet) method in Table 5. The speed of the GS-MIONet method is nearly 24\ntimes that of the original GS method.\n#Iterations Time (s) Speed up\nGS 672230 1652 /\nGS-MIONet 20963 69 ×23.9\nTable 5: Comparison of the GS iterative method and the GS-MIONet hybrid iterative\nmethod for the 2-d Poisson equation with inhomogeneous boundary condition. The\nsize of grid is 500 ×500. The speed of the GS-MIONet method is 23.9 times that of the original\nGS method.\n6 Conclusions\nWe have proposed a hybrid iterative method based on MIONet for PDEs, which combines the\ntraditional numerical iterative solver and the neural operator, and further systematically analyzed\nits theoretical properties, including the convergence condition, the spectral behavior, as well as the\nconvergence rate, in terms of the errors of the discretization and the model inference. We have\nshown the theoretical results for Richardson (damped Jacobi) and Gauss-Seidel. An upper bound of\nthe convergence rate of the hybrid method w.r.t. the correction period Mis given, which indicates\nan optimal Mto make the hybrid iteration converge fastest. Several numerical examples including\nthe hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to\nverify our theoretical results. In addition, we tested the hybrid iteration utilizing the multigrid\nmethod, it still reflects an excellent acceleration effect, unless the level of multigrid is very high.\nFurthermore, we have achieved the hybrid iteration for the Poisson equation with inhomogeneous\nboundary condition, which has not been studied in the work of HINTS. So far, we are able to deal\nwith the Poisson equation in which all the parameters are changing. As a meshless acceleration\nmethod, it is provided with enormous potentials for practice applications.\n280 0.5 11\n0.5\n0Input: k\n0.70.80.91.01.11.21.3\n0 0.5 11\n0.5\n0Input: f\n1.0\n0.5\n0.00.51.01.52.02.5\nu(0,0) u(1, 0) u(1,1) u(0, 1) u(0, 0)0.05\n0.04\n0.03\n0.02\n0.01\n0.000.010.020.03\nu(x,0)\nu(1,y) u(1x,1)\nu(0,1y)\nInput: g\n0 0.5 11\n0.5\n0Reference: u\n0.04\n0.02\n0.000.020.040.06\n0 0.5 11\n0.5\n0MIONet: u\n0.04\n0.02\n0.000.020.040.06Figure 10: An example of one prediction of MIONet for the 2-d Poisson equation with\ninhomogeneous boundary condition.\n29References\n[1] A. Brandt. Multi-level adaptive solutions to boundary-value problems. Mathematics of com-\nputation , 31(138):333–390, 1977.\n[2] A. Brandt. Algebraic multigrid theory: The symmetric case. Applied mathematics and com-\nputation , 19(1-4):23–56, 1986.\n[3] S. C. Brenner. The mathematical theory of finite element methods . Springer, 2008.\n[4] W. E, C. Ma, and L. Wu. Barron spaces and the compositional function spaces for neural\nnetwork models. arXiv preprint arXiv:1906.08039 , 2019.\n[5] W. E and B. Yu. The deep Ritz method: a deep learning-based numerical algorithm for solving\nvariational problems. Communications in Mathematics and Statistics , 6(1):1–12, 2018.\n[6] R. Eymard, T. Gallou¨ et, and R. Herbin. Finite volume methods. Handbook of numerical\nanalysis , 7:713–1018, 2000.\n[7] G. Gupta, X. Xiao, and P. Bogdan. Multiwavelet-based operator learning for differential\nequations. Advances in neural information processing systems , 34:24048–24062, 2021.\n[8] W. Hackbusch. Multi-grid methods and applications , volume 4. Springer Science & Business\nMedia, 2013.\n[9] J. He, X. Liu, and J. Xu. MgNO: Efficient parameterization of linear operators via multigrid.\narXiv preprint arXiv:2310.19809 , 2023.\n[10] J. Hu and P. Jin. Experimental observation on a low-rank tensor model for eigenvalue problems.\narXiv preprint arXiv:2302.00538 , 2023.\n[11] J. Huang, H. Wang, and H. Yang. Int-deep: A deep learning initialized iterative method for\nnonlinear problems. Journal of computational physics , 419:109675, 2020.\n[12] Z. Jiang, M. Zhu, D. Li, Q. Li, Y. O. Yuan, and L. Lu. Fourier-MIONet: Fourier-enhanced\nmultiple-input neural operators for multiphase modeling of geological carbon sequestration.\narXiv preprint arXiv:2303.04778 , 2023.\n[13] P. Jin, S. Meng, and L. Lu. MIONet: Learning multiple-input operators via tensor product.\nSIAM Journal on Scientific Computing , 44(6):A3490–A3514, 2022.\n[14] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-\ninformed machine learning. Nature Reviews Physics , 3(6):422–440, 2021.\n[15] N. Kovachki, S. Lanthaler, and S. Mishra. On universal approximation and error bounds\nfor Fourier neural operators. The Journal of Machine Learning Research , 22(1):13237–13312,\n2021.\n[16] S. Lanthaler, S. Mishra, and G. E. Karniadakis. Error estimates for DeepONets: A deep\nlearning framework in infinite dimensions. Transactions of Mathematics and Its Applications ,\n6(1):tnac001, 2022.\n30[17] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anand-\nkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint\narXiv:2010.08895 , 2020.\n[18] Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandku-\nmar. Neural operator: Graph kernel network for partial differential equations. arXiv preprint\narXiv:2003.03485 , 2020.\n[19] L. Lu, P. Jin, and G. E. Karniadakis. DeepONet: Learning nonlinear operators for identify-\ning differential equations based on the universal approximation theorem of operators. arXiv\npreprint arXiv:1910.03193 , 2019.\n[20] L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis. Learning nonlinear operators\nvia DeepONet based on the universal approximation theorem of operators. Nature machine\nintelligence , 3(3):218–229, 2021.\n[21] L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, and S. G. Johnson. Physics-informed neural\nnetworks with hard constraints for inverse design. SIAM Journal on Scientific Computing ,\n43(6):B1105–B1132, 2021.\n[22] J. Mandel. Algebraic study of multigrid methods for symmetric, definite problems. Applied\nmathematics and computation , 25(1):39–56, 1988.\n[23] C. Marcati and C. Schwab. Exponential convergence of deep operator networks for elliptic\npartial differential equations. SIAM Journal on Numerical Analysis , 61(3):1513–1545, 2023.\n[24] N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, and\nA. Courville. On the spectral bias of neural networks. In International Conference on Machine\nLearning , pages 5301–5310. PMLR, 2019.\n[25] M. A. Rahman, Z. E. Ross, and K. Azizzadenesheli. U-no: U-shaped neural operators. arXiv\npreprint arXiv:2204.11127 , 2022.\n[26] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep\nlearning framework for solving forward and inverse problems involving nonlinear partial dif-\nferential equations. Journal of Computational physics , 378:686–707, 2019.\n[27] B. Raoni´ c, R. Molinaro, T. Rohner, S. Mishra, and E. de Bezenac. Convolutional Neural\nOperators. arXiv preprint arXiv:2302.01178 , 2023.\n[28] J. W. Ruge and K. St¨ uben. Algebraic multigrid. In Multigrid methods , pages 73–130. SIAM,\n1987.\n[29] J. Shen, T. Tang, and L.-L. Wang. Spectral methods: algorithms, analysis and applications ,\nvolume 41. Springer Science & Business Media, 2011.\n[30] J. W. Siegel and J. Xu. High-order approximation rates for shallow neural networks with\ncosine and ReLUkactivation functions. Applied and Computational Harmonic Analysis , 58:1–\n26, 2022.\n[31] J. Sirignano and K. Spiliopoulos. DGM: A deep learning algorithm for solving partial differ-\nential equations. Journal of computational physics , 375:1339–1364, 2018.\n31[32] G. D. Smith. Numerical solution of partial differential equations: finite difference methods .\nOxford university press, 1985.\n[33] L. N. Trefethen and D. Bau. Numerical linear algebra , volume 181. Siam, 2022.\n[34] U. Trottenberg, C. W. Oosterlee, and A. Schuller. Multigrid . Elsevier, 2000.\n[35] S. Wang, H. Wang, and P. Perdikaris. Learning the solution operator of parametric partial\ndifferential equations with physics-informed DeepONets. Science advances , 7(40):eabi8605,\n2021.\n[36] Y. Wang, P. Jin, and H. Xie. Tensor neural network and its numerical integration. arXiv\npreprint arXiv:2207.02754 , 2022.\n[37] H. Wu, T. Hu, H. Luo, J. Wang, and M. Long. Solving High-Dimensional PDEs with Latent\nSpectral Models. arXiv preprint arXiv:2301.12664 , 2023.\n[38] J. Xu. Theory of multilevel methods , volume 8924558. Cornell University Ithaca, NY, 1989.\n[39] J. Xu and L. Zikatanov. Algebraic multigrid methods. Acta Numerica , 26:591–721, 2017.\n[40] Z.-Q. J. Xu, Y. Zhang, T. Luo, Y. Xiao, and Z. Ma. Frequency principle: Fourier analysis\nsheds light on deep neural networks. arXiv preprint arXiv:1901.06523 , 2019.\n[41] E. Zhang, A. Kahana, E. Turkel, R. Ranade, J. Pathak, and G. E. Karniadakis. A hybrid\niterative numerical transferable solver (HINTS) for PDEs based on deep operator network and\nrelaxation methods. arXiv preprint arXiv:2208.13273 , 2022.\n32"    },    {        "title": "2402.07185v2.Existence_of_an_equilibrium_with_limited_stock_market_participation_and_power_utilities.pdf",        "content": "Existence of an equilibrium with limited stock market\nparticipation and power utilities1\nPaolo Guasoni\nUniversit` a di Bologna,\nDublin City University\nKasper Larsen\nRutgers University\nGiovanni Leoni\nCarnegie Mellon University\nMarch 1, 2024\nAbstract : For constants γ∈(0,1) and A∈(1,∞), we prove exis-\ntence and uniqueness of a solution to the singular and path-dependent\nRiccati-type ODE\n\n\nh′(y) =1+γ\ny\u0000\nγ−h(y)\u0001\n+h(y)γ+\u0000\n(A−γ)eR1\ny1−h(q)\n1−qdq−A\u0001\nh(y)\n1−y, y∈(0,1),\nh(0) = γ, h (1) = 1 .\nAs an application, we use the ODE solution to prove existence of a\nRadner equilibrium with homogenous power-utility investors in the\nlimited participation model from Basak and Cuoco (1998).\nKeywords : Singular ODE, shooting technique, incomplete Radner equilibrium\n1We would like to thank G. Chabakauri, M. Larsson, J. Ma, I. Tice, A. Shadi Tahvildar-Zadehand,\nand J. Zhang for constructive comments. Kasper Larsen is corresponding author and has contract\ninformation: Email: KL756@math.rutgers.edu and mailing address: Department of Mathematics,\nRutgers University, Hill Center 330 - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-\n8019, USA.\n1arXiv:2402.07185v2  [q-fin.MF]  29 Feb 2024Declaration of interest: The research of G. Leoni was partially supported by the\nNational Science Foundation under Grant No. DMS 2108784. The research of P.\nGuasoni was partially supported by SFI (16/IA/4443).\nDeclaration of generative AI in scientific writing: No AI nor AI-assisted tools have\nbeen used.\nData availability statement: We do not analyze nor generate any datasets.\n21 Introduction\nThere is an extensive literature on nonlinear second-order differential equations of the\ntype\u0000\nφ(y)G(w′)w′\u0001′+φ(y)F(y, w, w′) = 0 , y∈(0,1). (1.1)\nWhen GandF(y, w,·) depend only on |w′|andφ(y) =yn−1with y:=|x|, equation\n(1.1) becomes the radial version of the differential equation\ndiv\u0000\nG(|Dw|)Dw\u0001\n+F(|x|, w,|Dw|) = 0 , (1.2)\nsee, for example, [16], [18], and [19]. In this case, solutions to (1.1), or related\ndifferential inequalities, are often used to prove comparison and maximum principles\nfor the corresponding differential equation (see [19]). Important examples are G(ξ) =\n1 and G(ξ) =|ξ|p−2,p >1, which yield the Laplace equation\n∆u+F(|x|, u,|Du|) = 0 , (1.3)\nand the p-Laplace equation\ndiv(|Du|p−2Du) +F(|x|, u,|Du|) = 0 , (1.4)\nrespectively. In (1.3) and (1.4), we use u(x) :=w(|x|) and y:=|x|.\nThe differential equation (1.1) also arises in the study of singular self-similar so-\nlutions to the porous media equation with absorption\n∂u\n∂t= ∆( um)−up,\nwhere one looks for solutions of the type\nu(x, t) =t−aw(t−b|x|),\nsee, for example, [3], [11], and [14].\nIn this paper, we are interested in a special form of (1.1), which arises from a\nsystem of coupled stochastic control problems in financial economics (discussed below\nand detailed in Section 3). For γ∈(0,1) and A∈(1,∞), we consider the second\n3order nonlinear differential equation\n\u0000\nφ(y)w′\u0001′−γ(1 +γ)yγ(1−y)1−γ−φ(y)\u0000\n(1−y)ew−A\u0001\n(w′)2= 0, (1.5)\nfory∈(0,1), where\nφ(y) :=y1+γ(1−y)γ, y∈(0,1).\nTo highlight the singularities at y= 0 and y= 1, we write the differential equation\n(1.5) as\nw′′(y) =γ(1 +γ)\ny(1−y)+(1 +γ)(2y−1)\ny(1−y)w′(y) +\u0000\n(1−y)ew(y)−A\u0001\nw′(y)2. (1.6)\nFor our application in Section 3 below, we are interested in a singular solution w∈\nC2([0,1)), which satisfies the boundary condition\nlim\ny↑1(1−y)w′(y) = 1 . (1.7)\nWe will see that a necessary and sufficient condition for the local existence of a smooth\nsolution for ynear 0 is that the boundary condition\nw′(0) = γ (1.8)\nholds. The main challenge of the paper is to determine the range of initial values\nw(0) = w0 (1.9)\nfor which equation (1.6) admits a global solution w(y) for y∈(0,1) and construct a\nunique solution that satisfies (1.7). Following [3] and [14], we use a shooting argument.\nTo be precise, we show that for all\n0≤w0< A−γ,\nthe differential equation (1.6) has a unique solution satisfying the boundary conditions\n(1.8), (1.9), and\nlim\ny↑1(1−y)w′(y) =γ\nA∈(0,1).\nThen, we show that for all w0sufficiently large, the solutions to the Cauchy problem\n4(1.6), (1.8), and (1.9) explode for y <1. Finally, we show that the initial value w0\nfor which (1.7) holds is given by the supremum of the set of initial values w0>0 for\nwhich there is a smooth solution to (1.6), (1.8), and (1.9) satisfying\nlim\ny↑1(1−y)w′(y)≤γ.\nThe main challenges with respect to previous work are that: (i) the ODE (1.6) is\nsingular at both y= 0 and y= 1, (ii) the ODE (1.6) is not variational, so standard\ntechniques (see, e.g., [15]) cannot be applied, (iii) the ODE (1.6) is of Riccati type\ndue to the presence of the term ( w′)2, (iv) there is exponential growth in w, and (v)\nwe are looking for singular solutions (see, e.g., [8] and [16]).\nTo state our main result, we rewrite (1.6) in terms of\nh(y) := (1 −y)w′(y), y∈(0,1).\nThen, the ODE (1.6) becomes\n\n\nh′(y) =1+γ\ny\u0000\nγ−h(y)\u0001\n+h(y)γ+\u0000\n(A−γ)eR1\ny1−h(q)\n1−qdq−A\u0001\nh(y)\n1−y, y∈(0,1),\nh(0) = γ, h (1) = 1 .(1.10)\nOur main mathematical result is:\nTheorem 1.1. Forγ∈(0,1)andA∈(1,∞), there exists a unique solution h∈\nC1([0,1])of(1.10) satisfying γ≤h≤1.\nOur motivation for studying (1.10) comes from the limited stock-market participa-\ntion model in [2]. We use Theorem 1.1 to prove the existence of a Radner equilibrium\nwhen both investors have identical power utility functions with common relative risk-\naversion coefficient γ∈(0,1) and common time-preference parameter β > 0. Our\nanalysis is significantly more involved than the log-log-utility model originally devel-\noped in [2]. Because of the log-utility assumption placed on the restricted investor,\nthe model in [2] is explicitly solvable and no ODE is needed.\nThere are several existing model extensions of [2]. [10] proves existence of equilib-\nrium bubbles (i.e., models where the equilibrium stock-price process is a strict local\nmartingale) and shows that equilibrium uniqueness can fail in models with multiple\nstocks. In another extension, [17] proves the existence of an equilibrium when the\n5unrestricted investor has a power-utility function. The existence proofs in [2], [10],\nand [17] all rely crucially on the restricted investor having a logarithmic utility func-\ntion. This is because the optimal policy for an investor with a logarithmic utility\nfunction is available in closed form. [21] uses coupled BSDEs to prove equilibrium\nexistence for investors with heterogenous exponential utilities. Finally, our existence\nproof also puts some of the experimental numerics for homogenous investors reported\nfor power-power utilities in [4] on a solid mathematical foundation.\nWe use the mathematical setting from [2], which falls under the theory of incom-\nplete Radner equilibria. Incompleteness stems from the restricted investor’s inability\nto hold stocks. Utilities defined on the positive real line — such as log and power\n— restrict consumption processes to be nonnegative, which complicates the under-\nlying mathematical structure. In continuous-time settings with noise generated by\nBrownian motions, incomplete Radner equilibria are originally discussed in [6] but no\ngeneral existence result is available.2\n2 ODE existence\n2.1 Auxiliary ODE results\nLemma 2.1. Letγ∈(0,1). Then, we have:\n1. The function\nφ(y) :=y1+γ(1−y)γ, y∈[0,1], (2.1)\nsatisfies\nφ′(y)\nφ(y)=1 +γ\ny−γ\n1−y, y∈(0,1). (2.2)\n2. The function\nϕ(y) :=1\nφ(y)Zy\n0φ(t)\ntdt, y ∈(0,1),\n2The counterpart of complete equilibrium models is fully developed; see, e.g., Chapter 4 in [13]\nand [1].\n6can be extended to ϕ∈ C1([0,1))with\nϕ(0) =1\n1 +γ, ϕ′(0) =γ\n(1 +γ)(2 + γ). (2.3)\nProof. 1: Follows by computing derivatives.\n2: By Taylor’s formula centered at 0, we have\nφ(t)\nt=tγ(1−t)γ\n=tγ−γt1+γ+o(t1+γ),\n1\n(1−y)γ= 1 + yγ+o(y).\nTherefore,\nϕ(y) =1\nφ(y)Zy\n0φ(t)\ntdt=1+yγ+o(y)\ny1+γZy\n0\u0000\ntγ−γt1+γ+o(t1+γ)\u0001\ndt\n=\u0000\n1+yγ+o(y)\u0001\u00101\n1+γy1+γ−γ\n2+γy2+γ+o(y2+γ)\u0011\ny1+γ\n=1\n1+γ+γ\u0010\n1\n1+γ−1\n2+γ\u0011\ny+o(y).\nThis shows that ϕ(y) can be extended continuously to y= 0 with the boundary values\nin (2.3). For y >0, by (2.2), the previous calculation, and Taylor’s formula, we have\nϕ′(y) =−φ′(y)\nφ(y)2Zy\n0φ(t)\ntdt+1\nφ(y)φ(y)\ny\n=−\u0010\n1+γ\ny−γ\n1−y\u0011\nϕ(y) +1\ny\n=−\u0010\n1+γ\ny−γ−yγ+o(y)\u0011\u0010\n1\n1+γ+γ\u00001\n1+γ−1\n2+γ\u0001\ny+o(y)\u0011\n+1\ny\n=γ\n(1+γ)(2+γ)+o(1)\n=ϕ′(0) + o(1).\nHence, ϕ’s extension is in C1([0,1)). ♢\nTheorem 2.2. Leta: [0,1]→Rbe a continuous function, γ∈(0,1), and define\na0(y) :=γ(1 +γ)\ny, a 1(y) :=(2γ+ 1)y−(1 +γ)\ny, y∈(0,1]. (2.4)\n7Then, the singular Riccati ODE\nf′(y) =a0(y) +a1(y)\n1−yf(y) +a(y)\n1−yf(y)2, y > 0, (2.5)\nwith the boundary condition f(0) = γ, has a strictly positive local solution in C1([0, δ])\nin the sense that there exist δ∈(0,1\n2]and a function f: [0, δ]→(0,∞)with f∈\nC1([0, δ])satisfying (2.5) fory∈(0, δ).\nProof. Step 1/6: We eliminate the linear term in (2.5) by multiplying both sides of\n(2.5) by φfrom Lemma 2.1 to see\n\u0000\nφ(y)f(y)\u0001′=a0(y)φ(y) +a(y)\n1−yφ(y)f(y)2, y > 0.\nIntegrating both sides and using φ(0) = 0 and f(0) = γ, yields\nφ(y)f(y) =Zy\n0φ(t)\u0012\na0(t) +a(t)\n1−tf(t)2\u0013\ndt, y > 0.\nHence, if a local solution fof (2.5) exists, it satisfies the integral equation\nf(y) =1\nφ(y)Zy\n0φ(t)\u0012\na0(t) +a(t)\n1−tf(t)2\u0013\ndt, y > 0.\nThe next steps construct a fixed point for this integral equation for y∈[0, δ] for some\nδ∈(0,1\n2].\nStep 2/6: For 0 < δ≤1\n2to be chosen later and an arbitrary R >0, we consider the\nclosed convex set\nX:={f∈ C([0, δ]) :∥f−γ∥∞≤R}, (2.6)\nwhere ∥f∥∞:= max y∈[0,δ]|f(y)|. For f∈ X and 0 < y≤δ, we define Tby\nT(f)(y) :=1\nφ(y)Zy\n0φ(t)\u0010\na0(t) +a(t)\n1−tf(t)2\u0011\ndt. (2.7)\nWe claim that lim y↓0T(f)(y) exists and\nlim\ny↓0T(f)(y) =γ. (2.8)\n8To see (2.8), we use L’Hospital’s rule, (2.2), and (2.7) to get\nlim\ny↓0φ(y)\nφ′(y)\u0014\na0(y) +a(y)\n1−yf(y)2\u0015\n= lim\ny↓0γ(1 +γ) +ya(y)\n1−yf(y)2\n1 +γ−yγ\n1−y\n=γ.\nWe extend Tcontinuously to y= 0 by defining T(f)(0) := γso that T:X → C ([0, δ]).\nStep 3/6: We claim that if δ >0 is sufficiently small, then\nT(X)⊆ X. (2.9)\nConsider the continuous function\nψ(y, z) :=a(y)\n1−yz2, y∈[0,1\n2], z∈[γ−R, γ+R]. (2.10)\nBecause ψis continuous, there exists a constant M > 0 such that\n|ψ(y, z)| ≤M,∀(y, z)∈[0,1\n2]×[γ−R, γ+R]. (2.11)\nThe constant Mdepends on the function a(y) and RbutMdoes not depend on δ.\nProvided δ∈(0,1\n2) satisfies\nδ≤δ1:=R(2 +γ)\n2M2γ,\n9forf∈ X andy∈[0, δ], we have the following estimate\n1\nφ(y)Zy\n0φ(t)\f\f\fa(t)\n1−tf(t)2\f\f\fdt\n=1\nφ(y)Zy\n0φ(t)\f\fψ\u0000\nt, f(t)\u0001\f\fdt\n≤M\ny1+γ(1−y)γZy\n0t1+γ(1−t)γdt\n≤M2γ\ny1+γZy\n0t1+γdt\n=M2γ\n(2 +γ)y\n≤R\n2.(2.12)\nOn the other hand, L’Hospital’s rule, (2.1), and (2.4) give\nlim\ny↓01\nφ(y)Zy\n0φ(t)a0(t)dt=γ.\nHence, there exists δ2>0 such that\n∀y∈(0, δ2) :\f\f\f\f1\nφ(y)Zy\n0φ(t)a0(t)dt−γ\f\f\f\f≤R\n2. (2.13)\nTaking δ≤min{δ1, δ2,1\n2}, it follows from the estimates (2.12) and (2.13) that\n|T(f)(y)−γ| ≤R,∀y∈[0, δ],∀f∈ X.\nTherefore, the inclusion (2.9) holds.\nStep 4/6: This step proves that the operator Tfrom step 2 is equi-continuous (even\nbetter, Tturns out to be an equi-Lipschitz operator). For 0 < y 1< y 2≤δand\n10f∈ X, the formula for a0(y) in (2.4), (2.7), and (2.10) give\n|T(f)(y2)−T(f)(y1)|\n≤(1 +γ)γ\f\f\f\f1\nφ(y2)Zy2\n0φ(t)\ntdt−1\nφ(y1)Zy1\n0φ(t)\ntdt\f\f\f\f\n+\f\f\f\f1\nφ(y2)Zy2\n0φ(t)ψ\u0000\nt, f(t)\u0001\ndt−1\nφ(y1)Zy1\n0φ(t)ψ\u0000\nt, f(t)\u0001\ndt\f\f\f\f\n≤(1 +γ)γ\f\f\f\f1\nφ(y2)Zy2\n0φ(t)\ntdt−1\nφ(y1)Zy1\n0φ(t)\ntdt\f\f\f\f\n+1\nφ(y2)Zy2\ny1φ(t)|ψ\u0000\nt, f(t)\u0001\n|dt+\f\f\f\f1\nφ(y2)−1\nφ(y1)\f\f\f\fZy1\n0φ(t)|ψ\u0000\nt, f(t)\u0001\n|dt\n=:I+II+III.(2.14)\nWe estimate each of the three terms separately. Lemma 2.1 ensures ϕ∈ C1([0, δ]),\nand so, the Mean-Value Theorem gives\nI≤(1 +γ)γ(y2−y1)∥ϕ′∥∞.\nThe bound in (2.11) and y2≤δ≤1\n2give\nII≤M1\ny1+γ\n2(1−y2)γZy2\ny1t1+γ(1−t)γdt\n≤M2γ\ny1+γ\n2Zy2\ny1t1+γdt\n≤M2γ(y2−y1).\nSimilar, the bound in (2.11) and 0 < y 1< y 2≤δ≤1\n2give\nIII≤|φ(y1)−φ(y2)|\nφ(y2)φ(y1)MZy1\n0φ(t)dt\n≤22γ|φ(y1)−φ(y2)|\ny1+γ\n2y1+γ\n1MZy1\n0t1+γdt\n≤22γ|y1−y2|\ny1+γ\n2|φ′(z)|My 1\n≤22γ|y1−y2|\nyγ\n2|φ′(z)|M,\n11where the constant z∈(y1, y2) is produced by the Mean-Value Theorem. For arbi-\ntrary t∈(y1, y2), we have\n|φ′(t)|=|(1 +γ)tγ(1−t)γ−γt1+γ(1−t)γ−1|\n≤(1 +γ)yγ\n2+γy1+γ\n221−γ.\nHence,\nIII≤22γ\u0000\n1 +γ+γ2γ−1\u0001\nM|y1−y2|.\nCombining the estimates for I,II, and IIIgives a uniform (in f) Lipschitz constant\nLsuch that\n|T(f)(y2)−T(f)(y1)| ≤L|y2−y1|,∀0< y 1< y 2≤δ,∀f∈ X.\nUsing (2.8), it follows that this inequality holds also for y1= 0. Because Lis inde-\npendent of f∈ X,Tis equi-continuous.\nStep 5/6: The family {T(f) :f∈ X} is equi-bounded (step 3) and equi-continuous\n(step 4). Hence, by the Ascoli–Arzel´ a theorem, Tis compact. Then, the Schauder\nfixed point theorem produces a function f∈ X such that f(y) =T(f)(y) for all\ny∈[0, δ]. In particular, f(0) = T(f)(0) = γ, and\nf(y) =1\nφ(y)Zy\n0φ(t)\u0012\na0(t) +a(t)\n1−tf(t)2\u0013\ndt, y ∈[0, δ]. (2.15)\nSince the right-hand side is continuously differentiable for y >0, it follows that f\nis continuously differentiable for y∈(0, δ). We need to show that fis continuously\ndifferentiable at y= 0. Let ϕbe the function in Lemma 2.1. Then,\nf(y) = (1 + γ)γϕ(y) +1\nφ(y)Zy\n0φ(t)a(t)\n1−tf(t)2dt, y ∈(0, δ].\nBecause ϕ∈ C1([0,1\n2]), it remains to show that the second term on the right-hand\nside is also in C1([0, δ]). To this end, we define\nω(y) :=1\nφ(y)Zy\n0φ(t)a(t)\n1−tf(t)2dt, y ∈(0, δ].\n12Then,\nω′(y) =−φ′(y)\nφ(y)2Zy\n0φ(t)a(t)\n1−tf(t)2dt+a(y)\n1−yf(y)2, y∈(0, δ),\nwhich is continuous. L’Hopital’s rule and the definition of φ(y) in (2.1) ensure that\nthe integral term in ω′(y) has a finite limit as y↓0. The continuity of a(y) and\nf(y) aty= 0 ensures that the second term in ω′(y) also has a finite limit as y↓0.\nTherefore, ω∈ C1([0, δ]).\nStep 6/6: The local solution f(y) for y∈[0, δ] is necessarily strictly positive. This\nis because f(0) = γ >0 and should there be a point y0∈(0, δ) with f(y0) = 0, then\nf′(y0) =γ(1+γ)\ny0>0, and so fcan never become zero.\n♢\nRemark 2.1.Because the solution in Theorem 2.2 belongs to C1([0, δ]), we know that\nf(y) =γ+Ry\n0f′(t)dtfor all y∈[0, δ]. However, because the coefficient functions\nin (2.5) have singularities, we cannot use (2.5) to writeRy\n0f′(t)dtas a sum of three\nindividual integrals. For example, the definition of a0(y) in (2.4) implies that the first\nterm satisfiesRy\n0a0(t)dt=∞for all y >0.\nTheorem 2.3 (Comparison) .Letδ∈(0,1]and let a, b: [0, δ]7→Rbe continuous\nfunctions with a(y)≤b(y)for all y∈[0, δ]. Assume f, g: [0, δ]7→Rsolve the Cauchy\nproblems\n\n\nf′(y) =a0(y) +a1(y)\n1−yf(y) +a(y)\n1−yf(y)2, y∈(0, δ),\nf(0) = γ,\nand\n\n\ng′(y) =a0(y) +a1(y)\n1−yg(y) +b(y)\n1−yg(y)2, y∈(0, δ),\ng(0) = γ.\nThen, f(y)≤g(y)for all y∈[0, δ]. Furthermore, if a(y)< b(y)for all y∈[0, δ],\nthen f(y)< g(y)for all y∈(0, δ].\nProof. We argue by contradiction and assume there exists y1∈(0, δ] such that f(y1)>\ng(y1). Then, the function F(y) :=f(y)−g(y) satisfies F(0) = 0 and F(y1)>0. We\n13define\ny0:= sup {y∈[0, y1] :F(y) = 0} ∈[0, y1].\nBecause Fis continuous, we have y0∈[0, y1),F(y0) = 0, and F(y)>0 for all\ny∈(y0, y1]. For y∈(y0, y1), we have\nF′(y) =a1(y)\n1−yF(y) +a(y)\n1−yf(y)2−b(y)\n1−yg(y)2\n≤2γ+ 1\n1−yF(y) +b(y)\n1−y\u0000\nf(y)2−g(y)2\u0001\n=1\n1−yF(y)\u0010\n2γ+ 1 + b(y)\u0000\nf(y) +g(y)\u0001\u0011\n≤M\n1−yF(y), M := max\ny∈[y0,δ]\u0010\n2γ+ 1 + b(y)\u0000\nf(y) +g(y)\u0001\u0011\n.\nBecause F(y0) = 0, Gronwall’s inequality yields the contradiction\nF(y)≤F(y0)eRy\ny0M\n1−qdq= 0, y∈(y0, y1).\nWhen a < b , the integral representation (2.15) and f≤ggive\nf(y) =1\nφ(y)Zy\n0φ(t)\u0010\na0(t) +a(t)\n1−tf(t)2\u0011\ndt\n<1\nφ(y)Zy\n0φ(t)\u0010\na0(t) +b(t)\n1−tf(t)2\u0011\ndt\n≤1\nφ(y)Zy\n0φ(t)\u0010\na0(t) +b(t)\n1−tg(t)2\u0011\ndt\n=g(y), y∈(0, δ],\nwhere we have used f >0.\n♢\nProposition A.1 in Appendix A contains additional properties of the solution of\nthe ODE defined in (2.16) below.\nTheorem 2.4. Letγ∈(0,1)anda3∈(−∞,−γ]. Then, there exists a function\n14f∈ C([0,1])∩ C1([0,1))withf(y)∈[0,1]fory∈[0,1]that satisfies\nf(y) =1\ny1+γ(1−y)γZy\n0\u0010\nγ(1−γ)\u0000\nx(1−x)\u0001γ+a3x1+γ\n(1−x)1−γf(x)2\u0011\ndx, y ∈(0,1).\n(2.16)\nFurthermore, the function fsatisfies the ODE\nf′(y) =a0(y) +a1(y)\n1−yf(y) +a3\n1−yf(y)2, y∈(0,1), (2.17)\nand the boundary conditions\nf(0) = γ, f (1) =−γ\na3≤1. (2.18)\nFinally, when a3∈(−∞,−γ), we have f(y)∈[0,1)fory∈[0,1]and when a3∈\n[−1,−γ], we have f(y)≥γfory∈[0,1].\nProof. Step 1/4 (Global existence): We can rewrite (2.17) as\nf′(y) =a0(y) +f(y)\n1−y\u0000\na1(y) +a3f(y)\u0001\n, y∈(0,1). (2.19)\nTheorem 2.2 produces local existence and so let fbe a local solution of (2.17), which\nis strictly positive. To see that f(y) exists and is uniformly bounded for y∈[0,1), we\nargue by contradiction. If fwere to oscillate at some point y∞∈(0,1] in the sense\nlim inf\ny↑y∞f(x)<lim sup\ny↑y∞f(x) =∞,\nthen we could find an increasing sequence ( yn)n∈Nof local maxima such that yn↑y∞,\nf′(yn) = 0, and f(yn)→ ∞ . Because a3<0 and a1(y∞)∈R, we see from (2.19) that\nlimn→∞f′(yn) =−∞, which is a contradiction. Alternatively, if we have continuous\nexplosion in the sense\nlim\ny↑y∞f(y) =∞,\nwe get from (2.19) that lim y↑y∞f′(y) =−∞, which is again a contradiction because\nit implies that fis decreasing near y∞while f(y∞) =∞. All in all, f(y) exists and\nis uniformly bounded for y∈[0,1).\nIt remains to show that lim y↑1f(y) exists and do this by considering oscillations.\n15To this end, first assume there exists a sequence ( yn)n∈N⊂(0,1) monotonically\nincreasing to y= 1 such that f′(yn) = 0 for all n∈N. Because fis bounded, we can\nextract a subsequence if necessary to ensure that ℓ:= lim n→∞f(yn) exists in [0 ,∞).\nThe ODE in (2.19) yields\n0 = lim\nn→∞\u0010\na0(yn) +f(yn)\n1−yn\u0000\na1(yn) +a3f(yn)\u0001\u0011\n= (1 + γ)γ+ lim\nn→∞f(yn)\n1−yn\u0010\nγ+a3f(yn)\u0011\n,(2.20)\nwhich forces ℓ∈ {0,−γ\na3}. To see ℓ=−γ\na3, we argue by contradiction and assume\nℓ= 0. In that case, (2.20) yields\nlim\nn→∞ℓ−f(yn)\n1−yn=−lim\nn→∞f(yn)\n1−yn= 1 + γ >0, (2.21)\nwhich is impossible because f(y) is nonnegative for all y∈[0,1). So should f(y)\noscillate as y↑1, the sequence of corresponding max and min values f(yn),n∈N,\nconverges to −γ\na3. Because f(y) is uniformly bounded for y∈[0,1), the alternative is\nthat f(y) is monotone as y↑1 in which case lim y↑1f(y) obviously exists. All in all,\nthis shows that f∈ C([0,1])∩ C1([0,1)).\nStep 2/4 (Boundary values): By multiplying (1 −y) in (2.17), we get\n(1−y)f′(y) = (1 −y)a0(y) +f(y)\u0000\na1(y) +a3f(y)\u0001\n, y∈(0,1), (2.22)\nBecause f(1) = lim y↑1f(y), we see from (2.22) that\nlim\ny↑1(1−y)f′(y) =f(1)\u0000\nγ+a3f(1)\u0001\n=:c.\nTo see (2.18), we argue by contradiction. First, we assume c >0. For ϵ∈(0, c), we\nletyϵ∈(0,1) be such that\n∀y∈(yϵ,1) :c−ϵ≤(1−y)f′(y)≤c+ϵ.\nFory∈(yϵ,1), we get\nf(y)−f(yϵ) =Zy\nyϵf′(q)dq≥(c−ϵ)Zy\nyϵ1\n1−qdq.\n16By using the Monotone Convergence Theorem and f(1)<∞, this gives a contradic-\ntion. A similar argument rules out c <0. Therefore, c= 0 and so f(1)∈ {0,−a3\nγ}.\nTo rule out f(1) = 0, we again argue by contradiction and assume f(1) = 0. The\nproperty\nlim\ny↑1\u0000\na1(y) +a3f(y)\u0001\n=γ >0,\nallows us to find y0∈(0,1) such that\n∀y∈(y0,1) :a1(y) +a3f(y)>0. (2.23)\nBecause f >0 and a0>0, it follows from (2.22) that f′(y)>0 for y∈(y0,1). This\ngives the contradiction\n0 =f(1) = lim\ny↑1f(y)≥f(y0)>0.\nAll in all, we have f(1) =−a3\nγas claimed in (2.18).\nStep 3/4 (Global upper bound): We claim that f(y)≤1 for all y∈[0,1).\nBecause fis continuous on [0 ,1], there exists y1∈[0,1] such that\nf(y1) = max\ny∈[0,1]f(y).\nIfy1= 0, then f(y)≤f(0) = γ < 1 for all y∈[0,1], and the claim is proved.\nSimilarly, if y1= 1, then f(y)≤f(1) = −γ\na3≤1 for all y∈[0,1]. The only\nremaining case left to consider is y1∈(0,1) with f′(y1) = 0. In this case, the unique\npositive solution of (2.19) with f′(y1) = 0 is\nf(y1) =2a0(y1)(1−y1)\n−a1(y1) +p\na1(y1)2−4a3a0(y1)(1−y1). (2.24)\nand we claim that f(y1)≤1. This claim is proven by proving the inequality\n2a0(y)(1−y) +a1(y)≤p\n(a1(y))2−4a3a0(y)(1−y), y∈(0,1), (2.25)\nAssume there is y∈(0,1) such that (otherwise there is nothing to prove)\n0≤2a0(y)(1−y) +a1(y) =γ+ (2γ2−1)(1−y)\ny.\n17Squaring both sides of (2.25) produces the requirement\n0≥y\u0000\na3−γ2+γ+ 1\u0001\n+γ2−1 =y(a3+γ)−(1−γ2)(1−y). (2.26)\nThe inequality (2.26) holds because a3≤ −γ.\nWhen a3∈(−∞,−γ), we have f(1)<1. Then, for any maximizer y1∈(0,1)\nwith f′(y1) = 0, it suffices to prove that the inequality in (2.25) is strict. This follows\nbecause the left-hand-side of (2.26) is strictly negative for all y∈(0,1).\nStep 4/4 (Global lower bound): Leta3∈[−1,−γ]. To show f≥γ, we argue\nby contradiction and assume there is a point y1∈[0,1] such that f(y1)< γ and\nf′(y1) = 0. Because f(0) = γandf(1) = −γ\na3≥γ, it must be that y1∈(0,1).\nThe unique positive solution of f′(y1) = 0 is as in (2.24). The contradictory property\nf(y1)≥γis equivalent to\n2a0(y1)(1−y1) +γa1(y1)≥γp\na1(y1)2−4a3a0(y1)(1−y1). (2.27)\nThe left-hand-side of (2.27) equals\n2a0(y1)(1−y1) +γa1(y1) =γ(γ+ 1−y1)\ny1>0.\nSquaring both sides of (2.27) produces the equivalent property a3+ 1≥0, which\nholds by assumption. ♢\nRemark 2.2.For given constants y0∈(0,1) and f0∈(0,1), we can argue as in the\nabove proof of Theorem 2.4 and construct f∈ C([y0,1])∩C1([y0,1)) with f(y)∈[0,1]\nfory∈[y0,1] that satisfies the ODE\nf′(y) =a0(y) +a1(y)\n1−yf(y) +a3\n1−yf(y)2, y∈(y0,1), (2.28)\nand the boundary conditions\nf(y0) =f0, f(1) =−γ\na3. (2.29)\nIndeed, since the right-hand-side of (2.28) is locally Lipschitz for y∈[y0, y0+δ0] with\ny0∈(0,1−δ0), standard existence theory for ODEs gives a local solution f(y) for\ny∈[y0, y0+δ] for some δ∈(0, δ0). By reasoning as in Step 1, we see that f(y) exists\n18fory∈[y0,1) and is uniformly bounded. Step 2 still holds. Step 3 holds by using\nf(y0) =f0<1 in place of f(0) = γ <1.\nFinally, assuming that f0∈[γ,1) and a3∈[−1,−γ], Step 4 still produces f(y)≥γ\nfor all y∈[y0,1]. ♢\n2.2 Local ODE existence for all ξ\nFor constants σ2\nD, ξ, A, and a continuous function h: [0,1]7→R, we define the coeffi-\ncient functions (consistent with (2.4))\na0(y) :=γ(1 +γ)\ny, y∈(0,1],\na1(y) :=(2γ+ 1)y−(1 +γ)\ny, y∈(0,1],\na2(h, y) :=ξ\nσ2\nDeRy\n0h(q)−1\n1−qdq−A, y ∈(0,1).(2.30)\nTheorem 2.5. Letγ∈(0,1),σ2\nD>0, and A >1. For ξ0>0, there exists δ0∈(0,1\n2]\nsuch that\n1. For all ξ >0with|ξ−ξ0| ≤1, the integral equation\nh(y) =1\ny1+γ(1−y)γZy\n0\u0010\na0(x)(1−x)γ+a2(h, x)\n(1−x)1−γh(x)2\u0011\nx1+γdx, y ∈(0,1),\n(2.31)\nhas a local unique solution hξ∈ C1([0, δ0])andhξsatisfies the boundary value\nproblem\n\n\nh′(y) =a0(y) +a1(y)\n1−yh(y) +a2(h,y)\n1−yh(y)2, y∈(0, δ0),\nh(0) = γ.(2.32)\n2. For all ξi>0with|ξi−ξ0| ≤1fori∈ {1,2}, comparison holds for (2.32) in\nthe sense that 0< ξ1< ξ2implies hξ1(y)< h ξ2(y)for all y∈(0, δ0],\nProof. 1. The proof of local existence only requires minor modifications to the proof\nof Theorem 2.2. For 0 < δ≤1\n2andR >0 with R+γ≤1, let f∈ X withXdefined\n19in (2.6). We modify the operator Tin (2.7) by defining\nT(f)(y) :=1\nφ(y)Zy\n0φ(t)\u0012\na0(t) +a2(f, t)\n1−tf(t)2\u0013\ndt, y ∈(0, δ]. (2.33)\nTo replace the estimates in (2.12) and (2.14), we first need the bound\n|a2(f, t)|=\f\f\fξ\nσ2\nD(1−t)eRt\n0f(q)\n1−qdq−A\f\f\f\n≤(1 +ξ0\nσ2\nD)(1−t)e(R+γ)Rt\n01\n1−qdq+A\n≤(1 +ξ0\nσ2\nD)(1−t)1−R−γ+A\n≤1 +ξ0\nσ2\nD+A, t ∈[0, δ].(2.34)\nTherefore,\n|a2(f, t)|f(t)2≤\u0000\n1 +ξ0\nσ2\nD+A\u0001\n(R+γ)2≤1 +ξ0\nσ2\nD+A, t ∈[0, δ]. (2.35)\nThe estimate in (2.35) implies that the analogous of (2.12) and (2.14) hold with M\nreplaced by 1+ξ0\nσ2\nD+A. With these modifications, the rest of the local existence proof\nis identical.\nWe start by proving uniqueness for y∈[0, y1] for some y1∈(0, δ0]. To this end, we\nlethand˜hbe two solutions of (2.31) and set g:=h−˜h. Because h(0) = ˜h(0) = γ,\nit suffices to prove g(y) = 0 for y∈[0, y1]. Moreover, because γ < 1, by taking\ny1∈(0,1) small enough, we can assume h(y)≤1 and ˜h(y)≤1 for y∈[0, y1]. For\ny∈(0, y1], we have\n|g(y)| ≤1\ny1+γ(1−y)γZy\n0x1+γ\n(1−x)1−γ\f\f\fa2(h, x)h(x)2−a2(˜h, x)˜h(x)2\f\f\fdx\n≤1\ny1+γ(1−y)γZy\n0x1+γ\n(1−x)1−γ\f\f\fa2(h, x)−a2(˜h, x)\f\f\fh(x)2dx\n+1\ny1+γ(1−y)γZy\n0x1+γ\n(1−x)1−γ|a2(˜h, x)|\f\f\fh(x)2−˜h(x)2\f\f\fdx.(2.36)\nWe consider the last two integrals in (2.36) separately. The Mean-Value Theorem\ngives the inequality\n|ea−eb| ≤e|a|+|b||a−b|, a, b ∈R.\n20Therefore, the first integral in (2.36) is bounded by\n1\ny1+γ(1−y)γZy\n0x1+γ\n(1−x)1−γξ(1−x)\nσ2\nD\f\f\feRx\n0h(q)\n1−qdq−eRx\n0˜h(q)\n1−qdq\f\f\fdx\n≤ξ\nσ2\nD(1−y1)γZy\n0eRx\n0h(q)+˜h(q)\n1−qdqZx\n0|g(q)|\n1−qdqdx\n≤ξ\nσ2\nD(1−y1)γZy\n01\n(1−x)2Zy\n0|g(q)|\n1−qdqdx\n≤ξ\nσ2\nD(1−y1)γ+2+1Zy\n0|g(q)|dq.(2.37)\nFor the second integral in (2.36), the estimate in (2.34) ensures that a2(˜h, x) is\nuniformly bounded by some constant c1>0. Therefore,\n1\ny1+γ(1−y)γZy\n0x1+γ\n(1−x)1−γ|a2(˜h, x)|\f\fh(x)2−˜h(x)2\f\fdx\n≤c1\n(1−y1)Zy\n0\u0000\nh(x) +˜h(x)\u0001\f\fh(x)−˜h(x)\f\fdx\n≤2c1\n(1−y1)Zy\n0\f\fg(x)\f\fdx.(2.38)\nBy combining (2.36) with the two estimates (2.37) and (2.38) we get\n|g(y)| ≤c2Zy\n0\f\fg(x)\f\fdx, y ∈[0, y1],\nfor some constant c2>0. Because g(0) = 0, Gronwall’s inequality produces g(y) = 0\nfor all y∈[0, y1].\nNext, we prove uniqueness for y∈[y1, δ0]. The function [ y1, δ0]∋y7→Ry\n0h(q)\n1−qdq\ntransforms the ODE (2.32) into a second-order ODE. Because y1>0, there is no\nsingularity and standard uniqueness arguments apply.\n2. To argue by contradiction, we assume there exists y0∈(0, δ0] such that h1(y0)≥\nh2(y0). First, because ξ1< ξ2, there exists y1∈(0,1) such that\n∀y∈[0, y1] :ξ1(1−y)eRy\n0h1(q)\n1−qdq< ξ2(1−y)eRy\n0h2(q)\n1−qdq.\nThe strict comparison in Theorem 2.3 gives h1(y)< h 2(y) for y∈[0, y1]. Second, we\n21define\ny2:= sup {y∈[0, δ0] :∀q∈(0, y]h1(q)< h 2(q)} ∈[y1, y0].\nThe continuity of h1andh2gives h1(y2) =h2(y2). Because ξ1< ξ2, we have\n∀y∈[0, y2] :ξ1(1−y)eRy\n0h1(q)\n1−qdq< ξ2(1−y)eRy\n0h2(q)\n1−qdq.\nThe strict comparison in Theorem 2.3 gives the contradiction h1(y)< h 2(y) for y∈\n(0, y2]. ♢\n2.3 Global ODE existence for ξsmall\nTheorem 2.6. Letγ∈(0,1),σ2\nD>0, and A >1. For a constant ξsatisfying\n0≤ξ\nσ2\nD< A−γ, (2.39)\nthe integral equation (2.31) has a global solution hξ∈ C([0,1])∩ C1([0,1))with 0<\nhξ≤1. Furthermore, hξsatisfies the boundary value problem (2.32) globally, as well\nas the boundary condition\nh(1) = c0, c0:=γ\nA∈(0, γ). (2.40)\nProof. Letfbe as in Theorem 2.4 for the constant\na3:=ξ\nσ2\nD−A <−γ <0. (2.41)\nStep 1/3 (Global Existence): Let [0 , δ1) be the maximal interval of existence for\nh=hξproduced by Theorem 2.5(1). First, we claim h(y)≤1 for y∈[0, δ1). To\nsee this, we argue by contradiction and assume there exists y1∈[0, δ1) such that\nh(y1)>1. Because h(0) = γ <1, it must be that y1∈(0, δ1). We define\ny0:= sup {y∈[0, y1) :h(y)≤1} ∈(0, y1).\nThe continuity of hgives\n∀y∈[0, y0] :h(y)≤1 and h(y0) = 1 .\n22Therefore, for y∈[0, y0], we have\na2(hξ, y) =ξ\nσ2\nDeRy\n0h(q)−1\n1−qdq−A≤ξ\nσ2\nD−A=a3. (2.42)\nTheorem 2.4 gives an ODE solution fwith f <1. The comparison result in Theorem\n2.3 gives the contradiction\n1 =h(y0)≤f(y0)<1.\nBecause h(y)≤1 for all y∈[0, δ1), (2.42) holds and we can again use Theorems\n2.3 and 2.4 to see\nh(y)≤f(y)<1, y∈[0, δ1).\nTherefore, δ1= 1.\nStep 2/3 (Boundary values): We start by writing\n(1−y)eRy\n0h(x)\n1−xdx=eRy\n0h(x)−1\n1−xdx.\nSince h≤1, the integralR1\n0h(x)−1\n1−xdxexists in [ −∞,0], and so the following limit\nexists\nL:= lim\ny↑1(1−y)eRy\n0h(x)\n1−xdx=eR1\n0h(x)−1\n1−xdx∈[0,1]. (2.43)\nAs in Step 1 in the proof of Theorem 2.4, assume first that there exists a sequence\n(yn)n∈N⊂(0,1) monotonically increasing to y= 1 such that h′(yn) = 0 for all\nn∈N. Because h≤1, we can extract a subsequence, if necessary, to ensure that\nℓ:= lim n→∞h(yn)∈[0,1] exists. The ODE in (2.32) yields\n0 = lim\nn→∞\u0012\na0(yn) +h(yn)\n1−yn\u0000\na1(yn) +a2(h, y)h(yn)\u0001\u0013\n= (1 + γ)γ+ lim\nn→∞h(yn)\u0000\na1(yn) +a2(h, yn)h(yn)\u0001\n1−yn.\n23This implies that\n0 = lim\nn→∞h(yn)\u0000\na1(yn) +a2(h, yn)h(yn)\u0001\n=ℓ\u0012\nγ+\u0010ξ\nσ2\nDL−A\u0011\nℓ\u0013\n.\nHence, either ℓ= 0 or ℓ=ℓ1where\nℓ1:=γ\nA−ξ\nσ2\nDL. (2.44)\nWe argue by contradiction to rule out ℓ= 0. We assume ℓ= 0 to get\n0 = lim\nn→∞\u0012\na0(yn) +h(yn)\n1−yn\u0000\na1(yn) +a2(h, yn)h(yn)\u0001\u0013\n= (1 + γ)γ+ lim\nn→∞h(yn)\n1−ynγ.\nThis is a contradiction because his positive.\nBased on the above, should hoscillate as y↑1, the sequence of corresponding max\nand min values h(yn) would converge to ℓ1given in (2.44). Therefore, ℓ:= lim y↑1h(y)\nexists and equals ℓ1. Because h≤1, the alternative is that his monotone near y= 1,\nin which case the limit ℓ:= lim y↑1h(y) exists. To show ℓ=ℓ1when his monotone\nnear y= 1, we follow the argument in Step 2 in the proof of Theorem 2.4 but replace\n(2.22) with\n(1−y)h′(y) = (1 −y)a0(y) +h(y)a1(y) +a2(h, y)h(y)2, y∈(0,1).\nTaking limits yields\nlim\ny↑1(1−y)h′(y) =ℓγ+\u0010ξ\nσ2\nDL−A\u0011\nℓ2.\nAs in the proof of Theorem 2.4, we get ℓγ+\u0010\nξ\nσ2\nDL−A\u0011\nℓ2= 0 and so either ℓ= 0 or\nℓ=ℓ1forℓ1in (2.44). We rule out ℓ= 0 as in Step 2 of the proof of Theorem 2.4.\nTherefore, when h(y) is monotone near y= 1, we also have ℓ=ℓ1.\nLet us show that ℓ=c0where c0is from (2.40). Sinceξ\nσ2\nD< A−γandL∈[0,1],\n24we have that ℓ1from (2.44) satisfies\nℓ1=γ\nA−ξ\nσ2\nDL≤1. (2.45)\nWe consider two cases. First, if ℓ1<1, then taking ε >0 so small that η:=ℓ1+ε <1,\nthere exists y1∈(0,1) such that\n∀y∈[y1,1] :h(y)≤η <1.\nIt follows that h(y)≤ηfor all y∈[y1,1], while h(y)≤1 for all y∈[0, y1]. Hence, for\nally∈[y1,1],\n0≤(1−y)eRy\n0h(q)\n1−qdq\n≤(1−y)eRy1\n01\n1−qdqeRy\ny1η\n1−qdq\n=1−y\n1−y1\u00121−y1\n1−y\u0013η\n,\nwhich converges to zero as y↑1. Therefore, when ℓ1<1, we have L= 0 and so\nℓ=c0.\nSecond, assume ℓ1= 1. From (2.44), we see that ℓ1= 1 implies\nξL\nσ2\nD=A−γ.\nTherefore, the upper bound in (2.39) gives L > 1, which contradicts (2.43). Thus,\nℓ1= 1 cannot happen and ℓ=c0.\nTo see that h′∈ C([0,1)), we can repeat the argument from Step 2 of the proof of\nTheorem 2.2.\nStep 3/3: To see h(y)>0 for all y∈[0,1], the boundary values (2.32) imply that\nit suffices to show that there is no point y∈(0,1) such that\nh(y) =h′(y) = 0 . (2.46)\nHowever, the ODE (2.32) makes (2.46) impossible because a0(y)>0 for all y∈[0,1].\n♢\n252.4 Explosion for ξbig\nLemma 2.7. Letγ∈(0,1),σ2\nD>0, and A > 1. For y0∈(0,1), there exists\nξ(y0)∈(0,∞)such that\n∀ξ≥ξ(y0) :hξ(y0) =∞.\nProof. Lety0∈(0,1) be arbitrary and assume to the contrary that for all ξ >0, the\nlocal solution hξ(y) of (2.32) exists for y∈[0, y0] and that lim ξ↑∞hξ(y0)<∞(this\nlimit exists because Theorem 2.5 ensures that hξ(y0) is increasing in ξ). We consider\nξ >0 such that\nξ\nσ2\nD>A\n1−y0. (2.47)\nBy (2.31), the ODE for gξ(y) :=hξ(y)(1−y)γy1+γgives\ng′\nξ(y) =a0(y)(1−y)γy1+γ+a2(hξ,y)\u0000\n(1−y)y\u00011+γgξ(y)2\n=γ(1−γ)(1−y)γyγ+ξ\nσ2\nDeRy\n0hξ(q)−1\n1−qdq−A\n\u0000\n(1−y)y\u00011+γgξ(y)2\n≥γ(1−γ)(1−y)γyγ+ξ\nσ2\nD(1−y)−A\n\u0000\n(1−y)y\u00011+γgξ(y)2\n≥γ(1−γ)(1−y)γyγ+ξ\nσ2\nD(1−y0)−A\n\u0000\n(1−y)y\u00011+γgξ(y)2, y∈(0, y0),(2.48)\nwhere the first inequality uses hξ>0. Integrating (2.48) and using (2.47) and gξ(0) =\n0 give\ngξ(y0\n2)≥γ(1−γ)Zy0\n2\n0(1−q)γqγdq > 0. (2.49)\nTo create a contradiction, we define the constant\ncξ:=ξ\nσ2\nD(1−y0)−A\n\u0000\n(1−y0)y0\n2\u00011+γ>0.\nBecause cξis affine in ξ, we can increase ξ >0 if needed such that ξsatisfies both\n26(2.47) and\n2≤cξy0γ(1−γ)Zy0\n2\n0(1−q)γqγdq. (2.50)\nThe Riccati equation\n\n\nf′(y) =cξf(y)2, y >y0\n2,\nf(y0\n2) =gξ(y0\n2),\nhas the explicit solution\nf(y) =gξ(y0\n2)\n1−cξgξ(y0\n2)(y−y0\n2), y∈[y0\n2,1\ncξgξ(y0\n2)+y0\n2).\nThe solution fexplodes at\ny=1\ncξgξ(y0\n2)+y0\n2≤y0, (2.51)\nwhere the inequality follows from (2.49) and (2.50). For y∈[y0\n2, y0], (2.48) gives\ng′\nξ(y)≥ξ\nσ2\nD(1−y0)−A\n\u0000\n(1−y)y\u00011+γgξ(y)2≥cξgξ(y)2. (2.52)\nThen, comparison gives us that gξ(y)≥f(y) for y≥y0\n2. Therefore, gξalso explodes\nat or before yin (2.51). ♢\n2.5 Lipschitz estimates\nForξ >0, we denote by hξa local solution of the boundary value problem (2.32) and\ndefine the constant\nyξ:= inf{y >0 :hξ(y) = 1} ∧1∈(0,1], (2.53)\n27where inf ∅:=∞anda∧b:= min {a, b}fora, b∈R. We also define the decreasing\nfunction\nFξ(y) :=ξ\nσ2\nDeRy\n0hξ(q)−1\n1−q, y∈[0, yξ]. (2.54)\nLemma 2.8. Letγ∈(0,1),σ2\nD>0,A >1,y0∈(0,1), and ¯ξ >0. Then, there exist\nconstants M1>0andM2>0such that\n|hξ1(y)−hξ2(y)| ≤M1y|ξ1−ξ2|, y∈[0, yξ1∧yξ2∧y0], ξ 1, ξ2∈[0,¯ξ],(2.55)\n|Fξ1(y)−Fξ2(y)| ≤M2|ξ1−ξ2|, y∈[0, yξ1∧yξ2∧y0], ξ 1, ξ2∈[0,¯ξ].(2.56)\nProof. Forξ1, ξ2∈[0,¯ξ], we let h1andh2be the corresponding local ODE solutions\nof (2.32)\nh′\ni(y) +1 +γ\ny\u0000\nhi(y)−γ\u0001\n=γ\n1−yhi(y) +a2(ξi, hi, y)\n1−yhi(y)2, y≤yξi,\nwhere we have augmented the a2function from (2.30) by writing\na2(ξi, hi, y) :=ξi\nσ2\nDeRy\n0hi(q)−1\n1−qdq−A, y ≤yξi.\nSubtracting and multiplying by y1+γyield\ny1+γ\u0000\nh′\n1(y)−h′\n2(y)\u0001\n+yγ(1 +γ)\u0000\nh1(y)−h2(y)\u0001\n=y1+γγ\n1−y\u0000\nh1(y)−h2(y)\u0001\n+y1+γa2(ξ1, h1, y)−a2(ξ2, h2, y)\n1−yh1(y)2+y1+γa2(ξ2, h2, y)\n1−y\u0000\nh1(y)2−h2(y)2\u0001\n,\nfory≤yξ1∧yξ2. Computing the yderivative gives\n\u0010\ny1+γ\u0000\nh1(y)−h2(y)\u0001\u0011′\n=y1+γγ\n1−y\u0000\nh1(y)−h2(y)\u0001\n+y1+γa2(ξ1, h1, y)−a2(ξ2, h2, y)\n1−yh1(y)2+y1+γa2(ξ2, h2, y)\n1−y\u0000\nh1(y)2−h2(y)2\u0001\n.\nBecause hi∈ C1([0, yξi]) and hi(0) = γ, we can integrate from 0 to ywith y∈\n28(0, yξ1∧yξ2] to get\ny1+γ\u0000\nh1(y)−h2(y)\u0001\n=γZy\n0t1+γ\n1−t\u0000\nh1(t)−h2(t)\u0001\ndt\n+Zy\n0t1+γa2(ξ1, h1, t)−a2(ξ2, h2, t)\n1−th1(t)2dt\n+Zy\n0a2(ξ2, h2, t)t1+γ\n1−t\u0000\nh1(t)2−h2(t)2\u0001\ndt\n=Zy\n0t1+γ\n1−t\u0010\nγ+a2(ξ2, h2, t)\u0000\nh1(t) +h2(t)\u0001\u0011\u0000\nh1(t)−h2(t)\u0001\ndt\n+Zy\n0t1+γa2(ξ1, h1, t)−a2(ξ2, h2, t)\n1−th1(t)2dt.(2.57)\nThe bound hi(t)≤1 for t∈[0, yξ1∧yξ2] gives\n\f\f\f\fa2(ξ2, h2, t)\n1−t\u0000\nh1(t) +h2(t)\u0001\f\f\f\f≤2ξ2\nσ2\nD+A\n1−y0≤2¯ξ\nσ2\nD+A\n1−y0=:M0, t∈[0, yξ1∧yξ2∧y0].\nBecause\na2(ξ1, h1, t)−a2(ξ2, h2, t) =(1−t)(ξ1−ξ2)\nσ2\nDeRt\n0h1(q)\n1−qdq+(1−t)ξ2\nσ2\nD\u0010\neRt\n0h1(q)\n1−qdq−eRt\n0h2(q)\n1−qdq\u0011\n,\nthe Mean-Value Theorem applied to the function\n[0,1]∋θ7→eRt\n0θh1(q)+(1−θ)h2(q)\n1−qdq,\ngives θ(t)∈[0,1] such that\n|a2(ξ1, h1, t)−a2(ξ2, h2, t)|\n1−th1(t)2\n≤|ξ1−ξ2|\nσ2\nD+ξ2\nσ2\nD\f\f\f\feRt\n0θ(t)h1(q)+(1−θ(t))h2(q)\n1−qdqZt\n0h1(q)−h2(q)\n1−qdq\f\f\f\f\n≤|ξ1−ξ2|\nσ2\nD+ξ2\nσ2\nD(1−t)2Zt\n0|h1(q)−h2(q)|dq, t ∈[0, yξ1∧yξ2∧y0].(2.58)\n29Therefore, for y∈[0, yξ1∧yξ2∧y0], we have by (2.57)\ny1+γ|h1(y)−h2(y)| ≤\u0012γ\n1−y0+M0\u0013Zy\n0t1+γ|h1(t)−h2(t)|dt\n+1\nσ2\nD|ξ1−ξ2|Zy\n0t1+γdt\n+ξ2\nσ2\nD(1−y0)2Zy\n0t1+γdtZy\n0|h1(q)−h2(q)|dq.\nIn turn, for y∈[0, yξ1∧yξ2∧y0], we have\n|h1(y)−h2(y)| ≤\u0012γ\n1−y0+M0+ξ2y\nσ2\nD(1−y0)2\u0013Zy\n0|h1(q)−h2(q)|dq\n+1\nσ2\nD|ξ1−ξ2|y.\nFory∈[0, yξ1∧yξ2∧y0], Gronwall’s inequality gives (2.55) because\n|h1(y)−h2(y)| ≤|ξ1−ξ2|\nσ2\nDye\u0012\nγ\n1−y0+M0+ξ2\nσ2\nD(1−y0)2\u0013\ny\n≤M1y|ξ1−ξ2|, M 1:=1\nσ2\nDe\u0012\nγ\n1−y0+M0+¯ξ\nσ2\nD(1−y0)2\u0013\ny0.(2.59)\nFinally, to see (2.56), we use (2.58) to see for y∈[0, yξ1∧yξ2∧y0]\n|Fξ1(y)−Fξ2(y)| ≤|ξ1−ξ2|\nσ2\nD(1−y) +ξ2\nσ2\nDZy\n0|h1(q)−h2(q)|dq\n≤|ξ1−ξ2|\nσ2\nD+¯ξ\nσ2\nDM1|ξ1−ξ2|Zy\n0qdq\n=M2|ξ1−ξ2|, M 2:=1 +1\n2¯ξM1\nσ2\nD.\n♢\nThe uniqueness claim in Theorem 1.1 follows from the following result.\nCorollary 2.9. Letγ∈(0,1),σ2\nD>0, and A >1. For all ξ >0, the boundary value\nproblem (2.32) has a unique local solution.\nProof. Existence follows from Theorem 2.5. To prove uniqueness, we let hi:Ii→\n(0,∞),i∈ {1,2}, be two solutions where Ii⊆[0,1] is their maximal interval of\n30existence. We let yibe the right-end point of Ii. Because hi(0) = γ <1, there exists\ny0∈(0, y1∧y2) such that hi(y)<1 for all y∈[0, y2]. By applying (2.55) with\nξ1:=ξ2:=¯ξ:=ξ, we see that h1(y) =h2(y) for all y∈[0, y0]. Because y0>0, there\nis no singularity and standard uniqueness arguments imply I1=I2andh1(y) =h2(y)\nfor all y∈I1. ♢\n2.6 Global ODE existence for critical ξ\nThe existence claim in Theorem 1.1 follows from the following result after we note\nthat the ODE (2.32) coincides with the ODE in (1.10). This property follows because\nthe integral in (2.60) below ensures\n(A−γ)eR1\ny1−hξ0(q)\n1−qdq=ξ0\nσ2\nDeRy\n0hξ0(q)−1\n1−qdq, y∈[0,1].\nLemma 2.10. Letγ∈(0,1),σ2\nD>0,A >1, and define the set\nΞ :=\b\nξ >0 :hξ∈ C([0,1])∩ C1([0,1))solves (2.32) withhξ(0) = γandhξ(1)≤γ\t\n.\nThen, the following properties hold:\n1.Ξis a non-empty and bounded subset of (0,∞).\n2. For ξ∈Ξ,hξ(y)<1for all y∈[0,1].\n3.ξ0:= sup Ξ ̸∈Ξ.\n4.hξ0∈ C([0,1])∩ C1([0,1))solves (2.32) withhξ0(1) = 1 .\n5.R1\n01−hξ0(q)\n1−qdq∈(0,∞)and the integral is given by\nlim\ny↑1eRy\n0hξ0(q)−1\n1−qdq=σ2\nD(A−γ)\nξ0∈(0,1). (2.60)\n6.hξ0(y)≥γfor all y∈[0,1].\n7.hξ0∈ C1([0,1])withh′\nξ0(1) =1−γ2\nA−1>0.\nProof. 1. The set Ξ is not empty by Theorem 2.6. The set Ξ is bounded from above\nby Lemma 2.7.\n312. Let h=hξfor some ξ∈Ξ. We argue by contradiction and assume there is\nˆy∈[0,1] such that\nh(ˆy) = max\ny∈[0,1]h(y)≥1.\nBecause h(0) = γandh(1)≤γ, we have ˆ y∈(0,1) and so\nh(ˆy)> γ, h′(ˆy) = 0 , h′′(ˆy)≤0.\nInserting h′(ˆy) = 0 into the ODE (2.32) produces the relation\nξ\nσ2\nDeRˆy\n0h(q)−1\n1−qdq=h(ˆy)\u0000\nAˆyh(ˆy) + 1 + γ−(2γ+ 1)ˆy\u0001\n−γ(1 +γ)(1−ˆy)\nˆyh(ˆy)2.\nFurthermore, inserting this expression into the second derivative of the ODE (2.32)\nyields\n0≥ˆy2(1−ˆy)2h′′(ˆy)\n=−γ(1 +γ)(1−ˆy)2+\u0010\n1 +γ+ ˆy\u0000\nγ(3 +γ)ˆy+ ˆy−(1 +γ)(2 + γ)\u0001\u0011\nh(ˆy)\n+ ˆy\u0000\n1 +γ−ˆy(1 +A+ 2γ)\u0001\nh(ˆy)2+Aˆy2h(ˆy)3.\nFirst, because A >1 and h(ˆy)≥1,\nAˆy2\u0000\nh(ˆy)3−h(ˆy)2\u0001\n≥ˆy2\u0000\nh(ˆy)3−h(ˆy)2\u0001\n.\nTherefore,\n0≥ −γ(1 +γ)(1−ˆy)2+\u0010\n1 +γ+ ˆy\u0000\nγ(3 +γ)ˆy+ ˆy−(1 +γ)(2 + γ)\u0001\u0011\nh(ˆy)\n+ ˆy\u0000\nγ+ 1−y(2γ+ 2)\u0001\nh(ˆy)2+ ˆy2h(ˆy)3\n=\u0000\nh(ˆy)−γ\u0001\u0010\n1 +γ+ (1 + γ)\u0000\nh(ˆy)−2\u0001\nˆy+\u0000\nh(ˆy)−1\u0001\u0000\nh(ˆy)−1−γ\u0001\nˆy2\u0011\n.\nSecond, dividing by\u0000\nh(ˆy)−γ\u0001\n>0 implies the contradiction\n0≥1 +γ+ (1 + γ)\u0000\nh(ˆy)−2\u0001\nˆy+\u0000\nh(ˆy)−1\u0001\u0000\nh(ˆy)−1−γ\u0001\nˆy2\n= (1 + γ)(1−ˆy) + (1 + γ)\u0000\nh(ˆy)−1\u0001\nˆy+\u0000\nh(ˆy)−1\u00012ˆy2−γ\u0000\nh(ˆy)−1\u0001\nˆy2\n>0.(2.61)\n32In (2.61), the strict inequality follows from\n(1 +γ)(1−ˆy)>0,\u0000\nh(ˆy)−1\u00012ˆy2≥0,\nand the sum of the remaining two terms in (2.61) satisfies the bound\n\u0000\nh(ˆy)−1\u0001\u0000\n(1 +γ)ˆy−γˆy2\u0001\n=\u0000\nh(ˆy)−1\u0001\u0000\nˆy+γˆy(1−ˆy)\u0001\n≥0.\n3. We argue by contradiction and assume ξ0∈Ξ.\nStep 1/4: Given ϵ∈(0,1−γ), there exists y0∈(0,1) such that\n∀y∈(y0,1) : hξ0(y)< γ+ϵ <1.\nBecause hξ0≤1, the function from (2.54), i.e.,\nFξ0(y) :=ξ0\nσ2\nDeRy\n0hξ0(q)−1\n1−q, y∈[0,1),\nis non-increasing. Therefore, the limit Fξ0(1) := lim y↑1Fξ0(y) exists and is zero be-\ncause\nFξ0(1) = lim\ny↑1Fξ0(y0)eRy\ny0hξ0(q)−1\n1−qdq\n≤lim\ny↑1Fξ0(y0)eRy\ny0γ+ϵ−1\n1−qdq\n= lim\ny↑1Fξ0(y0)\u00121−y0\n1−y\u0013γ+ϵ−1\n= 0.\nBecause A >1, by increasing y0if necessary, we can therefore also assume y0satisfies\n∀y∈(y0,1) : Fξ0(y)<A−1\n2. (2.62)\nStep 2/4: Forξ∈(ξ0, ξ0+ 1), Theorem 2.5 gives a local solution hξ(y) for y∈[0, yξ]\nwhere yξis defined in (2.53). Lemma 2.8 with y0∈(0,1) from the previous step and\n¯ξ:=ξ0+ 1 gives a constant M1>0 such that\n|hξ(y)−hξ0(y)| ≤M1|ξ−ξ0|, y≤yξ∧y0, ξ∈(ξ0, ξ0+ 1).\n33Because hξ0(y)<1 for all y∈[0,1], we can consider ξ∈(ξ0, ξ0+ 1) such that\nM1|ξ−ξ0|+ sup\ny∈[0,1]hξ0(y)<1. (2.63)\nFor such ξandy∈[0, yξ∧y0], we have\nhξ(y)≤ |hξ(y)−hξ0(y)|+hξ0(y)\n≤M1|ξ−ξ0|+hξ0(y)\n<1.\nTherefore, whenever ξ∈(ξ0, ξ0+ 1) satisfies (2.63), we have yξ> y 0.\nStep 3/4: In addition to (2.63), this step creates an additional restriction on ξ∈\n(ξ0, ξ0+ 1). First, let Fξbe defined as in (2.54). Lemma 2.8 with y0∈(0,1) from the\nprevious step and ¯ξ:=ξ0+ 1 gives a constant M2>0 such that\n|Fξ(y)−Fξ0(y)| ≤M2|ξ−ξ0|, y≤yξ∧y0, ξ∈(ξ0, ξ0+ 1).\nForξsatisfying (2.63), the previous step ensures yξ> y 0and so the bound in (2.62)\ngives\nFξ(y0)≤ |Fξ(y0)−Fξ0(y0)|+Fξ0(y0)\n≤M2|ξ−ξ0|+A−1\n2.\nThis shows that we can lower ξ∈(ξ0, ξ0+ 1) such that, in addition to (2.63), we also\nhave\nFξ(y0)< A−1. (2.64)\nStep 4/4: Finally, this step creates a contradiction using comparison of ODE solu-\ntions. Let ξ∈(ξ0, ξ0+ 1) and y0∈(0,1) be as in the previous step. Using (2.64), the\nfunction in (2.54) satisfies\n∀y∈[y0, yξ] :Fξ(y) =Fξ(y0)eRy\ny0hξ(q)−1\n1−q≤Fξ(y0)< A−1.\n34This bound gives the ODE inequality\nh′\nξ(y) =a0(y) +hξ(y)a1(y) +\u0000\nFξ(y)−A\u0001\nhξ(y)\n1−y\n≤a0(y) +hξ(y)a1(y)−hξ(y)\n1−y, y∈(y0, yξ).\nUsing a3:=−1 in Theorem 2.4 yields an ODE solution fof\n\n\nf′(y) =a0(y) +f(y)a1(y)−f(y)\n1−y, y∈(y0,1),\nf(y0) =hξ(y0).\nThe standard comparison principle gives hξ(y)≤f(y) for y∈[y0, yξ]. Theorem 2.4\nand Remark 2.2 give f(1) = γandf(y)<1 for y∈[y0,1] and so we have yξ= 1. All\nin all, we have the contradiction ξ > ξ 0andξ∈Ξ because\nhξ(1)≤f(1) = γ.\n4. Let ξn↑ξ0:= sup Ξ and let hξndenote the corresponding solution of the ODE\n(2.32). Because of hξ’s monotonicity in ξ, we can define\nhξ0(y) := lim\nn→∞hξn(y), y∈[0,1),\nwhich satisfies 0 ≤hξ0(y)≤1 for y∈[0,1). By taking limits in the ODE (2.32), we\nsee that the ODE\nh′\nξ0(y) =a0(y) +hξ0(y)a1(y) +\u0000\nFξ0(y)−A\u0001\nhξ0(y)\n1−y, y∈(0,1),\nholds. We can argue as in first part of Step 2 in the proof of Theorem 2.6 to show\nthat hξ0(y) cannot oscillate as y↑1 and so hξ0(1) := lim y↑1hξ0(y) exists. Because\nξ0̸∈Ξ, it must be that hξ0(1)> γ.\nTo see hξ0(1) = 1, the bound hξ0≤1 allows us to argue as in the second part\nof Step 2 in the proof of Theorem 2.6. Here, it is shown that hξ0(1) = ℓ1where ℓ1\nis defined in (2.44). The argument following (2.45) shows that ℓ1<1 implies the\ncontradiction\nhξ0(1) = c0< γ < h ξ0(1),\n35where c0is defined in (2.40). Because hξ0(1)≤1, the only alternative is ℓ1= 1.\n5. This proof is similar to Step 2 in the proof of Theorem 2.6. Because 0 ≤hξ0≤1,\nthe limit lim y↑1eRy\n0hξ0(q)−1\n1−qdqexists in [0 ,1], which allows us to use L’Hospital’s rule\nbelow. The integral representation (2.31) of hξ0yields\n1\ny1+γZy\n0x1+γ(1−x)γ\u0012\na0(x) +a2(hξ0, x)\n1−xhξ(x)2\u0013\ndx=hξ0(y)(1−y)γ.\nBecause hξ0(y) is continuous at y= 1 with a finite limit, the right-hand-side converges\nto 0 as y↑1. L’Hospital’s rule give us\nhξ0(1) = lim\ny↑11\ny1+γ(1−y)γZy\n0x1+γ(1−x)γ\u0012\na0(x) +a2(hξ0, x)\n1−xhξ0(x)2\u0013\ndx\n= lim\ny↑1y1+γ(1−y)γ\u0010\na0(y) +a2(hξ0,y)\n1−yhξ0(y)2\u0011\n(1 +γ)yγ(1−y)γ−γy1+γ(1−y)γ−1\n= lim\ny↑1y\u0000\n(1−y)a0(y) +a2(hξ0, y)hξ0(y)2\u0001\n(1 +γ)(1−y)−γy\n=1\nγ\u0012\nA−ξ0\nσ2\nDlim\ny↑1eRy\n0hξ0(q)−1\n1−qdq\u0013\nhξ0(1)2.\nBecause hξ0(1) = 1, this implies (2.60).\nThe bounds 0 ≤hξ0≤1 give\n0≤eR1\n0h(x)−1\n1−xdx≤1. (2.65)\nBecause hξ0(0) = γ∈(0,1) and hξ0(y) is continuous at y= 0, the two inequalities in\n(2.65) are strict.\n6. Because hξ0≤1, the limit in (2.60) gives the monotonicity property\neRy\n0hξ0(q)−1\n1−qdq≥eR1\n0hξ0(q)−1\n1−qdq=σ2\nD(A−γ)\nξ0, y∈[0,1]. (2.66)\nTherefore,\na2(hξ0, y) =ξ0\nσ2\nDeRy\n0hξ0(q)−1\n1−qdq−A≥ −γ.\n36Using a3:=−γin Theorem 2.4 produces an ODE solution fof\n\n\nf′(y) =a0(y) +f(y)a1(y)−γf(y)\n1−y, y∈(0,1),\nf(0) = γ,\nBecause 1 + a3>0, Theorem 2.4 also gives fwith f≥γ. The comparison result in\nTheorem 2.3 gives the lower bound hξ0≥f≥γ.\n7. We define\ng(y) :=1−hξ0(y)\n1−y, y∈(0,1).\nStep 1/4: To get an a priori estimate, this step assumes h′\nξ0(y) is continuous at\ny= 1. The ODE in (2.32) and (2.60) imply the representation\nh′\nξ0(y) =1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+hξ0(y)2(A−γ)eR1\nyg(q)dq−1\n1−y+γhξ0(y)g(y).(2.67)\nBecause h′\nξ0(y) is continuous at y= 1, L’Hopital’s rule gives\ng(1) = (1 + γ)(γ−1) + ( A−γ) lim\ny↑1eR1\nyg(q)dq−1\n1−y+γg(1)\n=γ2−1 + (A−γ)g(1) + γg(1).\nWe can solve to see g(1) =1−γ2\nA−1>0. The next steps prove that h′\nξ0(y) is indeed\ncontinuous at y= 1.\nStep 2/4: This step rules out that g(y) can monotonically explode to ∞asy↑1.\nWe argue by contradiction. The Mean-Value Theorem and g′(y)>0 for ynear 1\nyields θ(y)∈(y,1) such that\neR1\nyg(q)dq−1 =eR1\nyg(q)dqg\u0000\nθ(y)\u0001\n(1−y)≥eR1\nyg(q)dqg(y)(1−y),\nforynear 1. The ODE in (2.67) gives\nh′\nξ0(y)≥1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+\u0010\nhξ0(y)2(A−γ)eR1\nyg(q)dq+γhξ0(y)\u0011\ng(y), (2.68)\n37forynear 1. By differentiation, we see\ng′(y) =1\n1−y\u0000\ng(y)−h′\nξ0(y)\u0001\n(2.69)\nand so g′(y)>0 implies that g(y)≥h′\nξ0(y) for ynear 1. The inequality (2.68) gives\ng(y)≥1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+\u0010\nhξ0(y)2(A−γ)eR1\nyg(q)dq+γhξ0(y)\u0011\ng(y), (2.70)\nforynear 1. Because g≥0, we have eR1\nyg(q)dq≥1 and so the inequality (2.70) gives\n1 +γ\ny\u0000\nhξ0(y)−γ\u0001\n≥\u0000\nhξ0(y)2(A−γ) +γhξ0(y)−1\u0001\ng(y),\nforynear 1. By taking limits as y↑1 and using A >1 give the contradiction\n(1 +γ)(1−γ) = lim\ny↑11 +γ\ny\u0010\nhξ0(y)−γ\u0011\n≥(A−1) lim\ny↑1g(y) =∞.\nStep 3/4: This step considers oscillations. The ODE (2.67) and (2.69) give\n−(1−y)g′(y) =1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+hξ0(y)2(A−γ)eR1\nyg(q)dq−1\n1−y\n+\u0000\nγhξ0(y)−1\u0001\ng(y), y∈(0,1).(2.71)\nDifferentiating (2.71) at a point y∈(0,1) with g′(y) = 0 gives\n(1−y)2yg′′(y) = (1 + γ)\u0000\nγ−hξ0(y)\u0001\n−hξ0(y)2(A−γ)\u0010\neR1\nyg(q)dq−1\u0011\n+\u0012\nhξ0(y)\u0010\ny(A−γ)eR1\nyg(q)dq\u0000\nhξ0(y)−2\u0001\n+ 2Ay−γ\u0011\n+γ−(γ+ 3)y+ 2−γ(1−y)yg(y)\u0013\ng(y), y∈(0,1).(2.72)\nFirst, assume that ( yn)n∈Nis a sequence of local maxima for gwith yn↑1 such\nthat\n∀n∈N:g′(yn) = 0 , g′′(yn)≤0.\n38We can extract a subsequence ( ynk)k∈N⊆(yn)n∈Nsuch that\nlim\nk→∞g(ynk) = lim sup\nn→∞g(yn)∈[0,∞].\nBecause hξ0(1) = 1, we have\nlim\nk→∞(1−ynk)ynkg(ynk) = lim\nk→∞ynk\u0000\n1−hξ0(ynk)\u0001\n= 0,\nand so replacing ywith ynkin (2.72) and taking limits give\n0≥lim\nk→∞(1−ynk)2ynkg′′(ynk) =γ2−1 + (A−1) lim\nk→∞g(ynk).\nThis gives the upper bound for g’s local maxima\nlim sup\nn→∞g(yn)≤1−γ2\nA−1. (2.73)\nSecond, assume that ( yn)n∈Nis a sequence of local minima for gwith yn↑1 such\nthat\n∀n∈N:g′(yn) = 0 , g′′(yn)≥0.\nThen, similarly to (2.73), we have the lower bound for g’s local minima\nlim inf\nn→∞g(yn)≥1−γ2\nA−1. (2.74)\nFinally, combining the two bounds (2.73) and (2.74) shows that should g(y) oscil-\nlate as y↑1, the function gwill still have limit lim y↑1g(y) =g(1) =1−γ2\nA−1as in Step\n1.\nStep 4/4: This step shows that hξ0(y) is continuously differentiable at y= 1. Step\n2 and Step 3 ensure that that h′\nξ0(1) exists and that g(y) is continuous at y= 1. It\n39isg’s continuity which allows us to use L’Hopital’s rule to see\nlim\ny↑1γ+\u0000\n(A−γ)eR1\nyg(q)dq−A\u0001\nhξ0(y)\n1−y= lim\ny↑1(A−γ)\u0000\neR1\nyg(q)dq−1\u0001\nhξ0(y) +γ\u0000\n1−hξ0(y)\u0001\n1−y\n= (A−γ) lim\ny↑1eR1\nyg(q)dq−1\n1−y+γlim\ny↑11−hξ0(y)\n1−y\n= (A−γ)g(1) + γg(1)\n=Ag(1).\nThen, the ODE (2.67) and hξ0(1) = 1 give\nlim\ny↑1h′\nξ0(y) = (1 + γ)(γ−1) +Ag(1) = γ2−1 +Ag(1). (2.75)\nAnother application of L’Hopital’s rule gives\ng(1) = lim\ny↑11−hξ0(y)\n1−y= lim\ny↑1h′\nξ0(y) =γ2−1 +Ag(1). (2.76)\nBy solving (2.76), we get g(1) =1−γ2\nA−1>0 and so (2.75) gives lim y↑1h′\nξ0(y) =1−γ2\nA−1.♢\n40Figure 1 presents a numerical illustration.\nFigure 1: ODE solutions hξnforξn↑sup Ξ. The parameters are γ:= 0.5, σD:=\n0.2, A:= 2, and ξn∈ {0.15,0.152,0.1522,0.15223 ,0.152232 }.\n0.2 0.4 0.6 0.8 1.0y0.20.40.60.81.0h(y)\n3 Application to incomplete Radner equilibrium\ntheory\nOn a probability space (Ω ,F,P), we let ( Bt)t≥0be a Brownian motion generating the\nraw filtration ( F0\nt)t≥0withF0\nt:=σ(Bs)s∈[0,t]and assume F=∨t≥0F0\nt. We define the\naugmented filtration as Ft:=F0\nt∨F’sP-nullsets for t≥0. For t≥0, we write Et[·]\nas short-hand notation for the conditional expectation E[·|Ft].\nThroughout this section, let hbe the solution in Theorem 1.1.\n3.1 Autonomous state process Y\nWe define the drift function µYand and volatility function σYas\nµY(y) :=σ2\nD(1−y)\u0000\n1 +γ+ 2γyh(y)−2y(1 +γ)\u0001\n2yh(y)2, y∈(0,1],\nσY(y) :=σD1−y\nh(y), y∈[0,1].(3.1)\nThe next subsections use an autonomous state process Y= (Yt)t≥0valued in [0 ,1]\nwith dynamics\ndYt=µY(Yt)dt+σY(Yt)dBt, Y 0∈(0,1). (3.2)\n41It is µY(y)’s singularity at y= 0 and the singularities of both µY(y) and σY(y) at\ny= 1 that are problematic in the following analysis. The boundary behavior of the\nratio\nµY(y)\nσY(y)2=1 +γ+ 2γyh(y)−2y(1 +γ)\n2y(1−y), y∈(0,1), (3.3)\nasy↓0 and y↑1 classifies Y’s boundary behavior. We have the limits\nlim\ny↓0yµY(y)\nσY(y)2=1 +γ\n2>0,\nlim\ny↑1(1−y)µY(y)\nσY(y)2=−1\n2(1−γ)<0.(3.4)\nExample 5.4 in [5] illustrates that the singularities1\nyasy↓0 and1\n1−yasy↑1 in (3.4)\nrequire a careful error analysis. In particular, only identifying the limits in (3.4) is\ninsufficient to classify the boundary points y∈ {0,1}for the SDE (3.2). The next\nsubsections establish the following result.\nTheorem 3.1. Letγ∈(0,1),σ2\nD>0, and A >1. For any initial value Y0∈[0,1),\nthere exists a strong solution Y, pathwise unique, of the SDE (3.2) for the coefficients\ndefined in (3.1).\n3.1.1 Left-boundary point for Y\nWe start with the left-boundary point y= 0.\nLemma 3.2. Letγ∈(0,1),σ2\nD>0, and A >1. The functions in (3.1) satisfy\nyµY(y)\nσY(y)2=b0+b1(y)y, b 0:=1 +γ\n2, b 1(y) :=2γh(y)−1−γ\n2(1−y), y∈(0,1).(3.5)\nThe boundary point y= 0is inaccessible and entrance. Furthermore, the SDE (3.2)\nhas a nonnegative weak solution, unique in law, starting from Y0= 0.\nProof. Because b1is uniformly lower bounded on any interval [0 , a] fora∈(0,1), we\ncan define a constant M∈Ras\nM:= inf\ny∈[0,a]b1(y).\n42As in Eq. (2.12) in [5], we define the function\nρ(y) := exp\u0012\n2Za\nyµY(z)\nσY(z)2dz\u0013\n, y∈(0, a]. (3.6)\nThe decomposition in (3.5) gives\nρ(y) = exp\u0012\n2Za\nyµY(z)\nσY(z)2dz\u0013\n= exp\u0012\n2Za\ny\u0010b0\nz+b1(z)\u0011\ndz\u0013\n≥exp\u0012\n2Za\ny\u0010b0\nz+M\u0011\ndz\u0013\n= exp\u0012\n2b0\u0010\nlog(a)−log(y)\u0011\u0013\nexp\u0000\n2M(a−y)\u0001\n= exp\u0012\nlog\u0010\u0000a\ny\u00012b0\u0011\u0013\nexp\u0000\n2M(a−y)\u0001\n=\u0010a\ny\u00112b0exp\u0000\n2M(a−y)\u0001\n.(3.7)\nFollowing Eq. (2.13) in [5], we define the scale function\ns(y) :=−Za\nyρ(z)dz, y ∈(0, a]. (3.8)\nBecause b0≥1\n2, we get s(0) := lim y↓0s(y) =−∞. Therefore, Theorem 8A in [9]\nensures that y= 0 is inaccessible.\nNext, we first calculate the derivative\nρ′(y) =∂\n∂yexp\u0012\n2Za\ny\u0010b0\nz+b1(z)\u0011\ndz\u0013\n=−2ρ(y)\u0010b0\ny+b1(y)\u0011\n.\nThe lower bound (3.7) gives\nyρ(y)≥y\u0000a\ny\u00012b0exp\u0000\n2M(a−y)\u0001\n. (3.9)\nBecause b0>1\n2, the lower bound (3.9) gives lim y↓0yρ(y) =∞.\n43Second, two applications of L’Hopital’s rule yield\nlim\ny↓0|s(y)|\nρ(y)= lim\ny↓01\nρ(y)Za\nyρ(z)dz\n=−lim\ny↓01\nρ′(y)ρ(y)\n= lim\ny↓0y\n2\u0000\nb0+b1(y)y\u0001\n= 0,\nlim\ny↓0|s(y)|\nyρ(y)= lim\ny↓01\nyρ(y)Za\nyρ(z)dz\n=−lim\ny↓01\nρ(y) +yρ′(y)ρ(y)\n=−lim\ny↓01\n1−2y\u0010\nb0\ny+b1(y)\u0011\n=1\n2b0−1∈(0,∞),\nwhere the last line follows from b0>1\n2. Because b1is bounded and h≥γ, we have\nσY(y)≥σD(1−y)\nγand\nZa\n01 +|µY(y)|\nρ(y)σY(y)2|s(y)|dy≤Za\n0\u00121\nρ(y)σY(y)2+b0\ny+|b1(y)|\nρ(y)\u0013\n|s(y)|dy <∞.\nTheorem 8C in [9] gives us that y= 0 is not natural (equivalently, y= 0 is an entrance\npoint) becauseZa\n01\nρ(y)σY(y)2|s(y)|dy <∞.\nFinally, Theorem 2.16 in [5] shows that Y0= 0 produces a nonnegative weak\nsolution, unique in law, of the SDE (3.2).\n♢\n3.1.2 Right-boundary point for Y\nLemma 3.3. Letγ∈(0,1),σ2\nD>0, and A >1. Then, the boundary point y= 1is\ninaccessible, natural, and attracting.\n44Proof. We let a∈(0,1) be arbitrary and consider the function3\nρ(x) := exp\u0012\n−2Zx\naµY(y)\nσY(y)2dy\u0013\n= exp\u0012Zx\na\u0010\n(2y−1)(1+ γ)−2γyh(y)\n(1−y)y\u0011\ndy\u0013\n= exp\u0012Zx\na\u0010(2y−1)(1+ γ)−2γy−2γy\u0000\nh(y)−1\u0001\n(1−y)y\u0011\ndy\u0013\n= exp\u0012Zx\na\u00102−1\ny(1+γ)−2γ\u0000\nh(y)−1\u0001\n1−y\u0011\ndy\u0013\n, x∈[a,1).(3.10)\nTaylor’s formula for1\nycentered at y= 1 produces1\ny= 1+(1 −y)+o(1−y). Therefore,\nZx\na1\n(1−y)ydy=Zx\na\u00121\n1−y+ 1 +o(1−y)\n1−y\u0013\ndy, x ∈[a,1].\nThis allows us to rewrite (3.10) as\nρ(x) = exp\u0012Zx\na\u00101−γ−2γ\u0000\nh(y)−1\u0001\n1−y\u0011\ndy−(1 +γ)\u0010\nx−a+Zx\nao(1−y)\n1−ydy\u0011\u0013\n= exp\u0012Zx\na1−γ\n1−ydy+ 2γZx\na1−h(y)\n1−ydy−(1 +γ)\u0010\nx−a+Zx\nao(1−y)\n1−ydy\u0011\u0013\n=\u00101−a\n1−x\u00111−γ\nexp\u0000\nC(x)\u0001\n,(3.11)\nwhere we have defined the function\nC(x) := 2 γZx\na1−h(y)\n1−ydy−(1 +γ)\u0010\nx−a+Zx\nao(1−y)\n1−ydy\u0011\n, x∈[a,1].\nLemma 2.10(5) ensures that C(x) is a continuous function for x∈[a,1] and so\nc1:= min\nx∈[a,1]C(x)∈R, c 2:= max\nx∈[a,1]C(x)∈R. (3.12)\nBecause γ∈(0,1), we see from (3.11) that ρ(x) is integrable at x= 1 and so the\n3The definition of ρin (3.10) coincides with the definition of ρin (3.6).\n45function4\ns(y) :=Zy\naρ(x)dx, y ∈[a,1), (3.13)\nsatisfies s(1) := lim y↑1s(y)<∞.\nBecause Lemma 2.10(6) ensures h≥γ, we have\nσY(y)2=σ2\nD(1−y)2\nh(y)2≤σ2\nD(1−y)2\nγ2, y∈[0,1].\nThe Mean-Value Theorem gives θ(y)∈[y,1] such that\ns(1)−s(y) =s′\u0000\nθ(y)\u0001\n(1−y)\n=ρ\u0000\nθ(y)\u0001\n(1−y)\n=\u00101−a\n1−θ(y)\u00111−γ\neC\u0000\nθ(y)\u0001\n(1−y)\n≥\u00101−a\n1−θ(y)\u00111−γ\nec1(1−y).\nFor some constant c3>0, by (3.11), (3.12), and h(1) = 1, we have\ns(1)−s(y)\nρ(y)σY(y)2≥c3(1−y)\u0000\n1−θ(y)\u00011−γ(1−y)1+γ\n≥c3(1−y)\n(1−y)1−γ(1−y)1+γ\n=c3\n1−y,(3.14)\nwhich is not integrable for ynear 1. Therefore, Theorem 8A in [9] gives that the\nboundary point y= 1 is inaccessible.\nFrom (3.11), we ρ(x)≥c4(1−x)γ−1for some constant c4>0. This gives the\nlower bound\ns(y)\nρ(y)σY(y)2≥c5Ry\na(1−x)γ−1dx\n(1−y)1+γ=c5(1−a)γ−(1−y)γ\nγ(1−y)1+γ\nfor some constant c5. Because this lower bound is not integrable for ynear 1, Theorem\n4The definition of sin (3.13) coincides with the definition of sin (3.8).\n468C in [9] gives that y= 1 is natural (i.e., y= 1 is not entrance).\nFinally, Theorem 8D in [9] gives that y= 1 is attracting because s(1)<∞and\nthe lower bound\n1\nρ(y)σY(y)2≥c61\n(1−y)1+γ\nis not integrable for ynear 1, where c6is another constant.\n♢\n3.1.3 Proof of strong SDE existence of Y\nThe proof of the next result uses a variation of Le Gall’s uniqueness argument based\non local times (see, e.g., Exercise 3.7.12 in [12]).\nProof of Theorem 3.1. ForY0∈(0,1), the Engelbert and Schmidt conditions (i.e.,\nnon-degeneracy and local integrability) hold and ensure that a weak solution exists,\nunique in law, see, e.g., Theorem 5.5 in [12]. For Y0= 0, there is also a weak solution,\nunique in law, by Lemma 3.2.\nTo upgrade these weak solutions to strong solutions it suffices to prove pathwise\nuniqueness locally. To this end, for n∈Nsuch that Y0∈[0,1−1\nn], we define the\nstopping times τn:= inf{t≥0 :Yt= 1−1\nn}and will prove that strong uniqueness\nholds for t∈[0, τn]. Because h, h′∈ C([0,1)) with h≥γ, the formula\nσ′\nY(y) =−σD(1−y)h′(y) +h(y)\nh(y)2, y∈[0,1−1\nn],\nshows that σY(y) is Lipschitz for y∈[0,1−1\nn]. Pathwise uniqueness follows from\nProposition IX.3.2 in [20] because the local-time condition L0(Y1−Y2) = 0 holds by\nσY’s Lipschitz property and Corollary IX.3.4 in [20].\n♢\n3.2 Investors’ maximization problems\nThere is a single consumption good that serves as our model’s num´ eraire (i.e., all\nprices are real). As in [2], the stock pays dividends at rate D= (Dt)t≥0given by the\n47geometric Brownian motion\ndDt:=Dt\u0000\nµDdt+σDdBt\u0001\n, D 0>0, (3.15)\nwhere D0>0,µD∈R, and σD>0 are exogenous constants.5\nThere are two assets: a money market account with price process S(0)= (S(0))t≥0\nand a stock with price process S= (St)t≥0paying dividends at rate D. We conjecture\n(and verify) that there exist Itˆ o processes ( S(0), S) with dynamics\ndS(0)\nt=rtS(0)\ntdt, S(0)\n0:= 1, (3.16)\ndSt= (Strt−Dt)dt+StσS,t(κtdt+dBt), S 0∈(0,∞), (3.17)\nwhere S0is a constant and ( r, κ, σ S) are stochastic processes. Below, ( S0, r, κ, σ S) will\nbe endogenously determined in equilibrium. To ensure that (3.16) is well-defined, we\nrequire r∈ L1\nloc.6To ensure that (3.17) is well-defined, we require σS∈ L2\nlocand\nκ∈ L2\nloc.7\nAs in [2], we consider a model with only two investors (equivalently, two groups of\nhomogenous investors), and both investors are price takers. Investor 1 can trade the\nstock whereas investor 2 cannot hold or trade the stock. Investor 1’s self-financing\nwealth process is defined as\ndX1,t:=rtX1,tdt+θ1,tStσS,t(κtdt+dBt)−c1,tdt,\nX1,0:=θ(0)\n1,0−+S0∈R,(3.18)\nwhere the nonnegative consumption-rate process c1∈ L1\nlocand the number of shares\nof stock held θ1are investor 1’s control processes (both c1andθ1are endogenously\ndetermined in equilibrium). In (3.18), the exogenous constant θ(0)\n1,0−denotes investor\n1’s endowed holdings in the money market account. Furthermore, for simplicity, we\nnormalize the stock supply to one share outstanding so that X1,0in (3.18) reflects\ninvestor 1’s endowed stock holdings being θ1,0−= 1.\n5As in [2], the investors do not receive personal income and so Dis the model’s aggregate\nconsumption-rate process.\n6A process Q= (Qt)t≥0belongs to Lp\nloc,p∈ {1,2}, ifQis progressively measurable and satisfies\nP(RT\n0|Qt|pdt <∞) = 1 for all T >0.\n7Unlike the literature on endogenous market completeness (see, e.g., [1]), we do not require\nσS̸= 0.\n48The dynamics of investor 2’s self-financing wealth process are simpler because she\ncannot access the stock market at any time. Investor 2’s self-financing wealth process\nis defined by\ndX2,t:=rtX2,tdt−c2,tdt,\nX2,0:=θ(0)\n2,0−∈(0,∞).(3.19)\nIn (3.19), the nonnegative process c2∈ L1\nlocis investor 2’s consumption-rate process\nand is endogenous, and θ(0)\n2,0−is investor 2’s endowed holdings in the money market\naccount (exogenous). The process c2is endogenously determined in equilibrium and\nmust be such that the dynamics (3.19) are well-defined. Because the money market\naccount is in zero-net supply, the investors’ endowed money market holdings satisfy\nθ(0)\n1,0−+θ(0)\n2,0−= 0. The restriction on the initial money market holdings θ(0)\n2,0−=\n−θ(0)\n1,0−>0 in (3.19) is taken from footnote 5 in [2].\nAs in [2], investor 1 and investor 2 are assumed to have identical preferences.\nHowever, unlike the log-log utilities in [2], our model uses a common time preference\nparameter β >0 and a common constant relative risk aversion coefficient γ∈(0,1).8\nThe investors’ maximization problems are\nsup\nθ1,c1∈A11\n1−γE\u0014Z∞\n0e−βtc1−γ\n1,tdt\u0015\n,sup\nc2∈A21\n1−γE\u0014Z∞\n0e−βtc1−γ\n2,tdt\u0015\n, (3.20)\nwhere the admissible sets A1andA2in (3.20) are defined as:\nDefinition 3.4 (Admissibility) .We deem progressively measurable processes ( θ1, c1)\nadmissible for investor 1 and we write ( θ1, c1)∈ A 1ifc1,t≥0,t≥0, and the solution\nof (3.18) is nonnegative, i.e., X1,t≥0 for all t≥0. Likewise, we deem a progressively\nmeasurable process c2admissible for investor 2 and we write c2∈ A 2ifc2,t≥0,t≥0,\nand the solution of (3.19) is nonnegative, i.e., X2,t≥0 for all t≥0. ♢\n3.3 Radner equilibrium\nBecause both investors are price takers, the equilibrium is referred to as a Radner\nequilibrium. This equilibrium concept is standard and can be found in, e.g., Chapter\n8There are two natural extensions of our setting: (i) extend the model to different powers for\nthe two investors, and (ii) extend the model to γ∈(1,∞). However, both such extensions produce\nmore complicated governing ODEs than (1.10) and so we leave these extensions to future research.\n494 in the textbook [13].\nWe recall that the stock supply is in a net supply of one, the money market account\nis in zero net supply, and the aggregate consumption rate is Din (3.15).\nDefinition 3.5 (Radner equilibrium) .A constant S0>0 and processes ( r, κ, σ S) and\n(ˆθ1,ˆc1,ˆc2) constitute a Radner equilibrium if:\n(i) The processes ( ˆθ1,ˆc1)∈ A 1maximize (3.20) for i= 1; that is,\n(ˆθ1,ˆc1)∈argmax(θ,c)∈A11\n1−γEhZ∞\n0e−βtc1−γ\ntdti\n<∞. (3.21)\n(ii) The process ˆ c2∈ A 2maximizes (3.20) for i= 2; that is,\nˆc2∈argmaxc∈A21\n1−γEhZ∞\n0e−βtc1−γ\ntdti\n<∞. (3.22)\n(iii) The stock and consumption markets clear in the sense\nˆθ1,t= 1 and ˆ c1,t+ ˆc2,t=Dt,for all t≥0. (3.23)\n♢\nWalras’ law ensures that clearing in the stock market and clearing in the goods\nmarket imply clearing in the money market account. In other words, the two clearing\nconditions in (3.23) are sufficient for clearing the money market account too.\nThe next theorem gives existence of a Radner equilibrium under the additional\nrestriction θ(0)\n2,0−<g(0)\nD0where gsolves the below ODE (3.25). This restriction is a\ngeneralization of the restriction in Eq. (6) in [2]. In the following, we specialize\nTheorem 1.1 to the constant Adefined as\nA:=2β+σ2\nD−(1−γ)(2µ−γσ2\nD)\nσ2\nD. (3.24)\nThis particular expression for Acomes from equating the below ODE (3.25) with the\nODE (1.10). The restriction A >1 comes from (4.1) below. Instead of hξ0, we use h\nto denote the function produced by Theorem 1.1 in the following.\nTheorem 3.6. Letγ∈(0,1),σ2\nD>0, and β >1\n2(1−γ)(2µD−γσ2\nD).\n501. There exists a nonincreasing and nonnegative solution g∈ C1([0,1])∩ C2([0,1))\nof the ODE\n\n\nβg(y) = (1 −γ)µDg(y)−1\n2(1−γ)γσ2\nDg(y) +µY(y)g′(y)\n+1\n2σY(y)2g′′(y) + (1 −γ)σDσY(y)g′(y) + (1 −y)1−γ, y∈(0,1),\ng′(0) = 0 , g(1) = 0 .\n(3.25)\n2. For θ(0)\n2,0−∈\u0000\n0,g(0)\nD0\u0001\n, letY0∈(0,1)solve g(Y0)D0(1−Y0)γ=θ(0)\n2,0−. Then, there\nexists a Radner equilibrium in which the equilibrium interest rate process is rt=\nr(Yt)∈ L1\nlocand the equilibrium market price of risk process is κt=κ(Yt)∈ L2\nloc\nfor the deterministic functions\nr(y) :=β+γµD−1\n2γ(γ+ 1)σ2\nD−γ(γ+ 1)σ2\nD(1−y)\n2yh(y)2, y∈(0,1),\nκ(y) :=γσD\u00121−y\nyh(y)+ 1\u0013\n, y∈(0,1),(3.26)\nand the equilibrium consumption-rate processes in (3.21) and(3.22) are\nˆc1,t:=DtYt,ˆc2,t:=Dt(1−Yt), t≥0. (3.27)\nProof of Theorem 3.6(1). Lethbe the solution produced by Theorem 1.1 for Ade-\nfined in (3.24) and insert\ng(y) :=2\nξ0e−Ry\n0h(q)\n1−qdq(1−y)−γ, y∈[0,1), (3.28)\nand insert σYandµYfrom (3.1) into the ODE (3.25) to recover the ODE (1.10).\nNext, we verify the boundary conditions in (3.25). The representation (3.28)\nmeans that the boundary condition g(1) = 0 in (3.25) requires\nlim\ny↑1ξ0eRy\n0h(q)\n1−qdq(1−y)γ=∞, (3.29)\nBecause h(1) = 1 > γ, Eq. (3.29) holds. To see this, we pick ϵ >0 and yϵ∈(0,1)\n51such that\n∀y∈(yϵ,1) :h(y)> γ+ϵ.\nFory∈(yϵ,1), we have\neRy\n0h(q)\n1−qdq(1−y)γ≥eRy\nyϵh(q)\n1−qdq(1−y)γ\n≥eRy\nyϵγ+ϵ\n1−qdq(1−y)γ\n=(1−yϵ)γ+ϵ\n(1−y)ϵ.\nPassing y↑1 gives the boundary condition g(1) = 0. Because gdefined in (3.28) has\nderivative\ng′(y) =2\nξ0e−Ry\n0h(q)\n1−qdq 1\n(1−y)1+γ\u0000\nγ−h(y)\u0001\n, y∈(0,1), (3.30)\nwe see that the boundary condition g′(0) = 0 in (3.25) holds because h(0) = γ.\nFinally, because h(y)≥γfory∈[0,1], we see from (3.30) that g′(y)≤0 for\ny∈(0,1).\nThe proof of Theorem 3.6(2) is given in the next section.\n♢\n4 Proof of Theorem 3.6(2)\nThe duality techniques we use in the following proofs are standard and have been used\nin various contexts, see, e.g., the textbook [13]. Because the optimization problems\n(3.20) are on an infinite time horizon t∈[0,∞), there is a nontrivial tranversality\ncondition as t→ ∞ ; see, e.g., Section 9D in [7]. As we shall see, the tranversality\ncondition is ensured by the parameter restriction A > 1 for Ain (3.24), which is\nequivalent to β >1\n2(1−γ) (2µD−γσ2\nD). Among other things, the restriction A > 1\nensures that\nEhZ∞\n0e−βtD1−γ\ntdti\n=D1−γ\n0\nβ−1\n2(1−γ) (2µD−γσ2\nD)<∞, (4.1)\nwhere the geometric Brownian motion Dis from (3.15).\n524.1 State-price densities\nFor an arbitrary process η∈ L2\nloc, the process\ndZη,t:=−dZη,t\u0000\nr(Yt)dt+ηtdBt\u0001\n, Z η,0∈(0,∞), (4.2)\nis called a state-price density. Such density processes underly the duality theory\nwe use to prove Theorem 3.6(2) below. Any state-price density Zηhas the defining\nproperty that the process Zη,tXi,t+Rt\n0Zη,uci,udu,t≥0, is a supermartingale for any\nadmissible consumption process ciwhere i= 1 or i= 2 and Xiis the corresponding\nwealth process in (3.18) or (3.19).\nIn the below proofs, we establish the following first-order condition for investor 1\nˆZ1,t:=e−βt(DtYt)−γ=e−βtˆc−γ\n1,t, t≥0, (4.3)\ndˆZ1,t=−ˆZ1,t\u0000\nr(Yt)dt+κ(Yt)dBt\u0001\n, (4.4)\nwhere we inserted ˆ c1:=DYfrom (3.27). The dynamics in (4.4) follow from Itˆ o’s\nlemma and\ndˆc1,t=dDtYt\n=Dt\u0000\nµY(Yt) +µDYt+σDσY(Yt)\u0001\ndt+Dt\u0000\nσY(Yt) +σDYt\u0001\ndBt.(4.5)\nSimilarly, we establish the following first-order condition for investor 2\nˆZ2,t:=e−βt\u0000\nDt(1−Yt)\u0001−γ=e−βtˆc−γ\n2,t, t≥0, (4.6)\ndˆZ2,t=−ˆZ2,t\u0012\nr(Yt)dt−γσD1−h(y)\nh(y)dBt\u0013\n, (4.7)\nwhere we inserted ˆ c2:=D(1−Y) from (3.27). The dynamics in (4.7) follow from\nItˆ o’s lemma and\ndˆc2,t=dDt(1−Yt)\n=Dt\u0000\nµD−µY(Yt)−µDYt−σDσY(Yt)\u0001\ndt+Dt\u0000\nσD−σY(Yt)−σDYt\u0001\ndBt.\n(4.8)\nWhile the dtterms of bothdˆZ1\nˆZ1anddˆZ2\nˆZ2are identically equal to −r(Y), the dBterms\ndiffer. This is because investor 2 cannot access the stock market, which renders the\n53model incomplete.\n4.2 Investor 2’s optimization problem\nLetgdenote the function from Theorem 3.6(1) and define rt:=r(Yt) and κt:=κ(Yt)\nusing the funtions in (3.26).\nLemma 4.1. Letγ∈(0,1),σ2\nD>0, and β >1\n2(1−γ)(2µD−γσ2\nD).\n1. For Y0∈(0,1), we have\nEt\u0014Z∞\nte−βu\u0000\nDu(1−Yu)\u00011−γdu\u0015\n=e−βtg(Yt)D1−γ\nt, t≥0. (4.9)\n2. When Y0∈(0,1)satisfies g(Y0)D0(1−Y0)γ=θ(0)\n2,0−, investor 2’s value function\nis\nsup\nc2∈A21\n1−γE\u0014Z∞\n0e−βuc1−γ\n2,udu\u0015\n=1\n1−γE\u0014Z∞\n0e−βu\u0000\nDu(1−Yu)\u00011−γdu\u0015\n=1\n1−γg(Y0)D1−γ\n0.(4.10)\nProof. 1: For Y0∈(0,1) and n∈Nsuch that Y0∈(1\nn,1−1\nn), we define the nonde-\ncreasing sequence of stopping times τn(ω) := inf\b\ns≥t:Ys(ω)̸∈(1\nn,1−1\nn)\t\n,t≥0\nandω∈Ω, where inf ∅:=∞. Lemma 4.1 and Lemma 4.2 ensure that both y= 0\nandy= 1 are inaccessible and so P\u0000\nYt∈(0,1)∀t≥0\u0001\n= 1. We replace τnwith\nτn∧n, to get that τnis bounded by nuniformly in ω∈Ω, while lim n→∞τn(ω) =∞\ncontinues to hold.\nThe ODE (3.25) and Itˆ o’s lemma produce the following dynamics\nd\u0010\ne−βsg(Ys)D1−γ\ns+Zs\n0e−βu\u0000\nDu(1−Yu)\u00011−γdu\u0011\n=e−βsD1−γ\nsn\u0010\n(1−γ)σDg(Ys) +σY(Ys)g′(Ys)\u0011\ndBs\n+\u0010\n−βg(Ys) + (1 −γ)µDg(Ys)−1\n2(1−γ)γσ2\nDg(Ys) +µY(Ys)g′(Ys)\n+1\n2σY(Ys)2g′′(Ys) + (1 −γ)σDσY(Ys)g′(Ys) + (1 −Ys)1−γ\u0011\ndso\n,\n=e−βsD1−γ\ns\u0010\n(1−γ)σDg(Ys) +σY(Ys)g′(Ys)\u0011\ndBs,(4.11)\n54fors∈[0, τn]. We integrate (4.11) over u∈[t, τn] to get the representation\ne−βτng(Yτn)D1−γ\nτn+Zτn\nte−βu\u0000\nDu(1−Yu)\u00011−γdu\n=e−βtg(Yt)D1−γ\nt+Zτn\nte−βuD1−γ\nu\u0010\n(1−γ)σDg(Yu) +σY(Yu)g′(Yu)\u0011\ndBu.(4.12)\nTaking conditional expectation and using the martingale property of the stopped dB\nintegral give\nEt[e−βτng(Yτn)D1−γ\nτn] +Et\u0014Zτn\nte−βu\u0000\nDu(1−Yu)\u00011−γdu\u0015\n=e−βtg(Yt)D1−γ\nt.\nThe Monotone Convergence Theorem gives\nlim\nn→∞Et[e−βτng(Yτn)D1−γ\nτn] +Et\u0014Z∞\nte−βu\u0000\nDu(1−Yu)\u00011−γdu\u0015\n=e−βtg(Yt)D1−γ\nt.(4.13)\nBecause Dis a geometric Brownian motion, we have\ne−βtD1−γ\nt=D1−γ\n0e−βt+(1−γ)(µD−1\n2σ2\nD)t+(1−γ)σDBt\n=D1−γ\n0e\u0000\n−β+(1−γ)(µD−1\n2σ2\nD+σDBt\nt)\u0001\nt.\nThe law of the iterated logarithm gives lim t→∞Bt\nt= 0. Furthermore, because β >\n(1−γ)(µD−1\n2σ2\nD), we have P-a.s. the limit lim t→∞e−βtD1−γ\nt= 0. Therefore, be-\ncause gis uniformly bounded, it suffices to show that the family of random variables\n(e−βτnD1−γ\nτn)n∈Nis uniformly integrable so that we can interchange limit and expec-\ntation in (4.13) to see P-a.s. the following transversality condition\nlim\nn→∞Et[e−βτng(Yτn)D1−γ\nτn] =Et[ lim\nn→∞e−βτng(Yτn)D1−γ\nτn] = 0.\nTo establish uniform integrability, we use the inequality β >(1−γ)\u0000\nµD−1\n2γσ2\nD\u0001\nto find ϵ >0 but so small such that\nβ >(1−γ)\u0010\nµD−1\n2σ2\nD+1\n2(1 +ϵ)(1−γ)σ2\nD\u0011\n. (4.14)\n55We can write\n(e−βtD1−γ\nt)1+ϵ\n=D(1+ϵ)(1−γ)\n0 e−(1+ϵ)βt+(1+ ϵ)(1−γ)(µD−1\n2σ2\nD+1\n2(1+ϵ)(1−γ)σ2\nD)t+(1+ ϵ)(1−γ)σDBt−1\n2(1+ϵ)2(1−γ)2σ2\nDt.\nThe bound in (4.14) and Cauchy-Schwartz give the inequality\nE[(e−βτnD1−γ\nτn)1+ϵ]\n=D(1+ϵ)(1−γ)\n0 E[e−(1+ϵ)\u0000\nβ−(1−γ)(µD−1\n2σ2\nD+1\n2(1+ϵ)(1−γ)σ2\nD)\u0001\nτn+(1+ ϵ)(1−γ)σDBτn−1\n2(1+ϵ)2(1−γ)2σ2\nDτn]\n≤D(1+ϵ)(1−γ)\n0 E[e(1+ϵ)(1−γ)σDBτn−1\n2(1+ϵ)2(1−γ)2σ2\nDτn]\n=D(1+ϵ)(1−γ)\n0 ,\nwhere the last equality uses Doob’s optional sampling theorem (recall that τnis a\nbounded stopping time and that epBt−1\n2p2tis a martingale for any p∈R). De la\nValle´ e Poussin criterion gives uniform integrability.\n2: As in (3.27), we define ˆ c2:=D(1−Y) and we claim that the corresponding wealth\nprocesses is given by\nˆX2,t:=1\nˆZ2,tEthZ∞\nte−βu\u0000\nDu(1−Yu)\u00011−γdui\n=Dtg(Yt)(1−Yt)γ, t≥0,(4.15)\nwhere ˆZ2is defined in (4.6). We need to show\ndˆX2,t=\u0010\nˆX2,tr(Yt)−ˆc2,t\u0011\ndt, ˆX2,0=θ(0)\n2,0−. (4.16)\nThe initial condition ˆX2,0=θ(0)\n2,0−holds by the assumption that Y0satisfies g(Y0)D0(1−\nY0)γ=θ(0)\n2,0−. By inserting ˆX2=Dg(Y)(1−Y)γfrom (4.15) and ˆ c2=D(1−Y) from\n(3.27) into the dynamics (4.16), the dynamics (4.16) follow as soon as we show\nd\u0010\nDtg(Yt)(1−Yt)γ\n| {z }\nˆX2,t\u0011\n=\u0010\nDtg(Yt)(1−Yt)γ\n| {z }\nˆX2,tr(Yt)−Dt(1−Yt)|{z}\nˆc2,t\u0011\ndt.(4.17)\nItˆ o’s lemma and the functions in (3.26) produce the dynamics (4.17).\nTo see that ˆ c2is optimal, we use the following standard duality argument. Set\n56U(x) :=1\n1−γx1−γand let V(t, y) be the Fenchel conjugate\nV(t, b) := sup\nx>0\u0000\ne−βtU(x)−xb\u0001\n=γ\n1−γe−β\nγtbγ−1\nγ>0, b≥0.\nIn the below inequalities, we use P(Yt≤1) = 1 for t≥0, so that\nV(t,ˆZ2,t) =γ\n1−γe−βt\u0000\nDt(1−Yt)\u00011−γ≤γ\n1−γe−βtD1−γ\nt,P-a.s.,\nwhich is integrable over ( t, ω)∈[0,∞)×Ω by (4.1). Let T∈(0,∞) and let c∈ A 2\nbe arbitrary with corresponding wealth process X2≥0 given in (3.19). Then,\nE\u0014ZT\n0e−βtU(ct)dt\u0015\n≤E\u0014ZT\n0\u0010\nV(t,ˆZ2,t\u0001\n+ˆZ2,tct\u0011\ndt\u0015\n≤E\u0014ZT\n0\u0010\nV(t,ˆZ2,t\u0001\n+ˆZ2,tct\u0011\ndt+ˆZ2,TX2,T\u0015\n≤E\u0014ZT\n0V(t,ˆZ2,t)dt\u0015\n+ˆZ2,0X2,0\n=E\u0014ZT\n0V(t,ˆZ2,t)dt\u0015\n+ˆZ2,0θ(0)\n2,0−,\nwhere the first inequality uses Fenchel’s inequality, the second inequality follows from\nX2,T≥0, and the last inequality is produced by the supermartingale property of the\nstate-price density ˆZ2. We pass T↑ ∞using the Monotone Convergence Theorem to\nget\nE\u0014Z∞\n0e−βtU(ct)dt\u0015\n≤E\u0014Z∞\n0V(t,ˆZ2,t)dt\u0015\n+ˆZ2,0θ(0)\n2,0−\n=E\u0014Z∞\n0V(t,ˆZ2,t)dt\u0015\n+ˆZ2,0ˆX2,0\n=E\u0014Z∞\n0\u0010\nV(t,ˆZ2,t) +ˆZ2,tˆc2,t\u0011\ndt\u0015\n=E\u0014Z∞\n0e−βtU(ˆc2,t)dt\u0015\n,\nwhere the second equality uses (4.15) and the third equality uses (4.6). ♢\n574.3 Investor 1’s optimization problem\nLetgdenoted the function from Theorem 3.6(1) and define rt:=r(Yt) and κt:=κ(Yt)\nusing the funtions in (3.26).\nLemma 4.2. Letγ∈(0,1),σ2\nD>0,β >1\n2(1−γ)(2µD−γσ2\nD), and assume Y0∈(0,1)\nsatisfies g(Y0)D0(1−Y0)γ=θ(0)\n2,0−.\n1. There exists a volatility process σS∈ L2\nlocsuch that the nonnegative processes\nSt: =ˆX1,t+ˆX2,t, t≥0,\nˆX1,t: =1\nˆZ1,tEthZ∞\nte−βu(DuYu)1−γdui\n, t≥0,(4.18)\nsatisfy (3.17) and\ndˆX1,t=r(Yt)ˆX1,tdt+StσS,t\u0000\nκ(Yt)dt+dBt\u0001\n−DtYtdt,\nˆX1,0=θ(0)\n1,0−+S0∈R.(4.19)\n2. Investor 1’s optimal consumption rate process is ˆc1,t:=DtYt,t≥0.\nProof. 1. The Martingale Representation Theorem produces an integrand ∆ ∈ L2\nloc\nsuch that the second equation in (4.18) can be rewritten as\nˆZ1,tˆX1,t+Zt\n0ˆZ1,uDuYudu=EthZ∞\n0e−βu(DuYu)1−γdui\n=EhZ∞\n0e−βu(DuYu)1−γdui\n+Zt\n0∆sdBs\n=ˆZ1,0ˆX1,0+Zt\n0∆sdBs.(4.20)\nItˆ o’s lemma produces the following dynamics of ˆX1in (4.18)\ndˆX1,t=d ˆZ1,tˆX1,t\nˆZ1,t!\n=\u0010\nˆX1,t\u0000\nr(Yt) +κ(Yt)2\u0001\n−DtYt+κ(Yt)∆t\nˆZ1,t\u0011\ndt+\u0010\nˆX1,tκ(Yt) +∆t\nˆZ1,t\u0011\ndBt.\n(4.21)\n58By defining the volatility process\nσS,t:=1\nSt\u0010\nˆX1,tκ(Yt) +∆t\nˆZ1,t\u0011\n, t≥0, (4.22)\nwe see that the dynamics (4.21) equal those in (3.18). In other words, dˆX1are wealth\ndynamics for investor 1 when using θ1,t:= 1 and c1,t:=DtYtfort≥0.\nThe initial value in (3.18) follows from the first formula in (4.18) and initial clearing\nin the money market (i.e., θ(0)\n1,0−+θ(0)\n2,0−= 0) because\nθ(0)\n1,0−+S0=θ(0)\n1,0−+ˆX1,0+ˆX2,0\n=θ(0)\n1,0−+ˆX1,0+θ(0)\n2,0−\n=ˆX1,0.(4.23)\nTo see that (4.22) ensures that the dynamics dSof the first equation in (4.18)\nagree with (3.17), we use (3.18) and ˆ c2=D(1−Y) to see\ndSt=dˆX1,t+dˆX2,t\n=r(Yt)ˆX1,tdt+StσS,t\u0000\nκ(Yt)dt+dBt\u0001\n−DtYtdt+\u0000\nr(Yt)ˆX2,t−ˆc2,t\u0001\ndt\n=rtStdt+StσS,t\u0000\nκ(Yt)dt+dBt\u0001\n−Dtdt.(4.24)\n2. To see ˆ c1:=DYandˆθ1:= 1 are optimal for investor 1, we again use the following\nstandard duality argument as we did in the proof of Lemma 4.1. Let T > 0 and let\n(c, θ)∈ A 1be arbitrary with corresponding wealth process X1given in (3.18). Then,\nE\u0014ZT\n0e−βtU(ct)dt\u0015\n≤E\u0014ZT\n0\u0010\nV(t,ˆZ1,t\u0001\n+ˆZ1,tct\u0011\ndt\u0015\n≤E\u0014ZT\n0\u0010\nV(t,ˆZ1,t\u0001\n+ˆZ1,tct\u0011\ndt+ˆZ1,TX1,T\u0015\n≤E\u0014ZT\n0V(t,ˆZ1,t)dt\u0015\n+ˆZ1,0X1,0,\nwhere the first inequality uses Fenchel’s inequality, the second inequality follows from\nX1,T≥0, and the last inequality is produced by the supermartingale property of the\nstate-price density ˆZ1. We pass T↑ ∞using the Monotone Convergence Theorem to\n59get\nE\u0014Z∞\n0e−βtU(ct)dt\u0015\n≤E\u0014Z∞\n0V(t,ˆZ1,t)dt\u0015\n+ˆZ1,0ˆX1,0\n=E\u0014Z∞\n0\u0010\nV(t,ˆZ1,t) +ˆZ1,tˆc1,t\u0011\ndt\u0015\n=E\u0014Z∞\n0e−βtU(ˆc1,t)dt\u0015\n,\nwhere the first equality uses (4.18) and the second equality uses (4.3). ♢\n4.4 Remaining proof\nProof of Theorem 3.6(2). The regularity conditions r(Yt)∈ L1\nlocandκ(Yt)∈ L2\nloc\nfollow from the functions in (3.26) and the fact that y= 0 and y= 1 are inaccessible\nby Lemma 3.2 and Lemma 3.3. Optimality of (3.27) and ˆθ1:= 1 follows from Lemma\n4.1 and Lemma 4.2. Obviously, these control processes satisfy the clearing conditions\nin (3.23). ♢\nA The failure of the comparison principle at y= 1\nThis appendix reports on the failure of the comparison principle at y= 1. Theorem\n2.5 uses Schauder’s fixed-point theorem to construct a local solution of (1.10) starting\nfrom y= 0. Alternatively, by redefining the set X=X(y0)⊂ C\u0000\n[y0,1]\u0001\nas those\ncontinuous functions h: [y0,1]→Rfor which\nsup\ny∈[y0,1)|1−h(y)|\n1−y≤1−γ2\nA−1, (A.1)\nfor a constant y0∈[0,1), it is possible to use Schauder’s fixed-point theorem to\nconstruct a local solution of (1.10) starting from y= 1. However, as the next lemma\nshows, the comparison principle, i.e., the conclusion of Theorem 2.3, can fail when we\ncompare ODEs starting at y= 1. This is the reason why we construct local solutions\nstarting for y= 0 in Theorem 2.5.\nIn the next proposition, the function f∈ C1([0,1))∩ C([0,1]) denotes the unique\n60solution produced by setting a3:=−γin Theorem 2.4 of the ODE\n\n\nf′(y) =1+γ\ny\u0010\nγ−f(y)\u0011\n+γf(y)1−f(y)\n1−y, y∈(0,1),\nf(0) = γ, f (1) = 1 .(A.2)\nProposition A.1. Letγ∈(0,1),A >1, letfsolve (A.2) , and let hξ0∈ C1([0,1])be\nas in Lemma 2.10. Then, we have\n1.f /∈ C1([0,1]).\n2.limy↑1f′(y) =∞,R1\n0|f′(q)|dq <∞, andR1\n01−f(q)\n1−qdq <∞.\n3.f(y)fails the comparison principle with hξ0(y)foryclose to 1.\nProof. 1. To see f /∈ C1([0,1]), we argue by contradiction. By using L’Hopital’s rule\nandf(1) = 1 in (A.2), we get\nf′(1) = (1 + γ)(γ−1) +γlim\ny↑1f(y)1−f(y)\n1−y=γ2−1 +γf′(1).\nThis equation has the unique negative solution f′(1) =−(1+γ). Theorem 2.4 ensures\nthat f≤1. However, because f(1) = 1, f′(1)<0 gives the contradiction f(y)>1\nforyclose to 1.\n2.Step 1/3: We start by proving lim inf y↑11−f(y)\n1−y=∞. Because f(1) = 1, the first\nterm on the right-hand-side of (A.2) has limit\nlim\ny↑11 +γ\ny\u0010\nγ−f(y)\u0011\n=−(1−γ)(1 + γ)<0. (A.3)\nSetκ:= (1−γ)(1 + γ)>0 and consider any ϵsuch that\n0< ϵ < κ.\nThe limit in (A.3) gives yϵ∈(0,1) such that the solution of the ODE (A.2) satisfies\nf′(y)≤ −κ+ϵ+γf(y)1−f(y)\n1−y,\n≤ −κ+ϵ+γ1−f(y)\n1−y, y∈(yϵ,1),(A.4)\n61where the second inequality uses f≤1. Because\nd\ndy\u00121−f(y)\n(1−y)γ\u0013\n=γ1−f(y)\n(1−y)1+γ−f′(y)\n(1−y)γ≥κ−ϵ\n(1−y)γ, (A.5)\nwe have for y∈(yϵ,1)\n1−f(y)\n(1−y)γ−1−f(yϵ)\n(1−yϵ)γ=Zy\nyϵd\ndq\u00121−f(q)\n(1−q)γ\u0013\ndq\n≥(κ−ϵ)Zy\nyϵ1\n(1−q)γdq\n=−κ−ϵ\n1−γ\u0010\n(1−y)1−γ−(1−yϵ)1−γ\u0011\n.(A.6)\nTo argue by contradiction, we assume there is an increasing sequence ( yn)n∈N⊂\n(yϵ,1) converging to 1 such that\nlim\nn→∞1−f(yn)\n1−yn= lim inf\ny↑11−f(y)\n1−y∈[0,∞).\nIn that case, we have\nlim\nn→∞1−f(yn)\n(1−yn)γ= lim\nn→∞1−f(yn)\n1−yn(1−yn)1−γ= 0.\nIn (A.6), we replace ybyynand pass n↑ ∞to see\n−1−f(yϵ)\n(1−yϵ)γ≥κ−ϵ\n1−γ(1−yϵ)1−γ.\nBy multiplying by −(1−yϵ)γ, we get the contradiction\n0≤1−f(yϵ)≤ϵ−κ\n1−γ(1−yϵ)<0,\nwhere the first inequality holds by Theorem 2.4.\nStep 2/3: To prove lim y↑1f′(y) =∞, it suffices to show lim inf y↑1f′(y) =∞. By\ntaking liminf in the ODE (A.2) and using f(1) = 1, we get\nlim inf\ny↑1f′(y) =−(1−γ)(1 + γ) +γlim inf\ny↑11−f(y)\n1−y=∞.\n62Step 3/3: Consider yϵ∈(0,1) such that f′(y)>0 for y∈(yϵ,1). Then, f(1) = 1\nand the Monotone Convergence Theorem give\nZ1\nyϵf′(q)dq= lim\ny↑1Zy\nyϵf′(q)dq= lim\ny↑1f(y)−f(yϵ) = 1−f(yϵ)<∞.\nBecause f(1) = 1, by increasing yϵif necessary, we can assumef(y)\nγ≥γ\n2fory∈(yϵ,1).\nThe ODE (A.2) gives\n1 +γ\nyf(y) +f′(y) =1 +γ\nyγ+γf(y)1−f(y)\n1−y\n≥γ\n21−f(y)\n1−y, y∈(yϵ,1).\nBecause the left-hand-side is integrable over y∈(yϵ,1), the nonnegative function\n1−f(y)\n1−yis also integrable over y∈(yϵ,1). For y∈(0, yϵ), the bounds γ≤f(y)≤1\ngive integrability of1−f(y)\n1−y.\n3. The ODE for hξ0in (2.32) and hξ0(q)≤1 for all q∈[0,1] give the lower bound\nh′\nξ0(y) =1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+hξ0(y)γ+\u0010\n(A−γ)eR1\ny1−hξ0(q)\n1−qdq−A\u0011\nhξ0(y)\n1−y\n≥1 +γ\ny\u0000\nγ−hξ0(y)\u0001\n+γhξ0(y)1−hξ0(y)\n1−y, y∈(0,1).(A.7)\nTo argue by contradiction, we assume that the comparison principle holds for ynear\n1. Based on (A.2) and (A.7), the property f(1) = hξ0(1) = 1 gives hξ0(y)≤f(y) for\ny∈[0,1] and so\n1−hξ0(y)\n1−y≥1−f(y)\n1−y≥0, y∈(0,1).\nHowever, Lemma 2.10(7) ensures that h′\nξ0(y) is continuous at y= 1 with limit1−γ2\nA−1\nand so we get the contradiction\n∞>1−γ2\nA−1=h′\nξ0(1) = lim\ny↑11−hξ0(y)\n1−y≥lim\ny↑11−f(y)\n1−y=∞.\n♢\n63References\n[1] R. M. Anderson, R. C. Raimondo, Equilibrium in continuous-time financial mar-\nkets: Endogenously dynamically complete markets, Econometrica 76 (2008), 841–\n907.\n[2] S. Basak, D. Cuoco, An equilibrium model with restricted stock market partici-\npation, Review of Financial Studies 11 (1998), 309–341.\n[3] H. Brezis, L. A. Peletier, D. A. Terman, A very singular solution of the heat\nequation with absorption. Arch. Rational Mech. Anal. 95 (1986), no. 3, 185–209.\n[4] G. Chabakauri, Asset pricing with heterogeneous preferences, beliefs, and portfolio\nconstraints, Journal of Monetary Economics 75 (2015), 21–34.\n[5] A. S. Cherny, H. J. Engelbert, Singular stochastic differential equations, Springer,\nLecture Notes in Mathematics (2004).\n[6] D. Cuoco, H. He, Dynamic equilibrium in infinite-dimensional economies with\nincomplete financial markets, working paper (1994).\n[7] D. Duffie, Dynamic asset pricing theory, 3rd Ed., Princeton University Press\n(2001).\n[8] M. Garc´ ıa-Huidobro, R. Man´ asevich, C. S. Yarur, On positive singular solutions\nfor a class of nonhomogeneous p-Laplacian-like equations, J. Differential Equations\n145 (1998), no. 1, 23–51.\n[9] I. Helland, One-dimensional diffusion processes and their boundaries, working\npaper (1996).\n[10] J. Hugonnier, Rational asset pricing bubbles and portfolio constraints, Journal\nof Economic Theory 147 (2012), 2260–2302.\n[11] R. G. Iagar, P. Lauren¸ cot, Existence and uniqueness of very singular solutions\nfor a fast diffusion equation with gradient absorption. J. Lond. Math. Soc. (2) 87\n(2013), no. 2, 509–529.\n[12] I. Karatzas, S. Shreve, Brownian motion and stochastic calculus, 2nd Ed.,\nSpringer (1988).\n64[13] I. Karatzas, S. Shreve, Methods of mathematical finance, Springer (1998).\n[14] G. Leoni, A very singular solution for the porous media equation ut= ∆( um)−up\nwhen 0 < m < 1, J. Differential Equations 132 (1996), no. 2, 353–376.\n[15] J. Mawhin, Probl` emes de Dirichlet variationnels non lin´ eaires. S´ eminaire de\nMath´ ematiques Sup´ erieures, 104. Presses de l’Universit´ e de Montr´ eal, Montreal,\nQC, 1987.\n[16] W.-M. Ni, J. Serrin, Nonexistence theorems for singular solutions of quasilinear\npartial differential equations. Comm. Pure Appl. Math. 39 (1986), no. 3, 379–399.\n[17] R. Prieto, Dynamic equilibrium with heterogeneous agents and risk constraints,\nworking paper (2013).\n[18] P. Pucci, J. Serrin, Continuation and limit properties for solutions of strongly\nnonlinear second order differential equations. Asymptotic Anal. 4 (1991), no. 2,\n97–160.\n[19] P. Pucci, J. Serrin, Maximum principles for elliptic partial differential equations.\nHandbook of differential equations: stationary partial differential equations. Vol.\nIV, 355–483, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.\n[20] D. Revuz, M. Yor, Continuous martingales and Brownian motion, 3rd Ed.,\nSpringer (1999).\n[21] K. Weston, Existence of an equilibrium with limited participation, Finance &\nStochastics (2024), to appear.\n65"    },    {        "title": "2402.07213v1.Non_Radial_Free_Geodesics__II__In_Spatially_Curved_FLRW_Spacetime.pdf",        "content": "arXiv:2402.07213v1  [gr-qc]  11 Feb 2024Non-Radial Free Geodesics. II. In Spatially Curved FLRW\nSpacetime\nOmar Nemoul1, Hichem Guergouri1, Jamal Mimouni1,2\n1Research Unit in Scientific Mediation, CERIST, Constantine , 25016, Algeria.\n2Laboratoire de Physique Mathematique et Subatomique (LPMS ), University of\nConstantine 1, Constantine, 25017, Algeria.\nContributing authors: omar.nemoul@umc.edu.dz ;hichem.guergouri@univ-bejaia.dz ;\njamalmimouni@umc.edu.dz ;\nAbstract\nThis paper presents an in-depth exploration of timelike fre e geodesics in spatially curved Friedmann-\nLemaˆ ıtre-Robertson-Walker (FLRW)spacetime. Aunifiedap proachfor thesegeodesics encompassing\nboth radial and non-radial trajectories across Euclidean, spherical, and hyperbolic geometries is\nemployed. Using the symmetry properties of the system, two c onstants of motion related to this\ndynamical system are derived. This treatment facilitates t he explicit computation of radial and\nangular peculiar velocities, along with the evolution of th e comoving radial distance and angle over\ntime. The study introduces three distinct methods for chara cterizing geodesic solutions, further\ndelving into the flatness limit, the null geodesic limit, the radial geodesic limit, the comoving geodesic\nlimit, and the circular orbits. Additionally, we analyze Ki lling vectors, reflecting the symmetries\ninherentinthedynamicalsystem.Thisstudyprovidesamore profoundunderstandingofthebehavior\nof free particles as observed from a comoving reference fram e.\nKeywords: Non-radial geodesics, FLRW spacetime\n1 Introduction\nTheΛCDMcosmologicalmodel[ 1]hasbeenstronglyreinforcedbyobservationsoftheCosmicMicro wave\nBackground (CMB) from missions like COBE [ 2], WMAP [ 3], and Planck [ 4], along with studies high-\nlighting the universe’s accelerating expansion rate [ 5,6]. These thorough explorations have consistently\nrevealed the dominance of a mysterious repulsive gravity known as d ark energy, driving the universe’s\naccelerating expansion. Additionally, the observable universe exhib its a remarkable degree of spatial\nhomogeneity, isotropy, and geometric flatness on large scales. Th ese characteristics are key to confirming\nthe Friedmann-Lemaˆ ıtre-Robertson-Walker (FLRW) metric as th e definitive model for describing our\nspacetime [ 7–10]. The study of geodesics is of crucial importance not only in general relativity [ 11] but\nalso in cosmology [ 12] , astrophysics, and quantum gravity. A special focus on radially f reely-falling test\nparticles in the absence of non-gravitational forces are extensiv ely explored in Refs. [ 13–20]. While sev-\neral studies [ 13,15–18] have mainly focused on solving geodesic equations, other researc h efforts [ 19,20]\nused the symmetry of the system to construct the exact solution . In our recent study [ 21], we explored\nthe free timelike geodesics for non-radial motion in spatially flat FLRW spacetime, and further applied\nthese insights to the ΛCDM model.\nBuilding on the results in Ref. [ 21], this paper broadens its scope to include all spaces of maximal\nsymmetry. Mathematically speaking, these spaces are character ized by their constant curvature k. It\nis important to note that the complete connected Riemannian manifo lds of a constant curvature can\nbe classified into the Euclidean ( k= 0), spherical ( k >0), and hyperbolic ( k <0) geometries, other\nspaces with the same property are isometric to one of these three classes, by the Killing-Hopf theorem\n[22,23]. Accordingly, we shall present a unified formulation for all types of timelike free geodesics,\n1encompassing both radial and non-radial trajectoriesacross diff erent geometric spaces. Initially, we start\nwith a thorough review of the problem in flat space, recalling key findin gs from Ref. [ 21] to lay the\ngroundwork for our extended analysis.\nNext, we examine the heart of this study, focusing on the determin ation of the comoving radial\nχand angular φmotion for a freely-falling test particle in generally curved FLRW spac etime. It is\nvery challenging to directly solve either the Euler-Lagrange equatio ns [24,25] or the geodesic equations\nfor the unknown functions ( χ,φ) and (t,χ,φ), respectively. The complexity primarily arises from the\nintricate nature of the coupled partial differential equations syst em that emerges when employing both\nthe Euler-Lagrange and geodesic equations method. Fortunately , the inherent symmetry of the physical\nsystem offers a key advantage in overcoming this obstacle by yielding two constants of motion used in\nour approach. The isotropic symmetry of the system concerning t he angular variable φ, gives rise to\nthe first constant Bk. The derivation of the second constant involves recognizing that t he magnitude\nof peculiar velocity is a 3D scalar vector. By extending its established expression for radial to non-\nradial motion, we successfully derive the second constant of motio nAk, which now accounts for both\nradial and angular components of the peculiar velocity. In Appendix C, we rigorously demonstrate that\nthe non-radial solutions satisfy the Euler-Lagrange equations. S ubsequently, we establish that the non-\nradial geodesics in curved space meet several limits: the flatness lim it, the radial geodesic limit, the null\ngeodesic limit, and the comovinggeodesic limit. Furthermore, we prov ide simpler alternative expressions\nfor the exact solutions χ(t) andφ(t). Our discussion then delves into the Killing vectors associated with\nthe symmetries of the dynamical system with determining their prec ise expressions.\nThroughout this paper, Greek letters µ,ν, etc., represent spacetime indices, taking values 0, 1, 2,\nand 3. Spatial indices, denoted by Latin letters i,j, etc., have values 1, 2, and 3. We adopt a metric\nsignatureof(+ ,−,−,−)in a spacetime coordinatesystem defined by cosmictime tand comovingspatial\ncoordinates xi. (x,y,z) and (χ,θ,φ) represent the comovingCartesian and spherical coordinate sys tems,\nrespectively. Moreover, our analysis employs a unit system where t he speed of light is normalized to\nc= 1.\n2 Non-Radial Geodesics in Spatially Flat FLRW Spacetime\nBefore addressing the non-radial geodesics for free motion in gen eral FLRW spacetime, it is useful to\nfirst examine the problem in the flat case. This was recently underta ken in our paper [ 21], where motion\nis constrained to the ( xy) plane by setting θ=π\n2. This constraint simplifies the spacetime interval to\nds2=dt2−γijdxidxj\n=dt2−a2(t)(dχ2+χ2dφ2). (1)\nwhereγijrepresentsthe 3Dspatial metric, while a(t) is the expansionscalarfactor, a function describing\nhow spatial dimensions scale over time. The action for a freely-falling particle is expressed as\nS[χ(t),φ(t)] =−m/integraldisplay\nds=−m/integraldisplay\ndt/radicalig\n1−a2(t)˙χ2−a2(t)χ2˙φ2. (2)\nThe corresponding Lagrangian is\nL(χ,˙χ,˙φ,t) =−m/radicalig\n1−a2(t)˙χ2−a2(t)χ2˙φ2. (3)\nTo ensure that the Lagrangian is expressed in energy units and to o btain the correct non-relativistic\nlimit in accordance with Newton’s law, we’ve included the factor “ −m”. The squared magnitude of the\npeculiar velocity is given by\nv2\npec(t) =a2(t)γij˙xi˙xj\n=a2(t)˙χ2+a2(t)χ2˙φ2. (4)\nSubsequently, two symmetries of this Lagrangian given by the Killing v ectors\nξA=±/radicaligg\nA2−B2\nχ2∂χ+B\nχ2∂φ, (5a)\n2ξB=∂φ, (5b)\nare used to derive two constants of motion\nA= sgn(A)a2(t)/radicaligg\n˙χ2+χ2˙φ2\n1−a2(t)˙χ2−a2(t)χ2˙φ2. (6a)\nB=a2(t)χ2˙φ/radicalig\n1−a2(t)˙χ2−a2(t)χ2˙φ2. (6b)\nIt can be seen that the geodesic solutiondoes not depend on sgn( A), where both “ A” and “−A”yield the\nsame physics. As delineated in Ref. [ 21], the exact solution can be characterized by three distinct cases:\n•In terms of (χi,φi,A,B,sgn(˙χi)):\nχ(t;χi,A,B,sgn(˙χi)) =/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftigg/integraldisplayt\nti|A|\na(t′)/radicalbig\na2(t′)+A2dt′+sgn(˙χi)/radicalbigg\nχ2\ni−B2\nA2/parenrightigg2\n+B2\nA2,(7a)\nφ(t;χi,φi,A,B,sgn(˙χi)) =/integraldisplayt\ntiB\nχ2(t′;χi,A,B,sgn( ˙χi))\na(t′)/radicalbig\na2(t′)+A2dt′+φi, (7b)\nwith the initial radial distance χi≥/vextendsingle/vextendsingleB\nA/vextendsingle/vextendsingle, the initial angle −π < φi≤π, arbitrary real constants of\nmotionAandB, whereA= 0⇒B= 0 (See Ref. [ 21]) and the sign of the initial comoving radial\nvelocity sgn(˙ χi) =±or 0.\n•In terms of (χi,φi,vχ,i,vφ,i):\nχ(t;χi,vχ,i,vφ,i) =/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/parenleftbigg/integraldisplayt\ntiai\na(t′)/radicaltp/radicalvertex/radicalvertex/radicalbtv2\nχ,i+v2\nφ,i\na2(t′)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i)dt′+χivχ,i/radicalBig\nv2\nχ,i+v2\nφ,i/parenrightbigg2\n+χ2\niv2\nφ,i\nv2\nχ,i+v2\nφ,i\n(8a)\nφ(t;χi,φi,vχ,i,vφ,i) =/integraldisplayt\ntiai\na(t′)χivφ,i\nχ2(t′;χi,vχ,i,vφ,i)/radicalBig\na2(t′)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i)dt′+φi, (8b)\nwithχi≥0,−π <φi≤π, and 0≤v2\nχ,i+v2\nφ,i≤1. Here,vχ,iandvφ,irepresent the initial radial and\nangular peculiar velocity components at a specific initial time ti, respectively. The notation ai=a(ti)\nis employed.\n•In terms of (χi,φi,vpec,i,ψi):\nχ(t;χi,vpec,i,ψi) =/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt\n/integraldisplayt\ntiai\na(t′)vpec,i/radicalig\na2(t′)(1−v2\npec,i)+a2\niv2\npec,idt′+χicosψi\n2\n+χ2\nisin2ψi,(9a)\nφ(t;χi,φi,vpec,i,ψi) =/integraldisplayt\ntiai\na(t′)χivpec,isinψi\nχ2(t′;χi,vpec,i,ψi)/radicalig\na2(t′)(1−v2\npec,i)+a2\niv2\npec,idt′+φi, (9b)\nwithχi≥0,−π <φi≤π, 0≤vpec,i≤1,−π <ψi≤π, and−π <ψi≤π. Here,vpec,irepresents the\nmagnitude of /vector vpec,i, andψiindicates the deviation angle of /vector vpec,ifrom the radial axis (deviation angle)\natt=ti. They are given by\nvpec,i=/radicalig\nv2\nχ,i+v2\nφ,i, (10a)\nψi= arctan/parenleftbiggvφ,i\nvχ,i/parenrightbigg\n. (10b)\nNote that in this case of flat space and for given constants of motio nAandB, the smallest possible\ndistance to which the free particle can approach the origin at t∗isχ∗=|B/A|, corresponding to ˙ χ∗= 0.\n32.1 Radial-Geodesic Limit\nThe geodesic for Radial Motion (RM) can be obtained by setting B= 0 in Eqs. ( 7), orvφ,iin Eqs. ( 8),\norψi= 0,πin Eqs. ( 9). It gives\nχ(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\nti1\na(t′)A/radicalbig\na2(t′)+A2dt′+χi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (11a)\nφ(t) =φi. (11b)\nHere, we use the redundancy of sgn( A) and choose to align it with sgn(˙ χi). This solution can be directly\nderived by starting from the following spacetime interval for radial motion\nds2\nRM=dt2−a2(t)dχ2. (12)\nThe action for radial motion only depends on χ(t) as\nSRM[χ(t)] =−m/integraldisplay\ndt/radicalbig\n1−a2(t)˙χ2. (13)\nThe corresponding Lagrangian is\nLRM(˙χ(t),t) =−m/radicalbig\n1−a2(t)˙χ2. (14)\nNotice that this Lagrangian exhibits a transnational symmetry in th e radial distance χ, leading to the\nfollowing conserved quantity\nARM=A/vextendsingle/vextendsingle\nB=0=a2(t)˙χ/radicalbig\n1−a2(t)˙χ2. (15)\nIn this case, the squared magnitude of the peculiar velocity v2\npecis given by\nv2\npec(t) =v2\nχ(t) =a2(t)˙χ2=A2\nRM\na2(t)+A2\nRM. (16)\nThe radial solution ( 11) corresponding to the action ( 13) has been derived in various works [ 13–20].\nIt is important to note that this treatment of the radial geodesic d oes not explicitly depend on the\ngeometry of the 3D space. Both flat, spherical, and hyperbolic spa ces yield the same solution ( 11) for\nradial motion. The only distinction among them lies in the expression of the scale factor a(t).\n2.2 Null and Comoving Geodesic Limits\nThe null geodesic can be found by having A,B→+∞in Eqs. ( 7), orv2\nχ,i+v2\nφ,i= 1 in Eqs. ( 8), or\nvpec,i= 1 in Eqs. ( 9). On the other hand, the comoving geodesic arises when setting all parameters\nA,B,vχ,i,vφ,i, andvpec,ito be zero.\nIn the next section, we shall generalize this analysis to spatially curv ed FLRW spacetime, not limited\nto flat space.\n3 Non-Radial Geodesics in generally Curved FLRW Spacetime\nWe now turn to a homogeneous and isotropic 3D space, whose arbitr ary spatial geometry is captured\nby the curvature parameter k. This parameter allows for three distinct geometrical spaces:\nSpherical space , k>0\nFlat space, k = 0\nHyperbolic space , k<0\nHere, the curvature parameter kis defined in terms of the curvature radius ρas follows\nk=K\nρ2, (17)\n4whereK= sgn(k) denotes the curvature sign, taking values -1, 0, and +1 for hype rbolic, flat, and spher-\nical geometries, respectively. The curvature parameter kcan also be expressed in terms of measurable\nquantities as\nk=−H2\n0Ωk,0\nc2, (18)\nwhereH0is the Hubble parameterat the present time, and cis the speed of light. Ω k,0representstoday’s\ncurvature density parameter, given by\nΩk,0= 1−Ωr,0−Ωm,0−ΩΛ,0. (19)\nHere,Ωr,0,Ωm,0,andΩ Λ,0representthe radiation,matter,andcosmologicalconstantden sityparameters\nat the present time, respectively. Now, to address the problem in a generally 3D space, one can simply\nreplaceχin the flat case with1√\nksin(√\nkχ) for spherical space, and1√\n|k|sinh(/radicalbig\n|k|χ) for the hyperbolic\ncase. Accordingly, let’s define the spacetime interval for motion re stricted to the ( xy) plane as\nds2\nk=dt2−γ(k)\nijdxidxj\n=dt2−a2(t)(dχ2+S2\nk(χ)dφ2), (20)\nwhereγ(k)\nijrepresents the 3D spatial metric for the three classes of maximally symmetric space ( K=\n0,−1,+1). TheSkfunction is defined as\nSk(χ) =\n\n1√\nksin(√\nkχ), k> 0\nχ, k = 0\n1√−ksinh(√\n−kχ), k<0, (21)\nwith its inverse S−1\nkfunction\nS−1\nk(α) =\n\n1√\nkarcsin(√\nkα), k> 0\nα, k = 0\n1√−karcsinh(√\n−kα), k<0. (22)\nWe define the Ckfunction as\nCk(χ) =dSk(χ)\ndχ=\n\ncos(√\nkχ), k> 0\n1, k = 0\ncosh(√\n−kχ), k<0, (23)\nwith its inverse C−1\nkfunction\nC−1\nk(α) =/braceleftigg1√\nkarccos(α), k> 0\n1√−karccosh(α), k<0. (24)\nand finally, the Tkfunction\nTk(χ) =Sk(χ)\nCk(χ)=\n\n1√\nktan(√\nkχ), k> 0\nχ, k = 0\n1√−ktanh(√\n−kχ), k<0, (25)\nwith its inverse T−1\nkfunction\nT−1\nk(α) =\n\n1√\nkarcctan(√\nkα), k> 0\nα, k = 0\n1√\n−karctanh(√\n−kα), k<0. (26)\n5They satisfy the following identities\nC2\nk(χ)+kS2\nk(χ) = 1, (27a)\n1+kT2\nk(α) =1\nC2\nk(α). (27b)\nThe first identity includes the Pythagorean trigonometric identity f ork>0 and the hyperbolic identity\nfork <0. In this context, the comoving radial distance χassumes real, positive values for k≤0.\nHowever, for k>0,χis confined to a compact range, specifically 0 ≤χ≤π√\nk. Additionally, the function\nCkis not invertible for k= 0, as it does not constitute an injective (one-to-one) mapping in t his case.\n3.1 Geodesics From Euler-Lagrange Equations\nLet us now formulate the action for a space generally curved, char acterized by the curvature parameter\nk. We have\nSk[χ(t),φ(t)] =−m/integraldisplay\ndsk=−m/integraldisplay\ndt/radicalig\n1−a2(t)˙χ2−a2(t)S2\nk(χ)˙φ2. (28)\nIn this context, the Lagrangian is expressed as\nLk(χ,˙χ,˙φ,t) =−m/radicalig\n1−a2(t)˙χ2−a2(t)S2\nk(χ)˙φ2. (29)\nThe squared magnitude of the peculiar velocity takes the form\nv2\npec(t) =a2(t)γ(k)\nij˙xi˙xj, (30)\nwhereγk\nijrepresents the 3D spatial metric. This can be decomposed into rad ial and angularcomponents,\nresulting in\nv2\npec(t) =v2\nχ(t)+v2\nφ(t), (31)\nwhere\nv2\nχ(t) =a2(t)˙χ2, (32a)\nv2\nφ(t) =a2(t)S2\nk(χ)˙φ2. (32b)\nFollowingthemethodologyusedintherecentworkRef.[ 21],wheretheexactsolutionfornon-radialtime-\nlike geodesics in the flat case were constructed, we apply the Euler- Lagrange equations for ( χ(t),φ(t)),\nyielding\nd\ndt\na2(t)˙χ(t)/radicalig\n1−a2(t)˙χ2(t)−a2(t)S2\nk(χ(t))˙φ2(t)\n=Sk(χ(t))Ck(χ(t))a2(t)˙φ2(t)/radicalig\n1−a2(t)˙χ2(t)−a2(t)S2\nk(χ(t))˙φ2(t),(33a)\nd\ndt\na2(t)S2\nk(χ(t))˙φ(t)/radicalig\n1−a2(t)˙χ2(t)−a2(t)S2\nk(χ(t))˙φ2(t)\n= 0. (33b)\n3.2 Determining the Constants of Motion\nIt is evident that the Lagrangian exhibits symmetry under any time- independent rotation δof the\nangular variable φ, as it is independent of φ\nφ→φ′=φ+δ. (34)\nFrom Noether’s theorem [ 26], this symmetry in the dynamical system implies the existence of a con -\nservation law, signified by a conserved quantity that remains invaria nt over time (constant of motion).\nFrom the Euler-Lagrange equation, we derive the constant of mot ion as\na2(t)S2\nk(χ)˙φ/radicalig\n1−a2(t)˙χ2−a2(t)S2\nk(χ)˙φ2=Bk, (35)\n6whereBkis an arbitrary real number, related to both the initial angular and r adial velocities of the\nparticle. It’s important to note that:\n•sgn(˙φ) = sgn(Bk),\n•Bk>0⇔φ(t) is an increasing function,\n•Bk<0⇔φ(t) is a decreasing function,\n•Radial Geodesics: Bk= 0⇔φ(t) is a constant function,\n•B0=B, which corresponds to the same constant of motion as in the flat ca se.\nBy inverting Eq. ( 35), we can obtain the square of the angular peculiar velocity v2\nφas\nv2\nφ(t) =a2(t)S2\nk(χ)˙φ2=B2\nk(1−a2(t)˙χ2)\na2(t)S2\nk(χ)+B2\nk. (36)\nTo identify the second constant of motion, we use the fact that th e magnitude of the peculiar velocity\nvpec(t) is a 3D scalar, invariant under spatial coordinate transformation s, as evident from its spatial\ncovariant form in Eq. ( 30). In our homogeneous and isotropic 3D space, any general geode sic curve can\nbe transformed to have a purely radial spatial part through appr opriate local spatial coordinates. This\nallows us to generalize the 3D scalar peculiar velocity expression ( 16) for radial motion to include all\ngeneral geodesic motions. Hence, we have\nv2\npec(t) =v2\nχ(t)+v2\nφ(t)\n=a2(t)˙χ2+a2(t)S2\nk(χ)˙φ2\n=A2\nk\na2(t)+A2\nk. (37)\nThe real number Akis now related to both the initial conditions for both radial and angula r peculiar\nvelocities. It is worth mentioning that:\n•sgn(Ak) is redundant for our physical motion, i.e., does not affect the phys ical nature of the motion\nwhere both “ Ak” and “−Ak” represent the same physics.\n•Comoving Geodesics: Ak= 0⇔vpec(t) = 0⇔vχ(t) = 0 andvφ(t) = 0\n•Ak= 0⇒Bk= 0, although the converse does not necessarily hold true.\n•Null Geodesics: vpec(t) = 1⇔Ak= +∞ ⇔Bk= +∞\n•A0=A, gives the same constant of motion for the flat case.\nWe have obtained a system of two equations, involving the comoving v elocity variables (˙ χ,˙φ). Solving\nthese two equations for (˙ χ,˙φ) gives\n˙χ(t) =sgn(˙χ(t))\na(t)/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nk−B2\nk\nS2\nk(χ(t))\na2(t)+A2\nk, (38a)\n˙φ(t) =Bk\nS2\nk(χ(t))\na(t)/radicalbig\na2(t)+A2\nk, (38b)\nand by corresponding the peculiar velocity components ( vχ,vφ) are\nvχ(t) =a(t)˙χ(t) = sgn(vχ(t))/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nk−B2\nk\nS2\nk(χ(t))\na2(t)+A2\nk, (39a)\nvφ(t) =a(t)Sk(χ)˙φ(t) =Bk\nSk(χ(t))/radicalbig\na2(t)+A2\nk. (39b)\nWe note that both radial and angular peculiar velocities decrease ov er time, approaching zero as the\nscale factor tends towards infinity.\n3.3 The initial Conditions for Peculiar Velocity Component s\nIt is clear from Eqs. ( 39a) and (39b) that in order to determine the peculiar velocity components vχ(t)\nandvφ(t) of the particle, the constants of motion AkandBkmust be fixed. Now, we will express Akand\nBkin terms of the initial radial vχ,i=vχ(ti) and angular vφ,i=vφ(ti) peculiar velocities at a chosen\n7initial time ti. By substituting t=tiinto Eqs. ( 39a) and (39b) and then inverting these equations for\nAkandBk, we obtain\nAk= sgn(A)ai/radicaligg\nv2\nχ,i+v2\nφ,i\n1−v2\nχ,i−v2\nφ,i, (40a)\nBk=aiSk(χi)vφ,i/radicalig\n1−v2\nχ,i−v2\nφ,i. (40b)\nSubstituting these derived expressions for AkandBkback into Eqs. ( 38) and (39), the equations for\n˙χ(t) and˙φ(t) are\n˙χ(t) = sgn(˙χ(t))ai\na(t)/radicaltp/radicalvertex/radicalvertex/radicalbtv2\nχ,i+v2\nφ,i−S2\nk(χi)v2\nφ,i\nS2\nk(χ(t))\na2(t)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i), (41a)\n˙φ(t) =ai\na(t)Sk(χi)vφ,i\nS2\nk(χ(t))/radicalig\na2(t)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i), (41b)\nand their corresponding peculiar components\nvχ(t) = sgn(vχ(t))ai\na(t)/radicaltp/radicalvertex/radicalvertex/radicalbtv2\nχ,i+v2\nφ,i−S2\nk(χi)v2\nφ,i\nS2\nk(χ(t))\na2(t)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i), (42a)\nvφ(t) =aiSk(χi)vφ,i\nSk(χ(t))/radicalig\na2(t)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i), (42b)\nwhere 0 ≤v2\nχ,i+v2\nφ,i≤1. These expressions provide the comoving velocity components (˙ χ,˙φ), and\ntheir corresponding peculiar velocities ( vχ,vφ) at any time t, in terms of the initial peculiar velocities\n(vχ,i,vφ,i) at an initial time ti. Additionally, using polar coordinates ( vpec,i,ψi) instead of ( vχ,i,vφ,i)\ncan parameterize the comoving and peculiar velocities more effective ly. This polar coordinate system is\nusseful for determining the comoving and peculiar velocity compone nts as\n˙χ(t) = sgn(˙χ(t))aivpec,i\na(t)/radicaltp/radicalvertex/radicalvertex/radicalbt1−S2\nk(χi)\nS2\nk(χ(t))sin2ψi\na2(t)(1−v2\npec,i)+a2\niv2\npec,i, (43a)\n˙φ(t) =aivpec,i\na(t)Sk(χi)\nS2\nk(χ(t))sinψi\n/radicalig\na2(t)(1−v2\npec,i)+a2\niv2\npec,i, (43b)\nand\nvχ(t) = sgn(vχ(t))aivpec,i/radicaltp/radicalvertex/radicalvertex/radicalbt1−S2\nk(χi)\nS2\nk(χ(t))sin2ψi\na2(t)(1−v2\npec,i)+a2\niv2\npec,i, (44a)\nvφ(t) =Sk(χi)\nSk(χ(t))aivpec,isinψi/radicalig\na2(t)(1−v2\npec,i)+a2\niv2\npec,i. (44b)\nNote that at t=ti, inward radial motion corresponds to ψi= 0, whereas outward radial motion relates\ntoψi=π.\n3.4 Geodesics Parametrization for General Motion\nGeodesic motion in curved spacetime can be found by integrating Eqs . (38a) and (38b). However, the\ntreatment for the radial χequation is non-trivial, this complexity is thoroughly explored in Appen dixA,\nwhere a detailed analysis and careful consideration are provided. N ow, building upon previous work in\n8this field, we establish three distinct methods to characterize a spe cific geodesic solution for a freely-\nfalling test particle under any real value of the curvature paramet erk, as described below:\n•(χi,φi,Ak,Bk,sgn(˙χi))initial conditions:\nUsing Eqs. ( 38a) and (38b), we express the comoving radial distance χ(as detailed in Appendix A) and\ntheangleφforafreetest-particleatanygiventime t,underfourspecificinitialconditions( χi,φi,Ak,Bk)\nand sgn(˙χi) as\nχ(t;χi,Ak,Bk,sgn(˙χi)) =S−1\nk/parenleftigg/radicaligg/parenleftbigg\n1−kB2\nk\nA2\nk/parenrightbigg\nS2\nk(Rk(t))+B2\nk\nA2\nk/parenrightigg\n, (45a)\nφ(t;χi,φi,Ak,Bk,sgn(˙χi)) =/integraldisplayt\ntiBk\nS2\nk(χ(t′;χi,Ak,Bk,sgn( ˙χi)))\na(t′)/radicalbig\na2(t′)+A2\nkdt′+φi. (45b)\nHere, the radial function Rkis given in terms of ( χi,Ak,Bk,sgn(˙χi)) as\nRk(t;Ak,Bk,sgn(˙χi)) =/integraldisplayt\nti|Ak|dt′\na(t′)/radicalbig\na2(t′)+A2\nkdt′+sgn(˙χi)T−1\nk\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χi)−B2\nk\nA2\nk\nC2\nk(χi)\n,(46)\nwith sgn(˙χi) =±or 0,Ak= 0⇒Bk= 0. By fixing AkandBk, the radial solution χ(t) is constrained\naccording to Eqs. ( 45a) and (46) for different curvature parameters kas follows\nχ(t)≥S−1\nk/parenleftig/vextendsingle/vextendsingle/vextendsingleBk\nAk/vextendsingle/vextendsingle/vextendsingle/parenrightig\n, k ≤0\nS−1\nk/parenleftig/vextendsingle/vextendsingle/vextendsingleBk\nAk/vextendsingle/vextendsingle/vextendsingle/parenrightig\n≤χ(t)≤π√\nk−S−1\nk/parenleftig/vextendsingle/vextendsingle/vextendsingleBk\nAk/vextendsingle/vextendsingle/vextendsingle/parenrightig\n, k>0. (47)\nInthiscontext,theinitialcomovingradialdistance χiisconstrainedas χ(t)byEq.( 47).Threeimportant\nremarks on Eqs. ( 46) and (47) are worth highlighting:\n1. For all geometrical cases, the trajectory for general motion has a lower bound defined by\nS−1\nk(|Bk/Ak|), which represents the minimum comoving distance that a freely-fa lling particle can\napproach the origin.\n2. The upper bound of these trajectories depends on the specific geometry:\n(a) Flat and Hyperbolic Geometries: In flat ( k= 0) and hyperbolic ( k<0) geometries, the general\ngeodesics are unbounded. There is no upper limit to the comoving dist anceχ(t) a free particle\ncan travel in these spaces.\n(b) Spherical Geometry: Conversely, in spherical geometry ( k >0), the upper bound of χ(t) is\nclearly defined. It is specified by π/√\nk−S−1\nk(|Bk/Ak|). This expression representsthe maximum\ndistance a free travelercan reachin spherical space. Additionally, for radial motion ( Bk= 0), the\nupper bound simplifies to π/√\nk. This indicates that the free particle can move to the antipodal\npoint of the sphere.\n3. The function Rkis termed the ‘radial function’ as it represents the radial distance of the freely-falling\nparticle along a radial geodesic, measured by the closest comoving p oint in the trajectory to the\norigin, i.e., the projected point of the origin onto the trajectory, w hich isS−1\nk(|Bk/Ak|) away from\nthe origin (for more detail see Sec. 4 in Ref. [ 21]).\n•(χi,φi,vχ,i,vφ,i)initial conditions:\nUsing Eqs. ( 41a) and (41b), the radial comoving distance χand the angle φfor a free test-particle at\nany given time t, under initial conditions ( χi,φi,vχ,i,vφ,i), are expressed as\nχ(t;χi,vχ,i,vφ,i) =S−1\nk\n/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftigg\n1−kS2\nk(χi)v2\nφ,i\nv2\nχ,i+v2\nφ,i/parenrightigg\nS2\nk(Rk(t))+S2\nk(χi)v2\nφ,i\nv2\nχ,i+v2\nφ,i\n, (48a)\nφ(t;χi,φi,vχ,i,vφ,i) =/integraldisplayt\ntiai\na(t′)Sk(χi)vφ,i\nS2\nk(χ(t′;χi,vχ,i,vφ,i))/radicalig\na2(t′)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i)dt′+φi.(48b)\n9The radial function Rkin terms of ( χi,vχ,i,vφ,i), takes the following form\nRk(t;χi,vχ,i,vφ,i) =/integraldisplayt\ntiai\na(t′)/radicaligg\nv2\nχ,i+v2\nφ,i\na2(t′)(1−v2\nχ,i−v2\nφ,i)+a2\ni(v2\nχ,i+v2\nφ,i)dt′+T−1\nk\nvχ,i|Tk(χi)|/radicalig\nv2\nχ,i+v2\nφ,i\n.\n(49)\nwhereχi≥0,−π<φi≤π, and 0≤v2\nχ,i+v2\nφ,i≤1.\n•(χi,φi,vpec,i,ψi)initial conditions:\nFor the initial conditions ( χi,φi,vpec,i,ψi), we can use Eqs. ( 43a) and (43b) to express χandφof a free\ntest-particle at any given time tas\nχ(t;χi,vpec,i,ψi) =S−1\nk/parenleftbigg/radicalig/parenleftbig\n1−kS2\nk(χi)sin2ψi/parenrightbig\nS2\nk(Rk(t))+S2\nk(χi)sin2ψi/parenrightbigg\n,(50a)\nφ(t;χi,φi,vpec,i,ψi) =/integraldisplayt\ntiai\na(t′)Sk(χi)vpec,isinψi\nS2\nk(χ(t′;χi,vpec,i,ψi))/radicalig\na2(t′)(1−v2\npec,i)+a2\niv2\npec,idt′+φi, (50b)\nwhere the radial function Rkis given in terms of ( χi,vpec,i,ψi) as\nRk(t;χi,vpec,i,ψi) =/integraldisplayt\ntiai\na(t′)vpec,i/radicalig\na2(t′)(1−v2\npec,i)+a2\niv2\npec,idt′+T−1\nk(|Tk(χi)|cosψi),(51)\nwhereχi≥0,−π<φi≤π, 0≤vpec,i≤1, and−π<ψi≤π.\nIn the first parametrization ( 45), whenAkandBkare fixed, the initial comoving radial distance\nχimust be constrained within a specific range given by Eq. ( 47). However, one can show that in\nthe second ( 48) and third ( 50) parametrizations, the initial comoving distance χiis independent of\nother initial parameters. Therefore, using these latter two form ulations is more practical for describing\nnon-radial solutions.\n3.5 The Flat Geodesic Limit\nSettingk= 0 in Eqs. ( 45), (48), and (50), reduces the expressions to the exact solutions found in\nequations ( 7), (8), and (9), respectively, corresponding to the non-radial geodesic solutio n in flat 3D\nspace. This demonstrates the consistency of the general formu lation with the well-known results in\nEuclidean geometry.\n3.6 The Null Geodesic Limit\nTo explore the null geodesic limit, we use the parametrization ( χi,φi,vpec,i,ψi) and setvpec,i= 1. This\nleads to the following solutions\nχ(t;χi,1,ψi) =S−1\nk/parenleftbigg/radicalig/parenleftbig\n1−kS2\nk(χi)sin2ψi/parenrightbig\nS2\nk(Rk(t))+S2\nk(χi)sin2ψi/parenrightbigg\n, (52a)\nφ(t;χi,φi,1,ψi) =/integraldisplayt\nti1\na(t′)Sk(χi)sinψi\nS2\nk(χ(t′;χi,1,ψi))dt′+φi. (52b)\nThe radial function Rkfor this limit is given as:\nRk(t) =/integraldisplayt\ntidt′\na(t′)+T−1\nk(|Tk(χi)|cosψi). (53)\n3.7 The Radial Geodesic Limit\nTo apply the radial geodesic limit, we use the parametrization ( χi,φi,Ak,Bk) and setBk= 0. The\ncomoving solution thus obtained is\nχ(t;χi,Ak,0,sgn(˙χi)) =|Rk(t)|, (54a)\n10φ(t;χi,φi,Ak,0,sgn(˙χi)) =φi. (54b)\nThe radial function Rkin this context is expressed as:\nRk(t) =/integraldisplayt\ntiAkdt′\na(t′)/radicalbig\na2(t′)+A2\nkdt′+χi. (55)\nwhere we set sgn( Ak) = sgn(˙χi). These Eqs.( 54a) and (54b) are identical to the solution ( 11a) and (11b)\nfor radial motion. Noting that the spacetime interval for radial mo tion in Eq. ( 12) has the same form\nacross flat, spherical, and hyperbolic geometries. It is unsurprisin g that the solution in the radial limit\ndoes not explicitly depend on the curvature parameter k. However, the dependence on geometry in this\ncase is implicitly manifested through the scale factor a(t).\n3.8 Circular Orbits\nThese are orbits with a constant radius over time, indicated by χ(t) =χi, resulting in a radial peculiar\nvelocity of vχ(t) =vχ,i= 0. It can be demonstrated, using Eqs ( 42a) and (48a), that circular orbits\nmust fulfill the following condition: S2\nk(χi) =1\nk. As expected, this condition uniquely holds for spherical\ngeometry where k >0, leading to χ(t) =χi=π\n2√\nk, describing a great circle trajectory that remains\nequidistant from the origin.\n4 Alternative Expression for the Comoving Radial Distance\nApplyingidentity ( 27a)inEq.( 45a),thecomovingradialdistancecanbesimplifiedusingthe Ckfunction\nfor cases where k/negationslash= 0. The expression for χ(t) is:\nχ(t) =C−1\nk/parenleftigg/radicaligg\n1−kB2\nk\nA2\nkCk(Rk(t))/parenrightigg\n. (56)\nHere, the radial function Rkis provided in Eq. ( 46),Ckand its inverse C−1\nkis defined in Eqs. ( 23)\nand (24). In terms of ( χi,φi,vχ,i,vφ,i), the comoving radial distance is:\nχ(t) =C−1\nk/parenleftigg/radicaligg\n1−kS2\nk(χi)v2\nφ,i\nv2\nχ,i+v2\nφ,iCk(Rk(t))/parenrightigg\n, (57)\nwhereRkis now specified by Eq. ( 49). Similarly, for initial conditions ( χi,φi,vpec,i,ψi),χ(t) is expressed\nas:\nχ(t) =C−1\nk/parenleftbigg/radicalig\n1−kS2\nk(χi)sin2ψiCk(Rk(t))/parenrightbigg\n, (58)\nwhereRkis outlined in Eq. ( 51). In this case where the exact solution is undefined for k= 0, we cannot\ndirectly set k= 0. Instead, we should consider the limit where χ,χi≪1√\nk, or take the direct limit as\nk→0. One can check that when this limit is applied in Eqs. ( 56), (57), and (58), we obtain the correct\nsolutions for the flat 3D space as outlined in Eqs. ( 7), (8), and (9), respectively.\n5 Alternative Expression for the Angle\nItis clearthat freegeodesicsin the comovingframecorrespondto straight-linegeodesicsinflat geometry,\ngreat circles in spherical geometry, and hyperbolic lines in hyperbolic geometry. As a result, φ(t) is\nexplicitly dependent only on the radial distance χ(t) and not on the scale factor a(t). Accordingly, we\ncan derive the geometric formula for the trajectory by eliminating a(t), which has been accomplished in\nAppendix B, where an alternative formula describing the real trajectory of t he freely-falling particle in\nflat, spherical, and hyperbolic geometries is presented. The traje ctory can be expressed in terms of φ(t)\nas\nφ(t) = sin(Bk)/bracketleftigg\nsgn(˙χ(t))arctan/parenleftigg/radicaligg\nA2\nkS2\nk(χ(t))−B2\nk\nB2\nkC2\nk(χ(t))/parenrightigg\n−sgn(˙χi)arctan/parenleftigg/radicaligg\nA2\nkS2\nk(χi)−B2\nk\nB2\nkC2\nk(χi)/parenrightigg/bracketrightigg\n+φi.\n(59)\n11In terms of ( χi,φi,vχ,i,vφ,i)\nφ(t) = sin(vφ,i)\nsgn(˙χ(t))arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χ(t))(v2\nχ,i+v2\nφ,i)−S2\nk(χi)v2\nφ,i\nC2\nk(χ(t))S2\nk(χi)v2\nφ,i\n−sgn(vχ,i)arctan\n/vextendsingle/vextendsingle/vextendsinglevχ,i\nvφ,i/vextendsingle/vextendsingle/vextendsingle\n|Ck(χi)|\n\n+φi.\n(60)\nIn terms of ( χi,φi,vpec,i,ψi)\nφ(t) = sin(ψi(π−ψi))\nsgn(˙χ(t))arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χ(t))−S2\nk(χi)sin2ψi\nC2\nk(χ(t))S2\nk(χi)sin2ψi\n−sgn/parenleftBigπ\n2−|ψi|/parenrightBig\narctan/parenleftbigg|cotψi|\n|Ck(χi)|/parenrightbigg\n+φi.\n(61)\nwhere we have used the sign relations:\nsgn(˙χi) = sgn(vχ,i) = sgn/parenleftigπ\n2−|ψi|/parenrightig\n, (62a)\nsgn(B) = sgn(vφ,i) = sgn(ψi(π−ψi)). (62b)\n6 Symmetry and Killing vectors\nBuilding on the analysis presented in subsection 4.I. of our recent wo rk [21], we deduce the Killing\nvectorsξAkandξBkassociated with the constants of motion AkandBk, respectively, as\nξAk=±/radicaligg\nA2\nk−B2\nk\nS2\nk(χ)∂χ+Bk\nS2\nk(χ)∂φ, (63a)\nξBk=∂φ. (63b)\nFourimportantpointsaretobehighlightedinthecontextofKillingvec torsandtheirrelationtodifferent\ngeometrical spaces:\n1. For the flat case k= 0, the constants of motion simplify to A0=A, and the associated Killing vector\nξA0=ξA, whereξAis expressed in Eq. ( 5a) for the flat case discussion.\n2. The Killing vector ξBkrelated to rotational symmetry, maintains the same form acrossa ll geometries:\nflat, spherical, and hyperbolic. This universality underscores the f undamental nature of the isotropic\nsymmetry in various curved spaces.\n3. It can be easily shown that the transformation generated by th ese Killing vectors ( 63a) and (63b)\nrender the Lagrangian ( 29) invariant.\n4. The derived Killing vectors ξAkandξBksatisfy the Killing equation, representing the isometry\ncondition for distance-preserving transformations between met ric spaces\n∇µξν+∇νξµ= 0 (64)\nHere,∇µis the covariant derivative, defined by the Christoffel symbols for t he spacetime interval.\n7 Conclusion\nIn conclusion, this analysis serves as a direct extension of our prior work presented in [ 21]. While our\nearlier study explored non-radial timelike geodesics within a spatially fl at FLRW spacetime, this work\nhas expanded it to encompass all maximally symmetric geometries: fla t, spherical, and hyperbolic. This\nbroader analysis provides a unified formulation, elegantly expresse d in terms of the curvature parameter\nk. The approach we followed is based on using the symmetry of the sys tem to construct two conserved\nquantitiesAkandBk, which makes for a straightforward and simpler determination of th e radial ˙χ\nand the angular ˙φvelocities, along with the evolution of the comoving radial distance χand the angle\nφ. This offers an advantageous alternative to traditional Euler-Lag range or even the geodesic equation\nmethods. In this framework, geodesics are fully determined by thr ee appropriately parameterized initial\nconditions ( χi,φi,Ak,Bk) as detailed in Eqs. ( 45), (χi,φi,vχ,i,vφ,i) in Eqs. ( 48), and (χi,φi,vpec,i,ψi) in\nEqs. (50). Furthermore, we have elucidated the flatness limit, the radial ge odesic limit, the null geodesic\nlimit, and the comoving geodesic limit for the established non-radial so lution. Our study of general\ngeodesic motion has produced alternative and simpler expressions f or the comoving radial solution χ(t),\nspecifically in Eqs. ( 56), (57), and (58). However, these expressions are only valid for non-flat spatial\ngeometries ( k/negationslash= 0). Moreover, we have derived alternative geometrical formulat ions for the angle φ(t),\n12presented in Eqs. ( 59), (60), and (61), which are only dependent on the comoving radial distance χ(t).\nLastly, we have identified the Killing vectors ( 63a) and (63b) corresponding to the conserved quantities\nAkandBk. Our paper provides deeper insights into the behavior of free part icles from our perspective,\nencompassing not only radial but also non-radial motions across va rious curved spacetime geometries.\nWe intend to extend this investigation to incorporate the gravitatio nal influence of galaxies and Earth’s\npeculiar velocity relative to the CMB.\nA Proving the Exact Solution for the Comoving Radial\nDistance\nThe comoving radial distance for free non-radial geodesic motion in a generally FLRW spacetime can\nbe determined by integrating Eq. ( 38a).This process involves separating the coordinate variable χfrom\nthe time coordinate t. Once separated, integration yields\n/integraldisplayχ\nχidχ′\n/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ′)=/integraldisplayt\ntisgn(˙χ(t′))dt′\na(t′)/radicalbig\na2(t′)+A2\nk. (65)\nTheleft-handsideofthisequationcanbeintegratedforeachgeom etricspaceusingthefollowingformulae\nfork>0 :/integraldisplaydχ/radicalbigg\nA2\nk−kB2\nk\nsin2(√\nkχ)=1\n|Ak|√\nkarctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtsin2(√\nkχ)−kB2\nk\nA2\nk\ncos2(√\nkχ)\n, (66a)\nfork= 0 :/integraldisplaydχ/radicalig\nA2\nk−B2\nk\nχ2=1\n|Ak|/radicaligg\nχ2−B2\nk\nA2\nk, (66b)\nfork<0 :/integraldisplaydχ/radicalbigg\nA2\nk+kB2\nk\nsinh2(√−kχ)=1\n|Ak|√\n−karctanh\n/radicaltp/radicalvertex/radicalvertex/radicalbtsinh2(√\n−kχ)+kB2\nk\nA2\nk\ncosh2(√\n−kχ)\n.(66c)\nThese integrals can be combined using the Sk,Ck, andT−1\nkfunctions given in Eqs. ( 21), (23), and (26),\nrespectively. We obtain\n/integraldisplaydχ/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ)=1\n|Ak|T−1\nk\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χ)−B2\nk\nA2\nk\nC2\nk(χ)\n. (67)\nBy using this integral along with the identities provided in Eqs. ( 27a) and (27b), Eq. (65) yields\nχ(t) =S−1\nk/parenleftigg/radicaligg/parenleftbigg\n1−kB2\nk\nA2\nk/parenrightbigg\nS2\nk(Rk(t))+B2\nk\nA2\nk/parenrightigg\n. (68)\nHere, the function Rkhas the following form\nRk(t) =/integraldisplayt\ntisgn(˙χ(t′))|Ak|dt′\na(t′)/radicalbig\na2(t′)+A2\nk+T−1\nk\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χi)−B2\nk\nA2\nk\nC2\nk(χi)\n. (69)\nAddressing the integral terms in Rkpresents a non-trivial challenge, mainly due to the complexities\ninvolved in dealing with sgn(˙ χ(t′)). By employing the method described in Appendix B of Ref. [ 21], we\n13effectively address this issue and demonstrate that\nRk(t) =/integraldisplayt\nti|Ak|dt′\na(t′)/radicalbig\na2(t′)+A2\nk+sgn(˙χi)T−1\nk\n/radicaltp/radicalvertex/radicalvertex/radicalbtS2\nk(χi)−B2\nk\nA2\nk\nC2\nk(χi)\n. (70)\nThis proves the expression for the comoving radial solution χ(t) as outlined in Eq. ( 45a).\nB Proving the Alternative Expression for the Angle\nNow, let us prove that the geodesic solution for the angle φ(t), as shown in Eq. ( 45b), is identical to the\nalternative geometrical formula presented in Eq. ( 59). Our aim is to show that φ(t) does not explicitly\ndepend on the scale factor a(t). By dividing Eqs. ( 38b) and (38a), we can eliminate the explicit relation\nwith the scale factor, resulting in the following expression:\n˙χ(t)\n˙φ(t)=sgn(˙χ(t))\nBkS2\nk(χ(t))/radicaligg\nA2\nk−B2\nk\nS2\nk(χ(t)). (71)\nUpon separating variables φandt, we arrive at\n/integraldisplayφ\nφidφ=Bk/integraldisplayt\ntisgn(˙χ(t′))˙χ(t′)\nS2\nk(χ(t′))/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ(t′))dt′. (72)\nApplying the method from Appendix C of Ref. [ 21] for the flat case to address the presence of sgn(˙ χ(t′))\ninside the integral, we find\nφ(t) =Bk/integraldisplayχ\nχisgn(˙χ)dχ′\nS2\nk(χ′)/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ′)+φi. (73)\nThe integral in the right-hand side of this equation can be performe d for each specific geometrical space\nusing the following formulae\nfork>0 :/integraldisplaykdχ\nsin2(√\nkχ)/radicalbigg\nA2\nk−kB2\nk\nsin2(√\nkχ)=1\n|Bk|arctan/parenleftigg/radicaligg\nA2\nksin2(√\nkχ)−kB2\nk\nkB2\nkcos2(√\nkχ)/parenrightigg\n, (74a)\nfork= 0 :/integraldisplaydχ\nχ2/radicalig\nA2\nk−B2\nk\nχ2=1\n|Bk|arctan/parenleftigg/radicaligg\nA2\nk\nB2\nkχ2−1/parenrightigg\n, (74b)\nfork<0 :/integraldisplay−kdχ\nsinh2(√\n−kχ)/radicalbigg\nA2\nk+kB2\nk\nsinh2(√−kχ)=1\n|Bk|arctan/parenleftigg/radicaligg\nA2\nksinh2(√\n−kχ)+kB2\nk\n−kB2\nkcosh2(√\n−kχ)/parenrightigg\n.(74c)\nThese integrals can be unified using the SkandCkfrom Eqs. ( 21) and (23), respectively, as follows\n/integraldisplaydχ\nS2\nk(χ)/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ)=1\n|Bk|arctan/parenleftBigg/radicalBigg\nA2\nkS2\nk(χ)−B2\nk\nB2\nkC2\nk(χ)/parenrightBigg\n. (75)\nBy using this integral, the expression for φ(t) in Eq. ( 73) can be proved to have the following form\nφ(t) = sin(Bk)/bracketleftigg\nsgn(˙χ(t))arctan/parenleftigg/radicaligg\nA2\nkS2\nk(χ(t))−B2\nk\nB2\nkC2\nk(χ(t))/parenrightigg\n−sgn(˙χi)arctan/parenleftigg/radicaligg\nA2\nkS2\nk(χi)−B2\nk\nB2\nkC2\nk(χi)/parenrightigg/bracketrightigg\n+φi.\n(76)\nThis proves the alternative expression for the angle φ(t) as expressed in Eq. ( 59).\n14C Checking that the Solutions Satisfy the Euler-Lagrange\nEquations\nWe now confirm that our solution for the comoving radial-angular velo city in Eqs. ( 38a) and (38b) for\ngenerally curved spacetime satisfy the Euler-Lagrange equations (33a) and (33b). Let’s begin with the\nfirst equation ( 33a) by computing its left-hand and right-hand sides to get the following\nLHS =d\ndt\na2˙χ/radicalBig\n1−a2˙χ2−a2S2\nk(χ)˙φ2\n= sgn(˙χ)d\ndt/bracketleftBigg/radicalBigg\nA2\nk−B2\nk\nS2\nk(χ)/bracketrightBigg\n= sgn(˙χ)B2\nk˙χCk(χ)\nS3\nk(χ)/radicalbigg\nA2\nk−B2\nk\nS2\nk(χ=1\naB2\nkCk(χ)\nS3\nk(χ)/radicalBig\na2+A2\nk,\n(77a)\nRHS =Sk(χ)Ck(χ)a2˙φ2\n/radicalBig\n1−a2˙χ2−a2S2\nk(χ)˙φ2=1\naB2\nkCk(χ)\nS3\nk(χ)/radicalBig\na2+A2\nk. (77b)\nThis confirms that the comoving velocities in Eqs. ( 38a) and (38b) indeed satisfy the Euler-Lagrange\nequation ( 33a). Now, by doing the same thing for the Euler-Lagrange equation ( 33b), it can be checked\neasily from the constant of motion definition in Eq. ( 35) that\nd\ndt\na2S2\nk(χ)˙φ/radicalig\n1−a2˙χ2−a2S2\nk(χ)˙φ2\n=d\ndtBk= 0, (78)\nwhichconfirmsthatoursolutioninEqs.( 38a)and (38b)alsosatisfiestheEuler-Lagrangeequation( 33b).\nConsequently,both Euler-Lagrangeequations( 33a)and(33b)arecheckedtofulfill oursolutionindicated\nby Eqs. ( 38a) and (38b).\nD Summary: Explicit Geodesic Solutions for General Motion\nin Spherical and Hyperbolic Spaces\nThe timelike geodesics for both radial and non-radial free motion in s patially spherical FLRW spacetime\n(k>0) are given by\nχ(t) =1√\nkarcsin\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalbt/parenleftigg\n1−kB2\nk\nA2\nk/parenrightigg\nsin2\n/integraldisplayt\nti√\nk|Ak|dt′\na(t′)/radicalig\na2(t′) +A2\nk+ sgn( ˙χi)arctan\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtsin2(√\nkχi)−kB2\nk\nA2\nk\ncos2(√\nkχi)\n\n+kB2\nk\nA2\nk\n,(79a)\nφ(t) = sin(Bk)\nsgn( ˙χ(t))arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nksin2(√\nkχ(t))−kB2\nk\nkB2\nkcos2(√\nkχ(t))\n−sgn( ˙χi)arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nksin2(√\nkχi)−kB2\nk\nkB2\nkcos2(√\nkχi)\n\n+φi.(79b)\nThe timelike geodesics for both radial and non-radial free motion in s patially hyperbolic FLRW\nspacetime ( k<0) are given by\nχ(t) =1√−karcsinh\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalbt/parenleftigg\n1 +kB2\nk\nA2\nk/parenrightigg\nsinh2\n/integraldisplayt\nti√−k|Ak|dt′\na(t′)/radicalig\na2(t′) +A2\nk+ sgn( ˙χi)arctanh\n/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtsinh2(√−kχi) +kB2\nk\nA2\nk\ncosh2(√−kχi)\n\n−kB2\nk\nA2\nk\n,\n(80a)\nφ(t) = sin(Bk)\nsgn( ˙χ(t))arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nksinh2(√−kχ(t)) +kB2\nk\n−kB2\nkcosh2(√−kχ(t))\n−sgn( ˙χi)arctan\n/radicaltp/radicalvertex/radicalvertex/radicalbtA2\nksinh2(√−kχi) +kB2\nk\n−kB2\nkcosh2(√−kχi)\n\n+φi.\n(80b)\nThese explicit relations are given by the constants of motion ( Ak,Bk). However, we can expressed them\nin terms of ( vχ,i,vφ,i) by using Eqs. ( 48), and alternatively, in terms of ( vpec,i,ψi) through Eqs. ( 50).\nReferences\n[1] Peebles, P.J.E.: Principles of Physical Cosmology vol. 27. Princeton university press, ??? (1993)\n[2] Smoot, G.F., et al.: Structure in the cobe differential microwave radiometer first-ye ar maps.\nAstrophysical Journal, vol. 396, no. 1, p. L1-L5 396, 1–5 (1992)\n[3] Hinshaw, G., et al.: Nine-year wilkinson microwave anisotropy probe (wmap) observat ions:\ncosmological parameter results. The Astrophysical Journal Sup plement Series 208(2), 19 (2013)\n15[4] Aghanim, N., et al.: Planck 2018 results-vi. cosmological parameters. Astronomy & A strophysics\n641, 6 (2020)\n[5] Perlmutter,S., et al.: Measurementsof ωandλfrom42high-redshift supernovae.The Astrophysical\nJournal517(2), 565 (1999)\n[6] Riess, A.G., et al.: Observational evidence from supernovae for an accelerating un iverse and a\ncosmological constant. The astronomical journal 116(3), 1009 (1998)\n[7] Friedmann, A.: ¨Uber die kr¨ ummung des raumes. Z. Phys. 10, 377–386 (1922)\n[8] Lemaitre, G.: A homogeneous universe of constant mass and incr easing radius accounting for the\nradial velocity of extra-galactic nebulae. In: A Source Book in Astr onomy and Astrophysics, 1900–\n1975, pp. 844–848. Harvard University Press, ??? (1979)\n[9] Robertson, H.P.: Kinematics and world-structure. Astrophysic al Journal, vol. 82, p. 284 82, 284\n(1935)\n[10] Walker, A.G.: On milne’s theory of world-structure. Proceedings of the London Mathematical\nSociety2(1), 90–127 (1937)\n[11] Einstein, A.: Die feldgleichungen der gravitation. Sitzungsberich te der K¨ oniglich Preußischen\nAkademie der Wissenschaften, 844–847 (1915)\n[12] Baumann, D.: Cosmology, part iii mathematical tripos. Universit y lecture notes 56, 34 (2014)\n[13] Whiting, A.B.: The expansion of space: Free particle motion and th e cosmological redshift. arXiv\npreprint astro-ph/0404095 (2004)\n[14] Davis, T.M., Lineweaver, C.H., Webb, J.K.: Solutions to the tethere d galaxy problem in an expand-\ninguniverse andthe observationof recedingblueshifted objects. American JournalofPhysics 71(4),\n358–364 (2003)\n[15] Grøn, Ø., Elgarøy, Ø.: Is space expanding in the friedmann univer se models? American Journal of\nPhysics75(2), 151–157 (2007)\n[16] Barnes, L.A., Francis, M.J., James, J.B., Lewis, G.F.: Joining the hub ble flow: Implications for\nexpanding space. Monthly Notices of the Royal Astronomical Socie ty373(1), 382–390 (2006)\n[17] Kerachian,M.:Uniformlyacceleratedtravelerinanflrwuniverse .PhysicalReviewD 101(8),083536\n(2020)\n[18] Vachon, G., Vanderwee, R., Faraoni, V.: Revisiting geodesic obse rvers in cosmology. The European\nPhysical Journal C 81, 1–9 (2021)\n[19] Cot˘ aescu,I.I.:Kinemaitcsinspatiallyflatflrwspacetimes.Chine sePhysicsC 45(10),105101(2021)\n[20] Nemoul, O., Guergouri, H., Mimouni, J.: Studying the behavior of ra dial free geodesics\ninλcdm model. International Journal of Geometric Methods in Modern Physics (2024).\nDOI:10.1142/S0219887824501342\n[21] Nemoul, O., Guergouri, H., Mimouni, J.: Non-Radial Free Geodesics . I. In Spatially Flat FLRW\nSpacetime. will be published\n[22] Killing,W.:Ueberdieclifford-klein’schenraumformen.Mathematisc heAnnalen 39,257–278(1891)\n[23] Hopf, H.: Zum clifford-kleinschen raumproblem. Mathematische A nnalen95(1), 313–339 (1926)\n[24] Euler,L.:Methodus InveniendiLineasCurvasMaximiMinimivePro prietateGaudentesSiveSolutio\nProblematis Isoperimetrici Latissimo Sensu Accepti vol. 1. Springe r, ??? (1952)\n[25] Lagrange, J.L.: Essai D’une Nouvelle Ethode Pour De’terminer les Maxima, et les Minima des\n16Formules Integrales Indefinies. De l’Imprimerie royale, ??? (1761)\n[26] Noether, E., Tavel, M.: Invariant variation problems. arXiv prep rint physics/0503066 (2005)\n17"    },    {        "title": "2402.07219v1.The_dependence_of_local_regularity_of_solutions_on_the_summability_of_coefficients_and_nonhomogenous_term.pdf",        "content": "arXiv:2402.07219v1  [math.AP]  11 Feb 2024The dependence of local regularity of solutions\non the summability of coefficients and\nnonhomogenous term∗\nZheng Li, Bin Guo†\nSchool of Mathematics, Jilin University, Changchun 130012, PR Chin a\nAbstract: In this paper, we mainly discuss the local regularity of the s olution to the\nfollowing problem\n/braceleftigg\n−div(A(x)∇u(x)) =f(x), x∈Ω,\nu(x) = 0, x ∈∂Ω,\nwhere Ω is a bounded domain in Rn. In particular, we are concerned with the connection\nbetween the regularity of the solution uand the integrability of the coefficient matrix A(x) as\nwell as the nonhomogeneous term f. To be more precise, our first result is to prove that the\nmaximum norm of ucan be controlled by /ba∇dblf/ba∇dblswithf∈Ls(Ω), s >nq\n2q−n, q >n\n2. Meanwhile,\nwe construct some counterexamples to illustrate the indexnq\n2q−nbeing sharp. Subsequently, we\ngive an improved upper bound for the maximum norm of u. Namely, there exists a positive\nconstant Csuch that\n/ba∇dblu/ba∇dbl∞≤C/ba∇dblf/ba∇dblnq\n2q−n/bracketleftigg\nlog/parenleftigg\n/ba∇dblf/ba∇dbls\n/ba∇dblf/ba∇dblnq\n2q−n+1/parenrightigg\n+1/bracketrightigg\n.\nSpecially, themaindifferenceofourapproachcomparedtothe argumentsof[8, 33]istoconstruct\ntwo classes of truncation functions to remove the assumptio n of the boundedness of u. Finally,\nbased on the previous results and Moser iteration argument, we derive the Harnack inequality\nofufrom which the H¨ older continuity of the solution follows. I n addition, we also find that the\nLebesgue space Ln\n2(Ω) to which the inverse of the smallest eigenvalue λ(x) of the matrix A(x)\nbelongs is essentially sharp in order to establish local bou ndedness and the H¨ older continuity of\nthe solution.\nKeywords: Nonuniformly elliptic equations; Bounded solutions; Harn ack inequality\n1 Introduction\nIn this article, we consider the following second order elli ptic equation in divergence form\n−div(A(x)∇u) =f, x∈Ω, (1.1)\nwhereΩisaboundeddomainin Rn,n≥2. Inordertomeasureellipticity of A(x), thecoefficient\nmatrixA(x) satisfies the following assumptions:\n∗The research is supported by NSFC (11301211) and NSF of Jilin Prov. (201500520056JH).\n†Corresponding author\nEmail address: bguo@jlu.edu.cn (Bin Guo)\n12\n(H1)A(x) : Ω→Rn×nis a measurable matrix whose entries aij(x) satisfy\naij(x) =aji(x),1≤i,j≤n;\n(H2) There exist λ(x) andµ(x) such that\nλ(x)|ξ|2≤A(x)·ξ·ξ≤µ(x)|ξ|2,∀ξ∈Rn,\nwhereλ(x) andµ(x) are nonnegative measurable functions and satisfy\nλ−1(x)∈Lq(Ω), µ(x)∈Lp(Ω),\nwithp, q >1 and1\np+1\nq<2\nn−1.\nWhen theequation (1.1) is uniformlyandstrictly elliptic, i.e.,λ−1(x) andµ(x) areessentially\nbounded, it is well known that weak solutions are H¨ older con tinuous. Roughly speaking, for\nthe planar case ( n= 2), the corresponding study dates back to the work of Morrey [21]. For\nmore works on the best H¨ older continuity exponent, the inte rested readers may refer to the\nwork of Wildman [32] and Piccinini and Spagnolo [26]. For the higher dimensions ( n≥3),\nH¨ older continuity of solutions was established in the late 1950’s by De Giorgi [10] and Nash\n[25]. H¨ older continuity also follows from the Harnack ineq uality, as demonstrated by Moser [22,\n23]. In particular, Trudinger [31] obtained that for any f∈Ls(Ω) with s >n\n2, there exists a\nconstant C >0 such that\n/ba∇dblu/ba∇dbl∞/lessorequalslantC/ba∇dblf/ba∇dbls, (1.2)\nfor any nonnegative weak solution u∈H1\n0(Ω) of (1.1). This proof relies on Moser iteration\nand the Sobolev inequality. Different from the method used in [ 31], Talenti [28] applied the\nrearrangement methodtoobtain thesameresultsandgave the explicit expressionoftheconstant\n“C”. After these pioneer works, there are many research activi ties regarding global regularity\ntheory for linear or nonlinear elliptic equations [15, 18, 1 9, 20, 27, 31] and references therein.\nOn the contrary, if λ−1(x) is unbounded, then the equation (1.1) is degenerate; If µ(x)\nis unbounded, then the equation (1.1) is singular. Speciall y, Murthy and Stampacchia [24],\nTrudinger [30] proved that weak solutions to (1.1) are local ly bounded and satisfy the Harnack\ninequality under the assumptions that λ−1(x)∈Lq(Ω) and µ(x)∈Lp(Ω) with1\np+1\nq<2\nn. It’s\nworth noting that their key point is to obtain the following S obolev inequality\n(ˆ\nΩ|u|γµ(x)dx)1\nγ/lessorequalslantC/parenleftbigˆ\nΩ|∇u|2λ(x)dx/parenrightbig1\n2, (1.3)\nforanyu∈H1\n0(Ω)andγ >2. Furthermore, BellaandSch¨ affner[2]alsoobtainedthesam eresults\nunder the optimal condition1\np+1\nq<2\nn−1,n/greaterorequalslant2 (refer to [1, 9, 17] for a recent generalization\nto scalar autonomous integral functionals with ( p,q)-growth). Compared to the arguments\ngiven in [24, 30], the key observation of [2] is to give the opt imal Caccioppoli inequality with\nrespect to cut-off function by using the Sobolev inequality i n then−1-dimensional spheres\nrather than the n-dimensional balls. In addition, Fabes, Kenig and Serapion i [12] gave some\nstronger conditions on the minimum and maximum eigenvalues ofA(x), i.e., Muckenhoupt class\nA2. These conditions will guarantee the validity of the follow ing scale invariant Sobolev and\nPoincar´ e inequalities like\n/parenleftbigˆ\nB|u−uB|γλdx/parenrightbig2\nγ/lessorequalslantCr2/parenleftbigˆ\nBλ−1dx/parenrightbig2\nγ−1ˆ\nB|∇u|2λdx, (1.4)3\nfor anyu∈H1(B) withB=B(y,r) andγ >2,uB:=ffl\nBudx=1\n|B|´\nBudx. For more related\nresults, we may refer to [5, 8, 13].\nRecently, for the Dirichlet problem of (1.1), Xu [33] improv ed (1.2) in the uniformly elliptic\ncase. Using Moser iteration technique, he obtained an upper bound in the logarithmic form of\n/ba∇dblf/ba∇dbls. Subsequently, Cruz-Uribe and Rodney [8] extended Xu’s res ult to the degenerate elliptic\nequations with f∈LA(Ω), where A(t) =tσ′log(e+t)q,(q > σ′>0).\nMotivated by mentioned works, we plan to investigate the rel ationship between the local reg-\nularity ofsolutions andtheintegrability oftheminimumei genvalue andthemaximumeigenvalue\nof the coefficient matrix A(x) as well as f. To the best of our knowledge, it is widely recognized\nthat the smoothness of the solutions depends on the smoothne ss of the coefficients matrix and\nthe data. For instance, for the absence of the term f, Zhong [34] constructed discontinuous\nsolutions for degenerate equations to give partial answer t o De Giorgi’s conjectures, which were\nraised by De Giorgi [11] in a talk in Lecce, 1995. In this direc tion, as far as we know, the classical\nresults are established by Trudinger [30] under the assumpt ion that1\np+1\nq<2\nn. Later, Bella and\nSch¨ affner [2] discussed the same problem under the optimal co ndition1\np+1\nq<2\nn−1. However,\nfor the present of the term f, such problem is seldom discussed. Naturally, some problem s arise:\n•Whether does the inequality similar as (1.2) hold for nonuni form elliptic problem?\n•What is a critical space of the nonhomogeneous term fin the Lebesgue class Ls(Ω)?\n•Whether or not is the Lebesgue space Ln\n2(Ω) to which λ−1(x) belongs is sharp to ensure\nthe H¨ older continuity of the solution?\nIn this paper, we gave full answers to three problems mention ed above. Here, we are inter-\nested in situations beyond any smoothness where aij(x) are assumed to be merely measurable\nfunctions satisfying conditions (H1), (H2), and fis any function from some Lebesgue space Ls.\nOur first finding is as follows: when q >n\n2,n≥2, we prove that the maximum of the weak\nsolution uof (1.1) can be controlled by /ba∇dblf/ba∇dbls, wheres >nq\n2q−n. Meanwhile, we can provide a\ncounterexample to illustrate thatnq\n2q−nis a critical exponent. More precisely, when q=n\n2, n≥2,\nwe observe that for any s∈[1,+∞), there exists f∈Ls(Ω) such that the weak solution uof\n(1.1) is unbounded. In this case, for n≥3, even when f∈L∞(Ω), there exists the solution\nsatisfying sup\nΩu= +∞. In addition, for the case of q∈/parenleftbign−1\n2,n\n2/parenrightbig\n, n≥3, we also find there\nexistsf∈L∞(Ω) such that sup\nΩu= +∞. On the contrary, when1\np+1\nq≥2\nn−1, n≥3, a\ncounterexample is provided in [3, 14], demonstrating that w henf= 0, the weak solution uof\n(1.1) satisfies sup\nΩu= +∞. Finally, in the scenario where n= 2,1\nq+1\np=2\nn−1, i.e.,p=q= 1,\nBella and Sch¨ affner [2] elucidated, using the maximum princi ple and Sobolev inequality in one\ndimension, that the weak solution uis locally bounded when f= 0. However, we demonstrate\nthat when f/\\e}atio\\slash≡0, for any s∈[1,+∞), there is a counterexample which shows that (1.2) is\nimpossible.\nNext we will state our main results. The first result provides an estimate of the local\nmaximum of |u|.\nTheorem 1.1. Assume that (H1)-(H2) hold, and f∈Ls(Ω)for anys >nq\n2q−n(q >n\n2). Then\nfor anyγ >0, there exists C=C(n,p,q,γ)∈[1,+∞)such that for any ball BR⋐Ω, R >0,\nthe solution uof(1.1)satisfies\nsup\nBθR|u| ≤C(Λ(BR))δp′\nγ(δ−1)\n×/parenleftigg\n1\n((1−θ)R)n\nγ(m∗+1)/ba∇dblu/ba∇dblLγ(BR)+1\n((1−θ)R)n\nγm∗+n\ns−2/ba∇dblf/ba∇dblLs(BR)/parenrightigg\n,(1.5)4\nwhere\nθ∈(0,1), δ= 2−1\nχ, χ:=2q\n(q+1)p∗,1\np∗= min/braceleftbigg1\n2+1\nn−1−1\n2p,1/bracerightbigg\n,\nm∗=p′δ\nδ−1max/braceleftbigg1\np+1\nq,2\nq/bracerightbigg\n, p′=p\np−1,\nΛ(BR) :=/parenleftbigg \nBRλ−qdx/parenrightbigg1\nq/parenleftbigg \nBRµpdx/parenrightbigg1\np\n+/parenleftbigg \nBRλ−qdx/parenrightbigg2\nq\n.\nRemark 1.1. We note thatnq\n2q−n→n\n2asq→+∞, which is consistent with the conclusion\nof the uniformly elliptic equations.\nRemark 1.2. As a byproduct, the corresponding global results would be pr oceeding as in\nthe proof of [31, Theorem 4.1].\nRemark 1.3. From the conclusion of Theorem 1.1, we found that the integra bility of f\ndepends not only on dimension n, but also on the integrability of λ−1. So, a natural question\narises: “What is a critical exponent of the nonhomogeneous term in the class Ls(Ω)?”\nBefore giving some answers to this problem, we first consider the special case A(x) =I.\nUnder such case, (1.1) reduces the Poisson equation. We first extendfto˜f, where\n˜f=/braceleftigg\nf, x∈B1={x= (x1,x2,···,xn)∈Rn;|x|=/radicalbig\nx2\n1+x2\n2+···+x2n<1},\n0, x∈Rn\\B1.\nThen, formally, the solution may be written as the following\nu(x) =C(n)ˆ\nRn˜f(y)|x−y|2−ndy, x∈B1.\nA simple computation shows the singular integral\nˆ\nB1/parenleftbigg1\n|x−y|/parenrightbiggs(n−2)\ns−1\ndy, x∈B1\nconverges if and only if s >n\n2. Further, the H¨ older’s inequality implies\nu(x)≤C(n)/ba∇dblf/ba∇dblLs(B1)\nˆ\nB1/parenleftbigg1\n|x−y|/parenrightbiggs(n−2)\ns−1\ndy\n1−1\ns\n.\nThe formal derivation sheds light on the fact that s >n\n2is the sufficient condition for the\nboundednessof |u|. Of course, we also obtain similar analytic conclusion from scaling viewpoint.\nIndeed, we set uλ(x) =u(λx) andfλ(x) =λ2f(λx) for any 0 < λ≪1. It is easy to verify that\n∆uλ=fλ(x) and/ba∇dblfλ(x)/ba∇dblLs(B1)=λ2−n\ns/ba∇dblf/ba∇dblLs(Bλ). It is also easy to verify that the necessary\ncondition for (1.2) is s >n\n2by rescaling technique. Therefore, it is well known that s >n\n2is the\nsufficient and necessary condition for (1.2). The argument ab ove is similar in spirit, though not\nin detail, to the examples in [6, 29]. This leads to a natural q uestion: is the exponentnq\n2q−nin\nTheorem 1.1 optimal?\nNext, we will demonstrate that the exponentnq\n2q−nin Theorem 1.1 is optimal. What’s more,\nwe will provide some counterexamples to illustrate the corr esponding results.5\nExample 1.1. Let Ω = B1\n4(0), with q >n\n2andp, qsatisfy1\nq+1\np<2\nn−1. HereA(x) is a\npositive semidefinite diagonal matrix, and λ(x) =µ(x) =|x|β/parenleftig\nlog1\n|x|/parenrightigθ\n, whereβ=n\nq<2 and\nθ=1\nq+1\n2−1\nn. Then there exists f∈Lnq\n2q−n(Ω) with n≥3, orf∈Ls(Ω), s∈[1,q\nq−1) with\nn= 2 such that the solution uof (1.1) is unbounded. To be precise, the functions f(x) and\nu(x) may be expressed as follows\nf(x) = (n+1)|x|β−1/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−θ|x|β−1/parenleftbigg\nlog1\n|x|/parenrightbiggθ−1ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−2|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy,\nand\nu(x) =ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−11\nlog1\n|y|dy.\nRemark 1.4. Whenn= 2 and q >1, we can only say that s >q\nq−1is almost sharp. We\nconjecture that the case of n= 2 and f∈Lq\nq−1(Ω) is different. However, at present, we can\nmake no further assertions regarding this.\nRemark 1.5. Example 1.1 indicates that if the assumption on λ−1remains unchanged and\nthe integrability of fbecomes weaker, then it will result in u /∈L∞(Ω).\nIn Theorem 1.1, we havenq\n2q−n→+∞asq→n\n2. The question arises whether /ba∇dblu/ba∇dbl∞can be\ncontrolled by /ba∇dblf/ba∇dbl∞whenq=n\n2,and1\np+1\nq<2\nn−1forn≥2. By the way, in this scenario, with\nq∗=2nq\n(q+1)n−2q= 2, the interpolation inequality in Theorem 1.1 becomes inv alid. However, we\nwill provide the following counterexample to demonstrate w henn≥3,q=n\n2and1\np+1\nq<2\nn−1,\nthere exists f∈L∞(Ω) such that uis unbounded, and when n= 2, for any s∈[1,+∞), we can\nfindf∈Ls(Ω) such that the maximum norm of ucannot be dominated by /ba∇dblf/ba∇dbls.\nExample 1.2. Let Ω = B1\n4(0), with q=n\n2andp, qsatisfy1\np+1\nq<2\nn−1. Here,A(x) is a\npositive semidefinite diagonal matrix, and λ(x) =µ(x) =|x|2/parenleftig\nlog1\n|x|/parenrightigθ\n, whereθ=5\n2n. Then\nforn≥3, there exists f∈L∞(Ω), and for n= 2, there exists f∈Ls(Ω) (1≤s <+∞) such\nthat the weak solution u∈H1(Ω,A) of (1.1) is unbounded.\nUnder our restriction1\np+1\nq<2\nn−1, we observe that q >n−1\n2. In the preceding discussion,\nwe focus on the case where q≥n\n2. Now we will demonstrate that for q∈/parenleftbign−1\n2,n\n2/parenrightbig\n,n≥3, there\nexistsf∈L∞(Ω) such that the weak solution uof (1.1) is unbounded.\nExample 1.3. Forn≥3, q∈/parenleftbign−1\n2,n\n2/parenrightbig\nandp, qsatisfy1\np+1\nq<2\nn−1. Considerthecasewhere\nΩ =B1\n4(0),A(x) is a positive semidefinite diagonal matrix, and λ(x) =µ(x) =|x|β/parenleftig\nlog1\n|x|/parenrightigθ\n,\nwithβ=n\nq>2,andθ >1\nq. Then there exists f∈L∞(Ω) such that sup\nΩu= +∞.6\nRemark 1.6. Example 1.2 and 1.3 demonstrate that for /ba∇dblu/ba∇dbl∞<+∞,Ln\n2(Ω) is the critical\nLebesgue space for λ−1in divergence equations.\nIn addition, for the scenario where1\np+1\nq≥2\nn−1andn≥3, please refer to [2, 3, 14] for related\nexamples. In particular, in [14, Theorem 2], an example is pr esented where the weak solution u\nof (1.1) is unbounded under the conditions1\np+1\nq>2\nn−1andf= 0 (A similar description was\nalso provided in [2, Remark 3.5]). Subsequently, Bella and S ch¨ affner [3] extended the example\nin [14] to the P-Laplacian equation when1\np+1\nq≥m\nn−1andn≥3. It is noteworthy that they\ndemonstrated in [2] that when n= 2, p=q= 1, f= 0, there exists /ba∇dblu/ba∇dbl∞<+∞. Expanding\non these findings, we establish that when f/\\e}atio\\slash≡0,q=2\nn−1(n≥3) andp= +∞, there exists\nf∈L∞(Ω) such that the solution uof (1.1) is unbounded. See Remark 6.1 below for further\ndetails.\nRemark 1.7. Whenn= 2,q= 1,p≥1,f/\\e}atio\\slash≡0, we are unable to establish any relationship\nbetween /ba∇dblu/ba∇dbl∞and/ba∇dblf/ba∇dbl∞. This is because in this case, the maximum principle and the M oser\niteration techniquearenotapplicable. Itmay benecessary toexploreanewapproachor consider\na counterexample in order to address this issue.\nOur second result is to give an improved upper bound, which ma y measure how much bigger\nthe associated space Ls(Ω) is than the optimal space Lnq\n2q−n(Ω).\nTheorem 1.2. Under the assumptions in Theorem 1.1, the solution u∈H1\n0(Ω,A)of problem\n(1.1)satisfies\n/ba∇dblu/ba∇dbl∞≤C/ba∇dblf/ba∇dbls0/bracketleftbigg\nlog/parenleftbigg/ba∇dblf/ba∇dbls\n/ba∇dblf/ba∇dbls0+1/parenrightbigg\n+1/bracketrightbigg\n, (1.6)\nwheres0:=nq\n2q−n,C=C/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq,Ω,n,q/parenrightig\n∈[1,+∞).\nRemark 1.8. Compared with [8, 33], our proof also works without any assum ption of the\nboundedness of u. Our main strategy is to give more precise Caccioppoli inequ ality for eαu+by\napplying truncation approaching techniques.\nThirdly, with the aid of Theorem 1.1, we establish the Harnac k inequality.\nTheorem 1.3. Assuming that (H1)-(H2) hold, q >n\n2andf∈Ls(Ω)for some s >nq\n2q−n. Let\nu∈H1(Ω,A)be a nonnegative solution in Ω. Then for any BR⊂Ω, the following inequality\nholds\nsup\nBR\n2u≤C/parenleftigg\ninf\nBR\n2u+R2−n\ns/ba∇dblf/ba∇dblLs(BR)/parenrightigg\n, (1.7)\nwhereCdepends on n, p, qandΛ(BR).\nFinally, we present some consequences of Theorem 1.3, which are now considered standard\nand therefore we only provide the statements without the pro of. In a strictly elliptic setting\nwhereΛ( BR)isuniformlybounded,theHarnackinequality impliesglob alH¨ oldercontinuity. Due\nto the dependence of the constant in (1.7) on Λ( BR), the global H¨ older continuity is generally\nnot true in the nonuniform case (see [2, 30]). However, Theor em 1.3 leads to the following local\nresult.7\nCorollary 1.1. Suppose that (H1)-(H2) hold, q >n\n2andf∈Ls(Ω)for some s >nq\n2q−n. Let\nu∈H1(Ω,A)be a weak solution of (1.1). Then for any r∈(0,R], we have\nsup\nBr\n2u−inf\nBr\n2u≤C/parenleftigr\nR/parenrightigα/braceleftigg/parenleftbigg1\nRnˆ\nBRuγdx/parenrightbigg1\nγ\n+R2−n\ns/ba∇dblf/ba∇dblLs(BR)/bracerightigg\n,\nwhereC=C(Λ(BR), n, p, q ),α=α(Λ(BR), n, p, q s )>0.\nRemark 1.9. Because Λ( BR) is not uniformly bounded for all ball BR⊂Ω, we only obtain\nthis local oscillation result. Please refer to [30] for corr esponding example.\nA direct consequence of Corollary 1.1 is the following local H¨ older continuity of u.\nCorollary 1.2. Letu∈H1(Ω,A)be a weak solution of (1.1). Suppose f∈Ls(Ω)for some\ns >nq\n2q−n. Thenu∈Cα\nloc(Ω)forα∈(0,1)depending on Λ(BR), n, p, q, s . Moreover, for any\nBR⊂Ω, the following inequality holds\n|u(x)−u(y)| ≤C/parenleftbigg|x−y|\nR/parenrightbiggα/braceleftigg/parenleftbigg1\nRnˆ\nBRuγdx/parenrightbigg1\nγ\n+R2−n\ns/ba∇dblf/ba∇dblLs(BR)/bracerightigg\n,\nfor anyx, y∈BR\n8, whereC=C(Λ(BR),n,p,q)≥1.\nOur main results are established through the Moser iteratio n. First, Theorem 1.1 demon-\nstrates the weak solutions to nonuniformly elliptic equati ons are bounded with optimal index\ntof. And we explain the relation between /ba∇dblu/ba∇dbl∞and the integrability of λ−1,µas well as fby\nExample 1.1-1.3. Next, in Theorem 1.2, we obtain critical up per bound /ba∇dblf/ba∇dbls0log/parenleftig\n/bardblf/bardbls\n/bardblf/bardbls0+1/parenrightig\nfor the weak solution uof the Dirichlet problem, which is smaller than the traditio nal upper\nbound/ba∇dblf/ba∇dbls. And then the ratio/bardblf/bardbls\n/bardblf/bardbls0measures how much bigger the space Ls(Ω) is than the\ncritical space norm Ls0(Ω).Finally, the Harnack inequality is established in Theorem 1 .3.\nThe rest of this paper is organized as follows. In Section 2, w e provide some preliminary\nknowledge, including some definitions and useful lemmas. In Section 3, we prove the Theorem\n1.1, by utilizing a variation of the Moser iteration when1\np+1\nq<2\nn−1, and carefully modifyingthe\nargument from [2]. Section 4 is dedication to proving Theore m 1.2. Our proof is a generalization\nof the argument in [33], and it requires us to address several technical obstacles. Section 5\ncontains the proof of Theorem 1.3. Finally, in Section 6, we e stablish Example 1.1. In a sense,\nExample 1.2, 1.3 can be considered as variants of Example 1.1 .\n2 Preliminaries\nIn this section, we introduce some notations and lemmas that will be used throughout the\npaper. In what follows, we denote by /ba∇dbl·/ba∇dbls(s≥1) the usual norm in Ls(Ω). And Cdenotes a\ngeneric positive constant, which may differ at each appearanc e, but whose value depends only\non the underlying parameters. If we want to specify this depe ndence, we will write, for instance,\nC(n,s), etc.\nThe spaces H1\n0(Ω,A) andH1(Ω,A) are, respectively, defined as the completion of C∞\n0(Ω)\nandC∞(Ω) with respect to the norm\n/ba∇dbl·/ba∇dblH1(Ω,A):= (B(·,·))1\n2,8\nwhere\nB(u,v) :=ˆ\nΩA·∇u·∇vdx+ˆ\nΩµuvdx,\nFor the related properties of the spaces H1(Ω,A) andH1\n0(Ω,A), please refer to [30, 31].\nHere, we will only recall the following chain rule.\nLemma 2.1.[31]LetF:R→Rbe uniformly Lipschitz-continuous with F(0) = 0. Then\nu∈H1\n0(Ω,A) (or∈H1(Ω,A)) implies F(u)∈H1\n0(Ω,A) (or∈H1(Ω,A)), and it holds that\n∇xF=F′(u)∇xua.e. Ω.\nNext, we introduce the definition of weak solution (subsolut ion, supersolution) of (1.1).\nDefinition 2.1. The function uis said to be a weak solution (subsolution, supersolution) o f\n(1.1)inΩif and only if u∈H1(Ω,A)and\nB(u,φ) =ˆ\nΩfφdx(≤0,≥0),∀φ≥0∈H1\n0(Ω,A). (2.1)\nMoreover, uis said to be a local weak solution of (1.1)inΩif and only if uis a weak solution\nof(1.1)inΩ′for any bounded open set Ω′⋐Ω.\nRemark 2.1. Sincef∈Ls(Ω), s >nq\n2q−n, the right-hand side of (2.1) is valid according to\nH¨ older’s inequality and Sobolev inequality.\nAt the end of this section, we introduce another important le mma, which was first proposed\nin [2], and provided improved Caccioppoli inequality by sel ecting the minimum points of the\nfunctional Jand applying the Sobolev inequality on spheres. For a detail ed proof, please refer\nto [2, Lemma 2.1].\nLemma 2.2.[2]Fixn≥2, andp≥1satisfyp >n−1\n2ifn≥3. For0< ρ < σ < +∞, let\nv∈W1,p∗(Bσ)with1\np∗= min{1\n2−1\n2p+1\nn−1,1}andµ∈Lp(Bσ),µ≥0satisfying µv2∈L1(Bσ).\nConsider\nJ(ρ,σ,v) := inf/braceleftbiggˆ\nBσµv2|∇η|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingleη∈C1\n0(Bσ), η≥0, η≡1inBρ/bracerightbigg\n.(2.2)\nThen there exists c= (n,p)∈[1,+∞)such that\nJ(ρ,σ,v)≤c(σ−ρ)−2n\nn−1/ba∇dblµ/ba∇dblLp(Bσ\\Bρ)/parenleftig\n/ba∇dbl∇v/ba∇dbl2\nLp∗(Bσ\\Bρ)+ρ−2/ba∇dblv/ba∇dbl2\nLp∗(Bσ\\Bρ)/parenrightig\n.(2.3)\n3 Proof of Theorem 1.1\nTo better understand the outline of our proof, we will divide this proof into three steps.\nStep 1. Establish the estimate of subsolution for θ=1\n4,R= 1andγ= 2p′, where\np′=p\np−1.i.e., ifuis subsolution of (1.1) for γ= 2p′, then we claim that\n/ba∇dblu+/ba∇dblL∞(B1\n4)≤C(Λ(B1))δ\n2(δ−1)/bracketleftigg/parenleftbiggˆ\nB1|u+|2p′dx/parenrightbigg1\n2p′\n+/ba∇dblf/ba∇dblLs(B1)/bracketrightigg\n. (3.1)\nTodothis,wefirstintroducethedefinitionofsometestfunct ion. Forβ≥1andm∈(0,+∞),\nwe define\nφm=/braceleftigg\n(u++k)β−kβ, u ≤m,\nβ(m+k)β−1(u−m)+(m+k)β−kβ, u > m,9\nwherek:=/ba∇dblf/ba∇dblLs(B1). Setφ=φmη2, whereη(x)≥0, η∈C∞\n0(B1), withη≡1 forx∈Bρ;\nη≡0 forx∈B1\\Bσ, where1\n4≤ρ < σ≤1\n2. Using (2.1), we obtain\nˆ\nΩA(x)∇u∇/parenleftbig\nφmη2/parenrightbig\ndx≤ˆ\nΩfη2φmdx. (3.2)\nThen, by means of (3.2), the conditions (H1)-(H2), Young’s i nequality, and convexity of φmin\nthe form of φm≤(u++k)φ′\nm, we get\nˆ\nΩ(A(x)∇u+∇u+)φ′\nmη2dx≤1\n2ˆ\nΩA(x)∇u+∇u+φ′\nmη2dx\n+8ˆ\nΩµ|∇η|2(u++k)2φ′\nmdx+ˆ\nΩfη2φmdx.(3.3)\nOnce again, using the elliptic condition A(x)∇u+∇u+≥λ(x)|∇u+|2, it follows from (3.3) that\nˆ\nΩλ|∇u+|2φ′\nmη2dx≤16ˆ\nΩµ|∇η|2(u++k)2φ′\nmdx+2ˆ\nΩfη2φmdx. (3.4)\nImposing H¨ older’s inequality on the last term of (3.4), we o btain\nˆ\nBσfφmη2dx≤ˆ\nBσ|f|\nk(u++k)φmη2dx≤/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u++k)1\n2φ1\n2mη/vextendsingle/vextendsingle/vextendsingle/vextendsingle2s′\ndx/parenrightigg1\ns′\n,(3.5)\nwheres′=s\ns−1.\nNext, we estimate the right-hand side of (3.5). Actually, q >n\n2impliesp∗<2s′=2s\ns−1<\nq∗:=2nq\n(q+1)n−2q. And then to combine Sobolev inequality and interpolation i nequality yields the\nfollowing inequalities\n/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u++k)1\n2φ1\n2mη/vextendsingle/vextendsingle/vextendsingle/vextendsingle2s′\ndx/parenrightigg1\ns′\n≤ε/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u++k)1\n2φ1\n2mη/vextendsingle/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightigg2\nq∗\n+1\nε/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u++k)1\n2φ1\n2mη/vextendsingle/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightigg2\np∗\n≤ε/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftbigg\n(u++k)1\n2φ1\n2mη/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2q\nq+1\ndx/parenrightiggq+1\nq\n+1\nε/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle(u++k)1\n2φ1\n2mη/vextendsingle/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightigg2\np∗\n, (3.6)\nforsufficientlysmall ε(whichwillbedeterminedlater). Furthermore, keepingH¨ o lder’sinequality\nand Cauchy-Schwarz inequality in mind, weestimate the first term of theright-hand sideof (3.6)\nas follows\n/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftbigg\n(u++k)1\n2φ1\n2mη/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2q\nq+1\ndx/parenrightiggq+1\nq\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nLq(Bσ)/parenleftiggˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftbigg\n(u++k)1\n2φ1\n2mη/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightigg\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nLq(Bσ)/parenleftigg\n2ˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftbigg\n(u++k)1\n2φ1\n2m/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nη2dx10\n+ 2ˆ\nBσλ|∇η|2(u++k)φmdx/parenrightbigg\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nLq(Bσ)/parenleftigg\n2ˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftbigg\n(u++k)1\n2φ1\n2m/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nη2dx\n+ 2ˆ\nBσµ|∇η|2(u++k)φmdx/parenrightbigg\n. (3.7)\nBy applying H¨ older’s inequality and substituting (3.5), ( 3.6), (3.7) into (3.4), we conclude\n/parenleftiggˆ\nBρ|η∇Gm(u)|2q\nq+1dx/parenrightiggq+1\nq\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nLq(Bσ)ˆ\nBσλ|∇Gm(u)|2η2dx\n≤C/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nLq(Bσ)ˆ\nBσµ|∇η|2(u++k)2φ′\nmdx\n+C/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble2\nLq(Bσ)/parenleftbiggˆ\nBσ/vextendsingle/vextendsingle(u++k)2φ′\nmη/vextendsingle/vextendsinglep∗\n2dx/parenrightbigg2\np∗\n,(3.8)\nwhereGm(u) =ˆu+\n0/vextendsingle/vextendsingleφ′\nm(s)/vextendsingle/vextendsingle1\n2ds.\nTo optimize the first term on the right-hand side of (3.8) with respect to η, we need to verify\nthatµ(u++k)2φ′\nmand (u++k)(φ′\nm)1\n2satisfy the conditions of Lemma 2.2. Indeed, it is not\nhard to find µ(u++k)2φ′\nm∈L1(Bσ) and (u++k)(φ′\nm)1\n2∈W1,p∗(Bσ) by virtue of u∈H1(Ω,A),\np∗<2q\nq+1as well as the properties of φ′\nm. Further Lemma 2.2 shows\ninf\nη/braceleftbiggˆ\nBσµ|∇η|2(u++k)2φ′\nmdx/bracerightbigg\n≤C(σ−ρ)−2n\nn−1/ba∇dblµ/ba∇dblLp(Bσ)/vextenddouble/vextenddouble/vextenddouble(u++k)/parenleftbig\nφ′\nm/parenrightbig1\n2/vextenddouble/vextenddouble/vextenddouble2\nW1,p∗(Bσ).\n(3.9)\nOn the other hand, when mgoes to infinity, we have\nlim\nm→+∞|∇Gm(u)|2= lim\nm→+∞/vextendsingle/vextendsingleG′\nm(u)/vextendsingle/vextendsingle2|∇u+|2≥β(u++k)β−1|∇u+|2a.e.,\n(u++k)2φ′\nm≤β(u++k)β+1,∀m∈[1,+∞).\nObviously, to use (3.8), (3.9) and η≡1 inBρderives the following inequality\n/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\n(u++k)β+1\n2/parenrightig/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightiggq+1\n2q\n≤C(β+1)(σ−ρ)−n\nn−1(Λ(Bσ))1\n2\n×/vextenddouble/vextenddouble/vextenddouble(u++k)β+1\n2/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bσ). (3.10)\nTo make Moser iteration argument work, we further prove that there exists δ >1 such that\nthe term /ba∇dbl(u++k)(β+1)δ\n2/ba∇dblW1,p∗(Bρ)may be dominated by /ba∇dbl(u++k)β+1\n2/ba∇dblW1,p∗(Bσ). Indeed, on\nthe one hand, choose δ= 2−1\nχ>1 withχ=2q\n(q+1)p∗>1, and apply H¨ older’s inequality with\nexponent2q\n(q+1)p∗to have\n/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\n(u++k)β+1\n2δ/parenrightig/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightigg1\np∗\n≤/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingleδ(u++k)β+1\n2(δ−1)∇/parenleftig\n(u++k)β+1\n2/parenrightig/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightigg1\np∗11\n≤δ/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\n(u++k)β+1\n2/parenrightig/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightiggq+1\n2q\n×/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingle(u++k)β+1\n2/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightiggδ−1\np∗\n. (3.11)\nOn the other hand, with the aid of Sobolev inequality W1,p∗(Bρ)֒→Lt(Bρ) where 1 ≤t≤np∗\nn−p∗,\nwe have\n/parenleftiggˆ\nBρ(u++k)β+1\n2δp∗dx/parenrightigg1\nδp∗\n≤C/vextenddouble/vextenddouble/vextenddouble(u++k)β+1\n2/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bρ). (3.12)\nThus, combining (3.10), (3.11) with (3.12), we have\n/vextenddouble/vextenddouble/vextenddouble(u++k)β+1\n2δ/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bρ)≤C(β+1)(Λ(B1))1\n2(σ−ρ)−n\nn−1/vextenddouble/vextenddouble/vextenddouble(u++k)β+1\n2/vextenddouble/vextenddouble/vextenddoubleδ\nW1,p∗(Bσ).(3.13)\nSetα=β+1\n2≥1, then (3.13) can be rewritten as\n/vextenddouble/vextenddouble/vextenddouble(u++k)αδ/vextenddouble/vextenddouble/vextenddouble1\nδα\nW1,p∗(Bρ)≤/parenleftig\nCα(Λ(B1))1\n2(σ−ρ)−n\nn−1/parenrightig1\nδα/ba∇dbl(u++k)α/ba∇dbl1\nα\nW1,p∗(Bσ).(3.14)\nWe further choose α=δm,ρm=1\n4+/parenleftbig1\n4/parenrightbigm+1,m∈N, and deduce the following recurrence\nrelations for the function u++k\n/vextenddouble/vextenddouble/vextenddouble(u++k)δm+1/vextenddouble/vextenddouble/vextenddouble1\nδm+1\nW1,p∗(Bρm+1)≤((Λ(B1)))1\n2δm+1δm\nδm+1/parenleftig\nC4n\nn−1/parenrightigm+1\nδm+1\n×/vextenddouble/vextenddouble/vextenddouble(u++k)δm/vextenddouble/vextenddouble/vextenddouble1\nδm\nW1,p∗(Bρm). (3.15)\nBy iteration, we find\n/ba∇dblu++k/ba∇dblL∞(B1\n4)≤C[(Λ(B1))]/summationtext∞\ni=01\n2δi+1/parenleftig\n4−n\nn−1δ/parenrightig/summationtext∞\ni=0i+1\nδi+1/ba∇dbl(u++k)/ba∇dblW1,p∗(B1\n2).(3.16)\nSubsequently, we estimate the right-hand side of (3.16). Us ing (3.8) with β= 1, ρ=1\n2, σ= 1,\nthe factp∗<2q\nq+1≤2≤2p′, and/ba∇dbl∇η/ba∇dblL∞(Bσ)≤c(σ−ρ)−1, we obtain\n\nˆ\nB1\n2|∇(u++k)|p∗dx\n1\np∗\n≤\nˆ\nB1\n2|∇(u++k)|2q\nq+1dx\nq+1\n2q\n≤(Λ(B1))1\n2/bracketleftigg/parenleftbiggˆ\nB1|u++k|2p′dx/parenrightbigg1\n2p′\n+/parenleftbiggˆ\nB1|u++k|p∗dx/parenrightbigg1\np∗/bracketrightigg\n.\nAnd then by H¨ older’s inequality, we get\n/parenleftbiggˆ\nB1|u++k|p∗dx/parenrightbigg1\np∗\n≤C/parenleftbiggˆ\nB1|u++k|2p′dx/parenrightbigg1\n2p′\n.12\nIn turn, combining the above inequalities with (3.16), we ob tain\n/ba∇dblu+/ba∇dblL∞(B1\n4)≤C(Λ(B1))δ\n2(δ−1)/parenleftbiggˆ\nB1|u++k|2p′dx/parenrightbigg1\n2p′\n≤C(Λ(B1))δ\n2(δ−1)/bracketleftigg/parenleftbiggˆ\nB1|u+|2p′dx/parenrightbigg1\n2p′\n+/ba∇dblf/ba∇dblLs(B1)/bracketrightigg\n, (3.17)\nwhich proves the claim.\nStep 2. The general case. Although the procedure is standard, we also give a sketch of\nour proof to make our article self-contained. For more detai ls, we may refer to [16]. First, let\n˜A(y) =A(Ry),˜λ(y) =λ(Ry),˜µ(y) =µ(Ry),˜u(y) =u(Ry),˜f(y) =R2f(Ry),˜k=/vextenddouble/vextenddouble/vextenddouble˜f/vextenddouble/vextenddouble/vextenddouble\nLs(B1),\nwherey∈B1. We apply what we just proved to ˜ uinB1and rewrite the result in terms of u,\nthen for any γ≥2p′, we derive\n/ba∇dblu+/ba∇dbl\nL∞/parenleftbigg\nB1\n4R/parenrightbigg≤C(Λ(BR))p′δ\nγ(δ−1)/parenleftigg\nR−n\nγ/parenleftbiggˆ\nBR|u+|γdx/parenrightbigg1\nγ\n+R2−n\ns/ba∇dblf/ba∇dblLs(BR)/parenrightigg\n,(3.18)\nwhereC=C(p,q,n,γ)≥1. Again, by the same argument as above, (3.18) clearly also h olds\nforBRreplaced by B(1−θ)R(x0), where x0∈BθR, we arrive at\n/ba∇dblu+/ba∇dbl\nL∞/parenleftbigg\nB1\n4(1−θ)R(x0)/parenrightbigg≤C/bracketleftbig\nΛ/parenleftbig\nB(1−θ)R(x0)/parenrightbig/bracketrightbigp′δ\nγ(δ−1)\n\n[(1−θ)R]−n\nγ/parenleftiggˆ\nB(1−θ)R(x0)|u+|γdx/parenrightigg1\nγ\n+[(1−θ)R]2−n\ns/ba∇dblf/ba∇dblLs(BR)/bracerightig\n≤C(Λ(BR))p′δ\nγ(δ−1)/bracketleftigg\n1\n(1−θ)n\nγ(m∗+1)R−n\nγ/parenleftbiggˆ\nBR|u+|γdx/parenrightbigg1\nγ\n+1\n(1−θ)n\nγm∗+n\ns−2R2−n\ns/ba∇dblf/ba∇dblLs(BR)/bracketrightigg\n, (3.19)\nwherem∗=p′δ\nδ−1max/braceleftig\n1\np+1\nq,2\nq/bracerightig\n,C=C(n,p,q,γ).\nNext, we will be in position to prove (1.5) for γ∈(0,2p′). First, we observe that\n/parenleftbiggˆ\nBR|u+|2p′dx/parenrightbigg1\n2p′\n≤ /ba∇dblu+/ba∇dbl2p′−γ\n2p′\nL∞(BR)/ba∇dblu+/ba∇dblγ\n2p′\nLγ(BR). (3.20)\nInaddition, applyingtheYoung’sinequalityto(3.20), and combiningthiswith(3.19)for γ= 2p′,\nwe obtain\n/ba∇dblu+/ba∇dblL∞(BθR)≤1\n2/ba∇dblu+/ba∇dblL∞(BR)+C(Λ(BR))δp′\nγ(δ−1)1\n(1−θ)n\nγ(m∗+1)R−n\nγ/ba∇dblu+/ba∇dblLγ(BR)\n+(Λ(BR))δ\n2(δ−1)1\n(1−θ)n\nγm∗+n\ns−2R2−n\ns/ba∇dblf/ba∇dblLs(BR), (3.21)13\nwhereC=C(n,p,q,γ). Then, set r:=θR(θR < R≤1, θ∈(0,1)) and apply (3.21) to get\n/ba∇dblu+/ba∇dblL∞(Br)≤1\n2/ba∇dblu+/ba∇dblL∞(BR)+C(Λ(BR))δp′\nγ(δ−1)1\n/parenleftbig\n1−r\nR/parenrightbign\nγ(m∗+1)R−n\nγ/ba∇dblu+/ba∇dblLγ(B1)\n+C(Λ(BR))δ\n2(δ−1)1\n/parenleftbig\n1−r\nR/parenrightbign\nγm∗+n\ns−2R2−n\ns/ba∇dblf/ba∇dblLs(B1)\n≤1\n2/ba∇dblu+/ba∇dblL∞(BR)+C(Λ(B1))δp′\nγ(δ−1)1\n(R−r)n\nγ(m∗+1)/ba∇dblu+/ba∇dblLγ(B1)\n+C(Λ(B1))δ\n2(δ−1)1\n(R−r)n\nγm∗+n\ns−2/ba∇dblf/ba∇dblLs(B1)\n≤1\n2/ba∇dblu+/ba∇dblL∞(BR)+C(Λ(B1))δp′\nγ(δ−1)\n×/parenleftigg\n1\n(R−r)n\nγ(m∗+1)/ba∇dblu+/ba∇dblLγ(B1)+1\n(R−r)n\nγm∗+n\ns−2/ba∇dblf/ba∇dblLs(B1)/parenrightigg\n,(3.22)\nwhereC=C(n,p,q,γ). Leth(r) :=/ba∇dblu+/ba∇dblL∞(Br), then (3.22) becomes\nh(r)≤1\n2h(R)+C(Λ(B1))δp′\nγ(δ−1)/parenleftigg\n1\n(R−r)n\nγ(m∗+1)/ba∇dblu+/ba∇dblLγ(B1)\n+1\n(R−r)n\nγm∗+n\ns−2/ba∇dblf/ba∇dblLs(B1)/parenrightigg\n.\nProceeding as a proof of Lemma 4.3 in [16], we may derive\n/ba∇dblu+/ba∇dblL∞(Bθ)≤C(Λ(B1))δp′\nγ(δ−1)\n×/parenleftigg\n1\n(1−θ)n\nγ(m∗+1)/ba∇dblu+/ba∇dblLγ(B1)+1\n(1−θ)n\nγm∗+n\ns−2/ba∇dblf/ba∇dblLs(B1)/parenrightigg\n,(3.23)\nwhereC=C(n,p,q,γ).\nStep 3. Consider /bardbl(−u)+/bardblL∞(Bθ).We consider uis supersolution of (1.1), and rewrite\nthe above result in terms of −u, then we could prove the claim.\nThe proof of Theorem 1.1 is complete.\nRemark 3.1. The proof of Theorem 1.1, clearly shows that the decrease in t he integrability\nofλ−1leads to an increase in the integrability index of fcompared to uniform case. However,\nthis does not imply that the integrability assumption of µis redundant. Indeed, by virtue of\np∗<2q\nq+1, we also obtain p >1\n2\nn−1−1\nq. Therefore, a sufficiently strong integrability assumption\nforµis indispensable for the continuation of the proof.\n4 Proof of Theorem 1.2\nIn this section, without loss of generality, we can assume th at/ba∇dblf/ba∇dblnq\n2q−n= 1 forf/\\e}atio\\slash≡0. We\nshallbeginwiththeLemma4.1. Thislemmaismainly basedon [ 8, Lemma2.22] and[33, Lemma\nB], but in comparison, we do not assume that uis bounded. Our idea of proof is motivated by\n[4, 7].14\nLemma 4.1. Assume that (H1)-(H2) hold, q >n\n2andf∈Ls(Ω)for some s >nq\n2q−n.Let\nube the non-negative, weak subsolution of (1.1)inΩwith zero Dirichlet boundary value. Then\nfor anyα∈/parenleftig\n0,4\nc2s/bardblλ−1/bardblq/parenrightig\n,wherecsis the sharp constant of the Sobolev inequality1, there exists\nC=C(cs,Ω,n,q)such that\n/parenleftbiggˆ\nΩeαq∗u\n2dx/parenrightbigg1\nq∗\n≤2α|Ω|1−(q∗)′\ns0\n/parenleftig\n2−cs/ba∇dblλ−1/ba∇dbl1\n2q√α/parenrightig2+|Ω|1\nq∗,\nwhereq∗=2nq\nn(q+1)−2q.\nProof.First we fix α∈/parenleftig\n0,4\nc2s/bardblλ−1/bardblq/parenrightig\nand define\nϕ(u) =φα\nN(u) =/braceleftigg\neαu−1, u ≤N,\nαeαNu+(1−αN)eαN−1, u > N.\nUsing Lemma 2.1, we have φα\nN(u)∈H1\n0(Ω,A).Substituting φα\nN(u) into (2.1) yields\nˆ\nΩA(x)∇u∇φα\nNdx=ˆ\nΩA(x)∇u∇u(φα\nN)′dx\n≤ˆ\nΩ|fφα\nN|dx≤ˆ\nΩ|f|/bracketleftbigg/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingle2\n+2/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg\ndx,∀N≥1.(4.1)\nThen from (H1)-(H2), Lemma 2.1 and (4.1), it infers\nˆ\nΩλ(x)|∇Fα\nN(u)|2dx≤ˆ\nΩA(x)∇u∇φα\nNdx\n≤ˆ\nΩ|f|/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingle2\ndx+2ˆ\nΩ|f|/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingledx, (4.2)\nwhereFα\nN=ˆu\n0/vextendsingle/vextendsingle(φα\nN)′(s)/vextendsingle/vextendsingle1\n2ds. In addition, according to the definition of φα\nN, we know\nlim\nN→∞|∇Fα\nN(u)| ≥√αeαu\n2|∇u|. (4.3)\nTherefore, together with H¨ older’s inequality, (4.3) and ( 4.2), we have\n/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\neαu\n2−1/parenrightig/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightbiggq+1\n2q\n=/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingle∇eαu\n2/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightbiggq+1\n2q\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/parenleftbiggˆ\nΩλ(x)/vextendsingle/vextendsingle/vextendsingle∇eαu\n2/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\n2\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/parenleftbiggα\n4ˆ\nΩλ(x)|∇Fα\nN(u)|2dx/parenrightbigg1\n2\n≤/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/bracketleftbiggα\n4ˆ\nΩ|f|/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingle2\ndx\n1Here the Sobolev inequality is /ba∇dblu/ba∇dblp∗≤cs/ba∇dbl∇u/ba∇dblp, wherep∗=np\nn−p, u∈W1,p\n0(Ω).15\n+α\n2ˆ\nΩ|f|/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingledx/bracketrightbigg1\n2\n. (4.4)\nOnce again, we apply Sobolev inequality and (4.4) to observe that the following inequality holds\n/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightbigg1\nq∗\n≤cs/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\neαu\n2−1/parenrightig/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightbiggq+1\n2q\n≤cs/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/bracketleftigg\nα\n4/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightbigg2\nq∗\n+α\n2/ba∇dblf/ba∇dbl(q∗)′/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightbigg1\nq∗/bracketrightigg1\n2\n, (4.5)\nwhereq∗=2nq\nn(q+1)−2q. Meanwhile, we notice that\n/ba∇dblf/ba∇dbl(q∗)′≤ |Ω|1−(q∗)′\ns0,\nand 1−cs/ba∇dblλ−1/ba∇dbl1\n2\nq√α\n2>0, so, it follows from (4.5) that\n/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightbigg1\nq∗\n≤2α/ba∇dblf/ba∇dbl(p∗)′\n/parenleftig\n2−cs/ba∇dblλ−1/ba∇dbl1\n2q√α/parenrightig2≤2α|Ω|1−(q∗)′\ns0\n/parenleftig\n2−cs/ba∇dblλ−1/ba∇dbl1\n2q√α/parenrightig2.(4.6)\nBy Minkowski’s inequality and (4.6), we get\n/parenleftbiggˆ\nΩeαq∗u\n2dx/parenrightbigg1\nq∗\n≤/parenleftbiggˆ\nΩ/vextendsingle/vextendsingle/vextendsingleeαu\n2−1/vextendsingle/vextendsingle/vextendsingleq∗\ndx/parenrightbigg1\nq∗\n+|Ω|1\nq∗\n≤2α|Ω|1−(q∗)′\ns0\n/parenleftig\n2−cs/ba∇dblλ−1/ba∇dbl1\n2q√α/parenrightig2+|Ω|1\nq∗. (4.7)\nThis completes the proof of Lemma 4.1.\nProof of Theorem 1.2. Without any loss ofgenerality, weassumesup\nΩu=/ba∇dblu/ba∇dbl∞, otherwise\nwe consider −u. Letϕ=/tildewiderφα\nN(u)˜η, where\n/tildewiderφα\nN(u) =/braceleftigg\neα\n2u+, u ≤N,\nα\n2eα\n2N(u−N)+eα\n2N, u > N,and ˜η≥0∈C∞\n0(Ω). (4.8)\nWe choose ϕas a test function in (2.1) and obtain\nˆ\nΩ/parenleftig\nA(x)∇u∇/tildewiderφα\nN/parenrightig\n˜ηdx+ˆ\nΩ(A(x)∇u∇˜η)/tildewiderφα\nNdx=ˆ\nΩf/tildewiderφα\nN˜ηdx.\nIn view ofˆ\nΩ/parenleftig\nA(x)∇u∇/tildewiderφα\nN/parenrightig\n˜ηdx≥0, we have\nˆ\nΩ/tildewiderφα\nNA(x)∇u∇˜ηdx≤ˆ\nΩf/tildewiderφα\nN˜ηdx. (4.9)16\nSinceC∞\n0(Ω)isdensein H1\n0(Ω,A), itisobviousthat(4.9)holdsforany ˜ η∈H1\n0(Ω,A). Therefore,\nwe may choose ˜ ηsatisfying\n˜η=ηβ\nN(u) =\n\n/parenleftig\neα\n2u+/parenrightigβ\n−1, u ≤N,\nαβ\n2eαβ\n2N(u−N)+eαN\n2−1, u > N,\nwhereβ≥1 as a test function. Hence, substituting ηβ\nNinto (4.9), we have\nˆ\nΩ(A(x)∇u+∇u+)/parenleftig\nηβ\nN/parenrightig′/tildewiderφα\nNdx≤ˆ\nΩf/tildewiderφα\nNηβ\nNdx. (4.10)\nUsing Lemma 2.1 and (H1), we rewrite (4.10) as\nˆ\nΩλ|∇˜G(u)|2dx≤ˆ\nΩf/tildewiderφα\nNηβ\nNdx,\nwhere˜G(u) =´u\n0/bracketleftbigg/parenleftig\nηβ\nN/parenrightig′\n(s)/tildewiderφα\nN(s)/bracketrightbigg1\n2\nds. A simple computation shows\nlim\nN→∞|∇˜G(u)|2≥8β\n(β+1)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\neα\n2u+/parenrightigβ+1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (4.11)\n/tildewiderφα\nNηβ\nN≤/parenleftig\neα\n2u+/parenrightigβ+1\n,∀N≥1. (4.12)\nLetω=eα\n2u+, then together with (4.10)-(4.12), we obtain\nˆ\nΩλ/vextendsingle/vextendsingle/vextendsingle∇ωβ+1\n2/vextendsingle/vextendsingle/vextendsingle2\ndx≤α(β+1)2\n8βˆ\nΩfωβ+1dx. (4.13)\nWith the aid of the Sobolev inequality and (4.13), we deduce\n/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightigq∗\ndx/parenrightbigg1\nq∗\n≤/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2−1/parenrightigq∗\ndx/parenrightbigg1\nq∗\n+|Ω|1\nq∗\n≤cs/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/parenleftbiggˆ\nΩλ/vextendsingle/vextendsingle/vextendsingle∇ωβ+1\n2/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\n2\n+|Ω|1\nq∗\n≤cs/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq√α(β+1)\n2√2β/ba∇dblf/ba∇dbl1\n2s/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightig2s′\ndx/parenrightbigg1\n2s′\n+|Ω|1\nq∗−1\n2s′/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightig2s′\ndx/parenrightbigg1\n2s′\n=/bracketleftbigg\ncs/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq√α(β+1)\n2√2β/ba∇dblf/ba∇dbl1\n2s+|Ω|1\nq∗−1\n2s′/bracketrightbigg\n×/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightig2s′\ndx/parenrightbigg1\n2s′\n, (4.14)\nwhere 2s′< q∗due tos > s0=/parenleftig\nq∗\n2/parenrightig′\n. Because ofβ+1\n2√β≥1, (4.14) may be rewritten as the\nfollowing form\n/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightigq∗\ndx/parenrightbigg2\nq∗(β+1)\n≤/bracketleftbigg\ncs/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq√α(β+1)\n2√2β/ba∇dblf/ba∇dbl1\n2s+|Ω|1\nq∗−1\n2s′/bracketrightbigg2\nβ+117\n×/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightig2s′\ndx/parenrightbigg1\ns′(β+1)\n≤/bracketleftbigg\nC√α(β+1)√β/parenleftbigg/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/ba∇dblf/ba∇dbl1\n2s+1/parenrightbigg/bracketrightbigg2\nβ+1\n×/parenleftbiggˆ\nΩ/parenleftig\nωβ+1\n2/parenrightig2s′\ndx/parenrightbigg1\ns′(β+1)\n. (4.15)\nHereC=C(n,p,q,s, Ω)≥1. Now, if we setβ+1\n2=χi, i∈Nin (4.15), where χ=q∗\n2s′>1, then\n(4.15) becomes\n/ba∇dblω/ba∇dblχiq∗≤/bracketleftbigg√αC/parenleftbigg/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble1\n2\nq/ba∇dblf/ba∇dbl1\n2s+1/parenrightbigg/bracketrightbigg1\nχi\nχi\nχi/ba∇dblω/ba∇dbl2χis′\n≤/bracketleftig\nCα/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig/bracketrightig/summationtexti\nj=01\n2χjχ/summationtexti\nj=0j\nχj/ba∇dblω/ba∇dbl2s′. (4.16)\nAnd then, taking i→+∞yields\n/ba∇dblω/ba∇dbl∞≤/bracketleftig\nCα/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig/bracketrightigχ\n2(χ−1)χχ\n(χ−1)2/ba∇dblω/ba∇dbl2s′, (4.17)\nwhereC=C(n,p,q,s, Ω). Using Lemma 4.1 and H¨ older’s inequality, (4.17) can be r ewritten as\n/ba∇dblω/ba∇dbl∞≤C/bracketleftig\nα/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig/bracketrightigχ\n2(χ−1)/parenleftbiggˆ\nΩω2s′dx/parenrightbigg1\n2s′\n≤C/bracketleftig\nα/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig/bracketrightigχ\n2(χ−1)/parenleftigg/parenleftbiggˆ\nΩeα\n2q∗udx/parenrightbigg1\nq∗\n+|Ω|1\nq∗/parenrightigg\n≤C/bracketleftig\nα/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig/bracketrightigχ\n2(χ−1)\nα|Ω|1−(q∗)′\ns0\n/parenleftig\n2−cs/ba∇dblλ−1/ba∇dblq√α/parenrightig2+|Ω|1\nq∗\n.\nBy the definition of ω, we further obtain\neα\n2/bardblu+/bardbl≤C/bracketleftbig\nα/parenleftbig\n/ba∇dblλ−1/ba∇dblq/ba∇dblf/ba∇dbls+1/parenrightbig/bracketrightbigχ\n2(χ−1)\nα|Ω|1−(q∗)′\ns0\n(2−cs/ba∇dblλ−1/ba∇dblq√α)2+|Ω|1\nq∗\n.(4.18)\nObviously, (4.18) is equivalent to\n/ba∇dblu/ba∇dbl∞≤C/bracketleftig\nlog/parenleftig/vextenddouble/vextenddoubleλ−1/vextenddouble/vextenddouble\nq/ba∇dblf/ba∇dbls+1/parenrightig\n+1/bracketrightig\n, (4.19)\nwhereC=C(n,p,q,s, Ω)≥1. This completes the proof.\nRemark 4.1. We need to point out some differences below. Firstly, in the pro of of Lemma\n4.1, we construct a truncation function that satisfies the co ndition of Lemma 2.1 to bypass some\ndifficulties from the global boundedness of u, as described in [8, 33]. Secondly, In the proof of\nTheorem 1.2, we construct a Caccioppoli inequality for eαu+to avoid the transformation in [8,\n33] which essentially depends on the bounded condition.185 Harnack Inequality — Proof of Theorem 1.3\nToendtheproofof Theorem1.3, wefirstgive akey lemma, namel y, weak Harnack inequality.\nLemma 5.1. (Weak Harnack inequality) Assume that (H1)-(H2) hold, q >n\n2andf∈Ls(Ω)\nfor some s >nq\n2q−n. Letu∈H1(A,Ω)be a nonnegative supersolution in Ω. Then for any\nBR⊂Ω,γ∈(0,q∗\n2)and any 0< θ < τ < 1, there exists a constant Csuch that\n/parenleftbigg1\nRnˆ\nBτRuγdx/parenrightbigg1\nγ\n≤C/parenleftbigg\ninf\nBθRu+R2−n\ns/ba∇dblf/ba∇dblLs(BR)/parenrightbigg\n,\nwhereq∗=2nq\nn(q+1)−2q,C≤c1ec2Λ(BR),c1=c1(γ,n,p,q,s,τ,θ )≥1,c2=c2(γ,n,p,q)>0.\nRemark 5.1. Theorem 1.3 may be concluded by combining Theorem 1.1 with Le mma 5.1.\nThe prove of Lemma 5.1 is based on the strategy of [2, Theorem 4 .1]. Even though experts\nmight already anticipate how to adapt the argument of [2], fo r the completeness of the article,\nwe give a detailed proof.\nProof of Lemma 5.1. To better comprehend the structure of our proof, we will divi de our\nproof into three steps. And without loss of generality, we se tR= 1 here.\nStep 1. We claim that\nexp/parenleftbigg \nBτlog(u+h)dx/parenrightbigg\n≤C/parenleftbigg\ninf\nBθu+/ba∇dblf/ba∇dblLs(B1)/parenrightbigg\n, (5.1)\nwhereC=C(n,τ,θ,Λ(B1),p,q,s).\nFirst, we take φ=ϕ\nu+hin (2.1), where h:=/ba∇dblf/ba∇dblLs(B1)andϕ≥0∈H1\n0(B1,A), then we have\nˆ\nB1/parenleftbigg1\nu+hA(x)∇u∇ϕ−ϕ\n(u+h)2A(x)∇u∇u/parenrightbigg\ndx≥ˆ\nB1fϕ\nu+hdx. (5.2)\nDue toϕ\n(u+h)2A(x)∇u∇u≥0, (5.2) becomes\nˆ\nB1A(x)∇/parenleftbigg\nlogk\nu+h/parenrightbigg\n∇ϕdx≤ −ˆ\nB1fϕ\nu+hdx. (5.3)\nSet logk\nu+h=vandf\nu+h=˜f, and rewrite (5.3) as\nˆ\nB1A(x)∇v∇ϕdx≤ −ˆ\nB1˜fϕdx, (5.4)\nwhere/ba∇dbl˜f/ba∇dblLs(B1)≤1. Then (5.4) implies that vis the weak subsolution of −div(A(x)∇v) =−˜f.\nTo give a priori estimate for v, we choose ϕ=ϕmη2in (5.4), where\nϕm=/braceleftigg\n(v++1)β−1, v < m,\nβ(m+1)β−1(v−m)+(m+1)β−1, v≥m,\nandη∈C∞\n0(Bτ), η≡1 inBρ;η≡0 inBτ\\Bσand 0< θ≤ρ < σ≤τ≤1, by the same\nprocedure of Theorem 1.1, we have\nsup\nBθv≤C(Λ(B1))δp′\nq∗(δ−1)/parenleftigg\n1\n(τ−θ)n\nq∗(m∗+1)/ba∇dblv/ba∇dblLq∗(Bτ)+1\n(τ−θ)nm∗\nq∗+n\ns−2/parenrightigg\n,(5.5)19\nwhereq∗=2nq\nn(q+1)−2q. In the sequel, we will estimate the term /ba∇dblv/ba∇dblLq∗(Bτ)in (5.5). Let φ=η2\n1\nu+h\nin (2.1), where η1∈C∞\n0(B1) andη1≡1 inBτ;η1≡0 inB1\\B1+τ\n2. After direct calculation,\nwe have\nˆ\nB1η2\n1A(x)∇v∇vdx+ˆ\nB12η1A(x)∇v∇η1dx≤ −ˆ\nB1˜fη2\n1dx,\nand then using the Young’s inequality and H¨ older’s inequal ity yields\nˆ\nB1λ|∇v|2η2\n1dx≤Cˆ\nB1µ|∇η1|2dx+|B1|1\ns′\n≤C/ba∇dblµ/ba∇dblLp(B1)\n(1−τ)2|B1|1\np′+|B1|1\ns′. (5.6)\nSubsequently, we let k= exp/parenleftigffl\nBτlog(u+h)dx/parenrightig\nsatisfyingffl\nBτvdx= 0, and apply Poincar´ e\ninequality to get\n/ba∇dblv/ba∇dblLq∗(Bτ)≤C/ba∇dbl∇v/ba∇dbl\nL2q\nq+1(Bτ)≤ /ba∇dblλ−1/ba∇dbl1\n2\nLq(Bτ)/ba∇dblλ1\n2∇v/ba∇dblL2(Bτ). (5.7)\nSubstituting (5.6) into (5.7) and it is not hard to verify\nsup\nBθv≤C(Λ(B1))δp′\nq∗(δ−1)+1\n2/parenleftigg\n1\n(τ−θ)n\nq∗(m∗+1)(1−τ)+1\n(τ−θ)nm∗\nq∗+n\ns−2/parenrightigg\n.\nFinally, (5.1) follows from the definitions of vandk.\nStep 2. We claim that there exist p0=p0(τ,θ,Λ(B1),p,s,n,q )>0 andC=C(τ,θ,Λ(B1),\np,s,n,q)>0 such that\n\nˆ\nB3τ+θ\n4|u+h|p0dx\n1\np0\n≤Cexp/parenleftbigg \nBτlog(u+h)dx/parenrightbigg\n. (5.8)\nFirst ofall, wetake φ=η21\nu+h/parenleftbig\n(ω++1)β+(2β)β/parenrightbig\nin(2.1), where ω= logu+h\nk=−v, β≥1,\nη≡1 inBρ;η1≡0 inBτ\\Bσandη∈C∞\n0(Bσ), with3τ+θ\n4≤ρ < σ≤τ <1, and notice\n∇φ=2η\nu+h/parenleftig\n(ω++1)β+(2β)β/parenrightig\n∇η−η2\n(u+h)2/parenleftig\n(ω++1)β+(2β)β/parenrightig\n∇u\n+βη2\nu+h(ω++1)β−1∇(ω++1).\nThen, we obtain\nˆ\nBση2\n(u+h)2/parenleftig\n(ω++1)β+(2β)β/parenrightig\nA(x)∇u∇udx\n−βˆ\nBση2\nu+h(ω++1)β−1A(x)∇u∇(ω++1)dx\n≤ˆ\nBσ2η\nu+h/parenleftig\n(ω++1)β+(2β)β/parenrightig\nA(x)∇u∇ηdx−ˆ\nBσfη2\nu+h/parenleftig\n(ω++1)β+(2β)β/parenrightig\ndx.(5.9)20\nSubsequently, using Young’s inequality, we can estimate th e first term on the right-hand side of\n(5.9) as follows\nˆ\nBσ2η\nu+h/parenleftig\n(ω++1)β+(2β)β/parenrightig\nA(x)∇u∇ηdx\n≤1\n4ˆ\nBση2(ω++1)β−1\n(u+h)2A(x)∇u∇udx+8ˆ\nBσ(ω++1)β+1A(x)∇η∇ηdx\n+1\n4ˆ\nBση2\n(u+h)2(2β)βA(x)∇u∇udx+8ˆ\nBσ(2β)βA(x)∇η∇ηdx\n≤1\n4ˆ\nBση2(ω++1)β\n(u+h)2A(x)∇u∇udx+8ˆ\nBσ(ω++1)β+1A(x)∇η∇ηdx\n+1\n4ˆ\nBση2\n(u+h)2(2β)βA(x)∇u∇udx+8ˆ\nBσ(2β)βA(x)∇η∇ηdx\n=1\n4ˆ\nBση2\n(u+h)2/parenleftig\n(ω++1)β+(2β)β/parenrightig\nA(x)∇u∇udx\n+8ˆ\nBσ(ω++1)β+1A(x)∇η∇ηdx+8ˆ\nBσ(2β)βA(x)∇η∇ηdx. (5.10)\nBy definition of ω+, we observe that the first term on the right-hand side in (5.10 ) can be\nabsorbed into the first term on the left-hand side of (5.9). Th is leads to the following inequality\nˆ\nBση2/bracketleftbigg3\n4/parenleftig\n(ω++1)β+(2β)β/parenrightig\n−β(ω++1)β−1/bracketrightbigg\nA(x)∇(ω++1)∇(ω++1)dx\n≤8ˆ\nBσ(ω++1)β+1A(x)∇η∇ηdx+8ˆ\nBσ(2β)βA(x)∇η∇ηdx\n+ˆ\nBσ|˜f|η2/parenleftig\n(ω++1)β+(2β)β/parenrightig\ndx.\nOwing to ( ω++1)β+(2β)β≥2β(ω++1)β−1, the previous inequality reduces\nˆ\nBσβη2(ω++1)β−1A(x)∇(ω++1)∇(ω++1)dx\n≤16ˆ\nBσ(ω++1)β+1A(x)∇η∇ηdx+16ˆ\nBσ(2β)βA(x)∇η∇ηdx\n+2ˆ\nBσ|˜f|η2/parenleftig\n(ω++1)β+(2β)β/parenrightig\ndx. (5.11)\nWith the help of (H2) and H¨ older’s inequality in (5.11), we o btain\n4β\n(β+1)2ˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\nη2dx≤16ˆ\nBσµ|∇η|2(ω++1)β+1dx\n+(2β)β/parenleftbigg\n16/ba∇dblµ/ba∇dblLp(Bσ)|B1|1\np′1\n(σ−ρ)2+2|B1|1\ns′/parenrightbigg\n+2ˆ\nBσ|˜f|η2(ω++1)β+1dx. (5.12)\nHence, following the lines of proof of (3.5)-(3.7), we concl ude\nˆ\nBσ|˜f|η2(ω++1)βdx≤/parenleftbiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingleη(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsingle2s′\ndx/parenrightbigg1\ns′21\n≤ε/parenleftiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle∇/parenleftig\n(ω++1)β+1\n2η/parenrightig/vextendsingle/vextendsingle/vextendsingle2q\nq+1dx/parenrightiggq+1\nq\n+1\nε/parenleftbiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle(ω++1)β+1\n2η/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightbigg2\np∗\n≤2ε/ba∇dblλ−1/ba∇dblq/bracketleftbiggˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\nη2dx\n+ˆ\nBσµ|∇η|2(ω++1)β+1dx/bracketrightbigg\n+1\nε/parenleftbiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle(ω++1)β+1\n2η/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightbigg2\np∗\n. (5.13)\nBy setting ε=β\n2(β+1)2/bardblλ−1/bardblLq(Bσ)and absorbing the first term on the right-hand side of (5.13)\ninto the left-hand side of (5.12), it is easy to calculate tha t\nˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\nη2d≤16(β+1)2\nβˆ\nBσµ|∇η|2(ω++1)β+1dx\n+2/ba∇dblλ−1/ba∇dblq(β+1)4\nβ2/parenleftbiggˆ\nBσ/vextendsingle/vextendsingle/vextendsingle(ω++1)β+1\n2η/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightbigg2\np∗\n+32(β+1)2(2β)β−1/ba∇dblµ/ba∇dblLp(Bσ)1\n(σ−ρ)2. (5.14)\nFurthermore, minimizing the right hand of (5.14) for η(|x|)∈C∞\n0(Bτ), withη(|x|)≡1 inBρ;\nη(|x|)≡0 inBτ\\Bσ, and using Lemma 2.2, we obtain\nˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\nη2dx≤/bracketleftigg\n16(β+1)2\nβ(σ−ρ)2n\nn−1/ba∇dblµ/ba∇dblLp(Bσ)+2/ba∇dblλ−1/ba∇dblLq(Bσ)(β+1)4\nβ2/bracketrightigg\n×/vextenddouble/vextenddouble/vextenddouble(ω++1)β+1\n2/vextenddouble/vextenddouble/vextenddouble2\nW1,p∗(Bσ)\n+32(β+1)2(2β)β−1/ba∇dblµ/ba∇dblLp(Bσ)1\n(σ−ρ)2. (5.15)\nThen, we replace ubyωin inequalities (3.10)-(3.12) and use (5.15) to get\n/vextenddouble/vextenddouble/vextenddouble(ω++1)β+1\n2δ/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bρ)≤CΛ(B1)1\n2/parenleftigg\nβ+1\n(σ−ρ)n\nn−1√β+(β+1)2\nβ/parenrightigg/vextenddouble/vextenddouble/vextenddouble(ω++1)β+1\n2/vextenddouble/vextenddouble/vextenddoubleδ\nW1,p∗(Bσ)\n+(2β)β−1\n2(β+1)Λ(B1)1\n21\nσ−ρ/parenleftiggˆ\nBρ/vextendsingle/vextendsingle/vextendsingle(ω++1)β+1\n2/vextendsingle/vextendsingle/vextendsinglep∗\ndx/parenrightiggδ−1\np∗\n.\n(5.16)\nKeeping Young’s inequality in mind, we find that the second te rm on the right-hand side of\n(5.16) has the following estimate\n/vextenddouble/vextenddouble/vextenddouble(ω++1)β+1\n2δ/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bρ)≤CΛ(B1)δ\n2(δ−1)/parenleftigg\nβ+1\n(σ−ρ)n\nn−1√β+(β+1)2\nβ+(σ−ρ)−δ\nδ−1/parenrightigg22\n×/vextenddouble/vextenddouble/vextenddouble(ω++1)β+1\n2/vextenddouble/vextenddouble/vextenddoubleδ\nW1,p∗(Bσ)+/bracketleftig\n(2β)β−1\n2(β+1)/bracketrightigδ\n. (5.17)\nSetα=β+1\n2≥1, then, (5.17) is rewritten as\n/vextenddouble/vextenddouble/vextenddouble(ω++1)αδ/vextenddouble/vextenddouble/vextenddouble1\nαδ\nW1,p∗(Bρ)≤C1\nαδΛ(B1)1\n2α(δ−1)/parenleftigg\nα\n(σ−ρ)n\nn−1√2α−1+α2\n2α−1+(σ−ρ)δ\nδ−1/parenrightigg1\nδα\n×/ba∇dbl(ω++1)α/ba∇dbl1\nα\nW1,p∗(Bσ)+4α.\nTherefore, we set α=δm,ρm=3τ+θ\n4+1\n2m+2(τ−θ),m∈N, and build up the following iterative\nprocess\n/vextenddouble/vextenddouble/vextenddouble(ω++1)δm+1/vextenddouble/vextenddouble/vextenddouble1\nδm+1\nW1,p∗(Bρm+1)≤C1\nδm+1Λ(B1)1\n2(δ−1)δm/parenleftig\n2δ\nδ−1+n\nn−1δ/parenrightigm+1\nδm\n×/vextenddouble/vextenddouble/vextenddouble(ω++1)δm/vextenddouble/vextenddouble/vextenddouble1\nδm\nW1,p∗(Bρm)+4δm, (5.18)\nwhich implies\n/vextenddouble/vextenddouble/vextenddouble(ω++1)δm+1/vextenddouble/vextenddouble/vextenddouble1\nδm+1\nW1,p∗(Bρm+1)≤CΛ(B1)δ\n2(δ−1)2/parenleftig\n2δ\nδ−1+n\nn−1δ/parenrightigδ\n(δ−1)2\n×/parenleftig\n/ba∇dblω++1/ba∇dblW1,p∗(Bρ0)+δm/parenrightig\n, (5.19)\nwhereCdepends on n, p, q, τ, θ .\nFinally, recall that ω=−v, together with (5.6) and (5.7), and we get\n/ba∇dblω/ba∇dblW1,p∗(Bτ)=/ba∇dblv/ba∇dblW1,p∗(Bτ)≤C/ba∇dblv/ba∇dbl\nW1,2q\nq+1(Bτ)≤C(τ,Λ(B1), p, q,s).\nThen for any i > p∗, there exists m0∈Nsuch that δm0p∗≤i≤δm0+1p∗, (5.19) can be rewritten\nas\n/ba∇dblω+/ba∇dbl\nLi/parenleftbigg\nB3τ+θ\n4/parenrightbigg≤C1Λ(B1)C2i,\nwhereC1=C1(τ,p,q,s,n,θ ) andC2=C2(n,p,q). We further let p0=/parenleftbig\n2C1Λ(B1)C2e/parenrightbig−1\nsatisfying\npi\n0/ba∇dblω+/ba∇dbli\nLi(B3τ+θ\n4)\ni!≤1\n2i,\nthus we have\n\nˆ\nB3τ+θ\n4/parenleftbiggu+h\nk/parenrightbiggp0\ndx\n1\np0\n≤\nˆ\nB3τ+θ\n4ep0ω+dx+|B1|\n1\np0\n≤\n∞/summationdisplay\ni=0p0/ba∇dblω+/ba∇dbli\nLi(B3τ+θ\n4)\ni!+|B1|\n1\np0\n≤(2+|B1|)1\np0.23\nThe proof of the inequality (5.8) is complete.\nStep 3. We claim that for any γ∈[p0,q∗\n2), there exists C=C(p,q,Λ(B1),γ,n)≥1 such\nthat\n/parenleftbiggˆ\nBθ(u+h)γdx/parenrightbigg1\nγ\n≤C\nˆ\nB3τ+θ\n4(u+h)p0dx\n1\np0\n, (5.20)\nwhereC≤C3Λ(B1)C4/parenleftBig\n1\np0−1\nγ/parenrightBig\nwithC3=C3(γ,n,p,q,τ,θ )∈[1,+∞) andC4=C4(γ,n,p,q)∈\n[1,+∞).\nTo complete the proof of (5.20), we take φ=η2(u+h)βwithβ∈(−1,0) in (2.1), and\n0<τ+3θ\n4≤ρ < σ≤τ+θ\n2<1,η≡1 inBρ;η1≡0 inBτ+θ\n2\\Bσ, we obtain\nβˆ\nBσ(u+h)β−1η2A(x)∇u∇(u+h)dx+2ˆ\nBση(u+h)βA(x)∇u∇ηdx≥ˆ\nBσfη2(u+h)βdx,\nthen by Young’s inequality, we get\n−βˆ\nBσ(u+h)β−1η2A(x)∇u∇(u+h)dx≤ −ˆ\nBσfη2(u+h)βdx\n+2ˆ\nBση(u+h)βA(x)∇u∇ηdx\n≤ −β\n2ˆ\nBσ(u+h)β−1η2A(x)∇u∇(u+h)dx\n−ˆ\nBσfη2(u+h)βdx\n−8\nβˆ\nBσµ|∇η|2(u+h)β+1dx. (5.21)\nAbsorbing the first term on the right-hand side into the left- hand side of (5.21), we obtain\n−4β\n(β+1)2ˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(u+h)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\ndx≤ˆ\nBσ|f|η2(u+h)βdx\n−8\nβˆ\nBσµ|∇η|2(u+h)β+1dx.\nNext, recalling (5.13) and replacing ω++1 byu+h, the above inequality may be rewritten as\nˆ\nBσλ/vextendsingle/vextendsingle/vextendsingle∇(u+h)β+1\n2/vextendsingle/vextendsingle/vextendsingle2\ndx≤16(β+1)2\n|β|ˆ\nBσµ|∇η|2(u+h)β+1dx\n+2/ba∇dblλ−1/ba∇dblq(β+1)4\nβ2/parenleftbiggˆ\nBσ(u+h)(β+1)p∗\n2dx/parenrightbigg2\np∗\n.(5.22)\nIn turn, keeping Lemma 2.2 and H¨ older’s inequality in mind, we get\n/vextenddouble/vextenddouble/vextenddouble∇(u+h)β+1\n2/vextenddouble/vextenddouble/vextenddouble\nL2q\nq+1(Bρ)≤C/parenleftigg\nβ+1√β+(β+1)2\n|β|/parenrightigg\nΛ(B1)1\n2\n(σ−ρ)n\nn−1/vextenddouble/vextenddouble/vextenddouble(u+h)β+1\n2/vextenddouble/vextenddouble/vextenddouble\nW1,p∗(Bσ).(5.23)24\nSetα=β+1\n2∈/parenleftbig\n0,1\n2/parenrightbig\n, then (3.11), (3.12) and (5.23) imply\n/vextenddouble/vextenddouble/vextenddouble(u+h)αδ/vextenddouble/vextenddouble/vextenddouble1\nαδ\nW1,p∗(Bρ)≤/parenleftig\nCΛ(B1)1\n2/parenrightig1\nαδ\n(σ−ρ)n\n(n−1)αδ/parenleftbigg2α√1−2α+4α2\n1−2α/parenrightbigg1\nαδ\n/ba∇dbl(u+h)α/ba∇dbl1\nα\nW1,p∗(Bσ).\nTakingα=α0\nδmwhereα0∈/parenleftbig\n0,1\n2/parenrightbig\n, andρm=τ+θ\n2+1\n2m(θ−τ)\n4, we get\n/vextenddouble/vextenddouble/vextenddouble(u+h)α0\nδm−1/vextenddouble/vextenddouble/vextenddoubleδm−1\nα0\nW1,p∗(Bρm−1)≤/parenleftigg\nCΛ(B1)1\n2\n(τ−θ)/parenrightiggδm−1\nα0\nα0\nδm/radicalig\n1−2α0\nδm+4α2\n0\nδ2m/parenleftbig\n1−2α0\nδm/parenrightbig\nδm−1\nα0\n×2mnδm−1\nα0(n−1)/vextenddouble/vextenddouble/vextenddouble(u+h)α0\nδm/vextenddouble/vextenddouble/vextenddoubleδm\nα0\nW1,p∗(Bρm).\nThen by iteration, we obtain\n/ba∇dbl(u+h)α0/ba∇dbl1\nα0\nW1,p∗(Bρ0)≤/parenleftigg\nCΛ(B1)1\n2α2\n0\n(τ−θ)/parenleftbig\n1−2α0\nδ/parenrightbig/parenrightiggδm−1\n(δ−1)α0/parenleftigg\n2n\nn−1\nδ/parenrightigg1\nα0/parenleftBig(m+1)\nδ−1δm+1−δm+1−1\n(δ−1)2/parenrightBig\n×/vextenddouble/vextenddouble/vextenddouble(u+h)α0\nδm/vextenddouble/vextenddouble/vextenddoubleδm\nα0\nW1,p∗(Bρm). (5.24)\nSubsequently, we fix m∈Nsatisfying2pα0\nδm(p−1)≤p0≤2pα0\nδm−1(p−1), and apply H¨ older’s inequality\nto (5.22) to estimate the right-hand of (5.24) as follows\n/vextenddouble/vextenddouble/vextenddouble∇(u+h)α0\nδm/vextenddouble/vextenddouble/vextenddouble\nLp∗(Bρm)≤C/vextenddouble/vextenddouble/vextenddouble∇(u+h)α0\nδm/vextenddouble/vextenddouble/vextenddouble\nL2q\nq+1/parenleftbigg\nBτ+θ\n2/parenrightbigg\n≤C5/vextenddouble/vextenddouble/vextenddouble(u+h)α0\nδm/vextenddouble/vextenddouble/vextenddouble\nL2p\np−1/parenleftbigg\nB3τ+θ\n4/parenrightbigg≤C6/ba∇dblu+h/ba∇dblα0\nδm\nLp0(3τ+θ\n4),\nwhereC5,C6depend on τ,θ,α0,Λ(B1),n,p,q,s . Further, inserting the above inequality into\n(5.24), one has\n/ba∇dbl(u+h)α0/ba∇dbl1\nα0\nW1,p∗(Bρ0)≤CΛ(B1)pδ\n(p−1)p0(δ−1)−1\n2α0(δ−1)/ba∇dblu+h/ba∇dbl\nLp0/parenleftbigg\nB3τ+θ\n4/parenrightbigg.(5.25)\nOnce again, by Sobolev inequality, (5.22) and (5.23), we find\n/ba∇dbl(u+h)α0/ba∇dblLq∗(Bθ)≤C/ba∇dbl(u+h)α0/ba∇dbl\nW1,2q\nq+1(Bθ)≤C/ba∇dbl(u+h)α0/ba∇dbl\nW1,p∗/parenleftbigg\nBτ+3θ\n4/parenrightbigg,(5.26)\nwhereC=C(τ,θ,n,Λ(B1),q,p,α 0). Then a combination of (5.25) and (5.26) yields the desired\nclaim (5.20).\nThis completes the proof of Lemma 5.1.\n6 Proof of Example 1.1 – 1.3\nIn this section we first construct Example 1.1 to demonstrate thats=nq\n2q−nis optimal. Our\nexample is intuitively straightforward.25\nProof of Example 1.1. Let Ω be the ball B=B1\n4(0). Define\nA(x) =\na1(x) 0 ···0\n0a1(x)···0\n............\n0 0 ···a1(x)\n, a1(x) =|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθ\n,\nand\nf(x) = (n+1)|x|β−1/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−θ|x|β−1/parenleftbigg\nlog1\n|x|/parenrightbiggθ−1ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−2|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy,\nwhereβ=n\nq<2,θ=1\n2+1\nq−1\nn.\nIt is easy to observe that\nu(x) =ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−11\nlog1\n|y|dy\nis the weak solution of (1.1) and u(0) = +∞. To make the argument rigorous, we exploit some\ntechnique from [8] to show that\nˆ\nΩA(x)∇u∇ϕdx=ˆ\nΩfϕdx,∀ϕ≥0∈C∞\n0(Ω).\nFor each k≥2, letχkbe aC∞, non-negative and radial function satisfying\nχk(x) =/braceleftigg\n0,|x| ≤2−k−2,\n1,2−k−1≤ |x|<1\n4.\nDefinefk=χkf. Notice that each fkis continuous, and\nuk=χkˆ\nB1\n2(0)|y|1−n(|x|+|y|)−11\nlog1\n|y|dy∈C2(Ω)\nis the solution of −div(A(x)∇uk) =fk, in the weak sense,\nˆ\nΩA(x)∇uk∇ϕdx=ˆ\nΩfkϕdx,∀ϕ≥0∈C∞\n0(Ω).\nThe strong convergence fk→finLs(Ω) implies\nˆ\nΩA(x)∇uk∇ϕdx→ˆ\nΩA(x)∇u∇ϕdx, k→ ∞.26\nNext, wewillverifythat λ−1(x)∈Lq(Ω). Indeed,byusingapolarcoordinatetransformation,\nwe obtain\nˆ\nB1\n2(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleq\ndx=αnˆ1\n2\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nrβ/parenleftbig\nlog1\nr/parenrightbigθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleq\nrn−1dr=αnˆ1\n2\n01\n/parenleftbig\nlog1\nr/parenrightbigθqrn−1−βqdr\n=αnˆ∞\n21\n(logt)θq/parenleftbigg1\nt/parenrightbiggn+1−βq\ndt=αnˆ∞\n21\n(logt)θq1\ntdt <+∞,\nwhereαnis the surface area of an n-dimensional sphere.\nTo complete the proof of Example 1.1, we will demonstrate tha tf∈Lnq\n2q−n(Ω) when n≥3.\nIt is sufficient to demonstrate one term of fthat belongs to Lnq\n2q−n(Ω), the remaining terms can\nbe handled in a similar manner. For example, we will show that\n|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy∈Lnq\n2q−n/parenleftig\nB1\n4(0)/parenrightig\n.\nIn fact, we know\nˆ\nB1\n4(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenq\n2q−n\ndx\n=ˆ\nB1\n4(0)|x|βnq\n2q−n/parenleftbigg\nlog1\n|x|/parenrightbiggθnq\n2q−n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenq\n2q−n\ndx.(6.1)\nThen, we will estimate\nˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy. (6.2)\nWe decompose (6.2) into two terms\nˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy=ˆ\nB1\n2(0)∩{|y|<|x|1\n2}|y|1−n(|x|+|y|)−31\nlog1\n|y|dy\n+ˆ\nB1\n2(0)∩{|y|≥|x|1\n2}|y|1−n(|x|+|y|)−31\nlog1\n|y|dy\n=I1+I2.\nBy using a polar coordinate transformation, we find I1may be estimated as follows\nI1=ˆ\nB1\n2(0)∩{|y|<|x|1\n2}|y|1−n(|x|+|y|)−31\nlog1\n|y|dy≤αnˆ|x|1\n2\n0(|x|+r)−31\nlog1\nrdr\n≤2αn\nlog1\n|x|ˆ|x|1\n2\n0(|x|+r)−3dr=αn\nlog1\n|x|/parenleftbigg\n|x|−2−/parenleftig\n|x|1\n2+|x|/parenrightig−2/parenrightbigg\n≤αn\nlog1\n|x||x|−2.(6.3)27\nSimilarly, for I2, we also estimate\nI2≤αnˆ1\n2\n|x|1\n2(|x|+r)−31\nlog1\nrdr≤αn\nlog2/parenleftig\n|x|+|x|1\n2/parenrightig−3ˆ1\n2\n|x|1\n2dr\n=αn\nlog2/parenleftig\n|x|+|x|1\n2/parenrightig−3/parenleftbigg1\n2−|x|1\n2/parenrightbigg\n≤αn\nlog2|x|−3\n2/parenleftbigg1\n2−|x|1\n2/parenrightbigg\n=αn\n2log2|x|−3\n2.(6.4)\nBy using (6.2)-(6.4), (6.1) has the following estimate\nˆ\nB1\n4(0)|x|βnq\n2q−n/parenleftbigg\nlog1\n|x|/parenrightbiggθnq\n2q−n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenq\n2q−n\ndx\n≤cˆ\nB1\n4(0)|x|βnq\n2q−n/parenleftbigg\nlog1\n|x|/parenrightbiggθnq\n2q−n/parenleftigg\n1\nlog1\n|x||x|−2/parenrightiggnq\n2q−n\ndx\n+cˆ\nB1\n4(0)|x|βnq\n2q−n/parenleftbigg\nlog1\n|x|/parenrightbiggθnq\n2q−n/parenleftig\n|x|−3\n2/parenrightignq\n2q−ndx\n=cˆ1\n4\n0r(β−2)nq\n2q−n+n−1/parenleftbigg\nlog1\nr/parenrightbiggnq(θ−1)\n2q−n\ndr+cˆ1\n4\n0r(β−3\n2)nq\n2q−n+n−1/parenleftbigg\nlog1\nr/parenrightbiggnqθ\n2q−n\ndr\n=I3+I4,\nwhereconly depends on n, q. Now, we estimate I3. Notice\n(β−2)nq\n2q−n+n−1 =(n\nq−2)nq\n2q−n+n−1 =−1,\nnq(θ−1)\n2q−n=n−q−2nq\n2(2q−n)<−1,\nthen, a simple computation shows I3<+∞.\nSimilarly, due to\n(β−3\n2)nq\n2q−n+n−1 =nq\nq−n\n2−1>−1,\nit is not hard to prove that\nI4<+∞.\nThus we arrive at\nˆ\nB1\n4(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|x|β/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenq\n2q−n\ndx <+∞.\nFor the case n= 2, by a similar argument, we could also verify f∈Ls(B1\n4(0)), for s∈[1,q\nq−1).\nProof of Example 1.2. We take β= 2 in the proof of Example 1.1, and we have\na1(x) =|x|2/parenleftbigg\nlog1\n|x|/parenrightbiggθ\n,28\nf(x) = (n+1)|x|/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n+θ|x|/parenleftbigg\nlog1\n|x|/parenrightbiggθ−1ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−2|x|2/parenleftbigg\nlog1\n|x|/parenrightbiggθˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy,\nwhereθ=5\n2n. We observe\nu(x) =ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−11\nlog1\n|y|dy\nis the weak solution of (1.1), and u(0) =∞. The rest of procedure is similar to Example 1.1.\nFinally, we find f∈L∞(B1\n4(0)) when n≥3 andf∈Ls(B1\n4(0)),s∈[1,+∞) whenn= 2.\nProof of Example 1.3. We setβ=n\nq∈/parenleftig\n2,2n\nn−1/parenrightig\n, θ= 1 in Example 1.1. Let\na1(x) =|x|n\nq/parenleftbigg\nlog1\n|x|/parenrightbigg2\nq\n,\nf(x) = (n+1)|x|n\nq−1/parenleftbigg\nlog1\n|x|/parenrightbiggˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−|x|n\nq−1ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−21\nlog1\n|y|dy\n−2|x|n\nq/parenleftbigg\nlog1\n|x|/parenrightbiggˆ\nB1\n2(0)|y|1−n(|x|+|y|)−31\nlog1\n|y|dy,\nandf∈L∞(B1\n4(0)). Then we take\nu(x) =ˆ\nB1\n2(0)|y|1−n(|x|+|y|)−11\nlog1\n|y|dy,\nby a similar argument as above, we could verify uis the weak solution of (1.1).\nRemark 6.1. It’s interesting to see that Example 1.3 could be extended di rectly to the case\nof1\np+1\nq=2\nn−1, n≥3. Indeed, when we take q=2\nn−1, p= +∞, we find that there exists\nf∈L∞(Ω), which makes the weak solution uunbounded.\nReferences\n[1] P. Bella, M. Sch¨ affner, On the regularity of minimizers for scalar in tegral functionals with (p,q)-\ngrowth, Anal. PDE 13 (7) (2020) 2241–2257.\n[2] P.Bella, M. Sch¨ affner, LocalboundednessandHarnackinequa lityforsolutionsoflinearnonuniformly\nelliptic equations, Comm. Pure Appl. Math. 74 (3) (2021) 453–477.\n[3] P. Bella, M. Sch¨ affner, Local boundedness for p-Laplacian with degenerate coefficients, Math. Eng.\n5 (5) (2023) 1-20.29\n[4] S. Chanillo, Y.Y. Li, Continuity of solutions of uniformly elliptic equatio ns inR2, Manuscr. Math.\n77 (4) (1992) 415–433.\n[5] S. Chanillo, R.L. Wheeden, Harnack’s inequality and mean-value ineq ualities for solutions of degen-\nerate elliptic equations, Comm. Partial Differential Equations 11 (10 ) (1986) 1111–1134.\n[6] A. Cianchi, Maximizing the L∞norm of the gradient of solutions to the Poisson equation, J. Geom.\nAnal. 2 (6) (1992) 499–515.\n[7] A. Cianchi, Strong and weak type inequalities for some classical op erators in Orlicz spaces, J. Lond.\nMath. Soc. (2) 60 (1) (1999) 187–202.\n[8] D. Cruz-Uribe, S. Rodney, Bounded weak solutions to elliptic PDE w ith data in Orlicz spaces, J.\nDifferential Equations 297 (2021) 409–432.\n[9] G. Cupini, P. Marcellini, E. Mascolo, Local boundedness of weak so lutions to elliptic equations with\np,q-growth, Math. Eng. 5 (3) (2023) Paper No. 065, 28 pp.\n[10] E. De Giorgi, Sulla differenziabilit` a e l’analiticit` a delle estremali degli integrali multipli regolari,\nMem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (3) (1957) 25–43.\n[11] E. De Giorgi, Congetture sulla continuit` a delle soluzioni di equaz ioni lineari ellittiche autoaggiunte\na coefficienti illimitati, Unpublished, 1995.\n[12] E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solu tions of degenerate elliptic equa-\ntions, Comm. Partial Differential Equations 7 (1) (1982) 77–116.\n[13] B. Franchi, C.E. Guti´ errez, R.L. Wheeden, Two-weight Sobole v-Poincar´ e inequalities and Harnack\ninequality fora classofdegenerateelliptic operators, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.\nRend. Lincei (9) Mat. Appl. 5 (2) (1994) 167–175.\n[14] B. Franchi, R. Serapioni, F. Serra Cassano, Irregular solution s of linear degenerate elliptic equations,\nPotential Anal. 9 (3) (1998) 201–216.\n[15] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations o f second order, Second edition.\nGrundlehren der mathematischen Wissenschaften [Fundamental P rinciples of Mathematical Sci-\nences], 224. Springer-Verlag, Berlin, 1983.\n[16] Q. Han, F.H. Lin, Elliptic partial differential equations, Courant L ecture Notes in Mathematics,\nvol. 1, New York University, Courant Institute of Mathematical Sc iences; American Mathematical\nSociety, New York; Providence, RI, 1997.\n[17] J. Hirsch, M. Sch¨ affner, Growth conditions and regularity, an optimal local boundedness result,\nCommun. Contemp. Math. 23 (3) (2021) Paper No. 2050029, 17 pp .\n[18] O.A. Ladyˇ zhenskaya, N.N. Ural’tseva, Linear and quasilinear ellip tic equations, Translated from\nthe Russian by Scripta Technica, Inc. Translation editor: Leon Ehr enpreis Academic Press, New\nYork-London 1968.\n[19] V.G. Maz’ja, Some estimates of solutions of second-order elliptic equations, Dokl. Akad. Nauk SSSR\n137 (1961) 1057–1059.\n[20] V.G. Maz’ja, Weak solutions of the Dirichlet and Neumann problems , Trudy Moskov. Mat. Obˇ sˇ z. 20\n(1969) 137–172.\n[21] C.B. Morrey, On the solutions of quasi-slinear elliptic partial differ ential equations, Trans. Amer.\nMath. Soc. 43 (1) (1938) 126–166.\n[22] J.Moser,AnewproofofDeGiorgi’stheoremconcerningthereg ularityproblemforellipticdifferential\nequations, Comm. Pure Appl. Math. 13 (1960) 457–468.\n[23] J.Moser,OnHarnack’stheoremforelliptic differentialequation s, Comm.PureAppl. Math.14(1961)\n577–591.\n[24] M.K.V. Murthy, G. Stampacchia, Boundary value problems for so me degenerate-elliptic operators,\nAnn. Mat. Pura Appl. 80 (4) (1968) 1–122.\n[25] J.Nash, Continuityofsolutionsofparabolicandellipticequations ,Amer.J.Math.80(1958)931–954.\n[26] L.C. Piccinini, S. Spagnolo, On the H¨ older continuity of solutions o f second order elliptic equations\nin two variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (3) (1972) 39 1–402.\n[27] G. Stampacchia, Le probl` eme de Dirichlet pour les´ equations e lliptiques du second ordre ` a coefficients\ndiscontinus, (French) Ann. Inst. Fourier (Grenoble) 15 (1965) 1 89–258.30\n[28] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola N orm. Sup. Pisa Cl. Sci. 3 (4) (1976)\n697–718.\n[29] E.V. Teixeira, J.M. Urbano, Geometric tangential analysis and sh arp regularity for degenerate pdes,\nin Harnack inequalities and nonlinear operators, Springer INdAM Ser . Springer, cham 46 (2021)\n175–192.\n[30] N.S. Trudinger, On the regularity of generalized solutions of linea r, non-uniformly elliptic equations,\nArch. Rational Mech. Anal. 42 (1971) 50–62.\n[31] N.S. Trudinger, Linear elliptic operators with measurable coefficie nts, Ann. Scuola Norm. Sup. Pisa\nCl. Sci. 27 (3) (1973) 265–308.\n[32] K.O. Widman, On the H¨ older continuity of solutions of elliptic partia l differential equations in two\nvariables with coefficients in L∞, Comm. Pure Appl. Math. 22 (1969) 669–682.\n[33] X.S. Xu, Logarithmic up bounds for solutions of elliptic partial diffe rential equations, Proc. Amer.\nMath. Soc. 139 (10) (2011) 3485–3490.\n[34] X. Zhong, Discontinuous solutions of linear, degenerate elliptic e quations, J. Math. Pures Appl. 90\n(9) (2008) 31–41."    },    {        "title": "2402.07248v1.Depth_Separations_in_Neural_Networks__Separating_the_Dimension_from_the_Accuracy.pdf",        "content": "arXiv:2402.07248v1  [cs.LG]  11 Feb 2024Depth Separations in Neural Networks:\nSeparating the Dimension from the Accuracy\nItay Safran1, Daniel Reichman2, and Paul Valiant1\n1Purdue University\n2Worcester Polytechnic Institute\nAbstract\nWe prove an exponential separation between depth 2 and depth 3 neural networks, when approximat-\ning anO(1)-Lipschitz target function to constant accuracy, with resp ect to a distribution with support in\n[0,1]d, assuming exponentially bounded weights. This addresses a n open problem posed in Safran et al.\n[20], and proves that the curse of dimensionality manifests in depth 2 approximation, even in cases where\nthe target function can be represented efficiently using dep th 3. Previously, lower bounds that were used\nto separate depth 2 from depth 3 required that at least one of t he Lipschitz parameter, target accuracy or\n(some measure of) the size of the domain of approximation sca le polynomially with the input dimension,\nwhereas we fix the former two and restrict our domain to the uni t hypercube. Our lower bound holds for\na wide variety of activation functions, and is based on a nove l application of an average- to worst-case\nrandom self-reducibility argument, to reduce the problem t o threshold circuits lower bounds.\n1 Introduction\nThere is significant empirical evidence suggesting that dep th plays a crucial role in the practical success of\ndeep learning [8, 5]. From a purely theoretical perspective , while depth 2 neural networks are known to be\nuniversal approximators for continuous functions over com pact domains [1, 6, 9], it is now well-established\nthat depth may be necessary to obtain a compact representati on of certain target functions [3, 22, 16, 2, 26,\n10, 19, 20, 25, 18, 15, 21]. Perhaps the most stark such exampl es are when separating depth 2 from depth\n3 – following the seminal work of Eldan and Shamir [3], many wo rks have demonstrated depth separations\nwhere depth can be exponentially more beneficial than width, even when increased by just one: There exists\na function f:Rd→Rthat can be approximated to accuracy ε >0using a network of depth 3 and\nwidthpoly(d,1/ε), yet any depth 2 network which approximates fto the same accuracy would require\nexponentially many more neurons (in dor1/ε).\nDespite the growing number of settings where such separatio ns can be shown, known results often\ninvolve contrived settings with complicated distribution s and oscillatory target functions. On the other hand,\nfunctions arising in machine learning settings are often sm ooth, with bounded fluctuations. Moreover, it\nwas recently shown that in some examples constructed to show depth separations with neural networks, the\nsame property which prevents approximation using the shall ow architecture, may also prove detrimental for\noptimization, even when using a network which is sufficientl y deep to express the target function efficiently\n[11, 12]. In light of this, several recent works have shifted their focus to devising depth separation results\nfor functions that are arguably more natural [19, 20, 18, 17, 15, 21]. The rationale is that such results could\n1be better aligned with learning problems arising in applica tions, rather than contrived examples that may not\nbe representative of broader classes of functions.\nIn Safran et al. [20], the authors observe, given a depth 2 low er bound for neural network approxima-\ntion, that the Lipschitz constant of the target function and the target accuracy parameters, can be traded off\nby rescaling the function by a multiplicative factor. Moreo ver, one can also strengthen the accuracy lower\nbound, at the cost of dilating the domain of approximation, b y using a change of variables (see [20, Theo-\nrem 9] for a precise statement). This important observation suggests, that if either one of these parameters\nscales with the input dimension d(as is the case in all known separation results), then it is no t possible\nto pinpoint whether the true cause of the difficulty in the app roximation stems from the input dimension\nitself, or from the remaining parameters that were forced to scale with it. In other words, it is not clear if\nknown approximation lower bounds for depth 2 are a manifesta tion of the curse of dimensionality, or if the\ndifficulty lies somewhere else. To study the root cause for th e hardness of approximation, the authors pose\nthe following question:\nCan we show a superpolynomial depth 2 vs. depth 3 neural netwo rk separation result in terms\nof the dimension d, for approximating O(1)-Lipschitz functions up to constant accuracy ε, on a\ndomain of bounded radius (all independent of d)?[20]\nTheir main result, is that perhaps surprisingly, if one cons iders radial target functions that are com-\nmonly used to show such depth separations, then an exponenti al dependence on dis not possible. This is\ndemonstrated by providing a general approximation result w herein width poly(d)suffices.\nMotivated by the above question, in this work, we prove a sepa ration result between depth 2 and depth\n3, where the target function is O(1)-Lipschitz, the domain of approximation is contained insid e the unit\nhypercube, and the separation is exponential even if the tar get accuracy is an absolute constant. We point\nout that for technical reasons, our result does not quite res olve the above question, since the unit hypercube\nhas radius which scales as√\nd. However, our result does provide a separation where the eff ects of the\nparameters other than the input dimension are minimal compa red to other results in the literature (see Table 1\nfor a comparison). Moreover, despite not being bounded in a b all of constant radius, the unit hypercube is\na natural domain to study approximation problems. This is su pported by the fact that its d-dimensional\nLebesgue measure is constant (namely, the volume of the doma in is independent of d), whereas the unit ball\nneeds to be rescaled by a factor of Θ(√\nd)to have a constant volume. So at least in some sense, our domai n\nof approximation is more independent of the input dimension than a domain of constant radius.\nOur proof technique is quite different than those commonly u sed in the literature. While it is common to\nuse some technical analysis tool such as Fourier spectrum an alysis [3] or spherical harmonics [2], our lower\nbound relies on a reduction to threshold circuits, where an e xact computation lower bound for the IP mod\n2 function is used. The key ingredient in our proof is a reduct ion from an average-case complexity, where\nintuitively, only a constant portion of the inputs are class ified correctly; to a worst-case complexity, where\nwe are able to construct a network which is capable of classif ying allthe inputs correctly. This construction\nutilizes the randomization of the input in a manner which pre serves its output value, yet induces sufficient\nrandomness so as to result in a concentration of measure whic h provides a worst-case guarantee. This\nis achieved by a careful analysis of properties of the probab ility distribution induced by our randomized\nself-reduction, and much of the proof of the lower bound is de dicated to this part.\nWe point out that our separation holds for a wide family of act ivation functions, which also includes\ncommonly used functions like the ReLU, threshold and sigmoi dal activations. Similarly to other lower\nbounds in the literature that are shown with respect to compa ctly supported distributions, our result assumes\na mild exponential upper bound on the magnitude of the weight s of the approximating network. In contrast,\n2Table 1: Several known lower bounds for approximation using depth 2 neural networks, where the target\nfunction is scaled to be O(1)-Lipschitz, and for the sake of comparison, the distributio n is scaled to either\ncontain the unit hypercube (for compactly supported distri butions), or normalized to have most of its prob-\nability mass contained within the unit hypercube (non-comp actly supported distributions). An asterisk∗\ndenotes that the result in Daniely [2] is not for a radial func tion, but can be reduced to one taking the form\nx/ma√sto→1\n2πd2.5sin(2πd2.5/⌊ard⌊lx/⌊ard⌊l2), which oscillates in any direction from the origin (see [20, 23]). In contrast, our\nresult is non-oscillatory in all such directions, but it can be seen rather as an extension of a Boolean function\nto the unit hypercube. Safran et al. [20] use a reduction to th e main result of Eldan and Shamir [3] to show a\nlower bound for a non-oscillatory function, but the cost of r emoving this oscillation is that the accuracy gets\nworse by a factor of d−2. To the best of our knowledge, our lower bound, which is highl ighted in bold, is the\nonly constant accuracy lower bound in this setting. Since in many machine learning applications the input\ndimension is quite large, we have that lower bounds that scal e as1/poly(d)are too permissive, in contrast\nto our constant lower bound.\nReference Distribution Function Accuracy\nEldan and Shamir [3] Radial, heavy-tailed Radial, oscillatory Ω/parenleftbig\nd−4/parenrightbig\nDaniely [2] Radial∗, compactly supported Radial∗, oscillatory Ω/parenleftbig\nd−2/parenrightbig\nSafran et al. [20] Radial, heavy-tailed, Radial, non-oscillatory Ω/parenleftbig\nd−6/parenrightbig\nVenturi et al. [25] Product, heavy-tailed Product, oscillatory Ω/parenleftbig\nd−4/parenrightbig\nThis paper Product, compactly supported Non-oscillatory Ω(1)\nlower bounds that do not impose any restrictions on the magni tude of the weights, rely on the technique in\nEldan and Shamir [3], and use heavy-tailed data distributio ns. To the best of our knowledge, it remains a\nmajor open problem to show a superpolynomial separation res ult between depth 2 and depth 3, with respect\nto a distribution with bounded support, and when allowing un bounded weights.\nThe remainder of this paper is structured as follows: After p resenting our contributions in this paper\nin more detail below, we discuss related work in the literatu re. In Sec. 2we present the notation used\nthroughout this paper, followed by a formal construction of our separation setting, the set of assumptions\nused in our results, and the main result in this paper. Sec. 3details our depth 2 lower bound, as well as\nprovides a sketch of the proof. Sec. 4details our depth 3 positive approximation result, as well a s provides\na concrete example of the construction for the ReLU activati on function.\nOur contributions\n• We prove that under mild assumptions on the activation func tionσ(Assumption 1), a depth 2 neu-\nral network with σactivations cannot well-approximate a certain O(1)-Lipschitz function fd(see\nEq. ( 1)). It is shown that there exists a simple distribution with s upport in [0,1]2d, such that fdcannot\nbe approximated to better than constant accuracy with respe ct to this distribution, unless the width of\nthe network scales as Ω/parenleftBig\nexp(Ω(d))\npoly(C)/parenrightBig\n(Thm. 3.1).\n• We prove that under different, mild assumptions on the acti vation function σ(Assumption 2), a depth\n3 neural network with σactivations can approximate the same function fdto arbitrary accuracy in an\nL∞sense, using width that scales polynomially with the target accuracy and d(Thm. 4.1).\n• Combining our lower and upper bounds, we obtain our main res ult (Thm. 2.1), which demonstrates\n3the manifestation of the curse of dimensionality in depth 2 a pproximation, even if the target function\nis easy to approximate using depth 3. This is in contrast with previously known results that only imply\nan exponential lower bound in a high-accuracy regime where t he target accuracy scales as 1/poly(d).\nRelated work\nSeparating depth 2 from depth 3, continuous, L2lower bound setting. For concreteness, let us fo-\ncus here on the setting of separating depth 2 from depth 3 in a c ontinuous, L2lower bound setting. The\nseminal work of Eldan and Shamir [3] provided the first expone ntialL2lower bound for depth 2 neural\nnetworks. The authors construct an approximately radial, o scillatory target function, and use a Fourier spec-\ntrum analysis argument to prove their unconstrained lower b ound (imposing no bounds on the magnitude\nof the weights). As an artifact of this technique, a superpos ition argument is required to guarantee that the\ntarget function cannot be approximated with fewer than expo nentially many neurons. This, however, also\nforces the Lipschitz constant and the number of oscillation s to scale as poly(d). Venturi et al. [25] adapt\nthe technique of Fourier spectrum analysis to a setting with product target functions and distributions, rather\nthan radial ones. Due to this technique, both works [3, 25] re quire the target accuracy to scale as Ω/parenleftbig\nd−4/parenrightbig\nfor\nthe curse of dimensionality to manifest. Safran et al. [21] a lso use Fourier spectrum analysis to show a depth\nseparation result between depth 2 and depth 3, for approxima ting the non-oscillatory 1-Lipschitz maximum\nfunction on the domain [0,1]d, with respect to the uniform distribution. However, their s eparation is only\npolynomial in magnitude, and for accuracy which scales as 1/poly(d)rather than a constant.\nDaniely [2] used a simple and elegant harmonic analysis tech nique, to show a depth 2 vs. depth 3\nseparation, using exponentially bounded weights, for osci llatory target functions, over a product distribution\non two spheres (that can be reduced to a radial distribution – see [20, 23] for more details). With some\ncareful analysis, it can be shown that with this technique, t he approximation error required for the curse\nof dimensionality to manifest is Ω/parenleftbig\nd−2/parenrightbig\n. [18, 15] prove depth 2 approximation lower bounds for non-\noscillatory target functions, by using the main result in Da niely [2], and decoupling the dependence of the\ninput dimension from the accuracy in the lower bound. Howeve r, both results require the accuracy to scale\nasd−2for the curse of dimensionality to manifest, a property inhe rited by the proof technique being used.\nSafran and Shamir [19], Safran et al. [20] use reductions to t he main result of Eldan and Shamir [3]\nto derive exponential depth 2 lower bounds for non-oscillat ory functions. Safran and Shamir [19] show\nthis for non-Lipschitz ellipsoid indicator functions, and Safran et al. [20] show this for a 1-Lipschitz radial\nfunction. Like the previous results using Fourier spectrum analysis, these results require accuracy scaling\nwithΩ/parenleftbig\nd−4/parenrightbig\nto have exponential dependence on the input dimension, but a dditionally, simplifying the\nfunction to be non-oscillatory results in an additional d−2factor, for an overall accuracy lower bound of\nΩ/parenleftbig\nd−6/parenrightbig\n. In contrast, our lower bound is also for a non-oscillatory f unction, but it is already exponential for\nconstant accuracy.\nConnection between neural networks and threshold circuits .The connection between approximation\nlower bounds when using neural networks and threshold circu its has been studied in multiple works recently.\nMartens et al. [13] study lower bounds for depth 2 neural netw orks with non-threshold activation functions,\nby constructing a threshold network which computes the same real-valued function, and applying known\nthreshold circuit lower bounds. Their work differs from our in a few ways: (i) While we use a similar\nreduction technique, we only approximate non-threshold ac tivations using thresholds rather than compute\nthe exact same real function; (ii) we focus on approximation on continuous domains rather than the discrete\nuniform distribution over the Boolean hypercube; and most i mportantly, (iii) we consider a weaker notion\n4ofaverage-case approximation, whereas their work deals with exact computation of Boolean functions.\nMukherjee and Basu [14] derive sub-linear size lower bounds for neural networks by showing reductions to\nknown threshold circuit lower bounds. Vardi and Shamir [23] show barriers for achieving depth separations\nin neural networks by using reductions to open problems in th reshold circuits, and applying known “natural\nproof barrier” results, showing that such separations woul d solve long-standing open problem that are widely\nbelieved to be very difficult. Since their results only apply to networks of depth 4or more, they do not apply\nin the setting investigated in this paper, which to the best o f our knowledge does not solve any open problem\nin circuit complexity. Vardi et al. [24] use communication c omplexity to derive size lower bounds for ReLU\nnetworks which compute IP mod 2. While this lower bound appli es to any depth, it is for a network size\nwhich is sublinear in the input dimension, whereas we use a si milar distribution, but in order to show an\nexponential separation between depth 2 and depth 3.\nProgress on the open question posed in Safran et al. [20]. To the best of our knowledge, the only\nworks that made progress with the open question posed in Safr an et al. [20], are [20, 7]. Safran et al. [20]\nobserve that the target accuracy, Lipschitz parameter of th e target function, and the radius of the domain of\napproximation, can all be traded off with polynomial factor s. As one of their main contributions, the authors\nshow that when all three parameters are held constant, then a n exponential separation between depth 2 and\ndepth 3 is not possible for radial functions, by proving a pos itiveL∞approximation result in this setting\nwhere width polynomial in dsuffices. This indicates that in order to resolve the questio n, one must consider\nnon-radial target functions. Hsu et al. [7] consider the que stion of how many randomly initialized ReLU\nneurons are required for the approximation of arbitrary fun ctions with a constant Lipschitz parameter, with\nrespect to the uniform distribution over [0,1]d, and with high probability. Their main positive approximat ion\nresult is that perhaps surprisingly, if the target accuracy is fixed, then a depth 2, width poly(d)random\nReLU network will approximate any O(1)-Lipschitz function with high probability (hence – there ex ists a\nnetwork with this approximation). However, since this resu lt is with respect to a uniform L2approximation\nrather than an L∞approximation, it does not imply that lower bounds exponent ial indare not possible\nfor distributions on [0,1]dthat are different from the uniform distribution. Indeed, o ur lower bound is for\na distribution with support in [0,1]2d, that has more probability mass concentrated closer to the B oolean\nhypercube.\n2 Setting and Main Result\nIn this section, we formally define our setting and present th e notation and terminology used throughout the\npaper, before turning to present our assumptions and main re sult.\n2.1 Preliminaries and notation\nNotation and terminology. We let[n]be shorthand for the set {1,...,n}. We denote vectors using\nbold-faced letters (e.g. x) and matrices or random variables using upper-case letters (e.g.X). Given a\nvectorx= (x1,...,x d)∈Rd, we define round(x) = (round( x1),...,round(xd)), whereround(x)\nroundsxto the nearest integer. We let U(A)denote the uniform distribution on a set A⊆Rd. We define\nthe Boolean functions AND :{0,1}2→ {0,1},AND(x,y) =x·y; andIPd:{0,1}2d→ {0,1},\nIPd(x,y) =/an}⌊ra⌋ketle{tx,y/an}⌊ra⌋ketri}htmod 2 . Given a function f:R→R, we define its total variation on the interval\n[a,b]⊆RasVb\na(f):= supP/summationtextnP\ni=1|f(xi)−f(xi−1)|, where the supremum is taken over all the possible\npartitions of the interval [a,b].\n5Neural networks and threshold circuits. We consider fully connected, feed-forward neural networks ,\ncomputing functions from RdtoR. Aσ-network consists of layers of neurons. In every layer excep t for\nthe output neuron, an affine function of the inputs is compute d, followed by a computation of the non-linear\nactivation function σ:R→R. The single output neuron simply computes an affine transfor mation of\nits inputs. Each layer with a non-linear activation is calle d ahidden layer , and the depth of a network\nis defined as the number of hidden layers plus one. The width of a network is defined as the number of\nneurons in the largest hidden layer, and the sizeof the network is the total number of neurons across all\nlayers. Analogously, we define a threshold network as aσ-network which employs the threshold activation\nfunction; namely, where σ(x) = 1 for allx≥0.5, andσ(x) = 0 otherwise. We make the distinction\nbetween a threshold network and a threshold circuit , where in a threshold circuit, the output neuron also has\na non-linear activation rather than a linear activation as i s the case for threshold networks.\n2.2 Formal construction\nWe begin with defining the distribution used to show our separ ation result. Let\nAd:= ([0,0.25]∪[0.75,1])d\nbe the support of our distribution. We define our distributio n to be the uniform distribution over the set\nA2d; namely, the distribution U(A2d). It is interesting to note that this distribution can also be seen as an\ninterpolation between the two distributions U([0,1]d)andU({0,1}d), where as discussed in the related work\nsubsection, the former is too spread to show constant accura cy lower bounds for O(1)-Lipschitz functions,\nand the latter is a discrete rather than a continuous distrib ution – the setting where neural networks are\ntypically being used. We define our hard to approximate funct ionfd: [0,1]2d→Ras\nfd(x,y):= IPd(round(x),round(y))·min\ni∈[d]{4|xi−0.5|,4|yi−0.5|,1}, (1)\nwhere the inputs satisfy x,y∈[0,1]d. It is straightforward to verify that fd(x,y) = IP d(round(x),round(y))\nfor allx,y∈ Ad. Moreover, it is easy to see that fdis4-Lipschitz on [0,1]2d:1It equals 0if there\nexistsi∈[d]such that xi= 0.5oryi= 0.5, so it suffices to prove that it is 4-Lipschitz on each\none of the 22ddisjoint sub-hypercubes of side length 0.5that are contained in [0,1]2d. For each such\nsub-hypercube, if IPdequals0, thenfdequals0on the sub-hypercube, and it is clearly 4-Lipschitz.\nOtherwise, IPdequals1on the sub-hypercube, and therefore for all x,yin this sub-hypercube we have\nfd(x,y) = min i∈[d]{4|xi−0.5|,4|yi−0.5|,1}= 4min i∈[d]{|xi−0.5|,|yi−0.5|,0.25}, which is 4-\nLipschitz since the minimum function is 1-Lipschitz.\n2.3 Assumptions\nBefore we can present our main result in this paper, we will fir st formally state and discuss our assumptions.\nWe begin with formally stating our assumption on the family o f activation functions for which our lower\nbound holds.\n1We remark that while we define fdto be4-Lipschitz on [0,1]2d, this is in fact not necessary, since our distribution is onl y\nsupported on A2d⊂[0,1]2d, and therefore it does not matter how fdbehaves outside of this set. Nevertheless, we extend it to be\nLipschitz on [0,1]2dto highlight its simple structure compared to some of the oth er functions that were used to separate depth 2\nfrom depth 3.\n6Assumption 1 (Lower bound) .The activation function σis (Lebesgue) measurable and satisfies\n|σ(x)|≤Cσ(1+|x|ασ)\nand\nVb\na(σ)≤Cσ(1+(|a|+|b|)ασ)\nfor allx∈R,a < b and for some constants Cσ,ασ, which depend solely on σ.\nThe boundedness of |σ(x)|is a standard assumption when proving approximation lower b ounds, and\nit is also used in Eldan and Shamir [3] for example. Since our p roof is based on a reduction to threshold\ncircuits, our technique fails if we consider certain activa tion functions that are highly oscillatory, since there\nare such pathological activations that can be used to comput e any Boolean function f:{0,1}d→{0,1},\neven with just a single neuron. In light of this, we also make a n assumption that the total variation of the\nactivation function is polynomially bounded on a compact do main. Such an assumption is very mild, and\nholds for essentially any activation function which is used in practice.\nHaving discussed our lower bound assumption, we now move on t o state and discuss our upper bound\nassumption.\nAssumption 2 (Upper bound) .There exists a constant cσwhich depends solely on σsuch that the following\nholds: For all R >0and anyL-Lipschitz function f: [−R,R]→R, and for any δ, there exist scalars\na,{αi,βi,γi}w\ni=1, wherew,|a|,|αi|,|βi|,|γi|≤cσRL\nδfor alli∈[w], such that the function\nh(x) =a+w/summationdisplay\ni=1αi·σ(βix−γi)\nsatisfies\nsup\nx∈[−R,R]|f(x)−h(x)|≤δ.\nThe above is a slight modification of Assumption 2 in Eldan and Shamir [3], and is satisfied by many\nstandard activation functions that are used in the literatu re, which in particular include threshold, ReLU and\nsigmoidal activations (see Lemma A.4in the appendix which implies that the threshold activation satisfies\nthis property, and see Appendix A in Eldan and Shamir [3] for a proof for the ReLU activation). We point out\nthat we also require an additional mild requirement in our as sumption, that the weights of the approximating\nnetworkhare bounded in magnitude. This is in order to control the magn itude of the weights in our\napproximation of fd, and get a valid separation that requires weights of polynom ials magnitude, which\nstands in contrast to our lower bounds, where polynomially b ounded weights imply that width exponential\nindis necessary.\n2.4 Main result\nWe are ready to present our main theorem in this paper:\nTheorem 2.1. Consider the sequence of distributions {U(A2d)}∞\nd=1, and the sequence of O(1)-Lipschitz\nfunctions fd:R2d→Rdefined in Eq. ( 1). Then for all C >0and sufficiently large d, we have for any\nactivation function σthat satisfies both Assumption 1and Assumption 2, that the following hold\n7•For any depth 2 σ-networkNd:Rd→R, with weights bounded in magnitude by C, we have\nE\nx,y∼U(Ad)/bracketleftBig\n(fd(x,y)−Nd(x,y))2/bracketrightBig\n>1\n400,\nunlessNdhas width at least Ω/parenleftBig\nexp(Ω(d))\npoly(C)/parenrightBig\n.\n•For allε >0, there exists a depth 3, width poly (d,1/ε),σ-networkN′\nd, such that\nsup\nx,y∈Ad/vextendsingle/vextendsinglefd(x,y)−N′\nd(x,y)/vextendsingle/vextendsingle≤ε,\nwhere the asymptotic notation hides constants that depend s olely onσ.\nThe above theorem is an immediate consequence of our lower bo und (Thm. 3.1) and upper bound\n(Thm. 4.1), which will be presented in detail in the following section s.\n3 Lower Bound\nIn this section, we present our lower bound for the approxima tion of the function fd. Thereafter, we provide\na proof sketch which conveys the main technical ideas behind the result. We begin with formally stating our\nlower bound as follows.\nTheorem 3.1. Suppose that σsatisfies Assumption 1, and letN:R2d→Rbe a depth 2 σ-network with\nweights bounded by Cthat satisfies\nEx,y∼U(Ad)/bracketleftBig\n(N(x,y)−fd(x,y))2/bracketrightBig\n≤1\n400.\nThen,Nhas width\nΩ/parenleftbiggexp(Ω(d))\npoly(C)/parenrightbigg\n,\nwhere the asymptotic notation hides constants that depend s olely onσ.\nThe proof of the above theorem relies on an average- to worst- case reduction in a neural network setting,\nfollowed by a reduction to threshold circuits. Given a netwo rk which approximates fdwell, we can use it\nto construct a network which achieves similar accuracy, but with margins that can be made arbitrarily more\nuniform due to a concentration of measure argument. The crux of our proof is identifying a construction that\nre-randomizes the input sufficiently well, effectively obt aining a strong enough concentration of measure,\nwhile also doing so in a manner which maintains the output val ue of the function. Thereafter, a very careful\ntechnical analysis is required to establish that the accura cy lost due to this re-randomization process is at\nmost a constant. We refer the reader to Subsection 3.1for a more detailed proof sketch of the theorem, and\nto Appendix A.1for the full proof.\nWe point out that the constant1\n400is arbitrary, and our proof technique is capable of improvin g this to be\narbitrarily close to the constant1\n16(at the cost of increasing the constants hidden in the asympt otic notation).\nSince a trivial approximation of a single constant neuron (w hich returns the value 0.5) yields accuracy1\n4, this\nindicates that our analysis is very tight, and that adding ev en exponentially many neurons does not improve\nupon the trivial approximation by much.\n8We remark that our weight boundedness assumption is mild, si nce our lower bound remains exponential\nind, as long as Cis in itself not exponential in d. In such a case, where the weights required for expressing\nfdmust have exponential magnitude, it is known that stable gra dient descent must run for exponentially\nmany iterations in order for the weights to reach such a magni tude (see Safran and Lee [18] for a more\nformal result of this kind). This suggests that even if a netw ork of size polynomial in dcan approximate\nfdwell, then in practice, learning such a representation usin g standard techniques is not tractable, and it is\ntherefore of lesser interest from a practical perspective.\n3.1 Techniques, and proof sketch of Thm. 3.1\nIn this subsection, we detail the key ideas behind the proof o f our lower bound. The reader is referred to\nAppendix A.1for the full proof.\n3.1.1 Step 1: From a continuous to a discrete distribution\nWe begin with assuming that we have a depth 2, σ-networkN, which approximates fdto accuracy1\n400,\nwhereσsatisfies Assumption 1, and our goal is to lower bound its width. By our assumption, t his network\nprovides a good approximation with respect to the continuou s distributionU(A2d). Since our aim here is\nto eventually use lower bounds from threshold circuits to ge t a lower bound on the width of N, we aim to\nreduce this lower bound over the continuous distribution to the discrete distribution U({0,1}2d). To this end,\nwe use a similar argument to the one used in Vardi et al. [24, Pr op. 6.1], who observe that the distribution\nU(A2d)can be decomposed into the sum of the two distributions U([0,0.25]2d)andU({0,0.75}2d). By\nusing Markov’s inequality on the randomness induced by the f ormer continuous component, we have that\nwith positive probability, we can find a depth 2 neural networ kN′, which has width similar to N; and a\ndiscrete sub-cube with side length 0.75, which is contained in A2d; such thatN′approximates IPdto an\naverage square loss of1\n399. By performing linear operations in the hidden layer of N′, which do not affect\nits size, we are able to modify it without changing its archit ecture, making it approximate IPdeffectively\nwith respect to the distribution U({0,1}2d).\n3.1.2 Step 2: From average- to worst-case using randomizati on\nIn the previous step, we constructed a depth 2, σ-networkN′, which approximates IPduniformly over\n{0,1}2d, to an average squared error of1\n399. Intuitively, this means that N′computes IPdwell in an average-\ncase sense, since by Markov’s inequality, for at least a cons tant fraction of the inputs x,y∈Ad, we have that\nround(N′(x,y)) = IP d(x,y). Our aim is now to use N′to construct a depth 2, σ-networkN′′, such that\nround(N′′(x,y)) = IP d(x,y)holds for all inputs x,y∈Ad. To this end, we use a randomization scheme\non the input, where we map it to a higher dimensional space, an d use a higher dimensional architecture of\nN′. The purpose of this scheme is to alter the input in a manner wh ich induces as much randomness as\npossible, but while also keeping its output unchanged. This is achieved by identifying the following three\ndifferent alterations:\n• Using the identity\nIPd(x,y) = IP d(x+x′,y+y′)+IPd(x′,y′)+IPd(x+x′,y′)+IPd(x′,y+y′) mod 2 ,\nwhich holds for all x,y,x′,y′∈{0,1}d, we can generate uniformly random binary vectors x′,y′∈\n{0,1}d, and replace xandywith the vectors (x+x′,x′,x+x′,x′)and(y+y′,y′,y′,y+y′),\n9respectively, while keeping the IP mod 2 output unchanged. T his additional randomness is useful\nfor handling cases where the inputs have a lot of structure (e .g., when x,yare the all-zero or all-one\nvectors).\n• We can pad our modified inputs with O(d)many uniformly generated bits, such that pairs where\nxi=yi= 1 are conditioned to be an even number. Due to this conditionin g, we have that the inner\nproduct mod 2 value is unchanged. This padding greatly incre ases the randomness of the input.\n• Lastly, since addition mod 2 is commutative, we can sample a random permutation, and apply it to our\nmodified input, once again altering our input while keeping i ts IP mod 2 value unchanged. This allows\nus to add valuable randomness to our input, since permuted ve ctors with an almost equal number of\npairs of the form xi=yi= 0,xi= 0,yi= 1,xi= 1,yi= 0,xi=yi= 1 are much closer to\nuniformly sampled vectors.\nCombining the above into a single randomization process, we have a new, higher dimensional input (but\nwith at most a linear blow-up), which has the same IP mod 2 valu e, yet whose probability distribution is\nclose (in a certain sense) to a uniformly random input from th e higher dimensional space. The bulk of the\ntechnical analysis in our proof consists of proving that thi s process results in a distribution close enough to\nuniform, so as to incur at most a constant additional loss in o ur approximation. This requires a very careful\nand tight analysis of the distribution.\nAfter constructing the architecture which performs the abo ve randomization process, we repeat the pro-\ncess polynomially many times, we concatenate the obtained h idden layers which perform this computation,\nand we use the output neuron to average their outputs. This re sults in a concentration of measure, which\nmakes the approximation error much more uniformly spread ov er the Boolean hypercube. By a union bound\nand the probabilistic method, we can find realizations of our random construction that achieve this, and mod-\nify our network to incorporate these realizations without c hanging the architecture of the network N′′. This\ncan be done since the operations of inverting a bit of the inpu t or permuting the inputs are linear operations\nthat can be absorbed in the weights of the hidden layer.\n3.1.3 Step 3: Constructing a threshold circuit computing IPd\nFollowing the previous two steps, we now have a depth 2, σ-networkN′′, such that round(N′′(x,y)) =\nIPd(x,y)for allx,y∈{0,1}d. The final step in our proof is to use this architecture to cons truct a depth 2\nthreshold circuit with the same property. To this end, we firs t need to approximate σto arbitrary accuracy\nusing a depth 2 threshold network on a compact domain. This ca n be done by constructing a piecewise\nlinear approximation as follows: Beginning with the leftmo st point in the domain of approximation, we\nchoose a constant function which coalesces with σat this point, and we extend it until its distance from σ\ndeviates from our target accuracy, in which case we make a ‘ju mp’ to a different constant value, continuing\nin this manner until the approximation is complete. Since ea ch such jump increases the variation of the\nconstructed approximation, and since σis of bounded variation due to Assumption 1, we have that we\ncannot perform too many such jumps; namely, σcan be approximated by a piecewise linear function with\nnot too many discontinuities. Replacing each σwith a moderately sized depth 2 threshold network, we\nobtain a moderately sized depth 2 threshold network N′′′that satisfies round(N′′′(x,y)) = IP d(x,y).\nLastly, we turn this threshold network into a threshold circ uit by adding a threshold activation on the output\nneuron. It is a classic result in circuit complexity that IPdcannot be computed by a depth 2 threshold circuit\nwith bounded weights, unless its width is exponential in d[4]. This implies a lower bound on the width of\nN, from which the theorem follows.\n104 Upper Bound\nHaving presented our lower bound, in this section, we turn to complement it with an upper bound for depth 3\nnetworks. We also provide a concrete example in the case wher eσis the ReLU activation function, in which\nour required width and magnitude of the weights provides a st ronger result than the bounds guaranteed in\nour theorem.\nWe now formally state our upper bound result below.\nTheorem 4.1. Letε >0, and suppose that σsatisfies Assumption 2. Then, there exists a depth 3, width\nO/parenleftBig\nd2\nε/parenrightBig\nσ-networkN, with weights bounded in magnitude by O/parenleftbigd\nε2/parenrightbig\n, such that\nsup\n(x,y)∈A2d|N(x,y)−fd(x,y)|≤ε.\nThe proof of the theorem, which appears in Appendix A.2, utilizes the fact that fdcan be approximated\nefficiently by composing two different simple functions. Si nce a depth 3 network has two hidden layers\nwith non-linear σactivations, we are able to use each hidden layer to compute e ach function, and obtain the\ndesired approximation.\nWe remark that we provide our positive approximation result in terms of the L∞norm rather than\ntheL2norm with respect to the uniform distribution supported on A2das our lower bound does. Since\nL∞approximation is more stringent than L2, this provides a more general result which implies an L2\napproximation of fd(·,·)with respect to anydistribution supported on A2d.\nTo give a more concrete example of an approximation obtained by our theorem, below we specify how\nthis construction can be done when σis the ReLU activation.\nExample 4.2. Let[x]+:= max{0,x}denote the ReLU activation function, and define\ntd(x):= [z]++d/summationdisplay\ni=12·(−1)i[z−i]+.\nThen, the network given by\nN(x,y):=td/parenleftBiggd/summationdisplay\ni=1[4xi+4yi−5]+−[4xi+4yi−6]+/parenrightBigg\nsatisfies\nsup\nx,y∈Ad|N(x,y)−fd(x,y)|= 0.\nIt is straightforward to verify that Nis a depth 3, width 2dReLU network, with weights bounded in\nmagnitude byO(d). Moreover, a simple computation shows that Ncoalesces with fdonA2d. Lastly, this\napproximation is also sparse, in the sense that it requires o nlyO(d)neuron connections (see Fig. 1for a\nvisualization of the functions computed in each layer).\nFollowing the works [11, 12], which show that some functions that were used to prove approximation\nlower bounds for neural networks cannot be learned efficient ly using standard methods, Safran and Lee [18]\nhave shown an optimization-based separation result where t he deeper architecture can provably learn the\nefficient representation from finite data, using standard te chniques such as gradient descent. It is interesting\nto note that Example 4.2provides a simple, linear in size and sparse approximation o ffd. This simplicity\n1100.20.40.60.8100.51\n00.51\n(a)0 0.5 1 1 .5 2 2 .5 3 3 .5 400.51\n(b)\nFigure 1: Computing fdusing a depth 3 ReLU network. Subfigure 1aplots the function (x,y)/ma√sto→\n[4x+4y−5]+−[4x+4y−6]+, which equals AND(round( x),round(y))for allx,y∈A1. Subfig-\nure1bplots the function x/ma√sto→td(x), defined in Example 4.2. When composing the latter with a sum of\nthe former, iterating over all pairs of coordinates xi,yi, we obtain a function that coalesces with fdonA2d.\nBest viewed in color.\nis much desired, since it may suggest that similarly to Safra n and Lee [18], learning this representation\nfrom finite data using a standard learning algorithm is tract able, and despite the simplicity of this ReLU\napproximation, our lower bound for this function provides a separation which is stronger than previously\nknown results. We leave the study of proving such a stronger o ptimization-based separation result as an\nintriguing future work direction.\nAcknowledgements.\nPaul Valiant is partially supported by NSF award CCF-212780 6. We thank Srikanth Srinivasan for useful\ndiscussions.\nReferences\n[1] George Cybenko. Approximation by superpositions of a si gmoidal function. Mathematics of control,\nsignals and systems , 2(4):303–314, 1989.\n[2] Amit Daniely. Depth separation for neural networks. In Conference on Learning Theory , pages 690–\n696, 2017.\n[3] Ronen Eldan and Ohad Shamir. The power of depth for feedfo rward neural networks. In Conference\non learning theory , pages 907–940. PMLR, 2016.\n[4] Andr´ as Hajnal, Wolfgang Maass, Pavel Pudl´ ak, Mario Sz egedy, and Gy¨ orgy Tur´ an. Threshold circuits\nof bounded depth. Journal of Computer and System Sciences , 46(2):129–154, 1993.\n[5] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. De ep residual learning for image recog-\nnition. In Proceedings of the IEEE conference on computer vision and pa ttern recognition , pages\n770–778, 2016.\n12[6] Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Mu ltilayer feedforward networks are univer-\nsal approximators. Neural networks , 2(5):359–366, 1989.\n[7] Daniel Hsu, Clayton H Sanford, Rocco Servedio, and Emman ouil Vasileios Vlatakis-Gkaragkounis.\nOn the approximation power of two-layer networks of random r elus. In Conference on Learning\nTheory , pages 2423–2461. PMLR, 2021.\n[8] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep lea rning. nature , 521(7553):436–444, 2015.\n[9] Moshe Leshno, Vladimir Ya Lin, Allan Pinkus, and Shimon S chocken. Multilayer feedforward net-\nworks with a nonpolynomial activation function can approxi mate any function. Neural networks , 6(6):\n861–867, 1993.\n[10] Shiyu Liang and Rayadurgam Srikant. Why deep neural net works for function approximation? arXiv\npreprint arXiv:1610.04161 , 2016.\n[11] Eran Malach and Shai Shalev-Shwartz. Is deeper better o nly when shallow is good? Advances in\nNeural Information Processing Systems , 32, 2019.\n[12] Eran Malach, Gilad Yehudai, Shai Shalev-Schwartz, and Ohad Shamir. The connection between ap-\nproximation, depth separation and learnability in neural n etworks. In Conference on Learning Theory ,\npages 3265–3295. PMLR, 2021.\n[13] James Martens, Arkadev Chattopadhya, Toni Pitassi, an d Richard Zemel. On the representational\nefficiency of restricted boltzmann machines. Advances in Neural Information Processing Systems , 26,\n2013.\n[14] Anirbit Mukherjee and Amitabh Basu. Lower bounds over b oolean inputs for deep neural networks\nwith relu gates. arXiv preprint arXiv:1711.03073 , 2017.\n[15] Eshaan Nichani, Alex Damian, and Jason D Lee. Provable g uarantees for nonlinear feature learning in\nthree-layer neural networks. arXiv preprint arXiv:2305.06986 , 2023.\n[16] Tomaso Poggio, Hrushikesh Mhaskar, Lorenzo Rosasco, B rando Miranda, and Qianli Liao. Why and\nwhen can deep-but not shallow-networks avoid the curse of di mensionality: a review. International\nJournal of Automation and Computing , 14(5):503–519, 2017.\n[17] Yunwei Ren, Mo Zhou, and Rong Ge. Depth separation with m ultilayer mean-field networks. arXiv\npreprint arXiv:2304.01063 , 2023.\n[18] Itay Safran and Jason Lee. Optimization-based separat ions for neural networks. In Conference on\nLearning Theory , pages 3–64. PMLR, 2022.\n[19] Itay Safran and Ohad Shamir. Depth-width tradeoffs in a pproximating natural functions with neural\nnetworks. In International conference on machine learning , pages 2979–2987. PMLR, 2017.\n[20] Itay Safran, Ronen Eldan, and Ohad Shamir. Depth separa tions in neural networks: what is actually\nbeing separated? In Conference on Learning Theory , pages 2664–2666. PMLR, 2019.\n[21] Itay Safran, Daniel Reichman, and Paul Valiant. How man y neurons does it take to approximate\nthe maximum? In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discre te Algorithms\n(SODA) , pages 3156–3183. SIAM, 2024.\n13[22] Matus Telgarsky. Benefits of depth in neural networks. I nConference on learning theory , pages\n1517–1539. PMLR, 2016.\n[23] Gal Vardi and Ohad Shamir. Neural networks with small we ights and depth-separation barriers. arXiv\npreprint arXiv:2006.00625 , 2020.\n[24] Gal Vardi, Daniel Reichman, Toniann Pitassi, and Ohad S hamir. Size and depth separation in approxi-\nmating benign functions with neural networks. In Conference on Learning Theory , pages 4195–4223.\nPMLR, 2021.\n[25] Luca Venturi, Samy Jelassi, Tristan Ozuch, and Joan Bru na. Depth separation beyond radial functions.\nThe Journal of Machine Learning Research , 23(1):5309–5364, 2022.\n[26] Dmitry Yarotsky. Error bounds for approximations with deep relu networks. Neural Networks , 94:\n103–114, 2017.\nA Proofs\nA.1 Proof of Thm. 3.1\nBefore we can prove the theorem, we will first state and prove s everal auxiliary lemmas that will be used\nlater on. In what follows, we use the notation /⌊ard⌊lD/⌊ard⌊l2to denote the L2norm of a discrete distribution D.\nNamely, for a random variable X∼D sampled from a sample space X, we have\n/⌊ard⌊lD/⌊ard⌊l2:=/summationdisplay\nx∈X/parenleftbigg\nP\nx∼D[X=x]/parenrightbigg2\n.\nThe following lemma provides an upper bound on certain subse ts of theL2norm of the distribution we\nanalyze in our reduction scheme.\nLemma A.1. Ford,D positive integers divisible by 4, and given d1,d2,d3,d4≥0withd1+d2+d3+d4=d\nthen we have:\n/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbigD\nD1,D2,D3,D4/parenrightbig2\n/parenleftbigD+d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig≤e4\nD/summationtext4\ni=1(di−d\n4)2·/parenleftbigg\n1+d\nD/parenrightbigg3/2\n·4D−d(2)\nProof. We start by comparing(Di+di)!\nDi!to(D/4+di)!\nD/4!. ForDi≥D/4the ratio of these two expressions equals\n/producttextDi\nj=D/4+1j+di\nj≤/producttextDi\nj=D/4+1D/4+di\nD/4= (1 +4di\nD)Di−D/4. On the other hand, for Di< D/4that ratio\nof our two expressions equals/producttextD/4\nj=Di+1j\nj+di≤/producttextD/4\nj=Di+1D/4\nD/4+di= (1 +4di\nD)Di−D/4. Thus in all cases,\n(Di+di)!\nDi!≤(1+4di\nD)Di−D/4(D/4+di)!\nD/4!. We apply this inequality to bound the terms on the left hand s ide of\nEq. ( 2):\n/parenleftbigD\nD1,D2,D3,D4/parenrightbig\n/parenleftbigD+d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig=D!\n(D+d)!(D1+d1)!\nD1!(D2+d2)!\nD2!(D3+d3)!\nD3!(D4+d4)!\nD4!(3)\n≤/parenleftbigD\nD/4,D/4,D/4,D/4/parenrightbig\n/parenleftbigD+d\nD/4+d1,D/4+d2,D/4+d3,D/4+d4/parenrightbig4/productdisplay\ni=1(1+4di\nD)Di−D/4\n14We will evaluate the sum in Equation 2and using the identity\n(p1+p2+p3+p4)D=/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbiggD\nD1,D2,D3,D4/parenrightbigg\npD1\n1pD2\n2pD3\n3pD4\n4\nalong with the above equation we can write,\n/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbigD\nD1,D2,D3,D4/parenrightbig2\n/parenleftbigD+d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig\n=/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbiggD\nD1,D2,D3,D4/parenrightbigg/parenleftbigD\nD1,D2,D3,D4/parenrightbig\n/parenleftbigD+d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig\n≤/parenleftbigD\nD/4,D/4,D/4,D/4/parenrightbig\n/parenleftbigD+d\nD/4+d1,D/4+d2,D/4+d3,D/4+d4/parenrightbig/parenleftBigg4/productdisplay\ni=1(1+4di\nD)−D/4/parenrightBigg/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbiggD\nD1,D2,D3,D4/parenrightbigg4/productdisplay\ni=1(1+4di\nD)Di\n=/parenleftbigD\nD/4,D/4,D/4,D/4/parenrightbig\n/parenleftbigD+d\nD/4+d1,D/4+d2,D/4+d3,D/4+d4/parenrightbig/parenleftBigg4/productdisplay\ni=1(1+4di\nD)−D/4/parenrightBigg\n(4+4/summationtext\nidi\nD)D\nSince the second derivative of the function log(x)is≥−1for inputs x≥1, we use the Lagrange\nremainder form of the Taylor expansion around any ℓ≥1to lower bound the logarithm function for all\nx≥1aslog(x)≥log(ℓ)+1\nℓ(x−ℓ)−1\n2(x−ℓ)2. We use this to bound the last two terms above, bounding\nlog(x)forx= 1+4di\nD, centering our approximation at ℓ= 1+d\nD, and using the fact that/summationtext\nidi=d:\n/parenleftBigg4/productdisplay\ni=1(1+4di\nD)−D/4/parenrightBigg\n(4+4/summationtext\nidi\nD)D≤\n4/productdisplay\ni=1/parenleftBigg\nelog(1+d\nD)+4di−d\nD(1+d\nD)−1\n2(4di−d\nD)2/parenrightBigg−D/4\n(4+4/summationtext\nidi\nD)D\n= 4De/summationtext4\ni=12(di−d\n4)2\nD\nWe then turn to the first term, which we reexpress as\n/parenleftbigD\nD/4,D/4,D/4,D/4/parenrightbig\n/parenleftbigD+d\nD/4+d1,D/4+d2,D/4+d3,D/4+d4/parenrightbig=D!\n(D+d)!(D/4+d1)!\n(D/4)!(D/4+d2)!\n(D/4)!(D/4+d3)!\n(D/4)!(D/4+d4)!\n(D/4)!\nWe bound (D/4+di)!by reexpressing it via the Gamma function as Γ(1+D/4+di). We then use the\nLagrange remainder form of the Taylor expansion of the funct ionf(x) := logΓ( x)aroundℓ= 1+D\n4+d\n4\nto conclude that there exists ybetweenℓandxsuch that\nf(x) =f(ℓ)+(x−ℓ)f′(ℓ)+1\n2f′′(y)(x−ℓ)2\nThe second derivative of the logΓ function at xis bounded byx+1\nx2≤x+1\nx2−1=1\nx−1. Thus, letting\nxi= 1+D/4+di, we have, that there exist yibetweenxiandℓfor which\n4/productdisplay\ni=1(D/4+di)!≤4/productdisplay\ni=1ef(ℓ)+(xi−ℓ)f′(ℓ)+1\n2f′′(yi)(xi−ℓ)2\n= (D/4+d/4)!4e1\n2/summationtext4\ni=1f′′(yi)(di−d\n4)2\n≤(D/4+d/4)!4e2\nD/summationtext4\ni=1(di−d\n4)2\n15where the equality makes use of the fact that the middle (line ar inxi) term vanishes since/summationtext4\ni=1(xi−ℓ) = 0 ;\nthe last inequality makes use of the fact that both xi,ℓ≥1+D/4, and thus, since yiis between these, we\nhaveyi≥1+D/4, and hence our bound on the second derivative of logΓ(y)is at most4\nD.\nThus, overall, we have shown\n/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbigD\nD1,D2,D3,D4/parenrightbig2\n/parenleftbigD+d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig≤4De/summationtext4\ni=12(di−d\n4)2\nDD!(D/4+d/4)!4\n(D+d)!(D/4)!4e2\nD/summationtext4\ni=1(di−d\n4)2\n= 4DD!(D/4+d/4)!4\n(D+d)!(D/4)!4e4\nD/summationtext4\ni=1(di−d\n4)2\nFinally, we bound the ratio of factorials as follows. Define t he function fon integer input f(j):=\n4−4j/parenleftbig4j\nj,j,j,j/parenrightbig\n= 4−4j(4j)!\nj!4. We will show that for any integers k≥jwe havef(j)\nf(k)≤(j/k)3/2. We have\nf(j)\nf(j+1)=(4j+4)3\n(4j+1)(4j+2)(4j+3). Expressing each term in the denominator as a weighted avera ge of4jand\n4j+4and applying the (weighted) AM-GM inequality to each term on the denominator separately, we thus\nlower bound the denominator by (4j)3/2(4j+4)3/2and thus upper boundf(j)\nf(j+1)≤(4j+4)3/2\n(4j)3/2= (j+1\nj)3/2.\nMultiplying this bound for all numbers between jandkyieldsf(j)\nf(k)≤(j/k)3/2as claimed.\nThusD!(D/4+d/4)!4\n(D+d)!(D/4)!4≤/parenleftbigD+d\nD/parenrightbig3/2. And we have derived Eq. ( 2) as claimed.\nThe following lemma bounds the moment generating function o f a certain distribution, which arises in\nour analysis of the reduction scheme.\nLemma A.2. For any dimension dand any vectors x,y∈{0,1}d, consider the process of picking random\nvectorsx′,y′←{0,1}dand outputting\nA:= (x+x′,x′,x+x′,x′),\nB:= (y+y′,y′,y′,y+y′),\nwhere all additions are mod 2. Among these 4dentries, let d1count the number of indices j∈{1,...,4d}\nwhereAj=Bj= 0; letd2count the entries where Aj= 0,Bj= 1; letd3count the entries where\nAj= 1,Bj= 0; and letd4count the entries where Aj=Bj= 1. Then\nE\nd1,d2,d3,d4/bracketleftBig\nes/summationtext4\ni=1(di−d)2/bracketrightBig\n≤/parenleftbigg1\n1−24ds/parenrightbigg2\nProof. Our overall analysis technique will be to first compute the mo ment generating function (MGF) of the\ndistribution of (d1,d2,d3,d4), centered at its mean, (d,d,d,d). Explicitly, since we have a 4-dimensional\ndistribution, the MGF has 4 parameters:\nM(t1,t2,t3,t4):=E\nd1,d2,d3,d4/bracketleftBig\ne/summationtext4\ni=1ti(di−d)/bracketrightBig\nCrucially, the MGF of the sum of independent random vectors m ultiplies; thus we will compute the\nMGF for the contribution to (d1,d2,d3,d4)from each location Aj,Bjseparately.\nAfter we have computed the MGF, we will use this to compute the quantity in the lemma statement,\nEd1,d2,d3,d4/bracketleftBig\nes/summationtext4\ni=1(di−d)2/bracketrightBig\n. Since the MGF of the Gaussian N(0,2s)equalsest2, the MGF for the cor-\nresponding 4-dimensional Gaussian, after a slight change o f variables, yields the relation that, for any\nd1,d2,d3,d4:\nE\n(t1,t2,t3,t4)←N(0,2s)[e/summationtext4\ni=1ti(di−d)] =es/summationtext4\ni=1(di−d)2\n16and thus that\nE\n(t1,t2,t3,t4)←N(0,2s)[M(t1,t2,t3,t4)] =E\nd1,d2,d3,d4[es/summationtext4\ni=1(di−d)2] (4)\nConsider, for some index j, the 4 cases for the pair xj,yj, and the 4-tuple of pairs of entries (xj+\nx′\nj,yj+y′\nj),(x′\nj,y′\nj),(xj+x′\nj,y′\nj),(x′\nj,yj+y′\nj)they induce in A,B .\nIfxj= 0,yj= 0, then the 4-tuple of pairs (xj+x′\nj,yj+y′\nj),(x′\nj,y′\nj),(xj+x′\nj,y′\nj),(x′\nj,yj+\ny′\nj)produced in the vectors A,B contains simply 4 copies of the randomly chosen pair of bits (x′\nj,y′\nj).\nThus the contribution to the counts d1,d2,d3,d4will be uniformly randomly chosen among the 4-tuples\n(4,0,0,0),(0,4,0,0),(0,0,4,0),(0,0,0,4). The moment generating function of this uniform distributi on\nover 4 possibilities is just the sum of the 4 terms in the defini tion of the MGF; as usual, we center the MGF\nof this portion of the distribution at its mean, which here is (1,1,1,1):\nMj(t1,t2,t3,t4) =1\n4e(4−1)t1+(0−1)t2+(0−1)t3+(0−1)t4+1\n4e(0−1)t1+(4−1)t2+(0−1)t3+(0−1)t4\n+1\n4e(0−1)t1+(0−1)t2+(4−1)t3+(0−1)t4+1\n4e(0−1)t1+(0−1)t2+(0−1)t3+(4−1)t4\nWe can loosely bound this by e6/summationtext4\ni=1t2\nias follows. Each of the 4 exponential terms is the exponentia l\nof the inner product of (t1,t2,t3,t4)with a vector vof length√\n12. Thus, for any unit 4-vector u, we\nhave that the exponential e/an}bracketle{t(t1,t2,t3,t4),v/an}bracketri}hthas second derivative in direction uthat is at most 12times the\nvalue of the exponential e/an}bracketle{t(t1,t2,t3,t4),v/an}bracketri}ht. Since this property—of having directional second derivat ive in\ndirection uthat is at most 12 times the function value—is preserved unde r scaling a function by a positive\nconstant, and is preserved by function addition, we conclud e that this property applies to the entire function\nMj(t1,t2,t3,t4). Namely, for any vector u, the second derivative of Mj(t1,t2,t3,t4)in the direction uis at\nmost12Mj(t1,t2,t3,t4). SinceMj(t1,t2,t3,t4)has value 1 at the origin, and 0 gradient, we can bound its\nvalue at any multiple xuby solving the differential equation f(0) = 1,f′(0) = 0,f′′(x)≤12f(x), to yield\nf(x)≤cosh(√\n12x)≤e(12/2)x2. Since this bounds M(t1,t2,t3,t4)when(t1,t2,t3,t4) =xufor any unit\nvectoru, we conclude that M(t1,t2,t3,t4)≤e6/summationtext4\ni=1t2\ni.\nMoving on to the next case, if xj= 0,yj= 1case, then the 4-tuple of pairs (xj+x′\nj,yj+y′\nj),(x′\nj,y′\nj),(xj+\nx′\nj,y′\nj),(x′\nj,yj+y′\nj)contains 2 copies of (x′\nj,y′\nj)and 2 copies of (x′\nj,1 +y′\nj). Thus the contribution to\nthe counts d1,d2,d3,d4will be uniformly chosen among the 4-tuples (2,2,0,0),(0,0,2,2), where there\nare only 2 possibilities in this case. The moment generating function of this uniform distribution over 2\npossibilities is just the sum of the 2 terms in the definition o f the MGF, which as usual we center at the\nmean,(1,1,1,1):\nMj(t1,t2,t3,t4) =1\n2e(2−1)t1+(2−1)t2+(0−1)t3+(0−1)t4+1\n2e(0−1)t1+(0−1)t2+(2−1)t3+(2−1)t4\nThis equals cosh(/an}⌊ra⌋ketle{t(t1,t2,t3,t4),(1,1,−1,−1)/an}⌊ra⌋ketri}ht)≤e/an}bracketle{t(t1,t2,t3,t4),(1,1,−1,−1)/an}bracketri}ht2/2≤e(||(1,1,−1,−1)||2\n2/2)/summationtext4\ni=1t2\ni,\nnamelye2/summationtext4\ni=1t2\ni.\nThe next case, if xj= 1,yj= 0 , is analogous to the previous case. Here the 4-tuple of pairs\n(xj+x′\nj,yj+y′\nj),(x′\nj,y′\nj),(xj+x′\nj,y′\nj),(x′\nj,yj+y′\nj)contains 2 copies of (x′\nj,y′\nj)and 2 copies of\n(1+x′\nj,y′\nj). Thus the contribution to the counts d1,d2,d3,d4will be uniformly chosen among the 4-tuples\n(2,0,2,0),(0,2,0,2). The moment generating function of this uniform distributi on over 2 possibilities is\njust the sum of the 2 terms in the definition of the MGF, which as usual we center at the mean, (1,1,1,1):\nMj(t1,t2,t3,t4) =1\n2e(2−1)t1+(0−1)t2+(2−1)t3+(0−1)t4+1\n2e(0−1)t1+(2−1)t2+(0−1)t3+(2−1)t4\n17Analogously to the previous case, this is at most e2/summationtext4\ni=1t2\ni.\nFinally, in the case xj= 1,yj= 1, then the 4-tuple of pairs (xj+x′\nj,yj+y′\nj),(x′\nj,y′\nj),(xj+\nx′\nj,y′\nj),(x′\nj,yj+y′\nj)equals(1 +x′\nj,1 +y′\nj),(x′\nj,y′\nj),(1 +x′\nj,y′\nj),(x′\nj,1 +y′\nj)and thus always con-\ntributes(1,1,1,1)to the 4 counts. Thus the moment generating function centere d at(1,1,1,1)is, trivially,\nMj(t1,t2,t3,t4) = 1 .\nPutting together the pieces: since the moment generating fu nction is simply the product of its contribu-\ntion from each index jfrom 1 to d, and in all cases the we bounded this contribution by e6/summationtext4\ni=1t2\ni, we have\nthat the overall MGF of all dindices is bounded as\nM(t1,t2,t3,t4)≤e6d/summationtext4\ni=1t2\ni\nWe now bound the desired quantity in this lemma via Eq. ( 4). We thus have\nE\n(t1,t2,t3,t4)←N(0,2s)[M(t1,t2,t3,t4)]≤ E\n(t1,t2,t3,t4)←N(0,2s)[e6d/summationtext4\ni=1t2\ni]\n=/integraldisplay\nR41\n√\n2π·2s4e−1\n4s/summationtext4\ni=1t2\nie6d/summationtext4\ni=1t2\nidt1dt2dt3dt4\n=/radicalBigg\n1\n1/(4s)−6d\n2·2s4\n=/parenleftbigg1\n1−24ds/parenrightbigg2\nas desired.\nThe following proposition establishes an L2upper bound on the distribution induced by the randomiza-\ntion process in our reduction.\nProposition A.3. For any dimension dand any vectors x,y∈{0,1}d, letD= 100d. Pick random vectors\nx′,y′←{0,1}d; pick random vectors x′′,y′′←{0,1}Dsuch that there are an even number of indices\nj∈{1,...,D}wherex′′\nj=y′′\nj= 1; and pick a random permutation τof4d+Delements. The claim is\nthat the pair of length 4d+Dvectors\nX:=τ(x+x′,x′,x+x′,x′,x′′),\nY:=τ(y+y′,y′,y′,y+y′,y′′),\nconsidered as a distribution Dover domain{0,1}2(4d+D), hasL2norm at most 8·2−(4d+D), which is 8\ntimes the L2norm of the uniform distribution over this domain.\nProof. Among the 4dentries of the pair A= (x+x′,x′,x+x′,x′),B= (y+y′,y′,y′,y+y′)let\nd1count the number of indices where Aj=Bj= 0; letd2count the entries where Aj= 0,Bj= 1;\nletd3count the entries where Aj= 1,Bj= 0; and letd4count the entries where Aj=Bj= 1. Thus\nd1+d2+d3+d4= 4d.\nSinceD=/summationtext\nd1,d2,d3,d4PrD[d1,d2,d3,d4]D|d1,d2,d3,d4, by the triangle inequality we have\n||D||2≤/summationdisplay\nd1,d2,d3,d4Pr\nD[d1,d2,d3,d4]||D|d1,d2,d3,d4||2 (5)\nFor the random vectors x′′,y′′, letD1,D2,D3,D4count the number of indices j∈{1,...,D}respec-\ntively where x′′\nj= 0,y′′\nj= 0; wherex′′\nj= 0,y′′\nj= 1; wherex′′\nj= 1,y′′\nj= 0; and where x′′\nj= 1,y′′\nj= 1.\nThusD1+D2+D3+D4=D.\n18Thus there will be D1+d1total indices where Xj= 0,Yj= 0, etc. Given these total counts D1+\nd1,D2+d2,D3+d3,D4+d4, the total number of rearrangements of these columns is the m ultinomial/parenleftbigD+4d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig\n; and the random permutation τwill choose a uniformly random one of these\narrangements.\nThe probability of the random vectors x′′,y′′having counts exactly D1,D2,D3,D4would be exactly\n4−D/parenleftbigD\nD1,D2,D3,D4/parenrightbig\nif we were not conditioning on D4being even. The probability of D4being even is≥1\n2,\nsince the difference between the number of instantiations w hereD4is even versus D4is odd can be exactly\ncomputed as/summationtext\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftbigD\nD1,D2,D3,D4/parenrightbig\n(−1)D4= (1+1+1−1)D≥0. Thus, by the law of\nconditional probability, we have\n4−D/parenleftbiggD\nD1,D2,D3,D4/parenrightbigg\n=P[D1,D2,D3,D4]\n=P[D1,D2,D3,D4|D4is even]P[D4is even]+P[D1,D2,D3,D4|D4is odd]P[D4is odd]\n≥1\n2P[D1,D2,D3,D4|D4is even],\ntherefore the probability of D1,D2,D3,D4with even D4is≤2·4−D/parenleftbigD\nD1,D2,D3,D4/parenrightbig\n.\nThus, overall, the random process D|d1,d2,d3,d4can be described as picking D1,D2,D3,D4with proba-\nbility≤2·4−D/parenleftbigD\nD1,D2,D3,D4/parenrightbig\n, and then uniformly splitting this probability among the/parenleftbigD+4d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig\npossible rearrangements of these columns. We point out that , for a distribution where, for different indices\nj, we have pjprobability mass uniformly divided among njelements, its squared L2norm is/summationtext\njnj(pj\nnj)2=\n/summationtext\njp2\nj\nnj. Thus bound the L2norm of our conditional distribution as\n||D|d1,d2,d3,d4||2≤/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\n(D1,D2,D3,D4):/summationtext\niDi=D/parenleftBig\n2·4−D/parenleftbigD\nD1,D2,D3,D4/parenrightbig/parenrightBig2\n/parenleftbigD+4d\nD1+d1,D2+d2,D3+d3,D4+d4/parenrightbig\nThis expression is exactly 2·4−Dtimes the square root of the expression bounded in Lemma A.1, if in\nLemma A.1we reparameterize das4d. Namely,\n||D|d1,d2,d3,d4||2≤2·e2\nD/summationtext4\ni=1(di−d)2·/parenleftbigg\n1+4d\nD/parenrightbigg3/4\n·2−(4d+D)\nCombining this bound with Eq. ( 5) (the triangle inequality), we have\n||D||2≤E\nd1,d2,d3,d4[||D|d1,d2,d3,d4||2]≤2/parenleftbigg\n1+4d\nD/parenrightbigg3/4\n·2−(4d+D)E\nd1,d2,d3,d4[e2\nD/summationtext4\ni=1(di−d)2]\nThe right hand side is exactly the form of Lemma A.2, applied with s=2\nD, thus yielding our overall\nbound of\n||D||2≤2/parenleftbigg\n1+4d\nD/parenrightbigg3/4\n·2−(4d+D)/parenleftBigg\n1\n1−48d\nD/parenrightBigg2\nForD≥100d, we see that||D||2≤8·2−(4d+D), as desired.\nThe following lemma guarantees that activation functions w hich satisfy Assumption 1, can be simulated\nto arbitrary accuracy, using depth 2 threshold networks of p olynomial width.\n19Lemma A.4. Letσbe an activation function that satisfies Assumption 1. Then for all δ >0, there exists a\ndepth 2 threshold network Nof width at mostpoly(R)\nδand weights of magnitude at mostpoly(R)\nδ, such that\nsup\nx∈[−R,R]|N(x)−σ(x)|≤δ.\nProof. We will first construct a piecewise constant function that ap proximates σto anL∞distance of δ, and\nthen we will compute this piecewise constant function preci sely using a depth 2 threshold network.\nLetx1:=−Randy1:=σ(x1). We define the set\nA1:={a∈[−R,R] :∀x≤a,|σ(x)−y1|≤δ}.\nNote that A1/ne}ationslash=∅sincex1∈A1, and define x′\n1=x1andx2:= supA1. We now split our analysis into\nthree cases, depending on the continuity properties of σatx2.\n1. Suppose that σis continuous at x2from the right. By the definition of A1we have that x1< x2,\nsince ifx1=x2by contradiction, due to right continuity, we can find an inte rval[x1,b)forb > x1\nclose enough to x1, such that|σ(x)−y1|≤δfor allx∈[x1,b), which contradict the definition of\nA1. We define y2:=σ(x2), and we have by the definition of A1thatsupx∈[x1,x2)|σ(x)−y1|≤δ.\nMoreover, if σis continuous at x2from both sides then this implies that |y2−y1|=δ, and if it is only\ncontinuous from the right this implies that |y2−y1|≥δ. This can be seen to hold true since if we\nhad|y2−y1|< δby contradiction, then from right continuity there exists a n interval [x2,b)for some\nb > x2such that|σ(x)−y1|< δfor allx∈[x2,b), which contradicts the definition of A1. We can\nnow define the set\nA2:={a∈[x2,R] :∀x∈[x2,a],|σ(x)−y2|≤δ},\nand note that A2/ne}ationslash=∅sincex2∈A2. Definex′\n2=x2, we can conclude that in this case, we have an\ninterval[x1,x2)such that|σ(x)−y1|≤δfor allx∈[x1,x2), that|σ(x2)−y2|= 0≤δ, and that\n|σ(x′\n1)−σ(x′\n2)|≥δ.\n2. Suppose that σis not continuous at x2from the right, but it is either continuous at x2from the left or\nx1=x2, which in both cases implies that |σ(x2)−y1|≤δ. It must also hold that |σ(x)−y1|≤δ\nfor allx∈[x1,x2]. Definey2:= limx→x2+σ(x), note that by the definition of A1andy2, we have\nsimilarly to the previous case that |y2−y1|≥δ, and letx′\n2be close enough to x2from the right so\nthat from right continuity we get |y2−σ(x′\n2)|≤0.25δ, and thus|σ(x′\n1)−σ(x′\n2)|≥0.75δ. We can\nnow define the set\nA2:={a∈(x2,R] :∀x∈(x2,a],|σ(x)−y2|≤δ},\nwhich is not empty since (x2,x′\n2]⊆A2.\n3. Suppose that x1< x2and thatσhas a discontinuity at x2from both sides, and note that together\nwith the previous two items, this covers all possibilities. Then, since σis of bounded variation, we\nhave that both limits at the two sides exist and are finite. Nex t, ifx2∈A1, then this implies that\n|σ(x2)−y1|≤δ. We can now define y2:= limx→x2+σ(x)and proceed in the same manner as in\nItem 2to define A2andx′\n2. Otherwise, we have that x2/∈A1, which implies that |σ(x2)−y1|> δ.\nWe define y2:=σ(x2)which trivially implies |σ(x2)−y2|= 0≤δ, and proceed to define x′\n2and\nA2as in Item 1.\nDefinex3:= supA2, we can continue in this manner and define sequences of target valuesy1,y2,... and\npointsx1,x2,...,x′\n1,x′\n2,...such that for all i, we have\n20•|σ(x)−yi|≤δfor allx∈(xi,xi+1).\n•|σ(xi)−yi−1|≤δor|σ(xi)−yi|≤δ.\n•|σ(xi)−σ(xi+1)|≥δand|σ(xi)−σ(x′\ni)|≤0.25δ, which imply|σ(x′\ni)−σ(x′\ni+1)|≥0.5δ.\nWe now bound the length nof the sequences required to get R∈An; namely, the number of piecewise\nconstant segments required to approximate σto accuracy δuniformly on [−R,R]. We have by Assumption 1\nthatσhas total variation at most Cσ(1+2R)ασ. On the other hand, we have from the above properties that\nthe partition x′\n1,x′\n2,...,x′\nnof[−R,R]has total variation at least\nn−1/summationdisplay\ni=1/vextendsingle/vextendsingleσ(x′\ni+1)−σ(x′\ni)/vextendsingle/vextendsingle≥0.5(n−1)δ.\nCombining these two inequalities, we obtain 0.5(n−1)δ≤Cσ(1+2R)ασ, implying n=poly(R)\nδ.\nIt now only remains to approximate a piecewise linear functi on, withpoly(R)\nδconstant segments, us-\ning a threshold network with a similar number of neurons. Defi ney0= 0, it is easy to verify that this\napproximation is given by the expression\nn/summationdisplay\ni=1ξi(yi−yi−1)σthresh(ξi(x−xi)−0.5)−0.5(ξi−1)(yi−yi−1),\nwhereξi∈{−1,1}is chosen according to the discontinuity of our piecewise co nstant function at the point\nxi. Namely, by setting wi:=ξi,bi:=−ξixi−0.5,vi:=ξi(yi−yi−1)andb0:=−0.5/summationtextn\ni=1(ξi−1)(yi−\nyi−1), we obtain a width nthreshold network\nN(x):=n/summationdisplay\ni=1viσthresh(wix+bi)+b0,\nsatisfying\nsup\nx∈[−R,R]|σ(x)−N(x)|≤δ.\nLastly, by Assumption 1, we have that|yi|≤poly(R)for alli∈[n], implying thatNhas weights of\nmagnitude at mostpoly(R)\nδ.\nThe following proposition guarantees that functions on the Boolean hypercube, computed by depth\n2 networks, which employ activation functions that satisfy Assumption 1, can be simulated to arbitrary\naccuracy using depth 2 threshold networks of width polynomi al in the size of the network.\nProposition A.5. Letδ >0, suppose that σsatisfies Assumption 1, and letf:{0,1}d→Rbe a function\ncomputed by a depth 2, width m σ-network, with weights bounded by C. Then there exists a depth 2\nthreshold networkNof widthm2poly(d,C)\nδand weights bounded in magnitude bypoly(d,C)\nδ, such that\nmax\nx∈{0,1}d|f(x)−N(x)|≤δ.\n21Proof. Let\nf(x):=m/summationdisplay\ni=1viσ(/an}⌊ra⌋ketle{twi,x/an}⌊ra⌋ketri}ht+bi)+b0\nbe the function computed by the network f. We first observe that by our weight boundedness assumption,\nwe have for all i∈[m]that|/an}⌊ra⌋ketle{twi,x/an}⌊ra⌋ketri}ht+bi| ≤ /⌊ard⌊lwi/⌊ard⌊l/⌊ard⌊lx/⌊ard⌊l+C≤(d+ 1)C. We now use Lemma A.4\nto approximate each σ-neuron in fto accuracyδ\nmC, and obtain mdepth 2, widthmpoly(d,C)\nδthreshold\nnetworksN1,N2,...,Nm, such that\n|Ni(x)−σ(/an}⌊ra⌋ketle{twi,x/an}⌊ra⌋ketri}ht+bi)|≤δ\nmC\nfor alli∈[m]. Define the network Ngiven by\nN(x):=m/summationdisplay\ni=1viNi(x)+b0,\nfix anyx∈{0,1}dand compute\n|f(x)−N(x)|≤m/summationdisplay\ni=1vi|σ(/an}⌊ra⌋ketle{twi,x/an}⌊ra⌋ketri}ht+bi)−Ni(x)|≤m/summationdisplay\ni=1Cδ\nmC=δ.\nLastly, sinceNis a linear combination of depth 2 neural networks, each of wh ich is of width at most\nmpoly(d,C)\nδand weights of magnitudepoly(d,C)\nδ, we have thatNis in itself a depth 2, widthm2poly(d,C)\nδ\nthreshold network, with weights bounded bypoly(d,C)\nδas required.\nWith the above lemmas and propositions at hand, we are finally ready to prove the theorem.\nProof of Thm. 3.1.First, for any natural dand realC >0, definew(d,C)to be the minimal width required\nfor a depth 2 σ-network with weights bounded in magnitude by Cto approximate fdto accuracy at most\n1\n400. Our goal is therefore to derive a lower bound on w(d,C).\nLetD:= 100d, we first assume that we are given a depth 2, width w(D+4d,C)σ-networkN, with\nweights bounded by C, that satisfies\nE\nx,y∼U(AD+4d)/bracketleftBig\n(N(x,y)−fD+4d(x,y))2/bracketrightBig\n≤1\n400,\nand we will show that this implies the existence of a σ-network of a similar size and with a similar magnitude\nof the weights, which gives a similar accuracy when approxim atingIPD+4duniformly over the Boolean\nhypercube.\nUsing the law of total expectation, we can break the above exp ectation into two iterated expectations as\nfollows\nE\nc∼U([0,0.25]2D+8d)/bracketleftbigg\nE\nz∼U({0,0.75}2D+8d)/bracketleftBig\n(N(z+c)−fD+4d(z+c))2/bracketrightBig/bracketrightbigg\n≤1\n400.\nForc∼U/parenleftbig\n[0,0.25]2D+8d/parenrightbig\n, we can define the non-negative random variable\nXc:= E\nz∼U({0,0.75}2D+8d)/bracketleftBig\n(N(z+c)−fD+4d(z+c))2/bracketrightBig\n.\n22This allows us to rewrite the former inequality more compact ly as\nE\nc∼U([0,0.25]2D+8d)[Xc]≤1\n400. (6)\nUsing Markov’s inequality on Xc, and by virtue of Eq. ( 6), we have\nP\nc∼U([0,0.25]2D+8d)/bracketleftBigg\nXc<400\n399E\nc∼U([0,0.25]2D+8d)[Xc]≤1\n399/bracketrightBigg\n≥1−399\n400>0.\nNamely, there must exist some c∈[0,0.25]2D+8dsuch that\nE\nz∼U({0,0.75}2D+8d)/bracketleftBig\n(N(z+c)−fD+4d(z+c))2/bracketrightBig\n=Xc≤1\n399. (7)\nNext, we have by the definition of fthatfD+4d(z+c) =fD+4d(z) =fD+4d/parenleftbig4\n3z/parenrightbig\n= IPD+4d/parenleftbig4\n3z/parenrightbig\nfor all\nc∈[0,0.25]2D+8dand allz∈{0,0.75}2D+8d. Moreover, there exists a σ-networkN′of the same depth\nand width asN, such thatN(z+c) =N′/parenleftbig4\n3z/parenrightbig\nfor allc∈[0,0.25]2D+8dand allz∈{0,0.75}2D+8d. This\nholds true since shifting the input by a constant vector cmerely shifts the biases in the first hidden layer\nby the same vector, and scaling the inputs by a multiplicativ e constant4\n3is equivalent to multiplying the\nweights of the hidden neurons by 0.75. Since both operations are linear, they can be absorbed in th e hidden\nlayer without changing the architecture of N′, and where the magnitude of the weights is now at most 2C.\nThe above observations, the definition of Xc, and Eq. ( 7), imply the existence of some c∈[0,0.25]2D+8d\nthat satisfies\nE\nx,y∼U({0,1}D+4d)/bracketleftBig/parenleftbig\nN′(x,y)−IPD+4d(x,y)/parenrightbig2/bracketrightBig\n= E\nz∼U({0,0.75}2D+8d)/bracketleftBigg/parenleftbigg\nN′/parenleftbigg4\n3z/parenrightbigg\n−IPD+4d/parenleftbigg4\n3z/parenrightbigg/parenrightbigg2/bracketrightBigg\n= E\nz∼U({0,0.75}2D+8d)/bracketleftBig\n(N(z+c)−fD+4d(z+c))2/bracketrightBig\n≤1\n399. (8)\nLetn∈Nto be determined later, we now construct a neural network N′′:RD+4d→Rthat will achieve\na small margin on all the inputs x,y∈{0,1}dsimultaneously. The network will consist of nblocks, where\neach of which has the architecture of N′, and is re-randomized using the following process for every j∈[n]:\n• We sample two binary vectors x′\nj,y′\nj∈{0,1}duniformly at random.\n• We sample two binary vectors x′′\nj,y′′\nj∈ {0,1}Duniformly at random, until the number of pairs\n(xj,i,yj,i)that are both one is an even number for every j∈[n].\n• We concatenate our sample into two binary vectors (x+x′,x′,x+x′,x′,x′′)and(y+y′,y′,y′,y+\ny′,y′′).\n• We sample a permutation τj:RD+4d→Runiformly at random, and use this permutation to permute\nthe previous two binary vectors, denoting the results as ˆxjandˆyj, respectively.\n23• We modify each N′\njto simulate their computation on the inputs ˆxj,ˆyj. For the coordinates receiving\nx+x′ory+y′as input, this can be done by keeping the weights in the hidden layer unchanged in\nthe case where their corresponding coordinate in x′was drawn as 0, and by composing the weights\nwith the transformation xi/ma√sto→1−xiwhenever x′\ni= 1. Since this is a linear transformation of the\ninput, it can be absorbed into the weights of the first hidden l ayer without changing the architecture\nofN′\nj. Lastly, multiplying the weights of the hidden layer with th e permutation matrix which corre-\nsponds to the sampled permutation τj, which is also a linear transformation, also allows us to kee p\nthe architecture of N′\njunchanged.\nWe now turn to bound the approximation error in absolute valu e of the networkN′when approximating\nIPD+4d. Fix some x,y∈{0,1}d, and letDdenote the distribution over the set {0,1}2(D+4d)which is\ninduced by the randomness in picking x′\nj,y′\nj∈{0,1}d,x′′\nj,y′′\nj∈{0,1}Dand a uniformly chosen permuta-\ntionτjof[D+4d], as described in the above random construction of N′\nj. Cauchy-Schwarz, combined with\nEq. ( 8) and Proposition A.3’s bound on/⌊ard⌊lD/⌊ard⌊l2yields\nE(ˆx,ˆy)∼D/bracketleftbig/vextendsingle/vextendsingleN′(ˆx,ˆy)−IPD+4d(ˆx,ˆy)/vextendsingle/vextendsingle/bracketrightbig\n=/summationdisplay\nˆx,ˆy∈{0,1}D+4dP\nD[X=ˆx,Y=ˆy]·/vextendsingle/vextendsingleN′(ˆx,ˆy)−IPD+4d(ˆx,ˆy)/vextendsingle/vextendsingle\n≤||D|| 2/radicalBigg/summationdisplay\nˆx,ˆy∈{0,1}D+4d(N′(ˆx,ˆy)−IPD+4d(ˆx,ˆy))2\n≤8/radicalbigg\n1\n399<0.41. (9)\nNext, we formally define N′′as the network\nN′′(x,y):=1\nnn/summationdisplay\nj=1N′\nj(ˆxj,ˆyj).\nNote that sinceN′′is a linear combination of depth 2 networks, it is in itself a d epth 2 network. Moreover,\nwe remark that this random process always preserves the valu e of the inner product. To see this, fix any\nx′,y′,x′′,y′′,τchosen according to the above random process. Then by taking equalities that are mod 2,\nwe have\nIPD+4d(τ(ˆx),τ(ˆy)) = IP D+4d(ˆx,ˆy)\n= IPd(x+x′,y+y′)+IPd(x′,y′)+IPd(x+x′,y′)+IPd(x′,y+y′)+IPD(x′′,y′′)\n= IPd(x+x′,y+y′)+IPd(x′,y′)+IPd(x+x′,y′)+IPd(x′,y+y′)\n= IPd(x,y+y′)+IPd(x,y′) = IP d(x,y).\nwhere the first equality follows from the fact that permutati ons preserve sums, the second equality is by the\ndefinition of our construction of ˆx,ˆy, the third equality is due to x′′andy′′always having an even number\nof pairsxi= 1,yi= 1 which implies IPD(x′′,y′′) = 0 , and the last two equalities follow from basic\nproperties of the inner product mod 2. Thus, we have\nIPd(x,y) = IP D+4d(ˆxj,ˆyj)\nfor allx,y∈{0,1}dand allj∈[n]. Having definedN′′, we now bound the approximation error in absolute\nvalue which it achieves on an arbitrary input x,y∈{0,1}d, over the randomness induced by the previously\n24described process. Compute\n/vextendsingle/vextendsingleN′′(x,y)−IPd(x,y)/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nj=1N′\nj(ˆxj,ˆyj)−E/bracketleftbig\nN′\n1(ˆx1,ˆy1)/bracketrightbig\n+E/bracketleftbig\nN′\n1(ˆx1,ˆy1)/bracketrightbig\n−IPD+4d(ˆx1,ˆy1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nj=1N′\nj(ˆxj,ˆyj)−E/bracketleftbig\nN′\n1(ˆx1,ˆy1)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+E/bracketleftbig/vextendsingle/vextendsingleN′\n1(ˆx1,ˆy1)−IPD+4d(ˆx1,ˆy1)/vextendsingle/vextendsingle/bracketrightbig\n,\nwhere the expectation is taken over the randomness in sampli ng(ˆxj,ˆyj)fromD. Using Eq. ( 9), we can\nupper bound the above by\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nj=1N′\nj(ˆxj,ˆyj)−E/bracketleftbig\nN′\n1(ˆx1,ˆy1)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+0.41. (10)\nNext, we will use Hoeffding’s inequality to upper bound the a bsolute value term above, but first we\nwill derive an upper bound on the magnitude of the output of th e networkN′\nj. We have that each neuron\ninN′\njhas weights of magnitude at most 2C, therefore, from Assumption 1we get that the output of each\nhidden neuron is upper bounded by poly(d,C)over the domain{0,1}D+4d. This implies that the output\nof the output neuron is at most B:= poly(d,C)w(D+ 4d,C), since the width of N′\njisw(D+ 4d,C)\nand its output neuron has weights bounded by C. We now use this bound, and the fact that N′\nj(ˆxj,ˆyj),\nj= 1,...,n are independent with respect to the randomness in sampling x′\nj,x′′\nj,y′\nj,y′′\nj,τj, to invoke\nHoeffding’s inequality, yielding\nP\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn/summationdisplay\nj=1N′\nj(ˆxj,ˆyj)−E/bracketleftbig\nN′\n1(ˆx1,ˆy1)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle>0.04\n≤2exp/parenleftbigg\n−0.5n0.042\nB2/parenrightbigg\n.\nSettingn= 2500B2d= poly(d,C,w(D+4d,C)), we can upper bound the above by 2exp(−2d), which\nis strictly less than 2−2dfor alld≥2. Taking a union bound over all 22dpossibilities for the inputs\n(x,y)∈{0,1}2d, we have by substituting our Hoeffding bound in Eq. ( 10), that with positive probability,\nN′′satisfies|N′′(x,y)−IPd(x,y)| ≤0.41 + 0.04 = 0.45for all inputs (x,y)∈ {0,1}2d. By the\nprobabilistic method, this implies the existence of partic ular realizations of the random variables\nx′\n1,...,x′\nn,x′′\n1,...,x′′\nn,y′\n1,...,y′\nn,y′′\n1,...,y′′\nn,τ′\n1,...,τ′\nn\nwith this property. Define the neural network N′′′as the network obtained from substituting these variables\nwith the above realizations, and note that doing so merely de creases the input dimension of the network,\nwhile keeping the computation in the hidden layer linear, wh ich thus does not change the architecture of\nN′′′and maintains its width.\nTo conclude the derivation so far, we have shown the existenc e of a depth 2 σ-networkN′′′, which has\nwidthpoly(d,C,w(D+4d,C)), and satisfies\nmax\n(x,y)∈{0,1}2d/vextendsingle/vextendsingleN′′′(x,y)−IPd(x,y)/vextendsingle/vextendsingle<0.45.\nWe now use Proposition A.5withδ= 0.04to obtain a threshold network ¯NfromN′′′, having width\npoly(d,C,w(D+4d,C))and weights of magnitude at most poly(d,C), such that for all x,y∈{0,1}d\n/vextendsingle/vextendsingle¯N(x,y)−IPd(x,y)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle¯N(x,y)−N′′′(x,y)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleN′′′(x,y)−IPd(x,y)/vextendsingle/vextendsingle≤0.45+0.04 = 0.49.\n25Constructing a threshold circuit from ¯N, which employs a threshold activation on its output neuron, we\nobtain a threshold circuit which computes IPd(·,·). We lower bound the width of the circuit by using the\nfollowing fact adapted from Hajnal et al. [4] which appears i n Martens et al. [13], and is stated here in a\nslightly modified manner for the sake of completeness.\nFact A.6. For a depth 2 threshold network Nof widthmand weights bounded in magnitude by C, that\nsatisfies\nmax\n(x,y)∈{0,1}2d|N(x,y)−IPd(x,y)|≤0.5−δ\nfor someδ∈(0,0.5), we have\nm≥Ω/parenleftBigg\nδ2d/3\nC/parenrightBigg\n.\nSubstituting δ= 0.01, the above fact and our expression for the width of ¯Nimply the inequality\npoly(d,C,w(D+4d,C))≥Ω/parenleftBigg\n2d/3\npoly(d,C)/parenrightBigg\n.\nSimplifying the above by using more asymptotic notation and absorbing terms that are polynomial in dinto\nthe exponent, we have\nw(D+4d,C)≥Ω/parenleftBigg\n2Ω(d)\npoly(C)/parenrightBigg\n.\nRecall that D= 100d. Letting d′:= 104dwhich implies d= Ω(d′)and performing a change of variables,\nthe theorem follows.\nA.2 Proof of Thm. 4.1\nLetε >0. First, if ε >1\n2, then we have\nsup\nx,y∈Ad/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2−fd(x,y)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1\n2.\nNamely, a network which computes a constant function satisfi es our requirements. We can thus assume\nfrom now on that ε≤1\n2.\nNext, we define a few auxiliary functions that will be used in t he construction of our approximation. Let\ng1(z):=\n\n0, z∈(−∞,5],\nz−5, z∈(5,6),\n1, z∈[6,∞),\nand Let\ng2(z):=/braceleftBigg\nzmod 1,⌊z⌋even,\n1−(zmod 1),⌊z⌋odd.\nLetδ1:=ε\n2d, we have from Assumption 2and the fact that g1(·)is1-Lipschitz, that there exists some depth\n2σ-network h1, of widthO/parenleftbigd\nε/parenrightbig\nand weights bounded by O/parenleftbig1\nε/parenrightbig\n, which satisfies\n|g1(z)−h1(z)|≤ε\n2d,∀z∈[0,8]. (11)\n26Likewise, there exists some depth 2 σ-network h2, of widthO/parenleftbigd\nε/parenrightbig\nand weights bounded by O/parenleftbig1\nε/parenrightbig\n, which\nsatisfies\n|g2(z)−h2(z)|≤ε\n2,∀z∈[−2d−1,2d+1]. (12)\nNow, it is easy to verify that\ng1(4x+4y) = AND(round( x),round(y))∀x,y∈A1,\nimplying\nd/summationdisplay\ni=1g1(4xi+4yi) =/an}⌊ra⌋ketle{tround(x),round(y)/an}⌊ra⌋ketri}ht ∀x,y∈Ad,\nand thus\ng2/parenleftBiggd/summationdisplay\ni=1g1(4xi+4yi)/parenrightBigg\n= IPd(round(x),round(y)) =fd(x,y)∀x,y∈Ad.\nWe now define the network Nwhich approximates fd(x,y)well on the setAd. For all x,y∈Ad,\ndefine\nN(x,y):=h2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n.\nWe construct a depth 3 neural network that computes the above function as follows:\n• The first hidden layer will consist of dcopiesh1,i,i= 1,...,d , of the depth 2, width O/parenleftbigd\nε/parenrightbig\nnetwork\nh1. Eachh1,iwill receive (xi,yi)as input; thus, the weight assigned to the coordinates xi,yiis set to\n4, and the weight assigned to the remaining coordinates is set to zero. Note that the width of the first\nlayer is thereforeO/parenleftBig\nd2\nε/parenrightBig\n, and the magnitude of its weights is bounded by O/parenleftbig1\nε/parenrightbig\n.\n• The second hidden layer will consist of a single copy of the n etwork computing the function h2. This\nimplies that the width of this layer is O/parenleftbigd\nε/parenrightbig\n. Each neuron in this layer will assign the same set of\nweights for each incoming output from the output neurons of t he components h1,i,i= 1,...,d .\nSince this sum is a linear operation performed on these outpu t neurons, we can effectively absorb\nthe output neurons into the hidden neurons in the second laye r, and thus avoid adding a third hidden\nlayer. Note that by absorbing weights of magnitude at most O/parenleftbig1\nε/parenrightbig\nin a layer with weights bounded\nbyO/parenleftbigd\nε/parenrightbig\n, we get that the weights in the second hidden layer are of magn itude at mostO/parenleftbigd\nε2/parenrightbig\n.\nConcluding our construction of N, we have that its width is O/parenleftBig\nd2\nε/parenrightBig\n, due to the first hidden layer; and its\nmagnitude of the weights is O/parenleftbigd\nε2/parenrightbig\n, due to the second hidden layer.\nIt is therefore only left, given arbitrary x,y∈Ad, to upper bound the expression\n|N(x,y)−fd(x,y)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleh2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n−g2/parenleftBiggd/summationdisplay\ni=1g1(4xi+4yi)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleh2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n−g2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n−g2/parenleftBiggd/summationdisplay\ni=1g1(4xi+4yi)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (13)\n27We begin with upper bounding the first absolute value term. By virtue of Eq. ( 11), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled/summationdisplay\ni=1h1(4xi+4yi)−d/summationdisplay\ni=1g1(4xi+4yi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤d/summationdisplay\ni=1|h1(4xi+4yi)−g1(4xi+4yi)|≤ε\n2, (14)\nimplying\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled/summationdisplay\ni=1h1(4xi+4yi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled/summationdisplay\ni=1g1(4xi+4yi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+ε\n2≤d/summationdisplay\ni=1|g1(4xi+4yi)|+1\n4≤d+1\n4,\nfor allx,y∈A. The above and Eq. ( 12) imply\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleh2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n−g2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ε\n2. (15)\nMoving on to bound the second absolute value term in Eq. ( 13), we have by the fact that g2(·)is1-Lipschitz\nthat\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg2/parenleftBiggd/summationdisplay\ni=1h1(4xi+4yi)/parenrightBigg\n−g2/parenleftBiggd/summationdisplay\ni=1g1(4xi+4yi)/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingled/summationdisplay\ni=1h1(4xi+4yi)−d/summationdisplay\ni=1g1(4xi+4yi)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ε\n2,\nwhere in the second inequality we used Eq. ( 14). Plugging the above and Eq. ( 15) back in Eq. ( 13), we get\n|N(x,y)−fd(x,y)|≤ε,\nfor allx,y∈Ad, as desired.\n28"    },    {        "title": "2402.07251v1.Physics_Informed_Neural_Networks_with_Hard_Linear_Equality_Constraints.pdf",        "content": "Physics-Informed Neural Networks with Hard Linear\nEquality Constraints\nHao Chena, Gonzalo E. Constante Floresa, Can Lia,∗\naDavidson School of Chemical Engineering, Purdue University, 480 W. Stadium\nAve, West Lafayette, 47907, IN, USA\nAbstract\nSurrogate modeling is used to replace computationally expensive simulations.\nNeural networks have been widely applied as surrogate models that enable\nefficient evaluations over complex physical systems. Despite this, neural net-\nworks are data-driven models and devoid of any physics. The incorporation of\nphysics into neural networks can improve generalization and data efficiency.\nThe physics-informed neural network (PINN) is an approach to leverage\nknown physical constraints present in the data, but it cannot strictly satisfy\nthem in the predictions. This work proposes a novel physics-informed neural\nnetwork, KKT-hPINN, which rigorously guarantees hard linear equality con-\nstraints through projection layers derived from KKT conditions. Numerical\nexperiments on Aspen models of a continuous stirred-tank reactor (CSTR)\nunit, an extractive distillation subsystem, and a chemical plant demonstrate\nthat this model can further enhance the prediction accuracy.\nKeywords: Surrogate modeling, Physics-informed neural network, Artificial\nintelligence\n1. Introduction\nWith the development of computational capability and capacity, com-\nputer simulations have attained widespread use across various fields, facili-\ntating high-fidelity modeling and analysis of complex systems or processes.\n∗Corresponding author at: Davidson School of Chemical Engineering, Purdue Univer-\nsity, USA.\nEmail address: canli@purdue.edu (Can Li)\nSubmitted preprintarXiv:2402.07251v1  [cs.LG]  11 Feb 2024These simulations are governed by a large number of algebraic equations,\nordinary or partial differential equations (ODEs or PDEs), or a combina-\ntion of both differential and algebraic equations (DAEs). These equations\nare derived from fundamental principles and mechanistic laws, such as the\nphysical laws in thermodynamics and transport phenomena. High-fidelity\nmodels with these equations can serve as digital representations of the physi-\ncal systems in the real world. However, the physically accurate representation\nis accompanied by a heightened mathematical complexity that elevates the\ncomputational expense of simulation. This impedes the use of high-fidelity\nphysical models especially in applications where it is essential to simulate a\nsystem repeatedly in a timely manner.\nTo efficiently generate simulation outputs, data-driven approaches have\nsought to substitute a high-fidelity physical model with a surrogate model\n(Misener and Biegler, 2023; Bhosekar and Ierapetritou, 2018; Bradley et al.,\n2022; Williams and Cremaschi, 2021), A surrogate model stands for a reduced-\norder model that aims for a computationally efficient approximation at the\ncost of a certain level of accuracy. This approach provides a more practical\nmeans of inferring a system’s responses under a great variety of conditions.\nAmong them, feed-forward neural networks have been proven to be capa-\nble of approximating any continuous function theoretically. As a universal\napproximator, it has demonstrated effectiveness in capturing complex and\nnonlinear relationships in high-dimensional space, with remarkable advance-\nments in computer vision and natural language processing (LeCun et al.,\n2015).\nHence, neural networks have received substantial attention as surrogate\nmodels, which facilitate their deployment in time-sensitive and large-scale\napplications (Bradley et al., 2022; Dias and Ierapetritou, 2020; Kim and\nBoukouvala, 2020; Mohammadi et al., 2022). For instance, incorporating\nsimulators or underlying equations into optimization can become exceedingly\nexpensive, whereas neural networks offer a more efficient alternative. On the\none hand, black-box optimization that enables the use of simulators without\nan explicit algebraic form is not practical when dealing with a high degree of\nfreedom. Even with access to high-fidelity physical models, optimization re-\nmains challenging despite advances in state-of-the-art solvers. The algebraic\nequations often give rise to nonconvex mixed-integer nonlinear programming\n(MINLP), and the DAEs introduce optimization problems with ODE or PDE\nconstraints, which can be intractable. On the other hand, neural networks\nwith ReLU activation layers can be integrated as surrogate models within\n2mixed-integer linear programming (MILP) problems to replace the nonlinear\nfunctions (Tsay et al., 2021).\nOn the flip side, the drawback of neural networks is the lack of physical\ninterpretability. This limitation originates from two factors. First, the in-\ntricate architecture of neural networks is composed of neurons hierarchically\narranged in multiple layers. It is challenging for humans to understand the\nrole of each neuron due to the complexity of hierarchical representations. Sec-\nond, as a data-driven method, neural networks are black-box models whose\nparameters are not physics-informed. Training a neural network is equiva-\nlent to minimizing an objective function, often referred to as a loss function,\nover these parameters. While this process updates parameters by evaluating\nthe error between model prediction and ground truth, it does not take into\naccount any physics. Therefore, the model prediction cannot conform to any\nphysical laws. The lack of physical interpretability can hinder their appli-\ncation in high-stakes decision-making problems, where undetected errors or\nmisunderstandings within the model could potentially lead to catastrophic\nlosses.\nTo mitigate this drawback, the physics-informed neural network (PINN)\nwas developed to leverage the physical constraints that data are subject to\nin particular application domains. In the prevalent PINN method, a penalty\nterm is incorporated into the loss function to account for the violation of con-\nstraints (Karniadakis et al., 2021). This can be considered a multi-objective\noptimization problem, where the goal is to minimize the prediction errors and\nviolation of physics simultaneously at the cost of each other. However, even\nwith the sacrifice of some prediction accuracy, the PINN method still can-\nnot guarantee the exact satisfaction of physical constraints. In other words,\nPINN only strikes a balance between approximating the ground truth and\nfavoring the first principles during the training process. Hence, this is also\nnamed “soft constraint”, as they are not strictly satisfied.\nThe inclusion of soft constraints in PINN still cannot avoid physically\ninconsistent results, rendering it unsuitable for certain applications. For\ninstance, PINN cannot strictly enforce conservation laws when trained to\nrepresent a chemical unit. In practical scenarios, it is necessary to partition\na large system into distinct units, each of which is then replaced with sur-\nrogate models (Henao and Maravelias, 2011). While this strategy mitigates\nthe error resulting from the high dimensionality of a large system, it also in-\ntroduces discrepancies among these units. Although the impact of violating\nconstraints might be negligible in an isolated unit, the errors of intermediate\n3variables can propagate throughout the entire system and be magnified over\ntime (Ma et al., 2022a). Therefore, the violation of mass balance becomes\nunacceptable when multiple surrogate models are interconnected.\nThis calls for the development of PINN architectures that embed hard\nconstraints (hPINN) rather than soft constraints. A few studies claiming\nto enforce hard constraints fail to provide rigorous mathematical guarantees\n(Dener et al., 2020; Lu et al., 2021). The realization of “hard constraints”\nin these architectures heavily relies on trial-and-error tuning of hyperparam-\neters. Hyperparameters that are set without adequate experience can easily\nlead to the failure of hard constraints and even poor predictions. In this\nwork, we develop a novel PINN architecture with two non-trainable layers.\nThe two layers equivalently represent an orthogonal projection of model pre-\ndictions onto a feasible region of predefined linear equality constraints. This\nprojection can be formulated as a quadratic program (QP) and analytically\nsolved by the KKT conditions within the architecture. We hence refer to it as\nKKT-hPINN, as it is grounded in an analytical solution that always satisfies\nhard linear equality constraints in both training and testing processes. It does\nnot require additional hyperparameters and does not increase computational\ncost. Results from three case studies show that KKT-hPINN can outperform\nneural networks and PINN when serving as unit-level, subsystem-level, and\nplant-level surrogate models in process system engineering.\nThe remainder of the paper is organized as follows. In Section 2, we begin\nwith a literature review of data-driven methods used for surrogate models,\nas well as their pros and cons. Section 3 describes the proposed neural net-\nwork architecture, KKT-hPINN, that strictly imposes hard linear equality\nconstraints. We present the case studies in Section 4 and demonstrate its\nsuperiority over unconstrained neural networks and soft-constrained PINN.\nIt is observed that KKT-hPINN not only imposes hard linear equality con-\nstraints in practice but also improves the predictive capability. In the end,\nSection 5 concludes the essential characteristics of the KKT-hPINN.\n2. Literature review\nNumerous studies have been conducted to formulate data-driven approaches\nfor modeling input-output relationships, with some being applied in chem-\nical engineering. This section seeks to summarize the surrogate modeling\ntechniques that have been extensively explored in this field, as well as some\nrelated work about PINN.\n4Automated Learning of Algebraic Models (ALAMO) is a computational\nmethodology to learn algebraic models by selecting the best subset of regres-\nsors (Wilson and Sahinidis, 2017; Cozad et al., 2014). By minimizing metrics\nthat are common in information theory and statistical theory, ALAMO learns\nalgebraic functions that take into account both subset fitness and complexity.\nThe outcome is in an enclosed functional form, which is expressed as a linear\ncombination of basis functions. This approach yields a more tractable model\nand requires less data. ALAMO is distinguished as one of the surrogate\nmodeling techniques developed by chemical engineers, featuring its success\nin reaction kinetics parameter estimation (Wilson and Sahinidis, 2019; Na\net al., 2021) and process optimization (Ma et al., 2022a,b).\nGaussian process regression (GP) is a non-parametric and probabilistic\napproach that does not estimate parameters for a particular function. In-\nstead, GP directly characterizes the distribution of the mapping function\nitself, providing the mean and covariance of model predictions. The function\nis regarded as a Gaussian process, wherein any finite set of random variables\nconforms to a joint Gaussian distribution. GP capitalizes on its capacity to\ninfer the conditional predictive distribution using simulation data, circum-\nventing a training process (Rasmussen and Williams, 2005). Incorporation of\nGP into optimization has been widely explored (Wiebe et al., 2022; Schweidt-\nmann et al., 2021; Quirante et al., 2015). It has shown promising results in a\nbroad range of applications including model predictive control (MPC) (Paul-\nson et al., 2021; Bonzanini et al., 2021), erosion prediction (Dai et al., 2022),\nexperimental design (Olofsson et al., 2018), biotechnology (Mehrian et al.,\n2018), pharmaceutical manufacturing (Boukouvala and Ierapetritou, 2013),\nand feasibility analysis (Boukouvala and Ierapetritou, 2012). Introducing\nphysics-based knowledge into GP has also been investigated and showed im-\nprovements in prediction performance, but this still depends on penalization\nand soft constraints (Kim et al., 2023). A recent study also proposed a lin-\nearly constrained GP such that any samples and prediction can automatically\nfulfill known linear operator constraints (Jidling et al., 2017).\nNeural network is another popular approach characterized by its high\nflexibility and adaptability. The structure and mechanisms of NN will be\nexplained in detail in Section 3. Neural networks have exhibited remark-\nable competence in approximating deterministic relationships between in-\nputs and outputs across various domains. Breakthrough developments have\nbeen found in computer vision (Krizhevsky et al., 2012; Szegedy et al.,\n2015), speech recognition (Hinton et al., 2012), natural language process-\n5ing (Sutskever et al., 2014), as well as protein structure prediction (Jumper\net al., 2021). These applications harness neural networks to discover un-\nknown relationships from data. On the other hand, neural networks have\nalso been leveraged to substitute known physical models. In process sys-\ntem engineering, some works have been dedicated to accurately representing\nchemical processes (Misener and Biegler, 2023; Ma et al., 2022b; Goldstein\net al., 2022; Henao and Maravelias, 2011). The growing interest in surrogate\nmodels of chemical processes can be attributed to recent studies into the\nmixed-integer programming (MIP) formulations for neural networks (Fis-\nchetti and Jo, 2018). This exploration paves the way to integrate process\noptimization with the expressive power of neural networks, thereby facilitat-\ning decision-making over complex systems (Grimstad and Andersson, 2019;\nAnderson et al., 2020; Tsay et al., 2021; Schweidtmann and Mitsos, 2019).\nAn open-source package, OMLT, designed to automate the transformation\nof trained neural networks has been made available (Ceccon et al., 2022).\nPINN not only learns surrogate models as a class of function approxima-\ntors but also integrates the physical constraints (Karniadakis et al., 2021).\nHence, PINN has found extensive attention in surrogate modeling for multi-\nphysics problems in which data are usually expensive. Unlike classic numer-\nical methods, PINN typically targets ill-posed problems, where data under\ncertain conditions are not precisely known due to difficulties in measure-\nments in the real world. To address uncertainties stemming from imperfect\ndata and stochastic physical systems, PINN has demonstrated its success\nin inferring solutions for both forward and inverse problems that involve\npartial differential equations (PDEs) (Raissi et al., 2019), and uncovering\nhidden physics in partially known stochastic ordinary differential equations\n(SDEs) (O’Leary et al., 2022). Domain-specific applications include colloidal\nself-assembly (Nodozi et al., 2023), fluid mechanics (Raissi et al., 2020), ther-\nmodynamics (Cai et al., 2021) and Model Predictive Control (MPC) (Zheng\nand Wu, 2023; Zheng et al., 2023; Alhajeri et al., 2022). These works leverage\nautomatic differentiation and embed differential equations into the loss of a\nneural network as soft constraints.\nIn parallel, a few studies have also attempted to integrate hard constraints\ninto PINN but lack a mathematical guarantee of strict satisfaction (Dener\net al., 2020; Lu et al., 2021). They adapt the augmented Lagrangian method\nto the training process, namely aug-Lag hPINN. In this approach, the inner\nloop addresses unconstrained optimization problems with conventional opti-\nmizers, while the outer loop updates the Lagrangian multipliers and penalty\n6Figure 1: Fully connected feed-forward neural network architectures. The neuron iat\nthe (l)th layer is the linear combination of neurons at the ( l−1)th layer followed by a\nnonlinear activation σ, which is z(l)\ni=σ(P\njw(l−1)\nijz(l−1)\nj +b(l−1)\ni0). The solid red lines\nrepresent w(l−1)\nij andb(l−1)\ni0respectively.\nfactors. Inappropriate initial values may lead to ill-conditioned optimization\nor being stuck at poor local minima. Technically, aug-Lag hPINN requires\nsufficient training time until the loss of constraints violation falls below a\nspecific tolerance in every inner loop. This is achieved by a predetermined\nnumber of sub-iterations, which results in increased computing time. This\napproach relies on the assumption that the violation decreases to nearly zero\nwithin the inner loops.\n3. Methodology\n3.1. Neural Network Structure\nA fully connected feed-forward neural network can be expressed as a\nmulti-layer perceptron (MLP) model that maps input xto output ythrough\na number of hidden layers as shown in Figure 1. The forward equation in\nmatrix form of a feed-forward neural network with L+ 1 layers is defined as\nz(l)=σ(W(l−1)z(l−1)+b(l−1))∀l= 0, ..., L (1)\n7where z(l)denotes the output of the l-th layer, z(0)=x∈ RN0is the input,\nandz(L)=ˆ y∈ RNLis the output. In every layer, the input neurons from the\nprevious layer, z(l−1), are weighted by W(l−1)and biased by b(l−1), followed by\na nonlinear activation function σ(·). This results in the neurons, z(l)∈ RNl,\non the l-th layer.\nThe nature of training a neural network over Ntraining data points is to\nminimize a loss function over the weights and biases {Wl,bl}L−1\nl=0. The mean\nsquared error (MSE) is one of the widely used loss functions for regression\ntasks. Denoting these trainable parameters as Θ={Wl,bl}L−1\nl=0, the solution\nto this minimization problem, θ∗= argminθ∈ΘJ(θ) where J(θ) is the loss\nfunction, can be iteratively approximated by a wide range of well-established\nalgorithms. For first-order gradient-based optimizers, the learning direction\nis the gradient of the loss function with respect to parameters. The gradients\ncan be accessed by chain rule and retrieved through automatic differentiation.\n3.2. Hard Linearly-Constrained Neural Network Architecture\nIn certain applications, the relationships between some inputs xand out-\nputsymay not be completely unknown. For instance, xandycan contain\nsome variables representing inflows and outflows that are subject to a mass or\nenergy balance. The conservation between them is typically accessible using\nengineering principles and can be expressed as linear equality constraints,\nAx+By=b. In the present work, we propose a physics-informed neural\nnetwork architecture that allows these linear equality constraints to always\nhold in both training and testing processes.\nBefore introducing the algebraic representation of the architecture, we\nfirst provide the geometric intuition. This specialized architecture can be\nviewed as a neural network connected to a corrector that adjusts the pre-\ndictions deviating from linear equality constraints shown in Figure 2. For a\ngiven input ˆx, the output from a neural network prediction ˆymay not sat-\nisfy the linear equality constraints. The corrector will output the orthogonal\nprojection of ˆyonto the feasible region, denoted as ˜y.\nSpecifically, two separate, non-trainable, and fully connected layers are\nadded to a neural network as the corrector, guiding its prediction to the\nclosest feasible point. One layer is a linear transformation of specific input ˆ x\nwith fixed weights A∗and fixed bias b∗, the other layer transforms the output\nˆ ypredicted from the original model NN( Θ,ˆ x) with a projection matrix B∗.\nThe definitions of A∗,b∗, and B∗will be specified in Theorem 1. We name\n8Aˆx+By=b˜yˆyFigure 2: Illustration: for a given input ˆ x∈ RN0, neural network prediction ˆ y∈ RNLis\northogonally projected to be ˜ y∈ RNLthat satisfies a system of linear equality constraints\nAˆx+By=b. As an illustrative example, a hyperplane is used here to represent a single\nconstraint where A∈ R1×N0,B∈ R1×NL,b∈ R1.\nKKT-hPINN\nNN\nFigure 3: Grey block: illustration of NN architectures; Blue block: illustration of KKT-\nhPINN architectures consisting of trainable layers and two additional non-trainable pro-\njection layers (blue layers). The non-trainable parameters A∗,b∗andB∗can be explicitly\ncalculated from (2).\n9this architecture KKT-hPINN, as it is a hard linearly constrained physics-\ninformed neural network derived from the KKT conditions.\nTheorem 1. Given any input ˆ x∈ RN0, a neural network model ˆ y=\nNN(Θ,ˆ x) :RN0→ RNL, and prior knowledge about mequality constraints\nfor the input ˆ xand the ground truth y:Aˆx+By=bwhere A∈ Rm×N0,b∈\nRm,B∈ Rm×NLandrank(B) =m, the orthogonal projection of ˆ yonto a\nfeasible region represented by a system of linear equality constraints, ˜ y, has\nthe following analytical solution:\n˜ y=A∗ˆ x+B∗ˆ y+b∗, (2)\nwhere\nA∗=−BT(BBT)−1A\nB∗=I−BT(BBT)−1B\nb∗=BT(BBT)−1b\nProof. Correcting predictions that deviate from constraint satisfaction with\nminimum Euclidean distance can be conceptualized as an orthogonal projec-\ntion onto the feasible region of physical constraints, as depicted in Figure\n2. This can be formulated as the following quadratic programming problem\nafter the forward calculation of ˆ yin each iteration:\n˜ y= argminy1\n2∥y−ˆ y∥2s.t.Aˆ x+By=b (3)\nwhere A,Bandbare constant matrices and vector involved in certain prior\nknowledge. That is, instead of feeding the prediction ˆ ydirectly into the loss\nfunction for backpropagation, we first project it onto a feasible region where\nthe linear equality is strictly satisfied.\nBecause of the convexity of problem 3, the optimal primal and dual so-\nlutions ˜ yandλ∗can be found by solving the KKT conditions below.\n\u0014I BT\nB 0\u0015\u0014˜ y\nλ∗\u0015\n=\u0014ˆ y\nb−Aˆ x\u0015\nIn the neural network setting, the optimal solution ˜ yin (3) can be expressed\nas the sum of outputs from two additional projection layers with fixed pa-\nrameters A∗,B∗andb∗.\n10This architecture design presents a distinctive methodology with key fea-\ntures outlined in the following remarks:\nRemark 1 (Applicability to other architectures) .The proposed projection\nlayer can be applied to any existing neural network architectures, such as\nconvolutional neural networks and recurrent neural networks, instead of being\nlimited to MLP.\nRemark 2 (Compatibility to PINN) .The KKT-hPINN can be compatible\nwith the PINNs, i.e., enforcing hard linear equality constraints and soft non-\nlinear constraints simultaneously. This compatibility arises because the hard\nconstraints are introduced using a different architecture, while the soft con-\nstraints are embedded into the loss function.\nRemark 3 (Difference with post projection) .It is worth noting that em-\nbedding the projection layers within the architecture should be distinguished\nfrom trivially applying orthogonal projection to the predictions obtained from\na neural network in the testing process. The latter strategy, namely post-\nprojection during testing, does not take into account the ground truths of\nthe output in the training data when the performing projection step and thus\nmay compromise the accuracy of the predictions. In contrast, the presence of\nprojection layers not only imposes hard constraints but also changes how the\nmodel learns in every iteration of the training process. A quantitative expla-\nnation is that the projection layers will change the gradient descent direction,\nleading to different parameters for KKT-hPINN compared to neural networks\nwithout the projection layers.\nRemark 4 (Loss functions) .The KKT-hPINN architecture changes the loss\nfunction so that the learning direction, the gradient of the loss function, is\nalso changed. When the MSE is selected as the criterion of the loss function,\nredefine z(l):= [1 ,z(l)]∈ RNl+1and concatenate bias b(l)with weight W(l)\nat each layer as θ(l)= [b(l),W(l)]for notation simplicity, the optimization\nproblems solved in the training of the NN, PINN, and KKT-hPINN can be\n11expressed as\nNN: minΘ1\n2NNX\ni=1∥NN(Θ,xi)−yi∥2(4)\nPINN: minΘ1\n2NNX\ni=1(∥NN(Θ,xi)−yi∥2+λ∥Axi+BNN(Θ,xi)−b∥2)\n(5)\nKKT-hPINN: minΘ1\n2NNX\ni=1∥A∗xi+B∗NN(Θ,xi) +b∗−yi∥2(6)\nKKT-hPINN enforces the model to update the parameter in a way such that\ninherent linear patterns present within the simulation data are always satis-\nfied. It is anticipated that the convergence can be improved by this character-\nistic of KKT-hPINN.\n4. Case Studies\nThis section covers case studies of a CSTR unit, a DME-DEE chemi-\ncal plant, and an extractive distillation subsystem. The performance of the\nneural network (NN), the physics-informed neural network (PINN), the neu-\nral network with post projection in Remark 3 (NNPost), and the proposed\nKKT-hPINN are examined and compared.\nCase CSTR Plant Distillation\nHidden layers 2 2 2\nNeurons/layer 12 32 32\nLearning rate 10−410−410−4\nBatch size 16 16 16\nSamples ∼1500 ∼1200 ∼5000\nTable 1: Neural network architectures and hyperparameter settings\n12Experiment setup. The datasets are generated from simulators constructed\nin Aspen Plus V.11. The Aspen simulator for the CSTR unit is customized\nwith a built-in model, while those for the chemical plant and the extractive\ndistillation subsystem can be found in the supplementary files of Ma et al.\n(2022b). The preliminary data obtained from Aspen may not be valid due\nto unconvergence or numerical issues during the simulations. To ensure va-\nlidity, all the data marked with severe errors are eliminated. After that, the\ndata are filtered if they violate a certain tolerance for specific linear equality\nconstraints in the case studies. The tolerances for the data of the CSTR unit,\nthe DME-DEE chemical plant, and the extractive distillation unit are set to\nbe 10−8, 10−6, and 10−8respectively.\nThe data from the feasible simulations is stored and scaled using maxi-\nmum absolute scaling, i.e., dividing the data by the maximum absolute value\nof each variable in the dataset. The surrogate models are built using the\nPyTorch package in Python (Paszke et al., 2019). The neural network ar-\nchitectures used in the case studies, along with the hyperparameter settings,\nare documented in Table 1. NN, PINN, NNPost, and KKT-hPINN adopt\nthe same settings in each case study.\nThe models are trained and evaluated with respect to Root Mean Squared\nError (RMSE). During training, the parameters are updated in every epoch\nto minimize the RMSE value between the current prediction and ground\ntruth. A validation dataset is used to evaluate the extent of overfitting in\nmodels during each epoch. This assesses the RMSE value without influencing\nparameter updates. By default, 20% samples in Table 1 are used for valida-\ntion. A well-trained model is expected to maintain a negligible gap between\ntraining and validation RMSE values, with both values being consistently\nlow.\nThe well-trained model is then applied to an unseen test dataset, where\nthe RMSE values can vary inherently in different runs, because training a\nneural network cannot guarantee global optimality. To ensure the robust-\nness of the results, this process is repeated 10 times. Besides, generating\nhigh-fidelity data can be challenging and time-consuming in some fields. To\ninvestigate the effect of the size of the training dataset, models are trained\nusing a different number of training samples and compared in terms of their\nimprovements relative to a tranditional neural network.\n13RX\nB EB\nEFigure 4: Process flowsheet of the CSTR unit.\nVariables Description\nx1 Temperature of the RX unit\nx2 Molar flow rate of Benzene in the B stream\nx3 Molar flow rate of Ethylene in the E stream\ny1 Molar flow rate of Ethylbenzene in the EB stream\ny2 Molar flow rate of Benzene in the EB stream\ny3 Molar flow rate of Ethylene in the EB stream\nAx+By=b(\nx2−x3−y2+y3= 0,Reactants consumption\nx2−y1−y2= 0, EB production\nTable 2: Variables and constraint of the CSTR unit.\n4.1. CSTR Unit\n1To illustrate the construction of a KKT-hPINN, a simple simulator of\nCSTR as shown in Figure 4 is considered here. Suppose a CSTR converts\nreactants, Benzene (B) and Ethylene (E), into the product, Ethylbenzene\n(EB), and the process follows a first-order chemical reaction:\nB+E→EB (7)\nPure reactants are fed at a steady state. The CSTR has fixed volume\nand working pressure. The variables left for design are the molar flow rate of\nbenzene and ethylene, and the working temperature of CSTR. The goal is to\n1The proposed model and case studies are made available on our GitHub repository\nat:https://github.com/li-group/KKThPINN.git\n140 200 400 600 800 1000\nEpochs0.00000.00250.00500.00750.01000.01250.01500.01750.0200RMSE\nNN train\nNN val\nPINN train\nPINN val\nKKT-hPINN train\nKKT-hPINN val\n0 200 400 600 800 1000\nEpochs108\n107\n106\n105\n104\n103\n102\n101\nViolationFigure 5: Learning curve of the CSTR surrogate models. Left: training RMSE and\nvalidation RMSE (RMSE values for PINN are outside the limits); Right: the magnitude\nof violation of the linear equality constraints.\nleverage the neural network as a universal approximator to train a surrogate\nmodel that predicts the molar flow of ethylene, benzene, and ethylbenzene\nin the output stream. In practice, underlying equations of a physical model\nare usually much more complicated than the equation of reaction kinetics for\nthis CSTR unit, which necessitates the use of a surrogate model.\nTwo linear equations related to conservation properties are applicable\nto the input and the output streams: From the stoichiometry in Equation\n(7), it can be readily deduced that the consumption of benzene is equal to\nthe consumption of ethylene and the production of ethylbenzene, which is\nexpressed in the Table 2. It should noted that linear equality constraints\nembedded here should be independent according to Theorem 1. Therefore,\ndependent constraints, such as the relationship between the consumption\nof ethylene and the production of ethylbenzene, are not included. With the\nprior knowledge of A,Bandbin Table 2, the KKT-hPINN is able to find the\nprojection matrices and keep projecting the predictions onto feasible regions.\nComparison of training and validation loss. The trend of RMSE over epochs\nin Figure 5 illustrates the generalizability of NN, PINN and KKT-hPINN\nmodels to the validation dataset. The RMSE value of the KKT-hPINN be-\ncomes lower than that of the NN within just 1000 epochs. For the PINN,\nthe penalty term in the loss function in the (5) is excluded from RMSE cal-\n15PLUG1M1H1\nD1D3\nD4PLUG2\nH2\nD2\nB1 M2B3\nS1B2\nD2-OUT\nWATER+OHFEEDD1-IN\nWATERFIRS-OUT\nD1-TOP\nRX-IN\nDEE+METH\nDEE\nMETHRX-OUTS11\nS12\nRECY3\nRECY2DME\nS10\nRECY1\nPURGEFigure 6: Process flowsheet of the DME-DEE chemical plant.\nculation for a fair comparison. The presence of soft constraints dramatically\nhinders convergence of the PINN, leading to RMSE values that are notably\noutside the limits depicted in this figure.\nComparison of the violation of constraints. By substituting the predictions\ninto the linear equality constraints, Figure 5 also indicates the extent to\nwhich they violate the conservation laws that are established as constraints.\nIt is found that the stoichiometry does not hold in the NN model, with the\nresidual of constraints being around 10−1. Given the fact that the data has\nbeen scaled, 10−1is a significant violation. Even though much higher RMSE\nvalues are found in the PINN, it does result in a lower violation fluctuating\naround 10−4because of the presence of penalty term in the loss function. On\nthe contrary, the violation in the KKT-hPINN is always controlled at 10−7,\nwhich adequately meets practical needs. It should be noted that PyTorch\ntensors operate as float32 by default, which could potentially result in a lower\nnumerical precision. This accounts for the negligible decrease in the violation\nvalue from the tolerance of 10−8to 10−7.\n4.2. DME-DEE Chemical Plant\nThis case study focuses on a chemical plant where methanol, ethanol, and\nwater are used as feed materials to manufacture DME and DEE as shown in\nFigure 6. The feed is pre-processed to remove waste water and then trans-\nferred into reactor units. Following this, the products and reactants are\nseparated and refined, with the heavier components recycled back into the\n16preprocessing units. For illustrative simplicity, we assume that all the units\nhave been designed. In other words, the entire system is fixed and we aim to\nbuild a surrogate model that predicts the outflow given the inflow and recy-\ncling, treating the system as a black box. Nevertheless, it is straightforward\nto conclude that the flow-in is equal to the flow-out by the mass balance in\nTable 3, and this serves as the single hard constraint in this model.\nVariables Description\nx1 Mass flow rate of methanol in the FEED stream\nx2 Mass flow rate of ethanol in the FEED stream\nx3 Mass flow rate of water in the FEED stream\nx4 Total mass flow rate of the PURGE stream\ny1 Total mass flow rate of the DME stream\ny2 Mass flow rate of DME in the DME stream\ny3 Total mass flow rate of the DEE stream\ny4 Mass flow rate of DEE in the DEE stream\ny5 Total mass flow rate of the WATER stream\nAx+By=bn\nx1+x2+x3−x4−y1−y3−y5= 0,Mass balance\nTable 3: Variables and constraint of the DME-DEE chemical plant.\nComparison of training and validation loss. Figure 7 shows that all the mod-\nels can be generalized to the validation dataset. The deterministic relation-\nship in this dataset is easier to represent, as all models yield very low RMSE\nvalues. In spite of this, KKT-hPINN is still trained to a lower loss in the\nsame number of epochs.\nComparison of the violation of constraints. The magnitude of violation of\nmass balance is strictly maintained around 10−7in the KKT-hPINN model.\nOn the other hand, the predictions of NN and PINN cause higher viola-\ntions, both around 10−3. In this case, PINN does not exhibit a significant\nimprovement over NN. This implies that determining the penalty hyperpa-\nrameter for soft constraints heavily depends on trial and error. Inappropriate\nhyperparameters can result in slow convergence or significant violations.\n170 200 400 600 800 1000\nEpochs0.00000.00050.00100.00150.00200.00250.0030RMSENN train\nNN val\nPINN train\nPINN val\nKKT-hPINN train\nKKT-hPINN val\n0 200 400 600 800 1000\nEpochs108\n107\n106\n105\n104\n103\n102\n101\nViolationFigure 7: Learning curve of the DME-DEE chemical plant surrogate models. Left: training\nRMSE and validation RMSE (RMSE values for PINN are outside the limits); Right: the\nmagnitude of violation of the linear equality constraints.\nTable 4: Average RMSE ( ×10−3) over 10 runs in DME-DEE Chemical Plant test dataset\nin 1000 epochs. Overall: RMSE for all variables; Constrained: RMSE for the variables\ninvolved in the linear equality constraints; Unconstrained: RMSE for the variables not\ninvolved in the linear equality constraints.\nModel Overall Constrained Unconstrained\nNN 1 .039±0.262 0 .986±0.257 1 .090±0.254\nNNPost 1 .015±0.249 0 .964±0.251 1 .086±0.252\nPINN 4 .203±0.333 4 .187±0.271 4 .188±0.620\nKKT-hPINN 0 .767±0.082 0 .713±0.073 0 .842±0.091\n180.2 0.3 0.4\nUnused training sample ratio0.000000.000250.000500.000750.001000.001250.001500.001750.00200RMSE2.3%\n26.2%2.4%\n25.1%-0.5%\n24.9%NN\nNNPost\nKKT-hPINNFigure 8: Bar: Average RMSE of KKT-hPINN and NN post-project over 10 runs in DME-\nDEE Chemical Plant test dataset. 20%, 30% and 40% samples in the dataset are not used\nin the training process. Percentage: improvement of KKT-hPINN and NNPost relative to\nNN\nCOLUMN\nCOL-REC\nFEED RICH-SOLC7\nTOLUENE\nLEAN-SOLSOLVENT\nFigure 9: Process flowsheet of the extractive distillation subsystem.\n19Test performance. In Table 4, it is observed that KKT-hPINN remains the\nsuperior model among the others with the lowest test RMSE values. An in-\nteresting observation is that KKT-hPINN not only improves the prediction of\noutputs that are subject to the constraints but also lowers the loss of outputs\nthat are not involved in the constraints. This suggests that the addition of\ntwo projection layers provides the optimizer with a more physically feasible\nlearning direction, resulting in the parameters that cause smaller errors.\nExperiments with different training dataset sizes are conducted 10 times\nfor each. The result in Figure 8 shows that, regardless of the number of train-\ning samples, the KKT-hPINN model always decreases the overall test loss\nby around 25% relative to the NN. On the other hand, the post-projection\ncannot bring any significant improvement and sometimes may even cause de-\ncline. This aligns with Remark 3, highlighting the sharp distinction between\npost-projection after the training process and continuous projection during\nthe training process. In addition, the PINN greatly deteriorates the model\nprediction so that it is not compared in this figure.\n4.3. Extractive Distillation Subsystem\nHere, a case study of an extractive distillation subsystem as shown in\nFigure 9 is presented, in which the design includes operating specifications\nfor two distillation columns and the flow rate of the solvent. The Feed stream\nis flowing at a constant rate and consists of a 50/50 mixture of n-heptane and\ntoluene. Phenol, a solvent, is used to aid in the separation of the azeotropic\nmixture of n-heptane and toluene. The first distillation column is primarily\nresponsible for the separation of n-heptane, which is collected at the top.\nFollowing that, the second distillation column primarily separates toluene,\nwhile the phenol solvent is recovered at the bottom. The internal design of\ndistillation columns, including the number of stages and the type of reboiler\nand condenser, has already been established. However, the optimal operating\nspecifications have not been determined. In this case, the distillate rate and\nreflux ratio are chosen to specify the distillation columns. The goal is to train\na surrogate model that predicts heat duties and flow rates of each component\nin the distillate streams. As no chemical reaction occurs in the distillation\nprocess, the sum of the molar flow rate of n-heptane, toluene and phenol is\nalways equal to the distillate rate as shown in Table 5.\nComparison of training and validation loss. All three models reach the plateau\nin about 1000 epochs, and KKT-hPINN achieves a lower RMSE value. As\n20Variables Description\nx1 Molar flow rate of phenol in the SOLVENT stream\nx2 Reflux ratio of COLUMN column\nx3 Distillate rate of COLUMN column\nx4 Reflux ratio of COL-REC column\nx5 Distillate rate of COL-REC column\ny1 Molar flow rate of n-heptane in the C7 stream\ny2 Molar flow rate of toluene in the TOLUENE stream\ny3 Condenser heat duty of COLUMN column\ny4 Reboiler heat duty of COLUMN column\ny5 Condenser heat duty of COL-REC column\ny6 Reboiler heat duty of COL-REC column\ny7 Molar flow rate of toluene in the C7 stream\ny8 Molar flow rate of phenol in the C7 stream\ny9 Molar flow rate of n-heptane in the TOLUENE stream\ny10 Molar flow rate of phenol in the TOLUENE stream\nAx+By=b(\nx3−y1−y7−y8= 0,C7 fractions\nx5−y2−y9−y10= 0,TOLUENE fractions\nTable 5: Variables and constraint of the extractive distillation subsystem.\nobserved in Figure 10, the RMSE values of these models exhibit considerable\ndifferences. The hard constraints still decrease the losses of KKT-hPINN,\nimproving its ability to generalize effectively to noise-free simulation data.\nComparison of the violation of constraints. The result in Figure 10 shows\nthat the KKT-hPINN still adheres strictly to the hard constraints, whereas\nviolations of PINN fluctuate near 10−4. Even though the penalty hyperpa-\nrameter chosen in this run can reduce the violation values to 10−4in a small\nnumber of epochs, they reach a plateau for the remaining epochs.\nTest performance. The Table 6 also indicates that PINN and KKT-hPINN\nhave the highest and the lowest RMSE values respectively. This suggests\n210 200 400 600 800 1000\nEpochs0.0000.0020.0040.0060.0080.010RMSENN train\nNN val\nPINN train\nPINN val\nKKT-hPINN train\nKKT-hPINN val\n0 200 400 600 800 1000\nEpochs108\n107\n106\n105\n104\n103\n102\n101\nViolationFigure 10: Learning curve of the extractive distillation surrogate models. Left: training\nRMSE and validation RMSE (RMSE values for PINN are outside the limits); Right: the\nmagnitude of violation of the linear equality constraints.\nTable 6: Average RMSE ( ×10−3) over 10 runs in extractive distillation test dataset in 1000\nepochs. Overall: RMSE for all variables; Constrained: RMSE for the variables involved\nin the linear equality constraints; Unconstrained: RMSE for the variables not involved in\nthe linear equality constraints.\nModel Overall Constrained Unconstrained\nNN 4 .288±0.894 5 .070±1.209 2 .676±0.215\nNNPost 4 .283±0.894 5 .064±1.209 2 .676±0.215\nPINN 15 .549±2.039 18 .050±2.389 10 .623±2.171\nKKT-hPINN 3 .504±0.202 4 .090±0.239 2 .359±0.208\n220.2 0.3 0.4\nUnused training sample ratio0.0000.0010.0020.0030.0040.005RMSE0.1%2.8%0.1%\n13.5%0.1%\n18.3%NN\nNNPost\nKKT-hPINNFigure 11: Bar: Average RMSE over 10 runs in extractive distillation test dataset. 20%,\n30% and 40% samples in the dataset are not used in the training process. Percentage:\nimprovement of KKT-hPINN and NNPost relative to NN\nthat the choice of an appropriate penalty value for PINN is critical, while\nKKT-hPINN guarantees hard constraints using an analytical solution.\nIt is interesting that the RMSE values of all three models are almost at\nthe same level when only 20% of the data is not used for training in Figure 11.\nThis may be explained by the fact that the dataset in this particular example\ncontains a larger number of samples that are relatively sufficient for NN to\nrepresent the subsystem and become as competent as KKT-hPINN. However,\nwhen data are in shortage and hard to represent, projecting unconstrained\npredictions onto feasible regions where the data are known to reside becomes\nbeneficial again.\n5. Conclusions\nProcess optimization necessitates a more advanced surrogate model that\ncan strictly enforce hard constraints and exhibit superior accuracy. A physics-\ninformed neural network with hard linear equality constraints, KKT-hPINN,\nis proposed in this work. By introducing projection layers that embody an\nanalytical solution derived from KKT conditions, hard linear equality con-\nstraints are rigorously guaranteed. They guide the parameters to update\ntowards a more physically consistent and more accurate solution. Three\ncase studies have been presented to illustrate its utilization as unit-level,\nsubsystem-level, and plant-level surrogate models in process systems. In all\n23cases, it is observed that the loss decreases more rapidly in KKT-hPINN,\nleading to higher accuracy compared to other neural networks. More impor-\ntantly, all numerical results align with the mathematical guarantee of KKT-\nhPINN and distinguish it from the neural network and PINN. Overall, KKT-\nhPINN is a straightforward approach that is suitable for high-fidelity surro-\ngate modeling and maximizes the use of evident conservation constraints.\nKKT-hPINN preserves the end-to-end benefits of neural network and po-\ntentially mitigates the error propagation caused by soft constraints, which\nrenders it a prospective technique in the realm of process optimization.\nReferences\nAlhajeri, M.S., Abdullah, F., Wu, Z., Christofides, P.D., 2022. Physics-\ninformed machine learning modeling for predictive control using noisy data.\nChemical Engineering Research and Design 186, 34–49. doi: https://doi.\norg/10.1016/j.cherd.2022.07.035 .\nAnderson, R., Huchette, J., Ma, W., Tjandraatmadja, C., Vielma, J.P.,\n2020. Strong mixed-integer programming formulations for trained neu-\nral networks. Mathematical Programming 183, 3–39. doi: 10.1007/\ns10107-020-01474-5 .\nBhosekar, A., Ierapetritou, M., 2018. Advances in surrogate based mod-\neling, feasibility analysis, and optimization: A review. Computers &\nChemical Engineering 108, 250–267. doi: https://doi.org/10.1016/j.\ncompchemeng.2017.09.017 .\nBonzanini, A.D., Paulson, J.A., Makrygiorgos, G., Mesbah, A., 2021. Fast\napproximate learning-based multistage nonlinear model predictive con-\ntrol using gaussian processes and deep neural networks. Computers &\nChemical Engineering 145, 107174. doi: https://doi.org/10.1016/j.\ncompchemeng.2020.107174 .\nBoukouvala, F., Ierapetritou, M.G., 2012. Feasibility analysis of black-box\nprocesses using an adaptive sampling kriging-based method. Computers\n& Chemical Engineering 36, 358–368. doi: https://doi.org/10.1016/j.\ncompchemeng.2011.06.005 .\n24Boukouvala, F., Ierapetritou, M.G., 2013. Surrogate-based optimization of\nexpensive flowsheet modeling for continuous pharmaceutical manufactur-\ning. Journal of Pharmaceutical Innovation 8, 131–145. doi: 10.1007/\ns12247-013-9154-1 .\nBradley, W., Kim, J., Kilwein, Z., Blakely, L., Eydenberg, M., Jalvin, J.,\nLaird, C., Boukouvala, F., 2022. Perspectives on the integration between\nfirst-principles and data-driven modeling. Computers & Chemical En-\ngineering 166, 107898. doi: https://doi.org/10.1016/j.compchemeng.\n2022.107898 .\nCai, S., Wang, Z., Wang, S., Perdikaris, P., Karniadakis, G.E., 2021. Physics-\ninformed neural networks for heat transfer problems. Journal of Heat\nTransfer 143. doi: 10.1115/1.4050542 .\nCeccon, F., Jalving, J., Haddad, J., Thebelt, A., Tsay, C., Laird, C.D.,\nMisener, R., 2022. Omlt: Optimization & machine learning toolkit. J.\nMach. Learn. Res. 23.\nCozad, A., Sahinidis, N.V., Miller, D.C., 2014. Learning surrogate models for\nsimulation-based optimization. AIChE Journal 60, 2211–2227. doi: https:\n//doi.org/10.1002/aic.14418 .\nDai, W., Mohammadi, S., Cremaschi, S., 2022. A hybrid modeling frame-\nwork using dimensional analysis for erosion predictions. Computers &\nChemical Engineering 156, 107577. doi: https://doi.org/10.1016/j.\ncompchemeng.2021.107577 .\nDener, A., Miller, M.A., Churchill, R.M., Munson, T.S., Chang, C.S., 2020.\nTraining neural networks under physical constraints using a stochastic aug-\nmented lagrangian approach. ArXiv abs/2009.07330.\nDias, L.S., Ierapetritou, M.G., 2020. Integration of planning, scheduling\nand control problems using data-driven feasibility analysis and surrogate\nmodels. Computers & Chemical Engineering 134, 106714. doi: https:\n//doi.org/10.1016/j.compchemeng.2019.106714 .\nFischetti, M., Jo, J., 2018. Deep neural networks and mixed integer linear\noptimization. Constraints 23, 296–309. doi: 10.1007/s10601-018-9285-6 .\n25Goldstein, D., Heyer, M., Jakobs, D., Schultz, E.S., Biegler, L.T., 2022. Mul-\ntilevel surrogate modeling of an amine scrubbing process for co2capture.\nAIChE Journal 68, e17705. doi: https://doi.org/10.1002/aic.17705 .\nGrimstad, B., Andersson, H., 2019. Relu networks as surrogate models in\nmixed-integer linear programs. Computers & Chemical Engineering 131,\n106580. doi: https://doi.org/10.1016/j.compchemeng.2019.106580 .\nHenao, C.A., Maravelias, C.T., 2011. Surrogate-based superstructure opti-\nmization framework. AIChE Journal 57, 1216–1232. doi: https://doi.\norg/10.1002/aic.12341 .\nHinton, G., Deng, L., Yu, D., Dahl, G.E., rahman Mohamed, A., Jaitly, N.,\nSenior, A., Vanhoucke, V., Nguyen, P., Sainath, T.N., Kingsbury, B., 2012.\nDeep neural networks for acoustic modeling in speech recognition: The\nshared views of four research groups. IEEE Signal Processing Magazine\n29, 82–97. doi: 10.1109/MSP.2012.2205597 .\nJidling, C., Wahlstr¨ om, N., Wills, A., Sch¨ on, T.B., 2017. Linearly constrained\ngaussian processes, in: Advances in Neural Information Processing Sys-\ntems, Curran Associates, Inc.\nJumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger,\nO., Tunyasuvunakool, K., Bates, R., ˇZ´ ıdek, A., Potapenko, A., Bridgland,\nA., Meyer, C., Kohl, S.A.A., Ballard, A.J., Cowie, A., Romera-Paredes,\nB., Nikolov, S., Jain, R., Adler, J., Back, T., Petersen, S., Reiman, D.,\nClancy, E., Zielinski, M., Steinegger, M., Pacholska, M., Berghammer, T.,\nBodenstein, S., Silver, D., Vinyals, O., Senior, A.W., Kavukcuoglu, K.,\nKohli, P., Hassabis, D., 2021. Highly accurate protein structure prediction\nwith alphafold. Nature 596, 583–589. doi: 10.1038/s41586-021-03819-2 .\nKarniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S.,\nYang, L., 2021. Physics-informed machine learning. doi: 10.1038/\ns42254-021-00314-5 .\nKim, J., Luettgen, C., Paynabar, K., Boukouvala, F., 2023. Physics-based\npenalization for hyperparameter estimation in gaussian process regression.\nComputers & Chemical Engineering 178, 108320. doi: https://doi.org/\n10.1016/j.compchemeng.2023.108320 .\n26Kim, S.H., Boukouvala, F., 2020. Surrogate-based optimization for mixed-\ninteger nonlinear problems. Computers & Chemical Engineering 140,\n106847. doi: https://doi.org/10.1016/j.compchemeng.2020.106847 .\nKrizhevsky, A., Sutskever, I., Hinton, G.E., 2012. Imagenet classification\nwith deep convolutional neural networks, in: Advances in Neural Informa-\ntion Processing Systems, Curran Associates, Inc.\nLeCun, Y., Bengio, Y., Hinton, G., 2015. Deep learning. Nature 521, 436–\n444. doi: 10.1038/nature14539 .\nLu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., Johnson, S.G., 2021.\nPhysics-informed neural networks with hard constraints for inverse design.\nSIAM Journal on Scientific Computing 43, B1105–B1132. doi: 10.1137/\n21M1397908 .\nMa, K., Sahinidis, N.V., Amaran, S., Bindlish, R., Bury, S.J., Griffith, D.,\nRajagopalan, S., 2022a. Data-driven strategies for optimization of inte-\ngrated chemical plants. Computers & Chemical Engineering 166, 107961.\ndoi:https://doi.org/10.1016/j.compchemeng.2022.107961 .\nMa, K., Sahinidis, N.V., Bindlish, R., Bury, S.J., Haghpanah, R., Ra-\njagopalan, S., 2022b. Data-driven strategies for extractive distillation unit\noptimization. Computers & Chemical Engineering 167, 107970.\nMehrian, M., Guyot, Y., Papantoniou, I., Olofsson, S., Sonnaert, M., Mis-\nener, R., Geris, L., 2018. Maximizing neotissue growth kinetics in a\nperfusion bioreactor: An in silico strategy using model reduction and\nbayesian optimization. Biotechnology and Bioengineering 115, 617–629.\ndoi:https://doi.org/10.1002/bit.26500 .\nMisener, R., Biegler, L., 2023. Formulating data-driven surrogate models for\nprocess optimization. Computers & Chemical Engineering 179, 108411.\ndoi:https://doi.org/10.1016/j.compchemeng.2023.108411 .\nMohammadi, S., Williams, B., Cremaschi, S., 2022. Surrogate Mod-\neling and Surrogate-Based Optimization with Stochastic Simulations.\nElsevier. volume 49. pp. 31–40. doi: https://doi.org/10.1016/\nB978-0-323-85159-6.50005-1 .\n27Na, J., Bak, J.H., Sahinidis, N.V., 2021. Efficient bayesian inference using\nadversarial machine learning and low-complexity surrogate models. Com-\nputers & Chemical Engineering 151, 107322. doi: https://doi.org/10.\n1016/j.compchemeng.2021.107322 .\nNodozi, I., O’Leary, J., Mesbah, A., Halder, A., 2023. A physics-informed\ndeep learning approach for minimum effort stochastic control of colloidal\nself-assembly, in: 2023 American Control Conference (ACC), IEEE. pp.\n609–615.\nO’Leary, J., Paulson, J.A., Mesbah, A., 2022. Stochastic physics-informed\nneural ordinary differential equations. Journal of Computational Physics\n468, 111466. doi: https://doi.org/10.1016/j.jcp.2022.111466 .\nOlofsson, S., Deisenroth, M.P., Misener, R., 2018. Design of Experi-\nments for Model Discrimination using Gaussian Process Surrogate Mod-\nels. Elsevier. volume 44. pp. 847–852. doi: https://doi.org/10.1016/\nB978-0-444-64241-7.50136-1 .\nPaszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen,\nT., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Kopf, A., Yang,\nE., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B.,\nFang, L., Bai, J., Chintala, S., 2019. Pytorch: An imperative style, high-\nperformance deep learning library, in: Advances in Neural Information\nProcessing Systems, Curran Associates, Inc.\nPaulson, J.A., Shao, K., Mesbah, A., 2021. Probabilistically robust bayesian\noptimization for data-driven design of arbitrary controllers with gaussian\nprocess emulators, in: 2021 60th IEEE Conference on Decision and Control\n(CDC), pp. 3633–3639. doi: 10.1109/CDC45484.2021.9683046 .\nQuirante, N., Javaloyes, J., Ruiz-Femenia, R., Caballero, J.A., 2015. Op-\ntimization of Chemical Processes Using Surrogate Models Based on a\nKriging Interpolation. Elsevier. volume 37. pp. 179–184. doi: https:\n//doi.org/10.1016/B978-0-444-63578-5.50025-6 .\nRaissi, M., Perdikaris, P., Karniadakis, G., 2019. Physics-informed neural\nnetworks: A deep learning framework for solving forward and inverse prob-\nlems involving nonlinear partial differential equations. Journal of Compu-\ntational Physics 378, 686–707. doi: 10.1016/j.jcp.2018.10.045 .\n28Raissi, M., Yazdani, A., Karniadakis, G.E., 2020. Hidden fluid mechanics:\nLearning velocity and pressure fields from flow visualizations. Science 367,\n1026–1030. doi: 10.1126/science.aaw4741 .\nRasmussen, C.E., Williams, C.K.I., 2005. Gaussian Processes for Machine\nLearning. The MIT Press. doi: 10.7551/mitpress/3206.001.0001 .\nSchweidtmann, A.M., Bongartz, D., Grothe, D., Kerkenhoff, T., Lin, X., Na-\njman, J., Mitsos, A., 2021. Deterministic global optimization with gaussian\nprocesses embedded. Mathematical Programming Computation 13, 553–\n581. doi: 10.1007/s12532-021-00204-y .\nSchweidtmann, A.M., Mitsos, A., 2019. Deterministic global optimization\nwith artificial neural networks embedded. Journal of Optimization Theory\nand Applications 180, 925–948. doi: 10.1007/s10957-018-1396-0 .\nSutskever, I., Vinyals, O., Le, Q.V., 2014. Sequence to sequence learning with\nneural networks, in: Advances in Neural Information Processing Systems,\nCurran Associates, Inc.\nSzegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan,\nD., Vanhoucke, V., Rabinovich, A., 2015. Going deeper with convolu-\ntions, in: 2015 IEEE Conference on Computer Vision and Pattern Recog-\nnition (CVPR), IEEE Computer Society, Los Alamitos, CA, USA. pp. 1–9.\ndoi:10.1109/CVPR.2015.7298594 .\nTsay, C., Kronqvist, J., Thebelt, A., Misener, R., 2021. Partition-based for-\nmulations for mixed-integer optimization of trained relu neural networks,\nin: Advances in Neural Information Processing Systems, Curran Asso-\nciates, Inc.. pp. 3068–3080.\nWiebe, J., Cec´ ılio, I., Dunlop, J., Misener, R., 2022. A robust approach\nto warped gaussian process-constrained optimization. Mathematical Pro-\ngramming 196, 805–839. doi: 10.1007/s10107-021-01762-8 .\nWilliams, B., Cremaschi, S., 2021. Selection of surrogate modeling techniques\nfor surface approximation and surrogate-based optimization. Chemical\nEngineering Research and Design 170, 76–89. doi: https://doi.org/10.\n1016/j.cherd.2021.03.028 .\n29Wilson, Z.T., Sahinidis, N.V., 2017. The alamo approach to machine\nlearning. Computers & Chemical Engineering 106, 785–795. doi: https:\n//doi.org/10.1016/j.compchemeng.2017.02.010 .\nWilson, Z.T., Sahinidis, N.V., 2019. Automated learning of chemical reaction\nnetworks. Computers & Chemical Engineering 127, 88–98. doi: https:\n//doi.org/10.1016/j.compchemeng.2019.05.020 .\nZheng, Y., Hu, C., Wang, X., Wu, Z., 2023. Physics-informed recurrent neu-\nral network modeling for predictive control of nonlinear processes. Jour-\nnal of Process Control 128, 103005. doi: https://doi.org/10.1016/j.\njprocont.2023.103005 .\nZheng, Y., Wu, Z., 2023. Physics-informed online machine learning and\npredictive control of nonlinear processes with parameter uncertainty. In-\ndustrial & Engineering Chemistry Research 62, 2804–2818. doi: 10.1021/\nacs.iecr.2c03691 .\n30"    },    {        "title": "2402.07272v1.On_Thermodynamic_Stability_of_Black_Holes__Part_II__AdS_Family_of_Solutions.pdf",        "content": "On Thermodynamic Stability of Black Holes.\nPart II: AdS Family of Solutions\nV. Avramova, H. Dimova,b, M. Radomirova, R. C. Rashkova,c, and T. Vetsova\naDepartment of Physics, Sofia University,\n5 J. Bourchier Blvd., 1164 Sofia, Bulgaria\nbThe Bogoliubov Laboratory of Theoretical Physics, JINR,\n141980 Dubna, Moscow region, Russia\ncInstitute for Theoretical Physics, Vienna University of Technology,\nWiedner Hauptstr. 8–10, 1040 Vienna, Austria\nv.avramov,h_dimov,radomirov,rash,vetsov@phys.uni-sofia.bg\nAbstract\nThis study is aimed at providing a thorough analysis of the classical thermodynamic\nstability of black holes within the Anti-de Sitter (AdS) family. We utilize the Nambu\nbracket formalism to calculate local heat capacities and employ Sylvester’s criterion in the\nmass-energy ensemble to determine both local and global thermodynamic stability regions.\nWe explicitly show that all studied black holes have regions of both global and local stabil-\nity. Highlighting the crucial role of the cosmological constant, we establish the conditions\nnecessary for the existence of thermodynamically stable black hole configurations. Our\nwork highlights the applicability of classical thermodynamics in understanding black hole\nphysics, while acknowledging the potential deviations that may impact the observables of\nthe system beyond the classical level.\nContents\n1 Introduction 2\n2 Thermodynamic stability of Schwarzschild-AdS black hole 3\n3 Thermodynamic stability of Reissner-Nordström-AdS black hole 4\n3.1 The Reissner-Nordström-AdS black hole thermodynamics 5\n3.2 Global thermodynamic stability of RNAdS black hole 6\n3.3 Local thermodynamic stability of RNAdS black hole 7\n3.4 Davies curves and phase transitions of RNAdS 7arXiv:2402.07272v1  [gr-qc]  11 Feb 20244 Thermodynamic stability of Kerr-AdS black hole 8\n4.1 Kerr-AdS thermodynamics 8\n4.2 Global thermodynamic stability of Kerr-AdS black hole 9\n4.3 Local thermodynamic stability of Kerr-AdS black hole 10\n4.4 Critical points and phase transitions of Kerr-AdS black hole 11\n5 Thermodynamic stability of Kerr-Newman-AdS black hole 11\n5.1 Thermodynamics in Kerr-Newman-AdS spacetime 11\n5.2 Global thermodynamic stability of Kerr-Newman-AdS black hole 13\n5.3 Local thermodynamic stability of Kerr-Newman-AdS black hole 13\n6 Conclusion 17\nA Hessian of the mass of the KNAdS black hole 17\nB Section planes in the state space of KNAdS 18\nB.1 Recovering RNAdS by j= 0section 18\nB.2 Recovering Kerr-AdS by q= 0section 19\n1 Introduction\nThermodynamics of black holes [1–10] presents a promising avenue for directly testing our grav-\nitational theories. This stems from the inherent nature of thermal systems to be described by a\nfew observable parameters, a trait which carries over to black holes. These macroscopic param-\neters include mass, entropy, charge, angular momentum, and additional variables dependent on\nthe underlying gravitational model.\nIn reality these parameters can be heavily influenced by the surrounding environment. Ev-\nidence of such influence arises from the recent discovery of gravitational waves produced by\nblack hole mergers [11] and the apparent images of the supermasive black holes in the center of\nM87 and our own galaxy [12–14]. This influence can arise from various factors, such as matter\naccretion onto the event horizon [15–19], star collisions, or through radiative processes such as\nHawking radiation [6].\nIn these scenarios, a natural question arises - under what conditions does the system achieve\nthermodynamic equilibrium with its surroundings? Despite the well-established criteria for equi-\nlibrium in standard thermodynamics [20–24], this question remains largely unexplored, even for\nsimple black hole configurations. There are only a few instances where generic criteria have been\napplied to black holes, for instance [25–34].\nOur work aims to complement these studies by employing the complete strict classical criteria\nfor thermodynamic stability of the Anti-de Sitter (AdS) family of black hole solution (see for\nexample [35] and references therein). The advantage of our approach is the unified treatment\nof both local and global thermodynamic stability of such and similar systems [33,34]. We\nemphasize the importance of using natural parameters for a given thermodynamic representation\nand caution against relying solely on non-generic or partial stability criteria.\nAdS black holes provide solutions to Einstein’s equations in the presence of a negative cosmo-\nlogical constant. They include four primary backgrounds: Schwarzschild-AdS (SAdS), Reissner-\nNordström-AdS (RNAdS), Kerr-AdS, and Kerr-Newman-AdS (KNAdS) black holes. The ex-\nploration of these solutions spans a broad field of study, driven by their relevance to diverse\ndisciplines. A prominent example is the well-known AdS/CFT correspondence [36], where AdS\nblack holes are crucial in exploring various aspects of this duality, shedding light on the connec-\ntion between quantum field theory and gravity at finite temperature.\n2Moreover, AdS black holes serve as invaluable tools in exploring phenomena beyond tradi-\ntional gravitational theory. Their study facilitates investigations into topics like black hole ther-\nmodynamics [37], phase transitions in quantum field theory [38], and the behavior of strongly\ncorrelated systems in condensed matter physics [39]. The adaptability of AdS black holes broad-\nens their utility to cosmological models [40–43], providing frameworks for comprehending the\ndynamicsoftheearlyuniverseandtheimplicationsofnegativecosmologicalconstantsingravita-\ntional physics. Consequently, a coherent and systematic analysis of their properties, particularly\nconcerning thermodynamic stability, becomes imperative.\nThis paper constitutes the second part of a series dedicated to investigating the dynamic and\nthermodynamiccharacteristicsofdiverseblackholes. Herein, weemployoneofthemostrigorous\ncriteria for assessing global classical thermodynamic stability, known as the Sylvester criterion,\nwhich ensures the positive definiteness of the mass-energy Hessian form. This criterion arises\nfrom the strict convexity property of the energy potential in thermodynamics. When extended\nto black hole systems, classical criteria merely hint at the limitations of classical thermodynam-\nics when considering stability against fluctuations or spontaneous processes. Beyond this realm,\nquantum laws or deviations from classical setups may also manifest. For instance, certain scenar-\nios suggest potential violations of the third law of thermodynamics by black holes [10,44], which\nholds significance in the context of string theory and similar approaches to quantum gravity.\nOur analysis of the thermodynamic stability of the AdS family of black hole solutions es-\ntablishes in details the boundaries and the applicability of classical thermodynamic stability for\nAnti-de-Sitter black holes, where the cosmological constant is present as a true constant, which is\nthe main goal of this paper. Upon extension of this analysis, a natural question arises: how will\nclassical stability be influenced if the cosmological constant transitions into a thermodynamic\nparameter [45–48]? The ramifications of this scenario will be explored in the forthcoming third\ninstallment of this research series of works.\nEmploying a bottom-up approach, we commence with the simplest solutions and proceed\ntowards more intricate configurations. The organization of the text is outlined as follows: In\nSection 2, we explore the thermodynamic stability of the Schwarzschild-AdS black hole, demon-\nstrating how the inclusion of a cosmological constant results in a stable black hole configuration.\nMoving to Section 3, we analyze the RNAdS scenario, where the presence of an electric charge\nimposes additional constraints on the regions of thermodynamic stability, albeit maintaining\nglobally stable sectors. Section 4 presents a comprehensive analysis of the thermodynamic sta-\nbility in the presence of angular momentum, identifying both local and global stable regions,\nalongwiththeidentificationofpossibleDaviescurvesandphasetransitionpointsinbothRNAdS\nand Kerr-AdS cases. Finally, in Section 5, we unveil a significantly more intricate structure in the\nthermodynamic phase space of the KNAdS black hole. Here, distinct Davies surfaces correspond\nto the divergences of heat capacities, alongside higher-dimensional regions of local and global\nthermodynamic stability. Additionally, we illustrate how the previous solutions are embedded\nwithin the KNAdS state space (refer to Appendix B).\n2 Thermodynamic stability of Schwarzschild-AdS black hole\nThe metric of the Schwarzschild-anti-de-Sitter black hole solution (SAdS), with negative cosmo-\nlogical constant1Λ =−3/l2, is written by2[37,38]:\nds2=−\u0012\n1−2M\nr+r2\nl2\u0013\ndt2+\u0012\n1−2M\nr+r2\nl2\u0013−1\ndr2+r2\u0000\ndθ2+ sin2θ dϕ2\u0001\n,(2.1)\n1We will adopt units for which κ= 8π, thus G= 1.\n2In static spherical coordinates. In the limit l→ ∞one recovers the Schwarzschild case.\n3where l >0is the radius of the AdS space and Mis the mass of the black hole. The event\nhorizon is the real solution to the cubic equation r3+l2r−2l2M= 0with M > 0andl >0. The\nthermodynamics of the SAdS black hole is defined by its mass M, entropy Sand temperature\nT, given by\nM=1\n2r\nS\nπ\u0012\n1 +S\nπl2\u0013\n, T =∂M\n∂S=1\n4√\nπS\u0012\n1 +3S\nπl2\u0013\n, dM =TdS. (2.2)\nIt will be useful to introduce the following rescaled parameters:\nm=2M\nl, τ = 2πlT, s =S\nπl2, (2.3)\nhence the SAdS thermodynamics becomes\nm=√s(1 +s), τ =∂m\n∂s=1 + 3 s\n2√s, dm =τds. (2.4)\nIn the mass-energy representation m=m(s)there is only one condition for classical global\nthermodynamic stability, which follows from the\nHss≡∂2m\n∂s2=3s−1\n4s3/2>0⇒ s >1\n3. (2.5)\nThis suggests that the SAdS solution can be thermodynamically stable only if the black hole has\nentropy S > πl2/3. On the other hand, the heat capacity of the Schwarzschild-AdS black hole,\nC=∂m\n∂τ=∂m\n∂s1\n∂τ\n∂s=2s(1 + 3 s)\n3s−1>0⇒ s >1\n3, (2.6)\ndetermines its local thermodynamic stability. It is evident that both conditions for global and\nlocal thermodynamic equilibrium coincide for s >1/3.\nTaking the above analysis into account one can consider the SAdS solution to be locally and\ngloballystablefromclassicalthermodynamicstandpoint,whichisincontrasttotheSchwarzschild\ncase ( l→ ∞) [33]. Hence, the introduction of a finite cosmological parameter lleads to thermo-\ndynamically stable black hole configuration. This, however, is not a trivial result. For S < πl2/3,\nthe Schwarzschild-AdS black hole is classically unstable, thus can radiate. The heat capacity\ndiverges on the curve S=πl2/3, thus the point S=πl2/3is a Davies phase transition point.\nA note of caution should be marked here. If one considers positive semi-definiteness of the\nHessian Hss≥0there will be a contradiction between the local and the global thermodynamic\nstability, because the point S=πl2/3will be a part of the global region of stability, while it\ncorresponds to a divergence in the heat capacity. This is an explicit example of why one should\nuse the strong positive definiteness of the Hessian Hss>0and not the weaker semi-definite one.\nThe explanation of the stability regions in the context of Schwarzschild Anti-de Sitter (SAdS)\nis as follows. In the case of stable black hole configurations, the criterion S=πr2\n+> πl2/3\nimplies r2\n+> l2/3, indicating that the black hole horizon’s radius is comparable to the AdS space\nradius. On the other hand, for unstable configurations, r2\n+< l2/3, suggesting that observable\nastrophysical black holes with significantly smaller radii will emit radiation. The critical radius\nr2\n+=l2/3is the Davies point separating thermodynamically stable and unstable regions, a.i. it\nrepresents the local minimum of the Hawking temperature, Tmin=√\n3/(2πl).\n3 Thermodynamic stability of Reissner-Nordström-AdS black\nhole\nWe investigate the Reissner-Nordström-AdS black hole solution, where the inclusion of an elec-\ntric charge imposes supplementary constraints on the regions of thermodynamic stability. Our\n4analysis discerns both local and global stable regions, and also elucidates potential Davies curves\nand phase transition points. Moreover, we underscore the necessity of accounting for the reduc-\ntion of the thermodynamic phase space when processes are constrained by certain parameters,\nthus limiting cases should be approached with caution.\n3.1 The Reissner-Nordström-AdS black hole thermodynamics\nThe metric of the Reissner-Nordström-AdS (RNAdS) black hole is written by [49–51]:\nds2=−\u0012\n1−2M\nr+Q2\nr2+r2\nl2\u0013\ndt2+\u0012\n1−2M\nr+Q2\nr2+r2\nl2\u0013−1\ndr2+r2\u0000\ndθ2+ sin2θdϕ2\u0001\n,(3.1)\nwhere Mis the mass and Q∈Ris the electric charge of the RNAdS. The event horizon r+can\nbe determined as a real solution to the quartic equation r4+l2r2−2l2Mr+l2Q2= 0. This\nequation also provides the inner Cauchy horizon r−. The presence of a black hole is established\nby the condition r+> r−, which is related to the critical mass parameter M∗of the extremal\nRNAdS black hole ( r+=r−) as defined in [51]:\nM > M ∗=l\n3√\n6 r\n12Q2\nl2+ 1 + 2!\u0012r\n12Q2\nl2+ 1−1\u00131/2\n. (3.2)\nSimilarly to the SAdS case we can introduce the following rescaled thermodynamic parame-\nters3:\nm=2M\nl, τ = 2πlT, s =S\nπl2, ϕ = 2Φ , q =Q\nl, (3.3)\nwhere Tis the Hawking temperature, Sis the entropy and Φis the electric potential of the\nsolution. In these notations the thermodynamics of the RNAdS black hole takes the form:\nm=s2+s+q2\n√s, τ =∂m\n∂s\f\f\f\f\nq=3s2+s−q2\n2s3/2, ϕ =∂m\n∂q\f\f\f\f\ns=2q√s, dm =τds+ϕdq.(3.4)\nHereweassumetheenergyrepresentation, hence m,τandϕbecomefunctionsof (s, q). Imposing\npositive entropy s >0and positive temperature τ >0, we find\nq2< s(1 + 3 s). (3.5)\nThis condition coincides with (3.2), which makes (3.5) sufficient for the existence of the RNAdS\nsolution. Therefore the physical region is above the blue hyperbola in ( s, q) space (Fig.1) given\nby the following equation\n3s2+s−q2= 0. (3.6)\nWe can write the existence condition (3.5) together with (3.4) also as:\n(1 +s)√s < m < 2(1 + 2 s)√s,0< τ <1 + 3 s\n2√s,|ϕ|<2√\n1 + 3 s. (3.7)\n3Our scales differ from those in [52] by some factors of πor 2.\n5Figure 1: The existence of the black hole is defined above the blue hyperbola (b >0). Above the blue\nhyperbola and outside the green ellipse (b >0, g > 0)the RNAdS black hole can be globally stable.\nInside the green ellipse (g <0), which corresponds to the Davis curve for Cϕ, the black hole is unstable\nfor processes with constant ϕ. Inside the orange ellipse (o <0)– the Davis curve for Cq– the black\nhole is unstable for processes with constant q.\n3.2 Global thermodynamic stability of RNAdS black hole\nWe can now study the conditions for global thermodynamic stability of the RNAdS black hole.\nFor this purpose it would be useful to define the following functions:\nb(s, q) = 3 s2+s−q2, (3.8)\ng(s, q) = 3 s2−s+q2, (3.9)\no(s, q) = 3 s2−s+ 3q2, (3.10)\nwherethechosenletters( b-blue, g-green, o-orange)correspondtothecolorsofthecurvesdepicted\non Fig 1. The Hessian of the mass in (s, q)space is given by\nH= \nHssHsq\nHqsHqq!\n= ∂2m\n∂s2∂2m\n∂s∂q\n∂2m\n∂q∂s∂2m\n∂q2!\n= o(s,q)\n4s5/2−q\ns3/2\n−q\ns3/22√s!\n. (3.11)\nThe conditions for global thermodynamic stability require positive definite Hessian, i.e.\nHss=o(s, q)\n4s5/2>0,Hqq=2√s>0,detH=g(s, q)\n2s3>0. (3.12)\nOne notes that Hqqis always positive, while Hssis positive only if\no(s, q)>0. (3.13)\nThis condition is satisfied outside the orange ellipse, given by o(s, q) = 0(Fig, 1), which can be\nidentified as the Davies critical curve for Cq(see Eq.(3.18)). The determinant of the Hessian is\npositive if\ng(s, q)>0, (3.14)\nwhich is fulfilled outside the green ellipse, defined by g(s, q) = 0. The latter corresponds to the\nDavies curve for Cϕ(3.17). The conditions for global thermodynamic stability should agree with\nthe condition for the existence (3.5) of the black hole. This leads to\ns(1−3s)< q2< s(1 + 3 s), (3.15)\nwhichresidesintheregionabovethebluehyperbola (b >0)andoutsidethegreenellipse (g >0).\nThe equation (3.15) defines the parametric region that characterizes the global thermodynamic\n6stability of the RNAdS solution. Understanding the significance of this region, we can visualize\nthe macrostate of the black hole as a point within the (s, q)space. This point can either move\nwithinthe (s, q)spacethroughcertainprocessesorremainfixed, maintainingequilibriumwithits\nsurroundings. However, outside the region of global stability, the point cannot remain fixed, and\nthe black hole will undergo spontaneous changes in its macrostate, preventing it from reaching\nglobal equilibrium.\n3.3 Local thermodynamic stability of RNAdS black hole\nThe relevant heat capacities of the RNAdS black hole in (s, q)space are given by\nCm=τ∂s\n∂τ\f\f\f\f\nm=τ{s, m}s,q\n{τ, m}s,q=s(3s2+s−q2)\n3s2+q2, (3.16)\nCϕ=τ∂s\n∂τ\f\f\f\f\nϕ=τ{s, ϕ}s,q\n{τ, ϕ}s,q=2s(3s2+s−q2)\n3s2−s+q2, (3.17)\nCq=τ∂s\n∂τ\f\f\f\f\nq=τ{s, q}s,q\n{τ, q}s,q=2s(3s2+s−q2)\n3s2−s+ 3q2. (3.18)\nLet us consider a processes with constant mass. One can see that Cmis always positive\nin the region of existence (3.5), thus the black hole is locally stable there with respect to Cm.\nFurthermore, outside the green ellipse (g >0)(Fig. 1) the black hole can be globally stable.\nHowever, inside the green ellipse (g <0)the equilibrium is only local with respect to processes\nwith constant mass.\nNext, we consider processes with constant electric potential. The condition Cϕ>0matches\nthe region of global stability (3.15), hence above the blue hyperbola (b >0)and outside the\ngreen ellipse (g >0)the RNAdS black hole can be locally and globally stable. Inside the green\nellipse Cϕ<0, hence the black hole is thermodynamically unstable. Note that the green ellipse\n(g= 0)is a Davies curve for Cϕ, thus defining the locus of possible phase transitions.\nFinally, we consider processes with constant electric charge. In this case, the region of local\nstability Cq>0is located outside the orange ellipse (o >0). In the sector outside the green\nellipse (g > 0)the black hole can be locally and globally stable. Between the two ellipses\n(o >0, g < 0)the black hole could maintain only a local equilibrium. Inside the orange ellipse\ntheblackholeisunstable Cq<0. NotethattheorangeellipseisDaviescurvefor Cq, representing\npossible phase transitions.\nAll of the heat capacities are positive in the region of global stability (3.15) (above the blue\nhyperbola (b >0)and outside the green ellipse (g >0)(Fig. 1). However, we showed that the\nRNAdS solution also admits some partial regions of local thermodynamic stability, located inside\nthe green ellipse. Once again global stability is assured due to the presence of a cosmological\nconstant.\n3.4 Davies curves and phase transitions of RNAdS\nThe critical Davies curves mark the points where heat capacities exhibit divergences or change\nsigns ( C→0,±∞). Specifically, for Cm, one observes:\nCm→+0forb(s, q)→+0,(τ→+0), (3.19)\nIt is important to note that the limit above is not direct. This is due to the fact that at fixed\nmass the parameters sandqare not independent of each other, i.e. they satisfy (3.4)\nm=const =s2+s+q2\n√s→2(1 + 2 s)√s,when b(s, q)→+0. (3.20)\n7However, one notes that this particular value of the mass corresponds to its upper limit from\nthe condition of existence (3.7). Hence the limit b(s, q)→+0indicates a true phase transition\npoint. The curve b(s, q) = 0corresponds to the blue hyperbola as shown on Figure 1.\nMoreover, examining the limit q= 0presents no issue, as both mass and heat capacity\nremain finite and regular:\nm= (1 + s)√s, C m=1 + 3 s\n3, (3.21)\nwhich corresponds to the SAdS scenario, albeit with a different heat capacity Cmcompared\nto (2.6). This discrepancy arises from our approach to this point via a specific process with\nan associated heat capacity Cm, influenced by the presence of a charge. In the absence of any\ncharge ( q= 0), the system can indefinitely persist in the SAdS state, characterized by its distinct\nheat capacity as defined in (2.6). Consequently, we can relax the existence conditions (3.7) to\nencompass the lower limit of mass and the upper limit of temperature:\n(1 +s)√s≤m < 2(1 + 2 s)√s, 0< τ≤1 + 3 s\n2√s. (3.22)\nSimilar analysis can be done on the other two heat capacities. For example, considering a\nprocess with constant electric potential, one has\nCϕ→\n\n+0, b(s, q)→+0,(τ→+0),\n+∞, g(s, q)→+0,\n−∞, g(s, q)→ −0.(3.23)\nConsequently, considering processes with constant charge one finds\nCq→\n\n+0, b(s, q)→+0,(τ→+0),\n+∞, o(s, q)→+0,\n−∞, o(s, q)→ −0.(3.24)\nAll heat capacities converge to zero along the blue hyperbola (3.6), where the black hole\ntemperature approaches absolute zero, τ→0. This signifies the extremal scenario4, which re-\nmains unattainable through a finite number of processes due to the third law of thermodynamics.\nMoreover, Cϕ→+∞as we approach the outer boundary of the green ellipse (g→+0), and to\nnegative infinity when approaching from the inside (g→ −0). Additionally, Cqdiverges towards\npositive (negative) infinity when approaching the outer (inner) edge of the orange ellipse.\n4 Thermodynamic stability of Kerr-AdS black hole\nIn this section, we provide an in-depth analysis of the thermodynamic stability of Kerr-AdS\nblack holes with angular momentum. We identify both local and global stable regions, as well\nas highlight possible Davies curves and phase transition points.\n4.1 Kerr-AdS thermodynamics\nThe Kerr-AdS metric is given by (5.1) at Q= 0, [53,54]. Its thermodynamics can be written in\nthe form:\nm=p\n(1 +s)(j2+s2+s3)√s, τ=s2(1 +s)(1 + 3 s)−j2\n2p\ns3(1 +s)(j2+s2+s3), ω=j√1 +sp\ns(j2+s2+s3),(4.1)\n4It’s worth noting that extremal cases may hold significance in other theories such as modified gravity or\nstring theory, where violations of the third law of thermodynamics can occur [9,44].\n8where\nm=2M\nl, τ = 2πlT, s =S\nπl2, ω =lΩ, j =2J\nl2, dm =τds+ωdj. (4.2)\nIn this case, the mass m, the temperature τand the angular velocity ωare functions of (s, j).\nIt would be useful to define the following functions:\nb(s, j) =s2(1 +s)(1 + 3 s)−j2, (4.3)\ng(s, j) =s2(1 +s)\u0000\n2p\ns(1 +s)−(1 +s)\u0001\n−j2, (4.4)\no(s, j) =s4(1 +s)3(3s−1) + 6 j2s2(1 +s)2(1 + 2 s) +j4(3 + 4 s). (4.5)\nThe region of existence of Kerr-AdS black hole in ( s, j) space is defined by s >0andτ >0,\nwhich lead to the relation\nj2< s2(1 +s)(1 + 3 s). (4.6)\nIt is satisfied above the blue curve b(s, j) = 0(Fig.2). The condition for existence (4.6) together\nwith (4.1) lead to the following restrictions on the parameters of the black hole\n(1 +s)√s≤m < (1 +s)p\ns(2 + 3 s),0< τ≤1 + 3 s\n2√s,|ω|<s\n(1 +s)(1 + 3 s)\ns(2 + 3 s).(4.7)\nFigure 2: The existence of the black hole is defined above the blue curve (b>0). Above the green\ncurve (g>0)the system can be in global equilibrium. Below the green curve (g<0), which corresponds\nto the Davis curve for Cω, the black hole is unstable for processes with constant ω. Inside the orange\ncurve ( 0<0) – the Davis curve for Cj, the black hole is unstable for processes with constant j.\n4.2 Global thermodynamic stability of Kerr-AdS black hole\nThe Hessian of the mass in (s, j)space yields5\nH= \nHssHsj\nHjsHjj!\n= ∂2m\n∂s2∂2m\n∂s∂j\n∂2m\n∂j∂s∂2m\n∂j2!\n, (4.8)\nwith components given by:\nHss=o(s, j)\n4q\ns5(1 +s)3(j2+s2+s3)3,Hjj=\u0012s(1 +s)\nj2+s2+s3\u00133/2\n, (4.9)\n5See [55] for grand canonical treatment of stability.\n9Hsj=Hjs=−j\u0000\nj2+ 3s2(1 +s)2\u0001\n2q\ns3(1 +s) (j2+s2+s3)3,detH=g(s, j)\n4s3(1 +s) (j2+s2+s3)2.(4.10)\nThe conditions for global thermodynamic stability require\nHss>0,Hjj>0,detH>0. (4.11)\nOne notes that Hjjis always positive, while Hssis positive when the numerator is positive, i.e.\no(s, j)>0. (4.12)\nThis relation is fulfilled outside the closed orange curve, given by o(s, j) = 0(Fig.2). Further-\nmore, the determinant of the Hessian detHis positive when its numerator is positive:\ng(s, j)>0. (4.13)\nThis condition is satisfied above the green curve, given by g(s, j) = 0. From (4.12) and (4.13)\nwe conclude that the Kerr-AdS black hole can reside in global equilibrium above the green curve\n(g>0), e.i.:\nj2< s2(1 +s)\u0000\n2p\ns(1 +s)−(1 +s)\u0001\n. (4.14)\n4.3 Local thermodynamic stability of Kerr-AdS black hole\nThe heat capacities of the Kerr-AdS black hole in (s, j)space are given by\nCm=τ∂s\n∂τ\f\f\f\f\nm=τ{s, m}s,j\n{τ, m}s,j=s(1 +s)b(s, j)\ns2(1 +s)2(1 + 6 s) +j2(1 + 2 s), (4.15)\nCω=τ∂s\n∂τ\f\f\f\f\nω=τ{s, ω}s,j\n{τ, ω}s,j=2s3(1 +s)2b(s, j)\ns4(1 +s)3(3s−1)−2j2s2(1 +s)2−j4, (4.16)\nCj=τ∂s\n∂τ\f\f\f\f\nj=τ{s, j}s,j\n{τ, j}s,j=2s(1 +s)(j2+s2+s3)b(s, j)\no(s, j). (4.17)\nIn the region of existence (4.6) of the black hole Cm>0, hence we have local stability with\nrespect to processes with constant mass. Above the green curve one may have local and global\nequilibrium altogether (Fig. 2). Between the blue and the green curves the black hole is only in\nlocal equilibrium.\nUnder a process with constant angular velocity one has Cω>0within the region of global\nstability (4.14), given above the green curve. Between the blue and the green curves the black\nhole is unstable, Cω<0.\nFinally, we consider processes with constant angular momentum. In this case, the region of\nlocal stability Cj>0is located outside the closed orange curve (o>0). In the sector above\nthe green curve (g>0)the black hole can be locally and globally stable. Between the blue and\nthe green curve and outside the closed orange curve ( b>0,g<0ando>0)the black hole\ncould maintain only a local equilibrium. Inside the closed orange curve (o<0)the black hole is\nunstable Cj<0.\nFinally, one notes that all of the heat capacities are positive in the region of global stability\n(above the green curve).\n104.4 Critical points and phase transitions of Kerr-AdS black hole\nThe critical curves of the Kerr-AdS black hole are defined by:\nCm→0,b(s, j)→+0, C j→\n\n0,b(s, j)→+0,\n∞,g(s, j)→+0,\n−∞,g(s, j)→ −0,Cω→\n\n0,b(s, j)→+0,\n∞,o(s, j)→+0,\n−∞,o(s, j)→ −0.(4.18)\nAll of the heat capacities go to zero on the blue curve (4.3), where the temperature of the black\nhole is zero. Furthermore, Cω→ ∞ (−∞)when we approach the green curve from above (below).\nAdditionally, Cj→ ∞ (−∞)when we approach the closed orange curve (4.5) from the outside\n(inside). Finally the limit j→0is a regular limit towards SAdS black hole.\n5 Thermodynamic stability of Kerr-Newman-AdS black hole\nWe are now ready to consider the classical thermodynamic stability of the more general case of\nKerr-Newman-AdS black hole (KNAdS).\n5.1 Thermodynamics in Kerr-Newman-AdS spacetime\nThe metric of the Kerr-Newman-AdS black hole6is given by [35,54,56]:\nds2=−∆r\nρ2\u0012\ndt−asin2θ\nΞ−dϕ\u00132\n+ρ2\n∆rdr2+ρ2\n∆θdθ2+∆θsin2θ\nρ2\u0012\nadt−r2+a2\nΞ−dϕ\u00132\n,(5.1)\nwhere the following notations have been adopted:\nρ2=r2+a2cos2θ,Ξ−= 1−a2\nl2,Ξ+= 1 +a2\nl2, (5.2)\n∆r= (r2+a2)\u0012\n1 +r2\nl2\u0013\n−2µr+Q2,∆θ= 1−a2\nl2cos2θ. (5.3)\nHere aisthespecificangularmomentum, Qisthespecificchargeparameter, µisthespecificmass\nparameter as defined in Eq. (5.5). Additionally, lis the AdS radius related to the cosmological\nconstant by Λ =−3/l2. For a2< l2andµ > µ ∗the metric (5.2) represents AdS black hole with\nan event horizon located at r=r+, where r+is the largest solution of ∆r= 0. The parameter\nµ∗is the critical mass parameter defined by the extremal case r+=r−[35]:\nµ∗=l\n3√\n6\u0012r\nΞ2\n++12\nl2(a2+Q2) + 2Ξ +\u0013\u0012r\nΞ2\n++12\nl2(a2+Q2)−Ξ+\u00131/2\n.(5.4)\nThe thermodynamics of the KNAdS black hole in mass-energy representation has been pre-\nsented in [35], where the mass M, the entropy S, the angular momentum J, and the charge Q\nof the solution can be written by\nM=µ\nΞ2\n−, S =π(r2\n++a2)\nΞ−, J =aµ\nΞ2\n−, Q =Q\nΞ−. (5.5)\nThe Hawking temperature T, the angular velocity Ωand the charge potential Φyield:\nT=r+\n4π(r2\n++a2)\u0012\nΞ++3r2\n+\nl2−a2+Q2\nr2\n+\u0013\n,Ω =a\nr2\n++a2\u0012\n1 +r2\n+\nl2\u0013\n,Φ =Qr+\nr2\n++a2.(5.6)\n6It reduces to the standard KN solution for l→ ∞.\n11The fundamental relation M=M(S, J, Q )can be obtained by solving Eqs. (5.5) with respect\ntor+, a, µ,Q, i.e.\nr+=p\nl2M2S−J2(πl2+S)√πlM, a =J\nM, µ =(J2−l2M2)2\nl4M3,Q=Q\u0012\n1−J2\nl2M2\u0013\n.(5.7)\nInserting these expressions back into Eqs. (5.5) and (5.6) one obtains7the mass-energy thermo-\ndynamic representation of KNAdS system:\nM2=S\n4π+π\n4S(4J2+Q4) +Q2\n2+J2\nl2+S\n2πl2\u0012\nQ2+S\nπ+S2\n2π2l2\u0013\n, (5.8)\nT=∂M\n∂S\f\f\f\f\nJ,Q=1\n8πM\u0014\n1−π2\nS2(4J2+Q4) +2\nl2\u0012\nQ2+2S\nπ\u0013\n+3S2\nπ2l4\u0015\n, (5.9)\nΩ =∂M\n∂J\f\f\f\f\nS,Q=πJ\nMS\u0012\n1 +S\nπl2\u0013\n, (5.10)\nΦ =∂M\n∂Q\f\f\f\f\nS,J=πQ\n2MS\u0012\nQ2+S\nπ+S2\nπ2l2\u0013\n. (5.11)\nConsequently, the first law for KNAdS takes the form\ndM=TdS+ ΩdJ+ ΦdQ. (5.12)\nAssuming l >0andl̸=∞we can rescale the thermodynamic parameters:\nm=2M\nl, τ = 2πlT, s =S\nπl2, ω =lΩ, j =2J\nl2, ϕ = 2Φ , q =Q\nl,(5.13)\nwhich transform the KNAdS thermodynamics into:\ndm=τds+ωdj+ϕdq,\nm=p\nj2(1 +s) + (s2+s+q2)2\n√s, (5.14)\nτ=∂m\n∂s\f\f\f\f\nj,q=(s2+s+q2)(3s2+s−q2)−j2\n2s3/2p\nj2(1 +s) + (s2+s+q2)2, (5.15)\nω=∂m\n∂j\f\f\f\f\ns,q=j(1 +s)√sp\nj2(1 +s) + (s2+s+q2)2, (5.16)\nϕ=∂m\n∂q\f\f\f\f\ns,j=2q(s2+s+q2)√sp\nj2(1 +s) + (s2+s+q2)2. (5.17)\nLet us look at the possible restrictions on the parameters. For τ >0one finds\nj2<(s2+s+q2)(3s2+s−q2). (5.18)\nOn the other hand, the condition µ > µ ∗becomes\n9 (m2−j2)2\nq\n6(√\nX−j2−m2)\u0010√\nX+ 2(j2+m2)\u0011>1, (5.19)\nwhere\nX=j4+ 12q2\u0000\nm2−j2\u00012+ 14j2m2+m4and X > 0. (5.20)\nInserting mfrom (5.14) the condition (5.19) reduces to (5.18). Therefore, we consider (5.18) as\nthe sufficient condition for the existence of the KNAdS black hole. It is fulfilled above the blue\nsurface depicted on all of the figures below.\n7For the extended phase space thermodynamics see [35], where ℓ2=−3/Λ,P=−Λ/(8π),Θ =−V/(8π)and\nthe mass of the black hole M=E+PVis now identified as the enthalpy of spacetime. See also [57].\n125.2 Global thermodynamic stability of Kerr-Newman-AdS black hole\nAfter identifying the region of existence we can study the conditions for global thermodynamic\nstability of the KNAdS black hole. The Hessian of the mass in (s, j, q )space is defined by\nH=\nHssHsjHsq\nHjsHjjHjq\nHqsHqjHqq\n=\n∂2m\n∂s2∂2m\n∂s∂j∂2m\n∂s∂q\n∂2m\n∂j∂s∂2m\n∂j2∂2m\n∂j∂q\n∂2m\n∂q∂s∂2m\n∂q∂j∂2m\n∂q2\n, (5.21)\nThe conditions for global thermodynamic stability require Hto be positive definite matrix, i.e.\nHss>0,Hjj>0,Hqq>0,∆s>0,∆j>0,∆q>0,detH>0,(5.22)\nwhere the explicit components of Hand its principle minors ∆s,j,qare listed in Appendix A. To\nanalyzing these conditions we have to define the following functions:\nB(s, j, q ) =A1A2−j2, (5.23)\nP(s, j, q ) =A3\n1(3A1−4s) + 2j2B1+j4(3 + 4 s), (5.24)\nO(s, j, q ) =A4\n1\u0002\nA1+ 2s(s−1)\u0003\n+ 2j2A1B2+j4(A1+ 2q2)(3 + 4 s), (5.25)\nR(s, j, q ) =A3\n1(3A1−4s)(1 + s) + 2j2A1\u0002\nq2(1 + 3 s)−s(1 +s)2\u0003\n−j4(1 +s), (5.26)\nG(s, j, q ) =A4\n1\u0002\nA1+ 2s(s−1)\u0003\n(1+s)−2j2A2\n1\u0002\nA1+sA2−2s3\u0003\n−j4(A1+ 2q2)(1+s).(5.27)\nWe choose their letters to signify the color of the surface they represent when equal to zero as\ndepicted on the figures below, namely: B– blue, P– purple, O– orange, R– red, G– green.\nHere we also define the following positive quantities:\nA1=s2+s+q2>0, (5.28)\nA2= 3s2+s−q2>0, (5.29)\nB1=q4(3+2 s) + 2q2s(2+3 s) + 3s2(1+s)2(1+2 s)>0, (5.30)\nB2=q4(5 + 2 s) + 2q2s(4s(s+ 2) + 3) + 3 s2(1 +s)2(1 + 2 s)>0. (5.31)\nLet us first look at Hssfrom (A.2). It is positive outside the purple closed surface P= 0,\nwhere P >0. On the other hand, Hjj,Hqqand∆sare positive everywhere, while ∆jis positive\noutside ( O > 0) the closed orange surface O= 0, and ∆qis positive above ( R > 0) the red\nsurface R= 0. Finally, detHis positive above ( G > 0) the green surface G= 0. The region of\nglobal stability should be consistent with the region of existence (5.18) of the black hole, which\nis above the blue curve ( B > 0). Therefore, above both blue and green surfaces\nG(s, j, q )>0and B(s, j, q )>0, (5.32)\none can achieve global equilibrium. Below the green surface (G <0)KNAdS can be considered\nas only locally stable or unstable against fluctuations of the parameters.\n5.3 Local thermodynamic stability of Kerr-Newman-AdS black hole\nThe KNAdS black hole possesses various nontrivial heat capacities, and we will provide complete\nexpressions for each of them below to illustrate their definitions.\nFirst we consider all heat capacities at fixed mass min(s, j, q )space:\nCm,ω=τ∂s\n∂τ\f\f\f\f\nm,ω=τ{s, m, ω }s,j,q\n{τ, m, ω }s,j,q=A1\u0000\nA1A2−j2\u0001\ns(1 +s)\nA2\n1(q2+ 3s2)(1+s) +j2\u0002\nsA1+q2(1+2 s)\u0003, (5.33)\n13Cm,j=τ∂s\n∂τ\f\f\f\f\nm,j=τ{s, m, j }s,j,q\n{τ, m, j }s,j,q=A1\u0000\nA1A2−j2\u0001\ns\nA2\n1(q2+ 3s2) +j2\u0002\nA1+s(1+2 s)\u0003, (5.34)\nCm,ϕ=τ∂s\n∂τ\f\f\f\f\nm,ϕ=τ{s, m, ϕ }s,j,q\n{τ, m, ϕ }s,j,q=(A1+ 2q2)\u0000\nA1A2−j2\u0001\ns(1 +s)\nA1\u0000\nB2−2A2\n1\u0001\n+j2(A1+ 2q2)(1+2 s), (5.35)\nCm,q=τ∂s\n∂τ\f\f\f\f\nm,q=τ{s, m, q }s,j,q\n{τ, m, q }s,j,q=\u0000\nA1A2−j2\u0001\ns(1 +s)\nq4(1+2 s) + 2q2s2+s2(1+s)2(1+6 s) +j2(1+2 s).(5.36)\nAll of them are positive in the range of existence of the black hole (5.18), which is above the\nblue surface (Fig. 3). The local stability is defined in the region between the blue and the green\nsurface ( B > 0andG <0). Above these two surfaces ( B > 0andG >0) the black hole can be\nglobally stable (5.32).\nFigure 3: The blue surface represents the classically forbidden extremal case. Above the green surface\nthe KNAdS black hole can be globally stable. One note also that the green surface corensponds to the\nDavies surface of Cω,ϕ.\nThe next set of local heat capacities are defined at constant angular velocity, namely:\nCω,j=τ∂s\n∂τ\f\f\f\f\nω,j=τ{s, ω, j}s,j,q\n{τ, ω, j}s,j,q=A1\u0000\nA1A2−j2\u0001\ns(1 +s)\nA2\n1\u0002\nq2(2+s)+s(1+s)(1+3 s)\u0003\n+j2\u0002\nq2(2+3 s)+s(1+s)2\u0003,(5.37)\nCω,ϕ=τ∂s\n∂τ\f\f\f\f\nω,ϕ=τ{s, ω, ϕ}s,j,q\n{τ, ω, ϕ}s,j,q=2A3\n1\u0000\nA1A2−j2\u0001\ns(1 +s)\nG(s, j, q ), (5.38)\nCω,q=τ∂s\n∂τ\f\f\f\f\nω,q=τ{s, ω, q}s,j,q\n{τ, ω, q}s,j,q=2A2\n1\u0000\nA1A2−j2\u0001\ns(1 +s)\nR(s, j, q ). (5.39)\nThe heat capacity Cω,jis positive in the range of existence (5.18). It has the same regions of\nlocal and global stability as in the case m=const, i.e. the local stability is between the blue\nand the green surface. The global equilibrium (5.32) can be achieved above these two surfaces\n(Fig. 3).\nThe region of positive Cω,ϕcoincides with the region of global stability (5.32). Between the\nblue and the green surface ( B > 0andG < 0), the black hole is unstable ( Cω,ϕ<0). Above\nthese two surfaces the black hole can be globally or locally stable.\n14Finally, the region of positive Cω,qis above the blue and the red surface ( B > 0andR >0).\nBetween these two surfaces ( B > 0andR < 0), the black hole is unstable ( Cω,q<0). In the\nregion between the blue, the red and the green surfaces ( B > 0, R > 0andG <0), one has only\nlocal stability. Above the blue and the green surface (5.32) the black hole can be globally stable\n(Fig. 4).\nFigure 4: Combined surfaces from Fig. 3 and the Davies surface for Cω,q(the red surface).\nThe next couple of heat capacities are given by\nCj,ϕ=τ∂s\n∂τ\f\f\f\f\nj,ϕ=τ{s, j, ϕ}s,j,q\n{τ, j, ϕ}s,j,q=2s\u0002\nA3\n1+j2(A1+ 2q2)(1+s)\u0003\u0000\nA1A2−j2\u0001\nO(s, j, q ),(5.40)\nCj,q=τ∂s\n∂τ\f\f\f\f\nj,q=τ{s, j, q}s,j,q\n{τ, j, q}s,j,q=2s\u0002\nA2\n1+j2(1 +s)\u0003\u0000\nA1A2−j2\u0001\nP(s, j, q ). (5.41)\nOne notes that Cj,ϕ>0is positive above the blue surface ( B > 0) and outside the closed orange\nsurface ( O >0). Inside the closed orange surface ( O <0), the black hole is unstable ( Cj,ϕ<0).\nBetween the blue and the green surface and outside the closed orange surface ( B > 0, G < 0and\nO > 0) one has only local stability. Above the blue and the green surface ( B > 0andG > 0)\nthe black hole can be in global equilibrium (Fig. 5).\n15Figure 5: Combined surfaces from Fig. 3 and the Davies surface for Cj,ϕ(the orange surface).\nThe heat capacity Cj,qis positive above the blue surface and outside the closed purple surface\n(B > 0andP >0). Insidetheclosedpurplesurface( P <0),theblackholeisunstable( Cj,q<0).\nBetween the blue and the green surface and outside the closed purple surface ( B > 0, G < 0and\nP > 0) one has only local stability. Above the blue and the green surface (5.32) the black hole\ncan be in globally stable (Fig. 6),\nFigure 6: Combined surfaces from Fig. 3 and the Davies surface for Cj,q(the purple surface).\nThe final admissible heat capacity is given by Cϕ,q:\nCϕ,q=τ∂s\n∂τ\f\f\f\f\nϕ,q=τ{s, ϕ, q}s,j,q\n{τ, ϕ, q}s,j,q=A1\u0000\nA1A2−j2\u0001\ns(1 +s)\nA2\n1\u0002\nq2(2+s)+s(1+s)(1+3 s)\u0003\n+j2\u0002\nq2(2+3 s)+s(1+s)2\u0003,(5.42)\nwhich coincides with Cω,j.\nIn summary, all of the heat capacities are positive in the region of global stability, above\nthe blue and the green surfaces ( B > 0andG > 0). However, we showed that below the green\nsurface there are several regions of local thermodynamic stability, defined by the properties of the\ncorresponding heat capacities. This concludes the full classical local and global thermodynamic\nstability analysis of the KNAdS black hole.\n166 Conclusion\nIn summary, our investigation is focused on the thermodynamic stability of AdS black holes,\nemploying the robust Sylvester criterion within the mass-energy representation. This criterion,\ngrounded in the strict convexity property of the energy potential in thermodynamics, has illumi-\nnatedtheboundariesofclassicalthermodynamicswhenappliedtoAdSsystems. Ouranalysishas\nexplicitly revealed that AdS black holes exhibit regions of global stability, primarily influenced\nby the presence of the cosmological parameter l, effectively stabilizing the system. Moreover, by\nsetting l→ ∞, we can recover the results of [33].\nUtilizing the Nambu bracket formalism for local heat capacities, we have identified regions\nof local stability for thermodynamic processes, corresponding to specific restrictions imposed\non the system. In the AdS-Schwarzschild scenario, the region of local thermodynamic stability\naligns with the global one, albeit achievable only for black holes with event horizons comparable\nto the AdS radius. However, in other black hole cases such as RNAdS, Kerr-AdS, and KNAdS,\nregions of local stability extend beyond the boundaries of the global one.\nWhile some of our findings may have already been presented in existing literature, they have\nbeen scattered across various texts (see for example [25–32,46,58–67]). Nonetheless, to our\nknowledge, a comprehensive classical analysis of the thermodynamic stability of AdS black hole\nsystems has not been presented with such detail before. Our approach underscores the applica-\nbility of classical thermodynamics to black hole physics, while acknowledging the potential for\ndeviations from classical setups that could significantly influence classical observables. Despite\nthis, we anticipate classical thermodynamics to remain valid even in many exotic or complex\ncases due to its inherent rigidity.\nAboutfurtherdevelopments, weareadvancingtoexploretheimpactofconvertingthecosmo-\nlogicalconstantintoathermodynamicparameteranditwillbethefocalpointoftheforthcoming\nthird installment of our research. Additionally, investigating generic stability criteria for other\nblack hole solutions, including hairy black holes, lower and higher-dimensional black holes, reg-\nular black holes, those with extended thermodynamics, and related systems, holds significant\npromise for future inquiry.\nAcknowledgments\nThe authors would like to express their gratitude to S. Yazadjiev for his invaluable comments\nand discussions. H. D. thankfully acknowledges the support by the program “JINR-Bulgaria”\nof the Bulgarian Nuclear Regulatory Agency. V. A. gratefully acknowledges the support by\nthe Simons Foundation and the International Center for Mathematical Sciences in Sofia for the\nvarious annual scientific events. M. R., R. R. and T. V. were fully financed by the European\nUnion- NextGeneration EU, through the National Recovery and Resilience Plan of the Republic\nof Bulgaria, project BG-RRP-2.004-0008-C01.\nA Hessian of the mass of the KNAdS black hole\nThe Hessian of the mass in (s, j, q )space is\nH=\nHssHsjHsq\nHjsHjjHjq\nHqsHqjHqq\n=\n∂2m\n∂s2∂2m\n∂s∂j∂2m\n∂s∂q\n∂2m\n∂j∂s∂2m\n∂j2∂2m\n∂j∂q\n∂2m\n∂q∂s∂2m\n∂q∂j∂2m\n∂q2\n. (A.1)\n17with explicit components\nHss=A3\n1(3A1−4s) + 2j2B1+j4(3 + 4 s)\n4q\ns5\u0002\nA2\n1+j2(1 +s)\u00033, (A.2)\nHjj=A2\n1(1 +s)q\ns\u0002\nA2\n1+j2(1 +s)\u00033, (A.3)\nHqq=2\u0002\nA3\n1+j2(A1+ 2q2)(1 + s)\u0003\nq\ns\u0002\nA2\n1+j2(1 +s)\u00033, (A.4)\nHsj=−jA1\u0002\nA1+sA2+ 2s(1 + 2 s)\u0003\n+j3(1 +s)\n2q\ns3\u0002\nA2\n1+j2(1 +s)\u00033, (A.5)\nHsq=−q\u0002\nA3\n1−j2(A2−2s2)(1 + 2 s)\u0003\nq\ns3\u0002\nA2\n1+j2(1 +s)\u00033, (A.6)\nHjq=−2jqA 1(1 +s)q\ns\u0002\nA2\n1+j2(1 +s)\u00033, (A.7)\nwhere\nA1=s2+s+q2, A 2= 3s2+s−q2, (A.8)\nB1=q4(3+2 s) + 2q2s(2+3 s) + 3s2(1+s)2(1+2 s), (A.9)\nB2=q4(5 + 2 s) + 2q2s(4s(s+ 2) + 3) + 3 s2(1 +s)2(1 + 2 s). (A.10)\nThe principle minors are\n∆s=\f\f\f\f\fHjjHjq\nHqjHqq\f\f\f\f\f,∆j=\f\f\f\f\fHssHsq\nHqsHqq\f\f\f\f\f,∆q=\f\f\f\f\fHssHsj\nHjsHjj\f\f\f\f\f. (A.11)\n∆s=2A3\n1(1 +s)\ns\u0002\nA2\n1+j2(1 +s)\u00032, (A.12)\n∆j=A4\n1\u0002\nA1+ 2s(s−1)\u0003\n+ 2j2A1B2+j4(A1+ 2q2)(3 + 4 s)\n2s3\u0002\nA2\n1+j2(1 +s)\u00032, (A.13)\n∆q=A3\n1(3A1−4s)(1 + s) + 2j2A1\u0002\nq2(1 + 3 s)−s(1 +s)2\u0003\n−j4(1 +s)\n4s3\u0002\nA2\n1+j2(1 +s)\u00032. (A.14)\nThe determinant of the Hessian is\ndetH=A4\n1\u0002\nA1+ 2s(s−1)\u0003\n(1+s)−2j2A2\n1\u0002\nA1+sA2−2s3\u0003\n−j4(A1+ 2q2)(1+s)\n2q\ns7\u0002\nA2\n1+j2(1 +s)\u00035.(A.15)\nB Section planes in the state space of KNAdS\nB.1 Recovering RNAdS by j= 0section\nIf we superimpose Fig. 4 and Fig. 6, it becomes evident that the red surface and the purple\nclosed surface come into contact along a single ellipse situated in the j= 0plane. This ellipse\n18precisely corresponds to the orange ellipse (3.10), illustrated in Figure 1 in the RNAdS scenario,\nresiding at the intersection of the j= 0plane traversing the red and purple surfaces.\nExamining Fig. 5, within the same j= 0intersection plane, we observe that the green and\norange surfaces intersect at a single ellipse. This ellipse corresponds to the green ellipse (3.9)\nfeatured in Figure 1 in the RNAdS context.\nFinally, the blue surface (visible in all figures) intersects the j= 0plane at the blue hyperbola\ndepicted in Fig. 1.\nB.2 Recovering Kerr-AdS by q= 0section\nExamining the q= 0plane, one observes that the green and blue surfaces intersect this plane,\nforming the green and blue curves as illustrated in Fig. 2.\nUpon merging Fig. 5 and Fig. 6, it becomes apparent that the orange closed surface and the\npurple closed surface come into contact, forming a closed orange curve depicted in Fig. 2.\nReferences\n[1] J. D. Bekenstein, “Black holes and the second law,” Lett. Nuovo Cim. 4(1972) 737–740.\n[2] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7(1973) 2333–2346.\n[3] J. D. Bekenstein, “Generalized second law of thermodynamics in black hole physics,”\nPhys. Rev. D 9(1974) 3292–3300.\n[4] J. D. Bekenstein, “Statistical Black Hole Thermodynamics,” Phys. Rev. D 12(1975)\n3077–3085.\n[5] S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26\n(1971) 1344–1346.\n[6] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43(1975)\n199–220. [Erratum: Commun.Math.Phys. 46, 206 (1976)].\n[7] S. W. Hawking, “Black Holes and Thermodynamics,” Phys. Rev. D 13(1976) 191–197.\n[8] J. M. Bardeen, B. Carter, and S. W. Hawking, “The Four laws of black hole mechanics,”\nCommun. Math. Phys. 31(1973) 161–170.\n[9] P. C. W. Davies, “Thermodynamics of Black Holes,” Proc. Roy. Soc. Lond. A 353(1977)\n499–521.\n[10] P. C. W. Davies, “Thermodynamics of black holes,” Reports on Progress in Physics 41\nno. 8, (Aug, 1978) 1313.\n[11]LIGO Scientific, Virgo Collaboration, B. P. Abbott et al., “Observation of\nGravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116no. 6,\n(2016) 061102, arXiv:1602.03837 [gr-qc] .\n[12]Event Horizon Telescope Collaboration, K. Akiyama et al., “First M87 Event Horizon\nTelescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett.\n875(2019) L1, arXiv:1906.11238 [astro-ph.GA] .\n[13]Event Horizon Telescope Collaboration, P. Kocherlakota et al., “Constraints on\nblack-hole charges with the 2017 EHT observations of M87*,” Phys. Rev. D 103no. 10,\n(2021) 104047, arXiv:2105.09343 [gr-qc] .\n19[14]Event Horizon Telescope Collaboration, K. Akiyama et al., “First Sagittarius A*\nEvent Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the\nCenter of the Milky Way,” Astrophys. J. Lett. 930no. 2, (2022) L12.\n[15] J. P. Luminet, “Image of a spherical black hole with thin accretion disk,” Astron.\nAstrophys. 75(1979) 228–235.\n[16] D. N. Page and K. S. Thorne, “Disk-Accretion onto a Black Hole. Time-Averaged\nStructure of Accretion Disk,” Astrophys. J. 191(1974) 499–506.\n[17] K. S. Thorne, “Disk accretion onto a black hole. 2. Evolution of the hole.,” Astrophys. J.\n191(1974) 507–520.\n[18] G. Gyulchev, P. Nedkova, T. Vetsov, and S. Yazadjiev, “Image of the\nJanis-Newman-Winicour naked singularity with a thin accretion disk,” Phys. Rev. D 100\nno. 2, (2019) 024055, arXiv:1905.05273 [gr-qc] .\n[19] G. Gyulchev, P. Nedkova, T. Vetsov, and S. Yazadjiev, “Image of the thin accretion disk\naround compact objects in the Einstein–Gauss–Bonnet gravity,” Eur. Phys. J. C 81\nno. 10, (2021) 885, arXiv:2106.14697 [gr-qc] .\n[20] I. Bazarov, F. Immirzi, and A. Hayes, Thermodynamics . Pergamon Press, 1964.\n[21] H. Callen, Thermodynamics and an Introduction to Thermostatistics . Student Edition.\nWiley India Pvt. Limited, 2006.\n[22] W. Greiner, D. Rischke, L. Neise, and H. Stöcker, Thermodynamics and Statistical\nMechanics . Classical Theoretical Physics. Springer New York, 2012.\n[23] R. Swendsen, An Introduction to Statistical Mechanics and Thermodynamics: Second\nEdition. Oxford Graduate Texts. Oxford University Press, 2020.\n[24] S. Blundell and K. Blundell, Concepts in Thermal Physics . OUP Oxford, 2010.\n[25] B. P. Dolan, “Thermodynamic stability of asymptotically anti-de Sitter rotating black\nholes in higher dimensions,” Class. Quant. Grav. 31(2014) 165011, arXiv:1403.1507\n[gr-qc].\n[26] B. P. Dolan, “On the thermodynamic stability of rotating black holes in higher\ndimensions-a comparision of thermodynamic ensembles,” Class. Quant. Grav. 31no. 13,\n(2014) 135012, arXiv:1312.6810 [gr-qc] . [Erratum: Class.Quant.Grav. 31, 199601\n(2014)].\n[27] A. K. Sinha, “Black holes as possible dark matter,” in Dark Matter , M. L. Smith, ed.,\nch. 5. IntechOpen, Rijeka, 2021.\n[28] A. K. Sinha, “Thermal Fluctuations Of Stable Quantum ADS Kerr-Newman Black Hole,”\narXiv:1608.08359 [gr-qc] .\n[29] A. K. Sinha and P. Majumdar, “Thermal stability of charged rotating quantum black\nholes,” Mod. Phys. Lett. A 32no. 37, (2017) 1750208, arXiv:1512.04181 [gr-qc] .\n[30] A. K. Sinha, “Thermal fluctuations and correlations among hairs of a stable quantum\nblack hole: Some examples,” Mod. Phys. Lett. A 33no. 33, (2018) 1850190,\narXiv:1707.00687 [gr-qc] .\n20[31] A. K. Sinha, “Fluctuating quasi-stable quantum-charged rotating black holes,” Mod. Phys.\nLett. A35no. 16, (2020) 2050136.\n[32] A. K. Sinha, “Thermal stability criteria of a generic quantum black hole,” in New Ideas\nConcerning Black Holes and the Universe , E. Tatum, ed., ch. 3. IntechOpen, Rijeka, 2019.\n[33] V. Avramov, H. Dimov, M. Radomirov, R. C. Rashkov, and T. Vetsov, “On\nThermodynamic Stability of Black Holes. Part I: Classical stability,” arXiv:2302.11998\n[gr-qc].\n[34] V. Avramov, H. Dimov, M. Radomirov, R. C. Rashkov, and T. Vetsov, “Thermodynamic\nstability of ACGL Chern-Simons Black Hole and Optimal Processes,” Ann. U. Craiova\nPhys.33(2023) 78–97.\n[35] M. M. Caldarelli, G. Cognola, and D. Klemm, “Thermodynamics of Kerr-Newman-AdS\nblack holes and conformal field theories,” Class. Quant. Grav. 17(2000) 399–420,\narXiv:hep-th/9908022 .\n[36] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,”\nAdv. Theor. Math. Phys. 2(1998) 231–252, arXiv:hep-th/9711200 .\n[37] S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in anti-De Sitter\nSpace,” Commun. Math. Phys. 87(1983) 577.\n[38] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge\ntheories,” Adv. Theor. Math. Phys. 2(1998) 505–532, arXiv:hep-th/9803131 .\n[39] S. A. Hartnoll, A. Lucas, and S. Sachdev, Holographic quantum matter . MIT Press, 2016.\narXiv:1612.07324 [hep-th] .\n[40] A. Hebecker and J. March-Russell, “Randall-Sundrum II cosmology, AdS / CFT, and the\nbulk black hole,” Nucl. Phys. B 608(2001) 375–393, arXiv:hep-ph/0103214 .\n[41] L. Randall and R. Sundrum, “A Large mass hierarchy from a small extra dimension,”\nPhys. Rev. Lett. 83(1999) 3370–3373, arXiv:hep-ph/9905221 .\n[42] L. Randall and R. Sundrum, “An Alternative to compactification,” Phys. Rev. Lett. 83\n(1999) 4690–4693, arXiv:hep-th/9906064 .\n[43] T. Shiromizu and D. Ida, “Anti-de Sitter no hair, AdS / CFT and the brane world,” Phys.\nRev. D64(2001) 044015, arXiv:hep-th/0102035 .\n[44] I. Aref’eva and I. Volovich, “Violation of the Third Law of Thermodynamics by Black\nHoles, Riemann Zeta Function and Bose Gas in Negative Dimensions,” arXiv:2304.04695\n[hep-th] .\n[45] D. Kastor, S. Ray, and J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,”\nClass. Quant. Grav. 26(2009) 195011, arXiv:0904.2765 [hep-th] .\n[46] B. P. Dolan, “The cosmological constant and the black hole equation of state,” Class.\nQuant. Grav. 28(2011) 125020, arXiv:1008.5023 [gr-qc] .\n[47] B. P. Dolan, “Pressure and volume in the first law of black hole thermodynamics,” Class.\nQuant. Grav. 28(2011) 235017, arXiv:1106.6260 [gr-qc] .\n21[48] M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, “Black Hole Enthalpy and an\nEntropy Inequality for the Thermodynamic Volume,” Phys. Rev. D 84(2011) 024037,\narXiv:1012.2888 [hep-th] .\n[49] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Charged AdS black holes\nand catastrophic holography,” Phys. Rev. D 60(1999) 064018, arXiv:hep-th/9902170 .\n[50] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Holography,\nthermodynamics and fluctuations of charged AdS black holes,” Phys. Rev. D 60(1999)\n104026, arXiv:hep-th/9904197 .\n[51] J. Louko and S. N. Winters-Hilt, “Hamiltonian thermodynamics of the Reissner-Nordstrom\nanti-de Sitter black hole,” Phys. Rev. D 54(1996) 2647–2663, arXiv:gr-qc/9602003 .\n[52] R. Banerjee, S. Ghosh, and D. Roychowdhury, “New type of phase transition in Reissner\nNordström–AdS black hole and its thermodynamic geometry,” Phys. Lett. B 696(2011)\n156–162, arXiv:1008.2644 [gr-qc] .\n[53] G. W. Gibbons, M. J. Perry, and C. N. Pope, “The First law of thermodynamics for\nKerr-anti-de Sitter black holes,” Class. Quant. Grav. 22(2005) 1503–1526,\narXiv:hep-th/0408217 .\n[54] B. Carter, “Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations,”\nCommun. Math. Phys. 10no. 4, (1968) 280–310.\n[55] R. Monteiro, M. J. Perry, and J. E. Santos, “Semiclassical instabilities of Kerr-AdS black\nholes,” Phys. Rev. D 81(2010) 024001, arXiv:0905.2334 [gr-qc] .\n[56] J. F. Plebanski and M. Demianski, “Rotating, charged, and uniformly accelerating mass in\ngeneral relativity,” Annals Phys. 98(1976) 98–127.\n[57] M. T. N. Imseis, A. Al Balushi, and R. B. Mann, “Null hypersurfaces in\nKerr–Newman–AdS black hole and super-entropic black hole spacetimes,” Class. Quant.\nGrav.38no. 4, (2021) 045018, arXiv:2007.04354 [gr-qc] .\n[58] B. P. Dolan, “Compressibility of rotating black holes,” Phys. Rev. D 84(2011) 127503,\narXiv:1109.0198 [gr-qc] .\n[59] B. P. Dolan, “Black holes and Boyle’s law — The thermodynamics of the cosmological\nconstant,” Mod. Phys. Lett. A 30no. 03n04, (2015) 1540002, arXiv:1408.4023 [gr-qc] .\n[60] S. Carlip and S. Vaidya, “Phase transitions and critical behavior for charged black holes,”\nClass. Quant. Grav. 20(2003) 3827–3838, arXiv:gr-qc/0306054 .\n[61] B. M. N. Carter and I. P. Neupane, “Thermodynamics and stability of higher dimensional\nrotating (Kerr) AdS black holes,” Phys. Rev. D 72(2005) 043534, arXiv:gr-qc/0506103 .\n[62] A. Sahay, T. Sarkar, and G. Sengupta, “Thermodynamic Geometry and Phase Transitions\nin Kerr-Newman-AdS Black Holes,” JHEP04(2010) 118, arXiv:1002.2538 [hep-th] .\n[63] R. Banerjee, S. K. Modak, and S. Samanta, “Second Order Phase Transition and\nThermodynamic Geometry in Kerr-AdS Black Hole,” Phys. Rev. D 84(2011) 064024,\narXiv:1005.4832 [hep-th] .\n[64] H. Liu, H. Lu, M. Luo, and K.-N. Shao, “Thermodynamical Metrics and Black Hole Phase\nTransitions,” JHEP12(2010) 054, arXiv:1008.4482 [hep-th] .\n22[65] Y.-D. Tsai, X. N. Wu, and Y. Yang, “Phase Structure of Kerr-AdS Black Hole,” Phys.\nRev. D85(2012) 044005, arXiv:1104.0502 [hep-th] .\n[66] D. Roychowdhury, Phase transition in black holes . PhD thesis, Calcutta U., 2013.\narXiv:1403.4356 [gr-qc] .\n[67] D. Kubiznak and R. B. Mann, “Black hole chemistry,” Can. J. Phys. 93no. 9, (2015)\n999–1002, arXiv:1404.2126 [gr-qc] .\n23"    },    {        "title": "2402.07274v1.Constant_mean_curvature_graphs_with_prescribed_asymptotic_values_in___mathbb_E___1_τ__.pdf",        "content": "CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC\nVALUES IN E(−1, τ)\nANDREA DEL PRETE\nAbstract. In the homogeneous manifold E(−1, τ),for 0 < H <1\n2,we prove the existence of entire H-\ngraphs which are asymptotic to a rectifiable curve of the asymptotic boundary. We also find necessary\nand sufficient conditions for the existence of H-graphs over unbounded domains having prescribed, possibly\ninfinite boundary data.\n1.Introduction\nConstant mean curvature surfaces in simply connected homogeneous 3-manifolds have been a subject of\nextensive research by various authors over the past two decades. In particular, considerable attention has\nbeen devoted to 3-manifolds with an isometry group of dimension at least 4. These 3-manifolds, excluding\nthe hyperbolic space H3, can be classified in a 2-parameter family denoted as E(κ, τ), where κandτare real\nnumbers. This classification depends on the property of the E(κ, τ)-spaces to admit a Riemannian submersion\nπ:E(κ, τ)→M(κ) over the complete simply connected Riemannian surface with constant curvature κwhose\nfibers are the integral curves of the unique unitary Killing vector field ξ∈X(E) such that τis the bundle\ncurvature (that is ∇Xξ=τX∧ξfor all vector fields X∈X(E),where ∇is the Levi-Civita connection of E\nand∧is the cross product in Ewith respect to a chosen orientation, see [11]).\nThis geometric setup naturally prompts the question of investigating graphs, i.e. sections of the Riemannian\nsubmersion obtained deforming a fixed zero section through the flow lines, with constant mean curvature\nover specific domains of M(κ). Such an approach can be considered as a non-parametric version of the\nconstant mean curvature condition (see Section 4).\nIn this paper we focus on the case κ=−1 and investigate graphs with constant mean curvature Hthat\nwill be referred as an H-graphs or, whenever His not desired to be explicit, CMC graphs.\nOne of the pioneering works in this setting is due to Nelli and Rosenberg in H2×R=E(−1,0) (see [16]),\nwho solved the following problems.\n•The existence of a unique entire minimal graph with arbitrarily prescribed asymptotic values.\n•The Jenkins–Serrin problem over relatively compact domains, that is, to find necessary and sufficient\nconditions that guarantee the existence and uniqueness of a solution for the Dirichlet problem for the\nminimal surface equation with possibly infinite boundary data.\nSo far, no result with an analogous statement as in [16] has been given about CMC graphs having prescribed\nasymptotic boundary value. In the classical model, i.e. the one where the zero section is the minimal surface\ninvariant by rotation (see Section 2), such a result cannot be achieved due to the Slab Theorem (see [8]),\nwhich implies that any entire function satisfying the constant mean curvature equation must diverges. The\naim of this paper is to show that, if we assume the zero section to have constant mean curvature H, it is\npossible to prove a statement analogous to the one of Nelli and Rosenberg [16] about the existence of entire\nH-graphs in E(−1, τ).\nDipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit `a dell’Aquila, Via Vetoio Loc.\nCoppito – 67100 L’Aquila (Italy)\nE-mail address :andrea.delprete@graduate.univaq.it .\n2020 Mathematics Subject Classification. 53A10, 53C30, 53C42, 53C45.\nKey words and phrases. Homogeneous 3-manifolds, Constant mean curvature graphs, Asymptotic boundary.\n1arXiv:2402.07274v1  [math.DG]  11 Feb 20242 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nIn particular, for any rectifiable curve Γ in ∂∞E(−1, τ) =π−1(∂∞H2) that is the graph of a function of\n∂∞H2, we construct an entire graph Σ of subcritical constant mean curvature Hthat is asymptotic to Γ.\nThis can be done by changing the model of E(−1, τ) so that the plane {z= 0}has constant mean curvature\n0< H < 1/2. In such a way, we have a direct control over the symmetries of the graph Σ. Indeed, by\nmeans of the Maximum Principle one can prove that Σ inherits the symmetries of the curve Γ. We cannot\ncover the case H= 1/2 because of the half-space property given in [18, 12]. Moreover, due to our result, a\nhalf-space theorem for entire H-graphs with H < 1/2 cannot be proven.\nConcerning the Jenkins–Serrin problem, several extensions have been achieved. A Jenkins–Serrin problem\nfor CMC graphs in H2×Rover relatively compact domains of H2was proved by Hauswirth, Rosenberg and\nSpruck in [9] and by Eichmair and Metger in [5] in a general product space M×R. Furthermore, in [4], the\nauthor, together with Manzano and Nelli, proves an analogous result for minimal graphs in a general Killing\nsubmersion, which include the space E(−1, τ).In [13] Mazet, Rodriguez and Rosenberg proved in H2×R\na Jenkins–Serrin type result for minimal surfaces over unbounded domains of H2, allowing the asymptotic\nboundary of such domains to contain open subsets of ∂∞H2. Later, Melo proved the same result in the\nspaceE(−1, τ) [15]. A result for H-graphs in H2×R, similar to [13], was proved by Folha and Melo in [6].\nThey solved the Jenkins–Serrin problem for H-graphs with 0 < H < 1/2 over an unbounded domain whose\nasymptotic boundary does not contain open subsets of ∂∞H2. We use the constructed barriers to extend\nthe results in [6, 15], showing that in the model we consider we are able to solve the Jenkins–Serrin problem\nover a larger family of unbounded domains Ω of H2. In particular, we solve the following Dirichlet problem:\n\n\nH(u) =H inD,\nu=fi onCi,\nu= +∞ onAi,\nu=−∞ onBi,\nu=F on Γ,\nwhere Ai, Bi, Ciare such that\nAi∪Bi∪Ci=∂Ω, κ g(Ai) = 2 H, κ g(Bi) =−2Hand κg(Ci)≥2H,\nΓ is the asymptotic boundary of D, that is Γ ⊂∂∞H2, and fi∈ C∞(Ci) and F∈ C∞(Γ).\nThe paper is organized as follows. Section 2 is a description of the basic properties of the space E(−1, τ). In\nSection 3 we recall the constant mean curvature graphs that are invariant with respect to a one-parameter\ngroup of isometries of E(−1, τ) studied by Pe˜ nafiel in [19] and we study their asymptotic behaviour. In\nSection 4 we introduce the new model for E(−1, τ) in which the plane {z= 0}has constant mean curvature\nH∈(0,1/2). In Section 5, we use the surfaces studied in Section 3 to construct barriers for the Dirichlet\nProblem for the constant mean curvature equation. Finally, the main results of this paper are given in\nSections 6 and 7, where, respectively, we prove the existence of entire H-graphs with prescribed asymptotic\nboundary and the Jenkins–Serrin problem.\n2.The space E(−1, τ)\nThe space E(−1, τ) is the homogeneous 3-dimensional manifold defined as Ω ×R, endowed with the metric\nds2\nλ=λ2\u0000\ndx2+dy2\u0001\n+h\ndt−2τλ\u0010\u00001\nλ\u0001\nydx−\u00001\nλ\u0001\nxdy\u0011i2\n. (2.1)\nwhere Ω =\b\n(x, y)∈R2|λ(x, y)>0\t\nandλ∈ C∞(Ω) satisfies ∆(log( λ))−λ2= 0.\nNotice that\u0000\nΩ, λ2(dx2+dy2)\u0001\nis isometric to the hyperbolic plane H2with a constant sectional curvature\nof−1 and the Killing submersion is π(x, y, t ) = ( x, y) with ∂tbeing the unitary Killing field satisfying the\nidentity ∇X∂t=τX∧∂tfor any X∈X(E). In this context, we call vertical every vector field parallel\nto∂t,and horizontal every vector field orthogonal to ∂t. In this model, the mean curvature of the graph\nΣu= (x, y, u (x, y))⊂Ω×Rof the function u∈ C2(Ω) is given in the divergent form as\n2H= div \nGup\n1 +∥Gu∥2!\n, (2.2)CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 3\nwhere div( ·) and ∥ · ∥ are the divergence and norm of H2in the conformal metric λ2(dx2+dy2),and\nGu=\u0010\nux−2τλy\nλ\u0011\n∂x\nλ2+\u0000\nuy+2τλx\nλ\u0001∂y\nλ2is the generalized gradient.\nDepending on the choice of λ,we have two classical models to describe E(−1, τ) :\n•Ifλ(x, y) =λH(x, y) =1\ny, we get the half-plane model for H2and the half-space model HforE(−1, τ),\nwhere the vertical boundary is ( {y= 0} ∪ {∞} )×R≡(R∪ {∞} )×R.\n•Ifλ(x, y) =λC(x, y) =2\n1−x2−y2, we obtain the Poincar´ e Disk model for H2and the cylindrical model C\nforE(−1, τ),withS(1)×Ras vertical boundary.\nUsing the complex coordinate z=x+iy, the isometries between this two models are given by lifting the\nM¨ obius transformation of H2(see [1] for more details):\nψ: H → C\n(x, y, t )7→\u0010\nz−i\nz+1, t−4τarctan\u0010\nx\ny+1\u0011\u0011\n,\nand\nψ−1: C → H\n(x, y, t )7→\u0010\ni(z+1)\n1−z, t−4τarctan\u0010\ny\n1−x\u0011\u0011\n.(2.3)\nSome isometries of E(−1, τ),can be easily described using either CorH:\n•Vertical translation: in both models\nφ1: (x, y, t )7→(x, y, t +c),∀c∈R\n•Hyperbolic translation along an horizontal geodesic: using the half-space model H,the hyperbolic\ntranslation along the horizontal geodesics in {x=x0}is given by the map\nφ2: (x, y, t )7→(x0+c(x−x0), cy, t ),∀c∈R.\n•Parabolic translation: using the half-space model H,the hyperbolic translation along the horocycle\n{y=y0}is given by the map\nφ3: (x, y, t )7→(x+c, y, t ),∀c∈R.\n•Rotation: in the cylinder model C,rotations with respect to the t-axis are Euclidean rotations with\nrespect to the t-axis, that are,\nφ4: (x, y, z )7→(xcos(θ)−ysin(θ), xsin(θ) +ycos(θ), t),∀θ∈R.\nIn the cylinder model C, the surface {t= 0}is the rotational minimal surface Uknown as umbrella; in the\nhalf-space model H, the surface {t= 0}is the minimal surface, S,invariant with respect to hyperbolic and\nparabolic translation.\n3.Constant mean curvature surfaces invariant by a one-parameter group of isometries\nand their asymptotic behaviour\nIn this section, we investigate certain CMC surfaces that are invariant under either rotations or hyperbolic\ntranslations. These surfaces are vertical graphs of functions of Ω, in either the cylinder or the half-space\nmodel. The study of such surfaces has been conducted previously in [22] and [21] for τ= 0 and in [2] and\n[19] for τ̸= 0. In particular, our interest focus on their asymptotic behaviour with respect to the surface\n{t= 0}in each model, and the relation between SandU.\nTo describe the surfaces invariant by the rotation φ4we will use the cylinder model Cand the geodesic\npolar coordinates(\nx(ρ, θ) = tanh\u0000ρ\n2\u0001\ncos(θ),\ny(ρ, θ) = tanh\u0000ρ\n2\u0001\nsin(θ),\nof the base H2of the Killing submersion. A simple computation implies that there exists a one-parameter\nfamily of rotational H-surfaces, parametrized by\nRH\nd(ρ, θ) =\u0000\ntanh\u0000ρ\n2\u0001\ncos(θ),tanh\u0000ρ\n2\u0001\nsin(θ), vH\nd(ρ)\u0001\n,4 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nwhere\nvH\nd(ρ) =Zρ\n∗(2Hcosh( r) +d)q\n1 + 4 τ2tanh2\u0000r\n2\u0001\nq\nsinh2(r)−(2Hcosh( r) +d)2dr, (3.1)\nandd∈R.As it is shown in [19], depending on the choice of d,we can obtain three type of surfaces:\n(a) Embedded H-\nannulus\n (b)Entire rotational H-graph\n(c) Immersed H-\nannulus\nFigure 1. Generating curves of rotational H-surfaces.\n•ifd >−2H,thenRH\nd(ρ, θ) is the upper half of a proper embedded annulus symmetric with respect to\nthe slice {t= 0}(see Figure 1a);\n•ifd=−2H,thenRH\nd(ρ, θ) is an entire vertical graph contained in {t≥0}and tangent to {z= 0}at\nthe point (0 ,0,0) (see Figure 1b);\n•ifd <−2H,thenRH\nd(ρ, θ) is the half of a properly immersed (and non-embedded) annulus, symmetric\nwith respect to slice {t= 0}(see Figure 1c).\nTo describe the surfaces invariant by an hyperbolic translation φ2, we use the half-space model Hand its\ngeodesic polar coordinates(\nx(ϕ, θ) =eϕcos(θ),\ny(ϕ, θ) =eϕsin(θ),\nwith θ∈(0, π),and we study without loss of generality the hyperbolic translation that fixes the horizontal\ngeodesics in {x0= 0}.Hence, as it is shown in [2, Lemma 3.2], there exists a one-parameter family of\nH-surfaces invariant by hyperbolic translation, parametrized by\nΘH\nd(ϕ, θ) =\u0000\neϕcos(θ), eϕsin(θ), ud(ϕ)\u0001\n,\nwhere\nuH\nd(θ) =Zθ\n∗|d−2Hcot(ν)|p\n1 + 4 τ2cos2(ν)q\n1−sin2(ν)(d−2Hcot(ν))2dν−2τθ, (3.2)\nandd≥0.As it is shown in [2, Theorem 3.4], depending on the choice of d,denoting by θ1= arcsin\u0010\n2H√\nd2+4H2\u0011\nandθ2= arcsin\u0010\n1√\nd2+4H2\u0011\n,we can obtain three type of surfaces:\n•ifd∈[0,√\n1−4H2),then uH\ndis defined for any θ∈(0, π) and\nlim\nθ→0+uH\nd= lim\nθ→π−uH\nd= +∞\n(see Figure 2a);CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 5\n(a)The case d∈[0,√\n1−4H2)\n (b)The case d=√\n1−4H2,H̸=1\n2√\n2\n(c)The case d >√\n1−4H2\nFigure 2. H-surfaces invariant by hyperbolic translation.\n•ifd=√\n1−4H2,then uH\ndis defined for any θ∈\u0000\n0,π\n2+θ1\u0001\n∪\u0000π\n2+θ1, π\u0001\nand\nlim\nθ→0−uH\nd(θ) = lim\nθ→π+uH\nd(θ) = + ∞,\nlim\nθ→(π\n2+θ1)−˙uH\nd(θ) = lim\nθ→(π\n2+θ1)+˙uH\nd(θ) = + ∞,\n(see Figure 2b);\n•ifd >√\n1−4H2,then uH\ndis defined for any θ∈(0, θ1+θ2]∪[π−(θ2−θ1), π), that is, it has two\ndisconnected component, and\nlim\nθ→0−uH\nd(θ) = lim\nθ→π+uH\nd(θ) = + ∞\nlim\nθ→(θ1+θ2)−˙uH\nd(θ) = lim\nθ→(π−(θ2−θ1))+˙uH\nd(θ) = + ∞\n(see Figure 2c).\nIn [2, Theorem 3.4] the authors state that, when d=√\n1−4H2,\nlim\nθ→(π\n2+θ1)−uH\nd(θ), lim\nθ→(π\n2+θ1)+uH\nd(θ)\nare finite and that, when d >√\n1−4H2,\nlim\nθ→(θ1+θ2)−uH\nd(θ), lim\nθ→(π−(θ2−θ1))+uH\nd(θ)\nare finite. In the following proposition, we focus on the case d=1√\n2and prove that in this case the\nstatement cannot be true. We also prove in Proposition 3.2 that, for any H∈\u0010\n0,1\n2√\n2\u0011\n∪\u0010\n1\n2√\n2,1\n2\u0011\n, there\nexists d >√\n1−4H2such that either\nlim\nθ→(θ1+θ2)−uH\nd(θ) or lim\nθ→(π−(θ2−θ1))+uH\nd(θ)\ndiverges to + ∞or−∞.6 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nProposition 3.1. Ifd=√\n1−4H2andH=1\n2√\n2,then\nlim\nθ→(π\n2+θ1)−uH\nd(θ) = +∞and lim\nθ→(π\n2+θ1)+uH\nd(θ) =−∞.\nProof. Let Σ Hbe the vertical cylinder of constant mean curvature Hparametrized by\nΣH(ϕ, t) =\u0000\ncos\u0000\nθ1+π\n2\u0001\neϕ,sin\u0000\nθ1+π\n2\u0001\neϕ, t\u0001\n.\nSuppose that there exist k∈R,such that lim\nθ→θ−\n0uH\nd(θ) =k,then ΘH\nd(ϕ, θ) and Σ H(φ, t) are two surfaces of\nconstant mean curvature H,with mean curvature vector pointing in the same direction, which are tangent\nalong the curve (cos( θ0)eϕ,sin(θ0)eϕ, k) (see Figure 3). This contradicts the Boundary Maximum Principle.\nFigure 3. ΣHis tangent to ΩH\ndif lim\nθ→(θ1+π\n2)−uH\nd(θ) =k.\nThe same argument is used to prove that lim\nθ→θ+\n0uH\nd(θ) =−∞,and concludes the proof. □\nProposition 3.2. IfH >1\n2√\n2,there exists d∗∈(√\n1−4H2,+∞)such that\nlim\nθ→(π−(θ2−θ2))+uH\nd∗(θ) =−∞.\nIfH <1\n2√\n2,there exists d∗∈(√\n1−4H2,+∞)such that\nlim\nθ→(θ1+θ2)−uH\nd∗(θ) = +∞.\nProof. Notice that\nlim\nd→+∞(θ1+θ2) = 0 , lim\nd→+∞(π−(θ2−θ1)) =π\nand that\nlim\nd→√\n1−4H2(θ1+θ2) = lim\nd→√\n1−4H2(π−(θ2−θ1)) =π\n2+ arcsin(2 H) =π−arccos(2 H).\nIfH >1\n2√\n2,then\narccos(2 H) +π\n2> π−arccos(2 H) =π\n2+ arcsin(2 H).\nHence, by continuity, there exists d∗∈(√\n1−4H2,+∞) such that\nπ−(θ2−θ1) = arccos(2 H) +π\n2.CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 7\nAnalogously, if H <1\n2√\n2,then\narccos(2 H) +π\n2< π−arccos(2 H) =π\n2+ arcsin(2 H),\nand, by continuity, there exists d∗∈(√\n1−4H2,+∞) such that\nθ1+θ2= arccos(2 H) +π\n2.\nThe result follows applying the same argument of Proposition 3.1. □\nTo study the asymptotic behaviour, we compute the maximum of the height growth of each graph with\nrespect to the slide {t= 0}as a function of geodesic radius of the base H2.The study of RH\ndis easy, since\nthe growth function vH\ndis already defined as a function of the geodesic radius of H2.So, we only need to\nstudy its behaviour for ρ→ ∞ .An easy computation implies that\n˙vH\nd(ρ)≃2H√\n1 +τ2\n√\n1−4H2+2(d+ 4dτ2+ 8H(−1 + 4 H2)τ2)\n(1−4H2)3\n2√\n1 + 4 τ2e−ρ+o\u0000\ne−ρ\u0001\n,\nand integrating this function we get\nvH\nd(ρ)≃2H√\n1 +τ2\n√\n1−4H2ρ−2(d+ 4dτ2+ 8H(−1 + 4 H2)τ2)\n(1−4H2)3\n2√\n1 + 4 τ2e−ρ+o\u0000\ne−ρ\u0001\n. (3.3)\nIn a similar way we study the growth of ΘH\nd. First we have to notice that growth function uH\ndis not\nwritten with respect to the geodesic radius of the base, so we need to use the change θ(ρ) = 2 arctan ( eρ).\nThis implies that\n˙uH\nd(ρ) =2eρ\n1 +e2ρ\n|d−2Hcot (2 arctan ( eρ))|p\n1 + 4 τ2cos2(2 arctan ( eρ))q\n1−(dsin (2 arctan ( eρ))−2Hcos (2 arctan ( eρ)))2−2τ\n.\nIn this case, since we are allowing ρto be negative, we need study separately the growth function for ρ→+∞\nandρ→ −∞ . By a simple computation, we get\n˙uH\nd(ρ)≃\n\n2H√\n1+τ2√\n1−4H2+\u0010\n−4τ+dq\n1+4τ2\n(1−4H2)3\u0011\ne−ρ+o(e−ρ) for ρ→+∞,\n−2H√\n1+τ2√\n1−4H2+\u0010\n−4τ+dq\n1+4τ2\n(1−4H2)3\u0011\neρ+o(e−ρ) for ρ→ −∞ ,\nand integrating this function, we get that\nuH\nd(ρ)≃2H√\n1 +τ2\n√\n1−4H2|ρ|+Ke−|ρ|+o(e−|ρ|) (3.4)\nfor|ρ| →+∞andK= 4τ−dq\n1+4τ2\n(1−4H2)3ifρ >0 and K=−4τ+dq\n1+4τ2\n(1−4H2)3ifρ <0.\nThe last thing to check is the asymptotic relation between UandS, that will allow us to compare RH\nd1and\nΘH\nd2.To do so, we use (2.3) to write Uin the half-space model Hand compare it to S,that is {t= 0}.So,\nin the cylinder model C,we parametrize U, that is {t= 0},by\nU(x, y) =\u0012−1 +x2+y2\nx2+ (y+ 1)2,−2x\nx2+ (y+ 1)2,0\u0013\n,\nwith ( x, y)∈\b\nR2|y >0\t\n.Hence, the image of Uby the isometry (2.3) is\n(ψ−1◦ U)(x, y) =\u0010\nx, y,4τarctan\u0010\nx\ny+1\u0011\u0011\n(see [1] for more details).\nIn particular we have that the vertical distance between UandSis bounded (see Figure 4). This implies\nthat, fixed H∈\u0002\n0,1\n2\u0001\n,for any d1, d2∈Rthe asymptotic vertical distance between RH\nd1and ΘH\nd2is bounded\nby a constant.8 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nFigure 4. The umbrella surface Uin the half-space model H.\n4.A new model for E(−1, τ)\nIn this subsection we describe a new cylindrical model and a new half-space model for E(−1, τ),in which\nthe slice {t= 0}has constant sectional curvature 0 < H <1\n2.\nIn general, if U⊂R2is a simply connected domain endowed with the conformal metric λ2(x, y)(dx2+dy2)\nandU×Ris such that π:U×R→Uis a unitary Killing submersion with bundle curvature τ, then U×R\nis endowed with the metric\nλ2(dx2+dy2) + [λ(a(x, y)dx+b(x, y)dy) +dt]2,\nwhere a, b∈ C∞(U) satisfy\n(λb)x−(λa)y\n2λ2=τ.\nIf Σ = ( x, y, d (x, y)),ford∈ C∞(U),is a complete graph, the map\nI:U×R→ U×R\n(x, y, t )7→(x, y, t−d(x, y))(4.1)\nis an isometry that allows to see Σ as the graph {z= 0}inU×Rendowed with the metric\nλ2(dx2+dy2) + [λ(a′(x, y)dx+b′(x, y)dy) +dt]2\nThe relation between aanda′andbandb′is given by the uniqueness of Killing submersions (see [10,\nTheorem 2.6]) and reads\nb′=b+dy\nλ, a′=a+dx\nλ.\nFor any H∈\u0002\n0,1\n2\u0003\n, applying this technique to E(−1, τ),described with the cylindrical model C,and the\nsurface RH\n−2H(ρ, θ), we get that E(−1, τ) is isometric to Ω ×R,endowed with the metric\nds2=λC(x, y)(dx2+dy2) +\u0002\nλC(x, y)\u0002\naH\nC(x, y)dx+bH\nC(x, y)dy\u0003\n+dt\u00032, (4.2)\nwith\naH\nC(x, y) = 2 yτ+ 2Hxq\n1+4(x2+y2)τ2\n1−4H2(x2+y2),\nbH\nC(x, y) =−2xτ+ 2Hyq\n1+4(x2+y2)τ2\n1−4H2(x2+y2),\nwhere the graph {t= 0}is rotationally invariant and has constant mean curvature H.\nIn this model the mean curvature equation of the graph of a function u:U⊂Ω→Ris given by\nH(u) =1\n2div \nGup\n1 +∥Gu∥2!\n, (4.3)\nwhere div( ·) and∥·∥are respectively the divergence and norm of H2with the Poincar´ e disk metric λ2\nC(dx2+\ndy2),and\nGu=1\nλ2\nC\u0002\n(ux−λCaH\nC)∂x+ (uy−λCbH\nC)∂y\u0003\n.CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 9\nFor any H∈\u0002\n0,1\n2\u0003\n, applying this technique to E(−1, τ),described with the half-space model H,and the\nsurface ΘH\n0(φ, θ), we get that E(−1, τ) is isometric to Ω ×R,endowed with the metric\nds2=λH(x, y)(dx2+dy2) +\u0002\nλH(x, y)\u0002\naH\nH(x, y)dx+bH\nH(x, y)dy\u0003\n+dt\u00032, (4.4)\nwith\naH\nH(x, y) =2y\nx2+y2\u0012\n(x2+y2+y)τ+Hx√\ny2+x2(1+4τ2)√\n(1−4H2)x2+y2\u0013\n,\nbH\nH(x, y) =2x\nx2+y2\u0012\n−yτ−Hx√\ny2+x2(1+4τ2)√\n(1−4H2)x2+y2\u0013\n,\nwhere the graph {t= 0}is invariant by hyperbolic translation and has constant mean curvature H.\nIn this model the mean curvature equation of the graph of a function u:U⊂Ω→Ris given by\nH(u) =1\n2div \nGup\n1 +∥Gu∥2!\n, (4.5)\nwhere div( ·) and∥·∥are respectively the divergence and norm of H2with the half-plane metric λ2\nH(dx2+dy2),\nand\nGu=1\nλ2\nH\u0002\n(ux−λHaH\nH)∂x+ (uy−λHbH\nH)∂y\u0003\n.\nRegardless of the model we are using, if D⊂Ω is an open domain and u∈ C2(D)∩ C0(∂D) satisfies\nH(u) =H∈R,then, integrating div\u0012\nGu√\n1+∥Gu∥2\u0013\ngives\n2HA(D) =Z\n∂D*\nGup\n1 +∥Gu∥2, ν+\n, (4.6)\nwhere A(D) is the hyperbolic area of Dandνis the outer normal to ∂D.The integral in the right side of\nthe equation is called fluxofuacross ∂D.Ifγ⊂∂Dis diffeomorphic to a segment we can define the flux of\nuacross γas follows:\nDefinition 4.1. Let Γ ⊂Dany smooth curve such that γ∪Γ bound a simply connected domain ∆ Γ.Then\nwe define the flux of uacross γas\nFlux u(γ) = 2 HA(∆Γ)−Z\nΓ*\nGup\n1 +∥Gu∥2, ν+\nNotice that this definition does not depend on the choice of Γ .\nWe now state two Lemmas that we need to construct the barriers and to prove Theorem 7.5. The proof is\nanalogous to the case τ= 0,proved in [9, Section 5].\nLemma 4.2. Letube such that P(u) = 0 in a bounded domain Dand let Γbe a piecewise C1curve in ∂D.\nThen:\n2HA(D) =Z\n∂D*\nGup\n1 +∥Gu∥2, ν+\nand\f\f\f\f\fZ\nΓ*\nGup\n1 +∥Gu∥2, ν+\f\f\f\f\f≤ |Γ|.\nLemma 4.3. LetDbe a domain bounded in part by an arc γand let {un}be a sequence of solutions of\nP(un) = 0 inD,such that each {un}is continuous on γ.Then\n(1) if the sequence tends to +∞uniformly on compact subsets of γwhile remaining uniformly bounded on\ncompact subsets of D,we have\nlim\nn→∞Z\nγ*\nGunp\n1 +∥Gun∥2, ν+\n=|γ|;\n(2) if the sequence tends to −∞ uniformly on compact subsets of γwhile remaining uniformly bounded on\ncompact subsets of D,we have\nlim\nn→∞Z\nγ*\nGunp\n1 +∥Gun∥2, ν+\n=−|γ|.10 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\n5.Construction of the barriers\nIn this section we build the asymptotic barriers that will be used in the proof of the Main Theorems. For\nE(−1, τ),we consider the half-space model with {t= 0}having constant mean curvature H, described by\nΩ×Rendowed with the metric (4.5). The study done in Section 3 implies that, in this model, the results\nof Propositions 3.1 and 3.2 can be read as follows:\nTheorem 5.1. For any c∈R,letγc⊂Ωbe the curve γc(t) =\u0010\n−2H√\n1−4H2t+c, t\u0011\nt>0and denote by\nD−\nc=\u001a\n(x, y)∈Ω|x <−2H√\n1−4H2y+c\u001b\n, D+\nc=\u001a\n(x, y)∈Ω|x >−2H√\n1−4H2y+c\u001b\n.\nThen\n•IfH=1\n2√\n2,inD±\ncthere exists a solution ω±to the asymptotic Dirichlet problem\n\n\nP(ω±) = 0 inD±\nc,\nω±= 0 in{y= 0},\nω±=±∞ inγc.\n•If1\n2√\n2< H <1\n2,inD−\ncthere exists a solution ω−to the asymptotic Dirichlet problem\nP(γc,−) =\n\nP(ω−) = 0 inD−\nc,\nω−= 0 in{y= 0},\nω−=−∞ inγc.\n•If0< H <1\n2√\n2,inD+\ncthere exists a solution ω+to the asymptotic Dirichlet problem\nP(γc,+) =\n\nP(ω+) = 0 inD+\nc,\nω+= 0 in{y= 0},\nω+= +∞ inγc.\nSo, it remains to prove that, when1\n2√\n2< H <1\n2,the problem P(γc,+) admits a solution ω+and that,\nwhen 0 < H <1\n2√\n2,the problem P(γc,−) admits a solution ω−.\nFirst, we recall a useful result proved in [20, Theorem 3.3]:\nTheorem 5.2. LetD⊂H2be a domain bounded in part by a C2arcγand let u∈ C2(D)be a solution\nofP(u) = 0 .Ifutend to +∞for any approach to interior points of γ,then γhas geodesic curvature\nκg(γ) = 2 H,while if utends to −∞ for any approach to interior points of γ,then γhas geodesic curvature\nκg(γ) =−2H.\nDenoting by θ1= arcsin (2 H), let γH\nc⊂Ω be the curve\u0000\ncos\u0000\nθ1+π\n2\u0001\nt+c,sin\u0000\nθ1+π\n2\u0001\nt\u0001\nt>0,and define\nD−\nH,c=n\n(x, y)∈Ω|x <cot\u0010\nθ1+π\n2\u0011\ny+co\n,\nD+\nH,c=n\n(x, y)∈Ω|x >cot\u0010\nθ1+π\n2\u0011\ny+co\n.\nIn this model, the surface ΘH√\n1−4H2has two components Θ−and Θ+,defined respectively in D−\nH,candD+\nH,c.\nUp to apply a vertical translation, Θ−is the graph of a negative function s−∈ C∞(D−\nH,c) satisfying\n\n\nP(s−) = 0 in D−\nH,c,\ns−= 0 in {y= 0},\n∂s−\n∂ν=−∞ inγH\nc,CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 11\nwhere νis the outer normal to γH\nc.Analogously, Θ+is the graph of a positive function s+∈ C∞(D+\nH,c)\nsatisfying\n\nP(s+) = 0 in D+\nH,c,\ns+= 0 in {y= 0},\n∂s+\n∂ν= +∞inγH\nc.\nUsing s+ands−as barriers we can prove the following result.\nTheorem 5.3. If1\n2√\n2< H <1\n2,then there exists a solution ω+to the problem P(γc,+).If0< H <1\n2√\n2,\nthen there exists a solution ω−to the problem P(γc,−).\nProof. Suppose first that1\n2√\n2< H <1\n2.A simple computation implies that γc⊂D−\nH,c.LetB(r) be the\ngeodesic disks centered in γc(1) with geodesic radius rand denote by D+\nr=D+∩B(r).For any n∈N,[3,\nTheorem 1] implies the existence of a solution ωn\nrto the problem\n\n\nP(ωn\nr) = 0 in D+\nr,\nω+\nn=n in∂D+\nr∩∂D+,\nω+\nn= 0 in ∂B(r)∩D+.\nThe maximum principle implies that, if we fix n,for any r1> r2we get ωn\nr1> ωn\nr2inD+\nr2.\nWe need to prove that, for any r∈R+andn∈N,we have ωn\nr< s+inD+\nH,c∩D+\nr.Assume that for\nsome nandr, ωn\nr> s+in a subdomain of D+\nH,c∩D+\nr.Leta∈R+be a sufficiently large constant such\nthatωn\nr−a < s+inD+\nH,c∩D+\nr.Sending a→0,the maximum principle implies that there exists a positive\nconstant a∗>0 such that ωn\nr−a∗≤s+in∂(D+\nH,c∩D+\nr) and the points where ωn\nr−a∗=s+are contained\ninγH\nc.We get a contradiction since∂s+\n∂ν= +∞while∂(ωn\nr−a∗)\n∂ν<+∞inγH\nc.\nSo we get that, fixed n∈N,{ωn\nr}ris an increasing sequence converging to a solution ωnof the problem\n\n\nP(ωn) = 0 in D+,\nωn=n inγc,\nωn= 0 in {y= 0}.\nNotice that {ωn}is an increasing sequence of solutions of P(ωn) = 0 bounded below by {t= 0}in all D+\nand above by s+inD+\nH,c.Hence, applying the Monotone Convergence Theorem, there exists ω+= lim\nn→∞ωn\ndefined in the convergence set U ⊂ D+.Since in D+\nH,cthe sequence has an upper bound, the divergence\nsetVwill be contained in D+\\D+\nH,c.Theorem 5.2 implies that ∂V ∩∂Uis the union of smooth arcs with\nconstant geodesic curvature 2 H, each one convex towards U.\nSince γcis the only smooth curve of constant geodesic curvature 2 Hcontained in D+\\D+\nH,cwith the right\norientation, the result follows by proving that ∂V ∩∂Uis smooth.\nAssume first that two arcs γ1, γ2⊂∂V ∩∂Umeet at a convex corner pofU.Letp1(resp. p2) a point in\nγ1(resp. γ2) sufficiently close to p,such that |˜γ1|=|˜γ2|=ε, where ˜ γ1(resp. ˜ γ2) is the part of γ1(resp.\nγ2) between pandp1(resp. p2) and let γ3be the geodesic arc connecting p1andp2(notice that γ3⊂ U\nsince pis a convex corner of ∂U). We denote by ∆ the triangle bounded by ˜ γ1∪˜γ2∪γ3.Equation (4.6) and\nDefinition 4.1 implies that\n2HA(∆) = Flux ωn(˜γ1) + Flux ωn(˜γ2) + Flux ωn(γ3), (5.1)\nwhile Lemma 4.3 implies that\nlim\nn→∞Flux ωn(˜γ1) =|˜γ1|=ε, lim\nn→∞Flux ωn(˜γ2) =|˜γ2|=ε. (5.2)\nFrom (5.1), (5.2) and Lemma 4.2, we deduce that\n2HA(∆)≥2ε− |γ3|.12 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nEquivalently, we have\n2HA(∆)\n|γ3|≥2ε\n|γ3|−1.\nA contradiction is given by the fact that lim\nε→0A(∆)\n|γ3|= 0,while the left-hand side of the equation remains\nuniformly positive.\nAssume now that two arcs γ1, γ2⊂∂V ∩∂Umeet at a convex corner pofV.Letp1be a point in the\ninterior of γ1,let Γ be a curve in Ujoining pandp1and denote by ∆ the domain bounded by Γ ∪γ1.In\nparticular u= (ω+)|∆is a solution to the Dirichlet problem\n\n\nP(u) = 0 in ∆ ,\nu=ω+ in Γ,\nu= +∞ inγ1.\nApplying an ambient isometry φ,such that φ(·) =·∗,we can assume that pis in the interior of ∆∗andΓ is\nin the interior of U,in particular we have that U ∩∆∗̸=∅andV ∩∆∗̸=∅.Hence, for a >0 sufficiently large,\nthe maximum principle implies that φ(u) +a > ωnin ∆∗for any n∈N.So, the Monotone Convergence\nTheorem implies that ∆∗⊂ U,but this contradicts the fact that V ∩∆∗̸=∅.\nSuppose now that 0 < H <1\n2√\n2.A simple computation implies that both γcand ˜γcare contained in D−\nH,c\n(where ˜ γcis the reflection of γcwith respect to the geodesic {x=c}inH2). We denote by ˜Dthe reflection\nofD+with respect to the geodesic {x=c}(clearly D−\nH,c⊂˜D). Let B(r) be the geodesic disks centered in\n˜γc(1) of geodesic radius rand denote by D−\nr=˜D∩B(r).For any n∈N,[3, Theore 1] implies the existence\nof the solution ωn\nrto the problem\n\n\nP(ωn\nr) = 0 in D−\nr,\nω+\nn=−n in∂D−\nr∩∂˜D,\nω+\nn= 0 in ∂B(r)∩˜D.\nAs above, it is easy to show that for any n∈N,there exists ωn= lim\nr→∞ωn\nrand that {ωn}nis a strictly\ndecreasing sequence of solutions of P(ωn) = 0 defined in ˜D.Hence, applying the Monotone Convergence\nTheorem, there exists ω−= lim\nn→∞ωndefined in the convergence set U ⊂D+.Since in D+\nH,cthe sequence has\nan upper bound, the divergence set Vwill be contained in ˜D\\D−\nH,c.Theorem 5.2 implies that ∂V ∩∂Uis\nthe union of smooth arcs of constant geodesic curvature 2 H, each one convex towards U.\nAgain, since γcis the only smooth curve of constant geodesic curvature 2 Hcontained in ˜D\\D−\nH,cwith the\nright orientation, the result follows by proving that ∂V ∩∂Uis smooth.\nIfγ1, γ2⊂∂V ∩∂Uare two arcs meeting at a convex corner pofV,we use the same argument as above.\nSo, assume that two arcs γ1, γ2⊂∂V ∩∂Umeet at a convex corner pofU.Letp1(resp. p2) a point in\nγ1(resp. γ2) sufficiently close to p,such that |˜γ1|=|˜γ2|=ε, where ˜ γ1(resp. ˜ γ2) is the part of γ1(resp.\nγ2) between pandp1(resp. p2) and let γ3be the geodesic arc connecting p1andp2(notice that γ3⊂ U\nsince pis a convex corner of ∂U). We denote by ∆ the triangle bounded by ˜ γ1∪˜γ2∪γ3.Equation (4.6) and\nDefinition 4.1 implies that\n2HA(∆) = Flux ωn(˜γ1) + Flux ωn(˜γ2) + Flux ωn(γ3), (5.3)\nwhile Lemma 4.3 implies that\nlim\nn→∞Flux ωn(˜γ1) =−ε, lim\nn→∞Flux ωn(˜γ2) =−ε. (5.4)\nFrom (5.1), (5.2) and Lemma 4.2, we deduce that, for εsufficiently small,\n2HA(∆)≤ −2ε+|γ3|<0,\nwhich gives a contradiction and completes the proof. □CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 13\nIf we describe E(−1, τ) with the model Ω ×Rendowed with the metric (4.2), applying rotations with\nrespect to the t-axis and parabolic translations, Theorems 5.1 and 5.3 guarantee the existence of asymptotic\nbarriers as follows:\nFigure 5. Asymptotic barriers.\nTheorem 5.4. Letγ⊂Ωbe a curve of constant geodesic curvature 2Hwhich separates the disk Ωin two\nconnected domains. We denote by D+the convex component and D−the other one. Then, there exists two\nfunctions ω+andω−solving the Dirichlet problems\n\n\nP(ω+) = 0 inD+,\nω+= +∞ inγ,\nω+=f+in∂D+\\γ,and\n\nP(ω−) = 0 inD−,\nω−=−∞ inγ,\nω−=f−in∂D−\\γ,\nwhere f+∈ C∞(∂D+\\γ)andf−∈ C∞(∂D−\\γ)are bounded functions.\n6.Existence of entire H-graphs\nIn this section we consider the model of E(−1, τ) given by Ω ×Rendowed with the metric in (4.2), in\nwhich {t= 0}is a rotational surface of constant mean curvature H∈\u0000\n0,1\n2\u0001\n.In this setting we prove the\nexistence of H-graph having prescribed asymptotic boundary value. The proof is inspired to the one of [16,\nTheorem 4], where the authors prove an analogous result for τ=H= 0.We point out that an existence\nresult for subcritical CMC graphs in H2×Rwas already prove in [7], we prove a similar result avoiding the\nuse of Scherk graphs over ideal domains in its proof.\nTheorem 6.1. LetΓbe a rectifiable curve in ∂Ω×R, that is the vertical graph of the function f∈ C(∂Ω).\nThen, there exists an entire H-graph having Γas asymptotic boundary. Such graph is unique.\nProof. Assume first that Γ is differentiable. For n∈N, letDn⊂Ω be the disk centered at the origin with\nEuclidean radius 1 −1\nn+1,so that {Dn}nis an exhaustion of Ω .Since ∂Dnhas geodesic curvature greater\nthan 2 Hfor any n∈N,for any g∈ C(∂Dn), there exists an H-graph u∈ C2(Dn) such that u≡gin∂Dn.\nLetηn:∂Dn→∂Ω be such that ηn(x, y) =\u0000n+1\nnx,n+1\nny\u0001\nandfn=f◦ηn.Hence, fn(∂Dn) is a sequence\nof curves lying between {t= min f}and{t= max f}and converging to Γ for n→ ∞ .Denote by unthe\nsolution of the Dirichlet problem for constant mean curvature HinDnwith fnas boundary values and by\nMnits graph. The maximum principle implies that min f≤un≤max ffor any n∈N,thus, Compactness\nTheorem implies that there exists a subsequence converging to a solution u: Ω→Runiformly on compact\nsubsets of Ω, and denote by Mits graph. We have that Γ is contained in the asymptotic boundary ∂MofM14 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nby construction, so it remain to prove that if p∈∂Ω×R\\Γ,then p /∈∂M. Assume that there exist p∈∂M\nlying below Γ ,without loss of generality we can assume p= (1,0, f(1,0)−2ε).Letε1>0 be such that\nmax\nq∈∂Ωε1(f((1,0))−f(q))< ε,where ∂Ωε1={(x, y)∈∂Ω| |y|< ε1}.Letγbe the curve of constant geodesic\ncurvature 2 Hthat has ∂(∂Ωε1) and that is concave with respect to the domain bounded by γand∂Ωε1.Let\nk∈Rbe a constant such that ω−(1,0) +k=f(1,0)−ε.Notice that if ϵ1is sufficiently small, up to apply a\nrotation with respect to the t-axis, we get that ω−+k < f in∂Ωε1.By construction, the graph of ω−+klies\nbelow Γ nfor any nsufficiently large, and so the maximum principle implies that Mnis above the graph of\nω−+k. Hence the limit of Mn, that is M,cannot contain pin its asymptotic boundary. If plies above Γ, we\ncan apply the same argument using ω+instead of ω−. The uniqueness follows by the maximum principle.\nNow, if Γ is rectifiable, we consider two families of differentiable curves approximating Γ respectively from\nabove and from below, and use the argument in [17, Correction 4(a)] and this conclude the proof. □\nRemark 6.2.The result of this theorem is consistent with the Slab Theorem in [8]. Indeed, moving back\nto the classical cylinder model Cwith{t= 0}being rotational and minimal, we have that all the solutions\ngiven applying Theorem 6.1 are asymptotic to the entire rotational H-graph, that is, they diverge to + ∞\napproaching the asymptotic vertical boundary of C.Furthermore, the previous result shows that a half-space\nTheorem analogous to [18, Theorem 1] and [12, Theorem 3] cannot be proven for H <1\n2and that the\nhypothesis of [14, Theorem 2] are sharp.\n7.Asymptotic Jenkins–Serrin result\nAgain we consider the model of E(−1, τ) given by Ω ×Rendowed with the metric in (4.2), in which {t= 0}\nis a rotational surface of constant mean curvature H∈\u0000\n0,1\n2\u0001\n.In this section we prove a Jenkins–Serrin\ntype result for subcritical constant mean curvature graphs over unbounded domains of H2.When τ= 0 this\nresult was proven by Mazet, Rodriguez and Rosenberg (see [13, Theorems 4.9, 4.12]) for H= 0 and by Folha\nand Melo (see [6, Theorems 3.1, 3.2]) for 0 < H <1\n2when the asymptotic boundary of the domain consists\nonly of a finite number of isolated points of the asymptotic boundary of H2.We extend these results to the\ncaseτ̸= 0 allowing the asymptotic boundary of the domains to contain part of the asymptotic boundary of\nH2.\nDefinition 7.1 (Jenkins–Serrin domain) .A domain D ⊂H2is called a Jenkins–Serrin domain if its\nboundary ∂∞Dconsists of a finite number of arcs AiandBiof constant geodesic curvature 2 Hand−2H\nrespectively, a finite number of arcs Ciof geodesic curvature kg(Ci)≥2Hand a finite number of open arcs\nDiat∂∞H2,together with their endpoints, which are called the vertices of D.\nWe say that a Jenkins–Serrin domain Disadmissible if\nA) neither two Aior two Bimeets at a convex corner of D,nor at an asymptotic vertex;\nB) denoting by B∗\nithe reflection of BiinH2with respect to the geodesic passing through the endpoints of\nBi, for any i∈I,the interior of B∗\nidoes not intersect D.\nIfDis an admissible Jenkins–Serrin domain, we call extended Jenkins–Serrin domain the domain D∗whose\nboundary ∂∞D∗consists of the union of the arcs Ai,B∗\ni,CiandDi.\nRemark 7.2.Condition (A) of admissibility is a necessary condition that can be deduced by the proof of\nTheorem 5.3. Condition (B), is necessary to define the extended Jenkins–Serrin domain, where is possible\nto solve a Dirichlet problem with finite boundary data.\nDefinition 7.3 (Jenkins–Serrin Problem) .Letfi∈ C(Ci) and gi∈ C0(Di).Then, we call Jenkins–Serrin\nProblem the Dirichlet Problem\nPJS(D) :=\n\nu= +∞in∪Ai,\nu=−∞ in∪Bi,\nu=fi inCi,\nu=gi inDi.CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 15\nDefinition 7.4 (Inscribed polygon) .LetDbe an admissible domain. We say that Pis an admissible\ninscribed polygon if P⊂ D ∪ ∂D,its edges have constant geodesic curvature ±2Hand all the vertices of P\nare vertices of D.\nFor each ideal vertex piofDat∂∞H2,we consider a horocycle Hiatpi.Assume Hiis small enough so\nthat it does not intersect bounded edges of ∂∞DandHi∩Hj=∅for every i̸=j.LetFibe the convex\nhorodisk with boundary Hi.Each Aimeets exactly two horodisks. Denote by ˜Aithe compact arc of Ai\nwhich is the part of Aioutside the two horodisks; we define |Ai|= as the length if ˜Ai.LetP⊂ D be an\ninscribed polygon and denote by DP⊂ D the domain bounded by P.For each arc ηj∈P,we define ˜ ηjand\n|ηj|in the same way.\nγ(P) =X\nj∈J|ηj|, α (P) =X\nAi⊂P|Ai|and β(P) =X\nBi∈P|Bi|.\nFor any family of horocycles H={Hi},denote by Fithe convex horodisk bounded by Hi,˜DP=S\ni(DP∪Fi)\nandDH\nP=DP\\˜DPand we define\n˜A(DP) :=A(DH\nP) +A(˜DP).\nNotice that this definition makes sense, since A(˜Ω) is finite (see [6, Section 3]).\nTheorem 7.5. LetDbe an admissible Jenkins–Serrin domain. Assume that both {Ci}and{Di}are empty.\nThen, there exists a solution to the Dirichlet problem PJS(D)if and only if for some choice of horocycles at\nthe vertices we have\nα(∂D) =β(∂D) + 2H˜A(D),\nand for any inscribed polygon P,\nα(P)< γ(P) + 2H˜A(DP) and β(P)> γ(P)−2H˜A(DP). (7.1)\nIfDis relatively compact, the solution is unique up to vertical translation.\nIf either {Ci}or{Di}is not empty, there exists a solution to PJS(D)if and only if for some choice of\nhorocycles at the vertices, the inequalities in (7.1) are satisfied for any polygon Pinscribed in D.IfDis\nrelatively compact, the solution is unique.\nSketch of the proof. Since the Jenkins–Serrin result relies on the existence of upper and lower barriers and\ncomputation of areas and lengths in the base H2,the proof of the theorem above will be equal to the one\nof [9, Theorems 7.11 and 7.12] (for Drelatively compact) and [6, Theorems 3.1 and 3.2] (for Dunbounded\nand{Di}=∅).\nIfDis unbounded and {Di} ̸=∅,we built a sequence of solutions as done in the proof of [13, Theorem 4.9]\nfor the minimal case. Consider the extended Jenkins–Serrin domain D∗.Forn∈N, letDn⊂Ω be the\ndisk centered at the origin with Euclidean radius 1 −1\nn+1,and let Dn=Dn∩ D∗,so that {Dn}nis an\nexhaustion of D∗.Letηn:∂Dn→∂Ω be such that ηn(x, y) =\u0000n+1\nnx,n+1\nny\u0001\n,denote by Di,n=η−1\nn(Di) and\ndefine gi,n=gi◦ηn:Di,n→R.Denote by Ci,n=Ci∩Dn, Ai,n=Ai∩DnandBi,n=B∗\ni∩Dn,so that\n∂Dn=∪Ai,n∪Bi,n∪Ci,n∪Di,n∪Γn,where Γ n= (∂Dn∩ D∗)\\(∪\niDi,n).For any m, n∈N,we can solve\nthe Dirichlet problem\n\nP(um,n) = 0 in Dn,\num,n=m inAi,n,\num,n=−m inBi,n,\num,n=fi,m inCi,n,\num,n=gi,n,m inDi,n,\num,n= 0 in Γ n,\nwhere the function fi,m(resp. gi,n,m) denotes the function fi(resp. gi,n) truncated above and below by m\nand−m,respectively. Notice that, by the maximum principle we have that −m≤um,n≤m.Hence, as\nin the proof of Main Theorem 6.1, the existence of the barriers ω+andω−and the compactness theorem16 CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ)\nimplies that, up to pass to a subsequence, {um,n}converges uniformly on compact subsets of D∗to an\nH-graph um:D∗→Rwith boundary data\n\n\nP(um) = 0 in Dn,\num=m inAi,\num=−m inBi,\num=fi,m inCi,\num=gi,m inDi,\nwhere the function fi(resp. gi,m) denotes again the function fi,m(resp. gi) truncated above and below by\nmand−m,respectively. For any Di⊂∂Dthere exists a curve ˆDihaving the same endpoints points of Di\nand geodesic curvature κg(ˆDi)≥2Hsuch that the domain ˆDibounded by Di∪ˆDiis concave and ˆDi⊂ D.\nUsing ω+andω−,we can find two functions ˆ g+\ni,ˆg−\ni∈ C(ˆDi) such that ˆ g−\ni< um<ˆg+\niinˆDi,for any m∈N.\nNotice that D \\(∪\niˆDi) is an admissible Jenkins–Serrin domain with {Di}=∅there exist u+, u−∈ C∞(ˆD)\nsatisfying\n\nP(u+) = 0 ,inD \\(∪\niˆDi),\nu+= +∞ inAi,\nu+=−∞ inBi,\nu+=fi inCi,\nu+= ˆg+\ni inˆDiand\n\nP(u−) = 0 ,inD \\(∪\niˆDi),\nu−= +∞ inAi,\nu−=−∞ inBi,\nu−=fi inCi,\nu−= ˆg−\ni inˆDi.\nThe Maximum principle finally implies that u−≤um≤u+inD \\(∪\niˆDi),for any m∈N.Hence, applying\nthe compactness theorem, we get that {um}admits a subsequence that converges to the solution uof the\nJenkins–Serrin problem in every compact subset of D.The fact that uconverges to the function giinDi\nfollows by applying the same argument of the proof of Theorem 6.1. □\nReferences\n[1] Jes´ us Castro-Infantes. On the asymptotic Plateau problem in fSL2(R).J. Math. Anal. Appl. 507, no. 2 (2022), 125831.\n[2] Qing Cui, Albet˜ a Mafra and Carlos Pe˜ nafiel. Immersed hyperbolic and parabolic screw motion surfaces in the space\n]PSL 2(R, τ).Geom. Dedicata 178(2015), 297–322.\n[3] Marcos Dajczer and Jorge Herbert de Lira. Killing graphs with prescribed mean curvature and Riemannian submersions.\nAnn. Inst. Henri Poincare (C) Anal. Non Lineaire 26, no. 3 (2009), 763–775.\n[4] Andrea Del Prete, Jos´ e M. Manzano and Barbara Nelli. The Jenkins–Serrin problem in 3-manifolds with a Killing vector\nfield. Preprint (available at arXiv:2306.12195) (2023).\n[5] M. Eichmair, J. Metzger. Jenkins–Serrin-type results for the Jang equation. J. Differential Geom. ,102(2016), 207–242.\n[6] Abigail Folha and Sofia Melo. The Dirichlet problem for constant mean curvature graphs in H2×Rover unbounded domains.\nPac. J. Math. 251, no. 1 (2011), 37–65.\n[7] Abigail Folha and Harold Rosenberg. Entire constant mean curvature graphs in H2×R.Pac. J. Math. 316, no. 2 (2022),\n307–333.\n[8] Laurent Hauswirth, Ana Menezes and Magdalena Rodr´ ıguez. Slab theorem and halfspace theorem for constant mean cur-\nvature surfaces in H2×R.Rev. Mat. Iberoam. 39, no. 1 (2023), 307–320.\n[9] Laurent Hauswirth, Harold Rosenberg and Joel Spruck. Infinite Boundary Value Problems for Constant Mean Curvature\nGraphs in H2×RandS2×R.Am. J. Math. 131no. 1 (2009), 195–226.\n[10] Ana M. Lerma and Jos´ e M. Manzano. Compact stable surfaces with constant mean curvature in Killing submersions. Ann.\nMat. Pura Appl. 196(2017), 1345–1364.\n[11] Jos´ e M Manzano. On the classification of Killing submersions and their isometries. Pac. J. Math 270, no. 2 (2014), 367–392.\n[12] Laurent Mazet. The half space property for cmc 1 /2 graphs in E(−1, τ).Calc. Var. Partial Differ. Equ. 52(2015), 661–680.\n[13] Laurent Mazet, M Magdalena Rodr´ ıguez, and Harold Rosenberg. The Dirichlet problem for the minimal surface equation,\nwith possible infinite boundary data, over domains in a Riemannian surface. Proc. Lond. Math. Soc. 102, no. 6 (2011),\n985–1023.\n[14] Laurent Mazet and Gabriela A. Wanderley. A half-space theorem for graphs of constant mean curvature 0 < H < 1/2 in\nH2×R.Ill. J. Math. 59, no. 1 (2015), 43–53.\n[15] Sofia Melo. Minimal graphs in ]PSL 2(R) over unbounded domains. Bull. Braz. Math. Soc., New Series 45(2014), 91–116.\n[16] Barbara Nelli and Harold Rosenberg. Minimal surfaces in H2×R.Bull. Braz. Math. Soc., New Series 33, no. 2 (2002),\n263–292.CONSTANT MEAN CURVATURE GRAPHS WITH PRESCRIBED ASYMPTOTIC VALUES IN E(−1, τ) 17\n[17] Barbara Nelli and Harold Rosenberg. Erratum to “Minimal surfaces in H2×R”.Bull. Braz. Math. Soc., New Series 38\n(2007), 661–664.\n[18] Barbara Nelli and Ricardo Sa Earp. A halfspace theorem for mean curvature H= 1/2 surfaces in H2×R.J. Math. Anal.\nAppl. 365(2010), 167–170.\n[19] Carlos Pe˜ nafiel. Invariant surfaces in ]PSL 2(R, τ) and applications. Bull. Braz. Math. Soc., New Series 43(2012), 545–578.\n[20] Harold Rosenberg, Rabah Souam and Eric Toubiana. General curvature estimates for stable H-surfaces in 3-manifolds\napplications. J. Differ. Geom. 84, no. 3 (2010), 623–648.\n[21] Ricardo Sa Earp. Parabolic and hyperbolic screw motion surfaces in H2×R.J. Aust. Math. Soc. 85, no. 1 (2008), 113–143.\n[22] Ricardo Sa Earp and Eric Toubiana. Screw motion surfaces in H2×RandS2×R.Ill. J. Math 49, no.4, (2005), 1323–1362."    },    {        "title": "2402.07287v2.Solutions_of_generalized_supergravity_equations_with_the_BTZ_black_hole_metric.pdf",        "content": "TTMP manuscript No.\n(will be inserted by the editor)\nSolutions of generalized supergravity equations with the BTZ black hole\nmetric\nAli Eghbalia, Simin Ghasemi-Sorkhabib, Adel Rezaei-Aghdamc\n1Department of Physics, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran\nReceived: date / Accepted: date\nAbstract\nWe proceed to investigate the solutions of generalized super-\ngravity equations (GSE) in three dimensions. Our candidate\nis the metric of BTZ black hole. It is shown that only the\ncases with J=M=0 and J=0,M̸=0 of the BTZ metric\nsatisfy the GSE. In the former, we find a family of solutions\nincluding the field strength Hrϕt=2r/l, the cosmological\nconstant Λ=−1/l2, one-form Zµand a vector field which\nis obtained to be a linear combination of the directions of\nthe time translation and rotational symmetries. In the lat-\nter, the solutions possess the same field strength as before,\nwhile the cosmological constant Λ, one-form Zµand vector\nfield Iwill be different from the previous case. Finally, we\nshow that the charged black string solution found by Horne\nand Horowitz, which is Abelian T-dual to the the BTZ black\nhole solution, can be considered as a solution for the GSE.\nKeywords: Generalized supergravity equations; Standard su-\npergravity; BTZ metric\n1 Introduction\nSupergravity is a modern field theory combining the prin-\nciples of supersymmetry and general relativity. The 10- di-\nmensional supergravity theory describes the dynamics of mass-\nless string excitations and arises in string theory as a low-\nenergy effective theory. Recently, new string backgrounds\nhave been discovered that satisfy a more general set of mo-\ntion equations than ordinary supergravity. These equations,\nwhich are a generalization of the standard supergravity equa-\ntions, are called the GSE. It’s worth noting that the primary\naCorresponding author: eghbali978@gmail.com\nbs.ghassemi.s@gmail.com\ncrezaei-a@azaruniv.ac.irdifference between standard supergravity and GSE is the ab-\nsence of a scalar dilaton. Arutyunov, et al. [1] introduced the\nGSE in order to investigate integrable deformations of the\nAdS 5×S5type II superstring sigma model [2, 3]1, which\nis closely related to non-Abelian T-duality transformations\n[10–13]. This generalized system in string theory includes\nadditional vector fields Iin addition to the standard compo-\nnent fields of type IIB supergravity. Up to now, the corre-\nsponding classical action has not been discovered, and only\nthe equations of motion are presented. Additionally, Ref.\n[14] presents a solution of standard supergravity with a lin-\near dilaton that has been mapped to a GSE solution through\na formal T-duality transformation along a specific direction.\nThese findings emphasize the importance of treating solu-\ntions of standard supergravity and GSE equally in the con-\ntext of string theory, as T-duality is a fundamental symme-\ntry in string theory. As a great progress in the recent study\nof string theory, Tseytlin and Wulff [15] demonstrated that\nthe GSE can be reproduced by solving the κ-symmetry con-\nstraints. In fact, their results show that the κ-symmetry of the\nGreen-Schwarz action requires the background supergravity\nfields to satisfy the GSE.\nLater, the GSE attracted the attention of many researchers,\nin such a way that it was shown that [16] the whole bosonic\npart of the type II GSE can be reproduced from the T-duality\ncovariant equations of motion of the double field theory when\none chooses a non-standard solution of the strong constraint.\nAdditionally, in Ref. [16], the Weyl invariance of the bosonic\nsigma model on a generalized gravity background has been\n1In order to construct the Yang-Baxter deformations of AdS 5×S5, one\nmust employ the classical r-matrix as a solution of the homogeneous\nclassical Yang-Baxter equation [4]. It has been shown that the Yang-\nBaxter deformed background of AdS 5×S5satisfies the equations of\nmotion of type IIB supergravity if the classical r-matrix satisfies the\nunimodularity condition [5] (see, also, [6–9]). Otherwise, the back-\nground is a solution of the GSE.arXiv:2402.07287v2  [hep-th]  16 Feb 20242\nshown by using the doubled formalism. We think that the\nresults of work of Ref. [16] have provided positive evidence\nthat superstring theories defined on solutions of the GSE are\nWeyl invariant (see, also, [17]). In Ref. [18], without intro-\nducing a T-duality manifest formulation of string theory, it\nhas been constructed a local counterterm that cancels out\nthe Weyl anomaly of bosonic string theory defined in gen-\neralized supergravity backgrounds; the results of their work\nsupport the Weyl invariance of string theory in generalized\nsupergravity backgrounds. Also, it has been shown that the\nequations of motion of generalized supergravity can be fol-\nlowed from the sigma model once the Killing vector Iis\nidentified with the trace of the structure constants [12]. In\nthis regard, for the Bianchi spacetimes, it was shown that\n[12] the non-Abelian T-duals with respect to non-semisimple\ngroups are solutions to generalized supergravity.\nAs a spin off from this progress, it would be interesting\nto present new solutions for the GSE in three dimensions.\nIn this work, we obtain some new solutions for the GSE,\nincluding special cases of the BTZ metric ( J=M=0 and\nJ=0,M̸=0). Notably, when we select the vector field I\nbased on our perspective, it becomes evident that the cases\nwith J̸=0,M=0 and J̸=0,M̸=0 do not satisfy the GSE.\nAs an interesting result, we show that the charged black\nstring solution found by Horne and Horowitz [19], which\nis Abelian T-dual to the the BTZ black hole solution, can be\nconsidered as a solution for the GSE.\nThe structure of the paper is as follows. We start with\na short overview of the GSE in section 2, and introduce our\nnotation. Next, in section 3, after the introduction of the BTZ\nblack hole metric, we consider the BTZ black hole solutions\nin the context of the low energy approximation. Section 4\ncontains the original results of the work: we look into the\nBTZ metrics in the context of the GSE and show that only\nthe cases J=M=0 and J=0,M̸=0 of the BTZ metric can\nsatisfy the GSE. In section 5, the charged black string solu-\ntion is considered as a solution for the GSE. Finally, conclu-\nsion is reported in section 6.\n2 A brief review of the GSE\nThe subject of this section is a brief overview of the GSE. In\nthe absence of Ramond-Ramond fields, these equations in D\ndimensions take the following form [1]\nRµν−1\n4HµρσHρσ\nν+ (∇µXν+∇νXµ) =0, (1)\n1\n2∇λHλµν−XλHλµν−∇µXν+∇νXµ=0, (2)\nR−1\n12H2+4∇µXµ−4XµXµ−4Λ=0, (3)\nwhere RµνandRare the respective Ricci tensor and Gauss\ncurvature that are calculated from the metric Gµν, and Λis the cosmological constant. Here, the D-dimensional in-\ndices µ,ν,...of coordinates xµare raised or lowered with\nthe metric Gµν. The covariant derivative ∇µis the conven-\ntional Levi-Civita connection associated with Gµν. The field\nstrength Hµνρcorresponding to anti-symmetry tensor field\nB is defined as\nHµνρ=∂µBνρ+∂νBρµ+∂ρBµν. (4)\nIn addition, Xµis defined to be Xµ=Iµ+Zµin which I=\nIµ∂µis a vector field, while Z=Zµdxµis a one-form. They\nsatisfy [1]\nLIGµν=0, (5)\nLIBµν=0, (6)\n∇µZν−∇νZµ+IλHλµν =0, (7)\nIλZλ=0, (8)\nwhereLstands for the Lie derivative. The conventional\ndilaton is included in Zµas follows:\nZµ=∂µΦ+BνµIν. (9)\nHere, Φis a scalar dilaton field hidden within Zµ. Note that\nif we set Iµ=0, then one gets that Xµ=∂µΦ, and thus, the\nGSE reduce to the standard supergravity equations.\nIn this work, we present new solutions for the GSE (1)-\n(3) together with (5)-(8), including special cases of the BTZ\nmetric, field strength H, one-form Zµ, and an appropriate\nvector field I. In this manner, we will derive a family of so-\nlutions for the cases J=0,M=0 and J=0,M̸=0 within\nthe BTZ metric. As mentioned earlier, the case Iµ=0 of the\nGSE corresponds to the standard supergravity equations. It\nhas already been shown that [20] a slight modification of\nthe BTZ black hole solution yields an exact solution to the\nstandard supergravity equations. In the next section, after a\nbrief overview of the BTZ black hole, we consider the BTZ\nmetric in the context of the low energy approximation.\n3 The BTZ black hole metric as a solution of the\nstandard supergravity equations\nFirst of all, let us introduce the metric of BTZ black hole.\nThe BTZ black hole, discovered by Banados, Teitelboim,\nand Zanelli [21], is a 2 +1-dimensional solution of Ein-\nstein’s equations with negative cosmological constant, mass,\nangular momentum, and charge. Unlike its higher-dimensional\ncounterparts, the BTZ black hole is asymptotically anti-de\nSitter and lacks a curvature singularity at the origin. The line\nelement for the black hole solutions is as follows [21]:\nds2= (M−r2\nl2)dt2−J dt d ϕ+r2dϕ2\n+ (r2\nl2−M+J2\n4r2)−1dr2, 0≤ϕ≤2π. (10)3\nwhere the radius lis related to the cosmological constant by\nl= (−Λ)−1/2. The constants of motion, denoted by Mand\nJ, represent the mass and angular momentum of the BTZ\nblack hole, respectively. The line element (10) describes a\nblack hole solution with outer and inner horizons at r=r+\nandr=r−, respectively,\nr±=l\u0000M\n2\u00011/2\u001a\n1±\u0012\n1−J2\nM2l2\u00131/2\u001b1/2\n. (11)\nThe mass Mand angular momentum Jare related to r=r±\nbyM= (r2\n++r2\n−)/l2andJ=2r+r−/l2. Solutions with\n−1<M<0 and J=0 describe point particle sources with\nnaked conical singularities at r=0. The metric with J=0\nandM=−1 can be recognized as ordinary anti-de Sitter\nspace, and it is separated by a mass gap from the case where\nJ=0 and M=0. The vacuum state, representing empty\nspace, is obtained by letting the horizon size go to zero. This\ncorresponds to M→0, which requires J→0. It’s worth not-\ning that the metric for the J=0 and M=0 black hole is\nnot the same as the AdS 3metric, which has negative mass,\nM=−1.\nAs mentioned in the previous section, a modified BTZ\nblack hole solution was obtained to a 2 +1-dimensional string\ntheory with a matter source given by anti-symmetric B-field\nwith the contribution of the cosmological constant Λ[20].\nThere, Horowitz and Welch showed that very solution to\n3-dimensional general relativity with Λ<0 can be con-\nsidered as a solution to the standard supergravity equations\nwith a constant dilaton field, Λ=−1/l2and field strength\nHµνρ=2εµνρ/l, where εµνρstands for the volume form in\nthree dimensions2. In addition, it was shown that [20] the\nsolution of BTZ black hole (10) along with a constant dila-\nton field, Λ=−1/l2and field strength Hrϕt=2r/lsatisfy\nthe equations of motion of the standard supergravity. It is\nworth mentioning that 3-dimensional black hole solutions\nto (10) do not exist assumed that Hµνρ=0.\n4 Solutions of the GSE with the BTZ metric\nIn what follows we shall look into the BTZ solutions in the\ncontext of the GSE. We show that the cases J=M=0 and\nJ=0,M̸=0 of the BTZ metric can be considered as so-\nlutions of the GSE. As mentioned in section 3, the Mand\nJare the mass and angular momentum of the BTZ black\nhole, respectively. They are appeared due to the time trans-\nlation symmetry and rotational symmetry of the metric, cor-\nresponding to the Killing vectors ∂/∂tand∂/∂ϕ, respec-\ntively. On the other hand, relation (5) is called Killing equa-\ntion, which in terms of a Killing vector field Ka=Kµ\na∂µit\ncan be written as\n∂µKλ\naGλν+Kλ\na∂λGµν+∂νKλ\naGµλ=0. (12)\n2Note that a special property of three dimensions is that the field\nstrength Hµνρmust be proportional to the volume form εµνρ.Accordingly, the vector field Ican be a Killing vector or\na linear combination of the Killing vectors corresponding\nto metric (10). The Killing vectors Kacorresponding to the\nBTZ metric can be derived by solving Killing equation (12),\ngiving us six linearly independent vectors. Here, to construct\nan appropriate vector field Iwe use the Killing vectors of the\nBTZ metric. We assume that the choice of\nI=α1K1+···+α6K6, (13)\nfor some constants αi, can be a suitable candidate for solv-\ning the GSE (1)-(3) together with (5)-(8). It should be noted\nthe fact that the BTZ solutions must be single-valued in the\nangular direction. Therefore, one should be careful in choos-\ning the vector field I. As the first example, we look into the\ncase J=0,M=−λ2<0 of (10) in full detail.\n4.1 Solutions with J=0, M=- λ2<0\nIn this subsection we shall investigate the solutions of the\nGSE for the case J=0,M=−λ2<0 of the BTZ metric.\nBefore proceeding to do this, let us first write down the BTZ\nmetric for the case J=0,M=−λ2<0. Using relation (10),\nit is given by\nds2=−(λ2+r2\nl2)dt2+ (λ2+r2\nl2)−1dr2+r2dϕ2.(14)\nThe Killing vectors of the metric (14) can be derived by\nsolving Killing equations. They are then read off\nK1=∂\n∂ϕ,\nK2=p\nλ2l2+r2hrl\nλ2l2+r2cos(λt\nl)sin(λϕ)∂\n∂t\n+λsin(λt\nl)sin(λϕ)∂\n∂r+1\nrsin(λt\nl)cos(λϕ)∂\n∂ϕi\n,\nK3=p\nλ2l2+r2h\n−rl\nλ2l2+r2sin(λt\nl)sin(λϕ)∂\n∂t\n+λcos(λt\nl)sin(λϕ)∂\n∂r+1\nrcos(λt\nl)cos(λϕ)∂\n∂ϕi\n,\nK4=p\nλ2l2+r2h\n−rl\nλ2l2+r2cos(λt\nl)cos(λϕ)∂\n∂t\n−λsin(λt\nl)cos(λϕ)∂\n∂r+1\nrsin(λt\nl)sin(λϕ)∂\n∂ϕi\n,\nK5=p\nλ2l2+r2hrl\nλ2l2+r2sin(λt\nl)cos(λϕ)∂\n∂t\n−λcos(λt\nl)cos(λϕ)∂\n∂r+1\nrcos(λt\nl)sin(λϕ)∂\n∂ϕi\n,\nK6=−l2∂\n∂t. (15)\nFortunately, for integer values of λ, the resulting Killing\nvectors are single-valued in the angular direction. Now, we\napply the Killing vectors (15) to construct an appropriate4\nvector field by using formula (13). As we will see, our so-\nlutions are classified into two special cases. In both cases of\nsolutions, the anti-symmetry tensor field Bis considered to\nbeB=−r2/l dt∧dϕ. Then, one can employ formula (5)\nto calculate the field strength corresponding to this B-field,\ngiving us\nH=−2r\nldr∧dt∧dϕ. (16)\nHere, the forms of our solutions including the metric (14)\nand field strength (16) are given by the following two cases\nI and II:\n•Case I: In this case, the GSE (1)-(3) together with (5)-(8)\nare fulfilled with the metric (14) and field strength (16) if the\nvector field I, one-form Zand cosmological constant Λcan\nnow be expressed in the following forms\nI=−α6l2∂\n∂t,Z=α6lr2dϕ,Λ=−1\nl2+λ2α62l4.(17)\nUsing these, the dilaton field is constant, and the compo-\nnents of Xµread off\nXt=α6l2(λ2+r2\nl2),Xϕ=α6lr2,Xr=0. (18)\n••Case II: In this case of solutions, the vector field Iis a\nrotational symmetry, which together with one-form ZandΛ\nare given as follows:\nI=α1∂\n∂ϕ,Z=α1l(λ2+r2\nl2)dt,Λ=−1\nl2+λ2α12l2.(19)\nThen, one finds that\nXt=α1l(λ2+r2\nl2),Xϕ=α1r2,Xr=0. (20)\nThe corresponding dilaton field to this case of the solutions\nis time-dependent, Φ=c0+λ2α1lt.\nLet us now discuss the solutions for all positive and neg-\native values of M, when Jis zero. The BTZ metric with\nJ=0,M̸=0 is given by\nds2= (M−r2\nl2)dt2+ (r2\nl2−M)−1dr2+r2dϕ2. (21)\nIn this case, the solutions are similar to those of the case\nJ=0,M=−λ2, so that one should use Minstead of −λ2\nin the solutions (17) and (19). Then, the solutions take the\nforms\nI=−α6l2∂\n∂t,Z=α6lr2dϕ,Λ=−1\nl2−Mα62l4,(22)\nand\nI=α1∂\n∂ϕ,Z=α1l(r2\nl2−M)dt,Λ=−1\nl2−Mα12l2,(23)\nand in the same way for the components of Xµ.4.2 Solution with J=0, M=0\nThe BTZ metric with J=0,M=0 is simply found by using\nthe formula (10), giving us\nds2=−r2\nl2dt2+l2\nr2dr2+r2dφ2. (24)\nIn order to construct the vector field I, one may determine\nthe Killing vectors corresponding to the metric (24) similar\nto what was done in (15). Analogously, only non-zero com-\nponent of the tensor field Bis considered to be Bϕt=r2/lfor\nwhich the field strength is calculated to be as in (16). Now,\nwe solve the GSE with the metric (24), field strength (16)\nand a constant dilaton field. The equations are then satisfied\nif the vector field I, one-form ZandΛhave the following\nforms3\nI=−α6l2∂\n∂t+α3∂\n∂ϕ,\nZ=α3r2\nldt+α6lr2dϕ,\nΛ=−1\nl2. (25)\nUsing the first two relations and also the B-field Bϕt=r2/l,\none then finds that the components of Xµare\nXt= (α3+α6l)r2\nl,\nXϕ= (α3+α6l)r2,\nXr=0. (26)\nAt the end of this section let us compare the solutions\nwith J=0, M=0 and J=0, M ̸=0. As showed in the above, for\nthe case of J=0, M=0 we obtained the vector Ias a linear\ncombination of the directions of the time translation and ro-\ntational symmetries, while this is not true for the case of J=0,\nM̸=0. In the case M ̸=0 of the solutions, if one uses a linear\ncombination of (22) and (23) for the vector Iand one-form\nZ, then, equation (8) becomes α1α6l3M=0. This means\nthat one of the constants α1orα6should be zero. Therefore,\nunlike the case of J=0, M=0, a linear combination of (22)\nand (23) cannot be a solution for the GSE.\n5 Abelian target space dual of the BTZ as a solution of\nthe GSE\nOne of the interesting issues in the context of GSE is that to\ninvestigate the solutions of the GSE under the T-duality. The\n3Note that our solution including the metric (24), Bϕt=r2/land re-\nlations given by equation (25) together with a constant dilaton field\nmay be related to a GSE solution found in equation (4.12) of Ref.\n[22]. There, it has been obtained a 10-dimensional solution for the GSE\n(with a non-zero Killing vector I) whose metric is locally AdS 3×S3×\nT4. If we look at the AdS 3part of solution (4.12) of [22], it may be\nlocally the same as our BTZ solution.5\nT-duality symmetry is one of the most interesting properties\nof string theory connecting seemingly different backgrounds\nin which the strings propagate. We say that the duality is\nAbelian if it is constructed on an Abelian isometry group\n[23]. By making use of the Buscher’s duality transforma-\ntions [23], it was shown that the BTZ black hole solution dis-\ncussed at the end of section 3 is, under the Abelian T-duality,\nequivalent to the charged black string solution discussed in\nRef. [19]. There, Horne and Horowitz found a family of so-\nlutions to low energy string theory describing charged black\nstrings in three dimensions. This family of solutions is given\nby\nd˜s2=−\u0000\n1−M\nˆr\u0001\ndˆt2+\u0010\n1−Q2\nMˆr\u0011\ndˆx2\n+\u0000\n1−M\nˆr\u0001−1\u0010\n1−Q2\nMˆr\u0011−1l2dˆr2\n4ˆr2, (27)\n˜H=Q\nˆr2dˆr∧dˆt∧dˆx, (28)\n˜Φ=−1\n2ln(lˆr), (29)\nwhere M=r2\n+/landQ=J/2 are the respective the mass\nand charge of the black string. It can be easily shown that\nthe above metric admits two Killing vectors ∂/∂ˆtand∂/∂ˆx.\nHere, we shall show that the metric (27) and field strength\n(28) can satisfy the GSE, if the vector field I, one-form Z,\ndilaton field and cosmological constant express as follows:\nI=α1∂\n∂ˆt,\nZ=−1\n2ˆrdˆr+α1(M\nQ−Q\nˆr)dˆx,\nΦ=c0−1\n2ln ˆr+α1M\nQˆx,\nΛ=2\nl2+2α12(M2−Q2)\nQ2. (30)\nIn addition to this solution, one can see that the GSE (1)-(3)\ntogether with (5)-(8) are fulfilled with the metric (27) and\nfield strength (28) if the vector field I, one-form Z, dilaton\nfield and cosmological constant can now be expressed in the\nfollowing forms\nI=α2∂\n∂ˆx,\nZ=−1\n2ˆrdˆr+α2Q(1\nˆr−1\nM)dˆt,\nΦ=c0−1\n2ln ˆr−α2Q\nMˆt,\nΛ=2\nl2+2α22(M2−Q2)\nM2. (31)\n6 Conclusion\nIn this study, we have obtained some new solutions for the\nGSE in three dimensions. Our solutions include the specialcases J=0,M=0 and J=0,M̸=0 of the BTZ metric\ntogether with the field strength H=−(2r/l)dr∧dt∧dϕ.\nIn each case we have calculated the the vector field I, one-\nform Zand the cosmological constant Λ. To determine the\nvector field Iwe have used the Killing vectors corresponding\nto the BTZ metrics. By comparing the solutions with J=\n0,M=0 and J=0,M̸=0, we concluded that unlike the\ncase of J=0,M=0, a linear combination of (22) and (23)\ncannot be a solution for the GSE. However, we have derived\na family of solutions as presented in relations (22), (23) and\n(25). Notably, our results confirm that these solutions are\nsingle-valued in the angular direction. On the other hand, we\nhave checked that the cases J̸=0,M=0 and J̸=0,M̸=0\nfor the BTZ metric do not satisfy the GSE.\nWhile we considered the BTZ metrics, one can easily\nrepeat for other geometric metrics such as Thurston geome-\ntries. As mentioned in section 3, the BTZ metric with J=0\nandM=−1 represents ordinary anti-de Sitter space ( AdS 3),\nwhich is nothing but one of Lorentzian Thurston geometries.\nTherefore, we have investigated that the AdS 3space can be\nconsidered as a solution for the GSE. Finally, as an inter-\nesting result, we have shown that the charged black string\nsolution (27), which is Abelian T-dual to the the BTZ black\nhole solution, can be also a solution for the GSE. In fact, this\nresult helps to answer the question whether the solutions of\nthe GSE are, under the Abelian T-duality, preserved. It will\nbe an interesting future direction to be addressed.\nAcknowledgements This work has been supported by the research\nvice chanceller of Azarbaijan Shahid Madani University under research\nfund No. 1402 /231.\nReferences\n1. G. Arutyunov, S. Frolov, B. Hoare, R. Roiban, A. A.\nTseytlin, Nucl. Phys. B 903, 262 (2016)\n2. F. Delduc, M. Magro, B. Vicedo, Phys. Rev. Lett. 112,\n051601 (2014)\n3. F. Delduc, M. Magro, B. Vicedo, J. High Energy Phys.\n10, 132 (2014)\n4. I. Kawaguchi, T. Matsumoto, K. Yoshida, J. High Energy\nPhys. 04, 153 (2014)\n5. R. Borsato, L. Wulff, J. High Energy Phys. 10, 045\n(2016)\n6. T. Araujo, I. Bakhmatov, E. O Colgain, J. Sakamoto, M.\nM. Sheikh-Jabbari, K. Yoshida, Phys. Rev. D 95, 105006\n(2017)\n7. T. Araujo, I. Bakhmatov, E. O Colgain, J. Sakamoto, M.\nM. Sheikh-Jabbari, K. Yoshida, J. Phys. A 51, 235401\n(2018)\n8. T. Araujo, E. O Colgain, J. Sakamoto, M. M. Sheikh-\nJabbari, K. Yoshida, Eur. Phys. J. C 77, 739 (2017)6\n9. I. Bakhmatov, O. Kelekci, E. O Colgain, M. M. Sheikh-\nJabbari, Phys. Rev. D 98, 021901 (2018)\n10. D. Orlando, S. Reffert, J. I. Sakamoto, K. Yoshida, J.\nPhys. A 49, 445403 (2016)\n11. B. Hoare, A. A. Tseytlin, J. Phys. A 49, 494001 (2016)\n12. M. Hong, Y . Kim, E. O. Colgain, Eur. Phys. J. C 78,\n1025 (2018)\n13. R. Borsato, L. Wulff, J. High Energy Phys. 08, 027\n(2018)\n14. B. Hoare, A. A. Tseytlin, J. High Energy Phys. 10, 060\n(2015)\n15. L. Wulff, A. A. Tseytlin, J. High Energy Phys. 06, 174\n(2016)\n16. J. Sakamoto, Y . Sakatani, K. Yoshida, PTEP 053B07\n(2017)17. Y . Sakatani, S. Uehara, K. Yoshida, J. High Energy\nPhys. 04, 123 (2017)\n18. J. J. Fernandez-Melgarejo, J. Sakamoto, Y . Sakatani, K.\nYoshida, Phys. Rev. Lett. 122, 111602 (2019)\n19. J. Horne and G. Horowitz, Nucl. Phys. B 368, 444\n(1992)\n20. G. Horowitz, D. Welch, Phys. Rev. Lett. 71, 328 (1993)\n21. M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett.\n69, 1849 (1992)\n22. Y . Sakatani, Prog. Theor. Exp. Phys. 073B04 (2019)\n23. T. Buscher, Phys. Lett. B 194, 59 (1987); Phys. Lett. B\n201, 466 (1988)."    },    {        "title": "2402.07331v2.Fundamental_Problems_on_Bounded_Treewidth_Graphs__The_Real_Source_of_Hardness.pdf",        "content": "Fundamental Problems on Bounded-Treewidth Graphs:\nThe Real Source of Hardness\nBarış Can Esmer∗Jacob Focke∗Dániel Marx∗Paweł Rzążewski†\nAbstract\nIt is known for many algorithmic problems that if a tree decomposition of width tis\ngiven in the input, then the problem can be solved with exponential dependence on t. A\nline of research initiated by Lokshtanov, Marx, and Saurabh [SODA 2011] produced lower\nbounds showing that in many cases known algorithms already achieve the best possible\nexponential dependence on t, assuming the Strong Exponential-Time Hypothesis (SETH).\nThe main message of this paper is showing that the same lower bounds can already be\nobtained in a much more restricted setting: informally, a graph consisting of a block of\ntvertices connected to components of constant size already has the same hardness as a\ngeneral tree decomposition of width t.\nFormally, a (σ, δ)-hubis a set Qof vertices such that every component of Qhas size\nat most σand is adjacent to at most δvertices of Q. We explore if the known tight lower\nbounds parameterized by the width of the given tree decomposition remain valid if we\nparameterize by the size of the given hub.\n•For every ε > 0, there are σ, δ > 0such that Independent Set (equivalently\nVertex Cover ) cannot be solved in time (2−ε)p·n, even if a (σ, δ)-hub of size\npis given in the input, assuming the SETH. This matches the earlier tight lower\nbounds parameterized by width of the tree decomposition. Similar tight bounds are\nobtained for Odd Cycle Transversal ,Max Cut ,q-Coloring , and edge/vertex\ndeletions versions of q-Coloring .\n•For every ε >0, there are σ, δ > 0such that △-Partition cannot be solved in time\n(2−ε)p·n, even if a (σ, δ)-hub of size pis given in the input, assuming the Set Cover\nConjecture (SCC). In fact, we prove that this statement is equivalent to the SCC,\nthus it is unlikely that this could be proved assuming the SETH.\n•ForDominating Set , wecanproveanon-tightlowerboundrulingout (2−ε)p·nO(1)\nalgorithms, assuming eitherthe SETH or the SCC, but this does not match the\n3p·nO(1)upper bound.\nThus our results reveal that, for many problems, the research on lower bounds on the\ndependence on tree width was never really about tree decompositions, but the real source\nof hardness comes from a much simplerstructure.\nAdditionally, we study if the same lower bounds can be obtained if σandδare fixed\nuniversal constants (not depending on ε). We show that lower bounds of this form are\npossible for Max Cut and the edge-deletion version of q-Coloring , under the Max 3-Sat\nHypothesis (M3SH). However, no such lower bounds are possible for Independent Set ,\nOdd Cycle Transversal , and the vertex-deletion version of q-Coloring : better than\nbrute force algorithms are possible for every fixed (σ, δ).\n∗CISPA Helmholtz Center for Information Security. The first author is also affiliated with the Saarbrücken\nGraduate School of Computer Science, Saarland Informatics Campus, Germany. Research of the third author\nis supported by the European Research Council (ERC) consolidator grant No. 725978 SYSTEMATICGRAPH.\n†Warsaw University of Technology, Faculty of Mathematics and Information Science and University of\nWarsaw, Institute of Informatics, pawel.rzazewski@pw.edu.pl .\n1arXiv:2402.07331v2  [cs.CC]  19 Feb 2024Contents\n1 Introduction 1\n2 Technical Overview 6\n2.1 q-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.2 Vertex Deletion to q-Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.3 Edge Deletion to q-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.4 Covering, Packing, and Partitioning . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.5 Triangle Partition and Triangle Packing . . . . . . . . . . . . . . . . . . . . . . 11\n3 Preliminaries 14\n4q-Coloring 16\n5 Vertex Deletion to q-Coloring 18\n5.1 Hardness for for q-ColoringVD . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n5.2 Simple Algorithm for q-ColoringVD . . . . . . . . . . . . . . . . . . . . . . . 28\n6 Edge Deletion to q-Coloring 28\n6.1Max CSP — Hardness under M3SH . . . . . . . . . . . . . . . . . . . . . . . . 29\n6.2 Realizing Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n6.3 Hardness for Max Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n6.4 Hardness for q-ColoringED . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37\n6.5 Simple Algorithm for q-ColoringED . . . . . . . . . . . . . . . . . . . . . . . 40\n7List- q-Coloring andList- q-ColoringVD with gadgets of constant de-\ngree faster than brute force 41\n7.1 Faster algorithm for List- q-Coloring . . . . . . . . . . . . . . . . . . . . . . . 41\n7.2 Faster algorithm for List- q-ColoringVD . . . . . . . . . . . . . . . . . . . . . 42\n8 Equivalent Hypotheses: Set Covering/Packing/Partitioning 46\n8.1 Basic Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46\n8.2 Structured Families of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n8.3 Reductions Based on Structured Families . . . . . . . . . . . . . . . . . . . . . 52\n9△-Packing and△-Partition 55\n9.1 Reducing Set Partition to△-Partition . . . . . . . . . . . . . . . . . . . . 55\n9.2 Reducing △-Packing toSet Packing . . . . . . . . . . . . . . . . . . . . . . 58\n10 Dominating Set 62\nReferences 641 Introduction\nStarting with the work of Lokshtanov, Marx, and Saurabh [24], there is a line of research\ndevoted to giving lower bounds on how the running time of parameterized algorithms can\ndepend on treewidth (or more precisely, on the width of a given tree decomposition) [4,7,8,\n11–13,19,28,29,32,33]. The goal of this paper is to revisit the fundamental results from [24]\nto point out that previous work could have considered a simplerparameter to obtain stronger\nlower bounds in a more uniform way. Thus, in a sense, this line of research was never really\nabout treewidth; a fact that future work should take into account.\nSuppose we want to solve some algorithmic problem on a graph Ggiven with a tree\ndecompositionofwidth t. FormanyNP-hardproblems, standarddynamicprogramtechniques\nor meta theorems such as Courcelle’s Theorem [5] show that the problem can be solved in time\nf(t)·nO(1)for some computable function f[10, Chapter 7]. In many cases, the running time is\nactually ct·nO(1)for some constant c >1, where it is an obvious goal to make the constant as\nsmall as possible. A line of work started by Lokshtanov, Marx, and Saurabh [24] provides tight\nconditional lower bounds for many problems with known ct·nO(1)-time algorithms. The lower\nbounds are based on the Strong Exponential-Time Hypothesis, formulated by Impagliazzo,\nPaturi, and Zane [16,17].\nStrong Exponential-Time Hypothesis (SETH).There is no ε >0such that for every\nk, every n-variable instance of k-Satcan be solved in time (2−ε)n·nO(1).\nThe goal of these results is to provide evidence that the base cof the exponent in the\nbest known ct·nO(1)-time algorithm is optimal: if a (c−ε)t·nO(1)-time algorithm exists for\nanyε >0, then SETH fails. The following theorem summarizes the basic results obtained by\nLokshtanov, Marx, and Saurabh [24].\nTheorem 1.1 ([24]).If there exists an ε >0such that\n1.Independent Set can be solved in time (2−ε)t·nO(1), or\n2.Dominating Set can be solved in time (3−ε)t·nO(1), or\n3.Max Cut can be solved in time (2−ε)t·nO(1), or\n4.Odd Cycle Transversal can be solved in time (3−ε)t·nO(1), or\n5.q-Coloring can be solved in time (q−ε)t·nO(1)for some q⩾3, or\n6.Triangle Partition can be solved in time (2−ε)t·nO(1),\non input an n-vertex graph Gtogether with a tree decomposition of width at most t, then the\nSETH fails.\nAlready in [24] it is pointed out that many of the lower bounds remain true even in the\nmore restricted setting where the input is not a tree decomposition, but a path decomposition.\nThis raises the following natural questions:\n•How much further can we restrict the input and still obtain the same lower bounds?\n•What is the realstructural source of hardness in these results?\nIn this paper, we show that many of these lower bounds remain true in a much more restricted\nsetting where a block of pvertices is connected to constant-size components. Additionally, we\n1demonstrate that our results are very close to being best possible, as further restrictions of\nthe structure of the graphs allow better algorithms.\nWe say that a set Qof vertices is a (σ, δ)-hubofGif every component of G−Qhas at most\nσvertices and each such component is adjacent to at most δvertices of QinG1. Our goal is\nto prove lower bounds parameterized by the size of a (σ, δ)-hub given in the input, where σ\nandδare treated as constants. One can observe that a (σ, δ)-hub of size pinGcan be easily\nturned into a tree decomposition of width less than p+σ, hence the treewidth of Gis at most\np+σ. Therefore, any lower bound parameterized by the size pof hub immediately implies a\nlower bound parameterized by the width of the given tree decomposition. We systematically\ngo through the list of problems investigated by Lokshtanov, Marx, and Saurabh [24], to see\nif the same lower bound can be obtained with this parameterization. Our results show that,\nin most cases, the results remain valid under parameterization by hub size. However, new\ninsights, techniques and arguments are needed; in particular, we require different complexity\nassumptions for some of the statements.\nColoring problems and relatives. Let us first consider the q-Coloring problem: given\na graph G, the task is to find a coloring of the vertices of Gwith qcolors such that adjacent\nvertices receive different colors. Given a (σ, δ)-hub Qof size p, we can try all possible q-\ncolorings on Qand check if they can be extented to every component of G−Q. Assuming\nσandδare constants, this leads to a qp·nO(1)algorithm. Our first result shows that this\nis essentially best possible, assuming the SETH; note that this result immediately implies\nTheorem 1.1(5).\nTheorem 1.2. Letq⩾3be an integer.\n1. For every σ, δ⩾1,q-Coloring onn-vertex graphs can be solved in time qp·nO(1)if a\n(σ, δ)-hub of size pis given in the input.\n2. For every ε >0, there exist integers σ, δ⩾1such that if there is an algorithm solving\nin time (q−ε)p·nO(1)every n-vertex instance of q-Coloring given with a (σ, δ)-hub\nof size at most p, then the SETH fails.\nTheq-ColoringED problem is an edge-deletion optimization version of q-Coloring :\ngiven a graph G, the task is to find a set Xof edges of minimum size such that G\\Xhas\naq-coloring. We show that qp·nO(1)running time is essentially optimal for this problem as\nwell.\nTheorem 1.3. Letq⩾2be an integer.\n1. For every σ, δ⩾1,q-ColoringED onn-vertex graphs can be solved in time qp·nO(1)\nif a(σ, δ)-hub of size pis given in the input.\n2. For every ε >0, there exist integers σ, δ⩾1such that if there is an algorithm solving in\ntime (q−ε)p·nO(1)every n-vertex instance of q-ColoringED given with a (σ, δ)-hub\nof size at most p, then the SETH fails.\n1This notion is related to component order connectivity , which is the size of the smallest set Qof vertices\nsuch that deleting Qleaves components of size not larger than some predefined constant σ[1,3,6,14,20,22,25,\n27,34,35,38]. Our definition has the additional constraint on the neighborhood size of each component. As\nwe often refer to the set Qitself (not only its smallest possible size) and we want to make the constants σ,δ\nexplicit, the terminology (σ, δ )-hub is grammatically more convenient than trying to express the same using\ncomponent order connectivity.\n2Forq⩾3, the lower bound of Theorem 1.2 for q-Coloring immediately implies the\nsame lower bound for the more general problem q-ColoringED . Observe that for q= 2,\ntheq-ColoringED problem is equivalent to the Max Cut problem: deleting the minimum\nnumber of edges to make the graph bipartite is equivalent to finding a bipartition with the\nmaximum number of edges going between the two classes. Thus the lower bound for Max\nCutis needed to complete the proof of Theorem 1.3.\nLet us consider now the vertex-deletion version q-ColoringVD , where given a graph G,\nthe task is to find a set Xof vertices of minimum size such that G−Xhas a q-coloring\n(equivalently, we want to find a partial q-coloring on the maximum number of vertices). For\nthisproblem, abruteforceapproachwouldneedtoconsider (q+1)ppossibilitiesona (σ, δ)-hub\nof size p: each vertex can receive either one of the qcolors, or be deleted.\nTheorem 1.4. Letq⩾1be an integer.\n1. For every σ, δ⩾1,q-ColoringVD onn-vertex graphs can be solved in time (q+ 1)p·\nnO(1)if a(σ, δ)-hub of size pis given in the input.\n2. For every ε >0, there exist integers σ, δ⩾1such that if there is an algorithm solving in\ntime (q+1−ε)p·nO(1)every n-vertex instance of q-ColoringVD given with a (σ, δ)-hub\nof size at most p, then the SETH fails.\nObservethat Vertex Cover isequivalentto 1-ColoringVD andOdd Cycle Transver-\nsalis equivalent to 2-ColoringVD . Furthermore, Independent Set andVertex Cover\nhave the same time complexity (due to the well-known fact that minimum size of a vertex\ncover plus the maximum size of an independent set is always equal to the number of vertices).\nThus the definition of q-ColoringVD gives a convenient unified formulation that includes\nthese fundamental problems.\nPackingproblems. Givenagraph G,theTriangle Partition (denotedby △-Partition\nfor short) problem asks for a partition of the vertex set into triangles. Triangle Packing\n(denotedby △-Packing )isthemoregeneralproblemwherethetaskistofindamaximum-size\ncollection of vertex-disjoint triangles. Given a tree decomposition of width t, Theorem 1.1(6)\nshows that 2t·nO(1)is essentially the best possible running time. It seems that the same\nlower bound holds when parameterizing by the size of a hub, but the source of hardness is\nsomehow different. Instead of assuming the SETH, we prove this lower bound under the Set\nCover Conjecture (SCC) [9,10]. In the d-Set Cover problem, we are given a universe U\nof size nand a collection Fof subsets of U, each with size at most d. The task is to find a\nminimum-size collection of sets whose union covers the universe.\nSet Cover Conjecture (SCC).For all ε > 0, there exists d⩾1such that there is no\nalgorithm that solves every ⩽d-Set Cover instance (U,F)in time (2−ε)n·nO(1)where\nn=|U|.\nWe actually show that the lower bounds for △-Partition /△-Packing areequivalent to\nthe SCC.\nTheorem 1.5. The following three statements are equivalent:\n•The SCC is true.\n3•For every ε >0, there are σ, δ > 0such that △-Partition on an n-vertex graph cannot\nbe solved in time (2−ε)p·nO(1), even if the input contains a (σ, δ)-hub of size p.\n•For every ε >0, there are σ, δ > 0such that △-Packing on an n-vertex graph cannot\nbe solved in time (2−ε)p·nO(1), even if the input contains a (σ, δ)-hub of size p.\nIdeally, one would like to prove lower bounds under the more established conjecture: the\nSETH. However, Theorem 1.5 shows that it is no shortcoming of our technique that we prove\nthe lower bound based on the SCC instead. If we proved statement 2 or 3 under the SETH,\nthen this would prove that the SETH implies the SCC, resolving a longstanding open question.\nDominatingSet. Givenan n-vertexgraphwithatreedecompositionofwidth t, aminimum\ndominating set can be computed in time 3t·nO(1)using an algorithm based on fast subset\nconvolution[36,37]. ByTheorem1.1(2), thisrunningtimecannotbeimprovedto (3−ε)t·nO(1)\nfor any ε >0, assuming the SETH. Can we get a (3−ε)p·nO(1)algorithm if a hub of size p\nis given in the input? We currently have no answer to this question. In fact, we do not even\nhave a good guess whether or not such algorithms should be possible. What we do have are\ntwo very simple weaker results that rule out (2−ε)p·nO(1)algorithms, with one proof based\non the SETH and the other proof based on the SCC.\nTheorem 1.6. For every ε >0, there are σ, δ > 0such that Dominating Set on an n-vertex\ngraph with a (σ, δ)-hub of size pgiven in the input cannot be solved in time (2−ε)p·nO(1),\nunlessboththe SETH and the SCC fail.\nTheorem 1.6 suggests that, if there is no (3−ε)p·nO(1)time algorithm for Dominating\nSet, then perhaps the matching lower bound needs a complexity assumption that is stronger\nthan both the SETH and the SCC.\nUniversal constants for σandδ?The lower bounds in Theorems 1.2–1.6 are stated in\na somewhat technical form: “for every ε >0, there are σandδsuch that . . .”. The statements\nwould be simpler and more intuitive if they were formulated in a setting where σandδare\nuniversal constants, say, 100. Can we prove statements that show, for example, that there is\nno(2−ε)p·nO(1)algorithm, where pis the size of a (100,100)-hub given in the input?\nTheanswertothisquestioniscomplicated. Forthevertex-deletionproblem q-ColoringVD\n(which includes Vertex Cover andOdd Cycle Transversal ) there are actually better\nthan brute force algorithms for fixed constant values of σandδ.\nTheorem 1.7. For every q⩾3andσ, δ > 0, there exists ε >0with the following property:\nevery instance (G, L)ofq-ColoringVD withnvertices, given with a (σ, δ)-hub of size p, can\nbe solved in time (q+ 1−ε)p·nO(1).\nThus Theorem 1.7 explains why the formulation of Theorem 1.4 needs to quantify over σ\nandδ, and cannot be stated for a fixed pair (σ, δ).\nOn the other hand, for the edge-deletion problem q-ColoringED (which includes Max\nCut), we can prove stronger lower bounds where σandδare universal constants. However,\nwe need a complexity assumption different from the SETH.\nAn instance of Max 3-Satis a CNF formula φwith at most three literals in each clause.\nWe ask for the minimum number of clauses that need to be deleted in order to obtain a\nsatisfiable formula. Equivalently, we look for a valuation of the variables which violates the\n4minimumnumberofclauses. Clearly, aninstanceof Max 3-Satwithnvariablescanbesolved\nin time 2n·nO(1)by exhaustive search. It is a notorious problem whether this running time\ncan be significantly improved, i.e., whether there exists an ε >0such that every n-variable\ninstance of Max 3-Satcan be solved in time (2−ε)n.\nMax 3-Sat Hypothesis (M3SH).There is no ε >0such that every n-variable instance of\nMax 3-Satcan be solved in time (2−ε)n·nO(1).\nUnder this assumption, we can prove a lower bound where δ= 6andσis a constant\n(depending only on q).\nTheorem 1.8. For every q⩾2there is an integer σsuch that the following holds. For every\nε >0, no algorithm solves every n-vertex instance of q-ColoringED that is given with a\n(σ,6)-hub of size p, in time (q−ε)p·nO(1), unless the M3SH fails.\nFor the case q= 2we even show a slight improvement over Theorem 6.3 — in this case it\nsuffices to consider instances with a constant σandδ= 4.\nFor△-Partition , we do not know if the lower bound of Theorem 1.5, ruling out (2−ε)p·\nnO(1)running time under the SCC, remains valid for some fixed universal σandδindependent\nofε. Note that the proof of Theorem 1.5 provides a reduction from △-Partition tod-Set\nPacking for some d. It is known that d-Set Packing over a universe of size ncan be solved\nin time (2−ε)n·(n+m)O(1)with some ε >0depending on d[2,21,31]. However, our reduction\nfrom△-Partition tod-Set Packing chooses din a way that it cannot be used to reduce\nthe case of a fixed σandδto ad-Set Packing problem with fixed d. It seems that we would\nneed to understand if certain generalizations of d-Set Packing can also be solved in time\n(2−ε)n·(n+m)O(1)for fixed d. The simplest such problem would be the generalization\nofd-Set Packing where the sets in the input are partitioned into pairs and the solution is\nallowed to use at most one set from each pair.\nDiscussion. Given the amount of attention to algorithms on tree decompositions and the\nnumber of nontrivial techniques that were developed to achieve the best known algorithms,\nit is a natural question to ask if these algorithms are optimal. Even though understanding\ntreewidth is a very natural motivation for this line of research, the actual results turned out\nto be less related to treewidth than one would assume initially: the lower bounds remain\nvalid even under more restricted conditions. Already the first paper on this topic [24] states\nthe lower bounds in a stronger form, as parameterized by pathwidth or by feedback vertex\nset number (both of which are bounded below by treewidth). Some other results considered\nparameters such as the size of a set Qwhere every component of G−Qis a path [18] or\nhas bounded treewidth [15]. However, our results show that none of these lower bounds got\nto the fundamental reason why known algorithms on bounded-treewidth graphs cannot be\nimproved: Theorems 1.2–1.5 highlight that these algorithms are best possible already if we\nconsider a much more restricted problem setting where constant-sized gadgets are attached\nto a set of hub vertices. Moreover, Theorems 1.2 and 1.4 are likely to be best possible: as\nwe have seen, for coloring and its vertex-deletion generalizations, σandδcannot be made a\nconstant independent from ε(Theorem 1.7). Therefore, one additional conceptual message\nof our results is understanding where the hardness of solving problems on bounded-treewidth\ngraphs really stems from, by reaching the arguably most restricted setting in which the lower\nbounds hold.\n5The success of Theorems 1.2–1.4 (for coloring problems and relatives) suggests that possi-\nbly all the treewidth optimality results could be revisited and the same methodology could be\nused to strengthen to parameterization by hub size. But the story is more complicated than\nthat. For example, for △-Packing , the ground truth appears to be that the lower bound\nparameterized by width of the tree decomposition can be strengthened to a lower bound\nparameterized by hub size. However, proving the lower bound parameterized by hub size\nrequires a different proof technique, and we can do it only by assuming the SCC — and for\nall we know this is an assumption orthogonal to the SETH. In fact, we showed that the lower\nbound for △-Packing is equivalent to the SCC, making it unlikely that a simple proof based\non the SETH exists. For Dominating Set , we currently do not know how to obtain tight\nbounds, highlighting that it is far from granted that all results parameterized by width of tree\ndecomposition can be easily turned into lower bounds parameterized by hub size.\nAnother important aspect of our results is the delicate way they have to be formulated,\nwith the values of σandδdepending on ε. Theorem 1.8 shows that in some cases it is possible\nto prove a stronger bound where σandδare universal constants, but this comes at a cost\nof choosing a different complexity assumption (M3SH). Thus, there is a tradeoff between the\nchoice of the complexity assumption and the strength of the lower bound. In general, it seems\nthat the choice of complexity assumption can play a crucial role in these kind of lower bounds\nparameterized by hub size. This has to be contrasted with the case of parameterization by\nthe width of the tree decomposition, where the known lower bounds are obtained from the\nSETH (or its counting version).\nIt would be natural to try to obtain lower bounds parameterized by hub size for other\nalgorithmicproblemsaswell. Thelowerboundsobtainedinthispaperforvariousfundamental\nproblems can serve as a starting point for such further results. Concerning the problems\nstudied in this paper, we leave two main open questions:\n•ForDominating Set , can we improve the lower bound of Theorem 1.6 to rule out\n(3−ε)p·nO(1)algorithms, under some reasonable assumption? Or is there perhaps an\nalgorithm beating this bound?\n•For△-Partition /△-Packing , can we improve the lower bound of Theorem 1.5 such\nthatσandδare universal constants? Or is it true perhaps that for every fixed σandδ,\nthere is an algorithm solving these problems in time (2−ε)p·nO(1)for some ε >0?\n2 Technical Overview\nIn this section, we overview some of the most important technical ideas in our results.\n2.1 q-Coloring\nThe algorithmic statement in Theorem 1.2 is easily obtained via a simple branching procedure.\nFor the hardness part, we use a lower bound of Lampis [23] for constraint satisfaction problems\n(CSP) as a starting point: for any ε >0and integer d, there is an integer rsuch that there\nis no algorithm solving CSP on nvariables of domain size dandr-ary constraints in time\n(d−ε)n. Therefore, to prove Theorem 1.2(2), we give a reduction that, given an n-variable\nCSP instance where the variables are over [q]and the arity of constraints is some constant r,\ncreates an instance of q-Coloring having a hub of size roughly n.\n6First, we introduce a set of nmain vertices in the hub, representing the variables of the\nCSP instance. We would like to represent each r-ary constraint with a gadget that is attached\nto a set Sofrvertices. We will first allow our gadgets to use lists that specify to which colors\ncertain vertices are allowed to be mapped. In a second step we then remove these lists.\nA bit more formally, we say that an r-ary q-gadgetis a graph Jtogether with a list\nassignment L:V(J)→2[q]andrdistinguished vertices x= (z1, . . . , z r)from J. The vertices\nz1, . . . , z rare called portals. Alist coloring of(J, L)is an assignment φ:V(J)→[q]that\nrespects the lists L, i.e., with φ(v)∈L(v)for all v∈V(J).\nA construction by Jaffke and Jansen [18] gives a gadget that enforces that a set of vertices\nforbids one prescribed coloring. We use this statement to construct a gadget extends precisely\nthe set of colorings that are allowed according to some relation.\nProposition 2.1. Letq⩾3andr⩾1be integers, and let R⊆[q]rbe a relation. Then there\nexists an r-aryq-gadget F= (F, L, (z1, . . . , z r))such that\n•the list of every vertex is contained in [q],\n•for each i∈r, it holds that L(zi) = [q],\n•{z1, . . . , z r}is an independent set,\n•for any ψ:{z1, . . . , z r} →[q], coloring vertices z1, . . . , z raccording to ψcan be extended\nto a list coloring of (F, L)if and only if (ψ(z1), . . . , ψ (zr))∈R.\nThen, by introducing one gadget per constraint and attaching it to the vertices of the hub,\nfrom the qnpossible behaviors of the hub vertices, only those can be extended to the gadgets\nthat correspond to a satisfying assignment of the CSP instance. Note that gadgets are allowed\nto use lists and they model the relational constraints using list colorings. So the final step to\nobtain a reduction to q-Coloring is to remove these lists. This can be done using a standard\nconstruction, where a central clique of size qis used to model the qcolors, and a vertex vof\nthe graph is adjacent to the ith vertex of the clique, whenever i /∈L(v).\n2.2 Vertex Deletion to q-Coloring.\nSimilarly to q-Coloring , the algorithmic statement in Theorem 1.4 is easily obtained via\na simple branching procedure. However, for q-ColoringVD , we need to consider q+ 1\npossibilities at each vertex: assigning to it one of the qcolors, or deleting it. This leads\nto the running time (q+ 1)p·nO(1). The hardness proof is also similar, but this time we\nhave to give a reduction that, given an n-variable CSP instance where the variables are over\n[q+ 1]and the arity of constraints is some constant r, creates an instance of q-ColoringVD\nhaving a hub of size roughly n. Intuitively, we are using deletion as the (q+ 1)-st color: the\n(q+ 1)npossibilities for these vertices in the q-ColoringVD problem (coloring with qcolors\n+deletion) correspond to the (q+ 1)npossible assignments of the CSP instance. To enforce\nthis interpretation, we attach to these vertices small gadgets representing each constraint. We\nattach a large number of copies of each such gadget, which means that it makes no sense for\nan optimum solution to delete vertices from these gadgets and hence deletions occur only in\nthe hub. This means that we can treat the vertices of the gadgets as “undeletable”.\nWe would like to use again the construction from Proposition 2.1 to create gadgets that\nenforce that a set of vertices has one of the prescribed colorings/deletions. A gadget can force\nthe deletion of a vertex if its neighbors are colored using all qcolors. However, there is a\nfundamental limitation of this technique: deleting a vertex is always better than coloring it.\n7That is, a gadget cannot really force a set Sof vertices to the color “red”: from the viewpoint\nof the gadget, deleting some of them and coloring the rest red is equally good. In other words,\nit is not true that every relation R⊆[q+ 1]rcan be represented by a gadget that allows only\nthese combinations of qcolors + deletion on a set Sofrvertices.\nTo get around this limitation, we use a grouping technique to have control over how many\nvertices are deleted. Let us divide the nvariables into M=n/bblocks B1,. . .,BMof\nsizebeach. Let us guess the number fiof variables in Bithat receive the value q+ 1in a\nhypothetical solution; that is, we expect fideletions in block Biof central vertices. Instead\nof just attaching a gadget to a set Sof at most rvertices, now each gadget is attached to the\nat most rblocks containing S. Besides ensuring a combination of values on Sthat satisfies\nthe constraint, the gadget also ensures that each block Biit is attached to has at least the\nguessed number fiof deletions. This way, if we have a solution with exactlyPM\ni=1fideletions,\nthen we know that it has exactly fideletions in the i-th block. Therefore, if a gadget forces\nthe deletion of fivertices of Biand forces a coloring on the remaining vertices of Bi, then we\nknow that that block has exactly this behavior in the solution.\n2.3 Edge Deletion to q-Coloring\nLet us turn our attention to the edge-deletion version now. Similarly to the vertex-deletion\nversion, the algorithmic results are simple, thus we discuss only the hardness proofs here.\nAs starting point for all our reductions, we use a CSP problem with domain size qthat\nnaturally generalizes Max 3-Sat: the task is to find an assignment of variables that satisfies\nthe maximum number of constraints. For q= 2, the hardness of this problem follows from\nthe SETH and the M3SH. For q⩾3, we prove a new tight lower bound based on M3SH.\nForq⩾3, the lower bound of Theorem 1.3 (hardness of q-ColoringED under the SETH)\nalready follows from our result for findingaq-coloring without deletions (Theorem 1.2). So,\nin order to complete the proof of Theorem 1.3, we give a reduction from the CSP problem\nwith q= 2to2-ColoringED (i.e,Max Cut ), which shows hardness under SETH. As the\ngadgets of Proposition 2.1 work only for q⩾3, we need to design new gadgets using only 2\ncolors for this case.\nThe same reduction can be used to establish the lower bound from Theorem 1.8 (hardness\nofq-ColoringED under the M3SH) in the q= 2case. For the q⩾3case, we present a\nreduction from the CSP problem with domain size qtoq-ColoringED . Here we can once\nagain use the gadgets from Proposition 2.1.\nIn all cases, as the gadgets we design may use lists, we establish respective lower bounds\nfor the list coloring problem on the way. In a second step, we then show how to remove the\nlists.\nMaxCSP — Hardness under the M3SH For some positive integers dandr, we define\nMax ( d,r)-CSP: Given vvariables over a q-element domain and a set of nrelational con-\nstraints of arity 3, the task is to find an assignment of the variables such that the maximum\nnumber of constraints are satisfied. The problem can be solved in time qv·nO(1)by brute\nforce. For q= 2, the problem is clearly a generalization of Max 3-Sat, hence the M3SH\nimmediately implies that there is no (q−ε)v·nO(1)algorithm for any ε >0. We show that the\nM3SH actually implies this for any q⩾2. This might also be a helpful tool for future work.\nTheorem 2.2. Ford⩾2and any r⩾3, there is no algorithm solving every n-variable\ninstance of Max ( d,r)-CSPin time (d−ε)n·nO(1)forε >0, unless the M3SH fails.\n8In order to show Theorem 2.2, if qis a power of 2, then a simple grouping argument works:\nfor example, if q= 24= 16, then each variable of the CSP instance can represent 4 variables\nof the Max 3-Satinstance, and hence it is clear that a (q−ε)valgorithm would imply a\n(2−ε)4valgorithm for a Max 3-Satinstance with 4vvariables.\nThe argument is not that simple if qis not a power of 2, say q= 15. Then a variable\nof that CSP instance cannot represent all 16 possibilities of 4 variables of the Max 3-Sat\ninstance, and using it to represent only 3 variables would be wasteful. We cannot use the\nusual trick of grouping the CSP variables such that each group together represents a group\nofMax 3-Satvariables: then each constraint representing a clause would need to involve\nnot only 3 variables, but 3 blocks of variables, making δlarger than 3. Instead, for each\nblock of 4 variables of the Max 3-Satinstance, we randomly choose 15 out of the 16 possible\nassignments, and use a single variable of the CSP instance to represent these possibilities. An\noptimum solution of a 4v-variable Max 3-Satinstance “survives” this random selection with\nprobability (15/16)v. Thus a (15−ε)v·nO(1)time algorithm for the CSP problem would\ngive a randomized (16/15)v·(15−ε)v·nO(1)= (16 −ε′)v·nO(1)= (2−ε′′)4v·nO(1)time\nalgorithm for Max 3-Sat, violating (a randomized version of) the M3SH. Furthermore, we\nshow in Section 6.1 that the argument can be derandomized using the logarithmic integrality\ngap between integer and fractional covers in hypergraphs.\nRealizing Relations using Lists Recall that an r-aryq-gadget is a graph with lists in\n[q]andrspecified portal vertices. For our hardness proofs, we reduce from Max ( q,r)-CSP,\nand we use gadgets to “model” the relations in [q]r. We say that an r-aryq-gadgetrealizes\na relation R∈[q]rif there is an integer ksuch that (1) for each d∈R, if the portals are\ncolored according to d, then it requires precisely kedge deletions to extend this to a full list\ncoloring of the gadget, and (2) extending a state that is notinRrequires strictly more than k\nedge deletions. We say that such a gadget 1-realizes R, if for each state outside of Rit takes\nprecisely k+ 1edge deletions to extend this state. So, this is a stronger notion in the sense\nthat now the violation cost is the same for all tuples outside of R. Moreover, with a 1-realizer\nin hand, by identifying copies of this gadget with the same portal vertices one can freely adjust\nthe precise violation cost — this works as long as the portals form an independent set and\ntherefore no multiedges are introduced in the copying process.\nFor our treatment of the case q⩾3, we again use Proposition 2.1 to show that arbitrary\nrelations over a domain of size qcan be realized. As Proposition 2.1 is for the decision problem\nwithout deletions, it does not help for the case q= 2, i.e., for Max Cut /2-Coloring . In this\ncase, we need a different approach to show that every relation over a domain of size 2can be\nrealized. For 2-Coloring , a single edge is essentially a “Not Equals”-gadget as the endpoints\nhave to take different colors or otherwise the edge needs to be deleted. Starting from this, we\nshow how to model OR-relations of any arity. With these building blocks we then obtain the\nfollowing result.\nTheorem 2.3. For each r⩾1, and R⊆[2]r, there is an r-ary2-gadget that 1-realizes R.\nRemoving the Lists Note that gadgets may use lists and therefore, on the way, we first\nobtain the following lower bounds for the respective list coloring problems.\nTheorem 2.4. For every q⩾2andε >0, there are integers σandδsuch that if an algorithm\nsolves in time (q−ε)p·nO(1)every n-vertex instance of List- q-ColoringED that is given\nwith a (σ, δ)-hub of size p, then the SETH fails.\n9Theorem 2.5. For every q⩾2, there is a constant σqsuch that, for every ε > 0, if an\nalgorithm solves in time (q−ε)p·nO(1)every n-vertex instance of List- q-ColoringED that\nis given with a (σq,3)-hub of size p, then the M3SH fails.\nIn a second step, we show how to remove the lists by adding some additional object of size\nroughly q(a central vertex or a q-clique for q= 2orq⩾3, respectively). This addition is\nthen considered to be part of the hub, thereby increasing the size of the hub by some constant.\nHowever, this modification means that for the other gadgets the number of neighbors in the\nhub increases slightly. This is irrelevant for the SETH-based lower bound, but it leads to a\nslight increase in the universal constant δthat we obtain for our M3SH-based lower bounds\nfor the coloring problems without lists.\n2.4 Covering, Packing, and Partitioning\nTheorem 1.5 gives lower bounds for △-Partition and△-Packing based on the Set Cover\nConjecture. This hypothesis was formulated in terms of the d-Set Cover problem. For\nour purposes, it is convenient to consider slightly different covering/partitioning problems.\nTo facilitate our reductions and as a tool for future reductions of this type, we establish\nequivalences between eight different covering type problems. Before we make this more formal\nin Theorem 2.6, let us briefly introduce the corresponding problems.\nFirst, we use =d-Set Cover and⩽d-Set Cover to distinguish between the problem for\nwhich the sets have size exactly dor at most d, respectively. For △-Partition , it is more nat-\nural to start a reduction from the partitioning problems =d-Set Partition or⩽d-Set Par-\ntition, in which the task is to find pairwise disjoint sets that cover the universe. The ⩽d-Set\nPartition problem can be considered as a decision problem. However, we can also consider\nthe corresponding optimization problem in which the task is to minimize the number of se-\nlected sets, and we use ⩽d-Set Partition (#Sets) to denote this problem. Further variants\nare the optimization problems =d-Set Packing (#Sets) and⩽d-Set Packing (#Sets) ,\nin which we need to select the maximum number of pairwise disjoint sets. For ⩽d-Set Pack-\ning, an equally natural goal is to maximize the total size of the selected sets (for =d-Set\nPacking , this is of course equivalent to maximizing the number of selected sets). So we use\n⩽d-Set Packing (Union) to denote the packing problem in which the union/total size of\nthe selected sets is maximized.\nGiven the large number of variants of d-Set Cover , one may wonder how they are related\nto each other. In particular, does the SCC imply lower bounds for these variants? There are\nobvious reductions between some of these problems (e.g., from =d-Set Cover to⩽d-Set\nCover) and there are also reductions that are not so straightforward. We fully clarify this\nquestion by showing that choosing anyof these problems in the definition of the SCC leads to\nan equivalent statement. Thus in our proofs to follow we can choose whichever form is most\nconvenient for us. Knowing this equivalence could prove useful for future work as well.\nTheorem 2.6. Suppose that for one of the problems below, it is true that for every ε >0,\nthere is an integer dsuch that the problem cannot be solved in time (2−ε)n·nO(1), where n\nis the size of the universe. Then this holds for all the other problems as well. In particular,\nany of these statements is equivalent to the SCC.\n1.=d-Set Cover\n2.=d-Set Partition\n103.=d-Set Packing (#Sets)\n4.⩽d-Set Cover\n5.⩽d-Set Partition\n6.⩽d-Set Partition (#Sets)\n7.⩽d-Set Packing (#Sets)\n8.⩽d-Set Packing (Union)\nTo make the statements about relationships between the problems from the list in The-\norem 2.6 more concise, it will be convenient to introduce some shorthand notation. Let\nA={Ad}d⩾1andB={Bd}d⩾1be two families of problems where AdandBdbelong to the\nlist in Theorem 2.6. To shorten notation, we speak of an n-element instance if the universe\nUof an instance has size n. We say that Ais2n-hard if the following lower bound holds\nFor each ε > 0there is some d⩾1such that no algorithm solves Adon all\nn-element instances in time (2−ε)n·nO(1).\nUsing this language, the SCC states that {⩽d-Set Cover }d⩾1is2n-hard. To establish\nTheorem 2.6, we show reductions, stating that if Ais2n-hard then Bis2n-hard as well.\nSpelled out this means:\nSuppose for each ε >0there is some d⩾1such that no algorithm solves Adon\nalln-element instances in time (2−ε)n·nO(1).\nThen, for each ε >0there is some d′⩾1such that no algorithm solves Bd′on all\nn-element instances in time (2−ε)n·nO(1).\nThis shows that this is really a relationship between two classes of problems, and not\nnecessarily a relationship between AdandBdfor the same value d. To make this distinction\nexplicit, we write =∗-Set Cover if we refer to the class of problems {=d-Set Cover }d⩾1.\nWe use analogous notation for the other problems on the list. For example, a simple ob-\nservation is that if =∗-Set Cover is2n-hard then so is ⩽∗-Set Cover as the latter is a\ngeneralization of the former. The reductions we use to prove Theorem 2.6 are illustrated in\nFigure 1.\n2.5 Triangle Partition and Triangle Packing\nNow let us discuss the proof of Theorem 1.5 that can be found in Section 9. The proof\nconsists of two main steps: (1) a reduction from =∗-Set Partition to△-Partition , and\n(2) a reduction from △-Packing to⩽∗-Set Packing (#Sets) (see Figure 2). Recall that\nby Theorem 2.6, assuming the SCC, all of =∗-Set Partition ,⩽∗-Set Packing (#Sets) ,\nand⩽∗-Set Cover are2n-hard. Finally, △-Partition trivially reduces to △-Packing , so\nindeed, the statements in Theorem 1.5 are equivalent.\nReducing Set Partition to△-Partition .We start with step (1), i.e., reducing an\ninstance (U,F)of=d-Set Partition to an equivalent instance Gof△-Partition . With a\nsimple technical trick we can ensure that dis divisible by 3.\nThe main building block used in the reduction is the so-called △-eqgadget. For fixed d,\nit is a graph with ddesignated vertices called portals. The gadget essentially has exactly two\ntriangle packings that cover all non-portal vertices:\n11Obs. 8.1 Lem. 8.11\nObs. 8.2\nObs. 8.1Obs. 8.3 Lem. 8.4Obs. 8.5 Lem. 8.6\nLem. 8.9 Lem. 8.10\n⩽∗-Set Partition (#Sets)\n⩽∗-Set Cover⩽∗-Set Packing (Union)\n=∗-Set Cover=∗-Set Partition=∗-Set Packing (#Sets)⩽∗-Set Packing (#Sets)\n⩽∗-Set Partition\nFigure 1: An overview of the proof of Theorem 2.6. An arrow from AtoBindicates an\nimplication stating that if Ais2n-hard then Bis2n-hard as well.\n•one that also covers all portals (i.e., is actually a triangle partition), and\n•one that covers no portal .\nNow the construction of Gis simple: we introduce the set Qcontaining one vertex for each\nelement of U, and for each set S∈ Fwe introduce a copy of the △-eqgadget whose portals\nrepresent elements of Sand are identified with corresponding vertices from Q. It is straightfor-\nward to verify that there is F′⊆ Fthat partitions Uif and only if Ghas a triangle partition:\nthe sets from F′correspond to △-eqgadgets whose non-portal vertices are covered in the first\nway. Note that Qis a(σ, d)-hub of G, where σis the number of vertices of the △-eqgadget,\ni.e., is a constant that depends only on d.\nReducing △-Packing toSet Packing .Now let us consider a graph Ggiven with a\n(σ, δ)-hub Qof size p, and an integer t. We will show that a hypothetical fast algorithm for\n⩽d-Set Packing (#Sets) can be used to determine whether Ghas a triangle packing of\nsize at least t.\nFor simplicity of exposition, assume that Ghas no triangles contained in Q; dealing with\nsuch triangles is not difficult but would complicate the notation. We say that a component C\nofG−Qisactivein some triangle packing Πif there is a triangle in Πthat intersects both\nCandQ. Note that for any triangle packing there are at most pactive components.\nWe would like to guess components that are active for some (unknown) solution Π. How-\never, this results in too many branches. We deal with it by employing color-coding and\nreducing the problem to its auxiliary precolored variant . Suppose for a moment that we are\n12=\u0003-Set Partition\n4-Partition\n4-Packing\n\u0014\u0003-Set Packing (#Sets)\n\u0014\u0003-Set Cover\n(1)\n(2)\n(trivial)Figure 2: An overview of the reductions in the proof of Theorem 1.5. The two dashed arrows\nrefer to 2n-hardness reductions from Theorem 2.6. To establish these two connections, note\nthat we actually utilize all reductions shown in Figure 1, except for the simple Oberserva-\ntions 8.1 and 8.5. The arrows annotated with (1) and (2) denote the reductions proved in\nSection 9.\ngiven a coloring ψof components of G−Qintop/ccolors, where cis a large constant, with\na promise that at most ccomponents in each color are active in Π.\nFor a color i∈[p/c], letCidenote the set of components of G−Qcolored ibyψ. The\ncontribution of the color itoΠis the number of triangles that intersect vertices in components\nfromCi. Note that the size of Πis the sum of contributions of all color (since we assumed\nthat there are no triangles contained in Q). What can be said about the contribution of i?\nCertainly picking a maximum triangle packing in the graph consisting only of components\nfromCiis a lower bound. Let Xidenote the number of triangles in such a triangle packing\nand note that Xican be computed in polynomial time as each component of G−Qis of\nconstant size. Moreover, for each active component C∈ Ci, there are at most σtriangles that\nintersect both CandQ(as each of them has to use a distinct vertex from C). As, by the\npromise on ψ, there are at most cactive components in Ci, we observe that the contribution\nofiis at most Xi+cσ. We exhaustively guess the contribution of each color by guessing the\noffset qiagainst Xi; it gives a constant number of options per color. We reject guesses where\nthe total contribution of all colors, i.e., the number of all triangles packed, is less than t.\nFor each color i, we enumerate all sets S⊆Qthat are candidates for these vertices of Q\nthat form triangles with vertices from components of Ci; call such sets i-valid. An i-valid set\nSmust satisfy the following two conditions. First, the size of Sis at most 2cσ, as there are\nat most cσvertices in active components from Ciand each such vertex belongs to a triangle\nwith at most two vertices from Q. Second, there exists a triangle packing ΠSin the graph\ninduced by Stogether with components of Cisuch that\n•at most celements from Ciare active in ΠS(this follows from the promise on ψ), and\n•the number of triangles in ΠSis at least Xi+qi(by our guess of qi).\nIt is not difficult to verify that i-valid sets can be enumerated in polynomial time, where the\ndegree of the polynomial depends on candσ.\nNow we are ready to construct an instance (U,F, p/c)of⩽d-Set Packing (#Sets) . The\nuniverse UisQ∪ {ai|i∈[p/c]}, i.e., it consists of the hub of Gand one extra vertex per\ncolor. For each i-valid set S, we include in Fthe set S∪ {ai}. Again, one can verify that F\ncontains p/cpairwise disjoint sets if and only if Ghas a packing of ttriangles that agree both\nwith ψand with the guessed values of qi’s.\n13By adjusting c, we can ensure that the whole algorithm works in time (2−ε′)p·|V(G)|O(1),\nfor some ε′>0, provided that we have a fast algorithm for ⩽d-Set Packing (#Sets) .\nThe only thing left is to argue how we obtain the coloring ψsatisfying the promise. Here\nwe usesplitters introduced by Naor, Schulman, and Srinivasan [30]. Informally, a splitter is\na family of colorings of a “large set” X, such that for each “small subset” Y ⊆ Xthere is a\ncoloring that splits Yevenly. In our setting, the “large set” Xis the set of all components of\nG−Qand the “small subset” Yis the set of all active components with respect to some fixed\n(but unknown) solution; recall that there are at most psuch active components. Since our\ncolorings use p/ccolors, we are sure that there is some ψfor which at mostp\np/c=ccomponents\nin each color are active. Calling the result of Naor, Schulman, and Srinivasan [30], we can\nfind a small splitter Ψ, and then just exhaustively try every coloring ψ∈Ψ. Again, carefully\nadjusting the constants, we can ensure that the overall running time is (2−ε)p· |V(G)|O(1),\nfor some ε >0.\n3 Preliminaries\nFor an integer k, by[k]we denote {1, . . . , k }. For a set X, by2Xwe denote the family of all\nsubsets of X.\nGraph theory. LetGbe a graph. By V(G)andE(G)we denote, respectively, the vertex\nset and the edge set of G. Let X⊆V(G), byG[X]we denote the graph induced by X. By\nG−Xwe denote the graph obtained form Gby removing all vertices in Xalong with incident\nedges, i.e., G[V(G)\\X]. For a set X⊆E(G), byG\\Xwe denote the graph obtained by\nremoving all edges in X, i.e., (V(G), E(G)\\X). A vertex is isolatedif its neighborhood is\nempty.\nTreewidth and (σ, δ)-hubs. Consider a graph with a (σ, δ)-hub Qof size p. Introducing\na bag that contains Qthat is the center of a star whose leaves are Q∪Cifor each connected\ncomponent CiofG−Q. Then this is a tree decomposition of Gof width at most p+σ−1.\nWe state this observation formally.\nObservation 3.1. For some σ, δ⩾1, letGbe a graph given with a (σ, δ)-hub of size p. One\ncan obtain a tree decomposition of width less than p+σin time polynomial in the size of G.\nVariants of (list) colorings. In Section 1, we defined the q-Coloring problem as well\nas its vertex and edge deletion variant. We also use a generalization of vertex colorings that\ninclude lists. Formally, the List- q-Coloring problem takes as input a graph Gtogether with\nalistfunction L:V(G)→2[q]. The task is then to compute a q-coloring φ:V(G)→[q]that\nrespects lists L, i.e., with φ(v)∈L(v)for all v∈V(G). We say that such an assignment φis\naproper list coloring of(G, L).\nWealsousethecorrespondingvertexandedgedeletionvariant. Inthe List- q-ColoringVD\n(resp. List- q-ColoringED ) problem we ask for a smallest set Xofvertices(resp.edges)\nsuch that G−X(resp. G\\X) admits a proper list coloring that respects the lists L.\nNote that List- q-ColoringVD andList- q-ColoringED are optimization problems.\nSometimes it will be convenient to consider their corresponding decisions versions, when we\nare additionally given an integer kand we ask whether the instance graph can be modified into\n14a yes-instance of by removing at most kvertices/edges. In general, we will show algorithms\nfor the optimization version, and lower bounds for the decision version.\nGadgets. LetJbe a graph together with a list assignment L:V(J)→2[q]andrdistin-\nguished vertices x= (x1, . . . , x r)from J. Then we refer to the tuple J= (J, L,x)as an r-ary\nq-gadget. We might not specify rnorqin case they are clear from the context. The vertices\nx1, . . . , x rare called portals.\nDefinitionsforSetCovering,Partitioning,andPackingProblems. Forreferenceand\nto be clear about the distinctions between the different problems that appear in Section 8, we\nnow list all their definitions.\n=d-Set Cover\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size exactly dsuch\nthatU=SF, an integer t.\nQuestion: Is there a collection of at most tsets from Fwhose union is U?\n⩽d-Set Cover\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size at most dsuch\nthatU=SF, an integer t.\nQuestion: Is there a collection of at most tsets from Fwhose union is U?\n=d-Set Partition\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size exactly dsuch\nthatU=SF.\nQuestion: Is there a collection of ( |U|/d) sets from Fthat is a partition of U?\n⩽d-Set Partition\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size at most dsuch\nthatU=SF.\nQuestion: Is there a collection of sets from Fthat is a partition of U?\n⩽d-Set Partition (#Sets)\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size at most dsuch\nthatU=SF, an integer t.\nQuestion: Is there a collection of at most tsets from Fthat is a partition of U?\n=d-Set Packing (#Sets)\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size exactly d, an\ninteger t\nQuestion: Is there a collection of at least tsets from Fthat are pairwise disjoint?\n⩽d-Set Packing (#Sets)\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size at most d, an\ninteger t.\nQuestion: Is there a collection of at least tsets from Fthat are pairwise disjoint?\n15⩽d-Set Packing (Union)\nInput:A set Uof elements, a set family F ⊆ 2Uwhere each set has size at most d, an\ninteger t.\nQuestion: Is there a collection of sets from Fthat are pairwise disjoint and whose union\nhas size at least t?\n4q-Coloring\nThe goal of this section is to show Theorem 1.2. Its algorithmic part is very simple: Exhaus-\ntively enumerate all possible q-colorings of the hub Q; this results in qpbranches. Then, for\neach component CofG−Q, we check whether the coloring of the neighbors in Ccan be\nextended to a proper coloring of C. Note that this can be performed by brute-force, as the\nnumber of vertices in Cis at most σ, i.e., a constant. As the number of such components C\nis at most n, the overall running time is qp·nO(1).\nThe hardness part of Theorem 1.2 is restated in the following theorem.\nTheorem 4.1. For every q⩾3andε >0, there are integers σandδsuch that no algorithm\nsolves every n-vertex instance of q-Coloring , given with a (σ, δ)-hub of size p, in time (q−\nε)p·nO(1), unless the SETH fails.\nThe proof uses the following constraint satisfaction problem as a starting point. For\nintegers dandr, an instance of (d, r)-CSPis a pair (V,C), where Vis the set of variables and\nCis the set of constraints . Each constraint Cinvolves a sequence of variables, whose length,\ncalled the arityofC, is at most r. The constraint Cenforces some relation RC⊆[d]arity( C)\non the variables involved. We ask for a mapping V → [d]whichsatisfies every constraint, i.e.,\nthe sequence of images of variables involved in C∈ Cbelongs to RC.\nLampis showed the following lower bound for (d, r)-CSP.\nTheorem 4.2 (Lampis [23]) .For every d⩾2andεthere exists r, such that there is no\nalgorithm solving every N-variable instance of (d, r)-CSPin time (d−ε)N·NO(1), unless the\nSETH fails.\nBefore we proceed to the proof of Theorem 4.1, let us recall following lemma by Jaffke and\nJansen [18, Lemma 14] which will be a crucial tool used to build gadgets (defined in Section 3).\nLemma 4.3 (Jaffke, Jansen [18]) .Letq⩾3andr⩾1be integers, For any c∈[q]rthere\nexists an r-ary gadget Fc= (Fc, L,(x1, x2, . . . , x r)), where lists Lare contained in [q], such\nthat for every d= (d1, . . . , d r)∈[q]rthere exists a proper list coloring φofFcin which for\nalliit holds that φ(xi)̸=diif and only if d̸=c.\nThe following statement is a consequence of Lemma 4.3.\nProposition 2.1. Letq⩾3andr⩾1be integers, and let R⊆[q]rbe a relation. Then there\nexists an r-aryq-gadget F= (F, L, (z1, . . . , z r))such that\n•the list of every vertex is contained in [q],\n•for each i∈r, it holds that L(zi) = [q],\n•{z1, . . . , z r}is an independent set,\n•for any ψ:{z1, . . . , z r} →[q], coloring vertices z1, . . . , z raccording to ψcan be extended\nto a list coloring of (F, L)if and only if (ψ(z1), . . . , ψ (zr))∈R.\n16Proof.We start the construction with introducing an independent set Z={z1, . . . , z r}and\nwe set L(zi) := [ q]for every i∈[r].\nLet¯R:= [q]r\\Rand consider ¯c∈¯R. We call Lemma 4.3 to obtain a gadget F¯c=\n(F¯c, L,(x1. . . , x r)). For each i∈[r], we make xiadjacent to zi. We repeat this for every\n¯c∈¯R. This concludes the construction of F.\nConsider a coloring ψ:Z→[q]such that (ψ(z1), . . . , ψ (zr))∈R. Consider ¯c∈¯R. We\nneed to show that ψcan be extended to a coloring of F¯c. Note that (ψ(z1), . . . , ψ (zr))̸=¯cas\n¯c/∈R. Thus by Lemma 4.3 called for (d1. . . , d r) = (ψ(z1), . . . , ψ (zr))we note that that there\nexists a proper list coloring of F¯csuch that the color of each xiis different than ψ(zi). This\nis the sought-for coloring of F¯c.\nConsider a coloring ψ:Z→[q]such that ¯c:= (ψ(z1), . . . , ψ (zr))/∈R. For contradiction,\nsuppose that there exists a proper list coloring φofFthat extends ψ. Consider the gadget\nF¯c= (F¯c, L,(x1, . . . , x r)). Clearly, for all i∈[r], it holds that φ(xi)̸=ψ(zi). However, the\nexistence of such a coloring contradicts Lemma 4.3.\nNow we are ready to prove Theorem 4.1.\nProof of Theorem 4.1. Letq⩾3andε >0. For contradiction, suppose that for every σ, δ\nthere is an algorithm that solves every n-vertex instance of q-Coloring , given with a (σ, δ)-\nhub of size p, in time (q−ε)p·nO(1).\nLetrbe the value given by Theorem 4.2 for d=qandε. Consider an instance I= (V,C)\nof(q, r)-CSPand let n:=|V|. Note that without loss of generality we may assume that every\nconstraint in Chas at least one satisfying assignment. Furthermore, we may assume that\nthere are no two constraints with exactly the same set of vertices: otherwise we can replace\nall such constraints C1, C2, . . . , C mwith a single one whose set of satisfying assignments is the\nintersection of the sets of satisfying assignments of C1, C2, . . . , C m. Note that this means that\n|C|⩽nr.\nDefining an equivalent instance of List- q-Coloring .As the first step, we will define\nan instance (G′, L)ofList- q-Coloring .\nWe start with introducing an independent set Y={y(v)|v∈ V}and each vertex from Y\nhas list [q]. The vertex y(v)is meant to represent v, i.e., the coloring of y(v)corresponds to\nthe valuation of v.\nNow consider a constraint C∈ C. Let v1, v2, . . . , v ℓbe the variables of C, where ℓ:=\narity( C)⩽r. Let RCbe the relation enforced by ConVC. We call Proposition 2.1 for RCto\nobtain an ℓ-ary gadget\nF(C) = (F(C), L,(z1, . . . , z ℓ)).\nThe list of every vertex is contained in [q], and the interface vertices are pairwise non-adjacent\nand have lists [q]. For each i∈[ℓ], we identify ziwith y(vi). We repeat this for every C∈ C.\nThis completes the construction of (G′, L).\nIt is straightforward to verify that (G′, L)is a yes-instance of List- q-Coloring if and\nonly if Iis satisfiable. Indeed, the coloring of vertices from Yexactly corresponds to the\nvaluation of variables of I. For each C∈ C, a coloring of the interfaces of the gadget F(C)\nintroduced for Ccan be extended to a list coloring of the whole gadget if and only if the\ncoloring of interfaces corresponds to an assignment of variables that satisfies C.\n17Defining an equivalent instance of q-Coloring .The modification of (G′, L)into an\nequivalentinstanceof q-Coloring isstandard. Weintroduceacliqueonvertices a1, a2, . . . , a q.\nFor each u∈V(G′)and each i∈[q], we add an edge uaiif and only if i /∈L(u).\nAgain, it is straightforward to verify that this construction simulates the lists in L. Indeed,\nifφis a proper q-coloring of G, then by symmetry we may assume that φ(ai) =ifor every\ni∈[q]. Thus, for any vertex u∈V(G′)∩V(G)such that i /∈L(i), we have that φ(u)̸=i.\nStructure of G.The graph G′hasn+q+|C|·h(q, r) =O(nr)vertices, where h(q, r)is the\nsize of the largest gadget given by Proposition 2.1. Note that qis a constant and ris also a\nconstant (depending on qandε).\nNow consider the set Q=Y∪ {a1, a2, . . . , a q}, clearly |Y|=n+q. Note that every\ncomponent of G−Yis (a subgraph of) one of the gadgets given by Proposition 2.1, i.e., its\nsize is at most h(q, r). Furthermore, each such component is adjacent to at most r+qvertices\nfrom Q. Thus Qis a(σ, δ)-hub, where σ:=h(q, r)andδ:=r+qare constants depending\nonly on qandε.\nRunning time. Note that calling our hypothetical algorithm for q-Coloring onG′we can\nsolveIin time\n(q−ε)|Q|· |V(G′)|O(1)= (q−ε)n·nO(1),\nwhich, by Theorem 4.2, contradicts the SETH.\n5 Vertex Deletion to q-Coloring\nIn this section we prove Theorem 1.4. The algorithmic part is simple and it is covered in\nSection5.2. Themainworkisshowingthelowerbound,andthisisdoneinformofTheorem5.1\nin Section 5.1.\n5.1 Hardness for for q-ColoringVD\nWe prove the following lower bound for q-ColoringVD , which is identical to Theorem 1.4(2)\nTheorem 5.1. For every q⩾1andε >0, there exist integers σ, δ⩾1such that if there is\nan algorithm solving in time (q+ 1−ε)p·nO(1)every n-vertex instance of q-ColoringVD\ngiven with a (σ, δ)-hub of size at most p, then the SETH fails.\nTo that end, we start by showing that solving (d, r)-CSPfor certain structured instances\nis still hard. Let us define what we mean by structured.\nDefinition 5.2. For all d, r, b, N ∈Nandf:\u0002N\nb\u0003\n→ {0, . . . , b }, anN-variable instance (V,C)\nof(d, r)-CSPis called (b,f)-structured if the following holds\n•Nis divisible by bandVis partitioned intoN\nbblocks V1, . . . , V N/b, each of size b.\n•the scope of each constraint C∈ Cincludes, for each i, either all or none of the variables\nfrom the block Vi.\n•There are two types of constraints, i.e., C=Csign∪ Csatsuch that\n–Csigncontains a constraint Cifor each i∈\u0002N\nb\u0003\n. The constraint Cimakes sure that\nexactly fivariables from Viare set to d.\n18–For each C∈ Csat, if the scope of Ccontains Vi, then exactly fivariables of Viare\nset to d. Furthermore, no two constraints C, C′∈ Csathave exactly the same set of\nvariables.\nNote that the last property of Csatimplies that |Csat|⩽Nrand thus |C|⩽N/b+Nr. The\nastute reader might wonder why we need Csign, as these constraints are already implied by\nCsat. However, their special structure will be exploited in our reduction.\nLemma 5.3. For all N, d∈Nandε >0, there exists b, r∈Nsuch that there is no algorithm\nsolving every ( b,f)-structured N-variable (d, b·r)-CSPinstance where f:\u0002N\nb\u0003\n→ {0, . . . , b },\nin time (d−ε)N·NO(1), unless the SETH fails.\nProof.Letd∈Nandε∈R>0. Let rbe the value in the Theorem 4.2 for dandε. Let bbe\nthe smallest integer such that (b+ 1)1/b<1 +ε/d; note that bis a constant (again, depending\nondandε).\nLetI= (V,C)be an N-variable instance of (d, r)-CSP. We assume that there is some\nfixed total order on the set V. We can further assume that the number Nof variables is\ndivisible by b, as otherwise we can add at most b−1dummy variables. Then we partition V\nintoM=N/bblocks V1, . . . , V M, each of size b.\nLetr′=b·r; note that r′depends on dandε. Now we will construct a family Iof instances\nof(d, r′)-CSP, such that Iis satisfiable if and only if at least one instance in Iis satisfiable.\nConstruction of I.For an assignment φ:V → [d], itssignature is the vector f=\n(f1. . . , f M)∈ {0, . . . , b }M, where for each i∈[M]the value of fiis the number of vari-\nables from Vimapped by φtod. Note that the number of possible signatures is at most\n(b+ 1)M= (b+ 1)N/b=\u0010\n(b+ 1)1/b\u0011N\n<(1 +ε/d)N. (1)\nConsider one such signature f= (f1, . . . , f M). We include into Ithe instance If= (V,Cf)\nof(d, r′)-CSPdefined as follows. The set of constrains Cfwill be partitioned into two subsets,\ndenoted by Csign\nfandCsat\nf.\nFirst, for each block Vi, we add to Csign\nfthe constraint Ci\nfincluding all variables from Vi,\nwhich is satisfied if and only if exactly fivariables from Vihave value d.\nThen, for each constraint C∈ C, let Cfbe the constraint involving all variables in all\nblocks intersected by Csuch that Cfis satisfied by some assignment if an only if:\n•its projection to the variables from Csatisfies C,\n•for each block Viinvolved in Cf, exactly fivariables from Viare mapped to d.\nWe include each such constraint CfinCsat\nf. Now, if Csat\nfcontains several constraints C1, . . . , C p\nwith exactly the same set of variables, we replace them with a single constraint whose corre-\nsponding relation is the intersection of the relations forced by C1, . . . , C p. This completes the\ndefinition of If.\nNote that each constraint from Cfhas arity at most r′=b·rand thus Ifis an N-variable\ninstance of (d, r′)-CSP. Moreover, since bdivides Nand by the defnition of Cf, we observe\nthatIfis (b,f)-structured. Intuitively, constrains in Csign\nfmake sure that a solution has the\ncorrect signature, and constraints in Csat\nfmake sure that it satisfies I.\n19Equivalence of instances. First, observe that Ifis satisfiable if and only if Iis satisfied\nby some assignment with signature f. Consequently, since Icontains the instance Iffor every\npossible signature f, we conclude that Iis satisfiable if and only if Icontains a satisfiable\ninstance.\nRunningtime. Nowletusestimatetherunningtime. Foreachsignature f,theconstruction\nofIftakes polynomial time. Suppose there exists an algorithm solving every ( b,f)-structured\nN-variable instance of (d, b·r)-CSP, where f:\u0002N\nb\u0003\n→ {0, . . . , b }, in time (d−ε)N·NO(1).\nThen, given any N-variable instance Iof(d, r)-CSP, we can construct Iffor all possible\nsignatures f, where each Ifcan be solved in time (d−ε)N·NO(1). By (1), the number of\npossible signatures is at most (1 +ε/d)N. Consequently, Ican be solved in time\n\u0010\n1 +ε\nd\u0011N\n·(d−ε)N·NO(1)=\u0012\nd−ε2\nd\u0013N\n·NO(1)=\u0000\nd−ε′\u0001N·NO(1).\nwhere ε′:=ε2/d. By Theorem 4.2, this is not possible unless the SETH fails.\nIn the following, we will show lower bounds for the decision variant of the q-ColoringVD\nproblem. We assume that, along with the instance, we are given an integer kand we ask\nwhether the optimum solution (i.e., the set of vertices to delete) has size at most k. Clearly,\nsuch a lower bound implies the lower bound for the optimization version, as kcan be assumed\nto be bounded by the number of vertices of the instance.\nIt turns out that the characteristics of the problem differ between the cases q∈ {1,2}\nandq⩾3: this is because q-Coloring is polynomial-time solvable in the former case and\nNP-hard in the latter one. The proof of our lower bound is split into Lemmas 5.4 and 5.5.\nLemma 5.4. Forq∈ {1,2}andε >0, there exist σ, δ > 0depending on qandεsuch that\nthere is no algorithm solving all n-vertex instances of q-ColoringVD given with a (σ, δ)-hub\nof size at most p, in time (q+ 1−ε)p·nO(1), unless the SETH fails.\nProof.Letq∈ {1,2}andε >0. Let bandrbe the values given by Lemma 5.3 for d:=q+ 1\nandε.\nSuppose for a contradiction that for all constant σ, δ∈N, there exists an algorithm solving\nalln-vertex instances of q-ColoringVD , given with a (σ, δ)-hub of size at most p, in time\n(q+ 1−ε)p·nO(1).\nConsider a ( b,f)-structured N-variable (d, b·r)-CSPinstance I= (V,C), where f:\u0002N\nb\u0003\n→\n{0, . . . , b }. We will define an instance (G, k)ofq-ColoringVD that is equivalent to I, and\nsolving it with the hypothetical algorithm mentioned before, would allow us to solve Iin time\n(d−ε)N·NO(1). By Lemma 5.3 this contradicts the SETH.\nNote that without loss of generality we may assume that every constraint in Chas at least\none satisfying assignment, as otherwise Iis clearly a no-instance.\nConstruction of (G, k).We start by introducing the set Y=S\nv∈V{y(v)}ofvariable\nvertices. For each v∈ V, the vertex y(v)represents the variable v. The intended meaning is\nthat coloring y(v)with a color c∈[q]corresponds to assigning the value ctov. Deleting y(v)\nthen corresponds to assigning the value q+ 1tov.\nConsider a constraint C∈ C. Let VCbe the set of variables involved in Cand let RCbe\nthe relation enforced by ConVC. Let ℓCbe the size of RC, i.e., the number of valuations of\nVCthat satisfy C; recall that ℓC⩾1. We will create a gadget gadgetCrepresenting C.\n20We introduce a clique cliqueCwith verticesS\nr∈R{x(r)}, where x(r)corresponds to the\nassignment r.\nIn the case of q= 2, for each edge x(r)x(r′)ofcliqueC, we add two additional vertices and\nconnect them both to x(r)andx(r′).\nNext, we add edges between the vertices of cliqueCandY. Consider r∈R. For q= 1and\nallv∈VC, we connect x(r)andy(v)ifrsetsvto 2. For q= 2and for all v∈VCwe proceed\nas follows:\n•ifrsetsvto 1, we connect x(r)andy(v),\n•ifrsetsvto 2, then we introduce a two-edge path with endvertices x(r)andy(v),\n•and finally, if rsetsvto 3, we introduce both an edge and a two-edge path between x(r)\nandy(v).\nThis completes the description of gadgetC. Now, for each C∈ C, we introduce N+ 1\ncopies of the gadget. The j-th copy will be denoted by gadget(j)\nC, and we will use the same\nconvention when refering to the vertices of that copy. Finally, if q= 2, we introduce a single\n“global” vertex hand make it adjacent to every vertex clique(i)\nC, for all C∈ Candi∈[N+ 1].\nLetGbe the resulting graph. By Y′we denote the set of vertices that are notin any gadget(i)\nC\nforC∈ Candi∈[N+ 1], i.e., Y′=Yifq= 1, and Y′=Y∪ {h}ifq= 2.\nLet us define\nk′:=N/bX\ni=1fiand k:= (N+ 1)· X\nC∈C(ℓC−1)!\n+k′.\nThis completes the definition of the instance (G, k)ofq-ColoringVD .\nEquivalence of instances. We will now show that Iis satisfiable if and only if Gbecomes\nq-colorable after deleting at most kvertices.\nSuppose that there exists a satisfying assignment ψ:V → [q+ 1]. We will find a set Xof\nsize at most ksuch that G−Xisq-colorable. Note that for q= 1this means that Xis a\nvertex cover, while for q= 2this means that Xis an odd cycle transversal.\nFor all v∈ V, the vertex y(v)is included in Xif and only if ψ(v) =q+ 1. Furthermore,\nforq= 2, the vertex his not included in X. Since Iis (b,f)-structured, this means that\n|X∩Y′|=k′.\nConsider a constraint C∈ C. As ψsatisfies C, we obtain that ψrestricted to VCcorre-\nsponds to some element of RC, sayr. Now consider j∈[N+ 1]and the gadget gadget(j)\nC. We\ninclude in Xall vertices from clique(j)\nC, except the vertex x(r)(j). In total, we obtain that\n|X\\Y′|= (N+ 1)· X\nC∈C(ℓC−1)!\n=k−k′.\nSumming up, we have |X|=k. It is straightforward to verify that for q= 1, the set Xis\na vertex cover of G, i.e., the graph G−Xhas a proper 1-coloring. Similarly, for q= 2, the\nsetXis an odd cycle transversal: To see this, we can define a proper 2-coloring of G−Xas\nfollows. For each v∈ V, ifψ(v)∈ {1,2}, then we color y(v)with the color ψ(v). The vertex\nhreceives color 1, and the remaining vertices from clique(i)\nC, forC∈ Candi∈[N+ 1], receive\n21color 2. The vertices introduced for edges of clique(i)\nCreceive color 1. Finally, the endvertices\nof two-edge paths joining vertices from Ywith the vertices from clique(i)\nCare all colored 2, so\nwe can color the middle vertices with color 1.\nSumming up, there exists a set of vertices of size kwhose removal from Gresults in a\ngraph that is q-colorable. Thus (G, k)is a yes instance of q-ColoringVD .\nNow suppose that there exists a set X⊆V(G)of size at most kand a proper q-coloring\nφofG−X. We claim that for each constraint C∈ Candj∈[N+ 1]the following two\nproperties hold:\n(P1) At least ℓC−1vertices from gadget(j)\nCbelong to X.\n(P2) If exactly ℓC−1vertices from gadget(j)\nCbelong to X, then all of them belong to clique(j)\nC.\nForq= 1, the properties are clear: any vertex cover of a clique with ℓCvertices must\ncontain at least ℓC−1vertices. Furthermore, property (P2) follows trivially, as gadget(j)\nC\ncontains no vertices that are not in clique(j)\nC.\nNow consider the case q= 2and let us first argue about property (P1). For contradiction\nsuppose that Xcontains at most ℓC−2vertices from gadget(j)\nC; in particular there are two\nvertices from clique(j)\nCthat are not in X. On the other hand, Xmust contain at least ℓC−2\nvertices from clique(j)\nCas otherwise G−Xhas a triangle and thus is not bipartite. Therefore\nno other vertices from gadget(j)\nCare included in X. However, recall that for every edge of\nclique(j)\nCwe introduced two vertices adjacent with both endpoints of this edge. Thus G−X\ncontains a triangle, a contradiction.\nNow let us argue about (P2). For contradiction suppose that Xcontains ℓC−1vertices\nfrom gadget(j)\nC, but only at most ℓC−2of them are in clique(j)\nC. Repeating the argument from\nthe previous paragraph, this means that exactly ℓC−2vertices from clique(j)\nCare in X. Since\nXmight contain at most one of the vertices adjacent to both vertices from clique(j)\nC−X, we\nconclude that G−Xcontains a triangle, a contradiction.\nNote that property (P1) implies that |X\\Y′|⩾(N+ 1)·\u0000P\nC∈C(ℓC−1)\u0001\n=k−k′.\nWithoutlossofgeneralitywecanassumethatforeach C∈ Candeveryvertex xofgadgetC,\neither the copy of xin each gadget(j)\nCis included in X, or none of them. Furthermore, for\nq= 2, the coloring φrestricted to each copy of gadgetC(after removing X) is exactly the\nsame. Indeed, otherwise we can pick a copy whose intersection with Xis the smallest and\nuse this solution (i.e., vertices in Xand the q-coloring of the remaining vertices) on all other\ncopies, obtaining another solution with at most the same number of deleted vertices. Notice\nthat if for some C′∈ C, the set Xcontains more than ℓC′−1vertices from each copy of\ngadgetC′, then\n|X|=|X\\Y′|+|X∩Y′|⩾(N+1)· X\nC∈C(ℓC−1)!\n+(N+1)+ |X∩Y′|=k−k′+(N+1)> k,\na contradiction. Thus, for each C∈ Cand each j∈[N+ 1], the set Xcontains exactly\nℓC−1vertices from gadget(j)\nCand, by property (P2), all these vertices belong to clique(j)\nC. In\nparticular this implies that |X\\Y′|=k−k′.\nFinally, we set the valuation ψ:V → [q+ 1]according to the coloring of vertices in Y:\nv∈ Vis assigned the value i∈[q]ify(v)/∈Xandφ(v) =i, and it is assigned the value q+ 1\nify(v)∈X.\n22We claim that ψsatisfies all constraints in C. First, let us consider a constraint C∈ Csign\ncorresponding to block Vifori∈[N/b]. Consider a copy gadget(j)\nCofgadgetC; recall that the\nsolution (i.e., the intersection with Xand the coloring of the remaining vertices) is the same\nfor each copy. Since ℓC−1vertices were deleted from clique(j)\nC, there is a unique vertex xin\nclique(j)\nC−X. Let r∈RCbe the assignment corresponding to x. Since C∈ Csign, exactly fi\nvariables have the value q+ 1inr. Note that for each such variable v∈Vi, the vertex y(v)\nmust be included in X. Indeed, if q= 1, then y(v)xis an edge of G. Ifq= 2, then y(v)\nandxare contained in a triangle whose third vertex is not in X. Thus, for each i∈[N/b],\nat least fivertices from {x(v)|v∈V|i}must be included in X. SincePN/b\ni=1fi=k′and\n|X\\Y′|=k−k′, we conclude that |X∩Y|=k′. Consequently, for each i∈[N/b]exactly\nfivertices from {x(v)|v∈Vi}are in X. This implies that exactly fivariables from Viare\nmapped to q+ 1and thus Cis satisfied. Furthermore, we have that in case q= 2the vertex\nhdoes not belong to X. Without loss of generality we may assume that in this case we have\nφ(h) = 1(otherwise we can switch colors 1and2inφ).\nNow consider C∈ Csatand again let r∈RCbe the assignment corresponding to the\nunique vertex xofclique(1)\nC−X. By the argument in the previous paragraph we observe that\nfor each v∈VCwe have ψ(v) =q+ 1whenever rmaps vtoq+ 1. In particular, if q= 1,\nthis shows that Cis satisfied. So consider the case that q= 2, we need to argue that values\n1and2are assigned according to r. As φ(h) = 1, we have that φ(x) = 2. Note that for\nevery v∈VCsuch that rmaps vto 1 we have an edge joining xandxandy(v), and thus\nφ(y(v)) = 1. Simlarly, every v∈VCsuch that rmaps vto 2 is joined with xwith a two-edge\npath whose middle vertex is not in X, so we have φ(y(v)) = 2. Consequently, all variables\nfrom VCare assigned according to rand thus Cis satisfied.\nSumming up, indeed (G, k)is a yes-instance of q-ColoringVD if and only if Iis satisfi-\nable.\nStructure of the instance. Let us bound the number of vertices of G. We will only\nconsider the case that q= 2, the case of q= 1is similar but simpler. We have\n|V(G)|⩽|Y′|+ (N+ 1)·X\nC∈C\u0012\n(ℓC−1)·2 +\u0012ℓC−1\n2\u0013\n·2\u0013\n⩽(N+ 1) + ( N+ 1)·(N/b+Nbr)(2dbr+d2br) =NO(1),\nas|C|=|Csat|+|Csign|⩽N/b+Nbr,ℓC⩽dbr, and d, b, rare constants.\nDefine Q:=Y′, we clearly have Q=N+ 1. Note that connected components of G−Q\nare precisely copies of gadgetCforC∈ C. The number of vertices of each such component is\nat most σ:= (ℓC−1)·2 +\u0000ℓC−1\n2\u0001\n·2⩽d2br, and each such gadget is connected to at most\nδ:=br+ 1vertices of Q. Thus Qis a(σ, δ)-hub, where σandδdepend only on d, b, r, which\nin turn depend only on qandε.\nRunning time. The construction of the instance (G, k)takes time polynomial in N. Also,\nGhas a (σ, δ)-hub of size at most p:=N+ 1. Using the hypothetical algorithm that solves\nalln-vertex instances of q-ColoringVD instances with a (σ, δ)-hub of size at most pin time\n(q+ 1−ε)p·nO(1), we get an algorithm that solves Iin time\n(q+ 1−ε)p·nO(1)= (q+ 1−ε)N·NO(1)\n23which, by Lemma 5.3, contradicts the SETH.\nNow let us move to the proof of our lower bound for q-ColoringVD forq⩾3.\nLemma 5.5. Forq⩾3andε >0, there exist σ, δ > 0depending on qandεsuch that there is\nno algorithm solving all n-vertex instances of q-ColoringVD given with a (σ, δ)-hub of size\nat most p, in time (q+ 1−ε)p·nO(1), unless the SETH fails.\nProof.Letbandrbe the values given by Lemma 5.3 for d=q+ 1andε. Consider an\nN-variable ( b,f)-structured (q+ 1, b·r)-CSPinstance Iwith variables Vand constraints C,\nfor some f:\u0002N\nb\u0003\n→ {0, . . . , b }andN∈N. Again we may assume that each constraint has at\nleast one satisfying assignment.\nAuxiliary relations. For a tuple\nz= (z1,1, . . . , z 1,q, z2,1, . . . , z 2,q, . . . , z b,1, . . . , z b,q)∈[q]bq,\nbyRainbow (z)we denote the set of those ifor which {zi,1, . . . , z i,q}= [q]. For j∈ {0, . . . , b },\nwe define Rbj⊆[q]bqconsisting of all tuples zsuch that |Rainbow (z)|=j. The important\nproperty is that Rbjis invariant under permutations of the domain.\nFor each i∈[q+ 1]we define a q-element vector γ(i):\n•ifi= 1, then γ(i) = (2 ,2,3. . . , q ),\n•ifi∈[2, q], then γ(i) = (1 ,1,2, . . . , i −1, i+ 1, . . . , q ),\n•ifi=q+ 1, then γ(i) = (1 ,2, . . . , q ).\nThe crux is that if i⩽q, then the set of values appearing in γ(i)is exactly [q]\\ {i}, and if\ni=q+ 1, then the set of values appearing in γ(i)is exactly [q].\nNow consider a relation R⊆[q+1]ℓ. Let Force (R)⊆[q]qℓbe the relation defined as follows.\nForeach (a1, a2, . . . , a ℓ)∈R,weaddto Force (R)theℓq-elementsequence (γ(a1), γ(a2), . . . , γ (aℓ))\nover[q].\nNow we can proceed to our reduction.\nIntermediate step: an instance of the list variant. Let us first construct an instance\nof thelistvariant of q-ColoringVD , i.e., a triple (G′, L, k), where G′is a graph whose every\nvertex uis equipped with a list L(u)⊆[q]of admissible colors (the option of deleting a vertex\nis always available). We want to decide whether one can obtain a yes-instance of list q-coloring\nby deleting at most k:=PN/b\ni=1fivertices from G′.\nWe start by introducing the set Y=S\nv∈V{y(v)}of vertices, each with list [q]. For each\nv∈ V, the vertex y(v)represents the variable v. The intended meaning is that coloring y(v)\nwith a color c∈[q]corresponds to assigning the value ctov. Deleting y(v)then corresponds\nto assigning the value q+ 1tov.\nConsider a constraint C∈ Csignrelated to the block Vi. We call Proposition 2.1 for Rbfi\n— here is where we use the fact that q⩾3— to obtain a bq-ary gadget\nF(C) = (F(C), L,(z1,1, . . . , z 1,q, z2,1, . . . , z 2,q, . . . , z b,1, . . . , z b,q)),\nwith all lists contained in [q]and interface vertices forming an independent set. If vis the\nj-th variable in Vi, we make vertices zj,1, . . . , z j,qadjacent to y(v). We repeat the above step\nN+ 1times, i.e., for each i∈[N/b]we introduce N+ 1copies of the gadget F(C).\n24Now consider a constraint C∈ Csat. We proceed similarly as in the previous case. Denote\nthe arity of Cfbyℓ(where ℓ⩽r′) and let R⊆[q+ 1]r′be the relation enforced by C(i.e.,\nthe set of satisfying assignments).\nWe call Proposition 2.1 for the relation Force (R)to obtain an ℓq-ary gadget\nF(C) = (F(C), L,(z1,1, z1,2, . . . , z 1,q, z2,1, . . . , z 2,q, . . . , z ℓ,1, . . . , z ℓ,q)),\nwith all lists contained in [q]. Ifvis the j-th variable of C, we make the vertices zi,1, . . . , z i,q\nadjacent to y(v).\nThis completes the construction of (G′, L, k).\nEquivalence of instances. We claim that (G′, L, k)is a yes-instance of the list variant of\nq-ColoringVD if and only if Iis satisfiable.\nFirst, suppose that there exists a satisfying assignment φ:V → [q+ 1]ofI. We assign\ncolors to vertices of Yas described above: for v∈ V, ifφ(v)⩽q, then we color y(v)with\ncolor φ(v). Ifφ(v) =q+ 1, then the vertex y(v)is deleted.\nSince all constraints in Csignare satisfied, we note that we delete exactly fivariables from\neach block Vi, and thus in total we delete k=PN/b\ni=1fivertices from Y. Thus we need to\nargue that the coloring of the remaining vertices of Ycan be extended to the coloring of all\ngadgets, without deleting any further vertices.\nConsider a constraint C∈ Csignrelated to the block Viand a copy of the gadget F(C)\nwith the portals z1,1, . . . , z 1,q, . . . , z b,1, . . . , z b,q. Let j∈[b]andvbe the j-th variable of C.\nWe color the vertices zj,1, . . . , z j,qso that the vector of colors appearing on these vertices is\nγ(φ(v)). Note that by the definition of γ, the color of y(v)does not appear on its neighbors.\nFinally, by the definition of Rbfi, we can extend this coloring of the portals of the gadget to\nthe coloring of the whole gadget, without deleting any vertices from it.\nFor contraints C∈ Csatwe proceed in a similar way. Let Rbe the relation enforced by\nCon its variables. Consider the gadget F(C)introduced for Cand let z1,1, . . . , z ℓ,qbe its\nportals, where ℓ= arity( C). For each j, the vertices zj,1, . . . , z j,qreceive the vector of colors\nγ(φ(v)), where vis the j-th variable involved in C. Now the coloring of portals of the gadget\nis in Force (R), so it can be extended to the coloring of the whole gadget without removing\nany additional vertices.\nNow suppose that there is a set X⊆V(G)of size at most kand a proper list coloring ψ\nofG′−X. Note that we can safely assume that, for any C∈ Csign, allN+ 1copies of F(C)\nreceive exactly the same coloring. Indeed, if this is not the case, we can recolor all copies\nin the same way as the one in which there are the fewest deleted vertices, obtaining another\nsolution to the problem. As N+ 1> N⩾PN/b\ni=1fi=k, we conclude that for all i∈[N/b], no\nvertex from F(C)is deleted.\nWe set the valuation of variables of Iaccording to the coloring of Y. Consider a variable\nv∈ V. Ify(v)/∈X, we set the value of vtoψ(y(v)), and otherwise we set the value of vto\nq+1. Note that at most |X|⩽kvariables were mapped to q+1. We claim that this valuation\nsatisfies all constraints of I.\nFirst, consideraconstraint C∈ Csignrelatedtotheblock Vi, andacopyof F(C)introduced\nforC. Recall that no vertex from the gadget was deleted, thus the coloring of the portals of\nthe gadget must respect the relation Rbfi. In other words, for fivariables vofCit holds that\nall colors from [q]appear in the neighborhood of y(v). This means that such a vertex y(v)\n25must be deleted. Consequently, for each i∈[N/b], the set Xcontains at least fivertices y(v)\nforv∈Vi.\nSumming up, we conclude that |X|=kand that all the deleted vertices are in Y. Fur-\nthermore, exactly fivertices corresponding to variables from Viare deleted, which means that\nCis satisfied by our valuation.\nNow consider a constraint C∈ Csatof arity ℓ. Consider the gadget F(C)introduced for C\nand denote its portals by z= (z1,1, . . . , z ℓ,q). Recall that by the previous argument no vertex\nfrom the gadget is deleted. Thus the coloring of the portals of the gadget belongs to Force (R),\nwhere Ris the relation enforced by C. Let (a1, . . . , a p)∈Rbe such that (γ(a1), . . . , γ (ap))is\nthe coloring of the portals of the gadget.\nLetvbe the j-th variable of Cand let ibe such that v∈Vi. Recall that all variables from\nViappear in C.\nBy the definition of γ, ifaj=q+ 1, then j∈Rainbow( z)and so y(v)is certainly in X.\nConsequently, the value of the variable vwas set to q+ 1. Now consider the case that aj⩽q,\nsayaj=c. Then either ψ(y(v)) =c(and thus vgets the value c) ory(v)∈X. However, recall\nthat by the definition of C, for exactly fivariables from Vitheir index appears in Rainbow( z).\nNote that this is notensured by any relation in Csignand this is why C∈ Csatstill needs to\ncheck consistency with f.\nOn the other hand, exactly fivertices fromS\nv′∈Vi{y(v′)}are in X. Thus y(v)is not in\nXand therefore the value of vis set to c. Summing up, the valuation of the variables of Cis\nexactly (a1, . . . , a p), which means that Cis satisfied.\nLet us point out the special structure of the constraints in Csignwas not used yet and so far\nwe could have obtained the same outcome by introducing N+ 1copies of each gadget related\nto a contraint in Csat.\nConstruction of (G, k).Now let us modify the instance (G′, L, k)into an equivalent in-\nstance (G, k)ofq-ColoringVD .\nFor a graph eGwith lists eL:V(eG)→2[q]and a set V′⊆V(eG), bysimulating lists of V′by\nKwe denote the following operation. We introduce a clique Kwith vertices c1, . . . , c q, and\nfor every vertex uofeGand every i∈[q], we make uadjacent to ciif and only if i /∈eL(u).\nThe typical way of turning an instance of list q-coloring to an equivalent instance of q-\ncoloring is to introduce a “global” q-clique Kand simulate lists of all vertices of the graph with\nK; recall e.g., the proof of Theorem 4.1. However, in the vertex deletion variant this is no\nlonger that simple, as we have to prevent deleting vertices from this central clique. Note that\nwe cannot make them undeletable by introducing (say roughly the size of G) many copies\nof the clique and connecting them in an appropriate way, as this would create an instance\nwithout a (σ, δ)-hub for constant σ,δ.\nLet us discuss how to modify (G′, L, k)into(G, k). Note that the vertices from Yalready\nhave lists [q]so they do not have to be simulated. For each constraint C∈ Csignand for each\ncopy of the gadget F(C)introduced for C, we introduce a private q-clique and simulate the\nlists of all vertices of the gadget by this clique.\nNext we introduce a single q-clique K, and the lists of all the remaining vertices, i.e., the\nvertices from the gadgets F(C)introduced for C∈ Csat, are simulated by K. This completes\nthe construction of G.\nObserve that Kis meant to “synchronize” the colorings of the gadgets F(C)forC∈ Csat.\nHowever, the colorings of the gadgets F(C)forC∈ Csignare not synchronized, i.e., each clique\n26introduced for these gadgets might receive a different permutation of colors.\nLet us argue that (G′, L, k)is a yes-instance of the list variant of q-ColoringVD if and\nonly if (G, k)is a yes-instance of q-ColoringVD .\nFirst suppose that there is a set X⊆V(G′)of size at most kand a proper q-coloring ψof\nG′−Xthat respects lists L. Note that we can delete from Gexactly the same vertices X, and\nuse exactly the same coloring ψon the remaining vertices from V(G)∩V(G′), and then extend\nthis partial coloring to the additionally introduced q-cliques according to the description of\nthe operation of simulating lists.\nOn the other hand, suppose that there is a set X⊆V(G)of size at most kand a proper\nq-coloring ψofG−X. We aim to show that ψrespects lists Lon vertices from V(G)∩V(G′).\nSimilarly as we did in the previous paragraph when analyzing the properties of (G′, L, k)\nwe observe that no vertex from any gadget introduced for C∈ Csign, along with its private\nq-clique, was deleted. Indeed, this is because there are N+ 1copies of each gadget. Consider\ni∈[N/b]andtheconstraint C∈ Csignintroducedfortheblock Vi. Next,recallthattherelation\nRbfienforced by Cis invariant to permuting the set of colors. Consequently, simulating lists\nby a local copy of Kqis sufficient to ensure that the coloring ψof the vertices of the gadget\nsatisfies lists L. This is exactly the point where we use the special structure of constrains in\nCsign.\nNow, similarly as in the previous case, we can argue that X⊆Y. Consequently, the\nvertices from K, i.e., the common copy of Kqintroduced for all gadgets F(C)forC∈ Csat,\nare not deleted. Thus they simulate the lists of all vertices within the gadgets, i.e., the coloring\nψrespects lists L.\nStructure of G.The number of vertices of Gis at most (below h1(·), h2(·)are some func-\ntions)\nn=|Y|+|K|+X\nC∈Csign(|V(F(C)|+q)·(N+ 1) +X\nC∈Csat|V(F(C))|\n⩽N+q+N·(h1(b, q) +q)·(N+ 1) + |C| ·h2(b, q, r )\n⩽N+q+N(N+ 1)·(h1(b, q) +q) +Nr·h2(b, q, r )\n=O(Nr+1) =NO(1),\nasq, b, rare constants.\nSetQ=Y∪K, note that |Q|=N+q. There are two types of components of G−Q:\n(i) (subgraphs of) copies of the gadgets F(C)introduced for C∈ Csign, including the local\nq-clique, or (ii) (subgraphs of) copies of the gadgets F(C)introduced for C∈ Csat. Using the\nnotation from the formula above, the components of type (i) have size at most h1(b, q) +q\nand each of them attaches to bvertices from Q, and the components of type (ii) have size at\nmost h2(b, q, r )and each of them attaches to at most br+qvertices from Q. Since rdepends\nonqandε, we conclude that Qis a(σ, δ)-hub for some σ,δdepending only on qandε.\nRunning time. Now let us estimate the running time. The construction of (G, k)takes\npolynomial time. Furthermore, Ghas a (σ, δ)-hub of size at most p:=N+O(1). This implies\nthat if there is an algorithm solving all n-vertex instances of q-ColoringVD with a (σ, δ)-hub\nof size at most pin time (q+ 1−ε)p·nO(1), then we get an algorithm solving all N-variable\n27(b,f)-structured instances of (q+ 1, b·r)-CSPin time\n(q+ 1−ε)p·nO(1)= (q+ 1−ε)N·NO(1),\nwhich, by Lemma 5.3, contradicts the SETH.\nNow, Theorem 5.1 follows from combining Lemma 5.4 and Lemma 5.5.\n5.2 Simple Algorithm for q-ColoringVD\nThe following algorithm solves the q-ColoringVD problem for q⩾1, which also includes\ntheVertex Cover problem for q= 1andOdd Cycle Transversal forq= 2. It is worth\nnoting that this algorithm can be modified to use treewidth as the parameter.\nTheorem 5.6. For all integers q, σ, δ ⩾1, every n-vertex instance of q-ColoringVD can be\nsolved in time (q+ 1)p·nO(1)if a(σ, δ)-hub of size pis given in the input.\nProof.As input to q-ColoringVD , consider a graph Gwith nvertices, given along with a\n(σ, δ)-hub Qof size p. We exhaustively guess the set X⊆Qof hub vertices that are deleted\nand a q-coloring fofQ−X. This results in at most (q+ 1)pbranches.\nNow, for each component CofG−X, we compute the minimum number of vertices to\nbe deleted from C, so that the remaining vertices admit a proper q-coloring that extends f.\nWe output the coloring with the fewest vertices deleted in total (i.e., from Qand from the\ncomponents of G−Q).\nRecall that each component CofG−Qis of size at most σ, i.e., a constant. Thus, the\noptimum extension of the coloring Q−XtoCcan be computed in constant time. Since the\nnumber of components Cis at most n, we conclude that the running time of each branch is\nbounded by polynomial. Summing up, the overall running time is (q+ 1)p·nO(1).\n6 Edge Deletion to q-Coloring\nNow let us move to the edge-deletion counterpart of q-ColoringVD . The goal of this section\nis to prove Theorems 1.3 and 1.8.\nThe algorithmic part Theorem 1.3(1) is again simple and stated in Section 6.5 for com-\npleteness. For the hardness result Theorem 1.3(2), the case q⩾3already follows from our\nresult Theorem 1.2 for the decision problem. However, the case q= 2(i.e., the result for Max\nCut) is the missing piece, and so it remains to show the following hardness result.\nTheorem 6.1. For every ε >0, there are integers σandδsuch that no algorithm solves every\nn-vertex instance of Max Cut that is given with a (σ, δ)-hub of size p, in time (2−ε)p·nO(1),\nunless the SETH fails.\nOur second main result covered in this section, Theorem 1.8, states that under the M3SH\nthe lower bound from Theorem 1.3 holds even for universal constants σandδ, i.e., for σand\nδthat no longer depend on ε. For the case of Max Cut , we show that constant σandδ= 4\nsuffice.\nTheorem 6.2. There is an integer σsuch that, for every ε >0, no algorithm solves every\nn-vertex instance of Max Cut that is given with a (σ,4)-hub of size p, in time (2−ε)p·nO(1),\nunless the M3SH fails.\n28Forq⩾3, the respective lower bound for q-ColoringED can be obtained for constant σ\nandδ= 6.\nTheorem 6.3. For every q⩾3there is an integer σsuch that the following holds. For every\nε >0, no algorithm solves every n-vertex instance of q-ColoringED that is given with a\n(σ,6)-hub of size p, in time (q−ε)p·nO(1), unless the M3SH fails.\nThe combination of Theorems 6.2 and 6.3 gives Theorem 1.8.\nThis section is structured as follows. First, in Section 6.1, we establish a lower bound for\nMax-CSP (Theorem 2.2) based on the M3SH. This will be the starting point for our M3SH-\nbased lower bounds. Second, in order to bridge from the coloring problems to Max-CSP we\nshow how to model arbitrary relations with the coloring problems via gadget constructions.\nForq⩾3we can rely on the previously established Proposition 2.1. For q= 2,we have to do\nsome additional work, and this is done in Section 6.2. Third, in Section 6.3, we consider the\ncaseq= 2— here we show Theorems 6.1 and 6.2 in a unified proof. Finally, in Section 6.4,\nwe treat the case q⩾3and show Theorem 6.3. In both Section 6.3 and Section 6.4, we show\nintermediateresultsforthe List- q-ColoringED problem. Infact, wefirstshowthefollowing\nresults for list colorings, and subsequently show how to remove the lists in the reductions.\nTheorem 2.4. For every q⩾2andε >0, there are integers σandδsuch that if an algorithm\nsolves in time (q−ε)p·nO(1)every n-vertex instance of List- q-ColoringED that is given\nwith a (σ, δ)-hub of size p, then the SETH fails.\nTheorem 2.5. For every q⩾2, there is a constant σqsuch that, for every ε > 0, if an\nalgorithm solves in time (q−ε)p·nO(1)every n-vertex instance of List- q-ColoringED that\nis given with a (σq,3)-hub of size p, then the M3SH fails.\n6.1Max CSP — Hardness under M3SH\nLet us introduce a constraint-deletion variant of (d, r)-CSP. An instance of Max ( d,r)-CSP\nis the same as in (d, r)-CSPand we ask for a smallest possible number of constraints that\nneed to be deleted in order to obtain a yes-instance of (d, r)-CSP.\nClearly, using a brute-force approach we can solve an n-variable instance of Max ( d,r)-\nCSPin time dn·nO(1). Let us explore the consequences of M3SH to the complexity of\nMax ( d,r)-CSP. Clearly, Max 3-Satis a special case of Max ( 2,3)-CSP, which means that\nthere is no ε >0such that for any r⩾3,Max ( 2,r)-CSPwith nvariables can be solved in\ntime (2−ε)n·nO(1), unless the M3SH fails.\nTheorem 6.4. Forp⩾1and any r⩾3, there is no algorithm solving every n-variable\ninstance of Max ( 2p,r)-CSPin time (2p−ε)n·nO(1), for ε >0, unless the M3SH fails.\nProof.Note that it is sufficient to prove the lower bound for r= 3. Consider an instance I1of\nMax 3-Satwith variables Vand the set of clauses C, where |V|=n. By introducing fewer\nthan pdummy variables, we can assume that nis divisible by p. Let N=n/p. We partition\nVintoNblocks of size p. Denote these blocks by V1, V2, . . . , V N.\nNow we are going to create an instance I2ofMax ( 2p,3)-CSP. For each i∈[N], we\nintroduce a new variable xi. We bijectively map each possible valuation of variables in Vito\none of 2ppossible values of xi.\nNow consider a clause Cof the Max 3-Satinstance, involving variables v, v′, v′′, such\nthatv∈Vi,v′∈Vi′, and v′′∈Vi′′. We introduce a constraint involving variables xi, xi′, and\n29xi′′, which is satisfied by exactly those values that correspond to valuations that satisfy C.\nWe repeat this step for every clause C; note that the arity of each constraint is at most 3.\nIt is clear that the created instance I2ofMax ( 2p,3)-CSP is equivalent to the original\ninstance I1ofMax 3-Sat. Now let us argue about the running time. For contradiction,\nsuppose we can solve I2in time (2p−ε)N·NO(1)for some ε >0, i.e., (2p)δN·NO(1)for some\nδ <1depending on pandε. This means that I1can be solved in time (2p)δN·NO(1)=\n2δn·nO(1)= (2−ε′)n·nO(1)for some ε′>0. This contradicts the M3SH.\nLemma 6.5. Letd⩾3and let d′andpbe integers such that d′= 2pandd⩽d′. If\nthere is ε >0such that every n-variable instance of Max ( d,3)-CSPcan be solved in time\n(d−ε)n·nO(1), then there is ε′>0depending on dandεsuch that every n-variable instance\nofMax ( d′,3)-CSPcan be solved in randomized time (d′−ε′)n·nO(1).\nProof.Consider an instance I1ofMax ( d′,3)-CSPwith nvariables. Let φbe the (unknown)\noptimum solution, i.e., the one that violates the minimum number of constraints; let this\nnumber be k.\nFor each variable of I1independently, uniformly at random we discard d′−dpossible\nvaluations. More precisely, for each variable vwe select a subset D(v)⊂[d′]of size dand\nset it as the domain of v. We modify each constraint to be satisfied by only these valuations\nthat for each variable vpick a value from D(v); note that it is possible that some constraints\nbecome not satisfiable at all. This way we have created an instance I2ofMax ( d,3)-CSP.\nNote that, while in I2we only consider a subset of the valuations of I1the fact whether\nsome clause is violated or not is preserved. In particular, if φsatisfies the domains of all\nvariables in I2, then it is an optimal solution of I2.\nNote that the probability that φsatisfies all domains is\u0000d\nd′\u0001n. Thus, using standard\ncalculations we observe that if we repeat our random experiment (d′\nd)ntimes, then the success\nprobability is constant.\nSo the algorithm is as follows: we randomly create (d′\nd)ninstances of Max ( d,3)-CSP\nby discarding some valuations as previously described, and each of them is solved with our\nhypothetical algorithm. We return the best of found solutions; with constant probability this\nsolution will be an optimal solution for I1.\nLet us compute the running time (below δ, δ′<1andε′>0are some constants depending\nondandε):\n\u0012d′\nd\u0013n\n·(d−ε)n·nO(1)=\u0012d′\nd\u0013n\n·dδn·nO(1)=d′δ′n·nO(1)= (d′−ε′)n·nO(1).\nWe remark that we can make the success probability arbitrarily close to 1 by adding more\niterations of sampling, which results in a blow-up in the running time by a constant factor.\nNow let us discuss how to derandomize the argument in Lemma 6.5. Instead of repeated\nrandom sampling, we want to enumerate a family F, such that\n•each set in Fis of the form D1×. . .×Dn, where each Diis ad-element subset of [d′],\n•the union of all sets in Fcovers [d′]n, and\n•Fcan be enumerated in time roughly\u0010\nd′\nd\u0011n\n.\nThen we could replace the random sampling step by testing each member of Fseparately.\n30Notice that the idea above is actually about finding a small (approximate) solution to a\ncertain Set Cover instance. Recall that an instance of Set Cover consists of a universe\nUand a family Sof subsets of U, for which we can assume without loss of generality thatS\nS∈SS=U. The task is to find a minimum-sized family S′⊆ Ssuch thatS\nS∈S′S=U.\nFor integers d′⩾dandn, byS(d′, d, n)we denote the family consisting of sets of the form\nD1×D2×. . .×Dn, for all possible choices of D1, . . . , D n∈\u0000[d′]\nd\u0001\n. Thus, in this notation, our\ntask is to look for a small subfamily of S(d′, d, n)which covers [d′]n.\nLemma 6.6. For every n >1andd′⩾d, theSet Cover instance ([d′]n,S(d′, d, n))has a\nsolution of size\u0010\nd′\nd\u0011n\n·nO(1).\nProof.For an instance (U,S)ofSet Cover , afractional solution is a function f:S → [0,1],\nsuch that for every u∈Uwe haveP\nS∈S\nu∈Sf(S)⩾1. The weight of a fractional solution is\nP\nS∈Sf(S).\nWe claim that ([d′]n,S(d′, d, n))admits a fractional solution of weight at most\u0010\nd′\nd\u0011n\n.\nIndeed, set f(S) =\u0000d′−1\nd−1\u0001−nfor every S∈ S(d′, d, n). First, note that each element of [d′]nis\ncovered by exactly\u0000d′−1\nd−1\u0001nsets in S(d′, d, n), so indeed fis a fractional solution. Its weight is\nX\nS∈S(d′,d,n)f(S) = \u0000d′\nd\u0001\n\u0000d′−1\nd−1\u0001!n\n=\u0012d′\nd\u0013n\n.\nNext, we recall a well-known fact that the integrality gap for Set Cover is bounded by\na logarithmic function of the size of the universe [26]. This, combined with the observation\nfrom the previous paragraph, implies that ([d′]n,S(d′, d, n))has an integral solution of size\u0010\nd′\nd\u0011n\n·nO(1).\nNote that Lemma 6.6 is not sufficient for our derandomization procedure, as we need to\nbe able to find this solution efficiently.\nLemma 6.7. Letδ >0andd′⩾d⩾2be constants. For every n, there is a family F ⊆\nS(d′, d, n)of size at most\u0010\nd′\nd+δ\u0011n\nthat covers [d′]nand can be enumerated in time O(|F|).\nProof.In what follows we assume that nis sufficiently large, as otherwise we can find the\noptimum solution using brute force. The exact lower bound on ndepends on d, d′, and δand\nfollows from the reasoning below.\nLetcbe the constant hidden in the O(·)-notation in Lemma 6.6, i.e., such that for every\nn, the instance ([d′]n,S(d′, d, n))has an integral solution of size at most\u0010\nd′\nd\u0011n\n·nc. Let mbe\nthe minimum integer such that mc/m<1 +δd/d′. Note that mdepends only on δ, d, and d′,\nascis an absolute constant.\nFor simpilicity let us assume that mdivides n; the argument below can be easily modified\nto the general case by making the last block smaller. Let F′be the family given by Lemma 6.6\nfor([d′]m,S(d′, d, m )); note that we can compute it by brute force as mis a constant. We\nobserve that the family\nF=F′× F′×. . .× F′\n| {z }\nn/m times\n31covers [d′]n.\nLet us estimate the size of F. It is upper-bounded by\n|F′|n/m⩽\u0012\u0012d′\nd\u0013m\n·mc\u0013n/m\n=\u0012d′\nd\u0013n\n·(mc/m)n<\u0012d′\nd\u0013n\n·\u0012\n1 +dδ\nd′\u0013n\n=\u0012d′\nd+δ\u0013n\n.\nClearly the family Fcan be enumerated in time proportional to its size.\nWe can now prove Theorem 2.2, which we restate for convenience.\nTheorem 2.2. Ford⩾2and any r⩾3, there is no algorithm solving every n-variable\ninstance of Max ( d,r)-CSPin time (d−ε)n·nO(1)forε >0, unless the M3SH fails.\nProof.Ifdis a power of 2, then the result follows already from Theorem 6.4. So assume that\ndis not a power of 2, and let d′be the smallest power of 2 which is greater than d. We reduce\nfromMax ( d′,r)-CSPconsider an instance with nvariables.\nFor contradiction, suppose that there is some ε >0and some algorithm solving every\nn-variable instance of Max ( d,r)-CSPin time (d−ε)n·nO(1). Let δ:=εd′\nd2and let Fbe\nthe family given by Lemma 6.7 for this value of δ. We proceed similarly as in the proof of\nLemma 6.5: for each set in Fwe construct a corresponding instance of Max ( d,r)-CSPand\nsolve it using our hypothetical algorithm. As Fcovers [d′]n, we know that for some set in F\nwe will find the optimum solution for the original instance.\nLet us estimate the running time:\nO\u0012\u0012d′\nd+δ\u0013n\u0013\n+\u0012d′\nd+δ\u0013n\n·(d−ε)n·nO(1)=\u0012\nd′+δd−εd′\nd−εδ\u0013n\n·nO(1)=\u0012\nd′−ε2d′\nd2\u0013n\n·nO(1).\nBy Theorem 6.4, this contradicts the M3SH.\n6.2 Realizing Relations\nWe start by defining what it means for a gadget to realize a relation. Recall from the definition\nin Section 3 that a gadget may use lists.\nDefinition6.8 (Realizingarelation) .Forsomepositiveinteger qandr, letR⊆[q]r. Consider\nsome r-aryq-gadget J= (J, L,x), and some d∈[q]r. By cost ed(J,d)we denote the size of a\nminimum set of edges Xthat ensure that there is a proper list q-coloring φof(J\\X, L)with\nφ(x) =d. Then Jrealizesthe relation Rif there is an integer ksuch that, for each d∈[q]r,\ncost ed(J,d) =kifd∈R, and cost ed(J,d)> k, otherwise. We say that Jω-realizes Rfor\nsome integer ω⩾1if additionally, for each d/∈Rwe have cost ed(J,d) =k+ω.\nLemma 6.9. For each r⩾1andR⊆[2]rthere is an r-ary2-gadget that realizes R.\nProof.First note that a single edge x1x2whose endpoints have each have a list L(x1) =\nL(x2) ={1,2}enforces that its endpoints are mapped to different colors. So essentially this\nis a gadget that realizes the relation NEQ = {(1,2),(2,1)}.\nFor an integer p⩾2, byORpwe denote the relation [2]p\\ {2p}(the intuition behind the\nnotation is clear when we interpret 1astrueand2asfalse).\nClaim 6.9.1. There exists a gadget JOR2whose portals are pairwise non-adjacent that 2-\nrealizes OR2.\n32Proof of Claim. We start the construction of the gadget with a 5-cycle with consecutive\nvertices v1, v2, v3, v4, v5. We set the list of v5to{2}; the lists of vifori∈[4]are{1,2}. The\nportals of the created gadget JOR2arev1andv4, see also Figure 3a.\nIt is straightforward to verify that for d∈OR2precisely one edge has to be deleted and\nhence cost ed(JOR2,d) = 5 α+ 1, while for (2,2)we have to delete precisely three edges and\nhence cost ed(JOR2,(2,2)) = 5 α+ 3. ◁\nClaim 6.9.2. For every fixed integer ω⩾1there exists a gadget Jω\nOR2whose portals are\npairwise non-adjacent that 2ω-realizes OR2.\nProof of Claim. We introduce ωcopies of the gadget JOR2from Claim 6.9.1, let the portals\nof the i-th copy be xi, yi. We identify all xiinto a new vertex xand all yiinto a new vertex y;\nnote that the portals of JOR2are non-adjacent so this step does not introduce multiple edges.\nThe obtained graph with xandyas portals is our gadget Jω\nOR2.\nFord∈OR2, let cost ed(JOR2,d) =α′. It is straighforward to observe that for d∈OR2we\nhave cost ed(Jω\nOR2,d) =ωα′, andfor ¯d/∈OR2wehave cost ed(Jω\nOR2,¯d) =ω·(α′+2) = ωα′+2ω.\n◁\nClaim 6.9.3. For every p⩾3, there exists a gadget JORprealizing ORpwhose portals are\npairwise non-adjacent.\nProof of Claim. Letωbe an integer whose value will be specified later.\nWe introduce three pairwise disjoint sets of vertices X:=Sp\ni=1{xi},Y:=Sp\ni=1{yi}, and\nZ:=Sp\ni=1{zi}For each i∈[p]we add a copy of Jω\nOR2with portals identified with xiandyi,\nand another copy with portals identified with yiandzi. Next, for all distinct i, j, we add a\ncopy of Jω\nOR2with portals identified with zi, zj.\nNext, we add a new vertex uwith list {1}that is adjacent to every vertex in X∪Y∪Z.\nThis completes the construction of the gadget. Its portals are y1, . . . , y p. A schematic picture\nof the construction for p= 3is depicted in Figure 3b.\nWe say that a copy of Jω\nOR2is satisfied if the mapping of its portals satisfies OR2. Imagine\nthat ωis sufficiently large so that every copy of Jω\nOR2must be satisfied. Because of the\nuniversal vertex u, we have to delete one additional edge for every vertex in X∪Y∪Zthat\nis mapped to 1. Intuitively, we want to map as many of these vertices as possible to 2. Since\nall copies of Jω\nOR2must be satisfied, at most p+ 1vertices from X∪Y∪Zcan be mapped\nto2: precisely one vertex from each pair {xi, yi}is mapped to 2, and potentially one vertex\nfrom the set Z. The maximum number of 2s is only possible if at least one vertex from Y\nis mapped to 1. In other words, this happens if and only if the mapping of portals of JORp\nsatisfies ORp.\nFord∈OR2, let cost ed(JOR2,d) =α′. Ifd∈ORp, then cost ed(JORp,d) =α′′, where\nα′′:= (2p−1)+α′·\u0000\u0000p\n2\u0001\n+ 2p\u0001\n. By setting ω=α′′+1we ensure that we cannot afford making\neven one copy of Jω\nOR2unsatisfied. ◁\nFor an integer p⩾2, and d∈[2]p, byR̸=dwe denote the relation [2]p\\ {d}.\nClaim 6.9.4. For every p⩾2and every d∈[2]pthere exists a gadget JR̸=drealizing R̸=d\nwhose portals are non-adjacent.\nProof of Claim. Letd= (d1, . . . , d p). Introduce a copy of JORpgiven by Claim 6.9.3. Let its\nportals be x1, . . . , x p. Let I={i∈[p]|di= 1}. For each i∈I, introduce a vertex yithat is\nadjacent only to xi.\n33(a) The gadget realizing OR 2constructed in\nClaim 6.9.1.\n(b) The gadget realizing OR 3constructed in\nClaim 6.9.3.\nFigure 3: Gadgets constructed in the proof of Lemma 6.9. Black lines correspond to edges,\nand red lines denote copies of Jω\nOR2. Blue vertices are portals.\nThis completes the construction of JR̸=d. For i∈[p], itsi-th portal is xi(ifi /∈I) oryi\n(ifi∈I). It is straightforward to verify that JR̸=dsatisfies all required properties. ◁\nFinally, we show that using previously obtained gadgets we can realize any relation R.\nClaim 6.9.5. For every p⩾2and every R⊆[2]p, there exists a gadget JRrealizing Rwhose\nportals are non-adjacent.\nProof of Claim. We start with introducing vertices x1, . . . , x p, which will be the portals of our\ngadget. For each ¯d∈ {a, b}p\\Rwe call Claim 6.9.4 to obtain JR̸=¯dand identify the respective\nportals of the gadget with x1, x2, . . . , x p. It is straightforward to verify that such a gadget\nindeed realizes R. ◁\nThis completes the proof.\nWe strengthen the previous result by balancing out the violation costs.\nTheorem 2.3. For each r⩾1, and R⊆[2]r, there is an r-ary2-gadget that 1-realizes R.\nProof.Let¯R:= [2]r\\R, and let ¯d1, . . . , ¯d|¯R|be an enumeration of the elements of ¯R.\nWe define a relation R′in[2]r×[2]|¯R|as follows: A tuple (x1, . . . , x r, y1, . . . , y|¯R|)is inR′\nif and only if one of the following holds:\n1.(x1, . . . , x r)∈Randyi= 1for all i∈[|¯R|], or\n2. for some i∈[|¯R|],(x1, . . . , x r) =¯di,yi= 2, and yj= 1for all j̸=i.\nAccording to Lemma 6.9, there is a gadget Jthat realizes R′. Let Z1, . . . , Z rbe the portals\nofJthat belong to the respective entries x1. . . , x r, and let z1, . . . , z|¯R|be those that belong\ntoy1, . . . , y|¯R|, respectively. By definition of a realization, there is some integer αsuch that\nfor each d∈R′, we have cost ed(J,d) =α.\nWe modify Jto form a new gadget J′by attaching to each portal zia new vertex viwith\nlistL(vi) ={2}. Now interpret this modified gadget J′as a gadget with portals Z1, . . . , Z r.\nWe claim that J′1-realizes R: Whenever Z1, . . . , Z rare mapped to a state in R, mapping all\nof the zis to1requires only αedge deletions in Jand no further edge deletions outside as the\nvi’s can be mapped to 2. However, if Z1, . . . , Z rare in a state ¯di∈¯Rthen αedge deletions\nare required in J, and they are sufficient if ziis mapped to 2. However, this requires one more\n34edge deletion between ziandvi, both of which are mapped to 2. At the same time α+ 1edge\ndeletions are sufficient to extend a state ¯di, so all violations of Rhave the same cost α+ 1.\nSo we have shown that J′1-realizes R. Note that, according to Lemma 6.9, the portals of\nJare non-adjacent. Consequently, so are the portals of J′. Thus, we can introduce ωcopies\nofJ′on the same set of portals Z1, . . . , Z rto obtain a gadget that ω-realizes R.\n6.3 Hardness for Max Cut\nWe now consider the problem Max Cut , and our goal is to prove Theorems 6.1 and 6.2. To\nthis end we first show respective results for the list version of Max Cut , that is, we show\nTheorems 2.4 and 2.5 for the case q= 2. We do this in a unified proof that only in the end\nplugs in two different assumptions based on the SETH or the M3SH, respectively. Afterwards\nwe show how to model the lists using the nonlist version to obtain Theorems 6.1 and 6.2.\nProof of Theorems 2.4 and 2.5 for q= 2.Consider an instance Iof the decision version of\nMax ( 2,r)-CSPwith Nvariables V, clauses C={C1, . . . , C m}, and clause deletion budget\nz⩽mfor the number of clauses that can be violated. For each clause CiinCletRibe the\ncorresponding relation of arity ri⩽r.\nAninstanceof List- 2-ColoringED .Wedefineaninstance (G, L)ofList- 2-ColoringED .\nWe consider its decision version and also define a corresponding edge deletion budget z′.\nFor each variable v∈ V, the graph Gcontains a vertex y(v)with L(v) ={1,2}. Let\nY={y(v)|v∈ V}. Let us consider each Rias a relation in [2]ri. From Theorem 2.3,\nwe know that for each relation Rithere is a gadget Jithat 1-realizes Ri. Note that by the\ndefinition of realization these gadgets may use lists. Let αibe the required edge deletions\nwithin Jifor a state that satisfies Ri, i.e., for each d∈Riwe have cost ed(Ji,d) =αi. (By the\ndefinition of realizing, αidoes not depend on d.) Since Jiis a1-realizer, each state outside\nofRirequires αi+ 1edge deletions within Ji. Also note that we can compute αiin constant\ntime. For each i∈[m], the graph Gcontains the gadget Ji, where the portals of Jiare those\nvertices y(v1),y(v2),y(v3)for which (v1, v2, v3)is the scope of Ri. We set z′:=z+Pm\ni=1αi.\nEquivalence of instances. Suppose there is a satisfying assignment f:V → { 0,1}that\nviolates at most zclauses from C. Consider the assignment hthat maps y(v)to1whenever\nf(v) = 1, and that maps y(v)to2whenever f(v) = 0. By construction, in order to ensure that\nhcan be extended to a list 2-coloring, it suffices to make αi+ 1edge deletions in each gadget\nJifor which Ciis violated by f, and αiedge deletions for each of the remaining gadgets. Thus,\nit suffices to make a total of z′edge deletions. The reverse direction is also straight-forward.\nStructure of the constructed instance. The gadgets Jias well as the integers αidepend\nonly on the arity kof the relations. Consequently, for fixed k, the size of Gis|Y|+O(1) =\nNO(1). Let Q=Y. Each connected component of G−Qis a subgraph of a gadget Jiand\nhaskneighbors in Q(kportals of Ji). Thus, the size of such a component depends only on\nkand consequently Qis a(σ, k)-hub of size N+ 1for some σdepending only on k.\nRuntime. Asthesizeof Gispolynomialin N,thehypotheticalalgorithmfor List- 2-ColoringED\nwould require (2−ε)(N+1)·NO(1)= (2−ε)N·NO(1)time to solve Max ( 2,r)-CSPon instance\n35I. Note that Max ( 2,r)-CSPwith zero deletion budget clearly generalizes the decision prob-\nlemr-Sat. Thus, the hypothetical algorithm contradicts the SETH for some rdepending on\nε(and consequently for some σandδdepending on ε). Moreover, according to Theorem 2.2,\nthe hypothetical algorithm also contradicts the M3SH even for k= 3(and consequently for\nsome universal constant σandδ= 3) to prove Theorems 2.4 and 2.5 for q= 2.\nWe will now continue the previous reduction by removing the lists in the constructed\ninstance of List- 2-ColoringED to obtain a result for Max Cut .\nProof of Theorems 6.1 and 6.2. As in the proof of Theorems 2.4 and 2.5 for q= 2, consider\nan instance Iof the decision version of Max ( 2,r)-CSPwith Nvariables V, and then consider\nthe equivalent instance (G, L, z′)of the decision version of List- 2-ColoringED .\nConstructing the instance of Max Cut .Let us adjust the instance (G, L, z′)to an\nequivalent instance (G∗,|E(G∗)| −z′)of the decision version of Max Cut . Since all vertices\ninYhave list {1,2}, each vertex with list {1}or list {2}is part of some gadget Ji. To\nconstruct the graph G∗, we add a new vertex Ato the graph G. For each vertex vofGthat\nhas a list L(v) ={1}, we introduce a set of αi+ω+ 1(parallel) 3-vertex paths from vtoA\n(each such path contains the vertex v, one inner vertex, and the vertex A). Similarly, for each\nvertex vofGinJithat has a list L(v) ={2}, we introduce αi+ 2(parallel) 4-vertex paths\nfrom vtoA. This finishes the definition of G∗.\nEquivalence of instances. To show that the two instances are equivalent, first assume\nthat there is a set Xof at most z′edges such that there is a 2-coloring φofG\\Xthat\nrespects the lists in L. Then φcan be extended to a 2-coloring of G∗\\Xwith φ(A) = 1. This\nensures that all of the introduced 3-vertex paths have endpoints that are mapped to 1, and\nall4-vertex paths have endpoints that are mapped to different colors — and consequently all\nof these paths can be 2-colored without further edge deletions. If we interpret the two color\nclasses as the two different parts of a cut, this gives a cut of size |E(G∗)|−|X|⩾|E(G∗)|−z′.\nIn the opposite direction, suppose that there is a cut of size at least |E(G∗)| −z′. Let X\nbe the at most z′non-cut edges. Then there exists a 2-coloring φofG∗\\X(mapping vertices\non one side of the cut to 1, and the other side to 2). Without loss of generality, by renaming\nthe colors, we can assume that h(A) = 1. Suppose that for some vertex v,φ(v)/∈L(v), say\nφ(v) = 2butL(v) ={1}(the other case is analogous). As L(v)̸={1,2}the vertex vis\npart of some gadget Ji. Consequently, Xmust contain at least one edge from each of the\n3-vertex-paths connecting vandA. This gives a total of αi+ 2such edge deletions. However,\nit only requires a set Fof at most αi+ 1edge deletions on Jito ensure that the mapping\nφ(Y)can be extended to a 2-coloring of Ji\\F. Moreover, in this case φsatisfies the lists\nLon the vertices of Ji, which means that any 4-vertex paths that connects some vertex vof\nJitoAcan be 2-colored without further edge deletions. Summarizing, this shows that there\nexists a 2-coloring that respects the lists Land requires at most z′edge deletions.\nStructure of the constructed instance. Recall that the gadgets Jias well as the integers\nαidepend only on the arity kof the relations of I. Consequently, for fixed k, the size of G∗is\n|Y|+O(1) = NO(1). Let Q=Y∪ {A}. Each connected component of G∗−Qis a subgraph\nof a gadget Jitogether with at most αi+ 2pending paths per vertex of Ji. Note that each of\nthese components has at most k+ 1neighbors in Q(kportals of Jiplus the vertex A). Thus,\n36the size of such a component depends only on kand consequently Qis a(σ, k+ 1)-hub of size\nN+ 1for some σdepending only on k.\nRuntime. As the size of G∗is polynomial in N, the hypothetical algorithm for Max Cut\nwould require (2−ε)(N+1)·NO(1)= (2−ε)N·NO(1)time to solve Max ( 2,r)-CSPon instance\nI. Recall that Max ( 2,r)-CSPwith zero deletion budget generalizes the decision problem\nr-Sat. Thus, the hypothetical algorithm contradicts the SETH for some rdepending on ε\n(and consequently for some σandδdepending on ε). Moreover, according to Theorem 2.2,\nthe hypothetical algorithm also contradicts the M3SH even for k= 3, which implies that Q\nas defined in the previous paragraph is a (σ,4)-hub for constant σ. This proves Theorems 6.1\nand 6.2.\n6.4 Hardness for q-ColoringED\nNow we consider the problem q-ColoringED forq⩾3, and our goal is to prove Theorem 6.3.\nAgain, we first show respective results for the list coloring, that is, now we show Theorems 2.4\nand 2.5 for the case q⩾3, which also finishes the proofs of these two results. As before (but\nusing different gadgets), we show how to remove the lists to obtain Theorem 6.3, as desired.\nProof of Theorems 2.4 and 2.5 for q⩾3.With Theorem 2.2 in mind, for some fixed r⩾3,\nletI= (V,C)be an instance of the decision version of Max ( q,r)-CSPwith Nvariables and\ndeletion budget z⩽|C|.\nAn instance of List- q-ColoringED .We now define an instance (G, L, z′)of the deci-\nsion version of List- q-ColoringED . For each variable v∈ Vwe introduce a variable vertex\ny(v). Let Y={y(v)|v∈ V}. For a constraint in C, letR⊆[q]ℓbe the corresponding relation\nof arity ℓ⩽r, and let (v1, v2, . . . , v ℓ)∈ Vℓbe the scope of R.\nAccording to Proposition 2.1, for every d= (d1, d2, . . . , d ℓ)∈[q]ℓ, there is an ℓ-ary\ngadget Jd= (J, L,{z1, z2, . . . , z ℓ})with lists Lcontained in [q]such that any q-coloring\nψof the vertices z1, z2, . . . , z ℓcan be extended to a list q-coloring of Jdif and only if\n(ψ(z1), ψ(z2), . . . , ψ (zℓ))̸=d. Let γd⩾1be the minimum number of edge deletions in\nJdthat are required to extend dto a proper list coloring of this gadget; note that γdcan be\ncomputed in constant time as the number of vertices in Jis bounded by a function of ℓand\nq, i.e., a constant.\nLetP:=Qr\nℓ=1Q\np∈[q]ℓγp. Note that Pdepends only on qandr. For each vector d\nofℓ⩽relements from [q]withd/∈R, we introduce P/γ dcopies of Jdto balance out the\ndifferent violation costs for different d(and different R), and we identify the respective portals\nz1, z2, . . . , z ℓof each of these copies with the variable vertices y(v1), y(v2), . . . , y (vℓ). LetJR\nbe the union of the copies of all the gadgets Jdthat we introduced for this constraint (over all\nvectors d). We repeat this process for each constraint in C. This forms the graph Gtogether\nwith the lists L. Finally, we set z′=P·z.\nEquivalence of instances. Suppose there is a set of constraints C′⊆ Cwith|C′|⩽zsuch\nthat there is a satisfying assignment f:V → [q]for(V,C \\ C′). Consider the coloring φof\nYwith φ(y(v)) = f(v)for each v∈ V. Consider a constraint in Cwith the corresponding\nrelation Rand scope (v1, v2, . . . , v ℓ)∈ Vℓ. Iffsatisfies Cthen φcan be extended to a proper\nlist coloring of all the gadgets in JR(with zero required additional edge deletions). However,\n37iffviolates Cthend:= (φ(v1), φ(v2), . . . , φ (vℓ))is not in Rand consequently extending φ\nto a proper list coloring of Jdrequires γdedge deletions for each copy of Jd, for a total of P\nrequired edge deletions. Thus, φcan be extended to a proper list coloring of G\\Xwhere X\nis a set of P· |C′|⩽P·z=z′edges (where we use the fact that Pis independent of R).\nNow suppose there is a minimum-size set of edges Xwith|X|⩽z′such that G\\Xhas\na proper list q-coloring. Assume that φis such a coloring. Consider some gadget Jdthat is\npart of G. Let d= (d1, d2, . . . , d ℓ)and without loss of generality let (y(v1), y(v2), . . . , y (vℓ))\nbe the portals of Jd. If(φ(y(v1)), φ(y(v2)), . . . , φ (y(vℓ)))̸=dthen Xdoes not contain any\nedges of Jd(because of the minimality of X), whereas otherwise Xhas to contain precisely\nγdedges from Jd, and this holds for each of the P/γ dcopies of Jd. Thus, |X|is a multiple\nofPand the assignment fwith f(v):=φ(y(v))satisfies all but |X|/P⩽zconstraints in C.\nStructure of the constructed instance. The size of the gadgets Jdand the violation\ncosts γdare upper-bounded by some function in qandr. Consequently, the size of at most P\ncopies of the gadget Jdis upper-bounded by some function h(q, r). We can assume that no\ntwo constraints have scopes of variables such that one scope completely contains the other.\nSo we can assume that there are at most (N+ 1)rdifferent constraints in C. Consequently,\nthe size of Gis at most |Y|+ (N+ 1)r·h(q, r) =NO(1). Let Q=Y. Then each component\nofG−Qis a subgraph of a gadget of the form Jd. Hence the size of such a component\ndepends only on qandr, and each component has at most rneighbors in Q(the portals of\nJd). Therefore, the set Sis a(σ, r)-hub of size Nfor some σdepending only on qandr.\nRuntime. Asthesizeof Gispolynomialin N,thehypotheticalalgorithmfor List- q-ColoringED\nwould require (q−ε)N·NO(1)time to solve Max ( q,r)-CSPon instance I. This contradicts\nthe SETH for some rdepending on εaccording to Theorem 4.2. It also contradicts the M3SH\neven for r= 3according to Theorem 2.2, which means that the set Qas defined in the previous\nparagraph is a (σ,3)-hub. This proves Theorems 2.4 and 2.5 for q⩾3.\nWe will now continue the previous reduction by removing the lists in the constructed\ninstance of List- q-ColoringED .\nProof of Theorem 6.3. Forq⩾3, consider an instance I= (V,C)of the decision version of\nMax ( q,r)-CSPwith Nvariables Vand deletion budget z, and then consider the equivalent\ninstance (G, L, z′)ofList- q-ColoringED constructed in the proof of Theorems 2.4 and 2.5\nforq⩾3.\nConstructing the instance of q-ColoringED .We modify the instance (G, L, z′)into\nan equivalent instance (G∗, z′)ofq-ColoringED . Note that for each vertex in Ythe corre-\nsponding list is [q], so we only have to take care of lists of vertices that are non-portal vertices\nof some gadget of the form Jd, for some d= (d1, d2, . . . , d ℓ)∈[q]ℓ. Let γd⩾1be the mini-\nmum number of edge deletions in Jdthat are required to extend the state dof the portals to\na proper list coloring of this gadget.\nThe high-level idea is to simulate lists using a copy of the q-clique, similarly as in the proof\nof Theorem 5.1. However, again we need to be careful as deleting edges incident to this clique\n(either inside the clique or joining the clique with vertices of G) could destroy the structure\nof solutions.\n38We first introduce qnew vertices k1, . . . , k q. They will be used as a “global copy of a\nq-clique” Now we would like to turn {k1, . . . , k q}into a clique whose edges cannot be deleted.\nAs we cannot do it directly, instead of edges we will introduce many copies of gadgets that\nbehave “like an edge.” More specifically, for each distinct i, j∈[q], where i < j, we introduce\nz′+ 1copies of the following inequality gadget . We introduce a new (q−1)-clique Ki,jand\nmake it fully adjacent to ki(i.e., the set V(Ki,j)∪ {ki}induces a q-clique in G∗). Next, we\nintroduce a new vertex xi,jand make if fully adjacent to Ki,j(i.e., the set V(Ki,j)∪ {xi,j}\ninduces a q-clique in G∗). Note that this enforces that in any proper q-coloring, the color of\nxi,jis the same as the color of ki, and no further constraints are introduced. Finally, we make\nxi,jadjacent to kj, so that the color of kjmust be different than the color of ki.\nNow consider some d= (d1, d2, . . . , d ℓ)∈[q]ℓand a copy of a gadget Jd. We introduce\nγd+1“private” q-cliques Cp\ndwith vertex labels cp\n1, . . . , cp\nq(forp∈[γd+1]) . For i∈[ℓ], and for\neach p∈[γd+ 1], we introduce an edge between kdiand each vertex of Cp\ndwith the exception\nofcp\ndi. Then, for each vertex vofJdandj /∈L(v)and for each p∈[γd+ 1], we introduce an\nedge between vandcp\nj. This completes the construction of G∗.\nEquivalence of instances. Now let us show that the two instances are indeed equivalent.\nSuppose there is a proper list coloring ψofG\\Xfor some set of edges Xfrom G. We claim\nthat there is an extension φofψthat is a proper q-coloring of G∗\\X. To this end, we need\nto show how to extend ψto the cliques of the form Cp\nd, to the vertices k1, . . . , k q, and to\ninequality gadgets. For a clique of the form Cp\ndwe color the corresponding vertices cp\n1, . . . , cp\nq\nby setting φ(cp\ni) =i. For each i∈[q], we set φ(ki) =i. For each i, j∈[q]and each copy of\nthe inequality gadget, we color Ki,jarbitrarily using colors [q]\\ {i}andxi,jreceives the color\ni. Observe that a vertex kiis never adjacent to a vertex cp\niwith the same index i. Moreover,\na vertex cp\niis only adjacent to some vertex vinJdifi /∈L(v). Hence φis a proper coloring\nofG∗\\Xifψis a proper list coloring of G\\X.\nFor the other direction let φbe a proper q-coloring of G∗\\Xfor some minimum-size set\nof edges Xfrom G∗with|X|⩽z′. Note that since for each pair of i, j∈[q]we introduced\nz′+ 1copies of the inequality gadget between kiandkj, there is always one copy that is not\naffected by X. Thus we can assume that vertices k1, . . . , k qreceive pairwise distinct colors.\nBy permuting colors we can assume that for each i∈[q]we have φ(ki) =i.\nNow consider some d= (d1, . . . , d ℓ)and a copy of Jd= (J, L,{z1, . . . , z ℓ}). Let X′denote\nthose edges from Xthat have at least one endvertex in Jor in some clique of the form Cp\nd\nintroduced for Jd. We aim to understand where X′lies.\nClaim 6.9.1. Letπbe some permutation of [q]and let Jπ\nd= (J, Lπ,{z1, . . . , z ℓ}), where for\neach vertex vofJ,Lπ(v):={π(i)|i∈L(v)}. Let ψ:{z1, . . . , z ℓ} → [q]be a coloring of\nz={z1, . . . , z ℓ}. Then\n•Ifψ(z)̸=π(d)then ψcan be extended to a proper list coloring of Jπ\nd.\n•Ifψ(z) =π(d)thenγdedge deletions are required and sufficient to extend ψto a proper\nlist coloring of Jπ\nd.\nProof of Claim. This follows directly from the properties of Jdaccording to Proposition 2.1,\nif to each coloring of Jdwe apply the permutation π. ◁\nNote that Claim 6.9.1 implies that regardless of the coloring of z1, . . . , z ℓ, we can make\nthe copy of Jdq-colorable by removing at most γdedges from this gadget (i.e., with both\n39endverticesin J). Inparticular,thismeansthatwecansafelyassumethat |X′|⩽γd(otherwise\nwe can obtain another, better solution). Since we introduced γd+ 1local cliques Cp\ndforJd,\nthere is always at least one such clique whose vertices are not incident with any edge from\nX′. Since all local cliques have the same neighborhood, we can safely assume that the edges\nfrom X′are not incident to any of these cliques. In particular, all edges from X′have both\nendvertices in J, i.e., they are present in G.\nFurthermore, again, without loss of generality, we can assume that the corresponding\nvertices of each clique Cp\ndreceive the same color in φ, i.e., for all p, p′∈[γd]andi∈[q]we\nhave φ(cp\ni) =φ(cp′\ni). Since the colors of vertices of Cp\ndare pairwise distinct, we conclude that\nthere is a permutation πof[q]such that φ(cp\ni) =π(i), for every i, p. Summing up, we observe\nthat the lists enforced by the local cliques on the vertices of JdareLπ, where the definition of\nLπis as in Claim 6.9.1. Note that this implies that without loss of generality we may assume\nthat Xcontains either 0 (if φ(z)̸=π(d)) orγd(ifφ(z) =π(d)) edges from J– otherwise\nwe could obtain a better solution. Furthermore, observe that the edges joining {k1, . . . , k q}\nwith vertices from the local cliques imply that for each diwe have π(di) =di. Consequently,\n|X∩E(J)|= 0ifφ(z) =dand|X∩E(J)|=γdifφ(z) =d.\nBut now, by the definition of Jdandγd, we can remove |X∩E(J)|edges from Jand\nproperly color all non-portal vertices in a way that this coloring respects lists L. Repeating\nthis argument for each copy of Jd, we conclude that G\\Xadmits a proper q-coloring that\nrespects lists L.\nStructure of the constructed instance. The size of at most Pcopies of the gadget Jd\ntogether with Pcopies of a clique of size qis upper-bounded by some function h(q, r). We\ncan assume that there are at most (N+ 1)rdifferent constraints in C. Further, note that\nw=z′+ 1 = P·z+ 1⩽P·(N+ 1)r∈NO(1). Consequently, the size of G∗is at most\n|Y|+(N+1)r·h(q, r)+(w+1)·q3=NO(1). Let Q=Y∪{k1, . . . , k q}. Then each component\nofG∗−Qis of one of two forms. The first possibility is that it is a subgraph of a gadget of\nthe form Jdtogether with the corresponding cliques C1\nd, . . . , Cγd+1\nd, in which case the size of\nsuch a component depends only on qandr, and the component has at most 3 +rneighbors\ninQ, three in Y, and rin{k1, . . . , k q}. The second possibility is that such a component is an\ninequality gadget, i.e., it consists of qvertices and attaches to two vertices of Q. Therefore,\nthe set Qis a(σ,3 +r)-hub of size N+qfor some σdepending only on q.\nRuntime. Asthesizeof G∗ispolynomialin N,thehypotheticalalgorithmfor q-ColoringED\nwould require (q−ε)(N+q)·NO(1)= (q−ε)N·NO(1)time to solve Max ( q,r)-CSPon instance\nI. This contradicts the M3SH even for r= 3according to Theorem 2.2, which implies that\nthe set Qdefined in th eprevious paragraph is a (σ,6)-hub. This proves Theorem 6.3.\n6.5 Simple Algorithm for q-ColoringED\nThe following algorithm solves the q-ColoringED problem for q⩾2, which also includes\ntheMax Cut problem for q= 2. It is worth noting that this algorithm can be modified to\nuse treewidth as the parameter.\nTheorem6.10. For all integers q⩾2andσ, δ⩾1, every n-vertex instance of q-ColoringED\ncan be solved in time qp·nO(1)if a(σ, δ)-hub of size pis given in the input.\n40Proof.As input to q-ColoringED , consider a graph Gwith nvertices, given along with a\n(σ, δ)-hub Qof size p.\nWe start by guessing the coloring of Q. Let f:Q→[q]denote this coloring. Consider now\na component G′ofG\\Q, and let Odenote the graph induced by the vertices of G′and their\nneighbors in Q. Since G′has constant size and is adjacent to constantly many vertices in Q,\nwe can determine in constant time the minimum number of edges to delete from O, such that\nthe remaining graph admits a q-coloring.\nThe number of components in G\\Qis bounded above by a polynomial in n, and therefore,\nin polynomial time, we can find the optimum solution that agrees with Xandf. Moreover,\nsince there is always an (X, f)corresponding to the optimal solution, an algorithm that tries\nall possible Xandfgives the correct output. The running time of such an algorithm is upper\nbounded by qp·nO(1).\n7List-q-Coloring andList-q-ColoringVD with gadgets of\nconstant degree faster than brute force\nRecall that, assuming the SETH, we proved lower bounds for q-Coloring ,q-ColoringVD ,\nandq-ColoringED , parameterized by the size of a (σ, δ)-hub of the instance graph G, where\nσandδare constants depending on ε >0. In all cases, we were able to exclude an algorithm\nwithrunningtime (f(q)−ε)p·nO(1),forany ε >0,where f(q)isanfunctionof q(i.e.,either qor\nq+1),pis the size of a (σ, δ)-hub, and nis the number of vertices of G. Furthermore, assuming\nthe M3SH, we have a somewhat stronger lower bound for q-ColoringED where σandδare\nabsolute (and small) constants independent of ε. For q-Coloring andq-ColoringVD , we\nhave only a weaker form of the lower bound, where the values of σandδdepend on ε.\nIn this section, we show that these latter lower bounds cannot be strengthened so that\nσandδare absolute constants; in fact, we prove that already δon its own cannot be an\nabsolute constant. Specifically, we prove the following algorithmic results that hold even for\nthe respective coloring problems with lists.\nTheorem 7.1. For every q⩾3and every constant δthere exists εwith the following property:\nFor every constant σ, every instance (G, L)ofList- q-Coloring withnvertices, given with\na(σ, δ)-hub of size p, can be solved in time (q−ε)p·nO(1).\nTheorem 7.2. For every q⩾3and every constant δthere exists εwith the following property:\nFor every constant σ, every instance (G, L)ofList- q-ColoringVD with nvertices, given\nwith a (σ, δ)-hub of size p, can be solved in time (q+ 1−ε)p·nO(1).\n7.1 Faster algorithm for List- q-Coloring\nAs a warm-up, let us start with Theorem 7.1, whose proof is quite straightforward.\nProof of Theorem 7.1. LetXbe a (σ, δ)-hub of G. IfXis a constant-size set, we can guess\nits coloring exhaustively, and then adjust the lists of the neighbors of vertices in Xas follows.\nIfx∈Xis guessed to be colored with color i, then ican be removed from lists of all vertices\nin the neighborhood of X. After that we can safely remove Xfrom the graph, obtaining\nan equivalent instance, which can be solved in polynomial time as its every component is of\nconstant size. Thus from now on assume that |X|=pis sufficiently large.\n41We first check if there is a vertex with an empty list; if so, we immediately reject the\ninstance. Then we check whether there exists a component AofG−Xwith no neighbor in\nX. If such an Aexist, we solve the instance G[A]ofList- q-Coloring : it can be done in\nconstant time as |A|⩽σ. If it is a yes-instance, then we can proceed with the equivalent\ninstance G−A; it is a no-instance, then (G, L)is a no-instance and we reject it.\nConsider a component AofG−X, and let Γ(A)denote its neighborhood in X; recall that\nwe can assume that Γ(A)̸=∅. Let r=|Γ(A)|and recall that 1⩽r⩽δ.\nLetFbe the set of all colorings of Γ(A)with qcolors, respecting lists L, that can be\nextended to a proper list coloring of G[A∪Γ(A)]. Note that |F|⩽qrandFcan be computed\ninconstanttimeas |A∪Γ(A)|⩽δ+σ. If|F|=qr, theneverycoloringof Γ(A)canbeextended\nto the vertices of A. Thus we can remove Afrom G, obtaining an equivalent instance. So\nassume that |F|⩽qr−1.\nWe exhaustively guess a coloring of Γ(A)that belongs to F. In each branch we adjust the\nlists of the neighbors of vertices in Γ(A)as previously, and then remove A∪Γ(A)from the\ngraph, obtaining an equivalent instance.\nThenumberofleavesintherecursiontree, measuredasthefunctionof p, isupper-bounded\nby the recursive formula\nT(p)⩽(qr−1)·T(p−r).\nProceeding by induction we can show that\nT(p)⩽(qr−1)p/r·\u0010\nqδ−1\u0011(p−r)/δ\n⩽\u0010\nqδ−1\u0011p/δ\n,\nwhere the last inequality can be verified by a standard yet quite tedious calculation.\nAs every node in the recursion tree is processed in polynomial time, the total running time\nis bounded by T(p)·nO(1)⩽\u0000\nqδ−1\u0001p/δ·nO(1). Now Theorem 7.1 follows by observing that\u0000\nqδ−1\u00011/δ< q.\n7.2 Faster algorithm for List- q-ColoringVD\nNowletusproceedtothe List- q-ColoringVD problem. Itwillbemoreconvenienttoreduce\nit to a certain auxiliary variant of CSP. Before we formally define it, let us introduce some\nmore notation.\nLetXbe a set and let qbe a positive integer. By F×\nX,qwe denote the set of all functions\nf:X→[q]∪ {×}, where ×is a special symbol. For a function f∈ F×\nX,q, we let ||f||denote\n|f−1(×)|.\nWe define a binary relation ⪯onF×\nX,qas follows: for all f, f′∈ F×\nX,q, we say that f′⪯f\nif for all x∈Xeither f(x) =f′(x), orf(x) =×. In other words, fwas obtained from f′\nby mapping some (possibly empty) subset of elements of Xto×. Note that (F×\nX,q,⪯)is a\npartially ordered set and its unique maximum element is f×\nX, where f×\nX∈ F×\nX,qis the function\nthat maps every element of Xto×. We write f′≺fiff′⪯fandf′̸=f.\nForconstants q, r, aninstanceof (q, r)-CSP-with-Wildcard isatriple (V,C,cost), where\n•Vis the set of variables, each with domain [q]∪ {×},\n•Cis a multiset of subsets of V, each of size at most r,\n•costis a function that maps each pair (C, f)where C∈ C, f∈ F×\nC,qto a non-negative\ninteger and satisfies the following:\n42(wildcard property) For all C∈ Cand all f, f′∈ F×\nC,q, iff′⪯f,\nthen it holds that cost(C, f)⩽cost(C, f′).\nThe aim of (q, r)-CSP-with-Wildcard is to find an assignment φ:V → [q]∪ {×},\nminimizing the total cost defined as follows:\ntotal-cost(φ) =||φ||+X\nC∈Ccost(C, φ|C).\nIt is fairly straightforward to reduce an instance of List- q-ColoringVD with a (σ, δ)-hub of\nsizepto an instance of (q, δ)-CSP-with-Wildcard with pvariables.\nLemma 7.3. Letσ, δbe constants. For an instance (G, L)ofList- q-ColoringVD , given\nwith a (σ, δ)-hub of size p, in polynomial time we can compute an instance (V,C,cost)of\n(q, δ)-CSP-with-Wildcard with pvariables, such that the values of optimum solutions of\nboth instances are equal.\nProof.Without loss of generality assume that Gis connected. Let Xbe a (σ, δ)-hub of G,\nand let us enumerate the vertices of Xas{x1, x2, . . . , x p}. Recall that we can safely assume\nthat for each xj, the set L(xj)is non-empty, and clearly |L(xj)|⩽q. For each j∈[p], letρj\nbe an arbitrary surjective function from [q]toL(xj).\nThe set Vhaspelements v1, . . . , v p, each with domain [q]∪ {×}. For each i∈[p], the\nvariable virepresents xi. Assigning the value j∈[q]tovicorresponds to mapping xitoρi(j),\nwhile assigning ×tovicorresponds to deleting xi.\nNow consider a component AofG−X, and let Γ(A)denote its neighborhood in X. Recall\nthat|Γ(A)|⩽δ. We add the set C={vi|xi∈Γ(A)}toC. Finally, for every f∈ F×\nC,q, we\nsetcost(C, f)to be the minimum number of vertices from Athat need to be deleted so that\nthe mapping of Γ(A)corresponding to f(i.e., if f(vi) =j∈[q], then xiis mapped to ρi(j),\nand if f(vi) =×, then xiis deleted) can be extended to a proper list coloring of the subgraph\nofGinduced by the non-deleted vertices from A∪Γ(A).\nIt is possible that there is more than one component of G−Xwith exactly the same\nneighborhood in X. In this situation Ccontains multiple copies of C(recall that it is defined\nto be a multiset) and the values of the cost function of each copy correspond to a distinct\ncomponent of G−X.\nIt is clear that the created instance of (q, δ)-CSP-with-Wildcard is equivalent to the\noriginal instance (G, L)ofList- q-ColoringVD , i.e., the optimal solutions have the same\nvalue.\nFinally, let us argue that (V,C,cost)can indeed be computed in polynomial time. Observe\nthat the number of components of G−XisO(|V(G)|). Furthermore, for each such component\nAthere are at most (q+ 1)δassignments of variables corresponding the vertices of Γ(A).\nFinally, for each component Aand each assignment fwe can compute the minimum cost of\nextending ftoG[A∪Γ(A)]in constant time, as the size of Ais at most σ, i.e., a constant.\nClearly a brute-force approach to solving an instance (q, r)-CSP-with-Wildcard with\nnvariables requires time (q+ 1)n·nO(1). Indeed, we enumerate all possible assignments\nφ:V → [q]∪ {×}, for each of them we compute total-cost(φ), and we return the assignment\nwhose total cost is minimum. In the next theorem we show that actually we can solve the\nproblem faster.\n43Theorem 7.4. Letq, rbe constants. Any n-variable instance of (q, r)-CSP-with-Wildcard\ncan be solved in time ((q+ 1)r−1)n/r·nO(1).\nProof.Let(V,C,cost)denote an instance of (q, r)-CSP-with-Wildcard . Consider C∈ C.\nRecall that f×\nC∈ F×\nC,qis the function mapping every element of Cto×. By the wildcard prop-\nerty we observe that cost(C, f×\nC) = minf∈F×\nC,qcost(C, f). For brevity, we denote cost(C, f×\nC)\nbyc×(C).\nClaim 7.4.1. For each C∈ Cone of the following holds:\na) for all f∈ F×\nC,qit holds that cost(C, f) = c×(C), or\nb) there exist f, f′∈ F×\nC,qsuch that f′≺fand cost(C, f) +||f||⩽cost(C, f′) +||f′||.\nProof of Claim. Suppose that a) does not hold and let f′∈ F×\nC,qbe such that cost(C, f′)⩾\nc×(C) + 1and||f′||is maximum possible. Note that by the definition of c×(C)we know that\nf′̸=f×\nC, i.e., there exists v∈Csuch that f′(v)̸=×. Let fbe obtained from f′by mapping\nvto×. Note that f′≺f. Clearly ||f||=||f′||+ 1and, by the maximality of f′, we have\ncost(C, f) = c×(C). Thus we have\ncost(C, f) +||f||=c×(C) +||f′||+ 1⩽cost(C, f′) +||f′||,\nwhich completes the proof of claim. ◁\nClaim 7.4.2. LetC, f, f′satisfy property b) in Claim 7.4.1. Then there exists an optimum\nsolution φfor(V,C,cost)such that φ|C̸=f′.\nProof of Claim. Letφ′be an optimum solution for (V,C,cost). Ifφ′|C̸=f′, we are done, so\nsuppose otherwise. Define φ:V → [q]∪ {×}as follows:\nφ(v) =(\nφ′(v)ifv /∈C,\nf(v)ifv∈C.\nNote that as φ′|C=f′≺f, we have φ′≺φ. Let us compute the total cost of φ.\ntotal-cost(φ) =||φ||+X\nC′∈Ccost(C′, φ|C′)\n=|φ−1(×)\\C|+|φ−1(×)∩C|+cost(C, φ|C) +X\nC′∈C\\{ C}cost(C′, φ|C′)\n(1)=|φ′−1(×)\\C|+||f||+cost(C, f) +X\nC′∈C\\{ C}cost(C′, φ|C′)\n(2)\n⩽|φ′−1(×)\\C|+||f||+cost(C, f) +X\nC′∈C\\{ C}cost(C′, φ′|C′)\n(3)\n⩽|φ′−1(×)\\C|+||f′||+cost(C, f′) +X\nC′∈C\\{ C}cost(C′, φ′|C′)\n=||φ′||+X\nC′∈Ccost(C′, φ′|C′) = total-cost(φ′).\n44where step (1) follows as φ′|V\\C=φ|V\\C, step (2) follows from the wildcard property, and step\n(3) follows from the properties of C, f, f′given by Claim 7.4.1 b). Since φ′is an optimal solu-\ntionandthusitstotalcostisminimumpossible, weconcludethat total-cost(φ) = total-cost(φ′)\nand thus φis also an optimal solution. ◁\nIn the beginning of (each recursive call of) our algorithm, we exhaustively apply the\nfollowing two reduction rules. First, if there is some v∈ V \\S\nC∈C, then we remove it from\nV. Note that an optimal solution for the reduced instance can be extended to an optimal\nsolution for the original one by mapping vto any element of [q]. Second, if there are C, C′∈ C\nsuch that C′⊆C, we can obtain an equivalent instance by removing C′fromCand, for\neach g∈ F×\nC,q, increasing the value of cost(C, g)bycost(C′, g|C′). After applying the second\nreduction rule exhaustively we can assume that the sets in Care pairwise incomparable. It is\nclear that the reduction rules can be applied in polynomial time.\nNow we are ready to describe our algorithm. If n=|V|⩽r, we solve the problem in\nconstant time by brute force in constant time, as qandrare constants. Otherwise, pick any\nC∈ Cand denote |C|byr′. Clearly r′⩽r. Consider two possibilities given by Claim 7.4.1.\nIf possiblity a) applies to C, then notice that we can safely remove CfromC. Indeed, an\noptimal solution of (V,C \\ {C},cost|C\\{C})is also an optimal solution of (V,C,cost)and the\ntotal cost of this solution in the latter instance is equal to its total cost in the former instance\nplus c×(C). So assume that possibility b) applies and let f, f′be as in Claim 7.4.1 b).\nWe guess the valuation of variables from C, considering all possibilities except for f′. By\nClaim 7.4.2 we know that it is safe to ignore f′as there is always an optimum solution whose\nvaluation restricted to Cis not f′.\nThisresultsin (q+1)r′−1branches. Consideronesuchbranchcorrespondingtoavaluation\nef∈ F×\nC,q. The instance considered in this branch is (V′,C′,cost′), defined as follows. The set\nV′of variables is V \\C; note that it is non-empty as |C|⩽randn > r. The set C′is defined\nasC′:={eC∩ V′|eC∈ C}. Note that ∅/∈ C′, since otherwise the second reduction rule could\nhave been applied. For each C′∈ C′and for each g∈ F×\nC′,qwe set\ncost′(C′, g) =X\neC∈C\neC∩V′=C′cost(eC,(g∪ef)|eC),\nwhere by g∪efwe denote the valuation obtained by assigning the values of variables from\nCaccording to ef, and the values of variables from C′according to g; note that these sets\nare disjoint. Since we aim to solve (V,C,cost), the value of an optimum solution found for\n(V′,C′,cost′)must be increased by ||ef||. It is straightforward to verify that the above recursive\nprocedure is correct. Now let us argue about the running time.\nSince the local computation at each recursive call can be performed in polynomial time, in\norder to bound the time complexity we need to estimate the number of leaves in the recursion\ntree. Let T(n)denote this value for an instance with nvariables. We aim to show that\nT(n)⩽((q+ 1)r−1)n/r.\nIfn⩽r, then T(n) = 1⩽((q+ 1)r−1)n/r. Suppose now that n > rand for every n′< n\nwe have T(n′)⩽((q+ 1)r−1)n′/r. We obtain the following recursive inequality:\nT(n)⩽\u0010\n(q+ 1)r′−1\u0011\n·T(n−r′).\nApplying the inductive assumption, we get that\nT(n)⩽\u0010\n(q+ 1)r′−1\u0011\n·((q+ 1)r−1)(n−r′)/r⩽((q+ 1)r−1)n/r,\n45similarly as in the proof of Theorem 7.1. Summing up, the total running time is bounded by\nT(n)·nO(1)⩽((q+ 1)r−1)n/r·nO(1).\nNow Theorem 7.2 follows by combining Lemma 7.3 with Theorem 7.4 and observing that\n((q+ 1)r−1)1/r< q+ 1.\n8 Equivalent Hypotheses: Set Covering/Packing/Partitioning\nIn this section we prove Theorem 2.6.\nRecall that the Set Cover Conjecture (SCC) asserts that for all ε >0, there exists d⩾1\nsuch that there is no algorithm that solves every ⩽d-Set Cover instance (U,F)in time\n(2−ε)|U|· |U|O(1)[9].\nWe show that, for a whole list of problems, similar lower bound statements are actually all\nequivalenttotheSCC.Therespectiveproblemswereinformallyintroducedintheintroduction,\nSection 1. Formally, they are defined in Section 3. At a high level, we consider covering,\npacking and partition problems and distinguish between problem variants, where every set of\nthe family has size exactly dor at most d, and we also consider (whenever it makes sense)\nwhether the number of selected sets or the size of their union is maximized.\nAll discussed problems can be solved in time 2n·nO(1), where nis the size of the universe,\nby quite straightforward dynamic programming. We now show that these problems are all\nequivalent to the SCC in the following sense: a faster than standard dynamic programming\nalgorithm for any of those problems violates the SCC. Similarly, if the SCC fails, then there\nis afaster than standard dynamic programming algorithm for each of the problems in the list.\nTheorem 2.6. Suppose that for one of the problems below, it is true that for every ε >0,\nthere is an integer dsuch that the problem cannot be solved in time (2−ε)n·nO(1), where n\nis the size of the universe. Then this holds for all the other problems as well. In particular,\nany of these statements is equivalent to the SCC.\n1.=d-Set Cover\n2.=d-Set Partition\n3.=d-Set Packing (#Sets)\n4.⩽d-Set Cover\n5.⩽d-Set Partition\n6.⩽d-Set Partition (#Sets)\n7.⩽d-Set Packing (#Sets)\n8.⩽d-Set Packing (Union)\nFigure 1 gives an overview of the reductions that we prove in order to show Theorem 2.6.\n8.1 Basic Reductions\nRecall from Section 2.4 that we say that a problem class A={Ad|d⩾1}(such as ⩽∗-Set\nCover) is2n-hard if the following statement holds.\nFor each ε > 0, there is some d⩾1such that no algorithm solves Adon all\nn-element instances in time (2−ε)n·nO(1).\n46LetB={Bd|d⩾1}beanotherproblemclass. Inthefollowingsectionsweprovemanyreduc-\ntions of the form “If Ais2n-hard then Bis2n-hard”. We will usually prove the contrapositive\nstatement:\nSuppose for some ε >0it holds that for every d⩾1, there is an algorithm that\nsolves Bdonn-element instances in time (2−ε)n·nO(1).\nThen there exists ε′>0such that for every d⩾1, there is an algorithm that\nsolves Adonn-element instances in time (2−ε′)n·nO(1).\nAs⩽d-Set Cover (resp., ⩽d-Set Packing (#Sets) ) is more general than =d-Set\nCover(resp., =d-Set Packing (#Sets) ), we immediately observe the following.\nObservation 8.1. If=∗-Set Cover is2n-hard, then ⩽∗-Set Cover is2n-hard; and if\n=∗-Set Packing (#Sets) is2n-hard, then ⩽∗-Set Packing (#Sets) is2n-hard.\nSimilarly,since tdisjointsetsofsize dcoverexactly t·delements, ⩽d-Set Packing (Union)\nproblem is a generalization of =d-Set Packing (#Sets) .\nObservation8.2. If=∗-Set Packing (#Sets) is2n-hard, then ⩽∗-Set Packing (Union)\nis2n-hard.\nNext, we observe that =d-Set Partition can be easily reduced to =d-Set Cover . Now\nconsider a set family Fover the universe Usuch that each set in Fhas size d. We observe\nthat each set partition of (U,F)is a set cover of size n/d, and vice versa each set cover of size\nn/dcan only contain disjoint sets, and is therefore a set partition. Thus, =d-Set Cover can\nbe used to solve =d-Set Partition .\nObservation 8.3. If=∗-Set Partition is2n-hard, then =∗-Set Cover is2n-hard.\nAnother simple argument gives the following.\nLemma 8.4. If⩽∗-Set Cover is2n-hard, then ⩽∗-Set Partition (#Sets) is2n-hard..\nProof.Given a ⩽d-Set Cover instance (U,F, t), create a ⩽d-Set Partition (#Sets)\ninstance (U,F′, t)where F′:={A|A̸=∅, A⊆S, S∈ F} .\nFirst, suppose Uhas a cover S={S1, . . . , S r} ⊆ Fof size r⩽t. Without loss of generality\nwe can assume that it is inclusion-wise minimal, i.e., for every i∈[r]there exists u∈Sithat\nis not covered by S \\Si.\nDefine sets A1, . . . , A ras follows. We set A1=S1. Then, for i⩾2, we set Ai=\nSi\\S\ni<iAi. It is straightforward to see that\u0010S\n1⩽i⩽rAi\u0011\n=\u0010S\n1⩽i⩽SiSi\u0011\n. Furthermore,\nthe sets A1, . . . , A rare pairwise disjoint, and they are nonempty by minimality of S. Since\nfor1⩽i⩽r,Aiis a subset of Siat the end of the algorithm described above, we have\n{A1, . . . , A r} ⊆ F′. Consequently, this set is a partition of Uof size r⩽t.\nSecond, suppose that Uhas a partition S′⊆ F′of size r⩽t. Since each set in F′is a\nsubset of some set in F, there is a corresponding set cover SofUof size r⩽t.\nLetUbe some universe and Fa family of size- dsubsets of U. Clearly, (U,F)has a set\npartition if and only if it has a set packing of size (at least) |U|/d. So we obtain the following\nobservation.\n47Observation 8.5. If=∗-Set Partition is2n-hard, then =∗-Set Packing (#Sets) is\n2n-hard.\nIn order to prove the following reduction we again use the trick that we also employed to\nshow Lemma 5.3. Namely, we split the universe of the instance into blocks and then guess the\nnumber of unused elements for each block.\nLemma 8.6. If⩽∗-Set Packing (Union) is2n-hard, then ⩽∗-Set Partition (#Sets)\nis2n-hard.\nProof.Let(U,F, t)be an instance of ⩽d-Set Packing (Union) . Let n=|U|. Let bbe the\nsmallest integer such that (2b+ 2)1/b<(1 + ε/2). This choice will become clear later on.\nWithout loss of generality, we assume that nis divisible by b(as otherwise we can add at\nmost b−1dummy elements that are in none of the sets from F). We split the elements from\nUinto pairwise disjoint blocks U1, . . . , U n/bof size beach.\nSuppose we are given an integer xi∈ {0, . . . , b }for each block Uithat specifies how many\nelements of this block are notcovered in a hypothetical ⩽d-Set Packing (Union) solution.\nWe refer to the vector x= (x1, . . . , x n/b)as the signature of this solution. Note that each\nsolution has precisely one signature.\nConsider the following algorithm:\n1. Define the set U′=U∪ {ai|i∈[n/b]}, where aiis a new element introduced for each\nblock.\n2. Iterate over all possible signatures. For each such signature x, build a new set system\nFxthat contains all sets from Fand, in addition, for each iand each set S⊆Uiof size\nxi, the set S∪ {ai}.\n3. For d′=b+ 1, run the assumed algorithm for ⩽d′-Set Partition on the instance\n(U′,Fx, t+n/b).\nFirst note that by the fact that a set S∪{ai}has size xi+ 1⩽b+ 1 = d′,(U′,Fx, t+n/b)\nis, in fact, an instance of ⩽d′-Set Partition (#Sets) . The idea is that for each block Ui,\na partition of U′contains precisely one set that contains the new element ai. This set holds\nall elements that are not covered by a corresponding packing of sets from Fwith signature\nx. Hence the partition corresponding to such a packing contains one additional set per block.\nSo it is straight-forward that above algorithm solves ⩽d-Set Packing (Union) .\nLetn′=|U′|=n+n/b. There are at most (b+ 1)n/bdifferent signatures; and for each\nsignature the algorithm for ⩽d′-Set Partition takes time (2−ε)n′·n′O(1). Thus the total\nruntime is bounded by\n(b+ 1)n/b·(2−ε)n′·n′O(1)⩽(b+ 1)n/b·(2−ε)n+n/bnO(1)\n⩽(2b+ 2)n/b·(2−ε)n·nO(1)\n⩽(1 +ε/2)n·(2−ε)n·nO(1)\n= (2−ε2/2)n·nO(1).\nForε′=ε2/2this gives the correct runtime.\n488.2 Structured Families of Sets\nSeveral subsequent reductions use similar ideas. Therefore we introduce some notation and\nprove a few technical lemmas to avoid redundancy as much as possible. Let Ube a set of n\nelements and F ⊆ 2Ube a set family where each set has size at most d.\nLet(F1, . . . ,Fd)be the partition of Faccording to set size, i.e., let Ficontain all size- i\nsets of F. We now define a vector that represents a guess of how many sets of each size are\nchosen (in some solution). Let r= (r1, . . . , r d)∈ {0, . . . ,\u0000n\nd\u0001\n}dsatisfy\n1.Pd\ni=1ri⩽n, and\n2.ri⩽|Fi|.\nWe say that ris anF-signature and define w(r) =Pd\ni=1rito be the weightofr. For a\npacking/partition S ⊆ FofU, we say that Srespects rifScontains exactly risets from Fi.\nIntuitively, ouraimistomodify (U,F)suchthatallsetshavethesamesizewhilepreserving\nthe original packings/partitions. To this end, for each F-signature rand integer c⩾1, we will\ndefine a new set system (Ur,Fr)such that every set in Frhas size (c·d! + 1). Moreover, we\nwill show that there is a one-to-one relationship between the packings (partitions) of (U,F)\nthat respect r, and packings (partitions) of (Ur,Fr). We say that (Ur,Fr)is the (c,r)-join of\n(U,F).\nThe idea is to merge, for each i∈[d], selections of ai:=c·d!\nidisjoint sets from Fito obtain\nnew sets of size ai·i=c·d!\ni·i=c·d!(now the same size for all i). A selection of ri/aiof these\nnew sets then corresponds to a selection of riof the original sets. In order to avoid divisibility\nissues, we introduce some dummy sets and work with the vector s= (s1, . . . , s d)∈Zd\n⩾0with\nsi:=\u0018ri\nai\u0019\n·ai(fori∈[d]),\nSo, coming from ri,siis the next-largest integer that is divisible by ai.\nForIr:={i∈[d]|ri̸= 0}, and an index i∈Ir, we define [si−ri]new sets of dummy\nelements of size ieach:\nAr,i\nj:={ai,j\n1, . . . , ai,j\ni}forj∈[si−ri].\nWe assume that the sets Ar,i\njare all disjoint sets of new elements (outside of U). For a given\nsignature r, define the set of dummy elements\nNr:=[\ni∈Ir[\nj∈[si−ri]Ar,i\nj.\nNote that\n|Nr|=|[\ni∈Ir[\nj∈[si−ri]Ar,i\nj|⩽d·c·d! =O(1) (2)\nsince|Ir|⩽dandsi−ri⩽ai.\nFor each i∈Ir, the idea is to extend the part Fiby the new size- isets\nFr,i\nadd:={Ar,i\nj|j∈[si−ri]}.\n49Then, for each i∈Ir, we consider the sets that can be obtained as a union of aipairwise\ndisjoint sets from Fi∪ Fr,i\nadd, formally,\nGr\ni:=\u001a[\nj∈[ai]Vj|V1, . . . , V aiare pairwise disjoint sets from Fi∪ Fr,i\nadd\u001b\n.\nCrucially, for each i∈Irand set X∈ Gr\ni, it holds that |X|=ai·i=c·d!\ni·i=c·d!.\nUltimately, the idea is that all of these sets Xwill form a new collection of sets whose elements\nnow all have the same size.\nAs previously mentioned, for each i, we are ultimately interested in selections of precisely\nr′\ni:=ri\naiof the new size- aisets. To ensure that no more than r′\nisets are picked, we introduce\nr′\ninew elements Ei:=\u001a\nei\n1, . . . , ei\nr′\ni\u001b\n, and extend each set in our new collection of sets Gr\niby\none of these elements as follows:\nFor each i∈Ir, let\nZr\ni:=n\n{e} ∪X|e∈Ei, X∈ Gr\nio\n.\nWith this modification, each selection of disjoint sets from Zr\nihas size at most |Ei|=r′\ni. Let\nEr:=S\ni∈[d]Eiand let αr:=P\ni∈Irr′\nibe the target number of sets that we aim to select from\nthe new collection of sets.\nWe finish the definition of the (c,r)-join (Ur,Fr)by setting\nFr:=[\ni∈IrZr\ni,and Ur:=U∪Nr∪ Er.\nFor the bound on the size of this new instance, we observe that\n|Er|=X\ni∈Irr′\ni=dX\ni=1\u0018ri\nai\u0019\n⩽d+dX\ni=1ri\nai=d+dX\ni=1ri·i\nc·d!⩽d+dX\ni=1ri\nc(d−1) !⩽d+n\nc(d−1) !,\nwhere the last step follows because ris anF-signature. Moreover, each set in Frhas size\nc·d! + 1and consequently\n|Ur|=|U|+|Nr|+|Er|=n+O(1) + d+n\nc·(d−1) !=n+n\nc·(d−1) !+O(1).(3)\nNext we show that these structured instances preserve packings and partitions.\nLemma 8.7. LetFbe a collection of subsets of a universe U, each with size at most d. Let\nc⩾1be an integer. For each F-signature r, the following two statements are equivalent\n1.Uhas a packing S ⊆ Fsuch that |S ∩ F i|=rifori∈[d].\n2.Urhas a packing Sr⊆ F rof size αrsuch that (Nr∪ Er)⊆ S r.\nMoreover, it holds that [\nA∈SrA!\n\\(Nr∪ Er) = [\nA∈SA!\n.\nProof.Letrbe an F-signature.\n50First direction: 1. implies 2. Suppose Uhas a packing S ⊆ Fwith|S ∩ F i|=rifor\ni∈[d]. For i∈[d], we set Si:=S ∩ F iandsi:=l\nri\naim\n·ai. Let Ibe the set of indices ifor\nwhich Siis nonempty. For i∈I, consider the set Si∪ Fr,i\naddand observe that its size is equal\ntosibecause |Fr,i\nadd|=si−ri. Then, for each i∈I, partition Si∪ Fr,i\naddinto groups of size ai\nto define new sets of size ai·i=c·d!. Here we use the fact that the sets in Si∪ Fr,i\naddare\npairwise disjoint. Let Di\n1. . . , Di\nsibe an enumeration of the sets in Si∪ Fr,i\nadd. To group the\nsets, we set\nS′\ni:=\u001a\nDi\n(j−1)·ai+1∪. . .∪Di\nj·ai|j∈[si/ai]\u001b\n.\nObserve that each set in S′\nihas size ai·i=c·d!,|S′\ni|=si\nai=l\nri\naim\n,S′\ni⊆ Gr\ni, and importantly\nthat sets in S′\niare pairwise disjoint. Moreover,\n[\nA∈S′\niA=[\nA∈(Si∪Fr,i\nadd)A. (4)\nThen we add to each set in S′\nisome distinct element from Ei. Note that this operation is\nwell-defined since |Ei|=|S′\ni|=l\nri\naim\n. Explicitly, for an enumeration T1, . . . , Tlri\naimof the sets\ninS′\ni, let\nTi:=\u001a\n{ei\nj} ∪Tj|1⩽j⩽\u0018ri\nai\u0019\u001b\n.\nIt follows that each element of Tihas size c·d! + 1,|Ti|=l\nri\naim\n,Ti⊆ Z(r)\ni, and importantly\nthat sets in Tiare pairwise disjoint. Moreover,\n[\nA∈TiA=Ei∪\n[\nA∈S′\niA\n. (5)\nBy (4) and (5), it follows that Sr:=S\ni∈ITisatisfies\n1.Sr⊆ F r,\n2. sets in Srare pairwise disjoint,\n3. the union of the sets in Sris\n[\nA∈SrA=[\ni∈I[\nA∈TiA= [\nA∈SA!\n∪Nr∪ Er.\nThis implies that Sr⊆ F ris a packing of Ur. Finally, the size of Sris\n|Sr|=X\ni∈I|Ti|=X\ni∈I\u0018ri\nai\u0019\n=dX\ni=1\u0018ri\nai\u0019\n=αr.\n51Second direction: 2. implies 1. Now suppose that Urhas a packing Sr⊆ F rof size αr\nsuch that (Nr∪Er)⊆ S r. Let Ir:={i∈[d]|ri̸= 0}. Observe that for i∈Ir,|Sr∩Zr\ni|⩽l\nri\naim\nbecause each set in Zr\nicontains an element of Ei. However, this implies that Sr∩ Zr\nihas\nexactly r′\ni:=l\nri\naim\nelements as |Sr|=αr.\nSo, for each i∈Ir, there are pairwise disjoint sets Xi\n1, . . . , Xi\nr′\ni∈ Gr\nisuch that\nSr∩ Zr\ni=\u001a\n{ei\nj} ∪Xi\nj|j∈[r′\ni]\u001b\n.\nFurthermore, by definition of the set Gr\ni, each Xi\njis a union of aidisjoint sets of size i\nthat belong to the set Fi∪ Fr,i\nadd. Let κ(Xi\nj)denote these aisets. For each i∈Ir, let\nSi={A|A∈κ(Xi\nj), j∈[r′\ni]}. Then we have Si⊆ F i∪Fr,i\nadd,|Si|=l\nri\naim\n·ai=si, and the sets\ninSiare pairwise disjoint. Recall that Srhas the property that (Nr∪ Er)⊂ S r. Therefore,\nfor each i∈Irit holds that Fr,i\nadd⊆ Si. Since |Fr,i\nadd|=si−ri, it also holds that\n|Si\\ Fr,i\nadd|=|Si| − |Fr,i\nadd|=si−(si−ri) =ri.\nFinally, for S:=S\ni∈Ir\u0010\nSi\\ Fr,i\nadd\u0011\n, it follows that\n|S ∩ F i|=|Si\\ Fr,i\nadd|=ri.\nFurthermore, it is straightforward to verify that\u0000S\nA∈SA\u0001\n=\u0000S\nA∈SrA\u0001\n\\(Nr∪ Er).The\ncombination of this observation and the fact that Sris a packing of Urimplies that Sis a\npacking of U.\nNote that the assertion in Lemma 8.7 remains valid when considering partitions instead\nof packings. This is a consequence of Ur=U∪Nr∪ Erand the second part of Lemma 8.7.\nCorollary 8.8. LetFbe a collection of subsets of a universe U, each with size at most d. Let\nc⩾1be an integer. For each F-signature r, the following two statements are equivalent\n1.Uhas a partition S ⊆ Fsuch that |S ∩ F i|=rifori∈[d].\n2.Urhas a partition Sr⊆ F r.\n8.3 Reductions Based on Structured Families\nLemma8.9. If⩽∗-Set Partition (#Sets) is2n-hard, then ⩽∗-Set Partition is2n-hard.\nProof.Let(U,F, t)be an n-element instance of ⩽d-Set Partition (#Sets) for some d⩾1\nand let cbe the smallest integer such that (2−ε)1\nc·(d−1) !<(1 +ε\n2). Moreover, let Bbe an\nalgorithm that solves ⩽h-Set Partition forh:= (c·d! + 1)in time (2−ε)n·nO(1). Finally\ndefine ε′:=ε2\n2.\n52Algorithm for ⩽d-Set Partition (#Sets) and its correctness. Given an instance\n(U,F, t)of⩽d-Set Partition (#Sets) , the algorithm starts by guessing an F-signature r\nsuch that w(r)⩽t. For each such r, the algorithm constructs the (c,r)-join of (U,F), denoted\nby(Ur,Fr). Note that each set in Frhas size (c·d! + 1) = h, hence (Ur,Fr)is an instance of\n⩽h-Set Partition . Finally, the algorithm returns YESif any of the calls B(Ur,Fr)returns\nYES, otherwise it returns NO.\nTo establish the correctness of the algorithm, assume that (U,F, t)is a YES-instance.\nThen Uhas a partition S ⊆ Fof size at most t. Construct the F-signature rby defining the\ncoordinate ri:=|S ∩ F i|for1⩽i⩽dand observe that w(r) =|S|⩽t. By definition, S\nsatisfies Item 1 in Corollary 8.8 for this specific F-signature r, therefore it holds that Urhas a\npartition Sr⊆ F r. Therefore, B(Ur,Fr)and in turn the algorithm defined above returns YES.\nOn the other hand, if (U,F, t)is aNO-instance, then Uhas no partition S ⊆ Fof size at\nmost t. Therefore for any F-signature rguessed by the algorithm, Item 1 in Corollary 8.8 will\nnot hold. This implies that Item 2 does not hold as well. As a result B(Ur,Fr)returns NO.\nAll in all, the algorithm described above returns NOas well.\nRunning time. The number of F-signatures is at most\u0000n\nd\u0001d⩽nd2=nO(1). Constructing\nthe instance (Ur,Fr)takes polynomial time in n. Therefore, the total running time is\nnO(1)·max\nF-signature r(2−ε)|Ur|=nO(1)·(2−ε)n+n\nc·(d−1) !\n=nO(1)·(2−ε)n·\u0010\n1 +ε\n2\u0011n\n=nO(1)·\u0012\n2−ε2\n2\u0013n\n=nO(1)·\u0000\n2−ε′\u0001n\nwhere the first equality holds by (3) and the second equality holds because of the definition\nofc.\nLemma 8.10. If⩽∗-Set Partition is2n-hard, then =∗-Set Partition is2n-hard.\nProof.Letd⩾1and(U,F)be an instance of ⩽d-Set Partition . Moreover, let cbe the\nsmallest integer such that (2−ε)1\nc·(d−1) !<(1 +ε\n2)andBbe an algorithm that solves =h-Set\nPartition forh:= (c·d! + 1)in time (2−ε)n·nO(1). Finally define ε′:=ε2\n2.\nAlgorithm for ⩽d-Set Partition and its correctness. Given an instance (U,F)of\n⩽d-Set Partition , the algorithm starts with guessing an F-signature r. For each such r,\nthe algorithm constructs the (c,r)-join of (U,F), denoted by (Ur,Fr). Note that each set in\nFrhas size (c·d! + 1) = h, hence (Ur,Fr)is an instance of =h-Set Partition . Finally, the\nalgorithm returns YESif any of the calls B(Ur,Fr)returns YES, otherwise it returns NO.\nTo show that the algorithm is correct, assume that (U,F)is aYES-instance. Then Uhas\na partition S ⊆ F. Construct the F-signature rby defining ri:=|S ∩ F i|for1⩽i⩽d. By\ndefinition, Ssatisfies Item 1 in Corollary 8.8 for this specific F-signature r, therefore it holds\nthatUrhas a partition Sr⊆ F r. Therefore B(Ur,Fr)and in turn the algorithm defined above\nreturns YES.\nOn the other hand, if (U,F)is aNO-instance, then Uhas no partition Ssuch that S ⊆ F.\nTherefore for any F-signature rthe algorithm tries, Item 1 in Corollary 8.8 will not hold.\n53This implies that Item 2 does not hold as well, consequently B(Ur,Fr)returns NO. Hence, the\nalgorithm described above returns NO.\nRunning time. The number of F-signatures is at most\u0000n\nd\u0001d⩽nd2=nO(1). Constructing\nthe instance (Ur,Fr)takes polynomial time in nas well. Therefore, the whole running time\nbecomes\nnO(1)·max\nF-signature r(2−ε)|Ur|=nO(1)·(2−ε)n+n\nc·(d−1) !\n=nO(1)·(2−ε)n·\u0010\n1 +ε\n2\u0011n\n=nO(1)·\u0012\n2−ε2\n2\u0013n\n=nO(1)·\u0000\n2−ε′\u0001n\nwhere the first equality holds by (3) and the second equality holds because of the definition\nofc.\nLemma 8.11. If⩽∗-Set Packing (#Sets) is2n-hard, then =∗-Set Packing (#Sets) is\n2n-hard.\nProof.Let(U,F, t)be an instance of ⩽d-Set Packing (#Sets) for some d⩾1and let cbe\nthe smallest integer such that (2−ε)1\nc·(d−1) !<(1 +ε\n2). Moreover, let Bbe an algorithm that\nsolves =h-Set Packing (#Sets) forh:= (c·d! + 1)in time (2−ε)n·nO(1). Finally define\nε′:=ε2\n2.\nAlgorithm for ⩽d-Set Packing (#Sets) and its correctness. Given an instance\n(U,F, t)of⩽d-Set Packing (#Sets) , the algorithm starts with guessing an F-signature\nrof weight at least t. For each such r, the algorithm constructs the (c,r)-join of (U,F),\ndenoted by (Ur,Fr). Note that each set in Frhas size (c·d! + 1) = h, hence (Ur,Fr, αr)is\nan instance of =h-Set Packing (#Sets) . Finally, the algorithm returns YESif any of the\ncallsB(Ur,Fr, αr)returns YES, otherwise it returns NO.\nTo show that the algorithm is correct, assume that (U,F, t)is aYES-instance. Then Uhas\na packing S ⊆ Fof size at least t. Construct the F-signature rby defining ri:=|S ∩ F i|for\n1⩽i⩽dand observe that w(r) =|S|⩾t. By definition, Ssatisfies Item 1 in Lemma 8.7\nfor this specific F-signature r, therefore it holds that Urhas a packing Sr⊆ F rof size αr.\nTherefore B(Ur,Fr, αr)and in turn the algorithm defined above returns YES.\nOn the other hand, if (U,F, t)is aNO-instance, then Uhas no packing S ⊆ Fof size at\nleast t.Therefore for any F-signature rthe algorithm tries, Item 1 in Lemma 8.7 will not hold.\nThis implies that Item 1 does not hold as well, consequently B(Ur,Fr, αr)returns NO. Hence,\nthe algorithm described above returns NO.\nRunning time. The number of F-signatures is at most\u0000n\nd\u0001d⩽nd2=nO(1). Constructing\nthe instance (Ur,Fr, αr)takes polynomial time in nas well. Therefore, the whole running\n54time becomes\nnO(1)·max\nF-signature r(2−ε)|Ur|=nO(1)·(2−ε)n+n\nc·(d−1) !\n=nO(1)·(2−ε)n·\u0010\n1 +ε\n2\u0011n\n=nO(1)·\u0012\n2−ε2\n2\u0013n\n=nO(1)·\u0000\n2−ε′\u0001n\nwhere the first equality holds by (3) and the second equality holds because of the definition\nofc.\n9△-Packing and△-Partition\nIn this section we give the proof of Theorem 1.5. In Theorem 9.3 we show that a fastalgorithm\nfor△-Partition violates the SCC, where the technical details will be discussed in Section 9.1.\nSimilarly, in Section 9.2 we prove Theorem 9.4, which shows that if the SCC fails, then there\nexists afastalgorithm for △-Packing problem. All in all, since △-Packing problem is a\ngeneralization of △-Partition , Theorems 9.3 and 9.4 together imply that Theorem 1.5 holds.\n9.1 Reducing Set Partition to△-Partition\nToproceed, werequireatechnicalresultassertingthatwecanassumethevalueof din=d-Set\nPartition to be divisible by 3.\nLemma 9.1. Suppose for some ε >0it holds that, for every d⩾1there is an algorithm that\nsolves every n-element instance of =(3·d)-Set Partition in time (2−ε)n·nO(1). Then for\nevery d⩾1there is an algorithm that solves every n-element instance of =d-Set Partition\nin time (2−ε)n·nO(1).\nProof.Let(U,F)be an n-element instance of =d-Set Partition for some d⩾1. Note\nthat we can assume that ddivides n. Moreover, let Bbe an algorithm that solves =h-Set\nPartition forh:= 3·din time (2−ε)n·nO(1).\nThe algorithm is very similar to the ones given in Lemmas 8.9 to 8.11. Given an instance\n(U,F)of=d-Set Partition , let0⩽s⩽2denote the integern\ndmodulo 3. The algorithm\nconstructs eFby incorporating (3−s)pairwise disjoint sets of size d, each of which is also\ndisjoint from U, into the existing set F. LetAdenote the newly introduced sets and define\nR:=SA. The new universe U′is equal to U∪R, i.e., it has n+d(3−s)⩽n+ 3delements.\nFinally, the algorithm constructs F′by adding the union of all pairwise disjoint sets A, B,\nandCwhere A, B, C ∈eF. Observe that each set in F′has size equal to 3·d. In the end, the\nalgorithm returns YESifB(U′,F′)returns YES.\nTo show that the algorithm is correct, assume that (U,F)is aYES-instance. Then, Uhas\na partition S ⊆ F. Observe that the size of S ∪ Ais equal to 0 modulo 3 since |A|= 3−s.\nGroup the sets in S ∪ Ainto groups of size 3 and take the union of the sets in each group.\nLetS′be the resulting set. It is easy to verify that S′⊆ F′and the union of the sets in S′is\nequal to U∪R=U′. Hence (U′,F′)is also a YES-instance and the algorithm returns YES.\n55Similarly, if (U,F)is a no instance, meaning that Uhas no partition S ⊆ F, adding sets\ndisjoint from those in Fdoes not create a set system where the universe can be partitioned.\nHence, since B(U′,F′)returns NO, our algorithm returns NOas well.\nConstructing the instance (U′,F′)takes polynomial time in n. Therefore, the whole run-\nning time becomes\nnO(1)·(2−ε)|U′|=nO(1)·(2−ε)n+3·d=nO(1)·(2−ε)n.\nThe following corollary can be easily deduced from Theorem 2.6 and Lemma 9.1.\nCorollary 9.2. For all ε >0, there exists d⩾1such that there is no algorithm that solves\nevery n-element instance of =(3·d)-Set Partition in time (2−ε)n·nO(1), unless the SCC\nfails.\nTheorem 9.3. For all ε >0, there exists σ, δ⩾1such that there is no algorithm that solves\nevery n-vertex instance of △-Partition given with a (σ, δ)-hub of size at most pin time\n(2−ε)p·nO(1), unless the SCC fails.\nProof.Suppose there exists ε >0such that for all σ, δ⩾1there exists an algorithm that\nsolves every n-vertex △-Partition instance given with a (σ, δ)-hub of size at most pin time\n(2−ε)p·nO(1). Fix arbitrary d⩾1and denote r= 3×d. Consider an instance (U,F)\nof=r-Set Partition , where U= [n]. In the following, we will describe an algorithm that\nsolves (U,F)in time (2−ε)n·nO(1), which will contradict the SCC by Corollary 9.2.\nEquality Gadget. A△-eqgadgetis a graph with a set of designated vertices called portals\nthat has exacly two triangle packings that cover all non-portal vertices:\n•one that also covers all portals (i.e., it is a triangle partition in the gadget), and\n•one that covers no portal.\nA△-eqgadget behaves similarly to gadgets that we introduced for variants of q-Coloring ,\nbutthistimeforthe △-Partition problem. Whenconstructinganinstance Gof△-Partition ,\nonly the portal vertices of the gadget will have neighbors outside the gadget. Consequently,\nin any triangle packing of G, all portals of the △-eqgadget will be in the same state: either\nthey are all covered by triangles contained inside the gadget, or none of them is covered by\nsuch a triangle. This explains why we use the name equality gadget .\nNow let us show how to construct a △-eqgadget Zwith rportals and 4rvertices in\ntotal; see Figure 4. Introduce four sets of vertices P={p0, . . . , p r−1},Q={q0, . . . , q r−1},\nA={a0, . . . , a r−1}, and B={b0, . . . , b r−1}. All arithmetic iterations on indices of these\nvertices are performed modulo r.\nThe vertices from AandBform a cycle with concecutive vertices a1, b1, a2, b2, . . . , a r, br.\nThen, for each i∈[r], we create a triangle on vertices ai, bi, pi; letP1denote the set of such\ntriangles. Similarly, for each i∈[r], we create a triangle qi, bi, ai+1; letP2denote set of these\ntriangles. Finally, for each i∈[r\n3−1], we create a triangle on vertices q3i+1, q3i+2, q3i+3; call\nthe set of these triangles P3. This completes the construction of ZandPis the set of portal\nvertices.\nLet us argue that Zindeed has the property of a △-eqgadget. Let Pbe a triangle packing\ninZthat covers all non-portal vertices, i.e., A∪B∪Q. Suppose Pcontains a triangle from\n56a1\nbr\nar\n...\nb1\na2\np2\na4\np1\nq1\np3\n: : :\nqr\npr\nq3\nb2\na3\nb3\nq2Figure 4: The construction of the △-eqgadget with rportal vertices. The blue, orange, and\nviolet triangles, with dotted, dashed, and dotted-dashed edges respectively, belong to P1,P2,\nandP3respectively.\nP1, say, pi, ai, bifor1⩽i⩽r. In that case, since Pcovers ai+1, it must also contain the\ntriangle pi+1, ai+1, , bi+1. By induction, it is easy to show that Pshould include all triangles\ninP1. The remaining vertices, i.e., Qmust be covered by the triangles in P3. Therefore, in\nthis case all the portal vertices of Zare covered by P.\nAssume now that Phas no triangle from P1. In particular, no portal vertices are covered\nbyP. We need to argue that such Pexists and is unique. Note that in order to cover all\nnon-portal vertices, Pmust contain all the triangles in P2. Hence the constructed graph is\nindeed a △-eqgadget.\nConstruction of the graph G.Given a =r-Set Partition instance (U,F), let us create\nan instance Gof△-Partition as follows. First, create nvertices VU={v1, . . . , v n}, where\neach vicorresponds to i∈U. Then, for each S∈ F, create a △-eqgadget, denoted by △-\neq(S), with rportal vertices, and identify the portal vertices of △-eq(S)with corresponding\nvertices of VU. Let Gbe the resulting graph obtained by applying these operations. Note that\nGhasO(nr)vertices and VUis a(σ, δ)-hub of Gof size n, where σ= 4randδ=r.\nEquivalence of instances. Suppose Uhas a partition S ⊆ F. For each S∈ S, define\nVS={vi|i∈S}. Note that since Sis a partition of U, sets VSform a partition of the\nvertices VU. For S∈ S, letTSbe a triangle partition of △-eq(S)which covers all its portal\nvertices, i.e., VS; it exists by the definition of △-eqgadget. For S∈ F \\S, letTSbe a triangle\npartition of △-eq(S)which covers no portal vertices; again, it exists by the definition of △-eq\ngadget. Note thatS\nS∈FTSis a triangle partition of the whole graph G.\nNow, suppose Ghas a triangle partition T. Let us call a copy △-eq(S)of the △-eqgadget\ninGactiveif there is no triangle that intersects the gadget, but it not contained in the gadget.\nRecall that this means that each portal of △-eq(S)is a part of some triangle not contained in\n△-eq(S). Define S={S∈ F | △ -eq(S)is active }. With a similar reasoning as in the previous\nparagraph, it is straightforward to verify that Sis a partition of U. Summing up, (U,F)is a\nYES-instance of =(3·d)-Set Partition if and only if Gis aYES-instance of △-Packing .\n57Running Time. Building the graph Gtakes time polynomial time in n. Moreover, as\nmentioned before, GhasO(nr)vertices and VUis a(4r, r)-hub of Gof size n. Therefore, the\nassumed algorithm for =r-Set Partition , one can solve the instance (U,F)in time\n(2−ε)|VU|· |V(G)|O(1)= (2−ε)n·nO(1).\nThis completes the proof.\n9.2 Reducing △-Packing toSet Packing\nThe goal of this section is to prove Theorem 9.4.\nTheorem 9.4. Suppose the SCC fails. Then there exists ε >0such that, for all σ, δ⩾1,\nthere is an algorithm that solves every n-vertex instance of △-Packing given with a (σ, δ)-hub\nof size pin time (2−ε)p·nO(1).\nIn order to simplify the description of the proof, we split it into two steps. First, we show\nhow we can use a fast algorithm for ⩽d-Set Packing (#Sets) to solve an auxiliary variant\nof the problem called c-Precolored- △-Packing . Then, we use some color-coding idea to\nshow how an algorithm for c-Precolored- △-Packing can be used to solve △-Packing .\nLet us start with some definitions and notation. Let Gbe a graph and Q⊆V(G). Let\nCdenote the set of components of G−Qand let Ddenote the set of triangles of Gthat are\ncontained in Q. Let Πbe a collection of pairwise vertex-disjoint triangles in G. We say that\nC∈ Cisactive(with respect to Π) if there is a triangle in Πthat intersects both CandQ.\nA triangle in Disactiveif it belongs to Π.\nFor a function ψ:C ∪ D → N, byCi(resp., Di) we denote C ∩ψ−1(i)(resp., D ∩ψ−1(i)).\nFor a constant c, an instance of c-Precolored- △-Packing is a quadruple (G, t, Q, ψ ),\nwhere Gis graph, tis an integer, Qis a subset of V(G), and ψis a function that maps\nelements of C ∪Dto numbers in\b\n1, . . . ,\u0006\n|Q|/c\u0007\t\n(hereCandDare defined as in the previous\nparagraph). We call the numbers in the codomain of ψcolorsand call ψacoloring; we\nemphasize that this is an arbitrary coloring and has no correctness criterion. We ask whether\nGadmits a packing of at least ttriangles such that, for each color i, at most celements of\nCi∪ Diare active (the sets CiandDiare defined with respect to ψ).\nLemma 9.5. Suppose there is ε′>0such that for all d,⩽d-Set Packing (#Sets) with\nn-element universe can be solved in time (2−ε′)n·nO(1).\nThen there is ε′′>0such that for every σ, δ⩾1there is c0depending only on ε′′andσ, so\nthat for every c⩾c0, every instance (G, t, Q, ψ )ofc-Precolored- △-Packing , where Qis\na(σ, δ)-hub of size p, can be solved in time (2−ε′′)p· |V(G)|O(1).\nProof.Letσ, δbe fixed constants. Without loss of generality assume that σ⩾2. Let\nε′′=ε′/4 (6)\nand let c0be the smallest integer that satisfies both\n(c0σ+ 1)1/c0⩽(1 +ε′′) (7)\n(2−ε′)1/c0⩽(1 +ε′′); (8)\n58and pick any c⩾c0; clearly csatisfies these inequalities too. This choice will become clear\nlater in the proof. Note that ε′′depends only on ε′andc0depends only on ε′andσ.\nLet(G, t, Q, ψ )be an instance of c-Precolored- △-Packing , where Qis a(σ, δ)-hub of\nsizep. By introducing at most c−1isolated vertices to Q, we can assume that cdivides\np; note that these dummy vertices contribute to the running time only by a constant factor.\nWe will use the assumed algorithm for ⩽d-Set Packing (#Sets) ford= 2cσ+ 1to solve\n(G, t, Q, ψ )in time (2−ε′′)p· |V(G)|O(1).\nLetℓ=p/cdenote the number of colors used by ψ. The sets C,DandCi,Diover all i∈[ℓ]\nare defined as previously.\nFix some (unknown) optimum solution Π, i.e., a largest triangle packing in Gthat has at\nmost cactive elements in each color. A contribution of a component C∈ Cis the number\nof triangles in Πthat intersect C(they can either be contained in C, or have some vertices\ninCand some in Q). The contribution of a triangle in Dis 1 if this triangle is in Πand 0\notherwise. A contribution of a color i∈[ℓ]is the total contribution of all elements in Ci∪ Di.\nIn the algorithm, for each color i, we would like to exhaustively guess the contribution\nof this color. However, the numbers involved might be very large, as Cimight contain many\ncomponents. Thus we are going to do this indirectly.\nFix a color i∈[ℓ]. Let Xibe the number of triangles of a maximum triangle packing of\nthe graph induced by the components in Ci. Note that Xican be computed in time linear\nin|V(G)|as each component in Cihas size at most σand there are at most |V(G)|such\ncomponents. A maximum triangle packing can be obtained as the union of maximum triangle\npackings of the individual components.\nNote that the contribution of the color iis at least Xiand at most Xi+cσ. Indeed, on one\nhand, we can pick Xitriangles that are contained in the components in Ci, without interfering\nwith any object of any other color. On the other hand, each active triangle in Dicontributes\n1 to the sum, while each of at most σvertices of each active component in Cican be present in\nat most one triangle intersecting Q. Since there at at most cactive objects, we get the upper\nbound.\nSo, instead of guessing the contribution of each color i∈[ℓ]directly, we guess the offset\nqiof this value against Xi. Formally, the algorithm iterates over all tuples of the form q=\n(q1, . . . , q ℓ)∈ {0, . . . , cσ }ℓthat satisfyP\ni∈[ℓ](Xi+qi)⩾t. Note that if no such tuple exists,\nthe discussion above yields that (G, t, Q, ψ )is a NO-instance. Observe that the number of\nchoices for qis at most\n(cσ+ 1)ℓ= (cσ+ 1)p/c(7)\n⩽(1 +ε′′)p. (9)\nFix one such tuple q= (q1, . . . , q ℓ)and consider a color i∈[ℓ]. We say that a subset\nS⊆Qof vertices from the hub is i-validif it has size at most 2cσand the graph induced by\nStogether with the components in Cihas a triangle packing ΠSwith the following properties:\n•at most celements of Di∪ Ciare active w.r.t. ΠS,\n•all triangles of ΠScontained in Qare in Di,\n•the number of triangles in ΠSis at least Xi+qi.\nIntuitively, a set Sisi-valid if it is compatible with the choice of qi. The size bound comes\nfrom the fact that each active triangle in Diuses three vertices from Q, while the triangles\nintersecting each active component from Ciintersect at most 2σvertices of Q. As σ⩾2and\nthere are at most cactive elements in Di∪ Ci, the bound follows.\n59Claim 9.5.1. For each color iand offset qi, the i-valid sets can be enumerated in time poly-\nnomial in |V(G)|.\nProof of Claim. There are at most O(p2cσ) =|V(G)|O(1)subsets of Qof size at most 2cσ. As\neach such candidate set Sis of constant size, in polynomial time we can enumerate all possible\ntriangle packings in which every triangle intersects S. Furthermore, in polynomial time, each\nsuch packing can be extended in an optimal way by picking triangles contained in Ci; this is\npossible as each component in Ciis of constant size. It is straightforward to verify that Sis\ni-valid if and only if any of the packings obtained that way satisfies the conditions for ΠS.◁\nFor each tuple q, we can now define an instance (U,Fq, ℓ)of⩽d-Set Packing (#Sets)\nas follows. The universe Uof the instance consists of all vertices in the hub Qtogether with\na distinct element aiper each color i∈[ℓ], so in total |U|=p+ℓ=p+p/c. The set system\nFqcontains, for each color i∈[ℓ]and for each i-valid set S⊆Q, the set S∪ {ai}. Note\nthat each set in Fqis of size at most d= 2cσ+ 1, and the instance can be constructed in\ntime polynomial in |V(G)|. The algorithm iterates over all such instances and executes the\npreviously-mentioned algorithm for ⩽d-Set Packing (#Sets) on each of these instances. If\nin any of the calls we obtain a positive answer, we report that (G, t, Q, ψ )is aYES-instance,\nand otherwise we reject it.\nCorrectness The correctness of the algorithm follows from the following two claims.\nClaim9.5.2. If there is qfor which (U,Fq, ℓ)is aYES-instance of ⩽d-Set Packing (#Sets) ,\nthen (G, t, Q, ψ )is aYES-instance of c -Precolored- △-Packing .\nProof of Claim. LetF∗⊆ F qbe a family ℓpairwise disjoint sets. Since each set in Fqcontains\none of the elements {a1, . . . , a ℓ}, the solution F∗contains, for each color i, precisely one set\nof the form Si∪ {ai}. Note that Simay be empty. For each such set, let ΠSibe a triangle\npacking defined as above. We claim thatS\ni∈[ℓ]ΠSiis a triangle packing that witnesses that\n(G, t, Q, ψ )is aYES-instance.\nClearly the triangles within one set ΠSiare pairwise disjoint. Now consider two triangles\nfrom distinct sets, say ΠiandΠj. As sets of components CiandCjare pairwise disjoint, these\ntwo triangles may overlap only on Q, which means that Si∩Sj̸=∅. However, this is not\npossible as sets in Fare pairwise disjoint.\nNext, by the conditions in the definition of i-valid set, we observe that at most celements\nfromCi∪Diare active. Finally, the total number of triangles is at leastP\ni∈[ℓ](Xi+qi)which\nis at least tby the choice of q. ◁\nClaim 9.5.3. If(G, t, Q, ψ )is a YES-instance of c -Precolored- △-Packing , then there is\nqfor which (U,Fq, ℓ)is aYES-instance of ⩽d-Set Packing (#Sets) .\nProof of Claim. Suppose there exists a triangle packing ΠofGof size at least t, that has\nat most cactive elements of each color. Fix color i∈[ℓ]and let Πibe the subfamily of\nΠconsisting of triangles that either are in Di, or intersect Ci. Clearly the sets Πover all\ni∈[ℓ]form a partition of Π. Recall that we safely assume that Xi⩽|Πi|⩽Xi+cσDefine\nqi:=|Πi| −Xi. Note that q= (q1, . . . , q ℓ)is one of the tuples considered by our algorithm.\nFori∈[ℓ], letSi⊆Qbe the set consisting of vertices of triangles in Πi. The packing Πiis\na witness that Siis an i-valid set (with respect to q). Furthermore, sets Si∪{ai}are pairwise\ndisjoint. Thus, (U,Fq, ℓ)is aYES-instance of ⩽d-Set Packing (#Sets) . ◁\n60Runtime The algorithm iterates over all possible choices of q, for each choice it builds an in-\nstance (U,Fq)inpolynomialtime,andthencallstheassumedalgorithmfor ⩽d-Set Packing (#Sets) .\nAs|U|and|Fq|are bounded by a polynomial function of |V(G)|, and by (9), the total running\ntime is bounded by\n(1 +ε′′)p·(2−ε′)|U|· |V(G)|O(1)⩽(1 +µ)p·(2−ε′)p(1+1/c)· |V(G)|O(1)\n⩽\u0010\n(1 +ε′′)·(2−ε′)·(2−ε′)1/c\u0011p\n· |V(G)|O(1)(8)\n⩽\u0000\n(1 +ε′′)2·(2−ε′)\u0001p· |V(G)|O(1)\n(6)\n⩽(2−ε′′)p· |V(G)|O(1),\nas claimed. This completes the proof.\nNow let us argue that we can reduce solving △-Packing (where the instance is given with\na hub) to c-Precolored- △-Packing for suitably chosen c.\nIn this reduction we use some color coding idea. We will use the so-called splitters. For\nintegers N,p, and ℓwith N⩾p, an(N, p, ℓ )-splitter Ψis a family of functions from [N]to[ℓ]\nsuch that, for each subset Sof[N]with size |S|=p, there is an ψ∈Ψthat assigns colors from\n[ℓ]as evenly distributed as possible to the elements of S— formally, for (pmod ℓ)colors j∈[ℓ]\nit holds that |ψ−1(j)|=⌈p/ℓ⌉, and for the remaining colors we have |ψ−1(j)|=⌊p/ℓ⌋. The\nmethods of constructing splitters were introduced by Naor, Schulman, and Srinivasan [30].\nTheorem 9.6 ([30, Theorem 3 (ii)]) .Letℓ=ω(p)and let f(p, ℓ) = (2 πp/ℓ)ℓ/2eℓ2/(12p). An\n(N, p, ℓ )-splitter of size s=O(f(p, ℓ)1+o(1)logN)can be computed in time polynomial in s\nandN. Here the o(1)in the exponent is for ℓ/√pgoing to infinity.\nCorollary 9.7. For every ε >0there is c1such that for each N,pwith p⩽N,c⩾c1, an\n(N, p,⌈p/c⌉)-splitter can be computed (and iterated over) in time (1 +ε)p·NO(1).\nProof.Letc1be the smallest integer such that (2πc1)4/c1⩽(1 + ε). Choose N, p⩽N,\nandc⩾c1. Note that ℓ:=⌈p/c⌉=ω(√p). Thus, by Theorem 9.6, there is an (N, p, ℓ )-\nsplitter of size s=O(f(p, ℓ)1+o(1)logN)that can be computed in time polynomial in Nand\ns, where f(p, ℓ) = (2 πp/ℓ)ℓ/2eℓ2/(12p). The o(1)in the exponent is for ℓ/√p=⌈p/c⌉/√p\ngoing to infinity, i.e., it is for pgoing to infinity. If pis smaller than some constant the size\nof the splitter is trivially O(logN). So we can assume that pis sufficiently large such that\nf(p, ℓ)1+o(1)⩽f(p, ℓ)2andp > c. Since p > cwe have ⌈p/c⌉⩽2p/c. Then\nf(p, ℓ) = (2 πp/ℓ)ℓ/2eℓ2/(12p)⩽(2πc)p/ce4p2/(12pc2)⩽(2πc)2p/c.\nUsing the fact that (2πc)4/c⩽(1 +ε), it follows that\nf(p, ℓ)1+o(1)⩽f(p, ℓ)2⩽(2πc)4p/c⩽(1 +ε)p.\nSo we can compute an (N, p, ℓ )-splitter and then iterate over its members in time (1 +ε)p·\nNO(1).\nLet us proceed to the proof of the following result.\nLemma 9.8. Suppose there is ε′′>0such that for every σ, δ⩾1there is c0depending only on\nε′′andσ, so that for every c⩾c0, every instance (G, t, Q, ψ )ofc-Precolored- △-Packing ,\nwhere Qis a(σ, δ)-hub of size p, can be solved in time (2−ε′′)p· |V(G)|O(1).\nThen there is ε >0such that for every σ, δ⩾1, every n-vertex instance of △-Packing given\nwith a (σ, δ)-hub of size pcan be solved in time (2−ε)p·nO(1).\n61Proof.Define ε=ε′′/3. Let c0be as in the assumption of the lemma, and let c1be as in\nCorollary 9.7 (for ε). Let c= max( c0, c1).\nFixσandδ,andconsideran n-vertexinstance (G, t)of△-Packing ,givenwitha (σ, δ)-hub\nQof size p. Again, by adding at most c−1isolated vertices to the hub, we can assume that\ncdivides p. Define ℓ=p/c. LetCbe the set of components of G−Q, and let Dbe the set of\ntriangles contained in Q. Define N=|C+D|=O(n3).\nLetΨbe the (N, p, ℓ )-splitter given by Corollary 9.7 for N, p, c. We claim that (G, t)is\naYES-instance of △-Packing if and only if there exists ψ∈Ψsuch that (G, t, Q, ψ )is a\nYES-instance of c-Precolored- △-Packing .\nThe backwards implication is trivial; let us show the forward one. Suppose there is a\ntriangle packing ΠinG, consisting of at least ttriangles. Let C∗andD∗be the elements of C\nandD, respectively, that are active with respect to Π. Notice that |C∗∪D∗|⩽p. Indeed, each\nactive element of C ∪ Duses at least one vertex of Qand these vertices are pairwise distinct.\nLetSbe a subset of D ∪ Cof size exactly pthat contains C∗∪ D∗.\nWe interpret elements from Ψas functions from C+D(recall that |C+D|=N) to\n[ℓ] = [p/c]. By the definition of the splitter, there is ψ∈Ψin which each color appears exactly\np\np/c=ctimes on elements of S. Thus, for each color ithere are at most cactive elements of\nD ∪ C. Consequently, (G, t, Q, ψ )is aYES-instance of c-Precolored- △-Packing .\nNow the algorithm is simple. First, it uses Corollary 9.7 to compute Ψ. Then it it-\nerates over all ψ∈Ψ, and calls the assumed algorithm for the instance (G, t, Q, ψ )of\nc-Precolored- △-Packing . If any the calls succeeds, the algorithm reports a YES-instance,\nand otherwise it reports a NO-instance. The correctness follows from the reasoning above.\nThe running time is determined by the number of elements of Ψtimes the time needed to\ncall the algorithm for each instance of c-Precolored- △-Packing , thus it is bounded by\n(1 +ε)p·(2−ε′′)p·nO(1)= (2−ε′′+ 2ε′′/3−ε′′2/3)p·nO(1)⩽(2−ε)p·nO(1),\nas claimed. This completes the proof.\nNow Theorem 9.4 follows directly from the combination of Theorem 2.6, Lemma 9.5, and\nLemma 9.8.\n10 Dominating Set\nIn this section we turn our attention to Dominating Set parameterized by the hub size.\nWe show two quite simple reductions that exclude an algorithm with running time (2−\nε)p· |V(G)|O(1)(where the instance Gis given with a hub of size p) under two complexity\nassumptions: the SETH and the SCC.\nTheorem 10.1. For every ε >0there exists δsuch that the Dominating Set problem on\nn-vertex instances given with a (3, δ)-hub of size pcannot be solved in time (2−ε)p·|V(G)|O(1),\nunless the SCC fails.\nProof.Assume the SCC. Given some ε >0, letdbe the constant given by the SCC for this ε.\nWe reduce from ⩽d-Set Cover , let(U,F)be an instance where |U|=nand|F|=m. We\nstart the construction of the instance GofDominating Set with introducing a set Ywhich\ncontains a vertex yifor every i∈U. Next, for every set F∈ F, we proceed as follows. We\nintroduce a three-vertex path with consecutive vertices aF, bF, cF. Then we add an edge aFyi\n62if and only if i∈F. Denote A={aF|F∈ F},B={bF|F∈ F}, and C={cF|F∈ F}.\nThis completes the construction of G.\nNote that this construction can be performed in time polynomial in n(recall that dis\na constant and thus mis polynomial in n) and the constructed graph has n+ 3m=O(nd)\nvertices. Furthermore, the set Yis a(3, d)-hub of size n.\nWe claim that Ghas a dominating set of size at most k+mif and only if Fcontains at\nmost ksets that cover U.\nFirst, suppose that there is a subfamily F′⊆ Fof size at most ksuch that U=SF′.\nDefine X=B∪{aF|F∈ F′}. Clearly Xis of size k+m. Let us show that it is a dominating\nset.\nNotice that for each F∈ F, the vertices aF, bF, cFare dominated by bF∈X. Now\nconsider any yi∈Y. AsF′covers U, there is F∈ F′such that i∈F. This means that\naF∈Xandyiis dominated by aF.\nFor the other direction, suppose that Ghas a dominating set Xof size at most k+m. We\nclaim that we can assume that B⊆X⊆A∪B. Indeed, consider F∈ F. Notice that in order\nto dominate cF, the set Xmust contain at least one of bF, cF. However, if it contains cF, we\ncan obtain a solution of at most the same size by replacing cFwith bF(ifbF/∈X) or removing\ncF(otherwise). Thus we may assume that B⊆XandC∩X=∅. Now, for contradiction,\nsuppose that there is yi∈Y∩X. The vertex yican only dominate itself or some set of vertices\ninA. However, B⊆Xalready dominates all vertices in A. Hence, X\\ {yi}dominates every\nvertex of G, except possibly yi. So consider any F∈ Fsuch that i∈F; recall that such F\nexists as U=SF. Then the set X\\ {yi} ∪ {aF}is a dominating set of size at most |X|.\nDefine F′={F|aF∈X}; clearly |F′|⩽k. We claim thatSF′=U. Suppose there is\ni∈Uthat is not inSF′. However, this means that yiis not dominated by X, a contradiction.\nSumming up, a hypothetical algorithm that solves Dominating Set onGin time (2−ε)n·\n|V(G)|O(1)can be used to solve the instance (U,F)of⩽d-Set Cover in time (2−ε)n·nO(1),\ncontradicting the SCC.\nIn the ⩽d-Hitting Set problem the instance is a pair (U,F), where Uis a set called the\nuniverse andFis a family of subsets of U, each of size at most d. We ask for a minimum\nhitting set , i.e., a minimum-sized subset of Uthat intersects every element of F.\nTheorem 10.2 (Cygan et al. [9]) .For every ε >0there exists dsuch that the ⩽d-Hitting\nSetwith universe of size ncannot be solved in time (2−ε)n·nO(1), unless the SETH fails.\nWith an argument similar to the proof of Theorem 10.1, we can show the following result.\nTheorem 10.3. For every ε >0there exists δsuch that the Dominating Set problem on\nn-vertex instances given with a (2, δ)-hub of size pcannot be solved in time (2−ε)p·|V(G)|O(1),\nunless the SETH fails.\nProof.Letε >0and let dbe the constant given for that εby Theorem 10.2. We reduce from\n⩽d-Hitting Set . Let (U,F)be a corresponding instance with |U|=n.\nWe construct Gas follows. First, we introduce three sets Y={yi|i∈U},A={ai|i∈\nU}, and B={bi|i∈U}. For each i∈Uwe add edges yiaiandaibi. Next, for each\nF∈ F, we add a vertex zF. We add the edge yizFif and only if i∈F. This completes the\nconstruction of G. Note that |V(G)|=O(nd)andYis a(2, d)-hub.\nWe claim that Ghas a dominating set of size at most n+kif and only if (U,F)admits\na hitting set of size at most k. Let U′⊆Ube a hitting set of size at most k. We define\n63X=A∪{yi|i∈U′}. Clearly Xdominates X∪A∪B. Suppose that there is some zFwhich\nis not adjacent to any vertex in X. This means that Fdoes not intersect U′, a contradiction.\nNow suppose that Ghas a dominating set Xof size at most n+k. Similarly as in the\nproof of Theorem 10.1 we can assume that A⊆X⊆A∪Y. Define U′={i|yi∈X}; clearly\n|U′|⩽k. We claim that U′is a hitting set. Indeed, if there is some F∈ Fwhich is not\nintersected by U′, its corresponding vertex zFis not dominated by X.\nThus a hypothetical algorithm that solves Dominating Set onGin time (2−ε)n·\n|V(G)|O(1)can be used to solve the instance (U,F)of⩽d-Hitting Set in time (2−ε)n·nO(1).\nBy Theorem 10.2, this contradicts the SETH.\nReferences\n[1] Jørgen Bang-Jensen, Eduard Eiben, Gregory Z. Gutin, Magnus Wahlström, and Anders\nYeo. Component order connectivity in directed graphs. Algorithmica , 84(9):2767–2784,\n2022. doi:10.1007/s00453-022-01004-z .\n[2] Andreas Björklund. Exact Covers via Determinants. In Jean-Yves Marion and Thomas\nSchwentick, editors, 27th International Symposium on Theoretical Aspects of Com-\nputer Science , volume 5 of Leibniz International Proceedings in Informatics (LIPIcs) ,\npages 95–106, Dagstuhl, Germany, 2010. Schloss Dagstuhl – Leibniz-Zentrum für In-\nformatik. URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.\nSTACS.2010.2447 ,doi:10.4230/LIPIcs.STACS.2010.2447 .\n[3] F. Boesch, D. Gross, and C. Suffel. Component order connectivity. In Proceedings of the\nTwenty-ninth Southeastern International Conference on Combinatorics, Graph Theory\nand Computing (Boca Raton, FL, 1998) , volume 131, pages 145–155, 1998.\n[4] Glencora Borradaile and Hung Le. Optimal dynamic program for r-domination problems\nover tree decompositions. In Jiong Guo and Danny Hermelin, editors, 11th Interna-\ntional Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26,\n2016, Aarhus, Denmark , volume 63 of LIPIcs, pages 8:1–8:23. Schloss Dagstuhl - Leibniz-\nZentrum für Informatik, 2016. doi:10.4230/LIPIcs.IPEC.2016.8 .\n[5] Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite\ngraphs.Inf. Comput. , 85(1):12–75, 1990. doi:10.1016/0890-5401(90)90043-H .\n[6] Christophe Crespelle, Pål Grønås Drange, Fedor V. Fomin, and Petr A. Golovach. A\nsurvey of parameterized algorithms and the complexity of edge modification. Comput.\nSci. Rev. , 48:100556, 2023. doi:10.1016/j.cosrev.2023.100556 .\n[7] Radu Curticapean, Nathan Lindzey, and Jesper Nederlof. A tight lower bound for\ncounting Hamiltonian cycles via matrix rank. In Artur Czumaj, editor, Proceedings\nof the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA\n2018, New Orleans, LA, USA, January 7-10, 2018 , pages 1080–1099. SIAM, 2018.\ndoi:10.1137/1.9781611975031.70 .\n[8] Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting per-\nfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Robert\n64Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium\non Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016 , pages\n1650–1669. SIAM, 2016. doi:10.1137/1.9781611974331.ch113 .\n[9] Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio\nOkamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On Problems\nas Hard as CNF-SAT. ACM Transactions on Algorithms , 12(3):41:1–41:24, May 2016.\ndoi:10.1145/2925416 .\n[10] Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin\nPilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms . Springer,\n2015. doi:10.1007/978-3-319-21275-3 .\n[11] László Egri, Dániel Marx, and Paweł Rzążewski. Finding list homomorphisms from\nbounded-treewidth graphs to reflexive graphs: a complete complexity characterization. In\nRolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of\nComputer Science, STACS 2018, February 28 to March 3, 2018, Caen, France , volume 96\nofLIPIcs, pages 27:1–27:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.\ndoi:10.4230/LIPIcs.STACS.2018.27 .\n[12] Jacob Focke, Dániel Marx, Fionn Mc Inerney, Daniel Neuen, Govind S. Sankar, Philipp\nSchepper, and Philip Wellnitz. Tight complexity bounds for counting generalized dom-\ninating sets in bounded-treewidth graphs. In Nikhil Bansal and Viswanath Nagara-\njan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms,\nSODA 2023, Florence, Italy, January 22-25, 2023 , pages 3664–3683. SIAM, 2023.\ndoi:10.1137/1.9781611977554.ch140 .\n[13] Jacob Focke, Dániel Marx, and Paweł Rzążewski. Counting list homomorphisms from\ngraphs of bounded treewidth: tight complexity bounds. In Joseph (Seffi) Naor and\nNiv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete\nAlgorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12,\n2022, pages 431–458. SIAM, 2022. doi:10.1137/1.9781611977073.22 .\n[14] Daniel Gross, L. William Kazmierczak, John T. Saccoman, Charles L. Suffel, and Anto-\nnius Suhartomo. On component order edge connectivity of a complete bipartite graph.\nArs Comb. , 112:433–448, 2013.\n[15] Falko Hegerfeld and Stefan Kratsch. Towards exact structural thresholds for parameter-\nized complexity. In Holger Dell and Jesper Nederlof, editors, 17th International Sym-\nposium on Parameterized and Exact Computation, IPEC 2022, September 7-9, 2022,\nPotsdam, Germany , volume 249 of LIPIcs, pages 17:1–17:20. Schloss Dagstuhl - Leibniz-\nZentrum für Informatik, 2022. doi:10.4230/LIPIcs.IPEC.2022.17 .\n[16] Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT.J. Comput.\nSyst. Sci. , 62(2):367–375, 2001. doi:10.1006/jcss.2000.1727 .\n[17] Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have\nstrongly exponential complexity? J. Comput. Syst. Sci. , 63(4):512–530, 2001. doi:\n10.1006/jcss.2001.1774 .\n65[18] Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of\ngraph coloring problems. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos,\neditors,Algorithms and Complexity - 10th International Conference, CIAC 2017, Athens,\nGreece, May 24-26, 2017, Proceedings , volume 10236 of Lecture Notes in Computer Sci-\nence, pages 345–356, 2017. doi:10.1007/978-3-319-57586-5\\_29 .\n[19] Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural parameters,\ntightbounds, andapproximationfor (k, r)-center.Discret. Appl. Math. , 264:90–117, 2019.\ndoi:10.1016/j.dam.2018.11.002 .\n[20] Lawrence William Kazmierczak. On the relationship between connectivity and component\norder connectivity . ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Stevens\nInstitute of Technology. URL: http://gateway.proquest.com/openurl?url_ver=\nZ39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:\npqdiss&rft_dat=xri:pqdiss:3088817 .\n[21] Mikko Koivisto. Partitioning into sets of bounded cardinality. In Jianer Chen and Fe-\ndor V. Fomin, editors, Parameterized and Exact Computation, 4th International Work-\nshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected\nPapers, volume 5917 of Lecture Notes in Computer Science , pages 258–263. Springer,\n2009. doi:10.1007/978-3-642-11269-0\\_21 .\n[22] Mithilesh Kumar and Daniel Lokshtanov. A 2lk kernel for l-component order connec-\ntivity. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on\nParameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Den-\nmark, volume 63 of LIPIcs, pages 20:1–20:14. Schloss Dagstuhl - Leibniz-Zentrum für\nInformatik, 2016. doi:10.4230/LIPIcs.IPEC.2016.20 .\n[23] Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM J. Discret. Math. ,\n34(3):1538–1558, 2020. doi:10.1137/19M1280326 .\n[24] Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of\nbounded treewidth are probably optimal. ACM Trans. Algorithms , 14(2):13:1–13:30,\n2018. doi:10.1145/3170442 .\n[25] Daniel Lokshtanov, Pranabendu Misra, M. S. Ramanujan, Saket Saurabh, and Meirav\nZehavi. Fpt-approximation for FPT problems. In Dániel Marx, editor, Proceedings of the\n2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference,\nJanuary 10 - 13, 2021 , pages 199–218. SIAM, 2021. doi:10.1137/1.9781611976465.14 .\n[26] L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics ,\n13(4):383–390, 1975. URL: https://www.sciencedirect.com/science/article/pii/\n0012365X75900588 ,doi:https://doi.org/10.1016/0012-365X(75)90058-8 .\n[27] Dániel Marx, Pranabendu Misra, Daniel Neuen, and Prafullkumar Tale. A framework\nfor parameterized subexponential algorithms for generalized cycle hitting problems on\nplanar graphs. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete\nAlgorithms (SODA) , pages 2085–2127. [Society for Industrial and Applied Mathematics\n(SIAM)], Philadelphia, PA, 2022. doi:10.1137/1.9781611977073.83 .\n66[28] Dániel Marx, Govind S. Sankar, and Philipp Schepper. Degrees and gaps: Tight complex-\nity results of general factor problems parameterized by treewidth and cutwidth. In Nikhil\nBansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium\non Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow,\nScotland (Virtual Conference) , volume 198 of LIPIcs, pages 95:1–95:20. Schloss Dagstuhl\n- Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPIcs.ICALP.2021.95 .\n[29] Dániel Marx, Govind S. Sankar, and Philipp Schepper. Anti-factor is FPT parameterized\nby treewidth and list size (but counting is hard). In Holger Dell and Jesper Nederlof,\neditors,17th International Symposium on Parameterized and Exact Computation, IPEC\n2022, September 7-9, 2022, Potsdam, Germany , volume 249 of LIPIcs, pages 22:1–22:23.\nSchloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. doi:10.4230/LIPIcs.IPEC.\n2022.22.\n[30] Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal\nderandomization. In 36th Annual Symposium on Foundations of Computer Science, Mil-\nwaukee, Wisconsin, USA, 23-25 October 1995 , pages 182–191. IEEE Computer Society,\n1995. doi:10.1109/SFCS.1995.492475 .\n[31] Jesper Nederlof. Finding Large Set Covers Faster via the Representation Method. In\nPiotr Sankowski and Christos Zaroliagis, editors, 24th Annual European Symposium on\nAlgorithms (ESA 2016) , volume 57 of Leibniz International Proceedings in Informat-\nics (LIPIcs) , pages 69:1–69:15, Dagstuhl, Germany, 2016. Schloss Dagstuhl – Leibniz-\nZentrum für Informatik. URL: https://drops.dagstuhl.de/entities/document/10.\n4230/LIPIcs.ESA.2016.69 ,doi:10.4230/LIPIcs.ESA.2016.69 .\n[32] Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full complexity classification of\nthe list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni,\nGrzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on\nAlgorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference) , volume\n173ofLIPIcs, pages74:1–74:24.SchlossDagstuhl-Leibniz-ZentrumfürInformatik, 2020.\ndoi:10.4230/LIPIcs.ESA.2020.74 .\n[33] Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of the graph homomor-\nphism problem for bounded-treewidth graphs. SIAM J. Comput. , 50(2):487–508, 2021.\ndoi:10.1137/20M1320146 .\n[34] Masataka Shirahashi and Naoyuki Kamiyama. Kernelization algorithms for a generaliza-\ntion of the component order connectivity problem. J. Oper. Res. Soc. Japan , 66(2):112–\n129, 2023.\n[35] Dekel Tsur. Faster parameterized algorithms for two vertex deletion prob-\nlems. Theoretical Computer Science , 940:112–123, 2023. URL: https:\n//www.sciencedirect.com/science/article/pii/S0304397522006569 ,doi:https://\ndoi.org/10.1016/j.tcs.2022.10.044 .\n[36] Johan M. M. van Rooij. Fast algorithms for join operations on tree decompositions. In\nFedor V. Fomin, Stefan Kratsch, and Erik Jan van Leeuwen, editors, Treewidth, Kernels,\nand Algorithms - Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th\n67Birthday , volume 12160 of Lecture Notes in Computer Science , pages 262–297. Springer,\n2020. doi:10.1007/978-3-030-42071-0\\_18 .\n[37] Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic program-\nming on tree decompositions using generalised fast subset convolution. In Amos Fiat\nand Peter Sanders, editors, Algorithms - ESA 2009, 17th Annual European Symposium,\nCopenhagen, Denmark, September 7-9, 2009. Proceedings , volume 5757 of Lecture Notes\nin Computer Science , pages566–577.Springer, 2009. doi:10.1007/978-3-642-04128-0\\\n_51.\n[38] M. Yatauro. Component order connectivity and vertex degrees. Congr. Numer. , 220:195–\n205, 2014.\n68"    },    {        "title": "2402.07340v1.Random_Geometric_Graph_Alignment_with_Graph_Neural_Networks.pdf",        "content": "Random Geometric Graph Alignment with Graph Neural Networks\nSuqi Liu SUQIL @MED .HARVARD .EDU\nHarvard University\nMorgane Austern MAUSTERN @FAS.HARVARD .EDU\nHarvard University\nAbstract\nWe characterize the performance of graph neural networks for graph alignment problems in the\npresence of vertex feature information. More specifically, given two graphs that are independent\nperturbations of a single random geometric graph with noisy sparse features, the task is to recover\nan unknown one-to-one mapping between the vertices of the two graphs. We show under certain\nconditions on the sparsity and noise level of the feature vectors, a carefully designed one-layer\ngraph neural network can with high probability recover the correct alignment between the vertices\nwith the help of the graph structure. We also prove that our conditions on the noise level are tight up\nto logarithmic factors. Finally we compare the performance of the graph neural network to directly\nsolving an assignment problem on the noisy vertex features. We demonstrate that when the noise\nlevel is at least constant this direct matching fails to have perfect recovery while the graph neural\nnetwork can tolerate noise level growing as fast as a power of the size of the graph.\nKeywords: graph neural network, random geometric graph, graph alignment, entity alignment,\nassignment problem\n1. Introduction\nGraph neural networks (GNNs) (Kipf and Welling, 2016) have seen broad application in many im-\nportant domains with graph-structured data since its inception including social networks (Hamilton\net al., 2017), computational biology (Fan et al., 2019), chemistry (Gilmer et al., 2017), and knowl-\nedge graphs (Schlichtkrull et al., 2018). A standard graph neural network model (also called mes-\nsage passing neural network) consists of traditional multilayer perceptrons (MLPs) injected with a\nmessage passing step that aggregates the hidden representations from neighboring vertices in the\ngraph.\nDespite their wide popularity and dominant performance in graph-based tasks, the theoretical\nunderstanding of GNNs is only emerging in recent years (Jegelka, 2022). Most of the works focus\non traditional learning theory aspects such as representation power (Loukas, 2020) and generaliza-\ntion (Scarselli et al., 2018). A few of them touch on more classic graph-theoretic problems such as\ngraph isomorphism (Morris et al., 2019) and graph properties (Garg et al., 2020). However, there\nstill is a huge divide between the theory of GNNs and how they are used in the real world. The ben-\nefits and limitations of GNNs in many scenarios they are commonly applied to remain enigmatic.\nOur aim is to take a step towards closing this gap by investigating a common use case of GNNs. To\nthis end, we focus on the problem of aligning two sparsified geometric graphs together with noisy\nobservations of their vertex features. This type of question broadly exists in real life. For example,\nimagine that we have access to two different social networks that are actually built around the same\ngroup of people, together with inaccurate features of each node, but the node correspondence is notarXiv:2402.07340v1  [cs.LG]  12 Feb 2024LIUAUSTERN\nknown to us (think of Instagram and Facebook). Our goal is to recover the underlying node cor-\nrespondence from the networks and node features. We remark that this seemingly simplistic entity\nalignment task has numerous instantiations such as cross-lingual knowledge graph alignment (Wang\net al., 2018), molecular network comparison (Sharan and Ideker, 2006) and indeed GNNs achieve\nthe state-of-the-art performance.\nSpecifically, we look into a probabilistic generative model for the networks. We observe two\nnoisy versions GandG′of a single random geometric graph. Additionally for each graph, we also\nobserve some vertex feature information. These vertex features are sparse random vectors perturbed\nby independent Gaussian noise. In the previous example of social networks these feature vectors\ncan be thought of as user respective profiles. In the rest of the introduction, we define our model\nformally and state our main results.\n1.1. Model\nOur model is a generalization of the random intersection graph, a special case of the more general\nfamily of random geometric graphs. We first introduce several notations necessary for defining the\nmodel. A graph Gonnvertices is denoted by a pair G= (X, E)where X= (x1, . . . ,xn)is the\nlist of vertex features and Eis the set of undirected edges. That is, vertices iandjare connected\nby an undirected edge if and only if {i, j} ∈E.\nDefinition 1 (Random intersection graph) The random intersection graph G0= (X, E0)is de-\nfined as follows. A d-dimensional feature vector xiis associated to each vertex i. Given a sparsity\nparameter s≪dthe entries of xifollow independent Bernoulli distribution with parameter s/d.\ni.e., P(xi(j) = 1) = s/dand P(xi(j) = 0) = 1 −s/dindependently. For each pair of vertices\ni, j∈[n], i̸=j,{i, j} ∈E0if and only if\n⟨xi,xj⟩ ≥t\nfor a fixed threshold t≥1. We denote the graph by RIG( n, d, s, t ).\nTraditionally, a random intersection graph is defined by nrandom subsets of a total of del-\nements, and sets are connected if their intersection is nonempty (Singer, 1996). It is clear that\nthis corresponds to the case when t= 1 in our general definition of RIG. Written in the form of\nDefinition 1, it can also easy to notice that RIG is a special case of random inner (dot) product\ngraphs (Young and Scheinerman, 2007) specifically when the features are Bernoulli random vari-\nables. In contrast to Erd ¨os–R ´enyi model, the edges in RIG are regulated by the underlying sparse\nbinary feature vectors, as with other random geometric graphs (Penrose, 2003).\nWe do not directly observe the original graph G0; Rather, we have access to two noisy and\nincomplete versions of it, in which we observe perturbed feature vectors and keep only a subset of\nthe edges. The procedure is established to mimic the data pattern in real networks.\nDefinition 2 (Noisy and incomplete RIG) Given a graph G0= (X, E0)∼RIG( n, d, s, t ), we\nassume that the observed graph G= (Y, E)is generated by the following process.\n1. For each vertex i∈[n], the observed feature vector yiis generated by\nyi=xi+ϵi\nwhere ϵi∼ N(0, σ2Id)is independent noise for some σ≥0.\n2RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\n2. For each pair of vertices 1≤i̸=j≤n, an edge {i, j} ∈Ebetween those two vertices is\nobserved with probability qonly if {i, j} ∈E0. In other words,\nP({i, j} ∈E| {i, j} ∈E0) =q.\nWe denote the graph by NIRIG( X, σ, q).\nWe create a pair of correlated NIRIG as follows. We first sample a truth graph G0= (X, E0)∼\nRIG( n, d, s, t ). Then we have two copies of NIRIG( X, σ, q)independently from G0along with a\nvertex permutation. That is to say,\nG= (Y, E)∼NIRIG( X, σ, q)and G′= (Y′, E′)∼NIRIG( π∗(X), σ, q)\nindependently for an unknown permutation π∗∈ P(n), the set of all permutations on [n]. Our goal\nis to recover the true permutation π∗from the observations G= (Y, E)andG′= (Y′, E′).\n1.2. Main results\nThe vanilla graph neural network (also called graph convolutional network or message passing\nneural network) iteratively applies the following propagation rule (from layer ltol+ 1) to each\nvertex iinG= (X, E):\nxl+1\ni=η\u00121\n|Ni|X\nj∈NiWlxl\ni\u0013\nwhere Ni:={j∈[n] :{i, j} ∈E}is the neighborhood of i,ηis some nonlinear activation\nfunction, and Wlis the trainable weight parameter.\nWe employ a specially-designed one-layer graph neural network to find the matching between\nG= (Y, E)andG′= (Y′, E′). Define the neighborhood of vertex iinGandG′respectively as\nNi:={j∈[n] :{i, j} ∈E}and N′\ni:={j∈[n] :{i, j} ∈E′}.\nWe apply a message passing layer to the observations YandY′followed by a thresholding function\nη(u) = 1\u001a\nu≥1\n2\u001b\n.\nHence for each vertex i≤nthe output of the one-layer graph neural network on graph GandG′\nare respectively\nzi=η\u0012s\nt|Ni|X\nj∈Niyj\u0013\nandz′\ni=η\u0012s\nt|N′\ni|X\nj∈N′\niy′\nj\u0013\n.\nThe output of the graph neural network is finally used to find the matching π⋆. This is done by\nsolving an assignment problem between ZandZ′:\nˆπ= arg min\nπ∈P(n)\u0012\nℓ(π):=nX\ni=1∥zi−z′\nπ(i)∥2\u0013\n= arg max\nπ∈P(n)nX\ni=1⟨zi,z′\nπ(i)⟩ (1)\nwhere P(n)is the permutation group. This problem can be solved in polynomial time with combi-\nnatorial optimization algorithms such as the Hungarian algorithm (Kuhn, 1955).\n3LIUAUSTERN\nO md\nn1n1n2qm\ns\nsqm\nσ2s2\nO md\nn1n1n2qm\ns\nsqm\nσ2s2\nFigure 1: Phase diagram of perfect recovery for t= 1(left) and t= 2(right). Here σ2≍1/√n.\nThe RIG is parametrized by the sparsity parameter sof the underlying features. However, it\ncan be directly translated to properties of the underlying graph G0. When n,d, and tare fixed, the\nedge density of the graph is indeed determined by sin a nontrivial way. To make this dependence\nexplicit, we define another parameter m∈[1, n]such that\ns=s\ntd\u0012m\nn\u00131/t\n. (2)\nIt will be demonstrated by a later lemma (Lemma 18) that mroughly represents the order of the\naverage degree in RIG( n, d, s, t )when tis constant. For simplicity we keep t≥1fixed in our\ndiscussion. This notably implies using (2) that we have s≲√\nd. We do not put other restrictions\non the parameters we consider.\nOur main finding is that when the parameters n,d,m,σ, and qsatisfy certain conditions, the\nsolution found by (1) recovers the true permutation π∗with high probability.\nTheorem 3 (Perfect recovery) Let the matching problem be defined previously and we solve it\nusing (1). With probability going to 1asn→ ∞ we recover both the true vertex features and\nmatching if\nmin\u001a\ns,qm\ns,qm\nσ2s2\u001b\n≫logn+ log d.\nNote that finer and more detailed results can be found in Theorem 19 and 20.\nRemark 4 The parameter regime is specified by a mixture of sandmfor cleanness. However,\nas defined in (2),sis determined by mand vice versa. Since we are more interested in graph\nproperties, we choose the more natural parameter m. Replacing swith the definition in (2), we\nimmediately have\nmin\u001as\nd\u0012m\nn\u00131/t\n,s\nq2m2\nd\u0012n\nm\u00131/t\n,qm\nσ2d\u0012n\nm\u00131/t\u001b\n≫logn+ log d.\n4RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nSince qalso affects the edge density of the graph but in a less sophisticated way (only through sub-\nsampling), for the interest of disucssion, we also fix it to be a constant. Now we are ready to present\nan interplay among n,d,m, andσin several graph density regimes. Notably for different graph den-\nsities mwe identify the permissible range for d, within which perfect graph recovery is feasible, con-\ntingent on the noise level σ2being adequately small. We similarly derive the maximum size the noise\nσ2can take for which perfect recovery is still possible. We list the results in Table 1 when t= 1and\nt= 2form=alog2+ϵn, ϵ > 0(sparse) and m=bnα,0< α < 1(intermediate) , α= 1(dense)\nwhere a, b > 0are absolute constants.\nTable 1: Recovery conditions for different thresholds tand sparsity levels m.\nm (logn)2+ϵnα\ndfort= 1 (logn)ϵn≫d≫n\n(logn)ϵn1+α\n(logn)2≫d≫(logn)2n1−α\nσ2fort= 1 σ2≪1\n(logn)1−ϵ σ2≪nα\n(logn)3\ndfort= 2√n(logn)1+3\n2ϵ≫d≫√n(logn)1−ϵ\n2n1+3α\n2\n(logn)2≫d≫(logn)2(√n)1−α\nσ2fort= 2 σ2≪1\n(logn)1−ϵ σ2≪nα\n(logn)3\nRemark 5 Note that only the last term in the bound depends on the noise parameter σ. All previous\nterms are functions of n,d, and m. If we consider only the intermediate regime when m≍nαand\nd≍nβ, we can visualize the theorem as a diagram in the space of manddas in Figure 1. The\nentire colored region is specified by the first two terms (and m≤n) without σ. The term that\ninvolves σ“cuts through” the region and changes as σvarying at different order of n. The dark\nblue region is where perfect recovery is possible for the GNN.\nWe also have the following negative result that proves the conditions when the GNN cannot\nrecover the matching perfectly.\nTheorem 6 (Impossibility for perfect recovery) Assume that\nmin\u001a\ns,qm\ns\u001b\n≥c\nfor a constant c >0. Then there exist a contant δ >0such that P(ˆπ̸=π∗)≥δif\nqm\nσ2s2≤Cδ\nfor some constant Cδ>0.\nRemark 7 We see that the conditions assumed here are also required for perfect recovery. These\nregularity conditions are there to guarantee that the vertices are unique and that the signal is strong\nenough in the graph. The major difference between the possibility and impossibility bounds is the\n5LIUAUSTERN\ntermqm\nσ2s2which is also the only term that involves the noise parameter σ. For perfect recovery, it is\nrequired to be ≫logn+ log d, while for impossibility, it has to be bounded. This implies that the\nbound on the signal parameter is tight up to logarithmic factors. Notably combined with the perfect\nrecovery results, in the dense regime m≍nαwe show the algorithm can tolerate noise level σthat\ncan grow as a polynomial function of n.\nInstead of using the graph neural network, one can obtain a matching by solving an alignment\nproblem directly from the noisy vertex features YandY′:\n˜π= arg min\nπ∈P(n)˜ℓ(π):=nX\ni=1∥yi−y′\nπ(i)∥2= arg min\nπ∈P(n)nX\ni=1⟨yi,y′\nπ(i)⟩. (3)\nWe call this method the linear model. We note that this model has been widely used in for alignment\nproblems and also attracted a lot of theoretical analysis under different settings. However, as we will\nshow in the following theorem that when the noise is large, or the dimension is high, directly solving\nthe assignment problem will not achieve perfect recovery with at least constant probability.\nTheorem 8 (Impossibility for perfect recovery with vertex features) Ifσ2≥2s\u0000\n1 +K\nn\u0001−2for\nanyK > 4we have that\nP(˜π=π∗)≤e−K−4\n4.\nHence if σ2≫swe have that the probability of perfect recovery will converge to 0. If instead\n1\n4σ2≤s≤d\n2andσ2≫s√dlognthen\nlim\nn→∞P(˜π=π∗) = 0 .\nNote that finer and more detailed results are available in Proposition 31 and 32.\nRemark 9 The impossibility result is split into two regimes: the signal-to-noise ratio being large\nand small. We are more interested in how these bounds compare to the perfect recovery by the graph\nneural network. In order to make the comparison, we translate the parameter stom, although it\ndoes not have a clear physical meaning for aligning with vertex features. In one scenario the\nmessage is prominent: When σis at least a constant, in all parameter regimes that GNN has perfect\nrecovery, the linear model would fail.\n1.3. Numerical experiments\nWe verify our theoretical results by numerical experiments. All graphs are generated using the\nmodel introduced in section 1.1. We evaluate the features by two metrics: alignment error which\ncounts the percentage of mistakes the solution of the assignment problem makes and relative dis-\ntance of recovered features and the true features. The results are shown in Figure 2.\nWe see that the GNN model consistently outperforms the linear model particularly under large\nnoise and moderate size of neighborhood. Note that for the parameters chosen, the original graph\nis already very sparse. The average degree is about 56compared to n= 4000 nodes. This explains\nwhy the error for GNN is high when qis small.\n6RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\n(a)\n (b)\n(c)\n (d)\nFigure 2: Comparison for GNN and linear model. In (a) and (b), alignment error is evaluated. In\n(c) and (d), alignment cost is evaluated. Parameters are n= 4000 ,d= 200 ,s= 10 ,\nt= 3. For (a) and (c), q= 0.8is fixed and σranges from 0.1to1in0.1increments. For\n(b) and (d), σ= 0.4andqchanges from 0.1to1in0.1increments.\n1.4. Related work\nRandom graph matching has attracted large amounts of research in recent years (Ding et al., 2021;\nMao et al., 2021; Wu et al., 2022). However, the majority of the literature study the correlated\nErd¨os–R ´enyi model where two samples are created independently from a same Erd ¨os–R ´enyi graph\nthrough edge-resampling. This way of introducing noise is similar to our subsampling process.\nThe correlated stochastic block model introduced by Racz and Sridhar (2021) can be viewed as\nan intersection graph in one dimension. A recent paper by Wang et al. (2022) considers matching\nrandom geometric graphs but only assume all pairwise dot products or Euclidean distances are\nobserved under the Gaussian setting. One major difference that sets our model apart is that we are\nconsidering sparse binary features with noisy observation. Nevertheless, we would like to comment\nthat the main goal of our work is not to solve the graph matching problem optimally but rather to\nshow that the effectiveness of GNNs in practice is indeed supported by theoretical analysis.\n7LIUAUSTERN\nMatching two random point clouds has a long history in probability and combinatorics (Ajtai\net al., 1984). More recently, Kunisky and Niles-Weed (2022) studied matching recovery for Gaus-\nsian perturbed Gaussian vectors in various regimes by solving an assignment problem. Notably our\nfeature generating process resembles the one investigated by the authors. Our work builds upon the\nsame method but shows theoretically that with the help of incomplete observations from geometric\ngraphs, modern machine learning methods is capable of achieving much better results. Again, our\npurpose is not to provide better recovery bounds on different settings but to supply a theoretical\nunderstanding of why graph neural networks work well in the alignment problem.\n2. Notations and preliminaries\nWe start the discussion with a few notations used throughout the paper and show a few lemmas that\nprovide characterizations of NIRIG.\n2.1. Notations\nWe use bold uppercase letters to denote matrices and the corresponding lowercase letters to denote\ntheir rows. For example, X∈ Rn×dis ann-by-dreal matrix with rows (x1, . . . ,xn). For a vector\nx,x(k)stands for its kth entry. The p-norm of a vector is denoted by ∥·∥pand it defaults to 2-norm,\nor the Euclidean norm, when pis not specified. The inner product between two d-dimensional\nvectors xandyis written as ⟨x,y⟩:=Pd\nk=1x(k)y(k). The notation |·|is generally overloaded.\nWhen it is applied to a number a,|a|is the absolute value of a. And when applied to a set S,\n|S|denotes its cardinality. We use [n]:={1, . . . , n }to denote the set of all natural numbers up\nton. For a set S,Scstands for its complement. Gaussian (normal) distribution with mean µand\nvariance σ2is written as N(µ, σ2)and correspondingly the d-dimensional with mean vector µand\ncovariance matrix Σis written as N(µ,Σ).Φ(x), x∈ Ris the distribution function of a standard\nnormal distribution. We also use Φc(x):= 1−Φ(x)to represent the upper tail.\n2.2. Preliminaries\nIn this section we present some preliminary results on the tails of different random variables. These\nwill be used later in our proofs.\nProposition 10 (Chernoff bound) LetX1, . . . , X nbe independent Bernoulli random variables\nandN=Pn\ni=1Xibe their sum. Denote E[N] =µthe expected value of the sum. Then for any\nδ≥0,\nP(N≥(1 +δ)µ)≤exp\u0012\n−δ2µ\n2 +δ\u0013\nand for any 0< δ < 1,\nP(N≤(1−δ)µ)≤exp\u0012\n−δ2µ\n2\u0013\n.\nThe following lower bound for the tail of a nonnegative random variable is due to Paley and\nZygmund (1932) (see also Kahane (1985, Inequality II, p. 8)).\nProposition 11 (Paley–Zygmund) LetZbe a nonnegative random variable with finite variance.\nFor0≤θ≤1,\nP(Z≥θ E[Z])≥(1−θ)2 E[Z]2\nE[Z2].\n8RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nWe exploit this to deduce the following tail lower bound for the dot product of Gaussian vectors.\nProposition 12 (Lower bound for Gaussian inner product) Letx,y∼ N (0,Id)be two inde-\npendent standard Gaussian random vectors. We have that for any u≤√\nd/16, the following holds\nP(⟨x,y⟩ ≥u√\nd)≥(1−e−4u2)2e−22u2.\nProof Denote Z= exp( t⟨x,y⟩)for0< t < 1/2. By the proof of (Liu and R ´acz, 2023b,\nProposition 2.14), we have\nE[Z] = E[et⟨g1,g2⟩] =dY\ni=1E[etg1,ig2,i] = (1 −t2)−d/2.\nHence this implies that for every t <1\n4the following holds\nE[Z2] = E[e2t⟨g1,g2⟩] = (1 −4t2)−d/2.\nBy Paley–Zygmund (Proposition 11), for all θ∈(0,1)we have\nP\u0012\n⟨x,y⟩ ≥ −d\n2tlog(1−t2) +1\ntlogθ\u0013\n≥(1−θ)2\u00121−t2\n√\n1−4t2\u0013−d\n.\nBy taking t= 4u/√\ndandθ= exp( −4u2)foru≤√\nd/16, we have that\n−d\n2tlog(1−t2) +1\ntlogθ=−d3/2\n8ulog\u0012\n1−16u2\nd\u0013\n−u√\nd≥u√\nd\nwhere we used log(1−x)≤ −x, and\n\u00121−t2\n√\n1−4t2\u0013−d\n=\u0012\n1 + 2 t2\u00121 +t2/2\n1−4t2\u0013\u0013−d/2\n≥\u0012\n1 +44u2\nd\u0013−d/2\n≥e−22u2\nwhere we used (1 +x/d)d≤ex.\nProposition 13 (Lower bound for lazy random walk tail) LetSn=Pn\ni=1Xibe a lazy random\nwalk such that P(Xi= 1) = P(Xi=−1) = pand P(Xi= 0) = 1 −2p. Assume pn≥1. Then\nfor0≤u≤pn/β , there exist constants cβ, Cβ>0that depend only on βsuch that\nP(Sn≥u)≥cβexp\u0012\n−Cβu2\npn\u0013\n.\nProof LetYi∈ {− 1,1}fori∈[n]be i.i.d. Rademacher random variables, i.e., P(Yi= 1) =\nP(Yi=−1) = 1 /2. Let Zi∼Bern(2 p), i∈[n]be i.i.d. Bernoulli random variables. Then, we\ncan write Xi=YiZi. Let Nn=Pn\ni=1Zibe the sum of Zi’s. Then given Nn,Snis a sum of Nn\ni.i.d. Rademacher random variables. By (Zhang and Zhou, 2020, Corollary 4), for any β >1, there\nexists constants cβ, Cβ>0, such that for all 0≤u≤Nn/β,\nP(Sn≥u|Nn)≥cβexp\u0012\n−Cβu2\nNn\u0013\n.\n9LIUAUSTERN\nBy Chernoff bound (Proposition 10),\nP\u0012\nNn≤pn\n2\u0013\n≤exp\u0012\n−pn\n8\u0013\n.\nTherefore,\nP(Sn≥u)≥ P\u0012\nSn≥u, N n≥pn\n2\u0013\n= P\u0012\nSn≥u\f\f\f\fNn≥pn\n2\u0013\u0012\n1− P\u0012\nNn≤pn\n2\u0013\u0013\n≥cβexp\u0012\n−Cβ8u2\npn\u0013\u0012\n1−exp\u0012\n−pn\n8\u0013\u0013\n.\nThe claim directly follows.\n3. Auxiliary lemmas\nA necessary condition for perfect recovery is that the feature vector for each vertex is unique. The\nfollowing proposition tells us that this happens with high probability in RIG when sis sufficiently\nlarge.\nProposition 14 (No two vectors are the same) Ifs≥(1 +c) lognfor some c >0, then no two\nvectors in {x1, . . . ,xn}are the same with probability at least 1−n−2c.\nProof As the entries xi(k),xj(k)i.i.d∼Bern(s\nd), we obtain that\nP(xi(k)̸=xj(k)) =2s\nd.\nHence, by independence of the entries, we have\nP(xi=xj) = P(xi(k) =xj(k)∀k≤d) =\u0012\n1−2s\nd\u0013d\n.\nHence by a union bound argument we obtain that\nP(∃i̸=j≤ns.t.xi=xj)≤X\ni<jP(xi=xj)≤\u0012n\n2\u0013\u0012\n1−2s\nd\u0013d\n≤e−2(s−logn).\nThe claim directly follows.\nSincexi(k), k∈dare i.i.d. Bernoulli random variables with parameter s/d, applying Chernoff\nbound (Proposition 10) to |Si|=Pd\nk=1xi(k)directly gives us the following lower and upper\ndeviation bounds for |Si|.\nLemma 15 For all i∈[n]and any δ≥0,\nP(|Si| ≥(1 +δ)s)≤exp\u0012\n−δ2s\n2 +δ\u0013\n.\n10RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nLemma 16 For all i∈[n]and0< δ < 1,\nP(|Si| ≤(1−δ)s)≤exp\u0012\n−δ2s\n2\u0013\n.\nDefine the event\nEi:=\u001as\n2≤ |Si| ≤2s\u001b\n. (4)\nLemma 16 and 15 immediately suggest that\nP(Ec\ni)≤exp\u0012\n−s\n8\u0013\n+ exp\u0012\n−s\n3\u0013\n≤2 exp\u0012\n−s\n8\u0013\n. (5)\nIn the proofs, we use a lot of conditional arguments. Most frequently, s/2≤ |Si| ≤2s, which\nhappens with high probability from previous lemmas, is usually assumed. The following elementary\ninequality becomes handy in such situations.\nProposition 17 For any two events AandB,\nP(A)≤ P(A|B) + P(Bc).\nProof By law of total probability,\nP(A) = P(A∩B) + P(A∩Bc) = P(A|B) P(B) + P(A|Bc) P(Bc)≤ P(A|B) + P(Bc),\nwhere we used the fact that the probability measure is less than or equal to 1.\nFor all vertex i∈[n], let\nFi:=\u001a1\n2t+2·mq≤ |Ni| ≤2t+2et·mq\u001b\n. (6)\nLemma 18 The following holds for all i∈[n]when n≥(12t)tm,\nP(Fc\ni)≤2 exp\u0012\n−s\n8\u0013\n+ 2 exp\u0012\n−mq\n2t+6\u0013\n.\nProof By definition the neighborhood size is given by |Ni|=Pn\nj=1 1{j∈Ni}. Moreover\nwe remark that conditionally on xi, the edges\u0000\n1{j∈Ni}\u0001\nj∈[n]are independent and identically\ndistributed. We denote by pi:= P(j∈Ni|xi)the edge probability. Hence we remark that\nconditionally on xi,|Ni|is a sum of i.i.d. Bernoulli random variables with parameter pi.\nWe first obtain upper and lower bounds for piconditionally the event Eidefined in (4) holding.\nFor the rest of the proof, all probabilities are conditioned on Eiand we omit it from the condition for\nsimplicity of presentation. We note that ⟨xi,xj⟩=Pd\nk=1xi(k)xj(k)is aBinom( |Si|,s\nd)random\nvariable. Recall that the edges are a subsample of the original graph with probability q. Hence the\nconditional probability that j∈Niis given by\nP(j∈Ni|xi) = P(j∈Ni| ⟨xi,xj⟩ ≥t) P(⟨xi,xj⟩ ≥t|xi) =q P(⟨xi,xj⟩ ≥t|xi)\n=q|Si|X\nk=t\u0012|Si|\nk\u0013\u0012s\nd\u0013k\u0012\n1−s\nd\u0013|Si|−k\n=:pi.\n11LIUAUSTERN\nThen we can lower bound piby\npi=q|Si|X\nk=t\u0012|Si|\nk\u0013\u0012s\nd\u0013k\u0012\n1−s\nd\u0013|Si|−k\n≥q\u0012|Si|\nt\u0013\u0012s\nd\u0013t\u0012\n1−s\nd\u0013|Si|−t\n(a)\n≥q\u0012|Si|s\ntd\u0013t\u0012\n1−|Si|s\nd\u0013\n,\nwhere we used the elementary inequalities\u0000n\nk\u0001\n≥(n/k)kand(1−x)r≥1−rxforx≤1in(a).\nHence, when s/2≤ |Si| ≤2sandd≥4s2(or equivalently n≥(4t)tm), we have\npi≥q\n2t\u0012s2\ntd\u0013t\u0012\n1−2s2\nd\u0013(a)\n≥1\n2t+1·mq\nn(7)\nwhere to get (a)we used (2). We also have that for s/2≤ |Si| ≤2sandd≥12s2/t(or equivalently\nn≥(12)tm),\npi=q|Si|X\nk=t\u0012|Si|\nk\u0013\u0012s\nd\u0013k\u0012\n1−s\nd\u0013|Si|−k(a)\n≤q|Si|X\nk=t\u0012es|Si|\nkd\u0013k\n≤q|Si|X\nk=t\u00122es2\ntd\u0013k\n≤q∞X\nk=t\u00122es2\ntd\u0013k\n≤2q\u00122es2\ntd\u0013t(b)= 2t+1et·mq\nn,(8)\nwhere we used\u0000n\nk\u0001\n≤(en\nk)kin(a)and the equality (b)is by (2). Together with Proposition 10, we\nconclude that\nP(|Ni| ≥2t+2et·mq) = P\u0012\n|Ni| ≥2t+2et·mq\n(n−1)pi·(n−1)pi\u0013\n(a)\n≤ P\u0012\n|Ni| ≥\u0012\n1 +n+ 1\nn−1\u0013\n(n−1)pi\u0013\n≤exp\u0012\n−(n+ 1)2pi\n2n\u0013\n(b)\n≤exp\u0012\n−\u0012\n1 +1\nn\u00132mq\n2t+2\u0013\n≤exp\u0012\n−mq\n2t+2\u0013\nwhere we used (8) in (a)and (7) in (b), and\nP\u0012\n|Ni| ≤1\n2t+2·mq\u0013\n= P\u0012\n|Ni| ≤1\n2t+2·mq\n(n−1)pi·(n−1)pi\u0013\n(c)\n≤ P\u0012\n|Ni| ≥\u0012\n1−n−2\n2n−2\u0013\n(n−1)pi\u0013\n≤exp\u0012\n−(n−2)2pi\n8(n−1)\u0013\n(d)\n≤exp\u0012\n−(n−2)2mq\n2t+4n(n−1)\u0013\n≤exp\u0012\n−mq\n2t+6\u0013\nwhere we used (7) in (c)and (8) in (d).\nThe lemma directly follows from Proposition 17 combined with (4).\n12RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\n4. Proofs of perfect recovery results\nSince ˆπis the minimizer of the empirical risk ℓ, we have ℓ(ˆπ)≤ℓ(π∗). Without loss of generality,\nwe assume that π∗is the identity transformation in the proofs hence π∗(i) =ifor all i∈[n]. By\ntriangle inequality, we have\nℓ(ˆπ)≤ℓ(π∗) =nX\ni=1∥zi−z′\ni∥2≤nX\ni=1(∥zi−xi∥+∥z′\ni−xi∥)2≤2nX\ni=1∥zi−xi∥2+2nX\ni=1∥z′\ni−xi∥2.\n(9)\nWe will show that with high probability bothPn\ni=1∥z′\ni−xi∥2andPn\ni=1∥zi−xi∥2are equal to\nzero. This will directly imply that with high probability the empirical risk ℓ(ˆπ)is also zero.\nTheorem 19 (Perfect recovery for vertex features) LetG= (Y, E)∼NIRIG( X, σ, q)be de-\nfined as in section 1.1 and let Z= (z1, . . . ,zd)be the output of the graph neural network described\nin section 1.2. Then for n≥(12t)tm,\nP(∃is.t.xi̸=zi)≤nd\u0012\n3 exp\u0012\n−mq\n5·2t+7s\u0013\n+ 3 exp\u0012\n−s\n175\u0013\n+ exp\u0012\n−mq\n2t+7s2σ2\u0013\u0013\n.\nBy Proposition 14, with probability at least 1−n2e−2sall vertex features X= (x1, . . . ,xn)are\nunique. Therefore, π∗is the unique minimizer of (1). Since by Theorem 3 we know that ℓ(ˆπ) = 0\nwith high probability, we immediately have the following theorem by Proposition 17.\nTheorem 20 (Prefect recovery for vertex matching) Letˆπbe the solution of (1). Then for n≥\n(12t)tm,\nP(ˆπ̸=π∗)≤n2exp(−2s)+2nd\u0012\n3 exp\u0012\n−mq\n5·2t+7s\u0013\n+3 exp\u0012\n−s\n175\u0013\n+exp\u0012\n−mq\n2t+7s2σ2\u0013\u0013\n.\nThe rest of this section is devoted to proving Theorem 19. The proof is centered around events\ndefined as follows. For all i∈[n]andk∈[d], let\nAi,k:=\u001as\nt|Ni|X\nj∈Nixj(k) +s\nt|Ni|X\nj∈Niϵj(k)≤1\n2\u001b\n. (10)\nThe next four lemmas concern events that will directly lead to bounds on Ai,k.\nLemma 21 For all i∈[n],\nP\u0012s\nt|Ni|X\nj∈Niϵj(k)≥1\n4\f\f\f\f|Ni|\u0013\n≤exp\u0012\n−t2|Ni|\n32s2σ2\u0013\n.\nProof By definition of ϵj,ϵj(k)are i.i.d. N(0, σ2)forj∈Ni. Hence,\nP\u0012s\nt|Ni|X\nj∈Niϵj(k)≥1\n4\f\f\f\f|Ni|\u0013\n= Φc\u0012tp\n|Ni|\n4sσ\u0013\n.\nBy the upper tail bound of standard normal distribution (see, e.g., (Wainwright, 2019, (2.7)),\nΦc\u0012tp\n|Ni|\n4sσ\u0013\n≤exp\u0012\n−t2|Ni|\n32s2σ2\u0013\n.\nThe claim directly follows.\n13LIUAUSTERN\nLemma 22 For all i∈[n], when n≥8tm,\nP\u0012s\nt|Ni|X\nj∈Nixj(k)≥1\n4\f\f\f\fxi(k) = 0 ,|Ni|\u0013\n≤exp\u0012\n−t|Ni|\n24s\u0013\n.\nProof First we remark that knowing xi(k) = 0 , the coordinates (xj(k))j∈Niare i.i.d. Bern( s/d)\nrandom variables. Hence E[P\nj∈Nixj(k)| |Ni|] =s|Ni|/d. Therefore by taking δ=td\n4s2in\nProposition 10, we have\nP\u0012s\nt|Ni|X\nj∈Nixj(k)≥1\n4\f\f\f\fxi(k) = 0 ,|Ni|\u0013\n≤exp\u0012\n−\u0012td\n4s2−1\u00132\n·\u0012td\n4s2+ 1\u0013−1\n·s|Ni|\nd\u0013\n.\nForn≥8tm, we havetd\n4s2≥2. Hence,\n\u0012td\n4s2−1\u00132\n·\u0012td\n4s2+ 1\u0013−1\n·s|Ni|\nd≥\u0012td\n8s2\u00132\n·\u00123td\n8s2\u0013−1\n·s|Ni|\nd=t|Ni|\n24s.\nThe lemma directly follows.\nLemma 23 For all i∈[n],\nP\u0012s\nt|Ni|X\nj∈Ni\u0012t\ns−xj(k)\u0013\n≥1\n4\f\f\f\fxi(k) = 1 ,|Ni|\u0013\n≤exp\u0012\n−t|Ni|\n640s\u0013\n+ exp\u0012\n−s\n175\u0013\n.\nProof We consider a vertex j∈Ni. Since j∈Ni,⟨xi,xj⟩=P\nk∈Sixj(k)≥t. Given Si,\u0000\nxj(k)\u0001\nk∈Siare identical (but not independent) Bernoulli random variables. Hence P(xj(k) = 1|\nj∈Ni, k∈Si) =βare the same for all k∈Si. Since\nE\u0014X\nk∈Sixj(k)\f\f\f\fSi, j∈Ni\u0015\n=X\nk∈SiP(xj(k) = 1|j∈Ni, k∈Si) =β|Si| ≥t,\nwe have β≥t/|Si|. By Lemma 15 with δ= 1/4, we have\nP\u0012\n|Si| ≤5\n4s\u0013\n≥1−exp\u0012\n−s\n175\u0013\n. (11)\nNow conditioned on xi(k) = 1 ,\u0000\nxj(k)\u0001\nj∈Niare independent Bernoulli random variables with\nparameter β. By taking δ= 1−3t\n4βin Proposition 10, we obtain that\nP\u0012s\nt|Ni|X\nj∈Ni\u0012t\ns−xj(k)\u0013\n≥1\n4\f\f\f\fxi(k) = 1 ,|Ni|,|Si|\u0013\n= P\u0012X\nj∈Nixj(k)≥3t|Ni|\n4s=3t\n4sβ|Ni|β\f\f\f\fxi(k) = 1 ,|Ni|,|Si|\u0013\n≤exp\u0012\n−1\n2\u0012\n1−3t\n4sβ\u00132\n|Ni|β\u0013\n≤exp\u0012\n−1\n2\u0012\n1−3|Si|\n4s\u00132t|Ni|\n|Si|\u0013\n.\n14RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nTherefore,\nP\u0012s\nt|Ni|X\nj∈Ni\u0012t\ns−xj(k)\u0013\n≥1\n4\f\f\f\fxi(k) = 1 ,|Ni|,|Si| ≥5\n4s\u0013\n≤exp\u0012\n−t|Ni|\n640s\u0013\n.\nThe claim directly follows from applying Proposition 17.\nThe following two lemmas are the major building blocks of our proof. Intuitively, the probability\nthat the neuron in GNN is activated when the true signal is 1or the neuron is deactivated when the\ntrue signal is 0is very small if the parameters satisfy certain conditions.\nLemma 24 For all k∈[d], we have when n≥8tm,\nP(Ac\ni,k|xi(k) = 0 ,|Ni|)≤exp\u0012\n−t|Ni|\n24s\u0013\n+ exp\u0012\n−t|Ni|\n32s2σ2\u0013\n.\nProof We further define the following two events:\nB0:=\u001as\nt|Ni|X\nj∈Nixj(k)≤1\n4\u001b\nand B1:=\u001as\nt|Ni|X\nj∈Niϵj(k)≤1\n4\u001b\n. (12)\nThen we have Ai,k⊃B0∩B1. By a union bound, we obtain\nP(Ac\ni,k|xi(k) = 0 ,|Ni|)≤ P(Bc\n0|xi(k) = 0 ,|Ni|) + P(Bc\n1|xi(k) = 0 ,|Ni|).\nUsing Lemma 22, we have\nP(Bc\n0|xi(k) = 0 ,|Ni|)≤exp\u0012\n−t|Ni|\n24s\u0013\n.\nAnd Lemma 21 gives\nP(Bc\n1|xi(k) = 0 ,|Ni|) = P(Bc\n1| |Ni|)≤exp\u0012\n−t2|Ni|\n32s2σ2\u0013\n.\nThe lemma is proved by combining the above displays.\nLemma 25 For all k∈[d],\nP(Ai,k|xi(k) = 1 ,|Ni|)≤exp\u0012\n−t|Ni|\n640s\u0013\n+ exp\u0012\n−s\n175\u0013\n+ exp\u0012\n−t2|Ni|\n32s2σ2\u0013\n.\nProof We similarly define the two events:\nB0:=\u001as\nt|Ni|X\nj∈Ni\u0012t\ns−xj(k)\u0013\n≤1\n4\u001b\nand B1:=\u001a\n−s\nt|Ni|X\nj∈Niϵj(k)≤1\n4\u001b\n.\nRearranging the terms, we have Ac\ni,k⊃B0∩B1. Hence by a union bound\nP(Ai,k|xi(k) = 1 ,|Ni|)≤ P(Bc\n0|xi(k) = 1 ,|Ni|) + P(Bc\n1|xi(k) = 1 ,|Ni|).\n15LIUAUSTERN\nUsing Lemma 23, we arrive at\nP(Bc\n0|xi(k) = 1 ,|Ni|)≤exp\u0012\n−t|Ni|\n640s\u0013\n+ exp\u0012\n−s\n175\u0013\n.\nAnd applying Lemma 21 to −ϵi’s we have\nP(Bc\n1|xi(k) = 1 ,|Ni|)≤exp\u0012\n−t2|Ni|\n32s2σ2\u0013\n.\nWe obtain the claim by putting together the above displays.\nThe next lemma combines the previous two and proves an upper bound for making a mistake in\nthe vertex features.\nLemma 26 For all i∈[n]we have\nP(zi̸=xi)≤3dexp\u0012\n−mq\n5·2t+7s\u0013\n+ 3dexp\u0012\n−s\n175\u0013\n+dexp\u0012\n−mq\n2t+7s2σ2\u0013\n.\nProof We first note that by a union bound\nP(zi̸=xi)≤dX\ni=1P(zi(k)̸=xi(k)).\nBy the law of total probability, we have\nP(zi(k)̸=xi(k)) = P(zi(k) = 1|xi(k) = 0) × P(xi(k) = 0)\n+ P(zi(k) = 0|xi(k) = 1) × P(xi(k) = 1))\n≤max{ P(zi(k) = 1|xi(k) = 0) , P(zi(k) = 0|xi(k) = 1)}\nBy Proposition 17, using Lemma 24 and 18, we obtain that\nP(zi(k) = 1|xi(k) = 0) ≤ P\u0012\nzi(k) = 1\f\f\f\fxi(k) = 0 ,|Ni| ≥mq\n2t+2\u0013\n+ P\u0012\n|Ni| ≥mq\n2t+2\u0013\n≤3 exp\u0012\n−mq\n2t+7s\u0013\n+ exp\u0012\n−mq\n2t+7s2σ2\u0013\n+ 2 exp\u0012\n−s\n8\u0013\n.\nSimilarly with Lemma 25,\nP(zi(k) = 0|xi(k) = 1) ≤ P\u0012\nzi(k) = 0\f\f\f\fxi(k) = 1 ,|Ni| ≥mq\n2t+2\u0013\n+ P\u0012\n|Ni| ≥mq\n2t+2\u0013\n≤3 exp\u0012\n−tmq\n5·2t+7s\u0013\n+ 3 exp\u0012\n−s\n175\u0013\n+ exp\u0012\n−t2mq\n2t+7s2σ2\u0013\n.\nTherefore, by combining the above two displays, we obtain that\nP(zi(k)̸=xi(k))≤3 exp\u0012\n−mq\n5·2t+7s\u0013\n+ 3 exp\u0012\n−s\n175\u0013\n+ exp\u0012\n−mq\n2t+7s2σ2\u0013\n.\n16RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nThe claim directly follows.\nWith Lemma 26 in place, Theorem 3 directly follows from a union bound:\nP(∃is.t.xi̸=zi)≤nX\ni=1P(xi̸=zi)\n≤nd\u0012\n3 exp\u0012\n−mq\n5·2t+7s\u0013\n+ 3 exp\u0012\n−s\n175\u0013\n+ exp\u0012\n−mq\n2t+7s2σ2\u0013\u0013\n.\nThis concludes our proof for the perfect recovery.\n5. Proofs of the impossibility results for perfect recovery\nLemma 27 For all vertices i∈[n],\nP(zi(k) = 1|xi(k) = 0 ,X,|Ni|)≥Φc\u0012tp\n|Ni|\n2sσ\u0013\nwhere Φc(x):= 1−Φ(x)is the upper tail of standard normal distribution.\nProof Define Ai,kas in (10) by\nAi,k:=\u001as\nt|Ni|X\nj∈Nixj(k) +s\nt|Ni|X\nj∈Niϵj(k)≤1\n2\u001b\n.\nWe remark that\nP(zi(k) = 1|xi(k) = 0 ,X,|Ni|) = P(Ac\ni,k|xi(k) = 0 ,X,|Ni|)\n= P\u0012s\nt|Ni|X\nj∈Nixj(k) +s\nt|Ni|X\nj∈Niϵj(k)≥1\n2\f\f\f\fxi(k) = 0 ,X,|Ni|\u0013\n(a)\n≥ P\u0012s\nt|Ni|X\nj∈Niϵj(k)≥1\n2\f\f\f\fxi(k) = 0 ,X,|Ni|\u0013\n(b)= P\u0012s\nt|Ni|X\nj∈Niϵj(k)≥1\n2\f\f\f\f|Ni|\u0013\n= Φc\u0012tp\n|Ni|\n2sσ\u0013\n,\nwhere (a)is due to the fact thats\nt|Ni|P\nj∈Nixj(k)≥0is nonnegative and (b)is by independence\nof the noise.\nLemma 28 For a pair of vertices iandj,\nP(δi(k)−δj(k) = 1 ,δ′\ni(k)−δ′\nj(k) = 1| Mk\ni,j,|Ni|,|Nj|,|N′\ni|,|N′\nj|)\n≥1\n4Φc\u0012tp\n|Ni|\n2sσ\u0013\nΦc\u0012tp\n|N′\ni|\n2sσ\u0013\u0012\n1−exp\u0012\n−t|Nj|\n24s\u0013\n−exp\u0012\n−t|N′\nj|\n24s\u0013\u0013\n.\n17LIUAUSTERN\nProof By conditional independence of the random variables,\nP(δi(k)−δj(k) = 1 ,δ′\ni(k)−δ′\nj(k) = 1| Mk\ni,j,|Ni|,|Nj|,|N′\ni|,|N′\nj|)\n= EX[ P(δi(k) = 1| Mk\ni,j,X,|Ni|) P(δ′\ni(k) = 1| Mk\ni,j,X,|N′\ni|)\n× P(δj(k) = 0 ,δ′\nj(k) = 0| Mk\ni,j,X,|Nj|,|N′\nj|)]\n≥Φc\u0012tp\n|Ni|\n2sσ\u0013\nΦc\u0012tp\n|N′\ni|\n2sσ\u0013\nP(δj(k) = 0 ,δ′\nj(k) = 0| Mk\ni,j,|Nj|,|N′\nj|)\nwhere we used Lemma 27 in the inequality.\nP(δj(k) = 0 ,δ′\nj(k) = 0| Mk\ni,j,|Nj|,|N′\nj|)\n≥ P\u0012s\nt|Nj|X\nl∈Njxl(k)≤1\n4,s\nt|N′\nj|X\nl∈N′\njxl(k)≤1\n4,\ns\nt|Nj|X\nl∈N′\njϵl(k)≤1\n4,s\nt|N′\nj|X\nl∈N′\njϵ′\nl(k)≤1\n4\f\f\f\fMk\ni,j,|Nj|,|N′\nj|\u0013\n= P\u0012s\nt|Nj|X\nl∈Njϵl(k)≤1\n4\f\f\f\f|Nj|\u0013\nP\u0012s\nt|N′\nj|X\nl∈N′\njϵ′\nl(k)≤1\n4\f\f\f\f|Nj|\u0013\n× P\u0012s\nt|Nj|X\nl∈Njxl(k)≤1\n4,s\nt|N′\nj|X\nl∈N′\njxl(k)≤1\n4\f\f\f\fxj(k) = 0 ,|Nj|,|N′\nj|\u0013\n.\nSinces\nt|Nj|P\nl∈Njϵl(k)is a sum of independent centered Gaussian random variables,\nP\u0012s\nt|Nj|X\nl∈Njϵl(k)≤1\n4\f\f\f\f|Nj|\u0013\n≥ P\u0012s\nt|Nj|X\nl∈Njϵl(k)≤0\f\f\f\f|Nj|\u0013\n=1\n2.\nThe same holds fors\nt|N′\nj|P\nl∈N′\njϵ′\nl(k). By a union bound and Lemma 22,\nP\u0012s\nt|Nj|X\nl∈Njxl(k)≤1\n4,s\nt|N′\nj|X\nl∈N′\njxl(k)≤1\n4\f\f\f\fxj(k) = 0 ,|Nj|,|N′\nj|\u0013\n≥1− P\u0012s\nt|Nj|X\nl∈Njxl(k)≥1\n4\f\f\f\fxj(k) = 0 ,|Nj|\u0013\n− P\u0012s\nt|N′\nj|X\nl∈N′\njxl(k)≥1\n4\f\f\f\fxj(k) = 0 ,|N′\nj|\u0013\n≥1−exp\u0012\n−t|Nj|\n24s\u0013\n−exp\u0012\n−t|N′\nj|\n24s\u0013\n.\nPutting the above displays together, we hence proved the lemma.\nDenote i ▷ ◁ j the event\n{⟨zi,z′\nj⟩+⟨zj,z′\ni⟩ ≥ ⟨zi,z′\ni⟩+⟨zj,z′\nj⟩}.\n18RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nIt is clear that when i ▷ ◁ j happens, perfect recovery is not possible since by swapping iandj, the\nlossℓin (1) is always not bigger. The following proposition shows this and therefore concludes the\nproof for impossibility of perfect recovery.\nProposition 29 Assume that min{s,qm\ns} ≥candqm\nσ2s2≤Cfor constants c, C > 0. Then there is\na constant δ(c, C)>0such that\nP(i ▷ ◁ j )≥δ(c, C).\nProof The definition of i ▷ ◁ j reads\n⟨zi,z′\ni⟩+⟨zj,z′\nj⟩ ≤ ⟨zi,z′\nj⟩+⟨zj,z′\ni⟩\nwhich by definition of zi,zjand of z′\ni,z′\njmeans that\n⟨xi+δi,xi+δ′\ni⟩+⟨xj+δj,xj+δ′\nj⟩ ≤ ⟨xi+δi,xj+δ′\nj⟩+⟨xj+δj,xi+δ′\ni⟩,\nwhere δi:=zi−xiandδj:=zj−xj. Rearranging the terms, we have\n∥xi−xj∥2+⟨xi−xj,δi−δj⟩+⟨xi−xj,δ′\ni−δ′\nj⟩ ≤ −⟨ δi−δj,δ′\ni−δ′\nj⟩.\nObserve that\n⟨δi−δj,δ′\ni−δ′\nj⟩=X\nk∈Si∪Sj(δi(k)−δj(k))(δ′\ni(k)−δ′\nj(k))\n| {z }\nD0\n+X\nk∈(Si∪Sj)c(δi(k)−δj(k))(δ′\ni(k)−δ′\nj(k))\n| {z }\nD1.\nWe will bound both terms on the right-hand side. In this goal, we recall the event Ei={s\n2≤\n|Si| ≤2s}defined in (4). Using (5) we know that the event Ei∩Ejhappens with probability at least\n1−2e−s/8. We notice that when cEi∩ Ejholds we have\n∥xi−xj∥2≤ |Si|+|Sj| ≤4s.\nBy definition of xi’s and zi’s, we know that\n⟨xi−xj,δi−δj⟩ ≤2dX\nk=1|xi(k)−xj(k)|(a)= 2∥xi−xj∥2≤2(|Si|+|Sj|)≤8s\nwhere to obtain (a)we used the fact that as (xi(k))and(xj(k))are Bernoulli random variables\n∥xi−xj∥2=Pd\nk=1|xi(k)−xj(k)|. Recall that Siis the support of xi. First note that for every k\nwe have |δi(k)−δj(k)|a.s\n≤2. Hence we obtain that on event Ei∩ Ej\n|D0| ≤4|Si∪Sj| ≤4(|Si|+|Sj|)≤16s.\nPutting the above together, this implies that when Ei∩ Ejholds we have\n\f\f∥xi−xj∥2+⟨xi−xj,δi−δj⟩+⟨xi−xj,δ′\ni−δ′\nj⟩+D0\f\f≤28s. (13)\n19LIUAUSTERN\nWe now aim at bounding D1. In this goal, we let b∈ {− 1,0,1}dbe the vector such that\nb(k):=(\n(δi(k)−δj(k))(δ′\ni(k)−δ′\nj(k))ifk∈(S1∪S2)c.\n0 otherwise.\nMoreover for k∈(S1∪S2)c, by definition we have xi(k) =xj(k) = 0 . For ease of notation we\ndefine the events\nMk\ni,j:={xi(k) =xj(k) = 0}.\nWe will first see that P(b(k) = 1| Mk\ni,j) = P(b(k) =−1| Mk\ni,j)for all k∈(S1∪S2)c. We will\nthen lower bound the probability of b(k) =±1conditioned on a few events.\nRecall the definition of Ei,Ejin (4) and the events defined in (6), and let\nFi:=\u001a1\n2t+2·mq≤ |Ni| ≤2t+2et·mq\u001b\nandF′\ni:=\u001a1\n2t+2·mq≤ |N′\ni| ≤2t+2et·mq\u001b\n.\nFurther let Hi,j:=Ei∩ Ej∩ Fi∩ Fj∩ F′\ni∩ F′\nj.\nAs conditionally on Mk\ni,j∩ H i,jthe laws of δi(k),δj(k)∈ {0,1}are the same, by symmetry\nwe have\nP(δi(k)−δj(k) = 1| Mk\ni,j,Hi,j) = P(δi(k)−δj(k) =−1| Mk\ni,j,Hi,j).\nBy using Lemma 28, we obtain that\np:= P(b(k) = 1| Mk\ni,j,Hi,j)\n= P(δi(k)−δj(k) = 1 ,δ′\ni(k)−δ′\nj(k) = 1| Mk\ni,j,Hi,j)\n+ P(δi(k)−δj(k) =−1,δ′\ni(k)−δ′\nj(k) =−1| Mk\ni,j,Hi,j)\n≥1\n2Φc\u0012ct√mq\nsσ\u00132\u0012\n1−2 exp\u0012\n−Ctmq\ns\u0013\u0013\nfor constants ct, Ct>0that only depend on t. Since for k∈(Si∪Sj)cthe entries xi′(k), i′∈Ni\nis independent of Nihence zi(k)only depend on the size of the neighborhood. Therefore, b(k)’s\nare independent of each other. Applying Proposition 13 toP\nk∈(S1∪S2)cb(k)we obtain that\nP\u0012X\nk∈(S1∪S2)cb(k)≥28s\f\f\f\fHi,j\u0013\n≥cβexp\u0012\n−Cβs2\np(d−4s)\u0013\n.\nFurther use the bound (13), we have that\nP(i ▷ ◁ j ) = P(i ▷ ◁ j| Hi,j) P(Hi,j)≥ P\u0012X\nk∈(S1∪S2)cb(k)≥28s\f\f\f\fHi,j\u0013\nP(Hi,j)\n≥cβexp\u0012\n−Cβs2\np(d−4s)\u0013\nP(Hi,j).\nBy a union bound, we know that P(Hi,j)is bounded away from 0if for a constant c >0,\nmin{s, mq} ≥c.\n20RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nTherefore, P(▷ ◁)is bounded away from 0ifpis bounded away from 0which is satisfied if\nmq\ns≥c′andmq\ns2σ2≤C\nfor constants c′, C > 0. The lemma is hence proved.\n6. Proofs of impossibility results for noisy feature alignment\nIn Proposition 31 we will prove that if σ2≥4s\u0000\n1 +K\nn\u0001−2then the probability of perfect recovery\nis bounded away from 0. We state the following information-theoretic proposition before we move\nto the proof.\nProposition 30 Letk≤nbe any arbitrary integer and π∈ P(n)be an arbitrary permutation.\nLetπ(Y) = (yπ(1), . . . ,yπ(k))be the noisy observations of π(X) = (xπ(1), . . . ,xπ(k))as defined\nin Definition 2. Choose G= (g1, . . . ,gk)∈ Rk×dto be an independent Gaussian random matrix\nwith i.i.d. entries gi(j)∼ N(0, σ2). Then, the total variation distance satisfies\nTV(π(Y),G|X)≤1\n2σ∥vec(X)∥\nwhere we denoted\nTV(π(Y),G|X):= sup\nA∈B(Rn×d)| P(π(Y)∈A|X)− P(G∈A)|.\nProof By Pinsker’s inequality, we know that\nTV(π(Y),G|X)≤r\n1\n2KL(π(Y)∥G|X).\nWe remark that by definition of π(Y), we have vec(π(Y))∼ N (vec(Xπ), σ2Ikd). Hence using\nthe formula for the KL divergence between two Gaussian vectors (see, e.g., (Wainwright, 2019,\nExercise 15.13(b))), we have\nKL(π(Y)∥G|X) =1\n2σ2∥vec(π(X))∥2=1\n2σ2∥vec(X)∥2.\nWith the previous proposition in place, we are ready to prove the results. First denote by i ▷ ◁ j\nthe following event:\n{⟨yi,y′\nj⟩+⟨yj,y′\ni⟩ ≥ ⟨yi,y′\ni⟩+⟨yj,y′\nj⟩}.\nNote that i ▷ ◁ j is equivalent to\n{∥yi−y′\nj∥2+∥yi−y′\nj∥2≤ ∥yi−y′\ni∥2+∥yj−y′\nj∥2}.\nHence when i ▷ ◁ j happens, perfect recovery is not possible since if we swap iandj, the loss ℓin\n(3) is not increased.\n21LIUAUSTERN\nProposition 31 Let˜πbe the solution of (3). Ifσ2≥2s\u0000\n1 +K\nn\u0001−2for any K > 4we have that\nP(˜π=π∗)≤e−K−4\n4.\nHence if σ2≫swe have that the probability of perfect recovery will converge to 0.\nProof First by triangle inequality and Proposition 30 we have\nTV((y1,y2),(y2,y1)|x1:2)\n≤TV((y1,y2),(g1,g2)|x1:2) + TV(( y2,y1),(g1,g2)|x1:2)\n≤1\nσ(∥x1∥2+∥x2∥2)1/2.\nThis directly implies that if we write the following set\nA:={(a,b) :⟨a,y′\n2⟩+⟨b,y′\n1⟩ ≥ ⟨a,y′\n1⟩+⟨b,y′\n2⟩}\nthen we have\n| P((y1,y2)∈A|x1:2)− P((y2,y1)∈A|x1:2)| ≤1\nσ(∥x1∥2+∥x2∥2)1/2.\nNow by Jensen inequality we have\n| P((y1,y2)∈A)− P((y2,y1)∈A)|\n≤ E[| P((y1,y2)∈A|x1:2)− P((y2,y1)∈A|x1:2)|]\n≤1\nσE[(∥x1∥2+∥x2∥2)1/2]≤1\nσp\nE[∥x1∥2] + E[∥x2∥2](a)=√\n2s\nσ(14)\nwhere to get (a) we used the fact that ∥x1∥2,∥x2∥2∼Binom( d,s\nd).\nNow note that for continuous random variables (y1,y2)and(y′\n1,y′\n2),\nP(⟨y1,y′\n2⟩+⟨y2,y′\n1⟩=⟨y1,y′\n1⟩+⟨y2,y′\n2⟩) = 0\nThe events {(y1,y2)∈A}and{(y2,y1)∈A}are disjoint except for a zero-probability event.\nThis implies that\n1 = P((y1,y2)∈A) + P((y2,y1)∈A)(a)\n≤2 P((y1,y2)∈A) +√\n2s\nσ\nwhere (a)is a consequence of (14). Hence when σ2≥2s\u0000\n1 +K\nn\u0001−2for some δ >0, we obtain\nthat\nP((y1,y2)∈A)≥K\n2n.\nThis implies that\nP(˜π(1)̸= 1or˜π(2)̸= 2)≥K\n2n.\n22RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nUsing the fact that the random variables ( 1{˜π(i)̸=ior˜π(i+ 1)̸=i+ 1})iodd\ni≤n−1are independent\nwe obtain that\nP(∃is.t.˜π(i)̸=i) = P(∃iodd s.t. ˜π(i)̸=ior˜π(i+ 1)̸=i+ 1)\n≥1−\u0012\n1−K\n2n\u0013n/2−2\n≥1−exp\u0012\n−K−4\n4\u0013\n.\nHence we proved the desired result.\nThe following proposition complements the previous proposition and shows that when σ2≤s\nperfect recovery is still not possible ifσ2√\nd\nsis sufficiently large.\nProposition 32 Suppose that1\n4σ2≤s≤d\n2. Assume that ˜πis the solution of (3). Ifσ2≥√\n21s√\nd\nand then\nP(˜π̸=π∗)≥1\n8.\nIfs≫bnandσ2≫s√dbnfor a sequence (bn)diverging to infinity at a rate slower than bn=\nO(logn)then\nlim\nn→∞P(˜π=π∗) = 0 .\nProof Recall that i ▷ ◁ j if\n⟨yi,y′\ni⟩+⟨yj,y′\nj⟩ ≤ ⟨yi,y′\nj⟩+⟨yj,y′\ni⟩;\nwhich by definition of yi,yjand of y′\ni,y′\njmeans that\n⟨xi+ϵi,xi+ϵ′\ni⟩+⟨xj+ϵj,xj+ϵ′\nj⟩ ≤ ⟨xi+ϵi,xj+ϵ′\nj⟩+⟨xj+ϵj,xi+ϵ′\ni⟩.\nRearranging the terms yields i ▷ ◁ j if and only if\n∥xi−xj∥2+⟨xi−xj,ϵi−ϵj+ϵ′\ni−ϵ′\nj⟩ ≤ −⟨ ϵi−ϵj,ϵ′\ni−ϵ′\nj⟩.\nDefine the events\nA0:=\u001a\n−⟨ϵi−ϵj,ϵ′\ni−ϵ′\nj⟩ ≥(a1+a2)s\u001b\n, (15)\nA1:=\u001a\n∥xi−xj∥2≤a1s\u001b\n,\nA2:=\u001a\n⟨xi−xj,ϵi−ϵj+ϵ′\ni−ϵ′\nj⟩ ≤a2s\u001b\n.\nWe first remark that\nP(i ▷ ◁ j )≥ P(A0∩A1∩A2) = P(A0)− P(A0∩Ac\n1)− P(A0∩A1∩Ac\n2)\n≥ P(A0)− P(Ac\n1)− P(A1∩Ac\n2)≥ P(A0)− P(Ac\n1)− P(Ac\n2|A1).\nHence to obtain the desired result we will bound P(A0)from below and P(Ac\n1), P(Ac\n2|A1)from\nabove.\n23LIUAUSTERN\nWe first focus on the first result. In this goal, we remark that ⟨ϵi−ϵj,ϵ′\ni−ϵ′\nj⟩=Pd\nk=1(ϵi(k)−\nϵj(k))(ϵ′\ni(k)−ϵ′\nj(k))is a sum of i.i.d. random variables. Hence to lower bound P(A0)we can use\nthe central limit theorem. We choose a1= 7 anda2= 14 in (15). Moreover, for all k∈[d]we\ndefine\nb(k):= (ϵi(k)−ϵj(k))(ϵ′\ni(k)−ϵ′\nj(k)).\nBy definition of ϵiandϵj, we know that\nE[b(k)] = 0 ,\nE[b(k)2] = E[(ϵi(k)−ϵj(k))2] E[(ϵ′\ni(k)−ϵ′\nj(k))2] = 4σ4,\nE[|b(k)|3] = E[|ϵi(k)−ϵj(k)|3] E[|ϵ′\ni(k)−ϵ′\nj(k)|3] =64σ6\nπ.\nTherefore, by Berry–Esseen theorem (Tyurin, 2010), we have\n\f\f\f\fP(A0)−Φc\u001221s\n2σ2√\nd\u0013\f\f\f\f≤4\nπ√\nd.\nSince d≥(21s/σ2)2ands≥4σ2, we obtain that\nP(A0)≥Φc\u001221s\n2σ2√\nd\u0013\n−4\nπ√\nd≥Φc\u00121\n4\u0013\n−4\n42π≥3\n10−1\n30=4\n15.\nWe now aim to upper bound P(Ac\n1). Since\nE∥xi−xj∥2= 2d·s\nd·\u0012\n1−s\nd\u0013\n= 2s\u0012\n1−s\nd\u0013\n,\nusing δ= 6in Proposition 10 yields\nP(Ac\n1)≤exp\u0012\n−δ2E∥xi−xj∥2\n2 +δ\u0013\n= exp\u0012\n−9s\u0012\n1−s\nd\u0013\u0013\n≤exp\u0012\n−9\n2\u0013\nford≥2s.\nFinally we remark that conditional on xiandxj, the random variable ⟨xi−xj,ϵi−ϵj+ϵ′\ni−ϵ′\nj⟩\nis distributed as N(0,4∥xi−xj∥2σ2). Hence, by Gaussian concentration, we obtain that\nP(Ac\n2|xi,xj)≤exp\u0012\n−49s2\n2∥xi−xj∥2σ2\u0013\n≤exp\u0012\n−98s\n∥xi−xj∥2\u0013\n.\nThen, we have\nP(Ac\n2|A1)≤exp(−7).\nPutting them together, we have when d≥(21s)2/σ4\nP(i ▷ ◁ j )≥4\n15−exp\u0012\n−9\n2\u0013\n−exp(−7)≥1\n4.\nNow by a union bound argument we know that\nP(i ▷ ◁ j )≤ P(ˆπ(i)̸=i) + P(ˆπ(j)̸=j) = 2 P(ˆπ(i)̸=i).\n24RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nThis directly implies the first result.\nWe will now establish the second one. In this goal we note that as we have assumed that s≫bn\nwe know that\nmax\u0012 √sbn\n16(1 +p\ns/bn),√sbn\n2(2p\ns/bn+ 1)\u0013\n∼max\u0012bn\n16,bn\n4\u0013\n,\n11(1 + σ+p\ns/bn)2sbn\n2dσ4∼11s2\n2dσ4.\nMoreover we have assumed that bn≫s2\ndσ4hence there exists constants C1, C2>0such that for n\nlarge enough we have\nexp\u0012\n−C2\n2√sbn\n16(C1+p\ns/bn)\u0013\n+ exp\u0012\n−√sbnC2\n1\n2(2p\ns/bn+C1)\u0013\n≥1\n2exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\nand\nexp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n≤1−√\n3\n2.\nSeta1:=C1√bn√s+ 1anda2=C2σ√bn√s. Using Proposition 12 with u= (a1+a2)s/√\ndwe have\nP(A0)≥\u0012\n1−exp\u0012\n−(a1+a2)2s2\ndσ4\u0013\u00132\nexp\u0012\n−11(a1+a2)2s2\n2dσ4\u0013\n≥\u0012\n1−exp\u0012\n−(C1+C2σ+p\ns/bn)2sbn\ndσ4\u0013\u00132\nexp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n∼\u0012\n1−exp\u0012\n−s2\ndσ4\u0013\u00132\nexp\u0012\n−11s2\n2dσ4\u0013\n.\nWrite δ=a1−1then we have\nP(Ac\n1)≤exp\u0012\n−δ2E∥xi−xj∥2\n2 +δ\u0013\n= exp\u0012\n−(a1−1)2\n1 +a1s\u0012\n1−s\nd\u0013\u0013\n(a)\n≤exp\u0012\n−(a1−1)2\n2(1 + a1)s\u0013\n≤exp\u0012\n−√sbnC2\n1\n2(2p\ns/bn+C1)\u0013\n∼exp\u0012\n−bnC2\n1\n4\u0013\nwhere (a) is due to the assumption that d≥2s. Now by once again, conditionally on xiandxj, the\nrandom variable ⟨xi−xj,ϵi−ϵj+ϵ′\ni−ϵ′\nj⟩is distributed as N(0,4∥xi−xj∥2σ2). Hence we also\nhave\nP(Ac\n2|xi,xj)≤exp\u0012\n−a2\n2s2\n16∥xi−xj∥2σ2\u0013\nand hence that\nP(Ac\n2|A1)≤exp\u0012\n−a2\n2s\n16a1σ2\u0013\n≤exp\u0012\n−C2\n2√sbn\n16(C1+p\ns/bn)\u0013\n∼exp\u0012\n−C2\n2bn\n16\u0013\n.\n25LIUAUSTERN\nHence, for nlarge enough we have\nP(i ▷ ◁ j )≥exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n×\u0012\u0012\n1−exp\u0012\n−(C1+C2σ+p\ns/bn)2sbn\ndσ4\u0013\u00132\n−1\n2\u0013\n≥1\n4exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n∼1\n4exp\u0012\n−11s2\n2dσ4\u0013\n.\nMoreover we remark that by writing\np0:=1\n4exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n∼1\n4exp\u0012\n−11s2\n2dσ4\u0013\nwe obtain that for nlarge enough we have\nP(∃i, js.t.i ▷ ◁ j )≥ P(∃odd≤n−1s.t.i ▷ ◁ j )\n(a)\n≥1−( P(1▷ ◁2))⌊(n−1)/2⌋\n≥1−\u0012\n1 +n\n8exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\u0013−1\nWe have assumed thats2\n2dσ4≪bn=O(log(n))which implies that\n\u0012\n1 +n\n8exp\u0012\n−11(C1+C2σ+p\ns/bn)2sbn\n2dσ4\u0013\n∼1\n4exp\u0012\n−11s2\n2dσ4\u0013\u0013−1\n=o(1).\nHence we obtain the desired result.\n7. Open problems\nIn this paper, we showed various possibility and impossibility results for perfect recovery under\nthe noisy sparse feature setting. The same questions can be asked for other feature distributions, for\nexample, Gaussian features that are the center object of study by Kunisky and Niles-Weed (2022), or\nspherical distribution arising from the high-dimensional random geometric graphs (Devroye et al.,\n2011; Bubeck et al., 2016; Brennan et al., 2020; Liu and R ´acz, 2023a). However, it would be\nless clear what role message passing plays under these settings. Nevertheless averaging over the\nneighbors should still be effective thanks to the symmetry of the distributions.\nPerfect recovery is the main objective of this work. It would also be interesting and more\nimportantly useful in practice to study partial recovery for both vertex features and the unknown\nalignment. There has been quite a lot of research for partial recovery in either aligning Gaussian\nfeatures (Kunisky and Niles-Weed, 2022) or random graph matching (Cullina et al., 2019). How-\never, these questions still remain open for the current model. The techniques developed in the proofs\nof our perfect recovery results should bring us closer to the answers.\n26RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nReferences\nMikl ´os Ajtai, J ´anos Koml ´os, and G ´abor Tusn ´ady. On optimal matchings. Combinatorica , 4:259–\n264, 1984.\nMatthew Brennan, Guy Bresler, and Dheeraj Nagaraj. Phase transitions for detecting latent geome-\ntry in random graphs. Probability Theory and Related Fields , 178(3-4):1215–1289, 2020.\nS´ebastien Bubeck, Jian Ding, Ronen Eldan, and Mikl ´os Z R ´acz. Testing for high-dimensional\ngeometry in random graphs. Random Structures & Algorithms , 49(3):503–532, 2016.\nDaniel Cullina, Negar Kiyavash, Prateek Mittal, and H Vincent Poor. Partial recovery of Erd ¨os-\nR´enyi graph alignment via k-core alignment. Proceedings of the ACM on Measurement and\nAnalysis of Computing Systems , 3(3):1–21, 2019.\nLuc Devroye, Andr ´as Gy ¨orgy, G ´abor Lugosi, and Frederic Udina. High-Dimensional Random\nGeometric Graphs and their Clique Number. Electronic Journal of Probability , 16(none):2481 –\n2508, 2011.\nJian Ding, Zongming Ma, Yihong Wu, and Jiaming Xu. Efficient random graph matching via degree\nprofiles. Probability Theory and Related Fields , 179:29–115, 2021.\nWenqi Fan, Yao Ma, Qing Li, Yuan He, Eric Zhao, Jiliang Tang, and Dawei Yin. Graph neural\nnetworks for social recommendation. In The World Wide Web Conference , pages 417–426, 2019.\nVikas Garg, Stefanie Jegelka, and Tommi Jaakkola. Generalization and representational limits of\ngraph neural networks. In International Conference on Machine Learning , pages 3419–3430.\nPMLR, 2020.\nJustin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural\nmessage passing for quantum chemistry. In International Conference on Machine Learning ,\npages 1263–1272. PMLR, 2017.\nWill Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs.\nInAdvances in Neural Information Processing Systems , volume 30, 2017.\nStefanie Jegelka. Theory of graph neural networks: representation and learning , volume 7, pages\n5450–5476. EMS Press, 2022.\nJean-Pierre Kahane. Some random series of functions , volume 5. Cambridge University Press,\n1985.\nThomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional net-\nworks. In International Conference on Learning Representations , 2016.\nHarold W Kuhn. The hungarian method for the assignment problem. Naval Research Logistics\nQuarterly , 2(1-2):83–97, 1955.\nDmitriy Kunisky and Jonathan Niles-Weed. Strong recovery of geometric planted matchings. In\nProceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages\n834–876. SIAM, 2022.\n27LIUAUSTERN\nSuqi Liu and Mikl ´os Z R ´acz. Phase transition in noisy high-dimensional random geometric graphs.\nElectronic Journal of Statistics , 17(2):3512–3574, 2023a.\nSuqi Liu and Mikl ´os Z R ´acz. A probabilistic view of latent space graphs and phase transitions.\nBernoulli , 29(3):2417–2441, 2023b.\nAndreas Loukas. What graph neural networks cannot learn: depth vs width. In International\nConference on Learning Representations , 2020.\nCheng Mao, Mark Rudelson, and Konstantin Tikhomirov. Random graph matching with improved\nnoise robustness. In Conference on Learning Theory , pages 3296–3329. PMLR, 2021.\nChristopher Morris, Martin Ritzert, Matthias Fey, William L Hamilton, Jan Eric Lenssen, Gaurav\nRattan, and Martin Grohe. Weisfeiler and Leman go neural: Higher-order graph neural networks.\nInProceedings of the AAAI Conference on Artificial Intelligence , volume 33, pages 4602–4609,\n2019.\nRaymond EAC Paley and Antoni Zygmund. On some series of functions,(3). In Mathematical\nProceedings of the Cambridge Philosophical Society , volume 28, pages 190–205. Cambridge\nUniversity Press, 1932.\nMathew Penrose. Random Geometric Graphs . Oxford University Press, 05 2003.\nMiklos Racz and Anirudh Sridhar. Correlated stochastic block models: Exact graph matching with\napplications to recovering communities. Advances in Neural Information Processing Systems ,\n34:22259–22273, 2021.\nFranco Scarselli, Ah Chung Tsoi, and Markus Hagenbuchner. The vapnik–chervonenkis dimension\nof graph and recursive neural networks. Neural Networks , 108:248–259, 2018.\nMichael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max\nWelling. Modeling relational data with graph convolutional networks. In The Semantic Web: 15th\nInternational Conference, ESWC 2018, Heraklion, Crete, Greece, June 3–7, 2018, Proceedings\n15, pages 593–607. Springer, 2018.\nRoded Sharan and Trey Ideker. Modeling cellular machinery through biological network compari-\nson. Nature Biotechnology , 24(4):427–433, 2006.\nKaren B Singer. Random intersection graphs . PhD thesis, Johns Hopkins University, 1996.\nIlya S Tyurin. An improvement of upper estimates of the constants in the lyapunov theorem. Russian\nMathematical Surveys , 65(3):201–202, 2010.\nMartin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint . Cambridge\nSeries in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019.\nHaoyu Wang, Yihong Wu, Jiaming Xu, and Israel Yolou. Random graph matching in geometric\nmodels: the case of complete graphs. In Conference on Learning Theory , pages 3441–3488.\nPMLR, 2022.\n28RANDOM GEOMETRIC GRAPH ALIGNMENT WITH GRAPH NEURAL NETWORKS\nZhichun Wang, Qingsong Lv, Xiaohan Lan, and Yu Zhang. Cross-lingual knowledge graph align-\nment via graph convolutional networks. In Proceedings of the 2018 Conference on Empirical\nMethods in Natural Language Processing , pages 349–357, 2018.\nYihong Wu, Jiaming Xu, and H Yu Sophie. Settling the sharp reconstruction thresholds of random\ngraph matching. IEEE Transactions on Information Theory , 68(8):5391–5417, 2022.\nStephen J Young and Edward R Scheinerman. Random dot product graph models for social net-\nworks. In International Workshop on Algorithms and Models for the Web-Graph , pages 138–149.\nSpringer, 2007.\nAnru R Zhang and Yuchen Zhou. On the non-asymptotic and sharp lower tail bounds of random\nvariables. Stat, 9(1):e314, 2020.\n29"    },    {        "title": "2402.07441v2.Fully_Dynamic_Geometric_Vertex_Cover_and_Matching.pdf",        "content": "arXiv:2402.07441v2  [cs.CG]  14 Feb 2024Fully Dynamic Geometric Vertex Cover and Matching\nSujoy Bhore *Timothy M. Chan†\nAbstract\nIn this work, we study two fundamental graph optimization pr oblems, minimum vertex\ncover (MVC) and maximum-cardinality matching (MCM), for in tersection graphs of geometric\nobjects, e.g., disks, rectangles, hypercubes, etc., in d-dimensional Euclidean space. We consider\nthe problems in fully dynamic settings, allowing insertion s and deletions of objects.\nWe develop a general framework for dynamic MVC in intersecti on graphs, achieving sub-\nlinear amortized update time for most natural families of ge ometric objects. In particular, we\nshow that -\n• For a dynamic collection of disks in R2or hypercubes in Rd(for constant d), it is possible\nto maintain a(1+ε)-approximate vertex cover in polylog amortized update time. These\nresults also hold in the bipartite case.\n• For a dynamic collection of rectangles in R2, it is possible to maintain a (3\n2+ε)-approximate\nvertex cover in polylog amortized update time.\nAlong the way, we obtain the first near-linear time static alg orithms for MVC in the above two\ncases with the same approximation factors.\nNext, we turn our attention to the MCM problem. Although our M VC algorithms automat-\nically allow us to approximate the size of the MCM in bipartit e geometric intersection graphs,\nthey do not produce a matching. We give another general frame work to maintain an approxi-\nmate maximum matching, and further extend the approach to ha ndle non-bipartite intersection\ngraphs. In particular, we show that -\n• For a dynamic collection of (bichromatic or monochromatic ) disks in R2or hypercubes in\nRd(for constant d), it is possible to maintain a (1+ε)-approximate matching in polylog\namortized update time.\n1 Introduction\nDynamic geometric algorithms form a vital area of research i n Computational Geometry. In dy-\nnamic settings, the input is subject to discrete changes ove r time, that is, insertions and dele-\ntions of geometric objects. Over the years, dynamic algorit hms have been developed for a va-\nriety of geometric problems, including convex hull [ OvL81 ,Cha01 ,BJ02 ,Cha10 ,Cha20b ], clos-\nest/farthest pair [ Epp95 ,Cha20a ], connectivity [ Cha06 ,AC09 ,CPR11 ,KKK+22], width and other\n*Department of Computer Science & Engineering, Indian Insti tute of Technology Bombay, Mumbai, India.\nEmail: sujoy@cse.iitb.ac.in\n†Department of Computer Science, University of Illinois at U rbana-Champaign. Email: tmc@illinois.edu. Work\nsupported in part by NSF Grant CCF-2224271.\n1measures [ Cha01 ,Cha03b ,Cha20b ], coresets [ Cha09 ], and many others. More recently, much at-\ntention has turned towards NP-hard geometric optimization problems—building dynamic d ata\nstructures that can maintain solutions for an optimization task at hand, in truly sublinear update\ntime. Since it is only possible to maintain some approximate solution for an NP-hard problem,\nthe broad goal is to explore the trade-offs between update ti me and approximation ratio. Op-\ntimization problems that are solvable in polynomial time bu t do not have subquadratic exact\nalgorithms may also benefit from allowing approximation. Sev eral fundamental geometric op-\ntimization problems have been studied in dynamic settings o ver the last few years, for instance,\nset cover [ ACS+22,CH21 ,CHSX22 ,KLR+23], piercing or hitting set [ ACS+22,AHRS24 ], and inde-\npendent set [ BCIK21 ,HNW20 ,BLN22 ,BNTW23 ].\nIn this work, we focus on two other fundamental optimization problems: minimum vertex\ncover (MVC) and maximum-cardinality matching (MCM). Let G=(V,E)be an undirected graph.\nA subset of vertices S⊆Vis a vertex cover if each edge has at least one endpoint in S. An MVC\nrefers to a vertex cover with the minimum cardinality. A simp le algorithm that greedily computes\na maximal matching1and outputs its vertices gives a 2-approximation for MVC. This is likely to\nbe the best possible for general graphs, as the problem is har d to approximate within factor 2−o(1)\nunder the unique games conjecture [ KR08 ].\nMVC stands out as one of the most important problems, due to its connection to other funda-\nmental problems. For example, in the exact setting, the maxi mum independent set (MIS) problem\nis equivalent to MVC, as the complement of an MIS is an MVC. Howeve r, from the approxima-\ntion perspective, a sharp dichotomy exists between these pr oblems; for instance, MIS on general\ngraphs is much harder to approximate2. Another closely related classical optimization on graphs\nis MCM, where the objective is to find a matching with the largest n umber of edges. The size of\nan MVC is obviously at least as large as the size of an MCM; for bipa rtite graphs, they are well-\nknown to be equal. MCM is polynomial-time solvable: the class ical algorithm by Hopcroft and\nKarp [ HK73 ] runs in O(m√n)time for bipartite graphs with nvertices and medges, and Vazi-\nrani’s algorithm [ Vaz94 ] achieves the same run time for general graphs. By recent bre akthrough\nresults [ CKL+22], MCM can be solved in m1+o(1)time for bipartite graphs. Earlier, Duan and\nPettie [ DP10 ] obtained O(m)-time(1+ε)-approximation algorithms for general graphs.\nGeometric MVC & MCM. We focus on the intersection graph of a collection L={ℓ1,...,ℓn}of\ngeometric objects in d-dimensional Euclidean space Rd(fordconstant). In the intersection graph\nGL=(VL,EL)ofL, each object ℓi∈Lis represented by a vertex vi∈VL, and any pair of intersecting\nobjectsℓi,ℓj∈Lcorresponds to an edge (vi,vj)∈EL. A wide range of optimization problems have\nbeen studied over the years for various families of geometri c intersection graphs, and better results\nare often possible in such geometric graphs. Concerning MVC, the problem remains NP-hard,\neven for intersection graphs of unit disks [ CCJ90 ], but there exist PTAS s for intersection graphs\nof disks, squares, or other “fat” objects in any constant dim ension [ EJS05 ]. For rectangles3in the\nplane, Bar-Yehuda, Hermelin, and Rawitz [ BYHR11 ] obtained a(3\n2+ε)-approximation algorithm.\nBar-Yehuda et al.’s work made use of Nemhauser and Trotter’s standard LP-based kernelization\n1A matching in an input graph Gis a subset of edges M⊆Esuch that no two edges in Mshare a common endpoint.\nA matching Mis maximal if for every edge (u,v)∈E∖M, eitheruorvis matched in M.\n2No polynomial time algorithm can achieve an approximation f actorn1−ε, unless P=ZPP [Zuc07 ]. Throughout this\npaper,εdenotes an arbitrarily small positive constant.\n3All rectangles and boxes are axis-aligned by default in this paper.\n2for vertex cover [ NTJ75 ], which allows us to approximate the MVC by flipping to an MIS ins tance.\nFollowing the same kernelization approach, Har-Peled [ Har23 ] noted that by using the known\nQQPTAS4for MIS for rectangles [ CE16 ], one can immediately obtain a QQPTAS for MVC for\nrectangles. Similarly, by using the known QPTAS for MIS for polygons in the plane [ AHW19 ], one\ncan obtain a QPTAS for MVC for polygons.\nMCM on geometric intersection graphs has also been studied. E frat, Itai, and Katz [ EIK01 ]\nshowed how to compute the maximum matching in bipartite unit disk graphs in O(n3/slash.left2logn)\ntime. Their algorithm works for other geometric objects; fo r example, it runs in O(n3/slash.left2polylogn)\ntime for bipartite intersection graphs of arbitrary disks, by using known dynamic data structures\nfor disk intersection searching [ KKK+22]. Bonnet, Cabello, and Mulzer [ BCM23 ] improved the\nrunning time to O(nω/slash.left2)for certain cases, e.g., translates of a convex object, or ge ometric objects\nwith low “density”, where ω<2.38is the matrix multiplication exponent. Recently, Har-Pele d and\nYang [ HY22 ] presented near-linear time (1+ε)-approximation algorithms for MCM in (bipartite\nor non-bipartite) intersection graphs of arbitrary disks, among other things.\nDynamic geometric MVC & MCM. In this paper, we are interested in the dynamic version of\nMVC and MCM problems, for geometric intersection graphs, subj ect to insertions and deletions\nof objects inL. The goal is to obtain a data structure that can efficiently mai ntain an approximate\nMVC or MCM in GLat any point in time, with truly sublinear update times for bo th insertions\nand deletions.\nProblem 1. Given a dynamic collection of geometric objects in Rd, is it possible to maintain a\n(1+ε)-approximate MVC in truly sublinear update time?\nProblem 2. Given a dynamic collection of geometric objects in Rd, is it possible to maintain a\n(1+ε)-approximate MCM in truly sublinear update time?\nDynamic MVC & MCM are well-studied problems in the dynamic grap h algorithms literature,\nunder edge updates; see [ OR10 ,GP13 ,BS15 ,BS16 ,PS16 ,Sol16 ,BHI18 ,BK19 ,Beh23 ,BKSW23 ,\nABR24 ] for a non-exhaustive list. However, none of these results p articularly apply to our sce-\nnario, since the insertion/deletion of a single object corr esponds to a vertex update, which may\nrequire many edge updates. A natural barrier for designing d ynamic graph algorithms under\nvertex updates is the maximum degree of the input graph, whic h may be Ω(n). For geometric\nintersection graphs, however, the challenge is to break suc h barriers by not explicitly maintaining\nthe graph itself, which may have m=Ω(n2)size in the worst case.\nAlthough geometric versions of dynamic set cover, dynamic h itting set, and dynamic indepen-\ndent set have received much attention recently (as we have me ntioned), we are not aware of any\nprior work on dynamic geometric vertex cover or dynamic geom etric matching.\nFor unit disks in the plane, or more generally, fat objects of roughly the same size in Rd(for\nconstant d), it is not difficult to maintain a (1+ε)-approximate MVC in constant update time, by\nusing the standard shifted-grid technique [ HM85 ]. Moreover, this technique extends to maintain\na(1+ε)-approximate non-bipartite MCM in constant update time (see also [ HY22 ] in the static\n4We refer to a(1+ε)-approximation algorithm with running time of the form nO(polylog n)ornO(poly(loglogn)), for\nany fixed ε>0, as a QPTAS orQQPTAS respectively.\n3setting). However, the problems become much more challengi ng for more general families of\nobjects, such as disks of arbitrary radii, or arbitrary (non -fat) rectangles.\nIn this work, we resolve Problems 1and 2in an affirmative sense for many types of geometric\nobjects:\nOur contributions to MVC. We develop a general framework for dynamic geometric MVC (The -\norem 1), which implies all the specific results listed in Table 1.\nObjects Approx. Ratio Amortized update time Reference\nDisks in R21+ε O(2O(1/slash.leftε2)logO(1)n) Corollary 1\nRectangles in R2 3\n2+ε O(2O(1/slash.leftε2)logO(1)n) Corollary 2\nFat boxes in Rd1+ε O(2O(1/slash.leftεd)logO(1)n) Corollary 3\nBipartite disks in R21+ε O((1/slash.leftε7)logO(1)n) Corollary 4\nBipartite boxes (non-fat) in Rd1+ε O((1/slash.leftε7)logO(1)n) Corollary 5\nTable 1: Summary of results on dynamic MVC for intersection graphs of geometric objects.\nThese results on geometric vertex cover are notable when com pared against recent results on\nrelated dynamic geometric problems such as set cover and ind ependent set:\n• The polylogarithmic update times in Table 1are significantly better than known results for\ndynamic geometric set cover. For example, even for the simpl e case of squares in R2, the\nbest update time for set cover was near√n[CHSX22 ]; for the case of disks in R2, the best\nupdate time was near n12/slash.left13[CH21 ] (and was only for maintaining the approximate size but\nnot a cover).\n• Many of the results in Table 1achieve the ideal approximation ratio , namely, 1+ε. In con-\ntrast, previous dynamic algorithms for geometric set cover [CH21 ,CHSX22 ] were all in the\nregime of O(1)approximation ratio. Similarly, for dynamic geometric MIS, recent algo-\nrithms for squares or hypercubes [ HNW20 ] and disks [ BNTW23 ] all achieved O(1)approx-\nimation only.\n• Our framework is general and applicable to more types of objects. Though not stated in the\ntable, we can handle any class of semialgebraic objects with constant description complexity\nin sublinear time with approximation ratio 2+ε(or1+εin the bipartite cases). In contrast,\nthe known results for dynamic geometric set cover [ CH21 ,CHSX22 ] were limited to certain\nspecific types of objects; for example, they did not extend to b alls inR3or rectangles in R2.\nThere were no nontrivial results for dynamic MIS for rectangl es with constant approxima-\ntion ratio.\nA prerequisite to a (1+ε)-approximate dynamic algorithm with polylogarithmic upda te time\nis the existence of a near-linear-time static algorithm. MIS for disks (or fat objects) admits static\nPTAS s [EJS05 ,Cha03a ], but not an EPTAS5under standard hypotheses in parameterized complex-\nity [Mar05 ]; similarly, set cover for disks admits static PTAS s [MR10 ] but not an EPTAS [Mar05 ].\n5AnEPTAS refers to a PTAS with running time O(nc)for a constant cindependent of ε.\n4In contrast, an EPTAS for MVC for disks (or fat objects) can be obtained by known tech niques\nusing the aforementioned Nemhauser–Trotter kernel [ NTJ75 ]. This explains why in some sense,\ngeometric vertex cover is actually a better problem to study in dynamic settings than geometric\nindependent set or geometric set cover.\nHowever, a static EPTAS for MVC for disks (or fat objects) that runs in near linear time has not\nbeen explicitly given before, to the best of our knowledge. S imilarly, the static (3\n2+ε)-approximation\nalgorithm for MVC for rectangles by Bar-Yehuda, Heremelin, a nd Rawitz [ BYHR11 ] runs in poly-\nnomial time but not in near linear time. In deriving our dynam ic results, we obtain the first static\nnear-linear-time algorithms in these cases, which may be of independent interest.\nOur contributions to MCM. Next, we focus on MCM. Our approximation algorithms for MVC\nin bipartite geometric intersection graphs allows us to aut omatically approximate the size of the\nMCM in such bipartite graphs. However, it does not compute a ma tching. We develop an-\nother general framework for dynamic geometric MCM (Theorem 2), and further extend it to non-\nbipartite intersection graphs. The framework implies the s pecific results in Table 2:\nObjects Approx. Ratio Amortized update time Reference\nBipartite disks in R21+ε O((1/slash.leftε3)logO(1)n) Corollary 6\nBipartite boxes in Rd1+ε O((1/slash.leftε3)logO(1)n) Corollary 7\nDisks in R21+ε O(2O(1/slash.leftε)logO(1)n) Corollary 8\nBoxes in Rd1+ε O(2O(1/slash.leftε)logO(1)n) Corollary 9\nTable 2: Summary of results on dynamic MCM for intersection graphs of geometric objects.\nIn particular, we succeed in efficiently dynamizing the known static near-linear-time (1+ε)-\napproximation MCM algorithms for (bipartite or non-biparti te) disks by Har-Peled and Yang [ HY22 ].\nOur framework and techniques for MVC. Our framework for MVC has only 2 requirements on\nthe geometric object family:\n(i) the existence of an efficient data structure for dynamic intersection detection (maintaining a\nset of input objects under insertions and deletions so that w e can quickly decide whether a\nquery object intersects any of the input objects), and\n(ii) the existence of an efficient static approximation algorithm .\nIf the data structure in (i) takes polylogarithmic query and update time and the algorithm in\n(ii) takes near linear time ignoring polylogarithmic facto rs, then the resulting dynamic algorithm\nachieves polylogarithmic (amortized) update time, and its approximation ratio is the same as the\nstatic algorithm, up to 1+εfactors.\nIn some sense, the framework is as general as possible: Requi rement (ii) is an obvious pre-\nrequisite (as we have already noted); in fact, we only requir e a static algorithm for the case when\nthe optimal value is promised to be at least n/slash.left2roughly. Requirement (i) is also natural, as some\nkind of geometric range searching data structures is provab ly necessary. For example, consider\nthe case of disks in R2. Any dynamic approximate MVC algorithm needs to recognize wh ether the\n5MVC size is zero, and so must know whether the intersection gra ph is empty. Dynamic disk range\nemptiness (maintaining a set of input points under insertio ns and deletions so that we can quickly\ndecide whether a query disk contains any of the input points) can be reduced to this problem, by\ninserting all the input points, and repeatedly inserting a q uery disk and deleting it. (This explains\nwhy for our result on disks, we need multiple logarithmic fac tors in the update time, since the\nbest algorithm for dynamic disk range emptiness requires mu ltiple logarithmic factors [ Cha10 ],\nalthough the current best algorithm for dynamic disk inters ection detection requires still more\nlogarithmic factors [ KKK+22].)\nOur framework for MVC is established by combining a number of i deas. The initial idea is\na standard one: periodic rebuilding . The intuition is that when the optimal value is large, we can\nafford to do nothing for a long period of time, without hurtin g the approximation ratio by much.\nThis idea has been used before in a number of recent works on dy namic geometric independent set\n(e.g., [ CIK21 ]), dynamic piercing (e.g., [ AHRS24 ]), and dynamic graph algorithms (e.g., [ GP13 ]),\nalthough the general approach of periodic rebuilding is com monplace and appeared in much\nearlier works in dynamic computational geometry (e.g., [ Cha01 ,Cha03b ]).\nThe main challenge now lies in the case when the optimal value OPT is small. Our idea is\nto compute Nemhauser and Trotter’s standard LP-based kerne l for MVC, and then run a static\nalgorithm on the kernel. Unfortunately, we are unable to sol ve the LP exactly in sublinear time.\nHowever, by using the multiplicative weight update (MWU) method, we show that it is possible to\nsolve the LP approximately in time roughly linear in OPT (and thus sublinear in nin the small OPT\ncase), by using the intersection detection data structure p rovided by (i). We further show that an\napproximate solution to the LP is still sufficient to yield a ke rnel with approximately the same size.\nThe idea of using MWU in the design of dynamic geometric algori thms was pioneered in Chan\nand He’s recent work on dynamic geometric set cover [ CH21 ], and also appeared in subsequent\nwork on dynamic piercing [ AHRS24 ]; however, these works solved the LP in time superlinear in\nOPT, which is why their final update time bounds were significant ly worse.\nIn geometric approximation algorithms, the best-known app lication of MWU is geometric set\ncover or hitting set [ BG95 ]. Our work highlights its usefulness also to geometric vert ex cover—\nironically, the application to vertex cover is even simpler (and so our work might have additional\npedagogical value). Furthermore, the efficient implementat ion of MWU for dynamic geomet-\nric vertex cover turns out to lead to a nice, unexpected appli cation of another known technique:\nnamely, Eppstein’s data structure technique for generalized dynamic closest pair problems [ Epp95 ]\n(see also [ Cha20a ]).\nOur framework and techniques for MCM. For MCM, our framework only needs requirement (i).\nHere, we need different ideas, since kernels for matching do not seem to work as efficiently. In\nthe bipartite case, we use an approach based on Hopcroft and Ka rp’s classical matching algo-\nrithm [ HK73 ], which is known to yield good approximation after a constan t number of iterations.\nWe show how to implement the approximate version of Hopcroft and Karp’s algorithm in time\nroughly linear in OPT, by using the data structure from (i). P reviously, Efrat, Itai, and Katz [ EIK01 ]\nshowed how to implement Hopcroft and Karp’s algorithm faster using geometric data structures,\nbut their focus was on static algorithms whereas our goal is i n getting sublinear time. Thus, our\nadaptation of Hopcroft–Karp will be a little more delicate.\nIn the general non-bipartite case, we apply a technique by Lo tker, Patt-Shamir, and Pettie [ LPP15 ].\nwhich reduces non-bipartite to the bipartite case in the app roximate setting. Har-Peled and Yang [ HY22 ]\n6also applied the same technique to derive their static appro ximation algorithms for geometric non-\nbipartite MCM. We reinterpret this technique in terms of color-coding [AYZ95 ], which allows for\nefficient derandomization and dynamization, as well as a simp ler analysis.\n2 Minimum Vertex Cover\nIn this section, we study the MVC problem for dynamic intersec tion graphs of geometric objects.\n2.1 Approximating the LP via MWU\nFor a graph G=(V,E), afractional vertex cover is a vector(xv)v∈Vsuch that xu+xv≤1for alluv∈E\nandxv∈[0,1]for allv∈V. Itssizeis defined as∑v∈Vxv. Finding a minimum-size fractional vertex\ncover corresponds to solving an LP , namely, the standard LP r elaxation of the MVC problem.\nIt is known that this LP is equivalent to computing the MVC in a r elated bipartite graph, and\nthus can be solved exactly by known bipartite MCM algorithms— in fact, in time almost linear in\nthe number of edges by recent breakthrough results [ CKL+22]. However, there are two issues that\nprevent us from applying such algorithms. First of all, we ar e considering geometric intersection\ngraphs, which may have Ω(n2)number of edges; this issue could potentially be fixed by using\nknown techniques involving biclique covers to sparsify the graph (maximum matching in a bipar-\ntite graph then reduces to maximum flow in a sparser 3-layer gr aph [ FM95 ]). Second, we want\nefficient data structures that can solve the LP still faster, i nsublinear time when OPT is small.\nFor our purposes, we only need to solve the LP approximately. Our idea is to use a different\nwell-known technique: multiplicative weight update (MWU). The key lemma is stated below. The\nMWU algorithm and analysis here are nothing new (the descript ion is short enough that we opt to\ninclude it to be self-contained), and such algorithms have b een used before for static and dynamic\ngeometric set cover and other geometric optimization probl ems (the application to vertex cover\nturns out to be a little simpler). However, our contribution is not in the proof of the lemma, but in\nthe realization that MWU reduces the problem to designing a dy namic data structure (for finding\nmin-weight edges subject to vertex-weight updates), which geometric intersection graphs happen\nto possess, as we will see.\nLemma 1. We are given a graph G=(V,E). Suppose there is a data structure DSfor storing a vector\n(wv)v∈Vthat can support the following two operations in τtime: (i) find an edge uv∈Eminimizing\nwu+wv, and (ii) update a number wv.\nGiven a data structure DSfor the vector that currently has wv=1for allv∈V, we can compute a\n(1+O(δ))-approximation to the minimum fractional vertex cover in ̃O((1/slash.leftδ2)OPT⋅τ)time. Here, OPT\ndenotes the minimum vertex cover size.\nProof. Given a number z, the following algorithm attempts to find a fractional vertex cover of size\nat mostz(below,Wdenotes∑v∈Vwv):\nletwv=1for allv∈V, andW=n\nwhile there exists uv∈Ewithwu+wv<W/slash.leftzdo\nletuvbe such an edge\nW←W+δ(wu+wv),wu←(1+δ)wu,wv←(1+δ)wv\n7If and when the algorithm terminates, we have wu+wv≥W/slash.leftzfor alluv∈E. Thus, defining\nxv∶=min{zwv/slash.leftW,1}, we have xu+xv≥1for alluv∈E, and∑v∈Vxv≤z, i.e.,(xv)v∈Vis a fractional\nvertex cover of size at most z.\nWe now bound the number of iterations t. In each iteration, Wincreases by at most a factor of\n1+δ/slash.leftz. Thus, at the end,\nW≤(1+δ/slash.leftz)tn.\nWrite each wvas(1+δ)cvfor some integer cv. Let(x∗\nv)v∈Vbe an optimal fractional vertex cover\nof sizez∗. In each iteration, ∑v∈Vcvx∗\nvincreases by at least 1 (since we increment cuandcvfor the\nchosen edge uv, and we know x∗\nu+x∗\nv≥1). Thus, at the end, ∑v∈Vcvx∗\nv≥t. Since∑v∈Vx∗\nv=z∗, it\nfollows that maxv∈Vcv≥t/slash.leftz∗. Thus,\nW≥(1+δ)t/slash.leftz∗.\nTherefore,(1+δ)t/slash.leftz∗≤(1+δ/slash.leftz)tn≤eδt/slash.leftzn, implying(t/slash.leftz∗)ln(1+δ)≤δt/slash.leftz+lnn. So, ifz≥(1+δ)z∗,\nthent≤z∗lnn\nln(1+δ)−δ/slash.left(1+δ)=O((1/slash.leftδ2)z∗logn).\nNote that only O(t)=O((1/slash.leftδ2)z∗logn)of the numbers wvare not equal to 1, so the vector\n(xv)v∈Vcan be encoded in ̃O(z∗)space.\nWe can try different zvalues by binary or exponential search till the algorithm te rminates in\nO((1/slash.leftδ2)z∗logn)iterations. Each run can be implemented with O((1/slash.leftδ2)z∗logn)operations in\nDS. After each run, we reset all the modified values wvback to 1 by O((1/slash.leftδ2)z∗logn)update\noperations inDS.\n2.2 Kernel via Approximate LP\nOur approach for solving the vertex cover problem is to use a s tandard kernel by Nemhauser\nand Trotter [ NTJ75 ], which allows us to reduce the problem to an instance where t he number of\nvertices is at most 2OPT. Nemhauser and Trotter’s construction is obtained from the LP solution:\nthe kernel is simply the subset of all vertices v∈Vwithxv=1\n2.\nIn our scenario, we are only able to solve the LP approximatel y. We observe that this is\nstill enough to give a kernel of approximately the same size. We adapt the standard analysis of\nNemhauser and Trotter, but some extra ideas are needed. The p roof below will be self-contained.\nLemma 2. Letc≥1and0≤δ<γ<1\n4. Given a(1+O(δ))-approximation to the minimum fractional\nvertex cover in a graph G=(V,E), we can compute a subset K⊆Vof size at most(2+O(γ))OPT ,\ninO(OPT)time, such that a c(1+O(/radical.alt1\nδ/slash.leftγ))-approximation to the minimum vertex cover of Gcan be\nobtained from a c-approximation to the minimum vertex cover of G[K](the subgraph of Ginduced by K).\nHere, OPT denotes the minimum vertex cover size of G.\nProof. Let(xv)v∈Vbe the given fractional vertex cover. Let λ<γbe a parameter to be set later. Pick\na valueα∈[1\n2−γ−λ,1\n2−λ]which is an integer multiple of λ, minimizing/divides.alt0{v∈V∶α≤xv<α+λ}/divides.alt0.\nThis can be found in O(/divides.alt0{v∈V∶xv≥1\n2−γ−λ}/divides.alt0)≤O(OPT)time.6\nPartition Vinto 3 subsets: L={v∈V∶xv<α},H={v∈V∶xv>1−α}, andK={v∈V∶α≤\nxv≤1−α}. Note that/divides.alt0K/divides.alt0≤1\nα∑v∈Vxv≤(2+O(γ))(1+δ)OPT=(2+O(γ))OPT.\nLetSKbe ac-approximate minimum vertex cover of G[K]. We claim that S∶=SK∪His a\nc(1+O(/radical.alt1\nδ/slash.leftγ))-approximation to the minimum vertex cover of G.\n6This assumes an appropriate encoding of the vector (xv)v∈V, for example, the encoding from the proof of Lemma 1.\n8First,SK∪His a vertex cover of G, since vertices in Lcan only be adjacent to vertices in H.\nLetS∗be a minimum vertex cover of G, with/divides.alt0S∗/divides.alt0=OPT. Since K∩S∗is a vertex cover of\nG[K], we have/divides.alt0SK/divides.alt0≤c/divides.alt0K∩S∗/divides.alt0, and so/divides.alt0S/divides.alt0≤c(/divides.alt0K∩S∗/divides.alt0+/divides.alt0H/divides.alt0)=c(/divides.alt0S∗/divides.alt0+/divides.alt0H∖S∗/divides.alt0−/divides.alt0L∩S∗/divides.alt0).\nWe will upper-bound /divides.alt0H∖S∗/divides.alt0−/divides.alt0L∩S∗/divides.alt0. To this end, let L′={v∈V∶α≤xv<α+λ}, and define\nthe following modified vector (x′\nv)v∈V:\nx′\nv=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩xv−λifv∈H∖S∗\nxv+λifv∈(L∪L′)∩S∗\nxv otherwise.\nNote that(x′\nv)v∈Vis still a fractional vertex cover, since for each edge uv∈Ewithu∈H∖S∗and\nv∈(L∪L′)∩S∗, we have x′\nu+x′\nv=(xu−λ)+(xv+λ)≥1; on the other hand, for each edge uv∈Ewith\nu∈H∖S∗andv/slash.left∈(L∪L′)∩S∗, we have v∈S∗(sinceS∗is a vertex cover) and so v/slash.left∈L∪L′, implying\nthatx′\nu+x′\nv≥(1−α−λ)+(α+λ)≥1. Now,∑v∈Vxv−∑v∈Vx′\nv=λ(/divides.alt0H∖S∗/divides.alt0−/divides.alt0L∩S∗/divides.alt0−/divides.alt0L′∩S∗/divides.alt0). On the\nother hand,∑v∈Vxv−∑v∈Vx′\nv≤(1−1\n1+O(δ))∑v∈Vxv≤O(δ)/divides.alt0S∗/divides.alt0. It follows that/divides.alt0H∖S∗/divides.alt0−/divides.alt0L∩S∗/divides.alt0≤\nO(δ\nλ)/divides.alt0S∗/divides.alt0+/divides.alt0L′∩S∗/divides.alt0.\nBy our choice of α, we have/divides.alt0L′/divides.alt0≤O(1\nγ/slash.leftλ)⋅/divides.alt0{v∈V∶xv≥1\n2−γ−λ}/divides.alt0≤O(λ\nγ)/divides.alt0S∗/divides.alt0. We conclude\nthat/divides.alt0S/divides.alt0≤c(/divides.alt0S∗/divides.alt0+/divides.alt0H∖S∗/divides.alt0−/divides.alt0L∩S∗/divides.alt0)≤c(1+O(δ\nλ+λ\nγ))/divides.alt0S∗/divides.alt0. Choose λ=√γδ.\n2.3 Dynamic Geometric Vertex Cover via Kernels\nWe now use kernels to reduce the dynamic vertex cover for geom etric intersection graphs to a\nspecial case of static vertex cover where the number of objec ts is approximately at most 2OPT. We\nuse a simple, standard idea for dynamization: be lazy, and pe riodically recompute the solution\nonly after every εOPT updates (since the optimal size changes by at most 1 per up date). We\nobserve that the data structure subproblem of dynamic min-w eight intersecting pair, needed in\nLemma 1, is reducible to dynamic intersection detection by known te chniques.\nLemma 3. LetCbe a class of geometric objects, where there is a dynamic data st ructureDS0fornobjects in\nCthat can detect whether there is an object intersecting a query ob ject, and supports insertions and deletions\nof objects, with O(τ0(n))query and update time.\nThen there is a dynamic data structure DSfornweighted objects in Cthat maintains an intersecting\npair of objects minimizing the sum of the weights, under inser tions and deletions of weighted objects, with\ñO(τ0(n))amortized time.\nProof. Eppstein [ Epp95 ] gave a general technique to reduce the problem of dynamic cl osest pair to\nthe problem of dynamic nearest neighbor search, for arbitra ry distance functions, while increasing\nthe time per operation by at most two logarithmic factors (wi th amortization). Chan [ Cha20a ] gave\nan alternative method to achieve a similar result. The DSproblem can be viewed as a dynamic\nclosest pair problem, where the distance between objects uandviswu+wvif they intersect, and\n∞otherwise. Thus, our problem reduces to designing a data str ucture that can find a min-weight\nobject intersecting a query object, subject to insertions a nd deletions of objects ( dynamic min-weight\nintersection searching ).\nWe can further reduce this to the DS0problem ( dynamic intersection detection ) by a standard\nmulti-level data structuring technique [ AE99 ], where the primary data structure is a 1D search\ntree over the weights, and each node of the tree stores a secon dary data structure for dynamic\nintersection detection. Query and update time increase by o ne more logarithmic factor.\n9Theorem 1. Letc≥1,ε>0, and0≤δ<γ<1\n4. LetCbe a class of geometric objects with the following\noracles:\n(i) a dynamic data structure DS0fornobjects inCthat can detect whether there is an object intersecting\na query object, and supports insertions and deletions of obje cts, withO(τ0(n))query and update time;\n(ii) a static algorithm Afor computing a c-approximation of the minimum vertex cover of the intersectio n\ngraph ofnobjects inCinT(n)time, under the promise that n≤(2+O(γ))OPT , where OPT is the\noptimal vertex cover size.\nThen there is a dynamic data structure for nobjects inCthat maintains a c(1+O(/radical.alt1\nδ/slash.leftγ+ε))-\napproximation of the minimum vertex cover of the intersection graph, under insertions and deletions of\nobjects, iñO((1/slash.left(δ2ε))τ0(n)+(1/slash.leftε)T(n)/slash.leftn)amortized time, assuming that T(n)/slash.leftnis monotonically\nincreasing and T(2n)=O(T(n)).\nProof. Assume that b≤OPT<2bfor a given parameter b. Divide into phases with εbupdates each.\nAt the beginning of each phase:\n1. Compute a(1+δ)-approximation to the minimum fractional vertex cover by Le mma 1in\ñO((1/slash.leftδ2)b⋅τ0(n))amortized time using the data structure DSfrom Lemma 3. (A weight\nchange can be done by a deletion and an insertion. It is import ant to note that we don’t\nrebuildDSat the beginning of each phase; we continue using the same str uctureDSin the\nnext phase, after resetting the modified weights back to 1.)\n2. Generate a kernel Kwith size at most (2+O(γ))OPT by Lemma 2inO(b)time.\n3. Compute a c-approximation to the minimum vertex cover of the intersect ion graph of K\nby the static algorithm AinO(T((2+O(γ))OPT))=O(T(b))time, from which we obtain a\nc(1+O(/radical.alt1\nδ/slash.leftγ))-approximation of the minimum vertex cover of the entire int ersection graph.\nSince the above is done only at the beginning of each phase, th e amortized cost per update is\ñO((1/slash.leftδ2)b⋅τ0(n)+T(b)\nεb).\nDuring a phase, we handle an object insertion simply by inser ting it to the current cover, and\nwe handle an object deletion simply by removing it from the cu rrent cover. This incurs an additive\nerror at most O(εb)=O(εOPT). We also perform the insertion/deletion of the object in DS, with\ninitial weight 1, in ̃O(τ0(n))amortized time.\nHow do we obtain a correct guess b? We build the above data structure for each bthat is a\npower of 2, and run the algorithm simultaneously for each b(with appropriate cap on the run\ntime based on b).\n2.4 Specific Results\nWe now apply our framework to solve the dynamic geometric ver tex cover problem for various\nspecific families of geometric objects. By Theorem 1, it suffices to provide (i) a dynamic data\nstructureDS0for intersection detection queries, and (ii) a static algor ithmAfor solving the special\ncase of the vertex cover problem when n≤(2+O(γ))OPT.\n10Disks in R2.Intersection detection queries for disks in R2reduce to additively weighted Eu-\nclidean nearest neighbor search, where the weights (differ ent from the weights from MWU) are\nthe radii of the disks. Kaplan et al. [ KKK+22] adapted Chan’s data structure [ Cha20b ,Cha01 ] for\ndynamic Euclidean nearest neighbor search in R2and obtained a data structure for additively\nweighted Euclidean nearest neighbor search with polylogar ithmic amortized update time and\nquery time. Thus, DS0can be implemented in τ0(n)=O(logO(1)n)amortized time for disks\ninR2.\nIn Appendix A.1, we give a static (1+O(ε))-approximation algorithm Afor MVC for disks,\nunder the promise that n=O(OPT), with running time T(n)=̃O(2O(1/slash.leftε2)n). The algorithm is\nobtained by modifying a known PTAS for MIS for disks [ Cha03a ].\nApplying Theorem 1withc=1+O(ε),γ=Θ(1), andδ=ε2, we obtain our main result for\ndisks:\nCorollary 1. There is a dynamic data structure for ndisks inR2that maintains a(1+O(ε))-approximation\nof the minimum vertex cover of the intersection graph, under in sertions and deletions, in O(2O(1/slash.leftε2)logO(1)n)\namortized time.\nRectangles in R2.For rectangles in R2, dynamic intersection detection requires τ0(n)=O(logO(1)n)\nquery and update time, by standard orthogonal range searchi ng techniques (range trees) [ PS85 ,\nAE99 ].\n(Note: There is an alternative approach that bypasses Eppst ein’s technique and directly solves\ntheDSdata structure problem: We dynamically maintain a biclique cover , which can be done in\npolylogarithmic time for rectangles [ Cha06 ]. It is then easy to maintain the minimum weight of\neach biclique, by maintaining the minimum weight of each of t he two sides of the biclique with\npriority queues.)\nIn Appendix A.2, we give a static(3\n2+O(ε))-approximation algorithm for MVC for rectangles,\nwith running time T(n)=̃O(2O(1/slash.leftε2)n). The algorithm is obtained by modifying the method\nby Bar-Yehuda, Hermelin, and Rawitz [ BYHR11 ], and combining with our efficient kernelization\nmethod.\nApplying Theorem 1withc=3\n2+O(ε),γ=Θ(1), andδ=ε2, we obtain our main result for\nrectangles:\nCorollary 2. There is a dynamic data structure for naxis-aligned rectangles in R2that maintains a(3\n2+\nO(ε))-approximation of the minimum vertex cover of the intersectio n graph, under insertions and deletions,\ninO(2O(1/slash.leftε2)logO(1)n)amortized time.\nFat boxes (e.g., hypercubes) in Rd.For the case of fat axis-aligned boxes (e.g., hypercubes) in a\nconstant dimension d, dynamic intersection detection can again be solved with τ0(n)=O(logO(1)n)\namortized query and update time by orthogonal range searchi ng [PS85 ,AE99 ]. We can design\nthe static algorithm Ain exactly the same way as in the case of disks (since the metho d in Ap-\npendix A.1holds for fat objects in Rd), achieving T(n)=̃O(2O(1/slash.leftεd)n).\nCorollary 3. There is a dynamic data structure for nfat axis-aligned boxes in Rdfor any constant d\nthat maintains a(1+O(ε))-approximation of the minimum vertex cover of the intersectio n graph, under\ninsertions and deletions, in O(2O(1/slash.leftεd)logO(1)n)amortized time.\n11We can also obtain results for balls or other types of fat obje cts inRd, but because intersection\ndetection data structures have higher complexity, the upda te time would be sublinear rather than\npolylogarithmic.\nBipartite disks in R2.For the case of a bipartite intersection graph between two se ts of disks in\nR2, we have τ0(n)=O(logO(1)n)as already noted. In the bipartite case, the static algorith mAis\ntrivial: we just return the smaller of the two parts in the bip artition, which yields a vertex cover of\nsize at most n/slash.left2≤(1+O(ε))OPT under the promise that n≤(2+O(ε))OPT.\nApplying Theorem 1withc=1+O(ε),γ=ε, andδ=ε3, we obtain:\nCorollary 4. There is a dynamic data structure for two sets of O(n)disks inR2that maintains a(1+\nO(ε))-approximation of the minimum vertex cover of the bipartite i ntersection graph, under insertions and\ndeletions, in O((1/slash.leftε7)logO(1)n)amortized time.\nBipartite boxes in Rd.For the case of a bipartite intersection graph between two se ts of (not\nnecessarily fat) boxes in Rd, we have τ0(n)=O(logO(1)n)by orthogonal range searching. As\nalready noted, in bipartite cases, the static algorithm Ais trivial.\nCorollary 5. There is a dynamic data structure for two sets of O(n)axis-aligned boxes in Rdfor any con-\nstantdthat maintains a(1+O(ε))-approximation of the minimum vertex cover of the bipartite i ntersection\ngraph, under insertions and deletions, in O((1/slash.leftε7)logO(1)n)amortized time.\nWe did not write out the number of logarithmic factors in our r esults, as we have not attempted\nto optimize them, but it is upper-bounded by 3 plus the number of logarithmic factors in τ0(n).\n3 Bipartite Maximum-Cardinality Matching\nThe approach in the previous section finds a (1+O(ε))-approximation of the MVC of bipartite\ngeometric intersection graphs, and so it allows us to approx imate the size of the MCM in such\nbipartite graphs. However, it doesn’t compute a matching. I n this section, we give a different\napproach to maintain a (1+O(ε))-approximation of the MCM in bipartite intersection graphs. The\nfirst thought that comes to mind is to compute a kernel, as we hav e done for MVC, but for MCM,\nknown approaches seem to yield only a kernel of O(OPT2)size (e.g., see Gupta and Peng [ GP13 ]).\nInstead, we will bypass kernels and construct an approximat e MCM directly.\n3.1 Approximate Bipartite MCM via Modified Hopcroft–Karp\nIt is well known that Hopcroft and Karp’s O(m√n)-time algorithm for exact MCM in bipartite\ngraphs [ HK73 ] can be modified to give a (1+ε)-approximation algorithm that runs in near-\nlinear time, simply by terminating early after O(1/slash.leftε)iterations (for example, see the introduc-\ntion in [ DP10 ]). We describe a way to reimplement the algorithm in subline ar time when OPT is\nsmall by using appropriate data structures, which correspo nd to dynamic intersection detection\nin the case of geometric intersection graphs. Note that earl ier work by Efrat, Itai, and Katz [ EIK01 ]\nhas already combined Hopcroft and Karp’s algorithm with geom etric data structures to obtain\nstatic exact algorithm [ EIK01 ] for maximum matching in bipartite geometric intersection graphs\n(see also Har-Peled and Yang’s paper [ HY22 ] on static approximation algorithms). However, to\n12achieve bounds sensitive to OPT, our algorithm will work dif ferently (in particular, it will be DFS-\nbased instead of BFS-based).\nLemma 4. We are given an unweighted bipartite graph G=(V,E). Suppose there is a data structure DS0\nfor storing a subset S⊆Vof vertices, initially with S=∅, that supports the following two operations in τ0\ntime: given a vertex u∈V, find a neighbor of uthat is in S(if exists), and insert/delete a vertex to/from S.\nGiven a data structure DS0that currently has S=V, and given a maximal matching M0, we can\ncompute a(1+O(ε))-approximation to the maximum-cardinality matching in ̃O((1/slash.leftε2)OPT⋅τ0)time.\nHere, OPT denotes the maximum matching size.\nProof. The algorithm proceeds iteratively. We maintain a matching M. At the beginning of the\nℓ-th iteration, we know that the current matching Mdoes not have augmenting paths of length\n≤2ℓ−1. We find a maximal collection Γof vertex-disjoint augmenting paths of length 2ℓ+1. We\nthen augment Malong the paths in Γ. As shown by Hopcroft and Karp [ HK73 ], the new matching\nMwill then not have augmenting paths of length ≤2ℓ+1.\nAs shown by Hopcroft and Karp [ HK73 ], there are at least OPT −/divides.alt0M/divides.alt0vertex-disjoint augment-\ning paths, and so /divides.alt0M/divides.alt0≥(ℓ+1)(OPT−/divides.alt0M/divides.alt0), i.e., OPT≤(1+1\nℓ+1)/divides.alt0M/divides.alt0. Thus, once ℓreachesΘ(1/slash.leftε),\nwe may terminate the algorithm. Initially, we can set M=M0before the first iteration.\nIt suffices to describe how to find a maximal collection Γof vertex-disjoint augmenting paths\nof length 2ℓ+1in theℓ-th iteration, under the assumption that there are no shorte r augmenting\npaths. Hopcroft and Karp originally proposed a BFS approach, starting at the “exposed” vertices\nnot covered by the current matching M. Unfortunately, this approach does not work in our setting:\nbecause we want the running time to be near OPT, the searches n eed to start at vertices of M. We\nend up adopting a DFS approach, but the vertices of Mneed to be duplicated ℓtimes inℓ“layers”\n(this increases the running time by a factor of ℓ, but fortunately, ℓis small in our setting).\nLetVMbe the2/divides.alt0M/divides.alt0vertices of the current matching M. As is well known, OPT ≤2/divides.alt0M/divides.alt0. A\nwalkv0u1v1⋯uℓvℓuℓ+1inGis an augmenting walk of length 2ℓ+1ifv0/slash.left∈VM,v0u1/slash.left∈M,u1v1∈M,\n. . . ,vℓuℓ+1/slash.left∈M, anduℓ+1/slash.left∈VM. In such an augmenting walk of length ℓ, we automatically have\nv0≠uℓ+1(because Gis bipartite) and the walk must automatically be a simple pat h (because\notherwise we could short-cut and obtain an augmenting path o f length≤2ℓ−1). In the procedure\nEXTEND(v0u1v1⋯ui,ℓ)below, the input is a walk v0u1v1⋯uiwithv0u1/slash.left∈M,u1v1∈M, . . . ,vi−1ui/slash.left∈\nM, and the output is true if it is possible to extend it to an augm enting walk v0u1v1⋯uℓvℓuℓ+1of\nlength2ℓ+1that is vertex-disjoint from the augmenting walks generate d so far.\nMAXIMAL -AUG -PATHS(ℓ):\n1. letS=V∖VMandS1=⋯=Sℓ=VM\n2. for each u1∈S1do\n3. let v0be a neighbor of u1withv0∈S\n4. if v0does not exist then delete u1fromS1\n5. else EXTEND(v0u1,ℓ)\nEXTEND(v0u1v1⋯ui,ℓ):\n1. letvibe the partner of uiinM\n2. ifi=ℓthen\n3. let uℓ+1be a neighbor of vℓwithuℓ+1∈S\n4. if uℓ+1does not exist then delete uℓfromSℓand return false\n5. output the augmenting path v0u1v1⋯uℓvℓuℓ+1\n136. delete v0anduℓ+1fromS, andu1,...,uℓfrom all of S1,...,Sℓ, and return true\n7. for each neighbor ui+1ofviwithui+1∈Si+1∖{ui}do\n8. if EXTEND(v0u1v1⋯ui+1,ℓ)=true then return true\n9. delete uifromSiand return false\nNote that if it is not possible to extend the walk v0u1v1⋯uito an augmenting walk of length 2ℓ+\n1, then it is not possible to extend any other walk v′\n0u′\n1v′\n1⋯u′\niof the same length with u′\ni=ui. This\njustifies why we may delete uifromSiin line 9 of EXTEND (and similarly why we may delete uℓ\nfromSℓin line 4 of EXTEND , and why we may delete u1fromS1in line 4 of MAXIMAL -AUG -PATHS ).\nFor the running time analysis, note that each vertex may be a c andidate for uiin only one call\ntoEXTEND peri, because we delete uifromSi, either in line 9 if false is returned, or in line 6 if true\nis returned. Thus, the number of calls to EXTEND is at most O(ℓ/divides.alt0M/divides.alt0).\nWe use the given data structure DS0to maintain S. This allows us to do line 3 of MAXIMAL -AUG -PATHS .\nLine 1 of MAXIMAL -AUG -PATHS requires O(/divides.alt0M/divides.alt0)initial deletions from S. At the end, we reset Sto\nVby performing O(/divides.alt0M/divides.alt0)insertions of the deleted elements.\nWe also maintain S1,...,Sℓinℓnew instances of the data structure DS0. This allows us to\ndo lines 3 and 7 of EXTEND . Line 1 of MAXIMAL -AUG -PATHS requires O(ℓ/divides.alt0M/divides.alt0)initial insertions\ntoS1,...,Sℓ. We conclude that MAXIMAL -AUG -PATHS takesO(ℓ/divides.alt0M/divides.alt0⋅τ0)time. Hence, the overall\nrunning time of all ℓ=Θ(1/slash.leftε)iterations is O((1/slash.leftε2)/divides.alt0M/divides.alt0⋅τ0).\n3.2 Dynamic Geometric Bipartite MCM\nTo solve dynamic bipartite MCM for geometric intersection gr aphs, we again use a standard idea\nfor dynamization: be lazy, and periodically recompute the s olution only after every εOPT updates\n(since the optimal size changes by at most 1 per update).\nTheorem 2. LetCbe a class of geometric objects, where there is a dynamic data st ructureDS0fornobjects\ninCthat can find an object intersecting a query object (if exists), and supports insertions and deletions of\nobjects, with O(τ0(n))query and update time.\nThen there is a dynamic data structure for two sets of O(n)objects inCthat maintains a(1+O(ε))-\napproximation of the maximum-cardinality matching in the bip artite intersection graph, under insertions\nand deletions of objects, in ̃O((1/slash.leftε3)τ0(n))amortized time.\nProof. First, observe that we can maintain a maximal matching M0withO(τ0(n))update time:\nWe maintain the subset Sof all vertices not in M0in a data structure DS0. When we insert a new\nobjectu, we match it with an object in Sintersecting u(if exists) by querying DS0. When we delete\nan object u, we delete uand its partner vinM0, and reinsert v.\nAssume that b≤OPT<2bfor a given parameter b. Divide into phases with εbupdates each.\nAt the beginning of each phase, compute a (1+O(ε))-approximation of the maximum-cardinality\nmatching by Lemma 4iñO((1/slash.leftε2)b⋅τ0(n))time. (It is important to note that we don’t rebuild the\ndata structureDSforS=Vat the beginning of each phase; we continue using the same str ucture\nDSin the next phase, after resetting the modified Sback toV.)\nSince the above is done only at the beginning of each phase, th e amortized cost per update is\ñO((1/slash.leftε2)b⋅τ0(n)\nεb).\nDuring a phase, we handle an object insertion simply by doing nothing, and we handle an\nobject deletion simply by removing its incident edge (if exi sts) from the current matching. This\n14incurs an additive error at most O(εb)=O(εOPT). We also perform the insertion/deletion of the\nobject inDS0forS=V, iñO(τ0(n))time.\nHow do we obtain a correct guess b? We build the above data structure for each bthat is a\npower of 2, and run the algorithm simultaneously for each b.\n3.3 Specific Results\nRecall that for disks in R2as well as boxes in Rd, we have τ0(n)=O(logO(1)n). (Note that most\ndata structures for intersection detection can be modified to report a witness object intersecting\nthe query object if the answer is true.) Thus, we immediately obtain:\nCorollary 6. There is a dynamic data structure for two sets of O(n)disks inR2that maintains a(1+O(ε))-\napproximation of the maximum-cardinality matching in the bip artite intersection graph, under insertions\nand deletions, in O((1/slash.leftε3)logO(1)n)amortized time.\nCorollary 7. There is a dynamic data structure for two sets of O(n)axis-aligned boxes in Rdfor any\nconstantdthat maintains a(1+O(ε))-approximation of the maximum-cardinality matching in the bi partite\nintersection graph, under insertions and deletions, in O((1/slash.leftε3)logO(1)n)amortized time.\n4 General Maximum-Cardinality Matching\nIn this section, we adapt our results for bipartite MCM to gene ral MCM. We use a simple idea by\nLotker, Patt-Shamir, and Pettie [ LPP15 ] to reduce the problem for general graphs (in the approxi-\nmate setting) to finding maximal collections of short augment ing paths in bipartite graphs, which\nwe already know how to solve. (See also Har-Peled and Yang’s p aper [ HY22 ], which applied\nLotker et al.’s idea to obtain static approximation algorit hms for geometric intersection graphs.)\nThe reduction has exponential dependence in the path length ℓ(which is fine since ℓis small),\nand is originally randomized. We reinterpret their idea in t erms of color-coding [AYZ95 ], which\nallows for efficient derandomization, and also simplifies the a nalysis (bypassing Chernoff-bound\ncalculations). With this reinterpretation, it is easy to sh ow that the idea carries over to the dynamic\nsetting.\nWe begin with a lemma, which is a consequence of the standard c olor-coding technique:\nLemma 5. For anynandℓ, there exists a collection Z(n,ℓ)ofO(2O(ℓ)logn)subsets of[n]∶={1,...,n}\nsuch that for any two disjoint sets A,B⊆[n]of total size at most ℓ, we have A⊆ZandB⊆[n]∖Zfor\nsomeZ∈Z. Furthermore,Z(n,ℓ)can be constructed in O(2O(ℓ)nlogn)time.\nProof. As shown by Alon, Yuster, and Zwick [ AYZ95 ], there exists a collection H(n,ℓ)ofO(2O(ℓ)logn)\nmappings h∶[n]→[ℓ], such that for any set S⊆[n]of size at most ℓ, the elements in{h(v)∶v∈S}\nare all distinct. Furthermore, H(n,ℓ)can be constructed in O(2O(ℓ)nlogn)time. (This is related to\nthe notion of “ ℓ-perfect hash family”.)\nFor each h∈H(n,ℓ)and each subset I⊆[ℓ], add the subset Zh,I={v∈[n]∶h(v)∈I}toZ(n,ℓ).\nThe number of subsets is /divides.alt0H(n,ℓ)/divides.alt0⋅2ℓ≤2O(ℓ)logn. For any two disjoint sets A,B⊆[n]of total size\nat mostℓ, leth∈H(n,ℓ)be such that the elements in {h(a)∶a∈A}and{h(b)∶b∈B}are all distinct,\nand letI={h(a)∶a∈A}; thenA⊆Zh,IandB⊆[n]∖Zh,I.\nWe now present a non-bipartite analog of Lemma 4:\n15Lemma 6. We are given an unweighted graph G=(V,E), withV=[n]. LetZ(n,1/slash.leftε)be as in Lemma 5.\nSuppose there is a data structure DS∗\n0for storing a subset S⊆Vof vertices, initially with S=∅, that\nsupports the following two operations in τ0time: given a vertex u∈VandZ∈Z(n,1/slash.leftε), find a neighbor\nofuthat is in S∩Z(if exists) and a neighbor of uthat is in S∖Z(if exists); and insert/delete a vertex\nto/fromS.\nGiven a data structure DS0that currently has S=V, and given a maximal matching M0, we can\ncompute a(1+O(ε))-approximation to the maximum-cardinality matching in ̃O(2O(1/slash.leftε)OPT⋅τ0)time.\nHere, OPT denotes the maximum matching size.\nProof. As in the proof of Lemma 4, we iteratively maintain a current matching M, and it suffices\nto describe how to find a maximal collection Γof vertex-disjoint augmenting paths of length 2ℓ+1\nin theℓ-th iteration, under the assumption that there are no augmen ting paths of length ≤2ℓ−1.\nHowever, the presence of odd-length cycles complicates the computation of Γ.\nInitialize Γ=∅. We loop through each Z∈Z(n,1/slash.leftε)one by one and do the following. Let GZ\nbe the subgraph of Gwith edges{uv∈E∶u∈Z, v/slash.left∈Z}. We find a maximal collection of vertex-\ndisjoint augmenting paths of length 2ℓ+1inGZthat are vertex-disjoint from paths already selected\nto be inΓ; we then add this new collection to Γ. SinceGZis bipartite, this step can be done using\nthe MAXIMAL -AUG -PATHS procedure from the proof of Lemma 4. Since we are working with GZ\ninstead of G, when we find neighbors of a given vertex, they are now restrict ed to be in Zif the\ngiven vertex is in [n]∖Z, or vice versa; the data structure DS∗\n0allows for such queries. The only\nother change is that when we initialize S,S1,...,Sℓ, we should remove vertices that have appeared\nin paths already selected to be in Γ.\nAssume 2ℓ+2≤1/slash.leftε. We claim that after looping through all Z∈Z(n,1/slash.leftε), the resulting collec-\ntionΓof vertex-disjoint length- (2ℓ+1)augmenting paths is maximal in G. To see this, consider\nany length-(2ℓ+1)augmenting path v0u1v1⋯uℓvℓuℓ+1inG. There exists Z∈Z(n,1/slash.leftε)such that\nu1,...,uℓ+1∈Zandv0,...,vℓ/slash.left∈Z. Thus, the path must intersect some path in Γduring the itera-\ntion when we consider Z.\nWe can now obtain a non-bipartite analog of Theorem 2:\nTheorem 3. LetCbe a class of geometric objects, where there is a dynamic data st ructureDS0fornobjects\ninCthat can find an object intersecting a query object (if exists), and supports insertions and deletions of\nobjects, with O(τ0(n))query and update time.\nThen there is a dynamic data structure for O(n)objects inCthat maintains a(1+O(ε))-approximation\nof the maximum-cardinality matching in the intersection graph , under insertions and deletions of objects,\niñO(2O(1/slash.leftε)τ0(n))amortized time.\nProof. This is similar to the proof of Theorem 2, with Lemma 6replacing Lemma 4. The only\ndifference is that to support the data structure DS∗\n0, we maintain O(2O(1/slash.leftε)logn)parallel instances\nof the data structure DS0forS∩ZandS∖Z, for every Z∈Z(n,1/slash.leftε). This increases the update time\nby a factor of O(2O(1/slash.leftε)logn).\nWe have assumed that the input objects are labeled by integer s in[n]. When a new object is\ninserted, we can just assign it the next available label in [n]. When the number of objects exceeds\nn, we double nand rebuild the entire data structure from scratch. Similar ly, when the number of\nobjects is below n/slash.left4, we halve nand rebuild.\n16Corollary 8. There is a dynamic data structure for ndisks inR2that maintains a(1+O(ε))-approximation\nof the maximum-cardinality matching in the intersection graph , under insertions and deletions, in O(2O(1/slash.leftε)logO(1)n)\namortized time.\nCorollary 9. There is a dynamic data structure for naxis-aligned boxes in Rdfor any constant dthat\nmaintains a(1+O(ε))-approximation of the maximum-cardinality matching in the in tersection graph,\nunder insertions and deletions, in O(2O(1/slash.leftε)logO(1)n)amortized time.\n5 Conclusion\nIn this paper, we studied two fundamental graph optimizatio n problems, MVC and MCM, in fully\ndynamic geometric settings. For both of the problems, we dev eloped general frameworks based\non the dynamic intersection detection data structures and s tatic approximation algorithms, which\nare necessary ingredients for designing dynamic algorithm s for geometric intersection graphs.\nThese frameworks allowed us to obtain dynamic algorithms fo r a wide range of geometric inter-\nsection graphs, e.g., disks/rectangles in R2, hypercubes in Rd, etc. Moreover, our results extend\nto the bipartite versions of the problems as well. In this wor k, we primarily focused on the un-\nweighted case. Our approach does not readily extend to the we ighted setting. Obtaining efficient\ndynamic algorithms for both MVC and MCM in the weighted setting are interesting open prob-\nlems.\nReferences\n[ABR24] Amir Azarmehr, Soheil Behnezhad, and Mohammad Rogha ni. Fully dynamic match-\ning:(2−√\n2)-approximation in polylog update time. In Proceedings of the 35th An-\nnual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 3040–3061, 2024.\ndoi:10.1137/1.9781611977912.109 .3\n[AC09] Peyman Afshani and Timothy M. Chan. Dynamic connectiv ity for axis-parallel rect-\nangles. Algorithmica , 53(4):474–487, 2009. doi:10.1007/S00453-008-9234-7 .1\n[ACS+22] Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dy-\nnamic geometric set cover and hitting set. ACM Trans. Algorithms , 18(4):40:1–40:37,\n2022.doi:10.1145/3551639 .2\n[AE99] Pankaj K. Agarwal and Jeff Erickson. Geometric range s earching and its relatives. In\nB. Chazelle, J. E. Goodman, and R. Pollack, editors, Advances in Discrete and Computa-\ntional Geometry , pages 1–56. AMS Press, 1999. 9,11\n[AHRS24] Pankaj K. Agarwal, Sariel Har-Peled, Rahul Raychau dhury, and Stavros Sintos. Fast\napproximation algorithms for piercing boxes by points. In Proceedings of the 35th An-\nnual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 4892–4908, 2024.\ndoi:10.1137/1.9781611977912.174 .2,6\n[AHW19] Anna Adamaszek, Sariel Har-Peled, and Andreas Wies e. Approximation schemes\nfor independent set and sparse subsets of polygons. J. ACM , 66(4):29:1–29:40, 2019.\ndoi:10.1145/3326122 .3\n17[AYZ95] Noga Alon, Raphael Yuster, and Uri Zwick. Color-cod ing. J. ACM , 42(4):844–856, 1995.\ndoi:10.1145/210332.210337 .7,15\n[BCIK21] Sujoy Bhore, Jean Cardinal, John Iacono, and Grigor ios Koumoutsos. Dynamic geo-\nmetric independent set. In Abstracts of 23rd Thailand-Japan Conference on Discrete and\nComputational Geometry, Graphs, and Games (TJDCG) , 2021.arXiv:2007.08643 .2\n[BCM23] ´Edouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum matchings in ge-\nometric intersection graphs. Discrete & Computational Geometry , pages 1–30, 2023.\ndoi:10.1007/S00454-023-00564-3 .3\n[Beh23] Soheil Behnezhad. Dynamic algorithms for maximum m atching size. In Proceedings of\nthe 34th Annual ACM-SIAM Symposium on Discrete Algorithms (S ODA) , pages 129–162,\n2023.doi:10.1137/1.9781611977554.CH6 .3\n[BG95] Herv´ e Br¨ onnimann and Michael T. Goodrich. Almost op timal set covers in finite VC-\ndimension. Discret. Comput. Geom. , 14(4):463–479, 1995. doi:10.1007/BF02570718 .\n6\n[BHI18] Sayan Bhattacharya, Monika Henzinger, and Giuseppe F. Italiano. Deterministic fully\ndynamic data structures for vertex cover and matching. SIAM J. Comput. , 47(3):859–\n887, 2018. doi:10.1137/140998925 .3\n[BJ02] Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull. In Proceedings\nof the 43rd Symposium on Foundations of Computer Science (FOCS ), pages 617–626, 2002.\ndoi:10.1109/SFCS.2002.1181985 .1\n[BK19] Sayan Bhattacharya and Janardhan Kulkarni. Determini stically maintaining a (2+ε)-\napproximate minimum vertex cover in O(1/slash.leftε2)amortized update time. In Proceedings\nof the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 1872–\n1885, 2019. doi:10.1137/1.9781611975482.113 .3\n[BKSW23] Sayan Bhattacharya, Peter Kiss, Thatchaphol Saranu rak, and David Wajc. Dynamic\nmatching with better-than-2 approximation in polylogarit hmic update time. In Pro-\nceedings of the 34th Annual ACM-SIAM Symposium on Discrete Alg orithms (SODA) , pages\n100–128, 2023. doi:10.1137/1.9781611977554.CH5 .3\n[BLN22] Sujoy Bhore, Guangping Li, and Martin N¨ ollenburg. A n algorithmic study of fully\ndynamic independent sets for map labeling. ACM J. Exp. Algorithmics , 27:1.8:1–1.8:36,\n2022.doi:10.1145/3514240 .2\n[BNTW23] Sujoy Bhore, Martin N¨ ollenburg, Csaba D. T´ oth, an d Jules Wulms. Fully dy-\nnamic maximum independent sets of disks in polylogarithmic update time. CoRR ,\nabs/2308.00979, 2023. To appear in SoCG 2024. arXiv:2308.00979 .2,4\n[BS15] Aaron Bernstein and Cliff Stein. Fully dynamic match ing in bipartite graphs. In\nProceedings of the 42nd International Colloquium on Automat a, Languages, and Program-\nming (ICALP), Part I , volume 9134 of Lecture Notes in Computer Science , pages 167–179.\nSpringer, 2015. doi:10.1007/978-3-662-47672-7\\_14 .3\n18[BS16] Aaron Bernstein and Cliff Stein. Faster fully dynami c matchings with small approx-\nimation ratios. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete\nAlgorithms (SODA) , pages 692–711, 2016. doi:10.1137/1.9781611974331.CH50 .\n3\n[BYHR11] Reuven Bar-Yehuda, Danny Hermelin, and Dror Rawit z. Minimum ver-\ntex cover in rectangle graphs. Computational Geometry , 44(6-7):356–364, 2011.\ndoi:10.1016/J.COMGEO.2011.03.002 .2,5,11,24\n[CCJ90] Brent N. Clark, Charles J. Colbourn, and David S. Joh nson. Unit disk graphs. Discrete\nmathematics , 86(1-3):165–177, 1990. doi:10.1016/0012-365X(90)90358-O .2\n[CE16] Julia Chuzhoy and Alina Ene. On approximating maximu m independent set of rect-\nangles. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer\nScience (FOCS) , pages 820–829, 2016. doi:10.1109/FOCS.2016.92 .3\n[CH21] Timothy M. Chan and Qizheng He. More dynamic data struct ures for geometric\nset cover with sublinear update time. In Proceedings of the 37th International Sympo-\nsium on Computational Geometry (SoCG) , volume 189 of LIPIcs , pages 25:1–25:14, 2021.\ndoi:10.4230/LIPICS.SOCG.2021.25 .2,4,6\n[CH24] Timothy M. Chan and Zhengcheng Huang. Dynamic geometr ic connectivity in the\nplane with constant query time. CoRR , abs/arXiv:2402.05357, 2024. To appear in SoCG\n2024.arXiv:2402.05357 .23\n[Cha01] Timothy M. Chan. Dynamic planar convex hull operatio ns in near-logarithmaic amor-\ntized time. J. ACM , 48(1):1–12, 2001. doi:10.1145/363647.363652 .1,2,6,11\n[Cha03a] Timothy M. Chan. Polynomial-time approximation sc hemes for pack-\ning and piercing fat objects. J. Algorithms , 46(2):178–189, 2003.\ndoi:10.1016/S0196-6774(02)00294-8 .4,11,23\n[Cha03b] Timothy M. Chan. Semi-online maintenance of geomet ric optima and measures. SIAM\nJ. Comput. , 32(3):700–716, 2003. doi:10.1137/S0097539702404389 .2,6\n[Cha06] Timothy M. Chan. Dynamic subgraph connectivity with geometric applications. SIAM\nJ. Comput. , 36(3):681–694, 2006. doi:10.1137/S009753970343912X .1,11\n[Cha09] Timothy M. Chan. Dynamic coresets. Discret. Comput. Geom. , 42(3):469–488, 2009.\ndoi:10.1007/S00454-009-9165-3 .2\n[Cha10] Timothy M. Chan. A dynamic data structure for 3-d conv ex hulls\nand 2-d nearest neighbor queries. J. ACM , 57(3):16:1–16:15, 2010.\ndoi:10.1145/1706591.1706596 .1,6\n[Cha20a] Timothy M. Chan. Dynamic generalized closest pair: Revisiting Eppstein’s technique.\nInProceedings of the 3rd SIAM Symposium on Simplicity in Algorit hms (SOSA) , pages\n33–37, 2020. doi:10.1137/1.9781611976014.6 .1,6,9\n19[Cha20b] Timothy M. Chan. Dynamic geometric data structures via shallow cuttings. Discret.\nComput. Geom. , 64(4):1235–1252, 2020. doi:10.1007/S00454-020-00229-5 .1,2,\n11\n[CHSX22] Timothy M. Chan, Qizheng He, Subhash Suri, and Jie Xu e. Dynamic geometric set\ncover, revisited. In Proceedings of the 33rd Annual ACM-SIAM Symposium on Discrete Al-\ngorithms (SODA) , pages 3496–3528, 2022. doi:10.1137/1.9781611977073.139 .\n2,4\n[CIK21] Jean Cardinal, John Iacono, and Grigorios Koumoutsos . Worst-case efficient dy-\nnamic geometric independent set. In Proceedings of the 29th Annual European\nSymposium on Algorithms (ESA) , volume 204 of LIPIcs , pages 25:1–25:15, 2021.\ndoi:10.4230/LIPICS.ESA.2021.25 .6\n[CKL+22] Li Chen, Rasmus Kyng, Yang P . Liu, Richard Peng, Maximilian Probst Gutenberg, and\nSushant Sachdeva. Maximum flow and minimum-cost flow in almost -linear time.\nInProceedings of the 63rd IEEE Annual Symposium on Foundations of Computer Science\n(FOCS) , pages 612–623, 2022. doi:10.1109/FOCS54457.2022.00064 .2,7\n[CLP11] Timothy M. Chan, Kasper Green Larsen, and Mihai P˘ atras ¸cu. Orthogonal range\nsearching on the RAM, revisited. In Proceedings of the 27th ACM Symposium on Compu-\ntational Geometry (SoCG) , pages 1–10, 2011. doi:10.1145/1998196.1998198 .24\n[CPR11] Timothy M. Chan, Mihai P˘ atras ¸cu, and Liam Roditty. D ynamic connectivity:\nConnecting to networks and geometry. SIAM J. Comput. , 40(2):333–349, 2011.\ndoi:10.1137/090751670 .1\n[DP10] Ran Duan and Seth Pettie. Approximating maximum weig ht matching in near-linear\ntime. In Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer\nScience (FOCS) , pages 673–682, 2010. doi:10.1109/FOCS.2010.70 .2,12\n[EIK01] Alon Efrat, Alon Itai, and Matthew J Katz. Geometry help s in bottleneck matching and\nrelated problems. Algorithmica , 31:1–28, 2001. doi:10.1007/S00453-001-0016-8 .\n3,6,12\n[EJS05] Thomas Erlebach, Klaus Jansen, and Eike Seidel. Poly nomial-time approximation\nschemes for geometric intersection graphs. SIAM Journal on Computing , 34(6):1302–\n1323, 2005. doi:10.1137/S0097539702402676 .2,4\n[Epp95] David Eppstein. Dynamic Euclidean minimum spannin g trees and extrema of binary\nfunctions. Discret. Comput. Geom. , 13:111–122, 1995. doi:10.1007/BF02574030 .1,\n6,9\n[FM95] Tom´ as Feder and Rajeev Motwani. Clique partitions, gr aph compression\nand speeding-up algorithms. J. Comput. Syst. Sci. , 51(2):261–272, 1995.\ndoi:10.1006/JCSS.1995.1065 .7\n[GP13] Manoj Gupta and Richard Peng. Fully dynamic (1+ε)-approximate matchings. In Pro-\nceedings of the 54th Annual IEEE Symposium on Foundations of C omputer Science (FOCS) ,\npages 548–557, 2013. doi:10.1109/FOCS.2013.65 .3,6,12\n20[Har23] Sariel Har-Peled. Approximately: Independence im plies vertex cover. CoRR ,\nabs/2308.00840, 2023. arXiv:2308.00840 .3\n[HK73] John E. Hopcroft and Richard M. Karp. An n5/slash.left2algorithm for maximum matchings in\nbipartite graphs. SIAM J. Comput. , 2(4):225–231, 1973. doi:10.1137/0202019 .2,6,\n12,13\n[HM85] Dorit S. Hochbaum and Wolfgang Maass. Approximation sc hemes for covering and\npacking problems in image processing and VLSI. Journal of the ACM , 32(1):130–136,\n1985.doi:10.1145/2455.214106 .3\n[HNW20] Monika Henzinger, Stefan Neumann, and Andreas Wiese . Dynamic approximate\nmaximum independent set of intervals, hypercubes and hyper rectangles. In Proceed-\nings of the 36th International Symposium on Computational G eometry (SoCG) , volume 164\nofLIPIcs , pages 51:1–51:14, 2020. doi:10.4230/LIPICS.SOCG.2020.51 .2,4\n[HY22] Sariel Har-Peled and Everett Yang. Approximation al gorithms for maximum match-\nings in geometric intersection graphs. In Proceedings of the 38th International Sympo-\nsium on Computational Geometry (SoCG) , volume 224 of LIPIcs , pages 47:1–47:13, 2022.\ndoi:10.4230/LIPICS.SOCG.2022.47 .3,5,6,12,15\n[KKK+22] Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer,\nLiam Roditty, and Paul Seiferth. Dynamic connectivity in di sk graphs. In Proceedings\nof the 38th International Symposium on Computational Geome try (SoCG) , volume 224 of\nLIPIcs , pages 49:1–49:17, 2022. doi:10.4230/LIPICS.SOCG.2022.49 .1,3,6,11\n[KLPS86] Klara Kedem, Ron Livne, J´ anos Pach, and Micha Sharir. O n the union of Jordan regions\nand collision-free translational motion amidst polygonal obstacles. Discret. Comput.\nGeom. , 1:59–70, 1986. doi:10.1007/BF02187683 .24\n[KLR+23] Arindam Khan, Aditya Lonkar, Saladi Rahul, Aditya Subram anian, and Andreas\nWiese. Online and dynamic algorithms for geometric set cove r and hitting set. In\nProceedings of the 39th International Symposium on Computat ional Geometry (SoCG) , vol-\nume 258 of LIPIcs , pages 46:1–46:17, 2023. doi:10.4230/LIPICS.SOCG.2023.46 .\n2\n[KR08] Subhash Khot and Oded Regev. Vertex cover might be hard t o ap-\nproximate to within 2−ε. J. Comput. Syst. Sci. , 74(3):335–349, 2008.\ndoi:10.1016/J.JCSS.2007.06.019 .2\n[LPP15] Zvi Lotker, Boaz Patt-Shamir, and Seth Pettie. Impr oved distributed approximate\nmatching. J. ACM , 62(5):38:1–38:17, 2015. doi:10.1145/2786753 .6,15\n[LT80] Richard J. Lipton and Robert Endre Tarjan. Applicati ons of a planar separator theorem.\nSIAM J. Comput. , 9(3):615–627, 1980. doi:10.1137/0209046 .23,24\n[Mar05] D´ aniel Marx. Efficient approximation schemes for geom etric problems? In\nProceedings of the 13th Annual European Symposium on Algorit hms (ESA) , vol-\nume 3669 of Lecture Notes in Computer Science , pages 448–459. Springer, 2005.\ndoi:10.1007/11561071\\_41 .4\n21[MR10] Nabil H. Mustafa and Saurabh Ray. Improved results on ge omet-\nric hitting set problems. Discret. Comput. Geom. , 44(4):883–895, 2010.\ndoi:10.1007/S00454-010-9285-9 .4\n[NTJ75] George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties\nand algorithms. Mathematical Programming , 8(1):232–248, 1975. 3,5,8\n[OR10] Krzysztof Onak and Ronitt Rubinfeld. Maintaining a lar ge matching and a small ver-\ntex cover. In Proceedings of the 42nd ACM Symposium on Theory of Computing ( STOC) ,\npages 457–464. ACM, 2010. doi:10.1145/1806689.1806753 .3\n[OvL81] Mark H. Overmars and Jan van Leeuwen. Maintenance of co nfigurations in the plane.\nJ. Comput. Syst. Sci. , 23(2):166–204, 1981. doi:10.1016/0022-0000(81)90012-X .\n1\n[PS85] Franco P . Preparata and Michael I. Shamos. Computational Geometry: An Introduction .\nSpringer, 1985. 11,25\n[PS16] David Peleg and Shay Solomon. Dynamic (1+ε)-approximate matchings: A density-\nsensitive approach. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete\nAlgorithms (SODA) , pages 712–729, 2016. doi:10.1137/1.9781611974331.CH51 .\n3\n[Sol16] Shay Solomon. Fully dynamic maximal matching in con stant update time. In Pro-\nceedings of the 57th Annual IEEE Symposium on Foundations of C omputer Science (FOCS) ,\npages 325–334, 2016. doi:10.1109/FOCS.2016.43 .3\n[SW98] Warren D. Smith and Nicholas C. Wormald. Geometric se parator theorems & appli-\ncations. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer\nScience (FOCS) , pages 232–243, 1998. doi:10.1109/SFCS.1998.743449 .23\n[Vaz94] Vijay V . Vazirani. A theory of alternating paths and blossoms for proving correctness\nof theO(√\nVE)general graph maximum matching algorithm. Comb. , 14(1):71–109,\n1994.doi:10.1007/BF01305952 .2\n[Zuc07] David Zuckerman. Linear degree extractors and the i napproximability of\nmax clique and chromatic number. Theory Comput. , 3(1):103–128, 2007.\ndoi:10.4086/TOC.2007.V003A006 .2\n22A Minimum Vertex Cover: Fast Static Algorithms\nA.1 Disks in R2\nIn this subsection, we give a static, near-linear-time, (1+O(ε))-approximation algorithm Afor\nMVC for disks under the promise that n=O(OPT). As we are aiming for a (1+O(ε))-factor\napproximation, we may tolerate an additive error of O(εn)because of the promise. MVC with\nO(εn)additive error reduces to MIS with O(εn)additive error, by complementing the solution.\nPTAS s are known for MIS for disks, but allowing O(εn)additive error, there are actually EP-\nTASs that run in near linear time. For example, we can adapt an app roach by Chan [ Cha03a ] based\non divide-and-conquer via separators. Specifically, we can u se the following variant of Smith and\nWormald’s geometric separator theorem [ SW98 ]:\nLemma 7 (Smith and Wormald’s separator) .Givennfat objects in a constant dimension d, there is an\naxis-aligned hypercube B, such that the number of objects inside Band the number of objects outside Bare\nboth at most(1−β)nfor some constant β>0(dependent on d), and the objects intersecting ∂Bcan be\nstabbed by O(n1−1/slash.leftd)points. Furthermore, Bcan be constructed in O(n)time.\nProof. (The following is a modification of a proof in [ CH24 ], which is based on Smith and Wormald’s\noriginal proof [ SW98 ].)\nFor each object, first pick an arbitrary “reference” point ins ide the object. Let B0be the smallest\nhypercube that contains at leastn\n2d+1reference points. Let rbe the side length of B0. Lethbe a\nparameter to be set later. For each t∈{1\nh,2\nh,...,h−1\nh}, letBtbe the hypercube with a scaled copy\nofB0with the same center and side length (1+t)r. SinceBtcan be covered by 2dquadtree boxes\nof side length<r, the number of reference points inside Btis at most2dn\n2d+1.\nAn object of diameter ≤r/slash.lefthintersects ∂Btfor at most O(1)choices of t. Thus, we can find a\nvaluetinO(n)time such that ∂Btintersects O(n/slash.lefth)objects of diameter ≤r/slash.lefth. On the other hand,\nthe objects of diameter >r/slash.lefthintersecting ∂Btcan all be stabbed by O(hd−1)points, because of\nfatness. Set h=n1/slash.leftd, andBtsatisfies the desired properties with β=2d\n2d+1.\nOne remaining issue is how to find B0quickly. Let Cbe a sufficiently large constant. Form\naC×⋯×Cgrid, so that the number of reference points between any two c onsecutive parallel\ngrid hyperplanes is n/slash.leftC. This takes O(Cn)time by a linear-time selection algorithm. Round each\nreference point to its nearest grid point. The rounded point set is a multiset with O(Cd)distinct\nelements. Redefine B0as the smallest hypercube that contains at mostn\n2d+1rounded reference\npoints (multiplicity included), which can now be computed i nCO(1)time. For any box, the num-\nber of reference points inside the box changes by at most O(n/slash.leftC)after rounding. So, we just need\nto increase βbyO(1/slash.leftC).\nThe MIS algorithm is simple: we just recursively compute an in dependent set for the disks\ninsideBand an independent set for the disks outside B, and take the union of the two sets. If nis\nbelow a constant b, we solve the problem exactly by brute force in 2O(C)time. (This is analogous\nto Lipton and Tarjan’s original EPTAS for independent set for planar graphs [ LT80 ].)\nSince the optimal independent set can have at most O(√n)disks intersecting ∂B, the total\nadditive error satisfies a recurrence of the form\nE(n)≤⎧⎪⎪⎨⎪⎪⎩maxn1,n2≤(1−β)n∶n1+n2≤n(E(n1)+E(n2)+O(√n))ifn≥b\n0 ifn<b.\n23This yields E(n)=O(n/slash.left√\nb). We setb=1/slash.leftε2. The running time of the resulting static algorithm A\nisT(n)=̃O(2O(1/slash.leftε2)n).\nA.2 Rectangles in R2\nIn this subsection, we give a static, near-linear-time, (3\n2+O(ε))-approximation algorithm for MVC\nfor rectangles, by adapting Bar-Yehuda, Hermelin, and Rawi tz’s previous, polynomial-time, (3\n2+\nO(ε))-approximation algorithm [ BYHR11 ].\nTriangle-free case. We first start with the case when the intersection graph is tria ngle-free, or\nequivalently, when the maximum depth of the rectangles is at most 2. (The depth of a point q\namong a set of rectangles refers to the number of rectangles c ontaining q; the maximum depth\nrefers to the maximum over all q∈R2.)\nWe will design a static algorithm Afor MVC for rectangles, under the promise that n≤(2+\nO(ε))OPT.\nWe say that a rectangle rdominates another rectangle r′if∂rintersects ∂r′four times, with r\nhaving larger height than r′. Bar-Yehuda, Hermelin, and Rawitz’s algorithm proceeds as follows:\n1. Decompose Rinto 2 subsets R1andR2, such that there are no dominating pairs within each\nsubsetRi.\nThe existence of such a decomposition follows easily from Di lworth’s theorem, but for an\nexplicit construction, we can just define R1to contain all rectangles in Rthat are not dom-\ninated by any other rectangle in R, and define R2to beR∖R1. Correctness is easy to see\n(since the maximum depth is 2).\nWe can compute R1(and thus R2) by performing O(n)orthogonal range queries, after lifting\neach rectangle to a point in R4. In fact, computing R1corresponds to the maxima problem in\nR4, for which O(nlogn)-time algorithms are known [ CLP11 ].\n2. For each i∈{1,2}, compute a vertex cover SiofRiwhich approximates the minimum with\nadditive error at most εn.\nSince we tolerate εnadditive error, it suffices to approximate the MIS of Riwithεnadditive\nerror.\nBecause Rihas no dominating pairs, it forms a pseudo-disk arrangement (each pair inter-\nsects at most twice). It is known [ BYHR11 ,KLPS86 ] that the intersection graph of any set of\npseudo-disks that have maximum depth 2 and have no containme nt pair is in fact planar.\n(We can easily eliminate containment pair, by removing rect angles that contain another rect-\nangle; and we can detect such rectangles again by orthogonal range queries.) So, we can use\nknown near-linear-time EPTAS for MIS for planar graphs; for example, Lipton and Tarjan’s\ndivide-and-conquer algorithm via planar-graph separator s runs iñO(2O(1/slash.leftε2)n)time [ LT80 ].\n3. Return S, the smaller of the two sets: S1∪R2andS2∪R1.\nLetS∗be the minimum vertex cover. Then\n/divides.alt0S/divides.alt0≤1\n2(/divides.alt0S1/divides.alt0+/divides.alt0R2/divides.alt0+/divides.alt0S2/divides.alt0+/divides.alt0R1/divides.alt0)\n≤1\n2(/divides.alt0S∗∩R1/divides.alt0+/divides.alt0S∗∩R2/divides.alt0+/divides.alt0R1/divides.alt0+/divides.alt0R2/divides.alt0)+εn=1\n2(/divides.alt0S∗/divides.alt0+n)+εn≤(3\n2+O(ε))/divides.alt0S∗/divides.alt0\n24under the assumption that n≤(2+O(ε))/divides.alt0S∗/divides.alt0. The running time is T(n)=̃O(2O(1/slash.leftε2)n).\nAt this point, if we apply Theorem 1withc=3\n2+O(ε),γ=ε, andδ=ε3(andτ0(n)=\nO(logO(1)n)as noted in Section 2.4), we obtain:\nCorollary 10. There is a dynamic data structure for nrectangles in R2that maintains a(3\n2+O(ε))-\napproximation of the minimum vertex cover of the intersection graph, under insertions and deletions, in\nO(2O(1/slash.leftε2)logO(1)n)amortized time, under the assumption that the intersection g raph is triangle-free.\nGeneral case. We now design our final static algorithm Afor MVC for rectangles that avoids the\ntriangle-free assumption, again by building on Bar-Yehuda , Hermelin, and Rawitz’s approach:\n1. Remove vertex-disjoint triangles T1,...,Tℓso that the remaining set R′∶=R∖(T1∪⋯∪Tℓ)is\ntriangle-free.\nWe can implement this step greedily in polynomial time, but a faster approach is via a plane\nsweep. Namely, we modify the standard sweep-line algorithm to computing the maximum\ndepth of nrectangles in R2(similar to Klee’s measure problem) [ PS85 ]. The algorithm uses a\ndata structure for a 1D problem: maintaining the maximum dep th of intervals in R1, subject\nto insertions and deletions of intervals. Simple modificatio n of standard search trees achieve\nO(logn)time per insertion and deletion. As we sweep a vertical line ℓfrom left to right, we\nmaintain the maximum depth of the y-intervals of the rectangles intersected by ℓ. Whenℓ\nhits the left side of a rectangle, we insert an interval. When ℓhits the right side of a rectangle,\nwe delete an interval. When the maximum depth becomes 3, we re move the 3 intervals\ncontaining the maximum-depth point, which correspond to a t riangle in the intersection\ngraph, and continue the sweep. The total time is O(nlogn).\n2. Compute a vertex cover S′ofR′which is a(3\n2+O(ε))-approximation to the minimum. This\ncan be done by Corollary 10iñO(2O(1/slash.leftε2)n)time (we actually don’t need the full power of a\ndynamic data structure, since we just want a static algorith m for this step).\n3. Return S=S′∪TwithT∶=T1∪⋯∪Tℓ.\nLetS∗be the minimum vertex cover. The key observation is that S∗must contain at least 2 of\nthe 3 vertices in each triangle Ti, and so/divides.alt0S∗∩T/divides.alt0≥2\n3/divides.alt0T/divides.alt0. Thus,\n/divides.alt0S/divides.alt0=/divides.alt0S′/divides.alt0+/divides.alt0T/divides.alt0≤(3\n2+O(ε))/divides.alt0S∗∩R′/divides.alt0+3\n2/divides.alt0S∗∩T/divides.alt0≤(3\n2+O(ε))/divides.alt0S∗/divides.alt0.\nThe running time of our final static algorithm AisT(n)=̃O(2O(1/slash.leftε2)n).\n25"    },    {        "title": "2402.07443v1.The_I_O_Complexity_of_Attention__or_How_Optimal_is_Flash_Attention_.pdf",        "content": "The I/O Complexity of Attention, or\nHow Optimal is Flash Attention?\nBarna Saha∗Christopher Ye∗\nFebruary 13, 2024\nAbstract\nSelf-attention is at the heart of the popular Transformer architecture, yet suffers from\nquadratic time and memory complexity. In a recent significant development, FlashAttention\nshows that the I/O complexity of attention is the true bottleneck in scaling Transformers.\nGiven two levels of memory hierarchy, a fast cache (e.g. GPU on-chip SRAM) where computa-\ntion happens and a slow memory (e.g. GPU high-bandwidth memory) where the data resides,\nthe I/O complexity measures the number of accesses to the slow memory. FlashAttention is\nan I/O-aware algorithm for self-attention that requiresN2d2\nMI/O operations where Nis the\ndimension of the attention matrix, dis the head-dimension and Mis the size of cache. However,\nis this I/O complexity optimal? The known lower bound only rules out an I/O complexity of\no(Nd)when M= Θ( Nd), since the output of the attention mechanism that needs to be written\nin the slow memory is Ω(Nd). The main question that remained open after FlashAttention is\nwhether this I/O complexity is optimal for any value of M.\nWe resolve the above question in its full generality by showing an I/O complexity lower\nbound that matches the upper bound provided by FlashAttention for any values of M≥d2\nwithin any constant factors. Further, we give a better algorithm with lower I/O complexity\nforM < d2, and show that it is optimal as well. Moreover, our lower bounds do not rely on\nusing combinatorial matrix multiplication for computing the attention matrix. We show even\nif one uses fast matrix multiplication, the above I/O complexity bounds cannot be improved.\nWe do so by introducing a new communication complexity protocol for matrix compression,\nand connecting communication complexity to I/O complexity. To the best of our knowledge,\nthis is the first work to establish a connection between communication complexity and I/O\ncomplexity, and we believe this connection could be of independent interest and will find many\nmore applications in proving I/O complexity lower bounds in future.\n∗University of California, San Diego. The authors are partially supported by NSF grants 1652303, 1909046,\n2112533, and HDR TRIPODS Phase II grant 2217058.\n1arXiv:2402.07443v1  [cs.LG]  12 Feb 20241 Introduction\nTransformermodels[VSP+17]haveemergedasthearchitectureofchoiceforavarietyofapplications\nincluding natural language processing and computer vision. The self-attention module at the heart\nof the Transformer architecture requires quadratic time and memory complexity. Thus despite its\npopularity, Transformers are slow and memory-hungry. In a seminal paper, [DFE+22] proposes\nFlashAttention, an IO-aware algorithm for self-attention. Given two levels of memory hierarchy,\na fast cache (e.g. GPU on-chip SRAM) where computation happens and a slow memory (e.g.\nGPU high-bandwidth memory) where the data resides, the I/O complexity measures the number of\naccesses to the slow memory. [DFE+22] argues that I/O complexity is indeed the true bottleneck in\nachieving wall-clock speed up, and provide an algorithm that achieves an I/O complexity of O(N2d2\nM)\nwhere Nis the dimension of the attention matrix, dis the head-dimension and Mis the cache size.\nIn self-attention, three input matrices Q, K, V ∈RN×dare used to compute softmax( QKT)Vwhere\nA= exp( QKT)istheattentionmatrix. For M= Θ( Nd), theabove I/Ocomplexityboundbecomes\nNdwhich is also needed just to write the output of self-attention to slow memory. This leaves a\ntantalizing open question whether FlashAttention I/O complexity is optimal for any values of M.\nWe resolve this question in affirmative for any value of Mas long as M≥d2. Moreover, for\nM < d2, we give a better algorithm than FlashAttention and also show the bound is optimal.\nIndeed as we prove for M≤d2, the I/O complexity of self-attention is at least as high as the I/O\ncomplexity of computing matrix multiplication of two N×dandd×Nmatrices. For M≥d2which\nis also the more practical regime, our proof of optimality of FlashAttention brings in several new\nideas.\nFirst, we show that FlashAttention which uses combinatorial matrix multiplication has optimal\nI/Ocomplexitybyutilizingthefamous red-blue pebble game fromtheearlyeighties[HK81]. However,\nit may still be possible to improve FlashAttention by utilizing fast matrix multiplication [GU18,\nWXXZ23] to obtain lower I/O complexity. We rule out any such possibilities. We prove a general\nlower bound against all such algorithms. We introduce a new matrix compression problem, and\nshow that it has a high communication complexity. Next, we establish a connection between the\ncommunication complexity of a matrix compression protocol with the I/O complexity of attention.\nTo the best of our knowledge, this is the first work that shows a connection between communication\ncomplexity and I/O complexity, which may have further valuable theoretical consequences. We\nshow even when the input matrix entries are restricted to binary, the lower bound holds within\npolylogarithmic factors by utilizing Binary BCH codes [Hoc59, BR60] from coding theory to prove\nlower bounds against binary matrix compression.\n1.1 Related Work\nMany papers have studied attention, a key bottleneck of the transformer architecture [VSP+17]\nwhich has become the predominant architecture in applications such as natural language pro-\ncessing. Many approximate notions have been studied to reduce its compute and memory con-\nstraints [BMR+20, KVPF20, KKL20, ZGD+20, CDW+21, CLD+21, AS23, HJK+23]. Algorithms\nsuch as FlashAttention have made significant practical gains by designing I/O-aware algorithms\n[DFE+22, Dao23].\nThe role of learning under bounded space constraints has been studied primarily in the context\nof bounded memory, with applications including Online Learning [SWXZ22, PR23, PZ23], continual\nlearning[CPP22], convexoptimization[WS19,MSSV22,CP23], andothers[Raz19,SSV19,GLM20].\nIn this work, we instead consider learning in the context of a two-level memory hierarchy with a\nbounded cache . While the algorithm has unlimited memory, it can only access a small portion of\n2this memory at any given time, and is charged for every access to memory. We believe that the\ntask of minimizing I/O complexity is both theoretically interesting and practically significant, as\nI/O operations often form a key computational bottleneck in practice.\nThere is a long-line of work on I/O complexity [HK81, AV88, PS14a]. Within this body of work,\nmany papers have focused on the problem of matrix multiplication [BDHS12, BDH+12, PS14b,\nSHS15, BS17].\n1.2 Our Contributions\nWe study the I/O complexity of attention on a two-level memory hierarchy. The attention mech-\nanism takes inputs Q, K, V ∈RN×dand computes softmax( QKT)V(see Section 2.1) where A=\nexp(QKT)is the attention matrix. Our first result resolves the I/O complexity of any algorithm\ncomputing attention using the standard matrix multiplication algorithm. This answers an open\nquestion of [DFE+22] and shows that FlashAttention is optimal among this class of algorithms.\nTheorem 3.1. The I/O complexity of attention with standard matrix multiplication is\nΘ\u0012\nmin\u0012N2d√\nM,N2d2\nM\u0013\u0013\nWe divide the analysis into two cases, the large cache ( M≥d2) and small cache ( M < d2) case.\nWhen M < d2, we show that attention and matrix multiplication are equivalent. In the small cache\nsetting,N2d2\nM>N2d√\nM> N2, so we can explicitly write QKTto memory while computing atten-\ntion. Thus, any algorithm computing attention in fact computes QKT, establishing an equivalence\nbetween attention and matrix multiplication.\nIn the more interesting large cache case, when M≥d2, the upper bound is given by the\nbreakthrough FlashAttention algorithm [DFE+22]. In this setting, the I/O complexity approaches\nO(Nd)as cache size increases, providing significant practical improvements, despite the theoretical\ntime complexity remaining O(N2d). To prove a matching lower bound, we use the red-blue pebble\ngame of [HK81]. Executing an algorithm Aon a machine with cache size Mis equivalent to\nplaying the red-blue pebble game on the computational directed acyclic graph Gcorresponding\ntoA. Specifically, the partitioning lemma (Lemma 2.7) states that any successful execution of A\ncorresponds to a partition of Gwhere each part corresponds to a sub-computation of Awith no\nI/O operations (called a M-partition). Thus, it suffices to prove a lower bound on the size of any\nM-partition to obtain a I/O complexity lower bound for Aon a machine with cache size M. Our\nlower bound on the partition size then follows from a careful analysis of the computational graph\nof attention (Figure 1). Specifically, we show that any part Viin the partition can only compute\nM2\nd2+M≤2M2\nd2entries in the product QKT.\nWe then extend our lower bound in the large cache ( M≥d2) case to arbitrary algorithms for\nattention. More precisely, we show that even with fast matrix multiplication (FMM), Ω\u0010\nN2d2\nM\u0011\nI/O\noperations are required to compute attention.\nTheorem 4.8. Suppose Q, K∈FN×d\nqwhere q > N. The I/O complexity of attention (with any\nmatrix multiplication algorithm) is Ω\u0010\nmin\u0010\nN2d2\nM, N2\u0011\u0011\n.\nTo establish a general lower bound, we can no longer reason about a specific computational\ngraph. Instead, we appeal to communication complexity, a useful tool for proving lower bounds\nfor learning [DKS19, KLMY19], specifically space-bounded learning [CPP22]. To the best of our\nknowledge,thisisthefirstapplicationofcommunicationcomplexitytoI/Ocomplexitylowerbounds.\n3For simplicity, assume that Aperforms I/O operations in batches of size M(Lemma 4.1 shows\nthat this can be assumed without loss of generality). Within each batch, Acan only access the\ncache of size M, with no further I/O operations. In order to give a lower bound on the number of\nbatches, we argue that Acannot make too much progress in each batch. In the context of attention,\nwe measure progress as the number of entries computed in QKT(regardless of whether these entries\nare written to memory).\nFormally, we introduce the B-entry matrix compression problem (Definition 4.2) and argue\nthat any algorithm making too much progress on a given batch gives an efficient communication\nprotocol (Theorem 4.4). We then complete the argument by giving a lower bound on the communi-\ncation complexity of the matrix compression problem (Lemma 4.6). Our lower bound follows from\nconstructing input matrices satisfying strong linear independence constraints. For large q, this is\nachieved by Vandermonde matrices.\nFinally, we relax the constraint q > Nand consider the special case q= 2, i.e. Q, K, V are\nbinary matrices and obtain a lower bound tight up to polylogarithmic factors. Our lower bound\nuses Binary BCH codes [Hoc59, BR60] to construct matrices satisfying strong linear independence\nconstraints.\nTheorem 4.14. Suppose Q, K∈ {0,1}N×d. The I/O complexity of attention (with any matrix\nmultiplication algorithm) is Ω\u0010\nmin\u0010\nN2d2\nMlog2N, N2\u0011\u0011\n.\nIn the small cache setting ( M < d2), Theorem 4.17 gives a simple equivalence between attention\nand rectangular matrix multiplication.\n2 Preliminaries\nGiven amatrix A∈Rn×m, we indexanindividualentryas A[i, j]. The i-throwisdenoted A[i]while\nthej-th column is denoted A[∗, j].A[i1:i2, j1:j2]denotes a block of Aconsisting of entries (i, j)\nwhere i∈[i1, i2]andj∈[j1, j2]. Given a block size B, the block A[(i−1)·B+1 : i·B,(j−1)·B+1 :\nj·B]is denoted A(B)[i, j].\nAT, A−1denote the transpose and inverse of A.\nFor a vector v∈Rn, we similarly denote entries v[i], a contiguous block of entries as v[i1:i2],\nand the i-th block of size Basv(B)[i]. Let diag( v)denote the matrix D∈Rn×nwith D[i, i] =v[i].\n2.1 The Attention Mechanism\nGiven input matrices Q, K, V ∈RN×d, the attention mechanism computes D−1AVwhere the\nattention matrix A= exp( QKT)is computed by taking exponents entry-wise and D= diag( A·1)\nis the diagonal matrix containing row-sums of A.\n2.2 The Memory Hierarchy\nIn this work, we assume that the memory hierarchy has two levels: the small but fast layer (called\nthecache) and the large slow layer (called the memory). We assume that the memory is unbounded\nwhile the cache is bounded by some size constraint M. Furthermore, computations only occur on\nthe cache.\n42.3 Red-Blue Pebble Game\nWe require the red-blue pebble game of [HK81], designed to model computations on a two-level\nmemory hierarchy. Inspired by the pebble game often used to model space-bounded computation,\n[HK81] develops the red-blue pebble game to model I/O complexity. As with the standard peb-\nble game, the red-blue pebble game is played on a directed acyclic graph, where nodes represent\ncomputations and edges represent dependencies. Throughout the game, pebbles can be added to,\nremoved from, or recolored between red and blue in the graph, where red pebbles represent data in\ncache and blue pebbles represent data in slow memory. Given an upper bound on the number of\nred pebbles (cache size), the goal is to minimize the number of pebble recolorings (I/O operations)\nover the course of a computation. The game is defined formally below.\nDefinition 2.1 (Red-Blue Pebble Game [HK81]) .LetGbe a directed acyclic graph with a set of\ninputvertices containing at least all vertices with no parents, and a set of outputvertices containing\nat least all vertices with no children. A configuration is a pair of subsets of vertices, one containing\nall vertices with red pebbles, and the other containing all vertices with blue pebbles. Note that a\nvertex can have a red pebble, a blue pebble, both, or neither.\nTheinitial(resp.terminal) configuration is one in which only input (resp. output) vertices\ncontain pebbles, and all of these pebbles are blue. The rules of the red-blue pebble game are as\nfollows:\nR1(Input) A red pebble may be placed on any vertex with a blue pebble.\nR2(Output) A blue pebble may be placed on any vertex with a red pebble.\nR3(Compute) A red pebble may be placed on a vertex if all its parents have red pebbles.\nR4(Delete) A pebble may be removed from any vertex.\nAtransition is an ordered pair of configurations where the second can be obtained from the first\nfollowing one of the above rules. A calculation is a sequence of configurations, where each suc-\ncessive pair forms a transition. A complete calculation is a calculation that begins with the initial\nconfiguration and ends with the terminal configuration.\nIntypicalinstances,includingourpaper,theinputandoutputverticesaredisjoint. Furthermore,\nthe input vertices are typically exactly those with no parents, while the output vertices are exactly\nthose with no children. To model a bounded cache, we assume that there are at most Mred pebbles\non the graph at any given time, while any number of blue pebbles can be placed on the graph. Given\na graph G, we are interested in the I/O complexity.\nDefinition 2.2 (I/O Complexity [HK81]) .Given a graph Gand integer M, the I/O complexity\nQ(G, M )is defined by the minimum number of transitions according to rules R1andR2required\nby any complete calculation. When the underlying graph Gis clear, we omit Gand write Q(M).\nTo obtain I/O complexity lower bounds, [HK81] show that any complete calculation induces a\nM-partition. Roughly speaking, a M-partition partitions the vertex set such that each part can be\ncomputed completely using only 2MI/O operations. A lower bound on the size of any M-partition\nthen implies a lower bound on the I/O complexity of the computational graph.\nBefore defining the M-partition, we give the necessary definitions of dominator sets, minimum\nsets, and vertex subset dependence.\nDefinition 2.3 (Dominator Set) .LetGbe a directed acylic graph and S⊂Va subset. D⊂Vis\nadominator set ofSif for every path from an input vertex of Gto a vertex in Scontains a vertex\ninD.\n5Definition2.4 (MinimumSet) .LetGbe a directed acylic graph and S⊂Va subset. The minimum\nsetofS, denoted M, is the subset of vertices in Swith no children in S.\nDefinition 2.5 (Vertex Subset Dependence) .LetGbe a directed acyclic graph, with S, T⊂V\ndisjoint subsets. Tdepends onSif there exists an edge (s, t)∈Ewiths∈S, t∈T.\nWe now give the definition of an M-partition (called an S-partition in [HK81]).\nDefinition 2.6 (M-partition [HK81]) .LetGbe a directed acyclic graph. A family of subset {Vi}h\ni=1\nis aM-partition ofGif the following properties hold:\nP1(Partition) The sets Viare disjoint and V=Sh\ni=1Vi.\nP2(Dominator) For each Vi, there exists a dominator set Diof size at most M.\nP3(Minimum) For each Vi, the set of minimum vertices Mihas size at most M.\nP4(Acyclic) There is no cyclic dependence among vertex sets {Vi}h\ni=1.\nRoughly speaking, a M-partition partitions the vertex set such that each part can be computed\ncompletely using only 2MI/O operations. In particular, every node in a single part V′can be\nreached by placing red pebbles at the dominator vertices and using Rule R3. This can be thought\nof as the initial state of the cache at some point, and without any further reads from memory, every\nnode on V′can be computed. Then, after the computation, the values at the minimum vertices\ncan be written to memory. This corresponds to at most Mwrite operations. The following lemma\nrelates I/O complexity to the size of M-partitions.\nLemma 2.7 (Theorem 3.1 of [HK81]) .LetGbe a directed acyclic graph. Any complete calculation\nof the red-blue pebble game on G, using at most Mred pebbles, is associated with a 2M-partition of\nGsuch that,\nM·h≥Q(G, M )≥M·(h−1)\nwhere his the number of vertex subsets in the 2M-partition.\nIn particular, the partitioning lemma of [HK81] shows that it suffices to prove a lower bound on\nthe size of any 2M-partition to obtain a lower bound on the I/O complexity.\nLemma 2.8 (Lemma 3.1 of [HK81]) .For any directed acyclic graph G, let P(G, M )denote the\nminimum size of any M-partition of G. Then,\nQ(G, M )≥M·(P(G,2M)−1)\n2.4 Computational Graph for the Attention Mechanism\nFigure 1 gives a computational graph modelling the attention mechanism [DFE+22, Dao23]. In this\ncomputational graph, we assume that matrix products are computed using the standard algorithm,\nthat is, we compute (AB)ij=P\nkAikBkjfor all i, j.\nThe attention mechanism begins by computing the matrix product QKT. In the vertex set L1,\nthere are N2dvertices representing the values {QiℓKT\nℓj}i,j,ℓ. Then, each entry (QKT)ijis computed\nby summing the appropriate vertices in L1, i.e. the node (QKT)ij=P\nℓQiℓKT\nℓj. In particular, each\nentry (QKT)ijis connected to L1via a summation tree. Note that all nodes in the summation\ntrees are disjoint. The summation tree can be thought of as a balanced binary tree with dleaves,\nalthough the exact structure of the tree (i.e. order of summation) can be arbitrary. For a more\ndetailed description of the computational graph of the standard matrix multiplication algorithm,\nsee [HK81]. We define the set of level-1 vertices to be all nodes in the computational graph that\n6𝑸𝑳𝟏𝑲𝑸𝑲𝑻𝑨\n𝑫𝑫#𝟏𝑽𝑳𝟐𝑨𝑽𝑶Figure 1: Computational Graph for the Attention Mechanism.\nare in vertex sets L1, QKT, or one of the intermediate nodes in the N2summation trees between\nthe two layers. Figure 2 illustrates an example summation tree.\nThen, each entry in Ais computed by taking the exponent of the corresponding entry in QKT.\nEach node in Dis computed by summing over the rows of A. As above, this is realized in the graph\nbyNdisjoint summation trees and D−1is computed by taking the multiplicative inverse of each\nelement in D. The matrix product is computed as in the first step, first by storing all N2dproducts\nin the intermediate layer L2={AikVkj}i,k,jand computing AVviaNddisjoint summation trees.\nFinally, each node in Ois computed by scaling each entry of AVby the appropriate factor in D−1.\nAlthough we have (roughly) described the computational graph of FlashAttention-2 [Dao23]\nand there are different ways to implement attention, all algorithms begin by computing the product\nQKT, and our lower bounds will hold for any algorithm computing this matrix product.\n3 I/O Complexity of Attention\nInthissection, wepresentatightcharacterizationoftheI/Ocomplexityofanyalgorithmcomputing\nattention exactly using the standard matrix multiplication algorithm.\nTheorem 3.1. The I/O complexity of attention with standard matrix multiplication is\nΘ\u0012\nmin\u0012N2d√\nM,N2d2\nM\u0013\u0013\nAt the crossover point M=d2, observeN2d√\nM=N2d2\nM=N2. We define M≥d2as thelarge\ncacheregime, and M < d2as thesmall cache regime. We are primarily interested in the large cache\n7𝐿!𝑄!\"𝑄!#𝑄!$𝑄!%𝐾&\"𝐾&#𝐾&$𝐾&%𝑄!\"𝐾#\"𝑄!$𝐾#$𝑄!%𝐾#%𝑄!&𝐾#&𝑄𝐾!\"#\tFigure 2: A single summation tree with d= 4. Orange vertices denote inputs from Q. Yellow\nvertices denote inputs from K. Grey and green vertices denote level-1 vertices . Observe that these\nare disjoint for each summation tree. The green vertex specifically denotes an entry in QKT. Solid\nedges denote multiplications and dotted edges denote additions. Vertices in the blue box denote\nvertices in L1.\nregime, sincethisiswhereI/Ocomplexityissub-quadratic. ThisistheregimewhereFlashAttention\noutperforms standard implementations of attention, since we can avoid writing the entire N×N\nmatrix QKTto memory.\n3.1 Large Cache: M= Ω(d2)\nFor large M= Ω( d2), we show that the following result of [DFE+22] is optimal in terms of I/O\ncomplexity.\nTheorem 3.2 (Theorem 2 of [DFE+22]).FlashAttention has I/O complexity O\u0010\nN2d2\nM\u0011\n.\nTo prove a matching lower bound, we bound the number of level-1 vertices in each part of a\nM-partition. This gives a lower bound on the size of any partition, implying an I/O complexity\nlower bound via Lemma 2.8.\nLemma 3.3. Suppose M= Ω( d2)and let Pbe aM-partition of the computational graph in Figure\n1. Let V′∈ P. Then, V′contains at most O\u0010\nM2\nd\u0011\nlevel-1 vertices.\nProof.First, since V′has a dominator set D′of at most M, and each summation tree in the\ncomputationalgraphisdisjoint, thereareatmost Mlevel-1summationtreescontainingadominator\nvertex. Next, since V′has at most Mminimum vertices, disjointness again implies that at most M\nlevel-1 summation trees contain a minimum vertex. See Figure 3 for an example of V′containing\ndominator and minimum vertices.\nSuppose V′contains level-1 vertices not in any of the above 2Msummation trees (at most M\ntrees containing a dominator and another at most Mcontaining minimum vertices).\nConsider a tree Tcontaining level-1 vertices that does not intersect D′or contain a minimum\nvertex. We denote such a tree as extra. Since Tdoes not contain any minimum vertices, the root\nrofT, representing some entry (QKT)[i, j], must be in V′. There are 2dinput vertices with paths\ntor, namely the inputs {Q[i, ℓ], KT[ℓ, j]}d\nℓ=1. We denote the i-inputs of Tas the set {Q[i, ℓ]}d\nℓ=1\nand the j-inputs as the set {KT[ℓ, j]}d\nℓ=1. On each path, all vertices but the inputs are in T. Since\nTcontains no vertices in D′,D′must contain all 2dinput vertices. For example, in Figure 2, V′\n8(%&(%'(%((%))*&)*')*()*)!!\"\"#\"!!$\"#$!!%\"#%!!&\"#&*!\"!\"#\t(a){Qi2, Kj2, Qi1Kj1}are dominator vertices,\n{M}is the minimum vertex.\n𝑄!\"𝑄!#𝑄!$𝑄!%𝐾&\"𝐾&#𝐾&$𝐾&%𝑄!\"𝐾#\"𝑄!$𝐾#$𝑄!%𝐾#%𝑄!&𝐾#&𝑄𝐾!\"#\t(b){Qi1Kj1, Qi2Kj2}are dominator vertices,\n{QKT\nij}is the minimum vertex.\nFigure 3: Two examples of summation trees containing dominator and minimum vertices. Blue\nvertices denote elements of the partition Vi.\ncontains (QKT)[i, j]and none of the grey vertices, and therefore must contain all 2d= 8input\nvertices (orange and yellow).\nFor two extra trees Tij, Ti′j′whose roots correspond to entries (QKT)[i, j],(QKT)[i′, j′]respec-\ntively, observe that the i-inputs are equal if i=i′and disjoint otherwise. Similarly, the j-inputs\nare equal if j=j′and disjoint otherwise. Suppose V′contains level-1 vertices in Cextra level-1\nsummation trees. The Croots of these trees correspond to Centries in the matrix QKT. Suppose\ntheCentries have Idistinct values in the row index. Choose one entry in Cfor each row index.\nEach entry requires d i-input vertices to be placed in the dominator set, and furthermore these\ninput vertices are disjoint. Therefore, there are at least Idinput vertices from Qin the dominator\nsetD′. By a similar argument, if the Centries have Jdistinct values in the column index, there\nare at least Jdinput vertices from Kin the dominator set D′. In particular, if there are Cextra\ntrees, then there are at least (I+J)dinput vertices in the dominator set D′. Finally, we can bound\nCby observing that I+J≥max( I, J)≥√\nCsince IJ≥C. In particular,\n√\nCd≤(I+J)d≤ |D′| ≤M\nso that C≤M2\nd2.\nIn total, V′contains vertices in at most C+ 2Mlevel-1 trees. Each level-1 tree contains O(d)\nlevel-1 vertices so that V′contains at most,\nO\u0012M2\nd+Md\u0013\n=O\u0012M2\nd\u0013\nlevel-1 vertices, where we have used M= Ω( d2).\nUsing Lemma 2.8, we obtain a lower bound for computing attention using standard matrix\nmultiplication. In fact, since we have not used any other part of the computational graph, our lower\nbound holds for any algorithm computing QKTusing standard matrix multiplication.\nLemma 3.4. Suppose M= Ω( d2). Then, P(M) = Ω\u0010\nN2d2\nM2\u0011\nandQ(M) = Ω\u0010\nN2d2\nM\u0011\n.\nProof.Note that there are Ω(N2d)level-1 vertices. Since each part in the M-partition contains at\nmost O\u0010\nM2\nd\u0011\nlevel-1 vertices, the lower bound on the number of parts in the M-partition follows\nimmediately from Lemma 3.3.\n93.2 Small Cache: M=o(d2)\nWhen M=o(d2), we show that attention is equivalent to matrix multiplication, establishing a\nΘ\u0010\nN2d√\nM\u0011\nboundontheI/Ocomplexity. WefirstshowthatthereisanalgorithmwithI/Ocomplexity\nO\u0010\nN2d√\nM\u0011\n.\nTheorem 3.5. There is an algorithm computing attention with I/O complexity O\u0010\nN2d√\nM\u0011\n. Further-\nmore, this algorithm uses O\u0000\nN2d\u0001\ntime and O\u0000\nNd+N2\u0001\nspace.\nHigh Level Overview Our algorithm follows standard techniques for reducing the I/O complex-\nity of matrix multiplication. In Phase 1, we compute A= exp( QKT)andD= diag( A·1). First, we\ndivide Q, KintoN√\nM×d√\nMblocks of size√\nM×√\nMand proceed to compute QKTone√\nM×√\nM\nsize block at a time. In Lines 3 and 5, we iterate over blocks of QKT. In Lines 7 and 9, we compute\neach block of QKTby summing overd√\nMblock matrix products. After computing QKT, we apply\nexponents entry-wise and sum over rows in Lines 11 and 12 to compute the relevant blocks of A, D.\nIn Phase 2, we compute the matrix product D−1AV, storing the output in matrix O. We\nimagine D−1as a vector and partition intoN√\nMblocks of size√\nMand partition AintoN√\nM×N√\nM\nblocks of size√\nM×√\nM. Lines 16 and 18 iterates over blocks of O. Lines 20 and 22 compute a√\nM×√\nMblock of Oby summing overN√\nMmatrix products and scaling by the approprite block\nofD−1. This completes the overview of the algorithm.\n10Algorithm 1 SquareTilingAttention (Q, K, V, M )\nInput : Matrices Q, K, V ∈RN×d, Cache size M\nOutput: D−1AVwhere A= exp( QKT)andD= diag( A·1)\n1B←jp\nM/4k\n2Phase 1: Compute D, A\n3for1≤i≤ ⌈N/B⌉do\n4Initialize d(B)[i]←0Bin cache\n5for1≤j≤ ⌈N/B⌉do\n6 Initialize AB[i, j]←0B×Bin cache\n7for1≤ℓ≤ ⌈d/B⌉do\n8 Read Q(B)[i, ℓ]and(KT)(B)[ℓ, j]into cache\n9 Compute AB[i, j]←AB[i, j] +Q(B)[i, ℓ](KT)(B)[ℓ, j]in cache\n10 Delete Q(B)[i, ℓ]and(KT)(B)[ℓ, j]from cache\n11 Compute AB[i, j]←exp(AB[i, j])and write AB[i, j]into memory\n12 Compute d(B)[i]←d(B)[i] +AB[i, j]·1in cache\n13 Delete AB[i, j]from cache\n14Write d(B)[i]to memory and delete d(B)[i]from cache\n15Phase 2: Compute D−1AV\n16for1≤i≤ ⌈N/B⌉do\n17Read d(B)[i]into cache\n18for1≤j≤ ⌈d/B⌉do\n19 Initialize O(B)[i, j]←0B×Bin cache\n20 for1≤k≤ ⌈N/B⌉do\n21 Read A(B)[i, k]andV(B)[k, j]into cache\n22 Compute O(B)[i, j]←O(B)[i, j] + diag\u0000\nd(B)[i]\u0001−1A(B)[i, k]V(B)[k, j]\n23 Delete A(B)[i, k]andV(B)[k, j]from cache\n24 Write O(B)[i, j]to cache and delete O(B)[i, j]from cache\n25Delete d(B)[i]from cache\nCorrectness of Algorithm 1. We begin with Phase 1, showing that each block A(B)[i, j]is computed\ncorrectly. Fix a block A(B)[i, j]. For any indices i′∈[(i−1)B+ 1, iB]andj′∈[(j−1)B+ 1, jB],\nA[i′, j′] =dX\nℓ′=1Q[i, ℓ′]KT[ℓ′, j] =⌈d/B⌉X\nℓ=1ℓBX\nℓ′=(ℓ−1)B+1Q[i, ℓ′]KT[ℓ′, j]\nIn Line 9, we iterate over i′, j′, ℓ′, adding Q[i′, ℓ′]KT[ℓ′, j′]toA[i′, j′]. After iterating over 1≤\nℓ≤ ⌈d/B⌉,A(B)[i, j]is computed so that after applying expentry-wise, we write the correct block\nA(B)[i, j]into memory. Next, since dshould contain row-sums,\nd[i′] =NX\nj′=1A[i′, j′] =⌈N/B⌉X\nj=1jBX\nj′=(j−1)B+1A[i′, j′]\nwe correctly write d(B)[i]into memory. Throughout Phase 1, the number of items in cache is\n3B2+B≤4B2≤M.\n11Now, we proceed to Phase 2. Let D= diag( d) = diag( A·1). Similarly, for i′∈[(i−1)B+\n1, iB], j′∈[(j−1)B+ 1, jB]\nO[i′, j′] =1\nD[i′, i′]NX\nk′=1A[i′, k′]V[k′, j′] =⌈N/B⌉X\nk=1kBX\nk′=(k−1)B+1A[i′, k′]V[k′, j′]\nD[i′, i′]\nwhich is computed by the loop in Phase 2. In particular, Line 22 adds the appropriate value for each\nblock k. The overall size of cache required is B+ 3B2≤4B2≤M. Thus, Algorithm 1 correctly\ncomputes O=D−1AVwith cache size M.\nI/O Complexity of Algorithm 1. In Phase 1, for each iteration through i, j, ℓ, the algorithm reads\nO(B2)values from memory into cache. This dominates the I/O complexity of the algorithm. The\nI/O complexity of Phase 1 is therefore O\u0010\nN2d\nB3B2\u0011\n=O\u0010\nN2d\nB\u0011\n=O\u0010\nN2d√\nM\u0011\n.\nSimilarly for Phase 2, the I/O complexity is dominated by the reading A, Vinto the cache and\nthis has I/O complexity O\u0010\nN2d√\nM\u0011\n, thus bounding the overall I/O complexity.\nTime and Space Complexity of Algorithm 1. Since we use standard matrix multiplication, the over-\nall time complexity is O(N2d). The space required is O(Nd+N2)as the algorithm stores matrices\nQ, K, V, M and the vector d.\nWe now show that this is tight for M≤d2. We proceed by a reduction to the I/O complexity\nof matrix multiplication, invoking the following result.\nLemma 3.6 (Corollary 6.2 of [HK81]) .LetA∈Rm×kandB∈Rk×n. The standard algorithm for\nmatrix multiplication satisfies Q(M) = Ω\u0010\nmkn√\nM\u0011\n.\nTheorem 3.7. Suppose M=o(d2). Then, the I/O complexity of attention using standard matrix\nmultiplication is at least Ω\u0010\nN2d√\nM\u0011\nProof.The lower bound follows from a reduction to matrix multiplication. We take advantage of\nthe fact that if M≤d2,N2d√\nM≥N2, so the algorithm can afford to write the attention matrix\nAexplicitly to memory. Given an algorithm Afor attention, we have the following algorithm for\nmatrix multiplication. Given inputs Q, K, we execute Awith one modification: whenever an entry\nofQKTis computed for the first time, write this entry to memory. Over the course of the algorithm,\nthis computes QKTand adds at most N2additional I/Os, which any attention algorithm must use\nwhenever M < d2. We give the reduction in terms of the computational graph below.\nSuppose for contradiction there is an algorithm Acomputing attention with I/O complexity\no\u0010\nN2d√\nM\u0011\n. Consider Aas a complete calculation on the computational graph described in Figure 1.\nSinceAis a complete calculation and the set of input and output vertices are disjoint, every single\nvertex in the graph must have a pebble on it at some configuration in the calculation. Consider\nthen the algorithm Bwhich executes Awith the following modifications:\n1. Whenever a blue pebble is deleted from a vertex in QKT, do not delete.\n2. Whenever a red pebble is placed on a vertex in QKTfor the first time, place also a blue pebble\non this vertex.\nThe two properties guarantee that Bwill have at least the pebbles that Ahas, while any additional\npebbles must be blue, so that Brespects the constraint on the overall number of red pebbles at any\ngiven configuration. In particular, Bis a valid calculation that computes QKT. We now analyze\n12the I/O complexity of B. IfQAdenotes the I/O complexity of A, the additional writes due to the\nsecond rule imply an overall I/O complexity of,\nQA+N2=o\u0012N2d√\nM\u0013\nSinceBcomputes QKT, this contradicts Lemma 3.6.\n4 I/O Complexity of Attention with Fast Matrix Multiplication\nWe can in fact lower bound the I/O complexity of any algorithm computing attention exactly, in-\ncluding those using fast matrix multiplication. The only assumption we require is that the algorithm\ncomputes the matrix product QKTexplicitly (whether or not it writes the result to memory). Since\nwe do not make any further assumptions on the algorithm, the previous approach of analyzing a\ncomputational directed acyclic graph is not sufficient [HK81]. Instead, we relate I/O complexity to\ncompression lower bounds. As discussed previously, we are primarily interested in the large cache\nregime where M= Ω( d2).\n4.1 Large Cache: M= Ω(d2)\nGiven some algorithm Aand a sample execution, we split this computation into batches of roughly\nMI/O operations each using the framework of [PS14a].\nLemma 4.1. [Theorem 3 of [PS14a]] Suppose A′is an execution of algorithm Aon a machine with\ncache of size M. The execution A′can be split into Tepochs of at most MI/O operations, such\nthat in each epoch the algorithm Ahas access to a cache of size at most 2Mand no I/O operations.\nFurthermore, the I/O complexity of A′is at least (T−1)M.\nIntuitively, the algorithm uses the extra Mentries in the cache of size 2Mto create a buffer for\nthe next MI/O operations.\nProof.LetAbe any algorithm computing exact attention and A′an arbitrary execution of Aon\nmachine with a cache size of Mbits. Note that given A′all the decisions made by algorithm Aare\nalready taken.\nWe proceed to simulate the execution A′on a machine with cache size of 2Mso that the\ncomputation is split into epochs and I/O operations are performed only at the start and end of each\nepoch. Split the cache into two pieces, one block of size Mto simulate the cache of Aand one block\nas a buffer for I/O operations. At the start of the epoch, the simulation considers all of the Mnext\nI/O operations, performs all read I/Os by filling the buffer. During the epoch, any I/O operation\nis simulated by writing data between the two blocks of cache. Finally, at the end of the epoch, the\nsimulation takes all written entries in the buffer and writes them to memory. In particular, in each\nepoch, no I/O operations are performed so that the algorithm only has access to only a cache of\nsize2Mwith no I/O.\nFinally, in every epoch except for the last, MI/O operations are performed, so the I/O com-\nplexity of the execution A′is at least (T−1)M.\nTo show our lower bound, we will simplify the problem and assume the matrices Q, K, Vhave\nentries in some finite field Fq. This is similar to the “indivisibility\" assumptions of [AM98, PS14a],\nas the field size fixes some bound on the amount of information in a single cache entry. Otherwise,\n13if each cache entry can hold an arbitrary real number, then even when M= 1we could encode the\nentire matrices Q, Kin a single cache entry.\nUnder this assumption, we will assume the cache holds Melements of Fq, and we will corre-\nspondinglydefineI/Ocomplexityasthenumberoffinitefieldelementsreadandwrittenbetweenthe\nmemory hierarchy. In practice, matrices Q, Khave arbitrary real entries. Since our lower bounds\nonly need to consider algorithms computing QKT, we may restrict the inputs to some finite field Fq\nwithout loss of generality (choosing qlarge enough to avoid overflows under arithmetic operations)\nand we can assume qis of polynomial size since we do not need to consider the softmax operation.\nWe will separately consider the cases q= 2(i.e. every entry in the cache and the matrices is a single\nbit) and the case for an arbitrarily large finite field of size q.\n4.2 I/O Complexity and Compression\nWe will prove our I/O lower bound by arguing that any algorithm computing entries of QKT\namounts to an efficient compression protocol. First, we show a lower bound for any compression\nprotocol computing entries of QKT.\nDefinition 4.2 (Matrix Compression) .LetB≥0. In the B-entry matrix compression problem\nMatrixEntryCompressionBAlice is given input matrices Q, K ∈FN×d\nq. Alice must send a\nmessage to Bob so that Bob can compute at least Bentries in QKT.\nWe review the definition of one-way communication complexity below.\nDefinition 4.3 (One-Way Communication Complexity) .Letf:XA× X B→ Ybe an arbitrary\nfunction. Suppose Alice has x∈ X Aand Bob has y∈ X B. Aone-way communication protocol\ncomputing fis a pair of functions (E, D)such that D(E(x), y) =f(x, y)for all (x, y)∈ XA× X B.\nThecomplexity of the protocol is max (x,y)|E(x)|. Theone-way communication complexity offis\nthe complexity of the optimal protocol.\nOneWayCC (f) = min\n(E,D)max\n(x,y)∈XA×XB|E(x)|\nThe one-way communication complexity of MatrixEntryCompressionBisO(Blogq), since\nAlicecancompute QKTandtransmittherelevant Bentries. Instead, ifBobcomputes asquaresub-\nmatrix of QKT, Alice can instead send the relevant bits of Q, K, transmitting only O(d√\nBlogq)\nentries. We give a lower bound of Ω\u0010\nmin(d√\nBlogq, Blogq)\u0011\n, showing that this is essentially tight.\nOur key lemma states that any algorithm outputting many bits of QKTin one epoch (as\ndescribed in Lemma 4.1) gives an efficient compression protocol for MatrixEntryCompressionB.\nTheorem 4.4. LetA′(Q, K )be an execution of an algorithm Aon input Q, Kon a machine with\ncache of size M. Let Btbe the number of entries of QKTcomputed in the t-th epoch as described\nin Lemma 4.1 and B′(Q, K ) = max tBt. Define,\nB∗= min\nQ,KB′(Q, K )\nso that on every input Acomputes at least B∗entries of QKTon some epoch. Then, the one-way\ncommunication complexity of MatrixEntryCompressionB∗is at most 2Mlogq.\nProof.We will use Ato construct a compression protocol. Given input matrices Q, K, we execute\nAonQ, Kto obtain an execution A′. As described in Lemma 4.1, the execution A′can be split\n14into epochs, where in each epoch the algorithm Ahas access only to 2Mfinite field elements in\ncache and reads no other inputs from memory in this epoch. Let t∗be an epoch in which the\nalgorithm Acomputes B′(Q, K )entries of the product QKT. In particular, Alice can send to Bob\nthe state of the cache of size at most 2Mat the beginning of the t∗-th epoch, so that Bob computes\nB′(Q, K )≥B∗entries of QKT.\n4.3 Matrix Compression Lower Bounds\nIn this section, we prove a lower bound on the one-way communication complexity of the B-entry\nMatrixEntryCompression problem. More precisely, we will show an upper bound on the number\nof entries that can be computed given a message of size M. Our previous discussion then implies an\nupper bound on the number of bits computed in each epoch, therefore lower bounding the number\nof epochs and the I/O complexity.\nAs a warmup, we give a simple lower bound of√\nBlogq.\nLemma4.5. LetB≥0. Then, the one-way communication complexity of MatrixEntryCompressionB\nis at least√\nBlogq.\nProof.LetI⊂[N]2denote any set of indices of the computed entries of QKTwhere |I|=B.\nDefine RIto be the distinct row indices in IandCIto be the distinct column indices in Iso that\nI⊂RI×CI.\nWithout loss of generality, assume |RI| ≥ |CI|. We claim there are at least q|RI|distinct values in\nthe entries of QKTindexed by I. In particular, let Q, Kboth be matrices with non-zero values only\nin the first column. In K, we set every entry in the first column to 1. In this case, each row of QKT\nwill be the same, so we can assume the Bentries are |RI|entries in a column of QKT. Since we\ncan arbitrarily set the entries of Qindexed by RI, we can obtain q|RI|different outputs. Since there\nare at least q|RI|outputs, the message length Mmust be at least |RI|logq= max( |RI|,|CI|) logq\nin order to unambiguously determine the correct output. Then,\nB≤RICI≤max( RI, CI)2≤\u0012M\nlogq\u00132\nThus, M≥√\nBlogq.\nWe generalize this for matrices Q, Kof dimension N×dwith rank d≥1.\nLemma 4.6. Suppose Q, K∈FN×d\nqwith finite field Fqof size q > N. Then, the one-way commu-\nnication complexity of MatrixEntryCompressionBis at least min\u0012\ndq\nB\n2,B\n4\u0013\nlogq.\nWe show that for any set of Bindices, there are more than qMpossible values. Thus, a message\nlength of at least Mlogqis required to specify these entries exactly.\nProof.LetMdenote the maximum message length in the communication protocol. Let I= [N]2\ndenote the indices of QKTcomputed with B=|I|. Define RIto be the distinct row indices in I\nandCIto be the distinct column indices in Iso that I⊂RI×CI.\nFor each i∈RI, letRi={js.t.(i, j)∈I}be the computed entries in the i-th row of QKT.\nSimilarly, for each j∈CI, letCj={is.t.(i, j)∈I}be the computed entries in the j-th column\nofQKT. Next, define LR={is.t.|Ri| ≥d}andLC={js.t.|Cj| ≥d}. We also define SR=\n{(i, j)∈Is.t.i̸∈LR}andSC={(i, j)∈Is.t.j̸∈LC}.\n15First, suppose max(|LR|,|LC|)≥p\nB/2. Without loss of generality, assume |LR| ≥p\nB/2.\nSince q > N, we fix Kto be the Vandermonde matrix guaranteed by Lemma 4.7. In particular,\nevery subset of dcolumns in KTis linearly independent.\nFix some row i∈RI. We claim that there are at least qmin(|Ri|,d)distinct values in the Riindices\nof the i-th row. First, suppose |Ri| ≤d. By the construction of matrix Kand|Ri| ≤d, the columns\nofKTindexed by Riare linearly independent. Then, for any ⃗ v∈F|Ri|\nq, we can set the i-th row of\nQto be,\nQ[i] =⃗ v\u0000\nK[r1]TK[r2]T. . . K [r|Ri|]T\u0001−1\nwhere K[ri]is the ri-th row of KandRi={r1, . . . , r |Ri|}. In particular, this choice of Q[i]ensures\nthat the Rientries of QKTare exactly ⃗ vso that there are at least q|Ri|distinct values. Whenever\n|Ri|> d, we simply take an arbitrary subset of Riof size d, and use their linear independence to\nproceed with the same argument, thus obtaining the lower bound of qmin(|Ri|,d).\nFinally, note that the we can obtain these distinct values for each row in RIindependently, since\nwe have fixed Kas the Vandermonde matrix and for each row we only modify the entries of Q[i].\nIn particular, the total number of possible distinct values in all the entries of Iis at least,\nY\ni∈RIqmin(|Ri|,d)\nso that we obtain the following lower bound on the message length of the communication protocol\nin order to unambiguously specify Bentries,\nM≥\nX\ni∈RImin(|Ri|, d)\nlogq\n≥(d|LR|+|SR|) logq\n≥dr\nB\n2logq (1)\nwhere SR={(i, j)∈Is.t.i̸∈LR}and the final inequality follows from our assumption on LR. In\nparticular, this implies M≥dp\nB/2 logqas desired.\nFinally, we consider the case max(|LR|,|LC|)<p\nB/2. Then, the number of entries in LR×LC\nis at mostB\n2. Note that any entry of Inot in LR×LCmust be in SR∪SCand therefore,\nB\n2≤ |SR|+|SC|\nAssume without loss of generality |SR| ≥ |SC|. Equation 1 then implies M≥B\n4logq.\nIn our lower bound construction, we require matrices satisfying strong linear independence\nconstraints. Specifically, we require a N×dmatrix such that every subset of drows is linearly inde-\npendent. When the matrices have elements in a large finite field, this is obtained by Vandermonde\nmatrices.\nLemma 4.7. There is a N×dVandermonde matrix with entries in a finite field Fqof size q > N\nwhere every subset of drows is linearly independent.\n16Proof.Consider the N×dVandermonde matrix,\nV=\n1α1α2\n1. . . αd−1\n1\n1α2α2\n2. . . αd−1\n2...............\n1αNα2\nN. . . αd−1\nN\n\nwhere α1, α2, . . . , α Nare distinct elements in Fq, since we choose q > N. Then, for any subset of d\nrows, the determinant of this sub-matrix is,\n0̸=Y\n1≤i<j≤d(αki−αkj)\nwhere k1, . . . , k dare the indices of the drows. Since the determinant of this matrix is non-zero, the\nrows are linearly independent.\nWe are now ready to prove the main result of this section. For M≥d2, this matches the upper\nbound given by [DFE+22].\nTheorem 4.8. Suppose Q, K∈FN×d\nqwhere q > N. The I/O complexity of attention (with any\nmatrix multiplication algorithm) is Ω\u0010\nmin\u0010\nN2d2\nM, N2\u0011\u0011\n.\nProof.The theorem follows from combining Theorem 4.4 and Lemmas 4.1 and 4.6. Consider an\narbitrary algorithm Aand its best execution A′, on an input Q, K, V with B′(Q, K ) =B∗as\ndescribed in Theorem 4.4.\nSince q > Nwe apply Lemmas 4.1 and 4.6 and obtain\nmin \ndr\nB∗\n2,B∗\n4!\nlogq≤2Mlogq\nIn particular, the maximum number of entries of QKTcomputed in any epoch has the upper\nbound,\nB∗=O\u0012\nmax\u0012M2\nd2, M\u0013\u0013\nThen since the algorithm computes all N2values of QKT(regardless of whether these values\nare written to memory from cache), the number of epochs is at least,\nT= Ω\u0012\nmin\u0012N2d2\nM2,N2\nM\u0013\u0013\nThen, since the I/O complexity of Ais at least (T−1)M, this completes the lower bound.\nWhenever M≥d2, we obtain a lower bound matching the O\u0010\nN2d2\nM\u0011\nalgorithm of [DFE+22].\n174.4 Binary Matrix Compression Lower Bounds\nIn the previous section, we obtained a tight lower bound for the I/O complexity of attention when\nthe entries are allowed to come from a large finite field. We believe it is an interesting theoretical\nquestion to investigate the I/O complexity with entries in smaller finite fields. Specifically, we will\nconsider the binary finite field F2={0,1}.\nThe assumption q > Nwas only required to construct a matrix satisfying strong linear inde-\npendence constraints in Lemma 4.7. Even relaxing this constraint and considering q= 2, we can\nstill construct matrices satisfying fairly strong linear independence constraints using error correcting\ncodes. In fact, our previous use of Vandermonde matrices can be interpreted as using Reed-Solomon\ncodes by allowing for arbitrarily large finite fields. Recall that a linear code C ⊂ { 0,1}Nis the null-\nspace of the parity check matrix H. Using Binary BCH codes, we can obtain a similar result with\nbinary matrices.\nLemma 4.9. There exists a matrix K∈ {0,1}N×dsuch that every set of2d\nlog(N+1)−1rows is linearly\nindependent.\nBefore proving Lemma 4.9, we require several intermediate results. First, we state the following\nstandard lemma relating distance of linear codes to linear independence in the parity check matrix.\nLemma 4.10. A linear code has distance d, if and only if any (d−1)columns of the parity check\nmatrix is linearly independent and there exist dcolumns that are linearly dependent.\nProof.Consider a code Cof length n, dimension k, and distance d. Suppose there is a set of (d−1)\nlinearly dependent columns. Then, there is a vector xwith wt(x)≤d−1such that Hx= 0,\ncontradicting the minimum distance dofC. Since the distance of the code is d, there exists a vector\nxsuch that Hx= 0and wt(x) =d. In particular, there exists a subset of dindependent columns.\nTo prove the converse, note that the conditions on Himply there exists a codeword of weight d\nand no codeword of weight less than d, so that the minimum distance of the code is exactly d.\nNext, we use the fact that binary BCH codes are optimal high rate codes. Recall that a code\nwith parity check matrix Hof dimension d×Nhas dimension at least N−d.\nLemma 4.11. [Hoc59, BR60] For a length N= 2m−1and a distance s, there exists a code\nBCH[ N, s]with dimension at least N−\u0006s−1\n2\u0007\nlog(N+ 1).\nWe provide the definition of BCH codes. Recall an element α∈Fin a finite field is a primitive\nelement if it generates the multiplicative group F∗.\nDefinition 4.12 ([Hoc59, BR60]) .For length N= 2m−1, distance s, and primitive element\nα∈F∗\n2m, the binary BCH code is defined,\nBCH[ N, s] ={(c0, . . . , c N−1)s.t.c(α) =. . .=c(αs−1) = 0}\nwhere c(X) =c0+c1X+. . .+cN−1XN−1.\nThe proof of Lemma 4.11 is a standard exercise.\nProof of Lemma 4.11. Toshowalowerboundonthedimensionofthecode, wearguethattheparity\ncheckmatrix Hdoesnothavetoomanyrows. Webeginwithaweakerboundof N−(s−1) log( N+1).\nIn particular, we show that each constraint ccan be written as m= log( N+ 1)linear constraints.\n18We choose a basis β={β1= 1, β2, . . . , β m}ofFm\n2as a vector space. For any x∈Fm\n2, consider\nthe linear map x7→αxwhich can be written as x7→Mαxfor some matrix Mα∈Fm×m\n2where xis\nrepresented in the above basis. Then, the constraint c(α) = 0can be viewed as,\n\nc0\n0\n...\n0\n+Mα\nc1\n0\n...\n0\n+. . .+MαN−1\ncN−1\n0\n...\n0\n=\n0\n0\n...\n0\n\nThus, each constraint c(αi) = 0can be viewed as log(N+1)linear constraints. Since there are s−1\nsuch constraints, this ensures the parity check matrix has dimension (s−1) log( N+ 1)×N.\nWe now prove the improved bound. This follows from the fact that c(γ) = 0if and only if\nc(γ2) = 0. In particular,\u0004s−1\n2\u0005\nconstraints are redundant, leaving only\u0006s−1\n2\u0007\nrelevant constraints.\nIt remains to show c(γ2) = 0if and only if c(γ) = 0. Note that c(γ) = 0if and only if c(γ)2= 0.\nFurthermore, for α, β∈Fm\n2,(α+β)2=α2+β2. Then,\n0 =c(γ) =c(γ)2\n=c2\n0+ (c1γ)2+. . .+ (cN−1γN−1)2\n=c0+c1γ2+. . . c N−1γ2(N−1)=c(γ2)\nwhere we have used ci=c2\nifor all coefficients ci∈F2.\nWe now prove Lemma 4.9 and describe the construction of the desired matrix Ksatisfying\nstrong linear independence constraints.\nProof.We assume without loss of generality that N= 2m−1for some m. If not, we at most double\nNby choosing the minimum msuch that 2m−1≥Nand take any N-row sub-matrix.\nFrom the construction of the code BCH[ N, s], we observe that it has a parity check matrix Hof\ndimension\u0006s−1\n2\u0007\nlog(N+ 1)×N. Since the code has distance s, from Lemma 4.9, any set of s−1\nrows is linearly independent. In particular, if d=\u0006s−1\n2\u0007\nlog(N+ 1), we have,\ns≥2d\nlog(N+ 1)\ngiving the desired bound on s−1. Thus, we choose K=HT.\nGiven the construction of the matrix K, we can prove the following analogues of Lemma 4.6\nand Theorem 4.8. These results match the upper bound of Theorem 3.2 up to a O(log2N)factor.\nLemma 4.13. Suppose Q, Kare binary N×dmatrices. Then, the one-way communication com-\nplexity of MatrixEntryCompressionBis at least Ω\u0010\nmin\u0010\nd√\nB\nlogN, B\u0011\u0011\n.\nProof.The proof follows Lemma 4.6 closely, so we only point out the necessary modifications. Again\nletMdenote the maximum message length in the communication protocol and I= [N]2denote\nthe indices of QKTcomputed with B=|I|. Define RI, CI,{Ri}N\ni=1,{Cj}N\nj=1as in Lemma 4.6.\nWe modify LR={is.t.|Ri| ≥2d\nlog(N+1)−1}andLC={js.t.|Cj| ≥2d\nlog(N+1)−1}and define\nSRandSCas before.\nWe again consider first the case max(|LR|,|LC|)≥p\nB/2and assume |LR| ≥p\nB/2. Instead\nof the Vandermonde matrix, we fix Kto be the matrix guaranteed by Lemma 4.9. In particular,\nevery subset of2d\nlog(N+1)−1columns in KTis linearly independent.\n19Following an analogous argument as Lemma 4.6, we obtain the following lower bound on the\nmessage length of the communication protocol in order to unambiguously specify Bentries,\nM≥X\ni∈RImin\u0012\n|Ri|,2d\nlog(N+ 1)−1\u0013\n≥\u00122d\nlog(N+ 1)−1\u0013\n|LR|+|SR|\n= Ω \nd√\nB\nlogN!\n(2)\nwhere the final inequality follows from our assumption on LR.\nFinally, we consider the case max(|LR|,|LC|)<p\nB/2. Again following similar arguments as\nLemma 4.6 and Equation 2, we have,\nM≥max(|SR|,|SC|)≥B\n4\nso that Mis at least the minimum of the two bounds.\nWe now state the I/O complexity lower bound of attention given binary input matrices.\nTheorem 4.14. Suppose Q, K∈ {0,1}N×d. The I/O complexity of attention (with any matrix\nmultiplication algorithm) is Ω\u0010\nmin\u0010\nN2d2\nMlog2N, N2\u0011\u0011\n.\nProof.Following similar arguments as Theorem 4.8, we obtain,\nΩ \nmin \nd√\nB∗\nlogN, B∗!!\n≤2M\nwhere B∗is as defined in Theorem 4.4. In particular, the maximum number of entries of QKT\ncomputed in any epoch is at most,\nB∗=O\u0012\nmax\u0012M2log2N\nd2, M\u0013\u0013\nwhich gives the I/O complexity lower bound,\nΩ\u0012\nmin\u0012N2d2\nMlog2N, N2\u0013\u0013\nas desired.\n4.5 Small Cache: M=o(d2)\nIn the small cache setting, we proved an equivalence between attention and matrix multiplication in\nthe setting where both are computed using the standard algorithm. We do the same for algorithms\nusing fast matrix multiplication.\nLetQAtt(M)denote the I/O complexity of attention on a machine with cache size M. Let\nQM(a,b,c)(M)denote the I/O complexity of multiplying a a×bmatrix with a b×cmatrix on a\nmachine with cache size M. First, we show matrix multiplication is more expensive than attention.\n20Lemma 4.15. For all M,\nQAtt(M) =O\u0000\nQM(N,d,N )(M) +QM(N,N,d )(M)\u0001\nProof.First, weusethegivenalgorithmforrectangularmatrixmultiplicationtocompute QKTwith\nQM(N,d,N )I/O operations. Then, note that computing softmax( QKT)can be done in O(N2)time\nand therefore O(N2)I/O complexity as each operation can be performed with O(1)I/O operations.\nFinally, we use the given algorithm to compute softmax( QKT)Vwith QM(N,N,d ). Then, the overall\nI/O complexity is,\nQAtt(M) =O\u0000\nQM(N,d,N )+N2+QM(N,N,d )\u0001\n=O\u0000\nQM(N,d,N )+QM(N,N,d )\u0001\nsince the I/O complexity of both matrix products is at least N2, as either the input or output has\nsizeN2.\nWhen M≤d2, the two are equivalent.\nLemma 4.16. LetM≤d2. Then, QAtt(M) = Ω\u0000\nQM(N,d,N )(M)\u0001\n.\nProof.First, consider any two input matrices Q, KT. We simulate the attention algorithm with the\nfollowing modification: whenever an entry (QKT)ijis computed for the first time, we write this\nentry to memory. Our modified algorithm successfully computes the matrix product QKTusing\nat most O\u0000\nQAtt(M) +N2\u0001\nI/O operations. From Theorem 4.8, we have that whenever M≤d2,\nQAtt(M) = Ω( N2). As a result, we compute Attention with O(QAtt(M))I/O complexity.\nAs a corollary, we obtain the following equivalence in the small cache setting.\nTheorem 4.17. LetM≤d2. Then,\nQAtt(M) = Ω\u0000\nQM(N,d,N )(M)\u0001\nQAtt(M) =O\u0000\nQM(N,d,N )(M) +QM(N,N,d )(M)\u0001\n5 Conclusion\nWe have established tight I/O complexity lower bounds for attention. Our lower bound in fact\nholds for any algorithm computing matrix product QKT. We give a tight characterization of the\nI/O complexity of algorithms using standard matrix multiplication, answering an open question of\n[DFE+22].\nFurthermore, in the regime of practical interest, where cache size M≥d2is large, we extend our\nlower bound to algorithms using fast matrix multiplication, showing that FlashAttention is optimal\neven when fast matrix multiplication is allowed. Furthermore, we establish a connection between\ncommunication complexity and I/O complexity, which may be of independent theoretical interest.\nWe leave the problem of establishing tight I/O complexity bounds for attention in the small\ncache regime ( M≤d2) (equivalently rectangular matrix multiplication) as an interesting open\nproblem.\nAcknowledgements\nWe would like to thank Arya Mazumdar and Harry Sha for helpful discussions.\n21References\n[AM98] Lars Arge and Peter Bro Miltersen. On showing lower bounds for external-memory\ncomputational geometry problems. In External Memory Algorithms, Proceedings of a\nDIMACS Workshop , 1998. 13\n[AS23] Josh Alman and Zhao Song. Fast attention requires bounded entries. CoRR,\nabs/2302.13214, 2023. 2\n[AV88] Alok Aggarwal and Jeffrey Scott Vitter. The input/output complexity of sorting and\nrelated problems. Commun. ACM , 1988. 3\n[BDH+12] Grey Ballard, James Demmel, Olga Holtz, Benjamin Lipshitz, and Oded Schwartz.\nGraph expansion analysis for communication costs of fast rectangular matrix multipli-\ncation. In Mediterranean Conference on Algorithms MedAlg , 2012. 3\n[BDHS12] Grey Ballard, James Demmel, Olga Holtz, and Oded Schwartz. Graph expansion and\ncommunication costs of fast matrix multiplication. J. ACM, 59(6):32:1–32:23, 2012. 3\n[BMR+20] Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla\nDhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini\nAgarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya\nRamesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark\nChen, EricSigler, MateuszLitwin, ScottGray, BenjaminChess, JackClark, Christopher\nBerner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language\nmodelsarefew-shotlearners. In Neural Information Processing Systems, NeurIPS ,2020.\n2\n[BR60] R. C. Bose and Dwijendra K. Ray-Chaudhuri. On A class of error correcting binary\ngroup codes. Inf. Control. , 3(1):68–79, 1960. 2, 4, 18\n[BS17] Gianfranco Bilardi and Lorenzo De Stefani. The I/O complexity of strassen’s matrix\nmultiplication with recomputation. In Algorithms and Data Structures WADS , 2017. 3\n[CDW+21] Beidi Chen, Tri Dao, Eric Winsor, Zhao Song, Atri Rudra, and Christopher Ré. Scat-\nterbrain: Unifying sparse and low-rank attention. In Neural Information Processing\nSystems, NeurIPS , 2021. 2\n[CLD+21] Krzysztof Marcin Choromanski, Valerii Likhosherstov, David Dohan, Xingyou Song,\nAndreea Gane, Tamás Sarlós, Peter Hawkins, Jared Quincy Davis, Afroz Mohiuddin,\nLukasz Kaiser, David Benjamin Belanger, Lucy J. Colwell, and Adrian Weller. Rethink-\ningattentionwithperformers. In International Conference on Learning Representations,\nICLR, 2021. 2\n[CP23] Xi Chen and Binghui Peng. Memory-query tradeoffs for randomized convex optimiza-\ntion. InSymposium on Foundations of Computer Science, FOCS , 2023. 2\n[CPP22] Xi Chen, Christos H. Papadimitriou, and Binghui Peng. Memory bounds for continual\nlearning. In Symposium on Foundations of Computer Science, FOCS , 2022. 2, 3\n[Dao23] Tri Dao. Flashattention-2: Faster attention with better parallelism and work partition-\ning.CoRR, abs/2307.08691, 2023. 2, 6, 7\n22[DFE+22] Tri Dao, Daniel Y. Fu, Stefano Ermon, Atri Rudra, and Christopher Ré. Flashattention:\nFast and memory-efficient exact attention with io-awareness. In Neural Information\nProcessing SystemsNeurIPS , 2022. 2, 3, 6, 8, 17, 21\n[DKS19] Yuval Dagan, Gil Kur, and Ohad Shamir. Space lower bounds for linear prediction in\nthe streaming model. In Conference on Learning Theory, COLT , 2019. 3\n[GLM20] Alon Gonen, Shachar Lovett, and Michal Moshkovitz. Towards a combinatorial char-\nacterization of bounded-memory learning. In Neural Information Processing Systems,\nNeurIPS , 2020. 2\n[GU18] Francois Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using\npowers of the coppersmith-winograd tensor. In Artur Czumaj, editor, Proceedings of\nthe Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018,\nNew Orleans, LA, USA, January 7-10, 2018 , pages 1029–1046. SIAM, 2018. 2\n[HJK+23] Insu Han, Rajesh Jayaram, Amin Karbasi, Vahab Mirrokni, David P. Woodruff, and\nAmir Zandieh. Hyperattention: Long-context attention in near-linear time. CoRR,\nabs/2310.05869, 2023. 2\n[HK81] Jia-Wei Hong and Hsiang-Tsung Kung. I/o complexity: The red-blue pebble game. In\nSymposium on Theory of Computing STOC , 1981. 2, 3, 5, 6, 12, 13\n[Hoc59] Alexis Hocquenghem. Codes correcteurs d’erreurs. Chiffers, 2:147–156, 1959. 2, 4, 18\n[KKL20] Nikita Kitaev, Lukasz Kaiser, and Anselm Levskaya. Reformer: The efficient trans-\nformer. In International Conference on Learning Representations, ICLR , 2020. 2\n[KLMY19] Daniel Kane, Roi Livni, Shay Moran, and Amir Yehudayoff. On communication com-\nplexity of classification problems. In Conference on Learning Theory, COLT , 2019.\n3\n[KVPF20] Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pappas, and François Fleuret. Trans-\nformersarernns: Fastautoregressivetransformerswithlinearattention. In International\nConference on Machine Learning, ICML , 2020. 2\n[MSSV22] Annie Marsden, Vatsal Sharan, Aaron Sidford, and Gregory Valiant. Efficient convex\noptimization requires superlinear memory. In Conference on Learning Theory, COLT ,\n2022. 2\n[PR23] Binghui Peng and Aviad Rubinstein. Near optimal memory-regret tradeoff for online\nlearning. In Symposium on Foundations of Computer Science, FOCS , 2023. 2\n[PS14a] Rasmus Pagh and Francesco Silvestri. The input/output complexity of triangle enu-\nmeration. In Principles of Database Systems PODS , 2014. 3, 13\n[PS14b] Rasmus Pagh and Morten Stöckel. The input/output complexity of sparse matrix mul-\ntiplication. In European Symposium on Algorithms ESA , 2014. 3\n[PZ23] Binghui Peng and Fred Zhang. Online prediction in sub-linear space. In Symposium on\nDiscrete Algorithms, SODA , 2023. 2\n[Raz19] Ran Raz. Fast learning requires good memory: A time-space lower bound for parity\nlearning. J. ACM, 66(1):3:1–3:18, 2019. 2\n23[SHS15] Jacob Scott, Olga Holtz, and Oded Schwartz. Matrix multiplication i/o-complexity by\npath routing. In Symposium on Parallelism in Algorithms and Architectures SPAA ,\n2015. 3\n[SSV19] Vatsal Sharan, Aaron Sidford, and Gregory Valiant. Memory-sample tradeoffs for linear\nregression with small error. In Symposium on Theory of Computing, STOC , 2019. 2\n[SWXZ22] Vaidehi Srinivas, David P. Woodruff, Ziyu Xu, and Samson Zhou. Memory bounds for\nthe experts problem. In Symposium on Theory of Computing, STOC , 2022. 2\n[VSP+17] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N.\nGomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Neural\nInformation Processing Systems NeurIPS , 2017. 2\n[WS19] Blake E. Woodworth and Nathan Srebro. Open problem: The oracle complexity of\nconvex optimization with limited memory. In Conference on Learning Theory, COLT ,\n2019. 2\n[WXXZ23] Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu, and Renfei Zhou. New bounds\nfor matrix multiplication: from alpha to omega. CoRR, abs/2307.07970, 2023. 2\n[ZGD+20] ManzilZaheer, GuruGuruganesh, KumarAvinavaDubey, JoshuaAinslie, ChrisAlberti,\nSantiago Ontañón, Philip Pham, Anirudh Ravula, Qifan Wang, Li Yang, and Amr\nAhmed. Big bird: Transformers for longer sequences. In Neural Information Processing\nSystems, NeurIPS , 2020. 2\n24"    },    {        "title": "2402.07453v1.Bandit_Feedback_Online_Multiclass_Classification__Variants_and_Tradeoffs.pdf",        "content": "arXiv:2402.07453v1  [cs.LG]  12 Feb 2024Bandit-Feedback Online Multiclass Classification:\nVariants and Tradeoffs\nYuval Filmus1,2, Steve Hanneke3, Idan Mehalel1, and Shay Moran2,1,4,5\n1The Henry and Marilyn Taub Faculty of Computer Science, Technion, Israel\n2Faculty of Mathematics, Technion, Israel\n3Department of Computer Science, Purdue University, USA\n4Faculty of Data and Decision Sciences, Technion, Israel\n5Google Research, Israel\nFebruary 13, 2024\nAbstract\nConsider the domain of multiclass classification within the adversarial online setting.\nWhat is the price of relying on bandit feedback as opposed to full info rmation? To what\nextent can an adaptive adversary amplify the loss compared to an o blivious one? To what\nextent can a randomized learner reduce the loss compared to a det erministic one? We study\nthese questions in the mistake bound model and provide nearly tight answers.\nWe demonstrate that the optimal mistake bound under bandit feed back is at most O(k)\ntimes higher than the optimal mistake bound in the full information ca se, where krepresents\nthe number of labels. This bound is tight and provides an answer to an open question\npreviously posed and studied by Daniely and Helbertal [’13] and by Lon g [’17, ’20], who\nfocused on deterministic learners.\nMoreover,we present nearly optimal bounds of ˜Θ(k) on the gapbetween randomized and\ndeterministic learners, as well as between adaptive and oblivious adv ersaries in the bandit\nfeedback setting. This stands in contrast to the full information s cenario, where adaptive\nand obliviousadversariesareequivalent, and the gapin mistakeboun ds between randomized\nand deterministic learners is a constant multiplicative factor of 2.\nIn addition, our results imply that in some cases the optimal randomiz ed mistake bound\nis approximately the square-root of its deterministic parallel. Previo us results show that\nthis is essentially the smallest it can get.\n1Contents\n1 Introduction 3\n1.1 Main questions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Bounds for prediction with expert advice . . . . . . . . . . . . . . . . . . . . . . . 6\n1.3 Technical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n1.4 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2 Preliminaries 11\n3 The primal and dual games 13\n3.1 An optimal learner for the primal game . . . . . . . . . . . . . . . . . . . . . . . 15\n4 Prediction with expert advice 15\n4.1 Warm-up: deterministic learners . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n4.2 Randomized (Learner) and Adaptive (Adversary) . . . . . . . . . . . . . . . . . . 18\n4.3 Lower bounds for randomized learners . . . . . . . . . . . . . . . . . . . . . . . . 22\n4.4 Tight bounds for randomized learners . . . . . . . . . . . . . . . . . . . . . . . . 23\n5 The role of information 23\n5.1 Reduction from bandit to full-information feedback . . . . . . . . . . . . . . . . . 24\n5.2 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n5.3 An extension to the agnostic setting . . . . . . . . . . . . . . . . . . . . . . . . . 25\n6 The role of adaptivity 26\n7 The role of randomness 27\n8 Prediction without prior knowledge 29\n9 Open questions and future work 30\n9.1 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n9.2 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\n21 Introduction\nWe study the problem of online multiclass classification in t he mistake bound model, which\nmeasures the worst-case expected number of mistakes made by the learner in the entire game.\nThisproblem suggests a wideand appealinglandscapeof poss ibledifferent setups, each granting\nor preventing various resources from the learner or the adve rsary which generates the input.\nThe resources we discuss are the information provided by the adversary, the adaptivity of the\nadversary, and the randomness of the learner. We are interested in how the optimal mistake\nbound is affected by granting or preventing each of those resou rces. Below, we discuss these\nquestions and present our results.\n1.1 Main questions and results\nWe assume that a concept class H ∈ YXis given, where Xis adomainof instances and Yis\nalabel set. Unless stated otherwise, we restrict the learning scenari o to the realizable case, in\nwhich the input provided by the adversary is consistent with a concept from H.\nIn order to state the results, we define some notation. For a co ncept class H, letoptrand\nfull(H)\ndenotetheoptimalmistakeboundfor Hachievablewithfull-informationfeedback. Let optdet\nfull(H)\ndenote the same as optrand\nfull(H), with the additional restriction that the learner is deter ministic.\nIn the bandit feedback model, define optdet\nbandit(H) analogously to optdet\nfull(H). When the learner is\nallowed to be randomized, the mistake bound in the bandit fee dback model might change if the\ngame is played against an oblivious or adaptive adversary (w e define these types of adversaries\nbelow).1Therefore, we define the notations optobl\nbandit(H) andoptadap\nbandit(H) to denote the optimal\nmistake bounds of a randomized learner which receives bandi t feedback on its predictions, when\nthe adversary is oblivious or adaptive, respectively.\nUnless stated otherwise, Oand Ω notations hide universal constants that do not depend o n\nany parameter of the problem.\n1.1.1 Information\nIn the full-information feedback model, the learner learns the correct label at the end of each\nround, regardless of its prediction. In the bandit feedback model, significantly less information\nis provided at the end of each round: the adversary only revea ls whether the learner’s prediction\nwas correct or incorrect. This raises the following natural question studied e.g. by [AL99,DH13,\nDSBDSS15,Lon20].\nWhat is the price the learner pays for receiving only bandit f eedback?\nThe answer for this question is known for deterministic lear ners. [AL99] proved the upper\nbound\noptdet\nbandit(H) =O(optdet\nfull(H)·|Y|log|Y|) (1)\nfor every concept class H. A matching lower bound was given in [Lon20,Gen21], which sh owed\nthat for every natural k≥2 there exists a concept class Hwith label set of size |Y|=ksuch\nthat\noptdet\nbandit(H) = Ω(optdet\nfull(H)·|Y|log|Y|). (2)\nThey also find tight guarantees on the constants hidden in the O,Ω notations.\n1In the full-information model there is no difference between oblivious and adaptive adversaries. Essentially,\nthe reason is that in the full-information model the feedbac k is deterministic, even if the learner is randomized.\n3Finding an analogous result for randomized learners was rai sed as an open problem in\n[DH13].2We solve this open problem by proving the upper bound stated i n the following\ntheorem.\nTheorem 1.1 (Full-information vs. Bandit-feedback) .For every concept class Hit holds that\noptadap\nbandit(H) =O(optrand\nfull(H)·|Y|).\nFurthermore, for every natural k≥2there exists a concept class Hwith|Y|=ksuch that\noptobl\nbandit(H) = Ω(optrand\nfull(H)·|Y|).\nThe lower bound was noted e.g. in [DH13]. We complement it by p roving an upper bound\nthat holds forall classes. We prove Theorem1.1 inSection 5. Thetechniques developed to prove\nthis upper bound include new bounds for prediction with expert advice with bandit feedback,\nwhich is the main technical contribution of this work. These boundsarepresented in Section 1.2,\nand a technical overview of the proof can be found in Section 1 .3.\nA generalization to the agnostic setting. We also generalize results in the spirit of Theo-\nrem 1.1 to the agnostic case, in which the best hypothesis in c lass is inconsistent with the input\ninr⋆many rounds. Our results do not require that r⋆(or some bound r≥r⋆on it) is given\nto the learner. However, throughout most of the paper we assu me the stronger r-realizability\nassumption, in which an upper bound r≥r⋆is given to the learner. We explain in Section 8\nhow to remove this assumption using a standard “doubling tri ck”. Bounds for various setups\nin the agnostic setting are summarized in Table 1.\nIn the agnostic setting, learning algorithms are often meas ured by their expected regret.\nWhile in this work we measure algorithms by their mistake bou nd, note that a mistake bound\nof an algorithm is always at least as large as its regret. Ther efore, since our boundsdemonstrate\nno dependence on the number of rounds T, in some cases they provide improvements over the\nknown regret bound ˜O/parenleftbigg/radicalBig\nT|Y|optdet\nfull(H)/parenrightbigg\nof [DH13]. Specifically, when r⋆=O(optdet\nfull(H)), our\nresults imply a regret bound of O(|Y|optdet\nfull(H)), for all T.\n1.1.2 Adaptivity\nIn Section 1.1.1 we showed that randomness is necessary for o btaining optimal bounds on the\ncostofbanditfeedback. However, withinthesetupofarando mizedlearnerwhichreceives bandit\nfeedback, there are (at least) two types of adversaries to co nsider: an oblivious adversary which\nmust decide on the entire input in advance, and an adaptive ad versary that can decide on the\ninput on the fly. This raises the following natural question.\nWithin the bandit feedback model, what is the price the learn er pays for playing against\nan adaptive adversary?\nWe solve this question up to logarithmic factors. However, o ur nearly tight lower bounduses\npattern classes , which are a generalization of concept classes. In more deta il, a pattern class\nis a set of patterns p∈(X × Y)⋆which is downwards closed (i.e. closed under sub-patterns) .\nProving the lower bound in the theorem below using only conce pt classes or showing that this\nis not possible is an interesting and main problem left open b y this work.\n2The setting considered in [DH13], is, however, more general than the setting we consider. They consider an\nadversary which can decide on a few correct labels in each rou nd, and not only one. Generalizing this work to\ntheir multilabel setting is an interesting direction for fu ture work.\n4LearnerAdversary\nOblivious Adaptive\nRandomizedO/parenleftbig\n|Y|(optdet\nfull(H)+r⋆)/parenrightbig\nΩ/parenleftbig\n|Y|·optdet\nfull(H)+r⋆/parenrightbigΘ/parenleftbig\n|Y|(optdet\nfull(H)+r⋆)/parenrightbig\nDeterministicO/parenleftbig\n|Y|log|Y|(optdet\nfull(H)+r⋆)/parenrightbig\nΩ/parenleftbig\n|Y|(optdet\nfull(H)log|Y|+r⋆)/parenrightbig\nTable 1: Worst-case mistake bounds for learning concept cla sses with bandit feedback in the\nagnostic setting, in which the best concept in class is incon sistent with the feedback in r⋆\nmany rounds. No prior knowledge on r⋆is required. The bounds for the randomized setup are\nobtained by applying Theorem 5.3 to the upper bound in Theore m 5.1 and to the lower bounds\nin Lemma 4.11 and in [DH13]. The bounds for the deterministic setup are obtained by applying\nthe same theorem to the upper bound of [AL99], and to the lower bounds of Lemma 4.5 and\n[Lon20]. The same (up to constant) deterministic bounds wer e proved independently in [GT24]\n(however, their bounds are in terms of a given bound r≥r⋆).\nTheorem 1.2 (Oblivious vs. Adaptive Adversaries) .For every concept class Hit holds that\noptadap\nbandit(H) =O(optobl\nbandit(H)·|Y|log|Y|).\nFurthermore, for every natural k≥2there exists a pattern class Pwith|Y|=kand so that\noptadap\nbandit(P) = Ω(optobl\nbandit(P)·|Y|).\nThe upper bound is an immediate corollary of the upper bound ( 1). We prove the lower\nbound in Section 6.\nRemark 1.3 (Concept classes vs. Pattern classes) .Pattern classes are a more expressive gen-\neralization of concept classes (see a formal definition in Se ction 2). Most desirably, we would\nlike to prove upper bounds for all pattern classes, accompan ied with tight lower bounds that hold\nfor hard concept classes. We manage to do so in all of our resul ts (for the sake of simplicity,\nwe omitted it from the theorem statements), except for Theore m 1.2, in which the upper bound\ndoes hold for all pattern classes, but the lower bound holds o nly for hard pattern classes.\n1.1.3 Randomness\nIn Section 1.1.2 we discussed two different adversarial model s within the setup of randomized\nlearners. However, what happens if the learner cannot use ra ndomness? A folklore result on\nthe full-information feedback model states that for every c oncept class Hwith|Y|= 2 it holds\nthat\noptrand\nfull(H)≤optdet\nfull(H)≤2·optrand\nfull(H), (3)\nand that there are classes attaining each equality. Extendi ng this to |Y|>2 is straightforward\nby noting the following randomized-to-deterministic full -information algorithm conversion: If\nAis a randomized algorithm, then the mistake bound of the dete rministic algorithm A′who\nalways predicts the most probable prediction of Ais at most twice the mistake bound of A.\n5Indeed, whenever A′makes a mistake, Amakes a mistake with probability at least 1 /2. This\nraises the following natural question, also asked in [DH13] .\nWithin the bandit feedback model, what is the price the learn er pays for not using\nrandomness?\nWe resolve this question up to logarithmic factors.\nTheorem 1.4 (Randomized vs. Deterministic) .For every concept class Hit holds that\noptdet\nbandit(H) =O(optobl\nbandit(H)·|Y|log|Y|).\nFurthermore, for every natural k≥2there exists a concept class Hwith|Y|=ksuch that\noptdet\nbandit(H) = Ω(optadap\nbandit(H)·|Y|).\nThe upper bound is an immediate corollary of the upper bound ( 1). We prove the lower\nbound in Section 7.\nThe lower bound has a significant consequence on the problem o f finding a combinatorial\ndimension that quantifies the optimal mistake bound of rando mized learners in the bandit\nfeedback model. In the full-information model, the optimal deterministic mistake bound, which\nis captured precisely by the combinatorial Littlestone dimension [Lit88], quantifies the optimal\nmistake bound of randomized learners as well, as demonstrat ed in (3). In [DSBDSS15], a new\ncombinatorial dimension, coined the Bandit-Littlestone dimension , is introduced and proved to\ncapture the exact optimal mistake bound of deterministic le arners within the bandit feedback\nmodel. In [DH13], some hope is expressed for this dimension t o quantify the mistake bound\nof randomized learners as well, similarly to the case of full -information feedback. However, our\nlower bound shows that this is not the case. As we show in Secti on 7, the classes used in\nthe lower bound may be chosen such that optadap\nbandit(H) =O(|Y|), and thus optadap\nbandit(H) is only\nroughly/radicalBig\noptdet\nbandit(H). On the other hand, the upper bound in Theorem 1.4 together w ith the\nboundoptobl\nbandit(H)≥|Y|−1\n2(e.g. by [DH13]) shows that/radicalBig\noptdet\nbandit(H) is roughly the smallest\nthatoptobl\nbandit(H) can get.\n1.2 Bounds for prediction with expert advice\nA main technical result proved in this work, which is interes ting in its own right, is a nearly\noptimal randomized mistake bound for the problem of prediction with expert advice in ther-\nrealizable setting. In this problem, nexperts make deterministic predictions in every round, and\nit is promised that the best expert is inconsistent with the f eedback for at most rmany times\nthroughout the entire game. As in Section 1.1.1, theknowled ge ofris actually not required, and\nrcan be replaced with the actual number of inconsistencies of the best expert, r⋆. In Section 8,\nwe explain how to remove the assumption that ris given to the learner.\nThe learner should aggregate the experts’ predictions to ma ke their own (possibly random-\nized) predictions, while minimizing the expected number of mistakes made. This problem was\nextensively studied in the binary/full-information setti ngs (see, e.g., [Vov90,LW94,CBFHW96,\nCBFH+97,ALW06,MS10,BP19,FHMM23]) and some (asymptotically or even better than that)\ntight bounds were proved, in both deterministic and randomi zed (learner) settings. Already\nin early works on this problem, [Vov90,LW94] showed that the optimal mistake bound when\n|Y|= 2 is Θ(log n+r) for both randomized and deterministic learners.\n6LearnerAdversary\nOblivious Adaptive\nRandomizedO(k(logkn+r⋆))\nΩ(klogkn+r⋆)Θ(k(logkn+r⋆))\nDeterministic Θ(k(log(n/k)+r⋆+1))\nTable 2: Mistake bounds for prediction with expert advice . The size of the label set is k≥2,\nthere are n≥kexperts, and the best expert is inconsistent with the feedba ck forr⋆many times.\nNo prior knowledge on r⋆is required. The randomized bounds are due to Theorem 4.7 and\nLemmas 4.10 and 4.11. The deterministic bounds are stated in Theorem 4.1.\nIn this work, we consider this problem in the bandit feedback model. In fact, the main\ntool used to prove Theorem 1.1 is the following optimal (up to constant factors) bound on\noptadap\nbandit(n,k,r), which is the optimal mistake bound achievable by a randomi zed learner which\nplays against an adaptive adversary that provides bandit fe edback, when there are k≥2 many\nlabels,n≥kmany experts, and the best expert is inconsistent with the fe edback for at most\nr≥0 many times.\nTheorem 1.5. For every n≥k≥2andr≥0it holds that\noptadap\nbandit(n,k,r) = Θ(k(logkn+r)).\nThis bound generalizes the result optadap\nbandit(n,2,r) = Θ(log n+r) mentioned above.3In the\ncase where the adversary is oblivious, we prove the inferior lower bound\noptobl\nbandit(n,k,r) = Ω(klogkn+r)\nwhich is tight as long as r=O(logkn). Proving tight bounds against an oblivious adversary for\nall values of rremains open.\nWe also consider this problem in the deterministic (learner ) setting, for the sake of complete-\nness (we do not use the deterministic bound to prove any other results). We prove all bounds\nin Section 4, and summarize them in Table 2. Similarly to the r esults in Section 1.1.1, since our\nmistake bounds demonstrate no dependence on the number of ro undsT, they improve over the\nknown regret bound O/parenleftbig√Tklogn/parenrightbig\nof [ACBFS02] whenever r=O(logn).\n1.3 Technical overview\nIn this section, we explain the idea behind the proof of the up per bound optadap\nbandit(H) =\nO(optrand\nfull(H)·|Y|) in Theorem 1.1, which is our main technical contribution.\nThere are two main ingredients used in the proof of Theorem 1. 1:\n1. A reduction from the problem of learning a concept class Hto an instance of prediction\nwith expert advice with|Y|optdet\nfull(H)many experts, |Y|many labels, and where the best\nexpert is inconsistent with the feedback for at most optdet\nfull(H) many rounds.\n2. The upper bound for prediction with expert advice stated in Theorem 1.5.\n3When there are only 2 labels, full-information and bandit fe edback are the same.\n7Indeed, havingbothitems, wetaketheupperboundofitem(2) withtheparametersspecified\nin item (1), obtaining\noptadap\nbandit/parenleftBig\n|Y|optdet\nfull(H),|Y|,optdet\nfull(H)/parenrightBig\n=O/parenleftBig\n|Y|·optdet\nfull(H)/parenrightBig\n.\nBy item (1), this bound holds also for the problem of learning the concept class H. Since\noptdet\nfull(H)≤2optrand\nfull(H), we can replace optdet\nfull(H) withoptrand\nfull(H). It remains to sketch the\nideas behind the proofs of items (1) and (2), which we do in the following subsections. Of the\ntwo items, the proof of Item (2) is the main technical novelty .\n1.3.1 Proof idea of Item (1)\nIn the problem of prediction with expert advice , the expert predictions are generated in a com-\npletely adversarial fashion. That is, no assumptions are ma de on the way those predictions are\ngenerated. Therefore, any upper bound for prediction with expert advice holds in particular for\nthe case where the expert predictions are determined by an al gorithm chosen by the learner.\nWe can exploit this property to reduce the problem of learnin g a concept class Hto an instance\nofprediction with expert advice with|Y|optdet\nfull(H)many experts, |Y|many labels, and where the\nbest expert is inconsistent with the feedback for at most optdet\nfull(H) many rounds. The idea,\ndescribed below, is inspired by [Lon20,HLM21].\nLetAbe an optimal deterministic algorithm for learning Hgiven full-information feedback.\nWe run in parallel several copies of Aarranged in tree form. These copies will function as the\nexperts fed into an optimal algorithm for prediction with expert advice .\nInitially, there is a single copy of A. At every round, for each copy of A, if its prediction\nis consistent with the bandit feedback, we do nothing. Other wise, if the copy is at depth\nD <optdet\nfull(H), we split it into |Y|different copies at depth D+1, each “guessing” a different\nfull-information feedback for the problematic example; if the copy is at depth D, we do nothing.\nSince the tree has depth at most optdet\nfull(H), there are at most |Y|optdet\nfull(H)many copies of A.\nThe copy of Awhich always guessed correctly corresponds to an expert who se predictions are\ninconsistent with the feedback for at most optdet\nfull(H) many rounds.\nA formal statement (and proof) of this reduction can be found in Proposition 5.2.\n1.3.2 Proof idea of Item (2)\nIn the context of classification problems, the realizable ca se in which some concept from the\nlearned class accurately explains the correct classificati on is often simpler than the agnostic\ncase. Therefore, for the sake of understanding the proof of t he upper bound on optadap\nbandit(n,k,r)\nstated in Theorem 1.5, we first outline a proof for the upper bo und onoptadap\nbandit(n,k) :=\noptadap\nbandit(n,k,0). This proof has the same flavor of the proof for general r, but is simpler.\nWe then explain how to adapt the proof for general r.\nWhenr= 0, alllivingexperts (that is, experts which have been consistent with th e feedback\nso far) are identical: every expert which is inconsistent wi th the feedback in some round is\nimmediately eliminated, and its predictions need not be tak en into account any longer. By\napplying the law of total expectation, the optimal mistake b ound can thus be described by the\nfollowing optimization problem. Fix k≥2, and for every n≥1, letV(n) =optadap\nbandit(n,k). We\nhave\nV(n) = max\n/vector αmin\nπmax\ny∈Y\nπyV(αyn)+/summationdisplay\ny′/ne}ationslash=yπy′(1+V(αy′n))\n, (4)\n8where/vector α∈[0,1]kis ak-ary vector whose y’th entry specifies the fraction of living experts\npredicting y;πis the distribution used by the learner to draw the predictio n; andy∈ Yis the\ncorrect label chosen by the adversary.\nSince all living experts are identical, the natural intuiti on suggests that the optimal choice\nof/vector αis to let every entry in it be 1 /k, in which case the optimal choice of πwould be the uniform\ndistribution. In this case, it does not matter which ythe adversary chooses as the correct label.\nApplying this intuition to (4) results in the recurrence rel ation\nV(n)≤V(n/k)/k+(1−1/k)(1+V((1−1/k)n)). (5)\nSolving this recurrence relation gives V(n)≤klogb(k)n, where b(k) =kk\n(k−1)k−1, which\nimplies the statement in Theorem 1.5 since b(k)≥k. It remains to show that the intuition that\nled us from (4) to (5) is indeed correct. A natural approach wo uld be to directly prove that\nV(n), as defined in (4), satisfies V(n)≤klogb(k)nby induction on n. However, note that (4)\ncontains kdifferent recursive calls, so an inductive proof seems compli cated. To overcome this,\nwe use minimax duality, which significantly simplifies (4), a s we now explain.\nObserve that once /vector αis fixed, each round of the game is a zero-sum game between two\nrandomized parties.4Therefore, we can apply von Neumann’s minimax theorem and ob tain a\ndualgame which is equivalent to the primal(original) game in terms of its optimal mistake\nbound.5In the dual game, the adversary first chooses a distribution o ver the labels from which\nthe correct label is drawn, and then the learner chooses its p rediction. When the dual game is\nconsidered, the definition of V(n) from (4) becomes\nV(n) = max\n/vector α,πmin\ny∈Y[πyV(αyn)+(1−πy)(1+V((1−αy)n))]. (6)\nThe notation is the same as in (4), except that now πis the distribution used by the\nadversary to draw the correct label, and y∈ Yis the learner’s prediction. Eq. (6) contains only\ntwo recursive calls, making an inductive proof much easier ( but still quite technical).\nWe now explain how to adapt this approach to work for general r, whenris given. In\nSection 8, we explain how to remove the assumption that ris given to the learner. When r >0,\nthere is an inherent difference between the experts: different e xperts have been inconsistent\nwith the feedback for a different number of rounds, and so can affo rd a different number of\nfuture inconsistencies. Therefore, we need some mechanism that differentiates between experts\nwith different inconsistency budgets.\nInspired by weighted prediction techniques (See [CBL06, Section 2.1] and bibliographic re-\nmarks of Section 2), we rely on the fact that experts with high er budgets have higher potential\nto damage the learner, so we choose an expert potential function that matches the potential of\nan expert to damage the learner. As such, the potential shoul d be an increasing function of\nthe budget. We use exponential potentials and give an expert with budget i(corresponding to\nimore allowed inconsistencies) a potential of k2i. We now employ roughly the same argument\noutlined above for the realizable case. The main difference is thatV(·) depends on the total\npotential rather than on the number of living experts. As a re sult, the initial number of living\nexpertsnis replaced with the initial total potential n·k2r, which gives\nV(n·k2r)≤klogk(n·k2r) =klogkn+2kr,\nas stated in Theorem 1.5.\n4The adversary described in (4) is deterministic, but we can a llow it to be randomized without changing the\nvalue of V(n).\n5Similar usage of minimax duality appears in previous work, f or example [AABR09].\n91.4 Related work\nWe outline connections between previous work to the problem s in the focus of this work: the\nrole of various resources in realizable multiclass online l earning of concept classes, and prediction\nwith expert advice with bandit feedback.\n1.4.1 Information\nThe role of information in learning concept classes was prev iously studied in [AL99,DSBDSS15,\nDH13,Lon20]. To the best of our knowledge, [AL99] was the firs t to show that optdet\nbandit(H) =\nO(optdet\nfull(H)|Y|log|Y|) for every class H. [Lon20,Gen21] improved the constant in the upper\nbound of [AL99], and showed that it is in fact tight up to a 1+ o(1) factor by finding a sequence\nof concept classes demonstrating a matching separation bet weenoptdet\nbandit(H) andoptdet\nfull(H).\nOur work proves an analogous result for randomized learners (Theorem 1.1), with the exception\nthat we do not identify the exact leading constant in the opti mal mistake bound.\nTheagnosticcasewasstudiedin[DH13], whichprovedtheupp erbound ˜O/parenleftbigg/radicalBig\nT|Y|optdet\nfull(H)/parenrightbigg\non the optimal regret, where Tis the horizon, and showed that the upper bound is best-possi ble\nup to logarithmic factors. The PAC learning setting was stud ied in [DSBDSS15], which showed\nthat the price of bandit feedback is ˜O(|Y|) in this setting as well.\n1.4.2 Adaptivity\nIn the setting of full-information feedback, adaptive and o blivious adversaries are equivalent.\nIndeed, full-information feedback, in its essence, implie s that the feedback never depends on\nthe prediction drawn by the learner in a specific execution of the game. In the bandit feedback\nsetting, to the best of our knowledge, the existing literatu re on adaptive adversaries focuses on\nthe agnostic setting and analyze different notions of regret. Notable examples are [MOSW02,\nFM06,ADT12].\n1.4.3 Randomness\nIn the full-information feedback model, the Littlestone dimension [Lit88,DSBDSS15] captures\noptdet\nfull(H) precisely, and optrand\nfull(H) up to a multiplicative factor of 2. The paper [DSBDSS15]\nsuggests a new combinatorial dimension, the Bandit Littlestone dimension , and proves that it\ncaptures optdet\nbandit(H) precisely. The paper [RRS+23] shows that the Bandit Littlestone dimen-\nsion qualitatively characterizes learnability also in the agnostic and randomized setting, even\nwhen|Y|=∞.\nThe papers [DH13,RRS+23] ask whether the Bandit Littlestone dimension is a good qu anti-\ntative proxy for optobl\nbandit(H). Theorem 1.4 (and Theorem 7.1 in a more detailed version) sh ows\nthat this dimension is far from quantifying even optadap\nbandit(H).\n1.4.4 Prediction with expert advice\nTheproblem of prediction with expert advice in ther-realizable setting was introduced in[Vov90,\nLW94] for the binary case ( |Y|= 2). Since then, this problem has been well-studied, with\npapers spanning the past 30 years [CBFHW96,CBFH+97,ALW06,MS10,FHMM23]. The full-\ninformation setting for |Y|>2 and small rwas studied in [BP19].\nThe bandit feedback variation of the problem was introduced in [ACBFS02], in a more\ngeneral version that considers rewards instead of losses(ormistakes ). However, there is a\nmajordifferencebetweenhowtheydefine r-realizability andhowwedefineit(inthesettingofan\n10adaptiveadversary). Intheirdefinition, theremustbeanex pertwhichmakesatmost rmistakes,\nwhereas we only assume that there must be an expert whose pred ictions are inconsistent with\nthe bandit feedback at most rtimes.\nWhen the adversary is oblivious, both notions of r-realizability coincide (see Remark 2.1 for\nmore details). Therefore, it is possible that their techniq ues can be used to obtain tight bounds\nfor the oblivious setting for all values of r; our bounds for the adaptive setting are tight in the\noblivious setting only for small rvalues (see Theorem 4.12).\nUnfortunately, to the best of our knowledge, when convertin g rewards to losses and applying\nthe bounds of [ACBFS02], a dependence on the number of rounds Temerges (see for example\n[BCB+12, Theorem 3.1] and [CBL06, Theorem 6.10]), which is undesi rable when studying the\nmistake bound model (rather than the regret model). In this w ork we are mainly interested\nin the experts setting as a means for proving Theorem 1.1, and our proof specifically requires\nthe adaptive setting. We leave for future work the question o f finding mistake bounds against\noblivious adversaries which are tight for large values of r.\n2 Preliminaries\nLetXbe adomain, and let Ybe a countable label set. Unless stated otherwise, we assume that\nY= [k] for some natural k≥2. A pair ( x,y)∈ X × Y is called an example, and an element\nx∈ Xis called an instance or anunlabeled example . A function h:X → Yis called a hypothesis\nor aconcept. Ahypothesis class , orconcept class , is a non-empty set H ⊂ YX. We note here\nthat it is generally assumed that there exists an instance x∈ Xsuch that for every y∈ Ythere\nexistsh∈ Hsatisfying h(x) =y. This assumption is reasonable since when this is not the cas e,\nthe hypothesis class Hhas an essentially isomorphic class H′with label set strictly smaller than\nY(see [RRS+23]).\nWe will mostly consider the more general notion of pattern classes . Apatternp∈(X ×Y)⋆\nis a finite sequence of examples. A pattern class is a non-empt y set of patterns Pwhich\nisdownwards closed , meaning that for every p∈ P, every sub-pattern p′ofp(obtained by\nremoving examples arbitrarily) is in Pas well. Let p,qbe sequences (which can be either\npatterns or some other type of sequences). we denote by pIthe sub-sequence of pconsisting\nonly of the indices in I. The set Imight be given by a clear and well-known notation. For\nexample, if p=x1,...,x T, then for some t≤T, the notation p≤trepresents the sub-sequence\nx1,...,x t. Denote the concatenation of p,qbyp◦q. Ifpis a sub-sequence of q, we denote\np⊂q.\nNote that pattern classes generalize concept classes: for e very concept class Hwe may define\ntheinducedpattern class\nP(H) =\n\np∈(X ×Y)⋆: min\nh∈H/summationdisplay\n(x,y)∈p1[h(x)/\\⌉}atio\\slash=y] = 0\n\n. (7)\nInwords, P(H)containsallpatternsthatareconsistentwith H,andthusfromanonline-learning\nperspective there is no essential difference between HandP(H). Therefore, throughout this\nsection we consider only pattern classes.\nInthiswork, weconsider multiclass online learning of patternclasses inthe realizable setting.\nOnline learning is a repeated game between a learner and an ad versary. Each round tin the\ngame proceeds as follows:\n(i) The adversary sends an instance xt∈ Xto the learner.\n(ii) The learner predicts ˆ yt∈ Y(possibly at random).\n11(iii) The adversary provides (full-information or bandit) feedback to the learner.\nFull-information feedback means that the learner learns the correct label ytat the end of every\nround.Banditfeedback means that the learner only receives an indication whether ˆ yt=ytor\nnot. If the learner predicts ˆ ytdeterministically for every t, then the learner is deterministic .\nIn the bandit feedback setting, we model learners as functions Lrn: ((X ×{→,/\\⌉}atio\\slash→}×Y)⋆×\nX)→Π[Y] from the set of pairs of a feedback sequence and an instance, to the set Π[ Y] of all\nprobability distributions over Y. Forπ∈Π[Y] we denote the probability of yinπbyπy. In a\nfeedback sequence ( x1,op1,ˆy1),...,(xT,opT,ˆyT), the triplet ( xt,opt,ˆyt) represents the fact that\nin round tthe instance was xt, and that ˆ yt=ytifop=→, or ˆyt/\\⌉}atio\\slash=ytifop=/\\⌉}atio\\slash→. The learner’s\npredictions thus may depend on past feedback as well as on pas t and current instances. In the\nfull-information feedback model, op=→at all times, and so we omit it and use the pair ( xt,yt)\ninstead of the triplet ( xt,op,ˆyt).\nGiven a learner Lrnand a fixed input sequence of examples S= (x1,y1),...,(xT,yT), we\ndenote the expected number of mistakes that Lrnmakes when executed on the sequence Sby\nMfull(Lrn;S) if it receives full-information feedback, and by Mbandit(Lrn;S) if it receives bandit\nfeedback. We define the optimal mistake bound of Pwhen the adversary is oblivious to be\noptrand\nfull(P) = inf\nLrnsup\nS∈PMfull(Lrn;S),optobl\nbandit(P) = inf\nLrnsup\nS∈PMbandit(Lrn;S), (8)\nin the full information and bandit feedback models, respect ively.\nRemark 2.1 (Strong vs. Weak Realizability) .The restriction S∈ Pthat the supremum is\ntaken over is called strong realizability, which requires t hat the input sequence is taken from\nthe class P. On the other hand, unless stated otherwise, in this work we c onsider the weak\nrealizability assumption. Under this assumption, it is onl y required that for any learner, the\nfeedback provided by the adversary is consistent with some p attern from Pwith probability 1.\nWhen the adversary is oblivious and the input sequence Sis chosen beforehand, strong and weak\nrealizability coincide. Indeed, note that for any learner wh o never predicts a label with probability\n0, strong realizability must hold in order to satisfy weak rea lizability.\nThe adversary in the above definition is indeed oblivious to t he learner’s actions, in the\nsense that it must choose the target pattern S∈ Pbefore the beginning of the game. In\ncontrast, in the setting of the bandit feedback model we will also consider a stronger adaptive\nadversary which is allowed to choose the target pattern on th e fly, as long as it satisfies the\nweak realizability assumption, meaning that the feedback p rovided to the learner is consistent\nwith some p∈ Pwith probability 1. In addition, it is allowed to choose the c orrect label at\nrandom.6\nWhen the adversary is adaptive, the online learning game defi ned above operates as follows\nin every round t:\n(i) The adversary sends an instance xt∈ Xto the learner.\n(ii) The learner chooses a probability distribution π(t)overYand reveals it to the adversary.\n(iii) The adversary chooses a probability distribution τ(t)overY.\n(iv) The learner draws a prediction ˆ yt∈ Yfromπ(t)and reveals it to the adversary.\n(v) The adversary draws the correct label ytfromτ(t)and tells the learner if its prediction\nwas correct or not.\n6It is explained in the sequel that being randomized does not r eally help the adversary\n12We formalize an adaptive adversary as a pair of functions\nAdvx: (X ×{→,/\\⌉}atio\\slash→}×Y)⋆→ X,Advy: (X ×{→,/\\⌉}atio\\slash→}×Y)⋆×X ×Π[Y]→Π[Y].\nIn every round t,Advxis used in the first step to choose the instance xt, andAdvyis used in\nthe third step to determine the distribution from which the c orrect label yt∈ Yis drawn. In\nmore detail, let F∈(X ×{→,/\\⌉}atio\\slash→}×Y)⋆be the feedback sequence given to the learner before\nroundt. The adversary sends the instance xt=Advx(F) to the learner in the first step. In the\nthird step, the adversary computes the distribution τ(t)=Advy(F,xt,π(t)).\nA feedback sequence F∈(X ×{→,/\\⌉}atio\\slash→}×Y)⋆isrealizable byPif there exists a pattern in\nPthat is consistent with F. An adaptive adversary is consistent withPif for every feedback\nsequence F∈(X ×{→,/\\⌉}atio\\slash→}×Y)⋆which is realizable by Pand for every π∈Π[Y], the following\nholds. Let x=Advx(F). IfAdvy(F,x,π) =τ, then the feedback sequence F◦(x,→,y) is\nrealizable for every yin the support of τ.\nGiven a learner Lrnand an adaptive adversary Advwhich provides bandit feedback, we\ndenote the expected number of mistakes that Lrnmakes against AdvbyMbandit(Lrn;Adv). We\ndefine the optimal mistake bound of Pagainst an adaptive adversary which provides bandit\nfeedback by\noptadap\nbandit(P) = inf\nLrnsup\nAdvM(Lrn;Adv), (9)\nwhere the supremum is taken over all adaptive adversaries wh ich are consistent with P.\nWe denote by optdet\nfull(P) andoptdet\nbandit(P) the optimal deterministic mistake bounds7ofP\nwith full-information and bandit feedback, respectively. That is, in these settings we require\nthe additional restriction that Lrnmust be deterministic. We may sometimes refer to mistake\nbounds as the learner’s lossthroughout the entire game. Unless stated otherwise, the no tations\nO,Ω,Θ hide universal constants which do not depend on any paramet er of the problem.\n3 The primal and dual games\nIn this section we describe a game-theoretic formulation of optadap\nbandit, and apply the minimax\ntheorem to obtain a dual formulation which will be useful in S ection 4. We term the game\ndefined in Section 2 when played against an adaptive adversar y as the primal game . In the\ndual game , we basically just switch the order of the second and third st eps of the primal game\n(together with some other required but not meaningful minor changes). We can think of the\nprimal and dual games as the same game, only that after the adv ersary chooses xt, in the\nprimal game the learner chooses a distribution (over the lab els) first, and in the dual game\nthe adversary chooses a distribution first. Formally, the du al game operates as follows in each\nroundt:\n(i) The adversary picks an instance xt∈ Xand a probability distribution τ(t)overY. It\nreveals both to the learner.\n(ii) The learner chooses a probability distribution π(t)overY, draws a prediction ˆ yt∈ Yfrom\nπ(t), and reveals it to the adversary.\n(iii) The adversary draws the correct label ytfromτ(t)and tells the learner if its prediction\nwas correct or not.\n7One can verify that in the deterministic case, oblivious and adaptive adversaries are equivalent.\n13In this section, we show that the primal and dual games are equ ivalent in terms of their\noptimal mistake bounds. The motivation behind this is that t he target function of the opti-\nmization problem describing the optimal mistake bound in th e dual game will turn out to be\nsimpler. This will enable the analysis in Section 4, and migh t be of further interest. The equiva-\nlence will mainly follow from von Neumann’s minimax theorem [vN28]. Using minimax duality\nto analyze minimax target functions of online learning prob lems has proved useful in previous\nworks. Notable examples are [AABR09,RST10,RSS12].\nWe now get into the details. In the dual game, π(t)depends on the distribution τ(t)chosen\nby the adversary, and the distribution τ(t)does not depend on the distribution π(t). Therefore\nwe have to change the formal definitions of a learner and an adv ersary in the dual game. The\nadversary is defined as a single function Adv: (X ×{→,/\\⌉}atio\\slash→}×Y)⋆→(X ×Π[Y]). The learner\nis defined as a function Lrn: (X × {→,/\\⌉}atio\\slash→} × Y)⋆× X ×Π[Y]→Π[Y]. The optimal mistake\nbounds in the primal and dual games are defined in a dual manner :\noptadap\nbandit(P) = inf\nLrnsup\nAdvM(Lrn;Adv),optdadap\nbandit(P) = sup\nAdvinf\nLrnM(Lrn;Adv).(10)\nFirst, we provide explicit definitions for optadap\nbandit(P) andoptdadap\nbandit(P) in terms of simpler\nclasses, based on the instance and feedback provided by the a dversary in every round. For every\ninstance xdefine the class:\nPx={p∈ P: (x,y)◦p∈ Pfor some y∈ Y}.\nAdditionally, define the following classes for every pair of an instance and a label ( x,y):\nPx→y={p∈ P: (x,y)◦p∈ P},Px/ne}ationslash→y=Px\\Px→y.\nTo simplify presentation, we assume that in the primal game t he adversary is deterministic,\nand in the dual game the learner is deterministic. This is rea sonable since after the first player\nhas made its choice, the optimal choice of the second player i s deterministic. Let us now define\noptadap\nbandit(P) andoptdadap\nbandit(P) in terms of classes of the form Px→yandPx/ne}ationslash→y:\noptadap\nbandit(P) = sup\nxinf\nπsup\ny∈Y\nπy·optadap\nbandit(Px→y)+/summationdisplay\ny′/ne}ationslash=yπy′/parenleftBig\n1+optadap\nbandit(Px/ne}ationslash→y′)/parenrightBig\n,(11)\noptdadap\nbandit(P) = sup\nx,τinf\ny∈Y/bracketleftBig\nτy·optdadap\nbandit(Px→y)+(1−τy)/parenleftBig\n1+optdadap\nbandit(Px/ne}ationslash→y)/parenrightBig/bracketrightBig\n.(12)\nFor these equations to be well-defined, it is technically con venient to define optadap\nbandit(∅) =\noptdadap\nbandit(∅) =−∞. Now, by the law of total expectation, for every pattern clas sPthe above\nequations indeed define the optimal losses in the primal and d ual games. We can now prove the\nequivalence.\nLemma 3.1. For every pattern class P:\noptadap\nbandit(P) =optdadap\nbandit(P).\nProof.For a horizon (numberof rounds) T, letoptadap\nbandit(P,T),optdadap\nbandit(P,T) betheanalogue\nnotations to optadap\nbandit(P),optdadap\nbandit(P), with the additional restriction that the game is played\nforTrounds. We adapt (11) and (12) to those definitions in the obvi ous way. Note that\noptadap\nbandit(P) = sup\nT∈N,x∈Xoptadap\nbandit(Px,T),\n14and similarly for optdadap\nbandit(P). Therefore, to prove the lemma, it suffices to show that\noptadap\nbandit(Px,T) =optdadap\nbandit(Px,T) for all x,Tsuch that Px/\\⌉}atio\\slash=∅(whenPx=∅both are\n−∞for allT). Letx∈ Xbe such that Px/\\⌉}atio\\slash=∅. ForT= 0 we have optadap\nbandit(Px,0) = 0 =\noptdadap\nbandit(Px,0).\nFor the induction step, note that the expression for optdadap\nbandit(Px,T) suggested by (12) de-\nscribes a zero-sum game in which the adversary goes first, and both the learner and adversary\nchoose distributions over Yto draw a label from: the learner draws a prediction and the ad ver-\nsary draws the correct label. The target function is the opti mal loss to besuffered by the learner\nin the sequel, after the distributions are fixed. The adversa ry’s goal is to maximize it and the\nlearner’sgoalistominimizeit. Sincethisisazero-sumgam e, bytheminimaxtheorem[vN28]we\ncan let the learner go first instead, without changing the val ue ofoptdadap\nbandit(Px,T). Therefore\nwe can write optdadap\nbandit(Px,T) as:\ninf\nπsup\ny∈Y\nπy·optdadap\nbandit(Px→y,T−1)+/summationdisplay\ny′/ne}ationslash=yπy′(1+optdadap\nbandit(Px/ne}ationslash→y′,T−1))\n.\nBy the induction hypothesis, we can change every optdadap\nbandittooptadap\nbandit, which gives:\ninf\nπsup\ny∈Y\nπy·optadap\nbandit(Px→y,T−1)+/summationdisplay\ny′/ne}ationslash=yπy′(1+optadap\nbandit(Px/ne}ationslash→y′,T−1))\n.(13)\nBy applying (13), observe that optdadap\nbandit(Px,T) =optadap\nbandit(Px,T), as required.\n3.1 An optimal learner for the primal game\nThe optimal randomized algorithm achieving optadap\nbandit(P) suggests itself from (11). For com-\npleteness, we explicitly write it in Figure 1. The algorithm may be implemented using straight-\nforward dynamic programming for finite hypothesis classes, even in the r-realizable case where\nthe best hypothesis is inconsistent with the feedback in at m ostrmany indices.\nProposition 3.2. The algorithm BanditRandSOA is well-defined and optimal.\nProof.The choice of π(t)is well defined because (14) is a feasible linear optimizatio n problem,\nand as such it has a solution. The algorithm is optimal due to ( 11).\n4 Prediction with expert advice\nIn this section we are interested in optimal mistake bounds f or the bandit feedback setup of\nprediction with expert advice in ther-realizable setting. In this problem, nexperts are making\npredictions in each round, and it is promised that the best ex pert is inconsistent with the\nadversary’s feedback in at most rmany rounds. An adversary following this limitation is\ncalledr-consistent . Nothing else is assumed about how the experts decide on thei r predictions.\nFurthermore, in Section 8we explain howto remove theassump tionthat ris given tothelearner\nwithonly a constant factor degradation inthemistake bound . Based on theexperts’ predictions,\nthe learner should make its own prediction for each round, wh ile minimizing the expected\nnumber of mistakes it makes in the entire game. In the binary o r full-information feedback\nmodel, this problem is well-studied, and tight bounds were p roved for both deterministic and\nrandomized learners [Vov90,LW94,CBFHW96,CBFH+97,ALW06,BP19,FHMM23].\n15BanditRandSOA\nInput:A pattern class P.\nFort= 1,2,...\n1. Receive instance xt∈ X.\n2. Construct a distribution π(t)such that\nπ(t)∈argmin\nπ∈Π[Y]max\ny∈Y\nπyoptadap\nbandit(Px→y)+/summationdisplay\ny′/ne}ationslash=yπy′(1+optadap\nbandit(Px/ne}ationslash→y′))\n.(14)\n3. Draw the prediction ˆ ytfromπ(t).\n4. If the feedback is positive, update P:=Px→ˆyt, and otherwise update P:=Px/ne}ationslash→ˆyt.\nFigure 1: BanditRandSOA is an optimal randomized learner for online learning with ba ndit\nfeedback of pattern classes, where the adversary is allowed to be adaptive. It is inspired by\ntheRandSOA algorithm of [FHMM23], which is a randomized variant of Litt lestnoe’s [Lit88]\nwell-known SOAalgorithm.\nIn this work, we are mostly interested in the randomized (lea rner) and adaptive (adversary)\nsetup with bandit feedback, and the bound we prove for it play s a central role in the proof of\nTheorem 1.1. For the sake of better completeness and exhaust iveness, we consider the other\ndeterministic (learner) and oblivious (adversary) setups as well. Our results for all three setups\nare summarized in Table 2.\nWe adapt the notation for learning pattern classes. Suppose that there are k≥2 many\nlabels and n≥kmany experts, where the best expert is inconsistent with the feedback in\nat most r≥0 many rounds. We assume that n≥k, since if n < kthen having klabels is\nequivalent to having nlabels. When the learner is deterministic, the optimal mist ake bound\nis denoted by optdet\nbandit(n,k,r). When the learner is randomized, the optimal mistake bound is\ndenotedby optobl\nbandit(n,k,r) oroptadap\nbandit(n,k,r), dependingonwhethertheadversary isoblivious\nor adaptive, respectively. For the realizable case r= 0, we omit rfrom the notation.\n4.1 Warm-up: deterministic learners\nIn this section, we analyze optdet\nbandit(n,k,r) and prove the following theorem.\nTheorem 4.1. Letn≥k≥2,r≥0. Then:\noptdet\nbandit(n,k,r) = Θ(k(log(n/k)+r+1)).\nWe first prove the upper bound and then the lower bound.\nLemma 4.2. Letn≥k≥2,r≥0. Then for every α >1:\noptdet\nbandit(n,k,r)≤α\nα−1kln(n·αr/k)+k−1.\n16Proof.We describe a deterministic learner which makes at most the s tated number of mistakes.\nWe use a simple generalization of Weighted Majority [LW94]. A similar idea appears also in\n[Lon20].\nEvery expert is given an initial weight of αr. LetWtbe the total weight of experts in round\nt. In every round, predict the label that enjoys a weighted plu rality. If we make a mistake, then\nthe weight of every expert which voted for the predicted labe l is divided by α. If an expert\nreaches weight less than 1, we reduce its weight to 0, since it must mean that it made more\nthanrmistakes. We split the analysis into two epochs. The first epo ch is as long as Wt> k,\nand the second is afterwards.\nWe begin by analyzing the first epoch. By assumption, Wt> kat all times. On the other\nhand, whenever the learner makes a mistake, at least 1 /kof the weight is being divided by α.\nTherefore, if a mistake occurs in round tthen\nWt+1≤(1−1/k)Wt+Wt/(αk) =/parenleftbigg\n1−α−1\nαk/parenrightbigg\nWt.\nNow, ifm1is the total number of mistakes the learner makes as long as Wt> k, thenm1must\nsatisfy\nk < n·αr/parenleftbigg\n1−α−1\nαk/parenrightbiggm1\n≤n·αre−m1α−1\nαk.\nAfter some manipulations, we derive the bound\nm1<αk\nα−1ln(n·αr/k).\nWe now analyze m2, which is the number of mistakes in the second epoch. If Wt≤kthen\nthere are at most kexperts which have not yet made more than rmistakes. At least one of\nthem will never err again, and therefore in the worst case, th e learner will make m2≤k−1\nmore mistakes before eliminating all experts apart from the target expert. Summing m1+m2\ngives the stated bound.\nWe may now deduce the upper bound.\nCorollary 4.3. Letn≥k≥2,r≥0. Then:\noptdet\nbandit(n,k,r)≤e\ne−1k(ln(n/k)+r)+k−1.\nProof.Apply Lemma 4.2 with α=e.\nWe can also deduce an improved upper bound for the realizable (r= 0) case.\nCorollary 4.4. Letn≥k≥2. Then:\noptdet\nbandit(n,k)≤kln(n/k)+k−1.\nProof.We can easily see that optdet\nbandit(n,k)≤(1+ǫ)kln(n/k)+k−1for every ǫ >0 by applying\nLemma 4.2 with α→ ∞. There is also an explicit optimal learner for which this hol ds with\nǫ= 0, obtained by slightly changing the proof of Lemma 4.2.\nWe now turn to the lower bounds. We begin with the realizable c ase.\nLemma 4.5. Letn≥k≥3. Then:\noptdet\nbandit(n,k)≥(k/2−1)ln(n/k).\n17Proof.The adversary uses the following simple strategy. Let ntbe the number of alive experts\n(that is, experts which are consistent with the feedback giv en so far) at the beginning of round\nt. As long as nt> k, in every round tthe adversary splits the alive experts as evenly as possible\namong the klabels, and gives negative feedback to the learner. Note tha t the largest set in the\npartition consists of ⌈nt/k⌉ ≤nt/k+1<2nt/kexperts, since nt> k. Therefore, in every round\nat least a (1 −2/k)-fraction of the experts make it to the next round. The adver sary can thus\nforce at least mmany mistakes on the learner whenever msatisfies\n(1−2/k)m−1n > k,\nwhich holds if\ne−2\nk−2(m−1)n > k,\nby using the inequality 1 −x > e−x\nx−1that holds for all x <1. After some manipulations, the\ninequality reads\nm≤(k/2−1)ln(n/k)+1,\nwhich implies the stated bound.\nTo devise a lower bound for the r-realizable setting, we prove the following lemma. It was\nindependently proved also by [GT24].\nLemma 4.6. For every n≥k≥2,r≥0we have:\noptdet\nbandit(n,k,r)≥k(r+1)−1.\nProof.Suppose that there are n=kexperts. In every round for T=k(r+1)−1 rounds, the\nadversary lets expert ipredict the label i, and always provides negative feedback to the learner.\nLeti⋆be the label that the learner predicts the least number of tim es, and let i′/\\⌉}atio\\slash=i⋆. Since\nT=k(r+ 1)−1, it must hold that i⋆is predicted by the learner at most rtimes. In every\nround in which the learner predicts i⋆, set the true label to i′. In all other rounds set the true\nlabel toi⋆. By definition of the strategy, the best expert makes at most rmistakes.\nWe may now prove Theorem 4.1.\nProof of Theorem 4.1. The stated bounds are known when k= 2 by e.g. [Vov90,LW94]. For\nk≥3, the upper bound follows from Corollary 4.3, and the lower b ound follows from Lem-\nmas 4.5 and 4.6.\n4.2 Randomized (Learner) and Adaptive (Adversary)\nIn this section we provide a tight upper bound on optadap\nbandit(n,k,r) by proving the following\ntheorem.\nTheorem 4.7. Letn≥k≥2,r≥0. Then:\noptadap\nbandit(n,k,r)≤klogkn+2kr.\nWe will prove the theorem by analyzing the mistake bound of th e equivalent dual game, as\ndescribedin Section 3. Theprimal and dual games are describ edin Section 3 in terms of pattern\nclasses, so we first show that the problem of prediction with e xpert advice in the r-realizable\nsetting is in fact equivalent to learning an appropriate spe cific pattern class.\n18For every hypothesis class Handbudget function B:H →R, we define the pattern class\nP(H,B) =\n\np∈(X ×Y)⋆:\n∃h∈ H:/summationdisplay\n(x,y)∈p1[h(x)/\\⌉}atio\\slash=y]≤B(h)\n\n\n. (15)\nIn words, P(H,B) consists precisely of those patterns which are consistent with some h∈ H\neverywhere except for at most B(h) many places. We call B(h) thebudgetofh. Therefore, if\nBris the budget function that gives budget rto all hypotheses, then learning the pattern class\nP(H,Br) under the assumption of realizability is equivalent to lea rning the hypothesis class H\nunder the assumption of r-realizability.\nObserve that for a fixed number of labels k, the class of nexperts is simply the hardest con-\ncept class of size n, since every expert can predict any label in every round. Den ote this class\nbyUn,k, so that prediction with expert advice withnexperts and klabels in the r-realizable set-\nting is equivalent to learning the pattern class P(Un,k,Br) under the assumption of realizability.\nTherefore, we have\noptadap\nbandit(n,k,r) =optadap\nbandit(P(Un,k,Br)) =optdadap\nbandit(P(Un,k,Br))\ndue to Lemma 3.1.\nFor completeness, we formally define Un,k. This is the class of projection functions over the\nset ofk-ary vectors of length n,X= [k]n. That is, if the nfunctions are {h1,...,h n}then for\neveryx∈ Xwe have hi(x) =xi, wherexiis the value of the i’th entry of x.\nWe now define some notation used in the proof. As observed in [A LW06,BP19], since\nthe experts are identical, in every round of the game we only c are about how many of the\nexperts are still alive(which means they have not been inconsistent with the feedba ck for more\nthanrmany rounds), and among those which are alive, how many have b udgeti, for each\n0≤i≤r. Following [ALW06,BP19], we represent this information by an (r+ 1)-ary vector\n/vector m= (m0,...,m r), in which the number miindicates the number of living experts with budget\ni. We call this vector the stateof the game.\nFor any state /vector m, letV(/vector m) be the optimal loss in the dual (or primal) game for the patte rn\nclass representing the state /vector m, as defined in (15). That is, if P(Un,k,B/vector m) is the pattern class\nrepresenting the state /vector m, thenV(/vector m) =optdadap\nbandit(P(Un,k,B/vector m)). We call V(/vector m) thevalueof the\nstate/vector m. The goal is thus to upper bound the value of the initial state V(0,0,...,n).\nTowards this end, we use the following potential-based tech nique. An expert with budget i\nwill have a potential of k2i. This will allow us to bound V(/vector m) in terms of the total potential of\nthe experts, given by W(/vector m) =/summationtextr\ni=0mi·k2i.\nLetb(k) =kk\n(k−1)k−1. We will prove the the following.\nLemma 4.8. Letk≥2be the number of labels. For any state /vector mit holds that\nV(/vector m)≤k·logb(k)W(/vector m).\nIn order to prove Lemma 4.8, we will need the following techni cal claim, proved at the end\nof this section.\nLemma 4.9. Letk≥2, and let\nfk(β) := logb(k)(((1−β)/k2+β)(β/k2+(1−β))k−1)−1/k.\nThenfk(β)≤ −1for allβ∈[0,1].\nWe can now prove Lemma 4.8.\n19Proof of Lemma 4.8. The proof is by induction on the state, ordered in the obvious way: the\nstate (1,0,...,0) is the lowest, and the state (0 ,...,0,n) is the highest.\nIn order to be able to always apply the induction hypothesis, we must assume that the\nstate always decreases between rounds. Indeed, the only cas e where the state does not decrease\nbetween rounds is when all experts predict the same label y. In this case, an optimal adversary\nmust choose yas the correct label with probability 1, and thus an optimal l earner must predict\ny. Nothing is changed in such rounds, and therefore they may be removed, ensuring that the\ninduction hypothesis can always be applied.\nIn the base case, /vector m= (1,0,...,0). Therefore\nV(/vector m) = 0 =klogb(k)1 =klogb(k)W(/vector m).\nFor the induction step, consider the first round of the game, i n which the adversary draws the\ncorrect label from a fixed distribution τ. LetVy(/vector m) be the optimal number of mistakes made\nby the learner if it predicts ywhen the state of the game is /vector m. Let/vector m→ybe the state of the\ngame in the next round, if in the current round the correct lab el isy. Define /vector m/ne}ationslash→yin a similar\nway. Using (12), we may now write\nVy(/vector m) =τyV(/vector m→y)+(1−τy)(1+V(/vector m/ne}ationslash→y)).\nSince the learner is allowed to predict any y∈ Y(and specifically the ywhich minimizes Vy(/vector m)),\nit suffices to bound Vy(/vector m) for some yof our choice. We now consider two cases.\nCase 1. Suppose that there exists y∈ Ywhich satisfies\nV(/vector m→y)≥1+V(/vector m/ne}ationslash→y).\nThen:\nVy(/vector m)≤V(/vector m→y)≤klogb(k)W(/vector m→y)≤klogb(k)W(/vector m)\nby the induction hypothesis, concluding the first case.\nCase 2. Suppose that for every y∈ Y,\nV(/vector m→y)<1+V(/vector m/ne}ationslash→y).\nLety∈ Ybe such that τy≥1/k. Then by assumption,\nVy(/vector m)≤(1/k)V(/vector m→y)+(1−1/k)(1+V(/vector m/ne}ationslash→y)).\nThe induction hypothesis now implies that\nVy(/vector m)≤1−1/k+logb(k)W(/vector m→y)+(k−1)logb(k)W(/vector m/ne}ationslash→y). (16)\nLetβ∈[0,1] be the fraction of experts predicting y, weighted according to their potential.\nIf the true label is ythen aβ-fraction of the potential W(/vector m) is untouched, and a (1 −β)-fraction\nis multiplied by at most 1 /k2(experts with budget 1 have their potential multiplied by 0) , and\nso\nW(/vector m→y)≤((1−β)/k2+β)W(/vector m).\nSimilarly,\nW(/vector m/ne}ationslash→y)≤(β/k2+(1−β))W(/vector m).\n20Plugging this into (16), we have\nVy(/vector m)≤1−1/k+logb(k)((1−β)/k2+β)W(/vector m)+(k−1)logb(k)(β/k2+(1−β))W(/vector m),\nwhich after a bit of manipulation reads as\nVy(/vector m)≤klogb(k)W(/vector m)+1+logb(k)(((1−β)/k2+β)(β/k2+(1−β))k−1)−1/k.\nTo finish the proof, it remains to show that\nf(β) := logb(k)(((1−β)/k2+β)(β/k2+(1−β))k−1)−1/k\nis bounded from above by −1 for all β∈[0,1], which is the statement of Lemma 4.9.\nWe can now prove Theorem 4.7.\nProof of Theorem 4.7. Clearlyb(k)≥k. ApplyingLemma4.8ontheinitialstate /vector m= (0,...,0,n),\nwe get\noptadap\nbandit(n,k,r) =optdadap\nbandit(P(Un,k,Br)) =V(0,...,0,n)≤klogb(k)(n·k2r)≤klogkn+2kr,\nas desired.\n4.2.1 Proof of technical lemma\nIn this section we prove Lemma 4.9. Let us first restate it in a m ore convenient way. Define\ngk(β) =/parenleftbigg1−β\nk2+β/parenrightbigg/parenleftbiggβ\nk2+1−β/parenrightbiggk−1\n.\nThe lemma states that for all k≥2 andβ∈[0,1], we have\ngk(β)≤b(k)1/k−1,whereb(k) =kk\n(k−1)k−1.\nRoutine calculation shows that\ng′\nk(β) =/parenleftbiggβ\nk2+1−β/parenrightbiggk−11\nk2/parenleftbigg\n1−1\nk2/parenrightbigg/bracketleftbig\nk2−k+1−(k3−k)β/bracketrightbig\n.\nIt follows that gk(β) is maximized at\nβk=k2−k+1\nk3−k,\nat which point its value is\ngk(βk) =(k2+1)(k−1)k−1\nk3k.\nTherefore we need to show that for all k≥2,\n(k2+1)(k−1)k−1\nk3k≤b(k)−(1−1/k)=(k−1)(k−1)(1−1/k)\nkk−1.\nRearranging, we need to show that\n(k2+1)(k−1)(k−1)/k≤k2k+1.\nThis holds since k2+1≤2k2≤k3≤k2kand (k−1)(k−1)/k≤k.\n214.3 Lower bounds for randomized learners\nIn this section, we prove lower bounds for randomized learne rs. We prove a lower bound which\nis tight when the adversary is adaptive for all triplets n,k,r. We also prove a lower bound\nwhich is tight even when the adversary is oblivious, but only for all triplets n,k,rsatisfying\nr=O(logkn).\nWe first prove a near-optimal lower bound on optobl\nbandit(n,k).\nLemma 4.10. Letn≥k≥2. Then\noptobl\nbandit(n,k)≥k−1\n2⌊logkn⌋.\nProof.Assume without loss of generality that n=kmfor some integer m(otherwise replace n\nwith the largest power of kbelow it). We label the set of experts using the elements of [ k]m,\nthat is, for each y1,...,ym∈[k] there is an expert h(y1,...,ym). We choose the correct expert\nat random.\nWe label the instances by x1,...,x m, where the prediction of h(y1,...,ym) onxiisyi. The\nfixed sequence of instances is\nx1,...,x 1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nktimes,...,x m,...,x m/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nktimes.\nFor any randomized strategy of the learner, it will discover the correct label of xiafter\nhaving made at leastk−1\n2mistakes in expectation (see for example [DH13, Claim 2]). T herefore\nthe expected number of mistakes incurred by the learner is at leastk−1\n2m. Consequently, for\neach learner there is an expert h(y1,...,ym) on which it makes at least as many mistakes in\nexpectation.\nLet us now prove a lower bound for the r-realizable case. In contrast with the realizable\ncase, this bound holds only against an adaptive adversary.\nLemma 4.11. Letn≥k≥2,r≥0. Then\noptadap\nbandit(n,k,r)≥(k−1)r/2.\nProof.For better readability assume that ris even. Suppose that n=k. In every round, the\nadversary let expert ipredict the label i. We describe an adversary which chooses the correct\nlabels at random and forces the learner to make at least ( k−1)r/2 mistakes in expectation.\nTherefore, there exists an adversary which chooses the corr ect labels deterministically and\nachieving that. Before the game starts, The adversary draws a random sequence y1,...,ykr/2\nofkr/2 labels independently and uniformly. In round t, the adversary chooses ytas the correct\nlabel. The adversary ends the game when one of the following c onditions hold:\n1. The learner has predicted correctly in r/2 rounds.\n2. The game was played for kr/2 many rounds.\nFirst, we show that the learner makes at least the stated numb er of mistakes in expectation.\nThereare two cases to consider. If the game is finishedbecaus ekr/2 roundshave passed andthe\nlearner has not yet made r/2 correct predictions, then the learner has made at least ( k−1)r/2\nmistakes with probability one, concluding this case.\nIn the second case, we assume that the game is finished when the learner makes exactly\nr/2 correct predictions. Partition the rounds of the game into r/2 intervals: the first interval\nstarts at the first round. Each interval ends when the learner makes a correct prediction. The\n22expected number of rounds in each interval is stochatically dominated by a Geom(1 /k) random\nvariable. Since in every interval there is exactly a single c orrect prediction, the total expected\nnumber of mistakes in all intervals is ( k−1)r/2.\nIt remains to show that the adversary is r-consistent. Assume without loss of generality that\nthe best expert is not consistent with all correct predictio ns of the learner, which is at most r/2.\nNow, recall that the number of rounds in the game is at most kr/2. Therefore, there must be\na labeli⋆predicted by the learner in at most r/2 many rounds. The best expert is at least as\ngood as the expert i⋆, which predicts i⋆in all rounds. The adversary is thus r/2-consistent on\nrounds where the learner is correct, and r/2-consistent on rounds where the learner is incorrect.\nIn total, the adversary is r-consistent.\nIn the proof, we only used the adversary’s adaptivity to deci de on the number of rounds in\nthe game, and not on the correct labels. Therefore, one might conjecture that it is possible to\nuse the same technique for proving a lower bound also against an oblivious adversary. However,\nit is not possible; as we show in Proposition 6.2, it holds tha toptobl\nbandit(k,k,r) = Θ(k+r) for\nallk≥2,r≥0. The essential reason allowing Proposition 6.2 is that an o blivious adversary\nmust be r-realizable in a strong sense, as discussed in Remark 2.1.\n4.4 Tight bounds for randomized learners\nWe can now deduce tight bounds on optadap\nbandit(n,k,r), and bounds on optobl\nbandit(n,k,r) which are\ntight in the regime r=O(logkn).\nTheorem 4.12. Letn≥k≥2,r≥0. Then\noptadap\nbandit(n,k,r) = Θ(k(logkn+r)).\nFurthermore, if r≤C·logknfor some constant Cthen\noptobl\nbandit(n,k,r) = ΘC(klogkn)\nwhereΘChides a constant that depends on C.\nProof.The bound optadap\nbandit(n,k,r) = Θ((logkn+r)k) is implied by putting together the upper\nbound of Theorem 4.7, and the lower bounds of Lemmas 4.10 and 4 .11. The lower bound in\noptobl\nbandit(n,k,r) = ΘC(klogkn) forr≤C·logkneasily follows from Lemma 4.10, and the upper\nbound from Theorem 4.7.\n5 The role of information\nIn this section, we prove the following theorem, which impli es Theorem 1.1.\nTheorem 5.1 (Full information vs. bandit feedback) .For every pattern class P ⊂(X ×Y)⋆it\nholds that\noptadap\nbandit(P)≤6k·optrand\nfull(P).\nTheorem 5.1 will also allow us to extend Theorem 1.1 (which is formulated for concept\nclasses) to the r-realizable setting.\n235.1 Reduction from bandit to full-information feedback\nGiven Theorem 4.1, the main ingredient left in order to prove Theorem 5.1 is the following\nreduction from learning with bandit feedback to learning wi th full-information feedback. Let\nus briefly describe the idea, inspired by [Lon20,HLM21].\nThe upper bound we prove on optadap\nbandit(n,k,r) makes no assumptions on the way the ex-\nperts make their predictions. Therefore, it holds in partic ular for the case where the experts’\npredictions are determined by algorithms chosen by the lear ner. We can exploit this property\nto reduce learning with bandit feedback to learning with ful l-information feedback. Let Abe\na deterministic full-information learning algorithm that is guaranteed to make at most rmany\nmistakes against any adversary. Whenever Ais consistent with the feedback, we do nothing.\nWhenAis inconsistent with the feedback, we can be sure that it has m ade a mistake. We\nthus create kdifferent copies of A, each guessing a different full-information feedback, exact ly\none of which is correct. Given that Amakes at most rmany mistakes when provided with\nfull-information feedback, an optimal randomized algorit hm for the bandit feedback model will\nthus make at most optadap\nbandit(kr,k,r) =O(k·r) mistakes against an adaptive adversary. The\ndetails are made precise in the proof of the proposition belo w.\nProposition 5.2. LetAbe a deterministic learning algorithm for a pattern class P ⊂(X ×Y)⋆\nin the full-information setting, which makes at most rmany mistakes on any sequence S∈ P.\nThen,\noptadap\nbandit(P)≤3k·r.\nProof.We employ an optimal algorithm for prediction with expert ad vice, where the experts\nresult from running Aand translating the bandit feedback into a full-informatio n feedback in\nall possibly ways.\nAt each round twe will maintain a k-ary tree Tof depth at most r, each of whose leaves\nis labelled by a sequence of d≤rtagged examples ( t1,xt1,yt1),...,(td,xtd,ytd). Each internal\nnode has its outgoing edges labelled by the elements of Y. Initially, Tconsists of a single node\nlabelled by the empty sequence. Additionally, we communica te with an optimal learner Bfor\nthe problem of prediction with expert advice in the r-realizable setting, with krexperts indexed\nby [k]r.\nAt round t, the adversary sends an instance xtto the learner. The learner uses Ato generate\na prediction for each leaf: if the leaf is labelled by the sequ enceσ, then the corresponding\nprediction is A(σ,xt). These predictions are converted to expert predictions as follows. A leaf ℓ\nat depth dhas an “address” α∈[k]dformed by the labels of the edges on the path from the root.\nAll experts whose index extends αmake the same prediction as ℓ. These expert predictions are\nsent to the optimal learner B, which provides a label ˆ yt. The label is forwarded to the adversary,\nwhich responds with either →or/\\⌉}atio\\slash→.\nThe learner now updates the tree as follows. For each leaf in t he tree, it checks whether it\npredicts a label which is inconsistent with the adversary’s feedback. This can happen in two\nways: either the leaf predicted ˆ ytand the adversary feedback was /\\⌉}atio\\slash→, or the leaf’s prediction was\ndifferent from ˆ ytand the adversary feedback was →. For each such leaf, if the leaf is at depth\nr, then we mark it as bad(a leaf not marked as badis agood leaf ). Otherwise, suppose that\nthe leaf is labelled by σ. We add to the leaf kchildren, and label the y’th child by σ◦(t,xt,y).\nThis completes the description of the algorithm.\nLetS⋆∈ Pbe a pattern that the adversary is consistent with at the end o f the game. We\nsay that a label ( t1,xt1,yt1),...,(td,xtd,ytd) isconsistent with S⋆ifS⋆\nti= (xti,yti) fori∈[d].\nThe definition of the algorithm ensures that the following pr operties hold at the conclusion of\neach round t:\n241. Each good leaf at depth dcorresponds to an expert whose predictions are inconsisten t\nwith the bandit feedback for exactly drounds.\nThis is because we add children to a leaf precisely when its pr ediction is inconsistent with\nthe bandit feedback (unless it is already at maximal depth, i n which case we mark it as\nbad).\n2. For each good leaf whose label σis consistent with S⋆, if we run Aon the sequence σ,\nthen it makes a mistake at every round.\nIndeed, when we add a new leaf labelled σ◦(t,xt,yt), where S⋆\nt= (xt,yt), the prediction\nofAonxtmust have been incompatible with the bandit feedback, and in particular, with\nyt, which is compatible with the bandit feedback.\n3. There is always a good leaf whose label is consistent with S⋆.\nIndeed, consider any leaf ℓsatisfying this at time t−1, let its label be σ, and let S⋆\nt=\n(xt,yt).\nIfℓis at depth rthenAmust output yton input ( σ,xt) due to the preceding item (it\ncannot make more than rmistakes on any sub-sequence of S⋆). Consequently, ℓdoesn’t\nbecome bad.\nIfℓis at lower depth and its prediction is inconsistent with the bandit feedback, then its\nchild labelled σ◦(t,xt,yt) satisfies the required properties.\nIn particular, due to items (1) and (3) there is an expert whos e predictions are inconsistent\nwith the bandit feedback for at most rrounds. Using the algorithm given by Theorem 4.7\nresults in a learner whose loss is at most\noptadap\nbandit(kr,k,r)≤klogk(kr)+2kr= 3kr,\nas desired.\n5.2 Proof of Theorem 5.1\nProof of Theorem 5.1. We use Proposition 5.2 with Littlestone’s [Lit88] SOAalgorithm for the\nclassPas the full-information algorithm A. Originally, SOAwas formulated in terms of con-\ncept classes rather than pattern classes, but adapting it to pattern classes is straightforward\n(see e.g. [MSTY23]). Since the adversary is consistent with P, it is known that the SOAalgo-\nrithm will make at most 2 optrand\nfull(P) mistakes as long as it receives full-information feedback .\nProposition 5.2 now provides the stated bound.\n5.3 An extension to the agnostic setting\nIn Section 4 we considered the r-realizable setting, in which the adversary is only r-consistent\nwith the class of experts. The exact same setting can be consi dered when learning concept\nclasses in general. Furthermore, in Section 8 we explain how to remove the assumption that r\nis given to the learner.\nOur bounds hold for all pattern classes, allowing an immedia te adaptation of our results\nfor concept classes to the r-realizable setting. Indeed, as mentioned also in Section 4 , for every\nconcept class Hand a budget function B:H →Rwe can define the pattern class appearing in\n(15):\nP(H,B) =\n\np∈(X ×Y)⋆:\n∃h∈ H:/summationdisplay\n(x,y)∈p1[h(x)/\\⌉}atio\\slash=y]≤B(h)\n\n\n.\n25Choosing B=Br, whereBrgives the budget rto every h∈ H, simulates the class Hin\nther-realizable setting. To transform bounds for pattern class es in the realizable setting to\nbounds for concept classes in the r-realizable setting, we only need the following bounds on\noptdet\nfull(P(H,Br)), which were originally proved for the binary case k= 2, but can be extended\nin a straightforward manner to arbitrary k.\nTheorem 5.3 ([CBFHW96,AL99,FHMM23]) .For every concept class Handr≥0it holds\nthat\noptdet\nfull(P(H,Br)) = Θ(optdet\nfull(H)+r).\nThe bounds obtained using Theorem 5.3 are summarized in Tabl e 1.\n6 The role of adaptivity\nIn this section we prove the following result, which implies Theorem 1.2.\nTheorem 6.1 (Oblivious vs. adaptive adversaries) .For every natural k≥2there exists a\npattern class P ⊂(X ×Y)⋆with|Y|=kand so that\noptadap\nbandit(P) = Ω(k·optobl\nbandit(P)).\nThe proof employs the concept class Hwhich consists of all constant functions. This cor-\nresponds to the classical bandit problem, in which there are only labels. We use the notation\noptobl\nbandit(H,r) foroptobl\nbandit(P(H,Br)), and similarly for other setups.\nProposition 6.2. LetH ⊂ YXbe the hypothesis class consisting of all constant function s, and\nletr≥0. Ifk=|Y| ≥2then\noptobl\nbandit(H,r) = Θ(k+r).\nProof.The lower bound follows easily from [DH13, Claim 2]. For the u pper bound, we describe\na learner which makes at most 34( k+r) mistakes in expectation:\nFirst phase For 16(k+r) rounds, predict each label with probability 1 /k. If there are at least\n8(k+r)/kcorrect predictions, set yto be the plurality vote among all rounds in which\nthe learner guessed correctly, and move to the second phase. Otherwise, repeat the first\nphase.\nSecond phase In this phase, the learner always predicts y, switching back to the first phase\nonce it makes r+1 mistakes in the phase.\nInordertoanalyzetheexpectednumberofmistakes, weassum ing(withoutlossofgenerality)\nthat the input sequence is infinite, and let y⋆∈ Ybe the hypothesis which is r-consistent with\nthe input sequence chosen by the oblivious adversary.\nLet us analyze the expected number of mistakes. In the first ph ase, we consider 16( k+r)\nrounds, and therefore the expected numberof correct guesse s isµ=16(k+r)\nk. LetXbea random\nvariable counting the number of correct predictions in the fi rst phase. Since µ≥16, and since\nXis a sum of independent random variables taking values in {0,1}, applying Chernoff’s bound\nimplies that\nPr[X≤8(1+r/k)]≤Pr[X≤µ/2]≤e−µ\n8≤1/e2<1/4. (17)\nThere are at most rmany examples which are not labeled by y⋆. Therefore, in expectation,\nthere are at most r/kmany examples which are both correctly classified and whose l abel is not\n26given by h⋆. LetYbe a random variable counting all examples that are not label ed byy⋆from\nthe correctly classified examples in the first phase. Using Ma rkov’s inequality, we have:\nPr[Y≥X/2|X≥8(1+r/k)]≤E[Y]\nX/2≤r/k\n4r/k= 1/4. (18)\nIfy=y⋆then the learner will make at most rmore mistakes in the second phase.\nIn order to finish the proof we calculate the expected number o f times this two phase\nprocess will occur until that y=y⋆. In the worst case, by (17),(18) this number of times is\ndominated by a Geom/parenleftBig\n(3/4)2/parenrightBig\nrandom variable, which means that this process will occur at\nmost (4/3)2<2 times in expectation. Overall the expected number of mista kes is at most\n2(16(k+r)+r)<34(k+r).\nWe can now prove Theorem 6.1.\nProof of Theorem 6.1. Givenk≥2, we choose the pattern class P=P(H,Bk), where H\nconsists of all constant functions.\nLemma4.11showsthat optadap\nbandit(P) = Ω(k2), whileProposition6.2showsthat optobl\nbandit(P) =\nO(k), yielding the stated bound.\nTheproofofLemma4.11goesthroughwithanyhypothesisclas sofsizek, wherek=|Y|. We\nconclude that the worst-case dependence on rinoptobl\nbandit(H,r) is related to the size of H. For\nexample, every Hwith|H|=ksatisfiesoptobl\nbandit(H,r) = Θ(k+r). However, there are classes\nHwith|H|=krwhich satisfy optobl\nbandit(H,r) = Ω(kr) due to Lemma 4.10. This is in contrast\ntooptadap\nbandit(H,r): for any class H(of size at least k) it holds that optadap\nbandit(H,r) = Ω(kr). It\nwould be interesting to find tight bounds on the worst case optobl\nbandit(H,r) for the intermediate\ncases where |H|=kwforw∈(1,r).\n7 The role of randomness\nIn this section we prove the following result, which implies Theorem 1.4 by applying it with\nd=k.\nTheorem 7.1. For every d≥1andk≥2there exists a concept class H ⊂ YXwithY=\n{0,1,...,k}such that\n1.optdet\nfull(H) =d+1.\n2.optdet\nbandit(H) = Θ(d·k).\n3.optadap\nbandit(H) = Θ(d+k).\nProof.We first define the class H:=H(d,k). The domain is X={x1,...,x d·k}. For every\ny∈ {1,...,k}, defineHyto be the class of all functions that label all instances with y, except\nfor a set X′⊂ Xof at most dmany instances, which they label with 0. Let\nH=/uniondisplay\ny∈{1,...,k}Hy.\nWe start with Item 1. For the upper bound, the learner always a nswers 0 until it makes the\nfirst mistake and receives as feedback a positive label y. Then it always answers yand makes at\nmostdmore mistakes. For the lower bound, the adversary asks on x1,...,x d+1, and always tells\n27the learner that it made a mistake. In the first round, the adve rsary determines the true label\nto be either y= 1 ory= 2, depending on the learner’s prediction. In the other drounds, the\nadversary determines the true label to be either 0 or y, depending on the learner’s predictions.\nWe now prove Item 2. For the upper bound, the learner tries to a nswer 1 until it makes\nd+1 mistakes, then does the same with 2, and so on until it gets t ok. For the lower bound,\nthe input sequence of instances the adversary asks on is x1,...,x d·k, and the adversary always\nprovides negative feedback to the learner. The number of mis takes made by the learner is thus\nd·k, and it remains to find an appropriate target function that re alizes it. Since there are d·k\nrounds, there must be a label y⋆∈[k] predicted by the learner for a set X′of instances of size\nat mostd. We choose the target hypothesis h⋆to be the hypothesis that gives the label 0 to the\ninstances of X′, andy⋆to all other instances. The hypothesis h⋆belongs to Hby definition.\nFinally, we prove Item 3. The lower bound is straightforward by using [DH13, Claim 2]. We\nprove the upper bound by describing a randomized learner for Hthat makes at most 2( d+k)\nmistakes in expectation. The learner operates in two phases . In the first phase, for every\ninstance for which it has not yet received positive feedback , it picks 0 with probability half, and\nall other labels with probability 1 /(2k) each. Once the learner receives positive feedback which\nis given on a prediction different from 0, it enters the second p hase, in which it will always\npredict this label, and will make at most dmore mistakes on instances labeled with 0.\nIt remains to upper bound the expected number of mistakes in t he first phase. There are\ntwo types of mistakes:\n1. Rounds where the adversary chooses the label 0. We divide t hese rounds into d(or fewer)\nphases: the first one ends when the learner correctly guesses the first instance labeled 0;\nthe second one ends when the learner correctly guesses the se cond instance labeled 0; and\nso on. Once the learner has found dinstances labeled 0, the remaining instances must\nhave a positive label.\nLetXi(where 1 ≤i≤d) be the number of mistakes made in the i’th phase.\n2. Rounds where the adversary chooses a positive label. Let Ybe the number of mistakes\nin such rounds.\nEach ofX1,...,X dis stochastically dominated by a Geom(1 /2)−1 random variable (that\nis, a geometric random variable from which we subtract 1), an dYis stochastically dominated\nby a Geom(1 /2k)−1 random variable. It follows that the expected number of mis takes is at\nmost\nd(2−1)+(2k−1)< d+2k.\nTogether with the at most dadditional mistakes in the second phase, we conclude that th e\nlearner makes at most 2( d+k) mistakes in expectation.\nTheorem 7.1 provides a negative answer to an open question po sed by [DH13], who ask\nwhether optobl\nbandit(H) = Ω(optdet\nbandit(H)) for every class H. ForHdefined in the proof of The-\norem 7.1 with d=k, we see that optobl\nbandit(H) =O/parenleftbigg/radicalBig\noptdet\nbandit(H)/parenrightbigg\n. This separation is tight\nup to a logarithmic factor: the bounds optobl\nbandit(H) = Ω/parenleftBigoptdet\nbandit(H)\nklogk/parenrightBig\n(following from the up-\nper bound in [AL99]) and optobl\nbandit(H)≥k−1\n2(e.g. by [DH13]) imply that optobl\nbandit(H) =\nΩ/parenleftbigg/radicalBig\noptdet\nbandit(H)/logoptdet\nbandit(H)/parenrightbigg\nfor every class H.\nAnother interesting corollary of Theorem 7.1 is that a signi ficant separation between full-\ninformation and bandit feedback holds, in some cases, for de terministic learners but not for\nrandomized learners. Indeed, as long as d= Ω(k), we have that both optdet\nfull(H) andoptadap\nbandit(H)\nare of the order O(d), whileoptdet\nbandit(H) is of order Ω( d·k).\n28Algorithm DT\nInput:A concept class H; a learner AforHthat when given r≥r⋆enjoys a mistake\nbound of d1(r+d2), where d1,d2≥1 does not depend on r.\nInitialize: rM=d2.\nFort= 1,2,...\n1. If the best hypothesis was inconsistent with the feedback for at most rMmany\nrounds:\n(a) Predictas Apredictsundertheassumptionthatthebesthypothesisisin consis-\ntent with the feedback for at most rMmany rounds, and given all information\ngathered by Ain previous rounds.\n2. Otherwise:\n(a) Update rM:= 2·rM.\n(b) Restart A.\nFigure 2: The “doubling trick” algorithm DT.\n8 Prediction without prior knowledge\nWhen studying agnostic mistake bounds in previous sections , we assumed the r-realizability\nassumption, in which the number of inconsistencies the best hypothesis (or expert) has with\nthe feedback, r⋆, is given to the learner (or some decent upper bound on it). In this section,\nwe show how to remove this assumption, while suffering only a co nstant factor degradation\nin the mistake bound. To achieve this goal, we use a “doubling trick” in the same spirit of\nthe doubling trick used in [CBFH+97, Section 4.6]. The algorithm DT, described in Figure 2,\nreceives as input an algorithm Afor ther-realizable setting. That is, Arequires knowledge of\na decent upper bound r≥r⋆to work well. Algorithm DTconverts it to an algorithm which\nworks well without this prior knowledge, in the following wa y. It operates in phases, where in\nthe beginning of each phase it resets the memory of A, and guesses a bound rMonr⋆which is\nlarger than the value guessed in the previous phase. When the best expert in this phase reaches\nrM+1 inconsistencies, we deduce that our guess of rMwas incorrect, and move on to the next\nphase. Thus, once rM≥r⋆, it is guaranteed that DTenters its last phase. The values we choose\nforrMguarantee that the obtained mistake bound of DTis not much worse than of A, when\ngiven the knowledge of r⋆.\nLetDT(A) denote DTwhen executed with the r-realizable algorithm A.\nProposition 8.1. LetH ⊂ YXbe a concept class. Suppose that Ais a bandit feedback learning\nalgorithm for H, that when given a bound r≥r⋆, enjoys a mistake bound of d1(r+d2), where\nd1,d2≥1does not depend on r(but may depend on H). Then, algorithm DT(A), without any\nprior knowledge, enjoys a mistake bound of 10d1(r⋆+d2). Furthermore, if Ais deterministic\nthen so is DT(A).\nProof.We split the execution of DTto phases according to the value of rM. That is, in phase\ni, we have rM= 2id2, wherei≥0.\n29We now consider two cases. In the first case, suppose that r⋆≤d2. Then the initial value\nrM=d2is a correct guess, and thus DTsimply runs Awith this assumption, and enjoys a\nmistake bound of 2 d1d2≤2d1(r⋆+d2).\nIn the second case, let a⋆>0 be such that r⋆= 2a⋆d2. In the worst case, the values of rM\nchosen by DTduring its execution are 20·d2,21·d2,...,2a−1·d2,2a·d2, wherea=⌈a⋆⌉. Indeed,\nifh⋆is the best hypothesis, then it is inconsistent with the feed back for at most r⋆many times\nthroughout the entire run, and in particular, throughout th e last phase where rM≥r⋆. Since\nDTresetsAat the end of every phase, in each phase i,DTis guaranteed to make at most\nd1(2id2+d2) mistakes in expectation. Between phases, when the best hyp othesis is inconsistent\nwith the feedback for the ( rM+1)’th time, another mistake might be made by DT. By linearity\nof expectation, we see that the mistake bound of DTis at most\na+d1a/summationdisplay\ni=0(2id2+d2)≤a+d1a/summationdisplay\ni=02i+1d2≤5d12ad2.\nNow, since a−1≤a⋆, we have r⋆≥2a−1d2. Therefore:\n5d12ad2= 10d12a−1d2≤10d1r⋆.\nThe “furthermore” part of the proposition is obvious from th e definition of DT.\nThe mistake bounds proved in this work under the assumption t hat a bound r≥r⋆is given\nare typically of the form f(|Y|)(r+g(optdet\nfull(H),|Y|)), where f,gare functions to R. Therefore,\nwecanapplyProposition 8.1with d1=f(|Y|),d2=g(optdet\nfull(H),|Y|) toourbounds,andremove\nthe assumption that a bound r≥r⋆is given.\n9 Open questions and future work\nOur work leaves some interesting open questions and directi ons for future work.\n9.1 Open questions\nThe price of adaptivity for concept classes. Our proof of Theorem 6.1 uses pattern\nclasses. Can we prove it using only concept classes, similar ly to Theorem 7.1? If not, what is\nthe price of adaptivity when learning concept classes?\nThe exact role of randomness. There is a Θ(log k) gap between the lower and upper\nbounds in Theorem 1.4. What is the correct worst case price of not using randomness?\nThe agnostic setting with oblivious adversaries. Our mistake bounds in the agnostic\nsettingarenottight forlarge r⋆whentheadversaryis oblivious, bothfortheproblemoflear ning\na concept class (Table 1), and for prediction with expert advice (Table 2). It would beinteresting\nto obtain bounds that are tight for large r⋆against an oblivious adversary for both problems.\nOne possible approach towards proving such bounds for prediction with expert advice is to use\nthe techniques of [ACBFS02], and specifically the Exp4 algorithm.\n30A natural algorithm for the experts setting. Our randomized algorithm for prediction\nwith expert advice is optimal, but not very natural nor efficient, as it relies on t he calculation\nof minimax values. While the analysis of our algorithm emplo ys the well-known paradigm of\npotential-based weighted experts, the learning algorithm itself does not make any use of these\nweights to devise its predictions. This is in contrast to man y learning algorithms that integrate\nthe weights into the prediction process (See [CBL06, Sectio n 2.1] and bibliographic remarks in\nSection 2). Can we design a natural algorithm, in the spirit o f algorithms such as weighted\nmajority [LW94] and Exp4 [ACBFS02], that achieves the guarantees of Theorem 4.7, up to\nconstant factors?\n9.2 Future research\nThe multilabel setting. It would be interesting to generalize the results of this wor k to\nthe multilabel setting considered in [DH13], in which the ad versary is allowed to choose several\ncorrect labels in each round. When the adversary is adaptive , it is not hard to see that this is\nequivalent to the single-label setting considered in this w ork with a deterministic learner. What\nhappens when the adversary is oblivious?\nOther types of feedback. This work considers the full information and bandit feedbac k\nmodels. However, onecanthinkofothertypesoffeedback. Co nsiderforexamplethe comparison\nfeedback model: if the truelabel is yand the prediction is z, the adversary provides the feedback\nop∈ {<,=,>}such that zopy. An interesting direction for future research is to prove re sults\nin the spirit of this work for comparison feedback, or for oth er natural types of feedback.\nAcknowledgments\nWe thank Zachary Chase for many insightful discussions on th e problems in the focus of this\nwork.\nShay Moran is a Robert J. Shillman Fellow; he acknowledges su pport by ISF grant 1225/20,\nbyBSFgrant 2018385, byanAzrieli Faculty Fellowship, byIs rael PBC-VATAT, bytheTechnion\nCenter for Machine Learning and Intelligent Systems (MLIS) , and by the the European Union\n(ERC, GENERALIZATION, 101039692). Views and opinions expr essed are however those of\nthe author(s) only and do not necessarily reflect those of the European Union or the European\nResearch Council Executive Agency. Neither the European Un ion nor the granting authority\ncan be held responsible for them.\nReferences\n[AABR09] Jacob D. Abernethy, Alekh Agarwal, Peter L. Bartle tt, and Alexander Rakhlin.\nA stochastic view of optimal regret through minimax duality . InCOLT, 2009.\n[ACBFS02] Peter Auer, Nicolo Cesa-Bianchi, Yoav Freund, an d Robert E. Schapire. The\nnonstochastic multiarmed bandit problem. SIAM journal on computing , 32(1):48–\n77, 2002.\n[ADT12] Raman Arora, Ofer Dekel, and Ambuj Tewari. Online ba ndit learning against an\nadaptive adversary: from regret to policy regret. In Proceedings of the 29th In-\nternational Conference on Machine Learning , pages 1503–1510. Omnipress, 2012.\nURL:https://dl.acm.org/doi/10.5555/3042573.3042796 .\n31[AL99] Peter Auer and Philip M. Long. Structural results abo ut on-line learning models\nwith and without queries. Machine Learning , 36:147–181, 1999.\n[ALW06] Jacob Abernethy, John Langford, and Manfred K. Warm uth. Continuous experts\nand the binning algorithm. In International Conference on Computational Learn-\ning Theory , pages 544–558. Springer, 2006.\n[BCB+12] S´ ebastien Bubeck, Nicolo Cesa-Bianchi, et al. Regret a nalysis of stochastic and\nnonstochastic multi-armed bandit problems. Foundations and Trends ®in Ma-\nchine Learning , 5(1):1–122, 2012.\n[BP19] Simina Brˆ anzei and Yuval Peres. Online learning wit h an almost perfect expert.\nProceedings of the National Academy of Sciences , 116(13):5949–5954, 2019.\n[CBFH+97] Nicolo Cesa-Bianchi, Yoav Freund, David Haussler, Davi d P. Helmbold, Robert E.\nSchapire, and Manfred K. Warmuth. How to use expert advice. Journal of the\nACM (JACM) , 44(3):427–485, 1997.\n[CBFHW96] Nicolo Cesa-Bianchi, Yoav Freund, David P. Helmb old, and Manfred K. Warmuth.\nOn-line prediction and conversion strategies. Machine Learning , 25(1):71–110,\n1996.\n[CBL06] Nicolo Cesa-Bianchi and G´ abor Lugosi. Prediction, learning, and games . Cam-\nbridge university press, 2006.\n[DH13] Amit Daniely and Tom Helbertal. The price of bandit in formation in multiclass\nonline classification. In Conference on Learning Theory , pages 93–104. PMLR,\n2013.\n[DSBDSS15] AmitDaniely, SivanSabato, ShaiBen-David, and ShaiShalev-Shwartz. Multiclass\nlearnability and the ERM principle. J. Mach. Learn. Res. , 16(1):2377–2404, 2015.\n[FHMM23] Yuval Filmus, Steve Hanneke, Idan Mehalel, and Sha y Moran. Optimal prediction\nusing expert advice and randomized littlestone dimension. InCOLT, volume 195\nofProceedings of Machine Learning Research , pages 773–836. PMLR, 2023.\n[FM06] Daniela Pucci De Farias and Nimrod Megiddo. Combinin g expert advice in reac-\ntive environments. Journal of the ACM (JACM) , 53(5):762–799, 2006.\n[Gen21] Jesse Geneson. A note on the price of bandit feedback for mistake-bounded online\nlearning. Theoretical Computer Science , 874:42–45, 2021.\n[GT24] Jesse Geneson and Linus Tang. Bounds on the price of fe edback for mistake-\nbounded online learning. arXiv preprint arXiv:2401.05794 , 2024.\n[HLM21] SteveHanneke, RoiLivni, andShayMoran. Onlinelea rningwithsimplepredictors\nand a combinatorial characterization of minimax in 0/1 game s. InConference on\nLearning Theory , pages 2289–2314. PMLR, 2021.\n[Lit88] Nick Littlestone. Learning quickly when irrelevan t attributes abound: A new\nlinear-threshold algorithm. Machine learning , 2(4):285–318, 1988.\n[Lon20] Philip M. Long. New bounds on the price of bandit feed back for mistake-bounded\nonline multiclass learning. Theoretical Computer Science , 808:159–163, 2020.\n32[LW94] Nick Littlestone and Manfred K. Warmuth. The weighte d majority algorithm.\nInformation and computation , 108(2):212–261, 1994.\n[MOSW02] Neri Merhav, Erik Ordentlich, Gadiel Seroussi, an d Marcelo J Weinberger. On\nsequential strategies for loss functions with memory. IEEE Transactions on In-\nformation Theory , 48(7):1947–1958, 2002.\n[MS10] Indraneel Mukherjee and Robert E. Schapire. Learnin g with continuous experts\nusing drifting games. Theoretical Computer Science , 411(29-30):2670–2683, 2010.\n[MSTY23] Shay Moran, Ohad Sharon, Iska Tsubari, and Sivan Yo sebashvili. List online\nclassification. In The Thirty Sixth Annual Conference on Learning Theory , pages\n1885–1913. PMLR, 2023.\n[RRS+23] Ananth Raman, Vinod Raman, Unique Subedi, Idan Mehalel, and Ambuj\nTewari. Multiclass online learnability under bandit feedb ack.arXiv preprint\narXiv:2308.04620 , 2023.\n[RSS12] SashaRakhlin, OhadShamir, andKarthikSridharan. Relax andrandomize: From\nvalue to algorithms. Advances in Neural Information Processing Systems , 25, 2012.\n[RST10] Alexander Rakhlin, Karthik Sridharan, and Ambuj Te wari. Online learning: Ran-\ndom averages, combinatorial parameters, and learnability .Advances in Neural\nInformation Processing Systems , 23, 2010.\n[vN28] John von Neumann. Zur theorie der gesellschaftsspie le.Mathematische annalen ,\n100(1):295–320, 1928.\n[Vov90] Volodimir G. Vovk. Aggregating strategies. Proc. of Computational Learning\nTheory, 1990 , 1990.\n33"    },    {        "title": "2402.07491v1.Ultra_fast_Waveguide_MUTC_Photodiodes_over_220_GHz.pdf",        "content": "1\nUltra-fast Waveguide MUTC Photodiodes over 220\nGHz\nLinze Li , Luyu Wang , Tianyu Long , Zhouze Zhang , Juanjuan Lu , BaileChen ,Senior Member, IEEE\nAbstract —We present InP-based evanescently-coupled waveg-\nuide modified uni-traveling carrier photodiodes (MUTC-PDs)\nexhibiting a breakthrough in bandwidth. The optimization of\ncarrier transport and optical coupling is achieved through a\ndetailed discussion on the design of the cliff layer and waveg-\nuide layer. Addressing the parasitic capacitance challenge, we\nintroduce benzocyclobutene (BCB) beneath the PD electrodes,\neffectively overcoming the bandwidth bottleneck associated with\nthe RC time constant. Devices with sizes of 2 ×7µm2and 2×10\nµm2achieve 3-dB bandwidths over 220 GHz, along with external\nresponsivities of 0.161 A/W and 0.237 A/W, respectively. Notably,\nthe RF output power reaches a peak of -1.69 dBm at 215 GHz\nfor 2×15µm2PDs.\nIndex Terms —Photodiodes (PD), high-speed photodetectors,\nmodified uni-traveling-carrier (MUTC), THz generation.\nI. I NTRODUCTION\nTHE pursuit of ultra-high-speed data transmission has\nbeen one of the primary driving forces for research on\nultra-fast photodetectors. The advancement of communication\nfrequencies into the terahertz (THz) domain (0.1-1 THz) brings\nforth significant challenges, particularly in the generation and\ndetection of ultra-high-frequency signals for analog and digital\nphotonic applications [1]. In these systems, ultra-fast photodi-\nodes (PDs) play a vital role as the key component, typically\ndetermining the maximum allowable operating frequency and\ndynamic range [2]. For the generation of THz waves, photonic\ntechniques that based on high-speed photodetectors are con-\nsidered to be superior to conventional electronic techniques\nwith respect to wide frequency bandwidth, tunability, and\nstability [3]. Meanwhile, for the receiver end of the digital\ncommunication systems, integrating ultra-broadband photode-\ntectors is essential for the realization of next-generation 200\nGBaud interconnects [4, 5]. Among various types of high-\nspeed photodiodes, uni-traveling carrier photodiodes (UTC-\nPDs) [6, 7], utilizing only electrons as the active carriers in\nThis work was supported by the National Natural Science Foundation\nof China under Grant 61975121. The authors are grateful for the device\nfabrication support from the ShanghaiTech University Quantum Device Lab.\n(Corresponding author: Baile Chen.)\nLinze Li is with the School of Information Science and Technology,\nShanghaiTech University, Shanghai 201210, China, with Shanghai Engi-\nneering Research Center of Energy Efficient and Custom AI IC, Shanghai\n201210, China, with the Shanghai Institute of Microsystem and Information\nTechnology, Chinese Academy of Sciences, Shanghai 200050, China, and also\nwith the University of Chinese Academy of Sciences, Beijing 100049, China\n(e-mail: lilz@shanghaitech.edu.cn).\nLuyu Wang, Tianyu Long, Zhouze Zhang, Juanjuan Lu and Baile Chen\nare with the School of Information Science and Technology, ShanghaiTech\nUniversity, Shanghai 201210, China (e-mail: wangly4@shanghaitech.edu.cn;\nlongty2022@shanghaitech.edu.cn; zhangzhz@shanghaitech.edu.cn;\nlujj2@shanghaitech.edu.cn; chenbl@shanghaitech.edu.cn).the drift layer, stand out as highly promising structures for\napplications demanding both high speed and high power [8].\nSurface-illuminated PDs are capable of achieving impres-\nsive speed and saturation characteristics. The back illuminated\nmodified UTC-PDs (MUTC-PDs) reported in [9] exhibited\na wide bandwidth of 230 GHz by inductance peaking and\nreducing the diameter of PDs down to 3 µm. MUTC-PDs flip-\nchip bonded onto the diamond submount realized RF output\npower of -3 dBm at 150 GHz and -5.7 dBm at 165 GHz\n[10]. However, due to the vertical nature of carrier transport\nand optical coupling, surface-illuminated PDs face challenge\nin simultaneously achieving short carrier transit time and high\nresponsivity [11]. In contrast, waveguide PDs (WG-PDs) offer\na promising solution to the bandwidth-responsivity tradeoff by\ncoupling light laterally into the active region of the device\nthrough an optical waveguide. E. Rouvalis et al. reported\nevanescent-coupled UTC-PDs with a 170 GHz bandwidth and\na responsivity up to 0.27 A/W [12]. A waveguide integrated\nPD module with a 3dB-bandwidth of 145GHz and a respon-\nsivity of 0.4 A/W has been demonstrated [13].\nIn this work, we demonstrated evanescently-coupled waveg-\nuide MUTC-PDs with ultra-fast speed and high output power.\nThe cliff layer doping level was carefully designed to enhance\nthe electron transport and compensate for space charge ef-\nfect under high optical power injection. Evanescently optical\ncoupling was also optimized by tuning the refractive index\nand thickness of the waveguide layer. Furthermore, for better\nRC-limited bandwidth, we investigated the primary factors\ncontributing to the parasitic capacitance of the devices. By\nimplementing benzocyclobutene (BCB) under the coplanar\nwaveguide (CPW) electrodes, such capacitance was effectively\nreduced compared to the previous work [14]. To the best of our\nknowledge, this work obtained a record-high 3-dB bandwidth\nexceeding 220 GHz for 2 ×7µm2and 2×10µm2devices\nand a high saturation output power of -1.69 dBm at 215 GHz\nfor 2×15µm2device.\nII. D ESIGN AND FABRICATION OF DEVICE\nA. MUTC Structure Design\nThe designed epitaxial structure of the InGaAs/InP wafer\nwas grown on the semi-insulating InP substrate, as shown in\nFig. 1. The 180 nm absorption layer was partially depleted to\nbuild a high electric field across the hererojunction interface.\nAn 80 nm p-doped InGaAs undepleted absorption layer with\na step-grading profile(5 × 1017cm−3,1 × 1018cm−3, and\n2 × 1018cm−3) was implemented to form a quasi-electric\nfield which assists electron transport. Additionally, a 250 nmarXiv:2402.07491v1  [physics.app-ph]  12 Feb 20242\nFig. 1. Epi-layer structure of the InGaAs/InP photodiode.\nslightly n-doped InP drift layer was designed to serve as\nthe charge compensation layer. Heavily p-doped InGaAs and\nheavily n-doped InP layers were deposited as the p and n\ncontact,respectively. InGaAsP quaternary layers were grown\nat InGaAs/InP heterojunction interfaces to smooth the energy\nband discontinuities.\nA 30 nm cliff layer with the optimized doping level of\n2 × 1017cm−3was inserted in the drift layer. In order\nto investigate the influence of the cliff layer with different\ndoping level, we simulated the band diagram and electric field\ndistribution as the doping concentration of the cliff layer varies\nfrom 1 × 1017cm−3to 3 × 1017cm−3. Simulations are\nperformed by the Silvaco TCAD software at -2V bias and\n10 mA photocurrent for the 2 ×15µm2device. Increasing\nthe doping concentration in the cliff layer can significantly\ndecrease the conduction band barrier across the heterogeneous\ninterface in the depletion region, as shown in Fig.2 (a). This\nfavors electrons transport and alleviate the accumulation of\nelectrons at the InGaAs/InP interfaces. In addition, as depicted\nin Fig. 2 (b), the higher electric field induced by higher\ndoping level of cliff layer can also help electrons to cross\nthe heterogeneous inter-facial barriers and mitigate the effect\nof space charge effect at a high current level [15]. However,\nwhen the n-doping of the cliff layer increases to 3 × 1017\ncm−3, the electric field will be too low to sustain the fast\ntransport of electrons in InP drift layer. A doping level of 2 ×\n1017cm−3can maintain an electric field of 20-50 kV/cm in\nthe whole e-drift layer to enable electron overshoot [16] and\nensure a short carrier transit time.\nB. Optical coupling design\nA single-layer InGaAsP waveguide is designed for optical\ncoupling and the InP drift layer above acts as the cladding for\nit. For better saturation characteristics, longer coupling length\nis preferred to allow more uniform evanescently coupling\nof light from the passive waveguide to the active region\nof the photodiode. However, longer PD length would result\nin a higher junction capacitance, degrading the RC-limited\nbandwidth. To balance such trade-off, the largest length of the\nFig. 2. Simulated (a) energy band diagram and (b) electric field distribution\nfor the 2 ×15µm2device with different cliff layer doping levels at -2V bias\nand 10 mA photocurrent.\ndevice is set to be 15 um. In this work, the internal responsivity\nof the 2 ×15µm2device was simulated with respect to\ndifferent refractive index and thickness of the waveguide\nlayer, as shown in Fig. 3 (a). An optimal responsivity of\n1.1 A/W was achieved when the thickness and refractive\nindex of the waveguide layer is 0.4 µm and 3.41 (Q 1.3\nInGaAsP) respectively. Fig. 3 (b) shows the corresponding\noptical intensity as light propagates through the device.\nC. Fabricaiton\nThe device was fabricated into a triple-mesa structure,\nas shown in Fig. 4. Inductively coupled plasma (ICP) dry\netch process was conducted to achieve vertical sidewalls and\nprecise control of the size of p-mesa and waveguide mesa,\nwhile wet etch (H 3PO4:HCl = 1:3) was used for electrical\nisolation between n-mesas. Deposition of Ti/Pt/Au/Ti and\nGeAu/Ni/Au as p-contact and n-contact was followed by the\nrapid thermal annealing process at 360◦C to achieve the low\ncontact resistance.\nIn our previous work [14], the parasitic capacitance is the\nmajor limiting factor of RC response, which mainly consists\nof C conand C t, as shown in Fig. 5. C conis the capacitance3\nFig. 3. (a) Simulated internal responsivity versus refractive index and\nthickness of the waveguide layer for the 2 ×15µm2device. (b) Simulated\noptical intensity as light propagates through the structure.\nbetween the CPW signal pad and the n-contact layer [17].\nConsidering the stability of the air-bridge structure, the spacing\ndistance between these metals is limited to be around 1 µm,\nleading to a large C con. For a better RC limited bandwidth, 4\nµm thick Benzocyclobutene (BCB) was introduced under the\nCPWs (see Fig. 3) to eliminate the need for the previous air-\nbridge structure and effectively pad the signal electrode [18].\nSuch a non-suspended structure can also ensure stable and\nuniform contact between p-mesa and CPWs for devices with\nwidth down to 2 µm.\nCtis the capacitance introduced between electrodes of the\nCPW itself. The permittivity and thickness of the multilayered\nsubstrates have a significant impact on C t[19, 20]. As shown\nin Fig. 6, We employed Computer Simulation Technology\n(CST) EM Studio to simulate the curve of C tversus thickness\nof BCB on InP substrate. Due to the 4 µm thick BCB strip-\nsupporting layer, the C tis greatly reduced by more than half.\nIII. C HARACTERIZATION AND DISCUSSION\nA. DC and Optical Characteristics\nThe dark current versus bias voltage characteristics for\ndevices with different lengths are shown in Fig. 7. A typical\ndark current is around 100 pA under -5 V bias, which is at\nleast one order of magnitude lower than other InGaAs/InP\nbased MUTC-PDs [9, 10, 18]. Such low dark current can be\nattributed to the careful protection of the PD sidewalls and\nwaveguide mesa during the dry etch process.\nFig. 4. Front view SEM image of the fabricated waveguide MUTC-PD.\nFig. 5. Schematic diagram dipicting the main components of the parasitic\ncapacitance in the waveguide MUTC-PD.\nBCBInPCPW\nFig. 6. Simulated curve of C tversus thickness of BCB on InP substrate.\nInsert is the schematic diagram of simulated model.4\nFig. 7. Dark current versus bias voltage characteristic for the devices with\ndifferent lengths.\nTo calculate the internal responsivity of the devices, the\nfiber-to-waveguide coupling loss and waveguide propagation\nloss were experimentally measured to be 5.5 dB in total.\nThe assessment of fiber-to-waveguide coupling loss involved\nutilizing lensed fiber to couple light into and out of the test\nwaveguide, followed by a comparison of input and output\noptical power. The test waveguide featured a slight curve near\nthe waveguide facet to avoid collecting optical power outside\nthe waveguide. Propagation loss was evaluated by studying\nthe responsivity of devices with varying waveguide lengths.\nIn Fig 8, the external and internal responsivities of devices\nwith different lengths are illustrated. Devices with lengths\nof 7 µm, 10 µm and 15 µm show external responsivities\nof 0.161 A/W, 0.237 A/W and 0.312 A/W at 1550 nm\nwavelength. The associated internal responsivities are found to\nbe 0.567 A/W, 0.835 A/W, and 1.099 A/W, respectively. The\nsimulated internal responsivity versus the length of PDs using\nAnsys Lumerical FDTD is represented by the dashed line,\ndemonstrating a robust agreement with the measured results.\nB. RF Characteristics\nThe frequency response and the saturation performance of\nthe InGaAs/InP waveguide devices were investigated by an\noptical heterodyne setup [21]. Two tunable lasers with wave-\nlengths near 1.55 µm were combined to generate a heterodyne\nbeat note with a wide frequency tuning range from Megahertz\nto Terahertz. A lensed fiber with a 2- µm spot diameter was\nused to couple the modulated optical signal into the cleaved\nwaveguide facet with no anti-reflective coating. The output\nRF power was obtained by two measurement setups. From\nDC to 110 GHz, the RF signal was collected by the Rohde\n& Schwarz power meter (NRP-Z58) through the GSG Model\n110H probe. Frequency-dependent loss value introduced by\ncables, bias tee and GSG probe was calibrated by a vector\nnetwork analyzer (VNA) with corresponding calibration kit\n(85058E & CS-5 for DC to 67 GHz, WR10-VDI & CS-5\nfor 73.8 GHz to 110 GHz). For frequency above 110 GHz,\nFig. 8. Measured external (red circles) and internal (black circles) respon-\nsivities and simulated internal responsivities (dashed line) of the devices with\ndifferent lengths.\nFig. 9. Frequency response of the (a) 2 ×7µm2device at 2mA photocurrent,\n(b) 2×10µm2device at 3mA photocurrent and (c) 2 ×15µm2device at\n4mA photocurrent under -1.2 V bias.5\na VDI power meter (PM5) and a set of waveguide probes\ncovering 110 GHz to 220 GHz were used to measure the\nTHz power generation. The loss of the probes and waveguide\nconnectors provided by the manufacturer was de-embedded\nfrom the measurement results.\nIn Fig. 9, the frequency response of devices with lengths of\n7µm, 10 µm, and 15 µm is depicted at a -1.2 V bias voltage,\ncorresponding to photocurrents of 2 mA, 3 mA, and 4 mA,\nrespectively. The 2 ×7µm2and 2 ×10µm2devices\nshowcase ultra-fast speeds, with a power drop of less than 3 dB\nfrom DC to 220 GHz, indicating a 3-dB bandwidth exceeding\n220 GHz. Measurement beyond 220 GHz was not conducted\ndue to limitations in the current lab setup. The 2 ×15µm2\ndevice exhibits a lower bandwidth of 165 GHz, constrained\nby the RC response owing to a larger junction capacitance.\nAt a fixed frequency of 215 Ghz, the saturation performance\nof PDs with various lengths is characterized by measuring the\nRF output power as a function of the average photocurrent, as\nshown in Fig. 10 (a). Under -2 V bias, devices with lengths\nof 7µm, 10 µm and 15 µm reach output power of -3.70 dBm,\n-1.79 dBm and -1.69 dBm at 215 GHz, respectively. For\nphotocurrent below 5 mA, shorter devices output higher RF\npower due to better 3dB bandwidth. However, the 2 ×7µm2\ndevice saturates earlier at a photocurrent of 6.2 mA, leading\nto overall lower RF output power. Such a phenomenon is due\nto higher current densities as well as the in-homogeneous\nlight intensity distribution in small-sized devices. The RF\noutput power versus the average photocurrent of the 2 ×15\nµm2device under different bias voltages is shown in Fig. 10\n(b). At low light intensities, the difference between the RF\npower and the ideal power decreases with increasing light\nintensity, which can be attributed to faster electron transport\nresulting from self-induced quasi-field in the absorber [22].\nAt higher light intensities, saturation occurs due to growing\ncharge accumulation. The saturation current increases from 8.2\nto 10.6 mA when the bias voltage increases from 1.5 to 2.5 V ,\nsince higher electric field can suppress the space charge effect.\nThe device exhibits the best saturation performance under -2\nV bias, with a 1-dB compression point at 10.6 mA and output\npower of -1.69 dBm. Under -2.5 V , the electric field in the\ndrift layer is too high to enable electron’s overshoot and result\nin a lower bandwidth and output power.\nC. S11 Parameter Fitting\nThe S11 parameters up to 110 GHz were acquired through\non-wafer network analyzer measurements. These parame-\nters allows for the comprehensive characterization of the\nimpedance profile of the photodetectors. To extract the in-\ntrinsic equivalent circuit parameters, we employed parameter\nfitting within the Advanced Design System (ADS) software.\nThe equivalent circuit model illustrated in Fig. 11 (a) was\nused to represent the waveguide PDs. The model incorporates\nelements representing the series resistance Rs, CPW induc-\ntanceLt, and the parasitic capacitance Cp.CaandCcwere\ndesignated as the junction capacitance of the depleted InGaAs\nabsorber and the InP collector, while RaandRcrepresented\nthe junction resistances, correspondingly.\nFig. 10. (a) RF power versus photocurrent for the waveguide MUTC-PDs with\ndifferent lengths under -2V bias. (b) RF power versus photocurrent for the 2\n× 15µm2device under different bias voltages. The measurement frequency\nis fixed at 215 GHz.\nFig. 11. (a) Equivalent circuit model of the MUTC devices for S11 fitting.\n(b) Measured (blue line) and fitted (red line) S11 data of 2 × 7 µm2, 2 × 10\nµm2and 2 × 15 µm2devices with 50 MHz-110 GHz frequency range under\n-1.2 V bias.6\nTABLE I\nFITTING PARAMETERS\nArea ( µm2)Ca (fF) Cc (fF) Rs (Ω)Cp (fF) Lt (pH)\n2 × 7 17.2 5.5 23.3 4.4 49.9\n2 × 10 24.6 7.8 19.6 4.56 51.7\n2 × 15 36.9 11.8 16.1 4.38 57.7\nFig. 12. 3dB bandwidth and responsivity of waveguide coupled PDs based\non group III-V semiconductors reported in literature [12, 13, 23–30].\nThe alignment of the measured S11 (red line) with the\nfitted results (blue line) on the Smith Charts, as shown in\nFig. 11 (b), validated the accuracy of the circuit model. The\nextracted parameters are given in Table I. Notably, The Smith\nchart shows that the S11 values start at the open circuit point,\nindicating a high junction resistance ( RaandRc), which is\nset to be 100 M Ωand not presented in the table. Ccand\nCawere calculated separately using a parallel plate capacitor\napproximation, reflecting the dual-material depletion zones\ncharacteristic of the MUTC-PD.\nThe total parasitic capacitance Cp, which includes Ct,\nCcon, and other parasitic elements, maintained a consistent\nvalue of about 4.5 fF across the devices. In contrast to our\nearlier work where parasitic capacitance were approximately\n20 fF [14], the current study demonstrates a pronounced\ndecline to 4.5 fF. This significant reduction is largely owing\nto the application of BCB under the electrodes, contributing\nsubstantially to the enhanced bandwidth.\nFig. 12 summarize the state-of-the-art responsivity versus\n3-dB bandwidth of InGaAs/InP based WG-PDs. Our work\nshows the highest bandwidth along with a good responsivity.\nGiven that only a single waveguide layer is utilized for optical\ncoupling in this study, the incorporation of a diluted waveguide\nand a spot-size converter could further enhance coupling\nefficiency.\nIV. C ONCLUSION\nIn this work, we have reported InGaAs/InP based ultra-fast\nwaveguide MUTC-PDs operating at 1550 nm wavelength. The\ncliff layer doping level, as well as the refractive index andthickness of the waveguide layer, are meticulously adjusted to\nguarantee favorable RF and optical characteristics. Meanwhile,\nBCB is introduced to reduce the parasitic capacitance and\nfuther extend the bandwith of PDs. The 3 dB bandwidths of\nthe 2×7µm2and 2×10µm2devices surpass 220 GHz\nunder -1.2 V bias. At 215 GHz, an output RF power of -1.69\ndBm is measured, corresponding to a saturation current of 10.6\nmA. These promising findings suggest the potential for THz\ngeneration through InP photonics integrated circuits.\nREFERENCES\n[1] C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits\non the performance of RF-over-fiber links and their impact on\ndevice design,” IEEE Transactions on Microwave Theory and\nTechniques , vol. 54, no. 2, pp. 906–920, 2006.\n[2] J.-M. Wun, H.-Y . Liu, Y .-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B.\nHuang, and J.-W. Shi, “Photonic high-power continuous wave\nTHz-wave generation by using flip-chip packaged uni-traveling\ncarrier photodiodes and a femtosecond optical pulse generator,”\nJournal of Lightwave Technology , vol. 34, no. 4, pp. 1387–1397,\n2015.\n[3] T. Nagatsuma, H. Ito, and T. Ishibashi, “High-power RF pho-\ntodiodes and their applications,” Laser & Photonics Reviews ,\nvol. 3, no. 1-2, pp. 123–137, 2009.\n[4] T. Beckerwerth, P. Runge, and M. Schell, “High-Speed Pho-\ntodiodes for Power Efficient Data Transmission,” Journal of\nLightwave Technology , 2023.\n[5] T. Umezawa, A. Matsumoto, K. Akahane, A. Kanno, and N. Ya-\nmamoto, “Ultra-broadband photodetector module toward 200\nGHz using UTC-PD and frequency compensation technique,” in\nConference on Lasers and Electro-Optics/Pacific Rim . Optica\nPublishing Group, 2022, p. CTuP11E 05.\n[6] T. Ishibashi, T. Furuta, H. Fushimi, S. Kodama, H. Ito, T. Na-\ngatsuma, N. Shimizu, and Y . Miyamoto, “InP/InGaAs uni-\ntraveling-carrier photodiodes,” IEICE transactions on electron-\nics, vol. 83, no. 6, pp. 938–949, 2000.\n[7] T. Ishibashi and H. Ito, “Uni-traveling-carrier photodiodes,”\nJournal of Applied Physics , vol. 127, no. 3, 2020.\n[8] T. Ishibashi, T. Furuta, H. Fushimi, and H. Ito, “Photoresponse\ncharacteristics of uni-traveling-carrier photodiodes,” in Physics\nand Simulation of Optoelectronic Devices IX , vol. 4283. SPIE,\n2001, pp. 469–479.\n[9] Y . Tian, B. Xiong, C. Sun, Z. Hao, J. Wang, L. Wang, Y . Han,\nH. Li, L. Gan, and Y . Luo, “Ultrafast MUTC photodiodes over\n200 GHz with high saturation power,” Optics Express , vol. 31,\nno. 15, pp. 23 790–23 800, 2023.\n[10] C. Wei, X. Xie, Z. Wang, Y . Chen, Z. Zeng, X. Zou, W. Pan,\nand L. Yan, “150 GHz High-Power Photodiode by Flip-Chip\nBonding,” Journal of Lightwave Technology , 2023.\n[11] K. Wada, J. Liu, Y . Ishikawa, D. Ahn, D. Pan, P. Cai, and\nL. Kimerling, “High-speed photodetectors,” in Integrated Pho-\ntonics for Data Communication Applications . Elsevier, 2023,\npp. 123–157.\n[12] E. Rouvalis, M. Chtioui, F. Van Dijk, F. Lelarge, M. Fice,\nC. Renaud, G. Carpintero, and A. Seeds, “170 GHz uni-traveling\ncarrier photodiodes for InP-based photonic integrated circuits,”\nOptics express , vol. 20, no. 18, pp. 20 090–20 095, 2012.\n[13] P. Runge, F. Ganzer, J. Gl ¨asel, S. W ¨unsch, S. Mutschall, and\nM. Schell, “Broadband 145Ghz photodetector module target-\ning 200Gbaud applications,” in Optical Fiber Communication\nConference . Optica Publishing Group, 2020, pp. M2A–1.\n[14] L. Li, L. Wang, and B. Chen, “High-speed Waveguide Modified\nUni-Traveling Carrier Photodiodes with 130 GHz Bandwidth,”\nin2023 Opto-Electronics and Communications Conference\n(OECC) . IEEE, 2023, pp. 1–3.\n[15] Z. Li, H. Pan, H. Chen, A. Beling, and J. C. Campbell,\n“High-saturation-current modified uni-traveling-carrier photodi-7\node with cliff layer,” IEEE Journal of Quantum Electronics ,\nvol. 46, no. 5, pp. 626–632, 2010.\n[16] E. Chao, B. Xiong, C. Sun, Z. Hao, J. Wang, L. Wang, Y . Han,\nH. Li, J. Yu, and Y . Luo, “D-band MUTC photodiodes with\nflat frequency response,” IEEE Journal of Selected Topics in\nQuantum Electronics , vol. 28, no. 2: Optical Detectors, pp. 1–\n8, 2021.\n[17] B. Tossoun, J. Morgan, and A. Beling, “Ultra-low capacitance,\nhigh-speed integrated waveguide photodiodes on InP,” in Inte-\ngrated Photonics Research, Silicon and Nanophotonics . Optica\nPublishing Group, 2019, pp. IT3A–6.\n[18] M. Grzeslo, S. D ¨ulme, S. Clochiatti, T. Neerfeld, T. Haddad,\nP. Lu, J. Tebart, S. Makhlouf, C. Biurrun-Quel, J. L. F.\nEst´evez et al. , “High saturation photocurrent THz waveguide-\ntype MUTC-photodiodes reaching mW output power within the\nWR3. 4 band,” Optics Express , vol. 31, no. 4, pp. 6484–6498,\n2023.\n[19] G. Ghione and M. Goano, “Revisiting the partial-capacitance\napproach to the analysis of coplanar transmission lines on mul-\ntilayered substrates,” IEEE Transactions on microwave theory\nand techniques , vol. 51, no. 9, pp. 2007–2014, 2003.\n[20] T. Tanigawa, T. Onishi, S. Nagai, and T. Ueda, “12.5-Gbps\noperation of 850-nm vertical-cavity surface-emitting lasers with\nreduced parasitic capacitance by BCB planarization technique,”\nIEEE journal of quantum electronics , vol. 42, no. 8, pp. 785–\n790, 2006.\n[21] Y . Chen, Z. Xie, J. Huang, Z. Deng, and B. Chen, “High-\nspeed uni-traveling carrier photodiode for 2 µm wavelength\napplication,” Optica , vol. 6, no. 7, pp. 884–889, 2019.\n[22] N. Shimizu, N. Watanabe, T. Furuta, and T. Ishibashi, “Im-\nproved response of uni-traveling-carrier photodiodes by carrier\ninjection,” Japanese journal of applied physics , vol. 37, no. 3S,\np. 1424, 1998.\n[23] C. Renaud, M. Robertson, D. Rogers, R. Firth, P. Cannard,\nR. Moore, and A. Seeds, “A high responsivity, broadband\nwaveguide uni-travelling carrier photodiode,” in Millimeter-\nWave and Terahertz Photonics , vol. 6194. SPIE, 2006, pp.\n87–94.\n[24] C. Renaud, D. Moodie, M. Robertson, and A. Seeds, “High\noutput power at 110 GHz with a waveguide uni-travelling carrier\nphotodiode.” IEEE, 2007.\n[25] E. Rouvalis, C. C. Renaud, D. G. Moodie, M. J. Robertson, and\nA. J. Seeds, “Continuous wave terahertz generation from ultra-\nfast InP-based photodiodes,” IEEE Transactions on microwave\ntheory and techniques , vol. 60, no. 3, pp. 509–517, 2012.\n[26] M. Anagnosti, C. Caillaud, F. Blache, F. Jorge, P. Angelini, J.-F.\nParet, and M. Achouche, “Optimized high speed UTC photo-\ndiode for 100 Gbit/s applications,” IEEE Journal of Selected\nTopics in Quantum Electronics , vol. 20, no. 6, pp. 29–35, 2014.\n[27] V . Rymanov, A. St ¨ohr, S. D ¨ulme, and T. Tekin, “Triple transit\nregion photodiodes (TTR-PDs) providing high millimeter wave\noutput power,” Optics Express , vol. 22, no. 7, pp. 7550–7558,\n2014.\n[28] Q. Li, K. Sun, K. Li, Q. Yu, P. Runge, W. Ebert, A. Beling,\nand J. C. Campbell, “High-power evanescently coupled waveg-\nuide MUTC photodiode with >105-GHz bandwidth,” Journal of\nLightwave Technology , vol. 35, no. 21, pp. 4752–4757, 2017.\n[29] P. Runge, G. Zhou, F. Ganzer, S. Mutschall, and A. Seeger,\n“Waveguide integrated InP-based photodetector for 100Gbaud\napplications operating at wavelengths of 1310nm and 1550nm,”\nin2015 European Conference on Optical Communication\n(ECOC) . IEEE, 2015, pp. 1–3.\n[30] K. Kato, A. Kozen, Y . Muramoto, Y . Itaya, T. Nagatsuma, and\nM. Yaita, “110-GHz, 50%-efficiency mushroom-mesa waveg-\nuide pin photodiode for a 1.55- µm wavelength,” IEEE Photon-\nics Technology Letters , vol. 6, no. 6, pp. 719–721, 1994."    },    {        "title": "2402.07493v1.Representations_of_the__su_1_1___current_algebra_and_probabilistic_perspectives.pdf",        "content": "arXiv:2402.07493v1  [math.PR]  12 Feb 2024REPRESENTATIONS OF THE su(1,1)CURRENT ALGEBRA\nAND PROBABILISTIC PERSPECTIVES\nSIMONE FLOREANI, SABINE JANSEN, AND STEFAN WAGNER\nAbstract. We construct three representations of the su(1,1) current algebra:\nin extended Fock space, with Gamma random measures, and with negative bi-\nnomial (Pascal) point processes. For the second and third re presentations, the\nloweringandneutral operatorsaregenerators ofmeasure-v alued branchingpro-\ncesses (Dawson-Watanabe superprocesses) and spatial birt h-death processes.\nThe vacuum is the constant function 1 and iterated applicati on of raising op-\nerators yields Laguerre and Meixner polynomials. In additi on, we prove a\nBaker-Campbell-Hausdorff formula and give an explicit form ula for the action\nof unitaries exp( k+(ξ)−k−(ξ))exp(2i k0(θ)) on exponential vectors. We ex-\nplain how the representations fit in with a general scheme pro posed by Araki\nand with representations of the SL(2,R) current group with Vershik, Gelfand\nand Graev’s multiplicative measure.\nMSC2020: 81R10; 33C45; 60K35; 60H40\nKeywords : currentalgebras; infinite-dimensionalLiealgebras; ort hogonal poly-\nnomials; Gamma random measure; negative binomial (Pascal) point processes;\nMarkov processes\n1.Introduction\nThe generators for many interacting particle systems can be writt en with cre-\nation or annihilation operators, or raising and lowering operators. T his holds true\nnot only for bosons, fermions, or spin chains in quantum mechanics, but also for\nclassical systems that evolve according to some continuous-time M arkov process;\nthis helps uncover useful hidden symmetries [50]. Sometimes the ge nerators of two\ndifferent Markov semigroup ( Pt= exp(tL),t≥0) have the sameformal expression\nwith raising and lowering operators; the difference comes from differ ent representa-\ntions of the raising and lowering operators and their commutation re lations. These\nobservations lead to the algebraic approach to duality [15, 23, 28, 55].\nA concrete example is the following model for energy transport, se e Giardin` a,\nKurchan, Redig and Vafayi [23] and references therein. It is close ly related to a\nmodelbyKipnis, MarchioroandPresutti[31]. Thesimplestversionof theBrownian\nmomentum process on a finite 1D lattice {1,2,...,ℓ}is a continuous-time Markov\nprocess with state space Rℓand formal generator\nL=/summationdisplay\ni,j/parenleftBig\npi∂\n∂pj−pj∂\n∂pi/parenrightBig2\n.\nThe sum is over nearest-neighbor pairs {i,j}, say with periodic boundary condi-\ntions. (More general versions allow for open systems interacting w ith reservoirs at\nthe boundaries and for vector-valued momenta.) The interpretat ion is that at each\nsitei, sits a particle with momentum pi; two neighboring sites may exchange mo-\nmentum but the total kinetic energy1\n2/summationtext\nip2\niis conserved. The symmetric inclusion\nDate: 12 February 2024.\n12 S. FLOREANI, S. JANSEN, AND S. WAGNER\nprocessinstead describes particles hopping on the 1D lattice. The state spa ce isNℓ\n0\nand the formal generator is\n/parenleftbig˜Lf/parenrightbig\n(n) =/summationdisplay\ni,jni/parenleftbig\nnj+1\n2/parenrightbig/parenleftBig\nf(n1,...,n i−1,...,n j+1,...,n ℓ)−f(n1,...,n ℓ)/parenrightBig\n.\nBothLand˜Lcan be written as\n/summationdisplay\ni,j/parenleftBig\nk+\nik−\nj+k+\njk−\ni−2k0\nik0\nj+1\n8/parenrightBig\n(1)\nwith raising, lowering and neutral operators k±\ni,k0\njthat satisfy the su(1,1) com-\nmutation relations,\n[k−\ni,k+\ni] = 2k0\ni,[k0\ni,k±\ni] =±k±\ni.\nOperatorsassociated with different lattice sites commute. For the Brownian energy\nprocess,\nk+\ni=1\n2p2\ni, k−\ni=1\n2∂2\n∂p2\ni, k0\nif=1\n4/parenleftBig∂\n∂pipif+pi∂\n∂pif/parenrightBig\nFor the symmetric inclusion process,\nk+\nif(n) =nif(n1,...,n i−1,ni−1,ni+1,...,n ℓ),\nk−\nif(n) =/parenleftBig1\n2+ni/parenrightBig\nf(n1,...,n i−1,ni+1,ni+1,...,n ℓ),\nk0\nif(n) =/parenleftBig\nni+1\n4/parenrightBig\nf(n).\nNotice that the lowering operator k−\niannihilates the indicator /BD{n=0}that there is\nnoparticleatall. Thecommonalgebraicformula(1)helpsexplaindualitie sbetween\nthe two processes [23]. In turn, duality helps investigate transpor t phenomena\n[22, 31].\nHow can we extend the algebraic formalism to systems on Rd? Infinite-dimen-\nsional Lie algebras and current algebras provide the proper algebr aic framework;\nconcretely, we look for representations of the su(1,1) current algebra. Those key-\nwords may suggest a daunting background from quantum field theo ry, algebra or\ngeometry [20, 29], but the definition we shall adopt (Definition 2.1), lo osely follow-\ningAraki[5] and AccardiandBoukas[1]is asimple variantofthe familia rcanonical\ncommutation relations for bosons. In a previous article [19], we have already in-\ntroduced a family of operators that form a representation of the su(1,1) current\nalgebra, and we have showed how to deduce intertwining relations fo r consistent\nMarkov processes. Here we continue the analysis and place a stron ger emphasis on\nthe algebraic structure.\nWe give three representations of the su(1,1) current algebra. The first represen-\ntation is similar to the representation in Fock space of the canonical commutation\nrelations from quantum mechanics (Theorem 3.2). The other two re presentations\nareclosertoGaussianmodelsfortheEuclideanfreefields, inwhichHe rmite polyno-\nmials play a distinguished role [24]. The Hilbert spaces are L2-spaces with respect\nto some probability measure, the vacuum is the constant function 1 , and iterated\napplication of raising operators to the vacuum gives rise to orthogo nal polynomials.\nGaussian fields are replaced with Gamma random measures and negat ive binomial\npoint processes (also called Pascal point processes). Hermite poly nomials are re-\nplaced with Laguerre and Meixner polynomials (Theorem 3.7 and 3.9).\nIn addition, we prove an infinite-dimensional version of the Baker-C ampbell-\nHausdorff formula (Theorem 3.4) and we make explicit the action of a f amiliy\nof unitaries, analogous to the Weyl operators exp(i( c(f) +c†(f)) from quantum\nfield theory, on exponential vectors (Theorem 3.3). The theorem should yield aREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 3\nrepresentation of the current group for the universal cover of SU(1,1), but we\nrefrain from making this explicit.\nTo the best of our knowledge, the three representations are new , however the\ningredients that we use arenot and there areclosely related result s in the literature.\nThe first representation is similar to Boukas’s representation of Fe insilver’s finite-\ndifference algebra [12] and ´Sniady’s representation of commutation relations for\nquadratic white noise [52]. All these algebras are related: Accardi a nd Skeide\n[4] linked the representations of the finite-difference algebra and t he renormalized\nsquare of white noise, which in turn is related to to representations of the current\nalgebra ofsl(2,R) [3]; finally, as is well known, the groups SL(2,R) andSU(1,1)\nare isomorphic and so are their Lie algebras.\nTheextendedFockspaceusedinourfirstrepresentationalsoapp earsinchaosde-\ncompositions in non-Gaussianwhite noise analysis, see Kondratiev, d a Silva, Streit,\nand Us [33], Berezansky and Mierzejewski [10], Kondratiev and Lytv ynov [34].\nGamma random measures and negative binomial (Pascal) point proce sses, together\nwith raising, lowering, and neutral operators and infinite-dimension al Meixner and\nLaguerre polynomials have been intensely analyzed, see Lytvynov [3 8] and refer-\nences therein. Lytvynov also made the connection with the algebra of the square of\nthe white noise [40]. For an algebraic stance on Pascal white noise, se e Barhoumi,\nOuerdiane and Riahi [8].\nA slightly different series of articles goes back to Gelfand, Graev and Vershik\n[21]. Araki [5] proposed a general scheme for finding factorizable r epresentations of\ncurrent groups in Fock spaces—but did not implement it for SU(1,1) orSL(2,R).\nVershik, Gelfand and Graev however did investigate those groups [ 60]. For a full\naccount and further references we refer the reader to Graev a nd Vershik [58]. We\nonly mention this: A key role is played by a kind of generalized infinite-dim ensional\nLebesgue measure, defined as a specific σ-finite measure that is absolutely contin-\nuous with respect to the law of a Gamma random measure. Yet anoth er repre-\nsentation involves instead “quasi-Poisson spaces” [59]. These rep resentations are\ndifferent from ours, but in Appendices A and B we explain how our repr esentations\nrelate to Araki’s scheme and to the infinite-dimensional Lebesgue me asure.\nAll of the above-mentioned references have a strongly analytic fla vor to them.\nThis makes them hard to access for readers who may not wish to lear n about\ndistribution theory, nuclear spaces, rigged Hilbert spaces, or whit e noise analysis.\nTherefore we have strived to make the setup as simple as possible. W e keep topo-\nlogical assumptions to a minimum and we allow for non-smooth test fun ctions and\nreference measures with atoms.\nAnother distinctive feature of our presentation is the emphasis on probabilis-\ntic interpretations of operators. It turns out that the lowering a nd neutral op-\nerators in our second and third representations generate contin uous-time Markov\nprocesses that are of considerable intrinsic interest: measure-v alued branching pro-\ncesses (Dawson-Watanabe superprocesses) and spatial birth- death processes. This\nobservation, to the best of our knowledge, is new, not only for the current algebra\nbut also for the Lie algebra where the associated processes are diff usions (Laguerre\nand Bessel) and linear birth-death processes.\nThe article is organized as follows. First we recall some basics about t he group\nSU(1,1), define the kind of representations that we shall study, and pr ovide some\ncontext for our results (Section 2). Then we state our main result s on represen-\ntations of the su(1,1) current algebra (Section 3). The simpler representations of\nthesu(1,1) Lie algebra are given in Section 4. Finally we turn to the proofs of\nour main results (Section 5). In Appendices A and B we explain how our approach4 S. FLOREANI, S. JANSEN, AND S. WAGNER\nfits into Araki’s scheme [5] and to the representation of the SL(2,R) current group\nwith generalized Lebesgue measure sketched in Tsilevich, Vershik an d Yor [57].\n2.Preliminaries\nHere we gather some relevant background, most importantly, we d efine the rep-\nresentations of su(1,1) current algebras that we investigate in this article (Defini-\ntion 2.1). In addition, we provide some background that aids the und erstanding\nof our main results: basics on the group SU(1,1); representations of the canonical\ncommutation relations with Hermite and Charlier polynomials; how Gamm a and\nand negative binomial (Pascal) laws naturally appear; and why we may expect\nrelations with continuous-time Markov processes.\n2.1.The group SU(1,1).The group SU(1,1) consists of the complex 2 ×2 ma-\ntrices\nA=/parenleftbigga b\nba/parenrightbigg\n(2)\nwith|a|2−|b|2= 1. Every such matrix can be written as\nA(ξ,θ) = exp/parenleftbig\nξk+−ξk−)exp(2iθk0) (3)\nwhereξ∈C,θ∈R, and\nk+=/parenleftbigg0 i\n0 0/parenrightbigg\n, k−=/parenleftbigg0 0\ni 0/parenrightbigg\n, k0=/parenleftbigg1/2 0\n0−1/2/parenrightbigg\n.\nThe matrices satisfy the commutation relations\n[k−,k+] = 2k0,[k0,k±] =±k±. (4)\nThesu(1,1) algebra consists of the complex linear combinations of the matrice s\nk0,k+,k−. The exponential in (3) is easily computed, one has\nA(ξ,θ) =/parenleftBigg\ncosh|ξ|iξ\n|ξ|sinh|ξ|\n−iξ\n|ξ|sinh|ξ|cosh|ξ|/parenrightBigg/parenleftbiggeiθ0\n0 e−iθ/parenrightbigg\n. (5)\nBargmann [7] investigated representations of SU(1,1) in the context of the Lorentz\nand Poincar´ e groups from special relativity. The su(1,1) algebra also appears in\nquantum optics for pairs of photons: If aanda†are bosonic annihilation and\ncreation operators, then k−=1\n2a2,k+=1\n2(a†)2andk0=1\n2a†a+1\n4satisfy the\nsu(1,1) algebra [14, 16]. The operator exp(1\n2(ξa2−ξ(a†)2)) is the squeeze operator\n[51, Chapter 2.7].\n2.2.Current algebra. As is customary with bosonic creation and annihilation\noperators,weworkwithoperatorsindexedbyfunctions. Let( X,X)beameasurable\nspace andαbe aσ-finite reference measure on X; for example, Rdwith the Borel\nσ-algebra and the Lebesgue measure. We introduce an algebra of te st functions\nC=/braceleftbig\nϕ:X→C: bounded, measurable, α({x:ϕ(x)/\\e}atio\\slash= 0})<∞/bracerightbig\n.(6)\nThe following definition is modeled after a similar definition in Accardi and Boukas\n[2] for the algebra of the renormalized square of white noise. Araki [5] defines\nthesu(1,1) current algebra over Xas the set of su(1,1)-valued, bounded measur-\nable functions, equipped with the pointwise Lie algebra operations. O ur function-\nindexed operators k±(ϕ),k0(θ) represent the matrix-valued maps x/ma√sto→ϕ(x)k+,\nx/ma√sto→ϕ(x)k−,x/ma√sto→θ(x)k0withk±,k0the 2×2-matrices from Section 2.1.\nDefinition 2.1. A familyk±(ϕ),k0(ϕ),ϕ∈ Cof operators in a Hilbert space His\naFock representation of the su(1,1) current algebra withvacuum Ψ if:REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 5\n(i) The operators are defined on a common domain Dthat is dense in Hand\nthey map Dto itself, i.e., k#(ϕ)D ⊂ D.\n(ii)k+(ϕ)andk0(ϕ)are linear in ϕwhilek−(ϕ)is antilinear in ϕ.\n(iii) For all ϕ,θ∈ C,\n[k−(ϕ),k+(θ)] = 2k0(ϕθ)\n[k0(ϕ),k+(θ)] =k+(ϕθ),[k0(ϕ),k−(θ)] =−k−(ϕθ)(7)\nand[k#(ϕ),k#(θ)] = 0for#∈ {0,+,−}.\n(iv) For all f,g∈ Dandϕ∈ C,\n/a\\}bracketle{tf,k0(ϕ)g/a\\}bracketri}ht=/a\\}bracketle{tk0(ϕ)f,g/a\\}bracketri}ht,/a\\}bracketle{tf,k+(ϕ)g/a\\}bracketri}ht=/a\\}bracketle{tk−(ϕ)f,g/a\\}bracketri}ht. (8)\n(v)Ψ∈ D,||Ψ||= 1andk−(ϕ)Ψ = 0for allϕ∈ C.\n(vi)Ψis cyclic: the linear combinations of Ψandk+(ϕ1)···k+(ϕn)Ψwheren∈N\nandϕ1,...,ϕ n∈ C, are dense in F.\nAll our representations are factorizable —when test functions have disjoint sup-\nport, the associated operators commute. In addition, our three representations\nfrom Section 3 satisfy\nk0(ϕ)Ψ =/parenleftBig1\n2/integraldisplay\nϕdα/parenrightBig\nΨ\nfor allϕ∈ C. Thus, the measure1\n2αplays a role analogous to the Bargmann index\nfor the discrete series, minimal weight representations of SU(1,1) (see Section 4).\nWe remark that the three representations are unitarily equivalent . More pre-\ncisely, the isomorphism from [33, Equation (4.9)] associated with the chaos decom-\nposition of functionals of a Gamma random measure switches betwee n the repre-\nsentation in extended Fock spaces and the representation outline d in Section 3.3.\nSimilarly, the isomorphism from [38, Corollary 5.3], which relates to the c haos\ndecomposition of functionals associated with a Pascal point proces s, switches be-\ntweenthe representationin extended Fock spacesand the repre sentationintroduced\nin Section 3.4.\n2.3.Canonical commutation relations. The raising and lowering operators\nk+(f),k−(f) are similar to bosonic creation and annihilation operators c†(f),c(g)\nwith the canonical commutation relations [ c(f),c†(g)] =/a\\}bracketle{tf,g/a\\}bracketri}ht. The best known\nrepresentation, in quantum many-body theory, is in the Fock spac e of sequences of\nsymmetric functions ( fn)n∈N0. For better comparison with our results, we recall\ntwo additional representations. To keep things concrete in this se ctionXisRd\nequipped with its Borel σ-algebra and with the Lebesgue measure λ. The scalar\nproduct in L2(X,λ) is/a\\}bracketle{tf,g/a\\}bracketri}ht=/integraltextfgdλ.\n2.3.1.Gaussian white noise, Hermite polynomials. The setting of the first addi-\ntional representation is similar to that of the Euclidean quantum fre e field, we\nrefer to Glimm and Jaffe [24] for background and proofs. By the Boc hner-Minlos\ntheorem, there is a unique probability measure µonS′\nR, the space of real-valued\ntempered distributions on Rd, with/integraldisplay\neiϕ(f)µ(dϕ) = exp/parenleftBig\n−1\n2/integraldisplay\nf(x)2dx/parenrightBig\n.\nfor all real-valued Schwartz functions f. For alln∈Nand real-valued Schwartz\nfunctionsf1,...,f n, the vector( ϕ(f1),...,ϕ(fn)) is a Gaussianrandomvectorwith\ncovariance matrix C= (/integraltext\nfigjdλ)i,j=1,...,n. In particular\n•Iff1,...,f nhave disjoint support then ϕ(f1),...,ϕ(fn) are independent.\n•Each variable ϕ(f) has a normal distribution with mean zero and variance/integraltext\nf2dλ.6 S. FLOREANI, S. JANSEN, AND S. WAGNER\nThe Hilbert space is L2(S′\nR(Rd),µ), the space of complex-valued square-integrable\nfunctions. Let Dbe the set of functions F:S′\nR→Cof the form\nF(ϕ) =Pn/parenleftbig\nϕ(f1),...,ϕ(fn)/parenrightbig\nwithn∈N,f1,...,f nSchwartz functions, and Pn:Cn→Ca multivariate poly-\nnomial. The annihilation operators are\nc(f)F(ϕ) =/integraldisplay\nf(x)δ\nδϕ(x)F(ϕ)dx=n/summationdisplay\nj=1/a\\}bracketle{tf,fj/a\\}bracketri}ht∂Pn\n∂xj/parenleftbig\nϕ(f1),...,ϕ(fn)/parenrightbig\nand the creation operators are\nc†(g)F(ϕ) =ϕ(g)F(ϕ)−/integraldisplay\ng(x)δ\nδϕ(x)F(ϕ)dx\n=ϕ(g)F(ϕ)−n/summationdisplay\nj=1/a\\}bracketle{tfj,g/a\\}bracketri}ht∂Pn\n∂xj/parenleftbig\nϕ(f1),...,ϕ(fn)/parenrightbig\n.\nThe vacuum is the constant function /BD. Iterated application of creation operators\nto the vacuum gives rise to Hermite polynomials [24].\n2.3.2.Poisson point process. Charlier polynomials. The second additional repre-\nsentation comes from Poisson white noise analysis (Kondratiev, da S ilva, Streit,\nUs [33] and references therein). It is similar to the previous repres entation except\nthat the Gaussian measure µis replaced by the law of a Poisson point process, sup-\nported on a simpler space of point configurations—there is no need f or tempered\ndistributions.\nLetNbe the set of finite or countable sums of Dirac measures, equipped w ith\ntheσ-algebra generated by the counting variables η/ma√sto→η(B),B∈ X. The law\nπλof thePoisson point process with intensity measure λis the uniquely defined\nprobability measure on Nunder which\n•Each counting variable η/ma√sto→η(B),B∈ X, has a Poisson distribution with\nparameterλ(B)\nπλ/parenleftbig\n{η:η(B) =n}/parenrightbig\n= e−λ(B)λ(B)n\nn!.\n•For all disjoint B1,...,B n∈ X, the associated counting variables are inde-\npendent.\nMathematical physicists should think of πλas the grand-canonical Gibbs measure\nfor a classical ideal gas in which the expected number of particles in a regionBis\nλ(B). Whenλis Lebesgue measure, the ideal gas is homogeneous with density 1.\nWe work in the Hilbert space L2(N,πλ). Abbreviate η(f) =/integraltextfdηand letD\nbe the space of polynomials in the variables η(f) =/integraltext\nfdη. The annihilation and\ncreation operators are [33]\nc(f)F(η) =/integraldisplay\nf(x)/parenleftbig\nF(η+δx)−F(η)/parenrightbig\nλ(dx),\nc†(f)F(η) =/integraldisplay\nf(x)F(η−δx)η(dx)−F(η)/integraldisplay\nfdλ.\nThe vacuum Ψ is again the constant function 1, and iterated applicat ion of cre-\nation operators for indicators fgives rise to Charlier polynomials in the occupation\nnumbersη(B),B⊂X. For example, for a finite-volume set B⊂Rd,\n/parenleftbig\nc†( /BDB)/parenrightbign1(η) =Cn/parenleftbig\nη(B);λ(B)/parenrightbig\nwhere Cn(x;β) =xn+ lower order terms is a univariate Charlier polynomial, or-\nthogonal with respect to the measure ν({x}) = e−ββx/x! onN0[32].REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 7\n2.4.From Gauss and Poisson to Gamma and Pascal laws. Our second and\nthird representations of the su(1,1) current algebra rely on Gamma and Pascal\nlaws. Here we briefly motivate how these distributions arise, building o n results\nfrom quantum stochastic calculus. Let a†andabe the creation and annihilation\noperators for the harmonic oscillator in some Hilbert space Hwith vacuum ψ0, for\nexample,a=d\ndxanda†=x−d\ndxinL2(R,exp(−x2/2)dx), see the next subsec-\ntion. Write /a\\}bracketle{tf(A)/a\\}bracketri}htfor the vacuum expectation values /a\\}bracketle{tψ0,f(A)ψ0/a\\}bracketri}htof polynomial\nfunctionsfof an operator A. The following is known:\n(i) The vacuum expectations of a+a†are Gaussian:\n/a\\}bracketle{tf(a+a†)/a\\}bracketri}ht=1√\n2π/integraldisplay∞\n−∞f(z)exp(−z2/2)dz\nfor every polynomial f:R→R(just think of the harmonic oscillator).\n(ii)a†+a+ca†a, wherec∈R\\{0}, is associated with a Poisson law of parameter\nλ= 1/c2:\n/a\\}bracketle{tf(a+a†+ca†a)/a\\}bracketri}ht= e−λ∞/summationdisplay\nn=0λn\nn!f/parenleftbig\nc−1(n−λ)/parenrightbig\n.\nSee Hudson and Parthasarathy [26] and Meyer [41, Chapter IV.2( 5)]. A re-\nlated result is given by Ito and Kubo [27, Proposition 5.2].\n(iii)1\n2(a+a†)2goes with a Gamma distribution:\n/a\\}bracketle{tf/parenleftBig1\n2(a+a†)2/parenrightBig\n/a\\}bracketri}ht=1√π/integraldisplay∞\n0f(x)x−1/2e−xdx.\nThis follows from (i)—think x=z2/2.\n(iv)a2+(a†)2+βa†awithβ >2 goes with negative binomial (Pascal) distribution\nwith parameters α= 1/2andp∈(0,1) the unique solution of√p+√p−1=β:\nthere exist non-zero reals c1,c2such that\n/a\\}bracketle{tf/parenleftbig\na2+(a†)2+βa†a/parenrightbig\n/a\\}bracketri}ht= (1−p)α∞/summationdisplay\nn=0(α)n\nn!pnf/parenleftbig\nc1n+c2/parenrightbig\nwhere (α)0= 1, (α)n=α(α+1)···(α+n−1). See Accardi, Franz and Skeide\n[3] and the survey by Accardi and Boukas [1, Section 17]. A similar re sult\nholds forβ <−2.\n(v)a2+(a†)2+βa†awithβ∈(−2,2) is associated with a probability measure\nonR, the orthogonality measure for the Meixner-Pollaczek polynomials [ 3].\nRemembering that k+=1\n2(a†)2,k−=1\n2a2,k0=1\n2a†a+1\n4satisfy the commutation\nrelations of the su(1,1) algebra, we see that the operators\nX=k++k−+2k0, Y=k++k−+(√p−1+√p)k0(9)\nwithp∈(0,1), are naturally associated with Gamma and negative binomial (Pas-\ncal) laws. Our three representations give priority to different oper ators: roughly,\nwe diagonalize k0,XorY.\nCombinations of the type (9) are also motivated by the theory of or thogonal\npolynomials and Jacobi matrices, see Berezansky [9], Lytvynov [39 ] and references\ntherein. Consider for example the Laguerre polynomials pn(x) =L(α−1)\nn(x), or-\nthogonal with respect to the measure xα−1exp(−x)dxonR+and with leading\ncoefficient 1. They satisfy the three-term recurrence relation [3 2, Eq. (9.12.4)]\nxpn(x) =pn+1(x)+(2n+α)pn(x)+n(n+α−1)pn−1(x).8 S. FLOREANI, S. JANSEN, AND S. WAGNER\nEvery polynomial f(x) is a linear combination f(x) =/summationtext∞\nn=01\nn!g(n)pn(x), with at\nmost finitely many non-zero g(n)’s. When multiplied with x, it is\nxf(x) =∞/summationdisplay\nn=01\nn!pn(x)/parenleftBig\nng(n−1)+(2n+α)g(n)+(α+n)g(n+1)/parenrightBig\n.\nThus multiplication by xcorresponds to a linear transformation of the sequence of\ncoefficients, given by an infinite band-diagonal matrix, the Jacobi m atrix. Let us\ndefineJ+g(n) =ng(n−1),J0g(n) = (n+α/2)g(n),J−g(n) = (α+n)g(n+ 1),\nthen multiplication by xcorresponds to the action of J++2J0+J−in the space\nof sequences.\nFor infinite-dimensional orthogonal polynomials, the linear operato rsJ±andJ0\nare replaced by families of operators J±(ϕ) andJ0(ϕ), indexed by test functions\nϕ. The focus in this context is on the commutative family of operators J+(ϕ) +\nλJ0(ϕ) +J−(ϕ) (Jacobi fields ) and not so much on the full set of commutation\nrelations (see, however, [40]).\n2.5.Continuous-time Markov processes. Finally we sketch how continuous-\ntime Markov processes naturally enter the scene, using the Gauss ian model for the\nharmonic oscillator [41, Chapter III.2(2)]. Let µ=N(0,1) be the standard normal\ndistribution on R. The norm in L2(R,µ) is\n||f||2=1√\n2π/integraldisplay∞\n−∞|f(x)|2e−x2/2dx.\nThe densely defined operators a=d\ndx,a†=x−d\ndxsatisfy [a,a†] = id. Now, the\npoint we wish to highlight is this: Both the annihilation operator aandL=−/hatwiden,\nwith/hatwidenthe number operator\n/hatwiden=a†a=−/parenleftBigd2\ndx2−xd\ndx/parenrightBig\nare generators of continuous-time Markov processes. The oper atorL=−/hatwidengener-\nates an Ornstein-Uhlenbeck process for which the Gaussian measu reµis reversible.\nThe annihilation operator generates deterministic motion to the righ t,Xt=X0+t.\nThe creation operator corresponds to time-evolution of densities . Indeed exp( ta†)\nmapsf0toft(x) =f0(x−t)exp(tx). Thus, ifX0has probability density function\nρ0(x) =f0(x)exp(−x2/2)/√\n2π, thenXt=X0+thas probability density function\nρt(x) =ρ0(x−t) =ft(x)exp(−x2/2)/√\n2π.\nWe will encounter a similar structure for the second and third repre sentations\nof the current algebra (and of the Lie algebra, see Section 4): The lowering and\nthe neutral operator are related to continuous-time Markov pro cesses, concretely\nspatial birth-death processes and measure-valued branching pr ocesses.\n3.Main results\n3.1.Representation in extended Fock space. Let (X,X) be a Borel space, for\nexample, a complete metric separable space with its Borel σ-algebra. Let αbe a\nσ-finite measure on X.\n3.1.1.Extended Fock space. Define measures λnonXn, equipped with the product\nσ-algebra Xn, as follows. Let Snbe the set of permutations of {1,...,n}. Every\npermutation is a product of cycles with disjoint supports; the numb er of cycles is\ndenoted |σ|. Givenfn:Xn→Ca new function fσ\nn:X|σ|→Cis obtained by\nidentifying variables that belong to the same cycle. For example, for n= 3 andREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 9\nσ= (1 3)(2), we get fσ\n3(x,y) =f3(x,y,x). The measure λnonXnis the uniquely\ndefined measure for which/integraldisplay\nfndλn=/summationdisplay\nσ∈Sn/integraldisplay\nfσ\nndα|σ|\nfor all measurable fn:Xn→R+. For example, we have λ1=αand\n/integraldisplay\nf2dλ2=/integraldisplay\nf2(x,y)α(dx)α(dy) +/integraldisplay\nf(x,x)α(dx).\nLetL2(λn) =L2(Xn,Xn,λn) be the space of complex-valued, square-integrable\nfunctionsfnandL2\ns(λn) be the subspace of symmetric functions, i.e., for ev-\nery permutation σ∈Sn,fn(xσ(1),...,x σ(n)) =fn(x1,...,x n) forλn-almost all\n(x1,...,x n). Theinteracting orextended Fock space F[33] consists ofthe sequences\n(fn)n∈N0wheref0∈C,fn∈L2\ns(λn), and\n|f0|2+∞/summationdisplay\nn=11\nn!/integraldisplay\n|fn|2dλn<∞.\nThe extended Fock space is equipped with the scalar product\n/a\\}bracketle{t(fn)n∈N0,(gn)n∈N0/a\\}bracketri}ht=f0g0+∞/summationdisplay\nn=11\nn!/integraldisplay\nfngndλn.\nThevacuumis the sequence Ψ = (1 ,0,0,...). The symmetrized tensor product of\nfunctionsf1,...,f n:X→Cis\n/parenleftbig\nf1⊗s···⊗sfn/parenrightbig\n(x1,...,x n) =1\nn!/summationdisplay\nσ∈Snf1(xσ(1))···fn(xσ(n)).\nWe shall often identify a function Fn:Xn→Cwith the sequence (0 ,...,F n,0,...)\ninF. With this identification in mind, the following lemma is useful. Remember\nthe algebra Cof bounded functions supported on sets of finite measure.\nLemma 3.1. The space Dof finite linear combinations of the vacuum Ψand\nsymmetrized tensor products f1⊗s···⊗sfnof functions fi∈ Cis dense in F.\nThe lemma is proven in Section 5.1.\n3.1.2.Exponential vectors. Foru∈ Cwith sup |u|<1, we define a vector Eu∈F\nby\n(Eu)0= 1,(Eu)n(x1,...,x n) =n/productdisplay\ni=1u(xi).\nIn terms of the raising operators introduced below, Eu= exp(k+(u))Ψ, whence the\nnameexponential vector [41, Chapter IV.1]. (Mathematical physicists may be more\nfamiliar with the notion of coherent states ; for bosons, these are just the normalized\nexponential states, but for SU(1,1) the relation is more subtle, see the remark after\nTheorem 3.4 below.) We note\n/a\\}bracketle{tEu,Ev/a\\}bracketri}ht= exp/parenleftBig\n−/integraldisplay\nlog/parenleftbig\n1−u(x)v(x)/parenrightbig\nα(dx)/parenrightBig\n(10)\nwhere−log(1−z) =/summationtext∞\nj=1zj/jfor|z|<1. Eq.(10)alreadyappearedinBoukas[12,\n13], however Boukas’s Hilbert space is more abstract than our inter acting Fock\nspace.\nThe linear combinations of exponential vectors are dense in F, moreover if\nu1,...,u nare linearly independent in L2(X,α), then the associated exponential\nvectors are linearly independent in F. These properties are proven in the same10 S. FLOREANI, S. JANSEN, AND S. WAGNER\nway as their counterparts in the usual bosonic Fock space, we ref er to Meyer [41,\nChapter IV.1.3].\n3.1.3.Representation of the su(1,1)current algebra. Defineraising, lowering and\nneutral operators k+(ϕ),k−(ϕ),k0(ϕ) :D →F, indexed by ϕ∈ C, as follows: for\nn∈N,\n/parenleftbig\nk+(ϕ)f/parenrightbig\nn(x1,...,x n) =n/summationdisplay\ni=1ϕ(xi)fn−1(x1,...,x i−1,xi+1,...,x n),\n/parenleftbig\nk−(ϕ)f/parenrightbig\nn(x1,...,x n) = (n+1)/integraldisplay\nϕ(y)fn+1(y,x1,...,x n)α(dy)\n+n/summationdisplay\ni=1ϕ(xi)fn+1(x1,...,x n,xi),\n/parenleftbig\nk0(ϕ)f/parenrightbig\nn(x1,...,x n) =/parenleftBign/summationdisplay\ni=1ϕ(xi)+1\n2/integraldisplay\nϕdα/parenrightBig\nfn(x1,...,x n).\nForn= 0 we use the same expressions, with/summationtext0\ni=1(···) = 0. In particular,\n(k+(ϕ)f)0= 0, (k−(ϕ)f)0=/integraltext\nϕf1dα, and (k0(ϕ)f)0= (1\n2/integraltext\nϕdα)f0.\nSimilar operators already appear in ´Sniady [52] and Lytvynov [39]. ´Sniady stud-\nies them in the context of the renormalized algebra of the square of white noise,\nLytvynov investigates infinite-dimensional orthogonal polynomials and relates Ja-\ncobi fields to the square of white noise in [40]. Neither of them explicit ly comments\non the relation with the su(1,1) current algebra.\nTheorem 3.2. The operators k±(ϕ),k0(ϕ) :D →Fform a Fock representation of\nthesu(1,1)current algebra in Fwith vacuum Ψ = (1,0,0,...).\nThe operators k#(ϕ) may be written as\nk+(ϕ) =a†(ϕ), k−(ϕ) =a(ϕ)+b(ϕ), k0(ϕ) =/hatwiden(ϕ)+1\n2/integraldisplay\nϕdα\nwhere\na†(ϕ)f1⊗s···⊗sfn= (n+1)ϕ⊗sf1⊗s···⊗sfn,\na(ϕ)f1⊗s···⊗sfn=1\nnn/summationdisplay\ni=1/a\\}bracketle{tϕ,fi/a\\}bracketri}htf1⊗s···/hatwidefi···⊗sfn,\nb(ϕ)f1⊗s···⊗sfn=2\nn/summationdisplay\n1≤i<j≤n/parenleftbig\nϕfifj/parenrightbig\n⊗sf1⊗s···/hatwidefi···/hatwidefj···⊗sfn,\n/hatwiden(ϕ)f1⊗s···⊗sfn=n/summationdisplay\ni=1f1⊗s···⊗s(ϕfi)⊗s···⊗sfn.\nHere, as usual, /hatwidefimeans omission of fi. The scalar product is /a\\}bracketle{tϕ,f/a\\}bracketri}ht=/integraltext\nϕfdα. In\na†(ϕ),a(ϕ) we recognize the bosonic creation and annihilation operators; /hatwiden(ϕ) is\na smeared bosonic number operator (think/integraltext\nϕ(x)a†\nxaxdx). (Factors of√n+1 or\n1/√noften included in explicit formulas [46, Chapter X.7] are missing becaus e the\nscalar product in our (extended) Fock space has the factor 1 /n!.) The operators\nsatisfy\n[a(f),a†(g)] =/a\\}bracketle{tf,g/a\\}bracketri}ht,[/hatwiden(f),a†(g)] =a†(fg),[/hatwiden(f),a(g)] =−a(fg).(11)REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 11\n3.2.Unitary operators, Baker-Campbell-Hausdorff formula. Forξ,θ∈ C\nwith real-valued θ, define unitary operators, analogous to (3), by\nU(ξ,θ) = exp/parenleftbig\nk+(ξ)−k−(ξ)/parenrightbig\nexp/parenleftbig\n2ik0(θ)/parenrightbig\n. (12)\nThe second exponential on the right is a multiplication operator, with multiplier\nexp(i/integraltext\nθdα+ 2i/summationtextn\nj=1θ(xj)). The first exponential is by definition exp(i A) with\nAthe self-adjoint closure of1\ni(k+(ξ)−k−(ξ)). The latter operator is essentially\nself-adjoint on D; the proof with Nelson’s analytic vector theorem is similar to [19,\nLemma 4.1] and Reed and Simon [46, Theorem X.41(a)] and therefore omitted.\nThe unitaries U(ξ,θ) act on exponential vectors in a fairly explicit way. For\nz:X→ {ζ∈C:|ζ|<1}, let\nzξ(x) =z(x)+ξ(x)\n|ξ(x)|tanh|ξ(x)|\n1+z(x)ξ(x)\n|ξ(x)|tanh|ξ(x)|(13)\nand\nCξ(z) = exp/parenleftBigg\n−/integraldisplay\nlog/parenleftBig\ncosh|ξ(x)|+z(x)ξ(x)\n|ξ(x)|sinh|ξ(x)|/parenrightBig\nα(dx)/parenrightBigg\n.(14)\nAs explained by us in [19, Section 5], the equations are related to the a ction of\nSU(1,1) on the open unit disk by M¨ obius transforms, and to representa tions of\nSU(1,1) given by Bargmann [7, Section 9].\nTheorem 3.3. Letξ:X→C,θ:X→R,z:X→Cbe bounded functions in C\nwithsup|z|<1. Then\nexp/parenleftbig\nk+(ξ)−k−(ξ)/parenrightbig\nEz=Cξ(z)Ezξ,\nexp/parenleftbig\n2ik0(θ)/parenrightbig\nEz= exp/parenleftBig\ni/integraldisplay\nθdα/parenrightBig\nEexp(2iθ)z.\nThe proof of the first formula is similar to Proposition 5.1 in [19] and the refore\nomitted. The second formula is straightforward, we leave the deta ils to the reader.\nThe 2×2 matrices in SU(1,1) satisfy the Baker-Campbell-Hausdorff formula\nexp/parenleftBig\nξk+−ξk−/parenrightBig\n= exp/parenleftBigξ\n|ξ|(tanh|ξ|)k+/parenrightBig/parenleftBig1\ncosh|ξ|/parenrightBig2k0\nexp/parenleftBig\n−ξ\n|ξ|(tanh|ξ|)k−/parenrightBig\n.\n(15)\nThe unitaries U(ξ,0) satisfy a similar equation.\nTheorem 3.4 (Baker-Campbell-Hausdorff formula) .For allξ∈ Cand allf∈ D,\nwe have\nexp/parenleftBig\nk+(ξ)−k−(ξ)/parenrightBig\nf= exp/parenleftbig\nk+(v)/parenrightbig\nexp/parenleftbig\nk0(w)/parenrightbig\nexp/parenleftbig\n−k−(v)/parenrightbig\nf,(16)\nwherev:= /BD{ξ/\\e}atio\\slash=0}ξ\n|ξ|tanh|ξ|andw:=−2logcosh |ξ|.\nOn the right side exp( k0(w)) is a multiplication operator and exp( ±k±(v)) are\ndefined by exponential series that converge on the relevant doma ins (Lemma 5.2).\nThe theorem is proven in Section 5.2.\nRemark. The analogueof the Perelomov coherent state [14, 44] is Cξ= exp/parenleftbig\nk+(ξ)−\nk−(ξ)/parenrightbig\nΨ. Theorem3.4yields Cξ=λξexp(k+(v))Ψwithλξ= exp(−/integraltext\nlog(cosh|ξ|)dα)\nandvas given in the theorem. Thus, the coherent state Cξdiffers from the ex-\nponential state Evby a normalization and an additional reparametrization from ξ\ntov.12 S. FLOREANI, S. JANSEN, AND S. WAGNER\nAraki [5] emphasizes the role of expectation functionals g/ma√sto→E(g) for current\ngroups. In light of this, we mention as a corollary an explicit expressio n for the\nvacuum expectations.\nCorollary 3.5 (Vacuum expectations) .For allξ∈ Cand all real-valued θ∈ C,\n/a\\}bracketle{tΨ,U(ξ,θ)Ψ/a\\}bracketri}ht= exp/parenleftBig\ni/integraldisplay\nθdα−/integraldisplay/parenleftbig\nlogcosh|ξ|/parenrightbig\ndα/parenrightBig\n.\n3.3.Gamma random measure and Laguerre polynomials. Next comes a\nrepresentation in a probability space of random measures somewha t analogous to\nthe Gaussian model for the canonical commutation relations from S ection 2.3.1 as\nthe raising and lowering operators involve differential operators.\nLetMbe the space of s-finite measures on X(a measure is s-finite if it is a\ncountable sum of finite measures[36]). The space Mis equipped with the σ-algebra\ngenerated by the variables B/ma√sto→µ(B) withB∈ X. Given the σ-finite reference\nmeasureα, letραbe the uniquely defined probability measure on Msuch that:\n•For all disjoint B1,...,B n∈ X, the variables µ/ma√sto→µ(Bi),i= 1,...,n, are\nindependent.\n•For everyB∈ X, the variable µ/ma√sto→µ(B) has a Gamma distribution with\nshape parameter α(B) and scale parameter 1: if α(B)<∞,\nρα/parenleftbig\n{µ:µ(B)≤y}/parenrightbig\n=1\nΓ(α(B))/integraldisplayy\n0tα(B)−1e−tdt(y≥0).\nWhenα(B) = 0 or ∞, we ask that µ(B) = 0 or ∞,ρα-almost surely.\nThe probability measure ραis the law of a Gamma random measure . It is the\nlaw of a compound Poisson process [36, Chapter 15]: Let Π =/summationtext\niδ(Xi,Zi)be a\nPoisson point process on X×(0,∞) with intensity measure α(dx)z−1exp(−z)dz,\nthenξ=/summationtext\niZiδXihas lawρα[36, Example 15.6].\nWe work in the Hilbert space L2(M,ρα). LetDbe the space of polynomials in\nthe variables µ(ϕ) =/integraltext\nϕdµ, whereϕis in the algebra Cof test functions from (6).\nThusDconsists of the functions Fof the form\nF(µ) =Pn/parenleftbig\nµ(f1),...,µ(fn)/parenrightbig\n(17)\nwithn∈N,f1,...,f n∈ C, andPn:Cn→Ca multivariate polynomial with\ncomplex coefficients. Notice that if f∈ C, thenµ(|f|)<∞forρα-almost all\nµ∈ M, thusF(µ) is well-defined except possibly on ραnull sets on which we may\ndefine it to be zero.\nLemma 3.6. The space Dis dense in L2(M,ρα).\nThe lemma is a classical infinite-dimensional variant of a result due to R iesz\n[47]: For probability measures on R, uniqueness in the moment problem implies\ndensity of polynomials. For the reader’s convenience we give a self-c ontained proof\nin Section 5.3.\nFor a polynomial Fas in (17),µ∈Mandx∈X, set\nδF\nδµ(x)(µ) =d\ndtF(µ+tδx)/vextendsingle/vextendsingle/vextendsingle\nt=0=n/summationdisplay\nj=1fj(x)∂jPn/parenleftbig\nµ(f1),...,µ(fn)/parenrightbig\n,\nδ2F\nδµ(x)2(µ) =d2\ndt2F(µ+tδx)/vextendsingle/vextendsingle/vextendsingle\nt=0=n/summationdisplay\ni,j=1fi(x)fj(x)∂i∂jPn/parenleftbig\nµ(f1),...,µ(fn)/parenrightbigREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 13\nwhere∂jPnis the partial derivative of Pnwith respect to the j-th variable. Finally,\nwe define lowering, raising and neutral operators K#(ϕ) :D→D, whereϕ∈ C, by\nK−(ϕ)F(µ) =/integraldisplay\nϕ(x)δ2F\nδµ(x)2(µ)µ(dx) +/integraldisplay\nϕ(x)δF\nδµ(x)(µ)α(dx)\nK+(ϕ)F(µ) =/integraldisplay\nϕ(x)δ2F\nδµ(x)2(µ)µ(dx) +/integraldisplay\nϕ(x)δF\nδµ(x)(µ)(α−2µ)(dx)\n+(µ(ϕ)−α(ϕ))F(µ)\nK0(ϕ)F(µ) =−/integraldisplay\nϕ(x)δ2F\nδµ(x)2(µ)µ(dx)−/integraldisplay\nϕ(x)δF\nδµ(x)(µ)(α−µ)(dx)\n+1\n2α(ϕ)F(µ).\nRemark. For every real-valued test function ϕ∈ C,\nK+(ϕ)+K−(ϕ)+2K0(ϕ) =µ(ϕ). (18)\nThe right side is the multiplication operator with µ(ϕ). The above relation is in\nagreement with the univariate example (iii) from Section 2.4. It should remind the\nreader of the definition of Segal’s field operator Φ S(f) as a sum of bosonic creation\nand annihilation operators [46, Chapter X.7].\nForβ >0, let L(β−1)\nn(x),n∈N0, be monic Laguerre polynomials, i.e., the\npolynomials of degree nwith leading coefficient 1 that are orthogonal with respect\nto the measure xβ−1exp(−x)dxonR+.\nTheorem 3.7. The operators K±(ϕ),K0(ϕ) :D→D, withϕ∈ C, form a Fock\nrepresentation in L2(M,ρα)of thesu(1,1)current algebra with vacuum Ψ = /BD, the\nconstant function 1. Moreover,\nK+( /BDB1)n1···K+( /BDBℓ)nℓ/BD(µ) =ℓ/productdisplay\ni=1L(α(Bi)−1)\nni/parenleftbig\nµ(Bi)/parenrightbig\n(19)\nfor allℓ∈N, disjointB1,...,B ℓ∈ Xwith0< α(Bi)<∞,n1,...,n ℓ∈Nand\nρα-almost all µ∈M.\nThe theorem is proven in Section 5.3.\nWeremarkthatfornon-negative ϕ∈ C, theoperatorshaveprobabilisticinterpre-\ntations: The lowering operator K−(ϕ) can be interpreted as the formal generator\nof a measure-valued Markov process that is a variant of superpro cesses studied by\nStannat [54]. The process does not leave the distribution ραinvariant. The raising\noperatorK+(ϕ) describes the time-evolution of densities: if a process with genera -\ntorK−(ϕ)hasaninitiallaw π0=F0ραthatisabsolutelycontinuouswithrespectto\nραwith Radon-Nikodym derivative F0, then the law at time thas Radon-Nikodym\nderivative exp( tK+(ϕ))F0.\nMoreover, the process with formal generator −K0(ϕ) +1\n2α(ϕ) is known as\nDawson-Watanabe measure-valuedcontinuous-statebranch ing process , see[18, Equa-\ntion (1.1)] or [53, Equation (38)]. If ϕis constant to one and αis a finite measure,\nthe generator has spectrum −N0. Consequently, the corresponding Markov semi-\ngroup behaves analogously to the Ornstein-Uhlenbeck semigroup in the Gaussian\ncase [43, Equation (1.67)] or in the Poisson case [35, Proposition 4].\n3.4.Pascal point process and Meixner polynomials. Weconcludewitharep-\nresentation in a probability space of point configurations analogous to the Poisson-\nCharlier representation for the canonical commutation relations f rom Section 2.3.2.\nLetN=N(X) be the space of finite or countable sums of Dirac measures\n(including the zero measure) on the Borel space ( X,X). The space Nis equipped14 S. FLOREANI, S. JANSEN, AND S. WAGNER\nwith theσ-algebra Ngenerated by the counting variables B/ma√sto→η(B),B∈ X. For\na>0 the rising factorial (Pochhammer symbol) is\n(a)0= 1,(a)n=a(a+1)···(a+n−1).\nForp∈(0,1) andαaσ-finite measure αonX, letρp,αbe the distribution of a\nPascal point process, i.e., ρp,αis the uniquely defined measure under which:\n•The counting variables η(B1),...,η(Bk) are independent, for all k≥2 and\nall pairwise disjoint B1,...,B k∈ E.\n•For eachB∈ Ewith 0<α(B)<∞,\nρp,α/parenleftbig\n{η:η(B) =n}/parenrightbig\n= (1−p)α(B)pn\nn!/parenleftbig\nα(B)/parenrightbig\nn.\nIfα(B) = 0 or ∞thenη(B) = 0 or ∞respectively, ρp,αalmost surely.\nWe work in the Hilbert space L2(N,ρp,α). Functions are complex-valued and the\nscalarproductis /a\\}bracketle{tf,g/a\\}bracketri}ht=/integraltext\nfgdρp,α. LetDbethesetofpolynomialsinthevariables\nη(ϕ) =/integraltext\nϕdη, i.e.,Dis the set of functions of the form\nF(η) =Pn/parenleftbig\nη(ϕ1),...,η(ϕn)/parenrightbig\nwithn∈N,ϕ1,...,ϕ n∈ C, andPn:Cn→Ca multivariate polynomial.\nLemma 3.8. The space Dis dense in L2(N,ρp,α).\nThe proof of the lemma is similar to the proof of Lemma 3.6 and it is omitte d.\nWe introduce difference operators\nD+\nxF(η) =F(η+δx)−F(η),D−\nxF(η) =F(η−δx)−F(η)\nand the constant c=√p−1−√p. The backward difference D xF(η) is only defined\nwhenxbelongs to the configuration η, i.e.,η({x})≥1. ForF∈D, let\nK−(ϕ)F(η) =1\nc/parenleftBig/integraldisplay\nϕ(x)D+\nxF(η)(α+η)(dx)+/integraldisplay\nϕ(x)D−\nxF(η)η(dx)/parenrightBig\n,(20)\nK+(ϕ)F(η) =1\nc/parenleftBig\np/integraldisplay\nϕ(x)F(η+δx)(η+α)(dx)+1\np/integraldisplay\nϕ(x)F(η−δx)η(dx)/parenrightBig\n−1\nc/parenleftbig\n2η(ϕ)+α(ϕ)/parenrightbig\nF(η) (21)\nK0(ϕ)F(η) =−1\nc/parenleftBig√p/integraldisplay\nϕ(x)D+\nxF(η)(α+η)(dx)+1√p/integraldisplay\nϕ(x)D−\nxF(η)η(dx)/parenrightBig\n+1\n2α(ϕ)F(η). (22)\nRemark. Letusdenotetheoperatorinthefirstlineof (22)by /hatwideN(ϕ)sothatK0(ϕ) =\n1\n2α(ϕ)+/hatwideN(ϕ). An explicit computation yields, for real-valued ϕ∈ C,\nK+(ϕ)+K−(ϕ)+(√p+√p−1)/hatwideN(ϕ) =cη(ϕ)−√pα(ϕ).\nThe equation is also similar to relations between quadratic white noise, Pascal\nlaws and Meixner polynomials, see [1, Section 17], [3, Section 4] and ca se (iv) in\nSection 2.4.\nGivena>0 andp∈(0,1), letMn(x;a,p) be the Meixner polynomials of degree\nnwith leading coefficients 1, orthogonal with respect to the negative binomial law\nν({x}) = (1−p)a(a)xpx/x! onN0[32].REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 15\nTheorem 3.9. The operators K+(ϕ),K0(ϕ),K−(ϕ),ϕ∈ C, are a Fock repre-\nsentation in L2(N,N,ρp,α)of the current algebra of su(1,1)with vacuum /BD, the\nconstant function 1. Moreover, with c=√p−1−√p,\nK+( /BDB1)n1···K+( /BDBℓ)nℓ/BD(η) =ℓ/productdisplay\nj=1cnjMnj/parenleftbig\nη(Bj);α(Bj),p/parenrightbig\n(23)\nfor allℓ∈N, disjointB1,...,B ℓ∈ Xwith0< α(Bi)<∞,n1,...,n ℓ∈Nand\nρp,α-almost all η∈N.\nThe theorem is proven in Section 5.4.\nTo conclude, we remark that for non-negative ϕ, the operators are amenable to\na probabilistic interpretation. The processes that appear are var iants of the linear\nbirth-deathprocesseson N0calleddiscreteBesselanddiscreteLaguerreprocessesby\nMiclo and Patie [42], see also Section 4.3 below. The role of Meixner polyno mials\nand the Pascal law (among others) for linear birth-death process es was already\nstudied by Karlin and McGregor [30], see also Chapter 3 in Schoutens [4 9].\nThe lowering operator K−(ϕ) is the generator of a spatial birth-and-death pro-\ncess. The neutral operator K0(ϕ) is of the form1\n2α(ϕ)−L(ϕ) andL(ϕ) is the\ngenerator of a spatial birth and death process with reversible mea sureρp,α. Fur-\nthermore, when the measure αis finite so that the constant function /BDis inC, one\ncan show that L( /BD) has discrete spectrum −N0.\n4.The univariate case: three representations of su(1,1)\nOur three representations of the su(1,1) current algebra generalize much sim-\npler univariate counterparts that we recall here. The benefit is tw ofold. First,\nwe connect to standard representation theory for su(1,1) and univariate orthog-\nonal polynomials. Second, the probabilistic interpretation brings in w ell known\nprocesses: linear birth-death processes on N0and diffusions (squared Bessel and\nLaguerre) on R+.\n4.1.Representation in weighted ℓ2space.There are several classes of unitary\nirreducible representations of SU(1,1). The representations studied here all belong\nto what is called the “discrete class, minimal m,D+\nk” [7,§5.i] or “principal discrete\nseriesD+\nj” [37, Section C]. Within that class, the representation is uniquely cha r-\nacterized by a half-integer m0∈ {1\n2,1,3\n2,...}, sometimes called Bargmann index .\nThe neutral operator k0has simple eigenvalues m0,m0+ 1,...and the Casimir\noperator (k0)2−1\n2(k+k−+k−k+), analogous to total spin, is m0(1−m0) times the\nidentity operator.\nThe following is an example of such a representation. It is essentially t aken from\nGiardin` a, Kurchan, Redig and Vafayi [23]; it differs from the usual representations\ninℓ2(N0) by a similarity transformation. Remember the rising factorial ( α)0= 1,\n(α)n=α(α+ 1)···(α+n−1). Fixα >0 and define weights wα(n) = (α)n/n!.\nWe work in the Hilbert space ℓ2(N0,wα) and define operators k±,k0with common\ndense domain d, the space of sequences ( f(n))n∈N0that have at most finitely many\nnon-zero entries. The operators are given by\nk+f(n) =nf(n−1), k−f(n) = (α+n)f(n+1), k0f(n) =/parenleftBig\nn+α\n2/parenrightBig\nf(n).\nEquivalently, with en(j) =δn,j,\nk+en= (n+1)en+1, k−en= (α+n−1)en−1, k0en=/parenleftBig\nn+α\n2/parenrightBig\nen.(24)16 S. FLOREANI, S. JANSEN, AND S. WAGNER\nandk−e0= 0. One easily checks that k±andk0mapdinto itself and on d\nthey satisfy the commutation relations (4). Moreover, k0is symmetric on dand\n/a\\}bracketle{tf,k+g/a\\}bracketri}ht=/a\\}bracketle{tk−f,g/a\\}bracketri}htfor allf,g∈d.\nWhenαis integer, we get a representation of the group SU(1,1) with Bargmann\nindexm0=α/2. The representation maps the 2 ×2 matrixA(ξ,θ)∈SU(1,1) from\nEq. (5) to the unitary operator U(ξ,θ) = exp(ξk+−ξk−)exp(2iθk0). Whenαis not\ninteger, we get instead a representation of the universal coverin g group ofSU(1,1)\n[48]. (Notice that A(ξ,θ+2π) =A(ξ,θ) no matter what, but U(ξ,θ+2π) =U(ξ,θ)\nonly ifαis integer.)\n4.2.Representation with Laguerre polynomials. Letβ >−1. TheLaguerre\npolynomials in hypergeometric form are given by [32, Chapter 9.12]\nL(β)\nn(x) =(β+1)n\nn!n/summationdisplay\nj=0(−n)j\n(β+1)jxj\nj!=(−x)n\nn!+O(xn−1)\nthey satisfy the orthogonality relation,\n1\nΓ(β+1)/integraldisplay∞\n0L(β)\nn(x)L(β)\nm(x)xβe−xdx=δm,n(β+1)n\nn!. (25)\nForα>0, letℓ2(N0,wα) be the Hilbert space with weights wα(n) = (α)n/n! from\nSection4.3. Let µαbetheprobabilitymeasureon R+withdensityΓ( α)−1xα−1exp(−x)\nandL2(R+,µα)bethespaceofBorel-measurablesquare-integrablefunctions( complex-\nvalued). By the orthogonality relation (25), the linear operator\nU:ℓ2(N0,wα)→L2(R+,µα), f/ma√sto→Uf=∞/summationdisplay\nn=0f(n)L(α−1)\nn\nis an isometry that maps entoL(α−1)\nnand in particular e0to the constant function/BD. The linear hull of e0,...,e nis mapped to the space of polynomials of degree at\nmostn. Upon setting K#=U−1k#Uwe obtain a representation of the su(1,1)\nalgebra that satisfies K−/BD= 0 and\nK+Ln= (n+1)Ln+1, K−Ln= (α+n−1)Ln−1, K0Ln=/parenleftBig\nn+α\n2/parenrightBig\nK0Ln,(26)\ncompare (24). For better readability we have dropped the supers cripts on the\nLaguerre polynomials. The following proposition identifies K#as differential op-\nerators; it is similar to Proposition 4.13 in Groenevelt [25].\nProposition 4.1. We have\nK+=−x∂2\nx+(2x−α)∂x+(α−x),\nK−=−x∂2\nx−α∂x,\nK0=α\n2−/parenleftbig\nx∂2\nx+(α−x)∂x/parenrightbig\n.\nNotice that both L= (α\n2−K0) and−K−are infinitesimal generators of dif-\nfusions on R+. The operator Lis the generator of the Laguerre semigroup [6,\nSection 2.7.3] which is the Markov semigroup of the Cox-Ingersoll-Ross process\n(CIR) in mathematical finance to model interest rate movements [ 17]. It has the\nGamma distribution µαas areversiblemeasure. The loweringoperatorcorresponds\nto a squared Bessel process.\nProof.The generating function of our Laguerre polynomials is [32, Eq. (9.12 .10)]\nGt(x) =∞/summationdisplay\nn=0tnLn(x) = (1−t)−αexp/parenleftBigxt\nt−1/parenrightBig\n.REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 17\nIn view of Eq. (26), we have\n/parenleftbig\nK0−α\n2/parenrightbig\nGt(x) =∞/summationdisplay\nn=0tnnLn(x) =t∂tGt(x)\n=−/parenleftBig\nαt\nt−1+xt\n(t−1)2/parenrightBig\nGt(x)\n=−/parenleftBig\nxt2\n(t−1)2+(α−x)t\nt−1/parenrightBig\nGt(x)\n=−(x∂2\nx+(α−x)∂x)Gt(x).\nThis holds true for all sufficiently small t, the differential expression for K0follows.\nFor the raising operators:\nK+Gt(x) =∞/summationdisplay\nn=0tn(n+1)Ln+1(x) =∞/summationdisplay\nn=1ntn−1Ln(x) =∂tGt(x)\n=/parenleftBig\n−α\nt−1−x\n(t−1)2/parenrightBig\nGt(x)\n=/parenleftBig\n−xt2\n(t−1)2+(2x−α)t\nt−1+α−x/parenrightBig\nGt(x)\n=/parenleftBig\n−x∂2\nx+(2x−α)∂x+(α−x)/parenrightBig\nGt(x).\nFinally\nK−Gt(x) =∞/summationdisplay\nn=1tn(α+n−1)Ln−1(x)\n=αtGt(x)+t2∂tGt(x)\n=/parenleftBig\nαt−αt2\nt−1−xt2\n(t−1)2/parenrightBig\nGt(x)\n=/parenleftBig\n−αt\nt−1−xt2\n(t−1)2/parenrightBig\nGt(x)\n=−/parenleftbig\nx∂2\nx+α∂x/parenrightbig\nGt(x). /square\n4.3.Representation with Meixner polynomials. Forp∈(0,1) andα >0,\nthe Meixner polynomials are given by [32, Chapter 9.10]\nMn(x;α,p) =n/summationdisplay\nj=0(−n)j(−x)j\n(α)j1\nj!/parenleftBig\n1−1\np/parenrightBigj\n=xn\n(α)n/parenleftBig\n1−1\np/parenrightBign\n+O(xn−1)\nThey satisfy the orthogonality relation\n(1−p)α∞/summationdisplay\nx=0(α)x\nx!pxMn(x;α,p)Mm(x;α,p) =/parenleftBig(α)n\nn!pn/parenrightBig−1\nδm,n.(27)\nConsider the probability weights\nρp,α(x) = (1−p)−αpx\nx!(α)x(x∈N0)\nand the Hilbert space ℓ2(N0,ρp,α). By the orthogonality relation (27), the operator\nU:ℓ2(N0,wα)→ℓ2(N0,ρp,α), f/ma√sto→Uf=∞/summationdisplay\nn=0f(n)(α)n\nn!√pnMn(·;α,p).18 S. FLOREANI, S. JANSEN, AND S. WAGNER\nis an isometry. Let Mn(x;α,p) be the Meixner polynomials with leading coefficient\n1, then\nUen=1\nn!/parenleftbig√p−√p−1/parenrightbignMn(·;α,p).\nLet us define K#=U−1k#U, then we obtain operators with common domain the\nset of polynomials, that satisfy\nK+/BD= 0,(K+)n/BD= (√p−√p−1)nMn(·;α,p) (28)\nin analogy with k−e0= 0 and (k+)ne0=n!en. The following proposition identifies\nthe operators K#. We define forward and backward difference operators\n∂+f(x) =f(x+1)−f(x), ∂−f(x) =f(x−1)−f(x)\nwithf(−1) = 0. The proposition is similar to Proposition 4.7 in Groenevelt [25].\nProposition 4.2. For every polynomial f:N0→C,\nK+f(x) =−√p\n1−p/parenleftBig\np(α+x)f(x+1)+1\npxf(x−1)−(α+2x)f(x)/parenrightBig\n,\nK−f(x) =−√p\n1−p/parenleftBig\n(α+x)∂+f(x)+x∂−f(x)/parenrightBig\n,\nK0f(x) =α\n2f(x)−1\n1−p/parenleftbig\np(α+x)∂+f(x)+x∂−f(x)/parenrightbig\n.\nWe remark that up to additive and multiplicative constants, the lower ing and\nthe neutral operators are the generators of linear birth-death chains on N0. As\nmentioned after Theorem 3.9, those processes were already stud ied in connection\nwith orthogonal polynomials by Karlin and McGregor[30]; Miclo and Patie [42]\nprovided intertwining relations between these linear birth-death ch ains and squared\nBessel and Laguerre diffusions. Our representation clarifies the a lgebraic similarity\nof the processes: they admit the same formal expression with the su(1,1) algebra.\nProof of Proposition 4.2. The generating function of the Mexiner polynomials is\n[32]\nGt(x) =∞/summationdisplay\nn=0(α)n\nn!tnMn(x) =/parenleftBig\n1−t\np/parenrightBigx\n(1−t)−α−x.\nWe note\n∂+Gt(x) =Gt(x+1)−Gt(x) =/parenleftBig1−t/p\n1−t−1/parenrightBig\nGt(x) =1−1/p\n1−ttGt(x).\nand\n∂−Gt(x) =/parenleftBig1−t\n1−t/p−1/parenrightBig\nGt(x) =1/p−1\n1−t/ptGt(x).\nTherefore, in view of ( K0−α\n2)Mn=nMn, we get\n/parenleftbig\nK0−α\n2/parenrightbig\nGt(x) =∞/summationdisplay\nn=0(α)n\nn!tnnMn(x) =t∂tGt(x)\n=/parenleftBig\n−x\np/parenleftbig\n1−t\np/parenrightbig−1+(α+x)(1−t)−1/parenrightBig\ntGt(x)\n=−x\n1−p∂−Gt(x)−p(α+x)\n1−p∂+Gt(x).REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 19\nTurning to the lowering operators, we note first, K−M0= 0 and, for n≥1,\nK−(α)n\nn!tnMn=/parenleftbigt√p/parenrightbignK−Uen=/parenleftbigt√p/parenrightbignUk−en\n=/parenleftbigt√p/parenrightbign(α+n−1)Uen\n=/parenleftbigt√p/parenrightbign(α+n−1)(α)n−1\n(n−1)!√pn−1Mn−1\n=1√p(α)n\n(n−1)!tnMn−1.\nWe change the summation index in Gtfromntoℓ=n−1, note (α)n= (α)ℓ+1=\n(α+ℓ)(α)ℓand obtain\nK−Gt(x) =1√p∞/summationdisplay\nℓ=0(α)ℓ\nℓ!(α+ℓ)tℓ+1Mℓ(x)\n=1√p/parenleftbig\nαt+t2∂t/parenrightbig\nGt(x)\n=1√p/parenleftBig\nαt−xt2\np−t+(α+x)t2\n1−t/parenrightBig\nGt(x)\n=1√p/parenleftBig\n−x\n1−t/p+α+x\n1−t/parenrightBig\ntGt(x)\n=−√p\n1−p/parenleftbig\nx∂−Gt(x)+(α+x)∂+Gt(x)/parenrightbig\nThe above relations hold true for all sufficiently small t, the expressions for K0and\nK−follow. The expression for the raising operator is proven by similar ex plicit\ncomputations that we omit. Alternatively, the reader may check th e adjointness\n/a\\}bracketle{tf,K+g/a\\}bracketri}ht=/a\\}bracketle{tK−f,g/a\\}bracketri}hton a dense subspace of polynomials in ℓ2(N0,ρp,α). /square\n5.Proofs\nIn this section we prove our main results, namely the three represe ntationsof the\nsu(1,1) current algebra, the Baker-Campbell-Hausdorff formula, and t he explicit\naction of the unitaries exp( k+(ξ)−k−(ξ))exp(2ik0(θ)) on exponential vectors. In\naddition, we prove an integration by parts formula for the Gamma ra ndom measure\n(Proposition 5.6) that is of interest in its own right.\n5.1.Representation in extended Fock space. Proof of Theorem 3.2 .We\nstart with a lemma that helps prove the adjointness relations for th e lowering and\nraising operators in extended Fock space.\nLemma 5.1. For alln∈Nand non-negative or integrable f:Xn+1→C,\n/integraldisplay\nfdλn+1=/integraldisplay/parenleftBig/integraldisplay\nf(x1,...,x n,y)α(dy)+n/summationdisplay\ni=1f(x1,...,x n,xi)/parenrightBig\nλn(dx).(29)\nProof.We have to evaluate/summationtext\nσ∈Sn+1/integraltext\nfσdα|σ|. The permutations with σ(n+1) =\nn+1 are in one-to-one correspondence with permutations in Sn, they contribute\nthe first term on the right in (29). Any permutation with σ(n+1) =i≤nadmits a\ncycle decomposition with a single cycle containing n+1 andiand remaining cycles\ncontaining neither n+1 nori. Deletingn+1 from the cycle containing n+1 (e.g.,\nn+1→i→3→n+1 becomes i→3→i), we obtain the cycle decomposition\nof a permutation τ∈Snthat has the same number of cycles. These permutations\ncontribute the second term on the right in (29). /square20 S. FLOREANI, S. JANSEN, AND S. WAGNER\nProof of Lemma 3.1. Forn∈N, letVnbe the space of linear combinations of\nsymmetrized tensor products of indicators of sets Bj∈ Xwithα(Bj)<∞. It is\nenough to show that, for every n∈N, the space Vnis dense in L2\ns(λn).\nFixn∈N. Cartesian products B1×···×Bnof measurable sets with finite mass\nα(Bi)<∞form a generating π-system of the product σ-algebra Xn. Therefore,\nby the functional monotone class theorem [11, Theorem 2.12.9], the linear combi-\nnations of the constant function and indicators of Cartesian prod ucts are dense,\nwith respect to the supremum norm, in the space of bounded measu rable functions\nfromXntoC. It follows that the linear hull of Vnand the constant function 1 is\ndense, with respect to the supremum norm, in the space of bounde d, measurable\nsymmetric functions.\nIfαhas finite total mass, then λn(Xn)<∞as well and we have /BD∈Vn⊂\nL2\ns(λn). Furthermore, uniform convergence implies L2(λn)-convergence. Thus, ev-\nery bounded function in L2\ns(λn) can be approximated by a function in Vn. As the\nbounded functions are dense in L2\ns(λn), we find that Vnis dense.\nIfα(X) =∞, then the previous argument still applies to L2\ns(λn\n/BDCn), for every\nsetC∈ Xwithα(C)<∞(henceλn(Cn)<∞). Asαisσ-finite, every function\nf∈L2\ns(λn) is the limit in L2-norm of a sequence ( f /BDCn\nj)j∈Nwithα(Cj)<∞and\nthe proof of the lemma is easily concluded. /square\nProof of Theorem 3.2. By the expressions given above, the operators k#(ϕ) map\nsymmetrized tensor products to combinations of symmetrized ten sor products,\nhenceDis invariant. The common domain Dis dense by Lemma 3.1. Clearly\nk+(ϕ) andk−(ϕ) are linear in ϕwhilek−(ϕ) is antilinear in ϕ, moreover the vac-\nuum is a unit vector annihilated by all lowering operators. The vacuum is cyclic\nbecause repeated application of raising operators gives symmetriz ed tensor prod-\nucts, whose linear hull is precisely the dense space D.\nFor the commutation relations (7), we note\nb(ϕ)a†(ψ)f1⊗s···⊗sfn=2n/summationdisplay\nj=1(ϕψfj)⊗sf1···/hatwidefj···⊗sfn\n+2/summationdisplay\n1≤i<j≤n(ϕfifj)⊗sψ⊗sf1···/hatwidefi···/hatwidefj···⊗sfn\na†(ψ)b(ϕ)f1⊗s···⊗sfn=2/summationdisplay\n1≤i<j≤nψ⊗s/parenleftbig\nϕfifj/parenrightbig\n⊗sf1⊗s···/hatwidefi···/hatwidefj···⊗sfn\nhence\n/bracketleftbig\nb(ψ),a†(ϕ)/bracketrightbig\nf1⊗s···⊗sfn= 2/hatwiden(ψϕ)f1⊗s···⊗sfn\nand\n/bracketleftbig\nk−(ψ),k+(ϕ)/bracketrightbig\n=/bracketleftbig\na(ψ),a†(ψ)/bracketrightbig\n+/bracketleftbig\nb(ψ),a†(ψ)/bracketrightbig\n=/a\\}bracketle{tψ,ϕ/a\\}bracketri}ht+2/hatwiden(ψϕ) = 2k0(ψϕ).\nThe other commutation relations, as well as Eq. (11), are proven b y similar explicit\ncomputations that we omit; see also ´Sniady [52, Theorem 4].\nThe adjointness relation for the neutral operator is straightfor ward:k0(ϕ) is\na multiplication operator that multiplies with/summationtextn\ni=1ϕ(xi) +1\n2/integraltext\nϕdα, its adjoint\nmultiplies with the complex conjugate. For the raising and lowering ope rators, letREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 21\nϕ∈ Cand (Fn), (Gn) inD. We have\n1\n(n+1)!/integraldisplay\nGn+1(k+(ϕ)Fn)dλn+1\n=1\n(n+1)!/integraldisplay\nGn+1(x1,...,x n+1)/parenleftBign+1/summationdisplay\ni=1ϕ(xi)Fn(x1,...,/hatwidexi,...,x n+1)/parenrightBig\nλn+1(dx)\n=1\nn!/integraldisplay\nGn+1(x1,...,x n+1)ϕ(xn+1)Fn(x1,...,x n)λn+1(dx).\nWe apply Lemma 5.1 to f(x1,...,x n,y) =Gn+1(x1,...,x n,y)ϕ(y)Fn(x1,...,x n).\nThe inner integral on the right of (29) becomes\n/integraldisplay\nϕ(y)Gn+1(x,y)Fn(x)α(dy)+n/summationdisplay\ni=1ϕ(xi)Gn+1(x,xi)Fn(x)\nin which we recognize k−(ϕ)Gn+1Fn. Thus, we get\n1\n(n+1)!/integraldisplay\nGn+1(k+(ϕ)Fn)dλn+1=1\nn!/integraldisplay\nk−(ϕ)Gn+1Fndλn.\nSummation over n∈N0yields the required adjointness relation. See also ´Sniady\n[52, Theorem 3] or Lytvynov [39, Section 2]. /square\n5.2.Baker-Campbell-Hausdorff formula. Proof of Theorem 3.4. First we\ncheck that the product of exponentials exp( k+(v))exp(k0(w))exp(−k−(v)) in The-\norem 3.4 is well defined on D.\nLemma 5.2. Letf∈ Dandv,w∈ C. Then:\n(a)k−(v)nfvanishes for all except finitely many n, henceexp(−k−(v))f∈ D.\n(b)exp(k0(w))fis inD.\n(c) Ifsup|v|<1, then/summationtext∞\nn=01\nn!(k+(v))nfconverges absolutely in norm.\nThe condition sup |v|<∞is satisfied for |v|= tanh|ϕ|asϕ∈ Cis bounded.\nProof.Itisenoughtotreatfunctions f=k+(ϕ1)···k+(ϕm)Ψwithϕ1,...,ϕ m∈ C,\nm∈N0. Item (a) of the lemma follows from ( k−(ϕ))nf= 0 forn>m. For item\n(b), notice that\nexp/parenleftbig\nk0(w)/parenrightbig\nk+(ϕ1)···k+(ϕm)Ψ = exp/parenleftBig1\n2/integraldisplay\nwdα/parenrightBig\nk+(ϕ1ew)···k+(ϕmew)Ψ∈ D.\n(30)\nFor item (c), let C >0, 0< c <1 andB∈ Xbe such that α(B)<∞and\n|ϕj| ≤C /BDBforj∈ {1,...,m}and|v|<c /BDB. Then\n/parenleftbig\nk+(v)nf/parenrightbig\nℓ(x1,...,x ℓ)\n=δn+m,ℓ/summationdisplay\nσ∈Sn+mv(x1)···v(xn)ϕσ(n+1)(xn+1)···ϕσ(n+m)(xn+m).\nThus,\n/vextenddouble/vextenddoublek+(v)nf/vextenddouble/vextenddouble2≤1\n(n+m)!/parenleftBig\n(n+m)!cnCm/parenrightBig2\nλn+m(B)\nand\n∞/summationdisplay\nn=01\nn!/vextenddouble/vextenddouble(k+(v))nf/vextenddouble/vextenddouble≤Cm∞/summationdisplay\nn=0cn\nn!/parenleftBig\n(n+m)!(α(B))(n+m)/parenrightBig1/2\n<∞./square22 S. FLOREANI, S. JANSEN, AND S. WAGNER\nForthe proofofTheorem 3.4we adapt Truax[56], with extra caref orunbounded\noperators. For t≥0, set\nvt:= /BD{ϕ/\\e}atio\\slash=0}ϕ\n|ϕ|tanht|ϕ|, wt:=−2logcosht|ϕ|\nand define\nStf:= exp/parenleftbig\nk+(vt)/parenrightbig\nexp/parenleftbig\nk0(wt)/parenrightbig\nexp/parenleftbig\n−k−(vt)/parenrightbig\nf, f ∈ D.\nThe operator Stis well-defined on Dby Lemma5.2. We showthat t/ma√sto→Stfis norm-\ndifferentiable and solves the same differential equation as exp( t(k+(ϕ)−k−(ϕ))f.\nLemma 5.3. The mapt/ma√sto→Stfis norm-differentiable on R+, for every f∈ D,\nwith derivative\n∂tStf=/parenleftBig\nek+(vt)k+(∂tvt)ek0(wt)e−k−(vt)+ek+(vt)k0(∂twt)ek0(wt)e−k−(vt)\n−ek+(vt)ek0(wt)k−(∂tvt)e−k−(vt)/parenrightBig\nf.(31)\nNote that∂tvt,∂twt∈ C. The proof of the lemma is based on straightforward\nbut somewhat tedious bounds and it is omitted.\nThe following commutation relations help us express the right side of ( 31) with\nk+(ϕ) andk−(ϕ) only. We remarkthat k+(θ),k0(θ) andk−(θ) areclosablebecause\nthey havedensely defined adjoints. We use the samelettersfor th e closedoperators.\nLemma 5.4. For allv,w,θ∈ Cwithsup|v|<∞, and allf∈ D, the following\nholds true:\n(a)[k0(θ),exp(k+(v))]f= exp(k+(v))k+(θv)f.\n(b)exp(k+(v))fis in the domain of the closure of k−(θ)and\n/bracketleftbig\nk−(θ),ek+(v)/bracketrightbig\nf=/parenleftbig\n−k+(θv2)+2k0(θv)/parenrightbig\nek+(v)f.\n(c)ek0(w)k−(θ)f=k−(θe−w)ek0(w)f.\nProof.An induction over n, based on [ A,Bn+1] = [A,B]Bn+B[A,Bn] and the\ncommutation relations for k0(θ) andk+(v), yields\n[k0(θ),k+(v)n] =nk+(θv)k+(v)n−1\nfrom which one deduces\n[k0(θ),exp(k+(v))] = exp(k+(v))k+(θv). (32)\nNext we check by induction over nthat\n[k−(θ),k+(v)n] =n(n−1)k+(θv2)k+(v)n−2+2nk+(v)n−1k0(θv).(33)\nForn= 1 the previous relation reads [ k−(θ),k+(v)] = 2k0(θv), which holds true.\nFor the induction step from nton+1, we write\n[k−(θ),k+(v)n+1] =/bracketleftbig\nk−(θ),k+(v)/bracketrightbig\nk+(v)n+k+(v)/bracketleftbig\nk0(θ),k+(v)n/bracketrightbig\n= 2/bracketleftBig\nk0(θv),k+(v)n/bracketrightBig\n+2k+(v)nk0(θv)+k+(v)/bracketleftBig\nk0(θ),k+(v)n/bracketrightBig\n= 2nk+(θv2)k+(v)n−1+2k+(v)nk0(θv)+k+(v)/bracketleftBig\nk0(θ),k+(v)n/bracketrightBig\n=n(n+1)k+((θv2)k+(v)n−1+2(n+1)k+(v)nk0(vθ).REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 23\nThe inductive proof of (33) is complete. Now let f∈ DandN∈N. Eq. (33) yields\nk−(θ)N/summationdisplay\nn=01\nn!k+(v)nf\n=/parenleftBigN/summationdisplay\nn=01\nn!k+(v)n/parenrightBig\nk−(θ)f+/parenleftBigN−2/summationdisplay\nn=01\nn!k+(v)n/parenrightBig\nk+(θv2)f+2/parenleftBigN−1/summationdisplay\nn=01\nn!k+(v)n/parenrightBig\nk0(θv)f.\nThe right side converges in norm as N→ ∞by Lemma 5.2, hence so does the left\nside. It follows that exp( k+(v))fis in the domain of the closure of k−(θ) and we\nobtain/bracketleftbig\nk−(θ),ek+(v)/bracketrightbig\nf= ek+(v)/parenleftbig\nk+(θv2)+2k0(θv)/parenrightbig\nf.\nIn view of (32), we may also write\n/bracketleftbig\nk−(θ),ek+(v)/bracketrightbig\nf=/parenleftbig\n−k+(θv2)+2k0(θv)/parenrightbig\nek+(v)f.\nFor (c) it is enough to consider functions f=k+(ϕ1)···k+(ϕm)Ψ form∈N,\nϕ1,...,ϕ m∈ C. We write down the proof for m= 2, the general case is similar.\nWe have\nk−(θ)f=k−(θ)k+(ϕ1)k+(ϕ2)Ψ\n=/a\\}bracketle{tθ,ϕ1/a\\}bracketri}htk+(ϕ2)Ψ+/a\\}bracketle{tθ,ϕ2/a\\}bracketri}htk+(ϕ1)Ψ+2k+(ϕ1θϕ2)Ψ.\nBy (30),\nek0(w)k−(θ)f= e/integraltext\nwdα/2/parenleftBig\n/a\\}bracketle{tθ,ϕ1/a\\}bracketri}htk+(ϕ2ew)Ψ+/a\\}bracketle{tθ,ϕ2/a\\}bracketri}htk+(ϕ1ew)Ψ+2k+(ϕ1θϕ2ew)Ψ/parenrightBig\n.\nSimilarly,\nk−(ψ)ek0(w)f\n= e/integraltext\nwdα/2/parenleftBig\n/a\\}bracketle{tψ,ewϕ1/a\\}bracketri}htk+(ϕ2ew)Ψ+/a\\}bracketle{tψ,ewϕ2/a\\}bracketri}htk+(ϕ1ew)Ψ+2k+(ϕ1ewψϕ2ew)Ψ/parenrightBig\n.\nThe expressions are equal for ψ= exp(−w)θ. /square\nLemma 5.5. Letf∈ D. Then,Stfis in the domain of the closure of k+(ϕ)−\nk−(ϕ), denoted by k+(ϕ)−k−(ϕ)as well. Moreover,\n∂tStf=/parenleftbig\nk+(ϕ)−k−(ϕ)/parenrightbig\nStf.\nProof.In Lemma 5.4(b) we provedthat exp( k+(v))fis in the domain of the closure\nofk−(θ) by proving that k−(θ)/summationtextN\nn=0n!−1k+(v)nfconverges in norm as N→ ∞.\nThe same argument shows that exp( k+(v))falso belongs to the domain of the\nclosure ofk+(ϕ)−k−(ϕ). As exp( k0(wt)) and exp( −k−(vt)) mapDtoDby\nLemma 5.2, it follows that Stfis in th domain of the closure, for all f∈ D. By\nLemma 5.3, the derivative ∂tStfis a sum of three terms. The first is k+(∂tvt)Stf.\nThe second term is\nek+(vt)k0(∂twt)ek0(wt)e−k−(vt)f=/parenleftbig\nk0(∂twt)−k+(vt∂twt)/parenrightbig\nStf,\nwhere we have used Lemma 5.4(a). The third term is\n−ek+(vt)ek0(wt)k−(∂tvt)e−k−(vt)f\n=−ek+(vt)k−(e−wt∂tvt)ek0(wt)e−k−(vt)f\n=/parenleftBig\n−k+(e−wt∂tvtv2\nt)+2k0(e−wt∂vtvt)−k−(e−wt∂tvt)/parenrightBig\nStf.24 S. FLOREANI, S. JANSEN, AND S. WAGNER\nThis time we have used Lemma 5.4(c) and then (b). Altogether, we ge t\n∂tStf=/parenleftBig\nk+/parenleftBig\n∂tvt−(∂twt)vt−v2\nt(∂tvt)/parenrightBig\n+k0/parenleftBig\n∂twt+2vt(∂tvt)/parenrightBig\n−k−/parenleftbig\n(∂tvt)e−wt/parenrightbig/parenrightBig\nStf.\nThe proof is concluded with the differential equations\n∂tvt−(∂twt)vt−v2\nt(∂tvt)e−wt=ϕ,\n∂twt+2vt(∂tvt)e−wt= 0,\n(∂tvt)e−wt=ϕ. /square\nNow we haveall the ingredients to provethe Baker-Campbell-Hausd orffformula.\nProof of Theorem 3.4. As noted after Eq. (12), the closure Aof−i(k+(ϕ)−k−(ϕ))\nis self-adjoint. By Stone’s theorem, for every f∈ D, the function t/ma√sto→exp(itA)f\nis norm-differentiable with derivative ∂texp(itA)f= iAexp(itA)f. Hencegt:=\nexp(t(k+(ϕ)−k−(ϕ)))fis norm-differentiable with derivative\n∂tgt=/parenleftbig\nk+(ϕ)−k−(ϕ)/parenrightbig\ngt.\nThe mapt/ma√sto→Stfsatisfies the same differential equation by Lemma 5.5, moreover\nS0f=g0=f, thereforegt=Stffor allt≥0. In particular, t= 1 corresponds to\nthe Baker-Campbell-Hausdorff formula. /square\n5.3.Gamma random measure. Proof of Theorem 3.7. First we prove that\nthe space of polynomials Dis dense in L2(M,ρα). We actually prove a slightly\nstronger statement: the space of polynomials in the masses µ(B), withα(B)<∞,\nis dense.\nProof of Lemma 3.6. LetUbe the set of finite linear combinations of indicators/BDB, with non-negative coefficients, of sets with finite mass α(B)<∞. Foru∈U,\nletH0be the set of exponential maps Fu:N→R+,µ/ma√sto→exp(−µ(u)). Further let\nHbe the closureof H0, with respect tothe supremum norm, ofthe linearcombina-\ntions of exponentials. The set H0is closed with respect to pointwise multiplication\nand the vector space Hcontains the constant function 1 and is closed in L∞(M).\nTherefore, by the functional monotone class theorem [11, Theor em 2.12.9], Hcon-\ntains all bounded functions that are measurable with respect to σ(H0). In view\nofµ(B) =−logF/BDB(η), the set H0generates the full σ-algebra on M. Thus, H\ncontains all bounded measurable functions.\nAs a consequence, the linear combinations of exponentials µ/ma√sto→exp(−µ(u)) are\ndensein L∞(M)withrespecttouniformconvergence,henceafortioriin L2(M,ρα)\nwith respect to L2-norm. To conclude, we note that the exponential in turn can be\napproximatedin L2-norm by polynomials: Simply considera truncated exponential\nseries and bound the remainder, using that there exists ε=εu>0 such that/integraltext\nexp(εµ(u))ρα(dµ)<∞.\nWe obtain that polynomials in the variables µ(u) are dense in L2(M,ρα). Every\npolynomial in µ(u), withua linear combination of indicators, is alsoa a polynomial\nin massesµ(Bi),α(Bi)<∞. The lemma and the stronger claim stated before this\nproof follow. /square\nNext we prove an identity for Gamma random measures that will help u s prove\nthe adjointness relations (8). A straightforward integration by p art shows that for\nevery polynomial f:R→C, givena>0,\n/integraldisplay∞\n0xf′(x)xa−1e−xdx=/integraldisplay∞\n0(x−a)f(x)xa−1e−xdx.REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 25\nThegeneralizationforGammarandommeasuresisasfollows. Wewrite Eρα[F(µ)] =/integraltext\nF(µ)ρα(dµ).\nProposition 5.6. Letξbe a Gamma random measure with law ρα. For allϕ∈ C\nand all polynomials F∈D,\nEρα/bracketleftBig/integraldisplay\nϕ(x)δF\nδµ(x)(µ)µ(dx)/bracketrightBig\n=Eρα/bracketleftBig/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nF(µ)/bracketrightBig\n.\nProof.We consider first exponential functions F(µ) = exp(µ(f)) and exploit that\nthe Gamma random measure is compound Poisson. Let Π =/summationtext\niδ(Xi,Zi)be a\nPoisson point process on X×R+with intensity measure α(dx)z−1exp(−z)dzand\nξ=/summationtext\niZiδXi. By the Mecke equation for Π [36],\nE/bracketleftBig/summationdisplay\niG(Xi,Zi;Π)/bracketrightBig\n=/integraldisplay\nX/braceleftBig/integraldisplay∞\n0E/bracketleftBig\nG(x,z;Π+δ(x,z))/bracketrightBig\nz−1e−zdz/bracerightBig\nα(dx)\nfor all measurable G:X×R+×N(X×R+)→R+. Letϕandfbe non-negativetest\nfunctionsin C. Weassumethat ||f||∞issufficientlysmallsothat Eρα[exp(2µ(f))]<\n∞. The Mecke equation applied to\nG/parenleftBig\nx,z;/summationdisplay\njδ(xj,zj)/parenrightBig\n=zϕ(x)f(x)exp/parenleftBig/summationdisplay\njzjf(xj)/parenrightBig\nyields\nE/bracketleftBig\nξ(ϕf)eξ(f))/bracketrightBig\n=/integraldisplay\nX/braceleftBig/integraldisplay∞\n0E/bracketleftBig\nzϕ(x)f(x)ezf(x)eξ(f)/bracketrightBig\nz−1e−zdz/bracerightBig\nα(dx).(34)\nAn integration by parts gives\n/integraldisplay∞\n0f(x)ezf(x)e−zdz=−1+/integraldisplay∞\n0ezf(x)e−zdz.\nWe insert this identity on the right side in Eq. (34) (after changing th e order of\nintegration). This gives rise to two terms. The first, coming from −1, stays as is.\nFor the second term, we apply again the Mecke equation. At the end , we obtain\nE/bracketleftBig\nξ(ϕf)eξ(f))/bracketrightBig\n=−α(ϕ)E/bracketleftBig\neξ(f)/bracketrightBig\n+E/bracketleftBig\nξ(ϕ)eξ(f)/bracketrightBig\n.\nThe same identity holds true for freplaced by sfwiths∈(−1,1). We expand in\npowers ofs, equate coefficients on the left and right side, and obtain\nE/bracketleftBig\nξ(ϕf)nξ(f)n−1/bracketrightBig\n=E/bracketleftBig/parenleftbig\nξ(ϕ)−α(ϕ)/parenrightbig\nξ(f)n/bracketrightBig\n.\nThe compound Poisson process ξhas lawραand every polynomial is a linear\ncombination of monomials µ/ma√sto→µ(f)n. It follows that\nEρα/bracketleftBig\nµ(ϕf)P′/parenleftbig\nµ(f)/parenrightbig/bracketrightBig\n=Eρα/bracketleftBig/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nP/parenleftbig\nµ(f)/parenrightbig/bracketrightBig\nwhich is precisely the statement that we were looking for. /square\nLet us show right away how Proposition 5.6 implies the adjointness rela tions.\nLemma 5.7. The operators K#(ϕ)defined in Section 3.3 satisfy the adjointness\nrelations (8).\nProof.Pickf,ϕ∈ CandPa univariate polynomial. Set F(µ) =P(µ(f)) and\nG(µ) =P′(µ(f)). Notice that\nδ2F\nδµ(x)2(µ) =f(x)2P′′/parenleftbig\nµ(x)/parenrightbig\n=f(x)δG\nδµ(x)(µ)26 S. FLOREANI, S. JANSEN, AND S. WAGNER\nProposition 5.6 applied to ψ=ϕfandGthus yields\nEρα/bracketleftBig/integraldisplay\nϕ(x)δ2F\nδµ(x)2(µ)µ(dx)/bracketrightBig\n=Eρα/bracketleftBig/integraldisplay/parenleftbig\nµ(ϕf)−α(ϕf)/parenrightbig\nP′/parenleftbig\nµ(f)/parenrightbig/bracketrightBig\n=Eρα/bracketleftBig/integraldisplay\nϕ(x)δF\nδµ(x)(µ)(µ−α)(dx)/bracketrightBig\n,(35)\nThe statement extends, by linearity, to all polynomials F∈D, and it is equivalent\nto\nEρα/bracketleftBig\nk−(ϕ)F(µ)/bracketrightBig\n=Eρα/bracketleftBig/integraldisplay\nϕ(x)δF\nδµ(x)(µ)µ(dx)/bracketrightBig\n=Eρα/bracketleftBig/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nF(µ)/bracketrightBig\n.(36)\n(For the last equality we have used again Proposition 5.6.)\nNow letFandGbe two polynomials in Dandϕbe a test function in C. We\napply Eq. (36) to the function FG, keep in mind the product rule forthe variational\nderivativesδ\nδµ(x), and get\n/a\\}bracketle{tG,k−(ϕ)F/a\\}bracketri}ht+/a\\}bracketle{tk−(ϕ)G,F/a\\}bracketri}ht+2Eρα/bracketleftBig/integraldisplay\nϕ(x)δF\nδµ(x)δG\nδµ(x)(µ)µ(dx)/bracketrightBig\n=/a\\}bracketle{t/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nG,F/a\\}bracketri}ht.(37)\nOn the other hand Proposition 5.6 implies\nEρα/bracketleftBig/integraldisplay\nϕ(x)G(µ)δF\nδµ(x)(µ)µ(dx)/bracketrightBig\n=−Eρα/bracketleftBig/integraldisplay\nϕ(x)δG\nδµ(x)(µ)F(µ)µ(dx)/bracketrightBig\n+Eρα/bracketleftBig/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nGF(µ)/bracketrightBig\nand then, by an argument similar to (35),\nEρα/bracketleftBig/integraldisplay\nϕ(x)δG\nδµ(x)δF\nδµ(x)(µ)µ(dx)/bracketrightBig\n=−Eρα/bracketleftBig/integraldisplay\nϕ(x)δ2G\nδµ(x)(µ)F(µ)µ(dx)/bracketrightBig\n+Eρα/bracketleftBig/integraldisplay\nϕ(x)δG\nδµ(x)(µ)F(µ)(µ−α)(dx)/bracketrightBig\n=−/a\\}bracketle{tk−(ϕ)G,F/a\\}bracketri}ht+Eρα/bracketleftBig/integraldisplay\nϕ(x)δG\nδµ(x)(µ)F(µ)µ(dx)/bracketrightBig\n.\nWe combine this equation with (37) and get\n/a\\}bracketle{tG,k−(ϕ)F/a\\}bracketri}ht=/a\\}bracketle{tk−(ϕ)G,F/a\\}bracketri}ht−2Eρα/bracketleftBig/integraldisplay\nϕ(x)δG\nδµ(x)(µ)F(µ)µ(dx)]\n+/a\\}bracketle{t/parenleftbig\nµ(ϕ)−α(ϕ)/parenrightbig\nG,F/a\\}bracketri}ht.(38)\nThe right side is precisely /a\\}bracketle{tK+(ϕ)F,G/a\\}bracketri}ht. Thus we have proven the adjointness rela-\ntion for the raising and lowering operators. The symmetry of the ne utral operators\nis proven by similar computations that we omit. /square\nNow we have all we need to prove Theorem 3.7.\nProof of Theorem 3.7. (i)Thesetofpolynomials DisdenseinL2(M,ρα)byLemma3.6\nand one easily checks that K±(ϕ) andK0(ϕ) map polynomials to polynomials.\n(ii) ClearyK−(ϕ) is antilinear in ϕandK0(ϕ) andK+(ϕ) are linear in ϕ.\n(iii) It is enough to check the commutation relations on monomials F(µ) =\nµ(f)n(every other monomial is a linear combination of such simple powers, e .g.,\nµ(f)µ(g) =1\n4(µ(f+g)2−µ(f−g)2)). Thus, let F(µ) =P(µ(f)) withP(x) =xn.\nWe show first that the lowering operators commute. Explicit comput ations yield\nK−(ϕ)F(µ) =µ(ϕf2)P′′/parenleftbig\nµ(f)/parenrightbig\n+α(ϕf)P′/parenleftbig\nµ(f)/parenrightbigREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 27\nand\nK−(θ)K−(ϕ)F(µ) =µ(θf2)µ(ϕf2)P(4)/parenleftbig\nµ(f)/parenrightbig\n+/parenleftBig\n2µ(ϕθf3)+α(ϕf)µ(θf2)+α(θf)µ(ϕf2)/parenrightBig\nP(3)/parenleftbig\nµ(f)/parenrightbig\n+/parenleftBig\nα(θϕf2)+α(θf)α(ϕf)/parenrightBig\nP′′/parenleftbig\nµ(f)/parenrightbig\n.\nThe right side is symmetric in θandϕ, hence [K−(θ),K−(ϕ)]F= 0. Turning to\nthe commutation relation between the lowering and the neutral ope rators, we note\nthatK0(ϕ) =−K−(ϕ)+R(ϕ)+1\n2α(ϕ) with\nR(ϕ)F(µ) =/integraldisplay\nϕ(x)δF\nδµ(x)µ(dx) =µ(ϕf)P′/parenleftbig\nµ(f)/parenrightbig\n.\nTo lighten notation, we drop the variables µ(f) from the polynomial Pand its\nderivatives in the final expressions. We compute\nR(ϕ)K−(θ)F(µ) =/integraldisplay\nϕ(x)δ\nδµ(x)/parenleftBig\nµ(θf2)P′′/parenleftbig\nµ(f)/parenrightbig\n+α(θf)P′/parenleftbig\nµ(f)/parenrightbig/parenrightBig\nµ(dx)\n=µ(ϕθf2)P′′+µ(θf2)µ(ϕf)P(3)+α(θf)µ(ϕf)P′′\nand\nK−(θ)R(ϕ)F(µ) =/integraldisplay\nθ(x)δ2\nδµ(x)2µ(ϕf)P′/parenleftbig\nµ(f)/parenrightbig\nµ(dx)\n+/integraldisplay\nθ(x)δ\nδµ(x)µ(ϕf)P′/parenleftbig\nµ(f)/parenrightbig\nα(dx)\n= 2µ(θϕf2)P′′+µ(ϕf)µ(θf2)P(3)\n+α(θϕf)P′+α(θf)µ(ϕf)P′′.\nIt follows that\n[K0(ϕ),K−(θ)]F(µ) = [R(ϕ),K−(θ)]F(µ)\n=−µ(θϕf2)P′′−α(θϕ)P′\n=−K−(θϕ)F(µ).\nThe other commutation relations are proven by similar computations that we omit.\n(iv) The adjointness relations have been proven in Lemma 5.7.\n(v) The constant function 1 has norm 1 in L2(M,ρα) becauseραis a proba-\nbility measure. It is annihilated by the lowering operators because th e latter are\ndifferential operators.\n(vi) We show first that raising operators yield Laguerre polynomials. Letℓ∈N\nandB1,...,B ℓ∈ Xbe disjoint sets with α(Bi)<∞. Further let n1,...,n ℓ∈N.\nRemember the univariate operators K+from Proposition 4.1. The operator used a\nscalarα>0as aparameter, but now αis ameasure; we define K+\niasthe univariate\noperator with scalar α(Bi)>0. Write 1for the constant function N0∋n/ma√sto→1. An\ninduction over n1+···+nℓyields\nK+( /BDB1)n1···K+( /BDBℓ)nℓ/BD(η) =ℓ/productdisplay\ni=1/parenleftbig\n(−K+\ni)ni1/parenrightbig/parenleftbig\nµ(Bi)/parenrightbig\n.\nThe minus sign arises because we made slightly different choices when d efining\nK#andK#(ϕ): In the univariate case, we prioritized the definition through uni-\ntary equivalence, for the current algebra we changed signs so tha tK+( /BDB)n/BDis a\npolynomial with positive leading coefficient. Next we note\n(−K+\ni)ni(µ(Bi))ni= (−1)nini!L(α(Bi)−1)\nni/parenleftbig\nµ(Bi)/parenrightbig\n=L(α(Bi)−1)\nni/parenleftbig\nµ(Bi)/parenrightbig\n.28 S. FLOREANI, S. JANSEN, AND S. WAGNER\nEq. (19) in Theorem 3.7 follows. In particular, the linear hull of /BDand the iterates/producttextℓ\ni=1K+( /BDBi)ni/BDcontains all polynomials in occupation numbers µ(B), whereB\nruns through the sets of finite measure α(B)<∞. As noted at the beginning\nof this section, these polynomials are dense in L2(M,ρα), therefore the constant\nfunction 1 is cyclic. /square\n5.4.Pascal point process. Proof of Theorem 3.9. For the commutation rela-\ntions forK±(ϕ),K+(θ), it is convenient to introduce another family of operators\nk#(ϕ),ϕ∈ C, defined on the space Dof polynomials by\nk−(ϕ)F(η) =1√p/integraldisplay\nϕ(x)F(η+δx)(α+η)(dx),\nk+(ϕ)F(η) =√p/integraldisplay\nϕ(x)F(η−δx)η(dx),\nk0(ϕ)F(η) =F(η)/integraldisplay\nϕ(x)/parenleftBig\nη+1\n2α/parenrightBig\n(dx).\nThese operators are similar to their counterparts on the extende d Fock space. No-\ntice\nK−(ϕ) =1\nc/parenleftbigg√pk+(ϕ)+1√pk−(ϕ)−2k0(ϕ)/parenrightbigg\n,\nK+(ϕ) =1\nc/parenleftbigg1√pk+(ϕ)+√pk−(ϕ)−2k0(ϕ)/parenrightbigg\n,\nK0(ϕ) =1\nc/parenleftBig\n−k+(ϕ)−k−(ϕ)+/parenleftbig√p−1+√p/parenrightbig\nk0(ϕ)/parenrightBig\n.\nLemma 5.8. The operators k#(ϕ),ϕ∈ C, mapDto itself and on Dthey satisfy\nthe commutation relations (7).\nProof.Consider a monomial F(η) =η(f1)···η(fn), withn∈Nandf1,...,f n∈ C.\nThen\nk+(ϕ)F(η) =√p/integraldisplay\nϕ(x)n/productdisplay\ni=1/parenleftbig\nη(fi)−fi(x)/parenrightbig\nη(dx)\n=√p/summationdisplay\nI⊂[n](−1)n−|I|/parenleftBigg/productdisplay\ni∈Iη(fi)/parenrightBigg\nη/parenleftBig\nϕ/productdisplay\ni∈[n]\\Ifi/parenrightBig\nwhere [n] ={1,...,n}. It follows that k+(ϕ)Fis a polynomial. The constant\nfunction /BDismappedto√pη(ϕ), whichisapolynomialaswell. Aseverypolynomial\nisalinearcombinationof /BDandmonomialsofdegree n≥1,itfollowsthat k+(ϕ)D⊂\nD. The reasoning for k−(ϕ) andk0(ϕ) is similar and therefore omitted.\nFor the commutation relations, we note\nk−(ϕ)k+(ψ)F(η) =/integraldisplay\nϕ(x)/parenleftBig/integraldisplay\nψ(y)F(η+δx−δy)(η+δx)(dy)/parenrightBig\n(α+η)(dx),\nk+(ψ)k−(ϕ)F(η) =/integraldisplay\nψ(y)/parenleftBig/integraldisplay\nϕ(x)f(η+δx−δy)(α+η−δy)(dx)/parenrightBig\nη(dy).\nWhen we subtract the second line from the first, everything cance ls out except the\nDirac terms δx(dy) and−δy(dx) in the inner integrals. Hence\n[k−(ϕ),k+(ψ)]F(η) =/parenleftBig/integraldisplay\nϕ(x)ψ(x)(α+η)(dx)+/integraldisplay\nψ(y)ϕ(y)η(dy)/parenrightBig\nF(η)\n= 2k0(ϕψ)F(η).\nThe other commutation relations are proven by similar explicit comput ations, we\nleave the details to the reader. /squareREPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 29\nLemma5.9. The operators k#(ϕ)satisfy the adjointness relations (8)inL2(N,ρp,α).\nProof.The neutral operators k0(ϕ) is a multiplication operators which multiplies\nwithη(ϕ) +1\n2α(ϕ). The adjoint multiplies with the complex conjugate, i.e., with\nη(ϕ)+1\n2α(ϕ), hence /a\\}bracketle{tF,k0(ϕ)G/a\\}bracketri}ht=/a\\}bracketle{tk0(ϕ)F,G/a\\}bracketri}ht.\nFork±(ϕ), we exploit the relation\n/integraldisplay/integraldisplay\nF(x,η)η(dx)ρp,α(dη) =/integraldisplay/integraldisplay\nF(x,η+δx)p(α+η)(dx)ρp,α(dη).(39)\nItisvalidforallmeasurable F:N×X→R+, see[19, Proposition3.1]. Let ϕ∈ Cbe\nnon-negative and f,gbe non-negativepolynomials, e.g., f(η) =η(f1)···η(fn) with\nfi:X→R+inC, similarly for g. Eq. (39) applied to F(x,η) =ϕ(x)f(η+δx)g(η)\nyields/a\\}bracketle{tf,k+(ϕ)g/a\\}bracketri}ht=/a\\}bracketle{tk−(ϕ)f,g/a\\}bracketri}ht, compare [19, Lemma 3.2]. The relation is ex-\ntended to all polynomials and all ϕ∈ Cby taking linear combinations. /square\nNow we are equipped for the proof of Theorem 3.9.\nProof of Theorem 3.9. (i) The space Dof polynomials is dense in L2(N,ρp,α) by\nLemma3.8andtheoperators K#(ϕ), aslinearcombinationsoftheoperators k#(ϕ),\nmapDto itself by Lemma 5.8.\n(ii) Clearly K+(ϕ) andK0(ϕ) are linear in ϕandK−(ϕ) is antilinear in ϕ.\n(iii) The commutation relations are deduced from Lemma 5.8 as follows. The\ncommutator [ K−(ϕ),K+(θ)], timesc2, is a sum of three terms. First,\n/bracketleftBig√pk+(ϕ)+1√pk−(ϕ),1√pk+(θ)+√pk−(θ)/bracketrightBig\n=−2pk0(ϕθ)+2\npk0(ϕθ).\nSecond,\n/bracketleftBig√pk+(ϕ)+1√pk−(ϕ),−2k0(θ)/bracketrightBig\n= 2√pk+(ϕθ)−2√pk−(ϕθ).\nThird,\n/bracketleftBig\n−2k0(ϕ),1√pk+(θ)+√pk−(θ)/bracketrightBig\n=−2√pk+(ϕθ)+2√pk−(ϕθ).\nThe sum of these three expressions is\n−2c/parenleftbig\nk+(ϕθ)+k−(ϕθ)/parenrightbig\n+2c/parenleftbig√p−1+√p)/parenrightbig\nk0(ϕθ/parenrightbig\n= 2c2K0(ϕθ).\nThis completes the proof of [ K−(ϕ),K+(θ)] = 2K0(ϕθ). The proof of the other\ncommutation relations is similar.\n(iv) The adjointness relations for K#(ϕ) follow from the relations for k#(ϕ),\nproven in Lemma 5.9.\n(v) The constant function Ψ = /BDhas norm 1 in L2(N,ρp,α) becauseρp,αis a\nprobability measure. The function Ψ = /BDis annihilated by all lowering operators\nK−(ϕ) because D+\nx\n/BD(η) = D−\nx\n/BD(η) = 0 for all x,η.\n(vi) We show first that raising operators yield Meixner polynomials. Le tℓ∈N\nandB1,...,B ℓ∈ Xbe disjoint sets with α(Bi)<∞. Further let n1,...,n ℓ∈N.\nRemember the univariate operators K+from Proposition 4.2. The operator used a\nscalarα>0as aparameter, but now αis ameasure; we define K+\niasthe univariate\noperator with scalar α(Bi)>0. Write 1for the constant function N0∋n/ma√sto→1. An\ninduction over n1+···+nℓyields\nK+( /BDB1)n1···K+( /BDBℓ)nℓ/BD(η) =ℓ/productdisplay\ni=1/parenleftbig\n(−K+\ni)ni1/parenrightbig/parenleftbig\nη(Bi)/parenrightbig\n.30 S. FLOREANI, S. JANSEN, AND S. WAGNER\nThe minus sign arises because we made slightly different choices when d efining\nK#andK#(ϕ): In the univariate case, we prioritized the definition through uni-\ntary equivalence, for the current algebra we changed signs so tha tK+( /BDB)n/BDis a\npolynomial with positive leading coefficient. By Eq. (28),\n(−K+\ni)ni(η(Bi)) = (−1)ni/parenleftbig√p−√p−1/parenrightbigniMni/parenleftbig\nη(Bi);α(Bi),p/parenrightbig\n.\nTheprefactorinfrontoftheMeixnerpolynomialisprecisely cniandEq.(23)inThe-\norem3.9follows. Inparticular,thelinearhullof /BDandtheiterates/producttextℓ\ni=1K+( /BDBi)ni/BD\ncontainsallpolynomialsinoccupationnumbers η(B), whereBrunsthroughthesets\nof finite measure. By an argument completely analogous to the proo f of Lemma 3.6\ngiven in Section 5.3, this space is dense in L2(N,ρp,α). Thus Ψ = /BDis cyclic. /square\nAppendix A.Representation in standard Fock space. Araki’s scheme\nAraki’s factorizable representations of a current group Gare of the form [5]\nUgEf= exp(icg)exp/parenleftBig\n−1\n2||ϕg||2+/a\\}bracketle{tϕg,f/a\\}bracketri}ht/parenrightBig\nEQg(f−ϕg).\nThe unitaries Ugoperate in a Fock space Hfor which the single-particle Hilbert\nspace has the structure of a fiber integral h1=/integraltext⊕hdα. The exponential states Ef\nare dense. The unitaries Qgform a representation of Gin the one-particle Hilbert\nspaceh1, they are diagonal with respect to the fiber decomposition. The ph ase\ncg∈Rand the map g/ma√sto→ϕg∈h1have to satisfy some consistency conditions\n(cocycle conditions). The difficulty in Araki’s approach is to find cocyc les.\nHere we sketch how our representation in extended Fock space on Xmay be\nlifted to a representation in a Fock space on X×Nand how it fits Araki’s scheme.\nA.1.Isometric embedding into a standard Fock space. Chinese res tau-\nrant process. We start with a known isomorphism of the extended Fock space\nwith a subspace of a standard Fock space [33, 38]. Let Hbe a standard Fock space\nfor the one-particle Hilbert space L2(X×N,α⊗/summationtext∞\nn=11\nnδn). We include a factorial\nin the norm, i.e., the norm of an element F= (Fm)m∈N0ofHis\n||F||2=|F0|2\n+∞/summationdisplay\nm=11\nm!/integraldisplay/summationdisplay\nn1,...,nm/vextendsingle/vextendsingleFm/parenleftbig\n(x1,n1),(x2,n2),...(xm,nm)/parenrightbig/vextendsingle/vextendsingle21\nn1···1\nnmαm(dx).\nIn the picturesque language of the Chinese restaurant process [45, Chapter 3.1], a\nconfiguration ( x1,n1),...,(xm,nm) represents a seating plan in a restaurant with\nmtables; table no. iis located at place xiand hasnicustomers. (More prosaically,\nthink ofmpiles withniparticlessitting atopeachother ona location xi.) The total\nnumber of customers (particles) is n=n1+···+nmand we may list the locations\nof the customers by repeating the entry xi, for eachi= 1,...,m,nitimes:\n(xn1\n1xn2\n2···xnm\nm) = (x1,...x1,...,x m,...,x m)∈Xn1+···+nm.\nThis induces a mapping from the extended Fock space Fto the Hilbert space H:\nGiven an element f= (fn)n∈N0ofF, we define a new sequence F= (Fm)m∈N0by\nF0=f0, Fm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n=f/summationtextni(xn1\n1···xnm\nm).\nThe map U:F→H,f/ma√sto→Fis norm-preserving. It maps exponential states to\nexponential states: The image of Ez∈Fis given by F0= 0 and\nFm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n=m/productdisplay\ni=1z(xi)ni=m/productdisplay\ni=1f(xi,ni) (40)\nwithf(x,n) =z(x)n. That is, UEz=Efwhere Efis an exponential state in H.REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 31\nA.2.Lifting the representation. Theoperators k#(ϕ)fromSection3.1arelifted\nto operators K#(ϕ) inHwithUk#(ϕ) =K#(ϕ)U, as follows. The neutral operator\ninHis a multiplication operator\nK0(ϕ)Fm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n=/parenleftBig1\n2α(ϕ)+m/summationdisplay\ni=1niϕ(xi)/parenrightBig\nFm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n.\nThe lowering operator is\nK−(ϕ)Fm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n=m/summationdisplay\ni=1ϕ(xi)niFm/parenleftbig\n(x1,n1)...(xi,ni+1)...(xm,nm)/parenrightbig\n+/integraldisplay\nϕ(x)Fm+1/parenleftbig\n(x1,n1),...,(xm,nm),(y,1)/parenrightbig\nα(dy)\n(a new customer may join a table or open a new table). The raising ope rator is\nK+(ϕ)Fm/parenleftbig\n(x1,n1),...,(xm,nm)/parenrightbig\n=m/summationdisplay\ni=1ϕ(xi)ni/parenleftBig/BD{ni≥2}Fm/parenleftbig\n(x1,n1)...(xi,ni−1)...(xm,nm)/parenrightbig\n+ /BD{ni=1}Fm−1/parenleftbig\n(x1,n1).../hatwider(xi,1)...(xm,nm)/parenrightbig/parenrightBig\n(if asolodinerleaves, the number moftables decreases). Exponentiatingasin The-\norem 3.3, we should obtain a family of unitary operators in Hand a representation\nof the current group for the universal cover of SU(1,1). The action of\nU(ξ,θ) = exp/parenleftbig\nK+(ξ)−K−(ξ)/parenrightbig\nexp/parenleftbig\n2iK0(θ)/parenrightbig\non exponential states Efwithf(x,n) =z(x)n(see (40)) is\nU(ξ,θ)Ef=Cξ,θ(f)E/hatwidef,/hatwidef(x,n) =zξ,θ(x)n(41)\nwith\nCξ,θ(f) = exp/parenleftBig\ni/integraldisplay\nθdα−/integraldisplay\nlog/parenleftBig\ncosh|ξ|+e2iθzξ\n|ξ|sinh|ξ|/parenrightBig\ndα/parenrightBig\n(42)\nand\nzξ,θ(x) =e2iθ(x)z(x)+ξ(x)\n|ξ(x)|tanh|ξ(x)|\n1+e2iθ(x)z(x)ξ(x)\n|ξ(x)|tanh|ξ(x)|. (43)\nA.3.Matching the lifted representation to Araki’s scheme. We introduce\na single-table, non-spatial Hilbert space h=ℓ2(N,/summationtext∞\nn=11\nnδn), and operators\nk+f(n) =n /BD{n≥2}f(n−1),k−f(n) =nf(n+1),k0f(n) =nf(n).\nThese operatorssatisfy the su(1,1) commutation relations. The vacuum is ψ0(n) =\nδn,1, the Bargmann index is m0= 2 (because k0ψ0=2\n2ψ0). Set\nQξ,θ= exp(ξk+−ξk−)exp(2iθk0).\nWe use the same letters for the operators in L2(X×N,α⊗/summationtext∞\nn=11\nnδn) that act on\nthen-variable only. Further set\nϕξ,θ(x,n) =/parenleftBig\n−e2iθξ\n|ξ|tanh|ξ|/parenrightBign\n.32 S. FLOREANI, S. JANSEN, AND S. WAGNER\nProposition A.1. Letξ,θ∈ Cbe complex- and real-valued functions in C, re-\nspectively, and f∈L2(X×N,α⊗/summationtext∞\nn=11\nnδn). The unitary U(ξ,θ)acts on the\nexponential states Ef∈Has\nU(ξ,θ)Ef= exp/parenleftBig\ni/integraldisplay\nθdα−1\n2||ϕξ,θ||2+/a\\}bracketle{tϕξ,θ,f/a\\}bracketri}ht/parenrightBig\nEQξ,θ(f−ϕξ,θ).(44)\nProof sketch. Forξ≡0, the vector ϕξ,θvanishes and exp(2i θk0) is multiplication\nwith exp(2i θ(x)). Thus, the right side of Eq. (44) is exp(i α(θ))Eexp(2iθ)f, which\nis indeed the same as exp(2i K0(θ))Ef. Forθ≡0, one may replace ξbytξon\nboth sides in (44), differentiate with respect to t, and check that the differentiated\nequality holds true. The procedure is similar to the proof of Proposit ion 5.1 in our\narticle [19], we omit the details. The full statement is deduced by conc atenating\nU(ξ,θ) =U(ξ,0)U(0,θ). /square\nWe complement the proof sketch by a consistency check with Eqs. ( 41)–(43).\nForf(x,n) =z(x)nwith sup |z|<1, we compute\n/a\\}bracketle{tϕξ,θ,f/a\\}bracketri}ht=∞/summationdisplay\nn=11\nn/integraldisplay\nϕξ,θ(x,n)f(x,n)α(dx) =−/integraldisplay\nlog/parenleftBig\n1+zξ\n|ξ|tanh|ξ|/parenrightBig\ndα.\nWith the equality 1 −(tanhs)2= 1/(coshs)2, we get\n−1\n2||ϕξ,θ||2=1\n2/integraldisplay\nlog/parenleftbig\n1−(tanh|ξ|)2/parenrightbig\ndα=−/integraldisplay\nlog(cosh|ξ|)dα.\nIt follows that the exponential on the right side in (44) is equal to th e factor\nCξ,θfrom (41) and (42). It remains to check that EQξ,θ(f−ϕξ,θ)corresponds to/producttext\njzξ,θ(xj)njwithzξ,θ(x). For simplicity we only consider θ≡0 and real-valued\nξ. Checking the desired equality boils down to a computation in ℓ2(N,/summationtext∞\nn=11\nnδn):\nfort∈R,z∈Cwith|z|<1, and\nzt=z+tanht\n1+ztanht, ft(n) =zn\nt\none has\nd\ndtzn\nt=nzn−1\nt(1−z2\nt) = (k+−k−)ft(n)+δn,1.\nThe special case z=−tanhswith fixeds∈Rshows that gt(n) = (tanh(t−s))n\nsatisfies the same differential equality, therefore ∂t(ft−gt) = (k+−k−)(ft−gt)\nandft−gt= exp(t(k+−k−))(f0−g0). In the previous equality g0depends on\nsand the equality holds true for every s, in particular, s=−t. For this choice\ng0(n) = (−tanht)nandgt(n) = 0. Thus, we get\net(k+−k−)(f0−g0)(n) =ft(n) =zn\nt\nwhich is essentially the desired equality.\nAppendix B.Generalized Lebesgue measure and SL(2,R)\nHereweexplainhowourrepresentationswiththeGammarandommea surerelate\nto representations of the special linear group with the multiplicative measure. Let\nus first briefly describe the construction, following Tsilevich, Versh ik and Yor [57].\nWe assume that the measure αonXis finite so that the Gamma random measure\nwith lawραhas almost surely finite total mass, µ(X)<∞. The measure Lαis by\ndefinition the σ-finite measure on Mthat is absolutely continuous with respect to\nρα, with Radon-Nikodym derivative\ndLα\ndρα(µ) = exp(µ(X)).REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 33\nThe definition is motivated by a simple observation in the univariate cas e: The\nLebesgue measure on R+is absolutely continuous with respect to the Gamma\ndistribution e−xdxwith parameter 1, with Radon-Nikodym derivative exp( x).\nTheSL(2,R) current group over Xconsists of the bounded measurable maps\ntaking values in SL(2,R), the real 2 ×2 matrices with determinant 1. The lower\ntriangular subgroup consists of the elements g:X→SL(2,R) of the form\ng(x) =/parenleftbigga(x)−10\nb(x)a(x)/parenrightbigg\nwith bounded measurable maps a,b:X→R. The map ais bounded away from\nzero. Forµ∈M, the dilated measure Ma2µis given by Ma2µ(B) =/integraltext\nBa2(x)µ(dx).\nForF∈L2(M,Lα) andgas above, define\nUgF(µ) = exp/parenleftBig/integraldisplay\n|loga|dα+i/integraldisplay\nabdµ/parenrightBig\nF(Ma2µ).\nThe mapg/ma√sto→ Ugis a unitary representation of the triangular subgroup of the\nSL(2,R) current group ([57, Section 5] and references therein). The ac tion of the\nfull group is more complicated; in the univariate case it requires, in ad dition to the\nmultiplication and dilation operators, an integral operator [57].\nTo establish the connection with our representation from Section 3 .3, we first\nnotice that the spaces are unitarily equivalent. The map F/ma√sto→˜Fwith˜F(µ) =\nexp(µ(X)/2)F(µ) is a unitary isomorphism from L2(M,ρα) ontoL2(M,Lα). In\nL2(M,ρα), the unitary Ugbecomes\nUgF(µ) = exp/parenleftBig/integraldisplay\n|loga|dα+i/integraldisplay\nabdµ+1\n2/integraldisplay\n(1−a2)dµ/parenrightBig\nF(Ma2µ)\nFora(x)≡1, this is simply multiplication with exp(i µ(b)) hence, in view of (18),\nthis is the operator exp(i( K+(b) +K−(b) + 2K0(b))). Forb(x)≡0 anda(x) =\nexp(tϕ(x)) with real-valued bounded ϕ,UgmapsFto\nFt(µ) = exp/parenleftBig\nt/integraldisplay\nϕdα+1\n2/integraldisplay\n(1−e2tϕ)dµ/parenrightBig\nF(Mexp(2tϕ)µ).\nAn explicit computation for polynomial Fgives\n∂tFt(µ) = 2/integraldisplay\nϕ(x)δFt\nδµ(x)(µ)µ(dx)+/parenleftbig\nα(ϕ)−µ(ϕ)/parenrightbig\nFt(µ)\n=/parenleftbig\nK−(ϕ)−K+(ϕ)/parenrightbig\nFt(µ)\nandhence,theunitary Ugforgthediagonalmatrixwithdiagonalentriesexp( −tϕ(x)),\nexp(tϕ(x)), corresponds to exp( t(K−(ϕ)−K+(ϕ)).\nTo conclude, let k±,k0be the 2 ×2 matrices from Section 2.1. The correspon-\ndence\ni(k++k−+2k0)/ma√sto→/parenleftbigg0 0\n1 0/parenrightbigg\n, k−−k+/ma√sto→/parenleftbigg−1 0\n0 1/parenrightbigg\n,\ni(k++k−−2k0)/ma√sto→/parenleftbigg\n0 1\n0 0/parenrightbigg\nextends to an isomorphism from the real Lie algebra su(1,1) ontosl(2,R).\nThus, the representations with the multiplicative measure Lαagree with our\nrepresentations with the Gamma random measure. We pass from on e to the other\nby (1) mapping L2(M,Lα) toL2(M,ρα), and (2) exploiting the isomorphism be-\ntween the real Lie algebra su(1,1) andsl(2,R).34 S. FLOREANI, S. JANSEN, AND S. WAGNER\nAcknowledgements. S.F. acknowledges financial support from the Deutsche For-\nschungsgemeinschaft (DFG, German Research Foundation) unde r Germany’s Ex-\ncellence Strategy– EXC-2047/1 – 390685813. S.J. and S.W. were su pported un-\nder Germany’s excellence strategy EXC-2111-390814868. S.F. an d S.W. thank the\nHausdorff Institute for Mathematics (Bonn) for its hospitality dur ing the Junior\nTrimester Program Stochastic modelling in life sciences funded by the Deutsche\nForschungsgemeinschaft (DFG, German Research Foundation) u nder Germany’s\nExcellence Strategy - EXC-2047/1 - 390685813. The authors tha nk F. Redig for\nhis insights at the beginning of the project. S.W. thanks D. Span` o f or helpful\ndiscussions.\nReferences\n[1] L. Accardi and A. Boukas, Itˆ o Calculus and Quantum White Noise Calculus , Stochastic\nAnalysis and Applications (Berlin, Heidelberg) (F.E. Bent h, G. Di Nunno, T. Lindstrøm,\nB. Øksendal, and T. Zhang, eds.), Springer Berlin Heidelber g, 2007, pp. 7–51.\n[2] ,Quantum probability, renormalization and infinite-dimens ional∗-Lie algebras ,\nSIGMA Symmetry Integrability Geom. Methods Appl. 5(2009), Paper 056, 31.\n[3] L. Accardi, U. Franz, and M. Skeide, Renormalized squares of white noise and other non-\nGaussian noises as L´ evy processes on real Lie algebras , Comm. Math. Phys. 228(2002),\nno. 1, 123–150.\n[4] L. Accardi and M.Skeide, On the relation of the square of white noise and the finite diffe rence\nalgebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3(2000), no. 1, 185–189.\n[5] H. Araki, Factorizable representation of current algebra. Non commu tative extension of the\nL´ evy-Kinchin formula and cohomology of a solvable group wi th values in a Hilbert space ,\nPubl. Res. Inst. Math. Sci. 5(1969/70), 361–422.\n[6] D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators ,\nGrundlehren der mathematischen Wissenschaften [Fundamen tal Principles of Mathematical\nSciences], vol. 348, Springer, Cham, 2014.\n[7] V. Bargmann, Irreducible unitary representations of the Lorentz group , Ann. of Math. (2)\n48(1947), 568–640.\n[8] A. Barhoumi, H. Ouerdiane, and A. Riahi, Unitary representations of the Witt and sl(2,R)-\nalgebras through renormalized powers of the quantum Pascal white noise , Infin. Dimens.Anal.\nQuantum Probab. Relat. Top. 11(2008), no. 3, 323–350.\n[9] Y.M.Berezansky, On the theory of commutative Jacobi fields , Methods Funct. Anal.Topology\n4(1998), no. 1, 1–31.\n[10] Y.M. Berezansky and D.A. Mierzejewski, The structure of the extended symmetric Fock\nspace, Methods Funct. Anal. Topology 6(2000), no. 4, 1–13.\n[11] V.I. Bogachev, Measure theory , vol. 1, Springer, Berlin, 2007.\n[12] A. Boukas, Quantum stochastic analysis: A non-Brownian case , ProQuest LLC, Ann Arbor,\nMI, 1988, Thesis (Ph.D.)–Southern Illinois University at C arbondale.\n[13] ,An example of a quantum exponential process , Monatsh. Math. 112(1991), no. 3,\n209–215.\n[14] C. Brif, A. Vourdas, and A. Mann, Analytic representations based on SU(1,1)coherent states\nand their applications , J. Phys. A 29(1996), no. 18, 5873–5885.\n[15] G. Carinci, C. Giardin` a, and F. Redig, Duality for Markov processes: a Lie-algebraic ap-\nproach, Book, in preparation, 2024+.\n[16] G. Chiribella, G.M. D’Ariano, and P. Perinotti, Applications of the group SU(1,1)for quan-\ntum computation and tomography , Laser physics 16(2006), no. 11, 1572–1581.\n[17] J.C. Cox, J.E. Ingersoll, and S.A. Ross, A theory of the term structure of interest rates ,\nEconometrica 53(1985), no. 2, 385–407.\n[18] S.N. Ethier and R.C. Griffiths, The transition function of a measure-valued branching diffu -\nsion with immigration , Stochastic processes: A Festschrift in Honour of Gopinath Kallianpur,\nSpringer, New York, 1993, pp. 71–79.\n[19] S. Floreani, S. Jansen, and S. Wagner, Intertwinings for Continuum Particle Systems: An\nAlgebraic Approach , Preprint. arXiv:2311.08763, 2023.\n[20] J. Fuchs, Affine lie algebras and quantum groups: An introduction, with applications in\nconformal field theory , Cambridge University Press, 1995.\n[21] I.M. Gel’fand, M.I. Graev, and A.M. Vershik, Models of representations of current groups ,\nRepresentations of Lie groups and Lie algebras, Proc. Summe r Sch., Budapest 1971, Pt. 2,\n121-179 (1985)., 1985.REPRESENTATIONS OF THE su(1,1) CURRENT ALGEBRA 35\n[22] C. Giardin` a, J. Kurchan, and F. Redig, Duality and exact correlations for a model of heat\nconduction , J. Math. Phys. 48(2007), no. 3, 033301.\n[23] C. Giardin` a, J. Kurchan, F. Redig, and K. Vafayi, Duality and hidden symmetries in inter-\nacting particle systems , J. Stat. Phys. 135(2009), no. 1, 25–55.\n[24] J. Glimm and A. Jaffe, Quantum physics. a functional integral point of view , second ed.,\nSpringer-Verlag, New York, 1987.\n[25] W. Groenevelt, Orthogonal Stochastic Duality Functions from Lie Algebra R epresentations ,\nJ. Stat. Phys. 174(2019), no. 1, 97–119.\n[26] R.L. Hudson and K.R. Parthasarathy, Quantum Ito’s formula and stochastic evolutions ,\nComm. Math. Phys. 93(1984), no. 3, 301–323.\n[27] Y. Ito and I. Kubo, Calculus on Gaussian and Poisson white noises , Nagoya Math. J. 111\n(1988), 41–84.\n[28] S. Jansen and N. Kurt, On the notion(s) of duality for Markov processes , Probab. Surv. 11\n(2014), 59–120.\n[29] V.G. Kac, Infinite-dimensional Lie algebras , Cambridge University Press, 1990.\n[30] S. Karlin and J. McGregor, Linear growth birth and death processes , J. Math. Mech. 7(1958),\n643–662.\n[31] C. Kipnis, C. Marchioro, and E. Presutti, Heat flow in an exactly solvable model , J. Statist.\nPhys.27(1982), no. 1, 65–74.\n[32] R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and\ntheir q-Analogues , Springer Monographs in Mathematics, Springer, Berlin, 20 10.\n[33] Y.G. Kondratiev, J.L. da Silva, L Streit, and G.F. Us, Analysis on Poisson and gamma\nspaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(1998), no. 1, 91–117.\n[34] Y.G. Kondratiev and E. Lytvynov, Operators of gamma white noise calculus , Infin. Dimens.\nAnal. Quantum Probab. Relat. Top. 3(2000), no. 3, 303–335.\n[35] G. Last, Stochastic analysis for Poisson processes , Stochastic analysis for Poisson point pro-\ncesses. Malliavin calculus, Wiener-Itˆ o chaos expansions and stochastic geometry (G. Peccati\nand M. Reitzner, eds.), Bocconi Springer Ser., vol. 7, Bocco ni Univ. Press, 2016, pp. 1–36.\n[36] G. Last and M.Penrose, Lectures on the Poisson Process , Institute of Mathematical Statistics\nTextbooks, Cambridge University Press, 2017.\n[37] G. Lindblad and B. Nagel, Continuous bases for unitary irreducible representations of\nSU(1,1), Ann. Inst. H. Poincar´ e Sect. A (N.S.) 13(1970), 27–56.\n[38] E. Lytvynov, Orthogonal decompositions for L´ evy processes with an appl ication to the\ngamma, Pascal, and Meixner processes , Infin. Dimens. Anal. Quantum Probab. Relat. Top.\n6(2003), no. 1, 73–102.\n[39] ,Polynomials of Meixner’s type in infinite dimensions—Jacob i fields and orthogonal-\nity measures , J. Funct. Anal. 200(2003), no. 1, 118–149.\n[40] ,The square of white noise as a Jacobi field , Infin. Dimens. Anal. Quantum Probab.\nRelat. Top. 07(2004), 619–629.\n[41] P. Meyer, Quantum Probability for Probabilists , 2 ed., Lecture Notes in Mathematics, vol.\n1538, Springer, Berlin, 1995.\n[42] L. Miclo and P. Patie, On a gateway between continuous and discrete Bessel and Lagu erre\nprocesses , Ann. H. Lebesgue 2(2019), 59–98.\n[43] D. Nualart, The Malliavin calculus and related topics , second ed., Probability and its Appli-\ncations (New York), Springer-Verlag, Berlin, 2006.\n[44] A.M. Perelomov, Coherent states for arbitrary Lie group , Comm. Math. Phys. 26(1972),\n222–236.\n[45] J. Pitman, Combinatorial stochastic processes , Lecture Notes in Mathematics, vol. 1875,\nSpringer, Berlin, 2006, Lectures from the 32nd Summer Schoo l on Probability Theory held\nin Saint-Flour, July 7–24, 2002, With a foreword by Jean Pica rd.\n[46] M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analys is, self-\nadjointness , Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London,\n1975.\n[47] M. Riesz, Sur le probl` eme des moments et le th´ eor` eme de Parseval cor respondant. , Acta Litt.\nSci. Szeged 1(1923), 209–225.\n[48] W.R¨ uhland B.C.Yunn, Representations of the universal covering group of SU(1,1)and their\nbilinear and trilinear invariant forms , J. Mathematical Phys. 17(1976), no. 8, 1521–1530.\n[49] W. Schoutens, Stochastic processes and orthogonal polynomials , Lecture Notes in Statistics,\nvol. 146, Springer, New York, 2000.\n[50] G. Sch¨ utz and S. Sandow, Non-Abelian symmetries of stochastic processes: Derivati on of\ncorrelation functions for random-vertex models and disord ered-interacting-particle systems ,\nPhys. Rev. E 49(1994), no. 4, 2726.\n[51] M.O. Scully and M.S. Zubairy, Quantum optics , Cambridge University Press, 1997.36 S. FLOREANI, S. JANSEN, AND S. WAGNER\n[52] P.´Sniady,Quadratic bosonic and free white noises , Comm. Math. Phys. 211(2000), no. 3,\n615–628.\n[53] D. Span` o and A. Lijoi, Canonical correlations for dependent gamma processes , Preprint.\narXiv:1601.06079, 2016.\n[54] W. Stannat, Spectral properties for a class of continuous state branchi ng processes with\nimmigration , J. Funct. Anal. 201(2003), no. 1, 185–227.\n[55] A. Sturm, J.M. Swart, and F. V¨ ollering, The algebraic approach to duality: an introduction ,\nGenealogies of Interacting Particle Systems, Lect. Notes S er. Inst. Math. Sci. Natl. Univ.\nSingap., vol. 38, World Sci. Publ., Hackensack, NJ, 2020, pp . 81–150.\n[56] D.R. Truax, Baker-Campbell-Hausdorff relations and unitarity of SU(2)andSU(1,1)squeeze\noperators , Physical Review D (3) 31(1985), no. 8, 1988–1991.\n[57] N. Tsilevich, A. Vershik, and M. Yor, An infinite-dimensional analogue of the Lebesgue\nmeasure and distinguished properties of the gamma process , J. Funct. Anal. 185(2001),\nno. 1, 274–296.\n[58] A.M. Vershik and M.I. Graev, Integral models of representations of current algebras of s imple\nLie groups , Uspekhi Mat. Nauk 64(2009), no. 2(386), 5–72, Translation in Russian Math.\nSurveys 64(2009), no. 2, 205–271.\n[59] ,A Poisson model of the Fock space and representations of curr ent groups , Algebra\ni Analiz 23(2011), no. 3, 63–136, Translation in St. Petersburg Math. J .23(2012), no. 3,\n459–510.\n[60] A.M. Verˇ sik, I.M. Gel’fand, and M.I. Graev, Representations of the group SL(2,R), where\nRis a ring of functions , Uspehi Mat. Nauk 28(1973), no. 5(173), 83–128, Translation in\nRussian Math. Surveys 28(1973), no. 5, 87–132.\nUniversit ¨at Bonn, Institut f ¨ur Angewandte Mathematik, Abt. Stochastische Sys-\nteme mit Wechselwirkung, Endenicher Allee 60, 53115 Bonn, Ge rmany\nEmail address :sflorean@uni-bonn.de\nMathematisches Institut, Ludwig-Maximilians-Universit ¨at, 80333 M ¨unchen; Munich\nCenter forQuantumScience andTechnology (MCQST), Schelli ngstr. 4, 80799M ¨unchen,\nGermany.\nEmail address :jansen@math.lmu.de\nMathematisches Institut, Ludwig-Maximilians-Universit ¨at, 80333 M ¨unchen; Munich\nCenter forQuantumScience andTechnology (MCQST), Schelli ngstr. 4, 80799M ¨unchen,\nGermany.\nEmail address :swagner@math.lmu.de"    },    {        "title": "2402.07509v1.First_order_behavior_of_the_time_constant_in_non_isotropic_continuous_first_passage_percolation.pdf",        "content": "First-order behavior of the time constant in non-isotropic\ncontinuous first-passage percolation\nAnne-Laure Basdevant∗, Jean-Baptiste Gouéré†and Marie Théret‡\nAbstract\nLetNbe a norm on Rdwithd≥2and consider χa homogeneous Poisson point process on Rd\nwith intensity ε∈[0,∞). We define the Boolean model ΣN,εas the union of the balls of diameter 1\nfor the norm Nand centered at the points of χ. For every x,y∈Rd, LetTN,ε(x,y)be the minimum\ntime needed to travel from xtoyif one travels at speed 1outside ΣN,εand at infinite speed inside\nΣN,ε: this defines a continuous model of first-passage percolation, that has been studied in [10, 11]\nforN=∥·∥ 2, the Euclidean norm. The exact calculation of the time constant of this model\nµN,ε(x) := limn→∞TN,ε(0,nx)/nis out of reach. We investigate here the behavior of ε∝⇕⊣√∫⊔≀→µN,ε(x)\nnear 0, and enlight how the speed at which N(x)−µN,ε(x)goes to 0depends on xandN. For\ninstance, if Nis thep-norm forp∈(1,∞), we prove that N(x)−µ∥·∥p,ε(x)is of orderεκp(x)with\nκp(x) :=1\nd−d1(x)−1\n2−d−d1(x)\np,\nwhered1(x)is the number of non null coordinates of x. Together with the study of the time constant,\nwe also prove a control on the N-length of the geodesics, and get some informations on the number of\npoints ofχreally useful to those geodesics. The results are in fact more natural to prove in a slightly\ndifferent setting, where instead of centering a ball of diameter 1at each point of χand traveling at\ninfinite speed inside ΣNε, we put a reward of one unit of time at those points.\n1 Introduction and main results\n1.1 Background and motivations\nFirst-passage percolation on Zd.First-passage percolation was introduced by Hammersley and\nWelsh [13] in 1965 as a toy model to understand propagation phenomenon. Whereas the model of\npercolation questions if the propagation occurs or not, first-passage percolation describes at what speed\nthe propagation occurs. The media is represented by the graph Zd(d≥2). With each edge ebetween\nnearest neighbours in Zd, we associate a non-negative random variable τ(e)called the passage time of\ne,i.e., the time needed to cross the edge e. The family (τ(e))is chosen i.i.d. with common distribution\nfunctionF. This interpretation leads naturally to the definition of a random pseudo metric τonZdin\nthe following way. If π= (x0,e1,x1,...,en,xn)is a path between two vertices x0andxnofZd,i.e.,\nan alternating sequence of vertices (xi)and edges (ei)such thateiis precisely the edge of endpoints\nxi−1andxi, the passage time τ(π)ofπis defined by τ(π) =/summationtextn\ni=1τ(ei). For any couple of vertices\n(x,y)∈Zd×Zd, the minimal time needed to observe propagation from xtoyis thus defined as\nτ(x,y) = inf{τ(π)|π:x→y}where the infimum is taken over all paths πfromxtoy. We refer to\nthe surveys [2, 15] for an overview on classical results in this model. By Kingman’s ergodic subadditive\ntheorem, it is well known that under a moment assumption on F, for anyz∈Rd\\{0}, we have\nlim\nn→∞τ(0,⌊nz⌋)\nn=µF(z)a.s. and in L1, (1)\n∗LPSM - UMR CNRS 8001, Sorbonne Université, 4 place Jussieu, 75005 Paris, France,\nanne.laure.basdevant@normalesup.org\n†Institut Denis-Poisson - UMR CNRS 7013, Université de Tours, Parc de Grandmont, 37200 Tours, France, jean-\nbaptiste.gouere@lmpt.univ-tours.fr\n‡Modal’X - UMR CNRS 9023, UPL, Université Paris Nanterre, 92000 Nanterre, France, and FP2M, CNRS FR 2036,\nmarie.theret@parisnanterre.fr\n1arXiv:2402.07509v1  [math.PR]  12 Feb 2024where⌊nz⌋designates the coordinate-wise integer part of nz. The term µF(z)appearing here is a\nconstant, called the time constant of the model in the direction z, that depends on zbut also on the\ndimensiondof the underlying graph Zdand on the distribution Fof the passage times. The function\nz∝⇕⊣√∫⊔≀→µF(z)is either the null function or a norm on Rd. Moreover, under a stronger moment assumption\nonF, the convergence (1) towards the time constant is uniform in the directions: this result is known\nas theshape theorem (see [6, 15, 21]).\nProperties of F∝⇕⊣√∫⊔≀→µF(z).The study of the properties of µF(z)is a challenge, that has attracted\nattention of mathematicians over the last 50years. Let us roughly list some of the known properties of\nµF(z)as a function of the distribution F:\n•it is continuous for the topology of weak convergence (see [5, 7, 15])\n•it is strictly increasing (in a sense made clear in [18, 23])\n•µF= 0⇐⇒ P[τ(e) = 0]≥pc(d), wherepc(d)is the critical parameter of Bernoulli bond\npercolation on Zd(see [15]).\nIt worth noticing that the value of µF((1,0,..., 0))cannot be computed in any case when it is not null,\nexcept if the distribution is trivial: if τ(e) =aa.s. then µF(z) =a∥z∥1, where∥z∥1is the 1-norm ofz\ndefined by∥z∥1=/summationtextd\ni=1|zi|ifz= (z1,...,zd).\nBernoulli first-passage percolation on Zd.Since it is hard to understand how µF(z)depends on\nthe distribution F, it is relevant to look at particular and simple choices of distributions. One natural\nchoice among others is to consider a Bernoulli distribution with parameter 1−ε,i.e.,P[τ(e) = 0] =ε=\n1−P[τ(e) = 1]. Let us denote by µε(z)the corresponding passage time. Combining the results previously\nroughly stated, we know that ε∝⇕⊣√∫⊔≀→µε(z)is equal to∥z∥1atε= 0, is equal to 0atε=pc(d), and is\nstrictly decreasing with εon[0,pc(d)]. Chayes, Chayes and Durrett [4] investigate in dimension d= 2\nthe behavior of ε∝⇕⊣√∫⊔≀→µε((1,0))whenεgoes topc(2), and prove that it decays polynomially fast towards\n0, at a power that is linked with the speed of decreasing of the correlation length in the corresponding\npercolation model. The authors investigate in [3] at what speed µε(z)goes to∥z∥1whenεgoes to 0,\nand they prove that\nµε(z) =∥z∥1−C(z)ε1/d1(z)+o(ε1/d1(z)) (2)\nwhered1(z)is the number of non null coordinates of z, andC(z)is a constant whose dependence on z\nis partially explicit.\nBoolean model. Many variants of classical first-passage percolation on Zdcan be defined. One of\nthem is built on Rdfrom the Boolean model, that can be described as follows. Let Nbe a norm on Rd\nand denote by BN(c,r)the ball for the norm Ncentered at c∈Rdand of radius r∈(0,∞). Letξbe\na Poisson point process on Rd×(0,+∞)with intensity λ|·|⊗ν, whereλ∈(0,+∞),|·|designates the\nLebesgue measure on Rdandνdesignates a probability measure on (0,+∞). The Boolean model ΣN,λ,ν\nis defined by\nΣN,λ,ν =/uniondisplay\n(c,r)∈ξBN(c,r).\nRoughly speaking, it is built by throwing uniformly on Rdthe centers of the balls according to a Poisson\npoint process χof intensity λ|·|, and then adding around each center c∈χa ball of radius R(c)for the\nnormN, where the radii R(c)are chosen independently according to the distribution ν. The Poisson\npoint process of the centers of the balls, namely χ, can be properly defined as the projection of ξonRd.\nA common choice for the norm Nis of course the Euclidean norm that we designate by ∥·∥ 2. We refer\nto the book by Meester and Roy [20] for background on the Boolean model in the Euclidean case, and\nto books by Schneider and Weil [22] and by Last and Penrose [17] for background on Poisson processes.\n2Continuous first-passage percolation. A continuous model of first-passage percolation can be de-\nfined from the Boolean model by assuming that propagation occurs at speed 1outside ΣN,λ,ν, and at\ninfinite speed inside ΣN,λ,ν. Hence, the travel time ˜TN,λ,ν (π)of a pathπis theN-length ofπoutside\nΣN,λ,ν. Optimizing on polygonal paths from xtoyleads to the definition of the travel time ˜TN,λ,ν (x,y)\nbetweenxandyas the minimal time needed to see propagation from xtoy. More formal definitions are\ngiven in Section 1.2. In the case where Nis the Euclidean norm, this model was studied by the second\nand third authors in [10, 11], and previously introduced by Marchand and the second author in [9] as a\ntool to study another growth model considered by Deijfen [8]. By standard subadditive argument, it is\nwell known that for every z∈Rd, there exists a constant ˜µN,λ,ν (z)∈[0,1]such that\nlim\ns→∞˜TN,λ,ν (0,sz)\ns= ˜µN,λ,ν (z)a.s. and in L1.\nThis constant ˜µN,λ,ν (z)is still called the time constant in the direction z. By isotropy, for N=∥·∥ 2,\nthe time constant ˜µ∥·∥2,λ,ν(z)depends only on∥z∥2.\nProperties of (λ,ν)∝⇕⊣√∫⊔≀→˜µ∥·∥2,λ,ν(z).Fixz∈Rd\\{0}. Under some moment assumption on the\ndistribution νof the radii of the balls in the Boolean model, we know that\n•˜µ∥·∥2,λ,ν(z)>0if and only if the corresponding Boolean model Σ∥·∥2,λ,νis in some strongly sub-\ncritical percolation regime (see [10] for more details)\n•(λ,ν)∝⇕⊣√∫⊔≀→˜µ∥·∥2,λ,ν(z)is continuous under some domination assumptions (see [11] for more details).\nHowever, as on Zd, there is no way to calculate explicitly the value of ˜µ∥·∥2,λ,ν(z)when it is not null,\neven more so for the value of ˜µN,λ,ν (z)for general N. Similarly, it makes thus sense to consider specific\nchoices of parameters (λ,ν). Mimicking what was done on Zd, we restrict ourselves to the case where\nν=δ1/2andλ=εfor smallε,i.e., the diameter of the balls is equal to 1, and the Poisson point process\nχdescribing the centers of the balls has a small intensity ε. We write ˜µN,ε(z) = ˜µN,ε,δ 1/2(z)for short.\nAim of this work, part I. Whenε= 0,˜TN,0,δ1/2(0,z) =N(z), thus ˜µN,0(z) =N(z). The main goal\nof this work is to understand the behaviour of ε∝⇕⊣√∫⊔≀→˜µN,ε(z)near 0, for a given norm Nand a given point\nz∈Rd. With the choice N=∥·∥1, this setting is very close to Bernoulli first-passage percolation on Zd,\nand we can expect that the behaviour of ˜µ∥·∥1,ε(z)is still given by (2) at the first order in ε,i.e., that\n˜µ∥·∥1,ε(z)−∥z∥1is of orderε1/d1(z)whenεgoes to 0, whered1(z)is the number of non-null coordinates\nofz. However, for N=∥·∥ 2, the isotropy of the model implies necessarily a different behaviour. This\npaper aims also at understanding how the underlying norm Nimpacts the behaviour of ˜µN,ε(z)at the\nfirst order in ε.\nAn equivalent setting. The image of the homogeneous Poisson point process χonRdwith intensity\nεby the map x∈Rd∝⇕⊣√∫⊔≀→xε1/dis a homogeneous Poisson point process ΞonRdwith intensity 1,\nthus the image of ΣN,ε,δ 1/2by this map has the same distribution as ΣN,1,δε1/d/2. This implies that\n˜µN,ε(z) = ˜µN,ε,δ 1/2(z) = ˜µN,1,δε1/d/2(z). In what follows we adopt this point of view, i.e., the diameters\nof the balls are equal to ε1/d, and the centers of the balls are distributed according to a homogeneous\nPoisson point process Ξwith intensity 1. We write ˜TN,ε=˜TN,1,δε1/d/2for short.\nA different but close model. In this setting, since the diameter of each ball in the Boolean model is\nε1/dwith smallε, it can be useful to neglect first the possible overlaps between different balls, or the fact\nthat a path can travel inside the Boolean model without reaching the center of each ball it intersects.\nInstead, suppose that a path going through the center of a ball save a time ε1/d: it can be seen as a\nrewardthat the path earns. We define the alternative travel time TN,ε(π)of a pathπas theN-length\nofπ, minusε1/dtimes the number of rewards that πcollects. Optimizing on polygonal paths from xto\nyleads to the definition of the alternative travel time TN,ε(x,y)betweenxandyas the minimal time\nneeded to see propagation from xtoy. More formal definitions are given in Section 1.2. We emphasize\n3the fact that TN,εmay be negative. An alternative time constant can be defined in this setting for small\nenoughε: for every z∈Rd, there exists a constant µN,ε(z)such that\nlim\ns→∞TN,ε(0,sz)\ns=µN,ε(z)a.s. and in L1.\nSinceTN,ε(0,z)may be negative, the existence of µN,ε(z)is not a trivial consequence of standard sub-\nadditive arguments, and we refer to Theorem 1 for more details. Obviously, µN,0(z) = ˜µN,0(z) =N(z).\nAim of the work, part II. This alternative setting is interesting on its own, and the same questions\narise: what is the behaviour of µN,ε(z)at the first order in εwhenεgoes to 0, and how does it depends\nonN? Is it the same as the behaviour of ˜µN,ε(z)?\nAim of this work, part III. Our study of µN,ε(z)requires us to work with geodescis, i.e., paths of\nminimal travel time TN,ε. Our third and last goal is to understand some properties of those geodesics,\nin particular to prove bounds on their N-length and the number of rewards they collect, when εgoes to\n0.\n1.2 Definitions\nWe gather in this section all the definitions we need, even if some of them have been stated informally\nin the previous section.\nThe space Rd.Throughout the paper, d∈Ndesignates the dimension of the space, and we always\nassume that d≥2. Let·designate the standard scalar product on Rd. We consider different norms on\nRd. We write Nto designate a generic norm on Rd, whereas∥·∥pdesignates the p-norm, defined as\nusually: ifz= (z1,···,zd), then\nforp∈[1,∞),∥z∥p=/parenleftiggd/summationdisplay\ni=1|zi|p/parenrightigg1/p\n;forp=∞,∥z∥∞= max\n1≤i≤d|zi|.\nWe denote by BN(c,r)the closed ball for the norm Ncentered at c∈Rdand of radius r∈(0,∞), and\nwriteBN(r) =BN(0,r)for short. We designate by |·|the Lebesgue measure - usually on Rdor on\nRd−1, we omit the precision when it is not confusing, but write |·|kfor the Lebesgue measure on Rk\nif needed. For a subset AofRd, we denote by Ac=Rd\\Aits complement. Throughout the paper, u\ndesignates a vector of Rdsuch thatN(u) = 1. For such a vector u, that lies on the boundary of BN(1),\nwe can consider Ha supporting hyperplane of BN(1)atu,i.e., satisfying\n∀v∈H, N (u+v)≥N(u) = 1.\nSuch a supporting hyperplane always exists by convexity of BN(1)(but it may not be unique). For given\nuandH, we designate by u⋆the vector normal to Hand satisfying u·u⋆= 1.\nThe Poisson point process. LetΞbe a Poisson point process on Rdwith intensity|·|=|·|d. Let\nε>0. We consider two different models (see Tand ˜Tbelow). The points of Ξare seen as the locations\nof rewards of value ε1/d, or as centers of balls of diameter ε1/dfor the norm N. In this setting, we define\nthe Boolean model ΣN,εas\nΣN,ε=/uniondisplay\nc∈ΞBN/parenleftbigg\nc,ε1/d\n2/parenrightbigg\n.\nThepaths. Apolygonalpath πisafinitesequenceofdistinctpoints π= (x0,x1,...,xn),exceptmaybe\nthatx0=xn, and such that xi∈Ξfor alli∈{1,...,n−1}(we emphasize the fact that we do not require\nthatx0orxnbelong to Ξ). We consider the set Π(x,y)of polygonal paths π= (x0=x,x1,...,xn=y)\nfromxtoy. On occasions, we will discuss about generalized paths, that designate finite sequences of\ndistinct points π= (x0,x1,...,xn), except maybe that x0=xn(without consideration for the Poisson\n4point process Ξat all), and we denote by ˆΠ(x,y)the set of generalized paths from xtoy. Sometimes,\nwe will also need to consider the polygonal curve associated with π= (x0,...,xn)which we will denote\nby[π]. For a given segment [a,b]⊂Rd, we writeN([a,b]) =N(b−a)for theN-length of the segment\n[a,b]. By a slight abuse of notation, since [a,b]∩ΣN,εis a finite disjoint union of segments, we denote\nbyN([a,b]∩ΣN,ε)the finite sum of the N-lengths of the disjoint connected components of [a,b]∩ΣN,ε.\nWe defineN([a,b]∩Σc\nN,ε) =N([a,b])−N([a,b]∩ΣN,ε). For any path π= (x0,x1,...,xn), we denote\nbyN(π)the length of πfor the norm N,i.e.,\nN(π) =n/summationdisplay\ni=1N([xi−1,xi]) =n/summationdisplay\ni=1N(xi−xi−1).\nWe denote by N(π∩Σc\nN,ε)theN-length ofπoutside ΣN,ε,i.e.,\nN(π∩Σc\nN,ε) =n/summationdisplay\ni=1N([xi−1,xi]∩Σc\nN,ε).\nWe denote by ♯πthe number of distinct points in Ξ∩[π]. Note that for any generalized path, this\nquantity is a.s. finite and for a polygonal path π= (x0,x1,...,xn)such thatx0,xn/∈Ξ, we have a.s.\n♯π=n−1.\nThe model with balls. We first consider the model where a ball of diameter ε1/dis located at each\npoint of Ξ. Thetravel time ˜TN,ε(π)of a generalized path π= (x0,x1,...,xn)is\n˜TN,ε(π) =N(π∩Σc\nN,ε).\nWe define the travel time ˜TN,ε(x,y)betweenxandyas\n˜TN,ε(x,y) = inf\nπ∈ˆΠ(x,y)˜TN,ε(π).\nWe want to emphasize that the optimization can be made on paths instead of generalized paths, i.e.,\n˜TN,ε(x,y) = inf\nπ∈Π(x,y)˜TN,ε(π)\n(see Proposition 17). We denote by ˜γN,ε(x,y)ageodesic fromxtoyfor the time ˜TN,ε,i.e.,˜γN,ε(x,y)∈\nˆΠ(x,y)such that ˜TN,ε(x,y) =˜TN,ε(˜γN,ε(x,y)), and with minimal N-length among the possible geodesics\n(see Proposition 18 for the existence of a geodesic). For short, we write ˜γN,ε(z) := ˜γN,ε(0,z). The time\nconstant in this model is defined through standard subadditive arguments in the following way: for every\nz∈Rd, there exists a constant ˜µN,ε(z)≥0such that\nlim\ns→∞˜TN,ε(0,sz)\ns= ˜µN,ε(z)a.s. and in L1. (3)\nWhenNis thep-norm, we write ˜Tp,ε,˜γp,εand˜µp,εinstead of ˜T∥·∥p,ε,˜γ∥·∥p,εand˜µ∥·∥p,ε.\nThe model with rewards. We now consider the model where a reward of value ε1/dis located on\neach point of Ξ. Thealternative travel time TN,ε(π)of a pathπ= (x0,x1,...,xn)is\nTN,ε(π) =N(π)−ε1/d♯π.\nWe emphasize that TN,ε(π)may be negative, the terminology alternative travel time is chosen by analogy\nwith the model with balls but may be confusing. We define the alternative travel time TN,ε(x,y)between\nxandyby\nTN,ε(x,y) = inf\nπ∈ˆΠ(x,y)TN,ε(π).\nOur first result states that, for εsmall enough, a finite geodesic exists a.s. between any points x,y∈Rd\nand such geodesic can be chosen to be a polygonal path. Moreover, an alternative time constant can be\ndefined in this setting. Note that its existence is not trivial since the travel time may be negative.\n5Theorem 1. Forεsmall enough (depending on Nandd),\n(i) There exists a.s. for all x,y∈Rda finite polygonal path γN,ε(x,y)∈Π(x,y)fromxtoysuch that\nTN,ε(x,y) =TN,ε(γN,ε(x,y)).\nThe pathγN,ε(x,y)is called a geodesic from xtoy.\n(ii) For every z∈Rd, there exists a constant µN,ε(z)such that\nlim\ns→∞TN,ε(0,sz)\ns=µN,ε(z)a.s. and in L1. (4)\nMoreover, the application µN,ε(·)is a norm on Rd.\nNote that this theorem does not say anything about the uniqueness of the geodesics. For some norms\nN, they can in fact be non unique. For instance, consider the case N=∥·∥ 1andz= (1,..., 1): this\nnon uniqueness is a consequence of the fact that geodesics for the norm N=∥·∥ 1from 0tozare not\nunique either.\nFor short, we write γN,ε(z) :=γN,ε(0,z). To lighten the notations, when Nis thep-norm, we write\nTp,ε,γp,εandµp,εinstead ofT∥·∥p,ε,γ∥·∥p,εandµ∥·∥p,ε.\n1.3 Main results for a general norm N\nWe can now state the main results we obtain. We consider u∈Rdsuch thatN(u) = 1. The vector u\ngives a direction, and we want to study the behaviour of µN,ε(u). To do so, we need some tools that\ncapture the geometrical properties of the norm Nthat matter in our model. Let Hbe a supporting\nhyperplane of BN(1)atu,i.e.,\n∀v∈H, N (u+v)≥N(u) = 1.\nFor a given η≥0, let us consider the two following subsets of H:\nKη(u) :={v∈H:N(u+v)≤1 +η}andMη(u) :=Kη(u)∩(−Kη(u)).\nWe emphasize the fact that Kη(u)andMη(u)depends on ubut also on H, even if this dependence on\nHis not explicit in the notation. The sets Kη(u)andMη(u)are subsets of H, which is a hyperplane.\nWe thus write|Kη(u)|:=|Kη(u)|d−1(resp.|Mη(u)|:=|Mη(u)|d−1) to designate the (d−1)-dimensional\nLebesgue measure of Kη(u)(resp.Mη(u)). We define\nhu(η) :=η−d|Kη(u)|and ¯hu(η) =η−d|Mη(u)|.\nProposition 2. The functions η∝⇕⊣√∫⊔≀→hu(η)andη∝⇕⊣√∫⊔≀→¯hu(η)are continuous decreasing homeomorphism\nfrom (0,∞)to(0,∞). We denote by guand¯gutheir inverse functions. Note that guand¯guare also\ncontinuous decreasing homeomorphism from (0,∞)to(0,∞).\nWe can now consider the model with rewards located at the points of Ξ.\nTheorem 3. There exist constants C1,C2>0(depending only on Nandd) such that the following\nassertions hold.\n(i) Forεsmall enough, we have\n¯gu(C1ε−1)≤1−µN,ε(u)≤gu(C2ε−1).\n(ii) Recall that γN,ε(su)∈Π(0,su)designates any geodesic from 0tosufor the time TN,ε(0,su). For\nεsmall enough, we have, for any δ>0, a.s. for large s,\n(1−δ)ε−1/d¯gu(C1ε−1)s≤♯γN,ε(su)≤(1 +δ)ε−1/dgu(C2ε−1)s\nN(γN,ε(su))−s≤(1 +δ)gu(C2ε−1)s.\n6Notice that how small εhas to be in Points (i) and (ii) of Theorem 3 depends on Nanddbut also\nonuandH. This dependence appears for instance through the use of Lemma 10 (see how (10) implies\n(11) forηsmall enough).\nIt is a little bit frustrating not to be able to give a lower bound on N(γN,ε(su))in Theorem 3. In\nfact, such a lower bound exists in some specific cases, for directions uin whichBN(1)does not have a\n(d−1)-dimensional flat edge, see Proposition 15 in Section 5 for more details. However, we have no hope\nto obtain a lower bound in any direction, see Remark 27 for a concrete example.\nWe can now state the corresponding result for the model with balls centered at the points of Ξ,\ntogether with a comparison between the time constants in both models.\nTheorem 4. There exists some constant C2,C3>0(depending only on Nandd) such that the following\nassertions hold.\n(i) Forεsmall enough, we have\n¯gu(C3ε−1)≤1−˜µN,ε(u)≤1−µN,ε(u)≤gu(C2ε−1).\n(ii) Recall that ˜γN,ε(su)∈ˆΠ(0,su)designates any geodesic from 0tosufor the time ˜TN,ε(0,su), with\nminimalN-length among possible geodesics. For εsmall enough, we have, for any δ >0, a.s. for\nlarges,\nN(˜γN,ε(su))−s≤(1 +δ)gu(C2ε−1)s.\n(iii) Moreover, we have\nlim\nε→0µN,ε(u)−˜µN,ε(u)\ngu(C2ε−1)= 0.\nLet us emphasize the fact that we do not state a control on ♯˜γN,ε(su)in Theorem 4: indeed ˜γN,ε(su)∈\nˆΠ(0,su)is a generalized path, that do not have to travel between points of Ξ, thus the quantity ♯˜γN,ε(su)\ndoes not have a relevant signification. However, we do obtain a control on the number of balls of the\nBoolean model really useful to a geodesic if we look at geodesics inside a set of paths that are easier\nto deal with: for more details, we refer to Proposition 18 in Section 6.1) that state the existence of\na geodesic ˇγε(0,su)in a restrictive set ˇΠ(0,su)⊂Π(0,su), and to Equation (31) that gives an upper\nbound for♯ˇγε(0,su)- the lower bound on ♯ˇγε(0,su)is a straigthforward consequence of the upper bound\nonµN,ε(u), as in the proof of Corollary 12 in the model with rewards.\nFor specific choices of the norm N, namely when Nis thep-norm, it can be proved that the upper\nbound and lower bound appearing in Theorems 3 (i)and 4 (i)are of the same order in ε(see Section 1.4).\nIn this case, Assertion (iii)in Theorem 4 implies that Theorem 4 (i)is a consequence of Theorem 3 (i).\nHowever, it is not true in general, thus the different assertions in Theorem 4 must be stated separately.\n1.4 Main results for the p-norm\nWe now look at the particular case where Nis thep-norm, for a p∈[1,∞]. Letu= (u1,...,ud)∈Rd.\nWe define\nd1(u) =card({i∈{1,...,d}:ui̸= 0})\nd2(u) =d−d1(u)\nd3(u) =card({i∈{1,...,d}:|ui|=∥u∥∞}) (5)\nd4(u) =d−d3(u)\nwhere card (A)designates the cardinality of the finite set A. For a given p∈[1,∞], foru∈Rdsuch that\n∥u∥p= 1, we define\nκp(u) =\n\n1\nd1(u)ifp= 1,\n1\nd−d1(u)−1\n2−d2(u)\npifp∈(1,∞),\n1\nd4(u) + 1ifp=∞.(6)\n7Notice that for any p∈[1,∞], andu∈Rdsuch that∥u∥p= 1,d1(u)≥1andd−d1(u)−1\n2−d2(u)\np≥\nd−(d1(u)−1)−d2(u) = 1, thusκp(u)≤1.\nWe can rewrite Theorem 3 when Nis thep-norm, with the advantage that we can compute explicitly\nthe power of εthat appears at the first order in all our estimates, and notice that the lower bounds and\nupper bounds are of the same order.\nTheorem 5. There exist constants C′\ni=C′\ni(p,d,u )>0,i= 1,..., 6such that the following assertions\nhold.\n(i) Forεsmall enough, we have\nC′\n1εκp(u)≤1−µp,ε(u)≤C′\n2εκp(u).\n(ii) Recall that γp,ε(su)∈Π(0,su)designates any geodesic from 0tosufor the time Tp,ε(0,su). Forε\nsmall enough, we have a.s. for large s,\nC′\n3εκp(u)s≤ε1\nd♯γp,ε(su)≤C′\n4εκp(u)s\n∥γp,ε(su)∥p−s≤C′\n5εκp(u)s.\n(iii) Moreover, except if p= 1andd1(u) =d, or ifp=∞andd3(u) = 1, we have\nC′\n6εκp(u)s≤∥γp,ε(su)∥p−s.\nThe same holds concerning Theorem 4.\nTheorem 6. There exist constants C′′\ni=C′′\ni(p,d,u )>0,i= 1,..., 3such that the following assertions\nhold.\n(i) Forεsmall enough, we have\nC′′\n1εκp(u)≤1−˜µp,ε(u)≤C′′\n2εκp(u).\n(ii) Recall that ˜γp,ε(su)∈ˆΠ(0,su)designates any geodesic from 0tosufor the time ˜Tp,ε(0,su)with\nminimalp-length. For εsmall enough, we have a.s. for large s,\n∥˜γp,ε(su)∥p−s≤C′′\n3εκp(u)s.\n(iii) Moreover we have\nlim\nε→0µp,ε(u)−˜µp,ε(u)\nεκp(u)= 0.\nBeyond these applications of the previous results stated for a generic norm Nto thep-norms, some\nspecific properties of the p-norms allow us to obtain even the existence of the limit of (1−µp,ε(u))/εκp(u)\nwhenεgoes to zero, at least for some values of pand some specific directions u.\nTheorem 7. Suppose we are in one of the following cases:\n•p∈[1,∞]andu= (1,0,..., 0);\n•p∈{1,2}, whateveru∈Rdsuch that∥u∥p= 1;\n•p=∞andu∈Rdsuch that∥u∥∞= 1andd3(u)∈{1,2}.\nThen there exists a constant Kp(u)∈(0,∞)such that\nlim\nε→01−µp,ε(u)\nεκp(u)= lim\nε→01−˜µp,ε(u)\nεκp(u)=Kp(u).\nForp= 1andp= 2, we obtain the existence of the limit of (1−µp,ε(u))/εκp(u)whenεgoes to zero\nfor any direction u. Forp= 1, our results are similar to the ones previously obtained on the graph Zd\nfor the corresponding time constant µε(z), see Equation (2). For p= 2, we haveκ2(u) =2\nd+1whatever\nu, as prescribed by the isotropy of the model, and we obtain that\nµ2,ε(u) = 1−Cε2/(d+1)+o(ε2/(d+1))\nwhere for any u∈Rdsuch that∥u∥2= 1, withC=Kp(u)that does not depend on such a u.\n81.5 Organization of the paper\nThe paper is organized as follows.\nIn Section 2 we prove Theorem 1, i.e., we prove the existence of the time constant µN,ε, we state its\nbasic properties, and we also discuss the existence of geodesics in the model with rewards.\nSection 3 is devoted to the study of some geometrical properties of objects defined from the norm N.\nAmong other things, Proposition 2 is proved in this section.\nSections 4 and 5 are devoted to the study of the model with rewards at the points of Ξ,i.e., to the\nproof of Theorem 3. In Section 4, we use a greedy algorithm to construct a path from the origin towards\nthe direction u, whoseN-length is not too high, but that collects a certain amount of rewards: this gives\na lower bound on 1−µN,ε(u)and♯γN,ε(su)for larges. In Section 5, we prove that a path πfrom 0tosu\n(forslarge enough) whose N-length is upper-bounded cannot collect too much rewards (see Proposition\n13). We then use an initial rough upper bound on N(γN,ε(su)), together with a bootstrap argument, to\nstrengthen the control obtained in Proposition 13 and prove the desired upper bounds on 1−µN,ε(u),\n♯γN,ε(su)andN(γN,ε(su)).\nSection 6 is devoted to the study of the model with balls centered at the points of Ξ,i.e., to the proof\nof Theorem 4. First we prove in Section 6.1 that we can deal with geodesics that have nice properties.\nThen we compare ˜µN,ε(u)withµN,ε(u)in Section 6.2 to prove assertion (iii)of Theorem 4. Finally we\nadapt in Sections 6.3 and 6.4 the proofs written in the study of the previous model to this setting to\ncomplete the proof of assertions (i)and(ii)of Theorem 4.\nFinally in Section 7 we consider the case where N=∥·∥p, thep-norm, forp∈[1,∞]. First in Section\n7.1, for a given direction u∈Rdsuch that∥u∥p= 1, we choose adequately the supporting hyperplane\nHofB∥·∥p(1)atuwe consider. By computing an estimate of |Kη(u)|in Section 7.2 and an estimate of\n|Mη(u)|in Section 7.3, we can prove Theorems 5 and 6 in Section 7.4. By a monotonicity argument, we\nfinally prove Theorem 7 in Section 7.5.\n1.6 Notations\nFrom Section 2 to Section 6, we work with a fixed generic norm N. For that reason, we will omit the\nsubscriptNin our notations, since no confusion is possible (we write Tε,µε,γε(su)instead ofTN,ε,\nµN,ε,γN,ε(su)). In Section 7, we manipulate p-norms with different values of p, thus we put again the\nsubscriptpto recall the dependence on p.\n2 Existence and basic properties of µε\nFix a norm NonRdand consider the model where a reward of value ε1/dis located at each point of Ξ.\nRecall that the alternative travel time Tε(π)of a generalized path π= (x0,...,xn)is defined by\nTε(π) =N(π)−ε1/d♯π,\nwhere [π] :=/uniontextn\ni=1[xi−1,xi]is the polygonal curve with vertices (x0,...,xn)and♯πdenotes the number\nof distinct points of the a.s. finite set Ξ∩[π].\nThe first result states that minimizing the travel time over generalized paths does not get a better\nresult than minimizing over polygonal paths.\nLemma 8. For allx,y∈Rd,\nTε(x,y) := inf{Tε(π)|π∈ˆΠ(x,y)}= inf{Tε(π)|π∈Π(x,y)}.\nProof.Letπ1= (x=x0,...,xn=y)∈ˆΠ(x,y)be a generalized path. Let denote by a1,...,ak\nthe points of Ξwhich are in [π1]\\{x0,xn}ranked by their order of apparition in the curve [π1]. Then\nπ2= (x0,a1,...,ak,xn)belongsto Π(x,y). Wehavebyconstruction ♯π1≤♯π2andbytriangleinequality,\nN(π2)≤N(π1). This yields Tε(π2)≤Tε(π1).\nAs already noted, the alternative travel time Tε(π)of a polygonal path πmay be negative. Let\nus also note that the alternative travel time between xandydoes not satisfy the triangle inequality\n9Tε(x,y)≤Tε(x,z) +Tε(z,y). Hence, we cannot directly use classical theorems to prove the existence of\nfinite geodesic between two points of Rdor the existence of a time constant for this model. The following\nparagraphs are so devoted to rigorously prove the existence of this two objects. Even if the proofs are\nnot difficult nor very innovant, we give them for the sake of completeness.\nProof of Theorem 1 (i): Existence of a geodesic. Toprovethatafinitegeodesicexistsbetweentwopoints,\nwe need to show that paths with a large N-length cannot have a small travel time. Let denote by Π(0,⋆)\nthe set of polygonal paths starting from 0. For anys>0andk>1, we have\nP(∃π∈Π(0,⋆),N(π)≥ks,Tε(π)≤s)≤/summationdisplay\ni≥0P(∃π∈Π(0,⋆),#π=i,N(π)≥ks,Tε(π)≤s)\n=/summationdisplay\ni≥0P(∃π∈Π(0,⋆),#π=i,ks≤N(π)≤s+iε1/d)\n=/summationdisplay\ni≥(k−1)sε−1/dP(∃π∈Π(0,⋆),#π=i,N(π)≤s+iε1/d).\nSettingξ0= 0, we have, for β >0, using that 1x≥0≤eβx,\nP(∃π∈Π(0,⋆),#π=i,N(π)≤s+iε1/d) = P(∃(ξj)j≤i∈Ξ,i/summationdisplay\nj=1N(ξj−ξj−1)≤s+iε1/d)\n≤/integraldisplay\n(Rd)iexp(β(s+iε1/d−i/summationdisplay\nj=1N(ξj−ξj−1))dξ1...dξi\n≤exp(βs)/parenleftbigg/integraldisplay\nRdexp(β(ε1/d−N(z))dz/parenrightbiggi\n≤exp(βs)/parenleftbiggexp(βε1/d)\nβd/integraldisplay\nRdexp(−N(z))dz/parenrightbiggi\n.\nLet takeβsuch that1\nβd/integraltext\nRdexp(−N(z))dz< 1/4andε0small enough such that exp(βε1/d\n0)<2. We get\nthen, forε<ε 0,\nP(∃π∈Π(0,⋆),#π=i,N(π)≤s+iε1/d)≤exp(βs)\n2i. (7)\nIn particular, for all s>0, forε<ε 0,\nlim\nk→∞P(∃π∈Π(0,⋆),N(π)≥ks,Tε(π)≤s) = 0. (8)\nFixn∈Nand let us now prove that there exists a.s. a geodesic from xtoyfor anyx,y∈BN(n). Using\n(8), we get that there exists a.s. a (random) Ksuch that for any polygonal path πstarting from 0with\nN(π)≥Kn, we haveTε(π)≥4n. Let now ¯π= (x,x1,...,xk,y)be a polygonal path from xtoysuch\nthatN(¯π)≥(K+ 1)n. Thenπ= (0,x1,...,xk,y)is a polygonal path starting from 0and by triangle\ninequalityN(π)≥N(¯π)−N(x)≥Knand so that Tε(π)≥4n. Using that Tε(π)≤N(x) +Tε(¯π), we\ndeduce that Tε(¯π)≥3n. SinceTε(x,y)≤N(y−x)≤2n, we get in particular that, for all x,y∈BN(n),\ninf{Tε(π) :π∈Π(x,y)}= inf{Tε(π) :π∈Π(x,y),π⊂BN(K′)}\n= min{Tε(π) :π∈Π(x,y),π⊂BN(K′)}\nwithK′:= (K+ 2)n. The last infinimum is taken on a finite set of paths since there is a finite number\nof points of ΞinBN(K′). This implies the almost sure existence of a finite geodesic between xandy.\nThis holds for any n∈N, so we deduce the a.s. existence of a geodesic for any x,y∈Rd.\nProof of Theorem 1 (ii): Existence of the time constant. Let define, for x∈Rd\nJε(x) = inf{Tε(π) :π∈Π(x,⋆)}andXε(x,y) =Tε(x,y)−Jε(x)−Jε(y),\n10where Π(x,⋆)is the set of finite polygonal paths starting from x. Takingπ= (x), we note that Jε(x)≤0.\nMoreover, note that Equation (8) implies that Jε(x)andTε(x,y)are a.s. finite for εsmall enough so\nXε(x,y)is well defined. Moreover, since Jε(x)≤Tε(x,y), we get that Xε(x,y)is non-negative. Let us\nprove thatXε(x,y)satisfies the triangle inequality, i.e., forx,y,z∈Rd,Xε(x,z)≤Xε(x,y) +Xε(y,z).\nLetγε(y,x) = (y,a1,...,an−1,x)be a geodesic from ytoxandγε(y,z) = (y,b1,...,bm−1,z)be a\ngeodesic from ytoz. Seta0=b0=yandan=x,bm=z. Leti0,j0be two indices such that ai0=bj0\nand such{ai0+1,...,an}and{bj0+1,...,bm}are disjoints. Such indices exist since a0=b0=y. Let\nγ1:= (ai0,...,an−1,x)andγ2:= (bj0,...,bm−1,z)(γ1andγ2can be reduced to a point if i0=nor\nj0=m). Letγ′\n1:= (y,a1...,ai0)andγ′\n2:= (y,b1...,bj0). We have\nTε(x,y) =Tε(γ1) +Tε(γ′\n1) +ε1/d1{ai0∈Ξ}\nTε(y,z) =Tε(γ2) +Tε(γ′\n2) +ε1/d1{bj0∈Ξ}\n(the potential reward located at ai0=bj0is taking into account both in Tε(γ1)and inTε(γ′\n1)). Moreover,\nnoticing that (x,an−1,...,ai0=bj0,...,bm−1,z)is a polygonal path from xtozwith distinct vertices,\nwe get\nTε(x,z)≤Tε(γ1) +Tε(γ2) +ε1/d1{ai0∈Ξ}.\nBesides,Tε(γ′\n1)≥Jε(y)andTε(γ′\n2)≥Jε(y). So\nTε(x,y) +Tε(y,z)≥2Jε(y) +Tε(x,z).\nThis yields\nXε(x,y) +Xε(y,z)≥2Jε(y) +Tε(x,z)−Jε(x)−2Jε(y)−Jε(z) =Xε(x,z)\nand so the random variables (Xε(x,y),x,y∈Rd)satisfy the triangle inequality. In particular, for any\nu∈Rd, if we set for m≥n≥0,Xn,m:=Xε(nu,mu ), the process (Xn,m,0≤n≤m)is subadditive:\nXl,m≤Xl,n+Xn,mfor any 0≤l≤n≤m.\nTo apply Kingman’s subadditive ergodic theorem, we must also check that E(X0,n)is finite. We have\n0≤X0,n=Tε(0,nu)−Jε(nu)−Jε(0).\nUsing that Tε(0,nu)≤nN(u), we get\nE(X0,n)≤nN(u) + 2E(|Jε(0)|).\nRecall that Jε(0)≤0and we have for s≥0,\nP(Jε(0)≤−s)≤/summationdisplay\ni≥0P(∃π∈Π(0,⋆),#π=i,N(π)≤−s+ε1/di).\nNote that the bound obtain in (7) holds in fact also if s<0, so we get, for ε<ε 0,\nP(|Jε(0)|≥s)≤/summationdisplay\ni≥0exp(−βs)\n2i= 2 exp(−βs)\nwhich proves the integrability of Jε(0)forεsmall enough. Using the stationarity and the ergodicity of\nthe process, Kingman’s subadditive ergodic theorem [16] implies the existence of a limit\nµε(u) := lim\nn→∞X0,n\nn= inf\nn≥0E(X0,n)\nna.s. and in L1.\nNote that we have µε(u)≥0sinceX0,n≥0. Moreover, since, for any n≥0, the random variables\nJε(nu)have the same law and have finite expectation, we get that\nlim\nn→∞Jε(nu) +Jε(0)\nn= 0a.s. and in L1.\n11BN(1)\nBN(1+η)\nu+H\nu\nKη(u)\nBN(1)\nBN(1+η)\nu+H\nu\nMη(u)\nFigure 1: Illustration of the sets Kη(u)andMη(u)(note thatMη(u)is here strictly included in Kη(u)).\nSo we also get\nµε(u) = lim\nn→∞Tε(0,nu)\nna.s. and in L1.\nIt just remains to prove that this limit holds in fact for sgoing to infinity, s∈R. We write, for s>0,\nT(0,⌊s⌋u)−N(su−⌊s⌋u)\ns≤T(0,su)\ns≤T(0,⌊s⌋u) +N(su−⌊s⌋u)\ns\nT(0,⌊s⌋u)\n⌊s⌋⌊s⌋\ns−(s−⌊s⌋)N(u)\ns≤T(0,su)\ns≤T(0,⌊s⌋u)\n⌊s⌋⌊s⌋\ns+(s−⌊s⌋)N(u)\ns\nwhich yields\nµε(u) = lim\ns→∞Tε(0,su)\nsa.s. and in L1.\nIt remains to prove that for εsmall enough µε(·)is a norm. Triangle inequality and homogeneity are\nstraightforward. For the separation, using (7), we have, for εsmall enough and u∈Rdsuch that\nN(u) = 1,\nP/parenleftig\nTε(0,su)≤s\n2/parenrightig\n≤/summationdisplay\ni≥sε−1/d/2P/parenleftig\n∃π∈Π(0,⋆),#π=i,N(π)≤s\n2+iε1/d/parenrightig\n≤2 exp(βs/2)\n2sε−1/d/2\nwhich tends to 0 as stends to infinity if εis small enough (uniformly in u). In particular, this implies\nthat, for small enough ε, for allu∈BN(1),µε(u)≥1/2. Hence, we get that, for small enough ε,µε(·)\nis a norm and, in fact, for all u∈Rd,\nN(u)\n2≤µε(u)≤N(u).\n3 Some geometric results\nWe gather in this section the statement and proof of geometrical results. In particular, we establish a\nlink between|Kη(u)|and an integral, namely I+(η)(see Lemma 10), which is the quantity that appears\nin some of our forthcoming proofs.\nWe first prove Proposition 2.\n12Proof of Proposition 2. Letu∈Rdsuch thatN(u) = 1. LetHbe a supporting hyperplane of BN(1)at\nu. Letη≥0. We recall the following definitions (see Figure 3 for an illustration):\nKη(u) :={v∈H:N(u+v)≤1 +η}andMη(u) :=Kη(u)∩(−Kη(u))\nand\nhu(η) :=η−d|Kη(u)|and ¯hu(η) :=η−d|Mη(u)|.\nLetusprovethat huisacontinuousdecreasinghomeomorphismfrom (0,∞)to(0,∞). Thereadercan\ncheckthattheproofcanbeeasilyadaptedtoshowthat ¯huisalsoacontinuousdecreasinghomeomorphism\nfrom (0,∞)to(0,∞).\nFirst notice that v∈H∝⇕⊣√∫⊔≀→N(u+v)is convex. Indeed, for all v,v′∈H, for allλ∈[0,1], we have\nN(u+λv+ (1−λ)v′) =N(λ(u+v) + (1−λ)(u+v′))\n≤N(λ(u+v)) +N((1−λ)(u+v′)) =λN(u+v) + (1−λ)N(u+v′).\nBy convexity, for all λ∈[0,1]andη,η′>0we have\nλKη(u) + (1−λ)Kη′(u)⊂Kλη+(1−λ)η′(u)\nso\n|Kλη+(1−λ)η′(u)|1\nd−1≥|λKη(u) + (1−λ)Kη′(u)|1\nd−1.\nUsing Brunn-Minkowski’s inequality, we have\n|λKη(u) + (1−λ)Kη′(u)|1\nd−1≥λ|Kη(u)|1\nd−1+ (1−λ)|Kη′(u)|1\nd−1,\nso\nη∝⇕⊣√∫⊔≀→|Kη(u)|1\nd−1 (9)\nis concave and so, in particular, is continuous. This already proves that huis continuous. Let v∈H\nandη>0. For allλ∈[0,1], by convexity, we have\nN(u+λv)≤λN(u+v) + (1−λ)N(u) =λN(u+v) + (1−λ)\nthus\nN(u+v)−1≤η=⇒N(u+λv)−1≤λη,\nand so\nλKη(u)⊂Kλη(u).\nThis gives that for all 0<η1≤η2,\n/parenleftbiggη1\nη2/parenrightbiggd−1\n|Kη2(u)|≤|Kη1(u)|\nso the function ru(η) :=|Kη(u)|\nηd−1is non-increasing. This implies that hu(η) =ru(η)/ηis decreasing.\nMoreover, by triangle inequality, we have, for any u∈Rdsuch thatN(u) = 1,\n{v∈H:N(v)≤η}⊂Kη(u)⊂{v∈H:N(v)≤2 +η}.\nUsing that Nis equivalent to the euclidean norm, we get, for some constants c,c′depending only on d\nandN,\n{v∈H,∥v∥2≤cη}⊂Kη(u)⊂{v∈H,∥v∥2≤c′(2 +η)}.\nSinceHis(d−1)-dimensional, this gives that, for some A:=A(d,N)>0andB:=B(d,N)>0,\nAηd−1≤|Kη(u)|≤B(2 +η)d−1,\nand so\nAη−1≤hu(η)≤Bη−d(2 +η)d−1.\nThis implies in particular that hutends to +∞at 0 and to 0at infinity, and for η∈(0,1),\nAη−1≤hu(η)≤B′η−d.\n13We now state a geometrical lemma that will be useful in what follows.\nLemma 9. Foru∈Rdsuch thatN(u) = 1andHa supporting hyperplane of BN(1)atu, let denote by\nu⋆the vector normal to Hsuch thatu·u⋆= 1. There exist two constants c,c′>0depending only on d\nandNsuch that, for all u∈BN(1),\nc≤∥u⋆∥2≤c′.\nProof.Using that Nand||·|| 2are equivalent, there exist α,β > 0such that\nB∥·∥2(α)⊂BN(1)⊂B∥·∥2(β).\nSo, we get the lower bound on ∥u⋆∥2noticing that, for u∈BN(1),\n1 =u·u⋆≤∥u∥2∥u⋆∥2≤β∥u⋆∥2.\nMoreover,His a supporting hyperplane of BN(1)atuandu⋆is normal to HsoBN(1)is included in\n{v∈Rd:v·u⋆≤u·u⋆}. Applying this inequality to αu⋆/∥u⋆∥2∈B∥·∥2(α)⊂BN(1), we get\nα∥u⋆∥2=αu⋆\n∥u⋆∥2·u⋆≤u·u⋆= 1.\nFinally, we get\nβ−1≤∥u⋆∥2≤α−1.\nWe introduce now a function Idefined by an integral and check that this function is of the same\norder as the function huwhose inverse guappears in the statement of Theorem 3. We recall that for\nη≥0,hu(η)is defined by\nhu(η) =η−d|Kη(u)|.\nLemma 10. Foru∈Rdsuch thatN(u) = 1andHa supporting hyperplane of BN(1)atu, let denote\nbyu⋆the vector normal to Hsuch thatu·u⋆= 1and define, for η>0,\nI+(η) =/integraldisplay\nRd∩{x:x·u⋆>0}exp(−(N(x)−(1−η)x·u⋆))dx.\nThen there exists two constant c1,c2depending only on Nanddsuch that, for η>0, we have\nc1hu(η)≤I+(η)≤c2hu(η). (10)\nMoreover, define also, for η>0,\nI(η) =/integraldisplay\nRdexp(−(N(x)−(1−η)x·u⋆))dx,\nthen forηsmall enough, we have\nc1hu(η)≤I(η)≤2c2hu(η). (11)\nProof.Let∥·∥2be the Euclidean norm. Let ψ:=ψη:H→Hbe anaffinetransformation with positive\ndeterminant such that\nN(u+ψ(t))≤1 +ηfor allt∈Hsuch that∥t∥2≤1 (12)\nand\nN(u+ψ(t))≥1 +ηfor allt∈Hsuch that∥t∥2≥d−1. (13)\nLet us prove that such an application ψexists. The set\nKη(u) ={v∈H:N(u+v)≤1 +η}\n14iscompact, convex, withnon-emptyinterior. Thus, byJohn-LoewnerTheorem(seeforinstanceTheorem\nIII in [14]), there exists a centered ellipsoïde JofHand ac∈Hsuch that\nc+J⊂Kη(u)⊂c+ (d−1)J.\nLetBH(r) :=B∥·∥2,H(r)be the ball of Hof radiusrfor the Euclidean norm ∥·∥ 2. Letψ:H∝⇕⊣√∫⊔≀→H\nbe the linear function with positive determinant such that ψ(BH(1)) =Jandψ:=c+ψ. Then, we have\nψ(BH(1)) =c+J⊂Kη⊂c+ (d−1)J=ψ(BH(d−1))\nwhich shows that (12) and (13) hold. Moreover, we can find bounds on the determinant of ψ. Indeed,\nwe have\ndet(ψ) =|ψ(BH(1))|\n|BH(1)|=|ψ(BH(1))|\nVd−1\nwhereVd−1is the volume of the unit euclidean ball of Rd−1. Using that\n|ψ(BH(1))|≤|Kη(u)|≤|ψ(BH(d−1))|\nwe get\nVd−1det(ψ)≤|Kη(u)|≤(d−1)d−1Vd−1det(ψ) (14)\nRecall now that\nI+(η) =/integraldisplay\nRd∩{x:x·u⋆>0}exp(−(N(x)−(1−η)x·u⋆))dx.\nBy the change of variable1x=λu+vwithλ=x·u⋆andv∈H, we have\nI+(η) =1\n∥u⋆∥2/integraldisplay∞\n0/integraldisplay\nHexp(−(N(λu+v)−(1−η)λ))dvdλ\n=1\n∥u⋆∥2/integraldisplay∞\n0/integraldisplay\nHλd−1exp/parenleftbigg\n−ηλ−ηλN(u+w)−1\nη/parenrightbigg\ndwdλ.\nLetx=ληandw=ψ(t). Then\nI+(η) =1\n∥u⋆∥2η−ddet(ψ)G(η)\nwith\nG(η) =/integraldisplay∞\n0/integraldisplay\nHxd−1exp/parenleftbigg\n−x−xN(u+ψ(t))−1\nη/parenrightbigg\ndtdx.\nLetusprovethatthereexiststwoconstants c,c′>0(dependingonlyon Nandd)suchthatc≤G(η)≤c′\nforη>0. Let define\ngη(t) :=N(u+ψ(t))−1\nη.\nFirst recall that if t∈BH(1), thengη(t)≤1. Thus\nG(η)≥/integraldisplay∞\n0/integraldisplay\nBH(1)xd−1exp(−2x)dtdx =Vd−1/integraldisplay∞\n0xd−1exp(−2x)dx=Vd−1d!\n2d:=c.\nOn the other side, if ∥t∥2≥dthengη(t)≥1. Lett0=ψ−1(0). Note that gη(t0) = 0and so∥t0∥2≤d.\nUsing the convexity of gη, we get that for∥t∥2≥d,\ngη(t)≥∥t−t0∥2\nd+∥t0∥2≥∥t∥2\n2d−1\n2.\n1This change of variable can be written as follows. Let f1=u⋆/∥u⋆∥2and let (f2,...,fd)be an orthonormal basis\nofH, thus (f1,...,fd)is an orthonormal basis of Rd. Letx=/summationtextd\ni=1xifiandu=/summationtextd\ni=1uifithe decomposition of\nxanduin this basis. Consider now the basis (u,f2,...,fd)andx=λu+/summationtextd\ni=1vifithe decomposition of xin this\nbasis. Then x1=λu1and for every i∈{2,...,d}we havexi=x·fi=λui+vi. Thus the change of basis is given\nbyΨ : (λ,v2,...,vd)∝⇕⊣√∫⊔≀→(x1,x2,...,xd) = (λu1,v2+λu2,...,vd+λud). The Jacobian of ΨisJΨ(λ,v2,···,vd) =\ndetΨ′(λ,v2,...,vd) =u1=u·u⋆/∥u⋆∥2= 1/∥u⋆∥2.\n15Sincegηis non-negative on H, this lower bound holds in fact on H. This yields\nG(η)≤/integraldisplay∞\n0/integraldisplay\nHxd−1exp/parenleftbigg\n−x−x/parenleftbigg∥t∥2\n2d−1\n2/parenrightbigg/parenrightbigg\ndtdx\n≤/integraldisplay∞\n0/integraldisplay\nRd−1xd−1exp/parenleftbigg\n−x\n2−∥xt∥2\n2d/parenrightbigg\ndtdx\n≤/integraldisplay∞\n0exp(−x\n2)dx/integraldisplay\nRd−1exp/parenleftbigg\n−∥v∥2\n2d/parenrightbigg\ndv:=c′<∞.\nHence we get, using (14), that for η>0\nc\n∥u⋆∥2((d−1)d−1Vd−1)−1|Kη(u)|≤ηdI+(η)≤c′\n∥u⋆∥2(Vd−1)−1|Kη(u)|.\nWe conclude the study of I+(η)using Lemma 9 which bounds ∥u⋆∥2uniformly in u∈BN(1). We now\nstudyI(η)defined by\nI(η) =/integraldisplay\nRdexp(−(N(x)−(1−η)x·u⋆))dx.\nWe haveI(η) =I+(η) +I−(η)where\nI−(η) =/integraldisplay\nRd∩{x:x·u⋆<0}exp(−(N(x)−(1−η)x·u⋆))dx.\nNote that for η∈[0,1],0≤I−(η)≤/integraltext\nRdexp(−N(x))dx <∞soI−is bounded around 0. On the\ncontrary, since I+(η)is of the same order as hu(η) :=Kη(u)η−d, Proposition 2 implies that I+tends to\ninfinity at 0. So we get that I(η)∼I+(η)forηgoing to 0and so forηsmall enough, we have\nc1hu(η)≤I(η)≤2c2hu(η)\nas desired.\n4 Proof of the lower bound on 1−µε(u)\nThis section is devoted to the proof of a lower bound on 1−µε(u),i.e., the proof of Proposition 11 -\nand incidentally, as a corollary, we get a lower bound on ♯γε(su)for largestoo, see Corollary 12 below.\nTo do so, it is enough to exhibit a path from 0tosu(for a given direction uandslarge enough) whose\nN-length is not too high, but that gather enough rewards. This is done in Proposition 11 through a\ngreedy algorithm. This algorithm adds recursively to the path π= (0,x1···,xn−1)the closest point\nxnof the Poisson point process Ξ(in a certain sense) which is located in a cone with origin xn−1and\noriented in the direction u. This cone condition gives us a good upper bound on the N-length of the\npath we construct. However, we have to be careful in our procedure to be sure that the path πdoes not\ngo too far away from the prescribed direction u. We thus take care to compensate any gap previously\ncreated between the direction of xn−1and the prescribed direction uby choosing wisely the direction of\nthe next point xnthe path collects. At this stage, we need to deal with some symmetries, this is the\nreason why it is the set Mη(u), which is a symmetrized version of Kη(u), that arises in the proof.\nFix a direction u∈RdwithN(u) = 1,Ha supporting hyperplane of BN(1)atuand recall that\nMη(u) :={v∈H:N(u+v)≤1 +ηandN(u−v)≤1 +η}and ¯hu(η) :=η−d|Mη(u)|.\nProposition 2 states that η∝⇕⊣√∫⊔≀→¯hu(η)is a continuous decreasing homeomorphism from (0,∞)to(0,∞)so\nwe can define its inverse function ¯guwhich is also a continuous decreasing homeomorphism from (0,∞)\nto(0,∞). To prove the lower bound on 1−µε(u)given in (i) of Theorem 3, we prove in fact the following\nproposition.\nProposition 11. There exists a constant c >0(depending only on d,N) such that, for all direction\nu∈RdwithN(u) = 1, for allη∈(0,1), we have\nµε(u)≤1 +η−cε1/d|Mη(u)|1/d.\n16In particular, with C1:=/parenleftbig2\nc/parenrightbigd, we have that for εsmall enough,\nµε(u)≤1−¯gu/parenleftbig\nC1ε−1/parenrightbig\n.\nProof.To prove this proposition, we construct, using a greedy algorithm, a path πsfrom 0tosuwith\nlength close to s(1 +η)but going through a number of points of Ξlarger than c|Mη(u)|1/ds(see Figure\n2).\nWe first introduce some notation. For s≥0, let\nCη(s) :={x=λu+v: 0≤λ≤s, v∈H, N (λu+v)≤λ(1 +η),N(λu−v)≤λ(1 +η)}.\nHence,Cη(s)is a cone with axis uand of basis Mη(basis not necessarily orthogonal to its axis). Thus,\nifu⋆denotes the vector orthogonal to Hsuch thatu·u⋆= 1, we have2\n|Cη(s)|=1\nd∥u⋆∥2|Mη(u)|sd.\nIn the algorithm, we construct a path (0,x1,...,xn,su)from 0tosusuch thatxi+1is the first point\nofΞin the cone xi+Cη(∞). By construction, we prove that the length of such path cannot be much\nlarger than (1 +η)sand its number of points is at least of order |Mη(u)|1/ds. Optimizing on ηyields\nthen Proposition 11.\nRecall that Ξdenotes the points of a Poisson point process with intensity |·|onRd. By induction\nwe define a sequence (Xn)n≥0inRdwithX0= 0and if\nλn+1:= inf{s>0 : (Xn+Cη(s))∩Ξ̸=∅},\nwe setXn+1=Xn+λn+1u+Wn+1withWn+1∈Hand such that{Xn+1}= (Xn+Cη(λn+1))∩Ξ.\nWe defineSn=/summationtextn\ni=1λiandVn=/summationtextn\ni=1Wi. By the proprieties of Poisson point processes, (λi)i≥1\nand(Wi)i≥1are i.i.d. random variables and since Mη(u)is a symmetric set, W1has also a symmetric\ndistribution and so Vnis a symmetric random walk.\nLet us now define, for s>0,\nZ(s) := sup{n≥0 :Sn≤s}\nandconsiderthepath πs= (0,X1,...,XZ(s),su)(seeFigure2foranillustrationofthegreedyalgorithm).\nLet us find an upper bound of Tε(πs)by finding an upper bound of its length and a lower bound of Z(s),\nits number of points.\nThe sequence (λi)i≥1is i.i.d. and we have\nP(λ1≥t) = exp/parenleftbigg\n−1\nd∥u⋆∥2|Mη(u)|td/parenrightbigg\nso that\nE(λ1) =/parenleftbigg|Mη(u)|\nd∥u⋆∥2/parenrightbigg−1/d/integraldisplay∞\n0exp(−vd)dv:=|Mη|−1/d\n2cu\nwhere\n2cu:=/parenleftbigg1\nd∥u⋆∥2/parenrightbigg1/d/parenleftbigg/integraldisplay∞\n0exp(−vd)dv/parenrightbigg−1\n.\nSinceSn=/summationtextn\ni=1λi, a standard renewal theorem yields that, for slarge enough, we have\ncu|Mη(u)|1/ds≤Z(s)≤3cu|Mη(u)|1/ds. (15)\n2This can be proven using the same change of variable Ψ. Indeed,\n|Cη(s)|=/integraldisplay\nRd1x∈Cη(s)dx=1\n∥u⋆∥2/integraldisplay\nRd10≤λ≤s1v/λ∈Mη(u)dλdv\n=1\n∥u⋆∥2/integraldisplay\nRdλd−110≤λ≤s1w∈Mη(u)dλdw =1\n∥u⋆∥2|Mη(u)|sd\nd.\n17s.u\n0\nu\nu+H\nsu+H\nMη(u)\nX1\nX2\nX3Figure 2: Illustration of the greedy algorithm. The path constructed is drawn in blue. At each step, the\npath goes through the first point of Ξin the green cone in front of it.\nThis gives a lower bound for the number of points of πs. It remains now to find an upper bound for the\nlength ofπs. Note first that if we consider the path π′\ns= (0,X1,...,XZ(s)), we have by construction\nN(Xi+1−Xi)≤(1 +η)λi+1\nso\nN(π′\ns)≤(1 +η)SZ(s).\nThus\nN(πs)≤(1+η)SZ(s)+N(su−(SZ(s)u+VZ(s)))≤(1+η)SZ(s)+s−SZ(s)+N(VZ(s))≤(1+η)s+N(VZ(s)).\nLet us now prove that N(VZ(s))/stends a.s. to 0. The process (Vk)k≥0is a symmetric random walk and\nby construction, we have\nN(λ1u+V1)≤λ1(1 +η)\nwhich yields\nN(V1)≤λ1(1 +η) +N(λ1u)≤3λ1\nforη∈(0,1). This implies that E(N(V1))is finite. Using that for slarge enough Z(s)≤3cu|Mn|1/ds\na.s., the law of large number implies that\nlim\nn→∞N(VZ(s))\ns= 0a.s.\nHence, for all η∈(0,1), we have, for slarge enough,\nTε(0,su)≤Tε(πs)≤s/parenleftbigg\n1 +η+N(VZ(s))\ns−ε1/dcu|Mη(u)|1/d/parenrightbigg\n.\nWe deduce\nµε(u) = lim\ns→∞Tε(0,su)\ns≤1 +η−ε1/dcu|Mη(u)|1/d= 1−η/parenleftig\ncu/parenleftbig\nε¯hu(η)/parenrightbig1/d−1/parenrightig\n.\nLet definec1:= inf{cu,u∈BN(1)}and note, in view of Lemma 9, that c1>0. We get, for all u∈BN(1),\nµε(u)≤1−η/parenleftig\nc1/parenleftbig\nε¯hu(η)/parenrightbig1/d−1/parenrightig\n.\nThis proves the first part of Proposition 11. Taking\nη= ¯gu/parenleftigg/parenleftbigg2\nc1/parenrightbiggd\nε−1/parenrightigg\n,\n18(notice that η<1forεsmall enough), we obtain that c1/parenleftbig\nε¯hu(η)/parenrightbig1/d= 2, thus\nµε(u)≤1−η.\nLet us note that this implies a lower bound for the number of points taken by the geodesic, that we\nstate in the following Corollary.\nCorollary 12. LetC1=C1(d,N)be the constant given by Proposition 11. For εsmall enough, we\nhave, for any δ>0, a.s. for large s,\nε1/d♯γε(su)≥(1−δ)¯gu(C1ε−1)s, (16)\nwhereγε(su)denotes any geodesic from 0tosufor the time Tε(0,su).\nProof.Indeed, ifγε(su)denotes a geodesic from 0tosu, we have\nε1/d♯γε(su) =N(γε(su))−Tε(0,su)≥s/parenleftbigg\n1−Tε(0,su)\ns/parenrightbigg\n.\nUsing that 1−Tε(0,su)\nsconverges to 1−µε(u)and Proposition 11, we get that for any εsmall enough,\nfor anyδ>0, a.s. forslarge enough,\nε1/d♯γε(su)≥(1−δ)¯gu(C1ε−1)s.\n5 Proof of the upper bound on 1−µε(u)\nWe want now to find the lower bound for µε(u)of Theorem 3. In the course of the proof, we will also\nprove the upper bounds on the length and the number of rewards of the geodesic stated in Point (ii) of\nthe theorem. The idea of the proof is the following. For any geodesic γε(su)from 0 tosu, we have\nN(γε(su)) =Tε(0,su) +ε1/d♯γε(su)≤s+ε1/d♯γε(su). (17)\nAny upper bound on ♯γε(su)thus provides an upper bound on N(γε(su)). In the next proposition, we\nprove that, for any η>0small enough, for any slarge enough and any path πfrom 0 tosu, we have\nN(π)≤(1 +η)s=⇒♯π≤c|Kη(u)|1/ds. (18)\nIn particular, any good upper bound on N(γε(su))provides in turn an upper bound on ♯γε(su). We\nthen prove the following crude inequality (see (23)): for some constant c, almost surely, for any path π\nfrom 0 to a faraway point,\n♯π≤cN(π). (19)\nPlugging (19) in (17) gives a first upper bound on N(γε(su))(see (22)). Using repeatedly and alternately\n(17) and (18), we then get, after a finite number of steps, the upper bounds on N(γε(su))and♯γε(su)\ngiven in Theorem 3. From the upper bound on ♯γε(su)we deduce straightforwardly a lower bound on\nµε(u). We conclude this section by noticing that, at least under some added hypothesis, we can recover a\nlower bound on N(γε(su))(see Proposition 15) by a final last use of Proposition 13 and the lower bound\non♯γε(su)proved in Section 4 (see Corollary 12).\nWe start by proving the following proposition, that states that a path from 0tosuwith small\nN-length cannot collect too much rewards.\nProposition 13. Let consider the event\nE(η,s,u,c ) :={∃π∈Π(0,su),N(π)≤(1 +η)s,♯π≥c|Kη(u)|1/ds}.\n19Then there exists some c,c′>0depending only on dandNsuch that for any u∈BN(1)and any\nhyperplane Hwhich is a supporting hyperplane of BN(1)atu, there exists ¯η > 0(depending only on\nd,u,H) such that, for η<¯η, for alls>0, we have\nP[E(η,s,u,c )]≤2 exp[−sc′|Kη(u)|1/d]. (20)\nIn particular, for all η∈(0,¯η)there exists a.s. sηsuch that, for s≥sη, every path π∈Π(0,su)such\nthatN(π)≤(1 +η)ssatisfies♯π≤c|Kη(u)|1/ds.\nToprovethisproposition,wewillneedthefollowinglemmawhichroughlystatesthesamethingexcept\nthat we consider now a path from the origin 0to the hyperplane su+H. We denote by Π(0,su+H)\nthe set of all polygonal paths π∈Π(0,y)for somey∈su+H.\nLemma 14. ForA>0andHa supporting plane of BN(1)atu, define\nE(η,s,u,A,H ) :={∃π∈Π(0,su+H),∀x∈π,x·u⋆≤s,N(π)≤(1 +η)s,♯π≥As}.\nThen, for all s>0andη>0,\nP[E(η,s,u,A,H )]≤exp[−s(Alog 2−21+1/dI(η)1/dη)].\nWe first explain how Lemma 14 implies the proposition.\nProof of Proposition 13 using Lemma 14. A pathπ∈Π(0,su)can be decomposed into two paths π1\nandπ2whereπ1is the restriction of the path πuntil it intersects the hyperplane su+Handπ2is the\nintersection of the path πafterwards. Note that N(π) =N(π1) +N(π2)and moreover N(π1)≥s. Ifπ\nsatisfies the property appearing in the definition of the event E(η,s,u,c ), we thus get N(π2)≤ηsand\nmoreover we have either ♯π1≥c|Kη(u)|1/ds/2or♯π2≥c|Kη(u)|1/ds/2. Hence, if we define\nF(η,s,u,c ) :={∃π∈Π(su,⋆),N(π)≤ηs,♯π≥c\n2|Kη(u)|1/ds},\nwe get\nE(η,s,u,c )⊂E/parenleftig\nη,s,u,c\n2|Kη(u)|1/d,H/parenrightig\n∪F(η,s,u,c ).\nUsing (11) in Lemma 10, we see that we can choose csuch that for ηsmall enough we have\nc|Kη(u)|1/d\n2log 2≥η5I(η)1/d\nso that\nc|Kη(u)|1/d\n2log 2−21+1/dI(η)1/dη≥I(η)1/dη(5−4) =I(η)1/dη.\nThus by Lemma 14 we get\nP/bracketleftig\nE/parenleftig\nη,s,u,c\n2|Kη(u)|1/d,H/parenrightig/bracketrightig\n≤exp[−sI(η)1/dη)]≤exp[−sc1|Kη(u)|1/d]\nfor some constant c1. Moreover, let q=c\n2|Kη(u)|1/ds, then for any β >0,\nP[F(η,s,u,c )] = P[∃π∈Π(0,⋆),N(π)≤ηs,♯π≥q]\n≤P[∃(x1,...,xq)∈Ξ,q/summationdisplay\ni=1N(xi−xi−1)≤ηs]\n= exp(βηs)/parenleftbigg/integraldisplay\nexp(−βN(x))dx/parenrightbiggq\n= exp(βηs)/parenleftbigg1\nβd/integraldisplay\nexp(−N(x))dx/parenrightbiggq\n.\n20Choseβ0:=β0(d)such that the bracket above is equal to 1/2. Then\nP[F(η,s,u,c )]≤exp(β0ηs−qlog 2)\n= exp/parenleftig\n−ηs/parenleftigc\n2hu(η)1/dlog 2−β0/parenrightig/parenrightig\n,\nwherehu(η) =η−d|Kη(u)|. By Proposition 2, hu(η)tends to infinity as ηtends to 0, we choose ¯η:=\n¯η(d,u,H )such thatβ0≤c\n4hu(η)1/d(2 log 2−1)forη≤¯η. This yields\nP[F(η,s,u,c )]≤exp/bracketleftig\n−sc\n4|Kη(u)|1/d)/bracketrightig\n.\nThis implies that (20) holds with c′= min(c1,c/4). Moreover, if we set λn=n/(c|Kη(u)|1/d), we get\n∞/summationdisplay\nn=1P[E(η,λn,u,c)]≤2∞/summationdisplay\nn=1exp/bracketleftbigg\n−c′\ncn/bracketrightbigg\n.\nThus, Borel-Cantelli Lemma yields that for nlarge enough, every path π∈Π(0,λnu)such thatN(π)≤\n(1 +η)λnsatisfies♯π < c|Kη(u)|1/dλn=n. Then for s∈[λn,λn+1[, ifπis a path from 0tosusuch\nthatN(π)≤(1 +η)s, adding to πa straight line colinear to uyields a path from 0toλn+1usuch that\nN(π)≤(1 +η)λn+1. Thus we deduce that ♯π < n + 1. Since⌊c|Kη(u)|1/ds⌋=n, this implies in fact\nthat♯π≤c|Kη(u)|1/ds.\nWe now prove Lemma 14.\nProof of Lemma 14. Letq=⌈As⌉. Letπ= (0,x1,...,xq,xq+1)∈Π(0,su+H)wherex1,...,xqare\npoints of the Ξandxq+1−xqis colinear to u. Setx0= 0. Thus\nN(π) =q+1/summationdisplay\ni=1N(xi−xi−1) =q/summationdisplay\ni=1N(xi−xi−1) +|s−xq·u⋆|=q/summationdisplay\ni=1N(xi−xi−1) +s−xq·u⋆.\nThus\nN(π)≤(1 +η)s⇔q/summationdisplay\ni=1N(xi−xi−1)≤ηs+xq·u⋆.\nWe get for any α,β > 0,\nP[E(η,s,u,A,H )] = P(∃x1,...,xq∈Ξ,xq·u⋆≤s,q/summationdisplay\ni=1N(xi−xi−1)≤ηs+xq·u⋆)\n≤/integraldisplay\n(Rd)qexp/parenleftigg\nβ(s−xq·u⋆) +α/parenleftigg\nηs−q/summationdisplay\ni=1N(xi−xi−1) +xq·u⋆/parenrightigg/parenrightigg\ndx1...dxq.\nTakingβ=αη, we get\nP[E(η,s,u,A,H )]\n≤exp(2αηs)/integraldisplay\n(Rd)qexp/parenleftigg\n−α/parenleftiggq/summationdisplay\ni=1N(xi−xi−1)−(1−η)q/summationdisplay\ni=1(xi−xi−1)·u⋆/parenrightigg/parenrightigg\ndx1...dxq\n= exp(2αηs)/parenleftbigg/integraldisplay\nRdexp (−α(N(x)−(1−η)x·u⋆))dx/parenrightbiggq\n= exp(2αηs)/parenleftbiggI(η)\nαd/parenrightbiggq\n.\nWe conclude taking αsuch thatαd= 2I(η).\n21We can now prove the upper bound on the number of points and the length of a geodesic given in\n(ii) of Theorem 3, i.e., we can prove that there exists a constant C2(depending only on dandN) such\nthat ifγε(su)is a geodesic from 0tosu, then, forεsmall enough, we have for any δ>0, a.s., forslarge\nenough\nN(γε(su))≤s(1 + (1 +δ)gu(C2ε−1))and♯γε(su)≤(1 +δ)gu(C2ε−1)ε−1/ds. (21)\nProof of (21).We begin by proving that there exists some C > 0such that, for εsmall enough, we have\na.s. forslarge enough\nN(γε(su))≤s[1 +Cε1/d]. (22)\nWe use tools that come from the study of the greedy paths and greedy lattice animals. Let\nG(s) := sup/braceleftbigg♯π\nN(π):π∈Π(0,⋆), π̸⊂BN(s)/bracerightbigg\n≤sup/braceleftbigg♯π\nN(π):π∈Π(0,⋆)/bracerightbigg\n:=G.\nLetG(∞)be the increasing limit of G(s). We have the follonwing properties : G(∞)≤G,G(∞)is\nconstant a.s. and\nG(∞) =E[G(∞)]≤E[G]≤c/integraldisplay∞\n0δ1([r,+∞[)1/ddr:=C/4.\nThis is a consequence of (11)and Lemma 2.1 in [9] (which is the analog in a continuous setting of a\nresult by Martin [19] in a discrete setting), and these results have already been useful to study continuous\nfirst-passage percolation (see [10] Theorem 3.1 or [11] Theorem 13 and Corollary 14). Thus, a.s. for large\nenoughs, for allπ∈Π(0,⋆)such thatπ̸⊂BN(s/2)we have\n♯π≤(C/2)N(π). (23)\nIn particular, this holds for γε(su), a geodesic from 0tosu. Using that T(γε(su))≤s, we get\ns≥T(γε(su)) =N(γε(su))/bracketleftbigg\n1−ε1/d♯γε(su)\nN(γε(su))/bracketrightbigg\n≥N(γε(su))[1−Cε1/d/2]\nwhich gives N(γε(su))≤s[1−Cε1/d/2]−1≤s[1 +Cε1/d]forεsmall enough.\nLet now consider c>0such that the conclusion of Proposition 13 holds, i.e., for anyu∈BN(1), for\nηsmall enough we have a.s. that for slarge enough, any path π: 0∝⇕⊣√∫⊔≀→susuch thatN(π)≤(1 +η)sgo\ntrough less than c|Kη(u)|1/dpoints of Ξ. Let now define the sequence (ηn)n≥0by\nη0=Cε1/dandηn+1=c|Kηn(u)|1/dε1/d.\nThe function η∝⇕⊣√∫⊔≀→|Kη(u)|is continuous and increases with ηso the sequence (ηn)is monotonic. Assume\nthatεis small enough such that Cε1/d≤1andc|K1(u)|1/dε1/d≤1. We get then by induction that the\nthe sequence (ηn)remains in [0,1]so it converges to a solution of the equation\nη=c|Kη(u)|1/dε1/d,\ni.e, either to 0or toη∞:=gu((cdε)−1). Note that in both cases, if we fix some δ >0, we have, for n\nlarge enough, ηn≤(1 +δ)η∞.\nAssume moreover that εis small enough such that Proposition 13 holds for η0,i.e.,η0≤¯η. Let\nus prove by induction on nthat for all n≥1, there exists a.s. (a random) snsuch that, for s > sn,\n♯γε(su)≤ηnε−1/ds. We have seen that for slarge enough, N(γε(su))≤s[1 +η0]. Using Proposition\n13, we get that for slarge enough ♯γε(su)≤c|Kη0(u)|1/ds=η1ε−1/ds. Note that if the sequence\n(ηn)is non-decreasing, this directly implies that ♯γε(su)≤ηnε−1/ds. Thus we can assume that (ηn)is\nnon-increasing.\nAssume now that we proved that for slarge enough ♯γε(su)≤ηnε−1/ds. SinceT(γε(su))≤s, we\nget thatN(γε(su)) =T(γε(su)) +ε1/d♯γε(su)≤s(1 +ηn). Sinceηn≤η0≤¯η, we can again apply\nProposition 13 which yields that for slarge enough ♯γε(su)≤c|Kηn(u)|1/ds=ηn+1ε−1/ds. Hence, for\nalln≥0, we have a.s. for slarge enough\n♯γε(su)≤ηnε−1/ds\nand\nN(γε(su))≤s(1 +ηn).\nWe conclude using that for nlarge enough, ηn≤(1 +δ)η∞.\n22End of the proof of Theorem 3. Using (21) and the inequality Tε(0,su)≥s−ε1/d♯γε(su)we get that,\nforεsmall enough, we have for any δ>0, a.s. forslarge enough,\nTε(0,su)≥(1−(1 +δ)gu(cε−1))s.\nSinceµε(u) = lims→∞Tε(0,su)/s, we get for εsmall enough, for any δ>0,\n1−µε(u)≤(1 +δ)gu(cε−1).\nLettingδtends to 0, we get the upper bound in (i) of Theorem 3. The lower bound in (i) of Theorem 3\nhave been established in Proposition 11. Moreover we just proved the upper bounds in (ii) of Theorem\n3 and the lower bound on ♯γε(su)have been established in Corollary 12.\nAt this stage, we could hope to get a lower bound on N(γε(su)), using Proposition 13 and the\nlower bound on ♯γε(su)obtained in Theorem 3 (ii). However, in a general setting, we cannot control\nthe difference between guand¯gu, thus what we get is not very satisfying. Nevertheless, we can state\nthe following result, that will be useful for specific choices of the norm N. Consider the case where\n|K0(u)|= 0. This hypothesis corresponds to the fact that BN(u)does not have a (d−1)-dimensional\nflat edge in the direction of u. Notice that ˆℓu:η∝⇕⊣√∫⊔≀→|Kη(u)|is continuous and strictly increasing. Indeed\nˆℓuis obviously non-decreasing, and since ˆℓu(η) =ηdhu(η)the continuity of ˆℓuis a consequence of the\ncontinuity of hustated in Proposition 2. In the course of the proof of Proposition 2, we in fact proved\nthat (ˆℓu)1\nd−1is concave, see (9). Since (ˆℓu)1\nd−1is concave, non-decreasing and goes to infinity when ηgoes\nto infinity, this function has to be increasing, thus ˆℓuis increasing. If|K0(u)|= 0, then ˆℓuis a bijection\nfrom [0,∞)onto [0,∞). Let us denote by ℓuits inverse function. We can now state the following result.\nProposition 15. Suppose that|K0(u)|= 0, and denote by ℓu: [0,∞)→[0,∞)the inverse function\nofη∝⇕⊣√∫⊔≀→|Kη(u)|. Letcbe the constant appearing in Proposition 13, and C1the constant appearing in\nTheorem 3. Let denote by γε(su)a geodesic from 0tosu. Forεsmall enough, we have, for any δ∈(0,1],\na.s. for large s\nN(γε(su))−s≥ℓu/parenleftbig\n(1−δ)dc−dε−1¯gu(C1ε−1)d/parenrightbig\ns.\nProof of Proposition 15. Letcbe the constant appearing in Proposition 13, and C1the constant appear-\ning in Theorem 3. From the lower bound on ♯γε(su)obtained in Theorem 3 (ii), we know that for ε\nsmall enough, for any δ∈(0,1], we have a.s. for large s,\n♯γε(su)≥/parenleftbigg\n1−δ\n2/parenrightbigg\nε−1/d¯gu(C1ε−1)s. (24)\nDefineη:=ℓu/parenleftbig\n(1−δ)dc−dε−1¯gu(C1ε−1)d/parenrightbig\n. Since we assume that |M0(u)| ≤ |K0(u)|= 0, we get\nthat ¯hu(η) =η−d|Mη(u)|=o(η−d)asηtends to 0. So its inverse function ¯gusatisfies in turn that\n¯gu(x) =o(x−1/d)asxtends to infinity. Using that ℓu(0) = 0andη≤ℓu/parenleftbig\nc−dε−1¯gu(C1ε−1)d/parenrightbig\n, we get\nthatηtends to 0 as εtends to 0(uniformly with respect to δ∈(0,1]). Moreover, we have |Kη(u)|=\n(1−δ)dc−dε−1¯gu(C1ε−1)dby definition of ℓu. This leads to\nc|Kη(u)|1/d= (1−δ)ε−1/d¯gu(C1ε−1). (25)\nBy Proposition 13, we know that for εsmall enough, so that η <¯η, a.s., forslarge enough, every path\nπ∈Π(0,su)such thatN(π)≤(1 +η)ssatisfies♯π≤c|Kη(u)|1/ds. From (24) and (25), we get that a.s.,\nforslarge enough,\nN(γε(su))>(1 +η)s\nwhich ends the proof of Proposition 15.\nRemark 16. Foru∈RdwithN(u) = 1andHa supporting hyperplane of BN(1)atu, recall that\nΠ(0,su+H)is the set of paths from 0to somey∈su+H. We can define the time travel from 0to\nsu+Hby\nTε(su+H) := inf{Tε(π),π∈Π(0,su+H)}\n23and prove, for small enough ε, that\nµε(u,H) := lim\ns→∞Tε(su+H)\nsexists a.s. and in L1.\nNote that, in view of Lemma 14, one can use the same argument than above to show that 1−µε(u,H)≤\ngu/parenleftbig\nC2ε−1/parenrightbig\n. Moreover, to get an lower bound for this quantity, we can use a very similar greedy algorithm\nthan the one explained in Section 4 except that instead of looking at the next point of Ξin a cone of\nbasisMη, we look at the next point of Ξin a cone of basis Kη. The setKηnot being symmetric, we\ncannot control anymore the deviation of the path with respect to the direction u((Vn)n≥0is not anymore\na symmetric random walk) but the path created will still end at tome su+y, withy∈H, giving that\n1−µε(u,H)≥gu/parenleftbig\nC1ε−1/parenrightbig\n.Hence, we finally get\ngu/parenleftbig\nC1ε−1/parenrightbig\n≤1−µε(u,H)≤gu/parenleftbig\nC2ε−1/parenrightbig\n,\nwith the same function guappearing on both sides of the inequalities.\n6 Study of ˜µε(u)\n6.1 Some properties of the geodesics ˜γε(su)\nRecall that for x,y∈Rd,˜Tε(x,y)denotes the travel time between xandyin the first passage percolation\nmodel defined in Section 1.2\n˜Tε(x,y) := inf\nπ∈ˆΠ(x,y)˜Tε(π),\nwhere, for a path π,\n˜Tε(π) =N(π∩Σc\nε).\nIn this section, we prove that, for εsmall enough, geodesics and time constant exist for this model and\nbesides, we can impose some properties on the geodesics such that this model can be easily compared to\nthe model with rewards. We begin by proving that, as in the model with rewards, the infimum can in\nfact be taken only on polygonal paths.\nProposition 17. For everyx,y∈Rd, we have\n˜Tε(x,y) = inf\nπ∈ˆΠ(x,y)˜Tε(π) = inf\nπ∈Π(x,y)˜Tε(π).\nProof.Fixx,y∈Rdand consider a path π∈ˆΠ(x,y). We want to construct a path π′∈Π(x,y)such\nthat ˜Tε(π′)≤˜Tε(π).\nWe associate with the path πthe sequence (C1,...,Ck)of thekconnected components of Σεthat the\ncurve [π]intersects ranked by their order in apparition in [π](k∈Nmay be null). If there exists i0<i1\nsuchCi0=Ci1:=C, we can replace πbyˆπdefined as the concatenation of the three following paths:\n•the subpath π1ofπfromxto the first point ain[π]∩C;\n•the subpath π3ofπfrom the last point bin[π]∩Ctoy;\n•between those subpaths, a path π2betweenaandbsatisfying [π2]⊂C,i.e., that remains inside C.\nSince ˜Tε(π2) = 0, we have ˜Tε(ˆπ) = ˜Tε(ˆπ1) +˜Tε(ˆπ3)≤˜Tε(π), thus we can suppose that the connected\ncomponents (C1,...,Ck)are distinct and the path πsuccessively enters in the distinct connected com-\nponents (C1,...,Ck)and never go back to one of them once it has left it. For sake of clarity, we assume\nhere thatxandydoes not belong to Σε. We let the reader check that the proof below can be easily\nadapted if this condition does not hold.\nLet denote by (ai,1≤i≤k)the first point of [π]inCiand(bi,1≤i≤k)its last point (see Figure\n3). Then we have\nTε(π) =k/summationdisplay\ni=0N(ai+1−bi)\n24π\nπ′\na1\na2\nb2\nb1\nc1\nc2\nx\nx\ny\nd1\nd2Figure 3: The path π(in red) is a generalized path from xtoy. In blue, a polygonal path π′fromxto\nysuch thatTε(π′)≤Tε(π).\nwith the convention x=b0andy=ak+1. Letci∈Σ∩Cisuch thatai∈∂BN(ci,ε1/d/2)anddi∈Σ∩Ci\nsuch thatbi∈∂BN(ci,ε1/d/2). Letπi= (ci,xi,1,...,xi,ni,di)be a polygonal path ( i.e., with vertices in\nΞ) fromcitodiwhich remains in Ci. Consider now the path π′fromxtoywhich is the concatenation\nof the paths (πi,1≤i≤k), namely\nπ′:= (x,c1,x1,1,...,x 1,n1,d1,c2,...,dk,y).\nBy construction π′∈Π(x,y)and\nTε(π′) =N([x,c1]∩Σc\nε) +N([dk,y]∩Σc\nε) +k−1/summationdisplay\ni=1N([di,ci+1]∩Σc\nε)\n≤N(c1−x)−ε1/d\n2+N(y−dk)−ε1/d\n2+k−1/summationdisplay\ni=1/parenleftig\nN(ci+1−di)−ε1/d/parenrightig\n.(26)\nBut, by triangle inequality, we have\nN(ci+1−di)≤N(bi−di) +N(ai+1−bi) +N(ci+1−ai+1) =ε1/d+N(ai+1−bi),\nand\nN(c1−x)≤N(a1−x) +ε1/d\n2andN(y−dk)≤N(y−bk) +ε1/d\n2.\nSo we get that Tε(π′)≤Tε(π)as wanted.\nThe next step is now to prove the existence of geodesics and of a time constant for this model.\nHowever, the proofs are more classical in this case than for the model with rewards. Indeed, the random\nvariables (˜Tε(x,y),x,y∈Rd)are the travel times in a first passage percolation model, in particular,\nthey are non negative, satisfy the triangle inequality and are bounded by N(y−x). A classical use of\nKingman’s subadditive ergodic theorem enables then to prove the existence of the time constant: for\neveryz∈Rd, there exists a constant ˜µε(z)∈[0,N(z)]such that\nlim\ns→∞˜Tε(0,sz)\ns= ˜µε(z)a.s. and in L1.\nBesides, the Euclidean case N:=∥·∥2, has been studied in [10] where a condition is given to ensure that\n˜µε(·) := ˜µ2,ε(·)is strictly positive in terms of a percolation event for the Boolean model Σ∥·∥2,ε. This\nresult implies in particular that ˜µ2,ε(·)is a norm for small enough ε. Using the equivalence of the norm\nonRd, we have that ΣN,ε⊂Σ∥·∥2,cεfor somec >0so it is easy to deduce that, in fact, for any norm\nN, the function ˜µN,ε(·)is strictly positive for small enough ε.3When ˜µε:= ˜µN,ε(·)is a norm, a shape\n3It should be possible to characterize exactly the set of εsuch that ˜µN,ε(·)is a norm in terms of an event of percolation\nfor the Boolean model ΣN,εsimilar to the one given in [10].\n25theorem also holds:\nfor anyδ>0, fortlarge enough{z∈Rd,˜Tε(0,z)≤t}⊂(1 +δ)B˜µε(1)a.s.\nwhereB˜µε(1)denotes the unit ball for the norm ˜µε. This implies in particular that for any n∈Nand\nanyx,y∈BN(n), there exists a.s. some K > 0such that\n˜Tε(x,y) := inf{˜Tε(π) :π∈Π(x,y)}= min{˜Tε(π) :π∈Π(x,y),π⊂BN(K)},\nsince the last set of paths is a.s. finite. This gives the existence, for any x,y∈Rd, of a geodesic\nγε(x,y)∈Π(x,y),i.e., a polygonal path such that\n˜Tε(x,y) =˜Tε(γε(x,y)).\nOf course, such geodesic is not unique since we can modify it inside any connected component of Σε.\nTo compare the model with rewards and the first passage percolation model, we will need in fact more\nproperties on the geodesics we consider. So we introduce here a third set of paths, namely ˇΠε(x,y), and\nprove that the geodesics can be chosen inside this set. For a path π= (x=x0,x1,...,xn=y)from\nxtoy, let denote by (C1,...,Ck)the connected component of Σεthat the curve [π]intersects. Define\nˇΠε(x,y)as the set of paths from xtoysuch that\n(i)π∈Π(x,y).\n(ii) When [π]exits a connected component of Σεit never enters it again.\n(iii) for every connected component Ci, the subpath [π]∩Cican be written as πi= (ai,xi,0,...,xi,mi,bi)\nwheremi≥0,ai(resp.bi) is in the boundary of CiandBN(xi,0,ε1/d/2)(resp.BN(xi,mi,ε1/d/2))\nand for every j∈{0,...,mi−1},N(xi,j+1−xi,j)≤ε1/d.\nWe emphasize that we impose that mi≥0in the third point. This means that each time the path enters\na connected component CofΣε, it gets at least trough one point of Ξ∩C.\nProposition 18. Forεsmall enough there exists a.s., for every x,y∈Rd, a path ˇγε(x,y)∈ˇΠε(x,y)\nsuch that\n˜Tε(x,y) =˜Tε(ˇγε(x,y)).\nProof.Considerπ∈ˆΠ(x,y)a geodesic between xandy. We have already noticed that such path\nnecessarily satisfies Point (ii) of the definition of ˇΠε(x,y). We construct now of new path π′betweenx\nandyin the exact same way than in proof of Proposition 17 except that we impose moreover that the\npolygonal subpath πi= (ci,xi,1,...,xi,ni,di)created insideCibetweencianddisatisfiesN(xi,j+1−\nxi,j)≤ε1/d(such sequence of points necessarily exists since Ciis a connected component of balls of\ndiameterε1/d). Besides, since πis a geodesic, we necessarily have Tε(π) =Tε(π′)so in particular,\nInequality (26) must be an equality. This implies that [π′]cannot intersects other connected components\nofΣεthantheonealreadyintersectedby π. Sowededucethateachtime π′entersaconnectedcomponent\nCofΣε, it get at least trough one point of Ξ∩C. Hence,π′is indeed a path of ˇΠε(x,y).\n6.2 Comparison between ˜µε(u)andµε(u)\nThe comparison between ˜µε(u)andµε(u)is made in two steps. The first and easy step is to notice that,\nby looking at nice geodesics for the model with balls, we can prove that ˜Tε(0,x)≥Tε(0,x)for everyx,\nthus ˜µε(u)≤µε(u). In the second step, we have to prove that ˜Tε(0,x)−Tε(0,x)is not too big. When\na pathπ= (0,x1,...,xn)goes through points (xi)ofΞthat are at N-distance bigger than ε1/d, the\ncorresponding balls of the Boolean model do not overlap, thus ˜Tε(π)≤Tε(π). The travel time ˜Tε(π)\nmay be larger than Tε(π)only ifπgoes through balls of the Boolean model that overlap. The difference\n˜Tε(π)−Tε(π)can thus be controlled for a path π= (0,x1,...,xn)by obtaining an upper bound on the\nnumbers of couples of points among the (xi)that are at N-distance less than ε1/d(see Inequality (28)).\nThis upper bound is proved through Lemmas 19 and 20, using ideas that are quite similar to the ones\nused in Section 5.\nBy Proposition 18, for all x∈Rd,we know that there exists a geodesic ˜γε(x) = (0 =x0,...,xn=x)∈\nˇΠε(0,x)for˜Tε(0,x)such that, for all 0< i < n,xi∈Ξand ifxiandxi+1are in the same connected\n26component of Σε,N(xi+1−xi)≤ε1/d. Let (C1,...,Ck)be the distinct connected components of Σε\nthat the curve [˜γε(x)]intersects, and write [˜γε(x)]∩Ci= (ai,xi,0,...,xi,mi,bi)as in property (iii)of the\ndefinition of ˇΠε(0,x). Then\nN(˜γε(x)∩Σε) =k/summationdisplay\ni=1N(˜γε(x)∩Ci)\nand\nN(˜γε(x)∩Ci) =ε1/d\n2+mi−1/summationdisplay\nj=0N(xi,j+1−xi,j) +ε1/d\n2≤(mi+ 1)ε1/d.\nWe get that\nN(˜γε(x)∩Σε)≤ε1/d♯˜γε(x) (27)\nthus\n˜Tε(0,x)≥Tε(0,x).\nThis already yields that ˜µε(u)≥µε(u).\nLet now find an upper bound for ˜µε(u)−µε(u). For any polygonal path π= (x0,...,xq+1)with\nxi∈Ξfor1≤i≤q, we define\nY(π) :={j∈J1 :q+ 1K,∃i<j,N (xj−xi)≤ε1/d}.\nNote that if j /∈Y(π), thenBN(xj,ε1/d/2)does not intersect any other ball BN(xi,ε1/d/2)withi<j.\nHence, we see that\n˜T(π)≤N(π)−ε1/d(q−1−card(Y(π)))≤T(π) +ε1/d(2 +card(Y(π))).\nApplying this inequality to the geodesic γε(su)from 0tosuin the model with rewards studied previously,\nwe get\n˜T(γε(su))≤T(γε(su)) +ε1/d(2 +card(Y(γε(su)))). (28)\nWe obtain an upper bound for card (Y(γε(su)))by proving the following two lemmas.\nLemma 19. Forπ= (x0,...,xq+1)a path let define\nZ(π) :={j∈J1 :q+ 1K,N(xj−xj−1)≤5ε1/d}.\nThen we have a.s. for any geodesic γ, card (Y(γ))≤2card(Z(γ)).\nLemma 20. LetC2be such that Theorem 3 holds. Fix some direction uand some supporting plane H\nofBNatu. Then, for any λ > 0, there exists ε0>0thus that for ε < ε 0, we have, a.s. for slarge\nenough,\ncard(Z(γε(su)))≤λgu(C2ε−1)ε−1/ds.\nAssuming these lemmas hold, we get that for any λ>0, forεsmall enough we have a.s. for slarge\nenough,\n˜T(γε(su))≤T(γε(su)) + (2ε1/d+ 2λgu(C2ε−1)s)\nwhich implies that\nlim sup\nε→0˜µε(u)−µε(u)\ngu(C2ε−1)≤2λ\nand so Theorem 4 (iii)holds.\nProof of Lemma 19. To prove this inequality between the cardinal of Y(π)andZ(π), we in fact prove\nthat\ncard(Y(π)\\Z(π))≤card(Z(π)).\nLet us first note that for a path π= (0,x1,...,xq)withxi∈Ξfor1≤i≤q, we have a.s. (since 0/∈Ξ\na.s.)\nTε(π) =q/summationdisplay\ni=1/parenleftig\nN(xi−xi−1)−ε1/d/parenrightig\n.\n27Thus ifTε(π)≤−Kε1/dit implies in particular that card (Z(π))≥K. Let nowγ= (0,x1,...,xq,x)be\na finite geodesic from 0toxforTεand letj(1)<...<j (l)the indices such that\nY(γ)\\Z(γ) ={xj(1),...,xj(l)}.\nLet define, for i≤q, the truncated path γi= (0,x1,...,xi). Let us prove by induction on ithat\ncard(Z(γj(i)))≥i (29)\nwhich would imply that\ncard(Z(γ))≥card(Z(γj(l)))≥l=card(Y(γ)\\Z(γ)).\nWe have by definition xj(1)∈Y(γ)\\Z(γ),i.e., there exists some k<j (1)such thatN(xj(1)−xk)≤ε1/d\nbutN(xj(1)−xj(1)−1)≥5ε1/d. Using that\nε1/d≥N(xk−xj(1))≥Tε(xk,xj(1)) =Tε(xk,xj(1)−1) +Tε(xj(1)−1,xj(1))≥Tε(xk,xj(1)−1) + 3ε1/d\nwe get that\nTε(xk,xj(1)−1)≤−2ε1/d\nwhich implies in particular that\ncard(Z(γj(1)))≥card(Z((xk,...,xj(1)−1)))≥1.\nFixi, and assume now that for every m < i, we have card (Z(γj(m)))≥m. Letk < j (i)such that\nN(xj(i)−xk)≤ε1/d. Letm<isuch thatj(m)<k≤j(m+ 1)with the convention j(0) = 0. We have\nε1/d≥Tε(xk,xj(i)) =Tε(xk,xj(m+1)−1) +i/summationdisplay\np=m+1Tε(xj(p)−1,xj(p)) +i−1/summationdisplay\np=m+1Tε(xj(p),xj(p+1)−1).\nBy definition of the indexes j(p), we haveTε(xj(p)−1,xj(p))≥3ε1/dso we get\nTε(xk,xj(m+1)−1) +i−1/summationdisplay\np=m+1Tε(xj(p),xj(p+1)−1)≤ε1/d(1−3(i−m))≤−2(i−m)ε1/d.\nThis implies in particular that\ncard(Z((xk,...,xj(i))))≥i−m\nfor somek≥j(m). But, by hypothesis, we have\ncard(Z((0,...,xk−1)))≥card(Z((0,...,xj(m))))≥m\nso we get\ncard(Z(γj(i)))≥i.\nProof of Lemma 20. Fix someC2such that Theorem 3 holds. Fix some λ>0. Settingη∞:=gu(C2ε−1),\nwe want to prove that for εsmall enough, we have, a.s. for slarge enough\ncard(Z(γε(su)))≤λη∞ε−1/ds.\nRecall that we have a.s. for slarge enough N(γε(su))≤s(1+2η∞)and♯γε(su)≤2η∞ε−1/ds. Thus, it is\nsufficient to show that for slarge enough, for any path πfrom 0tosu+Hsuch thatN(π)≤s(1 + 2η∞)\nand♯π≤2η∞ε−1/ds, we have\n♯Z(π)≤λη∞ε−1/ds.\nLetq0:=λη∞ε−1/ds,q1:= 2η∞ε−1/ds,η:= 2η∞and set\nF(s) :={∃π∈Π(0,su+H),N(π)≤(1 +η)s,♯π≤q1,card(Z(π))≥q0}.\n28Proving thatF(s)does not occur a.s. for slarge enough will yield Lemma 20. To bound P(F(s)), we\nroughly use the same idea as in Section 5. For a sequence of points (xi)1≤i≤qinΞ, let denote by (ti)1≤i≤q\ntheir increments, i.e.,ti=xi−xi−1(with the convention t1=x1). Then, we have\nP[F(s)] =/summationdisplay\nq≤q1/summationdisplay\nZ⊂{1,...,q},|Z|≥q0P(AZ)\nwhere\nAZ:={∃(xi)1≤i≤q∈Ξq,xq·u⋆≤s,q/summationdisplay\ni=1N(ti)≤ηs+xq·u⋆,∀j∈Z,tj·u⋆≤5ε1/d}.\nUsing Chernov inequality, we have for all α,β > 0,\nP[AZ]\n≤/integraldisplay\n(Rd)qexp/parenleftigg\nβ(s−xq·u⋆) +α(ηs+xq·u⋆−q/summationdisplay\ni=1N(ti))/parenrightigg/productdisplay\nj∈Zexp/parenleftigα\n2(4ε1/d−(tj·u⋆))/parenrightig\ndt1...dtq.\nTakingβ=αη, we get\nP[AZ]≤exp(α(2ηs+ 2ε1/dcard(Z))/integraldisplay\n(Rd)qexp/parenleftigg\n−αq/summationdisplay\ni=1/bracketleftbigg\nN(ti)−/parenleftbigg\n1−η−1j∈Z\n2/parenrightbigg\n(ti·u⋆)/bracketrightbigg/parenrightigg\ndt1...dtq\n= exp(2α(ηs+ε1/dcard(Z))/parenleftbiggI(η)\nαd/parenrightbiggq−card (Z)/parenleftbiggI(1\n2+η)\nαd/parenrightbiggcard (Z)\n= exp(2α(ηs+ε1/dcard(Z))/parenleftigg\nI(η)1−card (Z)/qI(1\n2+η)card (Z)/q\nαd/parenrightiggq\n.\nWe know that limη→0I(η) = limη→0hu(η) = +∞whereasI(η)is bounded on [1/2,1]. Sinceη∞\ntends to 0 as εtends to 0, we deduce that for εsmall enough, I(η)≥I(1\n2+η)and so we get that for\ncard(Z)≥q0andq≤q1,\nI(η)1−card (Z)/qI/parenleftbigg1\n2+η/parenrightbiggcard (Z)/q\n≤I(η)1−q0/q1I/parenleftbigg1\n2+η/parenrightbiggq0/q1\n=I(η)1−λ/2I/parenleftbigg1\n2+η/parenrightbiggλ/2\n≤c1I(2η∞)1−λ/2\nforλ∈(0,1)andc1:= sup{I(x),x∈[1/2,1]}∨1. Using (11) in Lemma 10, which states that Iis of the\nsame order than hunear 0, we also get, for some constant c2, forεsmall enough,\nI(2η∞)≤c2hu(2η∞)≤c2hu(η∞) =c2C2ε−1.\nSo we get, for some constant c3>0,\nP[F(s)]≤/summationdisplay\nq0≤q≤q12qexp(2α(2η∞s+ε1/dq1)/parenleftbiggc3ε−1+λ/2\nαd/parenrightbiggq\n.\n≤/summationdisplay\nq≥q0exp(8αη∞s)/parenleftbigg2c3ε−1+λ/2\nαd/parenrightbiggq\nTakingαd= 4c3ε−1+λ/2,i.e.,α=c4ε−(1/d)+λ/(2d), we obtain\nP[F(s)]≤2 exp(8c4ε−(1/d)+λ/(2d)η∞s−q0log 2)\n= 2 exp(−sε−1/dη∞(λlog 2−8c4ελ/(2d))).\n29Hence, for any λ∈(0,1), ifεis small enough, this quantity decreases to 0 exponentially fast in s. Taking\nsn:=nε1/d/(λη∞)\nBorel-Cantelli Lemma yields that there exists a.s. n0such that, for n≥n0,F(sn)does not occur. Since\nfors∈[sn,sn+1[,⌊λη∞ε−1/ds⌋=⌊λη∞ε−1/dsn⌋, we deduce using the same argument as in proof of\nProposition 13, that in fact F(s)does not occur for slarge enough.\n6.3 Proof of the lower bound on 1−˜µε(u)\nWhen 1−µε(u)is of ordergu(C2ε−1)whenεgoes to 0, it is a straightforward consequence of Theorem\n4(iii)that the same holds for 1−˜µε(u). It will be the case for specific choices of the norm N, namely\nthep-norms, see Section 7. However, in general setting, the lower bound on 1−µε(u)given by Theorem\n3 is¯gu(C1ε−1), and there is no way to deduce the same lower bound for 1−˜µε(u)through Theorem 4\n(iii). It is however easy to adapt the greedy algorithm used to prove the lower bound on 1−µε(u)in\nSection 4 to get the same type of lower bound on 1−˜µε(u). The idea is to add an extra space between\ntwo consecutive points of the process Ξthat are collected by the greedy algorithm, to make sure that\nthe corresponding balls of diameter ε1/din the Boolean model do not intersect. Let us get into details.\nFix a direction u∈RdwithN(u) = 1,Ha supporting hyperplane of BN(1)atuand recall that\nMη:={v∈H:N(u+v)≤1 +ηandN(u−v)≤1 +η}and ¯hu(η) :=η−d|Mη(u)|.\nWe state the following analog of Proposition 11.\nProposition 21. There exists a constant c′>0(depending only on d,N) such that, for all direction\nu∈RdwithN(u) = 1, for allη∈(0,1), we have\n˜µε(u)≤1 +η−c′ε1/d|Mη|1/d.\nIn particular, we have, with C′\n1:=/parenleftbigg2\nc′/parenrightbiggd\n,\n˜µε(u)≤1−¯gu(C′\n1ε−1).\nProof.We use almost the same greedy algorithm as in the proof of Proposition 11 except that we impose\nalso that two consecutive points taken by the greedy path must be at distance at least 1. Recall that for\ns≥0\nCη(s) :={x=λu+v: 0≤λ≤s, v∈H, N (λu+v)≤λ(1 +η),N(λu−v)≤λ(1 +η)},\nand we have\n|Cη(s)|=1\nd∥u⋆∥2|Mη|sd.\nRecall that Ξdenotes the points of a Poisson point process with unit intensity on Rd. By induction we\ndefine a sequence (Xn)n≥0inRdwithX0= 0and if\nλn+1:= inf{s>1 : (Xn+Cη(s))∩Ξ̸=∅}\n(we emphasize the fact that we require that s>1), we setXn+1=Xn+λn+1u+Wn+1withWn+1∈H\nand such that Xn+1= (Xn+Cη(λn+1))∩Ξ. We define Sn=/summationtextn\ni=1λiandVn=/summationtextn\ni=1Wi. By the\nproperties of Poisson point processes, (λi)i≥1and(Wi)i≥1are i.i.d. random variables and since Mη(u)\nis a symmetric set, W1has also a symmetric distribution and so Vnis a symmetric random walk.\nLet us now define, for s>0,\nZ(s) := sup{n≥0 :Sn≤s}\nand consider the path πs= (0,X1,...,XZ(s),su). Let us find an upper bound of ˜Tε(πs)by finding an\nupper bound of its length and a lower bound of Z(s), its number of points.\n30The sequence (λi)i≥1is i.i.d. and we have for t≥1,\nP(λ1≥t) = exp/parenleftbigg\n−1\nd∥u⋆∥2|Mη(u)|(td−1)/parenrightbigg\nso that\nE(λ1) = 1 +/integraldisplay∞\n1exp/parenleftbigg\n−1\nd∥u⋆∥2|Mη(u)|(td−1)/parenrightbigg\ndu\n= 1 +/parenleftbigg|Mη(u)|\nd∥u⋆∥2/parenrightbigg−1/d/integraldisplay∞\n(|Mη(u)|/∥u⋆∥2)1/dexp(−vd)dv:= 1 +|Mη(u)|−1/d\n2c′u\nwhere\n2c′\nu:=/parenleftbigg1\nd∥u⋆∥2/parenrightbigg1/d/parenleftigg/integraldisplay∞\n(|Mη(u)|/∥u⋆∥2)1/dexp(−vd)dv/parenrightigg−1\n.\nUsing that for any η∈(0,1),Mη(u)⊂BN(0,3)we see that the (d−1)-dimensional volume |Mη(u)|of\nMη(u)is bounded by some constant c′\n1(depending only on dandN) forη∈(0,1). By Lemma 9, we\nalso know that c≤∥u⋆∥2≤c′for constants c,c′∈(0,∞)(depending only on dandN). We deduce that\nthere exist some constants c2,c3>0(depending only on dandNalso) such that for η∈(0,1),\nc2|Mη|−1/d≤E(λ1)≤c3|Mη|−1/d.\nSinceSn=/summationtextn\ni=1λi, a standard renewal theorem yields that, setting c4= 1/(2c3)andc5= 2/c2, fors\nlarge enough, we have a.s.\nc4|Mη|1/ds≤Z(s)≤c5|Mη|1/ds. (30)\nThis gives a lower bound for the number of points in πs. The rest of the proof of Proposition 21 is a\ncopy of the proof of Proposition 11 with Equation (30) instead of (15).\n6.4 Upper bound on N(˜γε(su))\nWe recall that ˜γε(su)∈ˆΠ(0,su)designates a generalized path that is a geodesic for ˜Tε(0,su)and has\nminimalN-length. To complete the proof of Theorem 4, it remains to prove (ii),i.e., the upper bound\nonN(˜γε(su)). In fact, if ˇγε(su)∈ˇΠε(0,su)designates a path that is a geodesic for ˜Tε(0,su), then\nˇγε(su)∈ˆΠ(0,su)too and by minimality we know that N(˜γε(su))≤N(ˇγε(su)). We will in fact prove an\nupper bound for N(ˇγε(su)). Since ˇγε(su)∈ˇΠε(0,su), Equation (27) states that\nN(ˇγε(su)∩Σε)≤ε1/d♯ˇγε(su)\nwhich gives that\nN(ˇγε(su))≤˜T(ˇγε(su)) +ε1/d♯ˇγε(su)≤s+ε1/d♯ˇγε(su).\nOne can check that the proof of (21) (which gives an upper bound for the length and the number of\npoints of the geodesic in the model with rewards) only use a similar inequality between N(γε(su))and\n♯γε(su). Mimicking this proof, we can so also obtain that, for any δ>1,\nN(ˇγε(su))≤s(1 +δgu(C2ε−1))and♯ˇγε(su)≤δgu(C2ε−1)ε−1/ds. (31)\n7p-norm\nIn this section, we suppose that Nis thep-norm for some p∈[1,∞]. We put a subscript pin our\nnotations to emphasize the dependence on p. For a given u∈Rd, we recall that the definition of di(u),\ni∈{1,..., 4}is given in (5). For a given p∈[1,∞]the definition of κp(u)is given in (6).\nThis section is organized as follows. First in Section 7.1 we specify for each p∈[1,∞]and each\nu∈Rdsuch that∥u∥p= 1what hyperplane Hwe consider as a supporting hyperplane of Bp(1)atu,\nand fix some notations. In this setting, we evaluate |Kη(u)|in Section 7.2. This must be done separately\nforp∈(1,∞),p= 1andp= +∞, since the estimations are based on a proper rescaling of Kη(u)\nwhich is not exactly of the same nature in those different situations. Then in Section 7.3 we compare\n|Mη(u)|to|Kη(u)|. This allows us to apply the results proved for a general norm Nto thep-norm in\nSection 7.4 to prove Theorems 5 and 6. Finally in Section 7.5 we prove Theorem 7, i.e., the existence of\nlimε→0(1−µε(u))/εκp(u), by an argument of monotonicity, at least for some pandu.\n317.1 Choice and description of the supporting hyperplane H\nWe have to deal separately with the cases p∈(1,∞),p= 1andp=∞. From now on, until the end of\nSection 7, we always consider that the hyperplane Hwe work with is the one described here.\nCasep∈(1,∞).ConsiderN=∥·∥pfor somep∈(1,∞). Letu∈Rdsuch that∥u∥p= 1. Let\nd1:=d1(u)be the number of coordinates of uthat are not null. By invariance of the model under\nsymmetries along the hyperplanes of coordinates and by permutations of coordinates, we can suppose\nthatu= (u1,...,ud1,0,..., 0)withui>0fori∈{1,...,d 1}. The ball Bp(1)is strictly convexe, and\nthe (unique) supporting hyperplane HofBp(1)atuis\nH:={v∈Rd:v·u⋆= 0}\nwhereu⋆= (up−1\n1,...,up−1\nd1,0,..., 0)satisfiesu·u⋆=/summationtextd1\ni=1up\ni= 1(u⋆is the dual vector of u). Note that\nwe haveH=H1⊕H2whereH1⊂Vect(e1,...,ed1)andH2=Vect(ed1+1,...,ed), for (e1,...,ed)the\ncanonical orthonormal basis of Rd. Forv∈H, we writev=v1+v2withv1= (v1,1,...,v 1,d1,0,..., 0)∈\nH1andv2= (0...,0,v2,d1+1,...,v 2,d)∈H2.\nCasep= 1.ConsiderN=∥·∥ 1and letu∈Rdsuch that∥u∥1= 1. We can suppose that u=\n(u1,...,ud1,0,..., 0)withui>0for alli∈{1,...,d 1}withd1=d1(u). In this case, a supporting\nhyperplane HofB1(1)atuis\nH:={v∈Rd:v·u⋆= 0}\nwhereu⋆= (1,..., 1,0,..., 0)satisfiesu·u⋆=/summationtextd1\ni=1ui= 1. We emphasize the fact that this hyperplane\nHis not the unique supporting hyperplane of B1(1)atuifd1<d, but this is the most natural choice of\nsupporting hyperplane since it is the most symmetric. As for p∈(1,∞), note that we have H=H1⊕H2\nwhereH1⊂Vect(e1,...,ed1)andH2=Vect(ed1+1,...,ed), for (e1,...,ed)the canonical orthonormal\nbasisof Rd. Forv∈H, wewritev1= (v1,1,...,v 1,d1,0,..., 0)∈H1andv2= (0...,0,v2,d1+1,...,v 2,d)∈\nH2.\nCasep=∞.ConsiderN=∥·∥∞and letu∈Rdsuch that∥u∥∞= 1. We can suppose that\nu= (1,..., 1,ud3+1,...,ud)withui∈[0,1)for alli∈{d3+ 1,...,d}withd3:=d3(u). In this case, a\nsupporting hyperplane HofB∞(1)atuis\nH:={v∈Rd:v·u⋆= 0}\nwhereu⋆= (1/d3,..., 1/d3,0,..., 0)(withd3non null coordinates) satisfies u·u⋆=/summationtextd3\ni=11/d3= 1. We\nemphasize the fact that this hyperplane His not the unique supporting hyperplane of B∞(1)atuif\nd3>1, but this is the most natural choice of supporting hyperplane since it is the most symmetric. Note\nthat we have H=H3⊕H4whereH3⊂Vect(e1,...,ed3)andH4=Vect(ed3+1,...,ed), for (e1,...,ed)\nthe canonical orthonormal basis of Rd. Forv∈H, we write v3= (v3,1,...,v 3,d3,0,..., 0)∈H3and\nv4= (0...,0,v4,d3+1,...,v 4,d)∈H4.\n7.2 Evaluation of |Kη(u)|\nThe aim of this section is to prove the following estimate of |Kη(u)|.\nProposition 22. ConsiderN=∥·∥pfor somep∈[1,∞]. Letu∈Rdsuch that∥u∥p= 1andHthe\nsupporting hyperplane of Bp(1)atuas defined in Section 7.1. There exist two constants A1:=A1(p,d,u ),\nA2:=A2(p,d,u ), such that for any η∈(0,1], we have\nA1ηγp(u)≤|Kη(u)|≤A2ηγp(u)\nwith\nγp(u) :=\n\nd1(u)−1\n2+d2(u)\npifp∈(1,∞),\nd2(u) =d−d1(u) ifp= 1,\nd3(u)−1 =d−(d4(u) + 1) ifp=∞.(32)\nThe proof of Proposition 22 is based on the simple idea that Kη(u), when properly rescaled with η,\nlooks roughly like a Euclidean ball in H. However, the good rescaling depends on p, and has to be done\nin a specific way for p= 1andp=∞.\n327.2.1p-norm with p∈(1,∞)\nThe good rescaling in ηin this case is of order η1/2inH1whereas it is of order η1/pinH2. We make it\nclear in the following lemma.\nLemma 23. Letp∈(1,∞)and\nˆKη(u) :={v∈H:∥u+η1/2v1+η1/pv2∥p\np≤1 +η}.\nThen, there exist η0>0,R1>R 0>0(depending on p,dandu) such that, for all η∈(0,η0),\nBH(R0)⊂ˆKη(u)⊂BH(R1)\nwhereBH(r) :={v∈H:∥v∥2≤r}.\nProof of Lemma 23. Let us find R0such thatBH(R0)⊂ˆKη(u)forη∈(0,1]. We have\n|1 +t|p= 1 +pt+O(t2).\nThus, there exists c>0, such that, for all t∈[−1,1],\n|1 +t|p≤1 +pt+ct2.\nLetRbe small enough such that R≤min{ui,i≤d1}anddRp+cR2/summationtext\ni≤d1up−2\ni≤1. Letv∈Hsuch\nthat∥v∥∞≤R. Then\n∥u+η1/2v1+η1/pv2∥p\np=η∥v2∥p\np+/summationdisplay\ni≤d1|ui+η1/2v1,i|p.\nIfη≤1, we have for all 1≤i≤d1,/vextenddouble/vextenddouble/vextenddouble/vextenddoubleη1/2v1,i\nui/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤R\nui≤1,\nhence we have\n|ui+η1/2v1,i|p≤up\ni+η1/2pv1,iup−1\ni+ηcv2\n1,iup−2\ni.\nUsing that v1·u⋆= 0and||u||p= 1, we get\n∥u+η1/2v1+η1/pv2∥p\np≤ηd∥v2∥p\n∞+ 1 +ηc/summationdisplay\ni≤d1v2\n1,iup−2\ni\n≤1 +η\ndRp+cR2/summationdisplay\ni≤d1up−2\ni\n\n≤1 +η\nwhich proves that v∈ˆKη(u). Using that∥v∥2≥∥v∥∞, we get that BH(R)⊂ˆKη(u).\nLet us now find η0>0andR1>0such that ˆKη(u)⊂BH(R1)forη∈(0,η0]. We have\n|1 +t|p= 1 +pt+p(p−1)\n2t2+O(t3).\nThus, there exists c>0such that, for all t∈R,\n|1 +t|p≥1 +pt+p(p−1)\n2t2−c|t|3.\nLetR> 0such that\nmin/parenleftbigg\nRp,p(p−1)\n2R2min\ni≤d1up−2\ni/parenrightbigg\n>2\nand letη0>0such thatR3cη1/2\n0/summationtext\ni≤d1up−3\ni= 1. Letv∈Hsuch that∥v∥∞=R. We have, for all\n1≤i≤d1,\n|ui+η1/2v1,i|p≥up\ni+η1/2pv1,iup−1\ni+ηp(p−1)\n2v2\n1,iup−2\ni−cη3/2|v1,i|3up−3\ni.\n33Using that v1·u⋆= 0and∥u∥p= 1, we get, for η∈(0,η0)\n∥u+η1/2v1+η1/pv2∥p\np≥η∥v2∥p\np+ 1 +ηp(p−1)\n2/summationdisplay\ni≤d1v2\n1,iup−2\ni−cη3/2/summationdisplay\ni≤d1|v1,i|3up−3\ni\n≥1 +η\n∥v2∥p\n∞+p(p−1)\n2∥v1∥2\n∞min\ni≤d1up−2\ni−∥v1∥3\n∞cη1/2\n0/summationdisplay\ni≤d1up−3\ni\n\n≥1 +η\n∥v2∥p\n∞+p(p−1)\n2∥v1∥2\n∞min\ni≤d1up−2\ni−R3cη1/2\n0/summationdisplay\ni≤d1up−3\ni\n\n>1 +η\nusing that R=∥v∥∞= max(∥v1∥∞,∥v2∥∞). Hence, for η∈(0,η0), we get that ˆKη(u)∩{v∈H:\n∥v∥∞=R}=∅. Since ˆKηis connexe and contains 0, we deduce that ˆKη(u)⊂{v∈H:∥v∥∞<R}for\nη∈(0,η0). Using that∥v∥2≤√\nd∥v∥∞, it implies that ˆKη(u)⊂BH(√\ndR).\nProof of Proposition 22, case p∈(1,∞).Recall that\nKˆη(u) ={v∈H:∥u+v∥p≤1 + ˆη}\nˆKη(u) ={v∈H:∥u+η1/2v1+η1/pv2∥p\np≤1 +η},\nthus\nv=v1+v2∈ˆKη(u)⇐⇒η1/2v1+η1/pv2∈Kˆη(u)with ˆη:= (1 +η)1/p−1.\nHence\n|ˆKη(u)|=|Kˆη(u)|η−γp(u)\nwith\nγp(u) :=d1(u)−1\n2+d2\np.\nLemma 23 implies that there exists constants A′\n1(p,d,u ),A′\n2(p,d,u )such that, for ηsmall enough,\nA′\n1≤|ˆKη(u)|≤A′\n2\nthus\nA′\n1ηγp(u)≤|Kˆη(u)|≤A′\n2ηγp(u).\nSince ˆη∼η/p, we also get the existence of constants A′′\n1(p,d,u ),A′′\n2(p,d,u )such that, for ˆη≤ˆη0small\nenough,\nA′′\n1ˆηγp(u)≤|Kˆη(u)|≤A′′\n2ˆηγp(u).\nSinceη∝⇕⊣√∫⊔≀→ηγp(u)isboundedawayfrom 0and∞on[ˆη0,1], weobtaintheexistenceofconstants A1(p,d,u ),\nA2(p,d,u )such that, for every η∈(0,1],\nA1ηγp(u)≤|Kη(u)|≤A′′\n2ηγp(u).\n7.2.2 1-norm\nThe good rescaling in ηis now of order 1inH1whereas it is of order ηinH2. We make it clear in the\nfollowing lemma.\nLemma 24. Let\nˆKη(u) :={v∈H:∥u+v1+ηv2∥1≤1 +η}.\nThen, there exist R1>R 0>0(depending on dandu) such that, for all η∈(0,1],\nBH(R0)⊂ˆKη(u)⊂BH(R1)\nwhereBH(r) :={v∈H:∥v∥2≤r}.\n34Proof of Lemma 24. As for the p-norm, it is sufficient to prove that for ∥v∥∞small enough, we have\nv∈ˆKη(u), and for∥v∥∞= 3for instance, we have v /∈ˆKη(u). We have\n∥u+v1+ηv2∥1=∥u+v1∥1+∥ηv2∥1=/summationdisplay\ni≤d1|ui+v1,i|+η∥v2∥1.\nIf∥v∥∞≤min(ui,i≤d1)∧(1/d), we have|ui+v1,i|=ui+v1,ifor all 1≤i≤d1, thus using that\n∥u∥1= 1andv1∈Hthus/summationtextd1\ni=1v1,i= 0, we get\n∥u+v1+ηv2∥1= 1 +η∥v2∥1≤1 +η,\nthusv∈ˆKη(u).\nAssume now that ∥v∥∞= 3. Either∥v1∥∞= 3: then, since∥u+v1∥1≥∥v1∥∞−∥u∥∞≥2, we get\n∥u+v1+ηv2∥1≥2.\nOr∥v2∥∞= 3: then using that v1∈Hand so∥u+v1∥1≥1, we get\n∥u+v1+ηv2∥1≥1 + 3η.\nIn both cases, v /∈ˆKη(u).\nProof of Proposition 22, case p= 1.It is similar to the proof in the case p∈(1,∞). We now have\n|ˆKη(u)|=|Kη(u)|η−d2(u):=|Kη(u)|η−γ1(u)\nwithγ1(u) :=d2(u), which yields similarly to the existence of constants A1(1,d,u),A2(1,d,u)such that,\nfor everyη∈(0,1],\nA1ηγ1(u)≤|Kη(u)|≤A2ηγ1(u).\n7.2.3∞-norm\nThe good rescaling in ηis now of order ηinH3whereas it is of order 1inH4. We make it clear in the\nfollowing lemma.\nLemma 25. Let\nˆKη(u) :={v∈H:∥u+ηv3+v4∥∞≤1 +η}.\nThen, there exist R1>R 0>0(depending on dandu) such that, for all η∈(0,1],\nBH(R0)⊂ˆKη(u)⊂BH(R1)\nwhereBH(r) :={v∈H:∥v∥2≤r}.\nProof of Lemma 25. If∥v∥∞≤min(1−ui,i>d 3)andv∈H, then for all i>d 3we have\n|ui+v4,i|≤ui+|v4,i|≤ui+∥v∥∞≤1,\nand for alli≤d3we have\n|ui+ηv3,i|≤ui+η|v3,i|≤1 +η∥v∥∞≤1 +η,\nthus\n∥u+ηv3+v4∥∞≤1 +η,\nand sov∈ˆKη(u).\nIf∥v∥∞= 2d, either∥v4∥∞= 2dand then\n∥u+ηv3+v4∥∞≥∥v4∥∞−1>1 +η,\nor∥v3∥∞= 2dand then since/summationtextd3\ni=1v3,i= 0, this implies that there exists some index i0≤d3such that\nv3,i0≥2, and then\n∥u+ηv3+v4∥∞≥ui0+ηv3,i0>1 +η.\nIn both cases, we conclude that v /∈ˆKη(u).\n35Proof of Proposition 22, case p=∞.In this case, we have\n|ˆKη(u)|=|Kη(u)|η−(d3−1):=|Kη(u)|η−γ∞(u)\nwithγ∞(u) :=d3(u)−1, which yields similarly to the existence of constants A1(∞,d,u),A2(∞,d,u)\nsuch that, for every η∈(0,1],\nA1ηγ∞(u)≤|Kη(u)|≤A2ηγ∞(u).\n7.3 Evaluation of |Mη(u)|\nWe now compare |Mη(u)|with|Kη(u)|.\nProposition 26. ConsiderN=∥·∥pfor somep∈[1,∞]. Letu∈Rdsuch that∥u∥p= 1andHthe\nsupporting hyperplane of Bp(1)atuas defined in Section 7.1. There exists ∆ = ∆(p,d,u )>0such that,\nfor allη∈(0,1],\n∆|Kη(u)|≤|Mη(u)|≤|Kη(u)|.\nProof.We write the proof for p∈(1,∞)but it can be easily adapted to the cases p= 1andp=∞. Let\nu= (u1,...,ud1,0,..., 0)withui>0and∥u∥p= 1,\nH:={v∈Rd:v·u⋆= 0}\nwhereu⋆= (up−1\n1,...,up−1\nd1,0,..., 0)as given in Section 7.1. Recall that Lemma 23 states that if\nˆKη(u) :={v∈H,||u+η1/2v1+η1/pv2||p\np≤1 +η},\nthen there exist η0>0,R1>R 0>0such that, for all η∈(0,η0),\nBH(R0)⊂ˆKη(u)⊂BH(R1)\nwithBH(r) :={v∈H:∥v∥2≤r}. Define ˆMη(u) := ˆKη(u)∩(−ˆKη(u)), we also have\nBH(R0)⊂ˆMη(u)⊂BH(R1)\nand so\n∆≤|ˆMη(u)|\n|ˆKη(u)|≤1\nfor some ∆(p,d,u )>0. Using that a change of variable yields that |Kˆη(u)|=ηγp(u)|ˆKη(u)|and\n|Mˆη(u)|=ηγp(u)|˜Mη(u)|with ˆη:= (1 +η)1/p−1, we also get\n∆≤|Mη(u)|\n|Kη(u)|≤1.\n7.4 Proof of Theorems 5 and 6\nProof.This is an application of Theorems 3, 4 and Proposition 15. Fix p∈[1,∞],u∈Rdsuch that\n∥u∥p= 1,Has given in Section 7.1 and recall that the definition of γp(u)is given in (32). By Proposition\n22, we know that there exist A1(p,d,u ),A2(p,d,u )such that for any η∈(0,1],\nA1ηγp(u)≤|Kη(u)|≤A2ηγp(u). (33)\nRecall that hu(η) =η−d|Kη(u)|, thus for any η∈(0,1],\nA1ηγp(u)−d≤hu(η)≤A2ηγp(u)−d.\n36By definition, gu=h−1\nu, thus the previous inequalities imply that for any xlarge enough (such that\ngu(x)≤1), we have\nB1x1\nγp(u)−d≤gu(x)≤B2x1\nγp(u)−d\nwithB1(p,d,u ) =A−1\nγp(u)−d\n2andB2(p,d,u ) =A−1\nγp(u)−d\n1. Applying these inequalities to x=Cε−1for\nsome constant C=C(d,p)we obtain that for εsmall enough,\nB′\n1ε1\nd−γp(u)≤gu(Cε−1)≤B′\n2ε1\nd−γp(u)\nfor some constants B′\ni=B′\ni(p,d,u ). Notice that by definition on γp(u)(see (32)) and κp(u)(see (6)), we\nhave\nκp(u) =1\nd−γp(u),\nthus we have just proved that for εsmall enough,\nB′\n1εκp(u)≤gu(Cε−1)≤B′\n2εκp(u). (34)\nSimilarly, combining Proposition 22 with Proposition 26, and recalling that ¯hu(η) =η−d|Mη(u)|and\n¯gu=¯h−1\nu, we obtain the existence of B′′\ni=B′′\ni(p,d,u ),i= 1,2, such that, for every εsmall enough,\nB′′\n1εκp(u)≤¯gu(Cε−1)≤B′′\n2εκp(u). (35)\nFinally, recall that in Proposition 15 we defined ℓu: [0,∞)→[0,∞)as the inverse function of ˆℓu:η∝⇕⊣√∫⊔≀→\n|Kη(u)|when|K0(u)|= 0. First, since K0(u) ={v∈H:∥u+v∥p=∥u∥p)}, the case|K0(u)|̸= 0\ncorresponds to the case where ubelongs to a (d−1)-dimensional flat edge of Bp(1). Forp∈(1,∞),\nBp(1)is strictly convex thus |K0(u)|= 0for every direction u. Forp= 1,|K0(u)|= 0if and only if\nd1(u)<d. Forp=∞,|K0(u)|= 0if and only if d3(u)≥2. We suppose from now on that we are in one\nof those cases, i.e., that|K0(u)|= 0. In other words, we restrict ourselves to the cases where γp(u)>0.\nEquation (33) tells us that for any η∈(0,1],\nA1ηγp(u)≤ˆℓu(η)≤A2ηγp(u).\nThis implies the existence of constants Ai(p,d,u ),i= 1,2, such that for every xsmall enough (so that\nℓu(x)≤1), we have\nA1x1\nγp(u)≤ℓu(x)≤A 2x1\nγp(u).\nIn particular, for constants C,C′depending on dandu, ifx=C′ε−1¯gu(Cε−1)d, we have for εsmall\nenough,\nx≥C′ε−1(B′′\n1εκp(u))d=A3ε−1+dκp(u)=A3εκp(u)γp(u)\nfor a constantA3=A3(p,d,u ). Sinceℓuis increasing, we get for εsmall enough that\nℓu(x)≥ℓu(A3ε−1+dκp(u))≥A 1(A3εκp(u)γp(u))1\nγp(u)=A4εκp(u)(36)\nfor a constantA4=A4(p,d,u ). Combining Theorems 3, 4 and Proposition 15 with Equations (34), (35)\nand (36) ends the proof of Theorems 5 and 6.\nRemark 27. It is not a surprise that our proof does not provide a lower bound for the N-length of the\ngeodesic in any cases. Consider for instance the case d= 2andN=∥·∥ 1:, chooseu∈Rdsatisfying\n∥u∥1= 1andd1(u) = 2, for instance u= (1/2,1/2). If we consider only oriented paths from 0to\nsu, going successively vertically to the north or horizontally to the east, those paths have a ∥·∥ 1-length\nequal tos. The maximal number Msof rewards, i.e., of points of the Poisson point process Ξ, that such\nan oriented path from 0tosucan collect has been introduced and studied by Hammersley [12], and is\nknown to be of order 1- in fact, this problem is even solvable, it was proved by Logan and Shepp and\nby Vershik and Kerov in 1977, and in a more probabilistic way by Aldous and Diaconis [1] in 1995,\nthat lims→∞Ms/s= 1. Thus, by restricting ourselves to oriented paths from 0tosu(whose∥·∥ 1-\nlength iss), it is already possible to obtain a quantity of rewards of order s, and thus a gain in time of\norderε1/ds=ε1/d1(u)s. Since our approach does not allow us to get something better than the order of\n1−µ1,ε(u)and♯γ1,ε(su), we have no hope it allows us to exclude that ∥γ1,ε(su)∥1could bes.\n377.5 Monotonicity\nTheorem 7 is now a direct consequence of the following proposition.\nProposition 28. Suppose we are in one of the following cases:\n(i)p∈[1,∞]andu= (1,0,..., 0);\n(ii)p∈{1,2}, whateveru∈Rdsuch that∥u∥p= 1;\n(iii)p=∞andu∈Rdsuch that∥u∥∞= 1andd3(u)∈{1,2}.\nThen the function1−µp,ε(u)\nεκp(u)increases with respect to ε.\nWe prove first (i) for p∈(1,∞)which also implies, by isotropy, that (ii) holds for the Euclidean\nnorm. We then prove (ii) for the 1-norm and in the last section, we establish (iii).\nRemark 29. In the realm of application of Proposition 28, we obtain the monotonicity of ε∝⇕⊣√∫⊔≀→1−µp,ε(u)\nεκp(u)\nas a consequence of a much stronger property. Indeed, by a rescaling argument, we express1−µp,ε(u)\nεκp(u)as\n1−µp,ε(u)\nεκp(u)= min\ns→∞inf\nπ=(x0...,xq+1)∈Π(0,su)q/summationdisplay\ni=1fp,d,u,s,π,i (ε)\nfor given functions fp,d,u,s,π,i. What we actually prove is the monotonicity of each one of those functions\nfp,d,u,s,π,i. The monotonicity of all the functions fp,d,u,s,π,i is not true for every p∈[1,∞]and every\nu∈Rdsuch that∥u∥p= 1. However, it doesn’t imply that the monotonicity of ε∝⇕⊣√∫⊔≀→1−µp,ε(u)\nεκp(u)is not true,\nonly that our approach cannot work.\n7.5.1p-norm with p∈(1,∞)\nConsider the p-norm with p∈(1,∞)and the direction e1= (1,0,..., 0). We want to prove that the\nfraction (1−µp,ε(e1))/εκp(e1)increases with respect to ε. To simplify notations, we write\nκ:=κp(e1) =1\nd−d−1\np=p\npd−d+ 1.\nLetH=Vect(e2,...,ed)and forx∈Rd\\H, we write x=λx(e1+tx)withλx∈R,tx∈H. For\nπ= (0,x1,...,xq,se1)a path from 0tose1, let define yi:=xi+1−xiwith the convention x0= 0,\nxq+1=se1. Then, a.s.,\nTp,ε(π) =q/summationdisplay\ni=0/parenleftig\n||yi||p−ε1/d/parenrightig\n+ε1/d\nand so\nµε(e1) = lim\ns→∞inf\nπ∈Π(0,su)/summationtextq\ni=0/parenleftbig\n||yi||p−ε1/d/parenrightbig\ns= lim\ns→∞inf\nπ∈Π(0,su)/summationtextq\ni=0/parenleftbig\n|λyi|(1 +||tyi||p\np)1/p−ε1/d/parenrightbig\ns.\nUsing that s=/summationtextλyi, we get\n1−µε(e1)\nεκ= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0λyi−|λyi|(1 +||tyi||p\np)1/p+ε1/d\nsεκ.\nLetψ:Rd∝⇕⊣√∫⊔≀→Rdbe the linear map defined for x=λ(e1+t)witht∈Hby\nψ(λ(e1+t)) :=ε−κ+1\ndλ(e1+εκ\npt).\n38Recall that κ=p/(pd−d+ 1)(we are here in the case d1= 1andd2=d−1). Using that His of\ndimensiond−1, we get det(ψ) =εαwith\nα=d/parenleftbigg\n−κ+1\nd/parenrightbigg\n+ (d−1)κ\np=κ/parenleftbigg\n−d+d−1\np/parenrightbigg\n+ 1 = 0.\nHence,ψpreserves the Lebesgue measure on Rdand soψ−1(Ξ)has the same law as Ξ. So we also have\n1−µε(e1)\nεκ= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0ε−κ+1\ndλyi−ε−κ+1\nd|λyi|(1 +||εκ\nptyi||p\np)1/p+ε1/d\nε−κ+1\ndsεκ\n= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0|λyi|(sgn(λyi)−(1 +εκ||tyi||p\np)1/p) +εκ\nsεκ.\nWe then conclude the proof using the following elementary lemma.\nLemma 30. For anyc≥0, the functions f:x∝⇕⊣√∫⊔≀→1−(1+cx)1/p\nxandg:=x∝⇕⊣√∫⊔≀→−1−(1+cx)1/p\nxare non-\ndecreasing on R+\nIndeed, assuming this lemma, the function (1−µε(e1))/εκis increasing with respect to εsince each\nterm of the previous sum increases with it.\nProof of Lemma 30. By a change a variable, it is sufficient to prove the case c= 1. Moreover since\ng(x) =f(x)−2\nx, it is sufficient to prove that fis non-decreasing. We have\nf′(x) =1\nx2/parenleftbigg\n(1 +x)1\np−1/parenleftbigg\n−x\np+ 1 +x/parenrightbigg\n−1/parenrightbigg\n.\nSo\nf′(x)≥0⇐⇒ 1 +/parenleftbigg\n1−1\np/parenrightbigg\nx≥(1 +x)1−1\np.\nThe last inequality holds for all x≥0by concavity of the function (1 +x)1−1/p.\n7.5.2 1-norm\nConsider the 1-norm. Let u= (u1,...,ud1,0,..., 0)withui>0and||u||1= 1. Recall the definition\nofH,H1andH2given in Section 7.1. For x∈Rd\\H, we writex=λx(u+t1,x+t2,x)withλx∈R,\nt1,x∈H1andt2,x∈H2. With the same convention as in the previous paragraph, we have\n1−µε(u)\nεκ= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0λyi−|λyi|||u+t1,yi+t2,yi||1+ε1/d\nsεκ.\nRecall that now we have κ:=κd(u) = 1/d1. Letψ:Rd∝⇕⊣√∫⊔≀→Rdbe the linear map defined for x=\nλ(u+t1+t2)witht1∈H1andt2∈H2by\nψ(λ(u+t1+t2)) :=ε−κ+1\ndλ(u+t1+εκt2).\nUsing that H2is of dimension d2, we get det(ψ) =εαwith\nα=d/parenleftbigg\n−κ+1\nd/parenrightbigg\n+d2κ= 1 + (d2−d)κ= 0.\nAs before,ψpreserves the Lebesgue measure and so we get, using that ||u+t1+t2||1=||u+t1||1+||t2||1,\n1−µε(u)\nεκ= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0|λyi|(sgn(λyi)−(||u+t1,yi||1+εκ||t2,yi||1)) +εκ\nsεκ\n= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0/parenleftbigg|λyi|(sgn(λyi)−||u+t1,yi||1)\nsεκ+|λyi|||t2,yi||1+ 1\ns/parenrightbigg\n.\n39Since, by definition of H,||u+t1,yi||1≥1, the numerator|λyi|(sgn(λyi)−||u+t1,yi||1)is non positive\nand so each term of the previous sum is again increasing with respect to ε.\n7.5.3∞-norm\nConsider the∞-norm. By symmetry, it is sufficient to prove the result for u= (1,u2,...,ud)oru=\n(1,1,u3,...,ud)withui∈[0,1). Recall the definition of H,H 3andH4given in Section 7.1. For\nx∈Rd\\H, we writex=λx(u+t3,x+t4,x)withλx∈R,t3,x∈H3andt4,x∈H4. With the same\nconvention as in the previous paragraph, we have\n1−µε(u)\nεκ= lim\ns→∞sup\nπ∈Π(0,su)q/summationdisplay\ni=0λyi−|λyi|||u+t3,yi+t4,yi||∞+ε1/d\nsεκ.\nRecall that now κ:=κd(u) = 1/(d4+1). Letψ:Rd∝⇕⊣√∫⊔≀→Rdbe the linear map defined for x=λ(u+t3+t4)\nwitht3∈H3andt4∈H4by\nψ(λ(u+t3+t4)) :=ε−κ+1\ndλ(u+εκt3+t4).\nAs before,ψpreserves the Lebesgue measure since d(−κ+1\nd) + (d3−1)κ= 0and so we get\n1−µε(u)\nεκ= lim\ns→∞sup\nπ:0→suq/summationdisplay\ni=0|λyi|(sgn(λyi)−||u+εκt3,yi+t4,yi||∞) +εκ\nsεκ.\nLet us check that if d3∈{1,2}, the function f(x) =1−||u+xt3+t4||∞\nxis non-decreasing on R∗\n+for any\n(t3,t4)∈H3×H4.\nIfd3= 1, thenH3={0}. So using that 1−||u+t4||∞≤0, the function fis indeed non-decreasing.\nIfd3= 2, thenH3=Vect(e2−e1), so there exists some c∈Rsuch thatt3=c(e2−e1)and by\nsymmetry, we can assume c≥0. Writingu=e1+e2+u4withu4∈H4, we have, for x≥0,\n||u+xt3+t4||∞= max(1 + cx,||u4+t4||∞)i.e.f(x) =−max/parenleftbigg\nc,||u4+t4||∞−1\nx/parenrightbigg\nand one can check that the function x∝⇕⊣√∫⊔≀→max(c,c′/x)is non-increasing on R∗\n+for any (c,c′)∈R+×R.\nReferences\n[1] D. Aldous and P. Diaconis. Hammersley’s interacting particle process and longest increasing subse-\nquences. Probab. Theory Relat. Fields , 103(2):199–213, 1995.\n[2] A. Auffinger, M. Damron, and J. Hanson. 50 years of first passage percolation. In University Lecture\nSeries, volume 68. American Mathematical Society, 2017.\n[3] A-L. Basdevant, J-B. Gouéré, and M. Théret. First-order behavior of the time constant in Bernoulli\nfirst-passage percolation. Ann. Appl. Probab. , 32(6):4535–4567, 2022.\n[4] J. T. Chayes, L. Chayes, and R. Durrett. Critical behavior of the two-dimensional first passage\ntime.Journal of Statistical Physics , 45(5):933–951, 1986.\n[5] J. T. Cox. The time constant of first-passage percolation on the square lattice. Adv. in Appl.\nProbab., 12(4):864–879, 1980.\n[6] J.T.CoxandR.Durrett. Somelimittheoremsforpercolationprocesseswithnecessaryandsufficient\nconditions. Ann. Probab. , 9(4):583–603, 1981.\n[7] J. T. Cox and H. Kesten. On the continuity of the time constant of first-passage percolation. J.\nAppl. Probab. , 18(4):809–819, 1981.\n40[8] M. Deijfen. Asymptotic shape in a continuum growth model. Adv. Appl. Probab. , 35(2):303–318,\n2003.\n[9] J-B. Gouéré and R. Marchand. Continuous first-passage percolation and continuous greedy paths\nmodel: Linear growth. Ann. Appl. Probab. , 18(6):2300–2319, 2008.\n[10] J-B. Gouéré and M. Théret. Positivity of the time constant in a continuous model of first passage\npercolation. Electron. J. Probab. , 22:21, 2017. Id/No 49.\n[11] J-B. Gouéré and M. Théret. Continuity of the time constant in a continuous model of first passage\npercolation. Ann. Inst. Henri Poincaré, Probab. Stat. , 58(4):1900–1941, 2022.\n[12] J. M. Hammersley. A few seedlings of research. Proc. 6th Berkeley Symp. Math. Stat. Probab.,\nUniv. Calif. 1970, 1, 345-394 (1972)., 1972.\n[13] J. M. Hammersley and D. J. A. Welsh. First-passage percolation, subadditive processes, stochastic\nnetworks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ.\nCalifornia, Berkeley, Calif , pages 61–110. Springer-Verlag, New York, 1965.\n[14] Fritz John. Extremum problems with inequalities as subsidiary conditions. Studies Essays, pres. to\nR. Courant, 187-204 (1948)., 1948.\n[15] H. Kesten. Aspects of first passage percolation. In École d’été de probabilités de Saint-Flour,\nXIV—1984 , volume 1180 of Lecture Notes in Math. , pages 125–264. Springer, Berlin, 1986.\n[16] J. F. C. Kingman. Subadditive ergodic theory. Ann. Probab. , 1:883–909, 1973.\n[17] G. Last and M. Penrose. Lectures on the Poisson Process . Institute of Mathematical Statistics\nTextbooks. Cambridge University Press, 2017.\n[18] R. Marchand. Strict inequalities for the time constant in first passage percolation. Ann. Appl.\nProbab., 12(3):1001–1038, 2002.\n[19] J. B. Martin. Linear growth for greedy lattice animals. Stochastic Processes Appl. , 98(1):43–66,\n2002.\n[20] R. Meester and R. Roy. Continuum percolation , volume 119 of Camb. Tracts Math. Cambridge:\nCambridge Univ. Press, 1996.\n[21] D. Richardson. Random growth in a tessellation. Proc. Cambridge Philos. Soc. , 74:515–528, 1973.\n[22] R. Schneider and W. Weil. Stochastic and integral geometry . Probab. Appl. Berlin: Springer, 2008.\n[23] J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann.\nAppl. Probab. , 3(1):56–80, 1993.\n41"    },    {        "title": "2402.07511v1.Global_subelliptic_estimates_for_geometric_Kramers_Fokker_Planck_operators_on_closed_manifolds.pdf",        "content": "arXiv:2402.07511v1  [math.AP]  12 Feb 2024Global subelliptic estimates for geometric\nKramers-Fokker-Planck operators on closed manifolds\nFrancis Nier*\nXingfeng Sang†\nFrancis White‡\nFebruary 13, 2024\nAbstract\nIn this article we reconsider the proof of subelliptic estim ates for Geometric Kramers-Fokker-\nPlanck operators, a class which includes Bismut’s hypoelli ptic Laplacian, when the base manifold\nis closed (no boundary). The method is significantly differe nt from the ones proposed by Bismut-\nLebeau in [BiLe] and Lebeau in [Leb1] and [Leb2]. As a new resu lt we are able to prove maximal\nsubelliptic estimates with some control of the constants in the two asymptotic regimes of high\n(b→0) and low ( b→+∞ ) friction. After a dyadic partition in the momentum variabl e, the analy-\nsis is essentially local in the position variable, contrary to the microlocal reduction techniques of\nthe previous works. In particular this method will be easier to adapt on manifolds with bound-\naries. A byproduct of our analysis is the introduction of a ve ry convenient double exponent Sobolev\nscale associated with globally defined differential operat ors. Applications of this convenient pa-\nrameter dependent functional analysis to accurate spectra l problems, in particular for Bismut’s\nhypoelliptic Laplacian with all its specific geometry, is de ferred to subsequent works.\nMSC2020: 35G15, 35H10, 35H20, 35R01, 47D06, 58J32, 58J50, 58J65, 60J 65\nKeywords: Geometric Kramers-Fokker-Planck operators, analysis on m anifolds, cotangent total\nspace, subelliptic estimates, global pseudodifferential calculus, commutation of unbounded opera-\ntors.\nContents\n1 Introduction 3\n1.1 Background and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Statement of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n1.4 Outline of the article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n*LAGA, Université de Sorbonne-Paris Nord (Paris XIII), 99 av enue J.B. Clément, F-93430 Villetaneuse.\nnier@math.univ-paris13.fr\n†LAGA, Université de Sorbonne-Paris Nord (Paris XIII), 99 av enue J.B. Clément, F-93430 Villetaneuse.\nsang@math.univ-paris13.fr\n‡Department of Mathematics, University of California Irvin e, 400 Physical Sciences Quad, Irvine CA-92697.\nfgwhite@uci.edu\n12 Reduction to a scalar operator 10\n2.1 Integration by parts and maximal accretivity . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n2.3 Changing locally the connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n3 Sobolev spaces 15\n3.1 First properties of ˜Wk,k∈N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n3.2 Pseudo-differential definition of ˜Ws,s∈R. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n3.3 Spaces ˜Ws1,s2(X;E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n4 A priori estimates on the scalar GKFP operator 25\n4.1 Dyadic partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n4.2 Partition with a 2ℓ-dependent grid in the open set Ω⊂Rd\nq. . . . . . . . . . . . . . . . . . 27\n4.3 Use of normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n4.4 From the local models to the global estimates for a fixed ℓ. . . . . . . . . . . . . . . . . . 32\n5 Euclidean Case 35\n6 Final proof of Theorem1.6 38\n6.1 Comparison of the local model /tildewiderPb,ℓ,mwith the euclidean case . . . . . . . . . . . . . . . . 38\n6.2 Comparison of the local models Pb,ℓ,mand/tildewiderPb,ℓ,m. . . . . . . . . . . . . . . . . . . . . . . 42\n6.3 Estimate for |p|∼2ℓlarge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44\n6.4 Estimate for |p|∼2ℓ≤Cb,g,ψ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46\n6.5 Lower bound for /ba∇dbl(Pb−iλ/b)u/ba∇dblL2+1/b2/ba∇dblu/ba∇dblL2. . . . . . . . . . . . . . . . . . . . . . . . . . 48\n7 Consequences and optimality of Theorem 1.6 49\n7.1 About b-dependent constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n7.2 Perturbative Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\n7.3 /tildewiderWs-versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\nA Comparison of harmonic oscillator hamiltonians 53\nB Complex Airy Operator 54\nC Result used to localize the operator 56\nDN−locand N−comp functional spaces 57\nE Some pseudo-differential calculus on X=T∗Q 58\nE.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58\nE.2 Global calculus when Q=Rd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60\nE.3 Localization and geometric invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65\nE.4 Globalization on Qand applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74\n21 Introduction\n1.1 Background and Motivations\nIn [Bis041][Bis042][Bis05] J.M. Bismut introduced the hyp oelliptic Laplacian which can be viewed\nalternatively as a deformation of Hodge theory on the cotang ent bundle or as a generalization to the\ncase of arbitrary degree differential forms of the Fokker-P lanck equation (often specified as Kramers-\nFokker-Planck equation in this case) associated with the La ngevin process. Very rapidly J.M. Bismut\nand G. Lebeau made in [BiLe] a careful analysis of families of such hypoelliptic Laplacians indexed\nby a parameteer b>0 , proving in particular the spectral convergence as b→0+of the hypoelliptic\nLaplacian to the Hodge Laplacian on the base manifold. It was in the mood of those times to develop\nthe accurate spectral analysis of parameter dependent non s elf-adjoint hypoelliptic operators and we\nrefer the reader to [Dav][DSZ][HerNi][HHS] for works in thi s direction which still have many devel-\nopments. In [BiLe] J.M. Bismut and G. Lebeau combined such in volved microlocal and semiclassical\nspectral analysis with the heavy geometrical construction of the hypoelliptic Laplacian and this was\ncontinued by S. Shen in [She], for the double parameter asymp totics where the first limit b→0+\nallows to recover a semiclassical Witten Laplacian on the ba se manifold let say with a parameter\nh→0+which leads to Morse theory as in [Wit][HeSj4][Zha]. In [Leb 1][Leb2], G. Lebeau introduced\na general class of non self-adjoint hypoelliptic operators for which accurate subelliptic estimates, a\ncorner stone of the spectral asymptotic analysis, can be pro ven.\nAbout the Witten Laplacian on the base manifold, it was reali zed in [LNV1][LNV2] that the in-\ntroduction of artificial boundary value problems, associat ed with a suitable cutting and gluing of\nthe manifold, was a very convenient and robust way for the gen eralization of the accurate spectral\nasymptotic analysis with a potential which can be more gener al than a Morse function. Such accu-\nrate spectral analysis was motivated by various questions r elated with molecular dynamics. This\nraised the question of understanding similar problems for t he Langevin process, where the arbitrary\ndegree form formulation, involves boundary value problems for the hypoelliptic Laplacian. The func-\ntional analysis was started in [Nie][NiSh] for fixed hypoell iptic Laplacians, which means with no\nparameter. The asymptotic spectral analysis with respect t o one parameter b>0 or two parameters\n(b,h)∈(0,+∞)2remains to be done.\nIn this direction, the microlocal reduction approach propo sed by in [BiLe][Leb1][Leb2] appears to be\nnot well adapted for a similar asymptotic analysis of bounda ry value problems. The present article\nproposes an alternative approach which relies on some local approximation of the hypoelliptic Lapla-\ncian on the cotangent T∗Qof a general compact Riemannian manifold Q, by the euclidean version.\nThis euclidean approximation relies on the use of normal coo rdinates around a point q0∈Qwhile\nthe Taylor expansion of the metric leads to controlled error terms in balls B(q0,rb,|p|) parametrized\nby the parameter b>0 and the size of momentum variable p∈T∗\nq0Q.\nWhile doing so we are able to provide a control of constants in the global subelliptic estimates not\nonly as b→0+but also as b→+∞ which can be of interest for further developments. The spect ral\nanalysis as b→0+proving the spectral convergence to the Hodge or Witten Lapl acian on the base\nmanifold, will be carried out in another text. Compared to th e works of [BiLe][She], the approach\nproposed here makes possible an easier extension to boundar y value problems and a more direct\nconnexion with standard tools of spectral analysis like Gru shin problems (see[SjZw]) and other re-\nlated works with similar spectral problems studied in [ReTa ][Dro][Smi]. In particular, global Sobolev\nscales ˜Ws1,s2(T∗Q) ,s1,s2∈R, are associated with a pair of commuting self-adjoint opera tors which\nmake use of the scalar horizontal Laplacian associated with a Sasaki type metric. This synthetic\n3formulation, of which the properties nevertheless rely on s ome global pseudodifferential techniques,\nwas inspired by [ReTa] and [BeBo]. Again, it will allow a rath er direct adaptation of techniques de-\nveloped for simpler spectral problems, namely scalar opera tors on a compact total space within the\nframework of standard Sobolev spaces, to the more involved f ramework of the hypoelliptic Laplacian\n(and possibly other global analysis problems on the total sp ace of the cotangent bundle X=T∗Q).\n1.2 General Framework\nIn this text we shall consider geometric Kramers-Fokker-Pl anck operators on X=T∗Q, where ( Q,g)\nis a smooth compact d-dimensional Riemannian manifold without boundary. Point s in Xwill be\ndenoted by x, and πX:X→Qwill denote the natural projection T∗\nqQ∋x/mapstoc〈a∇→q∈Q. Local coordinates\nonQwill be denoted by ( q1,...,qd) . We shall use Einstein’s convention of summing over repeat ed up\nand down indices. If q∈Qand ( U,q1,...qd) is a local coordinate system for Q, then an element of the\nfiber p∈T∗\nqQwill be written p=pjdqj, and ( q1,...,qd,p1,...,pd) will denote canonical coordinates\ninU×Rd∼T∗U⊂T∗Q. If local canonical coordinates ( q1,...,qd,p1,...,pd) have been fixed in a\nneighborhood T∗Uof a point x∈X, then we shall write x=(q,p) .\nThe metric gonQ, i.e. on the tangent bundle πTQ:TQ→Q, will be denoted by g(q)=tg(q)=/parenleftbig\ngjk(q)/parenrightbig\n1≤j,k≤dorg=gjk(q)dqjdqk. The corresponding dual metric on the cotangent bundle πT∗Q:\nT∗Q→Qwill be denoted by g−1(q)=(gjk(q))1≤j,k≤d. Let∇LCbe the Levi-Civita connection on the\ntangent bundle πTQ:TQ→Qassociated with the metric g. By abuse of notation, we shall also\ndenote by ∇LCthe connection on the tensor bundle\nT(k,ℓ)TQ=TQ⊗···⊗ TQ/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nktimes⊗T∗Q⊗···⊗ T∗Q/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nℓtimes(1.1)\ninduced by the Levi-Civita connection on TQfor every k,ℓ∈N. Ifq=(q1,...,qd) are local coordinates\nforQ, then the Christoffel symbols associated the Levi-Civita c onnection ∇LCare given by\nΓℓ\njk=1\n2gℓa/parenleftbig\n∂qjgak+∂qkga j−∂qagjk/parenrightbig\n,1≤j,k,ℓ≤d. (1.2)\nThe connection ∇LCgives rise to a global decomposition\nTX=THX⊕TVX, (1.3)\nwhere TVX=ker(dπX) is the vertical subbundle of TXassociated with the projection πXandTHX\nis the horizontal subbundle of TXdefined at each point x∈Xby\nTH\nxX={γ′(0) : there exists ǫ>0 and a smooth path γ: (−ǫ,ǫ)→Xsuch that\nγ(0)=xand∇LC\ndπX(γ′(t))γ(t)=0 for all −ǫ<t<ǫ}.(1.4)\nIn terms of local coordinates for X, the subbundles THXand TVXmay be described as follows. If\n(q1,...,qd,p1,...,pd) are local canonical coordinates for X, let\nej=∂\n∂qj+Γℓ\njk(q)pℓ∂\n∂pk,1≤j≤d, (1.5)\n4and let\n/hatwideej=∂\n∂pj,1≤j≤d. (1.6)\nTogether these vector fields form a local frame ( e1,...,ed,/hatwidee1,...,/hatwideed) for TXand locally we have\nTHX=span (e1,...,ed) (1.7)\nand\nTVX=span/parenleftig\n/hatwidee1,...,/hatwideed/parenrightig\n. (1.8)\nSince for every x=(q,p)∈Xthe differential dπX|xrestricts to a linear isomorphism TH\nxX→TqQ\nwhile TV\nxX∼=T∗\nqQ, the decomposition (1.3) yields the identifications\nTX∼=π∗\nX(TQ⊕T∗Q). (1.9)\nThe total tangent space TX=THX⊕TVXis equiped with the metric π∗\nX(g⊥⊕g−1), simply written\ng⊕g−1, by using the above identification. Clearly horizontal (res p. vertical) vector fields on X\nare given as sections in C∞(X;THX) (resp. C∞(X;TVX)). Specific subspaces of horizontal (resp.\nvertical) sections are provided by the fact that (1.9) induc es a natural imbedding\nig:C∞(Q;TQ⊕T∗Q)→C∞(X;π∗\nX(TQ⊕T∗Q))=C∞(X;TX)\nand we introduce\nC∞\nQ(X;THX)=ig(C∞(Q;TQ))⊂C∞(X;THX),\nresp. C∞\nQ(X;TVX)=ig(C∞(Q;T∗Q))⊂C∞(X;TVX).\nThese spaces C∞\nQ(X;THX) and C∞\nQ(X;TVQ) are C∞(Q;R) modules. Additionally on C∞(Q;TQ⊕\nT∗Q) aCk-norm can be fixed once and for all by using a finite partition of unity subordinate to an\nopen chart covering Q=∪J\nj=1Ωjwhile changing the atlas and the partition of unity gives an e quiv-\nalent norm. We therefore can speak of /ba∇dblT/ba∇dblCkforT∈C∞\nQ(X;THX) and T∈C∞\nQ(X;TVX) without\nspecifying its expression.\nIt will be convenient to use the following families of vector fields.\nDefinition 1.1. For any N∈Nand any k∈N, the set TH\nN,k(resp. TV\nN,k) is defined by\nTH\nN,k=/braceleftig\n(TH\n1,...,TH\nN)∈C∞\nQ(X;THX)N,∀j∈{1,...,N},/ba∇dblTH\nj/ba∇dblCk≤1/bracerightig\n,\nresp. TV\nN,k=/braceleftig\n(TV\n1,...,TV\nN)∈C∞\nQ(X;TVX)N,∀j∈{1,...,N},/ba∇dblTV\nj/ba∇dblCk≤1/bracerightig\n.\nBy duality, we also have the identifications\nT∗X∼=T∗Q⊕TQ. (1.10)\n5If (q1,...,qd,p1,...,pd) are local canonical coordinates for X, we let\nej=dqj,1≤j≤d, (1.11)\nand\n/hatwideej=dpj−Γℓ\njk(q)pℓdqk,1≤j≤d. (1.12)\nIt is clear that ( e1,...,ed,/hatwidee1,...,/hatwideed) is a local coframe for T∗X, and locally it is true that\n(THX)∗=span/parenleftig\ne1,...,ed/parenrightig\n(1.13)\nand\n(TVX)∗=span (/hatwidee1,...,/hatwideed). (1.14)\nWe also note that Xis naturally a symplectic manifold with respect to the usual symplectic form\nσgiven in local canonical coordinates ( q,p) by\nσ=d/summationdisplay\nj=1dpj∧dqj. (1.15)\nSince σ∧d/negat〉onslas〈=0, the manifold Xis orientable, and we orient Xso that every local canonical coordinate\nsystem ( q,p) is positively oriented. The volume form on Xfor the metric g⊕g−1is denoted by dvolX\nand given locally by\ndvolX=dq1∧···∧ dqd∧dp1∧···∧ dpd. (1.16)\nThe volume form dvolXis related to σby\ndvolX=1\nd!(−1)d(d+1)\n2σ∧d. (1.17)\nIn particular, if H∈C∞(X;R) and Yis the Hamilton vector field of Hwith respect to the symplectic\nform σ, i.e. Yis the unique smooth vector field on Xsuch that ιYσ= −dH, then the flow Φt=\nexp( tY) onXgenerated by Ypreserves dvolX. In this text, we will be primarily concerned with the\nsituation in which His the kinetic energy\nH(q,p)=1\n2|p|2\nq=1\n2gjk(q)pjpk. (1.18)\nIn this case, the Hamilton vector field YofHis given locally by\nY=gjk(q)pjek, (1.19)\nwhere ekis as in (1.5), and the projections of the integral curves of YtoQbyπXare precisely the\nsmooth geodesic curves of the metric g. We will use also the metric-dependent Japanese bracket\n〈p〉q=(1+|p|2\nq)1/2=(1+gi j(q)pipj)1/2, (1.20)\n6while the notation\n〈p〉=(1+|p|2)1/2=(1+δi jpipj)1/2(1+d/summationdisplay\ni=1p2\ni)1/2, (1.21)\nwill be used for the euclidean version.\nLetEπE− − →Qbe a smooth complex vector bundle over Qof complex dimension Nthat is equipped with\nan affine connection ∇Eand a Hermitian metric gE. LetE:=π∗\nXEπE− − →Xdenote the pullback bundle\nofEπE− − →Qby the map πX:X→Q. Locally, smooth sections uofEπE− − →Xhave the form\nu(x)=N/summationdisplay\nℓ=1uℓ(x)fℓ(q),x=(q,p)∈X, (1.22)\nwhere/parenleftbig\nf1,...,fN/parenrightbig\nis a smooth local frame for EπE− − →Qand u1,...,uℓare smooth locally defined\ncomplex-valued functions on X. We equip EπE− − →Xwith the pullback connection ∇E, which is de-\nfined using the decomposition (1.9) of TXby the relations\n/parenleftig\n∇E\neju/parenrightig\n(x)=N/summationdisplay\nℓ=1/bracketleftbigg\n(ejuℓ)(x)fℓ(q)+uℓ(x)∇E\n∂\n∂qjfℓ(q)/bracketrightbigg\n,\n/parenleftig\n∇E\n/hatwideeju/parenrightig\n(x)=N/summationdisplay\nℓ=1(/hatwideejuℓ)(x)fℓ(q)=N/summationdisplay\nℓ=1(∂pjuℓ)(x)fℓ(q),x=(q,p)∈X,1≤j≤d,(1.23)\nwhenever u∈C∞(X;E) is of the form (1.22). Because the connection ∇E\nTVis trivial for TV∈TXV,\nthe covariant derivative with respect to a vertical vector fi eld will be identified with the associated\nscalar first order differential operator . Accordingly the v ertical Laplacian and the vertical harmonic\noscillator, written locally as,\n∆p=/summationdisplay\n1≤j,k≤d1\n2gjk(q)∇E\n/hatwideej∇E\n/hatwideek=/summationdisplay\n1≤j,k≤d1\n2gjk(q)∂pj∂pk (1.24)\nO= −1\n2∆p+1\n2|p|2\nq. (1.25)\nare globally defined operators, which happen to be scalar dif ferential operators in the sense that in\nany local frame ( f1,...,fN) ofEπE→Q,\n∆p(N/summationdisplay\nℓ=1uℓ(x)fℓ(q))=N/summationdisplay\nℓ=1(∆puℓ)(x)fℓ(q) and O(N/summationdisplay\nℓ=1uℓ(x)fℓ(q))=N/summationdisplay\nℓ=1(Ouℓ)(x)fℓ(q).\nWe also equip the bundle Ewith the pulled back Hermitian metric gEdefined by\ngE(u,u′)=/summationdisplay\nℓ1,ℓ2uℓ1(x)u′\nℓ2(x)gE(fℓ1(q),fℓ2(q)),x=(q,p)∈X, (1.26)\nwhere u=/summationtextuℓ(x)fℓ(q) and u′=/summationtextu′\nℓ(x)fℓ(q) . Using the Hermitian metric gEonEand the volume\nform dvolXonX, we may introduce the Hilbert space L2(X;E) of square integrable sections of Eas\nfollows. The space L2(X;E) is the set of measurable sections usuch that\n〈u,u〉L2(X;E)=ˆ\nXgE\nx(u(x),u(x))dvolX(x)<+∞ , (1.27)\n7and it is a Hilbert space for the scalar product\n〈u1,u2〉L2(X;E)=ˆ\nXgE\nx(u1(x),u2(x))dvolX(x), (1.28)\nin which C∞\n0(X;E) is dense.\nBy recalling dvolX=dqdp , the operator Ois clearly self-adjoint with its maximal domain D(O)=/braceleftbig\nu∈L2(X,E),Ou∈L2(X;E)/bracerightbig\n, in which C∞\n0(X;E) is dense with the graph norm. It is also bounded\nfrom below byd\n2and/∇ad〉callow\nOis well defined.\nLet us now introduce some Sobolev type spaces, taking into ac count the different homogeneities of\nei,piand∂pi.\nDefinition 1.2. Fork∈Nand ua sufficiently regular section of E, we define\n/ba∇dblu/ba∇dbl˜Wk= sup\nN1+N2+N3\n2≤k\n(TH\n1,...,TH\nN1)∈TH\nN1,k\n(TH\n1,...,TV\nN2)∈TV\nN2,k/vextenddouble/vextenddouble/vextenddouble/vextenddouble〈p〉N3q∇E\nTH\n1...∇E\nTH\nN1∇E\nTV\n1...∇E\nTV\nN2u/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(X;E), (1.29)\nand we take\n˜Wk(X;E)=C∞\n0(X;E)/ba∇dbl·/ba∇dbl˜Wk. (1.30)\nThe space ˜Ws(X;E)is then defined for all s≥0by interpolation and for s<0by setting ˜Ws(X;E)=\n(˜W−s(X;E))∗.\nFinally the space ˜W1,s(X;E)is the space\n˜W1,s(X;E)=/braceleftig\nu∈Ws(X;E),/∇ad〉callow\nOu∈Ws(X;E)/bracerightig\nendowed with the norm /ba∇dbl/∇ad〉callow\nOu/ba∇dbl˜Ws.\nRemark 1.3. The supremum norm over the families of vector fields ensure th e geometrical global\nmeaning of the functional spaces ˜Wk(X;E)and therefore of ˜Ws(X;E)and ˜W1,s(X;E). It is not the most\nconvenient definition and in particular their Hilbert natur e is not obvious here. A more convenient\npresentation in terms of local coordinates and then the use o f a specific pseudo-differential calculus\npresented in Appendix E is detailed in Section 3.\nAlthough those spaces are modelled on Lebeau’s spaces in [Le b1][Leb2] they slightly differ, e.g. the\ncase s=1allows N2=2with two vertical derivatives bounded in L2.\nWe shall define geometric Kramers-Fokker-Planck operators as second order differential opera-\ntors acting on sections of the pullback bundle Ethat depend on a parameter b∈(0,∞). Our definition\nwill be slightly more general than that of Lebeau [Leb1][Leb 2] but in the same spirit.\nDefinition 1.4 (Geometric Kramers-Fokker-Planck Operator) .A Geometric Kramers-Fokker-Planck\n(abbreviated as GKFP) operator is a b-dependent operator P±,b+M(b)acting on C∞\n0(X;E)orS(X;E)\nwith\nP±,b=1\nb2O±1\nb∇E\nY,\nand ∀s∈R,M(b)∈L(˜W1,s(X;E);˜Ws(X;E))\nwhere Yis Hamilton vector field of the kinetic energy (1.18) with res pect to the symplectic form σand\nOis the vertical harmonic oscillator.\n8Actually the term M(b) will appear as a perturbative term for which the norm estima tes of\n/ba∇dblM(b)/ba∇dblL(˜W1,s;˜Ws)with respect to the parameter b>0 can be discussed afterwards. Actually all the\nanalysis focuses on the case M(b)=0 . The Hörmander Theorem about sum of squares and type II op-\nerators (see [Hor67]) provides the local hypoelliptic natu re of the geometric Kramers-Fokker-Planck\noperator P±,bfor every b∈(0,∞) . By following the method of Lebeau in [Leb1][Leb2] our aim i s to\nprovide accurate subelliptic estimates with the best regul arity exponents, that will also account for\nthe behavior of P±,bas either b→0+(the large friction limit) or b→∞ (the low friction limit).\n1.3 Statement of the Main Result\nRemember the following notion.\nDefinition 1.5. In a Hilbert space Hand densely defined operator A:D/mapstoc〈a∇→His called essentially\nmaximal accretive, if it is accretive, therefore closable, and if it admits a unique maximal accretive\nextension equal to its closure A:D(A)/mapstoc〈a∇→Hwith D(A)=D/ba∇dbl /ba∇dblA,/ba∇dblu/ba∇dbl2\nA=/ba∇dblu/ba∇dbl2\nH+/ba∇dblAu/ba∇dbl2\nH.\nThe main result of this paper is the following subelliptic es timate for geometric Kramers-Fokker-\nPlanck operators.\nTheorem 1.6. LetP±,b=1\nb2O±1\nb∇E\nY. There exists a constant Cg≥1determined by the geometric\ndata (g,E,gE,∇E)such that the operatorκb\nb2+P±,bis essentially maximal accretive on C∞\n0(X;E)(or\nonS(X;E)), when κb≥Cg(1+b2). IfP±,bdenotes its closure, the inequalities\nRe〈u,(κb\nb2+P±,b)u〉L2≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n. (1.31)\nand\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nP±,b−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+1\nb2/ba∇dblu/ba∇dblL2≥1\nCg(1+b)7/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/vextenddoubleO\nb2u/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb/parenleftig\n±∇E\nY−iλ/parenrightig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n+1\nb4/3/bracketleftigg\n||u||˜W2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/bracketrightigg\n1.2/parenrightbigg\n(1.32)\nhold for every u∈D(P±,b)and every (λ,b)∈R×(0,+∞).\nOther results involving the realizations of P±,bin the Sobolev spaces Ws(X;E) or other perturba-\ntive results will be deduced as corollaries in Section 7.\n1.4 Outline of the article\nIn Section 2 an elementary integration by part provides the fi rst a priori lower bound for Re〈u,P±,bu〉.\nThis implies that the analysis of P±,bcan be localized in the q-variable via partition of unity. Compar-\nison of different connections can be done locally which redu ces the problem to purely scalar operators\nand the then the essential maximal accretivity on C∞\n0(X;E) orS(X;E) is proved.\nThe Sobolev spaces ˜Ws1,s2(X;E) are then studied in Section 3. After a localization via part ition of\nunity, the Definition 1.2 is characterized in term of the suit able pseudodifferential calculus, local\nin the q-variable but global in the p-variable. The construction of this pseudodifferential ca lculus\n9relying on standard techniques, nevertheless to be adapted , is detailed in Appendix E. Section 3\nends with a very convenient global characterization of thes e Sobolev spaces ˜Ws1,s2(X;E) in terms of\nthe functional calculus of two geometrically defined commut ing self-adjoint operators, namely Oand\nW2=C−∆H+CO2where ∆His a scalar horizontal Laplacian.\nSection 4 is devoted to the localization process. A dyadic pa rtition of unity in the p-variable is used\nand then once the parameter 2jof the dyadic partition if fixed, a grid partition in the q-variable with\nthe spacing 2−jis introduced. Near point of the grid, a Taylor expansion of t he metric in normal\ncoordinates expresses the scalar GKFP operator as the eucli dean one with a (2j,b)-dependent error\nterm.\nIn Section 5 the maximal subelliptic estimate, where the exp onent 2/3 is obtained via the model\nproblem of the one dimensional complex Airy operator, is rec alled. Actually the uniform estimates\nwith respect to the parameters ( b,2j,λ)∈(0,+∞)2×Rare carefully checked.\nSection 6 gathers the local comparison of the scalar GKFP ope rator with the euclidean model of\nSection 4 with the uniform estimates of the euclidean model. Error terms due to the two partition\nof unities (dyadic in pand 2j-dependent grid in q) happen to be controlled by the lower bounds of\nthe parameter dependent euclidean model. While doing this, intermediate parameters of the grid\npartition must be tuned carefully according to the two regim es 2j>>1 or 2j≤C.\nSection 7 completes Theorem 1.6 with various consequences o r precisions. In particular the b-\ndependence of the perturbation M(b) in Definition 1.4, which allows the generalization of Theo-\nrem th:main is specified. A corollary is the ˜W0,s(X;E) version of Theorem 1.6, where a simple conju-\ngation reduces the perturbed operator in L2(X,dqdp ;E).\nThe Appendices gathers known material. A rather long paragr aph is about the global pseudodif-\nferential calculus on the total space X=T∗Q. As already said, it follows the general approach but\nthings have to be specified in particular for proving, via the Helffer-Sjöstrand formula, that func-\ntions of self-adjoint globally elliptic operators in this c lass are pseudodifferential operators with a\ngood asymptotic expansion.\n2 Reduction to a scalar operator\nHere we write first a priori estimates for Geometric Kramers- Fokker-Planck (GKFP) operators com-\ning from a simple integration by parts. The essential maxima l accretivity ofκb\nb2+P±,bis checked and\nall the perturbative terms coming from a partition of unity i n the q-variable will be shown to be of\nlower order with a uniform control of the constants w.r.t b. Similarly a local change of connection\nhappens to be of lower order and this reduces the problem to lo cal scalar GKFP operators.\n2.1 Integration by parts and maximal accretivity\nProposition 2.1. LetP±,b=1\nb2O±1\nb∇E\nY. There exists C0≥1, determined by the geometric data\n(g,E,∇E,gE), such that for all b>0,λ∈Rand for κb≥C0(1+b2)the inequality\nRe〈u,(κb\nb2+P±,b−iλ)u〉L2(X;E)≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0(X;E)+κb/ba∇dblu/ba∇dbl2\nL2(X;E)/bracketrightig\n(2.1)\nand/vextenddouble/vextenddouble/vextenddouble(κb\nb2+P±,b−iλ)u/vextenddouble/vextenddouble/vextenddouble2\nL2(X;E)≥κb\n16b4/bracketleftig\n/ba∇dblu/ba∇dbl2\nW1,0(X;E)+κb/ba∇dblu/ba∇dbl2\nL2(X;E)/bracketrightig\n(2.2)\nholds for all u∈C∞\n0(X;E).\n10Before proving this result let us specify the formal adjoint of∇E\nY. Start with the vector bundle\nπE:E→Qand the data ( ∇E,gE) and consider the dual connection with respect to gEgiven by\nX gE(s,s′)=gE(s,∇E\nXs′)+gE(∇E,∗\nXs,s′).\nThe unitary connection\n∇E,u=∇E+∇E,∗\n2\ndiffers from ∇Eby\n∇E,u−∇E=1\n2ω(∇E,gE)∈C∞(Q;T∗Q⊗End( E))\nWith the pull back we obtain with Y=gi j(q)piejwritten in a local canonical coordinates system\n∇E,u\nY−∇E\nY=1\n2gi j(q)piω(∇E,gE)(∂\n∂qj)=ai(q)pi,ai(q)∈End( Eq)\nWe also recall the formula\n∀v,w∈C∞\n0(X;E),∀T∈C∞(X;TX),ˆ\nXgE(v,∇E,u\nTw)dvolX=−ˆ\nXgE(∇E,u\nTv,w)+div(T)gE(v,w)dvolX\nwhile here dvolX=dqdp and div Y=0 .\nWe find that the formal adjoint of ∇E\nYis nothing but\n(∇E\nY)∗=−∇E\nY−gi j(q)piω(∇E,gE)(∂\n∂qj)=−∇E\nY−ai(q)pi. (2.3)\nProof of Proposition 2.1. Let∪J\nj=1Ωj=Qbe a finite open chart covering of Qand let/summationtextJ\nj=1̺j(q)2≡1\nbe a subordinate quadratic partition of unity, ̺j∈C∞\n0(Ωj;[0,1]) . Because ∇E\nY,P±,b,O, are at most\nfirst order differential operators in qwe get\n〈u,(κb\nb2+P±,b−iλ)u〉L2(X;E)=J/summationdisplay\nj=1〈uj,(κb\nb2+P±,b−iλ)uj〉L2(X;E)\n/ba∇dbl/∇ad〉callow\nOu/ba∇dbl2\nL2(X;E)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=/ba∇dblu/ba∇dbl2\n˜W1,0(X;E)+κb/ba∇dblu/ba∇dbl2\nL2(X;E)=〈u,Ou〉L2(X;E)+κb/ba∇dblu/ba∇dblL2(X,E)=J/summationdisplay\nj=1/ba∇dbl/∇ad〉callow\nOuj/ba∇dbl2\nL2(X;E)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=/ba∇dbluj/ba∇dbl2\n˜W1,0(X;E)+κb/ba∇dbluj/ba∇dbl2\nL2(X;E)\nfor all u∈C∞\n0(X;E) , by setting uj=̺ju∈C∞\n0(T∗Ωj;E) .\nWith canonical local coordinates ( q,p) inT∗Ωj, (2.3) implies\nRe〈uj,(κb\nb2+P±,b−iλ)uj〉L2(X;E)=〈uj,2κb−∆p+|p|2\nq\n2b2uj〉L2(X;E)∓1\nb〈uj,ai(q)piuj〉L2(X;E)\n≥1\n2b2/bracketleftig\n/ba∇dbluj/ba∇dbl2\n˜W1,0(X;E)+2κb/ba∇dblu/ba∇dbl2\nL2(X;E)/bracketrightig\n−C′\n0\nb/vextenddouble/vextenddoubleuj/vextenddouble/vextenddouble\nL2(X;E)/ba∇dbluj/ba∇dbl˜W1,0(X;E)\n≥1\n2b2/bracketleftig\n/ba∇dbluj/ba∇dbl2\n˜W1,0(X;E)+2κb/ba∇dblu/ba∇dbl2\nL2(X;E)/bracketrightig\n−1\n4b2/ba∇dbluj/ba∇dbl2\n˜W1,0(X;E)−2C′\n02/ba∇dbluj/ba∇dbl2\nL2(X;E),\n11for some C′\n0>0 determined by the geometric data ( g,E,gE,∇E) . Withκb\n2b2≥C01+b2\n2b2≥C0\n2, the first\ninequality (2.1) is proved for C0≥4C′\n02.\nUsing Cauchy-Schwarz inequality in the left hand side of (2. 1) yields\n/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2(X;E)/ba∇dblu/ba∇dblL2(X;E)≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0(X;E)+κb/ba∇dblu/ba∇dbl2\nL2(X;E)/bracketrightig\n.\nWe deduce at once /ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2(X;E)≥κb\n4b2/ba∇dblu/ba∇dblL2(X,E). The latter inequality multiplied by\n/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2(X;E)yields (2.2).\nCorollary 2.2. LetP±,b=1\nb2O±1\nb∇E\nYand let C0≥1be determined by the geometric data (g,E,∇E,gE)\naccording to Proposition 2.1 . For κb≥C0(1+b2)the operatorκb\nb2+P±,bis essentially maximal accretive\nonC∞\n0(X;E)and therefore on S(X;E).\nProof. Proposition 2.1 says that the operator (κb\nb2+P±,b) and its formal adjointκb\nb2+P∗\n±,bare accretive\nonC∞\n0(X;E) with the lower bound (2.1) .\nIt suffices to prove that the range (κb\nb2+P±b)C∞\n0(X;E) is dense in L2(X;E) . It is equivalent to\nu∈L2(X;E)\n(κb\nb2+P∗\n±,b)u=0∈D′(X;E)/bracerightigg\n⇒u=0.\nWith P∗\n±,b=P∓,b∓ai(q)biin local coordinates according to (2.3), Hörmander’s hypoe llipticity result\nfor type II operators (see [Hor67]) implies u∈C∞(X;E) . For χ∈C∞\n0(R;[0,1]) such that χ≡1 in a\nneighborhood of 0 and for ε>0 set uε=χ(ε|p|2\nq)u.\nThe above equation implies\n(κb\nb2+P∗\n±,b)uε=−/bracketleftig\nP∗\n±,b,χ(ε|p|2\nq)/bracketrightig\nu=−/bracketleftbigg\n−∆p\n2b2,χ(ε|p|2\nq)/bracketrightbigg\nu\nbecause Yf(|p|2\nq)=0 . The form of the last commutator allows to write\n(κb\nb2+P∗\n±,buε)=−/bracketleftbigg\n−∆p\n2b2,χ(ε|p|2\nq)/bracketrightbigg\n(1−˜χ(ε|p|2\nq))u\nwhere ˜χ∈C∞\n0(R;R) has a support included a neighborhood of 0 where χ≡1 and its derivatives\nvanish, while ˜χ≡1 in a smaller neighborhood of 0 . By taking the scalar product with uε, the\ninequality (2.1) for P∗\n±,bimplies\n1\n4b2/bracketleftig\n/ba∇dbluε/ba∇dbl2\n˜W1,0(X,E)+κb/ba∇dbluε/ba∇dbl2\nL2(X;E)/bracketrightig\n≤Cg,χ/ba∇dbluε/ba∇dbl˜W1,0(X;E)/ba∇dbl(1−˜χ(ε|p|2\nq))u/ba∇dblL2(X;E)\nand /radicaligg\nd\n2/ba∇dbluε/ba∇dblL2(X;E)≤/ba∇dbluε/ba∇dbl˜W1,0(X;E)≤4Cg,χb2/ba∇dbl(1−˜χ(ε)|p|2\nq)u/ba∇dblL2(X;E).\nLebesgue’s theorem for the limit ε→0 gives\n/radicaligg\nd\n2/ba∇dblu/ba∇dblL2(X;E)≤4Cg,χb2lim\nε→0/ba∇dbl(1−˜χ(ε|p|2\nq))u/ba∇dblL2(X;E)=0.\n122.2 Localization\nProposition 2.3. LetP±,b=1\nb2O±1\nb∇E\nYand fix Q=∪J\nj=1Ωja finite open chart covering of Q. Let\n/summationtextJ\nj=1̺j(q)2≡1be a subordinate quadratic partition of unity, ̺j∈C∞\n0(Ωj;[0,1]). There exists C0≥1,\ndetermined by the geometric data (g,E,∇E,gE), and now the partition of unity (̺j)1≤j≤J, such that\nfor all b>0,λ∈Rand for κb=C0(1+b2)the following equivalence of norms\n\n/vextenddouble/vextenddouble/vextenddouble(κb\nb2+P±,b−iλ)u/vextenddouble/vextenddouble/vextenddouble2\nL2(X;E)\n/summationtextJ\nj=1/vextenddouble/vextenddouble/vextenddouble(κb\nb2+P±,b−iλ)(̺ju)/vextenddouble/vextenddouble/vextenddouble2\nL2(X;E)\n±1\n≤4 (2.4)\nholds for all u∈C∞\n0(X;E).\nProof. It’s a straightforward application of Corollary C.2. We hav e to check the assumption (C.4)\nwhich says\n∀u∈C∞\n0(X;E),r\n2/summationdisplay\nj∈J/ba∇dbl(κb\nb2+P±,b−iλ)̺ju/ba∇dbl2\nL2(X;E)≥2/summationdisplay\nj1,j∈J/ba∇dbl[(κb\nb2+P±,b−iλ),̺j1]̺ju/ba∇dbl2\nL2(X;E)\n+4/summationdisplay\nj1,j2,j∈J/ba∇dbl[[(κb\nb2+P±,b−iλ),̺j2],̺j1]̺ju/ba∇dbl2\nL2(X;E),\nfor some r∈[0,1) . Because the operator P±,bis a first-order differential operator in the qvariable\nthe first commutator equals\n[κb\nb2+P±,b−iλ,̺j1]=±1\nbY̺j1=±1\nbgℓk(q)pk∂̺j1\n∂qℓ,\nforj1∈Jwhere the right-hand side is written local canonical coordi nate ( q,p). Moreover the double\ncommutators indexed by j1,j2∈Jall vanish.\nWe deduce the existence of C′\n0>0 such that\n∀u∈C∞\n0(X;E),/ba∇dbl[(κb\nb2+P±,b−iλ),̺j1]̺ju/ba∇dbl2\nL2(X;E)≤C′\n01\nb2/ba∇dbl|p|q̺ju/ba∇dbl2\nL2(X;E).\nand the summation over j1,j∈J, combined with the inequality (2.1), yields\n∀u∈C∞\n0(X;E),/summationdisplay\nj1,j∈J/ba∇dbl[(κb\nb2+P±,b−iλ),̺j1]̺ju/ba∇dbl2\nL2(X;E)≤C′\n0|J|16b2\nκb/summationdisplay\nj/ba∇dbl(κb\nb2+P±,b−iλ)̺ju/ba∇dbl2\nL2(X,E).\nWith C′\n0|J|16b2\nκb≤C′\n0|J|16\nC0, choosing C0≥1 large enough guarantees the assumption (C.4) with\nr=1\n2.\n2.3 Changing locally the connections\nWith Proposition 2.3 the analysis of P±,bcan be localized in a chart open domain Ωj. Additionally\nit can be assumed that there is a well defined local frame ( f1(q),...,fN(q)) of the restricted bundle\nE/vextendsingle/vextendsingle\nΩj. In this frame a trivial connection ∇E,jonE/vextendsingle/vextendsingle\nΩjand therefore a corresponding flat connection\n13onE/vextendsingle/vextendsingle\nT∗Ωjis defined by pull-back. The operator Pj\n±,bdefined locally can be identified with a scalar\noperator according to\nPj\n±,b/bracketleftigg\nN/summationdisplay\nℓ=1uℓfℓ/bracketrightigg\n=/parenleftbigg1\nb2O±1\nb∇E,j\nY/parenrightbigg/bracketleftigg\nN/summationdisplay\nℓ=1uℓfℓ/bracketrightigg\n(2.5)\n=N/summationdisplay\nℓ=1\n−gik(q)∂2\n∂pi∂pk+gik(q)pipk\n2b2(uℓ)±1\nbgik(q)piek(uℓ)\nfℓ. (2.6)\nProposition 2.4. Under the assumptions of Proposition 2.3 with and with the ad ditional condition\nthat Eis trivialized by a local frame (f1(q),...,fN(q))overΩjfor every j∈{1,...,J}, letPj\n±,bbe defined\nby(2.5). There exists C0≥1, determined by the geometric data (g,E,∇E,gE)and the partition of unity\n(̺j)1≤j≤J, such that for all b>0,λ∈Rand for κb=C0(1+b2)the following equivalence of norms\n\n/vextenddouble/vextenddouble/vextenddouble(κb\nb2+P±,b−iλ)u/vextenddouble/vextenddouble/vextenddouble2\nL2(X;E)\n/summationtextJ\nj=1/vextenddouble/vextenddouble/vextenddouble(κb\nb2+Pj\n±,b−iλ)(̺ju)/vextenddouble/vextenddouble/vextenddouble2\nL2(X;E)\n±1\n≤12 (2.7)\nholds for all u∈C∞\n0(X;E).\nProof. For a given j∈{1,...,J}and for v∈C∞\n0(Ωj;E) we have\n(P±,b−Pj\n±,b)(v)=±1\nb(∇E\nY−∇E,j\nY)(v)=±1\nbgik(q)piπ∗\nX(∇E\n∂\n∂qk−∇E,j\n∂\n∂qk)(v)\nwhere\n(∇E\n∂\n∂qk−∇E,j\n∂\n∂qk)[vℓ(q,p)fℓ(q)]=vℓ(q,p)Fℓ\nℓ′,k(q)fℓ′(q)\nwith Fℓ\nℓ′,k∈C∞(Ωj;R) .\nApplied to v=̺juthis gives the upper bound\n/ba∇dbl(P±,b−Pj\n±,b))(̺ju)/ba∇dbl2≤C′\n0\nb2/ba∇dbl|p|q(̺ju)/ba∇dbl2\nL2(X;E),\nwhere C′\n0depends only on the metric gEand the connection ∇Egiven on the vector bundle Eand is\nuniform with respect to j∈{1,...J}. The same argument as in the proof o Proposition 2.3 shows tha t\n/ba∇dbl(P±,b−Pj\n±,b)(̺ju)/ba∇dbl2\nL2(X;E)≤1\n4/ba∇dbl(κb\nb2+P±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E),\nwhen the constant C0inκb=C0(1+b2) is chosen large enough. The parallelogram identity implie s\n/ba∇dbl(κb\nb2+Pj\n±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E)≤3/ba∇dbl(κb\nb2+P±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E)\nand/ba∇dbl(κb\nb2+P±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E)≤3/ba∇dbl(κb\nb2+Pj\n±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E).\nBy summation over j∈{1,...,J}we obtain\n\n/summationtextJ\nj=1/ba∇dbl(κb\nb2+Pj\n±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E)/summationtextJ\nj=1/ba∇dbl(κb\nb2+P±,b−iλ)(̺ju)/ba∇dbl2\nL2(X;E)\n±1\n≤3.\nThe inequality (2.7) is obtained by taking the product with t he result (2.4) of Proposition 2.3.\n143 Sobolev spaces\nLike for the operator P±,bwe firstly reduce the characterization of u∈˜Wk(X;E) ,k∈N, to a local\nproblem with the possibility of replacing the connection ∇Eby a trivial connection in a given local\nframe. Then Ws(X;C) and its norm will be expressed in terms of the functional cal culus of pseudo-\ndifferential elliptic self-adjoint operator W2in the class OpS2\nΨ(Q;C) presented in Appendix E. Finally\ngeneral spaces ˜Ws1,s2(X;E) ,s1,s2∈Rare introduced by using the functional calculus of two commu t-\ning self-adjoint operators.\n3.1 First properties of ˜Wk,k∈N\nWe collect rather immediate consequences of the Definition 1 .2. The norm /ba∇dblu/ba∇dbl˜Wkis\n/ba∇dblu/ba∇dbl˜Wk= max\nN1+N2+N3\n2≤kPk,N1,N2,N3(u) (3.1)\nwhere Pk,N1,N2,N3(u) is the smallest constant C≥0 such that\n∀(TH\n1,...TH\nN1)∈C∞\nQ(X;THX)N1,∀(TV\n1,...,TV\nN2)∈C∞\nQ(X;TVX)N2,\n/ba∇dbl〈p〉N3q∇E\nTH\n1...∇E\nTH\nN1∇E\nTV\n1...∇E\nTV\nN2u/ba∇dblL2≤CN1/productdisplay\nn1=1/ba∇dblTH\nn1/ba∇dblCkN2/productdisplay\nn2=1/ba∇dblTV\nn2/ba∇dblCk.\nBecause\n〈p〉N3q∇E\nTH\n1...∇E\nTH\nN1∇E\nTV\n1...∇E\nTV\nN2u=∇E\nTH\n1...∇E\nTH\nN1〈p〉N3q∇E\nTV\n1...∇E\nTV\nN2u,\n/ba∇dblu/ba∇dbl˜Wkalso equals\n/ba∇dblu/ba∇dbl˜Wk= max\nN2+N3=j≤2k\nN1≤k−2jsup\n(TV\n1,...TV\nN2)∈C∞\nQ(X;TVX)N2Pk,N1,0,0(〈p〉N3q∇E\nTV\n1...∇E\nTV\nN2u)\n/producttextN2\nn2=1/ba∇dblTVn2/ba∇dblCk. (3.2)\nProposition 3.1. Letk∈N, letθ∈C∞(Q;End( E))and fix any Ck-norm on Q. The multiplication by\nπ∗\nXθ∈C∞(X;End( E)),π∗\nXθ(x)=π∗\nX(θ(πX(x))), is a bounded operator in ˜Wk(X;E)with\n∀u∈˜Wk(X;E),/ba∇dbl(π∗\nXθ)u/ba∇dbl˜Wk≤Ck/ba∇dblθ/ba∇dblCk/ba∇dblu/ba∇dbl˜Wk. (3.3)\nProof. Because 〈p〉N3q∇E\nTV\n1...∇E\nTV\nN2(π∗\nXθ)=(π∗\nXθ)〈p〉N3q∇E\nTV\n1...∇E\nTV\nN2and owing to (3.2) the problem is\nreduced to\nPk,N1,0,0((π∗\nXθ)v)≤Ck/ba∇dblθ/ba∇dblCkPk,N1,0,0(v)\nfor all N1∈{0,...,k}.\nIt is obviously true for N1=0 . If it is true for N1∈{0,...,k−1}then for TH\nN1+1∈C∞\nQ(X;TXH) with\nTN1+1=πX,∗TH\nN1+1∈C∞(Q;TQ) , we write\n∇E\nTH\nN1+1(π∗\nXθ)v=(π∗\nX(θ)∇E\nTH\nN1+1v)+[π∗\nX(∇End( E)\nTN1+1θ)]v.\n15We get\nPk−1,N1,0,0(∇E\nTH\nN+1(π∗\nXθ)v)≤Ck−1/bracketleftbigg\n/ba∇dblθ/ba∇dblCk−1Pk−1,N1,0,0(∇E\nTH\nN1+1v)+/ba∇dbl∇End( E)\nTN1+1θ/ba∇dblCk−1Pk−1,N1,0,0(v)/bracketrightbigg\n≤2Ck−1/ba∇dblθ/ba∇dblCkPk,N1+1,0,0(v).\nThe obvious inequality\nPk,N1+1,0,0(w)≤ sup\nTH\nN1+1∈C∞\nQ(X;THV)Pk−1,N1,0,0(∇E\nTH\nN1+1w)\n/ba∇dblTH\nN1+1/ba∇dblCk\nends the proof by induction.\nThe previous statement contains two particular cases which allow the local scalar characteriza-\ntion of u∈˜Wk(X;E) with a simpler norm.\nProposition 3.2. Fixk∈Nand consider two different connections ∇E,1and∇E,2on the vector bundle\nEπE→Qwith the associated connections ∇E,1and∇E,2onEπE→X. Let ˜Wk\n∇j(X;E)and/ba∇dbl /ba∇dbl ˜Wk\n∇jbe the\ncorresponding ˜Wk(X;E)spaces and norms according to Definition 1.2.\n1)The space Wk(X;E)is aC∞(Q;R)module and for any finite atlas Q=/uniontextJ\nj=1Ωjand for any subor-\ndinate partition of unity/summationtextJ\nj=1̺j(q)≡1, a section ubelongs to Wk(X;E)if and only if, for every\n1≤j≤J,̺ju∈Wk\nΩj−comp(T∗Ωj;E/vextendsingle/vextendsingle\nT∗Ωj)and the norm /ba∇dblu/ba∇dbl˜Wkis equivalent to max 1≤j≤J/ba∇dbl̺ju/ba∇dbl˜Wk.\n2)For two different connections ∇E,1and∇E,2, the two spaces ˜Wk\n∇1(X;E)and ˜Wk\n∇2(X;E)are equal and\nthe norms /ba∇dbl /ba∇dbl ˜Wk\n∇1and/ba∇dbl /ba∇dbl ˜Wk\n∇2are equivalent.\nProof. 1)Simply apply Proposition 3.1 with θ∈C∞(Q;R) . The definition of ˜Wk\nΩj−comp(T∗Ωj;E/vextendsingle/vextendsingle\nT∗Ωj)\nand the other statements are explained in Appendix D. Simply use the triangular inequality for\n/ba∇dblu/ba∇dbl˜Wk≤Jmax 1≤j≤J/ba∇dbl̺ju/ba∇dbl˜Wk.\n2)Remember ∇E,2\nT−∇E,1\nT=R(T)∈End( E) with/ba∇dblR(T)/ba∇dblCk≤Ck/ba∇dblT/ba∇dblCk+1for any T∈C∞(Q;TQ) . Let\nPj\nk,N1,N2,N3be the norms involved in (3.1)(3.2) for the two associated co nnections ∇E,ℓforℓ=1,2 . We\ncan make an induction proof with respect to N1∈{0,...,k}like in Proposition 3.1 after reducing the\nproblem to N2=N3=0 by noticing\n∀TV∈C∞\nQ(X;TXV),∇E,2\nTV=∇E,1\nTV\nand by using\n∇2\nTH−∇1\nTH=π∗\nX(R(T))\nfor any TH∈C∞\nQ(X;TXH) with πX,∗TH=T∈C∞(Q;TQ) .\nActually the induction proof relies on\nP2\nk−1,N1,0,0(∇E,2\nTH\nN1+1v)≤P2\nk−1,N1,0,0(∇E,1\nTH\nN1+1v)+P2\nk−1,N1,0,0([π∗(R(TH\nN1+1)]v)\n≤/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ninductionCk/bracketleftbigg\nP1\nk−1,N1,0,0(∇E,1\nTH\nN1+1v)+P1\nk−1,N1,0,0([π∗(R(TH\nN1+1)]v)/bracketrightbigg\n≤/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nProp. 3.1C′\nk/ba∇dblTH\nN1+1/ba∇dblCkP1\nk,N1+1,0,0(v),\n16and leads to /ba∇dblu/ba∇dbl˜Wk\n∇2≤C′′\nk/ba∇dblu/ba∇dbl˜Wk\n∇1. The result follows by symmetry.\nThe atlas ( Ωj)1≤j≤Jcan be chosen such that E/vextendsingle/vextendsingle\nΩjis trivial with a local frame ( f1\nj,...,fN\nj) ,fn\nj∈\nC∞(Ωj;E/vextendsingle/vextendsingle\nΩj) . Above Ωjthe connection ∇Ecan be replaced by the trivial connection ∇j,Egiven by\n∀T∈C∞(Q;TQ),∇j\nT(N/summationdisplay\nn=1vn(q)fn\nj)=N/summationdisplay\nn=1(Tvn)fn\nj.\nFor a section u=/summationtextN\nn=1un,j(x)fn\njofE/vextendsingle/vextendsingle\nT∗Ωjwe get\n∇j,E\nTH\n1...∇j,E\nTH\nN1∇j,E\nTV\n1...∇j,E\nTV\nN2u=N/summationdisplay\nn=1(TH\n1...TH\nN1TV\n1...TV\nN2vn,j)fn\nj.\nProposition 3.2 now implies that the˜Wk-norm of\nu=J/summationdisplay\nj=1̺j(q)u=J/summationdisplay\nj=1N/summationdisplay\nn=1un,jfn\nj\nwith un,j∈˜Wk\nΩj−comp(T∗Ωj;C)⊂˜Wk(X;C) , is equivalent to\nmax\n1≤j≤J\n1≤n≤N/ba∇dblun,j/ba∇dbl˜Wk(X;C).\nThe ˜Wk-spaces for k∈Nand their norms is are thus fully understood in a local scalar setting.\n3.2 Pseudo-differential definition of ˜Ws,s∈R\nThe end of Subsection 3.1 reduced the description of ˜Wk(X;E) to the local description of ˜Wk(X;C) . We\ncan thus focus on scalar sections, and we now give a pseudo-di fferential and a global characterization.\nWe need the pseudo-differential calculus in/uniontext\nm∈ROpSm\nΨ(Q;C) introduced in Appendix E. We recall\nthat a∈Sm\nΨ(Q;C) if in doubly canonical coordinates ( q,p,ξ,η) inT∗(T∗Ω) associated with the local\ncoordinates q=(q1,...,qd) on a chart open set Ω⊂Q, the uniform estimate\n|∂α\nq∂β\np∂γ\nξ∂δ\nηa(q,p,ξ,η)|≤Cα,β,γ,δ(1+|ξ|2+|p|4+|η|4)m−|γ|−|β|+|δ|\n2\n2\nand that, the quantization is the standard one,given by the l ocal kernel on T∗Ω×T∗Ω:\n[a(q,p,Dq,Dp)](q,p,q′,p′)=ˆ\nR2dei[(q−q′).ξ+(p−p′).η]a(q,p,ξ,η)dξdη\n(2π)2d.\nActually the general quantization of a∈Sm\nΨ(Q;C) is defined by introducing a partition of unity/summationtextJ\nj=1ˆ̺j(q)≡1 and cut-off functions ˆχ∈C∞\n0(Ωj;[0,1]) , ˆχ≡1 on supp ˆ̺j, and by setting aˆ̺,ˆχ(x,Dx)=\n/summationtextJ\nj=1(ˆ̺j(q)a)(x,Dx)◦ˆχj(q) , and the set of pseudo-differential operators is\nOpSm\nΨ=/braceleftbig\naˆ̺,ˆχ(x,Dx)+R,a∈Sm\nΨ(Q;C),R∈R(Q;C)/bracerightbig\n,\nwithR(Q;C)=L(S′(T∗Q;C);S(T∗Q;C)).It is proved in Appendix E that this pseudo-differential\ncalculus has the same properties as the usual pseudo-differ ential calculus, with a different homo-\ngeneity which takes into account the global estimates as p→∞ .\n17In canonical coordinates ( q,p) associated with the local coordinates ( q1,...,qd) onΩwe know the\ntwo frames ( e1,...,ed) (resp. ( ˆe1,..., ˆed)) ofT(T∗Ω)H(resp. T(T∗Ω)V) given by\nei=∂\n∂qi+Γk\niℓ(q)pk∂\n∂pℓ\nresp. ˆei=∂\n∂pi.\nAs differential operators, the locally defined operators ei,∂qi,pk∂pℓ,O=−gi j(q)∂pi∂pj+gi j(q)pipj\n2be-\nlong to OpS1\nΨ,Ω−loc(Ω;C) , while pk×,〈p〉q×and∂pkbelong to OpS1/2\nΨ,Ω−loc(Ω;C) .\nThus any TH∈C∞\nQ(X;TXH) is a differential operator that belongs to OpS1\nΨ(Q;C) , and any TV∈\nC∞\nQ(X;TXV) belongs to OpS1/2\nΨ(Q;C) .\nLet us introduce another operator which involves the scalar horizontal Laplacian ∆H. We follow\n[BeBo]: On Xwith the decomposition TX=TXH⊕TXV∼TQ⊕T∗Qgiven by the Levi-Civita con-\nnection associated with gwe put the riemannian metric g⊕⊥g−1and consider the associated to-\ntal Laplacian ∆x. The projection πX:X=T∗Q→Qis now a riemannian submersion with totally\ngeodesic fibers and the horizontal Laplacian ∆H=∆x−∆pequals in local canonical coordinates\n∆H=gi j(q)(eiej−Γk\ni j(q)ek).\nBecause the volume of g⊕⊥g−1is equal to the symplectic volume dqdp and∆xand∆pare symmetric\nonS(T∗Q;C) , the operator ∆His symmetric on S(T∗Q;C) for the L2(T∗Q,dqdp ;C) scalar product.\nBy introducing the adjoint differential operator e∗\ni=−∂i\nq−Γk\nii′pk∂pi′−Γ′\nii′and owing to the symmetry\nor by explicit computations with\n∂qigi j=∂qig−1(dqi,dqj)=g−1(−Γi\ni,kdqk,dqj)+g−1(dqi,−Γj\ni,kdqk)=−Γi\nikgk j−Γj\nikgik,\nthe horizontal Laplacian is also given by\n−∆H=e∗\nigi j(q)ej,\nwithout a divergence term because integrations are made wit h respect to the symplectic volume\ndqdp .\nDefinition 3.3. The operator W2is the closure in L2(X,dqdp ;C)of the differential operator Cg−\n∆H+CgO2:S(X;C)→S(X;C)⊂L2(X,dqdp ;C)forCg≥1large enough.\nNotice that because the flow exp( tei) sends isometrically T∗\nqQtoT∗\nexp(t∂qi)qQ, the commutations\n[ei,−∆p]=[ei,|p|2\nq]=[ei,O]=[∆H,O]=[W2,O]=0\nhold true on SΩ−loc(Ω;C) .\nAs a consequence of Appendix E we have a simple characterizat ion of ˜Ws(X;C) , in the case when\nE=Q⊗C. The general case can then be deduced either by the localizat ion at the end of the previous\nparagraph or by the approach proposed afterwards. Both are e quivalent.\n18Proposition 3.4. For any s∈R, the space ˜Ws(X;C)is characterized by\n˜Ws(X;C)=/braceleftbig\nu∈S′(X;C),∀A∈OpSs\nΨ(Q;C),Au∈L2(X,dqdp ;C)/bracerightbig\nForCg≥1large enough, Cg−∆H+CgO2:S(X;C)→L2(X,dqdp ;C)is a non negative essentially\nself-adjoint operator, with self-adjoint extension W2and D(W2)=˜W2(X;C).\nFor any s∈R,Ws=(W2)s/2is an elliptic operator in OpSs\nΨ(Q;C)and the norm on ˜Ws(X;C)can be\nchosen as /ba∇dblWsu/ba∇dblL2(X,dqdp ;C).\nProof. a)If we start from the above definition of ˜Ws(X;C) the problem is reduced to the ellipticity\nand the identification of the principal symbol of W2=Cg−∆H+CgO2. Because it is a differential\noperator/summationtextJ\nj=1ˆ̺j(q)W2ˆχj(q) , and we obtain W2=aˆ̺,ˆχ(x,Dx)+0 with a−a2∈S1\nΨ(Q;C) and\na2(x,Ξ)=Cg+|ξ+Γk\n..(q)pkη|2\nq+Cg\n4(|p|2\nq+|η|2\nq)2≥Cg+|ξ|2\nq−2|Γk\n..(q)pkη|2\nq+Cg\n4(|p|2\nq+|η|2\nq)2\nWe used the notation |τ|2\nq=gi j(q)τiτjforτ=ξ+Γk\n..(q)pkη,τ=pand|η|2\nq=gi j(q)ηiηj. The ellipticity\ncomes from\na2(x,Ξ)≥Cg+εg|ξ|2−1\nεg|p|2|η|2+Cgεg\n4(|p|4+|η|4)\nfor some εg>0 given by gand the fixed open covering/uniontextJ\nj=1Ωj. The ellipticity a2≥Cg+εg(|ξ|2+\n|p|4+|η|4) holds true if Cgε2\ng−1≥8ε2\ng.\nThe operator W2is symmetric with\n〈u,W2u〉=Cg/ba∇dblu/ba∇dbl2\nL2+J/summationdisplay\nj=1ˆ\nT∗Ωjgii′(q)/bracketleftig\n(eiθj(q)u)(ei′θj(q)u)−(∂qiθj)(∂qiθj)|u|2/bracketrightig\ndqdp+Cg/ba∇dblOu/ba∇dbl2\nL2\n(3.4)\nfor all u∈S(X;C) when/summationtextJ\nj=1θ2\nj(q)≡1 with θj∈C∞\n0(Ωj;[0,1]) . It is bounded from below by 1 for\nCg>0 large enough.\nIt suffices to apply Proposition E.22 of Appendix E.\nb)For the identification of the two definitions of ˜Ws(X;C) and the equivalence of the norms, it suffices\nto consider the case s=k∈N, because all the other cases will follow by interpolation an d duality.\nWe start from the Definition 1.2 and Proposition 3.2- 1)which says that the ˜Wk(X;C) norm of uis\nequivalent to\nmax\n1≤j≤J/ba∇dbl̺ju/ba∇dbl˜Wk\nwith/summationtextJ\nj=1̺j(q)≡1 ,̺j∈C∞\n0(Ωj;[0,1]) . Additionally the definition of ˜Wk\nΩj−comp(Ωj;C) ensures that it\nis independent of the choice of a coordinate system ( q1,...,qd) with equivalent norm for two different\nchoices. So let us work on T∗Ω=Ω×Rd⊂Rd\nq×Rd\npand let us consider functions u∈L2\nΩ−comp(T∗Ω;C)\nwith a Ω-support included in a fixed compact set K⊂⊂Ω(a neighborhood of supp ̺with ̺=̺j\nwhen Ω=Ωj). Any vector field TH∈C∞\nΩ−comp(T∗Ωd;TXH) (resp. TV∈C∞\nΩ−comp(T∗Ωd;TXV)) can be\nwritten\nTH=d/summationdisplay\ni=1ti(q)eiresp. TV=d/summationdisplay\ni=1ti(q)ˆei\nwith max 1≤i≤d/ba∇dblti/ba∇dblCk≍/ba∇dblTH/ba∇dblCk(resp. max 1≤i≤d/ba∇dblti/ba∇dblCk≍/ba∇dblTV/ba∇dblCk) .\n19We deduce\n/ba∇dbl〈p〉N3qTH\n1...TH\nN1TV\n1...TV\nN2u/ba∇dbl2\nL2\n/producttextN1\nn1=1/ba∇dblTHn1/ba∇dblCk/producttextN2\nn2=1/ba∇dblTVn2/ba∇dblCk≤CK,k max\n1≤i1,...,iN1≤d\n1≤j1,...,jN2≤d/ba∇dbl〈p〉N3qei1...eiN1ˆej1...ˆejN2u/ba∇dblL2\nwhere\n/ba∇dbl〈p〉N3qei1...eiN1ˆej1...ˆejN2u/ba∇dblL2=/ba∇dbl〈p〉N3q(χei1)...(χeiN1)(χˆej1)...(χˆejN2)u/ba∇dblL2\nfor some χ=χ(q)∈C∞\n0(Ω;[0,1]) such that χ≡1 in a neighborhood of K⊃Ω−supp u. Because\nχ(q)ei∈C∞\nΩ−comp(T∗Ω;TXH) and χ(q)ˆej∈C∞\nΩ−comp(T∗Ω;TXV) the right-hand side is bounded by\nCK,χ,k/ba∇dblu/ba∇dbl˜Wk.\nBy taking the supremum with respect to N1+N2+N3\n2≤k,TH\n1,...,TH\nN1∈C∞\nΩ−comp(T∗Ω;TXH) and\nTV\n1,...,TV\nN2∈C∞\nΩ−comp(T∗Ω;TXV) . The norm /ba∇dblu/ba∇dbl˜Wkforu∈L2(T∗Ω;C) with Ω-support included in\nK, is thus equivalent to\nmax\n1≤i1,...,iN1≤d\n1≤j1,...,jN2≤d\nN1+N2+N3\n2≤k/ba∇dbl〈p〉N3qei1...eiN1ˆej1...ˆejN2u/ba∇dblL2≍ max\n1≤i1,...,iN1≤d\nN1+|β|+N3\n2≤k/ba∇dbl〈p〉N3ei1...eiN1∂β\npu/ba∇dblL2 (3.5)\nwhere we have replaced 〈p〉q=(1+gi j(q)pipj)1/2by the equivalent quantity 〈p〉=(1+/summationtext\nip2\ni)1/2.\nFrom\n[ei,fk(q)pk]=(∂qifk)(q)pk+(fkΓℓ\nik)(q)pℓ,\nwe get by induction\nei1...eiN1∂β\np−∂qi1...∂qiN1∂β\np=/summationdisplay\n|α|≤N1−1\n|α|+|γ|+|β′|\n2=N1+|β|\n2fα,β′,γ(q)pγ∂α\nq∂β′\np\nand we deduce\nεN1/ba∇dbl〈p〉N3ei1...eiN1∂β\npu/ba∇dblL2−εN1/ba∇dbl〈p〉N3q∂qi1...∂qiN1∂β\npu/ba∇dblL2\n≤CK,kε max\n|α|≤N1−1\n|α|+N′\n3+|β′|\n2≤N1+|β|\n2εN1−1/ba∇dbl〈p〉N′\n3∂α\nq∂β′\npu/ba∇dblL2.\nChoosing ε=εK,kforεK,k>0 small enough implies that the norm /ba∇dblu/ba∇dbl˜Wkis equivalent to\nmax\n|α|+N3+|β|\n2≤k/ba∇dbl〈p〉N3∂α\nq∂β\npχ(q)u/ba∇dblL2or/radicaltp/radicalvertex/radicalvertex/radicalbt/summationdisplay\n|α|+N3+|β|\n2≤k/ba∇dbl〈p〉N3∂αq∂β\npχ(q)u/ba∇dbl2\nL2.\nBut according Appendix E and in particular Proposition E.7 , it is equivalent to the norm /ba∇dblWkχ(q)u/ba∇dblL2.\nLet us extend now this result to ˜Ws(Q;E) . For a self-adjoint non negative scalar operator A∈\nOpSm\nΨ(Q;C) , like W2with m=2 or Ws,s=m∈R, it is not possible to define directly its action on\nsections of E. However a localization technique makes it possible, up to l ower order corrections.\nWe fix, as we did in Appendix E, the atlas covering Q=/uniontextJ\nj=1Ωjby assuming that for every j∈\n{1,...,J}the two properties are satisfied:\n20• the open set ˜Ωj=/uniontext\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇essΩj′is a chart open set;\n• the restricted vector bundle E/vextendsingle/vextendsingle˜Ωjadmits an orthonormal frame ( f1\nj,...,fN\nj) for the metric gE.\nIf/summationtextJ\nj=1θ2\nj(q)≡1 is a quadratic partition of unity with θj∈C∞\n0(Ωj;[0,1]) we set\nAθ=J/summationdisplay\nj=1θj(q)◦Ascj◦θj(q),\nwhere Asc,jis the scalar pseudo-differential operator in the othonorm al local frame ( f1\nj,...,fN\nj) above\n˜Ωj:\nAsc,j(ukfk\nj(q))(q,p)=[Auk](q,p)fk\nj(q).\nWhen A=a̺,χ(x,Dx)+R=/summationtextJ\nj1=1(̺j1(q)a)(x,Dx)◦χj1(q)+R∈OpSm\nΨ(Q;C) we obtain\nAθ=J/summationdisplay\nj1=1̺j1(q)/summationdisplay\nΩj∩Ωj1/negat〉onslas〈=/emptysetst∇essθj(q)◦[a(x,Dx)]sc,j◦θj(q)◦χj1(q)+Rθ.\nIfUj1,j2(q) is the unitary matrix of ( f1\nj1,...,fN\nj1) in the frame ( f1\nj2,...,fN\nj2) , the operator\n/summationdisplay\nΩj∩Ωj1/negat〉onslas〈=/emptysetst∇essθj(q)◦[a(x,Dx)]sc,j◦θj(q)\nwith/uniontext\nΩj∩Ωj1/negat〉onslas〈=/emptysetst∇essΩj⊂˜Ωj1⊂Rd, is nothing but the operator ˜aj1(x,Dx) with aj1∈S(Ψm,gΨ;CN) given\nby/summationdisplay\nΩj∩Ωj1/negat〉onslas〈=/emptysetst∇essUj1,j(q)♯([(θj(q)a)♯θj(q)]⊗IdCN)♯Uj,j1(q),\nwhere we recall ( a♯b)(x,Dx)=a(x,Dx)◦b(x,Dx) .\nOwing to the exact chain rules Uj3,j1(q)=Uj3,j2◦Uj2,j1(q) and Uj3,j1(q)=Uj3,j2(q)♯Uj2,j1(q) and the\nexact commutation θj3(q)♯Uj2,j1(q)=Uj2,j1(q)♯θj3(q) , we can write\nAθ=J/summationdisplay\nj1=1(̺j1(q)aθ)(x,Dx)◦χj1(q)+Rθ=(aθ)̺,χ(x,Dx)+Rθ\nwith aθ∈Sm\nΨ(Q;End ,E) and Rθ∈R(Q;E) .\nAdditionally, if A=am,̺,χ(x,Dx)+Am−1,Am−1∈OpSm−1\nΨ(Q;C) , then Aθ=(am⊗IdE)̺,χ(x,Dx)+Aθ,m−1\nwith Aθ,m−1∈OpSm−1\nΨ(Q;EndE) . In particular the principal symbol does not depend on the c hosen\northonormal frames ( f1\nj,...,fN\nj)j=1,...,J.\nIfAis self-adjoint and elliptic, the same holds for Aθwith\nD(Aθ)=˜Wm(X;E)=/braceleftig\nu∈S′(X;E),∀j∈{1,...,J},u/vextendsingle/vextendsingle\nΩj=uk(x)fk\nj(q),uk∈˜Wm\nΩj−loc(T∗Ωj;C)/bracerightig\n=/braceleftbig\nu∈S′(X;E),∀B∈OpSm\nΨ(Q;EndE),Bu∈L2(X,dqdp ;E)/bracerightbig\n.\nWe can conclude with the following summary.\n21Proposition 3.5. Once the quadratic partition of unity/summationtextJ\nj=1θ2\nj(q)≡1and the local orthonormal\nframes (f1\nj,...,fN\nj),1≤j≤J, are fixed and for Cg≥Cg,θchosen large enough, the operator\nW2\nθ=J/summationdisplay\nj=1θj(q)(W2)sc,jθj(q)with D(W2\nθ)=/braceleftbig\ns∈L2(X,dqdp ;E),W2\nθs∈L2(X,dqdp ;E)/bracerightbig\nis self-adjoint and bounded from below by 1.\nFors∈R, the space ˜Ws(X;E)introduced in Definition 1.2 for s=k∈Nand then extended by interpo-\nlation and duality, equals\n˜Ws(X;E)=/braceleftbig\nu∈S′(X;E),∀B∈OpSs\nΨ(Q;EndE),Bu∈L2(X,dqdp ;E)/bracerightbig\n=/braceleftig\nu∈S′(X;E),(W2\nθ)s/2u∈L2(X,dqdp ;E)/bracerightig\n=/braceleftig\nu∈S′(X;E),∀j∈{1,...,J},u/vextendsingle/vextendsingle\nΩj=uk(x)fk\nj(q),Wsuk∈L2\nΩj−loc(T∗Ωj,dqdp ;E)/bracerightig\n=/braceleftig\nu∈S′(X;E),(Cs+(W|s|)θ)signsu∈L2(X,dqdp ;E)/bracerightig\n,\nwhere the constant Cs>0is chosen large enough.\nProof. We already know that W2=Cg−∆H+CgO2is elliptic, self-adjoint and bounded from below\nby 1 for Cg≥1 large enough with domain D(W2)=˜W2(X;C) .\nWith the previous discussion this proves that W2\nθis elliptic and self-adjoint. The same computation as\n(3.4) shows that W2\nθ≥1 : Actually the derivatives of the unitary matrix associate d with the change of\nframes, ∂qiUj1,j2(q) , bring lower order terms which are absorbed if Cg=Cg,θis chosen large enough.\nForW2\nθwith a scalar principal symbol a2⊗IdE≥1\nκΨ2⊗IdE, Proposition E.22 applies and ( W2\nθ)s/2=\nfs(W2\nθ) with fs∈S(〈t〉s/2,dt\n〈t〉2) is elliptic with the principal symbol fs(a2)⊗IdE=as/2\n2⊗IdE.\nThe local characterization with u/vextendsingle/vextendsingle\nΩj=uk(x)fk\nj(q) has been explained and with the reduction of the\nprevious paragraph and Proposition 3.4 it shows that D((W2\nθ)k/2) coincides with ˜Wk(X;E) when k∈N.\nThis ends the identifications of the general spaces ˜Ws(X;E) for s∈R.\nBecause W|s|=(W2)|s|/2=f|s|(W2) is elliptic with the principal symbol a|s|/2\n2fors/negat〉onslas〈=0 , (W|s|)θis elliptic\nwith the principal symbol a|s|/2\n2⊗IdE. It is self-adjoint with the same domain, ˜W|s|(X;E) , as ( W2\nθ)|s|/2.\nIt is bounded from below by Garding inequality. Adding a cons tant Csensures that ( Cs+(W|s|)θ) is\nbounded from below by 1 and invertible.\n3.3 Spaces ˜Ws1,s2(X;E)\nA priori Y=gi j(q)piejbelongs to OpS3/2\nΨ(Q;C) but it has locally some specific structure made of\nei∈OpS1\nΨ(Ω;C) and followed by a multiplication by pi. We start with a simple commutation result.\nProposition 3.6. The self-adjoint operator (W2\nθ,D(W2\nθ)=˜W2(X;E))modelled on W2=Cg−∆H+CgO2\nand introduced in Proposition 3.5 and the vertical harmonic oscillator Owith the maximal domain\nD(O)=/braceleftbig\nu∈L2(X,dqdp ;E),Ou∈L2(X,dqdp ;C)/bracerightbig\nmake a pair of strongly commuting self-adjoint\noperators: For any Borel functions f,g:R→C,f(W2\nθ)g(O)=g(O)f(W2\nθ)on the intersection of their\ndomain.\nProof. The space L2(X,dqdp ;E) is isomorphic to the direct integral´⊕\nQL2(Rd,dp)dvolg(q) after the\npointwise gauge transformation ( q,p)/mapstoc〈a∇→(q,g(q)−1/2.p) . In this direct integral decomposition the op-\nerator Ois nothing but´⊕\nQO dvolg(q) where O=/summationtextd\nj=1−∂2\npj+p2\nj\n2is the euclidean harmonic oscillator.\n22The associated unitary group eitOsatisfies eitO∂qie−itO=∂qi,eitOpie−itO=cos(t)pi−sin(t)Dpiand\neitODpie−itO=sin(t)pi+cos(t)Dpi. We deduce that for any t∈R,eitOis continuous from ˜W2(X;C)\ninto itself, and therefore as a scalar operator from ˜W2(X;E) into itself.\nBecause the unitary transform UΦ:L2(X;E)→´⊕\nQL2(Rd)dvolg(q) given by ( UΦu)(q,p′)=u(q,g(q)1/2.p)\nis a special case of Proposition E.13 (it suffices to consider locally the effect on the scalar components).\nIt is an isomorphism of W2(X;E)=D(W2\nθ) and eitOis continuous from D(W2\nθ) into itself for any t∈R.\nBecause the scalar operator W2andOcommute on S(X;C) we deduce that [ W2\nθ,O]=0 onS(X;E)\nwhich is a core for W2\nθ.\nWe have all the ingredients of [ABG] in order to conclude that\nadOW2\nθ=id\ndte−itOW2\nθeitO/vextendsingle/vextendsingle\nD(W2\nθ)=0,\nandW2\nθandOstrongly commute.\nThis leads to the introduction of the following, double inde xed, spaces.\nDefinition 3.7. For any s1,s2∈R, the space ˜Ws1,s2(X;E)is the space associated with the functional\ncalculus of the two commuting self-adjoint operators OandW2\nθand endowed with the Hilbert norm\n/ba∇dblu/ba∇dbl˜Ws1,s2=/ba∇dblOs1/2(W2\nθ)s2/2u/ba∇dblL2.\nIn particular the space ˜W1,s(T∗Q;E) of Definition 1.2 with the norm\n/ba∇dblu/ba∇dbl˜W1,s=/ba∇dblO1/2u/ba∇dbl˜Ws=/ba∇dbl(W2\nθ)s/2O1/2u/ba∇dblL2\nis the particular case s1=1 ,s2=s. Clearly the spaces ˜Ws1,s2(X;E) contain a finer description of the\nregularity properties. With\n/ba∇dblu/ba∇dbl2\n˜Ws1,s2=〈(W2\nθ)s2/2O(s1−1)/2u,O(W2\nθ)s2/2O(s1−1)/2u〉L2≤/ba∇dbl(W2\nθ)(s2+1/2)/2O(s1−1)/2u/ba∇dbl2\nL2=/ba∇dblu/ba∇dbl2\n˜Ws2+1/2,s1−1.\nfors1≥1 , we deduce the continuous embeddings ˜W0,s2+s1/2(X;E)⊂Ws1,s2(X;E) for s1≥0 and by\nduality ˜Ws1,s2(X;E)⊂˜Ws1,s2+s1/2(X;E) for s1<0 . We will essentially work with s1∈{0,1}.\nAs a first order differential operator with respect to qthe operator ∇E\nYcan be written\n∇E\nY=J/summationdisplay\nj=1θj(q)∇E\nYθj(q)=J/summationdisplay\nj=1θj(q)[gii′(q)pi′∇ei′/vextendsingle/vextendsingle\nT∗˜Ωj]θj(q),\nwhere gii′(q)pi∇E\nei′/vextendsingle/vextendsingle\nT∗˜Ωjis expressed with the local coordinates in ˜Ωj.\nWith the cut-off function ˜χj∈C∞\n0(˜Ωj;[0,1]) such that ˜χj≡1 in a neighborhood of supp θj′(q) when\nΩj∩Ωj′/negat〉onslas〈=/emptysetst∇ess, we can introduce the local scalar operator\nˆpj,i=˜χj(q)pi⊗IdE,ˆDj,i=˜χj(q)Dpi, (3.6)\nwhile ˆEj,i=θj(q)gii′(q)∇E\nei′θj(q)∈OpS1\nΨ(Q;EndE) (3.7)\nwith ˆEj,i−(θj(q)gii′(q)ei′θj(q)⊗IdE)∈OpS0\nΨ(Q;EndE). (3.8)\nWe have in particular\n∇E\nY=J/summationdisplay\nj=1d/summationdisplay\ni=1ˆEj,i◦ˆpj,i.\n23Proposition 3.8. Letˆpj,i,ˆDj,iand ˆEj,i,j∈{1,...,J},i∈{1,...,d}, be the operators defined by (3.6)\nand(3.7) . For any s∈Rwe have the estimates:\n•/ba∇dblˆpj,i/ba∇dblL(˜W1,s;˜W0,s)+/ba∇dblˆDj,i/ba∇dblL(˜W1,s;˜W0,s)≤Cg,s;\n•/ba∇dbl(W2\nθ)s/2ˆpj,i(W2\nθ)−s/2−ˆpj,i/ba∇dblL(˜W1,0;˜W0,1)+/ba∇dbl(W2\nθ)s/2ˆDj,i(W2\nθ)−s/2−ˆDj,i/ba∇dblL(˜W1,0;˜W0,1)≤Cg,s;\n•/ba∇dbl(W2\nθ)s/2∇E\nY(W2\nθ)−s/2−∇E\nY/ba∇dblL(˜W1,0;L2)≤Cg,s.\nProof. All the operators and commutators are well defined continuou s operators on S(X;E) . The\nestimates are then extended by density.\nForA=ˆpj,iorˆDj,iwe know A∈OpS1/2\nΨ(Q;E) while Aand ( W2\nθ)±s∈OpS±s\nΨ(Q;EndE) have scalar\nprincipal symbols. We deduce\n(W2\nθ)sA(W2\nθ)−s−A∈OpS−1/2\nΨ(Q;EndE)⊂L(˜W1,0(X;E);L2(X;E)),\nand/ba∇dblA/ba∇dblL(˜W1,s;˜W0,s)=/ba∇dbl(W2\nθ)s/2A(W2\nθ)−s/2/ba∇dblL(˜W1,0;L2)≤Cg,s.\nFor the second estimate we need a more accurate decompositio n of ( W2\nθ)s/2A(W2\nθ)−s/2−A. Let us write\nA=a(q,p,Dp) with the local coordinate writing, a(q,p,η)=˜χj(q)piwhen A=ˆpj,iand a(q,p,η)=\n˜χj(q)ηiwhen A=ˆDj,i, and let w(q,p,ξ,η)=(Cg+|ξ−Γk\n..pkη|2\ng+Cg/4(|p|2\ng+|η|2\ng)2)1/2be the principal\nscalar symbol of W2\nθ. If we forget the tensor product with Id E, we have\n(W2\nθ)±s−w±s(q,p,Dq,Dp)=R±s−1∈OpS±s−1\nΨ(Q;EndE)\nand\n(W2\nθ)sA(W2\nθ)−s−A=ws(q,p,Dq,Dp)◦A◦w−s(q,p,Dq,Dp)−A/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=A1,s+A2,s+Rs\nwith A2,s=(W2\nθ)s/2R−s−1A+Rs−1w−1(q,p,Dq,Dp)A∈L(˜W1,0;˜W0,1),\nand Rs=(W2\nθ)s/2[A,R−s−1]+Rs−1[A,w−s(q,p,Dq,Dp)]∈OpS−3/2\nΨ(Q;EndE)⊂L(˜W1,0;˜W0,1).\nBy pseudo-differential calculus the symbol of iA1,sequals\nws∂ηa.∂p(w−s)−ws∂pa.∂η(w−s)−ws∂qa.∂ξw−s+rs\n=s\n2d/summationdisplay\nk=1−(w−1∂ηka)♯(w−1∂pw2)+(w−1∂pka)♯(w−1∂ηkw2)+ws∂ξw−s♯∂qa+r′\ns\nwith rs,r′\ns∈S−3/2\nΨ(Q;C) .\nAn explicit computation shows that the operators ( w−1∂pkw2)(q,p,Dq,Dp) , (w−1∂pkw2)(q,p,Dq,Dp)\nand∂qa(q,p,Dp) belong to L(˜W1,0(X;E);L2(X;E)) .\nThe operators ( w−1∂ηka)(q,p,Dq,Dp) , (w−1∂pka)(q,p,Dq,Dp) belong to L(L2(X;E);˜W0,1(X;E)) .\nFinally the remainder r′\ns(q,p,Dq,Dp)∈OpS−3/2\nΨ(Q;C)⊂L(˜W1,0(X;E);˜W0,1(X;E)) .\nThis ends the proof of\n/ba∇dbl(W2\nθ)sA(W2\nθ)−s−A/ba∇dblL(˜W1,0;˜W0,1)≤Cg,sforA=ˆpj,korˆDj,i.\n24We split ( W2\nθ)∇E\nY(W2\nθ)−s−∇E\nYinto\nJ/summationdisplay\nj=1(W2\nθ)sˆEj,iˆpj,i(W2\nθ)−s−ˆEj,iˆpj,i=J/summationdisplay\nj=2/bracketleftbig\n(W2\nθ)sˆEj,i(W2\nθ)−s−ˆEj,i/bracketrightbig\n◦(W2\nθ)sˆpj,i(W2\nθ)−s\n+ˆEj,i◦/bracketleftbig\n(W2\nθ)sˆpj,i(W2\nθ)−s−ˆpj,i/bracketrightbig\nThe factor/bracketleftbig\n(W2\nθ)sˆEj,i(W2\nθ)−s−ˆEj,i/bracketrightbig\nbelongs to OpS0\nΨ(Q;EndE)⊂L(L2(X;E);L2(X;E)) while the op-\nerator ( W2\nθ)sˆpj,i(W2\nθ)−sbelongs to L(˜W1,0(X;E);L2(X;E)) .\nThe operator ( W2\nθ)sˆpj,i(W2\nθ)−s−ˆpj,ibelongs to L(˜W1,0(X;E);˜W0,1(X;E)) while the factor ˆEj,ibelongs\nto OpS1\nΨ(Q;EndE)⊂L(˜W0,1(X;E);L2(X;E)) .\n4 A priori estimates on the scalar GKFP operator\nIn this section we work directly with the localized scalar ve rsion of GKFP operators. The results of\nthis section will then applied to the operators Pj\n±,b’s of Subsection 2.3. From now on, we focus the\nanalysis to the case ±=+ , because the other case ±=− is the same, and we write simply Pj\nband all\nthe forthcoming related operators without the ±index.\nThe chart coordinates open set ΩinQis fixed and any coordinate system allows the identification\nT∗Ω=Ω×Rd⊂R2d\nq,p. The symplectic volume on Ω×Rd, is the usual Lebesgue measure dqdp and\nthe corresponding L2(Ω×Rd,dqdp ;C)-norm will be denoted simply by /ba∇dbl /ba∇dblL2. We consider a scalar\nGKFP operator\nPb=1\nb2O+1\nbY,b∈(0,∞),\nwith Y=gi j(q)pjei,O=−gi j(q)∂pi∂pj+gi j(q)pipj\n2,\nwith the domain\nD(Pb)=C∞\n0(Ω×Rd;C). (4.1)\nBy assuming Ω⊂⊂Ω1where Ω1is a bigger chart coordinates open subset of Q, we can assume\ng/vextendsingle/vextendsingle\nΩ1/2=˜g/vextendsingle/vextendsingle\nΩ1/2where ˜gis a riemannian metric on Rdwhich is euclidean outside a compact set, and\nΩ1/2is an open neighborhood of Ωsuch that Ω⊂⊂Ω1/2⊂⊂Ω1.\nAlternatively the local scalar GKFP operators Pbcan be introduced directly on Ω×Rd⊂R2dwith a\nmetric gwhich is a compactly supported perturbation of the euclidea n metric.\n4.1 Dyadic partition of unity\nBy following [Leb1][Leb2] or [BCD], let θ,˜θ∈C∞\n0(R) be such that supp (θ)⊂/bracketleftbig1\n4,4/bracketrightbig\n, supp/parenleftbig˜θ/parenrightbig\n⊂[0,4],\nand\n∀t∈[0,∞), ˜θ2(4t2)+∞/summationdisplay\nℓ=0θ2/parenleftig\n2−2ℓt2/parenrightig\n=1. (4.2)\nForx∈T∗Ωandℓ∈N∪{−1}, set\nθℓ(x)=/braceleftigg\nθ/parenleftig\n2−2ℓ|p|2\nq/parenrightig\n,ℓ>−1,\n˜θ(4|p|2\nq), ℓ=−1.(4.3)\n25The collection {θℓ}∞\nℓ=−1constitutes a quadratic dyadic partition of unity for T∗Ωin the sense that\n∀x∈T∗Ω,∞/summationdisplay\nℓ=−1θ2\nℓ(x)=1, (4.4)\nwith\nsupp (θℓ)⊂{2ℓ−1≤|p|q≤2ℓ+1}whenever ℓ>−1, (4.5)\nand\nsupp (θ−1)⊂{0≤|p|q≤1},and θ−1(x)=1 for 0≤|p|q≤1\n2. (4.6)\nNotice that because θℓis a function of |p|2\nq,θℓsatisfies\n∀ℓ∈N∪{−1},Yθℓ≡0. (4.7)\nWhen ( q,p) are canonical coordinates on T∗Ω=Ω×Rd, we also observe\n∀α∈Nd,∃Cα>0,∀ℓ∈N∪{−1},sup\nx∈T∗Ω|∂α\npθℓ(x)|≤Cα2−|α|ℓ. (4.8)\nProposition 4.1. There exists a constant Cg,θ,˜θ≥1depending only on the metric gand the functions\nθand ˜θso that\n1\n4/summationdisplay\nℓ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb−iλ\nb/parenrightbigg\nθℓu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤5\n2/summationdisplay\nℓ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb−iλ\nb/parenrightbigg\nθℓu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(4.9)\nholds for all u∈C∞\n0(Ω×Rd;C)and all (λ,b)∈R×R+when κb=Cg,θ,˜θ(1+b2).\nProof. Thanks to (4.7), we have the commutator identities\n/bracketleftbiggκb\nb2+Pb−iλ\nb,θℓ1/bracketrightbigg\n=1\nb2[O,θℓ1]=−1\nb2gi j(q)∂θℓ1\n∂pi∂\n∂pj−1\n2b2gi j(q)∂2θℓ1\n∂pi∂pj, (4.10)\nand\n/bracketleftbigg/bracketleftbiggκb\nb2+Pb−iλ\nb,θℓ1/bracketrightbigg\n,θℓ2/bracketrightbigg\n=1\nb2[[O,θℓ1],θℓ2]=−1\nb2gi j(q)∂θℓ1\n∂pi∂θℓ2\n∂pj. (4.11)\nfor any ℓ1,ℓ2∈N∪{−1}. From (4.8), (4.10), (4.11) and the integration by parts ine quality of Propo-\nsition 2.1, we deduce that there is a constant C′\ng,θ,˜θ≥1, depending only on the metric gand the\nfunctions θand ˜θsuch that κb=Cg,θ,˜θ(1+b2), with Cg,θ,˜θ=C0+32C′\ng,θ,˜θandC0≥1 fixed in Propo-\nsition 2.1, implies\n/summationdisplay\nℓ,ℓ1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbiggκb\nb2+Pb−iλ\nb,θℓ1/bracketrightbigg\nθℓu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤C′\ng,θ,˜θ/parenleftigg\n/summationdisplay\nℓ/parenleftbigg1\nb4/ba∇dblDpθℓu/ba∇dbl2\nL2+1\nb4/ba∇dblθℓu/ba∇dbl2\nL2/parenrightbigg/parenrightigg\n≤C′\ng,θ,˜θ/parenleftigg\n1\nκb+1\nκ2\nb/parenrightigg\n/summationdisplay\nℓ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb−iλ\nb/parenrightbigg\nθℓu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(4.12)\n26and\n/summationdisplay\nℓ,ℓ1,ℓ2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg/bracketleftbiggκb\nb2+Pb−iλ\nb,θℓ1/bracketrightbigg\n,θℓ2/bracketrightbigg\nθl/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤C′\ng,θ,˜θ\nb4/ba∇dblu/ba∇dbl2\nL2≤C′\ng,θ,˜θ\nκ2\nb/summationdisplay\nℓ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb−iλ\nb/parenrightbigg\nθℓu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(4.13)\nfor all λ∈R,b>0, and u∈C∞\n0(T∗Ω;C). The equivalence (4.9) then follows from Corollary C.2 wit h\nr=1\n2.\nFor every ℓ≥−1, we define the change of variable\nΦℓ:Ω×Rd→Ω×Rd\n(q,p)/mapstoc〈a∇→(q,2ℓp).\nThe change of variable in the integral give\n/ba∇dbl(κb\nb2+Pb−iλ\nb)θℓu/ba∇dblL2=/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)uℓ/ba∇dblL2 (4.14)\nwith uℓ(q,p)=2ℓd\n2θ(|p|2\nq)u(q,2ℓp) (for ℓ= −1 replace θby˜θ) and Pb,ℓ=Φ∗\nℓPb(Φ−1\nℓ)∗.After the\nchange of variable, operators are changed by\nPb,ℓ=1\nb2Oℓ+1\nbYℓ, (4.15)\nOℓ=Φ∗\nℓO(Φ−1\nℓ)∗=1\n2(2−2ℓgi j(q)DpiDpj+22ℓgi j(q)pipj) (4.16)\nand Yℓ=Φ∗\nℓY(Φ−1\nℓ)∗=2ℓgi j(q)pj(∂\n∂qi+Γm\nik(q)pm∂\n∂pk). (4.17)\nThe equivalence (4.9) can be rewritten as\n∀u∈C∞\n0(Ω×Rd;C),\n1\n4/summationdisplay\nℓ/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)uℓ/ba∇dbl2\nL2≤/ba∇dbl(κb\nb2+Pb−iλ\nb)u/ba∇dbl2\nL2≤5\n2/summationdisplay\nℓ/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)uℓ/ba∇dbl2\nL2,(4.18)\nwhere now uℓ∈C∞\n0(Sℓ,2;C) with\nSℓ,R=/braceleftbigg{x=(q,p)∈Ω×Rd,1\nR<|p|q<R}forℓ≥0\n{x=(q,p)∈Ω×Rd,|p|q<1} forℓ=−1,(4.19)\nfor any fixed R>1 .\n4.2 Partition with a 2ℓ-dependent grid in the open set Ω⊂Rd\nq\nHere the integer ℓ≥−1 is fixed and the localization will be done with some translat ion invariance in\nRdby using a regular grid with a spacing of size A2−ℓ.\nLet us start with the translation invariant partition of uni ty\n/summationdisplay\nm∈Zdψ2(q−m)≡1 with ψ∈C∞\n0(Rd;[0,1]). (4.20)\n27For any A>0 and ℓ∈Z,ℓ≥−1 , it can be written\n/summationdisplay\nm∈Zdψ2/parenleftbiggq−qm,ℓ,A\nA2−ℓ/parenrightbigg\n≡1 with qm,ℓ,A=A2−ℓm.\nAccordingly we set\nψm,ℓ,A(q)=ψ/parenleftbiggq−qm,ℓ,A\nA2−ℓ/parenrightbigg\nand we get\nSupp( ψm,ℓ,A)=qm,ℓ,A+A2−ℓSupp( ψ), (4.21)\n∀α,β∈Nd,|α|>0=⇒Dα\npDβ\nqψm,ℓ,A=0, (4.22)\n∀β∈Nd,∃Cβ,ψ>0,|A|β|2−ℓ|β|Dβ\nqψm,ℓ,A|≤Cβ,ψ. (4.23)\nProposition 4.2. LetPb,ℓand Sℓ,2be defined respectively by (4.15) and (4.19) forℓ∈Z,ℓ≥ −1.\nThere exists a constant Cg,ψ>0depending only on the metric gand the function ψsuch that\n1\n2/summationdisplay\nm∈Zd/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)ψm,ℓ,Au/ba∇dbl2\nL2−Cg,ψ\nA2b2/vextenddouble/vextenddouble/vextenddouble22ℓψm,ℓ,Au/vextenddouble/vextenddouble/vextenddouble2\nL2≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2\nand /ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2≤2/summationdisplay\nm∈Zd/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)ψm,ℓ,Au/ba∇dbl2\nL2+Cg,ψ\nA2b2/vextenddouble/vextenddouble/vextenddouble22ℓψm,ℓ,Au/vextenddouble/vextenddouble/vextenddouble2\nL2\nholds for all u∈C∞\n0(Sℓ,2;C)and for all (λ,b)∈R×(0,+∞).\nProof. The computation of commutators gives\n[κb\nb2+Pb,ℓ−iλ\nb,ψm1,ℓ,A]=1\nb[Yℓ,ψm1,ℓ,A]=1\nbgi j(q)2ℓpj∂ψm1,ℓ,A\n∂qi(q), (4.24)\n[[κb\nb2+Pb,ℓ−iλ\nb,ψℓ,m1,A],ψm2,ℓ,A]=1\nb[[Yℓ,ψm1,ℓ,A],ψm2,ℓ,A]=0. (4.25)\nBecause supp u⊂Sℓ,2⊂{x=(q,p),|p|q≤2}, the estimate (4.23) of the derivatives of ψm,ℓ,Aimplies\nthat the right-hand side of (4.24) satisfies\n/summationdisplay\nm1∈Zd/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nbgi j(q)2ℓpj∂ψm1,ℓ,A\n∂qi(q)ψm2,ℓ,Au/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤˜Cg,ψ24ℓ\n(Ab)2/vextenddouble/vextenddoubleψm2,ℓ,Au/vextenddouble/vextenddouble2\nL2\nfor all m1,m2∈Zdwith a constant ˜Cg,ψ>0 depending only on gandψ. Because the double com-\nmutator vanishes, we apply the formulas (C.2) and (C.3). Thi s yields the result with the constant\nCg,ψ=4˜C2\ng,ψ.\nSince\n{x=(q,p)∈Sℓ,2,q∈supp( ψℓ,m,A)}⊂(B(qm,ℓ,A,ˆCg,ψA2−ℓ)×Rd)∩Sℓ,2 (4.26)\nwhen u∈C∞\n0(Sℓ,2;C), the problem is thus reduced to finding lower bounds for/vextenddouble/vextenddouble/vextenddouble(κb\nb2+Pb,ℓ−iλ\nb)v/vextenddouble/vextenddouble/vextenddouble2\nL2\nwhen v∈C∞\n0/parenleftbig\nB/parenleftbig\nqm,ℓ,A,ˆCg,ψA2−ℓ/parenrightbig\n×Rd∩Sℓ,2/parenrightbig\n. When A2−ℓis small enough, (4.26) is contained in a\nq-ball with radius below the injectivity radius of the metric ˜gonRd, and normal coordinates around\nqm,ℓ,Acan be used. Note also that the ball B(qm,ℓ,A,ˆCg,ψA2−ℓ) can be equivalently taken for the\neuclidean metric or the metric ˜gby possibly adapting the constant ˆCg,ψ.\n284.3 Use of normal coordinates\nDue to (4.26), we are interested in/parenleftig\nκb\nb2+Pb,ℓ−iλ\nb/parenrightig\nvwhen v∈C∞\n0(B(qm,ℓ,A,ˆCg,ψA2−ℓ)×Rd)∩Sℓ,2. For\n2−ℓA≤εg,ψwith εg,ψsmall enough determined by the pair ( g,ψ) , we may introduce normal coordi-\nnates ˜q=(˜q1,..., ˜qd) centered at qm,ℓ,A. The associated canonical coordinates on B(0,ˆCg,ψA2−ℓ)×\nRd⊂T∗Rdwill be denoted by ( ˜q,˜p) with ˜p=(˜p1,..., ˜pd). Since Pb,ℓmaintains the same form (4.15)\nunder the coordinate change ( q,p)/mapstoc〈a∇→(˜q,˜p), we may assume without loss of generality when consid-\nering\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb,ℓ−iλ\nb/parenrightbigg\nv/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(4.27)\nfor 2−ℓA≤εg,ψand v∈C∞\n0(B(qm,ℓ,A,ˆCg,ψA2−ℓ)×Rd∩Sℓ,2;C) that qm,ℓ,A=0 and that the metric g\nsatisfies\n∀α∈Nd,∂α\nq/parenleftbig\ngi j(q)−δi j/parenrightbig\n=O(|q|(2−|α|)+). (4.28)\nWe note that since the Christoffel symbols for the metric gare given by\nΓℓ\nik(q)=1\n2gℓj(q)/parenleftbigg∂gji\n∂qk(q)+∂gjk\n∂qi(q)−∂gik\n∂qj(q)/parenrightbigg\n, (4.29)\nwe have\n∀α∈Nd,∂α\nqΓℓ\nik(q)=O(|q|(1−|α|+)). (4.30)\nin these coordinates. By taking εg,ψsmaller if necessary, we may restrict our attention to funct ions\nv∈C∞\n0(B(0,ˆC′\ng,ψA2−ℓ)×Rd∩S′\nℓ,4;C), where B(0,ˆC′\ng,ψA2−ℓ) denotes the Euclidean ball in Rdof radius\nˆC′\ng,ψA2−ℓcentered at the origin 0 ∈Rd,ˆC′\ng,ψ>0 is a constant depending only on gandψ, and\nS′\nℓ,4=/braceleftbig\n(q,p)∈R2d:1\n4<|p|<4/bracerightbig\n. Here|p|=(p2\n1+···+ p2\nd)1/2.\nLetg(q)=/parenleftbig\ngi j(q)/parenrightbig\n1≤i,j≤dextended to a function defined on the whole space Rdwith the same\nproperties. We now introduce the following non-symplectic change of coordinates on R2d:\nϕℓ,m: (q,p)/mapstoc〈a∇→/parenleftig\n2−ℓq,g(2−ℓq)p/parenrightig\n, (4.31)\nand the associated unitary map\nUℓ,m:L2(R2d;C)→L2(R2d;C)\nv/mapstoc〈a∇→2−ℓd\n2/radicalbig\ndet(g(2−ℓq))(v◦ϕℓ,m),\nwhich sends L2(B(0,ˆC′\ng,ψA2−ℓ)×Rd;C) into L2(B(0,ˆC′\ng,ψA)×Rd;C) .\nBy taking εg,ψsmaller if necessary, we can assume that the unitary map Uℓ,mand the pull-back\nϕ∗\nℓ,msend C∞\n0((B(0,ˆC′\ng,ψA2−ℓ)×Rd)∩S′\nℓ,4;C) into C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8;C).\nThe change of variables in the L2-norm gives\n/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)v/ba∇dblL2=/ba∇dbl/parenleftbigg\nUℓ,m(κb\nb2+Pb,ℓ−iλ\nb)U∗\nℓ,m/parenrightbigg\nUℓ,mv/ba∇dblL2, (4.32)\nwhere\nUℓ,mPb,ℓU∗\nℓ,m=/radicalig\ndet(g(2−ℓq))Pb,ℓ,m1/radicalbig\ndet(g(2−ℓq)),\n29with\nPb,ℓ,m=ϕ∗\nℓ,mPb,ℓ(ϕ−1\nℓ,m)∗.\nA straightforward computation shows that\nPb,ℓ,m=1\nb2Oℓ,m+1\nbYℓ,m, (4.33)\nUℓ,mPb,ℓU∗\nℓ,m=Pb,ℓ,m+/bracketleftbigg/radicalig\ndet(g(2−ℓq),κb\nb2+Pb,ℓ,m−iλ\nb/bracketrightbigg1/radicalbig\ndet(g(2−ℓq)(4.34)\nwhere\nOℓ,m:=ϕ∗\nℓ,mOℓϕℓ,m=Uℓ,mOℓU∗\nℓ,m=1\n2/summationdisplay\ni,j/parenleftbigg\ngi j(2−ℓq)Dpi\n2ℓDpj\n2ℓ+gi j(2−ℓq)2ℓpi2ℓpj/parenrightbigg\n(4.35)\nand\nYℓ,m:=ϕ∗\nℓ,mYℓϕℓ,m=22ℓδi jpj∂\n∂qi+fi j\nk(q,ℓ)pipj∂\n∂pk, (4.36)\nfi j\nk(q,ℓ) :=2ℓ/summationdisplay\nn′gn′,j(2−ℓq)/parenleftigg\n∂gkn′\n∂qi(2−ℓq)+/summationdisplay\nnΓn′\nin(2−ℓq)gnk(2−ℓq)/parenrightigg\n, (4.37)\nUℓ,mYℓU∗\nℓ,m=Yℓ,m+/bracketleftbigg/radicalig\ndet(g(2−ℓq),Yℓ,m/bracketrightbigg1/radicalbig\ndet(g(2−ℓq). (4.38)\nFrom (4.28) and (4.30) we know that\nsup\nq∈B(0,ˆC′\ng,ψA)|fi j\nk(q,ℓ)|≤C(1)\ng,ψA. (4.39)\nSince det( g)◦ϕℓ,m=det(g(2−ℓq)), we have\n/bracketleftbiggκb\nb2+Pb,ℓ,m−iλ\nb,/radicalig\ndet(g)◦ϕℓ,m/bracketrightbigg\n=1\nb/bracketleftbigg\nYℓ,m,/radicalig\ndet(g(2−ℓq))/bracketrightbigg\n=1\nbϕ∗\nℓ,m[Yℓ,/radicalbig\ndet(g)](ϕ−1\nℓ,m)∗\n=1\nb[Yℓ,/radicalbig\ndet(g)]◦ϕℓ,m\n=2ℓ\nbδkipk∂/radicalbig\ndet(g)\n∂qi◦ϕℓ,m. (4.40)\nis bounded by some constant C(2)\ng,ψA\nbon (B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8. With the identities (4.32) and (4.34)\nwe deduce\n| /ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)Uℓ,mv/ba∇dblL2−/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)v/ba∇dblL2|≤C(3)\ng,ψA\nb/ba∇dblUℓ,mv/ba∇dblL2. (4.41)\nProposition 4.3. There exists a constant Cg,ψ≥1, determined by the metric gand the function ψ,\nsuch that the inequalities\n1\n4/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)Uℓ,mv/ba∇dbl2\nL2≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)v/ba∇dbl2\nL2≤4/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)Uℓ,mv/ba∇dbl2\nL2, (4.42)\n30hold for all v∈C∞\n0((B(0,ˆC′\ng,ψA2−ℓ)×Rd)∩S′\nℓ,4;C), all (λ,b)∈R×(0,∞)when 2−ℓA≤1\nCg,ψandκb\nb2>\nCg,ψA\nb.\nAdditionally, we recall\nUℓ,mv∈C∞\n0((B(0,ˆC′\ng,ψA2−ℓ)×Rd)∩S′\nℓ,8;C),\nfor all v∈C∞\n0((B(0,ˆC′\ng,ψA2−ℓ)×Rd)∩S′\nℓ,4;C).\nProof. The same integration by parts as in the proof of Proposition 2 .1 now gives\n/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ)u/ba∇dblL2/ba∇dblu/ba∇dblL2≥1\n4b2/bracketleftig\nCg/ba∇dbl2−ℓDpu/ba∇dbl2\nL2+Cg/ba∇dbl2ℓ|p|u/ba∇dbl2\nL2+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n+1\nbRe〈u,fi j\nk(q,ℓ)pipj∂\n∂pku〉\n≥(κb\n4b2−C(4)\ng,ψA\nb)/ba∇dblu/ba∇dbl2\nL2\nfor all u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8;C) , owing to (4.39). For κb≥8C(4)\ng,ψAbwe deduce\n∀u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8;C),κb\n8b2/ba∇dblu/ba∇dbl≤/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)u/ba∇dbl.\nThe inequality (4.41) now implies\n|/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)Uℓ,mv/ba∇dblL2−/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)v/ba∇dblL2|≤8C(3)\ng,ψAb\nκb/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)Uℓ,mv/ba∇dblL2.(4.43)\nwhen κb≥8C(4)\ng,ψAb. It suffices to take Cg,ψ=max(8 C(4)\ng,ψ,16C(3)\ng,ψ) .\nThe next step is to replace the Oℓ,mwith the euclidean version defined by\n˜Oℓ=1\n2(δi j2−2ℓDpiDpj+22ℓ|p|2), (4.44)\nand we set according to (4.36)\n˜Pb,ℓ,m=1\nb2˜Oℓ+1\nbYℓ,m, (4.45)\nwith Yℓ,m=22ℓδi jpj∂\n∂qi+fi j\nk(q,ℓ)pipj∂\n∂pk. (4.46)\nProposition 4.4. There is a constant Cg,ψ≥1determined by the metric gand the function ψ, such\nthat the following inequalities\n/ba∇dbl/parenleftbiggκb\nb2+˜Pb,ℓ,m−iλ\nb/parenrightbigg\nu/ba∇dblL2−Cg,ψA22−2ℓ/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dblL2≤/ba∇dbl/parenleftbiggκb\nb2+Pb,ℓ,m−iλ\nb/parenrightbigg\nu/ba∇dblL2\nand\n/ba∇dbl/parenleftbiggκb\nb2+Pb,ℓ,m−iλ\nb/parenrightbigg\nu/ba∇dblL2≤/ba∇dbl/parenleftbiggκb\nb2+˜Pb,ℓ,m−iλ\nb/parenrightbigg\nu/ba∇dblL2+Cg,ψA22−2ℓ/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dblL2\nhold for all u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8,C), allℓ∈Z,ℓ≥−1, and all (λ,b)∈R×(0,+∞).\n31Proof. From the equality\nPb,ℓ,m−˜Pb,ℓ,m=1\nb2/parenleftbig\nOℓ,m−˜Oℓ/parenrightbig\n. (4.47)\nThe difference between Oℓ,mand ˜Oℓ,mis given by\nOℓ,m−˜Oℓ=1\n2/bracketleftig/parenleftig\ngi j(2−ℓq)−δi j/parenrightig\n2−2ℓDpiDpj+22ℓ/parenleftig\ngi j(2−ℓq)−δi j/parenrightig\npipj/bracketrightig\n. (4.48)\nA unitary change of scale replaces (2ℓp,2p−ℓDp) by ( p,Dp) and the problem is reduce to the com-\nparison of harmonic oscillator hamiltonians in Propositio n A.1 relying on global ellipticity. With\n|gi j(2−ℓq)−δi j|+|gi j(2−ℓq)−δi j|≤Cg,ψ,1A22−2ℓ, we obtain\n/ba∇dbl1\nb2/parenleftbig\nOℓ,m−˜Oℓ/parenrightbig\nu/ba∇dblL2≤Cg,ψA22−2ℓ/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dblL2, (4.49)\nfor some constant Cg,ψ>0 determined by ( g,ψ) . The two inequalities of this proposition follow.\nWe set h=1\n22ℓb,\nκb\nb2+˜Pb,ℓ,m−iλ\nb=22ℓ/parenleftbiggκbh\nb+ˆPb,h,f−ihλ/parenrightbigg\n(4.50)\nwhere\nˆPb,h,f=1\n2δi j(hDpi)(hDpj)+|p|2\n2b2+1\nbδi jpj∂\n∂qi+hfi j\nkpipj∂\n∂pk. (4.51)\nThe problem is now reduced to a careful study of the operator ˆPb,h,facting on C∞\n0((B(0,ˆC′\ng,ψA)×\nRd)∩S′\nℓ,8,C) . We will firstly consider the euclidean case in Section 5, wh eref=(fi j\nk)1≤i,j,k≤d=0 ,\nand the results for the general case will be obtained by some a ccurate perturbative argument in\nSubsection 6.1. Passing from the local to the global estimat es will be developed in the end of Section 6\nbut preliminary results are collected in the next paragraph .\n4.4 From the local models to the global estimates for a fixed ℓ\nWe work in the framework of Subsection 4.2 and Subsection 4.3 . Remember the notations and as-\nsumptions:\n• the partition of unity/summationtext\nm∈Zdψ2(q−m)≡1 and the notations ψm,ℓ,A(q)=ψ(q−qm,ℓ,A\n2−ℓA) ,qℓ,m,A=\nA2−ℓm;\n•Pb,ℓ=1\nb2Oℓ+1\nbYℓintroduced in (4.15)(4.16)(4.17) ;\n• the condition 2−ℓA≤εg,ψ=1\nCg,ψforεg,ψsmall enough which allows in Subsection 4.3 the use\nof normal coordinates centered at qℓ,m,A=A2−ℓm,m∈Zd, and the comparison between the\nmetric gwith the euclidean metric;\n• the unitary transform Uℓ,massociated with the change of variables ϕℓ,m: (q,p)/mapstoc〈a∇→/parenleftbig\n2−ℓq,g(2−ℓq)p/parenrightbig\nwritten in normal coordinates;\n• the operator Pb,ℓ,m=1\nb2Oℓ,m+1\nbYℓ,mintroduced in (4.33).\n32Proposition 4.5. With the above notations and assumptions, in particular 2−ℓA≤1\nCg,ψ, set for u∈\nC∞\n0(Sℓ,2,C)and for any m∈Zd,\nuℓ,m=Uℓ,m(ψℓ,m,Au)∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8,C).\nThe following inequalities\n/summationdisplay\nm∈Zd1\n8/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)−Cg,ψ\nA2b2/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d),\nand /ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≤/summationdisplay\nm∈Zd8/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)+Cg,ψ\nA2b2/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\nhold true as soon as κb≥Cg,ψAband the constant Cg,ψ≥1is chosen large enough.\nProof. Simply combine Proposition 4.2 and Proposition 4.3.\nA similar result can be written for the hamiltonian vector fie ld alone.\nProposition 4.6. With the same notations and assumptions as in Proposition 4. 5 the following in-\nequalities\n/summationdisplay\nm∈Zd1\n8/ba∇dbl1\nb(Yℓ,m−iλ)uℓ,m/ba∇dbl2\nL2(R2d)−Cg,ψ(1\nA2b2+1\nb222ℓ)/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≤/ba∇dbl1\nb(Yℓ−iλ)u/ba∇dbl2\nL2(R2d),\nand /ba∇dbl1\nb(Yℓ−iλ)u/ba∇dbl2\nL2(R2d)≤/summationdisplay\nm∈Zd8/ba∇dbl1\nb(Yℓ,m−iλ)uℓ,m/ba∇dbl2\nL2(R2d)+Cg,ψ(1\nA2b2+1\nb222ℓ)/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\nhold true as soon as the constant Cg,ψ≥1is chosen large enough.\nProof. In Proposition 4.2 the operator (κb\nb2+Pb,ℓ−iλ\nb) can be replaced by1\nb(Yℓ−iλ) and this produces\nthe same error termCg,ψ\nA2b2/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble2\nL2. In the same way, the calculation leading to (4.41) can be don e\nwith (κb\nb2+Pb,ℓ−iλ\nb) replaced by1\nb(Yℓ−iλ) . This second step leads to\n/ba∇dbl1\nb(Yℓ,m−iλ)Uℓ,mv/ba∇dbl2\nL2(R2d)≤2/ba∇dbl1\nb(Yℓ−iλ)v/ba∇dbl2\nL2(R2d)+2[C(3)\ng,ψ]2A2\nb2/ba∇dblUℓ,mv/ba∇dbl2\nL2(R2d)\nand\n/ba∇dbl1\nb(Yℓ−iλ)v/ba∇dbl2\nL2(R2d)≤2/ba∇dbl1\nb(Yℓ,m−iλ)Uℓ,mv/ba∇dbl2\nL2(R2d)+2[C(3)\ng,ψ]2A2\nb2/ba∇dblUℓ,mv/ba∇dbl2\nL2(R2d).\nWe conclude with\nA2\nb2/ba∇dblUℓ,mv/ba∇dbl2\nL2(R2d)≤2−2ℓA2\nb222ℓ/ba∇dbl22ℓUℓ,mv/ba∇dbl2\nL2(R2d)≤1\nb222ℓ/ba∇dbl22ℓUℓ,mv/ba∇dbl2\nL2(R2d).\nThe last result of this paragraph is about the comparison bet ween local and global estimates of\nthe ˜W2/3-norm appearing in the lower bound of Theorem 1.6. According to Proposition E.7- ii)in\nAppendix E, the ˜W2/3-norm of u∈C∞\n0(Ω×Rd;C) can be written\n/ba∇dblu/ba∇dbl2\n˜W2/3=/vextenddouble/vextenddouble/vextenddouble(˜O1)2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble|Dq|2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d),\n33expressed with operators constructed in the euclidean case . The change of scale ( p,Dp)→(2ℓp,2−ℓDp)\nleads to consider\n/ba∇dblu/ba∇dbl2\n˜W2/3,ℓ=/vextenddouble/vextenddouble/vextenddouble(˜Oℓ)2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble|Dq|2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d). (4.52)\nProposition 4.7. With the same notations and assumptions as in Proposition 4. 5 and with /ba∇dblu/ba∇dblW2/3,ℓ\ndefined by (4.52) , the following inequalities\n/ba∇dblu/ba∇dbl2\n˜W2/3,ℓ≤Cg,ψ/summationdisplay\nm∈Zd/ba∇dbl(˜Oℓ)2/3uℓ,m/ba∇dbl2\nL2(R2d)+/ba∇dbl|2ℓDq|2/3uℓ,m/ba∇dbl2\nL2(R2d)+1\nA4/3/vextenddouble/vextenddouble/vextenddouble22ℓ/3uℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\nhold true as soon as κb≥Cg,ψAband the constant Cg,ψ≥1is chosen large enough.\nProof. With/vextenddouble/vextenddouble〈Dq〉2/3u/vextenddouble/vextenddouble2\nL2(R2d)=〈u,(1−∆q)2/3u〉and (1−∆q)2/3=/bracketleftbig\n(1+|ξ|2)2/3/bracketrightbigWeyl(q,Dq) .\nand/summationtext\nm∈Zdψ2(q−m)≡1 , Weyl-Hörmander pseudo-differential calculus with the s tandard metric\ndq2+dξ2\n〈ξ〉2provides a constant Cψ≥1 such that\n/ba∇dbl〈Dq〉2\n3v/ba∇dbl2\nL2≤Cψ/summationdisplay\nm∈Zd/ba∇dbl〈Dq〉2\n3ψ(.−m)v/ba∇dbl2\nL2.\nBy setting h=A2−ℓ,ψ(q−hm\nh)=ψm,ℓ,A(q) ,v(q,p)=hd/2u(hq,p) and\nvm,ℓ,A(q,p)=ψm,ℓ,A(q)u(q,p)=h−d/2ψ(q−hm\nh)v(h−1q,p),\nthe above inequality becomes\n/ba∇dbl|hDq|2\n3u/ba∇dbl2\nL2≤Cψ/summationdisplay\nm∈Zd/ba∇dbl〈hDq〉2\n3vm,ℓ,A/ba∇dbl2\nL2≤2Cψ/bracketleftigg\n/summationdisplay\nm∈Zd/ba∇dbl|hDq|2\n3vm,ℓ,A/ba∇dbl2\nL2+/ba∇dblvm,ℓ,A/ba∇dbl2\nL2(R2d)/bracketrightigg\n.\nBy multiplying both sides of the inequality by h−4\n3=24ℓ/2\nA4/3, we get\n/ba∇dbl|Dq|2\n3u/ba∇dbl2\nL2≤2Cψ/summationdisplay\nm∈Zd/ba∇dbl|Dq|2\n3vm,ℓ,A/ba∇dbl2\nL2+1\nA4/3/ba∇dbl22ℓ/3vm,ℓ,A/ba∇dbl2\nL2(R2d).\nBy adding /ba∇dbl(˜Oℓ)u/ba∇dbl2\nL2(R2d==/summationtext\nm∈Zd/ba∇dbl(˜Oℓ)vm,ℓ,A/ba∇dbl2\nL2(R2d=, we deduce\n/ba∇dblu/ba∇dbl2\n˜W2/3,ℓ≤Cψ/bracketleftigg\n/summationdisplay\nm∈Zd/ba∇dblvm,ℓ,A/ba∇dbl2\n˜W2/3,ℓ+1\nA4/3/ba∇dbl22ℓ/3vm,ℓ,A/ba∇dbl2\nL2(R2d)/bracketrightigg\n,\nwhich is not exactly the seeked inequality expressed in term s of the um,ℓ,A. By setting ˜vm,ℓ,A=\n2ℓd/2vm,ℓ,A(q,2ℓp) we have on one side\n/ba∇dblvm,ℓ,A/ba∇dbl2\n˜W2/3,ℓ=/ba∇dbl˜vm,ℓ,A/ba∇dbl2\n˜W2/3,1\nwhile on the other side\n˜vm,ℓ,A(q,p)=2ℓd/2ψm,ℓ,A(q)u(q,2ℓp)=2ℓd/2[U−1\nℓ,mum,ℓ,A](q,2ℓp)\n=2ℓd\n/radicalbig\ndet(g(2ℓq))um,ℓ,A(2ℓq,g(2ℓq)−12ℓp)=[ˆUℓ,mˆum,ℓ,A](q,p)\nwith ˆum,ℓ,A(q,p)=2ℓdum,ℓ,A(2ℓq,2ℓp),[ˆUℓ,mw](q,p)=1/radicalbig\ndet(g(q))w(q,g(q)−1p).\n34Proposition gives /ba∇dblˆUℓ,mw/ba∇dbl˜W2/3,1≤C(1)\ng,ψ/ba∇dblw/ba∇dbl˜W2/3,1and\n/ba∇dbl˜vm,ℓ,A/ba∇dbl2\n˜W2/3,1≤[C(1)\ng,ψ]2/ba∇dblˆum,ℓ,A/ba∇dbl2\n˜W2/3,1≤[C(1)\ng,ψ]2/ba∇dbl(˜Oℓ)2/3um,ℓ,A/ba∇dbl2\nL2(R2d)+/ba∇dbl|2ℓDq|2/3um,ℓ,A/ba∇dbl2\nL2(R2d),\nwhile\n/ba∇dblvm,ℓ,A/ba∇dblL2(R2d)=/ba∇dbl˜vm,ℓ,A/ba∇dblL2(R2d)=/ba∇dblˆum,ℓ,A/ba∇dblL2(R2d)=/ba∇dblum,ℓ,A/ba∇dblL2(R2d).\nThis ends the proof after choosing the constant Cg,ψ≤1 large enough.\n5 Euclidean Case\nIn this section we consider the scalar euclidean case indexe d by two parameters b,h>0 with the\nKramers-Fokker-Planck operator\nˆPb,h,0=1\n2(−h2∆p+|p|2\nb2)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=ˆOb,h+1\nbd/summationdisplay\nj=1pj∂\n∂qj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=ip·Dq(5.1)\nonR2d=Rd\nq×Rd\npwhere ( q,p)=(q1,...,qd,p1,...,pd) .\nProposition 5.1. There exists a universal constant C≥1such that the inequality\nC/ba∇dbl(h\nb+ˆPb,h,0−ihλ)u/ba∇dbl2\nL2(R2d)≥/ba∇dbl(h\nb+ˆOb,h)u/ba∇dbl2\nL2(R2d)+/ba∇dbl(1\nbp·Dq−hλ)u/ba∇dbl2\nL2(R2d)\n+/ba∇dbl(|h\nbDq|2\n3+h\nb)u/ba∇dbl2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nh2|λ|/∇ad〉callow\nhb+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)(5.2)\nholds for all u∈C∞\n0(R2d;C), allλ∈Rand all (b,h)∈(0,∞)2.\nProof. A unitary change of scale transforms ( q,p,Dq,Dp) into (/radicalig\nb\nhq,/∇ad〉callow\nbhp,/radicalig\nh\nbDq,1/∇ad〉callow\nhbDp) and ˆPb,h,0\nintoh\nbˆP1,1,0. The problem is thus reduced to the case b=h=1 and the final result will be obtained\nafter multiplying both sides of the specific inequality by h2/b2.\nLet us introduce some simplified notations:\n• The partial Fourier transform with respect to the variable qis normalized as\nFq/mapstoc〈a∇→ξu(ξ,p)=ˆ\nRde−iq·ξu(q,p)dq,u∈C∞\n0(R2d). (5.3)\nIt is unitary from L2(R2d\nq,p,dqdp ;C) onto L2(R2d\nξ,p,dξ\n(2π)ddp;C) .\n• The operator ˆP1,1,0is simply denoted by ˆPand we set\nˆP=ˆP1,1,0=1\n2(−∆p+|p|2)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=O+ip·Dq\n/tildewideP:=Fq/mapstoc〈a∇→ξ◦ˆP◦/parenleftbig\nFq/mapstoc〈a∇→ξ/parenrightbig−1=ˆ⊕\nRd1\n2(−∆p+|p|2)+i(p·ξ)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=/tildewidePξdξ\n(2π)d, (5.4)\nin the direct integral decomposition L2(R2d\nξ,p,dξ\n(2π)ddp;C)=´⊕\nRdL2(Rd,dp;C)dξ\n(2π)d.\n35• The harmonic oscillator hamiltonian O=−∆p+|p|2\n2is decomposed as O=/summationtextd\nj=1Ojwith\nOj=−1\n2∂2\n∂p2\nj+1\n2p2\nj.\n• For any fixed ξ∈Rd, there exists an orthogonal matrix Rξ∈O(d) such that\nRT\nξ(ξ)=(|ξ|,0,...,0). (5.5)\nThe corresponding unitary pullback on functions ( RT\nξ)∗:L2(Rd)/mapstoc〈a∇→L2(Rd) is given by\n(RT\nξ)∗u(p)=u(RT\nξp),u∈L2(Rd),\nand we let ( Rξ)∗:=/parenleftig/parenleftig\nRT\nξ/parenrightig∗/parenrightig−1\nbe the inverse of/parenleftig\nRT\nξ/parenrightig∗\n. For ξ∈Rd, let\n/tildewidePξ,R:=(RT\nξ)∗◦/tildewidePξ◦(Rξ)∗=1\n2/parenleftbig\n−∆p+|p|2/parenrightbig\n+ip1|ξ| (5.6)\nbe the conjugation of /tildewidePξby (RT\nξ)∗. From (5.6), it is clear that\n/tildewidePξ,R=O1+ip1|ξ|+/summationdisplay\nj/negat〉onslas〈=1Oj. (5.7)\nWe now turn our attention to the topic of obtaining lower boun ds for the quantity\n/vextenddouble/vextenddouble(1+ˆP−iλ)u/vextenddouble/vextenddouble\nL2(R2d)(5.8)\nwhen u∈C∞\n0(R2d) and λ∈R. We begin by observing, via a straightforward calculation, that\n/ba∇dbl(1+O1+i(p1|ξ|−λ))u/ba∇dbl2\nL2(Rd)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg\n1−1\n2∂2\n∂p2\n1+i(p1|ξ|−λ)/parenrightigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublep2\n1\n2u/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2p+/ba∇dblp1u/ba∇dbl2\nL2(Rd)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddoublep1∂\n∂p1u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)−1\n2/ba∇dblu/ba∇dbl2\nL2(Rd)(5.9)\nfor any u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. Meanwhile, an integration by parts argument gives\n/ba∇dbl(1−1\n2∂2\n∂p2\n1+i(p1|ξ|−λ))u/ba∇dbl2\nL2(Rd)≥/ba∇dblu/ba∇dbl2\nL2(Rd)+1\n2/ba∇dbl∂\n∂p1u/ba∇dbl2\nL2(Rd)(5.10)\nfor every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. On the other hand, from Proposition B.1, we know that there\nis a universal constant C0>0 , such that\nC0/ba∇dbl1−1\n2∆p1+i(p1|ξ|−λ)u/ba∇dbl2\nL2(Rd)≥/ba∇dbl1\n2∆p1u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p1|ξ|−λ)u/ba∇dbl2\nL2(Rd)\n+(|ξ|2\n3+1)/ba∇dblu/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p1|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)(5.11)\n36for every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. Since\n/ba∇dbl(1+O1)u/ba∇dbl2\nL2(Rd)=1\n2/ba∇dblu/ba∇dbl2\nL2(Rd)+/ba∇dblp1u/ba∇dbl2\nL2(Rd)\n+/ba∇dblp2\n1\n2u/ba∇dbl2\nL2(Rd)+1\n2/ba∇dblp1∂\n∂p1u/ba∇dbl2\nL2(Rd)+/ba∇dbl∂\n∂p1u/ba∇dbl2\nL2(Rd)+/ba∇dbl1\n2∂2\n∂p2\n1u/ba∇dbl2\nL2(Rd)\nfor every u∈C∞\n0(Rd), we deduce from (5.9), (5.10), and (5.11) that there is a uni versal constant C1≥1\nsuch that\nC1/ba∇dbl(1+O1+i(p1|ξ|−λ))u/ba∇dbl2\nL2(Rd)≥/ba∇dbl(1+O1)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p1|ξ|−λ)u/ba∇dbl2\nL2(Rd)\n+(|ξ|2\n3+1)2/ba∇dblu/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p1|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)(5.12)\nfor every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R.\nNext, we observe that elementary algebraic manipulations g ive the following identity:\n(1+O)2−(1+O1)2=/parenleftigg\n/summationdisplay\nj/negat〉onslas〈=1Oj/parenrightigg2\n+2(1+O1)/summationdisplay\nj/negat〉onslas〈=1Oj≥/parenleftigg\n/summationdisplay\nj/negat〉onslas〈=1Oj/parenrightigg2\n. (5.13)\nA straightforward computation using (5.13) gives that\n(1+/tildewidePξ,R−iλ)∗(1+/tildewidePξ,R−iλ)=(1+O1−i(p1|ξ|−λ))(1+O1+i(p1|ξ|−λ))\n+(1+O)2−(1+O1)2(5.14)\nholds for every ξ∈Rdandλ∈R. As a consequence, we have\n/vextenddouble/vextenddouble(1+/tildewidePξ,R−iλ)u/vextenddouble/vextenddouble2\nL2(Rd)=/ba∇dbl(1+O1+i(p1|ξ|−λ))u/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\nj/negat〉onslas〈=1Oju/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)(5.15)\nfor every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. Combining (5.12) and (5.15) then leads to\nC1/ba∇dbl(1+/tildewidePξ,R−iλ)u/ba∇dbl2\nL2(Rd)≥/ba∇dbl/summationdisplay\nj/negat〉onslas〈=1Oju/ba∇dbl2\nL2(Rd)+/ba∇dbl(1+O1)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p1|ξ|−λ)u/ba∇dbl2\nL2(Rd)\n+(|ξ|2\n3+1)2/ba∇dblu/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)\n≥1\n2/ba∇dbl(1+O)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p1|ξ|−λ)u/ba∇dbl2\nL2(Rd)\n+(|ξ|2\n3+1)2/ba∇dblu/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)(5.16)\nfor every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. From (5.5), (5.6), (5.16), and the unitarity of/parenleftig\nRT\nξ/parenrightig∗\n, we see\nthat there is a universal constant C=2C1≥1 such that\nC/ba∇dbl(1+/tildewidePξ−iλ)u/ba∇dbl2\nL2(Rd)≥/ba∇dbl(1+O)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p·ξ−λ)u/ba∇dbl2\nL2(Rd)\n+(|ξ|2\n3+1)2/ba∇dblu/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)(5.17)\n37for every u∈C∞\n0(Rd),ξ∈Rd, and λ∈R. Using (5.4), (5.17) and the unitarity of Fq→ξ, we obtain\nC/ba∇dbl(1+ˆP−iλ)u/ba∇dbl2\nL2(Rd)≥/ba∇dbl(1+O)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(p·Dq−λ)u/ba∇dbl2\nL2(Rd)\n+/ba∇dbl(|Dq|2\n3+1)u/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n1+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)\nfor all u∈C∞\n0(R2d) and λ∈R. The change of scale introduced in the beginning of this proo f says for\nλreplaced by bλ\nC/ba∇dbl(1+b\nhˆPb,h,0−ibλ)u/ba∇dbl2\nL2(Rd)≥/ba∇dbl(1+b\nhˆOb,h)u/ba∇dbl2\nL2(Rd)+/ba∇dbl(1\nhp·Dq−bλ)u/ba∇dbl2\nL2(Rd)\n+/ba∇dbl(|/radicaligg\nb\nhDq|2\n3+1)u/ba∇dbl2\nL2(Rd)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n|bλ|\n1+|p|/∇ad〉callow\nhb\n2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Rd)\nfor all u∈C∞\n0(R2d) and λ∈R. Multiplying both sides by (h\nb)2ends the proof.\n6 Final proof of Theorem1.6\nWe now collect all the information given by the localization techniques of Section 4 and the accurate\nestimates for the euclidean model in Section 5. The first resu lt will be the derivation of the subelliptic\nestimate for the local model ˜Pb,ℓ,m=1\nb2˜Oℓ+1\nbYℓ,mintroduced in (4.45) at the end of Subsection 4.3\nfrom the subelliptic estimates for the euclidean model. The second result is about the other local\noperator Pb,ℓ,m=1\nb2Oℓ,m+1\nbYℓ,mintroduced in (4.33)(4.35)(4.36). These preliminary resu lts hold\nfor all momenta p∼2ℓand arbitrary values of the intermediate parameter A>0 introduced in the\ngrid partition. Then, in the third paragraph, we consider th e case of large momenta or large ℓand\nthe summation with respect to the grid index m∈Zdwill hold for A≥1 large enough. Here the\nintermediate parameter Awill be fixed to A=A∞(b)≥1 large enough according to the value of b>0\nand the geometric data.\nOnce A∞(b)≥1 is fixed, the fourth paragraph collects the information whe n the momentum p∼2ℓ\nis bounded by CA∞(b),b. For this part the term p×p×∂p, estimated by O(CA∞(b),b)∂p, is controlled\nby a simple integration by parts argument provided κbis large enough. The summation with respect\nto the grid index m∈Zdwill be done by choosing another value for the intermediate p arameter\nA=A0(b) with A0(b)>0 small enough according to the value of b>0 and the geometric data.\nFinally all of the summations with respect to ℓ≥−1 are carried out.\n6.1 Comparison of the local model /tildewiderPb,ℓ,mwith the euclidean case\nWe write general local subelliptic estimates for the local o perator ˜Pb,ℓ,m=1\nb2˜Oℓ+1\nbYℓ,mintroduced\nin (4.45) at the end of Subsection 4.3 and parametrized by the dyadic scale 2ℓ,b>0, the grid index\nm∈Zdand the constant grid scaling A>0 .\nProposition 6.1. LetC≥1be the universal constant given by Proposition 5.1 for the eu clidean metric.\nThere exists a constant C(3)\ng,ψ>0, depending only on the metric gand the function ψ, such that for all\n38(A,b)∈(0,+∞)2,κb≥C(3)\ng,ψAb+1implies the inequalities\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+˜Pb,ℓ,m−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n216b4/ba∇dbl22ℓu/ba∇dbl2\nL2(R2d)(6.1)\nand\n4C/parenleftig\n1+C(3)\ng,ψb2A2/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+˜Pb,ℓ,m−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2/parenleftbig\nκb+˜Oℓ/parenrightbig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb/parenleftbig\nYℓ,m−iλ/parenrightbig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ℓ\nb2Dq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+1\nb2\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d),\n(6.2)\nwhen either u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C)and (λ,ℓ)∈R×N,\nor u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n−1,8,C)and λ∈Rfor ℓ=−1.\nRemember the notations of Subsection 4.3\n•h=1\nb22ℓ,\n•κb\nb2+˜Pb,ℓ,m−iλ\nb=22ℓ/parenleftig\nκbh\nb+ˆPb,h,f−ihλ/parenrightig\n=1\nbh/parenleftig\nκbh\nb+ˆPb,h,f−ihλ/parenrightig\n,\n•˜Oℓ=22ℓˆOb,hwith ˆOb,h=−h2∆p+|p|2\nb2\n2,\n•1\nbYℓ,m=22ℓ(1\nbp·∂q+hfi j\nk(q,ℓ)pipj∂pk)=1\nbh(1\nbp·∂q+hfi j\nk(q,ℓ)pipj∂pk) ,\n•S′\n1,8={(q,p)∈R2d,1\n8≤|p|≤8}andS′\n0,8={(q,p)∈R2d,|p|≤8}.\nThe result of Proposition 6.1 is actually deduced from the sa me results for the operator ˆPb,h,f.\nThis will be done in two steps.\nProposition 6.2. There exists a constant C(2)\ng,ψ>0, depending on the metric gand the function ψ,\nsuch for all (A,b)∈(0,+∞)2and for κb≥1+C(2)\ng,ψAbthe inequalities\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2≥1\n28b2/ba∇dblu/ba∇dbl2\nL2(R2d)+1\n2/summationdisplay\nj/ba∇dblhDpju/ba∇dbl2\nL2(R2d)(6.3)\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≥1\n216b4/ba∇dblu/ba∇dbl2\nL2(R2d)+1\n29b2/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d). (6.4)\nhold true\nwhen either u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C)and (λ,ℓ)∈R×N,\nor u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n−1,8,C)and λ∈Rfor ℓ=−1.\n39Proof. A straightforward computation gives\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2=hκb\nb/ba∇dblu/ba∇dbl2\nL2(R2d)+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble|p|\nbu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+Re/angbracketleftbigg\nhfi j\nkpipj∂u\n∂pk,u/angbracketrightbigg\nL2, (6.5)\nand, owing to |p|≤8 and|fk\ni j(q)|≤C(1)\ng,ψAaccording to (4.39),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleRe/angbracketleftbigg\nhfi j\nkpipj∂u\n∂pk,u/angbracketrightbigg\nL2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingleh\n2/angbracketleftbigg/bracketleftbigg\nfi j\nkpipj,∂\n∂pk/bracketrightbigg\nu,u/angbracketrightbigg\nL2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−h\n2/angbracketleftigg\n/summationdisplay\nk/parenleftig\nfik\nkpi+fk j\nkpj/parenrightig\nu,u/angbracketrightigg\nL2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤8hC(1)\ng,ψA/ba∇dblu/ba∇dbl2\nL2(R2d)(6.6)\nfor any uis supported in ( B(0,ˆC′\ng,ψA)×Rd)∩S′\nℓ,8forℓ=0 orℓ=−1 .\nWhen ℓ=−1 and h=4\nbwe simply use\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2≥4κb−8C(1)\ng,ψAb\nb2/ba∇dblu/ba∇dbl2\nL2+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d)\n≥2\nb2/ba∇dblu/ba∇dbl2\nL2+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d),\nifκp≥1+C(2)\ng,ψAbandC(2)\ng,ψ=16C(1)\ng,ψ.\nWhen ℓ≥0 and supp u⊂(B(0,ˆC′\ng,ψA)×Rd)∩S0,8, we deduce with the lower bound |p| ≥2−3the\ninequality\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2≥(hκb\nb−hC(2)\ng,ψA\n2+1\n27b2)/ba∇dblu/ba∇dbl2\nL2+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d),\nwhere again C(2)\ng,ψ=16C(1)\ng,ψis determined by the metric gand the function ψ. Two cases are now\ndiscussed.\n•h≤1\n27C(2)\ng,ψAb2: we obtain\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2≥(hκb\nb+1\n28b2)/ba∇dblu/ba∇dbl2\nL2(R2d)+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d).\n•h≥1\n27C(2)\ng,ψAb2: the assumption κb≥C(2)\ng,ψAbimplies\nRe/angbracketleftbigg/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu,u/angbracketrightbigg\nL2≥(hκb\n2b+1\n27b2)/ba∇dblu/ba∇dbl2\nL2(R2d)+1\n2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d).\nThis ends the proof of (6.3) in both cases.\nThe second inequality (6.4) is deduced from (6.3) via the Cau chy-Schwarz inequality like in the proof\nof Proposition 2.1.\n40We are now able to give a lower bound for the operator ˆPb,h,f\nProposition 6.3. LetC≥1be the universal constant given by Proposition 5.1 for the eu clidean metric.\nThere exists a constant C(3)\ng,ψ>0, depending only on the metric gand the function ψ, such that for all\n(A,b)∈(0,+∞)2, and κb≥C(3)\ng,ψAb+1the inequality\nC/parenleftig\n1+C(3)\ng,ψA2b2/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκbh\nb+ˆOb,h/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\n4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q+hfi j\nk(q,ℓ)pipj∂pk−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleh\nbDq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+h\nb/parenrightigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nh2|λ|/∇ad〉callow\nhb+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)(6.7)\nholds\nwhen either u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C)and (λ,ℓ)∈R×N,\nor u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n−1,8,C)and λ∈Rfor ℓ=−1.\nProof. We start with the inequality\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,0−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)−/vextenddouble/vextenddouble/vextenddouble/vextenddoublehfi j\nkpipj∂u\n∂pk/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)(6.8)\nand\n/ba∇dblhfi j\nkpipj∂u\n∂pk/ba∇dbl2\nL2(R2d)≤[64C(1)\ng,ψA]2/summationdisplay\nj/ba∇dblhDpju/ba∇dbl2\nL(R2d), (6.9)\nwhich comes from |p|≤8 and (4.39) . With κb≥1 and C≥1 given by Proposition 5.1, the inequality\n(5.2) combined with\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(hκb\nb+Q−ihλ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble(h\nb+Q−ihλ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh(κb−1)\nbu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+2〈h(κb−1)\nbu,(h\nb+ˆOb,h)u〉\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≥0\nfor Q=ˆPb,h,0orQ=ˆOb,h=1\n2(−h2∆p+|p|2\nb2),\nimplies\nC/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,0−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆOb,h/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleh\nbDq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+h\nb/parenrightigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nh2|λ|/∇ad〉callow\nhb+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d).(6.10)\n41Combining (6.8)(6.9) and (6.10) implies\nC/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆOb,h/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleh\nbDq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+h\nb/parenrightigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nh2|λ|/∇ad〉callow\nhb+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n−C[64C(1)\ng,ψA]2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d).\n(6.11)\nPutting\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q+hfi j\nkpipj∂pk−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n−/vextenddouble/vextenddouble/vextenddoublehfi j\nkpipj∂pku/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n≥1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q+hfi j\nkpipj∂pk−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n−[64C(1)\ng,ψA]2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d)\ninto (6.11) gives\nC/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\n4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆOb,h/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\n4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nbp·∂q+hfi j\nkpipj∂pk−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleh\nbDq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+h\nb/parenrightigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\n2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nh2|λ|/∇ad〉callow\nhb+|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n−2C[64C(1)\ng,ψA]2/summationdisplay\nj/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d).\n(6.12)\nWhen κb≥C(2)\ng,ψAbthe inequality (6.4) provides\n/vextenddouble/vextenddoublehDpju/vextenddouble/vextenddouble2\nL2(R2d)≤29b2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigghκb\nb+ˆPb,h,f−ihλ/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\nand taking\nC(3)\ng,ψ=max( C(2)\ng,ψ,222[C(1)\ng,ψ]2)≥1,κb≥C(3)\ng,ψAb+1,\nyields the result.\n6.2 Comparison of the local models Pb,ℓ,mand/tildewiderPb,ℓ,m\nWe now deduce subelliptic estimates for the local operator Pb,ℓ,m=1\nb2Oℓ,m+1\nbYℓ,mintroduced in\n(4.33)(4.35)(4.36) from the one obtained in the previous pa ragraph for ˜Pb,ℓ,m. It is a consequence of\nthe upper bounds for the differences ( ˜Pb,ℓ,m−Pb,ℓ,m) and ( ˜Oℓ−Oℓ,m) studied in Proposition 4.4.\n42Proposition 6.4. There exists a constant C(4)\ng,ψ≥1such that for any A,b>0,κb≥C(4)\ng,ψ(1+A)(1+b)\nand22ℓ≥C(4)\ng,ψ(1+A)(1+b)A2imply\nC(4)\ng,ψ/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)u/ba∇dbl2\nL2(R2d)≥1\nb4/ba∇dbl22ℓu/ba∇dbl2\nL2(R2d)(6.13)\nand\nC(4)\ng,ψ(1+A)2(1+b)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb,ℓ,m−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2/parenleftbig\nκb+Oℓ,m/parenrightbig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb/parenleftbig\nYℓ,m−iλ/parenrightbig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2/parenleftbig\nκb+˜Oℓ/parenrightbig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ℓ\nb2Dq/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n3\n+1\nb2\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d),\n(6.14)\nwhen\nwhen either u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C)and (λ,ℓ)∈R×N,\nor u∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n−1,8,C)and λ∈Rfor ℓ=−1.\nProof. a)For the inequality (6.13) we must reconsider the proof of Pro position 6.2 after noticing\n(bh)Pb,ℓ,m=ˆPb,h,f+−h2(gi j(2−ℓq)−δi j)∂pi∂pj+(gi j(2−ℓq)−δi j)pipj/b2\n2.\nWith|g(2−ℓq)−Id|≤Cg,ψA22−2ℓwe get\n|〈u,−h2(gi j(2−ℓq)−δi j)∂pi∂pj+(gi j(2−ℓq)i j\nδ)pipj/b2\n2u〉|\n≤C′\ng,ψA22−2ℓ/bracketleftig\n/ba∇dblhDpu/ba∇dbl2\nL2(R2d)+/ba∇dblu/ba∇dbl2\nL2(R2d)/bracketrightig\nand\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle〈u,(bh)(κb\nb2+Pb,ℓ,m−iλ\nb)u〉−〈u,(hκb\nh+ˆPb,h,f−ihλ)u〉/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C′′\ng,ψA22−2ℓRe〈u,(hκb\nh+ˆPb,h,f−ihλ)u〉.\nThe lower bound for Re〈u,(bh)(κb\nb2+Pb,ℓ,m−iλ\nb)u〉and by Cauchy-Schwarz for /ba∇dbl(bh)(κb\nb2+Pb,ℓ,m−\niλ\nb)u/ba∇dbl2\nL2(R2d)are thus deduced from Proposition 6.2 when 22ℓ≥2C′′\ng,ψA2. The conditions and the\ninequality (6.13) are thus satisfied by taking C(4)\ng,ψ≥2C′′\ng,ψlarge enough.\nb)Let us consider (6.14). We recall the inequality (4.49)\n/ba∇dbl1\nb2/parenleftbig\nOℓ,m−˜Oℓ/parenrightbig\nu/ba∇dblL2(R2d)≤Cg,ψA22−2ℓ/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dblL2(R2d),\nwhich implies\n/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dbl2\nL2(R2d)≥1\n2/ba∇dbl/parenleftbiggκb\nb2+1\nb2Oℓ,m/parenrightbigg\nu/ba∇dbl2\nL2(R2d)\nand/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dbl2\nL2(R2d)≥1\n4/ba∇dbl/parenleftbiggκb\nb2+1\nb2Oℓ,m/parenrightbigg\nu/ba∇dbl2\nL2(R2d)+1\n4/ba∇dbl/parenleftbiggκb\nb2+1\nb2˜Oℓ/parenrightbigg\nu/ba∇dbl2\nL2(R2d)\n43as soon as 22ℓ≥Cg,ψA2\n/∇ad〉callow\n2−1.\nProposition 6.1 holds under the sufficient condition κb≥C(3)\ng,ψAb+1 which can be simplified into\nκb≥C(3)\ng,ψ(1+A)(1+b) while the left-hand side of (6.2) can be replaced by\n4CC(3)\ng,ψ(1+A)2(1+b)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+˜Pb,ℓ,m−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d).\nTherefore Proposition 6.1 and Proposition 4.4 say\n/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)u/ba∇dblL2(R2d)≥/parenleftbigg\n1−2/radicalig\nCC(3)\ng,ψ(1+A)(1+b)Cg,ψA22−2ℓ/parenrightbigg\n/ba∇dbl(κb\nb2+˜Pb,ℓ,m−iλ\nb)u/ba∇dblL2(R2d)\n≥1\n2/ba∇dbl(κb\nb2+˜Pb,ℓ,m−iλ\nb)u/ba∇dblL2(R2d)\nas soon as\n22ℓ≥4/radicalig\nCC(3)\ng,ψCg,ψ(1+A)(1+b)A2.\nWhen all these conditions are satisfied this proves the seeke d inequality with the upper bound\n64CC(3)\ng,ψ(1+A)2(1+b)2/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)u/ba∇dbl2\nL2(R2d).\nThe result is thus proved by choosing\nC(4)\ng,ψ≥max(4/radicalig\nCC(3)\ng,ψCg,ψ,64CC(3)\ng,ψ)\nwhere the right-hand side is larger than C(3)\ng,ψand for which C(4)\ng,ψ(1+A)(1+b)A2≥Cg,ψA2\n/∇ad〉callow\n2−1.\n6.3 Estimate for |p|∼2ℓlarge\nWe now prove subelliptic estimates for Pb,ℓby summing the local subelliptic estimates for Pb,ℓ,m. All\nthe error terms coming from the partition of unity/summationtext\nm∈Zdψ2(.−m)≡1 and studied in Subsections 4.2,\n4.3 and 4.4, happen to be relatively small enough when the par ameter A=A∞(b) is much larger than\n1 and ℓis large enough so that 22ℓ≫[A∞(b)]3×(1+b) .\nProposition 6.5. LetPb,ℓand Sℓ,2=S0,2⊂Ω×Rdbe defined respectively by (4.15) and (4.19) for\nℓ∈N.\nThere exists a constant C(5)\ng,ψ≥1such that A∞(b)=C(5)\ng,ψ(1+b),κb≥C(5)\ng,ψA∞(b)×(1+b)≥[C(5)\ng,ψ]2(1+b)2\nand22ℓ≥C(5)\ng,ψ[A∞(b)]3(1+b)=[C(5)\ng,ψ]4(1+b)4imply\n[C(5)\ng,ψ]3(1+b)4/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb,ℓ−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2(κb+Oℓ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb(Yℓ−iλ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\nb8/3/bracketleftbigg/vextenddouble/vextenddouble/vextenddouble(˜Oℓ)2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingleDq/vextendsingle/vextendsingle2\n3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)/bracketrightbigg\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d), (6.15)\nwhen u∈C∞\n0(S0,2;C),λ∈Rand b>0.\n44Proof. Our conditions and A=A∞(b)=C(5)\ng,ψ(1+b)≥1 and 22ℓ≥C(5)\ng,ψA3(1+b)≥C(5)\ng,ψA−2imply\n2−ℓA≤1/radicalig\nC(5)\ng,ψ≤1\nCg,ψwhen Cg,ψ≥1 is the constant of Proposition 4.5 and C(5)\ng,ψis chosen larger than\nC2\ng,ψ. With κb≥C(5)\ng,ψA3(1+b)≥Cg,ψAb, Proposition 4.5 says\n/summationdisplay\nm∈Zd1\n8/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)−Cg,ψ\nA2b2/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d),\nwith\n∀m∈Zd,uℓ,m=Uℓ,m(ψℓ,m,Au)∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C).\nBy choosing C(5)\ng,ψlarger than the constant 2 ×C(4)\ng,ψof Proposition 6.4, the condition κb≥C(5)\ng,ψA(1+b)≥\nC(4)\ng,ψ(1+A)(1+b) and the inequality (6.13) imply\n∀m∈Zd,C(4)\ng,ψ/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)≥1\nb4/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d,\nWithC(4)\ng,ψCg,ψb2\nA2≤1\n16when A≥C(5)\ng,ψ(1+b) and C(5)\ng,ψis chosen larger than 4/radicalig\nC(4)\ng,ψCg,ψ, we obtain\n1\n16/summationdisplay\nm∈Zd/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d),\nand the two lower bounds of /ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)of Proposition 6.4 can be used for every\nm∈Zd. The first one (6.13) already used gives now\n1\n32/summationdisplay\nm∈Zd/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)+1\nC(4)\ng,ψb4/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d)≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d).\nThe second one (6.14) with here 2 A≥(1+A) and combined with the second inequality of Proposi-\ntion 4.6 it implies\n210C(4)\ng,ψA3(1+b)/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥/ba∇dbl1\nb2(κb+Oℓ)u/ba∇dbl2\nL2(R2d)+/ba∇dbl1\nb(Yℓ−iλ)u/ba∇dbl2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+/summationdisplay\nm∈Zd/parenleftigg\n1\nC(4)\ng,ψb4−Cg,ψ\nA2b2−Cg,ψ\nb222ℓ/parenrightigg\n/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2(κb+˜Oℓ)uℓ,m/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+1\nb8/3/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle2ℓDq/vextendsingle/vextendsingle/vextendsingle2\n3uℓ,m/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d).\nMoreover κb≥C(5)\ng,ψA(1+b)≥(1+b)2implies1\nb2(κb+˜Oℓ)≥1 and\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2(κb+˜Oℓ)uℓ,m/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥1\nb8/3/vextenddouble/vextenddouble/vextenddouble˜O2/3\nℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d).\n45Proposition 4.7 implies\n/summationdisplay\nm∈Zd/ba∇dbl˜O2/3\nℓuℓ,m/ba∇dbl2\nL2(R2d)+/ba∇dbl|2ℓDq|2/3uℓ,m/ba∇dbl2\nL2(R2d)≥1\nCg,ψ/bracketleftig\n/ba∇dbl˜O2/3\nℓu/ba∇dbl2\nL2(R2d)+/ba∇dbl|Dq|2/3u/ba∇dbl2\nL2(R2d)/bracketrightig\n−/summationdisplay\nm∈Zd1\nA4/3/ba∇dbl22ℓ/3uℓ,m/ba∇dbl2\nL2(R2d)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≤/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d)\nThe complete error term in the lower bound is now bounded from below by\n1\nb4/summationdisplay\nm∈Zd/parenleftigg\n1\nC(4)\ng,ψ−Cg,ψb2\nA2−Cg,ψb2\n22ℓ−b4/3\nA4/3/parenrightigg\n/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d)\nwhich is non negative when A=C(5)\ng,ψ(1+b) and 22ℓ≥[C(5)\ng,ψ]4(1+b)4with C(5)\ng,ψlarge enough. The\ninequality (6.15) is then obtained by taking C(5)\ng,ψ≥28C(4)\ng,ψCg,ψ.\n6.4 Estimate for |p|∼2ℓ≤Cb,g,ψ\nWe have fixed the value of A∞(b)=C(5)\ng,ψ(1+b)≫1 in Proposition 6.5 in order to get a result for all\nℓ≥ℓb,g,ψ+1 with\n22ℓb,g,ψ+2≥[C(5)\ng,ψ]4(1+b)4≥22ℓb,g,ψ. (6.16)\nWe now consider all the bounded values of ℓ∈/braceleftbig\n−1,0,1,...,ℓb,g,ψ/bracerightbig\n. Here the error terms related with\nthe partition of unity/summationtext\nm∈Zdψ2(.−m)≡1 will be relatively small by taking a new, small enough, valu e\nfor the intermediate parameter A>0 denoted by A0(b) and the parameter κb≫1 .\nProposition 6.6. Letℓb,g,ψ∈Nbe such that (6.16) is satisfied. For all ℓ∈/braceleftbig\n−1,0,1,...,ℓb,g,ψ/bracerightbig\nletPb,ℓ\nand Sℓ,2⊂Ω×Rdbe defined respectively by (4.15) and(4.19) .\nTake A0(b)=1\n2C′\ng,ψ(1+b)where C′\ng,ψ=max( Cg,ψ,C(4)\ng,ψ),Cg,ψ≥1is given by Proposition 4.5 and Propo-\nsition 4.7 while C(4)\ng,ψ≥1is given by Proposition 6.4.\nThere exists a constant C(6)\ng,ψ≥1such that κb≥C(6)\ng,ψ(1+b)5implies\nC(6)\ng,ψ(1+b)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbiggκb\nb2+Pb,ℓ−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)≥/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb2(κb+Oℓ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb(Yℓ−iλ)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+1\nb8/3/bracketleftbigg/vextenddouble/vextenddouble/vextenddouble(˜Oℓ)2/3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingleDq/vextendsingle/vextendsingle2\n3u/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)/bracketrightbigg\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d), (6.17)\nwhen u∈C∞\n0(Sℓ,2;C),ℓ∈/braceleftbig\n−1,0,1,...,ℓb,g,ψ/bracerightbig\n,λ∈Rand b>0.\nProof. The proof has the same structure as the one of Proposition 6.5 . Choose now A=A0(b)=\n1\n2C′\ng,ψ(1+b), where the constant C′\ng,ψwill be fixed later in the proof.\nWith 2−ℓA≤2A≤1\nC′\ng,ψ(1+b)≤1\nCg,ψwhich combined with κb≥C(6)\ng,ψ(1+b)6≥b\n2(1+b)≥Cg,ψAbin Propo-\nsition 4.5, implies\n/summationdisplay\nm∈Zd1\n8/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)−Cg,ψ\nA2b2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤4[C′g,ψ]3(1+b)2\nb2/vextenddouble/vextenddouble/vextenddouble22ℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n×[C(5)\ng,ψ]8(1+b)8/ba∇dblu/ba∇dbl2\nL2(R2d)≤/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d),\n46with\nuℓ,m=Uℓ,m(ψℓ,m,Au)∈C∞\n0((B(0,ˆC′\ng,ψA)×Rd)∩S′\n0,8,C).\nfor all m∈Zdand all ℓ∈/braceleftbig\n−1,0,1,...,ℓb,g,ψ/bracerightbig\n.\nThe same integration by parts argument as for Proposition 2. 1 says that for κb≥C(6)\ng,ψ(1+b)6≥\nC(6)\ng,ψ(1+b)2with C(6)\ng,ψ≥1 large enough\n/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥κ2\nb\n16b4/ba∇dblu/ba∇dbl2\nL2(R2d)≥[C(6)\ng,ψ]2(1+b)10\n16b2/ba∇dblu/ba∇dbl2\nL2(R2d).\nThe bound 22ℓ≤[C(5)\ng,ψ]4(1+b)4implies\n[C(6)\ng,ψ]2(1+b)2\nb2/summationdisplay\nm∈Z/ba∇dbl22ℓuℓ,m/ba∇dbl2\nL2(R2d)≤[C(6)\ng,ψ]2[C(5)\ng,ψ]8(1+b)10\nb2/ba∇dblu/ba∇dbl2\nL2(R2d)\n≤16[C(5)\ng,ψ]8/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d).\nThe additional condition [ C(6)\ng,ψ]2≥8[C′\ng,ψ]3thus implies\n16[C(5)\ng,ψ]8/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥4[C′\ng,ψ]3[C(5)\ng,ψ]8(1+b)10\nb2/ba∇dblu/ba∇dbl2\nL2(R2d)+8[Cg,ψ]8\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥1+4/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)\nand\n16[C(5)\ng,ψ]8/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥/summationdisplay\nm∈Zd1\n8/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)\n+κ2\nb\n4b4/vextenddouble/vextenddoubleuℓ,m/vextenddouble/vextenddouble2\nL2(R2d)\nBecause 2ℓ≥1/2≥C(4)\ng,ψ(1+b)A≥C(4)\ng,ψ(1+A)(1+b)A2, the condition of Proposition 6.4 are satisfied.\nMultiplying the above inequality by 8 ×C(4)\ng,ψ4(1+b)2which is larger than 8 C(4)\ng,ψ(1+A)2(1+b)2≥1 ,\nleads to\n29C(4)\ng,ψ(1+b)2[C(5)\ng,ψ]8/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥/summationdisplay\nm∈ZdC(4)\ng,ψ(1+A)2(1+b)2/ba∇dbl(κb\nb2+Pb,ℓ,m−iλ\nb)uℓ,m/ba∇dbl2\nL2(R2d)\n+2κ2\nb(1+b)2\nb4/vextenddouble/vextenddoubleuℓ,m/vextenddouble/vextenddouble2\nL2(R2d)\nProposition 6.4 combined with Proposition 4.6 leads to\n29C(4)\ng,ψ(1+b)2[C(5)\ng,ψ]8/ba∇dbl(κb\nb2+Pb,ℓ−iλ\nb)u/ba∇dbl2\nL2(R2d)≥/ba∇dbl1\nb2(κb+Oℓ)u/ba∇dbl2\nL2(R2d)+/ba∇dbl1\nb(Yℓ−iλ)u/ba∇dbl2\nL2(R2d)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg1\nb2|λ|\n1+2ℓ|p|/parenrightbigg2\n3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)\n+/summationdisplay\nm∈Zd/parenleftigg\n2κ2\nb(1+b)2\nb4−Cg,ψ24ℓ\nA2b2−Cg,ψ24ℓ\nb222ℓ/parenrightigg\n/ba∇dbluℓ,m/ba∇dbl2\nL2(R2d)\n+1\nb8/3/bracketleftigg/vextenddouble/vextenddouble/vextenddouble˜O2/3\nℓuℓ,m/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsingle2ℓDq/vextendsingle/vextendsingle/vextendsingle2\n3uℓ,m/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(R2d)/bracketrightigg\n47where we used thatκb+˜Oℓ\nb2≥1 when κb≥C(6)\ng,ψ(1+b)6≥C(6)\ng,ψ(1+b)2, as in the proof of Proposition 6.5.\nProposition 4.7 implies\n/summationdisplay\nm∈Zd/ba∇dbl˜O2/3\nℓuℓ,m/ba∇dbl2\nL2(R2d)+/ba∇dbl|2ℓDq|2/3uℓ,m/ba∇dbl2\nL2(R2d)≥1\nCg,ψ/bracketleftig\n/ba∇dbl˜O2/3\nℓu/ba∇dbl2\nL2(R2d)+/ba∇dbl|Dq|2/3u/ba∇dbl2\nL2(R2d)/bracketrightig\n−/summationdisplay\nm∈Zd1\nA4/3/ba∇dbl22ℓ/3uℓ,m/ba∇dbl2\nL2(R2d)\n≥1\nCg,ψ/bracketleftig\n/ba∇dbl˜O2/3\nℓu/ba∇dbl2\nL2(R2d)+/ba∇dbl|Dq|2/3u/ba∇dbl2\nL2(R2d)/bracketrightig\n−24/3[C′\ng,ψ]4/3(1+b)4/3[C(5)\nb,ψ]8/3(1+b)8/3\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≤4[C′\ng,ψ]3[C(5)\ng,ψ]8(1+b)4/ba∇dbluℓ,m/ba∇dbl2\nL2(R2d)\nThe complete error term in the lower bound is now bounded from below by\n/summationdisplay\nm∈Zd/parenleftigg\nκ2\nb(1+b)2\nb4−4[C′\ng,ψ]3[C(5)\ng,ψ]8/parenleftbigg(1+b)10+(1+b)8\nb2+(1+b)4\nb8/3/parenrightbigg/parenrightigg\n/ba∇dbluℓ,m/ba∇dbl2\nL2(R2d)\nwhich is non negative as soon as\nκb(1+b)2\nb4≥12[C′\ng,ψ]3[C(5)\ng,ψ]8(1+b)10\nb2.\nA sufficient condition is κb≥C(6)\nb,ψ(1+b)5with C(6)\ng,ψ≥1 large enough.\nFor the final writing of the the inequality we also take C(6)\ng,ψ≥29Cg,ψC(4)\ng,ψ[C(5)\ng,ψ]8.\n6.5 Lower bound for /ba∇dbl(Pb−iλ/b)u/ba∇dblL2+1/b2/ba∇dblu/ba∇dblL2\nAfter the dyadic partition of unity for/summationtext∞\nℓ=−1θ2\nℓ(q,p)≡1 onΩ×Rdof Subsection 4.1 and by setting\nuℓ(q,p)=2ℓd/2θℓ(q,2ℓp)u(q,2ℓp)∈C∞\n0(S2,ℓ;C) for u∈C∞\n0(Ω×Rd;C),\nthe results of Proposition 6.5 and Proposition 6.6 give afte r a rescaling\nC(7)\ng,ψ(1+b)4/ba∇dbl(κb\nb2+Pb−iλ\nb)θℓu/ba∇dbl2\nL2(R2d)≥ /ba∇dbl (κb\nb2+O)θℓu/ba∇dbl2\nL2(R2d)+/ba∇dbl1\nb(Y−iλ)θℓu/ba∇dbl2\nL2(R2d)\n+1\nb8/3/ba∇dblθℓu/ba∇dbl2\n˜W2/3+1\nb8/3/ba∇dbl|λ|\n〈p〉θℓu/ba∇dbl2\nL2(R2d)\nfor all ℓ∈Z,ℓ≥−1 , as soon as κb≥C(7)\ng,ψ(1+b)5andC(7)\ng,ψ≥1 is chosen large enough. By summation\nwith respect to ℓ≥ −1 , Proposition 4.1 and the same result with Pb−iλ/breplaced by1\nb2Oand\nProposition 4.1 for /ba∇dbl /ba∇dblW2/3imply\nC(8)\ng,ψ(1+b)4/ba∇dbl(κb\nb2+Pb−iλ\nb)u/ba∇dbl2\nL2(R2d)≥ /ba∇dbl(κb+O)\nb2u/ba∇dbl2\nL2(R2d)+/ba∇dbl1\nb(Y−iλ)u/ba∇dbl2\nL2(R2d)\n+1\nb8/3/ba∇dblu/ba∇dbl2\n˜W2/3+1\nb8/3/ba∇dbl|λ|\n〈p〉u/ba∇dbl2\nL2(R2d)\n48for all u∈C∞\n0(Ω×Rd;C) as soon as κb≥C(8)\ng,ψ(1+b)5with C(8)\ng,ψ≥1 large enough.\nBy taking Cg,M≥C(8)\ng,ψlarge enough so that the comparison results of Proposition 2 .4 and Proposi-\ntion E.7- ii)can be applied with\nCg,M(1+b)2/ba∇dbl(κb\nb2+Pb−iλ\nb)u/ba∇dblL2(R2d)≥ /ba∇dbl(κb+O)\nb2u/ba∇dblL2(R2d)+/ba∇dbl1\nb(∇E\nY−iλ)u/ba∇dblL2(R2d)\n+1\nb4/3/ba∇dblu/ba∇dbl˜W2/3+1\nb4/3/ba∇dbl|λ|\n〈p〉u/ba∇dblL2(R2d)\nfor all u∈C∞\n0(X;E) , when κb≥Cg,M(1+b)5\nBy writing\nCg,M(1+b)2κb/parenleftbigg\n/ba∇dbl(Pb−iλ\nb)u/ba∇dblL2(R2d)+1\nb2/ba∇dblu/ba∇dblL2(R2d)/parenrightbigg\n≥Cg,M(1+b)2/ba∇dbl(κb\nb2+Pb−iλ\nb)u/ba∇dblL2(R2d)\nand by noticing that the factor of the left-hand side can be\nCg,M(1+b)2×Cg,M(1+b)5=Cg,M(1+b)7\nthis ends the proof of the inequality (1.32) in Theorem 1.6.\nThe essential maximal accretivity result for κb≥C0(1+b2) ,C0≥1 determined by ( g,E,gE,∇E) , was\nproved in Proposition 2.1 and Corollary 2.2.\n7 Consequences and optimality of Theorem 1.6\nIn this subsection we will discuss the optimality of the cons tants appearing in Theorem 1.6 as well\nas several consequences and extensions of the subelliptic e stimate.\n7.1 About b-dependent constants\nObviously the constant1\nCg(1+b)7of Theorem 1.6 can be replaced by a uniform constant when b∈]0,b0]\nfor some b0>0 . The question arises about the regime b→ +∞ . We do not claim that our lower\nbound is optimal with respect to basb→+∞ , but it cannot be written with a uniform constant.\nActually, we show here that scalar GKFP operators admit quas imodes at λ=0 of size O(b−2) as\nb→∞ .\nProposition 7.1. LetP±,b=1\nb2O±1\nbYbe the scalar GKFP operator on X=T∗Qwhere the Hermitian\nbundle is E=Q×Cwith∇Ethe trivial connection, gEis the usual pointwise Hermitian inherited from\nC, and M(b)=0. Then there exists u∈C∞\n0(X;E)=C∞\n0(X;C)with/ba∇dblu/ba∇dblL2(X;E)=1satisfying\n/vextenddouble/vextenddoubleP±,bu/vextenddouble/vextenddouble\nL2(X;E)≤Cb−2,b∈(0,∞), (7.1)\nfor some constant C>0independent of b.\nProof. Letu∈C∞(X;C) be any function of the form\nu(q,p)=ϕ(|p|2\nq),(q,p)∈X, (7.2)\n49where ϕ/negat〉onslas〈=0 belongs to C∞\n0(R;C). By multiplying uby a non-zero constant if necessary, we may\nensure that /ba∇dblu/ba∇dblL2(X;C)=1. Since uis a function of the kinetic energy, we have\nYu≡0. (7.3)\nHence\n/vextenddouble/vextenddoubleP±,bu/vextenddouble/vextenddouble\nL2(X;C)≤1\nb2/ba∇dblOu/ba∇dbl+1\nb/ba∇dblYu/ba∇dbl≤Cb−2,b∈(0,∞). (7.4)\nAn immediate consequence of Proposition 7.1 is that the best possible constant appearing on the\nright-hand side of (1.6) fails to be independent of bin general. Indeed, let P±,b=1\nb2O±1\nbYbe a scalar\nGKFP operator as in Proposition 7.1 and let C(b)>0 be the largest constant such that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nP±,b−iλ\nb/parenrightbigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+1\nb2/ba∇dblu/ba∇dblL2≥C(b)/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddoubleO\nb2u/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb/parenleftig\n∇E\nY−iλ/parenrightig\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+1\nb4/3/bracketleftigg\n||u||˜W2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/bracketrightigg/parenrightigg\n(7.5)\nholds for every u∈C∞\n0(X;C). Taking λ=0 in (7.5) gives\n/vextenddouble/vextenddoubleP±,bu/vextenddouble/vextenddouble\nL2(X;C)+1\nb2/ba∇dblu/ba∇dblL2(X;C)≥C(b)\nb4/3/ba∇dblu/ba∇dbl˜W2\n3(X;C)(7.6)\nIfu∈C∞\n0(X;C) is as in Proposition 7.1, then (7.1) and (7.6) together give\nC(b)≤Cb−2\n3,b∈(0,∞), (7.7)\nfor some C>0 independent of b. Because b−2/3→0 as b→∞ , there cannot exist a constant C0>0\nsuch that C(b)≥C0forb≫1 . In particular, C(b) cannot be constant with respect to b.\n7.2 Perturbative Estimate\nIn this subsection, we consider the stability of the subelli ptic estimate (1.6) under a general class of\nperturbations.\nProposition 7.2. LetP±,b=1\nb2O±1\nb∇E\nY, letM(b)∈L(˜W1,0(X;E);L2(X,dqdp ;E))satisfy\nM(b)=M1(b)+M0(b)\n/ba∇dblM1(b)/ba∇dblL(˜W1,0;L2)≤ν1(b)\nband/ba∇dblM0(b)/ba∇dblL(L2;L2)≤ν0/parenleftbigg\n1+1\nb2/parenrightbigg\n,\nand set\nP±,b,M=P±,b+M1(b)+M0(b).\nWe assume that ν1(b)andν0satisfy\n∀b∈(0,+∞),ν2\n1(b)b2≤Cg+8ν0\n16(1+b2),\n50where Cg≥1is the constant determined by (g,E,gE,∇E)in Theorem 1.6. For κb≥(Cg+16ν0)(1+b2),\nthe operatorκb\nb2+P±,b,Mis closable and its closureκb\nb2+P±,b,Mis maximal accretive with D(P±,b,M)=\nD(P±,b)and\n∀u∈D(P±,b,M),Re〈u,(κb\nb2+P±,b,M)u〉≥1\n8b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n.\nMoreover, the inequalities\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(P±,b,M−iλ\nb)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+1+b2\nb2/ba∇dblu/ba∇dblL2≥(1+b)−7\n8(Cg+16ν0)2/parenleftbigg\n/ba∇dblO\nb2u/ba∇dblL2+/ba∇dbl1\nb(±∇E\nY−iλ)u/ba∇dblL2\n+1\nb4/3/bracketleftigg\n||u||˜W2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/bracketrightigg/parenrightbigg\nand\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(P±,b,M−iλ\nb)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2+2κb\nb2/ba∇dblu/ba∇dblL2≥1\n4Cg(1+b)7/parenleftbigg\n/ba∇dblO\nb2u/ba∇dblL2+/ba∇dbl1\nb(±∇E\nY−iλ)u/ba∇dblL2\n+1\nb4/3/bracketleftigg\n||u||˜W2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/bracketrightigg/parenrightbigg\nhold for every u∈D(P±,b,M)and every (λ,b)∈R×(0,+∞).\nProof. Let us first check the accretivity of P±,b,MonC∞\n0(X;E) . For u∈C∞\n0(X;E) , write\nRe〈u,[κb\nb2+P±,b+M1(b)+M0(b)]u〉L2≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n−ν1(b)\nb/ba∇dblu/ba∇dblL2/ba∇dblu/ba∇dbl˜W1,0\n−ν01+b2\nb2/ba∇dblu0/ba∇dbl2\nL2\n≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n−1\n8b2/ba∇dblu/ba∇dbl2\n˜W1,0\n−(2ν2\n1(b)+ν01+b2\nb2)/ba∇dblu/ba∇dbl2\nL2\n≥1\n8b2/ba∇dblu/ba∇dbl2\n˜W1,0+κb+(Cg+8ν0)(1+b2)−16ν2\n1(b)b2\n8b2/ba∇dblu/ba∇dbl2\nL2\n≥1\n8b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\n.\nThis proves the accretivity of P±,b,Mwhich is therefore closable.\nFrom the inequality (1.31) for P±,bwe deduce for any λ∈R\n/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2/ba∇dblu/ba∇dblL2≥Re〈u,(κb\nb2+P±,b)u〉L2≥1\n4b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,0+κb/ba∇dblu/ba∇dbl2\nL2/bracketrightig\nand for all t>0\nt\n2/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dbl2\nL2≥1\n4b2/ba∇dblu/ba∇dbl2\n˜W1,0+κb−2t−1b2\n4b2/ba∇dblu/ba∇dbl2\nL2.\nThis inequality with t=1\n8ν2\n1(b)and\nκb−2t−1b2=κb−16ν2\n1(b)b2≥(Cg+16ν0)(1+b2)−16ν2\n1(b)b2≥0\n51gives\n1\n4/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dbl2\nL2≥ν2\n1(b)\nb2/ba∇dblu/ba∇dbl2\n˜W1,0≥/ba∇dblM1(b)u/ba∇dbl2\nL2.\nActually the inequality (1.31) also gives\n1\n4/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2−/ba∇dblM0(b)u/ba∇dblL2≥κb\n16b2/ba∇dblu/ba∇dblL2−ν0(1+b2)\nb2/ba∇dblu/ba∇dblL2≥(Cg−16ν0)(1+b2)\n16b2/ba∇dblu/ba∇dblL2≥0.\nUsed with λ=0 , we have two closed accretive operators A=(κ±,b\nb2+P±,b) and C=(κb\nb2+P±,b,M) such\nthat D=C∞\n0(X;E) is dense in D(A)=D(P±,b) and D(C)=D(P±,b,M) while\n∀u∈D,/ba∇dbl(A−C)u/ba∇dblL2≤3\n4/ba∇dblAu/ba∇dblL2,\nwith3\n4<1 and Amaximal accretive. Theorem X.50 of [ReSi] tells us that Cis maximal accretive as\nwell with D(C)=D(A) .\nWe can take κb=(Cg+16ν0)(1+b2) , and/ba∇dbl(A−C)u/ba∇dblL2≤3\n4/ba∇dbl(A−iλ)u/ba∇dblL2leads to\n2(Cg+16ν0)/bracketleftbig\n/ba∇dbl(P±,b,M−iλ)u/ba∇dblL2+1+b2\nb2/ba∇dblu/ba∇dblL2/bracketrightbig\n≥ /ba∇dbl (κb\nb2+P±,b,M−iλ)u/ba∇dblL2+κb\nb2/ba∇dblu/ba∇dblL2\n≥1\n4/ba∇dbl(κb\nb2+P±,b−iλ)u/ba∇dblL2+κb\nb2/ba∇dblu/ba∇dblL2\n≥1\n4/bracketleftbig\n/ba∇dbl(P±,b−iλ)u/ba∇dblL2+1\nb2/ba∇dblu/ba∇dblL2/bracketrightbig\n,\nand\n/ba∇dbl(P±,b,M−iλ)u/ba∇dblL2+2κb\nb2/ba∇dblu/ba∇dblL2≥1\n4/bracketleftbig\n/ba∇dbl(P±,b−iλ)u/ba∇dblL2+1\nb2/ba∇dblu/ba∇dblL2/bracketrightbig\nwhere, in both inequalities, the right-hand side is bounded from below by (1.32) of Theorem 1.6.\n7.3 /tildewiderWs-versions\nThe subellitpic estimate of Theorem 1.6 wich is concerned wi th the case L2(X,dqp;E)=˜W0(X;E)\ncan be extended to ˜Ws(X;E) subelliptic estimates for any s∈Ras follows.\nProposition 7.3. LetP±,b,M=1\nb2O±1\nb∇E\nY+M1(b)+M0(b)satisfy\n∀s∈R,/ba∇dblM1(b)/ba∇dblL(˜W1,s;˜Ws)≤ν1,s(b)\nb,/ba∇dblM0(b)/ba∇dblL(˜Ws;˜Ws)≤ν0,s1+b2\nb2\nwhenever\n∀b∈(0,+∞),ν2\n1,s(b)b2≤Cg+8ν0,s\n16(1+b2).\nFor every s∈R, there exists Cg,s≥1determined by the geometric data (g,E,gE,∇E)and the pair\n(ν0,s,ν1,s(.))such that, for κb≥Cg,s(1+b2), the operatorκb\nb2+P±,b,Mis closable in ˜Ws(X;E)and its\nclosureκb\nb2+P˜Ws\n±,b,Mis maximal accretive with D(P˜Ws\n±,b,M)=D(P˜Ws\n±,b)=W−s\nθD(P±,bL2\n)and\n∀u∈D(P˜Ws\n±,b,M),Re〈u,(κb\nb2+P˜Ws\n±,b,M)u〉˜Ws≥1\n8b2/bracketleftig\n/ba∇dblu/ba∇dbl2\n˜W1,s+κb/ba∇dblu/ba∇dbl2\nWs/bracketrightig\n.\n52Moreover, the inequalities\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(P˜Ws\n±,b,M−iλ\nb)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble˜Ws+1+b2\nb2/ba∇dblu/ba∇dbl˜Ws≥(1+b)−7\n8C2g,s/parenleftbigg\n/ba∇dblO\nb2u/ba∇dbl˜Ws+/ba∇dbl1\nb(±∇E\nY−iλ)u/ba∇dbl˜Ws\n+1\nb4/3/bracketleftigg\n||u||˜Ws+2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble˜Ws/bracketrightigg/parenrightbigg\nand\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(P˜Ws\n±,b,M−iλ\nb)u/vextenddouble/vextenddouble/vextenddouble/vextenddouble˜Ws+κb\nb2/ba∇dblu/ba∇dbl˜Ws≥1\n4Cg(1+b)7/parenleftbigg\n/ba∇dblO\nb2u/ba∇dbl˜Ws+/ba∇dbl1\nb(∇E\nY−iλ)u/ba∇dbl˜Ws\n+1\nb4/3/bracketleftigg\n||u||˜Ws+2\n3+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg|λ|\n〈p〉q/parenrightbigg2/3\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble˜Ws/bracketrightigg/parenrightbigg\nholds for every u∈D(P±,b,M)and every (λ,b)∈R×(0,+∞).\nProof. We use the pseudodifferential operator W2\nθintroduced in Propostion 3.5 which is self-adjoint\nwith D(W2\nθ)=˜W2(X;E) with an elliptic scalar principal symbol in S2\nΨ(Q;EndE) and for which we can\nwrite\n/ba∇dblu/ba∇dbl˜Ws1,s2=/ba∇dblOs1/2(W2\nθ)s2/2u/ba∇dblL2,\nfor any ( s1,s2)∈R2according to Definition 3.7.\nBecause Oand ( W2\nθ)s/2commute on S(X;E) , according to Proposition 3.6, when considering the\noperator P±,b,M:S(X;E)→˜Ws(X;E) we may instead work with the operator\n(W2\nθ)−s/2P±,b,M(W2\nθ)s/2=1\nb2O±1\nb∇E\nY±1\nb(W2\nθ)−s/2[∇E\nY,(W2\nθ)s/2]+(W2\nθ)−s/2(M1(b)+M0(b))(W2\nθ)s/2\ninitially defined from S(X;E)→L2(X;E) .\nThe assumptions ensure\n/ba∇dbl(W2\nθ)−s/2(M1(b)(W2\nθ)s/2/ba∇dblL(˜W1,0;L2)≤ν1,s(b)\nband/ba∇dbl(W2\nθ)−s/2(M0(b)(W2\nθ)s/2/ba∇dblL(L2;L2)≤ν0,s(1+1\nb2).\nWith ( W2\nθ)−s/2[∇E\nY,(W2\nθ)s/2]=(W2\nθ)−s/2∇E\nY(W2\nθ)s/2−∇E\nY, Proposition 3.8 tells us\n/ba∇dbl(W2\nθ)−s/2[∇E\nY,(W2\nθ)s/2]/ba∇dblL(˜W1,0;L2)≤˜Cg,s (7.8)\nfor some constant ˜Cg,s≥1 determined by ( g,E,gE,∇E) . It then suffices to apply Proposition 7.2\nwith M1(b) replaced by M1(b)±(W2\nθ)−s/2[∇E\nY,(W2\nθ)s/2] ,ν1(b) replaced by ν1,s(b)+˜Cg,sandν0byν0,s+\n2˜Cg,s.\nA Comparison of harmonic oscillator hamiltonians\nFor a positive definite symmetric matrix g=(gi j)1≤i,j≤d∈Mdd(R) with g−1=(gi j)1≤i,j≤dletOgde-\nnote the harmonic oscillator hamiltonian\nOg=−gi j∂pi∂pj+gi jpipj\n2\nD(Og)={u∈L2(Rd,dp),∀α,β∈Nd,|α|+|β|≤2,pα∂β\npu∈L2(Rd,dp)}=D(OId).\n53The following result is a consequence of the ellipticity of Ogin the Hörmander class S(〈p,η〉2,dp2+dη2\n〈p,η〉2)\ncombined with /ba∇dblOgu/ba∇dblL2≥d\n2/ba∇dblu/ba∇dblL2.\nProposition A.1. For two positive definite symmetric matrices g1and g2there exist two constants\nCg1,g2>0andCg1>0such that\n/parenleftigg/vextenddouble/vextenddoubleOg1u/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddoubleOg2u/vextenddouble/vextenddouble\nL2/parenrightigg±1\n≤Cg1,g2,\nand/vextenddouble/vextenddouble(Og2−Og1)u/vextenddouble/vextenddouble\nL2≤Cg1/ba∇dblg2−g1/ba∇dblMdd(R)/vextenddouble/vextenddoubleOg1u/vextenddouble/vextenddouble\nL2\nhold for all u∈D(OId).\nB Complex Airy Operator\nWe consider here the case of the one dimensional euclidean ca se of which the properties are due to\nthe fact that the complex Airy operator has a compact resolve nt and an empty spectrum.\nSet\nP1(ξ,λ)=i(p1ξ−λ)−1\n2∆p1\nwith ξ,λ∈R. It is maximal accretive with D(P1(ξ,λ))=/braceleftbig\nu∈L2(R,dp),P1(ξ,λ)u∈L2(R,dp)/bracerightbig\nin which\nC∞\n0(R) is dense (with respect to the graph norm).\nProposition B.1. There exists C0≥1, such that for all (ξ,λ)∈R2the inequality\nC0/ba∇dbl(1+P1(ξ,λ))u/ba∇dbl≥/ba∇dbl1\n2∆pu/ba∇dbl+/ba∇dbl(p1ξ−λ)u/ba∇dbl+(|ξ|2/3+1)/ba∇dblu/ba∇dbl+/ba∇dbl/parenleftbigg|λ|\n1+|p1|/parenrightbigg2\n3\nu/ba∇dbl, (B.1)\nholds for all u∈D(P1(ξ,λ)).\nProof. The lower bound\n∀u∈D(P1(ξ,λ)),/ba∇dbl(1+P1(ξ,λ))u/ba∇dbl2≥/ba∇dblu/ba∇dbl2+/ba∇dblP1(ξ,λ)u/ba∇dbl2\nis due to the accretivity of P1(ξ,λ) .\nWith\n/ba∇dblP1(ξ,λ)u/ba∇dbl2≥ /ba∇dbl1\n2∆p1u/ba∇dbl2+/ba∇dbl(λ−p1ξ)u/ba∇dbl2−|〈ξu,Dp1u〉|\n≥ /ba∇dbl1\n2∆p1u/ba∇dbl2+/ba∇dbl(λ−p1ξ)u/ba∇dbl2−[c4/ba∇dbl|ξ|2/3u/ba∇dbl2+1\n8/ba∇dbl∆p1u/ba∇dbl2]\n≥1\n2/ba∇dbl1\n2∆p1u/ba∇dbl2+/ba∇dbl(λ−p1ξ)u/ba∇dbl2−c′\n4/ba∇dbl|ξ|2/3u/ba∇dbl2.\nConsider now the lower bound of /ba∇dblP1(ξ,λ)u/ba∇dblby/ba∇dbl|ξ|2/3u/ba∇dbl. It is obviously true for ξ=0 . For ξ/negat〉onslas〈=0 , the\noperator\nP1(ξ,λ)=i(p−p0)ξ−1\n2∆p1,p0=λ\nξ,\n54is unitarily equivalent to |ξ|2/3P1(1,0)=|ξ|2/3(ip1−1\n2∆p1) and there exists c1>0 such that\n∀u∈D(P(ξ,λ)),/ba∇dblP1(ξ,λ)u/ba∇dbl≥c1|ξ|2/3/ba∇dblu/ba∇dbl.\nThere exists a constant C1≥1 such that\n∀(ξ,λ)∈R2,∀u∈D(P1(ξ,λ)),C1/ba∇dblP1(ξ,λ)u/ba∇dbl2≥/ba∇dblu/ba∇dbl2+/ba∇dbl1\n2∆p1u/ba∇dbl2+/ba∇dbl(p1ξ−λ)u/ba∇dbl2+/ba∇dbl|ξ|2/3u/ba∇dbl2.\nThe lower bound (B.1) is then obviously true for |λ|≤1 and it suffices to consider the case λ≥1 .\nTake a dyadic partition of unity χ2\n0(εp1)+/summationtext∞\nℓ=1χ2(ε2−ℓp1)≡1 with supp χ0∪supp χ⊂[−4,4] . We get\n/ba∇dblP1(ξ,λ)u/ba∇dbl2−∞/summationdisplay\nℓ=0/ba∇dblP1(ξ,λ)χℓu/ba∇dbl2≤Cχε2/bracketleftbig\n/ba∇dbl∂p1u/ba∇dbl2+/ba∇dblu/ba∇dbl2/bracketrightbig\n≤16C1Cχε2/ba∇dblP1(ξ,λ)u/ba∇dbl2\nfor some constant Cχ>0 determined by the pair ( χ0,χ) . By taking ε≤1\n8/∇ad〉callow\nC1Cχit suffices to consider\n/ba∇dblP1(ξ,λ)(χℓu)/ba∇dbl=1\n22ℓ/ba∇dblP1(ξ23ℓ,λ22ℓ)uℓ/ba∇dbl\nwith uℓ(p1)=2−ℓ/2(χℓu)(2−ℓp1) and supp uℓ⊂[−4ε−1,4ε−1] .\nThere are two cases\n•|λ22ℓ|≥2(4ε−123ℓ|ξ|) and then\n∀p1∈supp uℓ,|λ22ℓ−p1ξ23ℓ|≥|λ|22ℓ−4ε−123ℓ|ξ|≥|λ|22ℓ\n2\nThis implies\nC1/ba∇dbl1\n22ℓP(ξ23ℓ,λ22ℓ)uℓ/ba∇dbl2≥/ba∇dbl|λ|22ℓ\n2×22ℓuℓ/ba∇dbl2≥/ba∇dbl|λ|2\n2uℓ/ba∇dbl2\nFinally with |λ|≥1 and|λ|≥|λ|2/3≥/parenleftig\n|λ|\n2ℓ/parenrightig2/3\nwe obtain\n4C1/ba∇dbl1\n22ℓP(ξ23ℓ,λ22ℓ)uℓ/ba∇dbl2≥/ba∇dbl/parenleftbigg|λ|\n2ℓ/parenrightbigg2/3\nuℓ/ba∇dbl2,\nin this case.\n•|λ22ℓ|≤2(4ε−123ℓ|ξ|) and then the lower bound\nC1/ba∇dbl1\n22ℓP1(ξ23ℓ,λ22ℓ)uℓ/ba∇dbl2≥/ba∇dbl|ξ|2/3uℓ/ba∇dbl2\nimplies with |ξ|≥4−1ε|λ|\n2ℓ+1\nC1/ba∇dbl1\n22ℓP1(ξ23ℓ,λ2ℓ)uℓ/ba∇dbl2≥4−4/3ε4/3\n16/ba∇dbl/parenleftbigg|λ|\n2ℓ/parenrightbigg2/3\nuℓ/ba∇dbl2\nWe conclude with the uniform equivalence\nC−1\nε〈p〉≤2ℓ≤Cε〈p〉\non supp uℓwhere ε≤1\n8/∇ad〉callow\nC1Cψand all the other constants are actually universal constant s once the\npair ( χ0,χ) for the dyadic partition of unity is fixed.\n55C Result used to localize the operator\nIn this appendix Mis a manifold endowed with a volume density dvol and πE:E→Mis a smooth\ncomplex vector bundle endowed with a hermitian metric gE, so that L2(M;E) is well defined with\nthe norm /ba∇dbl /ba∇dblL2(M;E)simply denoted by /ba∇dbl /ba∇dbl.\nFor a differential operator Pacting on C∞\n0(M;E) and a function χ∈C∞(M) , the equality χP=\nPχ−/bracketleftbig\nP,χ/bracketrightbig\nand the triangular inequality give\n∀u∈C∞\n0(M;E),||Pχu||−||[P,χ]u||≤||χPu||≤|| Pχu||+||[P,χ]u||.\nIt then follows that\n∀u∈C∞\n0(M;E),1\n2||Pχu||2−||[P,χ]u||2≤||χPu||2≤2||Pχu||2+2||[P,χ]u||2.\nAfter three iterations with the additional assumptions tha t third order commutators vanish, which\nis relevant for differential operators of order less or equa l to 2 , we get the following statement.\nProposition C.1. LetPbe a differential operator acting on C∞\n0(M;E)and let χ1,χ2andχ3be three\nC∞functions such that\n[[[P,χ1],χ2],χ3]=0. (C.1)\nThe following inequalities hold for uinC∞\n0(M;E)\n||χ1χ2χ3Pu||2≤2||χ2χ3Pχ1u||2+4||χ3[P,χ1]χ2u||2+8||[[P,χ1],χ2]χ3u||2\nand\n||χ1χ2χ3Pu||2≥1\n2||χ2χ3Pχ1u||2−2||χ3[P,χ1]χ2u||2−4||[[P,χ1],χ2]χ3u||2.\nApplying the above proposition with a locally finite quadrat ic partition of unity ( χℓ)ℓ∈L,χℓ∈C∞\n0,/summationtext\nℓ∈Lχ2\nℓ≡1 , and summing over theindices ℓ1,ℓ2andℓ3leads to\n||Pu||2≤2/summationdisplay\nℓ1||Pχℓ1u||2+4/summationdisplay\nℓ1,ℓ2||[P,χℓ2]χℓ1u||2+8/summationdisplay\nℓ1,ℓ2,ℓ3||[[P,χℓ2],χℓ3]χℓ1u||2(C.2)\nand\n||Pu||2≥1\n2/summationdisplay\nℓ1||Pχℓ1u||2−2/summationdisplay\nℓ1,ℓ2||[P,χℓ2]χℓ1u||2−4/summationdisplay\nℓ1,ℓ2,ℓ3||[[P,χℓ2],χℓ3]χℓ1u||2(C.3)\nfor all u∈C∞\n0(M;E) .\nWith (C.2) and (C.3) we have\nCorollary C.2. Let(χℓ)ℓ∈Lbe a family of functions such that\n/summationdisplay\nℓ∈Lχ2\nℓ=1\nand let Pbe a second order differential operator such that\n∀u∈C∞\n0(M;E),r\n2/summationdisplay\nℓ1||Pχℓ1u||2≥2/summationdisplay\nℓ1,ℓ2||[P,χℓ2]χℓ1u||2+4/summationdisplay\nℓ1,ℓ2,ℓ3||[[P,χℓ2],χℓ3]χℓ1u||2(C.4)\nfor some r∈(0,1). Then\n∀u∈C∞\n0(M;E),(2+r)/summationdisplay\nℓ∈L||Pχℓu||2≥||Pu||2≥1−r\n2/summationdisplay\nℓ||Pχℓu||2. (C.5)\n56DN−locand N−comp functional spaces\nLetf:M→Nbe aC∞map from the manifold Mto the manifold N, let EπE→Mbe a vector bundle\nandF(M;E) be a locally convex space of sections continuously embedde d inD′(M;E) (abbreviated\nas a functional space of sections) such that for any χ∈C∞\n0(N;R) the multiplication by χ◦fis a\ncontinuous endomorphism of F(M;E) . The notation Ff−loc(M;E) will denote the set of sections sof\nEsuch that\n∀χ∈C∞\n0(N;R),[χ◦f]s∈F(M;E).\nOnce Ff−loc(M;E) is defined Ff−comp(M;E) is the set of sections s∈Ff−loc(M;E) such that there\nexists χ∈C∞\n0(N;R) with s=[χ◦f]s. For s∈Ff−loc(M;E) the f-support of sis defined by\nf−supp s=/intersectiondisplay\nF⊂N\nFclosed\ns/vextendsingle/vextendsinglef−1(N\\F)≡0F=f(supp s).\nFor a compact subset KofNandFf−K(M;E)=/braceleftbig\ns∈Ff−loc(M;E),f−supp s⊂K/bracerightbig\nand\nFf−comp(M;E)=/uniondisplay\nKcompact in NFf−K(M;E).\nWhen the topology on F(M;E) is known, the topology on Ff−loc(M;E) is the initial topology for the\ncollection of maps ( s/mapstoc〈a∇→[χ◦f]s)χ∈C∞\n0(N;R). This induces the topology on Ff−K(M;E) and the topology\nonFf−comp(M;E) is the inductive limit topology.\nWe will use this in a particular case.\nDefinition D.1. LetM=T∗NorM=T∗(T∗N)be endowed with the natural projection π:M→N\nand let F(M;E)⊂D′(M;E)be a functional space of sections of the vector bundle EπE→Mwhich is a\nC∞\n0(N;R)-module. We will use in both cases the notation FN−loc(M;E)=Fπ−loc(M;E),N−supp s=\nπ−supp s⊂N,FN−K(M;E)=Fπ−K(M;E),FN−comp(M;E)=Fπ−loc(M;E).\nWhen Nis a locally compact manifold, introducing a locally finite a tlas and a subordinate par-\ntition of unity/summationtext\ni∈Iχi(q)≡1 reduces the characterization of s∈FN−loc(M;E) ,M=T∗NorM=\nT∗(T∗N) to the meaning of s∈FΩ−loc(M;E) ,M=T∗Ω=Ω×RdorM=T∗(T∗Ω)=Ω×R3dand the\ninvariance of FΩ−loc(M;E) via a diffeomorphism φ:Ω→Ω. With the extension by 0 and the restric-\ntion, the embeddings FΩ−comp(Ω;E)⊂FN−comp(N;E)⊂FΩ′−loc(Ω′;E) hold for two different open sets\nΩandΩ′inN.\nExample:\nThe spaces SN−comp(T∗N;C) ,S′\nN−comp(T∗N;C) and their respective duals S′\nN−loc(T∗N;C) and\nSN−loc(T∗N;C) are well defined for any locally compact manifold N.\nIf additionally Nis compact SN−loc(T∗N;C)=SN−comp(T∗N;C) (resp. S′\nN−loc=S′\nN−comp) will be sim-\nply denoted S(T∗N;C) (resp. S′(T∗N;C)) .\nThe Schwartz kernel theorem for continuous maps C∞\n0(T∗N;C)→D′(T∗N;C) implies that any con-\ntinuous map from A:SN−comp(T∗N;C)→S′\nN−loc(T∗N;C) admits a kernel in KA∈S′\nN×N−loc(T∗(N×\nN);C) . Additionally Ais continous from S′\nN−comp(T∗N;C) toSN−loc(T∗N;C) if and only if its kernel\nKAbelongs to SN×N−loc(T∗(N×N);C) .\nOther N−loc and N−comp spaces are introduced in the text.\n57E Some pseudo-differential calculus on X=T∗Q\nThe manifold Qis either a compact manifold which can be endowed with any rie mannian metric or\nRd. The total space of the cotangent bundle is denoted by X=T∗Qand symbols of pseudo-differential\noperators are defined as functions of T∗X=T∗(T∗Q) .\nThe pseudo-differential calculus presented here and, of wh ich the global geometrical meaning is\nchecked, implements the idea that ∂qiis an operator of order 1 while pi×and∂piare of order 1/2 as\npresented in [Leb1][Leb2]. However our presentation, like our definition of the spaces ˜Wk(X;E) in\nDefinition 1.2 slightly differ from Lebeau’s approach (see R emark 1.3).\nE.1 Definitions and properties\nWe give here the definitions and state the main properties but their global geometrical meaning will\nbe consequences of the subsequent paragraphs.\nDefinition E.1. For any coordinate system (q1,...,qd)on a chart open set Ω⊂Q, the associated\ncanonical coordinates on T∗Ω⊂Xare(q1,...,qd,p1,...,pd)with p=pidqi∈T∗\nqQ. Accordingly\ndoubly canonical coordinates on T∗(T∗Ω)⊂T∗Xassociated with the coordinates (q1,...,qd)onΩ\nwill be (q,p,ξ,η)with (ξ,η)=ξidqi+ηidpi∈T∗\nq,pXfor the canonical coordinates (q,p)associated\nwith the coordinates (q1,...,qd). Those coordinates will be abbreviated as X=(x,Ξ)∈Ω×R3dwith\nx=(q,p)∈Ω×RdandΞ=(ξ,η)∈R2d.\nDefinition E.2. The set Sm\nΨ(Q;C)is the class of functions a∈C∞(T∗X;C)such that for any doubly\ncanonical coordinate system (q,p,ξ,η)onT∗(T∗Ω), where Ωis a chart open set in Q, the following\ninequalities hold:\n∀K⊂⊂Ω,∀(α,β,γ,δ)∈Nd,∃CK,α,β,γ,δ>0,∀(q,p,ξ,η)∈K×(Rd)3,\n|∂α\nq∂β\np∂γ\nξ∂δ\nηa(q,p,ξ,η)|≤CK,α,β,γ,δ(1+|ξ|2+|p|4+|η|4)m−|γ|−|β|+|δ|\n2\n2 .(E.1)\nThe intersection/intersectiontext\nm∈RSm\nΨ(Q;C)is denoted by S−∞\nΨ(Q;C). The topology on Sm\nΨ(Q;C)is given by the\nseminorms pK,α,β,γ,δ(a)which are the best constant CK,α,β,γ,δin the above inequality. For any open set\nΩ⊂Q, the spaces Sm\nΨ,Ω−loc(Ω;C)and Sm\nΨ,Ω−comp(Ω;C)are defined according to Appendix D . Finally\nthe equivalence relation a1∼a2means a1−a2∈S−∞\nΨ(Q;C).\nActually S−∞\nΨ(Q;C)=SQ−loc(T∗(T∗Q);C)=S(T∗(T∗Q);C) with the notations of Appendix D.\nDefinition E.3. The quantization of a symbol a∈Sm\nΩ−comp(Ω;C)is given by\na(q,p,Dq,Dp)u=ˆ\nΩ×R3dei[ξ(q−q′)+η.(p−p′)]a(q,p,ξ,η)u(q′,p′)dq′dp′dξdη∈S′\nΩ−comp(T∗Ω;C)\nfor any u∈S′\nΩ−comp(T∗Ω;C).\nThe global definition of a(q,p,Dq,Dp)fora∈Sm\nΨ(Q;C)is given by\na(q,p,Dq,Dp)u=N/summationdisplay\nn=1(χn(q)a)(q,p,Dq,Dp)(˜χn(q)u) (E.2)\n58for some partition of unity/summationtextN\nn=1χn≡1onQsubordinate to a finite atlas Q=/uniontextN\nn=1Ωnwith ˜χn∈\nC∞\n0(Ωn;[0,1]),˜χn≡1in a neighborhood of supp χn.\nThe set R(Q;C)of regularizing operators is L(S′(T∗Q;C);S(T∗Q;C)).\nThe set/braceleftbig\na(q,p,Dq,Dp)+R,a∈Sm\nΨ(Q;C),R∈R(Q;C)/bracerightbig\nis denoted OpSm\nΨ(Q;C)andOpS−∞\nΨ(Q;C)=R(Q;C)\nwith the equivalence relation A1∼A2inOpSm\nΨ(Q;C)iffA1=A2+Rwith R∈R(Q;C).\nThis pseudo-differential calculus has the same properties listed below as the classical pseudo-\ndifferential calculus and our approach relies on the global pseudo-differential calculus when Q=Rd\nrecalled in the next paragraph.\nProperties:\na)For any vector bundle C∞isomorphism Φ:T∗Q→T∗Q′given by ( q′,p′)=Φ(q,p)=(φ(q),L(q).p)\nwith L(q)∈GL(TqQ;Tφ(q)Q′) , the pull-back Φ∗defined by [ Φ∗a](x,Ξ)=a(Φ(x),tdΦ−1(x).Ξ)\nwith x=(q,p) andΞ=(ξ,η) defines a continuous isomorphism from Sm\nΨ(Q′;C) toSm\nΨ(Q;C) .\nAny a∈Sm\nΨ(Q;C) equals/summationtextN\nn=1χn(q)afor any finite partition of unity/summationtextN\nn=1χn(q)≡1 . The partic-\nular case where L(q)=tdφ(q)−1says that χna∈Sm\nΨ,Ωn−comp(Ωn,C) is independent of the choice\nof the coordinate system ( q1,...,qd) .\nb)When a∈Sm\nΨ,Ω−comp(Ω;C) and ˜χ1and ˜χ2are two elements of C∞\n0(Ω;[0,1]) such that ˜χ1−˜χ2≡\n0 in a neighborhood of Ω−supp a, the operator a◦[˜χ1−˜χ2] belongs to OpS−∞\nΨ(Q;C) . This\nproperty ensures that the quantization (E.2) is actually in dependent of the choice of the ˜χn\ncut-off functions as a map from Sm\nΨ(Q;C)/S−∞\nΨ(Q;C) to OpSm\nΨ(Q;C)/OpS−∞\nΨ(Q;C) .\nc)For any a∈Sm\nΨ(Q;C) the operator a(q,p,Dq,Dp) defined by (E.2) is continuous from S(X;C) to\nS(X;C) and from S′(X;C) toS′(X;C) .\nd)The set/uniontext\nm∈ROpSm\nΨ(Q;C) is an algebra for the composition product with\na1(q,p,Dq,Dp)◦a2(q,p,Dq,Dp)=[a1a2](q,p,Dq,Dp) mod OpSm1+m2−1\nΨ(Q;C),\n/bracketleftbig\na1(q,p,Dq,Dp),a2(q,p,Dq,Dp)/bracketrightbig\n=[1\ni{a1a2}](q,p,Dq,Dp) mod OpSm1+m2−2\nΨ(Q;C),\nforak∈Smk\nΨ(Q;C) and {a1,a2}=∂ξ.a1∂qa2+∂ηa1.∂pa2−∂qa1.∂ξa2−∂pa1.∂ηa2in doubly canon-\nical coordinates.\ne)For any family ( aj)j∈Nwith aj∈Sm−j\nΨ(Q;C) there exists a∈Sm\nΨ(Q;C) such that a−/summationtextJ\nj=0aj∈\nSm−J−1\nΨ(Q;C) , which is simply written a∼/summationtext\nj∈Naj.\nIn particular for any a∈OpSm\nΨ(Q;C) which is elliptic ( |a(q,p,ξ,η)|≥C−1(1+|ξ|2+|p|4+|η|4)m/2)\nthere exists b∼/summationtext∞\nn=0bjwith b0=1\nasuch that b(q,p,Dq,Dp)◦a(q,p,Dq,Dp)∼a(q,p,Dq,Dp)◦\nb(q,p,Dq,Dp)∼Id .\nf)For any a∈S0\nΨ(Q;C) ,a(q,p,Dq,Dp)∈L(L2(X,dqdp ;C)) with\n/ba∇dbla(q,p,Dq,Dp)/ba∇dblL(L2)≤C sup\n|α|+|β|+|γ|+|δ|≤Ndpα,β,γ,δ(a)\nfor some Nd∈Ndetermined by d=dimQ.\n59g)When Φ:T∗Q→T∗Q′is aC∞-isomorphism like in a)andUΦ:L2(X′,dq′dp′;C)→L2(X,dqdp ;C)\nis the unitary map defined by [ UΦu](q,p)=/radicalbig\n|det(dφ(q))det( L(q))|u(φ(q),L(q).p) then for any\na∈Sm\nΨ(Q′;C) ,UΦa(q′,p′,Dq′,Dp′)U−1\nΦ=b(q,p,Dq,Dp)∈OpSm\nΨ(Q;C) with b∼/summationtext∞\nj=0bjandb0=\n(Φ∗a) .\nWhen L(q)=tdφ(q)−1and q∈Ωa chart open set in Q, this result contains the fact that the\nspace of pseudo-differential operator OpSm\nΨ,Ω−comp(Ω;C) does not depend on the choice of coor-\ndinates on Qwith a functorial transformation of the principal symbol. W ith the local definition\nof the quantization (E.2) and b), the space OpSm\nΨ(Q;C) has a global geometric meaning.\nh)The vector bundle version of pseudo-differential operator sa(q,p,Dq,Dp)∈OpSm\nΨ(Q;π∗\n2(End( E)))\nacting on sections of π∗\nX(E) where X=T∗QπX→QandT∗Xπ2→Qare the natural projection and\nEπE→Qis a vector bundle over Q, is reduced to the case of matricial pseudo-differential op era-\ntors acting on CN-valued sections via the localized definition (E.2). It has t he same properties\nas the scalar pseudo-differential operators except for the principal symbol of a commutator. We\nwill use the abbreviation OpSm\nΨ(Q;EndE) for OpSm\nΨ(Q;π∗\n2(End( E))) .\ni)The seminorm topology on OpSm\nΨ(Q;C) and the continuity properties of ( A1,A2)→A1◦A2and of\nA→UΦ◦A◦U−1\nΦare discussed in Subsection E.4.\nE.2 Global calculus when Q=Rd\nLetT∗(T∗Rd)=R4d\nq,p,ξ,ηand consider the function\nΨ(q,p,ξ,η)=/radicalig\n1+|ξ|2+|η|4+|p|4.\nThe symbol class S(m,gΨ) is the set of a∈C∞(R4d;C) such that:\n∀α,β,γ,δ∈Nd,∃Cα,β,γ,δ>0,∀X=(q,p,ξ,η)∈R4d,|∂α\nq∂β\np∂γ\nξ∂δ\nηa(X)|≤Cα,β,γ,δm(X)Ψ(X)−|γ|−|β|+|δ|\n2\nand the topology on S(m,gΨ;C) is given by the family of the seminorms\npm,k(a)=/summationdisplay\n|α|+|β|+|γ|+|δ|≤k\nX∈R4d/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂α\nq∂β\np∂γ\nξ∂δ\nηa(X)\nm(X)Ψ−|γ|−|β|+|δ|\n2(X)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nWe follow the terminology of [Bon].\nProposition E.4. The metric gΨ=dq2+dξ2\nΨ2+dp2\nΨ+dη2\nΨonT∗R2d\nq,p=R4d\nq,p,ξ,ηis a splitted Hörmander\nmetric with the gain function Ψ(q,p,ξ,η).\nAdditionally it is geodesically temperate with gσ\nΨ=Ψ2gΨ.\nFor any s∈R, the function Ψsis agΨ-weight for any s∈Rand an elliptic symbol in S(Ψs,gΨ).\nProof. The properties gσ\nΨ=Ψ2gΨandgΨ,X(tx,−tΞ)=gΨ,X(tx,tΞ) (gΨis splitted) , Ψ≥1 (Hörmander\nuncertainty condition) and Ψs∈S(Ψs,gΨ;C) are obvious.\nThe inequality/parenleftbigggΨ,X\ngΨ,X′/parenrightbigg−±1\n≤max/braceleftbigg\n1,/parenleftbiggΨ(X)\nΨ(X′)/parenrightbigg±2\n,/parenleftbiggΨ(X)\nΨ(X′)/parenrightbigg±1/bracerightbigg\n, (E.3)\n60says that the slowness and (geodesic) temperance are proved when Ψis a slow and (geodesically)\ntempered weight for gΨ.\nSlowness: SetX=(q,p,ξ,η) and X′=(q′,p′,ξ′,η′) .For gΨ,X(X′−X)≤1\nR2let us prove/parenleftig\nΨ(X)\nΨ(X′)/parenrightig−1\n≤R2\nforR>1 large enough.\nThe assumption implies\n|˜ξ−˜ξ′|≤/radicalbig\n1+|˜ξ|2\nR=〈˜ξ〉\nRwith/braceleftigg˜ξ=1\n(1+|p|4+|η|4)1/2ξ\n˜ξ′=1\n(1+|p|4+|η|4)1/2ξ′.\nWe deduce\n|˜ξ′|2≤2|˜ξ|2+2|˜ξ′−˜ξ|2≤(1+2\nR2)〈˜ξ〉2,\nand\n|˜ξ|2≤2〈˜ξ′〉2+2\nR2〈˜ξ〉2.\nThis gives for R≥2 ,\n(1−2\nR2)〈˜ξ〉2≤2〈˜ξ′〉2≤2(1+2\nR2)〈˜ξ〉2\nand/parenleftbiggΨ(q,p,ξ,η)\nΨ(q′,p,ξ′,η)/parenrightbigg±1\n≤2(1+2\nR2).\nLet us consider now the quantity/parenleftbiggΨ(q′,p,ξ′,η)\nΨ(q′,p′,ξ′,η′)/parenrightbigg±1\nwhile noticing that the first result and the assumption gΨ,X(X−X′)≤1\nR2implies\n|p−p′|≤1\nRΨ(q,p,ξ,η)1/2≤/radicalbig\n2(1+2/R2)\nRΨ(q′,p,ξ′,η)1/2≤2\nR(1+|ξ′|2+|p|4+|η|4)1/4\nand|η−η′|≤/radicalbig\n2(1+2/R2)\nRΨ(q′,p,ξ′,η)1/2≤2\nR(1+|ξ′|2+|p|4+|η|4)1/4\nwith2\nR(1+|ξ′|2+|p|4+|η|4)1/4≤2〈ξ′〉1/2(1+|˜p|2+|˜η|2)1/2\nR\nby setting ˜p=〈ξ′〉−1/2pand ˜η=〈ξ′〉−1/2η. By using the same normalization for ( ˜p′,˜η′) we deduce that\nthe vectors Y=(˜p,˜η) and Y′=(˜p′,˜η′) satisfy\n|Y−Y′|≤2\nR〈Y〉\nand again/parenleftbiggΨ(q′,p,ξ′,η)\nΨ(q′,p′,ξ′,η′)/parenrightbigg±1/2\n=/parenleftbigg〈Y〉\n〈Y′〉/parenrightbigg±1\n≤2(1+8\nR2)\nwhen R/2>2 .\nWe deduce the uniform inequality\n/parenleftbiggΨ(X)\nΨ(X′)/parenrightbigg±1\n=/parenleftbiggΨ(q,p,ξ,η)\nΨ(q′,p,ξ′,η)/parenrightbigg±1\n×/parenleftbiggΨ(q′,p,ξ′,η)\nΨ(q′,p′,ξ′,η′)/parenrightbigg±1\n≤2(1+2\nR2)4(1+8\nR2)2≤212≤R\n61as soon as gΨ,X(X′−X)≤1\nR2ifR≥212.\nGeodesic Temperance: With gσ\nΨ≥dq2+dξ2+dp2+dη2, we get gσ\nΨ,X(X−X′)≥|X−X′|2and the\nsame inequality holds for the geodesic distance for gσ\nΨ,dσ\nΨ(X,X′)≥/vextendsingle/vextendsingleX−X′/vextendsingle/vextendsingle.\nFrom\nΨ(q′,p′,ξ′,η′)2≤1+|ξ′|2+(|p′|2+|η′|2)2≤1+2|ξ|2+2|ξ′−ξ|2+/parenleftbig\n2|p|2+2|p′−p|2+2|η|2+2|η′−η|/parenrightbig2\nwe deduce\nΨ(q′,p′,ξ′,η′)2≤64Ψ(q,p,ξ,η)2(1+|ξ′−ξ|2)(1+|p′−p|2+|η′−η|2)2\nand the symmetric version results from the exchange X↔X′. We obtain\n/parenleftbiggΨ(X)\nΨ(X′)/parenrightbigg±2\n≤64(1+|q−q′|2+|ξ−ξ′|2+|p′−p|2+|η′−η|2)3≤64(1+|X−X′|2)3.\nWith|X−X′|2≤min( gσ\nΨ,X(X−X′);dσ\nΨ(X,X′)2) , this proves that the weight Ψ, and the metric gΨ\nowing to (E.3), are geodesically tempered.\nAll the result of [HormIII]-Chap XVIII can be applied for the Weyl quantization aW(q,p,Dq,Dp)\nwhen a∈S(m,gΨ) and mis agΨ-weight . Because gΨis splitted the Weyl and standard quantiza-\ntions are equivalent and we recall a(x,Dx)=bW(x,Dx) with\na=eiDx.DΞ/2b=N−1/summationdisplay\nn=0(iDx.DΞ/2)n\nn!b+RN,+(b)\nb=e−iDx.DΞ/2a=N−1/summationdisplay\nn=0(−iDx.DΞ/2)n\nn!a+RN,−(a)\nwhere every n-th term is continuous from S(m,gΨ) toS(mΨ−n,gΨ) while the remainders are\ncontinous from S(m,gΨ) toS(mΨ−N,gΨ) . Accordingly if a1♯Wa2(resp. a1♯a2) denote the symbols of\naW\n1(x,Dx)◦aW\n2(x,Dx) (resp. a1(x,Dx)◦a2(x,Dx)) we know\na1♯Wa2(X)=eiσ(DX1,DX2)/2a1(X1)a2(X2)/vextendsingle/vextendsingle\nX1=X2=X\n=N−1/summationdisplay\nn=0(iσ(DX1,DX2)/2)n\nn!a1(X1)a2(X2)/vextendsingle/vextendsingle\nX1=X2=X+RW\nN(a1,a2),\nand respectively\na1♯a2(X)=eiDΞ1Dx2a1(X1)a2(X2)/vextendsingle/vextendsingle\nX1=X2=X=N−1/summationdisplay\nn=0(iDΞ1Dx2)n\nn!a1(X1)a2(X2)/vextendsingle/vextendsingle\nX1=X2=X+RN(a1,a2)\n=N−1/summationdisplay\nn=0/summationdisplay\n|α|≤n1\ni|α|α!∂α\nΞa1∂α\nxa2+RN(a1,a2),\nwhere every n-th term is is bilinear continuous from S(m1,gΨ)×S(m2,gΨ) toS(m1m2Ψ−n,gΨ) while\nthe remainder is bilinear continous from S(m1,gΨ)×S(m2,gΨ) toS(m1m2Ψ−N,gΨ) . Two differ-\nences: aW(x,Dx)∗=(¯a)W(x,Dx) remains true only modulo S(mΨ−1,gΨ) for the classical quantization\nwhile f(x)a(x,Dx)=(f a)(x,Dx) remains true only modulo S(mΨ−1,gΨ) for the Weyl quantization.\n62In [BoCh] were introduced the general Sobolev spaces H(m,g) for any Hörmander metric gand g-\nweight mas Hilbert spaces with the norms /ba∇dblu/ba∇dblH(m,g)=/ba∇dblMW(x,Dx)u/ba∇dblL2, where M∈S(m,g) is any\nfixed elliptic invertible operator.\nWe are concerned here with a simple case.\nDefinition E.5. Fors∈Rthe space ˜Ws(R2d;C)is nothing but H(Ψs,gΨ)with the norm\n/ba∇dblu/ba∇dbl˜Ws=/ba∇dbl(Ms)W(x,Dx)u/ba∇dblL2.\nwith Ms=(Cs+Ψ|s|)signsfor some Cs≥1.\nThe invertibility of Ms(x,Dx) :˜Ws(R2d;C)→L2(R2d,dqdp ;C) comes from ( Cs+Ψ|s|)−1♯W(Cs+\nΨ|s|)=1+O(1/Cs) inS(1,gΨ) .\nBecause gΨis geodesically tempered with gσ\nΨ=Ψ2gΨ, J.M. Bony provides us a simple version of\nBeals criterion in [Bon]. In our particular case the symbol c lass S+(1,gΨ) is nothing but the set of\na∈C∞(R2d;C) such that for any i∈{1,...,d}∂qia∈S(Ψ,gΨ) ,∂ξia∈S(1,gΨ) , while ∂piaand∂ηia\nbelong to S(Ψ1/2;gΨ) . An operator Aequals aW(x,Dx) with a∈S(1,gΨ) if and only if all the commu-\ntators adbW\n1(x,Dx)...adbW\nN(x,Dx)Aforbn∈S+(1,gΨ) are bounded in L(L2) .\nAlternatively, the simpler and original version of Beals cr iterion in [Bea] works here according\n[BoCh] (see [NaNi] for a detailed version of Remark 5.6 in [Bo Ch]) owing to the three properties\n• The metric is diagonal in the canonical basis BofR4d=T∗(R2d) , written as\nB=/braceleftig\n∂qi,∂pi,∂ξi,∂ηi,1≤i≤d/bracerightig\n,\nwhile the convex hull CX,B=/braceleftbig/summationtext\ne∈BtegΨ,X(e)−1/2e,(te)e∈B∈[−1,1]♯B/bracerightbig\nsatisfies\n∃r∈]0,1],∀X∈R4d=T∗(R2d),BgΨ,X(0,r)⊂CX,B⊂BgΨ,X(0,2).\n• If L(e)=(σ(e,X))Wfore∈Bwith σ=dη∧dp+dξ∧dqand Xthe radial vector field , we get\nL(∂qi)=−Dqi,L(∂pi)=−Dpi,L(∂ξi)=qiandL(∂ηi)=pi.\n• For a finite familly E=(en)1≤n≤Nof elements of Bthe weight mE(X) equals\nmE(X)=NE/productdisplay\nn=1gΨ,X(ek)1/2=Ψ(X)−N1−N2/2with/braceleftbiggN1=♯/braceleftbig\nk,ek∈/braceleftbig\n∂ξi/bracerightbig/bracerightbig\nN2=♯/braceleftbig\nk,ek∈/braceleftbig\n∂pi,∂ηi/bracerightbig/bracerightbig\nThe Beals criterion thus says that A=aW(q,p,Dq,Dp) with a∈S(Ψs,gΨ) for some gΨ-weight m, if\nand only if all the commutators\nadα\nqadβ\npadγ\nDqadδ\nDpA\ninitially defined as continuous operators on S(R2d;C) (or on S′(R2d;C)) actually belongs to\nL(˜Ws0(R2d;C);˜Ws0−s+|α|+|β|+|δ|\n2(R2d;C))\nfor some s0∈R(and equivalently for all s0∈R). Additionally the topology on S(Ψs;gΨ) is equiva-\nlently defined by the family of seminorms ( qΨs,k)k∈Nor (˜qΨs,k)k∈N,\nqΨs,k(A)= max\n|α|+|β|+|γ|+|δ|≤k/ba∇dbladα\nqadβ\npadγ\nDqadδ\nDpA/ba∇dbl\nL(L2;˜W−s+|α|+|β|+|δ|\n2).\nBeals criterion is especially convenient for the link betwe en a global pseudo-differential calculus and\nfunctional analysis.\n63Proposition E.6. LetA=aW(x,Dx)be a self-adjoint operator in L2(R2d,dqdp ;C)with an elliptic\n(and real) symbol a∈S(Ψµ,gΨ)(ellipticity means here a≥1\nCΨµuniformly on R2d) such that D(A)=\n˜Wµ(R2d;C). Then for any f∈S(〈t〉s,dt2\n〈t〉2;C), the operators f(A)and f(A)−f(a)W(q,p,Dq,Dp)are\npseudo-differential operators with symbols respectively inS(Ψµs,gΨ)and S(Ψµs−1,gΨ).\nIf additionally A≥CIdL2with C>0, then the same result holds for Asand As−(as)W(q,p,Dq,Dp).\nProof. The proof of Bony in [Bon]-Theorem 3.8 relies on Helffer-Sjö strand functional calculus for-\nmula very convenient with Beals criterion (seminorms on S(Ψs,gΨ) are expressed in terms of norms\nof commutators). We refer the reader to [Bon][DiSj][HeSj] o r to the end of Subsection E.4 for a more\ndetailed use of Helffer-Sjöstrand formula. The only thing w hich was not verified in [Bon], because\nit is about a more general framework, is the principal symbol statement. Actually we can focus on\nµ≥0 and when one knows that ( z−A)−1=bW\nz(x,Dx) with seminorms of bzestimated by pΨ−µ,k(b)≤\nCk〈z〉Nk\n|Imz|Nkit suffices to write/parenleftbig1\nz−a/parenrightbigW◦(z−A)=1+rW\nz(x,DX) with seminorms of rz∈S(Ψ−1,gΨ) es-\ntimated by pΨ−1,k(rz)≤C′\nk〈z〉N′\nk\n|Imz|N′\nk+1. We deduce/parenleftbig1\nz−a/parenrightbigW−(z−A)−1=cW\nz(x,Dx) with pΨ−µ−1,k(cz) es-\ntimated by〈z〉N′′\nk\n|Imz|N′′\nk+1. Inserting this into Helffer-Sjöstrand formula proves the result for s<0 , by\nsimple integration. For s≥0 , write f(A)=(i+A)NfN(A) with fN(t)=(i+t)−Nf(t)∈S(〈t〉s−N,dt2\n〈t〉2)\nandNlarge enough.\nThe former result provides us an easy way for comparing vario us simple definitions of the spaces\n˜Ws(R2d;C) and the equivalence of the norms.\nProposition E.7. Consider the self-adjoint operator A=1−∆2\nq+/parenleftig−∆p+|p|2\n2/parenrightig2\ninL2(R2d,dqdp ;C)with\ndomain D(A)=˜W2(R2d;C). Let/summationtext∞\nℓ=−1θ2\nℓ(t)≡1on[0,+∞)be a quadratic dyadic partition of unity\nlike in (4.2). Then the following squared norms on ˜Ws(R2d;C)equivalent with /ba∇dbl /ba∇dbl2\n˜Ws:\ni)/ba∇dblAs/2u/ba∇dbl2\nL2(R2d,dqdp ;C)for any s∈R;\nii)/ba∇dbl(−∆p+|p|2\n2)su/ba∇dbl2\nL2(R2d,dqdp ;C)+/ba∇dbl|Ds|su/ba∇dbl2\nL2(R2d,dqdp ;C)fors≥0;\niii)/summationtext∞\nℓ=−1/ba∇dblθℓ(|p|2)u/ba∇dbl2\n˜Ws;\niv)fors=k∈N,/summationtext\n|α|+|β|+|γ|\n2≤k/ba∇dbl∂α\nqpβ∂γ\npu/ba∇dbl2\nL2(R2d,dqdp ;C);\nv)fors=k∈N,/summationtext\n|α|+N3+|γ|\n2≤k/ba∇dbl〈p〉N3∂α\nq∂γ\npu/ba∇dbl2\nL2(R2d,dqdp :C);\nProof. The equivalence with i)is a direct application of Proposition E.6 because C+A=aW(q,p,Dq,Dp)\nwith a=C+1+|ξ|2+1\n4(|p|2+|η|2)2mod S(Ψ1,gΨ) which and ais elliptic for C>0 large enough. For\ns∈Rthe functional calculus says that /ba∇dblAs/2u/ba∇dblL2is equivalent to /ba∇dbl(C+A)s/2u/ba∇dblL2. But for C≥Cs>0\nlarge enough ( C+A)s/2=aW\ns(q,p,Dq,Dp) with aselliptic in S(Ψs,gψ) and/ba∇dbl(C+A)s/2u/ba∇dblL2is equiva-\nlent to/ba∇dblu/ba∇dbl˜Ws.\nThe statement ii)is actually a consequence of the functional calculus with (1 +t2+t′2)s≍t2s+t′2sfor\nallt≥0,t′≥d/2 when s≥0 is fixed .\nFors=kthe squared norms of iv)andv)are equivalent to 〈u,Bivu〉L2and〈u,Bvu〉with\nBiv=C+/summationdisplay\n|α|+|β|+|γ|\n2≤kDγ\npD2α\nqp2βDγ\np,Bv=C+/summationdisplay\n|α|+N3+|γ|\n2≤kDγ\npD2α\nq〈p〉2N3Dγ\np\n64which both have a symbol elliptic in S(Ψ2k,gΨ) for C>0 large enough. By Proposition E.6, the\noperators ( C+Biv)1/2and ( C+Bv)1/2have an elliptic symbol in S(Ψk,gΨ) for C>0 large enough and\nthe two norms of iv)andv)are equivalent to /ba∇dblu/ba∇dbl˜Wkfork∈N.\nFinally for iii), the partial Fourier Fq→ξtransform with respect to qsends L2(R2d,dqdp ;C) onto the\ndirect integral´\nRdL2(Rd,dp;C)dξ\n(2π)dand for s=k∈Nthe squared norm in ˜Wk(R2;R) is equivalent to\nNC,k(u)2=C/ba∇dblu/ba∇dbl2\nL2+/summationdisplay\n|α|+|β|+|γ|\n2≤k/ba∇dblξαpβ∂γ\npu/ba∇dbl2\nL2\nand to/summationtext\n2|α|+|β|+|γ|≤2NC,k−1(ξαpβ∂γ\npu)2. We are considering operators of order 2 in ∂pand Corol-\nlary C.2 can be applied with a recurrence with respect to k∈Nand gives\n1\nCk∞/summationdisplay\nℓ=−1NC,k(θℓ(|p|2)u)≤NC,k(u)2≤Ck∞/summationdisplay\nℓ=−1NC,k(θℓ(|p|2)u)2,\nfor all u∈C∞\n0(R2d;C) and by density for all u∈˜Wk(R2d;C) . The result for s∈Rfollows by interpola-\ntion and duality.\nRemark E.8. The equivalence with ii)could be done with a semi-classical calculus aW(/∇ad〉callow\nhp,/∇ad〉callow\nhDp)\nwith the semiclassical parameter h=1/∇ad〉callow\nC+|ξ|2with the symbol classes S(〈p,η〉µ1〈p〉µ2〈η〉µ3,dp2\n〈p〉2+dη2\n〈η〉2)\nwith 1+h2\n4(−∆p+|p|2)2=a1(/∇ad〉callow\nhp,/∇ad〉callow\nhDp),a1elliptic in S(〈p,η〉4,dp2\n〈p〉2+dη2\n〈η〉2)andχℓ(|p|2)∈S(1,dp2\n〈p〉2+\ndη2\n〈η〉2)(see e.g. [Rob] or [NaNi]). The necessity of two Hörmander me trics suggests the link with the\nsecond microlocalization of [BoLe]. The chosen elementary method suffices here.\nE.3 Localization and geometric invariance\nFor the localization in Ω×R3d,q∈Ω,Ωopen set of Rd, it is more convenient to work with the\nclassical quantization:\n[a(q,p,Dq,Dp)](q,p,q′,p′)=ˆ\nR2dei[ξ.(q−q′)+η.(p−p′)a(q,p,ξ,η)dξdη\n(2π)2d,\nfor which ̺(q)a(x,Dx)=[̺(q)a](x,Dx) .\nEvery a∈S′\nΩ−loc(Ω×R3d;C) gives rise to a kernel in S′\nΩ×Ω−loc(Ω×Ω×R2d;C) and therefore a continu-\nous operator AafromSΩ−comp(Ω×Rd;C) toS′\nΩ−loc(Ω×Rd;C) and a/mapstoc〈a∇→Aais a bijection. Consider the\nsymbol classe Sm\nΨ,Ω−loc(Ω;C) characterized by (E.1) with the associated space Sm\nΨ,Ω−comp(Ω;C) and the\nspaces ˜WsΩ−loc(Ω;C) ,˜Ws\nΩ−comp(Ω;C) . For two open sets ΩandΩ′ofRdand̺∈C∞\n0(Ω;C) we have the\nfollowing continuous embeddings when the letter EinE(Ω) stands for Sm\nΨ,˜Ws,SorS′:\nEΩ−comp(Ω)⊂ERd−comp(Rd)⊂EΩ′−loc(Ω′)\n̺(q)EΩ′−comp(Ω′)⊂̺(q)ERd−comp(Rd)⊂EΩ−comp(Ω).\nNotice also for •=loc or comp:\n/intersectiondisplay\ns∈R˜WsΩ−•(Ω×Rd;C)=SΩ−•(Ω×Rd;C) and/uniondisplay\ns∈R˜WsΩ−•(Ω×Rd;C)=S′\nΩ−•(Ω×Rd;C)\n65In particular symbols a∈Sm\nΨ,Ω−comp(Ω;C) can be viewed as symbols a∈S(Ψm,gΨ) . Therefore a(x,Dx)\ndefines a continuous operator from ˜Ws\nΩ−comp(Ω×Rd;C) to ˜Ws\nΩ−comp(Ω×Rd;C) . For χ∈C∞\n0(Ω;[0,1]) such\nthat χ≡1 on a neighborhood ΩχofΩ−supp awe have\na(x,Dx)χ(q) :˜Ws\nΩ−loc(Ω×Rd;C)→˜Ws\nΩ−comp(Ω×Rd;C)\nand a(x,Dx)χ(q)|˜Ws\nΩχ−comp=a(x,Dx)/vextendsingle/vextendsingle˜Ws\nΩχ−comp.\nFor two different choices of χandχ′which are equal to 1 in a neighborhood of Ω−supp a,a∈\nSm\nΨ,Ω−comp(Ω;C) , the differences a(x,Dx)χ(q)−a(x,Dx)χ′(q)=a(x,Dx)(χ(q)−χ′(q))=b(x,Dx) with b∈\nS(Ψ−∞,gΨ) and therefore is continuous from S′\nΩ′−loc(Ω′×Rd;C) toSΩ−comp(Ω×Rd;C) , where Ω′=Ω\nor any other open subset of Rd.\nDefinition E.9. For an open set Ωthe set of regularizing operators is\nR(Ω;C)=L(S′\nΩ−loc(Ω×Rd;C);SΩ−comp(Ω×Rd;C)).\nFor two operators A,Bcontinuous from SΩ−comp(Ω×Rd;C)toS′\nΩ−comp(Ω×Rd;C)the equivalence A∼B\nis defined by A−B∈R(Ω;C).\nThe set OpSm\nΨ(Ω;C)is the set of sums a(x,Dx)χ(q)+Rwith a∈Sm\nΨ,Ω−comp(Ω;C)⊂S(Ψm,gΨ),χ∈\nC∞\n0(Ω;[0,1])with χ≡1on a neighborhood of Ω−supp a, and R∈R(Ω;C). The set OpS−∞\nΨ(Ω;C)is\nR(Ω;C).\nWith the previous remarks/uniontext\nm∈ROpSm\nΨ(Ω;C) is an algebra and clearly/intersectiontext\nm∈ROpSm\nΨ(Ω;C)=R(Ω;C)=\nOpS−∞\nΨ(Ω;C) . Moreover if Aj=aj(x,Dx)χj(q)+Rj∈OpSm\nΨ(Ωj;C) for j=1,2 then A1◦A2∈OpSm\nΨ(Ω1∪\nΩ2;C) with A1◦A2∼a1(x,Dx)◦a2(x,Dx)χ(q)=(a1♯a2)(x,Dx)χ(q) for any χ∈C∞\n0(Ω1∩Ω2;C) such that\nχ≡1 on a neighborhood of Ω−supp a1∩Ω−supp a2.\nDefinition E.10. ForA∈L(SΩ−comp(Ω×Rd;C);S′\nΩ−loc(Ω×Rd;C)), the notation A∼/summationtext∞\nj=0aj(x,Dx)\nis thought of as a localized asymptotic sum, for a sequence aj∈Smj\nΨ,Ω−loc(Ω;C)with lim j→∞mj=−∞\nand (mj)j∈Ndecreasing. It means that for any pair ̺,χ∈C∞\n0(Ω;[0,1]), with χ≡1on a neighborhood\nofΩ−supp ̺, there exists a̺∈Sm0\nΨ(Ω;C)such that ̺(q)A∼a̺(x,Dx)χ(q)and for any J∈N,a̺−/summationtextJ\nj=0̺(q)aj∈SmJ+1\nΨ(Ω;C).\nNotice that if A=˜̺(q)A˜χ(q) for some ˜̺,˜χ∈C∞\n0(Ω;C) and aj=˜̺(q)ajfor all j∈N, the above\ncondition is reduced to a−/summationtextJ\nj=0aj∈SmJ+1(Ω;C) with aindependent of ̺. Moreover the localized\ndefinition means that we can always consider this simpler cas e.\nThe previous definition is justified by the following standar d result.\nProposition E.11. For any sequence aj∈S(Ψmj,gΨ)with mjdecreasing and lim j→∞mj= −∞ ,\nthere exists a∈S(Ψm0,gΨ)such that a−/summationtextJ\nj=0a∈S(ΨmJ+1,gΨ).\nForA∈L(SΩ−comp(Ω×Rd;C);S′\nΩ−loc(Ω×Rd;C)),A∼/summationtext∞\nj=0aj(x,Dx)is equivalent to the apparently\nweaker condition A−/summationtextJ\nj=0aj(x,Dx)∈L(˜W−µJ\nΩ−comp(Ω×Rd;C);˜WµJ\nΩ−loc(Ω×Rd;C))with limJ→∞µJ=−∞ .\nProof. The first statement can be reduced to the case where mj=m0−jby putting together bn=/summationtext\nm0−n−1<mj≤m0−naj. Then use the standard Borel summation in S(Ψm0,gΨ) by taking a=/summationtext∞\nn=0(1−\n66χ)/parenleftig\nΨ\nNn(1+pΨm0−n,n(bn))/parenrightig\nbnforχ∈C∞\n0(R;[0,1]) equal to 1 in a neighborhood of 0 and the sequence ( Nn)n∈N\nbeing increasing fast enough such that for every k∈N,\n∞/summationdisplay\nn=kpΨm0−k,k/bracketleftbigg\n(1−χ)/parenleftbiggΨ\nNn(1+pΨm0−n,n(bn)/parenrightbigg\nbn/bracketrightbigg\n<+∞ .\nFor the second statement, fix ̺,χ∈C∞(Ω;[0,1]) with χ≡1 in a neighborhood of ̺. Then take a̺∈\nS(Ψm0,gΨ) such that a̺−/summationtextj\nj=0̺(q)aj∈S(ΨmJ+1,gΨ) . For any J∈N, the difference D=̺(q)Aχ(q)−\na̺(x,Dx)χ(q) equals\nD=̺(q)Aχ(q)−a̺(x,Dx)χ(q)=̺(q)(A−J/summationdisplay\nj=0aj(x,Dx))χ(q)+(a̺(x,Dx)−mj/summationdisplay\nj=0(̺(q)aj)(x,Dx))χ(q)\nand belongs to L(˜W−µJ(R2d;C);˜Wµj(R2d;C)) for all J∈Nwith D=˜̺(q)D˜χ(q) for some pair ˜̺,˜χ∈\nC∞\n0(Ω;[0,1])⊂C∞(Rd;[0,1]) . This implies that D=̺(q)Aχ(q)−a̺(x,Dx)χ(q) is continuous from\nS′(R2d;C) toS(R2d;C) and with D=˜̺(q)D˜χ(q) it means D∈R(Ω;C) .\nIn particular when Ak∈OpSmk\nΨ(Ω;C) with Ak=ak(x,Dx)χk(q)+Rk,ak∈Smk\nΨ,Ω−comp(Ω;C) we can\nwrite as usual\nA1◦A2∼/summationdisplay\nα∈N2d1\ni|α|α!∂α\nξa1∂α\nxa2.\nWith the previous localization method, the global differen tial calculus of Subsection E.2 is well\ndefined with all its properties, if the following two conditi ons are satisfied:\n• the class of symbols Sm\nΨ,Ω−comp(Ω;C)⊂Sm\nΨ,Rd−comp(Rd;C) is sent onto Sm\nΨ,φ(Ω)−comp(φ(Ω;C)⊂\nSm\nΨ,Rd−comp(Rd;C) by the canonical transformation Φ∗:R4d\nq,p,ξ,η→R4d\nq,p,ξ,ηinduced by a diffeo-\nmorphism φ:Rd→RdwithΦ(q,p)=(φ(q),tdφ(q)−1.p) andΦ∗:T∗R2d→T∗R2d;\n• when UΦis the unitary transform in L2(R2d,dqdp ;C) given by\n(UΦu)(q,p)=u◦Φ(q,p)=u(φ(q),tdφ(q)−1.p)\nsatisfies UΦa(x,Dx)U−1\nΦ∼/summationtext∞\nn=0bn(x,Dx) in OpSm\nΨ(Ω,C) with bn∈Sm−n\nΨ,Ω−comp(Ω;C) ,b0=a◦Φ∗=\nΦ∗ainSm\nΨ,Ω−comp(Ω;C) , for any a∈Sm\nΨ,φ(Ω)−comp(φ(Ω);C) .\nIn particular all the sets Ω×Rd\npandΩ×R3d\np,ξ,ηcan be replaced by T∗ΩandT∗(T∗Ω) , with a natural\ngeometrical meaning.\nActually we will consider more general changes of variables onR2dwhich can be viewed as vector\nbundle isomorphisms of R2d=T∗Rd. We consider the following change of variables\n(˜q,˜p)=Φ(q,p)=(φ(q),L(q).p) (E.4)\nwhich is a C∞-diffeomorphism, a bijection such that dφ(q) and L(q) belong to GL d(R) for all qwith\nthe following estimates\n∀α∈Nd,∃Cα>0,/ba∇dbl∂α\nqdφ/ba∇dblL∞+/ba∇dbl∂α\nq(dφ)−1/ba∇dblL∞+/ba∇dbl∂α\nqL/ba∇dblL∞+/ba∇dbl∂α\nqL−1/ba∇dblL∞≤Cα. (E.5)\n67Note that Φ−1takes the same form with\nΦ−1(˜q,˜p)=(q,p)=(φ−1(˜q),[L(φ−1(˜q))]−1˜p).\nIt is given by a change of variable φ:Rd\nq→Rd\nqwhen L(q)=tdφ(q)−1.\nIts differential is given by\ndΦ(q,p)./parenleftbiggtq\ntp/parenrightbigg\n=/parenleftbiggdφ(q).tq\n(dL(q).p).tq+L(q).tp/parenrightbigg\n=/parenleftbiggdφ(q) 0\ndL(q).p L (q)/parenrightbigg/parenleftbiggtq\ntp/parenrightbigg\n.\nand\ntdΦ(q,p)−1=/parenleftbiggtdφ(q)−1−tdφ(q)−1t[L(q).p]tL(q)−1\n0tL(q)−1/parenrightbigg\n=/parenleftbiggL1(q) (L2(q).p)\n0 L3(q)/parenrightbigg\n,\nwhere all linear maps L1,L2and the bilinear map L2have uniformly bounded derivatives of any\norder with respect to q∈Rd.\nThe canonical transformation Φ∗:T∗R2d→T∗R2dis thus given by\n\n˜q\n˜p\n˜ξ\n˜η\n=Φ∗\nq\np\nξ\nη\n=\nφ(q)\nL(q).p\nL1(q).ξ+L2(q).p.η\nL3(q).η\n\nWith the transformation Φgiven by (E.4) we associate the unitary transform UΦinL2(R2d,dqdp ;C):\n(UΦu)(q,p)=|det(dφ(q))det( L(q))|1/2u(φ(q),L(q)p)=J1/2(q)u(φ(q),L(q)p). (E.6)\nLet us consider the simplest versions of spaces of functions .\nProposition E.12. For the tranformation Φgiven by (E.4) the following properties hold for any s∈R:\ni)The operator UΦ( resp. U−1\nΦ) is continuous from S(R2d;C)and from S′(R2d;C)onto itself. For\nany open subset Ω∈Rdit is continuous from S†\nφ(Ω)−•(φ(Ω)×Rd;C)(resp. S†\nΩ−•(Ω×Rd;C)) onto\nS†\nΩ−•(Ω×Rd;C)(resp. S†\nφ(Ω)−•(φ(Ω)×Rd;C)) where, with respective correspondence, S†stands\nforSorS′and•means locorcomp .\nii)The map a/mapstoc〈a∇→aΦ=a◦Φ∗is continuous from S(Ψs,gΨ)onto itself. For any open subset ΩinRd\nand any m∈R, it is continuous from Sm\nΨ,φ(Ω)−comp(Φ(Ω);C)onto Sm\nΨ,Ω−comp(Ω;R).\nProof. i)It suffices to consider J−1/2(q)UΦbecause |∂α\nqJ±1/2|≤Cαfor all α∈Nd. We write\n∂qi(J−1/2UΦu)(q,p)=∂qiφj(q)(∂qiu)(φ(q),L(q).p)+(∂qiL(q).p)j∂pju(φ(q),L(q).p)\n∂pj(J−1/2UΦu)(q,p)=L(q)j\nk(∂pku)(φ(q),L(q).p)\n1\nCΦ(1+|q|2+|p|2)≤1+|φ(q)|2+|L(q).p|2≤CΦ(1+|q|2+|p|2),\nwhere the last inequality is a consequence of 1 +|φ(q)|≤Cφ(1+|q|2) owing to |dφ|≤C0and its reverse\ninequality for φ−1.\nii)The formula for a◦Φ∗is\na◦Φ∗=a(φ(q),L(q).p,L1(q).ξ+L2(q).p.η,L3(q).η).\n68Let us first compare ΨandΨ◦Φ∗:\nΨ2◦Φ∗(q,p,ξ,η)=1+|L(q).p|4+|L1(q).ξ+L2(q).p.η|2+/vextendsingle/vextendsingleL3(q).η/vextendsingle/vextendsingle4\n≤CΦ(1+|p|4+|ξ|2+|p|2|η|2+|η|4)\n≤2CΦΨ2(q,p,ξ,η).\nApplied to Ψ2◦(Φ−1)∗, this provides the equivalence\n1\nC′\nΦΨ≤Ψ◦(Φ±1)∗≤C′\nΦΨ.\nThe operators ∂qi,∂pi,∂ξi,∂ηiapplied to a◦Φ∗are equivalent to the following C∞\nb(Rd\nq;R) linear\ncombinations (abbreviated as L.C) of elementary operators acting on a:\n∂qi:L.C of ∂qj,pk∂pj,ξi∂ξj,piηj∂ξℓ,∂ηj, which all are continuous from S(Ψs,gΨ) toS(Ψs,gΨ) .\n∂pi:L.C. of ∂pj,ηj∂ξk, which are continuous from S(Ψs,gΨ) toS(Ψs−1/2,gΨ) .\n∂ξi:L.C. of ∂ξjwhich are all continuous from S(Ψs,gΨ) toS(Ψs−1,gΨ) .\n∂ηi:L.C. of pj∂ξkand of ∂ηiwhich are all continuous from S(Ψs,gΨ) toS(Ψs−1/2,gΨ) .\nThis proves the continuity of a/mapstoc〈a∇→a◦Φ∗from S(Ψs,gΨ) toS(Ψs,gΨ) .\nLet us consider the functoriality of the transformation of t he quantization rule a/mapstoc〈a∇→a(x,Dx)χ(q)+\nRwith a∈Sm\nΨ,φ(Ω)−comp(Ω;C) ,χ≡1 in a neighborhood of φ(Ω)−supp aandR∈R(φ(Ω);C) .\nProposition E.13. For any A=a(x,Dx)χ(q)+R∈OpSm\nΨ(φ(Ω);C), the operator UΦAU∗\nΦis equal to\nbΦ(x,Dx)χ(Φ(q))+RΦwith RΦ∈R(φ(Ω);C)and bΦ∈Sm\nΨ,Ω−comp(Ω;C)and satisfies\nUΦAUΦ∼∞/summationdisplay\nn=0bn(x,Dx)\naccording to Definition E.10 with b0=a◦Φ∗. More precisely when Ωis a bounded open subset , with\nSm\nΨ,Ω−comp(Ω;C)⊂S(Ψm,gψ)and Sm\nΨ,φ(Ω)−comp(φ(Ω);C)⊂S(Ψm,gψ),\nbΦ=N−1/summationdisplay\nn=0bn+rΦ,N(a)\nwith a continuous map rΦ,N:S(Ψm,gψ)→S(Ψm−N,gψ)for every N∈N.\nRemark E.14. Except for the principal symbol this result does not say that the transformation a/mapstoc〈a∇→bΦ\ncorresponds bΦ=a◦Φ∗. It works exactly only for functions a(q,p)and in particular for the cut-\noff functions with respect to q. But when Ωis a bounded open subset of Rd,UΦa(x,Dx)χ(q)U−1\nΦ=\nbφ(x,Dx)χ(φ(q))defines a continuous operator from Sm\nΨ,φ(Ω)−comp(φ(Ω);C)toSm\nΨ,Ω−comp(Ω;C).\nProof. With the localization we can assume Ω=Rd,a∈S(Ψm;gΨ) ,Rd−supp acompact, and R=0 .\nWe introduce another cut-off function ˜χ∈C∞\n0(Rd;[0,1]) equal to 1 on Φ(Ω) when Ωis bounded and,\n69for a more general choice of Ω, equal to 1 in a neighborhood of supp χ.\nBecause the function J±1/2(q)∈S(1,gΨ) , the problem is reduced to the study of the operator\n(J−1/2(q)UΦ)˜χ(q)a(x,Dx)˜χ(q)(J−1/2(q)UΦ)−1\nof which the Schwarz kernel is given by the oscillating integ ral\nK(x,y)=ˆ\nR2dei[ξ.(φ(q)−φ(q′))+η.(L(q).p−L(q′).p′)]˜χ(φ(q))[a(φ(q),L(q).p,ξ,η)]˜χ(φ(q′))dξdη\n(2π)2d.\nMetrics and cut-off: OnR6d\nx,x′,Ξwith x=(q,p) ,x′=(q′,p′) andΞ=(ξ,η) we consider the metrics\nGf=dq2+dq′2+dp2\nf+dp′2\nf+dξ2\nf2+dη2\nf\nfor f=Ψ(q,p,ξ,η)=(1+|ξ|2+|p|4+|η|4)1/2and f=/tildewideΨ=(1+|ξ|2+|p|4+|p′|4+|η|4)1/2.\nThe metrics GΨandG/tildewideΨare slow (same proof as for gΨ) . But the slowness of gψimplies\n/parenleftbiggΨ(q,p,ξ,η)\nΨ(q′,p′,ξ,η)/parenrightbigg±1\n≤CΨwhen|p−p′|≤C−1\nΨΨ(q,p,ξ,η)1/2\nand therefore with the above notations\n|p−p′|≤1\nC′\nΨΨ1/2⇒/parenleftbiggΨ\n˜Ψ/parenrightbigg±1\n≤C′\nΨ\nfor some large enough constant C′\nΨ≥1 , and\nc∈S(Ψm,GΨ)⇔c∈S(/tildewideΨm,G˜Ψ) when supp c⊂/braceleftigg\n(x,y,Ξ)∈R6d,|p−p′|<1\nC′\nΨΨ1/2/bracerightigg\n.\nForθ∈C∞\n0(R;[0,1]) and ε>0 ,ε≤1\nC′\nΨfixed later consider the two cut-off functions\nΘ1(x,x′,Ξ)=Θ1(q,q′)=θ/parenleftig|q−q′|2\nε2/parenrightig\nand Θ2(x,x′,Ξ)=θ/parenleftig|p−p′|2\nε2Ψ/parenrightig\n.\nBy looking at the region 2n+10≤/tildewideΨ≤2n+12contained in a fixed shell for the rescaled variable\n(˜p,˜p′,˜ξ,˜η)=(2−n/2p,2−n/2p′,2−nξ,2−n/2η) , a homogeneity argument gives Θ2∈S(1,G˜Ψ)∩S(1,GΨ) .\nThe following properties become obvious when a∈S(Ψm,gΨ)\nb˜χ,φ=˜χ(φ(q))[a(φ(q),L(q).p,ξ,η)]˜χ(φ(q′))∈S(Ψm,GΨ),\nΘ1,Θ2,1−Θ2∈S(1,G/tildewideΨ)∩S(1,GΨ).\nWe now write the kernel K(x,x′) or the operator K:SRd−loc(R2d;C)→S′\nRd−comp(R2d;C) as\nK=Kdiag+K1+K2\nwith K1(x,x′)=ˆ\nR2dei[ξ.(φ(q)−φ(q′))+η.(L(q).p−L(q′).p′)](1−Θ1(q,q′))b˜χ,φ(x,x′,Ξ)dξdη\n(2π)2d\nK2(x,x′)=ˆ\nR2dei[ξ.(φ(q)−φ(q′))+η.(L(q).p−L(q′).p′)]Θ1(q,q′)(1−Θ2(x,x′,Ξ))b˜χ,φ(x,x′,Ξ)dξdη\n(2π)2d\nKdiag( x,x′)=ˆ\nR2dei[ξ.(φ(q)−φ(q′))+η.(L(q).p−L(q′).p′)]Θ1(q,q′)Θ2(x,x′,Ξ)b˜χ,φ(x,x′,Ξ)dξdη\n(2π)2d,\n70and use the same symbol Kdiag,K1,2for the associated operators SRd−loc(R2d;C)→S′\nRd−comp(R2d;C)\nwith uniformly controlled supports.\nNon stationary phase in q:For a given k∈N,N≥Nk,dintegrations by parts with\n/bracketleftig1\n|φ(q)−φ(q′)|2(φ(q)−φ(q′)).Dξ/bracketrightigN\neiξ.(φ(q)−φ(q′))=eiξ.(φ(q)−φ(q′)),\nand the lower bound\n∀(q,q′)∈supp(1−Θ1)∩V2\n˜χ,|φ(q)−φ(q′)|≥1\nC˜χ,φ, (E.7)\nforε≤1\nC˜χ,φ, small enough, and V˜χa compact neighborhood of supp ˜χ, implies that for all ( αj,βj,γj)∈\nN3d, 2|αj|+|βj|+|γj|≤k, for j=1,2 , the kernel\n(−1)α2+γ2∂α1qpβ1∂γ1\np∂α2\nq′(p′)β2∂γ2\np′K1(x,x′)\nof (∂α1qpβ1∂γ1\np)◦K1◦(∂α2qpβ2∂γ2\np) belongs to L2(R4d,dqdpdq′dp′;C) . Actually the estimates with powers\nofp′is deduced from our estimates with powers of pvia integration by parts with\nDηeiη.(L(q).p−L(q′).p′)=(L(q).p−L(q′).p′)eiη.(L(q).p−L(q′).p′).\nWe deduce that K1is Hilbert-Schmidt and therefore bounded operator from ˜W−k(R2d;C) to ˜Wk(R2d;C)\nfor any k∈C. With the fixed compact Ω-support, K1∈L(S′(R2d;C);S(R2d;C)) . It has a symbol in\nS(R4d;C)⊂S(Ψ−∞,gψ) with a compact support in Ω.\nKuranishi’s trick: Write\nξ.(φ(q)−φ(q′))+η.(L(q)p−L(q′)p′)=(ξ,η).(Φ(x)−Φ(x′))=(ξ,η)./bracketleftiggˆ1\n0dΦ((1−t)x+tx′)dt/bracketrightigg/parenleftbiggq−q′\np−p′/parenrightbigg\nand remember with xt=(1−t)x+tx′,qt=(1−t)q+tq′,pt=(1−t)p+tp′\nˆ1\n0dΦ(xt)dt=ˆ1\n0/parenleftbiggdφ(qt) 0\ndL(qt).ptL(qt)/parenrightbigg\ndt=/parenleftigg´1\n0dφ(qt)dt 0\n[´1\n0(1−t)dL(qt)dt].p+[´1\n0tdL(qt)dt]p′´1\n0L(qt)dt/parenrightigg\nBecause ( R2d\nq,q′−supp K)⊂supp ˜χ×supp ˜χ, we can fix ε≤1\nC˜χ,φsmall enough so that the inequalities\n∀(q,q′)∈suppΘ1∩V2\n˜χ,∀A∈Conv( dφ([q,q′]))∪Conv( L([q,q′])),|det(A)|≥1\nC˜χ,Φ, (E.8)\nwhere Conv( M([q,q′])) stands for the convex hull in Md(C) of the set M([q,q′])={M(qt,t∈[0,1])}⊂\nMd(C) , while (E.7) remains valid. We obtain for ( q,q′)\n[ξ.(φ(q)−φ(q′))+η.(L(q).p−L(q′).p′)]=/bracketleftbigg/parenleftbiggA(q,q′)B(q,q′).p+C(q,q′).p′\n0 D(q,q′)/parenrightbigg/parenleftbiggξ\nη/parenrightbigg/bracketrightbigg\n·/parenleftbiggq−q′\np−p′/parenrightbigg\nwith E(x,x′)=/parenleftbiggA(q,q′)B(q,q′).p+C(q,q′).p′\n0 D(q,q′)/parenrightbigg−1\n=/parenleftbiggA(q,q′)−1B′(q,q′).p+C′(q,q′).p′\n0 D(q,q′)−1/parenrightbigg\nand A,B,C,D,A−1,B−1,C′,D′∈S(1,dq2+dq′2;Md(R)).\n71We obtain\nK2(x,x′)=ˆ\nR2dei[ξ.(q−q′)+η.(p−p′)][Θ1(1−Θ2)b˜χ,φ](x,x′,E(x,x′).Ξ)|det(E−1)(q,q′)|dξdη\n(2π)2d,\nKdiag(x,x′)=ˆ\nR2dei[ξ.(q−q′)+η.(p−p′)][Θ1Θ2b˜χ,φ](x,x′,E(x,x′).Ξ)|det(E−1)(q,q′)|dξdη\n(2π)2d,\nwhere choosing ε≤1\nCψ,φ,˜χ, small enough, ensures\nΘ1(x,x′,E(x,x′).Ξ)=θ/parenleftig|q−q′|2\nε2/parenrightig\n=Θ1(q,q′)∈S(1,G/tildewideΨ),\nΘ2(x,x′,E(x,x′).Ξ)=θ/parenleftbigg|p−p′|2\nε2Ψ(0,p,A(q,q′).ξ+B′(q,q′).p.η+C′(q,q′).p′.η,D(q,q′)−1.η)/parenrightbigg\n∈S(1,G/tildewideΨ),\nb˜χ,φ(x,x′,E(x,x′).Ξ)=˜χ(q)˜χ(q′)a(φ(q),L(q).p,A(q,q′).ξ+B′(q,q′).p.η+C′(q,q′).p′.η,D(q,q′)−1.η)\n[(1−Θ2)b˜χ,φ](x,x′,E(x,x′).Ξ)∈S(/tildewideΨ|m|,dx2+dx′2+dΞ2),[Θ2b˜χ,φ](x,x′,E(x,x′).Ξ)∈S(/tildewideΨm,G/tildewideΨ).\nActually it suffices to check\nΨ2(0,p,E(x,x′).Ξ)≤C(1+(|ξ|+|p||η|+|p′||η|)2+|p|4+|p′|4+|η|4)≤C′/tildewideΨ2\nwith the symmetric version by applying the same result to Ψ2(0,p,E(x,x′)−1.Ξ) and then to use the\nequivalence/parenleftig/tildewideΨ\nΨ/parenrightig±1\n≤Cεwhen|p−p′|≤ε/tildewideΨ1/2owing to the slowness of G/tildewideΨ.\nNon stationary phase in p:Despite the bad a priori estimate of [(1 −Θ2)b˜χ,φ](x,x′,E(x,x′).Ξ) ,\nN≥Nk,dintegrations by parts for a given k∈Nwith\n/parenleftbigg1\n|p−p′|2(p−p′).Dη/parenrightbiggN\nei[ξ.(q−q′)+η.(p−p′)]=ei[ξ.(q−q′)+η.(p−p′)]\nand∀(x,x′)∈supp(1−Θ2)(.,.,E(.,.).),|p−p′|≥1\nCΨ,φ,˜χ/tildewideΨ1/2,\nleads to the property that the kernel of ( ∂α1qpβ1∂γ1\np)◦K2◦(∂α2qpβ2∂γ2\np) is Hilbert-Schmidt and therefore\nbounded in L2(R2d,dqdp ;C) for|αj|+|βj|+|γj|\n2≤k. We conclude as we did for K1that K2belongs to\nL(S′(R2d;C);S(R2d;C)) . It has a symbol in S(R4d;C)⊂S(Ψ−∞,gΨ) with a compact support in Ω.\nGauss transform : The kernel of Kdiagcan be written\nKdiag(x,x′)=b(x,Dx) with b(x,Ξ)=eiDΞ.Dx′/bracketleftbig\n[Θ1Θ2b˜χ,φ](x,x′,E(x,x′)Ξ)/bracketrightbig/vextendsingle/vextendsingle\nx=x′,\nwhere the metric G/tildewideΨis slow on R6d, the B-dual metric of G/tildewideΨforB=\n0 0 0\n0 01\n2IdR2d\n01\n2IdR2d 0\nis the de-\ngenerate metric GB=/tildewideΨ2dq′2+/tildewideΨdp′2+dξ2+/tildewideΨdη2. Fortunately G/tildewideΨisGB-temperate along the vector\nspace V0=/braceleftbig\n(x,x′,Ξ)∈R6d,x=x′/bracerightbig\nand/tildewideΨ/vextendsingle/vextendsingle\nV0can be replaced with Ψ. Theorem 18.4.11 of [HormIII]\ntells us that b∈S(Ψm,gΨ) with the asymptotic expansion\nb(x,Dx)∼∞/summationdisplay\nn=0bn(x,Dx)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nbn∈S(Ψm−n,gψ)\nand the first term b0(x,Ξ)=bφ,˜χ(x,x,Ξ)=a◦Φ∗(x,Ξ) .\n72Remark E.15. We could have used the general theory of global Fourier integ ral operators of J.M. Bony\nin [Bon]. At least when φ−IdRdand L−IdRdhave a compact support, this describes UΦas a Fourier\nintegral operator of which the global symbol is a section of t he fiber bundle of affine metaplectic opera-\ntorsM→Gφabove the graph of φ∗,GΦ=/braceleftbig\n(X,Φ∗(X)),X∈R4d/bracerightbig\nwith a value above (X0,Φ(X0)),\nx0=(q0,p0,ξ0,η0)which is the composition τφ(X0)UX0τ−x0where τ(x1,Ξ1)is the phase translation\nei(Ξ1.Dx−x1.DΞ)andUX0is a metaplectic representation of the linear symplectic ma pdΦ∗(X0).\nThe proposed methods mimics the classical techniques of pse udo-differential calculus for the metric\ndq2+dξ2\n〈ξ〉2modulo the localization process only in the q-variable and for specific linear transformations\nin the p-variable. It is informative from this point of view and prov ides a more explicit formulation\nfor the functoriality of the principal symbol.\nFrom Proposition E.12 and the definition of ˜Ws\nΩ−•(Ω;C),• =loc or comp , deduced from Defini-\ntion E.5, we obtain the following result.\nProposition E.16. Consider the unitary map UΦ:L2(R2d,dqdp ;C)→L2(R2d,dqdp ;C)given by\n(E.4)(E.5)(E.6) with the additional assumption that φ−IdRdand L−IdRdhave a compact support.\nThen for any s∈Rd,UΦandU−1\nΦare isomorphisms from ˜Ws(R2d;C)into itself, with Rd\nq−supp U±1\nΦu=\nφ∓1(Rd\nq−supp u)for every u∈˜Ws(R2d;C).\nProof. The support property is obvious from the definition (E.6) of UΦ. With the additional support\nassumption on Φ−IdR2d, we can write for any u∈˜Ws(R2d;C) ,UΦu=UΦχ1(φ(q))u+χ2(q)uforχ1,χ2∈\nC∞(Rd;[0,1]) ,χ1+χ2≡1 and supp χ1compact. From the Definition E.5 and the global pseudo-\ndifferential calculus in S(Ψ2s,gψ) remember\nC−1\nsRe〈u,M2s(x,Dx)u〉L2≤/ba∇dblu/ba∇dbl2\n˜Ws=/ba∇dblMW\ns(x,Dx)u/ba∇dbl2\nL2≤CsRe〈u,M2s(x,Dx)u〉L2,\nforMs=(Cs+Ψ|s|)signswith Cs≥1 large enough.\nWe deduce\n/ba∇dblUΦχ1(φ(q))u/ba∇dbl2\n˜Ws≤CsRe〈u,U∗\nφ(χ1(q)M2s)(x,Dx)UΦχ1(φ(q))u〉.\nBy Proposition E.13 we know\nU∗\nφ(χ1(q)M2s)(x,Dx)UΦχ1(φ(q))=b2s(x,Dx)◦χ1(φ(q))=cW\n2s(x,Dx)b2s,c2s∈S(Ψ2s,gΨ),\nand the pseudo-differential calculus in S(Ψ2s,gΨ) gives\n/ba∇dblUΦχ1(φ(q))u/ba∇dbl2\n˜Ws≤C′\ns/ba∇dblu/ba∇dbl2\n˜Ws.\nBy the triangular inequality UΦ:˜Ws(R2d;C)→˜Ws(R2d;C) is continuous and we conclude with U−1\nΦ=\nUΦ−1.\nAll this section gives a meaning to S†\nΩ−•(T∗Ω;C) ,Sm\nΨ,Ω−comp(Ω;C) ,R(Ω;C) , OpSm\nΨ(Ω;C) and\n˜Ws\nΩ−•(T∗Ω;C) ( with S†=SorS′and•meaning loc or comp) when Ωis a chart open set in the\ncompact manifold Q.\nThe topology, the continuity properties and the global elli pticity that we need will be better discussed\nin the global setting which avoids considerations of induct ive limit topologies.\n73E.4 Globalization on Qand applications\nLet us fix, an atlas covering of Q,Q=/uniontextJ\nj=1Ωjsuch that ˜Ωj=/uniontext\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇essΩj′is still a chart open set.\nWe take a finite partition of unity/summationtextJ\nj=1̺j(q)≡1 subordinate with the atlas covering Q=/uniontextJ\nj=1Ωj\nand cut-off functions χj∈C∞\n0(Ωj;[0,1]) such that χ1≡1 in a neighborhood of supp ̺j. Notice that\nχjχj′/negat〉onslas〈=0 implies Ωi∩Ωj/negat〉onslas〈=0 and in this case ̺j,̺j′,χj,χj′are cut-off functions in the coordinate\ncharts ˜Ωjand ˜Ωj′.\nThe spaces Sm\nΨ(Q;C) (resp. /tildewiderWs(T∗Q;C)) are the sets a=/summationtextJ\nj=1̺j(q)awith ̺j(q)a∈Sm\nΨ,Ωj−comp(Ωj;C)⊂\nS(Ψm,gΨ) (resp. ̺ja∈/tildewiderWs\nΩj−comp(T∗Ωj;C)) with the topologies given by\npm,k(a)=J/summationdisplay\nj=1p˜Ωj,Ψm,k(̺j(q)a),k∈N\nresp. /ba∇dbla/ba∇dbl2\n/tildewiderWs(Q;C)=J/summationdisplay\nj=1/ba∇dbl̺j(q)a/ba∇dbl2\n/tildewiderWs(T∗Ω;C).\nThe subscript ˜Ωjinp˜Ωj,Ψm,krefers to the choice of some local coordinates in ˜Ωj. But Proposition E.12\nand Proposition E.13 for the conjugation a/mapstoc〈a∇→UΦa(x,Dx)χ(q)U−1\nΦwithΦ(q,p)=(φ(q),tdφ(q)−1.p) ,\nsays that the seminorms p˜Ωj,Ψm,k(̺j(q)a) can be replaced by p˜Ωj′,Ψm,k(̺j(q)a) for any j′∈{1,...,J}\nsuch that Ωj⊂˜Ωj′.\nA vector bundle isomorphism on T∗Q, written locally as Φ: (q,p)/mapstoc〈a∇→(φ(q),L(q).p) with the associated\nunitary operator UΦ, gives rise to an isomorphism of the space ˜Ws(Q;C) according to the local result\nof Proposition E.16. This gives a first application, which is not exactly due to the pseudo-differential\ncalculus\nProposition E.17.\nFor any riemannian metric g=gi j(q)dqidqjonTQwith the dual metric gi j(q)dpidpjonT∗Q, if/summationtext∞\nℓ=−1θ2\nℓ(t)≡1is a quadratic dyadic partition of unity like (4.2) and|p|2\nq=gi j(q)pipj, then for every\ns∈Rthe squared norm /ba∇dblu/ba∇dbl2\n˜Ws(Q;C)is equivalent to/summationtext∞\nℓ=−1/ba∇dblθℓ(|p|2\nq)u/ba∇dbl2\n˜Ws(Q;C).\nProof. It suffices to use the local gauge transform given by Φ: (q,p)/mapstoc〈a∇→(q,g−1/2(q).p) with|p|2\nq=\ntpg−1(q)p=|g−1/2(q).p|2and to write\n∞/summationdisplay\nℓ=−1/ba∇dblθℓ(|p|2\nq)u/ba∇dbl2\n˜Ws≍∞/summationdisplay\nℓ=−1/ba∇dblU−1\nΦθℓ(|p|2\nq)u/ba∇dbl2\n˜Ws≍∞/summationdisplay\nℓ=−1/ba∇dblθℓ(|p|2)U−1\nΦu/ba∇dbl2\n˜Ws\nand Proposition E.7 gives\n∞/summationdisplay\nℓ=−1/ba∇dblθℓ(|p|2\nq)u/ba∇dbl2\n˜Ws≍/ba∇dblU−1\nΦu/ba∇dbl2\n˜Ws≍/ba∇dblu/ba∇dbl2\n˜Ws.\nThe intersection/intersectiontext\ns∈R/tildewiderWs(T∗Q;C) is nothing but S(T∗Q;C) .\nOnR(Q;C)=L(S′(T∗Q;C);S(T∗Q;C))∼S(T∗Q×T∗Q;C) , the Fréchet space topology is equiva-\nlently defined by the family of (semi)norms\nqk(R)=/ba∇dblR/ba∇dblL(/tildewiderW−k(T∗Q;C);/tildewiderWk(T∗Q;C)),k∈N.\n74Before giving an explicit family of seminorms on\nOpSm\nΨ(Q;C)=/braceleftigg\nJ/summationdisplay\nj=1(̺j(q)a)(x,Dx)◦χj(q)+R,a∈Sm\nΨ(Ω;C),R∈R(Ω;C)/bracerightigg\n,\nlet us check that any A∈OpSm\nΨ(Q;C) admits a canonical decomposition after fixing some cut-off\nfunction on Q×Q.\nAttention must be payed to the following point: although ̺j(q)aj′=̺j′(q)ajfor all pairs ( j,j′) allows\nto define a(x,Ξ)=/summationtextJ\nj′=1̺j′(q)aj′(x,Ξ) , the equality/summationtextJ\nj=1(̺j(q))aj(x,Dx)◦χj(q)−/summationtextJ\nj=1(̺j(q)a)(x,Dx)◦\nχj(q)=R∈R(Q;C) is true with R=0 only under the equality of the operators\n(̺j′(q)aj)(x,Dx)◦χj′(q)=(̺j(q)aj′)(x,Dx)◦χj(q)\nfor all pairs ( j,j′) .\nWith the subset ˜Ωj=/uniontext\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇essΩj′take a cut-off function ˜χj∈C∞\n0(˜Ωj;[0,1]) such that ˜χj≡1 on a\nneighborhood of/uniondisplay\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇esssupp χj⊃/uniondisplay\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇esssupp ̺j\nBecause for all j∈{1,...,J},̺j(q)χj(q′) and ˜χj(q)1Q\\˜Ωj(q′) vanish in a neighborhood of the diagonal\n∆Q={(q,q),q∈Q}, there exists Θ1∈C∞(Q×Q;[0,1]) such that Θ1≡1 in a neighborhood of ∆Qand\n̺j(q)Θ1(q,q′)=̺j(q)Θ1(q,q′)χj(q′)=̺j(q)˜χj(q)Θ1(q,q′)χj(q)\n=̺j(q)/bracketleftigg\nJ/summationdisplay\nj′=1̺j′(q)˜χj(q)/bracketrightigg\nΘ1(q,q′)χj(q′)\n=̺j(q)/bracketleftigg\nJ/summationdisplay\nj′=1̺j′(q)˜χj′(q)Θ1(q,q′)/bracketrightigg\nχj(q′), (E.9)\nwhere the equalities hold as multiplication operators on S′(T∗Ωj×T∗Ωj;C) . Additionally the func-\ntionΘ1can be chosen symmetric: Θ1(q,q′)=Θ1(q′,q) , and we set\nΘ2(q,q′)=1−Θ1(q,q′).\nFor any K∈L(S(T∗Q;C);S′(T∗Q;C)) , identified with its Schwartz kernel K(x,x′)∈S′(T∗Q×\nT∗Q;C) , we set\nKdiag(x,x′)=Θ1(q,q′)K(x,x′),Koff(x,x′)=Θ2(q,q′)K(x,x′),K=Kdiag+Koff. (E.10)\nNotice that K/mapstoc〈a∇→(Kdiag,Koff) is an isomorphism between S′(T∗Q×T∗Q;C) and the closed set of\nS′(T∗Q×T∗Q;C)×S′(T∗Q×T∗Q;C)\n/braceleftbig\n(K1,K2)∈S′(T∗Q×T∗Q;C)×S′(T∗Q×T∗Q;C),Θ1(q,q′)K2(x,x′)−Θ2(q,q′)K1(x,x′)=0/bracerightbig\n.\nWith (E.9), we have the additional properties\nKdiag=J/summationdisplay\nj=1̺j(q)◦Kdiag=J/summationdisplay\nj=1̺j(q)◦/bracketleftigg\nJ/summationdisplay\nj′=1(̺j′˜χj′)(q)◦K/bracketrightigg\ndiag◦χj(q)\nandJ/summationdisplay\nj=1/bracketleftbig\n(̺j(q)a)(x,Dx)◦χj(q)/bracketrightbig\ndiag=J/summationdisplay\nj=1̺j(q)◦/bracketleftigg\nJ/summationdisplay\nj′=1[(̺j′˜χj′)(q)a](x,Dx)/bracketrightigg\ndiag◦χj(q)\nfor some ˜χj∈C∞\n0(Ωj;[0,1]) such that ˜χj≡1 in a neighborhood of supp χj.\n75Proposition E.18. Every A=/summationtextJ\nj=1(̺j(q)a)(x,Dx)χq+R∈OpSm\nΨ(Q;C)admits a unique decomposition\nA=J/summationdisplay\nj=1(̺j(q)aA(x,Dx))χj(q)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=Adiag+RA/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=Aoff,\nwith aA∈Sm\nΨ(Q;C)and RA∈R(Q;C).\nAdditionally this provides a topological direct sum on OpSm(Q;C), because the map\nSm\nΨ(Q;C)×R(Q;C)→Sm\nΨ(Q;C)×R(Q;C)\n(a,R)/mapstoc〈a∇→(aA,RA),A=J/summationdisplay\nj=1(̺j(q)a)(x,Dx)χj(q)+R\nis continuous when Sm\nΨ(Q;C)×R(Q;C)is endowed with the seminorms (pm,k(a)+qk(R))k∈N.\nRemark E.19. When Ais a differential operator or more generally a local operato r with respect to\nq-variable, then we can write A=Adiagwith a vanishing remainder RA=0.\nProof. The decomposition of A=/summationtextJ\nj=1(̺j(q)a)(x,Dx)◦χj(q)+R=Aa+R\nA=(Aa+R)diag+(Aa+R)off=Aa,diag+Rdiag+Aa,off+Roff,\nwith Roff(x,x′)=Θ2(q,q′)R(x,x′)∈S(T∗Q×T∗Q;C),\nAa,off(x,x′)=J/summationdisplay\nj=1Aj,off(x,x′),Aj,off(x,x′)=Θ2(q,q′)[(̺j(q)a)(x,Dx)◦χj(q)](x,x′)\nRdiag=J/summationdisplay\nj=1̺j(q)◦/bracketleftigg\nJ/summationdisplay\nj′=1(̺j′˜χj′)(q)◦R/bracketrightigg\ndiag◦χj(q),\nand Aa,diag=J/summationdisplay\nj=1̺j(q)◦/bracketleftigg\nJ/summationdisplay\nj′=1[(̺j′˜χj′)(q)a](x,Dx)/bracketrightigg\ndiag◦χj(q).\nThe kernel of Aj,offwith coordinates in Ωjis\nˆ\nR2dei[ξ.(q−q′)+η.(p−p′)Θ2(q,q′)̺j(q)a(q,p,ξ,η)χj(q′)dξdη\n(2π)d.\nA non stationary phase argument with(q−q′)\n|q−q′|2Dξei[ξ.(q−q′)+η.(p−p′)=ei[ξ.(q−q′)+η.(p−p′)]with the factors\nϕj(q)θj(q′) implies that the map a/mapstoc〈a∇→Aj,off(x,x′) is continuous from S(Ψm,gΨ) toS(R4d;C) . Again\nwith the controlled support, the map a/mapstoc〈a∇→Aj,offis continuous from Sm\nΨ(Q;C) toR(Q;C) .\nThis proves that the map\n(a,A)/mapstoc〈a∇→Aoff=Roff+J/summationdisplay\nj=1Aj,off\nis continuous from Sm\nΨ(Q;C)×R(Q;C) toR(Q;C) .\nFor the diagonal part, let us first notice at the operator leve l that the sum with respect to j′is\nintroduced for\n̺j′(q)/bracketleftbig\n(̺j˜χj)(q)M/bracketrightbig\ndiag=̺j′(q)̺j(q)˜χj(q)˜χj′(q)[M]diagχj(q)χj′(q)\n76with M=RorM=(˜χj˜χj′(q)a)(x,Dx) .\nIt remains to identify these operators as the quantization o f symbols. Every term indexed by ( j,j′)\nfor a fixed j′∈{1,...,J}, has a kernel with a Q×Qsupport that is compact in ˜Ωj′טΩj′. We choose\ncoordinates in ˜Ωj′in order to compare the terms of the double sum for different v alues of j∈{1,...,J}.\nFor [( ̺j′˜χj′)(q)R]diagthe kernel\n(̺j′˜χj′)(q)R(x,x′)Θ1(q,q′)=̺j′(q)R(x,x′)Θ1(q,q′)χj′(q′)\nbelongs to S(Ωj′×Ωj×R2d;C) . With the factors ̺j′(q)χj′(q′) , it can be written aj′,reg(x,Dx) with\naj′,reg∈S(R4d;C)⊂S(Ψ−∞,gΨ) .\nForAa,diag, the kernel is localized in the same way and equals\nˆ\nR2dei[ξ.(q−q′)+η.(p−p′)]̺j′(q)a(q,p,ξ,η)Θ1(q,q′)χj′(q′)dξdη\n(2π)2d,\nWe obtain/bracketleftbig\n[̺j′˜χj′(q)a](x,Dx)/bracketrightbig\ndiag=bj′(x,Dx)\nwhere bj′is given by the Gauss transform\nbj′(x,ξ)=eiDΞ.Dx′̺j′(q)a(q,p,ξ,η)Θ1(q,q′)χj′(q′)/vextendsingle/vextendsingle\nX=X′=eiDξ.Dq′̺j′(q)a(q,p,ξ,η)Θ1(q,q′)χj′(q′)/vextendsingle/vextendsingle\nq′=q,\nwhere the variables ( p,η) are now simple parameters. We deduce that the map a/mapstoc〈a∇→bj′is continuous\nfrom S(Ψm,gΨ) toS(Ψm,gΨ) . With bj′(x,Dx)=˜χj′(q)bj′(x,Dx) we deduce that bj′∈Sm\nΨ,˜Ωj−comp(˜Ωj;C)⊂\nSm\nΨ(Q;C) .\nSo we have written\nRdiag=J/summationdisplay\nj=1(̺j(q)bj,reg)(x,Dx)◦χj(q),Adiag=J/summationdisplay\nj=1(̺j(q)bj)(x,Dx)◦χj(q)\nwith bj,reg∈S−∞\nΨ(Q;C),bj∈Sm\nΨ(Q;C)\nand∀j,j′∈{1,...,J},̺j′(q)(bj+bj,reg)=̺j(q)(bj′+bj′,reg),\nThe last identity follows the same strategy as at the operato r level except that we consider only left\nmultiplications by functions of q, which commute with the Gauss transform.\nDefinition E.20. According to Proposition E.18 the topology on OpSm\nΨ(Q;C)is equivalently defined\nby the family of (semi)norms (qm,k)k∈Nand(˜qm,k)k∈Nwith\nqm,k(A)=pm,k(aA)+qk(RA)with A=J/summationdisplay\nj=1̺j(q)aA(x,Dx)◦χj(q)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=Adiag+RA/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=Aoff\nand\n˜qm,k(A)=inf/braceleftigg\npm,k(a)+qk(R),A=J/summationdisplay\nj=1(̺j(q)a)(x,Dx)◦χj(q)+R,a∈Sm\nΨ(Q;C,R∈R(Q;C))/bracerightigg\n.\nDefinition E.21. We now write a̺,χ(x,Dx)=/summationtextJ\nj=1(̺j(q)a)(x,Dx)◦χj(q)fora∈Sm\nΨ(Q;C). A symbol\na∈Sm\nΨ(Q;C)is said to be elliptic if there exists κ≥1such that |a| ≥1\nκΨmforΨ≥κ. An operator\nA=a̺,χ(x,Dx)+R∈OpSm\nΨ(Q;C)is said to be elliptic if it admits a symbol awhich is elliptic.\n77The product of two operators A=a̺,χ(x,Dx)+R,A′=a′\n̺,χ(x,Dx)+R′∈OpSm\nΨ(Q;C) equals\nA◦A′=a̺,χ(x,Dx)◦a′\n̺,χ(x,Dx)+a̺,χ(x,Dx)◦R′+R◦a̺,χ(x,Dx)+R◦R′\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n∈R(Q;C)\nand the treatment of every non vanishing term ̺j(q)a(x,Dx)χj(q)̺j′(q)a(x,Dx)χj′(q) of the product\na̺,χ(x,Dx)◦a′\n̺,χ(x,Dx) can be studied in the chart open set ˜Ωjwith the expansion of the Gauss\ntransform eiDΞ1.Dx2a1(X1)a2(X2)/vextendsingle/vextendsingle\nX1=X2=XinS(Ψmm′,gψ) . We deduce that\nA◦A′=N/summationdisplay\nn=1bn(a,a′)̺,χ(x,Dx)+RN+1(A,A′)\nwith pm+m′−n,k(bn(a,a′))≤Cm,m′,n,kpm,ℓn,k(a)pm′,ℓn,k(A′) and the remainder estimated by\nqm+m′−N−1,k(RN+1(A,A′))≤C′\nm,m′,N+1,kqm,ℓN+1,k(A)qm,ℓN+1,k(A′),\nwhen pm,ℓ(a)≤Cm,ℓqm,ℓ(A) and pm,ℓ(a′)≤Cm,ℓqm,ℓ(A′) . According to Remark E.19, differential\noperators provide a wide family of examples where the latter condition holds true. And this can be\nextended for operators of which the Schwartz kernel is expli citly localized in a small neighborhood\nof the Q-diagonal.\nThe rough version of this continuity property says\nA◦A′=N/summationdisplay\nn=1Bn(A,A′)+RN+1(A,A′)\nwith qm+m′−n,k(Bn(A,A′))≤Cm,m′,n,kqm,ℓn,k(a)qm′,ℓn,k(A′) , and\nqm+m′−N−1,k(RN+1(A,A′))≤C′\nm,m′,N+1,kqm,ℓN+1,k(A)qm,ℓN+1,k(A′).\nSimilarly for a vector bundle isomorphism Φ:T∗Q→T∗Q, the conjugation by the associated unitary\ntransform A=a̺,χ(x,Dx)+R/mapstoc〈a∇→UΦAU−1\nΦcan be written\nUΦAU−1\nΦ=N/summationdisplay\nn=1[bn,Φ(a)]̺,χ(x,Dx)+RΦ,N+1(A)\nwith continuity estimates gathered from the local model tre ated in Proposition E.13. It can be written\nmore roughly as\nUΦAU−1\nΦ=N/summationdisplay\nn=1Bn,Φ(A)+RΦ,N+1(A)\nwith qm−n,k(Bn,Φ(A))≤CΦ,m,kqm,ℓn,k(A)\nand qm−N−1,k(RΦ,N+1(A))≤C′\nΦ,m,N+1qm,ℓN+1,k(A).\nBecause the function ( q,q′)/mapstoc〈a∇→˜χj(q)˜χj(q′)/bracketleftig/summationtext\nΩj′∩Ωj/negat〉onslas〈=/emptysetst∇ess̺j′(q)Θ1(q,q′)χj′(q′)−χj′(q)Θ1(q,q′)̺j′(q′)/bracketrightig\n, which\nis symmetric if Θ1(q,q′)=Θ1(q′,q) , has a compact support away from the diagonal ∆, the de-\ncomposition of the formal adjoint can be reduced to the local model with the formula b(x,Dx)∗=\n78[eiDΞ.Dxb](x,Dx) . The formal adjoint A∗of the operator A=a̺,χ(x,Dx)+R∈OpSm\nΨ(Q;C) can be writ-\nten\nA∗=N/summationdisplay\nn=0[bn]̺,χ(x,Dx)+RN+1(A)=N/summationdisplay\nn=0BN(A)+RN+1(A)\nwith b0=aand the estimates pm−n,k(bn)≤Cm,n,kpm,ℓn,k(a) and qm−n,k(Bn)≤Cm,n,kqm,ℓn,k(A) and\nqm−N−1,k(RN+1)≤Cm,N,kqm,ℓN+1,k(A) .\nAll the classical estimates can be decomposed in this way by g oing back to the Q=Rd-model. The\ncondition pm,ℓ(a)≤Cm,ℓqm,ℓ(A) holds true if one starts with an operator A=a̺,χ(x,Dx)+0 , exactly\ngiven by the local quantization rule with no regularizing gl obal remainder, and a/negat〉onslas〈∈S−∞\nΨ(Q;C) . For\nexample this is the case if Ais a differential operator. The estimates are then propagat ed via the\noperations .\nThe local Calderon-Vaillancourt theorem and our choice of t he norm qkonR(Q;C) gives at once the\nexistence of a kd∈Ndetermined by the dimension of Q, such that /ba∇dblA/ba∇dblL(L2)≤Cq0,kd(A) .\nSimilarly the Garding inequality says that if A∈Sm\nΨ(Q;C) has an elliptic non negative symbol a≥\n1\nκΨmforΨ≥κ, there exists Cκ≥1 and k1∈Nsuch that\n∀u∈S(T∗Q;C),Re〈u,Au〉L2≥1\nCκ/ba∇dblu/ba∇dbl2\n˜Wm/2−Cκqm,k1(A)/ba∇dblu/ba∇dbl2\n˜W(m−1)/2.\nAll these properties extend to OpSm\nΨ(Q;EndE) with the following constraint for the symbol of the\nadjoint: The reduction to a∈Sm\nΨ,Ω−comp(Ω;Md(C)) is done by chosing ˜Ωjsuch that the vector bun-\ndleE/vextendsingle/vextendsingle˜Ωjadmits an orthonormal frame ( f1\nj,...,fN\nj) for the metric gE. Then the adjoint of A=\nam,̺,χ(x,Dx)+Am−1∈OpSm\nΨ(Q;EndE) can be written\nA∗=(a∗\nm)̺,χ(x,Dx)+A′\nm−1with A′\nm−1∈OpSm−1\nΨ(Q;EndE)\nand this property is invariant by a change of orthonormal fra me.\nActually if U(q)∈UN(C) is the unitary matrix which represent another orthonormal frame ( ˜fN\nj,..., ˜fN\nj)\nin the frame ( fN\nj,...,fN\nj) for E/vextendsingle/vextendsingle˜Ωj, the symbol of am,̺,χ(x,Dx)+Am−1equals\nbm(x,Ξ)=U(q)am(x,Ξ)U−1(q)=U(q)am(x,Ξ)U∗(q) with b∗\nm(x,Ξ)=U(q)a∗\nm(x,Ξ)U∗(q).\nThe norms ( qm,k)k∈Nare very convenient for handling the ellipticity as it is don e in the case of the\nglobal pseudo-differential calculus on Rd. We focus here on the case of non negative elliptic operators .\nProposition E.22.\nLetA∈OpSm\nΨ(Q;EndE),m>0, be an elliptic operator with A=(am⊗IdE)̺,χ(x,Dx)+Am−1,am≥1\nκΨm\nforΨ≥κ,Am−1∈OpSm−1\nΨ(Q;EndE). If additionally Ais symmetric on S(T∗Q;E)then it is self-\nadjoint with D(A)=˜Wm(T∗Q;E), bounded from below, and its resolvent is compact.\nIn the case when m=2, ifA=a̺,χ(x,Dx)+R∈OpS2\nΨ(Q;EndE)fulfills the above conditions, then\nfor every f∈S(〈t〉s,dt2\n〈t〉2),s∈R, the operator f(A)−f(a2)̺,χ(x,Dx)belongs to OpS2s−1\nΨ(Q;EndE)while\nf(a2)∈S2s\nΨ(Q;EndE).\nProof. The first results are the standard ones.\nWe just show how Helffer-Sjöstrand formula can be used in thi s framework and we focus on the case\nm=2 . We write ain the form a=a2⊗IdE+a1with a1∈S1\nΨ(Q;EndE) and a2⊗IdEis simply written\n79a2.\nBy the Leibniz formula applied to 1 =(z−a2)−1×(z−a2) for z∈C\\R, the seminorms p−2,k(1\nz−a2) are\nestimated by\np−2,k/parenleftbigg1\nz−a2/parenrightbigg\n≤Ckp2,k(a)〈z〉k\n|Imz|k+1.\nBy taking Bz=/bracketleftig\n1\nz−a2+a1\n(z−a2)2/bracketrightig\n̺,χ(x,Dx)+0 , the second term of the expansion of Bz◦(z−A) , to be\nstudied in S1\nΨ(Q;EndE) , can be reduced to\n−1\n(z−a2)2a1◦a1+/summationdisplay\n|α1|+|α2|=11\nα1!α2!∂α1\nξ∂α2\nη1\n(z−a2).∂α1q∂α2p[a2+a1]\nand it is of order −2 with seminorms estimated like the seminorms p−2,k(1\nz−a2) . We deduce\nBz◦(z−A)=zBz−Bz◦A=(1−rL(z))\nwith q−2,k(rL(z))≤Ck〈z〉Nk\n|Imz|Nk+1. By left multiplying with/summationtextM\nn=0rL(z)nwe obtain\nM/summationdisplay\nn=0rL(z)n◦Bz◦(z−A)=1−rL(z)M+1,\nwith q(−M−1)2,k[rL(z)M+1]≤CM,k〈z〉NM,k\n|Imz|NM,k+1and/ba∇dblrL(z)M+1/ba∇dblL(L2;/tildewiderW(Mm−cd))≤C′\nM〈z〉NM\n|Imz|NM+1. In the right-\nhand side of\n(z−A)−1=M/summationdisplay\nn=0rL(z)n◦B(z)+(z−A)−1◦rL(z)M+1,\nall terms except the remainder term ( z−A)−1◦rL(s)M+1are known to be pseudo-differential operators\nrL(z)n◦B(z)∈OpS−2(n+1)\nΨ(Q;EndE) and q−2(n+1),k(rL(z)n◦B(z))≤Ck〈z〉Nn,k\n|Imz|Nn,k+1. But we can do the\nsame for the right-multiplication and obtain similarly:\n(z−A)−1=M/summationdisplay\nn=0Bz◦rD(z)n+rD(z)M+1(z−A)−1,\nwith the same upper bounds.\nWe deduce\n(z−A)−1=J/summationdisplay\nj=1/bracketleftbigg\n̺j(q)(1\nz−a2+a1\n(z−a2)2)/bracketrightbigg\n(x,Dx)χj(q)+2M/summationdisplay\nn=1bn(z)+rD(z)M+1(z−A)−1rL(z)M+1\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=rM(z)(E.11)\nwith q−2(n+1),k(bn)≤Cn,k〈z〉Nn,k\n|Imz|Nn,k+1,/ba∇dblrM/ba∇dblL(/tildewiderW−M+cd,/tildewiderWM−cd)≤CM〈z〉NM\n|Imz|NM+2.\nInserting (E.11) into the Helffer-Sjöstrand formula (see [ HeSj][DiSj]) gives\nf(A)=1\n2iπˆ\nC∂¯z˜f(z)(z−A)−1dz∧d¯z\nwhile f(a2)=1\n2iπˆ\nC∂¯z˜f(z)(z−a2)−1dz∧d¯z,f′(a2)=1\n2iπˆ\nC∂¯z˜f(z)(z−a2)−2dz∧d¯z,\nwith ˜f∈C∞(C;C),supp ˜f⊂/braceleftbig\nz∈C,|Imz|≤Cf〈z〉/bracerightbig\n,˜f/vextendsingle/vextendsingle\nR=f,\nand∀N∈N,∃CN>0,|∂¯z˜f(z)|≤CN|Imz|N\n〈z〉N〈z〉s−1\n80when f∈S(〈t〉s,dt2\n〈t〉) ,s<0 , we obtain by integration of the respective terms by choosi ngN≥\nmax/braceleftbig\nNn,k,NM/bracerightbig\nf(A)=[f(a2)]̺,χ(x,Dx)+[f′(a2)a1]̺,χ(x,Dx)+2M/summationdisplay\nn=1[βn]̺,χ(x,Dx)+RM\nwith βn∈S−2(n+1)\nΨ(Q;EndE) for n≥1 while f(a)∈S−2s\nΨ(Q;EndE) .\nLet us first conclude for s∈[−3/2,0[ . Take β∈S−4(Q;EndE) such that β∼/summationtext∞\nn=1βn. For every M∈N,\nf(A)=[f(a2)]̺,χ(x,Dx)+[f′(a2)a1]̺,χ(x,Dx)+β̺,χ(x,Dx)+Rβ,2M+1+RM\nwith Rβ,2M+1∈OpS−2(2M+1)\nΨ(Q;EndE) and RM∈L(/tildewiderW−M+cd;/tildewiderWM−cd) . By taking Marbritrarily large,\nthis says that f(A)−[f(a2)]̺,χ(x,Dx)−[f′(a2)a1]̺,χ(x,Dx)−β̺,χ(x,Dx) belongs to R(Q;EndE) .\nBecause s∈[−3/2,−0[ , we know f(a2)∈S2s\nΨ(Q;EndE) with 2 s≥ −3 and f′(a2)∈S2(s−1)(Q;EndE) ,\nf′(a2)a1∈S2s−1\nΨ(Q;EndE) , while β̺,χ∈S−4\nΨ(Q;EndE) ,−4≤−3−1≤2s−1 .\nNow for a general s<0 , simply write 〈t〉s=〈t〉s1n1with s1∈[−3/2,0[ and n1∈N. The composition of\npseudo-differential operators says that the principal sym bol of〈A〉s=〈A〉s1◦...◦〈A〉s1is〈a2〉s. Any\npower〈t〉s,s∈R, can be written 〈t〉2N〈t〉s′with s′<0 and N∈N. With〈t〉2N=(1+t2)N∈R[t] ,〈A〉sis\na pseudo-differential operator with principal symbol 〈a2〉sfor any s∈R. Finally a general function\nf∈S(〈t〉s;dt2\n〈t〉2) is written 〈t〉s+3/2fs(t) with fs∈S(〈t〉−3/2,dt2\n〈t〉2) .\nFor the End Eversion it suffices to notice that all the explicit computati ons above, are done essen-\ntially with scalar symbols .\nAcknowledgements: The two first authors benefited during this work from the suppo rt of the\nFrench ANR-Project QUAMPROCS \"Quantitative Analysis of Me tastable Processes\". Francis White\nis grateful to have received funding from the European Union ’s Horizon 2020 research and innova-\ntion programme under the Marie Sklodowska-Curie grant agre ement No 101034255.⋆⋆⋆⋆⋆\n⋆\n⋆\n⋆\n⋆⋆⋆⋆\nReferences\n[ABG] W. Amrein, A. Boutet de Monvel, V. Georgescu. C0-groups, commutator methods and spectral\ntheory of N-body hamiltonians Progress in Mathematics Vol. 135, Birkhäuser (1996).\n[BCD] H. Bahouri, J.Y. Chemin, R. Danchin. Fourier Analysis and Nonlinear Partial Differential\nEquations. Grundlehren der Mathematischen Wissenschaften 343. Sprin ger (2011).\n[Bea] R. Beals, Characterization of pseudodifferential op erators and applications. Duke Math. J.\nVol. 44 no. 1 (1977) pp. 45–57.\n[BeBo] L. Bérard-Bergery, J.P. Bourguignon. Laplacian and Riemannian submersion with totally\ngeodesic fibres. Global differential geometry and global an alysis, Proc. Colloq. Berlin 1979,\nLect. Notes Math. Vol. 838 (1981) pp. 30–35.\n[Bis041] J.M. Bismut. Le Laplacien hypoelliptique sur le fib ré cotangent C.R. Acad. Sci. Paris Sér.\nI, 338 (2004) pp 471–476.\n[Bis042] J.M. Bismut. Le Laplacien hypoelliptique. Sémina ire Equations aux Dérivées Partielles,\nExp. XXII, Ecole Polytechnique (2004).\n81[Bis05] J.M. Bismut. The hypoelliptic Laplacian on the cota ngent bundle. Journal of the American\nMath. Soc., Vol. 18 no. 2 (2005) pp 379–476.\n[BiLe] J.M. Bismut, G. Lebeau. The Hypoelliptic Laplacian and Ray-Singer Metrics . Annals of\nMathematics Studies 167 (2008).\n[Bon] J.M. Bony. Fourier Integral Operators and Weyl-Hörma nder Calculus. J.M. Bony, M. Morimoto\n(ed.), New trends in Microlocal Analysis, Tokyo. Springer ( 1997) pp. 3-21.\n[BoCh] J.M. Bony, J.Y. Chemin. Espaces fonctionnels associ és au calcul de Weyl-Hörmander. Bull.\nSoc. Math. France Vol. 122 no. 1 (1989) pp. 277-433.\n[BoLe] J.M. Bony, N. Lerner. Quantification asymptotique et microlocalisation d’ordre supérieur I.\nAnn. Scient. Ec. Norm. Sup., 4esérie 22 (1989) pp. 377-433\n[ChPi] J. Chazarain, A. Piriou. Introduction to the theory of linear partial differential e quations.\nStudies in Mathematics and its Applications Vol. 14 North-H olland (1982).\n[Dav] E.B. Davies. Semi-classical states for non self-adjo int Schrödinger operators. Comm. Math.\nPhys. 200, 35–41 (1999).\n[DiSj] M. Dimassi, J. Sjöstrand. Spectral asymptotics in the semi-classical limit. London Mathemat-\nical Society Lecture Note Series Vol. 268, Cambridge Univer sity Press (1999).\n[DSZ] N. Dencker, J. Sjöstrand, M. Zworski. Pseudospectra o f semi-classical (pseudo)differential\noperators. Comm. Pure Appl. Math. 57 (3), p. 384-415 (2004).\n[Dro] A. Drouot. Stochastic stability of Pollicott-Ruelle resonances. Commun. Math. Phys. Vol. 356\nno. 2, (2007) pp. 357-396.\n[HeSj] B. Helffer, J. Sjöstrand. Equation de Harper. Lectur e Notes in Physics Vol. 345 (1989) pp. 118–\n197.\n[HeSj4] B. Helffer and J. Sjöstrand. Puits multiples en limi te semi-classique IV - Étude du complexe\nde Witten -. Comm. Partial Differential Equations 10 (3) (19 85), pp. 245–340.\n[HHS] F. Hérau, M. Hitrik, and J. Sjöstrand. Tunnel effect an d symmetries for Kramers-Fokker-\nPlanck type operators. J. Inst. Math. Jussieu 10 (3) (2011) p p. 567–634.\n[HerNi] F. Hérau and F. Nier. Isotropic hypoellipticity and trend to the equilibrium for the Fokker-\nPlanck equation with a high-degree potential. Arch. Ration . Mech. Anal. 171(2) (2004) pp.\n151–218.\n[Hor67] L. Hörmander. Hypoelliptic second order different ial equations. Acta Math., Vol. 119 (1967)\npp. 147-171.\n[HormIII] L. Hörmander. The analysis of linear partial differential operators, III .Springer-Verlag\n(1985).\n[Leb1] G. Lebeau. Geometric Fokker-Planck equations. Port . Math. (N.S.) Vol. 64 no. 4 (2005),\npp. 469-530.\n82[Leb2] G. Lebeau Equations de Fokker-Planck géométriques. II. Estimations hypoelliptiques maxi-\nmales. Ann. Inst. Fourier. Vol. 57 no. 4 (2007) pp. 1285-1314\n[LNV1] D. Le Peutrec, F. Nier, C. Viterbo. Precise Arrhenius law for p-forms: The Witten Laplacian\nand Morse-Barannikov complex Ann. Henri Poincaré Vol. 14, N o 3 (2013) pp 567–610.\n[LNV2] D. Le Peutrec, F. Nier, C. Viterbo. Bar codes of persis tent cohomology and Arrhenius law for\np-forms. arXiv:2002.06949.\n[LiMa] J.L. Lions, E. Magenes. Non homogeneous boundary value problesm and applications. Vol. I.\nDie Grundleheren der mathematischen Wissenshaften, Band 1 82 Springer (1972).\n[NaNi] F. Nataf, F. Nier. Convergence of domain decompositi on me thods via semi-classical calculus.\nCommun. Partial Differ. Equations Vol. 23, no. 5-6 (1998) pp . 1007-1059.\n[Nie] F. Nier. Boundary conditions and subelliptic estimates for geometr ic Kramers-Fokker-Planck\noperators on manifolds with boundaries. Mem. Amer. Math. Soc. Vol. 252 no. 1200 (2018).\n[NiSh] F. Nier, S. Shen. Bismut hypoelliptic Laplacians for manifolds with boundaries.\narXiv:2107.01958 (2021).\n[ReSi] M. Reed and B. Simon. Method of Modern Mathematical Physics II . Academic press (1975).\n[Rob] D. Robert. Autour de l’approximation semiclassique. Progress in Mathematics Vol. 68,\nBirkhaüser (1987).\n[ReTa] Q. Ren, Z. Tao. Spectral asymptotics for kinetic brow nian motion on riemannian manifolds.\narXiv:2212.053994v3 (2023).\n[SjZw] J. Sjöstrand, M. Zworski. Elementary linear algebra for advanced spectral problems. Ann.\nInst. Fourier Vol. 57, no. 7, (2007) pp. 2095-2141.\n[She] S. Shen. Laplacien hypoelliptique, torsion analytiq ue et théorème de Cheeger-Müller. J. Funct.\nAnal Vol. 270 (2016) pp. 2817–2999.\n[Smi] H.F. Smith. Parametrix for a semiclassical subellipt ic operator. Analysis and PDE Vol. 13\nno. 8 (2020) pp. 2375–2398.\n[Wit] E. Witten. Supersymmetry and Morse inequalities. J. D iff. Geom. Vol. 17, no. 4 (1982) pp. 661–\n692.\n[Zha] W. Zhang. Lectures on Chern-Weil theory and Witten deformations. Nankai Tracts in Mathe-\nmatics. 4. World Scientific. xii, 117 p. (2001).\n[Zwo] M. Zworski. Semiclassical Analysis. Graduate Studies in Mathematics Vol. 138. American\nMathematical Society (2012).\n83"    },    {        "title": "2402.07515v1.Gravitational_Lensing_and_Clues_of__r__d___and__H__0___Tension__Viscous_Modified_Chaplygin_Gas_and_Variable_Modified_Chaplygin_Gas_in_Loop_Quantum_Cosmology.pdf",        "content": "Gravitational Lensing and Clues of rdand H0Tension: Viscous Modified Chaplygin Gas and\nVariable Modified Chaplygin Gas in Loop Quantum Cosmology\nRownak Kundu,1, 2,∗Ujjal Debnath,2, †Himanshu Chaudhary,3, 4, 5, ‡and G. Mustafa6, §\n1Department of Mathematics, Nalbari College, Nalbari-781335, Assam, India.\n2Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711 103, India.\n3Department of Applied Mathematics, Delhi Technological University, Delhi-110042, India\n4Pacif Institute of Cosmology and Selfology (PICS), Sagara, Sambalpur 768224, Odisha, India\n5Department of Mathematics, Shyamlal College, University of Delhi, Delhi-110032, India.\n6Department of Physics, Zhejiang Normal University, Jinhua 321004, People’s Republic of China,\nThis paper investigates the accelerated cosmic expansion in the late Universe by examining two\ndark energy models, namely viscous modified Chaplygin gas (VsMCG) and variable modified Chap-\nlygin gas (VMCG), within the framework of loop quantum cosmology alongside the ΛCDM model.\nOur main objective is to constrain fundamental cosmic parameters in these dark energy models with\nΛCDM model using 30 of the latest H(z)measurements obtained from the cosmic chronometers\nmethod (CC), including Type Ia Supernovae, Gamma-Ray Bursts (GRB), Quasars, and 24 uncorre-\nlated baryon acoustic oscillations (BAO) measurements from recent galaxy surveys across a redshift\nrange from 0.106 <z<2.33. Additionally, we include the most recent Hubble constant measure-\nment from Riess in 2022 to further enhance our constraints on fundamental cosmic parameters. In the\nΛCDM, VsMCG, and VMCG frameworks, Our investigation yields best-fit parameters for the Hub-\nble parameter ( H0) and sound horizon ( rd). The outcomes underscore a significant disparity between\n(H0) and rd) values using late-time observational measurements, reflecting the widely recognized H0\nand rdtensions. Further, we study the gravitational lensing optical depth of the two dark energy mod-\nels by plotting τ(zs)/Fvszsgraphs. The probability of finding gravitational lenses (optical depth) in\nboth our models increases with source redshift zs. The change in optical depth behaviour for different\nsets of parameter value constraints through data sets is also graphically analysed. Joint analysis of\nVsMCG and VMCG with standard cosmological model ΛCDM is also carried out to study the optical\ndepth behaviour. While the three models diverge in the early Universe, but for low redshift, they are\nfound to be indistinguishable from each other. Finally, by employing the Akaike information criteria\napproach for model comparison, our analysis indicates that none of the two dark energy models can\nbe dismissed based on the latest observational measurements.\nI. INTRODUCTION\nOur universe is undergoing accelerated expansion, a\nphenomenon supported by multiple independent stud-\nies [1–4] conducted over the past two decades. While\nthe actual cause of this accelerated phenomenon is still\ndebatable, many attribute it to huge negative pressure\nbuild-up in our present Universe. Many assume that\nsome kind of ‘mysterious’ energy must be at play here\nand have named it ‘dark energy’ (DE, hereafter). Dark\nenergy, or DE, almost occupies 70% of our Universe,\nso its study has been of immense importance. Over\nthe years, researchers have proposed numerous such\nDE candidates to explain our accelerating Universe,\nand among them, the cosmological constant Λhappens\nto be the simplest. The other candidates includes\n∗rownakkundu@gmail.com\n†ujjaldebnath@gmail.com\n‡himanshuch1729@gmail.com\n§gmustafa3828@gmail.comChaplygin gas [5, 6], quintessence [7, 8], phantom\nenergy [9], holographic dark energy [10, 11], among the\nnumerous others in literature [12]. Despite the diverse\nrange of proposed dark energy candidates, the key to\nunravelling the mysteries of dark energy lies not only in\ntheoretical constructs but also in empirical observations.\nObservational cosmology has thus become instrumental\nin discerning the nature of dark energy. Among the\nobservational probes employed by cosmologists, the\nstudy of the large-scale structure of the Universe stands\nout prominently. Observations of the large-scale struc-\nture, which include the spatial distribution of galaxies,\ngalaxy clusters, and cosmic voids, offer invaluable\ninsights into the underlying dynamics of cosmic evolu-\ntion [13]. By scrutinizing the cosmic web on vast scales,\ncosmologists can glean crucial information about the\ncomposition of the Universe, its expansion history, and\nthe influence of dark energy on cosmic structures [14].\nHowever, as observational techniques have advanced,\nthey have revealed a perplexing tension between\nearly-time and late-time cosmological observations.arXiv:2402.07515v1  [gr-qc]  12 Feb 20242\nThis tension manifests in discrepancies between mea-\nsurements obtained from the early Universe, such as\nthose derived from the cosmic microwave background\n(CMB) radiation [15] and those obtained from more\nrecent observations of the local Universe, such as those\nfrom Type Ia supernovae [16] and Baryon Acoustic\nOscillations (BAO). BAO plays a pivotal role, offering\na unique and powerful tool for understanding the\nlarge-scale structure of the Universe. These oscillations\nare imprints left on the distribution of matter in the\nearly Universe, originating from the coupled interaction\nbetween photons and baryons (protons and neutrons)\nbefore the era of recombination [17]. The significance\nof BAO lies in their ability to serve as standard rulers\non cosmological scales. As the Universe expands, these\nprimordial density fluctuations lead to characteristic\npatterns in the distribution of galaxies and cosmic mi-\ncrowave background radiation. The formation of BAO\ncan be traced back to the sound waves that propagated\nthrough the primordial plasma of the early Universe.\nAs the Universe expanded, the sound waves left behind\na distinctive pattern of overdense and underdense\nregions, creating a preferred scale known as the BAO\nscale or sound horizon [18]. This scale acts as a standard\nruler, allowing cosmologists to measure the geometry\nand expansion rate of the Universe over cosmic time.\nThe BAO scale, or sound horizon, serves as a crucial\ncosmological standard because its size is determined\nby fundamental cosmological parameters, such as the\ndensity of baryons and dark matter, as well as the speed\nof sound in the primordial plasma. Observations of\nBAO in the large-scale distribution of galaxies provide\na powerful constraint on these parameters and offer\ninsights into the nature of dark energy [19]. In the field\nof cosmology, the study of BAO and the sound horizon\nhas become an essential tool for precision cosmological\nmeasurements. Large galaxy surveys, such as the Sloan\nDigital Sky Survey (SDSS), have played a key role in\nmapping the distribution of galaxies and detecting\nthe BAO signal [20]. By analyzing the BAO scale in\nthe clustering of galaxies, researchers can infer the\nexpansion history of the Universe and shed light on\nthe mysterious components of dark matter and dark\nenergy. Thus, Baryon Acoustic Oscillations stand as a\ncornerstone in our quest to unravel the fundamental\nproperties and evolution of the cosmos.\nWhile BAO provide valuable insights about the\nlarge-scale structure of the Universe and the behavior of\ndark energy, the accelerating expansion of the cosmos\nremains a profound puzzle that has spurred researchers\nto explore alternative explanations. Many assume thatEinstein’s gravitational theory might be incomplete.\nSo, researchers modified the underlying gravity the-\nory (and hence, the name modified theory of gravity\n[21, 22]) to incorporate the accelerating phase of our\nUniverse. While these theories effectively explain the\nformation of structures and gravitational interactions\non a large scale, quantum gravity becomes necessary\nto understand behaviors at smaller scales. Further, the\nbackward evolution of our Universe in time results in\nthe collapse of our Universe into a single point with\ndiverging energy density. Such instances lead to the\nfailure of our classical gravity theory, rendering it in-\neffective in describing the unfolding events. Quantum\ngravity, with its distinct dynamics on smaller scales, is\nexpected to solve this dilemma. One such theory based\nupon quantum gravity is loop quantum cosmology\n(LQC, hereafter) [23, 24]. In recent years, several DE\nmodels have been studied in its framework. Jamil et al.\n[25] investigated the combination of modified Chaply-\ngin gas with dark matter within the LQC framework,\nresolving the cosmic coincidence problem. The authors\nin ref. [26], discovered that loop quantum effects could\nprevent future singularities in the FRW cosmology.\nGiven its importance, in our present study as well,\nwe considered two models of Chaplygin gas, namely,\nviscous modified Chaplygin gas (VsMCG) and variable\nmodified Chaplygin gas (VMCG) in LQC framework, to\nanalyze their optical depth behaviour by constraining\nthe model parameters to their best-fit values. Further,\nthe significance of the study of gravitational lensing\nalso has long been recognised in cosmology, and the\nphenomenal work of Refsdal et al. [27] in this context\nis worth mentioning. Over recent years, gravitational\nlensing emerged as an important tool for studying dark\nmatter, dark energy, and black holes, among many oth-\ners. Gravitational lensing optical depth represents the\nlikelihood of the formation of multiple images owing\nto the influence of gravitational lenses in our Universe.\nIt was first used by ref. [28, 29] and consequently by\nmany others [30–32], including Kundu et al. [33] to\nstudy and understand dark energy models. Our present\npaper uses this lensing phenomenon to qualitatively\nanalyse the two DE models against their constraint\nparameter values and record the changes graphically.\nThis methodology of studying the gravitational lensing\noptical depth of DE models can give us an estimation in\nunderstanding the large-scale structure of our Universe,\nthereby helping us better understand our models.\nFinally, our paper is organized as follows: Section\nII delves into the background equations of the two dark\nenergy models within the framework of loop quantum3\ncosmology. Section III outlines the methodology em-\nployed to constrain the parameters of the dark energy\nmodels using various datasets. Section IV is dedicated\nto the study of the lensing phenomenon, including the\nderivation of the equation for optical depth to be used\nin our study. In Section V, we present the outcomes of\nour study, and we conclude by discussing our findings\nin Section VI.\nII. BACKGROUND EQUATIONS: LOOP QUANTUM\nCOSMOLOGY\nIn loop quantum cosmology, for a isotropic and ho-\nmogeneous Universe the background equations are de-\nfined by:\nH2=1\n3 \n1−ρ\nρ1!\nρ, (1)\n˙H=−1\n2 \n1−2ρ\nρ1!\n(ρ+p) (2)\nwhere ρ1=√\n3\n16π2γ3G2¯his the critical loop quantum den-\nsity and γthe dimensionless “Barbero-Immirzi parame-\nter”. The conservation equation in LQC is given by:\n˙ρ+3H(ρ+p) =0 (3)\nAssuming the matter content of our Universe to be a\ncombination of DM and DE, the quantities ρand pde-\nfined in Eqs (1) and (2) will modify to ρ=ρm+ρd\nand p=pm+pd. With the consideration that both DMand DE are separately conserved, the equation (3) now\nseperates into two distinct equations as follows:\n˙ρm+3H(ρm+pm) =0, (4)\n˙ρd+3H(ρd+pd) =0, (5)\nSince, pressure in dark matter is very negligible (i.e.\npm=0), equation (4) gives ρm=ρm0(1+z)3. Here,\nρm0represent the density value at the present epoch.\nA. Viscous modified Chaplygin gas (VsMCG)\nThe expression for pressure in the case of viscous\nmodified Chaplygin gas (VsMCG) is given by [34]:\npd=Aρd−B\nρα\nd−3ζ0√ρdH (6)\nwhere A,Bandαare constants. Substituting pdfrom Eq\n(6) in Eq (5), we get:\nρd= \nB\n1+A−√\n3ζ0+C1\na3(1+α)(1+A−√\n3ζ0)! 1\n1+α\n(7)\nwhere C1is an integrating constant. The above expres-\nsion can be further re-written as:\nρd=ρd0{As+ (1−As)(1+z)3(1+α)(1+A−√\n3ζ0)}1\n1+α(8)\nwhere ρd0being the DE density value at the present\nepoch, As=B\n(1+A−√\n3ζ0)C1+Bsatisfying the conditions\n0<As<1 and 1 +A−√\n3ζ0>0, and ρ1+α\nd0=\n(1+A−√\n3ζ0)C1+B\n1+A−√\n3ζ0. Thus, we can write the Hubble param-\neter from equation (1) as follows:\nH2(z) =H2\n0\"\nΩm0(1+z)3+Ωd0\u001a\nAs+ (1−As)(1+z)3(1+α)(1+A−√\n3ζ0)\u001b 1\n1+α#\n×\n\"\n1−3H2\n0\nρ1\u001a\nΩm0(1+z)3+Ωd0\u001a\nAs+ (1−As)(1+z)3(1+α)(1+A−√\n3ζ0)\u001b 1\n1+α\u001b#\n(9)\nwhere Ωm0=ρm0\n3H2\n0andΩd0=ρd0\n3H2\n0are the dimension-\nless parameters all defined at the present epoch. Fur-\nther, substituting z=0 in equation (9) and equating theRHS to ‘1’ we get:\n\u0014\nΩm0+Ωd0\u0015\n×\"\n1−3H2\n0\nρ1\u001a\nΩm0+Ωd0\u001b#\n=1 (10)4\nwhich simplifies to\nΩd0=1\n6H2\n0\u0012\nρ1+q\nρ2\n1−12ρ1H2\n0\u0013\n−Ωm0 (11)\nB. Variable modified Chaplygin gas (VMCG)\nThe expression for pressure in the case of variable\nmodified Chaplygin gas (VMCG) is given by [35]:\npd=Cρd−D(a)\nρβ\nd(12)\nwhere 0 ≤β≤1 and C>0. Assuming, D(a) = D0a−n\nwhere D0>0 and n>0, we get from Eq (5):\nρd=\"\n3(1+β)D0\n{3(1+β)(1+C)−n}an+K\na3(1+C)(1+β)# 1\n1+β\n(13)where K(>0)is an arbitrary integrating constant and\n3(1+β)(1+C)−n>0. Also, here nmust be posi-\ntive, or else for a→∞we would have ρd→∞thereby\ncontradicting our case for expanding Universe. The ex-\npression for ρdabove can be further simplified to:\nρd=ρd0\u0014Cs\nan+1−Cs\na3(1+C)(1+β)\u00151\n1+β\n(14)\nwhere ρd0denotes the DE density value at the present\nepoch, Cs=1−K\nρ1+β\nd0and ρ1+β\nd0=K+3(1+β)D0\n3(1+β)(1+C)−n.\nThus, we can write the Hubble parameter from equation\n(1) as follows:\nH2(z) =H2\n0\"\nΩm0(1+z)3+Ωd0\u001a\nCs(1+z)n+ (1−Cs)(1+z)3(1+C)(1+β)\u001b 1\n1+β#\n×\n\"\n1−3H2\n0\nρ1\u001a\nΩm0(1+z)3+Ωd0\u001a\nCs(1+z)n+ (1−Cs)(1+z)3(1+C)(1+β)\u001b 1\n1+β\u001b#\n(15)\nwhere Ωd0is given in (11).\nIII. METHODOLOGY\nIn our study, we carefully picked a specific set of re-\ncent Baryon Acoustic Oscillation (BAO) measurements\nfrom various galaxy surveys, with the primary contri-\nbutions coming from observations made by the Sloan\nDigital Sky Survey (SDSS) [36–41]. We also included\nvaluable data from the Dark Energy Survey (DES) [42],\nthe Dark Energy Camera Legacy Survey (DECaLS)\n[43], and 6dFGS BAO [44] to enhance the diversity of\nour dataset. Recognizing the potential for correlations\namong selected data points, we took steps to address\nthis issue. While we intentionally crafted a subset to\nminimize highly correlated points, acknowledging and\ndealing with possible correlations within our chosen\ndata points was deemed crucial. For a robust estimation\nof systematic errors, we used mocks generated from\nN-body simulations to accurately determine covariancematrices. Given the diverse nature of measurements\nfrom various observational surveys, obtaining precise\ncovariance matrices between them posed a significant\nchallenge. To address this, we adopted the covariance\nanalysis approach outlined in [45], specifically using\nCii=σ2\ni. To simulate correlations within our chosen\nsubsample, we introduced non-diagonal elements into\nthe covariance matrix while maintaining symmetry.\nIncorporating non-negative correlations involved\nrandomly selecting up to 12 pairs of data points and\nassigning non-diagonal elements with magnitudes of\nCij=0.5σiσj. Here, σiand σjdenote the 1 σerrors\nof data points iand j. This approach allowed us to\neffectively represent correlations within 55% of our\nchosen BAO dataset. The variation of some cosmolog-\nical parameters of ΛCDM according to the number of\ncorrelated pairs is shown in Table I. Fig. 1 displays the\nposterior distributions for the ΛCDM Model with and\nwithout the incorporation of a test random covariance\nmatrix comprising fourteen components. The influence\nof this covariance matrix, ranging from null to fourteen5\nParameters ncorrelated pairs BAO BAO+R22\nΩm0 n = 0 0.270±0.015 0.266±0.015\nΩΛ0 0.725±0.011 0.728±0.0125\nΩm0 n = 7 0.264±0.016 0.261±0.015\nΩΛ0 0.730±0.013 0.733±0.011\nΩm0 n = 14 0.266±0.015 0.264±0.016\nΩΛ0 0.728±0.011 0.730±0.012\nTABLE I: The table shows slight variations in different\ncosmological parameters when comparing scenarios\nwith uncorrelated pairs ( n=0) to those with n=7 and\nn=14 randomly introduced correlated pairs.\ncomponents, appears negligible, closely resembling the\nuncorrelated dataset. Moving on to determining thebest-fit values for our cosmological model parameters,\nwe expanded the BAO dataset by incorporating thirty\nuncorrelated Hubble parameter measurements ob-\ntained through the cosmic chronometers (CC) method\ndiscussed in [46–49]. Additionally, we included the\nlatest Pantheon sample data on Type Ia Supernovae\n[50]. To broaden our observational scope further, we in-\ntroduced 24 binned quasar distance modulus data from\n[51], a set of 162 Gamma-Ray Bursts (GRBs) as outlined\nin [52], and the recent Hubble constant measurement\n(R22) [53] as an additional prior. In our analysis, we\nused a nested sampling approach implemented in the\nopen-source Polychord package [54], complemented by\nthe GetDist package [55] to present our results in a clear\nand informative manner.\nIV . GRA VITATIONAL LENSING\nThe differential probability that a distribution of\ngalaxies per unit redshift will be multiply imaged by a\nsource with redshift zsis [33]:\ndτ\ndzl=n(1+zl)3Scdt\ndzl(16)\nwhere ndenotes the co-moving number density of\nlenses, Sis the lens cross-section and cdt/dzlthe proper\ndistance interval. Further, following numerous work\n[29, 56] done previously, we assume an singular isother-\nmal sphere (SIS) profile for our lensing model. Thus, the\nexpression for lensing cross-section in an SIS profile is\ngiven by:\nS=16π3\u0012σ\nc\u00134\u0012DlDls\nDs\u00132\n(17)\nwhere σis the velocity dispersion, Ds,Dland Dls\nrepresents the angular diameter distances between\nobserver-source, observer-lens and lens-source respec-\ntively, given by:\nDz1z2=1√H0(1+z2)Zz2\nz11p\nH(z)dz (k=0)\n(18)\nand finally the proper distance interval reads:\ncdt\ndzl=c\n(1+zl)H(zl)(19)Now, in order to calculate npresent in Eq. (16), we use\nthe Schechter function given by:\ndn\ndL=n∗\ni\nL∗\ni \nL\nL∗\ni!γi\ne(−L/L∗\ni)(20)\nwhere idepends on the type of galaxy morphology (i.e.\nS,S0orE,n∗\nidenotes the characteristic number density,\nand l∗\nithe characteristic luminosity. Additionally, with\nthe help of the power-law relation L/L∗\ni= (σ/σ∗\ni)gi,\nequation (20) further reduces to\ndn\ndσ=n∗\ni\nσ∗\ni \nσ\nσ∗\ni!giγi+gi−1\ne(−σ/σ∗\ni)gigi (21)\nThus, the differential optical depth can now be obtained\nby integrating eqn. (16) from 0 to zsas follows:\nτ(zs) =\n∑\ni=S,S0,EFi\nZχs\n0\u0012χs−χl\nχs\u00132\nχ2\nldχl (22)\nwhere\nχi=p\nH0Zzi\n0dzp\nH(z)(23)\nis the co-moving distances, Firepresents the ability of\ntheith class of galaxy morphology to produce multiple\nimages, relying exclusively on the intrinsic and statisti-\ncal properties of the galaxies. Using equation (21), we6\n130140150160170\nrd(Mpc)607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\na)rdvsH0\n130140150160170\nrd(Mpc)0.680.700.720.740.760.78d0\n0.700.740.78\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 b)rdvsΩd0\n0.200.250.30\nm0\n607080H0\n607080\nH0BAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22\nc)H0vsΩd0\n0.200.250.30\nm0\n0.680.700.720.740.760.78d0\n0.700.740.78\nd0\nBAO\nBAO + Covariance\nBAO + R22\nBAO + Covariance + R22 d)Ωm0vsΩd0\nFIG. 1: Comparison of posterior distributions in ΛCDM with and without a randomly tested covariance matrix of\nfourteen components shows negligible impact. This renders the dataset nearly indistinguishable from an\nuncorrelated one. Covariance matrices beyond fourteen components exhibit minimal influence.\nmay calculate Fiin the form:\nFi=16π3σ4H−3\n0Z∞\n0n∗\ni\nσ∗\ni \nσ\nσ∗\ni!giαi+gi−1\ne(−σ/σ∗\ni)gigidσ\n=16π3σ4H−3\n0n∗\niΓ(αi+1)\n(24)Considering the uncertainties extensively addressed in\nthe literature [57–60], we will treat Fto be a normal-\nized factor, denoted as F≡∑i=S,S0,EFi. Therefore, from\nequations (22) and (24), we deduce a straightforward an-7\nMCMC Results\nModel Parameters Priors BAO BA0 + R22 CC + SC + BAO CC + SC + BAO + R22\nH0 [50,100] 69.089209±6.252595\n±4.39687473.906596±2.788642\n±1.35769769.854848±2.386935\n±1.25910071.616475±1.936192\n±1.003940\nΛCDM Model Ωm0 [0.,1.] 0.256796±0.069423\n±0.0257720.254859±0.068716\n±0.0269120.268654±0.028134\n±0.0128220.264407±0.030224\n±0.013179\nΩd0 [0.,1.] 0.734631±0.038211\n±0.0208880.736400±0.034343\n±0.0215100.724585±0.016999\n±0.0093730.728562±0.018197\n±0.009532\nrd(Mpc) [100,200] 149.504236±15.212831\n±10.037166139.447508±5.883881\n±2.914639146.543556±5.101856\n±2.598566143.299915±4.358875\n±2.218062\nrd/rf id [0.9,1.1] 1.008382±0.101188\n±0.0663440.940599±0.037308\n±0.0208060.990266±0.035810\n±0.0198260.967284±0.030444\n±0.015101\nH0 [50,100] 65.168094±6.617797\n±10.43446673.07201±0.978415\n±1.88979767.447204±2.514723\n±1.87803272.860428±1.040048\n±2.328246\nΩm0 [0.,1.] 0.365902±0.069461\n±0.1159900.305203±0.041952\n±0.0710960.346451±0.069036\n±0.1051640.289643±0.012301\n±0.025106\nVsMCG Model Ωd0 [0.,1.] 0.809753±0.133984\n±0.2344110.676112±0.101291\n±0.1902160.779877±0.158516\n±0.2390420.652845±0.028308\n±0.059635\nAs [0.,1.] 0.514312±0.491146\n±0.2791020.504348±0.457075\n±0.3242630.436312±0.416373\n±0.3195040.447568±0.430062\n±0.357146\nα [0.,1.] 0.683658±0.445186\n±0.2329320.628384±0.440275\n±0.2620420.665943±0.144111\n±0.0640400.667478±0.157306\n±0.074594\nA [2.,4.] 3.089772±1.049832\n±0.6497372.965524±0.891859\n±0.6020963.014177±0.952994\n±0.6997943.093859±1.043732\n±0.615597\nζ0 [1.5,2.5] 1.962968±0.442674\n±0.3107791.990460±0.453422\n±0.3498342.101817±0.555898\n±0.2973522.050346±0.511089\n±0.357694\nρ1 [1310000,1390000] 1351709.4±38561.9\n±28642.81349170.2±36390.3\n±26320.81349393.1±37491.8\n±26568.31353509.1+41653.0\n−25630.2\nrd(Mpc) [100,200] 149.571115±16.426449\n±8.752594140.757158±12.992500\n±8.454083147.559044±5.811905\n±2.723020141.158148±4.709648\n±2.299581\nrd/rf id [0.9,1.1] 1.009424±0.104352\n±0.0598810.990016±0.083577\n±0.0567650.995630±0.039894\n±0.0199540.993363±0.031652\n±0.016175\nH0 [50,100] 65.796665±6.461974\n±10.07087473.799085±1.325015\n±2.57486269.204717±2.413623\n±1.05603173.025435±1.390057\n±2.671485\nΩm0 [0.,1.] 0.286515±0.075623\n±0.1727560.197851±0.052402\n±0.1008700.229029±0.065811\n±0.1455010.181731±0.060864\n±0.119488\nVMCG Model Ωd0 [0.,1.] 0.700150±0.113069\n±0.1869130.803863±0.129291\n±0.2763530.801456±0.120813\n±0.2121900.717771±0.067980\n±0.116377\nn [0.5,1.5] 0.990322±0.007463\n±0.0347570.980549±0.017220\n±0.0487830.976961±0.018299\n±0.0514840.970191±0.026968\n±0.065479\nβ [0.8,1.4] 1.018741±0.086082\n±0.1150031.020759±0.084889\n±0.1128961.021356±0.085708\n±0.1157781.025479±0.097275\n±0.123108\nC [0.,0.3] 0.082870±0.057201\n±0.0794130.082870±0.057201\n±0.0794130.045065±0.036463\n±0.0443430.038790±0.033014\n±0.037825\nCs [0.,1.] 0.599738±0.361244\n±0.5458850.599738±0.361244\n±0.5458850.376236±0.293035\n±0.3648270.481279±0.380702\n±0.472094\nρ1 [1310000,1390000] 1351317.6±27841.6\n±39991.21349811.1±25642.0\n±36678.61354908.4±26738.7\n±42104.61350850.4±27481.4\n±37505.8\nrd(Mpc) [100,200] 148.683108±14.573363\n±8.949488139.365379±14.205048\n±9.762761147.906003±4.444459\n±2.386071141.623321±5.305695\n±2.638046\nrd/rf id [0.9,1.1] 1.002078±0.097008\n±0.0639681.000626±0.094807\n±0.0672230.995166±0.030789\n±0.0162010.994314±0.035566\n±0.019600\nTABLE II: The table presents constraints on cosmological parameters for the ΛCDM, VsMCG, and VMCG models at\na 95% confidence level.\nalytical expression for the optical depth of a point source\nat redshift zsin an FRW Universe with dark matter and\ndark energy components [33]:\nτ(zs) =F\n30χ3\ns (25)\nV . RESULTS\nFor the standard ΛCDM model, the posterior distri-\nbution of the key cosmological parameters is shown atthe 68% and 95% confidence levels in Fig. 2. Detailed\nresults from MCMC simulations are outlined in Table\nII. When the R22 prior is incorporated into the joint\ndataset, the optimal fitting value for H0is 71.50 ±0.882,\ndiverging [15] from but aligning closely with measure-\nments from the SNIe sample in [53]. In the absence\nof R22 priors with the Joint dataset, the estimated\nH0=69.837 ±1.204 corresponds more closely to the\nvalue in [15]. Figs 3 and 4 portray the 68% and 95%\nconfidence levels for crucial cosmological parameters\nin VsMCG and VMCG models. Across these models,\nwhen the R22 prior is included in the Joint dataset,\nthe optimal fitting value for H0diverges from [15] but8\n607080\nH00.700.750.80do\n0.150.200.250.30mo\n0.150.200.250.30\nmo\n0.700.750.80\ndo\nBAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22\nFIG. 2: The figure illustrates 1 σand 2 σregions using\nthe standard ΛCDM model to demonstrate the\nposterior distribution of various observational data\nmeasurements.\naligns more closely with the SNIe sample in [53]. Con-\nversely, without R22 priors and using the Joint dataset,\nthe estimated H0=69.837 ±1.204 aligns more closely\nwith [15]. The determined values for the matter density,\ndenoted as Ωm0, and the dark energy density, denoted\nasΩd0, in the ΛCDM and VMCG Models, appear to\nbe lower compared to the values documented in [15]\n(Ωm0=0.315±0.007, Ωd0=0.685±0.007). However,\nin the case of the VsMCG Model, it is higher than the\nvalue documented in [15]. However, this observation\nhas been documented in alternative studies [61, 62].\nIn the context of the Baryon Acoustic Oscillations\n(BAO) scale, it is defined by the cosmic sound horizon\nimprinted in the cosmic microwave background during\nthe drag epoch, denoted as zd. This epoch marks the\nseparation of baryons and photons. The BAO scale,\nrepresented by rd, is determined by the integral of the\nratio of the speed of sound ( cs) to the Hubble parameter\n(H0) over the redshift range from zdto infinity. The\nspeed of sound, cs, is given byqδpγ\nδρB+δργ, where δpγ\nis the pressure perturbation in photons, and δρBand\nδργare perturbations in the baryon and photon energy\ndensities, respectively. This expression is further sim-\nplified to1√\n3(1+R), with Rdefined as the ratio of baryon\ndensity perturbation to photon density perturbation(R≡δρB\nδργ=3ρB\n4ργ). Observational data from [15] provides\nthe redshift at the drag epoch as zd=1059.94 ±0.30.\nIn the case of a ΛCDM model, measurements from\n[15] estimate the BAO scale rdto be 147.09 ±0.26\nmegaparsecs (Mpc). In our exploration of the ΛCDM\nmodel, Fig 5a illustrates the posterior distribution\nof the contour plane for rd−H0. The BAO datasets\nprovide an estimated rdof 145.807 ±9.06 Mpc, aligning\nwith the reported findings in [63]. However, when we\nexclusively incorporate R22 prior into the BAO dataset,\nthe determined sound horizon at the drag epoch is\n138.345 ±2.45 Mpc. Examining the Joint dataset reveals\nan estimated BAO scale ( rd) of 145.811 ±2.34 Mpc,\nclosely aligning with the outcomes reported in [15].\nAdditionally, incorporating the R22 prior into the\ncomprehensive dataset results in rdof 142.591 ±1.49\nMpc, indicating proximity to the findings in [64]. Now,\nshifting to the VsMCG Model, Fig 5b presents the\nposterior distribution for the rd−H0contour plane. For\nthe BAO datasets, the resulting rdis 149.571 ±8.752\nMpc, consistent with the Planck results as reported\nin [63]. However, incorporating R22 prior exclusively\ninto the BAO dataset leads to a sound horizon at the\ndrag epoch of 140.757 ±8.454 Mpc. In the case of the\nJoint dataset, the determined rdis 147.559 ±2.72 Mpc,\naligning closely with the Planck results. Furthermore,\nintegrating the R22 prior into the full dataset results in\nrd=141.158 ±2.299 Mpc, showing proximity to the\nfindings presented in [64]. In the context of the VMCG\nModel, Fig 5c illustrates the posterior distribution of\nthe rd−H0contour plane. When considering BAO\ndatasets, the estimated BAO scale ( rd) is 148.683 ±8.949\nMpc, a result consistent with the findings reported in\n[63]. However, introducing the R22 prior into the BAO\ndataset alone yields a sound horizon at the drag epoch\nof 139.365 ±9.762 Mpc. In the case of the Joint dataset,\nthe calculated BAO scale ( rd) is 147.906 ±2.386 Mpc,\nwhich closely matches the results obtained from the\nPlanck mission. Additionally, when we include the\nR22 prior in the complete dataset, the resulting rdis\n141.623 ±2.638 Mpc, indicating a close agreement with\nthe findings reported in [64]. The results from ΛCDM,\nVsMCG and VMCG models exhibit tension with the rd\nvalue estimated by Planck. However, these results still\ndemonstrate agreement with the findings presented in\n[64] reports that by employing Binning and Gaussian\nmethods to combine measurements of 2D BAO and\nSNIa data, the values of the absolute BAO scale range\nfrom 141.45 Mpc to rd≤159.44Mpc (Binning) and\n143.35 Mpc to rd≤161.59Mpc (Gaussian). These\nfindings highlight a clear discrepancy between early\nand late-time observational measurements, analogous9\n50607080\nH01.3×1061.32×1061.34×1061.36×1061.38×1061.4×1061\n1.41.61.82.02.22.42.60\n234A0.00.20.40.60.81.0\n0.00.51.0AS0.40.60.81.0d0\n0.20.30.40.50.6m0\n0.20.30.40.50.6\nm0\n0.60.81.0\nd0\n0.00.51.0\nAS0.00.51.0\n234\nA1.52.02.5\n0\n1.321.38\n1\n×106BAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22\nFIG. 3: The figure illustrates the posterior distribution of various observational data measurements using the\nViscous Modified Chaplygin Gas model, highlighting the 1 σand 2 σregions.\nto the H0tension. It is noteworthy that our results\nare contingent on the range of priors for rdand H0,\ninfluencing the estimated values in the rd−H0contour\nplane. An interesting observation is that when we\nexclude the R22 prior, the results for H0and rdtend to\nalign with the Planck and SDSS results, mitigating the\ntension observed in the absence of this particular prior.Now, in the study of the gravitational lensing opti-\ncal depth, we use the analytical expression given by\neqn. (25). After constraining the model parameters\nof the DE models as discussed above and finding\ntheir best-fit values, we plot the graphs of τ(zs)/F\nagainst redshift zsfor both models. Figs. 6 and 7,10\n50607080\nH01.3×1061.32×1061.34×1061.36×1061.38×1061.4×1061\n1.01.21.41.6Cs0.000.050.100.150.20C0.91.01.11.2\n0.850.900.951.00n0.40.50.60.70.80.91.0d0\nm0\nm0\n0.60.81.0\nd0\n0.850.900.951.00\nn0.91.01.11.2\n0.00.10.2\nC1.01.21.41.6\nCs1.321.38\n1\n×106BAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22\n0.350.300.250.200.150.150.200.250.300.35\nFIG. 4: The figure illustrates the posterior distribution of various observational data measurements using the\nVariable Modified Chaplygin Gas model, highlighting the 1 σand 2 σregions.\nrespectively, show changes in the optical depth be-\nhaviour of the two DE models here w.r.t zs. We notice\nthat the probability of finding gravitational lenses\n(i.e. optical depth) increases for both the models with\nincreasing zsvalue. Further, from fig. 6, we observe\nthat the graphs of τ(zs)for parameter sets constrained\nthrough different data sets show similar behaviourexcept for the parameters constraint through BAO\nonly. However, for zs<1, they all coincide with each\nother. From these, we can conclude that the parameter\nvalues constraint through different data sets have\nvery little to no effect in detecting the distribution\nof gravitational lenses in our Universe. Again, if we\nanalyse fig. 7, we observe that parameters constrained11\n60 65 70 75 80\nH0130140150160170rd(Mpc)\nBAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22\na)ΛCDM Model\n60 65 70 75\nH0130140150160170rd\nBAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22 b) VsMCG Model\n60 70 80\nH0130140150160170rd\nBAO\nBAO + R22\nCC + SNIa + Q + GRB + BAO\nCC + SNIa + Q + GRB + BAO + R22 c) VMCG Model\nFIG. 5: The figure shows the posterior distribution of diverse observational data measurements within the rd−H0\ncontour plane using the ΛCDM, VsMCG and VMCG models. The shaded regions correspond to the 1 σand 2 σ\nconfidence plane.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nzs0.050.100.150.200.250.300.350.400.45(zs)\nF\nBAO\nBAO+R22\nCC+SC+BAO\nCC+SC+BAO+R22\nFIG. 6: Variation in optical depth behaviour of VsMCG\nmodel w.r.t zsfor parameters constrained through\ndifferent data sets given in Table II.\nthrough the combination of CC+SC+BAO +R22 give\na lower probability of finding gravitational lenses as\ncompared to the other three combinations. However,\nfor the other three combinations of data sets, viz.\nBAO ,BAO +R22,CC+SC+BAO , we can observe\nsmall deviations in τ(zs)for higher redshift values but\nforzs<1, the graph shows same lensing probability.\nThis implies that in this DE model, parameter values\nconstrained through CC+SC+BAO +R22 data sets\ngive a slightly better approximation of the distribution\nof gravitational lenses than the rest. Finally, we con-\nclude our study by plotting the optical depth of the\ntwo DE models considered here along with ΛCDM\nagainst source redshift zs. We observe from Fig. 8 that\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nzs0.050.100.150.200.250.30(zs)\nF\nBAO\nBAO+R22\nCC+SC+BAO\nCC+SC+BAO+R22FIG. 7: Variation in optical depth behaviour of VMCG\nmodel w.r.t zsfor parameters constrained through\ndifferent data sets given in Table II.\nalthough at the present epoch, the gravitational lensing\nprobability for the three models coincides with each\nother, however moving back in time (or higher redshift\nvalues), all the models highly diverge from each other\nwithΛCDM gives the highest lensing probability and\nVMCG the lowest.\nVI. DISCUSSIONS AND CONCLUSIONS\nAssuming our Universe to be filled with dark matter\nand dark energy combinations in a flat FRW Universe,\nwe reviewed two models of modified Chaplygin gas as\nthe fluid source. For the dark energy candidates, we12\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nRedshift (zs)0.10.20.30.40.5(z)/F\nCosmological Models Comparison\nCDM\nVsMCG\nVMCG\nFIG. 8: Variation in optical depth behaviour of ΛCDM\n(H0=71.67, Ωm0=0.26), VsMCG\n(H0=72.86, Ωm0=0.28, As=0.44, α=0.66, A=\n3.09, ζ=2.05, ρ1=1.35351 ×106) and VMCG\n(H0=74.02, Ωm0=0.18, n=0.97, β=1.02, C=\n0.03, Cs=0.48, ρ1=1.35085 ×106) model w.r.t source\nredshift zs.\nconsidered viscous modified Chaplygin gas (VsMCG)\nand variable modified Chaplygin gas (VMCG) in the\nframework of loop quantum cosmology. We determined\nthe respective Hubble parameter and, consequently,\nconstraint the associated model parameters with the\nhelp of the latest observational data. Our investiga-\ntion involved the selection of 24 Baryon Acoustic Os-\ncillation (BAO) points were deliberately chosen to min-\nimize correlations from an extensive dataset compris-\ning 333 BAO data points. The objective was to miti-\ngate inherent errors in the posterior distribution, which\ncould potentially stem from correlations between mea-\nsurements. Employing a method that mimics the intro-\nduction of random correlations into the covariance ma-\ntrix, we conducted a systematic procedure. Our subse-\nquent verification process attested that these introduced\ncorrelations do not lead to substantial alterations in the\nresulting cosmological parameters. Furthermore, we in-\ncorporated a variety of datasets into our analysis. This\ninvolved the inclusion of 30 uncorrelated data points\nfrom Cosmic Chronometers, a carefully selected set of\n40 points obtained from Type Ia supernovae, 24 points\nextracted from the Hubble diagram for quasars, and a\nconsiderable dataset consisting of 162 points derived\nfrom Gamma Ray Bursts. Additionally, we integrated\nthe most recent measurement of the Hubble constant\nconducted by researcher R22. Our recent investigation,conducted through a comprehensive array of observa-\ntional assessments, brings attention to the persistent ex-\nistence of the Hubble tension, although it is somewhat\nmitigated to a 2 σlevel for H0. By introducing the sound\nhorizon ( rd) as a free parameter, we have derived spe-\ncific values for H0and rdin various cosmological mod-\nels, including the Standard ΛCDM, VsMCG, and VMCG\nmodels. In the ΛCDM model, our analysis reveals H0=\n69.828145 ±1.009964 km/s/Mpc and rd=146.826212 ±\n2.201612 Mpc. For the VsMCG model, we obtain H0=\n67.447204 ±8.878032 km/s/Mpc and rd=147.559044 ±\n2.723020 Mpc. Similarly, in the case of VMCG, the re-\nsults indicate H0=69.204717 ±8.056031 km/s/Mpc\nand rd=147.906003 ±2.386071 Mpc. Crucially, our as-\nsessments highlight that the values of H0and rdbased\non low-redshift measurements align with early Planck\nestimates [15]. We also studied the gravitational lens-\ning optical depth of our two dark energy models with\nthe help of the constrained model parameter values.\nThe graph showing changes in optical depth is plot-\nted against their source redshift (fig. 6 & 7). For both\nmodels, the probability of finding gravitational lenses\nincreases with higher redshift values. However, for low\nredshift ( zs<1), the lensing probability in both the DE\nmodels coincides irrespective of the different sets of con-\nstrained parameter values obtained by considering dif-\nferent combinations of data sets. Lastly, we also carried\nout a joint analysis of the two DE models considered\nhere with that of ΛCDM model (fig. 8). We found that\nthe three models are indistinguishable for low redshift\nvalue ( zs<1); however, they highly diverge from each\nother in the early Universe. Finally, we conclude our\nstudy with a statistical evaluation of our cosmological\nmodels and utilize both the Akaike Information Crite-\nrion (AIC) and the Bayesian Information Criterion (BIC).\nThe AIC is expressed as [65, 66]: AIC =−2 ln(Lmax) +\n2k+2k(2k+1)\nNtot−k−1Here, Lmaxsignifies the maximum likeli-\nhood of the data, incorporating the entire dataset with-\nout the R22 prior. The parameters encompass Ntot, the\ntotal number of data points (in our instance, Ntot=\n296), and k, the number of parameters. For large Ntot,\nthis expression simplifies to:AIC ≃ − 2 ln(Lmax) +2k\nwhich is the conventional form of the AIC criterion [65].\nIn contrast, the Bayesian Information Criterion is for-\nmulated as [66]: BIC =−2 ln(Lmax) +klnNtot. By\napplying these criteria, we compute the AIC and BIC\nfor the standard ΛCDM, VsMCG, and VMCG models.\nThe obtained values for ΛCDM, VsMCG, and VMCG\nmodels are, respectively AIC = [277.38, 281.16, 281.12 ]\nand BIC = [277.59, 283.12, 282.34 ]. Despite the ΛCDM\nmodel displaying the best fit due to the lowest AIC,\nour collective AIC and BIC results lend support to all13\nthe tested models. This suggests that none of the mod-\nels can be dismissed based on the existing data. In\nthe evaluation of the VsMCG and VMCG models rel-\native to ΛCDM, we acknowledge that ΛCDM is em-\nbedded within both proposed extensions, differing by\n5 degrees of freedom. This distinction allows for the\napplication of standard statistical tests. The yardstick\nfor comparison is the reduced chi-square statistic, de-\nfined as χ2\nred=χ2/Dof, where Dof represents the de-\ngrees of freedom of the model, and χ2denotes theweighted sum of squared deviations with an equal num-\nber of runs for the three models, the statistic approxi-\nmates 1, expressed as:\u0012\nχ2\nDo fΛCDM,χ2\nDo f VsMCG,χ2\nDo f VMCG\u0013\n≈\n{0.961646, 0.971234, 0.954310 }. This comparative analy-\nsis offers valuable insights into the goodness of fit for\neach model, with values near 1 signifying a satisfactory\nalignment with the observed data. The findings suggest\nthat both the VsMCG and VMCG models provide viable\nalternatives to the standard ΛCDM model, with compa-\nrable goodness of fit and support from statistical criteria.\n[1] A. G. Riess, A. V . Filippenko, and et al., “Observational\nevidence from supernovae for an accelerating universe\nand a cosmological constant,” Astron. J. , vol. 116, no. 3,\np. 1009, 1998.\n[2] D. N. Spergel, L. Verde, and et al., “First-year Wilkin-\nson Microwave Anisotropy Probe (WMAP)* observa-\ntions: determination of cosmological parameters,” Astro-\nphys. J. Suppl. Ser. , vol. 148, no. 1, p. 175, 2003.\n[3] A. R. Liddle and D. H. Lyth, Cosmological inflation and\nlarge-scale structure . Cambridge university press, 2000.\n[4] E. Komatsu, J. Dunkley, and et al., “FIVE-YEAR WILKIN-\nSON MICROWAVE ANISOTROPY PROBE OBSERVA-\nTIONS: COSMOLOGICAL INTERPRETATION,” Astro-\nphys. J. Suppl. Ser. , vol. 180, pp. 330–376, feb 2009.\n[5] A. Kamenshchik, U. Moschella, and V . Pasquier, “An\nalternative to quintessence,” Physics Letters B , vol. 511,\nno. 2-4, pp. 265–268, 2001.\n[6] U. Debnath, A. Banerjee, and S. Chakraborty, “Role of\nmodified chaplygin gas in accelerated universe,” Classi-\ncal and Quantum Gravity , vol. 21, no. 23, p. 5609, 2004.\n[7] P . Peebles and B. Ratra, “Cosmology with a time-variable\ncosmological’constant’,” Astrophysical Journal, Part 2-\nLetters to the Editor (ISSN 0004-637X), vol. 325, Feb. 15,\n1988, p. L17-L20. NSF-supported research. , vol. 325, pp. L17–\nL20, 1988.\n[8] R. R. Caldwell, R. Dave, and P . J. Steinhardt, “Cosmolog-\nical imprint of an energy component with general equa-\ntion of state,” Physical Review Letters , vol. 80, no. 8, p. 1582,\n1998.\n[9] U. Alam, V . Sahni, T. Deep Saini, and A. A. Starobinsky,\n“Is there supernova evidence for dark energy metamor-\nphosis?,” Monthly Notices of the Royal Astronomical Society ,\nvol. 354, no. 1, pp. 275–291, 2004.\n[10] S. D. Hsu, “Entropy bounds and dark energy,” Phys. Lett.\nSect. B Nucl. Elem. Part. High-Energy Phys. , vol. 594, no. 1-\n2, pp. 13–16, 2004.\n[11] M. Li, “A model of holographic dark energy,” Phys. Lett.\nSect. B Nucl. Elem. Part. High-Energy Phys. , vol. 603, no. 1-\n2, pp. 1–5, 2004.\n[12] E. J. COPELAND, M. SAMI, and S. TSUJIKAWA, “DY-\nNAMICS OF DARK ENERGY,” Int. J. Mod. Phys. D ,\nvol. 15, pp. 1753–1935, nov 2006.[13] V . Springel, C. S. Frenk, and S. D. White, “The large-\nscale structure of the universe,” nature , vol. 440, no. 7088,\npp. 1137–1144, 2006.\n[14] A. Jenkins, C. Frenk, F. Pearce, P . Thomas, J. Colberg,\nS. D. White, H. Couchman, J. Peacock, G. Efstathiou, and\nA. Nelson, “Evolution of structure in cold dark matter\nuniverses,” The Astrophysical Journal , vol. 499, no. 1, p. 20,\n1998.\n[15] P . Collaboration, N. Aghanim, Y. Akrami, M. Ashdown,\nJ. Aumont, C. Baccigalupi, M. Ballardini, A. Banday,\nR. Barreiro, N. Bartolo, et al. , “Planck 2018 results. vi. cos-\nmological parameters,” 2020.\n[16] A. G. Riess, A. V . Filippenko, P . Challis, A. Clocchiatti,\nA. Diercks, P . M. Garnavich, R. L. Gilliland, C. J. Hogan,\nS. Jha, R. P . Kirshner, et al. , “Observational evidence from\nsupernovae for an accelerating universe and a cosmolog-\nical constant,” The astronomical journal , vol. 116, no. 3,\np. 1009, 1998.\n[17] D. J. Eisenstein and W. Hu, “Baryonic features in the mat-\nter transfer function,” The Astrophysical Journal , vol. 496,\nno. 2, p. 605, 1998.\n[18] P . J. Peebles and J. Yu, “Primeval adiabatic perturbation\nin an expanding universe,” Astrophysical Journal, vol. 162,\np. 815 , vol. 162, p. 815, 1970.\n[19] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro,\nM. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo,\nM. Tegmark, Z. Zheng, et al. , “Detection of the baryon\nacoustic peak in the large-scale correlation function of\nsdss luminous red galaxies,” The Astrophysical Journal ,\nvol. 633, no. 2, p. 560, 2005.\n[20] S. Alam, F. D. Albareti, C. A. Prieto, F. Anders, S. F.\nAnderson, T. Anderton, B. H. Andrews, E. Armengaud,\n´E. Aubourg, S. Bailey, et al. , “The eleventh and twelfth\ndata releases of the sloan digital sky survey: final data\nfrom sdss-iii,” The Astrophysical Journal Supplement Series ,\nvol. 219, no. 1, p. 12, 2015.\n[21] S. NOJIRI and S. D. ODINTSOV , “INTRODUCTION TO\nMODIFIED GRAVITY AND GRAVITATIONAL ALTER-\nNATIVE FOR DARK ENERGY,” Int. J. Geom. Methods\nMod. Phys. , vol. 04, pp. 115–145, feb 2007.\n[22] S. Capozziello and M. De Laurentis, “Extended Theories\nof Gravity,” Phys. Rep. , vol. 509, pp. 167–321, dec 2011.14\n[23] A. Ashtekar and J. Lewandowski, “Background inde-\npendent quantum gravity: a status report,” Classical and\nQuantum Gravity , vol. 21, no. 15, p. R53, 2004.\n[24] M. Bojowald, “Loop quantum cosmology,” Living Reviews\nin Relativity , vol. 11, pp. 1–131, 2008.\n[25] M. Jamil and U. Debnath, “Interacting modified chap-\nlygin gas in loop quantum cosmology,” Astrophysics and\nSpace Science , vol. 333, pp. 3–8, 2011.\n[26] X. Fu, H. Yu, and P . Wu, “Dynamics of interacting phan-\ntom scalar field dark energy in loop quantum cosmol-\nogy,” Physical Review D , vol. 78, no. 6, p. 063001, 2008.\n[27] S. Refsdal and H. Bondi, “The gravitational lens effect,”\nMonthly Notices of the Royal Astronomical Society , vol. 128,\nno. 4, pp. 295–306, 1964.\n[28] E. L. Turner, J. P . Ostriker, and J. R. Gott III, “The statistics\nof gravitational lenses-the distributions of image angular\nseparations and lens redshifts,” The Astrophysical Journal ,\nvol. 284, pp. 1–22, 1984.\n[29] J. R. Gott III, M.-G. Park, and H. M. Lee, “Setting limits\non q0 from gravitational lensing,” Astrophysical Journal,\nPart 1 (ISSN 0004-637X), vol. 338, March 1, 1989, p. 1-12. ,\nvol. 338, pp. 1–12, 1989.\n[30] Z.-H. Zhu, “Gravitational lensing statistical properties in\ngeneral frw cosmologies with dark energy component (s):\nanalytic results,” International Journal of Modern Physics D ,\nvol. 9, no. 05, pp. 591–600, 2000.\n[31] Z.-H. Zhu, “Gravitational lensing statistics as a probe of\ndark energy,” Modern Physics Letters A , vol. 15, no. 16,\npp. 1023–1029, 2000.\n[32] R. Freitas, S. Gonc ¸alves, and A. Oliveira, “Gravitational\nlenses in the dark universe,” Astrophysics and Space Sci-\nence, vol. 349, pp. 443–455, 2014.\n[33] R. Kundu, U. Debnath, and A. Pradhan, “Gravitational\nlensing: dark energy models in non-flat frw universe,”\nThe European Physical Journal C , vol. 83, no. 6, p. 553, 2023.\n[34] D. Aberkane, N. Mebarki, and S. Benchikh, “Viscous\nmodified chaplygin gas in classical and loop quan-\ntum cosmology,” Chinese Physics Letters , vol. 34, no. 6,\np. 069801, 2017.\n[35] U. Debnath, “Variable modified chaplygin gas and accel-\nerating universe,” Astrophysics and Space Science , vol. 312,\nno. 3-4, pp. 295–299, 2007.\n[36] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Bur-\nden, and M. Manera, “The clustering of the sdss dr7 main\ngalaxy sample–i. a 4 per cent distance measure at z= 0.15,”\nMonthly Notices of the Royal Astronomical Society , vol. 449,\nno. 1, pp. 835–847, 2015.\n[37] S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A.\nBlazek, A. S. Bolton, J. R. Brownstein, A. Burden, C.-\nH. Chuang, et al. , “The clustering of galaxies in the\ncompleted sdss-iii baryon oscillation spectroscopic sur-\nvey: cosmological analysis of the dr12 galaxy sample,”\nMonthly Notices of the Royal Astronomical Society , vol. 470,\nno. 3, pp. 2617–2652, 2017.\n[38] H. Gil-Mar ´ın, J. E. Bautista, R. Paviot, M. Vargas-Maga ˜na,\nS. de La Torre, S. Fromenteau, S. Alam, S. ´Avila, E. Burtin,\nC.-H. Chuang, et al. , “The completed sdss-iv extendedbaryon oscillation spectroscopic survey: measurement of\nthe bao and growth rate of structure of the luminous red\ngalaxy sample from the anisotropic power spectrum be-\ntween redshifts 0.6 and 1.0,” Monthly Notices of the Royal\nAstronomical Society , vol. 498, no. 2, pp. 2492–2531, 2020.\n[39] A. Raichoor, A. De Mattia, A. J. Ross, C. Zhao, S. Alam,\nS. Avila, J. Bautista, J. Brinkmann, J. R. Brownstein,\nE. Burtin, et al. , “The completed sdss-iv extended baryon\noscillation spectroscopic survey: large-scale structure cat-\nalogues and measurement of the isotropic bao between\nredshift 0.6 and 1.1 for the emission line galaxy sample,”\nMonthly Notices of the Royal Astronomical Society , vol. 500,\nno. 3, pp. 3254–3274, 2021.\n[40] J. Hou, A. G. S ´anchez, A. J. Ross, A. Smith, R. Neveux,\nJ. Bautista, E. Burtin, C. Zhao, R. Scoccimarro, K. S. Daw-\nson, et al. , “The completed sdss-iv extended baryon oscil-\nlation spectroscopic survey: Bao and rsd measurements\nfrom anisotropic clustering analysis of the quasar sam-\nple in configuration space between redshift 0.8 and 2.2,”\nMonthly Notices of the Royal Astronomical Society , vol. 500,\nno. 1, pp. 1201–1221, 2021.\n[41] H. D. M. Des Bourboux, J. Rich, A. Font-Ribera,\nV . de Sainte Agathe, J. Farr, T. Etourneau, J.-M. Le Goff,\nA. Cuceu, C. Balland, J. E. Bautista, et al. , “The completed\nsdss-iv extended baryon oscillation spectroscopic survey:\nbaryon acoustic oscillations with ly αforests,” The Astro-\nphysical Journal , vol. 901, no. 2, p. 153, 2020.\n[42] T. Abbott, M. Aguena, S. Allam, A. Amon, F. Andrade-\nOliveira, J. Asorey, S. Avila, G. Bernstein, E. Bertin,\nA. Brandao-Souza, et al. , “Dark energy survey year 3 re-\nsults: A 2.7% measurement of baryon acoustic oscilla-\ntion distance scale at redshift 0.835,” Physical Review D ,\nvol. 105, no. 4, p. 043512, 2022.\n[43] S. Sridhar, Y.-S. Song, A. J. Ross, R. Zhou, J. A. New-\nman, C.-H. Chuang, R. Blum, E. Gaztanaga, M. Landriau,\nand F. Prada, “Clustering of lrgs in the decals dr8 foot-\nprint: Distance constraints from baryon acoustic oscilla-\ntions using photometric redshifts,” The Astrophysical Jour-\nnal, vol. 904, no. 1, p. 69, 2020.\n[44] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-\nSmith, L. Campbell, Q. Parker, W. Saunders, and F. Wat-\nson, “The 6df galaxy survey: baryon acoustic oscilla-\ntions and the local hubble constant,” Monthly Notices of the\nRoyal Astronomical Society , vol. 416, no. 4, pp. 3017–3032,\n2011.\n[45] L. Kazantzidis and L. Perivolaropoulos, “Evolution of the\nfσ8 tension with the planck 15/ λcdm determination and\nimplications for modified gravity theories,” Physical Re-\nview D , vol. 97, no. 10, p. 103503, 2018.\n[46] M. Moresco, “Raising the bar: new constraints on the\nhubble parameter with cosmic chronometers at z 2,”\nMonthly Notices of the Royal Astronomical Society: Letters ,\nvol. 450, no. 1, pp. L16–L20, 2015.\n[47] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maras-\nton, L. Verde, D. Thomas, A. Citro, R. Tojeiro, and\nD. Wilkinson, “A 6% measurement of the hubble pa-\nrameter at z 0.45: direct evidence of the epoch of cos-15\nmic re-acceleration,” Journal of Cosmology and Astroparticle\nPhysics , vol. 2016, no. 05, pp. 014–014, 2016.\n[48] M. Moresco, L. Verde, L. Pozzetti, R. Jimenez, and\nA. Cimatti, “New constraints on cosmological parameters\nand neutrino properties using the expansion rate of the\nuniverse to z 1.75,” Journal of Cosmology and Astroparticle\nPhysics , vol. 2012, no. 07, p. 053, 2012.\n[49] M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti,\nG. Zamorani, M. Bolzonella, J. Dunlop, F. Lamareille,\nM. Mignoli, H. Pearce, et al. , “Improved constraints on\nthe expansion rate of the universe up to z 1.1 from the\nspectroscopic evolution of cosmic chronometers,” Journal\nof Cosmology and Astroparticle Physics , vol. 2012, no. 08,\npp. 006–006, 2012.\n[50] D. M. Scolnic, D. Jones, A. Rest, Y. Pan, R. Chornock,\nR. Foley, M. Huber, R. Kessler, G. Narayan, A. Riess,\net al. , “The complete light-curve sample of spectroscopi-\ncally confirmed sne ia from pan-starrs1 and cosmological\nconstraints from the combined pantheon sample,” The As-\ntrophysical Journal , vol. 859, no. 2, p. 101, 2018.\n[51] C. Roberts, K. Horne, A. O. Hodson, and A. D. Leg-\ngat, “Tests of λcdm and conformal gravity using grb and\nquasars as standard candles out to z∼8,”arXiv preprint\narXiv:1711.10369 , 2017.\n[52] M. Demianski, E. Piedipalumbo, D. Sawant, and L. Am-\nati, “Cosmology with gamma-ray bursts-ii. cosmography\nchallenges and cosmological scenarios for the accelerated\nuniverse,” Astronomy & Astrophysics , vol. 598, p. A113,\n2017.\n[53] A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout,\nS. Casertano, D. O. Jones, Y. Murakami, G. S. Anand,\nL. Breuval, et al. , “A comprehensive measurement of the\nlocal value of the hubble constant with 1 km s- 1 mpc- 1\nuncertainty from the hubble space telescope and the sh0es\nteam,” The Astrophysical journal letters , vol. 934, no. 1,\np. L7, 2022.\n[54] W. Handley, M. Hobson, and A. Lasenby, “Polychord:\nnested sampling for cosmology,” Monthly Notices of the\nRoyal Astronomical Society: Letters , vol. 450, no. 1, pp. L61–\nL65, 2015.[55] A. Lewis, “Getdist: a python package for analysing monte\ncarlo samples,” arXiv preprint arXiv:1910.13970 , 2019.\n[56] J. R. Gott III and J. E. Gunn, “The double quasar 1548+\n115a, b as a gravitational lens,” Astrophysical Journal, vol.\n190, p. L105 , vol. 190, p. L105, 1974.\n[57] E. L. Turner, “Gravitational lensing limits on the cosmo-\nlogical constant in a flat universe,” Astrophys. J. , vol. 365,\npp. L43—-L46, 1990.\n[58] M. Fukugita, T. Futamase, and M. Kasai, “A possible test\nfor the cosmological constant with gravitational lenses,”\nMon. Not. R. Astron. Soc. , vol. 246, p. 24P , 1990.\n[59] L. M. Krauss and M. White, “Gravitational lensing, finite\ngalaxy cores, and the cosmological constant,” Astrophys.\nJ., vol. 394, pp. 385–395, 1992.\n[60] C. S. Kochanek, “Analytic results for the gravitational\nlens statistics of singular isothermal spheres in general\ncosmologies,” Mon. Not. R. Astron. Soc. , vol. 261, no. 2,\npp. 453–463, 1993.\n[61] R. C. Nunes, S. K. Yadav, J. Jesus, and A. Bernui, “Cosmo-\nlogical parameter analyses using transversal bao data,”\nMonthly Notices of the Royal Astronomical Society , vol. 497,\nno. 2, pp. 2133–2141, 2020.\n[62] R. C. Nunes and A. Bernui, “Bao signatures in the 2-point\nangular correlations and the hubble tension,” The Euro-\npean Physical Journal C , vol. 80, pp. 1–8, 2020.\n[63] L. Verde, J. L. Bernal, A. F. Heavens, and R. Jimenez, “The\nlength of the low-redshift standard ruler,” Monthly Notices\nof the Royal Astronomical Society , vol. 467, no. 1, pp. 731–\n736, 2017.\n[64] T. Lemos, Ruchika, J. C. Carvalho, and J. Alcaniz, “Low-\nredshift estimates of the absolute scale of baryon acoustic\noscillations,” The European Physical Journal C , vol. 83, no. 6,\np. 495, 2023.\n[65] H. Akaike, “A new look at the statistical model identifica-\ntion,” IEEE transactions on automatic control , vol. 19, no. 6,\npp. 716–723, 1974.\n[66] G. Schwarz, “Estimating the dimension of a model,” The\nannals of statistics , pp. 461–464, 1978."    },    {        "title": "2402.07551v1.Pearcey_integrals__Stokes_lines_and_exact_baryonic_layers_in_the_low_energy_limit_of_QCD.pdf",        "content": "Pearcey integrals, Stokes lines and exact baryonic layers\nin the low energy limit of QCD\nSergio L. Cacciatori,1, 2Fabrizio Canfora,3, 4and Federica Muscolino3, 4\n1Universit` a dell’Insubria, Dipartimento di Scienza ed Alta Tecnologia, via Valleggio 11, 22100, Como, Italy\n2INFN, via Celoria 16, 20133, Milano, Italy\n3Universidad San Sebasti´ an, Facultad de Ingenier´ ıa, Arquitectura y Dise˜ no,\nsede Valdivia, General Lagos 1163, Valdivia 8420524, Chile\n4Centro de Estudios Cient´ ıficos (CECS), Casilla 1469, Valdivia, Chile\nThe first analytic solutions representing baryonic layers living at finite baryon density within\na constant magnetic field in the gauged Skyrme model are constructed. A remarkable feature of\nthese configurations is that, if the Skyrme term is neglected, then these baryonic layers in the\nconstant magnetic background cannot be found analytically and their energies grow very fast with\nthe magnetic field. On the other hand, if the Skyrme term is taken into account, the field equations\ncan be solved analytically and the corresponding solutions have a smooth limit for large magnetic\nfields. Thus, the Skyrme term discloses the universal character of these configurations living at\nfinite Baryon density in a constant magnetic field. The classical gran-canonical partition function\nof these configurations can be expressed explicitly in terms of the Pearcey integral. This fact\nallows us to determine analytically the Stokes lines of the partition function and the corresponding\ndependence on the baryonic chemical potential as well as on the external magnetic field. In this\nway, we can determine various critical curves in the ( µB−Bext) plane which separates different\nphysical behaviors. These families of inhomogeneous baryonic condensates can be also dressed with\nchiral conformal excitations of the solutions representing modulations of the layers themselves. Some\nphysical consequences are analyzed.\nCONTENTS\nI. Introduction 1\nA. Notation and characteristics of the model 3\n1. The properties of the gauged Skyrme\nmodel 3\n2. The Non-Linear Sigma Model vs. the\nSkyrme model 4\nII. Baryonic layers coupled to an external magnetic\nfield 4\nA. Baryonic layers solutions in the NL σM 5\n1. The energy density and baryonic charge 5\n2. The large external magnetic field limit 6\nB. Baryonic layers solutions in the Skyrme\nmodel 6\n1. The energy density and baryonic charge 6\n2. The thermodynamics of the universal\nsolution 7\n3. The dependence on the external field 8\n4. Free energy and susceptibility 11\nIII. Chiral conformal field theory of baryonic layers\nfrom the Skyrme model 12\nIV. Baryonic tubes coupled to an external magnetic\nfield 14\nA. Baryonic tubes solutions in the NL σM 15\nB. Baryonic tubes solutions in the Skyrme\nmodel 15\nV. Discussion and perspectives 16Acknowledgements 17\nA. The flavor-oscillating spaghetti 17\nB. On the equations of motion. 18\nC. General form of the Skyrme equations 19\n1. The Euler parameterization 19\n2. The exponential parameterization 20\nD. The caustic curve 20\nE. The Pearcey integral 22\nF. The piecewise-linear solutions 26\nG. The polarization effects 27\nReferences 28\nI. INTRODUCTION\nNon-perturbative effects of Quantum Chromodynamics\n(QCD) play a fundamental role in the low-energy pro-\ncesses of strong interactions since the color confinement\nprevents from describing the low energy limit of systems\nof quarks through perturbative approaches [1–3]. The\nunknown behavior of strong interactions in this regime\nmakes the understanding of certain phenomena a hard\ntask; it is the case, for instance, of the internal structure\nof Neutron Stars. Indeed, the intermediate layers of these\nobjects are known to be composed by particular states\nof baryonic matter, called Nuclear Pasta [4–11]. Here,arXiv:2402.07551v1  [hep-th]  12 Feb 20242\nbaryons appear to be deformed and organized in ordered\narrays with different shapes, which reminds of pasta (for\ninstance, they can form tubes, which reminds of spaghetti\nstates , or layers, which reminds of lasagna states , or balls,\nwhich remind of gnocchi states , and so on). Their forma-\ntion is due to the competition of non-perturbative QCD\neffects and Coulomb interactions. An analytical descrip-\ntion of these interactions is, by now, out of reach; for\nthis reason, they are generally studied employing numer-\nical simulations [4–7, 12, 13]. With these methods, it is\npossible to characterize the general structures of nuclear\npasta and some of its physical properties. The problem\nof the numerical simulations lies in the fact that the re-\nsults are limited by the necessity of a huge computational\npower; for instance, the introduction of external interac-\ntions and the study of these nuclear structures in curved\nspace-time is a hard task (see, for instance, [13]). Alter-\nnative methods require a more fundamental analysis of\nthe non-perturbative aspects of QCD. An example is pro-\nvided by the Lattice QCD (LQCD), in which the quark\nfields are represented by lattice sites and gluons by the\nlinks between them [14–23]. When the lattice is infinitely\nlarge, the non-perturbative effects arise. As shown in the\ncited works, LQCD leads to remarkable results, but the\nnon-linear nature of QCD prevents an analytical descrip-\ntion; moreover, the numerical results yield the so-called\nsign problem . For this reason, it is not very effective in\nthe study of the phase diagrams of QCD at finite tem-\nperatures.\nRecent results show that an analytical description of\nstructures of baryons is possible, considering the non-\nperturbative effects of QCD as properties of some topo-\nlogical spaces; this link is a known fact that has been\nstudied for decades by many authors [24–30]. A remark-\nable example is represented by the Skyrme model, which\nhas been shown to represent an Effective Field Theory\n(EFT) for strong interactions, providing a good descrip-\ntion of both single baryons and multi-baryonic systems\n[31–55]. The main difficult encountered in this repre-\nsentation is represented by the complicated shape of the\nequations of motion (known as the Skyrme equations ),\nwhich requires the introduction of an ansatz in order to\nobtain analytical solutions.\nOur interest focuses on nuclear pasta structures descrip-\ntion; in this scope, we refer to the works [39–55], in which\ntwo particular ans¨ atze have been introduced. The main\ngoal of these ans¨ atze is to reduce the Skyrme equations\nto one ODE, which allows describing the properties of\nthese baryonic structures through analytical tools. They\nare based on two parameterizations of the flavor group G;\nnamely, the exponential parameterization and the gener-\nalized Euler parametrization (for more details on the gen-\neralized Euler parameterization see, for instance, [56–58]\nand references therein). The employment of the first pa-\nrameterization is shown to describe a system in which the\nenergy and the baryonic charge concentrate in tubes and,\nfor this reason, are called baryonic tubes . Their shape can\nbe reconducted to the nuclear spaghetti states; therefore,the aforementioned ansatz is called spaghetti ansatz . The\nsecond is suitable for the description of baryonic layers ,\nwhich shape is conductible to the nuclear lasagna states;\nas for the previous case, the ansatz obtained using the\nEuler parameterization will be called lasagna ansatz . In\n[59], these ans¨ atze have been extended showing that they\ncan be used to describe the thermodynamical character-\nistics of nuclear pasta, obtaining interesting differences\nbetween the Non-Linear Sigma model (NL σM), which is\nobtained from the Skyrme model omitting the Skyrme\nterm, and the Skyrme model. More specifically, the\nNLσM allows for conformal solutions; whereas, in the\nSkyrme model, chiral conformal solutions are founded.\nThis is a remarkable result since it allows studying the\nthermodynamics of these baryonic structures from the\ncomputation of the partition function.\nThe techniques to construct analytic inhomogeneous con-\ndensates in the above references also work when the\nSkyrme model is minimally coupled to Maxwell field\n[54, 55]. The exact solutions describing baryonic layers\nand the tube together with the corresponding Maxwell\nfields generated by the U(1) Skyrme current have been\nfound by answering the following question: how should\none choose the ansatz for the Maxwell potential to keep\nthe solvability properties of the ungauged ansatz in a sec-\ntor with non-zero Baryonic charge? This question has a\nunique answer discussed in [55]. These exact solutions of\nthe gauged Skyrme - Maxwell system are Baryonic lay-\ners and tubes together with their electromagnetic field.\nHowever, there are important cases (from neutron stars\n[60] to heavy Ions collisions [61, 62]) in which the ex-\nternal electromagnetic fields are much stronger than the\nelectromagnetic fields generated by the inhomogeneous\nBaryonic condensates. When this happens, it is natu-\nral to ask: how intense external electromagnetic fields\ndeform the inhomogeneous Baryonic condensates of the\nlow energy limit of QCD? It could appear that when\nthe electromagnetic field is an external fixed background\nfield, it should be easier to handle the problem of find-\ning inhomogeneous condensates. Indeed, one may think\nthat the problem of analyzing only the gauged Skyrme\nfield equations in the given background field is easier than\nthe problem (analyzed and solved in [55]) of construct-\ning analytic baryonic layers and tubes together with their\nelectromagnetic fields generated in a self-consistent way.\nUnfortunately, this is not true as in those references [55]\nthe ansatz for the electromagnetic field is fixed uniquely\nby requiring a force-free condition [55]. On the other\nhand, when there is a strong external electromagnetic\nfield, it is necessary to adapt the ansatz for the SU(2)-\nvalued Skyrme field to the external fixed electromagnetic\nfield (which, in general, will not be force-free) and not\nthe other way around. Consequently, a relevant goal is\nto generalize the approach in [55] in order to describe\ninhomogeneous Baryonic condensates in strong external\nelectromagnetic fields. The fact that the solution to this\ntechnical problem is very far from obvious becomes quite\nclear if one observes that, until now, no exact analytic3\nBaryonic condensates in an external constant magnetic\nfield has been found. Therefore, one of the main goals\nof the present paper is to find explicitly inhomogeneous\nBaryonic condensates living within strong external elec-\ntromagnetic fields.\nThe present work aims to study the behavior of baryonic\nstructures able to describe the nuclear pasta behavior\nwhen an external magnetic field is applied. This analy-\nsis is motivated by the relevance in astrophysics of the\ninteractions between strong magnetic fields and baryonic\nstructures, such as in the Magnetars [63, 64]. We consid-\nered the spaghetti and lasagna ans¨ atze in both the NL σM\nmodel and the Skyrme model in order to depict the role\nof the Skyrme term when the soliton is exposed to exter-\nnal solicitations. Indeed, it is shown that the equations\nin NL σM lose coercivity when the external field becomes\nbig [65 ?]. Nevertheless, when the Skyrme term is added,\na solution can be always defined, at least in the case of\nbaryonic layers. In particular, in the latter, the solution\ncan be found using analytical tools. A remarkable prop-\nerty of this solution consists in its independence on the\nvalue of the external field; for this reason, we are going\nto call it as universal solution .\nA remarkable byproduct of our analysis is that we can\ncompute exactly the classical gran-canonical partition\nfunction associated with these families of Baryonic layers\nin a constant magnetic field in terms of the Pearcey in-\ntegral. The Pearcey integral is well known in optics and\nhas been studied in detail in connection with the Stokes\nphenomenon and resurgence [cite]. Thus, our exact so-\nlutions allow us to compute exactly the Stokes lines of\nthe gran-canonical partition functions in the µB−Bext\nplane. Taking into account the difficulties of lattice QCD\nwhen dealing with the Baryonic chemical potential, the\npresent analytic results which provide explicit expres-\nsions for critical lines separating different physical be-\nhaviors are especially important.\nA. Notation and characteristics of the model\nLet us introduce the main, general characteristics of the\nconsidered system. As anticipated in the previous sec-\ntions, our model is based on the definition of particular\nans¨ atze which allows us to solve analytically the Skyrme\nequations for a multi-baryonic system at finite density.\nThe last requirement is satisfied whether the system is\nconfined in a finite space volume (say, a box). Follow-\ning the notation introduced in [54, 55], the metric is de-\nscribed by\nds2=−dt2+L2\nrdr2+L2\nθdθ2+L2\nϕdϕ2. (1)\nHere, the variables r,θandϕrepresent dimensionless\nspace variables, with ranges\n0≤r≤2π , 0≤θ≤π , 0≤ϕ≤2π . (2)\nThe spatial dimensions are collected in the quantities Li,\nwhich take constant values and define the size of the box.The interaction with an external U(1) field is described\nby the Lagrangian\nL= Tr\u0014K\n2\u0012\nˆLµˆLµ+λ\n8ˆGµνˆGµν\u0013\u0015\n, (3)\nwhere\nˆLµ=Lµ−Aext\nµU−1[T, U],ˆGµν=h\nˆLµ,ˆLνi\n.(4)\nUrepresents the Skyrme field, which is given by a map\nU:R3+1→G, (5)\nwhere Gis a simple, compact Lie group, describing the\nunderlining flavor group of quarks. Lµis the ungauged\nleft current, which takes the form\nLµ=U−1∂µU=dim(G)X\ni=1Li\nµTi, Ti∈g. (6)\nHere gstands for the Lie algebra associated with the fla-\nvor group GandTiare the names of the generators of\ng. When G≡SU(2), the Lie algebra is denoted by su(2)\nandTiare represented by the Pauli matrices. Notice that\nwhen Aext\nµ→0, the Lagrangian matches the ungauged\none. The external field is defined along a specific direc-\ntion in the Lie algebra of the flavor group G, given by the\nelement T∈g≡Lie(G). Since the source of the cou-\npled field is considered to be external to the system, the\nvalue of Aext\nµdoes not undergo the equations of motions\n(or, better, it can be interpreted as a solution of some\nMaxwell equations). In particular, we are interested in\nthe behavior of nuclear pasta in the presence of a con-\nstant, strong magnetic field. In this scope, the potential\nAext\nµis taken as a linear function of the space variables\nwith Aext\n0= 0. This way, the electric and magnetic fields\nare defined respectively by\nEi=F0i= 0 and Bi=1\n2ϵijkFjk=const., (7)\nwhere Fµν=∂µAext\nν−∂νAext\nµis the field strength. As we\nare going to analyze in the following sections, the Skyrme\nequations are further simplified when the magnetic field\nundergoes some particular conditions. These conditions\nmust be specified case by case.\n1. The properties of the gauged Skyrme model\nThe contribution of the external field to the Skyrme equa-\ntions takes the general form\n∇µ\u0012\nˆLµ−λ\n4h\nˆGµν,ˆLνi\u0013\n= 0, (8)\nwhere ∇µrepresents the covariant derivative\n∇µVµ=∂µVµ+ [Lµ−Aext\nµU−1T3U, Vµ]\n=∂µVµ+ [ˆLµ−Aext\nµT3, Vµ]. (9)4\nWe know from [39–50, 54, 55] that the solutions to the\nungauged case can be found through the pasta ans¨ atze ,\nwhich are defined in such a way that the Skyrme equa-\ntions reduce to one ODE. Now, a question arises sponta-\nneously: Is it possible to find similar (or equal) ans¨ atze\nthat preserve the baryonic tubes or layers shapes also\nwhen the external field is present? As we will see in the\nfollowing sections, this question does not have a trivial\nanswer; in particular, when the treated case concerns the\nbaryonic tubes.\nThe difficulty of analyzing how the external field affects\nthe shape of nuclear pasta has been already emphasized\nin the introduction. In this scope, it is necessary to study\nthe behavior of the energy and baryonic densities.\nThe energy-momentum tensor is defined in the usual way\n(see, for instance, [54, 55]), substituting the uncoupled\nquantities with the hatted ones. Namely,\nTµν=−K\n2Tr\u0012\nˆLµˆLν−1\n2gµνˆLαˆLα\n+λ\n4(gαβˆGµαˆGνβ−1\n4gµνˆGαβˆGαβ)\u0013\n,(10)\nfrom which the energy density is derived ( ρE=T00).\nThe baryon charge is computed using\nB=Z\nVρB,with ρB=1\n24π2Tr(L ∧ L ∧ L ),(11)\nwhere Vis the spatial region spanned by the coordinates\nat any fixed time t. Here, a comment is in order. In [55] a\ngeneral definition of the gauge invariant baryonic charge\nis given, and it involves a non-trivial contribution of the\ngauge field, namely,\nˆρB=ρB+ 3εijk∂i[Aa\njTr(Lk(Ta+U−1TaU))\n−Aa\njAb\nkTr(TaU−1TbU)]. (12)\nHowever, in the present paper Aext\nµis a fixed external\nbackground, so it cannot be included in the definition of\nthe topological baryonic charge of the Skyrmion. For this\nreason, for the gauged baryonic density we take ˆ ρB=ρB.\nHowever, notice that the external field contribution still\nenters through the solution to the equations of motion.\n2. The Non-Linear Sigma Model vs. the Skyrme model\nWe underline the importance of the Skyrme term, which\ndefines the difference between the NL σM (which is repre-\nsented by the Skyrme model with λ= 0) and the Skyrme\nmodel. The Skyrme term was originally introduced to\nstabilize the topological structure of a single soliton de-\nfined in R3[31–35]. As studied in [54], this phenomenon\nis also observed when the structures of baryons are con-\nsidered over a finite space volume, defining a character-\nistic scale for nuclear pasta .\nOne of the aims of this paper is to analyze the role of\nthe Skyrme term when an external field is introduced.In the following section, this issue will be considered for\nboth the exponential and Euler parametrization. In both\ncases, for λ= 0 the equations of motion become un-\nsolvable when the external field is very strong (tends to\ninfinity). On the opposite, when λ̸= 0, the equations\nof motion are always solvable. This fact is particularly\nevident in the Euler parameterization, which admits an\nansatz that reduces the Skyrme equations to an ODE and\nkeeps the layer’s shape; this ansatz determines a univer-\nsal solution, in the sense that it is independent of the\nexternal field (thus, it exists when the external field is\nzero, finite or infinite). The case of the baryonic tubes is\nquite different and harder to be solved (at least, analyti-\ncally). It seems that the introduction of an external field\ndoes not allow for solutions that keep the baryonic tube\nstructure, leading to incompatible sets of equations.\nII. BARYONIC LAYERS COUPLED TO AN\nEXTERNAL MAGNETIC FIELD\nTo emphasize the role of the Skyrme term in the stability\nof the Skyrmion, we are going to compare the NL σM,\nwhich Lagrangian is defined by (3) with λ= 0, to the\nSkyrme model (with λ̸= 0). The Euler parameterization\nis very suitable for describing baryonic layers. Following\n[55], we make the ansatz\nU=eΦκeχζeΘκ, (13)\nwhere, for now, χ, Θ, and Φ are general functions of\nspace-time (a more specific ansatz on these functions will\nbe defined in the following steps). κis an element of g\nwhich can be written as\nκ=rX\nj=1(cjλj+c∗\nj˜λj), (14)\nwhere λjand˜λjare eigenmatrices of the simple roots\nαjand−αjrespectively. The quantities cjare complex\nnumbers that can be chosen arbitrarily in a set of al-\nlowed values, which aim to guarantee the periodicity of\nthe exponential of κ. The matrix ζbelongs to the Cartan\nsubalgebra of g(see [55, 57, 58]). With this choice of U,\nthe left current becomes\nLµ=e−ακe−ξκh\n∂µα(κ−ˆκ)\n+∂µξ(κ+ ˆκ) +∂µχfi\neξκeακ,(15)\nwhere\nα=1\n2(Θ−Φ), ξ =1\n2(Θ + Φ) (16)\nand ˆκ=e−χζκeχζ. As shown in [46, 53, 55], in the\nungauged model the baryonic layers are obtained when\nthe following orthogonality conditions are applied\n∂µΦ∂µΘ =∂µΦ∂µχ=∂µΘ∂µχ= 0. (17)5\nIfχ=χ(r), the equations of motions (independently on\nthe value of λ) reduce to one second order ODE (namely,\nχ(r)′′= 0). In the presence of an external field, we expect\nthat the ansatz (17) has to be modified to get explicitly\nsolvable equations. Furthermore, we cannot expect to\nbe able to reduce the equations just to one ODE. The\nlatter would be the optimal situation, but it could be in-\ncompatible with the presence of external fields. In what\nfollows, we will see in the case of the Euler parameteri-\nzation, the reduction of the equations of motion to one\nODE is possible but it is not obvious if it is the same\nwhen the baryonic tubes are considered.\nA. Baryonic layers solutions in the NL σM\nLet us consider the Lagrangian of the NL σM\nL=K\n2Tr\u0010\nˆLµˆLµ\u0011\n. (18)\nWhen the external field is zero and the conditions (17)\nare applied, the equations of motions simplify to\n∂µ∂µχ−sin(χ) (∂να∂να−∂νξ∂νξ) = 0 ,(19)\n∂µ∂µα= 0 and ∂µ∂µξ= 0. (20)\nIn [55], the functions αandξhave been chosen as\nα=q\n2θ−p\n2\u0012t\nLϕ−ϕ\u0013\n, (21)\nξ=q\n2θ+p\n2\u0012t\nLϕ−ϕ\u0013\n, (22)\nwhere pandqare integers. Moreover, χ=χ(r). This\nway, (20) are automatically satisfied and (19) reduces to\nχ′′(r) = 0.\nThe external field can be introduced by using (4), where\nwe choose T=κasU(1) gauge direction for the field Aµ.\nThis is equivalent to saying that the hatted quantities\nare obtained simply by replacing ∂µα→∂µα−Aµin the\nungauged terms. This remains true for the external field,\nwith Aµ=Aext\nµ(in Appendix C general equations for the\nSkyrme model have been obtained using the ansatz (13).\nThe analogous equations for the NL σM are obtained by\nimposing λ= 0). Now, one may wish to conserve the\nadvantages of the orthogonality conditions (17) also in\nthe gauged case. In this scope, we consider the following\nansatz for the external field\nAext\nµ∂µχ= 0. (23)\nHence, the gauged equations take the form\n∂µ∂µχ−sin(χ) (AνAν−∂νξ∂νξ) = 0 , (24)\n∂µAµ= 0 and ∂µ∂µξ= 0, (25)\nwhere Aµ=Aext\nµ−∂µα. Notice that now the second\nterm of (24) does not vanish when the choices (21) and\n(22) are applied, due to the presence of the external field.The condition (23) and the first equation of (25) are sat-\nisfied by the choice\nAext\nµ=\u0010\n0,0, Aext\nθ(r), Aext\nϕ(r)\u0011T\n, (26)\nwhere Aext\nθ(r) and Aext\nϕ(r) are linear functions of r, to\ngive a constant external magnetic field, see (7):\n⃗Bext=\u0010\n0,−∂rAext\nϕ(r), ∂rAext\nθ(r)\u0011T\n. (27)\nLet us define\nP(t, r, θ, ϕ ) =AνAν−∂νξ∂νξ. (28)\nWith our ansatz, P(t, r, θ, ϕ )≡P(r), where\nP(r) =Aext\nµAext,µ−q\nL2\nθAext\nθ+p\nL2\nϕAext\nϕ. (29)\nThus, the system of equations (24) and (25) becomes a\nsecond order ODE for the profile χ,\nχ′′(r)−sin(χ)P(r) = 0 . (30)\nNotice that P(r) is a second-order polynomial in r. (30)\ndoes not allow us to define an analytical expression of the\nsolution. Anyway, it is possible to define the expression\nfor the energy density and the baryonic density, which\nshows that the shape of the baryonic layer is conserved.\nMoreover, the total baryonic charge can be computed ex-\nactly, since it only depends on the boundary conditions.\n1. The energy density and baryonic charge\nThe energy density is given by (10). With our ansatz, it\ntakes the form\nρE=K\n2∥c∥2n8p2\nL2\nϕ+χ′2\nL2r+q2\nL2\nθ+ 4P(r) sin2\u0010χ\n2\u0011o\n.\n(31)\nThe baryonic density ρBis given by (11), which becomes\nρB=−∥c∥2\n4π2pq ∂ rcos(χ). (32)\nAs expected, it is a divergence, so its integral over the\nspace variables depends only on the boundary conditions,\nwhich have been fully studied in [55]. An explicit com-\nputation leads to\nB= 2Iη∥c∥2,with I=pqn, (33)\nwhere p,qandnare integers. This is the usual form for\nthe baryonic charge in the ungauged case, which takes\ninteger values (see [55]).6\n2. The large external magnetic field limit\nLet us look at the behavior of the ODE when the external\nfield becomes very large. Let us call\nAext\nθ=aθr+bθand Aext\nϕ=aϕr+bϕ. (34)\nWhen the coefficients aθandaϕbecomes much larger\nthan 1 /r, for nonzero r, the function (29) tends to\nP(r)→Pl(r) = (a2\nθ+a2\nϕ)r2. (35)\nThis way, equation (30) loses coercivity. We will see in\nthe next section that the introduction of the Skyrme term\nsaves the situation; in particular, it is always possible to\nfind an analytical solution for the Skyrme equations also\nfor big values of the external field.\nB. Baryonic layers solutions in the Skyrme model\nLet us consider the complete Lagrangian of the Skyrme\nmodel given in (3). The complete equations obtained for\nthe ansatz (13) are reported in Appendix C.\nIt is straightforward to show that the same choices de-\nfined in the previous section, summarized in the equa-\ntions (21), (22), (26) and χ=χ(r), can be used to reduce\nthe Skyrme equations to one ODE for the profile χ(r),\nwhich can be expressed as\nχ′′\u001a\n1 +λ\u0014\nP(r) +q2\n4L2\nθ\u0015\u001b\n−sin(χ)\u0012\n1−λ\n4χ′2\u0013\nP(r)\n−λsin(χ) cos( χ)n\u0012\nP(r) +q2\n4L2\nθ\u0013q2\n4L2\nθ\n−hq\n2L2\nθ\u0010\nAext\nθ−q\n2\u0011\n+p\n2L2\nϕAext\nϕi2o\n= 0,\n(36)\nwhere P(r) is the same as in (29). This equation can\nbe analytically solved if a further condition is imposed.\nIndeed, the last two lines of (36) are zero when\np/L2\nϕ=q/L2\nθ. (37)\nWhen this condition is applied, the first equation reducesto\nχ′′\nL2r\u001a\n1 +λ\u0014\nP(r) +q2\n4L2\nθ\u0015\u001b\n−sin(χ)\u0012\n1−λ\n4L2rχ′2\u0013\nP(r) = 0 . (38)\nThe solution is given by\nχ(r) =r\n4L2r\nλr+χ0, (39)\nwhere χ0is an integration constant. The boundary con-\nditions necessary for the existence of the solition require\nthat χ(0) = 0 and χ(2π) =π. The first condition im-\nposes that χ0= 0; moreover, the second condition is\nsatisfied whenλ\nL2\nr= 16, which means that the constant\nof the model λandLrare not free. We are going to call\n(39) as universal solution , since it does not depends on\nthe external field value.\nOther solutions, different from (39), can be considered,\nwhich do not require that λ/L2\nr= 16; these solutions are\npiecewise-linear, with slopep\n4L2r/λ, and zero elsewhere.\nThe only necessary condition in this case is that the slope\nof each linear part is bigger than the slope of the solution\nwhich does not present constant pieces. In the latter, we\nknow that a necessary condition is λ/L2\nr= 16; thus, in\nparticular, the slope becomesp\n1/4. In the piecewise-\nlinear solutions, the linear parts must undergo the con-\nditionp\n4L2r/λ >p\n1/4, which reads λ/L2\nr<16.\nIt is worth noticing here that all these solutions exist only\nwith λ̸= 0. Moreover, they are all independent of the\nvalue of the external field. This means that they always\nexist, even when the external field is zero or takes big val-\nues. As we are going to see in the next paragraph, this\nset of solutions defines baryonic layers; thus, it is remark-\nable the fact that the presence of the Skyrme term saves\nthe existence of baryonic layers when the external field is\nvery big. This is not an obvious result; for instance, as\nwe are going to observe, in the case of the baryonic tubes\nthe situation becomes more complicated.\nIn the following paragraphs, we are going to use this re-\nsult to analyze the main characteristics of the gauged\nbaryonic layers.\n1. The energy density and baryonic charge\nThe energy density associated with the solution of the\nSkyrme equations is\nρE=K\n2∥c∥2\u001a8p2\nL2\nϕ+q2\nL2\nθ+χ′2\nL2r+ 4P(r) sin2\u0010χ\n2\u0011\u001b\n+K\n2∥c∥2λ \n2p2\nL2\nϕ!\u001a\u0014\u0012\nP(r) +q2\n2L2\nθ\u0013\n−2Aρ∂ρξ\u0015\nsin2(χ) +χ′2\nL2r\u001b7\n+K\n2∥c∥2λ\u001a\u0014\u0012\nP(r) +q2\n4L2\nθ\u0013q2\n4L2\nθ−(Aρ∂ρξ)2\u0015\nsin2(χ) +\u0014\nP(r) sin2\u0010χ\n2\u0011\n+q2\n4L2\nθ\u0015χ′2\nL2r\u001b\n. (40)\nThe baryonic charge remains of the form (32); the contri-\nbution brought by the Skyrme term is considered through\nthe solution to the Skyrme equations. Nevertheless, the\nusual choice of the boundary conditions, defined by (C2),\nleaves the shape of Bunchanged.\n2. The thermodynamics of the universal solution\nAs discussed above, under certain conditions, it is al-\nways possible to find a universal solution to the Skyrme\nequations. Let us consider the case in which λ/L2\nr= 16.\nThus, the solution is represented by\nχ(r) =r\n2. (41)Moreover, when the only non-zero component of the ex-\nternal potential is the θ-component, Φ( u) is not necessar-\nily a linear function of u. Let us define Aext\nθ=hr+h′,\nwhere handh′are arbitrary constants. This way,\n⃗Bext= (0,0, h) (42)\nand\nP(r) =1\nL2\nθ\u0002\nh2r2+h(2h′−q)r+h′(h′−q)\u0003\n(43)\nCollecting the terms relative to the external field con-\ntribution and imposing that h′= 0 for simplicity, the\nenergy density takes the form\nρE=ρ0+hrρ 1+h2r2ρ2, (44)\nwhere\nρ0=K∥c∥2\u001ap2\n2L2\nϕ\u0014\n1 +λ\n8L2r+q2λ\n2L2\nθsin2\u0010r\n2\u0011\u0015\n+q2\n4L2\nθ\u0012\n1 +λ\n8L2r\u0013\n+1\n8L2r\u001b\n, (45)\nρ1=−K∥c∥2\u001ap2\n2L2\nϕ\u0014qλ\nL2\nθsin2\u0010r\n2\u0011\u0015\n+q\nL2\nθsin2\u0010r\n4\u0011\u0012\n1 +λ\n8L2r\u0013\u001b\n, (46)\nρ2=K∥c∥2\u001ap2\n2L2\nϕ\u0014λ\n2L2\nθsin2\u0010r\n2\u0011\u0015\n+1\nL2\nθsin2\u0010r\n4\u0011\u0012\n1 +λ\n8L2r\u0013\u001b\n. (47)\nIt is worth noticing that the presence of the terms pro-\nportional to sin2\u0000r\n4\u0001\ninρ1andρ2change the periodicity\nof the energy density, which, in particular, becomes 4 π-\nperiodic. This situation is represented in Figure 1, where\nthe energy density at different values of his depicted for\n0≤r≤2π. It appears evident that the energy density\nloses its periodicity. However, for small values of hit\nconserves approximately the shape of the layer. For this\nreason, we are going to consider this displacement from\nthe periodic solution as a perturbation. In future work,\nwe are going to analyze the impact of the back-reaction\non this behavior of the energy density: it is expected\nthat the skyrmion tends to conserve its shape and con-\ntrast the external field, maintaining the periodicity. The\ndependence of the total energy on the external field can\nbe formulated by integrating ρEover the volume of the\nbox. Namely,\nE=LrLθLϕZ\nρEdrdθdϕ. (48)\nAn explicit computation leads to the following form for\nthe total energy, where we used the condition p=qim-posed by Eq. (37)\nE(q;h) = 64 KL∥c∥2π24X\ni=1[Σi(h)]qi, (49)\nwhere\nΣ0(h) =π\n4L2r+h28π\nL2\nθ(π2+ 6), (50)\nΣ1(h) =−h\u00146\nL2\nθ(4 +π2)\u0015\n, (51)\nΣ2(h) =3π\nL2\nθ+6π\nL2\nϕπ+h216L2\nrπ\nL2\nϕL2\nθ2π2−3\n3, (52)\nΣ3(h) =−h16L2\nrπ2\nL2\nϕL2\nθ, (53)\nΣ4(h) =8L2\nr\nL2\nϕL2\nθπ (54)\nand ( q;h) indicates the dependence on the parameters q\nand on the coefficient of the gauge field h(the relation\nλ\n16L2r= 1 has been used).8\n0 1 2 3 4 5 6024681012\nrρEh=0\n0 1 2 3 4 5 6024681012\nrρEh=0h=0.01h=0.02h=0.03\nFIG. 1. Here is shown the energy density of the universal\nsolution of baryonic layers for different values of hwhen 0 ≤\nr≤2π. The upper Figure represents the uncoupled case, with\nh= 0 In the lower Figure, the uncoupled solution is compared\nto the other cases with h̸= 0. The parameters have been set\nto the following values: Lr=Lθ=Lϕ= 1, K= 2,∥c∥2= 1,\nandp=q= 1.\nThe baryonic charge is obtained by integrating the bary-onic density\nρB=∥c∥2\n8π2pqsin\u0010r\n2\u0011\n, (55)\nwhich gives\nB(p, q) =∥c∥2\n2pq (56)\nIn the most general case, the expression for the partition\nfunction can be written as follows\nZ=X\np,qexp{−β[E(p, q;h)−µBB(p, q)]}, (57)\nwhere β= 1/KBT(Trbeing the temperature) and µB\nis the chemical baryonic potential.\nBy assuming Lθ=Lϕ=Land the usual condition p=\nq, and using the expression (49) and (55), the partition\nfunction thus becomes\nZq= exp\u001a\n−β∥c∥2\n×\u0014\n2KLrL2π24X\ni=1Σi(h)qi−q2\n2µB\u0015\u001b\n.(58)\n3. The dependence on the external field\nWe want to analyze in detail the behavior of the partition\nfunction in terms of the external field. We can write Z\nin the form\nZ=+∞X\nq=−∞exp\u001a\n−β\nη3\n4\u0014\u0010\nη4q+η3\n4\u00114\n+\u0012\n˜η2η4−3\n8η2\n3\u0013\nη2\n4q2\n+\u0012\nη1η2\n4−1\n16η3\n3\u0013\nη4q+η0η3\n4−η4\n3\n256\u0015\u001b\n, (59)\nwhere\nηi= 2∥c∥2KLrL2π2Σi(h),fori= 0,1,2,3,4, (60)\n˜η2=∥c∥2h\n2KLrL2π2Σ2(h)−µB\n2i\n=η2−∥c∥2µB\n2. (61)\nStrictly speaking, our model is valid at low energies and,\ntherefore, for small values of the temperature, aka large\nβ. In this situation, for fixed values of all parameters,\nthere are just one or a few dominating terms of the se-\nries and all others are exponentially smaller. One has\nsimply to look for the absolute minimum of the quartic\nexponent as a function of the discrete variable q. On the\nother side, we need to consider such exponent as a func-\ntion of handµB, which makes our aim much harder toaccomplish. Nevertheless, we can observe that consider-\ning instead small values of βwould allow us to replace\nthe sum over qwith an integral, simpler to analyze from\nthis viewpoint. At least for large values of hand/or µB,\nas we will see, this can be analyzed using the usual sad-\ndle point approximation, more or less, the same saddles\nwe have just mentioned for the low-temperature case, at\nleast for values of energies such that the solitonic solu-\ntion still survives. For these reasons, our strategy will be9\nto pass to the description in the limit of the continuum,\nand only at the end we will comment on which changes\nwe do expect at low energies.\nAfter the substitution\nη4q−→q′−η3\n4, (62)\nthe partition function at low temperatures is therefore\ndescribed by the integral\nZ=Z+∞\n−∞dq′\nη4exp\u001a\n−a\u0002\nq′4+bq′2+cq′+d\u0003\u001b\n,(63)\nwhere\na=β\nη3\n4, b = ˜η2η4−3\n8η2\n3,\nc=η1η2\n4+η3\n3\n8−˜η2η4η3\n2,\nd=η0η3\n4−3η4\n3\n256+˜η2η4η3\n3\n16−η1η2\n4η3\n4. (64)\nSince the saddle points of the exponent in the integrand\nare relevant both in the discrete and in the continuum\nlimit, before a more systematic analysis of this integral,\nlet us first tell more about the stationary points. We\nexpect that different values of the parameters bandc\ncan determine regions of different phases of the baryonic\nlayers. On the other hand, since all parameters are real,\nthe contribution of the saddle points is influenced by their\nreal or complex nature in general. They are solutions of\nthe cubic equation\n4z3+ 2bz+c= 0, (65)\nwhich has discriminant ∆ = c2+(2b/3)3. For ∆ <0 there\nare 3 real solutions, while for ∆ >0 there are one real\nand 2 complex conjugate solutions. ∆ = 0 corresponds\nto the coalescence of at least two saddle points. It thus\ndetermines the caustic curve\nb3\n27+c2\n8= 0, (66)\nwhich separates different phases of the baryonic layer.\nIndeed, since both bandcdepend linearly on µBwith h-\ndependent coefficient, (66) determines a relation between\nthe chemical potential and the magnetic field, given by a\ncubic equation of the form\n0 =c1(h)[µ3\nB+c2(h)µ2\nB+c3(h)µB+c4(h)],(67)\nfor given functions cj(h). The detailed expression for this\ncubic can be found in App. D. Notice that for b >0 ∆ is\nalways positive so the caustic must belong in the b≤0\nregion. Moreover, in the equation, only h2appears ex-\nplicitly, so we can restrict to consider only positive values\nofh. The exact shape of the caustic curve thus depends\non the existence of real roots of the cubic, therefore, fromthe sign of its discriminant. In App. D, it is shown that\nthe discriminant is positive for 0 ≤h≤¯h, with\n¯h= 4η4\nˆη3s\nˆη1\nˆη3, (68)\nwhere ˆ ηj=ηj/h,j= 1,3, are constants. Therefore, for\nh < ¯hit appears only one branch of the caustic, while\na second double upper branch appears for h≥¯h, as\nrepresented in Figure II B 3.\nIn each portion of this phase space (63) may have dif-\nferent asymptotic behaviors when bandcare running\ntoward infinity in different directions, possibly mani-\nfesting the Stokes phenomenon . Let us briefly recall\nthat from a mathematical point of view the Stokes phe-\nnomenon appears each time one is solving a differen-\ntial equation (system) presenting at least one irregular\nsingular point. The asymptotic behavior of any solu-\ntion near the singular point has not a definite behav-\nior but depends on different angular sectors approach-\ning the point, separated by the so-called Stokes walls.\nA very simple explanation of this phenomenon can be\nfound in [66]. The prototype case of such phenomenon is\ngiven by the Airy function that, despite being an entire\nfunction, presents a complicated asymptotic behavior at\ninfinity, [67]. To fix our conventions, let us recall that\nwhat happens is that at infinity the Airy function takes\nthe form Ai(z) =c1u1(z) +c2u2(z), where cjare con-\nstants, while ujare polydromic functions. Thus, the uj\nare separately defined on a cut plane and transformed\naccording to a nontrivial monodromy after a change of\nphase of 2 πaround infinity. However, Aiis monodromic,\nhence, after such phase rotation, it must be described\nby a new combination of the uj, to preserve the mon-\nodromy. Therefore, the constant cjhas to change (dis-\ncontinuously, being constants) under such an operation.\nThis is essentially the Stokes phenomenon. During the\nphase rotation, one crosses lines that determine the main\nchange of the asymptotic behavior of the function: along\ncertain lines one has a maximal monotonic decrease or in-\ncrease of the modulus of the functions (essentially when\none of the two functions dominates), along other lines one\nhas an oscillating behavior (where the two functions have\ncomparable modulus). We call the former the anti-Stokes\nlines and the latter the Stokes lines . Ifuj∼eϕj, then\nanti-Stokes lines are defined by Im(ϕ1−ϕ2) = 0, while\nthe Stokes lines are defined by Re(ϕ1−ϕ2) = 0. In liter-\nature, one easily meets both these and the opposite con-\nvention (where the notion of Stokes and anti-Stokes are\ninterchanged). In practice, the (anti-)Stokes lines rep-\nresent walls, crossing which different saddle points enter\nthe game in determining the asymptotic behavior in a\nsaddle point approximation. This is largely studied in\nthe mathematical literature, whenever one has to tackle\nthe asymptotic expansion of solutions of differential sys-\ntems [68] [69] [70] [71], and it has several applications.\nFor example, it appears as a tool in Topological Quan-\ntum Field Theories (see e.g. [72], [73] and references10\nFIG. 2. Here we show the regions separated by the caustic. The lines represent the branches of the caustic and the shaded\nparts are the regions in which ∆ is negative. The parameters have been set to the following values: Lr=Lθ=Lϕ= 1,K= 2,\n∥c∥2= 1, and m= 1.\ntherein), in mirror symmetry and related topics [74], in\nChern-Simons theory [75], in Bridgeland stability [76], in\nQuantum Cohomology [77], etc.\nA particular field of application of interest for the present\napplication is resurgence theory [78]. In this case, indeed,\none uses Borel’s resummation methods to deal with di-\nverging infinite series, then determining quantitative re-\nlations between perturbative and non-perturbative data.\nSince saddle points typically catch non-perturbative in-\nformation, here is where the Stokes phenomenon plays a\nmajor role. In this way, Stokes lines provide a contact\nbetween perturbative and non-perturbative regimes and\nwe have to expect that they do not change sensitively\nin passing from one regime to the other. Since Skyrme\ntheory is a low-energy effective description of QCD, we\nthus expect that analyzing the Stokes phenomenon in\nthe Skyrme regime gives interesting information in the\nfull QCD regime and vice versa.\nTo analyze our specific case, it is convenient to consider\nthe change of variable q=a1\n4q′. In this way, (63) takes\nthe form\nZ=e−iπ\n8\nη4a1\n4P(x, y), (69)\nwhere\nP(x, y) =eiπ\n8Z+∞\n−∞dqexp\u001a\n−q4−xq2+iyq\u001b\n,(70)\nis the Pearcey’s integral, as redefined in [79], with\nx=a1\n2band y=ia3\n4c. (71)As we already mentioned, we are working for small val-\nues values of a. Therefore, in general, we expect xandy\nto be small. Nevertheless, nothing prevents us from con-\nsidering high values of hand/or µBenough to still have\nlarge values of xandy. Since these are exactly the cases\nwhere the Stokes phenomenon is expected to arise, here\nwe consider exactly such a situation. Since the Pearcey\nintegral has been largely studied, we can easily analyze\nits behavior in the regions of our interest by referring to\nthe suitable literature, see App. E.\nAsymptotic expansion for |x| → ∞ .When |x| → ∞\nwith bounded y, one gets a uniform qualitative behavior\nin the complex xplane, in the sense that Stokes lines do\nnot depend on y. A representation is provided in Fig. 3.\nAsymptotic expansion for |y| → ∞ .When |y| → ∞\nwith bounded x, we have an almost uniform qualitative\nbehavior, now in the complex yplane. Recall that the\nPearcey integral is an even function of y. A representa-\ntion of the Stokes lines is provided in Fig. 4.\nAsymptotic behavior around the caustic curve. In\nthis case, we get that near the cuspid, for large x, it is\ndominating an exponentially growing mode. Aside from\nit, two other modes, decaying exponentially at infinity,\nhave an oscillating character where ∆ >0, while are al-\ngebraic in ∆ <0. This behavior is quite different than\nthe one in [80], where he gets instead three oscillating\nterms with decay x−1\n2forα >0 and one oscillating term\nand two exponentially decaying for α < 0. This is the\nbehaviour at x=|x|e−iπ\n4in our coordinates. Notice that\nit is exactly this difference in the phase that makes the\ndifference. Indeed, the always oscillating mode in [80] is11\nIm(x)\nRe(x)ϕ=3\n4π\nϕ=−3\n4πanti-Stokes lineanti-Stokes line anti-Stokes lineStokes line\nStokes lineexponential grow\nexponential growalgebraic\nalgebraicalgebraic\nalgebraic\nFIG. 3. Stokes’ lines and anti-Stokes’ lines in the complex\nx-plane. Crossing the Stokes lines the function passes from\nan algebraic/oscillating behavior to an exponentially growing\nregime. In restricting to real variables we see that negative\nvalues exactly belong to an anti-Stokes line (maximally grow-\ning). Here ϕ= arg( x)\nIm(y)\nRe(y)ϕ=3\n8π ϕ=5\n8π\nϕ=1\n8π ϕ=7\n8π\nϕ=−3\n8π ϕ=−5\n8πϕ=−1\n8π ϕ=−7\n8πStokes line Stokes lineanti-Stokes line\nanti-Stokes lineanti-Stokes line\nanti-Stokes line\nFIG. 4. Stokes’ lines and anti-Stokes’ lines in the complex y-\nplane. The dark sectors correspond to exponential growth,\nwhile the light sector to exponential decay. The dashed\nlines are not Stokes lines but transition lines in the sense of\nPoincar´ e, see [79]. Here ϕ= arg( y). Notice that in in our spe-\ncific case y=ia3\n4cis purely imaginary, so in the exponential\ngrowing case, and, because of the symmetry, there is no trace\nof the Stokes transition in passing from positive to negative\nvalues.the one corresponding to the critical point z1, and it is os-\ncillating only for the above phase. As soon as x=|x|e−iϕ\nwith 0 < ϕ <π\n4(which is still in the region of the defi-\nnition of their expansion as well as ours), the two results\nagree, and this mode becomes exponentially growing.\nAn important remark. In our setting we considered\nthe extension of the xandyvariables to complex val-\nues. This led us to meet the Stokes phenomenon, which,\nhowever, appears to be related to physical situations only\nwhen we restrict ourselves to real values. However, our\nmodel is understood to describe low-energy QCD at fi-\nnite temperatures. It is a known fact that in a full non-\nperturbative setting, that is in lattice computation, a real\nchemical potential is problematic, and the solution con-\nsists of replacing it with a pure imaginary chemical po-\ntential [81]. More in general, in [82] it has been shown\nthat from the partition function with a purely imaginary\nchemical potential one can get a complete picture of the\nphase space. In [83], the same technique combined with\nnumerical computation has been used to get a complete\ndescription of the Roberger-Weiss transition. Therefore,\nwe conclude that the above discussions of the Pearcey in-\ntegral remain physically valid for generically complex x\nandyparameters since they just correspond to an imag-\ninary chemical potential.\n4. Free energy and susceptibility\nThe previous analysis of the partition functions allowed\nus to study the behaviors of the physical properties of the\nskyrmionic layers. In particular, in this section, we are\ngoing to analyze the free energy F(h) and the susceptibil-\nityχ(h) in terms of the external field h, the temperature\n1\nβ, and the magnetic moment µ.\nIn Fig. 5, different plots of the free energy in terms of the\nexternal field hand for different values of the magnetic\nmoment µare represented. Here, it is important to ob-\nserve that the function F(h) takes negative values when\nhbecomes big. It is necessary to point out that our\nmodel must be considered in the range of small values\nof the external field since we are neglecting the internal\nback-reaction of the skyrmion, which will be the subject\nof future works.\nIn Fig. 6, it is depicted the behavior of the susceptibility\nin terms of the external field for β= 10−4. It is evident\nthatχtakes positive values for small h. It can be due to\ndifferent reactions of the skyrmion at different values of\nthe external field, but it is worth remarking, as discussed\nin the perspectives, that we have not yet fully included\nthe back reaction here.12\nFIG. 5. In this picture is represented the Free Energy in terms of the external field for different values of the magnetic moment\nµ. Here, β= 10−4. It can be noticed that for certain values of the external field h, the free energy takes negative values.\nFIG. 6. In this picture is represented the susceptibility in terms of the external field for different values of the magnetic moment\nµ. Here, β= 10−4. For small values of h, it takes positive values and becomes negative after a certain limit.\nIII. CHIRAL CONFORMAL FIELD THEORY OF\nBARYONIC LAYERS FROM THE SKYRME\nMODEL\nIn [59], the authors introduce an extension of the ansatz\n(21) and (22), which consists in a generalization of the\nfunction Φ( t, ϕ). Indeed, let us observe that the orthogo-\nnality conditions (17) are preserved when Φ( u) is a gen-\neral function of u=t/Lϕ−ϕ. To keep track of the\ncontributions brought by this generalization, we write\nΦ(u) =˜Φ(u) +pu, (72)where ˜Φ(u) corresponds to the general function of uand\nthe last term has the usual form of the oldΦ. The intro-\nduction of the new term ˜Φ(u) leads to the definition of a\nchiral conformal field theory (CCFT), as stated in [59].\nIn this section, we want to study the contribution of the\nconformal part to the partition function under the ac-\ntion of the eternal field (in [59], this possibility has been\nanalyzed in the context without an external field).\nIn section II B, we defined some conditions that lead to\na significant simplification of the Skyrme equations. In\nparticular, we considered the condition (37). When the\nconformal term is added, a stronger condition is neces-\nsary. Indeed, the polynomial P(r) that appears in (36)13\n(and that have been introduced in (28)) takes a more\ngeneral form\nP(t, r, ϕ ) =Aext\nµAext,µ−2∂µαAext,µ, (73)\nwhere the dependence on tandϕis enclosed in Φ( u),\ncontained in α, and the dependence on r is due to Aext\nµ.\nDifferently from the other case (with ˜Φ = 0), the depen-\ndence on tandϕdoes not cancel. Therefore, also when\np/L2\nϕ=q/L2\nθ, the last term of (36) does not cancel. To\nre-conduce the Skyrme equation to the simplified form\n(38), we need the further condition Aext\nϕ= 0. This way,\nP(t, r, ϕ )→P(r) =Aext\nθ\nL2\nθ\u0010\nAext\nθ−q\u0011\n, (74)and the Skyrme equations are simplified to\nχ′′\nL2r\u001a\n1 +λ\u0014\nP(r) +q2\n4L2\nθ\u0015\u001b\n−sin(χ)\u0012\n1−λ\n4L2rχ′2\u0013\nP(r) = 0 . (75)\nNotice that also this case we can find the usual universal\nsolution analyzed in the previous section.\nLet us discuss how the baryonic charge, the energy den-\nsity, and the partition function generalize to the confor-\nmal case.\nFirst of all, it is straightforward to observe that the con-\nformal term does not contribute to the baryonic charge .\nIndeed, the baryonic density can be written as\nρCCFT\nB =ρB+∥c∥2\n8π2q∂ϕ˜Φ sin\u0010r\n2\u0011\n(76)\nwhere CCFT labels the conformal baryonic density and\nρBhas been defined in (F10). We can consider some\nperiodic conditions over the function ˜Φ. Namely,\n˜Φ(t, ϕ= 0) = ˜Φ(t, ϕ= 2π), (77)\n∂ϕ˜Φ(t, ϕ= 0) = ∂ϕ˜Φ(t, ϕ= 2π). (78)This way, the integral over ϕof˜Φ cancels.\nLet us discuss the generalization of the energy density .\nIt takes the usual general form defined by Eq. (44), but\nwith\nρ0=K∥c∥2\u001a\nI\u0014\n1 +λ\n8L2r+q2λ\n2L2\nθsin2\u0010r\n2\u0011\u0015\n+q2\n4L2\nθ\u0012\n1 +λ\n4L2r\u0013\n+1\n16L2r\u001b\n, (79)\n(80)\nwhere\nI=1\n4\u0010\n(∂t˜Φ)2+∂i˜Φ∂i˜Φ\u0011\n+∂t˜Φp\nLϕ+p2\n2L2\nϕ. (81)\nThus, the function ˜Φ(u) contributes only inside the factor\nI. The integration of the energy density over the volume\nleads to the following form of the total energy\nE(q;h, m) = 2 π2KLrLθLϕ∥c∥2\n·4X\ni=1h\ne0˜Σi(h) + Σ i(h)i\nqi, (82)\nwhere Σ i(h) take the same form as in eqs. (50) to (54)\nand\n˜Σ0(h) = 3 π+h28L2\nrπ\nL2\nθ\u00142π2−3\n3\u0015\n, (83)˜Σ1(h) =−h8L2\nrπ2\nL2\nθ(84)\n˜Σ2(h) =4L2\nr\nL2\nθπ, (85)\n˜Σ3(h) =˜Σ4(h) = 0 . (86)\nMoreover,\ne0=1\n2πZ2π\n0∂t˜Φ2+∂i˜Φ∂i˜Φdϕ. (87)\nNotice that ˜Φ(u) = 0 implies e0= 0. As already pro-\nposed in [59], the terms e0can be quantized, giving the\nfollowing form of the partition function14\nZ=∞X\nq=−∞exp\u001a\n−β∥c∥2\u0014\n2KLrL2π24X\ni=1Σi(h)qi−q2\n2µB\u0015\u001b\n×∞X\nn=0δ(n) exp\u001a\n−2βKL r∥c∥2π2n4X\ni=1˜Σi(h)qi\u001b\n, (88)\nwhere nand the sum over it derive from the quantization\nofe0andδ(n) is the degeneracy on each state n. Once\nmore, we consider the conditions Lϕ=Lθandp=q.\nIV. BARYONIC TUBES COUPLED TO AN\nEXTERNAL MAGNETIC FIELD\nIn the previous sections of this paper, we defined the an-\nalytical solutions of baryonic layers immersed in an ex-\nternal, constant magnetic field using the lasagna ansatz .\nIn this section, we guess whether this is also possible\nwhen the spaghetti ansatz is used. Unfortunately, we did\nnot succeed in finding analytical solutions to the Skyrme\nequations. Indeed, as we are going to explain in the\nfollowing sections, the terms introduced by the external\nfield prevented us from decoupling the Skyrme equations.\nAny tentative simplification (for instance, by imposing\nnew conditions of the external field of another ansatz on\nthe Skyrme field parameters) led to incompatibilities of\nthe Skyrme equations. This precludes us from giving a\nsatisfactory analysis of the physics of baryonic tubes un-\nder the action of an external field, but we were able to\noutline some important properties of the main quantities,\nsuch as the role of the Skyrme term and the behavior of\nthe energy density.\nAs it is known from [31–55], the exponential parameter-\nization is suitable to describe baryonic tubes. We recall\nhere the spaghetti ansatz , defined by\nU(t, r, θ, ϕ ) = exp( χ(r)τ1), (89)\nwhere τ1=⃗ n·⃗T=n1T1+n2T2+n3T3. The ansatz is\nspecified by\n⃗ n= (sin Θ cos Φ ,sin Θ sin Φ ,cos Θ) (90)\nΘ =qθ, Φ =p\u0012t\nLϕ−ϕ\u0013\n, p, q ∈N. (91)\nThe matrices Tidefine a basis of a three-dimensional sub-\nalgebra of g= Lie( G). They can be normalized in such\na way as to satisfy\n[Tj, Tk] =εjkmTm, Tr(TjTk) =−2IG,ρδjk,(92)\nwhere IG,ρis called the Dynkin index (see [55, 84]). This\nansatz satisfies the orthogonality conditions\n∂µΦ∂µΦ =∂µΦ∂µΘ =∂µΦ∂µχ=∂µΘ∂µχ= 0,(93)which are necessary to reduce the system of Skyrme equa-\ntions to one ODE. With a direct computation, one can\nshow that the introduction of an external field along\nT=T3contributes through the shift\n∂µΦ→∂µΦ +Aext\nµ. (94)\nLet us call Aµ=Aext\nµ+∂µΦ. It is worth noticing here\nthat this shift breaks part of the conditions (93). Indeed,\nAµAµ̸= 0 (before the translation, ∂µΦ was a null vec-\ntor). This does not allow us to uncouple the equations\nof Θ and χand, as a consequence, ∂µχ∂µΘ̸= 0. For this\nreason, it is not possible to consider the two functions\ndepending on different space-time variables; thus, let us\ndefine χ=χ(r, θ) and Θ = Θ( r, θ). One can try to sim-\nplify the equations by choosing Aext\nµin different ways.\nFor instance, the conditions\nAµ∂µΘ =Aµ∂µχ= 0. (95)\nallow keeping part of the conditions (93). The explicit\nform of the external field is given by\nAext\nµ= (0,0,0, Aϕ(r, θ))T, (96)\nwhich defines a magnetic field along θandr. Notice that,\nin this case, the magnetic field lies perpendicular to the\ntubes; thus, one can expect that it breaks the symmetry\nof the system. A second possibility is to choose Aext\nµin\nsuch a way to define a magnetic field along the tubes.\nFor example,\nAext\nµ= (0,0, Aθ(r),0)T. (97)\nNotice that in both cases the external field has been cho-\nsen in such a way that\n∂µAµ= 0. (98)\nThe problem with this choice lies in the fact that all the\nconditions (93) can be no longer satisfied, but it leads to\nan advantage. Indeed, as discussed better below, we want\nto conserve the baryonic tube shape of the soliton; in this\nscope, the energy density and the baryon density should\nbe constant in ϕ: we can always choose Φ = p(t/Lϕ−\nϕ) and one of the three Skyrme equations must become\ntrivial; otherwise, it is not possible to find a solution to\nthe system.15\nIn the following sections, we will not analyze both these\nsituations in detail, but we will describe the general prob-\nlem and the properties for λ= 0 and λ̸= 0. Moreover, in\nAppendix A we propose an alternative form of the exter-\nnal field, in which its direction in the Lie algebra of the\nflavor group depends on the space-time coordinates. In\nthis case, it becomes really simple to reduce the Skyrme\nequations to an analytically solvable ODE (in particu-\nlar, the usual universal solution appears). We called it\nThe flavor oscillating spaghetti . In any case, a better\ncharacterization of the baryonic tube solution requires,\nprobably, a better understanding of the spaghetti ansatz\ngeometry.\nA. Baryonic tubes solutions in the NL σM\nAs already defined above, the Skyrme model with λ= 0\ndefines the NL σM. The general equations can be written\nas (see Appendix C)\n∂µ∂µχ−sinχ\u0000\n∂µΘ∂µΘ + sin2ΘAµAµ\u0001\n= 0, (99)\n(1−cosχ)∂µ∂µΘ + sin χ∂µχ∂µΘ\n−(1−cosχ) sin Θ cos Θ AµAµ= 0,(100)\nsinχsin2Θ∂µχAµ\n+ 2(1−cosχ) sin Θ cos Θ ∂µΘAµ= 0,(101)\nwhere only the condition (98) on Aµhas been considered\n(in particular, it is independent on the choice (96) and\n(97). We can make the following observations.\nFirst of all , to get baryonic tubes, we can use the ansatz\nΦ = p(t/Lϕ−ϕ). Indeed, this cancels the dependence\nfrom ϕin the energy density. As already mentioned, the\nother functions cannot take the form of the ansatz (90),\nsince the terms introduced by the external field couples\nthe equations, but we can impose the following orthogo-\nnality conditions\n∂µχ∂µΦ =∂µΘ∂µΦ = 0 , (102)\nwhich means that χand Θ are functions of only randθ.\nIn this case, the energy density is\nρE=K∥c∥2\u001a\nAµAµ(1−cosχ) sin2Θ\n+1\n2∂µχ∂µχ+ (1−cosχ) (∂µΘ∂µΘ)\u001b\n.(103)Secondly , when the ansatz (96) is applied, the last equa-\ntion (101) is automatically satisfied, due to the conditions\n(93) and (98). On the other hand, when (97) is applied,\nall the three equations are non-trivial, but one of them\nshould be redundant , since the equation depends only on\nχand Θ.\nThirdly , when the external gauge field becomes big, the\nequation (99), (100) and (101) loose coercivity. This fact\nhas also been encountered in the baryonic layers case. We\nare going to show that the Skyrme term saves the consis-\ntency of the Skyrme equations and also for the spaghetti\nansatz.\nIt is worth mentioning here that the energy density di-\nverges for a growing external field. This is predictable\nsince the external fields contribute ti the energy of the\nsystems. This behavior has also been observed for the\nbaryonic layers. The main difference here consists in the\nfact that the solutions of the equations of motion neces-\nsarily depend on the external field (we need to remember\nthat the baryonic layers allow for a universal solution,\nindependent from the external field). Thus, one may ex-\npect that the tube configuration’s energy density (and so\nthe total energy) diverges faster than the layers config-\nuration’s energy. If this is the case, the baryonic tubes\nbecome less stable than the baryonic layers and a configu-\nration transition could appear. Those types of transitions\nwere also considered in [54].\nB. Baryonic tubes solutions in the Skyrme model\nThe contribution of the Skyrme term to the general equa-\ntions for the exponential parameterization is reported in\nAppendix C, where the condition (98) can be applied. It\nis straightforward to show that the first two observations\nconsidered in the previous section for the NL σM remain\nvalid. The main advantage of the introduction of the\nSkyrme term consists in the fact that the high limit of\nthe external field can now be considered. With a direct\ncomputation, one can observe that the Skyrme equations\ntake the following form\nsinχsin2Θ\u001a\nAµAµ+λ\n4\u0002\n(∂µχ∂µχ)(AνAν)−(∂µχAµ)2\u0003\n+λ(1−cosχ)\u0002\n(∂µΘ∂µΘ)(AνAν)−(∂µΘAµ)2\u0003\u001b\n−λ\n2∂µ\u0002\n(1−cosχ) sin2Θ(AνAν)∂µχ\u0003\n+λ\n2∂µ\u0002\n(1−cosχ) sin2Θ(∂νχAν)Aµ\u0003\n= 0, (104)\n(1−cosχ) sin Θ cos Θ\u001a\nAµAµ+λ\n4\u0002\n(∂µχ∂µχ)(AνAν)−(∂µχAµ)2\u0003\n+λ\n2(1−cosχ)\u0002\n(∂µΘ∂µΘ)(AνAν)−(∂µΘAµ)2\u0003\u001b16\n−λ\n2∂µ\u0002\n(1−cosχ)2sin2Θ(AνAν)∂µΘ\u0003\n+λ\n2∂µ\u0002\n(1−cosχ) sin2Θ(∂νΘAν)Aµ\u0003\n= 0, (105)\n∂µ\u001a\u0002\n(1−cosχ) sin2ΘAµ\u0003\n+λ\n4\u0002\n(1−cosχ) sin2Θ(∂νχ∂νχ)Aµ\u0003\n−λ\n4\u0002\n(1−cosχ) sin2Θ(∂νχAν)∂µχ\u0003\n+λ\n2\u0002\n(1−cosχ)2sin2Θ(∂νΘ∂νΘ)Aµ\u0003\n−λ\n2\u0002\n(1−cosχ)2sin2Θ(∂νΘAν)∂µΘ\u0003\u001b\n= 0. (106)\nAlso in this case, we tried different ways to find solutions\nto these equations, but we did not succeed.\nAs for the previous case, the energy may be considered\ngrowing faster than the solution with the lasagna ansatz.Thus, a transition between the two configurations could\nhappen. The point of transition could depend on the\nparameter λ. In this case, the energy density takes the\ngeneral form\nρE=K∥c∥2\u001a\nAµAµ(1−cosχ) sin2Θ\u0014\n1 +λ\n2(1−cosχ)∂µΘ∂µΘ +λ\n4∂µχ∂µχ\u0015\n+1\n2∂µχ∂µχ\n+ (1−cosχ)\u0014\n∂µΘ∂µΘ +λ\n4\u0000\n∂µχ∂µχ∂νΘ∂νΘ−(∂µχ∂µΘ)2\u0001\u0015\u001b\n. (107)\nV. DISCUSSION AND PERSPECTIVES\nWe have considered a comparison between the Skyrme\nmodel and the NL σM (which is represented by the\nSkyrme model with λ= 0) in the presence of an external\nelectromagnetic field. We used both the exponential and\nthe Euler parametrization for the ansatz. Interestingly,\nin both cases, the equations of motion for the pure NL σM\nbecome unsolvable when the external field is very strong\n(tends to infinity). Instead, for λ̸= 0, the equations of\nmotion are always solvable. Specifically, the Euler pa-\nrameterization ansatz reduces the Skyrme equations to\nan ODE and keeps the layer’s shape. This determines\na universal solution, that is independent of the external\nfield, therefore existing in all cases when the external field\nis zero, finite, or infinite. For baryonic tubes, the situa-\ntion is quite different since the equations of motion are\ndifficult to be solved analytically. However, it seems that\nintroducing an external field hampers the existence of\nsolutions that keep the baryonic tube structure, leading\nto incompatible sets of equations. One may wonder why\nthese topological solutions disappear in the limit when\nλ= 0. The answer is not new in nonlinear equations\nand can be read from equation (39). We see that the\nsolution is proportional to the inverse square root of the\ncoupling constant, so compensating the coupling in the\ninteractions and giving finite energy contribution terms\nindependent on λ.\nIn particular, we concentrated on the thermodynamics\nof the universal solution, when also a possibly complex\nBaryonic chemical potential is included. Very interest-\ningly, a quite rich Stokes phenomenon structure appears,\nmanifesting different phases of the Baryonic structure.Interestingly, such a manifestation of resurgence appears\nin such a concrete model. In particular, our calculations\ncome back to the Stokes phenomenology for Pearcey in-\ntegrals in all possible cases. To this end, we have approx-\nimated infinite sums with integrals. It will be interesting\nin future work to study the Stokes phenomenon directly\nfor the series. However, we have reasons to think of a\npriory that we should not expect any qualitative changes\nw.r.t. the integral case. Nevertheless, such computations\ncould become interesting for more precise quantitative\npredictions. To this end, however, we need first to fur-\nther improve our model by including some corrections we\nneglected here.\nOne of the most important issues is that we neglected\nback reactions in the presence of the external field. For\nexample, consider (31). This expression represents the\nenergy density inside the box. Assuming the topological\nconfiguration is not destroyed, only gauge transforma-\ntions compatible with the boundary conditions are al-\nlowed for the internal field, while arbitrary gauge trans-\nformations are allowed for the external gauge field. This\nmeans that an additional contribution must compensate\nfor the effect of the external field through polarization,\nthrough the distribution of charges along the boundary,\nand then generate a compensating internal field. We did\nnot include such polarization effects here, to avoid further\ntechnical complications to already involved mathemati-\ncal computations. We mean to study them separately in\nfuture work.\nAnother issue deserving attention regards the solutions\nin II B, for λ/L2\nr<16. In this case, the first deriva-\ntive in ris discontinuous and the second derivative gives\ndelta distributions at the discontinuities. Stokes’ theo-\nrem thus requires the insertion of brane sources located17\nat the discontinuities. This kind of solution therefore de-\nserves a more detailed study by including brane sources’\ncontributions. Here, we limited our analysis to the case\nreported in Appendix F, but further details going beyond\nthe aim of the paper will be the subject of future work.\nFinally, similar considerations as above can be done for\nthe analysis we presented in Sec. II B 4. Here the prob-\nlem is that the back-reaction may probably give an im-\nportant contribution to Fandχalso for small values\nofh, since it is expected to be of the order of the ex-\nternal field. Therefore, it is not really negligible. This\nissue deserves further investigation, presently under con-\nsideration. Furthermore, we preferred to leave the anal-\nysis of these physical quantities in terms of the temper-\nature for future work, with the aim of including also the\ncontribution back-reaction. Indeed, as already discussed\nabove, the internal reaction of the skyrmion is important\nin order to determine the value of the external field that\nbreaks our solutions, but also to study the possible phase\ntransitions and the behavior of the critical temperature\nin terms of the external field.\nACKNOWLEDGEMENTS\nWe thank Prabal Adhikari for the relevant discussions.\nF. C. has been funded by FONDECYT Grant 1240048.Appendix A: The flavor-oscillating spaghetti\nIn this section, we want to show how the baryonic tubes\ncan easily survive in an external field when the direc-\ntion in the Lie algebra of the flavor group of the external\nfield depends on the position in space-time. Indeed, as\ndiscussed in the paper, the choice of T=T3causes a\ntranslation of ∂µΦ→∂µΦ−Aext\nµ. Since ∂µΦ is a light-\nlike vector, which has been chosen to cancel some terms\nin the ungauged case, this translation causes the pres-\nence of those terms. A solution could be the choice of a\ndirection that does not lead to this type of translation.\nAn example is provided T=τ3, where τ3derives from\nthe notations introduced in [54, 55]. Namely,\nτ1= sin Θ(cos Φ T1+ sin Φ T2) + cos Θ T3, (A1)\nτ2=∂Θτ1= cos Θ(cos Φ T1+ sin Φ T2)−sin Θ T3,(A2)\nτ3=∂Φτ1\nsin Θ=−sin ΦT1+ cos Φ T2. (A3)\nRemember that τ1has also been used in the definition\nofUin (89). This way, the contribution of the external\nfield to the left current is accounted for\nˆLµ=U−1DµU=Lµ−Aext\nµU−1[τ3, U]. (A4)\nWith a direct computation, one finds that\nU−1[τ3, U] = sin χτ2−(1−cosχ)τ3. (A5)\nOn the other hand,\nLµ=∂µχτ1+∂µΦ sin Θ [sin χτ3+ (1−cosχ)τ2]\n+∂µΘ [sin χτ2−(1−cosχ)τ3].(A6)\nThus, the external field causes a translation of ∂µΘ. Let\nus call Aµ=Aext\nµ−∂µΘ. Using the conditions\n∂µΦ∂µΦ = 0 and Aext\nµ∂µΦ =Aext\nµ∂µχ= 0,(A7)\nthe equations are simplified to\n∂µ∂µχ\u0014\n1 +λ\n2AνAν(1−cosχ)\u0015\n− A µAµsinχ\u0012\n1−λ\n4∂νχ∂νχ\u0013\n= 0,(A8)\n∂µAµ= 0 and ∂µ∂µΦ = 0 . (A9)\nThe above conditions and the last two equations are au-\ntomatically satisfied with the usual choices\nAext\nµ= (0,0, Aext\nθ(r),0), (A10)\nΦ = Φ( u) and Θ = qθ. (A11)\nThis way, one can choose χ=χ(r) and reduce the equa-\ntions to a second-order ODE\nχ′′\u0014\n1 +λ\n2AνAν(1−cosχ)\u0015\n− A µAµsinχ\u0012\n1−λ\n4L2rχ′2(r)\u0013\n= 0,(A12)18\nwhich admits a universal solution, independent of the\nvalue of the external field.\nThe main problem linked to this type of description is\nthe fact that the direction of the external field in the\nspace of the Lie algebra of the flavor group depends on\nthe space-variable ϕ. Usually, the coefficients of the Tiis\nthe definition of τ1(which in the ansatz (89) have been\ncalled ni, as defined in (90)) are recognized with the three\ncharged pions, π±andπ0[31–35, 38]. It is easy to see\nthat when T=T3, the neutral pion corresponds to n3. In\nthis case, the flavor of the neutral pion oscillates , since\nthe direction of the external field oscillates . This is a\nstrange fact and, for this reason, we reported the treat-\nment of this model here in the Appendix, excluding it\nfrom the ”official” computations.Appendix B: On the equations of motion.\nFor the baryonic tubes, a simple calculation gives\nU−1∂µU=∂µχτ1+ sin χ∂µnjTj\n+ (1−cosχ)∂µnjnkϵjklTl. (B1)\nSimilarly,\nU−1T3U−T3=(1−cosχ)(n3njTj−T3)\n+ sin χϵ3jknjTk. (B2)\nAfter some manipulations we then get\nˆL=∂µχτ1+ sin χ\u0010\n∂µnj−ϵ3kjnkAext\nµ\u0011\nTj\n+ (1−cosχ)\u0010\n∂µnj−ϵ3kjnkAext\nµ\u0011\nnhϵjhlTl.(B3)\nOn the other hand, we have also\n∂µnj=∂µΘ∂Θnj+∂µΦϵ3kjnk, (B4)\nso that\nˆL=∂µχτ1+ sin χ\u0010\n∂µΘ∂Θnj+ϵ3kjnk(∂µΦ−Aext\nµ)\u0011\nTj+ (1−cosχ)\u0010\n∂µΘ∂Θnj+ϵ3kjnk(∂µΦ−Aext\nµ)\u0011\nnhϵjhlTl.\n(B5)\nTherefore, we can write ˆL=L[∂µΦ−Aext\nµ]. Hence, for\nVµ≡ˆLµ−λ\n4h\nˆGµν,ˆLνi\n, (B6)\nwe can write the equations of motion as\n∂µVµ+ [ˆLµ, Vµ] =Aext\nµ[T3, Vµ]. (B7)\nThe l.h.s. is thus obtained easily from the equations of\nmotion without external field, by shifting ∂µΦ, while in\nthe r.h.s., Vµis still a function of ∂µΦ−Aext\nµbut the\nextra term Aext\nµseems to break this scheme.\nHowever, we can prove that this is not the case in the\nfollowing way. The point is that in ∂µVµwe have to do\nthe shift before taking the derivative, but since Vµis a\nfunctional of Φ, through nj, after deriving we get new\n∂µΦ terms. We show that, because of gauge invariance\nof the action, these new terms pair with the r.h.s. of the\nabove equations giving once again the usual shift. To this\naim, let us first explore a bit the gauge transformations.\nThese are given by\nU7−→eαT3Ue−αT3, (B8)\nAext\nµ7−→Aext\nµ+∂µα. (B9)\nApplying this to ˆLgives eαT3ˆLe−αT3, so the lagrangian\nis gauge invariant. An elementary calculation shows thatthe gauge transformation of Usimply gives the transla-\ntion Φ 7−→Φ +α. This explains why Aµappears in the\ncombination ∂µΦ−Aext\nµinˆL: it is the only possible gauge\ninvariant combination. Now, let us write the functional\ndependence of Vµ:\nVµ=Vµ[χ,Θ,Φ, ∂µχ, ∂µΘ, ∂µΦ−Aext\nµ]. (B10)\nSince the combination ∂µΦ−Aext\nµis gauge invariant, and\ngiven the effect of gauge transformations, we can write\nVµ=eΦT3Vµ[χ,Θ,0, ∂µχ, ∂µΘ, ∂µΦ−Aext\nµ]e−ΦT3\n=eΦT3Vµ\n0e−ΦT3.\n(B11)\nTherefore,\n∂µVµ=eΦT3∂µVµ\n0e−ΦT3+eΦT3∂µΦ[T3, Vµ\n0]e−ΦT3,\n(B12)\nand so\n∂µVµ−Aext\nµ[T3, Vµ] =eΦT3∂µVµ\n0e−ΦT3\n+eΦT3(∂µΦ−Aext\nµ)[T3, Vµ\n0]e−ΦT3, (B13)\nas we wanted to prove. This shows that the equations\ngauged with Aext\nµare obtained from the ungauged ones\nby shifting ∂µby−Aext\nµeverywhere.19\nThe same calculations work for the baryonic layer solu-\ntions. In this case a gauge transformation shifts Φ →\nΦ +β, Θ→Θ−β, so only the combination αis affected\nby the gauge transformation, with α→α−β. Therefore,\nin this case, it necessarily must appear everywhere in the\ngauge invariant combination ∂µα+Aext\nµ.\nAppendix C: General form of the Skyrme equations\nIn this section, we report the general form of the Skyrme\nequations (i.e., with λ̸= 0) for both the Euler and expo-\nnential parameterization.\n1. The Euler parameterization\nIn the case of Euler parameterization, we remember that\nthe Skyrme field is written as\nU=eΦκeχheΘκ. (C1)Since we are considering the general equations, without\ntaking any ansatz on the function representing the Euler\nangles, χ, Θ, and Φ depend in a generalized way on the\nspace-time variables. The only ansatz is taken on the\nmatrices κandh, which are the same as the ones defined\nin Section II. As deeply analyzed in [55], the map (C1)\ndescribes a closed cycle when the boundary conditions on\nΦ, Θ and χare chosen in such a way that\n0≤Φ≤ησ2π,0≤Θ≤η2π, (C2)\n0≤χ≤nπ, (C3)\nwhere σ=η= 1 for odd-dimensional representations of\nthe elements of the Lie algebra gandσ=1\n2andη= 2\nfor even-dimensional representations (which means that\nσ=1\nη).\nWith these choices, the Skyrme equations take the form\n∂µ∂µχn\n1 +λh\nAµAµsin2\u0010χ\n2\u0011\n+∂µξ∂µξcos2\u0010χ\n2\u0011io\n−sin(χ)\u0012\n1−λ\n4∂µχ∂µχ\u0013\n(AνAν−∂νξ∂νξ)\n−λsin(χ) cos( χ)h\nAµAµ∂νξ∂νξ−(Aµ∂µξ)2i\n−λn\nsin2\u0010χ\n2\u0011\n∂µAµAν∂νχ+ cos2\u0010χ\n2\u0011\n∂µ∂µξ∂νξ∂νχ\n+ sin2\u0010χ\n2\u0011\n[Aµ∂µ(Aν∂νχ)−∂µχ∂µ(AνAν)] + cos2\u0010χ\n2\u0011\n[∂µξ∂µ(∂νξ∂νχ)−∂µχ∂µ(∂νξ∂νξ)]o\n−λ\n4sin(χ)h\n(Aµ∂µχ)2−(∂µξ∂µχ)2i\n= 0, (C4)\n4 sin\u0010χ\n2\u0011n\nsin\u0010χ\n2\u0011n\n∂µAµ\u0014\n1 +λ\n4∂νχ∂νχ\u0015\n−λ\n4[∂µ∂µχ∂νχAν+∂µχ∂µ(∂νχAν)− Aµ∂µ(∂νχ∂νχ)]o\n+ cos\u0010χ\n2\u0011nλ\n2sin(χ)∂µAµAν∂νξ−λ\n2sin(χ) [∂µ∂µξAνAν+∂µξ∂µ(AνAν)− A µ∂µ(Aν∂νξ)]\n+λcos(χ) [∂µχAµAν∂νξ−∂µχ∂µξAνAν] +b∂µχ∂µξoo\n−4 cos\u0010χ\n2\u0011n\ncos\u0010χ\n2\u0011n\n∂µ∂µξ\u0014\n1 +λ\n4∂νχ∂νχ\u0015\n−λ\n4[∂µ∂µχ∂νχ∂νξ+∂µχ∂µ(∂νχ∂νξ)−∂µξ∂µ(∂νχ∂νχ)]o\n+ sin\u0010χ\n2\u0011nλ\n2sin(χ)∂µ∂µξAν∂νξ−λ\n2sin(χ) [∂µAµ∂νξ∂νξ+Aµ∂µ(∂νξ∂νξ)−∂µξ∂µ(Aν∂νξ)]\n+λcos(χ) [∂µχ∂µξAν∂νξ−∂µχAµ∂νξ∂νξ]−∂µχAµoo\n= 0, (C5)\n4 sin\u0010χ\n2\u0011\u001a\ncos\u0010χ\n2\u0011n\n∂µAµ\u0014\n1 +λ\n4∂νχ∂νχ\u0015\n−λ\n4[∂µ∂µχ∂νχAν+∂µχ∂µ(∂νχAν)− Aµ∂µ(∂νχ∂νχ)]o\n−sin\u0010χ\n2\u0011nλ\n2sin(bχ)∂µAµAν∂νξ−λ\n2sin(χ) [∂µ∂µξAνAν+∂µξ∂µ(AνAν)− A µ∂µ(Aν∂νξ)]\n+λcos(χ) [∂µχAµAν∂νξ−∂µχ∂µξAνAν] +∂µχ∂µξoo\n+ 4 cos\u0010χ\n2\u0011n\nsin\u0010χ\n2\u0011n\n∂µ∂µξ\u0014\n1 +λ\n4∂νχ∂νχ\u0015\n−λ\n4[∂µ∂µχ∂νχ∂νξ+∂µχ∂µ(∂νχ∂νξ)−∂µξ∂µ(∂νχ∂νχ)]o\n−cos\u0010χ\n2\u0011nλ\n2sin(χ)∂µ∂µξAν∂νξ−λ\n2sin(χ) [∂µAµ∂νξ∂νξ+Aµ∂µ(∂νξ∂νξ)−∂µξ∂µ(Aν∂νξ)]\n+λcos(χ) [∂µχ∂µξAν∂νξ−∂µχAµ∂νξ∂νξ]−∂µχAµoo\n= 0, (C6)20\nwith α=1\n2(Θ−Φ),ξ=1\n2(Θ + Φ).\n2. The exponential parameterization\nThe general equations for the exponential parameteriza-\ntion are obtained considering the following form for the\nSkyrme field\nU(t, r, θ, ϕ ) = exp( χτ1), (C7)with, as defined in Section IV,\nτ1=⃗ n·⃗T=n1T1+n2T2+n3T3 (C8)\n⃗ n= (sin Θ cos Φ ,sin Θ sin Φ ,cos Θ) . (C9)\nAs for the baryonic layers, here, χ, Θ, and Φ are general\nfunctions of the space-time variables. Here, the bound-\naries are defined by\n0≤Φ≤2pπ, 0≤Θ≤qπ, (C10)\n0≤χ≤nπ, (C11)\nwith n,pandqintegers. The Skyrme equations are\n∂µ∂µχ−sinχ\u0000\n∂µΘ∂µΘ + sin2ΘAµAµ\u0001\n−λ\n4\u001a\nsinχ\u0002\n(∂µχ∂µχ)(∂νΘ∂νΘ)−(∂µχ∂µΘ)2\u0003\n+ sin χsin2Θ\u0002\n(∂µχ∂µχ)(AνAν)−(∂µχAµ)2\u0003\n+ 4 sin χ(1−cosχ) sin2Θ\u0002\n(∂µΘ∂µΘ)(AνAν)−(∂µΘAµ)2\u0003\n−2∂µ[(1−cosχ)(∂νΘ∂νΘ)∂µχ] + 2∂µ[(1−cosχ)(∂νχ∂νΘ)∂µΘ]\n−2∂µ\u0002\n(1−cosχ) sin2Θ(AνAν)∂µχ\u0003\n+ 2∂µ\u0002\n(1−cosχ) sin2Θ(∂νχAν)Aµ\u0003\u001b\n= 0, (C12)\n(1−cosχ)∂µ∂µΘ + sin χ∂µχ∂µΘ−(1−cosχ) sin Θ cos Θ AµAµ\n−λ\n4\u001a\n(1−cosχ) sin Θ cos Θ\u0002\n(∂µχ∂µχ)(AνAν)−(∂µχAµ)2\u0003\n+ 2(1−cosχ)2sin Θ cos Θ\u0002\n(∂µΘ∂µΘ)(AνAν)−(∂µΘAµ)2\u0003\n−∂µ[(1−cosχ)(∂νχ∂νχ)∂µΘ] + ∂µ[(1−cosχ)(∂νχ∂νΘ)∂µχ]\n−2∂µ\u0002\n(1−cosχ)2sin2Θ(AνAν)∂µΘ\u0003\n+ 2∂µ\u0002\n(1−cosχ) sin2Θ(∂νΘAν)Aµ\u0003\u001b\n= 0, (C13)\n∂µ\u001a\u0002\n(1−cosχ) sin2ΘAµ\u0003\n+λ\n4\u0002\n(1−cosχ) sin2Θ(∂νχ∂νχ)Aµ\u0003\n−λ\n4\u0002\n(1−cosχ) sin2Θ(∂νχAν)∂µχ\u0003\n+λ\n2\u0002\n(1−cosχ)2sin2Θ(∂νΘ∂νΘ)Aµ\u0003\n−λ\n2\u0002\n(1−cosχ)2sin2Θ(∂νΘAν)∂µΘ\u0003\u001b\n= 0. (C14)\nAppendix D: The caustic curve\nSince the parameters η1, η3depend linearly on the mag-\nnetic field h, we find convenient to introduce the con-\nstants ˜ ηi,i= 1,3, so that\nη1=−h˜η1, η 3=−h˜η3. (D1)\nIf we further introduce y≡y(µ, h) by\ny=η4˜η2−3\n32˜η2\n3h2, (D2)\nbecause of (61), yis linear in µB, and the cubic (67)\ntakes the form\ny3−27\n8h2˜η3\u0012\n˜η1η2\n4−˜η3\n3h2\n27\u0013\ny+27\n8\u0012\nh2˜η2\n1η4\n4+5\n32˜η1˜η3\n3η2\n4−˜η6\n3\n211\u0013\n= 0.(D3)\nThe discriminant of this cubic is\n∆y=36\n220h4(16˜η1η2\n4−˜η3\n3h2)3. (D4)\nIt is positive for small h, negative for large h, changes\nsign at (recalling ˜ ηj>0 for j= 1,3)\n¯h= 4s\n˜η1\n˜η3η4\n˜η3, (D5)\nand vanishes at h= 0. For h < ¯hthere is only one\nbranch, while a second branch appears at h≥¯h. Since21\nthere are no further zeros of ∆ y, the two branches cannot\nintersect. To understand the shape of the caustic in the\n(h, y) plane, it is convenient to determine explicitly an\nexpression for the solutions at h∼0,h→ ∞ andh∼¯h.\nForh= 0 the unique solution is y= 0 (triple degenerate).\nFor small h, one then immediately finds that the unique\nsolution is\ny0=−3\n2˜η2\n3\n1η4\n3\n4h2\n3+O(h2). (D6)\nFor the other cases, it is convenient to use Cardano’s\nformula\nyi=ωi−1\u0010\n−q\n2+p\n∆y\u00111\n3+ ¯ωi−1\u0010\n−q\n2−p\n∆y\u00111\n3,\n(D7)\ni= 1,2,3, where ω=1\n2(−1 +i√\n3), and we have written\n(D3) in the form y3+py+q= 0. For very large hwe\nhave\nq≈ −27\n214˜η6\n3h6, (D8)\n∆y≈ −36\n220˜η9\n3h10. (D9)\nTherefore,p\n|∆x|is smaller than qand we can write\nyi≈ωi−1\u0010\n−q\n2\u00111\n3 \n1−2ip\n|∆y|\n3q!\n+c.c., (D10)\nwhere c.c.means complex conjugate. More explicitly:\ny∞,1=3\n16˜η2\n3h2+O(1), (D11)\ny∞,2=−3\n32˜η2\n3h2 \n1−32\n˜η3\n2\n3√\n3h!\n+O(1), (D12)\ny∞,3=−3\n32˜η2\n3h2 \n1 +32\n˜η3\n2\n3√\n3h!\n+O(1). (D13)\nTo understand to which branch this solution belongs, it\nis sufficient to analyze the solutions in h >¯h, very near\nto¯h. Since now ∆ yis very small, the expressions (D10)\nare still valid but with different values of qand ∆ y:\nq\n2≈\u00129˜η1η2\n4\n2˜η3\u00133\n, (D14)\n∆y≈ −36\n217¯h3˜η3\n3(h−¯h)3. (D15)Therefore, we get\ny¯h,1=−9˜η1η2\n4\n˜η3+O((h−¯h)2), (D16)\ny¯h,2=9˜η1η2\n4\n2˜η3 \n1−√\n2˜η9\n2\n3¯h3\n2√\n366˜η3\n1η6\n4(h−¯h)3\n2!\n+O((h−¯h)2),\n(D17)\ny¯h,3=9˜η1η2\n4\n2˜η3 \n1 +√\n2˜η9\n2\n3¯h3\n2√\n366˜η3\n1η6\n4(h−¯h)3\n2!\n+O((h−¯h)2).\n(D18)\nWe see that y¯h,2andy¯h,3coincide at h=¯hso belong to\nthe new branch. y¯h,1necessarily belongs to the branch of\ny0. Since the two branches cannot intersect and y¯h,1is be-\nlowy¯h,2andy¯h,3, we infer that y∞,3belongs to the branch\nstarting from h= 0, while y∞,2belongs to the lower part\nof the second branch and y∞,1belongs to the upper part\nof the second branch. We considered the branch starting\nat¯has a unique branch, being connected, but we remark\nthat its upper and lower part form a cusp in h=¯h, since\nthere\ndy¯h,2\ndh\f\f\f\f\nh=¯h=dy¯h,3\ndh\f\f\f\f\nh=¯h= 0. (D19)\nFrom this analysis, we can easily reconstruct the picture\nin the ( h, µB)-plane simply by using (D2) and (61). It\nthen follows that the simple branch results to be above\nwhile the double branch is below. Notice also that from\n(D6) and (61) we get for the simple branch that it starts\nfrom\nµB= 4KLrL2π2Σ2(h, m) \n3π\nL2\nθ+6π\nL2\nϕ!\nwith an infinite positive slope. Finally, we can notice that\nthe caustic is necessarily confined in the region where b <\n0. By writing η2=η(0)\n2+η(1)\n2h2, where η(i)\n2are positive\nconstants, we see that such condition corresponds to\n∥c∥2\n2µB≥η(0)\n2+\u0012\nη(1)\n2−3\n8˜η2\n3\nη4\u0013\nh2. (D20)\nFrom the expressions relating the ηs to the Σ( h, m)s, we\nsee that the parenthesis is positive when\nΣ(1)\n2Σ4−3\n8˜Σ2\n3>0, (D21)\nwhere analogously to the ηs, we have defined Σ 3=:˜Σ3h\nand Σ 2:= Σ(0)\n2+ Σ(1)\n2h2. From the expressions (F12)–\n(F16), we see that the positiveness is never satisfied.\nFinally, it is interesting to notice that the lowest com-\nponent of the double branch is definitely decreasing. In-\ndeed, from the expression for y∞,1we get\n∥c∥2\n2µB,∞,1=η(0)\n2+\u0012\nη(1)\n2−9\n32˜η2\n3\nη4\u0013\nh2. (D22)22\nAs above, for m≥1 the expression in parenthesis results\nto be always negative, therefore, for large hsuch a lower\ncomponent becomes negative.\nThus, we see that the qualitative shape of the caustic\nin the ( h, µB) plane is universal, it does not depend on\nthe specific values of the parameters. A representative\npicture is shown in Figure 2.\nAppendix E: The Pearcey integral\nHere we recall the main properties of the Pearcey integral\n(70), expressed in the variable xandy, introduced in [79],\nrelated to the standard XandYvariables by x=Xe−iπ\n4,\ny=Y eiπ\n8. For any fixed value of xandy, we can expand\nthe exponentials e−xq2andeiyqin power series to get\nP(x, y) =1\n2eiπ\n8∞X\nn=0∞X\nm=0(−1)m\nm!(2n)!Γ\u0012m+n\n2+1\n4\u0013\nxmy2n,\n(E1)\nwhich converges in any compact polydisc. Therefore, the\ninteresting cases arise when x, ory, or both become very\nlarge. Even if a complete treatment is still lacking, these\ncases are well-studied in the literature, and we report\nhere the main results.\nLarge xexpansion. An exhaustive analysis of the\nasymptotic behavior of the Pearcey integral for large x\nat any fixed value for ycan be found in [79]. One has\nto distinguish two cases. For |arg(x)| ≤π\n2one finds the\ncomplete asymptotic expansion\nP(x, y)∼rπ\nxeiπ\n8S1(x, y), (E2)\nS1(x, y) =e−y2\n4x∞X\nm=0(−1)mam(y2/4x)\nm!x2m, (E3)am(z) =2−4mH4m(√z), (E4)\nwhere Hnis the n-th Hermite polynomial.\nFor|arg(−x)|<π\n2one finds\nP(x, y)∼rπ\nxeiπ\n8h\nσ(x)S1(x, y) +i√\n2ex2\n4S2(x, y)i\n,\n(E5)\nσ(x) =−sign arg( −x), (E6)\nS2(x, y) =∞X\nm=0(y2/8x)2m\nm!h\nP(2m, ξ) cosξ\n−Q(2m, ξ) sinξi\n, (E7)\nP(2m, ξ) =mX\nk=0(−1)k(2m+ 2k)!(2ξ)−2k\n(2k)!(2m−2k)!, (E8)\nQ(2m, ξ) =m−1X\nk=0(−1)k(2m+ 2k+ 1)!(2 ξ)−2k−1\n(2k+ 1)!(2 m−2k−1)!,(E9)\nξ=y\u0010\n−x\n2\u00111\n2. (E10)\nThe main ingredient here is the exponential ex2\n4in (E5).\nFrom it, we see that when |arg(−x)|<π\n4the Pearcey in-\ntegral grows exponentially far away from the origin, while\nafter crossing the half-lines arg( x) =±3\n4πit oscillates\nwith a dominating algebraic behavior. Such half-lines are\nStokes lines in the complex xplane, independently on y,\nasystays bounded. A constant discontinuity appears at\narg(x) =π, and for arg( z) =±π\n2we have a transition\nfrom a double function contribution ( S1+S2) to a sin-\ngle function contribution ( S1): these three half-lines are\nanti-Stokes lines[85].\nLarge yexpansion. The case of bounded xand large\n|y|is carefully developed in [86], [87]. Since the Pearcey\nintegral is even in y, it is sufficient to consider the region\narg(y)≤π\n2. In this case, one gets\nP(x, y) =\n\nP1(x, y) +P2(x, y) +O(e−1.38077|y|4\n3) if |arg(y)| ≤π\n8,\nP1(x, y) if −π\n2≤ |arg(y)|<−π\n8,\nP2(x, y) ifπ\n8<|arg(y)| ≤π\n2,(E11)\nwhere\nPk(x, y)∼p\nπ/3\n25\n6y1\n3exp\u0014\n3\u0010y\n4\u00114\n3e−(−1)k2iπ\n3−x\u0010y\n4\u00112\n3e−(−1)kiπ\n3+x2\n6\u0015\"mX\nn=0e(−1)k(2n+1)iπ\n6An(x)\ny2\n3n+O(1\ny2m+3)#\n,\n(E12)\nand\nAn(x) =nX\nr=⌊n+1\n2⌋2r−nX\nl=0(−1)n+lan,r,l(x)c2r+n−l(x), (E13)23\nan,r,l(x) =(−1)rxl24\n3(2r−n−l)\nk!(2r−n−l)!(n−r)!, (E14)\nc2r+n−l(x) =xn\n3n2n\n3⌊n\n2⌋X\nk=0\u00123\n2x2\u0013kn!\nk!(n−2k)!. (E15)\nFrom this, one can infer that there is a Stokes line along\nthe real half-line arg( y) = 0, and two anti-Stokes lines at\narg(y) =±3\n8π.\nLarge parameters near the caustic. This case is\nconsidered in [80]. The analysis, despite being valid also\nfor certain complex value, are not as complete as in the\nprevious case and allows us to do just a partial analysis,\nwhich is however interesting since it concerns the case\nwhen xandyare real. The key fact is that in such a sit-\nuation the number of saddle points contributing to the\nasymptotic expansion depends on the sign of the discrimi-\nnant ∆: when positive, only one saddle point contributes,\nwhile when negative, three saddle points contribute. The\ndelicate question is what happens at the caustic. Since\nthe discriminant can change sign only for negative b, it\nfollows that it is sufficient to consider the case of large\nnegative x. To use the results in [80], it is worth men-\ntioning that there is a further different convention for\nthe Pearcey integral. If we call P(X, Y) the standard\ndefinition and PK(X, Y) the one in [80], then we have\nP(Xe−iπ\n4, Y eiπ\n8) =P(X, Y) =1√\n2PK(X,Y√\n2).(E16)\nThe analysis in [80] (se also [88]) is done for real X, Y,\nwhile we are interested in real xand purely imaginary y,\nwhich correspond to complex XandY. In particular, we\nare exactly at the boundary of the region of the validity\nof their analysis, see the comment below formula (2.2) in\n[80]. Since we are not sure to stress their results to the\nboundary, it is convenient for us to repeat the calculation\nin our case.\nAs in [80], the interesting situation is near the points\nwhere the caustic changes sign, and, when xis real, also\nin our case, this happens when xis large and negative.\nThus, we change x→ −xand consider xpositive. The\ncaustic is along y2=8\n27x3, so we are interested in the\nlarge xbehaviour of\nQ(x, µ) =P(−x, µx3\n2), (E17)\nwhere we set\nµ=r\n8\n27−α, (E18)\nandαis a small real parameter. When αis positive there\nare three real critical points, while when αis negative\nthere are one real and two complex conjugate critical\npoints. It is convenient to do the calculations for positive\nαand then consider analytic continuation. After a simplechange of coordinates, we can write\nQ(x, µ) =eiπ\n8√xZ\nRe−x2(z4−z2+µz)dz. (E19)\nThe critical points of the quartic polynomial in zare\nz1=−r\n2\n3sin\u0010π\n3+ϕ\u0011\n, (E20)\nz2=−r\n2\n3sin (ϕ), (E21)\nz3=r\n2\n3sin\u0010π\n3−ϕ\u0011\n, (E22)\nwhere ϕis such that |ϕ| ≤π\n6and is defined by\nϕ=1\n3arcsin \nµr\n27\n8!\n. (E23)\nNotice that when α→0 then ϕ→π\n6and the two positive\nroots coincide at z2=z3= 1/√\n6. Taking into account\nthis fact, it is convenient to deform the integration con-\ntour to Γ = Γ 1∪Γ2, where Γ jare depicted in Fig 7.\nΓ2Γ2Γ2 Γ1Γ1Γ1\n0001√\n61√\n61√\n61√\n21√\n21√\n2−1√\n2−1√\n2−1√\n2\n−q\n2\n3−q\n2\n3−q\n2\n3\nFIG. 7. The integration path is deformed from the real axis to\nthe union of the two curves Γ 1and Γ 2. The red lines indicate\nthe displacing of the real solutions z1<0≤z2≤z3when ϕ\nvaries from 0 to π/6.\nFrom the picture, we see that in this way, we can always\ndeform Γ 1so that it passes through z2andz3, while Γ 1\npasses through z1. It is convenient to separate things in\nthis manner to have better control of what happens when\nthe two positive critical points merge.24\nWe first analyze the integral along Γ 1. Following [80], it is\nconvenient to consider a sequence of changes of variables.\nFirst, we introduce the shift z=t+1√\n6so that the\nexponent becomes\nz4−z2+µz=:g(t, α) +1\n9−α√\n6, (E24)\nwith\ng(t, α) =t4+4√\n6t3−αt. (E25)\nThen, we introduce a second change of variable u=u(t),\ndefined by\ng(t, α) =u3\n3−ζu+η. (E26)\nHere, we want the change of variables to be smooth at\nthe points tj=zj−1√\n6,j= 2,3. Taking the derivativeof the above expression w.r.t. u, we get\n∂t(t, α)dt\ndu=u2−ζ. (E27)\nThe only possibility fordt\nduto be finite and non-zero at\nt2andt3is that, correspondingly, u=±ζ1\n2. We choose\nthe plus sign to correspond to t3. Therefore, we get\nζ3\n2=3\n4[g(t2, α)−g(t3, α)], (E28)\nη=1\n2[g(t2, α) +g(t3, α)]. (E29)\nBy solving (E26), one then gets\nu(t, α) =2ζ1\n2sinψ, (E30)\nψ=1\n3arcsin3(η−g(z, α))\n2ζ3\n2. (E31)\nFrom this one can show that the change of variable is\nanalytic not only around the critical points but along\nthe whole path Γ 1−1√\n6. In the new coordinate uthe\ncontribution of the Γ 1curve to Qis therefore\nQ1(x, α) =eiπ\n8√xP1(x, α), (E32)\nwhere for large values of xwe have\nP1(x, α)∼e−x2\u0010\n1\n9−α√\n6+η\u0011∞X\nn=0x−2n\u0014\npn(α)Z\nCe−x2\u0010\nu3\n3−ζu\u0011\ndu+qn(α)Z\nCue−x2\u0010\nu3\n3−ζu\u0011\ndu\u0015\n, (E33)\nwhere pnandqnas exactly as defined in (3.7) of [80], and\nCis the path starting at ei2\n3π∞and ending at + ∞. Theintegrals along Care easily expressed in terms of Airy\nfunctions so that we get\nP1(x, α)∼ −e−x2\u0010\n1\n9−α√\n6+η\u0011π\nx2\n3\u0010\nBi(ζx4\n3) +iAi(ζx4\n3)\u0011∞X\nn=0x−2npn(α)\n−e−x2\u0010\n1\n9−α√\n6+η\u0011π\nx4\n3\u0010\nBi′(ζx4\n3) +iAi′(ζx4\n3)\u0011∞X\nn=0x−2nqn(α), (E34)\nwhere the prime indicates derivative w.r.t. the argument.\nIn particular, we find\np0(α) =1√\n233\n4+O(α), (E35)\nq0(α) =1\n4√\n231\n4+O(α). (E36)Now, let us consider the contribution of the path Γ 2. In\nthis case, it is convenient to introduce the coordinate t\nsuch that z=t+z1so that\nz4−z2+µz=z4\n1−z2\n1+µz1+1\n2t2(12z2\n1−2) + 4 t3z1+t4.\nIn looking for the steepest descent paths, after putting\nt=x+iy, we determine the curves where the polynomial25\ninthas zero imaginary part. This gives\n0 =−4y3(x+z1) +y(4x3+ 12x2z1+x(12z2\n1−2)).\n(E37)\nThis is easily solved, keeping into account that z2\n1≤1\n2.\nThe curves are depicted in Fig. 8.\nIm(t)\nRe(t)−z1−z1−z1 z2z2z2 z3z3z3\nFIG. 8. The red line represent the steepest descent line pass-\ning through z1(i.e. t= 0). The blue lines represent real\ncurves for the exponent. The dark region represents valleys\nat large twhile the light regions are the hills.\nWe see that it is not possible to deform Γ 2to the red\ncurve since it would need to cross hills. However, we can\ndeform it to the red curve until the point −z1and then\nto the upward dashed half-right line. In computing the\nsaddle point approximation we then see that for large x\nonly the red line component contributes to the integral,\nwhile the remaining part gives exponentially smaller re-\nsults. The contribution of the Γ 2curve to Qis therefore\nQ2(x, α) =eiπ\n8√xP2(x, α), (E38)where\nP2(x, α)∼e−x2h(z1)rπ\n6z2\n1−1h\n1+\n+1\nx2\u0012\n15z2\n1−3\n4\u0013\n+. . .i\n,(E39)\nh(z) =z4−z2+µz. (E40)\nPutting it all together, we are now able to analyze the\nasymptotic behavior of the Pearcey integral around the\ncaustic. To this aim, it is convenient to recall the follow-\ning asymptotic expressions for the Airy functions (see\n[89]):\nAi(s)∼1\n2√πs1\n4e−ξ∞X\nk=0(−1)kck\nξk,|arg(z)|< π, (E41)\nAi′(s)∼ −s1\n4\n2√πe−ξ∞X\nk=0(−1)kdk\nξk,|arg(z)|< π,\n(E42)\nBi(s)∼1√πs1\n4eξ∞X\nk=0ck\nξk,|arg(z)|<π\n3, (E43)\nBi′(s)∼ −s1\n4√πeξ∞X\nk=0dk\nξk,|arg(z)|<π\n3, (E44)\nwhere\nξ=2\n3s3\n2, (E45)\nand\nc0=d0= 1, d k=−6k+ 1\n6k−1ck, (E46)\nck=Γ(3k+ 1/2)\n(54)kk!Γ(k+ 1/2). (E47)\nWe will also need:\nBi(ze±π\n3i)∼r\n2\nπe∓π\n6i\nz1\n4h\nsin\u0010\nξ+π\n4∓i\n2log 2\u0011∞X\nk=0(−1)kc2k\nξ2k−cos\u0010\nξ+π\n4∓i\n2log 2\u0011∞X\nk=0(−1)kc2k+1\nξ2k+1i\n, (E48)\nBi′(ze±π\n3i)∼r\n2\nπe∓π\n6iz1\n4h\ncos\u0010\nξ+π\n4∓i\n2log 2\u0011∞X\nk=0(−1)kd2k\nξ2k+ sin\u0010\nξ+π\n4∓i\n2log 2\u0011∞X\nk=0(−1)kd2k+1\nξ2k+1i\n, (E49)\nwhich are valid when |arg(z)|<2\n3π. Let us write the final result in the form26\nQ(x, α)∼ −eiπ\n8√xe−x2\u0010\n1\n9−α√\n6+η\u0011\"\nπ√\n631\n41\nx2\n3(Bi(ζx4\n3) +iAi(ζx4\n3)) +31\n4π\n4√\n61\nx4\n3(Bi′(ζx4\n3) +iAi′(ζx4\n3))#\n+eiπ\n8e−x2h(z1)rπ\n6z2\n1−11√x+. . . . (E50)\nSince h(z1)<0 for any α, we see that the term in the\nsecond line ( Q2(x, α)) is always dominating. The first\nline, instead, contributes in different ways according to\nthe sign of α. For α >0,ζis real, and using the aboveformulas for the Airy functions we get\nQ1(x, α)∼ −eiπ\n8√xrπ\n61−√3ζ\n(3ζ)1\n4e−x2h(z3). (E51)\nHere h(z3) is positive, so this term decays exponentially\nlike a Gaussian.\nForα <0 instead oscillating modes appear. In this case,\nζ=|ζ|eiπ\n3and we get\nQ1(x, α)∼ −eiπ\n8√xe−x2\u0010\n1\n9−α√\n6+η\u0011\"√πeiπ\n4\n33\n4|ζ|1\n4sin\u00122\n3|ζ|3\n2x2+π\n4−i\n2log 2\u0013\n−√πe−iπ\n12|ζ|1\n4\n31\n44cos\u00122\n3|ζ|3\n2x2+π\n4−i\n2log 2\u0013#\n−eiπ\n8√xe−x2\u0010\n1\n9−α√\n6+η\u0011\ni√πe−iπ\n12\n2√\n631\n4 \n1−√\n6|ζ|1\n2eiπ\n6\n4!\ne−i2\n3|ζ|3\n2x2.\n(E52)\nFinally, for α= 0 we have ζ= 0 and η= 6−1\n23−1\n4, therefore we find\nQ1(x, α)∼ −e−iπ\n8e−x2 \n1\n9+1\n√\n631\n4!\nπ√\n631\n4 \n1\n31\n3Γ(1\n3)+i1\n32\n3Γ(2\n3)!\n1\nx1\n6. (E53)\nAppendix F: The piecewise-linear solutions\nIn this section, we want to give an insight into the case\nwhen the condition λ/16L2\nr= 1 is not satisfied.\nThe requirement to satisfy the imposed boundary condi-\ntions is\nλ\n16L2r<1. (F1)\nWith this condition, the universal solution takes the form\nχ(r) =\u001a\n0 for 0 ≤r≤r0(m),\nm\n2[r−r0(m)] for r > r 0(m),(F2)\nwhere m=p\n16L2r/λis the slope of the solution and\nr0(m) is the intersection between the non-zero part ofthe solution and the r-axis, which depends on m. In\nparticular, it takes values\nr0(m) = 2 π\u0012\n1−1\nm\u0013\n. (F3)\nWhen the condition λ/16L2\nr= 1 is satisfied, m= 1 and\nr0(1) = 0. This way, equation (F2) is a general formu-\nlation for the condition λ/16L2\nr≤1 (i.e., m≥1). It\nis worth noticing here that this is not the only solution,\nbut other solutions can be defined by alternating the con-\nstant parts and non-constant parts in an infinite number\nof combinations. It would correspond to pieces of bary-\nonic layers, leading to a discontinuous form of the energy\ndensity. For this reason, we considered only the solution\n(F2). In this more general case, the energy density takes\nthe form (44), but each ρihas an additional dependence27\non the parameter m. In particular, for 0 ≤r≤r0(m), the expressions are [90]\nρ0=K∥c∥2\u0012p2\n2L2\nϕ+q2\n4L2\nθ\u0013\n, (F4)\nρ1= 0, (F5)\nρ2= 0. (F6)\nOn the other hand, for r > r 0(m),\nρ0=K∥c∥2\u001ap2\n2L2\nϕ\u0014\n1 +λm2\n8L2r+q2λ\n2L2\nθsin2\u0012m(r−r0(m))\n2\u0013\u0015\n+q2\n4L2\nθ\u0012\n1 +λm2\n4L2r\u0013\n+m2\n16b2L2r\u001b\n, (F7)\nρ1=−K∥c∥2\u001ap2\n2L2\nϕ\u0014qλ\nL2\nθsin2\u0012m(r−r0(m))\n2\u0013\u0015\n+q\nL2\nθsin2\u0012m(r−r0(m))\n4\u0013\u0012\n1 +λm2\n8L2r\u0013\u001b\n, (F8)\nρ2=K∥c∥2\u001ap2\n2L2\nϕ\u0014λ\n2L2\nθsin2\u0012m(r−r0(m))\n2\u0013\u0015\n+1\nL2\nθsin2\u0012m(r−r0(m))\n4\u0013\u0012\n1 +λm2\n8L2r\u0013\u001b\n, (F9)\nThe baryonic density can be computed in a very similar\nway, giving\nρB=(0 for 0 ≤r≤r0(m)\n∥c∥2\n8π2mq∂ ϕΦ sin\u0010\nm(r−r0(m))\n2\u0011\nforr > r 0(m),\n(F10)\nwhich integral gives the usual integer value of equation\n(56). Thus, as for the energy density, the baryons accu-\nmulate in peaks that become sharper for bigger values\nofm, preserving the baryonic charge B. Notice that in\nthe region corresponding to the density (F4) there are no\nbaryons, but the energy density is nonzero (and indepen-\ndent of the external field). This energy thus correspond\nto mesonic fluctuations.\nThe total energy can be written as\nE(q;h, m) = 2 π2KLrLθLϕ∥c∥24X\ni=1Σi(h, m)qi,(F11)where, this time,\nΣ4(h, m) =8L2\nr\nL2\nϕL2\nθπ\nm3, (F12)\nΣ2(h, m) =3π\nL2\nθ+6π\nL2\nϕ\n+h216L2\nrπ\nL2\nϕL2\nθm3\u00142π2−3\n3m2+ 2π2\u0012\n1−1\nm\u0013\u0015\n, (F13)\nΣ0(h, m) =πm2\n4L2r\n+h28π\nL2\nθm\u0014\n3π2\u0012\n1−1\nm\u0013\n+1\nm\u0012\n2−1\nm\u0013\n+π2\nm2\u0015\n,(F14)\nΣ3(h, m) =−h16L2\nrπ2\nL2\nϕL2\nθm3\u0012\n2−1\nm\u0013\n, (F15)\nΣ1(h, m) =−h6\nL2\nθm\u00144\nm+π2\u0012\n2−1\nm\u0013\u0015\n. (F16)\nIt is straightforward to check that when m= 1 these\nquantities reduce to eqs. (50) to (54). The partition func-\ntion takes the form\nZ=X\nqexp\u001a\n−β∥c∥2\u0014\n2KLrL2π24X\ni=1Σi(h, m)qi−q2\n2µB\u0015\u001b\n, (F17)\nwhere, again, we used the condition (37) with Lθ=Lϕ=\nL. This allows us to determine the thermodynamic prop-\nerties of the baryonic layers. In this scope, we need to\nexplicitly compute the sums over q. This has already\nbeen studied for the ungauged case in [59].Appendix G: The polarization effects\nIn this section, we discuss the possibility of considering\nan electromagnetic field generated from the polarization\neffects induced by the external field. In this scope, we\nsuppose that to preserve its existence, the skyrmion gen-\nerates a backreaction that cancels the external magnetic28\nfield in the boundaries. This gives rise to an internal\nelectromagnetic field, whose behavior can be defined as\nsolutions to the Maxwell Equations.\nLet us choose the usual ansatz, with\nα=q\n2θ−p\n2\u0012t\nLϕ−ϕ\u0013\n, (G1)\nξ=q\n2θ+p\n2\u0012t\nLϕ−ϕ\u0013\n, (G2)\nχ=χ(r). (G3)Let us suppose that the induced internal field takes the\nform\nAµ=\u0012\n−Aϕ\nLϕ(r),0, Aθ(r), Aϕ(r)\u0013T\n. (G4)\nThis way, the Skyrme equation for the profile χ(r) be-\ncomes\n∂µ∂µχ\u001a\n1 +b2λ\u0014\u0012A2\nθ\nL2\nθ−q\nL2\nθAθ\u0013\nsin\u0012bχ\n2\u0013\n+q2\n4L2\nθ\u0015\u001b\n−bsin(bχ)\u0012\n1−b2λ\n4∂µχ∂µχ\u0013\u0012A2\nθ\nL2\nθ−q\nL2\nθAθ\u0013\n= 0,(G5)\nNotice that it admits a linear solution\nχ(r) =r\n4L2r\nb2λr. (G6)The Maxwell equations for the gauge field (G4) may be\nrewritten as follows\nA′′\nθ\nL2r−K\n2∥c∥2n\u0010q\n2−Aθ\u0011h\n8 sin2\u0010χ\n2\u0011\u0012\n1 +λ\n4∂νχ∂νχ\u0013\n−2λcos2(χ)q2\n4L2\nθio\n= 0, (G7)\nA′′\nα\nL2r−K\n2∥c∥2n\n(pq−Aα)h\n8 sin2\u0010χ\n2\u0011\u0012\n1 +λ\n4∂νχ∂νχ\u0013\n+ 2λsin2(aχ)q2\n4L2\nθio\n= 0, (G8)\nwhere, using the notation of [55],\nAα=pAθ+qAϕ. (G9)\nAs discussed in [55], these equations can be always re-\nconduced to a Hill equations form. In particular, when\nthe linear solution for χis considered, they become\nWittaker-Hill equations. Indeed, they may be written\nas\nA′′\ni+ [Λ i+ Γicos(2 ω) + ∆ icos(4 ω)]Ai= 0,(G10)\nwith i=θ, αand\nΛθ=K\n2∥c∥2\u0012λq2\n4L2\nθ−4\u0013\n,Γθ= 2K∥c∥2,∆θ=K\n2∥c∥2λq2\n4L2\nθ, (G11)\nΛα=−K\n2∥c∥2\u0012λq2\n4L2\nθ+ 4\u0013\n,\nΓθ= 2K∥c∥2,∆θ=K\n2∥c∥2λq2\n4L2\nθ, (G12)\nω=χ\n2. (G13)\n[1] Jeff Greensite. An introduction to the confinement prob-\nlem, volume 821. Springer, 2011.\n[2] Georges Ripka. Dual superconductor models of color con-\nfinement , volume 639. Springer Berlin, Heidelberg, 2004.\n[3] M. Shifman. Advanced topics in quantum field theory.: Alecture course . Cambridge Univ. Press, Cambridge, UK,\n2 2012.\n[4] A. S. Schneider, C. J. Horowitz, J. Hughto, and D. K.\nBerry. Nuclear “pasta” formation. Phys. Rev. C ,\n88(6):065807, 2013.29\n[5] C. J. Horowitz, D. K. Berry, C. M. Briggs, M. E. Caplan,\nA. Cumming, and A. S. Schneider. Disordered nuclear\npasta, magnetic field decay, and crust cooling in neutron\nstars. Phys. Rev. Lett. , 114(3):031102, 2015.\n[6] William G. Newton, Sarah Cantu, Shuxi Wang, Am-\nber Stinson, Mark Alexander Kaltenborn, and Ji-\nrina Rikovska Stone. Glassy quantum nuclear pasta in\nneutron star crusts. Phys. Rev. C , 105:025806, Feb 2022.\n[7] William G. Newton. A taste of pasta? Nature Physics ,\n9, 2013.\n[8] Jose A. Pons, Daniele Vigano’, and Nanda Rea. A highly\nresistive layer within the crust of X-ray pulsars limits\ntheir spin periods. Nature Phys. , 9:431–434, 2013.\n[9] C. O. Dorso, A. Strachan, and G. A. Frank. The nu-\ncleonic thermal conductivity of “pastas” in neutron star\nmatter. Nucl. Phys. A , 1002:122004, 2020.\n[10] Jorge A. Lopez, Claudio O. Dorso, and Guillermo A.\nFrank. Properties of nuclear pastas. Front. Phys. (Bei-\njing), 16(2):24301, 2021.\n[11] Igor R. Klebanov. Nuclear Matter in the Skyrme Model.\nNucl. Phys. B , 262:133–143, 1985.\n[12] C. J. Horowitz, J. Piekarewicz, and Brendan Reed. In-\nsights into nuclear saturation density from parity vio-\nlating electron scattering. Phys. Rev. C , 102(4):044321,\n2020.\n[13] M. E. Caplan, A. S. Schneider, and C. J. Horowitz. Elas-\nticity of Nuclear Pasta. Phys. Rev. Lett. , 121(13):132701,\n2018.\n[14] Mark G. Alford, Anton Kapustin, and Frank Wilczek.\nImaginary chemical potential and finite fermion density\non the lattice. Phys. Rev. D , 59:054502, 1999.\n[15] J. B. Kogut and D. K. Sinclair. Quenched lattice QCD\nat finite isospin density and related theories. Phys. Rev.\nD, 66:014508, 2002.\n[16] J. B. Kogut and D. K. Sinclair. Lattice QCD at finite\nisospin density at zero and finite temperature. Phys. Rev.\nD, 66:034505, 2002.\n[17] J. B. Kogut and D. K. Sinclair. The Finite tempera-\nture transition for 2-flavor lattice QCD at finite isospin\ndensity. Phys. Rev. D , 70:094501, 2004.\n[18] Silas R. Beane, William Detmold, Thomas C. Luu,\nKostas Orginos, Martin J. Savage, and Aaron Torok.\nMulti-Pion Systems in Lattice QCD and the Three-Pion\nInteraction. Phys. Rev. Lett. , 100:082004, 2008.\n[19] William Detmold, Martin J. Savage, Aaron Torok,\nSilas R. Beane, Thomas C. Luu, Kostas Orginos, and As-\nsumpta Parreno. Multi-Pion States in Lattice QCD and\nthe Charged-Pion Condensate. Phys. Rev. D , 78:014507,\n2008.\n[20] William Detmold, Kostas Orginos, Martin J. Savage, and\nAndre Walker-Loud. Kaon Condensation with Lattice\nQCD. Phys. Rev. D , 78:054514, 2008.\n[21] William Detmold and Brian Smigielski. Lattice QCD\nstudy of mixed systems of pions and kaons. Phys. Rev.\nD, 84:014508, 2011.\n[22] William Detmold, Kostas Orginos, and Zhifeng Shi. Lat-\ntice QCD at non-zero isospin chemical potential. Phys.\nRev. D , 86:054507, 2012.\n[23] G. Endr¨ odi. Magnetic structure of isospin-asymmetric\nQCD matter in neutron stars. Phys. Rev. D ,\n90(9):094501, 2014.\n[24] Erick J. Weinberg. Classical solutions in quantum field\ntheory: Solitons and Instantons in High Energy Physics .\nCambridge Monographs on Mathematical Physics. Cam-bridge University Press, 9 2012.\n[25] Gerard ’t Hooft. Gauge Fields with Unified Weak, Elec-\ntromagnetic, and Strong Interactions. In 1975 High-\nEnergy Particle Physics Divisional Conference of EPS\n(includes 8th biennial conf on Elem. Particles) , 9 1975.\n[26] S. Mandelstam. Vortices and Quark Confinement in Non-\nabelian Gauge Theories. Phys. Rept. , 23:245–249, 1976.\n[27] N. S. Manton and P. Sutcliffe. Topological solitons . Cam-\nbridge Monographs on Mathematical Physics. Cambridge\nUniversity Press, 2004.\n[28] Sidney R. Coleman. The Quantum Sine-Gordon Equa-\ntion as the Massive Thirring Model. Phys. Rev. D ,\n11:2088, 1975.\n[29] Edward Witten. Global Aspects of Current Algebra.\nNucl. Phys. B , 223:422–432, 1983.\n[30] Edward Witten. Current Algebra, Baryons, and Quark\nConfinement. Nucl. Phys. B , 223:433–444, 1983.\n[31] T. H. R. Skyrme. A new model for nuclear matter. Pro-\nceedings of the Royal Society of London. Series A, Math-\nematical and Physical Sciences , 226(1167):521–530, 1954.\n[32] T. H. R. Skyrme. Meson theory and nuclear matter.\nProceedings of the Royal Society of London. Series A,\nMathematical and Physical Sciences , 230(1181):277–286,\n1955.\n[33] T. H. R. Skyrme. A non-linear theory of strong interac-\ntions. Proceedings of the Royal Society of London. Series\nA, Mathematical and Physical Sciences , 247(1249):260–\n278, 1958.\n[34] T. H. R. Skyrme. A non-linear field theory. Proceedings\nof the Royal Society of London. Series A, Mathematical\nand Physical Sciences , 260(1300):127–138, 1961.\n[35] T. H. R. Skyrme. Particle states of a quantized meson\nfield. Proc. Roy. Soc. Lond. A , 262:237–245, 1961.\n[36] G.’t Hooft. A planar diagram theory for strong interac-\ntions. Nuclear Physics B , 72(3):461–473, 1974.\n[37] Edward Witten. Baryons in the 1n expansion. Nuclear\nPhysics B , 160(1):57–115, 1979.\n[38] Gregory S. Adkins, Chiara R. Nappi, and Edward Wit-\nten. Static Properties of Nucleons in the Skyrme Model.\nNucl. Phys. B , 228:552, 1983.\n[39] Fabrizio Canfora and Hideki Maeda. Hedgehog ansatz\nand its generalization for self-gravitating Skyrmions.\nPhys. Rev. D , 87(8):084049, 2013.\n[40] Fabrizio Canfora. Nonlinear superposition law and\nSkyrme crystals. Phys. Rev. D , 88(6):065028, 2013.\n[41] P. D. Alvarez, F. Canfora, N. Dimakis, and\nA. Paliathanasis. Integrability and chemical poten-\ntial in the (3 + 1)-dimensional Skyrme model. Phys.\nLett. B , 773:401–407, 2017.\n[42] Eloy Ayon-Beato, Fabrizio Canfora, and Jorge Zanelli.\nAnalytic self-gravitating Skyrmions, cosmological\nbounces and AdS wormholes. Phys. Lett. B , 752:201–\n205, 2016.\n[43] Fabrizio Canfora, Nikolaos Dimakis, and Andronikos\nPaliathanasis. Topologically nontrivial configurations in\nthe 4d Einstein-nonlinear σ-model system. Phys. Rev. D ,\n96(2):025021, 2017.\n[44] L. Avil´ es, F. Canfora, N. Dimakis, and D. Hidalgo. An-\nalytic topologically nontrivial solutions of the (3+1)-\ndimensional U(1) gauged Skyrme model and extended\nduality. Phys. Rev. D , 96(12):125005, 2017.\n[45] Fabrizio Canfora. Ordered arrays of Baryonic tubes in the\nSkyrme model in ( 3 + 1 ) dimensions at finite density.\nEur. Phys. J. C , 78(11):929, 2018.30\n[46] Fabrizio Canfora, Marcela Lagos, Seung Hun Oh, Julio\nOliva, and Aldo Vera. Analytic (3+1)-dimensional\ngauged Skyrmions, Heun, and Whittaker-Hill equations\nand resurgence. Phys. Rev. D , 98(8):085003, 2018.\n[47] Fabrizio Canfora, Seung Hun Oh, and Aldo Vera. Ana-\nlytic crystals of solitons in the four dimensional gauged\nnon-linear sigma model. Eur. Phys. J. C , 79(6):485, 2019.\n[48] Eloy Ay´ on-Beato, Fabrizio Canfora, Marcela Lagos, Julio\nOliva, and Aldo Vera. Analytic self-gravitating 4-\nBaryons, traversable NUT-AdS wormholes, flat space-\ntime multi-Skyrmions at finite volume and a novel tran-\nsition in the SU(3)-Skyrme model. Eur. Phys. J. C ,\n80(5):384, 2020.\n[49] Fabrizio Canfora, Marcela Lagos, and Aldo Vera. Crys-\ntals of superconducting Baryonic tubes in the low energy\nlimit of QCD at finite density. Eur. Phys. J. C , 80(8):697,\n2020.\n[50] Fabrizio Canfora, Stefano Carignano, Marcela Lagos,\nMassimo Mannarelli, and Aldo Vera. Pion crystals\nhosting topologically stable baryons. Phys. Rev. D ,\n103(7):076003, 2021.\n[51] Gonzalo Barriga, Fabrizio Canfora, Mat´ ıas Torres, and\nAldo Vera. Crystals of gauged solitons, force free plasma\nand resurgence. Phys. Rev. D , 103(9):096023, 2021.\n[52] Fabrizio Canfora. Non-linear composition and infinite\nconformal symmetry of topologically non-trivial solutions\nin (3 + 1)-dimensional Yang–Mills theory. Eur. Phys. J.\nC, 81(11):1032, 2021.\n[53] Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora,\nand Bianca L. Cerchiai. Analytic SU(N) Skyrmions at fi-\nnite Baryon density. Phys. Rev. D , 101(12):125011, 2020.\n[54] Sergio L. Cacciatori, Fabrizio Canfora, Marcela Lagos,\nFederica Muscolino, and Aldo Vera. Analytic multi-\nBaryonic solutions in the SU(N)-Skyrme model at finite\ndensity. JHEP , 12:150, 2021.\n[55] Sergio L. Cacciatori, Fabrizio Canfora, Marcela Lagos,\nFederica Muscolino, and Aldo Vera. Cooking pasta with\nLie groups. Nucl. Phys. B , 976:115693, 2022.\n[56] Todd Edward Tilma and G. Sudarshan. Generalized\nEuler angle parametrization for SU(N). J. Phys. A ,\n35:10467–10501, 2002.\n[57] Stefan Bertini, S. L. Cacciatori, and Bianca L. Cer-\nchiai. On the Euler angles for SU(N). J. Math. Phys. ,\n47:043510, 2006.\n[58] S. L. Cacciatori, F. Dalla Piazza, and A. Scotti. Compact\nLie groups: Euler constructions and generalized Dyson\nconjecture. Trans. Am. Math. Soc. , 369(7):4709–4724,\n2017.\n[59] Fabrizio Canfora, Diego Hidalgo, Marcela Lagos, Enzo\nMeneses, and Aldo Vera. Infinite conformal symme-\ntry and emergent chiral (super)fields of topologically\nnon-trivial configurations: From Yang-Mills-Higgs to the\nSkyrme model. Phys. Rew. D , 106:105016, 2022.\n[60] Victoria M. Kaspi and Andrei Beloborodov. Magnetars.\nAnn. Rev. Astron. Astrophys. , 55:261–301, 2017.\n[61] Dmitri E. Kharzeev, Larry D. McLerran, and Harmen J.\nWarringa. The Effects of topological charge change in\nheavy ion collisions: ’Event by event P and CP violation’.\nNucl. Phys. A , 803:227–253, 2008.\n[62] David d’Enterria et al. Opportunities for new physics\nsearches with heavy ions at colliders. In 2022 Snowmass\nSummer Study , 3 2022.\n[63] P. Esposito, N. Rea, and G. L. Israel. Magnetars: a short\nreview and some sparse considerations. Astrophys. SpaceSci. Libr. , 461:97–142, 2020.\n[64] S. Mereghetti. An introduction to the properties of\nmagnetars. AIP Conference Proceedings , 924(1):151–158,\n2007.\n[65] Vieri Benci and Donato Fortunato. Weighted sobolev\nspaces and the nonlinear dirichlet problem in unbounded\ndomains. Annali di Matematica Pura ed Applicata ,\n121:319–336, 1979.\n[66] R. E. Meyers. A simple explanation of the Stokes phe-\nnomenon. SIAM Review , 31 N.3:435–445, 1989.\n[67] Indeed, the phenomenon was discovered by G.G. Stokes\nin [91], [92] by studying the Airy integrals arising in the\ncomputation of the intensity of light near a caustic, [93].\n[68] M.H.Holmes. Introduction to Perturbation Methods .\nTexts in Applied Mathematics. Springer New York, NY,\n2013.\n[69] M.S.P. Eastham. The Asymptotic Solution of Linear Dif-\nferential Systems: Application of the Levinson Theorem .\nLondon Mathematical Society Monographs New Series.\nClarendon Press, Oxford, 1989.\n[70] Frank W.I. Olver. Asymptotics and Special Functions .\nCRC Press, New York, 1997.\n[71] R.Wong. Asymptotic Approximation of Integrals . Aca-\ndemic Press, New York, 1989.\n[72] Xia Gu and Babak Haghighat. Liouville conformal blocks\nand Stokes phenomena. 1 2023.\n[73] P. P. Boalch. Topology of the Stokes phenomenon. Pro-\nceedings of Symposia in Pure Mathematics , 103.1:55–100,\n2021.\n[74] Maxim Kontsevich and Yan Soibelman. Wall-crossing\nstructures in Donaldson-Thomas invariants, integrable\nsystems and Mirror Symmetry. Lect. Notes Union. Mat.\nItal., 15:197–308, 2014.\n[75] Edward Witten. Analytic Continuation Of Chern-Simons\nTheory. AMS/IP Stud. Adv. Math. , 50:347–446, 2011.\n[76] Tom Bridgeland and Valerio Toledano Laredo. Stability\nconditions and stokes factors. Inventiones mathematicae ,\n187(1):61–98, apr 2011.\n[77] D. Guzzetti. Stokes Matrices and Monodromy of the\nQuantum Cohomology of Projective Spaces. Comm.\nMath. Phys. , 207:341–383, 1999.\n[78] Victor Kowalenko. The Stokes Phenomenon, Borel Sum-\nmation and Mellin-Barnes Regularisation . Bertham Sci-\nence Publishers, Sharjah, U.A.E., 2018.\n[79] R. B. Paris. The asymptotic behaviour of Pearcey’s in-\ntegral for complex variables. Proc. Roy. Soc. Lond. A ,\n432:391–426, 1991.\n[80] D. Kaminski. Asymptotic expansion of the pearcey in-\ntegral near the caustic. SIAM journal on mathematical\nanalysis , 20, num.4:987–1005, 1989.\n[81] Mark Alford, Anton Kapustin, and Frank Wilczek. Imag-\ninary chemical potential and finite fermion density on the\nlattice. Phys. Rev. D , 59:054502, Jan 1999.\n[82] Felix Karbstein and Michael Thies. How to get from\nimaginary to real chemical potential. Phys. Rev. D ,\n75:025003, Jan 2007.\n[83] V. G. Bornyakov, N. V. Gerasimeniuk, V. A. Goy, A. A.\nKorneev, A. V. Molochkov, A. Nakamura, and R. N. Ro-\ngalyov. Numerical study of the roberge-weiss transition.\nPhys. Rev. D , 107:014508, Jan 2023.\n[84] E. B. Dynkin. Semisimple subalgebras of semisimple Lie\nalgebras. Trans. Am. Math. Soc. Ser. 2 , 6:111–244, 1957.\n[85] It is worth remarking here that our convention on Stokes\nlines is the opposite of that in [79].31\n[86] J.L. L´ opez and P.J. Pagola. The Pearcey integral in the\nhighly oscillatory region. Applied Mathematics and Com-\nputation , 275:404–410, 2016.\n[87] We agree with their conventions on Stokes lines.\n[88] Jakob J. Stamnes and Bjørn Spjelkavik. Evaluation of the\nfield near a cusp of a caustic. Optica Acta: International\nJournal of Optics , 30(9):1331–1358, 1983.\n[89] Milton Abramowitz and Irene A. Stegun. Handbook\nof Mathematical Functions with Formulas, Graphs, and\nMathematical Tables . Dover, New York City, ninth dover\nprinting, tenth gpo printing edition, 1964.\n[90] Once again, this is true up to the insertion of branesources at the boundaries, as required by the Stokes the-\norem (or, equivalently, by the boundary conditions re-\nquired to make the variational principle available).\n[91] G. G. Stokes. On the numerical calculation of a class of\ndefinite integrals end infinite series. Transaction of the\nCambridge Philosophical Society , IX(I):166–189, 1951.\n[92] G. G. Stokes. On the discontinuity of arbitrary constants\nwhich appear in divergent developments. Transaction of\nthe Cambridge Philosophical Society , X(I):105–128, 1958.\n[93] Georges Biddell Airy. On the intensity of light in the\nneighbourhood of a caustic. Transaction of the Cam-\nbridge Philosophical Society , VI(III):397–402, 1838."    },    {        "title": "2402.07552v1.Highly_efficient_channeling_of_single_photons_into_guided_modes_of_optical_nanocapillary_fibers.pdf",        "content": "arXiv:2402.07552v1  [quant-ph]  12 Feb 2024Highly efficient channeling of single photons into\nguided modes of optical nanocapillary fibers\nBashaiah Elaganuru1, Resmi M1, Ramachandrarao Yalla1*\n1School of Physics, University of Hyderabad, Hyderabad, 500 046,\nTelangana, India.\n*Corresponding author(s). E-mail(s): rrysp@uohyd.ac.in ;\nContributing authors: 18phph19@uohyd.ac.in ;19phph18@uohyd.ac.in ;\nAbstract\nWe report numerically the efficient channeling of single phot ons from a single\nquantum emitter into guided modes of optical nanocapillary fibers (NCFs). The\nNCF is formed of a liquid core optical nanofiber with inner and outer diame-\nters. We optimize the inner and outer diameters of the NCF fill ed with water\nmedium by placing a single dipole source (SDS) inside. The ma ximum channel-\ning efficiency of 52% is found when the radially polarized SDS is placed at the\ncenter of the NCF filled with the water medium. The optimum inn er and outer\ndiameters of the NCF are 100nm and 360nm for the emission wavelength of\n620nm, respectively. Additionally, we investigate the SDS pos ition dependence\ninside the NCF considering experimental ambiguity in placi ng a single quantum\nemitter inside the NCF. We found that the channeling efficienc y remains almost\nconstant for the water medium at the optimum condition. The p resent platform\nmay open a novel route for generating single photons in quant um technologies\nand detecting single cells in bio-sensing.\nKeywords: Optical nanofibers, Nanocapillary fibers, Single dipole sou rce, Single\nphotons\n1 Introduction\nIn quantum information processing and communication, single photo ns are employed\nas information carriers as they are the ideal choice. A quantum net work consists of\nquantum nodes for storing single photons and quantum channels fo r single photon\ntransmission. Single-mode optical fibers can serve as quantum cha nnels and single\n1quantum emitters as quantum nodes in a physical implementation of a quantum net-\nwork. Therefore, efficient channeling of single photons into a single- mode optical fiber\nis necessary. The generation of single photons nowadays is achieve d using various pro-\ntocols based on single atoms [ 1], single ions [ 2], single molecules [ 3], single quantum\ndots [4], single vacancy centers in nano-diamonds [ 5], single defect in silicon carbide\n[6], rare-earth-ion impurities in yttrium aluminum garnet/yttrium orth osilicate [ 6],\nand two-dimensional materials [ 6].\nHowever, the production of single photons is useless without an effic ient manip-\nulation and control manner. Efficiently channeling single photons fro m a single\nquantum emitter into a particular mode is a challenging task [ 7,8]. Various protocols\nhave been proposed and developed for efficiently collecting single pho tons. Examples\ninclude micropillar cavities [ 9], solid immersion lens [ 10], photonic crystal cavities [ 11],\ndiamond waveguides [ 12], plasmonic metal nanowires [ 13], optical nanofibers (ONFs)\n[14–24], and optical nanofiber tips (ONFTs)[ 20,25,26]. Other than ONF/ONFT,\nthe subsequent coupling of fluorescence photons into a single-mod e fiber reduces the\nactual collection efficiency due to the mode mismatch. Regarding ato ms and ions, it is\npossible to efficiently control the production of single photons on de mand. However,\nit requires complicated experimental setups and intermittent loadin g of atoms or\nions. Regarding solid-state quantum emitters, spectral diffusion a nd other broadening\nresult in photon distinguishability from the same quantum emitter/va rious emitters.\nCollecting single photons from quantum emitters in the high refractiv e index host\nmaterials is challenging.\nIn addition to efficient manipulation, efficient transmission lines betwee n quantum\nnodes are also essential for applications in quantum technologies. I n such an appli-\ncation, efficient channeling of single photons into single-mode optical fibers (SMFs),\nplaying the role of transmission lines, is required. To achieve this, nan ostructured\nsystems have been proposed and demonstrated, including sub-wa velength diameter\nsilica fibers termed ONFs and ONFTs [ 15–21]. Moreover, for application in quantum\nnetwork [ 7], in-line method i.e. fiber integrated light-matter interfaces such as those\nrealized by ONFs/ONFTs are advantageous since automatic channe ling to the SMF\nwas achieved [ 16–19]. However, in these efforts, a maximum channeling efficiency ( η)\nwas achieved up to 28% for the radially polarized single dipole source (S DS) [18,19].\nThe maximum η-value up to 20% has been experimentally demonstrated for single\nphotons from a single quantum dot directly into the SMF considering r andomly\noriented dipole [ 16]. Note that the single quantum dot was placed on the surface of\nthe ONF. It has been numerically shown that a maximum η-value of 38% from the\nradially polarized SDS to the SMF was achieved using the ONFT [ 20,25]. Note that\nthe SDS was placed on the facet of the tip. The feasibility of the simula tion results\nhas been experimentally shown by placing quantum dots on the facet of the ONFT\n[27]. Note that the interaction strength has been limited due to the pos itioning of the\nSDS on the surface of the ONF/ONFT.\nOne can enhance the η-value in two ways: one is creating a cavity on the ONF\nand the other is placing the single quantum emitter inside the ONF. Reg arding the\n2cavity on the ONF, the maximum η-value of 65% has been experimentally demon-\nstrated using a composite photonic crystal cavity on the ONF [ 28]. Note that a single\nquantum dot was placed on the surface of the ONF only, which limits th e interaction.\nRegarding placing the single quantum emitter inside, a new type of hollo w core fiber\nwith sub-micron diameter termed as capillary fibers has been propos ed and experi-\nmentally demonstrated [ 29]. However, these efforts demonstrated the η-value up to\n18% only. This is due to the thicker outer diameter of the capillary fibe r, leading\nto weak confinement of the field inside the capillary fiber. The system atic study of\nη-value for the hollow/liquid core ONF with inner and outer diameters te rmed optical\nnanocapillary fibers (NCFs) has not been reported yet. Note that the inner and outer\ndiameters of the NCF are in sub-wavelength. The position dependen ce of the SDS\ninside the NCF, keeping in view of experimental ambiguity, has not bee n reported yet.\nIn this paper, we use numerical simulations to report the efficient ch anneling of\nsingle photons from a single quantum emitter into guided modes of the optical NCF.\nWe perform numerical simulations to optimize the inner and outer diam eters of the\nNCF filled with liquid medium by placing the SDS inside, which was not possible\nin the case of the ONF. A maximum η-value of 52% is found when the radially\npolarized SDS is placed at the center of the NCF filled with water medium . For the\nemission wavelength of 620 nm, the optimum inner and outer diameter s of the NCF\nfilled with water are 100 nm and 360 nm, respectively. Additionally, we in vestigate\nthe position dependence of the SDS inside the NCF keeping in view of ex perimental\nambiguity. Simulated results show that the η-value remains almost the same for the\nwater medium while changing the position of the SDS inside the NCF.\n2 Methodology\nWe perform numerical simulations using Ansys lumerical finite differen ce time domain\n(Ansys, FDTD Package) platform [ 30,31]. First, we perform simulations to determine\ntheη-value into guided modes of the ONF in vacuum and liquid medium for refe rence.\nConceptual sketches of the ideas are shown in Figs. 1(a) and (b) for vacuum and\nliquid medium, respectively. We determine the η-value for different diameters of the\nONF (Dv/Dl) in both mediums. The SDS is placed on the surface of the ONF. We set\nthe polarization of the SDS to be radial/azimuthal/axial.We place the p ower monitor\n(M) sufficiently far from the SDS position to determine the coupled po wer (Pc).P\nandP0are the total power emitted by the SDS in the presence of the ONF a nd in\nthe vacuum environment, respectively. Thus, the channeling efficie ncy is defined as η=\nPc/P. We determine the η-value for different polarizations while sweeping the Dv/Dl-\nvalues of the ONF. For each case, we find the optimum Dv/Dl-values by maximizing\ntheη-value into the guided mode of the ONF.\nNext, we perform simulations to determine the η-value for the NCF based on\nthe optimum ONF diameters ( Dv/Dl). Conceptual sketches of the NCF-filled with\nvacuum and liquid medium are shown in Figs. 2(a) and (b), respectively. The SDS\nis placed inside the NCF. We determine the η-value for different polarizations while\n3(a) (b) \nM\nSDS\nM\n/gid00049\nSDSRadial\nAzimuthal\nAxialM\n/gid00041/gid00046/gid00041 /gid00046/gid00041\n/gid00039/gid00041\n/gid00049/gid00005 /gid00039/gid00005\nFig. 1 (a) and (b) show the conceptual sketches of the ideas for the o ptical nanofiber (ONF) placed\nin vacuum and liquid medium, respectively. DvandDlare the ONF diameters when the surrounding\nmedium is vacuum and liquid, respectively. ns,nv, andnlare refractive indices of silica, vacuum, and\nliquid, respectively. SDS and M denote the single dipole sou rce and monitor, respectively. The SDS\nis placed on the surface of the ONF. The insets show the radial , azimuthal, and axial polarizations\ncorresponding to the SDS directed perpendicular, tangent, and parallel to the cylindrical surface,\nrespectively.\nsweeping inner and outer diameters ( dinanddout) of the NCF in both mediums. The\nvacuum and liquid-filled NCFs are defined as hollow-core NCF and liquid-c ore NCF,\nrespectively. For each case, we find the optimum dinanddout-values by maximizing\ntheη-value into the guided mode of the NCF.\nONF, NCF, SDS, and M are placed inside the three-dimensional simulat ion area\nof 6×6×35µm3enclosed in the perfectly matched layers (PMLs) to avoid reflection s.\nThe length of the ONF/NCF is set at 45 µmso that they can be treated as infinitely\nlong. The maximum ONF/NCF diameter is chosen within the simulation reg ion. The\nSDS is placed 5 µmaway from the PML. The M is placed 25 µmaway from the SDS\nand 5µmaway from the PML. In the case of the ONF, the radial, azimuthal, an d\naxial polarizations corresponding to the SDS are directed perpend icular, tangent, and\nparallel to the cylindrical surface, respectively as clearly shown in t he inset of Fig. 1\n(b). In the case of the NCF, the radial and axial polarizations corr esponding to the\nSDS aredirected perpendicularand parallelto the cylinderaxis, res pectivelyasclearly\nshown in the insets of Fig. 2. We set the emission wavelength of the SDS at 620 nm,\ncorresponding to the emission wavelength of the quantum emitter [ 32]. In the case of\nthe ONF, refractive indices of a cylinder and surrounding mediums ar e set for silica\n4(a) (b) \nRadial\nSDS \n(b)\n /gid00031/gid00036/gid00041 /gid00031/gid00036/gid00041 \n/gid00031/gid00042/gid00048/gid00047 /gid00031/gid00049M/gid00041\nM\n/gid00046/gid00041\nRadial\nAxial AxialSDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS SDS \n/gid00042/gid00048/gid00047 /gid00041/gid00049/gid00049\n/gid00039/gid00039\n/gid00039\n(a)((((\n((\nMMM\n/gid00046/gid00046/gid00046/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00041/gid00046\n/gid00041/gid00041/gid00049\n/gid00041/gid00049/gid00041/gid00046\nFig. 2 (a) and (b) show the conceptual sketches of the ideas for the n anocapillary fiber (NCF)\nfilled with vacuum and liquid medium, respectively. din\nv/din\nlanddout\nv/dout\nlare the inner and outer\ndiameters of the NCF filled with vacuum/liquid, respectivel y.ns,nv, andnlare refractive indices\nof silica, vacuum, and liquid, respectively. SDS and M denot e the single dipole source and monitor,\nrespectively. The SDS is placed inside the NCF. The insets sh ow radial and axial polarizations\ncorresponding to the SDS directed perpendicular and parall el to the cylinder axis, respectively.\n(ns) and vacuum ( nv)/liquid ( nl), respectively. In the case of the hollow core NCF,\nrefractive indices of the outside of the cylinder, cylinder, and inside of the cylinder\nare set for vacuum ( nv), silica ( ns), and vacuum ( nv), respectively. In the case of the\nliquid core NCF, refractive indices of the outside of the cylinder, cylin der, and inside\nof the cylinder are set for vacuum ( nv), silica ( ns), and liquid ( nl), respectively.\n3 Results\nFigure3(a) shows the dependence of η-value as a function of the ONF diameter ( Dv)\nfor the SDS placed on the surface of the ONF. Horizontal and vert ical axes correspond\nto theDv-value and η-value, respectively. The SDS along the radial, azimuthal, and\n5(a) (b) \nFig. 3 (a) and (b) show summaries of η-values as a function of the optical nanofiber (ONF)\ndiameters ( Dv/Dl) for three polarizations when the ONF is placed in vacuum and water, respec-\ntively. Green/black squares, purple/red circles, and oran ge/blue triangles correspond to the radial,\nazimuthal, and axial polarizations, respectively.\naxial polarizations are shown by green squares, purple circles, and orange triangles,\nrespectively. One can readily see the maximum η-value is for the radial polarization.\nThis is due to the strong confinement of the field in the radial directio n. For the radial,\nazimuthal, and axial polarizations, we found the maximum η-value of 29%, 18%, and\n9%, respectively. The maximum η-value of 29% occurred at the Dv-value of 280 nm,\ncorresponding to the fiber size parameter of 1 .42.\nFigure3(b) shows the dependence of η-value as a function of the ONF diame-\nter (Dl) for the SDS placed on the surface of the ONF and the surrounding medium\nis water. Horizontal and vertical axes correspond to Dl-value and η-value, respec-\ntively. The SDS along the radial, azimuthal, and axial polarizations are shown by\nblack squares, red circles, and blue triangles, respectively. One ca n see that the maxi-\nmumη-value occurred for the radial polarization. For the radial, azimuth al, and axial\npolarizations, we found the maximum η-value of 8%, 7%, and 1%, respectively. The\nmaximum η-value of 8% occurred at the Dl-value of 430 nm, corresponding to the\nfiber size parameter of 2 .17.\nWe perform simulations to determine the η-value for the NCF outer diameter\nbased on the optimum η-value of ONF diameters in free space and water medium.\nThe NCF outer diameter is chosen as the average value of optimum dia meters of\nONF in both free space and water medium. Then, we sweep the inner d iameters while\nthe outer diameter is fixed at the average value. Then, we fix the inn er diameter of\nthe NCF at the value where there is no significant decay in the η-value and consider\nthe experimental realization of such a thin capillary hole. Then, we sw eep the outer\ndiameter of the NCF and find the optimum outer diameter of the NCF.\nFigure4(a) shows the dependence of η-value as a function of the NCF inner\ndiameters ( din) filled with vacuum/water. Note that the SDS is placed at the center\nof the NCF. The outer diameter ( dout) of the NCF is fixed at 360 nm. Based on the\nprevious simulation results, The dout-value is chosen as the average value of optimum\n6(a) (b) \nFig. 4 (a) The summary of η-values as a function of the inner diameter ( din) of the nanocapillary\nfiber (NCF) filled with vacuum and water for two polarizations . The outer diameter ( dout) of the NCF\nis fixed at 360 nm. (b) The summary of η-values as a function of dout-values of the NCF filled with\nvacuum and water for two polarizations. The din-value of NCF is fixed at 100 nm. Black squares and\nred circles denote the radial polarization in water and vacu um, respectively. Blue and green triangles\ndenote the axial polarization in water and vacuum, respecti vely.\nDvandDl-values.The din-valueofthe NCF variesfrom10nm to 300nm. The vertical\naxis corresponds to the η-value. Black squares and red circles represent the radial\npolarization of the SDS in water and vacuum, respectively. Blue and g reen triangles\nrepresent the axial polarization of the SDS in water and vacuum, re spectively. In both\ncases,one can readily see that the maximum value occurredfor the radialpolarization.\nFor the radial polarization, one can readily see that the maximum valu e occurred for\nthe water medium. We found the maximum η-value of 53%. Note that the η-value for\nthe radial and azimuthal polarizations are the same when the SDS is a t the center of\nthe NCF.\nFigure4(b) shows the dependence of η-value as a function of the NCF outer diam-\neter (dout). Thedin-value of the NCF is fixed at 100 nm considering the experimental\nfeasibility. The dout-value of the NCF is varied from 300 nm to 1000 nm. The verti-\ncal axis corresponds to the η-value. Black squares and red circles represent the radial\npolarization in water and vacuum, respectively. Blue and green trian gles represent the\naxial polarization in water and vacuum, respectively. One can readily see the max-\nimumη-value occurred for the radial polarization. We found the maximum η-value\nof 52% at dout-value of 360 nm corresponding to the fiber size parameter of 1 .82. In\nthe case of vacuum, the maximum η-value of 48% was found at dout-value of 370 nm\ncorresponding to the fiber size parameter of 1 .87.\nConsidering experimental feasibility, we investigate the choice of din-value. Figure\n5(a) shows the dependence of η-value as a function of the outer diameter ( dout)\nof the NCF. The din-value of the NCF is fixed at 250 nm. The dout-value of the\nNCF varies from 300 nm to 1000 nm. The vertical axis corresponds t o theη-value.\nBlack squares and red circles represent the radial polarization in wa ter and vacuum,\nrespectively. Blue and green triangles represent the axial polariza tion in water and\n7(a) (b) \nFig. 5 (a) The summary of η-values as a function of outer diameter ( dout) of the nanocapillary fiber\n(NCF) filled with vacuum and water for two polarizations. The inner diameter ( din) of the NCF is\nfixed at 250 nm. Black squares and red circles denote the radia l polarization in water and vacuum,\nrespectively. Blue and green triangles denote the axial pol arization in water and vacuum, respectively.\n(b) The dependence of η-value as a function of the position ( rin) of the single dipole source (SDS)\nfor three polarizations. The inner and outer diameters of th e NCF are fixed at 100 nm and 360 nm,\nrespectively. Black squares, purple diamonds, and blue tri angles are for the radial, azimuthal, and\naxial polarizations in the water medium, respectively. Red circles, orange stars, and green triangles\nare for the radial, azimuthal, and axial polarizations in th e vacuum, respectively. The inset shows the\nschematic for the position of the SDS.\nvacuum, respectively.One can readilysee the maximum η-value occurredfor the radial\npolarization. We found the maximum η-value of 47% at the dout-value of 380 nm\ncorrespondingto the fiber size parameter of 1 .92. In the case of vacuum, the maximum\nη-value of 31% was found at the dout-value of 440 nm corresponding to the fiber size\nparameter of 2 .23.\nAlso, we investigate the effect of the position ( rin) of the SDS inside the NCF by\nconsidering the experimental ambiguity in placing a single quantum emit ter. Figure 5\n(b) showsthe dependence of η-value asa function of rin-value. The dinanddout-values\nof the NCF are kept constant at 100 nm and 360 nm, respectively. T he horizontal\naxis corresponds to the rin-value, which varies from 0 nm (center) to 50 nm (inside\nedge) as shown in the inset of Fig. 5(b). The vertical axis corresponds to the η-value.\nBlack squares, purple diamonds, and blue triangles are for the radia l, azimuthal, and\naxial polarizations in the water medium, respectively. Red circles, or ange stars, and\ngreen triangles are for the radial, azimuthal, and axial polarizations in the vacuum,\nrespectively. In both cases, one can readily see that the η-value corresponding to the\nradial polarization is maximum. Note that in both cases of the axial po larization, the\nη-value is close to 1%.\n4 Discussion\nThe single-modecondition isdefined bythe parameter V=ka/radicalbig\nn2\n1−n2\n2<2.405,where\nais theONF radius, k=2π/λ,λiswavelengthofthe light,and n1andn2arerefractive\n8indices of the core and clad, respectively [ 14,33,34]. To satisfy the single-mode con-\ndition for the emission wavelength of 620 nm, the maximum ONF diamete rs (Dv/Dl)\nare 450 nm and 820 nm for the clad medium of vacuum and water, resp ectively. The\nmaximum DvandDl-values for single mode condition differ significantly. This is due\nto the significant difference in the refractive index between the silica and vacuum\ncompared to silica and water. As seen clearly in Figs. 3(a) and (b), that multimode\nbehavior appears at the expected ONF diameters, resulting in incre asing the η-value\nfor the thicker diameter of the ONF.\nAs seenin Figs. 3(a)and(b), theoptimum ONFdiametersin asingle-moderegime\ndiffer for three polarizations. The optimum Dv/Dl-values are 250/420 nm, 280/430\nnm, and 340/460 nm for azimuthal, radial, and axial polarizations, re spectively. This\nis due to the effective refractive index difference in three different p olarizations. In\nFig.3(a), the maximum η-value of 29% is found for the radial polarization, which is\nin good agreement with the previously reported works in Refs. [ 18,19]. The average\nη-value is 20%, assuming the randomly oriented dipoles, which is in good a greement\nwith the experimentally reported η-value in Ref. [ 16]. In Fig. 3(b), the maximum η-\nvalue of 8% is found for the radial polarization. The average η-value is 5%, considering\nthe randomly oriented dipoles for the experimental feasibility. The d ifference in the\nmaximum η-value is due to the weak field confinement around the ONF. However , the\nONF in water medium will find various applications for bio-sensing [ 35].\nAs seen in Figs. 4(a) and (b), one can readily see the maximum and minimum η-\nvalues are for the radial and axial polarizationsfor both mediums, r espectively. This is\ndue to the axialcomponent ( Ez) ofthe electricfield being lowat the center ofthe NCF\nas discussed in Ref. [ 33]. In Fig. 4(a), the maximum η-value for the radial polarization\nis almost the same for water and vacuum mediums up to din-value of 40 nm. However,\na significant change for η-value occurred for the din-value from 50 nm to 300 nm for\nthe vacuum compared to the water medium. This is due to the significa nt change in\nthe effective refractive index of the NCF while the din-value is high. In contrast, as\nseen in Fig. 4(b), one can readily see that as dout-value increases the η-value for axial\npolarizationincreases.This may be due to the coupling to multi-modes (higher-order).\nFor the radial polarization, a maximum η-value of 52% is realized at din-value of\n100 nm and dout-value of 360 nm of the NCF. In contrast, as seen in Fig. 5(a), one\ncan readily see the maximum η-value for the radial and axial polarizations get affected\nwhen the dinis fixed at 250 nm. For the radial polarization, a maximum η-value of\n47% is realized at the din-value of 250 nm and the dout-value of 380 nm of the NCF.\nThis is due to the weak field confinement inside the NCF because of the effective\nrefractive index change as the inner diameter increases. We also sim ulatedη-value for\nthe azimuthal polarization and found the same result as the radial p olarization. This\nis due to the placement of the SDS at the center in contrast to the S DS on the surface\nof the ONF. Note that η-value of 52% is almost two times the values compared to the\nSDS placed on the surface of the ONF [ 18,19]. The present value is almost 1.4 times\nthe value compared to the SDS placed on the facet of the ONFT. Com pared to other\nexisting fiber-based platforms, the NCF is a promising avenue. The a verageη-value is\n35%, assuming the randomly oriented dipoles, which is higher than the experimentally\nreported η-value for the ONF in Ref. [ 16].\n9Two key challenges are to be achieved regarding the experimental f easibility of the\npresent simulations. One is the fabrication of the NCF with such a sma ll capillary\nhole and the other is placing a single quantum emitter inside the NCF. Co mmercial\ncapillary optical fibers can be tapered to sub-wavelength diameter s to realize NCF\nusing aheatand pull technique[ 14,36,37]. The placementofasinglequantum emitter\ninside the NCF can be achieved by flowing through the center of the N CF by pushing\nthe liquid from one end of the NCF. Considering the experimental rea lization of the\nsimulation results, we choose water medium as quantum dots are diss olved in water.\nRegarding the sensitivity of the inner and outer diameters of the NC F to the η-value,\none can readily see in Figs. 4and Fig. 5(a) that the peak η-values occurred in rather\nbroad, not sharp. This implies that if there is any slight ambiguity in the inner and\nouter diameters of the NCF, it doesn’t affect the η-value significantly. This would give\nadditional freedom in the experimental realization.\nAs seen in Fig. 5(b), theη-value is almost unchanged while changing the rin-value\nfor the radial polarization. This suggests that the SDS position insid e the NCF is not\nnecessarily at the center to realize the maximum interaction i.e. maxim umη-value.\nThis would enable the easy experiments to place a single quantum emitt er inside the\nNCF, as it is challenging to control the position of a single quantum emit ter inside the\nNCF. Note that no significant changeoccursfor azimuthal polariza tionwhile changing\ntherin-values up to 50 nm. In any case, the maximum η-value of more than 50% can\nbe realized using the NCF.\nTheη-value can be further enhanced in two possible ways: one is creating a cavity\nstructure on the NCF [ 28,38–40], and the other is increasing the refractive index of\nthe medium and material. Regarding cavity, simulations suggest that η-value can be\nas high as 80%, which is higher than the experimentally reported value s of 65% if\nthe ONF is replaced with the NCF. Regarding increasing refractive ind ex, simulations\nsuggest that η-value can be higher than the present value if the silica is replaced with\nother high refractive index material like diamond/silicon nitride/gallium phosphate\n[26]. Also, the refractive index of the medium can affect the η-value if we replace water\nwith any higher refractive index liquid [ 29].\n5 Conclusion\nIn summary, we reported numerically the efficient channeling of single photons from\na single quantum emitter into guided modes of NCFs. The maximum chan neling effi-\nciency of 52% is found when the radially polarized dipole is placed at the c enter of\nthe NCF in the liquid (water) medium. The optimum inner and outer diame ters of\nthe NCF are 100 nm and 360 nm, respectively for the emission wavelen gth of 620\nnm. Additionally, we investigated the SDS position dependence inside t he NCF con-\nsidering experimental discrepancies in placing the single quantum emit ter inside the\nNCF. We found that the channeling efficiency remains almost the same for the water\nmedium at the optimum condition. The present platform may open a no vel route for\nquantum technologies and bio-sensing.\nAcknowledgments. RRY acknowledges financial support from the Scheme for\nTransformational and Advanced Research in Sciences (STARS) gr ant from the Indian\n10Institute of Science (IISc), Ministry of Human Resource Developm ent (MHRD) (File\nNo. STARS/APR2019/PS/271/FS) and the Institute of Eminence ( IoE) grant at the\nUniversity of Hyderabad, Ministry of Education (MoE) (File No. RC2- 21-019). RM\nacknowledgestheUniversityGrantsCommission(UGC) forthefinan cialsupport(Ref.\nNo.:1412/CSIR-UGC NET June 2019).\nAuthor contributions. BE performed the numerical calculations and plotted the\ngraphs. BE wrote the manuscript under the supervision of RRY. All authors reviewed\nthe manuscript. This work forms a part of the PhD thesis of BE.\nDeclarations\nEthical Approval. Not applicable. No human and/or animal studies have been\nexecuted.\nFunding. Topreparethismanuscript,fundingwasreceivedbyInstituteofE minence\n(IoE) grant at the University of Hyderabad, Ministry of Education (MoE) (File No.\nRC2-21-019).\nAvailability of data and materials. Data underlying the results presented in this\npaper are not publicly available at this time but may be obtained from th e authors\nupon reasonable request.\nConfict of interest. The authors declare no confict interests.\nCompeting interests. The authors declare no competing interests.\nReferences\n[1] Hood, C.J., Lynn, T., Doherty, A., Parkins, A., Kimble, H.: The atom- cavity\nmicroscope: Single atoms bound in orbit by single photons. Science 287(5457),\n1447–1453 (2000)\n[2] Barros, H., Stute, A., Northup, T., Russo, C., Schmidt, P., Blatt, R.: Determinis-\ntic single-photon source from a single ion. New Journal of Physics 11(10), 103004\n(2009)\n[3] Fleury, L., Segura, J.-M., Zumofen, G., Hecht, B., Wild, U.: Nonclass ical photon\nstatistics in single-molecule fluorescence at room temperature. Ph ysical review\nletters84(6), 1148 (2000)\n[4] Michler,P.,Imamo˘ glu,A., Mason,M.,Carson,P., Strouse,G., Bur atto,S.: Quan-\ntum correlation among photons from a single quantum dot at room te mperature.\nNature406(6799), 968–970 (2000)\n[5] Kurtsiefer, C., Mayer, S., Zarda, P., Weinfurter, H.: Stable solid- state source of\nsingle photons. Physical review letters 85(2), 290 (2000)\n11[6] Aharonovich,I.,Englund,D.,Toth,M.:Solid-statesingle-photon emitters.Nature\nphotonics 10(10), 631–641 (2016)\n[7] Kimble, H.J.: The quantum internet. Nature 453(7198), 1023–1030 (2008)\n[8] Bremer, L., Rodt, S., Reitzenstein, S.: Fiber-coupled quantum ligh t sourcesbased\non solid-state quantum emitters. Materials for Quantum Technolog y (2022)\n[9] Solomon, G., Pelton, M., Yamamoto, Y.: Single-mode spontaneous e mission from\na single quantum dot in a three-dimensionalmicrocavity. PhysicalRe view Letters\n86(17), 3903 (2001)\n[10] Schr¨ oder, T., G¨ adeke, F., Banholzer, M.J., Benson, O.: Ultrab right and efficient\nsingle-photon generation based on nitrogen-vacancy centres in n anodiamonds on\na solid immersion lens. New Journal of Physics 13(5), 055017 (2011)\n[11] Shambat, G., Provine, J., Rivoire, K., Sarmiento, T., Harris, J., Vu ˇ ckovi´ c, J.:\nOptical fiber tips functionalized with semiconductor photonic cryst al cavities.\nApplied Physics Letters 99(19), 191102 (2011)\n[12] Patel, R.N., Schr¨ oder,T., Wan, N., Li, L., Mouradian,S.L., Chen, E .H., Englund,\nD.R.: Efficient photon coupling from a diamond nitrogen vacancy cente r by\nintegration with silica fiber. Light: Science & Applications 5(2), 16032–16032\n(2016)\n[13] Akimov, A., Mukherjee, A., Yu, C., Chang, D., Zibrov, A., Hemmer, P., Park, H.,\nLukin, M.: Generation of single optical plasmons in metallic nanowires co upled\nto quantum dots. Nature 450(7168), 402–406 (2007)\n[14] Nayak,K.P.,Sadgrove,M.,Yalla,R.,LeKien,F.,Hakuta,K.:Nanofi berquantum\nphotonics. Journal of Optics 20(7), 073001 (2018)\n[15] Vetsch, E., Reitz, D., Sagu´ e, G., Schmidt, R., Dawkins, S., Rausc henbeutel, A.:\nOptical interface created by laser-cooled atoms trapped in the ev anescent field\nsurrounding an optical nanofiber. Physical review letters 104(20), 203603 (2010)\n[16] Yalla, R., Le Kien, F., Morinaga, M., Hakuta, K.: Efficient channeling o f fluores-\ncence photons from single quantum dots into guided modes of optica l nanofiber.\nPhysical review letters 109(6), 063602 (2012)\n[17] Fujiwara, M., Toubaru, K., Noda, T., Zhao, H.-Q., Takeuchi, S.: Hig hly effi-\ncient coupling of photons from nanoemitters into single-mode optica l fibers. Nano\nletters11(10), 4362–4365 (2011)\n[18] Le Kien, F., Gupta, S.D., Balykin, V., Hakuta, K.: Spontaneous emis sion of a\ncesiumatomnearananofiber:Efficientcouplingoflighttoguidedmode s.Physical\nReview A 72(3), 032509 (2005)\n12[19] Klimov, V.V., Ducloy, M.: Spontaneous emission rate of an excited a tom placed\nnear a nanofiber. Physical Review A 69(1), 013812 (2004)\n[20] Chonan, S., Kato, S., Aoki, T.: Efficient single-mode photon-coup ling device\nutilizing a nanofiber tip. Scientific reports 4(1), 4785 (2014)\n[21] Yonezu, Y., Wakui, K., Furusawa, K., Takeoka, M., Semba, K., Aok i, T.: Efficient\nsingle-photon coupling from a nitrogen-vacancy center embedded in a diamond\nnanowire utilizing an optical nanofiber. Scientific reports 7(1), 12985 (2017)\n[22] Morrissey, M.J., Deasy, K., Frawley, M., Kumar, R., Prel, E., Russe ll, L., Truong,\nV.G., Chormaic, S.N.: Spectroscopy,manipulation and trapping ofneu tral atoms,\nmolecules, and other particles using optical nanofibers: a review. S ensors13(8),\n10449–10481 (2013)\n[23] Yang,L., Zhou, Z., Wu, H., Dang,H., Yang,Y., Gao,J., Guo,X., Wan g,P., Tong,\nL.: Generating a sub-nanometer-confined optical field in a nanoslit w aveguiding\nmode. Advanced Photonics 5(4), 046003–046003 (2023)\n[24] Yalla,R., Kojima,Y.,Fukumoto, Y.,Suzuki, H.,Ariyada,O.,Shafi,K .M.,Nayak,\nK.P., Hakuta, K.: Integration of silicon-vacancy centers in nanodiam onds with an\noptical nanofiber. Applied Physics Letters 120(24) (2022)\n[25] Resmi, M., Bashaiah, E., Das, B., Yalla, R.: Efficient fiber-coupled sin gle photon\nsourceusing an optical nanofiber tip. In: Women in Optics and Photo nics in India\n2022, vol. 12638, pp. 107–109 (2023). SPIE\n[26] Das, B., Resmi, M., Bashaiah, E., Yalla, R.: Efficient single-mode coup ling design\nusing a silica/diamond nano-tip with a gold nanoparticle. In: Women in Op tics\nand Photonics in India 2022, vol. 12638, pp. 125–127 (2023). SPIE\n[27] Resmi, M., Elaganuru, B., Ramachandrarao, Y.: Channeling of fluo rescence pho-\ntons from quantum dots into guided modes of an optical nanofiber t ip. arXiv\n2401.16891 (2024)\n[28] Yalla, R., Sadgrove, M., Nayak, K.P., Hakuta, K.: Cavity quantum e lectrody-\nnamics on a nanofiber using a composite photonic crystal cavity. Ph ysical review\nletters113(14), 143601 (2014)\n[29] Faez, S., T¨ urschmann, P., Haakh, H.R., G¨ otzinger, S., Sandog hdar, V.: Coherent\ninteraction of light and single molecules in a dielectric nanoguide. Physic al review\nletters113(21), 213601 (2014)\n[30] Taflove, A., Hagness, S.C., Piket-May, M.: Computational electr omagnetics:\nthe finite-difference time-domain method. The Electrical Engineerin g Handbook\n3(629-670), 15 (2005)\n13[31] Schneider, J.B.: Understanding the finite-difference time-doma in method. School\nof electrical engineering and computer science Washington State U niversity 28\n(2010)\n[32] Shafi, K.M., Luo, W., Yalla, R., Iida, K., Tsutsumi, E., Miyanaga, A., Ha kuta,\nK.: Hybrid system of an optical nanofibre and a single quantum dot op erated at\ncryogenic temperatures. Scientific reports 8(1), 13494 (2018)\n[33] Le Kien, F., Liang, J., Hakuta, K., Balykin, V.: Field intensity distribu tions and\npolarization orientations in a vacuum-clad subwavelength-diameter optical fiber.\nOptics Communications 242(4-6), 445–455 (2004)\n[34] Okamoto, K.: Fundamentals of Optical Waveguides. Elsevier, ?? ? (2021)\n[35] Mauranyapin, N., Madsen, L., Taylor, M., Waleed, M., Bowen, W.: Ev anes-\ncentsingle-moleculebiosensingwithquantum-limitedprecision.Natur ePhotonics\n11(8), 477–481 (2017)\n[36] Ward,J.M.,O’Shea,D.G.,Shortt,B.J.,Morrissey,M.J.,Deasy,K.,N icChormaic,\nS.G.: Heat-and-pullrigforfibertaperfabrication.Review ofscient ificinstruments\n77(8) (2006)\n[37] Tong, L., Gattass, R.R., Ashcom, J.B., He, S., Lou, J., Shen, M., Ma xwell, I.,\nMazur, E.: Subwavelength-diameter silica wires for low-loss optical w ave guiding.\nNature426(6968), 816–819 (2003)\n[38] Yalla, R., Muhammed Shafi, K., Nayak, K.P., Hakuta, K.: One-sided c omposite\ncavity on an optical nanofiber for cavity qed. Applied Physics Lette rs120(7)\n(2022)\n[39] Yalla, R., Hakuta, K.: Design and implementation of a tunable compo site pho-\ntonic crystal cavity on an optical nanofiber. Applied Physics B 126(11), 187\n(2020)\n[40] Li, W., Du, J., Truong, V.G., Nic Chormaic, S.: Optical nanofiber-ba sed cavity\ninduced by periodic air-nanohole arrays. Applied Physics Letters 110(25) (2017)\n14"    },    {        "title": "2402.07565v1.Uniaxial_Compression_based_Recovery_Analyses_to_Describe_Polymer_Specific_and_Universal_Nanoindentation_Deformation_Phenomena.pdf",        "content": "Page 1 of 84 \n Title  \nUniaxial  Compression based Recovery \nAnalyses to Describe  \nPolymer -Specific and Universal  \nNanoindentation Deformation  Phenomena  \nFirst author: Prakash Sarkar  \nPhD student, Department of Metallurgical Engineering and Materials Science, Indian Institute  of \nTechnology Bombay, Mumbai - 400076, Maharashtra, India  \nORCID: 0000 -0002 -4404 -916X  \n \nCorresponding author: Hemant Nanavati  \nProfessor, Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai - \n400076, Maharashtra, India  \n*E-mail: hnanavati@iitb.ac.in  \nORCID: 0000 -0002 -5982 -6531  Page 2 of 84 \n Graphic for Table of Contents (TOC) Only  \n Page 3 of 84 \n Abstract  \nSharp tip nanoindentation of glassy polymers is a constrained, localized viscoelastoplastic \ndeformation. We interpret this complexity , in terms of the well -understood uniaxial deformation. \nFrom the uniaxial compression data in the literature, for PMMA, PC and crosslinked SU -8, we \nobtain their universal , yield -normalized recovery curves, with \n*\ny= , being one measure of the \ncorresponding strain state s (CSS). Nanoindentation recovery is determined from the 2sec constant \nrate unloading h-P data, modeled by a generalized power -law (variable power exponent). Comparing \nthese data -sets, yields the corre lation coefficient between the notional nanoindentation strain rate \nN( ~ ) hh\n and strain and true strain rate and strain, \nN t N t c   ==\n . The equivalent strain, , \nand the c value for any polymer , are within a narrow range, from the onset of indentation. \nCombining residual profiles via scanning probe microscopy with mathematical modeling of the \nindenter tip, provides the strain distribution beneath the tip. CSS measures examined  here,  indicate \npolymer -specific regions to regions common to glassy polymers , which are reached very early in the \nnanoindentation.  \nKeywords: Nanoindentation deformation profiles; Residual strain; equivalent nanoindentation strain ; \nGlassy p olymers; corresponding strain state . Page 4 of 84 \n 1. Introduction  \nNanoindentation is a potentially scale -bridging, relatively non -invasive technique , to determine \nmechanical properties of materials, particularly a t very low sample dimensions. The multiaxial , \nlocalized  and constrained deformation response of any material during n anoindentation , is very \ncomplex. This complexity is further manifested , particularly for indentation via sharp tips such as the \nBerkovich tip;  elastic, viscous and plastic contributions  occur si multaneously , from the instant of \nfirst contact. In addition, the contact regions share part of their deformation with the non -contact \nregion. Finally, the directions of the deformation and the resistance to volume change , are also \ndistributed.  \nLoad defor mation relationships are best understood in case of uniaxial and simple shear \ndeformations, along with the first level of complexity in terms of bending and torsion. These \nexperiments, have usually been carried out in the bulk state. However, this understa nding can form \nthe basis of extending our analysis of  nanoindentation deformation , if the elastic, viscous and plastic \naspects of the deformation can be separated . This premise  has been employed previously [1], [2], in \ndescrib ing th e phenomena during nanoindentation  deformation.  \nGlassy polymers are more pliant than other materials such as metals or cerami cs. In addition, their \ndeformation includes significant and simultaneous visco -plastic (VP) contributions, in addition to \nelastic contributions. These contributions need to be separated to the extent possible . We employ \nuniaxial compression as the basis to  identify aspects of nanoindentation deformation of glassy \npolymers. We consider both, thermosets (represented by the cross -linked epoxy, SU -8) and \nthermoplastics (represented by polycarbonate (PC) and Poly -methyl methacrylate (PMMA)).  \nNanoindentation char acterization methods essentially comprise employing the load -displacement \n(P-h) data to estimate the hardness, \nm c H P A=  (\nmP is the full load, and \ncA  is the contact area) and Page 5 of 84 \n the reduced modu lus, \n2rcE S A=  (\n()2\nN~1E−  for compliant materials) , where  \n and \nNE \nare the Poisson’s ratio  and elastic nanoindentation modulus of the indented sample . The stiffness, \nS , \nis the unloading  onset  slope, \nmPdP dh . The corresponding uniaxial compression properties are the \nyield stress, \ny  and the uniaxial modulus, E. The phenomena that are measured by \ny  and E, are \nbetter understood  than those measured by  \nH and \nrE . This work aims to interpret the complex \nnanoindentation  deformation phenomena , in terms of the uniaxial deformation phenomena, for \namorphous glassy polymers. The displacement controlled (DC) uniaxial compression data \nconsidered here, are at set strain -rates, and are obtained from the literature.  \nOne aspect  of expressing nanoindentation deformation in terms of uniaxial deformation, is its \ninterpretation  in terms of strain and strain rate. Mayo et al.  [3]–[5], have phenomenologically defined  \nhh\n as the nanoindentation strain rate,  \nN\n. Lucas et al.  [7], have phenomenologicall y defined the \ncorrelation between \nPP\n  with \nhh\n , for load -controlled (LC) nanoindentation experiments . In \nSection 5 , we first explain the necessity of considering LC nanoindentation with DC uniaxial \ncompression,  for our framework  here. \nSimultaneous yield and elastic deformation d uring nanoindentation of glassy polymers, are reflected \nin the instantaneous equivalent uniaxial strain. Therefore, as first described by Tabor [1], the residual \nnanoindentation strain post unloading , should correspond to that of the e quivalent uniaxial non-\nelastic strain. Then, the non -elastic (residual) strain, could be  related to  the imposed strain at \nmP , as \nshown in Fig. 1. Our analysis  applies this idea, and ensures that the total strain is the sum of the \nobjectively defined, recovered strain and the residual strain. Additionally , if the yield strain is a \nfunction of strain rate , we ensure simultaneous consistency with the equivalent strain rate.   Page 6 of 84 \n Our methodology involves topographical imaging via Scanning Pr obe Microscopy (SPM) , in order \nto examine the residual nanoindentation imprint on the material. We thus estimate recovered and \nresidual nanoindentation strain , for mapping with the corresponding uniaxial strains. This \ninterpretation of the equivalent strai n is provided in terms of the state of deformation, identified by \nrational metrics. The corresponding states of deformation (as defined by the recovery), suggest \nuniversal maps of uniaxial behaviours of glassy polymers, on which the limited regions of \nnano indentation can be located ; depending on the level of normalization of the corresponding states, \nthese regions may be polym er-specific, or common to all polymers.  Our framework provides insights \non the actual phenomena during nanoindentation deformation, w hich can be employed, for example, \nto develop more faithful frameworks for finite element investigations.  \n \nFigure  1: LC nanoindentation \n-Ph  Recovery in terms of DC uniaxial compression \n-  Recovery . \n2. Backgro und \nNanoindentation analyses broadly deal with estimating the nanoindentation properties , \nH and \n()()()()12 1 2 1\nN 2 1 1r c i iE S A E E  −−− = = − + −\n (\ni and \niE  are the Poisson’s ratio  and elastic \nmodulus of th e indenter,  respectively ), via the framework developed by Oliver and Pharr (OP) [8]–\n[10], for a range of elastoplastic  (EP) materials, whose moduli range from 68 GPa (Aluminum) to Page 7 of 84 \n 440 GPa (sapphire) . This framework is based on the developments by Love  [11] and Sneddon  [12] \n(LS) for ideal indenter indentations of ideally elastic materials.  \nPoisl et al.  [2], have  been the first to report the mapping of nanoindentation and uniax ial strain rates, \nas \nNc=\n , (\nc is the correlation coefficient ). They have examined  nanoindentation creep following \na step load on selenium (in the strain rate regime , where it exhibits Newtonian viscosity), and  have \nassumed Tabor’s phenomenological relationship, \n3y H= . Also,  assuming that \ncA  is that obtained \nby the conventional Oliver -Pharr (OP) framework  [8]–[10], they have found that  \nc ~ 0.09. This \nvalue has been considered for polymers as well [6], [13] . \nMaterial dependence of \nc  has been reported  [2], [14] –[19], by mapp ing the nanoindentation \nhh\n  to \nuniaxial creep rates, and recognizing that visco -elastic (VE) deformation and yie ld, occur \nsimultaneously during nanoindentation  [20]. Kermouche and co -workers [21]–[24], have synthesized  \nLC uniaxial deformation curves for various \nPP\n  values, at  \nmP  = 15 mN, employing published \nuniaxial m odulus and ideal yield strain formulations , along with the G’sell -Jonas model [25], for the \nflow stress. Subsequently, their nanoindentation data provides the strain -rate sensitivity parameter, \nenabling estimation of the representative strain s and strain rate s for the nanoindentation strain rate s, \nhh\n = 0.5\nPP\n . Attempts to understand the constrained deformation ( including frameworks in terms \nof uniaxial deformation [26][27]) via Berkovich nanoindentation, have been reported previously \n[28], [29]  as well . \nIn this report, we consider our nanoindentation experiments , where  the residual imprint s, are imaged \nvia SPM. The unloading curves, fit to a generalized power law (GPL), have been considered to \nprovide the hypothetical pure elastic recovery measure , commensurate with recovery from uniaxial \ndeformation . Uniaxial compression data for a range of DC deformation rates, have been sourced \nfrom the literature as follows: (i)  for cross -linked SU–8, we consider the DC micropillar compression Page 8 of 84 \n work [6], and (ii) for PC  and for PMMA , we consider the DC bulk compression data fro m multiple \nsources (for PC: [30], [31] , for PMMA:  [32]–[35]). The non -elasti c strain is computed systematically \nas per the schematic ( Fig. 1) for each stress -strain set, and is interpreted in terms of  the non -elastic \nresidual imprint depth.   \nIn order to express nanoindentation deformation in terms of equivalent uniaxial compressi ve strain, it \nis important to understand the anelastic deformation response in LC experiments. The effects are \nwell understood for uniaxial deformation, and there are a several reports [22]–[24], [36] –[42] on the \nVE effects during nanoindentation of polymers. We note here that these nanoindentation analyses, \nconsider \ncA  from OP -type methods, which , as we find in this work,  are inappropriate  for amorphous \nglassy polymer s.  \nThe common features in previous nanoindentation vs uniaxial deformation analyses, is that they \nrelate the creep during the hold period  of LC nanoindentation , and relate it to the same under \nsimulated uniaxial deformation, with one group [19] examining the LC hold period during \nexperimental deformation of fabricated micropillars. Subsequently, the variation of nanoindentation \nproperties with nanoindentation strain rate, is mapped to mechanical properties as function of \nuniaxial strain rate. The strain itself has not been considered,  because it is difficult to identify the \nabsolute strain during nanoindentation. We overcome this difficulty by employ ing experimentally \nobtained  knowledge of the elastically recovered strain, as our basis. Then, we need to determine only \nthe effective strain difference  between the imposed strain and the recovered strain , to obtain  the \nresidual strain .  \nWe discuss here,  the framework which combines the nanoindentation and uniaxial deformation data, \nto acquire insights into the nanoindentation response of amorphous glassy polymers .  Page 9 of 84 \n 3. Material  and sample preparation  \nThe materials we investigate here are amorphous glassy polymers , i.e., \n1. For thermosets, we have considered highly cross -linked SU -8 epo xy. For this, we begin with  SU-\n8 2050 grade resin, to fabricate thin film sample s on silicon wafer substrate, via a  standard photo -\nlithography process  [43], [44] . The fabrication steps are: (i)  RCA cleaning of 2\" dia. silicon wafe r, \n(ii) dehydration baking of wafer at 110°C for 30  min, (iii)  spin coating of SU -8 2050 grade at \noptimized parameters to obtain ~  22 µm thin film, (iv)  pre-exposur e baking at 65°C for 10  min \nand then 95°C for 5  min, (v)  UV-exposure for 13  s, (vi)  post-exposure baking (PB) at 95°C for \n30 min, and (vii)  hard baking (HB) at 150°C for 15  min.  \n2. Commercial PC  and PMMA , of unspecified grade, represent thermoplastics, and have been \nindented as received .  \n4. Nanoindentation experiments   \nWe carry out nanoindentation employing a  Hysitron  nano indenter (model: TI  Premier, Bruker \ncorporation™) with a Berkovich tip (radius ~  100 nm), at two levels of maximum load , \nmP  \n(1000  µN and 9000  µN), and at two levels of  relative  loading rate , \nPP\n  (0.1 /s and  1.0 /s). The \nloading is  followed by hold (70  s for cross -linked  SU-8, 100 s for PC , 150  s for PMMA ), conclud ing \nwith 2  s of unloading , for all indentations . Immediately after withdrawing the load, the residual \nindent impressions have been  consecutively imaged 3 times ( making them ~ 9 min apart)  via SPM , \nby scanning with the same tip, at a frequency of 0.8  Hz. We have performed four indents at each \nload.  Page 10 of 84 \n 5. Results and discussion  \nOur framework to understand nanoindentation deformation in terms of uniaxial compression, \nconsists of the following steps:  \na) Data from p revious DC uniaxial compression experiments  on highly cross -linked SU -8 \nmicropillars [6] and uniaxial bulk compression of PC [30], [31]  and PMMA  [32]–[35], have been \nre-examined , in terms of residual strains across various strain rates.  \nb) We have carried out  LC nanoindentation experiments at set \nPP\n  and have estimated for each \nindent, the hypothetical residual indent profile post -ideal elastic recovery, based on high rate \nunloading.  \nc) We have then combined the results of steps ‘a)’ and ‘b )’ above, to relate the indentation depth and \nthe notional indentation strain rate, to the equivalent uniaxial strain state and equivalent uniaxial \nstrain rate, respectively.  We ensure that all our analyses are based on the actual  load and \ndisplacement and  on the corresponding rates.  \nWe explain  first, the necessity of the apparent dichotomy – of carrying out nanoindentation in LC \nmode and interpreting it via uniaxial compression preformed in DC mode. Nanoindentation \nexperiments need to be  via LC mode, prima rily because:  \n1. The LC mode is a direct operation, whereas the DC mode is an indirect operation  \n2. In DC nanoindentation unloading, the indented material loses contact with the tip at \nfh  > 0.  \n3. In DC nanoindentation, the unload begins du ring the hold period, with \nmPdP dh→ . This \nmakes estimating \nS  difficult . However, h old is necessary to remove viscoelastic effects.  \nIn contrast, uniaxial compression data are required in DC mode because:  Page 11 of 84 \n 1. DC compres sion provides a clearly discernible \ny , enabling convenient identification of \nnanoindentation domains, in terms of the maxima, the inflection and the “flat”, constant stress  \nregion.  \n2. LC compression would lead to very high strain rate s, post yield. This would prevent assign ment \nof a clear strain rate to the uniaxial deformation, for relating to the equivalent nanoindentation \nstrain rate. In this framework , the recovery is measured in terms of strain.  Hence, t he actual strain \nand strain  rate in uniaxial compression are the relevant and essential parameters . \nWe begin with an examin ation , for our glassy amorphous polymers , of the uniaxial compression data \navailable in the literature.  \n5.1. Analysis of existing uniaxial compression data  \nDeta iled uniaxial compression experiments on highly cross -linked SU-8 micropillars  after toe -\ncorrection [45], over a strain rate range, 8.31×10-4 /s to 3.42×10-3 /s, have been reported previously \n[6]. These micropillars were fabricated via materials and methods similar to those employed for the \nnanoindentation experiments in this work.  \nFor PC and for PMMA, we  have examined bulk uniaxial compression experiments from various \nsources, over various ranges of strain rates. We recognize here, that the PC and PMMA examined in \nthe corresponding references, could be of grades different from those nanoindented by us.  \nOf the multiple sources for uniaxial compression data for PC, we co nsider only the data set [30], \n[31], encompassing a n engineering  strain rate range, 10-3 /s to 10-1 /s. Similarly, for PMMA, we \nconsider data at various set engineering strain rates (over the combined range, 10-4 /s to 1  /s) [32]–\n[35]. There are only limited data, at set engineering strain rate s (or displacement rate s). The \nengin eering strain rate basis, makes our analysis consistent with that for cross -linked SU-8. The \nother reason, as will be explained when mapping is described ( Section 5. 4), is that properties (\ny Page 12 of 84 \n and \nE ) at one t ype of strain rate (over the entire strain range) cannot be analytically mapped, \nexclusively, to another type of strain rate.  \nFrom t he DC uniaxial micro -compression (SU-8) and bulk compression (PC and PMMA) stress -\nstrain data , we generate normalized stres s-strain plots ( Fig. 2) for the three polymers, i.e., \n*\ny  =  \nvs. \n*\ny  = , for all rates. Here, \ny  is the maximum stress at yield, and \ny  is the corresponding \nstrain. We consider \n* , as a form of Corresponding Strain State ( CSS). We will discuss further \naspects of CSS measures, in subsequent sections.  \n \nFigure  2: CSS \n**-  curves  for cross -linked  SU-8 (black lines), PC (red lin es) and PMMA (blue \nlines). For clarity, we have shown the PMMA CSS curves, of only the data by Ames et \nal. [34]. \nThe variation of \ny  with \nln\n  is a well reported phenomenon; e.g., see [46], [47] . Similarly, a \nbroadly monoton ic, direct correlation is also expected between \nln\n  and the compressive modulus, \nE\n. Then, i n this work, we retain generality in our framework, by considering a linear relationship \nbetween \nln\n  and \ny (eqn. 1), (Fig. 3). \nlnypq=+\n          eqn. 1 \nThe linear fit parameters for the three polymers are listed in Table 1. Page 13 of 84 \n Based on the CSS measure, \n()()*\ny     =\n , we determine the non -elastic r esidual strain, \nne  [24], \n[48], [49] , from each of these curves. \nne  are obtained for the entire strain -range, by dropping a \nHookean line of slope \n()E\n , from the corresponding point on the stress -strain plot, as shown in \nFig. 1. Thus, the normalized residual strain is \n*\nne , and \n*\ne  is the normalized elastic strain  (eqn.  2).  \n \nFigure  3: Variation of \ny  with \nln\n  for cross -linked  SU-8, for PC and for PMMA. For PMMA, we \nhave shown only the data by Ames et al. [34]. \nTable 1: Linear fitted parameters for all polymers.  \n p q Constant \ny  \nCross -linked SU -8 9.46×10-4 5.23×10-2 0.045  \nPC 4.81×10-4 9.08×10-2 0.089  \nPMMA [34] 4.86×10-4 8.56×10-2 0.083  \nPMMA [32] 4.18×10-3 11.2×10-2 0.094  \nPMMA [33] -2.52×10-3 7.44×10-2 0.092  \n()\n()()* * * *\nne e\ny E     =−=−\n        eqn. 2 \nThere is a single source each, fo r PC and cross -linked SU -8 data. There are multiple sources for \nPMMA data. These enable examination of the effect of the grade in terms of specific features of the \nstress -strain plots, on the \n*\nne  vs. \n*  plots.  We combine Fig. 2 and eqn. 2, to estimate \n*\nne  for any Page 14 of 84 \n strain. For each of the three polymers, over the strain rate range of their respective data, CSS \nbehaviour ( Fig. 4(a)) containing polymer -specific aspects and aspects common to a ll polymers .  \nThese aspects include:  \n1. We find that qualitatively, the PC and PMMA correlations are similar to each other (no inflection \nexhibited) and different from the correlation for cross -linked SU -8 (exhibits inflection at \n*\n ~ 1.2). \n2. For CSS up to the elastic limit, \n*\n0  (~ 0.5 for cross -linked SU -8, ~ 0.3 for PC, and for PMMA \n[33],[34], \n*\n0 ~ 0.3, or  \n*\n0 ~ 0.4 [32]), there is no residual strain, i.e., \n*\nne0 . For any one source, \nover various strain rates, there is only a slight variation in \ny , and in \n*\n0 . Therefore, the \ncorresponding \n*\nne  vs. \n*  plots agree well with each other.  We note from the stress -strain data, \nthat the grades of PMMA samples from different sources [33],[34], differ in \ny , \ny, modulus \n(E), and the strain -hardening behaviour . However, these differences, as expected, do not affect the \ncorrespondence in their \n*\nne  plots. The qualitatively similar moduli, elastic limit and \ny , ensure \nthat the differences in the recoveries, due to the differences in greater post -yield stress (different \nhardening behaviour), does not significantly affect the unrecovered strain. However, differences \nin the elastic limit (PMMA sample s of refs.  [33],[34] vs. those  of ref.  [32] ), quantitatively shift \nthe \n*\nne  curve  laterally. Still, the plots of the two thermoplastics are broadly similar, in that, they \ndo not exhibit any inflection.  \n3. For CSS beyond the elastic limit, we define \n**\n0= − ; Fig. 4 (b) indicates that the normalized \nrecovery curves appear  to merge at high \n . However, they are polymer specific at low \n . Thus, \nwe find that the behaviour can be divided into two regions. In the first region, from \n*\n0  up to Page 15 of 84 \n \n* = \n*\nL~1.5 for cross -linked SU -8 and \n*  = \n*\nL~1.2 for PC and for PMMA, the relationship can \nbe fit to a polynomial in \n  as per eqn. 3..  \n6\n*\nne\n2i\ni\nik\n==\n           eqn. 3 \nFor the region, \n* >\n*\nL, the relationship is approximately linear; i.e.,  \n**\nne L Lmk=+\n          eqn. 4 \n \n(a)      (b) \nFigure  4: Fitted universal curve for CSS \n*\nne  for cross -linked  SU-8, PC and PMMA: Similar \ncorrelation for two PMMA samples  [33],[34] with similar elastic limit, \n*\n0 , but different \ny\n and different quantitative correlations in case of different \n*\n0  [32],[34] for PMMA : \n(a) vs. \n* . lateral shifts  correspond to effects of the elasticity limit, \n*\n0  and (b) vs \n**\n0= −\n. Broadly similar behavior after removal of the lateral shift, with polymer -\nspecific behavior at lower \n  \nDetails of the analyses are shown in Supporting Information  1. We note here that the  location of \n*\nL ,  \nis approximate, and has been identified visually. The choice of a linear relationship for \n* >\n*\nL, is to Page 16 of 84 \n facilitate the numerical analysis in section  5.4. The fitted po lynomial parameters ( eqn. 3) and the \nlinear fit parameters ( eqn. 4) are listed in Table 2. \nThe polynomial region beyond \n*  = 1, is typically the softening region, where the stress decreases \nduring extension. For PC and PMMA, \n*\nL  is effectively the inflection point in the plot. For SU -8, the \ninflection point is at \n*  ~ 1.2. The possible physical significance of \n*\nL  will be discussed in a \nsection s 5.4 and 5.6 .  \nTable  2: Fitted parameters values of universal residual strain curves.  \n \n6k  \n5k  \n4k  \n3k  \n2k  \nLk  \nLm  \nCross -linked SU -8 8.57 -25.58  26.21  -10.67  2.32 -0.65 1.00 \nPC -- -- -0.49 0.72 0.44 -0.77 1.10 \nPMMA [32] -- -- -0.407  0.337  0.812  -0.804  1.11 \nPMMA [33] -- -- 0.996  -2.50 2.328  -0.504  1.017  \nPMMA [34] -- -- 1.082  -2.505  2.216  -0.5885  1.056  \n5.2. Nanoindentation data  analysis  \nThis section comprises  two parts. In Part I, we analyse the the  \n-Ph  data. In Part  II, we examine  the \nSPM imaging data and discuss the “internal” upflow  (indicated by these data) and its consequences .  \nI. Analysis of Nanoindentation  h-P data  \nThe \n-Ph  curves for all indentation s at 0.1 /s and 1.0 /s \nPP\n  for all three polymers , indicate  excellent \nreproducibility ( Supporting Information  2). We consider here, the deformation response, in terms of \nthe unloading \n-Ph  data. For the unloading step, the tip begins from rest, and accelerates to the set \nunloading rate at ~  0.95\nmP ; deceleration begins on reaching ~  0.3\nmP , and the subsequent recovery \n(See Supporting Information  3), up to the residual depth (obtained from the nanoindentation data), \nfh\n, at \nP = 0 µN, includes significant  VE effects.  Page 17 of 84 \n To clearly separate out the various contributions to deformation, it is necessary to determine  the \nhypothetical non -elastic residual penetration depth, \nneh , obtained after  purely elastic recovery , \ne m neh h h=−\n (at the set unloading rate) . We suggest that for our experiments, the first part of the \nrecovery should be purely elastic (instant recovery) ; for this, we consider two options:  \n1. Modelling  the unloading \n-hP  data via generalized power law (GPL) model:   \nWe model the unloading \n-hP  data, only within the range, from ~  0.95\nmP , up to ~  0.3\nmP . \nConventional methods [8]–[10] employ the power law (PL) model  (written in log -log form as \n()ne ln lnP m h h=  −\n). In our work , the residual non -elastic displacement is based on 2s \nunloading time, and we will call it \nne 2h . For polymers  in this work , we recognize that 1/m, the \nslope  for the \n()ne2 lnhh−  vs \nlnP  correlation , varies with \nlnP , eithe r linearly or quadratically. \nTherefore,  we employ the GPL model,  \n()()()2ln 3 ln 22 1 0\nne2k P k P k\nh gP h++\n=+ . The fitted \nparameters are \n0k , \n1k, \n2k, \ng and \nne 2h . For the linear variation of \n1m  with \nlnP , \n2k = 0.  \n2. Estimating Data for Instant Unload:   \nThe variation in \n1m  with \nlnP , durin g the unloading over 2s, suggest s that the expected elastic \nunloading, is compromised by VE effects. We observe these  effects explicitly in the time -\ndependent recovery of the indentation profile, as described in Section 5.2.II. To circumvent these \neffects,  we consider  the top 10% of the constant rate unloading curve, i.e., from ~  0.95\nmP , up to \n~ 0.885\nmP . We model the unloading \n-hP  data in this range via the PL framework, i.e., \n()1\nne0mh gP h=+\n. Here,  we call  the residual non -elastic displacement, \nne0h , since it is based on \neffectively instant (zero time) unloading. We find that across all the polymers, as well both \nPP\n  \nand \nmP , m ~ 1.25,  which is consistent with the EP materials considered in the OP framework  \ndevelopment  [8]–[10].  Page 18 of 84 \n Figure 5 indica tes the two fits of the unloading \n-hP  data (from ~  0.95\nmP , up to ~  0.3\nmP , and from \n~ 0.95\nmP , up to ~  0.885\nmP ), and the estima tes for \nne 2h  and \nne0h . The accuracy of these method s is \nestablished by the very low relative residuals  (< 0.05%), throughout the corresponding fitting range s. \nThe fitted GPL parameters  for 2 s unloading as well  as the PL parameters for zero -time unloading,  \nare listed in Supporting Information  4. Table 3 lists the maximum displacement and the hypothetical \nresidual displacements . Both methods yield identical values for the stiffness, S. \n \nFigure 5: \n-Ph  curve with fitted equation for \nmP  = 1000  µN for cross -linked S U-8 (inset:  relative \nresiduals of 2s GPL fitting  and zero time PL fitting ) extending the fit s to the h-axis yields \nne0h\n and \nne 2h . \nTable 3: The maximum and hypothetical residual displacements and imprint size ( a) of the \nnanoindentations on glassy polymers  \n \nPP\n  (/s) \nmP  (µN)  \nne 2h  (nm)  \nne0h  (nm)  \nmh  (nm)  \nCross -linked \nSU-8 0.1 1000  209 ± 4 297 ± 1 420 ± 1 \n9000  704 ± 13 1023  ± 8 1374  ± 7 \n1.0 1000  200 ± 5 299 ± 5 423 ± 6 \n9000  536 ± 8 1021  ± 7 1366  ± 7 \nPC 0.1 1000  337 ± 11 432 ± 6 563 ± 4 \n9000  1144  ± 15 1440  ± 17 1860  ± 16 \n1.0 1000  348 ± 17 427 ± 7 559 ± 3 \n9000  981 ± 10 1393 ± 8 1816  ± 6 \nPMMA  0.1 1000  331 ± 10 372 ± 11 478 ± 8 \n9000  1039  ± 19 1195  ± 9 1537  ± 10 Page 19 of 84 \n The choice of the appropriate estimate of the elastic recovery (\nne0h  or \nne 2h ) vis-à-vis uniaxial \ndeformation, requires a comparison between the nanoindentation modulus, \nNE , and the uniaxial \ndeformation modulus, E. In addition to stiffness estimates, estimating \nNE  requires an accurate \nestimate of the actual, load -bearing contact area, Ac. This estimate is obtained from the SPM imaging \nof the residual nanoindentation imprint.  \nII. Analysis of SPM Data  \nAlong with the \n-Ph  data,  the post-unloading SPM images provide insights into the phenomena \nduring deformation.  We first examine the top view of the residual imprint after unloading, as shown \nin Fig. 6 for cross -linked SU -8 (imprints for all three polymers , are available in Supporting \nInformation  5). Figure  6 indicates the geometric contact area, \ncA , at full load; the projected area is \nan equilateral triangle ABC, of side length a.  \n \nFigure  6: Top view of residual imprint for cross -linked SU-8. Note the con tact area size at full load \n– equilateral ABC, of side a. \nThe image is of the residual imprint, which is due to t he residual non -elastic displacement \ncorresponding to viscoelastic -plastic (VEP) downflow beyond the elastic displacement. As described \nabove,  during unload  and over time  at P=0, part of this depth is recovered, but part remains as a \npermanent residual deformation. SPM imaging of residual imprint of the indent  requires ~  9 min Page 20 of 84 \n after unloading . Thus, the SPM image profile, corresponds to the inde nted region ~  6 min after \ncomplete unload.  Figure  7(a) is a representative image of a vertical SPM section.  The maximum \nresidual depth, \nrh , corresponds to the reduced residual indentation, after additional zero -load \nrecovery. We hav e plotted ( Fig. 7(b)) the vertical SPM sections with the origin at the \nz  = 0 level, at \nthe base of the depth -axis, i.e., at the centroid of horizontal tip section. We have indicated \nne 2h , \nfh  \nand \nrh , as per our experimental observations.  \nThe outermost tips of the permanent imprints, form the corners of the contact area. This principle is \nconsistent with our observation that the \na  value (c orner to corner distance  of the ABC in Fig. 6) is \nconstant for more than 12  hrs. after indentation [50], [51] . This is so even as the apex depth varies \nover that duration. As indicated by the SPM sections, we make the following observations:  \na) The tip of a real indenter is never an ideal point of size zero. There is a blunt depth, \nbh , which has \nbeen estimated ( Supporting Information  6) by modifying the fitting analysis of the contact area \ncalibration with the quartz standar d. \nb) The \na  values (Fig 6)  from the permanent imprints of the sharp edges of the Berkovich tip, yield \nthe geometric contact height, \n2 3 tan65.27c c bh h h a= + =  . Then, the sink -in depth, \nm sch h h  =−\n, where \nmm b h h h=+ . For the remainder of this report, for \nch  only, we remove the \nsuperscript \n  [52].  \nc) We observe that the re is pile-up (PU) along the facet, and that its maximum is farther away than \nthe distance corresponding to geometric contact with the tip. The maximum pile -up, \nPU\nmh , is ~  1% \nof \nmh  for cross -linked SU-8, ~ 8% o f \nmh  for PC and ~  3% of \nmh  for PMMA.  Page 21 of 84 \n d) SPM data indicates that \nPU\nm s hh  for PC, and \nPU\nm s hh  for PMMA and for cross -linked SU-8. \nThis suggests that for PC, during loading, a long with sink -in along the edge , pile -up occurs along \nthe facet .  \n  \nFigure  7: Residual indent  impression  for cross -linked SU-8 at \nmP  = 1000  µN and \nPP\n  = 0.1 /s \n(a) vertical  view, (b)  scan line which pass through the tip apex depth . \nWe find that the \n() m sfh h h−  ratios for our polymer materials, are lower than those obtained via \nthe OP -type methods [8]–[10]. Pile-up occurrence means that the sink -in along the facet is further \nreduced, as compared to that along the edge. We attribute th ese reduction s in sink -in to upf low, \nwhich we will discuss later in this section . \nm shh  is usually considered to be a measure of the \nelastic contribution to the deformation, and is realized from the loading step. In this work, we \nrecognize recovery as a measure of th e elastic contribution. We also note  that at the onset of unload \n(which occurs after an adequate hold period), there is no memory of the VE effects of the upflow  \nduring loading , except the instantaneous effects in terms of the contact area and the state of  residual \nstrain . Thus, r ecovery is governed by this distribution of deformation under the tip, and the contact at \nthat load. The strain in the layers beneath the tip, represent s the events at the tip apex, such as \ndisplacement and recovery, and the effect s of upflow, in terms of increased contact and absorbed \nstrain . Hence, the relevant comparison with elastic aspects of uniaxial compression would be based \n(b) \n(a) Page 22 of 84 \n on the recovery ; i.e., \ne m neh h h=−  (\nneh=\nne0h  or \nne 2h ), would be  the measure of the elastically \nrecovered strain on unloading . Then the effective elastic fraction of the total displacement is  \nemhh .  \nAdditionally , when there is upflow, for a given depth,  the contact increases and sink -in decreases. \nThus, the sink -in fraction decreases due to reasons other than additional downward deformation \nunder the indenter tip. Hence, for such cases, \nsh  cannot be considered as the sole measure of the \nelastic contribution. We can estimate the likely sink -in, \n()e\ne 2shh=− , had there been no upflow. \nThis interpretation enables an estimate of the upflow, \ne\nuf s sh h h=− . The superscript ‘e’  indicates \n“elastic recovery”.  \nTable 4 lists the \na  value s, the corresponding  \nch, the fractional elastic contributions and thus the \nfractional  upflow  (illustrated in Fig. 8). We find  that \nm chh  is independent of rate. It d epends \nslightly on \nmh  for SU -8, but not for PC and PMMA. However, \nemhh  depends on rate, particularly \nfor \nmP  = 9000  µN. The higher \nemhh  at higher rates, is consisten t with the expected, more solid -\nlike behaviour at such rates.   \nTable  4: Nanoindentation SPM Analysis – contact a values, \nm chh , recover y-based elastic \ncontribution estimates (\nemhh , assuming \nne ne2hh= ) and fractional upflow,  \nm ufhh . \n \nPP\n  (/s) \nmP  (µN)  \na  (µm) \nm chh  \nemhh  \nm ufhh\n310  \nCross -\nlinked \nSU-8 0.1 1000  2.84 ± 0.01 0.86 0.50 41 ± 4 \n9000  9.65 ± 0.03 0.92 0.49 99 ± 8 \n1.0 1000  2.83 ± 0.04 0.86 0.53 30 ± 3 \n9000  9.65 ± 0.02 0.93 0.61 112 ± 5 \nPC 0.1 1000  4.05 ± 0.03 0.93 0.40 75 ± 13 \n9000  13.24  ± 0.10 0.94 0.39 82 ± 4 \n1.0 1000  3.96 ± 0.03 0.91 0.43 44 ± 13 \n9000  13.00  ± 0.06 0.94 0.46 97 ± 8 \nPMMA  0.1 1000  3.46 ± 0.03 0.93 0.31 45 ± 7 \n9000  10.94  ± 0.03 0.94 0.32 56 ± 7 Page 23 of 84 \n \n \nFigure  8: Schematic of elastic and anelastic displacements considering existing upflow eve nts. \nIn our analyses , we employ the explicitly obtained Ac via SPM imaging, as opposed to the indirect \nmethod via the OP framework  [8]–[10]. We suggest that the upflow (not accounted for by the LS and \nOP development s) is a consequence of physically governed solid -like and fluid -like volume -\nconserving phenomena (illustrated in Fig. 9(a)-(b)). This extent of the upflow is possibly a  \ncharacteristic of compliant materials such as glassy polymers. Since upflow includes both, solid -like \nand fluid -like effects, in Table 4, we find that \nm ufhh  is not correlated with deformation rate. These \neffects are : \n(i) In terms of solid  deformation, the upflow corresponds to the Poisson effect. Friction -effects, \nreported in [53], [54] , which reduce the Poisson effect, exist  where the contact with the sample \nis horizontal (as would be expected in case of a flat punch indenter). In case of a Berk ovich \nindenter, the only horizontal contact, is the tangential point contact at the blunted tip apex. The \nPoisson effect solid deformation at the tip surface, is mostly away from the apex, and is directed \nlaterally , further away from the tip. Thus, this de formation displacement is not in contact with \nthe tip surface, precluding any consequences of surface friction on the upflow. It is only  some \nradial  distance  away from the contact, that this Poisson deformation “turns” upward, toward the \nunconstrained samp le surface above (the path of least resistance), resulting in upflow.  Page 24 of 84 \n (ii) In terms of the flow effect s, the volume -conserving upflow is manifested in terms of the \nrestricted slip (modified “no -slip” condition) at the tip surface, combined with the laminar -like \nupflow towards the  \nz = 0 surface, both being superimposed on the time -dependent anelastic \ndownward deformation along the indenter tip surface. This argument for “perfect friction” (no -\nslip between the indenter tip and the sample m aterial), also arises due to deformation of the \ncontact solid surface, as seen in the LS development  [11], [12] . This has been  further clarified by \nChaudhri et al.  [55][56], for conical indentation, which would be analogous  for Berkovich \nindentation. The y have inferred that the  deforming contact region of the surface, moves axially \ndownwards , along  with the tip surface .  \nIn the next section, we first determine the appropriate measure of nanoindentatio n recovery (i.e., \nwhether \nne 2h  or \nne0h ), based on the relationship between the uniaxial compressive modulus E and \nthe corresponding nanoindentation modulus, \nNE . This will enable develop ment of  a framework  \n(Section 5.4 ) to interpret the nanoindentation strain and strain rate in terms of the equivalent uniaxial \ncompression strain and strain rates, respectively.  \n  \nFigure  9: Schematic of  frictionless deformation and upflow event (a)  only due to upflow for flat \npunch (vertical surface contact), (b)  due to both, downflow and upflow for Berkovich tip \n(inclined surface contact). Here, \nuf\nch  and \nch  are the contact depths with and without \nupflow, respectively.  \n(a) \n(b) Page 25 of 84 \n 5.3. EN Effects on Mapping Nanoindentation Recovery  \nIn this section, we first determine the conventionally defined nanoindentation modulus, \nNE , as \ndescribed in Section 1 . This involves employing the estimate of \nmPS dP dh= , from the unloading \n-hP\n data, and \n()234cAa= , where a is determined by SPM imaging (Table 3). S estimates are \nidentical, whe ther from the GPL modelli ng of the unloading \n-hP  data in the range from ~  0.95\nmP , \nup to ~  0.3\nmP , or from the PL modelling of that data from ~  0.95\nmP , up to ~  0.885\nmP .  \nTable 5 indicates that \nNEE ~1.1 to 1.9.  The nanoindentation recovery measured in terms of \nindenter displacement, can be translated in terms of strain, via an indirect procedure involving \nlnh , \nas well as a correlation constant, c, as described in the next section. The equivalent uniaxial elastic \nrecovery (residual depth = \nne0h ), corresponds to a strain recovery (Fig.  1-b) at a greater slope \n(corresponding to \nNE ), than that for  the actual  uniaxial deformation (recovery slope ~  E).  \nIn order to circumvent potential error s due to the unavoidable mapping of unequal moduli , we \ndeliberately add consistently reproducible VE effects to the elastic recovery . We consider \nne 2h  as the \nmeasure of the non -elastic residual; i.e., we estimate \ne m ne2h h h=− , as the measure of the equivalent \nelastic recovery , for mapping to uniaxial compression. We further note that shorter fitting ranges, \ne.g., in case of shorter unloading times or the very small range to estimate \nne0h , could lead to \nuncertainty in these estimates. \nne 2h  is thus a reproducible equivalence correction. It is a balance, \nbetween having no VE effects , and accounting  for \nN 1 EE . \nWe explain this concept in terms of the elastic recovery and the stiffness. For the GPL framework \nwhich yields \nne 2h , and the unloading \n-hP  data correl ation yields \n()1\nmmPPS dP dh dh dP−\n== . Page 26 of 84 \n Considering the PL framework, \n()nemP B h h=− , \n() m m ne S mP h h=− . Then keeping \ncA , B and m \nfixed, the ratio of nanoindentation moduli measures, \nN0E  and \nN2E  (based on \nne0h  and \nne 2h , \nrespectively ), \n()()() N0 N2 m ne2 m ne0E E h h h h = − − . This enables comparison with \nNEE  \n(Table  5); i.e.,  \nN0 NEE=  and \nN2~ EE . See Figure  10. \nWe recognize that this comparison (\nN0 N 2EE  vs \nNEE ) is coarse over the three polymers; \nhowever, it is a reasonable approximation for the case of SU -8, where, both, the nanoindenta tion and \nuniaxial compression experiments have been carried out on the similarly synthesized polymers, \nemploying resins of similar grade. In subsequent analyses, we will drop the subscript “2”, and \nconsider \nne ne2hh= .  \n \nFigure 10:  The ch oice \nne ne2hh= , illustrated in terms of the uniaxial recovery slope – which should \nbe the uniaxial modulus, \nN2~EE . Page 27 of 84 \n Table 5: Effect of choice of recovery measure via -a-vis \nNEE   \n \nPP\n  (/s) \nmP  (µN)  \nNEE  \n() N0 N2EE  \nCross -linked \nSU-8 0.1 1000  1.87 ± 0.02 1.71 ± 0.02 \n9000  1.74 ± 0.01 1.91 ± 0.01 \n1.0 1000  1.83 ± 0.03 1.79 ± 0.00 \n9000  1.81 ± 0.01 2.42 ± 0.03 \nPC 0.1 1000  1.22 ± 0.01 1.71 ± 0.04 \n9000  1.07 ± 0.01 1.74 ± 0.04 \n1.0 1000  1.21 ± 0.01 1.58 ± 0.03 \n9000  1.12 ± 0.01 1.83 ± 0.03 \nPMMA  0.1 1000  1.39 ± 0.00 1.36 ± 0.00 \n9000  1.20 ± 0.01 1.46 ± 0.04 \n5.4. Correlation between  nanoindentation and compres sive strain s \nIn case of nanoindentation, from the beginning of the deformation, all aspects – elastic, viscous and \nplastic – contribute simultaneously. However, during rapid unloading, the purely elastic contribution \nis recovered first, followed by VE aspects at zero load, with the yield contribution remaining as the \nresidual at isothermal equilibrium. We exploit this sequencing , to phenomenologically map the \nevents , during the two deformation phenomena.  In Section 5.1, we have found that beyond the elastic  \nlimit, \n*\n0 , recovery from uniaxial deformation, remains partial, with \n*\nne  increasing monotonically \nwith imposed strain.  \nIn case  of DC uniaxial compression, the strain rate, \n\n , is propo rtional to the fixed displacement rate, \nh\n [6]. In case of LC nanoindent ation , the notional  strain rate,  \nN\n, is defined phenomenologically as \nhh\n [3]–[5], whic h can be correlated with \nPP\n  [7]. We also note that the nanoindentation strains , \nand residual  as well recovered strains , phenomenologically correspond to the true strain  scale , \nbecause the change in strain over time \n1t  to \n2t, is defined to be correlated with  \n() 12lnhh ; thu s, \n()() N N 1 2 1 2~ ~ ln t h h t t   −\n. In addition, the nanoindentation stress is considered in terms of \nthe instantaneous total projected contact area, which is analogous to the case of true stress. Page 28 of 84 \n Therefore, the uniaxial data and property correlations, which ar e in terms of engineering stress and \nstrain, need to be converted to true stress and strain  terms.   \nDuring nanoindentation loading , \nN~hh\n  decays  asymptotically  from a very high value  at \nh ~ 0 \n(shown in Appendix  A for cross -linked SU-8). However, f rom \ne~hh  (recall that \ne m neh h h=− ) \nonwards up to \nm hh= , \nhh\n  decreases by only ~  2%. We consider nanoindentation beyond  \neh, to \ncorrespond to the CSS beyond  \n*\ne. Similarly,  the corresponding residual strain at full  unload (at  \nneh), \nwould correspond to  the CSS at unload,  \n*\nne . Thus , for an y \nh  > \neh, \n() N N,e N,ne N,e e lnhh    = + = +\n.  \nThe normalized elastic limit  up to  \n*\n0, is effectively independent of uniaxial strain rate . We  then \nassume that this independence extends to the elastic ally rec overed CSS, \n*\ne. This  correspond s to \nN,e\n, attained  over the time period , during which,  there is a  significant variation  in nanoindentation \nstrain rate. Since this strain rate variation is inconsequential, w e wi ll employ this \nN,e  \ncorrespondence in the subsequent analysis.  \nAs a first approximation, we consider that the equivalent uniaxial compressive true strain , is \nproportional to the nanoindentation strain; i.e. , \ntNc= . This requires that, for  the corresponding \nstrain rates , \ntNc=\n  [2], [14] –[19]. Then , \nt t,e t,ne  =+ , means that  \nt,e t t,ne\nNN\nelnh\nh c c  −= − = −\n        eqn. 5 \nAs mentioned above in Section  5.1, one can analytically interchange between true strain and \nengineering strain as well as the corresponding rates , only at a given strain  (e.g.,  \ny). This is Page 29 of 84 \n because , true strain , \n() tln 1=+ , which means that engineering strain, \nt exp 1=− , and \nengineering strain rate, \nttexp  =\n . Thus , \n\n, and \ny as well, are  related to \nt\n  and \nt , \nsimultaneously. Using eqn. 1, \n() ttlnypq  = + +\n . Also, \n()()( )*\nt t t exp 1 ln pq    = − + +\n .  \nAt \nmP, the nanoindentation strain rate from the reference depth , \neh,  \n() me t\nN\nmelnhh\nt t c==−\n          eqn. 6 \nHence,  the nanoindentation strain ,  \nt t,ne t m\nN\nelnh\nc h c −= = +\n         eqn. 7 \nNext, w e correlate  the residual nanoindentation strain (corresponding to  \nneh ) with the residual \nuniaxial compression strain, \nne  (see Supporting Information  7 for details ). From eqn. 7, \n()*\nne ne\n**e ne1\n1c\ny\nyh\nh\n  + =+− \n         eqn. 8 \nWhen \n***\n0L  (combining eqn. 3 with eqn. 7),  \n() ( )()\n() ( )() ()6\n* t\nt t 0\ntt 2ne\n6 e* tt\nt t 0\nt t t t 2exp 11 lnln\nexp 1 exp 11 lnln lni\nici\ni\ni\nip q kpqh\nh\np q kp q p q  \n     =\n= − + + + − ++   =   −− + + + − −  + + + +\n\n eqn. 9 \nAgain, i f \ny is considered to be invariant with \nln\n , eqn.  9 simplifies to  Page 30 of 84 \n \n6\n* t\n0\n2ne\n6 e* tt\n0\n2exp 11\nexp 1 exp 11i\nyicy i\ni\nyi\nyy ik\nh\nh\nk\n=\n=−+−   =   −−+ − −  \n      eqn. 9a \nIf when \n*  > \n*\nL (combining eqn. 4 with eqn. 7),  \n()()\n()() ()t\nt t L L\ntt ne\nett\nt t L L\nt t t texp 11 lnln\nexp 1 exp 11 lnln lnc p q m kpq h\nh\np q m kp q p q\n    −+ + + +   ++  = −− + + + − +     + + + + \n  eqn. 10 \nIf \ny  is considered as invariant with \nln\n , eqn.  10 simplifies to  \nt\nLL\nne\nett\nLLexp 11\nexp 1 exp 11y c\ny\ny\nyymk\nh\nh\nmk\n −++ =   −−+ − +   \n      eqn. 10a \nIn eqns. 9, 9a, 10, 10a, \ny  and \n*\n0  are constant for a given polymer. The objective is to identify c, \nt\n \nand \nt  for a given nanoindentation.  Eqns. 6 and 7, relate \nt\n  and \nt  to the nanoindentation data, \nneh , \nmh\nand\neh , through c. Equation  7 is inc orporated into eqns. 9 and 10. The regime that the \nnanoindentation strain belongs to (whether \n*  < \n*\nL or \n*  > \n*\nL), is still unknown. Solving eqns. 6 \nand 9 simul taneously , yield a value set for \nt  and \nc ; eqns. 6 and 1 0, provide another value set for \nt  \nand c. The value of c for that nanoindentation, depends on which regime  contains the resu ltant \n*  \n(corresponding to the resultant \nt  values).  Page 31 of 84 \n Table  6 lists the  \n, \n\n and \nc  values , for the various experimental conditions for  the three  polymers.  \nThere is negligible pile -up in cross -linked SU-8; for PC  and PMMA , the contact pile -up ~ 0.015\nmh . \nHowever, we find that despite any pile -up, there is negligible variation between edge -side and facet \nside estimat es of \n , \n\n and \nc  values.  In addition, w e find that the uncertainty in the relationships \nbetween \ny  vs \nln\n , correspond to a relative uncertainty of ~  2.5% , in the \nc  values (listed in \nSupporting Information  8). A similar uncertainty exists while  considering \ny  vs \nln\n  to be constant.  \nTable  6 clearly indicates that the s trains for PC and PMMA are approximately twice those for cross -\nlinked SU-8. However, before examining in detail, the variations between polymers, w e first \nconsider \n  and \n\n , in terms of inter -related effects  of the two factors: (i)  relative loading rate, \nPP\n , \n(ii) maximum load, \nmP .  \nTable  6: \n , \n\n and \nc  values with respect to  tip apex  displ acement . \n \nPP\n  (/s) \nmP  (µN)  \n\n × 102 (/s) \n  × 102 \nc  \nCross -linked SU-8 0.1 1000  0.28 ± 0.00  6.21 ± 0.08  0.045  ± 0.000  \n9000  0.28 ± 0.0 0 6.44 ± 0.08  0.047  ± 0.000  \n1.0 1000  3.18 ± 0.29  6.31 ± 0.02  0.046  ± 0.000  \n9000  2.78 ± 0.03  5.86 ± 0.03  0.044  ± 0.000  \nPC 0.1 1000  0.53 ± 0.01  12.79  ± 0.38  0.083  ± 0.002  \n9000  0.46 ± 0.04  13.19  ± 0.32  0.085  ± 0.002  \n1.0 1000  5.01 ± 0.17  12.16  ± 0.5 8 0.079  ± 0.003  \n9000  4.50 ± 0.02  11.44  ± 0.07  0.077  ± 0.000  \nPMMA  [34] 0.1 1000  0.62 ± 0.01  14.57  ± 0.28 0.089 ± 0.00 1 \n9000  0.66 ± 0.0 2 14.76  ± 0.1 7 0.090  ± 0.000  \nPMMA [32]  0.1 1000  0.67 ± 0.02  16.98  ± 0.23 0.110 ± 0.00 1 \n9000  0.71 ± 0.0 1 17.21  ± 0.14  0.111 ± 0.00 0 \nPMMA [33] 0.1 1000  0.57 ± 0.03  14.16  ± 0.22 0.094  ± 0.00 1 \n9000  0.61 ± 0.0 1 14.34  ± 0.1 5 0.095  ± 0.00 0 \nFor a given \nPP\n , we find that \n\n  decreases marginally, when \nmP  increases by a factor of 9. This is \nbecause, for VE materials, the strain rate leads the stress, an d asymptotically decreases [6] towards \nthe Hertzian limit, \nhh\n  ~ 0.5\nPP\n  (shown in Appendix  A). \n\n  decreases with increase in  \nmP , Page 32 of 84 \n because the higher load provides additional time for \nN\n , to decrease towards the asymptote. In \naddition, as expected, \n\n  increase s correspond ing approximately , with the increase in  \nPP\n .  \nFor a given  \nmP, when \nPP\n  is increased by factor of 10, from 0.1  /s to 1.0  /s, \n decreases , although \nthe \nmh  values are comparable. The decrease in \n  is consistent with the increase in the relative \nelastic contribution  with strain rate , consistent with the trends in \nemhh , in Table  4. Table  4 also \nindicates  that for a given polymer,  the normalized downward anelastic deformation  metric , \nne mhh , \nis lowest at \nmP = 9000  µN and \nPP\n  = 1.0 /s, while the scaled upflow , \nm ufhh , are the  highe st. This \nresults in the lowest value s of \n  at this condition. We understand this phenomenon in terms of two \nphysical contributions.  \n(i) The volume -conserving flow in the direction towards the unconstrained surface is greater. This \nincludes the anelastic  (incompressible fluid -like) as well as the Poisson (solid -like) deformation. \nThe path of least resistance is from the region of high stress beneath the tip, to th e low -stress \nregion at the surface. The increase in upflow, absorbs part of the strain in the elements, in \nimmediate contact with the tip, which affects the strain distribution, as described below.  \n(ii) We recognize that during a slow loading step, there is ti me available for the  flow component of \nthe upper contact layers, to absorb part of the elastic deformation of the lower layers.  Hence, the \nincrease in load , compresses more the already yielded and therefore, more pliable contact layers   \nat lower rates (higher deformation gradient with depth) , than it does at higher rates  (Fig. 11(a)). \nRapid deformation compresses more layers (greater pervaded deformed depth), but does so \nrelatively equitably (lower deformation gradient with depth) , as shown in Fig. 11(b). \nConsequently, the strain in the layers below the tip, decreases graduall y, over a greater depth. \nHence, yield occurs up to a lesser depth , resulting in more layers that are elastically compressed  \n(i.e., strained less) , to account for the commensurate \nmh  .  Page 33 of 84 \n   \n  \nFigure  11: Schematic of the compressed layers under the tip apex at  \nmP  = 9000  N, for \nPP\n : \n(a) 0.1 /s: the pervaded deformati on region is smaller, with fewer deformed  layers , but \nwith greater (yield) deformation , (b) 1.0 /s: the pervaded deformation region is greater, \nwith more layers with recoverable, elastic deformation  \nFor the polymers i n Table  6, \nc values  ~ 0.044 to 0.047 for cross -linked SU-8, ~ 0.077 to 0.085 for \nPC, and ~ 0.09 to ~ 0.11 for P MMA . When \nmP  is increased by a factor of 9 from 1000  µN to \n9000  µN, the variation in \nc  is insignificant  for a ll polymers . When \nPP\n  is increased by a factor of \n10, from 0.1  /s to 1.0  /s, only f or PC,  does \nc  decrease with increase in \nPP\n .  \nThus, we suggest that for a given polymer, the \nc value depends on the nature  of strain ; i.e. , it \nprovides insights about  the nature of the deformation during nanoindentation . \nc is lower at lower \nstrains and if the material is more solid -like (less yielded) . Since the \nc  value does not vary with  \nmP, \nit is reached very early the during nanoindentation .  \nWe now examine closely, the effects of the physical phenomena on the estimates of  \nc. We \nrecognize that for all three  polymers, the same \nmP  values and \nPP\n  (i.e., \nN\n ) values , result in \ndifferent values of \n  and of \nln\n . These equiv alent strains indicate a state of strain ( CSS) which we \nwill consider next. The equivalent strain s in PC and in PMMA, are  greater than in cross -linked SU-8; \ni.e., PC and PMMA deformations contain greater anelastic contributions . Hence, we examine the \nc  \nvalues in terms of  the state of strain, characterized via various metrics of the CSS for that polymer.   \n(a) \n (b) Page 34 of 84 \n In order to examine the variations between polymers, a t this time, combined with the \n  defined in \nsection  5.1, we define another  anelastic CSS descriptor , \n()**\n01  = − . \n corresponds to the \nextent of the applied strain beyond the elastic limit, and \n*  values are simply the \n  values, scaled \nby the respective values at yield.  \nFor the polymers, we closely examine first, the plots of \n*\nne  \n**\ne ()=−  vs \n and \n*  (Fig. 12(a) \nand (b), respectiv ely), on which  we mark the equivalent CSS of our nanoindentation experiments.  \nFor the three  polymers, the nanoindentation deformation regimes are, \n()()()PMMA PC SU 8,−    , \nwhere for PMMA, \n()() **\n000.3 0.4==   .  \nThe nanoindentation regimes of the e quivalent CSS, \n*  and \n* , are located past the post -yield \ninflection. For cross -linked SU -8 the nanoindentation points are past the inflection, in the convex \nupward region. However, they are in the \n*  < \n*\nL region  (section 5.1) . The data for PC and PMMA \nare in the \n*  > \n*\nL region. We recognize that, in Fig  4, for the thermoplastics, the region where the \nsecond derivat ive is zero, is reached only at \n*  ~ \n*\nL. Thus, the thermoplastic polymer \nnanoindentation region is also past the post -yield inflection, as seen in Figs.  4 and 1 2 (discussed \nfurther in section 5.6) . Fig 12(a) suggests that there may be polymer -specific regimes on the \n  \nscale. By going to higher level of normalization, there could be a common regime for all polymers \non the \n*  scale  (Fig.  12(b)) . In this and in sub sequent analyses here, we need to consider that \n*\n0\n~0.3 (observed in a significant majority of the literature data) is representative for PMMA.  \nFor a given polymer, the CSS ranges , \n and \n* , are small, for a 9 -fold range of \nmP , and for a \ndecade -wide range of \nPP\n . Hence, it is likely that these CSS values are reached very early during Page 35 of 84 \n nanoindentation, for any amorphous glassy polymer. In ord er to understand the deformation \nphenomena further, we consider next, the equivalent uniaxial compression during nanoindentation, \nas function of relative radial distance,  \nmrr .  \n \n(a)      (b) \nFigure 1 2: Correlation s for CSS metrics  \n*\nne . The equivalent CSS of our nanoindentation \nexperiments are marked;  \n, \n, \n, \n represent  \nmP = 1000  µN and \nPP\n  = 0.1 /s, \n, \n, \n, \n represent  \nmP  = 9000  µN and \nPP\n  = 0.1 /s, \n, \n represent  \nmP  = 1000  µN and \nPP\n = 1.0 /s, and \n , \n represent  \nmP = 9000  µN and \nPP\n  = 1.0 /s. (a) Polymer -specific \nzones can  be determined with respect to \n  (b) There appears to be a common region for \nall polymers with respect to \n* . For PMMA , in case of different \n*\n0 , quantitative \ncorrelations vary with se parated nanoindentation regions  \n5.5. Deformation distribution under the indenter tip  \nThis section consists of two parts. We first estimate the nanoindentation profiles at full load, and \nafter the hypothetical elastic recovery. We then estimate the equival ent strains, based on the radially \ndistributed displacements and recoveries, employing the method described in Section 5.4.  Page 36 of 84 \n I. Estimate of state -specific topological profiles  \nDuring indentation, the surface of the region away from the tip, i.e., the non -contact region, \nundergoes radial deformation inwards, as well as axial deformation downwards. The LS \nanalyses  [11],  [12], [56] , describe  the response to indentation by a perfect conical indenter, of a \npurely elastic material, which recovers completely post -indentation. Based on the \n-Ph  data, \ntopological imaging and upflow analysis, we estimate next, the top ological profiles at specific \ndeformation states.  \nI.1. Profile as per SPM  imaging  \nThis profile is shown in Fig. 7(b). We call this profile the \nrh  profile, \n()p\nrhr . As indicated in the \nfigure, this profile can be fit to a power -law expression (details in Appendix  B). \nI.2. Pure elastic nanoindentation profile where hm = he, the elastic depth  \nWe estimate the profile for an ideal tip \n( 0)bh=  Berkovich nanoindentation, assuming that the \neh  \nvalue obtained above, corresponds to the maximum displacement for purely elastic nanoindentation \ndeformation. The corresponding sink -in depth would be \ne\nsh , as described above in Section 5. 2 II.  \nThe maximum depth profi le corresponds to maximum sample contact. Hence, the profile closely \nfollows the indenter geometry. As opposed to a single angle describing a cone, a Berkovich tip \nindenter, is described by a family of cones, with semi -angles spanning the range between the  two \nincluded angles, of the indenter. These two limiting angles for the Berkovich tip are the mid -facet \nsemi -apex angle of 65.27° and the remaining portion of the total included angle (142.30°), \ncorresponding to the edge angle, 142.30°  - 65.27°  = 77.03°. While traversing any facet, the \ninclination angle (to the vertical) decreases from the edge angle to the mid -facet angle, before \nsymmetrically increasing back to the edge angle at the opposite edge. Therefore, we will consider Page 37 of 84 \n our system in terms of a “Ber kovich cone” with a varying semi -angle and radius, from the edge to \nthe middle of the facet.  \nIn this case of ideal elastic indentation, the profile from \ne\nsh  to \neh , is in contact with the indenter tip. \nThe prof ile from \nz  = 0 to \nz  = \ne\nsh, corresponds to  the LS equation  [11], [12], [56] , for a conical \nindenter ( eqn. 11). \n()e 1 2 2 e\nee cot sinp\nsrh r r r r rr−= + − − \n       eqn. 11 \nAs described above, we consider two limiting values for cone half -angle: \n  = 77.03° in the edge \ndirection, and \n  = 65.27° in the facet middle direction. The contact radius in the edge direction, \ne, e e 2 3 tan65.27 3 2 tan65.27edr h h=  = \n. Along the facet -middle direction, \ne, e tan65.27farh= .  \nI.3. Estimate profile at \nmP  \nThe contact profile at \nmP  is up to \nmh . At \nmh , a further depth \nbh  below \nmh , the edge and facet lines \nmeet. For VEP deformation, t he contact profile would follow the ideal Berkovich tip from \nsh  to \nmh\n, as in the case of \ne\nsh  to \neh , above  (for purely elastic deformation) . However, for the real \nBerkovich tip  for VEP deformati on, the contact profile is truncated at \nmh .  \nThe profile from \nz  = 0 to \ns zh= , we consider the sink -in \n-zr  relationships ( eqn. 11) for our \nexperiment, to correspond  to radially outward -shifted \n-zr  relationships (i.e., by shifting the \nz -axes) \nfor the cones of depth \neh\n() ( ) e 2shh =−  [11], [12], [56] , such that the shifted \neh  cones \nshare the surface lines with the real Berkovich tip . \neh corresponds to the maximum  ideal elastic Page 38 of 84 \n displacement corresponding to \nsh , the geometric sink -in estimate via SPM imaging. We note here \nthat the entire deformation of the sunk -in region is recovered, and does not participate further in our \nanalyses.  \nI.4. P = 0 profile post hypothetical pure elastic recovery  \nAs described in Section 5. 2 I, our unloading \n-Ph  analysis indicates that a hypothetical purely elastic \nrecovery, would correspond to a residual apex depth, \nne 2h . We estimate the topological profile for \nthis hypothetical depth , based on the actual contact parameters and geometrical analysis of the SPM \nprofiles from two consecutive scans, as well as the \nmP  profile, determined above. There is  an \nexcellent correlation between the consecutive residual SPM profiles (\n()1p\nrhr , \n()2p\nrhr  and \n()3p\nrhr , \nsee Appendix  C) and the corresponding residual displacements (\n1rh , \n2rh , \n3rh ) of the tip apex \n(eqn. 12); i.e.,  \n()\n()()() ( )\n()() ( )1 1\n22;p p\nc r cr\np pcr c rh r h r hh\nhh h r h r− −=− −\n \n()\n()()() ( )\n()() ( )1 1\n33p p\nc r cr\np pcr c rh r h r hh\nhh h r h r− −=− −    eqn. 12 \nWe employ this proportional correlation, which is based on the geometrically obtained \nch , to \nestimate the profile for the hypothetical non -elastic residual imprint, \n() nephr  (eqn. 13), where \n()p\nrhr  \nis obtained by the power law fitting of the SPM section boundary ( Fig. 7(b), Appendix  B). \n()\n()()() ( )\n()() ( )nenepp\ncc\nppcr crh r h r hh\nhh h r h r− −=− −\n        eqn. 13 \nThese profiles are plotted in Fig. 13. Page 39 of 84 \n We devise an alternative method to estimate the \nneh  profile, without employing SPM imaging. Note \nthat we have estimated \neh  for tip apex displa cements. The ratio \n( )( ) e m m e m m 9000 1000 h h P h h P =  =\n. Hence, we interpolate the \neh  values between \nmP\n = 1000  µN and \nmP  = 9000  µN, via the power law equation, \nee e mmh B h= , to est imate the \neh  \nvalue for the displacement at any \nmrr  position  at \nmP  = 9000  µN. The fitted \neB  and \nem  values, are \nlisted in Appendix  D. We fi nd that at various radial positions, these power -law \nneh  values agree well \nwith those obtained via eqn.  13. \n \nFigure  13: Deformation profiles at \nmP , after elastic recovery (\nP  = 0), hyp othetical pure elastic \nnanoindentation, and as captured by SPM imagin g for cross -linked SU-8. Note that the \nindicated  \neh is the recovered displacement, (\nm nehh− ), transferred toward the surface to \nindicate elastic  nanoindentation. The hypothetical non -elastic residual profile exhibits an \nintermediate curvature (appears flat  at current magnification) between that of the \nacquired \nrh  profile and the linear contact profile at full load.  The shif ted axis intersects \nthe full load deformation profile at depth  \n() e 2s hh=− , such that the shifted \neh  \ncones share the surface lines with the real Berkovich tip . Page 40 of 84 \n II. Equivalent strain variation with radial distance from  tip apex  \nThe maximum radial distance  along the indenter tip , \nmr, at \nmP, is obtained by extending the contact \nprofile lines from the depth -axis at  \nmh, to \nz = 0. The maximum radial distances are \nm,edr  and \nm,far\n, along the edge and facet -middle directions, respectively (shown in Fig. 13). In the contact \nregion, the maximum load depths at various \nmrr  values , are equal to \n()() mm1h r r− . However, \ncomputing the strains and strain rates via the framework in the previous section s, (eqns. 1-10) \nrequires the value s of elastic deformation , at various fractional radial distances . The \nneh  profiles \ndetermined above, enable estimation of the  \neh values and then the  \nc values , at these radial positions . \nThe latter  are plotted in Fig. 14, and compared with the \nc  values at the tip apex for both loads.  The \ncombined plot of \nc  values, at various radial positions, at both loads  and loading rates , is available in \nSupporting Information  9. \nThe \nc  values decrease with rad ial distance from the depth axis (e.g., for cross -linked SU-8: from \n0.046 to 0.033 for  \nPP\n  = 0.1 /s and from 0. 045 to 0.022 for  \nPP\n  = 1.0 /s). The values are tabulated \nin Supporting Information  9. The decrease in \nc due to a 10 -fold increase in strain rate, is less than \nthe decrease due to radial distance (where the effective strain rate increase is only 4 -fold). Thus, the \nvariation in \nc  with \nmrr  is likely to be due to the variation in the nature of the deformation and the \nnature of the imposed stress. At the apex, the stress concentration and the degree of yield, are the \ngreatest. These decrease with increase in radial distance, so that the deformation is distributed more \nuniformly over the depth below the tip, at the corresponding radial position. This idea is consistent \nwith various FE studies of sharp -indenter nanoindentation on polymers (e.g., see [57], [58] ). Hence, \nas the nanoindentation displacement decreases with radial distance, the equivalent uniaxial strain \ndecreases by a progressively greater extent.  Page 41 of 84 \n \n \nFigure  14: The \nc  values for various se ctions of \nmrr  for cross -linked  SU-8, PC and PMMA . The \nlighter red and grey markers, correspond to SU-8 and PC data at \nPP\n  = 0.1 /s. \nConsidering the variation of \nc  with radial distance, provides the  variation s of \nc  with \n and \n  \nvalues  (Fig. 15(a)-(b)), which bridg e the gap between the apex regimes of all the polymers. Over all \npolymers, there is a clear trend of th e apex \nc  values with \n . However, the overall linear trend for  all \nvalues of  \nc, is clearer  with \n , than with \n . We note here that th e relationships of \nc  vs \n*  and \n* , \nare localized to individual polymers, with significantly different correlations across polymers (not \nshown).  \nThe foregoing discussion has considere d the nanoindentation deformation through the prism of \nuniaxial deformation – particularly in identifying the equivalent deformation ranges, via rationally \ndefined strain state metrics. There is a fundamental difference between the current work with respec t \nto earlier reports [2], [14] –[19], [21]–[24] of employing the perspective of uniaxial compression for \ninsights into nanoindentation deformation. Our approach incorporates rate effects, by directly \ndetermining the unrecovered strain for every strain over a range of strain rates. In addition, our \nanalyses are model -free, and interpret the phenomena in terms of the state of the deformation. We \nexplore next, the possible criteria, governing the material specific CSS ranges, on dimensionless \nuniaxial stress -strain plots.  Page 42 of 84 \n \n \n(a)      (b) \nFigure  15: Variation of \nc  values  for cross -linked SU -8, PC and PMMA with (a) \n, and (b) \n. For \n\n, there is an excellent common fi t for the tip apex only, for all polymers . For \n , the \nCSS measures at the tip apices correlate well  in separate local regimes , and are indicated \nby individual local fits. Considering the effect of radial position, bridges  the CSS regimes \nof all polymers , enabling  a global linear correlation.  \n5.6. Nanoindentation domains  vs equivalent uniaxial CSS \nIn this section, we  identify the nanoindentation domains , in the corresponding plots of stress -state vs \nstrain -state. We have already defined in Section 5. 1, a normaliz ation of  the stress, \n , by the yield \nconditions, to the corresponding dimensionless stress  state, \n*. Figure  16(a) contain s the plots of \n*  \nvs \n*. These plots are constructed via LOESS smoothening of Fig. 2. As mentioned above, the \n  \nrange has been listed in Table  1.  The stress state at the  elastic limit,  \n*\n0 = 0.63 for cross -linked SU-\n8, \n*\n0  = 0.55 for PC , and \n*\n0  = 0.57 for PMMA . \nIn order to account for the different elastic limits for the two polymers, similar to those for strain, we \ndefine additional stress state metrics, \n**\n0= −  and \n()**\n0 1  = − . We plot \n*  vs \n*  in \nFig. 16(b). The corresponding plots of \n  vs \n , are available in Supporting Information  10.  Page 43 of 84 \n The nanoindentation data are expectedly beyond the yield with respect to strain and various  CSS \nstrain metrics (Fig. 16(a) -(b)). In addition, we find that they are also beyond the post-yield inflection \npoints. These post -yield inflections on the \n*  vs \n* plots and \n*  vs \n*  plots, agree approximately \nwith the inflections  on the corresponding  \n*\nne vs \n*  (section 5.1) and  \n*\nne vs \n*  plots  (section 5.4). \nWe recall from section 5.1  that \n*\nL  are estimated visually. For PC and PMMA, there could be a \ncorrelation between these \n*\nL  values and the inflection points. Thus, despite the recovery correlation \ncurves being polymer -specific (in terms of \n*  and \n ), we find possible common ranges (in terms of \n*\n) of suitably des cribed scales of stress state and strain state, which are reached very early, during \nthe nanoindentation. Our findings in this work are based on thermoplastic as well as thermosetting  \namorphous glassy polymers, and are indicative of representative behaviou r for such materials. In the \nnext section, we summarize our findings, and provide our conclusions.  \n \n(a)       (b) \nFigure 1 6: LOESS smoothened curves for cross -linked SU -8, PC and PMMA  [34]. The equivalent \nstrain s of the nanoindentation experiments are beyond the post -yield inflection.  \n, \n, \n \nrepresent  \nmP  = 1000  µN and \nPP\n  = 0.1 /s, \n , \n, \n represent  \nmP  = 9000  µN and \nPP\n = 0.1 /s, \n , \n represent \nmP  = 1000  µN and \nPP\n  = 1.0 /s and \n , \n represent \nmP\n = 9000  µN and \nPP\n  = 1.0 /s. (a) \n* vs \n* (b) \n* vs \n* . \n(b) Page 44 of 84 \n 6. Conclusion s \nIn this investigation, we have employed the well understood uniaxial compression  phenomena (data \navailable in the literature) to interpret the complex material response durin g nanoindentation. We \nhave considered high ly cross -linked SU -8 to represent amorphous glassy thermosets and PC  and \nPMMA , to represent amorphous glassy thermoplastics.   \nRepresent ing the uniaxial stress -strain data over a range of engineering strain rates, \n\n , in terms of \nnormalized variables, \n*\ny  =  vs. \n*\ny  = , has provided  for each polymer, a universal \ncorrelation of residual (non-elastic) strain state, \n*\nne , vs. strain st ate, \n* . Thus, \n*  can be considered \nto be a measure of corresponding strain state, CSS. \nWe have combined nanoindentation experiments with SPM imaging  of residual imprint s, to obtain \nthe maximum depth, \nmh , the projected contact triangle side, \na . The ratio of the SPM -obtained \ncontact depth to the corresponding maximum depth, \nm~ 0.9chh , and is  broadly independent of \nPP\n and \nmP . Volume conserving material upflow gives rise to this high value. We have considered \na generalized power -law (GPL) model for the constant rate portion (2s unloading) of the unloading \ndata to estimate the corresponding hypothetical  non-elastic depth \nne 2h , and the stiffness, S. We note \nthat the recovery over 2s contains viscoelastic (VE) components. From this  enabled estimation of the \nconventionally defined nanoindentation modulus, \nNE , we find  \nNEE ~1.1 to 1.9  (E is uniaxial \ncompressive modulus) . Here,  we consider \n() m ne2hh−  to be the measure of instant elastic recovery, \nfor \nNE  to be reliably and reproducibly mapped to E. This further  enable s rational comparison \nbetween nanoindentation elastic recovery and uniaxial elastic recovery.  Page 45 of 84 \n We have then combined the uniaxial data as function of engineering strain rate, \n\n , and \nnanoindentation recovery data at the  two \nPP\n  values (which correlate with the nanoindentation \nstrain rates \nN~hh\n ). The recovery mapping is in terms of true strain \ntNc= . By simultaneously \nconsidering \ntNc=\n , we obtain the correlation coefficient, c; c ~ 0.044 to 0.047 for cross -linked \nSU-8, and c ~ 0.077 to 0.085 for PC , where the slight  variation is  with \nPP\n . For PMMA c ~ 0.09 \nwhen compared with uniaxial data having normalized elastic limit , \n*\n0~0.3 (which we consider to be \nrepresentative for the polymer) . c values are lower for greater deformation rates. At lower rates, there \nare a fewer deformed layers closer to the apex, but with a greater fraction having yield ed, in contrast \nto a greater number of elastically deformed layers at greater deformation rates.  \nWe have considered the Berkovich indenter as a family of cones with semi -angle range \ncorresponding to the Berkovich geometry. We have modified the LS analyses and to obtain t he \nprofile, \n() mphr  at \nmP . Proportional scaling analyses of consecutive SPM profiles to account for the \nVE effects, yield the likely non -elastic residual profiles, \n() nephr . Thus, we have  estimat ed the \nc  \nvalues for various radial positions (\nmrr  = 0.25, 0.50, 0.75) from the tip apex (where \nmrr  = 0). The \ndecrease  of the \nc  values with radial  positions is indicative of  the deformation distribution beneath \nthe tip; i.e. , in regions  radially away , the reduced stress concentrations result in a greater contribution \nof recoverable elastic deformation, leading to a lower equivalent uniaxial strain.  \nWe identify additional metrics for CSS, \n**\n0= − , \n()**\n01  = − . The radial variation in the \nc  \nvalues, bridge the \n  domains of the tip -apex equivalent strains, of the t hree polymers.  While the tip \napex \nc  values, vary linearly with \n , we find a common linear relationship between \nc  and \n .  \nWe have extended the development of dimen sionless stress -strain relationships beyond those \ndescribed  above, to various stress states metrics, as functions of the CSS. The equivalent \nnanoindentation strain and the CSS are always beyond the yield, as well as the post -yield inflection. Page 46 of 84 \n We thus locat e polymer specific zones (in terms of \n*  and \n ) and possible common zones (in terms \nof \n* ) of the nanoindentation points on rationally normalized uniaxial stress -strain curves . These \nzones and the polymer -specific c and \n  values , are reached very early during the indentation . \nAcknowledgments  \nWe would like to thank the Centre of Excellence in Nanoelectronics (CEN), IIT Bombay, for \nproviding SU-8 epoxy  and facili ties for the SU-8 sample  fabrication. The authors would like to \nacknowledge the Nanoindent er lab,  Central facility, IIT Bombay for the nanoindentation \nexperiments . The authors also acknowledge participation in preliminary discussions by Prof. Prita \nPant, D epartment of Metallurgical Engineering and Materials Science, IIT -Bombay.  \nFunding  \nStudent funding f or this project  was topped up by Prof. Prita Pant’s Research Development Fund at \nIIT-Bombay, RI/0106 -10000760 . \nAppendix A: Nanoindentation strain rate variat ion \nDuring nanoindentation loading, \nN~hh\n , decays asymptotically from a very high value. Hence, \nwe can fit \nlnP  vs \nt as \n() 1 2 3 4 ln 1 expP b t b b t b = + − − + , to obtain  \n() 1 2 3 31expdP Pb b b b tP dt P= = + −\n        eqn. A.1 \n1b\n, \n2b, \n3b and \n4b  are the fitting constants. Similar fitting (eqn.  A.1) is carried out between \nlnh  and \nt\n, to obtain \nhh\n , see Fig.  A.1(a)-(b) and Fig.  A.2(a)-(b). Page 47 of 84 \n   \nFigure A.1: Variation of norm alized rate s vs penetration depth for cross -linked SU-8 for \nmP\n = 1000  µN and set \nPP\n  is (a)  0.1 /s, (b)  1.0 /s. The elastic recovery (\neh ) positions, \nassumed to be rate independent, are indicated.  \n  \nFigure A.2: Variation of norm alized rate s vs loading time for cross -linked SU-8 for \nmP  = 1000  µN \nand set \nPP\n  is (a)  0.1 /s, (b)  1.0 /s. \nAppendix B: Power law fitting of residual image scan profile  \nThe SPM scan section corr esponding to the \nrh  profile (as shown in Fig.  B.1), includes the edge \ncorner, the tip apex position and the middle of the facet. The profiles can be expressed as power -law \nrelationships with radial distance, \nr , with the origins shifted to (i)  the position A at the  \nz = 0 level \nfor the edge side and (ii)  the maximum pile -up point, for the facet side. Thus , the resultant \nexpressions for the power law relationships are  (eqn.  B.1 and eqn.  B.2):  \n(a) \n (b) \n(a) \n (b) Page 48 of 84 \n \n() ,med\ned c ed z r r=−, for the edge side         eqn. B.1 \n()PU PU\nmmmfa\nfa z h r r+ = −\n, for the facet side        eqn. B.2 \nHere, \ned , \nfa , \nedm , \nfam  are the fitting constants (tabulated in Table  B.1), \nPU\nmh  is the maximum \npile-up height and \nPU\nmr  is the corresponding radial distance of maximum pile -up, \n, 3c edra=  and \n, 23c fara=\n.  \n \nFigur e B.1: Residual indent impression scan line with power law equation fitting for cross -linked  \nSU-8. \nTable  B.1: Power law fitting parameters of the scan profile obtained from residual imprints for \ncross -linked SU -8, PC and PMMA . \n \nPP\n  (/s) \nmP  (µN)  \ned  \nfa  \nedm  \nfam  \nCross -linked SU-8 0.1 1000  3.83 × 10-5 5.82 × 10-5 1.93 2.13 \n9000  7.99 × 10-5 2.18 × 10-5 1.68 2.17 \n1.0 1000 4.73 × 10-5 1.30 × 10-2 1.95 1.38 \n9000  8.97 × 10-6 9.78 × 10-7 1.97 2.55 \nPC 0.1 1000  1.99 × 10-4 6.13 × 10-3 1.80 1.50 \n9000  1.63 × 10-6 2.15 × 10-7 2.08 2.98 \n1.0 1000  6.72 × 10-4 4.94 × 10-4 1.59 1.98 \n9000  7.73 × 10-4 4.94 × 10-5 1.47 2.18 \nPMMA  0.1 1000  1.24 × 10-4 2.35 × 10-4 1.82 1.48 \n9000  6.71 × 10-6 1.04 × 10-6 1.24 1.86 Page 49 of 84 \n Appendix C: SPM  image  profile s vs. corresponding \nproportional profile s \nWe find an excellent correlation between the consecutive residual profiles and the correspondi ng \nresidual displacements of the tip apex ( eqns. 12 and 13) as shown in Fig.  C.1.  \n \nFigure C.1: SPM  image  profile for cross -linked SU-8 at 5 min, 12  min and 19  min and correspondi ng \nproportional profile for \nmP = 1000  µN and \nPP\n  = 1.0 /s. The proportional profile (open \ncircles) agrees very well with the corresponding scanned profile  (solid circles) . \nAppendix D \nThe fitted \neB  and \nem  values (Section 5.5 II) are liste d in Table  D.1. \nTable  D.1: Power law fitting parameters between \neh  and \nmh  for cross -linked SU -8, PC and PMMA . \n Parameters  \nPP\n (/s) \n0.1 1.0 \nCross -linked SU -8 \neB 0.70 0.47 \nem\n 0.94 1.02 \nPC \neB 0.45 0.46 \nem\n 0.98 0.99 \nPMMA  \neB 0.36 -- \nem\n 0.99 -- Page 50 of 84 \n References  \n[1] D. Tabor, The Hardness of Metals . New York, NY: Clarendon Press, 1951.  \n[2] W. H. Poisl, W. C. Oliver, and B. D. Fabes, “The relationship between indentation and \nuniaxial creep in amorphous selenium,” J. Mater. Res. , vol. 10, pp. 2024 –2032 , 1995, doi: \nhttps://doi.org/10.1557/JMR.1995.2024.  \n[3] M. J. Mayo and W. D. Nix, “A micro -indentation study of superplasticity in Pb, Sn, and Sn -38 \nwt% Pb,” Acta Metall. , vol. 36, pp. 2183 –2192, 1988, doi: 10.1016/0001 -6160(88)90319 -7. \n[4] M. J. Mayo, R. W. Siegel, A. Narayanasamy, and W. D. Nix, “Mechanical Properties of \nNanophase TiO2 as Determined by Nanoindentation,” J. Mater. Res. , vol. 5, pp. 1073 –1082, \n1990, doi: 10.1557/JMR.1990.1073.  \n[5] M. J. Mayo, R. W. Siegel, Y. X. Liao, and W. D. Nix, “Nanoin dentation of nanocrystalline \nZnO,” J. Mater. Res. , vol. 7, pp. 973 –979, 1992, doi: 10.1557/JMR.1992.0973.  \n[6] S. Chattaraj, P. Pant, and H. Nanavati, “Inter -relationships between mechanical properties of \nglassy polymers from nanoindentation and uniaxial co mpression,” Polymer (Guildf). , vol. 144, \npp. 128 –141, 2018, doi: 10.1016/j.polymer.2018.04.040.  \n[7] B. N. Lucas and W. C. Oliver, “Indentation power -law creep of high -purity indium,” Metall. \nMater. Trans. A , vol. 30, pp. 601 –610, 1999, doi: 10.1007/s11661 -999-0051 -7. \n[8] W. C. Oliver and G. M. Pharr, “An improved technique for determining hardness and elastic \nmodulus using load and displacement sensing indentation experiments,” J. Mater. Res. , vol. 7, \nno. 6, pp. 1564 –1583, Jun. 1992, doi: 10.1557/jmr.1992.1 564. Page 51 of 84 \n [9] W. C. Oliver and G. M. Pharr, “Measurement of hardness and elastic modulus by \ninstrumented indentation: advances in understanding and refinements to methodology,” J. \nMater. Res. , vol. 19, no. 1, pp. 3 –20, 2004, doi: 10.1557/jmr.2004.19.1.3.  \n[10] G. M. Pharr, W. C. C. Oliver, and F. R. R. Brotzen, “On the generality of the relationship \namong contact stiffness, contact area, and elastic modulus during indentation,” J. Mater. Res. , \nvol. 7, no. 3, pp. 613 –617, 1992, doi: 10.1557/JMR.1992.0613.  \n[11] A. E. H. H. Love, “Boussinesq’s problem for a rigid cone,” Q. J. Math. , vol. os -10, no. 1, pp. \n161–175, 1939, doi: 10.1093/qmath/os -10.1.161.  \n[12] I. N. Sneddon, “Boussinesq’s problem for a rigid cone,” Math. Proc. Cambridge Philos. Soc. , \nvol. 44, no. 4, pp. 492–507, Oct. 1948, doi: 10.1017/S0305004100024518.  \n[13] L. Malekmotiei, A. Samadi -Dooki, and G. Z. Voyiadjis, “Nanoindentation study of yielding \nand plasticity of poly(methyl methacrylate),” Macromolecules , vol. 48, pp. 5348 –5357, 2015, \ndoi: 10.1021/acs.m acromol.5b01064.  \n[14] M. E. Cordova and Y. L. Shen, “Indentation versus uniaxial power -law creep: a numerical \nassessment,” J. Mater. Sci. , vol. 50, pp. 1394 –1400, 2015, doi: 10.1007/s10853 -014-8699 -9. \n[15] N. J. Martinez and Y. L. Shen, “Analysis of indent ation -derived power -law creep response,” \nJ. Mater. Eng. Perform. , vol. 25, pp. 1109 –1116, 2016, doi: 10.1007/s11665 -016-1934 -6. \n[16] P. S. Phani, W. C. Oliver, and G. M. Pharr, “On the Measurement of Power Law Creep \nParameters from Instrumented Indentation ,” JOM , vol. 69, pp. 2229 –2236, 2017, doi: \n10.1007/s11837 -017-2535 -z. \n[17] C. Su, E. G. Herbert, S. Sohn, J. A. Lamanna, W. C. Oliver, and G. M. Pharr, “Journal of the \nMechanics and Physics of Solids Measurement of power -law creep parameters by Page 52 of 84 \n instrumente d indentation methods,” J. Mech. Phys. Solids , vol. 61, no. 2, pp. 517 –536, 2013, \ndoi: 10.1016/j.jmps.2012.09.009.  \n[18] H. Takagi, M. Dao, and M. Fujiwara, “Prediction of the constitutive equation for uniaxial \ncreep of a power -law material through instrume nted microindentation testing and modeling,” \nMater. Trans. , vol. 55, pp. 275 –284, 2014, doi: 10.2320/matertrans.M2013370.  \n[19] C. L. Wang, Y. H. Lai, J. C. Huang, and T. G. Nieh, “Creep of nanocrystalline nickel: A direct \ncomparison between uniaxial and na noindentation creep,” Scr. Mater. , vol. 62, pp. 175 –178, \n2010, doi: 10.1016/j.scriptamat.2009.10.021.  \n[20] B. J. Briscoe, L. Fiori, and E. Pelillo, “Nano -indentation of polymeric surfaces,” J. Phys. D. \nAppl. Phys. , vol. 31, pp. 2395 –2405, 1998, doi: https: //doi.org/10.1088/0022 -3727/31/19/006.  \n[21] P. Baral et al. , “Indentation creep vs. indentation relaxation: A matter of strain rate \ndefinition?,” Mater. Sci. Eng. A , vol. 781, p. 139246, 2020, doi: 10.1016/j.msea.2020.139246.  \n[22] G. Kermouche, J. L. Loube t, and J. M. Bergheau, “Cone indentation of time -dependent \nmaterials: The effects of the indentation strain rate,” Mech. Mater. , vol. 39, pp. 24 –38, 2007, \ndoi: 10.1016/j.mechmat.2006.02.005.  \n[23] G. Kermouche, J. L. Loubet, and J. M. Bergheau, “Extraction of stress -strain curves of elastic -\nviscoplastic solids using conical/pyramidal indentation testing with application to polymers,” \nMech. Mater. , vol. 40, no. 4 –5, pp. 271 –283, 2008, doi: 10.1016/j.mechmat.2007.08.003.  \n[24] G. Kermouche, J. L. Loubet, and J.  M. Bergheau, “A new index to estimate the strain rate \nsensitivity of glassy polymers using conical/pyramidal indentation,” Philos. Mag. , vol. 86, pp. \n5667 –5677, 2006, doi: 10.1080/14786430600778682.  \n[25] C. G’sell and J. J. Jonas, “Determination of the pl astic behaviour of solid polymers at constant Page 53 of 84 \n true strain rate,” J. Mater. Sci. , vol. 14, pp. 583 –591, 1979, doi: 10.1007/BF00772717.  \n[26] R. Anastasio, E. E. L. Maassen, R. Cardinaels, G. W. M. Peters, and L. C. A. van Breemen, \n“Thin film mechanical chara cterization of UV -curing acrylate systems,” Polymer (Guildf). , \nvol. 150, pp. 84 –94, 2018, doi: 10.1016/j.polymer.2018.07.015.  \n[27] T. S. Guruprasad, S. Bhattacharya, and S. Basu, “Size effect in microcompression of \npolystyrene micropillars,” Polymer (Guild f)., vol. 98, pp. 118 –128, 2016, doi: \n10.1016/j.polymer.2016.06.010.  \n[28] G. Z. Voyiadjis, L. Malekmotiei, and A. Samadi -Dooki, “Indentation size effect in amorphous \npolymers based on shear transformation mediated plasticity,” Polymer (Guildf). , vol. 137, pp. \n72–81, 2018, doi: 10.1016/j.polymer.2018.01.006.  \n[29] D. Vgenopoulos et al. , “Nanoindentation analysis of oriented polypropylene: Influence of \nelastic properties in tension and compression,” Polymer (Guildf). , vol. 151, no. June, pp. 197 –\n207, 2018, doi : 10.1016/j.polymer.2018.07.080.  \n[30] H. Zhu, H. Ou, and A. Popov, “A new phenomenological constitutive model for \nthermoplastics,” Mech. Mater. , vol. 157, p. 103817, 2021, doi: \n10.1016/j.mechmat.2021.103817.  \n[31] P. Yu, X. Yao, Q. Han, S. Zang, and Y. Gu, “A visco -elastoplastic constitutive model for \nlarge deformation response of polycarbonate over a wide range of strain rates and \ntemperatures,” Polymer (Guildf). , vol. 55, pp. 6577 –6593, 2014, doi: \n10.1016/j.polymer.2014.09.071.  \n[32] W. Hu, H. Guo, Y. Chen,  R. Xie, H. Jing, and P. He, “Experimental investigation and \nmodeling of the rate -dependent deformation behavior of PMMA at different temperatures,” Page 54 of 84 \n Eur. Polym. J. , vol. 85, pp. 313 –323, 2016, doi: 10.1016/j.eurpolymj.2016.10.036.  \n[33] Z. El -Qoubaa, L. Col ard, R. Matadi Boumbimba, and A. Rusinek, “Experimental Study and \nModelling of Poly (Methyl Methacrylate) and Polycarbonate Compressive Behavior from \nLow to High Strain Rates,” J. Dyn. Behav. Mater. , vol. 4, no. 2, pp. 179 –189, 2018, doi: \n10.1007/s40870 -018-0147 -5. \n[34] N. M. Ames, V. Srivastava, S. A. Chester, and L. Anand, “A thermo -mechanically coupled \ntheory for large deformations of amorphous polymers. Part II: Applications,” Int. J. Plast. , \nvol. 25, no. 8, pp. 1495 –1539, 2009, doi: 10.1016/j.ijplas.20 08.11.005.  \n[35] E. M. Arruda, M. C. Boyce, and R. Jayachandran, “Effects of strain rate, temperature and \nthermomechanical coupling on the finite strain deformation of glassy polymers,” Mech. \nMater. , vol. 19, no. 2 –3, pp. 193 –212, 1995, doi: 10.1016/0167 -6636(94)00034 -E. \n[36] N. Aleksy, G. Kermouche, A. Vautrin, and J. M. Bergheau, “Numerical study of scratch \nvelocity effect on recovery of viscoelastic -viscoplastic solids,” Int. J. Mech. Sci. , vol. 52, pp. \n455–463, 2010, doi: 10.1016/j.ijmecsci.2009.11.006.  \n[37] Z. Chen and S. Diebels, “Nanoindentation of soft polymers: Modeling, experiments and \nparameter identification,” Tech. Mech. , vol. 34, pp. 166 –189, 2014, doi: 10.24352/UB.OVGU -\n2017 -060. \n[38] C. Liu, S. Lee, L. Sung, and T. Nguyen, “Load -displacement re lations for nanoindentation of \nviscoelastic materials,” J. Appl. Phys. , vol. 100, p. 033503, 2006, doi: 10.1063/1.2220649.  \n[39] J. Menčík and L. Beneš, “Determination of viscoelastic properties by nanoindentation,” J. \nOptoelectron. Adv. Mater. , vol. 10, pp. 3288 –3291, 2008.  \n[40] G. Peng, Y. Ma, Y. Feng, Y. Huan, and C. Qin, “Nanoindentation creep of nonlinear Page 55 of 84 \n viscoelastic poly propylene,” Polym. Test. J. , vol. 43, pp. 38 –43, 2015, doi: \n10.1016/j.polymertesting.2015.02.006.  \n[41] G. Peng, T. Zhang, Y. Feng, and Y. Huan, “Determination of shear creep compliance of linear \nviscoelastic -plastic solids by instrumented indentation,” Polym. Test. , vol. 31, pp. 1038 –1044, \n2012, doi: 10.1016/j.polymertesting.2012.07.007.  \n[42] C. A. Tweedie and K. J. Van Vliet, “Contact creep compliance of viscoelastic materials via \nnanoindentation,” J. Mater. Res. , vol. 21, pp. 1576 –1589, 2006, doi: 10.1557 /jmr.2006.0197.  \n[43] J. D. Gelorme, R. J. Cox, and S. A. R. Gutierrez, “Photoresist composition and printed circuit \nboards and packages made therewith,” 1989 [Online]. Available: \nhttps://patents.google.com/patent/US4882245A/en%0Ahttp://www.google.com/paten ts?hl=en\n&lr=&vid=USPAT4882245&id=8943AAAAEBAJ&oi=fnd&dq=Photoresist+composition+an\nd+printed+circuit+boards+and+packages+made+therewith&printsec=abstract  \n[44] J. M. Shaw, J. D. Gelorme, N. C. LaBianca, W. E. Conley, and S. J. Holmes, “Negative \nphotoresists for optical lithography,” IBM J. Res. Dev. , vol. 41, pp. 81 –94, 1997, doi: \n10.1147/rd.411.0081.  \n[45] S. Chattaraj, P. Pant, D. N. Pawaskar, and H. Nanavati, “How many network chains of a \ndensely crosslinked glassy thermoset deform cooperatively at yield?,”  Polymer (Guildf). , vol. \n82, pp. 305 –318, 2016, doi: 10.1016/j.polymer.2015.11.023.  \n[46] H. Eyring, “Viscosity, plasticity, and diffusion as examples of absolute reaction rates,” J. \nChem. Phys. , vol. 4, pp. 283 –291, 1936, doi: 10.1063/1.1749836.  \n[47] W. D.  Cook, A. E. Mayr, and G. H. Edward, “Yielding behaviour in model epoxy thermosets - \nII. Temperature dependence,” Polymer (Guildf). , vol. 39, pp. 3725 –3733, 1998, doi: Page 56 of 84 \n 10.1016/S0032 -3861(97)10335 -4. \n[48] N. A. Stilwell and D. Tabor, “Elastic recovery of co nical indentations,” Proc. Phys. Soc. , vol. \n78, pp. 169 –179, 1961, doi: 10.1088/0370 -1328/78/2/302.  \n[49] M. Dao, N. Chollacoop, K. J. van Vliet, T. A. Venkatesh, and S. Suresh, “Computational \nmodeling of the forward and reverse problems in instrumented sha rp indentation,” Acta \nMater. , vol. 49, pp. 3899 –3918, 2001, doi: https://doi.org/10.1016/S1359 -6454(01)00295 -6. \n[50] D. Tabor, “A simple theory of static and dynamic hardness,” Proc. R. Soc. London. Ser. A. \nMath. Phys. Sci. , vol. 192, pp. 247 –274, 1948, do i: 10.1098/rspa.1948.0008.  \n[51] C. A. Tweedie and K. J. van Vliet, “On the indentation recovery and fleeting hardness of \npolymers,” J. Mater. Res. , vol. 21, pp. 3029 –3036, 2006, doi: 10.1557/jmr.2006.0377.  \n[52] “Literature considers hc as that obtained via  the Oliver -Pharr (OP) method. However, here, we \nconsider it to be that from SPM imaging.”.  \n[53] M. Mata and J. Alcalá, “The role of friction on sharp indentation,” J. Mech. Phys. Solids , vol. \n52, pp. 145 –165, 2004, doi: 10.1016/S0022 -5096(03)00075 -9. \n[54] C. M. Sanchez -Camargo, A. Hor, and C. Mabru, “A robust inverse analysis method for \nelastoplastic behavior identification using the true geometry modeling of Berkovich indenter,” \nInt. J. Mech. Sci. , vol. 171, p. 105370, 2020, doi: 10.1016/j.ijmecsci.2019.1 05370.  \n[55] M. Munawar Chaudhri, “The love equation for the normal loading of a rigid cone on an elastic \nhalf-space and a recent modification: A review,” Mater. Trans. , vol. 60, no. 8, pp. 1404 –1410, \n2019, doi: 10.2320/matertrans.MD201908.  \n[56] M. M. Chaud hri, “The Love equation for the normal loading of a rigid cone on an elastic half -Page 57 of 84 \n space and a recent modification: A review,” Mater. Trans. , vol. 60, pp. 1404 –1410, 2019, doi: \n10.2320/matertrans.MD201908.  \n[57] F. Bédoui, F. Sansoz, and N. S. Murthy, “Incid ence of nanoscale heterogeneity on the \nnanoindentation of a semicrystalline polymer: Experiments and modeling,” Acta Mater. , vol. \n56, no. 10, pp. 2296 –2306, 2008, doi: 10.1016/j.actamat.2008.01.021.  \n[58] H. Li, J. Chen, Q. Chen, and M. Liu, “Determining th e constitutive behavior of nonlinear \nvisco -elastic -plastic PMMA thin films using nanoindentation and finite element simulation,” \nMater. Des. , vol. 197, p. 109239, 2021, doi: 10.1016/j.matdes.2020.109239.  Page 58 of 84 \n Supp orting Information 1  \nUniversal curves for residual uniaxial strain  \nThe non -elastic residual strains in uniaxial compression, \nne , are obtained for the entire strain -range, \nby drawing a Hookean line of slope \n()E\n , from the correspondi ng point on the stress -strain plot, as \nshown in Fig.  1(a). We define, \n()()*\ny     =\n  and then determine \nne  from each \n-  curve, \nwhich are then normalized to \n()()*\nne y     =\n . The \n*\nne  vs \n*  computations are plotted in \nFig. S.1.1 for cross -linked SU-8, in Fig.  S.1.2 for PC and in Fig.  S.1.3-5 for PMMA. We find an \napproximately universal corresponding strain state ( CSS) behaviour, across the stra in rates examined \nhere.  \n1. Universal curve for cross -linked SU-8  \nWe have fit a polynomial equation (eqn.  S.1.1) for the data range of 0.5  < \n* < 1.5. \n()()()()()6 5 4 3 2* * * * * *\nne 6 5 4 3 2 0.5 0.5 0.5 0.5 0.5 k k k k k     = − + − + − + − + −\n  eqn. S.1.1 \nHence,  \n()()()()()*5 4 3 2* * * * * ne\n6 5 4 3 2 *6 0.5 5 0.5 4 0.5 3 0.5 2 0.5dk k k k k\nd    \n= − + − + − + − + −\n eqn. S.1.2 \nAt \n*  ~ 1.5, \n**\nnedd  ~ 1. Therefore, from eqn.  S.1.2, Page 59 of 84 \n \n \nFigure S.1.1: CSS \n*\nne vs \n* for cross -linked SU-8 over  all strain rates.  \n()()()()()5 4 3 2* * * * *\n6 5 4 3 26 0.5 5 0.5 4 0.5 3 0.5 2 0.5 1k k k k k    − + − + − + − + − =\n. Hence,  \n( ) 2 6 5 4 30.5 3 2.5 2 1.5k k k k k= − + + +\n        eqn. S.1.3 \nIncorporating the eqn.  S.1.3 into eqn.  S.1.1,  \n()()()()\n( ) ( )()6 5 4 3* * * * *\nne 6 5 4 3\n2*\n6 5 4 30.5 0.5 0.5 0.5\n        0.5 3 2.5 2 1.5 0.5k k k k\nk k k k    \n= − + − + − + −\n+ − + + + −\n    eqn. S.1.4 \nThe resultant fitting constant values are: \n6k  = 8.57, \n5k  = -25.58, \n4k  = 26.21, \n3k  = -10.67 and from \neqn. S.1.4, \n2k  = 2.32. Therefore, from eqn.  S.1.1, \n()()()()\n()6 5 4 3* * * * *\nne\n2*8.57 0.5 25.58 0.5 26.21 0.5 10.67 0.5\n        2.32 0.5    \n= − − − + − − −\n+−\n  eqn. S.1.5 \nWe have fit a linear equation for the data range of \n*  > 1.5, \n**\nne L Lmk=+ , \n**\nL ne Lkm = − . \nAt \n*  = 1.5 in eqn.  S.1.5, we obtain \n*\nne  = 0.85 and \nLm  = 1. Then, \nLk  = 0.85-1.5 = -0.65. Thus,  Page 60 of 84 \n \n**\nne 0.65=−          eqn. S.1.6 \n2. Universal curve for PC  \nIn the data range of 0.3  < \n* < 1.2, \n()()()4 3 2* * * *\nne 4 3 2 0.3 0.3 0.3 k k k   = − + − + −\n      eqn. S.1.7 \nHence,  \n()()()*32* * * ne\n4 3 2 *4 0.3 3 0.3 2 0.3dk k k\nd  \n= − + − + −\n     eqn. S.1.8 \n \nFigure S.1.2: CSS \n*\nne vs \n*  for PC  over all strain rate s. \nAt \n*  ~ 1.2, \n**\nnedd  ~ 1.10. Therefore, from eqn.  S.1.8,  \n()()()32\n4 3 24 1.2 0.3 3 1.2 0.3 2 1.2 0.3 1.10k k k− + − + − =\n. Therefore,  \n( ) 2 4 30.61 1.62 1.35k k k= − +\n         eqn. S.1.9 \nCombining eqn.  S.1.7 and eqn.  S.1.9, Page 61 of 84 \n \n()()( ) ( )()4 3 2* * * *\nne 4 3 4 3 0.3 0.3 0.61 1.62 1.35 0.3 k k k k   = − + − + − + −   eqn. S.1.10 \nWhen, \n*  > 1.2,  \n**\nne1.10 0.77 0.55= − =\n         eqn. S.1.11 \nThus, eqn.  S.1.10 becomes,  \n()()( ) ( )()4 3 2* * *\n4 3 4 3 0.3 0.3 0.61 1.62 1.35 0.3 0.55 k k k k  − + − + − + − =\n \n()()( ) ( )()4 3 2\n4 3 4 31.2 0.3 1.2 0.3 0.61 1.62 1.35 1.2 0.3 0.55k k k k − + − + − + − =\n. Hence,  \n34 1.8 0.16 kk=− −\n          eqn. S.1.12 \nFrom eqn.  S.1.9 and eqn.  S.1.12, \n( ) ( ) 2 4 40.61 1.62 1.35 1.8 0.16k k k= − + − −\n, \n240.83 0.81 kk = +  \nIncorporat ing the above values in eqn.  S.1.10,  \n()( )()( )()4 3 2* * * *\nne 4 4 4 0.3 1.8 0.16 0.3 0.83 0.81 0.3 k k k   = − + − − − + + −\n  eqn. S.1.13 \nFrom fitting, \n4k  = -0.49. Thus, \n3k  = 0.72, \n2k  = 0.44. Therefore,  \n()()()432* * * *\nne 0.49 0.3 0.72 0.3 0.44 0.3   =− − + − + −\n     eqn. S.1.14 \n3.A. Universal curve for PMMA [28]  \nIn the data range of 0.4  < \n* < 1.2, Page 62 of 84 \n \n()()()4 3 2* * * *\nne 4 3 2 0.4 0.4 0.4 k k k   = − + − + −      eqn. S.1.15 \nHence,  \n()()()*32* * * ne\n4 3 2 *4 0.4 3 0.4 2 0.4dk k k\nd  \n= − + − + −\n     eqn. S.1.16 \n \nFigure S.1 .3: CSS \n*\nne vs \n*  for P MMA [28]  over all strain rates.  \nAt \n*  ~ 1.2, \n**\nnedd  ~ 1.11. Therefore, from eqn.  S.1.16,  \n()()()32\n4 3 24 1.2 0.4 3 1.2 0.4 2 1.2 0.4 1.11k k k− + − + − =\n. Therefore,  \n4 3 2 2.05 1.92 1.6 1.11k k k+ + =\n \n( ) 2 4 30.695 1.28 1.2k k k= − +\n         eqn. S.1.17 \nCombining eqn.  S.1.15 and eqn. S.1.17, \n()() ( ) ( )()4 3 2* * * *\nne 4 3 4 3 0.4 0.4 0.695 1.28 1.2 0.4 k k k k   = − + − + − + −\n   eqn. S.1.18 Page 63 of 84 \n The re sultant fitting constant values are: \n4k  = -0.407 , \n3k = 0.337  and from eqn.  S.1.17, \n2k = 0.812 . \nTherefore, from eqn.  S.1.15, \n()()()4 3 2* * * *\nne 0.407 0.4 0.337 0.4 0.812 0.4   =− − + − + −\n    eqn. S.1.19 \nWe have fit a line ar equation for the data range of \n*  > 1.2, \n**\nne L Lmk=+ , \n**\nL ne Lkm = − . \nAt \n*  = 1.2 in eqn.  S.1.19, we obtain \n*\nne  = 0.53 and \nLm  = 1.11. Then, \nLk  = 0.53-1.33 = -0.804 . \nThus,  \n**\nne1.11 0.804=−\n          eqn. S.1.20 \n3.B. Universal curve for PMMA [29]  \nIn the data range of 0.3  < \n* < 1.2, \n()()()4 3 2* * * *\nne 4 3 2 0.3 0.3 0.3 k k k   = − + − + −\n      eqn. S.1.21 \nHenc e, \n()()()*32* * * ne\n4 3 2 *4 0.3 3 0.3 2 0.3dk k k\nd  \n= − + − + −\n     eqn. S.1.22 Page 64 of 84 \n \n \nFigure S.1.4: CSS \n*\nne vs \n*  for P MMA [29]  over all strain rates.  \nAt \n*  ~ 1.2, \n**\nnedd  ~ 1.02. Therefore, from eqn.  S.1.22,  \n()()()32\n4 3 24 1.2 0.3 3 1.2 0.3 2 1.2 0.3 1.02k k k− + − + − =\n. Therefore,  \n4 3 2 2.92 2.43 1.8 1.02k k k+ + =\n \n( ) 2 4 30.565 1.62 1.35k k k= − +\n        eqn. S.1.23 \nCombining eqn.  S.1.21 and eqn. S.1.23, \n()() ( ) ( )()4 3 2* * * *\nne 4 3 4 3 0.3 0.3 0.565 1.62 1.35 0.3 k k k k   = − + − + − + −\n  eqn. S.1.24 \nThe resultant fitting constant values are: \n4k  = 0.996 , \n3k = -2.50 and from eqn.  S.1.23, \n2k = 2.33. \nTherefore, from eqn.  S.1.21, \n()()()4 3 2* * * *\nne0.996 0.3 2.50 0.3 2.33 0.3   = − − − + −\n     eqn. S.1.25 \nWe have fit a linear equation for the data range of \n*  > 1.2, \n**\nne L Lmk=+ , \n**\nL ne Lkm = − . Page 65 of 84 \n At \n*  = 1.2 in eqn.  S.1.25, we obtain \n*\nne  = 0.716  and \nLm  = 1.02. Then, \nLk  = 0.716 -1.22 = -0.504. \nThus,  \n**\nne1.02 0.504=−\n          eqn. S.1.26 \n3.C. Universal curve for PMMA [30]  \nIn the data range of 0.3  < \n* < 1.2, \n()()()4 3 2* * * *\nne 4 3 2 0.3 0.3 0.3 k k k   = − + − + −\n      eqn. S.1.27 \nHence,  \n()()()*32* * * ne\n4 3 2 *4 0.3 3 0.3 2 0.3dk k k\nd  \n= − + − + −\n     eqn. S.1.28 \n \nFigure S.1.5 : CSS \n*\nne vs \n*  for P MMA [30] over all strain rates.  \nAt \n*  ~ 1.2, \n**\nnedd  ~ 1.06. Therefore, from eqn.  S.1.28,  \n()()()32\n4 3 24 1.2 0.3 3 1.2 0.3 2 1.2 0.3 1.06k k k− + − + − =\n. Therefore,  Page 66 of 84 \n \n4 3 2 2.92 2.43 1.8 1.06k k k+ + = \n( ) 2 4 30.587 1.62 1.35k k k= − +\n        eqn. S.1.29 \nCombining eqn.  S.1.27 and eqn.  S.1.29, \n()() ( ) ( )()4 3 2* * * *\nne 4 3 4 3 0.3 0.3 0.587 1.62 1.35 0.3 k k k k   = − + − + − + −\n  eqn. S.1.30 \nThe resultant fitting constant values are: \n4k  = 1.08, \n3k = -2.51 and from eqn.  S.1.29, \n2k = 2.22. \nTherefore, from eqn.  S.1.27, \n()()()4 3 2* * * *\nne1.08 0.3 2.51 0.3 2.22 0.3   = − − − + −\n     eqn. S.1.31 \nWe have fit a linear equation for the data range of \n*  > 1.2, \n**\nne L Lmk=+ , \n**\nL ne Lkm = − . \nAt \n*  = 1.2 in eqn.  S.1.31, we obtain \n*\nne  = 0.679 and \nLm  = 1.06. Then, \nLk  = 0.679-1.27 = -0.589. \nThus,  \n**\nne1.06 0.589=−\n          eqn. S.1.32 Page 67 of 84 \n Supporting Information  2 \nS.2.1. Load -displacement curves  \nThe \n-Ph  curves for cross -linked SU -8, PC and PMMA  are shown in Fig.  S.2.1(a)-(b), Fig.  S.2.2(a) -\n(b) and Fig.  S.2.3(a)-(b), respectively,  for two levels of maximum load,  \nmP , i.e., 1000  µN and \n9000  µN. \n  \nFigure  S.2.1: \n-Ph  curves for SU -8 for \nmP  = 1000  µN and 9000  µN and  \nPP\n  = (a) 0.1 /s, \n(b) 1.0 /s. \n  \nFigure  S.2.2: \n-Ph  curves  for PC for \nmP  = 1000  µN and 9000  µN and \nPP\n  = (a) 0.1 /s, \n(b) 1.0 /s. \n(b) \n(a) \n(a) \n(b) Page 68 of 84 \n \n \nFigure  S.2.3: \n-Ph  curves for PMMA for \nmP  = 1000  µN and 9000  µN and  \nPP\n  = 0.1 /s. Page 69 of 84 \n Supporting Information  3 \nUnloading rate variation during unloading  \nThe unloading step duration is fixed at 2  s. During unloading from Pm, the indenter tip begins from \nrest, accelerates to reach the set rate at 0.95\nmP , unloads at this rate up to 0.3\nmP , and then decelerates \nto zero rate at zero load. This is true for  all materials (including all the polymers examined here, and \nfor quartz), and for both maximum loads, \nmP  = 1000  µN and \nmP  = 9000  µN. The scaled unloading \nrate is plotted vs the sc aled instantaneous load in  Fig. S.4.1. \n \nFigure S. 3.1: Scaled unloading rate, \nmPP\n  vs scaled load, \nmPP , for cross -linked SU-8 for \nmP\n = 1000  µN and \nmP = 9000 µN. The acceleration region and deceleration region, \nare indication by their boundaries at 0.95\nmP  and 0.3\nmP , respectively.  Page 70 of 84 \n Supp orting Information 4  \nThe generalized power law (GPL) fitting has been described in Section 5. 2 I. The corr esponding \nfitted values are listed in Table  S.4.1. \nTable  S.4.1: GPL fit ted constants and their ranges . \nSample  \nPP\n  (/s) \nmP  (µN)  \n0k  \n1k  \n2k  \n1m (Value or range)  \nGPL  PL \nCross -\nlinked \nSU-8 0.1 1000  7.94 ± 0.76 -2.20 ± 0.23 0.16 ± 0.02 0.48 ± 0.00 \nto 0.62  ± 0.01 0.33 \n9000  9.55 ± 0.65 -1.96 ± 0.15 0.11 ± 0.01 0.40 ± 0.00 \nto 0.56  ± 0.01 0.34 \n1.0 1000  9.36 ± 0.80 -2.59 ± 0.24 0.19 ± 0.02 0.44 ± 0.01 \nto 0.62  ± 0.02 0.34 \n9000  5.89 ± 0.24 -1.20 ± 0.06 0.06 ± 0.00 0.32 ± 0.00 \nto 0.41  ± 0.00 0.36 \nPC 0.1 1000  5.12 ± 0.74 -1.28 ± 0.22 0.09 ± 0.02 0.46 ± 0.02 \nto 0.62  ± 0.03 0.35 \n9000  8.74 ± 0.67 -1.80 ± 0.14 0.10 ± 0.01 0.45 ± 0.01 \nto 0.59  ± 0.02 0.36 \n1.0 1000  9.93 ± 2.86 -2.78 ± 0.81 0.21 ± 0.06 0.49 ± 0.05 \nto 0.65  ± 0.11 0.38 \n9000  9.52 ± 2.25 -2.03 ± 0.48 0.11 ± 0.03 0.41 ± 0.04 \nto 0.50  ± 0.07 0.43 \nPMMA  0.1 1000  1.51 ± 0.01 -0.13 ± 0.01 -- 0.65 ± 0.00 \nto 0.79  ± 0.01 0.52 \n9000  0.89 ± 0.01 -0.03 ± 0.01 -- 0.57 ± 0.00 \nto 0.60  ± 0.00 0.54 Page 71 of 84 \n Supporting Information 5 \nS.5.1. Scanning probe microscopy images  \nThe SPM top view image are presented   \non cross -linked SU -8  \nat \nPP\n  = 0.1 /s for  \nmP = 1000  µN (Fig.  S.5.1(a)), for \nmP  = 9000  µN (Fig.  S.5.1(b)),   \nat \nPP\n  = 0.1 /s for  \nmP = 1000  µN (Fig.  S.5.2 (a)), for \nmP  = 9000  µN (Fig.  S.5.2 (b)),  \non PC   \nat \nPP\n  = 0.1 /s for  \nmP = 1000  µN (Fig.  S.5.3 (a)), for \nmP  = 9000  µN (Fig.  S.5.3 (b)) and   \nat \nPP\n  = 1.0 /s for  \nmP = 1000  µN ( Fig. S.5.4 (a)), for \nmP  = 9000  µN (Fig.  S.5.4 (b)),   \nand on PMMA    \nat \nPP\n  = 0.1 /s for  \nmP = 1000  µN (Fig.  S.5.5 (a)), for \nmP  = 9000  µN (Fig.  S.5.5 (b)). \n  \nFigure  S.5.1 : Top view of residual imprint for cross -linked SU-8 when \nPP\n  = 0.1 /s and \nmP\n = (a) 1000  µN, (b)  9000  µN.  \n(a) \n (b) Page 72 of 84 \n   \nFigure  S.5.2 : Top view of the residual imprint s for cross -linked SU -8 for \nPP\n  = 1.0 /s and \nmP\n = (a) 1000  µN, (b)  9000  µN.  \n  \nFigure  S.5.3 : Top view of the residual imprint s for PC for \nPP\n  = 0.1 /s and \nmP  = (a) 1000  µN, \n(b) 9000  µN. \n  \nFigure  S.5.4 : Top view of the residual imprint s for PC fo r \nPP\n  = 1.0 /s and \nmP  = (a) 1000  µN, \n(b) 9000  µN.  \n(a) \n(b) \n(a) \n (b) \n(a) \n (b) Page 73 of 84 \n   \nFigure  S.5.5 : Top view of the residual imprints for PMMA for \nPP\n  = 0.1 /s and  \nmP\n = (a) 1000  µN, (b)  9000  µN. \n(a) \n (b) Page 74 of 84 \n Supp orting Information 6 \nIndenter tip blunt height ( hb) estimate  \nThe \n-Ph  curves of the tip calibration on quartz standard, are employed to evaluate \nbh  at different \nusage days, for a given tip . For each \nmP , the known \nrE  = 69.6 ± 3.5 GPa, and \nH  = 9.3 ± 0.9 GPa, \nyield two \ncA  values, \n224ErcrA S E=  and \nmH\ncA P H= , respectively. We could choose either, and \nwe have considered the average.  \nThe Berkovich tip contact area , \n()()224.5c c bA h h h  + . Thus, for any calibration day, we obtain \nan estimate for hb. For any experiment day, hb varies linearly with the age of the tip; \nbh pt q=+ , \nwhere t is the duration  of tip usage, measured in days  (Fig.  S.6.1). \n \nFigure S. 6.1: Estimating tip blunt height, \nbh , as a function of days of tip usage.  Page 75 of 84 \n Supp orting Information 7  \nTrue strain  correlat ion with nanoindentation data  \nWe know that, \nt exp 1=−  and \nttexp  =\n . The premise of this work is:  \nFractional residual uniaxial strain = fractional residual nanoindentation strain, \nt,ne N,ne\ntN\n= .  \nNanoindentation residual strain  = \nt,e ne\nN,ne\nelnh\nhc=+  \nne\nt,ne t,e\nelnhch = +\n, \nne\nt,ne t,e\nelnhch  = − , \n()()ne\nne e\neln ln 1 ln 1hch  = + − +  \n()()** ne\nne e\neln ln 1 ln 1yyhch     = + − +\n, \n*\nne ne\n*e e1\n1c\ny\nyh\nh\n+ =+  . Therefore,  \n()*\nne ne\n**e ne1\n1c\ny\nyh\nh\n  + =+− \n         eqn. S.7.1 \n1. Cross-linked SU-8 true strain correlations  \nLinear fit of \ny  with \nln\n  yields \ny  = 9.46 × 10-4\nln\n + 5.23 × 10-2 = \nlnpq+\n  = \n() ttlnpq++\n . \nFrom eqn.  S.1.5, \n() ( )()()()\n()()6 5 4* * *\nne t t32**8.57 0.5 25.58 0.5 26.21 0.5\nln\n10.67 0.5 2.32 0.5pq  \n  \n− − − + −= + + \n− − + −\n Page 76 of 84 \n \n() ( )()()()\n()()6 5 4* * *\nt,ne t t32**8.57 0.5 25.58 0.5 26.21 0.5\nln 1 ln\n10.67 0.5 2.32 0.5pq  \n  \n − − − + −  = + + + \n − − + −  \n \nCombining the above with eqn.  S.7.1,  \n() ( )()()\n()()\n()\n() ( )()()\n()()\n()65**\n43**\ntt\n2*\nne\n65** e\n43* * *\ntt\n2*8.57 0.5 25.58 0.5\n1 ln 26.21 0.5 10.67 0.5\n2.32 0.5\n8.57 0.5 25.58 0.5\n1 ln 26.21 0.5 10.67 0.5\n2.32 0.5cpq\nh\nh\npq\n   \n\n\n    \n− − −\n\n + + + + − − −\n+− = − − −\n\n + + + − + − − −\n+−\n\n\n\n\n\n\n\n\n  eqn. S.7.2 \nFrom  eqn. S.1.6, \n()( )()\n()( )()*\nttne\n**ett1 ln 0.65\n1 ln 0.65cpqh\nh pq  \n   + + + − =  + + + − −\n . This yields,  \n() ( )()\n() ( )*\nttne\ne tt1 ln 0.65\n1 0.65 lncpqh\nh pq  \n+ + + − =+ + + \n       eqn. S.7.3 \n2. PC true strain correlations  \nLinear fit of \ny  with \nln\n  yields \ny  = 4.81 × 10-4\nln\n + 9.08 × 10-2 = \nlnpq+\n  = \n() ttlnpq++\n . \nFrom eqn.  S.1.14, \n()( )()()()432* * *\nne t t ln 0.49 0.3 0.72 0.3 0.44 0.3pq     = + + − − + − + −\n Page 77 of 84 \n \n()( )()()()432* * *\nne t tln 1 ln 0.49 0.3 0.72 0.3 0.44 0.3 pq        = + + + − − + − + −   \n \nBy incorporating the above into eqn.  S.7.3 , \n() ( )()()\n()\n() ( )()()\n()43**\ntt2*\nne\n43** e\n*\ntt2*0.49 0.3 0.72 0.3\n1 ln\n0.44 0.3\n0.49 0.3 0.72 0.3\n1 ln\n0.44 0.3cpq\nh\nh\npq\n\n\n\n  \n− − + −+ + + \n+− = − − + −+ + + − \n+−\n  eqn. S.7.4 \nFrom  eqn. S.1.11, \n() ( )()\n() ( )()*\nttne\n**ett1 ln 1.10 0.77\n1 ln 1.10 0.77cpqh\nh pq  \n   + + + − =  + + + − −\n . Hence,  \n()( )()\n()( )( )*\nttne\n*e tt1 ln 1.10 0.77\n1 ln 0.77 0.10cpqh\nh pq  \n  + + + − =+ + + − \n      eqn. S.7.5 \n3.A. PMMA [28]  true strain correlations  \nLinear fit of \ny  with \nln\n  yields \ny  = 4.18 × 10-3\nln\n + 0.1116  = \nlnpq+\n  = \n() ttlnpq++\n . \nFrom eqn.  S.1.19, \n()( )()()()4 3 2* * *\nne t t ln 0.407 0.4 0.337 0.4 0.812 0.4pq     = + + − − + − + −\n \n()( )()()()4 3 2* * *\nne t tln 1 ln 0.407 0.4 0.337 0.4 0.812 0.4 pq        = + + + − − + − + −   \n \nBy incorporating the above into eqn.  S.7.1, Page 78 of 84 \n \n() ( )()()\n()\n() ( )()()\n()43**\ntt2*\nne\n43** e\n*\ntt2*0.407 0.4 0.337 0.4\n1 ln\n0.812 0.4\n0.407 0.4 0.337 0.4\n1 ln\n0.812 0.4cpq\nh\nh\npq\n\n\n\n  \n− − + −+ + + \n+− = − − + −+ + + − \n+−\n  eqn. S.7.6 \nFrom  eqn. S.1.20, \n() ( )( )\n() ( )( )*\nttne\n**ett1 ln 1.11 0.804\n1 ln 1.11 0.804cpqh\nh pq  \n   + + + − =  + + + − −\n . Hence,  \n()( )( )\n()( )( )*\nttne\n*e tt1 ln 1.11 0.804\n1 ln 0.804 0.11cpqh\nh pq  \n  + + + − =+ + + − \n      eqn. S.7.7 \n3.B. PMMA [29] true strain correlations  \nLinear fit of \ny  with \nln\n  yields \ny  = -2.52 × 10-3\nln\n + 0.074  = \nlnpq+\n  = \n() ttlnpq++\n . \nFrom eqn.  S.1.25, \n()( )()()()4 3 2* * *\nne t t ln 0.996 0.3 2.50 0.3 2.33 0.3pq     = + + − − − + −\n \n()( )()()()4 3 2* * *\nne t tln 1 ln 0.996 0.3 2.50 0.3 2.33 0.3 pq        = + + + − − − + −   \n \nBy incorporating the above into eqn.  S.7.1, Page 79 of 84 \n \n() ( )()()\n()\n() ( )()()\n()43**\ntt2*\nne\n43** e\n*\ntt2*0.996 0.3 2.50 0.3\n1 ln\n2.33 0.3\n0.996 0.3 2.50 0.3\n1 ln\n2.33 0.3cpq\nh\nh\npq\n\n\n\n  \n− − −+ + + \n+− = − − −+ + + − \n+−\n  eqn. S.7.8 \nFrom  eqn. S.1.26, \n() ( )( )\n() ( )( )*\nttne\n**ett1 ln 1.02 0.504\n1 ln 1.02 0.504cpqh\nh pq  \n   + + + − =  + + + − −\n . Hence,  \n()( )( )\n()( )( )*\nttne\n*e tt1 ln 1.02 0.504\n1 ln 0.504 0.02cpqh\nh pq  \n  + + + − =+ + + − \n      eqn. S.7.9 \n3.C. PMMA [30 ] true strain correlations  \nLinear fit of \ny  with \nln\n  yields \ny  = 4.86 × 10-4\nln\n + 8.56 × 10-2 = \nlnpq+\n  = \n() ttlnpq++\n . \nFrom eqn.  S.1.31, \n()( )()()()4 3 2* * *\nne t t ln 1.08 0.3 2.51 0.3 2.22 0.3pq     = + + − − − + −\n \n()( )()()()4 3 2* * *\nne,t t t ln 1 ln 1.08 0.3 2.51 0.3 2.22 0.3 pq        = + + + − − − + −   \n \nBy incorporating the above into eqn.  S.7.1, Page 80 of 84 \n \n() ( )()()\n()\n() ( )()()\n()43**\ntt2*\nne\n43** e\n*\ntt2*1.08 0.3 2.51 0.3\n1 ln\n2.22 0.3\n1.08 0.3 2.51 0.3\n1 ln\n2.22 0.3cpq\nh\nh\npq\n\n\n\n  \n− − −+ + + \n+− = − − −+ + + − \n+−\n   eqn. S.7.10 \nFrom  eqn. S.1.32, \n() ( )( )\n() ( )( )*\nttne\n**ett1 ln 1.06 0.589\n1 ln 1.06 0.589cpqh\nh pq  \n   + + + − =  + + + − −\n . Hence,  \n()( )( )\n()( )( )*\nttne\n*e tt1 ln 1.06 0.589\n1 ln 0.589 0.06cpqh\nh pq  \n  + + + − =+ + + − \n      eqn. S.7.11 Page 81 of 84 \n Supp orting Information 8  \nWe find that the uncertainty in the linear relationship between \ny  vs \nln\n , correspond to an \nuncertainty of ~  2.5% in the \nc  values of cross -linked  SU-8. The \n\n, \n and \nc  values , are listed in \nTable  S.8.1 and in Table  S.8.2 for minimum and maximum value s respectively,  of the \ny  vs \nln\n  \nslope. \nTable  S.8.1: \n\n, \n and \nc  values  when \nmin,SU 8p−  = 2.39 × 10-4, \nmin,SU 8q−  = 4.73 × 10-2 for cross -\nlinked  SU-8. \nPP\n (/s) \nmP  (µN)  \n\n \n 102 (/s) \n  × 102 \nc \n0.1 1000  0.23 ± 0.00  6.24 ± 0.09  0.045  ± 0.000  \n9000  0.17 ± 0.00  6.74 ± 0.08  0.047  ± 0.001  \n1.0 1000  2.70 ± 0.25  6.04 ± 0.00  0.045  ± 0.000  \n9000  1.90 ± 0.00  5.78 ± 0.03  0.043  ± 0.000  \nTable  S.8.2: \n\n, \n and \nc  values when \nmax,SU 8p−  = 2.30 × 10-3, \nmax,SU 8q−  = 6.05 × 10-2 for cross -\nlinked  SU-8. \nPP\n (/s) \nmP  (µN)  \n\n \n 102 (/s) \n  × 102 \nc  \n0.1 1000  0.21 ± 0.00  6.03 ± 0.09  0.042  ± 0.000  \n9000  0.16 ± 0.00  6.37 ± 0.09  0.042  ± 0.001  \n1.0 1000  2.79 ± 0.27  6.01 ± 0.00  0.046  ± 0.000  \n9000  1.94 ± 0.03  6.18 ± 0.03  0.043  ± 0.000  Page 82 of 84 \n Supporting Information 9 \nTable  S.9.1, Table  S.9.2 and Table  S.9.3 list the \n\n , \n and \nc  values for 0.25, 0.50 and 0.75 \nmrr  \nposition s, respectively, for the all polymers . \nTable  S.9.1: \n\n, \n and \nc  values  for \nmrr  = 0.25 position  for cross -linked SU -8, PC and PMMA . \n \nPP\n  (/s) \nmP  (µN)  \n\n × 102 (/s) \n  × 102 \nc  \nCross -linked SU-8 0.1 1000  0.43 ± 0.00  6.23 ± 0.0 7 0.045 ± 0.000  \n9000  0.41 ± 0.00  6.49 ± 0.09  0.043  ± 0.000  \n1.0 1000  5.00 ± 0.18  5.98 ± 0.00  0.041  ± 0.000  \n9000  3.75 ± 0.25  5.72 ± 0.00  0.035  ± 0.000  \nPC 0.1 1000  0.97 ± 0.0 2 12.17  ± 0.29  0.077  ± 0.002  \n9000  0.65 ± 0.01  12.38  ± 0.34  0.073  ± 0.002  \n1.0 1000  6.81 ± 0.26  11.60  ± 0.37 0.068  ± 0.003  \n9000  6.13 ± 0.28  11.09  ± 0.59  0.064  ± 0.003  \nPMMA [30] 0.1 1000  1.08 ± 0.01  12.47  ± 0.22 0.084  ± 0.00 1 \n9000  0.39 ± 0.00  12.58  ± 0.28 0.079  ± 0.00 1 \nTable  S.9.2: \n\n, \n and \nc  values for  \nmrr  = 0.50 position for cross -linked SU -8, PC and PMMA . \n \nPP\n  (/s) \nmP  (µN)  \n\n × 102 (/s) \n  × 102 \nc  \nCross -linked SU-8 0.1 1000  0.72 ± 0.00  6.21 ± 0.07  0.042  ± 0.000  \n9000  0.62 ± 0.00  6.26 ± 0.09  0.040  ± 0.000  \n1.0 1000  8.42 ± 0.18  5.47 ± 0.00  0.037  ± 0.000  \n9000  4.67 ± 0.25  5.44 ± 0.00  0.028  ± 0.000  \nPC 0.1 1000  1.45 ± 0.01  10.59 ± 0.2 9 0.067 ± 0.002  \n9000  1.01 ± 0.01  11.84  ± 0.34  0.068  ± 0.002  \n1.0 1000  11.68  ± 0.26  9.95 ± 0.37  0.063  ± 0.003  \n9000  8.65 ± 0.28  10.43  ± 0.59  0.056  ± 0.003  \nPMMA [30] 0.1 1000  2.60 ± 0.01  11.11  ± 0.22 0.077  ± 0.00 1 \n9000  0.62 ± 0.00  11.87  ± 0.28 0.072  ± 0.00 1 Page 83 of 84 \n Table  S.9.3: \n\n, \n and \nc  values for  \nmrr  = 0.75 position for cross -linked SU -8, PC and PMMA . \n \nPP\n  (/s) \nmP  (µN)  \n\n × 102 (/s) \n  × 102 \nc  \nCross -linked SU-8 0.1 1000  2.42 ± 0.00  5.54 ± 0.07  0.038  ± 0.000  \n9000  1.11 ± 0.00  5.87 ± 0.09  0.033  ± 0.000  \n1.0 1000  10.37  ± 0.17  5.10 ± 0.00  0.032  ± 0.000  \n9000  7.33 ± 0.25  5.22 ± 0.00  0.022  ± 0.000  \nPC 0.1 1000  4.33 ± 0.01  10.29  ± 0.34  0.063  ± 0.002  \n9000  1.63 ± 0.01  10.29  ± 0.34  0.056  ± 0.002  \n1.0 1000  16.48 ± 0.26  9.45 ± 0.37  0.060  ± 0.003  \n9000  12.18  ± 0.28  9.10 ± 0.59  0.042  ± 0.003  \nPMMA [30] 0.1 1000  3.13 ± 0.01  9.98 ± 0.22 0.071  ± 0.00 1 \n9000  0.93 ± 0.00  11.12  ± 0.28 0.065  ± 0.001 Page 84 of 84 \n Supporting Information  10 \nS.10.1. Stress  state  vs corresponding  strain  state  \nWe have define d a stress  state, \n**\n0= − . We plot \n  vs \n  in Fig.  S.10.1.   \n \nFigure  S.10.1: \n vs \n  for cross -linked SU -8, PC and PMMA . The equivalent strains of the \nnanoindentation experiments are marked. \n , \n, \n represent \nmP  = 1000  µN and  \nPP\n = 0.1 /s, \n, \n, \n represent  \nmP  = 9000  µN and  \nPP\n  = 0.1 /s, \n, \n represent  \nmP\n = 1000  µN and  \nPP\n  = 1.0 /s and \n , \n represent  \nmP = 9000  µN and  \nPP\n  = 1.0 /s. \n "    },    {        "title": "2402.07593v1.Inverse_source_problems_for_coupled_parabolic_systems_from_measurements_of_one_internal_component.pdf",        "content": "INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS FROM\nMEASUREMENTS OF ONE INTERNAL COMPONENT\nCRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nAbstract. This paper is devoted to the study of inverse source problems for coupled systems of heat\nequations with constant or spatial–dependent coupling terms and whose internal measurements involve\na reduced number of observed states. The analysis is developed for two kind of systems: the first\none consists of parabolic equations with zero order coupling terms (or the so–called non–self–adjoint\nmatrix potential) and whose possibly space–dependent coefficients. The second one consists of parabolic\nequations with coupling in the diffusion matrix. In all configurations the source is decomposed in separate\nvariables, where the temporal part is known and scalar, whereas the spatial dependence is an unknown\nvector field. This work builds on previous methodologies for the recovery of source in scalar equations and\nStokes fluids, thus expanding the field to include coupled systems of second order parabolic equations.\nNumerical algorithms through the finite element method in 1D and 2D are performed. Several examples\nshowing that the algorithms make possible to recover space–dependent sources.\nKey words: Inverse source problems, Riesz bases, null controllability problem, Volterra equations,\nfinite element method.\nContents\n1. Main problems 1\n1.1. Introduction 1\n1.2. State of the art 3\n2. Preliminaries 4\n2.1. Spectral analysis 4\n2.2. Controllability 5\n2.3. Volterra equations 7\n3. Systems with constant coefficients 7\n4. Systems with space dependent coefficients 13\n5. Numerical results 17\n5.1. Control problem 17\n5.2. Inverse source problem 21\nAcknowledgements 26\nReferences 26\n1.Main problems\n1.1.Introduction. Inverse problems of determining coefficients or sources in coupled systems of partial\ndifferential equations have drawn an increasing interest during the last decade, specially in the case of\nparabolic or hyperbolic systems, although also in more complex systems that naturally appear in many\nbranches of science and engineering, including fluid mechanics, biology, medicine, among others. A\nchallenging task in inverse problems for coupled systems is related to whether it is possible to determine\n1arXiv:2402.07593v1  [math.OC]  12 Feb 20242 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nall sources (or coefficients) by a reduced number of measurements, where usually the measurements\nare given by all state variables on local subsets (either from boundary local subsets or internal local\nsubsets), in other words, the issue of measuring a less quantity of states than the number of sources\n(resp. coefficients). This subject is interesting from both theoretical and practical point of view.\nFrom a theoretical point of view, this question leads to different guidelines depending on types of\ncoupling. For instance, linear coupling in low–order terms might show matrix potentials with non–self\nadjoint operators, where tools such as perturbation theory, semigroups theory and spectral analysis are\nfrequently used in order to analyze those systems, or, even linear coupling in the main operator, which\ncould easily lead to degenerate operators. Obviously, the analysis of systems with nonlinear coupling\nterms turns out to be more complex than the previous one, what in its turn it requires knowledge in\napproximation theory as well as fixed point arguments. To continue, let us mention that, as explained in\n[34], the reconstruction of a general external source either from internal or boundary measurements is not\ndetermined uniquely, thus, by adding external forces into coupled systems, the inverse source problem\nbecomes solvable if some a priori knowledge is assumed, i.e., if the unknown source is a characteristic\nfunction [34], a point source [27], one part in the separation of variables [26].\nConcerning practical applications, there exist models in the real life involving partial data of physical\nquantities, for example, pressure estimation from velocity phase–contrast MRI [14], wireless communi-\ncation where only some components of the electric fields are measured [22], elastography [24], molecular\nmulti–photon transitions in laser fields [11], heat transfer [4], hybrid inverse problems [10].\nIn the present study, the source is decomposed in separate variables, where the temporal part is known\nand scalar, meanwhile, the spatial dependence is an unknown vector field. In fact, these external forces\nare associated to coupled systems of heat equations. Let us emphasize that our precise goal in the present\nwork will be to recover the spatial distribution of external forces in coupled parabolic systems from a\nreduced number of measured states in internal subsets. More precisely, our propose is to provide explicit\nformulas which show that it is possible to recover sources in coupled systems of heat equations from a\nlimited number of component of the state in small subdomains. Here, the word “recover” refers to two\nissues: the first one is that the measurements determine source reconstruction formulae of the coefficients\nassociated to the source f. The another one is to describe algorithms to compute the source terms.\nFurthermore, it is worth pointing out that this work is inspired on a previous methodology for recovering\nsources in scalar equations (heat equation [31], wave equation [46]) and Stokes fluids [30]. In that sense,\nwe now extend the existing results on scalar parabolic equations to coupled system of heat equations.\nTo be more precise, let us describe our inverse source problems from an abstract framework. Through-\nout the paper, Ω will be a nonempty bounded domain of Rd(d∈N) with smooth boundary ∂Ω,O⊂Ω\na nonempty open subset, which will denote the spatial subset of measurements. We denote by n∈N\nthe number of equations and by m∈Nthe number of observed components, with m < n . Let Abe\nan appropriate linear partial differential operator with domain D(A)⊂L2(Ω,K), time–independent, and\n(possibly) space–dependent coefficients, with K=RorC. The source term is of the form σ(t)F(x),\nwhere σis a known scalar function whereas F= (f1, . . . , f n)∗is unknown, and both in suitable spaces\nto define later on. In the present paper, we will focus on the following problems:\nProblem 1. The diagonal case with the same operator Aon each line, that is,\n\u001a\n∂tY+InAY+QY=σ(t)F(x)inΩ×(0, T),\nY(·,0) = 0 , inΩ,(1.1)\nwhere Q∈Mn(K)is a coupling matrix with possibly space–dependent coefficients and Inis the identity\nmatrix of size n. Here, we are interested in solving the following question: can we recover the source term\nF= (f1, . . . , f n)∗in system (1.1) from incomplete data, that is, from a reduced numbers of measurements\nof the solution in O×(0, T)?INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 3\nProblem 2. The case where the coupling is in the principal part:\n\u001a∂tY+DAY =σ(t)F(x)inΩ×(0, T),\nY(·,0) = 0 inΩ,(1.2)\nwith D∈Mn(K)a diffusion matrix with constant coefficients. In this case, Dis assumed to be di-\nagonalizable with positive eigenvalues. To this case, our question is: can we recover the source term\nF= (f1, . . . , f n)∗in system (1.2) from incomplete data, that is, from a reduced numbers of measurements\nof the solution in O×(0, T)?\n1.2.State of the art.\na) In relation to Problem 1 and under the structure of system (1.1) with A=−∆, the Bukhgeim–\nKlibanov method [36], which is based on Carleman estimates, have been employed to obtain some\nresults in the context on inverse coefficient problems. For instance: the articles [19, 13] proved\nLipschitz–type stability inequalities for 2 ×2 reaction–diffusion systems with a single observation\nacting on a subdomain. In addition, [20] gives an extension of those works to the nonlinear\ncase. Nevertheless, by considering hyperbolic systems in cascade formed by nequations, the\nrecent work [16] addresses uniqueness and stability (Lipschitz stability) aspects from internal\nmeasurements of all components of the solution except the last one. Another recent work is [23]\nfor two coupled Schr¨ odinger equations (that is, A=i∆ in (1.1)), where the authors treated the\nlogarithmic stability for determining two potentials from internal observations of one component\nof the solution. In contrast to the previous results, [23] combines Carleman estimates along with\nthe Fourier–Bros–Iagolnitzer transform in order to prove the logarithmic stability. The paper [18]\ndeals the identification problem of two discontinuous coefficients for a one–dimensional coupled\nparabolic system, observing only one component. In [18] the authors proved Carleman–type\ninequalities for theoretical issues, whereas the numerical results have been carried out using the\nfinite difference method for the temporal and spatial discretization jointly with the interior–point\nmethod (i.e., optimization problem).\nIn summary, the above articles show either identificability or stability properties for coupled\nhyperbolic or parabolic systems using incomplete measurements of their components. However,\nrespect to inverse source problems, to the best of our knowledge, we have: by considering a\ncascade system of ndegenerate parabolic equations with zero–order coupling terms and a general\nexternal force G=G(x, t) instead of σ(t)F(x), an identification problem is established in [5].\nMore precisely, under certain assumptions on Gand its temporal derivative, the article [5] proves\na Lipschitz–type stability (by means of Carleman inequalities) for determining the source Gby\nobservations in an subdomain of only one component and also using data of the ncomponents\nat the fixed positive time over the whole spatial domain. The paper closest to our research is [1],\nwhich considers two wave equations coupled in cascade. In fact, the identification and stability\nproblems of space–dependent sources from boundary incomplete observations are solved in [1].\nNevertheless, [1] does not provide insights upon exact reconstruction formulas to those sources.\nExplicit formulas of source reconstruction and their algorithms for coupled parabolic systems\nhave not been reported in the literature. Therefore, the first purpose of this paper is to fill that\ngap for systems (1.1) where Ais the Laplace operator.\nb) On Problem 2, as far as we know, there exist few works concerning inverse problems for coupled\nsystems such as (1.2). The recent paper [17] deals the coefficients determination problem for\nan transmitted diseases model (coupled parabolic equations in relation with the SIR model)\nwhose coupling is located in the principal term associated to the operator. The authors of [17]\napplied optimal control techniques with constrains in order to formulate and prove their main\nresult. Another recent work is [45], where the authors have proved Carleman inequalities to4 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nderive H¨ older stability for the inverse coefficient problem from internal observation data for a\nthree–dimensional system of two coupled heat equations with similar structure to (1.2).\nTo finish, we also refer to the articles [43, 44, 28] and their references therein, which are closely linked to\ninverse problems for coupled parabolic systems.\nAs mentioned, our approach follows the guidelines proposed in [46, 31] and [30] for the wave, heat\nand Stokes equations, respectively. Therefore, preliminary results on Volterra equations, as well as Riesz\nbases and controllability properties for coupled parabolic systems are given in section 2. In the remainder\nof this paper, our goal is threefold:\n•In section 3 we present source reconstruction formulas for parabolic systems whose coupling is a\nconstant coefficients matrix located either in the principal part of the operator or as a potential\nterm. In other words, we solve Problems 1, 2 for systems with constant coefficients and using\ninternal measurements in O×(0, T) from only one scalar component of the state, i.e., through\nyn|O×(0,T).\n•In section 4 we give a source reconstruction formula for one dimensional systems of two heat equa-\ntions with non–constant zero–order coupling term (i.e., potential matrix with space–dependent\ncoefficients). Roughly speaking, we solve Problem 1 in 1D for a 2 ×2–order parabolic system\nwith non–self–adjoint operators and from observation data y2|O×(0,T). The main difficulty relies\non the main operator, which is not self–adjoint, and therefore properties on the eigenfunctions\nassociated with non–self–adjoint matrix Sturm–Liouville operators are needed in order to provide\na source reconstruction result from only interior measurements and a reduced number of states.\n•Section 5 is focused on numerical issues and it is divided in two parts; the first one establishes\nnumerical results related to the controllability problem for coupled parabolic systems under the\nstructure of Problem 1. Those results are modeled in 1D and 2D by implementing the conjugate\ngradient method [32] using Lagrange finite elements. In the second part we present numerical\nreconstruction sources for Problem 1 in 1D and 2D. The spatial source reconstruction is displayed\nfor different configurations over the coupling parameters and observation data after developing\nan appropriate gradient descent algorithm.\n2.Preliminaries\nAs mentioned, in order to provide source reconstruction formulas for coupled systems of heat equations,\nwe shall mainly combine existing null controllability results for coupled parabolic systems with distributed\ncontrol, spectral properties for linear operators, and also integro–differential equations. Thus, this section\nis devoted to present these topics. Henceforth, Problems 1–2 will be analyzed by considering the operator\nA=−∆ with domain D(A) =H2(Ω)∩H1\n0(Ω).\n2.1.Spectral analysis. In this paragraph we describe the eigenvalues and eigenfunctions of the non–self\nadjoint operators L, L∗:D(L) =D(L∗) =H2(0, π)2∩H1\n0(0, π)2⊂L2(0, π)2→L2(0, π)2related to the\nfollowing Sturm–Liouville problem\n\u001aLΦ :=−∆Φ + V(x)Φ = λΦ in (0 , π),\nΦ(0, t) = Φ( π, t) = 0 ,(2.1)\nwhere\nV(x) =\u00120 0\nq(x) 0\u0013\n,and q∈L∞(0, π). (2.2)INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 5\nAdditionally, for every k∈N, we consider the normalized eigenfunctions of the Laplace operator with\nDirichlet boundary conditions over (0 , π), i.e.,\nφk(x) =r\n2\nπsin(kx),and Ik(q) :=πZ\n0q(x)φk(x)dx,∀k∈N. (2.3)\nThe next Lemma establishes biorthogonal Riesz bases associated to the operators LandL∗. All details\ncan be found in [25] and [9].\nLemma 1. Consider the families\nB=(\nΦ1,k=\u0012\n0\nφk\u0013\n,Φ2,k=\u0012\nφk\nψk\u0013\n:k∈N)\nandB∗=(\nΦ∗\n1,k=\u0012\nψk\nφk\u0013\n,Φ∗\n2,k=\u0012\nφk\n0\u0013\n,:k∈N)\n,\nwhere ψkis defined for all x∈(0, π)by\n\n\nψk(x) =αkφk(x)−1\nkxZ\n0sin(k(x−ζ))(Ik(q)φk(ζ)−q(ζ)φk(ζ))dζ,\nαk=1\nkπZ\n0xZ\n0sin(k(x−ζ))((Ik(q)φk(ζ)−q(ζ)φk(ζ)))φk(x)dζdx.(2.4)\nThen, one has\na)The spectrum of L∗andLare given by ρ(L∗) =ρ(L) ={k2:k∈N}.\nb)For every k∈N, the eigenvalue k2ofL∗has algebraic multiplicity 1. Moreover, in this case,\n\u001a\u0000\nL∗−k2Id\u0001\nΦ∗\n1,k=Ik(q)Φ∗\n2,k,\u0000\nL∗−k2Id\u0001\nΦ∗\n2,k= 0.(2.5)\nc)For every k∈N, the eigenvalue k2ofLhas algebraic multiplicity 1. Moreover, in this case,\n\u001a\u0000\nL−k2Id\u0001\nΦ1,k= 0,\u0000\nL−k2Id\u0001\nΦ2,k=Ik(q)Φ1,k.(2.6)\nd)The sequences BandB∗are biorthogonal Riesz basis of L2(0, π)2.\ne)The sequence B∗is a Schauder basis of H1\n0(0, π)2andBis its biorthogonal basis in H−1(0, π)2.\nRemark 1. Note that Lemma 1 shows a condition on the potential V(x)for which the root functions\nof the operators LandL∗form Riesz bases. Roughly speaking, the spectral theory for either regular or\nsingular Sturm–Liouville problems offers a wide variety of topics in order to analyze its eigenvalues and\neigenfunctions, see for instance [38, 41, 35] . However, we only incorporated the one–dimensional case\nof a2×2–order system with non–self adjoint matrix potential, which allows us to understand the main\npurpose of the spectral part in our strategy as well as to indicate the key points for future works. But\nthis is a very broad and independent area, which involves a delicate and deep analysis even in 1D for\nn×n–order systems. We do not touch on this area in our paper. Some impression of these issues can be\ngained from the papers [42, 21, 39] .\n2.2.Controllability. Depending on the structure of the coupling matrix Q∗and the control matrix B,\ndifferent results on null controllability for the adjoint system of (1.1) can be derived, see for instance6 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\n[33, 12, 29]. To our purpose, let us assume that Q∗∈L∞(Ω)n2andB∈Mn(R) have the structure:\nQ∗=\nq11 0 0 ··· 0\nq21q22 0··· 0\nq31q32q33··· 0\n...............\nqn1qn2··· qn,n−1qnn\nand B=diag(0,0, . . . , 0,1), (2.7)\nwhere\nqij≥q0>0 in an open set O0⊂O,∀i > j, i, j = 1, . . . , n. (2.8)\nThe following result holds from [33].\nLemma 2. Assume that Q∗∈L∞(Ω)n2andB∈Mn(R)are given by (2.7) and satisfy (2.8). Let\nτ∈(0, T]andΞ0∈L2(Ω)n. Then, there exists a control function U(τ)=U(τ)(Ξ0)∈L2(0, T;L2(O)n)\nsuch that the solution Ψof the problem\n\n−∂tΨ−∆Ψ + Q∗Ψ = 1 OBU(τ)inΩ×(0, τ),\nΨ = 0 on∂Ω×(0, τ),\nΨ(·, τ) = Ξ 0 inΩ,(2.9)\nsatisfies\nΨ(·,0) = 0 inΩ. (2.10)\nMoreover, there exists a positive constant C0depending only on ΩandOsuch that\n∥u(τ)\nn∥L2(0,T;L2(O))≤C0eC(τ)∥Ξ0∥L2(Ω)n, (2.11)\nwhere\nC(τ) =τ+1\nτ+ max\nj≤i\u0010\n|qij|2/(3(i−j)+3)+τ|qij|\u0011\n.\nRemark 2. It is worth mentioning that Lemma 2 was proved in [33, Theorem 1.2] for a forward system\ninstead of a backward system, however, the above configuration is more appropriate for solving our inverse\nsource problem, Problem 1. In addition, we also mention that the null controllability property with one\nscalar control for general complete matrices is not possible [33].\nConcerning the system (1.2), its associated null controllability problem reads: given an initial datum\nΞ0∈L2(Ω)n, we are looking for a control function UinL2(0, T;L2(O)n) such that the corresponding\nsolution Ψ to \n\n−∂tΨ−D∗∆Ψ = 1 OBU in Ω ×(0, T),\nΨ = 0 on ∂Ω×(0, T),\nΨ(·, T) = Ξ 0 in Ω ,(2.12)\nsatisfies the identity (2.10).\nFrom [8, Remark 20], the system (2.12) can be controlled to zero with only one scalar control ( m= 1)\nunder the following assumptions (sufficient and necessary conditions):\nA1) The diffusion matrix D∗is diagonalizable with positive real eigenvalues, i.e., for J=diag(di)n×n\nwith d1, d2, . . . , d n>0, one has D∗=P−1JP, with P∈Mn(R),detP̸= 0.\nA2)di̸=dj, fori̸=j, 1≤i, j≤n.\nA3)B= (b1, . . . , b n)∗∈Rnandbi̸= 0, for i= 1, . . . , n .\nThis information is summarized in the following Lemma 3.\nLemma 3. For any Ξ0∈L2(Ω)n, system (2.12) is null controllable with one scalar control if and only\nif the matrices D∗andBsatisfy (A1)–(A3).INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 7\nRemark 3. It is well known that the null controllability property for linear systems is equivalent to an\nobservability inequality for the associated adjoint system. In relation to Lemma 3, [7, page 271] proves\nsuch an observability inequality through Carleman estimates. In the setting of inverse problems, stability\naspects (Lipschitz–type) can be carried out using this comment, see remark 7 for additional information.\n2.3.Volterra equations. In this paragraph we recall technical results concerning scalar Volterra equa-\ntions of first and second kind that we will need later on. We invite readers to see [31] and references\ntherein for more details.\nLemma 4. Assume 0< t < τ < T ,σ∈W1,∞(0, τ)andη∈L2(0, τ;L2(Ω)). Then, there exists a unique\nθ∈H1(0, τ;L2(Ω)) satisfying the Volterra equation\nσ(0)∂tθ(x, t) +τZ\nt(σ(s−t)θ(x, s) +∂tσ(s−t)∂tθ(x, s))ds=η(x, t),\nθ(x, τ) = 0 .(2.13)\nFurthermore, there exists a constant C >0depending on ∥σ∥W1,∞(0,τ)such that\n∥θ∥H1(0,τ;L2(Ω))≤C∥η∥L2(0,τ;L2(Ω)). (2.14)\nNow, let us consider the operator K:L2(0, T;L2(Ω))→H1(0, T;L2(Ω)) defined by\n(Kv)(x, t) :=tZ\n0σ(s)v(x, t−s)ds. (2.15)\nThe following Lemma provides properties related to the operators KandK∗.\nLemma 5. Assume σ∈W1,∞(0, T). Then, there exist positive constants C1andC2depending only on\nΩ, Tand∥σ∥W1,∞(0,T)such that\nC1∥Kv∥H1(0,T;L2(Ω))≤ ∥v∥L2(0,T;L2(Ω))≤C2∥Kv∥H1(0,T;L2(Ω)). (2.16)\nFurthermore, the adjoint operator K∗:H1(0, T;L2(Ω))→L2(0, T;L2(Ω)) is given by\n(K∗θ)(x, t) =σ(0)∂tθ(x, t) +TZ\nt(σ(s−t)θ(x, s) +∂tσ(s−t)∂tθ(x, t))ds. (2.17)\n3.Systems with constant coefficients\nIn this section we present our first two main results associated to Problems 1–2 for the cases where\nthe coupling matrices are constants. To do that, let us consider the systems\n\n∂tY−∆Y+QY=σ(t)F(x) in Ω ×(0, T),\nY= 0 on ∂Ω×(0, T),\nY(·,0) = 0 in Ω ,(3.1)\nand \n\n∂tY−D∆Y=σ(t)F(x) in Ω ×(0, T),\nY= 0 on ∂Ω×(0, T),\nY(·,0) = 0 in Ω ,(3.2)\nwhere Q∈L∞(Ω)n2satisfies (2.7), (2.8) and Dsatisfies the assumptions (A1) and (A2).\nLet us observe that, thanks to the assumptions on the diffusion matrix Dand the potential matrix\nQ, for every source σ(t)F(x)∈L2(0, T;L2(Ω)n), system (3.1) (resp. system (3.2)) admits a unique8 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nweak solution Y∈C([0, T];L2(Ω)n)∩L2(0, T;H1\n0(Ω)n). Additionally, by considering σ∈W1,∞(0, T)\nand following, for example, the procedure given in [37, Chapter 3, Section 6 ], solvability in the space\nW2,1\n2(Ω×(0, T)) := L2(0, T;H2(Ω)n∩H1\n0(Ω)n)∩H1(0, T;L2(Ω)n) also holds for the systems (3.1) and\n(3.2).\nRemark 4. Before presenting the main results of this section, a common denominator to the systems\n(3.1) and(3.2) is the integral representation of their solutions, which are possible thanks to the linearity\nof those systems. More precisely, from the Duhamel principle, the solution Yfrom either of those systems\ncan be written by\nY(x, t) =tZ\n0σ(s)W(x, t−s)ds, (x, t)∈Ω×(0, T), (3.3)\nwhere Wsatisfies\n\n\n∂tW−∆W+QW = 0 in Ω×(0, T),\nW= 0 on ∂Ω×(0, T),\nW(·,0) =σ(0)F(·) in Ω.or\n\n∂tW−D∆W= 0 in Ω×(0, T),\nW= 0 on ∂Ω×(0, T),\nW(·,0) =σ(0)F(·)in Ω.(3.4)\nFurthermore, since ∂tY(x, t) =σ(0)W(x, t) +tR\n0∂tσ(t−s)W(x, s)ds, by evaluating at t=Tthe main\nequations of (3.1) and(3.2), we obtain the following identity:\nσ(0)W(x, T) +TZ\n0∂tσ(T−s)W(x, s)ds−∆Y(x, T) +TZ\n0σ(s)QW(x, T−s)ds=σ(T)F(x) (3.5)\nand\nσ(0)W(x, T) +TZ\n0∂tσ(T−s)W(x, s)ds−D∆Y(x, T) =σ(T)F(x). (3.6)\nAs mentioned, our inverse source problems are linked to null controllability properties for the adjoint\nsystems associated to (3.1) and (3.2), as well as spectral properties of the main operators. Thus, the\nrest of this section is dedicated to connect the identities (3.5) and (3.6) with those topics, and therefore,\ntwo relevant issues must be agreed. First, for the coupling matrices Q, D∈Mn(R) of (3.1) and (3.2),\nrespectively, their adjoint matrices Q∗, D∗∈Mn(R) must satisfy the controllability constraints (see (2.8)\nand assumptions (A1)–(A3)). Second, respect to the spectral part, since in this case the entries of Qand\nDare constants, it is enough to consider the L2–eigenfunctions and eigenvalues of the Laplace operator in\nΩ with Dirichlet boundary conditions. It will be denoted by {φk}k∈Nand{λk}k∈Nrespectively. Finally,\nin order to present our first result, some additional hypotheses must be considered.\nH1. Consider σ∈W1,∞(0, T)withσ(T)̸= 0. Furthermore, for some k∈N\naQ\nj,k(T) := \n1−λk\nσ(T)nX\ni=1mij(T)!\n̸= 0,∀i, j= 1, . . . , n, (3.7)\nwhere M= (mij(t)) =tZ\n0˜Φk(t)˜Φ−1\nk(s)σ(s)dsand˜Φkis a fundamental matrix associated to the linear\nordinary differential system: Z′+ (λkIn+Q)Z= 0.\nH2. Consider Lemma 2, where U(τ)= (u(τ)\n1, . . . , u(τ)\nn)∗is the control function associated to the system\n(2.9) extended by zero in (τ, T), and\nBU(τ)= (0,0, . . . , u(τ)\nn)∗andu(τ)\nn̸= 0. (3.8)INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 9\nH3. From H2, let Θ(τ):= (θ(τ)\n1, . . . , θ(τ)\nn)be a solution to ncopies of (2.13) with right–hand side given\nby(3.8), i.e., (see Lemma 4 and Lemma 5)\n(K∗Θ(τ))(x, t) = (η1, . . . , η n) =BU(τ). (3.9)\nOur first source reconstruction result is given in the following theorem.\nTheorem 1. Let H1–H3 be satisfied. Then, for every solution Y∈W2,1\n2(Ω×(0, T))to(3.1), the source\nF= (f1, . . . , f n)∗∈L2(Ω)nsatisfies the local reconstruction identity\nnX\nj=1aQ\nj,k(T)(fj, φk)L2(Ω)=−σ(0)\nσ(T)(yn,(θ(T)\n1,k)n)H1(0,T;L2(O))\n−1\nσ(T)TZ\n0∂tσ(T−s)(yn,(θ(s)\n1,k)n)H1(0,T;L2(O))ds−1\nσ(T)TZ\n0σ(T−s)(yn,(θ(s)\n2,k)n)H1(0,T;L2(O))ds.(3.10)\nProof. Due to that the proof essentially combines three different topics, we divide its proof in three steps.\nStep 1. Spectral representation. First, from the L2–eigenfunctions {φk}k∈Nof the Laplace operator with\nhomogeneous Dirichlet conditions, note that the solution of (3.1) can also be written as\nY(x, t) =X\nk∈NYk(t)φk(x), (3.11)\nwhere Yk(t) = (yk\n1(t), . . . , yk\nn(t))∗is the unique solution of the ordinary differential system\n\u001aY′\nk(t) + (λkIn+Q)Yk(t) =σ(t)Fk,\nYk(0) = 0 ,(3.12)\nwhere Fk= ((f1, φk)L2(Ω), . . . , (fn, φk)L2(Ω))∗=: (fk\n1, . . . , fk\nn)∗.\nBy solving (3.12), for every k∈N, we obtain\nYk(t) =M=(mij(t))n\ni,j=1\nz }| {\n tZ\n0˜Φk(t)˜Φ−1\nk(s)σ(s)ds!\nFk= nX\nj=1m1j(t)fk\nj,nX\nj=1m2j(t)fk\nj, . . . ,nX\nj=1mnj(t)fk\nj!∗\n, (3.13)\nwhere ˜Φkwas defined in (3.7) (see assumption H1).\nNow, let us define the vector Ξ k:= (φk, . . . , φ k)∈L2(Ω)nand consider the sequence B={Ξk}k∈N.\nBy multiplying the identity (3.5) by elements of Band integrating in space, we get\nσ(T)(F,Ξk)L2(Ω)n=σ(0)(W(T),Ξk)L2(Ω)n+TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds\n−(∆Y(T),Ξk)L2(Ω)n+TZ\n0σ(T−s)(QW(s),Ξk)L2(Ω)nds.(3.14)\nUsing (3.11) and the fact that {φk}k∈Nare eigenfunctions of the Laplace operator, the third term in the\nright–hand side of (3.14) can be transformed as follows:\n−(∆Y(T),Ξk)L2(Ω)n=−(Y(T),∆Ξk)L2(Ω)n=λk(Y(T),Ξk)L2(Ω)n=λknX\nj=1yk\nj(T). (3.15)10 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nThus, at this moment our reconstruction formula is given by\n(F,Ξk)L2(Ω)n=σ(0)\nσ(T)(W(T),Ξk)L2(Ω)n+1\nσ(T)TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds\n+λk\nσ(T)nX\nj=1yk\nj(T) +1\nσ(T)TZ\n0σ(T−s)(QW(s),Ξk)L2(Ω)nds,\nwhich is equivalent to (after taking into account (3.13)):\nnX\nj=1 \n1−λk\nσ(T)nX\ni=1mij(T)!\nfk\nj=σ(0)\nσ(T)(W(T),Ξk)L2(Ω)n\n+1\nσ(T)TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds+1\nσ(T)TZ\n0σ(T−s)(QW(s),Ξk)L2(Ω)nds.(3.16)\nStep 2. Controllability. Now, in order to replace the global terms ( W(s),Ξk)L2(Ω)nand ( QW(s),Ξk)L2(Ω)n\nby local terms in L2(0, s;L2(O)n), for every s∈(0, T], we use the null controllability property for adjoint\nsystems associated to (3.1). In other words, we apply the hypothesis H2 for every k∈N. Thus,\nif Ψ denotes the adjoint state of YandΨ := Q∗Ψ, Lemma 2 guarantees the existence of a control\nU(s)=U(s)(Ξk) =:U(s)\nk∈L2(0, s;L2(O)n) such that the systems\n\n\n−∂tΨ−∆Ψ + Q∗Ψ = 1 OBU(s)\nkin Ω ×(0, s),\nΨ = 0 on ∂Ω×(0, s),\nΨ(·, s) = Ξ k(·) in Ω(3.17)\nand\n\n−∂tΨ−∆Ψ +Q∗Ψ = 1 OQ∗BU(s)\nkin Ω ×(0, s),\nΨ = 0 on ∂Ω×(0, s),\nΨ(·, s) =Q∗Ξk(·) in Ω ,(3.18)\nsatisfy\nΨ(x,0) = Ψ(x,0) = 0 ,∀x∈Ω. (3.19)\nAdditionally, by multiplying the first system of (3.4) by Ψ solution of (3.17) and (3.19), and integrating\nby parts in L2(0, s;L2(Ω)n), we obtain (after extending U(s)\nkby zero at ( s, T))\n(W(s),Ξk)L2(Ω)n=−(W,1OBU(s)\nk)L2(0,s;L2(Ω)n)=−(W, BU(s)\nk)L2(0,T;L2(O)n). (3.20)\nAnalogously, for Ψ solution of (3.18) and (3.19) we have\n(QW(s),Ξk)L2(Ω)n=−(W,1OQ∗BU(s)\nk)L2(0,s;L2(Ω)n)=−(W, Q∗BU(s)\nk)L2(0,T;L2(O)n). (3.21)\nStep 3. Volterra equations. In this step, we essentially adapt subsection 2.3 to the case of systems of\nVolterra equations and we make the relation (3.9) established in H3 in order to ensure an appropriate\nconnexion with (3.20) and (3.21).\nFirst, we apply twice Lemma 4 in a vector form with data η1\nk:= 1OBU(s)\nk, and also with data\nη2\nk:= 1OQ∗BU(s)\nk. After that, Lemma 5 guarantees the identities\nK∗(Θ1\nk) = 1OBU(s)\nkand K∗(Θ2\nk) = 1OQ∗BU(s)\nk,∀k∈N.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 11\nThen, replacing the above relations in (3.20) and (3.21), we obtain\n(W(s),Ξk)L2(Ω)n=−(W, BU(s)\nk)L2(0,T;L2(O)n)=−(W, K∗(Θ1\nk))L2(0,T;L2(O)n),\n(QW(s),Ξk)L2(Ω)n=−(W, Q∗BU(s)\nk)L2(0,T;L2(O)n)=−(W, K∗(Θ2\nk))L2(0,T;L2(O)n).\nNote that, thanks to (3.9) and (3.8), we really have\n(W(s),Ξk)L2(Ω)n=−(W, K∗(Θ1\nk))L2(0,T;L2(O)n)=−(wn, K∗(θ1\nk)n)L2(0,T;L2(O))\n(QW(s),Ξk)L2(Ω)n=−(W, K∗(Θ2\nk))L2(0,T;L2(O)n)=−(wn, K∗(θ2\nk)n)L2(0,T;L2(O)).(3.22)\nFrom (2.15), it is easy to deduce that Y=KW, in particular yn=Kwn. Therefore, it follows\n(wn, K∗(θj\nk)n)L2(0,T;L2(O))= (yn,(θj\nk)n)H1(0,T;L2(O)), j = 1,2. (3.23)\nFinally, putting together (3.16), (3.22) and (3.23) we obtain the desired reconstruction formula (3.10),\nwhich completes the proof of Theorem 1. □\nRemark 5. Under the hypothesis H1, the reconstruction formula (3.10) holds for every k∈Nin the\nfollowing particular cases of time dependency σof the source (see [31, 30] ):\na)σ(t) =σ0constant.\nb)σa non–negative and increasing function.\nb)σ(t) = 1 +1\n2cos\u0010\n4t\nT−ε\u0011\nfort < T−ε, and σ(t) =3\n2fort > T−ε.\nNow, we establish the necessary hypothesis for solving Problem 2, which in our case is related to the\nsystem (3.2).\nH4. Consider σ∈W1,∞(0, T)withσ(T)̸= 0. Furthermore, for some k∈N\naD\nj,k(T) := \n1−λk\nσ(T)nX\nℓ=1 nX\ni=1diℓ!\nmℓj(T)!\n̸= 0,∀i, j= 1, . . . , n, (3.24)\nwhere M= (mij(t)) =tR\n0˜Φk(t)˜Φ−1\nk(s)σ(s)dsand˜Φka fundamental matrix associated to the ordinary\ndifferential system: Z′+λkDZ= 0.\nH5. Assume Lemma 3.\nH6. From H5, let Θ(τ):= (θ(τ)\n1, . . . , θ(τ)\nn)be a solution to ncopies of (2.13) with right–hand side given by\nBU(τ), where B∈Rnsatisfy the condition (A3). That is, we consider the following identity (see (2.13)\nand Lemma 2.17):\n(K∗Θ(τ))(x, t) = (η1, . . . , η n) =BU(τ). (3.25)\nOur second source reconstruction result is given in the following theorem.\nTheorem 2. Let H4–H6 be satisfied. Then, for every solution Y∈W2,1\n2(Ω×(0, T))to(3.1), the source\nF= (f1, . . . , f n)∗∈L2(Ω)nsatisfies the local reconstruction identity\nnP\nj=1aD\nj,k(T)(fj, φk)L2(Ω) =−σ(0)\nσ(T)(yn,(θk)n)H1(0,T;L2(O))\n−1\nσ(T)TR\n0∂tσ(T−s)(yn,(θk)n)H1(0,T;L2(O))ds.(3.26)12 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nProof. Following the structure of the proof of Theorem 1, we have again three steps.\nStep 1. Spectral representation. First, we consider the representation to the solution of (3.2) in a serie of\nthe form (3.11), where now Yk(t) = (yk\n1(t), . . . , yk\nn(t))∗is the unique solution of the ordinary differential\nsystem\u001aY′\nk(t) +λkDYk(t) =σ(t)Fk,\nYk(0) = 0 ,(3.27)\nwhere Fk= ((f1, φk)L2(Ω), . . . , (fn, φk)L2(Ω))∗=: (fk\n1, . . . , fk\nn)∗.\nNow, by solving (3.27) we obtain\nYk(t) =M=(mij(t))\nz }| {\n tZ\n0˜Φk(t)˜Φ−1\nk(s)σ(s)ds!\nFk= nX\nj=1m1j(t)fk\nj,nX\nj=1m2j(t)fk\nj, . . . ,nX\nj=1mnj(t)fk\nj!∗\n(3.28)\nwhere ˜Φkwas defined in (3.24) (see assumption H4).\nAgain, we consider the vector Ξ k:= (φk, . . . , φ k)∈L2(Ω)nand the sequence B={Ξk}k∈N. In\naddition, we consider the integral representation (3.3), where now Wsatisfies the system located in the\nright–hand side of (3.4). Thus, by multiplying the identity (3.6) by elements of Band integrating in\nspace, we get\nσ(T)(F,Ξk)L2(Ω)n=σ(0)(W(T),Ξk)L2(Ω)n\n+TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds−(D∆Y(T),Ξk)L2(Ω)n.\nNow, for every k∈N, the last term in the above identity can be transformed as follows:\n−(D∆Y(T),Ξk)L2(Ω)n=−(DY(T),∆Ξk)L2(Ω)n=λknX\nj=1nX\ni=1dijyk\nj(T).\nThen, at this moment our reconstruction formula is given by\n(F,Ξk)L2(Ω)n=σ(0)\nσ(T)(W(T),Ξk)L2(Ω)n\n+1\nσ(T)TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds+λk\nσ(T)nX\nj=1nX\ni=1dijyk\nj(T),\nor equivalently (taking into account (3.28))\nnX\nj=1 \n1−λk\nσ(T)nX\nℓ=1 nX\ni=1diℓ!\nmℓj(T)!\nfk\nj=σ(0)\nσ(T)(W(T),Ξk)L2(Ω)n\n+1\nσ(T)TZ\n0∂tσ(T−s)(W(s),Ξk)L2(Ω)nds.(3.29)\nStep 2. Controllability. In order to transform the global terms of (3.29) by local terms in the subdomain\nO×(0, T), we apply the null controllability property for parabolic systems whose coupling is in the main\noperator, see Lemma 3. Then, if Ψ denotes the adjoint state of Y, Lemma 3 allows to obtain a controlINVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 13\nfunction U(s)=U(s)(Ξk) =:U(s)\nk∈L2(0, s;L2(O)n) such that the system\n\n\n−∂tΨ−D∗∆Ψ = 1 OBU(s)\nkin Ω ×(0, s),\nΨ = 0 on ∂Ω×(0, s),\nΨ(·, s) = Ξ s(·) in Ω(3.30)\nsatisfy\nΨ(x,0) = 0 ,∀x∈Ω. (3.31)\nThus, by multiplying the second system of (3.4) by Ψ solution of (3.30) and (3.31), and integrating by\nparts in L2(0, s;L2(Ω)n), we obtain (after extending U(s)\nkby zero at ( s, T))\n(W(s),Ξk)L2(Ω)n=−(W,1OBU(s)\nk)L2(0,s;L2(Ω)n))=−(W, BU(s)\nk)L2(0,T;L2(Ω)n)). (3.32)\nIt is worth mentioning that the above identity is possible thanks to the fact that Dis a constant matrix\nand the operator A=−∆ is self–adjoint, since otherwise the analysis might be rather complex.\nStep 3. Volterra equations. Following the arguments of step 3 of the proof of Theorem 1, we use the\nrelation (3.25) given in assumption H6. More precisely, from (3.25), Lemma 4 and Lemma 5 we can\ndeduce\n(W(s),Ξk)L2(Ω)n=−(W, BU(s)\nk)L2(0,T;L2(Ω)n))=−(wn, K∗(θk)n)H1(0,T;L2(O)). (3.33)\nFinally, from (2.15) we have yn=Kwn, and in consequence\n−(wn, K∗(θk)n)H1(0,T;L2(O))=−(yn,(θk)n)H1(0,T;L2(O)). (3.34)\nTherefore, putting together (3.29), (3.33) and (3.34) we establish the formula (3.26). This completes the\nproof of Theorem 2. □\n4.Systems with space dependent coefficients\nIn this section, we are interested in the question of recovering the spatial dependence of a source for\nthe following coupled system of second–order parabolic equations\n\n\n∂tY+ (Lz}|{\n−∆ +Q(x))Y=σ(t)F(x) in (0 , π)×(0, T),\nY(0, t) =Y(π, t) = 0 in (0 , T),\nY(·,0) = 0 in (0 , π),(4.1)\nwhere L:H2(0, π)2∩H1\n0(0, π)2⊂L2(0, π)2→L2(0, π)2andQis given by\nQ(x) =\u00120 0\nq(x) 0\u0013\nand q∈L∞(0, π)∩W1,∞(˜O),˜O⊂O⊂(0, π).\nIn contrast to the previous section where the corresponding coupling terms are constants and the\nsymmetry of the self–adjoint operator A=−∆ allow to make auxiliar control systems for estimating\nglobal terms in an easy way (see step 2 inside the proof of Theorem 1 and Theorem 2), the strategy for\nrecovering the source F(x) of system (4.1) is very different. It mainly relies on three issues: a) properties\non the eigenfunctions associated with non–self–adjoint matrix Sturm–Liouville operators, namely, the\noperators L:=−∆ +Q(x) and L∗:=−∆ +Q(x)∗; b) a null controllability result related to the adjoint\nsystem of (4.1). The above regularity on the coefficient qis used in this part (see [25, theorem 1.1]); and\nagain, c) Volterra equations.\nOn the other hand, as mentioned in Remark 1, the spectral study for a system of n×nequations\nwith matrix operator Lis more difficult to investigate, involving concepts that are not considered in this\npaper. Furthermore, for a higher space dimension, an exhaustive revision from spectral theory [35] might14 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nbe useful in order to obtain a source reconstruction formula as in theorem below for more general systems\nto the presented in (4.1).\nBefore presenting the main theorem of this section, we will first consider some hypotheses.\nH7. Consider σ∈W1,∞(0, T)withσ(T)̸= 0. Furthermore, for some k∈N:\naL\nk(T) := σ(T) \n1−k2\nσ(T)TR\n0e−k2(T−s)σ(s)ds!\n̸= 0,\nbL\nk(T) := −Ik(q)\u0010\n1−k2TR\n0(T−s)e−k2(T−s)σ(s)ds\u0011\n.(4.2)\nH8. Consider Lemma 1.\nH 9. For any s∈(0, T], assume that the adjoint system associated to (4.1) with distributed control\n1OU(s)= (0,1Ou(s)\n2)satisfies the null controllability property (see [25]).\nH10. Consider Lemmas 4 and 5.\nTheorem 3. Let H7–H10 be satisfied. The, for any solution Y∈W2,1\n2((0, π)×(0, T))of(4.1), the\nsource F= (f1, f2)∈L2(0, π)2satisfies\naL\nk(T)\u0010\nfφk\n1+fψk\n1+fφk\n2\u0011\n+bL\nk(T)fφk\n1=−σ(0)(y2, θ(s)\nk)H1(0,T;L2(O))\n−TZ\n0∂tσ(T−s)(y2, θ(s)\nk)H1(0,T;L2(O))ds,(4.3)\nwhere fφk\n1:= (f1, φk)L2(0,π), fψk\n1:= (f1, ψk)L2(0,π)andfφk\n2:= (f2, φk)L2(0,π).\nProof. We will follow the strategy of the above section. The main novelty relies on the spectral part\nfor the operators LandL∗, and whose representation was given in subsection 2.1. Again, the proof is\ndivided in three steps.\nStep 1. Spectral property. The main ingredient in this step is the spectrum of the operators LandL∗.\nFirst, returning to (2.3), for every k∈N, we consider the normalized eigenfunctions φk(x) =q\n2\nπsin(kx)\nof the Laplace operator ∂xxwith Dirichlet boundary conditions over (0 , π).\nSecondly, thanks to Lemma 1, the solution to (4.1) can be represented in the form:\nY(x, t) =X\nk∈Nαk(t)Φ1,k(x) +βk(t)Φ2,k(x), (4.4)\nwhere αk(t) = ( Y(t),Φ1,k)L2(0,π)2, βk(t) = ( Y(t),Φ2,k)L2(0,π)2, and Φ 1,k,Φ2,kbelong to the family B,\nwhich constitutes a Riesz basis of L2(0, π)2. Moreover, the sequences {αk(T)}k∈Nand{βk(T)}k∈Ncan\nbe determined in explicit form, it is a consequence of the biorthogonal property between the families B\nandB∗as well as the relations (2.5) and (2.6). Thus, we can deduce the coupled system of ordinary\ndifferential equations\nd\ndt\u0012αk(t)\nβk(t)\u0013\n+\u0012k2Ik(q)\n0 k2\u0013\u0012αk(t)\nβk(t)\u0013\n=σ(t)\u0012(F,Φ∗\n1,k)L2(0,π)2\n(F,Φ∗\n2,k)L2(0,π)2\u0013\n,∀k∈N,\nwhere Ik(q) :=πR\n0q(x)φk(x)dx.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 15\nFrom the definitions of Φ∗\n1,kand Φ∗\n2,k(see Lemma 1), the explicit solution to the previous system\ncorresponds to:\n\n\nβk(t) =fφk\n1tZ\n0e−k2(t−s)σ(s)ds.\nαk(t) = (fψk\n1+fφk\n2)tZ\n0e−k2(t−s)σ(s)ds−Ik(q)tZ\n0e−k2(t−s)βk(s)ds,(4.5)\nwhere by simplicity fφk\n1:= (f1, φk)L2(0,π), fψk\n1:= (f1, ψk)L2(0,π)andfφk\n2:= (f2, φk)L2(0,π).\nOn the other hand, note that the integral representation (3.3) holds, as well as a similar system to\n(3.4). More precisely, for every ( x, t)∈(0, π)×(0, T) we have\nY(x, t) =tZ\n0σ(s)W(x, t−s)dsand\n\n∂tW+LW= 0 in (0 , π)×(0, T),\nW(0, t) =W(π, t) = 0 in (0 , T),\nW(·,0) = σ(0)F(·), in (0, π).(4.6)\nSince ∂tY(x, t) =σ(0)W(x, t) +tR\n0∂tσ(t−s)W(x, s)ds, then, by evaluating the main equations of (4.1)\nint=T, and multiplying by elements of the family B∗and integrating over (0 , π), we get\n(σ(T)F,Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=σ(0)(W(T),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2\n+TZ\n0∂tσ(T−s)(W(s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2ds\n+ (LY(T),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2.(4.7)\nUsing again Lemma 1 and (4.4), the last term in the right–hand side of (4.7) can be transformed as\nfollows:\n(LY(T),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=(Y(T), Ik(q)Φ∗\n2,k)L2(0,π)2+k2(αk(T) +βk(T))\n=(Ik(q) +k2)βk(T) +k2αk(T).(4.8)\nThus, at this moment our reconstruction formula is given by\nσ(T)(F,Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=σ(0)(W(T),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2\n+TZ\n0∂tσ(T−s)(W(s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2ds\n+ (Ik(q) +k2)βk(T) +k2αk(T).(4.9)\nwhere αk(T) and βk(T) have been obtained in (4.5).\nStep 2. Controllability. In relation to the proof of theorem 1 where the global terms are changed\nby using two adjoint systems and applying its respective controllability results, here, the global terms\n(w(·, s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2fors∈(0, T] are replaced by local terms on L2(0, s;L2(O))2by considering\nthe hypothesis H9. More precisely, we consider the function Φ∗\n1,k+Φ∗\n2,kas initial datum for the distributed\ncontrol system\n\n−∂tΨ +L∗Ψ = (0 ,1Ou(s)\n2) in (0 , π)×(0, s),\nΨ(0, t) = Ψ( L, t) = 0 in (0 , s),\nΨ(·,0) = Ξ 0(·) in (0 , π),(4.10)16 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nwhere Ψ satisfies\nΨ(·,0) = 0 in (0 , π).\nThus, raising as in the previous section, for every s∈(0, T] follows\n(W(·, s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=−(W,(0,1Ou(s)\n2))L2(0,s;L2(O)2)=−(w2, u(s)\n2)L2(0,T;L2(O)). (4.11)\nStep 3. Volterra equation. Letθ(s)\nkbe a solution to (2.13) with right–hand side given by 1 Ou(s)\n2. Using\n(2.13) and Lemma 5, we get K∗θ(s)\nk= 1Ou(s)\n2.\nReplacing this into (4.11), we have\n(W(·, s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=−(w2, u(s)\n2)L2(0,T;L2(O))=−(w2, K∗θ(s)\nk)L2(0,T;L2(O)). (4.12)\nAgain, from (2.15), y2=Kw2and in consequence\n(W(·, s),Φ∗\n1,k+ Φ∗\n2,k)L2(0,π)2=−(w2, K∗θ(s)\nk)L2(0,T;L2(O))=−(y2, θ(s)\nk)H1(0,T;L2(O))(4.13)\nFinally, putting together (4.5), (4.9) and (4.13), our reconstruction formula is\na(T)(fψk\n1+fφk\n2+fφ\n1) +b(T)fφ\n1=−σ(0)(y2, θ(s)\nk)H1(0,T;L2(O))\n−TZ\n0∂tσ(T−s)(y2, θ(s)\nk)H1(0,T;L2(O))ds,\nwhere\na(T) =σ(T) \n1−k2\nσ(T)TZ\n0e−k2(T−s)σ(s)ds!\nand\nb(T) =−Ik(q) \n1−k2TZ\n0e−k2(T−s) sZ\n0e−k2(s−τ)σ(τ)dτ!\nds!\n.\nThis completes the proof of theorem 3. □\nRemark 6. Note that, the advantage of using the linear combination Φ∗\n1,k+ Φ∗\n2,kin the above proof lies\nin the fact that we can recovery all coefficients for each term of F= (f1, . . . , f n)∗through subspaces of\nL2(0, π), i.e., for the source f1∈L2(0, π), we have L2=H1⊕H2, and it admits the representation\nf1=P\nk∈Nfφk\n1φk+fψk\n1ψk, where H1=⟨{φk:k∈N}⟩andH2=⟨{ψk:k∈N}⟩; meanwhile, since the\ncoupling occurs in the second equation, for f2∈L2(0, π), we only obtain PH1f2, where PH1represents\nthe orthogonal projector from L2(0, π)onto H1. Nevertheless, H1constitutes an orthonormal basis of\nL2(0, π), and therefore the source reconstruction formula (4.3) shows every term of Fin complete form.\nRemark 7. In concordance with the scalar case [31], if∂tσ(t) = 0 fort∈(T−ε, T)for some ε >0or,\nif∂tσ(t)decreases exponentially in (0, T), Lipschitz stability properties for the reconstruction formulas\ngiven in Theorems 1–3 also hold. It is easy to verify that from [31, Step 4, page 764] and therefore we\nhave decided to omit it. Nevertheless, the logarithmical stability problem linked to a more regular source\n(i.e., F∈(D(−∆ε))n) and a reduced number of local interior observations remains open.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 17\n5.Numerical results\n5.1.Control problem. In concordance to Section 2.2, Lemma 2, in this section we show numerical\nexamples to the approximate controllability problem for coupled parabolic systems. All examples are\ndevelopmented with a constant coupling matrix Q. The numerical control problem is solved by a variant\nof the conjugate gradient algorithm [32] with a tolerance of ϵ= 10−6as stopping criterion and using\nP1–Lagrange–type polynomials in the finite element scheme.\nLet us consider the following constrained optimization problem:\ninf\nU∈L2(O×(0,T))2 \n1\n2ZZ\nO×(0,T)|U(x, t)|2dxdt +k\n2∥Y(·, T)−YT∥2\nL2(Ω)!\n, (5.1)\nsubject to\n\n∂tY−ν∆Y+QY=U(x, t)1Oin Ω ×(0, T),\nY= 0 on ∂Ω×(0, T),\nY(·,0) = 0 in Ω .(5.2)\n5.1.1. 1D Examples. We consider system (5.2) in its one dimensional version. Here, the parameter are:\nΩ = (0 ,1), spatial step h= 2×10−2,T= 0.5 seconds, temporal step ∆ t= 10−2, diffusion coefficient\nν= 0.1, and three target functions YT\ni(x), i= 1,2,3 are defined as follows:\nYT\n1(x) =(\nyT\n1,1= 5x(1−x),\nyT\n1,2= 6x(1−x),(5.3)\nYT\n2(x) =(\nyT\n2,1= sin( x),\nyT\n2,2=−sin(x),(5.4)\nand\nYT\n3(x) =\n\nyT\n3,1=\n\n8\u0000\nx−1\n4\u0001\nif1\n4≤x≤1\n2\n8\u00003\n4−x\u0001\nif1\n2≤x≤3\n4\n0 elwewhere on (0 ,1)\nyT\n3,2=\n\n−2\u0000\nx−1\n4\u0001\nif1\n4≤x≤1\n2\n−2\u00003\n4−x\u0001\nif1\n2≤x≤3\n4\n0 elwewhere on (0 ,1)(5.5)\nTaking into account the target functions (5.3)–(5.5), Table 1 summarizes the numerical results to problem\n(5.1)–(5.2) for different configurations of control subdomains Oas well as coupling matrices Q= (qij)2×2.\nThe last column in Table 1 includes the relative error, by components, among the respective target\nfunction (either (5.3), (5.4), or (5.5)) and the final state Y(·, T).\nOn the other hand, in Figure 1 we display the different control functions U= (u1, u2) in relation to\nthe target functions (5.3)–(5.5). More precisely, the first column in Figure 1 shows the control function\nu1, the second column is related to the control function u2, and the third column in Figure 1 shows\napproximations between the target functions (5.3)–(5.5) and the states Y(x, T), x∈(0.1) respectively.\nNote that, in Figure 1, the approximation among the target functions (5.4)–(5.5) and its numerical states\nY(x, T), respectively, is better than whose between (5.3) and its numerical states Y(x, T). However,\nby increasing the measure of the subdomain O⊂(0,1) and therefore the control zone over u1, u2, we\nappreciate a best approximation between (5.3) and its numerical states Y(x, T).18 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\n(a)u1using yT\n1,1\n (b)u2using yT\n1,2\n (c)(yT\n1,1, yT\n1,2)\n(d)u1using yT\n2,1\n (e)u2using yT\n2,2\n (f)(yT\n2,1, yT\n2,2)\n(g)u1using yT\n2,1\n (h)u2using yT\n2,2\n (i)(yT\n2,1, yT\n2,2)\n(j)u1using yT\n3,1\n (k)u2using yT\n3,2\n (l)(yT\n3,1, yT\n3,2)\nFigure 1. First column shows the control function u1. The second column is related to\nthe control function u2, and the third column shows approximations between the target\nfunctions (5.3)–(5.5) and the states Y(x, T), x∈(0.1) respectively. The gray zone shows\nthe control zone O⊂(0.1).INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 19\nTarget function k O (q12, q21) \n∥y1(T)−yT\n1∥L2×(0,T)\n∥yT\n1∥L2×(0,T),∥y2(T)−yT\n2∥L2×(0,T)\n∥yT\n2∥L2×(0,T)!\n102(0.5,0.8) 0.1,0.1 2 .8×10−1,2.8×10−1\n104(0.5,0.8) 0.1,0.1 5 .0×10−2,5.0×10−2\n(5.3) 106(0.5,0.8) 0.1,0.1 8 .8×10−3,8.7×10−3\n106(0.1,0.4) 0.3,0.3 2 .1×10−2,2.0×10−2\n106(0.3,0.7) 0.4,0.4 3 .0×10−3,3.0×10−3\n102(0.5,0.8) 0.1,0.1 5 .5×10−1,5.5×10−1\n104(0.5,0.8) 0.1,0.1 1 .1×10−1,1.1×10−1\n(5.4) 106(0.5,0.8) 0.1,0.1 1 .4×10−2,1.4×10−2\n106(0.1,0.4) 0.3,0.3 3 .5×10−2,3.5×10−2\n106(0.3,0.7) 0.4,0.4 2 .6×10−3,2.6×10−3\n106(0.5,0.8) 0.1,0.1 3 .3×10−2,3.3×10−2\n(5.5) 106(0.1,0.4) 0.3,0.3 4 .4×10−2,4.4×10−2\n106(0.3,0.7) 0.4,0.4 9 .1×10−3,9.1×10−3\nTable 1. Numerical results to problem (5.1)–(5.2) for different configurations of control\nsubdomains Oas well as coupling matrices Q= (qij)2×2.\n5.1.2. 2D Examples. In this case, the optimal control problem (5.1)–(5.2) is defined on the square domain\nΩ = (0 ,1)×(0,1), whose spatial step is 2 ×10−2in each axis. The final time is T= 0.5 seconds with\na uniform time step ∆ t= 10−2, and the diffusion coefficient is ν= 0.1. In all experiments the penalty\ncoefficient is k= 106. In addition, let us consider the following target functions:\nYT\n4(x, y) =(\nyT\n4,1= sin( x) sin(y),\nyT\n4,2=−sin(x) sin(y)(5.6)\nYT\n5(x, y) =(\nyT\n5,1= 5xy(1−x)(1−y),\nyT\n5,2=−6xy(1−x)(1−y)(5.7)\nTable 2 summarizes the numerical results to problem (5.1)–(5.2) with different control subdomains Oas\nwell as coupling matrices Q= (qij)2×2, and taking into account the previous target functions (5.6)–(5.7).\nFigure 2 displays the numeric states in a final time Y(·, T) in comparison, by components, with the\ntarget functions (5.6)–(5.7). Figure 2 establishes results for two configurations for the control subdomain\nO, each one with its constant coupling matrix Q= (qij)2×2.\nIt is worth mentioning that similar approximate control results to the presented in Table 2 can be\nobtained for problem (5.1)–(5.2) with space dependent coefficients in Q. We have omitted those kind of\nexamples since our main goal is the inverse source problem and is not the control problem. In addition,\nthe above numerical results validate several theoretical results mentioned before, see for instance [33, 20].\nFinally, the extensive literature dealing the approximate control problem for linear parabolic system\nwhose coupling is a potential term is done, from a theoretical point of view of course, but it will be\nvery interesting to explore the above numerical framework for coupled hyperbolic system with similar\nstructure to Problem 1 because in that context some assumptions on the components in the coupling\nmatrix Qare imposed, see for instance [3] and [2].20 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\n(a)First component of Y(·, T)\n (b)Target function yT\n4,1\n(c)Second component of Y(·, T)\n (d)Target function yT\n4,2\n(e)First component of Y(·, T)\n (f)Target function yT\n5,1\n(g)Second component of Y(·, T)\n (h)Target function yT\n5,2\nFigure 2. Results in 2D among the state solution Y(·, T) and target functions (5.6)–\n(5.7). The first two rows for YT\n4with q12=q21= 0.4 and O= (0.3,0.7)2. The last two\nrows for YT\n5with q12=q21= 0.1 and O= (0.5,0.8)×(0.5,0.8).INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 21\nTarget function O (q12, q21) \n∥y1(T)−yT\n1∥L2×(0,T)\n∥yT\n1∥L2×(0,T),∥y2(T)−yT\n2∥L2×(0,T)\n∥yT\n2∥L2×(0,T)!\n(0.5,0.8)×(0.5,0.8) 0.1,0.1 2 .7×10−1,2.7×10−1\n(5.6) (0 .1,0.4)×(0.1,0.4) 0.3,0.3 4 .5×10−1,4.5×10−1\n(0.3,0.7)×(0.3,0.7) 0.4,0.4 4 .3×10−2,4.3×10−2\n(0.5,0.8)×(0.5,0.8) 0.1,0.1 3 .5×10−2,3.5×10−2\n(5.7) (0 .1,0.4)×(0.1,0.4) 0.3,0.3 7 .8×10−3,7.4×10−3\n(0.3,0.7)×(0.3,0.7) 0.4,0.4 2 .4×10−1,2.4×10−1\nTable 2. Numerical results in 2D to problem (5.1)–(5.2) for different configurations of\ncontrol subdomains Oas well as coupling matrices Q= (qij)2×2.\n5.2.Inverse source problem. In this section, we present several numerical tests related to Problem\n1 proved in Theorem 1 and Theorem 3. Since our theoretical results proved in previous sections are in\nRN, N > 1 when the coupling is given for a constant matrix Q, and 1D for the case where the coupling\nhas space dependent coefficients into the matrix Q, we perform numerical experiments in 1D and 2D with\ndifferent configurations.\nIn concordance with Problem 1, we recover the source term FR(x) = (fR\n1(x), fR\n2(x))Tas the solution\nof the following minimization problem:\nmin\nF∈L2(Ω)J(Y, F) =1\n2ZZ\nΩ×(0,T)|σ(t)F(x)|2dxdt +k\n22X\ni=1,i̸=jZZ\nO×(0,T)|yi−yi,obs|2+|∂tyi−∂tyi,obs|2dxdt (5.8)\nsubject to\n\n∂tY−ν∆Y+QY=σ(t)F(x) in Ω ×(0, T),\nY= 0 on ∂Ω×(0, T),\nY(·,0) = 0 in Ω .(5.9)\nwhere ( y1,obs, y2,obs) are observation data on the subdomain O×(0, T) (or also so called partial mea-\nsurements of the state Y= (y1, y2)T),Qis the coupling matrix with q12̸= 0, q21̸= 0, and σ(t) is a\nsinusoidal function for all experiments defined by (see [31, 30]): σ(t) := 1 +1\n2cos\u0010\n4πt\nT−t0\u0011\nfort < T−t0,\nandσ(t) :=3\n2ift≥T−t0, t0=T/10. The functional Jwas selected in this form by taking into account\nthe numerical results developed in this paper on the approximate control problem and also following\nsome ideas from [30]. Obviously, a rigorous answer must be linked to stability issues for Problem 1, and\nas far as we know, it is an open problem yet. The above minimization problem is solved approximating\nthe coupled parabolic system with P1–type finite elements in 1D and 2D over rectangular meshes. The\nFEM is implemented using the finite elements library FEniCS [6]. The minimization problem (5.8)–(5.9)\nis solved using a steepest descent method [15, 40], whose penalty term is k= 1200, a fixed step size in\n10−2, and a stopping criteria 10−5.\n5.2.1. Source reconstruction in 1D. We consider Ω = (0 ,1),T= 0.5,, and ν= 0.1. In this case 100\nLagrange–type elements were considered, and a time step ∆ t= 0.01 for the temporal discretization. We\nshow numerical reconstruction for two different kind of sources F(x) = (f1(x), f2(x)), namely:\na)F(x) = (f1(x), f2(x)) := (sin(2 πx),−sin(2πx)).22 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nb)F(x) = (f1(x), f2(x)) defined as follows:\nf1(x) :=\n\n8(x−0.1),ifx∈(0.1,0.35),\n8(0.6−x),ifx∈(0.35,0.6),\n0, other case ,and f2(x) :=f1(x−0.3)\nFigure 3 and Figure 4 display source reconstructions by considering observation domains O1= (0.5,0.9)\nandO2= (0.2,0.4)∪(0.6,0.8), as well as two scenarios of coupling terms. More precisely, Figure 3\ndescribes a scenario where the coupling matrix Qhas cubic polynomials in its components, i.e.,\nq12(x) =x(x−1)(5x−2) + 0 .5 and q21(x) =x(1−x)(5x−3) + 0 .5,\nmeanwhile, Figure 4 tackles a linear configuration whose coupling functions are q12(x) = 4 x−2 and\nq21(x) =−4x+ 2.\n5.2.2. Source reconstruction in 2D. In this case, we consider the unit square Ω = (0 ,1)2,T= 0.5, and\nν= 0.1. Here, we employ a regular mesh with 20 ×20 elements and a time step ∆ t= 0.01 for the\ntemporal discretization. In addition, we test two observation domains, namely, O1= (0.5,0.9)×(0.1,0.9)\nandO2= ((0 .2,0.4)∪(0.6,0.8))×(0.3,0.7). The source reconstruction F(x, y) = ( f1(x, y), f2(x, y)) is\ncarried out by considering the following two cases:\na)F(x, y) = (f1(x, y), f2(x, y)) := (sin(2 πx) sin(2 πy),−sin(2πx) sin(2 πy)).\nb)F(x, y) = (f1(x, y), f2(x, y)) defined by:\nf1(x, y) :=\n\n1,ifx∈(0.2,0.5),\n1,ify∈(0.2,0.5),\n0,other case ,f2(x, y) :=\n\n1,ifx∈(0.5,0.8),\n1,ify∈(0.5,0.8),\n0,other case .\nFigures 5 and 6 are associated to the numerical results to the source reconstruction in 2D using the\nabove functions. The modulus |F(x, y)|:=p\nf1(x, y)2+f2(x, y)2corresponds to the first column. In\naddition, the second column is the source reconstruction with measurements in both components of the\nstate Y= (y1, y2). Third column obeys to the reconstructed source by observing only the first component\nof the state, that means, observing only y1on the observatory O, and the last column is the reconstructed\nsource by observing only the second component of the state, that means, observing only y2on the\nobservatory O. Figure 5 summarizes the numerical experiments of the sinusoidal source reconstruction\nwhose coupling terms are q12= (4x−2)(4y+2) and q21= (4x+2)(4 y−2), and by using measurements of\nthe state in the observation domains O1(top) and O2(bottom). Analogously, Figure 6 has the numerical\nexperiments for the discontinuous source before defined and using q12= (x(x−1)(5x−2) + 0 .5))(y3−1)\nandq21= (x(1−x)(5x−3) + 0 .5)(y3−1) as coupling terms.\nThe relative error of each reconstruction is computed as the L2–norm of the difference between the\noriginal and the reconstructed source divided by the L2–norm of the original source, in percentage. The\nerror goes down in dependency of the observation domain O⊂Ω. To be precise, when the measure of the\nset Ω\\Otends to zero, i.e., µ(Ω\\O)→0, the relative error too. However, for the cases where µ(Ω\\O)>>0\nweaker reconstruction results are obtained. We think that a theoretical result associated to the logarithmic\nstability for Problem 1 or Problem 2 might lead to answers and therefore connections to numerical results\ndeveloped in several works on inverse source problems for parabolic systems [31, 30, 23]. In the same\nway, in view of the fact that there is no logarithmic stability for the continuous inverse problem, it will\nbe interesting to explore geometrical conditions on what is an optimal size for the observation domain O\nfor obtaining stronger results. The above comments are open problems as far as we know.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 23\n(a)rel. error 51 ,4%\n (b)rel. error 75 ,4%\n (c)rel. error 77 ,0%\n(d)rel. error 37 ,7%\n (e)rel. error 46 ,0%\n (f)rel. error 70 ,6%\n(g)rel. error 14 ,3%\n (h)rel. error 70 ,4%\n (i)rel. error 66 ,5%\n(j)rel. error 11 ,4%\n (k)rel. error 43 ,1%\n (l)rel. error 71 ,1%\nFigure 3. Source reconstructions using coupling terms q12(x) =x(x−1)(5x−2) + 0 .5\nandq21(x) =x(1−x)(5x−3)+0 .5. First column is the reconstruction of FR(x) observing\nboth components ( y1,obs, y2,obs). Second columns shows the reconstruction of FR(x)\nobserving the first component y1,obs, and the last one is the reconstruction observing the\nsecond component y2,obsuniquely. The highlighted area corresponds to the observation\ndomain O.24 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\n(a)rel. error 50 ,1%\n (b)rel. error 77 ,2%\n (c)rel. error 73 ,7%\n(d)rel. error 37 ,3%\n (e)rel. error 76 ,7%\n (f)rel. error 70 ,1%\n(g)rel. error 13 ,3%\n (h)rel. error 67 ,2%\n (i)rel. error 67 ,2%\n(j)rel. error 11 ,5%\n (k)rel. error 64 ,5%\n (l)rel. error 64 ,5%\nFigure 4. Source reconstructions using coupling terms q1(x) = 4 x−2 and q2(x) =\n−4x+ 2. First column is the reconstruction of FR(x) observing both components\n(y1,obs, y2,obs). Second columns shows the reconstruction of FR(x) observing the first\ncomponent y1,obs, and the last one is the reconstruction observing the second component\ny2,obsuniquely. The highlighted area corresponds to the observation domain O.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 25\n(A) Original source\n|F|(B) Observing y1, y2;\nrel. error 66,0%(C) Observing y1;\nrel. error 75,8%(D) Observing y2;\nrel. error 76,8%\n(A) Original source\n|F|(B) Observing y1, y2;\nrel. error 49,1%(C) Observing y1;\nrel. error 75,4%(D) Observing y2;\nrel. error 74,9%\nFigure 5. Source reconstruction for the sinusoidal source with measurements in O1\n(top), and measurements in O2(bottom).\n(A) Original source\n|F|(B) Observing y1, y2;\nrel. error 52,8%(C) Observing y1;\nrel. error 80,8%(D) Observing y2;\nrel. error 72,3%\n(A) Original source\n|F|(B) Observing y1, y2;\nrel. error 42,6%(C) Observing y1;\nrel. error 75,2%(D) Observing y2;\nrel. error 75,2%\nFigure 6. Source reconstruction for discontinuous source with measurements in O1\n(top), and measurements in O2(bottom).26 CRISTHIAN MONTOYA, IGNACIO BREVIS, AND DAVID BOLIVAR\nAcknowledgements\nIgnacio Brevis was supported by the Engineering and Physical Sciences Research Council (EPSRC),\nUK under Grant EP/W010011/1. The first and third authors acknowledge the internal project 1109–\n11090202021 EAFIT University.\nReferences\n[1] Fatiha Alabau-Boussouira, Piermarco Cannarsa, and Masahiro Yamamoto. Source reconstruction by partial measure-\nments for a class of hyperbolic systems in cascade. In Mathematical paradigms of climate science , volume 15 of Springer\nINdAM Ser. , pages 35–50. Springer, [Cham], 2016.\n[2] Fatiha Alabau-Boussouira, Jean-Michel Coron, and Guillaume Olive. Internal controllability of first order quasi-linear\nhyperbolic systems with a reduced number of controls. SIAM J. Control Optim. , 55(1):300–323, 2017.\n[3] Fatiha Alabau-Boussouira, Zhiqiang Wang, and Lixin Yu. A one-step optimal energy decay formula for indirectly\nnonlinearly damped hyperbolic systems coupled by velocities. ESAIM Control Optim. Calc. Var. , 23(2):721–749, 2017.\n[4] Oleg M Alifanov. Inverse heat transfer problems . Springer Science & Business Media, 2012.\n[5] Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, and Jawad Salhi. Lipschitz stability for some coupled degenerate\nparabolic systems with locally distributed observations of one component. Mathematical Control & Related Fields ,\n10(3):643, 2020.\n[6] M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N.\nWells. The FEniCS project version 1.5. Archive of Numerical Software , 3, 2015.\n[7] Farid Ammar-Khodja, Assia Benabdallah, C´ edric Dupaix, and Manuel Gonz´ alez-Burgos. A Kalman rank condition for\nthe localized distributed controllability of a class of linear parbolic systems. J. Evol. Equ. , 9(2):267–291, 2009.\n[8] Farid Ammar-Khodja, Assia Benabdallah, Manuel Gonz´ alez-Burgos, and Luz de Teresa. Recent results on the control-\nlability of linear coupled parabolic problems: a survey. Math. Control Relat. Fields , 1(3):267–306, 2011.\n[9] Farid Ammar Khodja, Assia Benabdallah, Manuel Gonz´ alez-Burgos, and Luz de Teresa. New phenomena for the null\ncontrollability of parabolic systems: minimal time and geometrical dependence. J. Math. Anal. Appl. , 444(2):1071–\n1113, 2016.\n[10] Guillaume Bal. Hybrid inverse problems and internal functionals. In Inverse problems and applications: inside out. II ,\nvolume 60 of Math. Sci. Res. Inst. Publ. , pages 325–368. Cambridge Univ. Press, Cambridge, 2013.\n[11] Andr´ e D Bandrauk. Molecules in laser fields . CRC Press, 1993.\n[12] Assia Benabdallah, Michel Cristofol, Patricia Gaitan, and Luz De Teresa. Controllability to trajectories for some\nparabolic systems of three and two equations by one control force. Math. Control Relat. Fields , 4(1):17–44, 2014.\n[13] Assia Benabdallah, Michel Cristofol, Patricia Gaitan, and Masahiro Yamamoto. Inverse problem for a parabolic system\nwith two components by measurements of one component. Appl. Anal. , 88(5):683–709, 2009.\n[14] Robert W Brown, Y-C Norman Cheng, E Mark Haacke, Michael R Thompson, and Ramesh Venkatesan. Magnetic\nresonance imaging: physical principles and sequence design . John Wiley & Sons, 2014.\n[15] R.L. Burden and J.D. Faires. Numerical Analysis . Brooks/Cole Cengage Learning, 2011.\n[16] Nicol´ as Carre˜ no, Roberto Morales, and Axel Osses. Potential reconstruction for a class of hyperbolic systems from\nincomplete measurements. Inverse Problems , 34(8):085005, 18, 2018.\n[17] Anibal Coronel, Fernando Huancas, and Mauricio Sep´ ulveda. Identification of space distributed coefficients in an\nindirectly transmitted diseases model. Inverse Problems , 35(11):115001, 20, 2019.\n[18] Michel Cristofol, Patricia Gaitan, Kati Niinim¨ aki, and Olivier Poisson. Inverse problem for a coupled parabolic system\nwith discontinuous conductivities: one-dimensional case. Inverse Probl. Imaging , 7(1):159–182, 2013.\n[19] Michel Cristofol, Patricia Gaitan, and Hichem Ramoul. Inverse problems for a 2 ×2 reaction-diffusion system using a\nCarleman estimate with one observation. Inverse Problems , 22(5):1561–1573, 2006.\n[20] Michel Cristofol, Patricia Gaitan, Hichem Ramoul, and Masahiro Yamamoto. Identification of two coefficients with\ndata of one component for a nonlinear parabolic system. Appl. Anal. , 91(11):2073–2081, 2012.\n[21] Fulya S ¸eref and O. A. Veliev. On non-self-adjoint Sturm-Liouville operators in the space of vector functions. Math.\nNotes , 95(1-2):180–190, 2014.\n[22] Oliver Dorn, Hugo Bertete-Aguirre, and George C. Papanicolaou. Adjoint fields and sensitivities for 3D electromagnetic\nimaging in isotropic and anisotropic media. In Inverse problems and imaging , volume 1943 of Lecture Notes in Math. ,\npages 35–65. Springer, Berlin, 2008.\n[23] Fangfang Dou and Masahiro Yamamoto. Logarithmic stability for a coefficient inverse problem of coupled Schr¨ odinger\nequations. Inverse Problems , 35(7):075006, 17, 2019.\n[24] Marvin M Doyley. Model-based elastography: a survey of approaches to the inverse elasticity problem. Physics in\nMedicine & Biology , 57(3):R35, 2012.INVERSE SOURCE PROBLEMS FOR COUPLED PARABOLIC SYSTEMS 27\n[25] Michel Duprez. Controllability of a 2 ×2 parabolic system by one force with space-dependent coupling term of order\none.ESAIM Control Optim. Calc. Var. , 23(4):1473–1498, 2017.\n[26] A. El Badia and T. Ha Duong. Some remarks on the problem of source identification from boundary measurements.\nInverse Problems , 14(4):883–891, 1998.\n[27] A. El Badia, T. Ha-Duong, and A. Hamdi. Identification of a point source in a linear advection-dispersion-reaction\nequation: application to a pollution source problem. Inverse Problems , 21(3):1121–1136, 2005.\n[28] Jishan Fan, Yu Jiang, and Gen Nakamura. Inverse problems for the Boussinesq system. Inverse Problems , 25(8):085007,\n10, 2009.\n[29] Enrique Fern´ andez-Cara, Manuel Gonz´ alez-Burgos, and Luz de Teresa. Controllability of linear and semilinear non-\ndiagonalizable parabolic systems. ESAIM Control Optim. Calc. Var. , 21(4):1178–1204, 2015.\n[30] Galina C. Garc´ ıa, Cristhian Montoya, and Axel Osses. A source reconstruction algorithm for the Stokes system from\nincomplete velocity measurements. Inverse Problems , 33(10):105003, 17, 2017.\n[31] Galina C. Garcia, Axel Osses, and Marcelo Tapia. A heat source reconstruction formula from single internal measure-\nments using a family of null controls. J. Inverse Ill-Posed Probl. , 21(6):755–779, 2013.\n[32] Roland Glowinski, Jacques-Louis Lions, and Jiwen He. Exact and approximate controllability for distributed parameter\nsystems , volume 117 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge,\n2008. A numerical approach.\n[33] Manuel Gonz´ alez-Burgos and Luz de Teresa. Controllability results for cascade systems of mcoupled parabolic PDEs\nby one control force. Port. Math. , 67(1):91–113, 2010.\n[34] Victor Isakov. Inverse source problems , volume 34 of Mathematical Surveys and Monographs . American Mathematical\nSociety, Providence, RI, 1990.\n[35] Tosio Kato. Perturbation theory for linear operators . Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint\nof the 1980 edition.\n[36] AL Bukhgeimand MV Klibanov. Global uniqueness of a class of multidimensional inverse problems. In Soviet Math.\nDokl, volume 24, pages 244–247, 1981.\n[37] O. A. Ladyˇ zenskaja, V. A. Solonnikov, and N.N. Ural’ceva. Linear and quasilinear equations of parabolic type . Trans-\nlated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical\nSociety, Providence, R.I., 1968.\n[38] B. M. Levitan and I. S. Sargsjan. Sturm-Liouville and Dirac operators , volume 59 of Mathematics and its Applications\n(Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian.\n[39] Boris Mityagin and Petr Siegl. Local form-subordination condition and Riesz basisness of root systems. J. Anal. Math. ,\n139(1):83–119, 2019.\n[40] J. Nocedal and S. Wright. Numerical Optimization . Springer-Verlag, 2006.\n[41] A. A. Shkalikov. Perturbations of self-adjoint and normal operators with discrete spectrum. Uspekhi Mat. Nauk ,\n71(5(431)):113–174, 2016.\n[42] O. A. Veliev. On nonselfadjoint Sturm-Liouville operators with matrix potentials. Mat. Zametki , 81(4):496–506, 2007.\n[43] Bin Wu, Ying Gao, Zewen Wang, and Qun Chen. Unique continuation for a reaction-diffusion system with cross\ndiffusion. J. Inverse Ill-Posed Probl. , 27(4):511–525, 2019.\n[44] Bin Wu and Jijun Liu. Determination of an unknown source for a thermoelastic system with a memory effect. Inverse\nProblems , 28(9):095012, 17, 2012.\n[45] Bin Wu and Jun Yu. H¨ older stability of an inverse problem for a strongly coupled reaction-diffusion system. IMA J.\nAppl. Math. , 82(2):424–444, 2017.\n[46] Masahiro Yamamoto. Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by\na control method. Inverse Problems , 11(2):481–496, 1995.\nC. Montoya, Escuela de Ciencias Aplicadas e Ingenier ´ıa, EAFIT, Medell ´ın Colombia\nEmail address :cdmontoyaz@eafit.edu.co\nI. Brevis, School of Mathematical Sciences, University of Nottingham, UK\nEmail address :ignacio.brevis@nottingham.ac.uk\nD. Bolivar, Escuela de Ciencias Aplicadas e Ingenier ´ıa, EAFIT, Medell ´ın Colombia\nEmail address :dboliva1@eafit.edu.co"    },    {        "title": "2402.07628v1.On_implicit_and_explicit_representations_for_1D_distributed_port_Hamiltonian_systems.pdf",        "content": "arXiv:2402.07628v1  [math.DS]  12 Feb 2024On implicit and explicit representations for 1D\ndistributed port-Hamiltonian systems∗\nAntoine Bendimerad-Hohl†1, Denis Matignon‡1, Ghislain Haine§1, and Laurent Lef` evre¶2\n1F´ ed´ eration ENAC ISAE-SUPAERO ONERA, Universit´ e de Toul ouse, France\n2Univ. Grenoble Alpes, Grenoble INP, LCIS, 26000 Valence, Fr ance\nAbstract\nFirst, two examples of 1D distributed port-Hamiltonian sys tems with dissipation, given in explicit\n(descriptor)form, areconsidered: theDzekstermodelfort heseepage ofundergroundwaterandananorod\nmodel with non-local viscous damping. Implicit representa tions in Stokes-Lagrange subspaces are for-\nmulated. These formulations lead to modified Hamiltonian fu nctions with spatial differential operators.\nThe associated power balance equations are derived, togeth er with the new boundary ports. Second,\nthe port-Hamiltonian formulations for the Timoshenko and t he Euler-Bernoulli beams are recalled, the\nlatter being a flow-constrained version of the former. Impli cit representations of these models in Stokes-\nLagrange subspaces and corresponding power balance equati ons are derived. Bijective transformations\nare proposed between the explicit and implicit representat ions. It is proven these transformations com-\nmute with the flow-constraint projection operator.\nKeywords: Distributed parameter systems, Implicit port-Hamiltonian systems , Constrained port-Hamil-\ntonian systems, Dzektser equation, non-local viscous dissipation , Timoshenko beam, Euler-Bernoulli beam\n1 Introduction\nPort-Hamiltonian systems (pHs) have been developped for the mod elling, (co-)simulation and control of\ncomplex multiphysics systems (see [18] for an introductive textboo k or [5] for applications of the approach to\nvarious domains, including mechanics, electromagnetism and irrever sible thermodynamics). This framework\nhas been extended to the case of distributed parameter systems (DPS) with boundary energy flows in the\nseminal paper [21]. The semi-group approach, when applied to linear d istributed parameter pHs, gives rise\nto particularly elegant and practical results in the 1D case (see [10] for an introduction), it has been recently\nextended to nD systems in [3]. Since 2002, the literature on distributed pHs has gro wn considerably, with\nboth theoretical and application papers (see [15] for a review).\nPort-Hamiltonian system dynamics with algebraic constraints have b een considered as well, leading to\nfinite-dimensional port-Hamiltonian Differential-Algebraic Equations systems (pH-DAEs, see for instance [1]\nfor linear descriptor systems in matrix representation or [12] for a review on control applications). From\na modeling or geometric perspective, these algebraic constraints a rise either from interconnection of the\nsubsystems composing the overall system, resulting in constraint s between effort and flow variables in the\n∗This work was partially supported by the AID from the French M inistry of the Armed Forces, and the IMPACTS project,\nentitled IMplicit Port-HAmiltonian ConTrol Systems, fund ed by the French National Research Agency (project ANR-21-C E48-\n001),https://impacts.ens2m.fr/\n†antoine.bendimerad-hohl@student.isae-supaero.fr\n‡denis.matignon@isae.fr\n§ghislain.haine@isae.fr\n¶laurent.lefevre@lcis.grenoble-inp.fr\n1underlying Dirac structure, or by implicit energy constitutive equat ions, resulting in constraints between\neffort and energy state variables assigned to live in some Lagrangian submanifold (see [19] or [20] for the\nnonlinear case).\nRecently, examples of distributed parameter models given in implicit fo rm, either singular or not, have\nbeen considered in the pHs settings (see for instance [22] for an imp licit formulation of the Allen-Cahn\nequation,[9]consideringtheDzektserequation(seepageofunde rgroundwater)or[7,8]consideringananorod\nwith non-local visco-elastic constitutive equations), along with str ucture-preserving numerical methods, see\n[2]. [11]extendboundarycontrolpHstoaclassofsystemswheret hevariationalderivativeoftheHamiltonian\nis replaced by a pair of reciprocal operators, generalizing – in the infi nite-dimensional settings – the implicit\ndefinition of the energy by a so-called Stokes-Lagrange subspace associated to the reciprocal operators.\nThis definition (of boundary pHs defined on a Stokes-Lagrange sub space) allows representing some of the\npreviously cited distributed parameter models given in implicit form. Sp ecifically, [11] consider the example\nof an elastic rod with non-local elasticity relation.\nIn this paper, we propose first an application of this Stokes-Lagra nge subspace approach to dissipative\nsystems, in the case of local dissipation only (Dzekster equation, s ubsection 3.1), and in the case with\nlocal and non-local dissipative ports (see the nanorod example in su bsection 3.2). In both cases, classical\nexplicit formulations and implicit representations using Stokes-Lagr ange operators are proposed. Note that\nin the considered implicit representations, spatial derivative opera tors are present inside the Hamiltonian as\nproposed for instance in [17], where infinite-dimensional pHs are defi ned on jet bundles. In [14], Boussinesq,\nelastic rod and Allen-Cahn equations are considered as examples of s uch systems. The authors propose a lift\nin the jet space where the Hamiltonian density only depends on the ex tended state variable. Then, geometric\nformulations with Stokes-Dirac structures are applicable.\nIn the second part of the paper, we derive similarly explicit and implicit r epresentations for the Timo-\nshenko and Euler-Bernoulli beams. The Euler-Bernoulli model may b e seen as flow-constrained version of\nthe explicit Timoshenko beam and is therefore an example of constra ined pHs with an implicit definition\nof the energy via a Stokes-Lagrange subspace. Bijective operat ors which transform explicit formulations\nto implicit ones are proposed for the Timoshenko beam model and for the Euler-Bernouilli reduced model.\nThese operators are similar to transformations between DAEs and geometric representations analyzed in\n[13], in the finite-dimensional LTI case. It is shown that the reductio n (from Timoshenko to Euler-Bernoulli\nmodels) and the bijective explicit-implicit transformations commute in this particular example. This allows,\nfor instance, to consider the implicit formulation for mathematical a nalysis, while the constrained explicit\nformulation (DAEs) is used for numerical computations.\n2 Preliminary results\n2.1 Variational derivatives\nTheorem 1. LetK:D(K)⊆L2(Ω,Rn)→L2(Ω,Rm)be a closed and densely-defined (differential) linear\noperator, and let\n∀α∈D(K),H(α) :=1\n2/integraldisplay\nΩ|K(α)|2dx,\nbe an energy functional.\nAssume that the following abstract Green’s identity holds f or allα∈D(K), β∈D(K†)\n/integraldisplay\nΩK(α)·β=/integraldisplay\nΩα·K†(β)+/angbracketleftγ(α),C(β)/angbracketrightU,Y, (1)\nwhereK†:D(K†)⊆L2(Ω,Rm)→L2(Ω,Rn)is another closed and densely-defined linear operator, call ed\ntheformal adjoint ofK,γ∈ L(D(K),U)is a (boundary) control operator on the Hilbert space U, and\nC ∈ L(D(K†),Y)its colocated observation operator, with Y:=U′.\nThen the weakvariational derivative of Hwith respect to αexists and is given as\nδw\nαH(α) =K†(K(α)).\n2Proof.\nLetα∈D(K),ε∈R,then, for all β∈C∞\nc(Ω,Rm) =:D(Ω), one has\nH(α+εβ) =1\n2/integraldisplay\nΩK(α+εβ)·K(α+εβ)dx\n=1\n2/integraldisplay\nΩ/parenleftbig\n|K(α)|2+2εK(α)·K(β)\n+ε2|K(β)|2/parenrightbig\ndx\n=H(α)+ε/integraldisplay\nΩα·K†(K(β))dx+o(ε).\nDefining the weak variational derivative in the sense of distribution ( thanks to (1))\n/angbracketleftδw\nαH(α),β/angbracketrightD(Ω)′,D(Ω):=/integraldisplay\nΩα·K†(K(β))dx,\ngives the result.\nCorollary 2. Given the Hamiltonians H1andH2, defined on H1(Ω,R),H2(Ω,R), respectively, by\nH1(α) =1\n2/integraldisplay\nΩ(∂xα)2dx,H2(α) =1\n2/integraldisplay\nΩ(∂2\nx2α)2dx,\ntheir weak variational derivatives are given by\nδw\nαH1(α) =−∂2\nx2α, δw\nαH2(α) =∂4\nx4α,\nrespectively.\nProof.\nTakeK=∂x,K†=−∂xin the first case, and K=K†=∂2\nx2in the second case.\n2.2 Operator transposition\nIn the following, the superscriptnotationson a variable xE, xIrefersto the explicit orimplicit representation\nof a system, respectively.\nLet us consideran operator Kas before, a set of distributed second order tensors κ∈L∞(Ω,Mn(R)),η∈\nL∞(Ω,Mm(R)), with∀x∈Ω,κ(x) =κ⊤(x)≥κ >0,η(x) =η⊤(x)≥η >0 almost everywhere, and a pHs\ndefined as/bracketleftbigg\n∂tαE\n1\n∂tαE\n2/bracketrightbigg\n=/bracketleftbigg\n0K†\n−K0/bracketrightbigg/bracketleftbigg\ne1\ne2/bracketrightbigg\n,/braceleftBigg\neE\n1=δα1HE,\neE\n2=δα2HE,\nwith the corresponding Hamiltonian defined as\nHE=1\n2/integraldisplay\nΩαE\n1·καE\n1+αE\n2·ηαE\n2dx.\nLet us now consider a second port-Hamiltonian system defined as\n/bracketleftbigg∂tαI\n1\n∂tαI\n2/bracketrightbigg\n=/bracketleftbigg0 Id\n−Id 0/bracketrightbigg/bracketleftbiggeI\n1\neI\n2/bracketrightbigg\n,/braceleftBigg\neI\n1=δαI\n1HI,\neI\n2=δw\nαI\n2HI,\n3with the corresponding Hamiltonian defined as\nHI=1\n2/integraldisplay\nΩαI\n1·καI\n1+K(αI\n2)·ηK(αI\n2)dx.\nFor the sake of readability, weak variational derivative will be denot ed with the same symbol as usual\nvariational derivative δin the sequel of the paper.\nFinally, let us define an operator G:L2(Ω,Rn)×D(K)→L2(Ω,Rn)×L2(Ω,Rm) and its †-companion\nG:=/bracketleftbiggId 0\n0K/bracketrightbigg\n,andG†:=/bracketleftbiggId 0\n0K†/bracketrightbigg\n,\nand the structure matrices of operators\nJE=/bracketleftbigg\n0K†\n−K0/bracketrightbigg\n,JI=/bracketleftbigg\n0 Id\n−Id 0/bracketrightbigg\n.\nWe then get the following theorem:\nTheorem 3. The previously defined operator Gallows passing from one representation to the other, the\ntransformation being given as\n\nGαI=αE,\neI=G†eE,\nJE=GJIG†.\nThe two systems are said to be equivalent .\nProof.\nA direct computation gives the results.\nRemark 4. Note that given such a transformation G, each state αI∈ker(G)does not participate in the\nHamiltonian, in particular they are removed by the transfor mation. This explains why when writing the\nwave equation in pH formulation, the deformation εis used instead of the displacement w: the Hamiltonian\nH=1\n2/integraltext\nΩp2+|grad(w)|2dxonly depends on the momentum pand deformation grad(w) =ε, hence the\ntransformation removing gradfrom the Hamiltonian yields (p,ε)as the state. And because the kernel of\ngradis the set of constant functions, rigid body motion is lost du ring the transformation.\n3 Two examples with damping\nIn§3.1, the seepage model is considered, and in §3.2, the example of the nanorod is studied. Throughout\nthis paper, the 1D domain Ω is defined as a bounded interval Ω = [ a,b].\n3.1 Dzektser\nFollowing [6], let us consider the following seepage model of undergrou nd water in 1D\n(Id−ε2∂2\nx2)∂th=α∂2\nx2h−β∂4\nx4h,\nwithα >0,β >0. The system admits a port-Hamiltonian representation given as\n\n\n\n(Id−ε2∂2\nx2) 0 0\n0 Id 0\n0 0 Id\n\nh\nF∇\nF∆\n=\n0∂x−∂2\nx2\n∂x0 0\n∂2\nx20 0\n\nh\nE∇\nE∆\n,\nwith the resistive relations\nE∇=αF∇,\nE∆=βF∆.(2)\n4Following [11] one can then define the Lagrange subspace operator s\nS= (Id−ε2∂2\nx2),P= Id,\nthe Dirac structure operator\nJ=\n0∂x−∂2\nx2\n∂x0 0\n∂2\nx20 0\n,\nand the resistive structure operator: R=/bracketleftbigg\nα0\n0β/bracketrightbigg\n. Finally, following [11, Section 5.], the Hamiltonian is\ngiven as\nH=1\n2/integraldisplay\nΩhS†Phdx+1\n2[ε2h∂xh]b\na,\n=1\n2/integraldisplay\nΩh2+ε2(∂xh)2dx.\nTheorem 5. (Dzektser Power balance) The power balance related to (2)reads:\nd\ndtH= [αh∂xh−βh∂3\nx3h+β∂xh∂2\nx2h]b\na\n−/integraldisplay\nΩ/parenleftbig\na(∂xh)2+β(∂2\nx2h)2/parenrightbig\ndx+[ε2∂xh∂th]b\na.\nMaking use of tr as the Dirichlet trace operator, one can then defin e (f∂,e∂) thepowerboundary port\nrelated to the Stokes-Dirac structure as\nf∂=tr(/bracketleftbig\nh, h, ∂ xh/bracketrightbig\n),\ne∂=tr(/bracketleftbigα∂xh,−β∂3\nx3h, β∂2\nx2h/bracketrightbig\n),\nand (χ∂,ε∂) theenergyboundary port related to the Stokes-Lagrange subspace as\nχ∂= tr(h), ε ∂= tr(ε2∂xh).\nThe power balance then reads\nd\ndtH= [f∂·e∂+d\ndt(χ∂)ε∂]b\na−/integraldisplay\nΩ(αF2\n∇+βF2\n∆)dx,\n≤[f∂·e∂+d\ndt(χ∂)ε∂]b\na.\nRemark 6. The previous inequality is the extension to the case of lossy systems of the equality established\nin [11] for the case of lossless systems.\nRemark 7. The energy boundary port (χ∂,ε∂)vanishes when ε→0, i.e., when the nonlocal term is removed,\nleading to a classical dissipative pHs.\n3.2 Nanorod\nLet us first begin by writing both versions of the Nanorod example an d then compare them.\n53.2.1 Explicit representation\nFollowing [7], the Hamiltonian of the system reads\nH:=1\n2/integraldisplay\nΩa2w2+ρA(∂tw)2+µρA/parenleftbig\n∂2\ntxw/parenrightbig2\n+/parenleftbig\nEA+µa2/parenrightbig\n(∂xw)2dx,\nand the state variable is given as\nz:= [w, ρA∂ tw, µρA∂2\ntxw, ∂xw, N]⊤,\nwithωis the displacement, ρA∂twthe momentum density, µρA∂2\ntxthe flow variable of the non locality, ∂xw\nthe strain and Nthe stress resultant. Let us now define E:= Diag(Id ,Id,Id,Id,0) and\nQ:=\na20 0 0 0\n01\nρA0 0 0\n0 01\nµρA0 0\n0 0 0 EA+µa20\n0 0 0 0 Id\n,\nwhich allows us to rewrite the Hamiltonian HasH=1\n2/integraltext\nΩz⊤E⊤Qz, with the algebraic property\nE⊤Q=Q⊤E.\nDefining furthermore\nJ:=\n0 Id 0 0 0\n−Id 0 0 0 ∂x\n0 0 0 −Id Id\n0 0 Id 0 0\n0∂x−Id 0 0\n,\nR:= Diag(0 ,b2,τdEA+µb2,0,0),\nwherebis the damping coefficient of the viscoelastic layer and τdthe viscous damping of the nanorod.\nThe dynamics of the system is given by\nE∂tz= (J −R)e, e=Qz. (3)\n3.2.2 Implicit representation\nLet us now write this system as an implicit port-Hamiltonian system, i.e., as a state-space representation\nwhere differential operators are present inside the Hamiltonian\n\n∂tw\n∂tε\n∂tp\nfd\nfσ\n=\n0 0 Id 0 0\n0 0 ∂x0 0\n−Id∂x0−Id∂x\n0 0 Id 0 0\n0 0 ∂x0 0\n\nF\nσ\nv\ned\neσ\n, (4)\nwithpthe momentum, wthe displacement, εthe strain, fdandedthe local and nonlocal dissipative ports,\nvthe velocity, Fthe force of the media applied to the nanorod, σthe nonlocal stress, and ed,eσthe local\nand nonlocal efforts linked to the dissipative port. The constitutive relations are\nF=aw,(Id−µ∂2\nx2)σ=Eε,\nv=p\nρA, ed=bfd,(Id−µ∂2\nx2)eσ=τdfσ.\n6Notice that (Id −µ∂2\nx2) is now inside the constitutive relations, and that two constitutive r elations have\nbecome non-local.\nLet us denote by z= (w,ε,p) the state and eS= (F,σ,v) the effort variable corresponding to the storage\nport.\nWe can now define the SandPmatrices as\nP:=\nId 0 0\n0 (Id−µ∂2\nx2) 0\n0 0 Id\n,S:=\na0 0\n0E0\n0 01\nρ\n. (5)\nAnd we get the following relation between state and effort variables\nS†z=P†eS.\nIn particular, we have that S†P=P†S, which shows that the correspondingHamiltonian is non-negative.\nMoreover, we can define the resistive matrices\nRL=/bracketleftbigg\nId 0\n0 (Id−µ∂2\nx2)/bracketrightbigg\n,RR=/bracketleftbigg\nb0\n0τd/bracketrightbigg\n,\nwhich yields the following implicit resistive structure\nR†\nLRR≥0,andRLer=RRfr,\nwithfr= (fd,fσ)⊤ander= (ed,eσ)⊤.\nLet us finally define the structure matrix\nJ=\n0 0 Id 0 0\n0 0 ∂x0 0\n−Id∂x0−Id∂x\n0 0 Id 0 0\n0 0 ∂x0 0\n,\nthen, the system becomes/bracketleftbigg\n∂tz\nfr/bracketrightbigg\n=J/bracketleftbigg\neS\ner/bracketrightbigg\n,/braceleftBigg\nRLer=RRfr,\nS†z=P†eS.\nLet us now write it using the image representation of the Lagrange subspace\n/bracketleftbiggP∂tξ\nfr/bracketrightbigg\n=J/bracketleftbiggeS\ner/bracketrightbigg\n,/braceleftBigg\nRLer=RRfr,\neS=Sξ.\nwithPξ=zbeing the latent space variable. Following [11, Section 5.], this represe ntation allows us to define\nthe corresponding Hamiltonian\nHI:=1\n2/integraldisplay\nΩξ⊤P†Sξdx+1\n2[µEξ2∂xξ2]b\na,\n=1\n2/integraldisplay\nΩaξ12+Eξ22+µE(∂xξ2)2+1\nρξ32dx.\nTheorem 8. (Nanorod) The power balance reads\nd\ndtHI= [µE∂tξ2∂xξ2+σv+eσv−µeσ∂xeσ]b\na\n−/integraldisplay\nΩbf2\nd+1\nτd(e2\nσ+µ(∂xeσ)2)dx. (6)\n7One can then identify the powerboundary ports (f∂,e∂) andenergyboundary ports (χ∂,ε∂) as\nf∂= tr(/bracketleftbigv, v,−µ1\nτd∂xeσ/bracketrightbig\n), e∂= tr(/bracketleftbigσ, eσ, eσ/bracketrightbig\n),\nχ∂= tr(ξ2), ε∂= tr(µE∂xξ2).\nThe power balance then reads\nd\ndtHI= [f∂·e∂+d\ndt(χ∂)ε∂]b\na\n−/integraldisplay\nΩbf2\nd+1\nτd(e2\nσ+µ(∂xeσ)2)dx.\n≤[f∂·e∂+d\ndt(χ∂)ε∂]b\na.\n4 Equivalent representations for classical beam models\nIn§4.1, explicit representations both for the Timoshenko and the Euler -Bernoulli beam are recalled, then in\n§4.2, implicit representations are derived, thus recovering the exam ples treated in the jet bundle formalism\nfirst presented in [17] for Timoshenko, and in [16, Example 5.1] for Eu ler-Bernoulli. Finally in §4.3, the\nequivalence between these representations is proved, and the pa ssage to the limit from Timoshenko to Euler-\nBernoulli well understood in a common setting.\n4.1 Explicit representations\n4.1.1 Timoshenko\nThe Timoshenko beam equation reads [4]\n/braceleftBigg\nρA∂2\ntw=T0∂2\nx2w+AκG∂ x(∂xw−φ),\nρI∂2\ntφ=EI∂2\nx2φ+AκG(∂xw−φ),\nwithρthe mass density, Athe crosssection area, T0the tension, κthe sheer coefficient, Gthe shear modulus,\nImoment of inertia, Ethe young’s modulus, wthe transverse displacement and φthe shear angle. Let us\nfirstly write it as an explicit port-Hamiltonian system\n\n∂tεw\n∂tpw\n∂tεφ\n∂tpφ\n∂tεw,φ\n=\n0∂x0 0 0\n∂x0 0 0 ∂x\n0 0 0 ∂x0\n0 0 ∂x0 Id\n0∂x0−Id 0\n\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:JE\nσw\nv\nσφ\nω\nN\n, (7)\nwithpwthe linear momentum, pφthe angular momentum, εw=∂xwthe deformation, εφ=∂xφthe spatial\nderivative of the shear angle, and εw,φ=∂xw−φthe difference between the deformation and the shear\nangle.\nWith the following constitutive relations\nv=pw\nρA, σw=T0εw, ω=pφ\nρI\nσφ=EIεφ, N=AκGεw,φ,\nand Hamiltonian\nHE:=1\n2/integraldisplay\nΩ/parenleftBigpw\nρA2\n+T0ε2\nw\n+pφ\nρI2\n+EIε2\nφ+AκGε2\nw,φ/parenrightBig\ndx.(8)\n8Moreover, one can define the matrices\nSE= Id,PE= Diag/parenleftbigT0,1\nρA, EI,1\nρI, AκG/parenrightbig\n,\nwithPE⊤SE=SE⊤PE>0.Finally one gets the following power balance:\nTheorem 9. (Timoshenko power balance - explicit case) The power balanc e of(7)reads\nd\ndtHE= [vσw+vN+ωσφ]b\na. (9)\nOne can identify the powerboundaryport variables f∂=tr(/bracketleftbigv v ω/bracketrightbig⊤),e∂=tr(/bracketleftbigσwN σφ/bracketrightbig⊤), tr being\nthe trace operator; yieldingd\ndtHE= [f∂·e∂]b\na.\n4.1.2 Euler–Bernoulli\nIn order to reduce the Timoshenko model to the Euler–Bernoulli mo del, one simply needs to transform the\nODE into a DAE by setting ∂tpφand∂tεw,φto zero in (7)\n\n∂tεw\n∂tpw\n∂tεφ\n0\n0\n=JE\nσw\nv\nσφ\nω\nN\n. (10)\nOne can then define the constrained Hamiltonian\nHE\nc=1\n2/integraldisplay\nΩpw\nρA2\n+T0ε2\nw+EIε2\nφdx, (11)\nand get the following power balance\nTheorem 10. (Euler-Bernoulli power balance - explicit case) The power b alance of (10)reads\nd\ndtHE\nc= [vσw−v∂xσφ+σφ∂xv]b\na. (12)\nOne can identify the powerboundary port variables fc\n∂= tr(/bracketleftbigv v ∂ xv/bracketrightbig⊤),ec\n∂=tr(/bracketleftbigσw−∂xσφσφ/bracketrightbig⊤).\nThis yieldsd\ndtHE= [fc\n∂·ec\n∂]b\na.\nRemark 11. One can solve the constraints analytically: ∂xσφ=−N, ∂ xv=ω.Which yields the following\nreduced system\n∂tεw\n∂tεφ\n∂tpw\n=\n0 0 ∂x\n0 0 ∂2\nx2\n∂x−∂2\nx20\n\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:JEr\nσw\nσφ\nv\n. (13)\nNote that by hypothesis, ∂tεw,φ=∂t(∂xφ−w) = 0, hence∂xφ−wis constant over time, however, this\nconstraint is not written explicitly in the previous set of e quations.\n4.2 Implicit representations\n4.2.1 Timoshenko\nLet us now write the Timoshenko beam equation in implicit form, i.e. by pu tting the differential operators\nin the Hamiltonian. \n∂tw\n∂tpw\n∂tφ\n∂tpφ\n=\n0 Id 0 0\n−Id 0 0 0\n0 0 0 Id\n0 0 −Id 0\n\nδwHI\nδpwHI\nδφHI\nδpφHI\n, (14)\n9with Hamiltonian\nHI:=1\n2/integraldisplay\nΩ/parenleftBig1\nρAp2\nw+1\nρIp2\nφ+T0(∂xw)2\n+EI(∂xφ)2+AκG(∂xw−φ)2/parenrightBig\ndx.(15)\nNotice that boundary terms will then appear in the constitutive rela tions! Moreover, this formulation allows\nfor direct access to the displacement winstead of its gradient εw. Let us now write the constitutive relations\nby using Corollary 2\n\nδwHI=−∂x(T0∂xw)−∂x(AκG(∂xw−φ)),\nδpwHI=pw\nρA,\nδφHI=−∂x(EI∂xφ)−AκG(∂xw−φ),\nδpφHI=pφ\nρI.\nWe can now define the structure matrix Jand the Lagrange subspace operators SandPas\nJI=\n0 Id 0 0\n−Id 0 0 0\n0 0 0 Id\n0 0 −Id 0\n,PI=\nId 0 0 0\n0 Id 0 0\n0 0 Id 0\n0 0 0 Id\n,\nSI=\nS1,10∂x(AκG·) 0\n01\nρA0 0\n−AκG∂ x·0S3,30\n0 0 01\nρI\n,\nwithSI\n1,1:=−∂x(T0∂x·)−∂x(AκG(∂x·)) andSI\n3,3:=−∂x(EI∂x·)+AκGId.\nThen, writing zI= (w,pw,φ,pφ)⊤, andeI= (δwH,δpwH,δφH,δpφH)⊤, we get\n/braceleftBigg\n∂tzI=JIeI,\nPI†eI=SI†zI.\nLet us finally write the system using the image representation\n/braceleftBigg\nPI∂tξI=JIeI,\neI=SIξI.\nNotice that, S†P=P†S ≥0 which comes from the non-negativity of the Hamiltonian (15). More over,\ndefining R=/bracketleftbig\nP S/bracketrightbig⊤, sinceSis invertible, the associated polynomial matrix of the operator R(s) verifies\nrank(R(s)) = 4∀s∈Cwhich gives the maximal reciprocity of the operator R(see [11] for details).\nTheorem 12. (Timoshenko power balance - implicit case) The power balanc e of(15)reads\nd\ndtHI= [∂tw(T0∂xw+AκG(∂xw−φ))\n+∂tφEI∂xφ]b\na.(16)\nOne can identify the energyboundary port variable\nχ∂= tr(/bracketleftbigw, w, φ/bracketrightbig⊤),\nε∂= tr(/bracketleftbigT0∂xw, AκG (∂xw−φ), EI∂ xφ/bracketrightbig⊤),\nthis yieldsd\ndtHI= [d\ndt(χ∂)·ε∂]b\na.\n104.2.2 Euler–Bernoulli\nLet us reduce the previouslydefined system by putting1\nρAp2\nφ= 0 andAκG(∂xw−φ)2= 0 in the Hamiltonian\n(15). One gets: ∂tpφ= 0, φ=∂xw. This allows us to define the constrained Hamiltonian as\nHI\nc=1\n2/integraldisplay\nΩ/parenleftbigg1\nρAp2\nw+T0(∂xw)2+EI(∂2\nx2w)2/parenrightbigg\ndx.\nThe constrained dynamic reads\n\nId 0 0 0\n0 Id 0 0\n∂x0 0 0\n0 0 0 0\n\n∂tw\n∂tpw\n∂tφ\n∂tpφ\n=\n0 Id 0 0\n−Id 0 0 0\n0 0 0 Id\n0 0 −Id 0\n\nδwHI\nδpwHI\nδφHI\nδpφHI\n. (17)\nTheorem 13. (Euler–Bernoulli power balance - Implicit case)\nd\ndtHI\nc= [∂tw(T0∂xw−∂x(EI∂2\nx2w))\n+∂t(∂xw)EI ∂2\nx2w]b\na.(18)\nOne can identify the energyboundary port variable\nχc\n∂= tr(/bracketleftbigw, w, ∂ xw/bracketrightbig⊤),\nεc\n∂= tr(/bracketleftbigT0∂xw,−∂x(EI ∂2\nx2w), EI∂2\nx2w/bracketrightbig⊤),\nthis yieldsd\ndtHI= [d\ndt(χc\n∂)·εc\n∂]b\na.\nRemark 14. Solving the two constraints ∂tpφand∂xw=φyields the reduced unconstrained system\n/bracketleftbigg\n∂tw\n∂tpw/bracketrightbigg\n=/bracketleftbigg\n0Id\n−Id0/bracketrightbigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:JIr/bracketleftbigg\nδwHI\nc\nδpwHI\nc/bracketrightbigg\n, (19)\ntogether with the constitutive relations\n/braceleftBigg\nδwHI\nc=−∂x(T0∂xw)+∂2\nx2(EI∂2\nx2w).\nδpwHI\nc=pw\nρA,\n4.3 From implicit to explicit formulations\n4.3.1 Timoshenko\nIn this subsection, we will denote by a zE,eEthe state and effort variables in the explicit formulation and by\nazI,eIthe state and effort in the implicit formulation. JE,SE,PEdenote the structure and constitutive\nmatrices in the explicit case and JI,SI,PIthe structure and constitutive matrices in the implicit case.\nLet us now describe a procedure that allows to pass from the implicit t o the explicit representation, this\nprocedure is similar to the one described in [13, Section 6.] but uses diff erential operatorsinstead of matrices.\nFirst let us define a transformation operator Gfrom (H1∩L2\n0)×L2×H1×L2to (L2)5, whereL2\n0is the\nspace of square-integrable functions with zero mean,/integraltext\nΩw= 0, as\nG:=\n∂x0 0 0\n0 Id 0 0\n0 0 ∂x0\n0 0 0 Id\n∂x0−Id 0\n.\n11Its inverse is actually well defined, thanks to Poincar´ e-Wirtinger’s inequality, since/integraltext\nΩw= 0. Physically\nspeaking, it means that one neglects rigid body motions of the beam. The inverse FofGis given by\nF(z) :=z/mapsto→\nx/mapsto→/parenleftBig/integraltextx\naz2dy−1\nb−a/integraltextb\naz2dy/parenrightBig\nz1\nz2−z5\nz3\n. (20)\nA direct computation then yields/braceleftBigg\nGzI=zE, eI=G†eE,\nzI=FzE,F†eI=eE.\nNote that a geometric interpretation might be useful to understand these transforma tions since states are\nusually described as vectors and co-states as covectors, hence they are contravariant and covariant, respec-\ntively. Then, one can compute the time derivative in the implicit case an d compare it to the explicit case\n∂tG(zI) =GJIeI=GJIG†eE.\nA direct computation shows that GJIG†=JE. Additionally, regarding the Lagrange subspace\nSE⊤zE=PE⊤eE.\nLetuspremultiplythisequationby G†andreplacethestateandeffortvariablesbytheirexplicitcounterp arts.\nWe get to\nG†SE⊤GzI=G†PE⊤F†eI,\nand we have G†SE⊤G=SI†andG†PE⊤F†=PI†.\n4.3.2 From Timoshenko to Euler–Bernoulli\nLet us denote\nΠE:=\nId 0 0 0 0\n0 Id 0 0 0\n0 0 Id 0 0\n0 0 0 0 0\n0 0 0 0 0\n, (21)\nthe (singular) operator that allows to constrain the system (7); w hen premultiplying the left side of equation\n(7) with ΠEone gets to the constrained system (10). In order to get the implic it counterpart ΠI, let us\napplyGto pass from (14) to (7), then ΠEto pass from (7) to (10), and finally Fto pass from (10) to (17).\nTherefore, let us compute FΠEG\nΠI:=FΠEG=\nId 0 0 0\n0 Id 0 0\n∂x0 0 0\n0 0 0 0\n.\nFinally, one can represent these transformations as a commutativ e diagram, shown in Fig. 1.\n4.3.3 Euler–Bernoulli\nLet us now apply the same procedure to the Euler–Bernoulli reducedcase, i.e., let us pass from (13) to (19).\nLet us denote by\nGr:=/bracketleftbiggId 0 0\n0∂x∂2\nx2/bracketrightbigg⊤\n,\n12(7),HE(14),HI\n(10),HE\nc (17),HI\ncG,F\nΠEΠI\nG,F\nFigure 1: Diagram of the transformations between beam models and representations\nthe transformation operator between the two reduced systems .\nThen, assuming εφ=∂xw, i.e., on the subspace where this constraint is satisfied, we have\nGr/bracketleftbiggw\npw/bracketrightbigg\n=\nεw\nεφ\npw\n,andG†\nr\nσw\nσφ\nv\n=/bracketleftbiggδwHI\nδpwHI/bracketrightbigg\n.\nAdditionally, one has: GrJI\nrG†\nr=JE\nr.\n5 Conclusion & Outlook\nImplicit Stokes-Lagrange representations have been proposed f or two examples of pHs, respectively with\nlocal and non local dissipation. The corresponding implicit Hamiltonian a nd power boundary and energy\nboundary port-variables have been derived.\nStokes-Lagrange representations have also been derived for th e Timoshenko and the Euler-Bernoulli models.\nTwo bijective transformations between the corresponding explicit and implicit models have been exhibited.\nSince the Euler-Bernoulli beam is a flow-constrained version of the T imoshenko beam, it has been checked\nthat the corresponding projection operatorscommute with the t ransformations between explicit and implicit\nrepresentations of both models.\nWe will now consider extending these results to 2D (Dzekster seapa ge model, Reissner-Mindlin and\nKirchoff-Love plate models) and 3D (Maxwell equations) examples. A nother avenue we want to explore\nconcernsthecomparativeanalysisofexplicitandimplicit formulations ,intermsoftheirnumericalproperties.\nReferences\n[1] Christopher Beattie, Volker Mehrmann, Hongguo Xu, and Hans Z wart. Linear port-Hamiltonian de-\nscriptor systems. Mathematics of Control, Signals, and Systems , 30:1–27, 2018.\n[2] Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lef` evre, and Denis Matignon. Implicit port-\nHamiltonian systems: structure-preserving discretization for th e nonlocal vibrations in a viscoelastic\nnanorod, and for a seepage model. IFAC-PapersOnLine , 56(2):6789–6795, 2023.\n[3] Andrea Brugnoli, Ghislain Haine, and Denis Matignon. Stokes-Dirac structures for distributed param-\neter port-Hamiltonian systems: An analytical viewpoint. Communications in Analysis and Mechanics ,\n15(3):362–387, 2023.\n[4] Michele Ducceschi and Stefan Bilbao. Conservative finite differen ce time domain schemes for the pre-\nstressed Timoshenko, shear and Euler–Bernoulli beam equations. Wave Motion , 89:142–165, 2019.\n[5] Vincent Duindam, Alessandro Macchelli, Stefano Stramigioli, and He rman Bruyninckx. Modeling and\ncontrol of complex physical systems: the port-Hamiltonian approach . SpringerScience & BusinessMedia,\n2009.\n13[6] E. S. Dzektser. Generalization of the equation of motion of grou nd waters with free surface. Dokl. Akad.\nNauk SSSR , 202(5):1031–1033, 1972.\n[7] Hanif Heidari and H Zwart. Port-Hamiltonian modelling of nonlocal lo ngitudinal vibrations in a vis-\ncoelastic nanorod. Mathematical and computer modelling of dynamical systems , 25(5):447–462, 2019.\n[8] HanifHeidariandHansZwart. Nonlocallongitudinalvibrationinan anorod,asystemtheoreticanalysis.\nMathematical Modelling of Natural Phenomena , 17:24, 2022.\n[9] Birgit Jacob and Kirsten Morris. On solvability of dissipative partial differential-algebraic equations.\nIEEE Control Systems Lett. , 6:3188–3193, 2022.\n[10] BirgitJacoband HansJZwart. Linear port-Hamiltonian systems on infinite-dimensional s paces, volume\n223. Springer Science, 2012.\n[11] BernhardMaschkeandArjanvan derSchaft. Linearboundar yport-Hamiltoniansystemswith implicitly\ndefined energy. arXiv preprint arXiv:2305.13772 , 2023.\n[12] Volker Mehrmann and Benjamin Unger. Control of port-Hamilto nian differential-algebraic systems and\napplications. Acta Numerica , 32:395–515, 2023.\n[13] VolkerMehrmannandArjan vanderSchaft. Differential–algebr aicsystemswith dissipativeHamiltonian\nstructure. Mathematics of Control, Signals, and Systems , pages 1–44, 2023.\n[14] Till Preuster, Manuel Schaller, and Bernhard Maschke. Jet sp ace extensions of infinite-dimensional\nHamiltonian systems. arXiv preprint arXiv:2401.15096 , 2024.\n[15] Ramy Rashad, Federico Califano, Arjan J van der Schaft, and S tefano Stramigioli. Twenty years of\ndistributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and\nInformation , 37(4):1400–1422, 2020.\n[16] M. Sch¨ oberl and K. Schlacher. Lagrangian and port-Hamilton ian formulation for distributed-parameter\nsystems. IFAC-PapersOnLine , 48(1):610–615, 2015.\n[17] Markus Sch¨ oberl and Andreas Siuka. Jet bundle formulation o f infinite-dimensional port-Hamiltonian\nsystems using differential operators. Automatica , 50:607–613, 2 2014.\n[18] Arjan van der Schaft, Dimitri Jeltsema, et al. Port-Hamiltonian systems theory: An introductory\noverview. Foundations and Trends ®in Systems and Control , 1(2-3):173–378, 2014.\n[19] Arjan van der Schaft and Bernhard Maschke. Generalized por t-Hamiltonian DAE systems. Systems &\nControl Letters , 121:31–37, 2018.\n[20] Arjan van der Schaft and Bernhard Maschke. Dirac and Lagra nge algebraic constraints in nonlinear\nport-Hamiltonian systems. Vietnam Journal of Mathematics , 48:929–939, 2020.\n[21] Arjan J van der Schaft and Bernhard M Maschke. Hamiltonian fo rmulation of distributed-parameter\nsystems with boundary energy flow. Journal of Geometry and Physics , 42(1-2):166–194, 2002.\n[22] Mohammed Yaghi, Fran¸ coise Couenne, Aur´ elie Galfr´ e, Laure nt Lef` evre, and Bernhard Maschke. Port-\nHamiltonian formulation of the solidification process for a pure subst ance: A phase field approach.\nIFAC-PapersOnLine , 55(18):93–98, 2022.\n14"    },    {        "title": "2402.07657v1.Evaluation_of_plasma_on_deflection_angle_and_shadow_by_a_black_hole_solution_immersed_in_perfect_fluid_in_Rastall_theory.pdf",        "content": "arXiv:2402.07657v1  [gr-qc]  12 Feb 2024Evaluation of plasma on deflection angle and shadow by a black hole solution immersed in\nperfect fluid in Rastall theory\nRiasat Ali,1,∗Xia Tiecheng,1,†Rimsha Babar,2,‡and Ali ¨Ovg ¨ un3, 4, §\n1Department of Mathematics, Shanghai University and Newtou ch Center for\nMathematics of Shanghai University, Shanghai-200444, Peo ple’s Republic of China\n2Department of Mathematics, GC University Faisalabad Layya h Campus, Layyah-31200, Pakistan\n3Physics Department, Eastern Mediterranean University,\nFamagusta, 99628 North Cyprus, via Mersin 10, Turkiye\n4Center for Theoretical Physics, Khazar University, 41 Mehs eti Street, Baku, AZ1096, Azerbaijan\n(Dated: February 13, 2024)\nThe present study examines the gravitational deflection of p articles in curved space-times im-\nmersed in perfect fluid in the context of Rastall theory, appl ying the Gibbons-Werner useful technique.\nIn its application to integral areas inside a four-dimensio nal space-time, the Gauss-Bonnet theorem\nstudies the computation expression of the deflection angle. The Gibbons-Werner technique has two\nlimitations in Rastall theory for space-times immersed in p erfect fluid: the integral region is generally\ninfinite and the integral complicates calculation, particu larly for complex space-times and extremely\naccurate solutions. An infinite region approach to Gibbons- Werner is proposed to avoid singular-\nity. For demonstrating the Gibbons-Werner method, we use a c omplete Rastall theory framework.\nIt studies black hole solutions like the dust field, radiatio n field, quintessence field, cosmological\nconstant field and phantom field using the Gibbons-Werner app roach. Additionally, we check the\ndeflection angle from these space-times under the influence o f plasma. Furthermore, we compute\nanalytically the influence of a plasma on a black hole shadow b y using a ray-tracing approach. In\nthe Hamiltonian equation, our model describes the plasma, s o the light ray motion equations are in-\ndependent of the plasma’s velocity. It is assumed that plasm a is a dispersive medium, pressure-less\nand non-magnetized. We investigate the perfect fluid in Rast all theory in further depth, where the\nplasma particle density corresponds to particle accumulat ion. The supermassive black hole shadows\nare explored in the case when plasma falls radially from infin ity onto a black hole.\nKeywords: Black hole solutions in Rastall theory; Gibbons- Werner technique; deflection angle; ray-tracing ap-\nproach; shadow\nI. INTRODUCTION\nIn general relativity (GR), one of the most significant fields of study is the movement of objects in intense grav-\nitational fields. An object’s path can be examined using the g eodesic expression of the intense gravitational field.\nBy considering it as a test particle, its gravitational field is ignorable. An important consideration in observing the\ntest particle’s velocity is the trajectory’s deflection. In general, there are two types of particles: massive particle s\nand massless particles, especially photons. Gravitationa l theories [ 1] such as GR are verified in significant part for\nmassless particles through gravitational deflection. Many techniques have been suggested [ 2]-[32] to study the de-\nflection angle (DA) of photons. For massive particles, they m ay function as representatives of all that exists. Later\non, Eddington’s detection justified [ 33] the deflection of light moving by the sun. Gravitational len sing has been\nextensively studied as a powerful tool in various fields of co smology and astronomy. For neutrinos, gravitons, neu-\ntrons and celestial events of high energy ( π-mesons, K-mesons, µons, etc.) are examples. There are also theoretically\nweakly associating massive particles and their axions [ 34]. Important details as to the source, lens, trajectory cont ext,\nand particles can actually be obtained from the gravitation al deflection of these significant particles [ 35–46]. The\nGibbons-Werner (GW) approach, originally proposed by Gibb ons-Werner in 2008 [ 46] and then modified by scien-\ntists in more recent times [ 47–53], helps in the geometric explanation and determination of t he DA for both massless\nand massive particles. The development of a four-dimension al spacetime that can be utilised to explain the motion\nand integral region of the particle on a two-dimensional Rie mann manifold, the trajectory of the particle, an auxiliary\n∗Electronic address: riasatyasin@gmail.com\n†Electronic address: xiatc@shu.edu.cn\n‡Electronic address: rimsha.babar10@gmail.com\n§Electronic address: ali.ovgun@emu.edu.tr2\ncircular curve, a radial upward curve that passes through th e source, and a radial outward curve that passes through\nthe observer are all part of the fundamental technique of the GW approach. The Gauss-Bonnet theorem (GBT) can\nbe applied to the integral region for the DA in the form of geom etric parameters.\nConsider a black hole (BH), which has a consistent spacetime structure but can be lighted by external electromag-\nnetic radiation with varying shapes, colours, and behaviou rs throughout time. The BH can be detected by many\nmathematical approaches, but its impact on light distribut ion can be observed. The Event Horizon Telescope Collab-\noration [ 54] appears as the first image of a BH. The image shows a continuou s spacetime structure illuminated by a\ntime-varying emission zone, encouraging more investigati on and explanation. Several of the concepts employed by\nthe Event Horizon Telescope (EHT), such as BH shadows, have b een given multiple explanations in the literature.\nIn [55], Synge calculated the angular radius of the Schwarzschild BH’s shadow using a static observer model. The\ninvestigation has been done on the various geometries of BH s hadows in Refs [ 56] -[101]. As the magnetic field\nparameter grows, the shadows of the Schwarzschild and Kerr B Hs immersed in the Melvin magnetic field grow and\nelongate in a horizontal direction. Black hole shadows can o nly be predicted by employing a ray-tracing approach\n[65,66,70] for light motion systems with non-integrability.\nHowever,the work has focused on the direct local effects of c osmic models on the known BH solutions. Babichev et\nal. [102] demonstrate that in a scenario with a phantom field, the accr eting particles of the phantom scalar field into\nthe central BH cause the BH mass to decrease. But this does hav e a broad influence. A modified metric that includes\nthe BHs surrounding spacetime can be used to determine the lo cal changes in the spacetime geometry surrounding\nthe fundamental BH. In this context, Kiselev [ 103] has achieved an analytically static, spherically symmetr ic solution\nto Einstein’s equations. The basis of the Rastall theory is r elated to high-curvature situations; hence, the physics\nof BHs may provide a suitable framework for further explorat ion of this theory. Thus, as a novel class of non-\nvacuum BH solutions to this theory, our goal in this study is t o find the surrounding Kiselev-like BH solutions. The\nproperties of the BHs surrounding fields, which are typicall y such as radiation, dust, or a dark energy component,\ndefine this solution [ 103,104]. The Kiselev every field solution’s special cases (a) Schwa rzschild BH encompassed by\nquintessence, BH encompassed by magnetised Ernst field and a Reissner-Nordstrom BH encircled by radiation and\ndust (b), as well as its phase changes and thermodynamical qu antities are examined in [ 104,105] and [ 106] describes\nthe dynamics of a neutral and a charged particle surrounding the Schwarzschild black encircled by quintessence\nmatter.\nIn order to avoid spacetime singularity, one of the main reas ons to research the BH surrounding field in Rastall\ntheory. Although there is a well-developed Rastall theory a t this point, there are numerous attempts, such as black\nrings field theory and BH geometry. The expectation that ther e should be an intrinsic extended structure in spacetime\nis acknowledged by almost all of this approach (Gibbons-Wer ner approach). Stable-orbit photons are in constant\nmotion around a BH; they are unable to leave the BH or travel to infinity. In this instance, the prograde stable photon\norbits correspond to the dark region that only partially eme rges from the main shadow. The BH shadow is half-\npanoramic (equatorial), due to the lack of backward unstabl e (stable) light rings. In this instance, the BH shadow\nturns into an equatorial, panoramic shadow without a grey ar ea for stable photon orbits. The expectation that there\nshould be an intrinsic extended structure in spacetime is ac knowledged by almost all of this approach (ray-tracing\napproach). It is studied that the BH shadows are immersed in c harge fields and that the BH shadow is affected by\nthe perfect fluid in the context of Rastall theory.\nThis is way the paper is formatted. Section 2 discusses an ove rview of the analytical technique for spherically\nsymmetric BHs in Rastall theory. Section 3 study the DA for BH s occupied by dust, quintessence, radiation, phantom\nfields and cosmological constant and also their graphical an alysis. In section 4, we investigate the results of DA and\ngraphical behavior in the background of plasma frame for ass ociated BHs. Section 5 comprises the shadow cast and\ncontour plots for BHs surrounded by dust, quintessence, rad iation, phantom fields and cosmological constant. In\nsection 6, we summarize the results of our work.\nII. INTRODUCTORY REVIEW OF BLACK HOLE IN THE BACKGROUND OF RA STALL THEORY\nWithin the framework of the Rastall theory of gravity, we see k the general non-vacuum spherically symmetric\nstatic uncharged BH solutions in this section. Utilizing Ra stall’s idea [ 107,108,110], we obtain the following for a\nspacetime where an energy-momentum source of Taboccupies the Ricci scalar Ras\nλR,b=Tab;a, (1)\nwith λrepresents the Rastall parameter, which is the conventiona l GR conservation law that has deviated. The\nRastall field equations can be expressed as\nkTab=Gab+kλgabR, (2)3\nwhere the gravitational constant of Rastall coupling is den oted by k. In the limit of λ→0 and k=8πG, these\nfield equations reduce to GR field equations, with Gis the gravitational constant of Newton coupling. To derive BH\nsolutions, we take the usual Schwarzschild coordinates gen eral spherical symmetric metric as [ 110]\nds2=−A(r)dt2+dr2\nA(r)+r2dΩ2, (3)\nwith the two-dimensional unit sphere is represented by the e quation dΩ2=dθ2+sin2θdφ2, and f(r)is a fundamen-\ntal metric function that depends on the radial coordinate. W e can derive non-vanishing components of the Rastall\ntensor, defined as Hab=Gab+kλgabR, can be followed by applying this metric, we get [ 110]\nH0\n0=G0\n0+kλR=−1\nAG00+kλR=1\nr2(´Ar−1+A)+kλR,\nH1\n1=G1\n1+kλR=AG 11+kλR=1\nr2(´Ar−1+A)+kλR,\nH2\n2=G2\n2+kλR=−1\nr2G22+kλR=1\nr2(´Ar+1\n2r2´´A)+kλR,\nH3\n3=G3\n3+kλR=−1\nr2sin2θG33+kλR=1\nr2(´Ar+1\n2r2´´A)+kλR. (4)\nIn this case, the Ricci scalar is defined as\nR=−1\nr2(r2´´A+4r´A+2+2A), (5)\nwith the derivative with regard to the radial coordinate r is represented by the prime sign. In relation to the Rastall\ntensor ( Ha\nb) non-vanishing components, then the following diagonal fo rm should be present in the entire energy-\nmomentum tensor that governs this spacetime.\nTa\nb=\nT0\n00 0 0\n0T1\n10 0\n0 0 T2\n20\n0 0 0 T3\n3\n, (6)\nwith it is satisfied with the Rastall tensor Ha\nbsymmetry features. With respect to solutions of Rastall ten sor, the\nequality conditions H0\n0=H1\n1and H2\n2=H3\n3must also have T0\n0=T1\n1and T2\n2=T3\n3, respectively. After that, a general\nenergy and momentum tensor T with these symmetry features ca n be generated in the manner shown as\nTa\nb=τa\nb+Ea\nb, (7)\nwith Ea\nbis the trace free of Maxwell tensor defined by\nEa\nb=2\nk(FaµFµ\nb−1\n4gabFµνFµν). (8)\nSo that Fabrepresents the anti-symmetric Faraday tensor obeying the c orresponding vacuum Maxwell expression as\nFab\n;a=0,\n∂[αFab] = 0. (9)\nGiven that the spacetime metric ( 3) has spherical symmetry, the only non-vanishing Faraday te nsor Fabof compo-\nnents is imposed to be F01=−F10. Next, using the equations in ( 9), one can get\nF01=Q\nr2, (10)\nwhere Qrepresents an integration constant behaving as an electros tatic charge, the only Maxwell tensor Ea\nbnon-\nvanishing components are given by equations ( 3), (8), and ( 10) as\nEa\nb=Q2\nkr4diagonal(−1,−1, 1, 1), (11)4\npossessing the symmetries in the Ha\nbtensor and evidently expressing an electrostatic field. Con versely, τa\nbdenotes\nthe surrounding field’s energy–momentum tensor, which is de fined [ 109] as\nτ0\n0=−ρs(r)\nτj\ni=−ρs(r)µ/parenleftbig−rjri\nrnrn−3νrjri\nrnrn+νδj\ni/parenrightbig\n. (12)\nWith respect to the arbitrary parameters µand ν, which correspond to the internal structure of the BH surrou nding\nthe field, this model of τa\nbimplies that the space component is proportionate to the tim e category, which represents\nthe energy density. The surrounding field in this case is indi cated by the subscript (s), which is usually any combi-\nnation of dust, radiation, quintessence, cosmological con stant, and phantom field. We can determine the isotropic\naverage over the angles by using [ 109] the\n<τj\ni>=1\n3ρsδj\ni=ρsδj\ni, (13)\nas it is assumed that <rjri>=µ\n3ρsδj\nirnrn. The barotropic expression of equilibrium for the surround ing field thus\nbecomes\nps=ρsωs,ωs=µ\n3, (14)\nwith ωsand ρsindicate the equation of the state parameter and the pressur e, respectively. The principle of the\naddition and linearity scenario assumed in reference [ 109] to find the free parameter νof the energy momentum\ntensor of the surrounding field as suggested is thus precisel y provided by the field expressions ( 4) with respect to\nthe entire energy-momentum tensor in ( 7), (11) and ( 12) as\nν=−3ωs+1\n6ωs. (15)\nThen, the following form can be used to derive the non-vanish ing components of the τabtensor\nτ0\n0=τ1\n1=−ρs\nτ2\n2=τ3\n3=1\n2(3ωs+1)ρs, (16)\nwhich in the Rastall tensor Ha\nbalso have identical symmetries. Consequently, all of Ha\nbsymmetry is admitted by our\nwhole constructed energy-momentum tensor in ( 7). The Rastall field equations τa\nbcan be regarded as the only energy-\nmomentum tensor of assistance. The solutions produced will thus explain the encircled uncharged BH solutions in\nthe Rastall theory context, which are not the same as the ones in GR. Within the context of this theory, the most\ngeneric class of static surrounding charged BH solutions ca n be obtained by including the Maxwell tensor Ea\nbin\nTa\nb. The field equations are solved and their general solution is obtained in the following. Next, we tackle the two\nuncharged/charged solutions. From the H0\n0=T0\n0and H1\n1=T1\n1components of the Rastall field expression, the\ndifferential equation that follows is obtained as\n1\nr2(r´A+A−1)−kλ\nr2(r2´´A+4r´A+2´A−2) =−Q2\nr4−kρs, (17)\nand H2\n2=T2\n2and H3\n3=T3\n3components are interpreted as\n1\nr2(r´A+1\n2r2´´A)−kλ\nr2(r2´´A+4r´A+2A−2) =−Q2\nr4+kρs\n2(1+3ωs), (18)\nAs a result [ 110], the two differential equations ( 17) and ( 18) allow us to analytically compute the one unknown\nfunctions A(r)as\nA(r) =1−2M\nr+Q2\nr2−Ns\nr1−3ωs−6kλ(1+ωs)\n1−3kλ(1+ωs), (19)\nwhere the BH mass and the surrounding field structure paramet er are represented by two integration constants,\nMand Ns, respectively. The parameters kand λrepresents the Rastall geometric parameters and ωsdenotes the5\nequation of state parameter of BH surrounding field. Keep in m ind that the characteristics of the surrounding field\nare represented by the integration constant Ns. Any combination of k,λand ωsparameters can accept various\npositive or negative Nsvalues. We recover the Reissner-Nordstr¨ om BH occupied by a surrounding field in GR,\nwhich was initially discovered by Kiselev [ 109], in the limit of λ→0 and k=8πGN. The metric in Eq. ( 19) is a\nnew static solution with some intriguing features. Now, we w ill study the surrounding BH by the dust radiation,\nquintessence, cosmological constant and phantom fields, as the sub-classes of the general solution of Eq. ( 19) and\ntheir interesting aspects in detail. The two differential e quations ( 17) and ( 18) allow us to analytically compute other\none unknown function ρs(r)as\nρs(r) =−3WsNs\nkr3(1+ωs)−12kλ(1+ωs)\n1−3kλ(1+ωs), (20)\nwhere the field structure parameter can be represented by\nWs(r) =−(1−4kλ)(kλ(1+ωs)−ωs)\n(1−3kλ(1+ωs))2. (21)\nKeep in mind that the characteristics of the surrounding fiel d are represented by the integration constant Ns. We\nhave ρs(r) =−3\nkWsNsr−3(1+ωs)where Ws=ωsforλ=0, in the GR limit. The BH metric ( 19) surrounded by the\ndust field in Rastall theory can be presented as\nds2=A(r)dt2+B(r)dr2+C(r)dθ2+D(r)dφ2, (22)\nwhere\nA(r) =1\nB(r)=1−2M\nr+Q2\nr2−Nd\nr1−6kλ\n1−3kλ,C(r) =r2,D(r) =r2sin2θ, (23)\nand Mis a mass of BH, Ndis the dust field structure parameter and Qis a charge of BH. This metric is not the\nsame as the metric of the charged BH in GR [ 109] that is encircled by a dust field. As one can see, the BH in the\ndust background in GR appears as a charged BH with an effectiv e mass Me=2M+Ndin the limit of λ→0\nand k=8πGN. We may observe that, for kλ/negationslash=0, the Rastall theory’s geometric parameters kand λcan be\ncrucial in determining different solutions with respect to GR. For kλ/negationslash=0, the Rastall correction term gives the BH a\ndistinct character that is incomparable to the mass and char ge terms, and it never behaves like the mass or charge\nterms. Due to the Rastall geometric parameters, the presenc e of such nontrivial character might significantly alter the\nthermodynamics, causal structure, and Penrose diagrams in comparison to those of GR. For this case, the geometric\nparameter Wdcan be defined as\nWd=kλ(1−4kλ)\n(1−3kλ)2. (24)\nNext, with respect to the weak energy condition denoted by th e relation WsNs≤0, we require Nd>0 for 0≤kλ<1\n4\nand Nd<0 for the field structure constant for kλ>1\n4. In this instance, Wdand ρdare essentially distinct from their\nGR versions and the density ρdcan be given as\nρd=3λ(1−4kλ)Nd\n(1−3kλ)2r−3−12kλ\n1−3kλ. (25)\nMoreover, the geometry of the Rastall theory allows us to der ive an effective equation of state parameter ωe f ffor the\nmodification term can be written as\nωe f f=1\n3/parenleftbigg\n−1+1−6kλ\n1−3kλ/parenrightbigg\n. (26)\nIt may be observed that, with the exception of the kλ=0 corresponding to the GR limit, ωe f fcan never be zero.\nTherefore, this theory’s solutions essentially diverge fr om GR’s. The Rastall solution of BH thermodynamical quan-\ntities are examined in [ 111,112]. Regarding Eq. ( 26), there are two distinct and intriguing groups. For the case ,\n1\n6<kλ<1\n3that implies to ωe f f≤1\n3. Here, dark energy is represented by an effective surroundi ng fluid with an\neffective equation of state parameter ωe, which results in an effective repulsive gravitational pul l. It is noteworthy6\nthat the effective equation of state ωe f fforkλ=2\n10and2\n9falls inside the quintessence range, whereas for kλ=3\n10, it\nfalls into the strong phantom range. This illustrates the id ea that, for a given k, the stronger the acceleration phase,\nor the stronger coupling gµνRin Rastall theory, the larger values of λ. When kλ<1\n6/uniontextkλ>1\n3implies to ωe f f≥ −1\n3is the result. Here, we have an effective surrounding fluid th at possesses the usual attractive gravitational effect and\nwhose equation of state parameter respects the strong energ y condition. Depending on the value of the effective\nequation of state parameter, this could lead to the universe expanding more slowly or even contracting.\nIII. DEFLECTION ANGLE IN NON-PLASMA MEDIUM\nThis section is based on the study of DA in a non-plasma frame f or a BH surrounded by dust field in Rastall theory.\nThe generalized GW technique [ 47,51] significantly improves the associated computation and pro vides a thorough\nframework for describing the GW approach for a wide range of s cenarios. We think that the GW method’s imple-\nmentation in BH physics will be made much easier by our study. When the GW approach is applied and developed\nfor spacetimes, there is an ill-defined infinite integral reg ion for some spacetimes expressing singular behavior, and\nthe computation required is complex. In particular, we stud y that the radial coordinate of the additional circular\narc can be chosen freely by carefully examining the integral s of geodesic curvature along the support circular arc\nand of Gaussian curvature throughout the integral region. A s a result, the ill-defined condition is resolved, and the\nintegral region without singular performance can be formed . With complex computation, we construct a simplified\nformula that allows us to determine the DA in a few steps. This formula is based on the free choice of the auxiliary\ncircular arc and the reduction of the integral of geodesic cu rvature along the path. The efficiency and effectiveness\nof our approach are persuasively validated when we finally co mpute the DA of particles in Kiselev spacetime in\nRastall gravity. Furthermore, we provide the DA of particle s for the charged solution in Rastall gravity for the first\ntime. The expression that follows can be used to find the DA for the Kiselev solution using GBT in the non-singular\ndomain region. When we consider the null condition ds2=0 at equatorial plane (θ=π\n2)for the metric ( 22), when\nthe tropical area is where the source, observer, and null pho ton are all located then the optical metric can be obtained\nas\ndt2=B(r)\nA(r)dr2+r2\nA(r)dφ. (27)\nWe rewrite the above metric in the form\ndt2=Y(r)dr2+Z(r)dφ, (28)\nwhere\nY(r) =B(r)\nA(r),Z=r2\nA(r). (29)\nIn order to calculate the Ricci scalar for the given metric, a t first, we compute the non-zero Christoffel symbols in the\nfollowing manner\nΓ1\n11=Z′(r)\n2Z(r)−Y′(r)\n2Y(r),Γ2\n12=Y(r)\nr−Y′(r)\n2,Γ1\n22=r2Y′(r)\n2Y(r)Z(r)−r\nZ(r), (30)\nwhere 1 and 2, represent the rand φcoordinates, respectively. The Ricci scalar is obtained as follows\nR=2Z′′(r)Y(r)Z(r)−Z′(r)Y′(r)Z(r)−Z′2(r)Y(r)\n2Y2(r)Z2(r). (31)\nThe Gaussian curvature can be derived by using the following formula\nK=R\n2. (32)\nUsing Eq. ( 33), the Gaussian curvature for a BH surrounded by dust field is c alculated as\nK≈2M\nr3+(1+6kλ)Nd\nr3−3Q2\nr4+3(2kλ−1)MN d\nr4+6MQ2\nr5+3(2−λ)NdQ2\n2r5+O(M2,Q3,λ2). (33)7\nThe DA is determined by applying the optical Gaussian curvat ure to a non-singular region We, bounded by ∂We=\nω˜g/uniontextUe. As an alternative, a non-singular domain with the Euler cha racteristic α(We) = 1 outside of the light\ntrajectory can be used. The GBT for this area can be expressed as [47]\n/integraldisplay /integraldisplay\nWeKdS+/contintegraldisplay\n∂Weκdt+∑\niǫi=2πα(We). (34)\nAsk=¯g(∇˙̟˙̟,¨̟)and ¯g(˙̟,˙̟) = 1,¨̟imply unite acceleration vector, and ǫiiexpresses the exterior angle at the\nithvertex, κrepresents geodesic curvature in the above expression. The associated jump angles drop toπ\n2ase→∞,\nyielding θO+θs→π. Thus,\n/integraldisplay /integraldisplay\nWeKdS+/contintegraldisplay\n∂Weκdt+ǫi=2πα(We), (35)\nhere, jump angle is represented by ǫi=π. Obtaining the geodesic curvature when e→∞in the following way\nκ(Ue) =| ∇˙Ue˙Ue|. (36)\nSince the radial component of geodesic curvature is compute d as\n(∇˙Ue˙Ue)r=˙Uφ\ne∂φ˙Ure+Γ1\n22(˙Uφ\ne)2. (37)\nFor large value of e,Ue:=r(φ) =e=constant, then the result is\nlim\ne→∞κ(Ue) = lim\ne→∞(∇˙Ue˙Ue)r→1\ne. (38)\nSince there is no topological defect in the geodesic curvatu re,K(Ue)→e−1. However, by using the optical metric\nEq. ( 27), it can be written as follows\nlime→∞dt=edφ. (39)\nUsing Eq. ( 38) and ( 39), we obtain\nK(Ue)dt=dφ. (40)\nNow, using Eq. ( 40), we can get the following equation\n/integraldisplay /integraldisplay\nWeKdS+/contintegraldisplay\n∂Weκdt=e→∞/integraldisplay /integraldisplay\nW∞KdS+/integraldisplayπ+ψ\n0dφ. (41)\nWe employ the straight-line approximation, r=b\nsinφ, where brepresents the impact parameter. The Gibbons and\nWerner approach uses the Gauss-Bonnet theorem to determine the DA as [ 47]\nψ≈ −/integraldisplayπ\n0/integraldisplay∞\nb\nsinφK/radicalig\ndet/vectorgdrd φ, (42)\nwhere/radicalbig\ndet/vectorgis computed as\n/radicalig\ndet/vectorg=3M\nr2+1\nr3−3Q2\n2r3+3MN d\n2r1+1−6kλ\n1−3kλ. (43)\nTo compute the angle of deflection, use the Gaussian curvatur e up to the leading order terms and obtain\nψ1≈4M\nb−2Nd\nb−12Ndkλ\nb−3Q2π\n4b2−3MN dπ\n2b2−3MN dkλπ\n2b2−4NQ2\n3b3+8MQ2\n3b3+2NdQ2kλ\n3b3+O(M2,Q4,λ2). (44)\nThe DA for a BH surrounded by a dust field is dependent on impact parameter b, BH mass M, charge Q, Rastall\ngeometric parameters k,λand dust field structure parameter Nd. We can also observe that the impact parameter\nis in an inverse relation with the angle ψ. Moreover, it has worth to mention here that, in the absence o f structure8\nparameter Nd=0 and Rastall parameters kλ=0, the above result reduce into DA of Reissner–Nordstr¨ om BH\n[115,116] as well as in the absence of charge Q=0, we recover the DA of Schwarzschild BH ψSch=4M/b[110].\nSch\nQ=0.1\nQ=0.5\nQ=1.0\nQ=1.5\n123456781234\nb\n\u00011(i)M=1,kλ=0.5,Nd=0.1\nSch\nkλ=0.1\nkλ=0.5\nkλ=1.0\nkλ=1.5\n123456781234\nbΨ1(ii)M=1,Q=0.5,Nd=0.1\nFIG. 1: Deflection angle ψ1versus impact parameter bwith variations of charge Qand Rastall parameters kλwith\nfixed mass M=1 and dust field structure parameter Nd=0.1\n.\nIn Fig. 1: the left panel shows the variations of charge Qwith fixed mass M=1, Rastall parameters kλ=0.5 and\ndust field structure parameter Nd=0.1 and right panel represents the variations of Rastall par ameters kλwith fixed\nmass M=1, charge Q=0.5 and dust field structure parameter Nd=0.1. In both panels the black curve gives the\nSchwarzschild solution. We can observe that the DA for risin g values of charge and Rastall parameters from a BH\nsurrounded by a dust field is decreasing via rising impact par ameter and it attains an asymptotically flat form till\nb→∞. A strong deflection can be observed for small value of impact parameter b=1.2. Graphically, we can also\nsee the inverse relationship between impact parameter band DA ψ1.\nA. BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD\nThe metric functions for a BH surrounded by a quintessence fie ld are given as\nA(r) =1\nB(r)=1−2M\nr+Q2\nr2−Nq\nr−1−2kλ\n1−kλC(r) =r2,D(r) =r2sin2θ. (45)\nUsing Eqs. ( 31) and ( 32), the Gaussian curvature for a BH surrounded by a quintessen ce field can be defined as\nK≈2M\nr3+4Q2Nq\nr3+18Q2kλNq\n4r3−3Q2\nr4+6MQ2\nr5+O(M2,Q4,λ2). (46)\nThe/radicalbig\ndet/vectorgfor the corresponding BH can be calculated as\n/radicalig\ndet/vectorg=3M\nr2+1\nr3−3Q2\n2r3+3Nq\n2r1+−1−2kλ\n1−kλ. (47)\nUsing Eqs. ( 46) and ( 47) into Eq. ( 42), we obtain the DA for BH surrounded by quintessence field as f ollows\nψ2≈4M\nb+8NqQ2\nb+9NqQ2kλ\nb−3Q2π\n4b2+8MQ2\n3b3+O(M2,Q4,λ2). (48)\nThe DA in non-plasma medium for a BH surrounded by a quintesse nce field depends on impact parameter b, BH\nmass M, charge Q, Rastall geometric parameters k,λand quintessential field structure parameter Nq. Furthermore,\nit is important to note that the above result reduces to the DA of Reissner–Nordstr¨ om BH in the absence of the\nstructure parameter Nq=0 and the Rastall parameters kλ=0. In addition, we recover the DA of Schwarzschild BH\nψSch=4M/bin the absence of charge Q=0.9\nSch\nQ=0.1\nQ=0.2\nQ=0.3\nQ=0.4\n123456780.51.01.52.02.53.03.54.0\nb\n\u00002(i)M=1,kλ=0.1,Nq=0.5\nSch\nkλ=0.1\nkλ=0.5\nkλ=1.0\nkλ=1.5\n123456781234\nbΨ2(ii)M=1,Q=0.3,Nq=0.5\nFIG. 2: Deflection angle ψ2via impact parameter bwith variations of charge Qand Rastall parameters kλwith fixed\nmass M=1 and quintessential field structure parameter Nq=0.5.\nIn Fig. 2: the left panel displays variations in charge Qwith fixed mass M=1, Rastall parameters kλ=0.1, and\nquintessential field structure parameter Nq=0.5, whereas Rastall parameters kλwith fixed mass M=1, charge\nQ=0.3, and quintessential field structure parameter Nq=0.5 are represented in the right panel. In both panels,\nthe Schwarzschild solution is indicated by the black curve. From a BH surrounded by a quintessential field, we\ncan see that the DA for rising values of charge and Rastall par ameters is decreasing via rising impact parameter,\nreaching an asymptotically flat form till b→∞. A significant deflection is seen for small value of impact par ameter.\nAdditionally, the inverse relationship between impact par ameter band DA ψ2is also observed.\nB. BLACK HOLE SURROUNDED BY THE RADIATION FIELD\nThe metric function for a BH surrounded by a radiation field is defined as\nA(r) =1\nB(r)=1−2M\nr+Q2−Nr\nr2, C(r) =r2, D(r) =r2sin2θ. (49)\nUtilizing Eqs. ( 31) and ( 32), the Gaussian curvature for a BH surrounded by the radiatio n field can be computed as\nfollows\nK≈2M\nr3−3Q2\nr4+3Nr\nr4+6MQ2\nr5−6MN r\nr5+4NrQ2\nr6+O(M2,Q4). (50)\nThe value of/radicalbig\ndet/vectorgfor associated BH is given by\n/radicalig\ndet/vectorg=3M\nr2+1\nr3−3Q2\n2r3+3Nr\n2r3. (51)\nUsing Eqs. ( 50) and ( 51) into Eq. ( 42), we derive the DA for BH surrounded by radiation field as\nψ3≈4M\nb−3Q2π\n4b2+3Nrπ\n4b2+8MQ2\n3b3−8MN r\n3b3+3NrQ2π\n8b4+O(M2,Q4). (52)\nThe DA in non-plasma frame for a BH surrounded by a radiation fi eld is dependent on impact parameter b, BH mass\nM, charge Qand radiation field structure parameter Nr. It is important to note that the above result reduces to the\nDA of Reissner–Nordstr¨ om BH in the absence of the structure parameter Nr=0. In addition, we recover the DA of\nSchwarzschild BH ψSch=4M/bin the absence of charge Q=0.10\nSch\nQ=0.5\nQ=1.0\nQ=1.5\nQ=2.0\n1 2 3 4 512345\nb\n\u00023(i)M=1=Nr\nSch\nNr=0.1\nNr=0.5\nNr=1.0\nNr=1.5\n123456780.51.01.52.02.53.03.54.0\nb\n\u00033(ii)M=1,Q=0.5\nFIG. 3: Deflection angle ψ3with respect to impact parameter bwith variations of charge Qand radiation field\nstructure parameter Nrwith fixed mass M=1.\nFig. 3: in the left panel charge Qvaried with fixed mass and the radiation field structure param eter M=1=Nr,\nwhile the right panel shows variations of radiation field str ucture parameter Nrwith fixed mass M=1 and charge\nQ=0.5. The black curve in both panels indicates the Schwarzsch ild solution. The DA with increasing charge and\nstructure parameter values is diminishing by growing impac t parameter from a BH surrounded by a radiation field,\nachieving an asymptotically flat form till b→∞. A notable deviation is observed when the impact parameter i s\nminimal. Furthermore, the impact parameter band DA ψ3are found to be inversely related. It is important to\nmention here that, at some points in left plot the light deflec ts with the same angle for different variations of charge.\nMoreover, a strong deflection can be observed in the presence of radiation field as compared to Schwarzschild case.\nC. BLACK HOLE SURROUNDED BY THE PHANTOM FILED\nThe metric functions for a BH surrounded by a phantom field are given by\nA(r) =1\nB(r)=1−2M\nr+Q2\nr2−Np\nr−3+2kλ\n1+kλ,C(r) =r2,D(r) =r2sin2θ. (53)\nConsidering Eqs. ( 31) and ( 32), the Gaussian curvature for a BH surrounded by the phantom fi eld can be given as\nfollows\nK≈2M\nr3+4kλM\nr3−3Q2\nr4−35NpkλQ2\n2r4+6MQ2\nr5−11NpM\nr5+O(M2,Q4,λ2). (54)\nThe/radicalbig\ndet/vectorgfor given BH is computed as\n/radicalig\ndet/vectorg=3M\nr2+1\nr3−3Q2\n2r3+3Np\n2r1+−3+2kλ\n1+kλ. (55)\nUtilizing Eqs. ( 54) and ( 55) into Eq. ( 42), we derive the DA for BH surrounded by phantom field as follow s\nψ4≈4M\nb+8Mkλ\nb−35NpkλQ2π\n8b2−3Q2π\n4b2+8MQ2\n3b3−44NpM\n9b3+O(M2,Q4,λ2). (56)\nThe DA in non-plasma medium for a BH surrounded by a phantom fie ld depends on impact parameter b, BH\nmass M, charge Q, Rastall geometric parameters k,λand phantom field structure parameter Np. It is significant\nto note that in the absence of the structure parameter Np=0 and the Rastall parameters kλ=0, the above result\nsimplifies to the DA of Reissner–Nordstr¨ om BH. Furthermore , in the absence of charge Q=0, we recover the DA of\nSchwarzschild BH ψSch=4M/b.11\nSch\nQ=0.1\nQ=0.5\nQ=1.0\nQ=1.5\n123456012345\nb\n\u00044(i)M=1,kλ=0.5=Np\nSch\nkλ=0.1\nkλ=0.5\nkλ=1.0\nkλ=1.5\n12345678051015\nbΨ4(ii)M=1,Q=0.1,Np=0.5\nFIG. 4: Deflection angle ψ4versus impact parameter bwith variations of charge Qand Rastall parameters kλwith\nfixed mass M=1 and phantom field structure parameter Np=0.5.\nFig. 4: in the left panel charge Qshows variations with fixed mass, Rastall parameters and the phantom field\nstructure parameter kλ=0.5=Np, while the right panel represents the variations of Rastall parameters kλwith\nfixed mass M=1, charge Q=0.1 and phantom field structure parameter Np=0.5. The Schwarzschild solution is\nshown by the black curve in both images. As the charge and Rast all parameters values increase, the DA decreases\ndue to an increasing impact parameter from a BH encircled by a phantom field. A strong deflection is shown when\nthe impact parameter is small. Moreover, an inverse relatio nship between the impact parameter band the DA ψ4is\ndiscovered. It is significant to note that, for various varia tions in charge, the light deflects at certain spots in the lef t\nplot with the same angle for Schwarzschild. Furthermore, a s ignificant deviation from the Schwarzschild scenario\ncan be seen when a phantom field is present.\nD. BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT\nThe metric functions for a BH surrounded by a cosmological co nstant is defined as\nA(r) =1\nB(r)=1−2M\nr+Q2\nr2−Ncr2,C(r) =r2,D=r2sin2θ. (57)\nConsidering Eqs. ( 31) and ( 32), the Gaussian curvature for a BH surrounded by the cosmolog ical constant is com-\nputed as\nK≈2M\nr3+Nc\nr3−3Q2\nr4−6MN c\nr4+6MQ2\nr5+6NcQ2\nr5+O(M2,Q4). (58)\nThe value of/radicalbig\ndet/vectorgfor BH surrounded by cosmological constant is calculated in the following way\n/radicalig\ndet/vectorg=3M\nr2+1\nr3−3Q2\n2r3+3rNc\n2. (59)\nUtilizing Eqs. ( 58) and ( 59) into Eq. ( 42), we investigate the DA for BH surrounded by cosmological co nstant as\nfollows\nψ5≈4M\nb+2Nc\nb−3MN cπ\n2b2−3Q2π\n4b2+8NcQ2\n3b3+8MQ2\n3b3+O(M2,Q4). (60)\nThe DA in non-plasma medium for a BH surrounded by a cosmologi cal constant depends upon impact parameter\nb, BH mass M, charge Qand cosmological field structure parameter Nc. The above conclusion simplifies to the DA\nof Reissner-Nordstr¨ om BH in the absence of the structure pa rameter Nc=0. Furthermore, we obtain the DA of\nSchwarzschild BH ψSch=4M/b, in the absence of charge Q=0.12\nSch\nQ=0.1\nQ=0.5\nQ=1.0\nQ=1.5\n123456780.51.01.52.02.53.03.54.0\nb\n\u00055(i)M=1,Nc=0.1\nSch\nNc=0.1\nNc=0.5\nNc=1.0\nNc=1.5\n123456780.51.01.52.02.53.03.54.0\nb\n\u00065(ii)M=1,Q=0.5\nFIG. 5: Deflection angle ψ5via impact parameter bwith variations of charge Qand cosmological field structure\nparameter Ncwith fixed mass M=1.\nFig. 5: in the left panel charge Qvaried with fixed mass and the radiation field structure param eter Nc=0.1,\nwhile the right panel gives variations of cosmological field structure parameter Ncwith fixed mass M=1 and\ncharge Q=0.5.\nThe Schwarzschild solution is shown in both panels by the bla ck curve. As the impact parameter grows from\na BH surrounded by cosmological field, the DA decreases with i ncreasing charge and structure parameter values,\nreaching an asymptotically flat form until b→∞. When the impact parameter is small, a strong deflection is se en.\nIn addition, an inverse relationship is established betwee n the impact parameter band the DA ψ5. It should be noted\nthat, for varying charges, the light deflects at certain spot s in the right plot at the same angle as compared to the\nSchwarzschild case, a significant deflection is shown in the p resence of cosmological field.\nIV . DEFLECTION ANGLE IN THE CONTEXT OF PLASMA MEDIUM\nThis section studies the computation of the DA in the context of plasma medium. Plasma is another important\ncomponent that influences a gravitational lens [ 66,117]. Refraction in plasma causes increased deflection. The\nrefractive index attempts to obtain auxiliary components a nd is especially important in the radio regime. Consider\nvbe the velocity of light as it travels through hot, ionised ga s to understand the impacts of plasma. The refractive\nindex, n(r) =c/vfor a BH surrounded by dust field is given by [ 51].\nn(r) =/radicaligg\n1−ω2e\nω2∞A(r), (61)\nwhere c=1 as well as ω2∞is the photon frequency measured by an observer at infinity an dω2eis the electron plasma\nfrequency. The optical metric in terms of plasma medium can b e defined as\ndσ2=gopt\nijdxidxj=n2/parenleftbiggB(r)\nA(r)dr2+r2\nA(r)dφ2/parenrightbigg\n. (62)\nThe optical Gaussian curvature for a BH surrounded by dust fie ld in terms of plasma frame can be calculated as\nfollows\nK≈2M\nr3+(1+6kλ)Nd\nr3+12kMλ\nr3−3Q2\nr4+3(2kλ−1)MN d\nr4+6MQ2\nr5+3(2−λ)NdQ2\n2r5+6Mω2\ne\nr3ω2∞+3Ndω2\ne\nr3ω2∞\n−15λkNdω2\ne\n2r3ω2∞−12MN dω2\ne\nr4ω2∞−5Q2ω2\ne\nr4ω2∞+30kMλNdω2\ne\nr4ω2∞+13NdQ2ω2\ne\nr5ω2∞+26Q2Mω2\ne\nr5ω2∞−15λkNdQ2ω2\ne\nr5ω2∞\n+O(M2,Q3,λ2,ω4\ne). (63)\nUsing the GBT, we calculate the bending angle. To do this, we u se the straight line approximation e=b\nsinφat zeroth\norder, and the DA is obtained as\nψ1≈ −/integraldisplayπ\n0/integraldisplay∞\nb/ sin φK/radicalig\ndet/vectorgdrd φ. (64)13\nThe DA of the BH surrounded by dust field in terms of plasma fram e for the leading order terms is computed as\nψ1≈4M\nb−2Nd\nb−12Ndkλ\nb+24Mkλ\nb−3Q2π\n4b2−3MN dπ\n2b2−3MN dkλπ\n2b2−4NQ2\n3b3+8MQ2\n3b3+2NdQ2kλ\n3b3+6Mω2e\nbω2∞\n+6Ndω2\ne\nbω2∞−15Ndkλω2\ne\nbω2∞−5Q2πλω2\ne\n4b2ω2∞+15MN dkλω2\ne\n2b2ω2∞−21MN dπω2\ne\n4b2ω2∞+52NdQ2ω2\ne\n9b3ω2∞+104MQ2ω2\ne\n9b3ω2∞−20NdQ2kλω2\ne\n3b3ω2∞\n+O(M2,Q4,λ2,ω4\ne). (65)\nThe DA for a BH surrounded by a dust field depends on impact para meter b, BH mass M, charge Q, Rastall geometric\nparameters kand λ, field structure parameters Ndas well as plasma frequencies ω∞&ωe. Moreover, it is important\nto mention here that, when, we ignore the plasma effects in Eq . (65) such that ω2\ne/ω2\n∞→0, then, we recover the\nresults of Eq. ( 44) in a non-plasma medium.\nωe2/ω∞2=0.1\nωe2/ω∞2=0.2\nωe2/ω∞2=0.3\nωe2/ω∞2=0.4\nωe2/ω∞2=0.5\n1 2 3 4 512345\nbΨ1\n(i)M =1,Q =0.5,k\u0007\n=0.1,Nd\n=0.2\nkλ=0.1\nkλ=0.2\nkλ=0.3\nkλ=0.4\nkλ=0.5\n12345678-0.50.00.51.01.52.02.5\nbΨ1(ii)M=1,Q=0.2,\be2/\b\t2=0.8,Nd=0.5\nFIG. 6: Deflection angle ψ1w.r.t impact parameter bwith variations of ratio of plasma frequencies ω2\ne/ω2\n∞and Rastall\nparameters kλwith fixed mass M=1.\nIn Fig. 6: the left panel shows the variations of ratio of plasma frequ encies ω2\ne/ω2\n∞with fixed mass M=1, charge\nQ=0.2, Rastall parameters kλ=0.1 and dust field structure parameter Nd=0.2 and right panel represents the\nvariations of Rastall parameters kλwith fixed mass M=1, charge Q=0.2, ratio of plasma frequencies ω2e/ω2∞=0.8\nand dust field structure parameter Nd=0.5. It is evident that when the impact parameter rises, the D A for increasing\nplasma frequencies and Rastall parameters from a BH encircl ed by a dust field decreases. A significant inclination\nis observable at low impact parameter values where we observ e the strongest deflection. The inverse link between\nimpact parameter band DA ψ1is also visually apparent.\nA. BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD\nThe Gaussian curvature for a BH surrounded by a quintessence field under the influence of plasma medium can\nbe given as\nK≈2M\nr3+4Q2Nq\nr3+9Q2kλNq\n2r3−3Q2\nr4+6MQ2\nr5+3Mω2e\nr3ω2∞+9NqkλQ2ω2e\nr3ω2∞+11NqQ2ω2e\nr3ω2∞−5Q2ω2e\nr4ω2∞+26MQ2ω2e\nr5ω2∞\n+O(M2,Q4,λ2,ω4\ne). (66)\nThe DA of a BH surrounded by the quintessence field in a plasma f rame for the leading order terms is calculated\nas\nψ2≈4M\nb+8NqQ2\nb+9NqQ2kλ\nb−3Q2π\n4b2+8MQ2\n3b3+6Mω2\ne\nbω2∞+22NqQ2ω2e\nbω2∞+18NqkλQ2ω2e\nbω2∞−5Q2πω2\ne\n4b2ω2∞\n+104MQ2ω2\ne\n9b3ω2∞+O(M2,Q4,λ2,ω4\ne). (67)14\nThe DA in plasma medium for a BH surrounded by a quintessence fi eld depends on impact parameter b, BH mass\nM, charge Q, Rastall geometric parameters kand λ, field structure parameters Nqas well as plasma frequencies ω∞\n&ωe. It has also worth to note that, when, we ignore the plasma eff ects in Eq. ( 67) such that ω2e/ω2∞→0, then, we\nrecover the results of Eq. ( 48) in a non-plasma medium.\nωe2/ω∞2=0.1\nωe2/ω∞2=0.2\nωe2/ω∞2=0.3\nωe2/ω∞2=0.4\nωe2/ω∞2=0.5\n1 2 3 4 50.51.01.52.02.53.0\nbΨ2(i)M=1=Nq,Q=0.5,k\n=0.1\nkλ=0.1\nkλ=0.5\nkλ=1.0\nkλ=1.5\nkλ=2.0\n1 2 3 4 51234567\nbΨ2(ii)M=1=Nq,Q=0.5,ωe2/ω\u000b2=0.8\nFIG. 7: Deflection angle ψ2versus impact parameter bwith variations in ratio of plasma frequencies ω2\ne/ω2\n∞and\nRastall parameters kλwith fixed mass, quintessential field structure parameter M=1=Ndand charge Q=0.5.\nIn Fig. 7: the left panel displays variations in ratio of plasma frequ encies ω2\ne/ω2\n∞with fixed mass, quintessential\nfield structure parameter M=1=Nd, charge Q=0.5 and Rastall parameters kλ=0.1, whereas variations in\nRastall parameters kλwith with fixed mass, quintessential field structure paramet erM=1=Nd, charge Q=0.5\nand ratio of plasma frequencies ω2e/ω2∞=0.8 are represented in the right panel. The DA for increasing plasma\nfrequencies and Rastall parameter values is shown to decrea se via growing impact parameter from a BH surrounded\nby a quintessential field, eventually achieving an asymptot ically flat form until b→∞. A notable deviation with\nstrong deflection is observed when the impact parameter is mi nimal. Furthermore, the inverse link between DA ψ2\nand impact parameter bis also noted.\nB. BLACK HOLE SURROUNDED BY THE RADIATION FIELD\nThe Gaussian curvature for the BH metric surrounded by a radi ation field in the presence of plasma can be calcu-\nlated as follows\nK≈2M\nr3−3Q2\nr4+3Nr\nr4+6MQ2\nr5−6MN r\nr5+4NrQ2\nr6+3Mω2e\nr3ω2∞−5Q2ω2e\nr4ω2∞+5Nrω2e\nr4ω2∞+26MQ2ω2e\nr5ω2∞−26MN rω2e\nr5ω2∞\n+20Q2Nrω2\ne\nr6ω2∞+O(M2,Q4,ω4\ne). (68)\nThe DA of a BH surrounded by the radiation field in a plasma medi um can be computed as\nψ3≈4M\nb−3Q2π\n4b2+3Nrπ\n4b2+8MQ2\n3b3−8MN r\n3b3+3NrQ2π\n8b4+6Mω2\ne\nbω2∞+5Nrπω2\ne\n4b2ω2∞−5Q2πω2\ne\n4b2ω2∞−104MN rω2\ne\n9b3ω2∞\n+104MQ2ω2\ne\n9b3ω2∞+15NQ2πω2\ne\n8b4ω2∞+O(M2,Q4,ω4\ne). (69)\nThe DA in plasma frame for a BH surrounded by a radiation field i s dependent on impact parameter b, BH mass M,\ncharge Q, radiation field structure parameter Nras well as plasma frequencies ω∞&ωe. It has also worth to mention\nhere that, when, we neglect the plasma effects in Eq. ( 69) i. e., ω2\ne/ω2\n∞→0, then, we obtain the results of Eq. ( 52) in\na non-plasma frame.15\fe2/ \f∞2=0.1\fe2/ \f∞2=0.2\fe2/ \f∞2=0.3\fe2/\f∞2=0.4\fe2/ \f∞2=0.5\n1 2 3 4 5123456\nb\n\r3(i)M=1,Q=0.5,Nr=0.1\nQ=0.2\nQ=0.4\nQ=0.6\nQ=0.8\nQ=1.0\n1234567812345678\nb\n\u000e3(ii)M=1,ωe2/ω∞2=0.8,Nr=0.1\nFIG. 8: Deflection angle ψ3via impact parameter bwith variations of plasma frequencies ratio ω2\ne/ω2\n∞and charge Q\nwith fixed mass M=1 and radiation field structure parameter Nr.\nFig. 8: in the left panel charge plasma frequencies ratio ω2\ne/ω2\n∞varies with fixed mass and the radiation field\nstructure parameter M=1, charge Q=0.5 and radiation field structure parameter Nr=0.1, whereas the right\npanel shows variations of charge Qwith fixed mass M=1, plasma frequencies ratio ω2e/ω2∞=0.8 and radiation\nfield structure parameter Nr=0.1. As the impact parameter grows from a BH surrounded by a ra diation field,\nthe DA with increasing charge and plasma frequencies ratio d iminishes, reaching an asymptotically flat form until\nb→∞. A significant deflection is shown when the impact parameter i s small. Moreover, an inverse relationship\nbetween the impact parameter band DA ψ3is discovered. A strong deflection occurs with the increasin g ratio of\nplasma frequencies.\nC. BLACK HOLE SURROUNDED BY THE PHANTOM FILED\nIn the presence of plasma frame, the curvature for a BH surrou nded by a phantom field can be computed in the\nfollowing way\nK≈2M\nr3+4kλM\nr3−3Q2\nr4−35NpkλQ2\n2r4+6MQ2\nr5−11NpM\nr5+3Mω2\ne\nr3ω2∞−4Mkλω2\ne\nr3ω2∞−5Q2ω2\ne\nr4ω2∞−55NpkλQ2ω2\ne\nr4ω2∞\n−36MN pω2\ne\nr5ω2∞+26MQ2ω2\ne\nr5ω2∞+O(M2,Q4,λ2,ω4\ne). (70)\nThe DA in a plasma medium of the BH surrounded by a phantom field is derived as\nψ4≈4M\nb+8Mkλ\nb−35NpkλQ2π\n8b2−3Q2π\n4b2+8MQ2\n3b3−44NpM\n9b3+6Mω2\ne\nbω2∞−8Mkλω2\ne\nbω2∞−5Q2πω2\ne\n4b2ω2∞\n−55NpkλQ2πω2\ne\n4b2ω2∞−16NpMω2\ne\nb3ω2∞+104MQ2ω2\ne\n9b3ω2∞+O(M2,Q4,λ2,ω4\ne). (71)\nThe DA in plasma medium for a BH surrounded by a phantom field de pends upon impact parameter b, BH mass\nM, charge Q, Rastall geometric parameters kand λ, field structure parameters Npas well as plasma frequencies ω∞\n&ωe. It has important to mention here that, when, we neglect the p lasma effects in Eq. ( 71) i. e., ω2\ne/ω2\n∞→0, then,\nwe obtain the results of Eq. ( 56) in a non-plasma medium.16\u000fe2/\u000f∞2=0.1\u000fe2/\u000f∞2=0.2\u000fe2/\u000f∞2=0.3\u000fe2/\u000f∞2=0.4\u000fe2/\u000f∞2=0.5\n1 2 3 4 51.01.52.02.53.0\nb\n\u00104(i)M=1,Q=0.3,k\u0011=0.1,Np=0.5\nkλ=0.1\nkλ=0.2\nkλ=0.3\nkλ=0.4\nkλ=0.5\n12345678\n-2\n-1012345\nbΨ4\n\u0012ii )M=1,Q=0.3,ωe2/ω\u00132=0.1,Np=0.5\nFIG. 9: Deflection angle ψ4w.r.t impact parameter bwith variations of plasma frequencies ratio ω2\ne/ω2\n∞and Rastall\nparameters kλwith fixed mass M=1, charge Q=0.5 and phantom field structure parameter Np=0.5.\nFig. 9: in the left panel charge plasma frequencies ratio ω2\ne/ω2\n∞show variations with fixed mass M=1, charge\nQ=0.3, Rastall parameters kλ=0.1 and the phantom field structure parameter Np=0.5, while the right panel\nrepresents the variations of Rastall parameters kλwith fixed mass M=1, charge Q=0.3, plasma frequencies\nratio ω2\ne/ω2\n∞=0.1 and phantom field structure parameter Np=0.5. Due to a rising impact parameter from a BH\nsurrounded by a phantom field, the DA falls as the values of the plasma frequencies ratio and Rastall parameters\nrise. When the impact parameter is modest, a significant defle ction is observed. Furthermore, a negative correlation\nis seen between the DA ψ4and the impact parameter b.\nD. BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT\nIn order to calculate the DA for BH surrounded by cosmologica l constant, we first compute the Gaussian curvature\nin the following way\nK≈2M\nr3+Nc\nr3−3Q2\nr4−6MN c\nr4+6NcQ2\nr5+6MQ2\nr5+3Mω2\ne\nr3ω2∞+3Ncω2\ne\nr3ω2∞−5Q2ω2\ne\nr4ω2∞+16NcQ2ω2\ne\nr5ω2∞+18MN cω2\ne\nr4ω2∞\n+26MQ2ω2\ne\nr5ω2∞+O(M2,Q4,ω4\ne). (72)\nIn a plasma frame, the DA of the BH surrounded by the cosmologi cal constant for the leading order terms is\ncalculated as\nψ5≈4M\nb+2Nc\nb−3MN cπ\n2b2−3Q2π\n4b2+8NcQ2\n3b3+8MQ2\n3b3+6Mω2e\nbω2∞+6Ncω2e\nbω2∞−5Q2πω2e\n4b2ω2∞+9MN cπω2e\n2b2ω2∞+4NcQ2ω2e\nb3ω2∞\n+104MQ2ω2\ne\n9b3ω2∞+O(M2,Q4,ω4\ne). (73)\nThe DA in plasma medium for a BH surrounded by a cosmological c onstant depends upon impact parameter b, BH\nmass M, charge Q, cosmological field structure parameter Ncas well as plasma frequencies ωe&ω∞. It has also\nworth to mention here that, when, we neglect the plasma effec ts in Eq. ( 73) i. e., ω2e/ω2∞→0, then, we get the results\nof Eq. ( 60) in a non-plasma frame.17\u0014e2/\u0014∞2=0.1\u0014e2/ \u0014∞2=0.2\u0014e2/ \u0014∞2=0.3\u0014e2/\u0014∞2=0.4\u0014e2/\u0014∞2=0.5\n1 2 3 4 51.01.52.02.53.03.54.0\nb\n\u00155(i)M=1,Q=0.3,Nc=0.5\nQ=0.1\nQ=0.2\nQ=0.3\nQ=0.4\nQ=0.5\n1 2 3 4 50246810\nb\n\u00165(ii)M=1,ωe2/ω∞2=0.8,Nc=0.5\nFIG. 10: Deflection angle ψ5via impact parameter bwith variations of plasma frequencies ratio ω2\ne/ω2\n∞and charge\nQwith fixed mass M=1 and cosmological field structure parameter Nc=0.5.\nFig. 10: in the left panel plasma frequencies ratio varies with fixed mass M=1, charge Q=0.3 and the radiation\nfield structure parameter Nc=0.5, while the right panel gives variations of charge Qwith fixed mass M=1,\ncosmological field structure parameter Nc=0.5 and plasma frequencies ratio ω2\ne/ω2\n∞=0.8.\nThe DA reduces with increasing plasma frequencies ratio and charge, achieving an asymptotically flat form until\nb→∞. This drop occurs as the impact parameter grows from a BH surr ounded by cosmological field. There is a\nsignificant strong deflection observed when the impact param eter is modest. Furthermore, a negative correlation is\nseen between the DA ψ5and the impact parameter b.\nV . EQUATIONS OF MOTION FOR LIGHT RAYS IN A NON-MAGNETIZED PLA SMA\nThis study examines the shadow cast by a BH surrounded by a dus t field given in Eq. ( 22) and the impact of\nRastall parameters on it. The core of the critical curve, als o known as the apparent boundary, is represented by a\nBH shadow. This curve has been optimised so that light rays th at are a part of it approach a bound orbit of photons\nasymptotically when a distant observer follows them back to the BH [ 66,118].\nFor a photon, the Hamilton-Jacobi method in the equatorial p lane θ=π\n2is given as\nH=1\n2/parenleftig\ngikjijk+ωp(r2)/parenrightig\n=1\n2/parenleftigg\n−y2\nt\nA(r)+y2r\nB(r)+y2\nφ\nD(r)+ωp(r2)/parenrightigg\n, (74)\nwhere jirepresents the momentum of the photon, yφ=jφis its angular momentum and yt=−jtis the photon’s\nenergy. The light rays are the solutions to Hamilton’s equat ions for a derivation of the Hamiltonian Eq. ( 74) from\nMaxwell’s equations with a two-fluid source as\n˙yi=−∂H\n∂xi,˙xi=−∂H\n∂yi. (75)\nThe above equations implies\n˙yt=−∂H\n∂t=0, ˙yφ=−∂H\n∂ψ=0, (76)\n˙yr=−∂H\n∂r=1\n2/parenleftigg\n−y2\ntA′(r)\nA(r)2+y2\nrB′(r)\nB(r)2+y2\nφD′(r)\nD(r)2−d\ndrωp(r2)/parenrightigg\n, (77)\n˙t=∂H\n∂yt=−yt\nA(r), (78)\n˙φ=∂H\n∂yφ=−yφ, (79)\n˙r=∂H\n∂yr=−yr\nB(r). (80)18\nIn this case, a prime denotes differentiation with regard to r, while a dot denotes differentiation with respect to an\naffine parameter λ.\nBy setting H=0 in Eq. ( 74), we get the result in this form\n0=−y2\nt\nA(r)+y2\nr\nB(r)+y2\nφ\nD(r)+ωp(r2). (81)\nFrom Eq. ( 76), it is followed that the ytand yφare constants of motion. We consider ω0=−jt. Given a fixed ω0\nand an asymptotically flat spacetime (e.g., A(r)→1 as r→∞), the gravitational redshift formula transforms the\nfrequency ωas measured by a static observer into a function of rin the given form\nω(r) =ω0/radicalbig\nA(r). (82)\nAs a result of Eq. ( 81), a light beam with constant speed ω0is limited to the area where\nω2\n0\nA(r)>ωp(r)2. (83)\nThe constraint ( 82) states that the photon frequency ω(r)must exceed the plasma frequency ωp(r)at a given place.\nFor light to propagate through a plasma, this is always true. In order to obtain the orbit equation, we use Eq. ( 78)\nand Eq. ( 79) and get\ndr\ndφ=˙r\n˙φ=D(r)yr\nB(r)yφ. (84)\nUsing yrfrom Eq. ( 81), we get the result in this form\ndr\ndφ=±/radicalbig\nD(r)/radicalbig\nB(r)/radicaltp/radicalvertex/radicalvertex/radicalbtω2\n0v(r)2\ny2\nφ−1, (85)\nwhere\nv(r)2=D(r)\nA(r)/parenleftigg\n1−A(r)ω2\np\nω2\n0/parenrightigg\n. (86)\nSince Xis the turning point of the trajectory, the conditiondr\ndφ/vextendsingle/vextendsingle/vextendsingle\nX=0 must hold. This equation connects Xto the\nconstant of motion,yφ\nω0as follows\nv(X)2=y2\nψ\nω2\n0. (87)\nThe shadow’s radius is defined by the original direction of th e photon’s light, which is asymptotically radial towards\nthe outer photon sphere. In order to calculate the shadow of t he BH surrounded by a dust field, we consider a light\nbeam projected with an angle γin the radial direction from the observer point r0. We define γin the following way\ncotγ=±√grr√gφφdr\ndφ/vextendsingle/vextendsingle/vextendsingle\nr=r0=±/radicalbig\nB(r)/radicalbig\nD(r)dr\ndφ/vextendsingle/vextendsingle/vextendsingle\nr=r0. (88)\nWe rewrite the orbit equation Eq. ( 85) after attaining a minimum radius Rby using Eq. ( 87) in the form\ndr\ndφ=±/radicalbig\nD(r)/radicalbig\nB(r)/radicaligg\nv(r0)2\nv(X)2−1. (89)\nUsing Eq. ( 89) into Eq. ( 88) for the angle γ, we get\ncot2γ=v(r0)2\nv(X)2−1. (90)19\nBy using the identity, we obtain\n1+cot2γ=1\nsin2γ. (91)\nUsing Eq. ( 91) into ( 90), we get the result\nsin2γ=v(X)2\nv(r0)2. (92)\nLight rays spiralling asymptotically towards a circular li ght orbit at radius rphdefine the boundary of the shadow γ.\nConsequently, X→rphin Eq. ( 92) provides the angular radius of the shadow as\nsin2γ=v(rph)2\nv(r0)2, (93)\nwhere v(r0)can be represented by the formula in Eq. ( 85) atr=r0. In numerous cases, it is possible to presume that\nthe observer is situated in an area with a negligible plasma d ensity. Then Eq. ( 85) implies\nv(r0)2=D(r0)\nA(r0), (94)\nand Eq. ( 93) can be written as follows\nsin2γ=A(r0)D(rph)\nA(rph)D(r0)/parenleftigg\n1−A(rph)ω2p(rph)\nω2o/parenrightigg\n. (95)\nAfter substituting the values of metric function into the ab ove equation, we obtain the result into following manner\nsin2γ=r2\nph/parenleftbigg\n2Mr0−Q2−r2\n0+Ndr1\n1−3kλ\n0/parenrightbigg/parenleftbigg\nQ2ω2\np−2Mrphω2\np+r2\nph(ω2\nP−ω2\n0)−Ndr1\n1−3kλ\nphω2\np/parenrightbigg\nr4\n0/parenleftbigg\nQ2−2Mrph+r2\nph−Ndr1\n1−3kλ\nph/parenrightbigg\nω2\n0. (96)\nThe shadow of the BH surrounded by a dust field is dependent on B H mass M, charge Q, Rastall parameters k,λ,\ndust field structure parameter Nd, plasma frequencies ωp,ω0, photon radius rphand observer radius r0.\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nαβ(i)M=1=r0,Q=0.3,ωp=0.1=kλ,ω0=0.5=Nd\n-3-2-10123-3-2-10123\nαβ(ii)M=1=rph,Q=0.3,ωp=0.1=kλ,ω0=0.5=Nd\nFIG. 11: Shadow of BH surrounded by dust field in celestial pla ne(α,β)for variations of photon radius rphand\nobserver radius r0.20\nFig. 11: the left panel shows the contour plots of shadow for variati ons of photon radius rph=0.5 (purple), 0.6\n(yellow), 0.7 (green), 0.8 (orange), 0.9 (blue) and 1 (red) w ith fixed values of mass M=1, observer radius r0=1,\ncharge Q=0.3, plasma frequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure\nparameter Nd=0.5. One can see that when the photon radius rises, the shadow radius size also increases.\nThe right panel gives the contour plots of shadow for differe nt choices of radius r0=0.5 (purple), 0.6 (yellow),\n0.7 (green), 0.8 (orange), 0.9 (blue), 1 (red) and fixed mass M=1, charge Q=0.3, photon radius rph=1, plasma\nfrequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure parameter Nd=0.5. It can be\nobserved that the size of shadow radius decreases with incre asing observers radius r0and it is bigger than photon\nradius.\nA. BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD\nAfter putting the values of metric function from Eq. ( 45) into Eq. ( 95), we get the result in the given form\nsin2γ1=r2\nph/parenleftbigg\n2Mr0−Q2−r2\n0+Nqr1+3kλ\n−1+kλ\n0/parenrightbigg/parenleftbigg\nQ2ω2\np−2Mrphω2\np+r2\nph(ω2\nP−ω2\n0)−Nqr1+3kλ\n−1+kλ\nphω2\np/parenrightbigg\nr4\n0/parenleftbigg\nQ2−2Mrph+r2\nph−Nqr1+3kλ\n−1+kλ\nph/parenrightbigg\nω2\n0. (97)\nThe shadow of the BH surrounded by a quintessence field is depe ndent on BH mass M, charge Q, Rastall parameters\nk,λ, quintessence field structure parameter Nq, plasma frequencies ωp,ω0, photon radius rphand observer radius\nr0.\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nαβ(i)M=1=r0,Q=0.3,ωp=0.1=kλ,ω0=0.5=Nq\n-3-2-10123-3-2-10123\nαβ(ii)M=1=rph,Q=0.3,ωp=0.1=kλ,ω0=0.5=Nd\nFIG. 12: Shadow of BH surrounded by quintessential field in ce lestial plane (α,β)for variations of photon radius rph\nand observer radius r0.\nFig. 12: the left panel represents the contour plots of shadow for va riations of photon radius rph=0.5 (purple),\n0.6 (yellow), 0.7 (green), 0.8 (orange), 0.9 (blue) and 1 (re d) with fixed values of mass M=1, observer radius r0=1,\ncharge Q=0.3, plasma frequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure\nparameter Nq=0.5. We can observe that when the photon radius rises, the sha dow radius size also increases.\nThe right panel gives the contour plots of shadow for differe nt choices of radius r0=0.5 (purple), 0.6 (yellow),\n0.7 (green), 0.8 (orange), 0.9 (blue), 1 (red) and fixed mass M=1, charge Q=0.3, photon radius rph=1, plasma\nfrequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure parameter Nq=0.5. It can be\nobserved that the size of shadow radius decreases with incre asing observers radius r0.21\nB. BLACK HOLE SURROUNDED BY THE RADIATION FIELD\nAfter substituting the values of metric function from Eq. ( 49) into Eq. ( 95), we obtain the result in the given form\nsin2γ2=r2\nph/parenleftbig\n2Mr0−Q2−r2\n0+Nr/parenrightbig/parenleftig\nQ2ω2p−2Mrphω2p+r2\nph(ω2\nP−ω2\n0)−Nrω2p/parenrightig\nr4\n0/parenleftig\nQ2−2Mrph+r2\nph−Nr/parenrightig\nω2\n0. (98)\nThe shadow of the BH surrounded by a phantom field depends upon BH mass M, charge Q, radiation field structure\nparameter Nr, plasma frequencies ωp,ω0, photon radius rphand observer radius r0.\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nαβ(i)M=1=r0,Q=0.3,ωp=0.1,ω0=0.5=Nr\n-3-2-10123-3-2-10123\nαβ(ii)M=1=rph,Q=0.3,ωp=0.1,ω0=0.5=Nr\nFIG. 13: Shadow of BH surrounded by radiation field in celesti al plane(α,β)for variations of photon radius rphand\nobserver radius r0.\nFig. 13: the left panel illustrate the contour plots of shadow for va riations of photon radius rph=0.5 (purple), 0.6\n(yellow), 0.7 (green), 0.8 (orange), 0.9 (blue) and 1 (red) w ith fixed values of mass M=1, observer radius r0=1,\ncharge Q=0.3, plasma frequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure\nparameter Nr=0.5. We can see that the shadow radius size increases along wi th the photon radius.\nThe right panel gives the contour plots of shadow for differe nt choices of radius r0=0.5 (purple), 0.6 (yellow),\n0.7 (green), 0.8 (orange), 0.9 (blue), 1 (red) and fixed mass M=1, charge Q=0.3, photon radius rph=1, plasma\nfrequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure parameter Nr=0.5. It is shown\nthat when the observers radius r0increases, the shadow radius’s size reduces.\nC. BLACK HOLE SURROUNDED BY THE PHANTOM FILED\nAfter putting the values of metric function from Eq. ( 53) into Eq. ( 95), we get the result as follows\nsin2γ3=r2\nph/parenleftbigg\n2Mr0−Q2−r2\n0+Npr5\n1+kλ\n0/parenrightbigg/parenleftbigg\nQ2ω2p−2Mrphω2p+r2\nph(ω2\nP−ω2\n0)−Npr5\n1+kλ\nphω2p/parenrightbigg\nr4\n0/parenleftbigg\nQ2−2Mrph+r2\nph−Npr5\n1+kλ\nph/parenrightbigg\nω2\n0. (99)\nThe shadow of the BH surrounded by a phantom field depends on BH mass M, charge Q, Rastall parameters k,λ,\nphantom field structure parameter Np, plasma frequencies ωp,ω0, photon radius rphand observer radius r0.22\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nαβ(i)M=1=r0,Q=0.3,ωp=0.1=kλ,ω0=0.5=Np\n-2 -1012-2-1012\nαβ(ii)M=1=rph,Q=0.3,ωp=0.1=kλ,ω0=0.5=Np\nFIG. 14: Shadow of BH surrounded by phantom field in celestial plane(α,β)for variations of photon radius rphand\nobserver radius r0.\nFig. 14: the left panel depicts the contour plots of shadow for diffe rent choices of photon radius rph=0.5 (purple),\n0.6 (yellow), 0.7 (green), 0.8 (orange), 0.9 (blue) and 1 (re d) with fixed values of mass M=1, observer radius r0=1,\ncharge Q=0.3, plasma frequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure\nparameter Np=0.5. It is evident that as the photon radius rises, so does the shadow radius size.\nThe right panel shows the contour plots of shadow for variati ons of radius r0=0.5 (purple), 0.6 (yellow), 0.7\n(green), 0.8 (orange), 0.9 (blue), 1 (red) and fixed mass M=1, charge Q=0.3, photon radius rph=1, plasma\nfrequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure parameter Np=0.5. It is\ndemonstrated that the size of the shadow radius decreases wi th increasing observers radius r0.\nD. BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT\nAfter using the values of metric function from Eq. ( 57) into Eq. ( 95), we acquire the result in the given form\nsin2γ4=r2\nph/parenleftbig\n2Mr0−Q2−r2\n0+Ncr4\n0/parenrightbig/parenleftig\nQ2ω2p−2Mrphω2p+r2\nph(ω2\nP−ω2\n0)−Ncr4\nphω2p/parenrightig\nr4\n0/parenleftig\nQ2−2Mrph+r2\nph−Ncr4\nph/parenrightig\nω2\n0. (100)\nThe shadow of the BH surrounded by a cosmological constant de pends on BH mass M, charge Q, cosmological field\nstructure parameter Nc, plasma frequencies ωp,ω0, photon radius rphand observer radius r0.23\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\nαβ(i)M=1=r0,Q=0.3,ωp=0.1,ω0=0.5=Nc\n-2 -10123-2-10123\nαβ(ii)M=1=rph,Q=0.3,ωp=0.1,ω0=0.5=Nc\nFIG. 15: Shadow of BH surrounded by cosmological field in cele stial plane (α,β)for variations of photon radius rph\nand observer radius r0.\nFig. 15: the left panel represents the contour plots of shadow for di fferent choices of photon radius rph=0.5\n(purple), 0.6 (yellow), 0.7 (green), 0.8 (orange), 0.9 (blu e) and 1 (red) with fixed values of mass M=1, observer\nradius r0=1, charge Q=0.3, plasma frequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field\nstructure parameter Nc=0.5. It is clear that the shadow radius size increases with th e photon radius.\nThe right panel shows the contour plots of shadow for variati ons of radius r0=0.5 (purple), 0.6 (yellow), 0.7\n(green), 0.8 (orange), 0.9 (blue), 1 (red) and fixed mass M=1, charge Q=0.3, photon radius rph=1, plasma\nfrequencies ωp=0.1, ω0=0.5, Rastall parameters kλ=0.1 and dust field structure parameter Nc=0.5. It is shown\nthat as the observers radius r0increases, the shadow radius’s size decreases.\nVI. CONCLUSIONS\nIn the framework of Rastall theory, we have studied generic c harged BH solutions surrounded by an ideal fluid.\nNext, we have looked more closely into the particular situat ions of BHs encircled by dust, quintessence, radiation,\nphantom fields and cosmological constant. In the generalize d theory, physical parameters are studied by applying\nthe weak energy condition, which signifies a positive energy density. An effective behaviour of the BH surrounding\nthe field is realized by comparing the solution in GR with the n ew term in the metric that arose from the Rastall\ntheory. The ray tracing and generalized GW methods offer a co mprehensive Rastall framework for characterizing\nstrategies for the Kiselev BH solution. In the non-plasma, p lasma and perfect fluid mediums, we have talked about\nthe Rastall solution and derived DA by using the GW technique . The Gaussian optical curvature is determined for\nthis purpose using an optical metric and we have then examine d the DA using GBT.\nFirstly, we investigated the DA for BHs surrounded by dust, q uintessence, radiation, phantom and cosmological\nfields in a non-plasma medium. We have concluded from our resu lts that BHs surrounded by dust, quintessence and\nphantom fields are dependent on impact parameter b, BH mass M, charge Q, Rastall geometric parameters k,λand\nfield structure parameters Nd,Nqand Np, respectively. Moreover, the DA for BHs surrounded by radia tion field and\ncosmological constant depends upon on impact parameter b, BH mass M, charge Qand field structure parameters\nNrand Nc, respectively. We also observed analytically and graphica lly that the impact parameter bis in an inverse\nrelation with the DA of corresponding BHs. Moreover, it has w orth to mention here that, in the absence of structure\nand Rastall parameters, the results of DA of our correspondi ng BHs reduce into DA of Reissner–Nordstr¨ om BH as\nwell as in the absence of charge Q=0, we recovered the DA of Schwarzschild BH ψSch=4M/b.\nWith the help of graphical interpretation, we showed that th e impact parameter bhas an inverse relation with\nDA. Moreover, by comparing the graphical conduct of DA of our corresponding BHs with Schwarzschild case,\nwe concluded that BHs surrounded by dust, quintessence field s and cosmological constants have less deflection\nas compared to Schwarzschild BH. In addition, we observed st rong deflection is seen for small value of impact24\nparameter in all plots of non-plasma medium.\nSecondly, we examined the DA for BHs surrounded by dust, quin tessence, radiation, phantom and cosmological\nfields in a plasma medium and concluded that the DA for BHs surr ounded by dust, quintessence and phantom\nfields depends upon impact parameter b, BH mass M, charge Q, Rastall geometric parameters kand λ, as well as\nplasma frequencies ω∞&ωeand field structure parameters Nd,Nqand Np, respectively. Moreover, the DA for BHs\nsurrounded by radiation field and cosmological constant dep ends upon on impact parameter b, BH mass M, charge\nQ, plasma frequencies ω∞&ωeand field structure parameters Nrand Nc, respectively. It has important to mention\nhere that, when, we neglect the plasma effects in our results i. e., ω2e/ω2∞→0, then, we obtained the results of DA\nin a non-plasma medium for corresponding BHs.\nAfter that, we analyzed the graphical results for DA in the pr esence of plasma for variations of plasma frequencies\nratio ω2e/ω2∞, Rasatll parameters kλand charge Q. Furthermore, from our graphical analysis, we concluded th at the\nDA for increasing plasma frequencies and Rastall parameter values is shown to decrease via growing impact param-\neter from a BH surrounded by a dust, quintessential and phant om field that eventually achieving an asymptotically\nflat form until b→∞. A notable deviation with strong deflection is observed when the impact parameter is minimal.\nFurthermore, the inverse link between DA and impact paramet erbis also noted in the presence of plasma medium.\nAt last, we investigated the shadows of BHs defined by the Kise lev solution in the Rastall theory. A Hamiltonian\nequation that is variable-separable and a light-ray motion equation are independent of the plasma’s velocity in the\npresence of a pressure-less and non-magnetized plasma. We e xplored the perfect fluid in Rastall theory, where\nparticle accumulation is correlated with the plasma freque ncy. The scenario in which plasma falls radially upon a\nBH from infinity is investigated in terms of dust, quintessen ce, radiation, phantom fields and cosmological constants\nencircled by a Rastall solution. Moreover, the shadow of our corresponding BHs depends on BH mass M, charge Q,\nRastall parameters k,λ, plasma frequencies ωp,ω0, photon radius rph, observer radius r0and their field structure\nparameters Nd,Nq,Nr,Npand Nc, respectively. We plotted the contour plots for correspond ing BHs for their\nshadows with variations of photon radius rphas well as observer radius r0. We concluded that the size of shadow\nradius increases with the rise in photon radius whereas the s ize of shadow decreases with rise in values of observer’s\nradius.\nThe presence of the conical singularity on the symmetric axi s in the background of Rastall spacetime is responsible\nfor the appearance of the BHs encircled by phantom fields, qui ntessence, cosmological constants, radiation, and dust.\nExamining how a conical singularity affects the DA and shado w variables is interesting. Future reports on this work\nwill be provided.\nAcknowledgement\nThe paper was funded by the National Natural Science Foundat ion of China 11975145. A. ¨O. would like to ac-\nknowledge the contribution of the COST Action CA21106 - COSM IC WISPers in the Dark Universe: Theory, as-\ntrophysics and experiments (CosmicWISPers) and the CA2211 3 - Fundamental challenges in theoretical physics\n(THEORY-CHALLENGES).\n[1] C. M. Will, The confrontation between general relativit y and experiment, Living reviews in relativity 17(2014) 1.\n[2] S. Weinberg, Principles and Applications of the General Theory of Relativity: Gravitation and Cosmology, Wiley (19 72).\n[3] A. Vilenkin, Cosmic strings as gravitational lenses, Th e Astrophysical Journal 282(1984) L51.\n[4] A. Vilenkin, Cosmic strings and domain walls, Physics re ports 121(1985) 263.\n[5] J.R. Gott III, Gravitational lensing effects of vacuum s trings-exact solutions, The Astrophysical Journal 288(1985) 422.\n[6] V . Bozza, Gravitational lensing in the strong field limit , Physical Review D 66(2002) 103001.\n[7] O. Wucknitz and U. Sperhake, Deflection of light and parti cles by moving gravitational lenses, Physical Review D 69(2004)\n063001.\n[8] W. Rindler and M. Ishak, Contribution of the cosmologica l constant to the relativistic bending of light revisited, P hysical\nReview D 76(2007) 043006.\n[9] J. Sultana, Contribution of the cosmological constant t o the bending of light in kerr–de sitter spacetime, Physical Review D\n88(2013) 042003.\n[10] A. Bhattacharya, R. Isaev, M. Scalia, C. Cattani and K.K . Nandi, Light bending in the galactic halo by rindler-ishak method,\nJournal of Cosmology and Astroparticle Physics 2010 (2010) 004.\n[11] A. Bhattacharya, G.M. Garipova, E. Laserra, A. Bhadra a nd K.K. Nandi, The vacuole model: new terms in the second orde r\ndeflection of light, Journal of Cosmology and Astroparticle Physics 2011 (2011) 028.25\n[12] C. Cattani, M. Scalia, E. Laserra, I. Bochicchio and K.K . Nandi, Correct light deflection in weyl conformal gravity, Physical\nReview D 87 (2013) 047503.\n[13] A. Mishra and S. Chakraborty, On the trajectories of nul l and timelike geodesics in different wormhole geometries, The\nEuropean Physical Journal C 78(2018) 1.\n[14] M. Ishak, Light deflection, lensing, and time delays fro m gravitational potentials and fermat’s principle in the pr esence of a\ncosmological constant, Physical Review D 78(2008) 103006.\n[15] M. Sereno, Influence of the cosmological constant on gra vitational lensing in small systems, Physical Review D 77(2008)\n043004.\n[16] E. Gallo and O.M. Moreschi, Gravitational lens optical scalars in terms of energy-momentum distributions, Physic al Review\nD83(2011) 083007.\n[17] V . Bozza and A. Postiglione, Alternatives to schwarzsc hild in the weak field limit of general relativity, Journal of Cosmology\nand Astroparticle Physics 2015 (2015) 036.\n[18] E.F. Boero and O.M. Moreschi, Gravitational lens optic al scalars in terms of energy–momentum distributions in the cosmo-\nlogical framework, Monthly Notices of the Royal Astronomic al Society 475(2018) 4683.\n[19] G. Crisnejo and E. Gallo, Expressions for optical scala rs and deflection angle at second order in terms of curvature s calars,\nPhysical Review D 97(2018) 084010.\n[20] D. Glavan and C. Lin, Einstein-gauss-bonnet gravity in four-dimensional spacetime, Physical Review letters 124(2020)\n081301.\n[21] R. Kumar, S.U. Islam and S.G. Ghosh, Gravitational lens ing by charged black hole in regularized 4d einstein–gauss– bonnet\ngravity, The European Physical Journal C 80(2020) 1.\n[22] K. S. Virbhadra and G. F. R. Ellis, Schwarzschild black h ole lensing, Physical Review D 62, 084003 (2000).\n[23] K. S. Virbhadra and G. F. R. Ellis, Gravitational lensin g by naked singularities, Physical Review D 65, 103004 (2002).\n[24] K. S. Virbhadra, Relativistic images of Schwarzschild black hole lensing, Physical Review D 79, 083004 (2009).\n[25] G. Lambiase, R. C. Pantig, D. J. Gogoi and A. ¨Ovg ¨ un, Investigating the connection between generalized uncertainty prin-\nciple and asymptotically safe gravity in black hole signatu res through shadow and quasinormal modes, The European\nPhysical Journal C 83, no.7, 679 (2023).\n[26] K. S. Virbhadra, Distortions of images of Schwarzschil d lensing, Physical Review D 106, no.6, 064038 (2022).\n[27] J. R. Nascimento, A. Y. Petrov, P . J. Porfirio and A. R. Soa res, Gravitational lensing in black-bounce spacetimes, Ph ysical\nReview D 102, no.4, 044021 (2020).\n[28] X. M. Kuang and A. ¨Ovg ¨ un, Strong gravitational lensing and shadow constrain t from M87* of slowly rotating Kerr-like\nblack hole, Annals of Physics 447, 169147 (2022).\n[29] X. M. Kuang, Z. Y. Tang, B. Wang and A. Wang, Constraining a modified gravity theory in strong gravitational lensing an d\nblack hole shadow observations, Physical Review D 106, no.6, 064012 (2022).\n[30] C. Furtado, J. R. Nascimento, A. Y. Petrov, P . J. Porf´ ır io and A. R. Soares, Strong gravitational lensing in a spacet ime with\ntopological charge within the Eddington-inspired Born-In feld gravity, Physical Review D 103, no.4, 044047 (2021).\n[31] Y. Kumaran and A. ¨Ovg ¨ un, Deflection Angle and Shadow of the Reissner–Nordstr ¨ om Black Hole with Higher-Order Mag-\nnetic Correction in Einstein-Nonlinear-Maxwell Fields, S ymmetry 14(2022) 2054.\n[32] K. Jafarzade, M.K. Zangeneh and F.S. Lobo, Shadow, defle ction angle and quasinormal modes of born-infeld charged bl ack\nholes, Journal of Cosmology and Astroparticle Physics 2021 (2021) 008.\n[33] F. W. Dyson, A.S. Eddington and C. Davidson, Ix. a determ ination of the deflection of light by the sun’s gravitational field,\nfrom observations made at the total eclipse of may 29, 1919, P hilosophical Transactions of the Royal Society of London,\nSeries A 220(1920) 291.\n[34] S. Dodelson, Gravitational lensing, Cambridge Univer sity Press (2017).\n[35] B.R. Patla, R.J. Nemiroff, D.H. Hoffmann and K. Zioutas , Flux enhancement of slow-moving particles by sun or jupite r: can\nthey be detected on earth?, The Astrophysical Journal 780(2013) 158.\n[36] J. Liu and M.S. Madhavacheril, Constraining neutrino m ass with the tomographic weak lensing one-point probabilit y dis-\ntribution function and power spectrum, Physical Review D 99(2019) 083508.\n[37] A. Accioly and R. Paszko, Photon mass and gravitational deflection, Physical Review D 69(2004) 107501.\n[38] A. Bhadra, K. Sarkar and K. Nandi, Testing gravity at the second post-newtonian level through gravitational deflect ion of\nmassive particles, Physical Review D 75(2007) 123004.\n[39] O.Y. Tsupko, Unbound motion of massive particles in the schwarzschild metric: Analytical description in case of st rong\ndeflection, Physical Review D 89(2014) 084075.\n[40] X. Liu, N. Yang and J. Jia, Gravitational lensing of mass ive particles in schwarzschild gravity, Classical and Quan tum\nGravity 33(2016) 175014.\n[41] G. He and W. Lin, Gravitational deflection of light and ma ssive particles by a moving kerr–newman black hole, Classic al\nand Quantum Gravity 33(2016) 095007.\n[42] G. He and W. Lin, Analytical derivation of second-order deflection in the equatorial plane of a radially moving\nkerr–newman black hole, Classical and Quantum Gravity 34(2017) 105006.\n[43] X. Pang and J. Jia, Gravitational lensing of massive par ticles in reissner–nordstrom black hole spacetime, Classi cal and\nQuantum Gravity 36(2019) 065012.\n[44] Z. H. Li, X. Zhou, W. J. Li and G. S. He, Gravitational defle ction of massive particles by a schwarzschild black hole in\nradiation gauge, Communications in Theoretical Physics 71(2019) 1219.\n[45] X. He, T. Xu, Y. Yu, A. Karamat, R. Babar, R. Ali, Deflectio n angle evolution with plasma medium and without plasma\nmedium in a parameterized black hole. Ann. Phys. 451(2023) 169247.26\n[46] X. He, S. Zhu, Y. Yu, A. Karamat, R. Babar, R. Ali, Deflecti on angle analysis under the influence of non-plasma medium an d\nplasma medium for regular black hole with cosmic string. Int . J. Geom. Methods Mod. Phys. 20, 2350205(2023).\n[47] G. Gibbons and M. Werner, Applications of the gauss–bon net theorem to gravitational lensing, Classical and Quantu m\nGravity 25(2008) 235009.\n[48] M. Werner, Gravitational lensing in the kerr-randers o ptical geometry, General Relativity and Gravitation 44(2012) 3047.\n[49] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura and H. Asada, Gravitational bending angle of light for finite distance and the\ngauss-bonnet theorem, Physical Review D 94(2016) 084015.\n[50] T. Ono, A. Ishihara and H. Asada, Gravitomagnetic bendi ng angle of light with finite-distance corrections in statio nary\naxisymmetric spacetimes, Physical Review D 96(2017) 104037.\n[51] G. Crisnejo and E. Gallo, Weak lensing in a plasma medium and gravitational deflection of massive particles using the\ngauss-bonnet theorem. a unified treatment, Physical Review D97(2018) 124016.\n[52] Z. Li and J. Jia, The finite-distance gravitational defle ction of massive particles in stationary spacetime: a jacob i metric\napproach, The European Physical Journal C 80(2020) 1.\n[53] R. Ali, M. Awais and A. Mahmood, Study of the Deflection An gle and Shadow of Black Hole Solutions in Non-Plasma and\nPlasma Mediums Under the Effect of Einstein-Gauss-Bonnet G ravity, Annalen der Physik 535(2023) 2300236.\n[54] Event Horizon Telescope Collaboration, Akiyama, K., A lberdi, A., et al. 2019a, The Astrophysical Journal Letters ,875, L2.\n[55] J. L. Synge, The Escape of Photons from Gravitationally Intense Stars, Monthly Notices of the Royal Astronomical So ciety\n131(1966) 463.\n[56] P . V . P . Cunha, C. A. R. Herdeiro and E. Radu, Fundamental photon orbits: black hole shadows and spacetime instabilit ies,\nPhysical Review D 96(2017) 024039.\n[57] M. Wang, S. Chen and J. Jing, Shadow casted by a Konoplya- Zhidenko rotating non-Kerr black hole, Journal of Cosmolog y\nand Astroparticle Physics 10(2017) 051.\n[58] Z. Li and C. Bambi, Measuring the Kerr spin parameter of r egular black holes from their shadow, Journal of Cosmology\nand Astroparticle Physics 1401 (2014) 041.\n[59] K. Hioki and K. I. Maeda, Measurement of the Kerr Spin Par ameter by Observation of a Compact Object’s Shadow, Physica l\nReview D 80(2009) 024042.\n[60] R. A. Konoplya, Shadow of a black hole surrounded by dark matter, Physics Letters B 795(2019) 1.\n[61] X. Hou, Z. Xu, M. Zhou and J. Wang, Black Hole Shadow of Sgr A in Dark Matter Halo, Journal of Cosmology and Astropar-\nticle Physics 07(2018) 015.\n[62] L. Amarilla, E. F. Eiroa, and G. Giribet, Null geodesics and shadow of a rotating black hole in extended Chern-Simons\nmodified gravity, Physical Review D 81(2010) 124045.\n[63] F. Long, S. Chen, M. Wang, and J.Jing, Shadow of a disform al Kerr black hole in quadratic DHOST theories, The European\nPhysical Journal C 80(2020) 1180.\n[64] H. C. D. L. Junior, P . V . P . Cunha, C. A. R. Herdeiro and L. C . B. Crispino, Shadows and lensing of black holes immersed in\nstrong magnetic fields, Physical Review D 104(2021) 044018.\n[65] M. Wang, S. Chen, J. Jing, Kerr Black hole shadows in Melv in magnetic field with stable photon orbits, Physical Review D\n104(2021) 084021.\n[66] V . Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, In fluence of a plasma on the shadow of a spherically symmetric bl ack\nhole, Physical Review D 92(2015) 104031.\n[67] S. Chen, M. Wang, J. Jing, Polarization effects in Kerr b lack hole shadow due to the coupling between photon and bumbl ebee\nfield, Journal of High Energy Physics 07(2020) 054.\n[68] A. Grenzebach, V . Perlick, and C. Lammerzahl, Photon Re gions and Shadows of Kerr-NewmanNUT Black Holes with a\nCosmological Constant, Physical Review D 89(2014) 124004.\n[69] V . Perlick, O. Y. Tsupko and G. S. BisnovatyiKogan, Blac k hole shadow in an expanding universe with a cosmological\nconstant, Physical Review D 97(2018) 104062.\n[70] M. Wang, S. Chen, J. Wang, J. Jing, Shadow of a Schwarzsch ild black hole surrounded by a Bach-Weyl ring, The European\nPhysical Journal C 80(2020) 110.\n[71] P . V . P . Cunha, N. A. Eiro, C. A. R. Herdeiro and J. P . S. Lem os, Lensing and shadow of a black hole surrounded by a heavy\naccretion disk, Journal of Cosmology and Astroparticle Phy sics03(2020) 035.\n[72] M. Wang, S. Chen, J. Wang, J. Jing, Effect of gravitation al wave on shadow of a Schwarzschild black hole, The European\nPhysical Journal C 81(2021) 509.\n[73] P . G. Nedkova, V . Tinchev, and S. S. Yazadjiev, Shadow of a Rotating Traversable Wormhole, Physical Review D 88(2013)\n124019.\n[74] G. Gyulchev, P . Nedkova, V . Tinchev and S. Yazadjiev, On the shadow of rotating traversable wormholes, The European\nPhysical Journal C 78 (2018) 544.\n[75] N. Ortiz, O. Sarbach and T. Zannias, Shadow of a naked sin gularity, Physical Review D 92(2015) 044035.\n[76] D. Dey, R. Shaikh, and P . S. Joshi, Shadow of nulllike and timelike naked singularities without photon spheres, Phys ical\nReview D 103(2021) 024015.\n[77] P . V . P . Cunha, C. A. R. Herdeiro, E. Radu and H. F. Runarss on, Shadows of Kerr black holes with scalar hair, Physical\nReview Letters 115(2015) 211102.\n[78] F. H. Vincent, E. Gourgoulhon, C. A. R. Herdeiro and E. Ra du, Astrophysical imaging of Kerr black holes with scalar ha ir,\nPhysical Review D 94(2016) 084045.\n[79] P . V . P . Cunha, J. Grover, C. A. R. Herdeiro, E. Radu, H. Ru narsson, and A. Wittig, Chaotic lensing around boson stars a nd\nKerr black holes with scalar hair, Physical Review D 94(2016) 104023.27\n[80] M. Wang, S. Chen, J. Jing, Shadows of Bonnor black dihole by chaotic lensing, Physical Review D 97(2016) 064029.\n[81] M. Wang, S. Chen, J. Jing, Chaotic shadow of a non-Kerr ro tating compact object with quadrupole mass moment, Physica l\nReview D 98(2016) 104040.\n[82] T. Johannsen, Photon Rings around Kerr and Kerr like Bla ck Holes, The Astrophysical Journal 777(2013) 170.\n[83] D. Nitta, T. Chiba and N. Sugiyama, Shadows of Multi-Bla ck Holes: Analytic Exploration, Physical Review D 86(2012)\n103001.\n[84] A. F. Zakharov, Constraints on a Tidal Charge of the Supe rmassive Black Hole in M87* with the EHT Observations in Apri l\n2017, Universe 8, no.3, 141 (2022).\n[85] S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A . Allahyari, P . Bambhaniya, D. Dey, S. G. Ghosh and P . S. Joshi ,et al.\nHorizon-scale tests of gravity theories and fundamental ph ysics from the Event Horizon Telescope image of Sagittarius A,\nClassical and Quantum Gravity 40, no.16, 165007 (2023).\n[86] S. Vagnozzi and L. Visinelli, Hunting for extra dimensi ons in the shadow of M87*, Physical Review D 100, no.2, 024020\n(2019).\n[87] A. Allahyari, M. Khodadi, S. Vagnozzi and D. F. Mota, Mag netically charged black holes from non-linear electrodyna mics\nand the Event Horizon Telescope, Journal of Cosmology and As troparticle Physics 02, 003 (2020).\n[88] M. Khodadi, A. Allahyari, S. Vagnozzi and D. F. Mota, Bla ck holes with scalar hair in light of the Event Horizon Telesc ope,\nJournal of Cosmology and Astroparticle Physics 09, 026 (2020).\n[89] A. Abdujabbarov, M. Amir, B. Ahmedov and S. G. Ghosh, Sha dow of rotating regular black holes, Physical Review D 93,\nno.10, 104004 (2016).\n[90] F. Atamurotov, A. Abdujabbarov and B. Ahmedov, Shadow o f rotating non-Kerr black hole, Physical Review D 88, no.6,\n064004 (2013).\n[91] M. Afrin, R. Kumar and S. G. Ghosh, Parameter estimation of hairy Kerr black holes from its shadow and constraints fro m\nM87*, Monthly Notices of the Royal Astronomical Society 504, 5927-5940 (2021).\n[92] R. Kumar and S. G. Ghosh, “Rotating black holes in 4 DEinstein-Gauss-Bonnet gravity and its shadow,” Journal of Cosmol-\nogy and Astroparticle Physics 07, 053 (2020).\n[93] F. Atamurotov, I. Hussain, G. Mustafa and A. ¨Ovg ¨ un, Weak deflection angle and shadow cast by the charged- Kiselev black\nhole with cloud of strings in plasma*, Chinese Physics C 47, no.2, 025102 (2023).\n[94] G. Mustafa, F. Atamurotov, I. Hussain, S. Shaymatov and A.¨Ovg ¨ un, Shadows and gravitational weak lensing by the\nSchwarzschild black hole in the string cloud background wit h quintessential field*, Chinese Physics C 46, no.12, 125107\n(2022).\n[95] A. Uniyal, R. C. Pantig and A. ¨Ovg ¨ un, “Probing a non-linear electrodynamics black hole w ith thin accretion disk, shadow,\nand deflection angle with M87* and Sgr A* from EHT,” Phys. Dark Univ. 40, 101178 (2023).\n[96] R. C. Pantig, L. Mastrototaro, G. Lambiase and A. ¨Ovg ¨ un, “Shadow, lensing, quasinormal modes, greybody bou nds and\nneutrino propagation by dyonic ModMax black holes,” Eur. Ph ys. J. C 82, no.12, 1155 (2022).\n[97] R. C. Pantig, A. ¨Ovg ¨ un and D. Demir, “Testing symmergent gravity through th e shadow image and weak field photon\ndeflection by a rotating black hole using the M87∗and Sgr. A∗results,” Eur. Phys. J. C 83, no.3, 250 (2023).\n[98] R. C. Pantig and A. ¨Ovg ¨ un, “Black Hole in Quantum Wave Dark Matter,” Fortsch. P hys. 71, no.1, 2200164 (2023).\n[99] B. Pulic ¸e, R. C. Pantig, A. ¨Ovg ¨ un and D. Demir, “Constraints on charged symmergent bla ck hole from shadow and lensing,”\nClass. Quant. Grav. 40, no.19, 195003 (2023).\n[100] G. Lambiase, L. Mastrototaro, R. C. Pantig and A. Ovgun , “Probing Schwarzschild-like black holes in metric-affine bumble-\nbee gravity with accretion disk, deflection angle, greybody bounds, and neutrino propagation,” JCAP 12, 026 (2023).\n[101] K. Okabayashi, N. Asaka and K. Nakao, Do black hole shad ows merge?, Physical Review D 102(2020) 044011.\n[102] E. Babichev, V . Dokuchaev and Yu. Eroshenko, Black Hol e Mass Decreasing due to Phantom Energy Accretion, Physical\nReview Letters 93(2004) 021102.\n[103] V .V . Kiselev, Quintessence and black holes, Classica l and Quantum Gravity 20(2003) 1187.\n[104] B. Majeed, M. Jamil and P . Pradhan, Thermodynamic Rela tions for Kiselev and Dilaton Black Hole, Advances in High\nEnergy Physics 2015 (2015) 124910.\n[105] R. Ali, Z. Akhtar, K. Bamba and M. U. Khan, Tunneling and thermodynamics evolution of the magnetised Ernst-like bla ck\nhole, General Relativity and Gravitation 55(2023) 28.\n[106] M. Jamil, S. Hussain and B. Majeed, Dynamics of particl es around a Schwarzschild-like black hole in the presence of\nquintessence and magnetic field, The European Physical Jour nal C 75(2015) 24.\n[107] P . Rastall, Generalization of the Einstein Theory, Ph ysical Review D 6(1972) 3357.\n[108] P . Rastall, A theory of gravity, Canadian Journal of Ph ysics 54(1976) 66.\n[109] V . V . Kiselev, Quintessence and black holes, Classica l and Quantum Gravity 20(2003) 1187.\n[110] Y. Heydarzade and F. Darabi, Black hole solutions surr ounded by perfect fuid in Rastall theory, Physics Letters B 771(2017)\n365.\n[111] R. Ali, M. Asgher and M. F. Malik, Gravitational analys is of neutral regular black hole in Rastall gravity. Modern P hysics\nLetters A 35, 2050225(2020).\n[112] R. Ali, K. Bamba, S. A. A. Shah and M. J. Saleem, Tunnelin g Analysis of Kerr-Newman Black Hole-Like Solution in Rasta ll\nTheory. International Journal of Modern Physics D 31(2022) 2250069.\n[113] Y. Huang, Z. Cao and Z. Lu, Generalized Gibbons-Werner method for stationary spacetimes, Journal of Cosmology and\nAstroparticle Physics 01(2024) 013.\n[114] W. Javed, S. Riaz, R. C. Pantig and A. ¨Ovg ¨ un, Weak gravitational lensing in dark matter and plasm a mediums for wormhole-\nlike static aether solution. The European Physical Journal C82(2022) 1057.28\n[115] Y. Guo and Y. G. Miao, Bounce corrections to gravitatio nal lensing, quasinormal spectral stability and gray-body factors of\nReissner-Nordstr¨ om black holes, Physical Review D 106(2022) 124052.\n[116] W. Javed, M. Atique, R. C. Pantig and A. ¨Ovg ¨ un, Weak Deflection Angle, Hawking Radiation and Greybo dy Bound of\nReissner-Nordstr¨ om Black Hole Corrected by Bounce Parame ter, Symmetry 15(2023) 148.\n[117] O. Y. Tsupko, Deflection of light rays by a spherically s ymmetric black hole in a dispersive medium, Physical Review D103,\nno.10, 104019 (2021).\n[118] V . Perlick and O. Y. Tsupko, Calculating black hole sha dows: Review of analytical studies, Physics Reports 947, 1-39 (2022)."    },    {        "title": "2402.07666v1.Approximating_the_Maximum_Independent_Set_of_Convex_Polygons_with_a_Bounded_Number_of_Directions.pdf",        "content": "Approximating the Maximum Independent Set of\nConvex Polygons with a Bounded Number of\nDirections\nFabrizio Grandoni /envel⌢pe/h⌢me\nIDSIA, USI-SUPSI, Switzerland\nEdin Husić /envel⌢pe/h⌢me\nIDSIA, USI-SUPSI, Switzerland\nMathieu Mari /envel⌢pe/h⌢me\nLIRMM, University of Montpellier, CNRS, Montpellier, France\nAntoine Tinguely /envel⌢pe\nIDSIA, USI-SUPSI, Switzerland\nAbstract\nIn the maximum independent set of convex polygons problem, we are given a set of nconvex polygons\nin the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons.\nHere we study a special case of the problem where the edges of the polygons can take at most d\nfixed directions. We present an 8d/3-approximation algorithm for this problem running in time\nO((nd)O(d4d)). The previous-best polynomial-time approximation (for constant d) was a classical nε\napproximationbyFoxandPach[SODA’11]thathasrecentlybeenimprovedtoa OPTε-approximation\nalgorithm by Cslovjecsek, Pilipczuk and Węgrzycki [SODA ’24], which also extends to an arbitrary\nset of convex polygons.\nOur result builds on, and generalizes the recent constant factor approximation algorithms for the\nmaximum independent set of axis-parallel rectangles problem (which is a special case of our problem\nwithd= 2) by Mitchell [FOCS’21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA’22].\n2012 ACM Subject Classification Theory of computation →Packing and covering problems\nKeywords and phrases Approximation algorithms, independent set, polygons\nDigital Object Identifier 10.4230/LIPIcs...\n©Fabrizio Grandoni, Edin Husić, Mathieu Mari and Antoine Tinguely;\nlicensed under Creative Commons License CC-BY 4.0\nLeibniz International Proceedings in Informatics\nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:2402.07666v1  [cs.CG]  12 Feb 2024F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:1\n1 Introduction\nThe Maximum Independent Set of Convex Polygons problem (MISP) is a natural geometric\npacking problem with many applications in map labeling [ 13,39], cellular networks [ 34],\nunsplittable flow [ 6], chip manufacturing [ 27], or data mining [ 18,33]. Given a set of n\nconvex polygons in the plane, the goal is to select a maximum number of them such that the\npolygons are pairwise non-overlapping.\nMISP is NP-hard [ 16,28], hence it makes sense to design approximation algorithms for it.\nDisappointingly, the best (polynomial-time) approximation ratio for MISP (more precisely\nfork-intersecting curves) has been nε[17], for any fixed constant ε >0. This ratio has\nrecently been improved to OPTε[12].\nApproximation Schemes. Interestingly, there is a quasi-polynomial time approximation\nscheme (QPTAS) for MISP [ 1] (if the polygons have at most quasi-polynomially many\nvertices in total). Thus, the problem is notAPX-Hard, assuming NP⊈DTIME (2polylog (n)),\nsuggesting that it should be possible to obtain a polynomial time approximation scheme\n(PTAS) for the problem.\nIf we assume that we are allowed to shrink the polygons by a factor 1−δfor an arbitrarily\nsmall constant δ, then there is a PTAS for the problem [ 40]. Note that here the output is\ncompared to the optimal solution without shrinking.\nWhen the input polygons are fat, e.g., regular polygons, then PTASes are known [ 9,15].\nAxis-parallel rectangles. A prominent special case of MISP that has attracted a lot of\nattention over the years is the maximum independent set of axis-parallel rectangles (MISR),\nwhere all the polygons are rectangles with their edges parallel with the axes. An O(logn)\napproximationforMISRwasgivenin[ 30,38]. Thiswasslightlyimprovedto O(logn/log logn)\nin [10], and substantially improved to O(log logn)in [7]. In a recent breakthrough result,\nMitchell [ 36] presented the first constant factor approximation algorithm with approximation\nratio 10, and later 3 +εin an updated version [ 37] with a considerably shorter case analysis.\nSubsequently, his approach was simplified and improved to a (2 +ε)-approximation algorithm\nby Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [ 21,22]. These approaches rely on a\ndynamic program that considers all the partitions of a bounding box containing the instance\ninto a number of containers with constant complexity (constant number of line segments).\nOur contribution. With the goal of better understanding the approximability of MISP, in\nthispaper, weconsiderthefollowingnaturalspecialcaseofMISP: d-MISPisthespecialcaseof\nMISP where the edges of the input polygons are parallel to a given set Dofd=|D|directions.\nNotice that MISR is equivalent to 2-MISP. Our main result is a constant approximation for\nd-MISP when dis a constant.\n▶Theorem 1. There exists an 8d/3-approximation algorithm for d-MISP running in time\nO((nd)O(d4d)).\nOur result builds on the approaches in [ 21,22,37], however we have to face several additional\ncomplications. In particular, already for d= 3the algorithm and its analysis deviates\nsubstantially from the known (polynomial-time) results in the literature about axis-aligned\nrectangles. An overview of our approach is given in Section 3.\nRelated Work. One can consider a natural weighted version of MISP, where each convex\npolygon has a positive weight, and the goal is to find an independent set of maximum total\nweight. The weighted version of MISR was studied in the literature, and the current-best\npolynomial time approximation factor is O(log logn)[8]. We remark that our approach,\nlikewise the approaches in [ 21,22,36], does not seem to extend to the weighted case. InXX:2 Approximating MISP with a bounded number of directions\nv1\nv2\nv3\nv4\nv5\nv6\nv7\nv8\nP\ne1(P)\ne2(P)\ne3(P)\ne4(P)\ne5(P)\ne6(P)\ne7(P)\ne8(P)\nFigure 1 A convex polygon in 4directions. The edge e3(P)is degenerate.\nparticular, finding a constant approximation for weighted MISR remains a challenging open\nproblem. We remark that the QPTAS in [ 1] extends to the weighted case, hence suggesting\nthat the weighted version of MISP might also admit a PTAS.\nMISR was also studied in terms of parameterized algorithms. Marx [ 35] proved that\nthe problem is W[1]-hard, which rules out the existence of an EPTAS. A parameterized\napproximation scheme for the problem is given in [23].\nA rectangle packing problem related to MISR is the 2D Knapsack problem. Here we\nare given an axis-parallel square (the knapsack ) and a collection of axis-parallel rectangles.\nThe goal is to pack a maximum cardinality (or weight) subset of rectangles in the knapsack\n(without rotations). 2D Knapsack admits a QPTAS [ 2] and a few constant approximation\nalgorithms are known [ 19,20,29]. Here as well, finding a PTAS is a challenging open\nproblem.\nBonsma et al. [ 6] established an intriguing connection between MISR and the Unsplittable\nFlow on a Path problem. A PTAS for the latter problem was recently obtained [ 24], closing\na very long line of research (see, e.g., [3, 4, 5, 6, 25, 26]).\n2 Preliminaries\nIn this paper, a (possibly closed) curveis always assumed to be a polygonal chain (or a\nsingleton point) and a polygonSis a bounded set with non-empty interior int(S)and whose\nboundary∂Sis a closed curve. We denote the closure ofSas¯S, so¯S=∂S∪int(S). We say\nthat two polygons S,T(with non-empty interior) touchifint(S)∩int(T) =∅but∂S∩∂T̸=∅\nandintersect ifint(S)∩int(T)̸=∅. A curveftouchesSiff∩int(S) =∅butf∩∂S̸=∅.\nA line segment or curve is called degenerate if it is a singleton point. A line segment\nor curve is assumed to be non-degenerate unless we explicitly state the opposite. For an\n(oriented) line segment e=ww′(resp. curve γ=w1w2···wk) we define the head ofe(ofγ)\nash(e) =w′(h(γ) =wk) and the tail ofe(ofγ)ast(e) =w(t(γ) =w1) and the interior of\ne(ofγ)asint(e) =e\\{h(e),t(e)}(int(γ) =γ\\{h(γ),t(γ)}). For a degenerate line segment\n(resp. curve), the head and the tail coincide with the line segment (resp. curve).\nFor a vector v= (x,y), letv⊥:= (y,−x)(which isvrotated clockwise orthogonally). For\na positive integer k, let[k]:={1,...,k}.\nInput. For a fixed positive integer d, the input of our problem is given by a set of\n(pairwise linearly independent) ddirection defining vectors D={v1,...,vd}⊆Z2and a\nsetIofnconvex polygons with edges oriented along the directions given in D. Polygons\nof this type are sometimes called d-discrete orientation polytopes ( d-DOPs) [31]. In this\npaper, we will more casually refer to them as (input) polygons; the significance of the word\n“polygon” will be clear from context. Without loss of generality, assume v1= (0,1)andF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:3\nthatv2,...,vdpoint to the left and are ordered by decreasing slope, see Figure 1. For\ni∈{d+ 1,..., 2d}, letvi:=−vi−d. The indices of the directions are counted modulo 2d,\ni.e.,i=i+ 2d=i−2d. More explicitly, each polygon P∈Iis encoded by 2dintegers\np1(P),...,p 2d(P)asP={x∈R2:x⊺v⊥\ni< pi(P),∀i∈[2d]}; and thus ¯P={x∈R2:\nx⊺v⊥\ni≤pi(P),∀i∈[2d]}. We assume that those linear inequalities are all tight, including\nredundant ones1, i.e.,ei(P):=¯P∩{x:x⊺v⊥\ni=pi(P)}̸=∅for everyi∈[2d].ei(P)is called\ntheedge ofPin direction vi. Then, for every i∈[2d],ei(P)andei+1(P)are incident and\nh(ei(P)) =t(ei+1(P)). Note that e1(P)e2(P)···e2d(P)forms a positively oriented closed\ncurve.\nGrid.LetL1be the set of all lines in directions v1,...,vdpassing through the vertices\nof the input polygons. In particular, all the edges (including the degenerate ones) of all\nthe polygons in the input lie on the lines in L1. Notice that|L1|≤2d2n. We recursively\ndefineVk, fork∈[2d]andLk, fork∈{2,..., 2d}as follows:Vkis the set of intersection\npoints of any two (non-parallel) lines in Lk, andLkis the set of all lines in directions D\npassing through points in Vk−1. We define the grid Gk= (Lk,Vk). Since|Vk|≤|Lk|2and\n|Lk|≤|Vk−1|·d, it follows that|Vk|≤(2d3n)2k. The gridG2dform the coordinate system of\nour algorithm: every geometric object appearing in the algorithm and the analysis lies on\nG2d. A line segment slies onGkifslies on some line in Lkand the extreme points of slie\nonVk. Similarly, a curve or polygon lies on Gkif all of its line segments do so.\nContainer. Consider the grid G1. LetC∗∈G1be a parallelogram that encloses all polygons\ninI; we callC∗thebounding box .2Acontainer (see Figure 2(a)) is a polygon on G2dwith\npositively oriented boundary s1f1s2f2...sκfκwhere 2≤κ≤5, such that:\ns1,s2,...,sκare disjoint and possibly degenerate parallelline segments on G2d(these\nwill later be called cutting lines ).\nFor allj∈[κ],fjis a simple curve on G2dconsisting of at most 2d+ 1line segments and\nt(fj) =h(sj)andh(fj) =t(sj+1)for everyj∈[κ](wheresκ+1=s1).\nFor allj∈[κ],int(sj)does not intersect with any other part of the boundary of the\ncontainer.\nFor alli,j∈[κ],i̸=j, the curves fiandfjmight touch but do not cross (defined below).\nIn particular, a container has at most 10d+10line segments. Let Cbe the set of all containers\nCwith int(C)⊆int(C∗). In particular, C∗is a container and C∗∈C. Abipartition of C∈C\nis a pair{C1,C2}⊆Csuch thatC1,C2split upC, i.e., int(C)\\(∂C1∪∂C2) =int(C1)∪int(C2)\nandC1andC2may touch but not intersect.\nCrossing curves. Two curves cross(see also Figure 2(b)) if each one of them contains\na connected subcurve w0w1···wkandq0q1···qk, respectively, which form a crossing, i.e.,\nifw0̸=q0,wk̸=qk,wi=qifor1≤i≤k−1and the (non-collinear) triangles w0q0w2\nandwtqtwt−2have the same orientation (i.e., are either both positively or both negatively\noriented).3For two curves formed by at most kline segments in total, it can be decided in\ntimeO(k3)whether there exists a crossing among them or not [ 11]. With this definition, it\nis guaranteed that every container has a well-defined interior [11].\n1An inequality is redundant if we can remove it from the definition of Pwithout affecting P.\n2Itcan, forexample, bechosenasaparallelogramdelimitedbytheleftmostandrightmostverticallinesand\nthe top and bottom v2-oriented lines in G1(i.e., the extension of e2(P′)whereP′=arg maxP∈Ip2(P)\nand the extension of ed+2(P′′)whereP′′= arg maxP∈Ipd+2(P)).\n3Any container is thus weakly simple according to the definitions in [ 14, Box 5.1] and [ 32]. The concept\nof weakly simple polygons is extensively discussed in [11].XX:4 Approximating MISP with a bounded number of directions\nf4\nf5\nf1\nf2\nf3\ns2\ns1\ns3\ns4\ns5\n(a)A container with κ= 5. The line segment s4on\nthe boundary of the container is degenerate. The\ncurvesf1andf5, as well as f1andf2, respectively,\ntouch on the green segments but do not cross.\nw0\nw1\nw2\nw4\nw3\nq0\nq1\nq2\nq3\nq4\nw0\n0\nw0\n1\nw0\n2\nw0\n4\nw0\n3\nq0\n0\nq0\n1\nq0\n2\nq0\n3\nq0\n4(b)The curves on the left touch without\ncrossing: the triangles w0q0w2andw4q4w2\nhave negative and positive orientation,\nrespectively. The curves on the right cross:\nthe triangles w′\n0q′\n0w′\n2andw′\n4q′\n4w′\n2are both\nnegatively orientated.\nFigure 2 A container with κ= 5. An illustration of crossing and non-crossing.\n3 Our Approach\nFirst, we present the algorithm in Section 3.1, and give an overview of the analysis in\nSections 3.2 and 3.3. The detailed analysis and proofs are given in the later sections.\n3.1 The algorithm\nOur algorithm is a dynamic program that generalizes the algorithm in [ 21]. Each cell of the\ndynamic program corresponds to a container C∈C. For each container, the dynamic program\ncomputes a set of disjoint polygons Dyn(C)⊆Ias follows. If Cencloses no polygon in I,\nsetDyn(C) =∅. IfCencloses exactly one polygon P∈I, set Dyn(C) ={P}. Otherwise,\nthe dynamic program goes through all bipartitions of Cand chooses the bipartition {C1,C2}\nthat maximizes|Dyn(C1)|+|Dyn(C2)|and sets Dyn(C) =Dyn(C1)∪Dyn(C2). The final\noutput of the algorithm is Dyn(C∗).\n▶Lemma 2 (Running time) .LetN=|V2d|be the number of points in the grid G2d.Dyn(C∗)\ncan be computed in time O/parenleftbig\nN20d+20/parenrightbig\n=O((nd)O(d4d)).\nProof.The boundary of each container can be identified by a sequence of 10d+ 10line\nsegments inG2d. There are therefore at most O/parenleftbig\nN10d+10/parenrightbig\ncontainers inC. As argued in\n[21], any bipartition {C1,C2}ofCis determined by the boundary between C1andC2, i.e.,\n∂C1∩∂C2, which is composed of at most 10d+ 10line segments. Thus, to compute Dyn(C),\nthe dynamic program does not consider more that O/parenleftbig\nN10d+10/parenrightbig\nbipartitions. This gives a total\nrunning time O/parenleftbig\nN20d+20/parenrightbig\n. The lemma follows since N=O((2d3n)4d), see Section 2. ◀\nIt is not hard to see that the output Dyn(C∗)is indeed an independent set, so we will\nfocus on showing that the algorithm has the claimed approximation guarantee.\n3.2 Analysis\nBy construction, the output solution Dyn(C∗)is the union of the solutions of two smaller\ncontainers, and so on. We represent this structure by a binary tree called recursiveF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:5\npartition defined below. We argue that Dyn(C∗)is the best solution among all the solutions\nrepresentable by a recursive partition. Then, we show the existence of a recursive partition\nthat respects the approximation factor claimed in Theorem 1.\n▶Definition 3. For a setR⊆I, arecursive partition ofRis a rooted tree Twith vertex\nsetVsuch that\nevery node u∈Vcorresponds to a pair (Cu,Pr(Cu))whereCu∈Cis a container, and\nPr(Cu)is the set of protected polygons ofRcontained in Cu,\nthe rootrofTcorresponds to (C∗,∅), i.e.,Cr=C∗andPr(Cr) =∅;\nevery internal node has two children u1,u2such that:Cu1andCu2form a bipartition of\nCu, and Pr(Cu)⊆Pr(Cu1)∪Pr(Cu2);\nfor every leaf uofT,Cucontains exactly one polygon Pu∈Ror no polygon inRat all;\nfor everyP∈R, there exists a leaf uofTsuch thatPlies inCu.\nClearly, ifR⊆Iadmits a recursive partition, it must be an independent set. It is easy\nto show by induction on the height of the tree that the output Dyn(C∗)admits a recursive\npartition, which leads to the following lemma.\n▶Lemma 4 ([21, Lemma 2.2]) .IfR⊆Iadmits a recursive partition, then |Dyn(C∗)|≥|R|.\nTherefore, Theorem 1 is a consequence of Lemma 2 and the following proposition.\n▶Proposition 5. LetOPTbe an optimal solution of an instance of MISP. There exists a\nrecursive partition for some set R⊆ OPTsuch that|R|≥3\n8d|OPT|.\n3.3 Informal overview of the proof of Proposition 5\nIntuitively, we construct the set Rby starting from an optimal solution OPTcontained in\nthe initial container (the bounding box) Cr=C∗andPr(Cr) =∅. Then, we will recursively\npartition the current container Cuinto two containers Cu1andCu2.Ris then defined as\nthe set of polygons of OPTthat are fully contained in the leaf containers. For a polygon P\ninCu, we say that Pislost (atCu)if it is neither contained in Cu1nor inCu2.\nBelow, one of the ddirections inDplays a special role: without loss of generality, we\nassume that this direction is vertical/vertical-up ( v1). The exact choice will be made later.\nAccountable polygons. We prove that there exists a subset ACC⊆OPT(theaccountable\npolygons) with at least3\n4d|OPT|polygons, such that for each polygon P∈ACClost during\npartitioning of some CuintoCu1andCu2we canchargea unique polygon P′∈OPTand\nP′lies in a leaf container of the recursive partition.\nWe next describe in more details the set of accountable polygons ACCand how protected\npolygons are defined. For technical reasons, we replace each original polygon P∈OPTwith\na new polygon ext(P)lying onG2dthat contains P(see Figures 3 and 4). The new set of\npolygons remains independent, and we will simply denote it by OPTin the following.\nLetP∈OPTand consider its edge e1(P)in direction vertical-up. Let P′∈OPTand\nconsider its edge ed+1(P′)in direction vertical-down. We say that PseesP′ife1(P)is\nnon-degenerate and h(ed+1(P′))∈int(e1(P))∪{t(e1(P))}, see Figure 4. We let the set ACC\nof accountable polygons be the polygons P∈OPTsuch thatPsees someP′∈OPT.\nIt is easy to show that each polygon is seen by at most one other polygon in OPT.\nPartitioning. ForC∈C, let OPT (C)be the set of polygons in OPTthat lie on int(C).\nOur construction is guided by a partitioning lemma which is stated later. Roughly speaking,\nletCbe a container with |OPT (C)|≥2, and let Pr(C)be the set of protected polygons inXX:6 Approximating MISP with a bounded number of directions\nC. The partitioning lemma states that Ccan be bipartitioned by a curve Γinto two smaller\ncontainersC1andC2such that\n(P1) Γcontains a vertical line segment ℓthat intersects all the polygons in OPT (C)that\nare intersected by Γ.\n(P2) Γdoes not intersect any polygon in Pr(C),\n(P3) Pr(C)⊆Pr(C1)∪Pr(C2).\nWe stress that the lemma does nothold for an arbitrary set Pr(C)(e.g., if we take Pr(C) =\nOPT(C)). The set of protected polygons in a container is defined below.\nCharging and protecting. The recursive partition which determines Ris defined by\nrepeatedly applying the partitioning lemma. During the construction of the recursive\npartition, we need to guarantee that the vertical line segments given by (P1) do not intersect\ntoo many polygons from OPT; this is the only possibility of “losing” some polygons. For this,\nwe use the set of accountable polygons ACC⊆OPT. Whenever we apply the partitioning\nlemma, the line ℓintersects some polygons in ACC. For eachP∈ACCthat is intersected by\nℓ, i.e., for each lost polygon P∈ACC, we charge exactly one polygon P′seen byP. By (P1),\nifℓintersectsP, then Γdoes not intersect P′. IfP′is not already an element of Pr(C)and\nthus an element of Pr(C1)∪Pr(C2), then we add the polygon P′to either Pr(C1)ifP′∈C1\nor to Pr(C2)ifP′∈C2. Moreover, if there is a polygon P′′∈OPT (C)that seesP, thenP′′\nis also added to either Pr(C1)orPr(C2).\nBy (P3), adding P′to one of Pr(C1)andPr(C2)means that the charged polygon P′\nwill remain protected. By (P2), P′will not be intersected by the curves in the following\napplications of the partitioning lemma. Therefore P′will be an element in R(our intended\nrecursive partition). Adding P′′to one of Pr(C1)andPr(C2)is also necessary, because the\npolygonPis already lost and if we were to lose P′′in one of the following steps, there might\nnot be a polygon which we could charge the loss of P′′to.\nWe conclude that for every polygon P∈ACClost in the partitioning of a container, we\ncan guarantee that a unique polygon P′seen byPis charged, and it will become the protected\npolygon in a leaf. At least half of the polygons in ACCare either lost or not, so there are at\nleast1\n2|ACC|polygons in the leaves. Proposition 5 follows since |ACC|≥3\n4d|OPT|.\n3.4 Comparison with previous work on MISR\nOverall, we follow the same high level approach as the papers on MISR [ 21,22,37]. Yet,\nto generalize the results on MISR to MISP, we encounter several technical difficulties. We\ndiscuss a few of the more prominent ones below.\nTo define the set ACC, we need the following property (later referred as (E3)): for every\nP∈OPTand every non-degenerate edge eofP,int(e)touches either another polygon\nP′∈OPTor the boundary of the bounding box. This property can be obtained by\n“maximally extending” OPTas in [21,37]. The difficulty here, unlike in the case of rectangles,\nis that naively extending the polygons can result in a grid of exponential size in n.\nFor MISR [ 21,37], the accountable polygons correspond to the non-nested polygons\n(both vertical and horizontal). It is essentially trivial to show that the number of non-nested\nrectangles is at least half of the optimal number of rectangles. In case of convex polygons,\nwe require a more careful argument to show that there are at least3\n4d|OPT|accountable\npolygons.\nTo obtain the partitioning lemma, we follow the same idea as in the case of axis-parallel\nrectangles but we need to work with significantly more complex objects. Firstly, the containersF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:7\nP\ne\ne\ne\nFigure 3 Illustration of the\nprocess of extending a polygon P.\nWe extend Pby moving the edge\neofPuntil int(e)touches another\npolygon in OPT.\nv1\nv2\nv3\nv4\nv5\nv6\nv7\nv8Figure4 Ablackarrowfrom PtoP′indicatesthat Psees\nP′with respect to the option (v1,t), i.e., direction vertical-\nup and tail. The blue (resp. red) corners represent the tails\n(resp. head) of all edges with direction vertical-down ( v5).\nThus, a polygon Psees a polygon P′if the vertical-up edge\nofPis touching the red corner of P′.\nwe work with have O(d)-times more line segments. Secondly, the containers that appear in\nour construction might not be simple (since some parts of the boundary may touch other parts\nof the boundary). These difficulties require more elaborate and more technical arguments.\n4 Charging options and accountable polygons\nLike the papers [21, 37] on MISR, first, we extend an optimum solution OPT.\n▶Definition 6. LetOPTbe an optimal solution of a MISP instance. We say that OPT′is\namaximal extension ofOPTif:\n(E1) OPT′is an independent set of (convex) polygons on G2dand enclosed in C∗.\n(E2)There exists a bijection ext:OPT→OPT′such thatP⊆ext(P)for everyP∈OPT.\n(E3)For everyP∈OPT′and every non-degenerate edge eofP,int(e)touches either\nanother polygon P′∈OPT′or∂C∗.\nIn Appendix A, we show that a maximal extension of OPTexists. The idea is to start\nwith OPTand whenever some non-degenerate edge eof a polygon Pdoes not satisfy (E3),\nthen we extend the polygon Pby moving the edge e“outside”, see Figure 3. We show that if\nwe extend all the edges of the same direction not satisfying (E3) together, then the maximal\nextension lies on the grid G2d.\nBy (E2) and (E1), it suffices to prove Proposition 5 for a maximal extension of OPT. (In\nparticular, (E1) implies that the polygons in OPT′have edges in the given ddirections.) The\npurpose of a maximal extension is (E3), which is helpful to bound the number of accountable\npolygons. For the rest of the paper, we assume that OPTis already “maximally extended”\nand thus satisfies (E3), and we work with the grid G2d.\nIn the rest of this section, by the term direction we mean a direction viwherei∈[2d],\nand say that edge eis of direction viif the points of the edge ecorrespond to t(e) +λ·vi,\nwithλ≥0. Acharging option is specified by a direction vi,i∈[2d]and a choice between t\nandh. LetO={vi}i∈[2d]×{t,h}be the set of the 2d·2 = 4dcharging options. We show\nthe existence of a charging option and a subset ACC⊆OPTof accountable polygons with\nrespect to this option such that (essentially) |ACC|≥3\n4d|OPT|.XX:8 Approximating MISP with a bounded number of directions\n▶Definition 7. LetP∈OPTand letebe the edge of Pin direction v=vi,i∈[2d].\nLetP′∈OPTande′be the (possibly degenerate) edge of P′of direction−v. For\na∈{t,h}, we say that PseesP′with respect to (v,a)ifeis non-degenerate and if\n¬a(e′)∈int(e)∪{a(e)}, where¬t=hand¬h=t. (See Figure 4.)\nWhenever there exists P′∈OPTand a charging option (v,a), such that PseesP′for\n(v,a)then we say that Pisaccountable for (v,a).\n▶Lemma 8. Let(v,a)∈Obe a charging option. Any polygon P′∈OPTis seen by at most\none other polygon P∈OPTwith respect to (v,a).\nProof.Assume that P′is seen byP1,P2∈OPTwith respect to (v,a). Lete1ande2be the\nedge in direction vofP1andP2, respectively. Then we have ¬a(e′)∈(int(e1)∪{a(e1)})∩\n(int(e2)∪{a(e2)}). Since int(e1)̸=∅andint(e2)̸=∅, it follows that int(e1)∩int(e2)̸=∅.\nThis implies that P1andP2intersect, thus P1=P2. ◀\nWe say that a polygon P∈OPTis acorner polygon in the bounding box C∗, if all but\none of the edges of Pare contained in the boundary of C∗. In particular, Pis a corner\npolygon if P=C∗. Similarly, if C∗is partitioned into two convex polygons, then both\nare corner polygons. Let Z⊆OPTbe the set of corner polygons in C∗. SinceC∗is a\nparallelogram, we have |Z|≤4, and the polygon C′=C∗\\(/uniontextZ)is convex.\n▶Lemma 9 (Good charging option) .Assume that OPTsatisfies (E3). Then, there exists a\ncharging option (v,a)∈Osuch that at least3\n4d|OPT\\Z|polygons in OPT\\Zare accountable\nwith respect to (v,a).\nProof.LetP∈OPTandcbe a vertex of P. Lete,e′be the two non-degenerate edges\nincident to cwherec=h(e) =t(e′). Denote with v(resp.v′) the direction of e(resp.e′).\n▷Claim 10. Suppose that eore′(or both) does not lie on the boundary of C∗. Then,Pis\naccountable with respect to (v,h)or(v′,t).\nProof.By (E3), each non-degenerate edge of Pnot contained in the boundary of the bounding\nbox, must touch some other polygon of OPTin its interior. By assumption either eore′\ndoes not lie on the boundary of C∗, without loss of generality, say e. ThenPtouches some\nP1∈OPTonint(e), i.e., int(e)∩e1̸=∅, wheree1is the edge of P1in direction−v(e1could\nbe degenerate). See Figure 5. If PseesP1with respect to (v,h), i.e.,t(e1)∈int(e)∪{h(e)}\nthen the claim is true, so assume that t(e1)/∈int(e)∪{h(e)}. This however implies c∈int(e1).\nSincec∈int(e1)andC∗is convex, it follows that e′is not on the boundary of C∗. Then,\nby (E3), there exists P2∈OPTthat touches Ponint(e′), i.e., int(e′)∩e2̸=∅, wheree2is\nthe edge of P2in direction−v′. IfPdoes not see P2with respect to (v′,t), thenc∈int(e2)\nby the same argument as before. So int(e1)andint(e2)intersect in cand thusP1andP2\nintersect (as e1ande2have different direction) which is a contradiction. Therefore, Pmust\nseeP2with respect to (v′,t). ◁\nConsiderP∈OPT\\Z. SincePis not a corner polygon in C∗, it has at least two\nconsecutive non-degenerate edges such that neither of them lies on ∂C∗. By Claim 10, every\nvertex ofPincident to one or both of these edges, provides a charging option for which Pis\naccountable. Thus, the total number of pairs (P,(v,a))withP∈OPT\\Zand(v,a)∈O\nsuch thatPis accountable with respect to (v,a)is at least 3|OPT\\Z|. Since|O|= 4d, thereF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:9\nPe e′cP1e1\nPe e′cP1\nP1c1\ne1\nFigure 5 Claim 10: the blue (red) corners represents the tail (head) of the edges in direction −v.\nexists an option (v,a)for which the number of accountable polygons in OPT\\Zis at least\n3\n4d|OPT\\Z|.4◀\n5 Recursive partitioning\nWithout loss of generality (by rotating and mirroring the initial instance if necessary), we\nassume that the option (v,a)satisfying Lemma 9 is vertical-up and tail, i.e., (v1,t). In other\nwords, for any P∈OPT, ife1(P)is non-degenerate and if there is a P′∈OPTsuch that\nh(ed+1(P′))∈int(e1(P))∪t(e1(P)), thenwesaythat PseesP′(andP′is seen byP)andthat\nPisaccountable . Lemma 9 states that there exists a subset ACC⊆OPT\\Zof accountable\npolygons such that |ACC|≥3\n4d|OPT\\Z|, consequently|Z|+|ACC|≥3\n4d|OPT|.\nWe will construct a recursive partition for a specific subset R⊆ OPT, such that|R|≥\n|Z|+1\n2|ACC|, which proves Proposition 5. Recall that OPT (C)denotes the set of polygons\ninOPTthat lie on int(C). Moreover, all of the polygons in OPTand the bounding box C∗\nlie on the gridG2d.\nHandling corner polygons. IfZ̸=∅, then we construct the first few nodes of the recursive\npartition as follows. Take any corner polygon P∈Z. Recall that the root rof the recursive\npartition corresponds to (C∗,∅). We add two children u1,u2torand partition C∗into\nthe containers Cu1=PandCu2=C∗\\P. Set Pr(Cu1),Pr(Cu2) =∅. By construction,\nOPT (Cu1) ={P}(sou1is a leaf in the final tree and OPT (Cu2) =OPT\\{P}. Notice that\nC∗\\Pis convex with at most five line segments since C∗is convex.C∗\\Phas five line\nsegments if Pis a triangle, and less if Phas more than three sides.) We recurse by treating\nCu2as the new bounding box.\nWe end up with a tree on |Z|+ 1leaves, where for one leaf u,Cuis a convex polygon\nsuch that OPT (Cu) =OPT\\Zand with at most eight line segments (since |Z|≤4) and\nPr(Cu) =∅. Each of the remaining |Z|leaves coincides with a unique element in Z. Thus, it\nsuffices to construct the recursive partition of OPT\\Zby treating Cuas the bounding box\nwith at most 8line segments. Equivalently, we assume Z=∅and allowC∗to have up to\neight line segments for the rest of this paper.\n5.1 The partitioning lemma – formal statement\nFor anyP∈OPT, let thetop ofPbe defined as the curve top(P) =e2(P)e3(P)···ed(P)\nand thebottom ofPas the curve bot(P) =ed+2(P)ed+3(P)···e2d(P). We define the bottom\n4If we could guarantee a maximal extension in which all the polygons have at least 4sides, then we\nwould improve3\n4dto1\nd. In particular, when d= 2we are in the case of axis-parallel rectangles and we\nobtain a 2d= 4-approximation algorithm. This is the same approximation factor achieved in [ 21,22,37]\nby charging each lost rectangle to one protected rectangle (the improved 2 +ϵfactor requires a more\ncomplex charging).XX:10 Approximating MISP with a bounded number of directions\nand top of the bounding box C∗in the same way. The following definitions are illustrated in\nFigure 6.\n▶Definition 11 (Top and bottom fences) .LetP,P′∈OPTbe two polygons such that Psees\nP′. Atop-fence is (a segment of) the curve top(P)h(e1(P))t(ed+1(P′)) top (P′)such that\nthe first and last line segment is not vertical. Symmetrically, a bottom-fence is (a segment\nof) the curve bot(P)t(e1(P))h(ed+1(P′)) bot (P′)such that the first and last line segment is\nnot vertical.\nIfP∈OPTdoes not see any polygon, then a segment of its bottom (or top) is also called\na bottom-fence (resp. top-fence).\nFor a vertical line segment (cutting line) s, we say that a fence emerges from sif one\nextreme point of the fence lies on s.\nTo prove the partitioning lemma, we further specialize the definition of a container (see\nSection 2)\n▶Definition 12 (Structured container) .A container Cwith∂C=s1f1s2f2···sκfκ,κ≤5,\nisstructured if thecutting lines s1,...,sκare vertical and the curves f1,...,fκare fences.\nWe say that a cutting line is a left cutting line if it is oriented downwards (or degenerate),\nandright cutting line if it is oriented upwards (or degenerate). In a structured container,\nthe left cutting lines (and thus right cutting lines) are consecutive (e.g., s1,...,sκ′are left\nandsκ′+1,...,sκare right cutting lines for some κ′∈[κ−1]).\n▶Definition 13 (Protected by fences) .LetCbe a structured container and sbe a (possibly\ndegenerate) cutting line on C. We say that a polygon P∈OPT (C)isprotected from the left\ninCviasifsis a left cutting line on ∂Cand\nthere exists a top-fence γhinCemerging from s, ending in h(e1(P)), and with top(P)⊆\nγh, and\nthere exists a bottom-fence γtinCemerging from s, ending in t(e1(P)), and with\nbot(P)⊆γt.\nWe say that Pis protected by fences γhandγt. Symmetrically, we say that a polygon\nP∈OPT(C)isprotected from the right in Cviasifsis a right cutting line on ∂Cand\nthere exists a top-fence σhinCemerging from s, ending int(ed+1(P)), and with top(P)⊆\nσh, and\nthere exists a bottom-fence σtinCemerging from s, ending in h(ed+1(P)), and with\nbot(P)⊆σt.\nWe say that Pis protected by fences σhandσt.A polygonP∈OPT (C)isprotected by\nfences inCif it is either protected from the left in Cor protected from the right in C.\nWe will show that each polygon in Pr(C)appearing in the construction of the recursive\npartition can be protected by fences inC.\nNext, we state the partitioning lemma, the proof is presented in Appendix B. The lemma\nholds only for structured containers. This matters for the construction of the recursive\npartition but it does not affect the algorithm, as it considers all possible containers.\n▶Lemma 14 (Partitioning lemma) .LetCbe a structured container such that |OPT (C)|≥2,\nand letPbe a set of polygons in Cprotected by fences. Then, there exists a curve Γsuch that\n(P1) ΓpartitionsCinto two structured containers C1,C2∈Cwith non-empty interiors.F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:11\nP2\nP3\nP4\nP9\nP12\nP10\nP5\nP8\nP6\nP11\nf2\nf3\nf4\nf1\ns1\ns2\ns3\ns4\nP1\nP7\nP14\nP16\nP15\nP13\nFigure 6 Example of a structured container with κ= 4. The black arrows represent “seeing”,\ntop-fences are green, bottom-fences are blue. The polygons P1,P3,P4,P5,P8are protected (only)\nfrom the left, P14,P16are protected (only) from the right, P9,P11are protected both from the left\nand from the right. Notice that the fences that protect P14(from the right) are not unique since\nP14seesP15andP16which are cut and touch s4, respectively. Note also that the bottom-fences\ntouchingP8,P11andP11,P13overlap.\n(P2)All the polygons in OPT (C)that are intersected by Γare intersected by one vertical\ncutting line ℓ⊆Γ.\n(P3) Γdoes not intersect any polygon protected by fences.\n(P4) Any polygon protected by fences in Cis protected by fences in either C1orC2.\n5.2 Construction and analysis of the recursive partition\nIn this section we prove Proposition 5, i.e., we provide a recursive partition for R⊆ OPT\nwith|R|≥1\n2|ACC|. (Recall that we already argued that we can assume Z=∅.) We give\nan iterative construction of a recursive partition with the help of the partitioning lemma.\nWe initialize a tree Twith root node r,Cr=C∗, and Pr(Cr) =∅. Then, iteratively,\nfor every childless node u∈V(T)with|OPT (Cu)|≥2, add two children u1,u2touand\nchooseCu1,Cu2∈Cas provided by (P1) in the partitioning lemma applied to CuandPr(Cu).\nDefine the set of protected polygons Pr(Cu1)andPr(Cu1)as follows.\n(A1) Set Pr(Cu1) = Pr(Cu)∩OPT(Cu1)andPr(Cu1) = Pr(Cu)∩OPT(Cu1).\n(A2)For eachP∈ACCthat is intersected by ℓ, i.e., eachP∈ACCthat is lost, if Psees\na polygonP′∈OPT (Cu)(ifPsees more than one polygon in OPT (Cu), choose one\nof them arbitrarily), add P′toPr(Cu1)ifP′is inCu1or to Pr(Cu2)ifP′is inCu2.\nMoreover, charge the loss of PtoP′.\n(A3)For eachQ′∈OPT (Cu)intersected by ℓfor which there is a polygon Q∈OPT (Cu)\nthat seesQ′, addQto either Pr(Cu1)orPr(Cu2)depending whether Qis inCu1or\nCu2.\nWe first show to that by this construction, a polygon is protected only if it is protected\nby fences.\n▶Lemma 15. LetP′∈Pr(Cu)for a nodeuofT. There exist fences that protect P′inCu.XX:12 Approximating MISP with a bounded number of directions\nP0\nP\ne1(P0)\n`u0\npx\npy\n\rx\nCu0\n\ry\nFigure 7 Illustration for the proof of Lemma 15: P′is protected by fences via (A2).\nProof.We first argue in the case that P′is protected for the first time, i.e., added to Pr(Cu)\nvia (A2) or (A3). Let u′be the parent of uinT.\nFirst assume that P′is protected via (A2). Let P∈ACC∩OPT (Cu′)be the polygon\nthat seesP′. By definition, Pis intersected by the cutting line ℓu′from (P1) during the\nbipartitioning of Cu′Letpxandpybe the two intersection points of ℓu′and∂P, wherepx\nis abovepy, see Figure 7. Since PseesP′, the curve γxontop(P)andtop(P′)frompx\ntoh(e1(P′))is a top-fence and the curve γyonbot(P)andbot(P′)frompytot(e1(P′))is\na bottom-fence. γxandγyboth emerge from ℓu′and thus protect P′from the left in Cu.\nHence,P′is protected by fences in Cu.\nThe argument is symmetric if P′is protected via (A3): there is a polygon Q∈OPT (Cu′)\nseen byP′that is intersected by the the cutting line ℓu′. Therefore, the curves on top(P′)\nandtop(Q)fromed+1(P′)toℓu′and of bot(P′)andbot(Q)fromed+1(P′)toℓu′form a pair\nof fences that protect P′from the right in Cu.\nIfP′is protected via (A1), then it has been protected for the first time in an ancestor of\nu, so the claim follows inductively from by (P3) and (P4). ◀\nWith (P3) and (P4), Lemma 15 implies that protected polygons are not lost and stay\nprotected, i.e., Pr(Cu)⊆Pr(Cu1)∪Pr(Cu2)for every interior node uinT. This in particular\nholds for every charged polygon. By the construction above, every charged polygon is\nprotected and charged only once by Lemma 8. To make our charging scheme work, we need\nto make sure that everylost accountable polygon provides one charge, which follows by (P2)\nand the following lemma.\n▶Lemma 16. LetP∈ACCbe a polygon that is intersected by the vertical line segment ℓu\nfor an internal node u∈T. Then there exists a polygon P′∈OPT(Cu)that is seen by P.\nProof.LetPbe the set of polygons seen by P. For the sake of contradiction, suppose that\nP∩OPT (Cu) =∅. If someP′∈Ppartially lies in Cu, i.e.,P′∩int(Cu)̸=∅, thenP′\nwas intersected by the vertical line ℓu′in an ancestor u′ofu, soPis protected via (A3).\nOtherwise, if all polygons in Plie outside of Cu, thene1(P)lies on a cutting line in ∂Cu.\nTherefore, top(P)andbot(P)form a top-fence and a bottom-fence, respectively, that protect\nPby fences in Cu. ◀\nProof of Proposition 5. By Lemma 9, we have |ACC|−|Z|≥3\n4d|OPT|−|Z|. Recall that\nwe have already assigned each polygon of Zto a unique leaf of T. By the charging scheme\ndescribed above and since a protected (and thus charged) polygon is never lost, we have a\nunique polygon contained in a leaf of Tfor each lost accountable polygon during the partition.\nThe proposition follows since at least half of the polygons in ACCare either lost, or at least\nhalf of the polygons in ACCare not lost. ◀F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:13\nReferences\n1Anna Adamaszek, Sariel Har-Peled, and Andreas Wiese. Approximation schemes for\nindependent set and sparse subsets of polygons. J. ACM, 66(4):29:1–29:40, 2019.\n2Anna Adamaszek and Andreas Wiese. A quasi-ptas for the two-dimensional geometric\nknapsack problem. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM\nSymposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015 ,\npages 1491–1505. SIAM, 2015.\n3Aris Anagnostopoulos, Fabrizio Grandoni, Stefano Leonardi, and Andreas Wiese. A mazing\n2+εapproximation for unsplittable flow on a path. In Chandra Chekuri, editor, Proceedings\nof the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014,\nPortland, Oregon, USA, January 5-7, 2014 , pages 26–41. SIAM, 2014.\n4Nikhil Bansal, Amit Chakrabarti, Amir Epstein, and Baruch Schieber. A quasi-ptas for\nunsplittable flow on line graphs. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual\nACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006 , pages 721–729.\nACM, 2006.\n5Nikhil Bansal, Zachary Friggstad, Rohit Khandekar, and Mohammad R. Salavatipour. A\nlogarithmic approximation for unsplittable flow on line graphs. ACM Trans. Algorithms ,\n10(1):1:1–1:15, 2014.\n6Paul S. Bonsma, Jens Schulz, and Andreas Wiese. A constant-factor approximation algorithm\nfor unsplittable flow on paths. SIAM J. Comput. , 43(2):767–799, 2014.\n7Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In\nProceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) ,\npages 892–901. SIAM, 2009.\n8Parinya Chalermsook and Bartosz Walczak. Coloring and maximum weight independent set\nof rectangles. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms\n(SODA), pages 860–868. SIAM, 2021.\n9Timothy M Chan. Polynomial-time approximation schemes for packing and piercing fat objects.\nJournal of Algorithms , 46(2):178–189, 2003.\n10Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent\nset of pseudo-disks. Discret. Comput. Geom. , 48(2):373–392, 2012.\n11Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In\nProceedings of the twenty-sixth annual ACM-SIAM Symposium on Discrete Algorithms , pages\n1655–1670. SIAM, 2014.\n12Jana Cslovjecsek, Michał Pilipczuk, and Karol Węgrzycki. A polynomial-time optε-\napproximation algorithm for maximum independent set of connected subgraphs in a planar\ngraph. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms\n(SODA), pages 625–638. SIAM, 2024.\n13Leila De Floriani, Paola Magillo, and Enrico Puppo. Applications of computational geometry\nto geographic information systems. Handbook of computational geometry , 7:333–388, 2000.\n14Erik D Demaine and Joseph O’Rourke. Geometric folding algorithms: linkages, origami,\npolyhedra . Cambridge university press, 2007.\n15Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for\ngeometric intersection graphs. SIAM J. Comput. , 34(6):1302–1323, 2005.\n16Robert J Fowler, Michael S Paterson, and Steven L Tanimoto. Optimal packing and covering\nin the plane are np-complete. Information processing letters , 12(3):133–137, 1981.\n17Jacob Fox and János Pach. Computing the independence number of intersection graphs. In\nProceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms\n(SODA), pages 1161–1165. SIAM, 2011.\n18Takeshi Fukuda, Yasuhiko Morimoto, Shinichi Morishita, and Takeshi Tokuyama. Data mining\nwith optimized two-dimensional association rules. ACM Transactions on Database Systems\n(TODS), 26(2):179–213, 2001.XX:14 Approximating MISP with a bounded number of directions\n19Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, Sandy Heydrich, Arindam Khan, and\nAndreas Wiese. Approximating geometric knapsack via l-packings. ACM Trans. Algorithms ,\n17(4):33:1–33:67, 2021.\n20Waldo Gálvez, Fabrizio Grandoni, Arindam Khan, Diego Ramírez-Romero, and Andreas\nWiese. Improved approximation algorithms for 2-dimensional knapsack: Packing into multiple\nl-shapes, spirals, and more. In 37th International Symposium on Computational Geometry\n(SoCG), volume 189, pages 39:1–39:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik,\n2021.\n21Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy Pittu,\nand Andreas Wiese. A 3-approximation algorithm for maximum independent set of rectangles.\nIn Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM\nSymposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA,\nJanuary 9 - 12, 2022 , pages 894–905. SIAM, 2022.\n22Waldo Gálvez, Arindam Khan, Mathieu Mari, Tobias Mömke, Madhusudhan Reddy, and\nAndreas Wiese. A (2 +ϵ)-approximation algorithm for maximum independent set of rectangles.\narXiv preprint arXiv:2106.00623 , 2021.\n23Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes\nfor independent set of rectangles and geometric knapsack. In Michael A. Bender, Ola Svensson,\nand Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019,\nSeptember 9-11, 2019, Munich/Garching, Germany , volume 144 of LIPIcs, pages 53:1–53:16.\nSchloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.\n24Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. A PTAS for unsplittable flow on\na path. In Stefano Leonardi and Anupam Gupta, editors, STOC ’22: 54th Annual ACM\nSIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022 , pages 289–302.\nACM, 2022.\n25Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. Unsplittable flow on a path: The\ngame! In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM\nSymposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA,\nJanuary 9 - 12, 2022 , pages 906–926. SIAM, 2022.\n26Fabrizio Grandoni, Tobias Mömke, Andreas Wiese, and Hang Zhou. A (5/3 + ϵ)-approximation\nfor unsplittable flow on a path: placing small tasks into boxes. In Ilias Diakonikolas, David\nKempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT\nSymposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018 ,\npages 607–619. ACM, 2018.\n27Dorit S Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing\nproblems in image processing and vlsi. Journal of the ACM (JACM) , 32(1):130–136, 1985.\n28Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of\nan intersection graph of rectangles in the plane. Journal of algorithms , 4(4):310–323, 1983.\n29Klaus Jansen and Guochuan Zhang. On rectangle packing: maximizing benefits. In J. Ian\nMunro, editor, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete\nAlgorithms (SODA) , pages 204–213. SIAM, 2004.\n30Sanjeev Khanna, S. Muthukrishnan, and Mike Paterson. On approximating rectangle tiling and\npacking. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms\n(SODA), pages 384–393. ACM/SIAM, 1998.\n31James T Klosowski, Martin Held, Joseph SB Mitchell, Henry Sowizral, and Karel Zikan.\nEfficient collision detection using bounding volume hierarchies of k-dops. IEEE transactions\non Visualization and Computer Graphics , 4(1):21–36, 1998.\n32Yoshiyuki Kusakari, Hitoshi Suzuki, and Takao Nishizeki. A shortest pair of paths on the\nplane with obstacles and crossing areas. International Journal of Computational Geometry &\nApplications , 9(02):151–170, 1999.\n33Brian Lent, Arun Swami, and Jennifer Widom. Clustering association rules. In Proceedings\n13th International Conference on Data Engineering , pages 220–231. IEEE, 1997.F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:15\n34Ewa Malesinska. Graph theoretical models for frequency assignment problems . Citeseer, 1997.\n35Dániel Marx. Efficient approximation schemes for geometric problems? In 13th Annual\nEuropean Symposium on Algorithms (ESA) , volume 3669, pages 448–459. Springer, 2005.\n36Joseph S. B. Mitchell. Approximating maximum independent set for rectangles in the plane.\nCoRR, abs/2101.00326, 2021. Version 1.\n37Joseph SB Mitchell. Approximating maximum independent set for rectangles in the plane.\nIn2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) , pages\n339–350. IEEE, 2022.\n38Frank Nielsen. Fast stabbing of boxes in high dimensions. Theor. Comput. Sci. , 246(1-2):53–72,\n2000.\n39Bram Verweij and Karen Aardal. An optimisation algorithm for maximum independent set\nwith applications in map labelling. In Algorithms-ESA’99: 7th Annual European Symposium\nPrague, Czech Republic, July 16–18, 1999 Proceedings 7 , pages 426–437. Springer, 1999.\n40Andreas Wiese. Independent set of convex polygons: From nϵto1 +ϵvia shrinking.\nAlgorithmica , 80(3):918–934, 2018.\nA Maximal extension\nFor convenience, we restate the definition of a maximal extension.\n▶Definition 6. LetOPTbe an optimal solution of a MISP instance. We say that OPT′is\namaximal extension ofOPTif:\n(E1) OPT′is an independent set of (convex) polygons on G2dand enclosed in C∗.\n(E2)There exists a bijection ext:OPT→OPT′such thatP⊆ext(P)for everyP∈OPT.\n(E3)For everyP∈OPT′and every non-degenerate edge eofP,int(e)touches either\nanother polygon P′∈OPT′or∂C∗.\nRecall thatL1is the set of all lines in directions v1,...,vdpassing through the corners of\ninput polygons, and that Vk, fork∈[2d]andLk, fork∈{2,..., 2d}are recursively defined\nas follows:Vkis the set of intersection points of any two (non-parallel) lines in Lk, andLkis\nthe set of all lines in directions Dspanned from all points in Vk−1. In particular, C∗∈G1\nandP∈G1for everyP∈OPT.\n▶Lemma 17. There exists a maximal extension of OPT.\nProof.We describe a process by which, sequentially, every edge of every polygon in OPTis\nextended at most once, such that the resulting instance is a maximal extension of the input\ninstance (so we have (E2) by construction). We use the following claim which is simple to\nprove.\n▷Claim 18. Two polygons P,P′∈Iintersect if and only if pi(P)>−pi+d(P′)for every\ni∈[2d].\nFirst, for every P∈OPT, we initialize D(P)as the set of all indices i∈[2d]such that\nei(P)is degenerate. We want to assure that for every i∈D(P),ei(P)stays degenerate\nduring the extension, unless it separates Pfrom another polygon, so we set pi(P) = +∞for\neveryi∈D(P).\nWe repeat the following sweep line algorithm for every i∈[2d]: for every polygon\nP∈OPTat a time, unless i∈D(P), continuously increase pi(P)(we say that we extendP\nin thei-th direction ) until one of the following eventsoccurs, see Figure 8.\n[S1]Ptouches some ˜P∈OPT(or∂C∗) but int(ei(P))does not overlap with ∂˜P(or∂C∗)\nandPintersects ˜P(or the exterior of C∗) ifpi(P)is increased any further.XX:16 Approximating MISP with a bounded number of directions\nvi\nP\nfP\nei1(P)\nei(P)\nei2(P)\nej(P)\nej+d(fP)\n[s3]\n[s1]\n(a)The extension finishes with [S3]after\n[S1].\nP\ncP\nei1(P)\nei(P)\nei2(P)\nei+d(bP)\n[s2](b)The extension finishes with [S2].\nFigure 8 The polygon Pis extended in the i-th direction. The position of ei(P)at each event is\nhighlighted. The point and line highlighted in red lie on Gi, all other points and lines lie on Gi−1.\n[S2] int(ei(P))overlaps with ∂ˆPfor some ˆP∈OPTor∂C∗.\n[S3]ei(P)becomes degenerate.\nTo shorten the analysis, we assume without loss of generality that at [S1]and[S2], we\nalways touch some polygon ˜PorˆP, respectively, instead of ∂C∗. Whenever we encounter\n[S1](assuming it does not happen at the same time as [S2]or[S3]), then it means that P\ntouches a (non-degenerate) edge ej+d(˜P)such thatj /∈{i,i+d}(and such that int(ej+d(˜P))\nintersectsPif we extend it further). Set pj(P) =−pj+d(˜P), removejfromD(P)and then\nkeep on increasing pi(P). When [S2]occurs, stop increasing pi(P). In case [S3]occurs, stop\nincreasingpi(P), additoD(P)and setpi(P) = +∞.\nIt is clear that the extension of ei(P)must finish by either [S2]or[S3], since otherwise\nPgrows indefinitely which contradicts P⊆int(C∗). On the other hand, Pcan be extended\nas long as [S2]or[S3]has not occurred. Whenever [S1]occurs, then at the moment of\ncontact, the edges ej+d(˜P)andej(P)overlap, see Figure 8(a). However since increasing\nPfurther in the i-th direction makes Pand ˜Pintersect, so we have pj(P)>−pj+d(˜P)by\nClaim 18. This means that the j-th constraint of Pis redundant and in particular that ej(P)\nis degenerate until the moment of contact between Pand ˜P. By setting pj(P) =−pj+d(˜P),\nwe can continue to extend Pin thei-th direction without causing an intersection between\nPand ˜P. Notice that starting from now, increasing pj(P)causes an intersection between\nPand ˜P. In particular, it is a non-redundant constraint. Therefore, ej(P)will never be\ndegenerate again once pi(P)is increased further.\nWe now argue that (E1) and (E3) hold by proving the two following claims.\n▷Claim 19. After the extension process, OPTlies onG2d.\nProof.We proceed by induction in the number of extensions, i.e., show that OPTlies onGk\nonce it has been extended in kdirections.\nAssume now that OPThas been extended in i−1directions, where i∈{1,..., 2d}. We\nfocus on the extension in the i-th direction of a single P∈OPT. Assume first that no polygon\nhas been extended in the i-th direction yet, meaning that P′∈Gi−1for everyP′∈OPTbyF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:17\nassumption. We show that at any event [S1],[S2]or[S3],P∈Giis maintained. Assume\ni /∈D(P)and leti1andi2be the indices of the two non-degenerate edges incident to ei(P)\n(w.l.o.g.h(ei1(P)) =t(ei(P)),t(ei2(P)) =h(ei(P))andi1< i < i 2). Notice during the\ncontinuous extension of Pin thei-th direction, ei1andei2get prolonged and all the edges\nei′(P), fori′/∈{i,i1,i2}∪D(P), are not altered.\nAt[S1], assume that ej+d(˜P)touchest(ei(P))(the argument is identical if it touches\nh(ei(P))) and therefore ej(P)as well. This means that i1<j <i. In particular, once we\nhave setpj(P) =−pj+d(˜P)and continue extending P,ej(P)is now a non-degenerate edge\nincident to ei(P)whileei1does not change any longer. We have ei1∈Li−1since it has only\nbe prolonged since the beginning of the extension, and ej(P)∈Li−1since it is collinear with\nej+d(˜P). Notice that t(ej(P)) =h(ei1(P))∈Vi−1. Now substitute i1=jand repeat the\nargument for the upcoming events.\nWhen we arrive at [S2], all edges of Plie onLi−1, sinceei(P)is collinear with ei+d(ˆP)∈\nLi−1. Every vertex of Plies onVi−1.\nAt[S3], we haveei′(P)∈Li−1for everyi′̸=i, and every vertex of Plies onVi−1. This\nimpliesei(P)∈Li.\nAssume now Pthat is not the first polygon to get extended in the i-th direction. If [S1]\noccurs and ˜Phas already been extended in the i-th direction, since j /∈{i,i+d}, we have\nej+d(˜P)∈Li−1and thusej(P)∈Li−1as well. Similarly, at [S2], we haveei+d(ˆP)∈Li−1\n(for anyP′∈OPT,ei+d(P′)stays intact since it is parallel to ei(P′), i.e.,i1̸=i+d,\ni2̸=i+d), soei(P)∈Li−1and every vertex of Plies onVi. For [S3], there is nothing more\nto show since is does not establish contact of Pwith a new polygon. ◁\n▷Claim 20. After the extension process, OPTsatisfies (E3).\nProof.Clearly, for any P∈OPT,i∈[2d], immediately after the extension of Pin thei-th\ndirection (assuming i /∈D(P)before the extension), ei(P)satisfies (E3). If the extension of P\nin thei-th direction has ended with [S2], then increasing pi(P)makesPandˆPintersect. This\nmeans that ei(P)is not degenerate, and int(ei(P))is non-empty. In particular, int(ei(P))\ntouches ˆPand satisfies (E3) at the end of the process. If the extension of Pin thei-th\ndirection has ended with [S3], then either ei(P)stays degenerate until the end (so (E3)\nis trivially satisfied) or it becomes non-degenerate during the extension of Pin another\ndirection than i. The latter case occurs at the event [S1]during the extension of Pin some\ndirectioni′∈[2d]\\{i,i+d}, whenei′(P)touches some ˜P∈OPTon the edge ei+d(˜P). Then,\nthe algorithm sets pi(P) =−pi+d(˜P)and continues the extension of Pin thei′-th direction,\nwhich prolongates ei(P). As argued above, at the end of the entire extension process, ei(P)\nis non-degenerate, and int(ei(P))touches ˜P, thus satisfying (E3). ◁\n◀\nB Proof of the partitioning lemma\n▶Lemma 14 (Partitioning lemma) .LetCbe a structured container such that |OPT (C)|≥2,\nand letPbe a set of polygons in Cprotected by fences. Then, there exists a curve Γsuch that\n(P1) ΓpartitionsCinto two structured containers C1,C2∈Cwith non-empty interiors.\n(P2)All the polygons in OPT (C)that are intersected by Γare intersected by one vertical\ncutting line ℓ⊆Γ.\n(P3) Γdoes not intersect any polygon protected by fences.\n(P4) Any polygon protected by fences in Cis protected by fences in either C1orC2.XX:18 Approximating MISP with a bounded number of directions\nProof.LetCbe a structured container with boundary ∂C=s1f1s2f2···sκfκ, where\ns1,...,sκare cutting lines and f1,...,fκare fences (to facilitate the proof, we write sκ+1=s1\nandfκ+1=f1). Moreover, for some index κ′withκ′<κ, the cutting lines s1,...,sκ′are\nleft cutting lines and sκ′+1,...,s 5are right cutting lines. We also assume that κis minimal,\ni.e.,fj∪sj+1∪fj+1never forms a fence (otherwise we merge these curves onto one fence ˜f\nands1f1···,sj˜fsj+2···fκis still the boundary of Cbut the index κdecreases). Without\nloss of generality, we assume that κ′≥⌈κ\n2⌉, that is, we assume that there are not less left\ncutting lines than right cutting lines, otherwise we mirror the instance vertically. We first\ntreat the case when κ= 5. In particular, we have κ′≥3. We will address the proof of cases\nwithκ≤4at the end of the proof since it suffices to adapt the arguments made for κ= 5.\nWithout loss of generality, we assume that Ccontains no spurs. A spuris a vertex\nwhose two incident edges overlap, this might happen if some line segment sjis degenerate\nand the two edges in fjandfj+1that are incident to sjoverlap. If the edges w′wand\nww′′form a spur around w, then replace those two edges with the new edge w′w′′and\nreiterate this procedure as long as there is a spur. Removing spurs this way does not decrease\nthe interior of a container nor create any new crossing. In particular, any polygon (with\nnon-crossing, positively oriented boundary) has non-empty interior if and only if its boundary\nis non-degenerate once all spurs are removed.\nSuppose there exists a polygon P0inCprotected from the left via s2; take such P0with\nthe right-most right edge (breaking ties arbitrarily). Since P0is protected (see Definition 13)\nfrom the left via s2, there exist a top-fence γhemerging from s2with endpoint h=h(e1(P0))\nand a bottom-fence γtemerging from s2with endpoint t=t(e1(P0)).\nNow, letphandptbe the two points that satisfy the following conditions:\n(i)phandptare in the same vertical line as handtsuch thatphis above or coincides with\nhandptis below or coincides with t.\n(ii)ph(resp.pt) lies either on ∂Cor on∂Ph\\(e1(Ph)∪ed+1(Ph))(resp.∂Pt\\(e1(Pt)∪\ned+1(Pt))), wherePh∈OPT(C)(resp.Pt∈OPT(C)) is protected in C.\n(iii)The semi-open vertical line segments hph\\{h}andtpt\\{t}do not contain any point\nthat satisfies condition (ii).\nIt is clear that these two points exist and are unique. We will construct Γby case analysis,\ndepending on the location of phandpt.\nBefore we proceed, let us first introduce some notation that allows us to define the\nboundaries of C1andC2in a compact form, see Figure 9. For i∈[κ],ri∈{si,fi}and\nc∈{h(Γ),t(Γ)}(as it will be defined below) such that c∈ri, we define r+\nito be the curve\nonrifromctoh(ri)andr−\nito be the curve on rifromt(ri)toc(r+\niorr−\nimight only\nconsist of a singleton point). In particular, ri=r−\nir+\niandt(r−\ni) =t(ri),h(r−\ni) =c=t(r+\ni)\nandh(r+\ni) =h(ri). For the oriented curve Γthat we will define below, let Γ−beΓbut with\ninverse orientation (i.e., h(Γ−) =t(Γ)andt(Γ−) =h(Γ)).\nWe first cover the simplest cases, namely when P0exists and int(e1(P0))⊆int(C), which\nare illustrated in Figure 10.\n(A1) Bothphandptlie onf1∪f2.This implies ph∈f1andpt∈f2sincef1lies above\nf2.5Choose Γ =phpt, thus∂C1= Γf+\n2s3···s1f−\n1and∂C2=f+\n1s2f−\n2Γ−.\n5For two objects A,B⊆R2, we say that Alies above (below) Bif for every (a,b)∈A×Bsuch thata\nandbhave the same x-coordinate, ahas larger (smaller) y-value than b. We define lying on the left or\nrightanalogously.F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:19\nf1\ns1\ns+\n2\ns3\ns4\ns5\nf2\nf3\nf4\nP0\nf+\n5\ns\u0000\n2\nf\u0000\n5\nh(s\u0000\n2) =t(s+\n2)\n=t(\u0000)\n=h(\u0000\u0000)\n\u0000\n\u0000\u0000\nC1\nC2\nh(f\u0000\n5) =t(f+\n5)\n=h(\u0000)\n=t(\u0000\u0000)\nFigure 9 Illustration of the notation used to construct C1andC2according to the location\nofphandpt. In this example, C1has boundary ∂C1= Γf+\n5s1f1s−\n2andC2has boundary ∂C2=\ns+\n2f2s3f3s4f4s5f−\n5Γ−.\n(A2)phorpt(or both) lies on/uniontext\ni≥3fi.Assume that pt∈fj,j≥3(ifptlies on multiple\nfences, choose jto be the smallest index). Choose Γ =γhhpt, thus∂C1= Γf+\njsj+1···,f1s−\n2\nand∂C2=s+\n2f2···f−\njΓ−.\n(A3)PhorPt(or both) exists and is protected from the right in C.Assume that\nPhis protected from the right. Let δhbe a bottom-fence starting at phand ending on some\nright cutting line sjthat protects Ph. Choose Γ =γhhphδh, thus∂C1= Γs+\njfj···f1s−\n2and\n∂C2=s+\n2f2···,fj−1s−\njΓ−.\n(A4) BothPhandPtexist and are protected from the left in C.From condition (ii),\nthere exists a bottom-fence δhfroms1tophwhich protects Phand terminating in phand\na top-fence δtfromsj(3≤j≤κ′) toptwhich protects Pt. Choose Γ =δhphptδt, thus\n∂C1= Γs+\njfj···f5s−\n1and∂C2=s+\n1f1s2···s−\njΓ−.\n(A5)Ptexists and is protected from the left in Candph∈f1; orPhexists and\nis protected from the left in Candpt∈f2.Without loss of generality, assume the\nfirst situation. Let δtbe the top-fence from sj(3≤j≤κ′) toptwhich protects Pt. Choose\nΓ =phptδt, thus∂C1= Γs+\njfj···s1f−\n1and∂C2=f+\n1s2···fj−1s−\njΓ−.\nNow, we discuss the instances where P0exists but int(e1(P0))̸⊆int(C), or equivalently,\nint(e1(P0))∩∂C̸=∅, see Figure 11. For (B1),Γcan be constructed straight away, while\n(B2)reduces to one of (A1)to(A5)by redefining phandptas an intermediate step.\n(B1)phptoverlaps a right cutting line sj.Assumeph∈sj. Choose Γ =γh, thus\n∂C1= Γs+\njfj···s1f1s−\n2and∂C2=s+\n2f2···s−\njΓ−.\n(B2) int(e1(P0))intersects a fence fj.Assume that fjis oriented from left to right.\nThereforet∈fj, ande1(P0)intersects the interior of the vertical line ˜eof a top-fence on\n∂C, thus ˜e=ed+1(˜P)for some ˜P∈OPT\\OPT (C); furthermore j /∈{1,5}. We redefine ph\nandph: ifj̸= 2, then setptandphast(˜e)orh, whichever lies lower. and then construct\nΓas in(A2), i.e., Γ =γhhptand∂C1,∂C2accordingly. For j= 2, setpt=t(˜e), and then\nrecompute phaccording to (i), (ii) and (iii) and additionally such that any point in the\nsemi-open vertical line segment t(˜e)ph\\{t(˜e)}satisfies (ii) (so phlies abovef2). Then, weXX:20 Approximating MISP with a bounded number of directions\nf1\ns1\ns2\nP0\ns3\ns4\ns5\nf2\nf3\nf4\nf5\nph\nh\nt\npt\n(a)Case(A1).\nf1\ns1\ns2\ns3\ns4\ns5\nf2\nf3\nf4\nf5\nph\nh\nt\npt\nP0 (b)Case(A2).\nf1\ns1\ns2\ns3\ns4\ns5\nf2\nf3\nf4\nf5\nPt\nh\nt\npt\nph\nPh\n\u000eh\nP0\n(c)Case(A3).\nf1\ns1\ns2\ns3\ns4\ns5\nf2\nf3\nf4\nf5\nph\nh\nt\npt\nP0\n\u000et\n\u000eh\nPt\nPh (d)Case(A4).\nf1\ns1\ns2\ns3\ns4\ns5\nf2\nf3\nf4\nf5\nph\nP0\npt\nh=t\nPt\n\u000et\n(e)Case(A5).\nFigure 10 Bipartitions of Cforint(e1(P0))⊆int(C). The areas int(C1)(orange) and int(C2)\n(yellow), are separated by the highlighted curve Γ(γh,γtin blue,ℓin green,δh,δtin orange).F. Grandoni, E. Husić, M. Mari and A. Tinguely XX:21\nf1\ns1\ns2\nP0\ns3\ns4\ns5\nf2\nf3\nf4\nf5\n(a)Case(B1).\nf1\ns1\ns2\nf5\nf2\nt\nee\nP0\nt(~e) =pt\nph\nfP\nh (b)Case(B2).\nFigure 11 Bipartitions of Cif the right side of P0touches∂C.int(C1)is orange, int(C2)is\nyellow.\napply the case analysis and construction of the cases (A1)to(A5). Notice that in any case,\nwe have int(phpt)⊆int(C).\nWe now consider instances where P0does not exist, see Figure 12. If s1ands3both\nlie on the right of s2, then it suffices to “shift” s2a little to the right, as defined in (C1).\nOtherwise, i.e., in (C3), we makeℓgoing down (or up) starting from the bottom (resp. top)\nofs2and reduce it to simpler cases, depending on the location of phandpt. There might also\noccur another very particular case, namely if there is one or two polygons that touches s2on\nits top and/or bottom (so ℓis degenerate) and is protected from the left by f2resp.f1, so\nthere is no “gap” above resp. below the polygon(s). We have to handle this case individually.\nLet us describe the latter case more formally. Assume that s3lies on the left of s2, and\nletP3be the polygon in OPT (C)(if it exists) with 1.h(s2)lies on the bottom (but not on\nthe side) of P32.P3is protected from the left via s33.for the top-fence µ3protectingP3,\nwe havef2⊆µ3. Ifs1lies on the left of s2, the polygon P1∈OPT (C), if it exists, touches\nt(s2)on its bottom (and the bottom-fence µ1⊇f1protectingP1), is defined symmetrically.\n(C1)P0does not exist and s1ands2both lie on the right of s2.Choose Γto be any\nvertical line segment ℓ(onG2d) witht(ℓ)∈f1andh(ℓ)∈f2which lies strictly on the right of\ns2and that does not intersect any polygon in OPT (C). Ifℓdoes not exist, then the leftmost\nP∈OPTbetweenf1andf2must touch s2and is therefore protected, which contradicts the\nnonexistence of P0. ThenC1andC2are constructed as in (A1), i.e.,∂C1= Γf+\n2s3···s1f−\n1\nand∂C2=f+\n1s2f−\n2Γ−.\n(C2)P0does not exist and either: s1lies on the right of s2andP3exists, or s3\nlies on the right of s2andP1exists, or both P1andP3exist.Let us first consider\nthe former occurrence, i.e., when s1lies on the right of s2andP3exists (the construction for\nthe second one is analogous). We assume that s2is non-degenerate or that the segment of\nµ3on the right of s2does not form a spur with f1. Indeed, otherwise s2can be dragged at\nthe right extreme point of the spur. Let ℓbe any vertical line segment (on G2d) strictly on\nthe right of s2witht(ℓ)∈f1andh(ℓ)∈µ3that does not intersect any polygon in OPT (C).\nℓexists by the same argument as in (C1). Letµ′\n3be the segment of µ3fromh(ℓ)toh(s2)\nand choose Γ =ℓµ′\n3, thus∂C1= Γf2s3···s1f−\n1and∂C2=f+\n1s2Γ−.\nWe construct Γsimilarly when both P1andP3exist. Again, we assume that s2is\nnon-degenerate or that the segments of µ1andµ3on the right of s2do not form a spur, since\notherwises2can be dragged at the right extreme point of that spur. Let ℓbe any vertical\nline segment (on G2d) strictly on the right of s2witht(ℓ)∈µ1andh(ℓ)∈µ3that does notXX:22 Approximating MISP with a bounded number of directions\ns1\ns2\nf1\n`\n(a)Case(C1).\nf2\nf1\nh=t=ph\ns1\ns2\nf3\npt\ns3\npt (b)Case(C3).\nf1\ns1\ns2\ns3\ns4\ns5\nf2\nf3\nf4\nf5\n\u00163\n\u00160\n3\n`\nP3\n(c)Case(C2), whenP3exists ands1lies\non the right of s2.\ns1\ns2\ns3\ns4\ns5\nf3\nf4\nf5\n\u00163\n\u00160\n3\n`\nP3\n\u00160\n1\n\u00161\nP1(d)Case(C2), when both P1andP3\nexist.\nFigure 12 Bipartitions of CifP0does not exist. The cutting line ℓ(green) lies on the right of\nint(C1)and on the left of int(C2).\nintersect any polygon in OPT (C). Letµ′\n3defined as above and µ′\n1be the segment of µ1from\nh(t(s2))tot(ℓ)and choose Γ =µ′\n1ℓµ′\n3, thus∂C1= Γf2s3···s1f1and∂C2=s2Γ−.\n(C3)P0does not exist and s1ors3(or both) lie on the left of s2, and without (C2)\nAssumes3lieson the left of s2, i.e.,f2is oriented from rightto left. Define h=t=ph=h(s2),\nγh=γt={h(s2)}andptaccording to (i), (ii), and such that any point on the semi-open\nvertical line segment h(s2)pt\\{h(s2)}satisfies (ii) (so ptlies belowh(s2)). If nowpt∈/uniontext\ni≥3fi\norPtis protected from the right, proceed as in (A2)or(A3), respectively. In the case that\nPtis protected from the left, let δtbe the top-fence which protects Pt(from the left cutting\nlinesj,3≤j≤κ′topt). Then choose Γ =phptδtand therefore ∂C1= Γs+\njfj···f1s1and\n∂C2=f2s2...s−\njΓ−(this case is similar to (A5)).\nFor every case, it is clear that ℓ= Γ∩phptdoes not intersect any protected polygon by the\ncriteria (i), (ii) and (iii). For every case, it is also easy to verify that Γ\\ℓis a union of fences.\nSince fences do not intersect any polygon in OPTby definition, (P2) and (P3) follow from\nthese observations. Notice also that (P4) follows from (P1) and (P3): For any P∈Pr(C)\nthat is protected in Cby fencesσh,σtvias, we haveP∈OPT (C1)orP∈OPT (C2)by\n(P1) and (P3). Assuming the former, then either ℓcrosses the fences protecting P, so then\nthe segments of σh,σtstarting from ℓform a new pair of fences in C1that protect PinC1\nviaℓ, or otherwise, σh,σtlie onC1and protect PinC1viasas before. We have P∈Pr(C1)\nin both cases. Therefore, to finish the proof (for κ= 5), we need to show (P1).\nIt is simple to verify that the polygons C1,C2constructed above are structured, i.e., thatF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:23\ntheir boundaries consist each of at most five disjoint vertical cutting lines and five fences\nthat alternate, for every case treated above. It remains to show that C1andC2are indeed\ncontainers, i.e., that they have non-empty interior and no crossings, and that they partition\nthe interior of C.\n▷Claim 21. BothC1andC2as constructed by Lemma 14 have non-empty interior.\nProof.We can argue either by proving int(Ci)̸=∅directly or by proving that ∂C1and\n∂C2are non-degenerate curves once all spurs are deleted. Since ∂CandΓby themselves\ncontain no spurs (by assumption and construction, respectively), for every construction of Γ,\nit suffices to show either Γ\\∂C̸=∅, or∂C1\\Γ̸=∅and∂C2\\Γ̸=∅. It is simple to verify\nthatphptdoes not form any spur with ∂C, so ifℓis non-degenerate, the claim holds as well.\nWe argue case by case.\n(A1).We have int(Γ) = int( ℓ)⊆int(C), which excludes any spur.\n(A2).Assumept∈fj,j≥3. If∂C1\\Γ =∅, we have Γ−=f+\njsj+1···,f1s−\n2, and thus the\ncontradiction that f5s1f1forms a fence. We have int(Ci)̸=∅sinceP0∈int(C2).\n(A3).AssumePhis protected from the right. We have ∂C2\\Γ̸=∅becauseγhlies (strictly)\naboveγt, which itself lies above f2, soγhcannot form any spur with f2. Similarly, we have\n∂C1\\Γ̸=∅becauseδhcannot form a spur with fj, sinceδh(which is a bottom-fence)\nlies (strictly) below fj.\n(A4).We havePt∈int(C1)andP0∈int(P0).\n(A5).Same argument as for (A4).\n(B1).We have Γ =γh⊆∂Cif and only if s+\njands−\n2are singletons and Γ−=fj···s1f1,\nwhich contradicts the assumption that f5s1f1cannot form a fence, thus Γ\\∂C̸=∅.\n(B2).The argument reduces to the cases (A1)to(A5).\n(C1).Same argument as for (A1).\n(C2).The claim follows from the assumption that f1andµ′\n3orµ′\n1andµ′\n3do not form any\nspur.\n(C3).We can assume pt=ph, otherwise ℓis non-degenerate. If pt∈fj,j≥3, then\nΓ ={h(s2)}, so∂C1\\Γ̸=∅and∂C2\\Γ̸=∅trivially hold. Else, if Ptexists (i.e., Γ =δt),\nthen∂C1\\Γ̸=∅becauses2∈∂C1, and∂C2\\Γ̸=∅becausef2\\Γ̸=∅(either because\nPtis protected from the right or because otherwise we are in (C2)).\n◁\n▷Claim 22. The respective boundaries of C1andC2contain no crossings.\nProof.For every case, the claim follows from one or many of those statements:\n(1) There is no crossing on Γ.\n(2) There are also no crossings on ∂C1\\Γand there are also no crossings on ∂C2\\Γ.\n(3)∂Candℓdo not cross.\n(4) Any fence does in Cdoes not cross ∂C.\nWe now prove those statements.\nFor all cases but (A4)and(C2)(when both P1andP3exist), Γis horizontally monotonic,\nso (1) is trivially true. For the two latter cases, the statement is true because δhlies above\nδtin(A4)andP1lies aboveP3in(C2).\n(2) holds because if there are two segments of ∂C1\\Γ(or∂C2\\Γ) that form a crossing,\nthen those two segments also lie and form a crossing on ∂C, which contradicts the assumption\nthatCis a structured container.XX:24 Approximating MISP with a bounded number of directions\n(3) holds because int(ℓ)⊆int(C).\nTo show (4), let γbe a fence in C. Ifγlies only on the top or bottom of one polygon\nP∈OPT (C)then a crossing between γand∂Cimplies that ∂Cto intersect P. However, P\nis protected in Cbyγ, which is a contradiction. If γlies on top or bottom of two polygons\nP,P′such thatPseesP′and one of them is protected in C, thenγand∂Cmust overlap\none1(P)∩ed+1(P′). However, this implies that Plies insideCandP′lies fully outside\nCor vice-versa, which is not possible: either both PandP′lie insideCor one of them is\nintersected by a cutting line of C.\n◁\n▷Claim 23. The polygons C1andC2partitionC.\nProof.For anyC′∈{C,C 1,C2}and any point w /∈∂C, letgwbe the vertical line with\nw∈gw, letwh\nC′(wt\nC′) be the lowest (highest) point on ∂C′∩gwstrictly above (below) w.\nLet alsofh\nC′(ft\nC′) be a fence on ∂C′withwh\nC′∈fh\nC′(wt\nC′∈ft\nC′). Notice that for every\nw∈int(C),wh\nCandwt\nCexist andfh\nC(ft\nC) is oriented from right to left (left to right). We\nalso have int(wh\nC′wt\nC′)⊆int(C′). We now argue that for every point w∈int(C)\\Γ, we have\nthat\n(1)wh\nC′andwt\nC′exist.\n(2)fh\nC′is oriented from right to left and ft\nC′is oriented from left to right.\nfor eitherC′=C1orC′=C2(but not both). Notice that since ∂C1∪∂C2=∂C∪Γby\nconstruction, for any point w∈int(C)\\Γ,wh\nC′andwt\nC′must exist for either C′=C1or\nC′=C2. We argue case by case.\n(A1).For anyw∈int(C)\\Γ, we havegw∩Γ =∅. Notice also that f1is oriented from right\nto left andf2is oriented from left to right. In particular, for any w∈int(C)\\Γ, we have\n(1) and (2) for C′=C2if and only if wh\nC∈f1,wt\nC∈f2andwlies on the left of Γ, and\n(1) and (2) for C′=C1otherwise.\n(A2).Assumept∈fj,j≥3, sofjis oriented from left to right. Notice that t(fj)as well as\nh(fi),t(fi)for2≤i≤j−1, lie on the left of hptand belowγh.\nFirst assume that wt\nC1exists, so either wt\nC1=wt\nC∈fi,i≥jori= 1, orwt\nC1∈γh. Either\nway,wlies above or on the right of γh, and in the latter case, so we have wh\nC1=wh\nC∈fi\nfori≥j+ 1ori= 1. and thus (1). We have that fh\nC1=fh\nCis oriented from right to\nleft (since (2) must hold for C′=C), andft\nC1=ft\nCis oriented from right to left; for the\nsame reason if wt\nC1=wt\nC, and because ft\nC1=γhis oriented from left to right otherwise.\nThis confirms (2) if wt\nC1exists.\nAssume now that wt\nC2exists, sowt\nC2=wt\nC∈/uniontext\n2≤i≤jfi. Ifwh\nC∈∂C2, we havewh\nC2=wh\nC\nand the claim is clear. Otherwise, int(wh\nCwt\nC)crossesγhsowh\nC2∈Γ−and (1) and (2)\nhold forC′=C2, since Γ−is oriented from right to left.\n(A3).Assume that Phis protected from the right and without loss of generality j= 5.\nNotice that Γis horizontally monotone. If wlies above Γ, thenwh\nC∈f1∪f5, therefore\nwh\nC1=wh\nCandwt\nC1∈Γexist, which confirms (1). Since Γ =γhis oriented from left to\nright andf1andf5are oriented from right to left, (2) holds as well. The same argument\ncan be used if wlies below Γ.\n(A4).Letj∈[κ′]such thatPtis protected via sj. We make a similar observation as in the\nargument of (A2):fjis oriented from left to right, t(fj),h(f5)as well ash(fi),t(fi)for\ni∈[j−1], lie on the left of phpt. Therefore, we can apply the same idea as for (A2):\nifwt\nC1exists, we have wh\nC1=wh\nC∈/uniontext\ni≥j+1fiorwh\nC1∈δt, which fulfils (1). (2) holdsF. Grandoni, E. Husić, M. Mari and A. Tinguely XX:25\nbecausefh\nCandδtare oriented from right to left and ft\nCandδhare oriented from left to\nright.\nIfwt\nC2exists, then either wh\nClies on the left of s1and sowh\nC2=wh\nC∈/uniontext\ni≤j−1or\nint(wh\nCwt\nC)crossesδhsowh\nC2∈Γ−. Either way, this implies (1). (2) is then simple to\nverify.\n(A5).Same argument as for (A2).\n(B1).Same argument as for (A3).\n(B2).The argument reduces to the cases (A1)to(A5).\n(C1).Same argument as for (A1).\n(C2).Assume first that P3exists and that s1lies on the right of s2. Clearly, for any\nw∈int(C)\\Γ, we have (1) and (2) for C′=C2if and only if wlies on the right of s2, on\nthe left ofℓand (strictly) between f1andµ′\n3and (1) and (2) hold for C′=C1otherwise.\nThe argument is similar if both P1andP3exist (substitute f1byµ′\n1).\n(C3).The argument reduces to the cases (A1)to(A5).\n◁\nInstances with κ≤4.We conclude the proof of the partitioning lemma by discussing\ninstances with κ≤4. Forκ= 4andκ= 3, we adhere to the construction of Γforκ= 5\nabove. The case analysis is identical as for κ= 5, except that some cases cannot occur (for\nexample,κ′= 2rules out (A4)).\nForκ= 2, we defineP0,h,t,ph,ptas above, except that P0is now protected via s1(since\ns2must be a right cutting line). If we have γh=f1andγt=f2, since|OPT (C)|≥2by\nassumption, this means that there is a polygon P′\n0∈OPT (C)that touches s1and seesP0,\nso we can set Γ =e1(P′\n0)∩ed+1(P0)and thus∂C1=∂P′\n0and∂C2=∂P0. Otherwise, we\nconstruct Γas before (again by substituting s2withs1andsjwiths2forj̸= 2).\nLastly, the proofs of Claim 21, Claim 22 and Claim 23 are essentially independent of κ\nand can be easily be adapted and verified for instances with κ≤4. ◀"    },    {        "title": "2402.07709v1.A_note_on_the_capacities_of_Lagrangian__p__sum.pdf",        "content": "arXiv:2402.07709v1  [math.SG]  12 Feb 2024A note on the capacities of Lagrangian p-sum\nFilip Bro´ ci´ c\nFebruary 13, 2024\nAbstract\nIn this short note, we construct an explicit embedding of the rescaling of the p-sum\nK⊕pK◦of the centrally symmetric convex domain Kand its polar K◦to the product\nK×K◦. The rescaling constant is sharp in some cases. Additionall y, we comment on the\nstrong Viterbo conjecture for K⊕pK◦.\n1 Introduction\nIn [Bro23], the author constructed a symplectic embedding e:B2n(4)→K×K◦from the\nball of capacity 4 to the product K×K◦, in the case when Kis the unit ball of an Euclidian\nnorm, and K◦={y∈Rn| /an}b∇acketle{ty,x/an}b∇acket∇i}ht,∀x∈K}is the polar set. In Proposition 2.2 we generalize\nthis construction to a larger class of domains. Our domains are cons tructed using an arbirtrary\nnorm/ba∇dbl·/ba∇dbl:Rn→[0,+∞) onRn. Let/ba∇dbl·/ba∇dbl∗be the dual norm to /ba∇dbl·/ba∇dbl, considered also on Rn.\nForp∈[1,∞) set/ba∇dbl(x,y)/ba∇dblp,∗:= (/ba∇dblx/ba∇dblp+/ba∇dbly/ba∇dblp\n∗)1/pand denote\nK⊕pK◦:={(x,y)∈R2n| /ba∇dblx/ba∇dblp+/ba∇dbly/ba∇dblp\n∗<1},\nthe unit ball in the norm /ba∇dbl(x,y)/ba∇dblp,∗, whereK:={x∈Rn| /ba∇dblx/ba∇dbl<1}is the unit ball in the\nnorm/ba∇dbl ·/ba∇dbl, andK◦is its polar set, which coincides with the unit ball in the norm /ba∇dbl· /ba∇dbl∗. For\np=∞we have/ba∇dbl(x,y)/ba∇dbl∞,∗= max{/ba∇dblx/ba∇dbl,/ba∇dbly/ba∇dbl∗}, i.e.,K⊕∞K◦=K×K◦. In Proposition 2.2, we\nconstruct a symplectic embedding from a rescaling of K⊕pK◦toK×K◦. Our construction\nimmediately implies estimates of the symplectic capacities of K⊕pK◦.\nDefinition 1.1. Symplectic capacity is a map c:O(R2n)→[0,+∞] from the set of open\nsubsets of R2nto non-negative real numbers such that:\n1.) (Monotonicity) If there is a symplectic embedding ψ:V → Uthenc(V)≤c(U);\n2.) (Rescaling) If λ>0 thenc(λU) =λ2c(U);\n3.) (Non-triviality) 0 <c(B2n(1))≤c(Z2n(1))<∞,\nwhereB2n(R) :={z∈R2n|π/ba∇dblz/ba∇dbl2≤R}, andZ2n(R) =B2(R)×R2n−2is the symplectic\ncylinder. Additionally, capacity is called normalized if it satisfies c(B2n(1)) =c(Z2n(1)) = 1.\nThep-sum of 2-dimensional Euclidian Lagrangian discs, and their Gromov w idth, was stud-\nied in [OR22]. Symplectic capacities of the symplectic p-sum were considered in [HKO23]. Here,\nwe prove the following theorem for all centrally symmetric convex do mainsK⊂Rn.\nTheorem 1. For every symplectic capacity c\nc(K⊕pK◦)≤Γ2(1+1\np)\nΓ(1+2\np)c(K×K◦).\n1Proof.This follows from Proposition 2.2, and the rescaling axiom for symplect ic capacities.\nMain examples of normalized capacities are Gromov width\nGr(U) = sup{R|e:B2n(R)→ U, e∗ωst=ωst},\nwhereωst=/summationtextdxi∧dyi, and cylindircal capacity\ncZ(U) = inf{R|e:U →Z2n(R), e∗ωst=ωst}.\nThe fact that these are (normalized) symplectic capacities is a refo rmulation of the famous Gro-\nmov’snon-squeezingtheorem(see [Gro85]). Otherexamples ofnor malizedsymplectic capacities\nare Ekhlan-Hofer capacity and Hofer-Zehnder capacity (see [EH8 9, HZ90]). They coincide on\nconvex sets, and for convex set X⊂R2nthey are equal to the minimal period of the Reeb orbit\non∂X, and we denote this quantity cEHZ(X). In [HK19], the capacity cEHZis calculated for\nconvex polytopes. It was conjectured in [Vit00, §5] that for any normalized symplectic capacity,\nifXis a convex set then\ncn(X)\nn!≤Vol(X).\nThis conjecture would follow from what is known as the strong Viterb o conjecture:\nConjecture 1 (Strong Viterbo conjecture) .All normalized capacities coincide on convex sets.\nIt is clear from the definition of a symplectic capacity, that every no rmalized capacity cis\nbetween Gromov width and cylindrical capacity, i.e., Gr(U)≤c(U)≤cZ(U). Hence, proving\nstrong Viterbo conjecture is equivalent to proving that Gr(X) =cZ(X) for every convex set X.\nBy projecting to a smart choice of a symplectic plane, it follows from [A AKO14, Remark\n4.2] that the cylindrical capacity of K×K◦satisfiescZ(K×K◦)≤4. The main result of\n[AAKO14] shows1thatcEHZ(K×K◦) = 4. Consequently, since every normalized capacity is\nbounded by the cylindrical capacity, we have cZ(K×K◦) = 4. From Theorem 1 we get\ncZ(K⊕pK◦)≤4Γ2(1+1\np)\nΓ(1+2\np).\nOstrover and Ramos proved in [OR22] that the p-sum of 2-dimensional Euclidian Lagrangian\ndiscs is a toric domain. They calculated the Gromov width of K⊕pK◦, forK={(x1,x2)∈\nR2|x2\n1+x2\n2≤1}. In the case p≥2 they showed:\nGr(K⊕pK◦) = 4Γ2(1+1\np)\nΓ(1+2\np).\nAs a corollary, we get:\nCorollary 1.1. Ifp≥2, the strong Viterbo conjecture holds for\nB2(1)⊕pB2(1) ={(x,y)∈R4| /ba∇dblx/ba∇dblp\nst+/ba∇dbly/ba∇dblp\nst<1}.\n1By proving this, they were able to relate Viterbo conjecture with Mahler’sconjecture from Convex Geometry,\nwhich states that Vol( K×K◦)≥4n/n!.\n2Thesame resultfollowsfrom[GHR22], where they provedthat the s trongViterboconjecture\nholds for monotone toric domains in R4. This result was generalized to arbitrary dimension in\n[CGH23]. However, for centrally symmetric and convex Kit is not known if the p-symK⊕pK◦\nis symplectomorphic to a toric domain in higher dimensions.\nOur construction also imples a new proof of [BK22, Corollary 5.4] for t he particular case\nwhenX=K⊕2K◦.\nTheorem 2.\n2+1\nn≤cEHZ(K⊕2K◦)≤π.\nProof.Using [AK17, Theorem 1.1], we have for a centrally symmetric convex s etT\n2+1/n\n/ba∇dblJ/ba∇dblT◦→T≤cEHZ(T),\nwhere/ba∇dblJ/ba∇dblT◦→T= supu,v∈T◦/an}b∇acketle{tJu,v/an}b∇acket∇i}ht(see also [GO16, Theorem 1.6]). For T=K⊕2K◦using\nLemma 2.4 we have JT◦=T, which further implies\nsup\nu,v∈T◦/an}b∇acketle{tJu,v/an}b∇acket∇i}ht= sup/braceleftBigg\nλ/an}b∇acketle{tux,ux/an}b∇acket∇i}ht+µ/an}b∇acketle{tuy,uy/an}b∇acket∇i}ht |/ba∇dblux/ba∇dbl2+/ba∇dbluy/ba∇dbl2\n∗≤1,\nλ2/ba∇dblux/ba∇dbl2+µ2/ba∇dbluy/ba∇dbl2\n∗≤1/bracerightBigg\n. (1)\nWhere (ux,uy) is a coordinate representation of u∈R2n=Rn×Rn. The equality in equation\n(1) follows from analyzing the critical points of the function\nF(u,v,λ,µ) =/an}b∇acketle{tJu,v/an}b∇acket∇i}ht−λ\n2(/ba∇dblu/ba∇dbl2\n2,∗−1)−µ\n2(/ba∇dblJv/ba∇dbl2\n2,∗−1).\nAlso, we have\n2(λ/ba∇dblux/ba∇dbl/ba∇dblux/ba∇dbl∗+µ/ba∇dbluy/ba∇dbl/ba∇dbluy/ba∇dbl∗)≤λ2/ba∇dblux/ba∇dbl2+/ba∇dblux/ba∇dbl2\n∗+µ2/ba∇dbluy/ba∇dbl2\n∗+/ba∇dbluy/ba∇dbl2≤2,\nwhich implies /ba∇dblJ/ba∇dblT◦→T≤1. On the other hand, for /ba∇dbluy/ba∇dbl= 1, setting u= (0,uy) and\nv= (∇N(uy),0) we get\n/an}b∇acketle{tJv,u/an}b∇acket∇i}ht=/an}b∇acketle{t∇N(uy),uy/an}b∇acket∇i}ht=/ba∇dbluy/ba∇dbl= 1,\nsince bothu,v∈T◦(see Lemma 2.2), we get /ba∇dblJ/ba∇dblT◦→T= 1.\nFor the other inequality, we use the symplectic embedding from Prop osition 2.1, together\nwith the result cEHZ(K×K◦) = 4 from [AAKO14]:\n4\nπcEHZ(K⊕2K◦) =cEHZ/parenleftbigg2√πK⊕2K◦/parenrightbigg\n≤cEHZ(K×K◦) = 4.\nAcknowledgements\nI am very thankful to my Ph.D. advisors Octav Cornea and Egor She lukhin for their con-\nstant support during my studies. This work benefited a lot from disc ussions with Dylan Cant,\nPazit Haim-Kislev, and Yaron Ostrover. I would also like to thank Viniciu s Ramos and Dan\nCristofaro-Gardiner for their comments on the earlier version of t his document. This research\nis partially supported by ´Etudes sup´ erieures et postdoctorales (ESP) scholarship and Fo ndation\nCourtois.\n32 Proofs\nWewillworkwithnormsthataresmooth(oratleast C1)awayfromtheorigin. Set N(x) :=/ba∇dblx/ba∇dbl,\nand denote ∇N(x) the gradient with respect to the standard inner product.\nProposition 2.1. There exists a relative symplectic embedding\ne:/radicalbig\n4/π(K⊕2K◦)→K×K◦.\nProof.Lete:B2(4)→(−1,1)×(−1,1) be a symplectic embedding from [Bro23, Lemma 3.7.].\nIt is of the form e(x,y) =/parenleftBig\nf(x),1\nf′(x)y/parenrightBig\n, where\nf(x) =2\nπarcsin/parenleftbigg/radicalbiggπ\n4x/parenrightbigg\n+x\n2/radicalbigg\n4\nπ−x2.\nFor higher dimensions we set ϕ(x) :=f(/bardblx/bardbl)\n/bardblx/bardblx. Sincefis odd and analytic ϕis smooth. It is\neasy to see that, for x/ne}ationslash= 0\nDϕ(x)h=/parenleftbigg\nf′(/ba∇dblx/ba∇dbl)−f(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dbl/parenrightbigg\n/an}b∇acketle{t∇N(x),h/an}b∇acket∇i}htx\n/ba∇dblx/ba∇dbl+f(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dblh,\nand\n/ba∇dblDϕ(x)h/ba∇dbl ≥f(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dbl/ba∇dblh/ba∇dbl−/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′(/ba∇dblx/ba∇dbl)−f(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dbl/vextendsingle/vextendsingle/vextendsingle/vextendsingle|/an}b∇acketle{t∇N(x),h/an}b∇acket∇i}ht|\n≥f(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dbl/ba∇dblh/ba∇dbl−/parenleftbiggf(/ba∇dblx/ba∇dbl)\n/ba∇dblx/ba∇dbl−f′(/ba∇dblx/ba∇dbl)/parenrightbigg\n/ba∇dbl∇N(x)/ba∇dbl∗/ba∇dblh/ba∇dbl\n≥f′(/ba∇dblx/ba∇dbl)/ba∇dblh/ba∇dbl. (2)\nThe second inequality follows fromf(t)\nt> f′(t) fort >0 and|/an}b∇acketle{t∇N(x),h/an}b∇acket∇i}ht| ≤ /ba∇dbl∇N(x)/ba∇dbl∗/ba∇dblh/ba∇dbl,\nand the last inequality follows from Lemma 2.2 where we show that /ba∇dbl∇N(x)/ba∇dbl∗= 1 . Since\nf′(t)>0 fort∈/bracketleftbigg\n0,2/radicalBig\nab\nπ/parenrightbigg\nwe get that Dϕ(x) is invertible. Now define symplectic embedding\ne:/radicalbig\n4/π(K⊕2K◦)→K×K◦as\ne(x,y) :=/parenleftbig\nϕ(x),(Dϕ(x)−1)∗y/parenrightbig\n.\nSince/ba∇dblDϕ(x)/ba∇dblop=/ba∇dblDϕ(x)∗/ba∇dblop≥f′(/ba∇dblx/ba∇dbl), wehave /ba∇dbl(Dϕ(x)−1)∗/ba∇dblop≤1/f′(/ba∇dblx/ba∇dbl), whichimplies\nthatIm(e)⊂K×K◦. To see that such embedding eis symplectic it is enough to note that\ne∗λst=λstwhereλst=/summationtextyidxi.\nMore generally, take two continuous functions g: [0,a]→[0,+∞) andh: [0,b]→[0,+∞),\nwhich are positive for x∈[0,a) (respectively x∈[0,b)) and/integraltexta\n0g(t)dt=/integraltextb\n0h(t)dt.Define a\nfunctionf: [0,a]→[0,b] implicitly by\n/integraldisplayx\n0g(t)dt=/integraldisplayf(x)\n0h(t)dt.\n4Iff′′≤0, the diffeomorphism ϕ:Bn(a)→Bn(b) defined by ϕ(x) := (f(/ba∇dblx/ba∇dbl)//ba∇dblx/ba∇dbl)xinduces\na symplectic embedding from the set B2n\ng:={(x,y)∈R2n| /ba∇dblx/ba∇dbl ≤a,/ba∇dbly/ba∇dbl∗≤g(/ba∇dblx/ba∇dbl)}to\nB2n\nh:={(x,y)∈R2n| /ba∇dblx/ba∇dbl ≤b,/ba∇dbly/ba∇dbl∗≤h(/ba∇dblx/ba∇dbl)}.\nTakingb= 1,h= 1 andg(t) :=/parenleftbigg\nΓ(1+2\np)\nΓ2(1+1\np)−tp/parenrightbigg1/p\nwe get:\nProposition 2.2. There exists a symplectic embedding e:/radicalBig\nΓ(1+2\np)\nΓ(1+1\np)(K⊕pK◦)→K×K◦.\nNote that the construction of an embedding works only for Kwith (at least) C1boundary.\nHowever, by approximating the non-smooth domains with smooth on es the conclusion of the\nTheorem 1 still holds. The embedding eis of class C1, if the boundary of Kis smooth,\none can take a smoothing of ϕnear the origin and construct a smooth symplectic embedding\ne:/radicalBig\nΓ(1+2\np)\nΓ(1+1\np)(K⊕pK◦)→(1+ǫ)K×K◦, for everyǫ>0.\nThe embeddings from Proposition 2.1 are not filling the volume, which we can see from the\nfollowing Lemma (see [BK22, Lemma 2.2.]).\nLemma 2.1.\nVol(K⊕pK◦) =Γ2(n\np+1)\nΓ(2n\np+1)Vol(K×K◦).\nProof.\nVol(K⊕pK◦) =/integraldisplay\n/bardblx/bardbl≤1/integraldisplay\n/bardbly/bardbl∗≤(1−/bardblx/bardblp)1/pdydx=/integraldisplay\n/bardblx/bardbl≤1Vol((1−/ba∇dblx/ba∇dblp)1/pK◦)dx\n= Vol(K◦)/integraldisplay\n/bardblx/bardbl≤1(1−/ba∇dblx/ba∇dblp)n/pdx. (3)\nHere, we have use that Vol( rK) =rnVol(K). By Fubini’s Theorem, we have\n/integraldisplay\n/bardblx/bardbl≤1(1−/ba∇dblx/ba∇dblp)n/pdx=/integraldisplay\n/bardblx/bardbl≤1/integraldisplay(1−/bardblx/bardblp)n/p\n0dtdx=/integraldisplay1\n0/integraldisplay\n/bardblx/bardbl≤(1−tp/n)1/pdxdt\n= Vol(K)/integraldisplay1\n0/parenleftBig\n1−tp\nn/parenrightBign/p\ndt, (4)\nBy settingt=un/pwe get\n/integraldisplay1\n0/parenleftBig\n1−tp\nn/parenrightBign\n2dt=n\np/integraldisplay1\n0(1−u)n\npun\np−1du=n\npB/parenleftbiggn\np+1,n\np/parenrightbigg\n=n\npΓ(n\np+1)Γ(n\np)\nΓ(2n\np+1)=Γ2(n\np+1)\nΓ(2n\np+1), (5)\nwhereB(a,b) =/integraltext1\n0(1−t)a−1tb−1dtis the Beta function, and Γ( a) =/integraltext∞\n0etta−1dtis the Gamma\nfunction. Combining (3), (4) and (5) we get\nVol(rK⊕pK◦) =Γ2(n\np+1)\nΓ(2n\np+1)Vol(K×K◦)r2n.\n5The following result is standard in Convex Geometry (see e.g. [Sch13, Remark 1.7.14]). We\ninclude the proof for the sake of completeness.\nLemma 2.2. The dual norm /ba∇dbl∇N(x)/ba∇dbl∗of the gradient ∇N(x) is equal to 1.\nProof.It is east to show that /ba∇dbl∇N(x)/ba∇dbl∗≥1. Consider a derivative with respect to sof the\nexpressionN(sx) =sN(x),s>0 we get /an}b∇acketle{t∇N(x),x/an}b∇acket∇i}ht=N(x) =/ba∇dblx/ba∇dbl.Now, from the definition of\nthe dual norm, wehave /an}b∇acketle{t∇N(x),x/an}b∇acket∇i}ht ≤ /ba∇dblx/ba∇dbl/ba∇dbl∇N(x)/ba∇dbl∗,which isananalogofthe Cauchy-Schwartz\ninequality, hence\n/ba∇dbl∇N(x)/ba∇dbl∗≥/an}b∇acketle{t∇N(x),x/an}b∇acket∇i}ht\n/ba∇dblx/ba∇dbl=/ba∇dblx/ba∇dbl\n/ba∇dblx/ba∇dbl= 1.\nWe will calculate /ba∇dbl∇N(x)/ba∇dbl∗= sup/bardbly/bardbl=1/an}b∇acketle{t∇N(x),y/an}b∇acket∇i}htusing Lagrange multipliers. Consider a\nfunctionF(y,λ) =/an}b∇acketle{ty,∇N(x)/an}b∇acket∇i}ht−λ(N(y)−1). The derivatives are\nDyF(y,λ)h=/an}b∇acketle{th,∇N(x)/an}b∇acket∇i}ht−λ/an}b∇acketle{th,∇N(y)/an}b∇acket∇i}ht=/an}b∇acketle{th,∇N(x)−λ∇N(y)/an}b∇acket∇i}ht,\nDλF(y,λ) =N(y)−1.\nSince∇N(sy) =∇N(y),s >0, and∇N(−y) =−∇N(y), critical points of Fcontain the set\n(y,λ)∈/braceleftBig/parenleftBig\nx\n/bardblx/bardbl,1/parenrightBig\n,/parenleftBig\n−x\n/bardblx/bardbl,1/parenrightBig\n,/parenleftBig\nx\n/bardblx/bardbl,−1/parenrightBig\n,/parenleftBig\n−x\n/bardblx/bardbl,−1/parenrightBig/bracerightBig\n.The other critical points are of form\n(y\n/bardbly/bardbl,λ), where ∇N(x) =λ∇N(y). For such yand ifλ>0, we will show in Lemma 2.3 that\n/angbracketleftbiggy\n/ba∇dbly/ba∇dbl,∇N(x)/angbracketrightbigg\n=/angbracketleftbiggx\n/ba∇dblx/ba∇dbl,∇N(x)/angbracketrightbigg\n.\nHence the maximum is achieved at/parenleftBig\nx\n/bardblx/bardbl,1/parenrightBig\nwhich implies /ba∇dbl∇N(x)/ba∇dbl∗=/angbracketleftBig\nx\n/bardblx/bardbl,∇N(x)/angbracketrightBig\n= 1.\nLemma 2.3. If∇N(x) =λ∇N(y) forλ>0 then/angbracketleftBig\ny\n/bardbly/bardbl,∇N(x)/angbracketrightBig\n=/angbracketleftBig\nx\n/bardblx/bardbl,∇N(x)/angbracketrightBig\n.\nProof.Since/ba∇dbl·/ba∇dblis a norm, the unit ball Bn(1) ={x∈Rn| /ba∇dblx/ba∇dbl ≤1}is a convex set. Now, we\nwill consider A:=Bn(1)∩Span{x,y}. This is again a convex set, with the property that the\nnormal vector to the ∂Aatx\n/bardblx/bardblis proportional to the normal vector aty\n/bardbly/bardbl. SinceAis convex,\nand∂Ais smooth, we have that the segment/bracketleftBig\nx\n/bardblx/bardbl,y\n/bardbly/bardbl/bracketrightBig\n⊂∂Ais contained in the boundary (see\nFigure 1). This further implies that /an}b∇acketle{ty\n/bardbly/bardbl,∇N(x)/an}b∇acket∇i}ht=/an}b∇acketle{tx\n/bardblx/bardbl,∇N(x)/an}b∇acket∇i}ht.\n/ba∇dblx/ba∇dbl= 1\nx\n/bardblx/bardbly\n/bardbly/bardbl∇N(x) ∇N(y)\nFigure 1: (Non)-convex domain\nLemma 2.4. If a convex body Kis the unit ball of the norm/radicalbig\n/ba∇dblx/ba∇dbl2+/ba∇dbly/ba∇dbl2∗then the dual\nbodyK◦is given by the norm/radicalbig\n/ba∇dblx/ba∇dbl2∗+/ba∇dbly/ba∇dbl2.\n6Proof.Dual of the norm /ba∇dbl(x,y)/ba∇dbl2,∗=/radicalbig\n/ba∇dblx/ba∇dbl2+/ba∇dbly/ba∇dbl2∗is equal to\n/ba∇dbl(x,y)/ba∇dbl∗\n2,∗= sup\n/bardbla/bardbl2+/bardblb/bardbl2∗=1/an}b∇acketle{ta,x/an}b∇acket∇i}ht+/an}b∇acketle{tb,y/an}b∇acket∇i}ht.\nApplying Lagrange multiplier theorem to the function\nF(a,b,λ) =/an}b∇acketle{ta,x/an}b∇acket∇i}ht+/an}b∇acketle{tb,y/an}b∇acket∇i}ht−λ\n2(/ba∇dbla/ba∇dbl2+/ba∇dblb/ba∇dbl2\n∗−1),\nwe get that the critical points satisfy λ/ba∇dbla/ba∇dbl∇N(a) =x, andλ/ba∇dblb/ba∇dbl∗∇N∗(b) =x. Using that\n/ba∇dbl∇N(a)/ba∇dbl∗= 1 and /ba∇dbl∇N∗(b)/ba∇dbl= 1 we get λ=/radicalbig\n/ba∇dblx/ba∇dbl2∗+/ba∇dbly/ba∇dbl2, which further implies\n/ba∇dbl(x,y)/ba∇dbl∗\n2,∗=/an}b∇acketle{ta,λ/ba∇dbla/ba∇dbl∇N(a)/an}b∇acket∇i}ht+/an}b∇acketle{tb,λ/ba∇dblb/ba∇dbl∗∇N∗(b)/an}b∇acket∇i}ht=/radicalbig\n/ba∇dblx/ba∇dbl2∗+/ba∇dbly/ba∇dbl2.\nReferences\n[AAKO14] S. Arstein-Avidan, R. Karasev, and Y. Ostrover. Fr om symplectic measurements to the Mahler\nconjecture. Duke Mathematical Journal , 163:2003–2022, 2014.\n[AK17] A. Akopyan and R. Karasev. Estimating symplectic cap acities from lengths of closed curves on the\nunit spheres. Preprint arXiv:1801.00242, 2017.\n[BK22] M. Berezovik and R. Karasev. Symplectic polarity and Mahler’s conjecture. Preprint\narXiv:2211.14630, 2022.\n[Bro23] F. Bro´ ci´ c. Riemannian distance and symplectic em beddings in the cotangent bundle. Preprint\narXiv:2303.12752, 2023.\n[CGH23] D. Cristofaro-Gardiner and R. Hind. On the agreemen t of symplectic capacities in high dimension.\nPreprint arXiv:2307.12125, 2023.\n[EH89] I. Ekeland and H. Hofer. Symplectic topology and hami ltonian dynamics. Math Z, 200:355–378,\n1989.\n[GHR22] J. Gutt, M. Hutchings, and V.G.B. Ramos. Examples ar ound the strong viterbo conjecture. J. Fixed\nPoint Theory Appl. , 24(41), 2022.\n[GO16] D. E. Gluskin and Y. Ostrover. Asymptotic equivalenc e of symplectic capacities. Comment. Math.\nHelv., 91(1):131–144, 2016.\n[Gro85] M. Gromov. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. , 82:307–347, 1985.\n[HK19] P. Haim-Kislev. On the symplectic size of convex poly topes.Geom. Funct. Anal. , 29:440–463, 2019.\n[HKO23] P. Haim-Kislev and Y. Ostrover. Remarks on symplect ic capacities of p-products. Int. J. of Math. ,\n34(4), 2023.\n[HZ90] H. Hofer and E. Zehnder. A new capacity for symplectic manifolds. In Paul H. Rabinowitz and\nEduard Zehnder, editors, Analysis, et Cetera , pages 405–427. Academic Press, 1990.\n[OR22] Y. Ostrover and V. G. B. Ramos. Symplectic embeddings of the lp sum of two discs. J. Topol. Anal. ,\n14:793–821, 2022.\n[Sch13] R. Schneider. Convex Bodies: The Brunn–Minkowski Theory . Encyclopedia of Mathematics and its\nApplications. Cambridge University Press, 2 edition, 2013 .\n[Vit00] C. Viterbo. Metric and isoperimetric problems in sy mplectic geometry. Journal of American Math-\nematical Society , 13:411–431, 2000.\nDepartment of Mathematics and Statistics, University of Mon treal, C.P. 6128 Succ. Centre-\nVille, Montreal, QC, H3C 3J7, Canada\nE-mail address: filip.brocic@umontreal.ca\n7"    },    {        "title": "2402.07714v1.Adaptive_Artificial_Immune_Networks_for_Mitigating_DoS_flooding_Attacks.pdf",        "content": "  Adaptive Artificial Immune Networks for Mitigating DoS flooding Attacks Jorge Maestre Vidala,∗, Ana Lucila Sandoval Orozcoa, Luis Javier Garc´ıa Villalbaa aGroup of Analysis, Security and Systems (GASS), Department of Software Engineering and Artificial Intelligence (DISIA), School of Computer Science, Office 431, Universidad Complutense de Madrid (UCM), Calle Profesor Jos´e Garc´ıa Santesmases s/n, Ciudad Universitaria, 28040 Madrid, Spain     Abstract Denial of service attacks pose a threat in constant growth. This is mainly due to their tendency to gain in sophistication, ease of implementation, ob- fuscation and the recent improvements in occultation of fingerprints. On the other hand, progress towards self-organizing networks, and the different tech- niques involved in their development, such as software-defined networking, network-function virtualization, artificial intelligence or cloud computing, fa- cilitates the design of new defensive strategies, more complete, consistent and able to adapt the defensive deployment to the current status of the network. In order to contribute to their development, in this paper, the use of artifi- cial immune systems to mitigate denial of service attacks is proposed. The approach is based on building networks of distributed sensors suited to the requirements of the monitored environment. These components are capable of identifying threats and reacting according to the behavior of the biolog- ical defense mechanisms in human beings. It is accomplished by emulating the different immune reactions, the establishment of quarantine areas and the construction of immune memory. For their assessment, experiments with public domain datasets (KDD’99, CAIDA’07 and CAIDA’08) and simula- tions on various network configurations based on traffic samples gathered by the University Complutense of Madrid and flooding attacks generated by the   ∗Tel. +34 91 394 76 38, Fax: +34 91 394 75 47 Email addresses: jmaestre@ucm.es (Jorge Maestre Vidal), asandoval@fdi.ucm.es (Ana Lucila Sandoval Orozco), javiergv@fdi.ucm.es (Luis Javier Garc´ıa Villalba) 2     tool DDoSIM were performed. Keywords: Anomalies, Artificial Immune System, Denial of Service, forecasting, Intrusion Detection System, network   1. Introduction By definition, Denial of Service (DoS) has the objective of disabling com- puter systems or networks. The DoS attacks with origin in multiple sources are referred as Distributed Denial of Service (DDoS) attacks. In recent years, the number of incidents related with these threats reported by the various organizations for cyber defense shows an alarming growth. According to the European Network and Information Security Agency (ENISA), between 2013 and 2014 an increase of 70% was observed (Marinos, L. and Sfakianakis, A. (2016)). In addition, they pose a threat that has begun to be used in or- der to achieve other objectives. These include disguising activities in rela- tionship with malware spreading, concealment of fraudulent money transfers (DoS (2016d)) or compromising anonymous networks, such as Tor or Freenet (Jansen et al. (2014)). This growth is attributed to various reasons: the first of them is that DDoS attacks are usually triggered by previously infected systems, which in most of the cases are part of botnets. The botnets have been adapted to be resilient against the classical detection schemes, thus allowing the construction and maintenance of larger collections of zombies and increasing their difficulty to be identified. (DoS (2016f)). Another im- portant reason is that attackers are able to take advantage of amplifying elements, in this way enhancing their potential to be harmful. To do this, they exploit vulnerabilities in protocol implementations on the intermediate network devices, particularly at DNS, NTP and SNMP. On the other hand, as the European Police Office (Europol) warns (DoS (2016g)), the DDoS is becoming increasingly linked with the organized crime. Rent botnets for execution of these attacks is a very profitable business on the black market, often supplied as Crimeware-as-a-Service (CaaS). Finally, offenders with lack of formation have a wide variety of tools for easily configuration and deploy- ment of flooding attacks. The black market also offers technical support, a situation that expands the range of user profiles which are able to attack with success. The most common DDoS methods are based on flooding. Because of this, they are the principal object of study in this research. The flooding attacks 3     modus operandi involves the injection of large volumes of traffic in order to saturate the victim systems (Zargar et al. (2013)). The popularity of this group of DoS attacks is mostly due to cheap price and simple implementation, in comparison with the good results they provide. Thus, there are lots of pro- posals aimed at their detection and mitigation. However few of them meet all the requirements to be effective in real use cases, emphasizing the needs of high true positive rates, unrepresentative false positive rates, low consump- tion of computational resources and real-time performance. In addition it is noteworthy that proposals of the literature seldom consider advantages of the new trends on networking. It is expected that the emerging networks, taking the example of 5G, increasingly move towards self-management. The use of novel technologies such as Software-defined networking (SDN), Network- Function Virtualization (NFV), Artificial Intelligence or Cloud computing, facilitates the design of Self-Organizing Networks (SON). This should encour- age the appearance of more interesting proposals that are capable of reacting with a much more comprehensive view of the problem being treated. To serve this cause, a strategy for detection and mitigation of DDoS flooding attacks is proposed. Therein the deployment of a sensor network that integrates an Artificial Immune System (AIS) inspired by the biological defense mechanisms of human beings is introduced. Unlike similar proposals, conventional bio-inspired methods for pattern recognition were not applied. Instead a combination of strategies for DDoS detection based on the study of variations of the entropy on the network traffic by thresholding, with the adaptation of the biological immune reactions is proposed. This makes it possible to apply real-time countermeasures, building an immune memory and establishment of quarantine areas, all in accordance with the current state of the protected network. In view of this, it is important to highlight the two major contributions of this paper: firstly, a new method for detect- ing DDoS that is able to forecast anomalies on the entropy of the traffic analyzed, and thereby recognition of flooding attacks is introduced. This is performed by representation of the entropy in time series and the definition of prediction intervals. On the other hand, a strategy for management of immune agents that implement the previously described detection system is proposed. Within this, the decisions are made as to when and how they will act and in what level of restriction, all this depending on the status of the network and orchestrated by an artificial immune approach. The proposal was implemented and evaluated in different experimental scenarios, which involved public domain datasets (KDD’99 (DoS (2016e)), 4     CAIDA’07 (DoS (2016b)) and CAIDA’08 (DoS (2016a))), simulations on var- ious network configurations, traffic monitored in the University Complutense of Madrid (UCM) and flooding attacks generated by the tool DDoSIM (DoS (2016c)). The paper is divided into seven sections, and the first of them is the present introduction. The background necessary for a better understanding of the approach is described in section II. The proposed AIS is introduced in section III. The novel DDoS detection method implemented in the vari- ous agents of the AIS is detailed in section IV. Experiments, datasets and methodology are described in section V. Results are discussed in section VI. Finally, conclusions and future work are presented in section VII.  2. Background The following describes all aspects necessary for understanding the pro- posal. Among them it is important to highlight those involving the charac- teristics of the DDoS flooding attacks and their countermeasures, a general review of the human being immune system and the different approaches with this in mind, in order to provide defenses against cybercrime.  2.1. Flooding threats: attacks and countermeasures According to (Wei et al. (2013)), there are two types of traffic injection able to compromise a system or network by flooding. The first one is based on the constant and continuous generation of large volumes of information, and is well known as high rate flooding. This is a method which is usually very visible that easily overflows the computing capacity of the victim. On the other hand, the victim may be compromised by less noisy attacks, which are able to exploit vulnerabilities in the various communication protocols. They are known as low rate flooding attacks, using as a typical example, the attacks with On/Off patterns addressed against the TCP (Tang et al. (2014)). In both cases, the malicious traffic may be sent to the victim in a direct or reflected way (Bhuyan et al. (2015); Anagnostopoulos et al. (2013)). Most efforts of the community to deal with denial of service assume these behaviors, and from them different methods for detection, mitigation and identification of sources are proposed. Their most important features and evaluation schemes are described below. The detection of the attacks is often necessary to conduct either of the other defensive reactions. This is based on the analysis of traffic that flows 5     through the protected environment, looking for signatures of previously known attacks or anomalous behaviors. For that purpose various techniques have been proposed, such as probabilistic models based on Markov (Shin et al. (2013)), Genetic Algorithms (GA) (Lee et al. (2012), Chaos theory (Chen et al. (2013)), CUSUM statistical analysis with wavelet transforms (Calle- gari et al. (2012)), forensic methods based on visualization (Cai et al. (2010)), fuzzy logic (Kumar and Selvakumar (2013)) or the study of variations on the traffic entropy (Ozcelik and Brooks (2015); Bhuyan et al. (2015)). In ap- proaches like (Zhou et al. (2014)), the problem of the similarity of the nature of the DDoS attacks in comparison with non-malicious events is studied. This is the case of the well-known situations such as flash crowds, which often occur when a large amount of legitimate users converge on a certain service in a short time interval. The total or partial reduction of the damage inflicted by the attacks is defined as mitigation. To do this, it is common to use honeypots (Heckman et al. (2013)), puzzles that recognize non-human users (Zhu et al. (2014)), bandwidth enlargement (Khanna et al. (2012)), filtering or the adoption of security protocols such as IPsec (DoS (2016g)). As can be observed, the set of mitigation actions may also contain prevention strategies. These are characterized by not having direct dependence on the attack detection. To find the compromised systems from which the malicious traffic is origi- nated is referred to as the identification of their sources. Ideally, its objective is to track the cybercriminal. However, given the administrative difficulties that this process involves (different Internet Service Providers (ISPs), prox- ies, data privacy legislations, etc.), and the recent advances on footprint occultation, this goal is often very hard to carry out successfully. Conse- quently, many of the proposals in the literature just focus on getting as close as possible to the attacker, in order to sanitize the largest amount of regions within the protected network. On the identification of sources, the packet traceback is the most frequent approach. In (Alenezi and Reed (2014)) this issue is discussed, a lot of current approaches are collected, and a new scheme for uniform tracking is proposed. The Passive IP Traceback (PIT) that by- passes the deployment difficulties of the conventional IP traceback techniques by investigation of ICMP error messages is proposed in (Yao et al. (2015)). Finally, in (Jeong and Lee (2014); Yao et al. (2015)) the influence of the characteristics of the network topology on the effectiveness of the strategies for packet marking is studied. The evaluation of defense systems against DoS/DDoS is a controversial 6     issue today. Over the years, various collections of traffic samples and method- ologies for assessment of these tools were proposed. They are mainly based on public domain datasets containing both legitimate traffic and DoS/DDoS threats. The malicious content usually corresponds with real traffic captures (KDD’99, DARPA’99, FIFA World Cup’98, etc.) or traffic generated by tools that imitate the behavior of the real attacks (D-ITG, Harpoon, Curl-loader, DDOSIM, etc.). In (Bhatia et al. (2014)) each of them is discussed in depth, and the lacks on the current verification methods are summarized. Thus they conclude that the traffic captures which were traditionally applied are outdated and widely disparate of the current traffic. For this reason, it is rec- ommended to consider only CAIDA 07 (DoS (2016b)) among them, labeled as the best of the bad solutions. On the other hand, the use of simulation tools entails the loss of realism. Also it is difficult to compare new proposals with those on the bibliography (Kumar and Selvakumar (2013)). This is be- cause every paper defines their own scenarios of experimentation according to their requirements.  2.2. Human beings immune system All living beings have developed multiple immune mechanisms, empha- sizing among them defenses of vertebrate species due to their sophistication. Many types of proteins, cells, organs and tissues form part of these systems, and they are related through an elaborate and dynamic network. As part of this more complex immune response, the human immune system, over time, adapts to recognize specific antigens, which is called adaptive immunity. The defense mechanisms compose the innate immunity, and usually are the first line of protection. Each component of the innate immunity is able to recognize and eliminate different types of antigens. They are mainly rejection reactions conducted by external physical barriers such as skin or mucous membranes, and internal defensive elements like Natural Killer (NK) cells or phagocytes. They lack immune memory, and are only effective against pathogens known as priori. On the other hand, the adaptive immunity presents specificity, i.e., after learning how to identify and reject an antigen, the knowledge gained allows it to react more firmly against the intruder, by generating new and stronger agents; but they can only act for mitigating the threat for which they were created. The most important cells involved in this process are lymphocytes and presenting cells, and the most important adaptive immune responses are humoral and cellular. In both of them take part agents responsible for 7     antigen recognition and disposal. Given they pose a good example of this process, are briefly explained below. • Humoral response. The antibodies detect and eliminate the threats by swallowing them. The remains of this process are captured by Th lymphocytes, and these stimulate the Tb lymphocytes to generate an even greater amount of antibodies specialized in recognizing the threat. Antigens never seen before are identified by the antibodies that have suffered small mutations in their construction process, allowing them to recognize different antigenic determinants. • Cellular response. The Th detect the intrusions. Then they attract Tc cells for disposal. As in the previous case, the remains stimulate the generation of Th and Tc, specializing them against the new threat. In addition, the recent Tc are often able to identify the antigen. In both reactions, increasing defensive measures is temporary. After a quarantine period, the system is regulated by removing the excess of agents. These are programmed to die by apoptosis, also known as cell death. Acquired immunity is the basis of the vaccination in human beings. When samples of an antigen are detected, the organisms deploy temporary and specific countermeasures for disposal. Therefore the response is faster and more effective than in the first contact. 2.3. Artificial Immune Systems The adaptation of biological defenses towards information security is usu- ally performed by deployment of multi-agent systems (Ou (2012)). In pio- neering approaches, such as (King et al. (2001); Harmer et al. (2002)), the main guidelines for the emulation of the activities carried out by immune cells were introduced. They often apply some of the four classical bio-inspired al- gorithms: negative selection, clonal selection, artificial immune networks and Danger Theory (DT), which are briefly described below. Negative selection is the process by which the immune agents learn to distinguish antigens from the cells of the organism itself. In (Zhaowen et al. (2012)) there is a good example of its application for detection of anomalies. But as the authors suggest, it poses a methodology that tends to generate high false positive rates, a situation that often leads to its complementation by clonal selection algorithms (Ligeiro (2014)). These are based on the as- sumption that every lymphocyte at its growth stage must be able to react 8     against a specific antigen before being released in the body. Clonal selection is an effective solution to problems of recognition and optimization (Castro and Zuben (2002)). In proposals like (Sheshtawi et al. (2010)), it is also applied for identifying malware, often assuming an important refining step over negative selection. The immune networks stem to the idea of extending clonal selection to networks. They are commonly implemented as a channel of interaction be- tween the different actors of AIS. In (Seresht and Azmi (2014)) there is an example of an immune network for connecting different agents that perform negative selection. In (Yang et al. (2011)) a similar deployment is adopted, but this time involving agents that apply Danger Theory. The Danger Theory takes into account the latest advances in medical research. Consequently and unlike their predecessors, it rejects the idea that organisms have the capacity to distinguish between own cells and antigens. Instead it postulated that the triggering of immune reactions is originated by warning signals sent from tissues in direct contact with the threat. Because of its novelty, it is one of the most common algorithms in the bibliography of recent years. In (Aickelin et al. (2003)) the bases for its adaptation to intrusion detection are defined. DT is applied to recognition of enumeration attacks in (Greensmith et al. (2010)), and it is also considered for intrusion detection based on studying the system calls of the protected environment in (Azmi and Pishgoo (2013)). Despite the predominance of these methods in the bibliography, not all AIS are based on such specific processes of the biological immune systems. Some approaches imitate the global behavior of the innate and adaptive immune responses on the vertebrate beings, without following predefined algorithms. This makes it easier to combine the defensive strategies used in each field, with the ideas provided by biological immune systems, by this way reaching solutions that best fit the real use cases. A good example of this is proposed in (Boukerche et al. (2007)), where this idea is implemented to detect anomalies in network traffic. To do this, six main classes of agents are considered, among which are distributed sensing, communication and reaction tasks. Their adaptive immune response involves the cloning of a greater number of antibody agents in the threatened regions, by this way increasing their presence on the protected environment. In (Chen (2010)) antibodies are mobile agents that swarm the network looking for indicators of damage. Another example is (Visconti and Tahayori (2011)), wherein when the immune agents identify a potentially harmful incidence, the adaptive 9     reaction is triggered. This entails the increase of the weights of the monitored features, thus gaining restrictiveness and specificity. Finally, in (Venkatesan et al. (2013)) a solution to some of the attacks that affect mobile networks is proposed. As in previous approaches, its behavior is based on cloning the agents that have been able to recognize a specific threat.  3. Biological immune reactions against DDoS To design a bio-inspired system for detecting and mitigating denial of service attacks requires being aware of certain aspects of both threats to be treated and features of the defensive deployment. In order to establish the limitations and goals of this approach, the following describes the most relevant assumptions and requirements that have been taken into account throughout its development.  • The denial of service attacks to be treated may originate in one or more sources. In addition, they acquire the ability to take alternative routes to the victim, adapted to network conditions, such as filters that drop malicious traffic or congestion. • Malicious traffic may harm the protected network in different places: proximities to victim nodes, intermediate network actors, or close to their source. Once a section is compromised, it is possible that a chain reaction leads to the compromise of the others. • When a region on the protected environment is victim of a flooding attack, it must be quarantined until any hint of aggression disappears. In this way it is possible to provide proactive perimeter security. • Although a region is in quarantine, the legitimate traffic should con- tinue flowing through it practically in its normal rhythm. In other words, the quarantine should not block sections of the network nor prejudice their quality of service in a representative manner. Other- wise, the attacks achieve their goal. • Mitigation actions should be performed as close as possible to the source of the threat. Thus the amount of network regions at risk is reduced, and the identification of the attacker is facilitated. 10     • Communications between the various immune agents must be per- formed through secure channels, in this way preventing that the at- tacker exploits them. • Immune agents must be able to be activated/deactivated according to the state of the protected network. • Defensive action must be proportionate to the risk to be treated. Thus the impact of the autoimmune reactions triggered by false positives is reduced. • In order to enhance the alert correlation processes and the tasks per- formed by human operators, a record must exist which collects the current state of the network, the different immune reactions and their effectiveness.  As can be observed, these premises meet an important part of the needs of the current networks. However there are other aspects that have been set aside, highlighting among them the fight against the various evasion strate- gies. These contain methods for disguising the source of the attacks or hinder the tracking of the flooding path (Fast-Flux, Domain Generation Algorithms (DGA), exploitation of anonymous networks, etc.) (Zargar et al. (2013); Al- Duwairi and Al-Hammouri (2014). On the other hand, there are algorithms designed for misleading the detection system and thus do not triggering coun- termeasures, such as those proposed in (Ozcelik and Brooks (2015)). Their consideration involves adding a lot of complexity to the proposal, and be- cause of this, it is out of the scope of the paper. Another important aspect that is also delegated to future work is to facilitate the interoperability with security protocols and data protection policies. We understand that analyz- ing obfuscated headers also means an important increment on its complexity, not being recommended for a first approach. Bearing this in mind, the ar- chitecture, behavior and properties of the proposal are described in depth below.  3.1. Architecture The proposed system has distributed architecture and its different actors assume the various roles of the biological immune systems. Its success de- pends mainly on two types of agents spread along the protected network: H detectors (DH) and A detectors (DA). DH are involved in the innate immune 11     response and in the adaptive response. Consequently they are capable of rec- ognizing and blocking new attacks, and they participate in the construction of the immunological memory. On the other hand, DA have the ability to detect and mitigate the attacks previously identified by DH assuming a very important role in the adaptive response.  \n Figure 1: Example of distribution of the different actors in the AIS  In Fig. 1 an example of their deployment is shown. Apart from DH and DA, other important components take part in the immune process, which are defined below. • Protected network. The protected network is defined as the network segment that connects the attacker with the victim, and where the AIS is operating. Therefore, it is the element to be protected and the main scenario of different immune reactions. • Intermediate nodes. Every node on intermediate subnets between the attacker and the victim that is not capable of detecting, mitigating or taking part in the immune reactions is defined as intermediate node. • H detectors. The immune system agents that perform innate reac- tions, and are capable of triggering adaptive reactions are defined as H detectors or DH. \nOrchestrator \n  Control Chanel Intermediate node Detector Detector \nVictim \n \n Network 12     • A detectors. The immune system agents that only perform adaptive reactions are defined as A detectors or DA. • Orchestrator. The orchestrator is the mediator between the immune agents. Its most important tasks are delimitation of the scope of the activation signals triggered at the immune responses, management of quarantine areas, separation of the AIS control data with respect to the protected network traffic and information gathering. • Protected connection. Every connection between nodes on the protected network over which the proposed AIS acts is defined as protected con- nection. • Control channel. The connections between the orchestrator and im- mune agents are referred as control channel. The distribution and cooperation of the various components of this ar- chitecture allows performing innate and adaptive responses to attacks. The deployment of an orchestrator on a different data plane from the protected network connections provides autonomy, prevents that the traffic generated by the AIS penalize QoS, and facilitates the adoption of security protocols, thus reducing the risk of packet poisoning or Man-in-the-Middle attacks. But despite its obvious benefits, it is optional, because in certain use cases it is not possible to have much control of the protected environment (law restric- tions, privacy, etc.). In this situation, the immune agents may have sufficient autonomy to do without it, which involves: being in direct contact with agents over which it has influence, saving a log with the performed actions and determining whether it belongs to a region in quarantine or not (and the consequences that this entails). Usually these actions can be carried out easily, as for example by the implementation of tunneling between agents for their communication, the use of counters to determine the quarantine period, etc. but not with the global overview of the protected network and the advantages that the orchestrator provides.  3.2. Artificial immune responses Each response conducted by the proposal is subject to conditions and events that emulate the behavior of the biological immune systems. With this in mind, the behavior of the AIS is mainly conducted by two different artificial immune responses: innate and adaptive. 13     3.2.1. Innate response As in biological systems, the innate immunity on the approach is the first line of the defense strategy. It aims to identify and mitigate new threats and protect H detectors of disablement by flooding. The process of innate immunity requires maintaining activated DH agents along the protected net- work. These sensors monitor the entire traffic flowing through them looking for suspicious anomalies. Therefore, detected attacks must present certain evident characteristics related to considerable fluctuations in the analyzed traffic distribution. Once a threat is identified, the mitigation measures con- sist mainly on the adoption of directives that restrict the communications with nodes, ports or services involved in the attack vector. We are aware that more sophisticated mitigation actions could be implemented, but their decision and development are delegated to future studies in order to facilitate a better understanding of this first approach. The innate response provides quick and efficient countermeasures, re- quiring no communication with the orchestrator prior to their launch. By recognition and elimination of pathogens before they enter into the system, the proposal innate response emulates the behavior of the immune system of human beings. This is because it acts in the same way as the various exter- nal physical barriers or cells, and without specificity. In addition, it should be noted how agents involved in this task act coincide with those of most conventional Intrusion Prevention Systems (IPS), i.e. IDS with the ability to apply basic countermeasures.  3.2.2. Adaptive response The adaptive response is the next defensive step in the proposal. It is trig- gered every time a DH agent recognizes a new threat, which implies that they must hold at least a short memory capable of storing their latest decisions. In this context, determining when an attack is Non-Seen-Before (NSB) im- plies it is not presence in the immune memory. Because of this, that memory has a very important role in the AIS decision-making. Determination of how the immune memory will be implemented is not usually a trivial problem. With this purpose, the two more intuitive schemes to take into account are those centralized or distributed. When the immune memory is centralized, is sustained by the orchestrator. In this case this component is responsible for determining whether an incidence is NSB, considering information provided by all the sensors. However, if it is distributed, each detector disposes only the information gathered for itself, or by a group of close sensors. Hence 14     an attack tagged as NSB by an agent could not be new for other detec- tors. This second approach provides greater autonomy to the agents and allows some of them to trigger different adaptive reactions against the same pathogen. Given that this is a more efficient strategy and a good way of covering separated regions of the protected network, it was implemented at the experimentation. Once the adaptive reaction is released, the DH that identified the attack sends activation signals to the DA agents in close proximity. The scope of these signals can be determined in different ways. It may affect the surround- ing neighbors, nodes on a certain region or network devices in specific routes previously established by the orchestrator. This also enables the application of Artificial Intelligence and Data Mining in order to research their optimal propagation, giving many possibilities for future works. For simplicity the first of the previously mentioned alternatives was implemented. Then the activated DA agents analyze traffic flowing through them. Un- like DH, their detection engines increase restrictiveness in proportion to the flood of the attack, usually acting much more stricter than DH. In this way it is prevented that the division of the attack flow reach the victim by al- ternative routes, assuming that when it is split, it becomes less noisy and hence more difficult to be detected. In order to prevent this measure result- ing in a substantial increase in the false positive rate, specificity is taken into account. To ensure specificity, they are only able to apply countermeasures against the threat that has activated them. Therefore, they can only take action against several attacks if they have been activated to mitigate each of them. Because of all of this, and as in nature, the artificial adaptive immune response involves the increase of the amount of effectives able to react against a certain triggering attack. At the end, the deployed countermeasures are effective for a certain period of time. While the threat persists, the immune response remains activated. If it is no longer visible, a quarantine period is activated. The quarantine is interrupted only upon detection of replicas of the intrusion (implying back to the previous state), or when the countdown expires. The network segments covered by a set of DA agents active against a specific threat and coordinated by the same DH sensor, are their quarantine region. 3.3. Implementation The behavior of the artificial immune responses is implemented in the following procedure: 15     1. At first, DH analyze all the traffic flowing through them. DA remain on standby waiting to be activated. 2. When the DH identify malicious traffic, the related connections will be blocked. This is an innate immune response. Then, the sensor warns the DA surrounding to proceed with the activation of countermeasures. In this way the adaptive immune response is triggered. The warnings contain information about their origin and the malicious flow charac- teristics. 3. When activated, the DA analyze the traffic from their potentially harm- ful sources. Unlike in DH, the level of restriction of their decision threshold are increased according to the characteristics of the trigger- ing attacks. They also block the identified threats. Thus, if the attacks cross alternative paths, they are also mitigated. 4. In the case that different DH emit activation signals for the same attack, the DA apply the most restrictive one. 5. The DA return to standby mode just in case a period of time without news of the attacker has passed, or triggering signals have not been received. 6. DH remain vigilant to face new incoming threats. An example of the behavior of the proposed AIS when dealing with DDoS flooding attacks is described in Fig. 2. It is part of the situation shown in Fig. 2a, where S is the source of the attack, T is the target, and each Ni is an intermediate node located at i. In the case that the sensor DH is unable to recognize the threat, it is propagated along the protected network via different routes and according to the load balancing policies, as shown in Fig. 2b. But if the attack is successfully detected, the innate immune response is initiated. Thus the traffic from S is discarded, slowing the advance of the intrusion, as shown in Fig. 2c. As a legitimate flow trying to reach its destination when some network incident block its route, the malicious traffic will try to reach the victim for alternative paths. Fortunately, when the threat was detected the adaptive immune response was also triggered. As shown in Fig. 2d, this has led to the activation of the neighboring DA agents. Consequently, the attack reaching the victim through the new connections is prevented. 16      \n  (a) Original Attack (b) Load Balancing  \n(c) Interception (d) Adaptive Response  Figure 2: Example of the AIS behavior   Despite its simplicity, this scheme has a large number of advantages. Firstly, it increases the defensive measures when a threat is recognized, pro- portional to the risk level. In addition, the specificity makes it only occur on specific connections, without affecting other traffic routes and reducing the impact of the false positives. Furthermore, the presence of two types of agents allows it to strategically adapt the defensive network to the monitor- ing environment. In this way the most powerful computers may act as DH, and the rest assume the role of DA. Note that a single node can assume both roles, or even deploy their own virtual network including several of them. On the other hand, the restriction level is self-regulated through deactivation of agents when the quarantine time passes, thus saving resources. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 17     3.4. Properties The main characteristics of the biological immune systems have been assumed and adapted to the fight against DDoS in the following ways:  • Innate and adaptive responses. At the innate response, the proposal has the capacity to react against threats which were not previously rec- ognized. Once detected, it strengthens the defensive measures against future replies as occurs in biological adaptive immune reactions. • Specificity. In nature each adaptive immune cell reacts against only one type of antigen (unlike the case with innate responses). Bearing this in mind, in this approach the innate response is common to any type of threat detected. However, the immune response affects only the triggering flooding attack. • Clonality. When unknown antigens are detected, the adaptive biologi- cal response clones the cells that have been able to be recognised. This increase the chance of identifying their replicas. The AIS behaves in a similar way when activating agents after the adaptive response is released. • Immune memory. Both in nature and the proposal, the deployed coun- termeasures remain active for a period of time. This enables them to react more effectively against future replicates. • Self-Regulation. Once the adaptive response is triggered, the human immune system is regulated by the apoptosis of most of the new cells. In the AIS, defensive measures are also reduced when a certain period of time is exceeded and no replications of the triggering threat are detected. • Autonomy. In both cases, once the entities of the defensive deployment are activated, they operate with complete autonomy. • Diversity. In nature, the set of immune agents must be able to detect any antigen. The same thing happens in the AIS, where specificity is applied only in the deployment of countermeasures. 18  n n \nL p    4. Detection of flooding attacks The following describes the most important aspects of detection of threats and identification of sources on the approach, which involves metrics, fore- casting, thresholding and recognition of compromised nodes.  4.1. Metric The monitored traffic is analyzed by studying the entropy of its distribu- tion. In particular, entropy fluctuations are analyzed looking for anomalous behaviors. The decision to use entropy variations over other detection meth- ods proposed in the bibliography is that, as demonstrated in (Ozcelik and Brooks (2015)), is much more effective. This is principally because its accu- racy depends less than those of the others on how the protected network is used. The traditional entropy was first adapted to the information theory by Shannon in 1948 (Shannon (1948)). It was considered as a measure of fluctuations on qualitative variables, and it is often defined as the degree of unpredictability on their behavior. Given the qualitative variable X, the finite set {x1, x2, · · · , xn} and their probabilities p1, p2, · · · , pn, the Shannon entropy was described by the following expression:  H(X) = L pi i=1  log 1   a pi = − L pi i=1  loga pi  (1) where loga b×logbx = logax. In addition, if the variable is deterministic then H(x) = 0 must be satisfied. This means that all the pi probabilities are 0, except for one, which has value 1. There are different generalizations of this entropy, adapted to the different use cases. The AIS applies that proposed by R`enyi. This decision was made considering studies like (Bhuyan et al. (2015)), where its effectiveness stands out from the rest of variations when applied on the detection of DDoS flooding attacks. R`enyi entropy is defined as follows:  1 H(X) = 1 − α  log2 n α i i=1  (2) where the parameter α indicates its order, such as α ≥ 0 and α /= 1. Note that the Shannon entropy is the particularly case proposed by R`enyi when α = 1. 19  t=0 t=0    In our approach, the variable X is defined as the volume of traffic flowing between two network nodes A and B, destined to port C. The P probability implies that every pi represents the frequency of occurrence in the monitored traffic of packets with certain source, destination and that are addressed to a specific port. Therefore it is fulfilled that:  ai  pi = No.packets  (3) where ai is the total amount of packets that met the previously described condition. In this proposal the entropy variations are treated as univariate time series of N observations, expressed a:  Hα(X)t=0, Hα(X)t=1, · · · , Hα(X)t=N ; (Hα(X)N ) (4)  4.2. Forecasting entropy variations When estimating the entropy of future observations it is taken into ac- count that Hα(X)N series may experiment changes in trend and seasonality over time. Additionally, the forecasting methods to consider should be ef- fective with few observations, and able to run efficiently in real time. For this reasons, to model the network behavior the triple exponential smooth- ing proposed by Holt-Winters has been chosen. This election is supported by publications like (Gardner and Dannenbring (1980); Groff (1973)), where it is shown that considering the trend and seasonality of series in which they are unrepresentative, leads to insignificant prediction errors. Furthermore, the time required to calculate their forecasts is considerably lower than in the autoregressive models. Note that in order to avoid confusion between the α range on R`enyi entropy and the α parameter on Holt-Winters, henceforth Hα(X) is summarized as H(X), and the coming alpha symbols will refer only to the forecasting adjustment. The Holt-Winters model allows performing the prediction of the next observation H(X)t+1 by analyzing three different components B, T and S. These are defined in the following recursive expression:  Bt = α(Hi − St−N ) + (1 − α)(Bt−1 + Tt−1) (5) Tt = β(Bt − Bt−1) + (1 − β)Tt−1 (6) St = γ(Ht − Bt) + (1 − γ)Bt−n (7) 20  N    where Bt is the base estimation at t, the estimation of the trend is Tt and the estimation of the seasonal factor is St. The forecasting parameters α, β and γ fall in the range 0 < α, β, γ < 1, and facilitate the adjustment of the smoothing.  The prediction Ht+1 is usually calculated by additive or multiplicative operations. This approach considers the additive version, since it is assumed that the seasonal pattern of the series is independent of its trend. Consequently the forecast is calculated as follows:  H(X)t+1 = Bt + Tt + St (8) Another important aspect to keep in mind is the initialization method of B0, T0 and S0 estimators. It is assumed that when no trend or seasonality is expected on the time series, the initialization of estimators based on the latest observations is preferable over the use of global measures. The imple- mented method is described in (Makridakis et al. (1998)), which has proven to behave particularly well in similar use cases. Namely, the last twenty-four observations are considered. The calculations performed are the following:  B0 = M¯1 (9) M¯2 − M¯1 T0 = (10) 12 St−12 = pt M (11) 1 where M1 summarizes the first twelve observations and M2 the last dozen. The adjustment of parameters α, β, γ is obtained by calculating the values minimizing the function Sum of the Squared Errors of prediction (SSE), defined as:  SSE(α, β, γ) = L(H(X)t − H\\(X) t=1  4.3. Definition of prediction intervals   t|t−1  )2 (12) For the evaluation of the variation of entropy according with X in the observation t, two thresholds are constructed: an upper threshold Tht and a lower threshold Tlt. From these the prediction interval of the sensor is defined in the same way as is usually performed when implementing Holt-Winters (Hyndman et al. (2005)). They are expressed as follows: 21  2    Tht(t) = p0 + K × ✓var(Et) (13) Tlt(t) = p0 − K × ✓var(Et) (14) where Et is the prediction error in t and p0 is the prediction of the last obser- vation. The prediction error is given by the difference between the forecast and the t observation. The variance V ar(Et) is calculated considering the prediction error of the last t observations. In addition, the thresholds include a parameter K, from which it is possible to adjust the sensitivity of the detec- tor. In the case of DH agents, the default value Z = α is assigned to K, thus relating the thresholds with the normal distribution of the series. Note that this is not a wrong decision considering publications as (Makridakis et al. (1998)), where it has shown that when the time series does not approach the normal distribution, the error is unrepresentative. Moreover the margin rate of both intervals is in the order 100(1 − α). On the other hand, as part of the adaptive response, the DA agents acquire the ability to increase their level of restriction based on the anomalous activities detected by DH. Hence the parameter k is determined by the following expression:   K(t) = K   prev (1 − V olatk ) (15) V olleg Where Kprev is the last setting of K, Vatk is the total volume of traffic mon- itored during the attack and Vleg is the total volume of traffic monitored at the previous legitimate situation. As an example, in Fig. 3, the time series associated with the entropy of the monitored traffic in part of one of the experiment and its prediction are shown. Legitimate traffic passes through the sensor until the observation at t = 56; then the injection of a large volume of traffic is observed. In Fig. 4, the prediction intervals of the sensor under the same circumstances are shown. When the attack is launched, both thresholds are exceeded. Then the agent reports of the incidence and drops the malicious packets, so the entropy back to their original values.  4.4. Identification of sources Before the deployment of mitigation measures, and in order to allow speci- ficity, it is important to identify the possible origins of the detected threat. 22     1  0.8  0.6  0.4  0.2  0 10 20 30 40 50 60 70 t  Figure 3: Example of forecasting of the entropy  1   0.8  0.6  0.4  0.2  0 10 20 30 40 50 60 70 t  Figure 4: Example of variations on the thresholds of prediction   With this purpose, the different p probabilities observed at t are studied. The following assumptions are taken into account:  • Attacks could be originated from different IP addresses. • Attacks may be directed against different destination IP addresses and ports. • The routes Source, Destination, Port with higher p are more likely to be involved in the attack  On this basis, the various instances of X are grouped in function of their p. Given the nature of the information to be processed, the implemented algorithm must be unsupervised. Of the many options available, k-means \nObservation Prediction \nTime Series Upper Threshold Lower Threshold Entropy Entropy 23     algorithm has been chosen and the K value has been adjusted by the elbow method. K-means was selected because is well known and it is especially re- silient regarding the presence of outliers and errors in distance measurements. The elbow method compensates its main drawback: the need to previously specify the number of clusters to be generated. This method launches k- means with different K values, until unrepresentative variations in the sum of the squared error between each member of the clusters with its central value occurs. Alternatives to this decision can be found in (Rui and Wunsch (2005)). At the end of the identification of sources, nodes grouped in the cluster with greater p area tagged as suspicious, and therefore are quaran- tined.  5. Experimentation In the implementation of the sensors, the observations are delimited by a fixed number of packets whose order of arrival is consecutive. A common alternative to this is their delimitation by considering time intervals. Both pose advantages and disadvantages, with the first choice being more efficient in the analysis of collections of previously captured traces, as is the case of most of the datasets analyzed (Ozcelik and Brooks (2015)). In addition, a sliding window of size N that gathers the observations involved in the calculation of the entropy was applied. This boundedness is important to ensure that the implemented algorithms are computable, avoiding the case where N → ∞. The proposal has been evaluated in two stages. Firstly, the accuracy of the various agents when dealing with DDoS flooding attacks was measured. On the other hand, several features related to the effectiveness of the deployment of the AIS were studied. The following describes each of them in detail.  5.1. Assessment of accuracy of the immune agents Given the controversy relating to the evaluation methods of the effec- tiveness of intrusion detection system for identification of DDoS attacks, the scheme proposed in (Kumar and Selvakumar (2013)) was applied. This involves the use of two well-known datasets: KDD’99 (DoS (2016e)) and CAIDA’07 (DoS (2016b)); and in the generation of flooding attacks with the tool DDoSIM (DoS (2016c)) in a real use case. The following describes the principal characteristics of each test and their application. 24     5.1.1. KDD’99 The KDD’99 (DoS (2016e)) is one of the most referenced methodologies in the bibliography, and according to (Bhatia et al. (2014)), possibly the only one that presents a dataset with reliable labeling. It was created in 1999, under the KDD Cup competition, and from captures of traffic provided by DARPA’98. The competition task was to build a network intrusion detec- tor, a predictive model capable of distinguishing between “bad” connections, called intrusions or attacks, and “good” normal connections. It provides 41 different features of legitimate and malicious traffic samples. The attacks fall into four main categories: DoS (Denial of Service, e.g. syn flood ), R2L (unauthorized access from a remote machine, e.g. guessing password ), U2R (unauthorized access to local superuser (root) privileges, e.g., buffer overflow attacks) and probing (surveillance and other probing, e.g., port scanning). Originally it separates a subset of the datasets for the training stages of ex- pert intrusion detection systems, and the rest for their evaluation. Since the proposed system requires no training, all samples have been applied in the evaluation process, with the exception of the first observations, necessary for initialization of the predictive models. It is noteworthy that the antiq- uity of the dataset and the discovery of irregularities in its content have led to its discrediting. An important part of the research community considers KDD’99 unrepresentative, mainly due to lack of heterogeneity in comparison with current networks, old class of intrusions, errors when data gathering, etc. As is discussed in (Viswanathan et al. (2013)), this leads to the mistake of consider that the experimental results are scalable to real monitoring en- vironments. But despite this, it remains one of the most used methodologies, mainly due to the administrative difficulties associated with the publication of new datasets and the fact that it was implemented in most of the previous proposals.  5.1.2. CAIDA’07/08 The CAIDA’07 dataset (DoS (2016b)) provided samples of traffic traces containing DDoS flooding attacks (mainly ICMP, SYN and HTTP) moni- tored in August 2007. They are divided into files with extension pcap and spaced at time intervals of five minutes. As described in their documenta- tion, after the capture, most of the non-malicious contents were removed. Therefore it provides a good battery of tests to evaluate the effectiveness of the detection systems when analyzing flooding attacks, thus allowing their hit rates to be calculated. However, it is also necessary to determine their 25     behavior when processing legitimate traffic, thereby calculating the false pos- itive rates. Consequently, the use of CAIDA’07 is often complemented with the passive collection of traffic traces CAIDA’08 (DoS (2016a)), as described in (Kumar and Selvakumar (2013)). The latter compiles usual and legiti- mate traffic monitored at the data centers Equinix of San Jose and Chicago (the same networks as CAIDA’07). Both represent the traditional evaluation scheme with greater similitude to recent networks, allowing the contrast of the results, and involving a more current context. In the evaluation of the proposal, were separated into traces of 15,000 packets, and the distinction of the class of flooding attacks has not been taken into account.  5.1.3. DDoSIM and UCM traffic This test scenario is a real use case. It combines the study of the habitual traffic on the subnet corresponding with the Faculty of Computer Science of the Complutense University of Madrid (UCM), with the analysis of flooding attacks injected by the tool DDoSIM (DoS (2016c)). The generated attacks act at application layer, and are based on the massive send of different HTTP and TCP requests. During its course, DDoSIM simulates the behavior of various zombie computers using random assignment of IP addresses, which are able to log into the victim servers. Once established, it proceeds to the flooding of requests. The captured traffic has been divided into two groups: legitimate and malicious. Both contain samples with traces of 40,000 packets in format pcap. The first one is applied to calculate the false positive rate of the approach. The other is to determine the hit rate.  5.2. Evaluation of the Artificial Immune System To evaluate the effectiveness of the deployment of the AIS, a simulator ca- pable of generating traffic distributions and different networks with different locations of DH and DA has been implemented. This is because none of the functional standards for the evaluation of similar systems provides a complete knowledge of the organization of various networks. In the generation of new networks, several parameters had been taken into account: number of nodes, legitimate traffic volume, branching component, and cyclic component. The last two determine the number of connections associated with each node and the number of cycles in the network when it is plotted as a graph of finite dimensions. On the other hand a tool for simulating flooding attacks has been developed. Given a network built by the previously described scheme, 26     the tool emulates the injection of a certain amount of packets from a source node to a victim side, according with (Bhatia et al. (2014)).  \n Figure 5: Construction of virtual networks for testing  In Fig. 5 a summary of the tasks involved in the creation of each of the networks in the experiments is shown. In a first step, the network topology and the distribution of agents are defined. The network is built according to the previously described parameters, and is represented by a graph where the vertices are its nodes, and the edges are its connections. Then the origin and the destination of the attacks are defined. The location of immune agents is defined by greedy graph coloring (Galinier and Hertz (2006)), where the two most frequent calculated colors represent the class of actors. Thus the amount of DA sensors dependent of each DH is regulated. The second level defined how the traffic is generated. The legitimate communications are randomly decided by taking into account the simulation parameters. The injection of traffic is carried out by the tool hping3. In the case of clean traffic, various protocols (FTP, HTTP, ICMP, etc.) and actions (transfer of files, requests, session maintenance, etc.) are performed. With all of this, a script that allows the deployment of the network in a virtualized environment is \n  \nScript \nLegitimate \n \n Generation \nVirtualization \nMalicious \nDistribution of Agents \n   27     built. Through its execution it is possible to assess the effectiveness of the AIS in the test. In the experiments, 220 different networks have been considered. For each of them, the behavior of denial of service attacks with different power, directed between each possible pair of source and destination nodes had been studied. Different features of the proposal have been evaluated in function of the location of the immune agents, the flooding power of the attacks, the congestion of the network, or the malicious traffic mitigated.  6. Results The results obtained when assessing the behavior of the agents separately and in the study of their capacity of collaboration according with the pro- posed AIS, are described below.  6.1. Detection of threats At the evaluation of the effectiveness of the artificial immune agents when detecting threats, the adjustment parameters of the detectors are the variable α that defines the order of the traffic entropy, and the number of packets per observation. Variations on the first of them have a similar result with the three benchmarks: When higher is the value, the higher the level of restriction of the sensor. In such a way that when α > 3, the deployment of the AIS is counterproductive because the false positive rate becomes excessively high (exceeding 20%). In general terms, the smaller the value of α, the lower the rate of false positives. For this reason it was considered α = 1 along the experimentation. Bearing this in mind, the following describes and discusses the results obtained by analyzing KDD’99, CAIDA’07/08 and DDoSIM injection against habitual UCM traffic.  6.1.1. KDD’99 In Fig. 6 the results obtained by analyzing the collection KDD’99 are shown. The X axis displays the number of packets per observation, whereas the Y axis indicates the True Positive Rate (TPR) and False Positive Rate (FPR). The hit rate has remained nearly unchanged in all the tests. However, the amount of errors in analyzing legitimate traffic is especially sensitive to the position in X; with fewer packets per sample, the system behaves more restrictively. Its ability in detection ceases to depend on it from 2,000 observations, at which it operates in saturation mode. This reason has led to study in more detail the two class of flooding attacks contained in more than 28  \n   100,000 packages within the dataset: smurf and neptune. The rest (back, teardrop, pod and land ) are not taken into account because KDD’99 does not provide enough information to initialize the predictive models in the cases where there are lots of packages per observation. The true positive rates in saturation are 98.66% (neptune) and 100% (smurf ), with a false positive rate of 1.42%.  1  FPR 0.8 TPR Neptune TPR Smurf 0.6  0.4  0.2  0 900 1400 1900 2400 Packets  Figure 6: Results when analyzing KDD’99   1   0.8   FPR TPR  0.6  0.4  0.2  0 50 100 150 200 Packets  Figure 7: Results when analyzing CAIDA’07 and CAIDA’08   6.1.2. CAIDA’07/08 In Fig. 7 the results obtained when analyzing the traces of CAIDA’07 and CAIDA’08 are shown. The behavior of the agents is very similar to the previous test. But this time, the detectors operate in saturation with fewer packets per observation; in particular, around 160. In this case the hit Rate Rate 29     rate is 98.11% and false positive is almost 1.91%. The difference between the results obtained in KDD’99 and CAIDA’07/08 is mainly due to the variations in the homogeneity of samples. In KDD’99 the traffic is older, and the scenario where the samples were gathered is more limited. Consequently, the legitimate samples are more like each other, thus reducing the tendency of the system to issue false positives.  6.1.3. DDoSIM and UCM traffic In Fig. 8 the results obtained in the experimentation with UCM traffic and DDoSIM are shown. They remain the behavior of the previous tests, reaching saturation in 2,000 packets per observation. However, the accuracy obtained is considerably worse: the hit rate is 92.3% and the false positive rate is 8.3%. The difference precision achieved demonstrates that the good results obtained when applying the evaluation functional standards are not scalable to real networks. This is because the homogeneity of the traffic analyzed is much higher, according to the current usage models. But despite the high false positive rate, the quality of service of the protected environment will not be affected. The proportional increase in the rigor with which act the agents of the adaptive immune response and the specificity of the approach, allow that most of the legitimate traffic involved in the emission of false positives reach their destination by alternative ways; this will be unusual when dealing with malicious traffic. The experimentation also emphasizes another important feature of the proposed method: most of the identified attacks have triggered alerts with proximity to the beginning and end of the malicious flooding. These are the observations where variations of entropy differ most from the legitimate traf- fic. When the attack is constant (as example, due to high rate attacks), the entropy tends to stabilize again, becoming invisible to the detector. Such be- havior is illustrated more clearly in Fig. 9. It shows the impact on the traffic entropy of a distributed attack generated by DDoSIM against the UCM net- work. The X axis displays the observations and the Y axis indicates the value of the normalized entropy. The attack starts from the observation 60. En- tropy anomalies are particularly visible on the following 15-20 observations, where the forecast exceeds the prediction thresholds, and thus the sensor re- ports an incidence. If mitigation measures (such as the packet drop applied in Fig. 2) are not adopted, the entropy is stabilized, albeit with much higher values. Most likely the attack will not be visible again until completion. At that time 15-20 observations reveal the descent of the entropy to their usual 30     values. In view of this behavior, and in order to prevent evasion strategies, the combination of detection and mitigation measures is required, as is driven by the proposed AIS.  1   0.8   FPR TPR  0.6  0.4  0.2  0 1100  1300  1500  1700  1900  2100  2300  2500 Packets  Figure 8: Results when analyzing traffic from DDOSIM and UCM   0.016  0.014  0.012  0.01  0.008  0.006  0.004  0.002  0 20 40 60 80 100 t  Figure 9: Evolution of the entropy with DDOSIM and UCM   6.2. Artificial immune reactions The behavior of the proposed systems taking into account the location of the agents, the flooding power of the attack, the legitimate congestion of the protected network and the mitigation capacity of the AIS are described below. \nObservation Upper Threshold Lower Threshold Entropy Rate 31  \n   6.2.1. Location of the immune agents In Fig. 10 the results when DH and DA are triggered in different place- ments are shown. Y axis indicates the TPR/FPR and the X axis the location of the agents. The latter is indicated by a value in range 0 to 1, which de- termines the distance in the path between source and victim, where 0 is the location of the source, and 1 the location of the victim. When new attacks are launched, the average TPR is 0.85 and the FPR is 0.072. There is a trend: near the ends, the TPR value approaches 1. However, at intermediate points the accuracy is reduced by up TPR is 0.70. This occurs in the points closest to the equidistant location between the ends. When the DA agents are activated by the immune adaptive response, the trend is repeated. Nevertheless the average TPR was increased by 7%. The greatest improvement is observed at intermediate distances. The worst TPR value is 0.82, which implies the improvement of 12%. The adaptive immunity response has low impact on the FPR, i.e., it increased 0.6% at the worst case. In view of these results, and assuming that most of the conventional IPS behave similarly to the innate response, it can be stated that the pro- posed AIS poses a significant improvement in the cases where they find more difficulties to operate adequately. This occurs at the nodes above halfway between the attacker and the victim, mainly because the malicious traffic can spread over a larger number of alternative paths; near the ends they tend to converge (close to the victim) or diverge (close to the attacker).  1  0.8   0.6   0.4 TPR DH TPR DA FPR DH FPR DA  0.2  0 0 0.2 0.4 0.6 0.8 1 Location  Figure 10: Results based on the location of the immune agents Rate 32  \n   6.2.2. Flooding Power In Fig. 11 the accuracy of the system depending on the flooding potency is observed. The Y axis details the TPR/FPR and the X axis indicates the power of the attacks. The last is represented in values in the range of 0 to 1, which correspond to the percentage of the bandwidth that aims to occupy, with 0 being no traffic injection and 1 means the complete saturation of the connections. As shown, when the attack is more powerful than 0.5, the accuracy in both responses is similar and close to 1. Therefore, changes are irrelevant. However, when less noisy attacks, they are more difficult to be detected. These cases are where the adaptive response improves the performance of the system. At the best case, the TPR has increased by 26.5% in the power range 0.3 to 0.4. In summary, the higher the power of the attacks, the greater the noise caused, and therefore the threats are easier to detect. When attacks are less visible, a more significant improvement of the proposed AIS on the conventional IPS is observed.  1   0.8   0.6  TPR DH TPR DA FPR DH FPR DA  0.4  0.2  0 0 0.2 0.4 0.6 0.8 1 Flooding Power  Figure 11: Results based on the flooding attack   6.2.3. Network Congestion In Fig. 12 the detection capability in function of the volume of legiti- mate traffic flowing through the networks is shown. The X axis details the TPR/FPF, and the Y axis indicated the legitimate traffic volume. The last is indicated with values in range 0 to 1. As it was done in the above test, X represents the saturation level of the network. Results are very similar to those in Fig. 11. When the traffic density is low, the proposed strategy is very accurate, as is the case of the conventional schemes. This is because the attacks are much more visible, taking a larger share of bandwidth. However, Rate 33  \n   when congestion is high, especially above 0.7, the hit rate decreases, and the false positive rate increases. In the same way as in the previous tests, this problem is reduced by the activation of the adaptive response. Particularly, this strategy results in an improvement of the TPR of 13.7% when greatest slope (congestion between 0.6 and 0.7).  1   0.8   0.6  TPR DH TPR DA FPR DH FPR DA  0.4  0.2  0 0 0.2 0.4 0.6 0.8 1 Congestion  Figure 12: Results based on the legitimate network congestion   1  0.8   0.6   0.4  TPR DH TPR DA FPR DH FPR DA  0.2  0 0 20 40 60 80 100 Nodes  Figure 13: Mitigation of attacks based on the number of compromised nodes   6.2.4. Mitigation In Fig. 13 the rate of attacks that have not reached their destinations, depending on the number of nodes in their paths, is observed. When acting solely the innate response, 81.4% of the threats had been blocked at the worst case. However, the adaptive response was able to prevent 95.5% of them, i.e., has improved accuracy by 14.1%. From the figure it also follows that the Rate Rate 34     longer the path, the greater the probability of success. This is because the load of the attack can be distributed more effectively. Bearing this in mind, it is possible to state that the larger the protected network, more relevant is the improvement obtained with this proposal. This is because the attacks that traverse larger networks often flow from greater amounts of nodes. A great effect is not achieved by avoiding their pass through few connections if there are many other routes available, as occur in the conventional IPS. But the adaptive response has the ability to spread rapidly over the network, thus increasing security measures in almost all available paths.  7. Conclusions In this paper a novel approach for detecting and mitigating DoS flooding attacks based on the emulation of the behavior of the immune system of the human beings has been proposed. It implied the design of different artificial immune agents and their distribution throw the protected network. The system has been evaluated in two stages. Firstly, the accuracy of the detection methods was measured taking into account the functional stan- dards KDD’99, CAIDA’07 and recent traffic of the UCM network compro- mised by the tool DDoSIM. The results obtained were satisfactory, empow- ering their collaboratively deployment. On the other hand, their efficiency as AIS has been evaluated. This has entailed its implementation on differ- ent virtualized networks and the assessment of their effectiveness based on different parameters. Regardless of the criteria from which the behavior of the AIS has been evaluated (location of the immune agents, flooding power, network congestion or mitigation), the adaptive response has always shown more effectiveness than the innate response. This results in a significant improvement in their ability to detect and mitigate attacks, without penal- ization in their error rates when processing legitimate traffic. The innate response behaves in the same way as the conventional IPS, so in the adap- tive reactions it is possible to study the raw effectiveness of the approach over the conventional mitigation schemes. Bearing these in mind, it is possible to confirm that the emulation of the biological immune responses is a very good way to enhance the classical countermeasures against DoS attacks. In view of these results, this proposal is promoting the initialization of new lines of research. The simplest of them is based on the improvement of metrics and forecasting methods. Others are introducing the addition of novel immune agents, thus allowing the AIS to better perform in more 35     complex use cases. But undoubtedly the most interesting are those that focus on its deployment at real uses cases. Any work in this regard will be especially interesting, given that the evaluation of the AIS has been mostly performed in simulated test scenarios. In particular, our research group is progressing towards its implementation on Software Defined Networks, and in adaptation to the anonymous network Tor.  References Anonymized Internet Traces 2008. http://www.caida.org/data/passive/ passive_2008_dataset.xml; 2016a. DDoS Attack 2007 Dataset. http://www.caida.org/data/passive/ ddos-20070804_dataset.xml; 2016b. DDoSIM Layer 7 DDoS Simulator. http://http://sourceforge.net/ projects/ddosim/; 2016c. Internet Security Threat Report 2014 Vo. 19. http://www.symantec.com; 2016d.  KDD Cup Dataset 1999. http://kdd.ics.uci.edu/databases/kddcup99/ kddcup99.html; 2016e. Security Threat Report 2014. http://www.sophos.com; 2016f. The Internet Organised Crime Threat Assessment (iOCTA. http://www. europol.europa.eu; 2016g. Aickelin, U., Bentley, P., Cayzer, S., Kim, J., McLeod, J.. Danger Theory: The Link between AIS and IDS. In: Proceedings of the 2nd International Conference on Artificial Immune Systems (ICARIS). Edinburgh, United Kingdom; 2003. p. 147–155. Al-Duwairi, B., Al-Hammouri, A.. Fast Flux Watch: A mechanism for online detection of fast flux networks. Journal of Advanced Research 2014;5(4):473–479. Alenezi, N., Reed, M.. Uniform DoS traceback. Computers & Security 2014;45(1):17–26. 36     Anagnostopoulos, M., Kambourakis, G., Kopanos, P., Louloudakis, G., Gritzalis, D.. DNS amplification attack revisited. Computers & Security 2013;39(B):475–485. Azmi, R., Pishgoo, B.. SHADuDT:secure hypervisor-based anomaly detec- tion based on danger theory. Computers & Security 2013;39(B):268–288. Bhatia, S., Schmidt, D., Mohay, G.. A framework for generating realistic traffic for Distributed Denial-of-Service attacks and Flash Events. Com- puters & Security 2014;40(1):95–107. Bhuyan,  M., Bhattacharyya,  D., Kalita,  J..  An empirical evaluation of information metrics for low-rate and high-rate DDoS attack detection. Pattern Recognition Letters 2015;51(1):1–7. Boukerche, A., Machado, R., Juca, K., Sobral, J., Notare, M.. An agent based and biological inspired real-time intrusion detection and se- curity model for computer network operations. International Journal of Computer Communications 2007;30:2649–2660. Cai, Y., Franco, R., Garc´ıa-Herranz, M.. Visual latency-based interac- tive visualization for digital forensics. Journal of Computational Science 2010;1(2):115–120. Callegari, C., Giordano, S., Pagano, M., Pepe, T.. Wave-cusum: improving cusum performance in network anomaly detection by means of wavelet analysis. Computers & Security 2012;31(5):727–735. Castro, L., Zuben, F.. Learning and Optimization Using the Clonal Selection Principle. IEEE Transactions on Evolutionary Computation 2002;6(3):239–251. Chen, B.. Agent-based artificial immune system approach for adaptive dam- age detection in monitoring networks. Journal of Network and Computer Applications 2010;33(6):633–645. Chen, Y., Ma, X., Wu, X.. DDoS detection algorithm based on prepro- cessing network traffic predicted method and chaos theory. IEEE Commu- nications Letters 2013;17(5):1052–1054. 37     Galinier, P., Hertz, A.. A survey of local search methods for graph coloring. Computers & Operations Research 2006;33(9):2547–2562. Gardner, E., Dannenbring, D.. Forecasting with Exponential Smoothing: Some Guidelines for Model Selection. Decision Sciences 1980;11(2):370– 383. Greensmith, J., Aickelin, U., Tedesco, G.. Information fusion for anomaly detection with the dendritic cell algorithm. Information Fusion 2010;11(1):21–34. Groff, G.. Empirical Comparison of Models for Short-Range Forecasting. Management Sciences 1973;20(1):22–31. Harmer, P., Williams, P., Gunsch, G., Lamont, G.. An artificial immune system architecture for computer security applications. IEEE Transactions on Evolutionary Computation 2002;6(3):250–280. Heckman, K., Walsh, M., Stech, F., O’Boyle, T., DiCato, S., Herber, A.. Active cyber defense with denial and deception: A cyber-wargame experiment. Computers & Security 2013;37:72–77. Hyndman, R., Koehler, A., Ord, J., Snyder, R.. Journal of Forecasting 2005;24:17–37. Jansen, R., Tschorsch, F., Johnson, A., Scheuermann, B.. The Sniper Attack: Anonymously Deanonymizing and Disabling the Tor Network. In: Proceedings of the 18th Symposium on Network and Distributed System Security (NDSS). San Diego, Ca, US; 2014. . Jeong, E., Lee, B.. An IP Traceback Protocol using a Compressed Hash Table, a Sinkhole Router and Data Mining based on Network Foren- sics against Network Attacks. Future Generation Computer Systems 2014;33(1):42–52. Khanna, S., Venkatesh, S., Fatemieh, O., Khan, F., Gunter, C.. Adaptive selective verification: an efficient adaptive countermeasure to thwart DoS attacks. IEEE/ACM Transactions on Netwowking 2012;20(3):715–728. King,  R., Russ,  S., Lambert,  A., Reese,  D.. An artificial immune sys- tem model for intelligent agents. Future Generation Computer Systems 2001;17(4):335–343. 38     Kumar, P., Selvakumar, S.. Detection of distributed denial of service attacks using an ensemble of adaptive and hybrid neuro-fuzzy systems. Computer Communications 2013;36(3):303–319. Lee, S., Kim, D., Lee, J., Park, J.. Detection of DDoS attacks using optimized traffic matrix. Computers & Mathematics with Applications 2012;63(2):501–510. Ligeiro, R.. Monitoring applications: An immune inspired algorithm for software-fault detection. Applied Soft Computing 2014;24:1095–1104. Makridakis, S., Wheelwright, S., Hyndman, R.. John Wiley and Sons, New York, 1998. Marinos, L. and Sfakianakis, A., . Threat Landscape 2014. http://www. enisa.europa.eu/; 2016. Ou, C.. Host-based intrusion detection systems adapted from agent-based artificial immune systems. Neurocomputing 2012;88(1):78–86. Ozcelik, I., Brooks, R.. Deceiving entropy based DoS detection. Computers & Security 2015;48(1):234–245. Rui, X., Wunsch, D.. Survey of clustering algorithms. IEEE Transactions on Neural Networks 2005;16(3):645–678. Seresht, N., Azmi, R.. MAIS-IDS: A distributed intrusion detection system using multi-agent AIS approach. Engineering Applications of Artificial Intelligence 2014;25:286–298. Shannon, C.. A mathematical theory of communication. Bell Systems Technical Journal 1948;27(3):379–423. Sheshtawi, K., Abdul-Kader, H., Ismail, N.. Artificial Immune Clonal Selection Classification Algorithms for Classifying Malware and Benign Processes Using API Call Sequences. International Journal of Computer Science and Network Security 2010;10(4):31–39. Shin, S., Lee, S., Kim, H., Kim, S.. Advanced probabilistic approach for network intrusion forecasting and detection. Expert Systems with Appli- cations 2013;40(1):315–322. 39     Tang, Y., Luo, X., Hui, Q., Chang, R.. Modeling the Vulnerability of Feedback-Control Based Internet Services to Low-Rate DoS Attacks. IEEE Transactions on Information Forensics and Security 2014;9(3):339–353. Venkatesan, S., Baskaran, R., Chellappan, C., Vaish, A., Dhavachelvan, P.. Artificial immune system based mobile agent platform protection. Com- puter Standards & Interfaces 2013;34(4):365–373. Visconti, A., Tahayori, H.. Artificial immune system based on interval type- 2 fuzzy set paradigm. Applied Soft Computing 2011;11(6):4055–4063. Viswanathan, A., Tan, K., Neuman, C.. Deconstructing the Assessment of Anomaly-based Intrusion Detectors. In: Proceedings of the 16th In- ternational Symposium on Research in Attacks, Intrusions and Defenses (RAID). Rodney Bay, St. Lucia; 2013. p. 286–306. Wei, W., Chen, F., Xia, Y., Jin, G.. A rank correlation based detection against distributed reflection DoS attacks. IEEE Communications Letteres 2013;17(1):173–175. Yang, H., Guo, J., Deng, F.. Collaborative RFID intrusion detection with an artificial immune system. Journal of Intelligent Information Systems 2011;36(1):1–26. Yao, G., Bi, J., Vasilakos, A.. Passive IP Traceback: Disclosing the Locations of IP Spoofers From Path Backscatter. IEEE Transactions on Information Forensics and Security 2015;10(3):471–484. Zargar, S., Joshi, J., Tipper, D.. A Survey of Defense Mechanisms Against Distributed Denial of Service (DDoS) Flooding Attacks. IEEE Communi- cations Surveys & Tutorials 2013;15(4):2046–2069. Zhaowen, L., Shan, L., Yan, M.. A Negative Selection Approach to Intrusion Detection. In: Proceedings of the 11th International Conference on Artificial Immune Systems (ICARIS). Taormina, Italy; 2012. p. 178– 190. Zhou, W., Jia, W., Wen, S., Xiang, Y., Zhou, W.. Detection and defense of application-layer DDoS attacks in backbone web traffic. Future Generation Computer Systems 2014;38:36–46. 40     Zhu, B., Yan, J., Bao, G., Yang, M., Xu, N.. Captcha as Graphical Passwords-A New Security Primitive Based on Hard AI Problems. IEEE Transactions on Information Forensics and Security 2014;9(6):891–904. "    },    {        "title": "2402.07716v1._Λ___rm_s__CDM_cosmology_from_a_type_II_minimally_modified_gravity.pdf",        "content": "ΛsCDM cosmology from a type-II minimally modified gravity\n¨Ozg¨ ur Akarsu,1,∗Antonio De Felice,2,†Eleonora Di Valentino,3,‡Suresh Kumar,4,§\nRafael C. Nunes,5, 6,¶Emre ¨Oz¨ ulker,1, 3,∗∗J. Alberto Vazquez,7,††and Anita Yadav4,‡‡\n1Department of Physics, Istanbul Technical University, Maslak 34469 Istanbul, Turkey\n2Center for Gravitational Physics and Quantum Information,\nYukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan\n3School of Mathematics and Statistics, University of Sheffield,\nHounsfield Road, Sheffield S3 7RH, United Kingdom\n4Department of Mathematics, Indira Gandhi University, Meerpur, Haryana 122502, India\n5Instituto de F´ ısica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre RS, Brazil\n6Divis˜ ao de Astrof´ ısica, Instituto Nacional de Pesquisas Espaciais,\nAvenida dos Astronautas 1758, S˜ ao Jos´ e dos Campos, 12227-010, SP, Brazil\n7Instituto de Ciencias F´ ısicas, Universidad Nacional Aut´ onoma de M´ exico, Cuernavaca, Morelos, 62210, M´ exico\nWe have successfully integrated Λ sCDM, a promising model for alleviating cosmological tensions,\ninto a theoretical framework by endowing it with a specific Lagrangian from the VCDM model, a\ntype-II minimally modified gravity. In this theory, we demonstrate that an auxiliary scalar field with\na linear potential induces an effective cosmological constant, enabling the realization of an abrupt\nmirror AdS-dS transition in the late universe through a piecewise linear potential. To eliminate\nthe sudden singularity in this setup and ensure stable transitions, we smooth out this potential.\nRealized within the VCDM theory, the Λ sCDM model facilitates two types of rapid smooth mirror\nAdS-dS transitions: (i) the agitated transition, associated with a smooth jump in the potential, where\nΛs, and consequently H, exhibits a bump around the transition’s midpoint; and (ii) the quiescent\ntransition, associated with a smooth change in the slope of the potential, where Λ stransitions\ngracefully. These transitions are likely to leave distinct imprints on the background and perturbation\ndynamics, potentially allowing the observational data to distinguish between them. This novel\ntheoretical framework propels Λ sCDM into a fully predictive model capable of exploring the evolution\nof the Universe including the late-time AdS-dS transition epoch, and extends the applicability of the\nmodel. We believe further research is crucial in establishing Λ sCDM as a leading candidate or guide\nfor a new concordance cosmological model.\nIntroduction – The Hubble constant ( H0) tension is the\nforemost challenge in contemporary precision cosmology.\nAlthough the cosmological constant (CC) problem [ 1,2]\nhas a longer history, the H0tension seems more relevant\nto low-energy physics, a domain previously thought to\nbe well understood. This may be signaling possible new\nphysics beyond the standard cosmological model, i.e.,\nΛCDM, if not stemming from unidentified systematics.\nThe persistence of the tension across various probes and\nover time diminishes the possibility of systematics in the\ndata, yet, despite the plethora of attempts, there is no\nresolution through new physics that is both observation-\nally and theoretically fully satisfactory or at least widely\naccepted. Moreover, addressing the H0tension while\nensuring compatibility with all available data, without\nexacerbating other less definitive discrepancies such as\ntheS8tension, remains a challenging task. See [ 3–7] for\nrecent reviews.\nThe Λ sCDM model [ 8–11] emerges as one of the promis-\ning models for addressing major cosmological tensions,\nviz.,H0,MB(Type Ia Supernovae absolute magnitude),\nandS8(growth parameter) tensions, along with some\nother less significant tensions, and stands as the most\neconomical model with this capability. It was inspired\nby a recent conjecture based on the findings on gradu-\nated dark energy (gDE) [ 8], proposing that the Universehas, around redshift z†∼2, undergone a rapid mirror\nanti-de Sitter (AdS) vacuum to a de Sitter (dS) vacuum\ntransition (a mirror AdS-dS transition, corresponding\nto a sign-switching CC while maintaining the magnitude\nbefore and after the transition). The suggested rapid\nnature of the sign-switching CC, along with its shift from\nnegative to positive values, presents challenges in identi-\nfying a concrete physical mechanism; with that said, the\nphenomenological success of Λ sCDM despite its simplic-\nity, strongly encourages the search for possible underlying\nphysical mechanisms. Recently, it was shown in [ 12] that\nalthough the AdS swampland conjecture suggests that\nAdS-dS transition in the late universe seems unlikely (due\nto the arbitrarily large distance between AdS and dS vacua\nin moduli space), it can be realized through the Casimir\nforces of fields inhabiting the bulk.1Furthermore, it was\ndemonstrated in [ 13] that in various formulations of GR,\nit is possible to obtain a sign-switching CC through an\noverall sign change of the metric. In this Letter , we demon-\nstrate that the Λ sCDM model [ 8–11] can be elevated to a\ntheoretically complete physical cosmology, offering a fully\npredictive description of our universe within a type-II\nminimally modified gravity framework [14–19].\n1It remains to be clarified whether the model in [ 12] is complete, as\nit appears to lack a prescription for determining the perturbations.arXiv:2402.07716v1  [astro-ph.CO]  12 Feb 20242\nRationale – The simplest form of the Λ sCDM model [ 9–\n11] was constructed phenomenologically by replacing the\nCC in ΛCDM with an abrupt sign-switching one at red-\nshift z†:Λ→Λs≡Λs0sgn[z†−z], where the transition\nemploys the signum function (sgn), and Λ s0>0 is the\npresent-day value of Λ s. Accordingly, the Friedmann equa-\ntion2can be written as 3 M2\nPH2=P\nIρI+ρΛs0sgn[a−a†],\nwhere His the Hubble parameter, ρIrepresents the energy\ndensities of each standard matter field, and ρΛs0= Λ s0M2\nP\nis the energy density of the present-day CC, with a†being\nthe scale factor, a, at the time of the transition. Given\nthat this model is remarkably simple, representing a min-\nimal deviation from ΛCDM, and demonstrates a far su-\nperior statistical fit to the available data compared to\nΛCDM [ 9–11], it suggests that the CC might indeed have\nundergone a sign switch sometime after recombination.\nThis implies the profound possibility that the Universe\nwas in an AdS vacuum phase until recently ( z∼2), a\ntheoretical sweet spot welcomed due to the AdS/CFT\ncorrespondence [ 20] and preferred by string theory and\nstring theory motivated supergravities [ 21]. However, to\nadvance Λ sCDM beyond a phenomenological model and\nestablish it as a predictive one, it is necessary to integrate\nit with a concrete theoretical mechanism/model that ex-\nplains the sign-switch in the CC, as well as to smooth\nout the abrupt behavior implemented by the signum func-\ntion since the discontinuity in this case leads to a type\nII (sudden) singularity [ 22].3One of the primary aims\nof this letter is to eliminate this singularity and provide\nprecise predictions about the impact of the transition\non cosmological observables. However, smoothing out\nthe transition introduces a new challenge: the need for a\nmodel or mechanism that allows such a transition, partic-\nularly one that can facilitate rapid growth of H, i.e., large\nvalues of |˙H|/(NH2) (equivalent to R, the Ricci scalar),\nwhere N, the lapse function, determines the nature of\nthe time variable t. In any case, for a smooth but rapid\nAdS-dS transition, the dynamics will induce a kick in\n˙H/(NH2), leading to a phase of super-acceleration of the\nUniverse during the transition, which could significantly\ninfluence the observables of the theory, and quantifying\nthis effect unequivocally will be an integral part of the\nfeatures of the model. Furthermore, it is acknowledged\nthat super-accelerating behavior can readily introduce\nghost-like degrees of freedom and/or instabilities in per-\nturbation dynamics. For this reason, incorporating such\na dynamic and smooth component into a minimal theory\nof gravity becomes a natural course of action, specifically\n2In this study, we focus exclusively on spatially flat and uniform\ncosmology; it can be readily extended to non-flat scenarios.\n3See [23] for a demonstration of the presence of a type II singularity\nin abrupt Λ sCDM and its minimal impact on the formation and\nevolution of cosmic bound structures, thereby preserving the\nmodel’s viability in this context.in the theory known as VCDM, which has only tensor\nmodes in the gravity sector and a predictive Lagrangian,\nand a smooth limit for an evolution that approaches (even\nexponentially fast) ΛCDM dynamics. For further explo-\nration of these aspects and other topics related to VCDM,\nsee Refs. [14–19].\nAbrupt ΛsCDM – As a warm-up, we first demonstrate\nthe realization of the abrupt Λ sCDM model [ 9–11], which\nis the simplest, idealized case achieved by replacing the\nusual CC in the ΛCDM model with\nΛ→ Λs≡Λs0sgn[a−a†], (1)\nwhere Λ s0>0 is the present-day ( a= 1) value of the CC.\nTo realize this within the VCDM framework [ 14] with\na potential, V(ϕ), we start by considering a continuous\npiece-wise linear potential with two segments of different\nslopes4around a critical point ϕ=ϕc:\nV(ϕ) =(\nαb(ϕ−ϕc)−β ,forϕ < ϕ c,\nαa(ϕ−ϕc)−β ,forϕ≥ϕc,(2)\nwhere the indices “b” and “a” denote the states before\nand after the critical point/transition, respectively. Using\nthis in the field equations (see Appendix A), we find a\ncorresponding effective CC (Λ sft=ρϕ/M2\nP) that shifts\n(sft) from an initial value to a final value as follows:\nΛsft=(\nΛsft,b=3\n4α2\nb−αbϕc−β ,forϕ < ϕ c,\nΛsft,a=3\n4α2\na−αaϕc−β , forϕ≥ϕc.(3)\nConsequently, a sudden change in the slope of the poten-\ntial induces an abrupt shift in the effective CC, ∆Λ sft≡\nΛsft,a−Λsft,b. And, the abrupt Λ sCDM, defined by\nΛs= Λ s0sgn[ϕc−ϕ], (4)\ncan be achieved by imposing Λ sft,a=−Λsft,b= Λ s0which\nrequires β= Λ s0+3\n4α2\nb−αbϕcand the final slope to be\nαa=2\n3ϕc+s\u00122\n3ϕc−αb\u00132\n+8\n3Λs0. (5)\nHere, the plus sign in front of the second fraction has\nbeen chosen such that ϕ(z= 0) > ϕc, which requires αa>\n2\n3ϕc+ 2H0, as derived from Eq. (15). An example of V(ϕ)\nand its corresponding Λ s(ϕ) for this kind of transition is\nillustrated in Fig. 1. Note that, there is a redundancy of\nparameters in this prescription; for a given ϕc, the exact\nsame Λ s(ϕ) can be achieved by infinitely many pairs of,\nsay,{αb, β}.\n4Refer to Appendix A for the rationale behind using linear poten-\ntial segments and the discussion on the possibility of a sudden\ntransition where the potential Vis discontinuous at ϕ=ϕc, not\njust in its slope. This scenario is also addressed later in the letter.3\nFIG. 1. Abrupt mirror AdS-dS transition. The upper panel\ndisplays the potential V(ϕ), and the lower panel the corre-\nsponding effective cosmological constant for the sudden step\ntransition. We have considered here αb=−1,ϕc=−5, and\nΛs0= 1, leading to αa≈ −0.48 and β≈ −3.25. The potential\nis not differentiable at transition. This model cannot be ac-\ncommodated within the framework of VCDM.\nWe have thus demonstrated that an abrupt Λ sCDM\ncan readily be realized within the VCDM framework.\nHowever, this abrupt version, despite describing the back-\nground dynamics for most of the redshift ranges, cannot\nbe fully integrated into VCDM. This is because a dis-\ncontinuity in the derivative of the potential at ϕcwould\ncause a discontinuity in H, leading to a singularity in\n˙H(a crucial term in perturbation theory, as shown in\nAppendix B). To address this issue, it becomes necessary\nto smooth out the transition. Consequently, we will now\ninvestigate a completely smooth ( C∞) model.\nSmooth ΛsCDM – In this section, we extend the above\nfindings by incorporating sigmoid functions, enabling\na smooth transition between two linear regimes. The\nsigmoid function, distinguished by its characteristic ‘S’-\nshaped curve, offers a gradual and continuous shift from\none value to another. Here, we utilize a typical sigmoid\nfunction called the logistic function; it is expressed as:\nS(ϕ) =1\n1 +e−2η(ϕ−ϕc), (6)\nwhere 2 η >0 determines the steepness of the curve dur-\ning the transition epoch, thus enabling control over the\nrapidity of the mirror AdS-dS transition. The parameterϕcnow represents the midpoint of this transition. For\ndetails on the rapidity/width of the transition epoch, see\nAppendix A.\nTo facilitate a quiescent smooth transition (see Ap-\npendix A where this term is made precise) from the pre-\nto the post-transition linear regimes, we employ the sig-\nmoid function in the following blended function:\nV= [αb(ϕ−ϕc)−β][1−S(ϕ)]+[αa(ϕ−ϕc)−β]S(ϕ),(7)\na formulation that is valid across all values of ϕ. This equa-\ntion ensures a smooth transition, with S(ϕ) progressing\nfrom 0 to 1 as ϕincreases across the critical value ϕc. Note\nthatS(ϕ), in this case, is specifically designed to facilitate\na smooth transition from the pre- to the post-transition\nlinear regimes, i.e., from the Λ s≈ −ΛdSphase to the\nΛs≈ΛdSphase, in accordance with the abrupt sign-switch\nsolution. Here we introduced a new term, Λ dS>0, that\nbounds5Λs(ϕ) =ρϕ/M2\nP, i.e., limϕ→±∞ ρϕ/M2\nP=±ΛdS.\nThe parameters must therefore satisfy the following rela-\ntions:\nβ=3α2\nbαa+\u0000\n4ΛdS−3α2\na\u0001\nαb+ 4Λ dSαa\n4αa−4αb,(8)\nϕc= (8Λ dS−3α2\na+ 3α2\nb)/(4αb−4αa), (9)\nwhere it is assumed that αb̸=αa, as required (see Ap-\npendix A). Finally, to determine the dynamics of our\nconstructed physical system, it is necessary to specify an\ninitial condition for ϕ. Specifically, to find H(z) and ϕ(z),\nwe must integrate the following differential equation:\ndϕ\ndz=−3\n2ρ+P\nM2\nPH(1 +z), (10)\nwhere H=1\n2V,ϕ−ϕ\n3, and an initial condition for ϕat\na certain reference redshift is required (see Appendix\nA). Given that the background evolution of each matter\ncomponent, ρI(and PI), is known as a function of z,\nthe expression for ρ+Pconsequently becomes a known\nfunction of z.\nWe have thus demonstrated that a smooth Λ sCDM\ncan also be realized within the VCDM framework using a\nsigmoid-based transition of the potential function, which\nensures the continuity of the function and its derivatives,\nmaking it ideally suited for, e.g., cosmological applications.\nFor example, a solution for V(ϕ) and its corresponding\nΛs(ϕ) is illustrated in Fig. 2. We now consider another\ntype of smooth Λ sCDM scenario permitted within the\nVCDM framework: the agitated smooth transition (see\nAppendix A where this term is made precise). Note that\nwithin VCDM, introducing a linear potential that sud-\ndenly changes slope is not the only method for achieving\n5For the abrupt sign switch, Λ dS= Λs0.4\nFIG. 2. Quiescent mirror AdS-dS transition. The upper\npanel displays the potential V(ϕ), and the lower panel the\ncorresponding effective cosmological constant for the sigmoid\ntransition. Since ϕ(z= 0) =3\n2αa−3H0, on considering\nαa<0 we find ϕ(z= 0) <0. Therefore, since ϕis a growing\nfunction of a, we also need ϕc< ϕ(z= 0) <0. We have then\nre-considered here the non-differentiable case of Fig. 1, and\nsmoothed it out with a sigmoid function. Hence, we have\nchosen αb=−1,αa≈ −0.48,β=−3.25,ϕc=−5 (the value\nat transition), η= 5, and Λ s0= 1.\nan abrupt shift in the effective CC. Another approach\ninvolves introducing a sudden shift from one constant\npotential value to another. Specifically, for an abrupt\nΛsCDM, one could consider a potential abruptly jump-\ning from Vb(ϕ) = const <0 to Va(ϕ) =−Vb(ϕ) at a\ncritical point. To create a smooth version of this sce-\nnario, we can again utilize S(ϕ); this can be achieved\nsimply by setting αb= 0 = αain Eq. (21), leading to\nV=−ΛdS[1−S(ϕ)] + Λ dSS(ϕ), i.e.,\nV(ϕ) =−ΛdStanh[ η(ϕ−ϕc)]. (11)\nBecause today ( z= 0), as the transition has already\noccurred, V(ϕ)≈ΛdS(or Λ s≈ΛdS), it follows that ϕ(z=\n0)≈ −3H0. Accordingly, we need to set ϕc< ϕ(z= 0) <\n0, which in turn implies, as can be shown, thatρϕ(ϕc)\nM2\nP>\n0. As discussed in Appendix A, this type of transition\ngenerally leads to large values of |V,ϕ|around ϕ=ϕc, and\ncorrespondingly, to large values of H2and|˙H|/(NH2).\nSpecifically, it is straightforward to show that while the\neffective CC eventually transitions from Λ s≈ −ΛdSto\nΛs≈ΛdS, around the critical point ( ϕ=ϕc), when the\nFIG. 3. Agitated mirror AdS-dS transition. The upper panel\ndisplays the potential V(ϕ), and the lower panel the corre-\nsponding effective cosmological constant for the tanh-sigmoid\ntransition. We have chosen here ϕc=−5,η= 5, and\n−Vint= Λ s0= 1. Since ϕis a growing function of a, we\nneed to consider ϕc< ϕ(z= 0) <0.\npotential is at its steepest, the effective CC will reach\nthe values of Λ s(ϕ∼ϕc)∼ηΛdSϕc+3\n4η2Λ2\ndS. Therefore,\nunless the parameters are extremely fine-tuned, the faster\nthe transition (i.e., the larger the η), the larger the bump\nexhibited by Λ s. An example of the typical shape of\nthe Λ sin this scenario can be observed in Fig. 3, which\ndeviates considerably from that in Fig. 2, potentially\nleading to different phenomenology distinguishable by\navailable observations. In particular, there is a noticeable\nlarge bump in Λ saround ϕ=ϕc, which generally expected\nto result in a corresponding large bump in H2around the\nmidway of the transition epoch. Therefore, while the two\ntypes of smooth Λ sCDM models we discussed share the\nsame asymptotic effective CC, they differ at the level of\nH2behavior during the transition epoch.\nSigmoid Λs(a)and reconstruction of the scalar field\npotential – We now explore an alternative approach to\nconstructing a smooth Λ sCDM. Instead of deriving Λ s(ϕ)\nfrom a potential, we directly implement a smooth one as\na function of scale a(or redshift z), corresponding to the\nintroduction of the Hubble function, given that the behav-\nior of the matter sector is established. With the matter\nsector fixed, the dynamics of ρϕ(a) are also automatically\ndefined. Knowing H(a), which yields the desired Λ s(a),\nenables us to immediately determine the background and5\nFIG. 4. Reconstructed quiescent mirror AdS-dS transition.\nThe upper panel shows the reconstructed field in redshift\nwith the corresponding potential evolution V(ϕ) for a given\nsigmoid Λ s(a) = Λ dStanh [ η(a−a†)]. Parameters chosen are\nϕ(a= 1) = −3H0,η= 10, Ω m0= 0.3, Ω r0= 9×10−5, and\na†= 1/(1 + 1 .8) (corresponding to z†= 1.8).\ncoefficients of the perturbation equations (26) without\nthe need for integration. For instance, by utilizing a\nwell-known smooth approximation of the signum function\n(sgnx≈tanhkx, for constant k >1), from Eq. (1), we\ndefine (with scaling):\nΛs(a) = Λ dStanh [ η(a−a†)], (12)\nwhere η >1 determines the rapidity of the transition and\nΛdS= Λ s0/tanh[ η(1−a†)]. For a fast transition (e.g., for\nη≳10) at around z†∼1.8, we can safely take Λ dS≈Λs0.\nA realistic example of this approach, specifically for ϕ(z)\nandV(ϕ), is presented in Fig. 4.\nIn this approach, we can straightforwardly further mod-\nify Λ s(a), as given in Eq. (12), to incorporate additional\nfeatures into the Λ sCDM model at various redshifts, be-\nyond the smooth mirror AdS-dS transition around z†∼2.\nHowever, we will leave the exploration of this possibility\nto future works. Having uniquely defined the Λ s(a) func-\ntion, we can now determine H2=P\nIϱI+ϱs(a) (where\nthe index Iruns over the standard matter components,\nandϱs≡Λs/3 represents the effective CC contribution),\nand consequently also the expressions for ˙H[see Eq. (27)].\nHence, with Λ s(a) specified, we have a complete model\nready for direct implementation into a Boltzmann code\n(see Appendix B).Conclusions – We have successfully integrated the\nΛsCDM model [ 8–11] into a novel theoretical framework\nby endowing it with a specific Lagrangian, that of a\ntype-II minimally modified gravity dubbed the VCDM\nmodel [ 14,15]. Consequently, Λ sCDM has now acquired\nthe status of a fully predictive model, applicable at any\ntime and in any configuration, even beyond the cosmolog-\nical context. In particular, we now have a fully predictive\nmodel for the entire evolution of the universe, including\nduring the AdS-dS transition epoch. This enhancement\nalso removes any ambiguity in applying the Λ sCDM model\nwhen confronting it with observational data.\nIn the VCDM framework, we have demonstrated that,\nmost generally, an auxiliary scalar field endowed with a\nlinear potential induces an effective cosmological constant,\nΛeff=const , in the Friedmann equation. This finding\nhas enabled us to realize an abrupt mirror AdS-dS transi-\ntion, as proposed in the abrupt Λ sCDM scenario [ 9–11],\nby defining a piece-wise linear potential with two seg-\nments. To eliminate the type II (sudden) singularity [ 22],\nwhich arises in the moment of an abrupt transition [ 23],\nand ensure stable transitions, we have smoothed out this\npiece-wise potential by implementing it into a blended\nfunction incorporating a sigmoid function. Our work re-\nveals that the Λ sCDM model, realized within the VCDM\ntheory, facilitates two types of rapid and smooth mirror\nAdS-dS transitions: (i)Agitated mirror AdS-dS transition ,\nwherein Λ s(and potentially H2as a consequence) exhibits\na bump around the midpoint of the transition from AdS\nto dS phase; and (ii)Quiescent mirror AdS-dS transition ,\nwherein Λ stransitions gracefully (monotonically) from\nAdS to dS phase. Although these two types of transitions\nshare the same cosmological constant values in the pre-\nand post-transition eras, and through this transition, the\nexpansion rate of the universe increases significantly from\nthe pre- to post-transition eras, they differ in their H2\nbehavior during the transition epoch. These distinct AdS-\ndS transitions are likely to leave unique imprints in both\nthe background and perturbation dynamics, potentially\nproviding observational insights into the transition epoch\nof the smooth Λ sCDM models. Consequently, observa-\ntional data may be capable of distinguishing between\nthese two types of transitions. Lastly, we have recon-\nstructed the potential for a given Λ s(a) described by a\nwell-known smooth approximation of the signum function\n(utilizing hyperbolic tangent), resulting in a quiescent\nmirror AdS-dS transition.\nIt is worth noting that integrating the Λ sCDM model\ninto the VCDM framework could have specific conse-\nquences on structures at cosmological scales. The pres-\nence of a smooth, fast transition typically results in a\ntemporary rise in the background value of |˙H|/(NH2)\nduring the transition epoch. And, the form of Hubble\nfunction, H, deviates non-negligibly from the baseline\nΛCDM model only during the transition epoch. The po-\ntential imprints of these deviations can be checked and6\nverified against observations. The Hubble parameter of\nthis model that aligns almost exactly with ΛCDM in\nthe pre- and post-transition eras, as demonstrated for\nthe abrupt Λ sCDM model in Refs. [ 9–11], allows for a\nbackground phenomenology that is in good agreement\nwith both high- and low-redshift cosmological data and is\ncapable of addressing major cosmological tensions, most\nnotably the H0tension, with remarkable success. It has\nbeen shown that these background dynamics can also\nalleviate the S8tension, related to the evolution of per-\nturbations, by modifying the H(z) term in the perturba-\ntion equations [ 10]. In the VCDM setup of the model,\nthe perturbation equations are defined at all times, in\ncontrast to previous Λ sCDM studies where the AdS-dS\ntransition epoch was not modeled, leaving its impact on\nobservables unpredicted. These equations are necessar-\nily different from those in GR, particularly during the\ntransition epoch, as seen in Eq. (26). In our new theoret-\nical framework, where Λ sCDM is augmented by VCDM,\ndifferent types of transitions may impact the predicted\nS8value differently. Therefore, a thorough study of the\nS8parameter, conducted by comparing all available data\nfrom related probes, will enable us to rigorously test the\nmodel and identify a family of potentials/transitions that\ncan simultaneously address the H0andS8tensions.\nThus, the integration of the Λ sCDM model into the\nVCDM framework elevates it to a fully predictive model\ncapable of exploring the evolution of the Universe,\nincluding the late-time AdS-dS transition epoch. This ad-\nvancement not only broadens its applicability but also has\nthe potential to deepen our understanding of cosmological\nphenomena, by giving a complete description of the\nbackground and perturbation evolution at all redshifts,\npaving the way to resolve the observational tensions\nplaguing the ΛCDM model. We believe further research is\nwarranted to establish the Λ sCDM model as a leading can-\ndidate or guide for a new concordance cosmological model.\n¨O.A. acknowledges the support of the Turkish Academy\nof Sciences in the scheme of the Outstanding Young\nScientist Award (T ¨UBA-GEB ˙IP). ¨O.A. is supported in\npart by TUBITAK grant 122F124. The work of A.D.F.\nwas supported by the Japan Society for the Promo-\ntion of Science Grants-in-Aid for Scientific Research\nNo. 20K03969 and by grant PID2020-118159GB-C41\nfunded by MCIN/AEI/10.13039/501100011033. E.D.V\nacknowledges support from the Royal Society through\na Royal Society Dorothy Hodgkin Research Fellowship.\nS.K. gratefully acknowledges support from the Science\nand Engineering Research Board (SERB), Govt. of India\n(File No. CRG/2021/004658). R.C.N. thanks the finan-\ncial support from the Conselho Nacional de Desenvolvi-\nmento Cient´ ıfico e Tecnologico (CNPq, National Coun-\ncil for Scientific and Technological Development) under\nproject No. 304306/2022-3, and the Funda¸ c˜ ao de Amparo\n` a pesquisa do Estado do RS (FAPERGS, Research Sup-port Foundation of the State of RS) for partial financial\nsupport under project No. 23/2551-0000848-3. E. ¨O. ac-\nknowledges the support by The Scientific and Techno-\nlogical Research Council of Turkey (T ¨UB˙ITAK) in the\nscheme of 2211/A National PhD Scholarship Program.\nJ.A.V. acknowledges the support provided by FOSEC\nSEP-CONACYT Investigaci´ on B´ asica A1-S-21925,\nUNAM-DGAPA-PAPIIT IN117723 and FORDECYT-\nPRONACES-CONACYT/304001/2019. A.Y. is sup-\nported by a Junior Research Fellowship (CSIR/UGC Ref.\nNo. 201610145543) from the University Grants Commis-\nsion, Govt. of India. This article/publication is based\nupon work from COST Action CA21136 – “Addressing\nobservational tensions in cosmology with systematics and\nfundamental physics (CosmoVerse)”, supported by COST\n(European Cooperation in Science and Technology).\nAppendix A: type-II minimally modified gravity\nIn the type-II minimally modified gravity introduced\nin [14] (later dubbed VCDM [ 15]), the standard cosmo-\nlogical constant, Λ, is promoted to a potential V(ϕ) of\na non-dynamical auxiliary field, ϕ, without introducing\nextra physical degrees of freedom. The action of this\ntheory is given by\nS=Sm+M2\nPZ\nd4x N√γ\u00141\n2\u0000\nR+KijKij−K2\u0001\n−V(ϕ) +λ2\nNγijDiDjϕ−3λ2\n4−λ(K+ϕ)\u0015\n,(13)\nwhere Smis the sum of standard matter Lagrangians,\nNis the lapse, and Kijis the extrinsic curvature (with\nK=γijKijas its trace) relative to the 3D metric γij\n(endowed with inverse γij, determinant γ, and a co-\nvariant derivative Di). Instead λ,λ2, and ϕare aux-\niliary fields. This theory of modified gravity breaks\nfour-dimensional diffeomorphism invariance, but retains\nthree-dimensional spatial diffeomorphism invariance and\ntime-reparametrization invariance. Consequently, on a\nhomogeneous and isotropic background, it can modify\nthe Hubble expansion rate while maintaining only two\ngravitational degrees of freedom, as in GR. In general,\nthis allows for a spectrum of possibilities typically much\nbroader than in scalar tensor theories. For the latter, gen-\nerally, the extra scalar degree of freedom imposes strong\nconstraints, both locally (viz., at solar system scales) and\nat cosmological scales. To evade the local constraints,\nthe scalar must be either very massive or shielded by\nnon-trivial dynamical mechanisms, such as chameleon or\nVainshtein effects. Additionally, constraining the cosmo-\nlogical background dynamics is necessary to avoid ghost\nand gradient instabilities.\nThe equations of motion for VCDM on a homogeneous7\nand isotropic background can be expressed as [14]:\nV=1\n3ϕ2−ρ\nM2\nP,dϕ\ndN=3\n2ρ+P\nM2\nPH,dρI\ndN+3(ρI+PI) = 0 ,\n(14)\nwhere N=ln(a/a0), with a0= 1 being the present-day\nvalue of the scale factor, and H= ˙a/(aN) is the Hubble\nexpansion rate, with a dot denoting differentiation w.r.t.\nthe time variable t, which is specified only after fixing\nN. For instance, when fixing N=a, the variable tcorre-\nsponds to the conformal time. It is important to note that\nϕis a growing function of a, meaning it increases as the\nuniverse expands ( H > 0). To derive the above equation,\nwe have assumed a cosmological expansion history for\nwhich H̸= 0, a condition expected to hold at least after\ninflation.6Additionally, we have defined ρ=P\nIρIand\nP=P\nIPI, where the sum includes all standard mat-\nter species, including the dark matter component, each\nsatisfying the usual continuity equation.\nProvided that ρ+P̸= 0, the following equation can be\nderived from the set of equations presented in Eq. (14):7\nϕ=3\n2V,ϕ−3H, (15)\nwhere V,ϕ≡dV\ndϕ. This equation can be used to specify\nthe initial condition for ϕtoday: in our model, where\nthe AdS-dS transition occurs in our past, V(z= 0) can\nbe approximated by a linear function of ϕ, namely V≈\nαϕ+β, thereby implying the initial condition ϕ(z= 0)≈\n3\n2α−3H0, for a chosen value of α. By combining the\nequations of motion, we express the Friedmann equation\nas follows;\n3M2\nPH2=ρ+ρϕ, (16)\nwhere\nρϕ≡M2\nP(V−ϕV,ϕ) +3\n4M2\nPV2\n,ϕ. (17)\nTaking derivatives of the expressions for ϕandHreveals\nthat\n˙H\nN=(ρ+P) (3V,ϕϕ−2)\n4M2\nP, (18)\nwhich deviates from the corresponding equation in GR if\nand only if V,ϕϕ̸= 0. Notably, if Vis a linear function of\nϕ, the standard ΛCDM dynamics are obtained. Examin-\ning Eq. (18), we find that it can be expressed in terms of\nan effective pressure defined by\nPϕ=−3\n2(ρ+P)V,ϕϕ−ρϕ, (19)\n6Choosing a different time variable is sufficient to describe certain\nphenomena, such as a bouncing solution [24].\n7The theory is also endowed with a shadowy mode, another\nauxiliary field, whose profile on this background is given by\nλ=−2\n3ϕ−2H.which makes it evident that Pϕ→ −ρϕ(mimicking a CC,\nρϕ→ρΛ= const) as V,ϕϕ→0 (approaching a linear\nfunction). Indeed, for scalar field to induce an effective\ncosmological constant, Λ eff= const, in the Friedmann\nequation, solving Eq. (17) reveals that:8\nρϕ/M2\nP= Λ eff⇒V(ϕ) =αϕ−3\n4α2+ Λeff.(20)\nThis result implies that an abrupt shift in Λ effat a critical\nvalue of ϕ=ϕcfrom one value to another is not necessarily\ncaused by a potential that has undergone an abrupt shift.\nFor instance, it is always possible to define a continuous\npiece-wise linear potential with two pieces, and as the\nscalar field rolls through it, it causes a shift in Λ effdue\nto the sudden change in the slope at ϕ=ϕc.\nIt is precisely these features of the VCDM [ 14,15]\nthat enable us to realize the Λ sCDM model [ 8–11] within\nthis framework, offering two main types of scenarios. In\nparticular, an abrupt mirror AdS-dS transition, (namely,\nan abrupt shift in Λ efffrom Λ AdS<0 to Λ dS=−ΛAdS>\n0) at ϕ=ϕc, is not necessarily caused by a potential that\nhas undergone an abrupt shift from Vb(ϕ) =−ΛdS<0\ntoVa(ϕ) = Λ AdS>0. Instead, in the general scenario,\nit can be caused by a potential transitioning between\ntwo separate linear regimes, namely, from Vb(ϕ) =αbϕ−\n3\n4α2\nb−ΛdSforϕ < ϕ ctoVa(ϕ) =αaϕ−3\n4α2\na+ Λ dSfor\nϕ≥ϕc, where ϕc< ϕ(z= 0). That is, an abrupt sign-\nswitch can occur due to either (i)an abrupt jump in\nthe value of the potential, (ii)a sudden change in the\nslope of the potential, or (iii) a combination of the first\ntwo. In neither of these cases is the potential smooth;\nifVa(ϕ=ϕc) =Vb(ϕ=ϕc), it is continuous, but not\nin its derivatives. If Va(ϕ=ϕc)̸=Vb(ϕ=ϕc), even\nthe potential itself is discontinuous. Note, on the other\nhand, that in any case, the Hubble function, H, would\nbe discontinuous at ϕ=ϕc, leading to a type II (sudden)\nsingularity in the past [22, 23].\nTo circumvent this issue, one can straightforwardly\ndevise a smooth potential by implementing a blended\nfunction [see, e.g., Eq. (7)] that ensures a smooth transi-\ntion between two linear regimes by incorporating a sig-\nmoid function, specifically, a potential that approximates\nV(ϕ)≈Vb(ϕ) for ϕ < ϕ c−∆ϕ/2 and transitions to\nV(ϕ)≈Va(ϕ) for ϕ > ϕ c+ ∆ϕ/2. Here, ∆ ϕdenotes the\nbreadth of the transition epoch centered around ϕc—in\nthe case of Eq. (7) or similar potentials for which ρϕ\nevolves at all times, the transition epoch can be defined\nas the period between the first violation of the condition\n|ΛdS−Λs(a)|/ΛdS< εuntil the end of the last violation\nfor a chosen small value of ε. The post- and pre-transition\n8The solution V(ϕ) =ϕ2\n3+β,βbeing a constant, should be\nexcluded as Eq. (15) results in H= 0, leading to a non-dynamical\nevolution of the Universe.8\neras correspond to when the two branches of the sigmoid\nfunction resemble almost straight lines. In these eras, the\neffective cosmological constant, Λ eff, is indistinguishable\nfrom a constant phenomenologically. Currently, as we\nare in the post-transition era, it is reasonable to assume\nthat Λ eff(z= 0) = Λ s0≈ΛdS, akin to the abrupt Λ sCDM\nscenario [ 8–11]. For example, in the logistic function\nprovided in Eq. (6), S(ϕ) approaches 0 or 1 at the 99\npercent level for ϕ=ϕc±2η−1. Using this approxima-\ntion, the width of the transition region can be identified\nas ∆ϕ≈4η−1.\nNow, for instance, incorporating aforementioned linear\npotentials in to the blended function given in Eq. (7), the\nresultant potential can be expressed as:\nV(ϕ) = [αb(ϕ−ϕc)−3\n4α2\nb+αbϕc−ΛdS] [1−S(ϕ)]\n+ [αa(ϕ−ϕc)−3\n4α2\na+αaϕc+ ΛdS]S(ϕ),(21)\nwhere S(ϕ) represents a sigmoid function, e.g., the logistic\nfunction given in Eq. (6). This formulation leads us\nto identify two possible types of smooth mirror AdS-\ndS transition, based on whether the piece-wise linear\npotential with two pieces to be smoothed out is continuous\nor discontinuous, namely, whether the original function\nsatisfies Vb(ϕ=ϕc)̸=Va(ϕ=ϕc) or not:\ni.Agitated mirror AdS-dS transition : In this type, the\npiece-wise linear potential with two pieces to be smoothed\nout is discontinuous since Vb(ϕ=ϕc)̸=Va(ϕ=ϕc). It\nis necessary to smooth out not only the possible sudden\nchange in slope at ϕ=ϕc, but also the jump in the\npotential at this point. Consequently, the smoothing pro-\ncess will reflect the discontinuity of the original potential.\nSpecifically, for a sufficiently fast transition, |V,ϕ|will\ntend to large values, resulting in significant increases in\nboth H2/H2\n0and|˙H|/(NH2) during the transition.\nii.Quiescent mirror AdS-dS transition : In this type, the\npiece-wise linear potential with two pieces to be smoothed\nout is continuous; Vb(ϕ=ϕc) =Va(ϕ=ϕc). Unlike the\nagitated transition, there is no jump in the original poten-\ntial to address; it is solely the sudden change in the slope\nof the potential at ϕ=ϕcto be smoothed out, and the\nchange in the slope need not be large. Consequently, |V,ϕ|\ngenerically remains controlled and does not necessarily\nincrease substantially during the transition. However,\n|V,ϕϕ|might be large, contributing to significant increases\nin|˙H|/(NH2) during the transition. For the quiescent\nmirror AdS-dS transition, the potential (21)requires the\nfollowing additional relations:\nϕc=3α2\nb−3α2\na+ 8Λ s0\n4(αb−αa)< ϕ0=3\n2αa−3H0,(22)\nwhere ϕ0=ϕ(z= 0). Assuming also that H2\n0>Λs0/3>\n0, we identify two possible sets of parameter solutions.\nThe first is\nA1=n\nH0≤αa\n4−αb\n4, αb< αao\n. (23)And, the second set of solutions is given by\nA2=nαa\n4−αb\n4< H 0, αb< αa,\n3\n2H0(αa−αb)−3\n8(αa−αb)2<Λs0\u001b\n.(24)\nIn both cases, αb<0 for αa= 0.\nAppendix B: perturbations analysis\nAt the level of the perturbation equations of motion, it\nhas been shown in [ 14] that all the equations, including\nthose for the matter fields, are, in form, identical to those\nin ΛCDM. For instance, the shear equation reads\nΨ = Φ −9\n2a2\nk2X\nI(ϱI+pI)σI, (25)\nwhere we have adopted the CLASS [25] notation; for each\nmatter field, ϱI=ρI/3M2\nP,pI=PI/3M2\nP, and kdenotes\nthe wave number of the modes. This equation is expressed\nin terms of the Newtonian-gauge invariant fields Φ, Ψ,\nand the shear component of each matter field σI. The\nonly modified equation is:\n˙Φ +aHΨ =3 [k2−3a2(˙H/a)]P\nI(ϱI+pI)θI\nk2[2k2/a2+ 9P\nK(ϱK+pK)],(26)\nwhere a dot here represents differentiation w.r.t. the con-\nformal time (i.e., N=a), and θIis the divergence of\ntheIthfluid velocity (see [26]). The quantity ˙His given\nin Eq. (27) [or, equivalently, in Eq. (18)]. For instance,\nfor a given profile of ϱϕ(a), we have\n˙H=a2\n2ϱs,a−3\n2aX\nI(ϱI+pI), (27)\n¨H=−2aH˙H−3\n2a˙p+ 2a3Hϱs,a+1\n2a4H ϱs,aa.(28)\nHere ˙p=P\nI˙pI, where only the ultra-relativistic species\ncontribute a non-zero value.\nIn Eq. (26), the sums over KandIonly include the\nstandard matter fields, excluding the ϕ-component. As\nthe theory is minimal, there is no extra propagating de-\ngree of freedom, and therefore, no additional dynamical\nequation is required by construction. We observe that\n˙Hdeviates from the GR expression only when V,ϕϕ̸= 0,\nwhich also influences the dynamics of the perturbation\nequations, as evidenced by Eq. (26). During a rapid tran-\nsition epoch in the Hubble parameter, we have shown\nthat at least |˙H| ≫NH2. Consequently, also the pertur-\nbations will be non-negligibly affected by this change in\ndynamics, albeit for a short interval in z. This suggests\nthat the mirror AdS-dS transition will generally impact\nthe perturbation dynamics as well, and the various types9\nof this transition would leave their distinguishing imprints\non the perturbations. Since the theory is minimal, the\nscalar field ϕ, by construction, is an auxiliary field and\ntherefore does not propagate, preventing it from becoming\nunstable. In the small-scale regime, all no-ghost condi-\ntions for both matter fields and gravitational waves are\ntrivially satisfied. Moreover, the propagation speeds of all\nmodes, including gravitational waves, match those in GR.\nIn the subhorizon regime, the growth of perturbations for\nthe dust field follows the same equation of motion as in\nGR (see, e.g., [14]).\n∗akarsuo@itu.edu.tr\n†antonio.defelice@yukawa.kyoto-u.ac.jp\n‡e.divalentino@sheffield.ac.uk\n§suresh.math@igu.ac.in\n¶rafadcnunes@gmail.com\n∗∗ozulker17@itu.edu.tr\n††javazquez@icf.unam.mx\n‡‡anita.math.rs@igu.ac.in\n[1]S. Weinberg, The Cosmological Constant Problem, Rev.\nMod. Phys. 61, 1 (1989).\n[2]P. J. E. Peebles and B. Ratra, The Cosmological Constant\nand Dark Energy, Rev. Mod. Phys. 75, 559 (2003), astro-\nph/0207347.\n[3]E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang,\nA. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, In\nthe realm of the Hubble tension—a review of solutions,\nClass. Quant. Grav. 38, 153001 (2021), 2103.01183.\n[4]L. Perivolaropoulos and F. Skara, Challenges for ΛCDM:\nAn update, New Astron. Rev. 95, 101659 (2022),\n2105.05208.\n[5]E. Abdalla et al. , Cosmology intertwined: A review of the\nparticle physics, astrophysics, and cosmology associated\nwith the cosmological tensions and anomalies, JHEAp 34,\n49 (2022), 2203.06142.\n[6]E. Di Valentino, Challenges of the Standard Cosmological\nModel, Universe 8, 399 (2022).\n[7]O. Akarsu, E. O. Colg´ ain, A. A. Sen, and M. M. Sheikh-\nJabbari, ΛCDM Tensions: Localising Missing Physics\nthrough Consistency Checks, (2024), 2402.04767.\n[8]O. Akarsu, J. D. Barrow, L. A. Escamilla, and J. A.\nVazquez, Graduated dark energy: Observational hints of\na spontaneous sign switch in the cosmological constant,\nPhys. Rev. D 101, 063528 (2020), 1912.08751.\n[9]O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, and J. A. Vazquez,\nRelaxing cosmological tensions with a sign switching cos-\nmological constant, Phys. Rev. D 104, 123512 (2021),\n2108.09239.\n[10]O. Akarsu, S. Kumar, E. ¨Oz¨ ulker, J. A. Vazquez, and\nA. Yadav, Relaxing cosmological tensions with a sign\nswitching cosmological constant: Improved results withPlanck, BAO, and Pantheon data, Phys. Rev. D 108,\n023513 (2023), 2211.05742.\n[11]O. Akarsu, E. Di Valentino, S. Kumar, R. C. Nunes, J. A.\nVazquez, and A. Yadav, Λ sCDM model: A promising\nscenario for alleviation of cosmological tensions, (2023),\n2307.10899.\n[12]L. A. Anchordoqui, I. Antoniadis, and D. Lust, Anti-de\nSitter→de Sitter transition driven by Casimir forces and\nmitigating tensions in cosmological parameters, (2023),\n2312.12352.\n[13]B. Alexandre, S. Gielen, and J. a. Magueijo, Overall\nsignature of the metric and the cosmological constant,\n(2023), 2306.11502.\n[14]A. De Felice, A. Doll, and S. Mukohyama, A theory\nof type-II minimally modified gravity, JCAP 09, 034,\n2004.12549.\n[15]A. De Felice, S. Mukohyama, and M. C. Pookkillath,\nAddressing H0tension by means of VCDM, Phys. Lett. B\n816, 136201 (2021), [Erratum: Phys.Lett.B 818, 136364\n(2021)], 2009.08718.\n[16]A. De Felice and S. Mukohyama, Weakening gravity for\ndark matter in a type-II minimally modified gravity, JCAP\n04, 018, 2011.04188.\n[17]A. De Felice, S. Mukohyama, and M. C. Pookkillath,\nStatic, spherically symmetric objects in type-II mini-\nmally modified gravity, Phys. Rev. D 105, 104013 (2022),\n2110.14496.\n[18]A. De Felice, K.-i. Maeda, S. Mukohyama, and M. C.\nPookkillath, Comparison of two theories of Type-IIa mini-\nmally modified gravity, Phys. Rev. D 106, 024028 (2022),\n2204.08294.\n[19]A. F. Jalali, P. Martens, and S. Mukohyama, Spherical\nscalar collapse in a type-II minimally modified gravity,\n(2023), 2306.10672.\n[20]J. M. Maldacena, The Large N limit of superconformal\nfield theories and supergravity, Adv. Theor. Math. Phys.\n2, 231 (1998), hep-th/9711200.\n[21]R. Bousso and J. Polchinski, Quantization of four form\nfluxes and dynamical neutralization of the cosmological\nconstant, JHEP 06, 006, hep-th/0004134.\n[22]J. D. Barrow, Sudden future singularities, Class. Quant.\nGrav. 21, L79 (2004), gr-qc/0403084.\n[23]E. A. Paraskevas, A. Cam, L. Perivolaropoulos, and\nO. Akarsu, Transition dynamics in the Λ sCDM model:\nImplications for bound cosmic structures, (2024),\n2402.05908.\n[24]A. Ganz, P. Martens, S. Mukohyama, and R. Namba,\nBouncing cosmology in VCDM, JCAP 04, 060,\n2212.13561.\n[25]D. Blas, J. Lesgourgues, and T. Tram, The Cosmic Linear\nAnisotropy Solving System (CLASS). Part II: Approxi-\nmation schemes, JCAP 2011 (7), 034, 1104.2933.\n[26]C.-P. Ma and E. Bertschinger, Cosmological perturba-\ntion theory in the synchronous and conformal Newtonian\ngauges, Astrophys. J. 455, 7 (1995), astro-ph/9506072."    },    {        "title": "2402.07764v1.Designing_for_effective_heat_transfer_in_a_solid_thermal_energy_storage_system.pdf",        "content": "1  Designing for effective heat transfer in a solid thermal energy storage system  Shomik Verma, Colin Kelsall, Kyle Buznitsky, Alina LaPotin, Ashwin Sandeep, Asegun Henry*   Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. * ase@mit.edu  1. Abstract  Thermal energy storage using sensible heating of a solid storage medium is a potential low-cost technology for long-duration energy storage. To effectively get heat in and out of the solid material, channels of heat transfer fluid can be embedded within the storage material. Here we present design principles to improve performance of channel-embedded thermal energy storage systems, and we apply these principles to a high-temperature system using graphite as the storage material and liquid tin as the heat transfer fluid. We first analyze the impact of geometry and material properties on the performance of the system, determining the ideal channel spacing and length to achieve high (dis)charge temperature uniformity. We then analyze how controlling the fluid flowrate, heating infrastructure, and heat engine can increase discharge power uniformity and accelerate charging. Finally, we model 100 high-temperature graphite storage blocks using a porous media approximation and implement the developed design principles to demonstrate significant improvement in performance for both discharging (constant discharge power for >90% of rated duration) and charging (>90% charged within 4 hours). Overall, the hierarchical design procedure presented here enables the design of cheap yet high-performing solid thermal energy storage systems.  Keywords: thermal energy storage, solid storage material, channel-embedded, performance improvements, discharge power uniformity, charging acceleration, grid-scale storage, porous media approximation  2. Introduction  Renewables such as solar produce clean but intermittent electricity. Large-scale penetration of solar requires long-duration energy storage (>10 hours) to reliably meet demand and provide cleanly generated electricity at night or on cloudy days.1 Energy systems modeling shows the energy capacity cost (ECC, dollars per amount of thermal energy stored in kWh) is an important metric for evaluating long-duration energy storage technologies, and that an ECC of less than $20/kWh is critical for cost-competitiveness of renewables with storage technologies.2 Thermal energy storage (TES) is one promising technology with low ECC.3 In this energy storage technology, electricity is stored in the form of heat. Electricity can be converted to heat through resistance heating or heat pumps, and heat can be stored 2  through latent or sensible heating of a material surrounded by insulation to retain the heat, and heat can be converted back to electricity using a variety of heat engines including turbines, thermoelectrics, or thermophotovoltaics.4,5  There are a few important considerations for the storage material. First, we must choose cheap materials that can store high amounts of heat because of the emphasis on low ECC. Second, we should maximize the amount of thermal energy stored per volume to reduce insulation costs. Lastly, we should maximize the operating temperature to ensure high heat to electricity conversion efficiency. As shown in Figure 1(a), the materials with the lowest cost per unit thermal energy, highest energy density, and highest operating temperature are generally solids and rely on sensible heating. These solid storage systems require a heat exchanger to be integrated into the storage material for charging or discharging.6 There are different types of heat transfer mechanisms, including a radiative heater relying on radiation between a hot emitter and the storage medium, direct Joule heating of the storage medium, and a fluid-based heat exchanger using a heat transfer fluid (HTF) to carry the heat. Radiative heating can have high heat transfer resistance, and direct Joule heating depends on the electrical properties of the storage medium reducing flexibility in material choices. In contrast, fluid-based heat exchange enables low heat transfer resistance and cheap storage materials, so is often preferred. The challenge of using a solid storage material is that it may perform poorly from a heat transfer perspective. Specifically, the system would ideally output a constant temperature (and power) with time, to operate like an electrochemical battery, be in alignment with how the grid is currently operated, help improve reliability, and take advantage of price spikes. Systems that use liquids as the storage material, in either one or two tanks, can generally output high-temperature heat as long as hot fluid remains in the tank.7–10 Solid storage systems can be significantly worse, because as the storage material cools, the outlet temperature of the HTF decreases, potentially reducing the amount of power discharged from the TES system.9 Despite being cheap and energy dense, this can put solid storage systems at a disadvantage when compared to other forms of energy storage, and it is therefore important to design the heat transfer infrastructure for maximum performance.  3   Figure 1: (a) Plot of cost vs. energy density of a variety of storage materials, labeled by the phase which stores the maximum amount of heat (assuming 500°C for sensible heating) without decomposing. Solids enable the lowest cost and highest energy densities, as well as other benefits including achieving high temperatures and limited corrosion. Data used to create this plot is presented in Section S1. (b) Thermal energy grid storage (TEGS) system considered in this work. The system can be divided into charging and discharging infrastructure. For charging, electricity from any desired source is used to heat up liquid tin to high temperatures (~2400ºC) with resistive heating. The hot tin then flows through graphite storage blocks to heat them up, storing the energy in the form of sensible heat. During discharge, cold (~1900ºC) tin flows through the graphite blocks, heating up, and the hot tin then flows to the power block featuring multi-junction thermophotovoltaic (TPV) cells which convert the heat to electricity. Graphite is the storage medium and liquid tin is the heat transfer fluid. (c) Side cross-section view of a graphite storage block with embedded liquid tin channels, and top view of a graphite storage block showing the tin channel inlets. Dashed lines show how the larger block can be split into 25 sub-blocks, each with only 1 channel. Several previous works have investigated different heat exchanger designs and performance improvement strategies. One simple heat exchanger design is to use particles of solid storage medium surrounded by a heat transfer fluid, as investigated by many researchers including ELSihy et al.,11 Rodriguez and Lemos,12,13 and Marti et al.,14 who also discuss performance improvements through materials and operational changes. However, this geometry relies on the storage material being available as particles, and often features high pressure drops in the fluid.15 For other heat exchanger designs, Wu et al. compared four different geometries, including a block with channels of HTF drilled into it (channel-embedded block), solid rectangular plates with HTF flowing around them (parallel-plates), a bundle of solid rods with HFT flowing around it (rod-bundle), and the packed bed configuration as discussed previously.16 They found that the packed-bed \n.........Electricity in from any source powers heatersElectricity outusing thermophotovoltaics (TPV)ChargingDischarging\nInsulated graphite thermal storage blockswith liquid tin heat transfer fluid\nTop viewSide (cross-section)view\n(b)(c)\n10°1100101102103104cost ($/kWh-th)102103energy density (kWh-th/m3)solidliquidPCM(grayed out liquids boil below 1200°C)\nSandCementConcreteIron oreSiFeGraphiteSaltAlFire bricksMgNaMineral oilKCaCuNi\nSnGaMgOCaO(a)4  configuration had the best outlet temperature uniformity, indicating optimization needs to be done for the other configurations.16 Among the other configurations, the channel-embedded block allows the most flexibility in the form of the storage medium, as channels for HTF flow simply need to be drilled within or placed around the storage material, instead of having to form the storage block into different shapes. Therefore, the channel-embedded block configuration is investigated in this work. Some previous studies have investigated how best to optimize these channel-embedded solid systems for charging or discharging performance, using various parameters. Jian et al. investigated the effects of diameter ratio, channel length, flowrate, and number of tubes on discharge performance, optimizing storage cost based on an analytical solution using a corrected lumped capacitance model.17 There have also been experimental validations of this lumped capacitance model.18,19 Prasad et al. investigated the effect of the number of tubes and flowrate on charge and discharge performance.20 While the studies above only considered a single block, large-scale TES systems will consist of many blocks operating in an interconnected way. Doretti et al. considered series vs. parallel arrangements of blocks, as well as the diabatic case of heat loss to the environment, finding the optimal configuration using the minimal number of blocks.21 Kuang et al. investigated the effect of increasing the flowrate with time to counterbalance the decreasing temperature in two series-connected rod-bundles, enabling the extension of the stable discharging time by 1.6 times.22 Despite these previous works, there are still significant gaps in understanding exactly what is required for effective design. Many works consider only charging or discharging in isolation but not how designing for one condition impacts the other. Previous works often use a corrected lumped-capacitance approximation, which may not be valid for the large Biot numbers associated with some HTFs (such as liquid metals), and which ignores axial conduction. Further, previous works often only choose one storage material, and the impact of varying material properties on performance is not made clear. Perhaps most importantly, no previous work has considered a multi-block configuration of the size required for grid-scale deployment. This configuration requires analysis of optimal flow distribution and heat transfer between the blocks, which is particularly relevant at high temperatures (>1500ºC).  Therefore, this study will focus on how to design for effective heat transfer in a large-scale, channel-embedded solid thermal energy storage system during both discharging and charging. We have developed general design principles to maximize performance, with performance metrics defined for constant-power discharge and accelerated charging. We then use these design principles to analyze a specific solid sensible thermal energy grid storage system called TEGS (Thermal Energy Grid Storage). While only the TEGS system is analyzed here, the design principles are applicable to many solid storage systems, as discussed throughout.  An overview of the TEGS system is shown in Figure 1(b). In this system, graphite is used as the storage medium, and can be found in the form of large channel-embedded blocks (as shown in Figure 1(c)) which may be connected in series or parallel. Liquid tin is used as the HTF. During charging, resistance heaters are used as the heat input from electricity, and during discharging, thermophotovoltaics (TPV) are the heat engine. TPV works by converting the light radiated by a heat source into electricity. The system 5  operates from a minimum temperature of 1900°C to a maximum of 2400°C, to maximize TPV efficiency and power density.23  There are many questions we would like our design principles to answer. How big and how many channels should there be per block? How do the storage blocks best interface with the charging and discharging infrastructure? How should the HTF flow between the blocks be configured? What is the best arrangement of blocks considering radiation between them? One way to answer these questions would be to model the entire storage system, including coupled fluid flow and heat transfer, and radiation between the blocks. However, this quickly becomes computationally intractable due to the large size (~1GWh) and long duration (~30 hours) of the systems of interest. Therefore, we have developed a hierarchical design and modeling methodology to determine design principles that result in optimal performance. We started with analyzing the impact of physical (material properties, geometry) and operational (flowrate) variables on the performance of a single HTF flow path, then apply the insights generated to a model of a multi-block grid-scale TES system.  3. Theory and preliminary calculations  Given a storage system, the duration of storage (hours) is defined as the thermal energy (Wh) divided by the power extracted (W). Thermal energy is defined as:  𝐸=𝑚!%𝑐\",!(𝑇)𝑑𝑇$!\"#,%$!&',%≈𝑚!𝑐\",!Δ𝑇!\t where 𝑚! is the total mass of the solid blocks, 𝑐\",!(𝑇) is the specific heat of the solid as a function of temperature, and 𝑇%&',! is the maximum operating temperature while 𝑇%(),! is the minimum operating temperature of the storage system. If the specific heat does not change significantly over the temperature range, it can be considered constant and the integral can be simplified, where Δ𝑇!=-𝑇%&',!−𝑇%(),!/. The thermal power extracted (or provided) is calculated by:   𝑄=𝑚̇*[𝑐\",*-𝑇*,+,-/⋅𝑇*,+,-−𝑐\",*-𝑇*,()/⋅𝑇*,()] ≈𝑚̇*𝑐\",*Δ𝑇*  (1)  where  𝑚̇* is the flowrate of HTF through the system, 𝑐\",*(𝑇) is the specific heat of the HTF as a function of temperature, 𝑇*,+,- is the HTF outlet temperature from the storage blocks, and 𝑇*,() is the HTF inlet temperature to the storage blocks. Again, if the specific heat can be considered constant, this expression can be simplified where 𝛥𝑇* is the difference in HTF outlet and inlet temperatures.  During discharge, hot HTF leaves the storage heat exchanger, is cooled by the heat engine, and returns to the storage system cooled (top right loop in Figure 1(b)). The temperature of cold HTF entering the storage -𝑇*,()/ is assumed to be the minimum operating temperature of the system (𝑇%(),!), and ideally the storage would heat up the HTF such that its outlet temperature -𝑇*,+,-/ is the highest operating temperature of the 6  system -𝑇%&',!/ for the entire discharge. This is because providing the maximum temperature enables higher heat engine efficiency, improving the round-trip efficiency of the thermal storage system. Constant temperature is optimal because, as mentioned previously, providing a constant power output enables operation like an electrochemical battery, improving reliability, maximizing revenue, and easing grid integration. Conversely, during charging, cold HTF leaves the storage system, is heated by the charging infrastructure, and hot HTF returns to the storage system (bottom left loop in Figure 1(b)). The temperature of the hot HTF entering the storage -𝑇*,()/ is assumed to be the maximum operating temperature of the system -𝑇%&',!/, and ideally the HTF would transfer all its heat to the storage medium such that its outlet temperature -𝑇*,+,-/ from the storage blocks is the lowest operating temperature -𝑇%(),!/. Thus, max-𝛥𝑇*/=𝑇%&',!−𝑇%(),!=𝛥𝑇! and 𝑄%&'=𝑚̇*𝑐\",*𝛥𝑇!.  We can now define a characteristic time scale 𝜏, non-dimensional time 𝑡∗, non-dimensional temperature as a function of time 𝛩(𝑡), and non-dimensional temperature as a function of non-dimensional time 𝛩∗(𝑡∗), for a given storage system:   𝜏=!\"!\"#=#$$%,$#̇'$%,'  (2)   𝑡∗=𝑡𝜏\t 𝛩(𝑡)=⎩⎪⎨⎪⎧𝑇(𝑡)−𝑇%()𝑇%&'−𝑇%()\t𝑑𝑢𝑟𝑖𝑛𝑔\t𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑇%&'−𝑇(𝑡)𝑇%&'−𝑇%()\t\t\t\t\t\t\t𝑑𝑢𝑟𝑖𝑛𝑔\t𝑐ℎ𝑎𝑟𝑔𝑒\t\t𝛩∗(𝑡∗)=𝛩(𝑡)\t In reality, the ideal outlet temperatures discussed above are not achievable. As a solid thermal storage medium discharges and cools, the outlet temperature of the HTF decreases. Conversely, during charging the HTF outlet temperature increases as the storage medium heats up. Thus, there is non-ideality in real system operation. To get some intuition for this phenomenon, we can simulate a real system as an example. We model a simplified version of the TEGS system (a single block of graphite as the solid storage medium and one flow channel liquid tin as the HTF), with the specific geometric, material, and operational parameters described in Section 5.1. In this simulation, we start with a graphite block at 𝛩=1 and flow tin in at 𝛩=0 until the entire block is at 𝛩=0. Figure 2 shows the idealized vs. realistic non-dimensional temperature profiles for tin outlet temperature during (dis)charge, for the system described above with different graphite thermal conductivities, to show the impact of material properties on performance. From this example, we see that while the higher thermal conductivity cases start to approach the ideal limit, there is still significant room for improvement.  7   Figure 2: Comparison of non-dimensional tin outlet temperature for the ideal (dis)charge profile of a two-tank thermal storage system or electrochemical battery (dashed line), and the non-ideal (dis)charge profiles of channel-embedded passive system (solid lines, the nondimensional (dis)charge temperature profiles as a function of non-dimensional time for the 3 thermal conductivities shown). The block is initially at Θ=1 everywhere, and tin enters at Θ=0 until the block is at Θ=0 everywhere. Note that the same curves are obtained using either the discharging or charging definition of 𝛩.  One simple strategy for improvement is over-sizing storage capacity such that the storage system is never discharged more than e.g., 25%. However, this directly increases costs, as large amounts of storage material are never fully utilized. It also reduces the charging speed as the system would always be at a high temperature, reducing the Δ𝑇 available for charging power. These effects are discussed further in Section S2. Because of these issues, oversizing is ruled out as a potential solution. Instead, we are interested in determining how performance can be maximized based on changes to geometry, operation, and block arrangement in the storage system. We first define a temperature-based figure of merit (FOM) as, 𝐹𝑂𝑀$=%Θ∗(𝑡∗)\t𝑑𝑡∗/0 If Θ∗(𝑡∗) was an idealized square curve, this integral would evaluate to 1, so this FOM serves as an indicator of how ideal the actual (dis)charge profile is. While this is a useful non-dimensional metric, it has some limitations, namely that it only considers temperature and not power. What we care about practically are (a) how long we can output a constant electrical power during discharge, and (b) how fast the system can charge. Thus, to be more explicit, we first define electrical discharge and charging power as 𝑃+,-=𝑄+,-𝜂1(!23&456 𝑄()=𝑃()𝜂23&456 where 𝜂1(!23&456 is the efficiency of the heat engine used to generate electricity (𝑃+,-) from the thermal power discharge from the storage (𝑄+,-), and 𝜂23&456 is the efficiency of the heaters used to generate heat (𝑄()) from electricity (𝑃()). We then define two power-based FOMs. For discharge,  𝐹𝑂𝑀7,1(!23&456=𝑡7!\"#𝜏 which compares the amount of time that the storage system can output the nominal electrical power -𝑡7,%&'\t/, to the rated storage duration (𝜏) as defined in Equation 2. We define another FOM for charging, 0.00.51.01.52.0t§0.00.20.40.60.81.0£§Two-tankIncreasing k\n0.00.51.01.52.0t§0.00.20.40.60.81.0£§idealk=5WmKk = 10WmKk = 30WmK(of solidstoragematerial)8  𝐹𝑂𝑀7,23&456=∫𝑚̇(𝑡)𝑐\",9)Δ𝑇(𝑡):0𝑑𝑡𝜂23&456𝐸 which compares the integrated electrical power input over the charging duration to the energy capacity, and is essentially the state of charge of the system after the charging duration. In this work, we seek to maximize these FOMs. Because modeling the entire storage system is computationally intractable due to the large system size (~1 GWh), long durations (~10 hours), and large range of geometry sizes (~1mm to ~1m), we propose a hierarchical design procedure to methodologically model and optimize performance, in increasing levels of fidelity to a real system, as summarized in Figure 3. First, as shown in Figure 1(c), each large block can be separated into individual sub-blocks with only 1 channel embedded. Even with this simplified geometry, there are many variables (as shown in Figure 3(a)), so we use dimensional analysis to understand the state space and evaluate the effects of physical (geometric and material) parameters on 𝐹𝑂𝑀$. In addition to the physical variables, operational variables such as flowrate and heat engine operation are also important. Thus, second, as shown in Figure 3(b), we simulate a single-channel cylinder during both discharging and charging to determine the impact of operational variables such as flowrate, heater power, and heat engine control on 𝐹𝑂𝑀7,1(!23&456 and 𝐹𝑂𝑀7,23&456. Finally, as shown in Figure 3(c), we model a large-scale storage system with multi-channel blocks using a porous media approximation to determine the effects of block interconnection, block arrangement, and radiation between the blocks on the power FOMs.  \n Figure 3: Schematic of the three phases of the hierarchical design procedure presented here. (a) Geometry optimization of a block with a single channel by varying the physical parameters, (b) operational optimization by varying flowrate and controlling the heater power and heat engine, and (c) large-scale model with multiple large blocks to understand the impact of flow configuration (how many blocks should the tin flow through before exiting), block arrangement (what is the impact of n and m on performance), and radiation (how does the high temperatures of the blocks impact performance). \n!Sncp,SnkSnNuSnUSnTin\nDSnToutDC!Ccp,CkCTC,init\nL\nJoule heatingPinChargingDischargingTPV controlPout!̇!\",$%&'()\n!̇!\",*+,$%&'()\n.........(a)(b)(c)Tin inlet\nRadiation Out?Continue?nblocksmblocks9  4. Results and discussion  4.1. Impact of storage medium physical parameters on performance  First, we utilize dimensional analysis to help determine the impact of geometric parameters and material properties on performance. We are particularly interested in (a) how to design the channel-embedded heat exchanger, namely the channel spacing and length, and (b) how the thermal properties of the storage medium affect performance. For this analysis, we simplify the geometry to consider a cylindrical storage block with a single HTF channel. Figure 3(a) shows this simplified geometry along with key parameters in the analysis. We consider a cylindrical HTF channel of diameter 𝐷* surrounded by solid with outer diameter 𝐷!, both having length L. The HTF flows at velocity 𝑈(𝑡) and has a convective heat transfer coefficient with the solid of ℎ. Material properties such as density, specific heat, and thermal conductivity of both fluid and solid are also needed. Conducting dimensional analysis on the 15 variables with 4 dimensions results in 11 dimensionless Π groups, all of which are provided in Section S3. Rearranging in terms of known variables, we find that  𝐹𝑂𝑀$=𝑓W𝐿𝐷!,𝐷!𝐷*,𝑘!𝑘*,𝑐\",!𝑐\",*,𝜌!𝜌*,𝑈𝐷*𝛼!,ℎ𝐷*𝑘*=𝑁𝑢;(,𝜌*𝑈𝐷*𝜇*=𝑅𝑒;(,𝜇𝑈<𝑘*Δ𝑇=𝐵𝑟a These 9 dimensionless groups define the state space of possible channel-embedded thermal storage designs. By varying these 9 parameters, we can explore the impact of different variables on the FOM. To demonstrate how this tool can be used in a real system, we consider the TEGS system with liquid tin as the HTF and graphite as the storage medium. From previous works,24 we know that the densities and specific heats of different grades of graphite do not vary significantly, while thermal conductivity does. Tin material parameters are considered constant in this analysis. Further, we assume the volume ratio of tin to graphite to be 1%, which enables favorable technoeconomics as the cost of tin is ~30 times higher than that of graphite, and higher volume fractions increase the cost per energy of the system.25 This 1% volume fraction implies =;%)>=;()=;()=100⇒;%;(=√101≈10 is a constant. Additionally, we assume laminar tin flow meaning constant 𝑁𝑢=3;(?(. This assumption is validated in Section S3.  Based on these assumptions, we can simplify the expression above such that 𝐹𝑂𝑀$ can be written as a function of 3 dimensional groups:  𝐹𝑂𝑀$=𝑓W𝐿𝐷!,𝑈𝐷*𝛼!,𝑘!𝑘*a\t Where @;% is an aspect ratio, A;(B% compares tin advection to graphite conduction, and ?%?( compares the thermal conductivities of the two materials. We now have a more manageable 3-dimensional state space. We can use a finite element model (as discussed in Section 5.1) of the simplified geometry to calculate values of 𝐹𝑂𝑀$ for points in this 3D 10  state space, and determine where it is optimal to operate. 2D slices of the 3D state space are shown in Figure 4(a), which shows 𝐹𝑂𝑀$ vs @;% and A;(B% for various values of ?%?(. Visually, we first note that we can quickly identify the region of design space we want to operate in, namely the high-FOM yellow region. This allows quick evaluation of whether a designed configuration will be high performing. However, we also note that these plots look very similar across different 𝑘 ratios, which is unexpected because we previously determined a lower solid thermal conductivity 𝑘! should reduce performance. Instead, we notice that the impact of 𝑘! comes in the x-axis, since 𝑘! also appears in 𝛼!=?%C%2*,%. Thus, for a given flow geometry, lower values of 𝑘! shift the operating point to the right, away from the high-FOM yellow region, as A;(B% increases.  \n Figure 4: (a) Results from dimensional analysis showing the complete parameter space of the channel-embedded thermal energy storage system called TEGS, showing 2D slices of !\" and #\"$ vs. 𝐹𝑂𝑀% for constant 𝑘&/𝑘'. (b) Dimensioned sweeps of the parameter space for a given graphite thermal conductivity (10 W/m/K) and storage duration of 30 hours. The solid line is an isocontour of FOM=0.9, and the star shows the optimal operating point for largest diameter and smallest length to achieve high FOM. (c) depicts a line of FOM vs. D corresponding to the blue dashed line in (b). (d) and (e) are equivalent plots for a storage duration of 10 hours. While this non-dimensional map is useful, it is not immediately clear how to use the map to design a high-performing system. We thus demonstrate an example of how the mapped design space can be used practically, to design the geometric parameters of the TEGS system. We first consider a storage duration (𝜏) of 30 hours and a graphite \n!\"!/$\"%\"/%!=0.01 \n%\"/%!=0.03 \n%\"/%!=0.1\n%\"/%!=0.3\n%\"/%!=10.00.20.40.60.81.0D (m)0.50.60.70.80.91.0FOMTFOMTvs. D for L = 10 m(c)(a)\n01020304050L (m)0.00.20.40.60.81.0FOMTFOMTvs. L for D = 0.2 m(d)0.000.250.500.751.00D (m)020406080100L (m)ø=30 hours, k=10 W/m/K\n0.0000.1010.2020.3030.4040.5050.6060.7070.8080.909FOMT\n0.0000.0420.0840.1260.1690.211U (m/s)0.9★(b)\nConstant L, vary DConstant D, vary L★★11  thermal conductivity (𝑘!) of 10 W/m/K. We then choose a length 𝐿 and graphite diameter 𝐷! (which also defines 𝐷* since ;%;( was fixed at 10) of the storage block. We can then use the duration 𝜏 and length 𝐿 to calculate the flow velocity 𝑈, by re-writing Equation 2 as  𝜏=𝜌!𝑉!𝑐\"!𝑚̇*𝑐\"*=𝜌!𝐿𝜋(𝐷!\t<−𝐷*<)𝑐\"!𝜌*𝑈𝜋𝐷*<𝑐\"*=𝜌!𝐿fg𝐷!𝐷*h<−1i𝑐\"!𝜌*𝑈𝑐\"*=100𝜌!𝜌*𝑐\",!𝑐\",*𝐿𝑈\t ⇒𝑈=W100𝜌!𝑐\",!𝜌*𝑐\",*ag𝐿𝜏h  Thus, using the input variables 𝜏,𝑘!,𝐿,\tand 𝐷!, we can calculate 𝑈 and therefore all 3 of the dimensionless groups, and can use the non-dimensional map to determine the 𝐹𝑂𝑀$. The results are shown in Figure 4(b). A few trends are evident. First, reducing the diameter of the graphite 𝐷! (and therefore the tin channel 𝐷*) improves 𝐹𝑂𝑀$, as seen in Figure 4(c). This can be explained by the shorter conduction distance in the graphite, so the heat can more easily enter/leave the graphite, improving its performance. However, if we were to only reduce the graphite diameter 𝐷! while keeping 𝐷* constant, this would hurt the energy density (energy per volume) and cost of the system, since the tin HTF would take up more volume. Instead, because we scale down the fluid diameter 𝐷* with 𝐷!, we retain the energy density and low cost of the system while improving performance. Although the FOM monotonically increases with reduced 𝐷*, the reduction in 𝐷* does cause an increase in pressure drop, which should be avoided due to higher parasitic power consumption of the motor, wear on mechanical parts, and higher leakage potential. We recall that pressure drop in a tube is defined as:  𝛥𝑃=𝜌9)𝑈<𝐿2𝐷*𝑓(𝑅𝑒)⇒𝛥𝑃D&%∝𝜇9)𝑈𝐿𝐷*<,Δ𝑃-,4E,4+,53∝𝜌9)𝑈<𝐿2𝐷*\t suggesting that we should not go to the extreme of 𝐷*→0 if we want to prevent high pressure drops. This demonstrates the tradeoff between low 𝐷* improving performance while increasing pressure drop. The second important trend is that increasing the tin channel length 𝐿 improves FOM, but this effect plateaus quickly, as seen in Figure 4(d). For the configuration considered here, the FOM stops improving after around 10m, suggesting that e.g. a 20m long channel-embedded block has equivalent thermal performance to 2 10m long blocks. This is important because, from the pressure drop definition above, it is beneficial to minimize L. We can therefore conclude that placing 𝑛 10m long pipes in parallel has equivalent thermal performance to a single 10𝑛 m long pipe, and the 𝑛 pipes in parallel have 4𝑛 (laminar) to 16𝑛 (turbulent rough) times lower pressure drop. Thus, optimizing the FOM with the constraint of minimizing pressure drop (minimizing L and maximizing D) suggests an optimal operating point of 𝐷9=20𝑐𝑚 (𝐷*=2𝑐𝑚) and 𝐿=10𝑚 to achieve an 𝐹𝑂𝑀$ of 90%. 12  This analysis is repeated for storage duration 𝜏=100 hours and graphite thermal conductivity 𝑘!=1\tW/m/K in Section S4 to demonstrate applicability to a different configuration. While only the TEGS system is considered here, the design procedure can be applied to any channel-embedded solid thermal storage system, by (a) applying the dimensional analysis derived here for the system in question and mapping the state space, (b) selecting a storage duration and material properties, and (c) determining the maximum diameter and minimum length that provides a high FOM (e.g. 0.9) based on the state space map. Now that an understanding of the effect of physical parameters has been developed, we can consider operational variables and how the storage block would be integrated with charge and discharge infrastructure.  4.2. Coupling the storage medium with (dis)charging infrastructure  Given a solid storage geometry from the previous section, we can now determine how the operation of the storage system impacts performance. There are two specific items to improve. First, output electrical power should ideally be constant over time, at the rated output power of the storage system, which is what utilities would expect and is what power plants and electrochemical batteries provide. However, as seen in Figure 2, temperature slowly decreases over time, potentially reducing power output. Second, since the intent of the storage system is to improve the reliability of variable renewable energy sources, the system should be able to charge quickly when renewable energy is available; it is anticipated that there will be greater value on the grid for a storage system that can charge faster than it discharges.26,27 Thus, the two reasons for trying to optimize operational variables are (a) achieving closer to constant electrical power output on discharge and (b) faster charging.  4.2.1. Constant power output during discharge  For discharging, while we could use 𝐹𝑂𝑀$ above, it is not fully appropriate for a few reasons. First, it only considers temperature and not power directly. Second, the setup of 𝐹𝑂𝑀$ assumes the HTF flowrate does not change with time, which is the simplest operation mode but may not be optimal. Third, the integral form obscures potentially important discharge dynamics – for example, a system that outputs 100% of rated power for 80% of rated duration has the same 𝐹𝑂𝑀$ as a system that outputs 80% of rated power for 100% of rated duration. However, the first system is more ideal because utilities expect the rated power output. Of course, the most ideal system would output 100% of rated power for 100% of rated duration; therefore, we define an FOM directly based on duration of constant power output, namely,  𝐹𝑂𝑀7,1(!23&456=𝑡7!\"#𝜏\t where 𝑡7,%&'\tis the amount of time that the storage system can output the nominal electrical power, and 𝜏 is the storage duration as in Equation 2.  13  We now introduce a technique to maximize 𝐹𝑂𝑀7,1(!23&456 for a general thermal storage system. First, if we want to output constant electrical power, we need to ensure constant thermal power is being output. We noted previously that the cause of thermal power decrease over time is the drop in outlet temperature due to the storage medium being cooled. To counteract this, as the outlet temperature (and therefore Δ𝑇*) drops, we could concurrently ramp up mass flowrate of the HTF, keeping output thermal power constant as defined by 𝑄(𝑡)=𝑚̇*(𝑡)𝑐\",*Δ𝑇*(𝑡). Thus, we have 𝑚̇*(𝑡)=𝑄%&'𝑐\",*Δ𝑇*(𝑡) We must bound 𝑚̇* however, to avoid extreme flowrates as Δ𝑇* drops to 0. Therefore, we define a maximum flowrate 𝑚̇%&' and a maximum flowrate factor 𝑓=%̇!\"#%̇((-H0). Thus, the flowrate is able to keep up with the drop in Δ𝑇* until 𝑚̇*(𝑡)=𝑚̇%&', and we can calculate the Δ𝑇* at that point as  Δ𝑇*,-346!=𝑄%&'𝑐\",*𝑚̇%&' where we have defined a threshold Δ𝑇*,-346! below which output power starts to decrease because the flowrate cannot increase further. This suggests that the thermal power output as a function of time is 𝑄+,-(𝑡)=o𝑄%&'\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t𝑤ℎ𝑒𝑛\tΔ𝑇*(𝑡)>Δ𝑇*,-346!𝑚̇%&'𝑐\",*Δ𝑇*\t(𝑡)\t\t\t𝑤ℎ𝑒𝑛\tΔ𝑇*(𝑡)<Δ𝑇*,-346! Higher values of 𝑚̇%&' (and therefore higher 𝑓) will result in lower values of Δ𝑇*,-346!, from the equation above. This then implies that higher 𝑓 results in longer durations of constant power output at 𝑄%&', since Δ𝑇*(𝑡) will be higher than Δ𝑇*,-346! for longer. To make this approach clear, we consider an example using the TEGS system, of a single cylindrical graphite block of length 𝐿=10𝑚 and diameter 𝐷J=0.2𝑚 (tin channel diameter 𝐷9)=0.02𝑚), with a graphite thermal conductivity of 10 W/m/K, based on the geometric optimization done in the previous section. We then consider a storage duration of 31.6 hours and a maximum flowrate factor 𝑓=%̇!\"#%̇'+!=5.6 (for demonstration purposes – we will consider additional values later). We can then numerically model the system while incorporating the approach above, with the results shown in Figure 5(a)-(c). As seen, as the outlet temperature of the tin HTF decreases (Figure 5(a)), we can increase the mass flowrate until a value of 5.6 times the flowrate at t=0 (Figure 5(b)). We can therefore keep the thermal power output constant for time 𝑡7,%&' (blue line in Figure 5(c)). 14   Figure 5: Demonstration of the operational principle of flow ramping, to counteract the lower temperature tin outlet of the storage medium. First considering a discharge duration of 31.6 hours and a flowrate factor of 5.6, (a) Tin outlet temperature from the storage medium, subtracted by the inlet temperature (1900ºC), (b) flowrate of tin as a function of time showing the ramping, (c) thermal power of the tin output from the storage medium (blue line) and electrical power output from the TPV (red line), (d) Normalized TPV area required to convert the thermal power to electrical power at 40% efficiency, (e) the impact of allowing the flowrate to increase to a certain factor on the discharging FOM, for 𝜏=31.6, with the triangle showing the f=5.6 case, (f) The percent of additional TPV area required to achieve the 𝐹𝑂𝑀(,*+&,-./01 shown in (e), with the triangle showing the f=5.6 case. (g) Discharge FOM for various storage durations and flowrate factors, with solid line an isocontour of 𝐹𝑂𝑀(,*+&,-./01=0.9, and the dashed line showing where 𝜏=31.6 lies. (h) Percent of additional TPV area required to achieve the 𝐹𝑂𝑀(,*+&,-./01\t𝑖𝑛\t(𝑔). While the approach above keeps output thermal power constant, we ultimately want to convert the thermal power to electrical power. Keeping electrical power constant is challenging, because as the discharging process progresses, the heat carried by the HTF becomes lower grade (lower temperature) and higher flowrate. The heat engine will likely have a harder time converting the lower grade heat to electricity. This suggests that the heat engine may need to be throttled at the beginning of discharge, meaning it would output less than its rated (maximum) power when temperature is maximum. To give an example of how a heat engine may follow this strategy, we consider the heat engine used in the example TEGS system, which is thermophotovoltaics (TPV). TPV converts the energy radiated by a heat source, in this case the liquid tin HTF, to electricity. However, as the temperature of the HTF drops, the intensity of emitted radiation decreases, meaning more TPV area is required to keep the generated electrical power constant. To demonstrate this, we develop a simple model: 𝑄+,-=𝑞\"w𝑇*,&K5(𝑡)x𝐴$7L(𝑡)=𝑃+,-𝜂$7L where 𝑞\"(𝑇) is the net (radiative) heat flux from the HTF to the TPV (modeled based on experimental data), 𝑇*,&K5(𝑡) is the 4th order average of HTF temperature as a function of time, and 𝐴$7L(𝑡) is the active area of the TPV as a function of time. As the HTF temperature decreases, 𝑞\"-𝑇*,&K5/ will also decrease, suggesting more TPV area will need to be exposed to the HTF to extract the same amount of thermal power from the tin (𝑄+,-) and generate the same electrical power (the thermal power 𝑄+,- is converted to electrical power 𝑃+,- at a TPV efficiency 𝜂$7L). In the schematic of the TEGS system shown in Figure 1(b), the TPVs are fully facing the liquid tin, but this could be changed with time – this analysis suggests the TPV stick should be initially retracted, then as the \n!!!\"#\"0510max flow factor (f)0.40.60.81.0FOMP,discharge102030tP,max, hrø= 31.6 hours(e)\n0510max flow factor (f)0255075100TPV area added (%)ø= 31.6 hours(f)\n0.9!=31.6\tℎ)(g)\n!=31.6\tℎ)(h)\n(a)(b)\n(c)(d)FOMP,discharge=0.92▼▼15  temperature of the HTF drops, the TPV stick should be inserted further to expose more TPV area to the HTF. This is demonstrated in Figure 5(d), where we see the TPV area exposed to the tin HTF increases with time to keep the electrical power generated constant (Figure 5(c), red line). The example above only considered a single maximum flowrate factor 𝑓, so next we consider multiple different values. As discussed previously, the benefit of increasing 𝑓 is longer duration of constant discharge power as the higher flowrate can counteract the lower HTF outlet temperature. However, the lower HTF outlet temperature results in a lower radiated heat flux, meaning more TPV area is required to convert the thermal power to electrical power. Therefore, there is a tradeoff between higher 𝐹𝑂𝑀7,1(!23&456 and higher cost of installing additional TPV cells. Figure 5(e) and (f) demonstrate this tradeoff. Using the same storage duration of 𝜏=31.6 hours, as we increase the flowrate factor, the 𝐹𝑂𝑀7,1(!23&456 increases. Particularly, we note that in the base case (flowrate factor 𝑓=1) we get an 𝐹𝑂𝑀7,1(!23&456 of 0.4. Increasing the flowrate factor to 3 (meaning the flowrate can increase to 3x the nominal flowrate) allows 𝐹𝑂𝑀7,1(!23&456 to reach 0.9 (equivalently, a constant power duration of 28.4 hours out of 31.6 hours). However, this requires approximately 40% more TPV cells (since the outlet temperature decreases with time), increasing the capital expenditure of the system. We have thus far only considered a storage duration of 31.6 hours, so next we expand our analysis to durations ranging from 10 to 100 hours, as shown in Figure 5(g) and (h). As seen, the systems with longer storage duration times require a lower flowrate factor 𝑓 to reach an FOM of 0.9, while systems with shorter duration require higher 𝑓 to reach an FOM of 0.9. This can be explained by the different outlet temperature profiles for different storage durations. The longer storage durations already have a more uniform discharge profile without any modifications (at 𝑓=1) so only minimal modifications are necessary. In contrast, shorter storage durations have a faster decline in outlet temperature as their nominal flowrate is higher, which makes achieving a constant power output more difficult. Next, we note that the percent increase in TPV area added (relative to 𝑓=1 for each storage duration) is independent of the storage duration. To understand this better, we consider a flowrate factor 𝑓 – we can calculate the threshold Δ𝑇*,-346!=72*,,'%̇((-H0)*=2-,,'%̇((-H0)M00°J2-,,'%̇((-H0)*=M00°J*. Because the TPV area is calculated based on Δ𝑇*,-346!, the additional TPV area required is independent of the storage duration and only depends on 𝑓. In summary, in this sub-section we introduced a technique for keeping discharge electrical power constant. First, we keep output thermal power constant by counterbalancing the dropping HTF outlet temperature with an increasing HTF mass flowrate. To convert this thermal power to electrical power, we throttle the heat engine at the beginning of discharge to allow for high power output later, when the HTF temperature is low and flowrate high. There is a tradeoff between performance and cost, which should be evaluated with technoeconomic analysis in a future work.  16  4.2.2. Accelerating charging   We can apply the same approach to the second objective of fast charging. For this objective the uniformity of power over time is not as important, rather we care about charging the system as fully as possible within the charging duration. We therefore define a charging FOM as the ratio between the electrical energy input over the charging duration to the energy capacity:  𝐹𝑂𝑀7,23&456=∫𝑚̇(𝑡)𝑐\",9)Δ𝑇(𝑡):0𝑑𝑡𝜂23&456𝐸  which is essentially the state of charge of the system after charging is complete. Since 𝜂23&456 depends on the type of heating infrastructure used and is generally fixed, for this analysis we will focus on the thermal power in the numerator. Maximizing 𝐹𝑂𝑀7,23&456 therefore corresponds to maximizing the amount of thermal power input into the system.  To motivate the above methodology for charging, if we keep the flowrate constant then as the outlet temperature of HTF from the solid storage increases (as the solid heats up), the amount of thermal power we can supply to the HTF through the heaters drops, since the Δ𝑇 between the HTF outlet temperature and the maximum operating temperature of the system decreases. Therefore, we can increase the flowrate to keep the thermal power input constant. We evaluate the effectiveness of the methodology presented in the context of the TEGS system. For this analysis, we consider a graphite block of the same dimensions as the previous subsection (length 𝐿=10𝑚, diameter 𝐷J=0.2𝑚, tin channel diameter 𝐷9)=0.02𝑚, determined from dimensional analysis). For the accelerated charging analysis, we first consider a short charging duration of 𝜏=5 hours, which corresponds to, for example, the amount of time abundant solar might be available during the day. The nominal flowrate for such a system would be 𝑚̇)+%=O/M32*,,'(M00°Q). To accelerate charging, we define a similar maximum flowrate 𝑚̇%&' and a flowrate factor 𝑓=%̇!\"#%̇'+!. The flowrate is allowed to increase with time as the outlet temperature drops, to keep the power input at its maximum (rated) value, until reaching this maximum flowrate. The results for the given system, with 5 hours of charging duration and various flowrate factors, are shown in Figure 6(a)-(d). Figure 6(e) shows how 𝐹𝑂𝑀7,23&456 varies for other charging durations and flowrate factors. 17   Figure 6: Demonstration of how flow ramping counteracts the lower tin Δ𝑇 and ensures charging at the maximum rated charging power. First considering a charge duration of 5 hours, (a) Δ𝑇 of the tin outlet vs. inlet to the storage medium, (b) flowrate of tin as a function of time, (c) thermal power of the tin output from the storage medium. (d) The impact of allowing the flowrate to increase to a certain factor on the charging FOM, for 𝜏=5ℎ. (e) Charge FOM for various storage durations and flowrate factors, with solid line an isocontour of 𝐹𝑂𝑀(,,-./01=0.9, and the dashed line showing where 𝜏=5 lies. As seen in Figure 6(d), increasing the flowrate factor from 1 to 5 improves 𝐹𝑂𝑀7,23&456\tfrom 75% to 90%, with nominal improvements above 5x. The specific response of the thermal storage system to various flowrate factors is shown in Figure 6(a)-(c). When considering various charging durations from 1 to 10 hours (Figure 6(e)), the accelerated charging is easier at longer durations since their charging FOMs are already high with no modifications (at 𝑓=1), similar to the phenomenon observed in discharge power uniformity. We note that for all durations >4 hours, we can nearly fully charge the system (>90%) with flowrate multipliers of 10 or lower. For shorter durations, full charge could be accomplished with higher flowrate factors, but other techniques such as changing the geometry of the channels might be required instead. In summary, accelerating charging by increasing HTF flowrate to establish a near-constant power input is promising, as it has the potential to reach almost complete charging in a short duration. Although it increases pressure drop, it requires minimal infrastructure to implement (only additional pumping throughput), so there is little tradeoff between performance and cost. Other options for accelerating charging are presented in Section S7, but are shown to be not effective for various reasons. In our analysis thus far, we have only considered a single channel-embedded cylinder. In the following section, we will analyze a larger block consisting of multiple channels and will add modeling of radiation between blocks, to determine how different configurations impact performance, as well as how to improve poorly performing configurations.    \n(a)(b)\n(c)\n!=5ℎ0.9(e)\n510max flow factor (f)0.40.60.81.0FOMP,charge(d)Increasing f18  4.3. Large-scale modeling of a thermal storage system  To analyze a large array of blocks, approximations must be made to reduce the computational expense of solving the full fluid and temperature profiles within each block. We employ a porous media approximation for each block. The details of the porous media approximation are provided in Section 5.2, with the essence being that the porous block is made to exhibit equivalent heat transfer and fluid flow as the channel-embedded block by matching porosity, permeability, and the solid-fluid heat transfer coefficient. The resulting approximation shows good agreement to the fully coupled channel-embedded model with 2-3 orders of magnitude faster computation (see Section S6). To show how this porous media approximation can be used for design, we consider the TEGS system with graphite as the storage medium and tin as the HTF. We consider large rectangular graphite blocks of dimensions 1m x 1m x 4m, which was determined by quotes from manufacturers and structural limitations of resting the blocks on porous insulating materials. From the previous dimensional analysis, we found that 0.02m tin channel diameter with 0.2m spacing is ideal – therefore each graphite block must have a 5x5 array of tin channels to allow a spacing of 0.2m (as shown in Figure 1(c)). We also noticed that the ideal tin flow path length was at least 10m, but since each block is only 4m high, blocks may have to be connected in series. Due to the high temperatures of the storage system (~2000°C), the blocks also emit significant radiation which must be included in the simulation. Radiation can detract from (dis)charge performance by helping make the system more isothermal, reducing the outlet temperature of the tin. We consider a grid-scale system of ~200 MWh of thermal energy storage capacity, corresponding to 100 blocks. Using the porous media approximation, we investigate 5 different block arrangement configurations, as shown in Figure 7.  \n Figure 7: Comparison of various block configurations for the large-scale grid storage simulation. 100 blocks of 1m x 1m x 4m are considered here. Insulation surrounds the blocks and is occasionally shown as transparent for clarity. (a) Shows all 100 blocks stacked vertically, (b) shows 100 blocks placed in a line, (c) shows a 10 x 10 grid of blocks with a single flow path connecting all of them, (d) shows a 10 x 10 grid of blocks with 10 flow paths, and (e) shows a 10 x 10 grid of blocks with 100 independent flow paths. \n...\n1m x 1m x 4m100 blocksConfiguration 1:vertical stack(a)\n...\n100 blocksConfiguration 2: horizontal string(b)\nConfiguration 3: all in series(c)\n10 blocks\n10 blocks\nConfiguration 4: 10 parallel paths(d)\nConfiguration 5: 100 parallel paths(e)\n.........\n..................\nInlet/outletBetween blocks10 blocks\n10 blocks10 blocks\n10 blocks19   The first configuration (Figure 7(a)) includes all blocks stacked vertically, which ensures blocks are thermally isolated from each other since each block radiates only to the surrounding insulation, but is impractical due to the immense height of the resulting system. The second configuration (Figure 7(b)) has all blocks arranged in a string, which ensures each block can only radiate to 2 adjacent blocks, limiting interaction. Configurations 1 and 2 require a significant amount of insulation due to their long aspect ratio, but they are optimal from a heat transfer standpoint. Next, we consider 3 more practical configurations using a grid of blocks. Configuration 3 (Figure 7(c)) arranges a 9x11 grid with all blocks in series such that the tin inlet is only on the top left block and the tin outlet is only on the bottom right (9x11 was used to ensure the inlet and outlet are maximally separated). Configuration 4 (Figure 7(d)) consists of a 10x10 grid with 10 parallel paths for tin flow, such that tin enters in the top row and exits in the bottom row. Configuration 5 (Figure 7(e)) features a 10x10 grid with all blocks in parallel, such that tin enters and exits from each block. Note that configurations 3, 4, and 5 have significantly less outer surface area than 1 and 2, reducing the amount of insulation required. Now, we consider discharging performance over a duration of 20 hours. The results for the tin outlet temperature as a function of time for the 5 different configurations are shown in Figure 8(a). As seen, configurations 1 and 2 have very similar performance. While no blocks can see each other in configuration 1, in configuration 2, 2 sides of each block are exposed to another block, but because each block is not significantly different in temperature from adjacent blocks, this has limited influence on temperature, so its performance is similar to configuration 1. Configurations 3 and 4 are also similar. In both cases, most blocks are exposed on all 4 sides to other blocks. The temperature difference between adjacent blocks is possibly much higher than in configuration 2, which helps explain the difference in performance. GIFs of the temperature evolution of the blocks for configurations 3 and 4 are present in Section S8. Configuration 5 is worse than configurations 3 and 4. This difference can be partially explained by the short tin flow path of 4m in configuration 5, which is less than the 10m ideal discovered through dimensional analysis. As seen, the discharge performance of the 10x10 grids needs to be improved. We thus employ the technique outlined in Section 4.2.1 of increasing flowrate and TPV area with time to improve the power uniformity during discharge. As seen in Figure 8(b), this methodology vastly improves performance and even makes the 10x10 case outperform the vertically stacked case. Notably, only a small increase in maximum flowrate (1 to 1.25) enables the configuration to outperform the vertically stacked case in terms of duration of constant discharge power, with only a minimal cost increase (additional pumping costs and ~10% more TPV area). With a maximum flowrate of 5x the nominal (requiring ~50% more TPV area), we achieve constant power output for ~90% of the 20-hour discharge cycle (18 hours), as seen in Figure 8(c).  20   Figure 8: Multi-block simulations of 100 blocks (200MWh of thermal energy capacity), considering 20 hours of discharge and 4 hours of charging. (a) Comparison of the outlet temperature of tin vs. time for the 5 block configurations described previously. The blue and green lines nearly overlap, as do the yellow and red lines. (b) The outlet power from the tin to the discharging infrastructure, for the 1x100 reference case (conf 2), as well as the 10x10 grid case (conf 4) implementing the flowrate ramping technique presented previously (for f = 1 through 10), showing the 10x10 grid can perform better than the 1x100 case when implementing flowrate ramping. (c) Demonstrates the impact of increasing maximum flowrate on the discharging FOM for the 10x10 grid. GIFs of temperature profile vs. time for configuration (3) and (4) during discharge are shown in Figure S5. (d) Comparison of the outlet temperature of tin vs. time for the 5 block configurations during charging. (e) The inlet power from the charging infrastructure to the tin, for various flowrate multipliers in configuration 4. (f) The impact of increasing flowrate ramping factor on the charging FOM for the 10x10 grid. Next, we consider charging performance over a duration of 4 hours. The tin outlet temperature as a function of time for the 5 different configurations are shown in Figure 8(d). As seen, the difference between the configurations is not as drastic as in discharging, suggesting that the faster rate of charging causes adjacent blocks to reach similar temperatures quicker, limiting the effect of radiation between blocks. However, the 10x10 grid with all blocks in parallel (configuration 5) still performs worse than all other cases, again demonstrating the importance of the dimensional analysis conducted previously showing the minimum tin path length of 10m. Further, it is evident that charging performance needs to be improved, since the tin outlet temperature has not reached 2400C within the 4 hours (which would indicate a fully charged system). For accelerating charging, we use the technique introduced in Section 4.2.2 of increasing flowrate above nominal as the temperature rises to create a constant power input. Figure 8(e) shows how increasing the flowrate multiplier f allows for longer durations of constant input power, accelerating the charging speed. Figure 8(f) shows that the block is nearly completely charged within 4 hours by allowing the flowrate to increase five-fold above nominal, which improves charging performance while only increasing pumping costs. Overall, these simulations show that for a grid of blocks, the choice of flow path is minimally important as long as the tin flow path length is higher than the minimum determined by dimensional analysis. If no modifications to flowrate were used, a thermally 05101520time (hrs)2100215022002250230023502400outlet T (±C)\n(1) 1x100 vertical blocks(2) 1x100 string of blocks(3) 10x10 grid, all series(4) 10x10 grid, 10 parallel paths(5) 10x10 grid, 100 parallel paths(a)\n246810flow multiplier0.00.20.40.60.81.0FOMP,chargeConf. 4(f)\n01234time (hrs)190020002100220023002400outlet T (±C)\n(1) 1x100 vertical blocks(2) 1x100 string of blocks(3) 10x10 grid, all series(4) 10x10 grid, 10 parallel paths(5) 10x10 grid, 100 parallel paths(d)f=1 for all casesf=1 for all cases\n01020time (hrs)2100215022002250230023502400outlet T (±C)\nConf. 1: 1x100 vertical blocksConf. 2: 1x100 string of blocksConf. 3: 10x10 grid, all seriesConf. 4: 10x10 grid, 10 parallel pathsConf. 5: 10x10 grid, 100 parallel paths246810flow multiplier0.00.20.40.60.81.0FOMP,dischargeConf. 4(c)\n01234time (hrs)0.60.81.01.21.41.61.8Power (MW)(e)Increasing f01234time (hrs)1.01.5Power (MW)Conf. 4: f = 1Conf. 4: f = 2.5Conf. 4: f = 5Conf. 4: f = 7.5Conf. 4: f = 1005101520time (hrs)0.200.250.300.35Power (MW)(b)\nIncreasing f\n05101520time (hrs)0.200.250.300.35Power (MW)Conf 2: f = 1Conf 4: f = 1Conf 4: f = 1.25Conf 4: f = 3Conf 4: f = 5Conf 4: f = 1021  isolated string of blocks would be ideal, but this would be impractical due to the drastically high insulation costs. Although the grid is worse due to radiation between the blocks, the techniques developed in this work can be utilized to enhance performance.  5. Experimental procedures  5.1. Numerical modeling of a single channel-embedded cylinder  COMSOL Multiphysics® was used for coupled fluid flow and heat transfer modeling. Low-quality graphite was used was the storage medium while liquid tin was used as the heat transfer fluid. The relevant material properties of graphite and tin are present in the table below. If ranges were provided, a temperature-dependent property was used in the model varying from 1900 to 2400C.  Material Property Value Low-quality graphite Thermal conductivity (𝑘) 10 W/m/K  Specific heat (𝑐\") 2000 J/kg/K  Density (𝜌) 1700 kg/m3 Liquid tin Thermal conductivity (𝑘) 60-65 W/m/K  Specific heat -𝑐\"/ 247-250 J/kg/K  Density (𝜌) 4800 kg/m3  Dynamic viscosity (𝜇) 0.001 Pa⋅s  An axially symmetric 2D model was used with the outer boundary being adiabatic. Other dimensions and flowrates were determined based on user inputs. As mentioned, laminar flow was used as the primary purpose was to accurately model heat transfer, and turbulent flow only minimally impacts heat transfer as detailed in Section S4.  For the example case presented in Section 3, a block of diameter 0.2m, length 10m, and channel diameter 0.02m was used, with a tin flowrate of 0.04 kg/s, and other material properties given above. During discharging we have assumed a constant 𝜂1(!23&456 of 0.4, and during charging we have assumed a constant 𝜂23&456 of 0.99.   5.2. Porous media approximation  Figure 9 shows how a block of 25 channels can be approximated as a single block with porosity.  22   Figure 9: Schematic of porous media approximation showing (a) Cross section of 1m x 1m x 4m graphite block with 25 tin channels embedded of diameter 2cm, and (b) porous media approximation of the same block, where two temperature fields are tracked in the entire block, with parameters matched to achieve equivalent fluid and thermal response. For a porous block to exhibit equivalent heat transfer and fluid flow as a channel-embedded block, we must match some parameters.28 First is the porosity, such that the volume fraction of liquid and solid is the same. Second is the permeability, which is tuned to match the pressure drop in the reference block. Namely, the porous media fluid flow adds an additional source term in the momentum equation:  𝑆=𝜅>/𝜇+W12𝜌|𝑢|√𝜅a𝑢\t where 𝜅 is the permeability, 𝜇 is the viscosity, and 𝑢 is the velocity. The permeability was tuned such that the pressure drops matched. Next, the porous media approximation keeps track of two temperature fields, one for fluid and one for solid. They are coupled with a heat source/sink term involving a relative heat transfer coefficient. The governing heat transfer equations in the solid and fluid, respectively, are:  (1−𝜖\")𝜌!𝑐\",!𝜕𝑇!𝜕𝑡+-−(1−𝜖\")𝑘!𝛻<𝑇!/=ℎ!*g𝐴𝑉h-𝑇*−𝑇!/\t\t𝜖\"𝜌*𝑐\",*𝜕𝑇*𝜕𝑡+𝜖\"𝜌*𝑐\",*(𝑢*⋅𝛻)𝑇*+(−𝜖\"𝑘*𝛻<𝑇*)=ℎ!*g𝐴𝑉h(𝑇!−𝑇*)\t where 𝜖\" is the porosity, 𝑠 subscripts indicate solid while 𝑓 indicates fluid. 𝐴 and 𝑉 are the pore surface area and volume, which are calculated directly from the geometry of the reference block. ℎ!* is the heat transfer coefficient between the solid and fluid. In our case, it includes convection and conduction resistances. Namely,  ℎ!*=1(𝑅2+)K+𝐶/𝑅2+)1)𝐴\t\ns = 0.20m, H = 4m\nsH\n!!\"=0.02&\n(a)(b)\n23  𝑅2+)K=1ℎ𝐴=1𝑁𝑢𝑘𝐷𝐴,\t\t𝑅2+)1=𝑙𝑛\tw𝑟+𝑟(x\t2𝜋𝑘𝐿\t Where 𝐶/ is a correction factor to both convert the cylindrical conduction resistance to rectangular and to convert the conduction resistance to a convection resistance, as was done in Xu et al.29 Note that because the lateral conduction is included in the heat transfer coefficient ℎ!* through the 𝑅2+)1 term, the thermal conductivity 𝑘! in the governing heat transfer equation is anisotropic, with 0 laterally and the nominal values axially, to avoid double-counting the conduction contributions. Once the above values are obtained, a coarse-grained porous media approximation may be used.  To validate this approximation, we consider the TEGS system, and model a large graphite block with 25 tin channels. We compare the tin outlet temperature vs. time for both the full geometry case and the porous media approximation for a variety of flowrates. This comparison is shown in Figure S6. As seen, the porous media approximation offers good agreement with 2-3 orders of magnitude faster computation.  5.3. Simulation parameters for large-scale modeling  For each of the configurations, we consider surface-to-surface radiation between blocks and the insulation, which is externally adiabatic. We use a spacing of 10cm between the blocks and insulation, and 20cm between blocks. We assume all surfaces have emissivity 0.9, which is slightly higher than the reported emissivity of graphite at high temperatures,30,31 so overestimates the heat transfer between blocks. Where possible, the flow configuration is defined such that if the outlet of one block is at the bottom (top), the inlet of the subsequent block is also at the bottom (top). This reflects the real-world flow conditions. Only a 10x10 grid is considered in this work as this is the most optimal grid to minimize outer surface area for insulation. Other grids could be considered by repeating the analysis and taking advantage of the faster simulation time of the porous medium approximation.   6. Conclusions and future work  In summary, this work introduces design principles for sensible heat solid thermal storage systems to improve their (dis)charge performance. We consider the channel-embedded configuration, which features channels of heat transfer fluid embedded within the solid storage medium. Typically, these systems may have poor performance, facing challenges in output power uniformity during discharge and slow charging rates. Further, if the sensible heating occurs at high temperatures as required for high round-trip efficiency, radiation between blocks can degrade performance. This suggests design optimization is necessary. However, grid-scale versions of this system are computationally intractable to model fully. Therefore, in this work we have utilized a hierarchical approach to design optimization, developing design principles for channel-embedded thermal storage systems at each step of the hierarchy. We then apply these 24  design principles to a specific system that uses graphite as the storage medium and liquid tin as the heat transfer fluid (called thermal energy grid storage or TEGS). First, we investigate the impact of geometric parameters on performance using dimensional analysis. We define an FOM based on temperature uniformity with time and derive the non-dimensional groups that impact this FOM for a general system. To demonstrate how this analysis can be used for a specific system, we consider the TEGS system, which allows us to simplify some terms and develop a relationship between storage medium diameter, heat transfer fluid flow path length, and FOM. Specifically, for a storage medium of thermal conductivity 10 W/m/K, we find a diameter of 20cm and length of 10m achieves a temperature FOM of 90% for a discharge duration of 30 hours. While we have generated these numbers for TEGS specifically, the dimensional analysis is generalizable to other channel-embedded thermal energy storage systems using different materials and storage durations. Next, we investigate how varying the operation of the storage system can help optimize discharging and charging performance. For discharge, we define an FOM based on the time the storage system can provide the nominal power compared to the rated storage duration. We find that increasing HTF flowrate as the outlet temperature drops enables near-constant power output, but requires throttling of the heat engine. Using the TEGS system as an example, we find that allowing flowrate to increase by 5x and TPV area to increase 50% allows >90% discharge FOM. For charging, we define an FOM based on the fraction of storage capacity able to be charged in a certain charging duration, i.e. the state of charge after the charging duration. We find that we can apply the same methodology from discharging of ramping up flowrate as temperature difference drops, to keep input power constant at its maximum value. For TEGS, allowing flowrate to ramp up by 5x allows 90% charging within 5 hours. Finally, we model the full grid-scale system and propose a porous media approximation that is calibrated to match the heat transfer and fluid flow properties of the fully coupled case to accelerate computation. We apply this to TEGS to find and consider 100 1m x 1m x 4m commercially available graphite blocks (corresponding to a thermal energy capacity of 200MWh) arranged in various configurations to determine the effect of radiation between the blocks. We first find that blocks should be connected in series to achieve the minimum tin flow path as determined by dimensional analysis (at least 10m). We then find that the 10x10 grid arrangement of blocks (minimizing insulation costs) underperforms compared to arrangements featuring blocks that are thermally isolated from each other. By utilizing the insights generated in previous parts, this thermal performance can be improved to near its maximum, and we demonstrate discharging and charging FOMs of >90%. In this work we have primarily studied heat transfer considerations and have briefly discussed their impact on cost where appropriate. A more rigorous technoeconomic analysis evaluating each proposed improvement to performance is warranted and should be the focus of a future study. Further, while we have considered discharging and charging in isolation, it is also important to consider the transition between the two. While a detailed analysis is outside the scope of this work, many results in the literature suggest that flow should be reversed between discharge and charge, to ensure thermoclines developed in the storage media are maintained.32–34 Longer-duration studies where the system is charged and discharge multiple times should be the subject of a future work. 25  Further, it would be useful to model the grid integration of such a thermal energy storage system with realistic power input/output profiles. This would involve further modeling and approximations to the large-scale model to enable longer duration simulations. Overall, this work develops design principles to improve the performance of cheap thermal energy grid storage systems, which in turn facilitates its adoption and helps increase renewables penetration for a cleaner future.  7. Data availability   Data and code are available on GitHub.35  8. Acknowledgements  This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Award Nos. 1745302 and 2141064.  9. Author contributions  Conceptualization, S.V., C.K., A.H.; Methodology, S.V., C.K., K.B., A.L., A.S., A.H.; Investigation, S.V.; Writing – Original Draft, S.V.; Writing – Review and Editing, S.V., K.B., A.L., A.S., A.H.; Supervision, A.H.  10. Declaration of interests  A.H. has founded a company for thermal energy storage based on the TEGS system discussed in this work, and therefore has a financial interest.    26  11. References  (1) Dowling, J. A.; Rinaldi, K. Z.; Ruggles, T. H.; Davis, S. J.; Yuan, M.; Tong, F.; Lewis, N. S.; Caldeira, K. Role of Long-Duration Energy Storage in Variable Renewable Electricity Systems. Joule 2020, 4 (9), 1907–1928. https://doi.org/10.1016/j.joule.2020.07.007. (2) Ziegler, M. S.; Mueller, J. M.; Pereira, G. D.; Song, J.; Ferrara, M.; Chiang, Y.-M.; Trancik, J. E. Storage Requirements and Costs of Shaping Renewable Energy Toward Grid Decarbonization. Joule 2019, 3 (9), 2134–2153. https://doi.org/10.1016/j.joule.2019.06.012. (3) Glatzmaier, G. Developing a Cost Model and Methodology to Estimate Capital Costs for Thermal Energy Storage; NREL/TP-5500-53066; National Renewable Energy Lab. (NREL), Golden, CO (United States), 2011. https://doi.org/10.2172/1031953. (4) Laughlin, R. B. Pumped Thermal Grid Storage with Heat Exchange. J. Renew. Sustain. Energy 2017, 9 (4), 044103. https://doi.org/10.1063/1.4994054. (5) Armstrong, R.; Chiang, Y.-M.; Gruenspecht, H. The Future of Energy Storage: Chapter 4 - Thermal Energy Storage; MIT Future of; Massachusetts Institute of Technology, 2022; pp 113–146. https://energy.mit.edu/wp-content/uploads/2022/05/The-Future-of-Energy-Storage.pdf. (6) Bauer, T. Chapter 1 - Fundamentals of High-Temperature Thermal Energy Storage, Transfer, and Conversion. In Ultra-High Temperature Thermal Energy Storage, Transfer and Conversion; Datas, A., Ed.; Woodhead Publishing Series in Energy; Woodhead Publishing, 2021; pp 1–34. https://doi.org/10.1016/B978-0-12-819955-8.00001-6. (7) Amy, C.; Seyf, H. R.; Steiner, M. A.; Friedman, D. J.; Henry, A. Thermal Energy Grid Storage Using Multi-Junction Photovoltaics. Energy Environ. Sci. 2019, 12 (1), 334–343. https://doi.org/10.1039/C8EE02341G. (8) Herrmann, U.; Kelly, B.; Price, H. Two-Tank Molten Salt Storage for Parabolic Trough Solar Power Plants. Energy 2004, 29 (5), 883–893. https://doi.org/10.1016/S0360-5442(03)00193-2. (9) Alva, G.; Lin, Y.; Fang, G. An Overview of Thermal Energy Storage Systems. Energy 2018, 144, 341–378. https://doi.org/10.1016/j.energy.2017.12.037. (10) Xie, B.; Baudin, N.; Soto, J.; Fan, Y.; Luo, L. Chapter 10 - Thermocline Packed Bed Thermal Energy Storage System: A Review. In Renewable Energy Production and Distribution; Jeguirim, M., Ed.; Advances in Renewable Energy Technologies; Academic Press, 2022; Vol. 1, pp 325–385. https://doi.org/10.1016/B978-0-323-91892-3.24001-6. (11) ELSihy, Els. S.; Liao, Z.; Xu, C.; Du, X. Dynamic Characteristics of Solid Packed-Bed Thermocline Tank Using Molten-Salt as a Heat Transfer Fluid. Int. J. Heat Mass Transf. 2021, 165, 120677. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120677. (12) Rodrigues, F. A.; de Lemos, M. J. S. Discharge Effectiveness of Thermal Energy Storage Systems. Appl. Therm. Eng. 2022, 209, 118232. https://doi.org/10.1016/j.applthermaleng.2022.118232. (13) Rodrigues, F. A.; de Lemos, M. J. S. Effect of Porous Material Properties on Thermal Efficiencies of a Thermocline Storage Tank. Appl. Therm. Eng. 2020, 173, 115194. https://doi.org/10.1016/j.applthermaleng.2020.115194. (14) Marti, J.; Geissbühler, L.; Becattini, V.; Haselbacher, A.; Steinfeld, A. Constrained Multi-Objective Optimization of Thermocline Packed-Bed Thermal-Energy Storage. Appl. Energy 2018, 216, 694–708. https://doi.org/10.1016/j.apenergy.2017.12.072. (15) Esence, T.; Bruch, A.; Molina, S.; Stutz, B.; Fourmigué, J.-F. A Review on Experience Feedback and Numerical Modeling of Packed-Bed Thermal Energy Storage Systems. Sol. Energy 2017, 153, 628–654. https://doi.org/10.1016/j.solener.2017.03.032. (16) Wu, M.; Li, M.; Xu, C.; He, Y.; Tao, W. The Impact of Concrete Structure on the Thermal Performance of the Dual-Media Thermocline Thermal Storage Tank Using Concrete as the Solid Medium. Appl. Energy 2014, 113, 1363–1371. https://doi.org/10.1016/j.apenergy.2013.08.044. (17) Jian, Y.; Falcoz, Q.; Neveu, P.; Bai, F.; Wang, Y.; Wang, Z. Design and Optimization of Solid Thermal Energy Storage Modules for Solar Thermal Power Plant Applications. Appl. Energy 2015, 139, 30–42. https://doi.org/10.1016/j.apenergy.2014.11.019. (18) Jian, Y.; Bai, F.; Falcoz, Q.; Xu, C.; Wang, Y.; Wang, Z. Thermal Analysis and Design of Solid Energy Storage Systems Using a Modified Lumped Capacitance Method. Appl. Therm. Eng. 2015, 75, 213–223. https://doi.org/10.1016/j.applthermaleng.2014.10.010. (19) Doretti, L.; Martelletto, F.; Mancin, S. A Simplified Analytical Approach for Concrete Sensible Thermal Energy Storages Simulation. J. Energy Storage 2019, 22, 68–79. https://doi.org/10.1016/j.est.2019.01.029. (20) Prasad, L.; Niyas, H.; Muthukumar, P. Performance Analysis of High Temperature Sensible Heat Storage System during Charging and Discharging Cycles; ndian Institute of Technology Bombay, Mumb, 2013; pp 1240–1247. (21) Doretti, L.; Martelletto, F.; Mancin, S. Numerical Analyses of Concrete Thermal Energy Storage Systems: Effect of the Modules’ Arrangement. Energy Rep. 2020, 6, 199–214. https://doi.org/10.1016/j.egyr.2020.07.002. (22) Kuang, R.; Huang, N.; Chen, G.; Tan, J.; Liu, J.; Shen, Y. Numerical Analysis of Discharging Stability of Basalt Fiber Bundle Thermal Energy Storage Tank. Energy Rep. 2022, 8, 13014–13022. https://doi.org/10.1016/j.egyr.2022.09.115. (23) LaPotin, A.; Schulte, K. L.; Steiner, M. A.; Buznitsky, K.; Kelsall, C. C.; Friedman, D. J.; Tervo, E. J.; France, R. M.; Young, M. R.; Rohskopf, A.; Verma, S.; Wang, E. N.; Henry, A. Thermophotovoltaic Efficiency of 40%. Nature 2022, 604 (7905), 287–291. https://doi.org/10.1038/s41586-022-04473-y. (24) Verma, S.; Adams, M.; Foxen, M.; Sperry, B.; Yee, S.; Henry, A. High-Temperature Thermal Conductivity Measurements of Macro-Porous Graphite. arXiv February 2, 2023. https://doi.org/10.48550/arXiv.2301.03440. (25) Kelsall, C. C.; Buznitsky, K.; Henry, A. Technoeconomic Analysis of Thermal Energy Grid Storage Using Graphite and Tin. ArXiv210607624 Phys. 2021. (26) Sepulveda, N. A.; Jenkins, J. D.; Edington, A.; Mallapragada, D. S.; Lester, R. K. The Design Space for Long-Duration Energy Storage in Decarbonized Power Systems. Nat. Energy 2021, 6 (5), 506–516. https://doi.org/10.1038/s41560-021-00796-8. (27) Eikeland, O. F. Enhancing Decision-Making in the Electric Power Sector with Machine Learning and Optimization. Doctoral thesis, UiT Norges arktiske universitet, 2023. https://munin.uit.no/handle/10037/31514 (accessed 2024-01-31). (28) Zhu, Q.; Pishahang, M.; Caccia, M.; Kelsall, C. C.; LaPotin, A.; Sandhage, K. H.; Henry, A. Validation of the Porous Medium Approximation for Hydrodynamics Analysis in Compact Heat Exchangers. J. Fluids Eng. 2022, 144 (8). https://doi.org/10.1115/1.4053898. 27  (29) Xu, B.; Li, P.-W.; Chan, C. L. Extending the Validity of Lumped Capacitance Method for Large Biot Number in Thermal Storage Application. Sol. Energy 2012, 86 (6), 1709–1724. https://doi.org/10.1016/j.solener.2012.03.016. (30) Shu, S.; Wu, T.; Wang, Z.; Zhang, Y.; Yang, Z.; Liang, H. Research on the Normal Emissivity of Graphite between 150 and 500 °C by an Infrared Camera for Nuclear Fusion Devices. Nucl. Mater. Energy 2022, 31, 101182. https://doi.org/10.1016/j.nme.2022.101182. (31) Ren, D.; Tan, H.; Xuan, Y.; Han, Y.; Li, Q. Apparatus for Measuring Spectral Emissivity of Solid Materials at Elevated Temperatures. Int. J. Thermophys. 2016, 37 (5), 51. https://doi.org/10.1007/s10765-016-2058-9. (32) Thermal Storage System Concentrating Solar-Thermal Power Basics. Energy.gov. https://www.energy.gov/eere/solar/thermal-storage-system-concentrating-solar-thermal-power-basics (accessed 2024-02-02). (33) Dickinson, J. S.; Buik, N.; Matthews, M. C.; Snijders, A. Aquifer Thermal Energy Storage: Theoretical and Operational Analysis. Géotechnique 2009, 59 (3), 249–260. https://doi.org/10.1680/geot.2009.59.3.249. (34) Riahi, S.; Saman, W. Y.; Bruno, F.; Belusko, M.; Tay, N. H. S. Impact of Periodic Flow Reversal of Heat Transfer Fluid on the Melting and Solidification Processes in a Latent Heat Shell and Tube Storage System. Appl. Energy 2017, 191, 276–286. https://doi.org/10.1016/j.apenergy.2017.01.091. (35) shomikverma/graphite-discharge: code for graphite discharge modeling project. GitHub. https://github.com/shomikverma/graphite-discharge (accessed 2023-07-26). (36) CRC Handbook of Chemistry and Physics, 100th Edition, 100th edition.; Rumble, J., Ed.; CRC Press: Boca Raton London New York, 2019. (37) Informatics, N. O. of D. and. NIST Chemistry WebBook. https://webbook.nist.gov/chemistry/ (accessed 2024-02-10). (38) Taler, D. Heat Transfer in Turbulent Tube Flow of Liquid Metals. Procedia Eng. 2016, 157, 148–157. https://doi.org/10.1016/j.proeng.2016.08.350. (39) Brockmann, H. Analytic Angle Factors for the Radiant Interchange among the Surface Elements of Two Concentric Cylinders. Int. J. Heat Mass Transf. 1994, 37 (7), 1095–1100. https://doi.org/10.1016/0017-9310(94)90195-3.     1  Supplementary Information for “Designing for effective heat transfer in a solid thermal energy storage system”  Shomik Verma, Colin Kelsall, Kyle Buznitsky, Alina LaPotin, Ashwin Sandeep, Asegun Henry*   Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. * ase@mit.edu  Table of Contents S1\tThermal\tenergy\tstorage\tmaterials\tdata\t..............................................................................\t2\tS2\tDe-rating\tthe\tstorage\tduration\t..............................................................................................\t2\tS3\tDimensional\tanalysis\t.................................................................................................................\t4\tS4\tIdentifying\toptimal\tgeometry\tfor\talternative\tconfiguration\t.......................................\t6\tS5\tChecking\tvalidity\tof\tlaminar\tflow\tapproximation\t...........................................................\t6\tS6\tChecking\tvalidity\tof\tporous\tmedia\tapproximation\t.........................................................\t7\tS7\tEvaluating\toptions\tfor\taccelerated\tcharging\t.....................................................................\t8\tS8\tGIFs\tof\ttemperature\tprofile\tvs.\ttime\tfor\tfull-scale\tsimulation\t..................................\t14\tS9\tCharge/discharge\tdynamics\t.................................................................................................\t14\t   2   S1. Thermal energy storage materials data  \n 1 https://www.statista.com/statistics/219339/us-prices-of-cement/ 2 https://www.forbes.com/home-improvement/foundation/concrete-slab-cost/ 3 https://www.statista.com/statistics/219381/sand-and-gravel-prices-in-the-us/#:~:text=In%20the%20United%20States%2C%20the,per%20metric%20ton%20in%202022. 4 https://tradingeconomics.com/commodity/iron-ore 5 https://medium.com/@keruirefractory/exploring-how-much-do-fire-bricks-cost-a93a4530dffd#:~:text=Based%20on%20the%20above%20factors%20and%20other,can%20cost%20between%20$5%20and%20$15%20each. 6 https://www.northerngraphite.com/about-graphite/graphite-pricing/ 7 https://www.sciencedirect.com/science/article/abs/pii/S2352152X17302864?via%3Dihub 8 https://bulknaturaloils.com/mineral-oil-usp-90-c1016.html 9 https://www.lme.com/en/metals/non-ferrous/lme-aluminium#Summary 10 https://tradingeconomics.com/commodity/magnesium 11 https://price.metal.com/Silicon 12 https://www.nasdaq.com/market-activity/commodities/hg:cmx 13 https://www.lme.com/en/Metals/Ferrous/LME-Steel-Scrap-CFR-Turkey-Platts 14 https://ycharts.com/indicators/nickel_price#:~:text=Nickel%20Price%20is%20at%20a,42.88%25%20from%20one%20year%20ago. 15 https://ycharts.com/indicators/tin_price 16 https://www.dailymetalprice.com/metalpricecharts.php?c=ga&u=kg&d=240 17 https://www.metal.com/search?keyword=calcium&type=price 18 https://www.metal.com/search?keyword=sodium&type=price 19 https://www.metal.com/search?keyword=potassium&type=price 20 https://www.chemanalyst.com/Pricing-data/quicklime-1505 21 https://businessanalytiq.com/procurementanalytics/index/silica-price-index/  Material data was obtained from the CRC handbook36 and NIST.37 Solid heat capacity was calculated based on solid-phase heat capacity (assuming a constant 𝑐\" and a temperature difference of 500°C. Same for liquid phase heat capacity. The phase was determined based on which phase is able to store the most amount of heat, using 500°C for sensible heating, or the latent heat. Many of the natural materials would decompose before melting or shortly after melting, so these were considered to remain solid.  S2. De-rating the storage duration  We could design a system with a nominal power output 𝑃%&' for 𝜏 hours, but only discharge it for 𝜏4&-61 hours, where 𝜏4&-61 is defined as  𝜏4&-61=𝑓4&-61𝜏 MaterialMelting TBoiling TCp solidMelting latent heat (kJ/kg)Cp liquidsolid heat capacityPCM heat capacityliquid heat capacitysolid densityliquid densityCostheat capacity valuephasecostenergy densityref for cost(°C)(°C)(kJ/kgK)(kJ/kgK)kJ/kgkJ/kgkJ/kgkg/m3kg/m3$/m3kJ/kg$/kWh-tkWh-t/m3Cement (composite)155029000.7375745.751.15368.75745.755751440187.2368.75solid1.26915153147.5001181 Concrete (composite)120029000.705238.11.28352.5238.16402400215.827338352.5solid0.91841347235.0001882 Sand (SiO2)170029000.71601.3350160650152016.72350solid0.11314277147.7778963 Iron ore (Fe3O4)160026000.675960.93355964503000360335solid1.28955121279.166894 Fire bricks (alumina)200030000.810001.9400100095028002520.09016400solid8.10028332311.111365 Graphite400040001.575021001575750solid3.59999712437.500356 Salt (NaCl)2048501.11781.4196.9219700216021002160700liquid5.14285303408.333667 Mineral oil-43001.67507.688302905507.68liquid24.8187636117.0485388 Al65822000.913961.18455396590270023755400590liquid12.2033801389.2364239 Mg65711101.053581.34525358607.02173815845040.2607.02liquid17.1987606267.08901410 Si142026000.7117870.913551787455232925704890.91787PCM4.230550621275.7204711 Cu108325000.392060.491952062458960802073472245liquid120.4897545.80599212 Fe120025000.462470.82230247410787469803937410liquid4.39024039794.9450813 Ni145227300.442930.7322029336589007810142400365liquid157.808093791.84785614 Sn23225000.21590.2443.475912072656990181625120liquid749.9994233.00018615 Ga29.824030.37800.391.77680195590760801653960195liquid5169.22663329.33359716 Ca85014390.632130.875315213437.5155013789300437.5liquid49.3713891167.46541217 Na97.58801.211131.2687.7251136309709272522630liquid14.857131162.2251318 K637620.75560.76228.55638186282819826381liquid217.32266187.630070119 CaO260029000.759411.13759413303340668375solid1.91999846347.91694520 MgO280036000.94503600540450solid1.19999904450.00036213   Where 𝑓4&-61∈(0,1].\tWe can quantitatively evaluate this improvement with a figure of merit (FOM) defined as 𝐹𝑂𝑀$=∫Θ(𝑡)*.\"/01:0𝑑𝑡∫𝑑𝑡*.\"/01:0=∫Θ∗\t*.\"/010(𝑡∗)𝑑𝑡∗∫𝑑𝑡∗*.\"/010\t≤1\t where Θ(𝑡) is the non-dimensionalized (dis)charge temperature as a function of time, and Θ∗(𝑡∗) is the non-dimensionalized (dis)charge temperature as a function of non-dimensionalized time (as shown in Figure 2), such that Θ?∗(𝑡∗)=Θ?(𝑡∗⋅𝜏)=Θ?(𝑡). We now see some benefits of de-rating – for example, if the storage system in Figure 2 with 𝑘=5 was only rated for 𝑓4&-61=0.25 instead of 1, 𝐹𝑂𝑀$ would be approximately 1. However, if we compare several cases where 𝜏4&-61 is constant, as we decrease 𝑓4&-61, 𝜏 must increase. In the example above, 𝜏 must be 4 times larger, which means 4 times as many graphite blocks would be required, directly increasing costs. This is shown in Figure S1, where (a) demonstrates the increase in FOM enabled by de-rating (increasing H while keeping U constant), while (b) shows another FOM based on cost (higher H for same U results in higher costs and lower FOM. When we combine these two FOMs, we notice that an 𝑓4&-61 value of 1 (dashed diagonal line) maximizes the combined FOM. The other downside of oversizing the system is that the charging efficacy is lowered. Because the system temperature is always high, the heat transfer into the block during charging is limited due to a smaller Δ𝑇. Therefore the system is unable to provide as much charging power, and the charging duration is lengthened. We are therefore interested in determining how performance can be maximized based on changes to geometry, operation, and block arrangement in the storage system, while keeping 𝑓4&-61=1. \n Figure S1: FOM of derating discharge  \n4  S3. Dimensional analysis  From the main text, we know that  𝐹𝑂𝑀$=𝑓g𝐿𝐷,𝑈𝐷𝛼J,𝑘9)𝑘Jh\t We first verify that these are the correct groups, as presented in Figure S2. We assume 𝜌2,𝑐\",J,\tand 𝑘9) are constant, so the 3 groups simplify to @;, A;?2, and /?2. Changing the values of 𝐿,𝐷,𝑈,𝑘J but keeping the dimensional groups the same resulted in the same tin outlet temperature profile.  \n Figure S2: validity of dimensional analysis (H=L)  \n5   Figure S3: Results from dimensional analysis showing the complete parameter space of the channel-embedded thermal energy storage system. (a) Sweeps of the parameter space for !\" vs. #\"2 for various values of 𝑘. (b) A single slice of the 3D parameter space, for 𝑘=10\t𝑊/𝑚𝐾, demonstrating how the parameter space could be sampled for constant channel spacing (i.e. graphite diameter, solid lines) or constant channel length (dashed lines). (c) Results of sampling the parameter space given D=0.2m and varying the channel length L, and (d) sampling the parameter space given L=10m and varying the channel spacing D, for different storage durations. For (c) and (d), kernel ridge regression interpolation was used to obtain finer-grain results.  Now that we have verified the dimensional analysis, we can map the design space by sweeping over the grouped parameters. Figure S3(a) shows 𝐹𝑂𝑀$ vs @; and A;? for various values of 𝑘 (reproduced from main text). We next conduct a deeper analysis of the 𝑘=10 case. We probe the design space for constant diameter 𝐷 (solid lines) or constant flow length 𝐿 (dashed lines), for a given storage duration 𝜏, as shown in Figure S3(b). To see how 𝜏 defined in Equation 2 translates to a velocity, we can re-write it as   𝜏=𝜌J𝑉J𝑐\"J𝑚̇9)𝑐\"9)=𝜌J𝐿𝜋(𝐷<−𝐷9)<)𝑐\"J𝜌9)𝑈𝜋𝐷9)<𝑐\"9)∝𝐿𝑈\t \n6  Then, we can plot how 𝐹𝑂𝑀$ varies as these parameters change, as shown in Figure S3(c) and (d). For example, Figure S3(d) shows that for a given tin flow path length of 10m, decreasing the diameter significantly improves 𝐹𝑂𝑀$.  S4. Identifying optimal geometry for alternative configuration  In the main text we set the storage duration 𝜏 to 30 hours and the graphite thermal conductivity to 10 W/m/K to find the ideal diameter of the channel-embedded cylinder and tin flow path length. If instead we were interested in a longer storage duration of 100 hours and an alternative material with a thermal conductivity of 1 W/m/K, the optimization would be different, as shown below. \n Figure S4:  S5. Checking validity of laminar flow approximation  In certain flow cases, the Reynolds number of the flow may be higher than 4000. We are interested in determining if this impacts the heat transfer. We first calculate the Reynolds number for liquid tin flowing through a 2cm pipe at 1m/s: 𝑅𝑒=𝜌𝑉𝐷𝜇=6184𝑘𝑔𝑚R⋅1.32𝑚𝑠⋅0.02[𝑚]0.0011𝑘𝑔𝑚⋅𝑠≈150,000\tWe then calculate a Prandtl and Peclet number as  \n7  𝑃𝑟=𝜈𝛼=𝑐\"𝜇𝑘=244⋅0.001160=0.0044 𝑃𝑒=𝑅𝑒𝑃𝑟=663 Using correlations from Taler,38 we find that  𝑁𝑢=5+0.025𝑃𝑒0.T=9.5 (const surf temp) 𝑁𝑢=4.82+0.0185𝑃𝑒0.T<U=8.8 (const heat flux) Which suggests that the convective heat transfer coefficient in turbulent flow is approximately twice that of laminar flow. Although this then suggests that the convective contribution to the heat transfer resistance in turbulent flow is halved – however, the dominant resistance is still conduction through the graphite block, which is approximately 2 orders of magnitude greater. To prove this, we calculate the discharge profiles for both laminar and turbulent flow, for various discharge durations. As shown in Figure S5, these two flow conditions have very similar heat transfer performance.  \n Figure S5: validity of laminar approximation  𝐹𝑂𝑀7,23&456=∫𝑚̇(𝑡)𝑐\",9)Δ𝑇(𝑡):0𝑑𝑡𝐸=∫𝑚̇(𝑡)𝑐\",9)𝛩(𝑡):0𝑑𝑡𝐸/Δ𝑇%&'=∫𝑚̇∗(𝑡∗)𝑐\",9)𝛩∗(𝑡∗)/0𝜏𝑑𝑡∗𝐸/Δ𝑇%&'  ∫𝑚̇∗(𝑡∗)𝑐\",9)𝛩∗(𝑡∗)/0𝑑𝑡∗𝐸/𝜏/Δ𝑇%&'=∫𝑚̇∗(𝑡∗)𝑐\",9)𝛩∗(𝑡∗)/0𝑑𝑡∗𝑃/Δ𝑇%&'=∫𝑚̇∗(𝑡∗)𝑐\",9)𝛩∗(𝑡∗)\t𝑑𝑡∗/0𝑚̇)+%𝑐\",'\t where 𝑚̇)+%=7!\"#2*,'(M00°J)=O/:2*,'(M00°J) and 𝑚̇∗(𝑡∗) is flowrate as a function of non-dimensional time, such that 𝑚̇∗(𝑡∗)=𝑚̇(𝑡).  S6. Checking validity of porous media approximation  \n8   Figure S6: validity of porous media approximation S7. Evaluating options for accelerated charging  From our dimensional analysis, we found that the limiting heat transfer resistance is conduction through the storage material, especially if its thermal conductivity is low. Therefore, we focus on improving heat conduction. Given that conduction heat transfer can be approximated by 𝑞\"=?V$@, we could either reduce the conduction length, or increase the Δ𝑇.  We consider 4 additional options for maximizing 𝐹𝑂𝑀7,23&456. The first two options rely on reducing conduction length, while the last two rely on increasing the Δ𝑇. For option 1, we could reducing the conduction distance in the storage material, either by (a) scaling down both the storage block diameter and the HTF channel diameter, or (b) keeping the HTF channel diameter constant while reducing the storage block diameter, For option 2, we similarly reduce the conduction distance in the storage material, but this time by relying on radiation to transfer heat between the HTF channel and a cylindrical shell of storage material (with the same volume as the original cylindrical block). For option 3, we could increase the inlet temperature of the HTF above the highest operating temperature of the storage material, to accelerate the heat conduction into the storage material by increasing the temperature difference. For option 4, like the methodology presented in the main text, we could increase the flowrate, but in contrast, always have the flowrate at a high value. This increases the power output throughout the charging process, so requires additional heating power, but may be more effective. We analyze all 4 options in the context of the TEGS system.  No new methodology is introduced for option (1) or (4). For option (2), we note that the radiative resistance between the tin pipe and the graphite block is potentially much smaller than the conduction resistance through the graphite block, due to the high temperatures of the system. By adding a gap between the tin tube and the cylindrical graphite shell, we can reduce the thickness of the shell while keeping the volume of graphite constant, reducing conduction resistance. For option (3), we keep the amount of thermal energy in the blocks the same, but increase the maximum tin inlet temperature, \n9  which allows greater charging power from Equation 1. If the amount of energy input exceeds the energy capacity of the blocks, the charging is considered complete and input power is set to 0.   To summarize, option 1 involves resizing the storage system, option 2 involves adding a spacing to exploit low radiative resistance, option 3 involves supercharging the tin inlet temperature above 2400ºC, and option 4 involves increasing the charging power. For option 1, we re-do the geometry optimization from Section 4.1 given shorter storage durations, namely 5 and 3 hours, as shown in Figure S7. \n Figure S7: Geometric parameter space for storage durations of (a) 5 hours and (b) 3 hours, similar Figure 4.  As shown, to achieve 90% 𝐹𝑂𝑀$, this would require a block diameter of ~8cm for 5 hours and ~6cm for 3 hours. These correspond to tin tube diameters of 8mm and 6mm, respectively. This would drastically increase the pressure drop, increase the velocity corresponding to an increased likelihood of cavitation, and potentially introduce surface tension effects at these small diameters. Therefore, this resizing is impractical to implement.  Detailed analysis of option 1(b) is not presented here as it would require a separate dimensional analysis since the fraction of ;,';2 is not assumed to be constant. Generally, from the analysis above, it is expected that increasing the channel diameter without increasing the graphite diameter would improve FOM. However, it would increase cost as more tin would be required per volume of graphite, and it would reduce energy density, as more overall volume would be required to have the same volume of graphite. Therefore, this option is not practical. The second option for fast charging is to add a spacing between the tin tube and graphite block and exploit the low radiative resistance at these high temperatures. The radiative resistance and conduction resistance within the graphite block can be written as  𝑅4&1=1𝜖𝜎(𝑇/<+𝑇<<)(𝑇/+𝑇<)2𝜋𝑟\"𝐿𝐹/<\t𝑅2+)1=𝑙𝑛\tw𝑟+𝑟(x\t2𝜋𝐿𝑘J\t\n10  Note that the convective resistance of the tin flow and the conductive resistance of the tin pipe are neglected here. The parameters of the optimized geometry are 𝑟\"=𝑟(=0.01𝑚, 𝑟+=0.2𝑚, and 𝐿=10𝑚. We can investigate the impact of adding a gap between the tin flow pipe and the graphite block, such that 𝑟(>𝑟\", by calculating the resistances. Note that 𝑟+ is adjusted as 𝑟( is increased to keep the graphite block volume 𝑉=𝜋(𝑟+<−𝑟(<)𝐿 constant, such that the thickness 𝑟+−𝑟( decreases for larger 𝑟(, as shown in Figure S8(a). 𝐹/< is calculated using the equal finite concentric cylinders formula found in Brockmann.39 We assume 𝑘J=10W%X,𝑇/=2400°𝐶,\tand 𝑇<=1900°𝐶. Figure S8(b) shows the radiative resistance, conductive resistance, and total resistance when sweeping over different values of gap size. As seen, adding a small gap increases the total resistance, but this effect is mitigated after a gap of ~3mm. Adding a larger gap reduces the total resistance, primarily due to the lower conductive resistance created since 4+4& is lower at high 𝑟(. Adding a gap of 5cm reduces the overall resistance by approximately half. While the resistance analysis is a useful guide, it assumes steady state. To investigate the transient performance, we model the full 3D geometry with surface-to-surface radiation and various gap sizes. These results are shown in Figure S8(c). As seen, the peak performance occurs at around 4cm gap. Below this value, the conduction resistance is too high, and above this value, too much heat leaks out of the top and bottom of the graphite block. Note that this effect could be mitigated by adding a radiation shield to reflect the lost heat. The tradeoff with adding this gap is that the storage system takes up more volume, which reduces the energy/power density and increases the amount of insulation required. Figure S8(d) quantifies this effect, showing the percent increase in volume and surface area above the nominal for various gap sizes. As seen, the slight benefit to adding a radiative gap is outweighed by the large increase in surface area. 11   Figure S8: (a) Analysis of contributions to heat transfer resistance, considering both radiation between the tin channels and the graphite block and the conduction through the graphite block. (b) The effect of the size of the gap between the tin channel and the graphite block on the charging FOM. (c) The tradeoff of increasing the gap size, namely the increase in volume and outer surface area of the storage section and therefore increase in insulation costs.  The third option is to supercharge the inlet tin to above 2400ºC. This increases the temperature difference between tin and graphite and therefore increases the heat flux. The charging process was simulated for various tin inlet temperatures, as shown in Figure S9.  \n12   Figure S9: (a) Effect on charging FOM of superheating the tin above 2400ºC during the charging process. Then, for an added temperature of 200K, (b) outlet vs. inlet temperature of tin to the storage media, showing charging completed after 4.25 hours indicated by inlet temperature dropping to outlet temperature, (c) lowest temperature of the storage medium as a proxy of charging completeness, and (d) power input as a function of time, with the average power required to fully charge shown in green and the actual power input in blue, and the corresponding charging FOM shown.  In this case, we keep the thermal energy capacity of the blocks constant, at 𝐸=𝑚𝑐\",J(500°𝐶), but allow the input power to increase to 𝑃=𝑚̇𝑐\",9)(500°𝐶+𝑇&1161). As seen in Figure S9(a), setting 𝑇&1161 to be 200°𝐶 allows 𝐹𝑂𝑀7,23&456 to reach ~1. Specifically, as seen in Figure S9(b), the block is fully charged in around 4.25 hours, after which the tin inlet temperature is dropped to the outlet temperature to prevent further charging. In this case “fully charged” means the average block temperature is 2400ºC, however after 4.25 hours there is still temperature variation in the block, as seen in Figure S9(c), which shows the temperature at the bottom right of the block, which is the lowest temperature. Therefore, additional time is required to reach thermal equilibrium at 2400ºC. Another potential problem with this approach is the high vapor pressure of tin at temperatures above 2400ºC and boiling of tin above 2600ºC. This could lead to loss of tin in the flow paths and condensation in other areas. It could also lead to cavitation at high velocities. The fourth option is to increase the flowrate and therefore charging power through the entire charging process. We evaluate the effectiveness of this methodology in the context of the TEGS system. For this analysis, we consider a graphite block of the same dimensions as the previous subsection (length 𝐿=10𝑚, diameter 𝐷J=0.2𝑚, tin channel diameter 𝐷9)=0.02𝑚, determined from dimensional analysis). For the accelerated \n13  charging analysis, we first consider a short charging duration of 𝜏=5 hours, which corresponds to, for example, the amount of time abundant solar might be available during the day. The nominal flowrate for such a system would be 𝑚̇)+%=O/M32*,,'(M00°Q). To accelerate charging, we define a new maximum flowrate 𝑚̇%&' and a flowrate multiplier 𝑓=%̇!\"#%̇'+!. In contrast to the constant power discharging technique, instead of ramping up the flowrate with time, the flowrate is kept at the maximum value 𝑚̇%&' to maximize power input. The results for the given system, with 5 hours of charging duration and various flowrate multipliers, are shown in Figure S10.  \n Figure S10: (a) Increase in 𝐹𝑂𝑀(,,-./01 as the flowrate increases, for a charging duration of 5 hours. Then, considering a flowrate multiplier of 10, (b) the lowest temperature of the storage medium as a function of time, as a proxy for charging completeness, (c) mass flowrate of tin as a function of time, showing it is constant at 10x the nominal, (d) power input as a function of time, where the green line is the average power input required to fully charge the system, and the blue line is the actual power input. (e) Conducting the analysis for various other charging durations and flowrate factors, with the solid black line indicating an isocontour of FOM=0.9 and the blue dashed line indicating a charging duration of 5 hours. (f) Increase in charging power associated with each flowrate multiplier, where the % increase is relative to the f=1 case for each duration. This increase in charging power will increase the charging infrastructure cost. As seen in Figure S10(a), increasing the flowrate from 1x to 10x the nominal improves 𝐹𝑂𝑀7,23&456\tfrom 75% to 98%. The specific response of the thermal storage system to 10x the nominal flowrate is shown in Figure S10(b)-(d). Figure S10(b) shows the temperature of the bottom of the graphite block (near the tin outlet), as an indication of the completeness of charging since the bottom charges the slowest. As seen, within 5 hours the temperature almost approaches the maximum value of 2400ºC. However, as seen in Figure S10(d), this requires a higher input power at the beginning, increasing the cost of heating infrastructure, specifically by a factor of 10 (from 30 to 300 kW).  Next, we consider various charging durations from 1 to 10 hours, as shown in Figure S10(e). Similar to discharge power uniformity, the accelerated charging is easier at longer durations since their charging FOMs are already high even with no modifications \n14  (at 𝑓=1). We note that for all durations >3 hours, we are able to nearly fully charge the system (>90%) with flowrate multipliers of 5 or lower. As mentioned previously, the increase in performance comes at a cost of increasing charging power, as shown in Figure S10(f). This may not be significant if the charging  infrastructure is cheap. While increasing HTF flowrate to establish a near-constant HTF temperature within the channel is effective, it increases pressure drop and requires additional pumps and heating power, so there is again a tradeoff between performance and cost.  While the options all enable increases in 𝐹𝑂𝑀7,23&456, they were deemed impractical for various reasons, which are summarized here. For option 1(a), the smaller tin tube diameter introduces issues with higher pressure drop, surface tension, and cavitation. For option 1(b), the smaller graphite diameter means less volume of storage material, which means additional volume is required which increases insulation costs and reduces energy density of the battery. For option 2, the cylindrical shell geometry similarly requires additional area for insulation (and lower energy density) as well as increasing the heat loss out of the system due to the gap between the tin channel and the block. For option 3, the higher tin temperature introduces issues with sublimation and boiling. For option 4, the increase in charging power cost is impractical.  S8. GIFs of temperature profile vs. time for full-scale simulation  GIFs of temperature profile vs. time for 10x10 grid, with 1 path vs. 10 parallel paths available here: https://github.com/shomikverma/graphite-discharge/blob/main/plots/GIF_discharge_animation.gif  S9. Charge/discharge dynamics  In this paper, we have discussed charge and discharge cycles separately, but another important consideration is the charge-discharge transition. For example, if the cost of electricity suddenly drops while discharging, the system may want to immediately switch to charging. To optimize this process, we introduce the concept of flow reversal, to ensure the outlet temperature of the storage blocks is optimized. For example, for uni-directional flow, halfway through discharge, the temperature of the end of the block is still relatively high. Thus, there is limited charging potential. In contrast, if the flow is reversed, halfway through discharge, the top of the block is likely cold, allowing full charging capability.  To analyze this phenomenon, we investigate each case (unidirectional vs. reversed) separately. For both cases, we consider a block designed for 19 hours of storage and model several cycles of 9.6 hours of discharging and 2.4 hours of charging. We use Equation 2 to find the power required for each of discharge or charge. To ensure constant power during discharge and charge, we use Equation 3 to vary the flowrate, using a maximum flowrate factor of 10. During discharge, we use an inlet tin temperature of 1900°C, while during charge we use an inlet tin temperature of 2400°C. For uni-directional flow, we keep the top of the block as the inlet and the bottom as the outlet. For flow reversal, we use the top of the block as the inlet for discharging, 15  and the bottom of the block as inlet for discharging. Figure S11 shows the results for 72 hours of simulation, for both uni-directional ((a)-(c)) and reversed ((d)-(f)) flow.  \n Figure S11: Plots of temperature, massflow, and power for (a)-(c) unidirectional and (d)-(f) reversed flow.  As seen, flow reversal allows quicker transition from hot outlet temperature to cold, and vice versa. Due to the flexibility in flowrate, the uni-directional case can achieve similar thermal power output/input as the reversal case. If flowrate was set to be constant, this would not be the case. Further, when converting thermal to electrical power, TPV power density and efficiency must be considered. Because flow reversal enables higher temperatures, the discharge power will be higher for a given TPV area. Other arguments for flow reversal include minimization of exergy destruction and limiting thermal cycling of the graphite blocks. In the unidirectional case, during charging hot tin is put in contact with cold graphite, destroying more exergy than if hot tin was put in contact with hot graphite. In the flow reversal case, one side of the graphite remains hot while the other remains cold, reducing the number of thermal cycles the graphite sees.  GIFs of unidirectional vs. reversed flow available here: https://github.com/shomikverma/graphite-discharge/blob/main/plots/GIF_charge_discharge_animation.gif   \n"    },    {        "title": "2402.07768v1.Measuring_friction_from_simulations_of_folded_graphene_sheets.pdf",        "content": "Friction in folded graphene\nMeasuring friction from simulations of folded graphene sheets\nCharlie M. Rawlins1and Gareth A. Tribello1\nCentre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen’s University Belfast, Belfast,\nBT7 1NN\n(*Electronic mail: c.rawlins@qub.ac.uk)\n(Dated: 13 February 2024)\nWe run molecular dynamics simulations of folded graphene sheets and present a procedure to measure the sliding\nfriction in these systems based on the rate of decay of a damped-harmonic oscillator. This procedure allowed us to\nstudy the affect the size, geometry and the temperature of the graphene sheet had on the ability to propagate the initial\nfold and the rate at which it settles to a final ’fully-folded’ equilibrium state. We offer simple rationalisations for the\nrelationships between the initial geometries of our simulations and the friction values that emerge.\nI. INTRODUCTION\nThe discovery of graphene has led to a flurry of new activ-\nity in physics, chemistry and materials research1. This novel\nmaterial has numerous unique properties many of which come\nabout because a graphene sheet is only one atom thick2. This\n2D geometry ensures that graphene has excellent in-plane me-\nchanical flexibility, which, because there are also strong inter-\nlayer van der Waals interactions between graphene sheets, en-\nsures graphene sheets can be scrolled3,4, folded5and stacked\ninto novel assemblies. These assemblies of graphene also\nhave interesting properties one of the most intriguing of which\nis the evidence of super lubricity6,7.\nRecently Annett and Cross have developed a method for\nsynthesising folded graphene8. They first disrupt the graphene\nstructure using a nanoindentor or AFM tip. Novel kirigami\nribbons then grow spontaneously from the disrupted region.\nAnnett and Cross have used continuum mechanics to ratio-\nnalise the geometry of the structures that form. However, this\nmodel provides no information on the atomic scale features\nthat affect the formation and growth of these ribbons. Atom-\nistic simulation thus has a clear role to play when it comes to\nbetter understanding this phenomenon.\nSimulating the formation of kirigami ribbons is difficult\nbecause when the carbon-carbon bonds break carbon likely\nforms bonds with gaseous species from the atmosphere. In-\ncorporating these bond formation events in simulations is ex-\ntremely difficult, which is likely why the results from early\nsimulations are mixed9–11. However, even though it is likely\nnot possible to simulate kirigami ribbon formation directly,\ncarefully performed simulation can still provide insight as it\nallows one to study geometries that are simpler than those\ngenerated by experiment. For example, in the case of mul-\ntilayer graphene, simulations have been performed where the\ntop layer of graphene has been laterally offset. The result-\ning simulations show that the top layer will retract to be in\nalignment with the lower layers as the aligned configuration\nis more energetically favourable12–15. Furthrmore, as it re-\nturns to this equilibrium state the graphene sheet will undergo\noscillatory motion16. The dynamics of the motion has been re-\nlated to factors such as commensurability17–19, temperature12\nand surface roughness20,21. This work has helped to provide\ninsight into the interlayer forces of graphene, however, in the\ncase of ribbon-formation, the fold plays an important role indetermining the directionality and motion of the sheet. This\nfold is noticeably absent from these simulations and experi-\nments.\nStudies have been done on the self-folding and self-\nscrolling of graphene sheets when an edge has been\ndeformed4. However, the motion of these kinds of folded and\nscrolled sheets has not been studied in great detail. Only the\ninitial and final structures of edge-deformed graphene sheets\nhave been investigated. The intermediate motions which gov-\nern the rate at which these structures form has not been ex-\nplored.\nIn this work, we study the behaviour of a folded sheet of\ngraphene in vacuum. We find that these sheets exhibit sim-\nilar motion to free-standing sheets and undergo damped har-\nmonic oscillation as they reach an energetically more stable\nconfiguration. The model of this motion can be described us-\ning parameters which depend on the graphene twist angle, the\ndimensions of the sheet and the temperature. The parameters\nused for describing the dampening of the system can be used\nto provide information on the amount of friction between the\nlayers.\nIn what follows we show that friction is lowered when\ngraphene sheets are non-commensurately stacked and when\ntemperature is reduced. The remainder of this paper is laid\nout as follows. We first describe how the initial graphene sheet\nis set-up and how the resulting structures are characterised in\nsection II. Secondly, we describe the procedure used to ex-\ntract useful data from the simulations and how we model the\ndata using a damped-harmonic oscillator in section III. In the\nfinal parts of the paper we then examine how different initial\nset-ups affect these parameters.\nII. INITIAL CONFIGURATION AND OUTCOMES\nThe approach we take in this work is similar to the approach\nadopted by Pereira Junior and Ribeiro Junior4. We perform\nMD simulations to investigate how unsupported, single-layer\ngraphene approaches equilibrium from an initial structure.\nHowever, instead of starting the simulation with a configura-\ntion in which one end of the sheet is rolled up to form a scroll,\nwe start our simulations from a folded configuration similar to\nthe one shown in figure 1.\nThe initial structure is defined by five parameters that de-arXiv:2402.07768v1  [physics.chem-ph]  12 Feb 2024Friction in folded graphene 2\nθ\nxxyzr1r2(a)\n(c)(b)\n(d)\nFIG. 1. Schematic representation of a folded graphene sheet viewed on the xyplane (a) and the xzplane (c). Inserts shows the graphene lattice\nand how the twist angle θ(b) and the fold parameters r1andr2(d) are defined.\nscribe the geometry of the fold and parameters that describe\nthe shape of the flat sheet. The first two of these parameters\nare the length ( lx) and width ( ly) of the sheet. A parameter\nthat defines the orientation of the fold relative to the periodic\narrangement of carbon atoms is also required. In this work\nwe use the twist angle ( θ) for this purpose, which is defined\nas the angle between the x-axis of the lab frame and the line\nof symmetry that, if cut, would give rise to an edge with an\n’arm-chair’ configuration. Cutting graphene sheets along non\narm-chair directions leads to structures in which some carbon\natoms only participate in only one covalent bond. We removed\nthese carbons prior to starting simulations and thereby ensured\nthat all the carbon atoms at the edge of the graphene were\nbound to at least two additional carbon atoms. The fold was\nthen formed parallel to the y-axis of the lab frame and defined\nby two parameters, the radius of curvature r1and the length\nof sheet on the top layer which overlaps with the bottom layer\nr2. These parameters are illustrated in figure 1.\nTo study the self-folding of graphene sheets, fully-atomistic\nMD simulations using the ReaxFF potential22–24, as imple-\nmented in the Large-scale Atomic/Molecular Massively Par-\nallel Simulator (LAMMPS25), using the parameter set pro-\nvided by Chenoweth, van Duin, and Goddard26for C/H/O.\nThe equations of motion were numerically integrated using\nthe velocity-Verlet integrator and a time-step of 0.125 fs. Un-\nless otherwise stated temperature was fixed to 300 K using a\nNosé-Hoover thermostat27with a relaxation time of 125 fs.\nWe ran simulations of a 200 by 100Å graphene sheet start-\ning for twist angles of 0, 10, 20 and 30 degrees and a range\nofr1(2 to 3Å ) and r2(10 to 40Å ) values. For initial testing\nwith θ=0, the range of r1values was from 1.5 up to 5Å .\nThese simulations were run until the system has equilibrated\nand the fluctuations in the potential energy were less than 2\nmeV/atom. We observed three different behaviours in these\nsimulations. When the r1parameter is large and the r2pa-\nrameter is small the sheet unfolds in the manner illustrated infigure 2. This behaviour makes sense as the overlap between\nthe top and bottom layers is small. The energetic cost associ-\nated with bending the sheet cannot, therefore, be compensated\nfor by the interlayer attraction.\nWhen r2is around 10 times larger than r1, then the top part\nof the folded sheet slides back and forth along the x-axis and\nover the bottom part of the sheet as illustrated in figure 3. This\nsame behaviour is consistently observed in simulations started\nwith different initial velocities, which suggests that the ener-\ngetic cost for folding can be compensated for by an increase\nin interlayer attraction. This result is consistent with the re-\nsults from experiments in which kirigami structures form. In\nthose experiments the energy released when the overlap be-\ntween graphene sheets increases drives carbon-carbon bond\nbreaking reactions. Consequently, it is hardly surprising that\noverlap between the top and bottom layers compensates for\nfolding. Also shown in figure 3 are slices along the xzplane\nfor each snapshot. These snapshots show that the spacing be-\ntween the top and bottom layers is not constant throughout\nthe length of the sheet. Close to the fold the sheet resem-\nbles a tennis-racket. However, when this part of the sheet is\nexcluded, we find that the interlayer spacing remains mostly\nconstant throughout the course of the simulation. This is high-\nlighted in figure 3(i), which shows that the interlayer spacing\n(the method that this is determined is described in the follow-\ning section)very quickly settles from the initial configuration\nto a value of about 3.3Å . The expected interlayer spacing of\n3.35Å for graphene is shown as a red line in this figure for\nreference.\nIn the intermediate region, neither regular folding or un-\nfolding is observed. We instead get a behaviour that we have\nchristened ’atypical folding’ (see Fig. 4) in which random, in-\nplane thermal oscillations shift the fold direction away from\nthex-axis. Analysing these trajectories using the methods that\nwill be explained in the later parts of this paper is difficult\nwhich is why we classify these trajectories differently fromFriction in folded graphene 3\n(a)t=0ps\n(b)t=5ps\n(c)t=20ps\n(d)t=50ps\nFIG. 2. VMD snapshots of a 200 Å by 100 Å sheet with initial configuration of r1=2.0,r2=10 and θ=0◦. This is an example of\n’unfolding’.\n050100150200250Time (ps)3.23.33.43.53.63.73.83.94Interlayer spacing (Å)\n(a)t=0ps\n(b)t=0ps\n(c)t=25ps\n(d)t=25ps\n(e)t=50ps\n(f)t=50ps\n(g)t=250ps\n(h)t=250ps(i)\nFIG. 3. VMD snapshots of a 200 Å by 100 Å sheet with initial configuration of r1=2.0,r2=20 and θ=0◦. This is an example of ’folding’.\nInset (i) is a plot of the interlayer spacing between the top and bottom layers as a function of time.\nthe ones in the previous paragraph.\nNo restraints were placed on the carbon atoms in any of\nour simulations. The sliding of the graphene layers over each\nother is thus accompanied by out of plane oscillations when\nboth folding and atypical folding occurs. In fact, for thelargest sheets we studied in work (400 Å by 200 Å ) when\nθ=10 (which we show in later sections to be a low friction\nset-up) these oscillations are large enough to cause the sheet\nto fold for a second time as illustrated in figure 5.\nFigure 6 summarises the relationships between the ini-Friction in folded graphene 4\n(a)t=0ps\n(b)t=40ps\n(c)t=350ps\nFIG. 4. VMD snapshots of a 300 Å by 150 Å sheet with initial configuration of r1=3.0,r2=20 and θ=0◦. This is an example of ’atypical\nfolding’.\ntial geometry and the final equilibrium structure the system\nadopts. This figure shows that, as discussed in the previous\nparagraphs, unfolding occurs when r2is small or r1is large. In\nother words, the folded structure is only retained when there is\nenough attraction between the graphene layers to compensate\nfor the energetic cost associate with the bending.\nFigure 6 also includes data from simulations of sheets with\nlarger lxandlyvalues. Specifically 300 by 150 Å and 400\nby 200 Å sheets. The results for these larger sheets are very\nsimilar to the results that were obtained for the smaller sys-\ntem. At first glance this result appears surprising as making\nthe sheet wider increases interactions between the layers and\nfurther stabilises the fold. However, increasing the width of\nthe sheet also increases the width of the fold. Having a wider\nfold increases the energy and, it would appear, counteracts the\nnegative terms that come from having more overlap.\nIII. ESTIMATING FRICTION\nAs shown in figure 3, as the sheet slides over itself the sys-\ntem eventually settles into a fully-folded configuration. In or-\nder for this to occur, there must be some dampening force to\nslow down the sliding sheet, which we can assume is due to\nthe friction between the top and bottom layers. Our contention\nin the remainder of this paper is that the strength of the damp-\nening force can be determined by examining how the distance\nbetween the short edges of the sheet ( din figure 7) changes\nwith time.\nThe method for determining dfrom our simulations is\nslightly more involved than figure 7 suggests. We are helped\nby the fact that the atoms in the bold parts of the graphene\nsheet shown in figure 7 always remain in the top/bottom layer\nof the graphene structure as the system oscillates back and\nforth. In other words, the atoms in these parts of the sheet can\nbe said to be ’safe’ as they never become part of the curved\nportion of the sheet. These bold parts of the sheet have a\nlength of σand the indices of the atoms within them can be\nidentified from the starting configuration. This identification\nis possible because z-coordinates of all the atoms in the ini-\ntial configurations satisfy zmin≤z≤zmax. Atoms that have\nFIG. 5. VMD snapshots of a 400 Å by 200 Å sheet with initial\nconfiguration of r1=2.0,r2=20 and θ=10◦. The figures show an\nexample where ’double folding’ starts after about t=150 ps.\nz=zminare part of the bottom layer, atoms with z=zmaxare\npart of top layer and atoms with zmin<z<zmaxare in the\nfold. For the calculation of dwe make a list of the indices of\nthe atoms that have z=zmaxin this initial structure. We then\nmake a second list that contains the indices of those atoms that\nhave z=zminand that have an xcoordinate that is within σof\nthe maximum xcoordinate for the atoms.\nWe introduce a new coordinate, S, for each atom in the\ngraphene sheet. This coordinate tells us the xposition the\natom would have if the graphene sheet were laid flat and is,\ntherefore, independent from the shape of the sheet. The S\nvalue for each atom is thus determined at the start of the\nsimulation and never changes. The Scoordinates for the\natoms in the bottom layer of the initial structure (i.e. thoseFriction in folded graphene 5\n0 1 2 3 4 5\nr1 (Å)051015202530354045r2 (Å)\n0 1 2 3 4 5\nr1 (Å)051015202530354045\n0 1 2 3 4 5\nr1 (Å)051015202530354045\n200 by 100 300 by 150 400 by 200\nFIG. 6. Summary of the final states of an edge-folded graphene\nsheet based on initial geometries. The calculations we report on here\nwere performed using sheets of dimensions 200 by 100Å (left panel),\n300 by 150Å (middle panel) and 400 by 200Å (right panel). Trajec-\ntories where folding, unfolding, atypical folding and double folding\nare represented by green circles, red diamonds, blue squares and ma-\ngenta triangles respectively. The center, solid symbols are for offset\nangles of θ=0◦, the smallest ring around this symbol correspond-\ning to θ=10◦, the middle sized ring for θ=20◦and the largest ring\nbeing for θ=30◦.\natoms with z=zmin) can be set equal to the xcoordinates\nof these atoms in the initial structure. For the remaining\natoms, we can measure the distance dSbetween the initial\nposition of atom a, whose Scoordinate is known, and one of\nits neighbouring atoms b, whose Scoordinate is not known as\ndS=p\n(xa−xb)2+ (za−zb)2. The Scoordinate for atom b\nis then set equal to Sb=Sa−dS. This procedure resembles\nthe method that is used to calculate geodesic distances in the\nISOMAP algorithm28.\nWhen calculating the quantities labelled dandrfrom the\ninstantaneous coordinates of the atoms we use the indices of\nthe ’safe’ atoms in the blue regions of figure 7 and the Sco-\nordinates that were determined from the initial structure. To\ncalculate rwe first calculate an Ns×Nmatrix of distances,\nD, where Nsis the number of ’safe’ atoms and Nis the total\nnumber of atoms in the structure. The distances in this matrix\nDare calculated using Pythagoras’ theorem and the instanta-\nneous coordinates of the atoms. We then calculate a second\nNs×Nmatrix Mas:\nMi j=(\nDi jif|Si−Sj|>πDi j\n2\n0 otherwise\nTransforming the distances in this way ensures that Mi jonly\ncontains distances between atoms in the top and bottom lay-\ners. Distances between atoms in the same layer or distances\nbetween atoms in one layer and the curve are excluded be-\ncause the geodesic distance between them |Si−Sj|will be less\nthan half the circumference of a sphere with diameter Di j.\nWe next introduce two vectors rtandrb. The first of these\nvectors measures the distances between each of theNs\n2’safe’atoms in the top layer and the nearest atom in the bottom layer.\nThe second vector does a similar measurement for the ’safe’\natoms in the bottom layer. The components of these vectors\nare set equal to the smallest non-zero element in the row of\nMthat corresponds to the ’safe’ atom of interest. From these\nvectors we compute the means ⟨rt⟩and⟨rb⟩for the distribu-\ntion of distances29.rin figure 7, the distance between the top\nand bottom layers, is then set equal to the smaller of these two\nmeans.\nIn calculating the vectors rtandrbwe also determine the\natom from the opposite layer that is nearest to each of the\n’safe’ atoms. These determinations are used when we calcu-\nlated. To calculate this quantity we select the set of ’safe’\natoms with the smaller average rvalue and compute the fol-\nlowing quantity:\ndi=Sm−Si−Sj\nfor each of them. In this expression, Siis the Scoordinate\nof the ith safe atom and Sjis the Scoordinate of the closest\natom from the opposite layer, which was determined from the\ncalculation of r.Smis the total length of the flattened graphene\nsheet. As figure 7 shows, diis thus what remains once the\ngeodesic distance between atom iand the edge of the sheet and\nthe geodesic distance between atom closest to atom ifrom the\nopposite sheet is subtracted from the total length of the sheet.\nTo arrive at a final scalar value for d, an average over all the\nindividual divalues is computed. We once again remove any\noutliers when computing this mean using the method that we\ndescribed when we explained how ris calculated.\nThe solid black line in the top panel of figure 8 shows how\nthe value of dchanges over the course of a simulation. The\nresult shown here is for a 300 ×150 Å graphene sheet, which\nwas started off in a configuration which had r1=2 År2=30\nÅ and θ=0. You can see that the dundergoes a series of\ndamped oscillations as would be expected given the snapshots\nfrom the trajectory that are shown in figure 3. Ye et al.21\nhave observed similar behaviours in systems composed of flat\ngraphene sheets that are stacked on top of each other and\nhave fitted the oscillations using a damped harmonic oscilla-\ntor model. The blue line in figure in the upper panel of figure\n8 shows what we obtain when we attempt to fit the data using\na similar damped harmonic oscillator model. You can clearly\nsee that there is close fit to the data from the early parts of\nthe simulation but that there is a large discrepancy between\nthe model and the data for large t. The difficulties in fitting\nthe data using this simple model suggest that the frequency\nof oscillations changes as the simulation progresses. We thus\nchose to refit the data (red line) using the five parameter model\nbelow:\nx(t) =Ae−γωtcos\u0010\nωeβωtt+φ\u0011\n. (1)\nAandφare the initial amplitude and a phase factor and will\nnot be discussed further in this work. ωis the initial frequency\nof the oscillations and γis the damping parameter. The βpa-\nrameter allows us to describe the changes in frequency with\ntime that were observed (the blue curves are fits using the\nmodel above with β=0). Figure 8 illustrates that you get aFriction in folded graphene 6\nSm0Si\nSjrd\nzminzmaxσ\nσx\nFIG. 7. Schematic to demonstrate how the distance dbetween the top and bottom edge were determined. Bold lines on the folded sheet\ncorrespond to the initial top, flat portion of the sheet and the equivalent portion of graphene on the bottom part of the fold.\n-150-100-50050100150200250Distance (Å)\n0 50 100 150 200 250 300 350\nTime (ps)-10-505Velocity (Å/ps)\nFIG. 8. Example of distance between the short edges of the sheet d\nand sliding velocity over time. The raw data is shown in black, with\nthe red dashed lines being the fit of a damped harmonic oscillator\ngiven by Eq. 1 for the top figure and Eq. 2 for the bottom. The blue\nline is a fit with β=0.\nmuch closer fit to the data when this βparameter is allowed to\nvary, which confirms that the oscillation period changes dur-\ning the simulation.\nIf the folded graphene system is really undergoing damped\noscillations that can be described using the model above, the\nlayers should be moving relative to each other at a velocities\ngiven by the following expression:\nv(t) =−Aωe−γωt\u0010\nγcos\u0010\nωeβωtt+φ\u0011\n+eβωt(βωt+1)sin\u0010\nωeβωtt+φ\u0011\u0011\n.(2)\nThis expression was obtained by differentiating equation 1.\nTo test that this expression can indeed be used to describe the\nvelocity of the oscillations we stored the velocities of all the\n’safe’ atoms. The average velocity for the ’safe’ atoms that\nare initially in the top layer was then computed along with the\naverage velocity for the ’safe’ atoms that were initially in the\nbottom layer. The black line in the bottom panel of figure 8\nshows how the difference between these two averages changes\nduring the simulation. You can see that this quantity also os-\ncillates, which is what one would expect from the model inequation 2. More interestingly, the blue line shows the result\nof fitting the time series of velocity differences from the sim-\nulation to equation 2 with β=0, while the red line shows the\nresult that is obtained when βis a free parameter. Once again\nyou see that a closer fit is obtained when a parameter is in-\ncluded in the model that allows oscillation period to change\nas the simulation progresses.\nWe observe that the fold moves relatively rapidly to a geom-\netry, which has the two layers separated by their equilibrium\ndistance and that oscillations start once this initial relaxation\nphase is completed. We thus argue that the γ,ωandβparam-\neters of our damped harmonic oscillator model are properties\nof the graphene and not properties of the initial configuration.\nTo demonstrate this fact we run multiple simulations with dif-\nferent initial values for r1andr2. All values of r1andr2that\nundergo folding according to figure 6 were used. The val-\nues that we quote for the parameters in the rest of this paper\nare found by averaging over these simulations. We also quote\nthe variance for the distribution of parameters and hence show\nthat the parameters values we obtain do not depend strongly\nonr1orr2. As discussed in the next section, the parameters\nof these models change in ways that can be easily explained\nwhen the shape of the graphene sheet is varied.\nIV. EFFECT OF SHEET DIMENSIONS\nThe top and bottom panels of figure 9 show the γandω\nparameter values that were obtained for simulations of sheets\nwith various widths that were all 300 Å long. Solid circles\nindicate the values that were obtained from fitting using equa-\ntion 1 and solid squares are the results that were obtained by\nfitting using equation 2. You can see that neither ωorγde-\npends strongly on the width of the sheet. This result is to\nbe expected as we have not treated the edges of the graphene\nmeaning that they are the only regions of high friction. In\nother words, the middle parts of the sheet only make a mod-\nest contribution to the total friction. Consequently, because\nincreasing the width only increases the distance between the\nedges, γremains constant. The distance the graphene sheet\nmoves during each oscillation only depends on the length of\nthe sheet. A wider sheet would also not be expected to move\nmore rapidly than a narrower sheet. The fact that the lowerFriction in folded graphene 7\npanel of figure 9 shows that the oscillation frequency, ω, does\nnot depend on the length of the sheet is thus not surprising.\nThe top and bottom panels of figure 10 show the γandω\nparameter values that were obtained for simulations of sheets\nwith various lengths that were all 150 Å wide. Once again,\nsolid circles and solid squares are the results obtained using\nequations 1 and 2. The bottom panel of figure 10 shows that\nωdecreases as the length of the sheet increases. Furthermore,\nthe dashed lines in this panel were generated by fitting these ω\nvalues to a function of the formM\nL+Cwhere Lis the length\nof the sheet and MandCare parameters. You can see that\nthis function fits the data on ωreasonably closely. We believe\nthatωdecreases with1\nLbecause when the sheet is longer it\nmoves a greater distance during each oscillation. As the ve-\nlocity of the sheet does not increase when the sheet gets larger,\ncompleting a single oscillation takes longer when the sheet is\nlarger.\nThe top panel of figure 10 shows that the γparameter in-\ncreases as the sheet grows longer. Furthermore, the increase\ninγis reasonably well described by a linear function of the\nlength of the sheet. This result is easy to explain. Increasing\nthe length of the sheet increases the lengths of the edges of\nthe graphene. The majority of the friction between the two\nlayers is known to come from interactions between the edges\nin the top and bottom layers of the sliding graphene sheet. γ\nshould thus increase when the length of the sheet is increased\nand should remain the same (as we saw in the top panel of\nfigure 9) when the width of the sheet is changed.\nFigure 11 shows a plot of the γparameter against the β\nparameter for all the simulations that were reported in figure\n9, 10 and 12 (which is shown in a later section). You can\nclearly see that there is a linear relationship between the γ\nandβparameters of the model. We thus argue that these two\nparameters are both providing equivalent information on the\nfriction between the two layers. In the remainder of the paper\nwe thus report γvalues only as the same conclusion would be\nreached through an analysis of β.\nV. EFFECT OF TWIST ANGLE\nThe previous section showed that the ωandβparameters\nthat emerge from our model are not particularly interesting.\nThere is a linear relationship between ωand the length of the\nsheet and the value of βis proportional to the value of γ. For\nthe remainder of this paper we will thus only consider the γ\nparameters that emerge from our model fitting.\nFigure 12 shows how the γparameter of the fit changes with\nthe twist angle of the initial geometry. As shown in figure 1,\nthis twist angle measures the angle between the xaxis of the\nlab frame and the armchair direction of the graphene sheet.\nFigure 12 shows that γis small when this angle is 10 or 20\ndegrees and that it is large when the angle is 0 or 30 degrees.\nThis result makes physical sense as the two graphene layers\nare commensurately stacked when the angle is 0 or 30 degrees.\nThe sort of non-commensurate stacking layers that is known\nto give rise to the super lubricious sliding6is only present for\nthe other two angles.\n100 200\nWidth (Å)0.150.20.25 γ\n100 200\nWidth (Å)0.030.040.05 ω  (ps-1)FIG. 9. γ(top figure) and ω(bottom figure) for sheets of various\nwidths, a length of 300 Å and θ=0◦. Circles correspond to dis-\nplacement data, squares from velocity data.\nWhen the angle between the two graphene layers is 0 de-\ngrees the sliding (long) edge of the graphene fold has a arm-\nchair termination. By contrast when the angle is 30 degrees\nthis edge has an zig-zag termination. Our results thus suggest\nthat arm-chair edges slide more easily against each other than\nzig-zag.\nThe results shown in figure 12 were obtained from simula-\ntions of a 300 ×150 Å graphene sheet. When we ran simula-\ntions with other sizes of graphene sheet the relative magnitude\nof the γparameters for different twist angles stayed the same.\nVI. EFFECT OF THERMOSTAT\nThe simulations described in the previous sections were run\nin the NVT ensemble so a thermostat acts upon the atomic ve-\nlocities. To check that this thermostat is not dissipating energy\nfrom the oscillations we performed NVE simulations of the\nfolded sheet. A short equilibration in the NVT ensemble of 5\nps was required to move the system away from the artificial,\ninitial configuration, which was very high in energy. Results\nfor the γparameters that were obtained from fitting the posi-\ntion and velocity curves using equations 1 and 2 for simula-\ntions run in both the NVT and NVE ensemble are provided inFriction in folded graphene 8\n200 300 400\nLength (Å)00.050.10.150.20.250.3 γ\n200 300 400\nLength (Å)00.010.020.030.040.050.060.07 ω  (ps-1)\nFIG. 10. γ(top figure) and ω(bottom figure) for sheets of various\nlengths, a width of 150 Å and θ=0◦. Legend the same as in Fig. 9.\nDashed lines in the top figure corresponds to displacement and ve-\nlocity data fitted to the formula γ=ML+C. Dashed lines in bottom\nfigure are the same but for a fitting formula ω=M\nL+C.\n0 0.05 0.1 0.15 0.2\n β00.10.20.30.40.5 γ \nFIG. 11. γvsβfor various fits of different sized sheets. Black\ncircles corresponds to parameters from fitting of displacement data,\nred squares correspond to fitting with velocity data.\n0 10 20 30\nOffset angle (degrees)00.10.20.30.40.5 γ FIG. 12. γvsθfor sheet dimensions 300 Å by 150 Å. Legend the\nsame as for Fig. 9.\nγx γv\nθ=0\nNVT 0.2022 ±0.0054 0.2248 ±0.0047\nNVE 0.1677 ±0.0081 0.1939 ±0.0056\nθ=10\nNVT 0.0798 ±0.0027 0.0893 ±0.0029\nNVE 0.0816 ±0.0045 0.0929 ±0.0061\nθ=20\nNVT 0.1773 ±0.0062 0.2057 ±0.0075\nNVE 0.1883 ±0.0108 0.2187 ±0.0128\nθ=30\nNVT 0.3990 ±0.0160 0.4772 ±0.0201\nNVE 0.3516 ±0.0227 0.4209 ±0.0269\nTABLE I. Fitting parameter γfrom the NVT and NVE ensembles\ntable I. For most of the geometries considered the γparam-\neters that emerge from the NVT simulations are within the\nnumerical errors of the estimates that emerge from the NVE\nsimulations. Even when larger discrepancies were identified\nwe found that any difference in the values γω- a quantity that\nis not unit less and that has units of inverse time - for the NVE\nand NVT simulations were within statistical errors. We thus\nascribe any apparently-statistically-significant differences in\ntheγvalues for NVT and NVE simulations in table I to issues\nwith the fitting procedure and concluded that the thermostat is\nnot dissipating energy from the oscillations.\nVII. EFFECT OF TEMPERATURE\nFigure 13 shows how the fitted γvalues from the model\nchange with temperature. This figure was generated by per-\nforming simulations on a 300 ×150 Å graphene sheet with a\ntwist angle of 0 degrees.\nThe level of variation in the damping parameter in figure\n13 is much lower than the variation that you see in figure 12.\nIn other words, the relative orientation of the two layers has a\nmuch greater effect on the friction than the temperature. Fig-\nure 13 shows, however, that the friction appears to increaseFriction in folded graphene 9\n200 300 400 500 600 700 800 900 1000 1100\nTemperature (Kelvin)0.180.20.220.240.26 γ \nFIG. 13. γvs temperature for sheet dimensions 300 Å by 150 Å\nsheet and θ=0◦.\nwith temperature. Yang et al.12suggest that friction will in-\ncrease with temperature as the graphene sheet fluctuates more\nat higher temperature. These increased fluctuations make the\nsurface appear rougher and thus more strongly suppress slid-\ning motions. Our simulations would suggest that this is a very\nsmall effect, however. Furthermore, when we plot the param-\neters that emerge from fitting using equation 2 we do not see\nthe same monotonic increase in γwith temperature.\nVIII. CONCLUSIONS\nWe have successfully developed a procedure to generate a\nfolded sheet of graphene of arbitrary size and fold orientation,\nhave used MD simulations to observe the sliding behaviour as\nthe sheet settles to a fully-folded state and have developed an\noscillator model to describe this sliding behaviour. This model\nwas then used to characterise the friction between the top and\nbottom layers of the folded sheet in terms of the decay param-\neterγ. While this decay parameter is not an absolute measure\nof the friction of sliding graphene, examining the changes in γ\nin different situations provides better qualitative understand-\ning of how friction behaves at this level. When varying the\ndimensions of the sheet, it was found that γscaled near lin-\nearly with sheet length, but had virtually no dependence of\nwidth. This implies the greatest source of friction comes from\nthe edges of the sheet, which is a very reasonable as nothing\nhas been done to treat the edge atoms in our simulations. Al-\ntering the offset angle θshows that there is a significant drop\nin friction when going from commensurate stacking to non-\ncommensurate stacking. Finally, increasing the temperature\ncauses a modest increase in sliding friction, which may be\ndue to the ’roughness’ of the sheet increasing with increased\nthermal fluctuation.\nOur results show that the procedure developed here to\nmodel and analyse the sliding friction of a folded sheet of\ngraphene produces results which are consistent with our cur-\nrent understanding of friction in these systems. These results\nshould provide some insights into the formation and propaga-tion of kirigami ribbons in graphene observed by Annett and\nCross8. In order to better compare with experiment, the pro-\ncedure will need to be updated in order to include a substrate,\ntearing mechanisms or even impurities in the system. Fortu-\nnately, the procedure is set up in such a way that these updates\nshould not be a significant undertaking. Our next priority is to\ninclude a substrate in the simulation in order to better delin-\neate the affects of graphene-graphene and graphene-substrate\ninteractions.\nACKNOWLEDGMENTS\nThis work was supported by a research grant from the De-\npartment for the Economy Northern Ireland under the US-\nIreland R&D Partnership Programme. The computing re-\nsources used for this publication were from the Northern Ire-\nland High Performance Computing (NI-HPC) service funded\nby EPSRC (EP/T022175). We thank Robert Carpick, Graham\nCross, Ashlie Martini, David Wilkins and Myrta Grüning for\nuseful discussions.\nDATA AVAILABILITY STATEMENT\nThe data that supports the findings of this study are avail-\nable from the corresponding author upon reasonable request.\nScripts for generating folded graphene sheets, input files for\nrunning on LAMMPS and scripts used to extract data from\nthe resulting simulations as described in section III will be\navailable on Github.\n1Y . Zhu, S. Murali, W. Cai, X. Li, J. W. Suk, J. R. Potts, and\nR. S. Ruoff, “Graphene and graphene oxide: Synthesis, proper-\nties, and applications,” Advanced Materials 22, 3906–3924 (2010),\nhttps://onlinelibrary.wiley.com/doi/pdf/10.1002/adma.201001068.\n2A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.\nGeim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–\n162 (2009).\n3L. M. Viculis, J. J. Mack, and R. B. Kaner, “A chemical route to carbon\nnanoscrolls,” Science 299, 1361 (2003).\n4M. L. Pereira Junior and L. A. Ribeiro Junior, “Self-folding and self-\nscrolling mechanisms of edge-deformed graphene sheets: a molecular dy-\nnamics study,” Phys. Chem. Chem. Phys. 23, 15313–15318 (2021).\n5J. Zhang, J. Xiao, X. Meng, C. Monroe, Y . Huang, and J.-M. Zuo, “Free\nfolding of suspended graphene sheets by random mechanical stimulation,”\nPhys. Rev. Lett. 104, 166805 (2010).\n6M. Dienwiebel, G. S. Verhoeven, N. Pradeep, J. W. M. Frenken, J. A. He-\nimberg, and H. W. Zandbergen, “Superlubricity of graphite,” Phys. Rev.\nLett.92, 126101 (2004).\n7Y . Liu, F. Grey, and Q. Zheng, “The high-speed sliding friction of graphene\nand novel routes to persistent superlubricity,” Scientific Reports 4, 4875\n(2014).\n8J. Annett and G. L. W. Cross, “Self-assembly of graphene ribbons by spon-\ntaneous self-tearing and peeling from a substrate,” Nature 535, 271–275\n(2016).\n9A. F. Fonseca and D. S. Galvão, “Self-tearing and self-peeling of folded\ngraphene nanoribbons,” Carbon (2018).\n10A. F. Fonseca and D. S. Galvão, “Self-driven graphene tearing and peel-\ning: A fully atomistic molecular dynamics investigation,” MRS Advances\n3, 463–468 (2018).\n11Z.-Z. He, Y .-B. Zhu, and H.-A. Wu, “Self-folding mechanics of graphene\ntearing and peeling from a substrate,” Frontiers of Physics 13, 138111\n(2018).Friction in folded graphene 10\n12J. Yang, Z. Liu, F. Grey, Z. Xu, X. Li, Y . Liu, M. Urbakh, Y . Cheng,\nand Q. Zheng, “Observation of high-speed microscale superlubricity in\ngraphite,” Phys. Rev. Lett. 110, 255504 (2013).\n13Z. Liu, J. Yang, F. Grey, J. Z. Liu, Y . Liu, Y . Wang, Y . Yang, Y . Cheng,\nand Q. Zheng, “Observation of microscale superlubricity in graphite,” Phys.\nRev. Lett. 108, 205503 (2012).\n14Q. Zheng, B. Jiang, S. Liu, Y . Weng, L. Lu, Q. Xue, J. Zhu, Q. Jiang,\nS. Wang, and L. Peng, “Self-retracting motion of graphite microflakes,”\nPhys. Rev. Lett. 100, 067205 (2008).\n15T. W. Ng, C. Y . Lau, E. Bernados-Chamagne, J. Z. Liu, J. Sheridan, and\nN. Tan, “Graphite flake self-retraction response based on potential seeking,”\nNanoscale Research Letters 7, 185 (2012).\n16I. V . Lebedeva, A. A. Knizhnik, A. M. Popov, Y . E. Lozovik, and\nB. V . Potapkin, “Interlayer interaction and relative vibrations of bilayer\ngraphene,” Phys. Chem. Chem. Phys. 13, 5687–5695 (2011).\n17Z. Xu, X. Li, B. I. Yakobson, and F. Ding, “Interaction between graphene\nlayers and the mechanisms of graphite’s superlubricity and self-retraction,”\nNanoscale 5, 6736–6741 (2013).\n18A. M. Popov, I. V . Lebedeva, A. A. Knizhnik, Y . E. Lozovik, and B. V .\nPotapkin, “Molecular dynamics simulation of the self-retracting motion of\na graphene flake,” Phys. Rev. B 84, 245437 (2011).\n19A. M. Popov, I. V . Lebedeva, A. A. Knizhnik, Y . E. Lozovik, and\nB. V . Potapkin, “Commensurate-incommensurate phase transition in bi-\nlayer graphene,” Phys. Rev. B 84, 045404 (2011).\n20Z. Ye, A. O. de-la Roza, E. R. Johnson, and A. Martini, “The role of\nroughness-induced damping in the oscillatory motion of bilayer graphene,”\nNanotechnology 25, 425703 (2014).\n21Z. Ye, A. O. de-la Roza, E. R. Johnson, and A. Martini, “Oscillatory motion\nin layered materials: graphene, boron nitride, and molybdenum disulfide,”\nNanotechnology 26, 165701 (2015).\n22A. C. T. van Duin, S. Dasgupta, F. Lorant, and W. A. Goddard, “Reaxff: A\nreactive force field for hydrocarbons,” The Journal of Physical ChemistryA105, 9396–9409 (2001).\n23J. E. Mueller, A. C. T. van Duin, and W. A. I. Goddard, “Development and\nvalidation of reaxff reactive force field for hydrocarbon chemistry catalyzed\nby nickel,” The Journal of Physical Chemistry C 114, 4939–4949 (2010).\n24H. M. Aktulga, S. A. P. J. C. Fogarty, and A. Y . Grama, “Parallel reac-\ntive molecular dynamics: Numerical methods and algorithmic techniques,”\nParallel Computing 38, 245–259 (2012).\n25A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M.\nBrown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D.\nNguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton,\n“LAMMPS - a flexible simulation tool for particle-based materials model-\ning at the atomic, meso, and continuum scales,” Comp. Phys. Comm. 271,\n108171 (2022).\n26K. Chenoweth, A. C. van Duin, and W. A. Goddard, “Reaxff reactive force\nfield for molecular dynamics simulations of hydrocarbon oxidation,” The\njournal of physical chemistry. A (2008), 10.1021/jp709896w.\n27W. G. Hoover, “Canonical dynamics: Equilibrium phase-space distribu-\ntions,” Phys. Rev. A 31, 1695–1697 (1985).\n28J. B. Tenenbaum, V . De Silva, and J. C. Langford, “A global geometric\nframework for nonlinear dimensionality reduction,” Science 290, 2319–\n2323 (2000).\n29To make the calculation of these mean values more robust we also calculate\nthe standard deviation σ. We assume that any distance that is more than 2 σ\nfrom the mean is an outlier, discard it and then recalculate the mean without\nincluding these outliers.\n30Z. Liu, J. Z. Liu, Y . Cheng, Z. Li, L. Wang, and Q. Zheng, “Interlayer\nbinding energy of graphite: A mesoscopic determination from deforma-\ntion,” Phys. Rev. B 85, 205418 (2012)."    },    {        "title": "2402.07789v1.Instability_of_periodic_waves_for_the_Korteweg_de_Vries_Burgers_equation_with_monostable_source.pdf",        "content": "INSTABILITY OF PERIODIC WAVES FOR THE KORTEWEG-DE\nVRIES-BURGERS EQUATION WITH MONOSTABLE SOURCE\nRAFFAELE FOLINO, ANNA NAUMKINA, AND RAM ´ON G. PLAZA\nAbstract. In this paper, it is proved that the KdV-Burgers equation with a\nmonostable source term of Fisher-KPP type has small-amplitude periodic trav-\neling wave solutions with finite fundamental period. These solutions emerge\nfrom a supercritical local Hopf bifurcation around a critical value of the wave\nspeed. Moreover, it is shown that these periodic waves are spectrally unstable\nas solutions to the PDE, that is, the Floquet (continuous) spectrum of the\nlinearization around each periodic wave intersects the unstable half plane of\ncomplex values with positive real part. To that end, classical perturbation\ntheory for linear operators is applied in order to prove that the spectrum of\nthe linearized operator around the wave can be approximated by that of a\nconstant coefficient operator around the zero solution, which intersects the\nunstable complex half plane.\n1.Introduction\nThe Korteweg-de Vries-Burgers (KdVB) equation,\nut+αuu x−µuxx+βuxxx= 0,\nwith µ > 0,β > 0, was first derived by Su and Gardner [33] for a wide class of\nGalilean invariant physical systems under the weak nonlinearity and long wave-\nlength approximations, resulting into a model equation which underlies a balance\nbetween both dispersive and dissipative effects. It has been used to describe the ap-\nproximate behavior of many physical phenomena such as weakly nonlinear plasma\nwaves with electron inertial effects [17], undular bores in shallow water [20], cascad-\ning processes in turbulence theory [30], wave propagation in an elastic tube filled\nwith a viscous fluid [19], the flow of liquids containing gas bubbles [34], waves in\nelectronegative plasma [1, 13] and the description of gravity waves in atmospheric\ndynamics [21], among many others. A combination of the (viscous) Burgers [8]\nand the Korteweg-de Vries [29] equations, the KdVB model is perhaps the simplest\nphysically relevant scalar equation containing a nonlinear transport or convection\nterm, uux, a dissipation or viscosity term, uxx, and a dispersive term, uxxx, into one\nsingle equation. In all the aforementioned applications, the study of traveling wave\nsolutions is a fundamental issue. The literature on nonlinear wave propagation for\nthe KdVB equation is very extensive and we do not pretend to make a review here.\nThe reader is referred to the classical review paper by Jeffrey and Kakutani [18]\nand to the references cited therein.\n2020 Mathematics Subject Classification. 35Q53, 35B10, 35B35, 35P05, 47A75.\nKey words and phrases. KdV-Burgers-Fisher equation, periodic waves, Hopf bifurcation, Flo-\nquet spectrum, spectral instability.\n1arXiv:2402.07789v1  [math.AP]  12 Feb 20242 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nIn this paper, we consider the KdVB equation with a monostable source term,\nalso known as the Korteweg-de Vries-Burgers-Fisher (KdVBF) equation [26],\nut+αuu x−µuxx+βuxxx=ru(1−u), (1.1)\ndescribing the dynamics of a real, scalar unknown quantity u=u(x, t), depending\non space, x∈R, and time, t > 0, and where α, β, µ, r ≥0 are non-negative\nconstants. Since we are interested in the interplay between dissipation, dispersion,\nnonlinear transport and reaction, we assume that all physical constants are positive,\nα, β, µ, r > 0. Under this assumption and without loss of generality, we normalize\nthe model by making the transformations\nx→µ\nβx, t →µ3\nβ2t, α →µ4\nβ3α, r →β2\nµ3r,\nin order to recast equation (1.1) as\nut+αuu x+uxxx=uxx+ru(1−u), (1.2)\nfor some fixed constants α >0 and r >0; this is the model equation that we work\nwith for the rest of the paper.\nThe KdVBF equation (1.1) was recently introduced by Ko¸ cak [26], who proved\nthe existence of traveling waves (such as solitons, kink and anti-kink waves) under\nthe presence of the reaction term\nf(u) =ru(1−u),\nalso known as a logistic reaction function. This particular form of the reaction\nterm was considered by Fisher [12] and by Kolomogorov, Petrovsky and Piskunov\n[28] in the context of reaction-diffusion equations modelling invasion fronts, and\nit is often used to model dynamics of populations with limited resources which\nsaturate into a stable equilibrium point associated to an intrinsic carrying capacity\n(in this case, the equilibrium state u= 1). Although the nomenclature is not\nuniform, it is often known under the names of monostable, logistic or Fisher-KPP\nreaction term. It is one of the most widely used production mechanisms giving\nrise to wave propagation and determining their dynamics. Other studies have also\nconsidered the KdVB equation with sources: Gao [13], for instance, derived the\nKdVB equation with a linear reaction term (a damping source) in the study of\nwaves in electronegative plasma. We conjecture that the KdVBF equation might\nbe derived in the long wavelength and weakly nonlinear approximations from a\nphysically significant system, albeit such derivation is, as far as we know, still\nunknown.\nThe purpose of this paper is to contribute to the analysis of nonlinear waves for\nthe KdVBF equation (1.1). We study for the first time the problem of existence\nof periodic waves. Instead of applying the common methods based on exploiting\nsymmetries, or on solving the differential profile equations on a case-by-case basis,\nwe prove the existence of periodic waves via application of local bifurcation methods\n[3, 4, 9, 35]. Profiting from the presence of the unstable equilibrium point of the\nreaction function, we prove that a family of small-amplitude periodic waves emerges\nfrom a supercritical local Hopf bifurcation around a critical value of the wave speed;\nthis is the first result of the paper, see Theorem 2.1 below. These waves have small\namplitude and bounded fundamental period. The amplitudes are of order of the\nsquare root of the distance between the speed and the critical speed. The instability\nof the equilibrium point of the reaction is responsible for the existence of the wavesINSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 3\nand for the change of stability of the equilibrium point as the wave speed crosses the\nbifurcation critical value. Some numerical approximations illustrate the emergence\nof this family of waves.\nNext, we study the stability of these bounded periodic waves as solutions to\nthe nonlinear evolution equation (1.2). As it is customary, we first linearize the\nequation around each periodic wave and study the resulting spectral problem. In\nthe case of periodic waves, the linearized operator has periodic coefficients, leading\nto the concept of the Floquet spectrum , see [14,22,23]. We then recast the problem\nof locating the continuous or Floquet spectrum as a point spectral problem on an\nappropriate periodic space via a Bloch-type transformation. Since the waves have\nsmall amplitude, we follow previous analyses [3,4,9,27] and prove that the spectrum\ncan be approximated by the spectrum of a constant coefficient operator around the\nzero solution. For that purpose, a key ingredient is to show that the perturbed\nproblem encompasses relatively bounded perturbations. The analysis is based on\nrecasting the spectral equations for the Bloch-type operators as a perturbation\nproblem, where the perturbation parameter is precisely the distance between the\nspeed and the critical speed. We then apply the classical perturbation theory\nfor linear operators (cf. [16, 24]) to show that the unstable point eigenvalue of\nthe unperturbed operator splits into neighboring curves of Floquet spectra of the\nunderlying small amplitude waves, proving in this fashion the spectral instability\nresult, which is the main result of the paper, see Theorem 4.1 below. Up to our\nknowledge, this is the first analysis of existence and stability properties of periodic\nwaves for the KdVBF equation (1.2).\nOn notation. We denote the real and imaginary parts of a complex number λ∈C\nby Re λand Im λ, respectively, as well as complex conjugation by λ∗. Standard\nSobolev spaces of complex-valued functions on the real line will be denoted as\nL2(R) and Hm(R), with m∈N, endowed with the standard inner products and\nnorms. For any fundamental period L > 0, we denote by L2\nper([0, L]) the Hilbert\nspace of complex L-periodic functions in L2\nloc(R) satisfying u(x+L) =u(x) a.e. in\nxand with inner product and norm given by\n⟨u, v⟩L2per=ˆL\n0u(x)v(x)∗dx, ∥u∥2\nL2per=⟨u, u⟩L2per.\nFor any m∈N, the periodic Sobolev space Hm\nper([0, L]) will denote the set of\nall functions u∈L2\nper([0, L]) whose weak derivatives up to order mbelong to\nL2\nper([0, L]). Their standard inner product and norm are given by ⟨u, v⟩Hmper=Pm\nj=0⟨∂j\nxu, ∂j\nxv⟩L2perand∥u∥2\nHmper=⟨u, u⟩Hmper, respectively.\n2.Small-amplitude periodic waves\nThis section is devoted to prove that small amplitude periodic wave solutions to\nequation (1.2) do exist. These waves emerge from a Hopf bifurcation thanks to the\nthe presence of an unstable equilibrium point of the reaction function.\n2.1.Existence of periodic waves. A periodic wave is a solution to (1.2) of the\nform\nu(x, t) =φ(x−ct), (2.1)\nwhere the profile function, φ=φ(ξ),φ:R→R, is at least of class C3and\nperiodic, with fundamental period L > 0, that is, φ(ξ+L) =φ(ξ) for all ξ∈R.4 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nHere ξ=x−ctdenotes the Galilean variable of translation, and c∈Ris the wave\nspeed. Upon substitution of the ansatz (2.1) into (1.2) one obtains the differential\nprofile equation,\n−cφ′+αφφ′+φ′′′=φ′′+rφ(1−φ), (2.2)\nwhere′=d/dξ.\nIn order to rewrite (2.2) as a first order system, let us define\nΦ =\nϕ1\nϕ2\nϕ3\n:=\nφ\nφ′\nφ′′\n,\nso that\nΦ′=F(Φ) :=\nϕ2\nϕ3\nϕ3+rϕ1(1−ϕ1)−αϕ1ϕ2+cϕ2\n. (2.3)\nNotice that F∈C∞(R3;R3) and the only equilibrium points in R3of system (2.3)\nareP0= (0,0,0) and P1= (1,0,0). The Jacobian of Fis given by\nA(Φ) := DΦF(Φ) =\n0 1 0\n0 0 1\nr(1−2ϕ1)−αϕ2c−αϕ11\n∈C∞(R3;R3×3).\nDenote this Jacobian evaluated at P0as\nA0:=A(P0) =\n0 1 0\n0 0 1\nr c 1\n.\nTherefore, its characteristic equation is\np(λ, c) := det( λI−A0) =λ2(λ−1)−λc−r= 0. (2.4)\nFor a fixed r >0, the complex roots of equation (2.4) depend on c∈Rand are\ndenoted as λj=λj(c),j= 1,2,3. In our setting, the speed value c∈Rwill play\nthe role of the bifurcation parameter. Notice that, evaluated at the critical speed\nc0:=−r, (2.5)\nthe characteristic polynomial is p(λ,−r) = (λ−1)(λ2+r) and the roots are\nλ1(c0) = 1 , λ 2(c0) =i√r, λ 3(c0) =−i√r.\nIf we denote η(c) = Re λ(c) and ζ(c) = Im λ(c) as the real and imaginary parts\nof any root λ(c) of (2.4), respectively, then upon substitution into (2.4) we have\n(η+iζ)2(η−1 +iζ)−c(η+iζ)−r= 0,\nfor all c∈R. Taking the real and imaginary parts of last equation yields G(η, ζ, c ) =\n0, with G:R3→R2,G= (G1, G2), and\nG1(η, ζ, c ) = (η2−ζ2)(η−1)−2ηζ2−cη−r= 0,\nG2(η, ζ, c ) =ζ(η2−ζ2) + 2ηζ(η−1)−cζ= 0.\nLet us denote ( η0, ζ0j, c0) := (Re λj(c0),Imλj(c0), c0) = (0 ,(−1)j√r,−r), for\nj= 2,3. Then, clearly, G1(η0, ζ0j, c0) =r−r= 0 and\nG2(η0, ζ0j, c0) =−(−1)jr3/2+ (−1)jr3/2= 0.INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 5\nMoreover, the Jacobian matrix of Gwith respect to the ( η, ζ)-variables is\nD(η,ζ)G=\u00123(η2−ζ2)−2η−c 2ζ−6ηζ\n−2ζ+ 6ηζ 3(η2−ζ2)−2η−c\u0013\n.\nHence, evaluating this Jacobian at ( η0, ζ0j, c0) we obtain\nJ0:=\u0000\nD(η,ζ)G\u0001\n|(η0,ζ0j,c0)=\u0012\n−2r (−1)j2√r\n−(−1)j2√r −2r\u0013\n, j = 2,3.\nNotice that det J0= 4r(r+ 1)>0. Henceforth, by the Implicit Function Theorem\nthere exist local functions\n(eηj,eζj)(c) = (Re λj(c),Imλj(c)),\ndefined on a neighborhood of c0such that eηj(c0) =η0,eζj(c0) =ζ0j, and there holds\nG(eηj(c),eζj(c), c) = 0 for all cin such neighborhood of c0. Moreover, ( eηj,eζj) are of\nclass C1and their derivatives are given by\nd\ndc\u0012eηj\neζj\u0013\n=−\u0014\n(D(η,ζ)G)−1\u0012\n∂cG1\n∂cG2\u0013\u0015\f\f\f\f\n(eηj(c),eζj(c),c).\nSince ∂cG1=−ηand∂cG2=−ζwe readily obtain\nd\ndc\u0012eηj\neζj\u0013\f\f\f\f\nc0=J−1\n0\u0012η0\nζ0j\u0013\n=1\n2r(r+ 1)\u0012r −(−1)j√r\n(−1)j√r r\u0013\u00120\n−(−1)j√r\u0013\n=1\n2(r+ 1)\u00121\n−(−1)j√r\u0013\n, j = 2,3.\nThis yields\nd\ndc\u0000\nReλj(c)\u0001\f\f\f\f\nc0=1\n2(r+ 1)>0, j = 2,3. (2.6)\nTo sum up, for fixed physical parameter values, α >0 and r >0, the first order\nsystem (2.3) is, with a slight abuse of notation, of the form Φ′=F(Φ, c) where\nc∈Ris interpreted as a bifurcation parameter, Fis smooth in both variables\nand it satisfies that, when evaluated at ( P0, c0) (where c0=−ris the critical\nbifurcation parameter), the Jacobian A0= (DΦF)|(P0,c0)has two simple purely\nimaginary eigenvalues, λ2(c0) and λ3(c0), with λj(c0) = (−1)ji ω0, for j= 2,3\nandω0:=√r >0, and no other eigenvalues with non zero real parts. Moreover,\ncondition (2.6) holds, yielding Re λj(c)>0 for c > c 0.\nUpon application of Hopf’s bifurcation Theorem (see Theorem 3.15 in [32], p.\n81, or Theorem 3.4.2 in [15], p. 151), we deduce the existence of 0 < ϵ0≪1\nsufficiently small such that the system has a unique (up to translations) family of\nclosed periodic orbit solutions, Φ ϵ= Φ ϵ(ξ), for each value of 0 < ϵ < ϵ 0. This\nfamily is parametrized by speed values of the form\nc(ϵ) =c0+ϵ=−r+ϵ.\nEach periodic orbit has a fundamental period given by Lϵ= 2π/ω 0+O(ϵ) =\n2π/√r+O(ϵ) and amplitude growing like |Φϵ|=O(√ϵ). Moreover, from the regu-\nlarity of the vector field, it follows that Φ ϵ∈C∞. Identifying the first component\nof each of these periodic orbits as a periodic traveling wave for equation (1.2), we\nhave thus proved the following existence result.6 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nTheorem 2.1 (existence of small amplitude periodic waves) .For any fixed param-\neter values r, α > 0, there exist ϵ0>0sufficiently small and a critical speed value\nc0=−rsuch that the KdV-Burgers-Fisher equation (1.2) has a family of smooth\nperiodic traveling wave solutions of the form u(x, t) =φϵ(x−c(ϵ)t), indexed by\nϵ∈(0, ϵ0), with fundamental period\nLϵ=2π√r+O(ϵ), (2.7)\nand with amplitude growing like\n|φϵ|,|(φϵ)′|=O(√ϵ). (2.8)\nFor any ϵ∈(0, ϵ0), the speed of propagation is given by c(ϵ) =−r+ϵ, and satisfies\nc(ϵ)→c0asϵ→0+. Finally, the periodic wave φϵis unique up to translations,\nfor each fixed ϵ∈(0, ϵ0).\nRemark 2.2. From condition (2.6) and standard bifurcation theory, a unique\ncurve of periodic orbits bifurcate into the region c(ϵ)> c0=−r. As a result, the\nlinearization at the rest point P0passes from having eigenvalues with Re λ1>0\nand Re λj<0,j= 2,3 when c=−r+ϵwith ϵ <0, to a unstable hyperbolic\nfocus with Re λj>0,j= 1,2,3, for c=−r+ϵwith ϵ >0. Hence, the change of\nqualitative behavior of the system corresponds to a supercritical Hopf bifurcation\nand the emerging periodic orbits for ϵ > 0 are stable. Of course, this notion\nof stable/unstable periodic orbit refers to the standard concept from dynamical\nsystems theory: an orbit is stable as a solution to system (2.3) for a constant\nvalue of cif any other nearby solution to the system with the same ctends to\nthe orbit under consideration. This definition of stability is completely unrelated\nto the concept of spectrally stable periodic wave , which refers to the dynamical\nstability of the traveling wave as a solution to the evolution PDE, see [3, 4] for\nfurther information and discussions. Actually, we shall prove that these orbits are\nspectrally unstable as periodic wave solutions to the KdVBF equation (1.2).\n2.2.Numerical approximation. In order to illustrate the emergence of small\namplitude periodic waves for the KdVBF equation (1.2), let us fix the physical\nparameters as r=α≡1. Then the dynamical system (2.3) in R3now reads\n\nϕ1\nϕ2\nϕ3\n′\n=\nϕ2\nϕ3\nϕ3+ϕ1(1−ϕ1)−ϕ1ϕ2+c(ϵ)ϕ2\n, (2.9)\nwhere the speed, or bifurcation parameter, is given by c(ϵ) =c0+ϵ=−1 +ϵand\nϵtakes values in a neighborhood of zero. As before,′=d/dξ, where ξ=x−c(ϵ)t\ndenotes the translation variable, once the value of the speed has been fixed. Figure 1\nshows the flow in the phase space of the solutions system (2.9) for c(ϵ) =−1±ϵwith\nϵ= 0.001. To that end, we used the standard tools provided by Mathematica ©.\nFor instance, Figure 1(a) depicts the flow when c=−1−0.001, that is, before\nthe bifurcation occurs. Notice that the origin is a hyperbolic saddle with both\nstable and unstable eigendirections. Figure 1(b) shows the flow in phase space of\nthe system (2.9) for c=−1 + 0 .001, right after the supercritical Hopf bifurcation\noccurs. The unique, small-amplitude periodic orbit that emerges as a result of such\nbifurcation is represented in red color.\nIt is to be observed that the small periodic orbit appearing in Figure 1(b) is, in\nfact, a numerical approximation. For that purpose, we used the NDSolve package,INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 7\n(a)c=−1−0.001\n (b)c=−1 + 0 .001\nFigure 1. Emergence of small-amplitude periodic waves for the Kd-\nVBF equation (1.2) with r=α= 1. Panel (a) shows the flow in\nthe three dimensional phase space of system (2.9) for a speed value of\nc(ϵ) =−1−ϵwith ϵ= 0.001. The arrows in light blue color show the\ndirection of the vector field. The origin is a hyperbolic saddle with both\nstable and unstable eigendirections. Panel (b) shows the flow in phase\nspace of system (2.9) for a speed value of c(ϵ) =−1 +ϵwith ϵ= 0.001,\nafter the supercritical bifurcation occurs. As a result, a small amplitude\nperiodic orbit (shown in red color) emerges around the origin (color on-\nline).\nin which a standard explicit Runge-Kutta method was implemented with an initial\npoint given by (0 ,0.01,0). Figure 2(a) shows a magnification of the flow in phase\nspace for the speed value c=−1+0.001 (the same as in Figure 1(b)), together with\nthe computed periodic orbit. Figure 2(b) contains the plot of the first component\nof the approximated periodic orbit, ϕ1, as a function of the translation variable\nξ=x−c(ϵ)t.\n3.The stability problem\nIn this section we make precise the notion of spectral stability of the periodic\nwaves under consideration and exhibit its equivalence to computing the Floquet\nspectrum of the linearized operator around the wave.\n3.1.Spectral stability. Fix a parameter value ϵ∈(0, ϵ0) and consider the unique\ntraveling wave solution to (1.2), u(x, t) =φϵ(x−c(ϵ)t), from Theorem 2.1. With a\nslight abuse of notation, let us make the transformation x7→x−c(ϵ)tin order to\nrewrite equation (1.2) as\nut−c(ϵ)ux+αuu x+uxxx=uxx+ru(1−u). (3.1)\nIn other words, we recast the problem into a Galilean coordinate frame under which\nthe profile φϵ=φϵ(x) is now a stationary solution to (3.1) (thanks to equation\n(2.2)). Motivated by the notion of spatially localized, finite energy perturbations8 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\n(a)c=−1 + 0 .001\n-6 -4 -2 0 2 4 6-0.010-0.0050.0000.0050.010 (b)φϵ(ξ) =ϕ1(ξ),ξ=x−c(ϵ)t\nFigure 2. Numerical approximation of the periodic orbit for c=−1+\n0.001. Panel (a) is a magnification of the flow Figure 1(b), together\nwith the numerical approximation of the emerging periodic orbit (in red\ncolor). Panel (b) shows the graph (in red) of the first component ϕ1of\nthe numerical approximation of the periodic orbit as a function of the\nGalilean variable of translation, ξ=x−c(ϵ)t(color online).\nin the Galilean coordinate frame in which the periodic wave is stationary, let us\nconsider solutions to (3.1) of the form\nu(x, t) =φϵ(x) +eλtv(x),\nwhere λ∈Candv∈L2(R). Upon substitution and using the profile equation (2.2),\nthe spatial perturbation vmust be a solution to the following nonlinear equation\nλv+αvvx+(αφϵ(x)−c(ϵ))vx+vxxx=vxx+\u0000\nr(1−2φϵ(x))−α(φϵ)′(x)\u0001\nv−rφϵ(x)v2.\nLinearizing last equation we obtain\nλv=−vxxx+vxx−(αφϵ(x)−c(ϵ))vx+\u0000\nr(1−2φϵ(x))−α(φϵ)′(x)\u0001\nv,\nthat is, we arrive at the following spectral problem for the linearized operator\naround the periodic wave,\nλv=Lϵv, (3.2)\nwhere, for each 0 < ϵ < ϵ 0, the linear operator Lϵis defined as\n\n\nLϵ:L2(R)→L2(R),\nD(Lϵ) =H3(R),\nLϵ:=−∂3\nx+∂2\nx+aϵ\n1(x)∂x+aϵ\n0(x)Id,(3.3)\nwith coefficients\naϵ\n1(x) :=c(ϵ)−αφϵ(x),\naϵ\n0(x) :=r(1−2φϵ(x))−α(φϵ)′(x),(3.4)INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 9\nfor all x∈R. Since the periodic wave is smooth and Lϵ-periodic, the coefficients\naϵ\nj,j= 0,1, and their derivatives are smooth, bounded and Lϵ-periodic. Notice\nthat the operator Lϵis densely defined in L2(R), with domain D(Lϵ) =H3(R).\nMoreover, in view that the coefficients clearly satisfy aϵ\nj∈W1,∞(R),j= 0,1, from\nLemma 3.1.2 in [23], p. 40, we conclude that Lϵis a closed operator in L2(R).\nLet us now recall some definitions from classical spectral theory. Let L:X→Y\nbe a densely defined, closed, linear operator with domain D(L)⊆XandX, Y\nBanach spaces. The resolvent set of L, denoted as ρ(L)|X, is defined as the set of\ncomplex numbers λ∈Csuch that L −λis invertible and ( L −λ)−1is a bounded\noperator. The complement of ρ(L)|Xis what we call the X-spectrum of Land we\ndenote it as σ(L)|X=C\\ρ(L)|X. Any complex number, λ∈C, is said to belong to\nthe point spectrum, σpt(L)|X, ifL −λis a Fredholm operator with index equal to\nzero and non-trivial kernel. Also, λbelongs to the essential (or continuous , in the\nperiodic case) spectrum, σess(L)|X, provided that either L −λis not Fredholm or\nit is Fredholm with non-zero index. Clearly, both the point and essential spectra\nare subsets of σ(L)|X. Moreover, since the operator is closed then there holds\nσ(L)|X=σpt(L)|X∪σess(L)|X, see [24], p. 167. The point spectrum comprises\ndiscrete eigenvalues with finite (algebraic) multiplicity. The reader is referred to\nKato [24] and Kapitula and Promislow [23] for further information.\nDefinition 3.1 (spectral stability) .The periodic traveling wave solution, φϵ, to\n(1.2) is spectrally stable if the L2-spectrum of the linearized operator around the\nwaveLϵsatisfies\nσ(Lϵ)|L2⊂ {λ∈C: Reλ≤0}.\nOtherwise, we say that it is spectrally unstable .\nRemark 3.2. The choice of L2as a base space is motivated by the notion of\nspatially localized, finite energy perturbations in the translation coordinate frame\nin which the wave is stationary.\n3.2.The Floquet spectrum. Since the coefficients of the differential operator Lϵ\nare periodic, it is well known from Floquet theory that Lϵhas no L2-point spectrum\nand that σ(Lϵ)|L2=σess(Lϵ)|L2(see Lemma 3.3 in [22], or Lemma 59, p. 1487,\nin [11]). Moreover, it is possible to parametrize the essential spectrum in terms of\nFloquet multipliers of the form eiθ∈S1,θ∈R(mod 2 π). For each θ∈(−π, π],\nlet us denote σθas the set of complex numbers λfor which there exists a bounded\nnon-trivial solution v∈L∞(R) to the quasi-periodic problem\nλv=Lϵv,\n∂j\nxv(Lϵ) =eiθ∂j\nxv(0), j = 0,1,2.(3.5)\nTheFloquet spectrum is defined as\nσF:=[\n−π<θ≤πσθ.\nIt turns out that the Floquet spectrum coincides with σ(Lϵ)|L2.\nLemma 3.3 (Floquet characterization of the spectrum) .σ(L)|L2=σF.\nProof. See the proof of Proposition 3.4 in Jones et al. [22]. □10 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nOne advantage of this characterization is that the purely essential spectrum\nσ(L)|L2can be recast as the union of partial spectral σθ. Moreover, for each fixed θ\nthe set σθis discrete because it is the zero set of an analytic function (see, e.g., [22],\np. 4649). Notice that if θ= 0 then the boundary conditions in (3.5) become\nperiodic and, consequently, σ0detects perturbations which are co-periodic.\nOne can further remove the dependence on θof the boundary conditions in (3.5)\nand pose the problem in a proper periodic space independently of (but indexed by)\nθ, by means of a Bloch-type transformation (cf. [23]). Indeed, let us define\ny:=πx\nLϵ, w (y) :=e−iθy/πv\u0010Lϵy\nπ\u0011\n. (3.6)\nThen, it is easy to show that for any given θ∈(−π, π] any solution to (3.5)\nis transformed to the solution to the following equation with periodic boundary\nconditions,\n\n\nλw=−1\nL3ϵ\u0000\niθ+π∂y\u00013w+1\nL2ϵ\u0000\niθ+π∂y\u00012w+1\nLϵaϵ\n1(Lϵy/π)\u0000\niθ+π∂y\u0001\nw+\n+aϵ\n0(Lϵy/π)w,\n∂j\nyw(π) =∂j\nyw(0), j = 0,1,2,\nwhere the coefficients aϵ\nj(·) are defined in (3.4). Therefore, multiplying last equation\nbyL3\nϵ>0 and defining the rescaled spectral parameter as\neλ:=L3\nϵλ,\nwe arrive at a family of spectral problems\neλw=Lϵ\nθw, (3.7)\nindexed by θ∈(−π, π], where the Bloch-type operators are defined as\n\n\nLϵ\nθ:L2\nper([0, π])→L2\nper([0, π]),\nD(Lϵ\nθ) =H3\nper([0, π]),\nLϵ\nθ=−\u0000\niθ+π∂y\u00013+Lϵ\u0000\niθ+π∂y\u00012+L2\nϵeaϵ\n1(y)\u0000\niθ+π∂y\u0001\n+L3\nϵeaϵ\n0(y)Id.\n(3.8)\nHere, we have defined\neaϵ\nj(y) :=aϵ\nj(Lϵy/π), y ∈[0, π], j = 0,1.\nSince the family has compactly embedded domains in L2\nper([0, π]) then their\nspectra consist entirely of discrete point eigenvalues, σ(Lϵ\nθ)|L2per=σpt(Lϵ\nθ)|L2per. In\nthis fashion, we relate the L2-spectrum of the operator in (3.3), which is purely\nessential, to the discrete point spectrum of the family of operators (3.8), defined\non a periodic Sobolev space. Moreover, since for any λin the spectrum we have\nReλ >0 if and only if Re eλ >0, we immediately obtain the following criterion for\ninstability.\nProposition 3.4 (spectral instability criterion) .For each ϵ∈(0, ϵ0)the periodic\nwave φϵis spectrally unstable if and only if there exists θ0∈(−π, π]for which\nσpt(Lϵ\nθ0)|L2per∩ {λ∈C: Reλ >0} ̸=∅.INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 11\n4.Spectral instability\nIn this section we follow recent stability analyses of periodic waves (cf. [3,4,9,27])\nto prove our main result, pertaining to the spectral instability of the waves of\nTheorem 2.1.\nTheorem 4.1 (spectral instability of small amplitude periodic waves) .There exists\nϵ1∈(0, ϵ0]such that every small-amplitude periodic wave φϵwith ϵ∈(0, ϵ1)from\nTheorem 2.1 is spectrally unstable, that is, the Floquet spectrum of the linearized\noperator around the wave intersects the unstable half plane, namely\nσ(Lϵ)|L2∩ {λ∈C: Reλ >0} ̸=∅.\nIn order to prove Theorem 4.1, we first need to pose the spectral problem (3.7)\nfor each operator in (3.8) as a spectral problem for a perturbed operator, where the\nperturbation parameter will depend on ϵ >0.\n4.1.Relatively bounded perturbations. In this section we recast the spec-\ntral equation (3.7) for each Bloch operator as a perturbation problem. For the\nforthcoming analysis, it is crucial that these perturbation operators are relatively\nbounded with respect to the operators of lower order. First, let us observe that,\nfrom Theorem 2.1, the periodic waves under consideration have fundamental period\nand amplitude given by\nLϵ=2π√r+O(ϵ) =:L0+O(ϵ),\nand\n|φϵ(x)|,|(φϵ)′(x)|=O(p\n|c(ϵ)−c0|) =O(√ϵ),\nrespectively, as ϵ→0+, and uniformly for all x∈[0, Lϵ]. Hence, we write\nLϵ=L0+√ϵL1,\nwhere L1:= (Lϵ−L0)/√ϵ=O(√ϵ) =O(1) as ϵ→0+and\nc=c(ϵ) =c0+√ϵc1,\nwhere c1:= (c(ϵ)−c0)/√ϵ=O(1). Note that, even though the speed and the\nfundamental period are perturbations of O(ϵ), for convenience and without loss of\naccuracy, we regard both quantities as perturbations of O(√ϵ), which is the size\nof the amplitude of the waves. Therefore, the first order coefficient of each Bloch\noperator in (3.8) can be written as\nL2\nϵeaϵ\n1(y) =L2\nϵaϵ\n1(Lϵy/π) = (L0+O(ϵ))2aϵ\n1(Lϵy/π)\n= (L0+O(ϵ))2(c(ϵ)−αφϵ(Lϵy/π))\n= (L0+O(ϵ))2(c0+O(ϵ)−αO(√ϵ))\n=L2\n0c0+√ϵb1(y),\nwhere\nb1(y) :=L2\nϵaϵ\n1(Lϵy/π)−L2\n0c0√ϵ=O(1),∀y∈[0, π].12 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nRecall that L0= 2π/√randc0=−r, so that L2\n0c0=−4π2<0. Likewise, the\ncoefficient of zeroth order is\nL3\nϵeaϵ\n0(y) =L3\nϵaϵ\n0(Lϵy/π) = (L0+O(ϵ))3aϵ\n1(Lϵy/π)\n= (L0+O(ϵ))3\u0000\nr(1−2φϵ(Lϵy/π))−α(φϵ)′(Lϵy/π)\u0001\n= (L0+O(ϵ))3(r+O(√ϵ))\n=L3\n0r+O(√ϵ)\n=L3\n0r+b0(y),\nwhere\nb0(y) :=L3\nϵaϵ\n0(Lϵy/π)−L3\n0r√ϵ=O(1),∀y∈[0, π].\nFinally, Lϵ=L0+O(ϵ) implies that\nb2:=Lϵ−L0√ϵ=O(1),\nis a uniformly bounded constant coefficient (indeed, b2≤C√ϵ≤Cfor 0 < ϵ≤ϵ0,\nsufficiently small). Let us denote\nε:=√ϵ∈(0,√ϵ0) = (0 , ε0).\nTo sum up, we recast the Bloch operators (3.8) in perturbation form as\nLϵ\nθ=−\u0000\niθ+π∂y\u00013+ (L0+εb2)\u0000\niθ+π∂y\u00012+\n+ (L2\n0c0+εb1(y))(iθ+π∂y) + (L3\n0r+εb0(y))Id,\n=L0\nθ+εL1\nθ,\nfor each θ∈(−π, π], where we have defined the operators Lj\nθ,j= 0,1, as\n\n\nL0\nθ:L2\nper([0, π])→L2\nper([0, π]),\nD(L0\nθ) =H3\nper([0, π]),\nL0\nθ=−\u0000\niθ+π∂y\u00013+L0\u0000\niθ+π∂y\u00012+L2\n0c0\u0000\niθ+π∂y\u0001\n+L3\n0rId,(4.1)\nand\n\nL1\nθ:L2\nper([0, π])→L2\nper([0, π]),\nD(L1\nθ) =H2\nper([0, π]),\nL1\nθ=b2\u0000\niθ+π∂y\u00012+b1(y)\u0000\niθ+π∂y\u0001\n+b0(y) Id,(4.2)\nfor each θ∈(−π, π]. Notice that L0\nθis a third order differential operator with\nconstant coefficients, whereas L1\nθis a second order operator with bounded coeffi-\ncients. Moreover, both operators are closed and densely defined in L2\nper([0, π]), with\nD(L0\nθ)⊂D(L1\nθ).\nIn this fashion, we have rewritten the spectral problem (3.7) as a perturbed\nspectral problem of the form\neλw= (L0\nθ+εL1\nθ)w, w ∈H3\nper([0, π]) =D(Lϵ\nθ). (4.3)\nBefore showing that this is a case of relatively bounded perturbations, we need\nto prove a preliminary lemma.INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 13\nLemma 4.2. For every u∈H3\nper([0, π])there hold the estimates\n∥uyy∥L2per≤2\n3δ3/2∥uyyy∥L2per+1\n3δ−3∥u∥L2per, (4.4)\n∥uy∥L2per≤1\n3δ3∥uyyy∥L2per+2\n3δ−3/2∥u∥L2per, (4.5)\nfor any arbitrary δ >0.\nProof. Integrating by parts and using the periodicity of u∈H3\nper([0, π]) and of its\nderivatives, one obtains\n∥uy∥2\nL2per=ˆπ\n0|uy|2dy= ((uy)∗u)|π\n0−ˆπ\n0(uyy)∗u dy≤ˆπ\n0|uyy||u|dy\n≤ ∥uyy∥L2per∥u∥L2per.\nSimilarly, we have\n∥uyy∥L2per≤ ∥uyyy∥1/2\nL2per∥uy∥1/2\nL2per≤ ∥uyyy∥1/2\nL2per∥uyy∥1/4\nL2per∥u∥1/4\nL2per,\nwhich yields, in turn,\n∥uyy∥3/4\nL2per≤ ∥uyyy∥1/2\nL2per∥u∥1/4\nL2per.\nThe last inequality is equivalent to\n∥uyy∥L2per≤ ∥uyyy∥2/3\nL2per∥u∥1/3\nL2per, (4.6)\nand, as a consequence,\n∥uyy∥L2per≤ ∥uyy∥1/2\nL2per∥u∥1/2\nL2per≤ ∥uyyy∥1/3\nL2per∥u∥1/6\nL2per∥u∥1/2\nL2per\n=∥uyyy∥1/3\nL2\nper∥u∥2/3\nL2\nper. (4.7)\nNext, apply Young’s inequality, ab≤ap/p+bq/qwith p−1+q−1= 1 and a, b > 0\nby taking a=δ∥uyyy∥2/3\nL2per,b=δ−1∥u∥1/3\nL2per,p= 3/2,q= 3 and any δ >0, to obtain\n∥uyyy∥2/3\nL2per∥u∥1/3\nL2per≤2\n3δ3/2∥uyyy∥L2per+1\n3δ−1∥u∥L2per.\nUpon substitution into (4.6) we get (4.4). Similarly, take a=δ∥uyyy∥1/3\nL2per,b=\nδ−1∥u∥2/3\nL2per,p= 3,q= 3/2 and apply Young ´s inequality to obtain\n∥uyyy∥1/3\nL2per∥u∥2/3\nL2per≤1\n3δ3∥uyyy∥L2per+2\n3δ−3/2∥u∥L2per.\nSubstitute last inequality into (4.7) to arrive at (4.5). The lemma is proved. □\nLet us now recall that if T,L:X→Yare linear operators in X,Y, Banach\nspaces, we say that Tis relatively bounded with respect to L(or simply L-bounded)\nprovided that D(L)⊂D(T) and that there exist constants a, b≥0 such that\n∥Tu∥ ≤a∥u∥+b∥Lu∥,\nfor all u∈D(L), see Kato [24], p. 190. The following lemma is the main result of\nthis section.\nLemma 4.3. For any fixed θ∈(−π, π], the operator L1\nθis relatively bounded with\nrespect to L0\nθ.14 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nProof. Since D(L0\nθ) =H3\nper([0, π])⊂H2\nper([0, π]) = D(L1\nθ), we only need to show\nthat there exist non-negative constants, a, b≥0, such that\n∥L1\nθu∥L2per≤a∥u∥L2per+b∥L0\nθu∥L2per, (4.8)\nfor each u∈H3\nper([0, π]). First, let us estimate\n∥L1\nθu∥L2per=\r\rb2(iθ+π∂y)2u+b1(y)\u0000\niθ+π∂y\u0001\n+b0(y)u\r\r\nL2per\n≤π2|b2|∥uyy∥L2per+π\u0000\n∥b1∥L∞+ 2|θ||b2|\u0001\n∥uy∥L2per+\n+\u0000\n∥b0∥L∞+|θ|∥b1∥L∞+θ2|b2|\u0001\n∥u∥L2per\n≤C2∥uyy∥L2per+C1∥uy∥L2per+C0∥u∥L2per,\nwith uniform constants\nC2:=π2|b2|, C 1:=π\u0000\n∥b1∥L∞+ 2|θ||b2|\u0001\n, C 0:=\u0000\n∥b0∥L∞+|θ|∥b1∥L∞+θ2|b2|\u0001\n,\nbecause |θ| ≤π. Using estimates (4.4) and (4.5) we arrive at\n∥L1\nθu∥L2per≤C3,δ∥uyyy∥L2per+C4,δ∥u∥L2per, (4.9)\nwhere\nC3,δ:=2\n3C2δ3/2+1\n3C1δ3, C 4,δ:=1\n3C2δ−3+2\n3C1δ−3/2+C0,\nare uniform positive constants for each δ >0.\nOn the other hand, we have the estimate,\n∥L0\nθu∥L2per=\r\r\r−\u0000\niθ+π∂y\u00013u+L0\u0000\niθ+π∂y\u00012u+L2\n0c0\u0000\niθ+π∂y\u0001\nu+L3\n0ru\r\r\r\nL2per\n=\r\r−π3uyyy+ (L0π2−3iθπ2)uyy+\u0000\n2iθπL 0+L2\n0c0π−3(iθ)2π\u0001\nuy+\n+\u0000\nL0(iθ)2+L2\n0c0iθ+rL3\n0−(iθ)3\u0001\nu\r\r\nL2per\n≥π3∥uyyy∥L2per− |b3|∥uyy∥L2per− |b4|∥uy∥L2per− |b5|∥u∥L2per,\nwhere\nb3:=L0π2−3iθπ2,\nb4:= 2iθπL 0+L2\n0c0π−3(iθ)2π,\nb5:=L0(iθ)2+L2\n0c0iθ+rL3\n0−(iθ)3,\nare uniformly bounded constants. Now, use estimates (4.4) and (4.5) to obtain\n∥L0\nθu∥L2per≥π3∥uyyy∥L2per− |b3|\u00002\n3δ3/2∥uyyy∥L2per+1\n3δ−3∥u∥L2per\u0001\n− |b4|\u00001\n3δ3∥uyyy∥L2per+2\n3δ−3/2∥u∥L2per\u0001\n− |b5|∥u∥L2per\n≥\u0000\nπ3−2\n3|b3|δ3/2−1\n3|b4|δ3\u0001\n∥uyyy∥L2per\n−\u0000\n|b5|+2\n3δ−3/2|b4|+1\n3|b3|δ−1/3\u0001\n∥u∥L2per.\nNow, choose 0 < δ≪1 sufficiently small such that\nC5,δ:=π3−2\n3|b3|δ3/2−1\n3|b4|δ3≥1\n3π3>0,\nand set\nC6,δ:=|b5|+2\n3δ−3/2|b4|+1\n3|b3|δ−1/3>0.INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 15\nThis implies that\n∥uyyy∥L2per≤1\nC5,δ∥L0\nθu∥L2per+C6,δ\nC5,δ∥u∥L2per.\nSubstituting into (4.9), we obtain\n∥L1\nθu∥L2per≤C3,δ\nC5,δ∥L0\nθu∥L2per+\u0010C3,δC6,δ\nC5,δ+C4,δ\u0011\n∥u∥L2per\n=b∥L0\nθu∥L2per+a∥u∥L2per,\nas claimed, with a=C3,δ/C5,δ>0 and b=C4,δ+C3,δC6,δ/C5,δ>0. The lemma\nis proved. □\n4.2.A quick review of perturbation theory for linear operators. For com-\npleteness, in this section we briefly recall the basic perturbation theory for a family\nof linear operators of the form L(κ) =L(0)+κL(1)+κ2L(2)+. . ., where κis a\ncomplex parameter. The reader is referred to the classical books by Hislop and\nSigal [16] (chapter 15) and Kato [24] (chapter VII) for more information. We de-\nscribe the fundamental conditions under which an eigenvalue λ0ofL(0)persists for\nκin a neighborhood of the origin. Let us first recall the definition of an analytic\nfamily of type (A) (see [24], p. 375, or [16], §15.4).\nDefinition 4.4. A family of linear closed operators L(κ) :X→Y, with X, Y\nBanach spaces, defined for κin an open complex domain B⊂Ccontaining the\norigin, is said to be analytic of type (A) if the domains are independent of κ, that\nis,D(L(κ))≡Dfor all κ∈B, and the mapping κ7→ L(κ)uis analytic in κ∈B\nfor each u∈D. In that case, L(κ) has a convergent Taylor expansion of the form\nL(κ)u=L(0)u+κL(1)u+κ2L(2)u+. . . ,\nconverging in a disk |κ|< Rinside Bindependent of ufor some radius R >0. Here\nL(0) =L(0)andL(j):D⊂X→Yare linear operators for each j∈N.\nThe following result is the general stability theorem for discrete eigenvalues of\nanalytic perturbations of a closed operator. Essentially it says that a discrete eigen-\nvalue splits into several eigenvalue branches under a perturbation. These branches\ntend to the original discrete eigenvalue when the perturbation parameter goes to\nzero.\nTheorem 4.5. LetL(κ)be an analytic family of type (A) about κ0= 0. Let λ0be\na discrete eigenvalue of L(0)=L(0). Then there exist families λℓ(κ),ℓ= 1, . . . , q ,\nof discrete eigenvalues of L(κ)such that:\n(i)λℓ(0) = λ0for all ℓ= 1, . . . , q, and the total algebraic multiplicity of the\neigenvalues λℓ(κ),ℓ= 1, . . . , q, is equal to the algebraic multiplicity of λ0;\n(ii)each family λℓ(κ)is analytic in κ1/pfor some positive integer p∈Z+, that\nis, each family has a Puiseux series expansion of the form\nλℓ(κ) =λ0+β1ωℓκ1/p+β2ω2ℓκ2/p+. . . ,\nfor some βj∈Cand where ω=e2πi/p, converging in a (possibly smaller)\ndisk|κ|< R′≤R. In particular, the branches are continuous in κnear the\norigin and λℓ(κ)→λ0when κ→0.\nProof. See Theorem 15.11, p. 155 in [16]. There are no negative powers of κ1/pso\nthat the branches are continuous at the origin, see Knopp [25]. □16 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nIn order to apply the previous perturbation result, let us first examine the per-\nturbed spectral problem (4.3) for the Bloch operator with θ= 0, namely, to\neλw= (L0\n0+εL1\n0)w, w ∈H3\nper([0, π]), (4.10)\nand consider its complexification:\n\n\neL0(κ) :=L0\n0+κL1\n0,\nD(eL0(κ)) =H3\nper([0, π]),\neL0(κ) :L2\nper([0, π])→L2\nper([0, π]),(4.11)\nwhere κ∈Bˆε={κ∈C:|κ|<ˆε}, for some ˆ ε >0 to be determined. That is,\nwe extend the perturbed spectral problem (4.10) to a complex open neighborhood\nof the origin. The family of operators in (4.11) has a common domain, D:=\nD(eL0(κ))≡H3\nper([0, π]), which is independent of κ∈Bˆε. Notice that the Bloch\noperators for θ= 0 are given by\n\n\nL0\n0:L2\nper([0, π])→L2\nper([0, π]),\nD(L0\n0) =H3\nper([0, π]),\nL0\n0=−π3∂3\ny+L0π2∂2\ny+L2\n0c0π∂y+L3\n0rId,(4.12)\nand\n\nL1\n0:L2\nper([0, π])→L2\nper([0, π]),\nD(L1\n0) =H2\nper([0, π]),\nL1\n0=b2π2∂2\ny+b1(y)π∂y+b0(y) Id.(4.13)\nWe start by making a straightforward observation.\nLemma 4.6. The Bloch operator L0\n0has a real positive discrete eigenvalue, eλ0=\nrL3\n0>0, which is associated to the constant eigenfunction w0(y)≡1/√π∈\nH3\nper([0, π]), with ∥w0∥L2per= 1.\nProof. Follows from a direct computation, as L0\n0w0=rL3\n0w0. Moreover, clearly\nw0∈H3\nper([0, π]) and ∥w0∥2\nL2per=´π\n0|w0(y)|2dy= 1. As we have already men-\ntioned, all eigenvalues of the Bloch operators (3.8) are discrete. The lemma is\nproved. □\nNext, we show that the family eL0(κ) is an analytic family of type (A). This is a\ndirect consequence of L1\n0being L0\n0-bounded.\nLemma 4.7. There exists ˆε > 0such that the family of operators (4.11) is an\nanalytic family of type (A) for κ∈Bˆε.\nProof. From Lemma 4.3, in the particular case with θ= 0 we already know that\nthe operator L1\n0is relatively bounded with respect to L0\n0on the space L2\nper([0, π]).\nTherefore, there exist a, b > 0 such that\n∥L1\n0u∥L2per≤a∥u∥L2per+b∥L0\n0u∥L2per,\nfor all u∈D(L0\n0) =H3\nper([0, π])⊂D(L1\n0) =H2\nper([0, π]). Thus, we can apply\nTheorem VII-2.6 and Remark VII-2.7 in Kato [24], pp. 377-378, to conclude that\nthere exists a radius ˆ ε >0 such that the family eL0(κ) =L0\n0+κL1\n0is analytic of\ntype (A) for |κ|<ˆε(in fact, ˆ ε < b−1). □INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 17\nAfter these preparations, we are able to apply perturbation theory for the family\nof linear operators.\nLemma 4.8. There exist ε1>0and families of discrete eigenvalues, eλℓ(κ),ℓ=\n1, . . . , q , of the operators eL0(κ) =L0\n0+κL1\n0for|κ|< ε1such that eλℓ(0) =eλ0=\nrL3\n0>0. Moreover, for each 0< ε < ε 1and|κ|< εthere holds\nσpt(L0\n0+κL1\n0)|L2per∩ {λ∈C:|λ−rL3\n0|< ζ(ε)} ̸=∅, (4.14)\nfor some radius 0< ζ(ε) =O(ε1/p)and some p∈Z+.\nProof. From Lemmata 4.7 and 4.6 we know that L0\n0+κL1\n0is an analytic family\nof type (A) for |κ|<ˆεand that eλ0=rL3\n0>0 is a discrete eigenvalue of L0\n0.\nHenceforth, we apply Theorem 4.5 to conclude that there exist families of discrete\neigenvalues, eλℓ(κ),ℓ= 1, . . . , q , of the operators eL0(κ) =L0\n0+κL1\n0, for|κ|<ˆεsuch\nthateλℓ(0) =eλ0=rL3\n0>0 and the families underlie Puiseux series expansions of\nthe form\neλℓ(κ) =eλ0+β1ωℓκ1/p+β2ω2ℓκ2/p+. . . ,\nwith ω=e2πi/pfor some p∈Z,p > 0, converging in a (possibly smaller) disk\n|κ|< ε 1≤ˆε. Each eλℓ(κ) is a discrete eigenvalue of eL0(κ) =L0\n0+κL1\n0with\neλℓ(0) =eλ0=rL3\n0>0 for each ℓ. Property (4.14) follows from continuity of the\nexpansion and that |eλℓ(κ)−eλ0| ≤C|κ|1/pfor|κ|small. This proves the result. □\nWe now prove the main result of the paper.\n4.3.Proof of Theorem 4.1. In view that ε=√ϵ, let us choose 0 < ϵ 1:=\nmin{ϵ0, ε2\n1}. Hence, from Lemma 4.8 we conclude that there exists θ0= 0∈(−π, π]\nfor which the spectrum of the perturbed Bloch operator with θ= 0 and ϵ∈(0, ϵ1)\nintersects the unstable half plane (in view that eλ0=rL3\n0>0):\nσpt(Lϵ\n0)|L2per∩ {λ∈C: Reλ >0} ̸=∅.\nTherefore, from the spectral instability criterion stated in Proposition 3.4 we obtain\nthe result. Notice that if we vary θ≈0 (within a small neighborhood of the origin),\nwe obtain actual curves of Floquet spectrum that remain in a neighborhood of the\nunstable half plane. This completes the proof of Theorem 4.1. □\n5.Discussion\nIn this paper, we have proved the existence of bounded periodic waves for the\nKdVBF equation (1.2). For that purpose we have verified that, for any fixed positive\nvalues of the physical parameters α > 0 and r >0, a family of small-amplitude\nand finite period waves emerges from a supercritical local Hopf bifurcation around\na critical value of the wave speed given by c0=−r. To illustrate the emergence of\nthe waves we conducted numerical approximations of both the waves and of the flow\nin phase space for fixed parameter values. In addition, we studied the stability of\nthe family of periodic waves as solutions to the PDE. In particular, we proved that\nthese waves are spectrally unstable or, in other words, that the Floquet spectrum\nof the linearized operator intersects the unstable complex half plane. Heuristically,\nthis instability result has a direct interpretation: the small amplitude waves shrink\nto the zero equilibrium solution when the small parameter tends to zero, ϵ→0, and\nthe linearized operator around the wave tends (formally) to a constant coefficient18 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\nlinearized Bloch operator around the origin. The latter has a spectrum determined\nby a dispersion relation that intersects the unstable half plane thanks to the positive\nsign of the reaction term at the rest state. We then invoke the classical perturbation\ntheory for linear operators to conclude.\nThis perturbative technique has been recently applied to study stability of small-\namplitude waves for scalar viscous balance laws [4], scalar Hamiltonian equations\n[27], hyperbolic systems of balance laws [3] and scalar reaction-diffusion equations\ncoupled to one ordinary differential equation [9]. This appears to be a general be-\nhavior and the method could be used to study the instability of small periodic waves\narising in other contexts. In a recent contribution, Chen and Duan [10] found the\nconditions (in a unified framework) to prove the perturbation results on the related\nspectra for a large class of second order differential operators with asymptotically\nconstant coefficients. Their result does not apply, however, to the KdVBF equa-\ntion, which is of third order. From a technical viewpoint, it is to be noticed that\nwe did not prove that the unstable eigenvalue of the unperturbed operator is sim-\nple or non-degenerate. Nonetheless, we directly applied perturbation theory for a\nhigher multiplicity eigenvalue for which the perturbation eigenvalue branches can\nbe represented by a Pusieux series which remain continuous in the limit. This al-\nlows us to conclude the result. This feature exhibits the flexibility of perturbation\ntheory of linear operators as a technique to examine instabilities of approximated\nwaves in general situations. Finally, the instability results of the present study may\nserve as the theoretical basis to demonstrate the orbital (nonlinear) instability of\nthe waves under consideration, as it happens in many other physical systems and\nmodel equations such as the KdV equation [31], viscous balance laws [2], the critical\nKdV and NLS models [6], KdV systems [5], and for general dispersive models [7],\njust to mention a few.\nAcknowledgements\nThe work of R. Folino was partially supported by DGAPA-UNAM, program\nPAPIIT, grant IA-102423. A. Naumkina was supported by CONAHCyT (Mexico),\nthrough a scholarship for graduate studies (CVU 1081529). The work of R. G. Plaza\nwas partially supported by DGAPA-UNAM, program PAPIIT, grant IN-104922.\nReferences\n[1]S. Ali Shan, S. Ali, and A. ur Rehman ,Nonplanar shocks in a warm electronegative plasma\nwith electron nonextensivity effects , Astrophys. Space Sci. 353(2014), pp. 151–162.\n[2]E.´Alvarez, J. Angulo Pava, and R. G. Plaza ,Orbital instability of periodic waves for\nscalar viscous balance laws , J. Evol. Equ. 24(2024), no. 7, pp. 1–35.\n[3]E.´Alvarez, R. Murillo, and R. G. Plaza ,Spectral instability of small-amplitude peri-\nodic waves for hyperbolic non-Fickian diffusion advection models with logistic source , Math.\nModel. Nat. Phenom. 17(2022), pp. 1–25. Paper No. 13.\n[4]E.´Alvarez and R. G. Plaza ,Existence and spectral instability of bounded spatially periodic\ntraveling waves for scalar viscous balance laws , Quart. Appl. Math. 79(2021), no. 3, pp. 493–\n544.\n[5]J. Angulo, O. Lopes, and A. Neves ,Instability of travelling waves for weakly coupled KdV\nsystems , Nonlinear Anal. 69(2008), no. 5-6, pp. 1870–1887.\n[6]J. Angulo Pava and F. Natali ,Stability and instability of periodic travelling wave solutions\nfor the critical Korteweg-de Vries and nonlinear Schr¨ odinger equations , Phys. D 238(2009),\nno. 6, pp. 603–621.\n[7]J. Angulo Pava and F. Natali ,On the instability of periodic waves for dispersive equations ,\nDiffer. Integral Equ. 29(2016), no. 9-10, pp. 837–874.INSTABILITY OF PERIODIC WAVES FOR THE KDV-BURGERS EQUATION 19\n[8]J. M. Burgers ,A mathematical model illustrating the theory of turbulence , in Advances in\nApplied Mechanics, R. von Mises and T. von K´ arm´ an, eds., Academic Press Inc., New York,\nN. Y., 1948, pp. 171–199.\n[9]S. Chen and J. Duan ,Instability of small-amplitude periodic waves from fold-Hopf bifurca-\ntion, J. Math. Phys. 63(2022), no. 11, pp. 112702, 1–18.\n[10]S. Chen and J. Duan ,Perturbation of the spectra for asymptotically constant differential\noperators and applications , Phys. D 448(2023), pp. 1–11. Paper No. 133735.\n[11]N. Dunford and J. T. Schwartz ,Linear operators. Part II: Spectral theory. Selfadjoint\noperators in Hilbert space , Wiley Classics Library, John Wiley & Sons Inc., New York, 1988.\n[12]R. A. Fisher ,The wave of advance of advantageous genes , Ann. Eugen. 7(1937), pp. 355–\n369.\n[13]D.-N. Gao ,Nonplanar ion acoustic solitary waves in an electronegative plasma by damped\nKorteweg–de Vries–Burgers equation , Chinese J. Phys. 77(2022), pp. 1789–1795.\n[14]R. A. Gardner ,On the structure of the spectra of periodic travelling waves , J. Math. Pures\nAppl. (9) 72(1993), no. 5, pp. 415–439.\n[15]J. Guckenheimer and P. Holmes ,Nonlinear oscillations, dynamical systems, and bifurca-\ntions of vector fields , vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York,\n1983.\n[16]P. D. Hislop and I. M. Sigal ,Introduction to spectral theory. With applications to\nSchr¨ odinger operators , vol. 113 of Applied Mathematical Sciences, Springer-Verlag, New\nYork, 1996.\n[17]P. N. Hu ,Collisional theory of shock and nonlinear waves in a plasma , Phys. Fluids 15\n(1972), no. 5, pp. 854–864.\n[18]A. Jeffrey and T. Kakutani ,Weak nonlinear dispersive waves: A discussion centered\naround the Korteweg-de Vries equation , SIAM Rev. 14(1972), pp. 582–643.\n[19]R. S. Johnson ,A non-linear equation incorporating damping and dispersion , J. Fluid Mech.\n42(1970), pp. 49–60.\n[20]R. S. Johnson ,Shallow water waves on a viscous fluid - the undular bore , Phys. Fluids 15\n(1972), no. 10, pp. 1693–1699.\n[21]R. S. Johnson ,A modern introduction to the mathematical theory of water waves , Cambridge\nTexts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.\n[22]C. K. R. T. Jones, R. Marangell, P. D. Miller, and R. G. Plaza ,Spectral and modu-\nlational stability of periodic wavetrains for the nonlinear Klein-Gordon equation , J. Differ.\nEqu. 257(2014), no. 12, pp. 4632–4703.\n[23]T. Kapitula and K. Promislow ,Spectral and dynamical stability of nonlinear waves ,\nvol. 185 of Applied Mathematical Sciences, Springer-Verlag, New York, 2013.\n[24]T. Kato ,Perturbation Theory for Linear Operators , Classics in Mathematics, Springer-\nVerlag, New York, Second ed., 1980.\n[25]K. Knopp ,Theory of Functions, Parts I and II , Dover Publications, New York, 1966.\n[26]H. Koc ¸ak ,Traveling waves in nonlinear media with dispersion, dissipation, and reaction ,\nChaos 30(2020), no. 9, pp. 093143, 6.\n[27]R. Koll ´ar, B. Deconinck, and O. Trichtchenko ,Direct characterization of spectral stabil-\nity of small-amplitude periodic waves in scalar Hamiltonian problems via dispersion relation ,\nSIAM J. Math. Anal. 51(2019), no. 4, pp. 3145–3169.\n[28]A. N. Kolmogorov, I. Petrovsky, and N. Piskunov ,Etude de l’´ equation de la diffusion\navec croissance de la quantit´ e de matiere et son applicationa un probleme biologique , Mosc.\nUniv. Bull. Math 1(1937), pp. 1–25.\n[29]D. J. Korteweg and G. de Vries ,On the change of form of long waves advancing in a\nrectangular canal, and on a new type of long stationary waves , Philos. Mag. (5) 39(1895),\nno. 240, pp. 422–443.\n[30]S. D. Liu and S. K. Liu ,KdV-Burgers equation modelling of turbulence , Sci. Sin A 35\n(1992), pp. 576–586.\n[31]O. Lopes ,A linearized instability result for solitary waves , Discrete Contin. Dyn. Syst. 8\n(2002), no. 1, pp. 115–119.\n[32]J. E. Marsden and M. McCracken ,The Hopf bifurcation and its applications , vol. 19 of\nApplied Mathematical Sciences, Springer-Verlag, New York, 1976.20 R. FOLINO, A. NAUMKINA, AND R. G. PLAZA\n[33]C. H. Su and C. S. Gardner ,Korteweg-de Vries equation and generalizations. III. Deriva-\ntion of the Korteweg-de Vries equation and Burgers equation , J. Math. Phys. 10(1969),\npp. 536–539.\n[34]L. V. Wijngaarden ,One-dimensional flow of liquids containing small gas bubbles , Ann.\nRev. Fluid Mech. 4(1972), no. 1, pp. 369–396.\n[35]Y. Zhou, Q. Liu, and W. Zhang ,Bounded traveling waves of the generalized Burgers-Fisher\nequation , Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23(2013), no. 3, pp. 1350054, 11.\n(R. Folino) Departamento de Matem ´aticas y Mec ´anica, Instituto de Investigaciones\nen Matem ´aticas Aplicadas y en Sistemas, Universidad Nacional Aut ´onoma de M ´exico,\nCircuito Escolar s/n, Ciudad Universitaria, C.P. 04510, Cd. de M ´exico (Mexico)\nEmail address :folino@mym.iimas.unam.mx\n(A. Naumkina) Departamento de Matem ´aticas y Mec ´anica, Instituto de Investiga-\nciones en Matem ´aticas Aplicadas y en Sistemas, Universidad Nacional Aut ´onoma de\nM´exico, Circuito Escolar s/n, Ciudad Universitaria, C.P. 04510, Cd. de M ´exico (Mex-\nico)\nEmail address :naumkinaanna75@gmail.com\n(R. G. Plaza) Departamento de Matem ´aticas y Mec ´anica, Instituto de Investigaciones\nen Matem ´aticas Aplicadas y en Sistemas, Universidad Nacional Aut ´onoma de M ´exico,\nCircuito Escolar s/n, Ciudad Universitaria, C.P. 04510, Cd. de M ´exico (Mexico)\nEmail address :plaza@aries.iimas.unam.mx"    }]